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In this work a theorical framework to apply the Poincaré compactification technique to locally Lipschitz continuous vector fields is developed. It is proved that these vectors fields are compactifiable in the $n$-dimensional sphere, though the compactified vector field can be identically null in the equator. Moreover, for a fixed projection to the hemisphere, all the compactifications of a vector field, which are not identically null on the equator are equivalent. Also, the conditions determining the invariance of the equator for the compactified vector field are obtained. Up to the knowledge of the authors, this is the first time that the Poincaré compactification of locally Lipschitz continuous vector fields is studied.
These results are illustrated applying them to some families of vector fields, like polynomial vector fields, vector fields defined as a sum of homogeneous functions and vector fields defined by piecewise linear functions.
title: |
Poincaré compactification for\
non-polynomial vector fields
---
<span style="font-variant:small-caps;">José Luis Bravo & Manuel Fernández</span>
<span style="font-variant:small-caps;">Antonio E. Teruel$^*$</span>
Introduction
============
The study of the asymptotic behaviour of the solutions of an autonomous ordinary differential system is usually carried out throughout the compactification of the phase space, that is, mapping the phase space to a compact manifold. In the Poincaré compactification, this compact manifold is the $n$-dimensional sphere $\mathbb{S}^n$ centered at the origin $O$, which has the advantage of identifying the different directions at infinity by its projections on the equator.
Denote $H_+$ to the upper hemisphere and $E$ the equator, that is, if $z=(z_1,\ldots,z_{n+1})$ denotes the coordinates of the point $z$ in $\mathbb{S}^{n}$, then $$H_+=\{z\in\mathbb{S}^n\colon z_{n+1}>0\},\quad E=\{z\in\mathbb{S}^n\colon z_{n+1}=0\}.$$ Consider also a diffeomorphism $$\mathbb{R}^{n}\overset{h}{\longrightarrow} H_+.$$ In the classical Poincaré compactification [@P], the diffeomorphism is defined by the stereographic projection of $\mathbb{R}^n$ onto the north hemisphere of the sphere $\mathbb{S}^{n}$ and locating the projection point at the center of the sphere. In other words, the diffeomorphism is defined identifying $\mathbb{R}^n$ as the hyperplane of $\mathbb{R}^{n+1}$ tangent to the sphere at the north pole $e^{n+1}=(0,\ldots,0,1)\in\mathbb{R}^{n+1}$, and defining the mapping $h$ that assigns to each point $x$ of $\mathbb{R}^n$ the intersection of $H_+$ and the line through $x$ and $O$.
Let $f$ be a vector field in $\mathbb{R}^n$ and denote $g$ its projection to $H_+$ by $h$. The vector field is called compactifiable if there is a regularization function $\rho$ (a change in the parametrization of time) such that $\rho g$ can be extended to ${H}_+\cup E$ with certain regularity (see [@LT] for more details).
The existence of an unique integral curve through a point $z_0$ on the sphere is equivalent to the existence of a unique solution of the initial value problem $ x'= f(x)$ with $x(0)=h^{-1}(z_0)$. Moreover (see [@ALGM73_1]), if $F$ is a vector field on $\mathbb{S}^{n}$ continuous and locally Lipschitz continuous, then for every point $z_0$ on the sphere there exists a unique integral curve of $F$ through the point $z_0$, and the integral curve is defined for every $t\in\mathbb{R}$. Therefore, we shall require that any compactified flow be locally Lipschitz on its compact domain.
The Poincaré compactification is applied to polynomial vector fields, with the usual projection, $h$, and the regularization function $\rho(z)=z_{n+1}^{N-1}$, where $N$ is the degree of the polinomial vector field. See e.g. [@MP; @PT; @PBG] for some recent papers using this technique. For polynomial Hamiltonian systems see [@DLLP]. It has also been extended to some families of vector fields, for instance, to rational vector fields [@VG] in a similar way to polynomial vector fields, or to quasi-homogeneous vector fields chosing a different projection $h$ and the same regularization function, but in this case $N$ is defined in terms of the sum of the degrees of the homogeneous functions (see e.g. [@CGP; @GPS]).
One can consider more general projections $h$ belonging to a certain class of admissible compactifications and wonder when the compactification obtained is equivalent to the classical one. Sufficient conditions for this has been obtained in [@EG] for polynomial vector fields, and has been generalized to quasi-homogeneous vector fields in [@M], in both cases using regularization functions of the form $\rho(z)=z_{n+1}^{N-1}$, for certain $N$.
In the present paper we study the Poincaré compactification of vector fields only assuming they are locally Lipschitz continuous. We study the existence of regularization functions depending only of the latitude of the point in the sphere, that is, a function $\rho$ only depending on $z_{n+1}$, such that the projected vector field can be extended to $H_+\cup E$ as a locally Lipschitz continuous vector field.
We prove that every locally Lipschitz continuous vector field is Poincaré compactifiable, but the furnished compactification could be zero on the equator. So, every point in the equator is a rest point and the compactification hides the dynamic at infinity. To avoid this situation we define non-null Poincaré compactifiable vector fields, and we prove the equivalence of any non-null compactification of a fixed vector field. Indeed, we obtain an explicit expression of the regularization function for any non-null compactifiable vector field.
The definition of compactification given in this paper does not imply the invariance of the equator, so we establish a characterization of non-null compactifiable vector fields with invariant equator, in this case, taking $h$ as the classical projection of Poincaré.
Next, we apply the obtained results to some families of vector fields, to give some thought to the compactification properties of three families of vector fields: Polynomial vector fields, polynomial-growth vector fields, and piecewise polynomial systems. For the first family we recover the clasical Poincaré result, and it brings out the fact that the compactified vector field is identically null on the equator if and only if the polynomial vector field of degree $N$ is $f(x)= q(x)x+R(x)$, where $q(x)$ is a scalar polynomial of degree $N-1$ and $R(x)$ is a polynomial of degree strictly lower than $N$. We also note that, like in [@GPS], the above resuls extends to the case $$f(x)= \sum_{l=0}^N f_{l}(x),$$ where $f_{l}(x)$ is a enough regular homogeneous function of degree $l$, i.e. $$f_{l}(\lambda x) = \lambda^l f_{l}(x), \quad \lambda\in\mathbb{R},\ x\in\mathbb{R}^n.$$
Vector fields that grows as polynomial as $\| x\| \to \infty$ can be compactified in a similar way to polynomial fields. Under hypotheses that guarantees the Lipschitz continuity of the vector field on the closed upper hemisphere, we establish the Poincaré compactification with invariant equator.
The last part is devoted to the compactification of piecewise polynomial vector fields. As a consequence of the results we obtain that piecewise linear (PWL) vector fields are non-null compactifiable vector fields and we characterize the invariance of the equator. These results are an extension of those presented in [@LT] to a general dimension phase space and to a vector fields with finite number of linear pieces.
The structure of the paper is as follows, in Section 2, we establish the general theory and in Section 3, we apply it to the three families above mentioned.
Poincaré compactifiable vector fields
=====================================
This Section deals with the Poincaré compactification of locally Lipschitz continuous vector fields defined on $\mathbb{R}^n$. Consider a differential equation$$\label{eq:vf}
x'=f(x),$$ where the vector field $f\colon \mathbb{R}^n\to\mathbb{R}^n$ is a locally Lipschitz continuous function. Let us project the vector field defined by to the upper hemisphere. To this end, we fix the diffeomorphism $h:\mathbb{R}^n\rightarrow H_{+}$.
The *projected vector field* is then given by the differential equation $$\label{eq:proj_vf}
z'=g(z):=Dh\left(h^{-1}(z)\right)f\left(h^{-1}(z)\right).$$ See Figure \[fig:prcmfig2\] for a graphical representation of the compactification process. Notice that in Figure \[fig:prcmfig2\] the sterographical projection $h$ is represented.
To study the behaviour near the equator, we introduce a new system of coordinates on $H_+\cup E$ minus the north pole $e_{n+1}$, using the diffeomorphism $$E\times [0,1) \longrightarrow (H_+ \cup E)\backslash\{e^{n+1}\},$$ defined by $$\label{eq:zdelta}
(z,\delta)\to z_\delta = z \sqrt{1-\delta^2}+ \delta e^{n+1}.$$ We note that varying $z$ and keeping $\delta$ constant then $(z,\delta)$ is a parallel of $H_+\cup E$, and varying $\delta$ and keeping $z$ constant then $(z,\delta)$ is a meridian.
Let $\pi:H_+ \cup E \rightarrow \mathbb{R}^n$ be the projection of the first $n$–coordinates, that is $\pi(z)=(z_1, \ldots, z_n)^T$. Then, if $(z_1,\ldots,z_{n+1})$ are the coordinates of $z_\delta$, $$z_{n+1}=\sqrt{1-\|\pi(z)\|^2}=\delta> 0.$$
![Poincaré compactification of a vector field $f$. The phase space $\mathbb{R}^n$ is identified with the hyperplane of $\mathbb{R}^{n+1}$ tangent to the unit sphere $\mathbb{S}^n$ at the north pole $e^{n+1}$. The stereographic projection $h$ maps the phase space onto the north hemisphere $H_+$ and induces the projected vector field $g$ on $H_+$. The compactified vector field $F_{\rho}$ is the Lipschitz continuous extension of $g$ to the equator $E$ after a regularizating chage of coordinates.[]{data-label="fig:prcmfig2"}](prcmfig2.eps)
(0,0) (-170,115)[$x=h^{-1}(z_{\delta})$]{} (-160,85)[$z_{\delta}$]{} (-156,34)[$z$]{} (-200,25)[$\pi(z_{\delta})$]{} (-175,55)[$\delta$]{} (-300,150)[$\mathbb{R}^n$]{} (-315,60)[$H_+$]{} (-290,20)[$E$]{}
\[def:compactifiable\] Let $f(x)$ be a locally Lipschitz continuous funtion in $\mathbb{R}^n$. We say that $x'=f(x)$ is a [*Poincaré compactifiable*]{} vector field by a projection $h$ if there exists a function $\rho:(0,1] \to \mathbb{R}^+= \{\delta \in\mathbb{R} : \delta>0\}$ such that the function $\rho(z_{n+1})g\left( z \right)$ admits a Lipschitz continuous extension to $H_+ \cup E$.
The function $\rho$ will be called *regularization function* and the extension to $H_+ \cup E$ of the regularized vector field $\rho(z_{n+1})g\left( z \right)$ will be called *the compactified vector field*.
Note that, in Definition 1, no condition on $\rho$ has been established but the fact that function $\rho(z_{n+1})g\left( z \right)$ admits a Lipschitz continuous extension to $H_+ \cup E$. Later on, and under additionally conditions for the vector field $f$, some properties on $\rho$ will be derived.
Thus, if $f$ is a compactifiable vector field and $g$ is given in , for every $z\in E$, there exists $$\begin{aligned}
v_{\rho}(z):=\lim_{\delta\to 0+} \rho(\delta)g(z\sqrt{1-\delta^2}+\delta e^{n+1}),\end{aligned}$$ and the compactified vector field writes as $$\label{eq:compvf}
F_\rho(z)=\begin{cases}
\rho(z_{n+1})g\left( z \right) & \text{ if } z_{n+1}>0, \\
v_{\rho}(z) & \text{ if } z_{n+1} =0,
\end{cases}$$ and it is Lipschitz continuous in $H_+ \cup E$.
For short, let us define the function $$\label{fun:g}
G:E\times (0,1] \to \mathbb{R}^n,\quad
G(z,\delta)=g(z_{\delta}),$$ where $z_{\delta}$ is given in . So, $$\begin{aligned}
v_{\rho}(z):=\lim_{\delta\to 0+} \rho(\delta)G\left(z,\delta\right).\end{aligned}$$
In the next Proposition we characterize the compactifiable vector fields in terms of the behaviour of the regularized vector field.
\[prop:Poincare-compactifiable\] A vector field $f(x)$ is Poincaré compactifiable for a projection $h$ if and only if there exists a function $\rho\colon (0,1]\to \mathbb{R}^+$ such that
- There exists $\lim_{\delta\to 0+} \rho(\delta)G(z,\delta)$, uniformly in $z\in E$.
- $\rho(z_{n+1})g\left( z \right)$ is globally Lipschitz continuous in $H_+$.
Assume that $f(x)$ is a Poincaré compactifiable vector field. Therefore, there exists a regularization function $\rho:(0,1]\to \mathbb{R}^+$ such that the compactified vector field $F_\rho(z)$, given in , is locally Lipschitz continuous on the compact manifold $H_+\cup E$. Hence, $F_\rho(z)$ it is globally Lipschitz on $H_+\cup E$ which proves (b). Moreover, since $F_\rho(z)$ is uniformly continuous in a compact set, $v_{\rho}(z)$ exists for every $z\in E$ and the limit is uniform in $z\in E$, which proves (a).
Conversely, assume that the statements (a) and (b) are satisfied. From statement (a) function $v_{\rho}(z)$ is well defined and we can define $F_\rho(z)$ as in . From statement (b), $F_\rho$ is globally Lipschitz continuous on $H_+$. Then we only need to check that $\| F_{\rho}(z)-F_{\rho}(\bar z)\|\leq L\|z-\bar{z}\|$ for $z\in E$, $\bar{z}\in H_+$ and for $z,\bar{z}\in E$, where $L$ is the global Lipschitz constant of $\rho(z_{n+1})g(z)$ in $H_+$.
Assume we are in the first case, let $z_{\delta}= z\sqrt{1-\delta^2}+\delta e^{n+1}$, then $$\begin{aligned}
\| F_{\rho}(z)-F_{\rho}(\bar z)\|&= \lim_{\delta\to 0^+}\| \rho(\delta)g(z_{\delta})-\rho(\bar z_{n+1})g(\bar z)\| \\
&\leq \lim_{\delta\to 0^+} L \|z_{\delta}-\bar{z}\|=L\|z-\bar z\|. \end{aligned}$$ Consider now the second case and assume $z,\bar z \in E$. Set $z_{\delta}= z\sqrt{1-\delta^2}+\delta e^{n+1}$ and $\bar z_{\bar\delta}=\bar z\sqrt{1-\bar\delta^2}+\bar\delta e^{n+1}$. From $$\begin{aligned}
\| F_{\rho}(z)-F_{\rho}(\bar z)\|& \leq
\| F_{\rho}(z)-\rho(\delta)g(z_{\delta})\| + \| F_{\rho}(\bar z)-\rho(\bar\delta)g(\bar z_{\bar\delta})\| + \| \rho(\delta)
g(z_{\delta})-\rho(\bar \delta)g(\bar z_{\bar\delta})\| \\
& \leq \| F_{\rho}(z)-\rho(\delta)g(z_{\delta})\|+ \| F_{\rho}(\bar z)-\rho(\bar\delta)g(\bar z_{\bar\delta})\|+L\|z_{\delta}-\bar z_{\bar\delta} \|\end{aligned}$$ we obtain $F_{\rho}(z)$ Lipschitz continuous in $H_+ \cup E$ using the uniform convergence in statement (a).
The next result proves that for every Lipschitz continuous vector field and every projection $h$, there always exists a regularization function $\rho(\delta)$ that compactifies the vector field, although the dynamic at infinity is trivial.
\[theorem:tc\] Every locally Lipschitz continuous vector field $f$ is Poincaré compactifiable for any projection $h$.
Given a locally Lipschitz continuous vector field $f$, we will prove that there exists a differentiable function $\rho:(0,1]\to \mathbb{R}^+$ such that $F_{\rho}$ is locally Lipschitz continuous in $H_+\cup E$.
The projected vector field $g$ defined in is a locally Lipschitz function provided $f$ is a locally Lipschitz continuous vector field and $h$ is a diffeomorphism. Then, function $G$ defined in is locally Lipschitz continuous. Let us denote by $L_G(\delta)$ the Lipschitz constant of the function $G$ on the compact set $E\times [\delta, 1]$.
We claim that there exists a function $\rho\in \mathcal{C}^1((0,1])$, such that $$\label{th1_key}
\max\left\{\rho(\delta),\rho'(\delta)\right\} m(\delta)\leq \delta, \quad \text{for every }\delta\in(0,1],$$ where $m(\delta)=\max\left\{1, L_G(\delta), \max_{(z,\sigma)\in E\times [\delta,1]} \|G(z,\sigma)\| \right\}$.
Since $E\times [\delta_1,1]\subset E \times [\delta_2,1]$ if $\delta_1>\delta_2$, $m(\delta)$ is a decreasing function. By linear interpolation at the nodes $$\left\{\left(\frac 1 k, \frac 1 {(k+1)m(1/(k+1))}\right)\right\}_{k=1}^{\infty},$$ we define a continuous, positive and strictly increasing function $\tilde{\rho}(\delta)$ such that $\tilde{\rho}(\delta)\leq \delta/m(\delta)$. Set $\rho(\delta)= \int_0^{\delta}\tilde{\rho}(r)\,dr$. Then $$0\leq \rho(\delta)=\int_0^\delta \tilde{\rho}(r)\,dr\leq \tilde{\rho}(\delta) \delta\leq \tilde{\rho}(\delta)\leq \delta/m(\delta).$$
Note that $v_{\rho}$ is identically null. Indeed by , $$\|v_{\rho}(y)\|=\lim_{\delta\to 0+} \rho(\delta)\|G\left(y,\delta\right)\|
\leq \lim_{\delta\to 0+} \rho(\delta) m(\delta)
\leq \lim_{\delta\to 0+} \delta =0.$$ Next, we prove that the function $F_{\rho}(z)$ is locally Lipschitz continuous. As the function is locally Lipschitz continuous in $H_+$, we only need to prove that it is locally Lipschitz continuous in a neighborhood of $E$. Let us consider a point of the equator and a neighborhood $U$ of this point. Let $z, \bar z \in U$. We are going to bound $\|F_{\rho}(z)-F_{\rho}(\bar z)\|/\| z-\bar z\|$ for $z, \bar z \in U$, $z \neq \bar z$.
If $z,\bar z \in E$, then the bound is $0$ since, according to , $F_{\rho}(z)=v_{\rho}(z)=0$ and $F_{\rho}(\bar z)=v_{\rho}(\bar{z})=0$.
Assume $z\in H_+$, $\bar z\in E$. There exist $z_e\in E$, $0<\delta<1$, such that $z=z_e \sqrt{1-\delta^2}+\delta e^{n+1}$. Since $F_{\rho}(\bar z)=0$, then $$\|F_{\rho}(z)-F_{\rho}(\bar z)\|=\|\rho(\delta) G(z_e,\delta)\|\leq \delta=|z_{n+1}-\bar z_{n+1}| \leq \|z- \bar z\|.$$
Finally, assume that $z,\bar z\in H_+$, that is, $z=z_e\sqrt{1-\delta^2}+ \delta e^{n+1}$, $\bar z=\bar z_e\sqrt{1-\bar\delta^2}+ \bar \delta e^{n+1}$, for certain $z_e,\bar z_e \in E$, $0<\delta, \bar\delta<1$. We may assume $\delta<\bar \delta$. Then $$\begin{aligned}
\frac{\|F_{\rho}(z)-F_{\rho}(\bar z)\|}{\|z-\bar z\|}&=\frac{\|\rho(\delta) G(z_e,\delta)-\rho(\bar\delta) G(\bar z_e,\bar\delta)\| }{\|z-\bar z\|}\\
&\leq \rho(\delta) \frac{\|G(z_e,\delta)-G(\bar z_e,\delta)\|}{\|z-\bar z\|} +
\frac{|\rho(\delta)-\rho(\bar\delta)|}{\|z-\bar z\|} \|G(\bar z_e,\bar\delta)\| \\
&\leq \rho(\delta) L_{G}(\delta) + \frac{ \rho'(\xi) (\bar\delta-\delta) m(\bar\delta)}{\|z-\bar z\|}\\
&\leq \rho(\delta) L_{G}(\delta) + \rho'(\xi) m(\bar\delta),\end{aligned}$$ where $\delta <\xi <\bar\delta$. Since $\rho(\delta)L_G(\delta)\leq \rho(\delta)m(\delta)\leq \delta$, and since $\rho'$ is an increasing function, from we obtain, $\rho'(\xi)m(\bar\delta)\leq \rho'(\bar\delta)m(\bar\delta)\leq \bar\delta$. Then $$\frac {\|F_{\rho}(z)-F_{\rho}(\bar z)\|}{\|z-\bar z\|} \leq \delta+\bar\delta \leq 2.$$
Non-null Poincaré compactification
----------------------------------
Theorem \[theorem:tc\] states that every locally Lipschitz continuous vector field is Poincaré compactifiable, but the compactified vector field it provides is identically null along the equator. In this subsection we study compactifications with a non-trivial dynamics in the equator.
We say that the compactification is *identically null* when $v_\rho \equiv 0$ and *non-null* otherwise. To emphasize the dependence of the compactification on the function $\rho$, we will say $f$ is a Poincaré compactifiable vector field by the regularization function $\rho$.
Next we prove that any two non-null Poincaré compatifications are topologically equivalent via the identity.
\[pro:nonnull\] A locally Lipschitz continuous vector field $f(x)$ is a non-null Poincaré compactifiable vector field for a projection $h$ if and only if there exist $\bar{z} \in E$ and $0<\bar{\delta}<1$ such that the following function is defined $$\bar{\rho}(\delta)=\begin{cases}
\|G(\bar{z},\delta)\|^{-1}, & \text{ if } 0<\delta<\bar{\delta}, \\
\|G(\bar{z},\bar{\delta})\|^{-1}, & \text{ if } \bar{\delta} \leq \delta \leq 1,
\end{cases}$$ and the vector field is non-null Poincaré compactifiable by the regularization function $\bar \rho$.
Assume that $f$ is a non-null Poincaré compactifiable vector field. Then there exists a regularization function $\rho\colon (0,1] \to \mathbb{R}_+$ such that $F_\rho(z)$ is locally Lipschitz continuous in $H_+ \cup E$ and the function $v_{\rho}(z)$ is not idendically null in $E$. Take $\bar{z} \in E$ such that $v_{\rho}(\bar{z})\neq0$. By continuity, there exists $\bar \delta$ such that $G(\bar{z},\delta)\neq0$ in $(0,\bar{\delta}]$. Therefore, the function $\bar{\rho}(\delta)$, given in the statement of the proposition, is well defined. Now, we are going to prove that the function $F_{\bar \rho}(z)$ given in , but with regularization function $\bar{\rho}$, is Lipschitz continuous in $H_+ \cup E$. Before that, let $z \in E$. Since $\rho(\delta)>0$ and $v_{\rho}(\bar z)\neq 0$, it follows that $$\lim_{\delta \to 0+} \frac{\rho(\delta) G(z,\delta)}{\|\rho(\delta) G(\bar{z},\delta)\|}=\frac{v_{\rho}(z)}{\|v_{\rho}(\bar z)\|}.$$ Therefore $$\begin{split}
v_{\bar{\rho}}(z) &= \lim_{\delta\to 0+} \bar{\rho}(\delta)G(z,\delta)\\
&=\lim_{\delta \to 0+} \frac{ G(z,\delta)}{\| G(\bar{z},\delta)\|}\\
&=\lim_{\delta \to 0+} \frac{\rho(\delta) G(z,\delta)}{\|\rho(\delta) G(\bar{z},\delta)\|}= \frac{v_{\rho}(z)}{\|v_{\rho}(\bar{z}) \|}.
\end{split}$$ We conclude that $v_{\bar \rho}$ is well defined and, since $\|v_{\bar{\rho}}(\bar{z})\|=1$, it is a non-null function.
To prove that $F_{\bar{\rho}}$ is Lipschitz continuous in $H_+ \cup E$ we note that, for $z_{n+1}\geq \bar{\delta}$ it follows that $$\label{campo-tilde1}
F_{\bar \rho}(z)=g(z)\, \| G(\bar z,\bar\delta) \|^{-1},
$$ and for $0<z_{n+1}<\bar{\delta}$ it follows that $$\label{campo-tilde2}
F_{\bar \rho}(z)=\frac{F_\rho(z)}{\left\|F_\rho\left(\bar{z} \sqrt{ 1- z_{n+1}^2}+z_{n+1}e^{n+1}\right)\right\|}.$$ Since $g(z)$ is globally Lipschitz continuous in $\{z \in \mathbb{S}:z_{n+1}\geq \bar\delta\}$, $F_\rho(z)$ is Lipschitz continuous in $H_+ \cup E$, and $$\lim_{z_{n+1}\to 0}\left\|F_\rho(\bar{z} \sqrt{ 1- z_{n+1}^2}+z_{n+1}e^{n+1})\right\|= \|v_{\rho}(\bar{z})\|>0,$$ we obtain that $F_{\bar \rho}(z)$ is the quotient of two Lipschitz continuous functions where the denominator does not vanish. Hence, $F_{\bar \rho}(z)$ is a Lipschitz continuous function in $H_+ \cup E$.
There exist locally Lipschitz continuous vector fields that are not non-null Poincaré compactifiable. Indeed, consider the vector field defined by the following differential equation $$x'=\cos(\|x\|)x,\quad x\in\mathbb{R}^n.$$ The critical points are $$\{x\in\mathbb{R}^n\colon \|x\|=\pi/2+k\pi\text{ for some }k\in\mathbb{Z}^+\}.$$ Let $x$ such that $\|x\|=\pi/2$, and consider the sequence $$\{\lambda_k x\}_{k\in\mathbb{Z}^+},\quad\text{where }\lambda_k=\frac{\frac{\pi}{2}+k\pi}{\frac{\pi}{2}}=1+2k.$$ Then $$h(\lambda_k x)=\left(\frac{\lambda_k x}{\sqrt{1+\|\lambda_k x\|^2}},\frac{1}{\sqrt{1+\|\lambda_k x\|^2}}\right)
\to \left(\frac{x}{\|x\|},0\right).$$ Since $f(\lambda_k x)=0$ for every $k\in\mathbb{Z}^+$, then $F_\rho(x/\|x\|,0)=0$. That is, the vector field is identically null in $E$.
Let $n=2$, and define the vector field such that for $(z_1,z_2,0)\in E$, and $0<\delta<1$, $$G((z_1,z_2,0),\delta)=\left(
\frac{\delta}{ \sqrt{(1-\delta^2)z_1+\delta^2} }
e^{z_1\frac{\sqrt{1-\delta^2}}{\delta}},
0,
-\frac{\sqrt{1-\delta^2}z_1}{\sqrt{(1-\delta^2)z_1+\delta^2}}
e^{z_1\frac{\sqrt{1-\delta^2}}{\delta}}
\right).$$
Then the compactified vector field is null on $E$, since if we assume that the vector field is non-null on $E$, by Proposition \[pro:nonnull\], there exists $\bar z=(\bar z_1,\bar z_2,0) \in E$ and $0<\bar\delta<1$ such that $f$ is compactifiable by $$\bar \rho(\delta)=\frac{1}{\|G(\bar z,\delta)\|}=e^{-\frac{\bar z_1\sqrt{1-\delta^2}}{\delta}}, \quad 0< \delta <\bar\delta.$$ Then $$v_{\bar \rho}(z)=\left(\lim_{\delta\to 0^+} e^{\frac{(z_1 - \bar z_1)\sqrt{1-\delta^2}}{\delta}},0\right).$$ In consequence, $v_{\bar \rho}(z)$ is not defined, when $z_1>\bar z_1$, which is a contradiction with the assumption that $f$ is compactifiable with regularization function $\bar{\rho}$.
Note that if we take other $\tilde z\in E$ and $\tilde\rho(\delta)=1/\|G(\tilde z,\delta)\|$, $$\lim_{\delta\to 0^+} \frac{\bar \rho(\delta)}{\tilde \rho(\delta)}=
\lim_{\delta\to 0^+} \frac{\|G(\tilde z,\delta)\|}{\|G(\bar z,\delta)\|}=
\lim_{\delta\to 0^+} e^{(\tilde z_1-\bar z_1)\frac{\sqrt{1-\delta^2}}{\delta}}=
\begin{cases}
0,\quad \text{if }\tilde z_1<\bar z_1,\\
\infty,\quad \text{if }\tilde z_1>\bar z_1.\\
\end{cases}$$ From this we conclude that the growth of the norm of the compactified vector field along the directions of $E$ are not equivalent.
In Section \[sec:3\], we will see that this does not happen for polynomial vector fields.
We say that two compactifications are [*equivalent*]{} if the compactified vector fields are topologically equivalent.
Any two non-null Poincaré compactification of a fixed locally Lipschitz continuous vector field are equivalent.
Note that for any regularization function $\rho$, defining $\bar{\rho}$ as in Proposition \[pro:nonnull\], the vectors fields $F_\rho$ and $F_{\bar\rho}$ are topologically equivalent since they are proportional by a positive function, see -. Moreover, the topological equivalence is the identity.
Invariance of the equator
-------------------------
The Poincaré compactification of polynomial vector fields with regularization function $\rho(z)=z_{n+1}^{N-1}$ produces a compactification such that $E$ is invariant by the flow of the vector field $F_{\rho}(z)$. This invariance is useful to further project the compactified vector field into the unit disk with differentiability. Nevertheless, the definition of Poincaré compactification used here does not imply this invariance.
In this subsection we study when the equator is invariant in the classical Poincaré compatification, i.e., when the projection is given by the stereographic projection $h:\mathbb{R}^n\rightarrow H_{+}$ defined by $$\label{eq:steregprj}
h(x)=
\left(
\frac{x_1}{ \sqrt{1+\|x\|^2} },\ldots,
\frac{x_n}{ \sqrt{1+\|x\|^2} },
\frac{1}{ \sqrt{1+\|x\|^2} }\right),$$ being its inverse $$h^{-1}(z)=\left(\frac{z_1}{z_{n+1}},\ldots,\frac{z_n}{z_{n+1}}\right).$$
Note that for any identically null compactification, the equator is always invariant as it consists of singular points. Therefore, we only need to study the invariance in the case of non-null compactifications.
Firstly, we obtain the expression of the compactified vector field in terms of the parametrization of $H_+ \cup E$ given in . Moreover, we recall that $g(z)=Dh({h}^{-1}(z))f(h^{-1}(z))$, where $$\begin{aligned}
Dh({h}^{-1}(z))&
=z_{n+1}\left( \begin{array}{c} I-\pi(z) \pi(z)^T \\ -z_{n+1}\pi(z)^T \end{array} \right), \end{aligned}$$ being $I$ the unit matrix of order $n$. Hence, the compactified vector field is $$\label{def:campo_sobre_S}
F_{\rho}(z_{\delta})=\rho(\delta) g(z_{\delta})=
\delta \rho(\delta) \left( \begin{array}{c} f(h^{-1}(z_\delta))-\pi(z_\delta) \pi(z_\delta)^Tf(h^{-1}(z_\delta)) \\ -\delta\pi(z_\delta)^Tf(h^{-1}(z_\delta)) \end{array} \right).$$ Then, $$\pi (F_{\rho}(z_{\delta}))= \delta\rho(\delta)\left(f(h^{-1}(z_{\delta}))- \langle \pi(z_{\delta}), f(h^{-1}(z_{\delta})) \rangle \pi(z_{\delta })\right)$$ where $\langle \cdot,\cdot \rangle$ is the ordinary scalar product in $\mathbb{R}^n$. The $n+1$ component of the vector field is $$\pi_{n+1}(F_{\rho}(z_{\delta}))= -\delta^2\rho(\delta)\langle \pi(z_{\delta}), f(h^{-1}(z_{\delta})) \rangle.$$ From this, we obtain $$\label{eq:pi_k}
\pi\left(F_{\rho}(z_{\delta})\right)=
\frac{\delta^2\rho(\delta)f(h^{-1}(z_{\delta}))+ \pi_{n+1}(F_{\rho}(z_{\delta}))\pi(z_{\delta})}{\delta}.$$
Notice that the equator $E$ is invariant under the flows of the compactified vector field $F_{\rho}(z)$ when the last coordinate of $F_{\rho}(z)$ is zero, i.e. $\pi_{n+1}(F_{\rho}(z))=0,$ for every $z\in E$. In the next result we characterize the invariance of the equator in terms of the vector field $f$.
\[theo:invariant\_inf\_manifold0\] Assume that $f$ is a non-null Poincaré compactifiable Lipschitz continuous vector field, and let $F_{\rho}$ be the compactified vector field. The equator $E$ is invariant under the flow of $F_{\rho}$, if and only if for every $z\in E$, $$\label{eq:theo:invariant}
\lim_{\delta\to 0^+} \delta^2\rho(\delta)f(h^{-1}(z_{\delta}))=0,$$ where $z_{\delta}$ is given by .
We shall prove that for each $z\in E$, $\pi_{n+1}\left(F_{\rho}(z)\right)=0$ if and only if holds.
Suppose that $z\in E$ is such that $$\pi_{n+1}\left(F_{\rho}(z)\right)=\lim_{\delta \to 0^+} \pi_{n+1}\left(F_{\rho}(z_{\delta})\right)=0.$$ As the limit of $\pi(F_{\rho}(z_{\delta}))$ exists as $\delta \to 0^+$ and the denominator in tends to zero, then $$0=\lim_{\delta\to 0^+} \delta^2\rho(\delta)f(h^{-1}(z_{\delta}))+ z_{\delta } \pi_{n+1}(F_{\rho}(z_{\delta}))
=\lim_{\delta\to 0^+} \delta^2\rho(\delta)f(h^{-1}(z_{\delta})).$$
Conversely, assume holds. That is $$\lim_{\delta\to 0^+}\delta^2 \rho(\delta)\langle f(h^{-1}(z_{\delta})), \pi(z) \rangle =0.$$ Then $$\begin{split}
\pi_{n+1}\left(F_{\rho}(z)\right)=&\lim_{\delta \to 0^+} \pi_{n+1}\left(F_{\rho}(z_{\delta})\right)=-
\lim_{\delta \to 0^+} \delta^2\rho(\delta)\langle \pi(z_{\delta}), f(h^{-1}(z_{\delta})) \rangle\\
=& - \lim_{\delta \to 0^+} \delta^2\rho(\delta)\langle \pi(z), f(h^{-1}(z_{\delta})) \rangle=0.
\end{split}$$
Families of non-null Poincaré compactifiable vector fiels {#sec:3}
=========================================================
In this section we discuss three applications of our results: compactification of polynomial vector fields, polynomial-growth vector fields and piecewise linear vector fields.
Polynomial vector fields
------------------------
In this subsection we apply previous results to polynomial vector fields, in order to show they provide the classical Poincaré compactification, but desingularizing the vector field.
Let $f$ be a polynomial in $x_1,\dots,x_n$ of degree $N$. Then $f(x)=\sum_{|\alpha|\leq N} f_{\alpha}x^{\alpha}$, where $\alpha=(\alpha_1,\dots,\alpha_n)$, $\alpha_k \in\mathbb{Z_+}$, $|\alpha|=\sum_{j=1}^n \alpha_j$, $x^{\alpha}= x_1^{\alpha_1}\dots x_n^{\alpha_n}$, and $f_{\alpha} \in\mathbb{R}^n$. Since $f$ has degree $N$, we are assuming that $\sum_{|\alpha|= N} f_{\alpha}x^{\alpha}$ is not identically null.
\[Poincare\] If $f$ is a polynomial vector field of degree $N$, then it is a compactifiable vector field by the regularization function $\rho(\delta)= \delta^{N-1}$, and the equator is invariant under the flow of the compatified vector field.
For any given $z\in E$ we consider $z_{\delta}=z\sqrt{1-\delta^2}+\delta e^{n+1}$, $\delta\in (0,1]$, and the vector field $F_{\rho}(z_{\delta})$ on $H_+$. By taking into account that $h^{-1 }(z_{\delta})=\pi(z) \sqrt{1-\delta^2}/\delta$, we have $$f(h^{-1}(z_{\delta}))=\sum_{|\alpha|\leq N} f_{\alpha} \pi(z)^{\alpha} \frac {(1-\delta^2)^{\frac{|\alpha |}{2}}}{\delta^{|\alpha |}} =
\frac {(1-\delta^2)^{\frac{N}{2}}}{\delta^N} \sum_{|\alpha|= N} f_{\alpha} \pi(z)^{\alpha} +R(z,\delta),$$ where $R(z,\delta)$ is a function with $\lim_{\delta\to 0^+} \delta^N R(z,\delta)=0$.
From , $$\begin{split}
\lim_{\delta\to 0^+} \pi(F_{\rho}(z_{\delta}))&= \sum_{|\alpha|=N} \left( f_{\alpha}\pi(z)^{\alpha} - \langle \pi(z), f_{\alpha}\pi(z)^{\alpha} \rangle \pi(z)\right) \\
\lim_{\delta\to 0^+} \pi_{n+1} (F_{\rho}(z_{\delta}))&=0,
\end{split}$$ which implies that the vector field is compatificable and the equator is invariant under the flow.
\[Poincare2\] Let $f$ be a polynomial field of degree $N$ which can be compactified by a regularization function $\rho(\delta)$.
- The equator is invariant under the compactified flow if and only if $\lim_{\delta\to 0} \rho(\delta)\delta^{2-N}=0$.
- If $\rho(\delta)=\delta^{N-1}$ and the compactification is identically null, then the vector field $f$ can be compactified by the regularization function $\tilde{\rho}(\delta)=\delta^{N-2}$, but in this case the equator is not invariant under the compactified flow.
- The compactification by $\rho(\delta)=\delta^{N-1}$ is identically null if and only if the vector field is $f(x)=q(x) x + R(x)$, where $q(x)$ is a scalar homogeneous polynomial of degree $N-1$, and $R$ is a polynomial of degree strictly lower than $N$.
\(a) From Theorem \[theo:invariant\_inf\_manifold0\] the equator is invariant under the flow if and only if for every $z \in E$, $$0=\lim_{\delta\to 0^+} \delta^2\rho(\delta)f(h_+^{-1}(z_{\delta}))=\lim_{\delta\to 0} \delta^{2-N} (1-\delta^2)^{N/2} \rho(\delta) \sum_{|\alpha|=N} f_{\alpha} \pi(z)^{\alpha}.$$ The result follows straightforward since there exists $z_0\in E$ such that $\sum_{|\alpha|=N} f_{k\alpha} \pi(z_0)^{\alpha}\neq 0$, provided that the vector field has degree $N$.
\(b) Since $$\begin{aligned}
\pi(F_{\rho}(z_{\delta}))&= \delta^N \left(f(h^{-1}(z_{\delta}))-\biggl\langle \pi(z_{\delta}), f(h^{-1}(z_{\delta})) \biggl\rangle \pi(z_{\delta })
\right) \\
&= \left( (1-\delta^2)^{\frac{N}{2}} \sum_{|\alpha|=N} f_{\alpha} \pi(z)^{\alpha} + \delta^N R(z,\delta)- \biggl\langle \pi(z_{\delta}),
(1-\delta^2)^{\frac{N}{2}} \sum_{|\alpha|=N} f_{\alpha} \pi(z)^{\alpha} \biggl\rangle \pi(z_{\delta})
\right),\end{aligned}$$ it follows from the hypothesis that $$\label{eq:cnula}
\begin{split}
\lim_{\delta\to 0^+} \pi(F_{\rho}(z_{\delta}))=
\sum_{|\alpha|=N} f_{\alpha} \pi(z)^{\alpha}
- \biggl\langle \pi(z),\sum_{|\alpha|=N} f_{\alpha} \pi(z)^{\alpha} \biggr\rangle \pi(z_{\delta})=0.
\end{split}$$ Taking into account that $\pi(F_{\tilde{\rho}}(z_{\delta}))=\pi(F_{\rho}(z_{\delta}))/{\delta}$ in a similar way we obtain $$\lim_{\delta\to 0^+} \pi(F_{\tilde\rho}(z_{\delta}))=
\sum_{|\alpha|=N-1} f_{\alpha} \pi(z)^{\alpha}
- \biggl\langle \pi(z),\sum_{|\alpha|=N-1} f_{\alpha} \pi(z)^{\alpha} \biggr\rangle \pi(z).$$ Moreover, $$\begin{aligned}
\lim_{\delta\to 0^+} \pi_{n+1} (F_{\tilde{\rho}}(z_{\delta}))&= - \lim_{\delta\to 0^+}
\biggl\langle \pi(z_{\delta}), (1-\delta^2)^{\frac{N}{2}} \sum_{|\alpha|=N} f_{\alpha} \pi(z)^{\alpha} + \delta^N R(z,\delta)\biggr \rangle \\
&=-\biggl\langle \pi(z), \sum_{|\alpha|=N} f_{\alpha} \pi(z)^{\alpha} \biggr\rangle .\end{aligned}$$ Therefore, the vector field is compactifiable by the regularization function $\tilde{\rho}(\delta)$.
Since $\sum_{|\alpha|=N} f_{\alpha} \pi(z)^{\alpha}$ is not identically null, the equator is not invariant under the flow $\pi(F_{\tilde{\rho}}(z_{\delta}))$.
\(c) We shall denote $f^N$ to the homogeneous part of degree $N$, that is $$f^N(x)=\sum_{|\alpha|=N} f_{\alpha} x^{\alpha}.$$ Assume that the compactified vector field is identically null by $\rho(\delta)=\delta^{N-1}$. By we have $$f^N(z)=\langle \pi(z), f^N(z) \rangle \pi(z), \quad z\in E.$$ Let $x\in \mathbb{R}^n$, $y=x/\|x\|$. Then $$f^N(x)= \frac{\left\langle x,f^N(x) \right\rangle}{\|x\|^2} x.$$ Necessarily, $q(x)=\frac{\left\langle x,f^N(x) \right\rangle}{\|x\|^2}$ is a homogeneous polynomial of degree $N-1$.
Conversely, let $f^N(x)=x q(x)$, where $q$ is a scalar homogeneous polynomial of degree $N-1$. For every $z \in E$, $\|\pi(z)\|=1$ and then $$\lim_{\delta\to 0^+} \pi(F_{\rho}(z_{\delta}))= q(\pi(z))\pi(z)- \left\langle \pi(z),q(\pi(z))\pi(z)\right\rangle \pi(z) =0.$$
In Corollaries \[Poincare\] and \[Poincare2\] we have only used that $$f(x)= \sum_{l=0}^N f_{l}(x),$$ where $f_{l}(x)$ is a locally Lipschitz continuous homogeneous function of degree $l$, i.e. $$f_{l}(\lambda x) = \lambda^l f_{l}(x), \quad \lambda\in\mathbb{R},\ x\in\mathbb{R}^n.$$ See [@GPS].
Polynomial-growth vector fields
-------------------------------
Let $\mathbb{S}^{n-1}=\{x\in\mathbb{R}^n : \|x\|=1 \}$. If $f$ is locally Lipschitz continuous, then so is $$(x,\delta) \in \mathbb{S}^{n-1} \times (0,1] \to f\left( \frac{x}{\delta}\right).$$ In the following result we will use the next hypotheses:
There exists $N \in\mathbb{N}$ such that $$\label{h:1}
(x,\delta) \in \mathbb{S}^{n-1} \times (0,1] \to \delta^N f\left( \frac{x}{\delta}\right).$$ is globally Lipschitz continuous and there exists $$\label{h:2}
\omega_N(x)= \lim_{\delta \to 0^+}\delta^N f\left( \frac{x}{\delta}\right), \quad \text{uniformly in } x\in\mathbb{S}^{n-1} .$$
Assume and are satisfied. Then the vector field $f(x)$ is Poincaré compactifiable by the regularization function $\rho(\delta)=\delta^{N-1}$. Moreover the equator is invariant.
Considering $$\delta^N f\left( \frac{\pi(z)\sqrt{1-\delta^2}}{\delta}\right)= \left( 1-\delta^2\right)^{N/2}\left( \frac{\delta}{\sqrt{1-\delta^2}}\right)^N f\left( \frac{\pi(z)\sqrt{1-\delta^2}}{\delta}\right),$$ we obtain that $$\delta^N f\left( \frac{\pi(z)\sqrt{1-\delta^2}}{\delta}\right)$$ is globally Lipschitz continuous in $ E \times (0,1] $ and $$\lim_{\delta\to 0}\delta^N f\left( \frac{\pi(z)\sqrt{1-\delta^2}}{\delta}\right)=\omega_N(\pi(z)), \quad\text{ uniformly in } E.$$
By we obtain $$\begin{split}
\pi (F_{\rho}(z_{\delta}))&= \delta^N\left(f\left( \frac{\pi(z)\sqrt{1-\delta^2}}{\delta}\right)- \biggl\langle \pi(z_{\delta}), f\left( \frac{\pi(z)\sqrt{1-\delta^2}}{\delta}\right) \biggr\rangle \pi(z_{\delta })\right),\\
\pi_{n+1}(F_{\rho}(z_{\delta}))&= -\delta^{N+1}\biggl\langle \pi(z_{\delta}), f\left( \frac{\pi(z)\sqrt{1-\delta^2}}{\delta}\right) \biggr\rangle .
\end{split}$$ Taking limit as $\delta \to 0^+$, we have $$\begin{aligned}
\lim_{\delta\to 0^+}\pi (F_{\rho}(z_{\delta}))&=\omega_N(\pi(z))-\biggl\langle \pi(z),\omega_N(\pi(z))\biggr\rangle \pi(z), \\
\lim_{\delta\to 0^+}\pi_{n+1} (F_{\rho}(z_{\delta}))&=0. \end{aligned}$$ being both limits uniform in $E$. By Proposition \[prop:Poincare-compactifiable\] we get that $F_{\rho}(z)$ admits a Lipschitz continuous extension to $H_+ \cup E$, that is, the vector field $x'=f(x)$ is Poincaré compactifiable. Moreover the equator is invariant.
Piecewise polynomial vector fields
----------------------------------
In this subsection we apply previous results to a class of vector fields showing polynomial behaviour near the infinity, the piecewise polynomial vector fields. Then, we restrict ourselves to the case where the maximum degree of the involved polynomials is equal $1$, that is, the piecewise linear (PWL) vector fields. PWL vector fields have attracted the attention of different authors since they appeared in the work of Andronov et al [@AVK66]. Nowadays, different works use these systems to produce simple exemples of very complicated dynamical objects or to provide results which are not easy to prove in a general framework, see [@LT; @BBCK07] and references therein. Moreover, PWL differential system are used to model real systems (electronic circuits, neuronal behaviours, etc ...) in a framework which is more friendly for the analysis and less expensive computationally, see [@BBCK07].
Consider finitely many connected subsets with non-empty interior $S_i \subset \mathbb{R}^n$, $i=0,\dots,p$, such that $S_{i}\cap S_{j}=\emptyset$ if $i\neq j$ and $\bigcup_{i=0}^{p} {S}_i=\mathbb{R}^n$.
A function $f:\mathbb{R}^n\to \mathbb{R}^n$ is a [*piecewise polynomial vector field*]{} if there exist polynomials $f_i\in\mathbb{R}[x_1,\ldots,x_n]$, such that $$f(x)=\sum_{i=0}^{p} \chi_{_{S_i}}(x) f_i(x),$$ where $\chi_{_{S_i}}$ is the characteristic funtion of the set $S_i$.
Every polynomial piecewise continuous vector field is compactifiable for any projection $h$.
We will prove that the vector field is locally Lipschitz continuous, and we conclude by Theorem \[theorem:tc\].
Let $B$ a ball in $\mathbb{R}^n$, and take $x,\bar x\in B$. There exists $0=\alpha_0<\ldots<\alpha_m=1$ such that if $$x^i=\alpha_i x+(1-\alpha_i) \bar x$$ then $x^i,x^{i+1}\in \bar S_{j_i}$, for certain $0\leq j_i\leq m$.
Since the polynomials are Lipschitz in $B$, let $L$ be the maximum of their Lipschitz constants. Then $$\|f(x)-f(\bar x)\|\leq \sum_{i=0}^{m-1} \left\|f(x^{i})-f(x^{i+1})\right\|\leq
L \sum_{i=0}^{m-1} \left\|x^{i}-x^{i+1}\right\|=L\|x-\bar x\|.$$
Even when the boundaries of the piecewise polynomial vector field are $(n-1)$-dimensional algebraic manifolds and move away from the origin in a way which can be handled, the difference between the degrees of the involved polynomials can force the compactified vector field to be identically null at infinity. Next we introduce an example showing this behaviour.
Given the piecewise polynomial system $$f((x_1,x_2))=\left\{
\begin{array}{ll}
(x_2,\,x_1) & x_1\leq -1,\\
(x_1+x_2+x_1^2,\, x_1+x_2+x_1x_2) & |x_1|\leq 1,\\
(2x_1+x_2,\, x_1+2x_2) & x_1>1,
\end{array}
\right.$$ the projected vector field defined over the hemisphere $H_+$ is given by $$g(z)=\left\{
\begin{array}{ll}
g_{-}(z) & z_1\leq -z_{n+1},\\
g_0(z) & |z_1|\leq z_{n+1},\\
g_+(z) & z_1 \geq z_{n+1},
\end{array}
\right.$$ where $$g_-(z)=\begin{pmatrix}
z_2-2z_1^2z_2\\
z_1-2z_1z_2^2\\
-2 z_1z_2z_3
\end{pmatrix},\quad
g_0(z)=\begin{pmatrix}
z_1+z_2+\frac{z_1^2}{z_3}-z_1^3-z_1z_2^2-2z_1^2z_2-\frac{z_1^4}{z_3}-\frac {z_1^2z_2^2}{z_3}\\
z_1+z_2+\frac{z_1z_2}{z_3}-z_1^2z_2-2z_1z_2^2-z_2^3-\frac{z_1^3z_2}{z_3}-\frac{z_1z_2^3}{z_3}\\
-(z_1^3+z_1^2z_3+z_1z_2^2+2z_1z_2z_3+z_2^2z_3)
\end{pmatrix},$$ and $$g_+(z)=\begin{pmatrix}
2 z_1 - 2 z_1^3 + z_2 - 2 z_1^2 z_2 - 2 z_1 z_2^2\\
z_1 + 2z_2 - 2z_1^2z_2 - 2z_1z_2^2 - 2z_2^3\\
-2z_1^2z_3 - 2z_1z_2z_3 - 2z_2^2z_3
\end{pmatrix}.$$ From the expression of $g_0(z)$, the regularization function must be $\rho(\delta)=\delta$. Therefore, the regularized vector field $\rho(z_{n+1})g(z)$ extends continuously to the equator, but it is identically null at the equator.
At this point, we restrict our attention to piecewise polynomial vector fields such that all the polynomial vector fields $f^i$ having the same degree $N$. In particular, we consider the case $N=1$, which corresponds with the family of the piecewise linear (PWL) vector fields.
Consider finitely many connected subsets with non-empty interior $S_i \subset \mathbb{R}^n$, $i=0,\dots,p$, such that $S_{i}\cap S_{j}=\emptyset$ if $i\neq j$, $\bigcup_{i=0}^{p} {S}_i=\mathbb{R}^n$ and $\sum_{ij}=\bar{S}_i \cap \bar{S}_j$ is either an $(n-1)$-dimensional manifold or is the empty set.
A function $f:\mathbb{R}^n\to \mathbb{R}^n$ is a [*piecewise linear vector field*]{} if there exist matrices of order $n$, $A_i$, and vectors, $b_i\in \mathbb{R}^n$, such that $$f(x)=\sum_{i=0}^{p} \chi_{_{S_i}}(x) (A_i x +b_i),$$ where $\chi_{_{S_i}}$ is the characteristic funtion of the set $S_i$.
Next, we rewrite PWL vector fields in a form, called the Lure’s form (see Lemma \[Lure\](c)-(d)), which can be considered suitable for the compactification process.
\[Lure\] Consider a continuous piecewise linear vector field $f$.
- Then $\sum_{ij}$ is an affine subspace of dimension $n-1$.
- Given two different boundaries $\sum_{ij}$ and $\sum_{i'j'}$, then $\sum_{ij} \cap \sum_{i'j'}= \emptyset$.
- There exist $\tau_1<\dots<\tau_p$, a continuous piecewise funtion $\varphi:\mathbb{R} \to \mathbb{R}$ given by $$\varphi(\sigma)=
\left\{
\begin{array}{ll}
\alpha_{0}\sigma+\beta_{0} & \sigma \leq \tau_{1},\\
\alpha_i\sigma+\beta_i & \sigma \in [\tau_i,\tau_{i+1}],\\
\alpha_{p}\sigma+\beta_{p} & \sigma \geq \tau_{p},
\end{array}
\right.$$ a $n\times n$ matrix $A$, and vectors $k,b\in \mathbb{R}^n$ such that $$f(x)=Ax + \varphi(k^Tx)b.$$
- There exists a linear change of coordinates $y=Mx$ such that $$f(y)=\bar{A}y+\varphi(e_1^Ty)\bar{b},$$ where $\bar{A}=M^{-1}AM$, $\bar{b}=M^{-1}b$ and $e_1$ is the first element of the cannonical base of $\mathbb{R}^n.$
Since the vector field is continuous, the boundary $\sum_{ij}$ can be written as the $x\in\mathbb{R}^n$ such that $(A_{i}-A_{j})x=(b_{j}-b_i)$, which proves (a).
Assuming that $\sum_{ij} \cap \sum_{i'j'}\neq \emptyset$, it follows that either $\sum_{ij'}$ or $\sum_{i'j}$ is a $n-2$-dimensional affin manifold, which contradicts the hypothesis, so we conclude statement (b).
From statements (a) and (b), it follows that there exists a vector $k\in \mathbb{R}^n$ such that every boundary $\sum_{ij}$ can be written as $\sum_{ij}=\{x\in \mathbb{R}^n: k^Tx=\tau_{ij}\}$.
Reindexing the regions and the boundaries if necessary, we consider $\tau_{1}<\dots<\tau_{p}$ and the boundaries $\sum_{i}=\{x\in \mathbb{R}^n: k^Tx=\tau_{i}\}$.
Denote $j$ to the index of the region $S_j$ containing the origin. Let $A=A_j$ and $b=b_j$. Therefore, the piecewise linear function $\varphi(\sigma)$ defined as $$\varphi(\sigma)=
\left\{
\begin{array}{ll}
\alpha_{0}\sigma+\beta_{0} & \sigma \leq \tau_{1},\\
\alpha_i\sigma+\beta_i & \sigma \in [\tau_i,\tau_{i+1}],\\
\alpha_{p}\sigma+\beta_{p} & \sigma \geq \tau_{p},
\end{array}
\right.$$ can be obtained from the following equation $$\begin{aligned}
b_k&=\beta_k\,b,\\
A_k&=A+\alpha_k\,b\,k^T,
\end{aligned}$$ since from the last equation we conclude that $(A_k-A_j)x=\alpha_k\,b\,\tau_k$ for $x\in \sum_k$, which proves statement (c).
The compactification of the continuous PWL vector fields has been also addressed in some papers. Nevertheless, every time just for a particular group of these vector fields [@LL18; @LT]. Next we consider the general case. Considering the PWL system in the Lure’s form given in Lemma \[Lure\](d), from , the projected vector field writes as $$z'=g(z)=
\left(
\begin{array}{c} I-\pi(z) \pi(z)^T \\ -z_{n+1}\pi(z)^T \end{array}
\right)
\left(A\, \pi(z) + z_{n+1}\varphi\left(\frac {z_1}{z_{n+1}}\right) b\right).$$ which is already defined in $H_+\cup E$, so, in this case, the regularization function is $\rho(\delta)=1$.
Notice that the boundary $\sum_i=\{x\in\mathbb{R}^n: \varphi(e_1^Tx)=\tau_i\}$ at the half-sphere $H_+\cup E$ is given by $z\in H_+\cup E$ such that $z_1-z_{n+1}\tau_i=0$ with $i=0,\dots,p$, which extends continuously to the equator $E$ as the $\mathbb{S}^{n-2}$ given by $z_1=0, z_{n+1}=0$. Moreover, for those $z\in H_+\cup E$ such that $z_1 \neq 0$ the compactified vector field at the equator writes as $$\label{PWL_equator}
z'=g(z)=
\left(
\begin{array}{c} I-\pi(z) \pi(z)^T \\ 0^T \end{array}
\right)
\left(A +\alpha_{i}\, b e_{1}^T \right)\pi(z),
\quad \text{with }i\in\{0,p\}.$$ Since $A_{0}=A +\alpha_{0}\, be_{1}^T$ and $A_{p}=A +\alpha_{p}\, be_{1}^T$ are the matrices of the linear systems defined in the external domains, these systems play a relevant role in the compactification process.
Consider a continuous PWL vector field in the Lure’s form $$f(x)=Ax+\varphi(e_1^T x)b,$$ and let $A_{0}$ and $A_{p}$ be the matrices of the linear systems in the external domains. The compactification of the vector field is identically null if and only if both matrices $A_{0}$ and $A_{p}$ are diagonalizable and the diagonal matrices are $\lambda_{0}I$ and $\lambda_{p}I$ repectively.
Let us consider $z\in H_+\cup E$ with $z_1>0$. The case $z_1<0$ follows in a similar way. Assuming that $A_{p}$ is diagonalizable and the diagonal matrix is $\lambda_{p}\,I$, it follows $A_{p}\pi(z)=\lambda_{p}\pi(z)$ for every $z$. From , the expression of the vector field at the equator is $$z'=g(z)=
\left( \begin{array}{c} ( I -\pi(z) \pi(z)^T) A_{p} \pi(z) \\0^T \end{array} \right)=
\left( \begin{array}{c} A_{p} \pi(z) - \lambda_{p} \pi(z) \|\pi(z)\|^2 \\0^T \end{array} \right),$$ which is identically null, since $\|\pi(z)\|=1.$
Conversely, suppose that the vector field is identically null, then $$( I -\pi(z) \pi(z)^T) A_{p} \pi(z) =0,$$ for every $z\in H_+\cup E$ with $z_1>0$. Hence, $$\begin{aligned}
A_{p} \pi(z) &=& \pi(z) \pi(z)^T A_{p} \pi(z)\\
&=&( \pi(z)^T A_{p} \pi(z)) \pi(z).\end{aligned}$$ Therefore $\pi(z)^T A_{p} \pi(z)$ is the eigenvalue of $\pi(z)$ for every $z$, which implies that every direcction is a eigenvector. We conclude that the matrix $A_{p}$ is diagonalizable and the diagonal matrix is $\lambda\, I$.
Acknowledgments {#acknowledgments .unnumbered}
===============
The three authors are supported by Ministerio de Economía y Competitividad through the project MTM2017-83568-P (AEI/ERDF, EU). The first and second authors are also partially supported by the Junta de Extremadura/FEDER grants numbers GR18023 and IB18023.
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| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'André Coimbra,'
- 'Charles Strickland-Constable'
title: 'Generalised Structures for $\mathcal{N}=1$ AdS Backgrounds'
---
Introduction {#sec:intro}
============
In the context of the study of flux compactifications of string theory, the problem of describing possible supersymmetric Anti-de Sitter solutions has acquired central importance with the discovery of the AdS/CFT correspondence [@Maldacena:1997re]. Substantial progress has been achieved in understanding the geometry of such backgrounds of M theory and Type II using the tools of $G$-structures (see for instance [@Gauntlett:2002sc; @Gauntlett:2002fz; @Martelli:2003ki; @Behrndt:2004bh; @Lukas:2004ip; @Gauntlett:2004zh; @Lust:2004ig; @Gauntlett:2005ww; @Gabella:2012rc]) and generalised geometry (for example in [@Minasian:2006hv; @Gabella:2009wu; @Gabella:2010cy; @Apruzzi:2013yva; @Apruzzi:2014qva; @Apruzzi:2015zna]).
Combining both approaches, it was shown in a recent work [@CSW4] that it is possible to characterise fully generic minimally supersymmetric compactifications to $D\geq4$-dimensional Minkowski space by a novel integrability condition, formulated in the language of $\operatorname{\mathit{E_{d(d)}}}\times{\mathbb{R}}^+$ generalised geometry [@chris; @PW; @CSW2] (throughout $d=11-D$ and $\operatorname{\mathit{E_{d(d)}}}$ is the real split form of the rank $d$ exceptional Lie group). Concretely, it was proven that the Killing spinor equations constrain precisely the intrinsic torsion of the generalised $G$-structure defined by the Killing spinor on the generalised tangent bundle. In other words, the compactification space must be the generalised analogue of a special holonomy manifold, as setting the supersymmetry variations of the fermions to zero is equivalent to demanding that the intrinsic torsion vanishes.
Let us briefly recall the key features of exceptional generalised geometry for the description of supersymmetric backgrounds of Type II and M theory. Given a Riemannian spin manifold $M$ with $d\leq 7$ dimensions for M theory or $d-1\leq 6$ dimensions for Type II,
- we introduce the generalised tangent bundle $E$, which enlarges the usual tangent bundle to also accommodate the symmetries of the supergravity gauge fields [@PW; @chris];
- the bundle $E$ has the structure group $\operatorname{\mathit{E_{d(d)}}}\times{\mathbb{R}}^+$, so we can construct generalised tensors associated to $\operatorname{\mathit{E_{d(d)}}}\times{\mathbb{R}}^+$ representations [@PW; @CSW2];
- there is a differential structure on $E$ described by the exceptional Dorfman bracket, which generates both diffeomorphisms and gauge transformations [@PW];
- we can define generalised connections ${{D}}_V$ to take derivatives of generalised tensors along generalised vectors $V\in{\Gamma(E)}$, and a natural notion of generalised torsion tensor determined by the Dorfman bracket [@CSW2];
- there exists a generalised metric on $E$, whose components unify all the bosonic fields in the theory, and which is invariant under transformations of the maximal compact subgroup $H_d\subset\operatorname{\mathit{E_{d(d)}}}$ which generalises orthogonal transformations [@PW; @chris];
- the double cover of this local group, $\operatorname{\mathit{\tilde{H}_d}}$, can be realised as a subgroup of the Clifford algebra $\operatorname{Cliff}(d;{\mathbb{R}})$, and so we can think of spinors as representations of $\operatorname{\mathit{\tilde{H}_d}}$ [@CSW2; @CSW3];
- the existence of a globally non-vanishing spinor $\epsilon$ on $M$ defines a subgroup $G\subset \operatorname{\mathit{\tilde{H}_d}}$ which stabilises it, ie. a generalised $G$-structure [@PW; @GO; @GO2; @CSW3];
- there exists a torsion-free generalised connection compatible with $G$ if and only if $M$ is the internal space of a minimally supersymmetric Minkowski flux background [@CSW4].
Some subtleties aside, this last result was obtained simply by looking at the relevant groups and representations, which are summarised in table \[table\].
$d$ $\operatorname{\mathit{\tilde{H}_d}}$ $G$ ${{W}_{\text{int}}}\simeq$ KSEs
----- --------------------------------------- --------------------------------------------------------------------- --------------------------------------------------------------------------------------------------
7 $\operatorname{\mathit{SU}}(8)$ $\operatorname{\mathit{SU}}(7)$ ${\mathbf{1}} + {\mathbf{7}} + {\mathbf{21}}+ {\mathbf{35}} + {\text{c.c.}}$
6 $ \operatorname{\mathit{USp}}(8)$ $\operatorname{\mathit{USp}}(6)$ $2\cdot{\mathbf{1}} + 2\cdot{\mathbf{6}} + 2\cdot{\mathbf{14}} + {\mathbf{14'}} + {\text{c.c.}}$
5 $\operatorname{\mathit{USp}}(4)^2$ $\operatorname{\mathit{USp}}(2)\cdot\operatorname{\mathit{USp}}(4)$ $2\cdot{({\mathbf{1}}, {\mathbf{4}})} + 2\cdot{({\mathbf{2}}, {\mathbf{4}})} + {\text{c.c.}}$
4 $\operatorname{\mathit{USp}}(4)$ $\operatorname{\mathit{USp}}(2)$ $4\cdot{\mathbf{1}} + 5\cdot{\mathbf{2}} + 2\cdot{\mathbf{3}} + {\text{c.c.}}$
: Generalised spin group $\operatorname{\mathit{\tilde{H}_d}}$; stabiliser group $G\subset\operatorname{\mathit{\tilde{H}_d}}$ of the Killing spinor; and the space of intrinsic torsions ${{W}_{\text{int}}}$ of the generalised $G$-structure, which was proven to match the decomposition of the Killing spinor equations (KSEs). []{data-label="table"}
It was further claimed in [@CSW4] that generic minimally supersymmetric AdS compactifications could be similarly described, now by keeping certain singlet components of the generalised intrinsic torsion as a non-zero constant, corresponding to the cosmological constant $\Lambda$. We thus have that
> *The minimally supersymmetric $D\geq 4$ AdS backgrounds are in one-to-one correspondence with weak generalised $G$ holonomy spaces, with singlet torsion given by the cosmological constant and where $G=\operatorname{\mathit{SU}}(7), \operatorname{\mathit{USp}}(6), \operatorname{\mathit{USp}}(2)\times\operatorname{\mathit{USp}}(4), \operatorname{\mathit{USp}}(2)$ in dimensions $D=4,5,6,7$ respectively.*
In the following we will clarify this statement and demonstrate it explicitly.
AdS backgrounds
===============
We consider generic supersymmetric flux compactifications of M theory and Type II string theory to four- and higher-dimensional AdS space. This means we have the warped metric ansatz $$\label{eq:g}
{\mathrm{d}}s^2 = {\mathrm{e}}^{2A}{\mathrm{d}}s^2(\text{AdS}_D) + {\mathrm{d}}s^2(M) ,$$ with $D\geq 4$ and where the warp factor $A$ is a scalar function of the internal coordinates. The internal space $M$ is a spin manifold with Riemannian metric $g$, of dimension $d$ in M theory and $d-1$ in Type II. To match the conventions of [@CSW2; @CSW3], we take $A=\Delta$ in M theory, so that $A=\Delta+\frac{1}{3}\phi$ in Type II, where $\phi$ is the dilaton, and the metric is in the string frame. For the fluxes we keep only the components consistent with the $D$-dimensional AdS symmetry. Fermion fields are set to zero and we work in the supergravity limit $\alpha'=0$.
Killing spinor equations
------------------------
In a supersymmetric AdS space one has Killing spinors $\eta$ which must satisfy $$\label{eq:ads-kse}
\begin{aligned}
{\nabla}_\mu \eta_{\pm} &= \tfrac12 {\mathrm{e}}^{\pm 2{\mathrm{i}}\theta} \Lambda \gamma_\mu \eta_{\mp}, &\quad \text{in } D = 4,\\
{\nabla}_\mu \eta^A &= \tfrac12 M^A{}_B \Lambda \gamma_\mu \eta^B, &\quad \text{in } D = 5,\\
{\nabla}_\mu \eta_\pm^A &= \tfrac12 (N^{\pm1}){}^A{}_B \Lambda \gamma_\mu \eta_\mp^A, &\quad \text{in } D = 6,\\
{\nabla}_\mu \eta^A &= \tfrac12 \Lambda \gamma_\mu \eta^A, &\quad \text{in } D = 7,\\
\end{aligned}$$ where ${\nabla}$ is the Levi–Civita connection in $\text{AdS}_D$, $A,B$ are the $\operatorname{\mathit{SU}}(2)$ indices of the symplectic Majorana and Majorana-Weyl spinors in $D=5,7$ and $D=6$ respectively, and $\pm$ subscripts denote chirality in even dimensions under the top gamma matrix $\gamma^{(D)}$.[^1]
In order to write these as R-symmetry covariant equations we have included a constant arbitrary phase $\theta$ in $D=4$, a constant $2\times2$ traceless Hermitian matrix, which squares to the identity, $M^A{}_B$ rotating the sympletic Majorana spinors in $D=5$, and for $D=6$, the matrix $N^A{}_B$ which is a constant element of $\operatorname{\mathit{SU}}(2)$. Usually these equations are written with particular values for $\theta, M,N$. One could, for example, rotate the spinors $\eta$ by R-symmetry transformations to chose $\theta = 0$, $M = \sigma^3$ and $N = {\mathbbold{1}}$. Doing so explicitly breaks the Minkowski R-symmetry group and allows us to directly obtain the surviving R-symmetry in AdS. For instance, of the full $U(1)$ R-symmetry in $D=4$ we see that only a residual $\mathbb{Z}_2$ would remain, while in $D=7$ the equation is actually invariant under the full $\operatorname{\mathit{SU}}(2)$, so the R-symmetry stays the same as in flat space.
We must now tensor these with the internal Killing spinors to obtain a supersymmetric solution of the full higher-dimensional theory. For concreteness we will focus on the M theory description, though our analysis covers the Type II cases straightforwardly [@CSW2; @CSW3; @CSW4].
Given spinors $\epsilon$ on the internal space $M$, we construct an eleven-dimensional spinor $\varepsilon^-$ as $$\begin{aligned}
\varepsilon^- &= \eta_+ \otimes \epsilon + \eta_- \otimes \epsilon^*, &\quad \text{in } D = 4,\\
\varepsilon^- &= \epsilon_{AB} \, \eta^A \otimes \epsilon^B, &\quad \text{in } D = 5,\\
\varepsilon^- &= \epsilon_{AB} \, \eta_+^A \otimes \epsilon_1^B
+ \epsilon_{AB} \, \eta_-^A \otimes \epsilon_2^B, &\quad \text{in } D = 6,\\
\varepsilon^- &= \epsilon_{AB} \, \eta^A \otimes \epsilon^B, &\quad \text{in } D = 7.\\
\end{aligned}$$ Using the definitions of [@CSW3] for the fermionic fields, we find the internal components of the Killing spinor equations for the eleven-dimensional supersymmetry parameter $\varepsilon^-$ can then be neatly written in all dimensions as $$\label{eq:AdS-susy-ferm}
\begin{aligned}
\Big[
{\nabla}_m + \tfrac{1}{288} F_{n_1 \dots n_4} \left(
\Gamma_m{}^{n_1 \dots n_4}
- 8 \delta_{m}{}^{n_1} \Gamma^{n_2 n_3 n_4}\right)
- \tfrac{1}{12} \tfrac{1}{6!} \tilde{F}_{mn_1 \dots n_6}
\Gamma^{n_1 \dots n_6}
\Big] \varepsilon^- & = 0,\\
\Big[
\slashed{{\nabla}} - \tfrac{1}{4} \slashed{F}
- \tfrac{1}{4} \slashed{\tilde{F}}
+ \tfrac{D-2}{2} (\slashed{{\partial}} \Delta)
\Big] \varepsilon^-
+ \tfrac{D-2}{2} \Lambda \varepsilon^+ & = 0,
\end{aligned}$$ where $F, \,{{\tilde{F}}}$ are the internal four- and seven-form fluxes respectively, $\Gamma_{m}$ are now $\operatorname{Cliff}(10,1)$ gamma matrices, and we define $\varepsilon^+$ by $$\label{eq:epsilon-plus}
\begin{aligned}
\varepsilon^+ &= {\mathrm{e}}^{-2{\mathrm{i}}\theta} \eta_+ \otimes \epsilon^*
+ {\mathrm{e}}^{2{\mathrm{i}}\theta} \eta_- \otimes \epsilon, &\quad \text{in } D = 4,\\
\varepsilon^+ &= -M_{AB} \, \eta^A
\otimes ( {\mathrm{i}}\gamma^{(6)} \epsilon^B), &\quad \text{in } D = 5,\\
\varepsilon^+ &= N_{AB} \, \eta_+^A \otimes \epsilon_2^B
+ (N^{-1})_{AB} \, \eta_-^A \otimes \epsilon_1^B, &\quad \text{in } D = 6,\\
\varepsilon^+ &= \epsilon_{AB} \, \eta^A \otimes (\gamma^{(4)}\epsilon^B),
&\quad \text{in } D = 7.\\
\end{aligned}$$ with $M_{AB} = \epsilon_{AC} M^C{}_B$. The reason for the choice of the superscripts $\varepsilon^{\pm}$ is that, as we will discuss momentarily (see also [@CSW3]), they can be viewed as conjugate representations of $\operatorname{\mathit{Spin}}(D-1,1)\times\operatorname{\mathit{\tilde{H}_d}}$. Similar variables were identified in the earlier works [@Martelli:2003ki; @Gauntlett:2004zh].
In the following we can actually skip the discussion of $D = 6, \,d=5$ since we are only interested in backgrounds with minimal supersymmetry, and there is no such compactification to AdS${}_6$ [@Nahm:1977tg]. Note that this is perfectly compatible with our generalised intrinsic torsion analysis – we can see in table \[table\] that $D=6$ is the only case where the torsion contains no singlets, and thus cannot possibly accommodate the cosmological constant. We will discuss backgrounds with higher supersymmetry in a forthcoming paper [@future].
We can now decompose the full eleven-dimensional Killing spinor equation to obtain the conditions on the internal spinor, and thus on the geometry of the internal manifold. It is convenient at this point to make a choice of R-symmetry frame for the external spinors. This allows us write the internal conditions in terms of complex internal spinors, but breaks the external R-symmetry. For the $D=4$ case, we take the R-symmetry frame with $\theta = 0$ and simply write the equations for the complex internal spinor $\epsilon$ and do not write the conjugate equations for $\bar\epsilon$. For the $D=5$ case, we perform an $\operatorname{\mathit{SU}}(2)$ rotation to diagonalise the matrix $M^A{}_B$ to become $\sigma^3$. We may then write equations for the first “half" $\epsilon \equiv \epsilon^1$ of the symplectic Majorana spinor $\epsilon^A$, but omit those for $\epsilon^2$, which follow by conjugating those for $\epsilon^1$. Similarly, we choose to write equations also only for $\epsilon \equiv \epsilon^1$ in the $D=7$ case, though this time we do not have to make any choice of R-symmetry frame to do so.
Decomposing thus leads to the internal equations $$\label{eq:ads-kse-int}
\begin{aligned}
{\nabla}_m \epsilon + \tfrac{1}{288}
(\gamma_m{}^{n_1 \dots n_4} - 8 \delta_{m}{}^{n_1} \gamma^{n_2 n_3 n_4})
F_{n_1 \dots n_4} \epsilon - \tfrac{1}{12} \tfrac{1}{6!} \tilde{F}_{mn_1 \dots n_6} \gamma^{n_1 \dots n_6} = 0, \quad & \text{in } d = 4,6,7,\\
\slashed{{\nabla}} \epsilon + (\slashed{{\partial}} \Delta) \epsilon - \tfrac{1}{4} \slashed{F}\epsilon - \tfrac{1}{4} \slashed{\tilde{F}} \epsilon +\Lambda\epsilon^* = 0,\quad & \text{in } d = 7,\\
\slashed{{\nabla}} \epsilon + \tfrac{3}{2}(\slashed{{\partial}} \Delta) \epsilon - \tfrac{1}{4} \slashed{F}\epsilon - \tfrac{1}{4} \slashed{\tilde{F}} \epsilon -\tfrac32\Lambda{\mathrm{i}}\gamma^{(6)}\epsilon = 0,\quad & \text{in } d = 6,\\
\slashed{{\nabla}} \epsilon + \tfrac{5}{2}(\slashed{{\partial}} \Delta) \epsilon - \tfrac{1}{4} \slashed{F}\epsilon - \tfrac{1}{4} \slashed{\tilde{F}} \epsilon +\tfrac52\Lambda\gamma^{(4)}\epsilon = 0,\quad & \text{in } d = 4.\\
\end{aligned}$$ These are then the AdS background Killing spinor equations we wish to examine.
As for the Minkowski case, these equations imply (see e.g. [@Lukas:2004ip; @Gauntlett:2004zh] for the cases of $d=6,7$) that the internal spinor is normalised as $$\label{eq:spinor-norm}
|| \epsilon ||{}^2 = \epsilon^\dagger \epsilon = {\mathrm{e}}^\Delta.$$
### Generic form in generalised geometry
Let us start by rewriting these in the compact language of generalised geometry, which makes their larger local $\operatorname{\mathit{\tilde{H}_d}}$ symmetry manifest. For notational convenience, a first step is to introduce the rescaled spinor variables which are more naturally adapted to the $\operatorname{\mathit{\tilde{H}_d}}$ symmetry [@CSW3] $$\begin{aligned}
\hat{\varepsilon}^\pm = {\mathrm{e}}^{-\Delta/2} {\varepsilon}^\pm ,
\end{aligned}$$ for the eleven-dimensional spinors and $$\begin{aligned}
\hat\epsilon^- = {\mathrm{e}}^{-\Delta/2} \epsilon ,
\end{aligned}$$ for the internal spinors.
A subtle point to note here is that, as is discussed in appendix **B** of [@CSW3], there are generically two ways of realising $\operatorname{\mathit{\tilde{H}_d}}$ in $\operatorname{Cliff}(d;{\mathbb{R}})$, related by taking $\gamma^a\rightarrow -\gamma^a$, leading to two generically inequivalent spinor bundles $S^+$ and $S^-$. For instance in $d=7$, we have $\tilde{H}_7=SU(8)$ and $S^-$ is associated to the ${\mathbf{8}}$ of $SU(8)$ and $S^+$ to the ${\mathbf{\bar{8}}}$. In even dimensions these are actually isomorphic, $S^+\simeq S^-$, with the isomorphism given by the top gamma $\gamma^{(d)}$, for example in $d=4$ we have that $\hat\epsilon^+=\gamma^{(4)}\hat\epsilon^-$. With this in mind, we can then introduce torsion-free generalised spin-connections $D$ and find that the Killing spinor equations for AdS backgrounds become, in manifestly $\operatorname{\mathit{\tilde{H}_d}}$-invariant form and for all dimensions $$\label{eq:gen-kse}
\begin{aligned}
{{D}}{\times_{J^-}} \hat\epsilon^- &= 0 ,\\
{{D}}{\times_{S^+}} \hat\epsilon^- &= -\tfrac{9-d}{2} \Lambda \hat\epsilon^+,
\end{aligned}$$ where ${\times_{X}}$ denotes projection to the $X$ representation and $J$ is the representation of the vector-spinor in $d$-dimensions. We list the precise forms of these generic equations in the next section.
For completeness, we note that one can also write in terms of undecomposed eleven-dimensional spinors. The group $\operatorname{\mathit{Spin}}(D-1,1)\times\operatorname{\mathit{\tilde{H}_d}}$ can be embedded in $\operatorname{Cliff}(10,1;{\mathbb{R}})$, again in two different ways related by the overall sign of the gamma matrices. Labelling the representations corresponding to the spinors $\hat{S}^\pm$ and those for the vector-spinors $\hat{J}^\pm$, as in [@CSW3], we have $$\label{eq:gen-kse-11d}
\begin{aligned}
{{D}}{\times_{\hat{J}^-}} \hat\varepsilon^- &= 0 ,\\
{{D}}{\times_{\hat{S}^+}} \hat\varepsilon^- &= -\tfrac{9-d}{2} \Lambda \hat\varepsilon^+.
\end{aligned}$$
Generalised structures with singlet torsion
-------------------------------------------
The result now follows almost immediately. In [@CSW4], it was shown that the left-hand side of matches exactly the intrinsic torsion $T_{\text{int}}$ of the generalised $G$-structure[^2] as listed in table \[table\]. The right-hand side, which simply vanished in the Minkowski case, now contains just the cosmological constant multiplying the ($G$-invariant) Killing spinor. Therefore, the Killing spinor equations are precisely equivalent to setting a singlet component of the intrinsic torsion to be proportional to $\Lambda$ and all other components to zero. In other words, the generalised connection that is compatible with the $G$-structure is not a torsion-free $D$ like in the Minkowski case, but $D+\Lambda$ instead, a connection with singlet torsion. One should thus think of the internal manifold as the generalised analogue of a manifold with weak special holonomy.
We can say a bit more about which singlet in the torsion in particular corresponds to the cosmological constant by looking in detail at each dimension. We skip $D=6$ since, as mentioned, there is no minimally supersymmetric AdS background there.
The problem is simplified by noting that in we find that the cosmological constant must lie in an $\operatorname{\mathit{\tilde{H}_d}}$ representation of the torsion that appears in the $S^+$ equation but not in the $J^-$ one (otherwise it would appear in the right-hand side of both equations).[^3]
Another observation is that one expects the relevant singlet in the intrinsic torsion to break the R-symmetry from the $D$-dimensional Minkowski group to the $D$-dimensional AdS group. For Minkowski compactifications, the R-symmetry group arises as the commutant of the $G$ structure group inside $\operatorname{\mathit{\tilde{H}_d}}$. We therefore look to identify a singlet which transforms under this commutant group, exactly as in , which is stabilised by the relevant AdS subgroup in the same way.
For $D=4$, the spinors $\hat\epsilon \equiv \hat\epsilon^-$ transform in the ${\mathbf{8}}$ of $\operatorname{\mathit{SU}}(8)$, while $\hat\epsilon^+ = \bar{\hat\epsilon}$ transform in the ${\mathbf{\bar8}}$. The Killing spinor equations become explicitly [@CSW3; @CSW4] $$\begin{aligned}
({{D}}{\times_{J^-}} \hat\epsilon^-){}^{[\alpha\beta\gamma]} &= {{D}}^{[\alpha\beta}\hat\epsilon^{\gamma]} = 0,\\
({{D}}{\times_{{S}^+}} \hat\epsilon^-){}_\alpha
&= -{{D}}_{\alpha\beta} \hat\epsilon^\beta = - \Lambda \bar{\hat\epsilon}_\alpha .
\end{aligned}$$ The representations of the torsion which appear in the second equation but not the first are the ${\mathbf{\bar{28}}}+{\mathbf{\bar{36}}}$ (ie. objects with, respectively, two lower anti-symmetric and symmetric indices). However, the ${\mathbf{\bar{28}}}$ also appears in the conjugate gravitino variation ${{D}}_{[\alpha\beta}\bar{\hat\epsilon}_{\gamma]} = 0$, so this cannot contain the cosmological constant term. The Killing spinor is stabilised by an $\operatorname{\mathit{SU}}(7)\subset\operatorname{\mathit{SU}}(8)$ subgroup and the decomposition of the ${\mathbf{\bar{36}}}$ contains a singlet, so this must be the cosmological constant. Now we note that the commutant of $\operatorname{\mathit{SU}}(7)$ in $\operatorname{\mathit{SU}}(8)$ is $U(1)$, and the singlet resulting from the symmetric two-index ${\mathbf{\bar{36}}}$ will carry a charge ${\mathbf{2}}$ under this $U(1)$. In fact, looking back at equations and , one can see that this singlet is essentially ${\mathrm{e}}^{-2 {\mathrm{i}}\theta} \Lambda$ from equation , but now viewed as transforming under the $U(1)$ commutant of $\operatorname{\mathit{SU}}(7)$. It will therefore be stabilised by a $\mathbb{Z}_2\subset U(1)$ subgroup, ie. precisely the R-symmetry group of $\mathcal{N}=1$ AdS${}_4$.
For $D = 5$, the generalised spin group is $\tilde{H}_6=\operatorname{\mathit{USp}}(8)$, with spinors transforming in the ${\mathbf{8}}$. In $\operatorname{\mathit{USp}}(8)$ indices, the Killing spinor equations are explicitly $$\label{eq:gen6-kse}
\begin{aligned}
({{D}}{\times_{J^-}} \hat\epsilon^-){}^{[\alpha\beta\gamma]}
&= {{D}}^{[\alpha\beta}\hat\epsilon^{\gamma]}
+ \tfrac13 C^{[\alpha\beta} {{D}}^{\gamma]}{}_\delta \hat\epsilon^\delta = 0,\\
({{D}}{\times_{{S}^+}} \hat\epsilon^-){}_\alpha
&= {{D}}_{\alpha\beta} \hat\epsilon{}^\beta
= \tfrac{3}{2} \Lambda \hat\epsilon_\alpha ,
\end{aligned}$$ where $C^{\alpha\beta}$ is the symplectic invariant. The only $\operatorname{\mathit{USp}}(8)$-irreducible component of torsion that constrains the second equation but not the first transforms in the ${\mathbf{36}}$.
The special holonomy group here is $\operatorname{\mathit{USp}}(6)\subset \operatorname{\mathit{USp}}(8)$, and the decomposition of the ${\mathbf{36}}$ does indeed contain three singlets, one of which which we can thus identify as the cosmological constant. The commutant subgroup in turn is $\operatorname{\mathit{USp}}(2)\simeq SU(2)$, and the $\operatorname{\mathit{USp}}(6)$ singlets transform in the adjoint ${\mathbf{3}}$ of $\operatorname{\mathit{SU}}(2)$. Note however, that the $\operatorname{\mathit{USp}}(6)$ structure stabilises not only one spinor $\hat\epsilon$, but also a second spinor $\hat\epsilon'$. These two spinors form a symplectic Majorana pair $\hat\epsilon^A$. However, here we can treat this index as the $\operatorname{\mathit{USp}}(2)$ index labelling the two spinors preserved by $\operatorname{\mathit{USp}}(6)$. In a generic $\operatorname{\mathit{USp}}(2)$ frame, we can then rewrite the second line of as $${{D}}_{\alpha\beta} (\hat\epsilon^A){}^\beta
= \tfrac{3}{2} \Lambda M^A{}_B (\hat\epsilon^B){}_\alpha ,$$ with the constant matrix $M^A{}_B$ as in . As there, the cosmological constant comes attached with the traceless Hermitian matrix $M^A{}_B$, and so is naturally an element of the triplet representation. Fixing this element will therefore break the $\operatorname{\mathit{SU}}(2)$ down to $U(1)$, the R-symmetry group of $\mathcal{N}=1$ AdS${}_5$.
Finally, in $D = 7$ spinors transform in the ${\mathbf{4}}$ of $\operatorname{\mathit{Spin}}(5)\simeq \operatorname{\mathit{USp}}(4)$. The Killing spinor equations can be written explicitly [@CSW3] as $$\begin{aligned}
({{D}}{\times_{{J}^-}} \hat\epsilon^-) &= 2(\gamma^j {{D}}_{ij} \hat\epsilon
- \tfrac15 \gamma_i \gamma^{jj'} {{D}}_{jj'} \hat\epsilon ) = 0,\\
({{D}}{\times_{{S}^+}} \hat\epsilon^-) &= -\gamma^{ij}{{D}}_{ij} \hat\epsilon
= -\tfrac{5}{2} \Lambda \hat\epsilon ,
\end{aligned}$$ where we are actually using the more familiar $SO(5)$ indices $i,j\dots$ and omitting spinor indices. The only component of the torsion that appears just in the second equation is a singlet of $\operatorname{\mathit{Spin}}(5)$. This obviously is still invariant under the Killing spinor stabiliser $\operatorname{\mathit{USp}}(2)\subset\operatorname{\mathit{USp}}(4)$ so it must be the cosmological constant. Clearly the singlet is also automatically invariant under the entire commutant subgroup $\operatorname{\mathit{USp}}(2)$, which indeed is the R-symmetry group of AdS${}_7$.
We should note that in M theory there are actually no smooth AdS${}_7$ backgrounds which are strictly $\mathcal{N}=1$ [@Acharya:1998db]. However, recently a family of genuinely $\mathcal{N}=1$ solutions in massive Type IIA theory was discovered [@Apruzzi:2013yva]. To describe these backgrounds as generalised structures, one would presumably need a slightly modified formulation of $\operatorname{\mathit{E_{d(d)}}}\times{\mathbb{R}}^+$ generalised geometry for the massive Type IIA theory, which would be beyond the scope of this paper. For M theory the only possibility is the $S^4$ solution with maximal $\mathcal{N}=2$ supersymmetry. We will discuss backgrounds with higher preserved supersymmetry in a subsequent paper [@future], but we remark that in this case the generalised structure group reduces to the identity, with the commutant being the entire $\operatorname{\mathit{USp}}(4)$. The singlet in the torsion that we identified above would not break this commutant group (since it is a singlet of $\tilde{H}_4=\operatorname{\mathit{USp}}(4)$), reflecting that the AdS R-symmetry is the full $\operatorname{\mathit{USp}}(4)$ group for $\mathcal{N}=2$. The generalised parallelisation on AdS${}_7\times S^4$ is presented in detail in [@Lee:2014mla], as an example of the generic appearance of this structure in maximally supersymmetric compactifications.
Discussion
==========
We have shown that spaces admitting the appropriate generalised $G$-structure with constant singlet torsion are precisely the minimal AdS flux backgrounds. This is summarised in table \[tab:AdS\].
$d$ $G$ $G_{\text{com}}$ R-symmetry ${T_{\text{int}}}$
----- ---------------------------------------------------------------------- ----------------------------------- ---------------------------------- ----------------------------------
7 $\operatorname{\mathit{SU}}(7)$ $ U(1)$ ${\mathbb{Z}}_2$ ${\mathbf{1}}_2$
6 $ \operatorname{\mathit{USp}}(6)$ $ \operatorname{\mathit{USp}}(2)$ $U(1)$ ${({\mathbf{3}}, {\mathbf{1}})}$
5 $\operatorname{\mathit{USp}}(2)\times\operatorname{\mathit{USp}}(4)$ $\operatorname{\mathit{USp}}(2)$ — no singlets
4 $\operatorname{\mathit{USp}}(2)$ $\operatorname{\mathit{USp}}(2)$ $\operatorname{\mathit{USp}}(2)$ ${({\mathbf{1}}, {\mathbf{1}})}$
: Generalised structure subgroups $G\subset \operatorname{\mathit{\tilde{H}_d}}$, commutant groups $G_{\text{com}}$ of $G$ in $\operatorname{\mathit{\tilde{H}_d}}$, AdS R-symmetry groups and non-vanishing generalised intrinsic torsion as representations of $G_{\text{com}}\times G$ for minimal supersymmetry in AdS backgrounds.[]{data-label="tab:AdS"}
We stress again that even though we have focused on the M theory case, the formalism is such that the results necessarily extend to Type II strings.
As a corollary, we remark that there exists a definition of generalised Ricci curvature [@CSW2], which given an arbitrary spinor $\epsilon$ reads schematically (see [@CSW3] for explicit definitions) $$\begin{aligned}
{R^{\scriptscriptstyle 0}}\cdot\epsilon = {{D}}{\times_{J}} ({{D}}{\times_{J}}\epsilon)+{{D}}{\times_{J}} ({{D}}{\times_{S}}\epsilon),\\
{R}\epsilon = {{D}}{\times_{S}} ({{D}}{\times_{J}}\epsilon)+{{D}}{\times_{S}} ({{D}}{\times_{S}}\epsilon),
\end{aligned}$$ where ${R}$ is the generalised Ricci scalar and ${R^{\scriptscriptstyle 0}}$ is the traceless part of the generalised Ricci. The vanishing of the full generalised Ricci corresponds to the Minkowski equations of motion [@CSW2]. Using the result of appendix **B** of [@CSW4], it is then easy to show that if the intrinsic torsion conditions hold, we have that $${R^{\scriptscriptstyle 0}}= 0, \quad {R}= (9-d)(10-d)\Lambda^2 ,$$ ie. we have the natural generalised analogue of the Einstein manifold condition. This is simply a more geometric (and much simpler to derive) restatement of the supergravity result that in a supersymmetric AdS background the equations of motion are satisfied [@Gauntlett:2002fz]. On the other hand, if we had not assumed that the singlet torsion $\Lambda$ was a constant but rather an arbitrary function, these equations would not hold. However, imposing them would then force $\Lambda$ to be constant [@Lukas:2004ip].
Another result that naturally generalises in this setting is of that of the cone spaces over the classical Sasaki–Einstein and weak $G_2$ spaces – well-known AdS backgrounds – which then become special holonomy manifolds. The same will happen here, now for generic backgrounds. Viewing the $D$-dimensional AdS space as a warped product of $(D-1)$-dimensional flat space and a line, implies that the cones over the spaces listed in table \[tab:AdS\] must all be special holonomy spaces for $E_{d+1(d+1)}\times{\mathbb{R}}^+$ generalised geometry [@CSW4].
An interesting avenue of further study would be to see if the same statements can be made in generalised geometries other than ones based on the exceptional $\operatorname{\mathit{E_{d(d)}}}$ groups, or whether any such statements can be made for $D\leq3$. Recent work [@AdS3foliations] showed that generic $\mathcal{N} =1$ AdS$_3$ solutions of M theory can be described as foliations of seven-dimensional spaces, and it would be interesting to make contact with that picture using generalised geometry. For example, one could examine the conditions for backgrounds with zero internal $F_{(4)}$ flux in $\operatorname{\mathit{Spin}}(8,8)\times{\mathbb{R}}^+$ generalised geometry [@CSC].
The most obvious extension of this work, however, would be to demonstrate an analogous statement for all supersymmetric AdS backgrounds of M and Type II theories, not just the ones with minimal supersymmetry. In forthcoming work [@future] we will show how the methods used in [@CSW4] for the Minkowski case can be expanded to deal with higher $\mathcal{N}$ backgrounds, at which point one could hope the classification of AdS will follow similarly to that outlined in this paper.
We would like to thank Dan Waldram for helpful discussions. C. S-C. is supported by a grant from the Foundational Questions Institute (FQXi) Fund, a donor advised fund of the Silicon Valley Community Foundation on the basis of proposal FQXi-RFP3-1321 (this grant was administered by Theiss Research). A. C. is supported by the German Research Foundation (DFG) within the Cluster of Excellence “QUEST”. C. S-C. would like to thank Imperial College London for hospitality and also the EPSRC Programme Grant EP/K034456/1 “New Geometric Structures from String Theory" for visitor support.
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[^1]: For more details on conventions on Clifford algebras, intertwiners, spinor decompostions, etc. please refer to appendix **B** of [@CSW3]. Here, for convenience, we choose a representation in which $(\gamma_m)^T = (\gamma_m)^* = - \gamma_m$ for $d=6,7$.
[^2]: Strictly, this holds assuming that the Killing spinor $\hat\epsilon^-$ has constant norm. However, this is always true by .
[^3]: For the $\operatorname{\mathit{\tilde{H}_d}}$ decomposition of the torsion representations see [@CSW2].
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We characterize measure spaces such that the canonical map ${\mathbf{L}}_\infty \to {\mathbf{L}}_1^*$ is surjective. In case of $d$ dimensional Hausdorff measure of a complete separable metric space $X$ we give two equivalent conditions. One is in terms of the order completeness of a quotient Boolean algebra associated with measurable sets and with locally null sets. Another one is in terms of the possibility to decompose space in a certain way into sets of nonzero finite measure. We give examples of $X$ and $d$ so that whether these conditions are met is undecidable in ZFC, including one with $d$ equals the Hausdorff dimension of $X$.'
address: |
School of Mathematical Sciences\
Shanghai Key Laboratory of PMMP\
East China Normal University\
500 Dongchuang Road\
Shanghai 200062\
P.R. of China\
and NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai\
3663 Zhongshan Road North\
Shanghai 200062\
China
author:
- Thierry De Pauw
bibliography:
- '/home/thierry/Documents/LaTeX/Bibliography/thdp.bib'
title: Undecidably semilocalizable metric measures spaces
---
[^1]
Foreword
========
Some questions pertaining to the calculus of variations would benefit from a useful description of the dual of the Banach space $\mathbf{BV}({{\mathbf{R}}^n})$ of functions of bounded variation in the sense of [E. De Giorgi]{}. The question occurs as Problem 7.4 is [@ARCATA]. Measures belonging to this dual space have been characterized by [N.G. Meyers]{} and [W.P. Ziemer]{} in [@MEY.ZIE.77]. A description of the other members was obtained (in a slightly different context) by [F.J. Almgren]{} in [@ALM.65.CH] under the Continuum Hypothesis and the particular description was proved to be independent of Zermelo-Fraenkel axioms by the present author in [@DEP.98]. Recently, following former work of [R.D. Mauldin]{}, [N. Fusco]{} and [D. Spector]{} have given a more precise description under the Continuum Hypothesis, [@FUS.SPE.18].
In [@DEP.98] the problem is shown to be related to describing the dual of the Banach space ${\mathbf{L}}_1({{\mathbf{R}}^n},{\mathscr{H}}^{n-1})$ where ${\mathscr{H}}^{n-1}$ denotes Hausdorff $n-1$ dimensional measure in ${{\mathbf{R}}^n}$. Here we will restrict to the case when $n=2$ and we shall aim for results in $\mathsf{ZFC}$. The notation ${\mathbf{L}}_1({{\mathbf{R}}^n},{\mathscr{H}}^{n-1})$ however is misleading as it assumes the problem to be independent of the underlying $\sigma$-algebra. As we shall see, this is not the case.
Let $(X,{\mathscr{A}},\mu)$ be a measure space. There is a natural linear retraction $$\label{eq.100}
\Upsilon : {\mathbf{L}}_\infty(X,{\mathscr{A}},\mu) \to {\mathbf{L}}_1(X,{\mathscr{A}},\mu)^*$$ which sends ${\mathbf{g}}$ to $\mathbf{f} \mapsto \int_X gf d\mu$ where $g$ and $f$ represent ${\mathbf{g}}$ and $\mathbf{f}$ respectively. In general $\Upsilon$ does not need to be injective or surjective. It has been understood for a long time that $\Upsilon$ is injective if and only if $(X,{\mathscr{A}},\mu)$ is [*semifinite*]{}. This means that each $A \in {\mathscr{A}}$ such that $\mu(A)=\infty$ admits a subset ${\mathscr{A}}\ni B {\subseteq}A$ with $0 < \mu(B) < \infty$. Of course every $\sigma$-finite measure space is semifinite. Yet the dependence upon the $\sigma$-algebra under consideration already occurs in the case of interest to us. The situation is the following.
1. If $X$ is a complete separable metric space and $0 < d < \infty$ then the measure space $(X,{\mathscr{B}}(X),{\mathscr{H}}^d)$ is semifinite. Here ${\mathscr{B}}(X)$ denotes the $\sigma$-algebra of Borel subsets of $X$ and ${\mathscr{H}}^d$ is the $d$ dimensional Hausdorff measure on $X$. In case $X={{\mathbf{R}}^n}$ this was proved by [R.O. Davies]{}, [@DAV.52] and in general by [J. Howroyd]{}, [@HOW.95].
2. According to [D.H. Fremlin]{}, [@FREMLIN.IV 439H] the measure space $({\mathbf{R}}^2,{\mathscr{A}}_{{\mathscr{H}}^1},{\mathscr{H}}^1)$ is not semifinite, where ${\mathscr{A}}_{{\mathscr{H}}^1}$ denotes the $\sigma$-algebra consisting of ${\mathscr{H}}^1$ measurable subsets of ${\mathbf{R}}^2$. This is based on the existence of <<large>> universally null subsets of $[0,1]$ established by [E. Grzegorek]{}, [@GRZ.81]. See also the article of [O. Zindulka]{} [@ZIN.12].
Nonetheless, recalling our work [@DEP.98] it is the surjectivity of $\Upsilon$ that is relevant for the existence of a certain integral representation of members of the dual of $\mathbf{BV}({\mathbf{R}}^2)$. Injectivity pertains to its uniqueness.
Under the assumption that $(X,{\mathscr{A}},\mu)$ is semifinite, a necessary and sufficient condition for the surjectivity of $\Upsilon$ has been known for a long time. It asks for the quotient Boolean algebra ${\mathscr{A}}/{\mathscr{N}}_\mu$ to be order complete, where ${\mathscr{N}}_\mu = {\mathscr{A}}\cap \{ N : \mu(N) = 0 \}$ is the $\sigma$-ideal of $\mu$ null sets. Semifinite measure spaces with this property are sometimes called [*Maharam*]{}, [@FREMLIN.II 211G]. A stronger condition sometimes called [*decomposable*]{}, generalizes the idea of $\sigma$-finiteness to possibly uncountable decomposition into sets of finite measure, together with a new condition called [*locally determined*]{} (that measurability be determined by sets of finite measure), see \[61\] for the definition of locally determined and [@FREMLIN.II 211E] for the definiton of decomposable. If the quotient $\sigma$-algebra ${\mathscr{A}}/{\mathscr{N}}_\mu$ is not too big then decomposability implies Maharam according to [E.J. McShane]{}, [@MCS.62] but not in general according to [D.H. Fremlin]{}, [@FREMLIN.II 216E].
If $X$ is a Polish space and ${\mathscr{B}}(X)$ denotes the $\sigma$-algebra consisting of its Borel subsets, and if the measure space $(X,{\mathscr{B}}(X),\mu)$ is decomposable, then it is $\sigma$-finite. I learned the <<counting argument>> to prove this from [D.H. Fremlin]{}, see \[55\]. In view of (1) above it shows that $({\mathbf{R}}^2,{\mathscr{B}}({\mathbf{R}}^2),{\mathscr{H}}^1)$ is not decomposable. Since decomposability is stronger in general than the surjectivity of $\Upsilon$, we need to argue a bit more to show that $({\mathbf{R}}^2,{\mathscr{B}}({\mathbf{R}}^2),{\mathscr{H}}^1)$ is not Maharam, see below. This observation calls for developing a criterion for the surjectivety of $\Upsilon$ without assuming that $(X,{\mathscr{A}},\mu)$ be semifinite in the first place. We do this in Section 4. Thus regarding the question whether $$\Upsilon : {\mathbf{L}}_\infty\left({\mathbf{R}}^2,{\mathscr{A}},{\mathscr{H}}^1\right) \to {\mathbf{L}}_1\left({\mathbf{R}}^2,{\mathscr{A}},{\mathscr{H}}^1\right)^*$$ is surjective or not, the situation is the following.
1. If ${\mathscr{A}}={\mathscr{B}}({\mathbf{R}}^2)$ then $\Upsilon$ is not surjective. Since $({\mathbf{R}}^2,{\mathscr{B}}({\mathbf{R}}^2),{\mathscr{H}}^1)$ is semifinite according to (1), and not $\sigma$-finite, it is not decomposable, \[55\]. The argument of [E.J. McShane]{}, \[mcshane\] does not show $({\mathbf{R}}^2,{\mathscr{B}}({\mathbf{R}}^2),{\mathscr{H}}^1)$ is not Maharam (the reason being that its completion is not locally determined). However we give below a simple argument to the extent that it is not Maharam, based on Fubini’s Theorem.
2. If ${\mathscr{A}}={\mathscr{A}}_{{\mathscr{H}}^1}$ then the surjectivity of $\Upsilon$ is undecidable in $\mathsf{ZFC}$. The consistency of its surjectivity is a consequence of the Continuum Hypothesis, \[CH.implies.ad\], \[ad.implies.semiloc\] and \[Riesz\]. The consistency of it not being surjective was first noted in [@DEP.98] although in a slightly different disguise. The idea is explained below.
The present paper grew out of the attempt to adapt the techniques used to prove (3) and (4) to the case where ${\mathbf{R}}^2$ is replaced with a small compact subset $X {\subseteq}{\mathbf{R}}^2$ – as small as it can possibly be, i.e. of Hausdorff dimension 1 (of course not of $\sigma$-finite ${\mathscr{H}}^1$ measure, for in that case $(X,{\mathscr{B}}(X),{\mathscr{H}}^1)$ and $(X,{\mathscr{A}}_{{\mathscr{H}}^1},{\mathscr{H}}^1)$ are both Maharam and $\Upsilon$ is surjective, \[sigmafin\]). Why however would the answer depend on the $\sigma$-algebra under consideration? In order to understand this, let us try to prove that $\Upsilon$ is surjective.
We know from the classical Riesz’ Theorem that $\Upsilon$ is surjective whenever $(X,{\mathscr{A}},\mu)$ is a finite measure space. This suggests to consider ${\mathscr{A}}^f_\mu = {\mathscr{A}}\cap \{ A : \mu(A) < \infty \}$ and for each $A \in {\mathscr{A}}^f_\mu$ the map $$\Upsilon^A : {\mathbf{L}}_\infty(A,{\mathscr{A}}_A,\mu_A) \to {\mathbf{L}}_1(A,{\mathscr{A}}_A,\mu_A)^*$$ where $(A,{\mathscr{A}}_A,\mu_A)$ is the obvious measure subspace. Thus $\Upsilon^A$ is an isometric linear isomorphism and given $\alpha \in {\mathbf{L}}_1(X,{\mathscr{A}},\mu)^*$ there exist $g_A \in {\mathbf{g}}_A \in {\mathbf{L}}_\infty(A,{\mathscr{A}}_A,\mu_A)$ such that $$(\alpha \circ \iota_A)(\mathbf{f}) = \int_A g_Af d\mu_A$$ whenever $f \in \mathbf{f} \in {\mathbf{L}}_1(A,{\mathscr{A}}_A,\mu_A)$, where $\iota_A : {\mathbf{L}}_1(A,{\mathscr{A}}_A,\mu_A) \to {\mathbf{L}}_1(X,{\mathscr{A}},\mu)$ is the obvious embedding. From the $\mu_A$ almost everywhere uniqueness of the Radon-Nikodým derivative $g_A$ we infer that if $A,A' \in {\mathscr{A}}^f_\mu$ then $\mu ( A \cap A' \cap \{ g_A \neq g_{A'} \}) = 0 $. Thus $(g_A)_{A \in {\mathscr{A}}^f_\mu}$ is what we call, from now on a [*compatible family*]{} of locally defined measurable functions and the question is whether it corresponds to a globally defined measurable function, i.e. whether there exists an ${\mathscr{A}}$-measurable $g : X \to {\mathbf{R}}$ such that $\mu(A \cap \{ g \neq g_A \}) = 0$ for every $A \in {\mathscr{A}}^f_\mu$. If such $g$ exists let us call it a [*gluing*]{} of the compatible family $(g_A)_{A \in {\mathscr{A}}^f_\mu}$. This is reminiscent of, and not entirely unrelated to the sheaf property of the functor $U \mapsto C(U)$ where $U$ is a subset of a topological space, see \[category\](Q3).
It turns out to be rather useful to notice that the question whether a gluing exists or not can be asked in a slightly more general setting since it depends on the measure $\mu$ only insofar as its $\mu$ null sets are involved. Thus a [*measurable space with negligibles*]{} $(X,{\mathscr{A}},{\mathscr{N}})$ consists of a measurable space $(X,{\mathscr{A}})$ and a $\sigma$-ideal ${\mathscr{N}}{\subseteq}{\mathscr{A}}$. Given any ${\mathscr{E}}{\subseteq}{\mathscr{A}}$ one can readily define the notion of a compatible family $(g_E)_{E \in {\mathscr{E}}}$ by asking that $E \cap E' \cap \{ g_E \neq g_{E'} \} \in {\mathscr{N}}$ whenever $E,E' \in {\mathscr{E}}$, and by saying that an ${\mathscr{A}}$-measurable function $g : X \to {\mathbf{R}}$ is a gluing of $(g_E)_{E \in {\mathscr{E}}}$ provided $E \cap \{ g \neq g_E \} \in {\mathscr{N}}$ for all $E \in {\mathscr{E}}$. One then shows, \[gluing\] that each compatible family admits a gluing if and only if each ${\mathscr{E}}{\subseteq}{\mathscr{A}}$ admits an ${\mathscr{N}}$ essential supremum $A \in {\mathscr{A}}$. This means that
1. For every $E \in {\mathscr{E}}$ one has $E {\thicksim}A \in {\mathscr{N}}$;
2. For every $B \in {\mathscr{A}}$, if $E {\thicksim}B \in {\mathscr{N}}$ whenever $E \in {\mathscr{E}}$, then $A {\thicksim}B \in {\mathscr{N}}$.
We say that a measurable space with negligibles is [*localizable*]{} if it has this property.
In this paper we characterize those measure spaces such that $\Upsilon$ is surjective, \[Riesz\]. To state this we first define $${\mathscr{N}}_\mu\left[{\mathscr{A}}^f_\mu\right] = {\mathscr{A}}\cap \left\{ N : \mu(A \cap N) = 0 \text{ for all } A \in {\mathscr{A}}^f_\mu \right\} \,.$$ It is a $\sigma$-ideal, whose members one is tempted to call [*locally $\mu$ null*]{}.
For any measure space $(X,{\mathscr{A}},\mu)$, the map $\Upsilon$ (recall ) is surjective if and only if the measurable space with negligibles $\left(X,{\mathscr{A}},{\mathscr{N}}_\mu\left[{\mathscr{A}}^f_\mu\right] \right)$ is localizable.
We call a measure space [*semilocalizable*]{} if it has this property – thus no semifiniteness is assumed. We study the connection with the notion of [*almost decomposable*]{} measure space introduced in [@DEP.98], \[ad.implies.semiloc\] and \[mcshane\] thereby generalizing to non semifinite measure spaces the classical theory briefly evoked above. We call a measure space $(X,{\mathscr{A}},\mu)$ [*almost decomposable*]{} if there exists a disjointed family ${\mathscr{G}}{\subseteq}{\mathscr{A}}^f_\mu$ such that $$\forall A \in {\mathscr{P}}(X) : ( \forall G \in {\mathscr{G}}: A \cap G \in {\mathscr{A}}) \Rightarrow A \in {\mathscr{A}}\,,$$ and $$\forall A \in {\mathscr{A}}: \mu(A) < \infty \Rightarrow \mu(A) = \sum_{G \in {\mathscr{G}}} \mu(A \cap G) \,.$$ Using an idea of [E.J. McShane]{}, [@MCS.62] and the fact that there are not too many equivalence classes of measurable sets with respect to a Borel regular outer measure on a Polish space, \[63\] we prove the following, \[71\].
Let $X$ be a complete separable metric space and $0 < d < 1$. For the measure space $(X,{\mathscr{A}}_{{\mathscr{H}}^d},{\mathscr{H}}^d)$ the following are equivalent.
1. The canonical map $\Upsilon$ is surjective;
2. $(X,{\mathscr{A}}_{{\mathscr{H}}^d},{\mathscr{H}}^d)$ is semilocalizable;
3. $(X,{\mathscr{A}}_{{\mathscr{H}}^d},{\mathscr{H}}^d)$ is almost decomposable.
Let us now consider the measure space $({\mathbf{R}}^2,{\mathscr{B}}({\mathbf{R}}^2),{\mathscr{H}}^1)$ in view of the notion of semilocalizability. We know it is not semilocalizable, (3) above, but we promised to show how this is a consequence of Fubini’s Theorem. Define the vertical sections $V_s = \{s\} \times {\mathbf{R}}$, $s \in {\mathbf{R}}$, and the horizontal sections $H_t = {\mathbf{R}}\times \{t\}$, $t \in {\mathbf{R}}$. Assume if possible that $A \in {\mathscr{B}}({\mathbf{R}}^2)$ is an ${\mathscr{N}}_{{\mathscr{H}}^1}\left[{\mathscr{B}}({\mathbf{R}}^2)^f_{{\mathscr{H}}^1}\right]$ essential supremum of the family $(V_s)_{s \in {\mathbf{R}}}$. It would then readily follow that
1. ${\mathscr{H}}^1(V_s {\thicksim}A) = 0$ for every $s \in {\mathbf{R}}$;
2. ${\mathscr{H}}^1(H_t \cap A) = 0$ for every $t \in {\mathbf{R}}$.
Indeed upon noticing that $V_s$ and $H_t$ have $\sigma$-finite ${\mathscr{H}}^1$ measure, (a) is a rephrasing of (i) above and (b) follows from (ii) applied with $B=A {\thicksim}H_t$. Applying Fubini’s Theorem twice would yield $${\mathscr{L}}^2({\mathbf{R}}^2 {\thicksim}A) = \int_{\mathbf{R}}{\mathscr{H}}^1( V_s {\thicksim}A) d{\mathscr{L}}^1(s) = 0$$ according to (a), and $${\mathscr{L}}^2({\mathbf{R}}^2 \cap A) = \int_{\mathbf{R}}{\mathscr{H}}^1( H_t \cap A) d{\mathscr{L}}^1(t) = 0$$ according to (b). In turn ${\mathscr{L}}^2({\mathbf{R}}^2)=0$, a contradiction. Clearly the same argument applies with ${\mathbf{R}}^2$ replaced by any Borel set $X {\subseteq}{\mathbf{R}}^2$ such that ${\mathscr{L}}^2(X) > 0$, to showing that $(X,{\mathscr{B}}(X),{\mathscr{H}}^1)$ is not semilocalizable.
There are two cases when the above argument is not conclusive:
1. when $A$ is not ${\mathscr{L}}^2$ measurable (because Fubini’s Theorem does not apply);
2. when ${\mathscr{L}}^2(X)=0$ (because no contradiction ensues).
With regard to case ($\alpha$) indeed, when we replace the $\sigma$-algebra ${\mathscr{B}}({\mathbf{R}}^2)$ by the larger ${\mathscr{A}}_{{\mathscr{H}}^1}$ then $({\mathbf{R}}^2,{\mathscr{A}}_{{\mathscr{H}}^1},{\mathscr{H}}^1)$ is consistently semilocalizable. This is a consequence of the Continuum Hypothesis and a much more general statement holds[^2], \[CH.implies.ad\]. As noticed in [@DEP.98] it turns out however that $({\mathbf{R}}^2 , {\mathscr{A}}_{{\mathscr{H}}^1}, {\mathscr{H}}^1)$ is [*also*]{} consistently [*not*]{} semilocalizable. Here is the reason why. We assume that $A \in {\mathscr{A}}_{{\mathscr{H}}^1}$ is an ${\mathscr{N}}_{{\mathscr{H}}^1}\left[{\mathscr{A}}_{{\mathscr{H}}^1}^f\right]$ essential supremum of the family $(V_s)_{s \in {\mathbf{R}}}$. For each $s \in {\mathbf{R}}$ we define $T_s = {\mathbf{R}}\cap \{ t : (s,t) \in V_s {\thicksim}A \}$, thus ${\mathscr{L}}^1(T_s)=0$ according to (i). Now choose $E {\subseteq}{\mathbf{R}}$ such that ${\mathscr{L}}^1(E) > 0$ and $E$ has least cardinal among all sets with nonzero Lebesgue measure, and let ${\mathsf{non}}({\mathscr{N}}_{{\mathscr{L}}^1})$ denote this cardinal. Assume that there exists $t \in {\mathbf{R}}{\thicksim}\cup_{s \in E} T_s$. Then for each $s \in E$, $t \not \in T_s$, i.e. $(s,t) \in H_t \cap A$. Therefore ${\mathscr{L}}^1(E) = 0$ according to (ii), a contradiction. Of course we can reach this contradiction only if ${\mathbf{R}}\neq \cup_{s \in E} T_s$, which depends upon how big $E$ is. We denote as ${\mathsf{cov}}({\mathscr{N}}_{{\mathscr{L}}^1})$ the least cardinal of a covering of ${\mathbf{R}}$ by ${\mathscr{L}}^1$ negligible sets. Thus if $\operatorname{\mathrm{card}}E = {\mathsf{non}}({\mathscr{N}}_{{\mathscr{L}}^1}) < {\mathsf{cov}}({\mathscr{N}}_{{\mathscr{L}}^1})$ then the argument goes through. It turns out that this strict inequality of cardinals (appearing in the so-called Cichoń diagram) is consistent with $\mathsf{ZFC}$, [@BARTOSZYNSKI.JUDAH Chapter 7] or [@FREMLIN.V.2 552H and 552G]. We will refer to this idea below as the <<vertical-horizontal method>>. This argument is from [@DEP.98] ; I learned it from [D.H. Fremlin]{}.
With regard to case ($\beta$) above we observe again that the Continuum Hypothesis implies that $(X,{\mathscr{A}}_{{\mathscr{H}}^1},{\mathscr{H}}^1)$ is semilocalizable for any compact set $X {\subseteq}{\mathbf{R}}^2$ regardless whether it has zero ${\mathscr{L}}^2$ measure or not, \[CH.implies.ad\]. The question is therefore whether $(X,{\mathscr{A}}_{{\mathscr{H}}^1},{\mathscr{H}}^1)$ is semilocalizable in $\mathsf{ZFC}$ or consistently not semilocalizable. The latter occurs when the <<vertical-horizontal method>> generalizes from $X = {\mathbf{R}}^2$ to $X$. For instance it clearly generalizes to $X = [a,b ] \times [c,d]$ but it is not instantly obvious how to proceed if ${\mathscr{L}}^2(X) = 0$.
In connection with this question we now mention a result due to [G. Alberti, M. Csörnyei]{} and [D. Preiss]{}. Given $X {\subseteq}{\mathbf{R}}^2$, say compact, we call $\tau : X \to {\mathbf{G}}({\mathbf{R}}^2,1)$ a [*weak tangent field*]{} to $X$ if for every $C^1$ curve $\Gamma {\subseteq}{\mathbf{R}}^2$ one has $\operatorname{\mathrm{Tan}}(\Gamma,x) = \tau(x)$ at ${\mathscr{H}}^1$ almost every $x \in \Gamma \cap X$. The above argument using Fubini’s Theorem shows that if ${\mathscr{L}}^2(X) > 0$ then $X$ does not admit an ${\mathscr{L}}^2$ measurable weak tangent field. On the other hand under the Continuum Hypothesis any compact $X {\subseteq}{\mathbf{R}}^2$ admits an ${\mathscr{H}}^1$ measurable weak tangent field. One considers indeed the ${\mathscr{N}}_{{\mathscr{H}}^1}$ compatible family of line fields $\tau_\Gamma : \Gamma \cap X \to {\mathbf{G}}({\mathbf{R}}^2,1) : x \mapsto \operatorname{\mathrm{Tan}}(\Gamma,x)$, corresponding to all $C^1$ curves $\Gamma {\subseteq}{\mathbf{R}}^2$. It is indeed compatible since $\operatorname{\mathrm{Tan}}(\Gamma_1,x)=\operatorname{\mathrm{Tan}}(\Gamma_2,x)$ for ${\mathscr{H}}^1$ almost every $x \in \Gamma_1 \cap \Gamma_2$ whenever $\Gamma_1$ and $\Gamma_2$ are two $C^1$ curves in ${\mathbf{R}}^2$. A gluing $\tau$ of $(\tau_\Gamma)_\Gamma$ will be a weak tangent field to $X$. The existence of such gluing ensues from the semilocalizability of $(X,{\mathscr{A}}_{{\mathscr{H}}^1},{\mathscr{H}}^1)$, which holds under the Continuum Hypothesis, \[CH.implies.ad\] and \[ad.implies.semiloc\]. However [G. Alberti, M. Csörnyei]{} and [D. Preiss]{} proved the following striking result in $\mathsf{ZFC}$, [@ALB.CSO.PRE.05] and [@ALB.CSO.PRE.10]. [*If ${\mathscr{L}}^2(X)=0$ then $X$ admits a Borel measurable weak tangent field.*]{} One may wonder if this is a consequence, in $\mathsf{ZFC}$ of the localizability of $(X,{\mathscr{B}}(X),{\mathscr{N}})$ for some $\sigma$-ideal ${\mathscr{N}}$, for example ${\mathscr{N}}_{pu}$ the $\sigma$-ideal consisting of purely $({\mathscr{H}}^1,1)$ unrectifiable subsets of ${\mathbf{R}}^2$[^3]. This however is not the case: We give in \[CR.1\] below an example of a <<purely rectifiable>> ${\mathscr{L}}^2$ negligible compact set $X {\subseteq}{\mathbf{R}}^2$ such that for all $\sigma$-algebra ${\mathscr{B}}(X) {\subseteq}{\mathscr{A}}{\subseteq}{\mathscr{P}}(X)$ and a large collection of $\sigma$-ideals – including ${\mathscr{N}}_{{\mathscr{H}}^1}$ and ${\mathscr{N}}_{pu}$ – the measure space with negligibles $(X,{\mathscr{A}},{\mathscr{N}})$ is consistently not localizable. Furthermore $X$ is nearly as small as it can be for this to happen: ${\mathscr{H}}^1 {
\hskip2.5pt{\vrule height7pt width.5pt depth0pt}
\hskip-.2pt\vbox{\hrule height.5pt width7pt depth0pt}
\, }X$ is not $\sigma$-finite but $X$ has Hausdorff dimension 1[^4]. See the last stated Theorem of this introduction.
Thus we ought to explain how the <<vertical-horizontal method>> described above, showing that if ${\mathsf{non}}({\mathscr{N}}_{{\mathscr{L}}^1}) < {\mathsf{cov}}({\mathscr{N}}_{{\mathscr{L}}^1})$ then $\left({\mathbf{R}}^2,{\mathscr{A}}_{{\mathscr{H}}^1},{\mathscr{H}}^1\right)$ is not semilocalizable, can be adapted to the case where ${\mathbf{R}}^2$ is replaced with some suitable subset $X {\subseteq}{\mathbf{R}}^2$. We give a rather general version below, \[abstract.theorem\]. First of all we make the useful observation that if $(S,{\mathscr{B}}(S),\sigma)$ is a probability space, $S$ is Polish and $\sigma$ is diffuse, then ${\mathsf{non}}({\mathscr{N}}_{\bar{\sigma}}) = {\mathsf{non}}({\mathscr{N}}_{{\mathscr{L}}^1})$ and ${\mathsf{cov}}({\mathscr{N}}_{\bar{\sigma}}) = {\mathsf{cov}}({\mathscr{N}}_{{\mathscr{L}}^1})$. This ensues from the Kuratowski Isomorphism Theorem, \[number.8\]. A careful inspection of the argument leads to the following, \[85\].
Let $0 < d < 1$ and let $C_d {\subseteq}[0,1]$ be the standard self-similar Cantor set of Hausdorff dimension $0 < d < 1$. Whether the measure space $(C_d \times C_d , {\mathscr{A}}_{{\mathscr{H}}^d}, {\mathscr{H}}^d)$ is semilocalizable is undecidable in $\mathsf{ZFC}$.
Incidentally, constructing a certain isomorphism in the category of measurable spaces with negligibles we are able to infer the following, \[pue.8\].
Whether the measure space $\left([0,1],{\mathscr{A}}_{{\mathscr{H}}^{1/2}},{\mathscr{H}}^{\frac{1}{2}}\right)$ is semilocalizable is undecidable in $\mathsf{ZFC}$.
Here the exponent $1/2$ reflects the nature of the argument, viewing the space $X$ as a product of a kind, where <<vertical>> sets $V_s$ and <<horizontal>> sets $H_t$ of the same size make sense, their intersections behaving according to some technical assumptions (see the statement of \[abstract.theorem\]). The sets $X = C_d \times C_d$ are purely $({\mathscr{H}}^1,1)$ unrectifiable, \[pue.1\](1) and therefore irrelevant to the question whether the existence of a weak tangent field for $X$ follows from a localizability property: Any map $X \to {\mathbf{G}}({\mathbf{R}}^2,1)$ is a weak tangent field if $X$ is purely $({\mathscr{H}}^1,1)$ unrectifiable, let alone the fact that $d=1$ is omitted above.
Therefore we next seek to apply the <<vertical-horizontal method>> to an ${\mathscr{L}}^2$ negligible compact set $X {\subseteq}{\mathbf{R}}^2$ which is not purely $({\mathscr{H}}^1,1)$ unrectifiable and prove that $(X,{\mathscr{A}}_{{\mathscr{H}}^1},{\mathscr{H}}^1)$ is not semilocalizable. Let us choose $X$ as small as possible, i.e. of Hausdorff dimension 1, say $X = C \times [0,1]$ where $C {\subseteq}[0,1]$ is a Cantor set of Hausdorff dimension 0. It is of course clear that $V_s = \{s\} \times [0,1]$, $s \in C$, can be chosen as our vertical sets, yet the choice $H_t = C \times \{t\}$, $t \in [0,1]$, will be of no use since ${\mathscr{H}}^1(H_t)=0$ and therefore no contradiction can ensue when implementing the <<vertical-horizontal method>>. Instead we proceed as follows to define $H_t$. Let $\mu$ be a diffuse probability measure on $C$ and let $f(t) = \mu([0,t])$, $t \in [0,1]$, be its distribution function (this is a version of the Cantor-Vitali devil staircase for our 0 dimensional set $C$). Consider the graph $G$ of the function $\frac{1}{2}f$ ; thus $G$ is a rectifiable curve, and intersects non ${\mathscr{H}}^1$ trivially the set $X$, \[number.6\]. We then define $H_t = G + t.e_2$, $t \in [0,1/2]$, where $e_2=(0,1)$. It turns out that these will successfully play the role of horizontal sets, the details are in section \[example\]. The following subsumes \[number.9\] and \[CR.1\].
Assume that
1. $C {\subseteq}[0,1]$ is some Cantor set of Hausdorff dimension 0;
2. $X = C \times [0,1]$;
3. ${\mathscr{A}}$ is a $\sigma$-algebra and ${\mathscr{B}}(X) {\subseteq}{\mathscr{A}}{\subseteq}{\mathscr{P}}(X)$;
4. ${\mathscr{N}}= {\mathscr{N}}_{{\mathscr{H}}^1}$ or ${\mathscr{N}}= {\mathscr{N}}_{pu}$.
It follows that the measurable space with negligibles $(X,{\mathscr{A}},{\mathscr{N}})$ is consistently not localizable.
Of course this particular set $X$ admits an obvious constant weak tangent field: $\tau(x) = \operatorname{\mathrm{span}}\{e_2\}$, $x \in X$. Notice in particular that for ${\mathscr{H}}^1$ almost every $x \in G \cap X$ the tangent line to $G$ at $x$ is vertical. This illustrates that localizability is indeed much stronger than the existence of a weak tangent field.
My thanks are due to [David H. Fremlin]{}, not only for his inspiring treatise <<Measure Theory>> [@FREMLIN.I; @FREMLIN.II; @FREMLIN.III; @FREMLIN.IV; @FREMLIN.V.1; @FREMLIN.V.2] but also for many helpful conversations. I am also indebted to [Francis Borceux]{} for useful conversations regarding \[category\].
Preliminaries
=============
A [*measurable space*]{} consists of a pair $(X,{\mathscr{A}})$ where $X$ is a set and ${\mathscr{A}}$ is a $\sigma$-algebra of subsets of $X$. Whenever $(X,{\mathscr{A}})$ and $(Y,{\mathscr{B}})$ are measurable spaces and $f : X \to Y$ we say that $f$ is [*$({\mathscr{A}},{\mathscr{B}})$ measurable*]{} provided $f^{-1}(B) \in {\mathscr{A}}$ for all $A \in {\mathscr{A}}$. In the particular case when $Y={\mathbf{R}}$ it is always understood that ${\mathscr{B}}= {\mathscr{B}}({\mathbf{R}})$ is the $\sigma$-algebra consisting of Borel subsets of ${\mathbf{R}}$ and we say that $f$ is [*${\mathscr{A}}$ measurable*]{} instead of $({\mathscr{A}},{\mathscr{B}}({\mathbf{R}}))$ measurable. We let $L_0(X,{\mathscr{A}})$ denote the collection of ${\mathscr{A}}$ measurable functions $X \to {\mathbf{R}}$. It is an algebra and a Riesz space (under the pointwise operations and partial order).
As usual a [*measure space*]{} $(X,{\mathscr{A}},\mu)$ consists in a measurable space $(X,{\mathscr{A}})$ and a measure $\mu$ defined on the $\sigma$-algbera. We let $L_1(X,{\mathscr{A}},\mu)$ denote the subspace of $L_0(X,{\mathscr{A}})$ consisting of those $f$ such that $|f|$ is $\mu$-summable. The corresponding space of equivalence classes with respect to equality $\mu$ almost everywhere is denoted ${\mathbf{L}}_1(X,{\mathscr{A}},\mu)$. If $f \in L_1(X,{\mathscr{A}},\mu)$ we let $f^\bullet$ denote its equivalence class in ${\mathbf{L}}_1(X,{\mathscr{A}},\mu)$. Thus $f \in \mathbf{f} \in {\mathbf{L}}_1(X,{\mathscr{A}},\mu)$ means that $f$ is an actual function representing the equivalence class $\mathbf{f}$, i.e. $f^\bullet = \mathbf{f}$.
If $(X,{\mathscr{A}},\mu)$ is a measure space and ${\mathscr{B}}{\subseteq}{\mathscr{A}}$ is a $\sigma$-algebra, we let $\mu|_{\mathscr{B}}$ denote the restriction of $\mu$ to ${\mathscr{B}}$. If $A \in {\mathscr{A}}$ we let $(A,{\mathscr{A}}_A,\mu_A)$ denote the measure space where ${\mathscr{A}}_A = {\mathscr{B}}\cap \{ B : B {\subseteq}A \}$ and $\mu_A = \mu|_{{\mathscr{A}}_A}$.
\[P1\] With a measure space $(X,{\mathscr{A}},\mu)$ we associate an outer measure $\bar{\mu}$ on $X$ by the usual formula $$\bar{\mu}(S) = \inf \{ \mu(A) : {\mathscr{A}}\ni A {\supseteq}S \} \,.$$
If $X$ is a set and $\phi$ an outer measure on $X$ we let ${\mathscr{A}}_\phi$ denote the $\sigma$-algebra consisting of those subsets of $X$ which are $\phi$ measurable in the sense of Caratheory.
\[P3\]
*If $(X,{\mathscr{A}},\mu)$ is a measure space and $\bar{\mu}$ is associated with it as in \[P1\] then*
1. ${\mathscr{A}}{\subseteq}{\mathscr{A}}_{\bar{\mu}}$;
2. For every $A \in {\mathscr{A}}$ one has $\bar{\mu}(A) = \mu(A)$;
3. For every $S {\subseteq}X$ there exists ${\mathscr{A}}\ni A {\supseteq}S$ such that $\bar{\mu}(S) = \mu(A)$.
Let $A \in {\mathscr{A}}$ and $S {\subseteq}X$. We ought to show that $\bar{\mu}(S) {\geqslant}\bar{\mu}(S \cap A) + \bar{\mu}(S {\thicksim}A)$. Let ${\mathscr{A}}\ni B {\supseteq}S$ and notice that $\mu(B) = \mu(B \cap A) + \mu(B {\thicksim}A) {\geqslant}\bar{\mu}(S \cap A) + \bar{\mu}(S {\thicksim}A)$. Since $B$ is arbitrary the proof of (1) is complete. Given $A \in {\mathscr{A}}$ and ${\mathscr{A}}\ni B {\supseteq}A$ we clearly have $\mu(A) {\leqslant}\mu(B)$ and, since $B$ is arbitrary $\mu(A) {\leqslant}\bar{\mu}(A)$. Letting $B=A$ proves the equality of conclusion (2) and we now turn to establishing (3). If $\bar{\mu}(S)=\infty$ then take $A = X$. If not choose ${\mathscr{A}}\ni A'_n {\supseteq}S$ such that $\mu(A'_n) {\leqslant}n^{-1} + \bar{\mu}(S)$, $n \in {\mathbb{N}}^*$, let $A_n = \cap_{m {\geqslant}n} A'_m$, $A = \cap_{n \in {\mathbb{N}}^*} A_n$ and notice that ${\mathscr{A}}\ni A {\supseteq}S$ and $\bar{\mu}(S) {\leqslant}\mu(A) = \lim_n \mu(A_n) {\leqslant}\lim_n \mu(A'_n) = \bar{\mu}(S)$.
\[24\] If $X$ is a Polish space and $\phi$ an outer measure on $X$ we say that $\phi$ is [*Borel regular*]{} if
1. ${\mathscr{B}}(X) {\subseteq}{\mathscr{A}}_\phi$, i.e. each Borel subset of $X$ is $\phi$ measurable;
2. For every $A {\subseteq}X$ there exists a Borel set $B {\subseteq}X$ such that $A {\subseteq}B$ and $\phi(A)=\phi(B)$.
When $B$ is associated with $A$ as in (2) we call it a [*Borel hull*]{} of $A$. In this case one readily checks that $\phi(B) {\leqslant}\phi(B')$ whenever $B' \in {\mathscr{B}}(X)$ and $A {\subseteq}B'$.
[ *If $X$ is a Polish space, $\phi$ is a Borel regular outer measure on $X$, and $\mu = \phi|_{{\mathscr{B}}(X)}$, then $\bar{\mu} = \phi$.* ]{}
This is a particular case of \[P3\](2).
\[26\] [ *If $(X,{\mathscr{B}}(X),\mu)$ is a measure space where $X$ is Polish then $\bar{\mu}$ is Borel regular and $\bar{\mu}|_{{\mathscr{B}}(X)} = \mu$.* ]{}
This is a particular case of \[P3\].
Let $X$ be a metric space and $0 < d < \infty$. Given $0 < \delta {\leqslant}\infty$ and $A {\subseteq}X$ we define $${\mathscr{H}}^d_{(\delta)}(A) = \inf \left\{ \sum_{i \in I} (\operatorname{\mathrm{diam}}A_i)^d : A {\subseteq}\cup_{i \in I} A_i, I \text{ is at most countable, and } \operatorname{\mathrm{diam}}A_i {\leqslant}\delta \right\} \,.$$ We further let $${\mathscr{H}}^d(A) = \lim_{\delta \to 0^+} {\mathscr{H}}^d_{(\delta)}(A) = \inf \left\{ {\mathscr{H}}^d_{(\delta)}(A) : 0 < \delta {\leqslant}\infty\right\} \,.$$ Thus ${\mathscr{H}}^d$ is a Borel regular outer measure on $X$. Notice that our definition differs from that of [@GMT 2.10.2] by a constant mutliplicative factor. This does not affect the results stated in the present paper, except for the specific constants in \[pue.6\] which are of no relevance otherwise to our concerns.
Measurable Spaces with Negligibles
==================================
Most of the material in this Section is either known, or folklore or both, with the possible exception of the Definitions and Facts in \[localized.negligibles\] and \[ideals\] needed in the next Section. I learned about the concept of measurable space with negligibles in D.H. Fremlin’s treatise on Measure Theory. Here I call localizable a measurable space with negligibles whose quotient Boolean algebra is order complete, an important class of examples being the $\sigma$-finite measures spaces, \[sigmafinite.localizable\]. The main property of localizable measurable spaces with negligibles needed in the remaining part of this paper is the possibility of gluing, in a globally measurable way, the locally almost everywhere defined measurable functions, \[gluing\]. We spell out the proof which is similar to the case of measure spaces, see [@FREMLIN.II 213N] for one direction. We close the Section with categorical remarks and suggestions for future work.
\[def.MSN\] A [*measurable space with negligibles*]{} consists of a triple $(X,{\mathscr{A}},{\mathscr{N}})$ where $(X,{\mathscr{A}})$ is a measurable space and ${\mathscr{N}}{\subseteq}{\mathscr{A}}$ is a $\sigma$-ideal of ${\mathscr{A}}$. The latter means that:
1. $\emptyset \in {\mathscr{N}}$;
2. If $A \in {\mathscr{A}}$, $B \in {\mathscr{N}}$ and $A {\subseteq}B$ then $A \in {\mathscr{N}}$;
3. If $(A_n)_{n \in {\mathbb{N}}}$ is a sequence in ${\mathscr{N}}$ then $\cup_{n \in {\mathbb{N}}} A_n \in {\mathscr{N}}$.
Given a measure space $(X,{\mathscr{A}},\mu)$ we define $${\mathscr{N}}_\mu = {\mathscr{A}}\cap \{ N : \mu(N) = 0 \}$$ so that clearly $(X,{\mathscr{A}},{\mathscr{N}}_\mu)$ is a measurable space with negligibles. Even though it seems the natural measurable space with negligibles associated with $(X,{\mathscr{A}},\mu)$, it is by no means the only one that will matter in this paper, see \[def.semilocalizable\]. Similarly if $\phi$ is an outer measure on a set $X$ then $${\mathscr{N}}_\phi = {\mathscr{P}}(X) \cap \{ N : \phi(N) = 0 \}$$ is a $\sigma$-ideal of ${\mathscr{P}}(X)$.
Given a measurable space with negligibles $(X,{\mathscr{A}},{\mathscr{N}})$ and $g \in L_0(X,{\mathscr{A}})$ we define $$\| g \|_{\mathscr{N}}= \inf \{ t : X \cap \{ x : |g(x)| > t \} \in {\mathscr{N}}\} \in [0,\infty]$$ and we say that $g$ is [*${\mathscr{N}}$ essentially bounded*]{} if $\|g\|_{\mathscr{N}}< \infty$. Letting $L_\infty(X,{\mathscr{A}},{\mathscr{N}})$ denote the collection of such functions and be equipped with the operations and partial order inherited from $L_0(X,{\mathscr{A}})$ one checks it is an algebra and a Riesz space. Furthermore $\|\cdot\|_{\mathscr{N}}$ is a seminorm defined on $L_\infty(X,{\mathscr{A}},{\mathscr{N}})$. One classically shows that $\|g\|_{\mathscr{N}}= 0$ if and only $X \cap \{ g \neq 0 \} \in {\mathscr{N}}$ and we let ${\mathbf{L}}_\infty(X,{\mathscr{A}},{\mathscr{N}})$ be the corresponding quotient space equipped with the corresponding norm. The following is established in exactly the same way as in the case of measure spaces.
[ *Given a measurable space with negligibles $(X,{\mathscr{A}},{\mathscr{N}})$, ${\mathbf{L}}_\infty(X,{\mathscr{A}},{\mathscr{N}})$ is both a Banach space and a Banach lattice.* ]{}
\[def.stone\] Let $(X, {\mathscr{A}},{\mathscr{N}})$ be a measurable space with negligibles. Forgetting about the stability of ${\mathscr{A}}$ and ${\mathscr{N}}$ under [*countable (rather than finite)*]{} operations we view ${\mathscr{A}}$ as a Boolean algebra and ${\mathscr{N}}$ as an ideal of ${\mathscr{A}}$. As such the quotient ${\mathscr{A}}_{\mathscr{N}}:= {\mathscr{A}}/ {\mathscr{N}}$ is a Boolean algebra as well.
Given an arbitrary Boolean algebra ${\mathbf{B}}$ we recall that its [*Stone representation*]{} ${\mathsf{Spec}}({\mathbf{B}})$ is a totally disconnected compact Hausdorff topological space of which the Boolean algebra of clopen sets is isomorphic to ${\mathbf{B}}$, see e.g. [@FREMLIN.III 311E and 311I]. By a totally disconnected topological space we mean one whose connected subsets are all singletons ; if the space is assumed to be compact Hausdorff this is equivalent to the existence of a basis for the topology consisting of clopen (closed and open) subsets.
\[LC\] Given a measurable space with negligibles $(X,{\mathscr{A}},{\mathscr{N}})$, the Banach spaces ${\mathbf{L}}_\infty(X,{\mathscr{A}},{\mathscr{N}})$ and $C({\mathsf{Spec}}({\mathscr{A}}_{\mathscr{N}}))$ are isometrically isomorphic.
Letting ${\mathbf{L}}_{\infty,s}(X,{\mathscr{A}},{\mathscr{N}})$ denote the linear subspace of ${\mathbf{L}}_\infty(X,{\mathscr{A}},{\mathscr{N}})$ corresponding to those simple functions $g \in L_\infty(X,{\mathscr{A}},{\mathscr{N}})$, i.e. those having finite range, we define $\Xi : {\mathbf{L}}_{\infty,s}(X,{\mathscr{A}},{\mathscr{N}}) \to C({\mathsf{Spec}}({\mathscr{A}}_{\mathscr{N}}))$ by the formula $$\Xi(u^\bullet) = \sum_{y \in u(X)} y {\mathbbm{1}}_{\operatorname{\mathrm{St}}(u^{-1}\{y\}^\bullet)}$$ where $\operatorname{\mathrm{St}}: {\mathscr{A}}_{\mathscr{N}}\to {\mathsf{Spec}}({\mathscr{A}}_{\mathscr{N}})$ is the Stone isomorphism and the superscript bullet denotes the equivalence class. Since each ${\mathbbm{1}}_{\operatorname{\mathrm{St}}(A^\bullet)}$, $A \in {\mathscr{A}}$, is continuous, $\Xi$ is well defined. It is easy to check that $\Xi$ is a linear isometry onto its image. The basic Approximation Lemma of measurable functions by simple functions implies that ${\mathbf{L}}_{\infty,s}(X,{\mathscr{A}},{\mathscr{N}})$ is dense in ${\mathbf{L}}_\infty(X,{\mathscr{A}},{\mathscr{N}})$, therefore $\Xi$ uniquely extends to a linear isometry $\hat{\Xi} : {\mathbf{L}}_\infty(X,{\mathscr{A}},{\mathscr{N}}) \to C({\mathsf{Spec}}({\mathscr{A}}_{\mathscr{N}}))$. Upon noticing that $\operatorname{\mathrm{im}}\Xi$ is a subalgebra of $C({\mathsf{Spec}}({\mathscr{A}}_{\mathscr{N}}))$ that contains the constant functions and that separates points we infer from the Stone-Weierstrass Theorem that $\operatorname{\mathrm{im}}\Xi$ is dense and in turn that $\hat{\Xi}$ is surjective.
\[def.ess.sup\] Let $(X,{\mathscr{A}},{\mathscr{N}})$ be a measurable space with negligibles and ${\mathscr{E}}{\subseteq}{\mathscr{A}}$. We say that $A \in {\mathscr{A}}$ is an [*${\mathscr{N}}$ essential supremum*]{} of ${\mathscr{E}}$ whenever the following holds:
1. For every $E \in {\mathscr{E}}$ one has $E {\thicksim}A \in {\mathscr{N}}$;
2. If $B \in {\mathscr{A}}$ is such that $E {\thicksim}B \in {\mathscr{N}}$ for every $E \in {\mathscr{E}}$, then $A {\thicksim}B \in {\mathscr{N}}$.
In particular if $A,A' \in {\mathscr{A}}$ are both ${\mathscr{N}}$ essential suprema of ${\mathscr{E}}$ it follows that $A \ominus A' \in {\mathscr{N}}$ where $\ominus$ denotes the symmetric difference of two sets. If $A$ verifies condition (1) but necessarily condition (2) we call it an [*${\mathscr{N}}$ essential upper bound*]{} of ${\mathscr{E}}$.
\[abstract.loc\] We say that a measurable space with negligibles $(X,{\mathscr{A}},{\mathscr{N}})$ is [*localizable*]{} whenever each family ${\mathscr{E}}{\subseteq}{\mathscr{A}}$ admits an ${\mathscr{N}}$-essential supremum.
*Given a measurable space with negligibles $(X,{\mathscr{A}},{\mathscr{N}})$ the following conditions are equivalent:*
1. $(X,{\mathscr{A}},{\mathscr{N}})$ is localizable;
2. The Boolean algebra ${\mathscr{A}}_{\mathscr{N}}$ is order complete;
3. The Stone space ${\mathsf{Spec}}({\mathscr{A}}_{\mathscr{N}}$) is extremally disconnected;
4. The Banach lattice $C({\mathsf{Spec}}({\mathscr{A}}_{\mathscr{N}}))$ is order complete;
5. The Banach space $C({\mathsf{Spec}}({\mathscr{A}}_{\mathscr{N}}))$ is isometrically injective.
That (1) be equivalent to (2) is routine verification. If ${\mathbf{B}}$ is a Boolean algebra then ${\mathbf{B}}$ is order complete if and only if ${\mathsf{Spec}}({\mathbf{B}})$ is extremally disconnected, see e.g. [@FREMLIN.III 314S]. We recall that a compact Hausdorff topological space $K$ is called extremally disconnected if the closure of any open set is open. Furthermore a compact Hausdorff space $K$ is extremally disconnected if and only if $C(K)$ ir order complete, see e.g. [@ALBIAC.KALTON Problems 4.5 and 4.6]. This shows the equivalence between (3) and (4). The equivalence between (4) and (5) is a consequence of the Goodner-Nachbin Theorem, see [@ALBIAC.KALTON 4.3.6].
An important class of examples of localizable spaces with negligibles is given below, with a proof for the reader’s convenience.
\[sigmafinite.localizable\] If $(X,{\mathscr{A}},\mu)$ is a $\sigma$-finite measure space then $(X,{\mathscr{A}},{\mathscr{N}}_\mu)$ is localizable (recall \[def.MSN\]).
If $(X,{\mathscr{A}},\mu)$ is not finite choose a partition $(X_n)_{n \in {\mathbb{N}}}$ of $X$ into members of ${\mathscr{A}}$ such that $0 < \mu(X_n) < \infty$ for every $n \in {\mathbb{N}}$ and define a measure $\nu$ on ${\mathscr{A}}$ by the formula $$\nu(A) = \sum_{n \in {\mathbb{N}}} 2^{-n} \mu(X_n)^{-1} \mu(X_n \cap A) \,,$$ $A \in {\mathscr{A}}$. Observing that ${\mathscr{N}}_\nu = {\mathscr{N}}_\mu$ and that $\nu(X)=2$ we conclude that the proposition follows from its special case when $(X,{\mathscr{A}},\mu)$ is finite.
We henceforth assume that $\mu(X) < \infty$. Let ${\mathscr{E}}{\subseteq}{\mathscr{A}}$ and define $${\mathscr{F}}= {\mathscr{A}}\cap \{ F : \mu(F \cap E)=0 \text{ for every } E \in {\mathscr{E}}\} \,.$$ Notice that ${\mathscr{F}}$ is a $\sigma$-ideal. Put $\tau = \sup \{ \mu(F) : F \in {\mathscr{F}}\} < \infty$. There exists a nondecreasing sequence $(F_n)_{n \in {\mathbb{N}}}$ in ${\mathscr{F}}$ such that $\mu(F_n) {\geqslant}\tau - (n+1)^{-1}$ for every $n \in {\mathbb{N}}$. Thus $F := \cup_{n \in {\mathbb{N}}} F_n \in {\mathscr{F}}$ and $\mu(F) = \tau$. In particular $\mu(G {\thicksim}F) = 0$ for every $G \in {\mathscr{F}}$ (for otherwise $\mu(F \cup (G {\thicksim}F)) > \mu(F)$ and $F \cup (G {\thicksim}F) \in {\mathscr{F}}$, a contradiction). We now claim that $A = X {\thicksim}F$ is an ${\mathscr{N}}_\mu$-essential supremum of ${\mathscr{E}}$. Indeed:
1. Given $E \in {\mathscr{E}}$, $\mu(E {\thicksim}A) = \mu(E \cap F) = 0$ since $F \in {\mathscr{F}}$;
2. If $B \in {\mathscr{A}}$ is such that $\mu(E {\thicksim}B)=0$ for every $E \in {\mathscr{E}}$ then $G = X {\thicksim}B \in {\mathscr{F}}$ and hence $0 = \mu(G {\thicksim}F) = \mu(A {\thicksim}B)$.
\[localized.version\] The first obstacle that comes to mind for a measurable space with negligibles $(X,{\mathscr{A}},{\mathscr{N}})$ to be localizable is that ${\mathscr{A}}$ is not required to be stable under arbitrary unions. If it were, then condition (1) of \[def.ess.sup\] would be obviously satsified with $A = \cup {\mathscr{E}}\in {\mathscr{A}}$. This is not the end of the story however as condition (2) may well fail for such choice of $A$. In fact we give an example below \[CR.2\] (the paragraph before (Q7)) of a measurable space with negligibles of the type $(X,{\mathscr{P}}(X),{\mathscr{N}})$ which is consistently not localizable. Worse yet we exhibit a proper $\sigma$-algebra ${\mathscr{B}}{\subseteq}{\mathscr{P}}(X)$ such that $(X,{\mathscr{A}},{\mathscr{A}}\cap {\mathscr{N}})$ is consistently non localizable whenever ${\mathscr{B}}{\subseteq}{\mathscr{A}}{\subseteq}{\mathscr{P}}(X)$ is a $\sigma$-algebra. This ruins the hope that with each measurable space with negligibles one can associate a localizable version of it by <<adding enough measurable sets>> to the given $\sigma$-algebra. Accepting to enlarge the base set $X$, see \[category\](4) for resurrecting some weak hope.
\[last.lemma\] [ *Assume $(X,{\mathscr{A}},{\mathscr{N}})$ and $(Y,{\mathscr{B}},{\mathscr{M}})$ are measurable spaces with negligibles and $f : X \to Y$ is a bijection such that $${\mathscr{A}}= {\mathscr{P}}(X) \cap \{ f^{-1}(B) : B \in {\mathscr{B}}\}$$ and $${\mathscr{N}}= {\mathscr{P}}(X) \cap \{ f^{-1}(M) : M \in {\mathscr{M}}\} \,.$$ It follows that $(X,{\mathscr{A}},{\mathscr{N}})$ is localizable if and only if $(Y,{\mathscr{B}},{\mathscr{M}})$ is localizable.* ]{}
This can be checked directly by routine verifications from the definition of essential supremum or by observing that the quotient Boolean algebras ${\mathscr{A}}_{{\mathscr{N}}}$ and ${\mathscr{B}}_{{\mathscr{M}}}$ are isomorphic and referring to \[abstract.loc\]. Such $f$ is an instance of an isomorphism in the category to be discussed in \[category\]. We will use this result in \[pue.8\] below.
If $(X,{\mathscr{A}})$ is a measurable space and $E \in {\mathscr{A}}$ we associate with it its subspace $(E,{\mathscr{A}}_E)$ where ${\mathscr{A}}_E = {\mathscr{P}}(E) \cap \{E \cap A : A \in {\mathscr{A}}\}$.
Let $(X,{\mathscr{A}},{\mathscr{N}})$ be a measurable space with negligibles and let ${\mathscr{E}}{\subseteq}{\mathscr{A}}$. A [*family subordinated to ${\mathscr{E}}$*]{} is a family $(g_E)_{E \in {\mathscr{E}}}$ such that
1. $g_E : E \to {\mathbf{R}}$ is ${\mathscr{A}}_E$-measurable for every $E \in {\mathscr{E}}$.
We further say that $(g_E)_{E \in {\mathscr{E}}}$ is [*compatible*]{} if also
1. For every pair $E_1,E_2 \in {\mathscr{E}}$ one has $E_1 \cap E_2 \cap \{ g_{E_1} \neq g_{E_2} \} \in {\mathscr{N}}$.
A [*gluing*]{} of a compatible family $(g_E)_{E \in {\mathscr{E}}}$ subordinated to ${\mathscr{E}}$ is a function $g : X \to {\mathbf{R}}$ such that
1. $g$ is ${\mathscr{A}}$-measurable;
2. $E \cap \{ g \neq g_E \} \in {\mathscr{N}}$ for every $E \in {\mathscr{E}}$.
\[gluing\] Let $(X,{\mathscr{A}},{\mathscr{N}})$ be a measurable space with negligibles. The following are equivalent.
1. $(X,{\mathscr{A}},{\mathscr{N}})$ is localizable.
2. For every ${\mathscr{E}}{\subseteq}{\mathscr{A}}$, every compatible family subordinated to ${\mathscr{E}}$ admits a gluing.
$(1) \Rightarrow (2)$ Let ${\mathscr{E}}{\subseteq}{\mathscr{A}}$ and let $(g_E)_{E \in {\mathscr{E}}}$ be a compatible family subordinated to ${\mathscr{E}}$. With each $q \in {\mathbb{Q}}$ and $E \in {\mathscr{E}}$ we associate $E_q = E \cap \{ g_E {\geqslant}q \} \in {\mathscr{A}}$. Thus given $q \in {\mathbb{Q}}$ the family $\{ E_q : E \in {\mathscr{E}}\}$ admits an ${\mathscr{N}}$ essential supremum which we denote as $A_q \in {\mathscr{A}}$. Define $\bar{g} : X \to [-\infty,+\infty]$ by the formula $\bar{g}(x) = \sup \{ q : x \in A_q\}$, $x \in X$, where as usual $\inf \emptyset = - \infty$. Notice that if $q \in {\mathbb{Q}}$ then $\{ \bar{g} > q \} = \cup_{\substack{r \in {\mathbb{Q}}\\r > q}}A_r \in {\mathscr{A}}$, thus $\bar{g}$ is ${\mathscr{A}}$-measurable.
Given $E \in {\mathscr{E}}$ we shall now establish that $$\label{eq.1}
E \cap \{ g_E \neq \bar{g} \} \in {\mathscr{N}}\,.$$ If $x \in E \cap \{ g_E < \bar{g}\}$ then there exists $q \in {\mathbb{Q}}$ such that $g_E(x) < q$ and $x \in A_q$. Accordingly, $$\label{eq.2}
E \cap \{ g_E < \bar{g} \} {\subseteq}\cup_{q \in {\mathbb{Q}}} E \cap (A_q {\thicksim}E_q ) \,.$$ Now if $q \in {\mathbb{Q}}$ and $E' \in {\mathscr{E}}$ then $$E'_q {\thicksim}(E^c \cup E_q) = E \cap \{ g_E < q \} \cap E' \cap \{ g_{E'} {\geqslant}q \} {\subseteq}E \cap E' \cap \{ g_E \neq g_{E'} \} \in {\mathscr{N}}\,.$$ Since $E' \in {\mathscr{E}}$ is arbitrary we infer that $${\mathscr{N}}\ni A_q {\thicksim}(E^c \cup E_q) = E \cap (A_q {\thicksim}E_q)$$ and it therefore ensues from that $$\label{eq.3}
E \cap \{ g_E < \bar{g} \} \in {\mathscr{N}}\,.$$
Next if $x \in E \cap \{ g_E > \bar{g} \}$ then there exists $q \in {\mathbb{Q}}$ such that $g_E(x) > q$ and $x \not\in A_q$. Consequently, $$\label{eq.4}
E \cap \{ g_E > \bar{g} \} {\subseteq}\cup_{q \in {\mathbb{Q}}} E \cap (E_q {\thicksim}A_q) \in {\mathscr{N}}\,.$$ It now follows from and that holds.
Finally we let $A = \{ \bar{g} \in {\mathbf{R}}\} \in {\mathscr{A}}$ and $g = \bar{g}.{\mathbbm{1}}_A$ (with the usual convention that $(\pm \infty).0=0$). Thus $g$ is ${\mathscr{A}}$-measurable and, for each $E \in {\mathscr{E}}$, $E \cap \{ g \neq g_E \} {\subseteq}E \cap \{ \bar{g} \neq g_E \} \in {\mathscr{N}}$ since $g_E$ is ${\mathbf{R}}$ valued. Whence $g$ is a gluing of $(g_E)_{E \in {\mathscr{E}}}$.
$(2) \Rightarrow (1)$ Let ${\mathscr{E}}{\subseteq}{\mathscr{A}}$ and define ${\mathscr{E}}^* = {\mathscr{E}}{\thicksim}{\mathscr{N}}$ as well as $${\mathscr{F}}= {\mathscr{A}}\cap \{ F : F \cap E \in {\mathscr{N}}\text{ for every } E \in {\mathscr{E}}^* \} \,.$$ Notice that ${\mathscr{F}}\cap {\mathscr{E}}^* = \emptyset$. Put ${\mathscr{G}}= {\mathscr{E}}^* \cup {\mathscr{F}}$ and define a family $(g_G)_{G \in {\mathscr{G}}}$ subordinated to ${\mathscr{G}}$ as follows. If $E \in {\mathscr{E}}^*$ then $g_E = {\mathbbm{1}}_E$, and if $F \in {\mathscr{F}}$ then $g_F = 0.{\mathbbm{1}}_F$. One easily checks that $(g_G)_{G \in {\mathscr{G}}}$ is a compatible family, thus it admits a gluing $g$ by assumption. Let $A = \{ g = 1 \} \in {\mathscr{A}}$. We ought to show that $A$ is an ${\mathscr{N}}$ essential supremum of ${\mathscr{E}}$.
First let $E \in {\mathscr{E}}$. If $E \in {\mathscr{N}}$ then clearly $E {\thicksim}A \in {\mathscr{N}}$. Otherwise $E \in {\mathscr{E}}^*$ and hence $E {\thicksim}A = E \cap \{ g \neq 1 \} = E \cap \{ g \neq g_E \} \in {\mathscr{N}}$. Suppose now that $B \in {\mathscr{E}}$ is such that $E {\thicksim}B \in {\mathscr{N}}$ for every $E \in {\mathscr{E}}$. Let $F = A {\thicksim}B \in {\mathscr{A}}$. Given $E \in {\mathscr{E}}$ we observe that $F \cap E = E \cap (A {\thicksim}B) {\subseteq}A {\thicksim}B \in {\mathscr{N}}$. Therefore $F \in {\mathscr{F}}$. It follows that $A {\thicksim}B = F \cap A = F \cap \{ g = 1 \} {\subseteq}F \cap \{ g \neq g_F \} \in {\mathscr{N}}$ and the proof is complete.
\[localized.negligibles\] Given a measurable space with negligibles $(X,{\mathscr{A}},{\mathscr{N}})$ and ${\mathscr{F}}{\subseteq}{\mathscr{A}}$ an aribtrary family, we define $${\mathscr{N}}[{\mathscr{F}}] = {\mathscr{A}}\cap \{ A : A \cap F \in {\mathscr{N}}\text{ for every } F \in {\mathscr{F}}\} \,.$$ The following are immediate consequences of the definition.
**
1. ${\mathscr{N}}[{\mathscr{F}}]$ is a $\sigma$-ideal in ${\mathscr{A}}$.
2. ${\mathscr{N}}{\subseteq}{\mathscr{N}}[{\mathscr{F}}]$.
3. If ${\mathscr{F}}_1 {\subseteq}{\mathscr{F}}_2$ then ${\mathscr{N}}[{\mathscr{F}}_1] {\supseteq}{\mathscr{N}}[{\mathscr{F}}_2]$.
4. If $g$ and $g'$ are both gluings of a compatible family $(g_E)_{E \in {\mathscr{E}}}$ subordinated to ${\mathscr{E}}$ then $\{ g \neq g' \} \in {\mathscr{N}}[{\mathscr{E}}]$.
\[ideals\] Let $(X,{\mathscr{A}},{\mathscr{N}})$ be a measurable space with negligibles. We say that ${\mathscr{I}}{\subseteq}{\mathscr{A}}$ is an [*ideal*]{} in ${\mathscr{A}}$ whenever the following holds:
1. $\emptyset \in {\mathscr{I}}$;
2. If $A \in {\mathscr{A}}$, $B \in {\mathscr{I}}$ and $A {\subseteq}B$ then $A \in {\mathscr{I}}$;
3. If $A_1,\ldots,A_N \in {\mathscr{I}}$ then $\cup_{n=1}^N A_n \in {\mathscr{I}}$.
One observes that each ${\mathscr{E}}{\subseteq}{\mathscr{A}}$ is contained in a smallest ideal which we denote as $\operatorname{\mathrm{ideal}}({\mathscr{E}})$. The reader will easily check that $$\operatorname{\mathrm{ideal}}({\mathscr{E}}) = {\mathscr{A}}\cap \{ A : \text{ there exist } E_1,\ldots,E_n \in {\mathscr{E}}\text{ such that } A {\subseteq}\cup_{n=1}^N E_n \} \,.$$ Therefore
**
1. ${\mathscr{N}}[{\mathscr{E}}] = {\mathscr{N}}[\operatorname{\mathrm{ideal}}({\mathscr{E}})]$.
2. $A \in {\mathscr{A}}$ is an ${\mathscr{N}}$ essential supremum of ${\mathscr{E}}$ if and only if $A$ is an ${\mathscr{N}}$ essential supremum of $\operatorname{\mathrm{ideal}}({\mathscr{E}})$.
There is no difficulty in showing that the latter is a consequence of the definition of essential supremum and of the following claim: If $C \in {\mathscr{A}}$ is such that $E {\thicksim}C \in {\mathscr{N}}$ for every $E \in {\mathscr{E}}$ then also $F {\thicksim}C \in {\mathscr{N}}$ for every $F \in \operatorname{\mathrm{ideal}}({\mathscr{E}})$.
Let $(X,{\mathscr{A}},{\mathscr{N}})$ be a measurable space with negligibles, and ${\mathscr{I}}{\subseteq}{\mathscr{A}}$ an ideal. A [*partition of unity relative to ${\mathscr{I}}$*]{} is a collection ${\mathscr{E}}{\subseteq}{\mathscr{I}}$ such that
1. ${\mathscr{E}}\cap {\mathscr{N}}= \emptyset$;
2. For every $E_1,E_2 \in {\mathscr{E}}$, if $E_1 \neq E_2$ then $E_1 \cap E_2 \in {\mathscr{N}}$;
3. For every $A \in {\mathscr{I}}{\thicksim}{\mathscr{N}}$ there exists $E \in {\mathscr{E}}$ such that $A \cap E \not\in {\mathscr{N}}$.
\[existence.pu\] Let $(X,{\mathscr{A}},{\mathscr{N}})$ be a measurable space with negligibles, and ${\mathscr{I}}{\subseteq}{\mathscr{A}}$ an ideal. There exists a partition of unity ${\mathscr{E}}$ relative to ${\mathscr{I}}$. Furthermore ${\mathscr{E}}\neq \emptyset$ in case ${\mathscr{I}}{\thicksim}{\mathscr{N}}\neq \emptyset$.
This is a routine application of Zorn’s Lemma.
\[magnitude\] Let $(X,{\mathscr{A}},{\mathscr{N}})$ be a measurable space with negligbles, ${\mathscr{I}}$ an ideal in ${\mathscr{A}}$, and $\kappa$ a cardinal. We say that $(X,{\mathscr{A}},{\mathscr{N}})$ [*has magnitude less than $\kappa$ relative to ${\mathscr{I}}$*]{} whenever for every ${\mathscr{E}}{\subseteq}{\mathscr{I}}$ with the following properties:
1. ${\mathscr{E}}\cap {\mathscr{N}}= \emptyset$;
2. For every $E_1,E_2 \in {\mathscr{E}}$, if $E_1 \neq E_2$ then $E_1 \cap E_2 \in {\mathscr{N}}$;
one has $\operatorname{\mathrm{card}}{\mathscr{E}}{\leqslant}\kappa$.
\[category\] Here we refer to [@BORCEUX.1] for the vocabulary of category theory, and we offer some questions stated in this language. We start by defining a category ${\mathsf{MSN}}$. Its objects are the measurable spaces with negligibles. A morphism from an object $(X,{\mathscr{A}},{\mathscr{N}})$ to an object $(Y,{\mathscr{B}},{\mathscr{M}})$ consists in a map $f : X \to Y$ which is $({\mathscr{A}},{\mathscr{B}})$ measurable and such that $f^{-1}(M) \in {\mathscr{N}}$ for every $M \in {\mathscr{M}}$. For instance if $(X,{\mathscr{A}},\mu)$ is a measure space, $(Y,{\mathscr{B}})$ a measurable space, and $f : X \to Y$ an $({\mathscr{A}},{\mathscr{B}})$ measurable map, we recall that the measure $\nu = f_*\mu$ is defined on ${\mathscr{B}}$ by the formula $(f_*\mu)(B)=\mu(f^{-1}(B))$, $B \in {\mathscr{B}}$, and we readily check that $f$ defines a morphism between $(X,{\mathscr{A}},{\mathscr{N}}_\mu)$ and $(Y,{\mathscr{B}},{\mathscr{N}}_\nu)$. On then infers the following from the Kuratowski Isomorphism Theorem, [@SRIVASTAVA 3.4.23]. If $X$ is an uncountable Polish space and $\mu$ is a diffuse Borel probability measure on $X$ then $(X,{\mathscr{B}}(X),{\mathscr{N}}_\mu)$ and $([0,1],{\mathscr{B}}([0,1]),{\mathscr{N}}_{{\mathscr{L}}^1})$ are isomorphic objects of ${\mathsf{MSN}}$, where ${\mathscr{L}}^1$ is the Lebesgue measure on the unit interval $[0,1]$. Without a separability assumption of the base space, Maharam’s Theorem [@FREMLIN.III 332B] gives a classification of probability spaces at the level of measure algebras but not as strong it seems as to describe isomorphism classes in the category ${\mathsf{MSN}}$. Examples of isomorphisms in the category ${\mathsf{MSN}}$ that are not obtained via the Kuratowski Isomorphism Theorem are used below in the proof of \[pue.8\].
Next we define a full subcategory ${\mathsf{LOC}}$ of ${\mathsf{MSN}}$ by letting its objects be those localizable measurable spaces with negligibles and we consider the corresponding forgetful functor ${\mathsf{F}}: {\mathsf{LOC}}\to {\mathsf{MSN}}$.
1. [ *It would be desirable to study the categorical properties of ${\mathsf{MSN}}$ and ${\mathsf{LOC}}$. We claim for instance that they both admit equalizers and coproducts, preserved by ${\mathsf{F}}$. We also claim that ${\mathsf{MSN}}$ admits products (but the corresponding $\sigma$-ideal in the product is not of <<Fubini type>>), yet ${\mathsf{LOC}}$ may not. Finally we claim that ${\mathsf{MSN}}$ admits coequalizers, yet ${\mathsf{LOC}}$ may not.* ]{}
We now consider the category ${\mathsf{Bool}}$ whose objects are Boolean algebras and whose morphisms are the homomorphisms of Boolean algebras. We also consider ${\mathsf{CBool}}$ the [*full*]{} subcategory of ${\mathsf{Bool}}$ whose objects are the order complete Boolean algebras, together with the corresponding forgetful functor ${\mathsf{CBool}}\to {\mathsf{Bool}}$. One then defines a contravariant functor ${\mathsf{A}}: {\mathsf{MSN}}\to {\mathsf{Bool}}$ in the following way. The image by ${\mathsf{A}}$ of an object $(X,{\mathscr{A}},{\mathscr{N}})$ is the quotient Boolean algebra ${\mathscr{A}}_{\mathscr{N}}$ considered already in \[def.stone\]. Given a morphism $f$ between $(X,{\mathscr{A}},{\mathscr{N}})$ and $(Y,{\mathscr{B}},{\mathscr{M}})$ we let ${\mathsf{A}}(f)$ be the homomorphism of Boolean algebras ${\mathscr{B}}_{\mathscr{M}}\to {\mathscr{A}}_{\mathscr{N}}$ that maps $B^\bullet$ to $f^{-1}(B)^\bullet$, $B \in {\mathscr{B}}$, and one checks it is well defined. We notice that ${\mathsf{A}}(f)$ is in fact more than a homomorphism of Boolean algebras: It is sequentially order continuous. It is even order continuous when for instance ${\mathscr{A}}_{\mathscr{N}}$ is assumed to have the so-called countable chain condition, which is the case when $(X,{\mathscr{A}},{\mathscr{N}})$ has magnitude less than $\aleph_0$ relative to ${\mathscr{A}}$ (an example being $(X,{\mathscr{A}},{\mathscr{N}}_\mu)$ with $(X,{\mathscr{A}},\mu)$ a $\sigma$-finite measure space). The functor ${\mathsf{A}}$ obviously lifts to a contravariant functor ${\mathsf{A}}_c : {\mathsf{LOC}}\to {\mathsf{CBool}}$, leading to the following commutative diagram where vertical arrows denote the forgetful functors. $$\label{diag.1}
\begin{CD}
{\mathsf{LOC}}& @>{{\mathsf{A}}_c}>> & {\mathsf{CBool}}\\
@VVV & & @VVV \\
{\mathsf{MSN}}& @>{{\mathsf{A}}}>> & {\mathsf{Bool}}\end{CD}$$
1. [ *As mentioned in the first paragraph of this number, Maharam’s Theorem pertains to a classification of some objects in ${\mathsf{CBool}}$ arising as measure algebras of probability spaces, but not necessarily a classification in ${\mathsf{LOC}}$. This raises the question of <<realization>> in ${\mathsf{LOC}}$ of certain morphisms in ${\mathsf{CBool}}$. Specifically if $u : {\mathscr{B}}_{\mathscr{M}}\to {\mathscr{A}}_{\mathscr{N}}$ is a sequentially order continuous homomorphism of Boolean algebras, under what conditions on $(X,{\mathscr{A}},{\mathscr{N}})$ and $(Y,{\mathscr{B}},{\mathscr{M}})$ does there exist a morphism $f$ between these two objects in ${\mathsf{MSN}}$ such that ${\mathsf{A}}(f)=u$? Inspired by the special case [@FRE.99] it may be that the key ingredients are the existence of a <<lifting>> in $(Y,{\mathscr{B}},{\mathscr{M}})$ and the existence of a compact class in ${\mathscr{P}}(X)$ that <<generates>> $(X,{\mathscr{A}},{\mathscr{N}})$ in an appropriate sense. See also [@FREMLIN.III Chapter 34]. It further raises the question of identifying classes of measurable spaces with negligibles that admit a <<lifting>>, beyond the case of measure spaces (where the situation is well understood).* ]{}
Next we let ${\mathsf{Comp_{tot}}}$ (resp. ${\mathsf{Comp_{extr}}}$) denote the category of totally disconnected (resp. extremally disconnected) Hausdorff compact topological spaces and their continuous maps. We complete the diagram above by adding the central horizontal arrows which are (contravariant) equivalences of categories. $$\begin{CD}
{\mathsf{LOC}}& @>{{\mathsf{A}}_c}>> & {\mathsf{CBool}}& @>>> &{\mathsf{Comp_{extr}}}& @>>> & {\mathsf{Ban_1}}\\
@VVV & & @VVV & & @VVV & & @VVV\\
{\mathsf{MSN}}& @>{{\mathsf{A}}}>> & {\mathsf{Bool}}& @>>> & {\mathsf{Comp_{tot}}}& @>>> & {\mathsf{Ban_1}}\end{CD}$$ The objects of the category ${\mathsf{Ban_1}}$ are the Banach spaces and its morphisms are the linear contractions. The horizontal contravariant functors in the right hand row map an object $K$ to $C(K)$, and $K$ is extremally disconnected if and only if $C(K)$ is an injective object of ${\mathsf{Ban_1}}$. The composition of the horizontal functors in the diagram associates contravariantly with $(X,{\mathscr{A}},{\mathscr{N}})$ the Banach space $C({\mathsf{Spec}}({\mathscr{A}}_{\mathscr{N}}))$. The isometric isomorphism described in \[LC\] is functorial: There exists a natural transformation between the composition of horizontal functors in the above diagram and the functors ${\mathsf{L}}_\infty : {\mathsf{MSN}}\to {\mathsf{Ban_1}}$ (resp. ${\mathsf{L}}_{\infty,c} : {\mathsf{MSN}}\to {\mathsf{Ban_1}}$) defined as follows. An object $(X,{\mathscr{A}},{\mathscr{N}})$ is mapped to ${\mathbf{L}}_\infty(X,{\mathscr{A}},{\mathscr{N}})$. Given a morphism $f$ between the objects $(X,{\mathscr{A}},{\mathscr{N}})$ and $(Y,{\mathscr{B}},{\mathscr{M}})$ we let ${\mathsf{L}}_\infty(f) : {\mathbf{L}}_\infty(Y,{\mathscr{B}},{\mathscr{M}}) \to {\mathbf{L}}_\infty(X,{\mathscr{A}},{\mathscr{N}})$ be the linear contraction (well) defined by ${\mathsf{L}}_\infty(f)(u^\bullet) = (u \circ f)^\bullet$.
1. [ *There is a subtelty in the definition of localizability that is worth pointing out, related with its sheaf-like quality. Even though $U \mapsto C(U)$ is a sheaf, and despite \[LC\] not all measurable spaces with negligibles are localizable. This can be likely expressed in terms of properties of the functor ${\mathsf{A}}$ related to <<subspaces>>.* ]{}
Somewhat related to the search for a <<localizable version>> of an arbitrary measurable space with negligibles (recall \[localized.version\]) is the fact that the vertical forgetful functor on the right hand row of has no left adjoint (see [@SOL.66] for a proof of the Gaifman-Hales Theorem to the extent that there exists no free, order complete Boolean algebra generated by an infinite set), thus this forgetful functor does not satisfy the <<small set condition>> in Freyd’s adjoint functor Theorem [@BORCEUX.1 3.3.3]. Yet one can consider the following process.
1. [ *Given a measurable space with negligibles $(X,{\mathscr{A}},{\mathscr{N}})$ we seek to produce a new one, localizable, in a canonical (and functorial) way. We start with ${\mathscr{A}}_{\mathscr{N}}$ and we consider its Dedekind completion, say ${\mathbf{A}}$, see e.g. [@FREMLIN.III 314T]. Since ${\mathbf{A}}$ is order $\sigma$-complete we can next associate its Loomis-Sikorski realization (see [@FREMLIN.III 314M] or [@SIKORSKI §29]), a measurable space with negligibles $(\hat{X},\hat{{\mathscr{A}}},\hat{{\mathscr{N}}})$ such that ${\mathbf{A}}$ and $\hat{{\mathscr{A}}}_{\hat{{\mathscr{N}}}}$ are isomorphic as Boolean algebras. What are the properties of the functor thus defined? Can one identify and understand this localizable version for instance in cases described in \[CR.2\](Q7)?* ]{}
Semifinite and Semilocalizable Measure Spaces {#MS}
=============================================
For a measure space $(X,{\mathscr{A}},\mu)$ the canonical map from ${\mathbf{L}}_\infty(X,{\mathscr{A}},\mu)$ to ${\mathbf{L}}_1(X,{\mathscr{A}},\mu)^*$ is in general neither injective nor surjective. It is known that injectivity is equivalent to semifiniteness of $(X,{\mathscr{A}},\mu)$. We identify here a condition equivalent to surjectivity, which we call semilocalizability.
\[upsilon\] Let $(X,{\mathscr{A}},\mu)$ be a measure space. We consider the map $$\Upsilon : {\mathbf{L}}_\infty(X,{\mathscr{A}},\mu) \to {\mathbf{L}}_1(X,{\mathscr{A}},\mu)^*$$ defined in the following way. Given ${\mathbf{g}}\in {\mathbf{L}}_\infty(X,{\mathscr{A}},\mu)$ and ${\mathbf{f}}\in {\mathbf{L}}_1(X,{\mathscr{A}},\mu)$ we let $\Upsilon({\mathbf{g}})({\mathbf{f}}) = \int_X gf d\mu$ for a choice of $g \in {\mathbf{g}}$ and $f \in {\mathbf{f}}$. In general $\Upsilon$ is neither injective nor surjective. In this Section we state a necessary and sufficient condition for $\Upsilon$ to be injective (namely that the measure space be semifinite), and a necessary and sufficient condition for $\Upsilon$ to be surjective (namely that the measure space be semilocalizable).
We say that a measure space $(X,{\mathscr{A}},\mu)$ is [*semifinite*]{} whenever the following holds: For every $A \in {\mathscr{A}}$ such that $\mu(A) = \infty$ there exists $B \in {\mathscr{A}}$ such that $B {\subseteq}A$ and $0 < \mu(B) < \infty$. Clearly all $\sigma$-finite measure spaces are semifinite. We recall that $\Upsilon$ is injective if and only if $(X,{\mathscr{A}},\mu)$ is semfinite and in that case $\Upsilon$ is an isometry, [@FREMLIN.II 243G(a)]. Furthermore if $(X,{\mathscr{A}},\mu)$ is semifinite then $\Upsilon$ is bijective if and only if $(X,{\mathscr{A}},{\mathscr{N}}_\mu)$ is localizable, [@FREMLIN.II 243G(b)][^5]. Below we give a necessary and sufficient condition for $\Upsilon$ to be surjective (not assuming that it be injective in the first place). This seems to be new.
Let $(X,{\mathscr{A}},\mu)$ be a measure space. We say that $A \in {\mathscr{A}}$ is [*purely infinite*]{} if for every $B \in {\mathscr{A}}$ such that $B {\subseteq}A$ one has $\mu(B) = 0$ or $\mu(B) = \infty$. Thus $(X,{\mathscr{A}},\mu)$ is semifinite if and only if there exists no purely infinite $A \in {\mathscr{A}}$. We define $$\begin{split}
{\mathscr{N}}_\mu & = {\mathscr{A}}\cap \{ A : \mu(A)=0 \} \\
{\mathscr{A}}^f_\mu & = {\mathscr{A}}\cap \{ A : \mu(A) < \infty \} \\
{\mathscr{A}}^{pi}_\mu & = {\mathscr{A}}\cap \{ A : A \text{ is purely infinite } \} \,,
\end{split}$$ and we abbreviate ${\mathscr{A}}^f = {\mathscr{A}}^f_\mu$ and ${\mathscr{A}}^{pi} = {\mathscr{A}}^{pi}_\mu$ when no confusion occurs, which is almost always. Clearly ${\mathscr{A}}^f$ is an ideal, whereas ${\mathscr{N}}_\mu$ and ${\mathscr{N}}_\mu \cup {\mathscr{A}}^{pi}$ are $\sigma$-ideals. In fact one easily checks that ${\mathscr{N}}_\mu[{\mathscr{A}}^f] = {\mathscr{N}}_\mu \cup {\mathscr{A}}^{pi}$ and also that ${\mathscr{N}}_\mu \cup {\mathscr{A}}^{pi} = {\mathscr{N}}_{\mu_{sf}}$ where $(X,{\mathscr{A}},\mu_{sf})$ is the semifinite version of $(X,{\mathscr{A}},\mu)$, see [@FREMLIN.II 213X(c)] and also \[sigmafin\] below.
Referring to \[localized.negligibles\] we consider the $\sigma$-ideal ${\mathscr{N}}_\mu[{\mathscr{A}}^f]$ which will play the major role in the present Section. When we need to refer to its members we call these [*locally $\mu$ null*]{}.
\[sigmafin\]
Let $(X,{\mathscr{A}},\mu)$ be a measure space. The following are equivalent:
1. $(X,{\mathscr{A}},\mu)$ is semifinite;
2. ${\mathscr{N}}_\mu = {\mathscr{N}}_\mu[{\mathscr{A}}^f]$.
It follows from \[localized.negligibles\](1) that (2) is equivalent to ${\mathscr{N}}_\mu[{\mathscr{A}}^f] {\subseteq}{\mathscr{N}}_\mu$. Therefore $\neg (2)$ is equivalent to ${\mathscr{N}}_\mu[{\mathscr{A}}^f] {\thicksim}{\mathscr{N}}_\mu \neq \emptyset$. One easily observes that ${\mathscr{N}}_\mu[{\mathscr{A}}^f] {\thicksim}{\mathscr{N}}_\mu = {\mathscr{A}}^{pi}$. Since $(X,{\mathscr{A}},\mu)$ is semifinite if and only if ${\mathscr{A}}^{pi}=\emptyset$ the proof is complete.
\[def.semilocalizable\] A measure space $(X,{\mathscr{A}},\mu)$ is called [*semilocalizable*]{} if the measurable space with negligibles $(X,{\mathscr{A}},{\mathscr{N}}_\mu[{\mathscr{A}}^f])$ is localizable.
\[Riesz\] Let $(X,{\mathscr{A}},\mu)$ be a measure space. The following are equivalent.
1. $(X,{\mathscr{A}},\mu)$ is semilocalizable.
2. $\Upsilon$ is surjective.
$(1) \Rightarrow (2)$ To each $E \in {\mathscr{A}}^f$ we associate the linear isometry ${\boldsymbol{\beta}}_E : {\mathbf{L}}_1(E,{\mathscr{A}}_E,\mu_E) \to {\mathbf{L}}_1(X,{\mathscr{A}},\mu)$ defined in the obvious way (extending ${\mathbf{f}}\in {\mathbf{L}}_1(E,{\mathscr{A}}_E,\mu_E)$ by zero outside of $E$), as well as the linear map ${\boldsymbol{\rho}}_E : {\mathbf{L}}_1(X,{\mathscr{A}},\mu) \to {\mathbf{L}}_1(E,{\mathscr{E}},\mu_E)$ (restricting ${\mathbf{f}}\in {\mathbf{L}}_1(X,{\mathscr{A}},\mu)$ to $E$). Thus $({\boldsymbol{\rho}}_E \circ {\boldsymbol{\beta}}_E)({\mathbf{f}}) = {\mathbf{f}}$ for every ${\mathbf{f}}\in {\mathbf{L}}_1(E,{\mathscr{A}}_E,\mu_E)$ and $({\boldsymbol{\beta}}_E \circ {\boldsymbol{\rho}}_E)({\mathbf{f}}) = {\mathbf{f}}$ for every ${\mathbf{f}}\in {\mathbf{L}}_1(X,{\mathscr{A}},\mu)$ such that $E^c \cap \{ f \neq 0 \} \in {\mathscr{N}}_\mu$, $f \in {\mathbf{f}}$. Given ${\boldsymbol{\alpha}}\in {\mathbf{L}}_1(X,{\mathscr{A}},\mu)^*$ and $E \in {\mathscr{A}}^f$ it follows that ${\boldsymbol{\alpha}}\circ {\boldsymbol{\beta}}_E \in {\mathbf{L}}_1(X,{\mathscr{A}}_E,\mu_E)^*$. Since $(E,{\mathscr{A}}_E,\mu_E)$ is a finite measure space the classical Riesz Representation Theorem yields an ${\mathscr{A}}_E$-measurable function $g_E : E \to {\mathbf{R}}$ such that $({\boldsymbol{\alpha}}\circ {\boldsymbol{\beta}}_E)({\mathbf{f}}) = \int_X g_E f d\mu_E$ for every $f \in {\mathbf{f}}\in {\mathbf{L}}_1(E,{\mathscr{A}}_E,\mu_E)$ and $\sup |g_E| {\leqslant}\| {\boldsymbol{\alpha}}\circ {\boldsymbol{\beta}}_E \| {\leqslant}\| {\boldsymbol{\alpha}}\|$. We shall now observe that the family $(g_E)_{E \in {\mathscr{A}}^f}$ is compatible. Let $E_1,E_2 \in {\mathscr{A}}^f$, $n \in {\mathbb{N}}^*$ and define $Z_n = E_1 \cap E_2 \cap \{ g_{E_1} {\leqslant}n^{-1} + g_{E_2} \}$. Thus $f_n={\mathbbm{1}}_{Z_n} \in L_1(X,{\mathscr{A}},\mu)$ and $$\begin{gathered}
{\boldsymbol{\alpha}}({\mathbf{f}}_n) = ({\boldsymbol{\alpha}}\circ {\boldsymbol{\beta}}_{E_1} \circ {\boldsymbol{\rho}}_{E_1})({\mathbf{f}}_n) = \int_{E_1} g_{E_1} f_n d\mu_{E_1} = \int_{Z_n} g_{E_1} d\mu \\ {\leqslant}n^{-1} \mu(Z_n) + \int_{Z_n} g_{E_2} d\mu = \int_{E_2} g_{E_2} f_n d\mu_{E_2} = ({\boldsymbol{\alpha}}\circ {\boldsymbol{\beta}}_{E_2} \circ {\boldsymbol{\rho}}_{E_2})({\mathbf{f}}_n) = {\boldsymbol{\alpha}}({\mathbf{f}}_n) \,.\end{gathered}$$ Therefore $\mu(Z_n)=0$, thus also $E_1 \cap E_2 \cap \{ g_{E_1} < g_{E_2} \} = \cup_{n \in {\mathbb{N}}^*} Z_n \in {\mathscr{N}}_\mu$, and in turn $E_1 \cap E_2 \cap \{ g_{E_1} \neq g_{E_2} \} \in {\mathscr{N}}_\mu {\subseteq}{\mathscr{N}}_\mu[{\mathscr{A}}^f]$. Since $(X,{\mathscr{A}},{\mathscr{N}}_\mu[{\mathscr{A}}^f])$ is localizable by assumption it follows from \[gluing\] that $(g_E)_{E \in {\mathscr{A}}^f}$ admits a gluing $\tilde{g} : X \to {\mathbf{R}}$. We let $Z = X \cap \{ |\tilde{g}| > \|{\boldsymbol{\alpha}}\| \} \in {\mathscr{A}}$. It ensues from our choice of a special representative $g_E$ that $E \cap Z {\subseteq}E \cap \{ \tilde{g} \neq g_E \} \in {\mathscr{N}}_\mu[{\mathscr{A}}^f]$ for every $E \in {\mathscr{A}}^f$. Hence the function $g = \tilde{g}.{\mathbbm{1}}_Z$ is also a gluing of $(g_E)_{E \in {\mathscr{A}}^f}$, and furthermore $\sup |g| {\leqslant}\|{\boldsymbol{\alpha}}\| < \infty$. Therefore ${\mathbf{g}}\in {\mathbf{L}}_\infty(X,{\mathscr{A}},\mu)$ and it remains to establish that $\Upsilon({\mathbf{g}}) = {\boldsymbol{\alpha}}$.
Let $f \in {\mathbf{f}}\in {\mathbf{L}}_1(X,{\mathscr{A}},\mu)$, define $A = \{ f \neq 0 \}$, and $A_n = \{ |f| {\geqslant}n^{-1} \}$, $n \in {\mathbb{N}}^*$. Thus $A = \cup_{n \in {\mathbb{N}}^*} A_n$ and $A_n \in {\mathscr{A}}^f$ for each $n \in {\mathbb{N}}^*$. Letting $f_n = f.{\mathbbm{1}}_{A_n}$ we notice that $({\mathbf{f}}_n)_{n \in {\mathbb{N}}^*}$ converges to ${\mathbf{f}}$ in ${\mathbf{L}}_1(X,{\mathscr{A}},\mu)$, whence $\lim_n {\boldsymbol{\alpha}}({\mathbf{f}}_n) = {\boldsymbol{\alpha}}({\mathbf{f}})$. We also notice that $gf_n \to gf$ as $n \to \infty$, everywhere, and that $|gf_n| {\leqslant}|gf| \in L_1(X,{\mathscr{A}},\mu)$, so that the Dominated Convergence Theorem applies to $(gf_n)_{n \in {\mathbb{N}}^*}$. Accordingly, $$\begin{split}
\lim_n {\boldsymbol{\alpha}}({\mathbf{f}}_n) & = \lim_n ({\boldsymbol{\alpha}}\circ {\boldsymbol{\beta}}_{A_n} \circ {\boldsymbol{\rho}}_{A_n} )({\mathbf{f}}_n) \\
& = \lim_n \int_{A_n} g_{A_n} f_n d\mu_{A_n} \\
& = \lim_n \int_{A_n} g f_n d\mu \quad
\text{(because $A_n \cap \{ g \neq g_{A_n} \} \in {\mathscr{N}}_\mu$ according to \ref{sigmafin})}\\
& = \lim_n \int_X g f_n d\mu \\
& = \int_X gf d\mu \\
& = \Upsilon({\mathbf{g}})({\mathbf{f}}) \,.
\end{split}$$
$(2) \Rightarrow (1)$ Let ${\mathscr{E}}{\subseteq}{\mathscr{A}}$. We ought to show that ${\mathscr{E}}$ admits an ${\mathscr{N}}_\mu[{\mathscr{A}}^f]$-essential supremum in ${\mathscr{A}}$. According to \[ideals\](5) there is no restriction to assume that ${\mathscr{E}}$ is an ideal. We will define some ${\boldsymbol{\alpha}}\in {\mathbf{L}}_1(X,{\mathscr{A}},\mu)^*$ associated with ${\mathscr{E}}$. We start by defining $\alpha(f) \in {\mathbf{R}}_+$ associated with $f \in L_1(X,{\mathscr{A}},\mu)$, $f {\geqslant}0$, by the following formula: $$\alpha(f) = \sup_{E \in {\mathscr{E}}} \int_E f d\mu \,.$$ We claim that the following hold:
1. For every $f \in L_1(X,{\mathscr{A}},\mu)^+$ one has $0 {\leqslant}\alpha(f) {\leqslant}\int_X f d\mu < \infty$;
2. For every $f_1,f_2 \in L_1(X,{\mathscr{A}},\mu)^+$ one has $\alpha(f_1+f_2) = \alpha(f_1) + \alpha(f_2)$;
3. For every $f \in L_1(X,{\mathscr{A}},\mu)^+$ and every $t {\geqslant}0$ one has $\alpha(t.f) = t.\alpha(f)$;
4. For every $f_1,f_2,f'_1,f_2' \in L_1(X,{\mathscr{A}},\mu)^+$ if $f_1-f_2 = f'_1-f'_2 {\geqslant}0$ then $\alpha(f_1) - \alpha(f_2) = \alpha(f'_1) - \alpha(f'_2)$.
Claims (a) and (c) are obvious. For proving (d) we notice that $\alpha(f_1) + \alpha(f'_2) = \alpha(f_1+f'_2) = \alpha(f'_1+f_2) = \alpha(f'_1) + \alpha(f_2)$, according to (b). Regarding (b) we first notice that $\alpha(f_1+f_2) {\leqslant}\alpha(f_1) + \alpha(f_2)$. Furthermore gievn ${\varepsilon}> 0$ there are $E_j \in {\mathscr{E}}$, $j=1,2$, such that $\alpha(f_j) {\leqslant}{\varepsilon}+ \int_{E_j} f_j d\mu$. Therefore $$\begin{gathered}
\alpha(f_1) + \alpha(f_2) {\leqslant}2{\varepsilon}+ \int_{E_1} f_1 d \mu + \int_{E_2} f_2 d \mu {\leqslant}2{\varepsilon}+ \int_{E_1 \cup E_2} f_1 d \mu + \int_{E_1 \cup E_2} f_2 d \mu \\ = 2{\varepsilon}+ \int_{E_1 \cup E_2} (f_1 + f_2) d\mu {\leqslant}2{\varepsilon}+ \alpha(f_1+f_2)\end{gathered}$$ because $E_1 \cup E_2 \in {\mathscr{E}}$. Since ${\varepsilon}> 0$ is arbitrary, claim (b) follows.
Now if $f \in L_1(X,{\mathscr{A}},\mu)$ we define $\alpha(f) = \alpha(f^+) - \alpha(f^-) \in {\mathbf{R}}$ – a definition compatible with the previous one when $f {\geqslant}0$. It easily follows from (d) that $\alpha$ is additive. Observing that $\alpha(-f) = - \alpha(f)$ when $f {\geqslant}0$, it follows from (c) that $\alpha$ is homogeneous of degree 1. In other words $\alpha$ is linear. Furthemore (a) implies that $|\alpha(f)| {\leqslant}\int_X |f|d\mu$. It is now clear that ${\boldsymbol{\alpha}}({\mathbf{f}}) = \alpha(f)$, $f \in {\mathbf{f}}\in {\mathbf{L}}_1(X,{\mathscr{A}},\mu)$, is well defined and that ${\boldsymbol{\alpha}}\in {\mathbf{L}}_1(X,{\mathscr{A}},\mu)^*$.
It ensues from the hypothesis that there exists $g \in {\mathbf{g}}\in {\mathbf{L}}_\infty(X,{\mathscr{A}},\mu)$ such that $$\int_X gf d\mu = {\boldsymbol{\alpha}}({\mathbf{f}}) = \alpha(f) = \sup_{E \in {\mathscr{E}}} f d\mu$$ for all $0 {\leqslant}f \in {\mathbf{f}}\in {\mathbf{L}}_1(X,{\mathscr{A}},\mu)^+$. We define $A = \{ g \neq 0 \} \in {\mathscr{A}}$ and we will next check that $A$ is an ${\mathscr{N}}_\mu[{\mathscr{A}}^f]$ essential supremum of ${\mathscr{E}}$.
Let $E \in {\mathscr{E}}$. Define $Z = E {\thicksim}A = E \cap \{ g = 0 \}$. Given $F \in {\mathscr{A}}^f$ let ${\mathbbm{1}}_{F \cap Z} \in L_1(X,{\mathscr{A}},\mu)^+$. Thus $$0 = \int_X g{\mathbbm{1}}_{F \cap Z} d\mu = \alpha({\mathbbm{1}}_{F \cap Z}) {\geqslant}\int_E {\mathbbm{1}}_{F \cap Z} d\mu = \mu(F \cap (E {\thicksim}A)) \,.$$ Since $F \in {\mathscr{A}}^f$ is arbitrary it follows that $E {\thicksim}A \in {\mathscr{N}}_\mu[{\mathscr{A}}^f]$.
We next claim that if $F \in {\mathscr{A}}^f$ then $F \cap \{ g < 0 \} \in {\mathscr{N}}_\mu$. Letting $Z_n = \{ g {\leqslant}- n^{-1} \}$, $n \in {\mathbb{N}}^*$, we notice that ${\mathbbm{1}}_{F \cap Z_n} \in L_1(X,{\mathscr{A}},\mu)^+$ whence $$- n^{-1} \mu(F \cap Z_n) {\geqslant}\int_X g {\mathbbm{1}}_{F \cap Z_n} d\mu = \alpha({\mathbbm{1}}_{F \cap Z_n}) {\geqslant}0 \,.$$ Thus clearly $\mu(F \cap Z_n)=0$ and, since $n \in {\mathbb{N}}^*$ is arbitrary $\mu(F \cap \{ g < 0 \})=0$.
Finally we assume that $B \in {\mathscr{A}}$ is such that $\mu(F \cap (E {\thicksim}B))=0$ for every $F \in {\mathscr{A}}^f$. We define $Z = A {\thicksim}B \in {\mathscr{A}}$. Let $F \in {\mathscr{A}}^f$ and notice once again that ${\mathbbm{1}}_{F \cap Z} \in L_1(X,{\mathscr{A}},\mu)^+$, therefore $$\int_X g {\mathbbm{1}}_{F \cap Z} d \mu = \alpha({\mathbbm{1}}_{F \cap Z}) = \sup_{E \in {\mathscr{E}}} \int_E {\mathbbm{1}}_{F \cap Z} d\mu.$$ Now given $E \in {\mathscr{E}}$ we observe that $E \cap F \cap Z = F \cap E \cap (A {\thicksim}B) {\subseteq}F \cap (E {\thicksim}B) \in {\mathscr{N}}_\mu$ by our assumption about $B$. Since $E \in {\mathscr{E}}$ is arbitrary we infer that $$\int_X g {\mathbbm{1}}_{F \cap Z} d\mu = 0 \,.$$ It follows from the previous paragraph and the defintion of $Z$ that $g > 0$, $\mu$ almost everywhere on $Z \cap F$. Consequently $\mu(F \cap Z)=0$ and the proof is complete.
\[dixmier\] Let $(X,{\mathscr{A}},{\mathscr{N}})$ be a measurable space with negligibles. Here we recall that if ${\mathbf{L}}_\infty(X,{\mathscr{A}},{\mathscr{N}})$ is isometrically isomorphic to a dual Banach space then $(X,{\mathscr{A}},{\mathscr{N}})$ is localizable – indeed in this case $C({\mathsf{Spec}}({\mathscr{A}}_{{\mathscr{N}}}))$ is isometrically isomorphic to a dual Banach space according to \[LC\], whence it is isometrically injective [@ALBIAC.KALTON 4.3.8(i)], and it remains to recall \[abstract.loc\]. However if $(X,{\mathscr{A}},{\mathscr{N}})$ is localizable then ${\mathbf{L}}_\infty(X,{\mathscr{A}},{\mathscr{N}})$ does not need to be isometrically isomorphic to a dual Banach space in general (see [@ALBIAC.KALTON Problems 4.8 and 4.9] for an example due to R. Dixmier), yet below we show the conditions are equivalent in the class of measurable spaces with negligibles of the type $(X,{\mathscr{A}},{\mathscr{N}}_\mu[{\mathscr{A}}^f])$ for some measure space $(X,{\mathscr{A}},\mu)$.
Let $(X,{\mathscr{A}},\mu)$ be a measure space. The following are equivalent.
1. $(X,{\mathscr{A}},{\mathscr{N}}_\mu[{\mathscr{A}}^f])$ is localizable;
2. ${\mathbf{L}}_\infty(X,{\mathscr{A}},{\mathscr{N}}_\mu[{\mathscr{A}}^f])$ is isometrically isomorphic to a dual Banach space.
In this case ${\mathbf{L}}_\infty(X,{\mathscr{A}},{\mathscr{N}}_\mu[{\mathscr{A}}^f])$ is isometrically isomorphic to ${\mathbf{L}}_1(X,{\mathscr{A}},\mu)^*$.
That $(2) \Rightarrow (1)$ follows from the general argument in \[dixmier\]. We henceforth assume that $(X,{\mathscr{A}},{\mathscr{N}}_\mu[{\mathscr{A}}^f])$ is localizable and we let $L_\infty(X,{\mathscr{A}})$ denote the linear space consisting of those [*bounded*]{}, ${\mathscr{A}}$ measurable functions $g : X \to {\mathbf{R}}$. In the exact same way as in \[upsilon\] we define a linear map $$\hat{\Upsilon} : L_\infty(X,{\mathscr{A}}) \to {\mathbf{L}}_1(X,{\mathscr{A}},\mu)^* \,.$$
We claim that $\ker \hat{\Upsilon}$ consists of those $g \in L_\infty(X,{\mathscr{A}})$ such that $S_g = \{ g \neq 0 \} \in {\mathscr{N}}_\mu[{\mathscr{A}}^f]$. If $g$ has this property and $f \in L_1(X,{\mathscr{A}},\mu)$ then $\{ f \neq 0 \} = \cup_{n \in {\mathbb{N}}^*} \{ n {\leqslant}|f| \}$ and since each $\{ n {\leqslant}|f| \} \in {\mathscr{A}}^f$ it follows that $\{ gf \neq 0 \} = \{ g \neq 0 \} \cap \{ f \neq 0 \} \in {\mathscr{N}}_\mu$ and in turn $\hat{\Upsilon}(g)(f) = \int_X gf d\mu =0$. The other way around we let $g \in \ker \hat{\Upsilon}$ and we define $S^\pm = \{ \pm g > 0 \}$ so that $\{ g \neq 0 \} = S^+ \cup S^-$. Given $A \in {\mathscr{A}}^f$ and letting $f = {\mathbbm{1}}_{A \cap S^+}$ we infer that $0 = \hat{\Upsilon}(g)(f) = \int_A g^+ d\mu$ thus $S^+ \cap A \in {\mathscr{N}}_\mu$. Thus $S^+ \in {\mathscr{N}}_\mu[{\mathscr{A}}^f]$, and similarly $S^- \in {\mathscr{N}}_\mu[{\mathscr{A}}^f]$.
Since $ {\mathbf{L}}_1(X,{\mathscr{A}},\mu)^*$ is linearly isomorphic to $L_\infty(X,{\mathscr{A}})/\ker \hat{\Upsilon}$, the claim being established we now easily infer that $ {\mathbf{L}}_1(X,{\mathscr{A}},\mu)^*$ is linearly isomorphic to ${\mathbf{L}}_\infty(X,{\mathscr{A}},{\mathscr{N}}_\mu[{\mathscr{A}}^f])$. It remains to show that the corresponding linear isomorphism associated with $\hat{\Upsilon}$ is an isometry. We leave the details to the reader.
Almost Decomposable Measure Spaces
==================================
In this Section we state basic facts on the notion of almost decomposable measure space introduced in [@DEP.98]. It is an appropriate generalization to non semifinite measure spaces of the notion of decomposable measure space (also called strictly localizable measure space). I learned \[CH.implies.ad\] from [@GMT 2.5.10] (in a different language than here). I learned the idea in \[55\] from D.H. Fremlin.
\[almost.decomposition\] Let $(X,{\mathscr{A}},\mu)$ be a measure space. An [*almost decomposition*]{} of $(X,{\mathscr{A}},\mu)$ is a family ${\mathscr{G}}{\subseteq}{\mathscr{A}}$ with the following properties:
1. $\forall G \in {\mathscr{G}}: \mu(G) < \infty$;
2. ${\mathscr{G}}$ is disjointed;
3. $\forall A \in {\mathscr{P}}(X) : ( \forall G \in {\mathscr{G}}: A \cap G \in {\mathscr{A}}) \Rightarrow A \in {\mathscr{A}}$;
4. $\forall A \in {\mathscr{A}}: \mu(A) < \infty \Rightarrow \mu(A) = \sum_{G \in {\mathscr{G}}} \mu(A \cap G)$.
We say that $(X,{\mathscr{A}},\mu)$ is [*almost decompsable*]{} if it admits an almost decomposition.
\[52\] Almost decomposable measure spaces generalize $\sigma$-finite measure spaces. In fact, assuming that $(X,{\mathscr{A}},\mu)$ is semifinite, if ${\mathscr{G}}$ is an almost decomposition of $(X,{\mathscr{A}},\mu)$ and ${\mathscr{G}}$ is (at most) countable then $\mu$ is $\sigma$-finite. Indeed $S = \cup {\mathscr{G}}\in {\mathscr{A}}$ (either because ${\mathscr{G}}$ is countable or according to \[almost.decomposition\](3)), and we ought to show that $\mu(X {\thicksim}S) = 0$. If $A \in {\mathscr{A}}$, $\mu(A) < \infty$ and $A {\subseteq}X {\thicksim}S$ then $\mu(A)=0$ according to \[almost.decomposition\](4). Since $\mu$ is semifinite this implies $\mu(Z {\thicksim}S)=0$.
\[ad.implies.semiloc\] If a measure space admits an almost decomposition then it is semilocalizable.
Let ${\mathscr{G}}$ be an almost decomposition of the measure space $(X,{\mathscr{A}},\mu)$ and let ${\mathscr{E}}{\subseteq}{\mathscr{A}}$ be an arbitrary family. With each $G \in {\mathscr{G}}$ we associate ${\mathscr{E}}_G = \{ G \cap E : E \in {\mathscr{E}}\} {\subseteq}{\mathscr{A}}_G$. Since $(G,{\mathscr{A}}_G,\mu_G)$ is a finite measure space, $(G,{\mathscr{A}}_G,{\mathscr{N}}_{\mu_G})$ is localizable according to \[sigmafinite.localizable\] and we let $A_G \in {\mathscr{A}}_G {\subseteq}{\mathscr{A}}$ be an ${\mathscr{N}}_{\mu_G}$ essential supremum of ${\mathscr{E}}_G$. We now define a subset of $X$ $$A = \cup_{G \in {\mathscr{G}}} A_G \,.$$ Since $A \cap G = A_G \in {\mathscr{G}}$ for every $G \in {\mathscr{G}}$ it follows from condition (3) of the definition of an almost decomposition that $A \in {\mathscr{A}}$.
We shall now show that $E {\thicksim}A \in {\mathscr{N}}_\mu[{\mathscr{A}}^f]$ for every $E \in {\mathscr{E}}$. Let $E \in {\mathscr{E}}$ and $F \in {\mathscr{A}}^f$. It follows that $$\begin{split}
\mu [ F \cap (E {\thicksim}A) ] & = \sum_{G \in {\mathscr{G}}} \mu [ F \cap G \cap (E {\thicksim}A) ] \quad\quad \text{(by \ref{almost.decomposition}(4))}\\
& {\leqslant}\sum_{G \in {\mathscr{G}}} \mu [ (E \cap G) {\thicksim}A_G ] \\
& = 0 \,.
\end{split}$$
Finally we assume that $B \in {\mathscr{A}}$ is such that $E {\thicksim}B \in {\mathscr{N}}_\mu[{\mathscr{A}}^f]$ for all $E \in {\mathscr{E}}$ and we ought to show that $A {\thicksim}B \in {\mathscr{N}}_\mu[{\mathscr{A}}^f]$. Let $F \in {\mathscr{A}}^f$. We must show that $\mu [ F \cap (A {\thicksim}B) ] = 0$. Notice that $$\begin{split}
\mu[F \cap (A {\thicksim}B) ] & = \sum_{G \in {\mathscr{G}}} \mu [ F \cap G \cap (A {\thicksim}B) ] \quad\quad \text{(by \ref{almost.decomposition}(4))} \\
& {\leqslant}\sum_{G \in {\mathscr{G}}} \mu [G \cap (A {\thicksim}B) ] \,.
\end{split}$$ Thus it suffices to establish that $\mu[G \cap (A {\thicksim}B)]=0$ for each $G \in {\mathscr{G}}$. Fix $G \in {\mathscr{G}}$ and let $B_G = B \cap G$. Thus $\mu [(E \cap G) {\thicksim}B_G] = \mu[ G \cap (E {\thicksim}B) ]=0$ for every $E \in {\mathscr{E}}$. Therefore $\mu(A_G {\thicksim}B_G) = 0$. Since $A_G {\thicksim}B_G = (A \cap G) {\thicksim}(B \cap G) = G \cap (A {\thicksim}B)$ the proof is complete.
\[CH.implies.ad\] Let $X$ be a Polish space and let $\phi$ be a Borel regular outer measure on $X$ (recall \[24\]). Assuming the Continuum Hypothesis, the measure space $(X,{\mathscr{A}}_{\phi},\phi)$ admits an almost decomposition.
In case $X$ is finite the conclusion clearly holds. We henceforth assume $X$ is infinite. In that case $\operatorname{\mathrm{card}}{\mathscr{B}}(X) = 2^{\aleph_0}$ (the upper bound follows from the fact that Borel sets are Suslin, and Suslin sets are continuous images of closed subsets of a particular Polish space, the Baire space, see e.g. [@SRIVASTAVA 3.3.18]). We abbreviate ${\mathscr{B}}= {\mathscr{B}}(X)$ and as usual ${\mathscr{B}}^f = {\mathscr{B}}\cap \{ B : \phi(B) < \infty \}$. It now follows from the Continuum Hypothesis that ${\mathscr{B}}^f$ admits a well-ordering $\preccurlyeq$ such that every initial segment ${\mathscr{B}}_B^f = {\mathscr{B}}^f \cap \{ C : C \preccurlyeq B \text{ and } C \neq B \}$, $B \in {\mathscr{B}}^f$, is at most countable and therefore $\cup {\mathscr{B}}_B^f \in {\mathscr{B}}$. With each $B \in {\mathscr{B}}^f$ we associate the Borel set $G_B = B {\thicksim}\cup {\mathscr{B}}_B^f$. We claim that ${\mathscr{G}}= \{ G_B : B \in {\mathscr{B}}^f\}$ is an almost decomposition of $(X,{\mathscr{A}}_{\phi},\phi)$. Conditions (1) and (2) of \[almost.decomposition\] are readily satisfied.
In order to check that \[almost.decomposition\](3) holds, we let $A {\subseteq}X$ be such that $A \cap G$ is $\phi$-measurable for each $G \in {\mathscr{G}}$ and we ought to show that $A$ is $\phi$-measurable. Let $S {\subseteq}X$ be arbitrary. We must establish that $$\phi(S) {\geqslant}\phi(S \cap A) + \phi(S {\thicksim}A) \,.$$ Clearly we may assume that $\phi(S) < \infty$. We choose ${\mathscr{B}}^f \ni B {\supseteq}S$ with $\phi(S)=\phi(B)$ and we number $G_0,G_1,G_2,\ldots$ the sets $G_C$ corresponding to $C \in {\mathscr{B}}^f$ with $C \preccurlyeq B$ and $C \neq B$. Thus $(G_n)_{n \in {\mathbb{N}}}$ is a disjointed sequence of Borel sets whose union is $B$. In turn $B \cap A = \cup_{n \in {\mathbb{N}}} (G_n \cap A)$ is $\phi$-measurable according to our hypothesis about $A$. Therefore $B {\thicksim}A = B \cap (B^c \cup A^c) = B \cap (B \cap A)^c$ is also $\phi$-measurable, whence $$\phi(S) = \phi(B) = \phi(B \cap A) + \phi(B {\thicksim}A) {\geqslant}\phi(S \cap A) + \phi(S {\thicksim}A) \,$$ and the proof of (3) is complete.
We turn to proving that condition \[almost.decomposition\](4) holds. Let $A {\subseteq}X$ be $\phi$-measurable and such that $\phi(A) < \infty$. Owing to the Borel regularity of $\phi$ there exists ${\mathscr{B}}^f \ni B {\supseteq}A$ such that $\phi(A)=\phi(B)$. Associate $(G_n)_{n \in {\mathbb{N}}}$ with $B$ as above. It follows that $A = \cup_{n \in {\mathbb{N}}} A \cap G_n$ and of course each $A \cap G_n$ is $\phi$-measurable. Therefore $$\phi(A) = \sum_{n \in {\mathbb{N}}} \phi(A \cap G_n) \,.$$ Furthermore if $C \in {\mathscr{B}}^f$, $C \neq B$ and $B \preccurlyeq C$ then $A \cap G_C {\subseteq}B \cap G_C = \emptyset$ and therefore $\phi(A \cap G_C)=0$. Consequently $$\phi(A) = \sum_{G \in {\mathscr{G}}} \phi(A \cap G) \,.$$
\[55\] Assume $X$ is a Polish space and $\mu$ a Borel measure in $X$. If the measure space $(X,{\mathscr{B}}(X),\mu)$ is semifinite and almost decomposable then it is $\sigma$-finite.
Assume if possible that $(X,{\mathscr{B}}(X),\mu)$ is semifinite and almost decomposable but not $\sigma$-finite. Letting ${\mathscr{G}}$ be an almost decomposition of ${\mathscr{G}}$ it would ensue from \[52\] that ${\mathscr{G}}$ is uncountable. Let ${\boldsymbol{\kappa}}= \operatorname{\mathrm{card}}{\mathscr{G}}$. It follows from the axiom of choice that there exists $A {\subseteq}X$ such that $A \cap G$ is a singleton for each $G \in {\mathscr{G}}$. Thus $\operatorname{\mathrm{card}}A = {\boldsymbol{\kappa}}$, and $A \in {\mathscr{B}}(X)$ according to \[almost.decomposition\](3). Now, $A$ being an uncountable Suslin subset of a Polish space, ${\boldsymbol{\kappa}}= \operatorname{\mathrm{card}}A = {\boldsymbol{\frak c}}$, see e.g. [@SRIVASTAVA 4.3.5]. Furthermore if $B \in {\mathscr{P}}(A)$ then for each $G \in {\mathscr{G}}$ the set $B \cap G$ is either empty or a singleton, therefore $B \in {\mathscr{B}}(X)$ as follows from \[almost.decomposition\](3). Consequently $2^{{\boldsymbol{\frak c}}} = 2^{{\boldsymbol{\kappa}}} = \operatorname{\mathrm{card}}{\mathscr{P}}(A) {\leqslant}\operatorname{\mathrm{card}}{\mathscr{B}}(X) = {\boldsymbol{\frak c}}$ (where the last equality was already recalled at the beginning of the proof of \[CH.implies.ad\]), in contradiction with G. Cantor’s Theorem that ${\boldsymbol{\frak c}}< 2^{{\boldsymbol{\frak c}}}$.
\[56\] Let $X$ be an uncountable separable complete metric space, and $0 < d < \infty$. It follows that either the measure space $(X,{\mathscr{B}}(X),{\mathscr{H}}^d)$ is $\sigma$-finite or it is not almost decomposable.
Indeed \[55\] applies because $(X,{\mathscr{B}}(X),{\mathscr{H}}^d)$ is semifinite according to J. Howroyd’s Theorem [@HOW.95] or [@FREMLIN.IV 471S].
Locally Determined Measure Spaces of Magnitude less than Continuum
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Almost decomposable measure spaces are semilocalizable, \[ad.implies.semiloc\] but the converse does not hold. A classical counter-example (in case of semifinite measure spaces) is due to D.H. Fremlin [@FREMLIN.II 216E]. However if the corresponding quotient Boolean algbera is <<not too large>> the converse holds. In case of semifinite measure spaces this is due to E. J. McShane [@MCS.62]. Here we deal with the non semifinite case, \[mcshane\].
\[61\] A measure space $(X,{\mathscr{A}},\mu)$ is called [*locally determined*]{} whenever the following holds: $$\forall A \in {\mathscr{P}}(X) : \left[ \forall F \in {\mathscr{A}}^f : A \cap F \in {\mathscr{A}}\right] \Rightarrow A \in {\mathscr{A}}$$ where as usual $${\mathscr{A}}^f = {\mathscr{A}}\cap \{ A : \mu(A) < \infty \} \,.$$
\[62\] Let $\phi$ be an outer measure on a set $X$ and assume that $\phi$ has measurable hulls, i.e. $$\left( \forall S \in {\mathscr{P}}(X) \right) \left( \exists A \in {\mathscr{A}}_\phi \right) : S {\subseteq}A \text{ and } \phi(S) = \phi(A) \,.$$ It follows that $(X,{\mathscr{A}}_\phi,\phi)$ is locally determined.
Let $A \in {\mathscr{P}}(X)$ and assume that $A \cap F$ is $\phi$ measurable whenever $F$ is $\phi$ measurable and $\phi(F) < \infty$. We ought to show that $A$ is $\phi$ measurable. It suffices to establish that $$\phi(S) {\geqslant}\phi(S \cap A ) + \phi(S {\thicksim}A)$$ whenever $S \in {\mathscr{P}}(X)$ and $\phi(S) < \infty$. Let $B \in {\mathscr{A}}_\phi$ be a $\phi$ meausurable hull of $S$. Thus $B \in {\mathscr{A}}_\phi^f$ so that $A \cap B \in {\mathscr{A}}_\phi$ by assumption, and hence also $B {\thicksim}A = B {\thicksim}(A \cap B) \in {\mathscr{A}}_\phi$. Therefore $$\phi(S) = \phi(B) = \phi(B \cap A) + \phi(B {\thicksim}A) {\geqslant}\phi(S \cap A) + \phi(S {\thicksim}A)$$ and the proof is complete.
\[63\] Assume that $X$ is a Polish space and that $\phi$ is a Borel regular outer measure on $X$ (recall \[24\]). It follows that $(X,{\mathscr{A}}_\phi,\phi)$ has magnitude (recall \[magnitude\]) less than ${\boldsymbol{\frak c}}$ (the power of continuum).
Let ${\mathscr{E}}{\subseteq}{\mathscr{A}}^f$ be as in \[magnitude\]. With each $E \in {\mathscr{E}}$ we associate a Borel hull $B_E \in {\mathscr{B}}(X)$ such that $E {\subseteq}B_E$ and $\phi(E)=\phi(B_E)$. Since $\phi(B_E) < \infty$ and both $E$ and $B_E$ are $\phi$ measurable, we infer that $\phi(B_E {\thicksim}E)=0$. We now claim that if $E_1,E_2 \in {\mathscr{E}}$ and $E_1 \neq E_2$ then $B_{E_1} \neq B_{E_2}$. Indeed, assuming $E_1 \neq E_2$, we see that $$B_{E_1} \cap B_{E_2} {\subseteq}(B_{E_1} {\thicksim}E_1) \cup (E_1 \cap B_{E_2}) {\subseteq}(B_{E_1} {\thicksim}E_1) \cup (E_1 \cap E_2) \cup ( B_{E_2} {\thicksim}E_2)$$ is $\phi$ negligible. Assuming if possible that $B_{E_1} = B_{E_2}$ it would ensue that $B_{E_1}$ is $\phi$ nelgigible, whence also $E_1$, a contradiction. In othe words the map ${\mathscr{E}}\to {\mathscr{B}}(X) : E \mapsto B_E$ is injective. Since $\operatorname{\mathrm{card}}{\mathscr{B}}(X) {\leqslant}{\boldsymbol{\frak c}}$ the proof is complete.
\[416\] Let $(X,{\mathscr{A}},\mu)$ be a measure space which is complete and locally determined, and let ${\mathscr{E}}$ be such that
1. ${\mathscr{E}}{\subseteq}{\mathscr{A}}^f$;
2. ${\mathscr{E}}$ is disjointed;
3. $( \forall A \in {\mathscr{A}}^f {\thicksim}{\mathscr{N}}_\mu)(\exists E \in {\mathscr{E}}) : A \cap E \not \in {\mathscr{N}}_\mu$.
It follows that ${\mathscr{E}}$ is an almost decomposition of $(X,{\mathscr{A}},\mu)$.
We start by proving that condition (4) of \[almost.decomposition\] is satisfied. Let $A \in {\mathscr{A}}^f$. If $A \in {\mathscr{N}}_\mu$ there is noting to prove, thus we henceforth assume that $\mu(A) > 0$. Define $${\mathscr{E}}_A = {\mathscr{E}}\cap \{ E : E \cap A \not \in {\mathscr{N}}_\mu \} \,.$$ We first claim that ${\mathscr{E}}_A$ is at most countable. Indeed if ${\mathscr{F}}{\subseteq}{\mathscr{E}}_A$ is finite then $$\sum_{E \in {\mathscr{F}}} \mu(E \cap A) = \mu \left( (\cup {\mathscr{F}}) \cap A \right) {\leqslant}\mu(A)$$ because ${\mathscr{F}}$ is disjointed. Letting ${\mathscr{E}}_{A,n} = {\mathscr{E}}_A \cap \{ E : \mu(E \cap A) {\geqslant}n^{-1} \}$ we infer that ${\mathscr{E}}_A = \cup_{n=1}^\infty {\mathscr{E}}_{A,n}$ and $\operatorname{\mathrm{card}}{\mathscr{F}}{\leqslant}n \mu(A) < \infty$. This completes the proof of the claim.
Define $$B = \cup_{E \in {\mathscr{E}}_A} E \cap A$$ and notice that $B \in {\mathscr{A}}$ because ${\mathscr{E}}_A$ is at most countable. Let $C = A {\thicksim}B$. Assume if possible that $\mu(C) > 0$. Since $\mu(C) {\leqslant}\mu(A) < \infty$ it follows from hypothesis (3) that the exists $E \in {\mathscr{E}}$ such that $\mu(E \cap C) > 0$. Thus $C \in {\mathscr{E}}_A$ and in turn $E \cap C {\subseteq}B$ so that $E \cap C = E \cap (C \cap B) = \emptyset$ by the definition of $B$, a contradiction. Thus indeed $\mu(C)=0$. Finally $$\mu(A) = \mu(B) = \sum_{E \in {\mathscr{E}}_A} \mu(E \cap A) = \sum_{E \in {\mathscr{E}}} \mu(E \cap A) \,.$$
It remains to establish that condition (3) of \[almost.decomposition\] holds. Since $(X,{\mathscr{A}},\mu)$ is locally determined it suffices to show the following: If $A \in {\mathscr{P}}(X)$ and $A \cap E \in {\mathscr{A}}$ for every $E \in {\mathscr{E}}$, then $A \cap F$ for every $F \in {\mathscr{A}}^f$. Fix $A \in {\mathscr{P}}(X)$ that meets this condition. Let $F \in {\mathscr{A}}^f$. We apply the preceding paragraph to $F$: $$F = \left( \cup_{E \in {\mathscr{E}}_F} E \cap F \right) \cup N$$ for some $N \in {\mathscr{N}}_\mu$. Therefore $$A \cap F = \left( \cup_{E \in {\mathscr{E}}_F} \left( (A \cap E) \cap F \right)\right) \cup (A \cap N) \,.$$ Now each $A \cap E \in {\mathscr{A}}$ by assumption, thus also $A \cap E \cap F \in {\mathscr{A}}$, $E \in {\mathscr{E}}$. Since ${\mathscr{E}}_F$ is at most countable, the first term in the union of the right hand side above belongs to ${\mathscr{A}}$. Furthermore $A \cap N \in {\mathscr{N}}_\mu {\subseteq}{\mathscr{A}}$ because $(X,{\mathscr{A}},\mu)$ is complete. Therefore $A \cap F \in {\mathscr{A}}$. Since $F \in {\mathscr{A}}^f$ is arbitrary we are done.
\[mcshane\] Assume that a measure space $(X,{\mathscr{A}},\mu)$:
1. is complete;
2. is locally determined;
3. has magnitude less than ${\boldsymbol{\frak c}}$;
4. is semilocalizable.
It follows that it is almost decomposable.
According to \[existence.pu\] applied with ${\mathscr{I}}= {\mathscr{A}}^f$ and ${\mathscr{N}}= {\mathscr{N}}_\mu$, there exists ${\mathscr{E}}{\subseteq}{\mathscr{A}}$ such that
1. ${\mathscr{E}}\cap {\mathscr{N}}_\mu = \emptyset$;
2. For every $E_1, E_2 \ in {\mathscr{E}}$, if $E_1 \neq E_2$ then $E_1 \cap E_2 \in {\mathscr{N}}_\mu$;
3. For every $A \in {\mathscr{A}}^f {\thicksim}{\mathscr{N}}_\mu$ there exists $E \in {\mathscr{E}}$ such that $A \cap E \not \in {\mathscr{N}}_\mu$.
By hypothesis (3), $\operatorname{\mathrm{card}}{\mathscr{E}}{\leqslant}{\boldsymbol{\frak c}}$ thus there exists an injective map $u : {\mathscr{E}}\to ]0,1]$. With each $E \in {\mathscr{E}}$ we associate $g_E = u(E) {\mathbbm{1}}_E$ which is ${\mathscr{A}}$ measurable, thus $(g_E)_{E \in {\mathscr{E}}}$ is a family subordinated to ${\mathscr{E}}$. It is compatible relative to $(X,{\mathscr{A}},{\mathscr{N}}_\mu[{\mathscr{A}}^f])$ because if $E_1, E_2 \in {\mathscr{E}}$ and $E_1 \cap E_2 \in {\mathscr{N}}_\mu$ then $$E_1 \cap E_2 \cap \{ g_{E_1} \neq g_{E_2} \} {\subseteq}E_1 \cap E_2 \in {\mathscr{N}}_\mu {\subseteq}{\mathscr{N}}_\mu[{\mathscr{A}}^f] \,.$$ By hypothesis (4) it therefore admits a gluing $g$, i.e. an ${\mathscr{A}}$ measurable function $g : X \to {\mathbf{R}}$ such that for every $E \in {\mathscr{E}}$, $$E \cap \{ g \neq g_E \} \in {\mathscr{N}}_\mu[{\mathscr{A}}^f] \,.$$ Now for $E \in {\mathscr{E}}$ we define $$G_E = E \cap g^{-1}\{u(E)\} \in {\mathscr{A}}\,.$$ If $E_1 \neq E_2$ then $G_{E_1} \cap G_{E_2} {\subseteq}g^{-1}\{u(E_1)\} \cap g^{-1}\{u(E_1)\} = \emptyset$ because $u$ is injective. Accordingly, ${\mathscr{G}}= \{ G_E : E \in {\mathscr{E}}\}$ is disjointed. Also, $\mu(G_E) {\leqslant}\mu(E) < \infty$ when $E \in {\mathscr{E}}$, thus ${\mathscr{G}}{\subseteq}{\mathscr{A}}^f$.
In order to conclude it remains only to establish that ${\mathscr{G}}$ verifies condition (3) of \[416\]. Assume to the contrary that there exists $A \in {\mathscr{A}}^f {\thicksim}{\mathscr{N}}_\mu$ such that $A \cap G_E \in {\mathscr{N}}_\mu$ for all $E \in {\mathscr{E}}$. Given $E \in {\mathscr{E}}$ we shall now show that $A \cap E \in {\mathscr{N}}_\mu$, in contradiction with (c) above, thereby completing the proof. Since $A \cap E = ( A \cap G_E) \cup ( A \cap (E {\thicksim}G_E))$ and $A \cap G_E \in {\mathscr{N}}_\mu$, it suffices to show that $A \cap (E {\thicksim}G_E) \in {\mathscr{N}}_\mu$. Recall that $E {\thicksim}G_E = E \cap \{ g \neq g_E \} \in {\mathscr{N}}_\mu[{\mathscr{A}}^f]$. Since $A \in {\mathscr{A}}^f$ we infer indeed that $A \cap (E {\thicksim}G_E) \in {\mathscr{N}}_\mu$.
Assume that $(X,{\mathscr{A}},\mu)$ is a complete, locally determined measure space and has magnitude less than ${\boldsymbol{\frak c}}$. The following are equivalent.
1. $\Upsilon$ is surjective;
2. $(X,{\mathscr{A}},\mu)$ is semilocalizable;
3. $(X,{\mathscr{A}},\mu)$ admits an almost decomposition.
Hausdorff Measures in Complete Separable Metric Spaces
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\[71\] Let $X$ be a complete separable metric space and $0 < d < \infty$. For the measure space $(X,{\mathscr{A}}_{{\mathscr{H}}^d},{\mathscr{H}}^d)$ the following are equivalent.
1. The canonical map $\Upsilon : {\mathbf{L}}_\infty(X,{\mathscr{A}}_{{\mathscr{H}}^d},{\mathscr{H}}^d) \to {\mathbf{L}}_1(X,{\mathscr{A}}_{{\mathscr{H}}^d},{\mathscr{H}}^d)^*$ is surjective;
2. $(X,{\mathscr{A}}_{{\mathscr{H}}^d},{\mathscr{H}}^d)$ is semilocalizable;
3. $(X,{\mathscr{A}}_{{\mathscr{H}}^d},{\mathscr{H}}^d)$ is almost decomposable.
That (1) and (2) be equivalent for any measure space was established in \[Riesz\], and that $(3) \Rightarrow (2)$ for any measure space was proved in \[ad.implies.semiloc\]. It remains to observe that $(2) \Rightarrow (3)$ under the present assumptions is a consequence of \[mcshane\]. The measure space $(X,{\mathscr{A}}_{{\mathscr{H}}^d},{\mathscr{H}}^d)$ is clearly complete. The outer measure ${\mathscr{H}}^d$ being Borel regular, it follows from \[62\] that $(X,{\mathscr{A}}_{{\mathscr{H}}^d},{\mathscr{H}}^d)$ is locally determined, and from \[63\] and the separability assumption of $X$ that it has magnitude less than ${\boldsymbol{\frak c}}$.
An Abstract Condition for the Consistency of not being Semilocalizable
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\[number.7\] Let $X$ be a set and let ${\mathscr{N}}{\subseteq}{\mathscr{P}}(X)$ be a $\sigma$-ideal in ${\mathscr{P}}(X)$. We recall the following cardinals associated with ${\mathscr{N}}$: $$\begin{split}
{\mathsf{non}}({\mathscr{N}}) & = \min \{ \operatorname{\mathrm{card}}S : S {\subseteq}X \text{ and } S \not \in {\mathscr{N}}\} \\
{\mathsf{cov}}({\mathscr{N}}) & = \min \{ \operatorname{\mathrm{card}}{\mathscr{C}}: {\mathscr{C}}{\subseteq}{\mathscr{N}}\text{ and } X = \cup {\mathscr{C}}\} \,.
\end{split}$$ Letting ${\mathscr{L}}^1$ denote the restriction of the Lebesgue [*outer measure*]{} to the interval $[0,1]$ we consider the corresponding cardinals ${\mathsf{non}}({\mathscr{N}}_{{\mathscr{L}}^1})$ and ${\mathsf{cov}}({\mathscr{N}}_{{\mathscr{L}}^1})$. These are part of the so-called Cichoń diagram, see [@FREMLIN.V.1 522]. Below we will use the fact that the strict inequality ${\mathsf{non}}({\mathscr{N}}_{{\mathscr{L}}^1}) < {\mathsf{cov}}({\mathscr{N}}_{{\mathscr{L}}^1})$ is consistent with $\mathsf{ZFC}$, see [@BARTOSZYNSKI.JUDAH Chapter 7] or [@FREMLIN.V.2 552H and 552G].
\[number.8\] Let $X$ be a Polish space and let $\mu$ be a diffuse probability measure defined on ${\mathscr{B}}(X)$. It follows that:
1. ${\mathsf{non}}({\mathscr{N}}_{\bar{\mu}}) = {\mathsf{non}}({\mathscr{N}}_{{\mathscr{L}}^1})$;
2. ${\mathsf{cov}}({\mathscr{N}}_{\bar{\mu}}) = {\mathsf{cov}}({\mathscr{N}}_{{\mathscr{L}}^1})$.
Since $\mu$ is diffuse and nonzero, $X$ is uncountable and therefore the Kuratowski Isomorphism Theorem applies, [@SRIVASTAVA 3.4.23]: There exists a bijection $f : X \to [0,1]$ such that both $f$ and $f^{-1}$ are Borel measurable, and $f_*\mu = \lambda$ where $\lambda = {\mathscr{L}}^1|_{{\mathscr{B}}([0,1])}$. We claim that: $$\text{{\it For every} } S {\subseteq}X : \bar{\mu}(S) = 0 \text{ {\it if and only if} } {\mathscr{L}}^1(f(S))=0 \,.$$ Assume that $\bar{\mu}(S)=0$. Since $\bar{\mu}$ is Borel regular, \[26\] there exists ${\mathscr{B}}(X) \ni B {\supseteq}S$ such that $\mu(B)=0$. As $f(B)$ is Borel one has ${\mathscr{L}}^1(f(B))=\lambda(f(B))=(f_*\mu)(f(B))=\mu(B)=0$ and therefore ${\mathscr{L}}^1(f(S))=0$ because $f(S) {\subseteq}f(B)$. The other way round one argues similarly, referring to the Borel regularity of ${\mathscr{L}}^1$.
We now prove (1). Assume $S {\subseteq}[0,1]$ and $S \not \in {\mathscr{N}}_{{\mathscr{L}}^1}$, i.e. ${\mathscr{L}}^1(S) > 0$. It follows from the claim above claim that $f^{-1}(S) \not \in {\mathscr{N}}_{\bar{\mu}}$. Since $\operatorname{\mathrm{card}}S = \operatorname{\mathrm{card}}f^{-1}(S)$ we infer that ${\mathsf{non}}({\mathscr{N}}_{\bar{\mu}}) {\leqslant}{\mathsf{non}}({\mathscr{N}}_{{\mathscr{L}}^1})$. The reverse inequality is proved in a similar fashion.
Let $(N_i)_{i \in I} {\subseteq}{\mathscr{N}}_{{\mathscr{L}}^1}$ be such that $[0,1] = \cup_{i \in I} N_i$. It follows from the claim above that $(f^{-1}(N_i))_{i \in I} {\subseteq}{\mathscr{N}}_{\bar{\mu}}$, and clearly $X = \cup_{i \in I} f^{-1}(N_i)$, thus ${\mathsf{cov}}({\mathscr{N}}_{\bar{\mu}}) {\leqslant}{\mathsf{cov}}({\mathscr{N}}_{{\mathscr{L}}^1})$. The reverse inequality follows similarly and the proof of (2) is complete.
\[abstract.theorem\] Assume that $(X,{\mathscr{A}},{\mathscr{N}})$ is a measurable space with negligibles and that $(S,{\mathscr{B}}(S),\sigma)$, $(T,{\mathscr{B}}(T),\tau)$ are probability spaces with $S$ and $T$ being Polish spaces. Furthermore assume the existence of maps $S \to {\mathscr{A}}: s \mapsto V_s$ and $T \to {\mathscr{A}}: t \mapsto H_t$ with the following properties.
1. For every $s \in S$ and every $t \in T$ one has $\emptyset \neq V_s \cap H_t \in {\mathscr{N}}$;
2. For every $Z \in {\mathscr{A}}$ one has:
1. For every $s \in S$ if $V_s \cap Z \in {\mathscr{N}}$ then $$\overline{\tau} ( T \cap \{ t : H_t \cap V_s \cap Z \neq \emptyset \} ) = 0 \,;$$
2. For every $t \in T$ if $H_t \cap Z \in {\mathscr{N}}$ then $$\overline{\sigma } ( S \cap \{ s : V_s \cap H_t \cap Z \neq \emptyset \} ) = 0 \,.$$
Under the consistent assumption that ${\mathsf{non}}({\mathscr{N}}_{{\mathscr{L}}^1}) < {\mathsf{cov}}({\mathscr{N}}_{{\mathscr{L}}^1})$ it follows that $(X,{\mathscr{A}},{\mathscr{N}})$ is not localizable.
Proceeding toward a contradiction, assume if possible that $(X,{\mathscr{A}},{\mathscr{N}})$ is localizable. The family ${\mathscr{E}}= \{ V_s : s \in S \} {\subseteq}{\mathscr{A}}$ would then admit an ${\mathscr{N}}$ essential supremum, say $A \in {\mathscr{A}}$. Thus,
1. For every $s \in S$: $V_s {\thicksim}A \in {\mathscr{N}}$;
2. For every $t \in T$: $H_t \cap A \in {\mathscr{N}}$.
Condition (A) readily follows from the definition of an essential supremum whereas condition (B) is established in the following manner. Fix $t \in T$ and consider the set $B = A {\thicksim}H_t$ and observe that for every $s \in S$ one has $V_s {\thicksim}B = (V_s {\thicksim}A) \cup (V_s \cap H_t) \in {\mathscr{N}}$ since the first term is a member of ${\mathscr{N}}$ by (A), and the second by hypothesis (1). As $s$ is arbitrary, $B$ is an ${\mathscr{N}}$ essential upper bound of ${\mathscr{E}}$. Therefore $A {\thicksim}B \in {\mathscr{N}}$ and it remains to notice that $A {\thicksim}B = A \cap H_t$.
With each $s \in S$ we now associate the set $$T_s = T \cap \{ t : H_t \cap V_s \cap A^c \neq \emptyset \} \,.$$ Since $V_s \cap A^c \in {\mathscr{N}}$ by (A), our hypothesis (2)(a) implies that $\overline{\tau}(T_s)=0$. Now let $E {\subseteq}S$ be such that $\overline{\sigma}(E) > 0$ and $$\operatorname{\mathrm{card}}E = {\mathsf{non}}({\mathscr{N}}_{\bar{\sigma}}) = {\mathsf{non}}({\mathscr{N}}_{{\mathscr{L}}^1}) < {\mathsf{cov}}({\mathscr{N}}_{{\mathscr{L}}^1}) = {\mathsf{cov}}({\mathscr{N}}_{\bar{\tau}}) \,,$$ where the second and last equalities follow from \[number.8\] and the strict inequality is our consistent assumption. We next define $F = \cup_{s \in E} T_s {\subseteq}T$. Since each $T_s \in {\mathscr{N}}_{\bar{\tau}}$ and $\operatorname{\mathrm{card}}E < {\mathsf{cov}}({\mathscr{N}}_{\bar{\tau}})$ it ensues that $F \neq T$. Pick $t \in T {\thicksim}F$. Thus for each $s \in E$ one has $t \not \in T_s$, i.e. $H_t \cap V_s \cap A^c = \emptyset$ and in turn (since $V_s \cap H_t \neq \emptyset$ by assumption (1)) $H_t \cap V_s \cap A \neq \emptyset$. Accordingly, $$E {\subseteq}S \cap \{ s : V_s \cap H_t \cap A \neq \emptyset \} \,.$$ Yet $H_t \cap A \in {\mathscr{N}}$ for every $t \in T$ by (B), thus $\overline{\sigma}(E)=0$ by assumption (2)(b), a contradiction.
\[84\] One checks (in a smiliar way as in \[85\] below) that \[abstract.theorem\] applies to the measurable space with negligibles $(X,{\mathscr{A}},{\mathscr{N}})$ where $X = [0,1] \times [0,1]$, ${\mathscr{A}}$ is a $\sigma$ algebra of subsets of $X$ containing the Borel subsets of $X$, and ${\mathscr{N}}$ consists of those members of ${\mathscr{A}}$ that are ${\mathscr{H}}^1$ negligible. The hypotheses are met with both $(S,{\mathscr{B}}(S),\sigma)$ and $(T,{\mathscr{B}}(T),\tau)$ being $([0,1],{\mathscr{B}}([0,1]),{\mathscr{L}}^1|_{{\mathscr{B}}([0,1])})$ and $V_s = \{s\} \times [0,1]$, $H_t = [0,1] \times \{t\}$, $s,t \in [0,1]$. Thus $V_s \cap H_t = \{(s,t)\}$ and the condition $V_s \cap H_t \cap Z \neq \emptyset$ is equivalent to $(s,t) \in Z$, and therefore (2)(a) (resp. (2)(b)) of \[abstract.theorem\] holds since the corresponding slice of $Z$ is assumed to be ${\mathscr{H}}^1$ negligible, and the projection on the second (resp. first) axis contracts ${\mathscr{H}}^1$ measure. The case when ${\mathscr{A}}$ consists of those ${\mathscr{H}}^1$ measurable subsets of $X$ was proved in [@DEP.98]. The notion of a measurable space with negligibles makes it a possibility to dispense altogether with a measure being defined on ${\mathscr{A}}$ and, consequently allows for our slightly more general statement here. The point being that the nature of the statement does not involve a measure. See also \[CR.2\](Q5).
\[85\] Let $0 < d < 1$ and let $C_d {\subseteq}[0,1]$ be a self-similar Cantor set of Hausdorff dimension $d$ described in [@MATTILA 4.10]. The measure space $(C_d \times C_d, {\mathscr{A}}_{{\mathscr{H}}^d},{\mathscr{H}}^d)$ is consistently not semilocalizable.
We let $X = C_d \times C_d$, we let $\phi$ be the Hausdorff ${\mathscr{H}}^d$ measure restricted to $X$ and ${\mathscr{A}}= {\mathscr{A}}_\phi$. We also put ${\mathscr{N}}= {\mathscr{N}}_\phi[{\mathscr{A}}_\phi^f]$ and we aim at checking that \[abstract.theorem\]applies to the measurable space with negligibles $(X,{\mathscr{A}},{\mathscr{N}})$. To this end we consider the probability spaces $(S,{\mathscr{B}}(S),\sigma)$ and $(T,{\mathscr{B}}(T),\tau)$ both equal to $(C_d,{\mathscr{A}}_{{\mathscr{H}}^d {
\hskip2.5pt{\vrule height7pt width.5pt depth0pt}
\hskip-.2pt\vbox{\hrule height.5pt width7pt depth0pt}
\, }C_d},{\mathscr{H}}^d {
\hskip2.5pt{\vrule height7pt width.5pt depth0pt}
\hskip-.2pt\vbox{\hrule height.5pt width7pt depth0pt}
\, }C_d)$. We further define $V_s = \{s\} \times C$ and $H_t = C \times \{t \}$, $s,t \in C$. These belong to ${\mathscr{A}}$ because they are Borel and $\phi$ is Borel regular. For each $s,t \in C$, $\emptyset \neq V_s \cap H_t = \{(s,t)\} \in {\mathscr{N}}_\phi {\subseteq}{\mathscr{N}}$ thus hypothesis (1) of \[abstract.theorem\] is verified. Now let $Z {\subseteq}X_d$ be ${\mathscr{H}}^d$ measurable, $s \in C$, and assume that $V_s \cap Z \in {\mathscr{N}}= {\mathscr{N}}_\phi[{\mathscr{A}}_\phi^f]$. Since ${\mathscr{H}}^d(V_s) = {\mathscr{H}}^d(C) < \infty$ and $V_s \cap Z = V_s \cap (V_s \cap Z)$ we instantly infer that ${\mathscr{H}}^d(V_s \cap Z)=0$. Furthermore, $$C \cap \{ t : H_t \cap V_s \cap Z \neq \emptyset \} = C \cap \{ t : (s,t) \in V_s \cap Z \} = \pi_1(V_s \cap Z)$$ and in turn $${\mathscr{H}}^d \left( C \cap \{ t : H_t \cap V_s \cap Z \neq \emptyset \}\right) {\leqslant}(\operatorname{\mathrm{Lip}}\pi_1)^d {\mathscr{H}}^d(V_s \cap Z) = 0 \,.$$ This proves that condition (2)(b) of \[abstract.theorem\] is satisfied in the present case. Part (b) is checked in a similar fashion.
Purely Unrectifiable Example {#section.pue}
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\[pue.1\] We are given a sequence $(\lambda_k)_{k \in {\mathbb{N}}}$ of positive real numbers such that $\lambda_0 = 1$ and $0 < \lambda_k < \frac{1}{2} \lambda_{k-1}$ for every $k {\geqslant}1$. We will define inductively a sequence $({\mathscr{X}}_k)_{k \in {\mathbb{N}}}$ of sets of squares in ${\mathbf{R}}^2$. We start with $X_{0,0} = [0,1] \times [0,1]$ and ${\mathscr{X}}_0=\{X_{0,0}\}$. We let ${\mathscr{X}}_k$ consists of $4^k$ closed squares: It contains four subsquares of each $S \in {\mathscr{X}}_{k-1}$, each having a vertex in common with $S$ and sidelength $\lambda_k$. We let $$X = \cap_{k=1}^\infty \cup {\mathscr{X}}_k \,.$$ It clearly follows from the definitions that $\cup {\mathscr{X}}_k {\subseteq}\cup {\mathscr{X}}_{k-1}$ and that ${\mathscr{X}}_k$ consists of $4^k$ pairwise disjoint nonempty compact sets. Consequently $X$ is (topologically) a Cantor space.
One next defines $C {\subseteq}[0,1]$ as $C = \cap_{k=0}^\infty \cup {\mathscr{C}}_k$, where $({\mathscr{C}}_k)_{k \in {\mathbb{N}}}$ is defined inductively as follows. ${\mathscr{C}}_0=\{[0,1]\}$. We let ${\mathscr{C}}_k$ consists of $2^k$ closed intervals: It contains two subintervals of each $I \in {\mathscr{C}}_{k-1}$, each having an endpoint in common with $I$ and length $\lambda_k$.
1. [*$X = C \times C$ and if ${\mathscr{L}}^1(C)=0$ then $X$ is purely $({\mathscr{H}}^1,1)$ unrectifiable.*]{}
The first assertion follows from the observation that $S \in {\mathscr{X}}_k$ if and only if $S = I \times J$ for some $I,J \in {\mathscr{C}}_k$. The second assertion follows from [@MATTILA 18.10(4)].
\[pue.2\] We observe that each $S \in {\mathscr{X}}_k$ is contained in a unique $T \in {\mathscr{X}}_{k-1}$. It will be convenient to number ${\mathscr{X}}_k = \{ X_{k,j} : j=0,\ldots,4^k-1 \}$ in such a way that $X_{k,j} {\subseteq}X_{k-1,\lfloor j/4 \rfloor}$, $k \in {\mathbb{N}}^*$, $j=0,\ldots,4^k-1$. This is readily feasible.
We next consider the sequence $({\mathscr{I}}_k)_{k \in {\mathbb{N}}}$ of subsets of $[0,1]$ defined as follows. We put $I_{0,0}=[0,1]$ and ${\mathscr{I}}_0=\{I_{0,0}\}$, and we let ${\mathscr{I}}_k$ consist of $4^k$ nonoverlapping compact subintervals of $[0,1]$, each of length $4^{-k}$, such that $[0,1] = \cup {\mathscr{I}}_k$. We notice that each $I \in {\mathscr{I}}_k$ is contained in a unique $J \in {\mathscr{I}}_{k-1}$. We choose a numbering of ${\mathscr{I}}_k = \{ I_{k,\ell} : \ell=0,\ldots,4^k-1 \}$ in such a way that $I_{k,\ell} {\subseteq}I_{k-1,\lfloor \ell/4 \rfloor}$, $k \in {\mathbb{N}}^*$, $\ell=0,\ldots,4^k-1$.
Given two integers $j,j' \in {\mathbb{N}}$ we say that $j'$ is a [*daughter*]{} of $j$ if $j = \lfloor j'/4 \rfloor$. We say that a sequence $(j_k)_{k \in {\mathbb{N}}}$ of nonnegative integers is a [*lineage*]{} if $j_{k-1}$ is a daughter of $j_k$ for every $k {\geqslant}1$. The following now follows from our choice of numbering.
1. [ *Let $(j_k)_{k \in {\mathbb{N}}}$ be a sequence of nonnegative integers. The sequence $(X_{k,j_k})_{k \in {\mathbb{N}}}$ (resp. $(I_{k,j_k})_{k \in {\mathbb{N}}}$) is decreasing if and only if $(j_k)_{k \in {\mathbb{N}}}$ is a lineage.* ]{}
\[pue.3\] Here we will define functions $j : {\mathbb{N}}\times X \to {\mathbb{N}}$ and $\ell : {\mathbb{N}}\times Y \to {\mathbb{N}}$ where $Y {\subseteq}[0,1]$ is to be described momentarily. Given $x \in X$ and $k \in {\mathbb{N}}$, there exists a unique $j(k,x) \in \{0,\ldots,4^{k}-1\}$ such that $x \in X_{k,j(k,x)}$. This is because the family ${\mathscr{X}}_k$ is disjointed. Furthermore $(X_{k,j(k,x)})_{k \in {\mathbb{N}}}$ is decreasing, i.e. $(j(k,x))_{k \in {\mathbb{N}}}$ is a lineage.
If $y \in [0,1]$ and $k \in {\mathbb{N}}$ there does not necessarily exist a [*unique*]{} $\ell \in \{0,\ldots,4^k -1 \}$ such that $y \in I_{k,\ell}$.
1. [ *For every $y \in [0,1]$ and every $k \in {\mathbb{N}}$ there exists a unique $\ell \in \{0,\ldots,4^k - 1 \}$ such that $y \in I_{k,\ell}$ if and only if $y \not \in D_k$ where $D_k = \{ j.4^{-k} : j=1,\ldots,4^k - 1 \}$.* ]{}
If instead $y \in \{ j.4^{-k} : j=1,\ldots,4^k -1 \}$ then there are exactly two such $\ell$’s
1. [ *Assume $y \in [0,1]$. There are at most two lineages $(\ell_k)_{k \in {\mathbb{N}}}$ such that $y \in I_{k,\ell_k}$ for every $k \in {\mathbb{N}}$.* ]{}
In order to prove this, assume $(\ell_k)_{k \in {\mathbb{N}}}$, $(\ell'_k)_{k \in {\mathbb{N}}}$ and $(\ell''_k)_{k \in {\mathbb{N}}}$ are three lineages, at least two of which are distinct, such that $y \in I_{k,\ell_k} \cap I_{k,\ell'_k} \cap I_{k,\ell''_k}$ for every $k \in {\mathbb{N}}$. Let $k_0$ be the least integer such that $\{\ell_{k_0},\ell'_{k_0},\ell''_{k_0}\}$ is not a singleton. Renaming the sequences if necessary we may assume $\ell_{k_0}\neq\ell'_{k_0}$. Since any three distinct members of ${\mathscr{I}}_{k_0}$ have empty intersection it follows that either $\ell''_{k_0}=\ell_{k_0}$ or $\ell''_{k_0}=\ell'_{k_0}$. Renaming again the sequences if necessary we may assume the first case occurs. It remains to observe, by induction on $m$ that $\ell''_{k_0+m} = \ell_{k_0+m}$, $m \in {\mathbb{N}}$. This is because of the two members of ${\mathscr{I}}_{k_0+m}$ that contain $y$, only one is contained in $I_{\ell''_{k_0+m-1}}$.
To close this number we define $D = \cup_{k \in {\mathbb{N}}^*} D_k$ and $Y= [0,1] {\thicksim}D$. Thus for every $y \in Y$ and every $k \in {\mathbb{N}}$ there exists a unique $\ell(k,y) \in \{0,\ldots,4^k - 1 \}$ such that $y \in I_{k,\ell(k,y)}$. It follows that $(I_{k,\ell(k,y)})_{k \in {\mathbb{N}}}$ is decreasing, hence $(\ell(k,y))_{k \in {\mathbb{N}}}$ is a lineage.
\[pue.4\] There exists a Borel isomorphism $f : X \to [0,1]$ and a countable set $E {\subseteq}X$ with the following properties.
1. For every $k \in {\mathbb{N}}$ and every $S \in {\mathscr{X}}_k$ there exists $I \in {\mathscr{I}}_k$ such that $f(S{\thicksim}E) {\subseteq}I$;
2. For every $k \in {\mathbb{N}}$ and every $I \in {\mathscr{I}}_k$ there exists $S \in {\mathscr{X}}_k$ such that $f^{-1}(I{\thicksim}D) {\subseteq}S$;
3. $f(E)=D$.
We start by defining a map $g : X \to [0,1]$. Given $x \in X$ we consider the lineage $(j(k,x))_{k \in {\mathbb{N}}}$ defined in \[pue.2\]. It follows from \[pue.2\](1) that $(I_{k,j(k,x)})_{k \in {\mathbb{N}}}$ is a decreasing sequence of compact intervals, whose $k^{th}$ term has length $4^{-k}$. Accordingly there exists $g(x) \in [0,1]$ such that $$\{g(x)\} = \cap_{k \in {\mathbb{N}}} I_{k,j(k,x)} \,.$$ Now for each $k \in {\mathbb{N}}$ and $j \in \{0,\ldots,4^k -1 \}$ we pick $y_{k,j} \in I_{k,j}$ arbitrarily, and we observe that $$g(x) = \lim_k \sum_{j=0}^{4^k-1} y_{k,j} {\mathbbm{1}}_{X_{k,j}}(x) \,,$$ $x \in X$. This shows that $g$ is Borel measurable.
Letting $D {\subseteq}[0,1]$ be defined as in \[pue.3\] and $E = g^{-1}(D)$ we infer that $g|_{X {\thicksim}E}$ is injective. Suppose indeed that $x,x' \in X$ are such that $g(x)=g(x') \not \in D$. It follows from \[pue.3\](1) and the definition of $g$ that $j(k,x) = j(k,x')$, hence $\|x-x'\| {\leqslant}\operatorname{\mathrm{diam}}X_{j,k(k,x)} =4^{-k} \sqrt{2}$, for all $k \in {\mathbb{N}}$, thus $x=x'$. We further claim that $g(X {\thicksim}E)=[0,1] {\thicksim}D$. If indeed $y \in [0,1] {\thicksim}D$ we consider the lineage $(\ell(k,y))_{k \in {\mathbb{N}}}$ defined in \[pue.3\], so that $(X_{k,\ell(k,y)})_{k \in {\mathbb{N}}}$ is a decreasing sequence according to \[pue.2\](1) and hence there exists $h(y) \in X$ such that $$\{h(y)\} = \cap_{k \in {\mathbb{N}}} X_{k,\ell(k,y)} \,.$$ Upon observing that $j(k,g(y))=\ell(k,y)$ it follows from the definiton of $g$ that $g(h(y))=y$. By definition of $E$, $h(y) \in X {\thicksim}E$. In other words $h$ is the inverse of the bijection $X {\thicksim}E \to [0,1] {\thicksim}D : x \mapsto g(x)$. Picking $x_{k,j} \in X_{k,j}$ arbitrarily, $k \in {\mathbb{N}}$, $j=0,\ldots,4^k-1$, we note that $$h(y) = \lim_k \sum_{j=0}^{4^k-1} x_{k,j} {\mathbbm{1}}_{I_{k,j}}(y) \,,$$ $y \in [0,1] {\thicksim}D$, thereby showing that $h$ is Borel measurable.
Next we infer from \[pue.3\](2) and the definition of $g$ that $g^{-1}\{y\}$ contains at most two members, $y \in D$. Since $D$ is countable it follows that so is $E = g^{-1}(D)$. Choose arbitrarily a bijection ${\varphi}: E \to D$ and define $f : X \to [0,1]$ by $$f : X \to [0,1] : x \mapsto \begin{cases}
g(x) & \text{ if } x \not \in E \\
{\varphi}(x) & \text{ if } x \in E \,.
\end{cases}$$ It is now obvious that $f$ is a bijection, and that both $f|_{X {\thicksim}E}$ and $f|_E$ are Borel isomoprhisms. Thus $f$ itself is a Borel isomorphism.
Let $k_0 \in {\mathbb{N}}$ and $S \in {\mathscr{X}}_{k_0}$. It readily follows from the definition of $f$ that $f(S {\thicksim}E) {\subseteq}g(S)$. Let $j_0$ be such that $S = X_{k_0,j_0}$. If $x \in S$ then $j(k_0,x)=j_0$ so that, by definition of $f$, $g(x) \in I_{k_0,j_0}$. This proves (1). Similarly let $I = I_{k_0,j_0} \in {\mathscr{I}}_{k_0}$. Clearly $f^{-1}(I {\thicksim}D) = h(I {\thicksim}D)$. If $y \in I {\thicksim}D$ then $\ell(k_0,y) = j_0$ whence $h(y) \in X_{k_0,j_0}$. This proves (2).
\[pue.5\] Recall the construction of $X$ in \[pue.1\]. For the remaining part of this section $C$ will be a self-similar Cantor set of Hausdorff dimension $0 < d {\leqslant}\frac{\log 2}{\log 3}$. Thus we let the family ${\mathscr{C}}_k$ consists of $2^k$ members of length $\lambda_k = \lambda^k$ where $d = \frac{\log 2}{\log \lambda^{-1}}$, i.e. $0 < \lambda {\leqslant}\frac{1}{3}$ (see e.g. [@MATTILA 4.10] or [@GMT 2.10.28]). We choose our notation to reminisce about this choice by letting $C_d$ denote the corresponding Cantor set, and $X_d = C_d \times C_d$.
\[pue.6\] For every $Z {\subseteq}X_d$ one has $$\left( \frac{1}{2} \right)^{1+\frac{d}{2}} {\mathscr{H}}^d(Z) {\leqslant}{\mathscr{H}}^{\frac{1}{2}}(f(Z)) {\leqslant}\left( \frac{2}{\lambda}\right)^d {\mathscr{H}}^d(Z) \,.$$ In particular ${\mathscr{H}}^d(Z)=0$ if and only if ${\mathscr{H}}^{\frac{1}{2}}(f(Z))=0$.
We start by observing that for every $k \in {\mathbb{N}}$ and every $S \in {\mathscr{X}}_k$, $I \in {\mathscr{I}}_k$ one has $$\label{eq.5}
\sqrt{\operatorname{\mathrm{diam}}I} = \sqrt{4^{-k}} = 2^{-k} = \lambda^{kd} = \left(\frac{1}{\sqrt{2}} \operatorname{\mathrm{diam}}S \right)^d$$ because $2^k (\lambda^k)^d=1$ by our choice of $\lambda$ and $d$, \[pue.5\].
Let $Z {\subseteq}X_d$ and let ${\varepsilon}> 0$. In order to prove the right hand inequality there is no restriction to assume that ${\mathscr{H}}^d(Z) < \infty$. There exists a finite or countable covering $(U_i)_{i \in I}$ of $Z$ in $X$ such that $\sum_i (\operatorname{\mathrm{diam}}U_i)^d < {\varepsilon}+ {\mathscr{H}}^d_{({\varepsilon})}(Z)$ and $\operatorname{\mathrm{diam}}U_i < {\varepsilon}$ for every $i \in I$. Note we may assume each $U_i$ is open and nonempty. Abbreviate $\delta_i = \operatorname{\mathrm{diam}}U_i$, $i \in I$. Choosing $x_i \in U_i$ and letting $U'_i = {\mathbf{U}}(x_i,\delta_i) \cap X$ we see that $\operatorname{\mathrm{diam}}U'_i {\leqslant}2 \delta_i$, $i \in I$. Choose $k_i \in {\mathbb{N}}$ such that $\lambda^{-(k_i+1)} {\leqslant}\operatorname{\mathrm{diam}}U'_i < \lambda^{-k_i}$. Thus $U'_i$ intersects some $X_{k_i,j_i}$, $j_i \in \{0,\ldots,4^{k_i}-1\}$, and since $\lambda {\leqslant}\frac{1}{3}$ it intersects only one of them. Therefore $U'_i {\subseteq}X_{k_i,j_i}$. Notice that $\operatorname{\mathrm{diam}}X_{k_i,j_i} = \lambda^{k_i} \sqrt{2} {\leqslant}2\sqrt{2}\lambda^{-1} \delta_i$. Now $Z {\subseteq}\cup_{i \in I} {\subseteq}\cup_{i \in I} X_{k_i,j_i}$ thus also $Z {\thicksim}E {\subseteq}\cup_{i \in I} (X_{k_i,j_i} {\thicksim}E)$ and it follows from \[pue.4\](1) that $f(Z {\thicksim}E) {\subseteq}\cup_{i \in I} I_{k_i,j'_i}$ for some integers $j'_i \in {\mathbb{N}}$, $i \in I$. In turn implies that $${\mathscr{H}}^{\frac{1}{2}}_{(\eta_{\varepsilon})}(f(Z {\thicksim}E)) {\leqslant}\sum_{i \in I} \sqrt{ \operatorname{\mathrm{diam}}I_{k_i,j'_i}} {\leqslant}\left( \frac{2}{\lambda}\right)^d \sum_{i \in I} ( \operatorname{\mathrm{diam}}U_i)^d {\leqslant}\left( \frac{2}{\lambda}\right)^d \left({\varepsilon}+{\mathscr{H}}^d_{({\varepsilon})}(Z) \right) \,,$$ where $\eta_{\varepsilon}= 2^{3d-1}\lambda^{-2d}{\varepsilon}^{2d}$. Since ${\varepsilon}> 0$ is arbitrary we infer that ${\mathscr{H}}^{\frac{1}{2}}(f(Z{\thicksim}E)) {\leqslant}2^d \lambda^{-d} {\mathscr{H}}^d(Z)$. As $f(Z) {\subseteq}f(Z {\thicksim}E) \cup f(E)$ and $f(E)=D$ is countable, ${\mathscr{H}}^d(f(E))=0$ and the right hand inequality follows.
Let $A {\subseteq}[0,1]$ and let ${\varepsilon}> 0$. In order to prove the left hand inequality we may of course suppose that ${\mathscr{H}}^{\frac{1}{2}}(A) < \infty$. There exists a covering $(U_i)_{i \in I}$ of $A$ in $[0,1]$ such that $\sum_{i \in I} \sqrt{ \operatorname{\mathrm{diam}}U_i} < {\varepsilon}+ {\mathscr{H}}^{\frac{1}{2}}_{({\varepsilon})}(A)$ and $\operatorname{\mathrm{diam}}U_i < {\varepsilon}$ for every $i \in I$. There is no restriction to assume that each $U_i$ is a nondegenerate interval. We abbreviate $\delta_i = \operatorname{\mathrm{diam}}U_i$ and we let $k_i \in {\mathbb{N}}$ be such that $4^{-(k_i+1)} {\leqslant}\delta_i < 4^{k_i}$. Notice that $U_i$ intersects at most two members of ${\mathscr{I}}_{k_i}$, i.e. there exist $J_{i,1},J_{i,2} \in {\mathscr{I}}_{k_i}$ such that $U_i {\subseteq}J_{i,1} \cup J_{i,2}$. Furthermore $\operatorname{\mathrm{diam}}J_{i,q} {\leqslant}4 \operatorname{\mathrm{diam}}U_i$, $i \in I$, $q=1,2$. Now observe that $$f^{-1}(A {\thicksim}D) {\subseteq}\cup_{i \in I} \left( f^{-1}(J_{i,1} {\thicksim}D) \cup f^{-1} (J_{i,2} {\thicksim}D) \right) {\subseteq}\cup_{i \in I} \left( X_{k_i,j_{i,1}} \cup X_{k_i,j_{i,1}}\right)$$ for some integers $j_{i,1},j_{i,2} \in \{0,\ldots,4^{k_i}-1\}$, according to \[pue.4\](2). Therefore it follows from that $$\begin{gathered}
{\mathscr{H}}^d_{(\eta_{\varepsilon})}(f^{-1}(A {\thicksim}D)) {\leqslant}\sum_{i \in I} \left( \left( \operatorname{\mathrm{diam}}X_{k_i,j_{i,1}} \right)^d + \left( \operatorname{\mathrm{diam}}X_{k_i,j_{i,2}} \right)^d\right) \\
{\leqslant}2^{1 + \frac{d}{2}}\sum_{i \in I} \sqrt{\operatorname{\mathrm{diam}}U_i} {\leqslant}2^{1 + \frac{d}{2}} \left( {\varepsilon}+ {\mathscr{H}}^{\frac{1}{2}}_{({\varepsilon})}(A) \right)\,,\end{gathered}$$ where $\eta_{\varepsilon}= 2^\frac{1}{d} (4{\varepsilon})^\frac{2}{d}$. Since ${\varepsilon}> 0$ is arbitrary, ${\mathscr{H}}^d(f^{-1}(A {\thicksim}D))=0$. Finally ${\mathscr{H}}^d(f^{-1}(A)) {\leqslant}2^{1+\frac{1}{d}}{\mathscr{H}}^{\frac{1}{2}}(A)$ because $f^{-1}(A) {\subseteq}f^{-1}(A {\thicksim}D) \cup f^{-1}(D)$ and $f^{-1}(D)=E$ is countable whence ${\mathscr{H}}^d(f^{-1}(D))=0$.
\[pue.7\] We say that a measure space $(X,{\mathscr{A}},\mu)$ is [*undecidably semilocalizable*]{} if the proposition <<$(X,{\mathscr{A}},\mu)$ is semilocalizable>> is undecidable in $\mathsf{ZFC}$. In case $X$ is a complete separable metric space and $0 < d < \infty$, the measure space $(X,{\mathscr{A}}_{{\mathscr{H}}^d},{\mathscr{H}}^d)$ is undecidably semilocalizable if and only if the proposition <<$(X,{\mathscr{A}}_{{\mathscr{H}}^d},{\mathscr{H}}^d)$ is almost decomposable>> is undecidable in $\mathsf{ZFC}$. This is a consequence of \[71\].
\[pue.8\] The measure space $\left([0,1],{\mathscr{A}}_{{\mathscr{H}}^{1/2}},{\mathscr{H}}^{\frac{1}{2}}\right)$ is undecidably semilocalizable.
It is consistently semilocalizable according to \[CH.implies.ad\]. Fix $0 < d {\leqslant}\frac{\log 2}{\log 3}$ arbitrarily. Let $f : X_d \to [0,1]$ be as before and define a $\sigma$-algebra of subsets of $X_d$ by the formula $${\mathscr{A}}= {\mathscr{P}}(X_d) \cap \left\{ f^{-1}(A) : A \in {\mathscr{A}}_{{\mathscr{H}}^{\frac{1}{2}}} \right\} \,.$$ We claim that ${\mathscr{B}}(X_d) {\subseteq}{\mathscr{A}}$. Indeed if $B {\subseteq}X_d$ is Borel then so is $f(B)$, according to \[pue.4\]. Thus $f(B) \in {\mathscr{A}}_{{\mathscr{H}}^{1/2}}$ and in turn $B = f^{-1}(f(B)) \in {\mathscr{A}}$. We also define $${\mathscr{N}}= {\mathscr{P}}(X_d) \cap \left\{ f^{-1}(N) : N \in {\mathscr{N}}_{{\mathscr{H}}^{1/2}}\left[{\mathscr{A}}_{{\mathscr{H}}^{1/2}}^f\right] \right\} \,.$$ It ensues from their construction that the measurable spaces with negligibles $(X_d,{\mathscr{A}},{\mathscr{N}})$ and $\left([0,1],{\mathscr{A}}_{{\mathscr{H}}^{1/2}},{\mathscr{N}}_{{\mathscr{H}}^{1/2}}\left[{\mathscr{A}}_{{\mathscr{H}}^{1/2}}^f\right] \right)$ are isomorphic in the category ${\mathsf{MSN}}$. According to \[last.lemma\] one is localizable if and only if the other one is. Reasoning as in the proof of \[85\] we will now show that the former is localizable. First we notice that $V_s = \{s\} \times C_d$ and $H_t = C_d \times \{t\}$, $s,t \in C_d$, indeed belong to ${\mathscr{A}}$ because ${\mathscr{B}}(X_d) {\subseteq}{\mathscr{A}}$. Let $Z \in {\mathscr{A}}$ and $s \in C_d$ be such that $V_s \cap Z \in {\mathscr{N}}$. This means that $f(V_s \cap Z) \in {\mathscr{N}}_{{\mathscr{H}}^{1/2}}\left[{\mathscr{A}}_{{\mathscr{H}}^{1/2}}^f\right]$. Since $f(V_s \cap Z) = f(V_s) \cap f(Z)$ and $f(V_s) \in {\mathscr{A}}_{{\mathscr{H}}^{1/2}}^f$ according to \[pue.6\] we infer that ${\mathscr{H}}^{\frac{1}{2}}(f(V_s \cap Z))=0$ and in turn ${\mathscr{H}}^d-V_s \cap Z)=0$ again thanks to \[pue.6\]. Reasoning as in \[85\] we conclude that ${\mathscr{H}}^d ( C \cap \{ t : H_t \cap V_s \cap Z \neq \emptyset \}) = 0$. Similarly the condition (2)(b) of \[abstract.theorem\] holds as well and the proof is complete.
Purely Rectifiable Example {#example}
==========================
\[number.1\] We let $\{0,1\}^{{\mathbb{N}}^*}$ be the Cantor space equipped with its usual topology and its usual Borel, probability, product measure $\lambda$. For each $j \in {\mathbb{N}}^*$ we let ${\mathscr{S}}_j$ be a collection of disjoint, compact subintervals of $[0,1]$, and we let $(\ell_j)_{j \in {\mathbb{N}}^*}$ be a sequence in $(0,1)$, with the following properties:
1. $\operatorname{\mathrm{card}}{\mathscr{S}}_j = 2^j$;
2. For every $T \in {\mathscr{S}}_j$ one has $\operatorname{\mathrm{card}}{\mathscr{S}}_{j+1} \cap \{ S : S {\subseteq}T \} = 2$;
3. For every $S \in {\mathscr{S}}_j$ one has ${\mathscr{L}}^1(S)=\ell_j$.
We then define $C = \cap_{j \in {\mathbb{N}}^*} \cup {\mathscr{S}}_j$. This way we can realize a set $C$ of any Hausdorff dimension $0 {\leqslant}d < 1$, see e.g. [@MATTILA 4.10 and 4.11]. We will be mostly interested in the case $d=0$. In any case we will henceforth assume that ${\mathscr{L}}^1(C)=0$.
For each $j \in {\mathbb{N}}^*$ we number the members of ${\mathscr{S}}_j$ as $S_{j,0},\ldots,S_{j,2^j-1}$ in such a way that $\max S_{j,k} < \min S_{j,k+1}$, $k=0,\ldots,2^j-2$. Thus $S_{j+1,2k} \cup S_{j+1,2k+1} {\subseteq}S_{j,k}$ for all $j \in {\mathbb{N}}^*$ and all $k=0,\ldots,2^j-1$. Now given $\xi \in \{0,1\}^{{\mathbb{N}}^*}$ we define inductively $(k_\xi(j))_{j \in {\mathbb{N}}^*}$ as follows: $k_\xi(1)=\xi(1)$ and $k_\xi(j+1)=2k_\xi(j)+\xi(j+1)$. In turn we define the usual coding of $C$, $${\varphi}: \{0,1\}^{{\mathbb{N}}^*} \to C$$ by letting ${\varphi}(\xi)$ be the only point of $[0,1]$ such that $$\{ {\varphi}(\xi) \} = \cap_{j \in {\mathbb{N}}^*} S_{j,k_\xi(j)} \,.$$ Thus ${\varphi}$ is a homeomorphism.
\[number.2\] We define a Borel probability $\mu$ measure on $[0,1]$ by the formula $\mu(B) = \lambda({\varphi}^{-1}(B \cap C))$, $B \in {\mathscr{B}}([0,1])$. For each $j \in {\mathbb{N}}^*$ we define a Borel probability measure $\mu_j$ on $[0,1]$ by the formula $$\mu_j = \left(\frac{1}{2^j \ell_j}\right) {\mathscr{L}}^1 {
\hskip2.5pt{\vrule height7pt width.5pt depth0pt}
\hskip-.2pt\vbox{\hrule height.5pt width7pt depth0pt}
\, }(\cup {\mathscr{S}}_j) \,.$$
\[number.3\] The sequence $(\mu_j)_{j \in {\mathbb{N}}^*}$ converges weakly\* to $\mu$.
First we let $S \in {\mathscr{S}}_j$ for some $j \in {\mathbb{N}}^*$. Observe that $\mu(S) = 2^{-j}$. If $k {\geqslant}j$ then $\mu_k(S) = 2^{-k} \ell_k^{-1} {\mathscr{L}}^1( S \cap \cup {\mathscr{S}}_k)= 2^{-k}\ell_k^{-1} (2^{k-j} \ell_k) = 2^{-j}$. In particular $\lim_k \mu_k(S) = \mu(S)$.
Next we let $U {\subseteq}[0,1]$ be relatively open. There exists a disjointed sequence $(S_n)_{n \in {\mathbb{N}}}$ of members of $\cup_{j \in {\mathbb{N}}^*} {\mathscr{S}}_j$ such that each $S_n {\subseteq}U$ and $$C \cap U = C \cap \left( \cup_{n \in {\mathbb{N}}} S_n \right) \,.$$ It suffices indeed to let $(S_n)_{n \in {\mathbb{N}}}$ be a numbering of ${\mathscr{T}}= \cup_{j \in {\mathbb{N}}^*} {\mathscr{T}}_j$ where $({\mathscr{T}}_j)_{j\in {\mathbb{N}}^*}$ is defined inductively as follows: ${\mathscr{T}}_1 = {\mathscr{S}}_1 \cap \{ S : S {\subseteq}U \}$ and ${\mathscr{T}}_{j+1} = {\mathscr{S}}_{j+1} \cap \{ S : S {\subseteq}U \text{ and } S \cap \cup_{k=1}^j \cup {\mathscr{T}}_k = \emptyset \}$. Now given ${\varepsilon}> 0$ there exists $N \in {\mathbb{N}}$ such that $$\sum_{n \in {\mathbb{N}}} \mu(S_n) {\leqslant}{\varepsilon}+ \sum_{n=0}^N \mu(S_n) \,.$$ Furthemore, $$\begin{gathered}
\mu(U) = \sum_{n \in {\mathbb{N}}} \mu(S_n) {\leqslant}{\varepsilon}+ \sum_{n=0}^N \mu(S_n) = {\varepsilon}+ \sum_{n=0}^N \lim_k \mu_k(S_n) \\ = {\varepsilon}+ \lim_k \mu_k \left( \cup_{n=0}^N S_n \right) {\leqslant}{\varepsilon}+ \liminf_k \mu_k(U) \,.\end{gathered}$$ Since ${\varepsilon}> 0$ is arbitrary it follows that $\mu(U) {\leqslant}\liminf_k \mu_k(U)$.
Recalling that $\mu([0,1]) = \mu_k([0,1])$ for all $k \in {\mathbb{N}}^*$ we infer that for every compact $K {\subseteq}[0,1]$, $$\begin{gathered}
\mu(K) = \mu([0,1]) - \mu([0,1]{\thicksim}K) {\geqslant}\mu_k([0,1]) - \liminf_k \mu_k([0,1]{\thicksim}K) = \limsup_k \mu_k(K) \,.\end{gathered}$$ The conclusion follows from Portmanteau’s Theorem.
\[number.4\] We associate with $\mu$ its distribution function $$f : [0,1] \to [0,1] : t \mapsto \mu([0,t])$$ and we observe that $f$ is continuous (because $\mu$ is diffuse) and nondecreasing. We also define $$F : [0,1] \to {\mathbf{R}}^2 : t \mapsto (t,f(t))$$ and we observe that the set $\Gamma = \operatorname{\mathrm{graph}}(f) = F([0,1])$ is $1$-rectifiable and ${\mathscr{H}}^1(\Gamma) < \infty$. This most easily follows from the <<bow-tie lemma>> (see e.g. [@DEP.05c 4.8.3] applied with $n=m+1=2$, $S=\Gamma$, $r=3$, $\sigma=\sin(\pi/4)$ and $W = \operatorname{\mathrm{span}}\{e_1+e_2\}$).
We will approximate $f$ by the functions $$f_j : [0,1] \to [0,1] : t \mapsto \mu_j([0,t])$$ which are nondrecreasing and Lipschitz. Given $t \in [0,1]$ we notice that $\operatorname{\mathrm{Bdry}}[0,t]=\{0,t\}$ is $\mu$-null, whence $$f(t) = \mu([0,t]) = \lim_j \mu_j([0,t]) = f_j(t) \,,$$ according to \[number.3\] and [@EVANS.GARIEPY §1.9 Theorem 1]. Thus the sequence $(f_j)_{j \in {\mathbb{N}}^*}$ converges pointwise to $f$.
We next record that each $f_j$ is differentiable ${\mathscr{L}}^1$ almost everywhere. In fact upon defining $$\sigma_j = \frac{1}{2^j \ell_j}$$ one has $$f_j'(t) = \begin{cases}
0 & \text{if } t \not \in \cup {\mathscr{S}}_j \\
\sigma_j & \text{if } t \in \operatorname{\mathrm{Int}}\cup {\mathscr{S}}_j \,.
\end{cases}$$
We finally define $$F_j : [0,1] \to {\mathbf{R}}^2 : t \mapsto (t,f_j(t))$$ and related to the Jacobian of $F_j$ we define $${\mathbf{c}}_j = 2^j \ell_j \sqrt{1 + \sigma_j^2} = \sigma_j^{-1} \sqrt{1 + \sigma_j^2} \,.$$ Since ${\mathscr{L}}^1(C) = 0$ we infer that $\limsup_j \sigma_j = \infty$ (for otherwise $f$ would be Lipschitz) and in turn $$\lim_j {\mathbf{c}}_j = \lim_j \sigma_j^{-1}\sqrt{1+\sigma_j^2} = 1 \,.$$
\[number.5\] For every $j \in {\mathbb{N}}^*$ and every Borel set $B {\subseteq}[0,1]$ one has $${\mathscr{H}}^1(F_j(B)) {\geqslant}{\mathbf{c}}_j \mu_j(B) \,.$$
Let $B {\subseteq}[0,1]$ be Borel and define $B' = B \cap (\cup {\mathscr{S}}_j)$. It follows from the definiton of $\mu_j$ that $$\mu_j(B) = \mu_j(B') = \sigma_j {\mathscr{L}}^1(B') \,.$$ Recalling \[number.4\] it follows from the <<area formula>> in this simple case (see e.g. [@EVANS.GARIEPY §3.3 Theorem 1] for the general case) $${\mathscr{H}}^1(F_j(B)) {\geqslant}{\mathscr{H}}^1(F_j(B')) = \int_{B'} \sqrt{1 + f_j'(t)^2}d{\mathscr{L}}^1(t) = \sqrt{1+\sigma_j^2} {\mathscr{L}}^1(B') \,.$$
\[number.6\] Let $S {\subseteq}[0,1]$ be any set. It follows that $${\mathscr{H}}^1(F(S)) {\geqslant}\frac{\bar{\mu}(S)}{\sqrt{2}} \,.$$
We start with the case when $S = [a,b] {\subseteq}[0,1]$ is a closed interval. Since $F(S) {\subseteq}F([0,1])$ is $1$-rectifiable (and compact) the following <<integral geometric inequality>> follows for instance from [@GMT 3.2.27] ($\pi_1$ and $\pi_2$ denote resp. the projection from ${\mathbf{R}}^2$ onto its first and second axis): $${\mathscr{H}}^1(F(S)) {\geqslant}\sqrt{a_1^2 + a_2^2}$$ where $$a_1 = \int_{\mathbf{R}}\operatorname{\mathrm{card}}( F(S) \cap \pi_1^{-1}\{x\} ) d{\mathscr{L}}^1(x)
= {\mathscr{L}}^1(S)
= b-a \,,$$ and $$a_2 = \int_{\mathbf{R}}\operatorname{\mathrm{card}}( F(S) \cap \pi_2^{-1}\{y\} ) d{\mathscr{L}}^1(y)
= {\mathscr{L}}^1(f(S))
= f(b)-f(a) \,.$$ Similarly the other inequality from [@GMT 3.2.27] applies to $F_j(S)$: $$a_{1,j} + a_{2,j} {\geqslant}{\mathscr{H}}^1(F_j(S))$$ where $$a_{1,j} = \int_{\mathbf{R}}\operatorname{\mathrm{card}}( F_j(S) \cap \pi_1^{-1}\{x\} ) d{\mathscr{L}}^1(x)
= {\mathscr{L}}^1(S)
= b-a \,,$$ and $$a_{2,j} = \int_{\mathbf{R}}\operatorname{\mathrm{card}}( F_j(S) \cap \pi_2^{-1}\{y\} ) d{\mathscr{L}}^1(y)
= {\mathscr{L}}^1(f_j(S))
= f_j(b)-f_j(a) \,.$$ Accordingly, $$\begin{split}
{\mathscr{H}}^1(F(S)) & {\geqslant}\sqrt{ (b-a)^2 + (f(b)-f(a))^2 } \\
& = \lim_j \sqrt{ (b-a)^2 + (f_j(b)-f_j(a))^2 } \quad\quad \text{(by \ref{number.4})} \\
& {\geqslant}\frac{1}{\sqrt{2}} \lim_j \left( (b-a) + (f_j(b)-f_j(a)) \right) \\
& {\geqslant}\frac{1}{\sqrt{2}} \lim_j \left( a_{1,j} + a_{2,j} \right) \\
& {\geqslant}\frac{1}{\sqrt{2}} \limsup_j {\mathscr{H}}^1(F_j(S)) \\
& {\geqslant}\frac{1}{\sqrt{2}} \limsup_j {\mathbf{c}}_j \mu_j(S) \qquad\qquad\qquad\qquad \text{(by \ref{number.5})} \\
& = \frac{\mu(S)}{\sqrt{2}} \qquad\qquad \text{(according to \ref{number.3} since $\mu(\operatorname{\mathrm{Bdry}}S)=0$)}
\end{split}$$ This completes the proof in case $S$ is a closed interval.
We now turn to the case when $S {\subseteq}[0,1]$ is Borel. To this end we define $\nu(B) = \sqrt{2}{\mathscr{H}}^1(F(B))$, $B \in {\mathscr{B}}([0,1])$. Since $F(B) \in {\mathscr{B}}({\mathbf{R}}^2)$ whenever $B \in {\mathscr{B}}([0,1])$ (because $F$ is continuous, hence Borel measurable, and injective, see [@SRIVASTAVA 5.4.5]) and since $B_1 \cap B_2 = \emptyset$ implies that $F(B_1) \cap F(B_2) = \emptyset$ it follows that $\nu$ is a measure on ${\mathscr{B}}([0,1])$. Thus $\mu$ and $\nu$ are two finite Borel measures on $[0,1]$ such that $\mu(I) {\leqslant}\nu(I)$ whenever $I {\subseteq}[0,1]$ is a closed interval. Since $\nu$ is also clearly diffused we infer that $\mu(I) {\leqslant}\nu(I)$ whenever $I = (m2^{-n},(m+1)2^{-n}]$, for some $n \in {\mathbb{N}}^*$ and $m=0,\ldots,2^n-1$. Since each relatively open set $U {\subseteq}(0,1]$ is the union of a disjointed sequence of such dyadic semi-intervals it follows that $\mu(U) {\leqslant}\nu(U)$. Finally the outer regularity of $\nu$ yields $\mu(B) {\leqslant}\nu(B)$ for all $B \in {\mathscr{B}}([0,1])$.
We come to the case when $S {\subseteq}[0,1]$ is arbitrary. We choose a Borel set $B_1 {\subseteq}[0,1]$ such that $S {\subseteq}B_1$ and $\bar{\mu}(S)=\mu(B_1)$, we choose a Borel set $B_2 {\subseteq}{\mathbf{R}}^2$ such that $F(S) {\subseteq}B_2$ and ${\mathscr{H}}^1(F(S)) = {\mathscr{H}}^1(B_2)$, we let $B_3 = F^{-1}(B_2) {\subseteq}[0,1]$ which is Borel as well, and finally we define $B = B_1 \cap B_3$. Since $F$ is injective and $F(S) {\subseteq}B_2$ we see that $S = F^{-1}(F(S)) {\subseteq}F^{-1}(B_2) = B_3$, thus $S {\subseteq}B_1 \cap B_3 = B$. Therefore $\bar{\mu}(S) {\leqslant}\bar{\mu}(B) = \mu(B) {\leqslant}\mu(B_1) = \bar{\mu}(S)$ and we conclude that $\bar{\mu}(S)=\mu(B)$. Similarly, from $S {\subseteq}B {\subseteq}B_3$ and the definition of $B_2$ we infer that $F(S) {\subseteq}F(B) {\subseteq}F(B_3) {\subseteq}B_2$ and in turn ${\mathscr{H}}^1(F(S)) {\leqslant}{\mathscr{H}}^1(F(B)) {\leqslant}{\mathscr{H}}^1(B_2) = {\mathscr{H}}^1(F(S))$ so that ${\mathscr{H}}^1(F(S)) = {\mathscr{H}}^1(F(B))$. Finally it follows from the previous paragraph that $${\mathscr{H}}^1(F(S)) = {\mathscr{H}}^1(F(B)) {\geqslant}\frac{\mu(B)}{\sqrt{2}} = \frac{\bar{\mu}(S)}{\sqrt{2}} \,.$$
\[number.9\] Assume that
1. $C {\subseteq}[0,1]$ is a Cantor set such as in \[number.1\] and $X = C \times [0,2] {\subseteq}{\mathbf{R}}^2$;
2. ${\mathscr{A}}$ is a $\sigma$-algebra of subsets of $X$ such that ${\mathscr{B}}(X) {\subseteq}{\mathscr{A}}{\subseteq}{\mathscr{P}}(X)$;
3. ${\mathscr{N}}{\subseteq}{\mathscr{A}}$ is a $\sigma$-ideal with the following property:
1. $\{x\} \in {\mathscr{N}}$ for every $x \in X$;
2. For every $A \in {\mathscr{A}}$ and every $1$-rectifiable set $M {\subseteq}{\mathbf{R}}^2$ if $A \cap M \in {\mathscr{N}}$ then ${\mathscr{H}}^1(A \cap M)=0$;
4. ${\mathsf{non}}({\mathscr{N}}_{{\mathscr{L}}^1}) < {\mathsf{cov}}({\mathscr{N}}_{{\mathscr{L}}^1})$.
It follows that $(X,{\mathscr{A}},{\mathscr{N}})$ is not localizable.
In this proof $e_1,e_2$ denotes the canonical basis of ${\mathbf{R}}^2$ and $\pi_1, \pi_2$ the canonical projections of ${\mathbf{R}}^2$ on its first and second axis respectively. The result will be obtained as a consequence of \[abstract.theorem\] applied to $(X,{\mathscr{A}},{\mathscr{N}})$ as in the statement, $(S,{\mathscr{B}}(S),\sigma) = (C,{\mathscr{B}}(C),\mu)$, $(T,{\mathscr{B}}(T),\tau) = ([0,1],{\mathscr{B}}([0,1]), {\mathscr{L}}^1)$, $$V_s = \{s\} \times [0,2] \in {\mathscr{B}}(X) {\subseteq}{\mathscr{A}}\,,$$ $s \in C$, and $$H_t = (\Gamma + t.e_2) \cap X \in {\mathscr{B}}(X) {\subseteq}{\mathscr{A}}\,,$$ $t \in [0,1]$, where $\Gamma = F([0,1])$.
We now check that condition (1) of \[abstract.theorem\] is satisfied. Let $s \in C$ and $t \in [0,1]$. Since $H_t$ is contained in the graph of a function and $V_s$ is contained in a vertical line, $V_s \cap H_t$ is either empty or a singleton, therefore a member of ${\mathscr{N}}$ according to our current hypothesis (3)(a). It is easy to see that $p_{s,t} = (s,f(s)+t) \in V_s \cap H_t$, so that $V_s \cap H_t \neq \emptyset$.
We next verify that condition (2)(a) of \[abstract.theorem\] is satisfied. Fix $s \in C$ and $Z \in {\mathscr{A}}$ such that $V_s \cap Z \in {\mathscr{N}}$. Observe that $$\begin{gathered}
[0,2] \cap \{ t : H_t \cap V_s \cap Z \neq \emptyset \} = [0,1] \cap \{ t : p_{s,t} \in V_s \cap Z \} \\ = [0,2] \cap \{ t : t \in \pi_2( V_s \cap Z ) - f(s) \} \,,\end{gathered}$$ and therefore $${\mathscr{L}}^1( [0,2] \cap \{ t : H_t \cap V_s \cap Z \neq \emptyset \}) {\leqslant}{\mathscr{H}}^1( V_s \cap Z)=0$$ where the last equality follows from our assumption (3)(b) because $V_s$ is 1 rectifiable.
Finally we ought to show that condition (2)(b) of \[abstract.theorem\] is satisfied. Let $t \in [0,1]$ and $Z \in {\mathscr{A}}$ be such that $H_t \cap Z \in {\mathscr{N}}$. Observe that $$C \cap \{ s : V_s \cap H_t \cap Z \neq \emptyset \} = C \cap \{ s : p_{s,t} \in H_t \cap Z \} = \pi_1(H_t \cap Z) \,.$$ Since $H_t \cap Z = (\Gamma + t.e_2) \cap Z$ and $\Gamma + t.e_2$ is 1 rectifiable, our hypothesis (3)(b) implies that ${\mathscr{H}}^1(H_t \cap Z)=0$. Abbreviating $E = \pi_1(H_t \cap Z) {\subseteq}C$ it ensues from \[number.6\] that $$0 = {\mathscr{H}}^1(H_t \cap Z) = {\mathscr{H}}^1(F(E)+t.e_2) = {\mathscr{H}}^1(F(E)) {\geqslant}\frac{\bar{\mu}(E)}{\sqrt{2}} \,,$$ and the proof is complete.
\[main.result\] Let $C {\subseteq}[0,1]$ be a Cantor set as in \[number.1\] and $X = C \times [0,2]$. It follows that $(X,{\mathscr{A}}_{{\mathscr{H}}^1},{\mathscr{H}}^1)$ is undecidably semilocalizable.
It is consistently semilocalizable according to \[CH.implies.ad\] and it is consistently not semilocalizable according to \[number.9\] applied with ${\mathscr{A}}= {\mathscr{A}}_{{\mathscr{H}}^1}$ and ${\mathscr{N}}={\mathscr{N}}_{{\mathscr{H}}^1}\left[ {\mathscr{A}}_{{\mathscr{H}}^1}^f\right]$.
Concluding Remarks and Open Questions
=====================================
\[CR.1\] One may apply \[number.9\] to other $\sigma$-ideals than ${\mathscr{N}}_{{\mathscr{H}}^1}$. For example let $${\mathscr{N}}_{pu} = {\mathscr{P}}({\mathbf{R}}^2) \cap \{ S : S \text{ is purely $({\mathscr{H}}^1,1)$ unrectifiable} \} \,.$$ Recall that a set $S {\subseteq}{\mathbf{R}}^2$ is called [*purely $({\mathscr{H}}^1,1)$ unrectifiable*]{} whenever ${\mathscr{H}}^1(S \cap M) = 0$ for every 1 rectifiable $M {\subseteq}{\mathbf{R}}^2$. It then follows from \[number.9\] that for any $\sigma$-algebra ${\mathscr{B}}(X) {\subseteq}{\mathscr{A}}{\subseteq}{\mathscr{P}}(X)$ the measurable space with negligibles $(X,{\mathscr{A}},{\mathscr{A}}\cap {\mathscr{N}}_{pu})$ is consistently not localizable.
\[CR.2\] We turn back to \[number.9\] applied with ${\mathscr{N}}_{{\mathscr{H}}^1}$. It follows that $(X,{\mathscr{B}}(X),{\mathscr{B}}(X) \cap {\mathscr{N}}_{{\mathscr{H}}^1})$ is consistently not semilocalizable. It further follows in $\mathsf{ZFC}$ from \[56\] that $(X,{\mathscr{B}}(X),{\mathscr{B}}(X) \cap {\mathscr{N}}_{{\mathscr{H}}^1})$ is not almost decomposable.
1. [*I do not know whether, in $\mathsf{ZFC}$, $(X,{\mathscr{B}}(X),{\mathscr{B}}(X) \cap {\mathscr{N}}_{{\mathscr{H}}^1})$ is not semilocalizable.*]{}
Notice that, under $\mathsf{CH}$, semilocalizability does not follow from \[CH.implies.ad\]. A more general version of this question is the following.
1. [*I do not know whether in \[55\] the word <<almost decomposable>> might be replaced by the word <<semilocalizable>> without affecting the validity of the statement.*]{}
Notice that \[mcshane\] does not seem to apply in this situation, for the following reason. If $(X,{\mathscr{B}}(X),\mu)$ is such that $X$ is Polish and $\mu$ is semifinite, then I do not see a reason that the completion of $(X,{\mathscr{B}}(X),\mu)$ be locally determined.
Another consequence of \[number.9\] is that there does not exist, in $\mathsf{ZFC}$, a <<localizable version>> of $(X,{\mathscr{B}}(X),{\mathscr{B}}(X) \cap {\mathscr{N}}_{{\mathscr{H}}^1})$ obtained by simply <<enlarging>> the given $\sigma$-algebra ${\mathscr{B}}(X)$ to another one ${\mathscr{B}}(X) {\subseteq}{\mathscr{A}}{\subseteq}{\mathscr{P}}(X)$. Instead it seems necessary to enlarge $X$ first. As stated in \[category\](Q4) it would be interesting to investigate the following.
1. [*Assuming one has defined a specific left adjoint functor to the forgetful functor ${\mathsf{LOC}}\to {\mathsf{MSN}}$, describe its effect on $(X,{\mathscr{B}}(X),{\mathscr{B}}(X) \cap {\mathscr{N}}_{{\mathscr{H}}^1})$. It would be particularly interesting to give a geometric interpretation of the process.*]{}
\[CR.3\] Here we ask whether the behavior exhibited by the specific set $X$ of section \[example\] is shared by other compact subsets of ${\mathbf{R}}^2$ of Hausdorff dimension 1 but non $\sigma$-finite ${\mathscr{H}}^1$ measure.
1. [*Let $X {\subseteq}{\mathbf{R}}^2$ be a compact set of Hausdorff dimension 1 and such that ${\mathscr{H}}^1(X \cap U) = \infty$ for every open set $U {\subseteq}{\mathbf{R}}^2$ with $X \cap U \neq \emptyset$. Is $(X,{\mathscr{A}}_{{\mathscr{H}}^1},{\mathscr{H}}^1)$ undecidably semilocalizable?*]{}
\[CR.4\] In regard to (Q8), of particular interest would be an example of such $X$ which is purely $({\mathscr{H}}^1,1)$ unrectifiable. Let us for instance consider the following set $X$, using the notations of \[pue.1\]. Choosing $\lambda_k = k.4^{-k}$ one checks that $X$ has Hausdorff dimension 1 and ${\mathscr{H}}^1(X \cap U) = \infty$ whenever $U {\subseteq}{\mathbf{R}}^2$ is open and $X \cap U \neq \emptyset$. Indeed ${\mathscr{H}}^d(X) = 0$ when $1 < d$, is a consequence of the definition of Hausdorff measure, and if $S \in {\mathscr{X}}_k$ for some $k \in {\mathbb{N}}^*$ then ${\mathscr{H}}^1(X \cap S) = \infty$ according to [@GMT 2.10.27] because $X \cap S = K \times K$ for some $K {\subseteq}[0,1]$ with ${\mathscr{H}}^\frac{1}{2}(K)=\infty$ according to [@GMT 2.10.28]. Also observe, as in \[pue.1\](1) that $X$ is purely $({\mathscr{H}}^1,1)$ unrectifiable.
1. [*With the set $X$ described here, is $(X,{\mathscr{A}}_{{\mathscr{H}}^1},{\mathscr{H}}^1)$ undecidably semilocalizable?*]{}
Viewing $X$ as a product as in section \[section.pue\] does not seem to be immediately helpful since it gives information about a Hausdorff measure essentially of dimension $1/2$. Trying to use the graphs of distribution functions as in section \[example\] is no more successful since these graphs are rectifiable and $X$ is purely unrectifiable ; their intersection will always be ${\mathscr{H}}^1$ null. One may also attempt to produce families $V_s$ and $H_t$ needed in \[abstract.theorem\] as random Cantor subsets of $X$: the $V_s$ using more often a specific set of three subsquares at each generation, and the $H_t$ using more often a distinct specific set of three subsquares at each generation. However random sets constructed this way tend to intersect too often, making it hard to guarantee condition (2) of \[abstract.theorem\].
\[CR.5\] In the notation of section \[section.pue\] and with the same restriction on $d$ as in the proof of \[pue.8\],
1. [*I do not know whether the measurable spaces with negligibles $(X_d,{\mathscr{A}}_{{\mathscr{H}}^d},{\mathscr{N}}_{{\mathscr{H}}^d})$ and $\left( [0,1],{\mathscr{A}}_{{\mathscr{H}}^{1/2}}, {\mathscr{N}}_{{\mathscr{H}}^{1/2}} \right)$ are isomorphic in the category ${\mathsf{MSN}}$.* ]{}
This boils down to deciding whether the $\sigma$-algebra ${\mathscr{A}}$ defined in the proof of \[pue.8\] coincides with the $\sigma$-algebra ${\mathscr{A}}_{{\mathscr{H}}^d}$. The fact the answer to this question is not known turned out to be no obstacle thanks to the freedom allowed in \[abstract.theorem\] regarding the $\sigma$-algebra ${\mathscr{A}}$.
\[CR.6\] Our last question here concerns \[pue.8\]. The proof, based on \[abstract.theorem\] seems to require a product structure that forces the dimension to be $1/2$.
1. [ *Let $0 < d < 1$ and $d \neq \frac{1}{2}$. Is the measure space $\left( [0,1], {\mathscr{A}}_{{\mathscr{H}}^d} , {\mathscr{H}}^d \right)$ consistently not semilocalizable?* ]{}
[^1]: The author was partially supported by the Science and Technology Commission of Shanghai (No. 18dz2271000).
[^2]: I learned it from [@GMT 2.5.10]. Unfortunately the presentation there does not allow for putting emphasis on the role played by the choice of a particular $\sigma$-algebra.
[^3]: Recall that $N {\subseteq}{\mathbf{R}}^2$ is purely $({\mathscr{H}}^1,1)$ unrectifiable if and only if ${\mathscr{H}}^1(N \cap \Gamma)=0$ for all $C^1$ curves $\Gamma$, and notice that a weak tangent field to $X$ is well defined ${\mathscr{N}}_{pu}$ almost everyhwere.
[^4]: Of course for each Borel set $Y {\subseteq}{\mathbf{R}}^2$ such that ${\mathscr{H}}^1 {
\hskip2.5pt{\vrule height7pt width.5pt depth0pt}
\hskip-.2pt\vbox{\hrule height.5pt width7pt depth0pt}
\, }Y$ is $\sigma$-finite the measure space $(Y,{\mathscr{B}}(Y),{\mathscr{H}}^1)$ is semilocalizable, \[sigmafinite.localizable\] and $Y$ admits the obvious Borel weak tangent field defined on its rectifiable part
[^5]: D.H. Fremlin calls localizable a measure space $(X,{\mathscr{A}},\mu)$ which is semifinite and such that $(X,{\mathscr{A}},{\mathscr{N}}_\mu)$ is (in the vocabulary introduced in the present paper) a localizable measurable space with negligibles.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Bugs that surface in mobile applications can be difficult to reproduce and fix due to several confounding factors including the highly GUI-driven nature of mobile apps, varying contextual states, differing platform versions and device fragmentation. It is clear that developers need support in the form of automated tools that allow for more precise reporting of application defects in order to facilitate more efficient and effective bug fixes. In this paper, we present a tool aimed at supporting application testers and developers in the process of **O**n-**D**evice **B**ug **R**eporting. Our tool, called **ODBR**, leverages the uiautomator framework and low-level event stream capture to offer support for recording and replaying a series of input gesture and sensor events that describe a bug in an Android application.'
author:
-
bibliography:
- 'ms.bib'
title: 'On-Device Bug Reporting for Android Applications'
---
Introduction {#sec:intro}
============
Background and Related Work {#sec:background}
===========================
The ODBR Bug Reporting Tool {#sec:approach}
===========================
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Angular momentum can be defined by rearranging the Komar surface integral in terms of a twist form, encoding the twisting around of space-time due to a rotating mass, and an axial vector. If the axial vector is a coordinate vector and has vanishing transverse divergence, it can be uniquely specified under certain generic conditions. Along a trapping horizon, a conservation law expresses the rate of change of angular momentum of a general black hole in terms of angular momentum densities of matter and gravitational radiation. This identifies the transverse-normal block of an effective gravitational-radiation energy tensor, whose normal-normal block was recently identified in a corresponding energy conservation law. Angular momentum and energy are dual respectively to the axial vector and a previously identified vector, the conservation equations taking the same form. Including charge conservation, the three conserved quantities yield definitions of an effective energy, electric potential, angular velocity and surface gravity, satisfying a dynamical version of the so-called first law of black-hole mechanics. A corresponding zeroth law holds for null trapping horizons, resolving an ambiguity in taking the null limit.'
author:
- 'Sean A. Hayward'
date: Revised 20th October 2006
title: Angular momentum conservation for dynamical black holes
---
Introduction
============
The theory of black holes appears finally to be reaching a stage of maturity in which it can be applied in the most interesting, distorted, dynamic situations, with appropriate definitions of relevant physical quantities. This article mainly concerns angular momentum and its conservation, which is the last major piece of what seems to be an essentially complete new paradigm for black holes. It therefore seems timely to review below the key ideas and results of what might be called the heroic, classical and modern eras.
The first solution which would nowadays be called a black hole was discovered by Schwarzschild [@Sch] within a few weeks of the final formulation of the field equations of General Relativity by Einstein [@Ein], as the external gravitational field of a point with mass $M$. Charge $Q$ was soon added by Reissner [@Rei] and Nordström [@Nor], but even the Schwarzschild solution was not properly understood for decades. Schwarzschild described the mass point as located at what is now understood as the horizon, despite its non-zero area $A$. Einstein & Rosen [@ER] realized that the spatial geometry had a wormhole structure, extending through a minimal surface. Oppenheimer & Snyder [@OS] constructed a model of stellar collapse in which the star collapses through the horizon. Finally Kruskal [@Kru], as reported in a paper actually written by Wheeler [@WF], described the entire space-time, whence it became clear that there was a trapped region inside the horizon, from which nothing could escape to the exterior. Angular momentum $J$ was added by Kerr [@Ker] and, including $Q$, by Newman et al. [@New]. Uniqueness theorems identify these as the only black holes which are stationary, asymptotically flat, electrovac solutions.
Wheeler [@Whe] is credited with coining the term “black hole” and Penrose [@Pen0] with defining event horizons, which became the accepted definition of black holes. Hawking [@Haw0] showed that the area of an event horizon was non-decreasing, $A'\ge0$. The result became known as a “second law”, due to inaccurate analogies with the laws of thermodynamics and the results summarized by Bardeen et al. [@BCH]: a “zeroth law” that surface gravity $\kappa$ is constant on a stationary black hole; a “first law” $$\delta E=\kappa\delta A/8\pi+\Omega\delta J+\Phi\delta Q\label{g0}$$ for perturbations of stationary black holes, where $\Omega$ is the angular speed and $\Phi$ the electric potential of the horizon, and the ADM energy $E$ measures the total mass of the space-time; and a “third law” that $\kappa\not\to0$ by positive-energy perturbations of stationary black holes. This summarizes the classical theory of black holes as described in textbooks [@HE; @MTW; @Wal].
The last results above are perhaps best described as black-hole statics, being properties of stationary black holes, specifically of Killing horizons, rather than of general event horizons. While adequate in some astrophysical situations, this classical theory is inapplicable to general dynamical processes, for instance black-hole formation, rapid evolution and binary mergers. A theory of black-hole dynamics is needed, with generalizations of all the above-mentioned quantities. Event horizons are not an appropriate platform, since they cannot be located by mortals, let alone admit physical measurements. A more practical way to locate a black hole is by a [*marginal surface*]{}, an extremal surface of a null hypersurface, where light rays are momentarily caught by the gravitational field. Marginal surfaces are used extensively in numerical simulations, where they have historically been called apparent horizons, though the textbook definition of the latter is different [@HE; @MTW; @Wal]. Here a hypersurface foliated by marginal surfaces will be called a [*trapping horizon*]{}.
A systematic treatment of trapping horizons [@bhd] distinguished four sub-classes, called future or past, outer or inner trapping horizons, with the future outer type proposed as the practical location of a black hole. Such a horizon was shown to have several expected properties of a black hole, assuming the Einstein equation and positive-energy conditions: the horizon is achronal, being null in a special case of quasi-stationarity, but otherwise being spatial; the area form ${*}1$ of the marginal surfaces is constant in the null case and increasing in the spatial case; and the marginal surfaces have spherical topology, if compact. The area law implies that the area $A=\oint_S{*}1$ of the marginal surfaces $S$ is non-decreasing, $$L_\xi A\ge0\label{a0}$$ where $L$ denotes the Lie derivative and $\xi$ the generating vector of the marginal surfaces. So a black hole grows if something falls into it, otherwise staying the same size.
Comprehensive treatments were subsequently given for spherical symmetry [@sph; @1st; @in], cylindrical symmetry [@cyl] and a quasi-spherical approximation [@qs; @SH; @gwbh; @gwe]. In each case, definitions were found for all the non-zero physical quantities mentioned above, providing prototypes of all except $J$ and $\Omega$. In addition, an effective energy tensor $\Theta$ for gravitational radiation was found, entering equations additively to the matter energy tensor $T$. The Einstein equations were decomposed into forms with manifest physical meaning, such as a quasi-Newtonian gravitational law, a wave equation for the gravitational radiation, and an energy conservation law which can be written in the form $$L_\xi
M=\oint_S{*}(T_{\alpha\beta}+\Theta_{\alpha\beta})k^\alpha\tau^\beta\label{ec0}$$ where $\tau$ in the normal dual of $\xi$ and $k$ is a certain vector, playing the role of a Killing vector, which is null on the horizon. Such equations actually hold not just on a trapping horizon, but anywhere in the space-time, energy conservation reducing at null infinity to the Bondi energy equation.
Contemporaneously, Ashtekar et al. and others [@ABF1; @ABF2; @AFK; @ABD; @ABL1; @Boo; @ABL2; @DKSS; @GJ1] developed a theory of null trapping horizons with various additional conditions, under the names non-rotating isolated horizons, non-expanding horizons, weakly isolated horizons, rigidly rotating horizons and (strongly) isolated horizons. Each is intended to capture the idea that the black hole is quasi-stationary in some sense. They gave definitions of all the relevant physical quantities and derived a generalized version of the so-called first law.
Subsequently, Ashtekar & Krishnan [@AK1; @AK2; @AK3] studied future spatial trapping horizons under the name dynamical horizons, giving classes of definitions of energy and angular momentum, deriving corresponding flux equations and obtaining a version of the so-called first law for $Q=0$. However, the “3+1” formalism used to describe spatial trapping horizons breaks down when the horizon becomes null, so that the isolated-horizon and dynamical-horizon frameworks were essentially distinct. Some connections were drawn, particularly for slowly evolving horizons by Booth & Fairhurst [@BF1; @BF2; @BF3]. Recently, Andersson et al. [@AMS] showed that a stable trapping horizon is, on any one marginal surface, either spatial or null everywhere, so that transitions between the two types happen simultaneously on a marginal surface. They and Ashtekar & Galloway [@AG] also obtained some existence and uniqueness results for trapping horizons.
A unified framework for any trapping horizon is provided by a dual-null formalism [@dn; @dne], which was used throughout the earlier studies [@bhd; @sph; @1st; @in; @cyl; @qs; @SH; @gwbh; @gwe]. The energy flux equation was then cast in a surface-integral form where the null limit could be taken [@bhd2; @bhd3]. Moreover, it was cast in the form of a conservation law (\[ec0\]), by identifying an effective energy tensor $\Theta$. The mass $M$, which might take any value on a given $S$ by choice of scaling of $k$, was chosen to be the irreducible mass or Hawking mass [@Haw] for consistency with the earlier studies. This corresponds to the simplest general definition of $k$, such that it becomes a unit vector for round spheres near infinity.
The main task here is to make similar refinements for angular momentum, as briefly described in shorter articles [@j15; @bhd4]. In particular, one desires not just a flux equation but a conservation law of the form $$L_\xi
J=-\oint_S{*}(T_{\alpha\beta}+\Theta_{\alpha\beta})\psi^\alpha\tau^\beta.\label{amc0}$$ Here $\psi$ should be an axial vector in some sense, playing the role of an axial Killing vector, with $J$ being the angular momentum about that axis. It turns out that natural restrictions on $\psi$ allow it to be uniquely specified under certain generic conditions. The angular momentum, initially a functional $J[\psi]$, is obtained directly from the Komar integral [@Kom] in terms of a 1-form $\omega$ known as the twist [@dne], which reduces to the 1-form used for dynamical horizons [@AK1; @AK2; @AK3]. It encodes the rotational frame-dragging predicted in the Lense-Thirring effect, thereby giving a precise meaning to the twisting around of space-time due to a rotating mass.
The null limit is more subtle for angular momentum than for energy, where the irreducible mass is uniquely defined for a null trapping horizon, energy conservation (\[ec0\]) reducing correctly to $L_\xi M=0$. The dual-null foliation becomes non-unique for a null hypersurface, with $\omega$ becoming non-unique. This is reflected in the fact that different 1-forms were used for isolated horizons [@ABF1; @ABF2; @AFK; @ABD; @ABL1; @Boo; @ABL2; @DKSS; @GJ1] and dynamical horizons [@AK1; @AK2; @AK3]. Neither 1-form is necessarily preserved along a null trapping horizon, but a certain linear combination is so preserved. However, all three 1-forms coincide if the non-uniqueness of the dual-null foliation for a null hypersurface is fixed in a certain way. Then $J[\psi]$ becomes unique for a null trapping horizon and conservation (\[amc0\]) reduces as desired to $L_\xi J=0$. Thus a consistent treatment of angular momentum naturally resolves the issue of the degeneracy of the null limit. It follows that a black hole cannot change its angular momentum without increasing its area.
The article is organized as follows. Section II summarizes the underlying geometry. Section III reviews trapping horizons and conservation of energy. Section IV derives angular momentum from the Komar integral and shows how restrictions on the axial vector can be used to construct a unique definition. Section V derives and discusses the conservation law. Section VI includes charge conservation and discusses local versus quasi-local conservation. Section VII describes the state space, defining the remaining physical quantities and deriving a dynamical version of the so-called first law. Section VIII considers null trapping horizons, deriving a zeroth law. Appendices concern (A) weak fields, (B) normal fundamental forms and (C) a Kerr example. The above discussion serves as a summary.
Geometry
========
General Relativity will be assumed, with space-time metric $g$. The geometrical object of interest is a one-parameter family $\{S\}$ of spatial surfaces $S$, locally generating a foliated hypersurface $H$. Labelling the surfaces by a coordinate $x$, they are generated by a vector $\xi=\partial/\partial x$, which can be taken to be normal to the surfaces, $\bot\xi=0$, where $\bot$ denotes projection onto $S$. A Hodge duality operation on normal vectors $\eta$, $\bot\eta=0$, yields a dual normal vector $\eta^*$ satisfying $$\bot\eta^*=0,\quad g(\eta^*,\eta)=0,\quad
g(\eta^*,\eta^*)=-g(\eta,\eta).\label{nd}$$ In particular, $$\tau=\xi^*$$ is normal to $H$, with the same scaling (Fig.\[normal\]). The coordinate freedom here is just $x\mapsto\tilde x(x)$ and choice of transverse coordinates on $S$, under which all the key formulas will be invariant. The generating vector $\xi$ may have any causal character, at each point. For instance, a future outer trapping horizon is spatial while growing, becomes null when quasi-stationary, and would become temporal if shrinking during evaporation [@bhd; @bhd2; @bhd3].
![A non-null hypersurface $H$ foliated by spatial surfaces $S$, with generating vector $\xi$ and its normal dual $\tau=\xi^*$. If $H$ becomes null, $\xi$ and $\tau$ coincide.[]{data-label="normal"}](jgrg15_3.eps){height="15mm"}
A dual-null formalism [@dn; @dne] describes two families of null hypersurfaces $\Sigma_\pm$, intersecting in a two-parameter family of spatial surfaces, including the desired one-parameter family. Some merits of the dual-null approach, apart from comparative ease of calculation, are that it is adapted both to marginal surfaces, defined as extremal surfaces of null hypersurfaces, and to radiation propagation, which makes it easier to identify terms arising due to gravitational radiation [@bhd2; @bhd3; @bhd4]. Relevant aspects of the formalism are summarized as follows.
Labelling $\Sigma_\mp$ by coordinates $x^\pm$ which increase to the future, one may take transverse coordinates $x^a$ on $S$, which for a sphere would normally be angular coordinates $x^a=(\vartheta,\varphi)$. Writing space-time coordinates $x^\alpha=(x^+,x^-,x^a)$ indicates how one may use Greek letters $(\alpha,\beta,\ldots)$ for space-time indices and corresponding Latin letters $(a,b,\ldots)$ for transverse indices. The coordinate basis vectors are $\partial_\alpha=\partial/\partial x^\alpha$ and the dual 1-forms are $dx^\alpha$, satisfying $\partial_\beta(dx^\alpha)=\delta^\alpha_\beta$. Coordinate vectors commute, $[\partial_\alpha,\partial_\beta]=0$, where the brackets denote the Lie bracket or commutator. Two coordinate vectors have a special role, the evolution vectors $\partial_\pm=\partial/\partial x^\pm$ which generate the dynamics, spanning an integrable evolution space. The corresponding normal 1-forms $dx^\pm$ are null by assumption: $$g^{-1}(dx^\pm,dx^\pm)=0.$$ The relative normalization of the null normals may be encoded in a function $f$ defined by $$e^f=-g^{-1}(dx^+,dx^-)\label{ff}$$ where the metric sign convention is that spatial metrics are positive definite. The transverse metric, or the induced metric on $S$, is found to be $$h=g+2e^{-f}dx^+\otimes dx^-\label{tm}$$ where $\otimes$ denotes the symmetric tensor product. There are two shift vectors $$s_\pm=\bot\partial_\pm$$ where $\bot$ is generalized to indicate projection by $h$. The null normal vectors $$l_\pm=\partial_\pm-s_\pm=-e^{-f}g^{-1}(dx^\mp)\label{nn}$$ are future-null and satisfy $$\begin{aligned}
&&g(l_\pm,l_\pm)=0,\quad g(l_+,l_-)=-e^{-f},\nonumber\\
&&l_\pm(dx^\pm)=1,\quad l_\pm(dx^\mp)=0,\quad\bot l_\pm=0.\label{ls}\end{aligned}$$ The metric takes the form $$\begin{aligned}
g&=&h_{ab}(dx^a+s_+^adx^++s_-^adx^-)\otimes\nonumber\\&&(dx^b+s_+^bdx^++s_-^bdx^-)
-2e^{-f}dx^+\otimes dx^-.\end{aligned}$$ Then $(h,f,s_\pm)$ are configuration fields and the independent momentum fields are found to be linear combinations of the following transverse tensors: $$\begin{aligned}
\theta_\pm&=&{*}L_\pm{*}1\label{ex}\\ \sigma_\pm&=&\bot L_\pm h-\theta_\pm h\label{sh}\\
\nu_\pm&=&L_\pm f\label{in}\\
\omega&=&{\textstyle\frac12}e^fh([l_-,l_+])\label{tw}\end{aligned}$$ where ${*}$ is the Hodge operator of $h$ and $L_\pm$ is shorthand for the Lie derivative along $l_\pm$. Then the functions $\theta_\pm$ are the null expansions, the traceless bilinear forms $\sigma_\pm$ are the null shears, the 1-form $\omega$ is the twist, measuring the lack of integrability of the normal space, and the functions $\nu_\pm$ are the inaffinities, measuring the failure of the null normals to be affine. The fields $(\theta_\pm,\sigma_\pm,\nu_\pm,\omega)$ encode the extrinsic curvature of the dual-null foliation. These extrinsic fields are unique up to interchange $\pm\mapsto\mp$ and diffeomorphisms $x^\pm\mapsto\tilde x^\pm(x^\pm)$ which relabel the null hypersurfaces. Further description of the geometry was given recently [@bhd3].
As described, the dual-null formalism is manifestly covariant on $S$, with transverse indices not explicitly denoted, while $\pm$ indices indicate the chosen normal basis [@dne; @bhd3]. Conversely, one can use a formalism which is manifestly covariant on the normal space, with transverse but not normal indices explicitly denoted [@BHMS]. Both types of formalism can seem obscure to the uninitiated, so indices will be explicitly denoted in longer formulas in this article, nevertheless being omitted where the meaning should be clear. Capital Latin letters $(A,B,\ldots)$ will be used for normal indices, when not denoted by $\pm$ in the dual-null basis. Then the configuration fields are $(h_{ab},f,s_A{}^b)$, the momentum fields are $(\theta_A,\sigma_{Abc},\nu_A,\omega_a)$ and the derivative operators are $(\bot L_A,D_a)$, where $D$ is the covariant derivative operator of $h$.
Since the normal space is not integrable unless $\omega=0$, it generally does not admit a coordinate basis. However, one may still take $dx^\pm$ as basis 1-forms, in which case the dual basis vectors are $l_\pm$, as follows from (\[ls\]), implying $l_A(dx^B)=\delta_A^B$. In this basis, the normal metric $$\gamma=g-h$$ has components which follow from (\[tm\]) as $$\gamma_{AB}=-e^{-f}(dx^+_Adx^-_B+dx^-_Adx^+_B)$$ and its inverse has components $$\gamma^{AB}=-e^f(l_+^Al_-^B+l_-^Al_+^B)$$ which can be used to lower and raise normal indices. Also useful is the binormal $$\epsilon_{AB}=e^{-f}(dx^+_Adx^-_B-dx^-_Adx^+_B) \label{bnd}$$ or its inverse $$\epsilon^{AB}=e^f(l_-^Al_+^B-l_+^Al_-^B). \label{bnu}$$ The mixed form $$\epsilon^A{}_B=l_+^Adx^+_B-l_-^Adx^-_B \label{bnm}$$ has components $\epsilon^\pm{}_\pm=\pm1$, $\epsilon^\pm{}_\mp=0$, so can be used to express the duality operation (\[nd\]) on normal vectors, extended to the dual-null foliation, as $$(\eta^*)^A=\epsilon^A{}_B\eta^B. \label{nd2}$$
The dual-null Hamilton equations and integrability conditions for vacuum Einstein gravity were derived previously [@dne], with matter terms added subsequently [@gwbh]. The components of the field equations which are relevant to angular momentum turn out to be the twisting equations $$\begin{aligned}
\bot L_\pm\omega_a&=&-\theta_\pm\omega_a\pm\textstyle{\frac12}D_a\nu_\pm
\mp\textstyle{\frac12}D_a\theta_\pm\mp\textstyle{\frac12}\theta_\pm
D_af \nonumber\\
&&\qquad\qquad{}\pm\textstyle{\frac12}h^{cd}D_d\sigma_{\pm ac} \mp8\pi
T_{a\pm}\label{twisting}\end{aligned}$$ where $T_{a\pm}=h_a^\gamma T_{\gamma\beta}l_\pm^\beta$ is the transverse-normal projection of the energy tensor $T$, and units are such that Newton’s gravitational constant is unity. The corresponding all-index version can be written using the binormal as $$\begin{aligned}
\bot
L_B\omega_a&=&-\theta_B\omega_a+\textstyle{\frac12}\epsilon^E{}_B(D_a\nu_E
-D_a\theta_E-\theta_ED_af \nonumber\\
&&\qquad\qquad\quad{}+h^{cd}D_d\sigma_{Eac}-16\pi T_{aE}).\end{aligned}$$
Trapping horizons and conservation of energy
============================================
Returning to a general foliated hypersurface $H$, a normal vector $\eta$ has components $\eta^\pm$ along $l_\pm$, so that $\eta=\eta^+l_++\eta^-l_-$, and its normal dual is $\eta^*=\eta^+l_+-\eta^-l_-$. In particular, the generating vector is $$\xi=\xi^+l_++\xi^-l_-\label{xi}$$ and its dual is $$\tau=\xi^+l_+-\xi^-l_-.\label{tau}$$ Since the horizon is given parametrically by functions $x^\pm(x)$, the components $\xi^\pm=\partial x^\pm/\partial x$ are independent of transverse coordinates: $$D\xi^\pm=0.\label{dxi}$$ It is also useful to introduce the expansion $$\theta_\eta=L_\eta\log({*}1)=\theta_A\eta^A$$ along a normal vector $\eta$, particularly the expansion $\theta_\xi$ along the generating vector.
A trapping horizon [@bhd; @1st; @bhd2; @bhd3] is a hypersurface $H$ foliated by marginal surfaces, where $S$ is marginal if one of the null expansions, $\theta_+$ or $\theta_-$, vanishes everywhere on $S$. Then $S$ is an extremal surface of the null hypersurface $\Sigma_+$ or $\Sigma_-$.
The recently derived energy conservation law [@bhd2; @bhd3] will be stated here for later comparison, modifying some notation. Assuming compact $S$ henceforth, the transverse surfaces have area $$A=\oint_S{*}1$$ and the area radius $$R=\sqrt{A/4\pi}$$ is often more convenient. The Hawking mass [@Haw] $$M=\frac
R2\left(1-\frac1{16\pi}\oint_S{*}\gamma^{AB}\theta_A\theta_B\right)\label{hm}$$ can be used as a measure of the active gravitational mass on a transverse surface. Assuming the null energy condition, $M$ is the irreducible mass $R/2$ of a future outer trapping horizon, $L_\xi M\ge0$, by the area law (\[a0\]). On a stationary black-hole horizon, $M$ reduces to the usual definition of irreducible mass for a Kerr-Newman black hole, namely the mass which must remain even if rotational or electrical energy is extracted. It is generally not the ADM energy, but an effective energy $E$ is defined in §VII which does recover the ADM energy in this case. Equality on a trapping horizon will be denoted by $\cong$, e.g. $R\cong2M$.
Mass or energy has a certain duality with time, e.g. there is a standard formula for energy if a stationary Killing vector exists. For a general compact surface, the simplest definition of such a vector which applies correctly for a Schwarzschild black hole is [@bhd2; @bhd3] $$k=(g^{-1}(dR))^*\label{time}$$ or $k^A=\epsilon^{AB}L_BR$. This vector actually was found to be the appropriate dual of $M$, in the sense of conservation laws for trapping horizons [@bhd2; @bhd3] and for uniformly expanding flows [@gr; @BHMS]. In either case, the energy conservation law can be written as $$L_\xi M\cong\oint_S{*}(T_{AB}+\Theta_{AB})k^A\tau^B \label{ec}$$ where $\Theta$ is an effective energy tensor for gravitational radiation. This determines only the normal-normal components of $\Theta$, as $$\begin{aligned}
\Theta_{\pm\pm}&=&||\sigma_\pm||^2/32\pi\label{Theta0}\\
\Theta_{\pm\mp}&=&e^{-f}|\omega\mp{\textstyle\frac12}Df|^2/8\pi\label{Theta1}\end{aligned}$$ where $|\zeta|^2=h^{ab}\zeta_a\zeta_b$ and $||\sigma||^2=h^{ac}h^{bd}\sigma_{ab}\sigma_{cd}$. Further discussion is referred to [@bhd2; @bhd3; @gr; @BHMS].
Angular momentum
================
The standard definition of angular momentum for an axial Killing vector $\psi$ and at spatial infinity is the Komar integral [@Kom] $$J[\psi]=-\frac1{16\pi}\oint_S{*}\epsilon_{\alpha\beta}\nabla^\alpha\psi^\beta.
\label{komar}$$ Now consider $\psi$ to be a general transverse vector, $\bot\psi=\psi$ (Fig.\[transverse\]). Since $\epsilon_{\alpha\beta}\psi^\beta=0$, the Komar integral can be rewritten via (\[bnu\]) as $$J[\psi]=\frac1{8\pi}\oint_S{*}\psi^a\omega_a \label{am}$$ where $\omega$ is the twist (\[tw\]). Since the twist encodes the non-integrability of the normal space, it provides a geometrical measure of rotational frame-dragging. It is an invariant of a dual-null foliation and therefore of a non-null foliated hypersurface $H$, so the twist expression for $J[\psi]$ is also an invariant. Appendix A shows that $J[\psi]$ recovers the standard definition of angular momentum for a weak-field metric [@MTW], with the twist being directly related to the precessional angular velocity of a gyroscope due to the Lense-Thirring effect. Thus the twist does indeed encode the twisting around of space-time caused by a rotating mass.
There are several definitions of angular momentum which are similar surface integrals of an axial vector contracted with a 1-form [@ABD; @AK1; @BY], the situation being clarified by Gourgoulhon [@Gou] and in Appendix B. Ashtekar & Krishnan [@AK1] gave a definition for a dynamical horizon which involves a 1-form coinciding with $\omega$. Brown & York [@BY] gave a definition which was stated only for an axial Killing vector $\psi$, involving a 1-form which is generally inequivalent to $\omega$, but can be made to coincide if adapted to a trapping horizon. Ashtekar et al.[@ABD; @ABL1] gave a definition for a type II (rigidly rotating) isolated horizon, using a 1-form which is generally inequivalent to $\omega$. However, it can be made to coincide with $\omega$ if the dual-null foliation is fixed in a natural way, as described in the penultimate section.
The above properties suggest (\[am\]) as a general quasi-local definition of angular momentum. However, if the transverse vector $\psi$ does not have properties expected of an axial vector, the physical interpretation as angular momentum is questionable. For instance, it would be natural to expect an axial vector to have integral curves which form a smooth foliation of circles, apart from two poles, assuming spherical topology for $S$. In the following, two conditions on $\psi$ with various motivations will be considered, which, taken together, determine $\psi$ uniquely in a certain generic situation. These conditions then yield a conservation law with the desired form (\[amc0\]), as described in the next section.
![A transverse vector $\psi$.[]{data-label="transverse"}](jgrg15_1.eps){height="3cm"}
Ashtekar & Krishnan [@AK2] proposed that $\psi$ has vanishing transverse divergence: $$D_a\psi^a\cong0. \label{div}$$ This condition holds if $\psi$ is an axial Killing vector, and can be understood as a weaker condition, equivalent to $\psi$ generating a symmetry of the area form rather than of the whole metric, since $L_\psi({*}1)={*}D_a\psi^a$. Alternatively, assuming that the integral curves of $\psi$ are closed, it can always be satisfied by choice of scaling of $\psi$, as discussed by Booth & Fairhurst [@BF2]. The original motivation was that the different 1-forms used for dynamical and isolated horizons, denoted here by $\omega$ and $\omega+\frac12Df$, will then give the same angular momentum $J[\psi]$, by the Gauss divergence theorem.
Spherical topology will be assumed henceforth for $S$, which follows from the topology law [@bhd] for outer trapping horizons, assuming the dominant energy condition. If there exist angular coordinates $(\vartheta,\varphi)$ on $S$ with $\psi=\partial/\partial\varphi$, completing coordinates $(x,\vartheta,\varphi)$ on $H$, then since coordinate vectors commute, $$L_\xi\psi\cong0. \label{lie}$$ This condition was proposed as a natural way to propagate $\psi$ along $H$ by Gourgoulhon [@Gou]. Now there is a commutator identity [@dne] $$L_\xi(D_a\psi^a)-D_a(L_\xi\psi)^a=\psi^aD_a\theta_\xi. \label{com}$$ Therefore assuming both conditions (\[div\])–(\[lie\]) forces $$\psi^aD_a\theta_\xi\cong0. \label{con}$$ This is automatically satisfied if $D\theta_\xi\cong0$, as in spherical symmetry or along a null trapping horizon. However, generically one expects $D\theta_\xi\not\cong0$ almost everywhere. It must vanish somewhere on a sphere, by the hairy ball theorem, but the simplest generic situation is that there are curves $\gamma$ of constant $\theta_\xi$ which form a smooth foliation of circles, covering the surface except for two poles (Fig.\[axial\]). Assuming so, since $\psi$ is tangent to $\gamma$, one can find a unique $\psi$, up to sign, in terms of the unit tangent vector $\hat\psi$ and arc length $ds$ along $\gamma$: $$\psi\cong\hat\psi\oint_\gamma ds/2\pi \label{un}$$ where the scaling ensures that the axial coordinate $\varphi$ is identified at 0 and $2\pi$. Then the angular momentum becomes unique up to sign, $J[\psi]=J$, the sign being naturally fixed by $J\ge0$ and continuity of $\psi$, corresponding to a choice of orientation.
![Curves $\gamma$ of constant expansion $\theta_\xi$.[]{data-label="axial"}](jgrg15_2.eps){height="25mm"}
For an axisymmetric space-time with axial Killing vector $\psi$, (\[div\]) holds, while (\[lie\]) holds if $\xi$ respects the symmetry, $L_\psi\xi=0$, so the above construction, if unique as assumed, yields the correct axial vector. In particular, the construction does work for a Kerr space-time, as described in Appendix C.
To summarize this section, the definition (\[am\]) of angular momentum can be made generically unique if the axial vector is a coordinate vector, (\[lie\]), and generates a symmetry of the area form, (\[div\]). The construction can be applied in any situation where $D\theta_\xi\not\cong0$ almost everywhere, though the physical interpretation as angular momentum seems to be safest in the case of two poles, which locate the axis of rotation. Then $J$ is proposed to measure the angular momentum about that axis.
Conservation of angular momentum
================================
The main result of this paper is that $$L_\xi J\cong-\oint_S{*}(T_{aB}+\Theta_{aB})\psi^a\tau^B \label{amc}$$ holds along a trapping horizon under the conditions (\[div\])–(\[lie\]), where $$\Theta_{aB}=-\frac1{16\pi}h^{cd}D_d\sigma_{Bac} \label{Theta2}$$ is thereby determined to be the transverse-normal block of the effective energy tensor for gravitational radiation. It can be shown by differentiating the angular momentum (\[am\]) using (\[lie\]) to give $$L_\xi
J\cong\frac1{8\pi}\oint_S{*}(\theta_\xi\psi^a\omega_a+\psi^aL_\xi\omega_a)$$ then expanding $\xi$ by (\[xi\]) and using the twisting equations (\[twisting\]) to express $L_\xi\omega$. The term in $\theta_\xi=\xi^+\theta_++\xi^-\theta_-$ cancels with the first term from (\[twisting\]), while the $D$ gradients may all be removed as total divergences due to (\[dxi\]), (\[div\]) and the fact that (\[con\]) reduces to $\psi^aD_a\theta_\mp\cong0$ on a trapping horizon with $\theta_\pm\cong0$. This leaves just $$\begin{aligned}
L_\xi
J&\cong&\frac1{8\pi}\oint_S{*}\psi^a\Big(\xi^+(\textstyle{\frac12}h^{cd}D_d\sigma_{+ac}-8\pi
T_{a+})\nonumber\\
&&\qquad\qquad\quad-\xi^-(\textstyle{\frac12}h^{cd}D_d\sigma_{-ac}-8\pi
T_{a-})\Big)\end{aligned}$$ which is an expanded form of (\[amc\]), noting (\[tau\]) and thereby identifying (\[Theta2\]).
Apart from the inclusion of $\Theta$, the conservation law (\[amc\]) is the standard surface-integral form of conservation of angular momentum, were $\psi$ an axial Killing vector. It thereby describes the increase or decrease of angular momentum of a black hole due to infall of co-rotating or counter-rotating matter respectively. The corresponding volume-integral form for a spatial horizon $H$, expressing the change $[J]_{\partial H}$ in $J$ between two marginal surfaces, follows as $$[J]_{\partial H}\cong-\int_H{\hat*}(T_{aB}+\Theta_{aB})\psi^a\hat\tau^B
\label{amcv}$$ where $\hat\tau=\tau/\sqrt{g(\xi,\xi)}$ is the unit normal vector and ${\hat*}1 ={*}\sqrt{g(\xi,\xi)}\wedge dx$ is the proper volume element. Although more familiar, as for the energy conservation law [@bhd2; @bhd3], this form becomes degenerate as the horizon becomes null, since ${\hat*}1\to0$ while $\hat\tau$ ceases to exist. Since this is a physically important limit, where a black hole is not growing, the surface-integral form (\[amc\]) is preferred.
The null shears $\sigma_{\pm bc}$ have previously been identified in the energy conservation law (\[ec\]) as encoding the ingoing and outgoing transverse gravitational radiation, via the energy densities (\[Theta0\]), which agree with expressions in other limits, such as the Bondi energy density at null infinity and the Isaacson energy density of high-frequency linearized gravitational waves [@bhd2; @bhd3]. So the expression (\[Theta2\]) implies that gravitational radiation with a transversely differential waveform will generally possess angular momentum density. One can see corresponding terms in the linearized approximation [@MTW], but they are set to zero by the transverse-traceless gauge conditions, which are “transverse” in a different sense to that used here. In any case, the conservation law shows that a black hole can spin up or spin down even in vacuum, at a rate related to ingoing and outgoing gravitational radiation.
The identification of the transverse-normal block (\[Theta2\]) of $\Theta$ appears to be new. Previous versions of angular momentum flux laws for dynamical black holes [@AK1; @AK2; @AK3; @BF1; @BF2; @BF3; @Gou] contain different terms, which are not in energy-tensor form, i.e. some tensor contracted with $\psi$ and $\tau$. They can be recovered by removing a transverse divergence from $\Theta_{aB}\psi^a\tau^B$, yielding $\sigma_\tau^{ad}D_d\psi_a/16\pi=\sigma_\tau^{ad}L_\psi h_{ad}/32\pi$, where $\sigma_\tau^{ad}=\tau^Bh^{ae}h^{cd}\sigma_{Bce}$ encodes the shear along $\tau$. Such terms have been described by analogy with viscosity [@Gou; @GJ2].
The conservation laws (\[ec\]) and (\[amc\]) take a similar form, expressing rate of change of mass $M$ and angular momentum $J$ as surface integrals of densities of energy and angular momentum, with respect to preferred vectors $k$ and $\psi$ which play the role of stationary and axial Killing vectors, even if there are no symmetries. Of the ten conservation laws in flat-space physics, they are the two independent laws expected for an astrophysical black hole, which defines its own spin axis and centre-of-mass frame, in which its momentum vanishes.
Quasi-local conservation laws
=============================
For an electromagnetic field with charge-current density vector $j$, the total electric charge $Q$ in a region $H$ of a spatial hypersurface is defined as $$[Q]_{\partial H}=-\int_H{\hat*}g(j,\hat\tau). \label{ccv}$$ The surface-integral form of conservation of charge follows by the same arguments relating (\[amc\]) and (\[amcv\]): $$L_\xi Q=-\oint_S{*}g(j,\tau). \label{cc}$$ As before, this is more general, since $H$ may have any signature. The conservation laws for energy (\[ec\]) and angular momentum (\[amc\]) evidently take the same form $$L_\xi M\cong-\oint_S{*}g(\tilde\jmath,\tau), \quad L_\xi
J\cong-\oint_S{*}g(\bar\jmath,\tau) \label{cl}$$ by identifying current vectors $$\tilde\jmath^B=-k_A(T^{AB}+\Theta^{AB}),
\quad\bar\jmath^B=\psi_a(T^{aB}+\Theta^{aB}).$$
The local differential form of charge conservation, $$\nabla\!_\alpha j^\alpha=0 \label{ccd}$$ where $\nabla$ is the covariant derivative of $g$, notably does not hold for $\tilde\jmath$ or $\bar\jmath$ in general. A weaker property holds, obtained as follows. First note that in any of the three conservation laws (\[ec\]), (\[amc\]) and (\[cc\]), $\xi$ and $\tau$ may be interchanged. Thus there are really two independent laws in each case. This can be understood from special relativity: if $\xi$ were causal, one would interpret them as expressing rate of change of energy, angular momentum or charge as, respectively, power, torque or current; while if $\xi$ were spatial, one would normally convert to volume-integral form and regard them as defining the energy, angular momentum or charge in a region. One can make either interpretation for a black hole, since $H$ would be generically spatial, but the marginal surfaces $S$ locate the black hole in a family of time slices.
Given two independent equations in the normal space, it follows that $$\epsilon^A{}_BL_AM\cong-\oint_S{*}1\wedge\tilde\jmath_B, \quad
\epsilon^A{}_BL_AJ\cong-\oint_S{*}1\wedge\bar\jmath_B.$$ Expressed in terms of the curl and divergence of the normal space, $$\hbox{curl}\,M\cong-\oint_S{*}1\wedge\tilde\jmath, \quad
\hbox{curl}\,J\cong-\oint_S{*}1\wedge\bar\jmath$$ whereas $$\nabla\!_\alpha\jmath^\alpha={*}\hbox{div}({*}1\wedge\jmath)$$ for any normal vector $\jmath$. Then $\hbox{div}\,\hbox{curl}\,=0$ yields $$\oint_S{*}\nabla\!_\alpha\tilde\jmath^\alpha
\cong\oint_S{*}\nabla\!_\alpha\bar\jmath^\alpha\cong0. \label{qlc}$$ This subtly confirms the view that energy and angular momentum in General Relativity cannot be localized [@MTW], but might be quasi-localized, as surface integrals [@Pen]. The corresponding conservation laws have indeed been obtained in surface-integral but not local form.
State space
===========
There are now three conserved quantities $(M,J,Q)$, forming a state space for dynamical black holes. Following various authors [@ABL1; @Boo; @AK1; @AK2; @AK3; @BF1], related quantities may then be defined by formulas satisfied by Kerr-Newman black holes, specifically those for the ADM energy $$E\cong\frac{\sqrt{((2M)^2+Q^2)^2+(2J)^2}}{4M} \label{e}$$ the surface gravity $$\kappa\cong\frac{(2M)^4-(2J)^2-Q^4}{2(2M)^3\sqrt{((2M)^2+Q^2)^2+(2J)^2}}
\label{sg}$$ the angular speed $$\Omega\cong\frac{J}{M\sqrt{((2M)^2+Q^2)^2+(2J)^2}} \label{as}$$ and the electric potential $$\Phi\cong\frac{((2M)^2+Q^2)Q}{2M\sqrt{((2M)^2+Q^2)^2+(2J)^2}}. \label{ep}$$ It would be preferable to have independently motivated definitions of these quantities, but so far this has been done only in spherical symmetry [@sph; @1st; @in], where there are natural definitions of $E$, $\kappa$ and $\Phi=Q/R$ which can be applied anywhere in the space-time, coinciding with the above expressions on the outer horizons of a Reissner-Nordström black hole.
In the dynamical context, $E\ge M$ is generally not the ADM energy, since there may be matter or gravitational radiation outside the black hole. Rather, it can be interpreted as the effective energy of the black hole, as follows. Defining the moment of inertia $I$ by the usual formula $J\cong I\Omega$ yields $$I\cong M\sqrt{((2M)^2+Q^2)^2+(2J)^2}\cong ER^2.$$ Expanding $E$ for $J\ll M^2$ and $Q\ll M$ yields $$E\approx M+\textstyle{\frac12}I\Omega^2+\textstyle{\frac12}Q^2/R.$$ The second and third terms are standard expressions for rotational kinetic energy and electrostatic energy. Thus the irreducible mass $M$ plays the role of a rest mass, with $E$ including contributions from rotational and electrical energy.
Returning to the general case, the above definitions satisfy the state-space formulas $$\kappa\cong8\pi\frac{\partial E}{\partial A}\cong{1\over{4M}}\frac{\partial
E}{\partial M}, \quad\Omega\cong\frac{\partial E}{\partial J},
\quad\Phi\cong\frac{\partial E}{\partial Q}. \label{pd}$$ There follows a dynamic version of the so-called first law of black-hole mechanics [@BCH]: $$L_\xi E\cong\frac{\kappa}{8\pi}L_\xi A+\Omega L_\xi J+\Phi L_\xi Q.$$ As desired, the state-space perturbations in the classical law for Killing horizons [@BCH], or the version for isolated horizons [@ABD; @ABL1; @Boo], have been replaced by the derivatives along the trapping horizon, thereby promoting it to a genuine dynamical law.
The rate of change of effective energy can also be written in energy-tensor form, $$L_\xi
E=\oint_S{*}\left((T_{\alpha\beta}+\Theta_{\alpha\beta})K^\alpha\tau^\beta-\Phi
j_\beta\tau^\beta\right)$$ where $$K=4M\kappa k-\Omega\psi$$ plays the role of the stationary Killing vector. Note that in the classical theory of stationary black holes, the state variables are usually taken to be $(E,J,Q)$, with the irreducible mass $M$ defined as a dependent variable. The dynamical theory reveals that $(M,J,Q)$ are the more basic variables, since they each satisfy a simple conservation law. Then the effective energy $E$ is defined as a dependent variable and therefore satisfies the above conservation law. This reflects a shift in emphasis from the classical to the dynamical theory: the so-called first law is a dependent result, obtained from more fundamental conservation laws for energy, angular momentum and charge.
Null trapping horizons and zeroth law
=====================================
A zeroth law for trapping horizons follows from the above, if one defines local equilibrium by the absence of relevant fluxes: $$g(\tilde\jmath,\tau)\cong g(\bar\jmath,\tau)\cong g(j,\tau)\cong0.\label{0th}$$ Then $(M,J,Q)$ are constant on the horizon and so is $\kappa$. In fact, these conditions do hold on a null trapping horizon under the dominant energy condition, as shown below. This treatment also turns out to be compatible with the definition of weakly isolated horizon [@AFK; @ABL1; @Boo; @ABL2; @DKSS; @GJ1].
Consider a null trapping horizon, assumed henceforth in this section to be given by $\theta_+\cong0$. The null focussing equation yields $T_{++}+||\sigma_+||^2/32\pi\cong0$, so the null energy condition, which implies $T_{++}\ge0$, yields [@bhd] $$T_{++}\cong0,\quad\sigma_+\cong0.\label{ed}$$ Thus the degenerate metric of the horizon is preserved along the generating vector. The dominant energy condition, which implies that the energy-momentum $P_\alpha=-T_{\alpha\beta}l_+^\beta$ is causal, further yields [@AFK] $$T_{+a}\cong0\label{amd}$$ since $P=-T_{++}dx^+-T_{+-}dx^--T_{+a}dx^a$ would otherwise be spatial.
On a null hypersurface $H$, one can take the null coordinate to be the generating coordinate, $x^+\cong x$, meaning that the shift vector vanishes, $s_+\cong0$, so that $\xi\cong\tau\cong l_+$. Since $k\cong-e^fL_-Rl_+$ (\[time\]), one finds $$g(\tilde\jmath,\tau)\cong e^fL_-R(T_{++}+\Theta_{++}),\quad
g(\bar\jmath,\tau)\cong\psi^a(T_{a+}+\Theta_{a+}).$$ These fluxes vanish by the above results (\[ed\])–(\[amd\]) and the expressions (\[Theta0\]), (\[Theta2\]) for components of $\Theta$. The other flux in (\[0th\]) vanishes due to the Maxwell equations [@AFK].
For a null trapping horizon, the dual-null foliation is not unique, so the question arises whether there is a natural way to fix it. An affirmative answer is given by noting that the above results also imply, via the twisting equations (\[twisting\]), $$L_+(\omega-{\textstyle\frac12}Df)\cong0.$$ This restriction on the dual-null geometry is suggestive of a proto-conservation law for angular momentum. Now the only normal fundamental form intrinsic to a null hypersurface is $\zeta_{-+}=\frac12Df+\omega$ of Appendix B, which was therefore used by Ashtekar et al. [@AFK] to define angular momentum for an isolated horizon. However, it is the other null normal fundamental form $\zeta_{+-}=\frac12Df-\omega$, depending on the dual-null foliation, which is preserved as above. Given that the general definition (\[am\]) of angular momentum involves $\omega$, it seems best to fix the unwanted freedom by $$Df\cong0.\label{fix}$$ Recalling the definition (\[ff\]) or (\[ls\]) of $f$, this condition fixes the normalization of the extrinsic null normal $l_-$ with respect to the intrinsic null normal $l_+$, which is always possible on a null hypersurface $H$. In fact, it is common simply to fix $f\cong0$. A similar normalization is also used in the context of null infinity.
Then the definition (\[am\]) of angular momentum becomes unambiguous on a null trapping horizon, coincides with the definition for isolated horizons [@ABD; @ABL1; @Boo], and is preserved along the horizon, assuming only that $\psi$ is a coordinate vector field (\[lie\]): $$L_\xi J\cong0.$$ Since the area law [@bhd] shows that $A$ is increasing unless $H$ is null everywhere on a given $S$, this answers, in the negative, a simply stated physical question: whether a black hole can change its angular momentum without increasing its area.
The above reasoning has largely recovered the notion of a weakly isolated horizon introduced by Ashtekar et al. [@AFK], except that the scaling freedom in $l_+$ has not been fixed. In more detail, Ashtekar et al.[@AFK] introduced a 1-form which will here be denoted by $\varpi$, defined by $$\hat\nabla_\alpha l_+^\beta=\varpi_\alpha l_+^\beta$$ where $\hat\nabla$ is the covariant derivative operator of $H$. The transverse and normal components are found as $$\bot\varpi=-\omega-{\textstyle\frac12}Df,\quad l_+^\alpha\varpi_\alpha=-\nu_+.$$ Since $\varpi$ is an invariant of $H$ and $l_+$, Ashtekar et al. [@AFK] demanded $$L_+\varpi\cong0$$ in order to define a weakly isolated horizon. The transverse part agrees with the above results, which also imply $D\nu_+\cong0$, while the normal part further fixes the inaffinity $\nu_+$ to be constant on $H$. This fixes the scaling of $l_+$ up to a constant multiple. Ashtekar et al. [@AFK] defined the surface gravity to be $$\hat\kappa\cong-\nu_+$$ which recovers the standard surface gravity of a Killing horizon if $l_+$ is the null Killing vector [@Wal]. Then the constancy of $\hat\kappa$ can also be interpreted as a zeroth law. This still leaves non-zero $\hat\kappa$ ambiguous up to a constant multiple, not necessarily agreeing with the definition (\[sg\]), which therefore fixes that freedom.
The above considerations appear to have a closed a gap in the general paradigm, concerning how a growing black hole ceases to grow. It seems that the generically spatial trapping horizon simply becomes null. It is difficult to find a practical formalism describing all cases without some degeneracy arising in the null case, but the dual-null formalism appears to be adequate; one fixes the additional freedom in the null case by (\[fix\]). In particular, no additional conditions need be imposed on the horizon itself, as compared with the variety of definitions of isolated horizons [@ABF1; @ABF2; @AFK; @ABD; @ABL1; @Boo; @ABL2; @DKSS; @GJ1]. Numerical evidence that such horizons exist in practice has been given by Dreyer et al. [@DKSS], who looked for and found approximately null trapping horizons, under the name non-expanding horizons.
Research supported by the National Natural Science Foundation of China under grants 10375081 and 10473007 and by Shanghai Normal University under grant PL609. Thanks to Abhay Ashtekar, Ivan Booth, Eric Gourgoulhon and Badri Krishnan for discussions.
Twist and weak fields
=====================
The twist may be calculated by first finding the shift vectors $s_\pm$, due to the form [@dne] $$\omega={\textstyle\frac12}e^fh\left([\partial_+,s_-]-[\partial_-,s_+]+[s_-,s_+]\right).$$ If it is more convenient to use an orthonormal basis $\{l_0,l_1\}$ of the normal space, $$\bot l_0=\bot l_1=0=g(l_0,l_1),\quad g(l_0,l_0)=-1=-g(l_1,l_1)\label{ob}$$ then $$\omega={\textstyle\frac12}h\left([l_0,l_1]\right)$$ follows by linear combinations from (\[am\]), or directly from the Komar integral (\[komar\]). If the basis is adapted to a coordinate basis via $$l_A=\partial_A-s_A,\quad s_A=\bot\partial_A$$ then $$\omega={\textstyle\frac12}h\left([\partial_1,s_0]-[\partial_0,s_1]+[s_0,s_1]\right).$$ In either case, the first step is to find the shift vectors.
The weak-field metric [@MTW], in standard spherical polar coordinates $(t,r,\vartheta,\varphi)$ adapted to the axis of rotation, is $$\begin{aligned}
g&\sim&-\left(1-\frac{2M}r\right)dt^2-\frac{4J}r\sin^2\vartheta
dtd\varphi\nonumber\\
&&+\left(1+\frac{2M}r\right)\left(dr^2+r^2(d\vartheta^2+\sin^2\vartheta
d\varphi^2)\right)\end{aligned}$$ where, in this appendix only, $M$ and $J$ denote the mass and angular momentum as defined in this approximation, obtained by linearizing the metric and neglecting higher powers of $1/r$. The inverse metric is $$\begin{aligned}
g^{-1}&\sim&-\left(1+\frac{2M}r\right)\partial_t^2-\frac{4J}{r^3}\partial_t\partial_\varphi\nonumber\\
&&+\left(1-\frac{2M}r\right)\left(\partial_r^2+\frac1{r^2}
\left(\partial_\vartheta^2+\frac{\partial_\varphi^2}{\sin^2\vartheta}\right)\right).\end{aligned}$$ Taking the transverse surfaces $S$ as those of constant $(t,r)$, one can read off the nonzero component of the shift 1-forms $s_{Ab}$ as $$s_{t\varphi}=g_{t\varphi}\sim-\frac{2J}r\sin^2\vartheta,\quad
s_t{}^\varphi\sim-\frac{2J}{r^3}.$$ Then the non-zero component of the twist is given by $$\omega^\varphi\sim{\textstyle\frac12}\partial_rs_t{}^\varphi\sim\frac{3J}{r^4},
\quad\omega_\varphi\sim\frac{3J}{r^2}\sin^2\vartheta.$$ The area form is $${*}1\sim r^2\sin\vartheta d\vartheta\wedge d\varphi$$ so that $${*}\omega_\varphi\sim3J\sin^3\vartheta d\vartheta\wedge d\varphi.$$ Standard integrals yield $$\oint_S{*}\omega_\varphi\sim8\pi J.$$ Since $\omega_a\psi^a=\omega_\varphi$ if $\psi=\partial/\partial\varphi$, this agrees with the general definition (\[am\]) of angular momentum.
A directly measurable quantity due to rotational frame-dragging is the precessional angular velocity [@MTW] $$\vec\Omega_{LT}\sim\frac{J}{r^3}\left(3(\hat z\cdot\hat r)\hat r-\hat z\right)$$ of a gyroscope due to the Lense-Thirring effect, where $\hat r$ is a unit vector in the direction of the gyroscope and $\hat z$ is a unit vector along the axis of rotation. Results of measurements of the effect due to the Earth by Gravity Probe B are expected soon. If the twist $\omega$ is formally converted to an angular velocity $\vec\Omega$ by $$\omega\sim\vec\Omega\times\hat r$$ then $$\vec\Omega\sim\left|\frac\omega{\sin\vartheta}\right|\hat z
\sim\frac{3J}{r^3}\hat z$$ does have the direction and relativistic dimensions of angular velocity. Then $$\vec\Omega_{LT}\sim(\vec\Omega\cdot\hat r)\hat r-{\textstyle\frac13}\vec\Omega.$$ Curiously, this is a linear transformation of $\vec\Omega$, by the same traceless tensor used in defining quadrupole moments [@MTW]. The puckish role of the factor of 3 in the above calculations is also noteworthy. In any case, it confirms that the twist does indeed encode the twisting around of space-time due to a rotating mass, in a directly measurable way.
For completeness, the agreement of the weak-field mass with the Hawking mass can also be checked, as follows. One needs to keep track of an extra power of $1/r$ in the area form $${*}1\sim\left(1+\frac{2M}r\right)r^2\sin\vartheta d\vartheta\wedge d\varphi.$$ Then the expansion 1-form $\theta_A$ has non-zero component $$\theta_r\sim\partial_r\log\left(\left(1+\frac{2M}r\right)r^2\right)
\sim\frac2r\left(1-\frac Mr\right).$$ Then $$\gamma^{AB}\theta_A\theta_B\sim\gamma^{rr}\theta_r\theta_r
\sim\frac4{r^2}\left(1-\frac{4M}r\right)$$ and $$\begin{aligned}
\oint_S{*}\gamma^{AB}\theta_A\theta_B&\sim&
\oint_S4\left(1-\frac{2M}r\right)\sin\vartheta d\vartheta\wedge d\varphi\nonumber\\
&\sim&16\pi\left(1-\frac{2M}r\right).\end{aligned}$$ Since $R\sim r$, this agrees with the Hawking mass (\[hm\]).
Normal fundamental forms
========================
Various definitions of angular momentum [@AK1; @ABD; @BY] are similar to (\[am\]), with $\omega$ replaced by a 1-form which is, implicitly or explicitly, a normal fundamental form. To clarify the situation, normal fundamental forms are reviewed below, referring to previous treatments [@dne; @Gou].
Writing the twist (\[tw\]) explicitly in components, $$2\omega_\alpha=e^fh_{\alpha\beta}(l_-^\gamma\nabla_\gamma
l_+^\beta-l_+^\gamma\nabla_\gamma l_-^\beta).$$ If $l_\pm$ are adapted to a coordinate basis via (\[nn\]), then commutativity $\nabla_{[\beta}\nabla_{\gamma]}x^\pm=0$ allows it to be written as $$2\omega_\alpha=e^fh_\alpha^\beta(l_-^\gamma\nabla_\beta
l_{+\gamma}-l_+^\gamma\nabla_\beta l_{-\gamma}).$$ This is the difference of two normal fundamental forms $\zeta_{\mp\pm}$ with components $$\zeta_{\mp\pm\alpha}=e^fl_\mp^\gamma h_\alpha^\beta\nabla_\beta l_{\pm\gamma}.$$ They are the independent normal fundamental forms, since the corresponding $\zeta_{\pm\pm}$ vanish. Their sum is $Df$, since the normalization in (\[ls\]) yields $$D_\alpha f=e^fh_\alpha^\beta(l_-^\gamma\nabla_\beta
l_{+\gamma}+l_+^\gamma\nabla_\beta l_{-\gamma}).$$ Then $$\zeta_{\mp\pm}={\textstyle\frac12}Df\pm\omega$$ and the normal fundamental forms are thereby encoded in $\omega$ and $Df$.
For an orthonormal basis (\[ob\]), there is just one independent normal fundamental form $\hat\zeta_{01}=-\hat\zeta_{10}$, given by $$\hat\zeta_{01\alpha}=l_0^\gamma h_\alpha^\beta\nabla_\beta l_{1\gamma},
\quad\hat\zeta_{10\alpha}=l_1^\gamma h_\alpha^\beta\nabla_\beta l_{0\gamma}$$ with corresponding $\hat\zeta_{00}$, $\hat\zeta_{11}$ vanishing. Under a boost transformation $$l_0\mapsto l_0\cosh\rho+l_1\sinh\rho,\quad l_1\mapsto l_0\sinh\rho+l_1\cosh\rho$$ which preserves the orthonormal conditions (\[ob\]), the normal fundamental form is generally not invariant: $$\hat\zeta_{01}\mapsto\hat\zeta_{10}-D\rho,\quad\hat\zeta_{10}\mapsto\hat\zeta_{10}+D\rho.$$ However, if the orthonormal basis is adapted to the dual-null basis by, e.g.$\sqrt2l_\pm=l_0+l_1$, then $$\hat\zeta_{01}=-\hat\zeta_{10}=\omega.$$ For spatial $H$, the same result is obtained by adapting the orthonormal basis by choosing $$l_0=\tau/\sqrt{g(\xi,\xi)},\quad l_1=\xi/\sqrt{g(\xi,\xi)}.\label{aon}$$ The missing information in $Df$ can be recovered by instead defining $$\begin{aligned}
\zeta_{01\alpha}&=&\frac{\tau^\gamma
h_\alpha^\beta\nabla_\beta\xi_\gamma}{g(\xi,\xi)},
\quad\zeta_{10\alpha}=\frac{\xi^\gamma
h_\alpha^\beta\nabla_\beta\tau_\gamma}{g(\xi,\xi)},\nonumber\\
\zeta_{00\alpha}&=&\frac{\tau^\gamma
h_\alpha^\beta\nabla_\beta\tau_\gamma}{g(\xi,\xi)},
\quad\zeta_{11\alpha}=\frac{\xi^\gamma
h_\alpha^\beta\nabla_\beta\xi_\gamma}{g(\xi,\xi)}.\end{aligned}$$ Then $$\zeta_{01}=-\zeta_{10}=\omega,\quad\zeta_{00}=-\zeta_{11}={\textstyle\frac12}Df.$$ These 1-forms nevertheless become degenerate when $\xi$ becomes null.
As shown explicitly by Gourgoulhon [@Gou], the 1-form used by Brown & York [@BY] to define angular momentum is, in this notation, $-\hat\zeta_{10}$. Therefore it coincides with $\omega$ if the orthonormal basis is adapted to a trapping horizon, but generally not if it is adapted to a foliation of spatial hypersurfaces intersecting the trapping horizon in the marginal surfaces. The 1-form used by Ashtekar & Krishnan [@AK1] to define angular momentum for dynamical horizons is also $-\hat\zeta_{10}$, this time explicitly adapted to the horizon by (\[aon\]), therefore coinciding with $\omega$ in this case. The 1-form used by Ashtekar et al.[@ABD; @ABL1; @Boo] to define angular momentum for isolated horizons is $\zeta_{-+}$, which generally does not coincide with $\omega$. However, it does coincide if the gauge freedom is fixed by (\[fix\]).
Kerr example
============
Consider a Kerr space-time in Boyer-Lindquist coordinates $(t,r,\vartheta,\varphi)$, with $S$ given by constant $(t,r)$ and $\xi=\partial/\partial r$. Then $${*}1=\sqrt{(r^2+a^2)^2-\Delta a^2\sin^2\vartheta}\sin\vartheta
d\vartheta\wedge d\varphi$$ where $\Delta=r^2-2mr+a^2$, $$\theta_\xi=\frac{2r(r^2+a^2)-(r-m)a^2\sin^2\vartheta}{(r^2+a^2)^2-\Delta
a^2\sin^2\vartheta}$$ and $$D\theta_\xi=\frac{2a^2(r^2+a^2)(r^3-3mr^2+a^2r+ma^2)\sin\vartheta\cos\vartheta}{((r^2+a^2)^2-\Delta
a^2\sin^2\vartheta)^2}d\vartheta.$$ If $a\not=0$, $D\theta_\xi$ is non-zero except at the poles, equator and isolated values of $r$, so the construction yields a unique continuous $\psi$, $\psi=\partial/\partial\varphi$.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this paper we present some new limit theorems for power variations of stationary increment Lévy driven moving average processes. Recently, such asymptotic results have been investigated in \[Ann. Probab. 45(6B) (2017), 4477–4528, Festschrift for Bernt [Ø]{}ksendal, Stochastics 81(1) (2017), 360–383\] under the assumption that the kernel function potentially exhibits a singular behaviour at $0$. The aim of this work is to demonstrate how some of the results change when the kernel function has multiple singularity points. Our paper is also related to the article \[Stoch. Process. Appl. 125(2) (2014), 653–677\] that studied the same mathematical question for the class of Brownian semi-stationary models.'
address: 'Department of Mathematics, , Aarhus, '
author:
-
-
title: A limit theorem for a class of stationary increments Lévy moving average process with multiple singularities
---
./style/arxiv-vmsta.cfg
Introduction {#sec1}
============
In recent years limit theorems and statistical inference for high frequency observations of stochastic processes have received a great deal of attention. The most prominent class of high frequency statistics are power variations that have been proved to be of immense importance for the analysis of the fine structure of an underlying stochastic process. The asymptotic theory for power variations and related statistics has been intensively studied in the setting of Itô semimartingales, fractional Brownian motion and Brownian semi-stationary processes, to name just a few; see for example [@BCP11; @BCP13; @BGJPS05; @C01; @JP12] among many others.
In the recent work [@BLP17; @BP17] power variations of stationary increments Lévy moving average processes have been investigated in details. These are continuous-time stochastic processes $(X_t)_{t \geq
0}$, defined on a probability space $(\varOmega, \mathcal{F}, \mathbb
{P})$, that are given by $$\label{Xmodel} X_t = \int_{-\infty}^{t}
\bigl(g(t - s) - g_0(-s) \bigr) \, {\textup{d}}L_s,$$ where $L = (L_t)_{t \in{\mathbb{R}}}$ is a symmetric Lévy process on ${\mathbb{R}}$ with $L_0 = 0$ and without Gaussian component. Moreover, $g, g_0 : {\mathbb{R}}\to{\mathbb{R}}$ are deterministic functions vanishing on $(-\infty, 0)$. The most prominent subclasses include Lévy moving average processes, which correspond to the setting $g_0 = 0$, and the linear fractional stable motion, which is obtained by taking $g(s) = g_0(s) = s^\alpha_+$ and $L$ being a symmetric $\beta$-stable Lévy process with $\beta\in(0,
2)$. The latter is a self-similar process with index $H=\alpha+1/\beta
$; see [@ST94].
We introduce the $k$th order increments $\Delta_{i, k}^{n} X$ of $X$, $k \in{\mathbb{N}}$, that are defined by $$\label{filter} \Delta_{i, k}^{n} X := \sum
_{j = 0}^k (-1)^j \binom{k}{j}
X_{(i - j) /
n}, \quad i \geq k.$$ For example, we have that $\Delta_{i, 1}^n X = \smash{X_{\frac{i}{n}}}
- \smash{X_{\frac{i - 1}{n}}}$ and $\Delta_{i, 2}^n X = \smash{X_{\frac
{i}{n}}} - 2 \smash{X_{\frac{i - 1}{n}}} + \smash{X_{\frac{i -
2}{n}}}$. The main statistic of interest is the power variation computed on the basis of $k$th order increments: $$\label{vn} V(X, p; k)_n := \sum_{i = k}^n
|\Delta_{i, k}^{n} X|^p, \quad p > 0.$$ A variety of asymptotic results has been shown for the statistic $V(p;k)_n$ in [@BLP17; @BP17]. The mode of convergence and possible limits heavily depend on the interplay between the power $p$, the form of the kernel function $g$ and the Blumenthal–Getoor index of $L$. We recall that the Blumenthal–Getoor index is defined via $$\label{BG} \beta:= \inf \Biggl\{r \geq0 : \int_{-1}^1
|x|^r \, \nu({\textup{d}}x) < \infty \Biggr\} \in[0, 2],$$ where $\nu$ denotes the Lévy measure of $L$. It is well known that $\sum_{s \in[0,1]} |\Delta L_s|^p$ is finite when $p > \beta$, while it is infinite for $p < \beta$. Here $\Delta L_s = L_s - L_{s-}$ where $L_{s-} = \lim_{u \uparrow s, u < s} L_u$. To formulate the results of [@BLP17; @BP17], we introduce the following set of assumptions on $g$, $g_0$ and $\nu$: **Assumption (A):** The function $g : {\mathbb{R}}\to{\mathbb{R}}$ satisfies the condition $$\label{kshs} g(t) \sim c_0 t^\alpha\quad\text{as } t
\downarrow0 \quad\text{for some } \alpha> 0\text{ and } c_0 \neq0,$$ where $g(t) \sim f(t)$ as $t\downarrow0$ means that $\lim_{t
\downarrow0} g(t) / f(t) = 1$. For some $w\in(0, 2]$, $\limsup_{t \to
\infty} \nu(x : |x| \geq t) t^{w} < \infty$ and $g - g_0$ is a bounded function in $L^{w}({\mathbb{R}}_+)$. Furthermore, $g$ is $k$-times continuous differentiable on $(0, \infty)$ and there exists a $\delta> 0$ such that $|g^{(k)}(t)| \leq K t^{\alpha- k}$ for all $t \in(0, \delta)$, $|g^{(k)}|$ is decreasing on $(\delta, \infty)$ and $g^{(j)} \in
L^w((\delta, \infty))$ for $j \in\{1, k\}$. **Assumption (A-log):** In addition to (A) suppose that $$\int_\delta^\infty|g^{(k)}(s)|^w
\big| \log\bigl(|g^{(k)}(s)|\bigr)\big| \, {\textup{d}}s < \infty.$$ Intuitively speaking, Assumption (A) says that $g^{(k)}$ may have a singularity at $0$ when $\alpha$ is small, but it is smooth outside of $0$. The theorem below has been proved in [@BLP17; @BP17]. We recall that a sequence of ${\mathbb{R}}^d$-valued random variables $(Y_n)_{n\geq1}$ is said to converge stably in law to a random variable $Y$, defined on an extension of the original probability space $(\varOmega, \mathcal{F}, \mathbb{P})$, whenever the joint convergence in distribution $(Y_n, Z) {\xrightarrow{\smash{d}}}(Y,Z)$ holds for any $\mathcal
F$-measurable $Z$; in this case we use the notation $Y_n {\xrightarrow{\smash{\mathcal{L}-s}}}Y$. We refer to [@AE78; @R63] for a detailed exposition of stable convergence.
\[th1\] Suppose that Assumption (A) holds, the Blumenthal–Getoor index satisfies $\beta< 2$ and $p > \beta$. If $w = 1$ assume that (A-log) holds. Then we obtain the following cases:
1. \[it:th1:1\] When $\alpha<k-1/p$ then we have the stable convergence $$\label{part1} \begin{aligned} n^{\alpha p} V(X, p;
k)_n &{\xrightarrow{\smash{\mathcal{L}-s}}}|c_0|^p \sum
_{m : T_m \in[0, 1]} |\Delta L_{T_m}|^p
V_m
\\
\quad\text{with} \quad V_m &= \sum_{l = 0}^{\infty}
|h_k(l + U_m)|^p, \end{aligned} $$ where $(T_m)_{m\geq1}$ denote the jump times of $L$, $(U_m)_{m \geq
1}$ is a sequence of independent identically (i.i.) $\mathcal{U}(0,1)$-distributed variables independent of $L$, and the function $h_k$ is defined by $$\label{def-h} h_k(x) = \sum_{j = 0}^k
(-1)^j \binom{k}{j} (x - j)_{+}^{\alpha} \quad
\text{with} \quad y_+ = \max\{y, 0\}.$$
2. \[it:th1:2\] When $\alpha=k-1/p$ and additionally $1/p+1/w>1$, then we have $$\label{part2} \begin{aligned} \frac{n^{\alpha p}}{\log(n)} V(X,p;k)_n
&{\stackrel{\mathbb{P}}{\longrightarrow}}| c_0 q_{k, \alpha} |^p \sum
_{s \in(0, 1]} |\Delta L_s|^p
\\
\quad\text{with} \quad q_{k, \alpha} &:= \prod_{j = 0}^{k - 1}
(\alpha- j). \end{aligned} $$
We remark that the first order asymptotic theory of [@BLP17 Theorem 1.1] includes two more regimes: an ergodic type limit theorem in the setting $p < \beta$, $\alpha< k - 1 / \beta$ and convergence in probability to a random integral in the setting $p
\geq1$, $\alpha> k - 1 / \max\{p, \beta\}$. However, in this paper we concentrate ourselves on results of Theorem \[th1\], which are quite non-standard in the literature. More specifically, our aim is to extend the theory of Theorem \[th1\] to kernels $g$ that exhibit multiple singularities. We call a point $x \in{\mathbb{R}}_+$ a singularity point when the $k$th derivative $g^{(k)}$ of $g$ explodes at $x$. Note that under Assumption (A) and condition $\alpha\leq k - 1 / p$ the function $g$ has only one singularity point at $x=0$. In practical applications a singularity point $x \in{\mathbb{R}}_+$ leads to a strong feedback effect stemming from the past jumps around the time $t-x$. Such effects has been discussed in the context of turbulence modelling in [@GP14].
We will show that the limits in Theorem \[th1\]\[it:th1:1\] and \[it:th1:2\] will be affected by the presence of multiple singularity points. More precisely, we will see that the increments $\Delta
_{i,k}^{n} X$ can be heavily influenced by the jumps of $L$ that happened in the past, and the time delay is determined by the singularity points of $g$. The obtained result is similar in spirit to the work [@GP14] that studied quadratic variation of Brownian semi-stationary processes under multiple singularities of the kernel $g$. Furthermore, we will prove that in general the stable convergence in Theorem \[th1\]\[it:th1:1\] only holds along a subsequence.
The paper is structured as follows. Section \[sec2\] presents the main results of the article. Proofs are collected in Section \[sec3\].
Main results {#sec2}
============
We consider stationary increments Lévy moving average processes as defined at and recall that the driving motion $L$ is a pure jump Lévy process with Lévy measure $\nu$. Now, we introduce the condition on the kernel function $g$: **Assumption (B):** For some $w \in(0, 2]$, $\limsup_{t
\to\infty} \nu(x : |x| \geq t) t^w < \infty$ and $g - g_0$ is a bounded function in $L^w({\mathbb{R}}_+)$. Furthermore, there exist points $0 =
\theta_0 < \theta_1 < \cdots< \theta_l$ such that the following properties hold:
1. \[it:B:1\] $g(t) \sim c_0 t^{\alpha_0}$ as $t \downarrow0$ for some $\alpha_0 > 0$ and $c_0 \neq0$.
2. \[it:B:2\] $g(t) \sim c_z |t - \theta_z|^{\alpha_z}$ as $t \to
\theta_z$ for some $\alpha_z > 0$ and $c_z \neq0$, and for all $z = 1,
\ldots, l$.
3. \[it:B:3\] $g \in C^k({\mathbb{R}}_+ \setminus\{\theta_0, \ldots, \theta
_l\})$.
4. \[it:B:4\] There exist $\delta$, $K > 0$ such that $|g^{(k)}(t)| \leq K |t - \theta_z|^{\alpha_z - k}$ for all $t \in
(\theta_z - \delta, \theta_z + \delta) \setminus\{\theta_z\}$, for any $z = 0, \ldots, l$. Furthermore, there exists a $\delta' > 0$ such that $|g^{(k)}|$ is decreasing on $(\theta_l + \delta', \infty)$ and $g^{(j)} \in L^w ((\theta_l + \delta', \infty))$ for $j \in\{1, k\}$.
Let us give some remarks on Assumption (B). First of all, conditions (B)\[it:B:1\] and (B)\[it:B:2\], which are direct extensions of , mean that for small powers $\alpha_z > 0$ the points $\theta_z$ are singularities of $g$ in the sense that $g^{(k)}(\theta_z)$ does not exist. On the other hand, condition (B)\[it:B:3\] states that there exist no further singularities. The parameter $w$ is by no means unique. It simultaneously describes the tail behaviours of the Lévy measure $\nu$ and the integrability of the function $|g^{(k)}|$, which exhibit a trade-off. When $L$ is $\beta
$-stable we always take $w=\beta$. Furthermore, Assumption (B) guarantees the existence of $X_t$ for all $t \geq0$. Indeed, it follows from [@RR89 Theorem 7] that the process $X$ is well-defined if and only if for all $t \geq0$, $$\label{sdkfhsd;kf} \int_{-t}^\infty\int
_{{\mathbb{R}}} \bigl(| f_t(s)x |^2 \wedge1
\bigr) \, \nu ({\textup{d}}x) \, {\textup{d}}s < \infty,$$ where $f_t(s) = g(t + s) - g_0(s)$. By adding and subtracting $g$ to $f_t$ it follows by Assumption (B) and the mean value theorem that $f_t$ is a bounded function in $L^w({\mathbb{R}}_+)$. For all $\epsilon> 0$, Assumption (B) implies that $$\int_{{\mathbb{R}}} \bigl(|y x|^2 \wedge1\bigr) \,\nu(
{\textup{d}}x) \leq K \bigl( {\mathbf{1}}_{\{|y| \leq
1\}} |y|^w + {\mathbf{1}}_{\{|y| > 1\}}|y|^{\beta+ \epsilon}
\bigr),$$ which shows since $f_t$ is a bounded function in $L^w({\mathbb{R}}_+)$.
\[rem1\] Recall the following well-known results about the power variation of a pure jump Lévy process $L$: $$V(L,p;k)_n {\stackrel{\mathbb{P}}{\longrightarrow}}\sum_{s \in[0,1]} |\Delta
L_s|^p < \infty$$ for any $k \geq1$ and any $p > \beta$. Let us now consider a simple stationary increments Lévy moving average process $X$ with $g_0 = 0$ and $g(x) =
{\mathbf{1}}_{[0, 1]}(x)$. In this case we may call the points $\theta_0 = 0$ and $\theta_1 = 1$ the singularities of $g$, although they do not precisely correspond to conditions (B)\[it:B:1\] and (B)\[it:B:2\], and we observe that $X_t = L_t - L_{t - 1}$. Hence, we obtain the convergence in probability $$V(X,p;k)_n {\stackrel{\mathbb{P}}{\longrightarrow}}\sum_{s \in[0, 1]} |\Delta
L_s|^p + \sum_{s \in[-1,
0]} |\Delta
L_s|^p$$ for any $k \geq1$ and any $p > \beta$. This result demonstrates that even in the simplest setting multiple singularities lead to a different limit.
It turns out that only the minimal powers among $\{\alpha_0,
\ldots, \alpha_l\}$ determine the asymptotic behaviour of the statistic $V(X, p; k)_n$. Thus, we define $$\label{Aalpha} \alpha:= \min\{\alpha_0, \ldots,
\alpha_l\} \qquad\text{and} \qquad \mathcal{A} := \{z :
\alpha_z = \alpha\}.$$ Furthermore, we introduce the notation $h_{k, 0} := h_k$ and $$\label{hkz} h_{k, z}(x) = \sum_{j = 0}^k
(-1)^j \binom{k}{j} |x - j|^{\alpha_z} \quad\text{for } z =
1, \ldots, l.$$ In the main result below we consider a subsequence $(n_j)_{j \in{\mathbb{N}}}$ such that the following condition holds: $$\label{nj} \lim_{j \to\infty} \{n_j
\theta_z\} = \eta_z \in[0, 1] \quad\text {for all } z
\in\mathcal{A},$$ where $\{x\}$ denotes the fractional part of $x \in{\mathbb{R}}$. Obviously, such a subsequence always exists since $\{n \theta_z\}$ is a bounded sequence. Sometimes we will require a stronger condition, which is analogous to Assumption (A-log): **Assumption (B-log):** Condition (B) holds and we have that $$\int_{\theta_l+\delta'}^{\infty} |g^{(k)}
(t)|^{w} \big|\log\bigl(|g^{(k)} (t)|\bigr)\big| \, {\textup{d}}t < \infty.$$ The main result of the paper is the following theorem.
\[th2\] Suppose that Assumption (B) holds, the Blumenthal–Getoor index satisfies $\beta< 2$ and $p > \beta$. If $w = 1$ assume that (B-log) holds. Recall the notations and . Then we obtain the following cases:
1. \[it:th2:1\] When $\max_{0\leq z \leq l} \alpha_z <k-1/p$ and condition holds, then we have the stable convergence $$\begin{aligned}
\label{part1.2} &n_j^{\alpha p}V(X,p;k)_{n_j} {\xrightarrow{\smash{\mathcal{L}-s}}}\sum
_{z \in\mathcal{A}} |c_z|^p \sum
_{m : T_m \in[-\theta_z, 1 - \theta_z]} |\Delta L_{T_m}|^p
V_m^z
\\
&\text{with} \quad V_m^z = \sum
_{r\in{\mathbb{Z}}} \big|h_{k,z}\bigl(r+1-\{U_m+
\eta_z\} \bigr)\big|^p.
\nonumber\end{aligned}$$ as $j \to\infty$, where $(U_m)_{m \geq1}$ is a sequence of i.i. $\mathcal
{U}(0,1)$-distributed variables independent of $L$.
2. \[it:th2:2\] Let $\alpha= \alpha_0 = \cdots= \alpha_l = k -
1/p$. Assume that the functions $f_z : {\mathbb{R}}_+ \to{\mathbb{R}}$ defined by $f_z(x)
= g(x) / |x - \theta_z|^{\alpha}$ are in $ C^k((\theta_z - \delta,
\theta_ z + \delta))$ for all $\delta< \max_{1 \leq j \leq l}(\theta_j
- \theta_{j - 1})$. If $1/p+1/w>1$, then we have $$\label{part2.2} \frac{n^{\alpha p}}{\log(n)} V(X, p; k)_n {\stackrel{\mathbb{P}}{\longrightarrow}}|q_{k, \alpha}
|^p \sum_{z = 0}^{l}
|c_z|^p (1 + {\mathbf{1}}_{\{z \geq1\}} ) \sum
_{m : T_m \in[-\theta
_z, 1 - \theta_z]} |\Delta L_{T_m}|^p,$$ where the constant $q_{k,\alpha}$ has been introduced in Theorem \[th1\]\[it:th1:2\].
We remark that the stable convergence in Theorem \[th2\]\[it:th2:1\] only holds along the subsequence $(n_j)_{j \geq1}$, which is seen from the form of the limit in that depends on $(\eta_z)$. The original statistic $n^{\alpha p} V(X, p; k)_{n}$ is tight, but does not converge except when $\theta_z \in{\mathbb{N}}$ for all $z
\in\mathcal{A}$. On the other hand, in Theorem \[th2\]\[it:th2:2\] we do not require to consider a subsequence.
Notice that the interval $ [-\theta_z, 1 - \theta_z]$, which appears in Theorem \[th2\], is the set $[0, 1]$ shifted by $\theta_z$ to the left. Given the discussion of Remark \[rem1\], such a shift in the limit is not really surprising. We recall that a similar phenomenon has been discovered in [@GP14] in the context of Brownian semi-stationary processes. These are stochastic processes $(Y_t)_{t
\geq0}$ defined by $$Y_t = \int_{-\infty}^{t} g(t - s)
\sigma_s \, {\textup{d}}W_s,$$ where $W$ is a two-sided Brownian motion and $(\sigma_t)_{t \in{\mathbb{R}}}$ is a cádlág process. When the kernel function $g$ satisfies conditions (B)\[it:B:1\] and (B)\[it:B:2\] along with some further assumptions, which in particular ensure the existence of $Y_t$, the authors have shown the following convergence in probability (see [@GP14 Theorem 3.2]): $$\frac{1}{n \tau_n^2} V(Y, 2; k)_n {\stackrel{\mathbb{P}}{\longrightarrow}}\sum
_{z \in\mathcal{A}} \pi_z \int_{-\theta_z}^{1 - \theta_z}
\sigma_s^2 \, {\textup{d}}s$$ where $\tau_n^2 = {\mathbb{E}}[(\Delta_{k, k}^n G)^2]$ with $G_t = \int_{-\infty
}^{t} g(t - s) \, {\textup{d}}W_s$, and the probability weights $(\pi_z)_{z \in
\mathcal{A}}$ are given by $$\pi_z = \frac{c_z^2 \| h_{k, z}\|^2_{L^2({\mathbb{R}})}}{\sum_{z \in\mathcal{A}}
c_z^2 \| h_{k, z} \|^2_{L^2({\mathbb{R}})}}.$$ Hence, we observe the same shift phenomenon in the integration region as in Theorem \[th2\].
Proofs {#sec3}
======
Throughout this section all positive constants are denoted by $C$ although they may change from line to line. We will divide the proof of Theorem \[th2\] into several steps. First, we will show the statements and for a compound Poisson process. In the second step we will decompose the jump measure of $L$ into jumps that are bigger than $\epsilon$ and jumps that are smaller than $\epsilon$. The big jumps form a compound Poisson process and hence the claim follows from the first step. Finally, we prove negligibility of small jumps when $\epsilon\to0$.
We start with an important proposition.
\[prop1\] Let $T = (T_1, \ldots, T_d)$ be a stochastic vector with a density $v :
{\mathbb{R}}^d \to{\mathbb{R}}_+$. Suppose there exists an open convex set $A \subseteq{\mathbb{R}}^d$ such that $v$ is continuously differentiable on $A$ and vanishes outside of $A$. Then, under condition , it holds that $$\label{mainprop} \bigl(\{n_j T + n_j \theta_z
\}\bigr)_{z \in\mathcal{A}} {\xrightarrow{\smash{\mathcal{L}-s}}}\bigl(\{U + \eta_z\}
\bigr)_{z \in\mathcal{A}} \quad\text{as } j \to\infty,$$ where $\{x\}$ denotes the fractional parts of the vector $x \in{\mathbb{R}}^d$ and $x + a$, $a \in{\mathbb{R}}$, is componentwise addition. Here $U = (U_1,
\ldots, U_d)$ consists of i.i. $\mathcal{U}(0, 1)$-distributed random variables defined on an extension of the space $(\varOmega, \mathcal{F},
\mathbb{P})$ and being independent of $\mathcal{F}$.
We first show the stable convergence $$\label{stabconv} \{ nT \} {\xrightarrow{\smash{\mathcal{L}-s}}}U.$$ This statement has already been shown in [@BLP17 Lemma 4.1], but we demonstrate its proof for completeness. Let $f:{\mathbb{R}}^d \times{\mathbb{R}}^d\to{\mathbb{R}}$ be a $C^1$-function, which vanishes outside some closed ball in $A\times{\mathbb{R}}^d$. We claim that there exists a finite constant $K > 0$ such that for all $\rho> 0$ $$\label{eq-est-f-g} D_\rho:= \Bigg| \int_{{\mathbb{R}}^d} f\bigl(x, \{
x/\rho\}\bigr) v(x) \, {\textup{d}}x - \int_{{\mathbb{R}}^k} \biggl(\int
_{[0, 1]^d} f(x,u) \, {\textup{d}}u \biggr) v(x) \, {\textup{d}}x \Bigg| \leq K \rho.$$ By used for $\rho= 1/n$ we obtain that $$\label{sdkfh} {\mathbb{E}}\bigl[f\bigl(T, \{n T\}\bigr)\bigr] \longrightarrow{\mathbb{E}}\bigl[f(T,
U)\bigr]\quad\text{as } n \to \infty.$$ Moreover, due to [@AE78 Proposition 2(D”)], implies the stable convergence$\{n T\} {\xrightarrow{\smash{\mathcal{L}-s}}}U$ as $n\to\infty$. Thus, we need to prove the inequality . Define $\phi(x,u) := f(x,
u) v(x)$. Then it holds by substitution that $$\begin{aligned}
\int_{{\mathbb{R}}^d} f\bigl(x, \{ x/\rho\}\bigr) v(x) \, {\textup{d}}x &= \sum
_{j \in{\mathbb{Z}}^d} \int_{(0, 1]^d}
\rho^d \phi(\rho j + \rho u, u) \, {\textup{d}}u\end{aligned}$$ and $$\begin{aligned}
\int_{{\mathbb{R}}^d} \biggl(\int_{[0, 1]^d}
f(x,u) \, {\textup{d}}u \biggr) v(x) \, {\textup{d}}x &= \sum_{j \in{\mathbb{Z}}^d}
\int_{[0, 1]^d} \biggl(\int_{(\rho j, \rho(j +
1)]} \phi(x, u) \,
{\textup{d}}x \biggr) \, {\textup{d}}u.\end{aligned}$$ Hence, we conclude that $$\begin{aligned}
D_\rho&\leq\sum_{j \in{\mathbb{Z}}^d} \int
_{(0, 1]^d} \Bigg|\int_{(\rho j,
\rho(j + 1)]} \phi(x, u) \, {\textup{d}}x -
\rho^d \phi(\rho j + \rho u , u) \Bigg| \, {\textup{d}}u
\\
&\leq\sum_{j \in{\mathbb{Z}}^d} \int_{(0, 1]^d} \int
_{(\rho j,\rho(j + 1)]} \big|\phi(x, u)- \phi(\rho j + \rho u, u) \big| \, {\textup{d}}x \, {\textup{d}}u.\end{aligned}$$ Using that $A$ is convex and open, we deduce by the mean value theorem that there exists a positive constant $K$ and a compact set $B\subseteq
{\mathbb{R}}^d\times{\mathbb{R}}^d$ such that for all $j\in{\mathbb{Z}}^d$, $x \in(\rho j, \rho(j
+ 1)]$ and $u \in(0,1]^d$ we have $$\big|\phi(x, u) - \phi(\rho j + \rho u, u) \big| \leq K \rho{\mathbf{1}}_B(x, u).$$ Thus, $D_\rho\leq K \rho\int_{(0, 1]^d} \int_{{\mathbb{R}}^d} {\mathbf{1}}_{B} (x, u) \,
{\textup{d}}x \, {\textup{d}}u$, which shows .
Now, we are ready to prove the statement . By and condition we conclude that $$\bigl(\{n_j T\}, \{n_j \theta_z \}
\bigr)_{z \in\mathcal{A}} {\xrightarrow{\smash{\mathcal{L}-s}}}(U, \eta _z)_{z \in\mathcal{A}} \quad
\text{as } j \to\infty.$$ Next, consider the map $f : {\mathbb{R}}^d \times{\mathbb{R}}^{l'} \to{\mathbb{R}}^{d \times l'}$, where $l'$ denotes the cardinality of $\mathcal A$, given by $$f(x, y_1, \ldots, y_{l'}) = \bigl(\{x + y_1
\}, \ldots, \{x + y_{l'}\}\bigr).$$ This map is discontinuous exactly in those points $x, y_1, \ldots,
y_{l'}$ for which $x_j + y_i \in{\mathbb{Z}}$ for some $i \in\{1, \ldots, l'\}$ and some $j \in\{1, \ldots, d\}$. Note that the probability of the limiting variable $(U, \eta_z )_{z \in\mathcal{A}}$ lying in the latter set is $0$. Hence, it follows from the continuous mapping theorem for stable convergence that $$f\bigl(\{n_j T\}, \bigl(\{n_j \theta_z\}
\bigr)_{z \in\mathcal{A}}\bigr) {\xrightarrow{\smash{\mathcal{L}-s}}}f\bigl(U,(\eta_z)_{z \in\mathcal{A}}
\bigr) = \bigl(\{U + \eta_z\}\bigr)_{z \in
\mathcal{A}}$$ as $j \to\infty$. Since $x = \{x\} + \lfloor x \rfloor$ we have the identity $\{x + y\} = \{\{x\} + \{y\}\}$ and the left-hand side becomes $$f\bigl(\{n_j T\}, \bigl(\{n_j \theta_z\}
\bigr)_{z \in\mathcal{A}} \bigr) = \bigl(\{n_jT + n_j
\theta_z \}\bigr)_{z \in\mathcal{A}},$$ which concludes the proof of Proposition \[prop1\].
Now, we introduce the notation $$\label{gin} g_{i,n}(x) = \sum_{j = 0}^{k}
(-1)^j \binom{k}{j} g\bigl((i - j)/n - x\bigr),$$ and observe the identity $$\Delta_{i,k}^n X= \int_{{\mathbb{R}}}
g_{i, n}(s) \, {\textup{d}}L_s.$$ The next lemma presents some estimates for the function $g_{i, n}$. Its proof is a straightforward consequence of Assumption (B) and the Taylor expansion.
\[lem1\] Suppose that Assumption (B) holds and let $z=1,\ldots, l$. Then there exists an $N \in{\mathbb{N}}$ such that for all $n\geq N$ and $i \in\{k, \ldots
, n\}$ the following hold:
1. \[it:lem1:1\] $|g_{i,n}(x)| \leq C (|i/n - x - \theta
_z|^{\alpha_z} + n^{-\alpha_z} )$ for all $x\in[\frac{i-2k}{n} - \theta
_z, \frac{i + 2k}{n} - \theta_z]$.
2. \[it:lem1:2\] $|g_{i,n}(x)| \leq Cn^{-k}|(i-k)/n-x - \theta
_z|^{\alpha_z-k}$ for all $x \in(\frac{i}{n}-\delta- \theta_z,\tfrac
{i-k}{n} - \theta_z)$ if $\alpha_z - k < 0$.
3. \[it:lem1:3\] $|g_{i,n}(x)| \leq C n^{-k} |(i-k)/n - x - \theta
_z|^{\alpha_z - k}$ for all $x \in(\frac{i + k}{n} - \theta_z, \frac{i
- k}{n} + \delta- \theta_z)$ if $\alpha_z - k < 0$.
4. \[it:lem1:4\] $|h_{k,z}(x)| \leq|x - k|^{\alpha- k}$ for all $x\geq k+1$ and $|h_{k,z}(x)| \leq|x + k|^{\alpha- k}$ for all $x
\leq-k - 1$, if $\alpha_z - k < 0$.
5. \[it:lem1:5\] For each $\varepsilon>0$ it holds that $$\begin{aligned}
n^k|g_{i,n}(s)| {\mathbf{1}}_{(-\infty, \frac{i}{n} - \varepsilon- \theta_l]}(s) &\leq
C_\varepsilon \bigl({\mathbf{1}}_{[-\theta_l - \delta', 1 - \theta_l]}(s)
\\
&\quad+ {\mathbf{1}}_{(-\infty, -\theta_l - \delta')}(s) |g^{(k)}(-s)| \bigr).\end{aligned}$$
Furthermore, similar estimates hold for $z = 0$ with obvious adjustments that account for the fact that $g$ and $h_{k,0}$ are both vanishing on $(- \infty, 0)$.
Proof of Theorem \[th2\] in the compound Poisson case {#sec3.1}
-----------------------------------------------------
In this subsection we assume that $L$ is a compound Poisson process. Recall that $(T_m)_{m \geq1}$ denotes the jump times of $L$. Let $\varepsilon> 0$ and consider $n_j \in{\mathbb{N}}$ such that $\varepsilon n_j
> 4k$. Define the set $$\begin{aligned}
\varOmega_\varepsilon= \bigl\{&\omega\in\varOmega: \text{for all $m \in{\mathbb{N}}$
with $T_m(\omega) \in[-\theta_l, 1]$ it holds} \\
&|T_{m}(\omega) - T_i(\omega)| > 2\varepsilon,
T_m(\omega) + \theta_{z} - \theta_{z'} \notin
\bigl[T_i(\omega) - 2 \varepsilon, T_i(\omega) + 2
\varepsilon\bigr]
\\
&\forall i \neq m \ \forall z, z' \in\{0, \ldots, l\} \text{ and $
\Delta L_s(\omega) = 0$}
\\
&\text{for all $s \in[- \varepsilon- \theta_z, -
\theta_z + \varepsilon ] \cup[1 - \varepsilon- \theta_z,
1 -\theta_z + \varepsilon]$} \ \forall z\in\{0, \ldots, l\} \bigr\}.\end{aligned}$$ Roughly speaking, on the set $\varOmega_\varepsilon$ the jump times in $[-\theta_l, 1]$ are well separated, their increments are outside a small neighbourhood of $\theta_{z} - \theta_{z'}$, and there are no jumps around the fixed points $-\theta_z$ and $1 - \theta_z$. In particular, it obviously holds that $\mathbb{P}(\varOmega_\varepsilon) \to
1$ as $\varepsilon\to0$.
Throughout the proof we assume without loss of generality that $0 \in
\mathcal A$. Now, we introduce a decomposition, which is central for the proof. Recalling the definition of $g_{i, n}$ at , we observe the identity $$\label{maindec} \Delta_{i, k}^n X = \sum
_{z \in\mathcal{A}} M_{i, n, \varepsilon, z} + \sum_{z \in\mathcal{A}^c}
M_{i, n, \varepsilon, z} + R_{i, n,
\varepsilon},$$ where for $z = 1, \ldots, l$ $$\begin{aligned}
M_{i, n, \varepsilon, 0} &= \int_{\frac{i}{n} - \varepsilon}^{\frac{i}{n}}g_{i, n}(s) \, {\textup{d}}L_s,
\qquad
M_{i, n, \varepsilon, z} = \int_{\frac{i}{n} - \theta_z - \varepsilon}^{\frac{i}{n} - \theta_z + \frac{\lfloor n\varepsilon\rfloor}{n}} g_{i,n}(s) \, {\textup{d}}L_s\\
R_{i ,n, \varepsilon} &= \int_{-\infty}^{\frac{i}{n} - \theta_l - \varepsilon}
g_{i, n}(s) \, {\textup{d}}L_s + \sum_{z = 1}^{l}\int_{\frac
{i}{n} - \theta_z + \frac{\lfloor n\varepsilon\rfloor}{n}}^{\frac
{i}{n} - \theta_{z - 1} - \varepsilon} g_{i, n}(s) \, {\textup{d}}L_s.\end{aligned}$$ It turns out that the first term $\sum_{z \in\mathcal{A}} M_{i, n,
\varepsilon, z}$ is dominating, while the other two are negligible.
### Main terms in Theorem \[th2\]\[it:th2:1\] {#sec3.1.2}
In this subsection we consider the dominating term in the decomposition . We want to prove that, on $\varOmega_\varepsilon$, then for $j \to\infty$ $$\label{toshow} n_j^{\alpha p} \sum_{i = k}^{n_j}
\Bigg| \sum_{z \in\mathcal{A}} M_{i, n_j, \varepsilon, z} \Bigg|^p {\xrightarrow{\smash{\mathcal{L}-s}}}\sum_{z \in\mathcal A} |c_z|^p \sum
_{m : T_m \in[-\theta_z, 1 - \theta_z]} |\Delta L_{T_m}|^p
V_m^z,$$ where the limit has been introduced in . Let us fix an index $z \in\mathcal{A}$. Then, on $\varOmega_\varepsilon$, for each jump time $T_m \in(-\theta_z, 1 - \theta_z]$ there exists a unique random variable $i_{m, z} \in{\mathbb{N}}$ such that $$T_m \in \biggr(\frac{i_{m, z} - 1}{n} - \theta_z,
\frac{i_{m, z}}{n} - \theta_z \biggl].$$ We also observe the following implication, which follows directly from the definition of the set $\varOmega_\varepsilon$: $$\text{On } \varOmega_\varepsilon, \text{ if } M_{i,n,\varepsilon,z} \neq 0
\text{ for some } z \in\mathcal{A} \implies M_{i,n,\varepsilon,z'} = 0 \text{ for any }
z'\neq z \text{ in } \mathcal{A}.$$ Indeed, this is the consequence of the definition of the term $M_{i, n,
\varepsilon, z}$ and the statement $$T_m(\omega) + \theta_z - \theta_{z'} \notin
\bigl[T_{m'}(\omega) - 2\varepsilon, T_{m'}(\omega) + 2
\varepsilon\bigr] \ \forall m'\neq m \ \forall z, z'
\in\{0, \ldots, l\},$$ which holds on $\varOmega_\varepsilon$. Hence, we conclude that $$n^{\alpha p}\sum_{i = k}^{n} \Bigg|\sum
_{z \in\mathcal{A}} M_{i, n,
\varepsilon, z} \Bigg|^p =
n^{\alpha p} \sum_{z \in\mathcal{A}} \sum
_{i = k}^{n} | M_{i, n, \varepsilon, z} |^p$$ on $\varOmega_\varepsilon$, and we obtain the representation $$\begin{aligned}
&n^{\alpha p}\sum_{i = k}^{n}
|M_{i, n, \varepsilon, z}|^p = V_{n,
\varepsilon, z} \quad\text{with}
\nonumber
\\
\label{Vnz} & V_{n, \varepsilon, z} = n^{\alpha p} \sum
_{m : T_m \in
(-\theta_z, 1 - \theta_z]} |\Delta L_{T_m}|^p \sum
_{u = -\lfloor
n\varepsilon\rfloor}^{\lfloor n\varepsilon\rfloor+ v_m^z} |g_{i_{m,
z} + u, n}(T_m)|^p,\end{aligned}$$ where $v_m^z$ are random variables taking values in $\{-2, -1, 0\}$ that are measurable with respect to $T_m$. If $z = 0$ then the sum above is one-sided, i.e. from $u = 0$ to $\lfloor n\varepsilon\rfloor
$, cf. [@BLP17 Eq. (4.2)]. Next, we observe the identity $$\{nT_m + n \theta_z\} = nT_m + n
\theta_z - \lfloor nT_m + n \theta_z
\rfloor= nT_m + n \theta_z - (i_{m, z} -1).$$ Due to Assumption (B), we can write $g(x)= c_z|x - \theta_z|^{\alpha}
f(x)$ with $f(x) \to1$ as $x \to\theta_z$, for any $z\in\mathcal{A}$ (for $\theta_0 = 0$ we need to replace $|x|^{\alpha}$ by $x_+^{\alpha
}$). This allows us to decompose $$\begin{aligned}
&n^{\alpha} g \biggl(\frac{i_{m, z} + u - r}{n} - T_m \biggr)\nonumber\\
&\quad = c_z n^{\alpha} \Big|\frac{i_{m, z} + u -r}{n} - T_m - \theta_z \Big|^{\alpha} f \biggl(\frac{i_{m, z} + u -r}{n} - T_m \biggr)\nonumber\\
&\quad = c_z | u - r + i_{m,z} - nT_m - n\theta_z |^{\alpha} f \biggl(\frac{u-r}{n} +n^{-1} (i_{m,z} - n T_m) \biggr)\nonumber\\
&\quad = c_z | u - r + 1 - \{nT_m + n\theta_z\} |^{\alpha} f \biggl(\frac{u-r}{n} +n^{-1} \bigl(n\theta_z + 1 - \{n T_m + n\theta_z\}\bigr) \biggr)\nonumber\\
&\quad = c_z | u - r + 1 - \{nT_m + n \theta_z\}|^{\alpha} f \biggl(\frac{u - r + 1 - \{n T_m + n\theta_z\}}{n} +\theta_z \biggr),\label{gdecomp}\end{aligned}$$ for any $m \in{\mathbb{N}}$, $0 \leq r \leq k$ and $z \in\mathcal{A}$. Since $f(x) \to1$ as $x \to\theta_z$, we find that for any $d \in{\mathbb{N}}$ $$\begin{aligned}
& \biggl(n_j^{\alpha} g \biggl(\frac{i_{m, z} + u - r}{n_j} -
T_m \biggr) \biggr)_{|u|, m \leq d,\, 0\leq r \leq k,\, z \in\mathcal A}
\\
&{\xrightarrow{\smash{\mathcal{L}-s}}}\bigl(c_z |u-r + 1 - \{U_m +
\eta_z\}|^{\alpha}\bigr)_{|u|, m \leq d,\, 0
\leq r \leq k,\, z \in\mathcal A},\end{aligned}$$ which holds due to condition , decomposition and Proposition \[prop1\] (for $\theta_0 = 0 $ we again need to replace $|x|^{\alpha}$ by $x_+^{\alpha}$). Hence, by continuous mapping theorem for stable convergence we deduce that $$\label{gstab} \bigl(n_j^{\alpha} g_{i_{m,z} +u, n_j}(T_m)
\bigr)_{|u|, m \leq d,\, z \in
\mathcal A} {\xrightarrow{\smash{\mathcal{L}-s}}}\bigl(c_z h_{k, z} \bigl(1 + u -
\{U_m + \eta_z\}\bigr) \bigr)_{|u|, m
\leq d,\, z \in\mathcal A}$$ as $j \to\infty$, which is a key result of the proof. We now define a truncated version of $V_{n, \varepsilon, z}$ introduced in : $$V_{n,\varepsilon,z,d} := n^{\alpha p} \sum_{\substack{m \leq d: \\ T_m
\in(-\theta_z, 1 - \theta_z]}} |
\Delta L_{T_m}|^p \Biggl(\sum_{u =
-\lfloor\varepsilon d \rfloor}^{\lfloor\varepsilon d \rfloor+ v_m^z}
|g_{i_{m, z} + u, n}(T_m)|^p \Biggr).$$ From and properties of stable convergence we conclude that $$\label{vndconv} (V_{n_j, \varepsilon, z, d} )_{z \in\mathcal{A}} {\xrightarrow{\smash{\mathcal{L}-s}}}(V_{\varepsilon
, z, d})_{z \in\mathcal{A}}
\quad\text{as } j \to\infty,$$ where $$V_{\varepsilon, z, d} = |c_z|^p \sum
_{\substack{m \leq d : \\ T_m \in
(-\theta_z, 1 - \theta_z]}} |\Delta L_{T_m}|^p \Biggl(\sum
_{u =
-\lfloor\varepsilon d \rfloor}^{\lfloor\varepsilon d \rfloor+ v_m^z} \big|h_{k,z} \bigl(1+u -
\{U_m +\eta_z\}\bigr)\big|^p \Biggr).$$ Applying a monotone convergence argument, we deduce the almost sure convergence $$\label{domconv} V_{\varepsilon, z, d} \uparrow V_z =
|c_z|^p \sum_{T_m \in(-\theta_z,
1 - \theta_z]} |\Delta
L_{T_m}|^p \biggl(\sum_{u \in{\mathbb{Z}}}
\big|h_{k,z} \bigl(1 + u - \{U_m + \eta_z\}
\bigr)\big|^p \biggr)$$ as $d \to\infty$, where the second sum on the right-hand side is finite, since $|h_{k, z}(x)| \leq C|x|^{\alpha- k}$ for large enough $|x|$ and all $z \in\mathcal A$, and $\alpha< k - 1/p$. In view of and , we are left to prove the convergence $$\lim_{d \to\infty} \limsup_{n \to\infty} |
V_{n, \varepsilon, z, d} - V_{n, \varepsilon, z} | = 0$$ on $\varOmega_{\varepsilon}$. Set $K_d = \sum_{m > d : T_m \in(-\theta_z,
1 - \theta_z]} |\Delta L_{T_m}|^p$ and observe that $K_d \to0$ as $d
\to\infty$, since $L$ is a compound Poisson process. Due to Lemma \[lem1\] we conclude that $|n^{\alpha} g_{i, n}(x)| \leq C \min\{1, |i/n
- x|^{\alpha- k}\}$ and thus $$|V_{n, \varepsilon, z, d} - V_{n, \varepsilon, z} | \leq C \biggl(K_d + \sum
_{|u| > \lfloor\varepsilon d \rfloor} |u|^{p(\alpha- k)} \biggr) \quad\text{for all
} z \in\mathcal{A},$$ and the latter converges to $0$ almost surely as $d \to\infty$, because $\alpha< k - 1/p$. Consequently, we have shown .
### Main terms in Theorem \[th2\]\[it:th2:2\] {#sec3.1.3}
We start with a simple lemma.
\[lem2\] Let $(a_i)_{i \in{\mathbb{N}}}$ be a sequence of positive real numbers such that$\lim_{i \to\infty} ia_i =1$. Then it holds that $$\lim_{n \to\infty} \frac{1}{\log(n)} \sum
_{i = 1}^{cn} a_i =1$$ for any fixed $c \in{\mathbb{N}}$.
Due to the assumption of the lemma, we have that $(a_i)_{i \in{\mathbb{N}}}$ is a bounded sequence and for each $\epsilon> 0$ there exists an $N =
N(\epsilon)$ with $$|a_i - i^{-1}| \leq\epsilon i^{-1} \quad
\text{for all } i \geq N.$$ It obviously holds that $\lim_{n \to\infty} \sum_{i = 1}^{cn} i^{-1} /
\log(n) = 1$. On the other hand, we obtain that $$\limsup_{n \to\infty} \frac{1}{\log(n)} \sum
_{i = N}^{cn} |a_i - i^{-1}|
\leq\epsilon\limsup_{n \to\infty} \frac{1}{\log(n)} \sum
_{i
= 1}^{cn} i^{-1} = \epsilon.$$ Since $\epsilon> 0$ is arbitrary, we conclude the statement of Lemma \[lem2\].
Now, we will again use the decomposition , which holds on $\varOmega_{\varepsilon}$, and treat each term $V_{n, \varepsilon
, z}$ separately. We consider $z \geq1$ and we will show that $$\label{apprbyh} \frac{1}{\log(n)}\sum_{u = -\lfloor n\varepsilon\rfloor}^{\lfloor
n\varepsilon\rfloor+ v_m^z}
\big|n^{\alpha}g_{i_{m, z} + u, n}(T_m) - c_z
h_{k, z}\bigl(u + 1 - \{nT_m + n \theta_z\}
\bigr) \big|^p \to0$$ as $n \to\infty$, for any $m \in{\mathbb{N}}$. Let us first consider the case $|u| \geq k$. Recall that we have assumed that $f_z(x) = g(x) / |x -
\theta_z|^{\alpha}$ is in $C^k((\theta_z - \delta, \theta_z + \delta))$ for any $\delta< \max_{1 \leq j \leq l}(\theta_j - \theta_{j- 1})$. Now, due to identity and Taylor expansion of order $k$, we obtain the bound (cf. [@BP17 Eqs. (4.8) and (4.9)]) $$\sum_{u = -\lfloor n\varepsilon\rfloor}^{\lfloor n\varepsilon\rfloor
+ v_m^z} \big|n^{\alpha}g_{i_{m,z} + u,n}(T_m)
- c_z h_{k,z}\bigl(u+1-\{ nT_m + n
\theta_z\}\bigr) \big|^p {\mathbf{1}}_{\{|u| \geq k \}} \leq C,$$ for any $\varepsilon< \max_{1 \leq j \leq l}(\theta_j - \theta_{j -
1})$. Since $|n^{\alpha} g_{i_{m, z} + u, n}(T_m)|$ is bounded for any $|u| < k$ due to Lemma \[lem1\], we deduce the convergence in .
Next, for large enough $|u|$ we observe the bounds $$|q_{k, \alpha}|^p a_u \leq\big|h_{k, z}
\bigl(u + 1 - \{nT_m + n \theta_z\}\bigr)\big|^p
\leq|q_{k, \alpha}|^p a_{u - k - 1} \quad\text{where }
a_u = |u|^{-1}.$$ Hence, by Lemma \[lem2\], we conclude the convergence $$\label{convforh} \frac{1}{\log(n)}\sum_{u = -\lfloor n\varepsilon\rfloor}^{\lfloor
n\varepsilon\rfloor+ v_m^z}
\big| h_{k, z}\bigl(u + 1 - \{nT_m + n \theta_z\}
\bigr) \big|^p \to2|q_{k, \alpha}|^p \quad\text{as } n\to
\infty.$$ The same statement holds for $z=0$, but the limit becomes $|q_{k,\alpha
}|^p$, since in this setting the sum is one-sided. We set $\| x\|_p^{p} = \sum_{i=1}^m |x_i|^p$ for any $x \in{\mathbb{R}}^m$ and $p>0$, and recall that $\| x\|_p$ is a norm for $p \geq1$. It holds that $$\label{ineq} \begin{aligned} |\| x\|_p^p -
\| y\|_p^p| &\leq\| x - y \|_p^p
\quad\text{ when } p \in(0, 1],
\\
|\| x\|_p - \| y\|_p| &\leq\| x - y\|_p
\quad\text{ when } p > 1. \end{aligned} $$ By , and , and taking into account the definition of $V_{n, \varepsilon, z}$ at , we readily deduce the convergence $$\frac{V_{n, \varepsilon,z}}{\log(n)} {\stackrel{\mathbb{P}}{\longrightarrow}}|q_{k, \alpha} c_z|^p (1 +
{\mathbf{1}}_{\{z\geq1\}}) \sum_{m : T_m \in[-\theta_z, 1 - \theta_z]} |\Delta
L_{T_m}|^p$$ as $n \to\infty$, and hence $$n^{\alpha p}\sum_{i = k}^{n} \Bigg|\sum
_{z = 0}^{l} M_{i, n,
\varepsilon, z}
\Bigg|^p {\stackrel{\mathbb{P}}{\longrightarrow}}|q_{k,\alpha} |^p \sum
_{z = 0}^l |c_z|^p (1 +
{\mathbf{1}}_{\{z \geq1\}} ) \sum_{m : T_m \in[-\theta_z, 1- \theta
_z]} |\Delta
L_{T_m}|^p$$ as $n \to\infty$, on $\varOmega_{\varepsilon}$.
### Negligible terms {#sec3.1.1}
Due to inequalities at , it suffices to show that on $\varOmega
_{\varepsilon}$ $$\label{neglconv} a_n \sum_{i = k}^{n}
|R_{i, n, \varepsilon}|^p {\stackrel{\mathbb{P}}{\longrightarrow}}0 \quad\text {and} \quad a_n
\sum_{i = k}^{n} |M_{i, n, \varepsilon,z}|^p
{\stackrel{\mathbb{P}}{\longrightarrow}}0 \quad\text{for } z \in \mathcal{A}^c,$$ as $n \to\infty$, where $a_n = n^{\alpha p}$ in Theorem \[th2\]\[it:th2:1\] and $a_n = n^{\alpha p}/ \log(n)$ in Theorem \[th2\]\[it:th2:2\], and this will prove that these terms do not affect the limits in Theorem \[th2\]. At this stage we notice that outside the singularity points the kernel function $g$ satisfies the same properties under Assumption (B) (resp. Assumption (B-log)) as under Assumption (A) (resp. Assumption (A-log)). Consequently, we can apply the estimates for the term $R_{i, n, \varepsilon}$ derived in [@BLP17 Eqs. (4.8) and (4.12)] and [@BP17 Section 4] under conditions (A) and (A-log) $$\begin{aligned}
\sup_{n \in{\mathbb{N}}, i = k, \ldots, n} n^k |R_{i, n, \varepsilon}| &< \infty
\text{ almost surely if } w \in(0, 1],
\\
\sup_{n \in{\mathbb{N}}, i = k, \ldots, n} \frac{n^k |R_{i, n, \varepsilon
}|}{(\log(n))^q} &< \infty\text{ almost
surely if } w \in(1,2],\end{aligned}$$ where $q$ is determined via $1 / q + 1 / w = 1$, since $R_{i, n,
\varepsilon}$ is only affected by the function $g$ outside the singularity points $\theta_z$. We readily conclude the first convergence at in the setting of Theorem \[th2\]\[it:th2:1\], because $\alpha<k-1/p$. It also holds in the setting of Theorem \[th2\]\[it:th2:2\], where for $w \in(1,2]$ we use the assumption that $1 / p + 1 / w > 1$.
Now, we show the second statement of , which is only relevant in the setting of Theorem \[th2\]\[it:th2:1\]. Since $\alpha_z < k -1/p$ for all $z$, we can apply to $\sum_{i = k}^{n}
|M_{i, n, \varepsilon, z}|^p$, $z \in\mathcal{A}^c$, the same techniques as for $\sum_{i = k}^{n} |M_{i, n, \varepsilon, z}|^p$, $z
\in\mathcal{A}$. Hence, using the same methods as in Section \[sec3.1.2\], we conclude that on $\varOmega_{\varepsilon}$ $$n^{\alpha p} \sum_{i = k}^{n}
|M_{i, n, \varepsilon, z}|^p = O_{\mathbb
{P}} \bigl(n^{p(\alpha- \alpha_z)}
\bigr) \quad\text{for all } z \in \mathcal{A}^c,$$ where the notation $Y_n = O_{\mathbb{P}} (a_n)$ means that the sequence $a_n^{-1} Y_n$ is tight. Since $\alpha_z > \alpha$ for all $z \in
\mathcal{A}^c$, we obtain the second statement of . The results of Sections \[sec3.1.2\]–\[sec3.1.1\] and the fact that $\varOmega_{\varepsilon} \uparrow\varOmega$ as $\varepsilon\to0$ imply the assertion of Theorem \[th2\] in the compound Poisson case.
Proof of Theorem \[th2\] in the general case {#sec3.2}
--------------------------------------------
Let now $(L_t)_{t\in{\mathbb{R}}}$ be a general symmetric pure jump Lévy process with Blumenthal–Getoor index $\beta$. We denote by $N$ the corresponding Poisson random measure defined by $N(A) := \#\{t \in{\mathbb{R}}:
(t, \Delta L_t) \in A\}$ for all measurable $A \subseteq{\mathbb{R}}\times({\mathbb{R}}\setminus\{0\})$. Next, we introduce the process $$X_t(m) = \int_{(-\infty, t] \times[-\frac{1}{m}, \frac{1}{m}]} x \bigl(g(t - s) -
g_0(-s) \bigr) \, N({\textup{d}}s, {\textup{d}}x),$$ which only involves small jumps of $L$. We will prove that $$\label{smalljumps} \lim_{m \to\infty} \limsup_{n \to\infty}
\mathbb{P} \bigl( a_n V\bigl(X(m), p; k\bigr)_n > \epsilon
\bigr) = 0 \quad\text{for any } \epsilon>0,$$ where $a_n = n^{\alpha p}$ in Theorem \[th2\]\[it:th2:1\] and $a_n
= n^{\alpha p}/ \log(n)$ in Theorem \[th2\]\[it:th2:2\]. First, due to Markov’s inequality and the stationary increments of $X_t(m)$, it follows that $$\mathbb{P}\bigl(a_n V\bigl(X(m), p; k\bigr)_n > \epsilon
\bigr) \leq\epsilon^{-1} a_n \sum
_{i = k}^{n} {\mathbb{E}}\bigl[| \Delta_{i, k}^n
X(m) |^p\bigr] \leq\epsilon^{-1} b_n {\mathbb{E}}\bigl[|
\Delta_{k, k}^n X(m) |^p\bigr],$$ where $b_n = n a_n$. Hence it is enough to prove that $$\label{eq:Ynm} \lim_{m \to\infty} \limsup_{n \to\infty} {\mathbb{E}}\bigl[| Y_{n, m} |^p\bigr] = 0 \quad \text{where} \quad
Y_{n, m} = b_n^{1 / p} \Delta_{k, k}^n
X(m).$$ Notice the representation $$Y_{n, m} = \int_{(-\infty, \frac{k}{n}] \times[-\frac{1}{m}, \frac
{1}{m}]} \bigl(b_n^{1/p}
g_{k, n}(s)\bigr) x \, N({\textup{d}}s, {\textup{d}}x).$$ Using this together with [@RR89 Theorem 3.3], will follow if $$\begin{aligned}
&\lim_{m \to\infty} \limsup_{n \to\infty}
\xi_{n, m} = 0 \quad\text {where} \quad\xi_{n, m} = \int
_{| x | \leq\frac{1}{m}} \chi_n(x) \, \nu({\textup{d}}x) \quad\text{and}
\\
&\chi_n(x) = \int_{-\infty}^{\frac{k}{n}} \bigl( |
b_n^{1 / p} g_{k,
n}(s) x |^p
{\mathbf{1}}_{\{| b_n^{1 / p} g_{k,n}(s) x | \geq1 \}}
\\
&\quad\ \qquad+ | b_n^{1 / p} g_{k, n}(s) x
|^2 {\mathbf{1}}_{\{| b_n^{1 /
p} g_{k,n}(s) x | < 1 \}} \bigr) \, {\textup{d}}s.\end{aligned}$$ Suppose there exists a constant $K \geq0$ such that for all large $n
\in{\mathbb{N}}$ $$\label{eq:chi_leq} \chi_n(x) \leq K \bigl(| x |^p + | x
|^2\bigr) \quad\text{for all $x \in[-1, 1]$},$$ then the dominated convergence theorem implies that $$\limsup_{m \to\infty} \Bigl[\limsup_{n \to\infty}
\xi_{n, m} \Bigr] \leq K \limsup_{m \to\infty} \int
_{| x | \leq\frac{1}{m}} \bigl(| x |^p + | x |^2\bigr)
\, \nu({\textup{d}}x) = 0,$$ using the assumption that $p > \beta$. We consider only in the case of Theorem \[th2\]\[it:th2:1\] as \[it:th2:2\] is very similar, see [@BP17]. In the case of \[it:th2:1\] then $\smash{b_n^{1/p}} = n^{\alpha+ 1 / p}$. For short notation define $\varPhi_p : {\mathbb{R}}\to{\mathbb{R}}_+$ as the function $$\varPhi_p(y) = | y |^2 {\mathbf{1}}_{\{| y | \leq1\}} + | y
|^p {\mathbf{1}}_{\{| y | > 1\}}, \quad y \in{\mathbb{R}}.$$ Note that $\varPhi_p$ is of modular growth, i.e. there exists a constant $K_p > 0$ depending only on $p$ such that $\varPhi_p(x + y) \leq K_p(\varPhi_p(x) + \varPhi_p(y))$ for any $x,y \in{\mathbb{R}}$. We consider the following decomposition $$\begin{aligned}
\chi_n(x) &= \int_{\frac{k}{n} - \frac{1}{n}}^{\frac{k}{n}} \varPhi
_p\bigl(n^{\alpha+ 1/p} g_{k, n}(s) x\bigr) \, {\textup{d}}s + \sum
_{z = 1}^{l} \int_{\frac{k}{n} - \theta_z - \frac{1}{n}}^{\frac{k}{n} - \theta_z + \frac
{1}{n}}
\varPhi_p\bigl(n^{\alpha+ 1/p} g_{k, n}(s) x\bigr) \, {\textup{d}}s
\\
&\quad+ \sum_{z = 1}^{l} \int
_{\frac{k}{n} - \theta_z + \frac
{1}{n}}^{\frac{k}{n} - \theta_{z-1} - \frac{1}{n}} \varPhi_p
\bigl(n^{\alpha+
1/p} g_{k, n}(s) x\bigr) \, {\textup{d}}s
\\
&\quad+ \int_{\frac{k}{n} - \theta_l - \delta}^{\frac{k}{n} - \theta_l
- \frac{1}{n}} \varPhi_p
\bigl(n^{\alpha+ 1/p} g_{k, n}(s)x\bigr) \, {\textup{d}}s
\\
&\quad+ \int_{-\infty}^{\frac{k}{n} - \theta_l - \delta} \varPhi _p
\bigl(n^{\alpha+ 1/p} g_{k, n}(s) x\bigr) \, {\textup{d}}s
\\
&=: I_0(x) + \sum_{z = 1}^{l}
I_{1, z}(x) + \sum_{z = 1}^{l}
I_{2,
z}(x) + I_3(x) + I_4(x).\end{aligned}$$ We treat the five types of terms separately.
#### Estimation of $I_0$
By Lemma \[lem1\] $$| g_{k, n}(x) | \leq K \bigl(| \tfrac{k}{n} - s |^{\alpha_0}
\bigr) \quad\text {for all $s \in\bigl[\tfrac{k}{n} - \tfrac{1}{n},
\tfrac{k}{n}\bigr]$}.$$ Since $\varPhi_p$ is increasing on ${\mathbb{R}}_+$ and $\alpha\leq\alpha_0$ it follows that $$I_0(x) \leq K \int_{0}^{\frac{1}{n}}
\varPhi_p\bigl(x n^{\alpha+ 1/p} s^{\alpha
_0} \bigr) \, {\textup{d}}s
\leq K \int_{0}^{\frac{1}{n}} \varPhi_p\bigl(x
n^{\alpha+ 1/p} s^{\alpha} \bigr) \, {\textup{d}}s.$$ By elementary integration it follows that $$\begin{aligned}
&\int_{0}^{\frac{1}{n}} | x n^{\alpha+ 1/p}
s^{\alpha} |^2 {\mathbf{1}}_{\{ | x
n^{\alpha+ 1/p} s^\alpha| \leq1\}} \, {\textup{d}}s
\\
&\quad\leq K \bigl(x^2 {\mathbf{1}}_{\{ | x | \leq n^{-1/p} \}} n^{2/p - 1} +
{\mathbf{1}}_{\{
| x | > n^{-1 / p} \}} | x |^{-1/\alpha} n^{-1 - 1/(\alpha p)}\bigr)
\\
&\quad\leq K\bigl(x^2 + | x |^p\bigr).\end{aligned}$$ The second term in $\varPhi_p$ is dealt with as follows: $$\int_{0}^{\frac{1}{n}} | x n^{\alpha+ 1/p}
s^{\alpha} |^p {\mathbf{1}}_{\{| x
n^{\alpha+ 1/p} s^{\alpha} | > 1\}} \, {\textup{d}}s \leq| x
|^p n^{\alpha p
+ 1} \int_{0}^{\frac{1}{n}}
s^{\alpha p} \, {\textup{d}}s = \frac{| x
|^p}{\alpha p + 1}.$$ Combining the two estimates above it follows that $I_0(x) \leq K(|x|^2
+ |x|^p)$.
#### Estimation of $I_{1, z}$
Similarly as for $I_0$, we have, using arguments as in part \[it:lem1:1\] of Lemma \[lem1\], that $$| g_{k,n}(s) | \leq K \sum_{j = 0}^{k}
| \tfrac{k - j}{n} - s - \theta _z |^{\alpha_z} \quad\text{for
all $s \in\bigl[\tfrac{k}{n} - \theta_z - \tfrac{1}{n},
\tfrac{k}{n} - \theta_z + \tfrac{1}{n}\bigr]$}.$$ Using the modular growth of $\varPhi_p$ it follows that $$\begin{aligned}
&\int_{\frac{k}{n} - \theta_z - \frac{1}{n}}^{\frac{k}{n} - \theta_z +
\frac{1}{n}} \varPhi_p
\bigl(n^{\alpha+ 1/p} g_{k, n}(s) x\bigr) \, d s
\\
&\quad\leq K_p \sum_{j = 0}^{k}
\int_{\frac{k}{n} - \theta_z - \frac
{1}{n}}^{\frac{k}{n} - \theta_z + \frac{1}{n}} \varPhi_p
\bigl(n^{\alpha+ 1/p} |\tfrac{k - j}{n} - s - \theta_z|^{\alpha_z}
x\bigr) \, {\textup{d}}s
\\
&\quad= K_p \sum_{j = 0}^{k}
\int_{-\frac{j}{n} - \frac{1}{n}}^{-\frac
{j}{n} + \frac{1}{n}} \varPhi_p
\bigl(n^{\alpha+ 1/p} |s|^{\alpha_z} x\bigr) \, {\textup{d}}s
\\
&\quad\leq K_p \int_{-\frac{k + 1}{n}}^{\frac{k + 1}{n}} \varPhi
_p\bigl(n^{\alpha+ 1/p} |s|^{\alpha} x\bigr) \, {\textup{d}}s
\\
&\quad= K_p \int_{0}^{\frac{k + 1}{n}}
\varPhi_p\bigl(n^{\alpha+ 1/p} |s|^{\alpha} x\bigr) \, {\textup{d}}s.\end{aligned}$$ As for $I_0$, we get $I_{1, z}(x) \leq K(|x|^2 + |x|^p)$.
#### Estimation of $I_{2, z}$
We decompose $I_{2, z}$ into three terms corresponding to whether we are close to the singularity $\theta
_z$ from the right or close to the singularity $\theta_{z-1}$ from the left or in between them, but bounded away from both. More specifically, we decompose as $$\begin{aligned}
I_{2,z}(x) &= \int_{\frac{k}{n} - \theta_z + \frac{1}{n}}^{\frac{k}{n}
- \theta_z + \delta}
\varPhi_p\bigl(n^{\alpha+ 1/p} g_{k, n}(s) x\bigr) \, {\textup{d}}s
\\
&\quad+ \int_{\frac{k}{n} - \theta_z + \delta}^{\frac{k}{n} - \theta_{z
- 1} - \delta} \varPhi_p
\bigl(n^{\alpha+ 1/p} g_{k, n}(s) x\bigr) \, {\textup{d}}s
\\
&\quad+ \int_{\frac{k}{n} - \theta_{z-1} - \delta}^{\frac{k}{n} - \theta
_{z - 1} - \frac{1}{n}} \varPhi_p
\bigl(n^{\alpha+ 1/p} g_{k, n}(s) x\bigr) \, {\textup{d}}s =:
I_{2,z}^{l}(x) + I_{2,z}^{b}(x) +
I_{2,z}^{r}(x).\end{aligned}$$ First we note that arguments similar to Lemma \[lem1\]\[it:lem1:3\] imply that $$| g_{k, n}(s) | \leq K n^{-k} |\tfrac{k}{n} - s -
\theta_z|^{\alpha_z -
k} \quad\text{for all $s \in\bigl[
\tfrac{k}{n} - \theta_z + \tfrac{1}{n}, \tfrac{k}{n} -
\theta_z + \delta\bigr]$.}$$ Using again that $\varPhi_p$ is decreasing on ${\mathbb{R}}_+$ it follows that $$\begin{aligned}
I_{2, z}^{l}(x) &\leq K \int_{\frac{k}{n} - \theta_z + \frac
{1}{n}}^{\frac{k}{n} - \theta_z + \delta}
\varPhi_p\bigl(n^{\alpha+ 1/p - k} | \tfrac{k}{n} - s -
\theta_z |^{\alpha_z - k} x\bigr) \, {\textup{d}}s
\\
&\leq K \int_{\frac{1}{n}}^{\delta} \varPhi_p
\bigl(n^{\alpha+ 1/p - k} |s|^{\alpha_z - k} x\bigr) \, {\textup{d}}s.\end{aligned}$$ If $\alpha_z = k - 1/2$ then $$\begin{aligned}
\int_{\frac{1}{n}}^{\delta} | x n^{\alpha+ 1/p - k}
s^{\alpha_z -
k}|^2 {\mathbf{1}}_{\{ | x^2 n^{\alpha+ 1/p - k} s^{\alpha_z - k} | \leq1 \}} \, {\textup{d}}s &\leq
x^2 n^{2(\alpha+ 1/p - k)} \int_{\frac{1}{n}}^{\delta}
s^{-1} \, {\textup{d}}s
\\
&\leq K x^2,\end{aligned}$$ where we used that $\alpha< k - 1/p$. For $\alpha_z \neq k - 1 / 2$ we have that $$\begin{aligned}
&\int_{\frac{1}{n}}^{\delta} | x n^{\alpha+ 1/p - k}s^{\alpha_z -k}|^2 {\mathbf{1}}_{\{| x n^{\alpha+ 1/p - k} s^{\alpha_z - k}| \leq1\}} \, {\textup{d}}s\\
&\quad \leq K\bigl(|x|^2 n^{2(\alpha+ 1/p - k)} + |x|^2 n^{2(\alpha- \alpha_z) + 2/p - 1} {\mathbf{1}}_{\{| x | \leq n^{-1/p} \}}\\
&\qquad+ |x|^{\frac{1}{k - \alpha_z}} n^{\frac{\alpha+ 1/p - k}{k - \alpha_z}} {\mathbf{1}}_{\{|x| > n^{-1/p}\}}\bigr)\\
&\quad \leq K\bigl(x^2 + |x|^p\bigr),\end{aligned}$$ where we used that $\alpha\leq\alpha_z < k - 1/p$. Moreover, $$\int_{\frac{1}{n}}^{\delta} |x n^{\alpha+ 1/p - k}
s^{\alpha_z - k}|^p {\mathbf{1}}_{\{| x n^{\alpha+ 1/p - k} s^{\alpha_z - k}| > 1\}} \, {\textup{d}}s \leq K
|x|^p.$$ The term $I_{2,z}^{r}$ is handled similarly. For the last term $I_{2,
z}^b$ we note that, since we are bounded away from both $\theta_{z-1}$ and $\theta_z$, there exists a constant $K > 0$ such that $$|g_{k,n}(s)| \leq K n^{-k} \quad\text{for all $s \in\bigl[
\tfrac{k}{n} - \theta_z + \delta, \tfrac{k}{n} -
\theta_{z-1} - \delta\bigr]$.}$$ This readily implies the bound $I_{2,z}^{b}(x) \leq K(x^2 + |x|^p)$.
#### Estimation of $I_3$
Arguments as in Lemma \[lem1\] imply that $$|g_{k, n}(s)| \leq K n^{-k}|\tfrac{k}{n} - s -
\theta_z|^{\alpha_l - k} \quad\text{for all $s \in\bigl[
\tfrac{k}{n} - \theta_l - \delta, \tfrac
{k}{n} -
\theta_l - \tfrac{1}{n}\bigr]$.}$$ One may then proceed as for the term $I_{2,z}^l$ above to conclude that $I_3(x) \leq K(x^2 + |x|^p)$.
#### Estimation of $I_4$:
First we decompose the integral into two sub-integrals: $$\begin{aligned}
\int_{-\infty}^{\frac{k}{n} - \theta_l - \delta} \varPhi_p
\bigl(n^{\alpha+
1/p} g_{k, n}(s) x\bigr) \, {\textup{d}}s &= \int
_{-\delta' - \theta_l}^{\frac{k}{n}
- \delta- \theta_l} \varPhi_p
\bigl(n^{\alpha+ 1/p} g_{k, n}(s)x \bigr) \, {\textup{d}}s
\\
&\quad+ \int_{-\infty}^{-\delta' - \theta_l} \varPhi_p
\bigl(n^{\alpha+ 1/p} g_{k, n}(s) x\bigr) \, {\textup{d}}s.\end{aligned}$$ In the first integral we are bounded away from $\theta_l$, hence $|g_{k,n}(s)| \leq K n^{-k}$ for all $s$ in the interval $[-\delta' -
\theta_l, \frac{k}{n} - \delta- \theta_l]$. For the latter integral note first that by Lemma \[lem1\]\[it:lem1:5\] $$\int_{-\infty}^{-\delta' - \theta_l} \varPhi_p
\bigl(n^{\alpha+ 1/p} g_{k,
n}(s) x\bigr) \, {\textup{d}}s \leq\int
_{-\infty}^{-\delta' - \theta_l} \varPhi _p
\bigl(n^{\alpha+ 1/p - k} |g^{(k)}(-s)| x\bigr) \, {\textup{d}}s.$$ Now $$\begin{aligned}
&\int_{\delta' + \theta_l}^{\infty} | x n^{\alpha+ 1/p - k} g^{(k)}(s) |^2 {\mathbf{1}}_{\{ |x n^{\alpha+ 1/p - k} g^{(k)}(s) | \leq1 \}} \, {\textup{d}}s \\
&\quad \leq|x n^{\alpha+ 1/p - k}|^2 \int_{\delta' + \theta_l}^{\infty}|g^{(k)}(s)|^2 \, {\textup{d}}s.\end{aligned}$$ Since $|g^{(k)}|$ is decreasing on $(\theta_l + \delta', \infty)$ and $g^{(k)} \in L^w((\theta_l + \delta', \infty))$ for some $w \leq2$ it follows that the last integral is finite. Lastly, we find for $x \in
[-1 , 1]$ that $$\begin{aligned}
&\int_{\theta_l + \delta'}^{\infty} |x n^{\alpha+ 1/p - k}
g^{(k)}(s)|^p {\mathbf{1}}_{\{|x n^{\alpha+ 1/p - k} g^{(k)}(s)| > 1\}} \, {\textup{d}}s
\\
&\quad \leq|x|^p n^{p(\alpha+ 1/p - k)} \int_{\delta' + \theta_l}^{\infty}
|g^{(k)}(s)|^p {\mathbf{1}}_{\{|g^{(k)}(s)| > 1 \}} \, {\textup{d}}s.\end{aligned}$$ By our assumptions the last integral is finite, indeed $$\int_{\delta' + \theta_l}^{\infty} |g^{(k)}(s)|^p
{\mathbf{1}}_{\{|g^{(k)}(s)| >
1\}} \, {\textup{d}}s \leq K_p \| g^{(k)}
\|_{L^w((\delta' + \theta, \infty
))}^w < \infty.$$
### Negligibility of small jumps
Now, we note that $X_t - X_t(m)$ is the integral , where the integrator is a compound Poisson process that corresponds to big jumps of $L$. Hence, we obtain the results of Theorem \[th2\] for the process $X-X(m)$ as in Section \[sec3.1\]. More specifically, under assumptions of Theorem \[th2\]\[it:th2:1\] it holds that $$n_j^{\alpha p}V\bigl(X - X(m), p; k\bigr)_{n_j} {\xrightarrow{\smash{\mathcal{L}-s}}}\sum_{z \in\mathcal A} |c_z|^p \sum
_{r : T_r \in[-\theta_z, 1 - \theta_z]} |\Delta L_{T_r}|^p
{\mathbf{1}}_{\{|\Delta L_{T_r}| > \frac{1}{m}\}} V_r^z$$ where $V_r^z$ has been defined at . The term on the right-hand side converges to the limit of Theorem \[th2\]\[it:th2:1\] as $m \to\infty$, since $$\sum_{r : T_r \in[-\theta_z, 1 - \theta_z]} |\Delta L_{T_r}|^p
< \infty\quad\text{for any $p > \beta$.}$$ Finally, using the decomposition $X=(X-X(m)) + X(m)$ and letting first $n_j \to\infty$ and then $m\to\infty$, we deduce the statement of Theorem \[th2\] by and the inequalities . This completes the proof.
The authors acknowledge financial support from the project “Ambit fields: probabilistic properties and statistical inference” funded by Villum Fonden.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Recently, a simple prescription to embed the global Peccei-Quinn (PQ) symmetry into a gauged $U(1)$ symmetry has been proposed. There, explicit breaking of the global PQ symmetry expected in quantum gravity are highly suppressed due to the gauged PQ symmetry. In this paper, we apply the gauged PQ mechanism to models where the global PQ symmetry and supersymmetry (SUSY) are simultaneously broken at around $\mathcal{O}(10^{11-12})$GeV. Such scenario is motivated by an intriguing coincidence between the supersymmetry breaking scale which explains the observed Higgs boson mass by the gravity mediated sfermion masses, and the PQ breaking scale which evades all the astrophysical and the cosmological constraints. As a concrete example, we construct a model which consists of a simultaneous supersymmetry/PQ symmetry breaking sector based on $SU(2)$ dynamics and an additional PQ symmetry breaking sector based on $SU(N)$ dynamics. We also show that new vector-like particles are predicted in the TeV range in the minimum model, which can be tested by the LHC experiments.'
author:
- Hajime Fukuda
- Masahiro Ibe
- Motoo Suzuki
- 'Tsutomu T. Yanagida'
bibliography:
- 'papers.bib'
title: |
Gauged Peccei-Quinn Symmetry\
[*– A Case of Simultaneous Breaking of SUSY and PQ Symmetry –*]{}
---
Introduction {#sec:intro}
============
The Peccei-Quinn (PQ) mechanism[@Peccei:1977hh; @Peccei:1977ur; @Weinberg:1977ma; @Wilczek:1977pj] provides us with a very successful solution to the strong $CP$ problem. The effective $\theta$-angle of QCD is canceled by the vacuum expectation value (VEV) of the pseudo-Nambu-Goldstone boson, the axion $a$, which results from spontaneous breaking of the global $U(1)$ Peccei-Quinn symmetry, $U(1)_{PQ}$.
The solution of the strong $CP$ problem based on a global symmetry is, however, not on the very firm theoretical ground. As the QCD anomaly explicitly breaks the $U(1)_{PQ}$ symmetry, it cannot be an exact symmetry by definition. Besides, it is also argued that all global symmetries are broken by quantum gravity effects [@Hawking:1987mz; @Lavrelashvili:1987jg; @Giddings:1988cx; @Coleman:1988tj; @Gilbert:1989nq; @Banks:2010zn]. The explicit breaking of the PQ symmetry easily spoils the success of the PQ mechanism.
In Ref.[@Fukuda:2017ylt], a simple prescription has been proposed, with which the global $U(1)_{PQ}$ symmetry is embedded into a “gauged" $U(1)$ symmetry, $U(1)_{gPQ}$. There, the anomalies of the gauged PQ symmetry are canceled between the contributions from two (or more) PQ charged sectors. With appropriate charge assignment of $U(1)_{gPQ}$, the PQ charged sectors are highly decoupled with each other, and a global $U(1)_{PQ}$ symmetry appears as an accidental symmetry. As a part of the gauge symmetry, the accidental $U(1)_{PQ}$ is also well protected from explicit breaking caused by quantum gravity effects. This prescription provides a concise generalization of previous attempts to achieve the PQ symmetry as an accidental symmetry resulting from (discrete) gauge symmetries [@Barr:1992qq; @Kamionkowski:1992mf; @Holman:1992us; @Dine:1992vx; @Dias:2002gg; @Carpenter:2009zs; @Harigaya:2013vja; @Harigaya:2015soa; @Redi:2016esr; @Duerr:2017amf].
In this paper, we apply the construction of the gauged PQ symmetry to a model in which the global PQ symmetry and supersymmetry are simultaneously broken at around $\mathcal{O}(10^{11-12})$GeV [@Feldstein:2012bu]. Such scenario is motivated by an intriguing coincidence between the supersymmetry breaking scale which explains the observed Higgs boson mass by the gravity mediated sfermion masses in the hundreds to thousands TeV range [@Okada:1990vk] and the PQ breaking scale which evades all the astrophysical and the cosmological constraints.[^1]
The organization of the paper is as follows. In section \[sec:prescription\], we summarize the supersymmetric version of the gauged PQ mechanism. In section \[sec:proto\], we construct a model in which supersymmetry and the PQ symmetry are broken simultaneously by $SU(2) $ strong dynamics. In section \[sec:model\], we apply the gauged PQ mechanism to the model of simultaneous symmetry breaking. The final section is devoted to our conclusions.
General Prescription of the Gauged PQ Mechanism {#sec:prescription}
===============================================
In this section, we briefly summarize a supersymmetric version of the gauged PQ mechanism [@Fukuda:2017ylt].
Would-be Goldstone and Axion Superfields
----------------------------------------
As a simple example, let us consider two global PQ symmetries $U(1)_{PQ_1}$ and $U(1)_{PQ_2}$, which are broken by the VEVs of $\Phi_1$, $\bar{\Phi}_1$ and $\Phi_2$, $\bar{\Phi}_2$, respectively. For instance, such vacuum is achieved by the superpotential, \[eq:simple super\] W=ł\_1 X\_1(2\_1 |\_1-\_1\^2)+ł\_2 X\_2(2\_2 |\_2-\_2\^2) . Here, $\Phi_i$ and $\bar{\Phi}_i$ $(i=1,2)$ have charges $\pm 1$ under $U(1)_{PQ_i}$ and have vanishing charges under $U(1)_{PQ_j}$ $(j\neq i)$, respectively. The superfields, $X_{1,2}$, have vanishing charges under both the PQ symmetries. The parameters $\l_{1,2}$ are coupling constants, and $\L_{1,2}$ are dimensionful parameters. After the spontaneous breaking of the PQ symmetries, $\Phi$’s lead to the Goldstone superfields $A_{1,2}$,[^2] $$\begin{aligned}
\label{eq:dec1}
\Phi_1 &=& \frac{1}{\sqrt 2}\L_1 e^{A_1/\L_1 }\ , \quad \bar\Phi_1 = \frac{1}{\sqrt 2}\L_1 e^{-A_1/\L_1 }\ , \\
\label{eq:dec2}
\Phi_2 &=& \frac{1}{\sqrt 2}\L_2 e^{A_2/\L_2 }\ , \quad \bar\Phi_2 = \frac{1}{\sqrt 2}\L_2 e^{-A_2/\L_2 }\ .\end{aligned}$$ By using the Goldstone superfields, the PQ symmetries are realized by, $$\begin{aligned}
A_1/\L_1 & \to& A_1/\L_1 + i \alpha_1 \ , \quad(\alpha_1 = 0-2\pi)\ , \\
A_2/\L_2 &\to& A_2/\L_2 + i \alpha_2 \ , \quad(\alpha_2 = 0-2\pi)\ .\end{aligned}$$
The PQ symmetries are communicated to the supersymmetric Standard Model (SSM) sector by introducing extra quark multiplets as in the KSVZ axion model [@Kim:1979if; @Shifman:1979if]. Throughout this paper, we assume that the extra multiplets form ${\bf 5}$ and $\bar{\bf 5}$ representations of the $SU(5)$ gauge group of the Grand Unified Theory (GUT). Let us suppose that $\Phi_{1,2}$ couple to $N_1$ and $N_2$ flavors of the KSVZ extra multiplets ${\bf 5}_i$, $\bar{\bf 5}_i$ $(i=1, 2)$, respectively, W=\_1 [**5**]{}\_1|[**5**]{}\_1 + |\_2 [**5**]{}\_2|[**5**]{}\_2 . Through the above coupling, both the global PQ symmetries are broken by the Standard Model anomaly. The anomalous breaking of the global PQ symmetries lead to the anomalous coupling of the Goldstone superfields, $$\begin{aligned}
\label{eq:anomSF}
W_{\rm anom} = \frac{1}{8\pi^2}\left(N_1\frac{A_1}{\Lambda_1}
-N_2 \frac{A_2}{\Lambda_2}\right) \sum_l W_l^{\a}W_{l\a}\ ,\end{aligned}$$ where, $W_l^{\a}$ $(l=1,2,3)$ denote the field strength superfields of the Standard Model gauge interactions.[^3] We normalize the gauge field strength so that the gauge kinetic functions are given by = \_F + h.c. , with \_l = + , where $g_l$ and $\theta_l$ are the gauge coupling constants and the vacuum angles of the corresponding gauge interactions.
An important observation here is that there is a linear combination of the PQ symmetries for which the Standard Model anomalies are absent. In fact, a $U(1)$ symmetry under which $\Phi_{1,2}$ have charges $q_1$ and $q_2$ is free from the Standard Model anomaly for $$\begin{aligned}
\label{eq:anomfree}
q_1N_1-q_2N_2=0\ .\end{aligned}$$ In the gauged PQ mechanism, we identify the anomaly-free combination to be a gauge symmetry $U(1)_{gPQ}$. The gravitational anomaly and the self-anomaly of the $U(1)_{gPQ}$ are canceled by adding $U(1)_{gPQ}$ charged singlet fields. Hereafter, we take $q_1$ and $q_2$ are both positive and relatively prime numbers without loss of generality.
In the gauged PQ mechanism, one of the linear combinations of $A_{1,2}$ is the would-be Goldstone supermultiplet, and the other combination corresponds to the physical axion superfield. To see how the physical axion is extracted, let us consider the Kähler potential of $\Phi$’s, K=\_1\^ e\^[-2q\_1gV]{}\_1+|\_1\^ e\^[2q\_1gV]{}|\_1 +\_2\^ e\^[-2q\_2 gV]{}\_2+|\_2\^ e\^[2 q\_2g V]{}|\_2 , where $V$ and $g$ are the $U(1)_{gPQ}$ gauge supermultiplet and the gauge coupling constant, respectively. Under the $U(1)_{gPQ}$ gauge transformation, the gauge field is shifted by, $$\begin{aligned}
2gV \to 2gV' = 2gV - i \Theta + i \Theta^\dagger \ ,\end{aligned}$$ with $\Theta$ being the gauge parameter superfield. By substituting Eqs.(\[eq:dec1\]) and (\[eq:dec2\]), the Kähler potential is reduced to K&=& \_1\^2 ([2q\_1gV - ]{}) + \_2\^2 ([2q\_2gV - ]{}) .
The physical axion and the would-be Goldstone superfields $A$ and $G$ are obtained by rearranging $A_{1,2}$ by $$\begin{aligned}
\left(
\begin{array}{cc}
A^{(\dagger)}
\\
G^{(\dagger)}
\end{array}
\label{eq:decomp}
\right)=
\frac{1}{\sqrt{q_1^2 \Lambda_1^2 + q_2^2\Lambda_2^2 }}\left(
\begin{array}{cc}
q_2 \Lambda_2 & -q_1 \Lambda_1 \\
q_1 \Lambda_1 & q_2 \Lambda_2
\end{array}
\right)
\left(
\begin{array}{cc}
A_1^{(\dagger)} \\
A_2^{(\dagger)}
\end{array}
\right)\ .\end{aligned}$$ By using $A$ and $G$, the Kähler potential is rewritten by, $$\begin{aligned}
\label{eq:AGK}
K &=&
\L_1^2 \cosh\left(
2q_1 \tilde V - \frac{2q_2}{m_V}\frac{\L_2}{\L_1} (A^{\dagger}+A)
\right)
+
\L_2^2 \cosh\left(
2q_2 \tilde V + \frac{2q_1}{m_V}\frac{\L_1}{\L_2} (A^{\dagger}+A)
\right) \ , \end{aligned}$$ where $$\begin{aligned}
\tilde V &=& V - \frac{g}{m_V}(G^\dagger+G) \ , \\
m_V &=& 2g\sqrt{q_1^2 \L_1^2 + q_2 \L_2^2}\ .\end{aligned}$$ The final expression of Eq.(\[eq:AGK\]) shows there is no bi-linear term which mixes $A$ and $\tilde{V}$. Therefore, we find that $A$ corresponds to the physical axion superfield, while $G$ is the would-be Goldstone superfield which is absorbed by ${V}$ in the unitarity gauge. It should be noted that the physical axion $A$ is invariant under the gauge $U(1)_{gPQ}$ transformation.
For a later purpose, let us discuss the domain and the effective decay constant of the axion. The domains of the imaginary parts of $A_{1,2}$ (corresponding to the phases of $\Phi_{1,2}$) are given by $$\begin{aligned}
\frac{\operatorname{Im}[A_{i}]}{\L_{i}} = \frac{a_{i}}{f_{i}} = [0, 2\pi)\ , \quad (i = 1,2)\ ,\end{aligned}$$ where $a_{i} = \sqrt{2}\operatorname{Im}[A_i]$ and $f_{i} = \sqrt{2}\Lambda_i$. When $q_{1}$ and $q_2$ are relatively prime integers, the gauge invariant axion interval is given by [@Fukuda:2017ylt], a = \[A\] = \[0, ). Accordingly, the global $U(1)_{PQ}$ symmetry is realized by $$\begin{aligned}
\frac{a}{F_a} \to \frac{a'}{F_a} = \frac{a}{F_a} + \delta_{PQ}\ , \quad (\delta_{PQ} = 0-2\pi)\ ,\end{aligned}$$ where $F_a$ is defined as an effective decay constant, \[eq:effF\] F\_a= = .
Accidental Global PQ Symmetry
-----------------------------
As argued in [@Hawking:1987mz; @Lavrelashvili:1987jg; @Giddings:1988cx; @Coleman:1988tj; @Gilbert:1989nq; @Banks:2010zn], global symmetries are expected to be broken by quantum gravity effects which are manifested by explicit breaking terms suppressed by the Planck scale. On the other hand, such explicit breaking does not appear for the $U(1)_{gPQ}$ as it is an exact gauge symmetry. Thus, question is how well the accidental global PQ symmetry (corresponding to the shift of $A$) is protected from explicit breaking by the $U(1)_{gPQ}$ symmetry. An important observation here is that the $U(1)_{gPQ}$ symmetry is not distinguishable from the global PQ symmetries in each sector. Thus, there are no explicit breaking terms of the global PQ symmetries which consist of the fields in each PQ symmetric sector. Therefore, the interaction terms which potentially ruin the global PQ symmetries are the gauge invariant operators consisting of the fields in multiple PQ symmetric sectors. In the following, we estimate how badly the accidental global PQ symmetry is broken.
In the present model, the lowest dimensional $U(1)_{gPQ}$ invariant operators which break the global PQ symmetries are given by, \[eq:explicitW\] W\~ ( \_1\^[q\_2]{}|\^[q\_1]{}\_2 + |\^[q\_2]{}\_1 \_2\^[q\_1]{}) , where $M_{PL}=2.4 \times 10^{18}$GeV denotes the reduced Planck scale. When supersymmetry is spontaneously broken in a separate sector, the above superpotential contributes to the axion potential through the supergravity effects,[^4] \[eq:explicitV\] V \~ m\_a\^2 F\_a\^2 ()\^2 + \_1\^[q\_2]{}\_2\^[q\_1]{}+ h.c. + , where $m_{3/2}$ denotes the gravitino mass. In the final expression, we use $\Phi_1^{q_2}\bar{\Phi}^{q_1}_2 = \Lambda^{q_2 + q+1}/2^{(q_2+q_1)/2} e^{ia/F_a}$, and the intrinsic $\theta$ angle of QCD is absorbed by the definition of the axion field. The first term represents the axion mass term due to the QCD effects [@Weinberg:1977ma], $$\begin{aligned}
m_a^2 \simeq \frac{m_u m_d}{(m_u+m_d)^2} \frac{m_\pi^2 f_\pi^2}{F_a^2}\ ,\end{aligned}$$ where $m_{u,d}$ are the $u$- and $d$-quark masses, $m_\pi$ the pion mass, and $f_\pi \simeq 93$MeV the pion decay constant.
As a result, the effective $\theta$ angle at the vacuum of the axion is given by, \_[eff]{}&& \_1\^[q\_2]{}\_2\^[q\_1]{}\
&\~& 10\^[66-6.4(q\_1+q\_2)]{}( )\^2 ( )\^4 ( )\^[q\_2]{} ( )\^[q\_1]{}. Thus, for $q_1+q_2 { \mathop{}_{\textstyle \sim}^{\textstyle >} }12$, $m_{3/2} = {\cal O}(10^6)$GeV, and $\L_{1,2} ={\cal O}(10^{12})$GeV, the explicit breaking terms of the global PQ symmetries are small enough to be consistent with the measurement of the neutron EDM, i.e. $\theta_{\rm eff} < 10^{-11}$ [@Baker:2006ts]. In this way, a [*high quality*]{} global PQ symmetry appears as an accidental symmetry in the gauged PQ mechanism.
Domain Wall Problem {#sec:domainwall}
-------------------
Before closing this section, let us briefly discuss the domain wall problem. The anomaly cancelation condition in Eq.(\[eq:anomfree\]) is generically solved by, $$\begin{aligned}
N_1 = N_{\rm GCD}\times q_2\ , \quad N_2 = N_{\rm GCD}\times q_1\ ,\end{aligned}$$ where $N_{\rm GCD} \in {\mathbb N}$ is the greatest common devisor of $N_{1,2}$. Then, the anomalous coupling in Eq.(\[eq:anomSF\]) is rewritten by, $$\begin{aligned}
\label{eq:anomSF3}
W_{\rm anom} = \frac{N_{\rm GCD}}{8\pi^2}
\frac{\sqrt{2} A}{F_a}
\sum_l W_l^{\a}W_{l\a}\ ,\end{aligned}$$ and hence, \[eq:anomSM\] \_[anom]{}=\_[l]{} F\^[l]{}\_\^[l]{} . It should be noted that the anomalous coupling of the axion respects a discrete symmetry, ${\mathbb Z}_{N_{\rm GCD}}$, $$\begin{aligned}
\frac{a}{F_a} \to \frac{a}{F_a} + \frac{2\pi k}{N_{\rm GCD}}\ , \quad k = 0,\,\cdots,\, N_{\rm GCD} - 1 \ .\end{aligned}$$ for $N_{\rm GCD} > 1$.
The ${\mathbb Z}_{N_{\rm GCD}}$ symmetry is eventually broken in the vacuum of the axion. Thus, the model with $N_{\rm GCD} > 1$ suffers from the domain wall problem if the global PQ symmetry is broken after inflation since the average of the axion field value in each Hubble volume is randomly distributed. To avoid the domain wall problem, spontaneous breaking of the global PQ symmetry is required to take place before inflation, which in turn requires a rather small inflation scale to avoid the axion isocurvature problem (see, e.g. Ref. [@Kawasaki:2013ae; @Kawasaki:2018qwp]).
For $N_{\rm GCD} = 1$, on the other hand, there is no discrete symmetry which is broken by the VEV of the axion field. Still, however, there can be domain wall problems when the global PQ breaking takes place after inflation. To see this problem, let us remember that there can be various types of cosmic string configurations formed at spontaneous symmetry breaking of the PQ symmetries. For example, when both the gauged and the global PQ symmetries are broken spontaneously after inflation, there can be cosmic string configurations in which either the phase of $\Phi_{1}$ or $\Phi_{2}$ takes $0-2\pi$ around configurations. It should be noted that those configurations are the global strings and not the local string. Thus, the string tensions diverge in the limit of infinite volume which is cut off by the Hubble volume. The local string, on the other hand, corresponds to the configurations in which the phases of $\Phi_{1}$ and $\Phi_2$ wind $q_1$ times and $q_2$ times simultaneously. With the $U(1)_{gPQ}$ gauge field winding simultaneously, the tension of the local string is finite even in the limit of infinite volume for the local string.
A striking difference between the global strings and the local strings is how the axion field winds around the strings. Around the local strings, only the would-be-Goldstone field winds, while the axion winds around the global strings. Thus, when the axion potential is generated at around the QCD scale, the axion domain walls are formed only around the global strings, while they are not formed around the local strings. Once the domain walls are formed around the global strings, they immediately dominate over the energy density of the universe, which causes the domain wall problem. Therefore, for the domain wall problems not to occur, the local strings should be formed preferentially at the phase transition.
The string tensions of the global strings and the local strings, however, depend on model parameters. Thus, there is no guarantee that only the local strings preferentially survive in the course of the cosmic evolution. As an example, let us consider a case with ${ \left\langle {\Phi_1} \right\rangle } \gg { \left\langle {\Phi_2} \right\rangle }$. In this case, the cosmic strings are formed at the first phase transition, i.e. ${ \left\langle {\Phi_1} \right\rangle }\neq 0$ with ${ \left\langle {\Phi_2} \right\rangle } = 0$. At this stage, strings around which the phase of $\Phi_1$ winds just once are expected to be dominantly formed. They are [*local*]{} because we can take an appropriate charge normalization for the $U(1)_{gPQ}$. As the temperature of the universe decreases, the string networks follow the scaling solution where the number of the cosmic strings in each Hubble volume becomes constant (see, e.g., Ref. [@Vilenkin:2000jqa]).
Once the temperature becomes lower than the scale of the second phase transition, i.e., ${ \left\langle {\Phi_2} \right\rangle }\neq 0$, the [*local*]{} strings formed at the first phase transition become no more the local strings.[^5] Besides, formations of the global strings of $\Phi_2$ are also expected at the second phase transition in which the phase of $\Phi_2$ winds just once. To form a genuine local string, it is required to bundle $q_1$ ex-local strings (formed by $\Phi_1$) and $q_2$ global strings (formed by $\Phi_2$) into a single string. However, the confluence of global strings into a local string is quite unlikely as there is no correlation between the nature of the cosmic strings in the adjacent Hubble volumes. Therefore, when ${ \left\langle {\Phi_{1}} \right\rangle } \gg { \left\langle {\Phi_2} \right\rangle }$, the domain wall problem is expected to be not avoidable even if $N_{\rm GCD} = 1$.[^6]
In summary, let us list up possibilities to avoid the domain wall problem. The first possibility is a trivial one where both the gauged and the global PQ symmetries are broken before inflation. This solution does not require $N_{\rm GCD}= 1$. In this possibility, there is a constraint on the Hubble scale during inflation from the axion isocurvature problem.
The next possibility is only applicable for $N_{\rm GCD} = 1$ with $q_1 = 1$ and $q_2 = N (>1)$. Here, it is assumed that the first phase transition (i.e. ${ \left\langle {\Phi_1} \right\rangle }\neq 0$) takes place before inflation while the second phase transition (i.e. ${ \left\langle {\Phi_2} \right\rangle } \neq 0$) occurs after inflation. In this second possibility, the [*local*]{} strings formed at the first phase transition are inflated away. The global strings formed at the second phase transition, on the other hand, do not cause the domain wall problem as each of the global string is attached to only one domain wall [@Hiramatsu:2010yn; @Hiramatsu:2012gg].
In addition to these two possibilities, there can be another possibility which is applicable for $N_{\rm GCD} = 1$ with ${ \left\langle {\Phi_1} \right\rangle } \sim { \left\langle {\Phi_2} \right\rangle }$. In this case, there can be a possibility where the local strings are preferentially formed at the phase transition. Besides, the axion domain wall attached to the global strings may have very short lifetime for $N_{DW} = 1$ even if they are formed. To confirm this possibility, detailed numerical simulations are required, which goes beyond the scope of this paper.
It should be noted that the second possibility (and the third possibility if numerically confirmed) is one of the advantages of the gauged PQ mechanism over the models in which the global PQ symmetry results from an exact discrete symmetry, such as ${\mathbb Z}_N$. In such models, the axion potential also respects the ${\mathbb Z}_N$ symmetry, and hence, the domain wall problem is not avoidable when the global PQ symmetry is spontaneously broken after inflation. In the gauged PQ models, on the other hand, it is possible that the global PQ symmetry is broken after inflation without causing the domain wall problem nor the axion isocurvature problem.
Dynamical supersymmetry/PQ symmetry Breaking {#sec:proto}
=============================================
In this section, we discuss a model of a simultaneous breaking of supersymmetry and the [*global*]{} PQ symmetry. As we are interested in solutions to the strong $CP$-problem without severe fine-tuning, it is natural to seek models in which the PQ breaking scale is generate by dynamical transmutation. Thus, in the following, we construct a model of a simultaneous supersymmetry/PQ symmetry breaking sector based on a strong dynamics. For now, we do not consider the gauged PQ mechanism which will be implemented in the next section.
Simultaneous Breaking of Supersymmetry and Global PQ Symmetry
-------------------------------------------------------------
As the simplest example of the dynamical supersymmetry breaking models, we consider a model of supersymmetry breaking based on $SU(2) $ gauge dynamics (the IYIT model) [@Izawa:1996pk; @Intriligator:1996pu]. The advantage of this model is that the nature of dynamical supersymmetry breaking is calculable by using effective composite states.
The model consists of four $SU(2) $ doublets, $Q_i~(i=1-4)$, and six singlets, $Z_{ij}=-Z_{ji}~(i,j=1-4)$. Those superfields couple via the superpotential \[eq:tree\] W\_[IYIT]{}=\^[kl]{}\_[ij]{}Z\^[ij]{}Q\_kQ\_l where $\lambda^{kl}_{ij}$ denote coupling constants with $\lambda^{kl}_{ij}=-\lambda^{kl}_{ji}=-\lambda^{lk}_{ij}$. The maximal non-abelian global symmetry of the IYIT model is $SU(4)$ flavor symmetry, $SU(4)_f$, which is broken by $\lambda^{kl}_{ij}$.
The superpotential Eq.(\[eq:tree\]) respects a global $U(1)_A$ symmetry with charges, $Z$’s$(+2)$, $Q$’s$(-1)$, and a continuous $R$-symmetry, $U(1)_R$, with $Z$’s$(+2)$, $Q$’s$(0)$ (Tab.\[tab:IYIT\]). The former is broken down to the discrete subgroup, $\mathbb{Z}_4$, by the $SU(2) $ anomaly, while the latter is free from the $SU(2) $ anomaly. As we seek a solution to the strong $CP$ problem not relying on global symmetries, we consider that the $\mathbb{Z}_4$ and $U(1)_R$ symmetries are accidental symmetries and are broken by Planck suppressed operators.
It should be noted, however, that a discrete subgroup of $U(1)_R$, $\mathbb{Z}_{NR}~(N>2)$, plays crucial roles in constructing the SSM. Without $\mathbb{Z}_{NR}~(N>2)$ symmetry, the VEV of the superpotential is expected to be of the order of the Planck scale. Such a large VEV of the superpotential, in turn, does not allow a supersymmetry breaking scale lower than the Planck scale due to the condition for the flat present universe. In addition, it is also known that $R$-symmetry (or at least an approximate $R$-symmetry) is relevant for supersymmetry breaking vacua to be stable[@Affleck:1984xz; @Nelson:1993nf]. Given its importance, we assume that the $\mathbb{Z}_{NR}~(N>2)$ symmetry is an exact discrete gauge symmetry [@Krauss:1988zc; @Preskill:1990bm; @Preskill:1991kd; @Banks:1991xj; @Ibanez:1991hv; @Ibanez:1992ji; @Csaki:1997aw].[^7] In this paper, we take the simplest possibility, ${\mathbb Z}_{4R}$, assuming a presence of an extra multiplet of the ${\bf 5}$, $\bar{\bf 5}$ representations of the $SU(5)$ GUT. The ${\mathbb Z}_{4R}$ symmetry is free from the Standard Model anomaly when the $R$-charges of the bilinear term of the Higgs doublets and that of the extra multiplets are vanishing [@Kurosawa:2001iq; @Lee:2010gv; @Fallbacher:2011xg; @Evans:2011mf].[^8]
In this model, we identify the global PQ symmetry with a $U(1)$ subgroup of $SU(4)_f$ (Tab.\[tab:IYIT\]). As it is a subgroup of $SU(4)_f$, the PQ symmetry is free from the $SU(2) $ anomaly. Under the global $U(1)_{PQ}$ symmetry, the superpotential is reduced to W\_[IYIT]{} = \_[12]{}\^[12]{} Z\^[12]{}Q\_1 Q\_2 + \_[34]{}\^[34]{} Z\^[34]{}Q\_3 Q\_4 + \^[kl]{}\_[ij]{}Z\^[ij]{}Q\_kQ\_l where $\lambda$’s are dimensionless coupling constants with $\tilde\lambda^{kl}_{ij} = 0$ for $ij = 12$, $34$ or $kl$ = $12$, $34$. Hereafter, we take $\lambda^{12}_{12} = \lambda^{34}_{34} = \lambda$ for simplicity,although it is straightforward to extend the following analysis for $\lambda^{12}_{12}\neq \lambda^{34}_{34}$. As we will see shortly, the PQ symmetry is spontaneously broken by the VEV of $Q_{1}Q_2$ and $Q_3Q_4$.
\[tab:IYIT\]
By assuming the KSVZ axion model, the PQ symmetry is communicated to the SSM sector through couplings to the KSVZ extra multiplets in ${\bf 5}$ and $\bar{\bf 5}$ representations of the $SU(5)$ GUT,[^9] \[eq:KSVZ1\] W = Q\_1 Q\_2 [**5**]{}|[**5**]{} . The PQ charges of the KSVZ extra multiplets are given in Tab.\[tab:IYIT\]. Hereafter, we assume that there are $N_f$ flavors of the KSVZ extra multiplets. Once ${ \left\langle {Q_{3}Q_4} \right\rangle }$ spontaneously breaks the PQ symmetry, the axion couples to the SSM sector via Eq.(\[eq:KSVZ1\]) and the extra multiplets obtain masses of ${\cal O}({ \left\langle {Q_3Q_4} \right\rangle }/M_{PL})$.
Now, let us discuss how supersymmetry and the PQ symmetry are broken spontaneously. Below the dynamical scale of $SU(2) $ dynamics, $\Lambda$, the IYIT model is well described by using the composite fields, $M_{ij} \sim Q_iQ_j$, with an effective superpotential, \[eq:IYIT\] W\_[eff]{} \~ŁZ\_- [M]{}\_+ + ŁZ\_+ [M]{}\_- + \_[ab]{} ŁZ\_0\^a [M]{}\_0\^b +[X]{}(2 [M]{}\_+ [M]{}\_- + [M]{}\_0\^a [M]{}\_0\^a - \^2) . Here, ${M}_{+} \sim Q_1Q_2/\L$ and ${M}_{-} \sim Q_3Q_4/\L$ denote the PQ charged mesons, while ${M}_{0}^a$ $(a= 1-4)$ are the PQ neutral mesons. The coupling constants $\tilde \lambda$ and the singlets $Z_0$’s are also rearranged accordingly. In the effective superpotential, the quantum modified constraint $2{M}_+ {M}_- + {M}_0 {M}_0 - \Lambda^2=0$ [@Seiberg:1994bz] is implemented by a Lagrange multiplier field $\cal X$.
By assuming that $\lambda$’s are perturbative, and $\lambda_{\pm}(=\lambda)$ are smaller than $\tilde\lambda$’s, the VEVs of $M_\pm$ are given by $$\begin{aligned}
\label{eq:vacuum}
\langle M_+ \rangle= \frac{1}{\sqrt 2} \Lambda ,
\quad
\langle M_-\rangle = \frac{1}{\sqrt 2} \Lambda\ .\end{aligned}$$ Other fields do not obtain VEVs of ${\cal O}(\Lambda)$.[^10] At this vacuum, the PQ symmetry is spontaneously broken by $\langle M_\pm \rangle$ while supersymmetry is broken by the VEVs of the $F$-components of $Z_{\pm}$, i.e., $$\begin{aligned}
\label{eq:FZ}
F_{Z_\pm} \sim \frac{1}{\sqrt 2} \lambda \Lambda^2 \ ,\end{aligned}$$ simultaneously.
Here, let us comment that the ${\mathbb Z}_{4R}$ is not enough to restrict the superpotential in the form of Eq.(\[eq:tree\]). In fact, there can be superpotential terms such as $Z_0^3$ or $Z_0 Z_+ Z_-$ without the $U(1)_A$ (or ${\mathbb Z}_4$) symmetry. As those terms make the supersymmetry breaking vacuum in Eqs.(\[eq:vacuum\]) and (\[eq:FZ\]) metastable, the coefficients of those terms should be rather suppressed to make the vacuum long lived. Such suppression can be achieved, for example, by assuming that a subgroup of ${\mathbb Z}_4$ and $U(1)_{PQ}$ is an exact symmetry where $Z_{0}$’s are charged but $Z_\pm$ are neutral.[^11] It is also possible to suppress the unwanted terms by extending the SU(2) dynamics of the IYIT sector into a conformal window by adding extra doublets [@Ibe:2005pj; @Ibe:2005qv; @Ibe:2007wp].
Axion Supermultiplet
--------------------
The degeneracy due to the PQ symmetry breaking is parametrized by the axion superfield $A$, $$\begin{aligned}
\label{eq:axionSF}
M_+ = \frac{1}{\sqrt 2}\Lambda e^{A/\Lambda} \ , \quad
M_-= \frac{1}{\sqrt 2}\Lambda e^{-A/\Lambda} \ , \end{aligned}$$ with which the PQ symmetry is realized by $$\begin{aligned}
A/\Lambda \to A/\Lambda + i \alpha \ , \quad (\alpha = 0 - 2\pi)\ .\end{aligned}$$ Here, we reduce the domain of the $U(1)_{PQ}$ rotation parameter from $\alpha = 0-4\pi$ to $\alpha = 0-2\pi$, since all the $SU(2) $ gauge invariant fields have the PQ charge of $\pm 2$ (see Tab.\[tab:IYIT\]). In other words, the sign changes of $Q$’s by a phase rotation with $\alpha = 2\pi$ can be absorbed by a part of $SU(2) $ transformation.
The effective Kähler potential and superpotential of $M_\pm$ and $Z_\pm$ are given by, $$\begin{aligned}
\label{eq:axionEFF}
K_{\rm eff} &\sim& |Z_+|^2 + |Z_-|^2 + | M_+|^2 + | M_+|^2 + \cdots\ , \\
W_{\rm eff} &\sim& \lambda
\L M_+ Z_- + \lambda \L M_- Z_+ \ ,\end{aligned}$$ where the ellipses denote the higher dimensional operators. By substituting the axion superfield, the effective theory is reduced to $$\begin{aligned}
K_{\rm eff} &\sim& | Z_+|^2 + | Z_-|^2 + \frac{1}{2}(A^\dagger + A)^2 +\cdots \ , \\
\label{eq:axionEFF2}
W_{\rm eff} &\sim& \frac{1}{\sqrt 2}\lambda \Lambda^2 (Z_- e^{A/\L} + Z_+e^{-A/\L}) \ ,\end{aligned}$$ with some irrelevant holomorphic terms omitted in the Kähler potential. The scalar potential is accordingly given by,[^12] $$\begin{aligned}
V &\sim& \l^2 \L^4 \cosh\left(\frac{A^\dagger + A }{\L}\right)
+ \frac{1}{2}\l^2 \L^4\left| Z_+ e^{-A/\L} - Z_- e^{A/\L}\right|^2 \\
&\sim& \l^2 \L^4 \cosh\left(\frac{A^\dagger + A }{\L}\right)
+ \l^2 \L^4|T|^2 \ .\end{aligned}$$ In the final expression, we rearranged the scalar fields by introducing complex scalar fields $S$ and $T$, $$\begin{aligned}
Z_+ &=& \frac{1}{\sqrt{2}}(S+T)e^{-A/\L}\ , \\
Z_- &=& \frac{1}{\sqrt{2}}(S-T)e^{A/\L}\ , \end{aligned}$$ so that the PQ symmetry is manifest in the scalar potential.
The above scalar potential shows that the complex scalar $T$ and the real component of $A$ (the saxion) obtain masses of $\lambda\Lambda$, around their origins. The complex scalar filed $S$ (the pseudo-flat direction) and the imaginary part of $A$ (the axion $a$), on the other hand, remain massless. The pseudo-flat direction eventually obtains a mass from the higher order terms in the Kähler potential. For perturbative $\lambda$ and $\tilde\lambda$, the mass is dominated by the one-loop contributions [@Chacko:1998si; @Ibe:2009dx; @Ibe:2010ym], $$\begin{aligned}
m_S^2 \simeq \frac{1}{32\pi^2} \left({\lambda^2 (2\log2-1)}
+ \frac{4\lambda^4}{3\tilde\lambda^2} \right)
\frac{F_S^2}{ \Lambda^2}\ , \quad (F_S = \lambda \Lambda^2)\ ,\end{aligned}$$ with which the pseudo-flat direction is stabilized at its origin.[^13]
The superpotential in Eq.(\[eq:axionEFF2\]) also shows that the fermion partners of $A$ (the axino) and $T$ obtain a Dirac mass of $\lambda \Lambda$, with each other. The fermion partner of $S$ corresponds to the goldstino which is absorbed into the gravitino by the super-Higgs mechanism.
Putting together, the model achieves dynamical breaking of supersymmetry and the PQ breaking simultaneously. The axion supermultiplet splits into a massless axion and massive saxion/axino with masses of the supersymmetry/PQ breaking scale. The axion couples to the SSM sector via the coupling in Eq.(\[eq:KSVZ1\]), i.e., $$\begin{aligned}
W
\sim \frac{\L^2}{\sqrt{2}M_{\rm PL}} e^{A/{\Lambda} } \,\bar{\mathbf 5}\,{\mathbf 5} \sim \frac{\L^2}{\sqrt{2}M_{\rm PL}} e^{{ia}/{f_a} } \,\bar{\mathbf 5}\,{\mathbf 5} \ ,\end{aligned}$$ where $a = \sqrt{2} \operatorname{Im}[A]$ denotes the axion field and $f_a = \sqrt{2}\Lambda$. After integrating out the extra KSVZ multiplets, the axion couples to the SM gauge fields through $$\begin{aligned}
\label{eq:anomSF2}
W_{\rm anom} = \frac{N_f}{8\pi^2}
\frac{A}{\Lambda}
\sum_l W_l^{\a}W_{l\a}\ ,\end{aligned}$$ with which the strong $CP$ problem is solved
Explicit Breaking of the PQ symmetry
------------------------------------
Now, let us discuss explicit breaking of the global PQ symmetry expected in quantum gravity. In this model, the most relevant terms which break the global PQ symmetry are given by, [^14] W \~ Z\_[+]{} (Q\_[1]{}Q\_[2]{})\^2 + Z\_-(Q\_3 Q\_4)\^2 \~ Z\_ [M]{}\_\^2 . with $\k$ being a dimensionless coupling constant.[^15] The corresponding symmetry breaking terms in the scalar potentials are given by, V\~m\_a\^2 f\_a\^2()\^2 + ()\^2 \^4 e\^[i ]{} + h.c. . Here, we inserted the VEVs of $M_\pm$ and those of $F$-terms of $Z_\pm$. Due to the explicit breaking, the VEV of the axion, and hence, the effective $\theta$ angle is shifted to \[eq:quality\] \_[eff]{} = &&. \[ł\] ()\^2 ()|\_\
&&.10\^[40]{} \[ł\] ()\^4 ( )\^6 |\_ . Thus, unless $\operatorname{Im}[\k\l]$ is finely tuned to be smaller than ${\cal O}(10^{-11})$, the effective $\theta$ angle is too large to be consistent with the measurement of the neutron electric dipole moment (EDM) [@Baker:2006ts].
Gauged PQ Extension of simultaneous breaking model {#sec:model}
==================================================
Let us now implement the gauged PQ mechanism to the model of the simultaneous breaking of supersymmetry and the PQ symmetry in section\[sec:proto\]. For that purpose, we introduce an additional sector based on $SU(3)$ dynamics which breaks a PQ symmetry spontaneously. In the following, we call this model the $SU(3)'$ model, and put primes on the superfields and the symmetry groups in this sector.
$SU(3)'$ PQ Symmetry Breaking Model {#sec:pqbreaking}
-----------------------------------
The $SU(3)'$ model consists of three flavors of the (anti-)fundamental representation of $SU(3)$, $Q'$, $\bar{Q}'$, and nine $SU(3)$ singlets, $Z'$. The charge assignment of the global symmetries is given in Tab.\[tab:23\]. Under these symmetries, they couple via the superpotential W\_[PQ]{}=’\^[kl]{}\_[ij]{}Z\^[’ij]{}Q’\_k|[Q]{}’\_l , where $\lambda'^{kl}_{ij}$ denote coupling constants with $(i,j,k,l=1-3)$. The baryon symmetry, $U(1)_B$, is identified with the global PQ symmetry, $U(1)'_{PQ}$, while the maximal flavor symmetry, $SU(3)_L\times SU(3)_R$, is completely broken by $\lambda'$’s.
In addition to the global $U(1)'_{PQ}$ symmetry, the superpotential possesses a continuous $R$-symmetry and a $U(1)_A'$ symmetry (broken down to a ${\mathbb Z}_6$ symmetry by the $SU(3)'$ anomaly) in Tab.\[tab:23\]. As discussed previously, however, we consider that only ${\mathbb Z}_{4R}$ is an exact symmetry, and assume that $U(1)_R$ and $U(1)_A'$ are accidental symmetries broken by higher dimensional operators.[^16]
[|c|Y|Y|Y||Y|Y|]{} & $Q' $ &$\bar{Q}'$ &$Z'$ &${\bf 5}' $ & $\bar{ {\bf 5}}' $\
$SU(3)'$ & $\bf{3}$ &$\bar{\bf{3}}$ &$\bf{1}$ &$\bf{1}$ &$\bf{1}$\
$U(1)'_{PQ}$ & $+1$ &$-1$ &$0$ &$3$ & $0$\
$U(1)_R$ & $0$ &$0$ &$+2$ &$r_5'$ & $r_{\bar 5}'$\
$U(1)_A'$ & $+1$ &$+1$ &$-2$ &$-3$ & $0$\
\[tab:23\]
Below the dynamical scale of $SU(3)'$, $\L'$, the $SU(3)'$ sector is well described by the composite mesons and baryons, $$\begin{aligned}
M' \sim Q'\bar{Q}'/\L'\ , \quad B_+' \sim Q'Q'Q'/\L'^2\ , \quad {B}_-' \sim \bar{Q}'\bar{Q}'\bar{Q}'/\L'^2\ ,\end{aligned}$$ with an effective superpotential, \[eq:effSU3\] W’\_[eff]{}&=&’ ’ Z’ M’ + [X]{}’ (B’\_+ B’\_[-]{}+[det]{}(M’)/’-\^[’2]{}) . Here, the second term implements the deformed moduli constraint by a Lagrange multiplier field $\cal X'$ [@Seiberg:1994bz]. The mesons are neutral under $U(1)_{PQ}'$ while the baryons have charges $\pm 3$.
From the superpotential in Eq.(\[eq:effSU3\]), we find that the PQ symmetry is spontaneously broken by the VEVs of $B_\pm$. Accordingly, the vacuum is parametrized by the Goldstone superfield $A'$,[^17] $$\begin{aligned}
\label{eq:axionSF2}
B_+' &=&\frac{1}{\sqrt 2} \L_2 e^{A'/\Lambda_2}\ , \\
B_-' &=& \frac{1}{\sqrt 2}\L_2 e^{-A'/\Lambda_2}\ ,\end{aligned}$$ with $\L_2 =
\sqrt{2}\L' $. By using $A'$, the PQ symmetry is non-linearly realized by $$\begin{aligned}
\frac{A'}{\Lambda_2} \to \frac{A'}{\Lambda_2} + i\alpha' \ , \quad (\alpha' = 0 - 2\pi)\ .\end{aligned}$$ As in the case of the IYIT sector, the domain of the PQ symmetry is reduced from $\alpha' = 0 - 6\pi$ to $\alpha' = 0 - 2\pi$ as the $SU(3)'$ invariant fields have the PQ charges of $\pm 3$.
The $U(1)'_{PQ}$ symmetry in this sector is also communicated to the SSM sector through the couplings to $N_f'$ flavors of the KSVZ extra multiplets, $\mathbf 5'$ and $\bar{\mathbf 5}'$. With the charge assignment in Tab.\[tab:23\], the baryons couple to the extra multiplets in the superpotential, \[eq:KSVZ2\] W = |[Q]{}’|[Q]{}’|[Q]{}’ [**5**]{}’|[**5**]{}’ \~ B\_-’[**5**]{}’|[**5**]{}’ . Once $U(1)'_{PQ}$ is broken, the axion obtains the anomalous coupling to the SSM gauge fields, while the extra multiplets obtain masses of ${\cal O}(\Lambda'^3/M_{\rm PL}^2)$.
Gauged PQ Symmetry
------------------
\[tab:gPQ\]
Now, we are ready to find out a model of the gauged PQ symmetry by combining the simultaneous supersymmetry and the PQ symmetry breaking model in section\[sec:proto\] and the PQ symmetry breaking model in subsection\[sec:pqbreaking\]. To apply the prescription in section\[sec:prescription\], let us first identify $\Phi_1$ with the meson operator $M_+$ in section\[sec:proto\] and $\Phi_2$ with the baryon operator $B'_+$, i.e., $$\begin{aligned}
\Phi_1 = M_+\ , \\
\bar\Phi_2\ = B_-' \ , \end{aligned}$$ and assign $U(1)_{gPQ}$ charges of $q_1$ and $-q_2$ to them (Tab.\[tab:gPQ\]).[^18] Then, the anomaly-free condition of the $U(1)_{gPQ}$ symmetry in Eq.(\[eq:anomfree\]) is given by, $$\begin{aligned}
\label{eq:anomfree2}
q_1 N_f - q_2 N_f' = 0 \ .\end{aligned}$$
Once the two sectors are put together by the gauged PQ symmetry, spontaneous breaking of the PQ symmetries in the two sectors lead to the would-be Goldstone and the axion superfield. The would-be Goldstone is absorbed into the massive $U(1)_{gPQ}$ gauge multiplet, and the saxion and the axino in the axion supermultiplet obtain masses of the order of the supersymmetry breaking scale. As a result, the simultaneous breaking model with the gauged PQ mechanism leaves only a light axion which couples to the Standard Model gauge fields via Eq.(\[eq:anomSM\]).
Accidental Global PQ Symmetry
-----------------------------
As discussed in the previous section, the global PQ symmetry can be explicitly broken by the $U(1)_{gPQ}$ invariant operator consisting of the fields in the two sectors. Among the explicit breaking terms, the most relevant ones are given by,[^19] \[eq:explicit\] W &\~& (Z\_+(Q\_[1]{}Q\_2)\^[q\_2-1]{} (|[Q]{}’|[Q]{}’|[Q]{}’)\^[q\_1]{} + Z\_+(Q\_[3]{}Q\_4)\^[q\_2-1]{} (Q’Q’Q’)\^[q\_1]{} ) ,\
&\~& (Z\_+ M\_+\^[q\_2-1]{} [B]{}\_-’\^[q\_1]{} + Z\_- M\_-\^[q\_2-1]{} [B]{}\_+’\^[q\_1]{} ) , with $\k$ being a dimensionless coupling constant. It should be noted that these terms are consistent with the ${\mathbb Z}_{4R}$ symmetry, and hence, no factor of $m_{3/2}$ is required unlike the terms in Eq.(\[eq:explicitW\]). These operators roughly contribute to the axion potential, V \~ m\_a\^2 F\_a\^2 ()\^2 + \^[2q\_2]{}’\^[3q\_1]{}+ h.c. + , where the VEVs of $M_\pm$, $B_\pm'$, and those of the $F$-terms of $Z_{\pm}$ are inserted, Therefore, in the simultaneous breaking model with the gauged PQ mechanism, the effective $\theta$ angle at the vacuum is given by, \[eq:theta3\] \_[eff]{}&&\
&\~& 10\^[77.5-6.4(3q\_1+2q\_2)]{}( )\^4 ( )\^[2q\_2]{} ( )\^[3q\_1]{} . Thus, for $3q_1+2 q_2 { \mathop{}_{\textstyle \sim}^{\textstyle >} }14$, the explicit breaking of the global PQ symmetries are small enough to be consistent with the measurement of the neutron EDM, i.e., $\theta_{\rm eff}<10^{-11}$ [@Baker:2006ts].
Mass Spectrum of the KSVZ Multiplets {#sec:mass}
------------------------------------
The KSVZ multiplets, $(\bar{\mathbf 5},\bar{\mathbf 5})$ and $(\bar{\mathbf 5}',\bar{\mathbf 5}')$ were introduced to communicate the PQ symmetries to the SSM sector. After PQ symmetry breaking, those extra multiplets obtain supersymmetric masses of the order of $$\begin{aligned}
m_{KSVZ} &\sim& \frac{\Lambda^2}{M_{\rm PL}} \ , \\
\label{eq:KSVZp}
m_{KSVZ}' &\sim& \frac{\Lambda'^3}{M_{\rm PL}^2} \ , \end{aligned}$$ respectively (see Eqs.(\[eq:KSVZ1\]) and (\[eq:KSVZ2\])). The scalar components of the KSVZ multiplets also obtain masses of ${\cal O}(m_{3/2})$ through supergravity effects. Thus, most of the KSVZ multiplets become heavy and beyond the reach of the LHC experiments except for the fermion components of $(\bar{\mathbf 5}',\bar{\mathbf 5}')$.[^20]
The KSVZ extra multiplets are assumed to couple to the SSM particle via, $$\begin{aligned}
W
&\sim&
\frac{\e}{M_{\rm PL}} Q_1 Q_2 {\mathbf 5}\, {\bar{\mathbf 5}}_{SM} +
\frac{\e'}
{M_{\rm PL}^2}
\bar{Q}'\bar{Q}'\bar{Q}'
{\mathbf 5}' \,{\bar{\mathbf 5}}_{SM} \ ,\\
&\sim&
\epsilon \,m_{KSVZ} \,{\mathbf 5}\, {\bar{\mathbf 5}}_{SM} +
\epsilon'\, m_{KSVZ}' \,{\mathbf 5}' \,{\bar{\mathbf 5}}_{SM} \ ,\end{aligned}$$ where ${\bar{\mathbf 5}}_{SM}$ denotes the SSM matter multiplet, and $\e^{(')}$ are coefficients. Here, we take $r_5 = r_5' = 1$ so that $\bar{\mathbf 5}$ and $\bar{\mathbf 5}'$ have the same $R$-charges with $\bar{\mathbf 5}_{SM}$. Through the mixing terms, the KSVZ extra multiplets decay immediately into the SSM particles.
Finally, let us note that there can be mixing terms between (${\mathbf 5}$, $\bar{\mathbf 5}$) and (${\bar{\mathbf 5}'}$, ${\bar{\mathbf 5}'}$) through, $$\begin{aligned}
W \sim \frac{1}{M_{\rm PL}} Q_1 Q_2 {\mathbf 5}\, {\bar{\mathbf 5}'} +
\frac{1}
{M_{\rm PL}^2}
\bar{Q}'\bar{Q}'\bar{Q}'
{\mathbf 5}' \,{\bar{\mathbf 5}} \ .\end{aligned}$$ Although these operators consist of the fields in the two PQ symmetric sectors, they are invariant under not only the gauged PQ symmetry but also under the global PQ symmetries. Thus, these terms do not affect $\theta_{\rm eff}$. They do not affect the KSVZ mass spectrum significantly neither. From these reasons, we neglect these mixing terms throughout this paper.
PQ Charges in the $SU(3)'$ Model
--------------------------------
For a given $q_1$ and $q_2$, there are upper limits on $\Lambda$ and $\Lambda'$ to achive a high-quality global PQ symmetry (see Eq.(\[eq:theta3\])). The dynamical scales are also constrained from below for an appropriate supersymmetry breaking scale and for heavy enough KSVZ extra multiplets. As a lower limit on the supersymmetry breaking scale, i.e., $\Lambda$, we require $$\begin{aligned}
\label{eq:10TeV}
m_{3/2} &\simeq& \frac{\lambda\Lambda^2}{\sqrt{3}M_{\rm PL}} \gtrsim 10\,{\rm TeV}\ , \end{aligned}$$ so that the observed Higgs boson mass, $m_H \simeq 125$GeV, is achieved by the gravity mediated sfermion masses of ${\cal O}(m_{3/2})$. As a lower limit on the KSVZ extra multiplets, we put $$\begin{aligned}
\label{eq:750GeV}
m_{KSVZ}' &\simeq& \frac{\Lambda'^3}{M_{\rm PL}} \gtrsim 750\,{\rm GeV}\ ,\end{aligned}$$ from the null results of the searches for a heavy $b$-type quark at the LHC experiments [@Aad:2015kqa; @Aad:2015gdg; @Aad:2014efa; @Aad:2015mba].
In Fig.\[fig:const1\], we show the charge choices for $SU(3)'$ model for $N_{\rm eff} = 1$ and $\lambda= 1$. The charges colored by blue are excluded, with which $\theta_{\rm eff}$ cannot be suppressed enough for $m_{3/2}\gtrsim {\cal O}(1)\,$TeV and $m_{\rm KSVZ}' \gtrsim 750$GeV.[^21] In the figure, we require $\theta_{\rm eff} \lesssim 10^{-10}$ given ${\cal O}(1)$ uncertainties of the coefficients of the explicit breaking terms. The figure shows that these constraints exclude relatively small charges as the suppression of the explicit breaking term relies on large PQ charges.
![*Constraints on charges $q_1$ and $q_2$ in the $SU(3)'$ model for $N_{\rm GCD} = 1$. The allowed charges are colored by light blue and orange, although the orange colored charges are allowed only for $\Lambda' \gtrsim M_{\rm GUT}$. The charges colored by blue lead to too large $\theta_{\rm eff}$ or too light KSVZ extra multiplets. The gauge coupling constants of the SSM blow up below the GUT scale for the charges colored by red. The black colored charges are excluded as they are not relatively prime.* []{data-label="fig:const1"}](charge.pdf){width="\linewidth"}
The perturbative coupling unification of the SSM gauge coupling constants also puts constraints on the charges. From the anomaly-free condition in Eq.(\[eq:anomfree2\]), $N_f$ and $N_f'$ are given by $$\begin{aligned}
N_{f} = N_{\rm GCD}\times q_2\ , \quad N_{f}' = N_{\rm GCD} \times q_1\ .\end{aligned}$$ As the extra multiplets contribute to the renormalization group evolutions of the SSM gauge coupling constants and make them asymptotically non-free, the perturbative unification puts upper limits on $N_f$ and $N_f'$, and hence, on $q_1$ and $q_2$.
In Fig.\[fig:const1\], we color the charges by red, with which $\theta_{\rm eff} \lesssim 10^{-10}$ is not compatible with the perturbative unification. Here, we use the renormalization group equation at the one-loop level and require that $g_{1,2,3} < 4\pi$ below the GUT scale, i.e., $M_{\rm GUT} \simeq 10^{16}$GeV. We also take the masses of the sfermions, the heavy charged/neutral Higgs boson, and the Higgsinos to be at the gravitino mass scale. The gaugino masses are assumed to be dominated by the anomaly mediation effects [@Giudice:1998xp; @Randall:1998uk] which are roughly given by (see, e.g. [@Bhattacherjee:2012ed]), $$\begin{aligned}
m_{\rm bino} &\simeq& 10^{-2}\times m_{3/2}\ , \\
m_{\rm wino} &\simeq& 3\times 10^{-3}\times m_{3/2}\ , \\
m_{\rm gluino} &\simeq& 2.5\times 10^{-2}\times m_{3/2}\ , \end{aligned}$$ although the constraints do not depend on them significantly as long as they are in the TeV range. The gravitino mass is take to be within $10\,{\rm TeV}\le m_{3/2}\le 10$PeV. These choices are motivated by the pure gravity mediation model in Refs.[@Ibe:2006de; @*Ibe:2011aa; @*Ibe:2012hu] (see also Refs.[@Giudice:2004tc; @ArkaniHamed:2004yi; @Wells:2004di; @ArkaniHamed:2012gw] for closely related models).[^22] In the renormalization group evolution, we also take into account an extra multiplet required for the anomaly free condition of the ${\mathbb Z}_{4R}$ symmetry, whose masses are also at the gravitino mass scale.
The figure shows that the requirement for perturbative unification excludes the charges with $q_2> 7$ ($N_{f} > 7$). This is expected as $N_f$ flavors of the KSVZ extra multiplets have masses of $10\,{\rm TeV}\lesssim m_{KSVZ} \lesssim 10$PeV.[^23] On the other hand, a large $q_1$ is allowed. This is because the explicit breaking terms are suppressed by $(\Lambda'/M_{\rm PL})^{3q_1}$, and hence, a high-quality global PQ is possible even for a large $\Lambda'$ as long as $q_1$ is large. For a large $\L'$, $m_{KSVZ}'$ also becomes large, with which the perturbative unification is possible even if $N_f' = q_1$ is large. It should be noted, however, that the effective field theory approach is no more reliable when $\Lambda'$ is too close to the Planck scale. In the figure, we color the charges by orange if they require a large $\Lambda'$, i.e., $10^{16}\,{\rm GeV} \lesssim \Lambda' \lesssim 10^{17}$GeV. For $N_{\rm GCD} \ge 2$, there are no appropriate charges with which $\theta_{\rm eff}< 10^{-10}$ and the perturbative unification are compatible.
Parameter Regions in the $SU(3)'$ Model
---------------------------------------
In Fig.\[fig:parameters\], we show the parameter regions for a given $q_1$ and $q_2$. In each panel, we take $m_{3/2}<10$PeV and $\lambda = 1$, $10^{-1}$, $10^{-2}$, respectively. The gray shaded region is excluded, as $\theta_{\rm eff} < 10^{-10}$ is not satisfied (see Eq.(\[eq:theta3\])). The perturbative unification is not achieved in the blue shaded region. The red shaded region is excluded for too light KSVZ extra multiplets, i.e., $m_{KSVZ}' \lesssim 750$GeV. The green dashed lines are contours of the effective decay constant in Eq.(\[eq:effF\]).
The figure shows that the dynamical scale $\Lambda'$ is tightly constrained from above to achieve $\theta_{\rm eff} < 10^{-10}$ for the minimum charge choice, i.e., $q_1=5$ and $q_2 =1$. This is understood as the explicit breaking terms are not effectively suppressed for rather small charges. As a result, the PQ breaking scales are required to be low to avoid large explicit breaking effects. The upper limit on $\Lambda'$ becomes tighter for a larger $\Lambda$ as is expected from Eq.(\[eq:theta3\]). Furthermore, as the dynamical scale $\Lambda$ becomes larger for a smaller $\lambda$, the upper limit becomes even tighter for a smaller $\lambda$ for a given $m_{3/2}$. The constraints from the perturbative unification are, on the contrary, weaker since $m_{KSVZ}$ becomes larger for a smaller $\lambda$ for a given $m_{3/2}$.
An interesting property of the minimum choice is that the model predicts the KSVZ extra multiplets ($\mathbf 5'$, $\bar{\mathbf 5}'$) in the TeV range. This feature reflects the suppressed fermion masses of the KSVZ extra multiplet in Eq.(\[eq:KSVZp\]) caused by the composite nature of the PQ breaking field, i.e., $B_-'$, with a tight upper limit on $\Lambda'$. Thus, the model with the minimum charge choice can be tested by searching for vector-like colored particles at the LHC experiments.
For $q_1=7$ and $q_2 = 1$, the upper limit on $\Lambda'$ is weaker than for the minimum choice. This is because the suppression factor of the explicit breaking term, $(\Lambda'/M_{\rm PL})^{3q_1}$, can be very small even for a rather large $\Lambda'$ due to a large exponent. The constraint form the perturbative unification is, on the contrary, tighter for a large $q_1$ as $N_f'$ is proportional to $q_1$. For a large $N_f'$, the masses of the KSVZ extra multiplets, $m_{KSVZ}'$, is required to be high to avoid the blow-up of the gauge coupling constants below the GUT scale.
![ *The constraints on the parameter reions for given PQ charges, $q_1$ and $q_2$. The gray region is excluded as $\theta_{\rm eff} < 10^{-10}$ is not satisfied. The perturbative unification is not achieved in the blue region. The red regions are excluded by $m_{KSVZ}' \gtrsim 750$GeV. The green lines are the contours of the effective decay constant $F_a$.* []{data-label="fig:parameters"}](param_5_1.pdf){width="\linewidth"}
![ *The constraints on the parameter reions for given PQ charges, $q_1$ and $q_2$. The gray region is excluded as $\theta_{\rm eff} < 10^{-10}$ is not satisfied. The perturbative unification is not achieved in the blue region. The red regions are excluded by $m_{KSVZ}' \gtrsim 750$GeV. The green lines are the contours of the effective decay constant $F_a$.* []{data-label="fig:parameters"}](param_7_1.pdf){width="\linewidth"}
![ *The constraints on the parameter reions for given PQ charges, $q_1$ and $q_2$. The gray region is excluded as $\theta_{\rm eff} < 10^{-10}$ is not satisfied. The perturbative unification is not achieved in the blue region. The red regions are excluded by $m_{KSVZ}' \gtrsim 750$GeV. The green lines are the contours of the effective decay constant $F_a$.* []{data-label="fig:parameters"}](param_1_7.pdf){width="\linewidth"}
For $q_1 =1$ and $q_2 = 7$, the upper limit on $\Lambda'$ is also weaker than the minimum choice for $\lambda = 1$ due to a strong suppression of the explicit breaking terms by $(\Lambda/M_{\rm PL})^{2 q_2}$. As the suppression factor is sensitive to $\Lambda$, the upper limit on $\Lambda'$ becomes very tight for a smaller $\lambda$ for a given gravitino mass.
In all cases, we find that the gravitino mass is required to be in the hundreds TeV or larger, and hence, the model can be consistent with the observed Higgs boson mass achieved by the gravity mediated sfermion masses. It is also notable that the dynamical scale $\Lambda'$ is larger than $\Lambda$ in the allowed parameter region. Therefore, both the accidental global PQ symmetry and supersymmetry are broken by the IYIT sector while the gauged PQ symmetry is mainly broken by the $SU(3)'$ sector. This feature is attractive as it explains the coincidence between the global PQ breaking scale and the supersymmetry breaking scale.
Before closing this subsection, let us comment on the axion dark matter abundance. The axion starts coherent oscillation when the Hubble expansion rate becomes comparable to the axion mass, which leads to the present axion dark matter density [@Turner:1985si], \_[axion]{}h\^20.2\^2\_[i]{}()\^[1.19]{} . Here, $\theta_{i}$ is the initial misalignment angle of the axion field. Thus, the axion can be a dominant component for dark matter of $F_a = {\cal O}(10^{12})$GeV, i.e., $\Omega_{\rm DM} \simeq 0.12$ [@Calabrese:2017ypx]. As the figures show, $F_a = {\cal O}(10^{12})$GeV is possible in a wide range of the parameter space. Therefore, the model based on $SU(3)'$ can be consistent with the axion dark matter scenario.[^24]
Cancellation of Self- and Gravitational Anomalies {#sec:anomcancel}
--------------------------------------------------
As mentioned in section \[sec:prescription\], the gravitational anomaly and the self-anomaly of $U(1)_{gPQ}$ are canceled by adding $U(1)_{gPQ}$ charged singlet fields. In this subsection, we show a concrete model of the anomaly cancelation.
In the IYIT sector and the $SU(3)'$ sector, the $U(1)_{gPQ}$ charged fields are paired with fields with opposite charges. Thus, the fields in these sectors do not contribute to the self-anomaly nor the gravitational anomaly. The charges of the KSVZ extra multiplets are, on the other hand, not paired, and hence, they contribute to the anomalies, $$\begin{aligned}
{\cal A}_{\rm self}^{\rm KSVZ}&=&-5 N_f q_1^3 + 5 N_f' q_2^3 \ , \\
{\cal A}_{\rm gravitational}^{\rm KSVZ} &=&-5 N_f q_1 +5 N_f' q_2 \ ,\end{aligned}$$ respectively. The easiest way to cancel the anomaly is to introduce $5N_f$ singlet superfields $Y$ with a charge $q_1$ and $5N_f'$ singlet superfields $Y'$ with a charge $-q_2$. The charges of $Y$’s and $Y'$’s are given in Tab. \[tab:gPQ\].
As the singlet fields do not have mass partners with opposite charges, the supersymmetric masses of them are generated only after $U(1)_{gPQ}$ breaking. The mass terms of $Y$’s are given by \[eq:mY\] W\~ (Q\_3Q\_4)\^2YY \~m\_[3/2]{}YY . Here, we take the ${\mathbb Z}_{4R}$ charge of $Y$’s to be $1$, so that their scalar and fermion components are odd and even under the $R$-parity, respectively. The factor $m_{3/2}$ encapsulates the effects of spontaneous breaking of the ${\mathbb Z}_{NR}$ symmetry. As a result, the fermionic components of $Y$’s obtain $$\begin{aligned}
m_Y \sim 3\,\times 10^{-7}\,{\rm eV} \left(\frac{m_{3/2}}{10^6\,{\rm GeV}}\right)\left(\frac{\Lambda}{10^{13}\,{\rm GeV}}\right)^4\ ,\end{aligned}$$ while the masses of scalar components are dominated by the gravity mediated soft masses of ${\cal O}(m_{3/2})$.
The supersymmetric masses of $Y'$’s are even smaller, \[eq:mYp\] W\~ (|[Q]{}’|[Q]{}’|[Q]{}’)\^2Y’Y’ \~m\_[3/2]{} ()\^6Y’Y’ . Here, we take the ${\mathbb Z}_{4R}$ charge of $Y'$’s to be $1$, and the factor $m_{3/2}$ encapsulates the effects of spontaneous breaking of ${\mathbb Z}_{NR}$ again. As a result, the fermionic components of $Y'$’s obtain, $$\begin{aligned}
m_{Y'} \sim 5\times 10^{-12}\,{\rm eV}
\left(\frac{m_{3/2}}{10^6\,{\rm GeV}}\right)
\left(\frac{\Lambda'}{10^{14}\,{\rm GeV}}\right)^6\end{aligned}$$ while the masses of the scalar components of $Y'$’s are dominated by the gravity mediated soft masses as in the case of $Y$’s.[^25]
If the light fermions are abundantly produced in the early universe, they contribute to the dark radiation and result in an unacceptably large number of effective neutrino species, $N_{\rm eff}$. To evade this problem, we assume that spontaneous breaking of $U(1)_{gPQ}$ takes place before the end of inflation. We also assume that the gauge superfields of $U(1)_{gPQ}$ are heavier than the reheating temperature after inflation. Furthermore, it is also assumed that the branching fraction of the inflaton into $Y$’s and $Y'$’s are suppressed. With these assumptions, we can achieve cosmologically consistent models where the self- and the gravitational anomalies are canceled by the $U(1)_{gPQ}$ charged singlets.
$SU(N)'$ Dynamical PQ Symmetry Breaking Model
---------------------------------------------
So far, we have considered the dynamical PQ breaking sector based on the $SU(3)$ gauge theory. There, the deformed moduli constraint plays an important role to break the global PQ symmetry (i.e., the baryon symmetry) spontaneously. In this subsection, we discuss the models of dynamical PQ breaking based on $SU(N)$ gauge theory other than $N = 3$. We call such models, the $SU(N)'$ dynamical PQ breaking model.
First, let us consider the $SU(2)'$ model. With four fundamental representations of $SU(2)'$, $Q'$, the model exhibits the deformed moduli constraint. In this model, there is no baryon symmetry, and the global PQ symmetry is identified with a subgroup of the maximal non-abelian group $SU(4)_f$ as in the case of the IYIT sector. Then, the global PQ symmetry breaking is achieved by introducing four PQ neutral singlet superfields, $Z'$.[^26]
In this model, the KSVZ extra multiplets coupling to the $SU(2)'$ sector obtain masses via, $$\begin{aligned}
W\sim \frac{1}{M_{\rm PL}} Q_1' Q_2' {\mathbf 5}'\,\bar{\mathbf 5}' \ ,\end{aligned}$$ leading to $$\begin{aligned}
m_{KSVZ'} \sim \frac{\Lambda'^2}{M_{\rm PL}}\ .\end{aligned}$$ Thus, the $SU(2)'$ model allows a rather small $\Lambda'$ compared with the $SU(3)'$ model to achieve $m_{KSVZ}' \gtrsim 750$GeV. It is even possible to be $\L' \ll\L$. The possibility of $\L' \ll \L$ is, however, not very attractive as the model does not explain a coincidence between the PQ breaking scale and the supersymmetry breaking scale.
Next, let us consider the $SU(N)'$ $(N>3)$ model. In this case, the global PQ symmetry is identified with the baryon symmetry which is broken by the deformed moduli constraint as in the case of the $SU(3)'$ model. As the mass terms of the KSVZ extra multiplets, $\mathbf 5'$ and $\bar{\mathbf 5}'$, are given by, $$\begin{aligned}
W\sim \frac{1}{M_{\rm PL}^{N-2}} (\bar{Q}'\cdots \bar{Q}') {\mathbf 5}'\,\bar{\mathbf 5}' \ ,\end{aligned}$$ the dynamical scale $\Lambda'$ should be much higher than $\Lambda$ to satisfy $m_{KSVZ}' \gtrsim 750$GeV. Here, $\bar{Q}\cdots \bar{Q}$ denotes the baryon operators of the $SU(N)'$ sector. The $SU(N)'$ models are very similar to the $SU(3)'$ model except for the dynamical scale $\Lambda'$, although we do not discuss details of the $SU(N)'$ model further.
Conclusions {#sec:cons}
===========
In this paper, we apply the gauged PQ mechanism to a model in which the global PQ symmetry and supersymmetry are broken simultaneously. As a concrete example, we considered models which consist of simultaneous supersymmetry/PQ symmetry breaking sector based on $SU(2)$ dynamics (the IYIT sector) and a dynamical PQ symmetry breaking sector based on $SU(N)$ dynamics (the $SU(N)'$ sector). As we have seen, the $SU(3)'$ model is particularly successful where the gauged PQ symmetry is mainly broken by the $SU(3)'$ sector while both the accidental global PQ symmetry and supersymmetry are broken by the IYIT sector. Thus, the model explains the coincidence between the favored values of the supersymmetry breaking scale and the PQ breaking scale.
Besides the model with the minimum charge choice, $q_1 = 5$ and $q_2 = 1$, predicts the KSVZ extra multiplets ($\mathbf 5'$, $\bar{\mathbf 5}'$) in the TeV range due to a tight upper limit on $\Lambda'$. It should be noted that the light KSVZ fermions are due to composite nature of the PQ breaking field with a tight upper limit on $\L'$ in the minimal model. The model also predicts non-vanishing effective $\theta$ angle. Thus, the model with the minimum charge choice can be tested by combining the searches for vector-like colored particles at the LHC experiments and future measurements of the neutron EDM.
Finally, let us comment an advantage of the gauged PQ mechanism over the models in which the high-quality global PQ symmetry results from an exact discrete symmetry, such as ${\mathbb Z}_N$. As we have discussed briefly in subsection \[sec:domainwall\], the gauged PQ mechanism with $N_{\rm GCD} = 1$, $q_1 = 1$ and $q_2 = N (>1)$ allow models which are free from both the domain wall problem and the axion isocurvature problem. The assumption here is that the first stage of the phase transition (i.e. ${ \left\langle {\Phi_1} \right\rangle }\neq 0$) takes place before inflation while the second stage of the phase transition (i.e. ${ \left\langle {\Phi_2} \right\rangle } \neq 0$) occurs after inflation. Then, the local strings formed at the first phase transition are inflated away, while the global strings formed at the second phase transition do not cause the domain wall problem as $\Phi_2$ couples to only one-flavor of the KSVZ extra multiplet.[^27] As the global PQ symmetry is broken after inflation, the model does not suffer from the axion isocurvature problem. This option is not available in the models with an exact discrete symmetry where the axion potential is also symmetric under the discrete symmetry.
The authors would like to thank K. Harigaya and K. Yonekura for useful discussion on domain wall problems. This work is supported in part by Grants-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science, and Technology (MEXT) KAKENHI, Japan, No. 25105011, No. 15H05889 and No. 16H03991 (M. I.), No. 26104001 and No.26104009 and No.16H02176 (T. T. Y.); 17H02878 (M. I. and T. T. Y.), and by the World Premier International Research Center Initiative (WPI), MEXT, Japan. The work of H.F. is supported in part by a Research Fellowship for Young Scientists from the Japan Society for the Promotion of Science (JSPS).
[^1]: For correspondence between the sfermion mass scale and the Higgs boson mass, see also [@Ellis:1990nz; @Okada:1990gg; @Haber:1990aw]. For constraints on the PQ breaking scale, see, e.g., [@Raffelt:2006cw; @Kawasaki:2013ae; @Patrignani:2016xqp].
[^2]: Here, we set the origins of $A_{i}$ at which $\Phi_{i} = \bar{\Phi}_i$, while ${ \left\langle {\Phi_i} \right\rangle }\neq { \left\langle {\bar{\Phi}_i} \right\rangle }$ for ${ \left\langle {A_{i}} \right\rangle }\neq 0$.
[^3]: Here, the gauge indices of $SU(3)_c$ and $SU(2)_L$ are suppressed, and the GUT normalization is used for $U(1)_Y$.
[^4]: In supergravity, a superpotential term $W_i$ directly appears in the scalar potential as V= (n\_i - 3)m\_[3/2]{}W\_i + h.c. , with $n_i$ being the mass dimension of $W_i$.
[^5]: The configuration of the gauge field formed at the first phase transition does not coincide with the one required for the local string with ${ \left\langle {\Phi} \right\rangle }_2 \neq 0$.
[^6]: As there is no corresponding discrete symmetry, the domain wall is not stable completely. For ${ \left\langle {\Phi_1} \right\rangle }\gg { \left\langle {\Phi_2} \right\rangle }$, however, the decay rate (i.e., the puncture rate and/or the rate of the breaking off) is highly suppressed.
[^7]: In Ref.[@Harigaya:2013vja], it is proposed to achieve the global PQ symmetry as an accidental symmetry protected by the exact discrete $R$-symmetry without relying on the gauged PQ mechanism.
[^8]: For GUT models which are consistent with the ${\mathbb Z}_{4R}$ symmetry, see, e.g., [@Izawa:1997he; @Harigaya:2015zea].
[^9]: The KSVZ extra multiplets should be distinguished the extra multiplets required to cancel the Standard Model anomaly of the ${\mathbb Z}_{4R}$ symmetry.
[^10]: The scalar components of $Z_\pm$ and ${\cal X}$ obtain small VEVs of ${\cal O}(m_{3/2})$.
[^11]: As this symmetry is not broken spontaneously at the vacuum, and hence, $Z_0$’s and $M_0$’s are predicted to be stable. Thus, the simultaneous breaking of the IYIT sector should take place before inflation to avoid the production of those stable particles if we assume the above symmetry.
[^12]: Throughout the paper, we use the same symbols to describe the superfields and their scalar components.
[^13]: Here, we neglect the one-loop contributions from the $U(1)_{gPQ}$ gauge interaction by assuming that the gauge coupling constant is small. The contributions from the gauge interaction, in fact, destabilize the origin of the pseudo flat direction [@Dine:2006xt; @Ibe:2009dx; @Ibe:2010ym].
[^14]: Here, we require that $U(1)_{PQ}$ is not broken by renormalizable interactions as a part of definition of the global symmetry.
[^15]: Lower dimensional operators which break the PQ symmetry, such as $Z_+^4/M_{PL}$, are forbidden by the ${\mathbb Z}_{4R}$ symmetry.
[^16]: Without $U(1)_A'$ (or ${\mathbb Z}_6$), the superpotential terms such as $Z'^3$ are allowed even if we assume the ${\mathbb Z}_{4R}$ symmetry. Such terms, however, do not change the following discussion.
[^17]: The origin of $A'$ is set at which $B'_+ = B'_-$, and ${ \left\langle {B'_+} \right\rangle } \neq { \left\langle {B'_-} \right\rangle }$ for ${ \left\langle {A'} \right\rangle } \neq 0$, accordingly.
[^18]: The $U(1)_{gPQ}$ charges of $Q_{1,2}$ and $\bar{Q}'$’s corresponds to $q_1/2$ and $ q_2/3$, respectively.
[^19]: There are lower dimensional operators which break the global PQ symmetry with $M_\pm$ replaced by $M_{PL} \times Z_{\pm}$ in Eq.(\[eq:explicit\]). The explicit breaking effects of those operators are comparable to the ones of Eq.(\[eq:explicit\]) due to suppressed $A$-term VEVs of $Z_\pm = {\cal O}(m_{3/2})$.
[^20]: The extra multiplet to achieve the ${\mathbb Z}_{4R}$ symmetry also obtains the mass of ${\cal O}(m_{3/2})$ from the $R$-symmetry breaking effects [@Casas:1992mk].
[^21]: The charges colored by blue are not changed even if the lower limits on $m_{3/2}$ and $m_{KSVZ}'$ are relaxed to $m_{3/2}\gtrsim 100$GeV and $m_{KSVZ}' \gtrsim 100$GeV.
[^22]: Here, the Higgsino mediation effects neglected for simplicity. Besides, the gaugino spectrum is deflected from the anomaly mediation in the presence of the KSVZ extra multiplets [@Harigaya:2013asa].
[^23]: If we restrict to $m_{3/2}<1$PeV, the constraint becomes tighter and the charges with $q_2 >5$ are excluded.
[^24]: For $m_{3/2} \gg {\cal O}(1)$PeV, the wino is expected to be heavier than ${\cal O}(1)$TeV, whose relic abundance exceeds the observed dark matter density. In such parameter region, we need to assume either a dilution mechanism of dark matter or $R$-parity violation.
[^25]: There are mass terms proportional to $YY'$ which can be lager than Eqs.(\[eq:mY\]) and (\[eq:mYp\]) depending on the parameters. Even in such cases, there remain light fermions with masses either $m_Y$ or $m_Y'$ as the numbers of $Y$’s and $Y'$’s are different.
[^26]: It is tempting to make the $SU(2)'$ sector also be the IYIT supersymmetry breaking sector by introducing six singlet fields, $Z'$’s, instead. In this case, however, supersymmetry and the gauged PQ symmetry are broken by the dynamics, while the global PQ symmetry is broken separately.
[^27]: In this case, the axion dark matter density is dominated by the axions produced by the decay of the string-domain wall networks, which requires $F_a = {\cal O}(10^{11})$GeV [@Hiramatsu:2010yn; @Hiramatsu:2012gg]. Such a rather low $F_a$ is, for example, achieved in the $SU(2)'$ model.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We present the identification of two previously known young objects in the solar neighbourhood as a likely very wide binary. [TYC9486-927-1]{},an active, rapidly rotating early-M dwarf, and [2MASSJ21265040$-$8140293]{},a low-gravity L3 dwarf previously identified as candidate members of the $\sim$45 Myr old Tucana Horologium association (TucHor). An updated proper motion measurement of the L3 secondary, and a detailed analysis of the pair’s kinematics in the context of known nearby, young stars, reveals that they share common proper motion and are likely bound. New observations and analyses reveal the primary exhibits Li 6708 Å absorption consistent with M dwarfs younger than TucHor but older than the $\sim$10Myr TW Hydra association yielding an age range of 10-45Myr. A revised kinematic analysis suggests the space motions and positions of the pair are closer to, but not entirely in agreement with, the $\sim$24 Myr old $\beta$ Pictoris moving group. This revised 10-45Myr age range yields a mass range of 11.6–15 M$_J$ for the secondary. It is thus likely [2MASSJ2126$-$8140]{}is the widest orbit planetary mass object known ($>$4500AU) and its estimated mass, age, spectral type, and $T_{eff}$ are similar to the well-studied planet $\beta$ Pictoris b. Because of their extreme separation and youth, this low-mass pair provide an interesting case study for very wide binary formation and evolution.'
author:
- |
N.R. Deacon[^1]$^{1}$, J.E. Schlieder$^{2,3}$, S.J. Murphy$^4$\
$^1$Centre for Astrophysics Research, University of Hertfordshire, College Lane, Hatfield, AL10 9AB, UK\
$^2$NASA Postdoctoral Program Fellow, NASA Ames Research Center, Moffett Field, CA, USA\
$^3$Max Planck Institute for Astronomy, Konigstuhl 17, Heidelberg, 69117, Germany\
$^4$Research School of Astronomy and Astrophysics, Australian National University, Canberra, ACT 2611, Australia\
bibliography:
- 'ndeacon.bib'
title: 'A nearby young M dwarf with a wide, possibly planetary-mass companion[^2]'
---
\[firstpage\]
planets and satellites: detection, stars: binaries: visual, stars: brown dwarfs, stars: pre-main-sequence
Introduction
============
Very wide orbit ($>$1000AU) extrasolar planets represent a currently small but previously unexpected population of massive gas giant companions to stars. To date four such objects have been discovered by direct imaging by a variety of routes. WD 0806-661B (@Luhman2011; 2500AU 6–9M$_{J}$; @Luhman2012) was discovered with a targeted observation of a nearby white dwarf, GUPscb (@Naud2014; 2000AU, 9–12M$_{J}$) was found with a targeted observation of a young, nearby moving group member, SR12C by observations of a T Tauri binary in $\rho$ Ophiuchus (@Kuzuhara2011; 1100AU, 6–20M$_J$), whilst Ross458C (@Goldman2010; 1160AU, 5–14M$_{J}$) was discovered in widefield survey data and then identified as having a common proper motion with its host binary.
In this work we present the identification of two previously known young objects in the solar neighbourhood, TYC 9486-927-1 and 2MASS J21265040-8140283, as a co-moving wide pair with a probable planetary mass secondary. During an examination of the literature we found that these two objects are separated by 217$"$ and have similar proper motions. Hence we attempted to better determine their properties to see if they were a likely young, bound system. Using revised astrometry and detailed kinematic analyses of nearby young stars and brown dwarfs, we have determined that this previously known young brown dwarf/free-floating planetary mass object and young low mass star are a likely widely separated bound pair. We also present new spectroscopic observations and re-examinations of literature and archival data to refine the age of the system and estimate that the secondary is likely planetary mass and may be the widest orbit exoplanet yet discovered.
TYC 9486-927-1 was observed by [@Torres2006] as part of the Search for Associations Containing Young stars (SACY) programme [@Torres2008]. They assigned a spectral type of M1 and measured a radial velocity of $v_{rad}=8.7\pm4.6$km/s from 10 observations. The large uncertainty is likely due to the star’s high rotational velocity ($v\sin i=43.5\pm1.2$km/s); suggesting it is either a single rapid rotator or a spectroscopic binary with blended lines. [TYC9486-927-1]{}also shows signs of activity in X-ray [@Thomas1998], H$\alpha$ emission [@Torres2006] and the UV (using [*GALEX*]{} data from @Martin2005 we find $\log F_{FUV}/F_{J}$=-2.49, $\log F_{NUV}/F_{J}$=-2.11). [2MASSJ2126$-$8140]{}is an L3 first identified by [@Reid2008] (although referencing @Cruz2009 as the discovery paper). Subsequently [@Faherty2013] classified it as a low gravity L3$\gamma$ (using the gravity classification system of @Cruz2009). Recent VLT/ISAAC observations by [@Manjavacas2014] find it is a good match to the young, L3 companion CD-35 2722B [@Wahhaj2011]. These authors also used the spectral indices of [@Allers2013] to confirm that the [2MASSJ2126$-$8140]{}is an L3 and shows low gravity spectral features. [@Manjavacas2014] also used the BT-Settl-2013 atmospheric models [@Allard2012] to derive $T_{eff}=1800\pm100$K, $\log g = 4.0\pm0.5$dex, albeit with better fits to super-solar metallicity models. [@Filippazzo2015] use photometry, a trigonometric parallax of 31.3$\pm$2.6mas (referenced to Faherty et al., in prep.) and evolutionary models to derive an effective temperature of 1663$\pm$35K. They also derived a mass of 23.80$\pm$15.19M$_J$ assuming a broad young age range of 10–150Myr. [@Gagne2014] listed [2MASSJ2126$-$8140]{}as a high probability candidate member of Tucana-Horologium association (TucHor) but noted that its photometric distance would be in better agreement with its TucHor kinematic distance if it were an equal mass binary.
TYC 9486-927-1 and 2MASS J21265040$-$8140293
============================================
Our parameters for both components of this system are listed in Table \[sys\_sum\]. Below we outline how these were derived.
Astrometry of 2MASS J21265040$-$8140293
---------------------------------------
Using 2MASS and WISE astrometry, [@Gagne2014] measured proper motions of $\mu_{\alpha}\cos\delta=46.7\pm1.3$mas/yr and $\mu_{\delta}=-107.8\pm10.4$mas/yr for [2MASSJ2126$-$8140]{}. This is deviant by 6$\sigma$ in the R.A direction from the UCAC4 measurements [@Zacharias2013] for [TYC9486-927-1]{}of $\mu_{\alpha}\cos\delta=58.9\pm1.5$mas/yr and $\mu_{\delta}=-109.4\pm1.0$mas/yr. Due to the very small uncertainty on [@Gagne2014]’s $\mu_{\alpha}\cos\delta$ measurement and the availability of newer datasets we recalculated the astrometric solution for [2MASSJ2126$-$8140]{}using data from 2MASS [@Skrutskie2006], the WISE All-Sky release[@Wright2010], one epoch of WISE post-cryo data, one epoch of the reactivated NEO-WISE mission [@Mainzer2011] and the DENIS survey [@Epchtein1994]. For each of the three WISE epochs we averaged the single exposure positional measurements to produce three datapoints. We assumed positional errors (84mas on both axes) from the quoted errors on the position for [2MASSJ2126$-$8140]{}in the WISE All-Sky data release Source Catalogue and applied these to all three of our WISE datapoints [^3]. For 2MASS we used the quoted positions and positional error and for DENIS we used the approach of [@Luhman2013], measuring the positional scatter on objects of similar brightness close to the target. This latter calculation yielded positional uncertainties of 100mas in both R.A. and Dec. which were applied to positions averaged from the different DENIS epochs. These measurements were combined in a least squares fit which resulted in proper motion measurements of $\mu_{\alpha}\cos\delta=49.3\pm9.7$mas/yr and $\mu_{\delta}=-105.5\pm6.6$mas/yr. These figures deviate by less than $1\sigma$ from the UCAC4 proper motion measurements for [TYC9486-927-1]{}from [@Zacharias2013]. Our proper motion fit along with those from [@Gagne2014] and the UCAC4 proper motion for [TYC9486-927-1]{}are shown in Figure \[pm\_fit\]. The congruent proper motions are readily apparent on the plane of the sky in Figure \[pm\_im\].
![\[pm\_fit\] Our proper motion fit for [2MASSJ2126$-$8140]{}using data from various infrared surveys. The solid line is our proper motion fit, the dashed line is the proper motion for [TYC9486-927-1]{}[@Zacharias2013] shifted so it matches our proper motion at the midpoint of our dataset and the dotted line is the [@Gagne2014] proper motion extrapolated from the 2MASS position.](./pm_plot.ps)
[lrc]{} Template&RV (km/s)&Cross-correlation power\
1800K model&10.5$\pm$1.1&18%\
LHS 2065 (M9)&7.4$\pm$1.8&23%\
LHS 2351 (M7)&8.4$\pm$1.4&24%\
LHS 292 (M6)&8.1$\pm$1.4&23%\
GL 644C (M7)& 7.6$\pm$1.5&24%\
Adopted value&8.4$\pm$2.1\
The radial velocity of 2MASS J21265040$-$8140293
------------------------------------------------
[2MASSJ2126$-$8140]{} was observed with the Phoenix spectrograph [@Hinkle2003] mounted to the Gemini-South telescope on UT 2009 October 29 (Programme GS-2009B-C-2, PI K. Cruz). The observations consisted of two AB pairs with each exposure lasting 1800s. The data was obtained in the H-band with the 0.34 arcsecond slit which provides a resolving power of approximately 50,000. Along with the science data, flat lamp and dark calibration exposures were obtained on the same night. These data were downloaded from the Gemini Archive and reduced using a series of custom IDL routines. We corrected for bad pixels then flat-fielded and dark subtracted the science frames using a median master flat and dark frame. We attempted to extract the one dimensional spectrum from both sky-subtracted AB pairs, but the trace was only detected by our software in one pair. OH night sky lines were used to solve the dispersion solution and establish a wavelength scale. The final extracted spectrum covered 1.5512 - 1.5577 microns and had a SNR$\sim$5. Prior to cross correlation with model and observed templates, the spectrum was flattened by dividing by a 4th order polynomial fit to the continuum to the continuum, flux normalized, and corrected for the barycentric velocity.
To measure the RV of 2MJ2126, we cross-correlated the spectrum with a 1800K, log(g)=4.0, solar metallicity BT Settl [@Allard2010] model spectrum (to match the parameters from @Manjavacas2014) and observed M6, M7, and M9 Keck/NIRSPEC template spectra from [@Prato2002]. Our analyses provide consistent RVs using each template, all four measurements are listed in Table \[sec\_rv\]. All measurements are in reasonable agreement with [TYC9486-927-1]{}’sRV (10.0$\pm$1.0km/s, see Section \[tyc\_char\]). We adopt a range of 8.4$\pm$2.1km/s for [2MASSJ2126$-$8140]{}’sradial velocity based on our measurements. We note that this radial velocity estimate comes from a very low signal to noise spectrum. However we include this measurement to demonstrate that we have analysed the available archive data for [2MASSJ2126$-$8140]{}and can find no data which suggests that it is not in a bound system with [TYC9486-927-1]{}.
Characterisation of TYC 9486-927-1 {#tyc_char}
----------------------------------
TYC 9486-927-1 was classified as an active M1 by [@Torres2006] who also detected a Lithium 6708Å feature with an equivalent width of 104mÅ. We observed TYC 9486-927-1 with FEROS (@Kaufer1999; R=48,000, 3600–9200Å)on 2012 October 7 using all the standard settings, reductions, and RV analyses detailed in sections 2.3.7 and 3.11 of [@Bowler2015]. We subsequently observed the star on 2015 August 26 and 2015 October 27 with the WiFeS instrument on the ANU 2.3m telescope at Siding Spring using the R7000 grating (@Dopita2007; R=7000, 5250–7000Å).The WiFeS instrument set up, data reduction and analysis, including the derivation of line widths and radial velocities, was the same as that described in [@Murphy2014]. Our FEROS spectrum showed emission in H$\alpha$, H$\beta$, H$\gamma$ and H$\delta$ and yielded measurements of $v_{rad}=10.0\pm1.0$km/s,$EW_{Li}=85\pm$15mÅ and $v\sin i=40.0\pm2.0$km/s, while the WiFeS observations measured $v_{rad}=10.7\pm1.0$km/s and 9.7$\pm$1.0km/s respectively with $EW_{Li}=90\pm$10mÅin both observations. Table \[TYC\_spec\] shows the spectroscopic properties of [TYC9486-927-1]{}from our observations and the literature. Notably, the star exhibits no RV variations outside of uncertainties on the scale of months to years, strongly suggesting it is not a close to equal mass spectroscopic binary system. We adopt a radial velocity of 10.0$\pm$1.0km/s for the star, in agreement with the FEROS and WiFeS data, and lower precision measurements of 8.7+/-4.6 km/s [@Torres2006] and 11.9$\pm$3.0km/s [@Malo2014]. To further examine the possible binary nature of [TYC9486-927-1]{},we performed 2D cross correlations on our FEROS spectrum with a variety of primary/secondary template mass ratio combinations. None of these tests yielded a reliable cross-correlation function with power larger than in the case of a single star.
To garner an improved spectral type, we made an additional observation of [TYC9486-927-1]{}with WiFeS on the 2015 November 28 with the lower resolution R3000 grating (R=3000, 5200-9800A). We visually compared the flux calibrated and telluric corrected spectrum to spectral type standards from the lists of E. Mamajek[^4] observed that night with the same instrument settings. Figure \[wifes\_spec\] shows that [TYC9486-927-1]{}has a spectral type between M2 (GJ 382) and M2.5 (GJ 381), inconsistent with the M1 spectral type reported by [@Torres2006]. To compliment the visual comparison we also measured several molecular spectral type indices recently calibrated by [@Lepine2013]. These are listed in Table \[TYC\_ind\] compared to measurements made from a low-resolution spectrum of TYC 9486-927-1 by [@Gaidos2014], who assigned a spectral type of M3 from visual inspection. Based on all available spectroscopic observations we assign a spectral type of M2.0$\pm$0.5 for [TYC9486-927-1]{}. This also agrees with photometric spectral types obtained from $V-J$ (M2.3, @Lepine2013) and $V-K_s$ (M1.8, @Pecaut2013) colours.
[@Elliott2015] imaged TYC 9486-927-1 with the VLT/NACO AO imager and found no companion, despite being able to detect an equal mass companion down to a projected separation of 3AU (at an assumed photometric distance of 36.3pc distance or 4.2AU adjusting that photometric distance for equal-mass binarity). These observations and our multi-epoch radial velocity data suggest that TYC 9486 is a single, rapidly rotating star rather than a spectroscopic or tight, visual binary. However, it is still possible that TYC 9486-927-1 is an equal mass binary with a face-on orbit and close separation.
[lllcccc]{} Source&Date (UT)&SpT&RV(km/s)&EW Li (mÅ)&EW H$\alpha$ (Å)&$v\sin i$(km/s)\
[@Torres2006]&2001-09-08&M1e&8.7$\pm$4.6&104&-5.6&43.5$\pm$1.2\
[@Malo2014]&2010-05-25&…&11.9$\pm$3.0&…&…&44.8$\pm$4.2\
[@Gaidos2014]&…&M3&…&…&-3.9&…\
This work, FEROS&2012-10-07&…&10.0$\pm$1.0&85$\pm$15&…&40.0$\pm$2.0\
This work, WiFeS&2015-08-26&…&10.7$\pm$1.0&90$\pm$10&-5.7$\pm$0.2&…\
&2015-10-27&…&9.7$\pm$1.0&90$\pm$10&-9.5$\pm$0.2&…\
&2015-11-28&M2$\pm$0.5&…&…&-6.0$\pm$0.5&…\
[lcccccccc]{} Source&CaH2&CaH3&TiO5&VO1&VO2&ColorM&Visual&Final\
[@Gaidos2014]&0.613(M1.2)&0.793(M1.6)&0.697(M1.2)&…&…&…&M3&\
This work&0.574(M1.7)&0.769(M2.1)&0.591(M2.2)&0.940(M2.7)&0.886(M2.4)&1.575(M3.1)&M2&M2\
### The age of TYC 9486-927-1
[TYC9486-927-1]{}has rapid rotation and coronal and chromospheric activity suggestive of a young age. We measured a weak lithium 6708Å absorption feature in both our FEROS (85$\pm$15mÅ)and WiFeS spectra (90$\pm$10mÅ, see Figure \[wifes\_spec\] inset)consistent with the 104mÅmeasurement of [@Torres2006]. The detection of lithium is an important age diagnostic for early M-dwarfs, as the element is typically depleted in the photospheres of such stars on time scales of <40 Myr (e.g. @Mentuch2008). To illustrate this, in Figure \[Li\_plot\] we show the Li I 6708ÅEWs of M1 and M2 type stars from the SACY sample [@DaSilva2009] for the TW Hydrae (TWA; 10$\pm$3Myr), $\beta$ Pic (24$\pm$3Myr) and TucHor (45$\pm$4Myr) associations with additional TucHor members from [@Kraus2014a] (all ages from @Bell2015) against the EW for [TYC9486-927-1]{}. It is clear that the [TYC9486-927-1]{}has stronger lithium absorption than stars of similar spectral type in TucHor, weaker absorption than TWA members but in reasonable agreement with $\beta$ Pic members. Based on this comparison we conclude that [TYC9486-927-1]{}is older than TWA and likely of similar age or younger than TucHor. Thus, our Li analysis suggests an age comparable to the $\beta$ Pic moving group, but we note that Li depletion in low-mass stars can be affected by initial conditions (rotation, episodic accretion) and we therefore adopt a conservative age range of 10–45 Myr.
![\[Li\_plot\] The lithium 6708Å absorption for [TYC9486-927-1]{}(solid line with dashed line error bars) plotted against M1 and M2 members [@DaSilva2009] of the TWA (10$\pm$3Myr), $\beta$ Pic (24$\pm$3Myr) and TucHor (45$\pm$4Myr) associations (all ages from @Bell2015) with additional TucHor members from [@Kraus2014a]. The solid data points with error bars show the mean and standard deviation of each association. Clearly [TYC9486-927-1]{}has a lithium absorption strength between members of TWA and TucHor and in agreement with $\beta$ Pic.](./TYC9486_Li_comp2.ps)
[lcc]{} &[TYC9486-927-1]{}&[2MASSJ2126$-$8140]{}\
Position (J2000)&21 25 27.52 $-$81 38 27.8$^a$&21 26 50.40 $-$81 40 29.3$^a$\
$\mu_{\alpha}\cos\delta$ (mas/yr)&58.9$\pm$1.5$^{b}$&49.3$\pm$9.7$^{c}$\
$\mu_{\delta}$ (mas/yr)&$-$109.4$\pm$1.0$^{b}$&$-$105.5$\pm$6.6$^{c}$\
$v_{rad}$ (km/s)&10.0$\pm$1.0$^{c}$&8.4$\pm$2.1$^{c*}$\
$J$ (mag.)&8.244$\pm$0.03$^{g}$&15.54$\pm$0.06$^{g}$\
$H$ (mag.)&7.563$\pm$0.027$^{g}$&14.40$\pm$0.05$^{g}$\
$K_s$ (mag.)&7.34$\pm$0.04$^{g}$&13.55$\pm$0.04$^{g}$\
$W1$ (mag.)&7.241$\pm$0.032$^{h}$&12.910$\pm$0.024$^{h}$\
$W2$ (mag.)&7.151$\pm$0.021$^{h}$&12.472$\pm$0.023$^{h}$\
$W3$ (mag.)&7.039$\pm$0.016$^{h}$&11.885$\pm$0.161$^{h}$\
$W4$ (mag.)&6.953$\pm$0.061$^{h}$&$>$9.357$^{h}$\
Age&10–45Myr$^{c}$&10–150Myr$^f$\
Mass&$\sim$0.4M$_{\odot}$$^{c}$&11.6–15.0M$_J$$^{c}$\
Distance (pc)&20.5-29.0$^{c}$&31.9$^{+2.9}_{-2.4}$$^d$\
Spectral Type&M2$^c$&L3$\gamma$$^{e}$\
Separation&\
&\
\
\
\
\
\
\
\
\
\
Photometric distances and moving group membership
-------------------------------------------------
[TYC9486-927-1]{}lacks a trigonometric parallax measurement and thus any determination of its kinematics (and hence moving group membership) requires photometric distance estimates. To estimate absolute near-IR magnitudes for [TYC9486-927-1]{}we used our measured spectral type of M2. We then derived an effective temperature of 3490K for [TYC9486-927-1]{}using the 5–30Myr young star $T_{eff}$ scale of [@Pecaut2013] and applied this to the evolutionary models of [@Baraffe2015] at four ages (10, 20, 30 and 40Myr) to estimate the absolute magnitudes. [TYC9486-927-1]{}’s $J$, $H$ and $K_s$ 2MASS photometry were then compared to the these absolute magnitudes to calculate distances, neglecting the likely negligible extinction. We took the mean of these estimates as the adopted distance for [TYC9486-927-1]{}for each age (see Table \[TYC\_dist\]). Binarity would change the photometric distances although our multi-epoch RV measurements and the high resolution imaging of [@Elliott2015] show no evidence of a close companion to [TYC9486-927-1]{}.
To compare to the trigonometric parallax quoted in [@Filippazzo2015] for [2MASSJ2126$-$8140]{}we used the young L dwarf photometric distance relations of [@Gagne2015a]. Following a similar process to that described above but adopting the scatter on the relations as the error on our distances. As the [@Gagne2015a] relations cover a wide range of ages (up to 125Myr) they also cover a wide range of luminosities for each spectral type due to young objects having inflated radii. Hence the photometric distances do not deviate randomly across bands but will be correlated. Thus we do not adopt a weighted mean distance but take the distance from the band with the lowest scatter ($W2$, d= 26.7$^{+5.7}_{-4.7}$pc). This distance, and those for TYC 9486-927-1 using the 10 and 20Myr [@Baraffe2015] models agree well with the trigonometric parallax presented by [@Filippazzo2015].
To test the membership of TYC 9486-927-1 in several well known moving groups in the solar neighbourhood, we imputed our radial velocity (10$\pm$1km/s), the UCAC4 proper motions and the positions, along with the distance estimates for each age, into the BANYANII young moving group membership probability estimation tool [@Malo2013; @Gagne2014]. We assumed a 20% error on our photometric distance estimates and that the objects were younger than 1Gyr. For [2MASSJ2126$-$8140]{}we used our proper motion, and both the photometric distance estimate and the literature trigonometric parallax. The results of these calculations are shown in Table \[TYC\_dist\]. They suggest that the system is unlikely to be a TucHor member. Over the range of estimated photometric distances in Table \[TYC\_dist\] , BANYAN II provides probabilities of $\beta$ Pic membership ranging from about 4.9 to 74%. The membership probability for [2MASSJ2126$-$8140]{}is on the lower end of this range when we allow the radial velocity to be unconstrained. To further investigate the potential moving group membership we plotted the Galactic $U,V,W$ velocities and $X,Y,Z$ positions for [TYC9486-927-1]{}and [2MASSJ2126$-$8140]{}(Figure \[kin\_plot\]). We find that the reason BANYAN discounts TucHor membership is due to the system being a significant outlier in the $Z$ coordinate. While $\beta$ Pic remains the most likely moving group (both in kinematics and in age) for this system to be associated with, a difference in the $U$ velocity precludes us from claiming this is a bona-fide $\beta$ Pic member.
[cclccccccclll]{} Object&SpT&Age&RV (km/s)&$d_J$&$d_H$&$d_K$&$d_{W1}$&$d_{W2}$&$d_{adopted}$&$p_{TH}$&$p_{BP}$&$p_{YF}$\
TYC 9486-927-1&M2&10Myr&10.0$\pm$1.0&29.1$\pm$5.8&28.9$\pm$5.8&29.0$\pm$5.8&…&…&29.0$\pm$5.8&0.0%&59.1%&40.2%\
&M2&20Myr&10.0$\pm$1.0&26.4$\pm$5.3&26.0$\pm$5.2&26.1$\pm$5.2&…&…&26.2$\pm$5.2&0.0%&74.0%&25.8%\
&M2&30Myr&10.0$\pm$1.0&22.8$\pm$4.6&22.3$\pm$4.5&22.5$\pm$4.5&…&…&22.6$\pm$4.5&0.0%&71.8%&28.18%\
&M2&40Myr&10.0$\pm$1.0&20.7$\pm$4.1&20.3$\pm$4.1&20.5$\pm$4.1&…&…&20.5$\pm$4.1&0.0%&55.6%&44.4%\
&…&…&10.0$\pm$1.0&…&…&…&…&…&31.9$^{+2.9}_{-2.4}$&1.6%&9.5%&88.3%\
2MASS J2126$-$8140&L3&$<$125Myr&…&26.9$^{+19.0}_{-11.1}$&26.2$^{+10.2}_{-7.4}$&24.9$^{+7.3}_{-5.7}$&26.7$^{+8.0}_{6.2}$&26.7$^{+5.7}_{-4.7}$&26.7$^{+5.7}_{-4.7}$&0.0%&29.8%&68.8%\
&…&…&…&…&…&…&…&…&31.9$^{+2.9}_{-2.4}$&0.3%&4.9%&91.7%\
&L3&$<$125Myr&8.4$\pm$2.1&26.9$^{+19.0}_{-11.1}$&26.2$^{+10.2}_{-7.4}$&24.9$^{+7.3}_{-5.7}$&26.7$^{+8.0}_{-6.2}$&26.7$^{+5.7}_{-4.7}$&26.7$^{+5.7}_{-4.7}$&0.0%&69.4%&30.6%\
&…&…&8.4$\pm$2.1&…&…&…&…&…&31.9$^{+2.9}_{-2.4}$&1.0%&34.8%&64.0%\
Probability of chance alignment
-------------------------------
While it appears that [TYC9486-927-1]{}and [2MASSJ2126$-$8140]{}have matching proper motions and distances and both show signs of youth it is possible that they are a chance alignment of unrelated young objects. To determine the likelihood of this, we modified the method of [@Lepine2007]. We first constructed a list of known and candidate young stars in nearby young moving groups from [@Torres2008], [@Shkolnik2009], [@Schlieder2012], [@Kraus2014a] and [@Malo2014]. We then offset the positions of these stars by two degrees and searched for common proper motion companions in the 2MASS-WISE proper motion catalogue of [@Gagne2015] around these offset positions. This should result in only chance alignments of unrelated objects. We ran this process 18 times, on each occasion offsetting the positions of our input sample by 2 degrees but changing the direction of the offset by one-ninth of a radian each time. In this way we were able to sample a much larger area for chance alignments and thus reduce statistical noise. The results are shown in Figure \[prob\_lepine\] and clearly show a low probability ($\sim5\%$) of chance alignment considering all objects in Gagne’s proper motion sample or only those with L dwarf-like colours ($J-K_s>1.2$). There is also the possibility of the chance alignment between two unbound objects in the same moving group. Hence we carried out a simulation to see how often two members of one group would fall close to each other on the sky and have photometric distance estimates within 10pc of each other. To accomplish this we generated a random realisation of each of the eight young moving groups described in [@Gagne2014] using the parameters provided by that work. We then ran this simulation 50,000 times and determined that there is $<$1% chance that two objects in the same moving group would lie as close together on the sky as [TYC9486-927-1]{}and [2MASSJ2126$-$8140]{}(see Figure \[prob\_lepine\] lowest panel). We assumed that all chance alignments between members of one moving group would have matching proper motions due to both components having the bulk space velocity of the moving group. Note we did not consider the number of pairings between members of different moving groups (i.e. the number of chance alignments between AB Dor and TWA members). Even accounting for a factor of 2 or 3 missing members in the groups (especially at the lowest masses), it is clear that TYC 9486-927-1 and 2MASS J2126-8140 are unlikely to be chance alignments inside the same group.
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![\[prob\_lepine\] [**Top two rows:**]{}The results of our chance alignment using a modification of the offset position method of [@Lepine2007]. The top row includes matches with all objects in [@Gagne2015]’s catalogue while the middle row only includes matches with objects with $J-K_s>1.2$. [**Bottom row:**]{} The probability of finding two members of the same moving group at a particular separation on the sky from each other and with distances which agree within 10pc. Clearly it is unlikely that our proposed pair (marked by a star or a red arrow) is a chance alignment.](gagne_offset_plot_2d.ps "fig:") ![\[prob\_lepine\] [**Top two rows:**]{}The results of our chance alignment using a modification of the offset position method of [@Lepine2007]. The top row includes matches with all objects in [@Gagne2015]’s catalogue while the middle row only includes matches with objects with $J-K_s>1.2$. [**Bottom row:**]{} The probability of finding two members of the same moving group at a particular separation on the sky from each other and with distances which agree within 10pc. Clearly it is unlikely that our proposed pair (marked by a star or a red arrow) is a chance alignment.](gagne_offset_plot_1d.ps "fig:")
![\[prob\_lepine\] [**Top two rows:**]{}The results of our chance alignment using a modification of the offset position method of [@Lepine2007]. The top row includes matches with all objects in [@Gagne2015]’s catalogue while the middle row only includes matches with objects with $J-K_s>1.2$. [**Bottom row:**]{} The probability of finding two members of the same moving group at a particular separation on the sky from each other and with distances which agree within 10pc. Clearly it is unlikely that our proposed pair (marked by a star or a red arrow) is a chance alignment.](gagne_offset_plot_2d_L.ps "fig:") ![\[prob\_lepine\] [**Top two rows:**]{}The results of our chance alignment using a modification of the offset position method of [@Lepine2007]. The top row includes matches with all objects in [@Gagne2015]’s catalogue while the middle row only includes matches with objects with $J-K_s>1.2$. [**Bottom row:**]{} The probability of finding two members of the same moving group at a particular separation on the sky from each other and with distances which agree within 10pc. Clearly it is unlikely that our proposed pair (marked by a star or a red arrow) is a chance alignment.](gagne_offset_plot_1d_L.ps "fig:")
![\[prob\_lepine\] [**Top two rows:**]{}The results of our chance alignment using a modification of the offset position method of [@Lepine2007]. The top row includes matches with all objects in [@Gagne2015]’s catalogue while the middle row only includes matches with objects with $J-K_s>1.2$. [**Bottom row:**]{} The probability of finding two members of the same moving group at a particular separation on the sky from each other and with distances which agree within 10pc. Clearly it is unlikely that our proposed pair (marked by a star or a red arrow) is a chance alignment.](r_hist.ps "fig:")
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Physical properties of 2MASS J21265040$-$8140293
================================================
In order to estimate the mass of [2MASSJ2126$-$8140]{}, we used a Monte Carlo simulation. We drew temperatures from a flat distribution from the 1700–1900K range quoted by [@Manjavacas2014] and ages from our flat 10–45Myr range. We then compared with the COND [@Baraffe2003] and [@Saumon2008] ($f_{sed}=2$) evolutionary models. The [@Baraffe2003] models yielded masses between 11.6$M_{Jup}$ and 14.7$M_{Jup}$ [^5] and the [@Saumon2008] models preferred solutions in the 13.3–15$M_{Jup}$ range. Hence we adopt a mass range of 11.6–15$M_{Jup}$ range for [2MASSJ2126$-$8140]{}placing it on the 13$M_{Jup}$ deuterium-burning dividing line between planets and brown dwarfs. A similar calculation drawing the age from a [@Bell2015] $\beta$ Pic Gaussian age distribution of 24$\pm$3Myr yields a mass range of 12–14$M_J$ . Such masses and ages make 2MASS J2126-8140 an interesting wide-orbit analogue to $\beta$ Pic b [@Lagrange2010], whose primary is a member of the eponymous moving group.. [@Morzinski2015] find that $\beta$ Pic b has a mass of 12.7$\pm$0.3 $M_{Jup}$ and $T_{eff}$ of 1708$\pm$23K whilst [@Bonnefoy2013] find a spectral type of L1$^{+1.0}_{-1.5}$, $T_{eff} = 1700\pm100$K, $\log g = 4.0 \pm 0.5$dex and mass of $10_{-2}^{+3} M_{Jup}$ from $T_{eff}$ and $9_{-2}^{+3} M_{Jup}$ from luminosity. These compare well with our derived age range, 11.6–15$M_{Jup}$ mass range and [@Manjavacas2014]’s $T_{eff}=1800\pm100$K and $\log g = 4.0\pm0.5$dex for [2MASSJ2126$-$8140]{}. We note that $\beta$ Pic b is commonly referred to as a planet and also has mass estimates straddling the deuterium burning limit. Our system is also a younger analogue of the AB Dor (149$^{+51}_{-19}$Myr; @Bell2015) member GU Psc A/b system [@Naud2014] with both components having similar masses.
The range of photometric distances for the system gives projected separations of 4450-5700AU and the trigonometric parallax of the secondary a separation of 6900AU, wider than any star-planet system in the exoplanet.eu database (<http://exoplanet.eu/>).
Conclusions
===========
In summary: we have identified two previously known young objects [TYC9486-927-1]{}and [2MASSJ2126$-$8140]{}as having common proper motion. We find that the photometric distances of the pair agree and that they are unlikely to be an alignment of two unrelated young objects. Using the strength of the lithium 6708Åfeature we find an age range of 10–45Myr for [TYC9486-927-1]{}yielding a mass of 11.6–15$M_J$ for [2MASSJ2126$-$8140]{}. We note that the system has a wider separation than any known star-planet system and that [2MASSJ2126$-$8140]{}is similar in age, mass and temperature to the known exoplanet $\beta$ Pic b.
Acknowledgments {#acknowledgments .unnumbered}
===============
This research made use of the SIMBAD database, operated at CDS, Strasbourg, France. This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. This publication also makes use of data products from NEOWISE, which is a project of the Jet Propulsion Laboratory/California Institute of Technology, funded by the Planetary Science Division of the National Aeronautics and Space Administration. The DENIS project has been partly funded by the SCIENCE and the HCM plans of the European Commission under grants CT920791 and CT940627. It is supported by INSU, MEN and CNRS in France, by the State of Baden-Württemberg in Germany, by DGICYT in Spain, by CNR in Italy, by FFwFBWF in Austria, by FAPESP in Brazil, by OTKA grants F-4239 and F-013990 in Hungary, and by the ESO C&EE grant A-04-046. Jean Claude Renault from IAP was the Project manager. Observations were carried out thanks to the contribution of numerous students and young scientists from all involved institutes, under the supervision of P. Fouqué, survey astronomer resident in Chile. Based on observations obtained at the Gemini Observatory (PID GS-2009B-C-2, acquired through the Gemini Science Archive), which is operated by the Association of Universities for Research in Astronomy, Inc., under a cooperative agreement with the NSF on behalf of the Gemini partnership: the National Science Foundation (United States), the National Research Council (Canada), CONICYT (Chile), the Australian Research Council (Australia), Ministério da Ciência, Tecnologia e Inovação (Brazil) and Ministerio de Ciencia, Tecnología e Innovación Productiva (Argentina). The research of J.E.S. was supported by an appointment to the NASA Postdoctoral Program at NASA Ames Research Center, administered by Oak Ridge Associated Universities through a contract with NASA. We thank the anonymous referee for the prompt and helpful review that improved the quality and clarity of this manuscript. The authors would like to the thank the Brass Monkey, Heidelberg for the opportunity to pit their wits against each other every week.
\[lastpage\]
[^1]: E-mail:n.deacon2@herts.ac.uk
[^2]: Based on observations made with the ESO/MPG 2.2m telescope at the La Silla Observatory under programme ID 090.A-9010.
[^3]: The WISE All-Sky Source Catalogue position is the average of multiple measurements at one of our epochs and thus the error on this averaged position will be representative of the error on our averaged position at each of our three WISE epochs.
[^4]: <http://www.pas.rochester.edu/~emamajek/spt/>
[^5]: The full range of masses derived, not a confidence interval.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In the recent years there has been an enormous development in the evaluation of higher order quantum corrections. An essential ingredient in the practical calculations is provided by vacuum diagrams, i.e. integrals without external momenta. In this paper a program package is described which can deal with one-, two- and three-loop vacuum integrals with one non-zero mass parameter. The principle structure is introduced and the main parts of the package are described in detail. Explicit examples demonstrate the fields of application.'
author:
- |
[Matthias Steinhauser]{}\
[ II. Institut für Theoretische Physik,]{}\
[ Universität Hamburg, D-22761 Hamburg, Germany]{}\
\
[ Institut für Theoretische Teilchenphysik,]{}\
[ Universität Karlsruhe, D-76128 Karlsruhe, Germany]{}
title: |
-3cm
DESY 00–124
TTP00–14
hep-ph/0009029
July 2000
.7cm [MATAD]{}: a program package for the computation of MAssive TADpoles .3cm
---
16.cm
[**PROGRAM SUMMARY**]{}
- [*Title of program:*]{} [MATAD]{}
- [*Available from:*]{}\
[ http://www-ttp.physik.uni-karlsruhe.de/Progdata/MATAD/1/ ]{}
- [*Computer for which the program is designed and others on which it is operable:*]{} Any work-station or PC where [FORM]{} is running.
- [*Operating system or monitor under which the program has been tested:*]{} UNIX, [FORM]{} 2.3
- [*No. of bytes in distributed program including test data etc.:*]{} $706000$
- [*Distribution format:*]{} ASCII
- [*Keywords:*]{} three-loop computations, vacuum integrals, computer algebra, automation of computations
- [*Nature of physical problem:*]{} Multi-loop integrals are needed for the evaluation of quantum corrections. An important class of loop diagrams is covered by so-called vacuum integrals which have no external momentum. [MATAD]{} can analytically compute those one-, two- and three-loop vacuum integrals where one mass scale is present.
- [*Method of solution:*]{} The method of integration-by-parts is used in order to obtain recurrence relations which reduce complicated integrals to a small set of so-called master integrals. They have to be evaluated once and for all. In addition a user interface is provided which makes it easy to put in complicated diagrams in a rather compact way.
- [*Restrictions on the complexity of the problem:*]{} The restrictions on the complexity are given by the hardware limitations of the computer and the limits on the size of the storage files inside [FORM]{}.
- [*Typical running time:*]{} The runtime strongly depends on the complexity of the diagram under consideration. It may vary form a few seconds to the order of a few weeks.
[**LONG WRITE-UP**]{}
Introduction
============
The high experimental precision reached at the electron-positron machines LEP (CERN) and SLC (SLAC) and the hadron collider TEVATRON (FERMILAB) requires from the theoretical side the evaluation of higher order quantum corrections. In the cases where perturbative methods are applied the quantum corrections can be expressed through an expansion in the coupling constant of the underlying theory. The individual terms can in turn be expressed through so-called Feynman diagrams, which are often classified as multi-leg or multi-loop diagrams. Vacuum integrals, i.e. integrals without external momenta, constitute an important sub-class and often serve as building blocks in complex calculations.
In general the momentum integration of the loop integrals is divergent in four space-time dimensions. At present the most practical method to cope with this problem in higher loop orders is based on Dimensional Regularization [@dimreg]. There, the four space-time dimensions are replaced by $D=4-2\varepsilon$ dimensions. Then the integrals are solved for a choice of $\varepsilon$ that renders them finite. Finally an expansion for $\varepsilon\to0$ is performed and the divergences manifest themselves as poles in $\varepsilon$.
Important progress in practical computations has been made roughly 20 years ago by establishing an algorithm for the evaluation of propagator-type diagrams up to three loops in the massless case [@CheTka81]. They are important if there is only one external momentum which sets the mass scale for the problem. The formulae have been implemented on the computer in the [FORM]{} [@form] package [MINCER]{} [@mincer]. In 1995 for the first time three-loop diagrams in the opposite limit, i.e. zero external momentum but massive lines, were systematically examined [@Bro92]. Usually these are denoted as vacuum or tadpole diagrams. In [@Bro92] all integrals contributing to the photon propagator have been considered. The main characteristics of this class of diagrams is that the massive line forms a closed loop. These considerations have been extended to the $W$ boson current correlators which led to one of the most prominent applications of three-loop vacuum integrals, namely the $\rho$ parameter at ${\cal
O}(\alpha\alpha_s^2)$ [@Avdrho; @CheKueSte95rho]. The remaining cases have been considered in [@Avd97; @Bro98; @FleKal99; @CheSte00]. Thus it is — at least in principle — possible to treat all problems where exactly one heavy mass is involved.
In this paper we want to present the program package [MATAD]{} which was designed for the computation of MAssive TADpoles at one-, two- and three-loop order as pictured in Fig. \[fig:tad123\]. Thereby each line may be massless or carry the mass $M$. In mathematical form the integrals to be solved by [MATAD]{} read $$\begin{aligned}
&&
\int\frac{{\rm d}^D p}{(2\pi)^D}
\frac{1}{(p^2+M^2)^{n}}
\,,
\nonumber\\
&&
\int\frac{{\rm d}^D p}{(2\pi)^D}\frac{{\rm d}^D k}{(2\pi)^D}
\frac{1}{(p_1^2+M_1^2)^{n_1}(p_2^2+M_2^2)^{n_2}(p_3^2+M_3^2)^{n_3}}
\,,
\nonumber\\
&&
\int\frac{{\rm d}^D p}{(2\pi)^D}\frac{{\rm d}^D k}{(2\pi)^D}
\frac{{\rm d}^D l}{(2\pi)^D}
\frac{1}{(p_1^2+M_1^2)^{n_1}(p_2^2+M_2^2)^{n_2}(p_3^2+M_3^2)^{n_3}} \times
\nonumber\\&&\qquad\qquad\qquad\qquad\qquad\qquad
\frac{1}{(p_4^2+M_4^2)^{n_4}(p_5^2+M_5^2)^{n_5}(p_6^2+M_6^2)^{n_6}}
\,,
\label{eq:inputints}\end{aligned}$$ with $M_i=0$ or $M_i=M$. These expressions correspond to the diagrams in Fig. \[fig:tad123\] where the momentum $p_i$ flows through the line $i$ as indicated by the arrow. $p_i$ can be expressed as a linear combination of the loop momenta. However, these relations are in our case not of interest. Note that the integrals in Eq. (\[eq:inputints\]) are defined in Euclidean space.
The key idea for the computation of tadpole integrals is based on the integration-by-parts method [@CheTka81] (see also Appendix \[app:ibp\]). It can be used for the derivation of recurrence relations which relate vacuum integrals with different denominator structures. The proper use of the recurrence relations allows the reduction of an arbitrary integral to simple ones, which can be solved by successively using one- and two-loop formulae, and a linear combination of a few so-called master integrals. Only for them a hard calculation is necessary. In the case of three-loop tadpole diagrams nine master integrals are needed.
At first sight the applications for vacuum integrals seem to be quite restricted. However, for diagrams involving several mass scales, which follow a certain hierarchy, it is very often advantageous to apply an asymptotic expansion [@Smi95] which allows for a systematic expansion in the inverse heavy scale. Then the multi-scale integrals are expressed as products of single scale ones. In [@Har:diss] and [@Sei:dipl] the rules for the so-called large-momentum and hard-mass procedure have been automated and computer programs, [LMP]{} [@Har:diss] and [EXP]{} [@Sei:dipl], have been developed. They generate for a given diagram all relevant sub-graphs together with the administrative files which govern the very calculation. [LMP]{} [@Har:diss] is written in [PERL]{} and can be applied to problems where one large momentum is involved. [EXP]{} [@Sei:dipl], written in [C++]{}, allows for a successive use of the large-momentum and hard-mass procedure and thus can deal with problems involving many scales. Both programs produce output which can be read into [MATAD]{} and [MINCER]{}. Thus the combination of both massive vacuum integrals and massless propagator-type diagrams is very powerful to attack problems involving several different mass scales. We want to mention that [MATAD]{} can be easily linked to a generator for Feynman diagrams. More details — in particular on the automation of the computation of Feynman diagrams — can be found in [@HarSte98].
The outline of the paper is as follows: In Section \[sec:structure\] the structure of [MATAD]{} and the way it works is described. With the help of this section the reader should be able to use [MATAD]{} for his own problems. Deeper insight into some selected parts is provided in Section \[sec:details\]. In Section \[sec:examples\] explicit examples are discussed and hints for the convenient usage of [MATAD]{} are given. In Appendix \[app:ibp\] the ideas of the integration-by-parts method are reviewed. Appendix \[app:topfile\] lists all massive/massless combinations which are implemented into the topology files and in Appendix \[app:not\] the notation of the input and output is described. Furthermore the results for the master integrals are listed and the switches for the input-file are described. Appendix \[app:files\] contains a list of all files of [MATAD]{} and, finally, in Appendix \[app:testrun\] the complete output of one of the considered examples is listed.
Structure and mode of operation {#sec:structure}
===============================
As [MATAD]{} is completely written in [FORM]{} [@form] its installation reduces to copying the individual files into the corresponding directories. In the main directory the following files appear:
form.set inc/ matadform prc/ problems/
The directories `inc` and `prc` contain the include-files and procedures, respectively. They are described in more detail in Section \[sec:details\] and Appendix \[app:files\]. `matadform` is a shell script which calls [FORM]{} in such a way that files from sub-directories can be included. It has to be adjusted by the user by simply specifying the corresponding paths. The file `form.set` contains [FORM]{}-specific settings which have to be adjusted according to the underlying platform. For details concerning the different switches we refer to the [FORM]{} manual [@form]. The user-specific files are all contained in the folder `problems`.
There are at least two files which should be provided by the user: [main<prb>]{} and [.dia]{} where stands for the name of the considered problem. The first one contains apart from some parameters essentially the information which diagram should be treated. Some explicit examples are given below. All the information about the diagrams, the projectors to be applied, etc.is contained in the file [.dia]{}. It is built up by [FORM]{} folds and splits into two parts so that its generic structure looks as follows
*--#[ TREAT0:
[...]
*--#] TREAT0:
*--#[ TREAT1:
[...]
*--#] TREAT1:
*--#[ TREAT2:
[...]
*--#] TREAT2:
*--#[ TREATMAIN:
[...]
*--#] TREATMAIN:
*
* in the following list each diagram is contained in a separate FORM fold
*
*--#[ d1l1:
[... diagram 1 ...]
#define TOPOLOGY "XY"
*--#] d1l1:
*--#[ d1l2:
[... diagram 2 ...]
#define TOPOLOGY "XY"
*--#] d1l2:
[...]
The first part consists of the first four folds — the so-called special treat files. They provide the possibility to interact at different stages and thus influence the computation. Whereas [TREAT0]{}, [TREAT1]{} and [TREAT2]{} are read before the recurrence relations are applied the content of [TREATMAIN]{} is read right before the results are stored to disk. The second part of [.dia]{} contains a list of all diagrams to be considered where each diagram is written in a separate fold. The name of these folds is arbitrary.
Once these two files are set up the calculation is simply initiated by calling the program [main<prb>]{} and the following steps are performed. They are also illustrated in Fig. \[fig:matad\].
1. Read global settings. They are partly contained in [inc/main.gen]{} and should not be modified. Others can be set by the user in the file [main<prb>]{}. They are described in Appendix \[app:settings\].
2. Read the input data for the diagram specified in [main<prb>]{} with the help of the variables [PRB]{}, [FOLDER]{} and [DIAGRAM]{}. The generic [FORM]{} command reads `#include problems/'PRB'/'FOLDER'.dia # 'DIAGRAM'`.
As a next step the file [treat.prc]{} is called and the following operations are performed.
1. Insert Feynman rules for functions appearing in the input. In a first step the fermions (encoded in the functions [S]{}, [SS]{}, …, cf. Appendix \[app:notin\]) are resolved. It is important to do this before any contraction of indices is done. Then the propagators and vertices are treated.
In the current version the QCD Feynman rules are implemented (except the four-gluon vertex; see Appendix \[app:notin\]). It is, however, straightforward to implement new vertices in the user-specific treat files.
2. Apply projector. This should be done in one of the special treat files. The optimal position depends on the integrals to be computed. From now on only scalar integrals without any free indices are present.
3. Expansion of the scalar denominators in the small quantities (mass and/or momentum).
4. Perform traces.
5. Do Wick rotation. This is done by multiplying each momentum by the imaginary unit (see also Appendix \[app:not\]). From now on the expression is defined in Euclidean space.
6. Apply derivatives in order to factorize the external momentum. In this context see also the variables [DALA12]{} and [DALAQN]{} in Appendix \[app:settings\].
As there is the possibility to interact at three different places — after the fermions are treated ([TREAT0]{}) and before and after the traces are performed ([TREAT1]{}, respectively, [TREAT2]{}) the order of the commands may slightly be varied by the user.
At this stage the scalar products in the numerator of the integrals should be formed by either only loop momenta or only external momenta (which then constitutes a trivial prefactor). In the denominator the (scalar) propagators may be raised to arbitrary power.
The next steps constitute the main part of [MATAD]{}.
1. Express the scalar products of the numerator in terms of the denominators. This produces a “1” in the numerator of the integrals. It might be that this step is very time and memory consuming.
2. Apply recurrence relations to reduce the number of different integrals to simpler ones and to a small set of master integrals.
3. Expand the result in $\varepsilon$ and store it in the directory\
`problems/'PRB'/results/'DIAGRAM'.res` under the name `'DIAGRAM'`.
An expansion in $\varepsilon$ is also done at various intermediate steps. Although poles of at most third order may appear for a three-loop vacuum integral terms up to order $\varepsilon^6$ have to be kept in the expansion as artificial poles may appear during the application of the recurrence relations (cf. step 10).
Steps 9 and 10 constitute the central part of [MATAD]{}. They heavily depend on the loop-order and the topology which has to be specified apart from the very diagram in the folds [d1l1]{}, [d1l2]{}, …(see above). Thus let us elaborate on this point in the following. In principle it suffices to define one input topology at one-, two- and three-loop order, where the number of internal lines amount to one, three and six, respectively (see Fig. \[fig:tad123\]). If one allows each line to be massless and carry the mass $M$ at the same time these three topologies are sufficient to cover all possible cases that can occur in the calculation of one-, two- and three-loop vacuum integrals. Note that a partial fractioning for terms like $$\begin{aligned}
\frac{1}{\left(p^2\right)^a\left(p^2+M^2\right)^b}
\,,\end{aligned}$$ where $a$ and $b$ are positive integers, leads to the same topologies with the only difference that now each line is either massless of massive. It is, however, not at all practical to rewrite the input to the notation of Fig. \[fig:tad123\] before generating the file [dia.<prb>]{}. On the contrary it is advantageous to enlarge the input topologies. Currently the topologies shown in Fig. \[fig:intop\] are implemented in [MATAD]{}. The momentum $p_i$ flowing through line $i$ can be expressed as a linear combination of the loop momenta. For our purpose these relations are, however, not of interest. The choice of the topologies was guided by the package [MINCER]{} [@mincer] and for convenience the same notation concerning the definition of the momenta $p_i$ has been adopted. The implemented massive/massless combinations of each topology are listed in Appendix \[app:topfile\]. After the declaration of the diagram in the folds [d1l1]{}, [d1l2]{}, …the corresponding topology is specified via
#define TOPOLOGY "XY"
where `XY` corresponds to one of the topologies of Fig. \[fig:intop\].
The notation concerning the momenta as introduced in Fig. \[fig:intop\] is quite convenient to be used for the input. However, the very recursion procedure is formulated for the three-loop topology of Fig. \[fig:tad123\] where the lines are either massive or massless. This leads to $14$ different cases which are classified in [@Avd97]. Thus before step 9 is performed the momenta are transformed from the notation of Fig. \[fig:intop\], which is used in the input, to the so-called basic topologies shown in Fig. \[fig:batop\]. For them — after decomposing the scalar products in the numerator into parts of the denominator — the very recursion procedure is performed. Note that at this stage all propagators may be raised to an arbitrary integer power.
Actually some of the three-loop diagrams (e.g. `BN3`) can be computed by the successive use of one- and two-loop procedures for massless propagator type diagrams or vacuum integrals, respectively. In such cases some of the (one- and two-loop) routines from [MINCER]{} [@mincer] are used for parts of the computation. The corresponding procedures are listed in Appendix \[app:files\]. For other cases (e.g. `E4` or `BN2`) simple relations reduce one of the lines to zero and the resulting diagram can again be solved easily. Only for the cases `D5, D4, DN, DM, E3, BN` and `BN1` the recursion procedure has to be applied until one arrives at master integrals. They coincide with the corresponding diagrams of Fig. \[fig:batop\] where all denominators are raised to power one. Only the one pictured in Fig. \[fig:master\_add\], which results from `BN1`, is needed in addition. From this diagram even the ${\cal O}(\varepsilon)$ part is required. The analytic expressions are given in Appendix \[app:master\]. It should be mentioned that the recurrence relations for `BM` are quite involved. However, for this topology no difficult master integral is needed. Note that the basic topologies `BM1` and `BM2` actually coincide with `BN2` and `BN3`, respectively. For convenience partly different codes had been written which actually turned out to be quite useful while debugging [MATAD]{}.
The recurrence relations leading to simple diagrams and master integrals for the three-loop topologies are derived in Refs. [@Bro92] and [@Avd97]. Except the case where all six lines are massive all of them are implemented into [MATAD]{}. The reason why this case is missing is simply that it was not yet needed for practical calculations. All master integrals are listed in Appendix \[app:master\].
In Fig. \[fig:simple\] the simple integrals are listed which also result from the recursion procedure. They can all be expressed in terms of $\Gamma$ functions by the successive use of results for massless one-loop two-point functions, $P_{ab}(Q)$, in combination with one- ($V_a$) and two-loop ($V_{abc}$) vacuum integrals. For convenience we want to list the explicit results for $P_{ab}(q)$, $V_a$ and $V_{abc}$ in Euclidean space: $$\begin{aligned}
P_{ab}(Q) &=& \int\frac{{\rm d}^Dp}{\left(2\pi\right)^D}
\frac{1}{p^{2a}\left(p+Q\right)^{2b}}
\nonumber\\
&=&
\frac{\left(Q^2\right)^{D/2-a-b}}{\left(4\pi\right)^{D/2}}
\frac{
\Gamma(a+b-D/2)
\Gamma(D/2-a)
\Gamma(D/2-b)
}{
\Gamma(a)
\Gamma(b)
\Gamma(D-a-b)
}
\,,
\nonumber\\
V_{a}&=& \int\frac{{\rm d}^Dp}{\left(2\pi\right)^D}
\frac{1}{\left(p^2+M^2\right)^{a}}
\,\,=\,\,
\frac{\left(M^2\right)^{D/2-a}}{\left(4\pi\right)^{D/2}}
\frac{
\Gamma(a-D/2)
}{
\Gamma(a)
}
\,,
\nonumber\\
V_{abc} &=&
\int\frac{{\rm d}^D p}{(2\pi)^D}\frac{{\rm d}^D k}{(2\pi)^D}
\frac{1}{(p^2+M^2)^{a}(k^2+M^2)^{b}(\left(p+k\right)^2)^{c}}
\nonumber\\
&=&
\frac{\left(M^2\right)^{D-a-b-c}}{\left(4\pi\right)^{D}}
\frac{
\Gamma(a+b+c-D)
\Gamma(a+c-D/2)
\Gamma(b+c-D/2)
\Gamma(D/2-c)
}{
\Gamma(a)
\Gamma(b)
\Gamma(a+b+2c-D)
\Gamma(D/2)
}
\,.
\nonumber\\
\label{eq:simpint}\end{aligned}$$
\
The vacuum integrals occurring at one- and two-loop level are quite simple. Actually most of them can be expressed in terms of $\Gamma$ functions for arbitrary exponents of the propagators and no recursion relations are needed (cf. Eq. (\[eq:simpint\])). Only for the two-loop integral in Fig. \[fig:tad123\] where all three lines carry the mass $M$ it is useful to implement simple recurrence relations which reduce the integrals to one master integral where all exponents are raised to the first power only. For three-loop calculations the result of this integral is needed up to ${\cal O}(\varepsilon)$ ([T1ep]{}, see Appendix \[app:master\]).
For definiteness we want to consider an explicit example. Let us consider the three-loop diagram of Fig. \[fig:ladder\] which we would like to expand up to fourth order in the external momentum, $q_1$. For simplicity we neglect the tensor structure and consider only the scalar integral obtained in the case when the full line represents a massive particle and the curly line a massless one. In the directory [problems/scalar/]{} one can find the following files
mainscalar results/ scalar.dia
where [results/]{} is a directory to store the result of the diagram. The file [scalar.dia]{} looks as follows
*--#[ TREAT0:
*--#] TREAT0:
*--#[ TREAT1:
*--#] TREAT1:
*--#[ TREAT2:
*--#] TREAT2:
*--#[ TREATMAIN:
*--#] TREATMAIN:
*--#[ scalar:
M^4
*s1m
*s2m
*s3m
*Dh(p4,q1)
*Dh(p5,q1)
*Dh(p6,q1)
/p7.p7
/p8.p8
;
#define TOPOLOGY "LA"
*--#] scalar:
In this case no special treat file is needed. The very diagram can be found in the fold [scalar]{} where the multiplication with $M^4$ is done in order to end up with a dimensionless expression. The external momentum is routed through the lines 4, 5 and 6 as can be seen in the arguments of the function `Dh`. The other massive propagators are denoted by `s1m`, `s2m` and `s3m`. The massless lines translate into the factors `1/p7.p7` and `1/p8.p8`. For more details on the notation we refer to Appendix \[app:notin\]. The file [mainscalar]{} looks as follows:
#define PRB "scalar"
#define DALAQN "q1"
#define GAUGE "0"
#define POWER "4"
#define CUT "0"
#define FOLDER "scalar"
#define DIAGRAM "scalar"
#-
#include main.gen
With the command
#define POWER "4"
we require an expansion up to fourth order in $q_1$ and
#define DALAQN "q1"
effectively factors out the terms $(q_1^2)^n$. The calculation is initiated with the command
> matadform problems/scalar/mainscalar
and after a few seconds the final result is displayed on the screen
scalar =
- 2 + 2*z3 - Q1.Q1*z3*M^-2 + 227/216*Q1.Q1*M^-2 + 1/2*Q1.Q1^2*z3*M^-4
- 1876/3375*Q1.Q1^2*M^-4;
As expected it is finite and contains three terms in the expansion in $q_1^2/M^2$. The result is stored to the directory [problems/scalar/results/]{} and can be read with the [FORM]{} command [load]{}. Details on the notation of the output and the used conventions can be found in Appendix \[app:not\].
Some details on the internal structure {#sec:details}
======================================
Once the recurrence relations are implemented it is in principle possible to compute diagrams of arbitrary complexity. However, in practice one arrives quite soon at the limits set by the soft- or hardware. The algebra language [FORM]{} is designed to deal with large expressions. Still it happens quite easily that in internal steps the expressions exceed several Megabytes and even approach the order of a few Gigabyte. This significantly slows down the performance and it is advantageous to implement several tricks. Some of them are described in this section.
$\bullet$ During the recurrence procedure it happens very often that whole blocks of commands have to be executed until the recursion has reached an end. Within [FORM]{} there are the commands
repeat;
[...]
endrepeat;
which in principle allows for such a construction. However, in practice it is not possible to use between `repeat` and `endrepeat` a command which forces [FORM]{} to combine identical terms like, e.g., `.sort`. For this reason a preprocessor variable, `NOR` (see also Appendix \[app:settings\]), has been introduced which in combination with the construction
#do i=1,'NOR'
[...]
#enddo
allows the use of `.sort` in intermediate steps of the recursion commands.
$\bullet$ At this point we should also mention that the procedure `ACCU` has been adopted from [MINCER]{} [@mincer]. It collects in the argument of the function `acc()` the polynomials in $\varepsilon$ and thus significantly reduces the number of terms. E.g., the expression
x1*x2 + 4*ep*x1*x2 + 12*ep^3*x1*x2
transforms after `#call ACCU{test}` to
acc(1 + 4*ep + 12*ep^3)*x1*x2
and instead of three only one term has to be treated in the following commands.
$\bullet$ Very often it happens that propagators of the type $1/(M^2-(p+q)^2)^n$ have to be expanded in the momentum $q$ and afterwards derivatives w.r.t. $q$ are applied in order to factor out powers of $q^2$. It turns out that it is very useful to expand in a first step only the part $2pq$ and keep the factors $q^2$ unexpanded in the form $1/(M^2-(p^2+q^2))^n$. Thus less terms have to be considered while the derivatives are applied. Afterwards the expansion in $q^2$ is performed. A related discussion can also be found in [@Tar95].
$\bullet$ The expansion of the scalar denominators in a small momentum is also very time consuming — especially for high values of ’[POWER]{}’. In order to do this in an effective way the variable [poco]{}, which is an abbreviation for “power counting”, is defined via
S poco(:'POWER');
after the declaration of the preprocessor variable [POWER]{}. This definition ensures that the terms involving [poco$^n$]{} with $n>$`'POWER'` are automatically set to zero. Depending on the problem [poco]{} should also be considered in the special treat files.
$\bullet$ The application of the recurrence relations can lead to spurious $1/\varepsilon$ poles (cf. Appendix \[app:ibp\]) which are in general quite dangerous if an expansion in $\varepsilon$ is performed in intermediate steps. In [MATAD]{} at most three $1/\varepsilon$ poles arise from the recurrence relations. Together with a possible $1/\varepsilon^3$ term from the three-loop integrals an expansion up to order $\varepsilon^6$ has to be done in intermediate steps in order to get the correct constant term.
$\bullet$ For quite a lot of applications the topology `BN` (cf. Fig \[fig:batop\]) plays a crucial role. The original recurrence procedure for this toplogy to master integrals was proposed in [@Bro92]: In a first step the exponent of three out of the four massive denominators are reduced to one. Then the exponents of the massless lines are reduced to zero. Finally the remaining line is treated and one arrives (apart from simple integrals) at an integral consisting of four massive lines connecting two vertices. It is connected to the corresponding master integral of Fig. \[fig:batop\] through a simple relation.
The equation involved in the last recursion step is quite involved. It generates from each term more than ten terms at each call. Note that the exponent of the last massive denominator gets increased by the proceeding steps. This enormously slows down the calculation — in some cases it makes it even impossible. The idea to circumvent this problem is based on the observation that the massless exponents can be reduced to zero even if none of the massive ones is reduced to one. The corresponding recurrence relations are short and thus one arrives with only little effort at three-loop integrals with four massive lines, $B_N(0,0,n_3,n_4,n_5,n_6)$ (cf. Fig. \[fig:master\_add\] where all lines are massive and the exponents of the denominators are given by $n_3$, $n_4$, $n_5$ and $n_6$). These diagrams are now treated in the following way: Temporarily an external momentum is introduced which flows through one of the lines. We choose line 3 for definiteness. In a second step the operator $\Box_q=\partial/\partial q^\mu \partial/\partial q_\mu$ is applied and $q$ is set to zero afterwards. This leads to an equation connecting $B_N(0,0,n_3,n_4,n_5,n_6)$, $B_N(0,0,n_3-1,n_4,n_5,n_6)$ and $B_N(0,0,n_3-2,n_4,n_5,n_6)$ $$\begin{aligned}
B_N(0,0,n_3,n_4,n_5,n_6) &=&
-\frac{1}{4M^2(n_3-2)(n_3-1)} \Box_q B_N(0,0,n_3-2,n_4,n_5,n_6)
\nonumber\\&&\mbox{}
+\frac{-2D+4(n_3-1)}{4M^2(n_3-1)} B_N(0,0,n_3-1,n_4,n_5,n_6)
\,,
\label{eq:BNrec}\end{aligned}$$ which can be applied until $n_3=3$. As the other indices are not affected the same procedure can be applied to them as well and one ends up with the integrals $B_N(0,0,1,1,1,1)$, $B_N(0,0,2,1,1,1)$, $B_N(0,0,2,2,1,1)$, $B_N(0,0,2,2,2,1)$ and $B_N(0,0,2,2,2,2)$. Both their values for $q=0$ and the result for the application of $(\Box_q)^n$ has to be known where the index $n$ depends on how often Eq. (\[eq:BNrec\]) had to be applied. The overall number of the integrals needed is still small and for most practical applications well below 100. They can be computed once and for all using the method of Ref. [@Bro92] and can be collected in a table. Currently the table contains all results up to $n=10$ and partly for $n=11$. In case the table is too small the original recurrence procedure [@Bro92] has to be used. This is done by defining the preprocessor variable [BNRECOLD]{} in the main file.
Examples {#sec:examples}
========
In this section we want to discuss some typical examples which can be treated with [MATAD]{}. In particular the content of the [TREAT]{} folds and the switches in [main.<prb>]{} shall be discussed in detail. In Section \[sec:photon\] two-loop corrections to the photon polarization function are considered and in Section \[sec:hgg\] the calculation of three-loop vertex corrections contributing to the Higgs decay into gluons is discussed. As a last example we consider the computation of the fermion propagator in the limit of a small external momentum.
\[sec:photon\]Photon polarization function
------------------------------------------
In this section only a calculation of two-loop diagrams is presented. However, we want to take this opportunity to show a concept which can also be used for the treatment of more complex problems. In particular it is shown how the complete calculation can be automated and one can ensure that all results are indeed up-to-date.
To be precise, we want to consider the two-loop diagrams induced by a massive quark to the photon polarization function. The problem shall be called [Pi]{}. Then the file [Pi.dia]{} looks as follows:
*
* problem: Pi
*
*--#[ TREAT0:
#message project out transversal part
multiply, (d_(mu1,mu2)-q1(mu1)*q1(mu2)/q1.q1)*deno(3,-2);
.sort
*--#] TREAT0:
*--#[ TREAT1:
*--#] TREAT1:
*--#[ TREAT2:
*--#] TREAT2:
*--#[ TREATMAIN:
*--#] TREATMAIN:
*
* 2-loop diagrams for problem Pi
*
*--#[ d2l1:
((-1)
*Dg(nu1,nu2,p5)
*S(mu1,q1,p1m,nu1,q1,p2m,mu2,p3m,nu2,p4m)
*1);
#define TOPOLOGY "T1"
*--#] d2l1:
*--#[ d2l2:
((-1)
*Dg(nu1,nu2,p4)
*S(mu1,q1,-p2m,mu2,p1m,nu2,p3m,nu1,p1m)
*1);
#define TOPOLOGY "T2"
*--#] d2l2:
*--#[ d2l3:
((-1)
*Dg(nu1,nu2,p4)
*S(nu1,p3m,nu2,p1m,mu2,-q1,-p2m,mu1,p1m)
*1);
#define TOPOLOGY "T2"
*--#] d2l3:
The function [deno(x,y)]{} means $1/(x+y\varepsilon)$ and can be used for denominators. [TREAT0]{} contains the projector to the transversal part and the three contributing diagrams are named [d2l1]{}, [d2l2]{} and [d2l3]{}. The external momentum is denoted by `q1`. The mass of the quarks, which in the final result appears as [M]{}, is introduced through adding the symbol [m]{} to the corresponding momentum which is defined through the topologies [T1]{} and [T2]{} in Fig. \[fig:intop\]. We want to assume that $q_1^2\ll M^2$ and thus perform an expansion in $q_1$. This is achieved with the notation `S(...,q1,p1m,...)`. For more details concerning the notion we refer to Appendix \[app:not\].
The file [mainPi]{} reads:
#define PRB "Pi"
#define PROBLEM0 "1"
#define DALAQN "q1"
#define GAUGE "xi"
#define POWER "4"
#define CUT "1"
#define FOLDER "Pi"
***#define DIAGRAM "d2l1"
#-
#include main.gen
For most of the specified variables we refer to Appendix \[app:settings\]. We only want to mention that a general gauge parameter `xi` is chosen with the command `#define GAUGE "xi"`. This allows for an explicit check that the final result, i.e. the sum of the three diagrams is gauge parameter independent. The definition `#define POWER "4"` requests for an expansion up to fourth order in `q1`. The definition of the variable `DIAGRAM` is commented as it will be defined during the call of [mainPi]{} (see below).
The computation of each diagram could be started separately and the results could be summed at the end. Instead we want to take the opportunity to present a method which is unavoidable for problems where a large number of diagrams contribute. In the following we want to present two more files which can easily be adopted to other problems. The first one, [makePi]{}, is a so-called [GNU make]{} file and could look as follows:
SHELL = /bin/sh
DIA2 = \
problems/Pi/results/d2l1.res\
problems/Pi/results/d2l2.res\
problems/Pi/results/d2l3.res
problems/Pi/results/Pi.2.res: $(DIA2)
matadform problems/Pi/comPi > problems/Pi/log/comPi.log
ii = $(notdir $(basename $@))
$(DIA2): problems/Pi/mainPi
if [ -f problems/Pi/results/$(ii).res ]; \
then rm problems/Pi/results/$(ii).res; fi
time matadform -d DIAGRAM=$(ii) \
problems/Pi/mainPi > problems/Pi/log/$(ii).log;
if [ -f problems/Pi/results/$(ii).res ]; \
then rm problems/Pi/log/$(ii).log; fi
For details concerning the individual commands we refer to the literature [@make]. For us it is only important that at the beginning the diagrams we want to compute are listed. Furthermore, after the line ‘`problems/Pi/results/Pi.2.res: $(DIA2)`’ the command is given which specifies what shall be done once the computation of the individual diagrams is finished: The diagrams are summed with the help of the program [comPi]{}
*
* comPi
*
#-
#include declare.matad
#define PRB "Pi"
.global
#do i=1,3
load problems/'PRB'/results/d2l'i'.res;
#enddo
g res'PRB'2 =
#do i=1,3
+ d2l'i'
#enddo
;
b ep;
print;
.store
#+
save problems/'PRB'/results/res'PRB'.2.res res'PRB'2;
.end
To initiate the calculation one simply has to specify the command
> make -f problems/Pi/makePi
where [make]{} has to call the [GNU]{} version [@make] of the [make]{} command. The final result is stored in the file [problems/Pi/results/resPi.2.res]{}. It reads
resPi2 =
+ ep^-1 * ( - 6*Q1.Q1 + 8/5*Q1.Q1^2*M^-2 )
+ ep * ( - 35/6*Q1.Q1 - 6*Q1.Q1*z2 + 8/5*Q1.Q1^2*M^-2*z2 + 3116/1215*
Q1.Q1^2*M^-2 )
+ 13/3*Q1.Q1 - 128/405*Q1.Q1^2*M^-2;
Indeed, as expected, there is no trace of the gauge parameter `xi`.
\[sec:hgg\]Higgs decay into two gluons
--------------------------------------
Party for convenience and partly for historical reasons the notation of the input topologies in Fig. \[fig:intop\] is closely connected to two-point functions. However, [MATAD]{} only deals with vacuum diagrams independent of the number of external legs. In this section we want to show that also problems involving at first sight three-point functions can be approached using [MATAD]{}.
Let us consider QCD corrections to the decay of the Standard Model Higgs boson into two gluons. It is convenient to construct an effective theory where the top quark is integrated out. Details on the theoretical background can be found in [@CheKniSte97hgg]. Here it shall only be mentioned that the coefficient function which contains the dependence on the mass of the top quark, $M_t$, can be computed from triangle diagrams as pictured in Fig. \[fig:hgg\]. According to their Lorentz structure the result can be written as follows $$\begin{aligned}
K(M_t) \, \left( q_1^\nu q_2^\mu - q_1 q_2 g^{\mu\nu} \right)
\,,
\label{eq:c1}\end{aligned}$$ where $q_1$ and $q_2$ are the momenta of the gluons with polarization vectors $\epsilon^\mu(q_1)$ and $\epsilon^\nu(q_2)$. Thus the vertex diagrams have to be expanded up to linear order both in $q_1$ and $q_2$ and an appropriate projector has to be applied in order to get $K(M_t)$.
\
The file containing the diagrams could look as follows
*
* problem: hgg
*
*--#[ TREAT0:
multiply, (
a*deno(2,-2)*(q1.q2*d_(mu,nu)-q2(nu)*q1(mu)-q2(mu)*q1(nu))
+b*deno(2,-2)*(-q1.q2*d_(mu,nu)+(3-2*ep)*q2(nu)*q1(mu)+q2(mu)*q1(nu))
);
.sort
*--#] TREAT0:
*--#[ TREAT1:
*--#] TREAT1:
*--#[ TREAT2:
*--#] TREAT2:
*--#[ TREATMAIN:
*--#] TREATMAIN:
*--#[ d3l335:
((-1)
*M
*Dg(nu1,nu2,p1)
*Dg(nu7,nu8,-p4)
*Dg(nu3,nu4,q1,-p1)
*Dg(nu5,nu6,q2,p1)
*S(-q1,-p3m,nu7,-q1,p5m,nu4,-p2m,nu6,q2,p5m,nu8,q2,-p3m)
*V3g(mu,q1,nu1,-p1,nu3,p1-q1)
*V3g(nu,q2,nu2,p1,nu5,-p1-q2)
*1);
#define TOPOLOGY "O4"
*--#] d3l335:
where the diagram [d3l335]{} corresponds to the one shown in Fig. \[fig:hgg\]. The fold [TREAT0]{} contains (up to an overall factor $(q_1.q_2)^{-2}$) the projector on the coefficients in front of the structures $g^{\mu\nu}$ and $q^\nu_1 q^\mu_2$ of Eq. (\[eq:c1\]). They are marked by the symbols `a` and `b`, respectively. Thus the transversality of Eq. (\[eq:c1\]) can be explicitly checked in the sum of all contributing diagrams (the result of a single diagram does in general not have a transverse structure).
The corresponding [main]{}-file is very similar to the one listed in Section \[sec:photon\]. The only difference (apart from replacing [Pi]{} by [hgg]{}) is that the commands
#define DALAQN "q1"
#define GAUGE "xi"
#define POWER "4"
#define CUT "1"
should be replaced by
#define DALA12 "1"
#define GAUGE "0"
#define POWER "2"
#define CUT "0"
The third line ensures that an expansion of the integrand up to second order in the external momenta is performed and the first one sets $q_1^2$ and $q_2^2$ to zero and factors out the scalar product $q_1 q_2$. `#define CUT "0"` sets $\varepsilon$ to zero in the final result as the terms of ${\cal O}(\varepsilon)$ are anyway not computed completely at three-loop order. In this example we choose Feynman gauge which is achieved with `#define GAUGE "0"`.
After calling [MATAD]{} it takes of the order of a minute to obtain the result:
d3l335 =
+ ep^-2 * ( 40*Q1.Q2*M^2*a + 344/9*Q1.Q2^2*a - 232/9*Q1.Q2^2*b )
+ ep^-1 * ( - 308/3*Q1.Q2*M^2*a - 3530/27*Q1.Q2^2*a + 1786/27*Q1.Q2^2*
b )
+ 60*Q1.Q2*M^2*z2*a + 734/3*Q1.Q2*M^2*a - 1936/9*Q1.Q2^2*z3*a + 1136/9*
Q1.Q2^2*z3*b + 172/3*Q1.Q2^2*z2*a - 116/3*Q1.Q2^2*z2*b + 46817/81*
Q1.Q2^2*a - 26239/81*Q1.Q2^2*b;
Note that the terms proportional to `Q1.Q2*M^2` cancel after adding all contributing diagrams.
\[sub:ferm\]Fermion propagator
------------------------------
In this example we compute the small-momentum expansion of a three-loop diagram which contributes to the fermion propagator. Recently these kind of diagrams have been considered in different kinematical regions in order to obtain the three-loop relation between the $\overline{\rm MS}$ and on-shell quark mass [@CheSte99; @CheSte00]. This subsection contains all relevant input information whereas the complete output is given in Appendix \[app:testrun\].
The fermion self energy, $\Sigma(q)$, can be decomposed into a scalar and vector part $$\begin{aligned}
\Sigma(q) &=& M \Sigma_S(q^2) + q\hspace{-.45em}/\,\Sigma_V(q^2)
\,,
\label{eq:sigma}\end{aligned}$$ where $q$ is the external momentum and $M$ is the mass of the quark. We are interested in the computation of the scalar functions $\Sigma_S$ and $\Sigma_V$.
The file `mainfp` looks as follows
#define PRB "fp"
#define PROBLEM0 "1"
#define DALAQN "q1"
#define GAUGE "0"
#define POWER "1"
#define CUT "0"
#define FOLDER "fp"
#define DIAGRAM "d3l79"
#-
#include main.gen
The variable `POWER` is defined in such a way that an expansion up to third order is performed. This leads to the constant and order $q^2/M^2$ terms for the functions $\Sigma_S$ and $\Sigma_V$ as $\Sigma(q)$ itself has mass dimension one. The corresponding projectors and the diagram to be computed look as follows (`fp.dia`):
*
* problem: fp
*
*--#[ TREAT0:
multiply, 1/4*(1/M + a * g_(1,q1)/q1.q1);
*--#] TREAT0:
*--#[ TREAT1:
*--#] TREAT1:
*--#[ TREAT2:
*--#] TREAT2:
*--#[ TREATMAIN:
*--#] TREATMAIN:
*--#[ d3l79:
((1)
*Dg(nu3,nu4,p5)
*Dg(nu5,nu6,-p6)
*Dg(nu1,nu2,-q1,-p1)
*S(nu2,-p1m,nu5,-p4m,nu4,-p3m,nu6,-p2m,nu3,-p1m,nu1)
*1);
#define TOPOLOGY "O4"
*--#] d3l79:
Note that in the fold [TREAT0]{} the vector part gets multiplied by [a]{} in order to distinguish $\Sigma_V$ from $\Sigma_S$ in the final result. The input for the diagram corresponds to the one pictured in Fig. \[fig:fp\]. The complete output appearing on the screen and the result can be found in the Appendix \[app:testrun\].
\
Acknowledgments {#acknowledgments .unnumbered}
===============
It is a pleasure for me to thank K.G. Chetyrkin, R. Harlander, J.H. Kühn and T. Seidensticker for many discussions and useful advice within the recent years when [MATAD]{} was developed. This work was supported in part by DFG under Contract Ku 502/8-1 ([*DFG-Forschergruppe “Quantenfeldtheorie, Computeralgebra und Monte-Carlo-Simulationen”*]{}) and by SUN Microsystems through Academic Equipment Grant No. 14WU0148. I am grateful to the department of Theoretical Particle Physics of the University of Karlsruhe for the pleasant atmosphere during a visit when a major part of this project was carried out.
Appendix {#appendix .unnumbered}
========
\[app:ibp\]Integration-by-parts
===============================
The method of integration-by-parts plays a fundamental role in the computation of multi-loop diagrams [@CheTka81]. For this reason we want to demonstrate its underlying idea by considering a typical example.
The integration-by-parts algorithm uses the fact that the $D$-dimensional integral over a total derivative is equal to zero: $$\begin{aligned}
\int{\rm d}^D p {\partial\over \partial p^\mu} f(p,\ldots) &=& 0\,.
\label{eqipgen}\end{aligned}$$ By explicitly performing the differentiations one obtains recurrence relations connecting Feynman integrals of different complexity. The proper combination of different recurrence relations allows any Feynman integral (at least single-scale ones) to be reduced to a small set of so-called master integrals. The latter ones have to be evaluated only once and for all, either analytically or numerically.
\
Let us for definiteness consider the scalar two-loop diagram of Fig. \[fig:triangle\]. The corresponding Feynman integral shall be denoted by $$I(n_1,\ldots,n_5) = \int{{\rm d}^D p\over (2\pi)^D}{{\rm d}^D k\over (2\pi)^D}
{1\over (p_1^2 + m_1^2)^{n_1}\cdots(p_5^2+m_5^2)^{n_5}}\,,$$ where $p_1,\ldots,p_5$ are combinations of the loop momenta $p,k$ and the external momentum $q$ (we work in Euclidean space here). $n_1,\ldots,n_5$ are called the indices of the integral. Consider the sub-loop defined by the lines 2, 3 and 5, and take its loop momentum to be $p=p_5$. If we then apply the operator $(\partial/\partial p_5)\cdot p_5$ to the [*integrand*]{} of $I$, we obtain a relation of the form (\[eqipgen\]), where $$f(p_5,\ldots) = \frac{p_5^{\mu}}
{(p_5^2+m_5^2)^{n_5} (p_2^2+m_2^2)^{n_2} (p_3^2+m_3^2)^{n_3}}
\,.$$ Performing the differentiation and using momentum conservation at each vertex one derives the following equation: $$\begin{aligned}
\Big[
- n_3 {\bf 3^+}\left({\bf 5^-}-{\bf 4^-}+m_4^2-m_5^2-m_3^2\right)
- n_2 {\bf 2^+}\left({\bf 5^-}-{\bf 1^-}+m_1^2-m_5^2-m_2^2\right)
\nonumber\\\mbox{}
+ D-2n_5-n_3-n_2+2n_5 m_5^2 {\bf 5^+}
\Big]\, I(n_1,\ldots,n_5) \,\,=\,\, 0
\,,
\label{eqtrianglerec}\end{aligned}$$ where the operators ${\bf 1^{\pm}}, {\bf 2^{\pm}}, \ldots$ are used in order to raise and lower the indices: ${\bf I^{\pm}}I(\ldots,n_i,\ldots)
= I(\ldots,n_i\pm 1,\ldots)$. In Eq. (\[eqtrianglerec\]), generally referred to as the triangle rule, it is understood that the operators to the left of $I(n_1,\ldots,n_5)$ are applied [*before*]{} integration. If the condition $m_5=0, m_3=m_4$ and $m_1=m_2$ holds, increasing one index always means to reduce another one. Therefore this recurrence relation may be used to shift the indices $n_1$, $n_4$ or $n_5$ to zero which leads to much simpler integrals.
The triangle rule constitutes an important building block for the general recurrence relations. The strategy is to combine several independent equations of the kind (\[eqtrianglerec\]) in order to arrive at relations connecting one complicated integral to a set of simpler ones. For example, while the direct evaluation of even the completely massless case for the diagram in Fig. \[fig:triangle\] is non-trivial, application of the triangle rule (\[eqtrianglerec\]) leads to $$I(n_1,\ldots,n_5) \,=\, \frac{1}{D-2n_5-n_2-n_3}
\Big[
n_2{\bf 2^+}\left({\bf 5^-} - {\bf 1^-}\right)
+
n_3{\bf 3^+}\left({\bf 5^-} - {\bf 4^-}\right)
\Big]\,I(n_1,\ldots,n_5)
\,.
\label{eqrecI}$$ Repeated application of this equation reduces one of the indices $n_1$, $n_4$ or $n_5$ to zero. For example, for the simplest case ($n_1=n_2=\ldots=n_5=1$) one obtains the equation pictured in Fig. \[fig:2loopIP\]: The non-trivial diagram on the l.h.s. is expressed as a sum of two quite simple integrals which can be solved by applying the one-loop formula. This example also shows a possible trap of the integration-by-parts technique. In general its application introduces artificial $1/\varepsilon$ poles which cancel only after combining all terms. They require the expansion of the individual terms up to sufficiently high powers in $\varepsilon$ in order to obtain, for example, the finite part of the original diagram. This point must carefully be respected in computer realizations of the integration-by-parts algorithm: One must not cut the series at too low powers because then the result goes wrong; keeping too many terms, on the other hand, may intolerably slow down the performance.
In our example, the l.h.s. in Fig. \[fig:2loopIP\] is finite, each term on the r.h.s., however, develops $1/\varepsilon^2$ poles. The first three orders in the expansion for $\varepsilon\to 0$ cancel, and the ${\cal O}(\varepsilon)$ term of the square bracket, together with the $1/\varepsilon$ in front of it, leads to the well-known result (omitting factors $1/16\pi^2$): $I(1,1,1,1,1)=6\zeta(3)/q^2$, where $q$ is the external momentum.
-------- ---------------------------------------------- -------- --- -------- ------------------------
=2.5cm $\displaystyle =\frac{1}{\varepsilon}\Bigg[$ =2.5cm — =2.5cm $\displaystyle \Bigg]$
-------- ---------------------------------------------- -------- --- -------- ------------------------
In general, the successive application of recurrence relations generates a huge number of terms out of a single diagram. Therefore, a calculation carried out by hand becomes very tedious and the use of computer algebra is essential.
\[app:topfile\]The topology files
=================================
This part of the appendix provides a complete list of those massive/massless combinations which are implemented in the topology files [inc/TOPOLOGY/topXY]{}. In the following tables for all three-loop topologies of Fig. \[fig:intop\] the lines are listed which have to be massive. All other lines may be massless or absent. In some cases some of the lines have to be completely absent, i.e. it is even not allowed for them to be massless. This is explicitly specified in the columns “absent”.
[| l|l || l|l || l|l || l|l|]{}\
massive & absent & massive & absent & massive & absent & massive & absent\
1 & — & 2 & — & 1,2 & — & &\
[| l|l || l|l || l|l || l|l|]{}\
massive & absent & massive & absent & massive & absent & massive & absent\
1,2 & — & 1,2,4 & — & 1,2,3,4 & — & 1,4 & —\
2,4 & — & 1 & — & 2 & — & 1,2,5 & —\
[| l|l || l|l || l|l || l|l|]{}\
massive & absent & massive & absent & massive & absent & massive & absent\
1,3 & — & 2 & — & 1,2,3 & — & 2,4 & —\
3,4 & — & 3 & — & 1,2 & — & &\
[| l|l || l|l || l|l || l|l|]{}\
massive & absent & massive & absent & massive & absent & massive & absent\
1,2,3,4,5 & — & 1,2,3 & — & 4,5 & — & &\
[| l|l || l|l || l|l || l|l|]{}\
massive & absent & massive & absent & massive & absent & massive & absent\
1,3,6,7 & — & 4,5,6,7 & — & & & &\
[| l|l || l|l || l|l || l|l|]{}\
massive & absent & massive & absent & massive & absent & massive & absent\
1,2,3,4,5,6 & — & 1,3,4,6,7,8 & — & 1,6,7 & — & 3,4,8 & —\
1,2,3 & — & 1,3,7,8 & 4,5,6 & 1,7 & 4,5,6 & 3,8 & 4,5,6\
[| l|l || l|l || l|l || l|l|]{}\
massive & absent & massive & absent & massive & absent & massive & absent\
1,2,3,4,5,6 & — & 1,2,3 & — & & & &\
[| l|l || l|l || l|l || l|l|]{}\
massive & absent & massive & absent & massive & absent & massive & absent\
1,2,3,4,5,6 & — & 1,2,3,4 & — & 5,6 & — & 2,3 & —\
[| l|l || l|l || l|l || l|l|]{}\
massive & absent & massive & absent & massive & absent & massive & absent\
1,2,3,4,5 & — & 2 & — & 1,3,4,5 & — & 3,4 & —\
3,4,5 & — & 4,5 & — & 1,3 & — & 1,5 & —\
[| l|l || l|l || l|l || l|l|]{}\
massive & absent & massive & absent & massive & absent & massive & absent\
1,2,3,4,6,7 & — & 1,2,3,4 & — & 6,7 & — & 1,2,6,7 & —\
1,2 & — & 1,5,6 & 3,4 & 1,5,7 & 3,4 & 5,6 & —\
5 & — & 5,7 & — & 1,5,7 & — & 7 & —\
1,5 & — & 1 & — & 1,7 & — & 6 & —\
1,6,7 & — & 1,2,7 & 3,4 & & & &\
[| l|l || l|l || l|l || l|l|]{}\
massive & absent & massive & absent & massive & absent & massive & absent\
1,2,3,4,7 & — & 3,4,7 & — & 1,2 & — & 7 & —\
[| l|l || l|l || l|l || l|l|]{}\
massive & absent & massive & absent & massive & absent & massive & absent\
1,2,6,7 & — & 1,2,6 & — & 7 & — & 1,3,5,6 & 7\
1,3,5 & — & 3,5 & — & 1,5 & — & 5 & —\
1,3,5,7 & — & 3,5,7 & — & 1,2 & — & 1 & —\
2 & — & 1,6 & — & 2,6 & — & 1,3,7 & 6\
3,7 & 6 & 2,6 & — & 1,3,5,7 & 6 & 1,5,7 & 6\
1,2,7 & 6 & 2,3 & 7 & 1,2,3 & 7 & 3,4 & 7\
3,4,5 & 7 & 1,3,4 & 7 & 1,2,5 & 7 & 1,3,4,5,6 & 7\
1,4,5,6 & 7 & 1,3,4,5 & 7 & 1,4,5 & 7 & 1,4 & 7\
3,4,6 & 7 & 1,2,3,4 & 7 & 2,3,4 & 7 & 1,2,3,4,5 & 7\
1,2,4,5 & 7 & 1,3,4,6 & 7 & 1,2,3,5 & 7 & 2,3,5 & 7\
3,4,5,6 & 7 & 4,5,6 & 7 & 2,3,4,5 & 7 & 2,4,5 & 7\
2,4 & 7 & 1,2,4 & 7 & 1,5 & 7 & 1,3 & 7\
3 & 7 & 2,5 & 7 & 1,2 & 7 & 4,5 & 7\
4 & 7 & 1,2,4,5,6 & 7 & 2,4,5,6 & 7 & 6 & 7\
4,6 & 7 & 3,5,6 & 7 & 3,6 & 7 & 1,5,6 & 7\
2,5,6 & 7 & 1,3,6 & 7 & 1,2,5,6 & 7 & 1,2,4,6 & 7\
2,4,6 & 7 & 5,6 & 7 & 1,4,6 & 7 & &\
[| l|l || l|l || l|l || l|l|]{}\
massive & absent & massive & absent & massive & absent & massive & absent\
1,2,3,6 & — & 1,3 & — & 3 & — & 3,5,6 & —\
[| l|l || l|l || l|l || l|l|]{}\
massive & absent & massive & absent & massive & absent & massive & absent\
1,2,3,5 & — & 2 & — & 1,3,5 & — & 3 & —\
5 & — & 1,3 & — & 1,5 & — & &\
[| l|l || l|l || l|l || l|l|]{}\
massive & absent & massive & absent & massive & absent & massive & absent\
1,2,3,5 & — & 1,3,5 & — & 2 & — & 5 & —\
3,5 & — & 1,3 & — & & & &\
\[app:not\]Some details on the notation
=======================================
\[app:notin\]Notation of the input
----------------------------------
In this subsection we describe the notation to be used for the input.
On one side of the following expressions the [FORM]{} notation is used whereas the other side displays the corresponding mathematical terms. It is always assumed that $p_i$ ($i=1,\ldots,9$) is a loop momentum and $q_i$ ($i=1,\ldots,3$) is a small momentum in which an expansion is performed.
If scalar integrals are to be computed with [MATAD]{} the following notation has to be used: $$\begin{aligned}
\begin{array}{llll}
\mbox{massless lines:} & \frac{1}{p_1^2},\frac{1}{p_2^2},\ldots &
\longrightarrow & \mbox{\verb|1/p1.p1|, \verb|1/p2.p2|},\ldots
\\
& \frac{1}{-(p_1+q_1)^2},\ldots &
\longrightarrow & \mbox{\verb|Dl(p1,q1)|},\ldots
\\
\mbox{massive lines:} & \frac{1}{M^2-p_1^2},\frac{1}{M^2-p_2^2},\ldots &
\longrightarrow & \mbox{\verb|s1m|, \verb|s2m|},\ldots
\\
& \frac{1}{M^2-(p_1+q_1)^2},\ldots &
\longrightarrow & \mbox{\verb|Dh(p1,q1)|},\ldots
\end{array}\end{aligned}$$ Fermions are treated with the help of the (non-commutative) function `S` and the function `Dg` can be used for the gluon propagator:
$$\begin{aligned}
\begin{array}{llllll}
\verb|S(mu)|
&=&
\gamma^\mu
\,,
&
\verb|S(p1)|
&=&
\frac{p_1\hspace{-.7em}/}{-p_1^2}
\,,
\\
\verb|S(p1m)|
&=&
\frac{M+p_1\hspace{-.7em}/}{M^2-p_1^2}
\,,
&
\verb|S(q1,p1)|
&=&
\frac{p_1\hspace{-.7em}/\, + q_1\hspace{-.7em}/}{-(p_1+q_1)^2}
\,,
\\
\verb|S(q1,p1m)|
&=&
\frac{M+p_1\hspace{-.7em}/\, + q_1\hspace{-.7em}/}{M^2-(p_1+q_1)^2}
\,,
\\
\verb|Dg(mu,nu,p1)|
&=&
\frac{-g^{\mu\nu} - \xi \frac{p_1^\mu p_1^\nu}{-p_1^2}}{-p_1^2}
\,,
&
\verb|Dg(mu,nu,q1,p1)|
&=&
\frac{-g^{\mu\nu} - \xi \frac{(p_1+q_1)^\mu (p_1+q_1)^\nu}
{-(p_1+q_1)^2}}{-(p_1+q_1)^2}
\,.
\end{array}
\label{eq:SDg}\end{aligned}$$
Instead of `S` also `SS`, `SSS` or `SSSS` may be used which is useful in case more than one fermion line is involved. The vertices between three gluons and gluon and ghosts are implemented through the following functions (where the colour factors are not taken into account): $$\begin{aligned}
\begin{array}{lll}
\verb|V3g(i1,p1,i2,p2,i3,p3)|
&=&
\left(p_2- p_1\right)^{i_3} g^{i_1 i_2}
+\left(p_3- p_2\right)^{i_1} g^{i_2 i_3}
+\left(p_1- p_3\right)^{i_2} g^{i_3 i_1}
\,,
\\
\verb|Vgh(i1,p1)| &=& -p_1^{i_1}
\,.
\end{array}
\label{eq:V3g}\end{aligned}$$ If small momenta are present they can simply be added to $p_1$, $p_2$ or $p_3$. In analogy to the expansion in a small momentum it is possible to introduce functions where an expansion in small masses can be performed.
In the above expressions instead of `q1` also `q2` or `q3` may be chosen as momenta in which an expansion is done. For the loop momenta `p1`, …, `p9` is allowed.
For the Feynman rules given above the user has to check the consistency with own sign conventions. In particular all unnecessary factors like, e.g., the strong coupling constant are omitted. Also the imaginary unit, $i$, is suppressed. Furthermore the Feynman rules are chosen in such a way that the vertices are proportional to $i$ and all propagators are proportional to $1/i$ which leads to convenient cancellations. Thus each diagram can be written as $$\begin{aligned}
i\left(\frac{1}{i} \int\frac{{\rm d}^D k_j}{\left(2\pi\right)^D}
\right)^l
\times[\mbox{expression formed by the Feynman rules
of~(\ref{eq:SDg}) and~(\ref{eq:V3g}) }]
\,,\end{aligned}$$ where $l$ is the number of loops. The factor $1/i$ in front of the integration momenta disappears after the Wick rotation. The overall factor $i$, which occurs for all diagrams and which is independent of the number of loops, is omitted.
Notation and conventions of the output
--------------------------------------
At this point some words about the notation of the output of [MATAD]{} are in order. As already mentioned, the final expressions are expanded in $\varepsilon$ where — following general $\overline{\rm MS}$ conventions — the factors $\gamma_E$ and $\ln4\pi$ have been dropped. Furthermore the factors $(1/16\pi^2)^l$ and $(1/M^{2\varepsilon})^l$ where $l$ is the number of loops are omitted. Concerning the remaining symbols the translation is given in the following table:
--------------------------------------------------- -------------------------------------------
$\varepsilon$ `ep`
Euclidean external momenta `Q, Q1, Q2,` …
$\zeta_2, \zeta_3, \ldots$ (Rieman zeta function) `z2, z3,` …
mass appearing in the integrals `M`
parametrization of the finite parts
of the master integrals `D5, D4,` … (see Appendix \[app:master\])
--------------------------------------------------- -------------------------------------------
Note that all momenta which occur in the output have to be interpreted in Euclidean space.
\[app:master\]Master integrals
------------------------------
[MATAD]{} does not insert the complete finite parts of the master integrals automatically. Instead they are parameterized by the symbols [D5]{}, [D4]{}, [DM]{}, [DN]{}, [B4]{}, ... where the notation is adopted from the one introduced in Fig. \[fig:batop\]. In the following list one can find the corresponding analytical expressions [@Bro92; @Bro98; @FleKal99; @CheSte00; @DavTau93][^1]
$$\begin{aligned}
\verb|D6| &=&
6\zeta_3
-17 \zeta_4
-4 \zeta_2\ln^2 2
+\frac{2}{3}\ln^4 2
+ 16 \mbox{Li}_4\left(\frac{1}{2}\right)
-4 \left[{\rm Cl}_2\left(\frac{\pi}{3}\right)\right]^2
\,,
\nonumber\\
\verb|D5| &=& 6\zeta_3-\frac{469}{27}\zeta_4
+\frac{8}{3}\left[{\rm Cl}_2\left(\frac{\pi}{3}\right)\right]^2
-16\sum_{m>n>0} \frac{(-1)^m \cos(2\pi n/3)}{m^3n}
\nonumber\\&\approx&
-8.2168598175087380629133983386010858249695
\,,
\nonumber\\
\verb|D4| &=&
6\zeta_3
-\frac{77}{12}\zeta_4
-6 \left[\mbox{Cl}_2\left(\frac{\pi}{3}\right) \right]^2
\,,
\nonumber\\
\verb|D3| &=&
6\zeta_3
-\frac{15}{4}\zeta_4
-6 \left[\mbox{Cl}_2\left(\frac{\pi}{3}\right) \right]^2
\,,
\nonumber\\
\verb|DM| &=&
6\zeta_3
-\frac{11}{2}\zeta_4
-4 \left[\mbox{Cl}_2\left(\frac{\pi}{3}\right) \right]^2
\,,
\nonumber\\
\verb|DN| &=&
6\zeta_3
- 4\zeta_2\ln^2 2
+ \frac{2}{3}\ln^4 2
- \frac{21}{2}\zeta_4
+ 16 \mbox{Li}_4\left(\frac{1}{2}\right)
\,,
\nonumber\\
\verb|B4| &=&
- 4 \zeta_2 \ln^2 2
+ \frac{2}{3}\ln^4 2
- \frac{13}{2} \zeta_4
+ 16 \mbox{Li}_4\left(\frac{1}{2}\right)
\,,
\nonumber\\
\verb|E3| &=&
- \frac{139}{3}
- \frac{\pi\sqrt{3} \ln^2 3}{8}
- \frac{17\pi^3\sqrt{3}}{72}
- \frac{21}{2}\zeta_2
+ \frac{1}{3}\zeta_3
\nonumber\\&&\mbox{}
+ 10\sqrt{3}{\rm Cl}_2\left(\frac{\pi}{3}\right)
- 6\sqrt{3} {\rm Im}\left[ {\rm
Li}_3\left(\frac{e^{-i\pi/6}}{\sqrt{3}}\right)\right]
\,,
\nonumber\\
\verb|S2| &=& \frac{4}{9\sqrt{3}} \mbox{Cl}_2\left(\frac{\pi}{3}\right)
\,,
\nonumber\\
\verb|OepS2| &=&
- \frac{763}{32}
- \frac{9\pi\sqrt{3}\ln^2 3}{16}
- \frac{35\pi^3\sqrt{3}}{48}
+ \frac{195}{16}\zeta_2
- \frac{15}{4}\zeta_3
+ \frac{57}{16}\zeta_4
\nonumber\\&&\mbox{}
+ \frac{45\sqrt{3}}{2} {\rm Cl}_2\left(\frac{\pi}{3}\right)
- 27\sqrt{3} {\rm Im}\left[ {\rm
Li}_3\left(\frac{e^{-i\pi/6}}{\sqrt{3}}\right)\right]
\,,
\nonumber\\
\verb|T1ep| &=&
- \frac{45}{2}
- \frac{\pi\sqrt{3} \ln^2 3}{8}
\nonumber\\&&\mbox{}
- \frac{35\pi^3\sqrt{3}}{216}
- \frac{9}{2}\zeta_2
+ \zeta_3
+ 6\sqrt{3}{\rm Cl}_2\left(\frac{\pi}{3}\right)
- 6\sqrt{3} {\rm Im}\left[ {\rm
Li}_3\left(\frac{e^{-i\pi/6}}{\sqrt{3}}\right)\right]
\label{eq:master}
\,,\end{aligned}$$
with $\mbox{Cl}_2(x) = \mbox{Im}[\mbox{Li}_2(e^{ix})]$. $\mbox{Li}_2$, $\mbox{Li}_3$ and $\mbox{Li}_4$ are the Di-, Tri- and Quadrilogarithm, respectively. `S2` both appears in the diagram of Fig. \[fig:master\_add\] and the two-loop diagram of Fig. \[fig:tad123\] where all three lines have the same mass. Their ${\cal O}(\varepsilon)$ parts, which can contribute to the finite part of the three-loop results, contain `OepS2` and `T1ep`, respectively. We should mention that those expressions of Eq. (\[eq:master\]) which coincide with an existing topology (e.g. [D5]{} $\leftrightarrow$ [topD5]{}) indeed agree with the finite part of the corresponding master integral. On the other hand [B4]{}, e.g., comprises the complicated parts of the finite part of the master integral corresponding to [topBN]{}, however, not the complete one. `D6` is not yet implemented into [MATAD]{} and listed for completeness only.
\[app:settings\]Parameters and switches in [main.<prb>]{}
---------------------------------------------------------------
In the following a brief description of the switches in the file [main<prb>]{} is given. In all cases the [FORM]{} syntax reads `#define <VAR> "<VAL>"` where `<VAR>` is the variable to be defined and `<VAL>` the corresponding value.
It is required to define at least the following five variables:
[DIAGRAM:]{}
: name of the diagram to be computed.
[FOLDER:]{}
: the file containing the special treat files and the diagrams is called [’FOLDER’.dia]{}.
[GAUGE:]{}
: determines the choice for the gauge parameter. The variable $\xi$ as defined in the function `Dg` in Eq. (\[eq:SDg\]) is replaced by the value of [GAUGE]{}. In particular `#define GAUGE "0"` corresponds to Feynman gauge and with `#define GAUGE "xi"` the calculation is performed for general gauge parameter.
[POWER:]{}
: determines the depth of the expansion in the small quantities.
[PRB:]{}
: name of problem. [PRB]{} corresponds to the name of the fold where the problem-dependent files can be found.
The definition of the remaining variables is optional:
[BNRECOLD:]{}
: if this variable is defined the original recurrence procedure of Ref. [@Bro92] is used for the topology [BN]{}.
[CUT:]{}
: determines the depth of the expansion in $\varepsilon$ of the final result. At one-, two- and three-loop order at most the terms of order $\varepsilon^2$, $\varepsilon^1$, respectively, $\varepsilon^0$ are reliable. The default value is “2”.
[DALA12:]{}
: expansion in $q_1 q_2$. If this variable is set $q_1^2$ and $q_2^2$ are set to zero and powers in $q_1 q_2$ are factored out. Currently only the expansion up to order $(q_1 q_2)^4$ is implemented. Note that only positive powers of $q_1 q_2$ can be treated.
[DALAQN:]{}
: apply d’Alembert operator, $\Box_q=\partial/\partial q^\mu \partial/\partial q_\mu$, in order to factor out powers in $q^2$. In the argument of [DALAQN]{} the momentum is specified with respect to which the derivatives are performed.
[NOR:]{}
: is an abbreviation for Number Of Recursions. As the naive use of the [repeat]{}–[endrepeat]{} construction significantly slows down the performance the most complicated procedures are buffered by a `#do`–`#enddo` construction. [’NOR’]{} constitutes the upper bound of the do-loop. The default value is “10”.
[PROBLEM0/1/2/MAIN:]{}
: If one of these variables is set a special treat file is read at the corresponding position. The value has to agree with the one defined in the file `<prb>.dia`.
[TIME:]{}
: If this variable is set the statistics is printed at various steps of the calculation.
There are more switches in [inc/main.gen]{}. However, they should not be modified as most of them are in an experimental stage and not sufficiently tested.
\[app:files\]List of files
==========================
In this appendix we provide a list of all files belonging to [MATAD]{}. The directory [inc]{} contains essentially the include-files. In particular some of the procedures are collected in files which are included at the very beginning. The tables for the topology [BN]{} are located in [inc/TABLEDAL]{} and [inc/TABLEREC]{}. The directories [inc/TREAT]{} and [inc/TOPOLOGY]{} essentially contain the files treating the individual (input and basic) topologies and in the folder [prc]{} some auxiliary procedures are collected.
form.set inc/ matadform prc/ problems/
inc:
TABLEDAL/ bnm2m expandDr nomBM reduceBM_2 tblBN
TABLEREC/ declare.matad expepgam nomBN reduceBN
TOPOLOGY/ denoexp expnomdeno nomdecomBN reduceBN1
TREAT/ expandBN main.gen recursion_2 reduceBN_2
bnbm2prc expandBNM matad.info redcut symmetryBM
bnbmprc expandBNM_2 matminprc redcutnomdeno symmetryBN
inc/TABLEDAL:
BNd.tbl BNd0.tbl
inc/TABLEREC:
BN.tbl BNn1.tbl BNn1n2.tbl BNn2.tbl
inc/TOPOLOGY:
topBE topBN topBU topE3 topM2 topN3 topO4.add topY3
topBM topBN1 topD4 topE4 topM3 topNO topT1 topempty
topBM1 topBN2 topD5 topL1 topM4 topO1 topT2
topBM2 topBN3 topDM topLA topM5 topO2 topY1
topBM_2 topBN_2 topDN topM1 topN2 topO4 topY2
inc/TREAT:
treat.dala12 treatbm2 treatbn2 treatd5 treate4 treatm4
treat.dalaav treatbm_2 treatbn3 treatdm treatm1 treatm5
treatbm treatbn treatbn_2 treatdn treatm2 treatn1
treatbm1 treatbn1 treatd4 treate3 treatm3 treatt1
prc:
aver.prc cutep.prc dalaqn.prc difvecsc.prc solveS.prc treat.prc
aver1.prc dala12sc.prc dalasc.prc pochtabl.prc tabBN.prc
Files taken over from [MINCER]{} {#files-taken-over-from-mincer .unnumbered}
--------------------------------
one.prc simplify.prc finish.prc tabtwo.prc
accu.prc dotwo.prc newtwo.prc two.prc
triangl2.prc triangle.prc pochtabl.prc
\[app:testrun\]Fermion propagator: output
=========================================
In this appendix we present the complete output of the example discussed in Section \[sub:ferm\]. Calling [MATAD]{}
> matadform problems/fp/mainfp
leads to
FORM version 2.3 Apr 24 1997
*
* mainfp
*
#define PRB "fp"
#define PROBLEM0 "1"
#define DALAQN "q1"
#define GAUGE "0"
#define POWER "1"
#define CUT "0"
#define FOLDER "fp"
#define DIAGRAM "d3l79"
#-
*~~ MATAD -- computation of MAssive TADpoles
*~~ read generic main file
*~~ read diagram
G dia=
#include problems/'PRB'/'FOLDER'.dia # 'DIAGRAM'
((1)
*Dg(nu3,nu4,p5)
*Dg(nu5,nu6,-p6)
*Dg(nu1,nu2,-q1,-p1)
*S(nu2,-p1m,nu5,-p4m,nu4,-p3m,nu6,-p2m,nu3,-p1m,nu1)
*1);
#define TOPOLOGY "O4"
*--#] d3l79:
#-
*~~ Treat the traces
*~~ Include special treat-file 0
*~~ Feynman rules for vertices and propagators:
*~~ gluon-ghost-ghost-vertex
*~~ 3-gluon-vertex
*~~ gluon propagator
*~~ ghost propagator
*~~ expand denominators
*~~ Dh
*~~ Dl
*~~ 1
*~~ 1
*~~ Change notation to p1,p2,...
*~~ Trace 1
*~~ Trace 2
*~~ Trace 3
*~~ Trace 4
*~~ treat DL(x)
*~~ Do Wick-rotation
*~~ Apply d Alembertian w.r.t. q1
*~~ average done
*~~ Expand Dr(p,q)
*~~ 1
*~~ q_i -> Q_i
*~~ include TOPOLOGY-file
*~~ this is topO4
*~~ Recursion of type d5
*~~ this is topD5
*~~ numerator
*~~ do recursion
*~~ - done
*~~ Recursion of type d4
*~~ this is topD4
*~~ numerator
*~~ do recursion
*~~ - done
*~~ Recursion of type dm
*~~ this is topDM
*~~ numerator
*~~ do recursion
*~~ - done
*~~ Recursion of type dn
*~~ this is topDN
*~~ numerator
*~~ do recursion
*~~ - done
*~~ Recursion of type e4
*~~ this is topE4
*~~ numerator
*~~ do recursion
*~~ - done
*~~ Recursion of type e3
*~~ this is topE3
*~~ numerator
*~~ do recursion
*~~ - done
*~~ Recursion of type bn_2
*~~ this is topBN_2
*~~ numerator
*~~ do recursion
*~~ Use table for BN
*~~ - done
*~~ Recursion of type bn1
*~~ this is topBN1
*~~ numerator
*~~ do recursion
*~~ - done
*~~ Recursion of type bn2
*~~ this is topBN2
*~~ numerator
*~~ do recursion
*~~ - done
*~~ Recursion of type bn3
*~~ this is topBN2
*~~ numerator
*~~ do recursion
*~~ - done
*~~ Recursion of type bm_2
*~~ this is topBM_2
*~~ numerator
*~~ do recursion
*~~ - done
*~~ Recursion of type bm1
*~~ this is topBM1
*~~ numerator
*~~ do recursion
*~~ - done
*~~ Recursion of type bm2
*~~ this is topBM2
*~~ numerator
*~~ do recursion
*~~ - done
*~~ Integration of the simple integrals
*~~ Recursion of type m1
*~~ this is topM1
*~~ Recursion of type m2
*~~ this is topM2
*~~ Recursion of type m3
*~~ this is topM3
*~~ Recursion of type m4
*~~ this is topM4
*~~ Recursion of type m5
*~~ this is topM5
*~~ perform integration
*~~ Recursion of type t1
*~~ this is treatn1
*~~ Recursion of type n1
*~~ this is treatn1
*~~ Simplify
*~~ Do the "rest"-integration
Time = 5.33 sec Generated terms = 23
d3l79 Terms in output = 23
Bytes used = 410
d3l79 =
+ ep^-3 * ( - 8/3 - 1/3*a )
+ ep^-2 * ( 56/3 - 20/3*a )
+ ep^-1 * ( 112/3 - 16*z3 + 19/2*z2*a - 20*z2 - 97/12*a )
+ 334/3 + 1215/2*S2*a - 1620*S2 + 16*D3*a - 40*D3 - 1141/3*z3*a + 2368/
3*z3 + 144*z4*a - 288*z4 + 57*z2*a - 156*z2 - 32*a*B4 - 77/6*a + 64*
B4;
save problems/'PRB'/results/'DIAGRAM'.res 'DIAGRAM';
.end
[99]{}
\#1\#2\#3[[*Act. Phys. Pol. *]{}[**B \#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Act. Phys. Austr. *]{}[**\#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Comm. Math. Phys. *]{}[**\#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Comp. Phys. Commun. *]{}[**\#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Eur. Phys. J. *]{}[**C \#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Fortschr. Phys. *]{}[**\#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Int. J. Mod. Phys. *]{}[**C \#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Int. J. Mod. Phys. *]{}[**A \#1**]{} (\#2) \#3]{} \#1\#2\#3[[*J. Comp. Phys. *]{}[**\#1**]{} (\#2) \#3]{} \#1\#2\#3[[*JETP Lett. *]{}[**\#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Mod. Phys. Lett. *]{}[**A \#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Nucl. Inst. Meth. *]{}[**A \#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Nucl. Phys. *]{}[**B \#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Nuovo Cim. *]{}[**\#1A**]{} (\#2) \#3]{} \#1\#2\#3[[*Phys. Lett. *]{}[**B \#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Phys. Reports* ]{}[**\#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Phys. Rev. *]{}[**D \#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Phys. Rev. *]{}[**\#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Phys. Rev. Lett. *]{}[**\#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Phys. Reports* ]{}[**\#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Prog. Theor. Phys. *]{}[**\#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Sov. J. Nucl. Phys. *]{}[**\#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Teor. Mat. Fiz. *]{}[**\#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Yad. Fiz. *]{}[**\#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Z. Phys. *]{}[**C \#1**]{} (\#2) \#3]{} \#1\#2\#3[[*Prog. Part. Nucl. Phys. *]{}[**\#1**]{} (\#2) \#3]{} \#1\#2\#3[[ibid. ]{}[**\#1**]{} (\#2) \#3]{}
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[^1]: Note that there is a misprint in Eq. (22) of Ref. [@CheSte00]: all four appearing Clausen functions must be squared and the coefficient of $\zeta_4$ in the second to last equation must read “$-22$” instead of “$-31/2$”.
| {
"pile_set_name": "ArXiv"
} |
In recent years, various compounds containing two-leg cuprate ladders have been synthesized[@elbiorice]. The ground states of these materials are characterized by strong electronic interactions with a variety of competing ground states. Theoretically, the focus has been on using the Hubbard[@noack] and $t-J$ models[@troyer] to understand the ground state properties of these materials. At half-filling, the two-leg ladders exhibit a spin-gap insulating phase, while away from half-filling $d_{x^2-y^2}$ pairing and charge density wave (CDW) correlations become enhanced[@hayward].
Recently, Zhang proposed a model with $SO(5)$ symmetry to relate the two phases of antiferromagnetism (AF) and $d$-wave superconductivity (dSC), even though these two phases have quite distinct ground states[@zhang]. Subsequently, several groups have constructed microscopic models which have an exact $SO(5)$ symmetry[@szh; @rabello; @henley; @burgess]. Furthermore, it has been argued that both the Hubbard[@meixner] and $t-J$ models[@eder] in two-dimensions (2D) have approximate $SO(5)$ symmetry.
In this paper, we examine one such microscopic Hamiltonian[@szh] using quantum Monte Carlo (QMC) and exact diagonalization (ED) techniques. The model we study can be changed adiabatically from the pure Hubbard model on a two-leg ladder to a model with $SO(5)$ symmetry, thus elucidating how the low energy spectrum evolves as the higher symmetry is approached. In this way, we can test whether the low lying excitations of the Hubbard ladder are well described by the $SO(5)$ picture.
The microscopic $SO(5)$ ladder we investigate here was first introduced by Scalapino, Zhang and Hanke (SZH)[@szh]. The SZH model contains only local interactions on a rung of the ladder and is therefore much easier to visualize and implement numerically than 2D $SO(5)$ models with long range interactions. The Hamiltonian is given by
$$\begin{aligned}
H = H_{hop} + H_{int}.\end{aligned}$$
$H_{hop}$ allows electrons to move along the ladder (in the $\hat{x}$ direction) as well as within a rung (in the $\hat{y}$ direction):
$$\begin{aligned}
H_{hop} = -t_{\parallel}{ \sum_{{\bf i},s}
(c^{\dagger}_{{\bf i},s}c_{{\bf i+\hat{x}},s}+H.c.)} \nonumber\\
-t_{\perp}{ \sum_{{\bf i},s}
(c^{\dagger}_{{\bf i},s}c_{{\bf i+\hat{y}},s}+H.c.)}\end{aligned}$$
where $c^\dagger_{{\bf i},s}$ creates an electron at site ${\bf i}$ with spin projection $s$. The interaction term, $H_{int}$, is given by
$$\begin{aligned}
H_{int} = U{ \sum_{{\bf{i}}}(n_{{\bf{i}} \uparrow}-1/2)( n_{{\bf{i}}
\downarrow}-1/2)+\mu\sum_{{\bf{i}},s}n_{{\bf{i}}s} } \nonumber \\
+ V \sum_{\bf i} (n_{\bf i}-1)(n_{\bf i+\hat{y}}-1)
+ J \sum_{\bf i} \vec{S}_{\bf i}\cdot\vec{S}_{\bf i+\hat{y}}\end{aligned}$$
where $n_{\bf i}$ is the number operator for site ${\bf i}$; $U$ is the Coulomb repulsion between electrons occupying the same site; $V$ is an interaction between electrons on a given rung; $J$ is a rung spin-spin interaction with
$$\begin{aligned}
\vec{S_{\bf i}} = \frac{1}{2}\sum_{s,s'} c^\dagger_{{\bf i},s}
\vec{\sigma}_{s,s'}c_{{\bf i},s'}.\end{aligned}$$
In order for this Hamiltonian to be manifestly $SO(5)$ symmetric, the condition of $J=4(U+V)$ must be imposed.
The phase diagram of the SZH model at half-filling has been obtained at strong coupling, i.e., $U,V>>t_\perp,t_\parallel$[@szh]. Predominately for $U>0$, the ground state is well described as a product of rung singlets, i.e., a spin-gap insulator. Doped holes in this ground state form $d_{x^2-y^2}$ rung pairs and exhibit power law pairing and CDW correlations. A phase transition occurs for positive values of $U$ from the singlet ground state into a triplet ground state along the line $V=-U$, where each rung is occupied by a triplet magnon or a doublet pair. Furthermore, the triplet ground state can be considered to be an $SO(5)$ generalization of the spin-one Heisenberg chain, i.e., a system with a finite excitation gap and short range correlations.
At weak couplings ($U,V << t_\perp,t_\parallel$), the phase diagram of the SZH model was obtained using a perturbative renormalization group analysis[@lin]. The resulting phase diagram, although consistent with the strong coupling result, contains new phases not found in the strong coupling limit. Furthermore, it was shown that two-leg Hubbard-like ladders flow to $SO(5)$ symmetry even when explicit symmetry breaking terms were included, such as longer range hoppings[@arrigoni].
In this paper, we explicitly turn off the spin-spin interaction by requiring that $J=0$[@qmcnote]. Thus, our model is $SO(5)$ symmetric [*only*]{} when $V=-U$, where the ground state at large couplings was found to have a degeneracy between the rung singlet and triplet $SO(5)$ multiplets. Consequently, we now have an adjustable model which interpolates from the Hubbard ladder to the $SO(5)$ ladder by varying $V$ from 0 to $-U$. We will study this model in order to investigate how the low lying excitations evolve as $V\rightarrow -U$, i.e., as the model approaches $SO(5)$ symmetry. Throughout, we will take the isotropic hopping case of $t = t_\perp = t_\parallel = 1$ and will only consider $U>0$.
In Zhang’s $SO(5)$ theory, the superspin vector takes on a fixed magnitude below some characteristic temperature, $T^*$, and its direction fluctuates between the AF and dSC phases. For the case in which the model is manifestly $SO(5)$ symmetric $(V=-U)$, $T^*$ is pushed to infinity and all of the components of the superspin vector will be equal. To test this, we use deterministic quantum Monte Carlo (QMC)[@qmc] to measure the first and fourth components of the superspin vector given by
$$\begin{aligned}
n_1(r) &=& \frac{(-1)^r}{2}(\Delta^\dagger + \Delta) \nonumber \\
&=& \frac{(-1)^r}{2}\bigl(-ic^\dagger(r) \sigma_y \
c^\dagger(r+\hat{y}) + h.c.\bigr)\end{aligned}$$
and
$$\begin{aligned}
n_4(r) = \frac{(-1)^r}{2}\bigl(c^\dagger(r) \sigma_z c(r)
-c^\dagger(r+\hat{y}) \sigma_z c(r+\hat{y})\bigr).\end{aligned}$$
Here $r$ is the rung index, and the spinor index on the $c$ operators has been suppressed, i.e., $c(r) =
(c_{\uparrow,r},c_{\downarrow,r})$.
In Fig. 1, the sum of $n_1$ and $n_4$ over all rungs of a $2\times 8$ and $2\times 12$ ladder are shown, i.e., $\langle n_{1(4)}\rangle =
\frac{1}{N_{rungs}}\sum_r \langle n^\dagger_{1(4)}(r) n_{1(4)}(0)
\rangle$. At half-filling $(\mu=0)$ for an intermediate strength coupling of $U=4t$, the expected behavior is seen: as $-V$ approaches the $SO(5)$ value of $U$, the correlations become equal. Note that even at half-filling a sign problem exists due to the new Hubbard-Stratonovich fields introduced by the $V$ term[@qmcV]. At small to intermediate coupling strengths, i.e., $U<6t$, reliable results can be obtained, even at the $SO(5)$ point when $V=-U$. However, when $U$ becomes large, and consequently $-V$ becomes large as the symmetric point is approached, the sign problem becomes unmanageable. Thus, only $U=4t$ with $\beta t = 2$ QMC results are shown[@temp].
Recall that at strong couplings along the phase transition line of $U=-V$, the ground state was found to have a degeneracy between the rung singlet and rung triplet $SO(5)$ multiplets. Therefore, the correlation functions should decay as a power law in distance, $r$, with some thermal exponential activation, i.e., $n_{1(4)}(r,\beta)\sim
\mbox{exp}(-\beta/\xi)/r^\alpha$ (where $\xi$ is the thermal correlation length). However, when $V$ is different from $-U$, the correlations should decay exponentially in $r$.
In order to test if this critical behavior could be seen, the correlations of the superspin components were measured as a function of distance along the ladder. In Fig. 2, we show the results obtained for the $2\times 12$ ladder with $U=4t$ at half-filling as $V$ approaches the $SO(5)$ value. In Fig. 2(a), a qualitative change in the correlations of $n_1(r)$ is clearly observed between the Hubbard ladder and the $SO(5)$ ladder, indicating a crossover from a power law to an exponential decay in $n_1(r)$ as the $SO(5)$ point is approached. In contrast, the change in the $n_4(r)$ correlations (Fig. 2(b)) is less pronounced.
It was not possible to cleanly extract the power law exponents from a log-log analysis of these correlations for several reasons. First, $U=4t$ is not strong enough to recover the exact degeneracy between $SO(5)$ multiplets which is present only at large coupling strengths. Also, at a relatively high temperature of $\beta t = 2$, the thermal correlation length tends to dominate the behavior of the superspin components[@noack]. Since reliable measurements at either larger couplings or lower temperatures could not be obtained, let us now turn to a zero-temperature exact diagonalization (ED) analysis to better understand the low energy spectrum[@ed].
Because of the large Hilbert space at half-filling, the system sizes which can be studied with ED are limited, and we will only present results for a $2\times 6$ ladder. On the other hand, an advantage of ED is that it allows for the direct calculation of the spin-gap, as well as dynamical spectra[@meixner; @eder] given by:
$$\begin{aligned}
\hat{O}({\bf q},\omega) = -\frac{1}{\pi}\mbox{Im}
\langle {\bf k}|
\hat{O}^\dagger_{-{\bf q}}
\frac{1}{\omega+E_{g.s.}-\hat{H}-i\epsilon}
\hat{O}_{{\bf q}}
|{\bf k}\rangle\end{aligned}$$
where $|{\bf k}\rangle$ is the ground state wave function with momentum ${\bf k}$, and $\hat{O}({\bf q},\omega)$ is an arbitrary translationally invariant operator.
What is the nature of the ground state of the $2\times 6$ ladder? In Fig. 3(a), the dynamical spin response, using $\hat{O}_{\bf q} =
\hat{S}^z_{\bf q} = \sum_j \mbox{exp}(-i{\bf q\cdot r_j})
\hat{S}^z({\bf r_j})$, is shown for $U=4t$[@spread]. Starting from the Hubbard ladder, $V=0$, we find that a spin-gap is present at half-filling[@noack]. In particular, as seen in Fig. 3, as $V$ approaches $-U$, the spin-gap remains robust and even increases for the $SO(5)$ ladder rather than going to zero as one might expect from the strong coupling ($J\rightarrow 0$) limit. The low lying spin excitations are dominated by a large peak in the spin response occurring at a momentum transfer of $(\pi,\pi)$. (Note that Figs. 3(a) and (b) show the momentum integrated spin excitation spectrum, $S(\omega) = \int d{\bf q} S({\bf q},\omega)$.) Furthermore, there is little qualitative difference between the spectra as the model moves toward the ladder of higher symmetry. Similar behavior is seen for $U=8t$ (Fig. 3(b)) where again a spin-gap is always present. An angular resolved examination of $S({\bf
q},\omega)$ shows that the dominant low energy peak in $S(\omega)$ is due to inter-band scattering processes, involving a momentum transfer of $\pi$ in the rung direction, whereas the quasi-continuous spectral weight at higher frequencies stems from intra-band processes.
The spin-gap can also be obtained by measuring the energy difference between the ground state in the $S^z=0$ sector and the ground state with total $S^z=1$ at a momentum of $(\pi,\pi)$, shown in Fig. 3(c) with $V=-U$. The value obtained in this way agrees exactly with the position of the lowest peak in $S(\omega)$. The gap increases, reaching its maximum around the intermediate coupling strength of $U=6t$, and then falls off as $1/U$. For comparison, the expected result from strong coupling given by $4t^2/U$ is shown in Fig. 3(c) as the solid line. Therefore, it is understandable why the power law behavior in the correlations was not seen from the QMC simulations. An effective spin-spin interaction, $J_{eff}=4t^2/U$, causes the system to have a spin gap with the ground state being well described by a product of rung singlets. Hence, no degeneracy between the two $SO(5)$ multiplets exists.
In order to explore the approximate symmetry of the Hubbard ladder and how it evolves as a function of $V$, the dynamic response of the $\pi$ operator, i.e., $\hat{O}_{\bf q} = \hat{\pi}_{\bf q} = \sum_{\bf r}
\mbox{exp}(-i{\bf q\cdot r}) \hat{\pi}({\bf r})$ with
$$\begin{aligned}
\hat{\pi}({\bf r}) = \frac{1}{2}\bigl(\hat{\pi}_x({\bf r}) +
i\hat{\pi}_y({\bf r})\bigr)
= \frac{-i}{2}(-1)^{r} c_\uparrow({\bf r}+\hat{y})c_\uparrow({\bf r}),\end{aligned}$$
was measured for the state with one electron pair more than the half-filled state. Fig. 4 shows a direct comparison at $U=4t$ and $U=8t$ between the low energy spin excitations and the $\pi$ resonance using the energy of the half-filled ground state as a reference. Clearly, both have a dominant low energy peak at the same excitation energy independent of the rung coupling $V$, indicating that the resulting final states are equivalent. Thus, an approximate $SO(5)$ symmetry is revealed. However, it should be noted that the $\pi$ resonance has significant spectral weight at higher energies (not shown in Fig. 4) for the Hubbard ladder, whereas all the weight shifts to the low energy peak as the symmetric ladder is approached[@pinote]. A thorough finite-size scaling analysis is expected to show that this low energy $\pi$ resonance survives in the thermodynamic limit as the length of the ladder is increased.
In this paper, we have numerically analyzed a model which can be varied by changing a rung interaction so that it interpolates between the Hubbard and the $SO(5)$ SZH model on a two-leg ladder. We find that the spin-gap vanishes as $U/t$ goes to infinity, and the ground state is well described by rung singlets for intermediate to large coupling strengths. This can be understood since the hopping along the chain and within a rung creates an effective spin-spin interaction $J_{eff}=4t^2/U$. This effective spin-spin interaction must be overcome by an even stronger negative value of the rung interaction, $-|V|>U$ in order to restore the degeneracy between rung singlets and triplets obtained as $U/t\rightarrow\infty$. Furthermore, no qualitative changes occur in the low lying excitations of the Hubbard ladder as the $SO(5)$ symmetric point is approached. Therefore, we conclude that the $\pi$ operators are approximate eigenoperators of the Hubbard ladder near half-filling, and hence, the low energy behavior of the Hubbard ladder in the intermediate to strong coupling range has an approximate $SO(5)$ symmetry.
The authors are deeply grateful to D. Scalapino, and have benefited from enlightening conversations with A. Sandvik, A. Moreo, R. Noack, S. R. White, E. Jeckelmann, E. Demler, B. Sugar, C. Martin, A. DeLia, R. Konik and S.-C. Zhang. D. Duffy acknowledges the support from the Dept. of Energy under grant DE-FG03-85ER451907. We thank the San Diego Supercomputing Center for providing us access to their facilities.
E. Dagotto and T. M. Rice, [*Science*]{}, [**271**]{}, 618 (1996).
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Shou-Cheng Zhang [*Science*]{} [**275**]{}, 1089 (1997).
D. Scalapino, S.-C. Zhang, and W. Hanke, cond-mat/9711117.
S. Rabello, H. Kohno, E. Demler, and S.-C. Zhang, cond-mat/9707027.
C. Henley, cond-mat/9707275.
C. P. Burgess, J. M. Cline, R. MacKenzie, and R. Ray, cond-mat/9707290.
Stefan Meixner, Werner Hanke, Eugene Demler, and Shou-Cheng Zhang, cond-mat/9701217.
R. Eder, W. Hanke, and S.-C. Zhang, cond-mat/9707233.
Hsiu-Hau Lin, Leon Balents, and Matthew P. A. Fisher, cond-mat/9801285.
E. Arrigoni and W. Hanke, cond-mat/9712143.
One reason for imposing the condition $J=0$ is to simplify the QMC simulation. The Hubbard-Stratonovich transformation of the spin-spin interaction results in a large number of auxiliary fields over which one must sample to obtain the ground state properties. As can be seen with just the $V$-term which introduces 4 additional fields, the sign problem becomes worse when $J$ or $V$ become comparable to $U$, even at half-filling.
R. Blankenbelcher, D. J. Scalapino, and R. L. Sugar, Phys. Rev. D [**24**]{}, 2278 (1981); J. E. Hirsch, Phys. Rev. B [**28**]{}, 4059 (1983); J. E. Hirsch, Phys. Rev. B [**31**]{}, 4403 (1985); S. R. White, D. J. Scalapino, R. L. Sugar, E Y. Loh, J. E. Gubernatis, and R. T. Scalettar, Phys. Rev. B [**40**]{}, 506 (1989).
See for example J. E. Gubernatis, D. J. Scalapino, R. L. Sugar, and W. D. Toussaint, Phys. Rev. B [**32**]{}, 103 (1985).
For the intermediate value of the coupling $U=4t$ presented here, we could reach lower temperatures of $\beta t = 3$ or 4. However, the sign problem grows with decreasing temperature.
C. Lanczos, J. Res. Natl. Bur. Stand. [**45**]{}, 255 (1950); D. G. Pettifor, D. L. Weaire (eds.): [*The Recursion Method and Its Applications*]{}, Springer Ser. Solid-State Sci., Vol. 58 (Springer, Berlin, Heidelberg 1985); E. Dagotto and A. Moreo, Phys. Rev. D. [**31**]{}, 865 (1985).
The resulting delta function peaks in the spin response are replaced by Lorentzians of the form $\delta(\omega)=\frac{1}{\pi}\frac{\epsilon}{\omega^2+\epsilon^2}$. The broadening used throughout this work was $\epsilon=0.1t$.
We have checked explicitly that at the $SO(5)$ point all of the weight of the $\pi$ operator goes into the lowest energy peak shown in Fig. 4.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
Recently, Dziembowski [*et al.* ]{}introduced the notion of *non-malleable codes* (NMC), inspired from the notion of non-malleability in cryptography and the work of Gennaro [*et al.* ]{}in 2004 on tamper proof security. Informally, when using NMC, if an attacker modifies a codeword, decoding this modified codeword will return either the original message or a completely unrelated value.
The definition of NMC is related to a family of modifications authorized to the attacker. In their paper, Dziembowski [*et al.* ]{}propose a construction valid for the family of all bit-wise independent functions.
In this article, we study the link between the second version of the Wire-Tap (WT) Channel, introduced by Ozarow and Wyner in 1984, and NMC. Using coset-coding, we describe a new construction for NMC w.r.t. a subset of the family of bit-wise independent functions. Our scheme is easier to build and more efficient than the one proposed by Dziembowski [*et al.* ]{}
author:
- 'Hervé Chabanne [^1] [^2]'
- 'Gérard Cohen [^3]'
- 'Jean-Pierre Flori'
- Alain Patey
bibliography:
- 'IEEEabrv.bib'
- 'biblio.bib'
title: 'Non-Malleable Codes from the Wire-Tap Channel[^4]'
---
=\[draw, minimum size=2em\]
Introduction
============
In cryptography, the non-malleability property [@DDN91] requires that it is impossible, given a ciphertext, to produce another different ciphertext so that the corresponding plaintexts are related to each other. Non-malleability under adaptive chosen-ciphertext attack (NM-CCA2) is one of the strongest computational security property that is required from an asymmetric encryption scheme (it is equivalent to indistinguishability under adaptive chosen-ciphertext attack (IND-CCA2)).
Recently, Dziembowski [*et al.* ]{}[@DPW10] proposed a transposition of the cryptographic definition of non-malleability to the field of coding theory. Informally, they define a NMC as a code such that, when a codeword is subject to modifications, its decoding procedure either corrects these errors and decodes to the original message or returns a value that is completely unrelated to the original message.
The property of non-malleability, as defined in [@DPW10], is subject to a choice of a family of modifications that we allow an adversary to make on the codewords. Dziembowski [*et al.* ]{}also proved that it is impossible for a code to be non-malleable w.r.t. the set of all possible modifications of codewords.
The motivation for NMC is tamperproofness. The authors of [@DPW10] were indeed much influenced by the work of Gennaro [*et al.* ]{}[@GLMMR04]. Non-malleability can be useful in real-life applications. Some storage devices may be assumed to be “read-proof” because of a sufficient amount of physical or algorithmic protections to prevent anyone from learning the data stored on them. However, even if one cannot read the data, injecting faults in the data and observing the way it affects functions using these data can help to recover them. Injecting faults can be done for instance using lasers [@SA02]. There exists an important literature on how to use Differential Fault Analysis to break cryptosystems (e.g. [@BDL97; @BCNTW04]).
Dziembowski [*et al.* ]{}studied deeply the non-malleability w.r.t. bit-wise independent tampering functions, [*i.e.* ]{}modifications that affect each bit of the codeword independently: flipping the bit or setting it to 0 or 1. This is typically what can be done using fault injections and, consequently, focusing on this family of tampering functions is worthwhile.
In [@DPW10], a construction for NMC w.r.t. all bit-wise independent functions is proposed. However, an implementable construction is left as an open problem. Our goal is to propose NMC that can be explicitly built. To this end, we exploit a relation that can be established between the model for NMC and the second version of the Wire-Tap channel [@OW84]. This allows us to prove how coset-coding can be used to build a NMC. Furthermore, the decoding procedure of linear-coset coding consists uniquely of one matrix-vector product. Our construction is thus computationally efficient. Moreover, unlike their solution, our procedure always decodes messages whereas theirs is closer to error detection and often returns an error symbol.
Organization of the Paper {#organization-of-the-paper .unnumbered}
-------------------------
In Section \[sec:nmc\], we explain and give the formal definitions for NMC as established in [@DPW10]. We describe the model of the WT channel in Section \[sec:wire\] and explain the use of coset-coding. We show how the second version of the WT channel and NMC w.r.t. bit-wise independent functions are related and prove why coset-coding can be used as a NMC in Section \[sec:scheme\]. We finally conclude in Section \[sec:conclu\].
Non-Malleable Codes {#sec:nmc}
===================
In this section, we intend to give an easy-to-understand description of NMC and their goals. All definitions come from [@DPW10]. In the following, we consider a randomized encoding function ${\mathrm{Enc}}: {\{0,1\}}^k \mapsto {\{0,1\}}^n$, which is associated to a deterministic decoding function ${\mathrm{Dec}}: {\{0,1\}}^n \mapsto {\{0,1\}}^k \cup \{\bot\}$, where $\bot$ means that the codeword cannot be decoded. Let ${\mathbb{F}}_2$ denote the field with two elements.
The Tampering Experiment
------------------------
Let us first introduce the situation considered in NMC. In this model, a source message $m$ is encoded using ${\mathrm{Enc}}$, in order to be later decoded using ${\mathrm{Dec}}$. The codeword $c = {\mathrm{Enc}}(m)$ is stored on a device or sent over a channel before being decoded. During this phase, an attacker applies some tampering function $f$ belonging to a given family of functions ${\mathcal{F}}\subset {\mathbb{F}_2^n}^{\mathbb{F}_2^n} $ . A tampered codeword $\tilde{c} = f(c)$ is thus obtained. This erroneous codeword is then decoded to $\tilde{m}={\mathrm{Dec}}(\tilde{c})$. This process is described in Figure \[fig:exp\].
\(a) [Enc]{}; (begin) \[left of=a,node distance=2cm, coordinate\] ; (b) \[right of=a\] [$f$]{}; (c) \[right of=b\] [Dec]{}; (end) \[right of=c, node distance=2cm\]; (begin) edge node [$m$]{} (a); (a) edge node [$c$]{} (b); (b) edge node [$\tilde{c}$]{} (c) ; (c) edge node [$\tilde{m}$]{} (end);
Now focus on the behaviour of the attacker, called Eve in the following. Eve applies a function $f \in {\mathcal{F}}$ to the codeword $c$, but she does not read $c$. In the real world, this can be seen as injecting faults on a device that you cannot read (e.g. a smart-card) using, for instance, a laser. In this experiment, Eve can however read the resulting decoded message $\tilde{m}$ and try to learn as much as possible about $m$ from $\tilde{m}$. Let us also specify that $f$ is a deterministic function and, furthermore, that Eve knows which function she has chosen in ${\mathcal{F}}$.
Defining Non-Malleability
-------------------------
Let us now give the formal definition of non-malleability. Let ${\mathcal{F}}$ be a family of tampering functions. For each $f \in {\mathcal{F}}$, we define a random variable ${\mathrm{Tamper}^f_s}$ corresponding to the tampering experiment described in the previous section:
$${\mathrm{Tamper}^f_s}= \left\{ \begin{array}{c} c \leftarrow_R {\mathrm{Enc}}(s), \tilde{c} = f(c) , \tilde{s} = {\mathrm{Dec}}(\tilde{c}) \\
\mathrm{Output}: \tilde{s}
\end{array} \right\}$$
The randomness is induced by the encoding function ${\mathrm{Enc}}$.
The *Non-Malleability* property is defined as follows:
Let $({\mathrm{Enc}},{\mathrm{Dec}})$ be a coding scheme, where ${\mathrm{Enc}}: {\{0,1\}}^k \mapsto {\{0,1\}}^n$ is random and ${\mathrm{Dec}}: {\{0,1\}}^n \mapsto {\{0,1\}}^k \cup \{\bot\}$ deterministic. Let ${\mathcal{F}}\subset {{\mathbb{F}}_2^n}^{{\mathbb{F}}_2^n}$ be a family of tampering functions.
We say that the coding scheme $({\mathrm{Enc}},{\mathrm{Dec}})$ is *non-malleable w.r.t. ${\mathcal{F}}$* if for each $f \in {\mathcal{F}}$, there exists a distribution ${\mathcal{D}}_f$ over ${\{0,1\}}^k \cup \{\bot,{\mathbf{same}}\}$ such that, $\forall s \in {\{0,1\}}^k$, we have:
$$\label{eq:tamper}
{\mathrm{Tamper}^f_s}\approx \left\{ \begin{array}{c} \tilde{s} \leftarrow {\mathcal{D}}_f \\
\mathrm{Output}
\left\{ \begin{array}{c} s \ \mathrm{if} \ \tilde{s} = {\mathbf{same}}\\ \tilde{s} \ \mathrm{otherwise}\end{array}\right.
\end{array} \right\}$$
where $\approx$ denotes computational or statistical indistinguishability.
Explaining the Definition
-------------------------
First, notice that the definition is relative to a family $\mathcal{F}$ of tampering functions, but the property of indistinguishability concerns each function $f$ separately. Non-malleability w.r.t. a family is in fact non-malleability w.r.t. each function in this family.
Now let us recall what we expect from a NMC. We want that, after the tampering experiment, either the codeword $\tilde{c}$ is well-decoded to the original message $s$ despite the tampering or the decoding procedure results in a value $\tilde{s}$ that is unrelated to the original message. That is the idea behind the distribution ${\mathcal{D}}_f$: either it returns the symbol ${\mathbf{same}}$, meaning that the decoding furnishes the original value or it returns a value $\tilde{s} \in {\{0,1\}}^k \cup \{\bot\}$. As ${\mathcal{D}}_f$ depends only on $f$ and not on the message $s$, in the latter case, the value returned in the second part of Equation (\[eq:tamper\]) is unrelated to $s$.
Basic Examples
--------------
We summarize here two examples developed in [@DPW10] that correspond to usual families of codes encompassed by the definition of NMC.
### Error Correction {#error-correction .unnumbered}
Let us assume that ${\mathcal{F}}$ is a family of tampering functions and $C$ an error-correcting code such that errors introduced by the application of a function $f \in {\mathcal{F}}$ on any codeword of $C$ can be corrected. Then $C$ is non-malleable w.r.t. ${\mathcal{F}}$. The distribution associated to every function $f \in {\mathcal{F}}$ is the constant distribution ${\mathcal{D}}_f = {\mathbf{same}}$, since erroneous codewords are always well-decoded.
### Error Detection {#error-detection .unnumbered}
The same idea can be applied to error-detecting codes. If there is a family ${\mathcal{F}}$ of tampering functions such that each $f \in {\mathcal{F}}$ introduces errors in every codeword that are detected by a code $C$, then $C$ is non-malleable w.r.t. ${\mathcal{F}}$. The distribution associated to every function $f \in {\mathcal{F}}$ is the constant distribution ${\mathcal{D}}_f = \bot$.
General (Im)Possibility Results
-------------------------------
### Impossibility {#impossibility .unnumbered}
As proven in [@DPW10], no code is non-malleable w.r.t. the set of all possible tampering functions ([*i.e.* ]{}${\mathcal{F}}= {{\mathbb{F}}_2^n}^{{\mathbb{F}}_2^n}$). Indeed there is, for instance, in ${\mathcal{F}}$ a function that decodes the codeword, “increments” the message ([*i.e.* ]{}adds 1 to its representation in ${\mathbb{F}}_2^k$) and re-encodes it. The result of the decoding of such a tampered codeword would always be $s+1$ and thus would be neither the original message $s$ nor an unrelated value.
### Possibility {#possibility .unnumbered}
In [@DPW10], the authors prove that for any bounded-sized family of tampering functions, there exists a NMC. Their result is summed up in the following theorem:
Let ${\mathcal{F}}\subset {{\mathbb{F}}_2^n}^{{\mathbb{F}}_2^n}$ be a family of tampering functions such that $n > \log(\log(|{\mathcal{F}}|))$. Then there exists a non-malleable code w.r.t. ${\mathcal{F}}$.
Bit-wise Independent Tampering
------------------------------
Bit-wise independent tampering is a special case of tampering where each bit of the codeword is tampered with independently. Formally a function $f: {\{0,1\}}^n \mapsto {\{0,1\}}^n$ is bit-wise independent if we can find $n$ independent functions $f_1,\ldots,f_n : {\{0,1\}}\mapsto {\{0,1\}}$ such that $\forall x \in {\{0,1\}}^n, f(x)=(f_1(x),\ldots,f_n(x))$. There are four possibilities for each $f_i$ which we denote by **keep**, **flip**, **0** and **1** (**keep** and **flip** are explicit, **0** (resp. **1**) is the function that sets a bit to 0 (resp. 1) regardless of what it was before).
In [@DPW10], a construction for a NMC w.r.t. the family of all bit-wise independent functions is introduced. It uses Linear Error-Correcting Secret-Sharing (LECSS) schemes [@CCGHV07] and Algebraic Manipulation Detection (AMD) codes [@CDFPW08]. Both are quite new tools and even the authors of [@DPW10] leave the explicit construction of LECSS codes as an “interesting open problem”. Furthermore, their solution is quite close to error detecting codes as it decodes to $\bot$ after a tampering in most cases[^5].
In Section \[sec:scheme\], we propose a new way to build NMC w.r.t. bit-wise independent functions. Our solution covers less tampering functions but uses more standard and efficient tools. Moreover, our scheme is neither error-correcting nor error-detecting (it never returns $\bot$) and so, to our opinion, is closer to the original definition of non-malleability, which is more generic than error detection or correction.
The Wire-Tap Channel {#sec:wire}
====================
In the following, a *$[n,k,d]$ linear code* denotes a subspace of dimension $k$ of ${\mathbb{F}}_2^n$ with minimal Hamming distance $d$.
Linear Coset Coding
-------------------
Coset coding is a random encoding used for both models of WT Channel. This type of encoding uses a $[n,k,d]$ linear code $C$ with a parity-check matrix $H$. Let $r = n-k$. To encode a message $m \in \mathbb{F}_2^r$, one chooses randomly an element among all $x \in \mathbb{F}_2^n$ such that $m =H ^t \! x$. To decode a codeword $x$, one just applies the parity-check matrix $H$ and obtains the syndrome of $x$ for the code $C$, which is the message $m$. This procedure is summed up in Figure \[fig:coset\].
The Wire-Tap Channel I
----------------------
The Wire-Tap Channel was introduced by Wyner [@Wyn75]. In this model, a sender Alice sends messages over a potentially noisy channel to a receiver Bob. An adversary Eve listens to an auxiliary channel, the WT channel, which is a noisier version of the main channel. It was shown that, with an appropriate coding scheme, the secret message can be conveyed in such a way that Bob has complete knowledge of the secret and Eve does not learn anything. In the special case where the main channel is noiseless, the secrecy capacity can be achieved through a linear coset coding scheme. We summarize the WT Chanel I in Figure \[fig:wire1\].
(0,0) node \[\] (a) [Alice]{} node \[right of=a, carre, node distance=1.8cm\] (b) [Enc]{} node \[right of=b, coordinate, node distance=.7cm\] (begintap) node \[right of=b, node distance=3cm,carre, draw\] (c) [small (or no) noise]{} node \[below of=c, node distance=2cm,carre\] (endtap) [big noise]{} node \[right of=c,node distance=3cm\] (d) [Bob]{} node \[right of=endtap,node distance=3cm\](e)[Eve]{};
\(a) edge node [$m$]{} (b); (b) edge node [$c$]{} (c); (c) edge node [$c'$]{} (d); (begintap) edge node (endtap); (endtap)edge node [$c''$]{}(e);
The Wire-Tap Channel II
-----------------------
Ten years later, Ozarow and Wyner introduced a second version of the WT Channel [@OW84]. In this model, both main and WT channels are noiseless. This time, the disadvantage for Eve is that she can only see messages with erasures: she has only access to a limited number of bits per codeword. She is however allowed to choose which bits she can learn. We summarize the Wire-Tap Chanel II in Figure \[fig:wire2\].
(0,0) node \[\] (a) [Alice]{} node \[right of=a, carre, node distance=1.8cm\] (b) [Enc]{} node \[right of=b, coordinate, node distance=.7cm\] (begintap) node\[right of = b, coordinate, node distance=2 cm\](c) node \[below of=c, node distance=2cm, carre\] (endtap) [erasures]{} node \[right of=b,node distance=6cm\] (d) [Bob]{} node \[below of=d,node distance=2cm\](e)[Eve]{};
\(a) edge node [$m$]{} (b); (b) edge node [$c$]{} (d); (begintap) edge node (endtap); (endtap)edge node [chosen bits of $c$]{}(e);
The encoding used in this model is again a coset coding based on a linear code $C$, as in the Wire Tap Channel I with a noiseless main channel. Let $d^{\bot}$ denote the minimal distance of the dual $C^{\bot}$ of $C$. One can prove (see [@Wei91] for instance) that, if Eve can access less than $d^{\bot}$ bits of a codeword, then she gains no information at all on the associated message.
Linear coset-coding for the WT channel can be efficiently implemented using LDPC codes [@TDCMM07; @STBM10].
From the Wire-Tap Channel to Non-Malleable Codes {#sec:scheme}
================================================
For our construction, we only deal with tampering functions that are bit-wise independent.
Motivations for Using Wire-Tap
------------------------------
Roughly speaking, in both models, codewords are modified either with random faults (WT I), adversary-controlled erasures (WT II) or an adversary-controlled tampering function (NMC). From these modified codewords or their decoding results, the adversary tries to learn information on the original messages.
The first WT is a little different from the other models because errors are random and so do not occur in the same number and bit positions every time. It could however be covered by the definition of NMC if every possible tampering caused by these random errors were included in the family of tampering functions taken into account by the code.
Let us now assume that we want to use a linear coset-coding scheme with a parity-check matrix $H$ as NMC. We cannot be protected against tampering functions that only add errors ([*i.e.* ]{}bit-wise independent functions where the only choices for each bit are **keep** or **flip**). To see why, let ${\mathcal{F}}$ be a family of such functions. Obviously, for each $f \in {\mathcal{F}}$, there is an error vector $e \in {\mathbb{F}}_2^n$ such that $\forall c \in {\mathbb{F}}_2^n, f(c)=c+e$. Let us follow the tampering experiment. Let $m \in {\mathbb{F}}_2^r$ be a source message and $c$ an encoding of $m$. Say $c$ is tampered to $\tilde{c}=c+e$. Decoding results in $\tilde{m}=H ^t \! c + H ^t \! e = m + H ^t \! e$. Thus, $\tilde{m}$ is always $m$ plus a constant offset. It is consequently related to $m$. Linear coset-coding cannot be non-malleable w.r.t. these “error-only” functions. There must me some **0** and **1** in the tampering.
This is why we consider WT II. Indeed, using **0** and **1** on some bits of the codewords is, in an information-theoretic sense, like having erasures at the corresponding locations, as we do not know what was originally there. As WT II guarantees that no information is leaked from erased codewords encoded using an appropriate coset-coding scheme, there will be no relation between the decoded tampered codeword and the original message. That is what motivates our proposal.
The Construction
----------------
As discussed before, we consider bit-wise independent functions where the sub-functions are not only **keep** or **flip**. Nevertheless, we authorize bit-flips because if the result of the tampering experiment is unrelated to the original message, then the result added to a constant offset will also be unrelated to this message.
We state the following theorem:
\[theo\] Let ${\mathcal{F}}\subset {{\mathbb{F}}_2^n}^{{\mathbb{F}}_2^n}$ be a family of bit-wise independent tampering functions such that:
$\forall f=(f_1,\ldots,f_n) \in {\mathcal{F}}, |\{i | f_i = \mathbf{0} \ \mathrm{or} \ f_i = \mathbf{1} \} | \geq D$.
Let $C$ be a $[n,k,d]$-linear code such that $D > n-d^{\bot}$, where $d^{\bot}$ is the minimal distance of its dual code $C^{\bot}$.
Then a linear coset-coding using $C$ is non-malleable w.r.t. ${\mathcal{F}}$.
Proof of Non-Malleability
-------------------------
Our proof of non-malleability is inspired from the proof of security of the WT II in [@Zem00].
Let us consider we are in the situation of Theorem \[theo\]. Let $f =(f_1,\ldots,f_n) \in {\mathcal{F}}$ be a tampering function. Let $S_{\textbf{01}}$ be the set of all positions $i$ such that $f_i = \mathbf{0}$ or $f_i=\mathbf{1}$. Let $S_{\mathbf{keep}}$ and $S_{\mathbf{flip}}$ be the equivalent sets for **keep** and **flip**. Let $e \in {\mathbb{F}}_2^n$ be such that $\forall i=1,\ldots,n, e_i=\chi_{S_{\mathbf{flip}}} (i)$ (where $\chi_A$ denotes the indicator function of a set $A$) and $\epsilon \in {\mathbb{F}}_2^n$ be such that $\epsilon_i=1$ if $f_i=\mathbf{1}$ and $\epsilon_i = 0$ otherwise. Let $h_1,...,h_n$ denote the columns of the parity-check matrix $H$.
Let $m \in {\mathbb{F}}_2^r$ be a message encoded to $c \in {\mathbb{F}}_2^n$. Let $\tilde{c}=f(c)$ and $\tilde{m}=H \tilde{c}$. We have
$$\begin{aligned}
\tilde{m} &=& \sum\limits_{i \in S_{\textbf{01}}} h_i \tilde{c}_i +
\sum\limits_{i \in S_{\textbf{keep}}} h_i \tilde{c}_i + \sum\limits_{i \in S_{\textbf{flip}}} h_i \tilde{c}_i \\
&=& \sum\limits_{i \in S_{\textbf{01}}} h_i \epsilon_i + \sum\limits_{i \in S_{\textbf{keep}}} h_i c_i + \sum\limits_{i \in S_{\textbf{flip}}} h_i (c_i+e_i) \\
&=& H ^t \! \epsilon + H ^t \! e + \sum\limits_{i \in S_{\textbf{keep}} \cup S_{\textbf{flip}}} h_i c_i\\
(\hspace{-0.35cm}&=& m + H ^t \! \epsilon + H ^t \! e - \sum\limits_{i \in S_{\textbf{01}}} h_i c_i )\end{aligned}$$
If we want $\tilde{m}$ to be unrelated to $m$, then we want $\sum\limits_{i \in S_{\textbf{keep}} \cup S_{\textbf{flip}}}h_i c_i$ to be unrelated to $m$. If the submatrix $H_{\mathbf{kf}}$ made of the columns $h_i$, $i \in S_{\textbf{keep}} \cup S_{\textbf{flip}}$ is of full rank $r=n-k$, then we gain no information on the corresponding bits of $m$, and all values are equiprobable. This is achieved in particular if $|S_{\textbf{keep}} \cup S_{\textbf{flip}}|<d^{\bot}$ (see chapter 9 of [@Zem00]).
If $D>n-d^{\bot}$, then $|S_{\mathbf{01}}| > n-d^{\bot}$, [*i.e.* ]{}$n -|S_{\textbf{keep}} \cup S_{\textbf{flip}}| > n-d^{\bot}$ or $|S_{\textbf{keep}} \cup S_{\textbf{flip}}|<d^{\bot}$. The condition of the previous paragraph is thus achieved if we use the parameters of Theorem \[theo\].
Let us define more formally the distribution $D_f$ associated to $f$. Let $K_i$, $i\in S_{\textbf{keep}} \cup S_{\textbf{flip}}$ be Bernoulli(1/2) distributions. Then $D_f = H ^t \! \epsilon + H ^t \! e + \sum\limits_{i \in S_{\textbf{keep}} \cup S_{\textbf{flip}}} h_i K_i$. This distribution and the result of the tampering experiment are identically distributed.
The coset-coding scheme used in Theorem \[theo\] is consequently non-malleable w.r.t. ${\mathcal{F}}$.
Going Further
-------------
### Towards a Larger Family of Tampering Functions {#towards-a-larger-family-of-tampering-functions .unnumbered}
When comparing our construction to the one of [@DPW10], one can relate the LECSS and our coset-coding scheme. The only requirement that is not fulfilled by linear coset-coding is a large distance. As the distance of linear coset-coding is 1, we cannot assume $d > n/4$ as they do. That is why we cannot directly modify this construction and replace LECSS with coset-coding in the description of the code and the proof of non-malleability.
Both LECSS and coset-coding ensure non-malleability when the number of $\mathbf{0}$ or $\mathbf{1}$ sub-functions of the tampering function is high enough. To deal with the case where the number of such functions is low, Dziembowski [*et al.* ]{}concatenated the LECSS with an AMD code. In such a case, the tampering function acts by adding an error following a fixed distribution ([*i.e.* ]{}independent of the codeword) and the decoding procedure results in $\bot$ with high probability because of the AMD code. Therefore, non-malleability is ensured. Following this idea, it might also be possible to encapsulate our coset-coding scheme within an error-detecting or an error-correcting code. Thus we would achieve non-malleability w.r.t. a larger family of functions. In particular, functions with a small number of $\mathbf{0}$ or $\mathbf{1}$ sub-functions which cannot be dealt with by coset-coding alone could be included. For the error-detecting case, using an AMD code as in [@DPW10] seems to be feasible. However, for the error-correcting case, it is not clear which kind of correction strategy to use to deal with the effects of the linear coset-coding scheme. Nevertheless, if such functions are the only ones of interest, one must be aware that an error correcting or an error detecting code is sufficient by itself.
### Relaxing the Notion of Non-Malleability {#relaxing-the-notion-of-non-malleability .unnumbered}
In the model for the WT II described in this paper, we require that Eve cannot obtain any bit of information on the messages sent over the channel. This strong security notion can be relaxed. Indeed, one could be satisfied even if Eve learned only a bounded amount of bits. This is possible if we consider generalized Hamming distances [@Wei91] instead of the dual distance $d^{\bot}$ of the code considered in the linear coset-coding scheme. For $i \in \mathbb{N}$, the generalized distance $d_i$ is such that if Eve cannot obtain more than $d_i$ bits per message, then she gains no more than $i-1$ bits of information per message. For instance, $d_1 = d^{\bot}$.
In the same spirit, one could relax the notion of non-malleability. After the tampering experiment, we could state that either the decoding procedure returns the original message or it enables to learn a bounded number of bits of information on this message. Using our construction, it is easy to build another scheme that would satisfy this requirement. One would only have to replace dual distances by generalized distances.
Conclusion {#sec:conclu}
==========
We established in this paper a parallel between Non-Malleable Codes and the Wire-Tap Channel. This relation enabled us to build an efficient non-malleable scheme, w.r.t. a family of bit-wise independent functions, that is neither error-correcting nor error-detecting.
Considering bit-wise independent tampering is a worthwhile first step for NMC. An interesting open problem would be now to build schemes that are non-malleable w.r.t. larger families of functions.
Acknowledgement {#acknowledgement .unnumbered}
===============
The authors would like to thank Julien Bringer for his helpful comments.
[^1]: IDentity & Security Alliance (The Morpho and Télécom ParisTech Research Center) Télécom ParisTech – 46, rue Barrault - 75013 Paris - France – Email: {chabanne, cohen, flori, patey}@telecom-paristech.fr
[^2]: Morpho – 11, boulevard Gallieni - 92130 Issy-Les-Moulineaux - France – Email: {herve.chabanne, alain.patey}@morpho.com
[^3]: CNRS-LTCI
[^4]: This work has been partially funded by the ANR SPACES project.
[^5]: In their proof of non-malleability, the authors of [@DPW10] distinguish different cases depending on the considered tampering function (more precisely its number $q$ of $\mathbf{0}$ and $\mathbf{1}$ sub-functions) and the *secrecy* $t$ of the LECSS scheme. When $t < q < n-t$, the tampering experiment always returns $\bot$ and when $q \le t$, the scheme is likely to often return $\bot$.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'For a given positive integer $t$ we consider graphs having maximal independent sets of precisely $t$ distinct cardinalities and restrict our attention to those that have no vertices of degree one. In the situation when $t$ is four or larger and the length of the shortest cycle is at least $6t-6$, we completely characterize such graphs.'
author:
- |
$^1$Bert L. Hartnell and $^2$Douglas F. Rall[^1] [^2]\
\
$^1$Department of Mathematics\
& Computing Science\
Saint Mary’s University\
Halifax, Nova Scotia, Canada\
\
$^2$Department of Mathematics\
Furman University\
Greenville, SC 29613 USA\
title: |
On graphs having maximal independent sets\
of exactly $t$ distinct cardinalities
---
[ maximal independent set, girth, cycle]{}\
Introduction
============
A well-covered graph (Plummer [@mdp1970]) is one in which every maximal independent set of vertices is of one cardinality and is hence a maximum independent set. Finbow, Hartnell and Whitehead [@fhw1994] defined the class ${{\cal M}}_t$ to consist of those graphs which have exactly $t$ different sizes of maximal independent sets. Finbow, Hartnell and Nowakowski [@fhn1993] proved that the well-covered graphs (the ${{\cal M}}_1$ collection) of girth (the length of a shortest cycle) 6 or more, with the exceptions of $K_1$ and $C_7$, have the property that every vertex has degree one or has exactly one vertex of degree one in its neighborhood. Thus, $C_7$ is the unique graph in ${{\cal M}}_1$ with girth at least 6 that has minimum degree at least two. The graphs in ${{\cal M}}_2$ of girth 8 or more have also been characterized ([@fhw1994]). There are precisely five graphs in ${{\cal M}}_2$ of girth at least 8 and minimum degree 2 or more, namely the cycles $C_8, C_9, C_{10}, C_{11}$ and $C_{13}$. This implies there are no ${{\cal M}}_1$ graphs of girth at least 8 with minimum degree 2 or more and no ${{\cal M}}_2$ graphs of girth 14 or more and having minimum degree at least 2. For related work on the class ${{\cal M}}_t$ see [@bh1998] and [@bh2009].
In this paper we investigate the graphs in ${{\cal M}}_t$ that have minimum degree at least 2 and higher girth and establish that the characterization of these in ${{\cal M}}_1$ and ${{\cal M}}_2$ is part of a general pattern. In particular, for $t \ge 3$ we show that among graphs with minimum degree at least 2, ${{\cal M}}_t$ does not contain a graph of girth at least $6t+2$ and that $C_{6t-4}, C_{6t-3}, C_{6t-2},
C_{6t-1}$ and $C_{6t+1}$ are the only exceptions for girth at least $6t-4$. Furthermore, if $t \ge 4$, then these cycles along with $C_{6t-6}$ are the only graphs in ${{\cal M}}_t$ that have minimum degree at least 2 and girth at least $6t-6$.
Let $G$ be a finite simple graph. A vertex of degree 1 is called a [*leaf*]{} and any vertex that is adjacent to a leaf is called a [*support vertex*]{}. If $C$ is a cycle in a graph $G$ and $u$ and $v$ belong to $C$, we let $uCv$ denote the shorter of the two $u,v$-paths that are part of $C$. For $A \subseteq V(G)$ and $u$ a vertex in $G$, $d(u,A)$ will denote the length of a shortest path in $G$ from $u$ to a vertex of $A$. We will use ${{\cal M}}(G)$ to denote the collection of all maximal independent sets of $G$ and we define the [*independence spectrum*]{} ([*spectrum*]{} for short) of $G$ to be the set ${{\cal S}}(G)=\{\,|I|\,:\,I\in {{\cal M}}(G)\}$. The class ${{\cal M}}_t$ consists of those graphs $G$ for which $|{{\cal S}}(G)|=t$. The spectrum is not necessarily a set of consecutive positive integers (e.g., ${{\cal S}}(K_{2,4,5})=\{2,4,5\}$), but for paths and cycles it is. We denote the set of positive integers between $p$ and $q$ inclusive by $[p,q]$. The following proposition is easy to establish.
\[pathcycle\] For each positive integer $n$ at least 3, $${{\cal S}}(C_n)=[\lceil{n/3}\rceil,\lfloor{n/2}\rfloor]\quad \mbox{and} \quad {{\cal S}}(P_n)
=[\lceil{n/3}\rceil,\lceil{n/2}\rceil]\,.$$ Hence, $C_n \in {{\cal M}}_t$ and $P_n \in {{\cal M}}_s$ where $t=\lfloor{n/2}\rfloor - \lceil{n/3}\rceil +1$ and $s= \lceil{n/2}\rceil -
\lceil{n/3}\rceil +1$.
The following lemma from [@fhw1994] will be used throughout—often without mention.
\[leftover\] [[@fhw1994]]{} If the graph $G$ belongs to ${{\cal M}}_t$ and $I$ is an independent set of $G$, then for every component $C$ of $G-N[I]$ there exists $k\le t$ such that $C \in {{\cal M}}_k$. In addition, $G-N[I] \in {{\cal M}}_r$ for some $r \le t$.
Lemma \[leftover\] will most often be used in the following way. We will find an independent set $I$ in a graph $G$ and demonstrate that $G-N[I]$ has a component that is in the class ${{\cal M}}_s$ for some $s>t$ and conclude that $G \not\in{{\cal M}}_t$. The following lemma will be used in that context with Lemma \[leftover\].
\[cycleplus1\] If a cycle $C$ is in ${{\cal M}}_t$ and a new vertex is added as a leaf adjacent to a single vertex of $C$, then the resulting graph belongs to ${{\cal M}}_{t+1}$.
[[**Proof. **]{}]{}Assume ${{\cal S}}(C)=[k,k+t-1]$. Let $H$ be the graph formed by adding a leaf $x$ adjacent to $y$. Let $u$ and $v$ be the neighbors of $y$ on $C$. Note that $\{ I \in {{\cal M}}(H)\,:\, y \in I\}=\{ J \in {{\cal M}}(C)\,:\, y \in J\}$, and because of the symmetry of the cycle, ${{\cal S}}(C)=\{|J|\,:\, J \in {{\cal M}}(C), y \in J\}$. Also, $\{ I \in {{\cal M}}(H)\,:\, u \in I\}=\{ J \cup\{x\}\,:\, J \in {{\cal M}}(C), u \in J\}$. This shows that $[k,k+t] \subseteq {{\cal S}}(H)$. If $H$ has a maximal independent set $A$ of size less than $k$, then $x\in A$ and neither $u$ nor $v$ is in $A$, for otherwise $A \cap C$ is a maximal independent set in $C$ of cardinality less than $k$. But now $A'=(A-\{x\})\cup \{y\} \in {{\cal M}}(C)$ and $|A'|<k$, a contradiction. Therefore, ${{\cal S}}(H)=[k,k+t]$. We conclude that $H \in {{\cal M}}_{t+1}$.
$\Box$
In the class of graphs with leaves there is no connection between girth and the size of the spectrum. This can be seen by the following general construction. Let $t\ge 2$ and $g\ge 3$ be integers. Let $H$ be the graph formed by adding a single leaf adjacent to each vertex of a cycle of order $g$. For a single vertex $x$ on the cycle attach a path $v_1, v_2, \ldots, v_{2t-3}$ to $H$ by making $x$ and $v_1$ adjacent. Then add two leaves adjacent to $v_i$ if $i$ is odd, and add one leaf adjacent to $v_j$ if $j$ is even. The resulting graph of order $2g+5t-7$ has girth $g$ and belongs to the class ${{\cal M}}_t$. (The spectrum of this graph is $[g+2t-3, g+3t-4]$.) For this reason we will henceforth consider only graphs having minimum degree at least 2. For ease of reference we denote the class of graphs that are in ${{\cal M}}_t$ and have no leaves (i.e., minimum degree at least 2) by ${{{{\cal M}}}^2}_t$. Note that ${{{{\cal M}}}^2}_t \subseteq {{\cal M}}_t$. In the course of several of our proofs we will show that some given graph is not in ${{{{\cal M}}}^2}_t$ by demonstrating it does not belong to ${{\cal M}}_t$.
The remainder of this paper is devoted to verifying the entries in the following table.
---------- ------------ ------------- ------------ ------------ ------------ ------------ ------------- ------------ -------------
$6t-6$ $6t-5$ $6t-4$ $6t-3$ $6t-2$ $6t-1$ $6t$ $6t+1$ $\ge 6t+2$
$t=1$ $\Delta$ $\Delta$ $\Delta$ $\emptyset$ $C_7$ $\emptyset$
$t=2$ $\Delta$ $\Delta$ $C_8$ $C_9$ $C_{10}$ $C_{11}$ $\emptyset$ $C_{13}$ $\emptyset$
$t=3$ $C_{12}$ $\Delta$ $C_{14}$ $C_{15}$ $C_{16}$ $C_{17}$ $\emptyset$ $C_{19}$ $\emptyset$
$t=4$ $C_{18}$ $\emptyset$ $C_{20}$ $C_{21}$ $C_{22}$ $C_{23}$ $\emptyset$ $C_{25}$ $\emptyset$
$t\ge 5$ $C_{6t-6}$ $\emptyset$ $C_{6t-4}$ $C_{6t-3}$ $C_{6t-2}$ $C_{6t-1}$ $\emptyset$ $C_{6t+1}$ $\emptyset$
---------- ------------ ------------- ------------ ------------ ------------ ------------ ------------- ------------ -------------
: Graphs of given girth in ${{{{\cal M}}}^2}_t$[]{data-label="summarytable"}
The entry for a given girth (written as a function of $t$) and a given value of $t$ should be interpreted as follows. If a specific graph is given, then this is the unique graph of that girth that belongs to ${{{{\cal M}}}^2}_t$. For example, $C_{15}$ is the only graph of girth 15 in ${{{{\cal M}}}^2}_3$. If $\emptyset$ appears, then there are no graphs of that girth in ${{{{\cal M}}}^2}_t$. When the entry is $\Delta$, then it is known that ${{{{\cal M}}}^2}_t$ contains at least one graph of that girth (and it is not just a cycle). Some of these type of entries have been verified in previous papers. For example, see [@fhn1993] and [@fhw1994] for ${{{{\cal M}}}^2}_1$ and ${{{{\cal M}}}^2}_2$, respectively.
Establishing Table Entries
==========================
We begin by showing that for a given positive integer $t$ the only graphs in ${{\cal M}}_t$ with large enough girth must have leaves. The next result was proved for well-covered graphs ($t=1$) in [@fh1983]. Proposition \[pathcycle\] shows it is sharp in terms of girth.
\[leaves\] Let $t$ be a positive integer. If $g(G)\ge 6t+2$ and $\delta(G) \ge 2$, then $G \in {{\cal M}}_r(G)$ for some $r>t$.
[[**Proof. **]{}]{}Assume $t\ge 2$. Let $G$ have girth at least $6t+2$ and minimum degree at least two. We will show that $G$ has maximal independent sets of at least $t+1$ different sizes. Choose a cycle $C=v_1,v_2,\ldots,v_s$ of minimum length in $G$.
Assume first that $s \ge 6t+4$ and let $P$ denote the path $v_3,v_4,\ldots,
v_{6t+1}$. Since $\delta(G) \ge 2$ and $g(G)=s$, each vertex $u\not\in C$ that is adjacent to a vertex of $P$ has another neighbor $u'$ that does not belong to $P$ and is not adjacent to any vertex of $P$. Choose one such neighbor $u'$ for each $u$ and let $J$ denote the set of these neighbors. By the girth restriction it follows that the set $I=J\cup\{v_1,v_{6t+3}\}$ is independent. (If $s=6t+2$, then proceed as above except let $I=J\cup \{v_1\}$.) However, $P$ is a component of $G-N[I]$ and by Proposition \[pathcycle\], $P \in {{\cal M}}_{t+1}$. Similar to the proof of Lemma \[leftover\] this implies that $G$ has maximal independent sets of at least $t+1$ different sizes.
If $s=6t+3$, let $P$ be the path $v_3,v_4,\ldots, v_{6t+2}$. The set $J$ is chosen as before, and now $G-N[J \cup \{v_1\}]$ has the path $P$ of order $6t$ as a component. By Proposition \[pathcycle\] it once again follows that $G$ has at least $t+1$ distinct sizes of maximal independent sets.
$\Box$
For any positive integer $t$ it follows from Proposition \[pathcycle\] that $C_{6t+1} \in {{\cal M}}_t$. In [@fhn1993] it was shown that $C_7$ is the only well-covered graph of girth 7 and minimum degree 2 or more. The following theorem shows the similar result is true for larger values of $t$.
\[girth6tand6t+1\] Let $t\ge 2$ be an integer. The cycle $C_{6t+1}$ is the only graph of girth $6t+1$ in ${{{{\cal M}}}^2}_t$, and ${{{{\cal M}}}^2}_t$ contains no graphs of girth $6t$.
[[**Proof. **]{}]{}By Proposition \[pathcycle\] the cycle of order $6t+1$ belongs to ${{{{\cal M}}}^2}_t$. Suppose $G$ is a graph not isomorphic to $C_{6t+1}$ such that $g(G)=6t+1$ and $\delta(G) \ge 2$. Then $G$ has an induced cycle $C$ of order $6t+1$, and $C$ has a vertex $w$ of degree at least 3. Since $g(G)=6t+1$ and $\delta(G)\ge 2$ we can find an induced path $w,a,b,c$, such that none of $a, b$ or $c$ belongs to $C$. Let $X =\{ u \in V(G)\, :\, d(u,C)=2 \}-N(a)$ and let $Y=\{ u \in V(G)\,:\, d(u,a)=2, d(u,w)=3\}$. For any two vertices on $C$ there is a path using part of $C$ of length at most $3t$ joining them. Since $g(G) \ge 13$ it follows that $Y$ is independent. Suppose two vertices $x_1,x_2 \in X$ are adjacent. Let $x_1,v_1,w_1$ and $x_2,v_2,w_2$ be paths in $G$ with $w_1$ and $w_2$ on the cycle $C$. Then the cycle $x_1,v_1,w_1Cw_2,v_2,x_2,x_1$ has length at most $3t+5$. But then $3t+5\ge 6t+1$, which implies that $t=1$, a contradiction. Finally, if a vertex in $X$ is adjacent to a vertex in $Y$, then a similar argument shows that $G$ has a cycle of length at most $3t+6$ which also leads to a contradiction.
Therefore, $X \cup Y$ is an independent set. One of the components of the graph $G-N[X \cup Y]$ is the cycle $C$ with a single leaf $a$ attached at the support vertex $w$. By Lemma \[cycleplus1\] this component is in ${{\cal M}}_{t+1}$. An application of Lemma \[leftover\] then shows that $G \not\in{{{{\cal M}}}^2}_t$.
Now let $G$ be a graph of girth $6t$, and as above find an induced cycle $C$ of length $6t$. This time let $X =\{ u \in V(G)\, :\, d(u,C)=2 \}$. This set is independent unless there is a cycle of the form $x_1,v_1,w_1Cw_2,v_2,x_2,x_1$ that has length at most $3t+5$. But this means $3t+5 \ge 6t$ contradicting our assumption that $t\ge 2$. Hence $X$ is independent. The cycle $C$ is one of the components of $G-N[X]$. Since $C_{6t} \in {{\cal M}}_{t+1}$, Lemma \[leftover\] implies that $G \not\in {{{{\cal M}}}^2}_t$.
$\Box$
By following a line of reasoning similar to the first part of the proof of Theorem \[girth6tand6t+1\] one can prove the following result. The proof is omitted. As noted earlier, Theorem \[fourgirths\] also holds for $t=2$. See [@fhw1994].
\[fourgirths\] Let $t \ge 3$ be a positive integer. For each integer $n$ such that $6t-4 \le n \le 6t-1$, the cycle $C_n$ is the unique graph of girth $n$ that belongs to ${{{{\cal M}}}^2}_t$.
We now establish the uniqueness (for $t \ge 3$) of the table entry corresponding to those graphs with no leaves whose shortest cycle has length $6t-6$ and which have maximal independent sets of exactly $t$ distinct cardinalities.
\[girth6t+1\] For each integer $t \ge 3$, the cycle $C_{6t-6}$ is the only graph of girth $6t-6$ that belongs to ${{{{\cal M}}}^2}_t$.
[[**Proof. **]{}]{}The cycle of order $6t-6$ is in ${{{{\cal M}}}^2}_t$ by Proposition \[pathcycle\]. Suppose that $G$ is a graph of girth $6t-6$ with no leaves. If $G$ is not $C_{6t-6}$, then we can find an induced cycle $C$ of length $6t-6$ in $G$ with $w,a,b,c$, $X$ and $Y$ defined as in the proof of Theorem \[girth6tand6t+1\]. The set $Y$ is independent because $g(G)\ge 12$, and $X$ is independent since $t \ge 3$. If some vertex of $X$ is adjacent to a vertex of $Y$, then $G$ contains a cycle of length at most $3t-3+6$. It follows that $3t+3 \ge g(G)= 6t-6$, or equivalently $t \le 3$.
If the set $X \cup Y$ is independent, then $G-N[X \cup Y]$ has a component isomorphic to a cycle of length $6t-6$ with a single leaf attached at $w$. By Lemma \[cycleplus1\] this component is in ${{\cal M}}_{t+1}$ and so it follows from Lemma \[leftover\] that $G \not\in{{\cal M}}_t$.
![Part of $G$[]{data-label="Fig1"}](spectrum.eps){width="10"}
Thus we may assume that $t=3$ and that $X \cup Y$ is not independent. Without loss of generality we may assume that $c$ from $Y$ is adjacent to $x_1$ such that $x_1\in X$ and $x_1,v_1,w_1$ is a path where $w_1$ is on the cycle $C$. See Figure \[Fig1\]. By using the fact that $C$ has length 12 and $g(G)=12$ we infer that the length of $wCw_1$ is 6. Let $X'=X-N(v_1)$ and let $Z=\{u\,:\,d(u,v_1)=2, d(u,w_1)=3, ux_1 \not\in E(G)\}$. It is clear that $Z$ is independent.
As above, if a vertex of $Z$ is adjacent to a vertex $h$ of $X'$, then if $d(h,w)>2$ a cycle of length at most 11 is present and if $d(h,w)=2$ then $G$ contains a cycle of length 10, contradicting $g(G)=12$. Suppose $z_1 \in Y\cap Z$, say $z_1=y$ as in Figure \[Fig1\]. Then $z_1 \not =c$, and $a,b,c,x_1,v_1,x_2,z_1,u,a$ is a cycle, contradicting the girth assumption. Similarly, since $G$ has no cycles of length 9, it follows that $Z \cup Y$ is independent.
The set $X' \cup Y \cup Z$ is independent, and one of the components of the graph $G-N[X' \cup Y \cup Z]$ is the cycle $C$ with a single leaf attached at vertices $w$ and $w_1$. But this component has spectrum $\{4,5,6,7,8\}$ from which it follows that $G\not\in{{\cal M}}_3$.
$\Box$
We now show that when $t \ge 4$ there is a “gap” at girth $6t-5$ among the leafless graphs. That is, if $G$ has minimum degree at least 2 and the shortest cycle of $G$ has order $6t-5$, then $G$ does not belong to ${{\cal M}}_t$.
\[lowergap\] For each integer $t$ at least 4, the class ${{{{\cal M}}}^2}_t$ contains no graphs of girth $6t-5$.
[[**Proof. **]{}]{}First observe that $C_{6t-5}\in {{\cal M}}_{t-1}$. Our approach will be similar as that pursued in earlier proofs, except that we will be attempting to isolate a cycle of length $6t-5$ with a path of order 5 attached as in Figure \[Fig2\]. It is easy to check, using either $\{a,c,e\}$ or $\{a,d\}$ together with all possible maximal independent sets of a path of order $6t-6$, that this component has spectrum $[2t,3t]$ and hence belongs to ${{\cal M}}_{t+1}$. This in turn implies via Lemma \[leftover\] that $G \not\in{{{{\cal M}}}^2}_t$.
![The cycle $C$ with attachments[]{data-label="Fig2"}](girth6t-5.eps){width="8"}
Suppose that $G$ has girth $6t-5$ and has minimum degree at least 2. Let $C$ be an induced cycle of length $6t-5$ in $G$. There must exist a vertex $w$ on $C$ having degree at least 3. For any two vertices on $C$ there is a path on $C$ joining them whose length is at most $3t-3$. Because of the girth and minimum degree assumptions on $G$ we can find a path $w,a,b,c,d,e$ as in Figure \[Fig2\]. Let $A=\{a,b,c,d,e\}$. Let $X=\{u\,:\,d(u,C)=2\}-N(a)$ and let $Y=\{u\,:\,u \not\in C, d(u,A)=2, d(u,w)\ge 2\}$.
As in previous proofs it is straightforward to show that $X$ is independent. Since $g(G)=6t-5\ge 19$ no pair of vertices in $Y$ can be adjacent. Suppose first that $X\cup Y$ is independent. The graph in Figure \[Fig2\] is a component of $G-N[X\cup Y]$. As remarked at the outset, this shows that $G \not\in{{{{\cal M}}}^2}_t$. We note that for $t\ge 5$, the girth restriction ensures that $X\cup Y$ is independent.
Now consider $t=4$. Thus $C$ is of length 19. Let $s_1$ and $s_2$ be the adjacent vertices on $C$ that are at distance 9 from $w$. If both $s_1$ and $s_2$ are of degree two, then $X \cup Y$ is independent or else a cycle of length 18 would exist in $G$. Assume then without loss of generality that $s_1$ has a neighbor $r$ that is not on $C$. Let $U=N(r)-\{s_1\}$. For each $u_i \in U$ choose a vertex $v_i \in N(u_i)-\{r\}$, and set $V=\{v_i\,:\, u_i\in U\}$. Similarly, let $B=N(a)-\{w\}$. For each $b_i \in B$ choose a vertex $c_i \in N(b_i)-\{a\}$, and set $D=\{c_i\,:\, b_i\in B\}$. Since $g(G)=19$ the set $V \cup D \cup (X-U)$ is independent, and one of the components of $G-N[V \cup D \cup (X-U)]$ is a cycle of order 19 with a single leaf $a$ adjacent to $w$ and a single leaf $r$ adjacent to $s_1$. This component belongs to ${{\cal M}}_5$ which proves that $G \not\in{{{{\cal M}}}^2}_4$ and establishes the theorem.
$\Box$
Concluding Remarks
==================
We have shown that for a positive integer $t\ge 4$ and for each possible value of girth at least $6t-6$, the class ${{{{\cal M}}}^2}_t$ either contains exactly one graph of that girth (the cycle) or contains no graphs of that girth. It is interesting to note that as $t$ grows there is an ever increasing gap—in terms of girth—between the unique graph of girth $6t-6$ in ${{{{\cal M}}}^2}_t$ and ones of smaller girth. For instance, we can show that ${{{{\cal M}}}^2}_{31}$ contains no graphs of girth $r$ for $131 \le r \le 179$. Hence the cycles $C_{180}, C_{182}, C_{183},
C_{184}, C_{185}$ and $C_{187}$ are the only leafless members of ${{\cal M}}_{31}$ that have girth at least 131. Thus the six cycles are quite special in ${{{{\cal M}}}^2}_t$.
[999999]{}
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R. Barbosa and B. L. Hartnell: The effect of vertex and edge deletion on the number of sizes of maximal independent sets. [*J. Combin. Math. Combin. Comput.*]{} [**70**]{}, 111–116 (2009)
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[^1]: Corresponding author: e-mail: doug.rall@furman.edu
[^2]: Research supported in part by the Wylie Enrichment Fund of Furman University.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The anisotropic 3D equation describing the pointlike particles diffusion in slender impermeable tubes of revolution with cross section smoothly depending on the longitudial coordinate is the object of our study. We use singular perturbations approach to find the rigorous asymptotic expression for the local particles concentration as an expansion in the ratio of the characteristic transversal and longitudial diffusion relaxation times. The corresponding leading-term approximation is a generalization of well-known Fick-Jacobs approximation. This result allowed us to delineate the conditions on temporal and spatial scales under which the Fick-Jacobs approximation is valid. A striking analogy between solution of our problem and the method of inner-outer expansions for low Knudsen numbers gas kinetic theory is established. With the aid of this analogy we clarify the physical and mathematical meaning of the obtained results.'
author:
- 'Sergey D. Traytak'
title: 'Asymptotic solution of the diffusion equation in slender impermeable tubes of revolution. I. The leading-term approximations'
---
[^1]
Introduction
============
The problem of approximate reduction of the time-dependent 3D equation describing the local concentration field $C\left( \mathbf{x},t\right) $ of pointlike particles diffusing in a tube of varying with the longitudinal coordinate $z$ cross section to an effective time-dependent 1D equation appeared to be fairly tricky. For the first time, following the main idea of Fick’s approach, such kind of 1D equation was derived in 1935 by Jacobs.[Jacobs:1967]{} Particularly for a channel with a shape of a surface of revolution the relevant 1D equation reads$$\frac{\partial c\left( z,t\right) }{\partial t}=\frac{\partial }{\partial z}D\left\{ A\left( z\right) \frac{\partial }{\partial z}\left[ \frac{c\left(
z,t\right) }{A\left( z\right) }\right] \right\} , \label{in1}$$where $D$ is the translational diffusion coefficient in space with no constraints, $A\left( z\right) =\pi \left[ r\left( z\right) \right] ^{2}$ is the area of the tube cross-section $\Sigma _{z}$ of radius $r\left( z\right)
$ at a given point $z$ of the symmetry axis. The corresponding reduced concentration $c\left( z,t\right) $ is calculated by the formula $$c\left( z,t\right) =\int\limits_{\Sigma _{z}}C\left( x,y,z,t\right) dxdy.
\label{in2}$$Nowadays the reduced diffusion equation (\[in1\]) is commonly referred to as the Fick-Jacobs equation (FJE). In addition we will call Eq. (\[in1\]) a classical form of the FJE. It is interesting that, if we do not take into account a few works on this subject, for decades the FJE remained almost unclaimed. Situation has been changed drastically after 1992 when well-known Zwanzig’s article renewed the problem and stimulated considerable interest to this topic.[@Zwanzig:1992] Soon it turned out that the problem on diffusion in a tube of varying cross section is of great importance for numerous applications dealing with artificial and natural transport processes and thence many researchers studied it within different facets of theory and applications.[Grathwohl:1998,Konkoli:2005,Berezhkovskii:2007,Dendrites:2008,Burada:2008,Ai:2008,Wang:2009,Mondal:2010,Dagdug:2010,Biess:2011,Dagdug:2011,Berezhkovskii:2012]{} Even now these kind of investigations are close to the top among hot research topics on the diffusion-influenced processes in confined regions.[@Barzykin:2013]
In his seminal paper Zwanzig drew attention to the fact that Jacobs derivation is rather heuristic and, besides, it is completely free of impermeable wall boundary condition that should be imposed on the solution of the original higher dimensional diffusion equation. Moreover, Jacobs did not present any reasons for choosing the center line of the tube.[Zwanzig:1992]{} Taking into account that the FJE has exactly the same mathematical structure as the Smoluchowski equation for diffusion in a 1D potential field, Zwanzig derived the FJE starting from the general diffusion equation with a potential.[@Zwanzig:1992] According to Zwanzig the FJE may be presented in the form$$\frac{\partial c\left( z,t\right) }{\partial t}=\frac{\partial }{\partial z}D\left\{ e^{-\frac{U\left( z\right) }{k_{B}T}}\frac{\partial }{\partial z}\left[ e^{\frac{U\left( z\right) }{k_{B}T}}c\left( z,t\right) \right]
\right\} . \label{in2a}$$In (\[in2a\]) $U\left( z\right) $ is so-called entropy potential defined as$$U\left( z\right) =-k_{B}T\ln A\left( z\right) ,$$where $k_{B}$ and $T$ are the Boltzmann constant and the absolute temperature. Assuming that the channel radius varies slowly with increasing of the longitudinal variable $z$ , i.e., $$\left\vert r^{^{\prime }}\left( z\right) \right\vert \ll 1, \label{in2b}$$hereafter in the paper $\psi ^{^{\prime }}\left( \varsigma \right) :=d\psi
\left( \varsigma \right) /d\varsigma $, he also proposed a generalized form of the FJE. For particular case of 3D tube the original diffusion constant $D $ in the FJE (\[in1\]) was replaced by a spatially dependent effective diffusion coefficient $$D_{Zw}\left( z\right) \approx \frac{D}{1+\frac{1}{2}\left[ r^{^{\prime
}}\left( z\right) \right] ^{2}}.$$Later, in 2001 Reguera and Rubi developed this idea using some nonequilibrium thermodynamics reasons and obtained the corrected FJE with $$D_{R-R}\left( z\right) \approx \frac{D}{\sqrt{1+\left[ r^{^{\prime }}\left(
z\right) \right] ^{2}}}$$in case of 3D symmetric tubes.[@Reguera:2001]
Further corrections to the FJE were obtained by Kalnay and Percus with the help of so-called mapping technique, which differs from Zwanzig’s entropy barrier theory.[@Kalnay:2005; @Kalnay2:2005; @Kalnay:2006] This mapping procedure has been performed for the anisotropic diffusion equation without a potential$$\frac{\partial C}{\partial t}=\nabla \cdot \left( \mathbf{D\cdot \nabla }C\right) , \label{in3}$$where $\mathbf{D}$ is the translational diffusion tensor matrix. As in Refs. 17 and 19 we suppose here that the diffusion matrix is diagonal with transverse isotropy, i.e., $\mathbf{D}=$diag$\left( D_{x},D_{y},D_{z}\right)
$, and that $D_{x}=D_{y}=D_{\perp }$ is the transverse and $D_{z}=D_{\parallel }$ is the longitudinal translational diffusion coefficient, respectively. In their study Kalnay and Percus assumed also that $$\varepsilon =D_{\parallel }/D_{\perp }\ll 1, \label{in4}$$which allowed them to suggest that the transverse concentration profile equilibrates quickly and so-called Zwanzig’s factorization[@Zwanzig:1992] holds true. Moreover, this is a quasi steady-state theory, i.e., field $C\left( \mathbf{x},t\right) $ is assumed to be an explicitly time-independent. Time dependence is presented in this function implicitly by functional of the reduced concentration $c\left( z,t\right) $ only. Under given assumptions in the limit $\varepsilon \rightarrow 0$ diffusion equation (\[in3\]) with reftecting wall condition is reduced to the corresponding FJE. In its turn for higher-order terms in $\varepsilon $ Kalnay and Percus derived a generalized 1D equation that contains all higher derivatives with respect to $z$ of the tube radius $r\left( z\right) $ and reduced concentration $c\left( z,t\right) $.[Kalnay:2005,Kalnay2:2005,Kalnay:2006]{} Kalnay and Percus drew attention to the fact that the problem “requires an analysis of the short-time behavior”, but they did not deal with this question.[@Kalnay:2005] The projection method has been used by Dagdug and Pineda to find more general effective diffusion coefficient for the FJE, describing the unbiased motion of pointlike particles in 2D slender tilted asymmetric channels of varying width formed by straight wall.[@Dagdug1:2012] In the subsequent paper of the same authors, to test the validity of obtained formulae, a comparison of these analytical results against Brownian dynamics simulation results were performed.[@Dagdug:2012]
The biased diffusion transport of pointlike particles under the influence of a constant and uniform force field in 2D and 3D narrow spatially periodic channels of varying cross section is also rather well investigated.[Rubi:2007,Martens:2011,Martens2:2011]{}
It is worth noting that on account of a mathematical difficulties in solving the original problem for arbitrary tube radius $r\left( z\right) $ an effective 1D description for the simplified case when the tubes composed of some number of contacting equal spheres[@Berezhkovskii:2008] or cylindrical sections of different diameters was investigated.[Barzykin:2009,Makhnovskii:2010]{} Further generalization of the previous research to the case of a periodically expanded conical tube was recently reported.[@Barzykin:2013] It is important that tubes of this shape may be utilized as a controlled drug release device.[@Barzykin:2013] The interested reader can find numerous references to the previous analytical and numerical studies in a recent paper by Kalnay.[@Kalnay:2013]
The analysis of the literature showed that, despite the great amount of publications devoted to the topic, rigorous mathematical study of the corresponding boundary value problem for all spatial and temporal scales is still missing. Thus, the purpose of this paper is twofold. Firstly using rigorous technique of the matched asymptotic expansions [Lagerstrom:1988,Ilin:1992]{} we consider the 3D anisotropic diffusion equation (\[in3\]) which describes the diffusion of pointlike particles into a tube with impermeable wall having the shape of a surface of revolution. Secondly accurate asymptotic solution of the original 3D problem for all spatial and temporal scales allows us to elucidate the role, physical and mathematical sense and lastly accuracy of the leading-term approximation and, particularly, the validity of the Fick-Jacobs approximation.
The paper is organized in the following way. Section II contains the full mathematical statement of the corresponding boundary value problem. In Sec. III by means of singular perturbations approach we give the detailed preliminary ideas for asymptotic solution of the posed problem. Section IV devotes to the asymptotic solution in the outer subdomain and, particularly, derivation of the Fick-Jacobs equation. In Sec. V and Sec. VI we study solution in spatial and temporal diffusive boundary layers, respectively, to derive, in particular, the appropriate boundary and initial conditions for solution of the Fick-Jacobs equation. Section VII presents determination of the corner asymptotic solutions. In Sec. VIII the main result of the paper the leading-term approximation is reported. This section also comprises criteria for validity of the Fick-Jacobs approximation and establishes a profound analogy of the problem at issue with the gas kinetic theory for low Knudsen numbers. Finally the main concluding remarks are made in Sec. IX. Some subsidiary classical mathematical facts are given in Appendix.
Statement of the problem
========================
Consider the pointlike particles diffusion in a 3D tube of length $L$, which wall is obtained by rotation of the line $$r\left( z\right) =r_{M}R\left( z/L\right) \label{sp1}$$around the $z$ axis (see Fig. \[fig:geom\]). We assume function $r\left( z\right) $ to be smooth enough and introduced the maximum value of this function $$r_{M}:=r\left( z_{M}\right) =\max_{z\in \left[ 0,L\right] }r\left( z\right)$$which fully characterizes the transverse size of the tube. It is evident that $$0<R\left( z/L\right) \leq 1=R\left( z_{M}/L\right)$$for all $z\in \left[ 0,L\right] $. By definition we shall call tube slender (narrow) when $$r_{M}\ll L.$$In the cylindrical coordinate system $\left( r,\phi ,z\right) $ connected with the $z$ axis the tube region is $$\Sigma :=\left\{ 0<r<r\left( z\right) \right\} \times \left( 0<z<L\right)
\times \left( 0<\phi <2\pi \right) .$$ A cross section of the tube at any fixed value $z$ is $\Sigma _{z}:=\left\{
0<r<r\left( z\right) \right\} \times \left( 0<\phi <2\pi \right) $ and $\partial \Sigma _{w}:=\left\{ r=r\left( z\right) ,\phi \in \left( 0,2\pi
\right) ,z\in \left[ 0,L\right] \right\} $ is the tube wall. Moreover we suppose that the local concentration of diffusing particles $C\left( \mathbf{x},t\right) $ possesses the axial symmetry and therefore actually we shall treat here the 2D time-dependent diffusion equation.
Thus the anisotropic diffusion equation (\[in3\]), for the chosen cylindrical coordinates reads $$\frac{\partial C}{\partial t}=D_{\perp }\frac{1}{r}\frac{\partial }{\partial
r}\left( r\frac{\partial C}{\partial r}\right) +D_{\parallel }\frac{\partial
^{2}C}{\partial z^{2}} \label{sp2}$$in the space-time domain $\Sigma _{t}:=\Sigma \times \left( t>0\right) $. It is clear that at the $z$ axis one should take into consideration conditions of regularity and axial symmetry of solution, respectively $$\lim_{r\rightarrow 0}C<\infty ,\qquad \lim_{r\rightarrow 0}\frac{\partial C}{\partial r}=0. \label{sp3}$$
On the wall of the tube $\partial \Sigma _{w}$ we impose the common reflecting boundary condition $$\left. \left( \mathbf{n\cdot j}\right) \right\vert _{\partial \Sigma _{w}}=0,
\label{sp4}$$where $\mathbf{n}$ being the outer-pointing unit normal with respect to $\partial \Sigma _{w}$ (see Fig. \[fig:geom\]) and $\mathbf{j=-D\cdot }\nabla C$ is the local diffusing flux of particles. One can see that the tube wall $\partial
\Sigma _{w}$ may be defined analytically as $$w\left( r,z\right) =r-r_{M}R\left( z/L\right) =0.$$It is well known that for all nonsingular points $\left( r,z\right) \in $ $\partial \Sigma _{w}$ ($\nabla w\left( r,z\right) \neq 0$) the unit normal may be calculated as $\mathbf{n=}\nabla w/\left\Vert \mathbf{\nabla }w\right\Vert $. Taking this into account we can rewrite condition (\[sp4\]) as the orthogonality condition in cylindrical coordinates $$\left. \left[ \frac{\partial C}{\partial r}-r_{M}\frac{D_{\parallel }}{D_{\perp }}\frac{\partial C}{\partial z}\frac{d}{dz}R\left( z/L\right) \right] \right\vert _{w=0}=0. \label{sp4a}$$
To complete the problem statement one has to impose the initial condition $$\left. C\right\vert _{t=0}=C_{0}\left( r,z\right) \qquad \text{ in }\Sigma
\label{sp5}$$and, for definiteness, Diriclet boundary conditions on the ends of the tube $$\left. C\right\vert _{z=0}=C_{1}\left( r,t\right) ,\quad \left. C\right\vert
_{z=L}=C_{2}\left( r,t\right) . \label{sp6}$$We assume that all given functions $C_{0}\left( r,z\right) $, $C_{1}\left(
r,t\right) $ and $C_{2}\left( r,t\right) $ are continuous in their domains of definition. Hence according to the maximum principle for the diffusion equation we see that $C\in \left[ C_{m},C_{M}\right] $, where $C_{m}:=\min_{\partial \Sigma _{t}}\left\{ C_{0}\left( r,z\right)
,C_{1}\left( r,t\right) ,C_{2}\left( r,t\right) \right\} $ and $C_{M}:=\max_{\partial \Sigma _{t}}\left\{ C_{0}\left( r,z\right)
,C_{1}\left( r,t\right) ,C_{2}\left( r,t\right) \right\} $.
One can see that, due to complex geometry of the tube wall, analytical solution of the posed boundary value problem (\[sp2\])-(\[sp6\]) is not feasible in general case. That is why the FJE corresponding to Eq. (\[sp2\])
$$\frac{\partial c\left( z,t\right) }{\partial t}=D_{\parallel }\frac{\partial
}{\partial z}\left\{ A\left( z\right) \frac{\partial }{\partial z}\left[
\frac{c\left( z,t\right) }{A\left( z\right) }\right] \right\} \label{sp7}$$
plays an important role in applications. Note that in a special case of round cylindrical channel the FJE (\[sp7\]) simplifies to the common 1D second Fick’s equation $$\frac{\partial c\left( z,t\right) }{\partial t}=D_{\parallel }\frac{\partial
^{2}c\left( z,t\right) }{\partial z^{2}}. \label{sp9}$$
Our main objective with this paper is to construct a rigorous iterative procedure for asymptotic solution of the problem (\[sp2\])-(\[sp6\]) in case of a slender tube. Particularly this solution entails straightforwardly the FJE (\[sp7\]) with appropriate initial and boundary conditions. Besides, we will find criteria for validity of the corresponding approximation $c\left( z,t\right) $.
Formulation as a singular perturbed problem
===========================================
Non-dimensionalization of the problem
-------------------------------------
In order to perform the asymptotic solution of the posed boundary value problem (\[sp2\])-(\[sp6\]), we need to nondimensionalize it. It is expedient to rewrite this problem for dimensionless variables using the following scales $$\rho =r/r_{M},\qquad \xi =z/L,\qquad \tau =t/t_{L},$$where $t_{L}=L^{2}/D_{\parallel }$ is the characteristic longitudinal time for the diffusion length $L$. Moreover it is also convenient to treat the normalized dimensionless local concentration $$u\left( \rho .\xi .\tau \right) =C\left( \rho .\xi .\tau \right) /C_{M}.$$Accordingly, Eq. (\[sp2\]) takes the form $$\left( \mathcal{L}_{\rho }+\epsilon \mathcal{L}_{F}\right) u=0,\quad \text{
in }\Sigma _{\tau }, \label{nd4}$$where the unperturbed operator $$\mathcal{L}_{\rho }:=-\frac{1}{\rho }\frac{\partial }{\partial \rho }\left(
\rho \frac{\partial }{\partial \rho }\right) \label{nd5}$$is the radial part of the 2D Laplacian in polar coordinates. For the notation convenience in Eq. (\[nd4\]) and hereafter we define the dimensionless 1D Fick operator $$\mathcal{L}_{F}:=\frac{\partial }{\partial \tau }-\frac{\partial ^{2}}{\partial \xi ^{2}}.$$It is evident that for this problem $\epsilon \mathcal{L}_{F}$ being the perturbation operator. We shall show below that as $\epsilon \rightarrow 0$ unperturbed operator $\mathcal{L}_{\rho }$ and perturbation $\epsilon
\mathcal{L}_{F}$ determine the fast and slow behavior of the desired solution $u\left( \rho .\xi .\tau \right) $, respectively.
Initial and boundary conditions now read $$\left. u\right\vert _{\tau =0}=g_{0}\left( \rho ,\xi \right) , \label{nd6}$$$$\left. u\right\vert _{\xi =0}=g_{1}\left( \rho ,\tau \right) ,\qquad \left.
u\right\vert _{\xi =1}=g_{2}\left( \rho ,\tau \right) , \label{nd7}$$$$\left. \left[ \frac{\partial u}{\partial \rho }-\epsilon \frac{\partial u}{\partial \xi }R^{^{\prime }}\left( \xi \right) \right] \right\vert _{\rho
=R\left( \xi \right) }=0, \label{nd8}$$$$\left. u\right\vert _{\rho =0}<\infty ,\qquad \left. \frac{\partial u}{\partial \rho }\right\vert _{\rho =0}=0. \label{nd9}$$Hereafter we denote $g_{\gamma }=C_{\gamma }/C_{M}$ ($\gamma =0,1,2$).
In Eq. (\[nd4\]) and boundary condition (\[nd8\]) we introduced a new dimensionless parameter $\epsilon $ describing well the system under study $$\epsilon =\epsilon _{s}^{2}\frac{D_{\parallel }}{D_{\perp }}, \label{nd10}$$where $$\epsilon _{s}=\frac{r_{M}}{L}\ll 1 \label{nd10a}$$is the slenderness ratio (the relative thickness of the tube) for a narrow tube.[@Wang:2009] Note that, to make the Zwanzig’s factorization more plausible, it is usually assumed that relationship (\[in4\]) holds true.[@Kalnay:2005; @Kalnay2:2005; @Kalnay:2006] However it often happens in applications that $D_{\parallel }/D_{\perp }>1$.[@Bihan:2003] In any case we consider here the slenderness ratio $\epsilon _{s}$ is small enough to make $\epsilon $ small. Furthermore there is another important physical meaning of the introduced parameter $\epsilon $. To clarify this we rewrite (\[nd10\]) as $$\epsilon =\frac{t_{tr}}{t_{L}}\ll 1, \label{nd11}$$where $t_{tr}=r_{M}^{2}/D_{\perp }$ is the characteristic transversal time for the diffusion length $r_{M}$. Physically inequality (\[nd11\]) means that the diffusive relaxation along the transversal direction occurs much faster than that along axis $z$. In this connection we can call $\epsilon $ a relaxation parameter.
Simple inspection shows that for $\epsilon \rightarrow 0$ the posed problem (\[nd4\])-(\[nd9\]) is a singularly perturbed one.[Lagerstrom:1988,Ilin:1992]{} Really, if we just set $\epsilon =0$ in ([nd4]{}) we obtain unperturbed equation with a general solution, which cannot satisfy nor initial nor boundary conditions (\[nd6\])-(\[nd8\]). To study this problem we shall apply method of matched asymptotic expansions, which proved to be a powerful tool for solution of many singularly perturbed problems concerning the diffusion-influenced processes.[Traytak:1990,TraytakCPL:1991,Traytak:2004,TraytakCM:2007]{}
Subdomains for the asymptotic solution. Diffusion boundary layers
-----------------------------------------------------------------
Taking into account singularity of the perturbed problem (\[nd4\])-([nd9]{}) one can see that the diffusion equation exhibits certain diffusion boundary layers, i.e., subdomains of rapid change in the solution and its derivatives. The location and thickness of the boundary layer depends on a small parameter inherent in the problem under consideration (in our case this is $\epsilon $). For instance, it follows immediately from the general form of perturbed operator and conditions (\[nd6\]) and (\[nd7\]) that our problem possesses one temporal and two spatial boundary layers.[Ilin:1992]{}
To facilitate an understanding of the further study we shall use another decomposition of the space-time domain $\Sigma _{\tau }=\left\{ 0<\rho
<R\left( \xi \right) \right\} \times \Omega $, where the semi-infinite strip $\Omega :=\left( 0<\xi <1\right) \times \left( \tau >0\right) $ is its 2D cross section. For the boundary of the domain $\Omega $ we have $\partial
\Omega =\partial \Omega _{\tau }\cup \partial \Omega _{\xi }$, where $$\partial \Omega _{\tau }:=\left\{ \left( \xi ,\tau \right) :\tau =0\right\}$$is the temporal boundary $$\partial \Omega _{\xi }=\partial \Omega _{0}\cup \partial \Omega _{1}$$is the spatial boundary comprising two connected components at the endpoints of the tube $$\partial \Omega _{0}:=\left\{ \left( \xi ,\tau \right) :\xi =0\right\}
,\quad \partial \Omega _{1}:=\left\{ \left( \xi ,\tau \right) :\xi
=1\right\} .$$
Consider the structure of the diffusion boundary layer (see Fig. \[fig:dom\]). It is clear that we can decompose the 2D domain of variables $\left( \xi ,\tau
\right) $ as follows: $$\Omega =\Omega ^{\left( 0\right) }\cup \Omega ^{\left( b\right) }.
\label{sd1}$$Here $\Omega ^{\left( 0\right) }$ is a subdomain, where one does not expect rapid change in the solution and its derivatives and so relevant solution depends on slow variables $\left( \xi ,\tau \right) $ only. On the other hand a subdomain $\Omega ^{\left( b\right) }$ is the diffusion boundary layer that abutted the boundary$\partial \Omega $. Usually in perturbations theory the boundary layer subdomain $\Omega ^{\left( b\right)
} $ and $\Omega ^{\left( 0\right) }$ are called inner and outer subdomains, respectively.[@Ilin:1992]
Structure of the perturbed equation (\[nd4\]) allows us to define entirely the boundary layer $$\Omega ^{\left( b\right) }=\Omega _{\xi }^{\left( b\right) }\cup \Omega
_{\tau }^{\left( b\right) }, \label{sd2}$$where $\Omega _{\tau }^{\left( b\right) }$ is the temporal boundary layer subdomain abutted the initial values part of the boundary $\partial \Omega
_{\tau }$. In decomposition (\[sd2\]) the spatial subdomain $\Omega _{\xi
}^{\left( b\right) }$ consists of two strips in the semi-vicinities of the endpoints $\xi =0$ and $\xi =1$, respectively $$\Omega _{\xi }^{\left( b\right) }=\Omega _{0}^{\left( b\right) }\cup \Omega
_{1}^{\left( b\right) }. \label{sd3}$$The subsequent partition may be obtained if we introduce the corner subdomains near vertices $\left( 0,0\right) $ and $\left( 1,0\right) $: $\Omega _{0}^{\left( 2\right) }=\Omega _{0}^{\left( b\right) }\cap \Omega
_{\tau }^{\left( b\right) }$ and $\Omega _{1}^{\left( 2\right) }=\Omega
_{1}^{\left( b\right) }\cap \Omega _{\tau }^{\left( b\right) }$, where we have intersection of spatial and temporal boundary layers. Whence we can represent a strip near the left endpoint $\xi =0$ as $\Omega _{0}^{\left(
b\right) }=\Omega _{0}^{\left( 1\right) }\cup \Omega _{0}^{\left( 2\right) }$ and similarly a strip near the right endpoint $\xi =1$ as $\Omega
_{1}^{\left( b\right) }=\Omega _{1}^{\left( 1\right) }\cup \Omega
_{1}^{\left( 2\right) }$. Finally for the problem under study we obtain the following five fold partition of the diffusion boundary layer (see Fig. \[fig:dom\]) $$\Omega ^{\left( b\right) }=\Omega _{0}^{\left( 1\right) }\cup \Omega
_{0}^{\left( 2\right) }\cup \Omega _{0}^{\left( 3\right) }\cup \Omega
_{1}^{\left( 1\right) }\cup \Omega _{1}^{\left( 2\right) }, \label{sd4}$$where $\Omega _{0}^{\left( 3\right) }:=\Omega _{\tau }^{\left( b\right)
}\backslash \left( \Omega _{0}^{\left( 2\right) }\cup \Omega _{1}^{\left(
2\right) }\right) $.
Analysis of the posed boundary value problem (\[nd4\])-(\[nd9\]) leads to the following asymptotic definitions of the subdomains at issue.
\(1) Outer subdomain for the slow spatial and temporal variables $\xi $ and $\tau $$$\Omega ^{\left( 0\right) }:=\left\{ \mathcal{O}\left( \sqrt{\epsilon }\right) <\xi ,\mathcal{O}\left( \epsilon \right) <\tau \right\} ;$$
\(2) Left boundary layer subdomain for the fast spatial variable and for slow time $\tau $ $$\Omega _{0}^{\left( 1\right) }:=\left\{ \xi <\mathcal{O}\left( \sqrt{\epsilon }\right) ,\tau \right\} ;$$
\(3) Right boundary layer subdomain for the fast spatial variable and for slow time $\tau $ $$\Omega _{1}^{\left( 1\right) }:=\left\{ 1-\xi <\mathcal{O}\left( \sqrt{\epsilon }\right) ,\tau \right\} ;$$
\(4) Left corner boundary layer subdomain for the fast spatial and temporal variables $$\Omega _0^{\left( 2\right) }:=\left\{ \xi <\mathcal{O}\left( \sqrt{\epsilon }\right) ,\tau <\mathcal{O}\left( \epsilon \right) \right\} ;$$
\(5) Right corner boundary layer subdomain for the fast spatial and temporal variables $$\Omega _1^{\left( 2\right) }:=\left\{ 1-\xi <\mathcal{O}\left( \sqrt{\epsilon }\right) ,\tau <\mathcal{O}\left( \epsilon \right) \right\} ;$$
\(6) Initial boundary layer: Low inner subdomain for the fast temporal variable and slow spatial coordinate $\xi $ $$\Omega _{0}^{\left( 3\right) }:=\left\{ \xi ,\tau <\mathcal{O}\left(
\epsilon \right) \right\} .$$Note that in order to reduce the original singular perturbed problem to a set of simpler regular problems it is necessary to use inner (rescaled) variable in the relevant subdomains. However, it is expedient to perform this procedure during investigation of the boundary value problem (\[nd4\])-(\[nd9\]) in appropriate subdomains of the diffusion boundary layer.
General form of asymptotic solution
-----------------------------------
Our aim is to find the leading-term asymptotic solution of the problem ([nd4]{})-(\[nd9\]) $u_{a}\left( \rho ,\xi ,\tau ;\epsilon \right) $ uniformly valid to order $\mathcal{O}\left( 1\right) $ in the whole domain $\Sigma _{\tau }$ (see Appendix).
It is convenient to divide the desired asymptotic solution $u$ in three parts: outer $u^{\left( 0\right) }$ (regular in $\Omega ^{\left( 0\right) }$), boundary layer $u^{\left( b\right) }$ and corner boundary layer $u^{\left( c\right) }$. Thus the asymptotic solution $u$ may be sought in the form $$u=u^{\left( 0\right) }+u^{\left( b\right) }+u^{\left( c\right) }.
\label{sd6}$$In its turn the boundary layer solution is $$u^{\left( b\right) }=u^{\left( 1\right) }+\widetilde{u}^{\left( 1\right)
}+u^{\left( 3\right) },$$where $u^{\left( 1\right) }$ and $\widetilde{u}^{\left( 1\right) }$ are the left and right boundary layer solutions given in $\Omega _{0}^{\left(
1\right) }$ and $\Omega _{1}^{\left( 1\right) }$, respectively; $u^{\left(
3\right) }$ is the initial boundary layer solution in $\Omega _{0}^{\left(
3\right) }$. Finally the corner boundary layer solution $u^{\left( c\right)
} $ naturally is divided into the sum $$u^{\left( c\right) }=u^{\left( 2\right) }+\widetilde{u}^{\left( 2\right) },$$where $u^{\left( 2\right) }$ and $\widetilde{u}^{\left( 2\right) }$ are the left and right corner boundary layer solutions in $\Omega _{0}^{\left(
2\right) }$ and $\Omega _{1}^{\left( 2\right) }$, respectively. To find explicit form of the above asymptotic solutions one should rewrite original boundary value problem in corresponding subdomains using so-called stretched (or inner) variables inherent in these subdomains (see Sec. V).
The relevant boundary and initial conditions must also take into account the discrepancies arising for the boundary layer solutions $u^{\left( 1\right) }$, $\widetilde{u}^{\left( 1\right) }$ and $u^{\left( 3\right) }$ due to the function $u^{\left( 0\right) }$. Furthermore the corner functions $u^{\left(
2\right) }$ and $\widetilde{u}^{\left( 2\right) }$ should correct discrepancies caused by functions $u^{\left( 1\right) }$, $\widetilde{u}^{\left( 1\right) }$ and $u^{\left( 3\right) }$. This procedure is represented by the diagram $$\begin{tabular}{lllll}
$u^{\left( 1\right) }$ & $\leftarrow $ & $u^{\left( 0\right) }$ & $\rightarrow $ & $\widetilde{u}^{\left( 1\right) }$ \\
$\downarrow $ & & $\downarrow $ & & $\downarrow $ \\
$u^{\left( 2\right) }$ & $\leftarrow $ & $u^{\left( 3\right) }$ & $\rightarrow $ & $\widetilde{u}^{\left( 2\right) }$\end{tabular}\ . \label{sd7}$$It is worth noting here that the above procedure is similar to that used in the reflections method.[@TraytakCM:2007] On the other hand the matching conditions describe exponentially small influence of the appropriate solutions in the opposite directions $$\begin{tabular}{lllll}
$u^{\left( 1\right) }$ & $\rightarrow $ & $u^{\left( 0\right) }$ & $\leftarrow $ & $\widetilde{u}^{\left( 1\right) }$ \\
$\uparrow $ & & $\uparrow $ & & $\uparrow $ \\
$u^{\left( 2\right) }$ & $\rightarrow $ & $u^{\left( 3\right) }$ & $\leftarrow $ & $\widetilde{u}^{\left( 2\right) }$\end{tabular}\ . \label{sd8}$$The explicit form of functions included in (\[sd6\]) will be found during the asymptotic solution iterative procedure.
Solution in the outer subdomain $\Omega ^{\left( 0\right) }$.
=============================================================
Zeroth-order outer approximation. The Fick-Jacobs equation
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According to common matched asymptotic expansions method technique consider first the diffusion equation for slow variables $\xi $ and $\tau $ in the outer subdomain $\Omega ^{\left( 0\right) }$.[@Lagerstrom:1988; @Ilin:1992] Let us look for the asymptotic solution to Eq. (\[nd4\]) with conditions (\[nd6\])-(\[nd9\]) in $\Omega ^{\left( 0\right) }$ as a regular perturbation expansion in the relaxation parameter $$u^{\left( 0\right) }\left( \rho ,\xi ,\tau \right)
=\sum\limits_{n=0}^{\infty }u_{n}^{\left( 0\right) }\left( \rho ,\xi ,\tau
\right) \epsilon ^{n}\text{ }\quad \text{as }\epsilon \rightarrow 0.
\label{t11}$$So outer subdomain $\Omega ^{\left( 0\right) }$ sometimes is called regular one. Note that, although here we limit ourselves by determination of the leading order term $\mathcal{O}\left( 1\right) $, using the proposed approach, one can find functions $u_{n}^{\left( 0\right) }\left( \rho ,\xi
,\tau \right) $ of any reasonable number $n$. Substitution of (\[t11\]) in Eq. (\[nd4\]) leads to the following iterative equations $$\mathcal{L}_{\rho }u_{0}^{\left( 0\right) }=0, \label{t12}$$$$\mathcal{L}_{\rho }u_{n}^{\left( 0\right) }=-\mathcal{L}_{F}u_{n-1}^{\left(
0\right) },\quad n\geq 1 \label{t13}$$and in its turn conditions (\[nd9\]) read $$\left. u_{n}^{\left( 0\right) }\right\vert _{\rho =0}<\infty ,\qquad \left.
\frac{\partial u_{n}^{\left( 0\right) }}{\partial \rho }\right\vert _{\rho
=0}=0,\quad n\geq 0. \label{t13a}$$Similarly, inserting (\[t11\]) into the reflecting wall condition ([nd8]{}), we get the following recurrence relations $$\left. \frac{\partial u_{0}^{\left( 0\right) }}{\partial \rho }\right\vert
_{\rho =R\left( \xi \right) }=0, \label{t13b}$$$$\left. \left[ \frac{\partial u_{n}^{\left( 0\right) }}{\partial \rho }-\frac{\partial u_{n-1}^{\left( 0\right) }}{\partial \xi }R^{^{\prime }}\left( \xi
\right) \right] \right\vert _{\rho =R\left( \xi \right) }=0,\quad n\geq 1.
\label{t13c}$$
One can see that the general solution to the quasi steady-state Eq. ([t12]{}) is $$u_{0}^{\left( 0\right) }\left( \rho ,\xi ,\tau \right) =u_{00}^{\left(
0\right) }\left( \xi ,\tau \right) +u_{01}^{\left( 0\right) }\left( \xi
,\tau \right) \ln \rho . \label{t14}$$Here $u_{00}^{\left( 0\right) }\left( \xi ,\tau \right) $ and $u_{01}^{\left( 0\right) }\left( \xi ,\tau \right) $ are unknown functions to be determined from the boundary conditions (\[t13a\]) and (\[t13c\]). With the aid of conditions (\[t13a\]) we see that $u_{01}^{\left( 0\right)
}\left( \xi ,\tau \right) \equiv 0$ and therefore $u_{0}^{\left( 0\right)
}\left( \rho ,\xi ,\tau \right) =u_{00}^{\left( 0\right) }\left( \xi ,\tau
\right) $, which automatically obeys condition (\[t13b\]).
Consider now the general iterative problem (\[t13\]), (\[t13a\]) and (\[t13c\]) for $n\geq 1$. It is clear that this problem may be insoluble because the unperturbed operator $\mathcal{L}_{\rho }$ is in spectrum [Vishik:1960]{} (see Appendix). This circumstance leads to the fact that the approximations in (\[t11\]) cannot be given explicitly, and they are determined by some unknown functions $u_{n}^{\left( 0\right) }\left( \rho
,\xi ,\tau \right) $. Let us find the solvability condition for the iterative problem (\[t13\]), (\[t13a\]) and (\[t13c\]). Multiplying Eq. (\[t13\]) by $\rho $ and integrating then with respect to $\rho $ from $\rho =0$ we arrive at $$\int\limits_{0}^{\rho }\rho \mathcal{L}_{F}u_{n-1}^{\left( 0\right) }d\rho
=\rho \frac{\partial u_{n}^{\left( 0\right) }}{\partial \rho }.$$Hence, utilizing the recurrence boundary condition (\[t13c\]), the desired solvability condition for $u_{n}^{\left( 0\right) }\left( \rho ,\xi ,\tau
\right) $ reads $$\int\limits_{0}^{R\left( \xi \right) }\rho \mathcal{L}_{F}u_{n-1}^{\left(
0\right) }d\rho =R\left( \xi \right) R^{\prime }\left( \xi \right) \left.
\frac{\partial u_{n-1}^{\left( 0\right) }}{\partial \xi }\right\vert _{\rho
=R\left( \xi \right) },\quad n\geq 1. \label{t15}$$It is important to underline that solvability condition (\[t15\]) eventually follows from the reflecting boundary condition (\[nd8\]) imposed on the tube wall $\partial \Sigma _{w}$. In specific case at $n=1$ from (\[t15\]) we get straightforwardly the following condition $$\mathcal{L}_{F}u_{0}^{\left( 0\right) }=2\frac{R^{\prime }\left( \xi \right)
}{R\left( \xi \right) }\frac{\partial u_{0}^{\left( 0\right) }}{\partial \xi
}. \label{t17a}$$One can readily see that obtained condition (\[t17a\]) is a dimensionless form of the FJE (see Sec. X).
For further treatment it is convenient to put down the dimensionless FJE (\[t17a\]) in a compact form $$\mathcal{L}_{FJ}u_{0}^{\left( 0\right) }=0, \label{t19a}$$introducing the dimensionless Fick-Jacobs operator $$\mathcal{L}_{FJ}:=\mathcal{L}_{F}-2\frac{R^{\prime }\left( \xi \right) }{R\left( \xi \right) }\frac{\partial }{\partial \xi }.$$Moreover the zeroth-order approximation in the outer solution $u_{0}^{\left(
0\right) }\left( \xi ,\tau \right) $ we can naturally call the Fick-Jacobs approximation (FJA). To complete the derivation of the FJA one needs to infer the appropriate initial and boundary conditions using the asymptotic solutions of the posed problem in the diffusion boundary layer $\Omega
^{\left( b\right) }$. Thus it follows from the above treatment that mathematically the FJE is nothing other than a condition of solvability for the function $u_{1}^{\left( 0\right) }\left( \rho ,\xi ,\tau \right) $. This feature of $u_{1}^{\left( 0\right) }\left( \rho ,\xi ,\tau \right) $ is common with the Hilbert approximation in the kinetic theory for low Knudsen numbers (see Sec. VIII).
Assuming that (\[t19a\]) holds true, it is clear that the general solution to inhomogenious Eq. (\[t13\]), which satisfies conditions (\[t13a\]) is $$u_{1}^{\left( 0\right) }\left( \rho ,\xi ,\tau \right) =u_{10}^{\left(
0\right) }\left( \xi ,\tau \right) -\frac{1}{4}\rho ^{2}\mathcal{L}_{F}u_{0}^{\left( 0\right) }, \label{t16}$$where $u_{10}^{\left( 0\right) }\left( \xi ,\tau \right) $ is an arbitrary function to be found during asymptotic solution. Hence it is important to note that the outer approximation of order $\mathcal{O}\left( \epsilon
\right) $ contains a term depending on the transversal variable $\rho $.
To end this subsection, we observe that solution $u\left( \rho .\xi .\tau
\right) $ is nonanalytic in the relaxation parameter $\epsilon $ and, therefore, regular expansion (\[t11\]) in powers of $\epsilon $ fails to give uniformly valid approximation in the whole domain $\Sigma _{\tau }$. In this connection we note that expansion (\[t11\]) is an analog of the Hilbert expansion for solution of the Boltzmann equation (see Sec. VIII for details). Thus, as we mentioned in Sec. III, to find the uniformly valid approximation, one has to solve appropriate boundary value problems concerning the diffusion boundary layers.
Forms of the Fick-Jacobs equation
---------------------------------
Analysis of the literature showed that the classical form of FJE (\[in1\]) (or in case of anisotropic diffusion (\[sp7\])) is the most common in theoretical and applied papers. However we believe that the most natural form of the FJE for the axially symmetric tubes is the following divergent form [@Kalnay2:2005] $$\frac{\partial u_{0}^{\left( 0\right) }}{\partial \tau }=\frac{1}{R\left(
\xi \right) ^{2}}\frac{\partial }{\partial \xi }\left[ R\left( \xi \right)
^{2}\frac{\partial u_{0}^{\left( 0\right) }}{\partial \xi }\right] .
\label{di4}$$It seems interesting that the right hand side of Eq. (\[di4\]) resembles the Laplacian action in conformally flat metric, which was rather widely investigated in theoretical physics.[@Turbiner:1984] The connection of this equation with classical FJE (\[sp7\]) is known.[@Kalnay2:2005]Utilizing the cylindrical coordinates we can write relation (\[in2\]) as follows: $$c\left( z,t\right) =2\pi \int\limits_{0}^{r\left( z\right) }C\left(
r,z,t\right) rdr \label{di5}$$and making notation $$C_{M}u_{0}^{\left( 0\right) }\left( z,t\right) =c_{0}^{\left( 0\right)
}\left( z,t\right) :=\frac{1}{A\left( z\right) }\lim_{\epsilon \rightarrow
0}c\left( z,t\right) \label{di6}$$we obtain the FJE in the classical form (\[in1\]).
The FJE in the form (\[di4\]) may be useful, g.e., to describe diffusion in a long conical tube of the radius given by linear function$$R\left( \xi \right) =a\xi +b, \label{di7}$$where $a$ and $b$ are some constants. Indeed, assuming for definiteness that $a$ and $b$ are positive, by means of substitution (\[di7\]) we can reduce the FJE (\[di4\]) to well-known spherically symmetric diffusion equation $$\frac{\partial u_{0}^{\left( 0\right) }}{\partial \tau }=a^{2}\frac{1}{R^{2}}\frac{\partial }{\partial R}\left[ R^{2}\frac{\partial u_{0}^{\left(
0\right) }}{\partial R}\right] \label{di8}$$in a hollow sphere $b<R<a+b$.[@Carslaw:1959]
To present one more example rewrite Eq. (\[di4\]) in the dimensional form $$R^{2}\left( z\right) \frac{\partial c_{0}^{\left( 0\right) }}{\partial t}=D_{\Vert }\frac{\partial }{\partial z}\left[ R^{2}\left( z\right) \frac{\partial c_{0}^{\left( 0\right) }}{\partial z}\right] . \label{dis3}$$One can see that multiplication of (\[dis3\]) by $\pi $ and integration with respect to $z$ from $0$ to any current $z$ gives $$\Phi \left( z,t\right) =\Phi \left( 0,t\right) -\int\limits_{0}^{z}\frac{\partial c_{0}^{\left( 0\right) }}{\partial t}dV, \label{dis4}$$where $dV=\pi R^{2}\left( z\right) dz$ and $$\Phi \left( z,t\right) :=-D_{\Vert }\pi R^{2}\left( z\right) \frac{\partial
}{\partial z}c_{0}^{\left( 0\right) }\left( z,t\right)$$is the total flux of diffusing particles through the cross section at point $z$. For the steady state flux $\Phi _{s}\left( z\right) $ ($t\gg t_{L}$) relationship (\[dis4\]) takes the simplest form $$\Phi _{s}\left( z\right) =\Phi _{s}\left( 0\right) . \label{dis5}$$Thus, the FJE is similar to continuity equation and Eq. (\[dis5\]) is an analog to known Bernoulli’s principle in ideal fluid dynamics. The latter fact supports the analogy between the FJA $u_{0}^{\left( 0\right) }$ and the Hilbert solution to the Boltzmann equation at low Knudsen numbers (see Sec. IV and below).
Eq. (\[t19a\]) represents one more convenient form of the FJE, which for dimensional variables reads $$\frac{\partial c_{0}^{\left( 0\right) }}{\partial \tau }-D_{\parallel }\frac{\partial ^{2}c_{0}^{\left( 0\right) }}{\partial z^{2}}=V\left( z\right)
\frac{\partial c_{0}^{\left( 0\right) }}{\partial z}. \label{dis6}$$Here we denote $V\left( z\right) :=V_{L}A^{-1}dA/dz$ the effective drift velocity of diffusing particles along the $z$ axis, and $V_{L}=D_{\parallel
}/L$ the characteristic longitudinal diffusion velocity. It seems that convective diffusion interpretation (\[dis6\]) appeared to be even more appropriate for investigation of the FJE than widely used entropy potential form (\[in2a\]). This ensues from the fact that nowadays mathematical theory of convective diffusion equation is thoroughly elaborated in all aspects.
Solution in the subdomains $\Omega _{0}^{\left( 1\right) }$ and $\Omega _{1}^{\left( 1\right) }$. Boundary conditions for the Fick-Jacobs equation
==================================================================================================================================================
Let us study the solution of the problem (\[nd4\])-(\[nd9\]) in the spatial diffusion boundary layer subdomains $\Omega _{0}^{\left( 1\right) }$ and $\Omega _{1}^{\left( 1\right) }$ attached to the endpoints ($\xi
=\left\{ 0,1\right\} $) (see Fig. \[fig:dom\]). In subdomains $\Omega _{0}^{\left(
1\right) }$ and $\Omega _{1}^{\left( 1\right) }$ we introduce so-called inner coordinates: new stretched spatial variables $\xi ^{\ast }=\xi /\sqrt{\epsilon }$ and $\widetilde{\xi }=\left( 1-\xi \right) /\sqrt{\epsilon }$, respectively, leaving slow time $\tau $ unscaled. So in the left and right subdomains $\Omega _{0}^{\left( 1\right) }$ and $\Omega _{1}^{\left(
1\right) }$ one has $\xi ^{\ast }=\mathcal{O}\left( 1\right) $ and $\widetilde{\xi }=\mathcal{O}\left( 1\right) $ as $\epsilon \rightarrow 0$. Asymptotic solutions to the problem (\[nd4\])-(\[nd9\]) behave similarly in $\Omega _{0}^{\left( 1\right) }$ and $\Omega _{1}^{\left( 1\right) }$, therefore, for definiteness we consider in detail the solution corresponding to the left subdomain $\Omega _{0}^{\left( 1\right) }$ only. Rewriting the boundary value problem (\[nd4\])-(\[nd9\]) in the inner coordinates $\left( \rho ,\xi ^{\ast },\tau \right) $ of $\Omega _{0}^{\left( 1\right) }$ for the inner solution $u^{\left( 1\right) }\left( \rho ,\xi ^{\ast };\tau
\right) $ we obtain $$\frac{\partial ^{2}u^{\left( 1\right) }}{\partial \xi ^{\ast 2}}-\mathcal{L}_{\rho }u^{\left( 1\right) }=\epsilon \frac{\partial u^{\left( 1\right) }}{\partial \tau }\qquad \text{ in }\Omega _{0}^{\left( 1\right) }, \label{bc1}$$$$\left. u^{\left( 1\right) }\right\vert _{\rho =0}<\infty ,\qquad \left.
\frac{\partial u^{\left( 1\right) }}{\partial \rho }\right\vert _{\rho =0}=0,
\label{bc3}$$$$\left. \left[ \frac{\partial u^{\left( 1\right) }\left( \rho ,\xi ^{\ast
};\tau \right) }{\partial \rho }-\sqrt{\epsilon }\frac{\partial u^{\left(
1\right) }}{\partial \xi ^{\ast }}R^{^{\prime }}\left( \sqrt{\epsilon }\xi
^{\ast }\right) \right] \right\vert _{\rho =R\left( \sqrt{\epsilon }\xi
^{\ast }\right) }=0. \label{bc4}$$
Let us observe that conditions on the $z$ axis (\[bc3\]) must be satisfied in all other subdomains of the boundary layer ($\Omega _{1}^{\left( 1\right)
}$, $\Omega _{0}^{\left( 2\right) }$, $\Omega _{1}^{\left( 2\right) }$, and $\Omega _{0}^{\left( 3\right) }$), so for brief henceforward we omit them later in the text. Due to the type of Eq. (\[bc1\]) subdomain $\Omega
_{0}^{\left( 1\right) }$ ($\Omega _{1}^{\left( 1\right) }$) is often termed as elliptic boundary layer.[@Ilin:1992]
Find now the appropriate boundary conditions for $u^{\left( 1\right) }$. Bearing in mind the derivation of uniformly valid approximation (\[sd6\]), consider the partial sum $$u^{\left( 0,1\right) }\left( \rho ,\xi ,\xi ^{\ast },\tau \right) =u^{\left(
0\right) }\left( \rho ,\xi ,\tau \right) +u^{\left( 1\right) }\left( \rho
,\xi ^{\ast };\tau \right) \label{bc4a}$$which is defined in $\Omega ^{\left( 0\right) }\cup \Omega _{0}^{\left(
1\right) }$ with appropriate matching conditions. It is worth noting that both outer $u^{\left( 0\right) }\left( \rho ,\xi ,\tau \right) $ and inner $u^{\left( 1\right) }\left( \rho ,\xi ^{\ast };\tau \right) $ solutions are approximations to the same solution but defined in outer $\Omega ^{\left(
0\right) }$ and inner $\Omega _{0}^{\left( 1\right) }$ subdomains, respectively. So the compound approximation $u^{\left( 0,1\right) }$ should obeys the left boundary condition (\[nd7\]) and satisfies the corresponding diffusion equation (\[nd4\]) in $\Omega ^{\left( 0\right)
}\cup \Omega _{0}^{\left( 1\right) }$. Substitution of approximation ([bc4a]{}) into Eq. (\[nd4\]) yields $$\left( \mathcal{L}_{\rho }+\epsilon \mathcal{L}_{F}\right) u^{\left(
0\right) }\left( \rho ,\xi ,\tau \right)$$$$=\left( \frac{\partial ^{2}}{\partial \xi ^{\ast 2}}-\mathcal{L}_{\rho
}-\epsilon \frac{\partial }{\partial \tau }\right) u^{\left( 1\right)
}\left( \rho ,\xi ^{\ast };\tau \right) .$$Hence, employing the fact, that $\xi $ and $\xi ^{\ast }$ are independent variables, we obtain Eqs (\[nd4\]) and (\[bc1\]). In its turn from the left boundary condition (\[nd7\]) we have $$\left. u^{\left( 1\right) }\right\vert _{\xi ^{\ast }=0}=g_{1}\left( \rho
,\tau \right) -\left. u^{\left( 0\right) }\left( \rho ,\xi ,\tau \right)
\right\vert _{\xi =0}. \label{bc4b}$$Missing right boundary condition for $u^{\left( 1\right) }$ may be found using the matching condition between inner and outer solutions [Ilin:1992]{}, i.e. $$\left. u^{\left( 0,1\right) }\left( \rho ,\xi ,\xi ^{\ast },\tau \right)
\right\vert _{\xi ^{\ast }\rightarrow \infty }\rightarrow u^{\left( 0\right)
}\left( \rho ,\xi ,\tau \right)$$that immediately leads to the desired boundary condition $$\left. u^{\left( 1\right) }\left( \rho ,\xi ^{\ast };\tau \right)
\right\vert _{\xi ^{\ast }\rightarrow \infty }\rightarrow 0. \label{bc5}$$Note that hereafter the limit as $\xi ^{\ast }\rightarrow \infty $ means that $\epsilon \rightarrow 0$ provided $\xi $ is fixed.
It is clear that inside $\Omega _{0}^{\left( 1\right) }$ the problem under study becomes regular, so we can seek the solution $u^{\left( 1\right)
}\left( \rho ,\xi ^{\ast },\tau \right) $ in the form of the expansion $$u^{\left( 1\right) }\left( \rho ,\xi ^{\ast },\tau \right)
=\sum\limits_{m=0}^{\infty }u_{m}^{\left( 1\right) }\left( \rho ,\xi ^{\ast
};\tau \right) \epsilon ^{m/2}\text{ }\quad \text{as }\epsilon \rightarrow 0,
\label{bc6}$$where $u_{m}^{\left( 1\right) }\left( \rho ,\xi ^{\ast };\tau \right) $ are so-called functions of the boundary layer.[@Ilin:1992] One can see that implicitly the dependence on slow time $\tau $ arises in the second order approximation ($m=2$) only. In this way we formally reduced the problem in the inner subdomain $\Omega _{0}^{\left( 1\right) }$ to the quasi steady-state problem (we have only parametric dependence upon time $\tau $) posed on the semi-infinite tube bounded at $\xi ^{\ast }=0$.
Hence for the zeroth-order approximation we gain the problem $$\frac{\partial ^{2}u_{0}^{\left( 1\right) }}{\partial \xi ^{\ast 2}}-\mathcal{L}_{\rho }u_{0}^{\left( 1\right) }=0,\qquad 0<\rho <R_{0},
\label{bc7}$$$$\left. u_{0}^{\left( 1\right) }\right\vert _{\xi ^{\ast }=0}=g_{1}\left(
\rho ,\tau \right) -\left. u_{0}^{\left( 0\right) }\right\vert _{\xi =0},
\label{bc8}$$$$\left. u_{0}^{\left( 1\right) }\right\vert _{\xi ^{\ast }\rightarrow \infty
}\rightarrow 0, \label{bc8a}$$$$\left. \frac{\partial u_{0}^{\left( 1\right) }}{\partial \rho }\right\vert
_{\rho =R_{0}}=0. \label{bc9}$$Therefore for $u_{0}^{\left( 1\right) }$ the reflecting wall condition ([bc4]{}) simplifies to the relevant condition (\[bc9\]) on the circular cylinder of constant radius $R_{0}:=R\left( 0\right) $.
The general solution to Eq. (\[bc7\]) satisfying the reflecting condition (\[bc9\]) is $$u_{0}^{\left( 1\right) }\left( \rho ,\xi ^{\ast };\tau \right)
=\sum\limits_{k=0}^{\infty }b_{k}\left( \tau \right) e^{-\lambda _{k}\xi
^{\ast }}\widehat{J}_{0}\left( \lambda _{k}\frac{\rho }{R_{0}}\right) ,
\label{bc10}$$where $\left\{ \widehat{J}_{0}\left( \lambda _{k}\rho /R_{0}\right) \right\}
_{k=0}^{\infty }$ is the complete orthonormal system defined in Appendix.
Matching condition (\[bc8a\]) leads to $b_{0}\left( \tau \right) =0$ that yields the desired boundary condition for the solution of the FJE $u_{0}^{\left( 0\right) }\left( \xi ,\tau \right) $ at the left endpoint ($\xi =0$) $$\left. u_{0}^{\left( 0\right) }\left( \xi ,\tau \right) \right\vert _{\xi
=0}=\left\langle g_{1},\widehat{J}_{0}\left( 0\right) \right\rangle _{\mathcal{H}_{0}}\widehat{J}_{0}\left( 0\right) =\frac{2}{R_{0}^{2}}\int\limits_{0}^{R_{0}}\rho g_{1}\left( \rho ,\tau \right) d\rho .
\label{bc11}$$For $k\geq 1$ unknown coefficients $b_{k}\left( \tau \right) $ are $$b_{k}\left( \tau \right) =\left\langle g_{1},\widehat{J}_{0}\left( \lambda
_{k}\frac{\rho }{R_{0}}\right) \right\rangle _{\mathcal{H}_{0}}.
\label{bc12}$$Similar treatment of the inner solution in the right subdomain $\Omega
_{1}^{\left( 1\right) }$ gives $$\widetilde{u}_{0}^{\left( 1\right) }\left( \rho ,\widetilde{\xi };\tau
\right) =\sum\limits_{k=0}^{\infty }\widetilde{b}_{k}\left( \tau \right)
e_{k}^{-\lambda _{k}\widetilde{\xi }}\widehat{J}_{0}\left( \lambda _{k}\frac{\rho }{R_{1}}\right) , \label{bc13}$$Hence $\widetilde{b}_{0}\left( \tau \right) =0$ and the second boundary condition for the FJA $u_{0}^{\left( 0\right) }\left( \xi ,\tau \right) $ at the right endpoint ($\xi =1$) is $$\left. u_{0}^{\left( 0\right) }\left( \xi ,\tau \right) \right\vert _{\xi
=1}=\left\langle g_{2},\widehat{J}_{0}\left( 0\right) \right\rangle _{\mathcal{H}_{1}}\widehat{J}_{0}\left( 0\right) =\frac{2}{R_{1}^{2}}\int\limits_{0}^{R_{1}}\rho g_{2}\left( \rho ,\tau \right) d\rho ,
\label{bc14}$$where $R_{1}:=R\left( 1\right) $ and $$\widetilde{b}_{k}\left( \tau \right) =\left\langle g_{2},\widehat{J}_{0}\left( \lambda _{k}\frac{\rho }{R_{1}}\right) \right\rangle _{\mathcal{H}_{1}},\quad k\geq 1. \label{bc15}$$
Solution in the subdomain $\Omega _{0}^{\left( 3\right) }$. Initial conditions for the Fick-Jacobs equation
===========================================================================================================
Now we dwell on the solution to problem (\[nd4\])-(\[nd9\]) in the initial diffusion boundary layer subdomain $\Omega _{0}^{\left( 3\right) }$ attached to the the initial time $\tau =0$ (see Fig. \[fig:dom\]). In this subdomain inner variables are $\xi $ and the stretched (fast) time $\tau ^{\ast }=\tau
/\epsilon $ ($\tau ^{\ast }=\mathcal{O}\left( 1\right) $ as $\epsilon
\rightarrow 0$). Using these variables in the original problem (\[nd4\])-(\[nd9\]) similarly to the previous case we arrive at $$\frac{\partial u^{\left( 3\right) }}{\partial \tau ^{\ast }}+\mathcal{L}_{\rho }u^{\left( 3\right) }=\epsilon \frac{\partial ^{2}u^{\left( 3\right) }}{\partial \xi ^{2}}, \label{ic1}$$$$\left. u^{\left( 3\right) }\right\vert _{\tau ^{\ast }=0}=g_{0}\left( \rho
,\xi \right) -\left. u^{\left( 0\right) }\left( \rho ,\xi ,\tau \right)
\right\vert _{\tau =0}, \label{ic2}$$$$\left. \left[ \frac{\partial u^{\left( 3\right) }}{\partial \rho }-\epsilon
\frac{\partial u^{\left( 3\right) }}{\partial \xi }R^{^{\prime }}\left( \xi
\right) \right] \right\vert _{\rho =R\left( \xi \right) }=0. \label{ic3}$$One can see that now we obtained effectively the boundary value problem for the infinitely long solid of revolution. According to the type of Eq. ([ic1]{}) the subdomain $\Omega _{0}^{\left( 3\right) }$ is often called as parabolic boundary layer.[@Ilin:1992] We also must add to (\[ic2\]) and (\[ic3\]) the matching condition for the inner solution $$\left. u^{\left( 3\right) }\left( \rho ,\tau ^{\ast };\xi \right)
\right\vert _{\tau ^{\ast }\rightarrow \infty }\rightarrow 0. \label{ic4}$$
The corresponding regular expansion of the inner solution in $\Omega
_{0}^{\left( 3\right) }$ reads
$$u^{\left( 3\right) }\left( \rho ,\xi ,\tau ^{\ast }\right)
=\sum\limits_{m=0}^{\infty }u_{m}^{\left( 3\right) }\left( \rho ,\tau ^{\ast
};\xi \right) \epsilon ^{m}\text{ }\quad \text{as }\epsilon \rightarrow 0.
\label{ic5}$$
Here the functions of the boundary layer $u_{m}^{\left( 3\right) }\left(
\rho ,\tau ^{\ast };\xi \right) $ depend upon $\xi $ as a parameter. So for the zeroth-order function $u_{0}^{\left( 3\right) }\left( \rho ,\tau ^{\ast
};\xi \right) $ we get the following boundary value problem $$\frac{\partial u_{0}^{\left( 3\right) }}{\partial \tau ^{\ast }}+\mathcal{L}_{\rho }u_{0}^{\left( 3\right) }=0, \label{ic6}$$$$\left. u_{0}^{\left( 3\right) }\right\vert _{\tau ^{\ast }=0}=g_{0}\left(
\rho ,\xi \right) -\left. u_{0}^{\left( 0\right) }\right\vert _{\tau =0},
\label{ic7}$$$$\left. \frac{\partial u_{0}^{\left( 3\right) }}{\partial \rho }\right\vert
_{\rho =R\left( \xi \right) }=0. \label{ic8}$$One can see that the obtained problem (\[ic6\])-(\[ic8\]) describes the diffusion into the infinite circular cylinder of radius $R\left( \xi \right)
$. It is clear that general solution to Eq. (\[ic6\]), which obey the reflecting condition (\[ic8\]), may be expressed as $$u_{0}^{\left( 3\right) }\left( \rho ,\tau ^{\ast };\xi \right)
=\sum\limits_{k=0}^{\infty }a_{k}\left( \xi \right) e^{-\lambda _{k}^{2}\tau
^{\ast }}\widehat{J}_{0}\left( \lambda _{k}\frac{\rho }{R\left( \xi \right) }\right) . \label{ic9}$$Setting $\tau ^{\ast }=0$, this implies that to satisfy the matching condition $$\left. u_{0}^{\left( 3\right) }\right\vert _{\tau ^{\ast }\rightarrow \infty
}\rightarrow 0 \label{ic10}$$we should impose $a_{0}\left( \xi \right) =0$ at that, utilizing initial condition (\[ic7\]), we have the desired initial condition for the FJA $$\left. u_{0}^{\left( 0\right) }\left( \xi ,\tau \right) \right\vert _{\tau
=0}=\left\langle g_{0},\widehat{J}_{0}\left( 0\right) \right\rangle _{\mathcal{H}_{\xi }}\widehat{J}_{0}\left( 0\right) =\frac{2}{R^{2}\left( \xi
\right) }\int\limits_{0}^{R\left( \xi \right) }\rho g_{0}\left( \rho ,\xi
\right) d\rho . \label{ic11}$$and expression for unknown coefficients $a_{k}\left( \xi \right) $ ($k\geq 1$)$$a_{k}\left( \xi \right) =\left\langle g_{0},\widehat{J}_{0}\left( \lambda
_{k}\frac{\rho }{R\left( \xi \right) }\right) \right\rangle _{\mathcal{H}_{\xi }}. \label{ic12}$$For the problem under study this formula gives the answer to the question posed by Kalnay and Percus: “Having projected 2D equation to the 1D one ..., one may ask the question: how then is the projected 1D initial density $P\left( x,0\right) $ related to the original $\rho \left( x,y,0\right) $, and is there some reasonable projection algorithm?”[@Kalnay:2005]
Solution in the corner subdomains $\Omega _0^{\left( 2\right)
} $ and $\Omega _1^{\left( 2\right) }$
=============================================================
It is clear that behavior of inner solution in the corner subdomains $\Omega
_0^{\left( 2\right) }$ and $\Omega _1^{\left( 2\right) }$ (see Fig. \[fig:dom\]) is similar, therefore, for briefness we give the detailed treatment of the relevant boundary value problem only in $\Omega _0^{\left( 2\right) }$. For this purpose we define inner variables $(\xi ^{*},\tau ^{*})$ and rewrite Eq. (\[nd4\]) for the corner boundary layer function $u^{\left( 2\right)
}\left( \rho ,\xi ^{*},\tau ^{*}\right) $ as
$$\frac{\partial ^{2}u^{\left( 2\right) }}{\partial \xi ^{\ast 2}}-\mathcal{L}_{\rho }u^{\left( 2\right) }=\frac{\partial u^{\left( 2\right) }}{\partial
\tau ^{\ast }},\quad 0<\rho <R_{0}. \label{mc1}$$
Similarly to the previous case the subdomain $\Omega _{0}^{\left( 2\right) }$ (or $\Omega _{1}^{\left( 2\right) }$) is also called as parabolic boundary layer.[@Ilin:1992] The reflecting wall condition (\[nd8\]) in $\Omega
_{0}^{\left( 2\right) }$ takes the form $$\left. \left[ \frac{\partial u^{\left( 2\right) }\left( \rho ,\xi ^{\ast
},\tau ^{\ast }\right) }{\partial \rho }-\sqrt{\epsilon }\frac{\partial
u^{\left( 2\right) }}{\partial \xi ^{\ast }}R^{^{\prime }}\left( \sqrt{\epsilon }\xi ^{\ast }\right) \right] \right\vert _{\rho =R\left( \sqrt{\epsilon }\xi ^{\ast }\right) }=0. \label{mc2}$$One can see immediately from geometry of the problem that desired solution $u^{\left( 2\right) }\left( \rho ,\xi ^{\ast },\tau ^{\ast }\right) $ does not affect directly to the outer solution $u^{\left( 0\right) }\left( \xi
,\tau \right) $ in $\Omega ^{\left( 0\right) }$. According to scheme ([sd7]{}) function $u^{\left( 2\right) }\left( \rho ,\xi ^{\ast },\tau ^{\ast
}\right) $ should be matched with $u^{\left( 1\right) }\left( \rho ,\xi
^{\ast },\tau \right) $ and $u^{\left( 3\right) }\left( \rho ,\xi ,\tau
^{\ast }\right) $ in order to correct discrepancies due to these functions for initial and boundary conditions, respectively $$\left. u^{\left( 2\right) }\right\vert _{\tau ^{\ast }=0}=-\left. u^{\left(
1\right) }\right\vert _{\tau =0},\qquad \left. u^{\left( 2\right)
}\right\vert _{\xi ^{\ast }=0}=-\left. u^{\left( 3\right) }\right\vert _{\xi
=0}. \label{mc3}$$The relevant matching conditions (see scheme (\[sd8\])) for the corner boundary layer function $u^{\left( 2\right) }\left( \rho ,\xi ^{\ast },\tau
^{\ast }\right) $ in $\Omega _{0}^{\left( 1\right) }$ and $\Omega
_{0}^{\left( 3\right) }$ are $$\left. u^{\left( 2\right) }\right\vert _{\tau ^{\ast }\rightarrow \infty
}\rightarrow 0,\qquad \left. u^{\left( 2\right) }\right\vert _{\xi ^{\ast
}\rightarrow \infty }\rightarrow 0. \label{mc5}$$One can see that obtained problem (\[mc1\])-(\[mc5\]) effectively describes the time-dependent diffusion in a semi-infinite circular cylinder of radius $R_{0}$.
Inside the corner boundary layer subdomain at issue $\Omega _{0}^{\left(
2\right) }$ we can seek the solution in the regular form $$u^{\left( 2\right) }\left( \rho ,\xi ^{\ast },\tau ^{\ast }\right)
=\sum\limits_{m=0}^{\infty }u_{m}^{\left( 2\right) }\left( \rho ,\xi ^{\ast
},\tau ^{\ast }\right) \epsilon ^{m/2}\text{ }\quad \text{as }\epsilon
\rightarrow 0. \label{mc7}$$Employing expressions (\[ic9\]) and (\[bc10\]) it is clear that the zeroth-order approximation $u_{0}^{\left( 2\right) }$ to the corner function $u^{\left( 2\right) }$ in $\Omega _{0}^{\left( 2\right) }$ governs by the problem $$\frac{\partial ^{2}u_{0}^{\left( 2\right) }}{\partial \xi ^{\ast 2}}-\mathcal{L}_{\rho }u_{0}^{\left( 2\right) }=\frac{\partial u_{0}^{\left(
2\right) }}{\partial \tau ^{\ast }},\quad 0<\rho <R_{0}, \label{mc8}$$$$\left. u_{0}^{\left( 2\right) }\right\vert _{\tau ^{\ast
}=0}=-\sum\limits_{k=1}^{\infty }b_{k}\left( 0\right) e^{-\lambda _{k}\xi
^{\ast }}\widehat{J}_{0}\left( \lambda _{k}\frac{\rho }{R_{0}}\right) ,
\label{mc9}$$$$\left. u_{0}^{\left( 2\right) }\right\vert _{\xi ^{\ast
}=0}=-\sum\limits_{k=1}^{\infty }a_{k}\left( 0\right) e^{-\lambda
_{k}^{2}\tau ^{\ast }}\widehat{J}_{0}\left( \lambda _{k}\frac{\rho }{R_{0}}\right) , \label{mc10}$$$$\left. u_{0}^{\left( 2\right) }\right\vert _{\xi ^{\ast }\rightarrow \infty
}\rightarrow 0, \label{mc11}$$$$\left. \frac{\partial u_{0}^{\left( 2\right) }}{\partial \rho }\right\vert
_{\rho =R_{0}}=0. \label{mc12}$$It is convenient to look for solution of the obtained boudary value problem (\[mc8\])-(\[mc12\]) by means of projection method with respect to the orthonormal basis $\left\{ \widehat{J}_{0}\left( \lambda _{k}\frac{\rho }{R_{0}}\right) \right\} _{k=0}^{\infty }$ (see Appendix). Multipling Eq. ([mc8]{}) by $\widehat{J}_{0}\left( \lambda _{k}\frac{\rho }{R_{0}}\right) $ and using formulae (\[Hs2a\]) and (\[Hs3\]) one gets$$u_{0k}^{\left( 2\right) }\equiv 0,\qquad k=0;$$$$\frac{\partial ^{2}u_{0k}^{\left( 2\right) }}{\partial \xi ^{\ast 2}}-\lambda _{k}^{2}u_{0k}^{\left( 2\right) }=\frac{\partial u_{0k}^{\left(
2\right) }}{\partial \tau ^{\ast }},\qquad k\geq 1;$$where $$u_{0k}^{\left( 2\right) }\left( \xi ^{\ast },\tau ^{\ast }\right)
=\left\langle u_{0}^{\left( 2\right) },\widehat{J}_{0}\left( \lambda _{k}\frac{\rho }{R_{0}}\right) \right\rangle _{\mathcal{H}_{0}}.$$Introducing for $k\geq 1$ a subsidiary function $$w_{k}\left( \xi ^{\ast },\tau ^{\ast }\right) =e^{\lambda _{k}^{2}\tau
^{\ast }}u_{0k}^{\left( 2\right) }\left( \xi ^{\ast },\tau ^{\ast }\right)$$we finally reduce problem (\[mc8\])-(\[mc12\]) to$$\frac{\partial ^{2}w_{k}}{\partial \xi ^{\ast 2}}=\frac{\partial w_{k}}{\partial \tau ^{\ast }}, \label{mc14}$$$$\left. w_{k}\right\vert _{\tau ^{\ast }=0}=-b_{k}\left( 0\right) e^{-\lambda
_{k}\xi ^{\ast }}, \label{mc14a}$$$$\left. w_{k}\right\vert _{\xi ^{\ast }=0}=-a_{k}\left( 0\right) ,\quad
\left. w_{k}\right\vert _{\xi ^{\ast }\rightarrow \infty }\rightarrow 0.
\label{mc15}$$One can easily derive that solution to the boundary value problem (\[mc14\])-(\[mc15\]) reads [@Carslaw:1959]$$w_{k}\left( \xi ^{\ast },\tau ^{\ast }\right) =-a_{k}\left( 0\right) \mbox{erfc}\left( \frac{\xi ^{\ast }}{2\sqrt{\tau ^{\ast }}}\right)$$$$-\frac{1}{2}b_{k}\left( 0\right) e^{\lambda _{k}^{2}\tau ^{\ast }}\left[
e^{-\lambda _{k}\xi ^{\ast }}\mbox{erfc}\left( \lambda _{k}\sqrt{\tau ^{\ast
}}-\frac{\xi ^{\ast }}{2\sqrt{\tau ^{\ast }}}\right) \right.$$$$\left. -e^{\lambda _{k}\xi ^{\ast }}\mbox{erfc}\left( \lambda _{k}\sqrt{\tau
^{\ast }}+\frac{\xi ^{\ast }}{2\sqrt{\tau ^{\ast }}}\right) \right] ,
\label{mc16}$$where $$\mbox{erfc}\left( \varsigma \right) =\frac{2}{\sqrt{\pi }}\int_{\varsigma
}^{\infty }e^{-\alpha ^{2}}d\alpha$$is the complementary error function. Hence for the zeroth-order corner function we have expansion$$u_{0}^{\left( 2\right) }\left( \rho ,\xi ^{\ast },\tau ^{\ast }\right)
=\sum\limits_{k=1}^{\infty }w_{k}\left( \xi ^{\ast },\tau ^{\ast }\right)
e^{-\lambda _{k}^{2}\tau ^{\ast }}\widehat{J}_{0}\left( \lambda _{k}\frac{\rho }{R_{0}}\right) . \label{mc17}$$It is obvious that function (\[mc17\]) also satisfies matching condition (\[mc5\]) as $\tau ^{\ast }\rightarrow \infty $.
The appropriate leading-term approximation for the inner solution $\widetilde{u}^{\left( 2\right) }\left( \rho ,\widetilde{\xi },\tau ^{\ast
}\right) $ in the corner subdomain $\Omega _{1}^{\left( 2\right) }$ may be founded with the help of above derivation. For this purpose in (\[mc7\])-(\[mc17\]) one should implement the following substitutions$$\xi ^{\ast }\rightarrow \widetilde{\xi },\quad R_{0}\rightarrow R_{1},\quad
a_{k}\left( 0\right) \rightarrow a_{k}\left( 1\right) ,\quad b_{k}\left(
0\right) \rightarrow \widetilde{b}_{k}\left( 0\right) . \label{mc18}$$Denoting by $\widetilde{w}_{k}\left( \widetilde{\xi },\tau ^{\ast }\right) $ the result of sunstitutions (\[mc18\]) in formula (\[mc17\]) we arrive at the zeroth-order right corner approximation $$\widetilde{u}_{0}^{\left( 2\right) }\left( \rho ,\widetilde{\xi },\tau
^{\ast }\right) =\sum\limits_{k=1}^{\infty }\widetilde{w}_{k}\left(
\widetilde{\xi },\tau ^{\ast }\right) e^{-\lambda _{k}^{2}\tau ^{\ast }}\widehat{J}_{0}\left( \lambda _{k}\frac{\rho }{R_{1}}\right) . \label{mc19}$$Note in passing that the limit as $\widetilde{\xi }\rightarrow \infty $ means that $\epsilon \rightarrow 0$ provided $\xi $ is fixed.
Leading-term approximation
==========================
Explicit form of the general asymptotic solution
------------------------------------------------
Inserting partial expansions (\[t11\]), (\[bc6\]), (\[mc7\]) and ([ic5]{}) into general formula (\[sd6\]) we have uniformly valid as $\epsilon
\rightarrow 0$ in the whole space-time domain $\Sigma _{\tau }$ asymptotic solution$$u\left( \rho ,\xi ,\tau ;\epsilon \right) =\sum\limits_{m=0}^{\infty
}\left\{ \left[ u_{m}^{\left( 0\right) }\left( \rho ,\xi ,\tau \right)
+u_{m}^{\left( 3\right) }\left( \rho ,\tau ^{\ast };\xi \right) \right]
\epsilon ^{m}\right.$$$$+\left[ u_{m}^{\left( 1\right) }\left( \rho ,\xi ^{\ast };\tau \right)
\right. +\widetilde{u}_{m}^{\left( 1\right) }\left( \rho ,\widetilde{\xi };\tau \right)$$$$+\left. u_{m}^{\left( 2\right) }\left( \rho ,\xi ^{\ast },\tau ^{\ast
}\right) \left. +\widetilde{u}_{m}^{\left( 2\right) }\left( \rho ,\widetilde{\xi },\tau ^{\ast }\right) \right] \epsilon ^{m/2}\right\} . \label{uva1}$$Finally, collecting here all leading terms, and denoting by $\mathbf{q}:=\left\{ \rho ,\xi ,\xi ^{\ast },\widetilde{\xi },\tau ,\tau ^{\ast
}\right\} $ the complete set of outer and inner variables inherent in the problem under consideration, we can rewrite expansion (\[uva1\]) in a compact form $$u\left( \rho ,\xi ,\tau ;\epsilon \right) =u_{a}\left( \mathbf{q}\right) +\mathcal{O}\left( \sqrt{\epsilon }\right) \text{ }\quad \text{as }\epsilon
\rightarrow 0, \label{uva2}$$where $u_{a}\left( \mathbf{q}\right) $ is the leading-term approximation uniformly valid in domain $\Sigma _{\tau }$ to order $\mathcal{O}\left(
1\right) $. Accordingly, using the obtained results, function $u_{a}\left(
\mathbf{q}\right) $ may be given as follows: $$u_{a}\left( \mathbf{q}\right) =u_{0}^{\left( 0\right) }\left( \xi ,\tau
\right)$$$$+\sum\limits_{k=1}^{\infty }a_{k}\left( \xi \right) e^{-\lambda _{k}^{2}\tau
^{\ast }}\widehat{J}_{0}\left( \lambda _{k}\frac{\rho }{R\left( \xi \right) }\right)$$$$+\sum\limits_{k=1}^{\infty }\left[ b_{k}\left( \tau \right) e^{-\lambda
_{k}\xi ^{\ast }}+w_{k}\left( \xi ^{\ast },\tau ^{\ast }\right) e^{-\lambda
_{k}^{2}\tau ^{\ast }}\right] \widehat{J}_{0}\left( \lambda _{k}\frac{\rho }{R_{0}}\right)$$$$+\sum\limits_{k=1}^{\infty }\left[ \widetilde{b}_{k}\left( \tau \right)
e^{-\lambda _{k}\widetilde{\xi }}+\widetilde{w}_{k}\left( \widetilde{\xi },\tau ^{\ast }\right) e^{-\lambda _{k}^{2}\tau ^{\ast }}\right] \widehat{J}_{0}\left( \lambda _{k}\frac{\rho }{R_{1}}\right) . \label{uva3}$$This formula constitutes the main result of the present paper. As an important consequence of formula (\[uva3\]) we infer that within the leading-term approximation the total flux through a tube cross section is entirely determined by the FJA $u_{0}^{\left( 0\right) }\left( \xi ,\tau
\right) $ and initial boundary layer function $u_{0}^{\left( 3\right)
}\left( \rho ,\tau ^{\ast };\xi \right) $.
Combining expressions (\[bc11\]), (\[bc14\]), (\[ic11\]) and utilizing the projector $\mathcal{P}_{\xi }$ defined by formula (\[Hs7\]) one can see that the FJA $u_{0}^{\left( 0\right) }\left( \xi ,\tau \right) $ is uniquely determined by the following 1D boundary value problem $$\mathcal{L}_{FJ}u_{0}^{\left( 0\right) }=0, \label{uva4}$$$$\left. u_{0}^{\left( 0\right) }\right\vert _{\tau =0}=\mathcal{P}_{\xi
}g_{0}, \label{uva4a}$$$$\left. u_{0}^{\left( 0\right) }\right\vert _{\xi =0}=\mathcal{P}_{0}g_{1},\quad \left. u_{0}^{\left( 0\right) }\left( \xi ,\tau \right)
\right\vert _{\xi =1}=\mathcal{P}_{1}g_{2}. \label{uva4b}$$Hence one can see that function $u_{0}^{\left( 0\right) }\left( \xi ,\tau
\right) $ is the zeroth-order in $\epsilon $ projection of 3D concentration on the unit zero eigenfunction of the unperturbed operator $\mathcal{L}_{\rho }$ (see Appendix), that is $$u_{0}^{\left( 0\right) }\left( \xi ,\tau \right) =\lim_{\epsilon \rightarrow
0}\mathcal{P}_{\xi }u\left( \rho ,\xi ,\tau ;\epsilon \right) . \label{uva5}$$It is important to note here that we cannot obtain the FJE (\[uva4\]) just by simple projection of the original 3D equation (\[nd4\]) on the zero eigenfunction of the unperturbed operator $\mathcal{L}_{\rho }$ because operators of differentiation with respect to $\xi $ and projection operator (depending on $\xi $) do not commute.[@Kalnay:2005]
Thus, provided one has solved problem (\[uva4\])-(\[uva4b\]) with respect to the FJA $u_{0}^{\left( 0\right) }\left( \xi ,\tau \right) $ the desired leading-term approximation $u_{a}\left( \mathbf{q}\right) $ is governed explicitly by formula (\[uva3\]).
Note in passing that according to expansion (\[uva1\]) the contribution from inner solutions in the spatial boundary layer ($u_{m}^{\left( 1\right)
} $ and $\widetilde{u}_{m}^{\left( 1\right) }$) is much more important than that from the solutions for the initial layer ($u_{m}^{\left( 3\right) }$), since the influence of the spatial boundary layer at $m=1$ gives a term of order $\mathcal{O}\left( \sqrt{\epsilon }\right) $.
Validity of the Fick-Jacobs approximation
-----------------------------------------
Let us delineate the conditions on temporal and spatial scales under which the FJA $u_{0}^{\left( 0\right) }\left( \xi ,\tau \right) $ is valid. One can see that the general condition for validity of the FJA reads$$\left\vert u_{a}-u_{0}^{\left( 0\right) }\right\vert /u_{0}^{\left( 0\right)
}\ll 1. \label{uva6}$$To obtain simple validity conditions first we observe that contribution of the corner functions $u_{0}^{\left( 2\right) }$ and $\widetilde{u}_{0}^{\left( 2\right) }$ to $u$ is certainly less than that from the functions of the diffusion spatial $u_{0}^{\left( 1\right) }$, $\widetilde{u}_{0}^{\left( 1\right) }$ and temporal $u_{0}^{\left( 3\right) }$ boundary layers. Therefore one can ignore in (\[uva2\]) the corrections due to corner functions.
It is also clear from (\[uva2\]) and (\[uva3\]) that solution $u_{0}^{\left( 3\right) }$ corresponds to initial stage of the concentration evolution in $\Omega ^{\left( 3,0\right) }:=\Omega _{0}^{\left( 3\right)
}\cup \Omega ^{\left( 0\right) }$ (see Fig. \[fig:dom\]), where there is a relaxation to the equilibrium with respect to the transversal variable $\rho $, that is$$\frac{\partial u_{a}\left( \mathbf{q}\right) }{\partial \rho }=0\quad \text{
in }\Omega ^{\left( 3,0\right) }.$$This process occurs by exponential damping law with times spectrum $t_{k}=t_{tr}\lambda _{k}^{-2}$, ($k\geq 1$) and the characteristic relaxation longitudinal time for homogenization of initially nonuniform in $r $ distribution of concentration is determined by the lowest eigenvalue $$t_{1}=\frac{t_{tr}}{\lambda _{1}^{2}}\approx 0.0681\cdot \frac{r_{M}^{2}}{D_{\perp }}\ll t_{tr}. \label{uva7}$$
Similarly it follows from expression (\[uva2\]) and form of solutions $u_{0}^{\left( 1\right) }$, $\widetilde{u}_{0}^{\left( 1\right) }$ (\[uva3\]) that characteristic thickness of the spatial diffusion boundary layers $l_{1}$ along $z$ axis in $\Omega _{0}^{\left( 1\right) }\cup \Omega ^{\left(
0\right) }$ and $\Omega _{1}^{\left( 1\right) }\cup \Omega ^{\left( 0\right)
}$ (see Fig. \[fig:dom\]) is $$l_{1}=\frac{r_{M}}{\lambda _{1}}\approx 0.2610\cdot r_{M}<r_{M}.
\label{uva8}$$One can see from (\[uva7\]) and (\[uva8\]) that there is a simple connection between values $t_{1}$ and $l_{1}$: $t_{1}=l_{1}^{2}/D_{\parallel
}$.
Accordingly, combining (\[uva7\]) and (\[uva8\]) the validity of the FJA is determined by the following temporal and spatial conditions $$t\gtrsim t_{tr}\gg t_{1},\qquad \ z\text{ (or }L-z\text{)}\gg r_{M}>l_{1}.
\label{uva9}$$This means that for temporal and spatial scales (\[uva9\]) the explicit dependence of the leading-term approximation $u_{a}\left( \mathbf{q}\right) $ on the initial distribution (\[sp5\]) and boundary conditions (\[sp6\]) depending on the transversal coordinate $r$ disappeared. In other words the FJA works well under a quasi steady-state regime with respect to the characteristic transversal time $t_{tr}$, i.e., when we can eliminate dependence on fast transversal variable $\rho $ and consider dependence only upon slow “hydrodynamic” variables $\xi $ and $\tau $.
Particularly, for the cylindric tube of radius $r_{M}$ all deviations from the “equilibrium function” $v_{0}$ (see Appendix) caused by the inital and boundary conditions depending on transversal coordinate $\rho $ are vanished within diffusion boundary layer subdomains. Thus the subdomain $\Omega
^{\left( 0\right) }$ becomes “the equilibrium region”, where solution $u^{\left( 0\right) }$ does not contain the dependence on $\rho $. It is clear that this situation occurs due to the wall condition $\left. \left(
\partial u/\partial \rho \right) \right\vert _{\rho =1}=0$.
For general case it is clear that at least in a vicinity near the wall $\partial u/\partial \xi \neq 0$ in $\Sigma _{\tau }$ hence and from the reflection boundary condition (\[nd8\]) we have$$\left. \frac{\partial u}{\partial \rho }\right\vert _{\rho =R\left( \xi
\right) }\propto R^{^{\prime }}\left( \xi \right) .$$Therefore $\partial u/\partial \rho \neq 0$ in a vicinity near the wall and a deviation of $u\left( \rho .\xi .\tau \right) $ from the eqiliblium value increases with increasing of function $\left\vert R^{^{\prime }}\left( \xi
\right) \right\vert $. Nevertheless condition $\left\vert R^{^{\prime
}}\left( \xi \right) \right\vert \ll 1$ is not important for validity of the leading-term approximation $u_{a}\left( \mathbf{q}\right) $.
To describe similar heat transfer problem in the semi-bounded cylinder Luikov wrote in his book [@Luikov:1967]: Since there is no loss of heat through the wall of the rod we can treat it as a solid, where heat spreads only in one direction (see p. 182 of Ref. 39 ). Then in this book he considered only 1D equation. It infers from our study that this statement correct only out of the corresponding spatial and temporal boundary layers, that is in the outer sundomain $\Omega
^{\left( 0\right) }$.
Thus, conditions (\[uva9\]) determine the temporal and spatial scales when the FJA holds true, that is $$u_{a}\left( \mathbf{q}\right) \approx u_{0}^{\left( 0\right) }\left( \xi
,\tau \right) .$$
Analogy to the gas kinetic theory
---------------------------------
Previously, using an analogy with the gas kinetic theory, we proposed a general kinetics equation to describe the kinetics of diffusion-controlled reactions in case of infinite system for all spatial and temporal scales and interpreted some results on diffusive interaction in dense arrays of absorbing particles.[@Traytak:1995DI; @Traytak:1996] It is appropriate to note that idea of the projection method suggested by Kalnay and Percus was inspired by analogy with kinetic theory as well. Concerning their method in Ref. 17 they claimed the following: “It reminds one of Bogolubov’s derivation of the generalized Boltzmann equation, expressing the $n$-particle densities as a functional of the one-particle one; here we reduce similarly the number of coordinates.”
For the problem under study we also revealed a profound analogy with the gas kinetic theory at low Knudsen numbers, that helped us to choose an adequate mathematical method to find the desired asymptotic solution. The analogy, we intend to establish, becomes even more profound in the isotropic diffusion case, i.e., when $D_{\parallel }=D_{\perp }=D$. Hence we consider this case and, moreover, for the sake of simplicity, here we dwell on 1D gas system only. Denoting by $T$ a typical time for the gas system of a typical length $L$, $w$ a typical molecular velocity, $\lambda $ the mean free path, and $t_{\lambda }$ the mean free time we have $$T=L/w,\quad t_{\lambda }=\lambda /w. \label{ab1}$$Using these scales one can put down linearized Boltzmann’s equation with respect to the distribution function $f\left( \upsilon ,\xi ,\tau \right) $ in the dimensionless form[@Cercignani:1969] $$\mathcal{Q}_{\upsilon }f+\text{Kn}\left( \frac{\partial f}{\partial \tau }+\frac{\partial }{\partial \xi }\upsilon f\right) =0, \label{ab2}$$where $\upsilon =v/w$ being the dimensionless velocity, $\tau =t/T$ is the dimensionless time, $-\mathcal{Q}_{\upsilon }$ is the linearized collision operator and the small parameter Kn is so-called Knudsen number $$\text{Kn}=\frac{t_{\lambda }}{T}=\frac{\lambda }{L}\ll 1. \label{ab3}$$
The analogy between problem (\[nd4\])-(\[nd9\]) and the relevant problem for Eq. (\[ab2\]) appeared to be striking. Simple comparison of Knudsen number (\[ab3\]) with relaxation parameter (\[nd11\]) shows that the mean free time $t_{\lambda }$ corresponds to the characteristic transversal time $t_{tr}$. The value $w_{tr}=D/r_{M}$ may be treated as a typical transversal diffusion velocity and, therefore, $r_{M}$ corresponds to the mean free path $\lambda $. In both cases the perturbation operators describe the particles transport (with the relevant local fluxes for particles diffusion $-\left( \partial /\partial \xi \right) f$ and for particles flow $\upsilon f$) and unperturbed operators $\mathcal{L}_{\rho }$ and $-\mathcal{Q}_{\upsilon }$ are in spectrum. Namely for $-\mathcal{Q}_{\upsilon }$ number $\lambda _{0}=0$ is a degenerate eigenvalue with five associated eigenfunctions, meanwhile for $\mathcal{L}_{\rho }$ there is only one eigenfunction associated with one trivial eigenvalue. That is why in both cases for the zeroth-order outer approximation we can derive only equations for the corresponding projections on the above eigenfunctions (see ([uva5]{})). Moreover, one can see that so-called Hilbert asymptotic solution (ideal liquid approximation) of Eq. (\[ab2\]) entirely corresponds to the FJA $u_{0}^{\left( 0\right) }$ and the normal region[@Cercignani:1969] is nothing more than $\Omega ^{\left( 0\right) }$. In the kinetic theory as in the problem under study these terms, however, cannot describe the solutions into initial and boundary layers which naturally arise in both cases as well.[@Cercignani:1969]
It is well known that the classical Chapman-Enskog method is widely used to reduce the Boltzmann kinetic equation to appropriate hydrodynamic and transport equations. Noteworthy that if we consider higher-order approximations of the Chapman-Enskog method, we obtain differential equations of higher order. Nevertheless, it is long known that the Chapman-Enskog expansion can bring in solutions, which are nonexistent. In order to overcome these difficulties the method of matching inner and outer expansions was also applied in kinetic theory.[@Cercignani:1969] It seems that the Kalnay-Percus mapping approach[Kalnay:2005,Kalnay2:2005,Kalnay:2006]{} resembles some features of the Chapman-Enskog method, but this question needs to be investigated.
However, the analogy at issue is limited. The reflecting boundary condition (\[nd8\]) plays an essential role in the diffusion problem (\[nd4\])-([nd9]{}). Exactly due to this condition in contrast to the unperturbed operator of kinetics theory $-\mathcal{Q}_{\upsilon }$, operator $\mathcal{L}_{\rho }$ (\[nd5\]) is not self-adjoint (see Appendix). Nevertheless, detected analogy enables us to elucidate a number of the features inherent in the asymptotic solution of the diffusion problem (\[nd4\])-(\[nd9\]) at small values of the relaxation parameter $\epsilon $.
Concluding remarks
==================
By means of matched asymptotic expansions approach we gained here the uniformly valid leading-term approximation (\[uva3\]) to solution of the 3D diffusion problem (\[nd4\])-(\[nd9\]) with respect to small relaxation parameter of the tube $\epsilon $ (\[nd11\]).
Suggested here derivation elucidates the mathematical sense of the Fick-Jacobs equation as the solvability condition of the lowest order ([t15]{}) for the first correction in $\epsilon $ to the Fick-Jacobs approximation. Asymptotic solution also shows that known quasi-cylindrical condition $\left\vert R^{^{\prime }}\left( \xi \right) \right\vert \ll 1$ is not necessary for validity of the leading-term approximation including Fick-Jacobs approximation. At the same time matching procedure automatically gave us an exact algorithm for determination of the missing initial and boundary conditions which must be imposed on the Fick-Jacobs approximation and its corrections.
The explicit form of the leading-term approximation allowed us to delineate the conditions on temporal and spatial scales (\[uva9\]) under which the Fick-Jacobs approximation is valid.
One of the most noteworthy features of all previously suggested zeroth-order corrections to the Fick-Jacobs approximation is the absence of dependence on the transverse coordinates. However, as we have shown, even the leading-term approximation comprises the boundary layers solutions explicitly depending on the transversal coordinate. We also proved that the outer approximation $u^{\left( 0\right) }\left( \rho ,\xi ,\tau \right) $ starting from orders $\mathcal{O}\left( \epsilon \right) $ contains terms explicitly depending on the transversal variable $\rho $ (see (\[t16\])).
A profound analogy between the problem under consideration and the method of inner-outer expansions for low Knudsen numbers gas kinetic theory is established. This analogy enables us to clarify the physical and mathematical meaning of the obtained results.
It is important to underline that contrary to other known approaches our derivation of the Fick-Jacobs equation was implemented straightforwardly within the scope of asymptotic method procedure without any additional assumptions. In this connection we believe that it is rather inexpedient to exploit any physical arguments during the solution of quite well posed mathematical problem.
Future extension of the present work may include the higher order in $\epsilon $ corrections to the solution considered here and also the case of tubes of other varying constraint geometry, e.g., without axial symmetry. The results obtained in this paper allow us to hope that the matched asymptotic expansions method may be successfully applied to many other problems concerning diffusion transport of pointlike particles in tubes of varing cross section. For example, the diffusion of particles undergoing the influence of interaction potential, partially penetrable boundary condition on the tube wall and its ends or diffusion equation with a source term may be considered by means of above method.
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
===============
This research has been partially supported by Le STUDIUM (Loire Valley Institute for Advanced Studies). We also personally thank Professors P. Vigny and N. Fazzalari for their interest in this study and Professor F. Piazza for useful discussions.
APPENDIX {#appendix .unnumbered}
========
For the sake of completeness we recall here some useful classical mathematical definitions and facts.
The boundary layer of a domain $\Sigma _{\tau }$ comprises the set of points from $\Sigma _{\tau }$ such that their distance to the boundary $\partial
\Sigma _{\tau }$ does not exceed some given magnitude $\delta >0$, which is called the thickness of the layer.
In theory of singular perturbed problems a function $u_{a}\left( \mathbf{x};\epsilon \right) $ is said to be an approximation to $u\left( \mathbf{x};\epsilon \right) $ uniformly valid in a domain $\Lambda \subset \mathbb{R}^{n}$ to order $\mathcal{O}\left( \zeta \left( \epsilon \right) \right) $ as $\epsilon \rightarrow 0$ if $$\lim_{\epsilon \rightarrow 0}\frac{\left\vert u\left( \mathbf{x};\epsilon
\right) -u_{a}\left( \mathbf{x};\epsilon \right) \right\vert }{\zeta \left(
\epsilon \right) }=0 \label{con1}$$uniformly for all $\mathbf{x}\in \Lambda $.[@Lagerstrom:1988] Here $\zeta \left( \epsilon \right) $ is so-called a gauge function.
Let us introduce the space $\mathcal{H}_{\xi }$ of twice continuously differentiable real-valued functions $v:\left( 0,R\left( \xi \right) \right)
\rightarrow \mathbb{R}_{+}$ given on the cross section of the tube $0<\rho
<R\left( \xi \right) $ at any fixed point $\left( \xi ,\tau \right) $. Additionally we assume that functions $v\in \mathcal{H}_{\xi }$ obey the Neumann boundary conditions $$\left. v\right\vert _{\rho =0}<\infty ,\qquad \left. \frac{\partial v}{\partial \rho }\right\vert _{\rho =0}=0, \label{Hs0a}$$$$\left. \frac{\partial v}{\partial \rho }\right\vert _{\rho =R\left( \xi
\right) }=0 \label{Hs0b}$$at fixed point $\left( \xi ,\tau \right) $.
Then for any two functions $f$, $g\in \mathcal{H}_{\xi }$ we can introduce the weighted $L_{\rho }^{2}$ scaler product with the weight function $\rho $ as $$\left\langle f,g\right\rangle _{\mathcal{H}_{\xi
}}:=\int\limits_{0}^{R\left( \xi \right) }\rho f\left( \rho \right) g\left(
\rho \right) d\rho . \label{Hs1}$$One can show that this defines the weighted Hilbert space $\mathcal{H}_{\xi
}:=L_{\rho }^{2}\left( \left( 0,R\left( \xi \right) \right) \right) $ with the norm [@Rektorys:1977] $$\left\Vert f\right\Vert _{\mathcal{H}_{\xi }}=\left[ \int\limits_{0}^{R\left( \xi \right) }\rho f^{2}\left( \rho \right) d\rho \right]
^{1/2}<\infty .$$
Consider the linear operator $\mathcal{L}_{\rho }:\mathcal{H}_{\xi
}\rightarrow C\left( 0,R\left( \xi \right) \right) $ defined by (\[nd5\]). One can see that operator $\mathcal{L}_{\rho }$ is self-adjoint in $\mathcal{H}_{\xi }$, that is$$\left\langle \mathcal{L}_{\rho }f,g\right\rangle _{\mathcal{H}_{\xi
}}=\left\langle f,\mathcal{L}_{\rho }g\right\rangle _{\mathcal{H}_{\xi }}.
\label{Hs2a}$$Hence there exists the nontrivial solution of the eigenvalue problem $$\mathcal{L}_{\rho }v=\lambda ^{2}v \label{Hs3}$$for $v\in \mathcal{H}_{\xi }$ under the Neumann boundary conditions ([Hs0a]{}) and (\[Hs0b\]).
It has real pure-point spectrum of eigenvalues $\left\{ \lambda _{k}\right\}
_{k=0}^{\infty }$ such that for all $k\geq 0$ we have the ordering $$0\leq \lambda _{0}<\lambda _{1}<...<\lambda _{k}<...$$at that $\lambda _{k}\rightarrow \infty $ as $k\rightarrow \infty $.
The associated eigenfunctions $v_{k}\in \mathcal{H}_{\xi }$ of the problem (\[Hs3\]) are $$v_{k}:=J_{0}\left( \lambda _{k}\frac{\rho }{R\left( \xi \right) }\right) .
\label{Hs3a}$$Here $J_{\nu }\left( \zeta \right) $ is Bessel’s function of the first kind of order $\nu $ which may be defined by its Maclaurin series[Arfken:2001]{}$$J_{\nu }\left( \zeta \right) =\sum\limits_{m=0}^{\infty }\frac{\left(
-1\right) ^{m}}{\Gamma \left( m+\nu +1\right) m!}\left( \frac{\zeta }{2}\right) ^{2m+\nu }, \label{Hs3b}$$where $\Gamma \left( \beta \right) $ is the gamma function.
Thus the eigenvalues $\lambda _{k}$ of the eigenvalue problem (\[Hs3\]) are determined by the transcendental equation $$J_{0}^{^{\prime }}\left( \lambda _{k}\right) =0 \label{Hs4}$$which follows from the Neumann condition (\[Hs0b\]). Taking advantage of the known relation $J_{0}^{^{\prime }}\left( \zeta \right) =-J_{1}\left(
\zeta \right) $ we infer that $\lambda _{k}$ are also the roots of the transcendental equation $$J_{1}\left( \lambda _{k}\right) =0. \label{Hs4a}$$It follows from expansion (\[Hs3b\]) and Eq. (\[Hs4a\]) that $\lambda
_{0}=0$ and $v_{0}=J_{0}\left( 0\right) =1$. One can see that for $\lambda
_{0}=0$ there is only one linear independent eigenfunction $v_{0}\equiv const
$.
We have and other eigenvalues , e.g. [@Abram:1972] $$\lambda _{1}\approx 3.8317,\text{ }\lambda _{2}\approx 7.0156,\text{ }\lambda _{3}\approx 10.1735,...\text{ }$$Eigenfunctions $\left\{ v_{k}\right\} _{k=0}^{\infty }$ form a complete orthogonal system in $\mathcal{H}_{\xi }$ and the orthogonality property for them holds[@Arfken:2001] $$\left\langle v_{k},v_{m}\right\rangle _{\mathcal{H}_{\xi }}=\delta
_{km}\left\Vert v_{k}\right\Vert _{\mathcal{H}_{\xi }}^{2}, \label{Hs5}$$where $\delta _{km}$ is the Kronecker delta and $$\left\Vert v_{k}\right\Vert _{\mathcal{H}_{\xi }}^{2}=\frac{1}{2}R^{2}\left(
\xi \right) J_{0}^{2}\left( \lambda _{k}\right) . \label{Hs5a}$$The corresponding orthonormal system $\left\{ \widehat{v}_{k}\right\}
_{k=0}^{\infty }$ is defined by $$\widehat{v}_{k}=v_{k}/\left\Vert v_{k}\right\Vert _{\mathcal{H}_{\xi }}.
\label{Hs6}$$
For any $\lambda _{k}$ there is only one normed eigenfunction $\widehat{v}_{k}\in \mathcal{H}_{\xi }$ therefore for any function $f\in L_{\rho
}^{2}\left( \left( 0,R\left( \xi \right) \right) \right) $ we have the Fourier series with respect to orthonormal basis $\left\{ \widehat{v}_{k}\right\} _{k=0}^{\infty }$, that is$$f=\sum\limits_{k=0}^{\infty }c_{k}\widehat{v}_{k}, \label{Hs6a}$$where $c_{k}=\left\langle f,\widehat{v}_{k}\right\rangle $. It is known that series (\[Hs6a\]) converges in $\mathcal{H}_{\xi }$, so the set of orthonormal functions $\left\{ \widehat{v}_{k}\right\} _{k=0}^{\infty }$ is complete.[@Rektorys:1977]
Note that existence of the trivial eigenvalue $\left\{ \lambda
_{0}=0\right\} $ for (\[Hs3\]) leads to a serious complication for solution of Eq. (\[nd4\]). Operator $\mathcal{L}_{\rho }$ is termed to be in spectrum if there is at least one trivial eigenvalue belongning to the spectrum of $\mathcal{L}_{\rho }$.[@Vishik:1960]
Consider a linear subspace $V$ of the Hilbert space $\mathcal{H}_{\xi }$ spanned on the zero eigenfunction $v_{0}$, that is $V=\left\{ v\in
V:v=\alpha v_{0},\alpha \in \mathbb{R}\right\} $. It is well known that there exists space $V^{\bot }$ orthogonal to $V$ such that $\left( V^{\bot
}\right) ^{\perp }=V$ and $\mathcal{H}_{\xi }=V\oplus V^{\perp }$. So we can define the linear orthogonal projection operator (projector) $\mathcal{P}_{\xi }:\mathcal{H}_{\xi }\rightarrow V$ that maps any $w\in \mathcal{H}_{\xi }$ to $v\in V$ is called the orthogonal projection onto $V$, i.e.$$\mathcal{P}_{\xi }w=\left\langle w,v\right\rangle _{\mathcal{H}_{\xi }}v.
\label{Hs6b}$$In this paper it is convenient to define the projector as follows: $$\mathcal{P}_{\xi }w=\left\langle w,\widehat{v}_{0}\right\rangle _{\mathcal{H}_{\xi }}\widehat{v}_{0}, \label{Hs7}$$where $\widehat{v}_{0}=\widehat{J}_{0}\left( 0\right) =\sqrt{2}/R\left( \xi
\right) $.
We observe here that simple Neumann condition (\[Hs0b\]) arises in the simplified problems corresponding to diffusion boundary layers. In general case of functions $f$ and $g$ obeying the reflecting boundary condition ([nd8]{}) the unperturbed operator $\mathcal{L}_{\rho }$ (\[nd5\]) is not self-adjoint, i.e. $$\left\langle \mathcal{L}_{\rho }f,g\right\rangle _{\mathcal{H}_{\xi }}\neq
\left\langle f,\mathcal{L}_{\rho }g\right\rangle _{\mathcal{H}_{\xi }}.
\label{Hs2}$$The latter property of the unperturbed operator $\mathcal{L}_{\rho }$ greatly complicates the original boundary value problem (\[nd4\])-([nd9]{}).
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[^1]: Electronic mail: sergtray@mail.ru
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The capacity of cellular networks can be improved by the unprecedented array gain and spatial multiplexing offered by Massive MIMO. Since its inception, the coherent interference caused by pilot contamination has been believed to create a finite capacity limit, as the number of antennas goes to infinity. In this paper, we prove that this is incorrect and an artifact from using simplistic channel models and suboptimal precoding/combining schemes. We show that with multicell MMSE precoding/combining and a tiny amount of spatial channel correlation or large-scale fading variations over the array, the capacity increases without bound as the number of antennas increases, even under pilot contamination. More precisely, the result holds when the channel covariance matrices of the contaminating users are asymptotically linearly independent, which is generally the case. If also the diagonals of the covariance matrices are linearly independent, it is sufficient to know these diagonals (and not the full covariance matrices) to achieve an unlimited asymptotic capacity.'
author:
-
bibliography:
- 'IEEEabrv.bib'
- 'ref.bib'
title: Massive MIMO Has Unlimited Capacity
---
Massive MIMO, ergodic capacity, asymptotic analysis, spatial correlation, multi-cell MMSE processing, pilot contamination.
Introduction {#sec-intro}
============
The Shannon capacity of a channel manifests the spectral efficiency (SE) that it supports. Massive MIMO (multiple-input multiple-output) improves the sum SE of cellular networks by spatial multiplexing of a large number of user equipments (UEs) per cell [@marzetta2010noncooperative]. It is therefore considered a key time-division duplex (TDD) technology for the next generation of cellular networks [@Larsson2014; @Andrews2014a; @Larsson2017a]. The main difference between Massive MIMO and classical multiuser MIMO is the large number of antennas, $M$, at each base station (BS) whose signals are processed by individual radio-frequency chains. By exploiting channel estimates for coherent receive combining, the uplink signal power of a desired UE is reinforced by a factor $M$, while the power of the noise and independent interference does not increase. The same principle holds for the transmit precoding in the downlink. Since the channel estimates are obtained by uplink pilot signaling and the pilot resources are limited by the channel coherence time, the same pilots must be reused in multiple cells. This leads to pilot contamination which has two main consequences: the channel estimation quality is reduced due to pilot interference and the channel estimate of a desired UE is correlated with the channels to the interfering UEs that use the same pilot. Marzetta showed in his seminal paper [@marzetta2010noncooperative] that the interference from these UEs during data transmission is also reinforced by a factor $M$, under the assumptions of maximum ratio (MR) combining/precoding and independent and identically distributed (i.i.d.) Rayleigh fading channels. This means that pilot contamination creates a finite SE limit as $M \to \infty$.
The large-antenna limit has also been studied for other combining/precoding schemes, such as the minimum mean squared error (MMSE) scheme. Single-cell MMSE (S-MMSE) was considered in [@hoydis2013massive; @Guo2014a; @Krishnan2014a], while multicell MMSE (M-MMSE) was considered in [@Ngo2012b; @EmilEURASIP17]. The difference is that with M-MMSE, the BS makes use of estimates of the channels from the UEs in all cells, while with S-MMSE, the BS only uses channel estimates of the UEs in the own cell. In both cases, the SE was proved to have a finite limit as $M \to \infty$, under the assumption of i.i.d. Rayleigh fading channels (i.e., no spatial correlation). In contrast, there are special cases of spatially correlated fading that give rise to rank-deficient covariance matrices [@Yin2013a; @Adhikary2013; @You2015a]. If the UEs that share a pilot have rank-deficient covariance matrices with orthogonal support, then pilot contamination vanishes and the SE can increase without bound. The covariance matrices ${\mathbf{R}}_1$ and ${\mathbf{R}}_2$ have orthogonal support if ${\mathbf{R}}_1 {\mathbf{R}}_2 = {\mathbf{0}}$. To understand this condition, note that for arbitrary covariance matrices $$\label{eq:simple-covariance-matrices}
{\mathbf{R}}_1 = \begin{bmatrix} a & c \\ c^\star & b \end{bmatrix} \quad {\mathbf{R}}_2 = \begin{bmatrix} d & f \\ f^\star & e \end{bmatrix}$$ every element of ${\mathbf{R}}_1 {\mathbf{R}}_2 $ must be zero. The first element is $ad+cf^\star$. If we model the practical covariance matrices of two randomly located UEs as realizations of a random variable with continuous distribution, then $ad+cf^\star=0$ occurs with zero probability.[^1] Hence, orthogonal support is very unlikely in practice, although one can find special cases where it is satisfied. The one-ring model for uniform linear arrays (ULAs) gives orthogonal support if the channels have non-overlapping angular support [@Yin2013a; @Adhikary2013; @You2015a], but the ULA microwave measurements in [@Gao2015a] show that the angular support of practical channels is highly irregular and does not lead to orthogonal support. In conclusion, practical covariance matrices do not have orthogonal support, at least not at microwave frequencies.
The literature contains several categories of methods for mitigation of pilot contamination, also known as *pilot decontamination*. The first category allocates pilots to the UEs in an attempt to find combinations where the covariance matrices have relatively different support [@Yin2013a; @Adhikary2013; @Li2013a; @You2015a]. This method can substantially reduce pilot contamination, but can only remove the finite limit in the unlikely special case when the covariance matrices have orthogonal support. The second category utilizes semi-blind estimation to separate the subspace of desired UE channels from the subspace of interfering channels [@Ngo2012a; @Mueller2014b; @Hu2016a; @Yin2016a; @Vinogradova2016a]. This method can fully remove pilot contamination if $M$ and the size of the channel coherence block go jointly to infinity [@Yin2016a]. Unfortunately, the channel coherence is fixed and finite in practice (this is why we cannot give unique pilots to every cell), thus we cannot approach this limit in practice. The third category uses multiple pilot phases with different pilot sequences to successively eliminate pilot contamination [@Zhang2014a; @Vu2014a], without the need for statistical information. However, the total pilot length is larger or equal to the total number of UEs, which would allow allocating mutually orthogonal pilots to all UEs and thus trivially avoiding the pilot contamination problem. This is not a scalable solution for networks with many cells. The fourth category is pilot contamination precoding that rejects interference by coherent joint transmission/reception over the entire network [@ashikhmin2012pilot; @Li2013b]. This method appears to achieve an unbounded SE, but this has not been formally proved and requires that the data for all UEs is available at every BS, which might not be feasible in practice.
In summary, it appears that pilot contamination is a fundamental issue that manifests a finite SE limit, except in unlikely special cases. We show in this paper that this is basically a misunderstanding, spurred by the popularity of analyzing suboptimal combining/precoding schemes, such as MR and S-MMSE, and focusing on unrealistic i.i.d. Rayleigh fading channels (as in the prior work [@Ngo2012b; @EmilEURASIP17] on M-MMSE). We prove that the SE increases without bound in the presence of pilot contamination when using M-MMSE combining/precoding, if the pilot-sharing UEs have asymptotically linearly independent covariance matrices. Note that ${\mathbf{R}}_1$ and ${\mathbf{R}}_2$ in are linearly independent if $[a \, b \,c]^{\Ttran}$ and $[d \, e \,f]^{\Ttran}$ are non-parallel vectors, which happens almost surely for randomly generated covariance matrices. Hence, our results rely on a condition that is most likely satisfied in practice—it is the *general case*, while prior works on the asymptotics of Massive MIMO have considered practically unlikely special cases. In contrast to prior work, no multicell cooperation is utilized herein and there is no need for orthogonal support of covariance matrices. In the conference paper [@Bjornson2017a], we proved the main result in a two-user uplink scenario.[^2] In this paper, we prove the result for both uplink and downlink in a general setting. Section \[section:two-user\] proves and explains the intuition of the results in a two-user setup, while Section \[section:multi-user\] generalizes the results to a multicell setup. The results are demonstrated numerically in Section \[sec:numerical-results\] and the main conclusions are summarized in Section \[section:conclusion\].
### Notation {#notation .unnumbered}
The Frobenius and spectral norms of a matrix ${\mathbf{X}}$ are denoted by $\| {\mathbf{X}} \|_F$ and $\| {\mathbf{X}} \|_2$, respectively. The superscripts $^{\Ttran}$, $^\star$ and $^{\Htran}$ denote transpose, conjugate, and Hermitian transpose, respectively. We use $\triangleq$ to denote definitions, whereas $\CN({\bf 0},{\bf R})$ denotes the circularly symmetric complex Gaussian distribution with zero mean and covariance matrix ${\bf R}$. The expected value of a random variable $x$ is denoted by $\mathbb{E}\{ x \}$ and the variance is denoted by $\mathbb{V}\{ x \}$. The $N \times N$ identity matrix is denoted by ${\mathbf{I}}_N$, while ${\mathbf{0}}_N$ is an $N \times N$ all-zero matrix and ${\mathbf{1}}_N$ is an $N \times 1$ all-one vector. We use $a_n \asymp b_n$ to denote $a_n -b_n \to_{n\to \infty}0$ (almost surely (a.s.)) for two (random) sequences $a_n$, $b_n$.
Asymptotic Spectral Efficiency in a Two-User Scenario {#section:two-user}
=====================================================
In this section, we prove and explain our main result in a two-user scenario, where a BS equipped with $M$ antennas communicates with UE $1$ and UE $2$ that are using the same pilot. This setup is sufficient to demonstrate why M-MMSE combining and precoding reject the coherent interference caused by pilot contamination. We consider a block-fading model where each channel takes one realization in a coherence block of $\tau_c$ channel uses and independent realizations across blocks. We denote by ${\bf h}_{k} \in \mathbb{C}^{M}$ the channel from UE $k$ to the BS and consider Rayleigh fading with ${\mathbf{h}}_{k} \sim \CN \left( {\mathbf{0}}, {\mathbf{R}}_{k} \right)$ for $k=1,2$, where ${{\mathbf{R}}_{k} \in \mathbb{C}^{M\times M}}$ with[^3] $\tr ({\mathbf{R}}_{k} ) > 0$ is the channel covariance matrix, which is assumed to be known at the BS. The Gaussian distribution models the small-scale fading whereas the covariance matrix ${\mathbf{R}}_{k}$ describes the macroscopic effects. The normalized trace ${\beta_{k}= \frac{1}{M} \tr \left( {\mathbf{R}}_{k} \right)}$ determines the average large-scale fading between UE $k$ and the BS, while the eigenstructure of ${\mathbf{R}}_{k}$ describes the spatial channel correlation. A special case that is convenient for analysis is i.i.d. Rayleigh fading with ${{\mathbf{R}}_{k} = \beta_{k} {\mathbf{I}}_{M}}$ [@Marzetta2016a], but it only arises in fully isotropic fading environments. In general, each covariance matrix has spatial correlation and large-scale fading variations over the array, represented by non-zero off-diagonal elements and non-identical diagonal elements, respectively.
Uplink Channel Estimation
-------------------------
We assume that the BS and UEs are perfectly synchronized and operate according to a TDD protocol wherein the data transmission phase is preceded by an uplink pilot phase for channel estimation. Both UEs use the same $\taupu$-length pilot sequence $\bphiu \in \mathbb{C}^{\taupu}$ with elements such that $\| \bphiu \|^2 = \bphiu^{\Htran} \bphiu = {1}$. The received uplink signal ${\mathbf{Y}}^{p} \in \mathbb{C}^{N\times \taupu}$ at the [BS]{} is given by $$\begin{aligned}
{\mathbf{Y}}^{p}= \sqrt{\rho^{\rm{tr}}} {\mathbf{h}}_{1} \bphiu^{\Ttran} + \sqrt{\rho^{\rm{tr}}} {\mathbf{h}}_{2} \bphiu^{\Ttran} + {\mathbf{N}}^{p}\end{aligned}$$ where $\rho^{\rm{tr}}$ is the normalized pilot power and ${\mathbf{N}}^{p} \in \mathbb{C}^{N\times \taupu}$ is the normalized receiver noise with all elements independently distributed as $\CN(0,1)$. The matrix ${\mathbf{Y}}^{p}$ is the observation that the [BS]{} utilizes to estimate ${\bf h}_1$ and ${\bf h}_2$. We assume that channel estimation is performed using the [MMSE]{} estimator given in the next lemma (the proof relies on standard estimation theory [@Kay_Book]).
\[theorem:MMSE-estimate\_h\_jli\] The [MMSE]{} estimator of ${\mathbf{h}}_{k}$ for $k=1,2$, based on the observation ${\mathbf{Y}}^{p} $ at the [BS]{}, is $$\label{eq:MMSEestimator_h}
\begin{split}
\!\!\hat{{\mathbf{h}}}_{k} = \frac{1}{ \sqrt{\rho^{\rm{tr}}} }{\mathbf{R}}_{k}
{\bf{Q}}^{-1} {\mathbf{Y}}^{p} \bphiu^{\star}
\end{split}$$ with ${\bf{Q}} = \frac{1}{\rho^{\rm{tr}}} \mathbb{E}\{ {\mathbf{Y}}^{p} \bphiu^{\star} ( {\mathbf{Y}}^{p} \bphiu^{\star} )^{\Htran} \} = {\mathbf{R}}_{1} + {\mathbf{R}}_{2} + \frac{1}{ \rho^{\rm{tr}}} {\mathbf{I}}_{M}$ being the normalized covariance matrix of the observation after correlating with the pilot sequence. The estimate $\hat{{\mathbf{h}}}_{k} $ and the estimation error $\tilde{{\mathbf{h}}}_{k}= {\mathbf{h}}_{k} - \hat{{\mathbf{h}}}_{k}$ are independent random vectors distributed as $\hat{{\mathbf{h}}}_{k} \sim \CN({\bf 0},{\mathbf{\Phi}}_{k})$ and $\tilde{{\mathbf{h}}}_{k} \sim \CN({\bf 0},{\mathbf{R}}_{k} - {\mathbf{\Phi}}_{k})$ with ${\mathbf{\Phi}}_{k} = {\mathbf{R}}_{k}
{\bf{Q}}^{-1} {\mathbf{R}}_{k}$.
Interestingly, the estimates $\hat {\bf h}_1$ and $\hat {\bf h}_2$ are computed in an almost identical way in : the same matrix ${\bf{Q}}$ is inverted and multiplied with the same observation ${\mathbf{Y}}^{p} \bphiu^{\star}/\sqrt{\rho^{\rm{tr}}} $. The only difference is that for $\hat {\bf h}_k$ there is a multiplication with the UE’s own channel covariance matrix ${\mathbf{R}}_{k}$ in , for $k=1,2$. The channel estimates are thus correlated with correlation matrix $
{\mathbf{\Upsilon}}_{12} = {\mathbb{E}}\{\hat{{\mathbf{h}}}_{1}\hat{{\mathbf{h}}}_{2}^{\Htran}\} = {\mathbf{R}}_{1}
{\bf{Q}}^{-1} {\mathbf{R}}_{2}$. If ${\mathbf{R}}_{1}$ is invertible, then we can also write the relation between the estimates as $\hat{{\mathbf{h}}}_{2} = {\mathbf{R}}_{2} {\mathbf{R}}_{1}^{-1} \hat{{\mathbf{h}}}_{1}$. In the special case of i.i.d. fading channels with ${\mathbf{R}}_{1}=\beta_{1} {\mathbf{I}}_{M}$ and ${\mathbf{R}}_{2}= \beta_{2} {\mathbf{I}}_{M}$, the two channel estimates are parallel vectors that only differ in scaling: $\hat{{\mathbf{h}}}_{2} = \frac{\beta_{2}}{\beta_{1}} \hat{{\mathbf{h}}}_{1}$. This is an unwanted property caused by the inability of the [BS]{} to separate UEs that have transmitted the same pilot sequence over channels that are identically distributed (up to a scaling factor). In the alternative special case of ${\mathbf{R}}_{1} {\mathbf{R}}_{2} = {\mathbf{0}}_M$, the two [UE]{} channels are located in orthogonal subspaces (i.e., have orthogonal support), which leads to zero correlation: ${\mathbf{\Upsilon}}_{12}= {\mathbf{0}}_M$. Consequently, it is theoretically possible to let two UEs share a pilot sequence without causing pilot contamination, if their covariance matrices satisfy the orthogonality condition ${\mathbf{R}}_{1} {\mathbf{R}}_{2} = {\mathbf{0}}_M$. As described in Section \[sec-intro\], none of these special cases occur in practice, therefore we will develop a general way to deal with the correlation of channel estimates caused by pilot contamination.
Uplink Data Transmission
------------------------
During uplink data transmission, the received baseband signal at the BS is ${\bf y} \in \mathbb{C}^{M}$, given by $
{\mathbf{y}}= \sqrt{\rho^{\rm {ul}}} {\mathbf{h}}_{1} s_1 + \sqrt{\rho^{\rm {ul}}} {\mathbf{h}}_{2} s_2 + {\mathbf{n}}
$, where $s_k\sim\CN(0,1)$ is the information-bearing signal transmitted by UE $k$, ${\mathbf{n}}\sim \CN({\mathbf{0}},{\bf I}_M)$ is the independent receiver noise, and $\rho^{\rm {ul}}$ is the normalized transmit power. The BS detects the signal from UE $1$ by using a combining vector ${\mathbf{v}}_1\in \mathbb{C}^{M}$ to obtain ${\mathbf{v}}_1^{\Htran}{\mathbf{y}}$. Using a standard technique (see, e.g., [@hoydis2013massive; @Marzetta2016a]), the ergodic uplink capacity of UE $1$ is lower bounded by $$\begin{aligned}
\label{eq:SE-uplink-twousers}
\mathsf{SE}_{1}^{\rm {ul}} = \left( 1 - \frac{\taupu}{\tau_c} \right) \mathbb{E} \left\{ \log_2 \left( 1 + \gamma_{1}^{\rm {ul}} \right) \right\} \quad \textrm{[bit/s/Hz] }\end{aligned}$$ where the expectation is with respect to the channel estimates. We refer to $\mathsf{SE}_{1}^{\rm {ul}}$ as an achievable SE. The instantaneous effective signal-to-interference-and-noise ratio (SINR) $\gamma_{1}^{\rm {ul}}$ in is
$$\begin{aligned}
\nonumber
\gamma_{1}^{\rm {ul}} &= \frac{ | {\mathbf{v}}_{1}^{\Htran} \hat{{\mathbf{h}}}_{1} |^2 }{{\mathbb{E}}\left\{ | {\mathbf{v}}_{1}^{\Htran} \tilde{{\mathbf{h}}}_{1} |^2 +
| {\mathbf{v}}_{1}^{\Htran} {{\mathbf{h}}}_{2} |^2
+ \frac{1}{\rho^{\rm {ul}}}{\mathbf{v}}_{1}^{\Htran}{\mathbf{v}}_{1}
\Big| \hat{\bf{h}}_{1},\hat{\bf{h}}_{2} \right\}}\\ &= \frac{ | {\mathbf{v}}_{1}^{\Htran} \hat{{\mathbf{h}}}_{1} |^2 }{
{\mathbf{v}}_{1}^{\Htran} \left( \hat{{\mathbf{h}}}_{2} \hat{{\mathbf{h}}}_{2}^{\Htran} + {\mathbf{Z}} \right) {\mathbf{v}}_{1} } \label{eq:gamma1}\end{aligned}$$
with $
{\mathbf{Z}} = \sum_{k=1}^{2} ({\mathbf{R}}_{k} - {\mathbf{\Phi}}_{k}) + \frac{1}{\rho^{\rm {ul}}} {\mathbf{I}}_M.
$ Since $\gamma_{1}^{\rm{ul}}$ is a generalized Rayleigh quotient, the SINR is maximized by [@Ngo2012b; @EmilEURASIP17] $$\begin{aligned}
\label{v_k_MMSE}
{\mathbf{v}}_1= \left( \sum_{k=1}^2\hat{{\mathbf{h}}}_{k} \hat{{\mathbf{h}}}_{k}^{\Htran} + {\mathbf{Z}} \right)^{-1} \hat{{\mathbf{h}}}_{1}.\end{aligned}$$ This is called MMSE combining since not only maximizes the instantaneous [SINR]{} $\gamma_{1}^{\rm {ul}}$, but also minimizes ${\mathbb{E}}\{ |x_{1} - {\mathbf{v}}_{1}^{\Htran} {\bf y} |^2 \, |{{\hat{\bf h}}_{1}},{{\hat{\bf h}}_{2}}\}$ which is the mean squared error (MSE) in the data detection (conditioned on the channel estimates). Plugging into yields $$\begin{aligned}
\label{eq:gamma1_MMSE}
\gamma_{1}^{\rm {ul}} &= \hat{{\mathbf{h}}}_{1}^{\Htran}\left( \hat{{\mathbf{h}}}_{2} \hat{{\mathbf{h}}}_{2}^{\Htran} + {\bf Z}\right)^{-1} \hat{{\mathbf{h}}}_{1}.\end{aligned}$$ We will now analyze the asymptotic behavior of $\mathsf{SE}_{1}^{\rm {ul}}$ and $\gamma_{1}^{\rm{ul}}$ as $M\to \infty$. To this end, we make the following technical assumptions:
\[assumption\_1\] For $k=1,2$, $\mathop {\liminf}\limits_M\frac{1}{{M}}\tr ( {\mathbf{R}}_{k} ) > 0$ and $ \mathop {\limsup}\limits_M \| {\mathbf{R}}_{k}\|_2 < \infty$.
\[assumption\_2\] For $\boldsymbol{\lambda} = [\lambda_1, \lambda_2]^{\Ttran} \in \mathbb{R}^2$ and $i=1,2$, $$\begin{aligned}
\label{eq:assumption_2_relaxed}
\mathop {\liminf}\limits_M \inf_{\{\boldsymbol{\lambda}: \, \lambda_i=1\}} \frac{1}{{M}} \left\| \lambda_1 {\mathbf{R}}_{1} + \lambda_2 {\mathbf{R}}_{2} \right\|_F^2 > 0.\end{aligned}$$
The first assumption is a well established way to model that the array gathers more energy as $M$ increases and also that this energy originates from many spatial dimensions [@hoydis2013massive]. In particular, it is a sufficient condition for asymptotic channel hardening; that is, $\| {\mathbf{h}}_{k} \|^2/ \mathbb{E}\{ \| {\mathbf{h}}_{k} \|^2 \} \to 1$ in probability as $M \to \infty$. The second assumption requires ${\mathbf{R}}_1$ and ${\mathbf{R}}_2$ to be *asymptotically* linearly independent, in the sense that if one of the matrices is scaled to resemble the other one, the subspace in which the matrices differ has an energy proportional to $M$. Note that this is a stronger condition than linear independence, defined as $\inf_{\{\boldsymbol{\lambda}: \, \lambda_i=1\}} \| \lambda_1 {\mathbf{R}}_{1} + \lambda_2 {\mathbf{R}}_{2} \|_F^2 > 0$ for $i=1,2$, which is satisfied even if the matrices only differ in one element. We will elaborate further on Assumption \[assumption\_2\] in Section \[subsec:interpretation\].
The following is the first of the main results of this paper:
\[theorem:MMSE\] If MMSE combining is used, then under Assumptions \[assumption\_1\] and \[assumption\_2\], the instantaneous effective SINR $\gamma_{1}^{\rm{ul}} $ increases a.s. unboundedly as $M\to \infty$. Hence, $\mathsf{SE}_{1}^{\rm {ul}}$ increases unboundedly as $M\to \infty$.
The proof is given in Appendix B.
\[rem:2-UE-UL\] From the proof in Appendix B, we can see that $\gamma_{1}^{\rm{ul}}/M$ has a non-zero asymptotic limit, which implies that the SE grows towards infinity as $\log_2(M)$. While Theorem \[theorem:MMSE\] only considers UE 1, one only needs to interchange the UE indices to prove that the SE of UE 2 also grows unboundedly as $M\to \infty$. Hence, an unlimited asymptotic SE is simultaneously achievable for both UEs. Since the SE is a lower bound on capacity, we conclude that the asymptotic capacity is also unlimited.
Observe that if ${\bf R}_1$ and ${\bf R}_2$ are linearly dependent, i.e., ${\mathbf{R}}_1 = \eta {\mathbf{R}}_2$, then Assumption \[assumption\_2\] does not hold. Under these circumstances, $\hat {{\mathbf{h}}}_{2} = \frac{1}{\eta}\hat{{\mathbf{h}}}_{1}$ and by applying Lemma \[MIL\] in Appendix A we obtain $$\begin{aligned}
\label{eq:gamma1_MMSE_linearly_dependent}
\gamma_{1}^{\rm {ul}} &= \frac{\hat{{\mathbf{h}}}_{1}^{\Htran}{\bf Z}^{-1}\hat{{\mathbf{h}}}_{1}}{1 + \frac{1}{\eta^2}\hat{{\mathbf{h}}}_{1}^{\Htran}{\bf Z}^{-1}\hat{{\mathbf{h}}}_{1}}\end{aligned}$$ from which, it is straightforward to show that $\gamma_1^{\rm {ul}} \asymp \eta^2$ (by dividing and multiplying each term by $M$ and using Lemma \[lemma3\] in Appendix A). This implies that $\mathsf{SE}_1^{\rm {ul}} $ converges to a finite quantity when $M\to \infty$, as Marzetta showed in his seminal paper [@marzetta2010noncooperative] for the special case of ${\bf R}_1=\eta {\bf R}_2={\bf I}_M$.
Downlink Data Transmission
--------------------------
During the downlink data transmission, the BS transmits the signal ${\bf x} \in \mathbb{C}^M$. This signal is given by ${\bf x} = \sqrt{\rho^{\rm {dl}} }{\bf w}_1\varsigma_1 + \sqrt{\rho^{\rm {dl}}}{\bf w}_2\varsigma_2$, where $\varsigma_k\sim\CN(0,1)$ is the information-bearing signal transmitted to UE $k$, $\rho^{\rm {dl}}$ is the normalized downlink transmit power, and ${\bf w}_k$ is the precoding vector associated with UE $k$. This precoding vector satisfies $\mathbb{E} \left\{\|{\bf w}_k\|^2\right\} =1$, so that $\mathbb{E} \left\{\|{\bf w}_k\varsigma_k\|^2\right\} =\rho^{\rm{dl}}$ is the downlink transmit power allocated to UE $k$. The received downlink signal $z_1$ at UE 1 is[^4] $$\begin{aligned}
\notag
z_1 &= \sqrt{\rho^{\rm {dl}}}{\mathbf{h}}_{1}^{\Htran} {{\mathbf{w}}}_{1} \varsigma_1 + \sqrt{\rho^{\rm {dl}}}{\mathbf{h}}_{1}^{\Htran} {{\mathbf{w}}}_{2} \varsigma_2+ n_1
\\\notag
&=\sqrt{\rho^{\rm {dl}}}{\mathbb{E}}\left\{{\mathbf{h}}_{1}^{\Htran} {{\mathbf{w}}}_{1}\right\}\varsigma_1 + \sqrt{\rho^{\rm {dl}}}({\mathbf{h}}_{1}^{\Htran} {{\mathbf{w}}}_{1} - {\mathbb{E}}\left\{{\mathbf{h}}_{1}^{\Htran} {{\mathbf{w}}}_{1}\right\})\varsigma_1 \\&\hspace{3.1cm}+ \sqrt{\rho^{\rm {dl}}} {\mathbf{h}}_{1}^{\Htran} {{\mathbf{w}}}_{2} \varsigma_2+ n_1\label{eq:received_signal_DL}\end{aligned}$$ where $n_1\sim \CN(0,1)$ is the normalized receiver noise. The first term in is the desired signal received over the deterministic average precoded channel ${\mathbb{E}}\left\{{\mathbf{h}}_{1}^{\Htran} {{\mathbf{w}}}_{1}\right\}$, while the remaining terms are random variables with unknown realizations. By treating these terms as noise in the signal detection [@hoydis2013massive; @Marzetta2016a], the downlink ergodic channel capacity of UE 1 can be lower bounded by $$\begin{aligned}
\label{eq:SE-downlink-twousers}
\mathsf{SE}_{1}^{\rm dl} = \left( 1 - \frac{\taupu}{\tau_c} \right) \log_2 \left( 1 + \gamma_{1}^{\rm dl} \right) \quad \textrm{[bit/s/Hz] }\end{aligned}$$ with the effective SINR $$\begin{aligned}
\label{eq:gamma1_DL}
\gamma_{1}^{\rm {dl}}
= \frac{ | {\mathbb{E}}\{{\mathbf{h}}_{1}^{\Htran} {{\mathbf{w}}}_{1} \} |^2 }{ {\mathbb{E}}\left\{| {\mathbf{h}}_{1}^{\Htran} {{\mathbf{w}}}_{2}|^2\right\} + {\mathbb{V}}\left\{ {\mathbf{h}}_{1}^{\Htran} {{\mathbf{w}}}_{1}\right\} + \frac{1}{\rho^{\rm dl}}}.\end{aligned}$$ Since UE $1$ only needs to know ${\mathbb{E}}\left\{{\mathbf{h}}_{1}^{\Htran} {{\mathbf{w}}}_{1}\right\}$ and the total variance of the second to fourth term in , the SE in is achievable in the absence of downlink channel estimation. In contrast to the uplink, there is no precoding that is always optimal [@Bjornson2014d]. However, motivated by uplink-downlink duality [@EmilEURASIP17], a reasonable suboptimal choice is the so-called MMSE precoding $$\begin{aligned}
\label{eq:Section3_precoding}
{\bf w}_k = \frac{{\mathbf{v}}_k}{\sqrt{{\mathbb{E}}\left\{\|{\mathbf{v}}_k\|^2\right\}}} = \sqrt{\vartheta_k} \left( \sum_{k=1}^2\hat{{\mathbf{h}}}_{k} \hat{{\mathbf{h}}}_{k}^{\Htran} + {\mathbf{Z}} \right)^{-1} \hat{{\mathbf{h}}}_{k}\end{aligned}$$ where ${\mathbf{v}}_k = ( \sum_{k=1}^2\hat{{\mathbf{h}}}_{k} \hat{{\mathbf{h}}}_{k}^{\Htran} + {\mathbf{Z}} )^{-1} \hat{{\mathbf{h}}}_{k}$ is MMSE combining and $\vartheta_k = ({\mathbb{E}}\left\{\|{\mathbf{v}}_k\|^2\right\})^{-1}$ is a scaling factor. The following is the second main result of this paper:
\[theorem:MMSE\_precoding\] If MMSE precoding is used, then under Assumptions \[assumption\_1\] and \[assumption\_2\] the effective SINR $\gamma_{1}^{\rm{dl}} $ increases unboundedly as $M\to \infty$. Hence, $\mathsf{SE}_{1}^{\rm {dl}}$ increases unboundedly as $M\to \infty$.
The proof is given in Appendix D.
This theorem shows that, under the same conditions as in the uplink, the downlink SE (and thus the capacity) increases without bound as $M\to \infty$. The asymptotic SE growth is proportional to $\log_2(M)$, since the proof in Appendix D shows that $\gamma_{1}^{\rm{dl}}/M$ has a non-zero asymptotic limit. UE 2 can simultaneously achieve an unbounded SE, which is proved directly by interchanging the UE indices.
Interpretation and Generality {#subsec:interpretation}
-----------------------------
Theorems \[theorem:MMSE\] and \[theorem:MMSE\_precoding\] show that the SE (and thus the capacity) under pilot contamination is asymptotically unlimited if Assumption \[assumption\_2\] holds. To gain an intuitive interpretation of this underlying assumption, recall from that $\hat{{\mathbf{h}}}_1 = {\mathbf{R}}_{1} {\mathbf{a}}$ and $\hat{{\mathbf{h}}}_2 = {\mathbf{R}}_{2} {\mathbf{a}}$, where ${\mathbf{a}}= \frac{1}{ \sqrt{\rho^{\rm{tr}}} } {\bf{Q}}^{-1} {\mathbf{Y}}^{p} \bphiu^{*} $ is the same for both UEs. Hence, $\hat{{\mathbf{h}}}_1$ and $\hat{{\mathbf{h}}}_2$ are (asymptotically) linearly independent when ${\mathbf{R}}_{1}$ and ${\mathbf{R}}_{2}$ are (asymptotically) linearly independent, except for special choices of ${\mathbf{a}}$. As illustrated in Fig. \[figureOrthogonality\], it is then possible to find a combining vector ${\mathbf{v}}_1$ (or precoding vector ${\mathbf{w}}_1$) that is orthogonal to $\hat{{\mathbf{h}}}_2$, while being non-orthogonal to $\hat{{\mathbf{h}}}_1$. Similarly, one can find ${\mathbf{v}}_2$ (and ${\mathbf{w}}_2$) such that ${\mathbf{v}}_2^{\Htran} \hat{{\mathbf{h}}}_1 = 0$ and ${\mathbf{v}}_2^{\Htran} \hat{{\mathbf{h}}}_2 \neq 0$. For example, if we define $\hat{{\mathbf{H}}} = [ \hat{{\mathbf{h}}}_1 \,\, \hat{{\mathbf{h}}}_2 ] \in \mathbb{C}^{M \times 2}$, then the zero-forcing (ZF) combining vectors $$\label{eq:basic-ZF}
\big[ {\mathbf{v}}_1 \,\, {\mathbf{v}}_2 \big]= \hat{{\mathbf{H}}} \left( \hat{{\mathbf{H}}}^{\Htran} \hat{{\mathbf{H}}} \right)^{-1}$$ satisfy these conditions. Note that $\hat{{\mathbf{H}}}^{\Htran} \hat{{\mathbf{H}}}$ is only invertible if the channel estimates (columns in $\hat{{\mathbf{H}}}$) are linearly independent. Using ZF as defined in , we get ${\mathbf{v}}_1^{\Htran} \hat{{\mathbf{h}}}_2 = 0$ and ${\mathbf{v}}_1^{\Htran} \hat{{\mathbf{h}}}_1 = 1$. If the channel estimates are also asymptotically linearly independent, it follows[^5] that $ \| {\mathbf{v}}_1 \|^2 \to 0$ as $M \to \infty$; that is, we can reject the coherent interference and get unit signal gain, while at the same time using the array gain to make the noise term $\frac{1}{\rho^{\rm {ul}}}{\mathbf{v}}_{1}^{\Htran}{\mathbf{v}}_{1} = \frac{1}{\rho^{\rm {ul}}} \| {\mathbf{v}}_1 \|^2$ vanish asymptotically. Since optimal MMSE combining (and also MMSE precoding) provides a higher SINR than the heuristic ZF scheme in , it also rejects the coherent interference while retaining an array gain that grows with $M$.
To further explain the implications of Assumption \[assumption\_2\], we provide the following three examples.
\[example1\] Consider a two-user scenario with $${\mathbf{R}}_1 = \begin{bmatrix} 2 {\mathbf{I}}_N & {\mathbf{0}} \\ {\mathbf{0}} & {\mathbf{I}}_{M-N} \end{bmatrix} \qquad {\mathbf{R}}_2 = {\mathbf{I}}_M$$ where the covariance matrices have full rank and are only different in the first $N$ dimensions. For any given $M$, we notice that the argument of for UE $i=1$ becomes $$\begin{aligned}
\notag
& \inf_{\lambda_2} \frac{1}{{M}} \| {\mathbf{R}}_{1} + \lambda_2 {\mathbf{R}}_{2} \|_F^2 \\&= \inf_{\lambda_2} \frac{N(2+\lambda_2)^2+(M-N)(1+\lambda_2)^2}{M} =\frac{(M-N)N}{M^2} \label{eq:example1}
\end{aligned}$$ where the infimum is attained by $\lambda_2 = -(M+N)/M$. Note that goes to zero as $M \to \infty$ if $N$ is constant, while it has the non-zero limit $(1-\alpha)\alpha$ if $N = \alpha M$, for some $0 < \alpha < 1$. In the latter case, the matrices $ \{ {\mathbf{R}}_1, {\mathbf{R}}_2 \}$ satisfy . Interestingly, although the covariance matrices are diagonal, they are still asymptotically linearly independent and the subspace in which they differ has rank $\min(N,M-N)= M \min(\alpha, (1-\alpha) ) $, which is proportional to $M$.
Let us further exemplify the interference rejection by considering ZF combining, which provides lower SINR than MMSE combining, but gives more intuitive expressions. Assume for the sake of simplicity that the channel realizations are such that $\frac{1}{ \sqrt{\rho^{\rm{tr}}} } {\bf{Q}}^{-1} {\mathbf{Y}}^{p} \bphiu^{*} = {\mathbf{1}}_M$, which gives $\hat{{\mathbf{h}}}_1 = {\mathbf{R}}_{1} {\mathbf{1}}_M = [2 {\mathbf{1}}_N^{\Ttran} \, \, {\mathbf{1}}_{M-N}^{\Ttran}]^{\Ttran}$ and $\hat{{\mathbf{h}}}_2 = {\mathbf{R}}_{2} {\mathbf{1}}_M = {\mathbf{1}}_M$. The ZF combining vectors are then given by $$\big[ {\mathbf{v}}_1 \,\, {\mathbf{v}}_2 \big]= \hat{{\mathbf{H}}} \left( \hat{{\mathbf{H}}}^{\Htran} \hat{{\mathbf{H}}} \right)^{-1} \!\!\!\!\!\!=
\begin{bmatrix} \frac{1}{N} {\mathbf{1}}_N & -\frac{1}{N} {\mathbf{1}}_{N} \\ -\frac{1}{M-N} {\mathbf{1}}_{M-N}& \frac{2}{M-N} {\mathbf{1}}_{M-N} \end{bmatrix}.$$ If we set $\rho^{\rm {ul}}=\rho^{\rm {ul}}=1$ for simplicity, the instantaneous effective SINR in for UE 1 becomes $$\begin{aligned}
\notag
\gamma_{1}^{\rm {ul}}
&= \frac{ | {\mathbf{v}}_{1}^{\Htran} \hat{{\mathbf{h}}}_{1} |^2 }{
| {\mathbf{v}}_{1}^{\Htran} \hat{{\mathbf{h}}}_{2} |^2 + \sum_{k=1}^{2} {\mathbf{v}}_{1}^{\Htran}({\mathbf{R}}_{k} - {\mathbf{\Phi}}_{k}) {\mathbf{v}}_{1} + \| {\mathbf{v}}_{1}\|^2}
\\&= \frac{ 1 }{ 0 + \frac{7}{4N} + \frac{4}{3(M-N)} + \frac{M}{N(M-N)} } \end{aligned}$$ where the coherent interference from UE 2 is zero. The remaining terms go asymptotically to zero if $N = \alpha M$, for $0 < \alpha < 1$, in which case $\gamma_{1}^{\rm {ul}}$ grows without bound, as expected from Theorem \[theorem:MMSE\].
![If the pilot-contaminated channel estimates are linearly independent (i.e., not parallel), there exists a combining vector ${\mathbf{v}}_1$ that rejects the pilot-contaminated interference from UE 2 in the uplink, while the desired signal remains due to ${\mathbf{v}}_1^{\Htran} \hat{{\mathbf{h}}}_1 \neq 0$. Similarly, if ${\mathbf{w}}_1 = {\mathbf{v}}_1/\sqrt{\mathbb{E}\{ \| {\mathbf{v}}_1 \|^2 \}}$ is used as precoding vector, then no pilot-contaminated coherent interference is caused to UE 2 in the downlink.[]{data-label="figureOrthogonality"}](figureOrthogonality.pdf){width=".45\columnwidth"}
In the second example, we consider a scenario where Assumption \[assumption\_2\] is not satisfied.
\[example2\] Channels with i.i.d. fading, where the covariance matrices are ${\mathbf{R}}_1 = \beta_1 {\bf I}_M$ and ${\mathbf{R}}_2 = \beta_2 {\bf I}_M$, are a notable case when the covariance matrices are not linearly independent. However, any such case is non-robust to perturbations of the matrix elements. Suppose we replace ${\mathbf{R}}_1$ with $$\label{eq:matrix-example-independence2}
{\mathbf{R}}_1 = \beta_1 \left[ {\begin{array}{*{20}{c}}
{{\epsilon _1}}&0& \cdots \\
0& \ddots &0\\
\vdots &0&{{\epsilon _M}}
\end{array}} \right]$$ where $\epsilon_{1},\ldots,\epsilon_{M}$ are i.i.d. positive random variables. This modeling is motivated by the measurement results in [@Gao2015b], which shows that there are a few dB of large-scale fading variations over the antennas in a ULA. For UE $i=1$, we have $$\begin{aligned}
\notag
\mathop {\liminf}\limits_M &\inf_{\lambda_2} \frac{1}{{M}} \| {\mathbf{R}}_{1} + \lambda_2 {\mathbf{R}}_{2} \|_F^2 \\\notag&= \mathop {\liminf}\limits_M \inf_{\lambda_2} \frac{1}{M}\sum_{m=1}^{M} (\beta_1 \epsilon_m+\lambda_2\beta_2)^2 \\ & \mathop {=}^{(a)} \mathop {\liminf}\limits_M \beta_1^2\frac{1}{M}\sum_{m=1}^{M} {\left( \epsilon_m-\frac{1}{M}\sum\limits_{n=1}^{M}\epsilon_n\right)^2} \notag\\&\mathop {=}^{(b)} \beta_1^2 \mathbb{E}\{ (\epsilon_m-\mathbb{E}\{ \epsilon_m \})^2\}
\end{aligned}$$ where $(a)$ is obtained from the fact that $\lambda_2 = -\frac{\beta_1}{\beta_2}\frac{1}{M}\sum_{n=1}^{M} {\epsilon_n}$ minimizes $\frac{1}{M}\sum_{m=1}^{M} (\beta_1 \epsilon_m+\lambda_2\beta_2)^2$ and $(b)$ follows from the strong law of large numbers. Note that $\mathbb{E}\{ (\epsilon_m-\mathbb{E}\{ \epsilon_m \})^2\}$ in the last expression is the variance of $ \epsilon_m$. Since every random variable has non-zero variance and $\beta_1>0$, we conclude that $ \{ {\mathbf{R}}_1, {\mathbf{R}}_2 \}$ satisfy and thus Assumption \[assumption\_2\] holds.
The key implication from Example \[example2\] is that all cases where ${\mathbf{R}}_1$ and ${\mathbf{R}}_2$ are equal (up to a scaling factor) are non-robust to random perturbations and thus anomalies. Since practical propagation environments are irregular and behave randomly (see the measurements reported in [@Gao2015a; @Gao2015b]), linearly dependent covariance matrices are not appearing in practice and Assumption \[assumption\_2\] is generally satisfied. In other words, it is fair to say that the uplink and downlink SEs grow without bound as $M \to \infty$ in general, while the special cases when it does not occur are of no practical importance. We end this subsection with a comparison with related work and a remark regarding acquisition of channel statistics.
\[examplenew\] Consider a BS with two distributed arrays of $M'=M/2$ antennas that serve two UEs having the covariance matrices $${\mathbf{R}}_1 = \begin{bmatrix} b_{11} {\mathbf{I}}_{M'} & {\mathbf{0}} \\ {\mathbf{0}} & b_{12} {\mathbf{I}}_{M'} \end{bmatrix} \quad {\mathbf{R}}_2 = \begin{bmatrix} b_{21} {\mathbf{I}}_{M'} & {\mathbf{0}} \\ {\mathbf{0}} & b_{22} {\mathbf{I}}_{M'} \end{bmatrix}$$ with $b_{11}, b_{12}, b_{21}, b_{22}>0$. These covariance matrices are (asymptotically) linearly independent if $b_{11} b_{22} \neq b_{12} b_{21}$, in which case the uplink and downlink SEs grow without bound with MMSE or ZF.
The exemplified setup is equivalent to the multicell joint transmission scenario considered in the pilot contamination precoding works [@ashikhmin2012pilot; @Li2013b] in which the heuristic vectors $$\label{eq:PCP}
\big[ {\mathbf{v}}_1 \,\, {\mathbf{v}}_2 \big]= \begin{bmatrix} \frac{1}{M'} {\mathbf{y}}^{p}_1 & {\mathbf{0}} \\ {\mathbf{0}} & \frac{1}{M'} {\mathbf{y}}^{p}_2 \end{bmatrix}
\begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix}^{-1}$$ are used for combining and precoding, and ${\mathbf{y}}^{p}_1,{\mathbf{y}}^{p}_2 \in \mathbb{C}^{M'}$ are obtained from the received pilot signals as $ [ ({\mathbf{y}}^{p}_1)^{\Ttran} \, ({\mathbf{y}}^{p}_2)^{\Ttran} ]^{\Ttran} = {\mathbf{Y}}^{p} \bphiu^{\star}/\rho^{\rm{tr}}$. These vectors are specifically designed to make $\big[ {\mathbf{h}}_1 \,\, {\mathbf{h}}_2 \big]^{\Htran}\big[ {\mathbf{v}}_1 \,\, {\mathbf{v}}_2 \big]\asymp {\bf I}_2$ as $M \to \infty$, and thus this method has the same asymptotic behavior as ZF in the special case of block-diagonal covariance matrices where each block is a scaled identity matrix. Note that the matrix inverse in only exists if $b_{11} b_{22} \neq b_{12} b_{21}$, which is again the condition for linear independence of the covariance matrices. Since pilot contamination precoding can only be applied in special multicell cooperation cases, MMSE combining/precoding is generally the preferable choice.
Theorems \[theorem:MMSE\] and \[theorem:MMSE\_precoding\] exploit the MMSE estimator and thus the BS needs to know the (deterministic) channel statistics. In particular, the BS can only compute the MMSE estimate $\hat {\bf h}_k$ in Lemma \[theorem:MMSE-estimate\_h\_jli\] if it knows ${\mathbf{R}}_{k}$ and also the sum ${\mathbf{R}}_{1}+{\mathbf{R}}_{2}$ of the two covariance matrices. In practice, ${\mathbf{R}}_{k}$ can be estimated by a regularized sample covariance matrix, given realizations of ${\mathbf{h}}_{k}$ over multiple resource blocks (e.g., different times and frequencies) where this channel is either observed in only noise [@Yin2013a; @Shariati2014a; @Sun2015a] or where some observations are regular pilot transmissions containing the desired channel plus interference/noise and some contain only the interference/noise [@Bjornson2016c]. It seems that around $M$ samples are needed to obtain a sufficiently accurate covariance estimate [@Bjornson2016c]. The covariance estimation can be further improved if the channels have a known structure. For example, [@Haghighatshoar2017a] provides algorithms for estimating the covariance matrices of channels that have limited angle-delay support that is also separable between users.
Achievable SE with Partial Knowledge of Covariance Matrices {#sec:approximate_MMSE_two_user}
-----------------------------------------------------------
If the BS does not have full knowledge of the covariance matrices, an alternative method for channel estimation is to estimate each entry of ${\mathbf{h}}_{k}$ separately, ignoring the correlation among the elements. This leads to the element-wise MMSE (EW-MMSE) estimator (called diagonalized estimator in [@Shariati2014a]) that utilizes only the main diagonals of ${\mathbf{R}}_{1}$ and ${\mathbf{R}}_{2}$. The diagonals can be estimated efficiently using a small number of samples, that does not need to grow with $M$ [@Bjornson2016c; @Shariati2014a].
\[lemma:EW-MMSE\] Based on the observation $[ {\mathbf{Y}}^{p} \bphiu^{*} ]_{i}$, the BS can compute the EW-MMSE estimate of the $i$th element of ${\mathbf{h}}_{k}$ as $$\begin{aligned}
[\hat{{\mathbf{h}}}_{k}]_i = \frac{1}{ \sqrt{\rho^{\rm{tr}}} }\frac{ [ {\mathbf{R}}_{k} ]_{ii}}{ [ {\mathbf{R}}_{1} ]_{ii} + [ {\mathbf{R}}_{2} ]_{ii} + \frac{1}{ {\rho^{\rm{tr}}} } }[ {\mathbf{Y}}^{p} \bphiu^{*}]_{i}.\end{aligned}$$
We may write $\hat{{\mathbf{h}}}_{k} $ in Lemma \[lemma:EW-MMSE\] in matrix form as $$\label{eq:G_101}
\hat{{\mathbf{h}}}_{k} = \frac{1}{ \sqrt{\rho^{\rm{tr}}} } {\bf D}_k \boldsymbol{\Lambda}^{-1} {\mathbf{Y}}^{p} \bphiu^{*}$$ where ${\bf D}_k \in \mathbb{R}^{M\times M}$ and $\boldsymbol{\Lambda}\in \mathbb{R}^{M\times M}$ are diagonal matrices with elements $\{[ {\mathbf{R}}_{k} ]_{ii}: i=1,\ldots,M\}$ and $\{[ {\mathbf{R}}_{1} ]_{ii} + [ {\mathbf{R}}_{2} ]_{ii} + \frac{1}{ {\rho^{\rm{tr}}} } : i=1,\ldots,M\}$, respectively. [Notice that Assumption \[assumption\_1\] implies that[^6] $ \liminf_M \frac{1}{{M}}\tr ( {\mathbf{D}}_{k} ) > 0$ and $ \limsup_M \| {\mathbf{D}}_{k}\|_2 < \infty$ for $k=1,2$]{}. To quantify the achievable SE when using EW-MMSE, similar to the downlink we exploit the use-and-then-forget SE bound [@Marzetta2016a], which is less tight than but does not require the use of MMSE channel estimation. The uplink ergodic capacity of UE 1 can be thus lower bounded by ${\underline{\mathsf{SE}}}_{1}^{\rm ul} = ( 1 - \frac{\taupu}{\tau_c} ) \log_2 ( 1 + {\underline{\gamma}}_{1}^{\rm ul} )$ \[bit/s/Hz\] with $$\begin{aligned}
\label{eq:G_102}
{\underline{\gamma}}_{1}^{\rm ul}
= \frac{ | {\mathbb{E}}\{{\mathbf{v}}_{1}^{\Htran} {{\mathbf{h}}}_{1}\} |^2 }{ {\mathbb{E}}\left\{| {\mathbf{v}}_{1}^{\Htran} {{\mathbf{h}}}_{2}|^2\right\} + {\mathbb{V}}\{ {\mathbf{v}}_{1}^{\Htran} {{\mathbf{h}}}_{1} \} + \frac{1}{\rho^{\rm ul}} {\mathbb{E}}\left\{\|{\mathbf{v}}_{1} \|^2\right\} }.\end{aligned}$$ This bound is valid for any channel estimation and any combining scheme. A reasonable choice for ${\mathbf{v}}_1$ is the approximate MMSE combining vector: $$\begin{aligned}
\label{eq:G_104}
{\mathbf{v}}_1= \bigg( \sum_{k=1}^2\hat{{\mathbf{h}}}_{k} \hat{{\mathbf{h}}}_{k}^{\Htran} + {\mathbf{S}}\bigg)^{-1} \hat{{\mathbf{h}}}_{1}\end{aligned}$$ where $\hat{{\mathbf{h}}}_{1},\hat{{\mathbf{h}}}_{2}$ are computed as in and ${\mathbf{S}}$ is diagonal and given by ${\mathbf{S}} = \sum_{k=1}^{2} \Big({\mathbf{D}}_{k} - {\mathbf{D}}_{k}\boldsymbol{\Lambda}^{-1}{\mathbf{D}}_{k}\Big) + \frac{1}{\rho^{\rm {ul}}} {\mathbf{I}}_M$. Note that is equivalent to the MMSE combining in when the covariance matrices are diagonal. We will now analyze how ${\underline{\gamma}}_{1}^{\rm ul}$ behaves asymptotically as $M\to \infty$ when ${\mathbf{v}}_1$ is given by . To this end, we impose the following assumption, which states that ${\mathbf{D}}_1$ and ${\mathbf{D}}_2$ are asymptotically linearly independent (i.e., the diagonals of ${\mathbf{R}}_1$ and ${\mathbf{R}}_2$ are asymptotically linearly independent).
\[assumption\_6\]For $\boldsymbol{\lambda} = [\lambda_1, \lambda_2]^{\Ttran} \in \mathbb{R}^2$ and $i=1,2$, $$\begin{aligned}
\label{eq:assumption_6}
\mathop {\liminf}\limits_M \inf_{\{\boldsymbol{\lambda}: \, \lambda_i=1\}} \frac{1}{{M}} \left\| \lambda_1 {\mathbf{D}}_{1} + \lambda_2 {\mathbf{D}}_{2} \right\|_F^2 > 0.\end{aligned}$$
The following is the third main result of this paper:
\[theorem:EW-MMSE\_precoding\] If ${\bf v}_1$ in is used with $\hat{{\mathbf{h}}}_{1},\hat{{\mathbf{h}}}_{2}$ given by , then under Assumptions \[assumption\_1\] and \[assumption\_6\], the SINR $\underline\gamma_{1}^{\rm{ul}} $ increases unboundedly as $M\to \infty$. Hence, ${\underline{\mathsf{SE}}}_{1}^{\rm ul}$ increases unboundedly as $M\to \infty$.
The proof is given in Appendix E.
As a consequence of this theorem, under Assumptions \[assumption\_1\] and \[assumption\_6\], the uplink SEs of UE 1 and UE 2 increase without bound as $M\to \infty$ even if the BS has only knowledge of the diagonal elements of the covariance matrices. A similar result can be proved for the downlink, using the methodology adopted in Appendix D for proving Theorem \[theorem:MMSE\_precoding\]. The details are omitted for space limitations.
Asymptotic Spectral Efficiency in Multicell Massive MIMO {#section:multi-user}
========================================================
We will now generalize the results of Section \[section:two-user\] to a Massive MIMO network with $L$ cells, each comprising a BS with $M$ antennas and $K$ UEs. There are $\taupu=K$ pilots and the $k$th UE in each cell uses the same pilot. Following the notation from [@hoydis2013massive], the received signal ${\bf y}_j \in \mathbb{C}^{M}$ at BS $j$ is $${\bf y}_j = \sum_{l=1}^{L} \sum_{i=1}^{K} \sqrt{\rho} {\mathbf{h}}_{jli} x_{li} + {\mathbf{n}}_j$$ where $\rho$ is the normalized transmit power, $x_{li}$ is the unit-power signal from UE $i$ in cell $l$, ${\mathbf{h}}_{jli} \sim \CN ({\mathbf{0}}, {\mathbf{R}}_{jli})$ is the channel from this UE to BS $j$, ${\mathbf{R}}_{jli} \in \mathbb{C}^{M \times M}$ is the channel covariance matrix, and ${\mathbf{n}}_j \sim \CN ({\mathbf{0}}, {\mathbf{I}}_{M})$ is the independent receiver noise at BS $j$. Using a total uplink pilot power of $\rho^{\rm{tr}}$ per UE and standard MMSE estimation techniques [@hoydis2013massive], BS $j$ obtains the estimate of ${\mathbf{h}}_{jli}$ as $$\begin{aligned}
\hat{{\mathbf{h}}}_{jli} = {\mathbf{R}}_{jli} {\mathbf{Q}}_{ji}^{-1} \bigg( \sum_{l'=1}^{L} {\mathbf{h}}_{jl'i} + \frac{1}{\sqrt{\rho^{\rm{tr}}}} {\mathbf{n}}_{ji} \bigg) \!\sim \!\CN \left( {\mathbf{0}}, {\mathbf{\Phi}}_{jli} \right)\end{aligned}$$ where ${\mathbf{n}}_{ji} \sim \CN ({\mathbf{0}}, {\mathbf{I}}_{M})$ is noise, ${\mathbf{Q}}_{ji} = \sum_{l'=1}^{L} {\mathbf{R}}_{jl'i} + \frac{1}{\rho^{\rm{tr}}} {\mathbf{I}}_{M}$, and $
{\mathbf{\Phi}}_{jli} = {\mathbf{R}}_{jli} {\mathbf{Q}}_{ji}^{-1} {\mathbf{R}}_{jli}$. The estimation error $\tilde{{\mathbf{h}}}_{jli} = {\mathbf{h}}_{jli} - \hat{{\mathbf{h}}}_{jli} \sim \CN \left( {\mathbf{0}}, {\mathbf{R}}_{jli}- {\mathbf{\Phi}}_{jli} \right)$ is independent of $\hat{{\mathbf{h}}}_{jli}$. However, the estimates $\hat{{\mathbf{h}}}_{j1i}, \ldots, \hat{{\mathbf{h}}}_{jLi}$ of the UEs with the same pilot are correlated as $
\mathbb{E}\{ \hat{{\mathbf{h}}}_{jni} \hat{{\mathbf{h}}}_{jmi}^{\Htran}\} = {\mathbf{R}}_{jni} {\mathbf{Q}}_{ji}^{-1} {\mathbf{R}}_{jmi}.
$
Uplink Data Transmission
------------------------
We denote by ${\bf v}_{jk} \in \mathbb {C}^{M}$ the receive combining vector associated with UE $k$ in cell $j$. Using the same technique as in [@hoydis2013massive; @Marzetta2016a], the uplink ergodic capacity is lower bounded by $$\label{eq:uplink-rate-expression-general}
\begin{split}
\mathsf{SE}_{jk}^{\rm {ul}} = \left( 1- \frac{\tau_p}{\tau_c} \right) \mathbb{E} \left\{ \log_2 \left( 1 + \gamma_{jk}^{\rm {ul}} \right) \right\} \quad \textrm{[bit/s/Hz] }
\end{split}$$ with the instantaneous effective SINR $$\begin{aligned}
\notag
\gamma_{jk}^{\rm {ul}} & = \frac{ | {\mathbf{v}}_{jk}^{\Htran} \hat{{\mathbf{h}}}_{jjk} |^2 }{{\mathbb{E}}\left\{
\!\sum\limits_{(l,i)\ne (j,k)} | {\mathbf{v}}_{jk}^{\Htran} {{\mathbf{h}}}_{jli} |^2
+| {\mathbf{v}}_{jk}^{\Htran} \tilde{{\mathbf{h}}}_{jjk} |^2+ \frac{{\mathbf{v}}_{jk}^{\Htran} {\mathbf{v}}_{jk} }{\rho^{\rm {ul}}}
\Big| \hat{\bf{h}}_{(j)} \right\}} \\&= \frac{ | {\mathbf{v}}_{jk}^{\Htran} \hat{{\mathbf{h}}}_{jjk} |^2 }{
{\mathbf{v}}_{jk}^{\Htran} \left( \sum\limits_{(l,i)\ne (j,k)} \hat{{\mathbf{h}}}_{jli} \hat{{\mathbf{h}}}_{jli}^{\Htran} + {\mathbf{Z}}_j\right) {\mathbf{v}}_{jk}
} \label{eq:uplink-instant-SINR}\end{aligned}$$ where ${\mathbb{E}}\{\cdot|{{\hat{\bf h}}_{(j)}}\}$ denotes the conditional expectation given the [MMSE]{} channel estimates available at BS $j$ and ${\mathbf{Z}}_j = \sum\nolimits_{l=1}^{L} \sum\nolimits_{i=1}^{K} ({\mathbf{R}}_{jli} - {\mathbf{\Phi}}_{jli}) + \frac{1}{\rho^{\rm {ul}}} {\mathbf{I}}_{M}$. As shown in [@Ngo2012b; @EmilEURASIP17], the instantaneous effective [SINR]{} in for [UE]{} $k$ in cell $j$ is maximized by $$\label{eq:MMSE-combining}
{\mathbf{v}}_{jk} = \Bigg( \sum\limits_{l=1}^L\sum\limits_{i=1}^K \hat{{\mathbf{h}}}_{jli} \hat{{\mathbf{h}}}_{jli}^{\Htran} + {\mathbf{Z}}_j \Bigg)^{\!-1} \!\! \hat{{\mathbf{h}}}_{jjk}.$$ We refer to this “optimal” receive combining scheme as multicell MMSE (M-MMSE) combining. The “multicell” notion is used to differentiate it from the single-cell MMSE (S-MMSE) combining scheme [@hoydis2013massive; @Guo2014a; @Krishnan2014a], which is widely used in the literature and defined as $$\begin{aligned}
\bar{{\mathbf{v}}}_{jk} =
\left( \sum\limits_{i=1}^{K} \hat{{\mathbf{h}}}_{jji} \hat{{\mathbf{h}}}_{jji}^{\Htran} + \bar{{\mathbf{Z}}}_j\right)^{-1} \!\! \hat{{\mathbf{h}}}_{jjk} \label{eq:S-MMSE-combining}\end{aligned}$$ with $\bar{{\mathbf{Z}}}_j = \sum\nolimits_{i=1}^{K}{\mathbf{R}}_{jji} \!-\! {\mathbf{\Phi}}_{jji} + \sum\nolimits_{l=1,l \neq j}^{L} \sum\nolimits_{i=1}^{K} {\mathbf{R}}_{jli}
+ \frac{1}{\rho^{\rm {ul}}} {\mathbf{I}}_{M}$. The main difference from is that only channel estimates in the own cell are computed in S-MMSE, while $\hat{{\mathbf{h}}}_{jli} \hat{{\mathbf{h}}}_{jli}^{\Htran} - {\mathbf{\Phi}}_{jli} $ is replaced with its average (i.e., zero) for all $l \neq j$. The computational complexity of S-MMSE is thus slightly lower than with M-MMSE (see [@EmilEURASIP17] for a detailed discussion). However, both schemes only utilizes channel estimates that can be computed locally at the BS and the pilot overhead is identical since the same pilots are used to estimate both intra-cell and inter-cell channels. The S-MMSE scheme coincides with [M-MMSE]{} when there is only one isolated cell, but it is generally different and does not suppress interference from interfering [UEs]{} in other cells. Plugging into yields $$\begin{aligned}
\label{eq:gammajk_MMSE}
\gamma_{jk}^{\rm {ul}} &= \hat{{\mathbf{h}}}_{jjk}^{\Htran} \Bigg( \sum\limits_{(l,i)\ne (j,k)}\hat{{\mathbf{h}}}_{jli} \hat{{\mathbf{h}}}_{jli}^{\Htran} + {\mathbf{Z}}_j \Bigg)^{\!-1}\hat{{\mathbf{h}}}_{jjk}.\end{aligned}$$ We want to analyze $\gamma_{jk}^{\rm {ul}}$ when $M\to \infty$. To this end, we make the following two assumptions.
\[assumption\_3\]As $M\to \infty$ $\forall j,l,i$, $\liminf_M \;\frac{1}{{M}}\tr ( {\mathbf{R}}_{jli}) > 0 $ and $
\limsup_M \;\| {\mathbf{R}}_{jli}\|_2 < \infty$.
\[assumption\_4\] For any UE $k$ in cell $j$ with ${\boldsymbol\lambda}_{jk} = [\lambda_{j1k}, \ldots,\lambda_{jLk}]^{\Ttran} \in \mathbb{R}^{L}$ and $\lalt=1,\ldots,L$ $$\begin{aligned}
\label{Condition2_Assumption4_new}
\liminf_M \inf_{\{{\boldsymbol\lambda}_{jk}: \, \lambda_{j \lalt k}=1\}} \frac{1}{{M}}\left\| \sum\limits_{l=1}^L\lambda_{jlk} {\mathbf{R}}_{jlk} \right\|_F^2 > 0.\end{aligned}$$
The following is the fourth main result of the paper:
\[theorem:M-MMSE\] If M-MMSE combining is used, then under Assumptions \[assumption\_3\] and \[assumption\_4\] the SINR $\gamma_{jk}^{\rm {ul}} $ increases a.s. unboundedly as $M\to \infty$. Hence, $\mathsf{SE}_{jk}^{\rm {ul}}$ increases unboundedly as $M\to \infty$.
The proof is given in Appendix F.
This theorem proves the remarkable result that, under Assumptions \[assumption\_3\] and \[assumption\_4\], the uplink SE of a multicell Massive MIMO network increases without bound as $M\to \infty$, despite pilot contamination. This is in sharp contrast to the finite limit in case of MR combining [@marzetta2010noncooperative] or any other single-cell combining scheme [@hoydis2013massive; @Guo2014a; @Krishnan2014a] and it is due to the fact that M-MMSE rejects the coherent interference caused by pilot contamination when Assumptions \[assumption\_3\] and \[assumption\_4\] hold. Note that these are the natural multicell generalizations of Assumptions \[assumption\_1\] and \[assumption\_2\], respectively. In particular, the condition says that the covariance matrices $\{{\mathbf{R}}_{jlk}: l=1,\ldots,L\}$ of the channels from the pilot-sharing UEs to BS $j$ are asymptotically linearly independent, which implies the same condition for the estimated channels $\{\hat {{\mathbf{h}}}_{jlk}: l=1,\ldots,L\}$. This condition is used in Appendix F to prove Theorem \[theorem:M-MMSE\] in a fairly simple way.
However, we stress that Theorem \[theorem:M-MMSE\] is valid also in a more general setting in which $\hat {{\mathbf{h}}}_{jjk}$ is asymptotically linearly independent of the estimates of all pilot-interfering UEs’ channels, but some of the interfering channel estimates can be written as linear combinations of other interfering channels. Let $\mathcal S_{jk} \subseteq \{\hat {{\mathbf{h}}}_{jlk}: \forall l\ne j\}$ denote a subset of the estimated interfering channels that form a basis for all interfering channels. Under these circumstances, we only need to take the estimates in $\mathcal S_{jk}$ into account in the computation of the combining vector ${\bf v}_{jk}$ in and the same result follows. To gain further insights into this, we notice (as done for the two-user case in Section \[subsec:interpretation\]) that one can find a receive combining vector that is orthogonal to the subspace spanned by $\mathcal S_{jk}$. This scheme exhibits an unbounded SE when $M\to \infty$ as it rejects the interference from all pilot-contaminating UEs (not only from those in $\mathcal S_{jk}$), while retaining an array gain that grows with $M$. We call this scheme multicell ZF (M-ZF) and define it as ${\mathbf{v}}_{jk} = \hat{{\mathbf{H}}}_{jk} \big( \hat{{\mathbf{H}}}_{jk}^{\Htran} \hat{{\mathbf{H}}}_{jk} \big)^{-1} {\mathbf{e}}_{1}$, where ${\mathbf{e}}_{1}$ is the first column of ${\mathbf{I}}_{|\mathcal S_{jk}|+1}$ (with $|\mathcal S_{jk}|$ being the cardinality of $\mathcal S_{jk}$) and $\hat{{\mathbf{H}}}_{jk}\in \mathbb{C}^{N\times (|\mathcal S_{jk}|+1) }$ is the matrix with $\hat {{\mathbf{h}}}_{jjk}$ in the first column and the channel estimates in $\mathcal S_{jk}$ in the remaining columns. Since M-MMSE combining is the optimal scheme, it has to exhibit an unbounded SE if this is the case with M-ZF.
Downlink Data Transmission
--------------------------
During downlink data transmission, the BS in cell $l$ transmits ${\mathbf{x}}_l = \sqrt{\rho^{\rm{dl}}}\sum_{l=1}^{K} {\mathbf{w}}_{li} \varsigma_{li} $, where $\varsigma_{li} \sim \CN(0,1)$ is the data signal intended for UE $i$ in the cell and $\rho^{\rm{dl}}$ is the normalized transmit power. This signal is assigned to a transmit precoding vector $ {\mathbf{w}}_{li} \in \mathbb{C}^{M}$, which satisfies $\mathbb{E} \{ \| {\mathbf{w}}_{li} \|^2 \} =1$, such that $\mathbb{E} \{ \| {\mathbf{w}}_{li} \varsigma_{li} \|^2 \} = \rho^{\rm{dl}}$ is the transmit power allocated to this UE. Using the same technique as in [@hoydis2013massive; @Marzetta2016a], the downlink ergodic channel capacity of UE $k$ in cell $j$ can be lower bounded by $\mathsf{SE}_{jk}^{\rm {dl}} = \big( 1- \frac{\tau_p}{\tau_c} \big) \log_2 ( 1 +
\gamma^{\mathrm{dl}}_{jk} )$ \[bit/s/Hz\] with $$\label{eq:downlink-SINR-expression-forgetbound}
\gamma^{\mathrm{dl}}_{jk}= \frac{ | \mathbb{E}\{ {\mathbf{h}}_{jjk}^{\Htran} {\mathbf{w}}_{jk}\} |^2 }{
\sum\limits_{l=1}^{L} \sum\limits_{i=1}^{K} \mathbb{E} \{ | {\mathbf{h}}_{ljk}^{\Htran} {\mathbf{w}}_{li} |^2 \}
- | \mathbb{E}\{ {\mathbf{h}}_{jjk}^{\Htran}{\mathbf{w}}_{jk} \} |^2 + \frac{1}{\rho^{\rm{dl}}} }.$$ Unlike $\gamma^{\mathrm{ul}}_{jk}$ in , which only depends on the own combining vector ${\bf v}_{jk}$, $\gamma^{\mathrm{dl}}_{jk}$ depends on all precoding vectors $\{{\mathbf{w}}_{li}\}$. The precoding should ideally be selected jointly across the cells, which makes precoding optimization difficult in practice. Motivated by the uplink-downlink duality [@EmilEURASIP17], it is reasonable to select $\{{\mathbf{w}}_{li}\}$ based on the M-MMSE combining vectors $\{{\mathbf{v}}_{jk}\}$ given by . This leads to M-MMSE precoding $$\begin{aligned}
\label{eq:MMMSE_precoding}
{\bf w}_{jk} = \sqrt{\vartheta_{jk}} {\mathbf{v}}_{jk} = \sqrt{\vartheta_{jk}} \Bigg( \sum\limits_{l=1}^L\sum\limits_{i=1}^K \hat{{\mathbf{h}}}_{jli} \hat{{\mathbf{h}}}_{jli}^{\Htran} + {\mathbf{Z}}_j \Bigg)^{\!-1} \hat{{\mathbf{h}}}_{jjk}\end{aligned}$$ with the normalization factor $\vartheta_{jk} = (\sqrt{{\mathbb{E}}\left\{\|{\mathbf{v}}_{jk}\|^2\right\}})^{-1}$. This is the fifth main result of the paper:
\[theorem:M-MMSE\_precoding\] If M-MMSE precoding is used, then under Assumptions \[assumption\_3\] and \[assumption\_4\] the SINR $\gamma_{jk}^{\rm {dl}} $ grows unboundedly as $M\to \infty$. Hence, $\mathsf{SE}_{jk}^{\rm {dl}}$ grows unboundedly as $M\to \infty$.
Despite being much more involved, the proof basically unfolds from the same arguments used for proving Theorem \[theorem:MMSE\_precoding\] and by exploiting the results of Appendix F for Theorem \[theorem:M-MMSE\].
This theorem shows that an asymptotically unbounded downlink SE is achieved by all UEs in the network, despite the suboptimal assumptions of M-MMSE precoding, equal power allocation, and no estimation of the instantaneous realization of the precoded channels. The only important requirement is that the channel estimates to the desired UEs are asymptotically linearly independent from the channel estimates of pilot-contaminating UEs in other cells. Section \[sec:numerical-results\] demonstrates numerically that the DL SE grows without bound as $M \to \infty$.
Approximate M-MMSE Combining and Precoding {#sec:approximate_MMSE}
------------------------------------------
In Section \[sec:approximate\_MMSE\_two\_user\], we have shown that the SE with the approximate M-MMSE scheme (that only utilizes the diagonals of the covariance matrices) grows unbounded as $M\to\infty$, in a two-user scenario. This result can be generalized to a multicell Massive MIMO network. Due to space limitations, we concentrate on the uplink. In particular, we assume that the signal of UE $k$ in cell $j$ is detected by using the approximate M-MMSE combining vector $$\label{eq:EW_MMSE-combining}
{\mathbf{v}}_{jk} = \Bigg( \sum\limits_{l=1}^L\sum\limits_{i=1}^K \hat{{\mathbf{h}}}_{jli} \hat{{\mathbf{h}}}_{jli}^{\Htran} + {\mathbf{S}}_j \Bigg)^{\!-1} \hat{{\mathbf{h}}}_{jjk}$$ where ${\mathbf{S}}_j = \sum_{l=1}^{L}\sum_{i=1}^{K} \Big({\mathbf{D}}_{jli} - {\mathbf{D}}_{jli}\boldsymbol{\Lambda}_{ji}^{-1}{\mathbf{D}}_{jli}\Big) + \frac{1}{\rho^{\rm {ul}}} {\mathbf{I}}_M$ is a diagonal matrix and the EW-MMSE estimate of ${\mathbf{h}}_{jli}$ is $$\begin{aligned}
\hat{{\mathbf{h}}}_{jli} = \frac{1}{ \sqrt{\rho^{\rm{tr}}} } {\bf D}_{jli} \boldsymbol{\Lambda}_{ji}^{-1} \left( \sum_{l'=1}^{L} {\mathbf{h}}_{jl'i} + \frac{1}{\sqrt{\rho^{\rm{tr}}}} {\mathbf{n}}_{ji} \right) \end{aligned}$$ where ${\mathbf{n}}_{ji} \sim \CN ({\mathbf{0}}, {\mathbf{I}}_{M})$ is noise and ${\bf D}_{jli} \in \mathbb{R}^{M\times M}$ and $\boldsymbol{\Lambda}_{ji}\in \mathbb{R}^{M\times M}$ are diagonal with elements $\{[ {\mathbf{R}}_{jli} ]_{nn}: n=1,\ldots,M\}$ and $\{ \sum_{l'=1}^{L} [{\mathbf{R}}_{jl'i} ]_{nn} + \frac{1}{ {\rho^{\rm{tr}}} }: n=1,\ldots,M\}$, respectively. Since ${\bf D}_{jli}$ and $\boldsymbol{\Lambda}_{ji}$ are diagonal, the computational complexity of EW-MMSE estimation is substantially lower than for MMSE estimation; see [@Shariati2014a] for details. Notice that the combining scheme in can be applied without knowing the full channel covariance matrices, as it depends only on the diagonal elements of $\{{\bf R}_{jli}:l=1,\ldots,L\}$. This is because the elements of $\hat{{\mathbf{h}}}_{jli}$ are estimated separately, without exploiting the spatial channel correlation. By using the use-and-then-forget SE bound [@Marzetta2016a], the uplink ergodic capacity of UE $k$ in cell $j$ can be lower bounded by ${\underline{\mathsf{SE}}}_{jk}^{\rm ul} = ( 1 - \frac{\taupu}{\tau_c} ) \log_2 ( 1 + {\underline{\gamma}}_{jk}^{\rm ul} )$ \[bit/s/Hz\] with $$\begin{aligned}
\notag
&{\underline{\gamma}}_{jk}^{\rm ul}= \\& \frac{ | \mathbb{E}\{ {\mathbf{v}}_{jk}^{\Htran} {\mathbf{h}}_{jjk}\} |^2 }{
\sum\limits_{l=1}^{L} \sum\limits_{i=1}^{K} \mathbb{E} \{ | {\mathbf{v}}_{jk}^{\Htran} {\mathbf{h}}_{jli} |^2 \}
- | \mathbb{E}\{ {\mathbf{v}}_{jk}^{\Htran}{\mathbf{h}}_{jjk} \} |^2 + \frac{1}{\rho^{\rm{ul}}} \mathbb{E}\{ \left\|{\mathbf{v}}_{jk}\right\|^2 \} }. \end{aligned}$$ We now want to understand how ${\underline{\gamma}}_{jk}^{\rm ul}$ behaves when $M\to \infty$ under the following assumption, which is the extension of Assumption \[assumption\_4\] to the case where only the diagonals of covariance matrices are used for channel estimation and receive combining:
\[assumption\_7\] For any UE $k$ in cell $j$ with ${\boldsymbol\lambda}_{jk} = [\lambda_{j1k}, \ldots,\lambda_{jLk}]^{\Ttran} \in \mathbb{R}^{L}$ and $\lalt=1,\ldots,L$ $$\begin{aligned}
\label{Condition2_Assumption4}
\liminf_M \inf_{\{{\boldsymbol\lambda}_{jk}: \, \lambda_{j \lalt k}=1\}} \frac{1}{{M}}\left\| \sum\limits_{l=1}^L\lambda_{jlk} {\mathbf{D}}_{jlk} \right\|_F^2 > 0.\end{aligned}$$
The following is the last main result of the paper:
\[theorem:approximate\_M-MMSE\] If approximate M-MMSE combining is used, then under Assumptions \[assumption\_3\] and \[assumption\_7\] the SINR ${\underline{\gamma}}_{jk}^{\rm ul}$ increases unboundedly as $M\to \infty$. Hence, $\underline{\mathsf{SE}}_{jk}^{\rm {ul}}$ increases unboundedly as $M\to \infty$.
The proof is omitted for space limitations, but follows along the lines of Theorem \[theorem:EW-MMSE\_precoding\].
This theorem shows that it is sufficient that the diagonals of the covariance matrices are asymptotically linearly independent and known at the BS to achieve an unbounded uplink SE (and thus an unlimited capacity). This condition is generally satisfied since small random variations in the elements of the covariance matrices are sufficient to achieve asymptotic linear independence, as illustrated by Example \[example2\]. An unbounded SE can be also proved in the downlink using similar methods (omitted for space reasons). This will be demonstrated numerically in the next section.
Numerical Results {#sec:numerical-results}
=================
The simulation results can be reproduced using the code at <https://github.com/emilbjornson/unlimited-capacity>. In this section, we will show numerically that an unlimited SE is achievable under pilot contamination. To this end, we first evaluate three ways to generate the channel covariance matrices and the resulting spatial correlation. For an arbitrary user, the covariance matrix ${\mathbf{R}}$ can be modeled by:
1\) One-ring model for a ULA with half-wavelength antenna spacing and average large-scale fading $\beta$ [@Adhikary2013]. For an angle-of-arrival (AoA) $\theta$ and many scatterers that are uniformly distributed in the angular interval $[\theta-\Delta,\theta+\Delta]$, the $(m,n)$th element of ${\mathbf{R}}$ is $[ {\mathbf{R}} ]_{m,n} = \frac{\beta}{2\Delta} \int_{-\Delta}^{\Delta} e^{ \pi \imath (n-m) \sin(\theta+\delta) } d\delta$.
2\) Exponential correlation model for a ULA with correlation factor $r \in [0,1]$ between adjacent antennas, average large-scale fading $\beta$, and AoA $\theta$ [@Loyka2001a], which leads to $[ {\mathbf{R}} ]_{m,n} = \beta r^{|n-m|} e^{\imath (n-m) \theta}$.
3\) Uncorrelated Rayleigh fading with average large-scale fading $\beta$ and independent log-normal large-scale fading variations over the array, which gives (similar to the perturbations considered in Example \[example2\]) $$\begin{aligned}
\label{eq:uncorrelated-fading-array-correlation-model}
{\mathbf{R}} = \beta \diag \left( 10^{f_1/10},\ldots, 10^{f_M/10} \right)\end{aligned}$$ where $f_m \sim \mathcal{N}(0,\sigma^2)$ and $\sigma$ denotes the standard deviation.
![Average eigenvalue distribution with $M=100$ and for three different channel covariance models, whereof one gives a rank-deficient covariance matrix and the others have full rank.[]{data-label="figureEigenvalues"}](simulationEigenvalues.pdf){width="\columnwidth"}
-4mm
Fig. \[figureEigenvalues\] shows the eigenvalue distribution with the three covariance models above, for $M=100$ antennas, uniformly distributed AoAs $\theta$ in $[-\pi,+\pi)$, $\beta=1$, $\Delta =15^\circ$, $r=0.5$, and $\sigma=2$. All three models create eigenvalue variations, but there are substantial differences. The one-ring model provides rank-deficient covariance matrices, where a large fraction of the eigenvalues is zero (this fraction is computed in [@Adhikary2013]). In contrast, the other two models provide full-rank covariance matrices with more modest eigenvalue variations. In the remainder, we consider the latter two models to emphasize that our main results only require linear independence between the covariance matrices, not rank-deficiency (which in special cases give rise to orthogonal covariance supports [@Yin2013a]).
![Multicell setup with two UEs per cell in the shaded cell-edge area. All UEs have similar AoAs to all BSs, which typically leads to similar covariance matrices and thus high pilot contamination.[]{data-label="figureSetup"}](simulationSetup.pdf){width=".75\columnwidth"}
Uplink
------
We consider the challenging symmetric setup in Fig. \[figureSetup\] with $L=4$ cells, $K=2$ UEs per cell, pilots of length $\taupu=K$, and coherence blocks of $\tau_c=200$ channel uses. The BSs are located at the four corners of the area and the UEs are all located at the cell edges and have similar but non-identical AoAs and distances to the BSs. Thus, the pilot contamination is very large in this setup. Note that the star-marked UEs share a pilot, while the plus-marked UEs share another pilot.
![Uplink SE as a function of $M$, for covariance matrices based on the exponential correlation model ($r=0.5$).[]{data-label="figureAntennas"}](figureAntennas.pdf){width="\columnwidth"}
The asymptotic behavior of the uplink SE is shown in Fig. \[figureAntennas\] using the exponential correlation model ($r=0.5$), with M-MMSE, S-MMSE, MR, and M-ZF, where the latter cancels interference between all UEs. The SE per UE is shown as a function of the number of antennas, in logarithmic scale. The average SNR observed at a BS antenna is set equal in the pilot and data transmission: $\rho^{\rm{ul}} \tr({\mathbf{R}}_{jli})/M =\rho^{\rm{tr}} \tr({\mathbf{R}}_{jli})/M$. It is $-6.0$dB for the intracell UEs and between $-6.3$dB and $-11.5$dB for the interfering UEs in other cells. Fig. \[figureAntennas\] shows that S-MMSE provides slightly higher SE than MR, but both converge to asymptotic limits of around 1bit/s/Hz as $M$ grows. In contrast, M-MMSE provides an SE that grows without bound. The instantaneous effective SINR grows linearly with $M$, which is in line with Theorem \[theorem:M-MMSE\], as seen from the fact that the SE grows linearly when the horizontal scale is logarithmic. M-ZF performs poorly because the channel estimates are so similar that full interference suppression removes most of the desired signal. In contrast, M-MMSE finds a non-trivial tradeoff between interference suppression and coherent combining of the desired signal, leading to superior SE. The reference curve “time splitting” considers the case when the 4 cells are active in different coherence blocks, to remove pilot contamination. MMSE combining is used and the SE grows without bound, but at a slower pace than with M-MMSE, due to the extra pre-log factor of $1/4$. Hence, even for a small system with $L=4$, it is inefficient to avoid pilot contamination by time splitting.
![Uplink with covariance matrices modeled by for $M=200$ and $K=2$. (a) The SE as a function of the standard deviation $\sigma$ of the large-scale fading variations. (b) The received power after receive combining with $\sigma = 4$ is separated into desired signal power and interference from UEs with the same or different pilot than the desired UE.[]{data-label="figureArrayFading"}](figureArrayFading.pdf){width="\columnwidth"}
![Uplink with covariance matrices modeled by for $M=200$ and $K=2$. (a) The SE as a function of the standard deviation $\sigma$ of the large-scale fading variations. (b) The received power after receive combining with $\sigma = 4$ is separated into desired signal power and interference from UEs with the same or different pilot than the desired UE.[]{data-label="figureArrayFading"}](figureBarDiagram.pdf){width="\columnwidth"}
Next, we consider the uncorrelated Rayleigh fading model in with independent large-scale fading variations over the array. The uplink SE with $M=200$ antennas and varying standard deviation $\sigma$ from $0$ to $5$ is shown in Fig. \[figureArrayFading\](a). M-MMSE provides no benefit over S-MMSE or MR in the special case of $\sigma=0$, where the covariance matrices are linearly dependent (i.e., scaled identity matrices). This is a special case that has received massive attention in academic literature, mainly because it simplifies the mathematical analysis. However, M-MMSE provides substantial performance gains over S-MMSE and MR as soon as we depart from the scaled-identity model by adding small variations in the large-scale fading over the array, which make the covariance matrices linearly independent. This is in line with what we demonstrated in Example \[example2\]. As the variations increase, the SE with M-ZF improves particularly fast and approaches the SE with M-MMSE. M-ZF will never be the better scheme since M-MMSE is optimal. The motivation behind this simulation is the measurement results reported in [@Gao2015b], which show large-scale variations of around 4dB over a massive MIMO array—this corresponds to $\sigma \approx 4$ in our setup.
Fig. \[figureArrayFading\](b) shows the received power (normalized by the noise power) after receive combining for an arbitrary UE when $\sigma = 4$. It is divided into the desired signal power, the interference from UEs using the same pilot, and the interference from UEs using a different pilot. The figure shows that MR and S-MMSE suffer from strong interference from the UEs that use the same pilot, since these schemes are unable to mitigate the coherent interference caused by pilot contamination. In contrast, M-MMSE and M-ZF mitigate all types of interference and receive roughly the same amount of interference from UEs with the same or different pilots. Note that the price to pay for the interference rejection is a reduction in desired signal power when using M-MMSE and M-ZF.
Downlink
--------
The setup in Fig. \[figureSetup\] is also used in the downlink wherein we set $\rho^{\rm{dl}}=\rho^{\rm{ul}}$ to get the same SNRs as in the uplink. We consider a setup with both spatial channel correlation and large-scale fading variations over the array, such that the EW-MMSE estimator is suboptimal but Assumption \[assumption\_7\] is satisfied. More precisely, we consider a combination of the exponential correlation model and : $[ {\mathbf{R}} ]_{m,n} = \beta r^{|n-m|} e^{\imath (n-m) \theta} 10^{(f_m +f_n) /20}$, where $\theta$ is the AoA, $r=0.5$ is used as correlation factor, and $f_1,\ldots,f_M \sim \mathcal{N}(0,\sigma^2)$ give independent large-scale fading variations over the array with $\sigma=4$.
![Downlink SE as a function of $M$ for $K=2$, when using either the MMSE estimator (with full covariance knowledge) or the EW-MMSE estimator (with known diagonals of the covariance matrices). The exponential correlation model with $r=0.5$ is used, but with large-scale fading variations over the array with $\sigma=4$.[]{data-label="figureDownlinkSimulation"}](figureDownlinkMMSE.pdf){width="\columnwidth"}
![Downlink SE as a function of $M$ for $K=2$, when using either the MMSE estimator (with full covariance knowledge) or the EW-MMSE estimator (with known diagonals of the covariance matrices). The exponential correlation model with $r=0.5$ is used, but with large-scale fading variations over the array with $\sigma=4$.[]{data-label="figureDownlinkSimulation"}](figureDownlinkEWMMSE.pdf){width="\columnwidth"}
The downlink SE is shown in Fig. \[figureDownlinkSimulation\] as a function of $M$, where Fig. \[figureDownlinkSimulation\](a) shows results with the MMSE estimator that uses the full channel covariance matrices and Fig. \[figureDownlinkSimulation\](b) shows results with the EW-MMSE estimator that only uses the diagonals of the covariance matrices. When using the EW-MMSE estimator, we consider the approximate M-MMSE scheme in and a corresponding approximation of S-MMSE, while M-ZF and MR are as before. The results in Fig. \[figureDownlinkSimulation\](a) with the MMSE estimator are similar to the uplink in Fig. \[figureArrayFading\](a): M-MMSE and M-ZF provide SEs that grow without bound, while the SEs with S-MMSE and MR converge to finite limits. In contrast to the uplink, M-MMSE and M-ZF precoding are both suboptimal in the downlink, but they can be shown to be asymptotically equal.[^7] Interestingly, the same behaviors are observed in Fig. \[figureDownlinkSimulation\](b) when using the EW-MMSE estimator, which is a suboptimal estimator that neglects the off-diagonal elements of the covariance matrices. This result is in line with Theorem \[theorem:approximate\_M-MMSE\]. There is a small SE loss (2%–4% for M-MMSE) compared to Fig. \[figureDownlinkSimulation\](a), but this is a minor price to pay for the greatly simplified acquisition of covariance information (estimating the entire diagonal is as simple as estimating a single parameter [@Shariati2014a; @Bjornson2016c]).
![Downlink SE as a function of $M$ for $K=10$ UEs that are uniformly distributed in the shaded cell edge area. The setup and covariance model are otherwise the same as in Fig. \[figureDownlinkSimulation\].[]{data-label="figureDownlinkSimulationK10"}](figureDownlinkMMSE_K10.pdf){width="\columnwidth"}
![Downlink SE as a function of $M$ for $K=10$ UEs that are uniformly distributed in the shaded cell edge area. The setup and covariance model are otherwise the same as in Fig. \[figureDownlinkSimulation\].[]{data-label="figureDownlinkSimulationK10"}](figureDownlinkEWMMSE_K10.pdf){width="\columnwidth"}
We now increase the number of UEs per cell to $K=10$, which leads to more interference but the same pilot contamination per UE. The UEs are uniformly and independently distributed in the cell-edge area, which is the shaded area in Fig. \[figureSetup\]. The channel model is the same as in the previous figure. The downlink SE per UE is shown in Fig. \[figureDownlinkSimulationK10\] when using either MMSE or EW-MMSE estimation. The results resemble the ones for $K=2$, but the curves are basically shifted to the right due to the additional interference. M-MMSE and M-ZF provide SEs that grow without bound, while the SE with S-MMSE and MR saturate, but more antennas are needed before reaching saturation.
Conclusions and Practical Implications {#section:conclusion}
======================================
We proved that the capacity of Massive MIMO systems increases without bound as $M \to \infty$ in the presence of pilot contamination, despite the previous results that pointed toward the existence of a finite limit. This was achieved by showing that the conventional lower bounds on the capacity increase without bound when using M-MMSE precoding/combining. These schemes exploit the fact that the MMSE channel estimates of UEs that use the same pilot are linearly independent, due to their generally linearly independent covariance matrices. For our results to hold, the covariance matrices can have full rank and minor eigenvalue variations are sufficient. There are special cases where the channel covariance matrices are linearly dependent, but these are not robust to minor perturbations of the covariance matrices. Hence, they are anomalies that will never appear in practice or be drawn from a random distribution, although they have frequently been studied in the academic literature. Since the SE of MR (also known as conjugate beamforming or matched filtering) generally has a finite limit, we conclude that this scheme is not asymptotically optimal in Massive MIMO. Note that our results do not imply that the pilot contamination effect disappears; there is still a performance loss caused by estimation errors and interference rejection, but there is no fundamental capacity limit.
Most of our results assume that the full covariance matrices of the channels are known, but this is not a critical requirement. Theorems \[theorem:EW-MMSE\_precoding\] and \[theorem:approximate\_M-MMSE\] proved that it is sufficient that the diagonals of the covariance matrices are known and linearly independent between pilot-sharing UEs; a condition that has been shown to hold for practical channels by the measurements in [@Gao2015b]. Such statistical information can be accurately estimated from only some tens of channel observations [@Bjornson2016c], whereof some contain the desired signal plus interference/noise and some contain only interference/noise.
The purpose of analyzing the asymptotic capacity when $M \to \infty$ is not that we advocate the deployment of BSs with a nearly infinite number of antennas—that is physically impossible in a finite-sized world and the conventional channel models will eventually break down since more power is received than was transmitted. The importance of asymptotics is instead what it tells us about practical networks with finite numbers of antennas. For example, consider a network with any finite number of UEs that each have a finite-valued data rate requirement. Our main results imply that we can always satisfy these requirements by deploying sufficiently many antennas, even in the presence of pilot contamination. In fact, it is enough to have two channel uses per coherence block (one for pilot, one for data) to deliver any capacity value to any finite number of UEs. The linear M-MMSE scheme is sufficient to achieve this in practice and interference can be treated as noise in the receivers, because the capacity lower bounds that we considered rely on such simplifications.
Appendix A – Useful Results {#appendix:useful-results .unnumbered}
===========================
\[lemma3\] Let ${\bf A} \in\mathbb{C}^{M\times M}$ and ${\bf x},{\bf y}\sim \CN ({\bf 0}, \frac{1}{M} {\bf I}_M)$. Assume that ${\bf A}$ has uniformly bounded spectral norm and that ${\bf x}$ and ${\bf y}$ are mutually independent and independent of ${\bf A}$. Then, $ {\bf x}^{\Htran}{\bf A}{\bf x} \asymp \frac{1}{M} \tr ({\bf A})$, ${\bf x}^{\Htran}{\bf A}{\bf y} \asymp 0$ and ${\mathbb{E}}\{|{\bf x}^{\Htran}{\bf A}{\bf x} - \frac{1}{M} \tr ({\bf A})|^p\}= \mathcal {O}({M^{-p/2}})$.
\[[@Marshall2011]\]\[lemma2\] For any positive semi-definite $M \times M$ matrices ${\mathbf{A}}$ and ${\mathbf{B}}$, it holds that $\frac{1}{M}\tr \left( {\mathbf{A}} {\mathbf{B}}\right) \le \| {\mathbf{A}} {\mathbf{B}} \|_2\le\| {\mathbf{A}} \|_2\|{\mathbf{B}} \|_2$, $\tr \left( {\mathbf{A}} {\mathbf{B}}\right) \le \| {\mathbf{A}} \|_2\tr \left( {\mathbf{B}}\right)$ and $ \tr \left( ({\mathbf{I}}+{\mathbf{A}})^{-1} {\mathbf{B}} \right) \geq \frac{1}{1+ \| {\mathbf{A}} \|_2} \tr ({\mathbf{B}})$.
\[Matrix inversion lemma\]\[MIL\] Let ${\bf A} \in\mathbb{C}^{M\times M}$ be a Hermitian invertible matrix, then for any vector ${\bf x}\in \mathbb{C}^{M}$ and any scalar $\rho \in \mathbb{C}$ such that ${\bf A} + \rho {\bf x}{\bf x}^{\Htran}$ is invertible ${\bf x}^{\Htran}({\bf A} + \rho {\bf x}{\bf x}^{\Htran})^{-1} = \frac{{\bf x}^{\Htran}{\bf A}^{-1}}{1 + \rho {\bf x}^{\Htran}{\bf A}^{-1}{\bf x}}$ and $({\bf A} + \rho {\bf x}{\bf x}^{\Htran})^{-1} = {\bf A}^{-1} - \frac{\rho{\bf A}^{-1}{\bf x}{\bf x}^{\Htran}{\bf A}^{-1}}{1 + \rho {\bf x}^{\Htran}{\bf A}^{-1}{\bf x}}$.
Let ${\bf U}, {\bf C},{\bf V}$ be matrices of compatible sizes, then if ${\bf C}$ is invertible $\left({\bf A} + {\bf U}{\bf C}{\bf V}\right)^{-1} = {\bf A}^{-1}-{\bf A}^{-1}{\bf U}\left({\bf C}^{-1} + {\bf V}{\bf A}^{-1}{\bf U}\right)^{-1}{\bf V}{\bf A}^{-1}$.
Appendix B – Proof of Theorem \[theorem:MMSE\] {#appendix:proof-theorem:MMSE .unnumbered}
==============================================
By applying Lemma \[MIL\], we may rewrite $\gamma_1^{\rm{ul}}$ in as $$\begin{aligned}
\label{eq:gamma_1.1}
\gamma_{1}^{\rm {ul}} = {M}\Bigg(\frac{1}{M}\hat{{\mathbf{h}}}_{1}^{\Htran}{\bf Z}^{-1}\hat{{\mathbf{h}}}_{1} - \frac{\Big|\frac{1}{M}\hat{{\mathbf{h}}}_{1}^{\Htran}{\bf Z}^{-1}\hat{{\mathbf{h}}}_{2}\Big|^2}{\frac{1}{M}+\frac{1}{M}\hat{{\mathbf{h}}}_{2}^{\Htran}{\bf Z}^{-1}\hat{{\mathbf{h}}}_{2}}\Bigg)\end{aligned}$$ by also multiplying and dividing each term by $M$. Under Assumption \[assumption\_1\] and using Lemma \[lemma3\] we have, as ${M \to \infty}$, that[^8] $$\begin{aligned}
\label{eq:beta_11}
\frac{1}{M}\hat{\bf h}_1^{\Htran}{\bf Z}^{-1}\hat{\bf h}_1 &\asymp \frac{1}{M}\tr ( {\mathbf{\Phi}}_{1}{\bf Z}^{-1} )\triangleq \beta_{11}\\\label{eq:beta_22}
\frac{1}{M}\hat{\bf h}_2^{\Htran}{\bf Z}^{-1}\hat{\bf h}_2 &\asymp\frac{1}{M}\tr ( {\mathbf{\Phi}}_{2}{\bf Z}^{-1} ) \triangleq \beta_{22}\\ \label{eq:beta_12}
\frac{1}{M}\hat{\bf h}_1^{\Htran}{\bf Z}^{-1}\hat{\bf h}_2 &\asymp\frac{1}{M}\tr ({\mathbf{\Upsilon}}_{12} {\bf Z}^{-1} ) \triangleq \beta_{12}.\end{aligned}$$ Note that $\beta_{11}$, $\beta_{22}$, and $\beta_{12}$ are non-negative real-valued scalars, since the trace of a product of positive semi-definite matrices is always non-negative. Using this notation, it follows from Assumption \[assumption\_1\] that[^9] $\liminf_M\beta_{22}>0$ and we obtain $$\begin{aligned}
\frac{\gamma_{1}^{\rm {ul}}}{M} \asymp \delta_1 \triangleq \beta_{11} - \frac{\beta_{12}^2}{ \beta_{22}}.
\label{eq:asympotitic_SINR}\end{aligned}$$ To proceed, notice that Assumption \[assumption\_2\] implies the following result, as proved in Appendix C.
\[cor:assumption3\] If Assumption \[assumption\_2\] holds, then for $\boldsymbol{\lambda} = [\lambda_1, \lambda_2]^{\Ttran} \in \mathbb{R}^2$ and $i=1,2$, $$\begin{aligned}
\notag
&\mathop {\liminf}\limits_M \inf_{\{\boldsymbol{\lambda}: \, \lambda_i=1\}} \\ &\frac{1}{{M}}\tr \Big( {\mathbf{Q}}^{-1}\big(\lambda_1{\mathbf{R}}_{1} +\lambda_2 {\mathbf{R}}_{2}\big) {\mathbf{Z}}^{-1} \big(\lambda_1{\mathbf{R}}_{1} +\lambda_2 {\mathbf{R}}_{2}\big) \Big) > 0. \label{eq:assumption_2}\end{aligned}$$
By expanding the condition in Corollary \[cor:assumption3\] for $i=1$, we have that $$\begin{aligned}
\label{eq:Appendix_B.1_3}
\liminf_M \inf_{\lambda_2}\left(\beta_{11} + {\lambda_2^2}\beta_{22} + 2\lambda_2\beta_{12}\right) >0.
\end{aligned}$$ By the definition of the $\liminf_M$ operator, $\liminf_M \beta_{22}>0$ holds if and only if every convergent subsequence has a non-zero limit, i.e., $\lim_M \beta_{22} >0$. This ensures that, for an arbitrary convergent subsequence, $$\begin{aligned}
\label{eq:Appendix_B.1_4}
\inf_{\lambda_2}\left(\beta_{11} + {\lambda_2^2}\beta_{22} + 2\lambda_2\beta_{12}\right) = \beta_{11} - \frac{\beta_{12}^2}{\beta_{22}} = \delta_1
\end{aligned}$$ where the infimum is attained by $\lambda_2 = \beta_{12}/\beta_{22}$. Substituting into , implies that $\liminf_M\delta_1>0$. Therefore, we have that $\gamma_{1}^{\rm {ul}}$ grows a.s. unboundedly and, thus, the first part of the theorem follows.
Since $\gamma_{1}^{\rm{ul}} $ grows a.s. unboundedly and the logarithm is a strictly increasing function, it follows that $\log_2(1+\gamma_{1}^{\rm{ul}})$ also grows a.s. without bound. Moreover, since the almost sure divergence of a sequence of non-negative random variables implies the divergence of its expected value, it follows that also $\mathsf{SE}_{1}^{\rm {ul}} = (1-\taupu/\tau_c) \mathbb{E} \left\{ \log_2 \left( 1 + \gamma_{1}^{\rm {ul}} \right) \right\}$ grows without bound.
Appendix C – Proof of Corollary \[cor:assumption3\] in Appendix B {#appendix:proof:cor:assumption3 .unnumbered}
=================================================================
Consider $i=1$ and notice that the argument on the left-hand side of is lower bounded as $$\begin{aligned}
\frac{ \frac{1}{M}\| {\mathbf{R}}_{1} + \lambda_2 {\mathbf{R}}_{2} \|_F^2 }{ ( \frac1{\rho^{\rm{tr}}} + \| {\mathbf{R}}_{1} + {\mathbf{R}}_{2}\|_2) ( \frac1{\rho^{\rm{ul}}} + \| \sum_{k=1}^{2} ({\mathbf{R}}_{k} - {\mathbf{\Phi}}_{k}) \|_2) } \label{eq:assumption_2_relaxed-derivation}\end{aligned}$$ by applying Lemma \[lemma2\] twice. The denominator of is bounded from above due to Assumption \[assumption\_1\] and independent of $\lambda_2$. This proves that Assumption \[assumption\_2\] is sufficient for to hold for $i=1$. The result for $i=2$ follows by interchanging the indices in the proof.
Appendix D – Proof of Theorem \[theorem:MMSE\_precoding\] {#appendix:proof-theorem:MMSE_precoding .unnumbered}
=========================================================
We begin by plugging into to obtain $$\begin{aligned}
\label{eq:gamma1_DL_11}
\gamma_{1}^{\rm {dl}} &= \frac{ | {\mathbb{E}}\left\{{\mathbf{h}}_{1}^{\Htran} {{\mathbf{v}}}_{1}\right\} |^2 }{ \frac{\vartheta_2}{\vartheta_1}{\mathbb{E}}\left\{| {\mathbf{h}}_{1}^{\Htran} {{\mathbf{v}}}_{2}|^2\right\} + {\mathbb{V}} \{ {{\mathbf{h}}}_{1}^{\Htran} {{\mathbf{v}}}_{1} \} + \frac{1}{\rho^{\rm dl}\vartheta_1}}.\end{aligned}$$ We need to characterize all the terms in and begin with ${\mathbb{E}}\left\{{\mathbf{h}}_{1}^{\Htran} {{\mathbf{v}}}_{1}\right\}$. Notice that ${\mathbb{E}}\left\{{\mathbf{h}}_{1}^{\Htran} {{\mathbf{v}}}_{1}\right\}={\mathbb{E}}\big\{\hat {{\mathbf{h}}}_{1}^{\Htran} {{\mathbf{v}}}_{1}\big\}$ since ${{\mathbf{v}}}_{1}$ is independent of the zero-mean error $\tilde{{\mathbf{h}}}_{1}$. Then, we can express $\hat{{\mathbf{h}}}_{1}^{\Htran} {{\mathbf{v}}}_{1}$ as $$\begin{aligned}
\label{eq:C.1}
\hat{{\mathbf{h}}}_{1}^{\Htran} {{\mathbf{v}}}_{1} =\frac{{\hat{{\mathbf{h}}}}_{1}^{\Htran}\left(\hat{{\mathbf{h}}}_{2} \hat{{\mathbf{h}}}_{2}^{\Htran} + {\mathbf{Z}} \right)^{-1} \hat{{\mathbf{h}}}_{1}}{{1 + {\hat{{\mathbf{h}}}}_{1}^{\Htran}\left(\hat{{\mathbf{h}}}_{2} \hat{{\mathbf{h}}}_{2}^{\Htran} + {\mathbf{Z}} \right)^{-1} \hat{{\mathbf{h}}}_{1}}}=\frac{\gamma_{1}^{\rm {ul}}}{1 + {\gamma_{1}^{\rm {ul}}}}\end{aligned}$$ by first applying Lemma \[MIL\] and then identifying $\gamma_{1}^{\rm {ul}}$ in in the numerator and denominator. Theorem \[theorem:MMSE\] proves that $\frac{\gamma_{1}^{\rm {ul}}}{M} \asymp \delta_1$ and applying this result to yields $\hat{{\mathbf{h}}}_{1}^{\Htran} {{\mathbf{v}}}_{1}\asymp 1$. By the dominated convergence theorem and the continuous mapping theorem [@Couillet_book], we then have that $|{\mathbb{E}}\{{\mathbf{h}}_{1}^{\Htran} {{\mathbf{v}}}_{1}\}|^2 \asymp1$.
Consider now the noise term $\frac{1}{\rho^{\rm dl}\vartheta_1} =\frac{{\mathbb{E}}\{\|{\mathbf{v}}_1\|^2\}}{\rho^{\rm dl}}$ where $\vartheta_1 = ({\mathbb{E}}\left\{\|{\mathbf{v}}_1\|^2\right\})^{-1}$. By applying Lemma \[MIL\] twice, we may rewrite $\|{\mathbf{v}}_1\|^2$ as $$\begin{aligned}
\notag
\|{\mathbf{v}}_1\|^2 &= \frac{\hat{{\mathbf{h}}}_{1}^{\Htran}\left(\hat{{\mathbf{h}}}_{2} \hat{{\mathbf{h}}}_{2}^{\Htran} + {\mathbf{Z}} \right)^{-2} \hat{{\mathbf{h}}}_{1} }{\left(1 +\gamma_{1}^{\rm {ul}}\right)^2} \\&= \frac{1}{M} \frac{\frac{1}{M} \hat{{\mathbf{h}}}_{1}^{\Htran}\left(\hat{{\mathbf{h}}}_{2} \hat{{\mathbf{h}}}_{2}^{\Htran} + {\mathbf{Z}} \right)^{-2} \hat{{\mathbf{h}}}_{1} }{\left(\frac{1}{M} + \frac{1}{M} \gamma_{1}^{\rm {ul}}\right)^2}.\label{eq:C_10}\end{aligned}$$ Let ${\rm{Re}}(\cdot)$ denote the real-valued part of a scalar. The numerator in can be expressed as $$\begin{aligned}
\notag\frac{1}{M}\hat{{\mathbf{h}}}_{1}^{\Htran}{\bf Z}^{-2}\hat{{\mathbf{h}}}_{1} &- 2\frac{{\rm{Re}}(\frac{1}{M}\hat{{\mathbf{h}}}_{1}^{\Htran}{\bf Z}^{-1}\hat{{\mathbf{h}}}_{2}\frac{1}{M}\hat{{\mathbf{h}}}_{2}^{\Htran}{\bf Z}^{-2}\hat{{\mathbf{h}}}_{1})}{\frac{1}{M} + \frac{1}{M}\hat{\bf h}_2^{\Htran}{\bf Z}^{-1}\hat{{\mathbf{h}}}_{2}} \\&+ \frac{\frac{1}{M}\hat{{\mathbf{h}}}_{2}^{\Htran}{\bf Z}^{-2}\hat{{\mathbf{h}}}_{2}|\frac{1}{M}\hat{{\mathbf{h}}}_{2}^{\Htran}{\bf Z}^{-1}\hat{{\mathbf{h}}}_{1}|^2}{\big(\frac{1}{M} + \frac{1}{M} \hat{\bf h}_2^{\Htran}{\bf Z}^{-1}\hat{{\mathbf{h}}}_{2}\big)^2}\end{aligned}$$ by applying again Lemma \[MIL\] twice. Under Assumption \[assumption\_1\] and by applying Lemma \[lemma3\], $$\begin{aligned}
\frac{1}{M}\hat{\bf h}_1^{\Htran}{\bf Z}^{-2}\hat{\bf h}_1 &\asymp \frac{1}{M}\tr ( {\mathbf{\Phi}}_{1}{\bf Z}^{-2} )\triangleq \beta_{11}^{\prime}\\
\frac{1}{M}\hat{\bf h}_2^{\Htran}{\bf Z}^{-2}\hat{\bf h}_2 &\asymp\frac{1}{M}\tr ( {\mathbf{\Phi}}_{2}{\bf Z}^{-2} ) \triangleq \beta_{22}^{\prime}\\
\frac{1}{M}\hat{\bf h}_1^{\Htran}{\bf Z}^{-2}\hat{\bf h}_2 &\asymp\frac{1}{M}\tr ({\mathbf{\Upsilon}}_{12} {\bf Z}^{-2} ) \triangleq \beta_{12}^{\prime}\end{aligned}$$ where $\beta_{11}^{\prime}$, $\beta_{22}^{\prime}$, and $\beta_{12}^{\prime}$ are non-negative real-valued scalars, since the trace of a product of positive semi-definite matrices is always non-negative. Therefore, we obtain $$\begin{aligned}
\label{eq:C_10_1}
\frac{1}{M}\hat{{\mathbf{h}}}_{1}^{\Htran}\left(\hat{{\mathbf{h}}}_{2} \hat{{\mathbf{h}}}_{2}^{\Htran} + {\mathbf{Z}} \right)^{-2} \hat{{\mathbf{h}}}_{1} &\asymp \beta_{11}^{\prime} - 2\frac{\beta_{12}\beta_{12}^{\prime}}{\beta_{22}}+\frac{\beta_{12}^2\beta_{12}^{\prime}}{(\beta_{22})^2} \triangleq \delta_1^{\prime} .\end{aligned}$$ Plugging into and using $\frac{\gamma_{1}^{\rm {ul}}}{M} \asymp \delta_1$ yields $M\|{\mathbf{v}}_1\|^2 \asymp \frac{\delta_1^{\prime}}{\delta_1^2}$ such that $$\begin{aligned}
\frac{1}{\rho^{\rm dl}\vartheta_1}=\frac{{\mathbb{E}}\left\{\|{\mathbf{v}}_1\|^2\right\}}{\rho^{\rm dl}}\asymp \frac{1}{M\rho^{\rm dl}} \frac{\delta_1^{\prime}}{\delta_1^2 }. \end{aligned}$$
Consider now the two terms ${\mathbb{V}} \{ {{\mathbf{h}}}_{1}^{\Htran} {{\mathbf{v}}}_{1} \}$ and $\frac{\vartheta_2}{\vartheta_1}{\mathbb{E}}\big\{|{{\mathbf{h}}}_{1}^{\Htran} {{\mathbf{v}}}_{2}|^2\big\}$. Similar to [@hoydis2013massive Eq. (47)], we can upper bound ${\mathbb{V}} \{ {{\mathbf{h}}}_{1}^{\Htran} {{\mathbf{v}}}_{1} \}$ as ${\mathbb{V}} \{ {{\mathbf{h}}}_{1}^{\Htran} {{\mathbf{v}}}_{1} \} \le 2{\mathbb{E}}\left\{\left|{{\mathbf{h}}}_{1}^{\Htran} {{\mathbf{v}}}_{1} - {\mathbb{E}}\left\{{{\mathbf{h}}}_{1}^{\Htran} {{\mathbf{v}}}_{1}\right\}\right|\right\} + {\mathbb{E}}\big\{\big|\tilde{{\mathbf{h}}}_{1}^{\Htran} {{\mathbf{v}}}_{1}\big|^2\big\}$. Notice that (by using ${\mathbb{E}}\left\{{{\mathbf{h}}}_{1}^{\Htran} {{\mathbf{v}}}_{1}\right\}\asymp1$ and the dominated convergence theorem) ${\mathbb{E}}\left\{\left|{{\mathbf{h}}}_{1}^{\Htran} {{\mathbf{v}}}_{1} - {\mathbb{E}}\left\{{{\mathbf{h}}}_{1}^{\Htran} {{\mathbf{v}}}_{1}\right\}\right|\right\} \asymp 0$ and $$\begin{aligned}
\notag
{\mathbb{E}}\big\{\big|\tilde{{\mathbf{h}}}_{1}^{\Htran} {{\mathbf{v}}}_{1}\big|^2\big\} &= {\mathbb{E}}\big\{{{\mathbf{v}}}_{1}^{\Htran}({\mathbf{R}}_{1} - {\mathbf{\Phi}}_{1}){{\mathbf{v}}}_{1}\big\}\\&\mathop \leq\limits^{(a)}\|{\mathbf{R}}_{1} - {\mathbf{\Phi}}_{1} \|_2{\mathbb{E}}\big\{\|{\mathbf{v}}_1\|^2\big\}\mathop \asymp\limits^{(b)}0\end{aligned}$$ where $(a)$ and $(b)$ follow from Lemma \[lemma2\] and ${\mathbb{E}}\big\{\|{\mathbf{v}}_1\|^2\big\} \asymp 0$ (since, as shown above, $\|{\mathbf{v}}_1\|^2 \asymp \frac{1}{M}\frac{\delta_1^{\prime}}{\delta_1^2} \asymp 0 $), respectively. Therefore, we have that ${\mathbb{V}} \{ {{\mathbf{h}}}_{1}^{\Htran} {{\mathbf{v}}}_{1} \} \asymp 0$. Finally, we consider $\frac{\vartheta_2}{\vartheta_1}{\mathbb{E}}\big\{|{{\mathbf{h}}}_{1}^{\Htran} {{\mathbf{v}}}_{2}|^2\big\}$. By using , , and $\liminf_M\beta_{11}>0$ (as follows from Assumption \[assumption\_1\]), we have that $$\begin{aligned}
\notag
{{\mathbf{h}}}_{1}^{\Htran} {{\mathbf{v}}}_{2}&\mathop =\limits^{(a)} \frac{{{\mathbf{h}}}_{1}^{\Htran} \left(\hat{{\mathbf{h}}}_{1} \hat{{\mathbf{h}}}_{1}^{\Htran} + {\mathbf{Z}} \right)^{-1} \hat{{\mathbf{h}}}_{2}}{1+\hat{{\mathbf{h}}}_{2}^{\Htran}\left(\hat{{\mathbf{h}}}_{1} \hat{{\mathbf{h}}}_{1}^{\Htran} + {\mathbf{Z}} \right)^{-1}\hat{{\mathbf{h}}}_{2}} \\& \mathop =\limits^{(b)} \frac{\frac{1}{M}{{\mathbf{h}}}_{1}^{\Htran}{\mathbf{Z}}^{-1}\hat{{\mathbf{h}}}_{2} - \frac{\frac{1}{M}{{\mathbf{h}}}_{1}^{\Htran}{\mathbf{Z}}^{-1}\hat{{\mathbf{h}}}_{1}\frac{1}{M}\hat{{\mathbf{h}}}_{1}{\mathbf{Z}}^{-1}\hat{{\mathbf{h}}}_{2}^{\Htran}}{\frac{1}{M}+\frac{1}{M}\hat{{\mathbf{h}}}_{1}^{\Htran}{\mathbf{Z}}^{-1}\hat{{\mathbf{h}}}_{1}}}{\frac{1}{M}+\frac{1}{M}\gamma_{2}^{\rm {ul}}}\notag\\&\mathop \asymp\limits^{(c)} \frac{\beta_{12} - \frac{\beta_{11}\beta_{12}}{\beta_{11}}}{\delta_2} = 0\end{aligned}$$ where $(a)$ and $(b)$ follow from Lemma \[MIL\] after identifying[^10] $\hat{{\mathbf{h}}}_{2}^{\Htran}\big(\hat{{\mathbf{h}}}_{1} \hat{{\mathbf{h}}}_{1}^{\Htran} + {\mathbf{Z}} \big)^{-1}\hat{{\mathbf{h}}}_{2}$ as $\gamma_{2}^{\rm {ul}}$ (by also dividing and multiplying by $M$), and $(c)$ follows by using , and the fact that $$\begin{aligned}
\frac{\gamma_{2}^{\rm {ul}}}{M} \asymp \delta_2 \triangleq \beta_{22} - \frac{\beta_{21}^2}{ \beta_{11}}
\label{eq:asympotitic_SINR_2}\end{aligned}$$ with $\liminf_M \delta_2 >0$ (which follows from the proof of Theorem \[theorem:MMSE\] by interchanging UE indices). By applying Lemma \[lemma3\], this implies ${\mathbb{E}}\big\{|{{\mathbf{h}}}_{1}^{\Htran} {{\mathbf{v}}}_{2}|^2\big\}\asymp 0$. [Observe now that $\frac{\vartheta_2}{\vartheta_1} \asymp \frac{\delta_{1}^\prime}{\delta_1^{2}}\frac{\delta_2^{2}}{\delta_{2}^\prime}$ where $\delta_{2}^\prime$ is obtained from $\delta_{1}^\prime$ by interchanging UE indices. Since all the quantities in $\delta_1^{\prime}$ are uniformly bounded (due to Assumption \[assumption\_1\]), $\liminf_M\delta_1>0$ (as proved in Appendix B) and $\liminf_M\delta_2<\infty$ (since from $\delta_2 < \beta_{22}$ and $\liminf_M \beta_{22} < \infty$ due to Assumption \[assumption\_1\]), we eventually have that $\frac{\vartheta_2}{\vartheta_1}{\mathbb{E}}\big\{|{{\mathbf{h}}}_{1}^{\Htran} {{\mathbf{v}}}_{2}|^2\big\}\asymp 0$. ]{}
Combining all the above results yields $$\begin{aligned}
\label{eq:C10_DL}
\frac{\gamma_{1}^{\rm {dl}}}{M} \asymp \rho^{\rm {dl}} \frac{\delta_1^2}{\delta_1^{\prime}}.\end{aligned}$$ Since all the quantities in $\delta_1^{\prime}$ are uniformly bounded and $\liminf_M\delta_1>0$, it follows that $\gamma_{1}^{\rm {dl}}$ grows unboundedly as $M \to \infty$. This implies that also $\mathsf{SE}_{1}^{\rm {dl}}$ grows unboundedly as $M\to \infty$, which can be proved by the same arguments as in the last paragraph of Appendix B.
Appendix E – Proof of Theorem \[theorem:EW-MMSE\_precoding\] {#appendix:proof-theorem:EW_MMSE_precoding .unnumbered}
============================================================
The EW-MMSE estimate $\hat{{\mathbf{h}}}_{k} $ and the estimation error $\tilde{{\mathbf{h}}}_{k}= {\mathbf{h}}_{k} - \hat{{\mathbf{h}}}_{k}$ are random vectors distributed as $\hat{{\mathbf{h}}}_{k} \sim \CN({\bf 0},{\boldsymbol \Sigma}_k)$ and $\tilde{{\mathbf{h}}}_{k} \sim \CN({\bf 0},{\tilde{\boldsymbol \Sigma}}_k)$ with ${\boldsymbol \Sigma}_k = {\mathbf{D}}_{k}
\boldsymbol{\Lambda}^{-1}{\mathbf{Q}}\boldsymbol{\Lambda}^{-1}{\mathbf{D}}_{k}$ and $\tilde {\boldsymbol \Sigma}_k = {\mathbf{R}}_{k} - {\mathbf{D}}_{k}
\boldsymbol{\Lambda}^{-1}{\mathbf{R}}_{k} - {\mathbf{R}}_{k}
\boldsymbol{\Lambda}^{-1}{\mathbf{D}}_{k} - {\boldsymbol \Sigma}_k$. Unlike with MMSE estimation, the vectors $\hat{{\mathbf{h}}}_{k} $ and $\tilde{{\mathbf{h}}}_{k}$ are correlated with ${\mathbb{E}}\{\hat{{\mathbf{h}}}_{k}\tilde{{\mathbf{h}}}_{k}^{\Htran}\} = {\mathbb{E}}\{\hat{{\mathbf{h}}}_{k}({{\mathbf{h}}}_{k} - \hat{{\mathbf{h}}}_{k})^{\Htran}\}= {\mathbf{D}}_{k}
\boldsymbol{\Lambda}^{-1}{\mathbf{R}}_{k} - {\boldsymbol \Sigma}_k$. Hence, ${{\mathbf{v}}}_{1}$ and $\tilde{{\mathbf{h}}}_{1}$ are also correlated. For later convenience, we also notice that ${\mathbb{E}}\{{{\mathbf{h}}}_{1}\hat{{\mathbf{h}}}_{1}^{\Htran}\} = {\mathbf{R}}_1\boldsymbol{\Lambda}^{-1}{\mathbf{D}}_{1} $, ${\mathbb{E}}\{{{\mathbf{h}}}_{1}\hat{{\mathbf{h}}}_{2}^{\Htran}\} = {\mathbf{R}}_1\boldsymbol{\Lambda}^{-1}{\mathbf{D}}_{2}$, and ${\mathbb{E}}\{\hat{{\mathbf{h}}}_{2}\hat{{\mathbf{h}}}_{1}^{\Htran}\} = {\mathbf{D}}_{2}
\boldsymbol{\Lambda}^{-1}{\mathbf{Q}}\boldsymbol{\Lambda}^{-1}{\mathbf{D}}_{1} = {\mathbf{\Theta}}_{21}$. We need to characterize all the terms in and begin with ${\mathbb{E}}\left\{{\mathbf{v}}_{1}^{\Htran} {{\mathbf{h}}}_{1}\right\}$. By applying Lemma \[MIL\] and by dividing and multiplying by $M$, we can express ${{\mathbf{v}}}_{1}^{\Htran}{{\mathbf{h}}}_{1}$ as $$\begin{aligned}
\label{eq:G.1}
{{\mathbf{v}}}_{1}^{\Htran}{{\mathbf{h}}}_{1} = \frac{\frac{1}{M}{\hat{{\mathbf{h}}}}_{1}^{\Htran}\left(\hat{{\mathbf{h}}}_{2} \hat{{\mathbf{h}}}_{2}^{\Htran} + {\mathbf{S}} \right)^{-1}{{\mathbf{h}}}_{1}}{{\frac{1}{M} + \frac{1}{M}{\hat{{\mathbf{h}}}}_{1}^{\Htran}\left(\hat{{\mathbf{h}}}_{2} \hat{{\mathbf{h}}}_{2}^{\Htran} + {\mathbf{S}} \right)^{-1} \hat{{\mathbf{h}}}_{1}}} = \frac{\frac{1}{M}\tilde \mu_{1}^{\rm ul}}{\frac{1}{M} + \frac{1}{M}{\mu}_{1}^{\rm ul}}.\end{aligned}$$ Notice that ${\mu}_{1}^{\rm ul}$ has the same form as $\gamma_{1}^{\rm {ul}}$ in , but with $\{\hat{{\mathbf{h}}}_{k}: k=1,2\}$ now given by . Under Assumption \[assumption\_1\] and by Lemma \[lemma3\],[^11] $$\begin{aligned}
\label{eq:alpha_11}
\frac{1}{M}\hat{\bf h}_1^{\Htran}{\mathbf{S}}^{-1}\hat{\bf h}_1 &\asymp \frac{1}{M}\tr ( {\mathbf{\Sigma}}_{1} {\mathbf{S}}^{-1}) = \frac{1}{M}\sum_{i=1}^M\frac{[{\mathbf{R}}_{1}]_{i,i}^2}{[{\mathbf{S}}]_{i,i}[{\mathbf{\Lambda}}]_{i,i}} \triangleq \alpha_{11}\\\label{eq:alpha_22}
\frac{1}{M}\hat{\bf h}_2^{\Htran}{\mathbf{S}}^{-1}\hat{\bf h}_2 &\asymp\frac{1}{M}\tr ( {\mathbf{\Sigma}}_{2} {\mathbf{S}}^{-1}) = \frac{1}{M}\sum_{i=1}^M\frac{[{\mathbf{R}}_{2}]_{i,i}^2}{[{\mathbf{S}}]_{i,i}[{\mathbf{\Lambda}}]_{i,i}} \triangleq \alpha_{22}\\
\frac{1}{M}\hat{\bf h}_1^{\Htran}{\mathbf{S}}^{-1}\hat{\bf h}_2 &\asymp\frac{1}{M}\tr ({\mathbf{\Theta}}_{21} {\mathbf{S}}^{-1}) \notag\\ &= \frac{1}{M}\sum_{i=1}^M\frac{[{\mathbf{R}}_{1}]_{i,i}[{\mathbf{R}}_{2}]_{i,i}}{[{\mathbf{S}}]_{i,i}[{\mathbf{\Lambda}}]_{i,i}} \triangleq \alpha_{12}. \label{eq:alpha_12}\end{aligned}$$ By applying the same line of reasoning as when analyzing $\gamma_{1}^{\rm {ul}}$ in Appendix B and exploiting the fact that $\liminf_M\alpha_{22}>0$ (which follows from Assumption \[assumption\_1\]), we obtain $\frac{\mu_{1}^{\rm ul}}{M} = \frac{1}{M}{\hat{{\mathbf{h}}}}_{1}^{\Htran}\left(\hat{{\mathbf{h}}}_{2} \hat{{\mathbf{h}}}_{2}^{\Htran} + {\mathbf{S}} \right)^{-1}{{\mathbf{h}}}_{1}\asymp \upsilon_1 \triangleq \alpha_{11} - \frac{\alpha_{12}^2}{\alpha_{22}}$. Note that $\liminf_M \upsilon_1 > 0$ under Assumption \[assumption\_6\]. This can be proved, as done in Appendix B for $\delta_1$, by expanding the condition reported in the corollary below (the proof unfolds from the same arguments as in Appendix C).
\[corollary\_6\] If Assumption \[assumption\_6\] holds, then for $\boldsymbol{\lambda} = [\lambda_1, \lambda_2]^{\Ttran} \in \mathbb{R}^2$ and $i=1,2$, $$\begin{aligned}
\notag
& \mathop {\liminf}\limits_M \inf_{\{\boldsymbol{\lambda}: \, \lambda_i=1\}} \\ &\frac{1}{{M}}\tr \Big( \boldsymbol{\Lambda}^{-1}{\mathbf{Q}}\boldsymbol{\Lambda}^{-1}\big(\lambda_1{\mathbf{D}}_{1} +\lambda_2 {\mathbf{D}}_{2}\big){\mathbf{S}}^{-1}\big(\lambda_1{\mathbf{D}}_{1} +\lambda_2 {\mathbf{D}}_{2}\big) \Big) > 0. \label{eq:assumption_6_expanded}\end{aligned}$$
As for $\tilde \mu_{1}^{\rm ul}$ in , we have that $$\begin{aligned}
\notag
\frac{1}{M}\tilde \mu_{1}^{\rm ul} &= \frac{1}{M}{\hat{{\mathbf{h}}}}_{1}^{\Htran}\left(\hat{{\mathbf{h}}}_{2} \hat{{\mathbf{h}}}_{2}^{\Htran} + {\mathbf{S}} \right)^{-1}{{\mathbf{h}}}_{1} \\&= \frac{1}{M} {\hat {{\mathbf{h}}}}_{1}^{\Htran}{\mathbf{S}}^{-1}{{\mathbf{h}}}_{1} - \frac{\frac{1}{M}{\hat {{\mathbf{h}}}}_{1}^{\Htran}{\mathbf{S}}^{-1}\hat{{\mathbf{h}}}_{2}\frac{1}{M}\hat{{\mathbf{h}}}_{2}^{\Htran}{\mathbf{S}}^{-1}{{\mathbf{h}}}_{1}}{\frac{1}{M}+\frac{1}{M}\hat{{\mathbf{h}}}_{2}^{\Htran}{\mathbf{S}}^{-1}\hat{{\mathbf{h}}}_{2}} \asymp \upsilon_1\end{aligned}$$ since the diagonal structure of the matrices $\boldsymbol{\Lambda}$, ${\mathbf{D}}_{1}$, ${\mathbf{D}}_{2}$, and ${\mathbf{S}}$ implies that $$\begin{aligned}
\notag
\frac{1}{M}\hat{\bf h}_1^{\Htran}{\mathbf{S}}^{-1}{\bf h}_1 &\asymp \frac{1}{M}\tr ( {\mathbf{R}}_{1}\boldsymbol{\Lambda}^{-1}{\mathbf{D}}_{1}{\mathbf{S}}^{-1}) \\&= \frac{1}{M}\sum_{i=1}^M\frac{[{\mathbf{R}}_{1}]_{i,i}^2}{[{\mathbf{S}}]_{i,i}[{\mathbf{\Lambda}}]_{i,i}} = \alpha_{11}\\\notag
\frac{1}{M}\hat{\bf h}_2^{\Htran}{\mathbf{S}}^{-1} {\bf h}_1 &\asymp\frac{1}{M}\tr ({\mathbf{R}}_{1} \boldsymbol{\Lambda}^{-1}{\mathbf{D}}_{2} {\mathbf{S}}^{-1}) \\&= \frac{1}{M}\sum_{i=1}^M\frac{[{\mathbf{R}}_{1}]_{i,i}[{\mathbf{R}}_{2}]_{i,i}}{[{\mathbf{S}}]_{i,i}[{\mathbf{\Lambda}}]_{i,i}} = \alpha_{12}.\end{aligned}$$ Applying these results to yields ${\mathbf{v}}_{1}^{\Htran} {{\mathbf{h}}}_{1} \asymp 1$ from which it follows that $|{\mathbb{E}}\{{\mathbf{v}}_{1}^{\Htran} {{{\mathbf{h}}}}_{1}\}|^2 \asymp 1$.
$$\begin{aligned}
\label{eq:sec_E_inverse_of_MMSE_matrix}
\left( {\bf A}_{j,\setminus k}+ \hat{{\mathbf{H}}}_{jk,\setminus j}\hat{{\mathbf{H}}}_{jk,\setminus j}^{\Htran}\right)^{-1} = {\bf A}_{j,\setminus k}^{-1} - {\bf A}_{j,\setminus k}^{-1}\hat{{\mathbf{H}}}_{jk,\setminus j}\left( {\bf I}_{L-1}+ \hat{{\mathbf{H}}}_{jk,\setminus j}^{\Htran}{\bf A}_{j,\setminus k}^{-1}\hat{{\mathbf{H}}}_{jk,\setminus j}\right)^{-1}\!\hat{{\mathbf{H}}}_{jk,\setminus j}^{\Htran}{\bf A}_{j,\setminus k}^{-1}.\end{aligned}$$
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$$\begin{aligned}
\frac{\gamma_{jk}^{\rm {ul}}}{M} = \frac{1}{M} \hat{{\mathbf{h}}}_{jjk}^{\Htran}{\bf A}_{j,\setminus k}^{-1}\hat{{\mathbf{h}}}_{jjk} -
\frac{1}{M} \hat{{\mathbf{h}}}_{jjk}^{\Htran} {\bf A}_{j,\setminus k}^{-1}\hat{{\mathbf{H}}}_{jk,\setminus j}\left( \frac{1}{M} {\bf I}_{L-1}+ \frac{1}{M}\hat{{\mathbf{H}}}_{jk,\setminus j}^{\Htran}{\bf A}_{j,\setminus k}^{-1}\hat{{\mathbf{H}}}_{jk,\setminus j}\right)^{-1}\!\!\!\frac{1}{M}\hat{{\mathbf{H}}}_{jk,\setminus j}^{\Htran}{\bf A}_{j,\setminus k}^{-1}\hat{{\mathbf{h}}}_{jjk}.\label{eq:appendixc_7}\end{aligned}$$
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$$\begin{aligned}
\notag
\frac{1}{M}{\bf A}_{j,\setminus k}^{-1} & = \frac{1}{M}\Big(\sum\limits_{l} \sum\limits_{i\ne k} \hat{{\mathbf{h}}}_{jli}\hat{{\mathbf{h}}}_{jli}^{\Htran} + {\mathbf{Z}}_j\Big)^{-1} = \frac{1}{M}\Big( \hat{{\mathbf{H}}}_{j,\setminus k} \hat{{\mathbf{H}}}_{j,\setminus k}^{\Htran} + {\mathbf{Z}}_j\Big)^{-1} \\ &= \frac{1}{M}{\mathbf{Z}}_j^{-1} - \frac{1}{M}{\mathbf{Z}}_j^{-1}\hat{{\mathbf{H}}}_{j,\setminus k} \left( \frac{1}{M} {\bf I}_{L(K-1)}+ \frac{1}{M}\hat{{\mathbf{H}}}_{j,\setminus k}^{\Htran}{\mathbf{Z}}_j^{-1}\hat{{\mathbf{H}}}_{j,\setminus k}\right)^{-1}\!\!\!\frac{1}{M}\hat{{\mathbf{H}}}_{j,\setminus k}^{\Htran}{\mathbf{Z}}_j^{-1}\label{eq:newInFigure}\end{aligned}$$
------------------------------------------------------------------------
Next, consider the noise term $\frac{1}{\rho^{\rm ul}} {\mathbb{E}}\left\{||{\mathbf{v}}_{1} ||^2\right\}$ for which $$\begin{aligned}
\|{\mathbf{v}}_1\|^2 &= \frac{\hat{{\mathbf{h}}}_{1}^{\Htran}\left(\hat{{\mathbf{h}}}_{2} \hat{{\mathbf{h}}}_{2}^{\Htran} + {\mathbf{S}} \right)^{-2} \hat{{\mathbf{h}}}_{1} }{\left(1 + \mu_1^{\rm ul}\right)^2} \\ &= \frac{1}{M} \frac{\frac{1}{M} \hat{{\mathbf{h}}}_{1}^{\Htran}\left(\hat{{\mathbf{h}}}_{2} \hat{{\mathbf{h}}}_{2}^{\Htran} + {\mathbf{S}}\right)^{-2} \hat{{\mathbf{h}}}_{1} }{\left(\frac{1}{M} + \frac{1}{M} \mu_{1}^{\rm ul}\right)^2} . \label{eq:G_10}\end{aligned}$$ Under Assumption \[assumption\_1\] and by Lemma \[lemma3\], $$\begin{aligned}
\frac{1}{M}\hat{\bf h}_1^{\Htran}{\bf S}^{-2}\hat{\bf h}_1 &\asymp \frac{1}{M}\tr ( {\mathbf{\Sigma}}_{1}{\bf S}^{-2} ) = \frac{1}{M}\sum_{i=1}^M\frac{[{\mathbf{R}}_{1}]_{i,i}^2}{[{\mathbf{S}}]_{i,i}^2[{\mathbf{\Lambda}}]_{i,i}}\triangleq \alpha_{11}^{\prime}\\
\frac{1}{M}\hat{\bf h}_2^{\Htran}{\bf S}^{-2}\hat{\bf h}_2 &\asymp\frac{1}{M}\tr ( {\mathbf{\Sigma}}_{2}{\bf S}^{-2} ) = \frac{1}{M}\sum_{i=1}^M\frac{[{\mathbf{R}}_{2}]_{i,i}^2}{[{\mathbf{S}}]_{i,i}^2[{\mathbf{\Lambda}}]_{i,i}} \triangleq \alpha_{22}^{\prime}\\ \notag
\frac{1}{M}\hat{\bf h}_1^{\Htran}{\bf S}^{-2}\hat{\bf h}_2 &\asymp\frac{1}{M}\tr ({\mathbf{\Theta}}_{21} {\bf S}^{-2} ) \\ &= \frac{1}{M}\sum_{i=1}^M\frac{[{\mathbf{R}}_{1}]_{i,i}[{\mathbf{R}}_{2}]_{i,i}}{[{\mathbf{S}}]_{i,i}^2[{\mathbf{\Lambda}}]_{i,i}}\triangleq \alpha_{12}^{\prime}\end{aligned}$$ where $\alpha_{11}^{\prime}$, $\alpha_{22}^{\prime}$, and $\alpha_{12}^{\prime}$ are non-negative real-valued scalars. [By applying Lemma \[MIL\] twice to the numerator in and by using the above results, we obtain]{} $$\begin{aligned}
\label{eq:G_10_1}
\frac{1}{M} \hat{{\mathbf{h}}}_{1}^{\Htran}\left(\hat{{\mathbf{h}}}_{2} \hat{{\mathbf{h}}}_{2}^{\Htran} + {\mathbf{S}} \right)^{-2} \hat{{\mathbf{h}}}_{1} \asymp \alpha_{11}^{\prime} - 2\frac{\alpha_{12}\alpha_{12}^{\prime}}{\alpha_{22}}+\frac{\alpha_{12}^2\alpha_{22}^{\prime}}{(\alpha_{22})^2} \triangleq \upsilon_1^\prime.\end{aligned}$$ Plugging into yields $M\|{\mathbf{v}}_1\|^2 \asymp \frac{\upsilon_1^{\prime}}{\upsilon_{1}^2}$ such that $
\frac{1}{\rho^{\rm ul}} {\mathbb{E}}\left\{||{\mathbf{v}}_{1} ||^2\right\}\asymp \frac{1}{M\rho^{\rm ul}} \frac{\upsilon_1^{\prime}}{\upsilon_1^2}.
$
As for ${\mathbb{V}} \{ {{\mathbf{v}}}_{1}^{\Htran} {{\mathbf{h}}}_{1} \}$, it can be easily proved (using the above results and Lemma \[lemma3\]), that ${\mathbb{V}} \{ {{\mathbf{v}}}_{1}^{\Htran} {{\mathbf{h}}}_{1} \} \asymp 0$. Consider now the interference term ${\mathbb{E}}\big\{| {{\mathbf{v}}}_{1}^{\Htran}{{\mathbf{h}}}_{2}|^2\big\}$. Using and , we have (by applying Lemma \[MIL\] and dividing and multiplying by $M$) that $$\begin{aligned}
\notag
{{\mathbf{v}}}_{1}^{\Htran}{{\mathbf{h}}}_{2}&= \frac{\frac{1}{M}\hat {{\mathbf{h}}}_{1}^{\Htran} \left(\hat{{\mathbf{h}}}_{2} \hat{{\mathbf{h}}}_{2}^{\Htran} +{\bf S} \right)^{-1} {{\mathbf{h}}}_{2}}{\frac{1}{M}+\frac{1}{M}\mu_1^{\rm {ul}}} \\&= \frac{\frac{1}{M}\hat{{\mathbf{h}}}_{1}^{\Htran}{\bf S}^{-1}{{\mathbf{h}}}_{2} - \frac{ \frac{1}{M} \hat{{\mathbf{h}}}_{1}^{\Htran}{\bf S}^{-1}\hat{{\mathbf{h}}}_{2}\frac{1}{M}\hat{{\mathbf{h}}}_{2}^{\Htran}{\bf S}^{-1}{{\mathbf{h}}}_{2}}{\frac{1}{M}+\frac{1}{M}\hat{{\mathbf{h}}}_{2}^{\Htran}{\bf S}^{-1}\hat{{\mathbf{h}}}_{2}}}{\frac{1}{M}+\frac{1}{M}\mu_1^{\rm {ul}}}\notag\\&\asymp \frac{\alpha_{12} - \frac{\alpha_{12}\alpha_{22}}{\alpha_{22}}}{\upsilon_1} = 0\label{eq:appendixF_interference}\end{aligned}$$ where we have used the fact that $\frac{1}{M}\hat{{\mathbf{h}}}_{1}^{\Htran}{\bf S}^{-1}{{\mathbf{h}}}_{2} \asymp \alpha_{12}$ and $
\frac{1}{M}\hat{{\mathbf{h}}}_{2}^{\Htran}{\bf S}^{-1}{{\mathbf{h}}}_{2}\asymp \alpha_{22}$. Applying Lemma \[lemma3\] to , we obtain ${\mathbb{E}}\big\{|{{\mathbf{v}}}_{1}^{\Htran} {{\mathbf{h}}}_{2}|^2\big\}\asymp0$.
Combining all the above results together yields $\frac{{\underline{\gamma}}_{1}^{\rm ul}}{M} \asymp \rho^{\rm{ul}}\frac{\upsilon_1^2}{\upsilon_1^{\prime}}$. Since all the components of $\upsilon_1^{\prime}$ in are uniformly bounded and $\liminf_M\upsilon_1>0$ (under Assumption \[assumption\_6\]), it follows that ${\underline{\gamma}}_{1}^{\rm ul}$ grows unboundedly as $M \to \infty$. Hence, ${\underline{\mathsf{SE}}}_{1}^{\rm ul}$ also grows without bound.
Appendix F – Proof of Theorem \[theorem:M-MMSE\] {#proof:Theorem4 .unnumbered}
================================================
We start by rewriting $\gamma_{jk}^{\rm{ul}}$ in as $$\begin{aligned}
\label{eq:appendixc_1}
\gamma_{jk}^{\rm {ul}} =
\hat{{\mathbf{h}}}_{jjk}^{\Htran} \Bigg( \underbrace{\sum_{l} \sum_{i\ne k} \hat{{\mathbf{h}}}_{jli}\hat{{\mathbf{h}}}_{jli}^{\Htran} + {\mathbf{Z}}_j}_{{\bf A}_{j,\setminus k}} + \underbrace{\sum\limits_{l\ne j} \hat{{\mathbf{h}}}_{jlk}\hat{{\mathbf{h}}}_{jlk}^{\Htran}}_{\hat{{\mathbf{H}}}_{jk,\setminus j}\hat{{\mathbf{H}}}_{jk,\setminus j}^{\Htran}}\Bigg)^{-1} \hat{{\mathbf{h}}}_{jjk} \end{aligned}$$ where ${\bf A}_{j,\setminus k} = \sum_{l} \sum_{i\ne k} \hat{{\mathbf{h}}}_{jli}\hat{{\mathbf{h}}}_{jli}^{\Htran} + {\mathbf{Z}}_j$ is independent of $\{\hat{{\mathbf{h}}}_{jlk}: l=1,\ldots L\} $ and $\hat{{\mathbf{H}}}_{jk,\setminus j} = [\hat{{\mathbf{h}}}_{j1k} \ldots \hat{{\mathbf{h}}}_{jj-1k} \, \hat{{\mathbf{h}}}_{jj+1k} \ldots \hat{{\mathbf{h}}}_{jLk}] \in\mathbb{C}^{M\times(L-1)}$ collects all vectors $\hat{{\mathbf{h}}}_{jlk}$ with $l\ne j$ (i.e., the channels of UEs that cause pilot contamination). By Lemma \[MIL\], we obtain at the top of the page. Plugging into and dividing both sides by $M$ leads to . By applying Lemma \[MIL\] once again, follows where $\hat{{\mathbf{H}}}_{j,\setminus k}\in\mathbb{C}^{M\times L(K-1)}$ denotes the matrix collecting all vectors $\hat{{\mathbf{h}}}_{jli}$ with $i\ne k$, which is independent of $\hat{{\mathbf{h}}}_{jlk}$ for any $j$ and $l$. Therefore, it follows that the first term in is such that $$\begin{aligned}
\notag
\frac{1}{M}\hat{{\mathbf{h}}}_{jjk}^{\Htran}{\bf A}_{j,\setminus k}^{-1}\hat{{\mathbf{h}}}_{jjk} & \mathop\asymp^{{(a)}} \frac{1}{M}\hat{{\mathbf{h}}}_{jjk}^{\Htran}{\mathbf{Z}}_j^{-1}\hat{{\mathbf{h}}}_{jjk} \\&\mathop\asymp^{{(b)}} \frac{1}{M}\tr ( {\mathbf{\Phi}}_{jjk} {{\mathbf{Z}}}_{j}^{-1} ) \triangleq \beta_{jk,jj}
\end{aligned}$$ where ${{(a)}}$ follows from Lemma \[lemma3\] since $\hat{{\mathbf{h}}}_{jjk}$ and $\hat{{\mathbf{H}}}_{j,\setminus k}$ are independent and thus $\frac{1}{M}\hat{{\mathbf{h}}}_{jjk}^{\Htran}{\mathbf{Z}}^{-1}\hat{{\mathbf{H}}}_{j,\setminus k} \asymp {{\mathbf{0}}}_{L(K-1)}$ (remember that $\hat{{\mathbf{H}}}_{j,\setminus k}$ collects the $L(K-1)$ vectors $\{\hat{{\mathbf{h}}}_{jli}\}$ with $i\ne k$), and ${{(b)}}$ follows from Lemma \[lemma3\] by recalling that $\hat{{\mathbf{h}}}_{jjk}\sim \CN \left( {\mathbf{0}}, {\mathbf{\Phi}}_{jjk} \right)$ where the matrices ${\mathbf{\Phi}}_{jjk} $ can be proved (using Lemma \[lemma2\]) to have uniformly bounded spectral norm due to Assumption \[assumption\_3\]. Using similar arguments, we have that the $l$th element of the row vector $\frac{1}{M}\hat{{\mathbf{h}}}_{jjk}^{\Htran} {\bf A}_{j,\setminus k}^{-1}\hat{{\mathbf{H}}}_{jk,\setminus j} \in \mathbb{C}^{1\times (L-1)}$ is such that $$\begin{aligned}
\notag
\left[\frac{1}{M}\hat{{\mathbf{h}}}_{jjk}^{\Htran} {\bf A}_{j,\setminus k}^{-1}\hat{{\mathbf{H}}}_{jk,\setminus j}\right]_l & \asymp \frac{1}{M} \hat{{\mathbf{h}}}_{jjk}^{\Htran} {\bf Z}_j^{-1}\hat{{\mathbf{h}}}_{jlk} \\&\hspace{-3cm}\asymp \frac{1}{M}\tr ({\mathbf{R}}_{jlk} {\mathbf{Q}}_{jk}^{-1} {\mathbf{R}}_{jjk}{{\mathbf{Z}}}_{j}^{-1} ) \triangleq \beta_{jk,lj}\label{eq:Section_Appendix_81}
\end{aligned}$$ for $l=1,2,\ldots,L-1$. Furthermore, the $(n,m)$th element of $\frac{1}{M} \hat{{\mathbf{H}}}_{jk,\setminus j}^{\Htran}{\bf A}_{j,\setminus k}^{-1}\hat{{\mathbf{H}}}_{jk,\setminus j}$ is $$\begin{aligned}
\notag
\frac{1}{M}\left[\hat{{\mathbf{H}}}_{jk,\setminus j}^{\Htran}{\bf A}_{j,\setminus k}^{-1}\hat{{\mathbf{H}}}_{jk,\setminus j}\right]_{n,m} &\asymp \frac{1}{M}\hat{{\mathbf{h}}}_{jnk}^{\Htran}{\mathbf{Z}}_j^{-1}\hat{{\mathbf{h}}}_{jmk} \\&\hspace{-3.5cm}\asymp \frac{1}{M}\tr ( {\mathbf{R}}_{jmk} {\mathbf{Q}}_{jk}^{-1} {\mathbf{R}}_{jnk}{{\mathbf{Z}}}_{j}^{-1}) \triangleq \beta_{jk,mn}.\label{eq:Section_Appendix_82}
\end{aligned}$$ For notational convenience, let us define ${\mathbf{b}}_{jk}\in \mathbb{R}^{L-1}$ and ${\mathbf{C}}_{jk} \in \mathbb{R}^{(L-1)\times (L-1)}$ with entries $$\begin{aligned}
\notag
\big[{{\mathbf{b}}_{jk}}\big]_l = \beta_{jk,lj} &= \vecoperator \left(\frac{1}{\sqrt M} {{\mathbf{Z}}}_{j}^{-1/2}{\mathbf{R}}_{jlk} {\mathbf{Q}}_{jk}^{-1/2}\right)^{\Htran}\\&\vecoperator \left(\frac{1}{\sqrt M}{{\mathbf{Z}}}_{j}^{-1/2}{\mathbf{R}}_{jjk} {\mathbf{Q}}_{jk}^{-1/2}\right)\end{aligned}$$ and $$\begin{aligned}
\notag
\big[{\mathbf{C}}_{jk} \big]_{l,n} = \beta_{jk,ln} &= \vecoperator \left(\frac{1}{\sqrt M}{{\mathbf{Z}}}_{j}^{-1/2}{\mathbf{R}}_{jlk} {\mathbf{Q}}_{jk}^{-1/2}\right)^{\Htran}\\&\vecoperator \left(\frac{1}{\sqrt M}{{\mathbf{Z}}}_{j}^{-1/2}{\mathbf{R}}_{jnk} {\mathbf{Q}}_{jk}^{-1/2}\right) \label{eq:C_jk}\end{aligned}$$ where we have used the fact that $\tr({\bf AB}) = \vecoperator({\bf A}^{\Htran})^{\Htran}\vecoperator({\bf B})$. In Appendix G, it is shown that, under Assumption \[assumption\_4\], the following corollary holds.
\[assumption\_4\_1\] If Assumption \[assumption\_4\] holds, then for any UE $k$ in cell $j$ with ${\boldsymbol\lambda}_{jk} = [\lambda_{j1k}, \ldots,\lambda_{jLk}]^{\Ttran} \in \mathbb{R}^{L}$ and $l'=1,\ldots,L$ $$\begin{aligned}
\notag
& \liminf_M \inf_{\{{\boldsymbol\lambda}_{jk}:\lambda_{jl'k}=1\}} \\ &\frac{1}{{M}}\tr \Bigg( {\mathbf{Q}}_{jk}^{-1} \Big( \sum\limits_{l=1}^L\lambda_{jlk} {\mathbf{R}}_{jlk}\Big) {\mathbf{Z}}_{j}^{-1} \Big(\sum\limits_{l=1}^L\lambda_{jlk} {\mathbf{R}}_{jlk}\Big) \Bigg) > 0 \label{Condition2_Assumption4_11}
\end{aligned}$$ and the matrix ${\mathbf{C}}_{jk}$ is invertible as $M\to \infty$.
Since ${\mathbf{C}}_{jk}$ is invertible as $M\to \infty$ under Assumption \[assumption\_4\], we have that $\frac{\gamma_{jk}^{\rm {ul}}}{M}$ in is such that$$\begin{aligned}
\label{eq:general result}
\frac{\gamma_{jk}^{\rm {ul}}}{M} \asymp \delta_{jk} \triangleq \beta_{jj,jk} - {\mathbf{b}}_{jk}^{\Htran}{\mathbf{C}}_{jk}^{-1}{{\mathbf{b}}_{jk}}.\end{aligned}$$ Expanding condition in Corollary \[assumption\_4\_1\] for $\lalt=j$ and using the definitions of ${\mathbf{b}}_{jk}$ and ${\mathbf{C}}_{jk}$ yield $$\begin{aligned}
\label{eq:Appendix_E.1_3}
\liminf_M \inf_{\overline{\boldsymbol\lambda}_{jk}} \left(\beta_{jj,jk} +2\overline{\boldsymbol\lambda}_{jk}^{\Ttran}{\mathbf{b}}_{jk}+ \overline{\boldsymbol\lambda}_{jk}^{\Ttran}{\mathbf{C}}_{jk}\overline{\boldsymbol\lambda}_{jk}\right) >0
\end{aligned}$$ with $\overline{\boldsymbol\lambda}_{jk} =[\lambda_{j1k}, \ldots,\lambda_{j(j-1)k}, \lambda_{j(j+1)k}, \ldots,\lambda_{jLk}]^{\Ttran} \in \mathbb{R}^{L-1}$. The invertibility of ${\mathbf{C}}_{jk}$ as $M\to \infty$ ensures that the infimum exists for sufficiently large $M$ and that it is given by $$\begin{aligned}
\notag
\inf_{\overline{\boldsymbol{\lambda}}_{jk}}\big(\beta_{jj,jk} + 2\overline{\boldsymbol{\lambda}}_{jk}^{\Ttran}{\mathbf{b}}_{jk}&+ \overline{\boldsymbol{\lambda}}_{jk}^{\Ttran}{\mathbf{C}}_{jk}\overline{\boldsymbol{\lambda}}_{jk}\big) \\&= \beta_{jj,jk} - {\bf b}_{jk}^{\Ttran}{\mathbf{C}}_{jk}^{-1}{\bf b}_{jk} = \delta_{jk}\label{eq:Appendix_E.1_4}\end{aligned}$$ where the infimum is attained by $\overline{\boldsymbol{\lambda}}_{jk} = {\mathbf{C}}_{jk}^{-1}{\bf b}_{jk}$. Substituting into implies that $\liminf_M\delta_{jk}>0$. Therefore, $\gamma_{jk}^{\rm {ul}}$ grows a.s. unboundedly and this implies that $\mathsf{SE}_{jk}^{\rm {ul}}$ grows unboundedly as $M\to \infty$, which can be proved as done in the last paragraph of Appendix B.
Appendix G – Proof of Corollary \[assumption\_4\_1\] in Appendix F {#appendix:invertibility-C .unnumbered}
==================================================================
The argument of the left-hand side of can be lower bounded by $$\begin{aligned}
\label{eq:AppendixG_99}
\frac{\frac{1}{{M}}\big\| \sum_{l=1}^L\lambda_{jlk} {\mathbf{R}}_{jlk} \big\|_F^2}{\big( \frac1{\rho^{\rm{tr}}} + \big\| \sum_{l=1}^L{\mathbf{R}}_{jlk} \big\|_2\big) \big( \frac1{\rho^{\rm{ul}}} + \big\| \sum_{l=1}^L \big({\mathbf{R}}_{jlk} - {\mathbf{\Phi}}_{jlk}\big) \big\|_2\big) }\end{aligned}$$ by applying Lemma \[lemma2\] twice. Notice that the denominator is bounded due to Assumption \[assumption\_4\] and independent of $\{\lambda_{ljk}\}$. Therefore, if holds, it follows from that also holds.
We now exploit to prove that ${\mathbf{C}}_{jk}$ is invertible for sufficiently large $M$. To this end, observe that ${\mathbf{C}}_{jk}$ with entries given by is a Gramian matrix obtained as the inner products of the vectors $\{{\bf u}_{jlk} = \vecoperator \big(\frac{1}{\sqrt M}{{\mathbf{Z}}}_{j}^{-1/2}{\mathbf{R}}_{jlk} {\mathbf{Q}}_{jk}^{-1/2}\big): \forall l\ne j\}$. Therefore, as $M$ grows large the matrix ${\mathbf{C}}_{jk}$ is invertible if and only if the vectors $\{{\bf u}_{jlk}: \forall l\ne j\} $ are asymptotically linearly independent. Notice that the condition in Corollary \[assumption\_4\_1\] for $\lalt=j$ can be rewritten in compact form as $$\begin{aligned}
\notag
\liminf_M \inf_{\{{\boldsymbol\lambda}_{jk}: \lambda_{jjk}=1\}}&\left({\bf u}_{jjk} + \sum\nolimits_{l\ne j}\lambda_{jlk}{\bf u}_{jlk}\right)^{\!\Htran}\\&\left({\bf u}_{jjk} + \sum\nolimits_{l\ne j}\lambda_{jlk}{\bf u}_{jlk}\right) > 0\label{eq:AppendixG_911}\end{aligned}$$ which implies that the vectors $\{{\bf u}_{jlk}: \forall l\}$ are asymptotically linearly independent. Since any subset of a finite set with linearly independent vectors is also linearly independent, ensures that $\{{\bf u}_{jlk}: \forall l\ne j\}$ are also asymptotically linearly independent. This proves that, under Assumption \[assumption\_4\], the Gramian matrix ${\bf C}_{jk}$ is invertible as $M\to \infty$ and this completes the proof.
\[[{width="1.0in" height="1.25in"}]{}\] [Emil Björnson]{}(S’07, M’12) received the M.S. degree in Engineering Mathematics from Lund University, Sweden, in 2007. He received the Ph.D. degree in Telecommunications from KTH Royal Institute of Technology, Sweden, in 2011. From 2012 to mid 2014, he was a joint postdoc at the Alcatel-Lucent Chair on Flexible Radio, SUPELEC, France, and at KTH. He joined Linköping University, Sweden, in 2014 and is currently Senior Lecturer and Docent at the Division of Communication Systems.
He performs research on multi-antenna communications, Massive MIMO, radio resource allocation, energy-efficient communications, and network design. He is on the editorial board of the <span style="font-variant:small-caps;">IEEE Transactions on Communications</span> and the <span style="font-variant:small-caps;">IEEE Transactions on Green Communications and Networking</span>. He is the first author of the textbooks *Massive MIMO Networks: Spectral, Energy, and Hardware Efficiency* (2017) and *Optimal Resource Allocation in Coordinated Multi-Cell Systems* (2013). He is dedicated to reproducible research and has made a large amount of simulation code publicly available.
Dr. Björnson has performed MIMO research for more than ten years and has filed more than ten related patent applications. He received the 2016 Best PhD Award from EURASIP, the 2015 Ingvar Carlsson Award, and the 2014 Outstanding Young Researcher Award from IEEE ComSoc EMEA. He has co-authored papers that received best paper awards at WCSP 2017, IEEE ICC 2015, IEEE WCNC 2014, IEEE SAM 2014, IEEE CAMSAP 2011, and WCSP 2009.
[Jakob Hoydis]{}(S’08–M’12) received the diploma degree (Dipl.-Ing.) in electrical engineering and information technology from RWTH Aachen University, Germany, and the Ph.D. degree from Supélec, Gif-sur-Yvette, France, in 2008 and 2012, respectively. He is a member of technical staff at Nokia Bell Labs, France, where he is investigating applications of deep learning for the physical layer. Previous to this position he was co-founder and CTO of the social network SPRAED and worked for Alcatel-Lucent Bell Labs in Stuttgart, Germany. His research interests are in the areas of machine learning, cloud computing, SDR, large random matrix theory, information theory, signal processing, and their applications to wireless communications. He is a co-author of the textbook *Massive MIMO Networks: Spectral, Energy, and Hardware Efficiency* (2017). He is recipient of the 2012 Publication Prize of the Supélec Foundation, the 2013 VDE ITG Förderpreis, and the 2015 Leonard G. Abraham Prize of the IEEE COMSOC. He received the IEEE WCNC 2014 best paper award and has been nominated as an Exemplary Reviewer 2012 for the IEEE Communication letters.
\[[{width="1.0in" height="1.25in"}]{}\] [Luca Sanguinetti]{}(SM’15) received the Laurea Telecommunications Engineer degree (cum laude) and the Ph.D. degree in information engineering from the University of Pisa, Italy, in 2002 and 2005, respectively. Since 2005 he has been with the Dipartimento di Ingegneria dell’Informazione of the University of Pisa. In 2004, he was a visiting Ph.D. student at the German Aerospace Center (DLR), Oberpfaffenhofen, Germany. During the period June 2007 - June 2008, he was a postdoctoral associate in the Dept. Electrical Engineering at Princeton. During the period June 2010 - Sept. 2010, he was selected for a research assistantship at the Technische Universitat Munchen. From July 2013 to October 2017 he was with Large Systems and Networks Group (LANEAS), CentraleSupélec, Gif-sur-Yvette, France.
Dr. Sanguinetti is currently serving as an Associate Editor for the <span style="font-variant:small-caps;">IEEE Signal Processing Letters</span>. He served as an Associate Editor for <span style="font-variant:small-caps;">IEEE Transactions on Wireless communications</span>, and as Lead Guest Editor of <span style="font-variant:small-caps;">IEEE Journal on Selected Areas of Communications</span> Special Issue on “Game Theory for Networks” and as an Associate Editor for <span style="font-variant:small-caps;">IEEE Journal on Selected Areas of Communications</span> (series on Green Communications and Networking). Dr. Sanguinetti served as Exhibit Chair of the 2014 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP) and as the general co-chair of the 2016 Tyrrhenian Workshop on 5G&Beyond. He is a co-author of the textbook *Massive MIMO Networks: Spectral, Energy, and Hardware Efficiency* (2017).
His expertise and general interests span the areas of communications and signal processing, game theory and random matrix theory for wireless communications. He was the co-recipient of two best paper awards: *IEEE Wireless Commun. and Networking Conference (WCNC) 2013* and *IEEE Wireless Commun. and Networking Conference (WCNC) 2014*. He was also the recipient of the FP7 Marie Curie IEF 2013 “Dense deployments for green cellular networks”.
[^1]: For any continuous random variable $x$, the probability that $x$ takes a particular realization is zero, while the probability that $x$ takes a realization in a certain interval can be non-zero. Hence, if $x =ad+cf^\star$ then $x=0$ occurs with zero probability.
[^2]: After submitting our conference paper [@Bjornson2017a], the related work [@Neumann2017a] appeared. That paper considers the mean squared error in the uplink data detection of a single cell with multiple UEs per pilot sequence. The authors show that the error goes asymptotically to zero when having linearly independent covariance matrices. However, the paper [@Neumann2017a] contains no mathematical analysis of the achievable SE.
[^3]: This assumption implies that there is non-zero energy received from and transmitted to each UE.
[^4]: For notational convenience, we treat ${\mathbf{h}}_{1}^{\Htran}$ and ${\mathbf{h}}_{2}^{\Htran} $ as the downlink channels, instead of ${\mathbf{h}}_{1}^{\Ttran}$ and ${\mathbf{h}}_{2}^{\Ttran}$. This has no impact on the SE since the difference is only in a complex conjugate.
[^5]: Notice that, by applying Lemma 3 in Appendix A, we have $\frac{1}{M}[\hat{{\mathbf{H}}}^{\Htran} \hat{{\mathbf{H}}}]_{nm} \asymp \frac{1}{M}\tr({\mathbf{R}}_{n}
{\bf{Q}}^{-1} {\mathbf{R}}_{m})$. If the channel estimates are asymptotically linearly independent, then $\frac{1}{M}\hat{{\mathbf{H}}}^{\Htran} \hat{{\mathbf{H}}}$ is invertible as $M\to \infty$ and thus $\| {\mathbf{v}}_1 \|^2 = \frac{1}{M}\big[(\frac{1}{M}\hat{{\mathbf{H}}}^{\Htran} \hat{{\mathbf{H}}})^{-1}]_{11}\asymp 0$.
[^6]: This easily follows by observing that $\tr({\bf R}_k) = \tr({\bf D}_k)$ and also that $[{\bf D}_k]_{ii} = [{\bf R}_k]_{ii} \le \left\|{\bf R}_k\right\|_2$ since ${\bf R}_k$ is Hermitian.
[^7]: For M-MMSE precoding in , ${\mathbf{Z}}_j$ has bounded spectral norm while $\sum_l \sum_i \hat{{\mathbf{h}}}_{jli} \hat{{\mathbf{h}}}_{jli}^{\Htran}$ has $LK$ eigenvalues that grow unboundedly as $M \to \infty$. As the impact of ${\mathbf{Z}}_j$ vanishes, the approach in [@Bjornson2014d] can be used to prove that M-MMSE approaches M-ZF asymptotically.
[^8]: Under Assumption \[assumption\_1\], ${\bf Q}^{-1} {\bf R}_i{\bf Z}^{-1}{\bf R}_k$ has uniformly bounded spectral norm, which can be easily proved using Lemma \[lemma2\].
[^9]: [This can be proved by similar arguments as in Appendix C, since $\tr ({\bf A}^2)\ge (\tr ({\bf A}))^2/\rm{rank}({\bf A})$ if ${\bf A}$ is Hermitian and ${\bf A}\ne {\bf 0}$.]{}
[^10]: The uplink SINR $\gamma_{2}^{\rm {ul}}$ of UE 2 is obtained from by interchanging UE indices.
[^11]: The expressions in – have been simplified by utilizing the fact that ${\mathbf{Q}}$ and ${\mathbf{\Lambda}}$ have the same diagonal elements and ${\mathbf{R}}_k$ and ${\mathbf{D}}_k$ have the same diagonal elements, for $k=1,2$.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We present results from observations made at 33 GHz with the Very Small Array (VSA) telescope towards potential candidates in the Galactic plane for spinning dust emission. In the cases of the diffuse [Hii]{} regions LPH96 and NRAO591 we find no evidence for anomalous emission and, in combination with Effelsberg data at 1.4 and 2.7 GHz, confirm that their spectra are consistent with optically thin free–free emission. In the case of the infra-red bright SNR 3C396 we find emission inconsistent with a purely non-thermal spectrum and discuss the possibility of this excess arising from either a spinning dust component or a shallow spectrum PWN, although we conclude that the second case is unlikely given the strong constraints available from lower frequency radio images.'
date: 'Accepted —; received —; in original form '
title: 'Constraints on spinning dust towards Galactic targets with the VSA: a tentative detection of excess microwave emission towards 3C396'
---
radiation mechanisms: general—radio continuum: ISM—dust, extinction—ISM: individual: 3C396
Introduction
============
A localized excess of emission in the microwave region was first detected in the $\it{COBE}$/DMR data and was initially attributed to free–free emission (Kogut et al. 1996a, 1996b). Since then this anomalous emission has been detected by a number of authors (de Oliveira-Costa et al. 2002, 2004; Banday et al. 2003; Finkbeiner et al. 2004; Watson et al. 2005; Fern[á]{}ndez-Cerezo et al. 2006), and has been nicknamed ‘$\it{Foreground}$ $\it{X}$’. Although initially ascribed to thermal bremsstrahlung in view of its strong correlation with thermal dust, low H$\alpha$ surface brightness measurements (Leitch et al. 1997) implied gas temperatures in excess of $10^6$K, which were rejected on energetic grounds by Draine & Lazarian (1998a). Its physical mechanism has yet to be constrained; the most popular interpretation is that of rapidly rotating dust grains, or $\it{spinning}$ $\it{ dust}$ (Draine & Lazarian 1998a, 1998b). Other mechanisms which have been proposed include magnetic dust emission (Draine & Lazarian 1999), flat spectrum synchrotron (Bennett et al. 2003b), and bremsstrahlung from very hot electrons (Leitch et al. 1997). Spinning dust causes an excess of emission in the 10 – 50GHz region of the spectrum, where the combination of synchrotron, bremsstrahlung and thermal dust emission is a minimum, and is problematic especially for CMB experiments which utilise this region to minimise foreground contamination and avoid atmospheric emission. Consequently the presence of a poorly constrained foreground such as this anomalous dust emission is a potentially serious problem for CMB observers and needs to be better understood in order to be correctly removed. This has led to several directed observations (Finkbeiner et al. 2004; Watson et al. 2005; Casassus et al. 2004, 2006; Dickinson et al. 2006) being made towards targets suggested by the theoretical predictions of Draine & Lazarian (1998b; hereinafter DL98b). Whilst these have been mainly directed at [Hii]{} regions and dark clouds, DL98b also suggest that it may be possible to detect $10 - 100$GHz emission from spinning dust in photodissociation regions. Here we present observations at 33GHz towards two [Hii]{} regions and one supernova remnant (SNR) for which infrared observations imply the presence of significant photodissociation regions (PDRs).
The Telescope {#telescopes}
=============
The VSA is a 14 element interferometer sited at the Teide Observatory, in Tenerife, at an altitude of 2400m. The VSA operates in a single 1.5GHz wide channel at a central frequency of 33GHz (Scott et al. 2003). The 14 antennas use HEMT amplifiers with typical system temperature $\approx$35K. In its new super-extended configuration the VSA uses mirrors of diameter 65cm. The horn-reflector antennas are mounted on a tilt table hinged east–west and each antenna individually tracks the observed field by rotating its horn axis perpendicularly to the table hinge, wavefront coherence being maintained with an electronic path compensator system (Watson et al. 2003).
The individual tracking of the VSA antennas allows for the filtering of contaminating signals. These may be celestial sources, such as the Sun and Moon, or ground-spill and other environment based spurious signals. The VSA also uses a ground-shield to minimise ground-spill. The consequences of this design are two-fold. First, the VSA is able to observe continuously and can filter out emission from the Sun and Moon when they are as close as $9^{\circ}$. Second, the VSA is unaffected by ground-spill contamination for fields within 35$^{\circ}$ of the zenith and so is able to make direct images of the sky, rather than employing the lead–trail approach of many other interferometers operating in the microwave band.
Observations {#sec:lphcomp}
============
Observations were carried out as single pointings with a primary beam of 72arcmin FWHM. The data were calibrated using TauA and CasA in accordance with VSA reduction procedures. This calibration, along with appropriate flagging and filtering of the data, was performed using the special purpose package [reduce]{} developed specifically for reduction of VSA observations. Details of the VSA reduction and calibration procedure can be found in Dickinson et al (2004) and references therein.
Many surveys at lower frequency do not have similar resolution to the super-extended VSA, making comparison difficult. Instead we extrapolate from the Effelsberg 100m telescope at 2.7GHz (F$\ddot{\rm{u}}$rst et al. 1990). These data are single dish maps with a circular beam of 4.3arcmin FWHM.
For a robust comparison the 2.7GHz Effelsberg maps were multiplied by the primary beam of the VSA and visibility sampling was performed in the $\it{uv}$–plane with a correction for the Effelsberg beam.
LPH96
=====
The [Hii]{} region LPH96 (RA = 06$^{\rm{h}}$ 36$^{\rm{m}}$ 40$^{\rm{s}}$, $\delta$ = +10$^{\circ}$ 46$\arcmin$ 28$\arcsec$, J2000) has been observed with the Green Bank 43m telescope between 5 and 10GHz (Finkbeiner et al., 2002) and was shown to have a rising spectrum consistent with that expected from spinning dust. However, a pointed observation made with the CBI telescope (Dickinson et al., 2006) at 31 GHz shows emission consistent with an approximately flat spectrum source with only little possibility of spinning dust.
At 33GHz the peak flux density of LPH96 is 1.100$\pm$0.030Jy beam$^{-1}$. The synthesized beam of the VSA towards LPH96 is 9.1$\times$6.3arcmin$^2$ and the source is slightly resolved with structure extending towards the north and southwest. The 2.7GHz data after sampling gives a peak flux density of 1.436$\pm$0.010 Jy beam$^{-1}$. Comparing this with the VSA flux density we find a spectral index $\alpha$ = 0.106$\pm$0.026, where the errors are calculated from the thermal noise outside the beam on the 2.7 and 33 GHz maps. The spectral index, $\alpha$, is here defined so that flux density scales as $\nu^{-\alpha}$. This calculation fails to take into account systematic errors; taking errors of 5 percent on the flux density scales at each frequency gives $\alpha$ = 0.106$\pm$0.065.
The VSA measurement shows no indication of the excess emission which would be expected at 33GHz from the warm neutral medium (WNM) spinning dust model of DL98b consistent with the Green Bank data. The result is more consistent with that found by the CBI telescope (Dickinson et al. 2006) who found emission at 31GHz consistent only with $\alpha = 0.06 \pm 0.03$. Dickinson et al. also note that the Galactic plane survey of Langston et al. (2000) failed to detect LPH96 at 14.35 GHz with a detection limit of 2.5Jy.
3C396
=====
The supernova remnant (SNR) 3C396 ($=$G39.2$-$0.3), is a shell-like remnant at radio frequencies, with a mean angular diameter of $7\farcm8$ (Patnaik et al.1990). Its spectral behaviour has been extensively studied in the radio, most notably by Patnaik, and has a non-thermal radio spectrum with $\alpha
\approx
0.42$ between $\sim 400$ MHz and $\sim 10$ GHz. Below 30 MHz catalogued flux densities for 3C396 are contaminated by the near by steep-spectrum pulsar, PSR 1900$+$0.5 (Manchester & Taylor 1981). A pulsar wind nebula near the centre of this SNR has been detected in X-rays (see Olbert et al. 2003), but the remnant has not been detected optically. Only a lower limit of 7.7 kpc for its distance is available from its [Hi]{} absorption observation (Caswell et al.1975). Patnaik et al. conclude that the neighbouring [Hii]{} region NRAO 591 is likely to be at a distance of $\simeq 14$ kpc.
VSA observation of 3C396
------------------------
3C396 was observed in August 2006 in a short observation of 1.8 hours. Mapping and clean-based deconvolution were performed using the [AIPS]{} package. The observation was not limited by thermal noise but rather by the dynamic range of the telescope and has rms noise of 34.0mJy. The [VSA]{} map is shown in Fig. \[fig:3c396vsa\]; it shows both 3C396 and also the [Hii]{} region NRAO 591 to the north–west of the remnant. The VSA beam towards 3C396 is $9.1\times7.7$arcmin$^2$. Interesting features include a protrusion to the north and a faint detection of the “blow-out” tail to the north–east, both of which are also present at longer radio wavelengths.
Fits were obtained using the [AIPS]{} task [JMFIT]{} by drawing a bounding box around both sources and fitting for two Gaussians and a base level. The peak flux density for 3C396 was found to be 3.21$\pm$0.29Jy beam$^{-1}$ and the integrated flux density 6.64$\pm$0.33Jy at 33GHz, with the peak at 19$^{\rm{h}}$01$^{\rm{m}}$43$\fs2$ +05$^{\circ}$22$'$22$\farcs 4$ (B1950). For the secondary source NRAO 591 we find a peak flux density of 1.71$\pm$0.09Jy beam$^{-1}$ and an integrated flux density of 2.40$\pm$0.12Jy centered at 19$^{\rm{h}}$00$^{\rm{m}}$47$\fs4$ +05$^{\circ}$31$'$06$\farcs 3$ (B1950). The errors here include contributions from the rms noise on the observation, the statistical error from the Gaussian fits, and a conservative 5% error from the flux calibration which dominates the overall value. A complete discussion of the VSA flux calibration may be found in Dickinson et al. (2004).
![VSA observation of 3C396 with the [Hii]{} region NRAO 591 also visible to the north–west. Contours are overlaid at $-$2, $-$1, 1, 2, 4, 8, 16, 32 and 64 $\sigma$.\[fig:3c396vsa\]](./3c396grey.ps){height="7.cm" width="7.cm"}
The integrated spectrum of 3C396
--------------------------------
Again we use observations from the Effelsberg 100m telescope to examine the emission at lower radio frequencies. Using Effelsberg data at 1.4 (Reich et al. 1997) and 2.7GHz we convolve the data at 2.7GHz to match the 1.4GHz resolution of 9.4arcmin and find flux densities of 14.9 $\pm$ 1.3 Jy and 11.4 $\pm$ 1.3 Jy at 1.4 and 2.7GHz, respectively. These values agree with those of Reich et al. (1990) and give a spectral index of $\alpha$ = 0.46, which is consistent with the mean of 0.45 found for shell-type SNRs (Green, 2004; 2006). After sampling the Effelsberg observation at 2.7GHz to match the [*uv*]{}–coverage of the super-extended VSA towards 3C396 we find a flux density of 10.90$\pm$0.51Jy. Our best fit index implies a flux density of 3.4Jy at 33GHz extrapolated from the Effelsberg data.
We investigate the spectrum of 3C396 using our own flux density and other published values; all errors are quoted to 1$\sigma$. Taken at face value the [VSA]{} observation of 3C396 would imply an index $\alpha^{2.7}_{33}$ = 0.20$\pm$0.02, broadly consistent with a region of thermal emission. Alternatively, if we assume the index $\alpha^{1.4}_{2.7}$ of 0.46 determined from the Effelsberg data, then the [VSA]{} flux density would imply an excess of emission seen at microwave frequencies towards this source. A large number of observations between 400MHz and 5GHz exist and to confirm this spectral index we compile a spectrum using flux densities taken from Patnaik et al. (1990) who made corrections to the original measurements to bring them onto the flux density scale of Baars et al. (1977). The integrated spectrum of 3C396 is shown in Fig. \[fig:3c396spec\]. Data from Patnaik et al. is shown as crosses, the data from Effelsberg as filled squares and the VSA data as an unfilled diamond. Performing a weighted least squares fit to the Patnaik and Effelsberg data we find an index of $\alpha = 0.42\pm0.03$. The VSA measurement at 33GHz is inconsistent with this spectrum, shown in Fig. \[fig:3c396spec\] as a solid line. Including the VSA data at 33GHz we find a spectral index of $\alpha = 0.32\pm0.02$, shown as a dashed line. However, $\chi^2$ values for the two fitted spectra show that the spectral index of 0.42 ($\chi^2_{\rm{red}} = 1.07$, 31 d.o.f., P(1.07) = 0.366) is a better fit to the data. Indeed a spectral index of 0.32 ($\chi^2_{\rm{red}} = 1.95$, 32 d.o.f., P(1.95) = 0.001) would be unusually flat for a supernova remnant with less than 7% of shell like or possible shell like supernova remnants having a spectral index of less than or equal to 0.32 (Green 2006). However, we hesitate to over-interpret this statistic since the catalogued spectral indices are by no means uniform in quality.
Since previous observations (Finkbeiner et al. 2004) have suggested the presence of spinning dust in [Hii]{} regions we have also compiled a spectrum for NRAO591. We use flux densities compiled by Patnaik et al. (1990) which are plotted in Fig. \[fig:nraospec\] and find a spectrum compatible with that of optically-thin free–free emission.
![Infra-red emission towards 3C396. Colourscale is IRAS 100$\mu$m data overlaid with the VSA observation at 33GHz. Contours are $-$2, $-$1, 1, 2, 4, 8, 16, 32 and 64$\sigma$.\[fig:100um\]](./3c396iras.ps){height="7.cm" width="7.cm"}
Discussion and Conclusions
--------------------------
For the source 3C396 we constrain the possible excess emission relative to the non-thermal contribution at 33GHz by subtracting a non-thermal model extrapolated from lower frequency. For the flux densities between 400MHz and 10GHz, shown in Fig. \[fig:3c396spec\], a best fit spectral index of $\alpha =
0.42\pm0.03$ was found. This index implies a flux density of $4.19\pm$0.11Jy at 33GHz, where the error is statistical only. This leaves $2.45\pm$0.35Jy that may be due to an anomalous component. If we include a 5 percent error on the flux calibration these errors are increased to 0.24 and 0.41Jy, respectively. The excess accounts for 37% of the total 33GHz flux density.
Radio recombination line observations (Anantharamaiah, 1985) put an upper limit on the emission measure of 3C396 of 280cm$^{-6}$pc and fits an electron temperature of 5000K for the gas. This implies an upper limit on the free–free emission from the SNR of 0.061Jybeam$^{-1}$ at 33GHz. The beam sizes used for this measurement are poorly matched to that of the VSA and may cause the values to be over-estimated. Taking this into account it can be seen that the free–free contribution is small compared to the non-thermal and can only account for $\sim$2% of the excess emission. In addition, models including both free–free and non-thermal contributions provide poor fits to the Patnaik et al. data.
The possibility of the VSA seeing a secondary radio source in projection towards 3C396 is small. Radio sources with flux densities $> 100$mJy are seen with a frequency of 0.2/deg$^2$. The possibility of a rising spectrum source at $>$ 5GHz reduces this number by a factor of 10 (Waldram et al., 2003; Cleary et al., 2005).
Olbert et al. (2003) report the presence of a small pulsar wind nebula (PWN) within the 3C396 SNR , although the authors note the lack of any corresponding radio feature in high resolution 20cm VLA images of the remnant (Dyer & Reynolds 1999). For this PWN (or plerion-like component), to account for the excess flux density seen at 33GHz it would be very obvious at 1.4GHz, where it would have to contribute approximately 1/6 of the total flux density, assuming a flat spectral index – Olbert et al. however suggest that its contribution is $\leq$1/25 of the total radio flux density at 1.4GHz.
The infra-red emission at 100$\mu$m towards 3C396 is shown in Fig. \[fig:100um\]. In this region of the Galactic plane it is not certain whether the emission is associated with the SNR remnant, or is merely a projection. Reach et al. (2006) suggest that higher resolution Spitzer data shows emission at 3.6 to 8$\mu$m which may be associated with the SNR. They find IRAC colours for these regions consistent with both PDRs and [Hii]{} regions.
If the excess emission seen at 33GHz from 3C396 is associated with these regions then it is possible it may arise from the dipole emission of Draine & Lazarian (1998a;1998b). However, the discrepancy in resolution between the Spitzer and VSA telescopes precludes a more detailed spatial analysis. This possibility is illustrated in Figure \[fig:3c396spec2\] where the data is shown with the WNM spinning dust model of DL98b. For clarity we have binned the data of Patnaik et al. at similar frequencies, and have excluded one point at 10.7GHz. In addition, we note that the WNM model of DL98b, where the column density towards 3C396 is determined from the full-sky magnitude map of Schlegel, Finkbeiner & Davis (1998), predicts a peak flux density at 33GHz of 3.5Jybeam$^{-1}$ at the resolution of the VSA due to spinning dust. This gives an integrated flux density of approximately 7Jy, using the ratio of peak to integrated flux found with the VSA. However, it is likely that this method overestimates the integrated flux density since the morphology of the thermal dust emission appears more compact than the emission seen with the VSA. It is however within a factor of 3 of the excess emission we see at 33GHz.
In conclusion, we have assessed the possibility of spinning dust emission at 33GHz towards the SNR 3C396. Apart from Cas A and Tau A, few SNR have been studied in the microwave region. Consequently, in order to confirm this possibility further measurements are required in the range 10–20GHz.
ACKNOWLEDGEMENTS {#acknowledgements .unnumbered}
================
We thank the staff of the Mullard Radio Astronomy Observatory, the Teide Observatory and the Jodrell Bank Observatory for their invaluable assistance in the commissioning and operation of the VSA. The VSA is supported by PPARC and the IAC. AS acknowledges the support of a PPARC studentship.
Part of the research described in this paper was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration.
We would also like to thank the anonymous referee for his careful reading of this paper.
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\[lastpage\]
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'An overview of existing nonparametric tests of extreme-value dependence is presented. Given an i.i.d. sample of random vectors from a continuous distribution, such tests aim at assessing whether the underlying unknown copula is of the [*extreme-value*]{} type or not. The existing approaches available in the literature are summarized according to how departure from extreme-value dependence is assessed. Related statistical procedures useful when modeling data with this type of dependence are briefly described next. Two illustrations on real data sets are then carried out using some of the statistical procedures under consideration implemented in the package [copula]{}. Finally, the related problem of testing the [*maximum domain of attraction*]{} condition is discussed.'
author:
- |
Axel Bücher\
\
\
\
\
- |
Ivan Kojadinovic\
\
\
\
bibliography:
- 'biblio.bib'
title: 'An overview of nonparametric tests of extreme-value dependence and of some related statistical procedures'
---
Introduction
============
By definition, the class of extreme-value copulas consists of all possible limit copulas of affinely normalized, componentwise maxima of a multivariate i.i.d. sample, or, more generally, of a multivariate stationary time series. As a consequence, extreme-value copulas can be seen as appropriately capturing the dependence between extreme or rare events. The famous extremal types theorem of multivariate extreme value theory leads to a rather simple characterization of extreme-value copulas: the class of extreme-value copulas merely coincides with the class of max-stable copulas (see Section \[bk:sec:found\] below for a precise definition). Other characterizations are possible, most of which are based on a parametrization by a lower-dimensional function or measure [see e.g. @bk:GudSeg10 for an overview]. Serial dependence of the underlying time series is explicitly allowed provided certain mixing conditions hold [@bk:Hsi89; @bk:Hus90].
The theory underlying extreme-value copulas motivates their use in combination with the famous [*block maxima*]{} method popularized in the univariate case in the monograph of [@bk:Gum58]: from a given time series, calculate (componentwise) monthly or annual or, more generally, block maxima, and consider the class of extreme-value copulas (or parametric subclasses thereof) as an appropriate model for the multivariate sample of block maxima. If the block size is sufficiently large, it is unlikely that the respective maxima within a block occur at the beginning or the end of the block, whence, even under weak serial dependence of the underlying time series, block maxima could be considered as approximately independent. In statistical practice, independence has usually been postulated hitherto. Applications of the block maxima method can also be found in contexts in which the underlying time series is not necessarily stationary, as is the case when seasonalities are present (for instance in some hydrological problems).
The use of extreme-value copulas is not restricted to the framework of multivariate extreme-value theory. These dependence structures can actually be a convenient choice to model any data sets with positive association. Moreover, many parametric submodels are available in the literature [see e.g. @bk:GudSeg10 for an overview].
Extreme-value copulas have been successfully applied in empirical finance and insurance [see e.g. @bk:LonSol01; @bk:CebDenLam03; @bk:McNFreEmb05], and environmental sciences [see e.g. @bk:Taw88; @bk:SalDeMKotRos07]. They also arise in spatial statistics in connection with max-stable processes in which they determine the underlying spatial dependence [see e.g. @bk:DavPadRib12; @bk:Rib13; @bk:RibSed13].
From a statistical point of view, it is important to test the hypothesis that the copula of a given sample is an extreme-value copula. When applied within the context of the block maxima method, a rejection of this hypothesis would indicate that the size of the blocks is too small and should be enlarged, or that the (broad) conditions of the extremal types theorem are not satisfied. When applied outside of the extremal types theoretical framework, tests of extreme-value dependence merely indicate whether the class of max-stable copulas is a plausible choice for modeling the cross-sectional dependence in the data at hand. If there is no evidence against this class, additional statistical procedures tailored for extreme-value copulas can be used to carry out the data analysis.
This chapter is organized as follows. A brief overview of the theory underlying extreme-value copulas is given in the second section. The third section provides a summary of the procedures available in the literature for testing whether the copula of a random sample from a continuous distribution can be considered of the extreme-value type or not. Rather detailed analyses of bivariate financial data and bivariate insurance data are presented next. They are accompanied by code for the statistical system [@bk:Rsystem] from the [copula]{} package [@bk:copula]. Finally, in the last section, the related issue of testing the maximum domain of attraction condition is discussed.
Mathematical foundations {#bk:sec:found}
========================
Consider a $d$-dimensional random vector $\mathbf{X} = (X_1,\dots,X_d)$, $d\ge 2$, whose marginal cumulative distributions functions (c.d.f.s) $F_1,\dots,F_d$ are assumed to be continuous. Then, by [@bk:Skl59]’s representation theorem, the c.d.f. $F$ of $\mathbf{X}$ can be written in a unique way as $$F(\mathbf{x}) = C \{ F_1(x_1),\dots,F_d(x_d) \}, \qquad \mathbf{x}=(x_1, \dots, x_d) \in \mathbb{R}^d,$$ where the function $C:[0,1]^d \to [0,1]$ is a copula, i.e., the restriction of a multivariate c.d.f. with standard uniform margins to the unit hypercube. The above display is usually interpreted in the way that the copula $C$ completely characterizes the stochastic dependence among the components of $\mathbf X$.
A $d$-dimensional copula $C$ is an [*extreme-value*]{} copula if and only if there exists a copula $C^*$ such that, for any $\mathbf{u} \in [0,1]^d$, $$\begin{aligned}
\label{bk:eq:maxdom}
\lim_{n \to \infty} \{ C^*(u_1^{1/n},\dots,u_d^{1/n}) \}^n = C(\mathbf{u}).\end{aligned}$$ The copula $C^*$ is then said to be in the [*maximum domain of attraction*]{} of $C$, which shall be denoted as $C^* \in D(C)$ in what follows.
Some algebra reveals that $\{ C^*(u_1^{\scriptscriptstyle 1/n},\dots,u_d^{\scriptscriptstyle 1/n}) \}^n$ is the copula, evaluated at $\mathbf{u} \in [0,1]^d$, of the vector of componentwise maxima computed from an i.i.d. sample $\mathbf Y_1, \dots, \mathbf Y_n$ with continuous marginal c.d.f.s and copula $C^*$. The latter fact motivates the terminology *extreme-value copula*. It is additionally very useful to note that $C$ is an extreme-value copula if and only if it is [*max-stable*]{}, that is, if and only if, for any $\mathbf{u} \in [0,1]^d$ and $r \in \mathbb N$, $r > 0$, $$\label{bk:eq:maxstability}
\{ C(u_1^{1/r},\dots,u_d^{1/r}) \}^r = C(\mathbf{u}).$$ The sufficiency follows by using, in combination with , the fact that, for any $\mathbf{u} \in [0,1]^d$ and $r \in\mathbb N$, $r > 0$, $$\Big[C^*\big\{ (u_1^{1/r})^{1/n}, \dots, (u_d^{1/r})^{1/n} \big\} \Big]^{1/n}
=
\Big[ \big \{ C^*(u_1^{1/(nr)}, \dots, u_d^{1/(nr)}) \big\}^{1/(nr)} \Big]^r.$$ The necessity is an immediate consequence of the fact $C \in D(C)$ for any max-stable copula $C$. Interestingly enough, it can be shown that a max-stable copula actually satisfies for any real $r>0$ [see e.g. @bk:Gal78 Lemma 5.4.1].
An alternative, more complex characterization, essentially due to [@bk:Pic81], is as follows: a copula $C$ is of the extreme-value type if and only if there exists a function $A$ such that, for any $\mathbf{u} \in (0,1]^d \setminus \{(1,\dots,1)\}$, $$\label{bk:eq:Pickands_charact}
C(\mathbf{u})
=
\exp \left\{ \left( \sum_{j=1}^d \log u_j \right) A \left(\frac{\log u_2}{\sum_{j=1}^d \log u_j}, \dots, \frac{\log u_{d}}{\sum_{j=1}^d \log u_j} \right) \right\},$$ where $A:\Delta_{d-1} \to [1/d,1]$ is the [*Pickands dependence function*]{} and $\Delta_{d-1} = \{(w_1,\dots,w_{d-1}) \in [0,1]^{d-1} : w_1 + \dots + w_{d-1} \le 1 \}$ is the unit simplex [see e.g. @bk:GudSeg12 for more details]. If relation is met, then $A$ is necessarily convex and satisfies the boundary condition $\max\{1- \sum_{j=1}^{d-1} w_j, w_1, \dots, w_{d-1}\} \le A(\mathbf w) \le 1$ for all $\mathbf w = (w_1,\dots,w_{d-1}) \in \Delta_{d-1}$. The latter two conditions are, however, not sufficient to characterize the class of Pickands dependence functions unless $d=2$ [see e.g. @bk:BeiGoeSegTeu04 for a counterexample].
Several other characterizations of extreme-value copulas are possible, for instance using the [*spectral measure of $C$*]{} [see e.g. @bk:GudSeg12 for details] or the [*stable tail dependence function*]{} [@bk:res13; @bk:ChaFouGenNes14].
Existing tests of extreme-value dependence {#bk:sec:tests}
==========================================
Let $\mathcal{EV}$ denote the class of extreme-value copulas. Given a random sample $\mathbf{X}_1,\dots,\mathbf{X}_n$ from a c.d.f. $C\{F_1(x_1),\dots,F_d(x_d)\}$ with $F_1,\dots,F_d$ continuous and $C,F_1,\dots,F_d$ unknown, tests of extreme-value dependence aim at testing $$\label{bk:eq:H0}
H_0 : C \in \mathcal{EV} \qquad \mbox{against} \qquad H_1 : C \not \in \mathcal{EV}.$$ The existing tests for $H_0$ available in the literature are all rank-based and therefore margin-free. They can be classified into three groups according to how departure from extreme-value dependence is assessed.
Approaches based on Kendall’s distribution {#bk:subsec:kendall}
------------------------------------------
The first class of approaches, which is also the oldest, finds its origin in the seminal work of [@bk:GhoKhoRiv98] and is restricted to the case $d=2$. Given a bivariate random vector $\mathbf{X} = (X_1,X_2)$ with c.d.f. $F$, continuous marginal c.d.f.s $F_1$ and $F_2$ and copula $C$, the tests in this class are based on the random variable $$W = F(X_1,X_2) = C \{ F_1(X_1), F_2(X_2) \}.$$ The c.d.f. of $W$ is frequently referred to as [*Kendall’s distribution*]{} and will be denoted by $K$ subsequently. When $C \in \mathcal{EV}$, [@bk:GhoKhoRiv98] showed that $$\label{bk:eq:Kendall_dist}
K(w) = \Pr(W \leq w) = w - (1 - \tau) w \log w, \qquad w \in (0,1],$$ where $\tau$ denotes [*Kendall’s tau*]{}. Whether $C$ is of the extreme-value type or not, it is known since [@bk:SchWol81] that $$\tau = 4 \int_{[0,1]^2} C(u_1,u_2) \mathrm{d} C(u_1,u_2) - 1 = 4 \mathrm{E}(W) - 1.$$ When $C \in \mathcal{EV}$, [@bk:GhoKhoRiv98] also obtained from that, for $k\in \mathbb N$, $
\mu_k := E(W^k) = (k \tau + 1)/(k+1)^2,
$ which for instance implies that $$\label{bk:eq:test1}
-1 + 8 \mu_1 - 9 \mu_2 = 0.$$ In order to test $H_0$ from a bivariate random sample $\mathbf{X}_1,\dots,\mathbf{X}_n$ with c.d.f. $C\{F_1(x_1),F_2(x_2)\}$ where $C,F_1,F_2$ are unknown, [@bk:GhoKhoRiv98] suggested to assess whether a sample version of the left-hand side of is significantly different from zero or not. Specifically, they considered the statistic $$\label{bk:eq:S2n}
S_{2n} = - 1 + \frac{8}{n(n-1)} \sum_{i \neq j} I_{ij} - \frac{9}{n(n-1)(n-2)} \sum_{i \neq j \neq k} I_{ij} I_{kj},$$ where $I_{ij} = \mathbf{1}(X_{i1} \leq X_{j1},X_{i2} \leq X_{j2})$. As shown by [@bk:GhoKhoRiv98], $S_{2n}$ is a centered $U$-statistic which, under the null hypothesis, converges weakly to a normal random variable. To carry out the test, [@bk:GhoKhoRiv98] proposed to estimate the variance of $S_{2n}$ using a jackknife estimator. The test based on $S_{2n}$ was revisited by [@bk:BenGenNes09] who proposed two alternative strategies to compute approximate p-values for $S_{2n}$. The three versions of the test are implemented in the function `evTestK` of the package [copula]{}.
The above approach was recently furthered by [@bk:DuNes13] who used the first three moments of Kendall’s distribution and the theoretical relationship $$\label{bk:eq:test2}
-1+4\mu_1+9\mu_2-16\mu_3 = 0$$ under the null instead of . The corresponding test statistic will subsequently be denoted by $S_{3n}$. An additional contribution of the latter authors was to find a counterexample to [@bk:GhoKhoRiv98]’s conjecture that $K$ has the form in if and only if $C \in \mathcal{EV}$. The latter implies that tests in this class are not consistent. Despite that fact, the Monte Carlo experiments reported in [@bk:KojYan10c] and in [@bk:DuNes13] suggest that tests based on $S_{2n}$ and its extension studied in [@bk:DuNes13] are among the most powerful procedures for testing bivariate extreme-value dependence.
Notice finally that additional extensions of the approach of [@bk:GhoKhoRiv98] were studied in [@bk:Que12] along with tests based on Cramér–von Mises-like statistics derived from the empirical process $\sqrt{n} (K_n - K_{\tau_n})$, where $K_n$ is the empirical c.d.f. of $\hat W_1,\dots,\hat W_n$ with $\hat W_i = F_n(X_{i1},X_{i2})$ and $F_n$ the empirical c.d.f. of $\mathbf{X}_1,\dots,\mathbf{X}_n$, and $K_{\tau_n}$ is defined as in with $\tau$ replaced by its classical estimator denoted $\tau_n$.
Approaches based on max-stability {#bk:subsec:max}
---------------------------------
The second class of tests proposed in the literature consists of assessing empirically whether holds or not. It was investigated in [@bk:KojSegYan11] for $d \geq 2$. The key ingredient is a natural nonparametric estimator of the unknown copula $C$ known as the [*empirical copula*]{} [see e.g. @bk:Rus76; @bk:Deh79; @bk:Deh81].
Given a sample $\mathbf{X}_1,\dots,\mathbf{X}_n$ from a c.d.f. $C\{F_1(x_1),\dots,F_d(x_d)\}$ with $F_1,\dots,F_d$ continuous and $C,F_1,\dots,F_d$ unknown, let $\hat U_{ij} = R_{ij}/(n+1)$ for all $i \in \{1,\dots,n\}$ and $j \in \{1,\dots,d\}$, where $R_{ij}$ is the rank of $X_{ij}$ among $X_{1j},\dots,X_{nj}$, and set $\mathbf{\hat U}_i = (\hat U_{i1},\dots, \hat U_{id})$. It is worth noticing that the scaled ranks $\hat U_{ij}$ can equivalently be rewritten as $\hat U_{ij} = n F_{nj}(X_{ij}) / (n+1)$, where $F_{nj}$ is the empirical c.d.f. computed from $X_{1j},\dots,X_{nj}$, the scaling factor $n/(n+1)$ being classically introduced to avoid problems at the boundary of $[0,1]^d$. The empirical copula of $\mathbf{X}_1,\dots,\mathbf{X}_n$ is then frequently defined as the empirical c.d.f. computed from the [*pseudo-observations*]{} $\mathbf{\hat U}_1,\dots,\mathbf{\hat U}_n$, i.e., $$\label{bk:eq:empcop}
C_n(\mathbf{u}) = \frac{1}{n} \sum_{i=1}^n \mathbf{1} ( \mathbf{\hat U}_i \leq \mathbf{u} ), \qquad \mathbf{u} \in [0,1]^d.$$ The inequalities between vectors in the above definition are to be understood componentwise.
To test empirically, [@bk:KojSegYan11] considered test statistics constructed from the empirical process $$\label{bk:test_process}
\mathbb{D}_{r,n}(\mathbf{u}) = \sqrt{n} \left[ \{ C_n(u_1^{1/r},\dots,u_d^{1/r}) \}^r - C_n(\mathbf{u}) \right], \qquad \mathbf{u} \in [0,1]^d,$$ for some strictly positive fixed values of $r$. The recommended test statistic is $$\label{bk:eq:T345n}
T_{3,4,5,n} = T_{3,n} + T_{4,n} + T_{5,n},$$ where $T_{r,n} = \int_{[0,1]^d} \{\mathbb{D}_{r,n}(\mathbf{u})\}^2 \mathrm{d} C_n(\mathbf{u})$. Approximate p-values for the latter were computed using a [*multiplier bootstrap*]{}. The test based on $T_{3,4,5,n}$ is implemented in the function `evTestC` of the package [copula]{}. It is not a consistent test either, because the validity of is assessed only for a small number of $r$ values.
Approaches based on the estimation of the Pickands dependence function {#bk:subsec:pick}
----------------------------------------------------------------------
Recall that $\mathbf{X}_1,\dots,\mathbf{X}_n$ is a random sample from a c.d.f. $C\{F_1(x_1),\dots,F_d(x_d)\}$ with $F_1,\dots,F_d$ continuous and $C,F_1,\dots,F_d$ unknown. If $C \in \mathcal{EV}$, it can be expressed as in . The third class of tests exploits variations of the following idea: given a nonparametric estimator $A_n$ of $A$ and using the empirical copula $C_n$ defined in , relationship can be tested empirically.
The first test in this class is due to [@bk:KojYan10c] who, for $d=2$ only, constructed test statistics from the empirical process $$\mathbb{E}_n(u_1,u_2) = \sqrt{n} \left( C_n(u_1,u_2) - \exp \left[ \log(u_1 u_2) A_n \left\{ \frac{\log(u_2)}{\log(u_1 u_2)} \right\} \right] \right),$$ for $(u_1,u_2) \in (0,1]^2 \setminus \{(1,1)\}$. The recommended statistic is $$\label{bk:eq:TnA}
T_n^A = \int_{[0,1]^2} \mathbb{E}_n(u_1,u_2)^2 \mathrm{d} C_n(u_1,u_2),$$ when $A_n$ is the rank-based version of the Capéraà–Fougères–Genest (CFG) estimator of $A$ studied in [@bk:GenSeg09]. The resulting test relies on a [*multiplier bootstrap*]{} and is implemented in the function `evTestA` of the package [copula]{}. A multivariate version of this test was studied in [@bk:Gud12] using the multivariate extension of the rank-based CFG estimator of $A$ investigated in [@bk:GudSeg12].
An alternative class of nonparametric multivariate rank-based estimators of $A$ was proposed in [@bk:BucDetVol11] and [@bk:BerBucDet13]. These are based on the minimization of a weighted $L^2$-distance between the logarithms of the empirical and the unknown extreme-value copula. To derive multivariate tests of extreme-value dependence, the latter authors reused the aforementioned $L^2$-distance to measure the difference between the empirical copula in and a plug-in nonparametric estimator of $C$ under extreme-value dependence based on . The corresponding test statistic is subsequently denoted by $T_{L^2,n}$.
We end this subsection by briefly summarizing a recent graphical approach due to [@bk:CorGenNes14]. Their idea, hitherto restricted to the bivariate case, is as follows: given a copula $C$, consider the transformation $T_C:(0,1)^2 \to (0,1) \times (0,\infty]$, defined by $$T_C(u_1,u_2) = \left( \frac{\log (u_2)}{\log (u_1u_2)}, \frac{\log \{ C(u_1,u_2) \} }{ \log(u_1u_2) } \right), \qquad (u_1,u_2) \in (0,1)^2.$$ If $C \in \mathcal{EV}$, representation holds and we have $\log \{ C(u_1,u_2) \} = \log(u_1u_2) A \{ \log(u_2)/\log(u_1u_2) \}$ for all $(u_1,u_2) \in (0,1)^2$, whence $\mathcal S_C = \{ T_C(u,v): (u,v) \in (0,1)^2\}$ coincides with the graph of $A$, i.e., with the set $\{(t,A(t)) : t \in (0,1) \}$. More generally, some thought reveals that $H_0$ is valid if and only if $\mathcal S_C$ is a convex curve. The latter observation suggests to test $H_0$ in by estimating the set $\mathcal S_C$ and visually assessing the departure of that estimated set from a convex curve. The estimator defined in [@bk:CorGenNes14], called the [*A-plot*]{}, is given by $$\hat {\mathcal S}_n =
\left\{
(\hat T_i, \hat Z_i) : \hat T_i = \frac{\log(\hat U_{i2}) }{ \log( \hat U_{i1} \hat U_{i2})}, \hat Z_i = \frac{\log \{ C_n(\hat U_{i1}, \hat U_{i2}) \} }{ \log(\hat U_{i1} \hat U_{i1})}, i \in \{1, \dots, n\}
\right\}.$$ Examples of A-plots when $C \in \mathcal{EV}$ and when $C \not \in \mathcal{EV}$ can be found in Figure 1 of [@bk:CorGenNes14]. When $C$ is of the extreme-value type, the previous authors proposed a B-spline smoothing estimator for the Pickands dependence function $A$ based on $\hat {\mathcal S}_n$. The latter is subsequently denoted by $A_n$ for simplicity (even though the estimator depends on several smoothing parameters). Additionally to a pure graphical check, the authors propose $$\label{bk:eq:resid}
T_n = \frac{1}{n} \sum_{i=1}^n \{ \hat Z_i - A_n (\hat T_i) \}^2,$$ a residual sum of squares, as a formal test statistic for $H_0$. The hypothesis is rejected for unlikely large values of $T_n$. Specifically, an approximate p-value for $T_n$ is computed by means of a [*parametric bootstrap*]{} procedure based on simulating from a copula with Pickands dependence function $A_n$.
Finite-sample performance of some of the tests
----------------------------------------------
The finite-sample performance of the tests reviewed in the preceding sections was investigated by various authors. Table \[bk:tab:evc\] below, taken from [@bk:CorGenNes14], gathers those results from [@bk:CorGenNes14], [@bk:KojYan10c], [@bk:DuNes13], and [@bk:BucDetVol11] that were obtained under the same experimental settings (notice that the Gumbel–Hougaard copula is the only extreme-value copula among those considered in the table). As noted by [@bk:CorGenNes14], no test is uniformly better than the others: each test, except the one based on $T_{L^2,n}$ from [@bk:BucDetVol11], is favored for at least one of the considered scenarios under $H_1$. For high levels of dependence (as measured by Kendall’s tau), the tests based on $S_{2n}$ and $S_{3n}$ described in Section \[bk:subsec:kendall\] seem to yield the most accurate approximation of the nominal level (here 5%). The tests whose approximate p-values are computed by means of a multiplier bootstrap, i.e., the tests based on $T_{3,4,5,n}$ defined in and on $T_n^A$ and $T_{L^2,n}$ introduced in Section \[bk:subsec:pick\], are quite conservative for such scenarios. From a computational perspective, the test based on $S_{2n}$ seems to be the fastest, while the one based on $T_n^A$ defined in is the most computationally intensive. Additional comparison of the tests based on $S_{2n}$, $T_{3,4,5,n}$ and $T_n^A$ (resp. $S_{2n}$ and $S_{3n}$) can be found in @bk:KojYan10c [Tables 1–3] [resp. @bk:DuNes13 Table 5].
[@bk:KojSegYan11] and [@bk:BerBucDet13] also present simulation results for $d > 2$, which are in favor of the test based on $T_{3,4,5,n}$ defined in . Preliminary results obtained in [@bk:Gud12] indicate that the multivariate extension of the test based on $T_n^A$ defined in is likely to outperform the test based on $T_{3,4,5,n}$ for several scenarios under $H_1$.
Some related statistical inference procedures {#bk:sec:related}
=============================================
Once it has been decided to use an extreme-value copula to model dependence in a set of multivariate continuous i.i.d. observations, a typical next step is to choose a parametric family $\mathcal{C}$ in $\mathcal{EV}$ and estimate its unknown parameter(s) from the data. As many parametric families of extreme-value copulas are available [see e.g. @bk:GudSeg10; @bk:RibSed13], it is of strong practical interest to be able to test whether a given family $\mathcal{C}$ is a plausible model or not for the data at hand. In other words, tests for $
H_0 : C \in \mathcal{C}
$ against $
H_1 : C \not \in \mathcal{C}
$ would be needed. Such goodness-of-fit procedures were investigated in the bivariate case by [@bk:GenKojNesYan11] who considered Cramér–von Mises test statistics based on the difference between a nonparametric and a parametric estimator of the Pickands dependence function. The Monte Carlo experiments reported in the latter work highlighted the fact that, unless the amount of data is very large, there is hardly any practical difference among the existing bivariate symmetric parametric families of extreme-value copulas, and that an issue of more importance from a modeling perspective is whether a symmetric or asymmetric family should be used. For that purpose, the specific test of symmetry for bivariate extreme-value copulas investigated in [@bk:KojYan12] can be used as a complement to the goodness-of-fit test studied in [@bk:GenKojNesYan11]. Both tests are available in the [copula]{} package.
When $d > 2$ but $d$ remains reasonably small (say $d \leq 10$), generic goodness-of-fit tests (that is, developed for any parametric copula family, not necessarily of the extreme-value type) could be used [see e.g. @bk:GenRemBea09; @bk:KojYan11 and the references therein]. In a higher dimensional context, one possibility consists of using the specific approach for extreme-value copulas proposed by [@bk:Smi90] in his seminal work on max-stable processes. It consists of comparing nonparametric and parametric estimators of the underlying [*extremal coefficients*]{} (which are functionals of the Pickands dependence function). The latter approach was recently revisited in [@bk:KojShaYan14].
Illustrations and code from the [copula]{} package {#bk:sec:illus}
==================================================
We provide two illustrations below. The first one concerns bivariate financial logreturns and exemplifies the key theoretical connection between multivariate block maxima and extreme-value copulas briefly mentioned in the introduction and Section \[bk:sec:found\]. The second illustration consists of a detailed analysis of the well-known LOSS/ALAE insurance data with particular emphasis on the effect and handling of ties.
Bivariate financial logreturns
------------------------------
As a first illustration, we considered daily logreturns computed from the closing values of the Dow Jones and the S&P 500 stock indexes for the period 1990-2004. The closing values are available in the [QRM]{} package [@bk:QRM] and can be loaded by entering the following commands into the terminal:
> library(QRM)
> data(dji)
> data(sp500)
Daily logreturns for the period under consideration were computed using the [timeSeries]{} package [@bk:timeSeries]:
> d <- na.omit(cbind(dji,sp500))
> rd <- returns(d)
The statistical procedures mentioned in the previous sections should not however be directly applied on the resulting bivariate daily logreturns as the latter are strongly serially dependent. To obtain observations that might exhibit extreme-value dependence and could be considered approximately i.i.d., we first formed the bivariate series of componentwise monthly maxima. The last step was performed using functions from the [timeSeries]{} and [timeDate]{} packages [@bk:timeDate]:
> by <- timeSequence(from=start(rd), to=end(rd), by="month")
> mrd <- aggregate(rd, by, max)
The resulting component series do not contain ties which is compatible with the implicit assumption of continuous margins:
> x <- series(mrd)
> nrow(x)
[1] 171
> apply(x, 2, function(x) length(unique(x)))
DJI SP500
171 171
After loading the [copula]{} package with the command [library(copula)]{} and setting the random seed by typing [set.seed(123)]{}, the test of extreme-value dependence based on $S_{2n}$ (resp. $T_{3,4,5,n}$, $T_n^A$) defined in (resp. , ) was applied using the command [evTestK(x)]{} (resp. [evTestC(x)]{}, [evTestA(x, derivatives=“Cn”)]{}) and returned an approximate p-value of 0.5737 (resp. 0.4191, 0.2423). In other words, none of the tests detected any evidence against extreme-value dependence thereby suggesting that the copula of componentwise block maxima, for blocks of length corresponding to a month, is sufficiently close to an extreme-value copula. Note that, as the tests are rank-based, they could have equivalently been called on the pseudo-observations computed from the monthly block maxima. The random seed was set (to ensure exact reproducibility) because the second and third tests involve random number generation as their p-values are computed using resampling.
For illustration purposes, we next formed monthly logreturns as follows:
> srd <- aggregate(rd, by, sum)
> x <- series(srd)
Proceeding as previously, it can be verified that the resulting component series do not contain ties which is compatible with the implicit assumption of continuous margins. Monthly logreturns being merely sums of daily logreturns, the underlying unknown bivariate distribution should be far from exhibiting extreme-value dependence. The tests of extreme-value dependence based on $S_{2n}$, $T_{3,4,5,n}$ and $T_n^A$ returned approximate p-values of 0.0003, 0.02 and 0.0005, respectively, confirming that there is strong evidence in the data against extreme-value dependence.
LOSS/ALAE insurance data
------------------------
The well-known LOSS/ALAE insurance data are very frequently used for illustration purposes in copula modeling [see e.g. @bk:FreVal98; @bk:BenGenNes09; @bk:KojYan10]. The two variables of interests are LOSS, an indemnity payment, and ALAE, the corresponding allocated loss adjustment expense. They were observed for 1500 claims of an insurance company. Following [@bk:BenGenNes09], the following study is restricted to the 1466 uncensored claims.
The data are available in the [copula]{} package, and can be loaded by typing [library(copula)]{} followed by [data(loss)]{}. The uncensored claims described in terms of LOSS and ALAE were obtained as follows:
> myLoss <- subset(loss, censored==0, select=c("loss", "alae"))
These data, consisting of 1466 bivariate observations, contain a non-negligible amount of ties, the variable LOSS being particularly affected:
> sapply(myLoss, function(x) length(unique(x)))
loss alae
541 1401
The presence of ties is incompatible with the implicit assumption of continuous margins. Indeed, combined with the assumption that the data are i.i.d. observations, continuity of the margins implies that ties should no occur. Yet, ties are present here as in many other real data sets. The latter could be due either to the fact that the observed phenomena are truly discontinuous, or to precision/rounding issues. As far as the LOSS/ALAE data are concerned, the latter explanation applies.
Among the tests briefly described in Section \[bk:sec:tests\], only that of [@bk:CorGenNes14] explicitly considers the case of discontinuous margins (see Section 6 in that reference). The remaining tests were all implemented under the assumption of continuous margins. For the test based on $S_{2n}$ defined in , [@bk:GenNesRup11] provide a heuristic explanation of the fact that, for discontinuous margins, $S_{2n}$ is not necessarily centered anymore under the null.
Given the situation, there are roughly four possible courses of action: (i) stop the analysis, (ii) delete tied observations, (iii) use average ranks for ties or (iv) break ties at random, sometimes referred to as [*jittering*]{} (which amounts to adding a small, continuous white noise term to all observations). Arguments for not considering solution (ii) are given in @bk:GenNesRup11 [Section 2]. To empirically study solutions (iii) and (iv), the latter authors carried out an experiment consisting of applying the test based on $S_{2n}$ defined in on binned observations from a bivariate Gumbel–Hougaard copula. More specifically, tied observations were obtained by dividing the unit square uniformly into bins of dimension 0.1 by 0.1 (resp. 0.2 by 0.2) resulting in at most 100 (resp. 25) different bivariate observations whatever the sample size. In such a setting, [@bk:GenNesRup11] observed that both solutions (iii) and (iv) led to strongly inflated empirical levels for the test based on $S_{2n}$.
The situation in terms of ties in the LOSS/ALAE data is however far from being as extreme as in the experiment of [@bk:GenNesRup11]. In addition, ties mostly affect the LOSS variable. This prompted us first to consider solution (iv) as implemented in [@bk:KojYan10].
#### Random ranks for ties
The idea consists of carrying out the analysis for many different randomizations (with the hope that this will result in many different configurations for the parts of the data affected by ties) and then looking at the empirical distributions (and not the averages) of the results (here the p-values of various tests).
For illustration purposes, we first detail the analysis for one randomization:
> set.seed(123)
> pseudoLoss <- sapply(myLoss, rank, ties.method="random") /
(nrow(myLoss) + 1)
As a next step, the tests of extreme-value dependence based on $S_{2n}$, $T_{3,4,5,n}$ and $T_n^A$ defined in , and , respectively, were applied by successively typing `evTestK(pseudoLoss)`, `evTestC(pseudoLoss)` and `evTestA(pseudoLoss, derivatives=Cn)`, resulting in approximate p-values of 0.8845, 0.468 and 0.4231, respectively. Hence, none of the three tests detected any evidence against extreme-value dependence. The following step consisted of fitting a parametric family of bivariate extreme-value copulas to the data. As discussed in Section \[bk:sec:related\], given the very strong similarities among the existing families of bivariate symmetric extreme-value copulas, the only issue of practical importance is to assess whether a symmetric or asymmetric family should be used. To do so, we applied the test developed in [@bk:KojYan12] by calling `exchEVTest(pseudoLoss)`, with a resulting p-value of 0.1653. The previous result suggested to focus on an exchangeable family such as the Gumbel–Hougaard. We then ran the goodness-of-fit test proposed in [@bk:GenKojNesYan11] by calling
> gofEVCopula(gumbelCopula(), pseudoLoss, method="itau", verbose=FALSE)
The resulting p-value of 0.2592 suggested to fit the Gumbel–Hougaard family:
> fitCopula(gumbelCopula(), pseudoLoss, method="itau")
fitCopula() estimation based on 'inversion of Kendall's tau'
and a sample of size 1466.
Estimate Std. Error z value Pr(>|z|)
param 1.44040 0.03327 43.29 <2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
To assess how different randomizations of the ties affect the results, the above analysis was repeated 100 times using the following code:
> randomize <- function()
+ {
+ pseudoLoss <- sapply(myLoss, rank, ties.method="random") /
(nrow(myLoss) + 1)
+ evtK <- evTestK(pseudoLoss)$p.value
+ evtC <- evTestC(pseudoLoss)$p.value
+ evtA <- evTestA(pseudoLoss, derivatives="Cn")$p.value
+ exevt <- exchEVTest(pseudoLoss)$p.value
+ gofevGH <- gofEVCopula(gumbelCopula(), pseudoLoss, method="itau",
verbose=FALSE)$p.value
+ fitGH <- fitCopula(gumbelCopula(), pseudoLoss, method="itau")
+ c(evtK=evtK, evtC=evtC, evtA=evtA, exevt=exevt, gofevGH=gofevGH,
+ est=fitGH@estimate, se=sqrt(fitGH@var.est))
+ }
> reps <- t(replicate(100, randomize()))
> round(apply(reps, 2, summary), 3)
evtK evtC evtA exevt gofevGH est se
Min. 0.868 0.430 0.353 0.092 0.191 1.441 0.033
1st Qu. 0.898 0.462 0.396 0.112 0.223 1.442 0.033
Median 0.914 0.475 0.411 0.120 0.235 1.442 0.033
Mean 0.913 0.474 0.411 0.122 0.236 1.442 0.033
3rd Qu. 0.928 0.489 0.425 0.129 0.248 1.443 0.033
Max. 0.955 0.525 0.464 0.162 0.292 1.444 0.033
The empirical distributions of the results show that the different randomizations did not affect the results qualitatively.
#### Average ranks for ties
We also considered solution (iii), that is, average ranks for ties. The p-values of the three tests of extreme-value dependence (applied in the same order as previously) were 0.6, 0.02 and 0, respectively. The p-values of the tests of exchangeability and goodness of fit were 0.12 and 0.18, respectively. The estimate of the parameter of the Gumbel–Hougaard copula was 1.446.
#### Random or average ranks for ties?
The previous computations illustrate that solutions (iii) and (iv) for dealing with ties can result in significantly different conclusions. To gain insight into which solution should be preferred, if any, we designed an experiment tailored to the LOSS/ALAE data. Specifically, we simulated a large number of samples of size $n=1466$ from a Gumbel–Hougaard copula with parameter value 1.446, as suggested by the aforementioned parametric fit. We then modified each sample so that its marginal empirical c.d.f.s evaluated at the respective observations coincide with those of the LOSS/ALAE data. For instance, in the original data, the 27th to the 49th smallest values of LOSS are equal. Each simulated sample was modified so that the 27th to the 49th smallest values of the first variable all get replaced by the 49th smallest observation. The same approach was used for the second variable of the generated samples. Solutions (iii) and (iv) were applied next to each modified sample prior to running the tests of extreme-value dependence, and the resulting p-values were compared with those obtained by applying the tests on the corresponding unmodified sample (that is, with no ties). The code used to carry out the experiment for the test based on $S_{2n}$ defined in is given below:
> mr.loss <- rank(myLoss[,1], ties.method="max")
> mr.alae <- rank(sort(myLoss[,2]), ties.method="max")
> test.func <- function(x) evTestK(x)$p.value
> do1 <- function()
+ {
+ x <- rCopula(1466, gumbelCopula(1.446))
+ y <- x[order(x[,1]),]
+ y[,1] <- y[mr.loss,1]
+ y <- y[order(y[,2]),]
+ y[,2] <- y[mr.alae,2]
+ z <- apply(y, 2, rank, ties.method="random")
+ c(test.func(x), test.func(y), test.func(z))
+ }
> res <- t(replicate(1000, do1()))
> summary(round(res[,1] - res[,3],3))
Min. 1st Qu. Median Mean 3rd Qu. Max.
-0.093000 -0.014000 0.001000 0.000033 0.014000 0.105000
> summary(round(res[,1] - res[,2],3))
Min. 1st Qu. Median Mean 3rd Qu. Max.
-0.54600 -0.25520 0.14000 0.06697 0.34750 0.52300
> apply(res, 2, function(x) mean(x <= 0.05))
0.046 0.107 0.047
For the test based on $S_{2n}$, the p-values computed from a continuous sample and the corresponding randomized sample are very close on average, the maximal deviation being relatively small. On the contrary, the p-values computed from a continuous sample are larger on average than the p-values computed from the corresponding sample involving average ranks, and the maximal deviation is very large. We also see that when solution (iii) is considered, the test based on $S_{2n}$ is way too liberal, confirming the findings of [@bk:GenNesRup11], while, when solution (iv) is used, the test holds its level well. A similar experiment was performed for the test based on $T_{3,4,5,n}$ (with 100 replications only) and the conclusions are of the same nature but more pronounced:
> summary(round(res[,1] - res[,3],3))
Min. 1st Qu. Median Mean 3rd Qu. Max.
-0.10400 -0.01425 -0.00150 -0.00050 0.01650 0.08700
> summary(round(res[,1] - res[,2],3))
Min. 1st Qu. Median Mean 3rd Qu. Max.
-0.0190 0.2700 0.4820 0.4536 0.6592 0.8730
> apply(res, 2, function(x) mean(x <= 0.05))
0.05 0.45 0.05
The previous experiment can be adapted to any data set containing ties and suggests that, in the case of the LOSS/ALAE data, solution (iv) is meaningful while solution (iii) should be avoided.
Testing the maximum domain of attraction condition
==================================================
The statistical framework considered in the three previous sections can be regarded as the “classical” setting of dependence modeling by copulas. As mentioned in the introduction, modeling a copula by an extreme-value copula, or testing extreme-value dependence within such a framework, is particularly sensible if there are reasons to assume that the data at hand are generated by some maxima-forming process. If this is not the case, or if the hypothesis of extreme-value dependence is rejected, it might still be reasonable to make the (mild) assumption that the copula of interest lies in the domain of attraction of some extreme-value copula. It is the aim of the present section to briefly discuss how the latter assumption could be tested.
A precise formulation of the problem is as follows: we observe a sample of $d$-dimensional i.i.d. vectors $\mathbf Y_1, \dots, \mathbf Y_n$ with c.d.f. $C^*\{G_1(y_1), \dots, G_d(y_d)\}$, where $G_1, \dots, G_d$ are assumed continuous and $C^*,G_1,\dots,G_d$ are unknown. We are interested in tests of $$\begin{aligned}
\label{bk:eq:evcondition}
H_0: C^* \in D(C) \text{ for some } C \in \mathcal{EV}
\quad \text{against}\quad
H_1: C^* \notin D(C) \text{ for any } C \in \mathcal{EV},\end{aligned}$$ where the notation $C^* \in D(C)$ is defined below . Notice that the analogue univariate problem (i.e., testing the null hypothesis that the underlying distribution of a given univariate i.i.d. sample lies in the maximum domain of attraction of some extreme-value distribution) was tackled in [@bk:DieDehHus02], [@bk:DreDehLi06] and [@bk:HusLi06], while, in the multivariate case, only very few (validated) methods seem available.
A rejection of the null hypothesis in gives indication that the stochastic dependence between componentwise block maxima formed from the $\mathbf Y_i$, no matter how large the blocks are, cannot be adequately described by an extreme-value copula. On the other hand, if the hypothesis is not rejected, it is promising to consider an extreme-value copula as a model provided the block size is sufficiently large. Also, in the latter case, one could make use of to obtain the approximation that, for a sufficiently large $r$, $C^*(\mathbf v) \approx \{ C(v_1^r, \dots, v_d^r) \}^{1/r}= C(\mathbf v)$ for all $\mathbf v \in [\mathbf t, \mathbf 1]$, with $\mathbf t =(t_1, \dots, t_d)$ close to $\mathbf 1$. This would imply that, at least in the upper tail, the copula $C^*$ can be well-approximated by an extreme-value copula $C$. A threshold model of that form was for instance considered in [@bk:LedTaw96] in a bivariate setting with generalized Pareto marginals.
A first promising approach to test $H_0$ in consists of comparing two estimators of $C$ (or its characterizing objects) with different backgrounds. Under the null hypothesis, these estimators should not differ too much. Based on the peak-over-threshold method and in the bivariate case, [@bk:EinDehLi06] developed a test based on an Anderson–Darling-type statistic between two estimators of the stable-tail dependence function $\ell:[0,1]^2 \to \mathbb R$ defined by $\ell(x,y) = |x+y| A(x/|x+y|)$, where $A$ denotes the Pickands dependence function of $C$. Critical values for the test were obtained by approximately simulating from the limiting random variable. To the best of our knowledge, this testing procedure is the only validated method for testing the (bivariate) maximum domain of attraction condition.
A heuristic approach to test the null hypothesis in in the bivariate case was described in [@bk:CorGenNes14]. Their method consists of considering a trimmed A-plot (see also Section \[bk:subsec:pick\]) defined by only including those points $(\hat T_i, \hat Z_i)$ in the set $\hat{\mathcal S}_n=\hat{\mathcal S}_n(\mathbf t)$ for which $\hat{\mathbf U}_i \in [\mathbf t, \mathbf 1]$, with some suitable threshold parameter $\mathbf t =(t_1, t_2) \in [0,1]^2$ close to $\mathbf 1$. Based on the trimmed A-plot, the approach briefly described in Section \[bk:subsec:pick\] can be followed to obtain a B-spline smoothing estimator of the Pickands dependence function corresponding to the limiting extreme-value copula $C$. Plotting the residual sum of squared errors defined in against the threshold $\mathbf t$ serves as a data-driven method for the choice of the threshold. For that particular choice, the A-plot as well as the testing procedure described in Section \[bk:subsec:pick\] can be used to assess heuristically whether the maximum domain of attraction condition holds or not.
Finally, the tests described in Section \[bk:sec:tests\] can be adapted to obtain simple heuristic procedures for testing$H_0$ in . Under the null hypothesis, given $\mathbf Y_1, \dots, \mathbf Y_n$, if we form $k$ (componentwise) block maxima from blocks of length $m$, $$\mathbf X_{i} = (X_{i1}, \dots, X_{id}), \quad X_{ij}= \max \{Y_{m(i-1)+1,j}, \dots, Y_{mi,j}\},$$ $i \in \{1, \dots, k\}$, $j \in \{1, \dots d\}$, where $km = n$ and $m$ is sufficiently large (if $n$ is not an integer multiple of $m$, then a negligible remainder block of length strictly smaller than $m$ occurs), then the copula of the block maxima $\mathbf X_i$ should (approximately) be an extreme-value copula. The tests described in Section \[bk:sec:tests\] could next be applied to $\mathbf X_1, \dots, \mathbf X_k$ to obtain an indication of whether the maximum domain of attraction condition holds or not. Another promising approach consists of adapting the approach in Section \[bk:subsec:max\] by only testing max-stability in the upper tail $[\mathbf t, \mathbf 1]$, with some suitable threshold parameter $\mathbf t =(t_1, t_2) \in [0,1]^2$ close to $\mathbf 1$. This could be done by integrating the square of the process in over the restricted set $[\mathbf t, \mathbf 1]$.
Precise asymptotic validations of these methods are, however, not available. A treatment of occurring bias terms from an undersized choice of the block length or the threshold parameter would be necessary, as for instance carried out in [@bk:BucSeg14] in an estimation framework for time series based on block maxima. Also, data-driven methods to choose the block length $m$ or the threshold parameter $\mathbf t$ would need to be developed.
Open questions and ignored difficulties
=======================================
Several issues dealt with in this chapter would need to be thoroughly investigated in future research. For instance, the suggested approach for handling ties in data sets for which it is actually reasonable to assume that the apparent discontinuities are only due to precision or rounding issues would need to be studied more in depth. While for the LOSS/ALAE data set, it seemed reasonable to break ties at random a large number of times, this may not be the case for other data sets in which the proportion of ties is significantly larger [see e.g. @bk:GenNesRup11 Section 4]. Yet, even more difficult appears to be the problem of testing extreme-value dependence from truly discontinuous observations such as count data. A promising starting point for adapting some of the statistical procedures described in this work to such a context is the recent work of [@bk:GenNesRem14] on the [*multilinear empirical copula*]{}.
With financial applications in mind in particular, tests of extreme-value dependence would also need to be adapted to multivariate stationary times series. The methods briefly described in this chapter all rely on the assumption that the observations at hand are serially independent which is hardly verified for many data sets of interest. Applying the discussed statistical procedures to (almost i.i.d.) standardized residuals from common time series models might be an option, but it is unclear whether the necessary additional estimation step affects the limiting null distribution of the test statistics or not. For that purpose, a starting point might be the work of [@bk:Rem10] where the asymptotics of the empirical copula process of standardized residuals are investigated. If the tests are to be applied on the stationary raw time series data, then their empirical levels will most likely be affected by the serial dependence present in the observations. In such a situation, the dependent multiplier bootstrap studied in [@bk:BucKoj14] could be used to adapt some of the reviewed tests of extreme-value dependence.
**Acknowledgments** The authors are grateful to Christian Genest, Johanna Nešlehová and an anonymous referee for their constructive comments on an earlier version of this chapter. This work has been supported in part by the Collaborative Research Center *Statistical Modeling of Nonlinear Dynamic Processes* (SFB 823, project A7) of the German Research Foundation (DFG).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Modulational dynamics of oscillatory travelling rolls in magnetoconvection is studied near the onset of a Hopf bifurcation. Using weakly nonlinear analysis, we derive an envelope equation of oscillatory travelling rolls in the plane perpendicular to an ambient vertical magnetic field.The envelope equation is the Davey-Stewartson (DS) equation with complex coefficients, from which we obtain criteria for the modulational (Benjamin-Feir) instability of oscillatory travelling rolls.'
author:
- |
Ken-ichi Matsuba,Kenji Imai and Kazuhiro Nozaki\
Department of Physics,Nagoya University,Nagoya 464-01,Japan
title: |
Nonlinear Modulation of Travelling Rolls\
in Magnetoconvection
---
addtoreset[equation]{}[section]{} = i["10]{}
Introduction
============
A variety of spatially and temporally periodic patterns is found in weakly nonlinear Boussinesq convection in an imposed vertical magnetic field [@clune]. However, nonlinear modulation of the periodic patterns in the horizontal plane has thus far received less attention. Two kinds of bifurcations are known to convective patterns: one is to steady patterns and the other is to oscillatory patterns (a Hopf bifurcation) [@clune].Near the onset of the former bifurcation, modulational dynamics of travelling rolls may be described by the Newell-Whitehead-Segel (NWS) equation in terms of the Newell-Whitehead (NW) orderings [@newell1]. Near the onset of a Hopf bifurcation, the same ordings may yield a more complicated envelope equation [@brand], which includes various terms of different orders.\
In this paper, introducing consistent orderings (different from the NW orderings) to an envelope equation near the onset of a Hopf bifurcation, we derive the DS equation with complex coefficients, where horizontal incompressible flows couple to oscillatory travelling rolls in magnetoconvection.The analysis of a instability of its spatially uniform solution yields criteria for the modulational (Benjamin-Feir) instability of oscillatory travelling rolls.
Hopf Bifurcation
================
Boussinesq convection in an imposed vertical magnetic field is described by the equations [@clune] $$\begin{aligned}
{\partial_t}{\mbox{\boldmath $u$}}+({\mbox{\boldmath $u$}}\cdot\nabla){\mbox{\boldmath $u$}}&=&-\nabla P+\sigma R\Theta{\mbox{\boldmath$z$}}+\sigma\zeta Q
(\nabla\times{\mbox{\boldmath $B$}})\times{\mbox{\boldmath $B$}}+\sigma\triangle{\mbox{\boldmath $u$}},\label{b1}\\
{\partial_t}\Theta+({\mbox{\boldmath $u$}}\cdot\nabla)\Theta&=&w+\triangle\Theta,\label{b2}\\
{\partial_t}{\mbox{\boldmath $B$}}+({\mbox{\boldmath $u$}}\cdot\nabla){\mbox{\boldmath $B$}}&=&({\mbox{\boldmath $B$}}\cdot\nabla)
{\mbox{\boldmath $u$}}+\zeta\triangle{\mbox{\boldmath $B$}},\label{b3}\end{aligned}$$ together with $$\nabla\cdot{\mbox{\boldmath $u$}}=\nabla\cdot{\mbox{\boldmath $B$}}=0.\label{b4}$$ Here ${\mbox{\boldmath $u$}}\equiv(u,v,w)$ is the dimensionless velocity in (x,y,z) coordinates, ${\mbox{\boldmath$z$}}$ is the unit vector in the vertical ($z$) direction, $\Theta$ is the dimensionless temperature deviation from the conduction state and ${\mbox{\boldmath $B$}}$ is the dimensionless magnetic field. The parameters are the Prandtl number $\sigma$, the Rayleigh number $R$, the Chandrasekhar number $Q$ and the ratio of ohmic to thermal diffusivity denoted by $\zeta$. The magnetic field is given by $${\mbox{\boldmath $B$}}={\mbox{\boldmath$z$}}+{\mbox{\boldmath $b$}},$$ where ${\mbox{\boldmath $b$}}\equiv(a,b,c)$. The boundary conditions are $${\partial_z}u={\partial_z}v=w=\Theta=a=b=0\qquad \mbox{at}\qquad z=0,1.$$ The linear stability analysis of the conduction state ${\mbox{\boldmath $u$}}=\Theta={\mbox{\boldmath $b$}}=0$ shows that oscillatory convection sets in, for $\zeta<1$, at $$R=R_0=\frac{(\pi^2+k^2)^3}{k^2}\frac{(\sigma+\zeta)(1+\zeta)}{\sigma}+
\frac{\pi^2+k^2}{k^2}\frac{\zeta(\sigma+\zeta)}{1+\sigma}\pi^2 Q.$$ Here k is the horizontal wavenumber determined by minimizing the critical Rayleigh number $R_0$. Thus $k=k_0$, where $$(\pi^2+k_0^2)^3-\frac{3}{2}\pi^2(\pi^2+k_0^2)^2=\frac{\sigma\zeta}
{2(1+\sigma)(1+\zeta)}\pi^4Q.$$
Envelope Equation
=================
In this section, we derive the equation which describes the nonlinear evolution of a slowly varying envelope of oscillatory travelling rolls near the critical Rayleigh number $R_0$. For values of $R$ close to $R_0$, $$R=R_0+{\epsilon}^2R^{(2)}\qquad ({\epsilon}\ll 1),$$ we investigate the weakly nonlinear evolution of the wavepacket centered at the critical wavenumber $k_0$ and the corresponding frequency $\omega_0=
\omega(k_0)$ by approximating ${\mbox{\boldmath $u$}},{\mbox{\boldmath $b$}},\Theta$ and $P$ as $$\begin{aligned}
{\mbox{\boldmath $u$}}&=&{\epsilon}{\mbox{\boldmath $u$}}_1(\xi,\eta,\tau)E\left(\begin{array}
{c}\cos(\pi z)\\ \cos(\pi z)\\ \sin(\pi z) \end{array}\right)+
(\mbox{c.c.})+{\epsilon}^2{\mbox{\boldmath $u$}}^{(2)}+{\epsilon}^3{\mbox{\boldmath $u$}}^{(3)}+\cdots,\label{ex1}\\
{\mbox{\boldmath $b$}}&=&{\epsilon}{\mbox{\boldmath $b$}}_1(\xi,\eta,\tau)E\left(\begin{array}{c}\sin(\pi z)\\
\sin(\pi z)\\ \cos(\pi z) \end{array}\right)+(\mbox{c.c.})+
{\epsilon}^2{\mbox{\boldmath $b$}}^{(2)}+{\epsilon}^3{\mbox{\boldmath $b$}}^{(3)}+\cdots,\label{ex2}\\
\Theta&=&{\epsilon}\Theta_1(\xi,\eta,\tau)E\sin(\pi z)+(\mbox{c.c.})
+{\epsilon}^2\Theta^{(2)}+{\epsilon}^3\Theta^{(3)}+\cdots,\label{ex3}\\
P&=&P_1(\xi,\eta,\tau)E\cos(\pi z)+(\mbox{c.c.})+{\epsilon}^2P^{(2)}+
{\epsilon}^3P^{(3)}+\cdots,\label{ex4}\\
E&\equiv& \exp[i(k_0x-\omega_0t)] ,\nonumber \end{aligned}$$ where (c.c.) denotes the complex conjugate of the previous term, and $$\xi={\epsilon}(x-\lambda t),\quad\eta={\epsilon}y,\quad\tau={\epsilon}^2t.\label{str}$$ The present ordering (\[str\]) of the stretched variables $\xi,\eta,\tau$ is the same as one introduced in the derivation of the DS equation [@davey]. Substituting the expansions (\[ex1\])-(\[ex4\]) into Eqs(\[b1\])- (\[b4\]) and using the method of multiple scales [@taniuti], we obtain from the leading order equations(linearized equations) $$\begin{aligned}
\Theta_1&=&L_\theta w_1,\quad L_\theta=(\kappa^2-i\omega_0)^{-1},\nonumber\\
c_1&=&\pi L_cw_1,\quad L_c=(\zeta\kappa^2-i\omega_0)^{-1},\nonumber\\
P_1&=&\pi L_pw_1,\quad L_p=-\sigma(R_0L_\theta/\kappa^2+\zeta QL_c),
\nonumber\\
u_1&=&i\frac{\pi}{k_0}w_1,\quad a_1=-i\frac{\pi}{k_0}c_1,\nonumber\\
b_1&=&v_1=0, \nonumber\end{aligned}$$ where $\kappa^2=k_0^2+\pi^2$ and the linear dispersion relation becomes
$$f\equiv \sigma\kappa^2-i\omega_0-\pi^2L_p-\sigma R_0L_\theta=0.$$
The second order field variables are expressed as follows $$\begin{aligned}
w^{(2)}&=&w_1^{(2)}E\sin(\pi z),\nonumber\\
\Theta^{(2)}&=&\Theta_1^{(2)}E\sin(\pi z)+(\mbox{c.c.})+\Theta_0^{(2)}
\sin(2\pi z),\label{theta2}\\
c^{(2)}&=&[c_1^{(2)}\cos(\pi z)+\bar c_2^{(2)}E]E+(\mbox{c.c.}),\\
P^{(2)}&=&[P_1^{(2)}\cos(\pi z)+\{P_2^{(2)}\cos(2\pi z)+{\bar P}_2^{(2)}\}E]E+
(\mbox{c.c.})\nonumber \\
& &+P_0^{(2)}\cos(2\pi z)+{\bar P}_0^{(2)},\\
u^{(2)}&=&u_1^{(2)}E\cos(\pi z)+(\mbox{c.c.})+u_0^{(2)}\cos(2\pi z)+{\bar u}_0^{(2)},
\\
a^{(2)}&=&a_1^{(2)}E\sin(\pi z)+(\mbox{c.c.})+a_0^{(2)}\sin(2\pi z),\\
v^{(2)}&=&v_1^{(2)}E\cos(\pi z)+\bar v_0^{(2)},\quad b^{(2)}=b_1^{(2)}E
\sin(\pi z),\label{vb2}\end{aligned}$$ where ${\bar u}_0^{(2)}$ and $\bar v_0^{(2)}$ are determined by $$\begin{aligned}
({\partial_\xi}^2+{\partial_\eta}^2){\bar u}_0^{(2)}&=&\frac{\pi^2}{k_0^2\lambda}(1-\sigma\zeta Q
\pi^2|L_c|^2){\partial_\eta}^2|w_1|^2,\label{u0}\\
({\partial_\xi}^2+{\partial_\eta}^2)\bar v_0^{(2)}&=&-\frac{\pi^2}{k_0^2\lambda}(1-\sigma\zeta Q
\pi^2|L_c|^2){\partial_\eta}{\partial_\xi}|w_1|^2.\nonumber \end{aligned}$$ The other second order amplitudes such as $\Theta_1^{(2)},\Theta_0^{(2)},c_1^{(2)} $ and so on are given in Appendix and give $$\lambda=[\partial_k \omega(k)]_{k=k_0}\equiv\partial_k \omega_0.$$ As shown in Appendix, the solvability condition for the third order variable $w_1^{(3)}$ yields the following equation of the envelope of the first order vertical fluid velocity $w_1$ : $$i\,{\partial_\tau}w_1+\frac{\partial_k^2\omega_0}{2}{\partial_\xi}^2w_1+\frac{\partial_k
\omega_0}{2k_0}
{\partial_\eta}^2w_1+\left(\frac{q}{\partial_{\omega_0}f}|w_1|^2-k_0
{\bar u}_0^{(2)}\right)w_1+\frac{\partial_{R_0}f}
{\partial_{\omega_0}f}R^{(2)}w_1=0, \label{DS}$$ where $\partial_k\omega_0$ is real, while $\partial_k^2\omega_0\equiv(\partial_k^2\omega)_{k=k_0}$ has a complex value in general and $$\begin{aligned}
q&=&\sigma R_0\frac{k_0^2}{2\kappa^2}L_\theta L_\theta'+\sigma Q\frac
{\zeta\pi^4}{2\zeta k_0^2-i\omega_0}\left[\frac{3k_0^2-\pi^2}{\kappa^2}
|L_c|^2-Lc^2\right]
\nonumber\\
& &-\frac{8i\sigma Q\pi^6}{\kappa^2(Q+4\pi^2)}L_c''L_c
\nonumber\\
& &+\frac{iQ\pi^2}{Q+4\pi^2}\left(\pi^2\sigma\zeta QL_c^2+\sigma R_0\frac{k_0^2}
{\kappa^2}L_\theta^2+\frac{k_0^2-3\pi^2}{\kappa^2}\right)L_c''.\label{q}
\end{aligned}$$ A coupled system of equations (\[DS\]) and (\[u0\]) is the DS equation with complex coefficients, in which the real field ${\bar u}_0^{(2)}$ represents $z$ independent incompressive horizontal flows varying slowly.\
Modulational Instability of Oscillatory Travelling Rolls
========================================================
For later conveniences, the DS equation with complex coefficients (a coupled system of equations (\[DS\]) and (\[u0\]) ) is rewritten as follows. $$i\,{\partial_t}\Psi+\alpha{\partial_x}^2\Psi+\beta{\partial_y}^2\Psi+(\gamma|\Psi|^2+su)\Psi=i
r\Psi, \label{DS1}$$ $$({\partial_x}^2+a{\partial_y}^2)u= (b{\partial_x}^2+c{\partial_y}^2)|\Psi|^2,\label{DS2}$$ where $\alpha,\beta$ and $\gamma$ are complex constants $(\alpha=\alpha'+
i\alpha'',\beta=\beta'+i\beta'',\gamma=\gamma'+i\gamma'')$, while $s,r,a,b$ and $c$ are real constants $(a=1,b=0,\beta''=0$ in the present case). A spatially uniform solution of Eqs.(\[DS1\]) and (\[DS2\]) is given by $$\Psi=\Psi_0\equiv\psi_0\exp(-i\Omega t),\qquad u=0,$$ where $|\psi_0|^2=r/\gamma''$ and $\Omega=-\gamma'|\psi_0|^2$. Setting $$\begin{aligned}
\Psi&=&\Psi_0+\psi_1(t)\exp[i({\mbox{\boldmath $k$}}\cdot{\mbox{\boldmath $r$}}-\Omega_1t)]+
\psi_2(t)\exp[i(-{\mbox{\boldmath $k$}}\cdot{\mbox{\boldmath $r$}}-\Omega_2t)],\\
u&=&u_1(t)\exp[i\{{\mbox{\boldmath $k$}}\cdot{\mbox{\boldmath $r$}}-(\Omega_1-\Omega)t\}]\\
& &+u_2(t)\exp[i\{-{\mbox{\boldmath $k$}}\cdot{\mbox{\boldmath $r$}}-(\Omega_2-\Omega)t\}]+(\mbox{c.c.}),
\end{aligned}$$ where ${\mbox{\boldmath $k$}}=(k_x,k_y), {\mbox{\boldmath $r$}}=(x,y), 2\Omega=\Omega_1+\Omega_2$ and linearizing Eqs.(\[DS1\]) and (\[DS2\]) with respect to $\psi_1$ and $\psi_2$, we have $$\begin{aligned}
\frac{d\psi_1}{dt}&=&[r-i\{\alpha k_x^2+\beta k_y^2-\Omega_1-(\gamma+\hat
\gamma)|\psi_0|^2\}]\psi_1
+i\hat\gamma\psi_0^2\psi_2^*,\label{l1}\\
\frac{d\psi_2^*}{dt}&=&[r+i\{\alpha^*k_x^2+\beta^*k_y^2-\Omega_2-(\gamma^*
+\hat\gamma^*)|\psi_0|^2\}]\psi_2^*
-i\hat\gamma^*{\psi_0^*}^2\psi_1,\label{l2}
\end{aligned}$$ where $\hat\gamma=\gamma+s(bk_x^2+ck_y^2)/(k_x^2+ak_y^2)$ and \* denotes the complex conjugate. The linear equations (\[l1\]) and (\[l2\]) have exponentially growing solutions if the following condition is satisfied. $$(\alpha'\gamma'+\alpha''\hat\gamma'')k_x^2+(\beta'\gamma'+\beta''
\hat\gamma'')k_y^2>0.\label{inst}$$ Since $a>0$ in the present case, Eq.(\[inst\]) yields the following instability criteria. $$\hat\alpha\equiv\alpha'(\gamma'+sb)+\alpha''\gamma''>0,\label{alpha}$$ or $$\hat\beta\equiv\beta'(\gamma'+sc/a)+\beta''\gamma''>0,\label{beta}$$ or $$(\hat\alpha-\hat\beta/a)^2+\hat s[2(\hat\alpha+\hat\beta/a)+\hat s]>0,
\quad\mbox{and}\quad \hat\alpha+\hat\beta/a+\hat s>0,\label{newcr}$$ where $\hat s=s(c/a-b)(\alpha'-\beta'/a)>0$. The criterion (\[alpha\]) or (\[beta\]) is essentially the same as the case of the two-dimensional complex Ginzburg-Landau equation (Eq.(\[DS1\]) with $s=0$). A new criterion (\[newcr\]) comes from the coupling between convetive rolls and horizontal imcompressible flows.If $a<0$, although this is not the case in magnetoconvection, a spatially uniform solution of Eqs. (\[DS1\]) and (\[DS2\]) is shown to be always modulatinally unstable.
Concluding Remarks
==================
In this paper, we have derived a envelope equation of oscillatory travelling rolls near a Hopf bifurcation, which is not a type of the NWS equation but the DS equation with complex coefficients. The NWS type equation has a defect if the group velocity at the critical (bifurcation) point does not vanish. That is, it consists of different order terms: the main term is linear and proportional to the group velocity, while the other interesting terms such as nonlinear terms are of higher order. Although our derivation is based on weakly nonlinear analysis with multiple scales similar to [@brand], the present ordering of stretched variables (\[str\]) is different from the NW ordering [@newell1] and yields the DS equation with complex coefficients which consists of the same order terms.\
Analyzing the modulational instability of a spatially independent oscillatory solution of the DS equation with complex coefficients, we obtain criteria of the modulational instability, which include not only the known criterion for the complex Ginzburg-Landau equation ,which was first given in [@lange], but also a new criterion due to the coupling between convective rolls and horizontal incompressible flows.\
[**Appendix**]{}
Higher Order Amplitudes
=======================
In terms of the first order amplitudes and $w_1^{(2)}$, the second order amplitudes defined in Eqs.(\[theta2\])-(\[vb2\]) are given by $$\begin{aligned}
\Theta_1^{(2)}&=&L_\theta w_1^{(2)}-i\dot L_\theta{\partial_\xi}w_1,\quad
c_1^{(2)}=\pi L_cw_1^{(2)}-i\pi\dot L_c{\partial_\xi}w_1, \\
P_1^{(2)}&=&\pi L_pw_1^{(2)}-i\pi \dot L_p{\partial_\xi}w_1,\quad
u_1^{(2)}=i\frac{\pi}{k_0}(w_1^{(2)}+\frac{i}{k_0}{\partial_\xi}w_1),\\
a_1^{(2)}&=&-i\frac{\pi}{k_0}(c_1^{(2)}+\frac{i}{k_0}{\partial_\xi}c_1),\quad
v_1^{(2)}=\frac{\pi^2}{k_0}{\partial_\eta}w_1,\quad
b_1^{(2)}=-\frac{\pi^2}{k_0}{\partial_\eta}c_1,\\
\dot L_\theta&=&\lambda\partial_{\omega_0} L_\theta+\partial_{k_0}L_\theta
,\quad\mbox{etc.},\end{aligned}$$ and $$\begin{aligned}
\Theta_0^{(2)}&=&-\frac{L_\theta '}{2\pi}|w_1|^2,\quad
u_0^{(2)}=-2Q\frac{\pi^2}{k_0(Q+4\pi^2)}L_c''|w_1|^2,\quad a_0^{(2)}=\frac
{2\pi}{Q\zeta}u_0^{(2)},\\
P_0^{(2)}&=&(1+\frac{\sigma R_0L_\theta'}{4\pi^2})|w_1|^2+\frac{\sigma
\zeta Q\kappa^2}{2k_0^2}|c_1|^2,\\
{\bar P}_0^{(2)}&=&\lambda {\bar u}_0^{(2)}-\frac{\pi^2}{k_0^2}|w_1|^2-\frac
{\sigma\zeta Q}{2}(1-\frac{\pi^2}{k_0^2})|c_1|^2,\\
P_2^{(2)}&=&-\frac{\sigma\zeta Q\kappa^2}{4k_0^2}c_1^2,\quad
{\bar P}_2^{(2)}=-\sigma\zeta Q[\frac{\pi}{2\zeta k_0^2-i\omega_0}w_1+\frac
{\kappa^2}{4k_0^2}c_1]c_1+\frac{\pi^2}{2k_0^2}w_1^2,\\
\bar c_2^{(2)}&=&\frac{\pi}{2\zeta k_0^2-i\omega_0}w_1c_1,\quad
Lc''=\mbox{Im}(L_c),\quad L_\theta'=\mbox{Re}(L_\theta), \end{aligned}$$ Lengthy but straightforward calculations of the third order terms in Eqs. (\[b1\])-(\[b4\]) give the following third order field variables (envelopes) proportional to $E$ in terms of $w_1$ and $w_1^{(2)}$. $$\begin{aligned}
\Theta_1^{(3)}&=&L_\theta w_1^{(3)}-i\dot L_\theta{\partial_\xi}w_1^{(2)}-(\ddot L_
\theta/2){\partial_\xi}^2w_1-L_\theta^2({\partial_\tau}-{\partial_\eta}^2)w_1\\
& &-ik_0L_\theta^2({\bar u}_0^{(2)}-u_0^{(2)}/2)w_1+\pi L_\theta
\Theta_0^{(2)}w_1,\\
c_1^{(3)}&=&\pi[L_cw_1^{(3)}-i\dot L_c{\partial_\xi}w_1^{(2)}-(\ddot L_c/2)
{\partial_\xi}^2w_1-L_c^2({\partial_\tau}-\zeta{\partial_\eta}^2)w_1]\\
& &+ik_0L_c[a_0^{(2)}/2-\pi L_c({\bar u}_0^{(2)}
+u_0^{(2)}/2)]w_1-\pi L_c{\bar c}_2^{(2)}w_{-1},\\
P_1^{(3)}&=&\pi[L_pw_1^{(3)}-i\dot L_p{\partial_\xi}w_1^{(2)}-(\ddot L_p/2){\partial_\xi}^2w_1
-1/(2k_0)\partial_{k_0}L_p{\partial_\eta}^2w_1\\
& &+i\partial_{\omega_0}L_p{\partial_\tau}w_1-(\sigma R^{(2)}/\kappa^2)L_\theta w_1]\\
& &+\sigma\zeta Q\pi[L_c-L_c^*(k_0^2-3\pi^2)/\kappa^2]\bar c_2^{(2)}w_{-1}\\
& &-[(i\sigma\zeta Q/2)\{(k_0^4+4k_0^2\pi
^2-\pi^4)/k\kappa^2\}L_ca_0^{(2)}+k_0\pi\partial_{\omega_0}L_p\bar u_0^{(2)}\\
& &-(ik_0\pi/2)\{\sigma\zeta QL_c^2-(\sigma R_0/\kappa^2)L_\theta^2-
(4/\kappa^2)\}u_0^{(2)}\\
& &+(\sigma R_0\pi^2/\kappa^2)L_\theta\Theta_0^{(2)}]w_1 ,\end{aligned}$$ and $$\begin{aligned}
fw_1^{(3)}&=&i\dot f{\partial_\xi}w_1^{(2)}-i\partial_{\omega_0}f{\partial_\tau}w_1
+(1/2)(\ddot f{\partial_\xi}^2w_1+k_0^{-1}\partial_{k_0}f{\partial_\eta}^2w_1)\nonumber
\\& &+(k_0\partial_{\omega_0}f{\bar u}_0^{(2)}-\partial_{R_0}f
R^{(2)})w_1-q|w_1|^2w_1,\label{A1}\end{aligned}$$ where $q$ is given in Eq.(\[q\]). Since $$f=\dot f=0,\qquad\frac{\partial_{k_0}f}{\partial_{\omega_0}f}=-\partial_k
\omega_0,\qquad\frac{\ddot f}{\partial_{\omega_0}f}=-\partial_k^2\omega_0,$$ Eq.(\[A1\]) gives Eq.(\[DS\]).\
[**Ackowledgement**]{}
One of authors (K.N.) wishes to thank Prof.N.Bekki,Nihon University,for his introduction to magnetoconvection.\
[99]{} T.Clune and E.Knobloch,Physica D 74(1994)151. A.C.Newell and J.A.Whitehead,J.Fluid Mech.38(1969)279,\
L.A.Segel, J.Fluid Mech. 38 (1969) 203. H.R.Brand,P.S.Lomdahl and A.C.Newell, Phys.Lett. 118A(1986) 67. T.Taniuti,Suppl.Prog.Theor.Phys. 55(1974)1. A.Davey and K.Stewartson,Proc.R.Soc.Lond.A.338(1974)101. C.G.Lange and A.C.Newell,SIAM J.Appl.Math.27(1974)441.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The PEP-N experiment requires a fast on-line luminosity monitor of modest accuracy plus an off-line method of determining integrated luminosity with accuracy of 0.01 for each pb$^{-1}$. We propose the PEP-2 monitor, based on observing single bremsstrahlung at zero degrees to the positron direction at collision for the former and the use of Bhabha scatters at polar angles $>$.03 radians for the latter requirement.'
author:
- 'Mark Mandelkern, University of California, Irvine, CA92697, USA'
title: |
\
Luminosity Measurement at PEP-N
---
On-Line Luminosity
==================
An on-line monitor is required for tuning and monitoring the machine. It is desirable that it provide a measurement with 10% or better accuracy, and fluctuations of less than 1% at a refresh time of less than 1 second. The PEP-2 monitor, based on observing single bremsstrahlung at zero degrees to the positron direction at collision, described in Ref. [@bib:field] seems appropriate.
Single bremsstrahlung, or radiative Bhabha scattering, has a differential cross section, integrated over electron and positron angles, of:
$$\frac{d\sigma}{d\omega}=\frac{4\alpha r_0^2}{\omega}
\frac{E-\omega}{\omega}
(V-2/3)[ln\frac{m}{q_{min}}-1/2]$$
where $V=\frac{E-\omega}{E}+\frac{E}{E-\omega}$ and $q_min=\frac{m}{4\gamma^2}\frac{\omega}{E-\omega}$. Here $E$ is the initial electron or positron energy, $\gamma=E/m$ and $r_0=e^2/m$. The angular distribution of the $\gamma s$ is strongly forward with angular width $\sim\gamma^{-1}$. $\frac{d\sigma}{d\omega}$ is a function only of $\omega/E$ so the flux of $\gamma s$ at $\sim 0^\circ$ to the LER is independent of $s$. For PEP-N conditions I have used the program BBBREM [@bib:bbbrem], provided by Lew Keller, to estimate the cross section for $\omega>400$ MeV radiation from the $e^+$ beam to be 76 mb.
The momentum transfer for this process can be remarkably small, corresponding to a very large impact parameter $\rho$ and leading to screening effects which must be taken into account. If we choose E=3 GeV and $\omega>300 MeV$, $q_{min}=0.4 10^{-9}$ MeV and $\rho_{max}=0.05 cm$ which is greater than the transverse size of the beams in PEP-N. The consequence is that the cross section is cut off at a momentum transfer $\sim q_{min}$. This problem has been treated by various authors and the following result by Burov and Derbenev is quoted by Ref. [@bib:blinov] for the case of a for a Gaussian beam density where the transverse beam size is smaller than characteristic impact parameters:
$$\frac{d\sigma}{d\omega}=\frac{4\alpha
r_0^2}{\omega}\frac{E-\omega}{\omega}
(V-2/3)[ln\frac{\Delta_y\Delta_z}{\lambda_C(\Delta_y+\Delta_z)}+
ln2+c/2+\frac{V-5/9}{V-2/3}]$$
where $c=0.577$ and $\Delta_y$ and $\Delta_z$ are the rms transverse beam dimensions. $\lambda_C$ is the electron Compton wavelength ($m^{-1}$). The sensitivity of this effective cross section to variation of the PEP-N beam is approximately a 3.5% increase for a doubling of the radius. Despite this modest sensitivity, the dependence on beam size and shape introduces uncertainty that is undesireable for an absolute luminosity measurement. The background to radiative Bhabhas at $0^{\circ}$ is synchrotron radiation and beam-gas bremsstrahlung. At PEP-II, a Cerenkov shower counter is used with a threshold sufficiently high to be immune to the SR. The beam-gas background is apparently not a problem.\
The interaction region should be designed so that such a monitor can be installed, which requires a clear aperture, suitable window, and space for the monitor. At PEP-2, the monitor is installed at 8m from the interaction point. We also want this monitor well downstream of the detector.\
Off-Line Luminosity
===================
The accurate and precise determination of integrated luminosity required for the experiment will be obtained from QED processes observed in the detector. We require a 1% or better measurement for each inverse picobarn of running. The available processes are Bhabha scattering and annihilations into muon pairs and gammas. We consider them individually in the context of the standard detector design. Our luminosity determination will be similar to that of BABAR, described for example in Touramanis’ talk at the 2/2001 BABAR Collaboration Meeting. The BABAR determination is based on wide-angle ($>45^{\circ}$) Bhabhas and muon pairs. The systematic error is contributed to by the Monte Carlo (1-2%) and cut stability (1%), for an overall 2%. The annihilation to 2 photons has a greater systematic uncertainty, at least 3%, since the event rate is sensitive to mass and the geometrical acceptance is less well defined (angles for photons are not measured as well as those for charged particles).\
In PEP-N the experimental situation is somewhat different. Since the calorimeter has relatively course spatial resolution ($\sigma \sim
2.5$ cm), it is not possible to accurately define the acceptance for photons, leading to an unacceptably large systematic error for the 2 photon annihilation rate. Since the luminosity is much smaller than for BABAR and we seek 1% uncertainties on a point-by-point basis, we must accept Bhabha and especially muon pair events at smaller polar angles, which requires good angular measurements at small angles to adequately define the acceptance. To obtain a 1% statistical error for each inverse pb we require $>10,000$ events for an integrated cross section of $>10$ nb. On the other hand the PEP-N detector is simpler and we may do better in the Monte Carlo simulation, which is the dominant error for the BABAR luminosity. In particular one particle for all Bhabha and muon pair events will be seen by the forward planar tracking chamber and electromagnetic and hadron calorimeters.\
Geometry
--------
These (approximate) geometrical parameters are taken from the current detector layout. The beam pipe is expected to have a 5 cm radius and the default is 2.5 mm of aluminum. We assume $4\pi$ tracking with 200 micron resolutiom for radii $<60$ cm , planar forward tracking with 200 micron resolution at z=120 cm with unhindered aperture of $\pm
23^{\circ}$, planar forward electromagnetic calorimetry at z=180 cm with $\pm 36^{\circ}$ aperture and planar forward hadron calorimetry at z=220 cm with $\pm 27 ^{\circ}$ aperture. The forward hadron calorimeter will be used for muon ID.\
Bhabhas
-------
Both electron and positron can be identified at all angles since we have nearly $4\pi$ tracking and electromagnetic calorimetry. In order to get adequate statistics we must take advantage of the large forward cross section and count events in which one particle strikes the forward tracking chamber and forward electromagnetic calorimeter. It will certainly be helpful to identify the backward electron as well. The cross section, as seen in Table \[tab:bhabha\] is well over 100 nb at all energies. For good control of systematics, it will be useful to define an acceptance at a relatively large positron angle. This avoids relying on events in which the e$^+$ passes very obliquely through the beam pipe and reduces the angular accuracy and precision required to define the acceptance. However we wish events in which the forward track passes directly into the forward tracking chamber, missing the barrel calorimeter, as shown for example in Fig. \[fig:geom\]. We give cross sections integrated between positron laboratory angles of 0.3 ($17.2^{\circ}$) and 0.4 ($22.9^{\circ}$). As seen in Figure \[fig:p2\], the corresponding electron appears at $28^\circ$-$40^\circ$ at $\sqrt
s=1.4$ GeV and $97^\circ$-$114^\circ$ at $\sqrt s=3$ GeV, and is detected in the barrel calorimeter which extends backward to $157^\circ$. We will not be limited statistically in the Bhabha measurement. The acceptance determination requires that we measure angles to about 1.5 mr which should be relatively straightforward using the well defined interaction point and the forward tracking chamber about 120 cm from the interaction point with spatial resolution $\sim 200 \mu$m. Multiple scattering is a consideration here. At $17.2^{\circ}$, the effective thickness of the 2.5 mm Al beam pipe is .095 radiation lengths for a rms multiple scattering angle of 1.1 mr. We can’t tolerate a much thicker beam pipe.\
Muon pairs
----------
The muon pair cross section is much smaller and to obtain adequate statistics we would have to accept events at much smaller angles. Table \[tab:muon\] gives the integrated cross section between laboratory angles of 0.1 ($5.7^{\circ}$) and 0.4 ($22.9^{\circ}$). Even so the statistics will be marginal at the largest center of mass energies. The smaller angles would then require more precise angular measurements for the acceptance determination, i.e. about 0.5 mr. However the multiple scattering for a very forward muon passing obliquely through the beam pipe is much larger, i.e. at $5.7^{\circ}$, the effective beam pipe thickness is about 28% of a radiation length and the rms multiple scattering angle is about 2 mr. A substantially thinner beam pipe would be required, or one with an angled window which is not obviously feasible at small angles. Muon pairs will be useful as a rough check of the Bhabha measurement but it will hard to obtain a precise luminosity because of statistical and systematic uncertainties.\
Conclusion
----------
Using Bhabhas, the PEP-N detector as proposed should produce integrated luminosity measurements with the desired 1-2% accuracy for individual points representing about 1 pb$^{-1}$ of integrated luminosity. Muon pairs will be useful as a check although the muon pair luminosity will not generally have the required statistical accuracy.
[9]{}
Ecklund, S; Field,C; Mazaheri,G. A fast luminosity monitor sustem for PEP-II. SLAC-PUB-8688, Oct 2000. Submitted to Nucl.Instrum.Meth.
Kleiss, R and Burkhardt, H. BBBREM-Monte Carlo simulation of radiative Bhabha scattering in the very forward direction. NIKHEF-H/94-01, Jan 1994 (hep-ph 9401333).
A.E.Blinov [*et al.*]{}. Luminosity measurement with the MD-1 Detector at VEPP-4. [*Nuc. Inst. Meth.*]{} A273 (1988), 31-39.
$e^-$ energy $E_{cm}$ $\theta^l_{min}$ $\theta^l_{max}$ cos($\theta^{cm}_{max}$) cos($\theta^{cm}_{min}$) $\sigma$(nb)
-------------- ---------- ------------------ ------------------ -------------------------- -------------------------- -------------- --
0.100 1.114 0.300 0.400 0.171 -0.120 280.499
0.200 1.575 0.300 0.400 0.477 0.222 174.436
0.300 1.929 0.300 0.400 0.618 0.404 152.080
0.400 2.227 0.300 0.400 0.699 0.517 143.612
0.500 2.490 0.300 0.400 0.752 0.594 139.480
0.600 2.728 0.300 0.400 0.789 0.650 137.151
0.700 2.946 0.300 0.400 0.816 0.692 135.706
0.800 3.150 0.300 0.400 0.837 0.725 134.748
0.900 3.341 0.300 0.400 0.854 0.752 134.080
1.000 3.521 0.300 0.400 0.868 0.774 133.596
: Cross sections for Bhabhas.
\[tab:bhabha\]
$e^-$ energy $E_{cm}$ $\theta^l_{min}$ $\theta^l_{max}$ cos($\theta^{cm}_{max}$) cos($\theta^{cm}_{min}$) $\sigma$(nb)
-------------- ---------- ------------------ ------------------ -------------------------- -------------------------- -------------- --
0.100 1.114 0.100 0.400 0.856 -0.120 62.416
0.200 1.575 0.100 0.400 0.925 0.222 25.226
0.300 1.929 0.100 0.400 0.950 0.404 14.103
0.400 2.227 0.100 0.400 0.962 0.517 9.093
0.500 2.490 0.100 0.400 0.969 0.594 6.372
0.600 2.728 0.100 0.400 0.974 0.650 4.721
0.700 2.946 0.100 0.400 0.978 0.692 3.641
0.800 3.150 0.100 0.400 0.981 0.725 2.895
0.900 3.341 0.100 0.400 0.983 0.752 2.358
1.000 3.521 0.100 0.400 0.985 0.774 1.958
: Cross sections for $\mu$ pairs.
\[tab:muon\]
{width="150mm"}
{width="150mm"}
{width="150mm"}
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We continue the research initiated in hep-th/0607215 and apply our method of conformal automorphisms to generate various ${{\cal N}}{=}4$ superconformal quantum many-body systems on the real line from a set of decoupled particles extended by fermionic degrees of freedom. The $su(1,1|2)$ invariant models are governed by two scalar potentials obeying a system of nonlinear partial differential equations which generalizes the Witten-Dijkgraaf-Verlinde-Verlinde equations. As an application, the ${{\cal N}}{=}4$ superconformal extension of the three-particle ($A$-type) Calogero model generates a unique $G_2$-type Hamiltonian featuring three-body interactions. We fully analyze the ${{\cal N}}{=}4$ superconformal three- and four-particle models based on the root systems of $A_1\oplus G_2$ and $F_4$, respectively. Beyond Wyllard’s solutions we find a list of new models, whose translational non-invariance of the center-of-mass motion fails to decouple and extends even to the relative particle motion.'
---
ITP–UH–17/07\
LMP-TPU–9/07\
[**N=4 superconformal Calogero models**]{} $
\textrm{\Large Anton Galajinsky\ }^{a} ,\quad
\textrm{\Large Olaf Lechtenfeld\ }^{b} ,\quad
\textrm{\Large Kirill Polovnikov\ }^{a}
$ 0.7cm ${}^{a}$ [*Laboratory of Mathematical Physics, Tomsk Polytechnic University,\
634050 Tomsk, Lenin Ave. 30, Russian Federation*]{}\
[Emails: galajin, kir @mph.phtd.tpu.edu.ru]{} 0.4cm ${}^{b}$ [*Institut für Theoretische Physik, Leibniz Universität Hannover,\
Appelstrasse 2, D-30167 Hannover, Germany*]{}\
[Email: lechtenf@itp.uni-hannover.de]{} 0.2cm
PACS: 04.60.Ds; 11.30.Pb; 12.60.Jv\
Keywords: superconformal Calogero model, nonlocal conformal transformations
0
[**1. Introduction**]{}\
Recently conformally invariant models in one dimension were investigated extensively [@town]–[@div]. On the one hand, the interest derives from the AdS/CFT correspondence. Although there has been considerable progress in understanding the AdS/CFT duality [@adc/cft], nontrivial examples of AdS${}_2$/CFT${}_1$ correspondence are unknown. On the other hand, the conformal group SO$(2,d{-}1)$ is the isometry group of anti de Sitter space AdS${}_d$. Since anti de Sitter space describes the near-horizon geometry of a wide class of extreme black holes (for a review see e.g. [@moh]), it was conjectured [@claus; @gibb] that the study of conformally invariant models in $d{=}1$ yields new insight into the quantum mechanics of black holes. This idea was pushed further in a series of papers [@mich]–[@str3], where some conformal mechanics on black-hole moduli spaces in $d{=}4$ and $d{=}5$ was constructed and investigated.
Particularly appealing in this context seems a proposal in [@gibb] that an ${{\cal N}}{=}4$ superconformal extension of the Calogero model [@calo] might provide a microscopic description of the extreme Reissner-Nordström black hole near the horizon. It should be stressed, however, that the Calogero model, which describes a pair-wise interaction of $n$ identical particles on the real line, is not the only multi-particle exactly solvable conformal mechanics available in $d{=}1$. More complicated systems describing three-particle and four-particle interactions were studied in [@wolf]–[@ruehl]. Since in the context of [@gibb] it is the structure of the conformal algebra which matters, a priori any multi-particle ${{\cal N}}{=}4$ superconformal mechanics seems to be a good starting point. A classification of (off-shell) $d{=}1$ supermultiplets is interesting in its own right because of features absent in higher dimensions (see e.g. [@ils]). In this connection the construction of multi-particle ${{\cal N}}{=}4$ superconformal models is relevant for possible couplings of $d{=}1$, ${{\cal N}}{=}4$ supermultiplets.
Several attempts have been made to construct an ${{\cal N}}{=}4$ superconformal extension of the Calogero model [@wyl]–[@bgl]. In [@wyl] conditions for $su(1,1|2)$ invariance were formulated, and some solutions were presented. In [@bgk] the problem was solved for a complexification of the Calogero model. In [@gal; @bgl] the construction of an ${{\cal N}}{=}4$ superconformal Calogero model was reduced to solving a system of nonlinear partial differential equations, which generalizes the Witten-Dijkgraaf-Verlinde-Verlinde equation known from two-dimensional topological field theory [@w; @dvv]. However, beyond the two-particle case only partial results were obtained.
In the present work we continue the research initiated in [@glp] and apply the method of unitary transformations to generate various $su(1,1|2)$ invariant quantum many-body systems, including an ${{\cal N}}{=}4$ superconformal extension of the Calogero model. In section 2 we discuss a specific unitary transformation, which maps a generic conformally invariant model of $n$ identical particles on the real line to a set of decoupled particles, with the interaction being pushed into a nonlocal conformal boost generator. In this description, an ${{\cal N}}{=}4$ supersymmetric extension is straightforward to construct as we demonstrate in section 3. Both the conformal boost generator and its superpartner are nonlocal in this picture. The inverse transformation then provides us with the interacting Hamiltonian. The closure of the superconformal algebra poses constraints on the interaction, which are detailed and partially solved in section 4. Our superconformal models are governed by two scalar potentials obeying certain homogeneity conditions and the Witten-Dijkgraaf-Verlinde-Verlinde-type equations of [@gal; @bgl]. Explicit three- and four-particle solutions to these “structure equations” for the two scalar potentials are discussed in section 5 and found to be based on certain root systems. Beyond the models found by Wyllard [@wyl], we present a list of solutions which break translation invariance not only for the center-of-mass motion but also for the relative motion. In section 6 we summarize our results and discuss possible further developments.
[**2. Conformal mechanics in a free nonlocal representation**]{}\
Let us consider a system of $n$ identical particles on the real line with a Hamiltonian of the generic form \[h\] H= p\_i p\_i + V\_B (x\^1, …, x\^n) , where $m$ stands for the mass of each particle. Throughout the paper a summation over repeated indices is understood. Later, the bosonic potential $V_B$ will get supersymmetrically extended to a potential $V$ including $V_B$.
For conformally invariant models the Hamiltonian $H$ is part of the $so(1,2)$ conformal algebra \[al\] \[D,H\]=-H , \[H,K\]2D , \[D,K\]=K , where $D$ and $K$ are the dilatation and conformal boost generators, respectively. Their realization in term of coordinates and momenta, subject to \[x\^i, p\_j\]=\^i , reads D=- (x\^i p\_i +p\_i x\^i) = D\_0 K= x\^i x\^i = K\_0 , where the $0$ subscript indicates the generators in the free model ($V_B{=}0$). The first relation in (\[al\]) restricts the potential via \[ucl\] (x\^i \_i +2)V\_B 0 , meaning that $V_B$ must be homogeneous of degree $-2$ for the model to be conformally invariant. In this paper we assume this to be the case. Two simple solutions to (\[ucl\]) are the free model of $n$ non-interacting particles, V\_B= 0 H\_0 = p\_i p\_i , and the Calogero model of $n$ particles interacting through an inverse-square pair potential, V\_B = \_[i<j]{} H = H\_0 + V\_B .
As the next step we study the behavior of a generic conformal multi-particle mechanics under a judiciously chosen conformal-algebra automorphism. Given the particular $so(1,2)$ element \[a\] A = H -2 D + 1 K for a real parameter $\a$, let us consider the unitary transformation \[bak\] T T’ = \^[ A]{}T\^[- A]{} on the $so(1,2)$ generators:[^1] HH’ &=& K ,\
DD’ &=& -D+ K ,\
KK’ &=& \^2 H-4D+4K .
Notice that in the previous consideration it is only the structure of the conformal algebra that matters. Therefore, an analogous map exists for the free theory defined by $(H_0,D_0,K_0)$: T\_0 T’\_0 = \^[ A\_0]{}T\_0\^[- A\_0]{} A\_0 = H\_0 -2 D\_0 + 1 K\_0 . This suggests the idea to combine the $A$-map with inverse $A_0$-map to link $H$ and $H_0$ in the following scheme: \[tilded\]
(H,D,K) (H’,D’,K’) =(H’\_0,D’\_0,K’\_0[+]{}\^2V\_B)\
(,,) :=(H\_0,D\_0,K\_0[+]{}\^2\_B) (H’\_0,D’\_0,K’\_0[+]{}\^2V\_B)
with the abbreviation \_B=\^[- A\_0]{}V\_B\^[ A\_0]{} = K\_0 + \^2\_B . We remark that the dimensionful parameter $\a$ simply takes care of the different dimensionalities of the $so(1,2)$ generators and drops out of the final results as was shown in [@glp]. For the remainder of the paper we set $m=1$.
Thus, with the help of a unitary operation one can transform a generic multi-particle conformal mechanics (\[h\]), (\[ucl\]) into a one describing a system of non-interacting particles. A peculiar feature of this correspondence is that the generator of special conformal transformations $\wK$ is [*nonlocal*]{} and effectively hides the interaction potential. In fact, the interaction has disappeared in the Hamiltonian $\wH$ but resurfaced in a nonlocal contribution to the conformal boost $\wK$. Hence, the price paid for the simplification of the dynamics is a nonlocal realization of the full conformal algebra in the Hilbert space of the quantized conformal mechanics.
As an example, let us consider the conformal Calogero model describing the inverse-square pair-wise interaction of $n$ identical particles of unit mass on the real line, \[calodg\] V\_B=\_[i<j]{}\^n , where $g$ is the coupling constant. For this model, a map of $H$ to $H_0$ similar to ours was constructed in [@pol]. However, the entire $so(1,2)$ algebra was not examined, and the nonlocal structure present in $\wK$ was not revealed there. The quantum mechanical scattering analysis of the conformal Calogero model was accomplished in [@poli], where it was argued that the particles merely exchange their asymptotic momenta without altering their values. The asymptotic wave function only picks up an energy-independent phase factor through the scattering process. Since the conformal Calogero particles are indistinguishable, their physics is that of $n$ free bosons. Thus, the general consideration presented above is in agreement with [@poli; @suth].
[**3. [[N]{}]{}=4 superconformal extension**]{}\
The unitary map of a generic multi-particle conformal mechanics to a set of decoupled particles considered in the previous section offers a novel way to constructing superconformal extensions. In our setting this amounts to properly adding fermionic degrees of freedom to a free system and modifying the nonlocal boost generator $\wK$ so as to close an ${{\cal N}}$-extended superconformal algebra. Application of the inverse unitary transformation to the set of free superparticles then produces a desired superconformal extension of the original interacting conformal mechanics. In this section we discuss the corresponding algebraic framework.
The bosonic sector of the ${{\cal N}}{=}4$ superconformal algebra $su(1,1|2)$ includes two subalgebras. Along with $so(1,2)$ considered in the previous section one also finds the $su(2)$ R-symmetry subalgebra generated by $J_a$ with $a=1,2,3$. The fermionic sector is exhausted by the $su(2)$ doublet supersymmetry generators $Q_\a$ and ${\bar Q}^\a$ as well as their superconformal partners $S_\a$ and ${\bar S}^\a$, with $\a=1,2$, subject to the hermiticity relations \^|Q\^\^|S\^ . The bosonic generators are hermitian. The non-vanishing (anti)commutation relations in our superconformal algebra read[^2] $$\begin{aligned}
\label{algebra}
&
[D,H] \= -\ic \hbar\, H\ , &&
[H,K] \= 2\ic \hbar\, D\ ,
\nonumber\\[4pt]
&
[D,K] \= +\ic \hbar\, K\ , &&
[J_a,J_b] \= \ic \hbar\, \epsilon_{abc} J_c\ ,
\nonumber\\[2pt]
&
\{ Q_\a, \bar Q^\b \} \= 2\hbar\, H {\d_\a}^\b\ , &&
\{ Q_\a, \bar S^\b \} \=
+2\ic\hbar\,{{(\s_a)}_\a}^\b J_a-2\hbar\,D{\d_\a}^\b-\ic\hbar\,C{\d_\a}^\b\ ,
\nonumber\\[2pt]
&
\{ S_\a\,,\, \bar S^\b \} \= 2\hbar\, K {\d_\a}^\b\ , &&
\{ \bar Q^\a, S_\b \} \=
-2\ic\hbar\,{{(\s_a)}_\b}^\a J_a-2\hbar\,D{\d_\b}^\a+\ic\hbar\,C{\d_\b}^\a\ ,
\nonumber$$ $$\begin{aligned}
&
[D,Q_\a] \= -\sfrac{1}{2} \ic\hbar\, Q_\a\ , &&
[D,S_\a] \= +\sfrac{1}{2} \ic \hbar\, S_\a\ ,
\nonumber\\[4pt]
&
[K,Q_\a] \= +\ic \hbar\, S_\a\ , &&
[H,S_\a] \= -\ic \hbar\, Q_\a\ ,
\nonumber\\[2pt]
&
[J_a,Q_\a] \= -\sfrac{1}{2} \hbar\, {{(\s_a)}_\a}^\b Q_\b\ , &&
[J_a,S_\a] \= -\sfrac{1}{2} \hbar\, {{(\s_a)}_\a}^\b S_\b\ ,
\nonumber\\[4pt]
&
[D,\bar Q^\a] \= -\sfrac{1}{2} \ic \hbar\, \bar Q^\a\ , &&
[D,\bar S^\a] \= +\sfrac{1}{2} \ic \hbar\, \bar S^\a\ ,
\nonumber\\[4pt]
&
[K,\bar Q^\a] \= +\ic \hbar\, \bar S^\a\ , &&
[H,\bar S^\a] \= -\ic \hbar\, \bar Q^\a\ ,
\nonumber\\[2pt]
&
[J_a,\bar Q^\a] \= \sfrac{1}{2} \hbar\, \bar Q^\b {{(\s_a)}_\b}^\a\ , &&
[J_a,\bar S^\a] \= \sfrac{1}{2} \hbar\, \bar S^\b {{(\s_a)}_\b}^\a\ .\end{aligned}$$ Here $\e_{123}=1$, and $C$ stands for the central charge.
Following the same strategy as in the previous section, we employ the conformal automorphism (\[bak\]) and its free inverse as indicated in (\[tilded\]), with $A$ being of the same form as in (\[a\]). It is very plausible that the new (tilded) generators differ from the free ones only in the instances of $K$, $S_\a$ and $\bar S^\a$, so we write (omitting the complex conjugates and suppressing the indices) \[stilded\]
H &= H\_0 ,\
D &= D\_0 ,\
K &= K\_0 + \^2 ,\
Q &= Q\_0 ,\
S &= S\_0 - ,\
J &= J\_0 ,
where the correction to $S_0$ is determined from the form of $\wK$ through the $[\wK,\wQ]$ commutator in (\[algebra\]), and we again use the notation =\^[- A\_0]{}T\^[ A\_0]{} . Note that we have written $V$ instead of $V_B$, anticipating fermionic and quantum contributions to the Hamiltonian \[Hcorr\] H = H\_0 + V V = V\_B + V\_F + O() . Given $V$, the $[H,S]$ and $[H,\bar S]$ commutators in (\[algebra\]) enforce an interacting part for the supersymmetry generators, \[Qcorr\] Q\_= Q\_[0]{} -\[S\_[0]{},V\] |Q\^= |Q\^\_0 - \[|S\^\_0,V\] , while all other generators $T$ remain free, i.e. DD\_0 ,KK\_0 ,SS\_0 JJ\_0 . This is the result of inverting the map (\[stilded\]) to return from the tilded generators $\wT$ to the original ones $T$. We shall, however, use the tilded generators (\[stilded\]) to find the form of $V$.
For a mechanical realization of the $su(1,1|2)$ superalgebra, one introduces fermionic degrees of freedom represented by the operators $\psi^i_\a$ and $\bar\psi^{i\a}$, with $i=1,\dots,n$ and $\a=1,2$, which are hermitian conjugates of each other and obey the anti-commutation relations[^3] {\^i\_, \^j\_}0 , { [|]{}\^[i]{}, [|]{}\^[j]{} }0 , {\^i\_, [|]{}\^[j]{} }=[\_]{}\^\^[ij]{} . In the extended space it is easy to construct the free fermionic generators associated with the free Hamiltonian $H_0=\frac{1}{2}p_ip_i$, namely (for $m{=}1$) \[QSfree\] [Q\_0]{}\_p\_i \^i\_ , |Q\_0\^p\_i |\^[i]{} \_x\^i \^i\_ , |S\_0\^x\^i |\^[i]{} , as well as $su(2)$ generators \[Jfree\] [J\_0]{}\_a = |\^[i]{} [[(\_a)]{}\_]{}\^\^i\_ . Notice that these are automatically Weyl-ordered. The free dilatation and conformal boost operators maintain their bosonic form \[DKfree\] D\_0= -(x\^i p\_i +p\_i x\^i) K\_0= 12 x\^i x\^i . In contrast to the bosonic case, the free generators $T_0$ fail to satisfy the full algebra (\[algebra\]). Even for $C{=}0$, the $\{Q,\bar S\}$ and $\{\bar Q,S\}$ anticommutators require corrections cubic in the fermions, which we can restrict to $Q$ and $\bar Q$ as in (\[Qcorr\]). Dimensional analysis reveals that the coefficients of these cubic terms have a dimension of length${}^{-1}$ and thus cannot be constants. It follows further that $H$ contains quadratic and quartic fermionic terms, which are collected in $V_F$ in (\[Hcorr\]). Hence, even for $V_B{=}0$ there does not exist a free mechanical representation of the algebra (\[algebra\]).
The generators $\wK$, $\wS$ and $\widetilde{\bar S}$ are nonlocal. Substituting their form (\[stilded\]) into the superconformal algebra (\[algebra\]) one gets a set of restrictions on the form of the operator $V$: \[restr\]
& \[K\_0,V\]0 ,=-V , \[J\_[0a]{},V\]0 ,\
& { S\_[0]{},\[S\_[0]{},V\]}=\^2 \^i\_\^i\_ , { [|S\_0]{}\^,\[[|S\_0]{}\^,V\] }=\^2 [|]{}\^[i]{}[|]{}\^[i]{} ,\
& {S\_[0]{},\[[|S\_0]{}\^,V\]}+2\^2 [[(\_a)]{}\_]{}\^J\_[0a]{} + \^2(\^i\_|\^[i]{}[-]{}|\^[i]{} \^i\_) - \^2 C [\_]{}\^ ,\
& {[|S\_0]{}\^,\[S\_[0]{},V\]}=-2\^2 [[(\_a)]{}\_]{}\^J\_[0a]{} - \^2(\^i\_|\^[i]{}[-]{}|\^[i]{} \^i\_) + \^2 C [\_]{}\^ ,\
& { \[S\_[0]{},V\],\[S\_[0]{},V\]}+ {Q\_[0]{},\[S\_[0]{},V\]}+ {Q\_[0]{},\[S\_[0]{},V\]}0 ,\
& { \[|S\_0\^,V\],\[|S\_0\^,V\]}+ {|Q\_0\^,\[|S\_0\^,V\]}+ {|Q\_0\^,\[|S\_0\^,V\]}0 ,\
& { \[S\_[0]{},V\],\[[|S\_0]{}\^,V\]}+ {Q\_[0]{},\[[|S\_0]{}\^,V\]}+ {[|Q\_0]{}\^,\[S\_[0]{},V\]}+ 2\^3 V [\_]{}\^0 ,\
& -\[H\_0+V,\[S\_[0]{},V\]\]0 , -\[H\_0+V,\[[|S\_0]{}\^,V\]\]0 .
Notice that the vanishing (anti)commutators discarded in (\[algebra\]) should be taken into account as they also give constraints on $V$. For obtaining (\[restr\]) the following identities are helpful: \[ident\]
& \_[0]{}2Q\_[0]{}+S\_[0]{} , && \_0\^2|Q\_0\^+ |S\_0\^ ,\
& \_[0]{}=-Q\_[0]{} ,&& \_0\^=-|Q\_0\^ ,\
& S\_[0]{}=\_[0]{}+2\_[0]{} ,&& |S\_0\^=\_0\^+2\_0\^ .
[**4. The structure equations**]{}\
Let us discuss the structure of solutions to the constraints (\[restr\]). The first line in (\[restr\]) implies that the potential $V=V_B+V_F+O(\hbar)$ transforms as a scalar under SU(2) and is a degree $-2$ homogeneous function of the $x^i$. It is straightforward to check that an ansatz for $V_F$ quadratic in $\p^i$ and $\bar\p^i$ fails to solve (\[restr\]). This is in contrast with ${{\cal N}}{=}2$ superconformal extensions [@glp; @fm]. Thus it seems natural to try a general ansatz quartic in the fermionic coordinates,[^4] \[ans\] VV\_B(x) + O\_1(x) + \^2 O\_2(x) +M\_[ij]{}(x) \^i\_\^[j]{} + 14 L\_[ijkl]{}(x) \^i\_\^[j]{}|\^[k]{}|\^l\_ , with completely symmetric unknown functions $M_{ij}$ and $L_{ijkl}$. Here, the symbol $\langle\dots\rangle$ stands for symmetric (or Weyl) ordering (for our conventions see appendix A), and the contributions $\hbar O_1(x)$ and $\hbar^2 O_2(x)$ were included to account for the ordering ambiguity present in the fermionic sector. The argument $x$ indicates dependence on $\{x^1,\ldots,x^n\}$.
Introducing the notations \[wy\] L\_[ijkl]{}x\^l =: -W\_[ijk]{} M\_[ij]{}x\^j =: Y\_i and substituting the ansatz (\[ans\]) into the constraints (\[restr\]), one obtains the following system of partial differential and algebraic “structure equations”, && L\_[ijkl]{}=\_i W\_[jkl]{}=\_j W\_[ikl]{} , M\_[ij]{} =-\_i Y\_j =-\_j Y\_i , \[lw1\]\
&& x\^i W\_[ijk]{}=-\_[jk]{} , x\^i Y\_i =-C , \[lw2\]\
&& M\_[ij]{}+W\_[ijk]{}Y\_k0 , W\_[ikp]{} W\_[jlp]{} = W\_[jkp]{} W\_[ilp]{} , \[lw3\] as well as a boundary condition on $Y_i$, \[bc\] 12Y\_i Y\_i = V\_B . Besides, one determines the quantum corrections as \[qcor\] O\_10 O\_2=18W\_[ijk]{} W\_[ijk]{} . In contrast to ${{\cal N}}{=}2$ superconformal models, here the algebra requires a nontrivial quantum correction. The explicit derivation of (\[lw1\])–(\[qcor\]) is tedious and most efficiently achieved using reordering relations given in appendix A.
Taking into account that $W_{ijk}$ is a completely symmetric function, from (\[lw1\]) one finds \[pot\]
W\_[ijk]{} = \_i\_j\_k F &&& L\_[ijkl]{}= \_i\_j\_k\_l F ,\
Y\_i = \_i U &&& M\_[ij]{} = -\_i\_j U ,
with two scalar potentials $F(x)$ and $U(x)$ to be determined. Thus, these scalars govern the ${{\cal N}}{=}4$ superconformal extension and obey the following system of nonlinear partial differential equations, && (\_i\_k\_p F)(\_j\_l\_p F)= (\_j\_k\_p F)(\_i\_l\_p F),x\^i \_i \_j \_k F=-\_[jk]{} , \[w1\]\
&& \_i\_j U -(\_i\_j\_k F)\_k U0 , (\_i U)(\_i U)V\_B , x\^i \_i U=-C . \[w2\] Notice that $F$ is defined modulo a quadratic polynomial while $U$ is defined up to a constant.
Wyllard [@wyl] obtained equivalent equations, but employed a different fermionic ordering. In contrast to his equations, $\hbar$ does not appear in (\[w1\]) or (\[w2\]), since our Weyl-ordering prescription matches smoothly to the classical limit. For the classical Calogero model similar equations were discussed in [@bgl].
The right-most equations in (\[w1\]) and (\[w2\]) are inhomogeneous with constants $\d_{jk}$ and $C$ (the central charge) on the right-hand side and display an explicit coordinate dependence. Furthermore, the second equation in (\[w1\]) can be integrated twice to obtain \[w3\] x\^i \_i F -2F + x\^i x\^i 0 , where we used the freedom in the definition of $F$ to put the integration constants – a linear function on the right-hand side – to zero. It is important to realize that the inhomogeneous term in this integrated equation does break translation invariance and excludes the trivial solution $F=0$ equivalent to a homogeneous quadratic polynomial. This effect is absent in ${{\cal N}}{=}2$ superconformal models, where the four-fermion potential term is not needed and, hence, $F$ does not appear [@glp]. This issue is also discussed in [@wyl].
To be more explicit, we extract the center-of-mass dynamics by splitting \[comsplit\] F = F\_[com]{}(X) + F\_[rel]{}(x) U = U\_[com]{}(X) + U\_[rel]{}(x) with the center-of-mass coordinate $X:=\frac1n\sum_{i=1}^n x^i$. If the [*relative*]{} particle motion is translation invariant (which need not be the case), then \_[i=1]{}\^n \_i F\_[rel]{} = 0 = \_[i=1]{}\^n \_i U\_[rel]{} and, applying $\sum_i\pa_i$ to (\[w3\]) and the last equation in (\[w2\]), we readily find XF”\_[com]{} - F’\_[com]{} = -nX XU”\_[com]{} + U’\_[com]{} = 0 , which are solved by \[comsol\] F\_[com]{} = - X\^2|nX| + X\^2 + U\_[com]{} = -g\_0|nX| + with free constants $\lambda$, $\mu$, $\nu$ and $g_0$. Clearly, in this case we may put to zero $U_{\rm com}$ but not $F_{\rm com}$, so that for $g_0{=}0$ we end up with a center-of-mass contribution V\_[com]{} = X\^[-2]{} + X\^[-2]{} \_\^|\^|\_ \_ := 1n\_[i=1]{}\^n\_\^i . Hence, one can separate a translation-invariant relative motion from the center-of-mass motion, but the latter is non-linear due to an $X^{-2}$ potential as enforced by the superconformal algebra (\[algebra\]).
Our attack on (\[w1\]) and (\[w2\]) begins with the homogeneity conditions \[w4\] (x\^i\_i - 2) F = -12x\^ix\^i x\^i\_i U = -C . The most general solution is the sum of a particular solution and the general solution to the homogeneous equations, (x\^i\_i - 2) F\_[hom]{} = 0 x\^i\_i U\_[hom]{} = 0 , which is spanned by the homogeneous functions of degree two and zero, respectively. For a particular solution to (\[w4\]), we make the ansatz \[Fansatz\] F = -\_[=0]{}\^d h\_12(z\^)\^2|z\^| U = -\_[=0]{}\^d g\_|z\^| with a certain number ($d{+}1$) of linear coordinate combinations z\^= n\^\_ix\^i z\^0 = nX \_i x\^i . The relative motion is translation invariant if $\sum_i n_i^\m=0$ for $\m{>}0$. Compatibility with the conditions (\[w4\]) directly yields \[hcond\] \_[=0]{}\^d h\_n\_i\^n\_j\^= \_[ij]{} \_[=0]{}\^d g\_= C . The second relation fixes the central charge, and the first relation amounts to a decomposition of the identity $(\d_{ij})$ into rank-one projectors. It turns out that the $g_\m$ are independent free couplings (if not forced to zero) while the $h_\m$ are not.
A [*minimal*]{} solution involves $d{+}1=n$ mutually orthogonal vectors $n^\m$ beginning with $\vec n^0=(1,1,\ldots,1)$ and normalized as \[minimal\] n\^n\^ [\_i]{} n\_i\^n\_i\^= h\_\^[-1]{} \^ . From (\[Fansatz\]) we derive \[YWform\] W\_[ijk]{} = -\_[=0]{}\^[n-1]{} h\_ Y\_i = -\_[=0]{}\^[n-1]{} g\_ , and for the minimal choice (\[minimal\]) the bosonic potential becomes \[minpot\] V\_B = 12\_[=0]{}\^[n-1]{} O\_2 = 18\_[=0]{}\^[n-1]{} , which demonstrates that the quantum corrections only renormalize the coupling constants, g\_\^2 \_\^2 := g\_\^2 + 14 \^2 .
It is instructive to first investigate small values of $n$. At $n{=}2$, relative translation invariance demands $\vec n^0{=}(1,1)$ and $\vec n^1{=}(1,-1)$ with $h_0{=}h_1{=}\frac12$, whence
& F\_[rel]{}=-14(x\^1[-]{}x\^2)\^2|x\^1[-]{}x\^2| U\_[rel]{}=-g\_1|x\^1[-]{}x\^2| ,\
& W\_ = -12( ) V\_B+\^2O\_2 = + .
Beyond $n{=}2$, minimal choices are no longer invariant modulo sign under all permutations of the positions $x^i$, but, due to the linearity of (\[w4\]), this can be remedied by finally summing over all permutations. The result is, in general, an overcomplete set of $d{+}1>n$ non-orthogonal vectors. In section 5 we shall find a non-minimal one-parameter set (in $F$) of $n{=}3$ solutions to all structure equations for the choice \[3vec\] n\^0=(1,1,1) , n\^1=(1,-1,0) , n\^2=(1,1,-2) . However, a nontrivial $U_{\rm rel}$ based either on $\vec n^1$ or on $\vec n^2$ appears only for two specific parameter values. One may recognize here the root system of $A_1\oplus G_2$, which is the even part of the root system of the Lie superalgebra $G_3$. In the same section, we will describe five one-parameter families of $n{=}4$ solutions based on (parts of) the $F_4$ root system. Here, only three discrete models have $U_{\rm rel}$ non-vanishing, but for two of these the relative particle motion is not translation invariant.
In order to discover these and other solutions to the structure equations, within our ansatz (\[Fansatz\])–(\[hcond\]) it remains to solve the two left-most equations in (\[w1\]) and (\[w2\]), \[YW\] \_i Y\_j - W\_[ijk]{}Y\_k = 0 W\_[ikp]{}W\_[jlp]{}W\_[jkp]{}W\_[ilp]{} for $Y_i=\pa_iU$ and $W_{ijk}=\pa_i\pa_j\pa_kF$. This is quite tough because of their nonlinearity, and we address them in the following section. Already we notice, however, that the full system of structure equations (\[w1\]) and (\[w2\]) can be attacked in two different ways. One possibility, pursued in subsection 5.1, is to start with a given conformal potential $V_B$, e.g. of Calogero form, find a corresponding $U$, hence $Y$, and then search for a solution $W$ to (\[YW\]) before integrating it to $F$. Alternatively, as in subsection 5.2, one can take a particular solution $F$ of the quadratic relations in (\[YW\]), then find a solution $Y$ to the first equation in (\[YW\]) and integrate it to $U$, thereby determining $V_B$ afterwards. The second strategy will yield ${{\cal N}}{=}4$ superconformal models generalizing the Calogero one. Finally, any full solution $(Y,W)$ also determines the $su(1,1|2)$ generators as
& Q\_= (p\_k+Y\_k)\_\^k + W\_[ijk]{}\^i\_\^[j]{}|\^k\_ ,\
& |Q\^= (p\_k-Y\_k)|\^[k]{} + W\_[ijk]{}\^[i]{}|\^[j]{}|\^k\_ ,\
& H 12 p\_ip\_i + 12 Y\_iY\_i + W\_[ijk]{}W\_[ijk]{} - \_iY\_j \^i\_\^[j]{} +14 \_iW\_[jkl]{} \^i\_\^[j]{}|\^[k]{}|\^l\_ ,
while the other generators are of bilinear form given in (\[QSfree\]), (\[Jfree\]) and (\[DKfree\]).
We conclude the section by observing a resemblance of the quadratic relations in (\[YW\]) or (\[w1\]) to an $n$-parametric potential deformation of an $n$-dimensional Fröbenius algebra [@dub], which plays an important role in two-dimensional topological field theory [@w; @dvv]. Let us recall that an $n$-dimensional commutative associative algebra $A$ with unit element $e$ is called a Fröbenius algebra if it is supplied with a non-degenerate symmetric bilinear form obeying (for a review see e.g. [@dub]) \[frob\] a b,c= a,b c a,b,c A . Choosing a basis $\{e_i\,|\,i=1,\ldots,n\}$ with $e_1=e$, one has e\_i,e\_j=\_[ij]{} e\_i e\_j f\_[ij]{}\^k e\_k , where $\eta_{ij}$ is the metric with inverse $\eta^{ij}$ and ${f_{ij}}^k$ are the structure constants. The commutativity and associativity of the algebra along with (\[frob\]) produce the constraints \[f1\] [f\_[ij]{}]{}\^kf\_[ji]{}\^k ,\^j\_i\^j ,\^k \_[kl]{}f\_[lj]{}\^k \_[ki]{} ,\^k [f\_[kl]{}]{}\^m f\_[lj]{}\^k [f\_[ki]{}]{}\^m . Thus, ${f_{ij}}^k \eta_{kl} = f_{ijl}$ is totally symmetric and subject to the quadratic relations above.
An $n$-parametric potential deformation of such a Fröbenius algebra is defined by a set of functions f\_[ijk]{}(x)=\_i \_j \_k F(x) descending from some scalar potential $F(x)$ with $x=\{x^1,\ldots,x^n\}$. To qualify as a deformation, these functions must satisfy the relations \[wit\] f\_[1ij]{}(x)=\_[ij]{} , \_i\_[jk]{}0 , \^[kn]{} f\_[ijk]{}(x) f\_[lmn]{}(x)=\^[kn]{} f\_[ljk]{}(x) f\_[imn]{}(x) , which represent nonlinear partial differential equations for $F(x)$. In the context of two-dimensional topological field theory, $F$ is known as the free energy, and (\[wit\]) is called the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equation [@w; @dvv]. An interesting link between the WDVV equation and differential geometry was established in [@dub]. Comparing (\[w1\]) with (\[wit\]), we see that our algebra does not have a distinguished element serving as a unit element. Instead, the metric arises from the second equation in (\[w1\]) by a contraction of $f_{ijk}$ with the coordinates $x^i$.
[**5. Solutions to the structure equations** ]{}\
Proving the integrability of the structure equations (\[w1\]) and (\[w2\]) is a difficult task. For the WDVV equations this was done rigorously only for the simpler case of a decomposable Fröbenius algebra [@dub]. So, instead of trying to find a formal proof, we shall consider a few explicit examples and outline a simple constructive procedure how to integrate the structure equations. Finally, we give all solutions of the three- and four-particle cases which fit in our ansatz (\[Fansatz\]) with $A_1\oplus G_2$ and $F_4$ positive root vectors, respectively.
[*5.1. Three-body ${{\cal N}}{=}4$ superconformal Calogero model*]{}\
In this subsection we construct a particular solution to (\[w1\]) and (\[w2\]) or, equivalently, (\[lw1\])–(\[bc\]), for the case of three-body Calogero model governed by the potential \[bounda\] V\_B=\_[i<j]{}\^3 , leading to $C{=}3g$ because we sum over three permutations. It is easy to construct a corresponding $U$ satisfying the second and third equation in (\[w2\]). The general solution reads U=-g \_[i<j]{}\^3 |x\^i[-]{}x\^j| + L(y,z) , where $L$ is an arbitrary function of the ratios y= z= subject to (\_i L)(\_i L) = g \_[ij]{} , and so we may put $L\equiv0$, which we do for simplicity. Models based on the potential $\ U=-g\sum_{i<j}\ln|x^i{-}x^j|\ $ we term ‘Calogero’.
Next we turn to the WDVV coefficients $W_{ijk}$, of which there are ten for $n{=}3$. The six linear relations in the first equation of (\[lw2\]) allow us to express the WDVV coefficients in terms of four objects. In order to find their explicit form, we integrate (\[w3\]) to F=- ( x\^i x\^i |x\^1| - [(x\^1)]{}\^2 (y,z)) , where $\Delta(y,z)$ is an unknown function to be determined below, and we have distinguished the $x^1$ coordinate. Triple differentiation of $F$ yields \[wdvv-c\]
& x\^1 W\_[111]{}=-1 - - + \_1 +\_2 +3\_3 +3\_4 ,x\^1 W\_[123]{}=y z( \_3 + \_4) ,\
& x\^1 W\_[112]{}= -y \_1 -y \_3 -2y \_4 , x\^1 W\_[113]{}= -z \_2 -2 z \_3 - z \_4 ,\
& x\^1 W\_[122]{}=-1+ y\^2 \_1 + y\^2 \_4 , x\^1 W\_[133]{}=-1+ z\^2 \_2 + z\^2 \_3 ,\
& x\^1 W\_[222]{}=-y\^3 \_1 , x\^1 W\_[223]{}=-z y\^2 \_4 ,x\^1 W\_[233]{}=-y z\^2 \_3 , x\^1 W\_[333]{}=-z\^3 \_2 ,
with four subsidiary functions \[sigmas\]
& \_1 =12 y\^3 + 3 y\^2 +3y ,&& \_2 =12 z\^3 + 3 z\^2 +3z ,\
& \_3 =12 y z\^2 + y z , && \_4 =12 z y\^2 + y z .
In order to complete the analysis, we examine the first equation of (\[lw3\]), which couples the two scalar potentials. It yields six linear algebraic equations for the WDVV coefficients, but only three are independent. Abbreviating \[abc\]
& a=(y \_2 -\_1) U , && b=(z \_3 -\_1) U , && m=( x\^1 \_2 \_2 +\_1 ) U ,\
& p=( x\^1 \_3 \_3 +\_1 ) U , && nx\^1 \_2 \_3 U ,
one finds \[sigma123\] \_1=--\_4 , \_2=-++\_4 , \_3=--\_4 .
In order to fix the last missing coefficient $\Sigma_4$, one is to analyze the WDVV equations, i.e. the second relation in (\[lw3\]). Using the explicit representation (\[wdvv-c\]) it is straightforward to verify that among the six nontrivial WDVV equations at $n{=}3$ only one is independent, namely \[w-f\] W\^[22p]{}W\^[33p]{}W\^[23p]{}W\^[23p]{} . With the help of (\[sigma123\]) this reduces to a linear equation, which determines $\Sigma_4$ as $$\begin{aligned}
\label{sigma4}
\Sigma_4\=\frac{1}{18y}\Bigl(\frac{9}{y-z}+\frac{6}{y+z+yz}-
\frac{2}{2y-z-yz}+\frac{4}{2z-y-yz}+\frac{1}{2yz-y-z}\Bigr)\ ,\end{aligned}$$ and therewith $\Sigma_1$, $\Sigma_2$ and $\Sigma_3$.
The fact that for the three-body problem the WDVV equation (\[w-f\]) turns out to be linear can be understood in a different way. One can extract from the WDVV equation linear consequences which, along with other equations in (\[lw1\])–(\[bc\]), already contain all the information in (\[w-f\]). Indeed, let us differentiate the middle equation in (\[w2\]), (\_j \_i U) (\_i U) =\_j V\_B , and contract the first equation in (\[w2\]) with $\pa_i U$, \_j V\_BW\_[ijk]{}(\_i U) (\_k U) . Now contracting the WDVV equation with $(\pa_i U)(\pa_j U)$ and taking into account the first equation in (\[w2\]) one gets the linear equations \[suplin\] (\_i \_k U)( \_j \_k U)-W\_[ijk]{}\_k V\_B0 . It is straightforward to verify that only one component in (\[suplin\]) is independent and contains just the same information as (\[w-f\]).
Having fixed the WDVV coefficients algebraically, we are now in a position to find the potential $F$. Substituting (\[sigma123\]) and (\[sigma4\]) into (\[sigmas\]), one obtains for the single function $\Delta$ a system of partial differential equations of the Euler type. The standard change of variables y=\^t z=\^s turns it into a system of partial differential equations with constant coefficients. The latter is readily integrated by conventional means (see e.g. [@smirnov]) and yields the following free energy, \[fe\]
F(x\^1,x\^2,x\^3) = & -16 (x\^1[+]{}x\^2[+]{}x\^3)\^2 |x\^1[+]{}x\^2[+]{}x\^3| +\
& -14 \_[i<j]{} (x\^i[-]{}x\^j)\^2 |x\^i[-]{}x\^j| + 1[36]{} \_[i<jikj]{} (x\^i[+]{}x\^j[-]{}2x\^k)\^2 |x\^i[+]{}x\^j[-]{}2x\^k| ,
revealing the values h\_0 = 13 ,h\_1 = 12 ,h\_2 = - in the ansatz (\[Fansatz\]) for the three types of roots in (\[3vec\]). The relative particle motion is translation invariant. Note that each sum contains three terms, so that the result is totally symmetric in $\{x^1,x^2,x^3\}$. Six constants of integration enter a polynomial quadratic in $x$, which can be discarded since $F$ is defined up to such a polynomial. The quantum correction to the Calogero potential finally reads \[n3qu\] O\_2 = 38(x\^1[+]{}x\^1[+]{}x\^3)\^[-2]{} + 14\_[i<j]{} (x\^i[-]{}x\^j)\^[-2]{} +1[12]{} \_[i<jikj]{} (x\^i[+]{}x\^j[-]{}2x\^k)\^[-2]{} . For the reader’s convenience we display the corresponding WDVV coefficients in appendix B.
The ${{\cal N}}{=}4$ superconformal extension of the three-particle Calogero system produced a unique $G_2$-type integrable model with one free coupling and particular three-body interactions [@wolf]. Despite the latter, we call this a Calogero model because its bosonic classical potential $V_B$ is just the ($A$-type) Calogero one. This terminology differs from the one of Wyllard [@wyl], who allowed for three-body interactions in $U$ and $V_B$ from the outset. Our model agrees with his second one-parameter solution.
[*5.2. A four-body ${{\cal N}}{=}4$ superconformal model*]{}\
In this section we consider the second strategy outlined after (\[YW\]) and construct a four-body ${{\cal N}}{=}4$ superconformal model starting from a solution $F$ to the WDVV equations. For $n{=}4$ we make the following ansatz for the potential $F$, \[n4ansatz\]
F(x\^1,x\^2,x\^3,x\^4) = &-12 h\_0(x\^1[+]{}x\^2[+]{}x\^3[+]{}x\^4)\^2 |x\^1[+]{}x\^2[+]{}x\^3[+]{}x\^4| +\
&-12 h\_1\_[j>i<k<lkjl]{} (x\^i[+]{}x\^j[-]{}x\^k[-]{}x\^l)\^2 |x\^i[+]{}x\^j[-]{}x\^k[-]{}x\^l|
where the permutation sum has three terms. Note that the chosen positive root vectors n\^0=(1,1,1,1) ,n\^1=(1,1,-1,-1) ,(1,-1,1,-1) ,(1,-1,-1,1) give translation-invariant relative motion and form an orthogonal set, i.e. we look at a minimal model with a $A_1{\oplus}A_1{\oplus}A_1{\oplus}A_1$ root system. Substituting the ansatz into (\[w3\]), one learns that \[ab\] h\_0h\_1=14 , in agreement with the minimal property $\ h_\m^{-1}=\vec n^\m{\cdot}\vec n^\m$ from (\[minimal\]). For the case at hand one finds twenty WDVV equations, which happen to be satisfied identically for the above value of $h_0$ and $h_1$.
Let us take the corresponding ansatz for $U$, U = -g\_0|x\^1[+]{}x\^2[+]{}x\^3[+]{}x\^4| - g\_1 \_[j>i<k<lkjl]{} |x\^i[+]{}x\^j[-]{}x\^k[-]{}x\^l| , where $g_0$ and $g_1$ play the role of two independent coupling constants. It is straightforward to verify that the first equation in (\[w2\]) holds without imposing any restrictions on the form of the coupling constants. The last equation in (\[w2\]) determines the value of the central charge as Cg\_0+3g\_1 , while the second equation in (\[w2\]) determines the form of the bosonic potential, V\_B &=& 2 g\_0\^2(x\^1[+]{}x\^2[+]{}x\^3[+]{}x\^4)\^[-2]{} + 2 g\_1\^2 \_[j>i<k<lkjl]{} (x\^i[+]{}x\^j[-]{}x\^k[-]{}x\^l)\^[-2]{} ,\
O\_2 &=& 12(x\^1[+]{}x\^2[+]{}x\^3[+]{}x\^4)\^[-2]{} + 12 \_[j>i<k<lkjl]{} (x\^i[+]{}x\^j[-]{}x\^k[-]{}x\^l)\^[-2]{} , in tune with the minimal expressions (\[minpot\]). Notice that $g_0$ and $g_1$ are independent and may be set to zero individually, but not their quantum corrections. This model was also found in [@wyl].
[*5.3. All ${{\cal N}}{=}4$ three- and four-particle models based on $A_1\oplus G_2$ and $F_4$*]{}\
Let us finally make a more systematic search for ${{\cal N}}{=}4$ superconformal three- and four-particle models, where the sums in (\[Fansatz\]) run over particular positive root systems and all coefficients are left open. We adopt our second solution strategy and first solve the WDVV equations. The resulting admissible values for the coefficients $h_\m$ already define all $U=0$ models, since a vanishing $U$ solves the first equation in (\[YW\]) trivially. We shall encounter a free parameter $t$ in the allowed values $h_\m(t)$, for special values of which it is possible to turn on some $g_\m$ in $U$, i.e. find nontrivial solutions to the first equation in (\[YW\]). Motivated by the already known solutions, we allow any positive root from $A_1\oplus G_2$ in the $n{=}3$ case and from $F_4$ in the $n{=}4$ case. The result of a computer analysis is given below.
--------------------------- ------ ------ -------- -------------- -------- ------------- -------- ------------
pos. root $\vec n^\m$ $\#$ type $g_\m$ $h_\m$ $g_\m$ $h_\m$ $g_\m$ $h_\m$
\[-10pt\] $(1,\ph1,\ph1)$ 1 – $\ti$ $\sf13$ $\ti$ $\ph\sf13$ $\ti$ $\ph\sf13$
\[4pt\] $(1,-1,\ph0)$ 3 S $0$ $\sf13{-}3t$ $\ti$ $\ph\sf12$ $0$ $-\sf16$
\[4pt\] $(1,\ph1,-2)$ 3 L $0$ $t$ $0$ $-\sf1{18}$ $\ti$ $\ph\sf16$
\[4pt\]
--------------------------- ------ ------ -------- -------------- -------- ------------- -------- ------------
-------------------------------- ------ ------ -------- ----------------- -------- -------------- -------- -------------- -------- --------------
pos. root $\vec n^\m$ $\#$ type $g_\m$ $h_\m$ $g_\m$ $h_\m$ $g_\m$ $h_\m$ $g_\m$ $h_\m$
\[-10pt\] $(1,\ph1,\ph1,\ph1)$ 1 S $0$ $\sf1{12}{-}2t$ $0$ $\sf14{-}6t$ $0$ $0$ $0$ $0$
\[4pt\] $(1,\ph1,-1,-1)$ 3 S $0$ $\sf1{12}{-}2t$ $0$ $\sf14{-}6t$ $0$ $0$ $0$ $0$
\[4pt\] $(1,\ph1,\ph1,-1)$ 4 S $0$ $\sf1{12}{-}2t$ $0$ $0$ $0$ $\sf14{-}6t$ $0$ $0$
\[4pt\] $(2,\ph0,\ph0,\ph0)$ 4 S $0$ $\sf1{12}{-}2t$ $0$ $0$ $0$ $0$ $0$ $\sf14{-}6t$
\[4pt\] $(2,\ph2,\ph0,\ph0)$ 6 L $0$ $t$ $0$ $t$ $0$ $t$ $0$ $t$
\[4pt\] $(2,-2,\ph0,\ph0)$ 6 L $0$ $t$ $0$ $t$ $0$ $t$ $0$ $t$
\[4pt\]
-------------------------------- ------ ------ -------- ----------------- -------- -------------- -------- -------------- -------- --------------
-------------------------------- ------ ------ -------- ------------------ -------- ---------------- -------- ---------------- -------- ----------------
pos. root $\vec n^\m$ $\#$ type $g_\m$ $\ \ h_\m\ \ \,$ $g_\m$ $\ \ h_\m\ \,$ $g_\m$ $\ \ h_\m\ \,$ $g_\m$ $\ \ h_\m\ \,$
\[-10pt\] $(1,\ph1,\ph1,\ph1)$ 1 S $\ti$ $\sf14$ $\ti$ $\sf14$ $0$ $0$ $0$ $0$
\[4pt\] $(1,\ph1,-1,-1)$ 3 S $0$ $\sf14{-}4t$ $\ti$ $\sf14$ $0$ $0$ $0$ $0$
\[4pt\] $(1,\ph1,\ph1,-1)$ 4 S $0$ $0$ $0$ $0$ $\ti$ $\sf14$ $0$ $0$
\[4pt\] $(2,\ph0,\ph0,\ph0)$ 4 S $0$ $0$ $0$ $0$ $0$ $0$ $\ti$ $\sf14$
\[4pt\] $(2,\ph2,\ph0,\ph0)$ 6 L $0$ $0$ $0$ $0$ $0$ $0$ $0$ $0$
\[4pt\] $(2,-2,\ph0,\ph0)$ 6 L $0$ $t$ $0$ $0$ $0$ $0$ $0$ $0$
\[4pt\]
-------------------------------- ------ ------ -------- ------------------ -------- ---------------- -------- ---------------- -------- ----------------
In these tables, $\#$ is the number of positive roots obtained by permuting the entries of the displayed vector, ‘type’ refers to short (S) or long (L) roots, and $\ti$ indicates a free coupling $g_\m$. The free parameter $t$ reflects the freedom of shifting the weights between the short and the long roots.
For $n{=}3$, all models have translation-invariant relative motion, and all (except model 1 for $t{=}0$ and $t{=}\sfrac19$) exploit the full $G_2$ root system through $F$. Model 1 has $U_{\rm rel}=0$, but models 2 and 3 with a nontrivial $U_{\rm rel}$ arise at the special values of $t=-\sfrac1{18}$ and $t=\sfrac16$, respectively. Model 2 was constructed in subsection 5.1, and all three indeed appear in [@wyl].
For $n{=}4$, only models 5 and 6 feature translation-invariant relative motion, and only model 1 uses all roots of $F_4$. Models 1 through 4 have $U=0$, and model 5 shows $U_{\rm rel}=0$, leaving models 6, 7 and 8 with a nontrivial $U_{\rm rel}$. The latter three arise at the special point $t=0$ of the corresponding models listed above them. Models 1 to 4 all intersect at $t=\sfrac1{24}$, but model 2 also agrees with model 5 at $t=0$ (where it becomes model 6). Model 6 was presented in subsection 5.2 and also by Wyllard [@wyl], who insisted in relative translation invariance. Furthermore, it is interesting to characterize the eight models (plus some special $t$ values) by the subalgebra of $F_4$ each root system generates:
model number 1 $t{=}0,\sf1{24}$ 2, 3, 4 $t{=}\sf1{24}$ 5 $t{=}\sf1{16}$ 6, 7, 8
---------------------------- ------- ------------------ -------------- ---------------- ------------------ ------------------ --------------
\[-10pt\] $\#$ pos. roots 24 12 16 12 10 7 4
\[4pt\] dimension 52 28 36 28 24 18 12
\[4pt\] subalgebra $F_4$ $D_4$ $B_4$ $D_4$ $A_1{\oplus}B_3$ $A_1{\oplus}A_3$ $A_1^4$
\[4pt\]
For the reader’s convenience, we finally display the bosonic potentials for the models 5–8:
&V\_5= + \_ + \_ + O(\^2,\
&V\_6= + \_ + O(\^2,\^4) ,\
&V\_7=\_ + O(\^2,\^4) ,\
&V\_8=\_ + O(\^2,\^4) ,
with $O(\psi^2,\psi^4)$ being Weyl ordered and $\ \widetilde{g}_\m^2=g_\m^2+\sfrac14\hbar^2$. The central charge is $\ C=\sum_\m\#_\m g_\m$.
[**6. Conclusion**]{}\
In this paper the transformation of generic conformal multi-particle mechanics into a non-interacting system with nonlocal conformal symmetry [@glp] was extended to accommodate ${{\cal N}}{=}4$ supersymmetry. This step facilitates the construction of new $su(1,1|2)$ invariant many-body systems. More concretely, for a potential ansatz quartic in the fermionic coordinates, the closure of the superalgebra gave rise to a set of “structure equations” (\[w1\]) and (\[w2\]) for two scalar (pre)potentials $U$ and $F$ determining the potential $V$, including quantum corrections.
For the $n$-body functions $U$ and $F$ we made an ansatz based on the choice of a root system, with couplings $g$ and $h$, respectively, for each kind of root. This reduced the structure equations to (\[YW\]) with (\[YWform\]), i.e. quadratic algebraic WDVV-type equations for $\pa\pa\pa F$ and linear differential equations for $\pa U$ in the $F$ background. We fully analyzed these equations for the case of three and four particles and found various solutions, based on the root systems of $A_1\oplus G_2$ and $F_4$, respectively. The $G_2$-type models are identical to those of Wyllard [@wyl], whereas in the $F_4$ case we extend his result (our model 6) by several other solutions featuring translationally non-invariant relative particle motion. Results based on higher-dimensional root systems will be reported elsewhere.
For three particles, the generality of our ansatz was proved by explicit integration of the structure equations (\[w1\]) and (\[w2\]). With a growing number of particles, this becomes rather involved because these equations are very rigid. For a general solution (unbiased by the root-system ansatz) beyond $n{=}3$ a more advanced technique is needed.
Turning to possible further developments, it would be interesting to generalize the present analysis to models exhibiting a $D(2,1|\alpha)$ symmetry and to the ${{\cal N}}{=}8$ superconformal models constructed recently in [@bikl; @div]. One may also attempt to construct an off-shell superfield description. Finally, it is an open question whether the integrability of ${{\cal N}}{=}4$ superconformal multi-particle models is tied to the root systems of certain Lie superalgebras.
[**Acknowledgments**]{}\
We are indebted to E. Ivanov, S. Krivonos, A. Nersessian, F. Toppan and N. Wyllard for useful discussions. A.G. thanks the Institut für Theoretische Physik at Leibniz Universität Hannover for the hospitality extended to him at different stages of this work. We are grateful to the Joint Institute for Nuclear Research at Dubna for providing a stimulating atmosphere during the workshop SQS ’07. The research was supported by RF Presidential grants MD-8970.2006.2, NS-4489.2006.2, INTAS grant 03-51-6346, DFG grant 436 RUS 113/669/0-3, RFBR grant 06-02-16346, RFBR-DFG grant 06-02-04012 and the Dynasty Foundation.
[**Appendix A**]{}\
Given fermionic operators $\p_1,\dots,\p_n$, the Weyl-ordered product is defined as follows, $$\begin{aligned}
&&
\langle\p_1\p_2\rangle\=\sfrac{1}{2}\bigl(\p_1\p_2-\p_2\p_1\bigr)\ ,\\[2pt]
&&
\langle\p_1\p_2\p_3\rangle\=\sfrac{1}{3}\bigl(\p_1\langle\p_2\p_3\rangle +
\p_2\langle\p_3\p_1\rangle + \p_3\langle\p_1\p_2\rangle)\ ,\\[2pt]
&&
\langle\p_1\p_2\p_3\p_4\rangle\=\sfrac{1}{4}\bigl(
\p_1\langle\p_2\p_3\p_4\rangle - \p_2\langle\p_3\p_4\p_1\rangle +
\p_3\langle\p_4\p_1\p_2\rangle - \p_4\langle\p_1\p_2\p_3\rangle \bigr)\end{aligned}$$ etc., such that for any two neighboring operators one has $$\langle\dots\p_i\p_j\dots\rangle\=-\langle\dots\p_j\p_i\dots\rangle\ .$$ For deriving (\[restr\]) it is convenient to pass from Weyl-ordered operators to $qp$-ordered ones. In particular, for completely symmetric functions $M_{ij}$ and $L_{ijkl}$ one has $$\begin{aligned}
&&
M_{ij}\,\langle\p^i_\a \bar\p^{j\a}\rangle \=
M_{ij}\,\p^i_\a \bar\p^{j\a} - \hbar\,M_{kk}\ ,\\[2pt]
&&
W_{ijk}\,\langle\p^{i\b} \p^j_\b \bar\p^k_\a\rangle \=
W_{ijk}\,\p^{i\b} \p^j_\b \bar\p^k_\a - \hbar\,W_{kki}\,\p^i_\a\ ,\\[2pt]
&&
W_{ijk}\,\langle\p^{i\a} \bar\p^j_\b \bar\p^{k\b}\rangle \=
W_{ijk}\,\p^{i\a} \bar\p^j_\b \bar\p^{k\b} + \hbar\,W_{kki}\,\bar\p^{i\a}
\ ,\\[2pt]
&&
L_{ijkl}\,\langle\p^{i\a} \p^j_\a \bar\p^k_\b \bar\p^{l\b}\rangle \=
L_{ijkl}\,\p^{i\a} \p^j_\a \bar\p^k_\b \bar\p^{l\b} -
2\hbar\,L_{ijkk}\,\p^i_\a \bar\p^{j\a} + \hbar^2\,L_{kkpp}\ .\end{aligned}$$
[**Appendix B**]{}\
Here we present the explicit form of the WDVV coefficients for the three-body ${{\cal N}}{=}4$ superconformal Calogero model (\[fe\]): $$\begin{aligned}
18 W_{112}&=&
+\frac{9}{x_1{-}x_2}-\frac{4}{2x_1{-}x_2{-}x_3}
+\frac{2}{2x_2{-}x_1{-}x_3}-\frac{1}{2x_3{-}x_1{-}x_2}
-\frac{6}{x_1{+}x_2{+}x_3}\ ,\\[2pt]
18 W_{113}&=&
+\frac{9}{x_1{-}x_3}-\frac{4}{2x_1{-}x_2{-}x_3}
-\frac{1}{2x_2{-}x_1{-}x_3}+\frac{2}{2x_3{-}x_1{-}x_2}
-\frac{6}{x_1{+}x_2{+}x_3}\ ,\\[2pt]
18 W_{122}&=&
-\frac{9}{x_1{-}x_2}+\frac{2}{2x_1{-}x_2{-}x_3}
-\frac{4}{2x_2{-}x_1{-}x_3}-\frac{1}{2x_3{-}x_1{-}x_2}
-\frac{6}{x_1{+}x_2{+}x_3}\ ,\\[2pt]
18 W_{123}&=&
+\frac{2}{2x_1{-}x_2{-}x_3}+\frac{2}{2x_2{-}x_1{-}x_3}
+\frac{2}{2x_3{-}x_1{-}x_2}-\frac{6}{x_1{+}x_2{+}x_3}\ ,\\[2pt]
18 W_{133}&=&
-\frac{9}{x_1{-}x_3}+\frac{2}{2x_1{-}x_2{-}x_3}
-\frac{1}{2x_2{-}x_1{-}x_3}-\frac{4}{2x_3{-}x_1{-}x_2}
-\frac{6}{x_1{+}x_2{+}x_3}\ ,\\[2pt]
18 W_{223}&=&
+\frac{9}{x_2{-}x_3}-\frac{1}{2x_1{-}x_2{-}x_3}
-\frac{4}{2x_2{-}x_1{-}x_3}+\frac{2}{2x_3{-}x_1{-}x_2}
-\frac{6}{x_1{+}x_2{+}x_3}\ ,\\[2pt]
18 W_{233}&=&
-\frac{9}{x_2{-}x_3}-\frac{1}{2x_1{-}x_2{-}x_3}
+\frac{2}{2x_2{-}x_1{-}x_3}-\frac{4}{2x_3{-}x_1{-}x_2}
-\frac{6}{x_1{+}x_2{+}x_3}\ ,\\[2pt]
18 W_{111}&=&
-\frac{9}{x_1{-}x_2}-\frac{9}{x_1{-}x_3}
+\frac{8}{2x_1{-}x_2{-}x_3}-\frac{1}{2x_2{-}x_1{-}x_3}
-\frac{1}{2x_3{-}x_1{-}x_2}-\frac{6}{x_1{+}x_2{+}x_3} ,\\[2pt]
18 W_{222}&=&
+\frac{9}{x_1{-}x_2}-\frac{9}{x_2{-}x_3}
-\frac{1}{2x_1{-}x_2{-}x_3}+\frac{8}{2x_2{-}x_1{-}x_3}
-\frac{1}{2x_3{-}x_1{-}x_2}-\frac{6}{x_1{+}x_2{+}x_3}\ ,\\[2pt]
18 W_{333}&=&
+\frac{9}{x_1{-}x_3}+\frac{9}{x_2{-}x_3}
-\frac{1}{2x_1{-}x_2{-}x_3}-\frac{1}{2x_2{-}x_1{-}x_3}
+\frac{8}{2x_3{-}x_1{-}x_2}-\frac{6}{x_1{+}x_2{+}x_3}\ .\end{aligned}$$ The quantum correction $\hbar^2 O_2=\sfrac{\hbar^2}{8} W_{ijk}W_{ijk}$ to the two-body Calogero potential was given in (\[n3qu\]) and involves three-body interactions.
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[^1]: $A$ was chosen such that the Baker-Haussdorff series in (\[bak\]) terminates at the third step [@glp].
[^2]: $\s_1$, $\s_2$ and $\s_3$ denote the Pauli matrices.
[^3]: Spinor indices are raised and lowered with the invariant tensor $\e^{\a\b}$ and its inverse $\e_{\a\b}$, where $\e^{12}=1$.
[^4]: The classical consideration in [@bgl] implies that (\[ans\]) is indeed the most general quartic ansatz compatible with the ${{\cal N}}{=}4$ superconformal algebra.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The Orion–Eridanus superbubble, formed by the nearby Orion high mass star-forming region, contains multiple bright H$\alpha$ filaments on the Eridanus side of the superbubble. We examine the implications of the H$\alpha$ brightnesses and sizes of these filaments, the Eridanus filaments. We find that either the filaments must be highly elongated along the line of sight or they cannot be equilibrium structures illuminated solely by the Orion star-forming region. The Eridanus filaments may, instead, have formed when the Orion–Eridanus superbubble encountered and compressed a pre-existing, ionized gas cloud, such that the filaments are now out of equilibrium and slowly recombining.'
author:
- |
Andy Pon, $^{1,2,3}$ Doug Johnstone, $^{4,3,2}$ John Bally, $^{5}$ & Carl Heiles $^6$\
$^1$School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, UK\
$^2$Department of Physics and Astronomy, University of Victoria, PO Box 3055 STN CSC, Victoria, BC V8W 3P6, Canada\
$^3$NRC-Herzberg Institute of Astrophysics, 5071 West Saanich Road, Victoria, BC V9E 2E7, Canada\
$^4$Joint Astronomy Centre, 660 North A’ohoku Place, University Park, Hilo, HI 96720, USA\
$^5$Department of Astrophysical and Planetary Sciences, University of Colorado, UCB 389 CASA, Boulder, CO 80389-0389, USA\
$^6$Astronomy Department, University of California, 601 Campbell Hall 3411, Berkeley, CA 94720-3411, USA
bibliography:
- 'ponbib.bib'
title: 'The Origin of Ionized Filaments Within the Orion–Eridanus Superbubble'
---
ISM: clouds – ISM: structure – ISM: Individual (Orion–Eridanus Superbubble) – ISM: Individual (Eridanus Filaments) – ISM: bubbles
INTRODUCTION {#intro}
============
The closest high-mass star-forming region to the Sun that is currently forming massive stars is the Orion star-forming region, which is located at a distance of 400 pc from the Sun [@Hirota07; @Menten07; @Sandstrom07]. The Orion star-forming region is surrounded by a highly elongated superbubble, with dimensions of 20$^\circ \times 45^\circ$ as seen in H$\alpha$ emission [@Bally08], that is referred to as the Orion–Eridanus superbubble.
The Eridanus side of the superbubble contains a very prominent hook-shaped H$\alpha$ feature that was first discovered on H$\alpha$ images and Palomar Observatory Sky Survey plates by @Meaburn65 [@Meaburn67]. @Johnson78 break this hook into three separate arcs, all of which are labelled in Fig. \[fig:dicicco\]. Arc A is the eastern half of the hook, Arc B is the western half of the hook, and Arc C is the southern extension of the hook. For the remainder of this paper, we will refer to these three arcs collectively as the Eridanus filaments. In this paper, all references to north or south refer to increasing or decreasing declination and references to east or west refer to increasing or decreasing right ascension, unless otherwise specified.
![Orion–Eridanus superbubble as seen in H$\alpha$. Labels for the various major components of the bubble have been added to the image from @DiCicco09. Arcs A, B, and C are collectively referred to as the Eridanus filaments.[]{data-label="fig:dicicco"}](fig1.eps){width="3in"}
The Eridanus filaments are clearly brighter, in H$\alpha$, than the superbubble wall in their vicinity. They are also remarkably bright, with H$\alpha$ intensities of the order of 25 Rayleighs, given that they are located almost 200 pc from the Orion star-forming region [@Haffner03]. In this paper, we examine the implications of the H$\alpha$ brightnesses and sizes of these filaments, with a particular eye towards whether the filaments are consistent with being in ionization equilibrium with the Orion star-forming region.
The H$\alpha$ emission from a radiatively excited region is proportional to the amount of ionizing flux absorbed by that region, because the H$\alpha$ emission is generated from the recombination of bare protons and electrons. As such, information on the density and geometry of the emitting region, as well as the strength of the ionizing radiation field, can be obtained from H$\alpha$ intensities and distributions (e.g. @Reynolds79 [@Heiles99]). Strong dynamical events can also imprint their signatures into the H$\alpha$ emission from a gas cloud, such that ionization modelling can provide a window into the dynamical history of the H$\alpha$ emitting region.
In Section \[properties\], we review and derive the general properties of the Eridanus filaments. In Section \[equilibrium\], we examine the criteria required for the Eridanus filaments to be in ionization equilibrium with the Orion star-forming region, while in Section \[alternative models\], we examine alternative possibilities for the ionization state of the filaments. We briefly discuss the possible origins of the filaments in Section \[origin\] and we summarize our primary findings in Section \[conclusions\].
GENERAL PROPERTIES {#properties}
==================
The distances to the Eridanus filaments are not well known. Studies of interstellar absorption features in stellar spectra towards the Eridanus half of the Orion–Eridanus superbubble reveal a wall of gas moving towards the Sun at a distance of approximately 180 pc, which has often been interpreted as the near wall of the superbubble [@Guo95; @Burrows96; @Welsh05]. If the filaments are associated with the near side of the Orion–Eridanus superbubble, they would thus be only 180 pc distant. There is also some evidence, see Appendix \[Arc A\], that Arc A and the back side of the Orion–Eridanus superbubble are located at a distance greater than 500 pc [@Boumis01; @Welsh05; @Ryu06].
The Eridanus features have been detected in numerous H$\alpha$ surveys, including that by the Wisconsin H-Alpha Mapper (WHAM; @Haffner03). All values for the intensities of the filaments quoted below will be from the WHAM survey, if not otherwise stated.
Arc A is the brightest of the Eridanus filaments, with a peak intensity of 70 Rayleighs and a typical intensity closer to 25 Rayleighs. Arcs B and C have slightly lower typical intensities of 15 and 10 Rayleighs, respectively. To the west of Arc B, the H$\alpha$ intensity drops to 1 Rayleigh, whereas the intensity is closer to 5 Rayleigh throughout the interior of the Orion–Eridanus superbubble to the east of Arc A.
It is slightly odd that while the bubble wall shows clear signs of limb brightening, with the north and south edges of the bubble wall appearing more prominently in Fig. \[fig:dicicco\], the filaments do not show any significant limb brightening. That is, the north and south ends of the filaments do not appear to be significantly brighter than the middle of the filaments.
There are numerous H [i]{} features that correlate well with integrated H$\alpha$ emission features (e.g, @Hartmann97 [@Arnal00; @Bajaja05; @Kalberla05]). In particular, H [i]{} filaments are detected just to the west of Arcs A, B, and C. The H [i]{} filament closest to Arc B lies approximately 3$^\circ$ to the west of the arc (e.g., @Verschuur92), which corresponds to a physical separation between 10 and 25 pc for distances of the filaments from the Sun between 200 and 500 pc. Zeeman splitting measurements indicate that a partially ionized region, with a line-of-sight magnetic field of the order of 10 $\mu$Gauss, lies between Arcs B and C and the nearest H [i]{} filament [@Heiles89]. This spatial morphology is consistent with the H$\alpha$ coming from the ionized interior edge of a shell and the H [i]{} coming from the neutral exterior of the shell that is shielded from the ionizing photons of the Orion star-forming region by the inner regions of the shell.
Densities and Depths {#densities and depths}
--------------------
The observable emission measure of the H$\alpha$ transition, EM, is dependent upon both the number density of the emitting gas, $n$, and the line-of-sight depth of the emitting gas, $R$, via $$\mbox{EM} = \int n^2 \mbox{d}R.
\label{eqn:EM}$$ Based upon the observed intensity of the filaments, it is thus possible to either calculate the depth of the filaments, given an estimate of the density of the filaments, or to calculate the density of the filaments, given an estimate of their depths.
Since all of the Eridanus filaments are associated with both H [i]{} and H$\alpha$ emission, it appears that the filaments all contain ionization fronts. Therefore, the filaments should have a temperature of approximately 8000 K [@Basu99], for which an H$\alpha$ intensity of 1 Rayleigh corresponds to an emission measure of 2.25 pc cm$^{-6}$ (e.g., @Haffner03).
We assume that the gas pressure in the filaments is equal to the gas pressure within the Orion–Eridanus superbubble, although @BisnovatyiKogan95 suggest that the pressure within a superbubble’s wall may be twice that of the interior of the bubble. We adopt a pressure range of 1-$5 \times 10^{4}$ K cm$^{-3}$ for the interior of the Orion–Eridanus superbubble, and thus the Eridanus filaments, based upon previous observations [@Burrows93; @Guo95; @Burrows96]. For this pressure range and a temperature of 8000 K, the number density of the filaments is of the order of 1-6 cm$^{-3}$.
Alternatively, if the expansion velocity of the superbubble is taken to be 15 km s$^{-1}$ [@Reynolds79] and it is assumed that the expansion provides a ram pressure on the interior of the bubble to match the thermal pressure in the bubble, the filaments would have a density of the order of 1 cm$^{-3}$.
These densities are consistent with the densities derived by @Reynolds79 and @Heiles89. @Reynolds79 derive an electron density of $1.1 \,(D / 400\mbox{ pc})^{0.5}$ cm$^{-3}$ for the Eridanus filaments while @Heiles89 derive a density of 6 cm$^{-3}$ for the H [i]{} filament tracing Arcs B and C. If the temperature of the filaments is slightly lower than 8000 K, as suggested by @Heiles00, or if the filament pressure is twice the interior pressure, as predicted by @BisnovatyiKogan95, the above-calculated densities would increase by approximately a factor of 2. Conversely, if there is a pressure gradient within the bubble, such that the bubble pressure is lower near the filaments [@Burrows96], the filament densities could be lower by a factor of a few.
Since the typical observed intensity of the filaments is 15-25 Rayleighs [@Haffner03], the depth of the ionized material in these filaments must be of the order of 1-50 pc, given the density range of 1-6 cm$^{-3}$.
Alternatively, if we make the assumption that the depths of the filaments are approximately equal to their widths, we can derive a density range for the filaments. Both Arcs A and B have angular widths of approximately 1$^\circ$.5, which, if the arcs are 180 pc distant, would correspond to physical widths of 5 pc. If the arcs are 500 pc distant, their widths would be closer to 13 pc. For a depth of 5 pc, the number density required to produce the observed emission measure is of the order of 3 cm$^{-3}$, while for a depth of 13 pc, the required density is between 1.5 and 2 cm$^{-3}$. These densities are consistent with the densities estimated from the interior pressure of the bubble.
Under the above assumption of cylindrical geometry for the filaments, such that their depths are equal to their widths, the column densities of these filaments along our line of sight would be between 4 and 8 $\times 10^{19}$ cm$^{-2}$. This corresponds to visual extinctions (A$_V$) between 0.01 and 0.05. Since this is only the column density of ionized gas in the filaments, it is somewhat expected that these columns are slightly lower than the column density of a few times 10$^{20}$ derived towards the southern half of Arc A from other methods [@Heiles89; @Burrows93; @Guo95; @Snowden95Burrows]. Since Arc A is roughly 25$^\circ$ long and Arc B is 15$^\circ$ long, the total mass of ionized gas in the filaments is approximately $3 \times 10^{2}$ M$_\odot$ if the filaments are 180 pc distant, and $3 \times 10^{3}$ M$_\odot$ if the filaments are 500 pc distant.
The peak intensity of Arc A, 70 Rayleighs [@Haffner03], requires an unrealistically large depth of 70 pc for a density of 1.5 cm$^{-3}$, but only requires a depth of 17.5 pc for a density of 3 cm$^{-3}$. At this higher density, the peak intensity would correspond to a column density of $2 \times 10^{20}$ cm$^{-2}$, which is still significantly smaller than the column density of $1.9 \times 10^{21}$ cm$^{-2}$ derived towards the northern half of Arc A by @Heiles00. The higher column density estimated by @Heiles00 could be due to neutral hydrogen also being present in Arc A, as would be expected if the ionizing photons from the Orion star-forming region are fully trapped within Arc A. The much larger peak intensity of Barnard’s Loop, 250 Rayleighs, is likely due to a further increase in the density of the loop, rather than being due to a very large line-of-sight depth.
IONIZATION EQUILIBRIUM MODELS {#equilibrium}
=============================
While there is some debate in the literature regarding whether Arcs A and B are part of the Orion–Eridanus superbubble (e.g., @Boumis01, see also Appendix \[Arc A\]), for this section, we make the assumption that the filaments are in ionization equilibrium with the Orion star-forming region. We will later drop this assumption in Section \[alternative models\]
Because H$\alpha$ emission is powered by the absorption of ionizing photons, H$\alpha$ intensity should be proportional to the energy of ionizing radiation absorbed. This proportionality, however, can be broken if there is another energy source powering the ionization of the emitting gas or if the emitting regions are not in ionization equilibrium. For a superbubble, the H$\alpha$ surface luminosity of the superbubble’s wall should vary roughly with the flux of incident ionizing photons arriving from the ionizing source wherever the ionization front occurs within the bubble wall, since all of the incident ionizing photons are absorbed within the wall at these locations.. As such, the H$\alpha$ flux in these regions should vary inversely with the square of the distance from the ionizing source. For regions where the ionization front lies outside of the superbubble, the H$\alpha$ intensity should depend upon both the ionizing flux reaching the wall and the fraction of the ionizing flux that the wall captures. Thus, the H$\alpha$ brightness of the wall should show a discontinuity where the ionization front breaks out, with the H$\alpha$ intensity beyond the point where the ionization front breaks out being lower than that predicted from the inverse square distance dimming seen before the ionization front breaks out.
For the Eridanus filaments to be brighter in H$\alpha$ than the parts of the bubble wall closer to the Orion star-forming region, the superbubble walls near the filaments must not fully trap the ionizing photons incident upon them and the filaments must trap a greater fraction of the ionizing photons than the surrounding walls. The inability of the walls near the filaments to fully trap the ionizing photons incident upon them is consistent with the ionization front breaking out of the Orion–Eridanus superbubble’s wall at the western ends of Barnard’s Loop, where there is a significant decrease in H$\alpha$ intensity. It is, however, unclear where the extra mass in the filaments required to trap significantly more ionizing photons would have come from, as most models of superbubbles would predict relatively low gas masses at such large distances from the driving source [@Basu99]. The increase in brightness of the filaments relative to the adjacent bubble wall could, alternatively, just be due to an increase in the line-of-sight depth of the emitting gas in the filaments. In this case, the filaments would have a larger surface area over which to absorb energy from the Orion star-forming region, such that their emission could be greater than that of the surrounding bubble wall.
Proceeding with the assumption that the relative brightnesses of the Eridanus filaments and the surrounding bubble wall can be explained by variable fractions of ionizing photons absorbed or by changes in line-of-sight depth, the absolute intensity of the filaments must still be explained. If the filaments are in ionization equilibrium, there must not only be a large enough energy flux in ionizing photons incident upon the filaments to provide the power for the H$\alpha$ emission, but there must also be enough material within the filaments to absorb a large enough fraction of this incident ionizing photon energy. H [i]{} emission associated with the Eridanus filaments is clearly detected, indicating that there is enough material in the filaments to fully absorb all of the incident ionizing photons from the Orion star-forming region. Thus, we concentrate in the following paragraphs on whether the Orion star-forming region has a large enough ionizing photon luminosity to explain the observed characteristics of the Eridanus filaments.
We introduce a geometric scaling parameter, $\gamma_r$, to relate the line-of-sight depth from the Sun through the ionized portions of the filaments, $R$, to the depth to which the ionizing photons from the Orion star-forming region have ionized the filaments as measured from the Orion star-forming region, $L$, via $$R = \gamma_r L.
\label{eqn:gammar}$$ For cylindrical symmetry, $\gamma_r = 1$ at the midpoint of the filaments. For the Eridanus filaments, we expect that $\gamma_r$ should be of order unity.
The depth to which ionizing photons can ionize material, $d$, is given by $$d = \frac{\Phi_*}{4 \pi \, n^2 \, \alpha_b \, s^2},
\label{eqn:iondepth}$$ where $\Phi_*$ is the total ionizing luminosity of the source, $n$ is the density of the gas, $\alpha_b$ is the recombination coefficient for hydrogen, and $s$ is the distance to the ionizing source.
Under the assumption that the filaments have fully trapped the ionizing photons from the Orion star-forming region, such that $d = L$, combining and simplifying Equations \[eqn:EM\] through \[eqn:iondepth\] yields an expression for the ionizing luminosity of the Orion star-forming region in terms of the emission measure from the filaments as $$\gamma_r \, \Phi_* = 4 \pi \, \mbox{EM} \, \alpha_b \, s^2 .
\label{eqn:gammarphi}$$
Based upon the total H$\alpha$ emission from the Orion–Eridanus superbubble region, @Reynolds79 calculate that the Orion star-forming region must have an ionizing luminosity of the order of $4 \times 10^{49}$ s$^{-1}$. @Odell11 also independently determine the ionizing luminosity of the Orion star-forming region to be $1.9 \times 10^{49}$ s$^{-1}$, based upon the O-star models of @Heap06 and the spectral classifications of stars in the Orion star-forming region by @Goudis82, and $2.5 \times 10^{49}$ s$^{-1}$ based upon the H$\beta$ brightness of Barnard’s Loop.
For a distance of 220 pc between the filaments and the Orion star-forming region, an ionizing luminosity of $4 \times 10^{49}$ s$^{-1}$, a geometric $\gamma_r$ value of 1, and a hydrogen recombination coefficient of $2.6 \times 10^{-13}$ cm$^3$ s$^{-1}$, the expected emission measure from the Eridanus filaments is only 9 pc cm$^{-6}$, which would correspond to 4 Rayleighs. This is a factor of 4-6 lower than the observed 15-25 Rayleigh intensity of the filaments. To obtain an H$\alpha$ intensity between 15 and 25 Rayleighs, the product of the ionizing luminosity of Orion and $\gamma_r$ must be increased by this factor of 4-6. It seems unlikely that the ionizing luminosity of Orion has been underestimated by this large of a factor such that if this ionization equilibrium model is correct, it is likely that $\gamma_r$ must be large.
By treating the filaments as portions of sheets, the value of $\gamma_r$ can be related to the geometry of the filaments. That is, the filaments have to be oriented such that cos($\phi$) / cos($\theta$) = $\gamma_r$, where $\theta$ is the angle between the surface normal of a filament and the line of sight between the filament and the Sun, and $\phi$ is the angle between the surface normal of the filament and the line of sight from the Orion star-forming region to the filament. For a $\gamma_r$ value of 4, $\theta$ must be at least 75$^\circ$, and is more likely to be close to 80$^\circ$. Kompaneets models of superbubbles [@Kompaneets60; @Basu99] have been shown to well reproduce the shapes of superbubbles [@MacLow89; @Basu99; @Stil09], and the best-fitting Kompaneets models to the Orion–Eridanus superbubble [@Pon14b] predict that $\theta$ should be between 5$^\circ$ and 30$^\circ$ if the filaments are part of the superbubble wall.
While tangential sightlines to thin rings can produce significant limb brightening, this cannot explain the large $\gamma_r$ value required for the filaments since the filaments are roughly just as bright at their midpoints as at their northern and southern edges. At the midpoint of the filaments, there is no added depth due to a tangential sightline through the ring traced out by the filaments.
The H$\alpha$ brightness of the superbubble wall away from the Eridanus filaments is of the order of 5 Rayleighs, which is the intensity predicted for the currently accepted ionizing photon luminosity of the Orion star-forming region. Thus, a model in which the walls are in ionization equilibrium with the Orion star-forming region can explain the wall brightness. Alternatively, if the bubble wall only has a column density of $3 \times 10^{18}$ cm$^{-2}$, as suggested by @Burrows93, much of this H$\alpha$ emission may be coming from gas outside of the bubble that has been ionized by the photons passing through the bubble wall, rather than coming from the bubble wall itself. @Odell11 also show that Barnard’s Loop is consistent with being ionized by the bright stars of the Orion star-forming region.
The depth to which ionizing photons penetrate can also be compared to the observed widths of the Eridanus filaments. As before, we introduce a geometric scaling parameter, $\gamma_t$, which relates the width of the filaments on the plane of the sky, $W$, to the depth to which the ionizing photons from the Orion star-forming region have ionized the filaments as measured from the Orion star-forming region via: $$W = \gamma_t L.
\label{eqn:gammat}$$ If the Eridanus filaments were a cylindrical ring, then $\gamma_t =1$.
Under the assumption that the filaments fully trap the ionizing photons from the Orion star-forming region, Equation \[eqn:iondepth\] can be re-written as $$\gamma_t \, \Phi_* = 4 \pi \, n^2 \, \alpha_b \, s^2 \, W.
\label{eqn:gammatphi}$$ Unlike in Equation \[eqn:gammarphi\], the above expression still maintains a dependence upon the density of the filaments.
We choose to evaluate Equation \[eqn:gammatphi\] with the density range derived from the bubble’s interior pressure. We do not use the smaller density range for the filaments that we previously derived by assuming that the line-of-sight depth of the filaments is equal to their width because we do not want to presuppose the geometry of the filaments. If we were to use such a cylindrical geometry approximation, Equation \[eqn:gammatphi\] would just reduce to Equation \[eqn:gammarphi\]. For the density range of 1-6 cm$^{-3}$ and a range of filament widths from 5 to 13 pc, the value of $\gamma_t \, \Phi_*$ ranges from $2 \times 10^{49}$ s$^{-1}$, consistent with the estimated ionizing luminosity for the Orion star-forming region, all the way up to $2 \times 10^{51}$ s$^{-1}$, a factor of 100 larger than expected.
While the observed widths of the filaments are consistent with the depth to which the ionizing photons from the Orion star-forming region are capable of penetrating through the filaments, it is only for the extreme edge of allowed parameter space that this consistency is achieved. That is, for agreement, the filament densities must be as low as possible and the filaments must be as close to the Sun as possible, such that the filament widths are as small as possible. In such a case, however, the filaments would still have a depth, $R$, roughly a factor of 5 larger than their widths in order to produce the required emission measure. In this case, the filaments would still have to be fairly highly elongated, edge-on sheets.
From a combination of dust continuum, H [i]{}, and H$\alpha$ data, @Heiles99 estimate that the electron density in Arc A is approximately 1 cm$^{-3}$ and find that Arc A must have a depth between 32 and 120 pc. Since the width of Arc A would be 14 pc at a distance of 400 pc, @Heiles99 also conclude that Arc A is not a filament, but rather, an edge-on sheet. @Heiles00, however, apply this same technique to Barnard’s Loop and find that Barnard’s Loop must also have a depth of 160 pc. @Heiles00 note that by modifying their assumed grain size distribution, they can reduce this depth by a factor of 4 and increase the electron density by a factor of 2 such that they conclude that the observed fluxes from Barnard’s Loop should still be consistent with a filamentary model. Such a similar change to the assumed grain size distribution of @Heiles99 may allow Arc A to also be consistent with a filamentary model.
The presence of additional material within the superbubble, such as dust photoablated from the Orion molecular clouds, would absorb additional UV photons and cause the UV radiation field to decrease faster than predicted from geometric effects alone. Such material within the bubble would not emit significant H$\alpha$ flux, as the amount of H$\alpha$ intensity emitted per unit column density is much lower at 10$^6$ K, as in the interior of the bubble [@Burrows93], than it is at 10$^4$ K, as in the bubble wall. Material within the bubble would reduce the UV flux reaching the Eridanus filaments, making it even more difficult to explain the Eridanus filaments as being in ionization equilibrium with the Orion star-forming region. Similarly, @Heiles00 estimate the reddening towards one location in Arc A, based upon the relative observed intensities of H$\alpha$, \[N [ii]{}\], and 2325 MHz radio continuum emission, and suggest that the unabsorbed H$\alpha$ intensity towards the arcs may be as much as a factor of 3.7 higher than observed. A larger H$\alpha$ intensity for the filaments would require even larger values of $\gamma_r \, \Phi_*$.
If the filaments were porous such that they were composed of small, denser pockets of gas, then large values of $\gamma_t$ could be obtained. Since the emission measure and ionization depth have the same dependence on density, decreasing the volume filling factor of the filaments, however, would have no impact on $\gamma_r$.
ALTERNATIVE MODELS {#alternative models}
==================
In Section \[equilibrium\], it was shown that if the Eridanus filaments are in ionization equilibrium and ionized by the Orion star-forming region, they would either have to be relatively edge-on sheets, inclined at an angle of about 80$^\circ$ to the plane of the sky, or the Orion star-forming region would have to have an ionizing luminosity much larger than previously estimated. It is somewhat unlikely that the ionizing luminosity of the Orion star-forming region has been underestimated by a factor of 5. While it is possible that the filaments are sheets that are nearly edge on over their entire lengths, such a configuration is highly constrained. As such, we also look for alternative models for the filaments where we drop the requirement that the filaments are in ionization equilibrium with the Orion star-forming region.
A secondary ionization source for the filaments, such as shocks, a recent UV flash, or the hot plasma filling the bubble [@Boumis01], might account for the size and brightness of the filaments. While strong shocks are capable of ionizing hydrogen, @Reynolds79 note, based on @Cox72 and @Raymond76, that a 20-30 km s$^{-1}$ shock would not ionize enough hydrogen to produce the observed H$\alpha$ brightness of the Eridanus filaments. A higher \[O [iii]{}\]-to-H$\alpha$ ratio than seen by @Reynolds79 towards the superbubble would also be expected for ionization due to hard UV, soft X-rays, or cosmic rays [@Bergeron71]. Furthermore, there are no stars on the Eridanus side of the bubble that would have had enough energy to have formed a separate bubble which Arc A might be a part of [@Heiles76].
Alternatively, the discrepancy between the observed thickness and depth of the Eridanus filaments and the depth to which ionizing photons from Orion can ionize material can be resolved if the Eridanus filaments are not equilibrium objects. Since the recombination rate of hydrogen is dependent upon the square of the density of the gas, a greater column of gas can be ionized by the same ionizing flux if the gas is at a lower density. If the Eridanus filaments were formed via the compression of a pre-existing, ionized gas cloud, the column of ionized gas within the final compressed filament would be larger than the column that could be currently ionized, and thus, the H$\alpha$ intensity from such a compressed filament would be larger than if the filament were in ionization equilibrium. A low-density gas cloud initially situated beyond the superbubble could have been ionized by the ionizing photons from the Orion star-forming region because the superbubble wall towards the Eridanus side of the Orion–Eridanus superbubble does not appear to fully trap ionizing photons. Since strong shocks can increase the density of a gas by up to a factor of 4, the shocks induced in a pre-existing cloud when the superbubble wall encounters the cloud could increase the emission coming from the cloud by a factor of 4, which is roughly what is needed to explain the observed brightness of the filaments. We thus suggest, as an alternative to the filaments being highly inclined sheets, that the Eridanus filaments might have formed from the compression of a pre-existing, ionized gas cloud by a strong shock and that the filaments are currently out of ionization equilibrium.
For a hydrogen recombination coefficient of $2.6 \times 10^{-13}$ cm$^3$ s$^{-1}$, the recombination time is approximately 1.2 Myr $\left(n / \mbox{cm}^{-3}\right)^{-1}$ such that for a density of the order of a few cm$^{-3}$, the recombination time is just slightly less than 10$^6$ yr. While this time-scale is shorter than the few Myr age of the superbubble, estimated from the observed ages of Orion subgroups [@Brown94], it is not unrealistically small for the Eridanus filaments to still be in a non-equilibrium phase.
@Reynolds79 estimate that the line-of-sight expansion velocity of the Orion–Eridanus superbubble is 15 km s$^{-1}$, based upon observed line splitting. This line-of-sight velocity must be considerably less than the average total expansion velocity of the superbubble, as an average speed of 35 km s$^{-1}$ is required for the superbubble to have expanded to its full 300 pc length, assuming an upper limit of 8 Myr for the age of the bubble based upon the time since the formation of Orion OB1b [@Brown94]. It is quite possible that the superbubble could be a factor of 2 or more younger, thereby requiring an average expansion speed greater than 50 km s$^{-1}$. Given a temperature of the order of 10$^6$ K, the internal sound speed of the superbubble should also be of the order of 100 km s$^{-1}$.
If the filaments are shock-compressed features, then their original sizes must have been at least roughly a factor of 4 larger, given that strong shocks induce a density increase of a factor of 4. That is, the original material that was compressed to form the Eridanus filaments must have been between 20 and 50 pc in size, depending upon the adopted distances of the filaments. For the age of the filaments to be less than a Myr, as required given the hydrogen recombination time, the original shock must have been travelling at a speed greater than 20 km / s, or greater than 50 km / s if the filaments are farther than 500 pc distant, in order for the shock to have travelled across the entire cloud in less than the recombination time. While this is faster than the current line-of-sight expansion velocity of the superbubble, it is well within the range of plausible expansion speeds for the superbubble.
Based upon the 25$^\circ$ length of Arc A, the superbubble radius at the location of the Eridanus filaments is between 40 and 110 pc, depending upon the distance of Arc A. If the cloud shocked by the expanding superbubble has been carried along the expanding wall of the superbubble at a speed of the order of 50 km / s, then the initial collision would have had to have occurred roughly 1-2 Myr previously to explain the current radial size of the ring formed by the filaments. This is slightly on the long side if the filaments are to still be recombining from the initial collision. However, this time-scale assumes expansion from the central axis of the superbubble and this time-scale would be reduced if the initial shocked cloud had a significant spatial extent or was offset from the central axis of the superbubble, as hinted at by the hook shape of the filaments.
ORIGIN
======
The origin of the significant column density enhancement in the Eridanus filaments is an open question. The Eridanus filaments may have formed when the superbubble impacted a pre-existing gas cloud and swept up the gas into a dense ring around the outside of the bubble, as suggested above. The original gas cloud that was compressed could have been related to previous star formation events that occurred in one of the older Orion subgroups and the formation mechanism of the Eridanus filaments may bear some similarities to the formation of bipolar rings in planetary nebula and supernova remnants, as such rings are believed to form when a fast outflow impacts a previously ejected shell of material [@Soker02]. The apparent thinness of the filaments on the plane of the sky could be partially due to the onset of a thermal instability (e.g., @Field65), in which case the filaments would be expected to be slightly cooler than the surrounding wall.
An alternative possibility is that the breaking out of the ionization front from the superbubble wall at an earlier time may have resulted in a pressure discontinuity that funnelled material into a ring at the height of the filaments, although it is unlikely that such a mechanism could operate on the appropriate time-scale and it is unclear if there was enough material in the top part of the bubble to account for the significant column density of the arcs. Similarly, hydrodynamic instabilities, such as those seen in the simulations of @MacLow89, can also create structure in a superbubble far from the driving source, but it is not clear how much material can be incorporated in these instabilities.
The Orion nebula cluster alone has photoevaporated a few times 10$^2$ M$_\odot$ of material over the last megayear [@Odell01]. Over the last 10 Myr, the Orion star-forming region should have photoablated between 10$^3$ and a few times 10$^4$ M$_\odot$ of material into the bubble interior and this additional mass injected into the bubble may be the material that has formed the filaments. It is, however, unclear how such photoablated material would become so well focused into filamentary structures.
The flux of ionizing photons coming from the Orion star-forming region has also likely been quite temporally variable, especially around the occurrences of supernovae. The Eridanus filaments may have formed at a much earlier time and have only recently been ionized by a recent burst of ionizing photons from the Orion star-forming region. Previous supernovae would also have ejected shells of material into the superbubble cavity and the filaments may just be the remnants of previous supernova explosions. More modelling, however, is required for all of the above-suggested possibilities.
CONCLUSIONS
===========
The Orion star-forming region is the closest high-mass star-forming region currently forming stars and it has blown a large 20$^{\circ}$ by 45$^{\circ}$ superbubble into the ISM. The superbubble contains very prominent filaments on the Eridanus side, referred to as the Eridanus filaments. We find that the Eridanus filaments have gas densities between 1 and 6 cm$^{-3}$ and contain between 300 and 3000 M$_\odot$ of ionized gas. Based upon the widths and H$\alpha$ intensities of these filaments, we find that if these filaments are in ionization equilibrium with the Orion star-forming region, then either the filaments must be edge-on sheets, with depths of a factor of roughly 5 larger than their widths, or that the ionizing luminosity of the Orion star-forming region must be a factor of about 5 larger than previously determined. We suggest, as an alternative, that the Eridanus filaments are non-equilibrium structures that are currently in the process of recombining after being formed from the compression of a pre-existing gas cloud due to the expansion of the superbubble.
ACKNOWLEDGEMENTS {#acknowledgements .unnumbered}
================
We would like to thank Dr Basu, Dr Vaidya, and Dr Caselli for useful discussions and comments. AP was partially supported by the Natural Sciences and Engineering Research Council of Canada graduate scholarship programme. DJ acknowledges support from a Natural Sciences and Engineering Research Council (NSERC) Discovery Grant. This research has made use of the Smithsonian Astrophysical Observatory (SAO) / National Aeronautics and Space Administration’s (NASA’s) Astrophysics Data System (ADS). The WHAM is funded by the National Science Foundation. We would also like to thank our anonymous referee for many useful changes to this paper.
NATURE OF ARC A {#Arc A}
===============
@Reynolds79 were the first to suggest that Arc A might be a part of a large superbubble created by the Orion star-forming region. This interpretation of Arc A being a filament associated with the superbubble, however, has recently come under question, and the precise nature of Arc A is not yet agreed upon (e.g., @Boumis01).
Part of the difficulty in deciphering the nature of Arc A is that there are gas clouds unrelated to the superbubble that also lie along the line of sight towards Arc A. In H [i]{} and continuum emission, there is significant emission extending from the Galactic plane to a Galactic latitude of approximately -40$^\circ$ along the northern (higher declination) edge of the superbubble (e.g., @Neugebauer84 [@Hartmann97; @Arnal00; @Bajaja05; @Kalberla05; @MivilleDeschenes05]). In the north-west corner of the loop formed by Arcs A and B, where there is little H$\alpha$ emission, there also lies the end of an 80$^\circ$ long H [i]{} filament that has a characteristic velocity relative to the local standard of rest (LSR) of -8 km s$^{-1}$, the western half of which is dubbed the Pisces Ridge [@Fejes73]. The 100 M$_\odot$, diffuse molecular cloud MBM 18, also known as L1569, is also along the line of sight to the middle of Arc A, near where a linear H$\alpha$ feature, lying along a Galactic longitude of 190$^\circ$, crosses Arc A (e.g., @Magnani85 [@Magnani86]). MBM 18 also, unfortunately, has a CO, centroid, LSR velocity between 8 and 10 km s$^{-1}$ [@Magnani85; @Penprase93; @Gir94; @Magnani00], which is similar to the 11 km s$^{-1}$ central velocity of the H$\alpha$ emission coming from Arc A [@Reynolds79].
There are also some weaker CO detections along the northern half of Arc A, at velocities between 7 and 12 km s$^{-1}$, but there are no CO detections along the southern half of Arc A or along Arc B [@Magnani00; @Aoyama02]. Due to the presence of MBM 18, the Pisces Ridge, and other features present near the northern half of Arc A, we suggest that focusing observational efforts on the southern half of Arc A, where there is little additional CO or 100 micron emission, may provide the best opportunity to constrain the properties of Arc A without foreground or background contamination.
In the following, we review the pertinent arguments for and against the association of Arc A with the Orion–Eridanus superbubble.
Proper motion {#proper motion}
-------------
### Against association with the superbubble {#proper motion against}
@Boumis01 derive an upper limit to the proper motion of Arc A of 0.13 arcsec yr$^{-1}$ by tracking the location of a sharp edge of Arc A near the middle of the Arc over a 45 yr time span. If Arc A is 200 pc distant, this proper motion would constrain the tangential motion of the arc to be less than 6 km s$^{-1}$. At distances of 400 and 600 pc, the arc would require tangential velocities less than 11 and 17 km s$^{-1}$, respectively. @Boumis01 assume that the tangential velocity of the bubble is roughly equal to the line-of-sight expansion velocity derived by @Reynolds79, 15 km s$^{-1}$, and derive a minimum distance to Arc A of 530 pc. Based upon this large distance, they suggest that Arc A might be unassociated with Arc B and the superbubble.
### For association with the superbubble {#proper motion for}
The conclusions of @Boumis01 are predicated upon the assumption that the radial velocity of Arc A is the same as the velocity of Arc A in the plane of the sky. It is not entirely clear that the tangential velocity of Arc A should be equivalent to the radial expansion velocity, as this assumption essentially prescribes the angle between the space velocity of Arc A and the line of sight. As pointed out in Section \[alternative models\], the average expansion velocity of the superbubble must have been much larger than the currently accepted 15 km s$^{-1}$ line-of-sight expansion velocity. Kompaneets model fits to the superbubble [@Pon14b] seem to indicate that Arc B may be the better candidate for detecting a proper motion, as the expansion motion of Arc B should be more perpendicular to the plane of the sky than for Arc A. No such proper motion study has yet to be conducted for Arc B.
Radial velocities {#velocities}
-----------------
### Against association {#velocities against}
In the @Reynolds79 H$\alpha$ data, two line components are detected with velocities, relative to the LSR, of 3 and -25 km s$^{-1}$. These two components have been interpreted as being the velocities of the near and far side of the superbubble. Similarly, the H [i]{} lines in the region of the superbubble show complex line structure and are often double peaked. @Menon57 identify one component at a V$_{\mbox{LSR}}$ of 12 km s$^{-1}$ and another at -5 km s$^{-1}$, with these two components again presumably belonging to the two different sides of the superbubble.
As for the filaments, @Reynolds79 suggest a characteristic velocity of 11 km s$^{-1}$ for Arc A and 3 km s$^{-1}$ for Arc B. In the Leiden/Argentine/Bonn Galactic H [i]{} Survey data set [@Hartmann97; @Arnal00; @Bajaja05; @Kalberla05], filamentary H [i]{} emission near Arc A can be seen at velocities from 20 km s$^{-1}$ down to approximately -2 km s$^{-1}$. At velocities between 0 and 10 km s$^{-1}$, the H [i]{} is slightly offset to the west of Arc A, while between 10 and 20 km s$^{-1}$, the H [i]{} is essentially coincident with Arc A. This is in contrast to @Reynolds79 who, using the 18 $<$ V$_{\mbox{LSR}} <$ 21 km s$^{-1}$ data of @Heiles76, place the H [i]{} filament slightly to the east of Arc A. The long H [i]{} filament that runs to the west of Arcs B and C has a velocity, relative to the LSR, of 9 km s$^{-1}$ and there also exists a second, fainter H [i]{} filament further to the west, which is centred closer to 5 km s$^{-1}$ [@Verschuur73].
While in both H [i]{} and H$\alpha$ studies the gas near Arc B has a velocity similar to the larger of the two velocity components detected for the bubble wall, thereby suggesting that Arc B is associated with the far wall of the superbubble, the H [i]{} and H$\alpha$ emitting gas regions near Arc A have velocities larger than either of the two components detected in H [i]{} and H$\alpha$ towards the Eridanus side of the bubble. It is unclear what is the cause of this velocity difference. For instance, it may be due to some residual momentum left over from the formation of the arc or to foreground or background material along the line of sight confusing the line centroid from Arc A, but one possible explanation is simply that Arc A is unassociated with the superbubble.
Taking 20 km s$^{-1}$ as the characteristic velocity for H [i]{} gas associated with Arc A, as done by @Welsh05, implies a distance of 2.2 kpc for the arc, based upon a standard Galactic rotation curve with a circular velocity of 220 km s$^{-1}$. The 10 km s$^{-1}$ velocity associated with the H$\alpha$ emission from Arc A corresponds to a distance of 1.1 kpc. Both of these distances are much larger than the distance to the Orion star-forming region.
### For association {#velocities for}
Fig. \[fig:whamcent\] shows the velocity centroids across the Eridanus filaments, as derived from the WHAM H$\alpha$ data [@Haffner03]. The velocity centroids towards Arc A are at slightly more positive velocities than Arc B, although there is a fairly prominent velocity gradient across the two arcs, with the gas along the outside edge of the arcs having centroid velocities more than 10 km s$^{-1}$ larger than the inner edge of the arcs. Arc B is more clearly defined in Fig. \[fig:whamcent\], possibly due to the presence of foreground material in the direction of Arc A. The presence of such strong gradients makes it very difficult to accurately assign a particular velocity to the gas in the filaments in order to compare the filament velocities to the bubble wall velocities.
If the filaments were formed when a pre-existing cloud was shocked and compressed by the expanding superbubble, then the velocity gradients in the filaments could be the kinematic signature left behind by such a collision. It is, however, not clear if such a gradient would be indicative of continuing compression of the filaments or a gradual re-expansion of the filaments. These velocity gradients may also be caused by other physical effects, such as bulk rotation of the filaments.
![Centroid velocities, with respect to the LSR, of H$\alpha$, from the WHAM survey [@Haffner03], are shown in the colour scale, while the contours show the integrated intensity of the H$\alpha$ line. The contours are logarithmically spaced with each contour representing a factor of 2 increase in integrated intensity. The lowest contour corresponds to an integrated intensity of 10 Rayleighs. Only the region around the Eridanus filaments is shown.[]{data-label="fig:whamcent"}](figa1.eps){height="3in"}
As noted by @Green93, because Orion is near $l$ = 180$^\circ$, the motion due to Galactic rotation of any gas cloud towards Orion will be mainly in the plane of the sky and thus, radial velocities do not provide a good measure of distance.
Arcs A and B appear to join smoothly in velocity space. There is no obvious discontinuity suggesting that the two arcs are located at significantly different distances.
Absorption studies {#absorption}
------------------
### Against association {#absorption against}
Towards the Eridanus side of the superbubble, absorption features are seen between -8 and -20 km s$^{-1}$ in stars more distant than 180 pc, which are usually interpreted as being due to the near, approaching side of the superbubble [@Guo95; @Burrows96; @Welsh05]. From colour excesses of stars with known distances, @Lallement14 also find two elongated clouds at a distance of 170 pc from the Sun in the direction of Barnard’s Loop.
While the near side of the superbubble is readily detected via absorption features in stellar spectra, there are no unambiguous detections of positive velocity gas with velocities at or above 20 km s$^{-1}$ [@Guo95; @Burrows96; @Welsh05]. Since Arc A is associated with H [i]{} gas at a velocity of approximately 20 km s$^{-1}$, @Welsh05 argue that Arc A must be more than 500 pc distant. Thus, Arc A would appear to be behind the superbubble. While the back wall of the superbubble has not been conclusively detected in absorption studies [@Guo95; @Burrows96; @Welsh05], the back side of the bubble has been estimated to be within 540 pc towards ($l$, $b$) = (215$^{\circ}$, -26$^{\circ}$) and within 465 pc towards ($l$, $b$) = (209$^{\circ}$, -37$^{\circ}$) based upon Lyman $\alpha$ to 21-cm ratios [@Savage72; @Heiles76; @Long77] .
### For association {#absorption for}
Towards the superbubble, absorption features are detected around a V$_{\mbox{LSR}}$ of 7 km s$^{-1}$ in the spectra of stars at distances greater than 140 pc and these absorption features are believed to be due to the expanding shell of the local bubble (e.g., @Frisch90 [@Burrows96; @Lallement03; @Welsh05]). This emission is unlikely to be associated with the superbubble, as the gas is moving towards the Orion star-forming region, rather than away from it. Conversely, it is unlikely that the H$\alpha$ emission detected at positive velocities is associated with the local bubble wall, as the local bubble does not contain an obvious ionizing radiation source.
The near side of the superbubble is detected at velocities between -8 and -20 km s$^{-1}$ in absorption studies (e.g., @Welsh05). This velocity range is more positive than the lower of the two H$\alpha$ velocity components, -25 km s$^{-1}$, but more negative than the lower H [i]{} velocity component, -5 km s$^{-1}$. If the higher of the two detected velocity components, in both H [i]{} and H$\alpha$, corresponds to the far side of the bubble and if the far bubble wall were to appear in absorption studies at a velocity intermediate to that in H [i]{} and H$\alpha$ studies, as the near side does, then it would be expected that the far bubble wall would appear at velocities between 3 and 12 km s$^{-1}$. Since the local bubble wall has a velocity of 7 km s$^{-1}$, it is likely that any absorption features due to the far wall of the superbubble would be readily confused with absorption due to the local bubble. Therefore, the lack of a detection of the far bubble wall and Arc A in absorption studies can be simply explained by confusion due to the local bubble wall. Furthermore, the H$\alpha$ centroid of Arc A, 11 km s$^{-1}$, is much lower than the 20 km s$^{-1}$ value adopted by @Welsh05 in their absorption line survey and absorption lines at 11 km s$^{-1}$ could be more readily confused with absorption due to the local bubble or even MBM 18, which has a velocity of 8-10 km s$^{-1}$. @Heiles79 also note that in their catalogue of H [i]{} shells and supershells, many have only one side detected.
Absorption lines from the back wall of the superbubble would also not be expected to be detected by Na [i]{} line surveys, such as that by @Welsh05, if the back wall were fully ionized, because neutral sodium traces gas with temperatures less than 1000K [@Lallement03].
H$\alpha$ {#halpha intensity}
---------
### Against association {#halpha intensity against}
There is an abrupt drop in H$\alpha$ intensity to the west of Arc B, along with a corresponding drop in H$_{2}$ emission. @Ryu06 interpret this intensity change as being due to Arc B absorbing most of the UV photons from the Orion star-forming region, although this may also be partly due to a significant decrease in the column density of gas along the line of sight to the west of Arc B. In contrast, the H$\alpha$ and H$_{2}$ intensities immediately to the west of Arc A are not significantly less than the intensities to the east of Arc A. From this, @Ryu06 conclude that Arc A does not lie along the line of sight between the Orion star-forming region and Arc B, as Arc A does not appear to be absorbing all of the UV photons travelling westward from the Orion star-forming region. They note that this is consistent with Arc A being unassociated with the superbubble.
As suggested above, the H$\alpha$ emission from the Eridanus filaments is not easily compatible with the filaments being in ionization equilibrium with the Orion star-forming region. If the emission is due to the gas recombining, instead of being from reprocessed ionizing radiation, then the filaments do not need to be associated with the Orion superbubble.
### For association {#halpha intensity for}
If Arcs A and B lie along different sides of the bubble, then the requirement that Arcs A and B lie along different sight lines from Orion can also easily be satisfied for models where Arc A is associated with the superbubble.
If the superbubble is split into four quadrants, centred on the Orion star-forming region, then the H$\alpha$ flux is roughly the same in each quadrant, as would be expected if the emission is entirely due to reprocessed ionizing radiation from the Orion star-forming region [@Reynolds79]. Not including Arc A’s flux would make the Eridanus side of the superbubble underluminous, although flux equality between all of the quadrants would only be expected if the ionizing photons from the Orion star-forming region never break out of the superbubble wall. While the current emission from Arc A may not be fully powered by ionizing photons from Orion, as would be the case if it were out of ionization equilibrium, some source must have initially ionized the gas and it is unclear what this source would be if not the Orion star-forming region.
Arcs A and B appear to join smoothly in the plane of the sky along their southern ends in H$\alpha$ emission. The H [i]{} filaments tracing these two arcs also appear to connect at their northern ends, although there are numerous potential background and foreground contaminating sources along the north ends of these filaments. If the two arcs are not associated with one another, a more obvious discontinuity between the arcs might be expected. Unlike Arc A, there has been little question in the literature about whether Arc B is associated with the Orion–Eridanus superbubble such that if Arc A is associated with Arc B, it is likely also associated with the superbubble.
X-rays {#xrays}
------
### Against association {#xrays against}
The most prominent difference between the 0.25 and 0.75 keV maps of the Orion–Eridanus superbubble is that the 0.25 keV emission extends south of the intersection of Arcs A and B while the 0.75 keV emission does not [@Davidsen72; @Williamson74; @Naranan76; @Long77; @Fried80; @Nousek82; @Singh82; @McCammon83; @Marshall84; @Garmire92; @Burrows93; @Guo95; @Snowden95Burrows; @Snowden97].
While there seems to be some consensus that the X-ray emission coming from regions with both 0.25 and 0.75 keV emission, dubbed the Eridanus X-ray Enhancement 1 (EXE1) by @Burrows93, is generated by hot plasma within the bubble, there is some controversy over the nature of the 0.25 keV southern extension, which was named the Eridanus X-ray Enhancement 2 (EXE2) by @Burrows93. @Brown95 suggest that EXE1 and EXE2 are both related to the superbubble and are not physically separate. The existence of EXE2 provides some evidence that the superbubble might extend south of Arc A, calling into question why an arc associated with the superbubble would stop half way along the bubble.
While Arc A appears to be associated with a decrease in soft-X-ray emission, the centroid velocities of H$\alpha$ and H [i]{} emission from Arc A are not compatible with the velocities of the near side of the bubble. Arc A also has a larger characteristic velocity than Arc B, suggesting that Arc A should be farther towards the back side of the superbubble than Arc B, which has velocities consistent with the back side of the bubble. Thus, these velocities seem to suggest that Arc A cannot be on the front side of the superbubble to absorb C-ray’s from the interior of the bubble and some other foreground clouds must be causing this absorption.
### For association {#xrays for}
@Burrows93 consider EXE1 and EXE2 to be physically separate and suggest that EXE2 could be a bubble blown by a single star, although they do not rule out the possibility that EXE2 is a small-scale blowout of the superbubble. @Snowden95Burrows note that the superbubble is on the edge of a larger scale 0.25 keV enhancement, see also @Snowden95Freyberg, and they interpret the EXE2 emission to be coming from a diffuse hot halo background. @Heiles99 suggest that the extended 0.25 keV emission is due to hot gas that has leaked out of the superbubble and subsequently cooled slightly. @Heiles99 further point out that no receding portion of the superbubble is detected in either H$\alpha$ or H [i]{} for locations between Arc A and B and suggest that this is due to there being a hole in the back of the superbubble through which this hot gas can escape. @Burrows93 find that EXE1 is slightly warmer than EXE2, with EXE1 having a temperature of $2.2 \times 10^6$ K and EXE2 having a temperature of $1.5 \times 10^6$ K. If the superbubble wall west of Barnard’s Loop is not fully trapping the ionizing radiation from the Orion star-forming region, EXE2 may just be gas heated by radiation that is penetrating the superbubble. All of these options would be consistent with the edge of the superbubble wall running through the intersection point of Arc A and B, as would be expected if Arc A were associated with the superbubble.
Arc A appears to be strongly associated with a decrease in soft-X-ray emission, even along the southern region of the arc, which would be expected if Arc A were between the Sun and the interior of the superbubble and absorbing the X-rays coming from the hot interior of the bubble (e.g., @Guo95 [@Snowden95Burrows; @Snowden97]). A model in which Arc A is on the near side of the bubble, at a distance of approximately 180 pc, would be fully consistent with this X-ray absorption.
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'Abbas Sadat${}^{*,1}$, Mengye Ren${}^{*,1,2}$, Andrei Pokrovsky${}^{3}$, Yen-Chen Lin${}^{4}$, Ersin Yumer${}^{1}$, Raquel Urtasun${}^{1,2}$ [^1] [^2] [^3] [^4] [^5]'
bibliography:
- 'refs.bib'
title: |
**Jointly Learnable Behavior and Trajectory Planning\
for Self-Driving Vehicles**
---
[^1]: \* Equal contribution
[^2]: $^{1}$Abbas Sadat, Mengye Ren, Ersin Yumer and Raquel Urtasun are with Uber Advanced Technologies Group, 661 University Avenue, Suite 720, Toronto, Ontario, Canada, M5G 1M1. Email: [{asadat,mren3,yumer,urtasun}@uber.com]{}.
[^3]: $^{2}$Mengye Ren and Raquel Urtasun are also with University of Toronto.
[^4]: $^{3}$Andrei Pokrovsky is with GraphCore. Work done at Uber.
[^5]: $^{4}$Yen-Chen Lin is with Massachusetts Institute of Technology. Work done at Uber.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
Understanding the relationship between mock modular forms and quantum modular forms is a problem of current interest. Both mock and quantum modular forms exhibit modular-like transformation properties under suitable subgroups of $\textnormal{SL}_2(\mathbb Z)$, up to nontrivial error terms; however, their domains (the upper half-plane $\mathbb H$, and the rationals $\mathbb Q$, respectively) are notably different. Quantum modular forms, originally defined by Zagier in 2010, have also been shown to be related to the diverse areas of colored Jones polynomials, meromorphic Jacobi forms, partial theta functions, vertex algebras, and more.
In this paper we study the $(n+1)$-variable combinatorial rank generating function $R_n(x_1,x_2,\dots,x_n;q)$ for $n$-marked Durfee symbols. These are $n+1$ dimensional multisums for $n>1$, and specialize to the ordinary two-variable partition rank generating function when $n=1$. The mock modular properties of $R_n$ when viewed as a function of $\tau\in\mathbb H$, with $q=e^{2\pi i \tau}$, for various $n$ and fixed parameters $x_1, x_2, \cdots, x_n$, have been studied in a series of papers. Namely, by Bringmann and Ono when $n=1$ and $x_1$ a root of unity; by Bringmann when $n=2$ and $x_1=x_2=1$; by Bringmann, Garvan, and Mahlburg for $n\geq 2$ and $x_1=x_2=\dots=x_n=1$; and by the first and third authors for $n\geq 2$ and the $x_j$ suitable roots of unity ($1\leq j \leq n$).
The quantum modular properties of $R_1$ readily follow from existing results. Here, we focus our attention on the case $n\geq 2$, and prove for any $n\geq 2$ that the combinatorial generating function $R_n$ is a quantum modular form when viewed as a function of $x \in \mathbb Q$, where $q=e^{2\pi i x}$, and the $x_j$ are suitable distinct roots of unity.
author:
- 'Amanda Folsom, Min-Joo Jang, Sam Kimport, and Holly Swisher'
title: Quantum modular forms and singular combinatorial series with distinct roots of unity
---
[^1]
Introduction and Statement of results {#intro}
=====================================
Background
----------
A *partition* of a positive integer $n$ is any non-increasing sum of positive integers that adds to $n$. Integer partitions and modular forms are beautifully and intricately linked, due to the fact that the generating function for the partition function $p(n):= \# \{\mbox{partitions of } n \}$, is related to Dedekind’s eta function $\eta(\tau)$, a weight $\frac12$ modular form defined by $$\begin{aligned}
\label{def_eta}
\eta(\tau) := q^{\frac{1}{24}}\prod_{n=1}^\infty (1-q^n). \end{aligned}$$ Namely, $$\label{p-eta}
1 + \sum_{n=1}^\infty p(n)q^n = \frac{1}{(q;q)_{\infty}} = q^{\frac{1}{24}}\eta(\tau)^{-1},$$ where here and throughout this section $q:=e^{2\pi i \tau}$, $\tau \in {\mathbb{H}}:= \{x + i y \ | \ x \in \mathbb R, y \in \mathbb R^+\}$ the upper half of the complex plane, and the $q$-Pochhammer symbol is defined for $n\in{\mathbb N}_0\cup\{\infty\}$ by $$(a)_n=(a;q)_n:=\prod_{j=1}^n (1-aq^{j-1}).$$ In fact, the connections between partitions and modular forms go much deeper, and one example of this is given by the combinatorial rank function. Dyson [@Dyson] defined the [*rank*]{} of a partition to be its largest part minus its number of parts, and the *partition rank function* is defined by $$N(m,n) := \# \{\mbox{partitions of } n \mbox{ with rank equal to } m \}.$$ For example, $N(7,-2)=2$, because precisely 2 of the 15 partitions of $n=7$ have rank equal to $-2$; these are $2+2+2+1$, and $3+1+1+1+1$.
Partition rank functions have a rich history in the areas of combinatorics, $q$-hypergeometric series, number theory and modular forms. As one particularly notable example, Dyson conjectured that the rank could be used to combinatorially explain Ramanujan’s famous partition congruences modulo 5 and 7; this conjecture was later proved by Atkin and Swinnerton-Dyer [@AtkinSD].
It is well-known that the associated two variable generating function for $N(m,n)$ may be expressed as a $q$-hypergeometric series $$\begin{aligned}
\label{rankgenfn} \sum_{m=-\infty}^\infty \sum_{n=0}^\infty N(m,n) w^m q^n = \sum_{n=0}^\infty \frac{q^{n^2}}{(wq;q)_n(w^{-1}q;q)_n} =: R_1(w;q),\end{aligned}$$ noting here that $N(m,0)=\delta_{m0}$, where $\delta_{ij}$ is the Kronecker delta function.
Specializations in the $w$-variable of the rank generating function have been of particular interest in the area of modular forms. For example, when $w= 1$, we have that $$\label{r1mock1}
R_1(1;q) = 1+ \sum_{n=1}^\infty p(n) q^n = q^{\frac{1}{24}}\eta^{-1}(\tau)$$ thus recovering , which shows that the generating function for $p(n)$ is (essentially[^2]) the reciprocal of a weight $\frac12$ modular form.
If instead we let $w=-1$, then $$\label{r1mock2}
R_1(-1;q) = \sum_{n=0}^\infty \frac{q^{n^2}}{(-q;q)_n^2} =: f(q).$$ The function $f(q)$ is not a modular form, but one of Ramanujan’s original third order mock theta functions.
Mock theta functions, and more generally mock modular forms and harmonic Maass forms have been major areas of study. In particular, understanding how Ramanujan’s mock theta functions fit into the theory of modular forms was a question that persisted from Ramanujan’s death in 1920 until the groundbreaking 2002 thesis of Zwegers [@Zwegers1]: we now know that Ramanujan’s mock theta functions, a finite list of curious $q$-hypergeometric functions including $f(q)$, exhibit suitable modular transformation properties after they are *completed* by the addition of certain nonholomorphic functions. In particular, Ramanujan’s mock theta functions are examples of *mock modular forms*, the holomorphic parts of *harmonic Maass forms*. Briefly speaking, harmonic Maass forms, originally defined by Bruiner and Funke [@BF], are nonholomorphic generalizations of ordinary modular forms that in addition to satisfying appropriate modular transformations, must be eigenfunctions of a certain weight $k$-Laplacian operator, and satisfy suitable growth conditions at cusps (see [@BFOR; @BF; @OnoCDM; @ZagierB] for more).
Given that specializing $R_1$ at $w=\pm 1$ yields two different modular objects, namely an ordinary modular form and a mock modular form as seen in and , it is natural to ask about the modular properties of $R_1$ at other values of $w$. Bringmann and Ono answered this question in [@BO], and used the theory of harmonic Maass forms to prove that upon specialization of the parameter $w$ to complex roots of unity not equal to $1$, the rank generating function $R_1$ is also a mock modular form. (See also [@ZagierB] for related work.)
If $0<a<c$, then $$q^{-\frac{\ell_c}{24}}R_1(\zeta_c^a;q^{\ell_c}) + \frac{i \sin\left(\frac{\pi a}{c}\right) \ell_c^{\frac{1}{2}}}{\sqrt{3}} \int_{-\overline{\tau}}^{i\infty} \frac{\Theta\left(\frac{a}{c};\ell_c \rho\right)}{\sqrt{-i(\tau + \rho)}} d\rho$$ is a harmonic Maass form of weight $\frac{1}{2}$ on $\Gamma_c$.
Here, $\zeta_c^a := e^{\frac{2\pi ia}{c}}$ is a $c$-th root of unity, $\Theta\left(\frac{a}{c};\ell_c\tau\right)$ is a certain weight $3/2$ cusp form, $\ell_c:={\textnormal{lcm}}(2c^2,24)$, and $\Gamma_c$ is a particular subgroup of $\textnormal{SL}_2(\mathbb Z)$.
In this paper, as well as in prior work of two of the authors [@F-K], we study the related problem of understanding the modular properties of certain combinatorial $q$-hypergeometric series arising from objects called $n$-marked Durfee symbols, originally defined by Andrews in his notable work [@Andrews].
To understand $n$-marked Durfee symbols, we first describe Durfee symbols. For each partition, the Durfee symbol catalogs the size of its Durfee square, as well as the length of the columns to the right as well as the length of the rows beneath the Durfee square. For example, below we have the partitions of $4$, followed by their Ferrers diagrams with any element belonging to their Durfee squares marked by a square $(\sqbullet)$, followed by their Durfee symbols. $$\begin{array}{ccccc}
4 & 3+1 & 2+2 & 2+1+1 & 1+1+1+1 \\
\begin{array}{lllll} \sqbullet & \bullet & \bullet & \bullet \end{array} & \begin{array}{lll} \sqbullet & \bullet & \bullet \\ \bullet & & \end{array} & \begin{array}{ll} \sqbullet & \sqbullet \\ \sqbullet & \sqbullet \end{array} & \begin{array}{ll} \sqbullet & \bullet \\ \bullet & \\ \bullet & \end{array} & \begin{array}{l}\sqbullet \\ \bullet \\ \bullet \\ \bullet \end{array} \\
\hspace{2mm} \left( \begin{array}{lll} 1 & 1 & 1 \\ & & \end{array} \right)_1 \hspace{2mm} & \hspace{2mm} \left( \begin{array}{ll} 1 & 1 \\ 1 & \end{array} \right)_1 \hspace{2mm} & \hspace{2mm} \left( \begin{array}{l} \\ \end{array} \right)_2 \hspace{2mm} & \hspace{2mm} \left( \begin{array}{ll} 1 & \\ 1 & 1 \end{array} \right)_1 \hspace{2mm} & \hspace{2mm} \left( \begin{array}{lll} & & \\ 1 & 1 & 1 \end{array} \right)_1 \hspace{2mm}\\
\end{array}$$
Andrews defined the [*rank*]{} of a Durfee symbol to be the length of the partition in the top row, minus the length of the partition in the bottom row. Notice that this gives Dyson’s original rank of the associated partition. Andrews refined this idea by defining $n$-marked Durfee symbols, which use $n$ copies of the integers. For example, the following is a $3$-marked Durfee symbol of $55$, where $\alpha^j,\beta^j$ indicate the partitions in their respective columns. $$\left(
\begin{array}{cc|ccc|c}
4_3 & 4_3 & 3_2 & 3_2 & 2_2 & 2_1 \\
& 5_3 & & 3_2 & 2_2 & 2_1
\end{array}
\right)_5
=:
\left(
\begin{array}{c|c|c}
\alpha^3 & \alpha^2 & \alpha^1 \\
\beta^3 & \beta^2 & \beta^1
\end{array}
\right)_5$$ Each $n$-marked Durfee symbol has $n$ ranks, one defined for each column. Let $\rm{len}(\pi)$ denote the length of a partition $\pi$. Then the $n$th rank is defined to be $\rm{len}(\alpha^n) - \rm{len}(\beta^n)$, and each $j$th rank for $1\leq j <n$ is defined by $\rm{len}(\alpha^j) - \rm{len}(\beta^j) -1$. Thus the above example has $3$rd rank equal to $1$, $2$nd rank equal to $0$, and $1$st rank equal to $-1$.
Let $\mathcal{D}_n(m_1,m_2,\dots, m_n;r)$ denote the number of $n$-marked Durfee symbols arising from partitions of $r$ with $i$th rank equal to $m_i$. In [@Andrews], Andrews showed that the $( n+1)$-variable rank generating function for Durfee symbols may be expressed in terms of certain $q$-hypergeometric series, analogous to (\[rankgenfn\]). To describe this, for $n\geq 2$, define where ${\boldsymbol{x}} = {\boldsymbol{x}}_n := (x_1,x_2,\dots,x_n).$ For $n=1$, the function $R_1(x;q)$ is defined as the $q$-hypergeometric series in (\[rankgenfn\]). In what follows, for ease of notation, we may also write $R_1({\boldsymbol{x}};q)$ to denote $R_1(x;q)$, with the understanding that ${\boldsymbol{x}} := x$. In [@Andrews], Andrews established the following result, generalizing (\[rankgenfn\]).
For $n\geq 1$ we have that $$\begin{aligned}
\label{durfgenand1}\sum_{m_1,m_2,\dots,m_n = -\infty}^\infty \sum_{r=0}^\infty \mathcal{D}_n(m_1,m_2,\dots,m_n;r)x_1^{m_1}x_2^{m_2}\cdots x_n^{m_n}q^r = R_n({\boldsymbol{x}};q).\end{aligned}$$
When $n=1$, one recovers Dyson’s rank, that is, $\mathcal D_1(m_1;r)=N(m_1,r)$, so that reduces to in this case. The mock modularity of the associated two variable generating function $R_1(x_1;q)$ was established in [@BO] as described in the Theorem above. When $n=2$, the modular properties of $R_2(1,1;q)$ were originally studied by Bringmann in [@Bri1], who showed that $$R_2(1,1;q) := \frac{1}{(q;q)_\infty}\sum_{m\neq 0} \frac{(-1)^{m-1}q^{3m(m+1)/2}}{(1-q^m)^2}$$ is a *quasimock theta function*. In [@BGM], Bringmann, Garvan, and Mahlburg showed more generally that $R_{n}(1,1,\dots,1;q)$ is a quasimock theta function for $n\geq 2$. (See [@Bri1; @BGM] for precise details of these statements.)
In [@F-K], two of the authors established the automorphic properties of $R_n\left({\boldsymbol{x}};q\right)$, for more arbitrary parameters ${\boldsymbol{x}} = (x_1,x_2,\dots,x_n)$, thereby treating families of $n$-marked Durfee rank functions with additional singularities beyond those of $R_n(1,1,\dots,1;q)$. We point out that the techniques of Andrews [@Andrews] and Bringmann [@Bri1] were not directly applicable in this setting due to the presence of such additional singularities. These singular combinatorial families are essentially mixed mock and quasimock modular forms. To precisely state a result from [@F-K] along these lines, we first introduce some notation, which we also use for the remainder of this paper. Namely, we consider functions evaluated at certain length $n$ vectors ${\boldsymbol{\zeta_n}}$ of roots of unity defined as follows (as in [@F-K]).
In what follows, we let $n$ be a fixed integer satisfying $n\geq 2$. Suppose for $1\leq j \leq n$, $\alpha_j \in \mathbb Z$ and $\beta_j \in \mathbb N$, where $\beta_j \nmid \alpha_j, \beta_j \nmid 2\alpha_j$, and that $\frac{\alpha_{r}}{\beta_{r}} \pm \frac{\alpha_{s}}{\beta_{s}} \not\in\mathbb Z$ if $1\leq r\neq s \leq n$. Let $$\begin{aligned}
\notag
{\boldsymbol{\alpha_n}} &:= \Big( \frac{\alpha_{1}}{\beta_{1}},\frac{\alpha_{2}}{\beta_{2}},\dots,\frac{\alpha_{n}}{\beta_{n}} \Big) \in \mathbb Q^n \\
\label{zetavec}
{\boldsymbol{\zeta_n}} &:=\big(\zeta_{\beta_{1}}^{\alpha_{1}},\zeta_{\beta_{2}}^{\alpha_{2}},\dots,\zeta_{\beta_{n}}^{\alpha_{n}}\big) \in \mathbb C^n.
\end{aligned}$$
We point out that the dependence of the vector $\boldsymbol{\zeta_n}$ on $n$ is reflected only in the length of the vector, and not (necessarily) in the roots of unity that comprise its components. In particular, the vector components may be chosen to be $m$-th roots of unity for different values of $m$.
The conditions stated above for $\boldsymbol{\zeta_n}$, as given in [@F-K], do not require $\gcd(\alpha_j, \beta_j) = 1$. Instead, they merely require that $\frac{\alpha_j}{\beta_j} \not\in \frac{1}{2}{\mathbb Z}$. Without loss of generality, we will assume here that $\gcd(\alpha_j, \beta_j) = 1$. Then, requiring that $\beta_j \nmid 2\alpha_j$ is the same as saying $\beta_j \neq 2$.
In [@F-K], the authors proved that (under the hypotheses for $\boldsymbol{\zeta_n}$ given above) the completed nonholomorphic function $$\label{Ahat}
\widehat{\mathcal A}(\boldsymbol{\zeta_n};q) = q^{-\frac{1}{24}}R_n(\boldsymbol{\zeta_n};q) + \mathcal A^-(\boldsymbol{\zeta_n};q)$$ transforms like a modular form. Here the nonholomorphic part $\mathcal A^-$ is defined by $$\label{def_A-}
\mathcal A^-(\boldsymbol{\zeta_n};q) := \frac{1}{\eta(\tau)}\sum_{j=1}^{n} (\zeta_{2\beta_j}^{-3\alpha_j}-\zeta_{2\beta_j}^{-\alpha_j})\frac{\mathscr{R}_3^-\left(\frac{\alpha_j}{\beta_j},-2\tau;\tau\right)}{\Pi_{j}^\dag({\boldsymbol{\alpha_n}})},$$ where $\mathscr{R}_3$ is defined in , and the constant $\Pi_j^{\dag}$ is defined in [@F-K (4.2), with $n\mapsto j$ and $k\mapsto n$]. Precisely, we have the following special case of a theorem established by two of the authors in [@F-K].
If $n\geq 2$ is an integer, then $ \widehat{\mathcal A}\!\left( {\boldsymbol{\zeta_n}};q \right)$ is a nonholomorphic modular form of weight $1/2$ on $\Gamma_{n}$ with character $\chi_\gamma^{-1}$.
Here, the subgroup $\Gamma_{n}\subseteq \textnormal{SL}_2(\mathbb Z)$ under which $\widehat{\mathcal A}({\boldsymbol{\zeta_n}};q)$ transforms is defined by $$\begin{aligned}
\label{def_Gammangroup}
\Gamma_{n}:=\bigcap_{j=1}^{n} \Gamma_0\left(2\beta_j^2\right)\cap \Gamma_1(2\beta_j),\end{aligned}$$ and the Nebentypus character $\chi_\gamma$ is given in Lemma \[ETtrans\].
Quantum modular forms
---------------------
In this paper, we study the quantum modular properties of the $(n+1)$-variable rank generating function for $n$-marked Durfee symbols $R_n({\boldsymbol{x}};q)$. Loosely speaking, a quantum modular form is similar to a mock modular form in that it exhibits a modular-like transformation with respect to the action of a suitable subgroup of $\textnormal{SL}_2(\mathbb Z)$; however, the domain of a quantum modular form is not the upper half-plan $\mathbb H$, but rather the set of rationals $\mathbb Q$ or an appropriate subset. The formal definition of a quantum modular form was originally introduced by Zagier in [@Zqmf] and has been slightly modified to allow for half-integral weights, subgroups of $\operatorname{SL_2}(\mathbb{Z})$, etc. (see [@BFOR]).
\[qmf\] A weight $k \in \frac{1}{2} \mathbb{Z}$ quantum modular form is a complex-valued function $f$ on $\mathbb{Q}$, such that for all $\gamma = {\left(\begin{smallmatrix}a&b\\ c&d \end{smallmatrix} \right)} \in \operatorname{SL_2}(\mathbb{Z})$, the functions $h_\gamma: \mathbb{Q} \setminus \gamma^{-1}(i\infty) \rightarrow \mathbb{C}$ defined by $$h_\gamma(x) := f(x)-\varepsilon^{-1}(\gamma) (cx+d)^{-k} f\left(\frac{ax+b}{cx+d}\right)$$ satisfy a “suitable" property of continuity or analyticity in a subset of $\mathbb{R}$.
The complex numbers $\varepsilon(\gamma)$, which satisfy $|\varepsilon(\gamma)|=1$, are such as those appearing in the theory of half-integral weight modular forms.
We may modify Definition \[qmf\] appropriately to allow transformations on subgroups of $\operatorname{SL_2}(\mathbb{Z})$. We may also restrict the domains of the functions $h_\gamma$ to be suitable subsets of $\mathbb{Q}$.
The subject of quantum modular forms has been widely studied since the time of origin of the above definition. For example, quantum modular forms have been shown to be related to the diverse areas of Maass forms, Eichler integrals, partial theta functions, colored Jones polynomials, meromorphic Jacobi forms, and vertex algebras, among other things (see [@BFOR] and references therein). In particular, the notion of a quantum modular form is now known to have direct connection to Ramanujan’s original definition of a mock theta function. Namely, in his last letter to Hardy, Ramanujan examined the asymptotic difference between mock theta and modular theta functions as $q$ tends towards roots of unity $\zeta$ radially within the unit disk (equivalently, as $\tau$ approaches rational numbers vertically in the upper half plane, with $q=e^{2\pi i \tau}, \tau \in \mathbb H$), and we now know that these radial limit differences are equal to special values of quantum modular forms at rational numbers (see [@BFOR; @BR; @FOR]).
Results {#sec_results}
-------
On one hand, exploring the quantum modular properties of the rank generating function for $n$-marked Durfee symbols $R_n$ in (\[durfgenand1\]) is a natural problem given that two of the authors have established automorphic properties of this function on $\mathbb H$ (see [@F-K Theorem 1.1] above), that $\mathbb Q$ is a natural boundary to $\mathbb H$, and that there has been much progress made in understanding the relationship between quantum modular forms and mock modular forms recently [@BFOR]. Moreover, given that $R_n$ is a vast generalization of the two variable rank generating function in - both a combinatorial $q$-hypergeometric series and a mock modular form - understanding its automorphic properties in general is of interest. On the other hand, there is no reason to a priori expect $R_n$ to converge on $\mathbb Q$, let alone exhibit quantum modular properties there. Nevertheless, we establish quantum modular properties for the rank generating function for $n$-marked Durfee symbols $R_n$ in this paper.
For the remainder of this paper, we use the notation $$\mathcal V_{n}(\tau) := \mathcal V({\boldsymbol{\zeta_n}};q),$$ where $\mathcal V$ may refer to any one of the functions $\widehat{\mathcal A}, \mathcal A^-, R_n,$ etc. Moreover, we will write $$\label{rel_AR}
\mathcal A_{n}(\tau) = q^{-\frac{1}{24}} R_n(\boldsymbol{\zeta_n};q)$$ for the holomorphic part of $\widehat{\mathcal A}$; from [@F-K Theorem 1.1] above, we have that this function is a mock modular form of weight $1/2$ with character $\chi_\gamma^{-1}$ (see Lemma \[Chi\_gammaForm\]) for the group $\Gamma_n$ defined in (\[def\_Gammangroup\]). Here, we will show that $\mathcal{A}_n$ is also a quantum modular form, under the action of a subgroup $\Gamma_{\boldsymbol{\zeta_n}} \subseteq \Gamma_{n}$ defined in , with quantum set $$\label{qSetDef} {Q_{\boldsymbol{\zeta_n}}}:= \left\{\frac{h}{k}\in {\mathbb Q}\; \middle\vert\; \begin{aligned} & \ h\in{\mathbb Z}, k\in{\mathbb N}, \gcd(h,k) = 1, \ \beta_j \nmid k\ \forall\ 1\le j\le n,\\&\left\vert \frac{\alpha_j}{\beta_j}k - \left[\frac{\alpha_j}{\beta_j}k\right]\right\vert > \frac{1}{6}\ \forall\ 1\le j\le n\end{aligned} \right\},$$ where $[x]$ is the closest integer to $x$.
\[rmk:closest\_int\] For $x \in \frac12 + \mathbb Z$, different sources define $[x]$ to mean either $x-\frac12$ or $x+\frac12$. The definition of ${Q_{\boldsymbol{\zeta_n}}}$ involving $[ \cdot ]$ is well-defined for either of these conventions in the case of $x\in \frac12 + \mathbb Z,$ as $\vert x - [x]\vert = \frac{1}{2}$.
To define the exact subgroup under which $\mathcal A_n$ transforms as a quantum automorphic object, we let $$\label{def_ell}\ell = \ell({\boldsymbol}{\zeta_n}):= \begin{cases} 6\left[\text{lcm}(\beta_1, \dots, \beta_{n})\right]^2 &\text{ if } 3 \nmid \beta_j \text{ for all } 1\leq j \leq n, \\ 2\left[\text{lcm}(\beta_1, \dots, \beta_{n})\right]^2 &\text{ if } 3 \mid \beta_j \text{ for some } 1\leq j \leq n, \end{cases}$$ and let $S_\ell:=\left(\begin{smallmatrix}1 & 0 \\ \ell & 1 \end{smallmatrix}\right)$, $T:=\left(\begin{smallmatrix}1 & 1 \\ 0 & 1 \end{smallmatrix}\right)$. We then define the group generated by these two matrices as $$\label{eqn:GroupDefn}
{\Gamma_{\boldsymbol{\zeta_n}}}:= \langle S_\ell, T \rangle.$$
We now state our first main result, which proves that $\mathcal A_n(x),$ and hence $e(-\frac{x}{24})R_n(\boldsymbol{\zeta_n};e(x))$ is a quantum modular form on $Q_{\boldsymbol{\zeta_n}}$ with respect to $\Gamma_{\boldsymbol{\zeta_n}}$. Here and throughout we let $e(x):=e^{2\pi ix}$.
\[thm\_main\_N0\] Let $n \geq 2$. For all $\gamma = \left(\begin{smallmatrix}a & b \\ c & d \end{smallmatrix}\right) \in {\Gamma_{\boldsymbol{\zeta_n}}}$, and $x\in {Q_{\boldsymbol{\zeta_n}}}$, $$H_{n,\gamma}(x) := \mathcal{A}_n(x) - \chi_\gamma (c x+ d)^{-\frac12}\mathcal{A}_n(\gamma x)$$ is defined, and extends to an analytic function in $x$ on $\mathbb{R} - \{\frac{-c}{d}\}$. In particular, for the matrix $S_\ell$, $$\begin{gathered}
\notag
H_{n,S_\ell}(x) = \frac{\sqrt{3}}{2} \sum_{j=1}^{n}\frac{(\zeta_{2\beta_j}^{\alpha_j} - \zeta_{2\beta_j}^{3\alpha_j})}{\displaystyle\Pi^\dag_j( {{\boldsymbol}{\alpha_n}})} \left[\sum_\pm \zeta_6^{\pm1}\int_{\frac{1}{\ell}}^{i\infty}\frac{g_{\pm\frac13+\frac12,-\frac{3\alpha_j}{\beta_j}+\frac12}(3\rho)}{\sqrt{-i(\rho+x)}}d\rho \right] \\
+\sum_{j=1}^{n}\frac{(\zeta_{2\beta_j}^{-3\alpha_j} - \zeta_{2\beta_j}^{-\alpha_j})}{\displaystyle\Pi^\dag_j( {{\boldsymbol}{\alpha_n}})} (\ell x+1)^{-\frac12}\zeta_{24}^{-\ell}\mathcal{E}_1\left(\frac{\alpha_j}{\beta_j},\ell;x\right),\end{gathered}$$ where the weight $3/2$ theta functions $g_{a,b}$ are defined in , and $\mathcal E_1$ is defined in .
As mentioned above, the constants $\Pi^\dagger_j$ are defined in [@F-K (4.2)]. With the exception of replacing $n\mapsto j$ and $k\mapsto n$, we have preserved the notation for these constants from [@F-K].
Our results apply to any $n\geq 2$, as the quantum modular properties in the case $n=1$ readily follow from existing results. Namely, proceeding as in the proof of Theorem \[qsProof\], one may determine a suitable quantum set for the normalized rank generating function in [@BO Theorem 1.1]. Using [@BO Theorem 1.1], a short calculation shows that the error to modularity (with respect to the nontrivial generator of $\Gamma_c$) is a multiple of $$\int_{x}^{i\infty} \frac{\Theta(\frac{a}{c};\ell_c \rho)}{\sqrt{-i(\tau+\rho)}}d\rho$$ for some $x\in\mathbb Q$. When viewed as a function of $\tau$ in a subset of $\mathbb R$, this integral is analytic (e.g., see [@LZ; @Zqmf]).
One could also establish the quantum properties of a non-normalized version of $R_1$ by rewriting it in terms of the Appell-Lerch sum $A_3$, and proceeding as in the proof of Theorem \[thm\_main\_N0\]. In this case, $R_1(\zeta_1;q)$ (where $\zeta_1=e(\alpha_1/\beta_1)$) converges on the quantum set $Q_{\zeta_1}$, where this set is defined by letting $n=1$ in .
The interested reader may also wish to consult [@CLR] for general results on quantum properties associated to mock modular forms.
In a forthcoming joint work [@FJKS], we extend Theorem \[thm\_main\_N0\] to hold for the more general vectors of roots of unity considered in [@F-K], i.e., those with repeated entries. Allowing repeated roots of unity introduces additional singularities, and the modular completion of $R_n$ is significantly more complicated. This precludes us from proving the more general case in the same way as the restricted case we address here.
Preliminaries {#prelim}
=============
Modular, mock modular and Jacobi forms
--------------------------------------
A special ordinary modular form we require is Dedekind’s $\eta$-function, defined in (\[def\_eta\]). This function is well known to satisfy the following transformation law [@Rad].
\[ETtrans\] For $\gamma={\left(\begin{smallmatrix}a&b\\ c&d \end{smallmatrix} \right)} \in \textnormal{SL}_2(\mathbb Z)$, we have that $$\begin{aligned}
\eta\left(\gamma\tau\right) = \chi_\gamma(c\tau + d)^{\frac{1}{2}} \eta(\tau), \end{aligned}$$ where $$\chi_\gamma := \begin{cases} e\left(\frac{b}{24}\right), & \textnormal{ if } c=0, d=1, \\
\sqrt{-i} \ \omega_{d,c}^{-1}e\left(\frac{a+d}{24c}\right), & \textnormal{ if } c>0,\end{cases}$$ with $\omega_{d,c} := e(\frac12 s(d,c))$. Here the Dedekind sum $s(m,t)$ is given for integers $m$ and $t$ by $$s(m,t) := \sum_{j \!\!\!\mod t} \left(\!\!\left(\frac{j}{t}\right)\!\!\right)\left(\!\!\left(\frac{mj}{t}\right)\!\!\right),$$ where $((x)) := x-\lfloor x \rfloor - 1/2$ if $x\in \mathbb R\setminus \mathbb Z$, and $((x)):=0$ if $x\in \mathbb Z$.
The following gives a useful expression for $\chi_\gamma$ (see [@Knopp Ch. 4, Thm. 2]): $$\label{Chi_gammaForm}
\chi_\gamma =
\left\{ \begin{array}{ll}
\big(\frac{d}{|c|} \big)e\left(\frac{1}{24}\left( (a+d)c - bd(c^2-1) - 3c \right)\right)
& \mbox{ if } c \equiv 1 \pmod{2}, \\
\big( \frac{c}{d} \big) e\left(\frac{1}{24}\left( (a+d)c - bd(c^2-1) + 3d - 3 - 3cd \right)\right)
& \mbox{ if } d\equiv 1\pmod{2},
\end{array}\right.$$ where $\big(\frac{\alpha}{\beta}\big)$ is the generalized Legendre symbol.
We require two additional functions, namely the Jacobi theta function $\vartheta(u;\tau)$, an ordinary Jacobi form, and a nonholomorphic modular-like function $R(u;\tau)$ used by Zwegers in [@Zwegers1]. In what follows, we will also need certain transformation properties of these functions.
\[thetaTransform\] For $u \in{\mathbb C}$ and $\tau\in\mathbb{H}$, define $$\label{thetaDef}\vartheta(u;\tau) := \sum_{\nu\in\frac{1}{2} + {\mathbb Z}} e^{\pi i \nu^2\tau + 2\pi i \nu\left(u + \frac{1}{2}\right)}.$$ Then $\vartheta$ satisfies
1. $\vartheta(u+1; \tau) = -\vartheta(u; \tau),$\
2. $\vartheta(u + \tau; \tau) = -e^{-\pi i \tau - 2\pi i u}\vartheta(u; \tau),$\
3. $\displaystyle \vartheta(u; \tau) = - i e^{\pi i \tau/4}e^{-\pi i u} \prod_{m=1}^\infty (1-e^{2\pi i m\tau})(1-e^{2\pi i u}e^{2\pi i \tau(m-1)})(1 - e^{-2\pi i u}e^{2\pi i m\tau}).$
The nonholomorphic function $R(u;\tau)$ is defined in [@Zwegers1] by $$R(u;\tau):=\sum_{\nu\in\frac12+{\mathbb Z}} \left\{\operatorname{sgn}(\nu)-E\left(\left(\nu+\frac{\operatorname{Im}(u)}{\operatorname{Im}(\tau)}\right)\sqrt{2\operatorname{Im}(\tau)}\right)\right\}(-1)^{\nu-\frac12}e^{-\pi i\nu^2\tau-2\pi i\nu u},$$ where $$E(z):=2\int_0^ze^{-\pi t^2}dt.$$
The function $R$ transforms like a (nonholomorphic) mock Jacobi form as follows.
\[Rtransform\] The function $R$ satsifies the following transformation properties:
1. $R(u+1;\tau) = -R(u; \tau),$\
2. $R(u; \tau) + e^{-2\pi i u - \pi i\tau}R(u+\tau; \tau) = 2e^{-\pi i u - \pi i \tau/4}$,\
3. $R(u;\tau) = R(-u;\tau)$,\
4. $R(u;\tau+1)=e^{-\frac{\pi i}{4}} R(u;\tau)$,\
5. $\frac{1}{\sqrt{-i\tau}} e^{\pi i u^2/\tau} R\left(\frac{u}{\tau};-\frac{1}{\tau}\right)+R(u;\tau)=h(u;\tau),$ where the Mordell integral is defined by $$\begin{aligned}
\label{def_hmordell} h(u;\tau):=\int_{\mathbb R}\frac{e^{\pi i\tau t^2-2\pi ut}}{\cosh \pi t} dt. \end{aligned}$$
Using the functions $\vartheta$ and $R$, Zwegers defined the nonholomorphic function $$\begin{aligned}
\label{AminusDef}
\mathscr R_3 (u, v;\tau) :=& \frac{i}{2} \sum_{j=0}^{2} e^{2\pi i j u} \vartheta(v + j\tau + 1; 3\tau) R(3u - v - j\tau - 1; 3\tau)\\
=& \frac{i}{2} \sum_{j=0}^{2} e^{2\pi i j u} \vartheta(v + j\tau; 3\tau) R(3u - v - j\tau; 3\tau),\nonumber\end{aligned}$$ where the equality of the two expressions in is justified by Proposition \[thetaTransform\] and Proposition \[Rtransform\]. This function is used to complete the level three Appell function (see [@Zwegers2] or [@BFOR]) $$\begin{aligned}
A_3(u, v; \tau) := e^{3\pi i u} \sum_{n\in{\mathbb Z}} \frac{(-1)^n q^{3n(n+1)/2}e^{2\pi i nv}}{1 - e^{2\pi i u}q^n},\end{aligned}$$ where $u,v \in \mathbb C$, as $$\begin{aligned}
\label{def_A3hat} \widehat{A}_3(u, v; \tau) := A_3(u, v;\tau) + \mathscr R_3(u, v;\tau). \end{aligned}$$
This completed function transforms like a (non-holmorphic) Jacobi form, and in particular satisfies the following elliptic transformation.
\[completeAtransform\] For $n_1, n_2, m_1, m_2\in{\mathbb Z}$, the completed level $3$ Appell function $\widehat{A}_3$ satisfies $$\widehat{A}_3(u + n_1\tau + m_1, v + n_2\tau + m_2; \tau) = (-1)^{n_1 + m_1}e^{2\pi i (u(3n_1 - n_2) - vn_1)}q^{3n_1^2/2 - n_1n_2}\widehat{A}_3(u, v; \tau).$$
The following relationship between the Appell series $A_3$ and the combinatorial series $R_n$ is proved in [@F-K].
Under the hypotheses given above on $\boldsymbol{\zeta_n}$, we have that $$R_n(\boldsymbol{\zeta_n};q) =\frac{1}{(q)_\infty} \sum_{j=1}^n\left(\zeta_{2\beta_j}^{-3\alpha_j}-\zeta_{2\beta_j}^{-\alpha_j}\right)\frac{A_3\left(\frac{\alpha_j}{\beta_j},-2\tau;\tau\right)}{\Pi^\dag_j( {\boldsymbol{\alpha_n}})}.$$
We also note that $$\begin{aligned}
\widehat{\mathcal A}_n\left( \tau \right) = \frac{1}{\eta(\tau)}\sum_{j=1}^{n} (\zeta_{2\beta_j}^{-3\alpha_j}-\zeta_{2\beta_j}^{-\alpha_j})\frac{\widehat{A}_3\left(\frac{\alpha_j}{\beta_j},-2\tau;\tau\right)}{\Pi_{j}^\dag({\boldsymbol{\alpha_n}})}. \end{aligned}$$
In addition to working with the Appell sum $\widehat{A}_3$, we also make use of additional properties of the functions $h$ and $R$. In particular, Zwegers also showed how under certain hypotheses, these functions can be written in terms of integrals involving the weight $3/2$ modular forms $g_{a,b}(\tau)$, defined for $a,b\in\mathbb R$ and $\tau \in \mathbb H$ by $$\begin{aligned}
\label{def_gab}
g_{a,b}(\tau) := \sum_{\nu \in a + \mathbb Z} \nu e^{\pi i \nu^2\tau + 2\pi i \nu b}.
\end{aligned}$$ We will make use of the following results.
\[prop\_Zg\] The function $g_{a,b}$ satisfies:
1. $g_{a+1,b}(\tau)= g_{a,b}(\tau)$,\
2. $g_{a,b+1}(\tau)= e^{2\pi ia} g_{a,b}(\tau)$,\
3. $g_{a,b}(\tau+1)= e^{-\pi ia(a+1)}g_{a,a+b+\frac12}(\tau)$,\
4. $g_{a,b}(-\frac{1}{\tau})= i e^{2\pi iab} (-i\tau)^{\frac32}g_{b,-a}(\tau)$.
\[thm\_Zh2\] Let $\tau \in \mathbb H$. For $a,b \in (-\frac12,\frac12)$, we have $$h(a\tau-b;\tau) = - e\left(\tfrac{a^2\tau}{2} - a(b+\tfrac12)\right) \int_{0}^{i\infty} \frac{g_{a+\frac12,b+\frac12}(\rho)}{\sqrt{-i(\rho+\tau)}}d\rho.$$
The quantum set {#quantumSet}
===============
We call a subset $S \subseteq {\mathbb Q}$ a [*quantum set*]{} for a function $F$ with respect to the group $G\subseteq \textnormal{SL}_2({\mathbb Z})$ if both $F(x)$ and $F(Mx)$ exist (are non-singular) for all $x\in S$ and $M\in G$.
In this section, we will show that ${Q_{\boldsymbol{\zeta_n}}}$ as defined in is a quantum set for $\mathcal{A}_n$ with respect to the group ${\Gamma_{\boldsymbol{\zeta_n}}}$. Recall that ${Q_{\boldsymbol{\zeta_n}}}$ is defined as
$$\begin{aligned}
{Q_{\boldsymbol{\zeta_n}}}:= \left\{\frac{h}{k}\in {\mathbb Q}\; \middle\vert\; \begin{aligned} & \ h\in{\mathbb Z}, k\in{\mathbb N}, \gcd(h,k) = 1, \ \beta_j \nmid k\ \forall\ 1\le j\le n,\\&\left\vert \frac{\alpha_j}{\beta_j}k - \left[\frac{\alpha_j}{\beta_j}k\right]\right\vert > \frac{1}{6}\ \forall\ 1\le j\le n\end{aligned} \right\},\end{aligned}$$
where $[x]$ is the closest integer to $x$ (see Remark \[rmk:closest\_int\]).
Moreover, recall that the “holomorphic part” we consider (see §\[sec\_results\]) is $\mathcal A_n(\tau) = q^{-\frac{1}{24}} R_n(\boldsymbol{\zeta_n}; q)$. To show that ${Q_{\boldsymbol{\zeta_n}}}$ is a quantum set for $\mathcal A_n(\tau)$, we must first show that the the multi-sum defining $R_n(\boldsymbol{\zeta_n}; \zeta_k^h)$ converges for $\frac{h}{k}\in{Q_{\boldsymbol{\zeta_n}}}$. In what follows, as in the definition of ${Q_{\boldsymbol{\zeta_n}}}$, we take $h\in{\mathbb Z}$, $k\in{\mathbb N}$ such that $\gcd(h,k) = 1$.
We start by addressing the restriction that for $\frac{h}{k}\in {Q_{\boldsymbol{\zeta_n}}}$, $\beta_j\nmid k$ for all $1 \le j \le n$.
For $\frac{h}{k}\in {\mathbb Q}$, all summands of $R_n(\boldsymbol{\zeta_n}; \zeta_k^h)$ are finite if and only if $\beta_j \nmid k$ for all $1\le j \le n$.
Examining the multi-sum $R_n(\boldsymbol{\zeta_n}; \zeta_k^h)$, we see that all terms are a power of $\zeta_k^h$ divided by a product of factors of the form $1 - \zeta_{\beta_j}^{\pm\alpha_j} \zeta_k^{hm}$ for some integer $m\ge 1$. Therefore, to have each summand be finite, it is enough to ensure that $1 - \zeta_{\beta_j}^{\pm\alpha_j} \zeta_k^{hm} \neq 0$ for all $m\ge 1$ and for all $1\le j \le n$. For ease of notation in this proof, we will omit the subscripts for $\alpha_j$ and $\beta_j$.
If $1 - \zeta_{\beta}^{\pm\alpha} \zeta_k^{hm} = 0$ for some $m\in{\mathbb N}$, we have that $$\pm\frac{\alpha}{\beta} + \frac{hm}{k} \in{\mathbb Z}.$$ Let $K = {\textnormal{lcm}}(\beta, k) = \beta\beta^\prime = kk^\prime$. Then $\pm\frac{\alpha}{\beta} + \frac{hm}{k} \not\in{\mathbb Z}$ is the same as $\pm\alpha\beta^\prime + hmk^\prime \not\in K{\mathbb Z}$. Since $K = kk^\prime$, if $k^\prime \nmid \alpha\beta^\prime$, this is always true and we do not have a singularity.
However, since $K = \beta\beta^\prime = kk^\prime$, if $k^\prime \vert \alpha\beta^\prime$, then $\frac{\beta\beta^\prime}{k} \vert \alpha\beta^\prime$. This implies that $\beta \vert \alpha k$ and that $\beta \vert k$ since $\gcd(\alpha, \beta) = 1$.
Therefore, if $\beta\nmid k$, it is always the case that $k^\prime\nmid \alpha\beta^\prime$, so for all $m\in{\mathbb N}$, $$\pm\frac{\alpha}{\beta} + \frac{hm}{k} \not\in{\mathbb Z}.$$
Now that we have shown that all summands in $R_n(\boldsymbol{\zeta_n}; \zeta_k^h)$ are finite for $\frac{h}{k}\in{Q_{\boldsymbol{\zeta_n}}}$, we will show that the sum converges.
\[qsProof\] For $\boldsymbol{\zeta_n}$ as in , if $\frac{h}{k} \in {Q_{\boldsymbol{\zeta_n}}}$, then $R_n(\boldsymbol{\zeta_n}; \zeta_k^h)$ converges and can be evaluated as a finite sum. In particular, we have that:
$$\begin{gathered}
\label{eqn_Rnconvsum}
R_n(\boldsymbol{\zeta_n}; \zeta_k^h) = \prod_{j=1}^n \frac{1}{1 - ((1-x_j^k)(1-x_j^{-k}))^{-1}}\\
\times \!\!\!\!\! \sum_{\substack{0 < m_1\le k\\ 0 \le m_2, \dots, m_n < k}} \frac{\zeta_k^{h[(m_1 + m_2 + \dots + m_n)^2 + (m_1 + \dots + m_{n-1}) + (m_1 + \dots + m_{n-2}) + \dots + m_1]}}{(x_1\zeta_k^h;\zeta_k^h)_{m_1} \left(\frac{\zeta_k^h}{x_1};\zeta_k^h\right)_{m_1} (x_2 \zeta_k^{hm_1};\zeta_k^h)_{m_2 + 1} \left(\frac{\zeta_k^{hm_1}}{x_2};\zeta_k^h\right)_{m_2+1}} \\
\times \frac{1}{(x_3 \zeta_k^{h(m_1 + m_2)};\zeta_k^h)_{m_3 + 1}\!\!\left(\frac{\zeta_k^{h(m_1 + m_2)}}{x_3};\zeta_k^h\!\right)_{\!m_3 + 1} \!\!\!\!\!\!\!\!\!\!\cdots(x_n \zeta_k^{h(m_1 + \dots + m_{n-1})};\zeta_k^h)_{ m_n+1} \!\!\left(\!\frac{\zeta_k^{h(m_1 + \dots + m_{n-1})}}{x_n};\zeta_k^h\!\right)_{\! m_n+1} },\end{gathered}$$
where $\boldsymbol{\zeta_n} = (x_1, x_2, \dots, x_n)$.
We start by taking $\frac{h}{k} \in {Q_{\boldsymbol{\zeta_n}}}$, and write $\zeta = \zeta_k^h$. For ease of notation, we will use $x_j$ to denote the $j$-th component in $\boldsymbol{\zeta_n}$, so $x_j = e^{2\pi i \alpha_j/\beta_j}$. Further, for clarity of argument, we will carry out the proof in the case of $n = 2$, with comments throughout about how the proof follows for $n > 2$. We have that $$\begin{aligned}
\nonumber
R_2((x_1,x_2); \zeta) =& \sum_{\substack{m_1 > 0\\ m_2\ge 0}} \frac{\zeta^{(m_1+m_2)^2 + m_1}}{(x_1\zeta;\zeta)_{m_1}(x_1^{-1}\zeta;\zeta)_{m_1}(x_2\zeta^{m_1};\zeta)_{m_2+1}(x_2^{-1}\zeta^{m_1};\zeta)_{m_2+1}}\\
\label{sumRearranged3} =&\sum_{M_1, M_2 \ge 0} \frac{1}{(1 - x_1^k)^{M_1} (1 - x_1^{-k})^{M_1} (1-x_2^k)^{M_2}(1-x_2^{-k})^{M_2}}\\ \label{sumRearranged2}
&\times \sum_{\substack{0 < s_1 \le k\\ 0\le s_2 < k}}\frac{\zeta^{(s_1+s_2)^2+s_1}}{(x_1\zeta;\zeta)_{s_1}(x_1^{-1}\zeta;\zeta)_{s_1}(x_2\zeta^{s_1};\zeta)_{s_2+1}(x_2^{-1}\zeta^{s_1};\zeta)_{s_2+1}},
\end{aligned}$$ where we have let $m_j = s_j + M_j k$ for $0 < s_1 \le k$, $0 \le s_2 < k$, and $M_j\in{\mathbb N}_0$, and have used the fact that $$(x\zeta^r;\zeta)_{s+Mk} = (1 - x^k)^M \; (x\zeta^r; \zeta)_s ,$$ which holds for any $M, r, s\in{\mathbb N}_0$. (We note that for $n > 2$, we proceed as above, additionally taking $0 \le s_j \le k-1$ for $j > 2$.) The second sum in is a finite sum, as desired. For the first sum in (\[sumRearranged3\]) we notice that we in fact have the product of two geometric series, each of the form $$\sum_{M_j\ge 0} \left(\frac{1}{ (1-x_j^k)( 1 - x_j^{-k})}\right)^{M_j}.$$ By definition, we have $x_j = \cos\theta_j + i\sin\theta_j$ where $\theta_j = \frac{2\pi\alpha_j}{\beta_j}$. Therefore, this sum converges if and only if $$\begin{aligned}
\vert 1 - x_j^k\vert \vert 1 - x_j^{-k}\vert =2 - 2\cos(k\theta_j) > 1 \iff \cos(k\theta_j) < \frac{1}{2}.\end{aligned}$$ For $\cos(k\theta_j) < \frac{1}{2}$, it must be that $k\theta_j = r + 2\pi M$ where $-\pi < r \le \pi$, $\vert r \vert > \frac{\pi}{6}$, and $M \in{\mathbb Z}$. This is equivalent to saying $$\left\vert \frac{\alpha_j}{\beta_j}k - \left[\frac{\alpha_j}{\beta_j}k\right]\right\vert > \frac{1}{6}\ \ \forall\ 1\le j\le n,$$ as in the definition of ${Q_{\boldsymbol{\zeta_n}}}$ in . Therefore, we see that for $\frac{h}{k}\in{Q_{\boldsymbol{\zeta_n}}}$, $R_2((x_1, x_2); \zeta)$ converges to the claimed expression in .
We note that by Abel’s theorem, having shown convergence of $R_2((x_1, x_2); \zeta)$, we have that $R_2((x_1, x_2); q)$ converges to $R_2((x_1, x_2); \zeta)$ as $q\to\zeta$ radially within the unit disc.
As noted, the above argument extends to $n > 2$. Letting $m_j = s_j + M_j k$ with $0 < s_1 \le k$ and $0 \le s_j < k$ for $j \ge 2$, rewriting as in , and then summing the resulting geometric series gives the desired exact formula for $R_n(\boldsymbol{\zeta_n}; \zeta)$.
To complete the argument that ${Q_{\boldsymbol{\zeta_n}}}$ is a quantum set for $R_n(\boldsymbol{\zeta_n}; \zeta)$ with respect to $\Gamma_{\boldsymbol{\zeta_n}}$, it remains to be seen that $R_n(\boldsymbol{\zeta_n}; \xi)$ converges, where $\xi = e^{2\pi i \gamma(\frac{h}{k})}$ for $\frac{h}{k}\in{Q_{\boldsymbol{\zeta_n}}}$ and $\gamma\in {\Gamma_{\boldsymbol{\zeta_n}}}$, defined in . For the ease of the reader, we recall from and that $$\begin{aligned}
{\Gamma_{\boldsymbol{\zeta_n}}}:= \left\langle \left(\begin{matrix} 1 & 1\\ 0 & 1\end{matrix}\right), \left(\begin{matrix} 1 & 0\\ \ell & 1\end{matrix}\right)\right\rangle,\end{aligned}$$ where $$\ell = \ell_{\beta} := \begin{cases} 6\left[\text{lcm}(\beta_1, \dots, \beta_{k})\right]^2 &\text{ if $\forall j$, $3\not\vert \beta_j$}\\ 2\left[\text{lcm}(\beta_1, \dots, \beta_{k})\right]^2 &\text{ if $\exists j$, $3 \vert \beta_j$.}\end{cases}$$ The convergence of $R_n(\boldsymbol{\zeta_n}; \xi)$ is a direct consequence of the following lemma.
\[setClosed\] The set ${Q_{\boldsymbol{\zeta_n}}}$ is closed under the action of ${\Gamma_{\boldsymbol{\zeta_n}}}$.
Since ${\Gamma_{\boldsymbol{\zeta_n}}}$ is given as a set with two generators, it is enough to show that ${Q_{\boldsymbol{\zeta_n}}}$ is closed under action of each of those generators.
Let $\frac{h}{k}\in{Q_{\boldsymbol{\zeta_n}}}$. Then ${\left(\begin{smallmatrix}1&1\\ 0&1 \end{smallmatrix} \right)}\frac{h}{k} = \frac{h + k}{k}$. Note that $\gcd(h+k, k) = \gcd(h,k) = 1$ and we already know that $k$ satisfies the conditions in the definition of ${Q_{\boldsymbol{\zeta_n}}}$. Therefore, ${\left(\begin{smallmatrix}1&1\\ 0&1 \end{smallmatrix} \right)}\frac{h}{k}\in{Q_{\boldsymbol{\zeta_n}}}$.
Under the action of ${\left(\begin{smallmatrix}1&0\\ \ell&1 \end{smallmatrix} \right)}$, we have $$\left(\begin{array}{cc}1 & 0 \\ \ell & 1\end{array}\right)\frac{h}{k} = \frac{h}{h\ell + k}.$$ We first note that $\gcd(h, h\ell + k) = \gcd(h,k) = 1$, and $\beta_j \nmid (h\ell + k)$ as $\beta_j \vert \ell$ and $\beta_j \nmid k$. It remains to check that $$\left\vert \frac{\alpha_j}{\beta_j}(h\ell + k) - \left[\frac{\alpha_j}{\beta_j}(h\ell + k)\right]\right\vert > \frac{1}{6}\ \forall\ 1\le j\le n.$$ We have that $$\begin{aligned}
\nonumber
\left\vert \frac{\alpha_j}{\beta_j} (h \ell + k) - \left[\frac{\alpha_j}{\beta_j}(h\ell + k)\right]\right\vert &= \left\vert \frac{\alpha_j h\ell}{\beta_j} + \frac{\alpha_j}{\beta_j}k - \left[\frac{\alpha_j h\ell}{\beta_j} + \frac{\alpha_j}{\beta_j}k\right]\right\vert\\
&= \left\vert\frac{\alpha_j}{\beta_j} k - \left[\frac{\alpha_j}{\beta_j} k\right]\right\vert > \frac{1}{6},\label{closestIntSimplification}\end{aligned}$$ where we can simplify as in since, by definition of $\ell$, $\frac{\alpha_j\ell}{\beta_j} \in{\mathbb Z}$. Thus, ${Q_{\boldsymbol{\zeta_n}}}$ is closed under the action of ${\Gamma_{\boldsymbol{\zeta_n}}}$.
Proof of Theorem \[thm\_main\_N0\] {#n0proof}
==================================
We now prove Theorem \[thm\_main\_N0\]. Our first goal is to establish that $H_{n,\gamma}$ is analytic in $x$ on $\mathbb{R} - \{\frac{-c}{d}\}$ for all $x\in {Q_{\boldsymbol{\zeta_n}}}$ and $\gamma = \left(\begin{smallmatrix}a & b \\ c & d \end{smallmatrix}\right) \in {\Gamma_{\boldsymbol{\zeta_n}}}$. As shown in Section \[quantumSet\], we have that $\mathcal A_n(x)$ and $\mathcal A_n(\gamma x)$ are defined for all $x\in{Q_{\boldsymbol{\zeta_n}}}$ and $\gamma\in {\Gamma_{\boldsymbol{\zeta_n}}}$. Note that it suffices to consider only the generators $S_\ell$ and $T$ of ${\Gamma_{\boldsymbol{\zeta_n}}}$, since $$H_{n,\gamma \gamma'}(\tau)= H_{n,\gamma'}(\tau) + \chi_{\gamma'}(C\tau+D)^{-\frac12} H_{n,\gamma}(\gamma'\tau)$$ for $\gamma = \left(\begin{smallmatrix}a & b \\ c & d \end{smallmatrix}\right)$ and $\gamma' = \left(\begin{smallmatrix}A & B \\ C & D \end{smallmatrix}\right)$.
First, consider $\gamma = T$. Then by definition, $\chi_T=\zeta_{24}$, and so $H_{n,T}(x) = \mathcal A_n(x) - \zeta_{24} \mathcal A_n(x+1)$. When we map $x \mapsto x +1$, $q=e^{2\pi i x}$ remains invariant. Then since the definition of $R_{n}(x)$ in can be expressed as a series only involving integer powers of $q$, it is also invariant. Thus $$\mathcal A_n(x+1)=e^{\frac{ -2\pi i (x+1)}{24}}R_{n}(x) = \zeta_{24}^{-1}\mathcal A_n(x),$$ and so $H_{n,T}(x)=0$.
We now consider the case $\gamma = S_\ell$. In this case using we calculate that $\chi_{S_\ell}=\zeta_{24}^{-\ell}$. Thus, $$H_{n,S_\ell}(x) = \mathcal A_n(x) - \zeta_{24}^{-\ell}(\ell x +1)^{-\frac12} \mathcal A_n(S_\ell x).$$ From the modularity of $\widehat{\mathcal A}_n$ we have that $\widehat{\mathcal A}_n(x) = \zeta_{24}^{-\ell}(\ell x +1)^{-\frac12}\widehat{\mathcal A}_n(S_\ell x)$. Thus and give that $$\label{eq:HviaA-}
H_{n,S_\ell}(x) =-\mathcal A_n ^-(x)+\zeta_{24}^{-\ell}(\ell x +1)^{-\frac12}\mathcal A_n^-(S_\ell x),$$ where $\mathcal A_n^-$ is defined in .
Using the Jacobi triple product identity from Proposition \[thetaTransform\] item (3), we can simplify the theta functions to get that $\vartheta\left(-2\tau ;3\tau\right) = iq^{-\frac23}\eta(\tau)$, $\vartheta\left(-\tau ;3\tau\right) = iq^{-\frac16}\eta(\tau)$, and $\vartheta\left(0;3\tau\right) = 0$. Thus, $$\begin{aligned}
\mathscr{R}_3\left( \frac{\alpha_j}{\beta_j},-2\tau ; \tau \right) = - \frac12 q^{-\frac23} \eta(\tau) \sum_{\delta=0}^1 e\left(\frac{\alpha_j}{\beta_j} \delta \right) q^{\frac{\delta}{2}} R\left(\frac{3\alpha_j}{\beta_j} + (2-\delta)\tau ; 3\tau \right). \end{aligned}$$ Using Proposition \[Rtransform\] item (2), we can rewrite $$R\left(\frac{3\alpha_j}{\beta_j} + 2\tau ; 3\tau \right) = 2e\left(\frac{3\alpha_j}{2\beta_j} \right) q^{\frac58} - e \left(\frac{3\alpha_j}{\beta_j} \right) q^{\frac12} R\left(\frac{3\alpha_j}{\beta_j} - \tau; 3\tau \right),$$ so that $$\begin{gathered}
\notag
\sum_{\delta=0}^1 e\left(\frac{\alpha_j}{\beta_j} \delta \right) q^{\frac{\delta}{2}} R\left(\frac{3\alpha_j}{\beta_j} + (2-\delta)\tau ; 3\tau \right) = \\
2e\left(\frac{3\alpha_j}{2\beta_j} \right) q^{\frac58} + e \left(\frac{2\alpha_j}{\beta_j} \right) q^{\frac12} \sum_{\pm} \pm e \left(\mp \frac{\alpha_j}{\beta_j} \right)R\left(\frac{3\alpha_j}{\beta_j} \pm \tau; 3\tau \right).\end{gathered}$$ Thus we see that $$\begin{gathered}
\label{eq:F-}
\mathcal A_n^-(\tau) = -\frac12 \sum_{j=1}^{n}\frac{(\zeta_{2\beta_j}^{-3\alpha_j} - \zeta_{2\beta_j}^{-\alpha_j})}{\displaystyle\Pi^\dag_{j} ( {{\boldsymbol}{\alpha_k}})} e \left(\frac{2\alpha_j}{\beta_j} \right) q^{-\frac16} \sum_{\pm} \pm e \left(\mp \frac{\alpha_j}{\beta_j} \right) R\left(\frac{3\alpha_j}{\beta_j} \pm \tau; 3\tau \right) \\
-q^{-\frac{1}{24}} \sum_{j=1}^{n}\frac{(\zeta_{2\beta_j}^{-3\alpha_j} - \zeta_{2\beta_j}^{-\alpha_j})}{\displaystyle\Pi^\dag_{j} ( {{\boldsymbol}{\alpha_k}})} e \left(\frac{3\alpha_j}{2\beta_j} \right).\end{gathered}$$ Now to compute $\mathcal A_n^-(S_\ell \tau)$ we first define $$\begin{aligned}
F_{\alpha,\beta}(\tau):= q^{-\frac16} \sum_{\pm} \pm e \left(\mp \frac{\alpha}{\beta} \right) R\left(\frac{3\alpha}{\beta} \pm \tau; 3\tau \right).\end{aligned}$$ Then by and we can write $$\begin{gathered}
H_{n,S_\ell}(\tau) = \frac12 \sum_{j=1}^{n}\frac{(\zeta_{2\beta_j}^{-3\alpha_j} - \zeta_{2\beta_j}^{-\alpha_j})}{\displaystyle\Pi^\dag_{j} ( {{\boldsymbol}{\alpha_k}})} e \left(\frac{2\alpha_j}{\beta_j} \right) \left[ F_{\alpha_j,\beta_j}(\tau) - \zeta_{24}^{-\ell}(\ell \tau +1)^{-\frac12}F_{\alpha_j,\beta_j}(S_\ell \tau) \right] \\
+\sum_{j=1}^{n}\frac{(\zeta_{2\beta_j}^{-3\alpha_j} - \zeta_{2\beta_j}^{-\alpha_j})}{\displaystyle\Pi^\dag_{j} ( {{\boldsymbol}{\alpha_k}})} (\ell\tau+1)^{-\frac12}\zeta_{24}^{-\ell}\mathcal E_1\left(\frac{\alpha_j}{\beta_j},\ell;\tau\right), \end{gathered}$$ where $$\label{def_mathcalE}
\mathcal E_1\left(\frac{\alpha}{\beta},\ell;\tau\right):=(\ell\tau+1)^{\frac12} \zeta_{24}^\ell q^{-\frac{1}{24}}e\left(\frac32 \frac{\alpha}{\beta} \right) - e\left(\frac{-S_\ell\tau}{24} \right)e\left(\frac32 \frac{\alpha}{\beta} \right).$$
Thus in order to prove that $H_{n,S_\ell}(x)$ is analytic on $\mathbb{R} - \{\frac{-1}{\ell}\}$ it suffices to show that for each $1\leq j \leq n$, $$\begin{aligned}
G_{\alpha_j,\beta_j }(\tau) := F_{\alpha_j,\beta_j}(\tau) - \zeta_{24}^{-\ell}(\ell \tau +1)^{-\frac12}F_{\alpha_j,\beta_j}(S_\ell \tau) \end{aligned}$$ is analytic on $\mathbb{R} - \{\frac{-1}{\ell}\}$. We establish this in Proposition \[prop\_Habanalytic\] below.
\[prop\_Habanalytic\] Fix $1\leq j \leq n$ and set $(\alpha, \beta) := (\alpha_j, \beta_j)$. With notation and hypotheses as above, we have that $$\begin{aligned}
G_{\alpha,\beta}(\tau) = \sqrt{3}\sum_{\pm}\mp e\left(\mp\frac16\right) \int_{\frac{1}{\ell}}^{i\infty}\frac{g_{\pm\frac13 + \frac12, \frac12-3\frac{\alpha}{\beta}}(3\rho)}{\sqrt{-i(\rho+\tau)}}d\rho,\end{aligned}$$ which is analytic on $\mathbb R - \left \{\frac{-1}{\ell}\right \}$.
Fix $1\leq j \leq n$ and set $(\alpha, \beta) := (\alpha_j, \beta_j)$. Define $m := \left[\frac{3\alpha}{\beta} \right] \in \mathbb{Z}$, $r\in (-\frac12, \frac12)$ so that $\frac{3\alpha}{\beta} =m + r$. We note that $r\neq \pm \frac12$ since $\beta\neq 2$. Using Proposition \[Rtransform\] (1), we have that $$\label{eq:F_jtau}
F_{\alpha,\beta}(\tau) = q^{-\frac16} \sum_{\pm} \pm e \left(\frac{\mp r}{3} \right) e \left( \frac{\mp m}{3} \right) (-1)^{m} R\left( \pm \tau + r; 3\tau \right).$$ Letting $\tau_\ell := -\frac{1}{\tau} - \ell$ we have $S_\ell \tau = \frac{-1}{\tau_\ell}$. Using Proposition \[Rtransform\] (5) with $u=\frac{r}{3} \tau_\ell \mp \frac13$ and $\tau \mapsto \frac{\tau_\ell}{3}$ we see that $$\begin{gathered}
\label{eq:h1}
R\left(r \mp \frac{1}{\tau_\ell} ; \frac{-3}{\tau_\ell} \right) = \\
\sqrt{ \frac{-i\tau_\ell}{3} } \cdot e\left(-\frac12\left(\frac{r\tau_\ell}{3} \mp \frac13\right)^2\left(\frac{3}{\tau_\ell}\right)\right)\left[h\left(\frac{r\tau_\ell}{3} \mp \frac13; \frac{\tau_\ell}{3} \right) - R\left(\frac{r\tau_\ell}{3} \mp \frac13; \frac{\tau_\ell}{3} \right) \right].\end{gathered}$$ Using Proposition \[Rtransform\] parts (1) and (4) we see that $R\left(\frac{r\tau_\ell}{3} \mp \frac13; \frac{\tau_\ell}{3} \right) = \zeta_{24}^\ell R\left( \frac{-r}{3\tau} \mp \frac13 ; \frac{-1}{3\tau} \right)$. Then using Proposition \[Rtransform\] (5) with $u=\mp \tau - r$ and $\tau \mapsto 3\tau$ we obtain that $$R\left(\frac{r\tau_\ell}{3} \mp \frac13; \frac{\tau_\ell}{3} \right) = \zeta_{24}^\ell \sqrt{-i(3\tau)} \cdot e\left(\frac{-\left(\mp\tau - r\right)^2}{6\tau} \right) \left[h\left( \mp\tau - r; 3\tau\right) - R\left( \mp\tau - r ; 3\tau \right) \right],$$ which together with and gives $$\begin{gathered}
\notag
F_{\alpha,\beta}(S_\ell \tau) = \\
e\left(\frac{1}{6\tau_\ell} \right) \sum_{\pm} \pm e \left(\frac{\mp r}{3} \right) e \left( \frac{\mp m}{3} \right) (-1)^{m}
\sqrt{ \frac{-i\tau_\ell}{3} } \cdot e\left(-\frac12\left(\frac{r\tau_\ell}{3} \mp \frac13\right)^2\left(\frac{3}{\tau_\ell}\right)\right) \cdot \\
\left[h\left(\frac{r\tau_\ell}{3} \mp \frac13; \frac{\tau_\ell}{3} \right) -
\zeta_{24}^\ell \sqrt{-i(3\tau)} \cdot e\left(\frac{-\left(\mp\tau - r\right)^2}{6\tau}\right) \left[h\left( \mp\tau - r; 3\tau\right) - R\left( \mp\tau - r ; 3\tau \right)\right] \right].\end{gathered}$$ By the definition of $r$ and $\ell$ we have that $\frac{r^2\ell}{6} \in \mathbb{Z}$. Simplifying thus gives that $$\begin{gathered}
F_{\alpha,\beta}(S_\ell \tau) = \sum_{\pm} \pm (-1)^{m} e\left( \frac{\mp m }{3}\right) e\left(\frac{r^2}{6\tau} \right) \sqrt{\frac{-i\tau_\ell}{3}} h\left( \frac{r\tau_\ell}{3} \mp \frac13 ; \frac{\tau_\ell}{3}\right) \\
- \sum_{\pm} \pm (-1)^{m} e\left( \frac{\mp m}{3}\right) e\left(\frac{\mp r}{3} \right) q^{-\frac16} \zeta_{24}^\ell (\ell\tau + 1)^{\frac12} \cdot h\left(\mp \tau - r ; 3\tau \right) \\
+ \sum_{\pm} \pm (-1)^{m} e\left( \frac{\mp m}{3}\right) e\left(\frac{\mp r}{3} \right) q^{-\frac16} \zeta_{24}^\ell (\ell\tau + 1)^{\frac12} \cdot R\left(\mp \tau - r ; 3\tau \right),\end{gathered}$$ and so using Proposition \[Rtransform\] (3) and the fact that $h(u;\tau)=h(-u;\tau)$ which comes directly from the definition of $h$ in , we see that $$\begin{gathered}
\notag
G_{\alpha,\beta}(\tau) = q^{-\frac16} \sum_{\pm}\pm (-1)^{m} e\left( \frac{\mp m}{3}\right) e\left(\frac{\mp r}{3} \right)h\left(\pm \tau + r ; 3\tau \right) \\
- \sum_{\pm}\pm (-1)^{m} e\left( \frac{\mp m}{3}\right) e\left(\frac{r^2}{6\tau}\right) \zeta_{24}^{-\ell} \sqrt{\frac{i}{3\tau}} \cdot h\left( \frac{r\tau_\ell}{3} \mp \frac13 ; \frac{\tau_\ell}{3}\right).\end{gathered}$$ We now use Theorem \[thm\_Zh2\] to convert the $h$ functions into integrals. Letting $a=\frac{\pm 1}{3}$, $b=-r$, and $\tau \mapsto 3\tau$ gives that $$h\left(\pm \tau + r ; 3\tau \right) = -q^{\frac16} \zeta_6^{\mp 1} e\left(\frac{\pm r}{3} \right) \int_{0}^{i\infty} \frac{g_{\pm\frac13 + \frac12, \frac12 -r}(z) dz}{\sqrt{-i(z+3\tau)}}.$$ Letting $a=r$, $b=\frac{\pm 1}{3}$, and $\tau \mapsto \frac{\tau_\ell}{3}$ gives that $$h\left( \frac{r\tau_\ell}{3} \mp \frac13 ; \frac{\tau_\ell}{3}\right) = -e\left(\frac{-r^2}{6\tau} \right) e\left(\frac{\mp r}{3} \right) e\left(\frac{-r}{2} \right) \int_{0}^{i\infty} \frac{g_{r + \frac12, \pm\frac13 + \frac12}(z) dz}{\sqrt{-i\left(z+\frac{\tau_\ell}{3} \right)}}.$$ Thus $$\begin{gathered}
\notag
G_{\alpha,\beta}(\tau) = -\sum_{\pm} \pm \zeta_6^{\mp1}(-1)^{m} e\left( \frac{\mp m}{3}\right) \int_{0}^{i\infty} \frac{g_{\pm\frac13 + \frac12, \frac12 -r}(z) dz}{\sqrt{-i(z+3\tau)}} \\
+\sum_{\pm} \pm \zeta_{24}^{-\ell}(-1)^{m} e\left( \frac{\mp m}{3}\right)e\left(\frac{\mp r}{3} \right) e\left(\frac{-r}{2} \right) \sqrt{\frac{i}{3\tau}} \int_{0}^{i\infty} \frac{g_{r+ \frac12, \pm\frac13 + \frac12}(z) dz}{\sqrt{-i\left(z+\frac{\tau_\ell}{3} \right)}}.\end{gathered}$$
By a simple change of variables (let $z=\frac{\ell}{3} - \frac{1}{z}$) we can write $$\label{eq:int_convert}
\int_{0}^{i\infty} \frac{g_{r + \frac12, \pm\frac13 + \frac12}(z) dz}{\sqrt{-i\left(z+\frac{\tau_\ell}{3} \right)}} = -\sqrt{-3\tau} \int_{\frac{3}{\ell}}^{0} \frac{g_{r + \frac12, \pm\frac13 + \frac12}\left(\frac{\ell}{3} - \frac{1}{z} \right) dz}{z^{\frac32}\sqrt{-i(z+3\tau)}}.$$ Moreover, using Proposition \[prop\_Zg\] we can convert $$\begin{gathered}
\label{eq:g_convert}
g_{r + \frac12, \pm\frac13 + \frac12}\left(\frac{\ell}{3} - \frac{1}{z} \right) = \zeta_{24} ^\ell \cdot g_{r-\frac12, \pm\frac13 + \frac12}\left( \frac{-1}{z}\right) \\
= -\zeta_{24} ^\ell e\left(\frac18 \right) e\left(\frac{\mp1}{6} \right) e\left(\frac{\pm r}{3} \right) e\left(\frac{r}{2} \right) z^{\frac32} \cdot g_{\pm\frac13 + \frac12, \frac12 - r}(z).\end{gathered}$$ Thus by and we have that $$\begin{aligned}
G_{\alpha,\beta}(\tau) &=- \sum_{\pm} \pm \zeta_6^{\mp1}(-1)^{m} e\left( \frac{\mp m}{3}\right) \int_{0}^{i\infty} \frac{g_{\pm\frac13 + \frac12, \frac12 -r}(z) dz}{\sqrt{-i(z+3\tau)}} \notag \\
&\hspace{.5in}- \sum_{\pm} \pm \zeta_6^{\mp1}(-1)^{m } e\left( \frac{\mp m}{3}\right) \int_{\frac{3}{\ell}}^{0}\frac{g_{\pm\frac13 + \frac12, \frac12 -r}(z) dz}{\sqrt{-i(z+3\tau)}} \notag \\
&= -\sum_{\pm} \pm \zeta_6^{\mp1}(-1)^{m} e\left( \frac{\mp m}{3}\right) \int_{\frac{3}{\ell}}^{i\infty}\frac{g_{\pm\frac13 + \frac12, \frac12 -r}(z) dz}{\sqrt{-i(z+3\tau)}}. \label{Hab_integral}\end{aligned}$$
To complete the proof, one can deduce from Proposition \[prop\_Zg\] (2) that for $m\in{\mathbb Z}$, $$\begin{aligned}
g_{a,b}(\tau)=e(m a)g_{a,b-m}(\tau).\end{aligned}$$ Applying this to with a direct calculation gives us $$G_{\alpha,\beta}(\tau) = \sqrt{3}\sum_{\pm}\mp e\left(\mp\frac16\right) \int_{\frac{1}{\ell}}^{i\infty}\frac{g_{\pm\frac13 + \frac12, \frac12-3\frac{\alpha}{\beta}}(3z)}{\sqrt{-i(z+\tau)}}dz,$$ which is analytic on $\mathbb{R} - \{\frac{-1}{\ell}\}$ as desired.
Conclusion
==========
We have proven that when we restrict to vectors $\boldsymbol{\zeta_n}$ which contain distinct roots of unity, the mock modular form $q^{-\frac{1}{24}} R_n(\boldsymbol{\zeta_n};q)$ is also a quantum modular form. To consider the more general case where we allow roots of unity in $\boldsymbol{\zeta_n}$ to repeat, the situation is significantly more complicated. In this setting, as shown in [@F-K], the nonholomorphic completion of $q^{-\frac{1}{24}} R_n(\boldsymbol{\zeta_n};q)$ is not modular, but is instead a sum of two (nonholomorphic) modular forms of different weights. We will address this more general case in a forthcoming paper [@FJKS].
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D. Zagier, *Quantum modular forms*, Quanta of maths, 659-675, Clay Math. Proc., 11, Amer. Math. Soc., Providence, RI, 2010.
S. Zwegers, [*Mock theta functions*]{}, Ph.D. Thesis, Universiteit Utrecht, 2002.
S. Zwegers, [*Multivariable Appell functions*]{}, Preprint.
[^1]: Acknowledgements: The authors thank the Banff International Research Station (BIRS) and the Women in Numbers 4 (WIN4) workshop for the opportunity to initiate this collaboration. The first author is grateful for the support of National Science Foundation grant DMS-1449679, and the Simons Foundation.
[^2]: Here and throughout, as is standard in this subject for simplicity’s sake, we may slightly abuse terminology and refer to a function as a modular form or other modular object when in reality it must first be multiplied by a suitable power of $q$ to transform appropriately.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We prove that if a homeomorphism of a closed orientable surface $S$ has no wandering points and leaves invariant a compact, connected set $K$ which contains no periodic points, then either $K=S={\mathbb{T}}^2$, or $K$ is the intersection of a decreasing sequence of annuli. A version for non-orientable surfaces is given.'
address: 'Universidade Federal Fluminense, Instituto de Matemática, Rua Mário Santos Braga S/N, 24020-140 Niteroi, RJ, Brasil'
author:
- Andres Koropecki
bibliography:
- 'tesis.bib'
title: Aperiodic invariant continua for surface homeomorphisms
---
Introduction
------------
By *aperiodic invariant continuum* we mean a compact connected set which is invariant by some homeomorphism of a compact surface, and which contains no periodic points. We are interested in describing aperiodic invariant continua of non-wandering homeomorphisms. This type of sets appear frequently when studying generic area-preserving diffeomorphisms, due to a result of Mather [@mather-area], which states that for such diffeomorphisms, the boundary of certain open invariant sets (see Definition \[def:regular\]) is a finite union of aperiodic continua. Thus, having good topological information about aperiodic invariant continua is helpful to describe the dynamics of a $C^r$-generic area-preserving diffeomorphism. An example of this is the work of Franks and Le Calvez in [@franks-lecalvez] in the case that the surface is a sphere.
Our main result is the following
\[th:continuo\] Let $f\colon S\to S$ be a homeomorphism of a compact orientable surface such that $\Omega(f)=S$. If $K$ is an $f$-invariant continuum, then one of the following holds:
1. $f$ has a periodic point in $K$;
2. $K$ is annular;
3. $K=S={\mathbb{T}}^2$;
By *annular continuum* we mean an intersection of a nested sequence of topological annuli (see Definition \[def:annular\]). When $S$ is non-orientable, a version of Theorem \[th:continuo\] holds, however with two extra cases: $K$ could be a non-separating continuum in a Möbius strip, and in the case that $K=S$, the surface could be ${\mathbb{T}}^2$ or the Klein bottle (Corollary \[coro:non-orientable\]).
An important result of [@franks-lecalvez] states that for a generic area-preserving diffeomorphism of the sphere, the stable and unstable manifolds of hyperbolic periodic points are dense. This fact was generalized to an arbitrary surface by Xia [@xia-area], and one of the main steps of his proof is obtaining a version of Theorem \[th:continuo\] which assumes generic conditions on the (area-preserving) diffeomorphism and is restricted to continua which are the closure of a particular kind of open sets. Thus Theorem \[th:continuo\] extends Xia’s result to general homeomorphisms without wandering points (which includes area-preserving homeomorphisms), with no additional hypothesis on the continuum and no genericity conditions.
A question that motivates studying aperiodic invariant continua is the following:
\[q:1\] What are the possible obstructions to the transitivity of a $C^r$-generic area-preserving diffeomorphism?
Bonatti and Crovisier proved in [@bonatti-crovisier] that a $C^1$-generic area-preserving diffeomorphism of a compact manifold (of any dimension) is transitive. However, in dimension $2$, it is known that this is not true in the $C^r$ topology if $r$ is large enough, because of the KAM phenomenon: There are open sets of diffeomorphisms where a $C^r$-generic element has an elliptic periodic point surrounded by invariant circles (see, for instance, [@douady]), and this is an obstruction to transitivity. Hence the question is: is this the only possible obstruction? In other words, does the non-transitivity of a $C^r$-generic area-preserving diffeomorphism imply the existence of elliptic periodic points?
Studying Question \[q:1\], aperiodic invariant continua appear naturally as boundaries of invariant open sets. Theorem \[th:continuo\] implies that the presence of annular periodic continua is a necessary condition for the non-transitivity of a generic area-preserving diffeomorphism. In fact, a consequence of Theorem \[th:continuo\] is that the familly of aperiodic invariant continua which are minimal with respect to the property of being annular is pairwise disjoint (we call these continua *frontiers*, see [@k-m] for details). This allows a sort of decomposition of the dynamics in terms of the aperiodic invariant continua. Similar concepts appear in the work of Jäger [@jaeger] (where the word *circloid* is used instead of frontier) when studying nonwandering homeomorphisms of the torus with bounded mean motion.
In [@k-m], these observations play a fundamental role in the proof of the following result: for any $r\geq 1$, given a $C^r$-generic pair of area-preserving diffeomorphisms of a compact surface, the *iterated function system* (or, equivalently, the action of the semi-group) generated by them is transitive.
We should mention that the basic idea of the proof of Theorem \[th:continuo\] is inspired by the analogous result from [@franks-lecalvez] in the case where the surface is a sphere.
This article is organized as follows. In §1-5 we recall some background and results about ideal boundary points, continua, Lefschetz numbers and indices and we prove some elementary facts; in §5 we prove our main theorem, and a corollary about rotation numbers is mentioned; in §6 we state a version of the theorem for non-orientable surfaces, with an outline of the proof.
Acknowledgments {#acknowledgments .unnumbered}
---------------
I am grateful to M. Nassiri for motivating this problem, as well as L. N. Carvalho, J. Franks and E. R. Pujals for useful discussions.
Ideal boundary, continua, and complementary domains
---------------------------------------------------
If $U$ is a non-compact surface, a *boundary representative* of $U$ is a sequence $P_1\supset P_2\supset\cdots$ of connected unbounded ([i.e. ]{}not relatively compact) open sets in $U$ such that $\operatorname{\partial}_U P_n$ is compact for each $n$ and for any compact set $K\subset U$, there is $n_0>0$ such that $P_n\cap K=\emptyset$ if $n>n_0$. Two boundary representatives $\{P_i\}$ and $\{P_i'\}$ are said to be equivalent if for any $n>0$ there is $m>0$ such that $P_m\subset P_n'$, and vice-versa. The *ideal boundary* of $U$ is defined as the set $\operatorname{\rm{b}_I}U$ of all equivalence classes of boundary representatives. We denote by $U^*$ the space $U\cup \operatorname{\rm{b}_I}U$ with the topology generated by sets of the form $V \cup V'$, where $V$ is an open set in $U$ such that $\operatorname{\partial}_U V$ is compact, and $V'$ denotes the set of elements of $\operatorname{\rm{b}_I}U$ which have some boundary representative $\{P_i\}$ such that $P_i\subset V$ for all $i$. We call $U^*$ the *ideal completion* of $U$. Any homeomorphism $f\colon U\to U$ extends to a homeomorphism $f^*\colon U^*\to U^*$ such that $f^*|_{U} = f$. If $U$ is orientable and $\operatorname{\rm{b}_I}U$ is finite, then $U^*$ is a compact orientable boundaryless surface. See [@richards] and [@ahlfors-sario] for more details.
From now on, $S$ will denote a compact orientable surface. Let $U$ be an open connected subset of $S$. For each $p^*\in \operatorname{\rm{b}_I}U$, we write $Z(p^*)$ for the set $\operatorname{cl}_S(\bigcap_{V} V\cap U)$ where the intersection is taken over all neighborhoods $V$ of $p^*$ in $U^*$. It is easy to see that $Z(p^*)$ is a compact, connected, nonempty set (see [@mather-cara]).
\[def:regular\] We say that $U\subset S$ is a *complementary domain* if it is a connected component of the complement of some compact connected subset of $S$.
The next proposition is a direct consequence of [@mather-area Lemma 2.3].
If $U$ is a complementary domain in $S$, then it has finitely many ideal boundary points.
If $\operatorname{\rm{b}_I}U$ is finite, for each $p^*\in \operatorname{\rm{b}_I}U$ we may choose a neighborhood $V$ of $p$ such that ${\overline}{V}$ is homeomorphic to a closed disk, and such that ${\overline}{V}\cap \operatorname{\rm{b}_I}U = \{p\}$. Thus $V{\setminus}\{p\}$ is a topological annulus in $S$. And, unless $U$ is a topological disk, the boundary of $V$ is an essential simple closed curve in $S$. From this, we have
\[pro:regular-surface\] If $U$ is a complementary domain in $S$, then $\operatorname{\rm{b}_I}U$ is finite, and there is a compact bordered surface $S_U\subset U$ such that $U{\setminus}S_U$ has finitely many connected components, each of which is homeomorphic to an open annulus.
\[coro:regular-boundary\] If $U$ is a complementary domain in $S$, then $\operatorname{\partial}U$ has finitely many connected components.
Choose $K\subset U$ such that $U{\setminus}K$ is a finite union of disjoint annuli. If $A$ is a connected component of $U{\setminus}K$, then $\operatorname{\partial}A{\setminus}U$ is connected (since it is $Z(p^*)$ for some $p^*\in \operatorname{\rm{b}_I}U$), and $\operatorname{\partial}U = \bigcup_A \operatorname{\partial}A{\setminus}U$, where the union is taken over all connected components $A$ of $U{\setminus}K$. Since these are finitely many, the claim follows.
We remark that the number of boundary components of $U$ may be smaller than the number of ideal boundary points, since the sets $Z(p^*)$, $p^*\in \operatorname{\rm{b}_I}U$ need not be disjoint.
### Continua
By a *continuum* we mean a compact connected set.
\[lem:inf-cc-disk\] Let $K$ be a continuum and $\mathcal{U}$ the family of all connected components of $S{\setminus}K$. Then all but finitely many elements of $\mathcal{U}$ are simply connected.
We consider two cases. First suppose that for some $U\in {\mathcal}{U}$, there is a simple closed curve $\gamma$ which is homotopically nontrivial in $U$ but trivial in $S$. Let $D$ be the topological disk bounded by $\gamma$ in $S$. Since $\gamma$ is nontrivial in $U$, there is some point of $K$ in $D$. Since $K$ is connected and $\gamma \subset S{\setminus}K$, it follows that $K\subset D$. Thus if $U'\in {\mathcal}{U}$ and $U'\neq U$, then $U'\subset D$. From this we conclude that $U'$ is simply connected. Indeed, if $\gamma'$ is a homotopically nontrivial simple closed curve in $U'$, then by a similar argument it bounds a disk $D'\subset D$ which intersects $K$, so $K\subset D'$. But this implies that $S{\setminus}D'\subset U'$ (because it is connected) so $U'=U$, a contradiction. Therefore, all but one element of ${\mathcal}{U}$ are simply connected.
Now suppose that for every $U\in {\mathcal}{U}$, if $\gamma$ is homotopically nontrivial in $U$ then it is also homotopically nontrivial in $S$, and assume that there are infinitely many complementary domains $U_1, U_2, \dots$ of $K$ which are not simply connected. For each $U_i$, let $\gamma_i$ be a simple homotopically nontrivial simple closed curve in $U_i$. By our assumption, $\gamma_i$ is also nontrivial in $S$. The curves $\{\gamma_i:i\in {\mathbb{N}}\}$ are pairwise disjoint, so there must be infinitely many of them in the same homotopy class of $S$. But if, say, $\gamma_1$, $\gamma_2$ and $\gamma_3$ are all homotopic and disjoint, there are two disjoint annuli $A_1$ and $A_2$ such that (up to reordering the indices) $\operatorname{\partial}A_1 = \gamma_1\cup \gamma_2$ and $\operatorname{\partial}A_2=\gamma_2\cup \gamma_3$. Since the boundary of $A_1$ contains points of two different connected components of $S{\setminus}K$, it is clear that $A_1$ must intersect $K$. Since $K$ is connected, it follows that $K\subset A_1$. But with the same argument we also conclude that $K\subset A_2$, a contradiction. This completes the proof.
### Annular continua
\[def:annular\] A continuum $K\subset S$ is said to be *annular* if it has a neighborhood $A\subset S$ homeomorphic to an open annulus such that $A{\setminus}K$ has exactly two components, both homeomorphic to annuli. We call any such $A$ an *annular neighborhood* of $K$.
This definition is equivalent to saying that $K$ is the intersection of a sequence $\{A_i\}$ of closed topological annuli such that $A_{i+1}$ is an essential subset of $A_i$ ([i.e. ]{}it separates the two boundary components of $A_i$), for each $i\in {\mathbb{N}}$.
Indices and Lefschetz number
----------------------------
If $f$ is a homeomorphism and $D$ is a closed topological disk without fixed points in its boundary, we denote by $\mathrm{Ind}_f(D)$ the fixed point index of $f$ in $D$. (see [@dold]). If there are finitely many fixed points of $f$ in $D$, then $\mathrm{Ind}_f(D)$ is equal to the sum of the Lefschetz indices of these fixed points. If $D_1,\dots,D_n$ are disjoint disks such that the set of fixed points of $f$ is contained in the interior of their union, then we have the Lefschetz formula: $$\sum_{i=1}^n \mathrm{Ind}_f(D_i) = L(f)$$ where $L(f)$ denotes the Lefschetz number of $f$.
\[lem:lefschetz\] Let $S$ be an orientable closed surface with Euler characteristic $\chi(S)\leq 0$. Then, for any homeomorphism $f\colon S\to S$ there is $n>0$ such that the Lefschetz number of $f^n$ is non-positive: $L(f^n)\leq 0$.
When $\chi(S)<0$, a proof can be found in [@xia-area]. If $\chi(S)=0$, then $S\simeq {\mathbb{T}}^2$, and the automorphism induced by $f$ on $H_1(S,{\mathbb{Q}})$ can be represented by a matrix $A\in \operatorname{\rm{SL}}(2,{\mathbb{Z}})$. It is well known that any such matrix is either periodic ($A^{n}=I$ for some $n>0$, so $\operatorname{\rm{tr}}(A^n)=2$), parabolic (and then $\operatorname{\rm{tr}}(A^2)= 2$) or hyperbolic (and then $\operatorname{\rm{tr}}(A^2)>2$). In either case, there is $n$ such that $L(f^n)= 2-\operatorname{\rm{tr}}(A^n)\leq 0$.
Wandering points
----------------
Given a homeomorphism $f\colon S\to S$, we say that a nonempty open set $U$ is *wandering* if $f^n(U)\cap U=\emptyset$ for all $n>0$ (or, equivalently, for all $n\neq 0$). We denote by $\Omega(f)$ the set of non-wandering points of $f$. That is, the (compact, invariant) set of points which have no wandering neighborhood.
\[rem:wandering\] We will use the following observations several times:
1. If $\Omega(f)=S$, then $\Omega(f^n)=S$. To see this, given a nonempty open set $U_0$ we can define recursively $U_{i+1}= f^{k_{i+1}}(U_i)\cap U_i$ where $k_{i+1}>0$ is chosen such that the intersection is nonempty. Then there are integers $i_1<i_2<\cdots<i_n$ such that $k_{i_1}=k_{i_2}=\cdots = k_{i_n} (\mathrm{mod}\, n)$, so that $k_{i_1}+\cdots+k_{i_n}=mn$ for some $m>0$, and it is easy to verify that $f^{mn}(U_0)\cap U_0\neq \emptyset$.
2. If $\Omega(f)=S$ and $\{U_i\}_{i\in{\mathbb{N}}}$ is a family of pairwise disjoint open sets which are permuted by $f$ ([e.g. ]{}the connected components of the complement of a compact periodic set) then each $U_i$ is periodic for $f$.
\[lem:index-bd\] Let $D\subset S$ be a topological open disk and $f\colon {\overline}{D}\to {\overline}{D}$ a homeomorphism. Suppose that there is a neighborhood of $\operatorname{\partial}D$ in ${\overline}{D}$ which does not contain the positive or the negative orbit of any wandering open set, and $f$ has no fixed points in $\operatorname{\partial}D$. Then the index of the set of fixed points of $f$ in $D$ is $1$. In other words, there is a closed topological disk $D'$ which contains all fixed points of $f$ in $D$, such that $\mathrm{Ind}_f(D')=1$.
Since it contains no fixed points, $\operatorname{\partial}D$ is not reduced to a single point. By a theorem of Cartwright and Littlewood [@cartwright-littlewood-2] (see also [@franks-lecalvez Proposition 2.1]), the extension $\hat{f}$ of $f|_D$ to the prime ends compactification $\hat{D}$ of $D$ has no fixed points in the boundary circle $\operatorname{\partial}{\hat{D}}$. Thus $\hat{f}$ is orientation-preserving, and $\mathrm{Ind}_{\hat{f}}(\hat{D})=1$, and since fixed points of $\hat{f}$ are in a compact subset of $D$, we can choose a closed disk $D'\subset D$ containing all fixed points of $\hat{f}$, so that $\mathrm{Ind}_{\hat{f}}(D')=\mathrm{Ind}_{\hat{f}}(\hat{D})=1$. But since $D'\subset D$ and $\hat{f}|_{D'}=f|_{D'}$, it follows that $\mathrm{Ind}_{\hat{f}}(D')=\mathrm{Ind}_{f}(D')$ and we are done.
Main theorem
------------
We begin with a brief outline of the proof. The idea is to generalize the index argument used in [@franks-lecalvez] for the case of the sphere. However, to do that we need to modify the underlying manifold: we consider the (possibly infinitely many) connected components of $S{\setminus}K$. The non-wandering hypothesis guarantees that these components are permuted by $f$. Since these components are complementary domains, they have finitely many ends. Next we “remove” every nontrivial component (except for a neighborhood of its boundary), leaving a bordered submanifold $N$ of $S$ which is a neighborhood of $K$. We can modify $f|_N$ obtaining a map which coincides with $f$ in a neighborhood of $K$, but which leaves the boundary of $N$ invariant. After collapsing the boundary circles of $N$ to points, we obtain a new compact surface containing $K$, and a homeomorphism which has no periodic points on $K$, and by a Lefschetz index argument we conclude that this surface can only be a sphere. From this we conclude easily that $K$ is annular.
If $K$ is an aperiodic invariant continuum and $K\neq S$, then Theorem \[th:continuo\] implies that $K$ is annular. Following [@franks-lecalvez §3] (using a small annular neighborhood $A$ of $K$, and lifting $f$ to the universal covering of $A$) one can define the rotation set $\rho_f(K)\subset {\mathbb{R}}$ (which is defined modulo integer translations). Now, with almost no modifications, the proof of [@franks-lecalvez Proposition 5.2] remains valid. Thus we obtain the following
If $f\colon S\to S$ is an area preserving homeomorphism and $K\subsetneq S$ is an invariant continuum with no periodic points, then $K$ is annular, $\rho_{f}(K)$ consists of a single irrational number $\alpha$, and the rotation numbers in the prime ends from both sides of $K$ coincide (up to a sign change) with $\alpha$.
### Proof of Theorem \[th:continuo\]
We may assume that $f$ is orientation-preserving (otherwise consider $f^2$ instead of $f$). If $K=S$ and $f$ has no periodic points, then $S={\mathbb{T}}^2$ by the Lefschetz theorem, and we are done. Now suppose that $f$ has no periodic points in $K$ and $K\neq S$. We need to show that $K$ is annular.
Consider the family ${\mathcal}{V}$ of connected components of $S{\setminus}K$ which are not topological disks, which is finite by Proposition \[lem:inf-cc-disk\]. Since open sets are nonwandering, each element of ${\mathcal}{V}$ is periodic by $f$. Choosing a power of $f$ instead of $f$ we may (and we do from now on) assume that each element of ${\mathcal}{V}$ is fixed by $f$.
Since each $V\in {\mathcal}{V}$ is a complementary domain, by Proposition \[pro:regular-surface\] we can choose a compact surface with boundary $S_V\subset V$ such that $V{\setminus}S_V$ has finitely many components, all of which are annuli.
Given $V\in {\mathcal}{V}$, the ideal boundary points of $V$ are periodic by $(f|_V)^*$, so by taking a power of $f$ instead of $f$ we may assume that they are in fact fixed. This implies that if $\gamma$ is a sufficiently small closed loop in $V$ which bounds a disk containing $p^*$ in $V^*$, then $f(\gamma)$ is homotopic to $\gamma$ in $V$ (and thus in S). Moreover, $f(Z(p^*)) = Z(p^*)$ for any $p^*$ in $\operatorname{\rm{b}_I}V$. Note also that $Z(p^*)\subset K$ for all $p^*\in \operatorname{\rm{b}_I}V$.
Let $A_1,\dots, A_n$ be the connected components of $V{\setminus}S_V$. Each $A_i$ is a topological annulus, whose boundary in $S$ is given by a loop $\gamma_i$ and the continuum $Z_i = {\overline}{A}_i\cap K$ (which is $Z(p^*)$ for some $p^*\in V^*$). Since $f(Z_i)=Z_i$, if $\sigma_i \subset A_i$ is an essential simple closed curve close enough to $Z_i$, we have that $f(\sigma_i)\subset A_i$. Since $f(\sigma_i)$ is homotopic to $\sigma_i$ in $A_i$, there exists a homeomorphism $h_i\colon A_i\to A_i$ which maps $f(\sigma_i)$ to $\sigma_i$ and which is the identity in a neighborhood of the boundary of $A_i$; furthermore, we may assume that $h_i(x)=f^{-1}(x)$ for $x\in f(\sigma_i)$ (see [@epstein]). Extending $h_i$ to the identity outside $A_i$, and letting ${\tilde}{f}=h_1\dots h_nf$, we get an orientation preserving homeomorphism such that ${\tilde}{f}(x)=f(x)$ for $x\in S{\setminus}\cup_i A_i$ and ${\tilde}{f}(\sigma_i)=\sigma_i$. If ${\tilde}{S}_V$ is the surface bounded by $\sigma_1,\dots, \sigma_n$ which intersects $S_V$, we have that ${\tilde}{f}({\tilde}{S}_V)={\tilde}{S}_V$ and ${\tilde}{f}$ is the identity on the boundary of ${\tilde}{S}_V$. We do this for each $V\in {\mathcal}{V}$, and finally we consider the boundaryless compact surface ${\tilde}{S}$ obtained by collapsing each boundary circle of $S{\setminus}\operatorname{\rm{int}}\cup_{V\in {\mathcal}{V}}{{\tilde}{S}_V}$ to a point, and the induced homeomorphism which we still call ${\tilde}{f}$, for which these points are fixed (see figure \[fig1\]). This new surface contains $S{\setminus}\cup{\mathcal}{V}$, and ${\tilde}{f}$ coincides with $f$ on that set. Each $V\in {\mathcal}{V}$ was replaced by a (finite) union of one or more invariant topological disks, and the boundary of each of these disks is contained in $K$ (and hence, it contains no periodic points).
Since ${\mathcal}{V}$ consists of all components of $S{\setminus}K$ which are not disks, from our construction we see that all components of ${\tilde}{S}{\setminus}K$ are topological disks.
Suppose that $\chi({\tilde}{S})\leq 0$. Then by Lemma \[lem:lefschetz\] there is $n$ such that $L({\tilde}{f}^n)\leq 0$. Let $D$ be a connected component of ${\tilde}{S}{\setminus}K$ such that $f^n(D)=D$. We know that ${\tilde}{f}^n$ coincides with $f^n$ in a neighborhood of $\operatorname{\partial}D\subset K$, so the fact that $f^n$ has no wandering points (and no fixed points in $K$) implies that the hypotheses of Lemma \[lem:index-bd\] hold. Hence, the fixed point index of ${\tilde}{f}^n$ in $D$ must be $1$ (in particular, $D$ contains a fixed point). From this, it follows that there are finitely many ${\tilde}{f}^n$-invariant components in ${\tilde}{S}{\setminus}K$. In fact, if there were infinitely many, then one could find a sequence of fixed points accumulating in $K$, which contradicts the aperiodicity of $K$. Moreover, we may assume that there is at least one such component (by starting with an appropriate power of $f$ instead of $f$).
Since ${\tilde}{f}^n$ has no fixed points in $K$, denoting the components of ${\tilde}{S}{\setminus}K$ which are ${\tilde}{f}^n$-invariant by $D_1,\,\dots,\, D_k$, we have from the Lefschetz formula $$L({\tilde}{f}^n) = \sum_{i=1}^k \mathrm{Ind}_{{\tilde}{f}^n}(D_i) = k \geq 1,$$ which contradicts our choice of $n$. From this we conclude that $\chi({\tilde}{S})>0$, hence ${\tilde}{S}$ is a sphere.
But then, since ${\tilde}{f}$ preserves orientation, $L({\tilde}{f}^m)=\chi({\tilde}{S})=2$ for all $m$. This implies that ${\tilde}{S}{\setminus}K$ consists of exactly two components $D_1$ and $D_2$. In fact, if there were more than two such components, it would be possible to choose $m$ such that ${\hat}{f}^m$ leaves three or more of those components fixed, so that, repeating our previous argument, $L({\tilde}{f}^m)\geq 3$, contradicting our previous claim.
Since $D_1$ and $D_2$ are topological disks, each of them is the union of an increasing sequence of closed topological disks, so that $K$ is the intersection of a decreasing sequence of annuli $\{A_n\}$. These annuli are eventually contained in any neighborhood of $K$, which means that, for some $n_0$, $\{A_n\}_{n\geq n_0}$ is a decreasing sequence of annuli in the original surface $S$, and $\cap_{n\geq n_0} A_n=K$. Thus $K$ is annular in $S$. This completes the proof.
Non-orientable case of Theorem \[th:continuo\]
----------------------------------------------
\[coro:non-orientable\] Let $f\colon S\to S$ be a homeomorphism of the closed non-orientable surface $S$, such that $\Omega(f)=S$. If $K$ is an $f$-invariant continuum, then one of the following holds:
1. $f$ has a periodic point in $K$;
2. $K$ is annular;
3. $K$ is the intersection of a nested sequence of Möbius strips;
4. $K=S=$ Klein bottle.
We consider the oriented double covering $\pi\colon \hat{S}\to S$, and a lift $\hat{f}\colon \hat{S}\to \hat{S}$ of $f$. Since $f$ has no wandering points, that must be true of $\hat{f}$ as well. In fact, if $\hat{U}\subset \hat{S}$ is a sufficiently small open set, then $\pi^{-1}(\pi(\hat{U})) = \hat{U}\cup \hat{U}'$ where the union is disjoint and $\hat{U}'$ is homeomorphic to $\hat{U}$. If $n> 0$ is such that $f^n(\pi(U))\cap \pi(U)\neq \emptyset$ then either $\hat{f}^n(\hat{U})\cap \hat{U}\neq \emptyset$ or $\hat{V}'=\hat{f}^n(\hat{U})\cap \hat{U}'\neq \emptyset$. If the latter case holds, then again $\pi^{-1}(\pi(\hat{V}'))$ is the disjoint union of $\hat{V}'$ and $\hat{V}$, where $\hat{V}\subset \hat{U}$, and there is $m>0$ such that $\hat{f}^m(\hat{V})\cap \hat{V}' \neq \emptyset$ (which implies that $\hat{f}^{m+n}(\hat{U})\cap \hat{U})\neq \emptyset$) or $\hat{f}^m(\hat{V})\cap \hat{V} \neq \emptyset$ (which implies $\hat{f}^m(\hat{U})\cap \hat{U}\neq \emptyset$), so $\hat{U}$ is nonwandering.
Now $\pi^{-1}(K)$ consists of either a unique connected $\hat{f}$-invariant set or two copies of $K$ which are invariant by $\hat{f}$ if the lift is chosen appropriately. Let $\hat{K}$ be one of those components (or the unique component if there is only one). If $K$ has no periodic points of $f$, then $\hat{f}$ cannot have a periodic point in $\hat{K}$, because periodic points of $\hat{f}$ project to periodic points of $f$. Thus we are in the setting of Theorem \[th:continuo\], and we conclude that either $\hat{K}$ is annular or $\hat{S}={\mathbb{T}}^2$. In the latter case, it follows that $S$ is a Klein bottle. In the former case, we have a decreasing sequence of topological annuli $\{\hat{A}_i\}_{i\in {\mathbb{N}}}$ such that $\hat{K} = \bigcap_i \hat{A}_i$. The sets $\hat{A}_i$ project to a decreasing sequence of neighborhoods $\{A_i\}_{i\in {\mathbb{N}}}$ of $K$, each of which is either homeomorphic to an annulus (in which case it projects injectively) or to a Möbius strip, and it is easy to see that $K = \bigcap_i A_i$. By taking a subsequence of $\{A_i\}_{i\in {\mathbb{N}}}$ if necessary, we see that either $(2)$ or $(3)$ must hold.
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abstract: 'It is shown that the $R$-parity violating decays of the lighter top squarks ($\lstop$) triggered by the lepton number violating couplings $\lambda^{\prime}_{i33}$, where the lepton family index i = 1-3, can be observed at the LHC via the dilepton di-jet channel even if the coupling is as small as 10$^{-4}$ or 10$^{-5}$, which is the case in several models of neutrino mass, provided it is the next lightest supersymmetric particle(NLSP) the lightest neutralino being the lightest supersymmetric particle(LSP). We have first obtained a fairly model independent estimate of the minimum observable value of the parameter ($P_{ij} \equiv BR(\widetilde t \ra l_i^+ b) \times BR(\widetilde t^* \ra l_j^- \bar b$)) at the LHC for an integrated luminosity of 10fb$^{-1}$ as a function of $\mlstop$ by a standard Pythia based analysis. We have then computed the parameter $P_{ij}$ in several representative models of neutrino mass constrained by the neutrino oscillation data and have found that the theoretical predictions are above the estimated minimum observable levels for a wide region of the parameter space.'
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[**Probing $R$-parity violating models of neutrino mass at the LHC via top squark decays** ]{} 0.4cm Amitava Datta$^{(a)}$[^1] and Sujoy Poddar$^{(a)}$[^2] 0.1cm [*$^{(a)}$ Indian Institute of Science Education and Research, Kolkata,\
HC-VII, Sector III, Salt Lake City, Kolkata 700 106, India.\
*]{}
Introduction {#intro4}
============
Neutrino oscillation experiments[@other] have confirmed that neutrinos indeed have very tiny masses, several orders of magnitude smaller than any other fermion mass in the Standard Model (SM). The tiny masses of the neutrinos, however small, provide evidences of new physics beyond the SM.
Neutrinos can be either Dirac fermions or Majorana fermions depending upon whether the theory is lepton number conserving or violating. In the SM, as originally proposed by Glashow, Salam and Weinberg, neutrinos are massless since right handed neutrinos and lepton number violating terms are not included.
Both $R$-parity conserving (RPC) or $R$-parity violating (RPV) minimal supersymmetric extension of the SM (MSSM)[@susyrev] are attractive examples of physics beyond the SM. In general the MSSM may contain RPC as well as RPV couplings. The latter include both lepton number and baryon number violating terms which result in catastrophic proton decays. One escape route is to impose $R$-parity as a symmetry which eliminates all RPV couplings. This model is generally referred to as the RPC MSSM. However, neutrino masses can be naturally introduced in this model only if it is embeded in a grand unified theory (GUT) [@GUT]. Tiny Majorana neutrino masses are then generated by the see-saw mechanism [@seesaw]. Proton decay is a crucial test for most of the models belonging to this type.
However, an attractive alternative for generating Majorana masses of the neutrinos without allowing proton decay is to impose a discrete symmetry which eliminates baryon number violating couplings from the RPV sector of the MSSM but retains the lepton number violating ones. The observation of neutrinoless double beta decay[@double] and absence of proton decay may be the hallmark of such RPV models of neutrino mass.
The GUT based models though very elegant have hardly any unambiguous prediction which may tested at the large hadron collider(LHC). In contrast the RPV models of neutrino mass are based on TeV scale physics and, consequently, have many novel collider signatures.
The observables in the neutrino sector not only depend on the RPV parameters but also on the RPC ones like the masses of the superpartners generically called sparticles. Thus the precise determination of the neutrino masses and mixing angles in neutrino oscillation experiments together with the measurement of sparticle masses and branching ratios (BRs) at collider experiments can indeed test the viability of the RPV models quantitatively. Moreover the collider signatures of this model are quite distinct from that of the RPC model. In this paper our focus will be on a novel signature of a RPV model of $\nu$ mass which can be easily probed at the early stages of the upcoming LHC experiments.
In the RPC models the lightest supersymmetric particle (LSP) decays into lepton number violating channels producing signals with high multiplicity but without much missing energy which are in sharp contrast with the signals in a typical RPC model. In RPV MSSM the sparticles other than the LSP can also directly decay via lepton number violating channels which may lead to spectacular collider signatures. However, in a typical model of neutrino mass consistent with the oscillation data such couplings turn out to be so small[@rpv] that the RPC decay of the sparticles overwhelm the RPV decays. Thus the LSP decay is the only signature of $R$-parity violation.
However, the scenario changes dramatically if we consider the direct RPV decay of the lighter top squark ($\lstop$) [@nmass; @biswarup; @naba; @shibu] with the assumption that $\lstop$ it is the next lightest supersymmetric particle(NLSP) while the lightest neutralino ($\lspone$) is the LSP. The theoretical motivation for the $\lstop$-NLSP scenario is the fact that it’s superpartner - the top quark- is much heavier than any other matter particle in SM. This large top mass ($m_t$) leads to a spectacular mixing effect in the top squark mass matrix which suppresses the mass of the lighter eigenstate [@susyrev]. We assume that $\lstop$-NLSP decays via the loop induced mode $\lstop \ra c \lspone$ [@hikasa] and the four body[@boehm] decay mode, which occurs only in higher order of perturbation theory, with significant BR. The validity of this assumption will be justified later. The RPV decays can now naturally compete with the RPC ones in spite of the fact that couplings underlying the former modes are highly suppressed by the $\nu$ oscillation data [@global].
The lighter top squark decays into a lepton and a $b$-jet via RPV couplings $\lambda_{i33} '$ are listed below: a) l\_i\^+ b ; b) l\_i\^- |b where $i$=1-3 corresponds to $e$, $\mu$ and $\tau$ respectively. Our signal consists of opposite sign dileptons(OSDL), two hard jets with very little $\met$. These modes dominate, e.g., in many RPV models where neutrino masses are generated at the one loop level by the $\lambda'_{i33}$ couplings, where i is the lepton index and 3 stands for quarks or squarks belonging to the third generation (see below).
We take the lowest order QCD cross section of top squark pair production which depends on $\mlstop$ only. Requiring that the significance of the signal over the SM background be at least 5 $\sigma$ level for an integrated luminosity of 10 $fb^{-1}$ or smaller, we can then put fairly model independent lower limits on the products of the BRs (PBRs) of the RPV decay modes in Eq. 1. In our analysis both the signal and the backgrounds are simulated with Pythia. As expected the range of $\mlstop$ which can be probed at the LHC is significantly larger compared to the reach of Tevatron RUN II[@cdf; @admgspd]. The details of our simulations will be presented in the next section.
In principle the viability of probing any RPV model of neutrino mass with the above characteristics at the LHC can be checked by computing PBRs in respective models, and comparing with the estimated lower limits. For the purpose of illustration we have considered in section 3 a model based on three bilinear RPV couplings ($\mu_i$) and three trilinear couplings ($\lambda'_{i33}$) at the weak scale [@abada] and have carried out the above check. It is gratifying to note that most of the parameter space allowed by the neutrino oscillation data can be probed by the early LHC experiments with an integrated luminosity of 10 fb$^{-1}$ (see section 3). Moreover, the constraints from oscillation data indicate that the $\lambda'_{i33}$ couplings should have certain hierarchical pattern leading to distinct collider signatures [@adspdsp] . This hierarchy among the couplings can be qualitatively tested by observing the relative sizes of signals involving different OSDL signals.
The summary, the conclusions and future outlooks are in the last section.
The signals and the SM backgrounds {#result}
===================================
The production and decay of the lighter top squark pairs are simulated by Pythia[@pythia]. Initial and final state radiation, decay, hadronization, fragmentation and jet formation are implemented following the standard procedures in Pythia. We have considered only the RPV decay modes of $\lstop$ via the couplings $\lambda '_{i33}$ ,i = 1-3 (Eq. 1) and in this section their BRs are taken to be free parameters. We have used the toy calorimeter simulation (PYCELL) in Pythia with the following criteria:
- The calorimeter coverage is $\vert \eta \vert < 4.5$. The segmentation is given by $\Delta \eta \times \Delta \phi = 0.09 \times 0.09$ which resembles a generic LHC detector.
- A cone algorithm with $\Delta R(j,j)$ = $\sqrt {\Delta\eta^2 +
\Delta\phi^2}= 0.5 $ has been used for jet finding.
- Jets are ordered in E$\mathrm{_T}$ and E$^{\mathrm{jet}}_{\mathrm{T,min}} = 30 $GeV.
Various combinations of OSLDs in the final state are selected as follows:
- Only tau leptons decaying into hadrons are selected provided the resulting jet has P$\mathrm{_T \ge 30}$ GeV and $\vert\eta \vert < 3.0$.
- Leptons $(l=e,\mu)$ are selected with P$\mathrm{_T \ge 20}$ GeV and $\vert\eta \vert < 2.5$.
The following selection criteria(SC) are used for background rejection :
- The $\tau$-jets are tagged according to the tagging efficiencies provided by the CMS collaboration[@cms](Fig. 12.9)(SC1). Hadronic BR of the $\tau$ is also included in the corresponding efficiency. For $e$ and $\mu$ SC1 is the lepton-jet isolation cut. We require $\Delta R(l,j) > 0.5$.The detection efficiency of the leptons are assumed to be approximately $ 100 \%$ for simplicity.
- Events with two isolated leptonic objects (e,$\mu$ or tagged $\tau$-jets ) are rejected if P$\mathrm{_T \le 150}$ GeV, where $l$ = $e$ or $\mu$ (SC2) or $E^{V(\tau)}_T$ $< $100 GeV, where $E^{V(\tau)}_T$ is the $E_T$ of the $\tau$ jet.
- We select events with exactly two jets other than the tagged $\tau$-jets (SC3). The event is rejected if the additional jets have P$\mathrm{_T \le 100}$ GeV (SC 4). [^3]
- Events with missing transverse energy ($\etslash) > 60$ GeV are rejected (SC5).
Through SC1 we have severely constrained the transverse momentum of two leptons $l=e,\mu$ to reject the leptons coming from the leptonic decays of the tau. Moreover such a strong cut reduces most of the SM backgrounds significantly. We have considered backgrounds from: $ W W, W Z, Z Z, t \bar t$, Drell-Yan (DY) and QCD events. The missing energy veto plays a crucial role to tame down $W W$ and $t \bar t$ backgrounds as they are rich in missing energy. Mistagging of light jets as $\tau$-jets is a major source of background to di-tau events. We have taken this into account. However, if we also employ $b$-tagging then this background can be brought under control to some extent.
In our work $b$ tagging has been implemented according to the following prescription. A jet with $\vert \eta \vert < 2.5$ matching with a B-hadron of decay length $ > 0.9 ~\mathrm{mm} $ has been marked $ tagged$. The above criteria ensures that $\epsilon _{b}
\simeq 0.5$ in $ t \bar t$ events, where $\epsilon _{b}$ is the single $b$-jet tagging efficiency (i.e., the ratio of the number of tagged $b$-jets and the number of taggable $b$-jets in $t \bar t$ events).
The leading order(LO) cross-sections for $\lstop-\lstop^*$ pairs presented in Table 1 are computed using calcHEP (version 2.3.7)[@calchep].
Signal 240 300 400 450 500
-------------- ------ ----- ----- ------ ------
$\sigma(pb)$ 14.6 4.8 1.1 0.58 0.32
: $\lstop$ - $\lstop^*$ pair production cross section ($\sigma$) at the LHC for different $\mlstop$.
In Table 2 we have presented the combined efficiencies of SC1 - SC5 in steps. The first column of Table 2 shows signals with different topology of final states. Here $e~e~X$, $\tau
\tau X$, $e \tau X$ and $ e \mu X$ represent final states without b-jet tagging. The cumulative efficiency of each SC for $\mlstop$ = 400 GeV is presented in the next five columns. However, we have not separately presented the efficiencies corresponding to final states with muons as we have assumed that both $e$ and $\mu$ are detected with approximately 100$\%$ efficiency.
Table 3 contains the effect of b-jet tagging on different final states. We have used the notations $0b$, $1b$ and $2b$ to specify signal events with zero, one and two tagged b-jets respectively. From this Table it is also evident that the efficiencies increase for larger $\mlstop$ since the $P_T$ cut on leptons become less severe. This compensates the fall of the cross section with increasing $\mlstop$ to some extent.
----------------------------------------------------------------------------------------------
$\lstop \lstopbar$ $\epsilon_1$ $\epsilon_2$ $\epsilon_3$ $ $\epsilon_5$
\epsilon_4$
-------------------- -------------- -------------- -------------- ------------- --------------
$e e X$ 0.93708 0.292239 0.087228 0.043344 0.032823
$\tau \tau X$ 0.251343 0.111546 0.033201 0.031554 0.008955
$e \mu X$ 0.94101 0.295239 0.088216 0.043415 0.033060
$e \tau X$ 0.474948 0.180945 0.053820 0.044793 0.016965
----------------------------------------------------------------------------------------------
: Efficiency table for $\mlstop =400 \gev.$
------------------ ---------- --------- --------- --------- ---------
$\mlstop(\gev)$ 240 300 400 450 500
$e~ e~ 0b$ 0.00032 0.00066 0.00189 0.00234 0.00255
$e~ e~ 1b$ 0.00121 0.00330 0.01116 0.01461 0.01580
$e~ e~ 2b$ 0.00176 0.00509 0.01984 0.02620 0.03121
$ e~ e ~X$ 0.00328 0.00905 0.03282 0.04315 0.04957
$\tau~ \tau~ 0b$ 0.00059 0.00073 0.00112 0.00091 0.00097
$\tau~ \tau~ 1b$ 0.00153 0.00284 0.00363 0.00351 0.00391
$\tau~ \tau~ 2b$ 0.00126 0.00226 0.00421 0.00450 0.00522
$ \tau~ \tau ~X$ 0.00338 0.00582 0.00896 0.00892 0.01098
$\tau~ e~ 0b$ 0.00045 0.00081 0.00148 0.00142 0.00142
$\tau~ e~ 1b$ 0.00135 0.00307 0.00667 0.00705 0.00717
$\tau~ e~ 2b$ 0.00126 0.00346 0.00882 0.00997 0.01078
$ \tau~ e ~X$ 0.00308 0.00734 0.01697 0.01843 0.01936
$\mu~ e~ 0b$ 0.000315 0.00067 0.00190 0.00235 0.00257
$\mu~ e~ 1b$ 0.00123 0.00334 0.01125 0.01469 0.01635
$\mu~ e~ 2b$ 0.00178 0.00512 0.01992 0.02625 0.03129
$ \mu~ e ~X$ 0.00332 0.00912 0.03306 0.04329 0.05021
------------------ ---------- --------- --------- --------- ---------
: Final efficiencies for different $\mlstop$ (including b-tagging if implemented).
--------------------------------------------------------------------------------------------------------------------------------------
$t \bar t$ $\epsilon_1$ $\epsilon_2$ $\epsilon_3$ $ $\epsilon_5$
\epsilon_4$
------------- ----------------------- ----------------------- ------------------------ ----------------------- -----------------------
$e e$ $7.63 \times 10^{-3}$ $2.22\times 10^{-5}$ $ 5.70 \times 10^{-6}$ $7.00 \times 10^{-7}$ $1.00 \times 10^{-7}$
$\tau \tau$ $4.76\times 10^{-4}$ $4.00\times 10^{-6}$ $1.50 \times 10^{-6}$ $1.00\times 10^{-6}$ $4.00 \times 10^{-7}$
$e \mu$ $7.74 \times 10^{-3}$ $2.01 \times 10^{-5}$ $6.01 \times 10^{-6}$ $6.80 \times 10^{-7}$ $5.0 \times 10^{-7}$
$e \tau$ $1.88 \times 10^{-3}$ $9.3\times 10^{-6}$ $2.95 \times 10^{-5}$ $ 9.50\times 10^{-7}$ $2.00 \times 10^{-7}$
--------------------------------------------------------------------------------------------------------------------------------------
: Efficiency table for $t \bar t$ process
In Table 4 we have shown the effect of cuts on the background from $t \bar t$ events. SC2 is very effective in reducing this background significantly. Moreover this background is accompanied by large amount of $\met$ and SC5 also reduces it significantly. Since $t \bar t$ decays contain two $b$ quarks, $b$- tagging is not very effective here and has not been included in Table 4.
---------------------------------------------------------------------------------------------------------------------------------
$QCD$ $\epsilon_1$ $\epsilon_2$ $\epsilon_3$ $ $\epsilon_5$
\epsilon_4$
------------- ----------------------- ------------------------ ----------------- ------------------------ -----------------------
$e e$ $1.16 \times 10^{-5}$ 0 0 0 0
$\tau \tau$ $9.10 \times 10^{-3}$ $4.02 \times 10^{-3} $ $1.05 $2.85 \times 10^{-4} $ $2.10 \times 10^{-4}$
\times 10^{-3}$
$e \mu$ $6.0 \times 10^{-6}$ 0 0 0 0
$ e\tau$ 0 0 0 0 0
---------------------------------------------------------------------------------------------------------------------------------
: Efficiency table for the $QCD$ process in the $ \hat p_T$ bin: 400 GeV $< \hat p_T <$ 1000 GeV.
Table 5 presents another important background arising from the $2 \ra
2$ processes due to pure $QCD$ interactions for 400 GeV $< \hat p_T <$ 1000 GeV, where $\hat p_T$ is the transverse momentum of the two partons in the final state . However, SC2 completely kills all backgounds except for those with the di-$\tau$ final states. The latter background, mainly due to mistagging of light flavour jets as $\tau$-jets, affect the di-$\tau$ signal very seriously . The mistagging probability has also been taken from [@cms] (Fig. 12.9).
This background is very large, as expected, since the $QCD$ cross-section is very large. The leading order cross-sections have been computed by Pythia in two $ \hat p_T$ bins : (i) 400 GeV $< \hat p_T <$ 1000 GeV and (ii) 1000 GeV$< \hat p_T <$ 2000 GeV . We have chosen the QCD scale to be $\sqrt{\hat s}$. The corresponding cross-sections being 2090$\pb$ and 10$\pb$ respectively. Beyond 2000 GeV the number of events are negligible. We shall discuss later how the visibility of the di-$\tau$ signal can be improved by employing $b$ tagging.
------------------ ---------- ------------ ------ ------------ ----------- ------
Final state $W^+W^-$ $W^{\pm}Z$ $ZZ$ $t \bar t$ QCD DY
$\sigma(pb)$ 73.5 33.4 10.1 400 2090,10.6 3400
$e~ e~ 0b$ 0.37 0.33 0.40 0.40 - -
$e~ e~ 1b$ - - - - - -
$e~ e~ 2b$ - - - - - -
$ e~ e $ 0.37 0.33 0.40 0.40 - -
$\tau~ \tau~ 0b$ - - - - 4218 -
$\tau~ \tau~ 1b$ - - - 0.80 143 -
$\tau~ \tau~ 2b$ - - 0.20 0.80 12 -
$ \tau~ \tau $ - - 0.20 1.60 4373 -
$\tau~ e~ 0b$ - - - - - -
$\tau~ e~ 1b$ - - - 0.40 - -
$\tau~ e~ 2b$ - - - 0.40 - -
$ \tau~ e $ - - - 0.80 - -
$\mu~ e~ 0b$ 0.37 - - 0.40 - -
$\mu~ e~ 1b$ - - - 0.80 - -
$\mu~ e~ 2b$ - - - 0.80 - -
$\mu~ e $ 0.37 - - 2.00 - -
------------------ ---------- ------------ ------ ------------ ----------- ------
: Total number of all types of backgrounds survived after all cuts .
In Table 6 we have computed the numerically significant backgrounds of all types for $\lum$ = 10 $\ifb$. Here ’-’ denotes a vanishingly small background. It is clear from this table that only $t \bar t$ and QCD backgrounds are relevant. The LO cross-sections in the second row of Table 6 except for the QCD processes have been computed using calcHEP(version 2.3.7)[@calchep]. Due to very strong cut on $P_T$ of highest two leptons SC2 DY type backgrounds become vanishingly small. Moreover, SC3 and SC4 finally reduce it to zero. Other backgrounds like $WW$, $WZ$ and $ZZ$ become vanishingly small mainly due to SC2.
The Product Branching Ratio (PBR) is defined as : P\_[ij]{} BR(l\_i\^+ b) BR(\^\* l\_j\^- |b) where $i$ or $j$ can run from 1-3 corresponding to $e$, $\mu$ and $\tau$ respectively. The Minimum Observable Product Branching Ratio(MOPBR $\equiv P_{ij}^{min}$) corresponds to $S/\sqrt(B) \geq 5$, where $S$ and $B$ are the number of signal and background events respectively. However, for a typical signal with negligible background we have required $S \geq 10$ as the limit of observability and MOPBR is computed accordingly.
For a given $\lum$ the MOPBR for each process is computed from Table 3 and Table 6 by following expression:\
$$\begin{aligned}
P_{ij}^{min}& = &5 \sqrt {\eta\lum \Sigma \sigma^b \eps^b }
\over \eta \lum \sigma(\lstop \lstopbar) \eps,\end{aligned}$$ where $P_{ij}$ is already defined in Eq. 2. $\sigma^b$ and $\eps^b$ (not to be confused with $\eps_b$, the $b$-jet tagging efficiency) denote the cross section and the efficiency of background of type $b$ . Similarly $\eps$ is the final efficiency for the signal. $\eta$ is 2 for $i \neq j$ and $\eta$ is 1 for $i = j$. The integrated luminosity $\lum$ is taken to be 10 fb$^{-1}$. The estimated MOPBRs are given in Table 7(without $b$-jet tagging) and Table 8 ( with two tagged $b$-jets).
We remind the reader that in Table 7 and Table 8 a signal is assumed to be observable if S $\geq$ 10 even if B is $\leq$ 4. In Table 7 and Table 8 a ’$\times$’ indicates that corresponding channel can not be probed.
Our conclusions so far have been based on LO cross sections. If the next to leading order corrections are included the $\lstop -
\lstop^*$ production cross section is enhanced by 30 - 40 % due to a K-factor[@NLO]. It is then clear from Eq. 3, that the estimated MOPBR would remain unaltered even if all significant background cross sections are enhanced by a factor of two due to higher order corrections.
$\mlstop(\gev)$ 240 300 400 450 500
-------------------- ---------- ---------- ---------- ---------- ----------
$P_{11}^{min}(\%)$ 2.1 2.3 2.8 4.0 6.3
$P_{33}^{min}(\%)$ $\times$ $\times$ $\times$ $\times$ $\times$
$P_{12}^{min}(\%)$ 1.0 1.1 1.4 2.0 3.4
$P_{13}^{min}(\%)$ 1.1 1.4 2.7 4.7 8.0
: Minimum value of PBR estimated from the sample without $b$ tagging .
$\mlstop(\gev)$ 240 300 400 450 500
-------------------- ----- ------ ------ ------ ----------
$P_{11}^{min}(\%)$ 3.9 4.1 4.5 6.6 10.0
$P_{33}^{min}(\%)$ 9.4 16.0 37.5 66.4 $\times$
$P_{12}^{min}(\%)$ 1.9 2.0 2.3 3.3 5.0
$P_{13}^{min}(\%)$ 2.7 3.0 5.2 8.6 14.5
: Minimum value of PBR estimated from the 2-$b$ tagged sample.
We present in Fig. 1 the distribution (unnormalised) of invariant mass of a electron -jet pair in the dielectron-dijet sample without b-tagging for $\mlstop=300\gev$. We first reconstruct invariant mass for all possible electron-jet pair. Among these pairs We select the two such that the difference in their invariant mass is minimum. We then plot the higher of the two invariant masses. This peak, if observed, would unambiguously establish the lepton number violating nature of the underlying decay. In contrast if neutralino decay is the only signal of $R$-parity violation, then this information may not be available. For example, if $\lspone \rightarrow \nu b \bar b$ is the dominant decay mode of the LSP via the $\lambda^{\prime}_{i33}$ coupling then the lepton number violating nature of the decay dynamics will be hard to establish.
In the next section we shall calculate the PBR for different signals in a realistic models of neutrino mass constrained by the neutrino oscillation data and examine whether the predictions exceed the corresponding MOPBR estimated in this section. Our main aim is to illustrate that the LHC experiments will be sufficiently sensitive to probe these models and not to make an exhaustive study of all possible models.
Model Calculations
==================
The collider signatures considered in the last section arise only in models with non-vanishing trilinear $\lambda'_{i33}$ type couplings at the weak scale. However, consistency with neutrino oscillation data require the introduction of more RPV parameters (bilinear superpotential terms, bilinear soft breaking terms etc)[@subhendu]. In fact the list of possible choices is quite long. It is expected that the constraints on the $\lambda'$ couplings in the most general model imposed by the $\nu$ - oscillation data will be considerably weaker and the observability of the resulting dilepton-dijet signal will improve. Thus we have restricted ourselves to models with a minimal set of parameters capable of explaining the oscillation data with rather stringent constraints on the $\lambda'$ couplings.
We work in a basis where the sneutrino vevs are zero. It is assumed that in this basis only three nonzero bilinear($\mu_i$) and three trilinear($\lbp_{i33}$) couplings, all defined at the weak scale, are numerically significant. In this framework the neutrino mass matrix receives contributions both at the tree and one loop level. It should be emphasised that the tree level mass matrix, which is independent of $\lambda_{i33}$ couplings, yields only two massless neutrinos. Thus the interplay of the tree level and one loop mass matrices is essential for consistency with the oscillation data.
The chargino-charge lepton, the neutralino - neutrino and other relevant mixing matrices in this basis may be found in [@subhendu]. In principle the diagonalization of these matrices may induce additional lepton number violating couplings which can affect the BRs of the top squark decays considered in this paper. For example, the RPC coupling $\lstop - t - \widetilde W_3$ may induce new RPV vertices through $\widetilde W_3 - \nu$ mixing. However, it was shown in[@adsp] that the new modes induced in this way would have negligible BRs. As a result the approximation that the decays of the top squark NLSP are driven by the $\lbp_{i33}$ couplings only is justified.
In addition to the RPV parameters the neutrino masses and mixing angles depends on RPC parameters. In this paper we shall use the following popular assumptions to reduce the number of free parameters in the RPC sector: i) At the weak scale the soft breaking mass squared parameters of the L and R-type squarks belonging to the third generation are assumed to be the same( the other squark masses are not relevant for computing neutrino masses and mixing angles in this model). ii) We shall also use the relation $M_2 \approx 2~ M_1$ at the weak scale as is the case in models with a unified gaugino mass at $M_G$. Here $M_1$ and $M_2$ are respectively the soft breaking masses of the U(1) and SU(2) gauginos respectively.
The tree level neutrino mass matrix and, hence, the predicted neutrino masses depends on the parameters of the gaugino sector(through the parameter $C$[@abada; @adspdsp]). They are $M_2$, $M_1$ , $\mu$ (the higgsino mass parameter) and tan $\beta=v_2/v_1$, where $v_1$ and $v_2$ are the vacuum expectation values (vevs) for the down type and the up type neutral higgs bosons respectively. We remind the reader that for relatively large tan $\beta$s the loop decay overwhelms the RPV decay [@shibu; @boehm]. We have, therefore, restricted ourselves to $tan \beta$ = 5-8. It is also convenient to classify various models of the RPC sector according to the relative magnitude of $M_2$ and $\mu$. If $M_1 < M_2$ $\ll$ $\mu$, then the lighter chargino ($\tilde\chi_1^{\pm}$), the LSP ($\tilde\chi_1^0$) and the second lightest neutralino ($\tilde\chi_2^0$) are dominantly gauginos. Such models are referred to as the gaugino-like model. On the other hand in the mixed model ($M_1$$ < M_2$$\approx$ $\mu$), $\tilde\chi_1^{\pm}$ and $\tilde\chi_2^0$ are admixtures of gauginos and higgsinos. In both the cases, however, $\lspone$ is purely a bino to a very good approximation. There are models with $M_1$,$M_2$ $\gg$ $\mu$ in which $\tilde\chi_1^{\pm}$, $\tilde\chi_1^0$ and $ \tilde\chi_2^0$ are higgsino - like and all have approximately the same mass ($\approx \mu$). It is difficult to accommodate the top squark NLSP in such models without fine adjustments of the parameters. Thus the LSP decay seems to be the only viable collider signature. One can also construct models wino or higgsino dominated LSPs. However, the $\lstop$-NLSP scenario cannot be naturally accommodated in these frameworks for reasons similar to the one in the last paragraph.
The one loop mass matrix, on the other hand, depends on the sbottom sector (through the parameter $K_2$ [@abada; @adspdsp]). This parameter decreases for higher values of the common squark mass for the third generation. From the structure of the mass matrix it then appears that for fixed C, identical neutrino masses and mixing angles can be obtained for higher values of the trilinear couplings if $K_2$ is decreased. Thus at the first sight it seems that arbitrarily large width of the RPV decays may be accommodated for any given neutrino data. This, however, is not correct because of the complicated dependence of the RPV and loop decay BRs of $\lstop$ on the RPC parameters and certain theoretical constraints. The common squark mass cannot be increased arbitrarily without violating the top squark NLSP condition. Of course larger values of the trilinear soft breaking term $A_t$ may restore the NLSP condition. But larger values of $A_t$ tend to develop a charge colour breaking( CCB ) minimum of the scalar potential [@ccb]. Finally the pseudo scalar higgs mass parameter $M_A$ can be increased to satisfy the CCB condition. But as noted earlier [@adspdsp] that would enhance the loop decay width as well and suppress the BRs of the RPV decay modes.
We have chosen the following RPC scenarios : A) The gaugino dominated model and B)The mixed type model. The choice of RPC parameters for model A) and model B) are:\
A) $M_1=195.0,~ M_2= 370.0,~ \mu=710.0,~ \tan\beta=6.0,~ A_t=1100.0,
~A_b=1000.0,~ M_{\tilde q}$(common squark mass )=$450.0$, $M_{\tilde l}$ (common slepton mass ) = $400.0$ and $M_A=500.0$ and B) $M_1=170.0,~ M_2= 330.0,~ \mu=320.0,~ \tan\beta=6.0,~ A_t=1045.0,~ A_b=1000.0,~ M_{\tilde q}$=$450.0$, $M_{\tilde l}$ = $400.0$ and $M_A=200.0$, where all masses and mass parameters are in $\gev$. Both the scenarios correspond to $\mlstop=240 \gev$ and $\lstop$ is the NLSP. It should be noted that the slepton mass is specified to ensure that the $\lstop$ is the NLSP. It does not affect the neutrino mass matrix.
Even if $\lstop$ is the NLSP the following modes may compete with the RPV decays and overwhelm it:\
a) t ; b) b W c) c ; d) f |f b In the parameter spaces we have worked with the mode a) is kinematically disallowed. The second mode is highly suppressed if the LSP is Bino dominated as is assumed in this analysis. Thus in the scenario under consideration only modes c) and d) may compete with RPV decays of $\lstop$. In this section we have computed the PBRs taking into account the competition among the above three modes.
Next we have randomly generated bilinear and trilinear RPV couplings, $\mu_i$ and $\lambda '_{i33}$. Then these parameters are constrained by $\nu$ oscillation data which allows very few sets of RPV parameters for the above RPC parameters. Most of the allowed trilinear RPV couplings lie within $10^{-4} - 10^{-5}$. Finally the relevant PBRs have been calculated in model (A) and (B).
In Table 9 we present several representative sets of trilinear RPV parameters allowed by $\nu$ oscillation data and the corresponding PBR. A ‘-’ indicates that the predicted PBR is negligible. As noted before two of the couplings turns out to be large while the third one is suppressed due to oscillation constraints. It turns out that the PBR’s involving the large couplings are larger than the corresponding MOPBRs estimated in the last section even without b-tagging(see Table 7). The only exception is $P_{33}$ which cannot be probed without b-jet tagging (see Table 8)
${\lambda^{\prime}}_{133} [\times 10^{-5}]$ ${\lambda^{\prime}}_{233} [\times 10^{-5}]$ ${\lambda^{\prime}}_{333} [\times 10^{-5}]$ $P_{11}$ $P_{22}$ $P_{33}$ $P_{12}$ $P_{23}$ $P_{13}$
--------------------------------------------- --------------------------------------------- --------------------------------------------- ---------- ---------- ---------- ---------- ---------- ----------
Model A
1.6 8.3 10.0 - 5.1 10.8 0.2 7.4 0.3
7.5 0.7 9.2 4.9 - 11.4 - 0.1 7.5
4.6 4.5 0.3 6.8 6.8 - 6.7 - -
Model B
11.9 0.99 15.0 4.2 - 10.6 - - 6.6
0.59 13.6 16.8 - 4.3 10.2 - 6.6 -
7.3 7.4 0.9 6.3 6.6 - 6.4 0.1 0.1
: Trilinear RPV couplings allowed by $\nu$ oscillation data and the corresponding PBRs computed in models A and B (see text) with $\mlstop = 240 \gev
$.
For larger $\mlstop$, there exists allowed RPV parameter space with observable PBRs at the early LHC runs. However, if we go beyond $\mlstop=500 \gev$ the di-tau channel cannot be probed even with $b$ -tagging. Nevertheless, observation of the $e-\tau$ and the $\mu -\tau$ channel will provide evidence for a relatively large $\lambda_{333}$. We present in Table 10 for $\mlstop=500 \gev$. The RPC parameters corresponding to a Gaugino model are chosen to be:\
$M_1=475.0,~ M_2= 860.0, ~\mu=1650.0,~ \tan\beta=6.0,~ A_t=995.0,~ A_b=1000.0,
~M_{\tilde q}$=$575.0$, $M_{\tilde l}$ = $525.0$ and $M_A=300.0$, where all masses and mass parameters are in $\gev$.
${\lambda^{\prime}}_{133} [\times 10^{-5}]$ ${\lambda^{\prime}}_{233} [\times 10^{-5}]$ ${\lambda^{\prime}}_{333} [\times 10^{-5}]$ $P_{11}$ $P_{22}$ $P_{33}$ $P_{12}$ $P_{23}$ $P_{13}$
--------------------------------------------- --------------------------------------------- --------------------------------------------- ---------- ---------- ---------- ---------- ---------- ----------
9.1 4.0 6.4 20.7 - 5.1 4.0 2.0 10.3
4.4 10.9 5.6 - 31.4 2.2 5.1 8.3 1.3
: Same as Table 9 for $\mlstop = 500 \gev$.
We have checked that even for $\mlstop > 500 \gev$ there exits RPV parameter space allowed by oscillation data which leads to observable dilepton -dijet signals in early LHC experiments.
------------------- ---- ------------------- ------ -------------------- -----
$P_{11} , P_{13}$ 56 $P_{22} , P_{23}$ 1376 $P_{33} , P_{23}$ 874
$P_{11} , P_{12}$ 15 $P_{22} , P_{12}$ 119 $P_{33} , P_{13}$ 27
$P_{11} $ 2 $P_{22} $ 664 $P_{22} , P_{23}$ 274
$P_{12} $ 10 $ * *$ 17 $ P_{11} , P_{13}$ 25
$ * *$ 9 $P_{23} $ 45
$ P_{33} $ 304
$ * * $ 537
------------------- ---- ------------------- ------ -------------------- -----
: Number of allowed solutions in the mixed model($\mlstop=240 GeV$) consistent with $\nu$ oscillation data which satisfy the MOPBR given in Table 7 and Table 8. The above numbers are estimated for $\lum = 10 \ifb$.
We have randomly generated $10^9$ sets of RPV parameters in the mixed model with $\mlstop = 240 GeV$. Out of these only 4354 are consistent with the $\nu$-oscillation data. These solutions can be further classified into three groups according to the highest value of $\lambda
^{\prime}_{i33}$. The three columns in Table 11 correspond to these groups. The first column in Table 11 contains detailed information about the flavour structure of the RPV couplings in the 92 solutions with the hierarchy $\lambda ^{\prime}_{133} > \lambda ^{\prime}_{233},
\lambda ^{\prime}_{333}$. The next few rows display the number of solutions with predicted PBRs in different channels above the observable limits as given in Table 8. For example, the third row indicates that signals in $e e + 2 jets$ and $e \tau +
2jets$ channels are observable with 10 fb $^{-1}$ of data in 56 solutions. These channels, if observed, would further reveal that $\lambda ^{\prime}_{133} > \lambda ^{\prime}_{333} > \lambda
^{\prime}_{233}$. On the other hand observable signals in $e e + 2 jets$ and $e \mu + 2 jets$ channels as given in the next row would indicate the hierarchy $\lambda
^{\prime}_{133}
> \lambda ^{\prime}_{233} > \lambda^{\prime}_{233}$.
If only one channel, say the $e e+2jets$, is observed one can conclude that $\lambda ^{\prime}_{133} >> \lambda ^{\prime}_{233}, \lambda
^{\prime}_{333}$ (see row 5). On the other the observation of the $e \mu + 2 jets$ signal only (see row 6) would indicate $\lambda
^{\prime}_{133} \approx \lambda ^{\prime}_{233} >> \lambda
^{\prime}_{233}$. The channel $e \mu + 2 jets$ dominates over the $e e + 2 jets$ or the $\mu \mu + 2 jets$ channel because of the factor of two which enhances the number of events when leptons of two different flavours with all possible charge combinations are observed. Finally the seventh row with ‘\*\*’ indicates that no signal can be observed with $\lum$ = 10 fb $^{-1}$
The information in the next two columns are presented following the format and similar inferences about the hierarchy of the $\lambda
^{\prime}_{i33}$ can be drawn from the lepton flavour content of the final states. We have verified that for $\lum = 100 \ifb$ all solutions well predict atleast one $P_{ij}$ above the corresponding $P_{ij}^{min}$.
Conclusion
==========
In conclusion we reiterate that the OSDL signals with same or different flavours of leptons (e, $\mu$ or tau-jets) plus two additional jets arising from RPV decays of $\lstop$ - $\lstop^*$ pairs produced at the LHC would be a promising channel for probing the RPV coupling$\lambda^{\prime}_{i33}$ (see Eq. 1 and the discussions following it). This is true in general if $\lstop$ happens to be the NLSP, which is a theoretically well motivated scenario. This signal is especially interesting in the context of RPV models of neutrino mass. A part of our analysis (section 2), however, is fairly model independent since the size of the signal is necessarily controlled by the production cross section of the $\lstop$ - $\lstop^*$ pair as given by QCD and the product branching ratio $P_{ij}$ (see Eq. 2). The model independent estimates of $P_{ij}^{min}$ (see Eq. 3) corresponding to observable signals for different $\mlstop$s (see Eq. 3) for an integrated luminosity of 10 $fb^{-1}$ are presented in Table 7 and Table 8 using the Monte Carlo event generator Pythia. We have optimized the cuts for $\mlstop=240$ GeV. However, for even larger values of $\mlstop$ the signal efficiencies increase for the same set of cuts keeping the background events almost negligible. Top squark masses in the vicinity of 500 $\gev$ yield observable signals in this channel for realistic models of $m_{\nu}$. Although our calculations are based on LO top squark pair production cross sections we emphasise that the inclusion of NLO corrections are likely to yield even larger estimates of $P_{ij}^{min}$ as argued in section 2.
We have further noted that inspite of the combinatorial backgrounds, the invariant mass distribution of the lepton (e or $\mu$)-jet pair shows a peak at $\mlstop$ (see Fig.1). This peak, if discovered, will clearly establish the lepton number violating nature of the underlying interaction. This may not be possible if neutralino decays happen to be the only RPV signal.
In models of $\nu $-mass, the underlying $\lambda^{\prime}$ couplings turn out to be very small. If $\lambda^{\prime}_{i33}$ contributes to the one loop $\nu$-mass matrix, it is typically of the order of 10$^{-4}$ - 10$^{-5}$ due to constraints imposed by the $\nu$-oscillation data. Even if $\lambda^{\prime}$ is so small the RPV decay of the $\lstop$-NLSP may have sizable BRs over a large region of the parameter space because the competing loop induced decay (Eq. 4c) or the four body decay (Eq. 4d) of $\lstop$ also have suppressed widths. For the purpose of illustration we have considered a specific model of $\nu$-mass[@abada] with parameters constrained by the $\nu$-oscillation data. It is interesting to note that in this model most of the theoretically predicted $P_{ij}$’s (Eq. 2) for several representative choices of RPC parameters turn out to be larger than the $P_{ij}^{min}$’s estimated in section 2 for $\lum$= 10 $fb^{-1}$. For larger $\lum$ almost all solutions yield $P_{ij}$’s at the observable level. The relative size of the observed final states with various lepton flavour contents will indicate the hierarchy among the $\lambda^{\prime}_{i33}$s for different $i$’s.
[**Acknowledgement**]{}: AD and SP acknowledge financial support from Department of Science and Technology, Government of India under the project No (SR/S2/HEP-18/2003).
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[^1]: adatta@iiserkol.ac.in
[^2]: sujoy$\_$phy@iiserkol.ac.in
[^3]: It is expected that this cut would also suppress the SUSY backgrounds due to, e.g., $\tilde q, \tilde g$ production.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We consider computation of permanent of a positive $(N\times N)$ non-negative matrix, $P=(P_i^j|i,j=1,\cdots,N)$, or equivalently the problem of weighted counting of the perfect matchings over the complete bipartite graph $K_{N,N}$. The problem is known to be of likely exponential complexity. Stated as the partition function $Z$ of a graphical model, the problem allows exact Loop Calculus representation \[Chertkov, Chernyak ’06\] in terms of an interior minimum of the Bethe Free Energy functional over non-integer doubly stochastic matrix of marginal beliefs, $\beta=(\beta_i^j|i,j=1,\cdots,N)$, also correspondent to a fixed point of the iterative message-passing algorithm of the Belief Propagation (BP) type. Our main result is an explicit expression of the exact partition function (permanent) in terms of the matrix of BP marginals, $\beta$, as $Z=\mbox{Perm}(P)=Z_{BP} \mbox{Perm}(\beta_i^j(1-\beta_i^j))/\prod_{i,j}(1-\beta_i^j)$, where $Z_{BP}$ is the BP expression for the permanent stated explicitly in terms of $\beta$. We give two derivations of the formula, a direct one based on the Bethe Free Energy and an alternative one combining the Ihara graph-$\zeta$ function and the Loop Calculus approaches. Assuming that the matrix $\beta$ of the Belief Propagation marginals is calculated, we provide two lower bounds and one upper-bound to estimate the multiplicative term. Two complementary lower bounds are based on the Gurvits-van der Waerden theorem and on a relation between the modified permanent and determinant respectively.'
address:
- '$^1$ Institute of Statistical Mathematics, 10-3 Midori-cho, Tachikawa, Tokyo 190-8562 Japan.'
- |
$^2$ Center for Nonlinear Studies and Theoretical Division, LANL, NM, 87545\
also New Mexico Consortium, Los Alamos, NM 87544.
author:
- Yusuke Watanabe$^1$ and Michael Chertkov$^2$
bibliography:
- 'permanent.bib'
- 'BP\_review.bib'
- 'zeta.bib'
- 'MishaPapers.bib'
title: 'Belief propagation and loop calculus for the permanent of a non-negative matrix'
---
Introduction
============
The problem of calculating the permanent of a non-negative matrix arises in many contexts in statistics, data analysis and physics. For example, it is intrinsic to the parameter learning of a flow used to follow particles in turbulence and to cross-correlate two subsequent images [@10CKKVZ]. However, the problem is $\# P$-hard [@79Val], meaning that solving it in a time polynomial in the system size, $N$, is unlikely. Therefore, when size of the matrix is sufficiently large, one naturally looks for ways to approximate the permanent. A very significant breakthrough was achieved with invention of a so-called Fully-Polynomial-Randomized Algorithmic Schemes (FPRAS) for the permanent problem [@04JSV]: the permanent is approximated in a polynomial time, with high probability and within an arbitrarily small relative error. However, the complexity of this FPRAS is $O(N^{11})$, making it impractical for the majority of realistic applications. This motivates the task of finding a lighter deterministic or probabilistic algorithm capable of evaluating the permanent more efficiently.
This paper continues the thread of [@08CKV; @10CKKVZ] and [@09HJ], where the Belief Propagation (BP) algorithm was suggested as an efficient heuristic of good (but not absolute) quality to approximate the permanent. The BP family of algorithms, originally introduced in the context of error-correction codes [@63Gal] and artificial intelligence [@88Pea], can generally be stated for any graphical model [@05YFW]. The exactness of the BP on any graph without loops suggests that the algorithm can be an efficient heuristic for evaluating the partition function or for finding a Maximum Likelihood (ML) solution for the Graphical Model (GM) defined on sparse graphs. However, in the general loopy cases one would normally not expect BP to work well, thus making the heuristic results of [@08CKV; @10CKKVZ; @09HJ] somehow surprising, even though not completely unexpected in view of existence of polynomially efficient algorithms for the ML version of the problem [@55Kuh; @92Ber], also realized in [@08BSS] via an iterative BP algorithm. This raises the questions of understanding the performance of BP: what it does well and what it misses? It also motivates the challenge of improving the BP heuristics.
An approach potentially capable of handling the question and the challenge was recently suggested in the general framework of GM. The Loop Series/Calculus (LS) of [@06CCa; @06CCb] expresses the ratio between the Partition Function (PF) of a binary GM and its BP estimate in terms of a finite series, in which each term is associated with the so-called generalized loop (a subgraph with all vertices of degree larger than one) of the graph. Each term in the series, as well as the BP estimate of the partition function, is expressed in terms of a doubly stochastic matrix of marginal probabilities, $\beta=(\beta_i^j|i,j=1,\cdots,N)$, for matching pairs to contribute a perfect matching. This matrix $\beta$ describes a minimum of the so-called Bethe free energy, and it can also be understood as a fixed point of an iterative BP algorithm. The first term in the resulting LS is equal to one. Accounting for all the loop-corrections, one recovers the exact expression for the PF. In other words, the LS holds the key to understanding the gap between the approximate BP estimate for the PF and the exact result. In section \[sec:one\] and section \[sec:LC\], we will give a technical introduction to the variational Bethe Free Energy (BFE) formulation of BP and a brief overview of the LS approach for the permanent problem respectively.
[**Our results.**]{} In this paper, we develop an LS-based approach to describe the quality of the BP approximation for the permanent of a non-negative matrix. (i) Our natural starting point is the analysis of the BP solution itself conducted in section \[sec:BP\]. Evaluating the permanent of the non-negative matrix, $P=((p_i^j)^{1/T}|i,j=1,\cdots,N)$, dependent on the temperature parameter, $ T \in [0,\infty]$, we find that a non-integer BP solution is observed only at $T>T_c$, where $T_c$ is defined by (\[CritEq\]). (ii) At $T>T_c$, we derive an alternative representation for the LS in section \[sec:Per\_BP\_Per\]. The entire LS is collapsed to a product of two terms: the first term is an easy-to-calculate function of $\beta$, and the second term is the permanent of the matrix, $\beta.*(1-\beta)=(\beta_i^j(1-\beta_i^j))$. (The binary operator $.*$ denotes the element-wise multiplication of matrices.) This is our main result stated in theorem \[LS\_new\], and the majority of the consecutive statements of our paper follows from it. We also present yet another, alternative, derivation of the theorem \[LS\_new\] using the multivariate Ihara-Bass formula for the graph zeta-function in subsection \[subsec:Ihara\]. (iii) Section \[sec:low\] presents two easy-to-calculate lower bounds for the LS. The lower bound stated in the corollary \[Gurvits\_bound\] is based on the Gurvits-van der Waerden theorem applied to $\mbox{Perm}(\beta.*(1-\beta))$. Interestingly enough this lower bound is invariant with respect to the BP transformation, i.e. it is exactly equivalent to the lower bound derived via application of the van der Waerden-Gurvits theorem to the original permanent. Another lower bound is stated in theorem \[second\_low\]. Note, that as follows from an example discussed in the text, the two lower bounds are complementary: the latter is stronger at sufficiently small temperatures, while the former dominates the large $T$ region. (iv) Section \[sec:up\] discusses an upper bound on the transformed permanent based on the application of the Godzil-Gutman formula and the Hadamard inequality. Possible future extensions of the approach are discussed in section \[sec:path\].
Background (I): Graphical Models, Bethe Free energy and Belief Propagation. {#sec:one}
===========================================================================
Permanent of a non-negative matrix, $P=((p_i^j)^{1/T}|i,j=1,\cdots,N) \quad (0\leq p_i^j,\ 0\leq T\leq\infty)$, is a sum over the set of permutations on $\{1,\ldots,N\}$, which can be parameterized via binary-component vectors, $\sigma$, corresponding to perfect matchings (PM) on the complete bipartite graph $K_{N,N}$: $$\left\{ \sigma=(\sigma_i^j) \in \{0,1\}^{N \times N} \Big|
\forall i:\ \sum_{j=1}^N \sigma_i^j=1, \quad \forall j:\ \sum_{i=1}^N\sigma_i^j=1
\right\}.$$ This binary interpretation allows us to represent the permanent as the partition function (PF), $Z$, of a probabilistic model over the set of perfect matchings. Each perfect matching, $\sigma$, is realized with the probability $$\begin{aligned}
\fl
\quad {\cal P}(\sigma)=\frac{1}{Z}P^{\sigma}; \quad
P^{\sigma} \equiv \prod_{(i,j) \in E} (p_i^j)^{\sigma_i^j/T},\
Z\equiv \sum_{\sigma : {PM}}(p_i^j)^{\sigma_i^j/T} = {{\rm Perm}}(P), \label{GM}\end{aligned}$$ where $E=\{ (i,j) |\ i,j=1,\ldots,N \}$ is the edges of $K_{N,N}$. In the zero-temperature limit, $T\to 0$, (\[GM\]) selects one special ML solution, $\sigma_*=\arg\max_{\sigma} P^{\sigma}$. (Here and below we assume that $P$ is non-degenerate, in the sense that at $T\to 0$, ${\cal P}(\sigma)\to 0$ for $\forall\ \sigma\neq\sigma_*$.)
For a generic GM, assigning (un-normalized) weight $P^{\sigma}$ to a state $\sigma$, one defines exact variational (called Gibbs, in statistical physics, and Kullback-Leibler in statistics) functional $$\begin{aligned}
{\cal F}\{b(\sigma)\}\equiv T
\sum_{ \sigma }b(\sigma)\ln\frac{b(\sigma)}{P^{\sigma}}. \label{Gibbs}\end{aligned}$$ One finds that under condition that the belief, $b(\sigma)$, understood as a proxy to the probability ${\cal P}(\sigma)$, is normalized to unity, $\sum_{\sigma \in PM} b(\sigma)=1$, the Gibbs functional is convex and it achieves its only minimum at $b(\sigma)={\cal P}(\sigma)$ and ${\cal F}\{ \mathcal{P} \}=-T\ln Z$.
BP method offers an approximation which is exact when the underlying GM is a tree. As shown in [@05YFW], the BP approach can also be stated for a general GM as a relaxation of the Gibbs functional (\[Gibbs\]). In this paragraph we briefly review the concept of [@05YFW] with application to the permanent problem. For the GM (\[GM\]), the BP approximation for the state beliefs becomes $$\begin{aligned}
b(\sigma)\approx b_{\it BP}(\sigma)=
\frac{\prod_i b_i(\sigma_i)\prod_j b^j(\sigma^j)}{
\prod_{(i,j) \in E} b_i^j(\sigma_i^j)},
\label{BP_Belief}\end{aligned}$$ where $\forall i,j$: $\sigma_i=(\sigma_i^j \in \{0, 1\}|j=1,\cdots, N)$ s.t. $\sum_j\sigma_i^j=1$ and $\sigma^j=(\sigma_i^j \in \{0, 1\}|i=1,\cdots,N)$ s.t. $\sum_i\sigma_i^j=1$, i.e. $\sigma_i$ and $\sigma^j$ each has only $N$ allowed states corresponding to allowed local perfect matchings for the vertices $i$ and $j$ respectively. The vertex and edge beliefs are related to each other according to $$\forall (i,j) \in E :\quad b_i^j(\sigma_i^j)= \sum\limits_{\sigma_i\setminus\sigma_i^j}b_i(\sigma_i)=
\sum\limits_{\sigma^j\setminus\sigma_i^j}b^j(\sigma^j),\label{rel}$$ and the beliefs, as probabilities, should also satisfy the normalization conditions: $$\forall (i,j) \in E :\quad b_i^j(1)+b_i^j(0)=1.\label{norm}$$ Note, that our notations for beliefs are not identical to ones used in [@05YFW]: the multi-variable beliefs, $b_i$, are associated with vertexes of $K_{N,N}$, and the single-variable beliefs, $b_i^j$ are associated with edges of the graph. Substituting (\[BP\_Belief\]) into (\[Gibbs\]) and approximating $\sum_{ \sigma \in PM} b(\sigma) f(\sigma_i^j)$ with $\sum_{\sigma_i^j} b_i^j(\sigma_i^j) f(\sigma_i^j)$ etc, one arrives at the BFE functional $$\begin{aligned}
\fl
{\cal F}_{\it BP}\{b_i^j(\sigma_i^j);b_i(\sigma_i);b^j(\sigma^j)\}
\equiv E-T S, \quad
E\equiv\sum_{(i,j)}b_i^j(1)\log(p_i^j),\label{FE}\\
\fl S\equiv\sum_{(i,j)}\sum\limits_{\sigma_i^j}
b_i^j(\sigma_i^j)\ln b_i^j(\sigma_i^j)
-\sum_i\sum\limits_{\sigma_i} b_i(\sigma_i)
\ln b_i(\sigma_i)
- \sum_j\sum\limits_{\sigma^j} b^j(\sigma^j)\ln b^j(\sigma^j).\label{S}\end{aligned}$$ Note that the BFE functional is bounded from below and generally non-convex, and thus finding the absolute minimum of the BFE is the main task of the BFE approximation. The BP approximation $Z_{BP}$ of the partition function is given by ${\cal F}_{BP}=-T\ln Z_{BP}$ at a minimum of the BFE.
Moreover, the variational formulation of (\[rel\],\[norm\],\[FE\],\[S\]) can be significantly simplified in our case; one can utilize (\[rel\],\[norm\]) and express $b_i(\sigma_i), b^j(\sigma^j)$ and $b_i^j(\sigma_i^j)$ solely in terms of the $\beta_i^j\equiv
b_i^j(1)$ variables, satisfying doubly-stochastic constraints $$\begin{aligned}
\forall (i,j) \in E : 0\leq \beta_i^j\leq 1; \quad
\forall i: \sum_j \beta_i^j=1; \quad \forall j: \sum_i \beta_i^j=1. \label{ds_cond}\end{aligned}$$ The entropy (\[S\]) becomes $$\begin{aligned}
\fl
S\{\beta_i^j\}&=
\sum_{(i,j)} \left( \beta_i^j \log \beta_i^j + (1-\beta_i^j) \log (1-\beta_i^j) \right)
-
\sum_i \sum_j \beta_i^j \log \beta_i^j
-
\sum_j \sum_i \beta_i^j \log \beta_i^j \nonumber \\
\fl
&= \sum_{(i,j)}\left((1-\beta_i^j)\ln(1-\beta_i^j)-\beta_i^j\ln\beta_i^j
\right). \label{ES}\end{aligned}$$ Therefore, the Bethe-Free energy approach applied to the GM (\[GM\]) results in minimization of the following Bethe-Free Energy (BFE) functional $$\mathcal{F}_{BP}\{ \beta \} = T \sum_{(i,j) \in E}
\left(
\beta_i^j\ln\frac{ \beta_i^j }{(p_i^j)^{1/T}}
- (1-\beta_i^j)\ln(1-\beta_i^j)
\right), \label{BFE}$$ over $\beta=(\beta_i^j)$ under the constraints (\[ds\_cond\]).
To analyze the minima of the BFE, we incorporate Lagrange multipliers $\mu_i, \mu^j$ enforcing the constraints in (\[ds\_cond\]). Looking for a stationary point of the Lagrange function over the $\beta$ variables, one arrives at the following set of quadratic equations for each (of $N^2$) variables, $\beta_i^j$ $$\forall (i,j) \in E :\quad
\beta_i^j(1-\beta_i^j)=(p_i^j)^{1/T}\exp\left(\mu_i+\mu^j\right). \label{BP1}$$ One observes that any solution of (\[ds\_cond\],\[BP1\]) at $T>0$, that contains at least one $\beta_i^j$ which is not integer, does not contain any integers among all $\beta_i^j$. In fact, our main focus will be on these non-integer (interior) solutions of (\[ds\_cond\],\[BP1\]). To find a solution of BP (\[ds\_cond\],\[BP1\]) one relies on an iterative procedure. For a description of a set of iterative BP algorithms convergent to a minimum of the BFE for the perfect matching problem we refer the interested reader to [@08CKV; @10CKKVZ; @09HJ].
[Note that just derived BP approximation differs from the so-called Mean-Field (MF) approximation corresponding to the following ansatz $$\begin{aligned}
b(\sigma)\approx b_{\it MF}(\sigma)=
\prod_{(i,j) \in E} b_i^j(\sigma_i^j),
\label{BP_MF}\end{aligned}$$ enforcing statistical independence of the edge beliefs. If one substitutes $b(\sigma)$ by $b_{\it MF}(\sigma)$ in (\[Gibbs\]) and also accounts for the normalization condition (\[norm\]), which may be understood here as one enforcing the “Fermi exclusion principle” for an edge $(i,j)$ to contribute a perfect matching, $\sigma_i^j=1$, the resulting expression for the MF free energy will turn into BP expression (\[BFE\]) with the first term there changing sign to $-$. One expects that BP approximation outperforms MF approximation in accuracy. Consider, for example, $N=10$ and $\beta_i^j= 1/N$, then the exact, BP and MF entropies are $\ln(10!) \approx 15.10$, $ 100(.9\ln(.9)-.1\ln(.1)) \approx 13.54$ and $100(-.9\ln(.9)-.1 \ln(.1)) \approx 32.50$, respectively. An intuitive explanation for MF overestimating the entropy term is related to the fact that MF ignores correlations related to competitions between neighboring edges for contributing a perfect matching. ]{}
Threshold Behavior of BP at Low Temperatures {#sec:BP}
============================================
As discovered in [@08BSS], at $T=0$, properly scheduled iterative version of BP converges efficiently to the ML solution of the problem. In this context it is natural to ask the question of how a non-integer solution of BP emerges with a temperature increase. To address this question, we first consider the following homogeneous example.
[\[example1\] Define a homogeneous weight model biased toward a perfect matching solution, $\sigma_i^j=\delta_i^j$ : $p_i^j=1$ if $i\neq j$ and $p_i^i=W \ (W>1)$. Looking for $\beta$ in the homogeneous form $$\beta_i^j(T)=
\left\{\begin{array}{cc}
1- \epsilon (N-1) &\mbox{ :if } i=j \\ \epsilon &\mbox{ :otherwise, }
\end{array}\right.
\label{example}$$ one observes that this ansatz for $\beta$ solves the BP (\[ds\_cond\],\[BP1\]) at $\epsilon$ equal to $\epsilon_{min}= (N-1-W^{1/T})/((N-1)^2
-W^{1/T})$. At $T=\infty$, the probabilities are uniform, i.e. $\beta$ from (\[example\]) with $\epsilon=\epsilon_{min}$ is $\beta_i^j=1/N$ for all $(i,j) \in E$. Now consider lowering the temperature and observe that at $T_c=\ln W/\ln (N-1)$ the nontrivial solution, with $\beta_i^j\neq 0,1$ for all $(i,j) \in E$, turns exactly into the isolated/trivial ML one, $\beta_i^j=\delta_i^j$. Obviously one finds that the BFE, ${\cal F}_{BF}$, considered as a function of $\epsilon$, achieves its minimum at $\epsilon=\epsilon_{min}$ if $T>T_c$. Exactly at $T=T_c$ this $\epsilon_{min}=0$ and the nontrivial solution merges into the isolated ML solution. The dependence of the BFE on $\epsilon$ for different $T$ (at some exemplary values of $N$ and $W$) is shown in figure \[fig\]a. The partition function can be calculated efficiently. Counting the configurations straightforwardly (in a brute force combinatorial manner), one derives $Z=\sum_{k=0}^N W^{(N-k)/T} {N \choose k} D_k$. The following recursion is used to evaluate the number of permutations coefficient, $D_k$: $\forall\ k \geq 2,\ \ D_k=(k-1)(D_{k-1}+D_{k-2}),\quad D_0=1,\quad D_1=0$. A comparison of $T\ln Z$ and $T\ln Z_{BP}$ as functions of $T$ is shown in figure \[fig\]b. ]{}
Returning to the case of an arbitrary nonnegative $P$, we discover that this phenomenon of the nontrivial solution splitting at some finite nonzero (!!) temperature from the ML configuration is generic.
\[prop:Tc\] For any non-negative matrix $P=((p_i^j)^{1/T}|i,j=1,\cdots,N)$ one finds a special (we call it critical) temperature, $T_c$, such that for $T>T_c+\varepsilon$ a nontrivial solution of BP, corresponding to a local non-saturated minimum of ${\cal F}_{BP}$, dominating the respective value corresponding to the maximum likelihood solution, is realized for at least a sufficiently small positive $\varepsilon$. This special solution coincides with the best perfect matching solution at $T=T_c$ and it does not exist for $T<T_c$. The critical temperature $T_c$ solves $$\det(P_i^j - 2 \sigma_{* i}^{\ j} P_i^{j})=0 , \label{CritEq}$$ where $\sigma_*$ is the ML configuration.
Our proof of the proposition is constructive. Let us look for a solution of the BP equations weakly deviating from the ML configuration $\sigma_*$. Without loss of generality we assume that $\sigma_{* i}^{\ j} = \delta_i^j$. We introduce $v_i^j=\beta_i^j(1-\beta_i^j)\ll 1$ and observe that a nontrivial solution, approaching the ML one at $v\to 0$, is $\beta_i^j=(1-(1-2\delta_i^j)[{1-4v_i^j}]^{1/2})/2$. Linearizing the normalization condition, over $v$ one derives, $\forall i: v_i^i= \sum_{j \neq i}v_i^j ; \quad
\forall j: v_j^j= \sum_{i \neq j}v_i^j$ On the other hand, the BP equation (\[BP1\]), complemented by the set of linear constraints on $v$, translates into, $\forall i: \ P_i^i U^i= \sum_{j \neq i} P_i^j U^j ; \quad
\forall j: \ P_j^jU_j= \sum_{i \neq j}P_i^j U_i$, where $U_i=\exp(\mu_i)$ and $U^j=\exp(\mu^j)$. Requiring that the later equations have a nontrivial solution (with nonzero $v$), one arrives at the critical temperature condition, (\[CritEq\]). It is then straightforward to verify that the extension of the nontrivial solution into the $T<T_c$ domain is unphysical (as some elements of the respective small $v$ solution are negative), while the BFE associated with the nontrivial solution for $T>T_c$ is smaller than the one corresponding to the ML perfect matching.
We conjecture that the non-integer solution of BP equations discussed in proposition \[prop:Tc\] extends beyond the small $T_c+\varepsilon$ vicinity of $T_c$, and this solution transitions smoothly at $T\to\infty$ into the obvious fully homogeneous solution, $\beta_i^j=1/N$ for all $(i,j)\in E$. Another plausible conjecture is that no other non-integer solutions exist at $T<T_c$; therefore when the non-integer solution discussed in the proposition emerges at $T=T_c$ it, in fact, gives a global minimum of the BFE.
Background (II): Loop Calculus and Series {#sec:LC}
=========================================
Here we consider $T>T_c$ where, according to the main result of the previous section, there exists a solution of (\[ds\_cond\],\[BP1\]) lying in the interior of the doubly-stochastic-matrix polytope. We assume that such a nontrivial solution of the BP equations is found.
As shown in [@06CCa; @06CCb], the exact partition function of a generic GM can be expressed in terms of a LS, where each term is computed explicitly using the BP solution. Adapting this general result to the permanent, bulky yet straightforward algebra leads to the following exact expression for the partition function $Z$ from (\[GM\]): $$\begin{aligned}
Z/Z_{BP}=z_{LS}; \qquad z_{LS}\equiv 1+\sum_{C \neq \emptyset} r_C, \nonumber \\
r_C \equiv \!\left(\prod_{i\in C}
(1-q_i)\right)\!\!
\left(\prod_{j\in C} (1-q^j)\right)\!\!\prod_{(i,j)\in C}
\frac{\beta_i^j}{1-\beta_i^j}\,.
\label{rC}
\end{aligned}$$ The variables $\beta$ are in accordance with (\[ds\_cond\],\[BP1\]) and $C$ stands for an arbitrary generalized loop, defined as a subgraph of the complete bipartite graph with all its vertices having a degree larger than 1. The $q_i$ (or $q^j$) in (\[rC\]) are the $C$-dependent degrees, i.e. $q_i=\sum_{j \mid (i,j)\in C} 1$ and $q^j=\sum_{i \mid (i,j)\in C}
1$. According to (\[rC\]), those loops with an even/odd number of vertices give positive/negative contributions $r_C$.
Loop Series as a Permanent {#sec:Per_BP_Per}
==========================
This section, explaining the main result of the paper, is split in two parts. In subsection \[subsec:main\] we give a simple derivation of a very compact representation for the LS (\[rC\]) following directly from the BFE formulation. Subsection \[subsec:Ihara\] contains an alternative derivation of this main formula from LS using the concept of the Ihara-Bass graph $\zeta$-function [@Idiscrete; @Bass].
We also find it appropriate here to make the following general remark. Even though discussion of the manuscript is limited to permanents, counting perfect matchings over $K_{N,N}$, all the results reported in this section allows straightforward generalizations to weighted counting of perfect matchings over arbitrary (and not necessarily bipartite) graphs.
Permanent representation for $Z/Z_{BP}$ {#subsec:main}
---------------------------------------
\[LS\_new\] For any non-integer solution of the BP equations, (\[ds\_cond\],\[BP1\]), the following is true: $${{\rm Perm}}(P)/Z_{BP}= {{\rm Perm}}(\beta.*(1-\beta)) \prod_{(i,j)\in E}(1-\beta_i^j)^{-1}, \label{perm}$$ where $A.*B$ is the element-by-element multiplication of the $A$ and $B$ matrices.
From the definition of the BFE, ${\cal F}_{BP}=-T\ln Z_{BP}$, and (\[ds\_cond\],\[BP1\]) one derives $$\fl
Z_{BP}
= \hspace{-2mm} \prod_{(i,j)\in E}\left[
(1-\beta_i^j) \Big( \frac{(p_i^j)^{1/T}}{\beta_i^j(1-\beta_i^j)} \Big)^{\beta_i^j}
\right]
= \hspace{-2mm}
\prod_{(i,j)\in E} \hspace{-1mm} (1-\beta_i^j) \prod_i \e^{- \mu_i} \prod_j \e^{- \mu^j}.$$ On the other hand (\[BP1\]) results in, ${{\rm Perm}}(P)= {{\rm Perm}}(\beta.*(1-\beta))$ $ \prod_i \exp (- \mu_i) \prod_j \exp (- \mu^j)$. Combining the two formulas we arrive at (\[perm\]).
[Note that if one considers expanding the permanent on the rhs of (\[perm\]) over the elements of the matrix $\beta.*(1-\beta)$, each element of the expansion will be positive, in the contrast with the LS of (\[rC\]). Moreover, the number of terms in the Perm-expansion is significantly smaller than in the original LS. ]{}
From LS to the permanent representations for $Z/Z_{BP}$ {#subsec:Ihara}
-------------------------------------------------------
Here we discuss the relation between the two complementary representations of $Z/Z_{BP}$, i.e. between the LS expression (\[rC\]) and the permanent formula (\[perm\]). We do this in two steps, stated in the two theorems presented consequently, one relating the LS to an average of a determinant, and another one expressing it via the permanent of $\beta.*(1-\beta)$.
\[LS as Average of Determinant\]\[thmA1\] Let $\vec{E}$ be the set of directed edges obtained by duplicating undirected edges $E$ of $K_{N,N}$. Define the edge-adjacency matrix $\mathcal{M}$ of the complete bipartite graph $K_{N,N}$ according to $\mathcal{M}_{{i \rightarrow j},{k \rightarrow l}}= \delta_{l,i}(1-\delta_{j,k})$. Let $x=(x_{{i \rightarrow j}})_{({i \rightarrow j}) \in \vec{E}}$ be the set of random variables that satisfies ${\langlex_{{i \rightarrow j}}\rangle}=0$, ${\langlex_{{i \rightarrow j}}x_{{j \rightarrow i}}\rangle}=1$ and ${\langlex_{{i \rightarrow j}}x_{{k \rightarrow l}}\rangle}=0 \quad (\{i,j\} \neq \{k,l\})$. (Here and below ${\langle\cdots\rangle}_x$ stands for the mathematical expectation over the random variables $x$.) Then the following relation holds: $z_{LS} ={\langle \det [ I - i \mathcal{B} \mathcal{M}] \rangle}_x$, where $\mathcal{B}={{\rm diag}}(\sqrt{\beta_i^j/(1-\beta_i^j)} x_{{i \rightarrow j}})$.
For a general undirected graph $G$, the Ihara-Bass formula [@Idiscrete; @Bass] states that $$\zeta_{G}^{-1}(u)= \det[I-u \mathcal{M}]= (1-u)^{|E|-|{V}|} \det[I
+u^2(\mathcal{D}-I)-u\mathcal{A} ], \label{IB}$$ where $\mathcal{A}$ is the adjacency matrix and $\mathcal{D}={{\rm diag}}{(q_i;i\in V)}$ is the degree matrix of $G$. If we take the limit $u \rightarrow \infty$, this formula implies $\det{\mathcal{M}}=(-1)^{|E|} \prod_{i \in V}(1-q_i)$. Expanding the determinant, one derives $$\det [ I - i \mathcal{B} \mathcal{M}]
=\sum_{\{ e_1,\ldots,e_n \} \subset \vec{E} } \det \mathcal{M}|_{ \{ e_1,\ldots,e_n \} }
(-i)^{k} \prod_{l=1}^n (\mathcal{B})_{e_l,e_l}. \label{DetExpansion}$$ Evaluating the expectation of each summand in (\[DetExpansion\]), one observes that it is nonzero only if $({i \rightarrow j}) \in \{ e_1,\ldots,e_n \}$ implies $({j \rightarrow i}) \in \{ e_1,\ldots,e_n \}$, thus arriving at $$\fl
{\langle \det [ I - i \mathcal{B} \mathcal{M}] \rangle}_x
= \sum_{C \subset E} (-1)^{|C|} \det \mathcal{M}|_C \prod_{(i,j) \in C}
\frac{\beta_i^j}{1-\beta_i^j} = 1+\sum_{\emptyset \neq C \subset E} r_C.$$
For the doubly stochastic matrix of BP beliefs, $\beta$, and LS defined in (\[rC\]), one derives $$z_{LS}= {{\rm Perm}}(\beta.*(1-\beta)) \prod_{(i,j)\in E}(1-\beta_i^j)^{-1}.$$
We use theorem \[thmA1\], choosing the random variables $x_i^j=x_{{i \rightarrow j}}=x_{{j \rightarrow i}}$ that take $\pm 1$ values with probability $1/2$. We also utilize a multivariate version of the Ihara-Bass formula from [@10WF] to derive the following expression for $z_{LS}$ proving the theorem $$\begin{aligned}
\fl
\det [ I - i \mathcal{B} \mathcal{M}] =
\begin{small}
\det
\left[\begin{array}{cc}
0& \sqrt{\beta.*(1-\beta) }.*x \\
(\sqrt{\beta.*(1-\beta) }.*x)^{T}& 0 \\
\end{array}\right]\end{small}
\prod_{(i,j) \in E}(1-\beta_i^j)^{-1}, \\
\fl
z_{LS}
= {\langle\det(\sqrt{\beta.*(1-\beta) }.*x)^2\rangle}_x \prod_{(i,j)}(1-\beta_i^j)^{-1}
= {{\rm Perm}}(\beta.*(1-\beta)) \prod_{(i,j)}(1-\beta_i^j)^{-1}.\nonumber\end{aligned}$$
Invariance of the Gurvits-van der Waerden lower bound and new Lower Bounds for the Permanent {#sec:low}
============================================================================================
Van der Waerden [@26vdW] conjectured that the minimum of the permanent over the doubly stochastic matrices is $N^N/N!$, and it is only attained when all entries of the matrix are $1/N$. Though the conjecture appears to be simple, it remained open for over fifty years before Falikman [@81Fal] and Egorychev [@81Ego] finally proved it. Recently Gurvits [@08Gur] found an alternative, surprisingly short and elegant proof, that also allowed a number of unexpected extensions of the Van der Waerden conjecture. We call it the Gurvits-van der Waerden theorem. (See e.g. [@09LS].) A simplified form of this theorem is as follows.
\[T:Gurvits\] For an arbitrary non-negative $N \times N$ matrix $A$, $$\begin{aligned}
\fl
{{\rm Perm}}(A) \geq {{\rm cap}}(p_A) \frac{N^N}{N!},\ \ \mbox{where}\ \
p_A(x) \equiv \prod_i \sum_j a_{i,j}x_j, \ \ {{\rm cap}}(p_A) \equiv \inf_{x \in \mathbb{R}^N_{>0}}
\frac{p_A(x)}{\prod_j x_j}. \nonumber\end{aligned}$$
We have found that the lower bound of the theorem \[T:Gurvits\] has a “good” property with respect to the BP transformation. As stated in theorem \[LS\_new\], BP transforms the permanent to another permanent. Therefore, applying theorem \[T:Gurvits\] to both sides of (\[perm\]), one naturally asks how do the two lower bounds compare? A somewhat surprising result is that the Gurvits-van der Waerden theorem is invariant with respect to the BP transformation. Namely, ${{\rm cap}}(p_P)= Z_{BP}* {{\rm cap}}(p_{\beta.*(1-\beta)}) \prod_{(i,j)\in E}(1-\beta_i^j)^{-1}$. The lower bound for ${{\rm Perm}}(\beta.*(1-\beta))$ based on the theorem \[T:Gurvits\] is
\[Gurvits\_bound\] $${{\rm Perm}}(\beta.*(1-\beta))\geq \frac{N!}{N^N} \prod_{(i,j)\in E}(1-\beta_i^j)^{\beta_i^j}$$
This bound is the result of a direct application of the inequality, $\sum_j \beta_i^j(1-\beta_i^j)x_j
\geq \prod_j \left[(1-\beta_i^j)x_j \right]^{\beta_i^j}$, to theorem \[T:Gurvits\].
We also obtain another lower bound which improves the bound of corollary \[Gurvits\_bound\] at sufficiently low values of the temperature. See figure \[fig\]c for an illustration.
\[second\_low\] For an arbitrary perfect matching $\Pi$ (permutation of $\{1,\ldots,N\}$), $${{\rm Perm}}(\beta.*(1-\beta)) \geq
2 \prod_{i} \beta_i^{\Pi(i)}(1-\beta_i^{\Pi(i)})$$
Without loss of generality, we assume that $\Pi$ is the identity permutation. From the positivity of entries and (\[ds\_cond\]), we have ${{\rm Perm}}(\beta.*(1-\beta)) \geq \prod_i \beta_i^i {{\rm Perm}}(X)$, where $X_{ij}=\delta_{i,j}+(1-2\delta_{i,j})\beta_i^j$. Since $\beta$ is a stochastic matrix, $\det X=0$, and thus ${{\rm Perm}}(X) \geq 2 \prod_i
(1-\beta_i^i)$.
Note, for the sake of completeness, that a comprehensive review of other bounds on permanents of specialized matrices (for example $0,1$ matrices) can be found in [@86LP].
New Upper Bound for Permanent {#sec:up}
=============================
\[up\] $${{\rm Perm}}(\beta.*(1-\beta)) \leq \prod_j({1-\sum_i (\beta_i^j)^2}).$$ \[T:upper\]
We use the Godzil-Gutman representation for permanents [@81GG] $${{\rm Perm}}(\beta.*(1-\beta)) = {\langle \det(\sqrt{\beta.*(1-\beta)}.*\sigma)^2 \rangle}_\sigma,
\label{GG}$$ where $\sigma_i^j=\pm 1$, with $i,j=1,\ldots,N$ are independent random variables taking values $\pm 1$ equal probability. Each row of the matrix $\sqrt{\beta.*(1-\beta)}.*\sigma$ has the squared Euclid norm ${\sum_i \beta_i^j (1-\beta_i^j)} = {1-\sum_i (\beta_i^j)^2}$. Therefore, the upper bound is obtained from the Hadamard inequality, $|\det (a_1,\ldots,a_n) | \leq {\parallel a_1 \parallel}\cdots
{\parallel a_n \parallel}$.
Path Forward {#sec:path}
============
We consider this study to be the beginning of further research along the following lines: (1) More detailed analysis of the BP solution. In particular, study of $T_c$, e.g. concerning its dependence on the matrix size; analysis of the BP solution dependence on temperature; and the construction of an iterative algorithm provably convergent to a nontrivial BP solution for $T>T_c$. (2) Explanation of the BP invariance with respect to the Gurvits-van der Warden lower bound. (3) Development of a deterministic and/or randomized polynomial algorithm for estimating the permanent with provable guarantees based on the loop calculus expression. (4) Numerical tests of the lower and upper bounds for realistic large scale problems.
We are thankful to Leonid Gurvits for educating us, through his course of Lectures given at CNLS/LANL, about existing approaches in the “mathematics of the permanent”. YW acknowledges support of the Students Visit Abroad Program of the Graduate University for Advanced Studies which allowed him to spend two months at LANL and he is also grateful to CNLS at LANL for its hospitality. Research at LANL was carried out under the auspices of the National Nuclear Security Administration of the U.S. Department of Energy at Los Alamos National Laboratory under Contract No. DE C52-06NA25396. MC also acknowledges partial support of NMC via the NSF collaborative grant, CCF-0829945, on “Harnessing Statistical Physics for Computing and Communications”.
References {#references .unnumbered}
==========
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The ground state and magnetization process of the mixed spin-(1,1/2) Ising diamond chain is exactly solved by employing the generalized decoration-iteration mapping transformation and the transfer-matrix method. The decoration-iteration transformation is first used in order to establish a rigorous mapping equivalence with the corresponding spin-1 Blume-Emery-Griffiths chain in a non-zero magnetic field, which is subsequently exactly treated within the framework of the transfer-matrix technique. It is shown that the ground-state phase diagram includes just four different ground states and the low-temperature magnetization curve may exhibit an intermediate plateau precisely at one half of the saturation magnetization. Our rigorous results disprove recent Monte Carlo simulations of Zihua Xin *et al.* \[Z. Xin, S. Chen, C. Zhang, J. Magn. Magn. Mater. 324 (2012) 3704\], which imply an existence of the other magnetization plateaus at 0.283 and 0.426 of the saturation magnetization.'
address:
- 'Department of Theoretical Physics and Astrophysics, Faculty of Science, P. J. Šafárik University, Park Angelinum 9, 040 01 Košice, Slovak Republic'
- 'Institute for Condensed Matter Physics, National Academy of Sciences of Ukraine, 1 Svientsitskii Street, 79011 L’viv, Ukraine'
author:
- Bohdan Lisnyi
- Jozef Strečka
title: 'Ground-state phase diagram and magnetization process of the exactly solved mixed spin-(1,1/2) Ising diamond chain'
---
Ising model ,diamond chain ,spin frustration ,magnetization plateau 05.50.+q ,75.10.Hk ,75.10.Jm ,75.30.Kz ,75.40.Cx
Introduction
============
During the last few years, a considerable research interest has been devoted to the frustrated magnetism of diamond spin chains that was originally initiated by the effort to clarify several unusual magnetic features of the natural mineral azurite Cu$_3$(CO$_3$)$_2$(OH)$_2$ like for instance a presence of the one-third magnetization plateau in the low-temperature magnetization process [@kik04; @kik05; @kik06; @jes11; @hon11]. To provide a comprehensive description of the overall magnetic behaviour of the azurite, which is basically affected by a mutual interplay between the geometric spin frustration and quantum fluctuations, one has to employ a rather sophisticated combination of the first-principle density-functional calculations supplemented with the extensive numerical calculations [@jes11; @hon11]. However, it is worthy of notice that some important vestiges of this intriguing magnetic behaviour can be traced back also from the relevant behaviour of the exactly tractable spin-1/2 Ising diamond chain [@val08].
On the other hand, the exactly solvable mixed-spin Ising diamond chains have received much less attention so far. To the best of our knowledge, the mixed spin-(1/2,1) Ising chains with the spin-1/2 nodal atoms and the spin-1 interstitial (decorating) atoms have been just marginally investigated as special liming cases of the exactly solved mixed-spin Ising-Heisenberg diamond chains [@jas04; @can06; @roj11]. Quite recently, Zihua Xin *et al.* [@xin12] have studied another version of the mixed spin-(1,1/2) Ising diamond chain with the spin-1 nodal atoms and the spin-1/2 interstitial atoms within the framework of numerical Monte Carlo simulations. The main purpose of this work is to provide the exact analytical solution for this mixed-spin Ising diamond chain, which will convincingly contradict a presence of the striking magnetization plateaus at 0.283 and 0.426 of the saturation magnetization theoretically predicted in Ref. [@xin12].
This paper is organized as follows. The model and basic steps of the exact method will be clarified in Sec. \[model\]. The most interesting results for the ground-state phase diagrams and magnetization process of the symmetric as well as asymmetric mixed-spin Ising diamond chain will be widely discussed in Sec. \[result\]. Finally, our paper will end up with several concluding remarks and future outlooks mentioned in Sec. \[conclusion\].
Model and its exact solution {#model}
============================
Let us begin by considering the mixed spin-(1,1/2) Ising diamond chain in a presence of the external magnetic field. The magnetic structure of the investigated model system is schematically illustrated in Fig. \[fig1\] together with its primitive unit cell. As one can see, the primitive unit cell in a shape of diamond spin cluster involves two nodal Ising spins $S_{k}$ and $S_{k+1}$ along with two interstitial Ising spins $\mu_{k,1}$ and $\mu_{k,2}$. The total Hamiltonian for the mixed spin-(1,1/2) Ising diamond chain in a presence of the external magnetic field reads $$\begin{aligned}
{\cal H} &=&J_1 \sum_{k=1}^{N} S_{k} (\mu_{k,1} + \mu_{k-1,2}) + J_2 \sum_{k=1}^{N} \mu_{k,1} \mu_{k,2} \nonumber \\
&& + J_3 \sum_{k=1}^{N} S_{k} (\mu_{k-1,1} + \mu_{k,2}) - h \sum_{k=1}^{N} (S_{k} + \mu_{k,1} + \mu_{k,2}),
\label{htot}\end{aligned}$$ where the nodal Ising spins $S_k = \pm 1, 0$, the interstitial Ising spins $\mu_{k,\alpha} = \pm 1/2$ ($\alpha = 1,2$) and the periodic boundary conditions $\mu_{0,1} \equiv \mu_{N,1}, \mu_{0,2} \equiv \mu_{N,2}$ are implied for convenience. The interaction constants $J_1$ and $J_3$ label the nearest-neighbour interactions between the nodal and interstitial Ising spins along sides of the primitive diamond unit cell, while the interaction term $J_2$ accounts for the diagonal interaction between the nearest-neighbour interstitial spins from the same primitive cell. Finally, the Zeeman’s term $h$ determines the magnetostatic energy of the nodal and interstitial Ising spins in the external magnetic field.
![A fragment from the mixed-spin Ising diamond chain. The nodal ($S_{k}$, $S_{k+1}$) and interstitial ($\mu_{k,1}$, $\mu_{k,2}$) Ising spins belonging to the $k$th primitive unit cell are marked. The Ising interactions $J_1$ and $J_3$ along sides of the diamond unit cell are equal (different) in the symmetric (asymmetric) diamond chain.[]{data-label="fig1"}](fig1.eps){width="0.8\columnwidth"}
For further manipulations, it is quite advisable to rewrite the total Hamiltonian (\[htot\]) as a sum over cell Hamiltonians $${\cal H} = \sum_{k=1}^N {\cal H}_k,
\label{hk}$$ whereas the cell Hamiltonian ${\cal H}_k$ involves all the interaction terms of the $k$th diamond unit cell $$\begin{aligned}
{\cal H}_k &=& J_1 (\mu_{k,1} S_{k} + \mu_{k,2} S_{k+1}) + J_2 \mu_{k,1} \mu_{k,2}
+ J_3 (\mu_{k,1} S_{k+1} + \mu_{k,2} S_{k}) \nonumber \\
&& - h (\mu_{k,1} + \mu_{k,2}) - \frac{h}{2} (S_{k} + S_{k+1}).
\label{cell}\end{aligned}$$ The factor $\frac{1}{2}$ by the last term of Eq. (\[cell\]) avoids a double counting of the Zeeman’s term for the nodal Ising spins, which is equally split into two different cell Hamiltonians including one and the same nodal Ising spin. The partition function of the mixed spin-(1,1/2) Ising diamond chain can be written in this form $$\begin{aligned}
{\cal Z} = \sum_{\{S_k \}} \prod_{k=1}^N \sum_{\mu_{k,1}} \sum_{\mu_{k,2}} \exp (-\beta {\cal H}_k)
= \sum_{\{S_k \}} \prod_{k=1}^N {\cal Z}_k,
\label{pfd}\end{aligned}$$ where $\beta=1/(k_{\rm B} T)$, $k_{\rm B}$ is the Boltzmann’s constant, $T$ is the absolute temperature and the symbol $\sum_{\{S_k \}}$ marks a summation over all possible spin configurations of the nodal Ising spins. After performing the latter two summations over spin states of two interstitial Ising spins $\mu_{k,1}$ and $\mu_{k,2}$ one gets the effective Boltzmann’s factor, which can be subsequently replaced with a simpler equivalent expression through the generalized decoration-iteration transformation [@fis59; @roj09; @str10] $$\begin{aligned}
{\cal Z}_k &=& \sum_{\mu_{k,1}} \sum_{\mu_{k,2}} \exp (-\beta {\cal H}_k)
= 2 \exp \left[ \frac{\beta h}{2} \left(S_{k} + S_{k+1} \right) \right] \nonumber \\
&& \times \left \{ \exp \left( - \frac{\beta J_2}{4} \right)
\cosh \left[\frac{\beta}{2} \left(J_1 + J_3 \right) \left(S_{k} + S_{k+1} \right) - \beta h \right] \right.
\nonumber \\
&& \left. \quad + \exp \left(\frac{\beta J_2}{4} \right)
\cosh \left[\frac{\beta}{2} \left(J_1 - J_3 \right) \left(S_{k} - S_{k+1} \right)\right] \right \}
\nonumber \\
&=& A \exp \left[\beta R S_k S_{k+1} + \frac{\beta D}{2} \left(S_k^2 + S_{k+1}^2 \right) + \beta Q S_k^2 S_{k+1}^2 \right.
\nonumber \\
&& \left. ~\qquad + \beta L \left(S_k S_{k+1}^2 + S_k^2 S_{k+1} \right) + \frac{\beta h_0 }{2}\left(S_k + S_{k+1} \right) \right].
\label{dit}\end{aligned}$$ As usual, the transformation parameters $A$, $R$, $D$, $Q$, $L$, and $h_0$ are given by a self-consistency condition of the decoration-iteration transformation (\[dit\]), which requires that this mapping transformation must be valid independently of the spin states of two nodal Ising spins $S_k$ and $S_{k+1}$ involved therein [@fis59; @roj09; @str10]. By substituting all nine available spin states of two nodal Ising spins $S_k$ and $S_{k+1}$ into the transformation formula (\[dit\]) one merely gets six different expressions for the effective Boltzmann’s factor ${\cal Z}_{S_k, S_{k+1}}={\cal Z}_k (S_k, S_{k+1})$ $${\cal Z}_{0,0} = 2 \exp \left( -\frac{\beta J_2}{4} \right)
\cosh \left(\beta h \right) + 2 \exp \left(\frac{\beta J_2}{4} \right),$$ $$\begin{aligned}
{\cal Z}_{ \pm 1,0} &{=}& 2 \exp \left( {\pm} \frac{\beta h}{2} \right)
\! \left \{ \exp \left( \frac{\beta J_2}{4} \right) \cosh \left[\frac{\beta}{2} \left(J_1 - J_3 \right) \right] \right.
\\
&& \qquad\qquad\quad
\left. {+} \exp \left( - \frac{\beta J_2}{4} \right)
\cosh \left[\frac{\beta}{2} \left(J_1 {+} J_3 \mp 2h \right) \right] \right \},\end{aligned}$$ $$\begin{aligned}
{\cal Z}_{1,1} {=} 2 \exp \left(\beta h \right) \! \left \{ \exp \left( \frac{\beta J_2}{4} \right)
{+} \exp\! \left(\! {-} \frac{\beta J_2}{4} \! \right) \cosh \left[\beta \left(J_1 {+} J_3 {-} h \right) \right] \right \},\end{aligned}$$ $$\begin{aligned}
{\cal Z}_{1,-1} {=} 2 \exp \left( \frac{\beta J_2}{4} \right) \cosh \left[\beta \left(J_1 {-} J_3 \right) \right]
{+} 2 \exp \left( - \frac{\beta J_2}{4} \right) \cosh \left( \beta h \right),\end{aligned}$$ $$\begin{aligned}
{\cal Z}_{-1,-1} &{=}& 2 \exp \left({-}\beta h \right) \left \{ \exp \left( \frac{\beta J_2}{4} \right) \right.
\nonumber\\
&& \qquad\qquad\quad
\left. {+} \exp \left( {-} \frac{\beta J_2}{4} \right) \cosh \left[\beta \left(J_1 {+} J_3 {+} h \right) \right] \right \},
\label{bf}\end{aligned}$$ which unambiguously determine so far unspecified mapping parameters through the relations $$\begin{aligned}
A = {\cal Z}_{0,0}, \quad
\beta R = \frac{1}{4} \ln \left(\frac{{\cal Z}_{1,1} {\cal Z}_{-1,-1}}{{\cal Z}^{2}_{1,-1}}\right), \nonumber\end{aligned}$$ $$\begin{aligned}
\beta D &=& \ln \left( \frac{{\cal Z}_{1,0} {\cal Z}_{-1,0}}{{\cal Z}^{2}_{0,0}}\right), \quad
\beta Q = \frac{1}{4} \ln \left( \frac{{\cal Z}_{1,1} {\cal Z}_{-1,-1}{\cal Z}^2_{1,-1}{\cal Z}_{0,0}^4}{{\cal Z}^4_{1,0}{\cal Z}^4_{-1,0}}\right),
\nonumber \\
\beta L &=& \frac{1}{4} \ln \left( \frac{{\cal Z}_{1,1} {\cal Z}^2_{-1,0}}{{\cal Z}_{-1,-1} {\cal Z}^2_{1,0}} \right),
\quad \beta h_0 = \ln \left( \frac{{\cal Z}_{1,0}}{{\cal Z}_{-1,0}}\right).
\label{mp}\end{aligned}$$ If the decoration-iteration transformation (\[dit\]) with the mapping parameters obeying the relations (\[bf\])-(\[mp\]) is now substituted into Eq. (\[pf\]) one in turn obtains a simple mapping correspondence $$\begin{aligned}
{\cal Z} (\beta, J_1, J_2, J_3, h) = A^N {\cal Z}_{\rm BEG} (\beta, R, D, Q, L, h_0),
\label{pf}\end{aligned}$$ which relates the partition function ${\cal Z}$ of the mixed spin-(1,1/2) Ising diamond chain to the partition function ${\cal Z}_{\rm BEG}$ of the corresponding spin-1 Blume-Emery-Griffiths (BEG) chain [@blu71; @kri74; @kri75] given by the effective Hamiltonian $$\begin{aligned}
{\cal H}_{\rm BEG} = &-& R \sum_{k=1}^{N} S_k S_{k+1} - D \sum_{k=1}^{N} S_k^2 - Q \sum_{k=1}^{N} S_k^2 S_{k+1}^2 \nonumber \\
&-& L \sum_{k=1}^{N} (S_k S_{k+1}^2 + S_k^2 S_{k+1}) - h_0 \sum_{k=1}^{N} S_k.
\label{beg}\end{aligned}$$ It is quite evident from the effective Hamiltonian (\[beg\]) that the mapping parameters $R$, $D$, $Q$, $L$, and $h_0$ represent the effective bilinear interaction, the single-ion anisotropy, the biquadratic interaction, the two-spin third-order interaction and the magnetic field of the corresponding spin-1 BEG chain.
The exact solution for the partition function of the generalized spin-1 BEG chain can easily be found by means of the transfer-matrix approach [@kri74; @kri75; @bax82]. Let us therefore merely recall the basic steps of this well-known procedure. The partition function of the spin-1 BEG chain can be first factorized into the following product $$\begin{aligned}
{\cal Z}_{\rm BEG} = \sum_{S_1} \sum_{S_2} \cdots \sum_{S_N} \prod_{k=1}^N {\rm T} (S_k, S_{k+1}),
\label{pfbeg}\end{aligned}$$ where the expression ${\rm T} (S_k, S_{k+1})$ is defined as $$\begin{aligned}
{\rm T} (S_k, S_{k+1}) &=& \exp \left[\beta R S_k S_{k+1}
+ \frac{\beta D}{2} \left(S_k^2 + S_{k+1}^2 \right) + \beta Q S_k^2 S_{k+1}^2 \right. \nonumber \\
&& \left. + \beta L \left(S_k S_{k+1}^2 + S_k^2 S_{k+1}\right) + \frac{\beta h_0}{2} \left(S_k {+} S_{k+1} \right)\right].
\label{tm}\end{aligned}$$ The relevant expression given by Eq. (\[tm\]) can be considered as the usual transfer matrix $$\begin{aligned}
{\rm T} (S_k, S_{k+1}) = \left( \begin{array}{ccc}
{\rm T} (1,1) & {\rm T} (1,0) & {\rm T} (1,-1) \\
{\rm T} (0,1) & {\rm T} (0,0) & {\rm T} (0,-1) \\
{\rm T} (-1,1) & {\rm T} (-1,0) & {\rm T} (-1,-1)
\end{array}
\right),
\nonumber\end{aligned}$$ whereas the sequential summation over spin states of the spin-1 BEG chain will correspond to a multiplication of the relevant transfer matrices. As a result, the partition function can easily be calculated with the help of the respective eigenvalues $\lambda_i$ of the transfer matrix with regard to $$\begin{aligned}
{\cal Z}_{\rm BEG} = \! \sum_{S_1 = \pm 1, 0} {\rm T}^N (S_1, S_1) = \mbox{Tr} \, {\rm T}^N = \sum_{i=1}^3 \lambda_i^N.
\label{pfbege}\end{aligned}$$ For completeness, let us quote the final expressions for the three transfer-matrix eigenvalues $$\begin{aligned}
\lambda_i = r + 2 \, {\rm sgn} (q) \, \sqrt{p} \cos \left[\phi + (i-1) \frac{2\pi}{3} \right],
\label{evtm}\end{aligned}$$ where $$\begin{aligned}
{\rm sgn} (q) = \left \{\begin{array}{rl}
-1, & ~q < 0
\\[3pt]
1, & ~q \geq 0
\end{array}\right. , \nonumber\end{aligned}$$ $$\begin{aligned}
r &=& \frac{1}{3} \left[1 + 2 \exp(\beta R + \beta D + \beta Q) \cosh \left(\beta h_0 + 2 \beta L \right) \right], \nonumber \\
p &=& \frac{1}{4} (r-1)^2 + \frac{1}{3} \exp(2 \beta R + 2 \beta D + 2 \beta Q) \sinh^2 \left(\beta h_0 + 2 \beta L \right) \nonumber \\
&& + \frac{1}{3} \exp(-2 \beta R + 2 \beta D + 2 \beta Q) + \frac{2}{3} \exp(\beta D) \cosh \left(\beta h_0 \right), \nonumber \\
q &=& r^3 - \exp(\beta R {+} 2 \beta D {+} \beta Q) \cosh \left(2 \beta L \right)
+ \exp(-\beta R {+} 2 \beta D {+} \beta Q) \nonumber \\
&& + r \exp(\beta D) \cosh \left(\beta h_0 \right) + (1 {-} r) \exp(2 \beta D + 2 \beta Q) \sinh (2 \beta R) \nonumber \\
&& - r \exp(\beta R + \beta D + \beta Q) \cosh (\beta h_0 + 2 \beta L), \nonumber \\
\phi &=& \frac{1}{3} \arctan \left(\frac{\sqrt{p^3 - q^2}}{q} \right).
\label{pqr}\end{aligned}$$ In the thermodynamic limit $N \to \infty$, the partition function as well as the associated free energy per site of the spin-1 BEG chain is simply given by the largest transfer-matrix eigenvalue $\lambda_{\rm max} = {\rm max} \{ \lambda_1, \lambda_2, \lambda_3 \}$ $$\begin{aligned}
f_{\rm BEG} = -\frac{1}{\beta} \lim_{N \to \infty} \frac{1}{N} \ln {\cal Z}_{\rm BEG} = -\frac{1}{\beta} \ln \lambda_{\rm max}.
\label{frebege}\end{aligned}$$
Our rigorous calculation for the partition function of the mixed spin-(1,1/2) Ising diamond chain is thus formally completed, since it is now sufficient to substitute the exact result (\[pfbeg\]) for the partition function of the corresponding spin-1 BEG chain into the relevant mapping relation (\[pf\]) between both partition functions. As a result, the reduced free energy of the mixed spin-(1,1/2) Ising diamond chain per unit cell reads $$\begin{aligned}
f = f_{\rm BEG} -\frac{1}{\beta} \ln A = -\frac{1}{\beta} (\ln A + \ln \lambda_{\rm max}),
\label{free}\end{aligned}$$ while the total magnetization normalized with respect to its saturation value readily follows from the relation $$\begin{aligned}
\frac{m}{m_s} = - \frac{1}{2} \frac{\partial f}{\partial h} = \frac{1}{2} \left[\frac{\partial \ln A}{\partial (\beta h)}
+ \frac{\partial \ln \lambda_{\rm max}}{\partial (\beta h)}\right].
\label{mag}\end{aligned}$$
Results and discussions {#result}
=======================
Now, let us proceed to a discussion of the most interesting results obtained for the mixed spin-(1,1/2) Ising diamond chain with the antiferromagnetic interactions ($J_1, J_2, J_3>0$), whose magnetic behaviour may be fundamentally affected by the geometric frustration triggered by the competing diagonal interaction $J_2$ present in the diamond-like units. It is quite evident that the magnetic properties of the mixed spin-(1,1/2) Ising diamond chain given by the Hamiltonian (\[htot\]) remain invariant under the transformation $(J_1, J_3) \to (J_3, J_1)$, which allows us to consider $J_1 \geq J_3$ without loss of generality. For simplicity, we will pass to the dimensionless parameters $J_2/J_1$, $J_3/J_1$, and $h/J_1$ by normalizing all the interaction parameters with respect to the exchange constant $J_1$, which will hereafter serve as the energy unit. While the former interaction parameter $J_2/J_1$ measures a degree of the geometric frustration inherent in the investigated mixed-spin diamond chain, the latter interaction parameter $J_3/J_1 \in [0,1]$ characterizes a degree of the asymmetry of both Ising couplings along sides of the diamond units.
{width="85.00000%"}
First, let us establish the ground-state phase diagram for the symmetric as well as the asymmetric version of the mixed spin-(1,1/2) Ising diamond chain. Depending on the interplay between the interaction parameters $J_2/J_1$, $J_3/J_1$, and $h/J_1$, one finds in total four different ground states to be further referred to as the antiferromagnetic state (AF), the nodal antiferromagnetic state (NAF), the unsaturated paramagnetic state (UPA), and the saturated paramagnetic state (SPA). The relevant ground states are unambiguously given by the following spin configurations quoted along with the respective ground-state energies per primitive unit cell $$\begin{aligned}
|\mbox{AF} \rangle &=& \left \{\begin{array}{l}
\prod\limits_{k=1}^N \left|S_k = -1 \right\rangle \, \left|\mu_{k,1} = \frac{1}{2} \right\rangle \, \left|\mu_{k,2} = \frac{1}{2} \right\rangle
\\
\prod\limits_{k=1}^N \left|S_k = 1 \right\rangle \, \left|\mu_{k,1} = -\frac{1}{2} \right\rangle \, \left|\mu_{k,2} = -\frac{1}{2} \right\rangle
\end{array}\right.,
\nonumber \\
{\cal E}_{\rm{AF}} &=& -J_1 + \frac{J_2}{4} - J_3,
\nonumber \\
|\mbox{NAF} \rangle &=& \left \{\begin{array}{l}
{\prod\limits_{k=1}^N \left|S_k = (-1)^{k} \right\rangle \,
\left|\mu_{k,1} = \frac{(-1)^{k+1}}{2} \right\rangle \, \left| \mu_{k,2} = \frac{(-1)^{k}}{2} \right\rangle}
\\
{\prod\limits_{k=1}^N \left|S_k = (-1)^{k+1} \right\rangle \,
\left|\mu_{k,1} = \frac{(-1)^{k}}{2} \right\rangle \, \left|\mu_{k,2} = \frac{(-1)^{k+1}}{2} \right\rangle}
\end{array}\right.,
\nonumber \\
{\cal E}_{\rm{NAF}} &=& -J_1 -\frac{J_2}{4} + J_3,
\nonumber \\
|\mbox{UPA} \rangle &=& \prod\limits_{k=1}^N \left|S_k = 1 \right\rangle \,
\left \{
\left|\mu_{k,1} = \frac{1}{2} \right\rangle \, \left|\mu_{k,2} = -\frac{1}{2} \right\rangle
\atop
\left|\mu_{k,1} = -\frac{1}{2} \right\rangle \, \left|\mu_{k,2} = \frac{1}{2} \right\rangle
\right.,
\nonumber \\
{\cal E}_{\rm{UPA}} &=& -\frac{J_2}{4} - h,
\nonumber \\
|\mbox{SPA} \rangle &=& \prod\limits_{k=1}^N \left|S_k = 1 \right\rangle \, \left|\mu_{k,1} = \frac{1}{2} \right\rangle \, \left|\mu_{k,2} = \frac{1}{2} \right\rangle,
\nonumber \\
{\cal E}_{\rm{SPA}} &=& J_1 + \frac{J_2}{4} + J_3 - 2 h.\end{aligned}$$ Apparently, the two-fold degenerate AF ground state corresponds to the antiferromagnetic ordering, at which the nodal Ising spins are aligned in opposite to the interstitial Ising spins. Even though the nodal spins have a twice as large magnetic moment as the interstitial spins, the total magnetization completely cancels out on behalf of a twice as large number of the interstitial spins.[^1] Another two spin configurations inherent to the NAF ground state involve the antiferromagnetic alignment of the interstitial spins from the same diamond unit, as well as, the antiferromagnetic alignment of the nodal spins from the nearest-neighbour diamond units. Under this condition, one observes an interesting doubling of the magnetic unit cell when comparing it with the primitive diamond unit cell of the mixed-spin Ising diamond chain. However, the most intriguing spin alignment can be found in the macroscopically degenerate UPA ground state, where the nodal spins are fully polarized by the external magnetic field and the interstitial spins from the same diamond unit occupy one out of two equiprobable antiferromagnetic states $|\mu_{k,1} = 1/2 \rangle \, |\mu_{k,2} = -1/2 \rangle$ and $|\mu_{k,1} = -1/2 \rangle \, |\mu_{k,2} =1/2 \rangle$. The macroscopic degeneracy of the UPA ground state is accordingly proportional to a total number of the interstitial spin pairs, which is also reflected in the respective value of the residual entropy ${\cal S}_{\rm res} = N k_{\rm B} \ln2$. At sufficiently high magnetic fields, one finally detects the trivial SPA ground state with a full alignment of all nodal as well as interstitial spins into the external magnetic field.
{width="85.00000%"}
The relevant ground-state phase diagram including all the available ground states is displayed in Fig. \[fig2\] for the particular case of the symmetric diamond chain (Fig. \[fig2\]a), as well as, the more general case of the asymmetric diamond chain (Fig. \[fig2\]b). Let us at first comment the ground-state phase diagram of the symmetric diamond spin chain. It is quite obvious from Fig. \[fig2\]a that the interaction term $J_2/J_1$, which is responsible for a geometric spin frustration, enhances a stability region of the UPA ground state, whereas there are two different scenarios of the magnetization process depending on whether $J_2/J_1 < 4$ or $J_2/J_1 \geq 4$. In the former case, the AF spin alignment is being the respective ground state at low enough fields, the UPA ground state develops in a range of moderate fields and finally, the SPA ground state is stabilized at sufficiently high fields. On the other hand, the AF spin arrangement does not exist in the ground state for the latter particular case with $J_2/J_1 \geq 4$. Under this condition, the macroscopically degenerate UPA spin arrangement always represent the respective ground state below the saturation field, which determines the field-induced transition towards the fully aligned SPA ground state. As far as the ground-state phase diagram of the more general asymmetric diamond chain (Fig. \[fig2\]b) is concerned, one observes the same general trends in the relevant ground-state phase diagram with exception of a presence of the additional possible ground state that corresponds to the NAF spin arrangement. It can be easily understood from Fig. \[fig2\]b that the existence of the NAF ground state is restricted to the parameter region, where $J_2/J_1 > 4 J_3/J_1$ and simultaneously $h/J_1 < 1 - J_3/J_1$. The former inequality implies that the NAF ground state replaces the AF one in a rather wide region of the parameter space when considering the highly asymmetric diamond chain with $J_3/J_1 << 1$, which is also the condition of a sufficiently high persistence of the NAF ground state with respect to the external magnetic field according to the latter inequality.
To provide an independent verification of the established ground-state phase diagram, let us examine in detail the magnetization process of the mixed spin-(1,1/2) Ising diamond chain by investigating the field dependence of the total magnetization at different temperatures. For illustration, Fig. \[fig3\]a depicts the total magnetization normalized with respect to its saturation value in dependence on the reduced magnetic field for the fixed values of the interaction parameters $J_2/J_1=4$ and $J_3/J_1 = 0.5$. It is noteworthy that the zero-temperature magnetization curve, which is in accordance with the ground-state phase diagram shown in Fig. \[fig2\]b, is plotted in Fig. \[fig3\]a by a thin solid line along with the finite-temperature magnetization curves calculated with the help of Eq. (\[mag\]). In agreement with the ground-state phase diagram (Fig. \[fig2\]b), the zero-temperature magnetization curve exhibits two abrupt magnetization jumps reflecting two consecutive field-induced transitions between NAF-UPA and UPA-SPA at the relevant transition fields $h_{\rm c1}/J_1 = 0.5$ and $h_{\rm c2}/J_1 = 3.5$, respectively. It can be clearly seen from Fig. \[fig3\]a that the low-temperature magnetization curves closely follow the displayed zero-temperature magnetization curve (see for instance the magnetization curve for $k_{\rm B} T/J_1 = 0.01$), which proves a correctness of the established ground-state phase diagram and disproves an existence of any further ground state that would be reflected in some additional magnetization plateau. Apparently, the observed magnetization jumps and magnetization plateaus at zero and one-half of the saturation magnetization are just gradually smudged upon increasing temperature. To summarize, the low-temperature magnetization curves of the mixed spin-(1,1/2) Ising diamond chain exhibit at most two different magnetization plateaus before reaching the saturation magnetization, which coincide with a presence of the AF or NAF ground state with zero total magnetization and of the UPA ground state with the total magnetization equal to a half of the saturation magnetization.
Next, let us focus on typical temperature dependences of the total magnetization as depicted in Fig. \[fig3\]b for the same set of the interaction parameters $J_2/J_1=4$ and $J_3/J_1 = 0.5$ at several values of the external magnetic field. If the magnetic field is lower than the first critical field $h<h_{\rm c1}$, then, the total magnetization exhibits a round temperature-induced maximum as a function of the temperature when falling towards zero at sufficiently low and high temperatures. Contrary to this, the total magnetization starts from the one-half of the saturation value for the mediate magnetic fields $h \in (h_{\rm c1}, h_{\rm c2})$. The magnetization then either shows a monotonous temperature-induced decline for the magnetic fields slightly above the first transition field (i.e. $h \gtrapprox h_{\rm c1}$), or the non-monotonous temperature dependence with a round maximum for the magnetic field slightly below the saturation field (i.e. $h \lessapprox h_{\rm c2}$). Finally, the total magnetization always exhibits a more or less steep temperature-induced decrease when starting from the saturation value for the magnetic fields stronger than the saturation field $h>h_{\rm c2}$. For completeness, let us quote that the total magnetization normalized with respect to its saturation value starts from the non-trivial values $0.34151$ and $2/3$ for two particular magnetic fields equal to the transition fields $h_{\rm c1}$ and $h_{\rm c2}$, respectively.
Last but not least, let us make a few comments on the magnetization curves of the mixed spin-(1,1/2) Ising diamond chain obtained by Zihua Xin *et al*. [@xin12] by employing the Monte Carlo simulations. Note that Xin and co-workers have reported in Ref. [@xin12] two additional intermediate magnetization plateaus at 0.283 and 0.426 of the saturation magnetization, which were ascribed to a presence of some metastable states in the low-temperature magnetization curves. It should be pointed out, however, that the standard Monte Carlo simulation based on the Metropolis algorithm has been employed in Ref. [@xin12], which should only give the stable states in thermal equilibrium rather than metastable states. From this perspective, one has to refute both striking magnetization plateaus at 0.283 and 0.426 of the saturation magnetization reported in Ref. [@xin12], since they evidently contradict the exact analytical results presented in this work. It is worthwhile to remark that Monte Carlo simulations were performed in Ref. [@xin12] at the unusually low temperature $k_{\rm B} T/J_1 = 0.0001$, which might indicate extremely long relaxation times needed for establishing the thermal equilibrium and hence, one may consider an insufficient number of Monte Carlo steps used for equilibration at a given very low temperature as the main reason for the above mentioned discrepancy.
Conclusion
==========
In the present article, the ground state and magnetization process of the mixed spin-(1,1/2) Ising diamond chain have been rigorously studied by combining the generalized decoration-iteration transformation with the transfer-matrix method. In particular, our attention was focused on possible magnetization scenarios leading to the intermediate magnetization plateaus and the overall nature of available ground states. It has been demonstrated that the ground-state phase diagram of the symmetric diamond chain totally consist of three different ground states, while the ground-state phase diagram of the more general asymmetric diamond chain includes in total four different ground states. It has been evidenced that two different magnetization scenarios may in principle occur for the symmetric as well as asymmetric mixed-spin diamond chain depending on a mutual interplay between the interaction parameters. It has been actually evidenced that the low-temperature magnetization curves of the symmetric and asymmetric diamond chains display at most two different magnetization plateaus, which manifest a presence of the AF or NAF ground states with zero total magnetization and/or UPA ground state with the total magnetization equal to a half of the saturation magnetization. Owing to this fact, our exact analytical calculations refute recent Monte Carlo simulations by Zihua Xin *et al*. [@xin12], which have predicted two additional striking magnetization plateaus at 0.283 and 0.426 of the saturation magnetization.
From the methodological point of view, we have adapted in the present work the generalized decoration-iteration transformation in order to establish a rigorous mapping correspondence of the investigated mixed spin-(1,1/2) Ising diamond chain with the effective spin-1 BEG chain. To the best of our knowledge, this form of the generalized mapping transformation has been adapted so far just for a calculation of the zero-field properties of the mixed spin-(1,3/2) Ising linear chain [@fir97; @fir03]. The present work thus brings a rather simple recipe to greatly extend this rigorous mapping approach, which may be further substantially generalized in order to investigate magnetic properties of several exactly soluble mixed spin-(1,$S$) Ising and Ising-Heisenberg diamond chains in a non-zero magnetic field. As a matter of fact, the present approach can be rather straightforwardly extended in order to account for the quite general spin numbers of the interstitial spins, the more general Heisenberg interaction between the interstitial spins, the single-ion anisotropy, the next-nearest-neighbour and/or biquadratic interaction between the nodal spins and so on.
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[^1]: Note that both sublattice magnetizations completely cancel out just when assuming the equal $g$-factors for two different magnetic entities as originally imposed in the Hamiltonian (\[htot\]). In the real magnetic substances, one should however expect at least some small difference between the relevant $g$-factors, which should result in a non-zero total magnetization and the ferrimagnetic rather than the antiferromagnetic spin arrangement.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We examine the potential of the COMPASS experiment at CERN to study color transparency via exclusive vector meson production in hard muon-nucleus scattering. It is demonstrated that COMPASS has high sensitivity to test this important prediction of perturbative QCD.'
---
2[(c)\^2]{} PS. [|\_|]{} 2[r\^2 ]{} 2[$Q^2$]{} 2c2[GeV$^2$/$c^2$]{} 2[\^2]{} \#1\#2\#3\#4[[\#1]{} [**\#2**]{}, \#3 (\#4)]{} 2 2pigamma
TAUP-2671-2001\
12 June 2001
\
[— Feasibility Study —\
]{}
$^{2}$ Sołtan Institute for Nuclear Studies, ul. Hoża 69, PL 00-681 Warsaw, Poland,
$^{3}$ School of Physics and Astronomy, R. and B. Sackler Faculty of Exact Sciences,\
Tel Aviv University, 69978 Ramat Aviv, Israel
[*E-mails: sandacz@fuw.edu.pl, oleg@fuw.edu.pl, murraym@tauphy.tau.ac.il, eip@tauphy.tau.ac.il*]{}
Introduction {#lab_sec_1}
============
(CT) is a phenomenon of perturbative QCD (pQCD), whose characteristic feature is that small color-singlet objects interact with hadrons with small cross sections [@low; @bb; @bfgms; @fms]. Cross section for the interaction of such small object, or small size configuration (SSC), with a hadron target has been calculated in QCD using the factorization theorem [@bbfs; @fms; @fks; @frs]. These QCD calculations confirmed the hypothesis of F. Low [@low] on smallness of the cross section for the interaction of SSC with a hadron, if the gluon density in the hadron is not very high (moderately small $x$). They also predict a related phenomenon of [*color opacity*]{} when the gluon density becomes very large (at small $x$) and SSC interacts with the hadron with large cross section [@bb; @fms].
The prerequisite for observing CT is to select a sample containing SSC’s via a hard process (i.e. with large $Q^{2}\!$, high $p^{}_{t}\/$, or large produced mass). To suppress non-perturbative (not SSC) background different additional restrictions should be imposed depending on the process. For instance for hard exclusive $\rho ^0$ leptoproduction, in addition to large $Q^2$, selection of the longitudinally polarized mesons is required. To ’measure’ the SSC-nucleon cross sections one should study absorption of the SSC propagating through nuclear matter. In order to clearly observe CT it is necessary that the SSC lives long enough to traverse distances larger than the size of the target nucleus. Another requirement is that SSC stays small while propagating through the nucleus. These requirements, which are quantified in terms of the coherence length and the formation length, are discussed later.
Various processes have been proposed to study CT phenomenon.
- Coherent vector meson ($J \! / \! \psi\/$, $\rho $, $\phi $) production at small $t$. For near-forward ($t \approx 0$) coherent production and complete CT, one expects that the cross section for the nuclear target is related to that for the nucleon target by $$\frac{{\rm d}\sigma}{{\rm d}t}(\gamma ^* A \rightarrow V A) \, |^{}_{t \approx 0} =
A^2 \frac{{\rm d}\sigma}{{\rm d}t}(\gamma ^* N \rightarrow V N) \, |^{}_{t \approx 0} \: .$$ Gluon shadowing/antishadowing in nuclei is neglected in this formula, but for moderately small $x (> 0.01)$ this effect is expected to be small. Usually in experiments the $t$-integrated coherent cross sections are measured, for which CT predicts an approximate $A^{4/3}$ dependence. Using more realistic nuclear form factors one predicts $A^{1.40}$ [@Sokolov].
Otherwise, if the color-singlet objects, while propagating in nuclear matter, interact with nucleons with large cross sections, the expected $A$-dependence is weaker; $A^{2/3}$ for the cross sections comparable or larger than the pion-nucleon cross section.
- Incoherent (quasi-elastic) vector meson ($J \! / \! \psi\/$, $\rho $, $\phi $) production on nuclei. For complete CT and neglecting effects of the gluon shadowing/antishadowing in the nuclei one expects $$\frac{{\rm d}\sigma}{{\rm d}t}(\gamma ^* A \rightarrow V N (A-1)) =
A \frac{{\rm d}\sigma}{{\rm d}t}(\gamma ^* N \rightarrow V N) \: .$$ Here, for the nuclear target the meson $V\/$ is produced on a single nucleon of the nucleus and $(A-1)$ denotes the system of spectator nucleons.
- Coherent or incoherent production of excited vector meson states $\psi '$ or $\rho '$. CT predicts the same $A$ dependence of $J \! / \! \psi\/$ and $\psi '$, or $\rho $ and $\rho '$. This contrasts with the naive expectations whereby one may expect larger absorption for excited mesons since they are larger.
- Coherent diffractive dissociation of hadrons or photons into high $p_t$ di-jets. Such process probes the small transverse-size component of the projectile wave-function as well as CT effects. For CT the $t$-integrated cross section of coherent diffractive production of high $p_t$ di-jets has the same $A$-dependence as for the processes (a). Using more realistic wave functions one predicts for asymptotically high energies $A^{\alpha }$ dependence, where $\alpha $ is in the range 1.45 – 1.60, depending on $p_t$ [@fms].
- Coherent vector meson production on light nuclei (deuteron, helium) in the large $t$ range, where the effects of double (multiple) scattering are important. CT will suppress the double (multiple) scattering contribution to the differential cross section ${\rm d}\sigma /{\rm d}t$ [@fms; @fgkss].
- $A$-dependence of the fraction of large rapidity gap (diffractive) events. CT predicts an $A$ independent fraction; otherwise it will grow with $A$ [@rapgap].
- Large $Q^2$ quasielastic $(e,e'p)$ scattering on nuclei, $e A \rightarrow e p (A-1)$. In the CT limit the cross section will be proportional to $A$, as for process (b) [@almu].
- Large $t$ quasielastic $(p,2p)$ scattering on nuclei, $p A \rightarrow p p (A-1)$. Also for this process, for complete CT the expected $A$ dependence is like for process (b) [@almu].
In searches for CT a commonly used quantity, measured in experiments, is the [*nuclear transparency*]{} $$T = \frac{\sigma^{}_{\! \! A}}{A \, \sigma^{}_{\! N}} \: ,$$ which is the ratio of the cross section per nucleon for a selected process on a nucleus $A$ to the corresponding cross section on a free nucleon. For the incoherent processes CT predicts $T \simeq 1$ independently of $A$. For the large absorption in the nuclear matter $T$ will be smaller than unity and will decrease with $A$. Although the nuclear transparency could be defined also for coherent processes, usually cross sections for coherent production on different nuclei are compared directly. As mentioned above, for $t$-integrated coherent cross sections CT predicts $\sigma^{}_{\! \! A} \propto A^{4/3} \!$, whereas for a larger absorption the $A$-dependence is weaker.
In the following discussion we concentrate on exclusive vector meson production (VMP), the processes (a)–(c), which could be studied at the COMPASS experiment [@compass; @bradamante]. Exclusive meson production on the nucleon, free or bound in the nucleus, can be viewed as proceeding according to the diagram shown in Fig. \[diag\]. Kinematic variables used in this paper are listed in Table \[kinvar\].
----------------------------------------- ------------------------------------------------------------------
$k$ four-momentum of the incident muon,
$k'$ four-momentum of the scattered muon,
$p$ four-momentum of the target nucleon,
$v$ four-momentum of the vector meson $V$,
$q = k - k'$ four-momentum of the virtual photon,
–$Q^2 = q^2$ invariant mass squared of the virtual photon $\gamma^{\ast} \:$,
$\nu =(p\cdot q)/M^{}_{\! p}$ energy of the virtual photon in the laboratory system,
$M^{}_{\! p}$ is the proton mass,
$x = Q^2/(2 M^{}_{\! p} \nu )$ Bjorken scaling variable,
$y = (p\cdot q)/(p\cdot k)$ fraction of the lepton energy lost in the laboratory system,
$W^2 = (p+q)^2$ total energy squared in the $\gamma^{\ast} \! - \! N$ system,
$t = (q-v)^2$ four-momentum transfer to the target,
$p_t^2$ transverse-momentum squared of the vector meson
with respect to the virtual photon direction,
$m^{}_{\! V} = (v^2)^{\frac{1}{2}}$ invariant mass of the vector meson $V$,
$M^{2}_{\! X} = (p+q-v)^2$ missing-mass squared of the undetected recoiled system,
$I = (M^{2}_{\! X} - M^{2}_{\! p})/W^2$ inelasticity,
$z = (p\cdot v)/(p\cdot q)$ fraction of the virtual photon energy in the laboratory sy-
stem taken by meson $V$,
$z \approx 1-I $
----------------------------------------- ------------------------------------------------------------------
: Kinematic variables used in the text. \[kinvar\]
One of predictions of pQCD is that at high $Q^2$ the longitudinally polarized virtual photons $\gamma^{\ast}_{L}\/$ fluctuate into hadronic components, e.g. $q\bar{q}$ pairs, whose transverse size $b = \linebreak \: \mid \! \bar{r}_{\! \perp q} - \bar{r}_{\! \perp \bar{q}} \! \mid $ decreases with $Q^2 \!$, $b \propto (Q^{2})^{-1/2} \!$. At large $Q^2$ the values of $b$ are significantly smaller than the size of the nucleon. For instance, at $Q^2 = 10 \: \rm{GeV}^2$ and at $x = 0.01$ $b_{u\bar{u}} \simeq b_{d\bar{d}} \simeq 0.3 \;$fm [@fks], to be compared with the size of $\rho $ meson $b_{\! \rho } \simeq 1.4 \;$fm. For transversely polarized photons, in addition to the small size fluctuations, non-perturbative large size components may be expected, even at reasonable large $Q^2$ $(\simeq 10 \:\rm{GeV}^2)$.
Another pQCD prediction is that the heavy quarks fluctuations, e.g. $c\bar{c}$, of the virtual photon have small transverse size already for quasi-real production; $b_{c\bar{c}} = 0.22 \;$fm at $Q^2 \simeq 0$.
The total cross section for the interaction of a small size $q\bar{q}$ pair with the nucleon is given in pQCD by the formula [@bbfs] $$\sigma _{q\bar{q}, \: N} = \frac{\pi^2}{3} \: b^2 \: \alpha_s(Q^2) \; x \; g(x, \: Q^2) \;,$$ where $g(x, \: Q^2)$ is the gluon distribution function in the nucleon. The cross section for the interaction of $q\bar{q}$ pair of $b = 0.3 \;$fm with the nucleon is about 3$\;$mb at $x = 10^{-2}\,$ [@fks]. At very small $x$ the gluon distribution $g\/$ in the nucleon increases, which leads to an increase of the cross section $\sigma _{q\bar{q}, \: N}$. For instance, for $b = 0.3 \;$fm $\sigma _{q\bar{q}, \: N}$ is about 18$\;$mb at $x = 10^{-5} \!$. Therefore, even for the small $q\bar{q}$ objects the cross section becomes large at sufficiently small $x$. This phenomenon is mentioned earlier [*color opacity*]{} and it could be studied at small $x$ at future electron-nucleus colliders.
In order to study CT for the exclusive production of light quark mesons (like $\rho $ or $\rho'$) one should select [*large*]{} $Q^2 \!$, [*moderately small*]{} $x$ and [*longitudinally*]{} polarized mesons. For similar studies for the production of $J \! / \! \psi\/$ or $\psi '$ one should just select [*moderately small*]{} $x$.
In studies of CT an important role is played by the [*coherence length*]{} and the [*formation length*]{}. The coherence length $l_c$ (sometimes referred to as propagation or interaction length) is defined as the distance traversed by the $q\bar{q}$ fluctuation of the virtual photon in the target nucleon/nucleus system and is given by $$l_c = \frac{2 \nu}{Q^2+M^2} = \frac{\beta }{M^{}_{\! N} \: x} \; .$$ Here $\beta = Q^2 / (Q^2+M^2)$, $M$ is the invariant mass of the $q\bar{q}$ fluctuation, and $M^{}_{N}\/$ is the mass of the nucleon. For $l^{}_{c}\/$ values smaller than the size of the nucleus the life-time of the $q\bar{q}$ fluctuation is short and a chance for the hadronic fluctuation to interact in nuclear matter decreases. In particular this significantly inhibits coherent production. For incoherent production the effects due to the small $l^{}_{c}\/$ values mimic the $Q^2$ dependence of the nuclear transparency predicted by CT [@RPW].
The formation length is the distance in the target system needed for the $q\bar{q}$ fluctuation, which scattered on a nucleon, to develop into a hadron $h$. It is equal to $$l^{}_{\! f} = \frac{\nu }{m^{}_{h} \: \Delta m} \; ,$$ where $m^{}_{h}\/$ is the mass of the hadron $h$ and $\Delta m$ is the mass difference between the hadron $h$ and its lowest orbital excitation. For small $l^{}_{\! f}\/$ values the $q\bar{q}$ fluctuation evolves quickly into a full-size final hadron and the absorption of the final hadron $h$ in the nucleus plays a role.
Experimentally the effects of the coherence length and the formation length were analysed in detail by the HERMES experiment for incoherent exclusive production of $\rho^{0}\/$ on nuclei [@HERMES] and were shown to play an important role at the small $l^{}_{c}\/$ values.
Therefore, for a clean demonstration of CT effects the optimal conditions are when the values of $l^{}_{c}\/$ and $l^{}_{\! f}\/$ exceed the size of the target nucleus. If the above is not feasible, the variation of nuclear transparency $T(A,Q^2)$ with $Q^2$ should be studied for different, fixed values of coherence length. This way, a change of $T(A)$ between low and high $Q^2$ values could be associated with the onset of CT, and not with varying $l^{}_{c}\/$.
The $t$-dependence of the cross section for exclusive meson production on the nucleon is approximately exponential, $e^{b(t-t_{min})}$, where $t_{min} \approx -M_p^2x^2(1+ M^2/Q^2)^2$ and $M$ is the mass of the produced meson. For soft QCD processes the slope parameter $b$ should be independent of $Q^2$, depending only on the energy. This results from the Regge model [@collins]. For hard VMP, if SSC’s are important, one predicts [@Fra98] that the slope $b$ will significantly decrease with increasing $Q^2$ approaching a universal value related to the nucleon radius. This is so because the transverse interquark distances in the SSC decrease with increasing $Q^2$. This prediction agrees with high energy data on hard diffractive vector meson production on the proton [@h1; @zeus; @Fra98b].
For VMP on a nucleus $A$ the $t$-dependence of the cross section is approximately reproduced by a sum of two exponential functions. The peak at the lowest $t$ values, with the slope proportional to the nucleus squared radius $< \! R^{2}_{\! A} \! >$, is mostly due to the coherent production, whereas at somewhat larger $t$ the incoherent production on quasi-free nucleons dominates and the slope is equal to that for the production on free nucleons. Applying cuts on $t$ allows to select samples of events, which are strongly dominated by either coherent or incoherent production. It was successfully demonstrated in Refs [@Sokolov; @NMCJpsi; @Wei97].
In order to observe maximal coherent exclusive meson production for which the whole nucleus contributes, it is necessary to satisfy the condition $$|t_{min}< \! R^{2}_{\! A} \! >\!/3\, | << 1 \:.$$ In the COMPASS experiment for low $x$ values in the range 0.006 – 0.02 and for $Q^2 \approx 2-10 ~\:\rm{GeV}^2$, one has $|t_{min}< \! R^{2}_{\! A} \! >\!/3\, |
\approx ~10x^2 A^{2/3}$ and the condition (7) is satisfied even for lead $(A=207)$.
Experimental searches for CT started more than a decade ago and encompass various processes: large $t$ quasielastic $(p, \: 2p)$ scattering [@AGS], large $Q^2$ quasielastic $(e, \: e'p)$ scattering [@SLAC], $J \! / \! \psi\/$ photoproduction [@Sokolov] and $J \! / \! \psi\/$ muoproduction [@NMCJpsi], exclusive $\rho^{0}$ leptoproduction [@E665; @nmc; @HERMES] and coherent diffractive dissociation of the pion into two high-$p^{}_{t}\/$ jets [@Wei97].
The pioneering studies of CT in large $t$ quasi-elastic proton scattering [@AGS] found a rise of the nuclear transparency as the beam energy increased from 6 to 9 GeV and a decrease at higher energies. The large $Q^2$ quasi-elastic electron scattering studies [@SLAC] did not show $Q^2$ dependence. The explanation of these results in terms of CT is still debatable.
Strong recent evidence for CT comes from Fermilab E791 experiment on the pendence of coherent diffractive dissociation of pions into two high-$p^{}_{t}\/$ jets [@Wei97]. Also the E691 results on $A$-dependence of coherent $J \! / \! \psi\/$ photoproduction [@Sokolov] and the NMC measurements of $A$-dependence of coherent and quasielastic $J \! / \! \psi\/$ muoproduction are consistent with CT. Measurements of the nuclear transparency for incoherent exclusive $\rho^{0}$ production by Fermilab experiment E665 give a hint for CT. However, due to the low statistics of that data at high $Q^2 \!$, it was not possible to disentangle effects of decreasing $l^{}_{c}\/$ at high $Q^2$ and to demonstrate CT unambiguously.
Experimental method {#lab_sec_2}
===================
We propose to study CT via [*exclusive vector meson production*]{} $\mu A \rightarrow \mu \, V A$ (coherent) and $\mu A \rightarrow \mu \, V N (A-1)$ (incoherent) on various nuclei $A$ and optionally also on a proton or deuteron target. As a primary objective we propose to study the production of the following mesons $V$: $\rho^{0} \!$, $J \! / \! \psi\/$, $\phi $, $\psi' (\psi (2S))$ and $\rho' (\rho (1450), \rho (1700))$. Also investigations of the production of other mesons will be possible. The preferable decay modes are those into the charged particles: $\rho ^0 \rightarrow \pi^{+} \pi^{-} \!$, $J \! / \! \psi \rightarrow e^{+} e^{-} / \mu^{+} \mu^{-} \!$, $\phi \rightarrow K^+ K^- \!$, $\psi ' \rightarrow J \! / \! \psi \, \pi^{+} \pi^{-} \!$, $\rho' \rightarrow \pi^{+} \pi^{+} \pi^{-} \pi^{-} \!$. Two or more nuclear targets will be used. An additional proton (deuteron) target would be beneficial.
Our proposal is to complement the initial setup of the COMPASS [@compass; @bradamante] for the muon run with the polarized target, by adding two thin nuclear targets of lead and carbon of 17.6 g/cm$^2$ each. The carbon target will be a cylinder 8 cm long and of 3 cm of diameter. The lead target will consist of 4 discs of 3 cm of diameter, distributed over length of 8 cm. Only one nuclear target will be exposed to the muon beam at a time, with frequent exchanges (every few hours) of different targets. The nuclear targets will be located downstream of the polarized target, at the end of the solenoid magnet and before the first tracking detector (first micromega chamber).
The high-intensity high-energy incident muon-beam will impinge on the polarized target and a downstream thin nuclear target. The momenta of the scattered muon and of the produced charged particles will be reconstructed in the two magnetic spectrometers, using the magnets SM1 and SM2, instrumented with micromega chambers, drift chambers, GEM detectors, straw chambers, multiwire proportional chambers and scintillating fibers. Adding a recoil detector which will surround the nuclear target and register slow particles emitted from it may not be possible for the present setup due to the limited space. However, it would be advantageous and possible for dedicated runs taken with a modified setup. In the following we assume that the recoil particle(s) remain(s) undetected, and in order to select exclusive events one has to rely only on the kinematics of the scattered muon and the produced meson.
The trigger will use the information from hodoscopes registering the scattered muons, and from calorimeters registering deposits of energy of the particles in the final state. For VMP reactions triggering only on the scattered muon is in principle possible. However, for better efficiency of the trigger, especially at small $Q^2 \!$, it will be useful for certain processes to require in addition a minimum energy deposit in the calorimeters.
The off-line selection of exclusive events for the production of different mesons will be similar to that described for $\rho^{0}$ production in Ref. [@nmc; @memo2000; @pmmpsv]. In particular the discrimination of non-exclusive events will be done by applying cuts on the inelasticity $I\/$ (for the definition see Table \[kinvar\]). In Fig. \[ange\] the inelasticity distribution is shown for the SMC $\rho ^0$ sample [@smcrho] for the events with the invariant mass in the central part of the $\rho ^0$ peak. For the inelasticity distribution the peak at $I=0$ is the signal of exclusive $\rho ^0$ production. Non-exclusive events, where in addition to detected fast hadrons, slow undetected hadrons were produced, appear at $I>0$. For the cut $-0.05 < I < 0.05$ defining the exclusive sample the amount of the residual non-exclusive background for the SMC experiment was up to about 10% at large $Q^2$. The kinematical smearing in $I$ and the width of the elastic peak in COMPASS is expected to be about the same (cf. Sect. 3.2) as that shown in Fig. \[ange\] for the SMC experiment. Although the smearing will be similar, we expect the level of the non-exclusive background to be lower in COMPASS due to the wider angular and momentum acceptance coverage for final state hadrons. In addition, with larger statistic in COMPASS it will be possible to apply more tight inelasticity cuts, further reducing the background. The effect of this residual background on various observables will be studied by varying the inelasticity cuts, similarly as was done in Ref. [@nmc1].
The selections of coherent or incoherent production will be done on a statistical basis, using the $t$-distribution; at the lowest $\mid \! t \! \mid $ values coherent events predominate, whereas at somewhat larger $\mid \! t \! \mid $ there is almost clean sample of incoherent events.
Separation of the $\rho^{0}$ samples with the enhanced content of longitudinally or transversely polarized mesons will be done by applying cuts on the measured angular distributions of pions from the decays of the parent $\rho^{0}$.
The minimal covered $Q^2$ range is expected to be $0.05 < Q^2 < 10 \:\rm{GeV}^2 \!$. For the ium and large $Q^2$ values $(Q^2 > 2\:\rm{GeV}^2)$ the range $0.006 < x < 0.1$ will be covered with good acceptance.
The basic observable for each process studied will be the ratio of the nuclear transparencies for lead and carbon, $R^{}_{\rm T} = T_{\rm{Pb}}/T_{\rm{C}} =
(\sigma _{\rm{Pb}}/A_{\rm{Pb}})/(\sigma _{\rm{C}}/A_{\rm{C}})$. Due to the proposed geometry of the targets, the acceptances will cancel in the ratio $R^{}_{\rm T}$. Also the absolute beam flux measurement will not be necessary for the ratio $R^{}_{\rm T}$, provided that the relative determination of the beam fluxes for the exposures with different target materials could be done, e.g. by counting DIS events originating in the polarized target. The measured ratio $R^{}_{\rm T}$ should be corrected for different losses of events in lead and carbon, which are due to the secondary interactions in the targets. They will be estimated from the MC simulations.
Simulation of exclusive $\rho^{0}$ events {#lab_sec_3}
=========================================
In this section we describe details of the simulation of exclusive coherent $(\mu A \rightarrow \mu \, \rho^{0} A)$ and incoherent $(\mu A \rightarrow \mu \, \rho^{0} N (A-1))$ $\rho ^0$ production in the COMPASS experiment with the carbon and lead targets. The simulations were done with a dedicated fast Monte Carlo program which generates deep inelastic exclusive $\rho^{0}$ events with subsequent decay $\rho ^0 \rightarrow \pi^{+} \pi^{-} \!$. At this stage there was no attempt to include any background in the event generators. Here we describe the event generator as well as the treatment of different experimental aspects: losses due to the secondary interactions of pions, propagation through the magnetic fields, angular and momentum resolutions, muon trigger acceptance, acceptance for final state pions and efficiency of tracks reconstruction.
Radiative corrections for exclusive $\rho ^0$ production are expected to be similar to those for the NMC experiment, which were in the range of 2% to 5% [@kurek]. These corrections were not included in the present simulations.
Event generator
---------------
First we present the used parameterization of the cross section for the production on the free nucleon, $\mu \, N \rightarrow \mu \, \rho^{0} N$, with the subsequent decay $\rho^{0} \rightarrow \pi^{+} \pi^{-}$: $$\sigma _{\mu N \rightarrow \mu \rho^{0} N} =
\Gamma^{}_{\! T}(Q^2 \! , \: \nu) \cdot
\sigma^{\rm tot}_{\gamma^{\ast} N \rightarrow \rho^{0} N}(Q^2 \! , \: \nu) \cdot
F(p^{2}_{t}, \: \cos \theta, \: \phi ) \; ,$$ where $\theta\/$ and $\phi\/$ are, respectively, the polar and azimuthal angles of $\pi^{+}\/$ from the decay, calculated in the parent $\rho^{0}$ center-of-mass system, with respect to the direction of flight of $\rho^{0} \!$, $\Gamma^{}_{\! T}\/$ is the flux of transverse virtual photons $$\Gamma^{}_{\! T} = \frac{\alpha (\nu -\frac{Q^2}{2M^{}_{\! p}})}{2\pi Q^2 E^{2}_{\! \mu }
(1 - \epsilon)}\; ,$$ $\alpha\/$ is the fine-structure constant, $E^{}_{\! \mu}\/$ the muon-beam energy and $\epsilon\/$ is the virtual photon polarization given by $$\epsilon = \frac{1 - \frac{\nu }{E_{\! \mu }} - \frac{Q^2}{4E^2_{\! \mu }}}
{1 - \frac{\nu }{E_{\! \mu }}
+ \frac{1}{2}(\frac{\nu }{E_{\! \mu }})^2
+ \frac{Q^2}{4E^2_{\! \mu }}} \; .$$
The virtual photon cross section was parametrized as $$\sigma^{\rm tot}_{\gamma^{\ast} N \rightarrow \rho^{0} N}(Q^2 \! , \: \nu) = 27.4 \:\rm{nb}
\cdot \mbox{\Huge (} \frac{6\: \rm{GeV}^2}{Q^2} \mbox{\Huge )}^{\! \! 1.96} .$$ This is the NMC parametrisation of the cross sections per nucleon for exclusive $\rho^{0}$ production on carbon [@nmc]. As the NMC data shows little $A$-dependence of the virtual photon tegrated) cross section per nucleon, we use the same parametrisation of $\sigma^{\rm tot}_{\gamma^{\ast} N \rightarrow \rho^{0} N}$ for carbon and proton targets. This parametrisation does not apply at small $Q^2$ values, namely at $Q^2 < 1\: \rm{GeV}^2\!$.
The function $F\/$ comprises the $p^{2}_{t}\/$ distributions of produced $\rho^{0}$ and the angular distributions of pions coming from its decay $$F = a^{}_{L} \cdot f^{}_{L} \, (p^{2}_{t}) \cdot W^{}_{\! L} \, (\cos \theta, \: \phi ) +
a^{}_{T} \cdot f^{}_{\! T} \, (p^{2}_{t}) \cdot W^{}_{\! T} \, (\cos \theta, \: \phi ) \; .$$ Here $a^{}_{L} = r^{04}_{00}$, $a^{}_{T} = 1 - r^{04}_{00}$, $r^{04}_{00} = r^{04}_{00} (Q^{2} \!, \: \nu )$ is the $\rho^{0}$ density matrix element, which can be identified as a fraction of longitudinally polarized (helicity = 0) $\rho^{0}$ mesons, and the indices $L\/$ and $T\/$ refer to longitudinally and transversely polarized $\rho^{0}$’s, respectively. The fraction $r_{00}^{04}$ can be expressed [@SW] by the ratio $R$ of the cross sections for exclusive production by longitudinal and transverse virtual photons. We use a parametrisation of $R$ given by [@schildknecht] which reproduce data on exclusive $\rho ^0$ production in a wide range of $Q^2 \!$.
The $p^{2}_{t}\/$ distributions are described by $$f^{}_{\! i} (p^{2}_{t}) = b^{}_{i} \: e^{ -b^{}_{i} \: p^{2}_{t}} \: ,$$ where $i = L\/$ or $T\/$, $b^{}_{L} = 4.5 + 4 \cdot (0.5/Q^2) \: \rm{GeV}^{-2}$ and $b^{}_{T} = 8.5 \: \rm{GeV}^{-2} \!$. These parameterizations weighted by the fractions of longitudinally and transversely polarized mesons allow to reproduce reasonably the values of the effective slope $b$ for the exclusive $\rho ^0$ production measured at HERA and in the fixed-target experiments in a wide range of $Q^2 \!$.
The angular distributions of the pions from $\rho^{0}$’s decays are given by $$W^{}_{\! i} \, (\cos \theta, \: \phi ) = \frac{1}{2\pi } \: \frac{3}{4} \:
\mbox{\Large \{} (1 - P^{}_{\! i}) \: + \:
(3 P^{}_{\! i} - 1) \cos^{2} \! \theta \mbox{\Large \}} \; ,$$ where $i = L\/$ or $T\/$, $P^{}_{\! L} = 0$ and $P^{}_{\! T} = 1$. For the nuclear targets we assume the same $W^{}_{\! i}$ distributions as for the proton as suggested by [@nmc]. Note that the distributions $f^{}_{\! i} \, (p^{2}_{t})$, $W^{}_{\! i} \, (\cos
\theta, \: \phi )$ and $F(p^{2}_{t}, \: \cos \theta, \: \phi )$ are normalized to unity.
The invariant mass of two decay pions was generated using the relativistic p-wave Breit-Wigner shape for the $\rho $ resonance [@jackson].
We relate the differential cross sections for the proton $$\mbox{\Huge (} \frac{{\rm{d}} \sigma^{}_{\! N}}{{\rm{d}} t}
\mbox{\Huge )}^{}_{\! \! i}
\equiv \Gamma^{}_{\! T} \cdot \sigma^{\rm tot}_{\gamma^{\ast} N \rightarrow
% \rho^{0} N} \cdot a^{}_{i} \cdot f^{}_{\! i}(t) \; ,
\rho^{0} N} \cdot a^{}_{i} \cdot f^{}_{\! i}(-t) \; ,$$ to these for coherent and incoherent production on the nucleus $A$ by $$\mbox{\Huge (} \frac{{\rm{d}} \sigma^{\rm coh}_{\! \! A}}{{\rm{d}} t}
\mbox{\Huge )}_{\! \! i} =
A^{2}_{\rm eff \!, \; coh} \cdot e^{< R^{2}_{\! A} > \, t/3} \cdot
\mbox{\Huge (} \frac{{\rm{d}} \sigma^{}_{\! N}}{{\rm{d}} t} \mbox{\Huge
)}^{}_{\! \! i} \;\: ,$$ $$\mbox{\Huge (} \frac{{\rm{d}} \sigma^{\rm inc}_{\! \! A}}{{\rm{d}} t}
\mbox{\Huge )}^{}_{\! \! i} =
A^{}_{\rm eff \!, \; inc} \cdot \mbox{\Huge (} \frac{{\rm{d}} \sigma^{}_{\! N}}{{\rm{d}} t}
\mbox{\Huge )}^{}_{\! \! i} \;\: ,$$ respectively. Here $< \! R^{2}_{\! A} \! >$ is the mean squared radius of the nucleus, $A^{}_{\rm eff \!, \; coh}$ and $A^{}_{\rm eff \!, \; inc}$ take account of nuclear screening for the coherent and incoherent processes, correspondingly. The cross section for incoherent exclusive meson production is summed over all final states of the recoiling system, i.e. it is given for the so called closure approximation. The suppression of the incoherent cross section at small $t$ due to the Pauli blocking is neglected here. We used the approximation $t - t^{}_{\rm min} \simeq - p^{2}_{t}\/$, where $ \mid \! t^{}_{\rm min} \! \mid$ is the minimal kinematically allowed $\mid \!
t \! \mid$ value for given $W^{2} \!$, $Q^{2} \!$, $m^{}_{\! V}\/$ and $M^{2}_{\! X\/}$.
We generated the cross sections for two models. For the complete color transparency model (CT model) we used $A^{}_{\rm eff \!, \; coh} = A^{}_{\rm eff \!, \; inc} = A\/$. In another model we assumed a substantial nuclear absorption (NA model) and used $A^{}_{\rm eff \!, \; coh} = A^{}_{\rm eff \!, \; inc} = A^{0.75 \!}$. These could be compared to different predictions of the vector meson dominance model (VMD), which vary depending on the $\rho $ meson-nucleon total cross section and on the coherence length $l_c$ (see e.g. [@KM]). The $A$ dependences of the NA model are in the range of predictions of VMD, except for incoherent production at large $l_c$, where VMD predicts stronger nuclear absorption.
Simulation of the experimental effects
--------------------------------------
The secondary hadronic interactions of the decay pions in the target were simulated. The assumed density of targets, $\rho^{}_{\rm tgt}\/$, was 2.2 g/cm$^3 \!$. The interaction length $\lambda^{\pi}_{\rm int}\/$ for pions in the target material, was assumed equal to 130 g/cm$^2$ for the carbon and 290 g/cm$^2$ for the lead target. For an exclusive $\rho ^0$ event to be reconstructible, it was required that none of the decay pions underwent an inelastic hadronic interaction.
The trajectories of charged particles were simulated taking into account the geometry of SM1 and SM2 magnets of the COMPASS experiment. Homogeneous fields inside both magnets were assumed. For the SM1 magnet the bending power $\int \! B \, {\rm{d}}l\/$ was assumed equal to 1.0 T$\cdot$m (independent of the beam energy), whereas for the SM2 magnet it was assumed equal to 2.3 T$\cdot$m for a 100 GeV muon beam energy, and 5.2 T$\cdot$m for a 190 GeV muon beam. After tracking of the produced charged particles through the detector, a flag was assigned to each particle telling how far it propagated in the COMPASS setup.
Kinematic smearing of the beam, of the scattered muon and of the charged hadrons was simulated. Assumed values of dispersions of measured particle momenta and angles are based on the experience of previous muon experiments at CERN, as well as on the results of studies at COMPASS. The relative error on the momentum, $\sigma (p)/p$, was assumed equal to 0.5% for beam tracks, 0.75% for the tracks passing only the first magnet and 0.44% for the tracks passing the second magnet. The error on the angle of a particle was assumed to be 0.15 mrad. For a 190 GeV beam and the kinematic cuts listed in Section \[lab\_sec\_3\] the resulting smearing of the inelasticity $I$ is about 0.018 and the smearing of the invariant mass of two pions is about 6 MeV.
To simulate the trigger acceptance a trajectory of the scattered muon behind the second magnet was calculated, and the hits in the muon hodoscopes H4 and H5 were checked. Each of these two hodoscopes consists in fact of a few different hodoscopes (namely of the Ladders, the Primed System, the Unprimed System) but for the purpose of the present analysis we will consider just two cases, corresponding to different trigger acceptances. First, we assumed that only the Ladders and the Primed System are available. This trigger is called the Medium $Q^{2}$ range Trigger (MT). If in addition the Unprimed System is also implemented, the Full $Q^2$ range Trigger (FT) will result. The $Q^2$ dependence of the trigger efficiency, $\epsilon^{}_{\rm tr}\/$, for these triggers was presented in [@memo2000]. For the MT trigger at 190 GeV beam $\epsilon^{}_{\rm tr}$ decreases quickly with $Q^2$ from 0.7 at $Q^2 = 2 \:\rm{GeV}^2$ to about 0.1 at $Q^2 = 10 \:\rm{GeV}^2 \!$. The acceptance is several times smaller for this trigger at 100 GeV beam. For the FT trigger the $Q^2$ dependence is weaker and the trigger acceptance is higher; it is always bigger than 0.5 for $Q^2 < 70 \:\rm{GeV}^2$ at 190 GeV beam energy and for $Q^2 < 20 \:\rm{GeV}^2$ at 100 GeV beam. In conclusion, for the MT trigger the data taking with the higher beam energy seems the only acceptable choice, whereas for the FT trigger running at both beam energies is feasible, although the covered $Q^2$ range is larger at the higher beam energy.
For a $\rho^0$ meson to be accepted it was required that each pion from its decay was emitted in the laboratory at an angle within the acceptance of the SM1 magnet, and that its momentum was bigger than 2$\;$GeV.
Based on the preliminary results of the track reconstruction by the programs developed at COMPASS, simple and rather conservative assumptions were used to simulate the efficiency of tracks reconstruction. For the tracks seen only in the first spectrometer the single track reconstruction efficiency was assumed equal to 0.8 for the momentum range $p > 2 \:\rm{GeV}$, and for the tracks observed also in the second spectrometer it was assumed equal to 0.95. The efficiency $\epsilon^{}_{\rm rec}\/$ to reconstruct all three tracks of the scattered muon and of two pions was assumed to be equal $0.7 \cdot 0.95^3 + 0.3 \cdot 0.95^2 \cdot 0.80$. This assumption was motivated by the observation that for 70% of accepted DIS exclusive $\rho^{0}$ events all three measured tracks are seen in the second spectrometer, while for the remaining 30% events the scattered muon and the fast pion are seen in both first and second spectrometers, whereas the slow pion is observed in the first spectrometer only.
Results on exclusive $\rho ^0$ production {#lab_sec_4}
=========================================
Due to the higher trigger efficiency and larger $Q^2$ range at higher beam energy we considered 190 GeV muon beam. The simulations were done independently for the carbon $(A = 12)$ and lead targets $(A = 207)$, and for two triggers (MT and FT). For each target and each trigger we assumed two different models describing the nuclear effects for exclusive $\rho^{0}$ production: CT model and NA model (cf. Section 3.1).
The kinematic range considered was the following: $$\label{EQ_MC5} 2 < Q^2 < 80\: \rm{GeV}^2 \: ,$$ $$\label{EQ_MC6}
35 < \nu < 170\: \rm{GeV} \: .$$ The upper cut on $\nu $ was chosen to eliminate the kinematic region where the amount of radiative events is large, whereas the lower one to eliminate the region where the acceptance for pions from $\rho^{0}$ decay is low.
The total efficiency $\epsilon^{}_{\rm tot}\/$ to observe exclusive $\rho^{0}$ events results from: the acceptance of the trigger $(\epsilon^{}_{\rm tr})\/$, acceptance to detect the pions $(\epsilon^{}_{\rm had})\/$, efficiency for tracks reconstruction $(\epsilon^{}_{\rm rec})\/$, cut on the invariant mass of two pions, $0.62 < M^{}_{\pi \pi } < 0.92 \: \rm{GeV}^2 \!$, used for the selection of the samples ($\epsilon^{}_{\rm mass}\/$), and efficiency for an event to survive the secondary interactions $(\epsilon^{}_{\rm sec})\/$. The contributions of all these effects to $\epsilon^{}_{\rm tot}\/$ are similar to those presented in [@memo2000] for the polarized target, except for the effects of the secondary interactions. The approximate value of $\epsilon^{}_{\rm sec}\/$ is equal to 0.87 for the carbon target and 0.94 for the lead target. The total efficiency $\epsilon^{}_{\rm tot}\/$ is about 0.48 for the FT trigger and about 0.30 for the MT trigger.
The total expected cross section for exclusive $\rho ^0$ production on the nucleon is $$\sigma^{\rm tot}_{\mu N \rightarrow \mu \rho^{0} N} = \int_{\nu^{}_{\rm min}}^{\nu^{}_{\rm max}}
\! \! \int_{Q^{2}_{\rm min}}^{Q^{2}_{\rm max}}
\! \! \sigma^{}_{\mu N \rightarrow \mu \rho^{0} N} (Q^{2} \!, \: \nu ) \:
{\rm d} Q^{\! 2} \, {\rm d} \nu \; ,$$ where the kinematic range was defined before. The value of $\sigma^{\rm tot}_{\mu N \rightarrow \mu \rho^{0} N}\/$ is 283 pb. For nuclear targets the corresponding values depend on $A$ and on the assumed model for nuclear absorption.
The expected muon beam intensity will be about $10^{8} \! / \! \rm{s}$ during spills of length of about 2$\:$s, which will repeat every 14.4$\:$s. With the proposed thin nuclear targets, each of 17.6 g/cm$^2$, the luminosity will be ${\cal{L}} = 12.6 \:\rm{pb}^{-1} \cdot \rm{day}^{-1} \!$. The estimates of the numbers of accepted events were done for a period of data taking of 150 days (1 year), divided equally between two targets. An overall SPS and COMPASS efficiency of 25% was assumed. The numbers of accepted events for the carbon and lead targets, assuming the two models for the nuclear absorption mentioned earlier, are given in Table \[rates\].
\[5mm\]\[3mm\][ ]{} [ $N^{}_{\rm C}$ ]{} [ $N^{}_{\rm Pb}$ ]{}
---------------------- ---------------------- -----------------------
\[5mm\]\[3mm\][CT]{} [70 000]{} [200 000]{}
\[5mm\]\[3mm\][NA]{} [28 000]{} [20 000]{}
: Numbers of accepted events for two considered models of the nuclear absorption. \[rates\]
The distributions of accepted exclusive events as a function of $x$, $Q^2$ and $\nu $ are similar to those presented in [@memo2000].
In Fig.$\;$\[pt2\] we present the $p^{2}_{t}\/$ distributions for both targets. We observe clear coherent peaks at small $p^{2}_{t}\/$ ($< 0.05 \:\rm{GeV}^2$) and less steep distributions for the incoherent events at larger $p^{2}_{t}\/$. The arrows at the top histograms indicate the cut $p^{2}_{t} > 0.1 \:\rm{GeV}^2 \!$, used to select the incoherent samples. The contribution from coherent events is negligible in these samples. For the middle and bottom histograms the arrows indicate the cut $p^{2}_{t} < 0.02 \:\rm{GeV}^2 \!$, used to select the samples which are dominated by coherent events — the so called coherent samples. For the samples defined by the latter cut the fraction of the incoherent events is at the level of up to 10%, depending on the nucleus and on the model for nuclear absorption. We plan also to use the standard method to determine coherent and incoherent components, by fitting $p_t^2$ distribution.
The effect of the kinematical smearing on $p^{2}_{t}\/$ may be seen by comparing the distributions for the generated events (middle row) to the ones for measured events (bottom raw) where the acceptance and smearing were included. The smearing of $p^{2}_{t}\/$ increases with increasing $p^{2}_{t}\/$; it is about $0.006 \:\rm{GeV}^2$ for the coherent samples ($p^{2}_{t} < 0.02 \:\rm{GeV}^2$) and about $0.03 \:\rm{GeV}^2$ for the incoherent samples ($p^{2}_{t} > 0.1 \:\rm{GeV}^2$).
The analysis of the $\rho $ decay distributions allows us to study spin-dependent properties of the production process [@SW], in particular the polarization of $\rho $. Usually the $\rho^{0}$ decay angular distribution $W(\cos \theta, \: \phi )$ is studied in the $s$-channel helicity frame, which is the most convenient for describing the $\rho $ decay after photo- and electroproduction [@angdis]. The $\rho ^0$ direction in the virtual photon-nucleon centre-of-mass system is taken as the quantization axis. The angle $\theta \/$ is the polar angle and $\phi\/$ the azimuthal angle of the $\pi^{+}\/$ in the $\rho^{0}$ centre-of-mass system. The $\cos \theta \/$ distributions for pions from $\rho^{0}$ decays are shown in Fig.$\;$\[costh\] for the lead target. The distributions for longitudinally (dashed lines) and transversely (dotted lines) polarized parent $\rho^{0}$’s are markedly different. Their sum is also indicated.
Usually fits to the combined $\cos \theta \/$ distributions are performed in order to determine the density matrix element $r^{04}_{00}$ (cf. Eq. 11), which can be identified as the probability that the $\rho ^0$ was longitudinally polarized. For exclusive $\rho^{0}$ production the approximate $s$-channel helicity conservation (SCHC) is observed [@zeus1; @h1], i.e. the helicity of $\rho^{0}$ is predominantly equal to that of the virtual photon. Assuming SCHC and using the fitted $r^{04}_{00}$ one can estimate the ratio $R = \sigma _L/\sigma _T$ for exclusive virtual photoproduction ([@SW]). In COMPASS we plan to measure $R$ as well, and study its $Q^2$- and $A$-dependence, which is expected to reflect the strength of nuclear absorption.
As $Q^2$ increases, the approach to the CT limit is expected to be different for VMP by longitudinally polarized virtual photons from that by transversely polarized photons. Therefore, in order to increase the sensitivity of the search for CT, we propose another method. It consists in studying $A$-dependence of the cross sections for samples with different $\rho ^0$ polarizations, which will be selected by cuts on $\cos \theta \/$. For instance, after applying the cut $\mid \! \cos \theta \! \mid > 0.7$ the fraction of accepted events with $\rho^{0}_{L}\/$ is (80–95)% depending on the simulation, whereas for the cut $\mid \! \cos \theta \! \mid < 0.4$ the fraction of events with $\rho^{0}_{T}\/$ is (75–92)%. For an approximate SCHC, such cuts will allow us to select the samples with enhanced contributions of the events initiated by the virtual photons of a desired polarization. Studies of the samples with [*different polarizations*]{} of the virtual photons are [*important*]{} for the clear demonstration of CT.
Another aspect which is important for CT studies, is the covered range of the coherence length $l^{}_{c}\/$ (cf. Section 1). In Fig. \[lcq2\] we present the plot of $l^{}_{c}\/$ vs. $Q^{2}\/$ for a sample of accepted events. The effects of initial and final state interactions in the nuclei vary at small $l_c$ values [@HERMES]. Therefore, it was suggested that in order to disentangle effects due to CT from those caused by the modified absorption at small $l_c$ values, one should study $A$- and $Q^2$-dependences of cross sections at fixed values of $l_c$. This approach will be possible, if large statistics data were available.
For a limited statistics, a possible solution to avoid the mentioned effects is to use the combined data at $l_c$ values exceeding the sizes if the target nuclei. The radius of the carbon nucleus is $< r^{2}_{\rm C} >^{1/2} = 2.5\:\rm{fm}$ and that of the lead nucleus is $< r^{2}_{\rm Pb} >^{1/2} = 5.5\:\rm{fm}$ [@radii]. Therefore, one may use the selection $l^{}_{c} > l^{\rm min}_{c} \simeq 2 \cdot < r^{2}_{\rm Pb} >^{1/2} = 11 \:\rm{fm}$. The value of $l^{\rm min}_{c}\/$ is indicated in Fig.$\;$\[lcq2\] by arrows. About a half of events survive the cut on $l^{}_{c}$. These events cover the range of $Q^{2} < 6 \:\rm{GeV}^2 \!$, which is expected to be sufficient to observe CT.
The estimated values and statistical precision of $R^{}_{\rm T}$, the ratio of the nuclear transparencies for lead and carbon, are presented for different $Q^{2}$ bins in Fig.$\;$\[ratcoh\] for $p^{2}_{t} < 0.02 \:\rm{GeV}^{2}$ and in Fig.$\;$\[ratinc\] for $p^{2}_{t} > 0.1 \:\rm{GeV}^{2} \!$. The $Q^{2}$ bins are specified in Table \[q2bins\]. Each figure comprises predictions for two models, CT and NA, and for 6 different samples of accepted events for each model. For each sample a set of “measurements” in different $Q^2$ bins is shown. Sets [**A**]{} and [**B**]{} were obtained using the standard selections for the MT and FT triggers, respectively. Four remaining sets were obtained for the FT trigger with additional selections: [**C**]{} with $\mid \! \cos \theta \! \mid < 0.4$, [**D**]{} with $\mid \! \cos
\theta \! \mid > 0.7$, [**E**]{} with $\mid \! \cos \theta \! \mid < 0.4$ and $l^{}_{c} > 11\:\rm{fm}$, and [**F**]{} with $\mid \! \cos \theta \! \mid > 0.7$ and $l^{}_{c} > 11 \:\rm{fm}$. Note that for sets [**E**]{} and [**F**]{} only three lower $Q^{2}$ bins appear (cf. Fig.$\;$\[lcq2\]). One expects large differences in $R^{}_{\rm T}$ for the two considered models. For coherent samples $R^{}_{\rm T} \approx 5$ for CT model and $\approx 1$ for NA model. At $Q^2 \simeq 5 \:\rm{GeV}^2$ the precision of the measurement of $R^{}_{\rm T}$ for coherent events will be better than 17%, even for the restricted samples [**E**]{} and [**F**]{}, thus allowing excellent discrimination between the two models for the nuclear absorption. For the incoherent events the power to discriminate models by $R^{}_{\rm T}$ measurements will be more limited.
------------------------------ ---------------------------------- ------------------- ---------------
\[5mm\]\[2mm\][Bin number]{} $Q^{2}$ bin $<Q^{2}>$ $<x>$
\[4mm\]\[3mm\][$[\rm{GeV}^2]$]{} $[\rm{GeV}^2]$
\[5mm\]\[3mm\][1]{} 2–3 2.4 0.016
\[5mm\]\[3mm\][2]{} 3–4 3.4 0.022
\[5mm\]\[3mm\][3]{} 4–6 4.8 0.031
\[5mm\]\[3mm\][4]{} 6–9 7.2 0.048
\[5mm\]\[3mm\][5]{} 9–12 10.2 0.072
\[5mm\]\[3mm\][6]{} 12–20 14.8 0.11
------------------------------ ---------------------------------- ------------------- ---------------
: $Q^{2}$ bins used for the determination of $R^{}_{\rm T}$. \[q2bins\]
Comparison with previous experiments {#lab_sec_5}
====================================
In this section we compare COMPASS capabilities to demonstrate CT to those of previous experiments in which exclusive $\rho ^0$ leptoproduction on nuclear targets was studied. We concentrate on this reaction, as among different exclusive VMD channels it has the largest cross section. For the comparison we have selected experiments which covered $Q^2$ range extending to large values, bigger than 2 GeV$^2$. This condition is satisfied for the following experiments: HERMES [@HERMES], NMC [@nmc] and E665 [@E665].
The published results from the HERMES experiment concern the incoherent exclusive $\rho ^0$ production on $^1{\rm H}$, $^2{\rm H}$, $^3{\rm He}$ and $^{14}{\rm N}$ targets. The electron beam energy was 27.5 GeV and the covered $Q^2$ and $l_c$ ranges are $0.4 < Q^2 < 5.5\: {\rm GeV}^2$ and $0.6 < l_c < 8$ fm. Like in all fixed-target experiments with lepton beams, the covered kinematic range is such that for a given $Q^2$ the average value of $l_c$ decreases with increasing $Q^2$. The highest values of the HERMES data where one may expect to observe the onset of CT are correlated with the small values of $l_c$ below 2 fm. The increase of the nuclear transparency observed for $^{14}{\rm N}$ is explained as due to the reduced coherence length, without resorting to CT.
Wider kinematic ranges were coverd by experiments using high-energy muon beams. NMC has published the data on coherent and incoherent exclusive $\rho ^0$ production on $^2{\rm H}$, C and Ca targets. The muon beam energy was 200 GeV and the data cover the ranges $2 < Q^2 < 25\: {\rm GeV}^2$ and $1 < l_c < 30$ fm. For the NMC kinematic range, the large values of $l_c$ which exceed the diameter of the largest target nucleus (Ca), $l_c > 7$ fm, correspond to $Q^2 < 5.5\: {\rm GeV}^2$. In principle, in this range it is possible to study $Q^2$ dependence of nuclear absorption in order to observe CT not obscured by effects of small $l_c$. However, due to the moderate statistics of the data, such detailed analysis was not possible for NMC.
The most favourable kinematical conditions were realised in the E665 experiment due to the high muon beam energy of 470 GeV. The experiment has published the data on incoherent exclusive $\rho ^0$ production on $^1{\rm H}$, $^2{\rm H}$, C, Ca and Pb targets. The data were taken in a wide $Q^2$ range, including also very small values $Q^2 > 0.1\: {\rm GeV}^2$, and in a wide range of $l_c$, $1 < l_c < 200$ fm. At small and moderate $Q^2$ values, $< 3$ GeV$^2$, which correlate with $l_c$ values exceeding the diameter of the lead nucleus, precise measurements of the nuclear transparency were obtained. They indicate strong nuclear absorption. At the highest $Q^2$ bin, $Q^2 > 3\: {\rm GeV}^2$, the nuclear transparency increases in qualitative agreement with CT. However, the data in this bin correspond to a wide range of $l_c$, $1 < l_c < 40$ fm, and therefore could be affected by effects due to the reduced coherence length. The low statistics in the large $Q^2$ bin did not allow more detailed studies.
The COMPASS data at medium and large $Q^2$ (FT trigger) will cover the kinematic range similar to that of the NMC data. The expected statistics for the carbon target will be about 2 orders of magnitude higher than that of the NMC data for the same target. This increase is in particular due to higher beam intensity and larger acceptance in the COMPASS experiment. Similarly like in the E665 experiment COMPASS will use also lead target and will extend measurements to small $Q^2$. The statistics of COMPASS data will be significantly higher also than that of E665. For example, for incoherent events at $Q^2 > 3\: {\rm GeV}^2$ the ratio of nuclear transparencies for lead and carbon will be determined with accuracy better than 3%, comparing to about 30% for E665. Another more sensitive method to study nuclear effects in exclusive $\rho ^0$ production, the one by using coherent production, will be employed in COMPASS as well. Due to large statistics splitting of COMPASS data in several $Q^2$ and $l_c$ bins as well as the selection of events with longitudinal or transverese $\rho ^0$ polarization will be possible.
Open questions {#lab_sec_6}
==============
In addition to the $\rho ^0$ production, we will study exclusive production of $J \! / \! \psi\/$, $\phi $, $\psi'$ and $\rho'$ and their ratios in order to demonstrate CT. For these channels the analysis will be similar to that for $\rho^{0} \!$, but the rates will be lower due to the smaller production cross sections. The estimates of the required period of data taking and of the target thicknesses are subject of further analysis.
We consider also a possibility to study CT via coherent production of hadron pairs with large relative transverse momenta. The experimental procedure would be similar to that used in the Fermilab experiment E791 [@Wei97], in which CT was observed in coherent diffraction of pions into high-$p_t$ di-jets. We will select a sample of events with high-$p_t$ hadron pair and an additional requirement that all observed charged hadrons carry at least 90% of the energy of $\gamma ^*$. An estimate of $t$ will be obtained using momenta of all measured charged hadron tracks. The expected resolution in $t$, which will be crucial for selection of the coherent sample, will be determined from a dedicated simulation and reconstruction of events with high-$p_t$ hadron pairs.
Finally we mention few experimental aspects which have to be further investigated.\
Here we propose to position the nuclear targets downstream of the polarized target. Then the CT physics could complement the standard COMPASS program by using the downstream nuclear targets simultaneously with the polarized target. Similar target configuration was already used during the 1985 running of the EMC experiment, when the polarized target was used together with several nuclear targets situated downstream of PT.
The effect of the downstream nuclear targets on the hadrons coming from the polarized target should be small. For instance, for the reconstructible tracks of hadrons from PT, only about 4% of slow pions from $D^{\ast}$ decays and about 1% of hadrons from $D^{0}$ decays will be in the acceptance of the proposed nuclear target [@adam]. For the scattered muon coming from PT and passing through the nuclear target, the additional multiple Coulomb scattering should result in a small increase of the errors of the reconstructed position of the event vertex.
Assuming a dedicated run with unpolarized targets, different target configurations for CT studies are possible. With more available space one could envisage targets at least about 4 times thicker than the thin targets presented earlier. Thus the four-fold reduction of the data taking time will be possible. With an even larger amount of available beam time, one could expose more nuclear targets of different materials, and in addition the liquid hydrogen or deuterium targets. The latter ones are also important for measurements of [*off-forward parton distributions*]{} (also referred to as [*skewed parton distributions*]{} or [*generalized parton distributions*]{}) [@off]. For a dedicated run with unpolarized targets, one could take advantage of a possibility of adding a recoil detector. This is crucial for certain processes, e.g. Deep Virtual Compton Scattering (DVCS), but in any case it will help to reduce the non-exclusive background for any exclusive process.
Conclusions {#lab_sec_7}
===========
We have simulated and analysed exclusive $\rho^{0}$ muoproduction $(\mu A \rightarrow \mu \, \rho^{0} A$ and $\mu A \rightarrow \mu \, \rho^{0} N (A-1))$ at the COMPASS experiment using thin nuclear targets of carbon and lead. For muon beam energy of 190 GeV and a trigger for medium and large $Q^{2}$, the covered kinematic range is $2 < Q^2 < 20\: \rm{GeV}^2$ and $35 < \nu < 170\: \rm{GeV}$.
Good resolutions in $Q^{2} \!$, $l^{}_{c}$, $t$ ($p^{2}_{t}$) and $\cos
\theta \/$ are feasible. An efficient selection of coherent or incoherent events is possible by applying cuts on $p^{2}_{t}$. In order to obtain the samples of events initiated with a probability of about 80% by either $\gamma^{\ast}_{L}\/$ or $\gamma^{\ast}_{T}\/$, the cuts on the $\rho^{0}$ decay angular distribution of $\cos \theta \/$ will be used. The search for CT could be facilitated by using the events with $l^{}_{c}\/$ values exceeding the sizes of the target nuclei. The fraction of such events is substantial and the covered $Q^{2}$ range seems sufficient to observe CT.
We showed high sensitivity of the measured ratio $R^{}_{\rm T}$ of nuclear transparencies for lead and carbon for different models of nuclear absorption. Good statistical accuracy of the measured $R^{}_{\rm T}$ may be achieved already for one year of data taking. These measurements, taken at different $Q^{2}$ intervals, may allow to discriminate between different mechanisms of the interaction of the hadronic components of the virtual photon with the nucleus.
In conclusion, the planned comprehensive studies of exclusive vector meson production on different nuclear targets at the COMPASS experiment should unambiguously demonstrate CT.
Acknowledgements {#lab_sec_8}
================
The authors gratefully acknowledge useful discussions with L. Frankfurt. We thank also S. Kananov for discussions on the ZEUS exclusive $\rho ^0$ data. The work was supported in part by the Israel Science Foundation (ISF) founded by the Israel Academy of Sciences and Humanities, Jerusalem, Israel and by the Polish State Committee for Scientific Research (KBN SPUB Nr 621/E-78/SPUB-M/CERN/P-03/DZ 298/2000 and KBN grant Nr 2 P03B 113 19). One of us (A.S.) acknowledges support from the Raymond and Beverly Sackler Visiting Chair in Exact Sciences during his stay at Tel Aviv University.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The Warm-Hot Intergalactic Medium (WHIM) arises from shock-heated gas collapsing in large-scale filaments and probably harbours a substantial fraction of the baryons in the local Universe. Absorption-line measurements in the ultraviolet (UV) and in the X-ray band currently represent the best method to study the WHIM at low redshifts. We here describe the physical properties of the WHIM and the concepts behind WHIM absorption line measurements of [H]{} and high ions such as [O]{}, [O]{}, and [O]{} in the far-ultraviolet and X-ray band. We review results of recent WHIM absorption line studies carried out with UV and X-ray satellites such as FUSE, HST, Chandra, and XMM-Newton and discuss their implications for our knowledge of the WHIM.'
author:
- 'P. Richter'
- 'F.B.S. Paerels'
- 'J.S. Kaastra'
date: 'Received: 20 September 2007; Accepted: 21 September 2007'
title: 'FUV and X-ray absorption in the Warm-Hot Intergalactic Medium '
---
Introduction {#Introduction}
============
As recent cosmological simulations imply, the temperature of the intergalactic medium (IGM) undergoes a significant change from high to low redshifts parallel to the proceeding of large-scale structure formation in the Universe (e.g., @cen1999; @dave2001). As a result, a substantial fraction of the baryonic matter in the local Universe is expected to reside in the so-called Warm-Hot Intergalactic Medium (WHIM). The WHIM represents a low-density ($n_{\rm H}\sim10^{-6}-10^{-4}$ cm$^{-3}$), high-temperature ($T\sim 10^5-10^7$ K) plasma that primarily is made of protons, electrons, [He]{}, and [He]{}, together with traces of some highly-ionised heavy elements. The WHIM is believed to emerge from intergalactic gas that is shock-heated to high temperatures as the medium is collapsing under the action of gravity in large-scale filaments (e.g., @valageas2002). In this scenario, part of the warm (photoionised) intergalactic medium that gives rise to the Ly$\alpha$ forest in the spectra of distant quasars (QSO) is falling in to the potential wells of the increasingly pronounced filaments, gains energy (through gravity), and is heated to high temperatures by shocks that run through the plasma.
Because of the low density and the high degree of ionisation, direct observations of the shock-heated and collisionally ionised WHIM are challenging with current instrumentation (in contrast to the photoionised IGM, which is easily observable through the Ly$\alpha$ forest). Diffuse emission from the WHIM plasma must have a very low surface brightness and its detection awaits UV and X-ray observatories more sensitive than currently available (see, e.g., @fang2005; @kawahara2006). The most promising approach to study the WHIM with observations at low redshift is to search for absorption features from the WHIM in FUV and in the X-ray regime in the spectra of quasars, active galactic nuclei (AGN) and other suited extragalactic background sources. As the WHIM represents a highly-ionised plasma, the most important WHIM absorption lines are those originating from the electronic transitions of high-ionisation state ions (hereafter referred to as “high ions”) of abundant heavy elements such as oxygen and carbon. Among these, five-times ionised oxygen ([O]{}) is the most valuable high ion to trace the WHIM at temperatures of $T\sim 3\times 10^5$ K in the FUV regime. In the X-ray band, the [O]{} and [O]{} transitions represent the key observables to trace the WHIM at higher temperatures in the range $3\times 10^5 < T < 10^7$. In addition to the spectral signatures of high ions of heavy elements the search for broad and shallow Ly$\alpha$ absorption from the tiny fraction of neutral hydrogen in the WHIM represents another possibility to identify and study the most massive WHIM filaments in the intergalactic medium with FUV absorption spectroscopy. Finally, for the interpretation of the observed WHIM absorption features in UV and X-ray spectra the comparison between real data and artificial spectra generated by numerical simulations that include realistic gas physics is of great importance to identify possible pitfalls related to technical and physical issues such as limited signal-to-noise ratios and spectral resolution, line-broadening mechanisms, non-equilibrium conditions, and others.
In this chapter, we review the physics and methodology of the UV and X-ray absorption measurements of warm-hot intergalactic gas at low redshift and summarise the results of recent observations obtained with space-based observatories. The outline of this chapter is the following. The ionisation conditions of the WHIM and the most important absorption signatures of this gas in the UV and X-ray band are presented in Sect.2. Recent UV absorption measurements of the WHIM at low redshift are discussed in Sect.3. Similarly, measurements of the WHIM in the X-ray are presented in Sect.4. In Sect.5 we compare the results from WHIM observations with predictions from numerical simulations and give an overview of WHIM measurements at high redshift. Finally, some concluding remarks are given in Sect.6.
Physical properties of the WHIM {#Physical properties of the WHIM}
===============================
WHIM ionisation conditions {#WHIM ionisation conditions}
--------------------------
The occurrence and characteristics of the WHIM absorption signatures in the FUV and X-ray band are determined to a high degree by the ionisation conditions in the gas. We briefly discuss the WHIM ionisation properties, as this is crucial for interpretation of the WHIM absorption lines in FUV and X-ray spectra that arise in such warm-hot gas. Generally, there are two processes that determine the ionisation state of warm-hot gas in the intergalactic medium: collisional ionisation caused by the high temperature of the gas in collapsed structures and photoionisation by the cosmic FUV background.
### Hydrogen {#Hydrogen}
By far most of the mass of the WHIM is in the form of ionised hydrogen. Therefore, understanding the processes that lead to the ionisation of hydrogen is essential for the interpretation of WHIM absorption lines and for a reliable estimate of the baryon content of warm-hot intergalactic gas. The ionisation potential of neutral hydrogen is $13.6$ eV and thus both ionisation by particle collisions and ionisation by high-energy photons contribute to the ionisation of [H]{} in warm-hot gas. We start with collisional ionisation, which is believed to dominate the ionisation of hydrogen at temperatures $>10^5$ K.
In collisional ionisation equilibrium (CIE) – the most simple approach to characterise the ionisation conditions in low-density, high-temperature plasmas – the ionisation fraction depends only on the gas temperature. If we ignore any charge-exchange reactions (which is justified in case of hydrogen), the neutral hydrogen fraction in CIE is simply the ratio between the recombination coefficient $\alpha_{\rm H}(T)$ and the collisional ionisation coefficient $\beta_{\rm H}(T)$: $$f_{\mbox{H\,{\scriptsize I}}, \rm coll}=\frac{\alpha_{\rm H}(T)}{\beta_{\rm H}(T)}.$$
Above gas temperatures of $\sim 1.5 \times 10^4$ K collisions by thermal electrons efficiently ionise hydrogen to a high degree, and already at $T\sim 3 \times 10^4$ K the neutral hydrogen fraction in the gas is less then one percent. For the temperature range that is characteristic for the WHIM, $T=10^5-10^7$ K, one can approximate the ionisation fraction in a collisional ionisation equilibrium in the way $$\log\,f_{\mbox{H\,{\scriptsize I}}, \rm coll}
\approx 13.9 - 5.4\,{\rm log}\,T + 0.33\,
({\rm log}\,T)^2.$$ where $T$ is in units K (@richter2006a; @sutherland1993). Thus, for WHIM gas with $T=10^6$ K the neutral hydrogen fraction in the gas in CIE is only $\sim 2.4 \times 10^{-7}$.
Next to particle collisions, photons with energies $>13.6$ eV contribute to the ionisation of the WHIM, in particular in the low-temperature WHIM tail at $\sim 10^5$ K and below. Such ionising photons in intergalactic space are indeed provided by the metagalactic ultraviolet (UV) background, originating from the hard radiation emitted by QSOs and AGN. Fig. \[fig:fig1\] shows the spectral shape of the UV background at $z=0$ (left panel) and the redshift-dependence of the hydrogen photoionisation rate from the UV background (right panel) based on the models by @haardt1996.
![ [*Left panel:*]{} Spectral shape of the metagalactic UV background at $z=0$ (from @haardt1996). Plotted is the flux of photons ($F_{\nu}=4 \pi J_{\nu}$) against the frequency $\nu$. The hydrogen ionisation edge is indicated with a dashed line. [*Right panel:*]{} Redshift-dependence of the hydrogen photoionisation rate $\Gamma$ from the UV background for the range $z=0$ to $z=5$. Adapted from @haardt1996. []{data-label="fig:fig1"}](fig1.ps){width="\hsize"}
Considering photoionisation, one generally can write for the neutral-hydrogen fraction in the gas: $$f_{\mbox{H\,{\scriptsize I}},\rm photo} =
\frac{n_{\mathrm e}\,\alpha_{\rm H}(T)}{\Gamma_{\mbox{H\,{\scriptsize I}}}},$$ where $\alpha_{\rm H}(T)$ denotes the temperature-dependent recombination rate of hydrogen, $n_{\mathrm e}$ is the electron density, and $\Gamma_{\mbox{H\,{\scriptsize I}}}$ is the photoionisation rate. $\Gamma_{\mbox{H\,{\scriptsize I}}}$ depends on the ambient ionising radiation field $J_{\nu}$ (in units ergcm$^{-2}$s$^{-1}$Hz$^{-1}$ sr$^{-1}$) in the WHIM provided by the metagalactic UV background (see Fig. \[fig:fig1\]): $$\Gamma_{\mbox{H\,{\scriptsize I}}} = 4\pi\,\int\limits_{\displaystyle{\nu_{\rm L}}}^{\displaystyle{\infty}}
\frac{\sigma_{\nu} J_{\nu}}{{\mathrm h}\nu}\,{\mathrm d}\nu
\approx 2.5\times10^{-14}\,J_{-23}\,{\rm s}^{-1}.$$ Here, $\nu_{\rm L}$ is the frequency at the Lyman limit and ${\sigma_{\nu}}$ denotes the photoionisation cross section of hydrogen, which scales with $\nu^{-3}$ for frequencies larger that $\nu_{\rm L}$ (see @kaastra2008 - Chapter 9, this volume). We have introduced the dimensionless scaling factor $J_{-23}$ which gives the metagalactic UV radiation intensity at the Lyman limit in units $10^{-23}$ ergcm$^{-2}$s$^{-1}$Hz$^{-1}$ sr$^{-1}$. For $z=0$ we have $J_{-23}\sim 1-2$, while for $z=3$ the value for $J_{-23}$ is $\sim 80$, thus significantly higher [@haardt1996]. Assuming $n_{\rm e} = n_{\rm H}$ and inserting a proper function for $\alpha_{\rm H}(T)$, we finally can write for the logarithmic neutral hydrogen fraction in a purely photoionised WHIM plasma $${\rm log}\,f_{\mbox{H\,{\scriptsize I}}, \rm photo}
\approx {\rm log}\,\left( \frac{16\,n_{\rm H}
\,T_4^{-0.76}}{J_{-23}}\right),$$ where $n_{\rm H}$ is the hydrogen volume density in units cm$^{-3}$ and $T_4$ is the temperature in units $10^4$ K. Thus, for purely photoionised intergalactic gas at $z=0$ with $n=5\times 10^{-6}$ and $T=10^6$ K we find that the neutral hydrogen fraction is $ f_{\mbox{H\,{\scriptsize I}}, \rm photo} \sim 2.4 \times 10^{-6}$. This is ten times higher than for CIE, indicating that collisions dominate the ionisation fraction of hydrogen in intermediate and high-temperature WHIM regions. However, note that at lower temperatures near $T=10^5$ K at the same density we have $f_{\mbox{H\,{\scriptsize I}}, \rm photo}
\sim f_{\mbox{H\,{\scriptsize I}}, \rm coll}$. Since this is the WHIM temperature regime preferentially detected by UV absorption features (e.g., [O]{} and broad Ly$\alpha$), photoionisation is important and needs to be accounted for when it comes to the interpretation of WHIM absorbers observed in the FUV. From a WHIM simulation at $z=0$ including both collisional ionisation and photoionisation (@richter2006b; see Fig. \[fig:fig2\]) find the following empirical relation between neutral hydrogen fraction and gas temperature for a WHIM density range between log $n_{\rm H}=-5.3$ and $-5.6$: $${\rm log}\,f_{\mbox{H\,{\scriptsize I}}}
\approx 0.75 - 1.25\,{\rm log}\,T.$$
![ The neutral hydrogen fraction, log $f_{\rm H\,I}
={\rm log} (n_{\rm H\,I}/n_{\rm H})$, in a WHIM simulation (photoionisation$+$collisional ionisation), is plotted as a function of the gas temperature, log $T$. The light gray shaded indicates cells in the density range log $n_{\rm H}=-5$ to $-7$. The dark gray shaded area refers to cells that have log $n_{\rm H}=-5.3$ to $-5.6$, thus a density range that is characteristic for WHIM absorbers. Adapted from @richter2006b. []{data-label="fig:fig2"}](fig2.ps){width="0.67\hsize"}
This equation may serve as a thumb rule to estimate ionisation fractions in WHIM absorbers at $z=0$ if the gas temperature is known (e.g., from measurements of the line widths; see Sect.2.2.1).
![ CIE ion fractions of selected high ions of oxygen ([O]{}, [O]{}, [O]{}; left panel) and neon ([Ne]{}, [Ne]{}; right panel) in the WHIM temperature range log ($T$/K$)=4.5-7.0$, based on calculations by @sutherland1993. []{data-label="fig:fig3"}](fig3.ps){width="\hsize"}
### Oxygen and other metals {#Oxygen and other metals}
While hydrogen provides most of the mass in the WHIM, the most important diagnostic lines to study this gas phase are from highly ionised metals such as oxygen, neon, carbon, magnesium, and others. Therefore, the understanding of the ionisation properties of the observed high ions of these elements is as important as for hydrogen. As for hydrogen, both collisional ionisation and photoionisation need to be considered. With its single electron, hydrogen can only be either neutral or fully ionised. Heavy elements, in contrast, have several electrons available and are – even at very high temperatures – usually only partly ionised. Thus, electronic transitions exist for such highly-ionised metals (“high ions”) in warm-hot gas. Of particular importance for observations of the WHIM are the high ionisation states of oxygen, [O]{}, [O]{}, and [O]{}, as they have strong electronic transitions in the UV ([O]{}) and at X-ray wavelengths ([O]{} & [O]{}) and oxygen is a relatively abundant element. Another important metal for observing warm-hot gas in the UV and X-ray band is neon ([Ne]{}, [Ne]{}, [Ne]{}, [Ne]{}). In collisional ionisation equilibrium, the ionisation state of these elements is determined solely by the temperature of the gas. For each element, the ionisation fractions of the ionisation states (e.g., four-times vs. five-times ionised) then are characterised by the respective ionisation potentials (IPs) of the individual ionisation levels. For instance, at $T\sim 1-3 \times 10^5$ K, a significant fraction of the oxygen is five-times ionised (O$^{+5}$ or [O]{}, IP$=138$ eV). Six-times ionised oxygen (O$^{+6}$ or [O]{}, IP$=739$ eV) and seven-times ionised oxygen (O$^{+7}$ or [O]{}, IP$=871$ eV) predominantly exist at higher temperatures in the range $3\times 10^5 - 3\times 10^6$ K and $3\times 10^6 - 10^7$ K, respectively. Fig. \[fig:fig3\] shows the ionisation fractions of the most important high ions of oxygen and neon, based on the CIE calculations of @sutherland1993; see also @kaastra2008 - Chapter 9, this volume. High ions of other elements such as carbon, nitrogen, silicon and magnesium are less important for WHIM observations as their observable transitions trace lower temperature gas (e.g., [C]{}, [Si]{}) or the abundance of these elements in the intergalactic medium are too low. It is important to note at this point, that the discussed relation between ionisation state/fraction and gas temperature explicitly assumes that the gas is in an ionisation [*equilibrium*]{}. This may not be generally the case in the WHIM, however, as the densities are generally very low. For instance, under particular non-equilibrium conditions the timescales for cooling, recombination, and ion/electron equilibration may differ significantly from each other (see for instance @bykov2008 - Chapter 8, this volume). In such a case, the presence of high ions such as [O]{} and/or measured high-ion ratios would [*not*]{} serve as a reliable “thermometer” for the WHIM gas. In addition, WHIM filaments most likely neither are isothermal nor do they have a constant particle density. In fact, as WHIM simulations demonstrate, WHIM absorbers seem to represent a mix of cooler photoionised and hotter collisionally ionised gas with a substantial intrinsic density range. The absorption features from high ions arising in such a multi-phase medium therefore are generally difficult to interpret in terms of physical conditions and baryon budget.
In view of the high energies required to produce the high ions of oxygen and neon in combination with the spectral shape of the metagalactic background radiation (see Fig. \[fig:fig1\]), photoionisation of high metal ions in the WHIM is less important than for hydrogen. However, for [O]{} photoionisation is important at low redshifts in WHIM regions with very low densities or in systems located close to a strong local radiation source (e.g., in [O]{} systems associated with the background QSO). Note that at high redshift, most of the intervening [O]{} appears to be photoionised owing to the significantly higher intensity of the metagalactic background radiation in the early Universe (see Sect.5.2).
-------- ------------- ------------------ ------------- ------- ---------------------
Ion \[X/H\]$^1$ Ionisation Absorption Band CIE temperature$^2$
potential \[eV\] lines \[Å\] range \[$10^6$ K\]
[O]{} $-3.34$ $138$ $1031.926$ FUV $0.2-0.5$
$1037.617$
[O]{} $-3.34$ $739$ $21.602$ X-ray $0.3-3.0$
[O]{} $-3.34$ $871$ $18.969$ X-ray $1.0-10.0$
[Ne]{} $-4.16$ $239$ $770.409$ EUV $0.5-1.3$
$780.324$
[Ne]{} $-4.16$ $1196$ $13.447$ X-ray $0.6-6.3$
-------- ------------- ------------------ ------------- ------- ---------------------
: Data on O and Ne high ions having observable absorption lines
\
$^1$ \[X/H\] is the $\log$ of the number density of element X relative to hydrogen for Solar abundances, taken here from @asplund2004.\
$^2$ CIE models from @sutherland1993.
WHIM absorption signatures in the UV and X-ray band {#WHIM absorption signatures in the UV and X-ray band}
---------------------------------------------------
### UV absorption {#UV absorption}
As indicated in the previous subsection, five-times ionised oxygen ([O]{}) is by far the most important high ion to trace the WHIM at temperatures of $T\sim 3\times 10^5$ K in the ultraviolet regime (assuming CIE, see above). Oxygen is a relatively abundant element and the two lithium-like $(1s^22s)\,^2S_{1/2}\rightarrow(1s^22p)\,^2P_{1/2,3/2}$ electronic transitions of [O]{} located in the FUV at $1031.9$ and $1037.6$ Å have large oscillator strengths ($f_{1031}=0.133, f_{1037}=0.066$). Next to [O]{}, [Ne]{} traces WHIM gas near $T\sim 7\times10^5$ K (in collisional ionisation equilibrium) and thus is possibly suited to complement the [O]{} measurements of the WHIM in a higher temperature regime. The two available [Ne]{} lines are located in the extreme ultraviolet (EUV) at $770.4$ Å ($f_{770}=0.103$) and $780.3$ Å ($f_{780}=0.051$), allowing us to trace high-column density WHIM absorbers at redshifts $z>0.18$ with current FUV satellites such as FUSE. However, as the cosmic abundance of [Ne]{} is relatively low, [Ne]{} is not expected to be a particularly sensitive tracer of the WHIM at the S/N levels achievable with current UV spectrographs. The same argument holds for the high ion [Mg]{}, which has two transitions in the EUV at even lower wavelengths ($\lambda\lambda$ 609.8, 624.9 Å). So far, only [O]{} and in one case [Ne]{} has been observed in the WHIM at low redshift (see Sect.3.2). Note that WHIM absorption features by [O]{} (and [Ne]{}) are mostly unsaturated and the line profiles are fully or nearly resolved by current UV instruments such as FUSE and STIS, which provide spectral resolutions of $R=\lambda/\Delta \lambda \approx 20,000$ and $45,000$, respectively. Table 1 summarises physical parameters of O and Ne high ions and their observable transitions in the UV and X-ray bands.
Four-times ionised nitrogen ([N]{}; I.P. is $98$ eV) also is believed to trace predominantly collisionally ionised gas at temperatures near $T\sim 2\times10^5$ K, but its lower cosmic abundance together with its deficiency in low metallicity environments due to nucleosynthesis effects (e.g., @pettini2002) makes it very difficult to detect in the WHIM. Other available strong high-ion transitions in the UV from carbon ([C]{} $\lambda\lambda$ 1548.2, 1550.8 Å) and silicon ([Si]{} $\lambda\lambda$ 1393.8, 1402.8 Å) are believed to trace mainly photoionised gas at temperatures $T<10^5$ K, but not the shock-heated warm-hot gas at higher temperatures.
Next to high-ion absorption from heavy elements, recent UV observations (@richter2004; @sembach2004; @lehner2007) have indicated that WHIM filaments can be detected in Ly$\alpha$ absorption of neutral hydrogen. Although the vast majority of the hydrogen in the WHIM is ionised (by collisional processes and UV radiation), a tiny fraction ($f_{\rm H\,I}<10^{-5}$, typically) of neutral hydrogen is expected to be present. Depending on the total gas column density of a WHIM absorber and its temperature, weak [H]{} Ly$\alpha$ absorption at column densities $12.5\leq$ log $N$([H]{})$\leq 14.0$ may arise from WHIM filaments and could be used to trace the ionised hydrogen component. The Ly$\alpha$ absorption from WHIM filaments is very broad due to thermal line broadening, resulting in large Doppler parameters of $b>40$ kms$^{-1}$. Such lines are generally difficult to detect, as they are broad and shallow. High resolution, high S/N FUV spectra of QSOs with smooth background continua are required to successfully search for broad Ly$\alpha$ absorption in the low-redshift WHIM. STIS installed on the HST is the only instrument that has provided such data, but due to the instrumental limitations of space-based observatories, the number of QSO spectra adequate for searching for WHIM broad Ly$\alpha$ absorption (in the following abbreviated as “BLA”) is very limited.
The $b$ values of the BLAs are assumed to be composed of a thermal component, $b_{\rm th}$, and a non-thermal component, $b_{\rm nt}$, in the way that $$b=\sqrt{b_{\rm th}\,^2+b_{\rm nt}\,^2}.$$
The non-thermal component may include processes like macroscopic turbulence, unresolved velocity-components, and others (see @richter2006a for a detailed discussion). The contribution of the thermal component to $b$ depends on the gas temperature: $$b_{\rm th} = \sqrt \frac{2kT}{m} \approx 0.13 \, \sqrt
\frac{T}{A}\, \rm{km\,s}^{-1},$$
where $T$ is in K, $k$ is the Boltzmann constant, $m$ is the particle mass, and $A$ is the atomic weight. For the shock-heated WHIM gas with log $T\geq5$ one thus expects $b_{\rm th}\geq40$ kms$^{-1}$. The non-thermal broadening mechanisms are expected to contribute to some degree to the total $b$ values in WHIM absorbers (see @richter2006a), so that the measured $b$ value of a BLA provides only an upper limit for the temperature of the gas.
### X-ray absorption {#X-ray absorption}
The highest ionisation phase of the WHIM will produce and absorb line radiation primarily in the He- and H-like ions of the low-$Z$ elements (C, N, O, Ne), and possibly in the L-shell ions of Fe. In practice, much of the attention is focused on oxygen, because of its relatively high abundance, and because the strongest resonance lines in He- and H-like O are in a relatively ’clean’ wavelength band. For reference, the Ly$\alpha$ transitions of [C]{}, [N]{}, [O]{}, and [Ne]{} occur at 33.7360, 24.7810, 18.9689, and 12.1339 Å, respectively (wavelengths of the $1s-2p_{1/2,3/2}$ doublet weighted with oscillator strength; @johnson1985. The He-like ions [C]{}, [N]{}, [O]{}, and [Ne]{} have their strongest transition, the $n=1-2$ resonance line, at 40.2674, 28.7800, 21.6015, and 13.4473 Å (@drake1988; see also Table 1). Data on the higher order series members can be found in @verner1996. As far as the Fe L shell ions are concerned, the most likely transition to show up would be the strongest line in Ne-like [Fe]{}, $n=2p-3d$ $\lambda 15.014$ Å. In addition, all lower ionisation stages of C, N, O, and Ne (with the exception of neutral Ne of course) can also absorb by $n=1-2$; the strongest of these transitions would be $1s-2p$ in [O]{} at 22.019 Å [@schmidt2004]. Likewise, the lower ionisation stages of Fe could in principle produce $n=2-3$ absorption.
The thermal widths of all these transitions will be very small, requiring resolving powers of order $R\sim 10\,000$ (C, N, O, Ne) for gas temperatures of order $10^6$ K to be resolved; for Fe, the requirement is even higher, by a factor $\sim 2$. As we will see, for practical reasons, these requirements exceed the current capabilities of astrophysical X-ray spectroscopy by a large factor. Due to the small Doppler broadening (ignoring turbulent velocity fields for now), the lines will rapidly saturate. For He- and H-like O resonance line absorption, saturation sets in at an equivalent width of order 1 mÅ (@kaastra2008 - Chapter 9, this volume), or column densities of order a few times $10^{14}$ ions cm$^{-2}$. The challenge, therefore, for X-ray spectroscopy presented by the IGM is to detect small equivalent width, near-saturation lines that are unresolved.
The baryon content of the WHIM as measured by UV and X-ray absorbers {#The baryon content of the WHIM as measured by UV and X-ray absorbers}
--------------------------------------------------------------------
One important result from absorption line measurements of the WHIM in the UV is the observed number density of WHIM absorbers, usually expressed as ${\mathrm d}N/{\mathrm d}z$, the number of absorbers per unit redshift. For instance, from recent measurements with FUSE and HST/STIS one finds for [O]{} absorbers and Broad Ly$\alpha$ absorbers at $z\approx0$ values of ${\mathrm d}N/{\mathrm d}z$([O]{}$)\approx 20$ and ${\mathrm d}N/{\mathrm d}z$(BLA$)\approx 30$ (see Sect.3.2). Currently, the WHIM absorber density is only measurable in the UV, since in the X-ray band both the observed number of WHIM absorption lines and the available redshift path for WHIM observations is too small to derive statistically significant values of ${\mathrm d}N/{\mathrm d}z$([O]{}) and ${\mathrm d}N/{\mathrm d}z$([O]{}).
A particularly interesting question now is, how the observed number density of high-ion lines or BLAs translates into an estimate of the cosmological baryon mass density of the WHIM, $\Omega_{\mathrm b}$(WHIM). To obtain such an estimate of the baryon content of the WHIM from UV and X-ray absorption measurements one has to consider two main steps. First, one needs to transform the observed column densities of the high ions (e.g., [O]{}, [O]{}, [O]{}) into a total gas column density by modelling the ionisation conditions in the gas. In a second step, one then integrates over the total gas column densities of all observed WHIM absorbers along the given redshift path and from that derives $\Omega_{\mathrm b}$(WHIM) for a chosen cosmology. Throughout the paper we will assume a $\Lambda$CDM cosmology with $H_0=70$ kms$^{-1}$Mpc$^{-1}$, $\Omega_{\Lambda}=0.7$, $\Omega_{\mathrm m}=0.3$, and $\Omega_{\mathrm b}=0.045$. For the first step the uncertainty lies in the estimate of the ionisation fraction of hydrogen of the WHIM. For this, it is usually assumed that the WHIM is in collisional ionisation equilibrium, but photoionisation and non-equilibrium conditions may play a significant role. In the case of using metal ions such as [O]{} the unknown oxygen abundance (O/H) of the gas introduces an additional uncertainty (see below) for the estimate of $\Omega_{\mathrm b}$(WHIM). For the second step, it is important to have a large enough sample of WHIM absorption lines and a sufficient total redshift path along [*different*]{} directions in order to handle statistical errors and the problem of cosmic variance. As mentioned earlier, these requirements currently are fulfilled only for the UV absorbers.
The cosmological mass density $\Omega_{\mathrm b}$ of [O]{} absorbers (and, similarly, for other high ions) in terms of the current critical density $\rho_{\rm c}$ can be estimated by $$\Omega_{\rm b}(\mbox{O\,{\scriptsize VI}})=\frac{\mu\,m_{\rm H}\,H_0}
{\rho_{\rm c}\,c}\,\sum_{ij}\,\frac{N(\mbox{O\,{\scriptsize VI}})_{ij}}
{f_{\mbox{O\,{\scriptsize VI}},ij}\,{\rm (O/H)}_{ij}\,\Delta X_j}.$$ In this equation, $\mu=1.3$ is the mean molecular weight, $m_{\rm H}=1.673 \times 10^{-27}$ kg is the mass per hydrogen atom, $H_0$ is the adopted local Hubble constant, and $\rho_{\rm c}=3H_0\,^2/8 \pi G$ is the current critical density. The index $i$ denotes an individual high-ion absorption system along a line of sight $j$. Each measured high-ion absorption system $i$ is characterised by its measured ion column density (e.g., $N$([O]{})$_{ij}$), the ionisation fraction of the measured ion (e.g., $f_{\mbox{O\,{\scriptsize VI}},ij}$), and the local abundance of the element measured compared to hydrogen (e.g., the local oxygen-to-hydrogen ratio, by number). Each line of sight $j$ has a characteristic redshift range $\Delta z$ in which high-ion absorption may be detected. The corresponding comoving path length $\Delta X$ available for the detection of WHIM high-ion absorbers then is given by: $$\Delta X_j=(1+z)^2\,[\Omega_{\Lambda}+\Omega_{\rm m}(1+z)^3]^{-0.5}\,\Delta z_j.$$ In analogy, we can write for the cosmological mass density of the BLAs: $$\Omega_{\rm b}{\rm (BLA)}=\frac{\mu\,m_{\rm H}\,H_0}
{\rho_{\rm c}\,c}\,\sum_{ij}\,\frac{N(\mbox{H\,{\scriptsize I}})_{ij}}
{f_{\mbox{H\,{\scriptsize I}},ij}\,\Delta X_j}.$$
As can be easily seen, the advantage of using BLAs for deriving the WHIM mass density is that the metallicity of the gas is unimportant for the determination of $\Omega_{\mathrm b}$. The disadvantage is, however, that the ionisation corrections are very large and uncertain, since they are determined indirectly from the BLA line widths (see Sect.2.2.1).
UV measurements of the WHIM {#UV measurements of the WHIM}
===========================
Past and present UV instruments {#Past and present UV instruments}
-------------------------------
The first and second generations of space based UV spectrographs such as [*Copernicus*]{} and the [*International Ultraviolet Explorer*]{} (IUE) did not have sufficient sensitivity to systematically study intervening absorption in the intergalactic medium along a large number of sightlines. The early low- and intermediate resolution spectrographs installed on the [*Hubble Space Telescope*]{} (HST), namely the [*Faint Object Spectrograph*]{} (FOS) and the [*Goddard High Resolution Spectrograph*]{} (GHRS), were used to study the properties of the local Ly$\alpha$ forest and intervening metal-line systems (e.g., @stocke1995; @shull1998). While intervening [O]{} absorption has been detected with these instruments (e.g., @tripp1998), the concept of a warm-hot intergalactic gas phase was not really established at that time. With the implementation of the high-resolution capabilities of the [*Space Telescope Imaging Spectrograph*]{} (STIS) installed on HST the first systematic analyses of WHIM [O]{} absorbers as significant low-redshift baryon reservoirs came out in 2000 (see @tripp2000), thus relatively soon after the importance of a shock-heated intergalactic gas phase was realised in cosmological simulations for the first time (e.g., @cen1999; @dave2001). The STIS echelle spectrograph together with the E140M grating provides a high spectral-resolution of $R\approx 45\,000$, corresponding to a velocity resolution of $\sim 7$ kms$^{-1}$ in the STIS E140M wavelength band between $1150$ and $1730$ Å (e.g., @kimble1998; @woodgate1998). An example for a STIS quasar spectrum with intervening hydrogen and metal-line absorption is shown in Fig. \[fig:fig4\]. Note that at the spectral resolution of the STIS E140M grating all intergalactic absorption lines (i.e., hydrogen and metal lines) are fully resolved. In 1999, the [*Far Ultraviolet Spectroscopic Explorer*]{} (FUSE) became available, covering the wavelength range between $912$ and $1187$ Å. Equipped with a Rowland-type spectrograph providing a medium spectral resolution of $R\approx 20\,000$ (FWHM$\sim20$ kms$^{-1}$) FUSE is able to observe extragalactic UV background sources brighter than $V=16.5$ mag with acceptable integration time and signal-to-noise (S/N) ratios (for a description of FUSE see @moos2000; @sahnow2000). With this resolution, FUSE is able to resolve the broader intergalactic absorption from the [H]{} Lyman series, while most of the narrow metal-line absorbers remain just unresolved. This is not a problem for [O]{} WHIM studies with FUSE, however, since the spectral resolution is very close to the actual line widths and the [O]{} absorption usually is not saturated. FUSE complements the STIS instruments at lower wavelengths down to the Lyman limit and consequently combined FUSE and STIS spectra of $\sim 15$ low redshift QSOs and AGN have been used to study the low-redshift WHIM via intervening [O]{} and BLA absorption (see @tripp2007 and references therein). Unfortunately, since 2006/2007 both STIS and FUSE are out of commission due to technical problems.
Fresh spectroscopic UV data from WHIM absorption line studies will become available once the [*Cosmic Origins Spectrograph*]{} (COS) will be installed on HST during the next HST service mission (SM-4), which currently is scheduled for late 2008. COS will observe in the UV wavelength band between $1150$ and $3000$ Å at medium resolution ($R\approx 20\,000$). COS has been designed with maximum effective area as the primary constraint: it provides more than an order of magnitude gain in sensitivity over previous HST instruments. Due to its very high sensitivity, COS thus will be able to observe [*hundreds*]{} of low- and intermediate redshift QSOs and AGN and thus will deliver an enormous data archive to study the properties of WHIM UV absorption lines systems in great detail (see also @paerels2008 - Chapter 19, this volume).
Intervening WHIM absorbers at low redshift {#Intervening WHIM absorbers at low redshift}
------------------------------------------
![ STIS spectrum of the quasar PG1259+593 in the wavelength range between 1300 and 1400 Å. Next to absorption from the local Ly$\alpha$ forest and gas in the Milky Way there are several absorption features that most likely are related to highly ionised gas in the WHIM. Absorption from five-times ionised oxygen ([O]{}) is observed at $z=0.25971$ and $z=0.31978$. Broad [H]{} Ly$\alpha$ and Ly$\beta$ absorption is detected at $z=0.08041$, $0.09196$, $0.10281$, $0.13351$, $0.14034$, $0.14381$, $0.14852$, $0.15136$, and $z=0.31978$. From @richter2004. []{data-label="fig:fig4"}](fig4.ps){width="\hsize"}
![ Examples for [H]{} and [O]{} absorption in two absorption systems at $z=0.23351$ and $z=0.6656$ towards PG0953+415 and H1821+643, respectively, plotted on a rest frame velocity scale (observed with STIS). Adapted from @tripp2007. []{data-label="fig:fig5"}](fig5.ps){width="\hsize"}
We start with the [O]{} absorbers which are believed to trace the low-temperature tail of the WHIM at $T<5\times 10^5$ K. Up to now, more than 50 detections of intervening [O]{} absorbers at $z<0.5$ have been reported in the literature (e.g., @tripp2000; @oegerle2000; @chen2000; @savage2002; @richter2004; @sembach2004; @savage2005; @danforth2005; @tripp2007). All of these detections are based on FUSE and STIS data. Fig. \[fig:fig5\] shows two examples for intervening [O]{} absorption at $z=0.23351$ and $z=0.26656$ in the direction of PG0953+415 and H1821+643, as observed with STIS. The most recent compilation of low-redshift intervening [O]{} absorbers is that of @tripp2007, who have analysed 16 sightlines toward low-redshift QSOs observed with STIS and FUSE along a total redshift path of $\Delta z\approx 3$. They find a total of 53 intervening [O]{} absorbers (i.e., they are not within $5000$ kms$^{-1}$ of $z_{\rm QSO}$) comprised of 78 individual absorption components[^1]. The measurements imply a number density of [O]{} absorbing systems per unit redshift of ${\mathrm d}N_{\mbox{O\,{\scriptsize VI}}}/{\mathrm d}z \approx 18\pm 3$ for equivalent widths $W_{\lambda}\geq30$ mÅ. The corresponding number density of [O]{} absorption [*components*]{} is ${\mathrm d}N_{\mbox{O\,{\scriptsize VI}}}/{\mathrm d}z \approx 25\pm 3$. These values are slightly higher than what is found by earlier analyses of smaller [O]{} samples [@danforth2005], but lie within the cited $2\sigma$ error ranges. The discrepancy between the measured [O]{} number densities probably is due to the different approaches of estimating the redshift path $\Delta z$ along which the [O]{} absorption takes place. If one assumes that the gas is in a collisional ionisation equilibrium, i.e., that $\sim 20$ percent of the oxygen is present in the form of [O]{} ($f_{\mbox{O\,{\scriptsize VI}}}\leq 0.2$), and further assumes that the mean oxygen abundance is $0.1$ Solar, the measured number density of [O]{} absorbers corresponds to a cosmological mass density of $\Omega_{\mathrm b}$([O]{})$\approx 0.0020-0.0030$ $h_{70}\,^{-1}$. These values imply that intervening [O]{} absorbers trace $\sim 5-7$ percent of the total baryon mass in the local Universe. For the interpretation of $\Omega_{\mathrm b}$([O]{}) it has to be noted that [O]{} absorption traces collisionally ionised gas at temperatures around $3 \times 10^5$ K (and also low-density, photoionised gas at lower temperatures), but not the million-degree gas phase which probably contains the majority of the baryons in the WHIM.
The recent analysis of @tripp2007 indicates, however, that this rather simple conversion from measured [O]{} column densities to $\Omega_{\mathrm b}$([O]{}) may not be justified in general, as the CIE assumption possibly breaks down for a considerable fraction of the [O]{} systems. From the measured line widths of the [H]{} Ly$\alpha$ absorption that is associated with the [O]{} Tripp et al.conclude that $\sim 40$ percent of their [O]{} systems belong to cooler, photoionised gas with $T<10^5$ K, possibly not at all associated with shock-heated warm-hot gas. In addition, about half of the intervening [O]{} absorbers arise in rather complex, multi-phase systems that can accommodate hot gas at relatively low metallicity. It thus appears that – without having additional information about the physical conditions in each [O]{} absorber – the estimate of the baryon budget in intervening [O]{} systems is afflicted with rather large systematic uncertainties.
In high-column density [O]{} systems at redshifts $z>0.18$, such desired additional information may be provided by the presence or absence of [Ne]{} (see Sect.2.2.1), which in CIE traces gas at $T\sim 7 \times 10^5$ K. Toward the quasar PG1259+593 @richter2004 have reported a tentative detection of [Ne]{} absorption at $\sim 2 \sigma$ significance in an [O]{} absorber at $z\approx 0.25$. The first secure detection of intervening [Ne]{} absorption (at $\sim 4 \sigma$ significance) was presented by @savage2005 in a multi-phase [O]{} absorption system at $z\approx 0.21$ in the direction of the quasar HE0226$-$4110. The latter authors show that in this particular absorber the high-ion ratio [Ne]{}/[O]{}$=0.33$ is in agreement with gas in CIE at temperature of $T\sim 5\times 10^5$ K. With future high S/N absorption line data of low-redshift QSOs (as will be provided by COS) it is expected that the number of detections of WHIM [Ne]{} absorbers will increase substantially, so that an important new diagnostic will become available for the analysis of high-ion absorbers.
One other key aspect in understanding the distribution and nature of intervening [O]{} systems concerns their relation to the large-scale distribution of galaxies. Combining FUSE data of 37 [O]{} absorbers with a database of more than a million galaxy positions and redshifts, @stocke2006 find that all of these [O]{} systems lie within $800\,h_{70}\,^{-1}$ kpc of the nearest galaxy. These results suggest that [O]{} systems preferentially arise in the immediate circumgalactic environment and extended halos of galaxies, where the metallicity of the gas is expected to be relatively high compared to regions far away from galactic structures. Some very local analogs of intervening [O]{} systems thus may be the [O]{} high-velocity clouds in the Local Group that are discussed in the next subsection. Due to apparent strong connection between intervening [O]{} systems and galactic structures and a resulting galaxy/metallicity bias problem it is of great interest to consider other tracers of warm-hot gas, which are independent of the metallicity of the gas. The broad hydrogen Ly$\alpha$ absorbers – as will be discussed in the following – therefore represent an important alternative for studying the WHIM at low redshift.
![ Broad Lyman $\alpha$ absorbers towards the quasars H1821+643 and PG0953+415 (STIS observations), plotted on a rest frame velocity scale. If thermal line broadening dominates the width of the absorption, these systems trace the WHIM at temperatures between $10^5$ and $10^6$ K. From @richter2006a. []{data-label="fig:fig6"}](fig6.ps){width="\hsize"}
As described in Sect.2.2.1, BLAs represent [H]{} Ly$\alpha$ absorbers with large Doppler parameters $b>40$ kms$^{-1}$. If thermal line broadening dominates the width of the absorption, these systems trace the WHIM at temperatures between $10^5$ and $10^6$ K, typically (note that for most systems with $T>10^6$ K BLAs are both too broad and too shallow to be unambiguously identified with the limitations of current UV spectrographs). The existence of [H]{} Ly$\alpha$ absorbers with relatively large line widths has been occasionally reported in earlier absorption-line studies of the local intergalactic medium (e.g., @tripp2001; @bowen2002). Motivated by the rather frequent occurrence of broad absorbers along QSO sightlines with relatively large redshift paths, the first systematic analyses of BLAs in STIS low-$z$ data were carried out by @richter2004 and @sembach2004. @richter2006a have inspected four sightlines observed with STIS towards the quasars PG1259+593 ($z_{\rm em}=0.478$), PG1116+215 ($z_{\rm em}=0.176$), H1821+643 ($z_{\rm em}=0.297$), and PG0953+415 ($z_{\rm em}=0.239$) for the presence of BLAs and they identified a number of good candidates. Their study implies a BLA number density per unit redshift of ${\mathrm d}N_{\rm BLA}/{\mathrm d}z \approx 22-53$ for Doppler parameters $b\geq40$ kms$^{-1}$ and above a sensitivity limit of log ($N$(cm$^{-2})/b($kms$^{-1}))\geq 11.3$. The large range for ${\mathrm d}N_{\rm BLA}/{\mathrm d}z$ partly is due to the uncertainty about defining reliable selection criteria for separating spurious cases from good broad Ly$\alpha$ candidates (see discussions in @richter2004 [@richter2006a] and @sembach2004). Transforming the number density ${\mathrm d}N_{\rm BLA}/{\mathrm d}z$ into a cosmological baryonic mass density, @richter2006a obtains $\Omega_{\mathrm b}$(BLA)$\geq 0.0027\,h_{70}\,^{-1}$. This limit is about 6 percent of the total baryonic mass density in the Universe expected from the current cosmological models (see above), and is comparable with the value derived for the intervening [O]{} absorbers (see above). Examples for several BLAs in the STIS spectrum of the quasar H1821+643 are shown in Fig. \[fig:fig6\].
More recently, @lehner2007 have analysed BLAs in low-redshift STIS spectra along seven sightlines. They find a BLA number density of ${\mathrm d}N_{\rm BLA}/{\mathrm d}z=30\pm4$ for $b=40-150$ kms$^{-1}$ and log $N$([H]{}$)>13.2$ for the redshift range $z=0-0.4$. They conclude that BLAs host at least 20 percent of the baryons in the local Universe, while the photoionised Ly$\alpha$ forest, which produces a large number of narrow Ly$\alpha$ absorbers (NLAs), contributes with $\sim 30$ percent to the total baryon budget. In addition, @prause2007 have investigated the properties of BLAs at intermediate redshifts ($z=0.9-1.9$) along five other quasars using STIS high- and intermediate-resolution data. They find a number density of reliably detected BLA candidates of ${\mathrm d}N_{\rm BLA}/{\mathrm d}z\approx14$ and obtain a lower limit of the contribution of BLAs to the total baryon budget of $\sim 2$ percent in this redshift range. The frequency and baryon content of BLAs at intermediate redshifts obviously is lower than at $z=0$, indicating that at intermediate redshifts shock-heating of the intergalactic gas from the infall in large-scale filaments is not yet very efficient. This is in line with the predictions from cosmological simulations.
The Milky Way halo and Local Group gas {#The Milky Way halo and Local Group gas}
--------------------------------------
One primary goal of the FUSE mission was to constrain the distribution and kinematics of hot gas in the thick disk and lower halo of the Milky Way by studying the properties of Galactic [O]{} absorption systems at radial velocities $|v_{\rm LSR}|\leq 100$ kms$^{-1}$ (@savage2000; @savage2003; @wakker2003). However, as the FUSE data unveil, [O]{} absorption associated with Milky Way gas is observed not only at low velocities but also at $|v_{\rm LSR}|>100$ kms$^{-1}$ [@sembach2003]. The topic of cool and hot gas in the halo of the Milky Way recently has been reviewed by @richter2006c. These detections imply that next to the Milky Way’s hot “atmosphere” (i.e., the Galactic Corona) individual pockets of hot gas exist that move with high velocities through the circumgalactic environment of the Milky Way. Such high-velocity [O]{} absorbers may contain a substantial fraction of the baryonic matter in the Local Group in the form of warm-hot gas and thus – as discussed in the previous subsection – possibly represent the local counterparts of some of the intervening [O]{} absorbers observed towards low-redshift QSOs.
From their FUSE survey of high-velocity [O]{} absorption @sembach2003 find that probably more than 60 percent of the sky at high velocities is covered by ionised hydrogen (associated with the [O]{} absorbing gas) above a column density level of log $N$([H]{})$=18$, assuming a metallicity of the gas of $0.2$ Solar. Some of the high-velocity [O]{} detected with FUSE appears to be associated with known high-velocity [H]{} 21 cm structures (e.g., the high-velocity clouds complex A, complex C, the Magellanic Stream, and the Outer Arm). Other high-velocity [O]{} features, however, have no counterparts in [H]{} 21 cm emission. The high radial velocities for most of these [O]{} absorbers are incompatible with those expected for the hot coronal gas (even if the coronal gas motion is decoupled from the underlying rotating disk). A transformation from the Local Standard of Rest to the Galactic Standard of Rest and the Local Group Standard of Rest velocity reference frames reduces the dispersion around the mean of the high-velocity [O]{} centroids (@sembach2003; @nicastro2003). This can be interpreted as evidence that [*some*]{} of the [O]{} high-velocity absorbers are intergalactic clouds in the Local Group rather than clouds directly associated with the Milky Way. However, it is extremely difficult to discriminate between a Local Group explanation and a distant Galactic explanation for these absorbers. The presence of intergalactic [O]{} absorbing gas in the Local Group is in line with theoretical models that predict that there should be a large reservoir of hot gas left over from the formation of the Local Group (see, e.g., @cen1999).
It is unlikely that the high-velocity [O]{} is produced by photoionisation. Probably, the gas is collisionally ionised at temperatures of several $10^5$ K. The [O]{} then may be produced in the turbulent interface regions between very hot ($T>10^6$ K) gas in an extended Galactic Corona and the cooler gas clouds that are moving through this hot medium (see @sembach2003). Evidence for the existence of such interfaces also comes from the comparison of absorption lines from neutral and weakly ionised species with absorption from high ions like [O]{} [@fox2004].
X-ray measurements of the WHIM {#X-ray measurements of the WHIM}
==============================
Past and present X-ray instruments {#Past and present X-ray instruments}
----------------------------------
With the advent of the [*Chandra*]{} and [*XMM-Newton*]{} observatories, high resolution X-ray spectroscopy of a wide variety of cosmic sources became feasible for the first time. Among the possible results most eagerly speculated upon was the detection of intergalactic absorption lines from highly ionised metals in the continuum spectra of bright extragalactic sources. After all, one of the most striking results from the [*Einstein*]{} observatory had been the detection of a very significant broad absorption feature at $\sim 600$ eV in the spectrum of PKS2155$-$304 with the Objective Grating Spectrometer [@canizares1984]. Ironically, if interpreted as intergalactic H-like O Ly$\alpha$ absorption, redshifted and broadened by the expansion of the Universe, the strength of the feature implied the presence of a highly ionised IGM of near-critical density, a possibility that has of course definitively been discounted since then.
The High Energy Transmission Grating Spectrometer (HETGS; @canizares2005) and the Low Energy Transmission Grating Spectrometer (LETGS; @brinkman2000) on [*Chandra*]{}, and the Reflection Grating Spectrometer (RGS) on [*XMM-Newton*]{} [@denherder2001] were the first instruments to provide sensitivity to weak interstellar and intergalactic X-ray absorption lines. The ’traditional’ ionisation detectors (proportional counters, CCD’s) do not have sufficient energy resolution for this application. But the angular resolution provided by an X-ray telescope can be used to produce a high resolution spectrum, by the use of diffracting elements. Laboratory X-ray spectroscopy is typically performed with crystal diffraction spectrometers, and the use of crystal spectrometers for general use in astrophysics was pioneered on the [*Einstein*]{} observatory (e.g. @canizares1979). The Focal Plane Crystal Spectrometer indeed detected the first ever narrow X-ray absorption line in a cosmic source, the $1s-2p$ absorption by neutral oxygen in the interstellar medium towards the Crab [@canizares1984]. Previous grating spectrometers (the Objective Grating Spectrometer on [*Einstein*]{} and the two Transmission Grating Spectrometers on [*EXOSAT*]{}) had only limited resolution and sensitivity. But there is no fundamental limit to the resolution of a diffraction grating spectrometer, and the high angular resolution of the [*Chandra*]{} telescope has allowed for high resolution spectroscopy using transmission gratings. The focusing optics on [*XMM-Newton*]{} have more modest angular resolution, but they are used with a fixed array of grazing incidence reflection gratings, which produce very large dispersion angles and thus high spectral resolution. The HETGS provides a spectral resolution of $\Delta\lambda = 0.0125$ Å over the $\approx 1.5-15$ Å band with the high line density grating, and $\Delta\lambda = 0.025$ Å over the $2-20$ Å band with the medium line density grating. The LETGS has $\Delta\lambda = 0.05$ Å over the $\approx 2-170$ Å band, while the RGS has $\Delta\lambda = 0.06$ Å over the $5-38$ Å band. These numbers translate to resolving powers of $R = 400-1500$ in the O K band, and with a sufficiently bright continuum source, one should be able to detect equivalent widths of order $0.1-0.5$ eV ($5-20$ mÅ), or below in spectra with very high signal-to-noise. The predictions for H- and He-like O resonance absorption line strengths are generally smaller than these thresholds, but not grossly so, and so a search for intergalactic O was initiated early on. Given that the current spectrometers are not expected to resolve the absorption lines, the only freedom we have to increase the sensitivity of the search is to increase the signal-to-noise ratio in the continuum, and it becomes crucial to find suitable, very bright sources, at redshifts that are large enough that there is a reasonable a priori probability of finding a filament with detectable line absorption.
As we will see when we discuss the results of the observational searches for X-ray absorption lines, the problem is made considerably more difficult by the very sparseness of the expected absorption signature. Frequently, when absorption features of marginal statistical significance are detected in astrophysical data, plausibility is greatly enhanced by simple, unique spectroscopic arguments. For instance, for all plausible parameter configurations, an absorbing gas cloud detected in [N]{} should also produce detectable [C]{} absorption; or, both members of a doublet should appear in the correct strength ratio if unsaturated. In the early stages of Ly$\alpha$ forest astrophysics, it was arguments of this type, rather than the crossing of formal statistical detection thresholds alone, that guided the field (e.g., @lynds1971). But for the ’X-ray Forest’ absorption, we expect a very different situation. The detailed simulations confirm what simple analytical arguments had suggested: in most cases, Intergalactic absorption systems that are in principle detectable with current or planned X-ray instrumentation will show just a single absorption line, usually the [O]{} $n=1-2$ resonance line, at an unknown redshift. When assessing the possibility that a given apparent absorption feature is ’real’, one has to allow for the number of independent trial redshifts (very roughly given by the width of the wavelength band surveyed, divided by the nominal spectral resolution of the spectrometer), and with a wide band and a high spectral resolution, this tends to dramatically reduce the significance of even fairly impressive apparent absorption dips. For instance, an apparent absorption line detected at a formal ’3$\sigma$ significance’ (or $p = 0.0015$ a priori probability for a negative deviation this large to occur due to statistical fluctuation) with [*Chandra*]{} LETGS in the $21.6-23.0$ Å band pales to ’1.7 $\sigma$’ if we assume it is the [O]{} resonance line in the redshift range $z = 0-0.065$; with $N \approx 30$ independent trials, the chances of not seeing a $3\sigma$ excursion are $(1-p)^N = 0.956$, or: one will see such a feature one in twenty times if one tries this experiment (we are assuming a Gaussian distribution of fluctuations here). If we allow for confusion with [O]{} Ly$\alpha$ at higher redshift, or even other transitions, the significance is even further reduced. And the larger the number of sources surveyed, the larger the probability of false alarm. Clearly, more reliable statistics on intervening X-ray absorbers and detections at higher significance are desired, but the required high-quality data will not be available until the next-generation X-ray facilities such as [*XEUS*]{} and [*Constellation X*]{} are installed (see @paerels2008 - Chapter 19, this volume).
Nevertheless, even with these odds, the above discussed high-ion measurements are important observations to do with the currently available instruments [*Chandra*]{} and [*XMM-Newton*]{}. Given the predicted strengths of the absorption lines (e.g., @chen2003; see also Sect.5.1), attention has naturally focused on a handful of very bright BL Lac- and similar sources. Below, we discuss the results of the searches. Note that the subject has recently also been reviewed by @bregman2007.
Intervening WHIM absorbers at low redshift {#Intervening WHIM absorbers at low redshift X-ray}
------------------------------------------
The first attempt at detecting redshifted X-ray O absorption lines was performed by @mathur2003 with a dedicated deep observation (470 ks) with the [*Chandra*]{} LETGS of the quasar H1821+643, which has several confirmed intervening [O]{} absorbers. No significant X-ray absorption lines were found at the redshifts of the [O]{} systems, but this was not really surprising in view of the modest signal to noise ratio in the X-ray continuum. Since it requires very bright continua to detect the weak absorption, it is also not surprising that the number of suitable extragalactic sources is severely limited. Early observations of a sample of these (e.g. S50836+710, PKS2149$-$306; @fang2001b; PKS2155$-$304, @fang2002) produced no convincing detections. Nicastro and his colleagues then embarked on a campaign to observe Mrk421 during its periodic X-ray outbursts, when its X-ray flux rises by an order of magnitude (e.g. @nicastro2005). The net result of this has been the accumulation of a very deep spectrum with the [*Chandra*]{} LETGS, with a total of more than 7 million continuum counts, in about 1000 resolution elements. @nicastroetal2005 have claimed evidence for the detection of two intervening absorption systems in these data, at $z = 0.011$ and $z = 0.027$. But the spectrum of the same source observed with the [*XMM-Newton*]{} RGS does not show these absorption lines [@rasmussen2007], despite higher signal-to-noise and comparable spectral resolution (Mrk 421 is observed by [*XMM-Newton*]{} for calibration purposes, and by late 2006, more than 1 Ms exposure had been accumulated). @kaastra2006 have reanalysed the [*Chandra*]{} LETGS data, and find no significant absorption. Other sources, less bright but with larger redshifts, have been observed (see for instance @steenbrugge2006 for observations of 1ES1028+511 at $z=0.361$), but to date no convincing evidence for intervening absorption has materialised.
Observations have been conducted to try and detect the absorption by intergalactic gas presumably associated with known locations of cosmic overdensity, centred on massive clusters. @fujimoto2004 attempted to detect absorption in the quasar LBQS 1228+1116, located behind the Virgo cluster. An [*XMM-Newton*]{} RGS spectrum revealed a marginal feature at the (Virgo) redshifted position of [O]{} Ly$\alpha$, but only at the $\sim 95$% confidence level. Likewise, @takei2007 took advantage of the location of X Comae behind the Coma cluster to try and detect absorption from Coma or its surroundings, but no convincing, strong absorption lines were detected in a deep observation with [*XMM-Newton*]{} RGS. The parallel CCD imaging data obtained with EPIC show weak evidence for [Ne]{} $n=1-2$ line emission at the redshift of Coma, which, if real, would most likely be associated with WHIM gas around the cluster, seen in projection (the cluster virial temperature is too high for [Ne]{}). In practice, the absence of very bright point sources behind clusters, which makes absorption studies difficult, and the bright foregrounds in emission, will probably make this approach to detecting and characterizing the WHIM not much easier than the random line-of-sight searches.
The conclusion from the search for intergalactic X-ray absorption is that there is no convincing, clear detection for intervening absorption. This is, in retrospect, not that surprising, given the sensitivity of the current X-ray spectrometers, the abundance of suitably bright and sufficiently distant continuum sources, and the predicted properties of the WHIM.
The Milky Way halo and Local Group gas {#The Milky Way halo and Local Group gas X-ray}
--------------------------------------
The first positive result of the analysis of bright continuum spectra was the detection of [O]{} and [O]{} $n=1-2$ resonance line absorption at redshift zero. @nicastro2002 first identified the resonance lines in the [*Chandra*]{} LETGS spectrum of PKS2155$-$304 ([O]{} $n=1-2$, $n=1-3$, [O]{} Ly$\alpha$, [Ne]{} $n=1-2$). @rasmussen2003 detected resonance absorption in the [*XMM-Newton*]{} RGS spectra of 3C273, Mrk421, and PKS2155$-$304. Since then, at least [O]{} $n=1-2$ has been detected in effectively all sufficiently bright continuum sources, both with [*Chandra*]{} and [*XMM-Newton*]{}; a recent compilation appears in @fang2006. Portions of a deeper spectrum that shows the zero redshift absorption are shown in Fig. \[fig:fig7\].
![[*Chandra*]{} LETGS spectrum of Mrk421. Crosses are the data, the solid line is a model. The labels identify $z \approx 0$ absorption lines in Ne, O, and C. The vertical tick marks indicate the locations of possible intergalactic absorption lines. From @williams2005. []{data-label="fig:fig7"}](fig7.ps){width="\hsize"}
@nicastro2002 initially interpreted the absorption as arising in an extended intergalactic filament. The argument that drives this interpretation is based on the assumption that [O]{}, [O]{}, and [O]{} are all located in a single phase of the absorbing gas. The simultaneous appearance of finite amounts of [O]{} and [O]{} only occurs in photoionised gas, not in gas in collisional ionisation equilibrium, and this requires that the gas has very low density (the photoionisation is produced by the local X-ray background radiation field; for a measured ionisation parameter, the known intensity of the ionising field fixes the gas density). The measured ionisation balance then implies a length scale on the order of $l \sim 10\ (Z_{0.1})^{-1}$ Mpc, where $Z_{0.1}$ is the metallicity in units 0.1 Solar. This is a very large length, and even for $0.3\ Z_{\odot}$ metallicity, the structure still would not fit in the Local Group (and it is unlikely to have this high a metallicity if it were larger than the Local Group). In fact, the absorption lines should have been marginally resolved in this case, if the structure expands with a fair fraction of the expansion of the Universe.
@rasmussen2003 constrained the properties of the absorbing gas by dropping the [O]{}, and by taking into account the intensity of the diffuse [O]{} and [O]{} line emission as measured by the Wisconsin/Goddard rocket-borne Quantum X-ray Calorimeter (XQC) experiment [@mccammon2002]. The cooling timescale of [O]{}-bearing gas is much smaller than that of gas with the higher ionisation stages of O, and this justifies the assumption that [O]{} is located in a different, transient phase of the gas. With only [O]{} and [O]{}, the medium can be denser and more compact, and be in collisional ionisation equilibrium. Treating the measured [O]{} emission line intensity as an upper limit to the emission from a uniform medium, and constraining the ionisation balance from the measured ratio of [O]{} and [O]{} column densities in the lines of sight to Mrk421 and PKS2155$-$304, Rasmussen et al. derived an upper limit on the density of the medium of $n_{\mathrm e} \lsim 2 \times 10^{-4}$ cm$^{-3}$ and a length scale $l \gsim 100$ kpc.
@bregman2007 favours a different solution, with a lower electron temperature and hence a higher [O]{} ion fraction. If one assumes Solar abundance and sets the [O]{} fraction to 0.5, the characteristic density becomes $n_e \sim 10^{-3}$ cm$^{-3}$, and the length scale ($l \sim 20$ kpc) suggests a hot Galactic halo, rather than a Local Group intragroup medium.
Arguments for both type of solution (a small compact halo and a more tenuous Local Group medium) can be given. The most direct of these is a measurement of the [O]{} line absorption towards the LMC by @wang2005 in the spectrum of the X-ray binary LMC X-3, which indicates that a major fraction of the [O]{} column in that direction is in fact in front of the LMC. @bregman2007 points out that the distribution of column densities of highly ionised O on the sky is not strongly correlated with the likely projected mass distribution of the Local Group, and that the measured velocity centroid of the absorption lines appears characteristic of Milky Way gas, rather than Local Group gas. On the other hand, a direct measurement of the Doppler broadening of the [O]{} gas, from the curve of growth of the $n=1-2$ and $n=1-3$ absorption lines in the spectrum of Mrk421 and PKS2155$-$304 (@williams2005, @williams2007), indicates an ion temperature of $T_{\rm i} \approx 10^{6.0-6.3}$ K (Mrk421) and $T_i \approx 10^{6.2-6.4}$ K (PKS2155$-$304), and these values favour the low-density, Local Group solution. Regardless, the prospect of directly observing hot gas expelled from the Galactic disk, or measuring the virial temperature of the Galaxy and/or the Local Group is exciting enough to warrant further attention to redshift zero absorption and emission.
Finally, the spectrum of Mrk421 shows the expected $z = 0$ innershell [O]{} absorption, at 22.019 Å, both with [*Chandra*]{} and with [*XMM-Newton*]{}. There has been some confusion regarding an apparent discrepancy between the [O]{} column densities derived from the FUV and from the X-ray absorption lines, in the sense that the X-ray column appeared to be significantly larger than the FUV column [@williams2005]. Proposed physical explanations for this effect involve a depletion of the lower level of the FUV transitions ($1s^2 2s$) in favour of (at least) $1s^2 2p$, which weakens the $\lambda\lambda 1032, 1038$ Åabsorption but does not affect the $1s-2p$ X-ray absorption. However, it requires very high densities to maintain a finite excited state population, and, more directly, the measured wavelength of the X-ray line is actually not consistent with the wavelength calculated for $1s-2p$ in excited [O]{}, off by about $0.03-0.05$ Å, on the order of a full resolution element of both the [*Chandra*]{} LETGS and the [*XMM-Newton*]{} RGS (Raassen 2007, private communication). The conclusion is that the discrepancy is due to an authentic statistical fluctuation in the X-ray spectrum – or, more ironically, to the presence of a weak, slightly redshifted [O]{} absorption line.
Additional aspects {#Additional aspects}
==================
Results from numerical simulations {#Results from numerical simulations}
----------------------------------
Cosmological simulations not only have been used to investigate the large-scale distribution and physical state of the warm-hot intergalactic medium, they also have been applied to predict statistical properties of high-ion absorption systems that can be readily compared with the UV and X-ray measurements (e.g., @cen2001; @fang2001a; @chen2003; @furlanetto2005; @tumlinson2005; @cen2006). Usually, a large number of artificial spectra along random sight-lines through the simulated volume are generated. Sometimes, instrumental properties of existing spectrographs and noise characteristics are modelled, too (e.g., @fangano2007). The most important quantities derived from artificial spectra that can be compared with observational data are the cumulative and differential number densities (${\mathrm d}N/{\mathrm d}z$) of [O]{}, [O]{}, [O]{} systems as a function of the absorption equivalent width. An example for this is shown in Fig. \[fig:fig8\]. Generally, there is a good match between the simulations and observations for the overall shape of the ${\mathrm d}N/{\mathrm d}z$ distribution (see also Sect. 3.2), but mild discrepancies exist at either low or high equivalent widths, depending on what simulation is used (see, e.g., @tripp2007). For the interpretation of such discrepancies it is important to keep in mind that the different simulations are based on different physical [*models*]{} for the gas, e.g., some simulations include galaxy feedback models, galactic wind models, non-equilibrium ionisation conditions, etc., others do not. For more information on numerical simulations of the WHIM see @bertone2008 - Chapter 14, this volume.
WHIM simulations also have been used to investigate the frequency and nature of BLAs at low redshift [@richter2006b]). As the simulations suggest, BLAs indeed host a substantial fraction of the baryons at $z=0$. From the artificial UV spectra generated from their simulation Richter et al.derive a number of BLAs per unit redshift of $({\mathrm d}{N}/{\mathrm d}z)_{\rm BLA}\approx 38$ for [H]{} absorbers with log $(N$(cm$^{-2})/b$(kms$^{-1}))\geq 10.7$, $b\geq40$ kms$^{-1}$, and total hydrogen column densities $N$([H]{}$)\leq 10^{20.5}$ cm$^{-2}$. The baryon content of these systems is $\sim 25$ percent of the total baryon budget in the simulation. These results are roughly in line with the observations if partial photoionisation of BLAs is taken into account (@richter2006a; @lehner2007). From the simulation further follows that BLAs predominantly trace shock-heated collisionally ionised WHIM gas at temperatures log $T\approx 4.4-6.2$. Yet, about 30 percent of the BLAs in the simulation originate in the photoionised Ly$\alpha$ forest (log $T<4.3$) and their large line widths are determined by non-thermal broadening effects such as unresolved velocity structure and macroscopic turbulence. Fig. \[fig:fig9\] shows two examples of the velocity profiles of BLAs generated from simulations presented in @richter2006b.
The results from the analysis of artificially generated UV spectra underline that the comparison between WHIM simulations and quasar absorption line studies indeed are quite important for improving both the physical models in cosmological simulations and the strategies for future observations of the warm-hot intergalactic gas.
![The differential number of intervening oxygen high-ion ([O]{}, [O]{},[O]{}) absorbers in the WHIM in a cosmological simulation is plotted against the equivalent width of the absorption (for details see @cen2006). While for [O]{} and [O]{} no significant observational results are available to be compared with the simulated spectra (see Sect.4.2), the predicted frequency of [O]{} absorbers is in good agreement with the observations (Sect.3.2). Adapted from @cen2006. []{data-label="fig:fig8"}](fig8.ps){width="0.75\hsize"}
![Two examples for BLA absorbers from the WHIM in a cosmological simulation are shown. The panels show the logarithmic total hydrogen volume density, gas temperature, neutral hydrogen volume density, and normalised intensity for [H]{} Ly$\alpha$ and [O]{} $\lambda 1031.9$ absorption as a function of the radial velocity along each sightline. From @richter2006b.[]{data-label="fig:fig9"}](fig9.ps){width="\hsize"}
WHIM absorbers at high redshift {#WHIM absorbers at high redshift}
-------------------------------
Although this chapter concentrates on the properties of WHIM absorbers at low redshift (as visible in UV and X-ray absorption) a few words about high-ion absorption at high redshifts ($z>2$) shall be given at this point. At redshifts $z>2$, by far most of the baryons are residing in the photoionised intergalactic medium that gives rise to the Ly$\alpha$ forest. At this early epoch of the Universe, baryons situated in galaxies and in warm-hot intergalactic gas created by large-scale structure formation contribute together with only $<15$ percent to the total baryon content of the Universe. Despite the relative unimportant role of the WHIM at high $z$, [O]{} absorbers are commonly found in optical spectra of high-redshift quasars (e.g., @bergeron2002; Carswell et al. @carswell2002; Simcoe et al. @simcoe2004). The observation of intervening [O]{} absorbers at high redshift is much easier than in the local Universe, since the absorption features are redshifted into the optical regime and thus are easily accessible with ground-based observatories. However, blending problems with the numerous [H]{} Ly$\alpha$ forest lines at high $z$ are much more severe than for low-redshift sightlines. Because of the higher intensity of the metagalactic UV background at high redshift it is expected that many of the [O]{} systems in the early Universe are photoionised. Collisional ionisation of [O]{} yet may be important for high-redshift absorbers that originate in galactic winds (see, e.g., @fangano2007). While for low redshifts the population of [O]{} absorbers is important for the search of the “mission baryons” that are locked in the WHIM phase in the local Universe, [O]{} absorbers at high redshift are believed to represent a solution for the problem of the “missing metals” in the early epochs of structure formation. This problem arises from the facts that at high redshift an IGM metallicity of $\sim 0.04$ is expected from the star-formation activity of Lyman-Break Galaxies (LBGs), while observations of intervening [C]{} absorption systems suggest an IGM abundance of only $\sim 0.001$ solar (@songaila2001; @scannapieco2006), thus more than one order of magnitude too low. Possibly, most of the missing metals at high $z$ are hidden in highly-ionised hot gaseous halos that surround the star-forming galaxies (e.g., @ferrara2005) and thus should be detectable only with high ions such as [O]{} rather than with intermediate ions such as [C]{}. Using the UVES spectrograph installed on the [*Very Large Telescope*]{} (VLT) @bergeron2005 have studied the properties of high-redshift [O]{} absorbers along ten QSO sightlines and have found possible evidence for such a scenario. Additional studies are required to investigate the nature of high-$z$ [O]{} systems and their relation to galactic structures in more detail. However, from the existing measurements clearly follows that the study of high-ion absorbers at large redshifts is of great importance to our understanding of the formation and evolution of galactic structures at high $z$ and the transport of metals into the IGM.
Concluding remarks {#Concluding remarks}
------------------
The analysis of absorption features from high ions of heavy elements and neutral hydrogen currently represents the best method to study baryon content, physical properties, and distribution of the warm-hot intergalactic gas in large-scale filaments at low and high redshift. However, the interpretation of these spectral signatures in terms of WHIM baryon content and origin still is afflicted with rather large systematic uncertainties due to the limited data quality and the often poorly known physical conditions in WHIM absorbers (e.g., ionisation conditions, metal content, etc.). Future instruments in the UV (e.g., COS) and in the X-ray band (e.g., [*XEUS*]{}, [*Constellation X*]{}) hold the prospect of providing large amounts of new data on the WHIM with good signal-to-noise ratios and substantially improved absorber statistics. These missions therefore will be of great importance to improve our understanding of this important intergalactic gas phase.
The authors thank ISSI (Bern) for support of the team “Non-virialized X-ray components in clusters of galaxies”. SRON is supported fiancially by NWO, the Netherlands Organization for Scientific Research.
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[^1]: I.e., some of the [O]{} systems have velocity sub-structure
| {
"pile_set_name": "ArXiv"
} |
[**Renormalization-Group flow for the field strength in scalar self-interacting theories**]{}\
\
[*INFN, Sezione di Catania*]{}\
[*Via S. Sofia 64, I-95123, Catania, Italy*]{}\
\
We consider the Renormalization-Group coupled equations for the effective potential $V(\phi)$ and the field strength $Z(\phi)$ in the spontaneously broken phase as a function of the infrared cutoff momentum $k$. In the $k \to 0$ limit, the numerical solution of the coupled equations, while consistent with the expected convexity property of $V(\phi)$, indicates a sharp peaking of $Z(\phi)$ close to the end points of the flatness region that define the physical realization of the broken phase. This might represent further evidence in favor of the non-trivial vacuum field renormalization effect already discovered with variational methods.\
0.5 cm Pacs 11.10.Hi , 11.10.Kk
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[**1.**]{} Renormalization Group (RG) techniques originally inspired to the Kadanoff-Wilson blocking procedure [@kad] represent a powerful method to approach non-perturbative phenomena in quantum field theory. A widely accepted technique consists in starting from a bare action defined at some ultraviolet cutoff $\Lambda$ and effectively integrating out shells of quantum modes down to an infrared cutoff $k$. This procedure provides a $k-$dependent effective action $\Gamma_k[\Phi]$ that evolves into the full effective action $\Gamma[\Phi]$ in the limit $k\to 0$.
The $k-$dependence of $\Gamma_k[\Phi]$ is determined by a differential functional flow equation that is known in the literature in slightly different forms [@weg; @nicol; @polch; @wet1; @mor1]. In particular, with the flows discussed in detail in Ref.[@wetreport] one starts form first principles and obtains a class of functionals that interpolates between the classical bare Euclidean action and the full effective action of the theory. However, some features, such as the basic convexity property of the effective action for $k \to 0$ [@tetradis; @litim2; @alexander; @zappala], are independent of the particular approach.
In this Letter, we shall study the coupled equations for the $k-$dependent effective potential $V_k(\phi)$ and field strength $Z_k(\phi)$, which naturally appear in a derivative expansion of $\Gamma_k[\Phi]$ (around the coordinate independent configuration $\Phi(x)=\phi$ ), using a proper-time infrared regulator. This approach generates evolution equations which are well defined truncations of the first-principle flows within a background field formulation [@litim12; @litim11]. Incidentally, as shown in [@litim12], they are also an approxmation to another exact flow (a generalised Callan-Symanzik flow) without background fields.
The equations for $V_k(\phi)$ and $Z_k(\phi)$, derived in Ref.[@bonanno], correspond to such first-principle flows up to additional terms explicitly determined in Ref.[@litim12]. An indication of the reliability of such an approximation is provided by the consistent computation of the critical exponents in scalar self-interacting theories for various numbers $D$ of the space-time dimensions and $N$ of field components [@bonanno]. At the same time, in Ref.[@zappala], it was shown that going beyond the approximation $Z=1$ is essential to reproduce successfully the energy gap between the exact ground state and the first excited state of the double well potential in the quantum-mechanical limit of the theory $D=1$. In addition, the coupled equations obtained by using the proper time regulator are not affected by singularities and/or ambiguities that instead appear using a sharp infrared cutoff [@io2l].
With these premises, assuming the possibility to neglect the additional terms of Ref.[@litim12] and adopting the same notations of Refs.[@zappala; @bonanno], we shall start our analysis considering the two equations \[vdim\] k[[V]{}]{} = ([k\^2]{})\^[D/2]{} e\^[-V”/(Z k\^2)]{} &&k[[Z]{}]{} = ([k\^2]{})\^[D/2]{} e\^[-V”/(Z k\^2)]{}\
&&( -[[Z”]{}]{}+[[(4+18D-D\^2) (Z’)\^2]{}]{} +[[(10-D)Z’V”’]{}]{}- [[Z(V”’)\^2]{}]{}) \[zdim\] where we have set $V=V_k(\phi)$, $Z=Z_k(\phi)$ and used the notation $V',Z'$,...to indicate differentiation with respect to $\phi$. In the following, we shall first analyze these two equations to understand the approach to convexity and obtain informations on $Z$ and finally provide a possible physical interpretation of our numerical results. 15 pt
[**2.**]{} For a numerical solution of Eqs.(\[vdim\]) and (\[zdim\]) it is convenient to use dimensionless variables defined as follows: $t={\rm ln}(\Lambda/k)$, $x=k^{1-D/2}\phi$, $V(t,x)=k^{-D} V_k(\phi)$ and $Z(t,x)=Z_k(\phi)$. By defining the first derivative of the effective potential $f(x,t)=\partial_x
V(t,x)$, Eqs.(\[vdim\],\[zdim\]) become \[fadim\] [[f]{}]{} =[[(D+2)]{}]{}f + [[(2-D)]{}]{} x[[f]{}]{} - [[1]{}]{} e\^[-f’/Z]{} \[zadim\] [[Z]{}]{}&=& [[(2-D)]{}]{} x[[Z]{}]{}+ [[1]{}]{} ( [[Z’]{}]{}e\^[-f’/Z]{})\
-[[e\^[-f’/Z]{}]{}]{} &(&[[f’(Z’)\^2]{}]{}+[[18D-D\^2-20]{}]{} [[(Z’)\^2]{}]{} +[[(4-D)Z’f”]{}]{}- [[(f”)\^2]{}]{}) It is easy to show that these two coupled equations can be transformed into the structure ($i,j$=1-3) P\_[ij]{} [[U\_j]{}]{} + Q\_i= [[R\_i]{}]{} where the components of the vector $U_i(x,t)$ are the unknown functions of the problem and where $P_{ij}$, $Q_i$ and $R_i$ can depend on $x,t, U_i, {{\partial
U_i}\over{\partial x}}$.
In this way, the numerical solution has been obtained with the help of the NAG routines that integrate a system of non-linear parabolic partial differential equations in the $(x,t)$ two-dimensional plane. The spatial discretisation is performed using a Chebyshev $C^o$ collocation method, and the method of lines is employed to reduce the problem to a system of ordinary differential equations. The routines contain two main parameters (the size of the discretisation grid $N_{\rm points}$ in the spatial dimension and the local accuracy $\Delta$ in the time integration) that can be changed to control the stability of the solution. In our case, after reaching sufficiently large values of $N_{\rm points}$ ($\sim 2000$) and sufficiently small values of $\Delta$ ($\sim 10^{-7}$), the numerical results remain remarkably stable for further variations of these parameters.
We shall focus on the quantum-field theoretical case $D=4$ assuming standard boundary conditions at the cutoff scale : i) a renormalizable form for the bare, broken-phase potential \[bare\] V\_()=-[[1]{}]{}M\^2\^2 + \^4 and ii) a unit renormalization condition for the derivative term in the bare action Z\_()=1 With this choice of the classical potential the problem is manifestly invariant under the interchange $\phi \to -\phi$ at all values of $k$.
We shall also concentrate on the weak coupling limit $\lambda=0.1$, fixing $M=1$ and $\Lambda=10$. In this way, one gets a well defined hierarchy of scales where the infrared region corresponds to the limit $k \ll M\ll \Lambda$.
Before addressing the full problem, we have considered the standard approximation of setting $Z=1$. In this case, we can find a good approximation to the exact solution of Eq.(\[fadim\]) for large $x$ and large $t$ as a cubic polynomial with $t-$dependent coefficients \[asy\] f\_[asy]{}(x,t) =A(t)x\^3+B(t)x\^2+C(t)x +D(t) This type of form, motivated by RG arguments, see e.g. [@litim3], becomes exact where $f'$ is large and positive so that $e^{-f'}\to 0$ yielding $A(t)=A_0$, $B(t)=B_0\exp(t)$, $C(t)=C_0\exp(2t)$, $D(t)=D_0\exp(3t)$. Higher powers $x^n$, with $n>3$, might also be inserted but they are suppressed by exponential terms $e^{-(n-3)t}$. At the same time, $f=0$ is also a solution of Eq.(\[fadim\]) and corresponds to a flat effective potential in $x$. As we shall see, the $k-$evolution, for $k\to 0$, turns out to be a sort of way to interpolate between $f=0$ and the asymptotic trend in Eq.(\[asy\]).
We show in Fig.1 the plots of $V'_k(\phi)\equiv {{\partial
V_k(\phi)}\over{\partial \phi}}$ at various values of the infrared cutoff down to the smallest value $k=0.05$ that can be numerically handled by our integration routine. This corresponds to a maximal value $t_{\rm max}\sim 5.3$ which is comparable to the highest value $t=5$ reported in Fig.1a of Ref.[@litim2]. In our problem, this seems to mark the boundary of the region that is very difficult to handle numerically.
As one can see, the approach to convexity is very clean, in full agreement with the general theoretical arguments of Ref.[@litim2]. It is characterized by an almost linear behaviour in the inner $\phi-$region that matches with the outer, asymptotic cubic shape discussed above. In this sense, the limitation in the maximal $t$ is just a numerical artifact of our integration method since from the physical point of view there should be no conceptual problem in the limit $t \to
\infty$. This is supported by a fit to the slope $S(k)$ of $V'_k(\phi)$ at $\phi=0$ in Fig.1, which suggests the following functional form ($a$ and $b$ being numerical coefficients obtained from the fit) S(k)\~a \[[exp]{}(-bk\^2) -1\] that vanishes in the limit $k \to 0$.
For small $k$, the minimum of the first derivative of the effective potential $\phi=\hat{\phi}(k)$ does not correspond to an analytic behaviour. This is a well known result of quantum field theory [@syma]: the Legendre-transformed effective potential is not an infinitely differentiable function.
Let us now consider the full problem defined by Eqs.(\[fadim\]) and (\[zadim\]). Again, for large $x$ and $t$, the pair $f=f_{\rm
asy}(x,t)$ and $Z=1$ provide a simultaneous solution. However, for finite values of $k$, where the potential is not yet convex downward and $f'<0$, the term $e^{{{|f'|}\over{Z}}}$ drives $Z$ to grow.
We show in Fig.2, the simultaneous solutions for $Z_k(\phi)$ (upper panel) and $V'_k(\phi)\equiv {{\partial V_k(\phi)}\over{\partial
\phi}}$ (lower panel) for various values of the infrared cutoff down to $k=0.07$ (again $t_{\rm max}\sim 5$) that represents, for the coupled problem, the point beyond which the integration routines no longer work. The strong peaking in $Z_k(\phi)$, to a very high accuracy, occurs at $\hat{\phi}(k)$ where $V'_k(\phi)$ has its minimum. On the basis of the general convexification property this point tends, for $k \to 0$, to the end point $\hat{\phi}(0)\equiv \phi_0$ of the flatness region that defines the physical realization of the broken phase. Finally, we report in Fig.3 the values of $Z_k(\phi)$ at $\phi=0$ (circles) and at the peak for $\phi=\hat{\phi}(k)$ (diamonds) in the range $0.07\leq k \leq 0.15$. Using some extrapolation forms, the value of the peak seems to tend, in the limit $k \to 0$, to a very large but finite value $Z_{k=0}(\phi_0)$.
15 pt [**3.**]{} Let us now explore a possible physical interpretation of the numerical results reported above. We start by observing that spontaneous symmetry breaking is usually considered a semi-classical phenomenon, i.e. based on a classical potential with perturbative quantum corrections. These, with our choice of the bare parameters $\lambda=0.1$ and $M=1$ in Eq.(\[bare\]), and our cutoff value $\Lambda=10$, are typically small for all quantities. In particular $Z$, in perturbation theory, is a non-leading quantity since its one-loop correction is ultraviolet finite. Therefore, the deviations from unity are expected to be very small.
Re-writing the $\Phi^4$ term in the standard form ${{\lambda_{\rm
st}}\over{4!}}\Phi^4$, with $\lambda_{\rm st}=2.4$, the perturbative prediction is \[zpert\] Z\_[pert]{}-1=[O]{}( [[\_[st]{}]{}]{})\~10\^[-2]{} This is also consistent with the assumed exact “triviality” property of the theory for $D=4$ [@book] that requires $Z \to 1$ in the continuum limit $\Lambda \to \infty$.
Now, let us compare this prediction with our $Z_k(\phi)$ in Fig.2. For large values of the infrared cutoff, when the effective potential is still smooth, we find $Z_k(\phi)\sim 1$ for all values of $\phi$, as expected. However, at smaller $k$, say $k <\delta$ with $\delta\sim 0.15$, there are large deviations from unity in the region of $\phi$ where the smooth form of the perturbative potential evolves into the typical non-analytical behaviour of the exact effective potential. This leads to the observed strong peaking phenomenon at the point $\hat{\phi}(k)$, where $V'_k(\phi)$ reaches its minimum value. This point, on the basis of the general convexification property of $V(\phi)$ tends, for $k \to 0$, to the value $\hat{\phi}(0)=\phi_0$, the end point of the flatness region that defines the physical realization of the broken phase.
If we express the full scalar field $\Phi(x)$ as (x)=+h(x) the above results indicate that the higher frequency components of the fluctuation field $h(x)$, those with 4-momentum $p_\mu$ such that $\delta \leq |p|\leq \Lambda$, represent genuine quantum corrections for all values of the background field $\phi$ in agreement with their perturbative representation as weakly coupled massive states.
On the other hand, the components with a 4-momentum $p_\mu$ such that $ |p| \leq \delta$, are non-perturbative for values of the background field in the range $0 \leq \phi \leq \hat{\phi}(|p|)$. In particular, the very low-frequency modes with $|p| \to 0$ behave non-perturbatively for all values of the background in the full range $0 \leq \phi \leq \phi_0$. They can be thought as collective excitations of the scalar condensate and cannot be represented as standard massive states.
The existence of a peculiar $p_\mu \to 0$ limit in the broken phase, for which one can give some general arguments [@consoli], finds support in the results of lattice simulations of the broken-symmetry phase (see Ref.[@cea]). There, differently from what happens in the symmetric phase, the connected scalar propagator deviates significantly from (the lattice version of) the massive single-particle form $1/(p^2+{\rm const})$ for $p_\mu \to 0$. In particular, looking at Figs. 7, 8 and 9 of Ref.[@stevenson], one can clearly see that, approaching the continuum limit of the lattice theory, these deviations become more and more pronounced but also confined to a smaller and smaller region of momenta near $p_\mu=0$.
This observation suggests that the existence of a non-perturbative infrared sector in a region $0\leq |p| \leq \delta$ might not be in contradiction with the assumed exact “triviality” property of the theory if, in the continuum limit, the infrared scale $\delta$ vanishes in units of the physical parameter $m$ associated with the massive part of the spectrum. This means to establish a hierarchy of scales $\delta \ll m \ll \Lambda$ such that ${{\delta}\over{m}}\to
0$ when ${{m}\over{\Lambda}} \to 0$.
If this happens, the region $0\leq |p| \leq \delta$ would just shrink to the zero-measure set $p_\mu=0$, for the continuum theory where $m$ sets the unit mass scale, thus recovering the exact Lorentz covariance of the energy spectrum since the point $p_\mu=0$ forms a Lorentz-invariant subset. In this limit, the RG function $Z_k(\phi)$ would become a step function which is unity for all finite values of $k$ (and $\phi$) and is only singular for $k=0$ in the range $0 \leq \phi \leq \phi_0$. In this way, one is left with a massive, free-field theory for all non-zero values of the momentum, and the only remnant of the non-trivial infrared sector is the singular re-scaling of $\phi$ (the projection of the full scalar field $\Phi(x)$ onto $p_\mu=0$).
This is precisely the scenario of Refs.[@alternative], where for $\Lambda \to \infty$ the re-scaling of the scalar condensate diverges logarithmically as $\sim \ln\Lambda$ and the re-scaling of the finite-momentum modes tends to unity. The existence of such a divergent re-scaling factor for the vacuum field would have potentially important phenomenological implications for the scalar sector of the standard model and for the validity of the generally accepted upper bounds on the Higgs boson mass [@alternative].
Of course, for a more precise comparison with the numerical results obtained in this Letter, one should study the value of the peak in $Z_k(\phi)$ at different values of the bare parameters to check the predicted logarithmic behaviour $Z_{\rm peak}\sim \ln (\Lambda)$ that, at the present, is just a conjecture suggested by previous works. In turn, this requires to improve on the present integration routines to extend the solution of the RG equations towards the point $k=0$ thus reducing the arbitrariness associated with different extrapolation forms.
At the same time, the existence of a peak in $Z_k(\phi)$ is an interesting feature that would deserve to go beyond the present numerical analysis. In this respect, since, according to [@laca], $V_k(\phi)$, for $k\to 0$, is only weakly dependent on the type of infrared regulator, we expect the peak of Z to be a physical phenomenon and not an artifact of the regulator. This can be checked by re-considering the problem from scratch at the level of the exact flow equations.
50 pt
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'An unambiguous definition of meson resonance masses requires a description of the associated phase shifts in terms of a manifestly unitary $S$-matrix and its complex poles. However, the commonly used Breit-Wigner (BW) parametrisations can lead to appreciable deviations. We demonstrate this for a simple elastic resonance, viz. $\rho(770)$, whose pole and BW masses turn out to differ by almost 5 MeV. In the case of the very broad $f_0(500)$ and $K_0^\star(700)$ scalar mesons, the discrepancies are shown to become much larger, while also putting question marks at the listed PDG BW masses and widths. Furthermore, some results are reviewed of a manifestly unitary model for meson spectroscopy, which highlight the potentially huge deviations from static model predictions. Finally, a related unitary model for production amplitudes is shown to explain several meson enhancements as non-resonant threshold effects, with profound implications for spectroscopy.'
address:
- 'Centro de Física e Engenharia de Materiais Avançados, Instituto Superior Técnico, Universidade de Lisboa, P-1049-001, Portugal'
- 'Centro de Física da UC, Departamento de Física, Universidade de Coimbra, P-3004-516, Portugal'
author:
- |
George Rupp\
Eef van Beveren
title: 'Dramatic implications of unitarity for meson spectroscopy[^1]'
---
Introduction
============
The most fundamental cornerstone of the PDG tables is the uniqueness of $S$-matrix pole positions of unstable particles, as a consequence of quantum-field-theory principles. Therefore, the unitarity property of the $S$-matrix should ideally be respected in whatever description of mesonic resonances in experiment, on the lattice, and in quark models. However, simple Breit-Wigner (BW) parametrisations that not always satisfy unitarity continue to be widely used in data analyses of mesonic processes. In this short paper, the resulting discrepancies will be studied for three elastic meson resonances, viz. $\rho(770)$, $f_0(500)$ (alias $\sigma$), and $K_0^\star(700)$ (alias $\kappa$).
Now, quark models usually treat mesons as permanently bound $q\bar{q}$ states, ignoring the dynamical effects of strong decay, be it real or virtual. Only a model that respects $S$-matrix unitarity of the decay products can be reliably compared to resonances in experiment. A few important results of models employed by us since long ago will be reviewed here. Finally, some predictions of a strongly related unitary model of productions processes, with far-reaching consequences for meson spectroscopy, will be briefly revisited.\
Pole mass vs. Breit-Wigner mass
===============================
Now we summarise very succinctly how to relate the pole mass of an elastic resonance to its typical Breit-Wigner (BW) mass, with some applications. A detailed derivation will be published elsewhere.
A $1\times1$ partial-wave $S$-matrix, being a function of the relative momentum $k$, can be written as [@Taylor] $S_l(k)=J_l(-k)/J_l(k)$, where $J_l(k)$ is the so-called Jost function. A resonance then corresponds to a simple pole in $S_l(k)$ for complex $k$ with positive real part and negative imaginary part, that is, a pole lying in the fourth quadrant of the complex $k$-plane. So the simplest ansatz for the $S$-matrix and thus for the Jost function is to write $J_l(k)=k-{k_{\mbox{\scriptsize pole}}}= k-(c-id)$, with $c>0 , d>0$. Note that this requires $S_l(k)$ to have a zero in the second quadrant, viz. for $k=-c+id$. But then the $S$-matrix cannot be unitary [@Kaminski], for real $k$, i.e., $S_l^\star(k)\neq S_l^{-1}(k)$. It will only satisfy unitarity if [@Taylor] the Jost function obeys $J_l^\star(k)=J_l(-k)$, for real $k$. Consequently, the Jost function should read [@Kaminski] $J_l(k)=(k-{k_{\mbox{\scriptsize pole}}})(k+{k_{\mbox{\scriptsize pole}}}^\star)=(k-c+id)(k+c+id)$. So $S_l(k)$ has a symmetric pair of poles in the 3rd and 4th quadrants, corresponding to an equally symmetric pair of zeros in the 1st and 2nd quadrants. Note that in the complex-energy plane, given by $E=2\sqrt{k^2+m^2}$ in the case of two equal-mass particles, this amounts to one pole and one zero lying symmetrically in the 4th and 1st quadrants, respectively. Since a $1\times1$ $S$-matrix can generally be written as $S_l(k)=\exp(2i\delta_l(k))=(1+i\tan\delta_l(k))/(1-i\tan\delta_l(k)$), we can use the unitary expression for the Jost function above to derive [@Kaminski]\
$$\tan\delta_l(k) \; = \; \frac{2k\,\mbox{Im}({k_{\mbox{\scriptsize pole}}})}{k^2-|{k_{\mbox{\scriptsize pole}}}|^2} \; = \;
\frac{2dk}{c^2+d^2-k^2} \; . \\[-1.5mm]
\label{tandelta}$$ When the partial-wave phase shift $\delta_l(k)$ reaches $90^\circ$, we get for the modulus of the corresponding amplitude , for ${k_{\mbox{\scriptsize max}}}^2=c^2+d^2$. The associated maximum energy $E_{\mbox{\scriptsize max}}=2\sqrt{{k_{\mbox{\scriptsize max}}}^2+m^2}=2\sqrt{c^2+d^2+m^2}$ is different from the maximum in a typical Breit-Wigner (BW) amplitude $T_l(E)\propto (E-{M_{\mbox{\scriptsize BW}}}+i{\Gamma_{\mbox{\scriptsize BW}}}/2)^{-1}$, which is called the BW mass and just given by the real part of the pole in the fourth quadrant of the complex-energy plane, viz.${M_{\mbox{\scriptsize BW}}}=2\sqrt{{k_{\mbox{\scriptsize BW}}}^2+m^2}=2\sqrt{c^2+m^2}$. Such a BW amplitude, in spite of being unitary in the case of an isolated resonance, can give rise to significant differences compared to $S$-matrix approaches.
Next we illustrate the consequences of these unitarity considerations in the simple case of the very well-known meson $\rho(770)$ [@PDG2018], which is an elastic $P$-wave resonance in $\pi\pi$ scattering. The PDG lists its mass and total width as [@PDG2018] $M_{\rho^0}=(775.26\pm0.25)$ MeV and $\Gamma_{\!\!\rho^0}=(147.8\pm0.9)$ MeV, where the width follows almost exclusively ($\approx100$%) from the decay mode $\rho^0\to\pi^+\pi^-$, with $m_{\pi^\pm}=139.57$ MeV.
In the following, we shall refer to BW mass (${M_{\mbox{\scriptsize BW}}}$) for the energy where the resonance’s phase shift passes through $90^\circ$ and so the modulus of the amplitude is maximum. This also holds for the standard BW amplitude given above, though in the latter case it corresponds to the real part of the resonance pole’s complex energy. In contrast, here we want to determine the difference between pole mass and (unitary) BW mass for the $\rho(770)$. After some lengthy yet straightforward algebra, we can express the pole mass explicitly in terms of the BW mass and the pole width as\
$${M_{\mbox{\scriptsize pole}}}\; = \; \sqrt{\sqrt{({M_{\mbox{\scriptsize BW}}}^2-4m^2)^2-4m^2\,{\Gamma_{\!\mbox{\scriptsize pole}}}^2}+4m^2-{\Gamma_{\!\mbox{\scriptsize pole}}}^2/4} \; .
\mbox{ } \\[-1mm]
\label{mp}$$ Note that it is not possible to write ${M_{\mbox{\scriptsize pole}}}$ as a simple closed-form expression in terms of both ${M_{\mbox{\scriptsize BW}}}$ and ${\Gamma_{\mbox{\scriptsize BW}}}$. Assuming for the moment that ${\Gamma_{\!\mbox{\scriptsize pole}}}\simeq{\Gamma_{\mbox{\scriptsize BW}}}$, we substitute in Eq. (\[mp\]) the PDG values for ${M_{\mbox{\scriptsize BW}}}$ and ${\Gamma_{\!\mbox{\scriptsize pole}}}$, which yields ${M_{\mbox{\scriptsize pole}}}=770.67$ MeV. This is 4.5 MeV lower than the PDG $\rho(770)$ mass of 775.25 MeV! Now we check whether indeed ${\Gamma_{\!\mbox{\scriptsize pole}}}\simeq{\Gamma_{\mbox{\scriptsize BW}}}$, by calculating the half-width of the $\rho(770)$ peak from the modulus squared of the amplitude $T_l(k)$, starting from Eq. (\[tandelta\]). The result is ${\Gamma_{\mbox{\scriptsize BW}}}=147.83$ MeV, so indeed very close to the assumed ${\Gamma_{\!\mbox{\scriptsize pole}}}=147.8$ MeV. Finally, we compare pole mass and width vs. BW mass and width for the very broad scalar mesons $f_0(500)$ and $K_0^\star(700)$ [@PDG2018]. As the latter resonance decays into $K\pi$, we must now deal with the unequal-mass case, which does not allow to derive simple expressions. Yet on the computer the real and imaginary parts of ${k_{\mbox{\scriptsize pole}}}$ can be simply obtained, allowing to derive ${M_{\mbox{\scriptsize BW}}}$ and ${\Gamma_{\mbox{\scriptsize BW}}}$ as before.
Let us now check what the consequences are for $f_0(500)$ and $K_0^\star(700)$. Their pole positions as well as BW masses and widths are listed in the PDG Meson Tables as [@PDG2018] $$\begin{aligned}
{l}
\hspace*{-5mm}
f_0(500): \;\;
\left\{\begin{array}{l}
{E_{\mbox{\scriptsize pole}}}\;=\;\left\{(475\pm75)-i(275\pm75)\right\} \mbox{MeV}\;,
\\[1mm]
{M_{\mbox{\scriptsize BW}}}\; = \; (475 \pm 75) \; \mbox{MeV} \;,\;\;
{\Gamma_{\mbox{\scriptsize BW}}}\; = \; (550 \pm 150) \; \mbox{MeV} \; ;
\end{array}\right. \\
\label{sigmapdg}
K_0^\star(700): \;\;
\left\{\begin{array}{l}
{E_{\mbox{\scriptsize pole}}}\;=\;\left\{(680\pm50)-i(300\pm40)\right\} \mbox{MeV}\;,
\\[1mm]
{M_{\mbox{\scriptsize BW}}}\; = \; (824 \pm 30) \; \mbox{MeV} \;,\;\;
{\Gamma_{\mbox{\scriptsize BW}}}\; = \; (478 \pm 50) \; \mbox{MeV} \; .
\end{array}\right.
\label{kappapdg}\end{aligned}$$ But using our equations imposed by elastic $S$-matrix unitarity, we obtain\
$$\begin{array}{rl}
f_0(500): & \!\!{M_{\mbox{\scriptsize BW}}}\; = \; (592 \pm 99) \; \mbox{MeV} \; , \;\;
{\Gamma_{\mbox{\scriptsize BW}}}\; = \; (555 \pm 152) \; \mbox{MeV} \; ; \\[1mm]
K_0^\star(700): & \!\!{M_{\mbox{\scriptsize BW}}}\; = \; (907 \pm 49) \; \mbox{MeV} \; , \;\;
{\Gamma_{\mbox{\scriptsize BW}}}\; = \; (709 \pm 122) \; \mbox{MeV} \; .
\label{skunitary}
\end{array}$$ The conclusion is that the PDG seems to underestimate the BW masses of both $f_0(500)$ and $K_0^\star(700)$, as well as the BW width of $K_0^\star(700)$. We stress again that here “BW” refers to the energy at which $\delta_l(E)=90^\circ$, in the context of the present simple pole model. Note that reality is more complicated, since the $f_0(500)$ resonance overlaps with $f_0(980)$ [@PDG2018] and $K_0^\star(700)$ with $K_0^\star(1430)$ [@PDG2018], besides the influence of Adler zeros on the amplitudes [@PLB572p1]. Nevertheless, the need for a uniform and unitary treatment of especially broad resonances in experimental analyses is undeniable.
To conclude this section, we note that calculating ${M_{\mbox{\scriptsize BW}}}$ for $f_0(500)$ and $K_0^\star(700)$ via the cross section instead of the amplitude’s modulus becomes already problematic, while no ${\Gamma_{\mbox{\scriptsize BW}}}$ can even be defined at all. Also, for inelastic resonances the mass discrepancy due to the use of a non-unitary parametrisation can become as large as 170 MeV in the case of $\rho(1450)$ [@PRD96p113004].\
Unitarity distortions of $q\bar{q}$ spectra
===========================================
Fully accounting for unitarity when describing meson resonances, or just computing mass shifts of $q\bar{q}$ states from real and virtual meson loops, can give rise to enormous distortions of confinement spectra [@CPC41p053104]. Moreover, it can even lead to the dynamical generation of additional states. This allowed the unitarised multichannel quark model of Ref. [@ZPC30p615] to predict for the first time a complete nonet of light scalar-meson resonances, whose predicted masses and widths are still compatible with present-day PDG limits [@PDG2018]. More recently, a strongly related model was formulated [@IJTPGTNO11p179] in $p$-space, called Resonance-Spectrum Expansion (RSE), resulting in a coupled-channel $T$-matrix for non-exotic meson-meson scattering diagrammatically given by\
-- -- -- -- --
-- -- -- -- --
\
Here, the wiggly lines represent a tower of bare $q\bar{q}$ states, which couple to two-meson channels via a [${}^{3\!}P_0$]{} vertex. For more details and closed-form multichannel expressions, see e.g. Ref. [@PRD84p094020]. Using the RSE formalism, a coupled-channel calculation of light and intermediate scalar mesons was carried out in Ref. [@APPS2p437], yielding the poles\
$f_0(500): \;(464-i217)$ MeV, $f_0(1370): \;(1335-i185)$ MeV;\
$f_0(980): \;(987-i29)$ MeV, $f_0(1500): \;(1530-i14)$ MeV;\
$a_0(980): \;(1023-i47)$ MeV, $a_0(1450): \;(1420-i185)$ MeV;\
$K_0^\star(700): \;(722-i266)$ MeV, $K_0^*(1430): \;(1400-i96)$ MeV.\
These results are close to those found in the $r$-space model of Ref. [@ZPC30p615]. Note again the generation of two scalar resonances for each bare $P$-wave $q\bar{q}$ state.
The possible doubling of resonances due to unitarisation becomes yet more peculiar in cases where it is not even clear which is the intrinsic one and which the dynamically generated state. For example, the $D_{s0}^\star(2317)$ [@PDG2018] scalar $c\bar{s}$ meson showed up as a dynamical state in a simple RSE model [@PRL91p012003] with only the $DK$ channel included, but as a strongly mass-shifted intrinsic state in the multichannel RSE calculation of Ref. [@PRL97p202001]. This cross-over is demonstrated in more detail for the $\chi_{c1}(2P)$ [@PDG2018] (old $X(3872)$) axial-vector $c\bar{c}$ state in Ref. [@EPJC73p2351], with being an intrinsic or dynamical state depending on fine details of the model’s parameters. Clearly, this ambiguity in the quark-model assignment of $D_{s0}^\star(2317)$ and $\chi_{c1}(2P)$, as well as of probably several other mesons, has severe implications for spectroscopy.\
Non-resonant peaks from unitary production amplitudes
=====================================================
Most meson resonances are nowadays not observed in meson-meson scattering, mainly extracted from meson-proton data, but rather in production processes, like e.g. $e^+e^-$ annihilation or $B$-meson decays. In these situations no initial $q\bar{q}$ annihilation takes place, as the starting point is already an isolated $q\bar{q}$ pair, resulting from a virtual photon in $e^+e^-$ or as a decay product from a heavier meson like e.g. $J/\psi$ or $B$. The corresponding production amplitude $P$ is defined [@AOP323p1215] in the RSE formalism as a non-resonant, lead term plus its infinite rescattering series via the above RSE $T$-matrix, i.e.,\
-- -- --
-- -- --
\
or $P_k=\mbox{Re}(Z_k)+i\sum_l Z_l T_{kl}$, with the $Z_k$ being purely kinematical functions related to the $q\bar{q}$–meson-meson vertex. In the RSE model of Ref. [@AOP323p1215], where the detailed expressions can be found, the $Z_k$ are spherical Hankel$^{(1)}$ functions and their real parts spherical Bessel functions. The $P_k$ components satisfy [@AOP323p1215; @EPL81p61002] the extended-unitarity relation $\mbox{Im}(P_k)=\sum_l T^\star_{kl}\,P_l$. Note that this can be rewritten in terms of purely imaginary functions $\tilde{Z}_k$, so without the inhomogeneous term, but then the real functions $i\tilde{Z}_k$ would necessarily include elements of the $T$-matrix itself and so not be purely kinematical anymore [@EPL84p51002].
There can be many applications of our production formalism in hadron spectroscopy. In Ref. [@PRD80p074001] several structures are analysed in $K^+K^-$, $D\bar{D}$, $B\bar{B}$, and $\Lambda_c\bar{\Lambda}_c$ data. The most dramatic conclusions are that $\Upsilon(10580)$ and $X(4660)$ (now called $\psi(4660)$ [@PDG2018]) are probably not genuine resonances but rather enhancements rising from the $B\bar{B}$ and $\Lambda_c\bar{\Lambda}_c$ thresholds, respectively.
Conclusions
===========
We have shown unitarity to be an essential constraint in analysing scattering data in order to allow an unambiguous determination of resonance parameters, even in the elastic case. On the other hand, in quark models a unitary description of meson resonances may lead to enormous deviations from the naive bound-state spectra, and moreover give rise to extra states not present in the bare spectra. Finally, unitarity also plays a fundamental role in production processes, by relating them to scattering and yielding threshold enhancements that may be mistaken for true resonances. The consequences for modern meson spectroscopy are far-reaching.\
[99]{} J. R. Taylor, *Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, *John Wiley & Sons, Inc., New York, London, Sydney, Toronto, 1972, pp. 477, ISBN 0-471-84900-6; see page 220.**
R. Kamiński, in: Seminar at CeFEMA/IST in Lisbon, May 29, 2018.
M. Tanabashi [*et al.*]{} \[Particle Data Group\], [*Phys. Rev. D*]{} [**98**]{}, 030001 (2018). D. V. Bugg, [*Phys. Lett. B*]{} [**572**]{}, 1 (2003) \[[*Erratum-ibid.*]{} [**595**]{}, 556 (2004)\]. E. Bartoš [*et al.*]{}, [*Phys. Rev. D*]{} [**96**]{}, 113004 (2017) (see Table II). G. Rupp, E. van Beveren, [*Chin. Phys. C*]{} [**41**]{}, 053104 (2017) \[arXiv:1611.00793 \[hep-ph\]\]. E. van Beveren [*et al.*]{}, [*Z. Phys. C*]{} [**30**]{}, 615 (1986) \[arXiv:0710.4067 \[hep-ph\]\]. E. van Beveren, G. Rupp, [*Int. J. Theor. Phys. Group Theor. Nonlin. Opt.*]{} [**11**]{}, 179 (2006) \[arXiv:hep-ph/0304105\]. S. Coito, G. Rupp, E. van Beveren, [*Phys. Rev. D*]{} [**84**]{}, 094020 (2011) \[arXiv:1106.2760 \[hep-ph\]\]. G. Rupp, S. Coito, E. van Beveren, [*Acta Phys. Pol. B Proc. Suppl.*]{} [**2**]{}, 437 (2009) \[arXiv:0905.3308 \[hep-ph\]\]. E. van Beveren, G. Rupp, [*Phys. Rev. Lett.*]{} [**91**]{}, 012003 (2003) \[arXiv:hep-ph/0305035\]. E. van Beveren, G. Rupp, [*Phys. Rev. Lett.*]{} [**97**]{}, 202001 (2006) \[arXiv:hep-ph/0606110\]. S. Coito, G. Rupp, E. van Beveren, [*Eur. Phys. J. C*]{} [**73**]{}, 2351 (2013) \[arXiv:1212.0648 \[hep-ph\]\]. E. van Beveren, G. Rupp, [*Ann. Phys.*]{} [**323**]{}, 1215 (2008) \[arXiv:0706.4119 \[hep-ph\]\]. E. van Beveren, G. Rupp, [*Europys. Lett.*]{} [**81**]{}, 61002 (2008) \[arXiv:0710.5823 \[hep-ph\]\]. E. van Beveren, G. Rupp, [*Europhys. Lett.*]{} [**84**]{}, 51002 (2008). E. van Beveren, G. Rupp, [*Phys. Rev. D*]{} [**80**]{}, 074001 (2009) \[arXiv:0908.0242 \[hep-ph\]\].
[^1]: Presented by G. Rupp at Workshop “Excited QCD 2019”, Schladming, Austria, Jan. 30 – Feb. 3, 2019.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We show that every Hilbert C$^*$-module $E$ is a JB$^*$-triple in a canonical way and establish an explicit expression for the holomorphic automorphisms of the unit ball of $E$.'
author:
- |
José M. Isidro [^1]\
Facultad de Matemáticas,\
Universidad de Santiago,\
Santiago de Compostela, Spain.\
[jmisidro@zmat.usc.es]{}
date: 'December 1, 2001'
title: 'Hilbert C$^*$-modules are JB$^*$-triples.'
---
Introduction
------------
Hilbert C$^*$-modules first appeared in 1953 in a work of Kaplansky [@KAPL] who worked only with modules over commutative unital C$^*$-algebras. In 1973 Paschke [@PAS] proved that most of the properties of Hilbert C$^*$-modules were valid for modules over an arbitrary C$^*$-algebra. About the same time Reiffel independently developed much of the same theory and used it to study representations of C$^*$-algebras. Since then the subject has grown and spread rapidly and now there is an extensive literature on the topic (see [@LAN2] for a systematic introduction). Many interesting developments have been made by Kasparov, who used Hilbert C$^*$-modules as the framework for K-theory. More recently Hilbert C$^*$-modules have been a useful tool in the C$^*$-algebraic approach to quantum groups. The geometry of Hilbert C$^*$-modules has been investigated by Solel in [@SOL], where the isometries of these Banach spaces have been characterized. See also [@AND].
On the other hand Kaup, searching for a metric-algebraic setting in which he could make the study of bounded symmetric domains in complex Banach spaces, introduced a class of complex Banach spaces called JB$^*$-triples. In 1983 he proved that, except for a biholomorphic bijection, every such a domain is the open unit ball of a JB$^*$-triple [@KAUR]. In 1981 he made the complete analytic classification of bonded symmetric domains in reflexive Banach spaces [@KAUS]. Since then the study of JB$^*$-triples has grown and spread considerably.
These two theories have developed independently from one another. Here we show that every Hilbert C$^*$-module is, in a canonical way, a JB$^*$-triple, a bridge between the two theories that may be useful in the study of the geometry of Hilbert C$^*$-modules. We also establish an explicit expression of the holomorphic automorphisms of the unit ball of a Hilbert C$^*$-module.
Hilbert C$^*$-modules
---------------------
We now introduce formally the objects we shall be studying. Let $A$ be a C$^*$-algebra (not necessarily unital or commutative) where the product is denoted by juxtaposition $xy$, the norm is $\Vert \cdot
\Vert_A$ and the $x\mapsto x^*$ stands for the conjugation. An [*inner product*]{} $A$-module is a complex linear space $E$ with two laws of composition $E\times A\to E$ (denoted by $(x,a)\mapsto x.a$) and $E\times E\to A$ (denoted by $(x,y)\mapsto
\langle x ,y\rangle$) such that the following properties hold:
1. With respect to the operation $(x,a)\mapsto x.a$, $E$ is a right $A$-module with a compatible scalar multiplication, that is, $\lambda (x.a)
=(\lambda x).a= x.(\lambda a)$ for all $x\in E$, $a\in A$ and $\lambda \in
\mathbb{C}$.
2. The inner product $(x,y)\mapsto\langle x,y\rangle$ satisfies $$\begin{aligned}
\langle x, \alpha y+\beta z\rangle&=&\alpha\langle x,y\rangle + \beta \langle
x,z\rangle, \\
\langle x, y.a\rangle&=&\langle x,y\rangle a\\
\langle y,x\rangle&=&\langle x,y\rangle^* \\
\langle x,x\rangle &\leq &0, \qquad \hbox{\rm if }\;\; \langle x, x\rangle =0
\;\; \hbox{\rm then}\;\; x=0.\end{aligned}$$ for all $x,yz\in E,$ all $\alpha ,\beta \in \mathbb{C}$ and all $a\in A$.
Note that in particular, the inner product is complex linear in the second variable while it is conjugate linear in the first. This convention is in line with the recent research literature. Let $E$ be an inner product $A$-module; then the Cauchy-Schwarz inequality $$\langle y,x\rangle\,\langle x,y\rangle \leq \Vert \langle x,x\rangle\Vert \,
\langle y,y\rangle,$$ holds, hence $\Vert x\Vert_E ^2\colon = \Vert \langle x,x\rangle\Vert_A $ is a norm in $E$ with respect to which the inner product and the module product are continuous, that is $$\Vert \langle x,y\rangle\Vert _A\leq \Vert x\Vert_E\,\Vert y\Vert_E,
\qquad \Vert x.a\Vert \leq \Vert x\Vert_E\, \Vert a\Vert_A.$$ To simplify the notation we shall use the same symbol $\Vert \cdot \Vert$ to denote the norms on $A$ and $E$. We can also define an $A$-valued [*norm*]{} by $\vert x\vert\colon = \langle x,x\rangle^{\frac{1}{2}}$ for $x\in E$ and we have $$\vert \langle x,y\rangle\vert \leq \Vert x\Vert \, \vert y\vert , \qquad
\vert \langle x,y\rangle \vert \leq \vert x\vert \,\Vert y\Vert,$$ and the module product is also continuous with respect to the new norm on $A$ since $$\Vert x.a\Vert \leq \Vert x\Vert \, \vert a\vert.$$ However the $A$-valued norm in an inner $A$-module $E$ need to be handled with care. For example it need not be the case $\vert x+y\vert \leq \vert x\vert
+\vert y\vert $. An inner product $A$-module $E$ which is a Banach space with respect to the norm $\Vert \cdot\Vert $ is called a [*Hilbert C$^*$-algebra module*]{}. Every C$^*$-algebra can be converted into a Hilbert C$^*$-algebra module by taking $E\colon = A$ with the natural module operation $x.a\colon =xa$ and the inner product $\langle a,b\rangle
\colon = a^*\,b$ for $x,a,b\in A$.
A linear map $f\colon E\to E$ is called an $A$-map if $f(x.a)=f(x).a$ holds for all $x\in E$ and $a\in A$, and we say that $f$ is [*adjointable*]{} if there exists an $A$-map $f^*\colon E\to E$ such that $$\langle f(x),y\rangle =\langle x, f^*(y)\rangle , \qquad x,y\in E.$$ In such a case $f$ is continuous (though the converse is not true!), $f^*$ is adjointable and $(f^*)^*=f$. We let $\mathcal{A}(E)\subset \mathcal{L}(E)$ denote the vector space of all adjointable $A$-module maps on $E$. In fact $\mathcal{A}(E)$ is a C$^*$-algebra in the operator norm since $\Vert f^*f\Vert = \Vert f\Vert^2$ holds for all $f\in \mathcal{A}(E)$. For $x,\,y\in E$ we define $\theta_{x,y}$ (also denoted by $x\otimes y^*$) by $$\theta _{x,y}(z)\colon = x.\,\langle y,z\rangle, \qquad z\in E,$$ Then $\theta_{x,y}$ is adjointable and $\theta_{x,y}^*=\theta_{y,x}$ (see [@LAN2] p. 9). For later reference we state the following
\[pos\] Let $E$ be a Hilbert C$^*$-module and let $f\colon E\to E$ be a bounded $A$-module map. Then $f$ is a positive element in the C$^*$-algebra $\mathcal{A}(E)$ if and only if $\langle x,f(x)\rangle\geq o$ for all $x\in E$.
We refer to [@LAN2] for background on Hilbert C$^*$-modules and for the proofs of the above results.
JB$^*$-triples.
---------------
For a complex Banach space $X$ denote by $\mathcal{L}(X)$ the Banach algebra of all bounded complex-linear operators on $X$. A complex Banach space $Z$ with a continuous mapping $(a, b, c)
\mapsto \{a, b, c\}$ from $Z\times Z\times Z$ to $Z$ is called a [*JB\*-triple*]{} if the following conditions are satisfied for all $a, b, c, d \in Z$, where the operator $a\square b\in \mathcal{L}(Z)$ is defined by $z\mapsto \{abz\}$ and $\lbrack\, , \, \rbrack$ is the commutator product:
1. $\{abc\}$ is symmetric complex linear in $a, c$ and conjugate linear in $b$.
2. $\lbrack a\square b , \, c\square d \rbrack = \{a,b,c\}\square d -
c\square \{d,a,b\}$ (called the Jordan identity. )
3. $a\square a$ is hermitian and has spectrum $\geq 0.$
4. $\Vert \{a,a,a\}\Vert = \Vert a\Vert ^3$.
If a complex vector space $Z$ admits a JB\*-triple structure, then the norm and the triple product determine each other. An [*automorphism*]{} is a linear bijection $\phi \in \mathcal{L}(Z)$ such that $\phi \{z,z, z\}= \{(\phi z), ( \phi z),(\phi z)\}$ for $z\in Z$, which occurs if and only if $\phi$ is a surjective linear isometry of $Z$.
Recall that every C\*-algebra $Z$ is a JB\*-triple with respect to the triple product $2\{abc\} \colon =(ab^*c+cb^*a)$. In that case, every projection in $Z$ is a tripotent and more generally the tripotents are precisely the partial isometries in $Z$. C$^*$-algebra derivations and C$^*$-automorphisms are derivations and automorphisms of $Z$ as a JB$^*$-triple though the converse is not true.
We refer to [@KAUR], [@UPM] and the references therein for the background of JB$^*$-triples theory.
Hilbert $C^*$-modules are JB$^*$-triples.
-----------------------------------------
For $a\in A$ fixed, we denote by $R_a \in
\mathcal{L}(E)$ the operator $x\mapsto x.a$ of right multiplication by $a$.
Every Hilbert C$^*$-module $E$ is a a JB$^*$-triple in a canonical way
Let $E$ be a Hilbert C$^*$-module over the C$^*$-algebra $A$ and define a triple product in $E$ by $$\label{tp}
2\,\{x,y,z\}\colon = x.\langle
y,z\rangle+z.\langle y,x\rangle, \qquad
x,y,z\in E.$$ It is clear that $\{\cdot , \cdot , \cdot \}$ symmetric complex linear in the external variables, and complex conjugate linear in the middle variable. It is a matter of routine calculation to check that the triple product satisfies the Jordan identity. On the other hand, for fixed $x\in E$ we have $$2 \,(x\Box x)\, z= x.\langle x, z\rangle +
z.\langle x,x\rangle , \qquad z\in E$$ which can be written in the form $x\Box x=\frac{1}{2}
(\theta_{x,x}+R_{\vert x\vert ^2})$. We show that the summands in the right hand side of the latter are hermitian elements in the algebra $\mathcal{L}
(E)$. Since $\mathcal{A}(E)$ is a closed complex subalgebra of $\mathcal{L}(E)$ and contains the unit element, it suffices to consider the numerical range of $\theta_{x,x}$ and $R_{\vert x\vert ^2}$ viewed as elements in the C$^*$-algebra $\mathcal{A}(E)$, and we have seen before that $\theta_{x,x}$ is selfadjoint. Clearly $$(\exp \,itR_{\vert x\vert ^2})\,(w)= w.(\exp \,it \vert
x\vert^2) \qquad w\in E,$$ and as $\exp \,it \vert x\vert^2$ is a unitary element in $A$, the operator $\exp \,itR_{\vert x\vert
^2}$ is an isometry of $E$ for all $t\in \mathbb{R}$, which shows that $R_{\vert x\vert ^2}$ is hermitian. For $y\in E$ we have $$\langle y,\; \theta_{x,x}(y)\rangle= \langle y,\;
x.\langle
x,y\rangle\rangle=\langle y,x\rangle\langle
x,y\rangle\geq 0$$ which by (\[pos\]) proves that $\theta_{x,x}\geq 0$ in $\mathcal{A}(E)$ hence also in $\mathcal{L}(E)$. Clearly $\vert x\vert^2\geq 0$ in $A$, hence its spectrum satisfies $\sigma_A(\vert
x\vert^2)\subset [0,\infty )$ and therefore $$\sigma_{\mathcal{L}(A)}(R_{\vert x\vert^2})\subset
\sigma_A(\vert x\vert^2)\subset [0,\infty )$$ Since the numerical range is the convex hull of the spectrum, $R_{\vert x\vert^2}\geq 0$ as we wanted to check.
Let us set $y\colon =\langle x,x\rangle \in
A$ for every $x\in E$. The definition of the norm in $E$ and the properties of the norm in the C$^*$-algebra $A$ yield $$\begin{aligned}
\Vert \{ x,x,x\}\Vert ^2&=&\Vert x.\langle
x,x\rangle\Vert ^2=
\Vert \langle x.\langle x,x\rangle ,\;
x.\langle x,x\rangle \rangle \Vert =\\
\Vert \langle x,x\rangle \,\langle x,x\rangle\,\langle
x,x\rangle \Vert &=& \Vert \{y,y,y\}\Vert = \Vert
y\Vert ^3 = \Vert \langle x,x\rangle \Vert ^3 = \Vert
x\Vert ^6\end{aligned}$$ which shows property 4. Finally, this is the unique JB$^*$-triple structure on $E$ since the triple product is determined by the norm of $E$.
Holomorphic automorphisms of the unit ball.
-------------------------------------------
Motivated by the deep formal analogy between Hilbert C$^*$-modules $E$ and Hilbert spaces $H$, we shall establish an explicit formula for the holomorphic automorphisms of the unit ball of $E$. Recall [@KAUR] that, for $c\in E$, the [*Bergmann operator*]{} of $E$ is given by $$B(c,c)(x)\colon = x-2(c\Box c)(x)+ Q_c^2(x), \qquad
x\in E.$$ In our case $$\begin{aligned}
2(c\Box c)(x) &=& 2\{c,c,x\}=c.\langle
x,x\rangle +x.\vert c\vert^2= c\otimes c^*
(x)+x.\vert c\vert^2,
\\
Q_c^2(x)&=& \{c ,\,Q_c(x),\,c\}= \{c,\;
c.\langle x,c\rangle ,\, c\}= \\
c.\langle c.\langle x,c\rangle ,
\,c\rangle &=&
c.\langle c,x\rangle \vert c\vert ^2=
(c\otimes c^*) (x.\vert c\vert^2).\end{aligned}$$ Therefore $$\begin{aligned}
B(c,c)(x)&=& x.(\One -\vert c\vert ^2 )+
(c\otimes c^*)\,\big( x.(\One -\vert c\vert ^2 )\big)
=\\
{}&=& ( \One -c\otimes c^* ) \, \big(
x.(\One -\vert c\vert ^2 ) \big) . \end{aligned}$$ Recall that $\One -c\otimes c^*$ and $\one -\vert
c\vert ^2$ are selfadjoint elements in the C$^*$-algebras $\mathcal{A}(E)$ and $A$ respectively, hence they have well defined square roots. We show the operator $$B_c(x)\colon = (\One -c\otimes c^*)^{\frac{1}{2}}
\,\big( x. (\one -\vert c\vert
^2)^{\frac{1}{2}}\big),
\qquad x\in E,$$ satisfies $B_c^2 =B(c,c)$. Indeed, since $\One
-c\otimes c^*$ is an $A$-linear map so is its square root and we have $$\begin{aligned}
&{}&B_c(B_c(x))= (\One -c\otimes c^*)^{\frac{1}{2}}\;
\big( B_c(x).\, (\one -\vert c\vert^2)^{\frac{1}{2}}
\big) = \\
&{}&\big( \,(\One -c\otimes c^*)^{\frac{1}{2}}\,
B_c(x)\,\big) .
(\one -\vert c\vert^2)^{\frac{1}{2}})= \\
&{}&\big( \,(\One -c\otimes c^*)^{\frac{1}{2}}\,
[\, (\One -c\otimes c^*)^{\frac{1}{2}} \,x. (\one
-\vert c\vert^2)^{\frac{1}{2}}\,]\big)\, .\,
(\one -\vert c\vert^2)^{\frac{1}{2}} =\\
&{}&(\One -c\otimes c^*) x (\one -\vert c\vert^2)=
B(c,c)(x) \end{aligned}$$ as we wanted to check.
Now we prove that for $c$ and $x$ in the open unit ball of $E$, $\one +\langle c,x\rangle$ is an invertible element in $A$. Indeed, let us denote by $\sigma (a)$ and $v(a)$ the spectrum and the numerical range of $a$ in the unital algebra $A$. Then $$\sigma (\one +\langle c,x\rangle )\subset v(\one
+\langle c,x\rangle )\subset 1+v(\langle c,x\rangle).$$ Since $\Vert \langle c.x\rangle \Vert \leq \Vert
c\Vert \, \Vert x\Vert <1$, the numerical range $v(\langle c,x\rangle )$ is contained in the open unit disc of $\mathbb{C}$, therefore $-1\notin
v(\langle c,x\rangle )$ and by the above $0\notin
\sigma (\one +\langle c,x\rangle )$. In particular, $$x.(\one +\langle c,x\rangle )^{-1}$$ is well defined in $A$. Recall [@KAUR] that for $c$ in the open unit ball of $E$, the [*transvection*]{} $g_c$ is the holomorphic automorphism of the open ball of $E$ given by $$g_c(x)\colon = c+ B_c \big( x(\One +c\Box
x)^{-1}\big),
\qquad \Vert x\Vert <1.$$ Replacing the expression of $B_c$ and $\One+c\Box x$ we get $$g_c(x) =c+ (\One -c\otimes c^*)^{\frac{1}{2}}\;[\,
x.(\one +\langle c,x\rangle )^{-1}\, (\one -\vert
c\vert ^2)^{\frac{1}{2}}\,]$$ an expression that can be restated in terms of projections and coincides with the well known formula for the transvections of the ball in a Hilbert space see ([@HAR] p. 21). By [@KAUR] every holomorphic automorphism $h$ of the unit ball of $E$ can be represented in the form $h= L \circ g_c$ for some surjective linear isometry of $E$ and some $c\in E$ with $\Vert c\Vert <1$.
Extreme points of the unit ball.
--------------------------------
For a complex Banach spce $E$, the set $\Extr B_E$ of extreme points in the unit ball $B_E$ of $E$ palys an important role in the study of the geometry of $E$. Obviously, we can replace a HIlbert C$^*$-module $E$ with its associated JB$^*$-triple in order to study the extreme points of the ball $B_E$. By ([@KAUP] prop. 3.5) we have $$\Extr B_E=\{c\in E \colon B(c,c)=0\}$$ that is, for $c\in E$ the condition $c\in \Extr B_E$ is equivalent to $( \One -c\otimes c^* ) \, \big(
x.(\One -\vert c\vert ^2 ) \big)$ for all $x\in E.$ Therefore we have two obvious families of extreme points given by $$\begin{aligned}
E.(\one -\vert c\vert) &=&\{0\}\Longrightarrow c\in \Extr B_E,\\
(\Id -c\otimes c^*)\,E&=&\{0\}\Longrightarrow c\in \Extr B_E.\end{aligned}$$ These two families may coincide (as it occurs when $E$ is Hilbert space) but in general they are different. We do not know whter every extreme point lies in one of the above families. Every extreme point is a tripotent, that is, it satisfies $c=\{c, c,c\}=
c.\langle c, c\rangle$.
It might be interesting to characterize Hilbert C$^*$-triples within the category of JB$^*$-triples.
[99]{}
[^1]: Supported by Ministerio de Educación y Cultura of Spain, Research Project PB 98-1371.
| {
"pile_set_name": "ArXiv"
} |
[**Complete BFT Embedding of Massive Theory with One- and Two-form Gauge Fields**]{}
0.5cm
[Seung-Kook Kim$^{}$[^1], Yong-Wan Kim$^{}$[^2] and Young-Jai Park$^{}$[^3]]{}
[$^{1}$ Department of Physics, Seonam University,\
Namwon, Chonbuk 590-170, Korea\
$^{2}$ Department of Physics and Institute for Science and Technology,\
Sejong University, Seoul 143-747, Korea\
$^{3}$Department of Physics and Basic Science Research Institute,\
Sogang University, C.P.O. Box 1142, Seoul 100-611, Korea\
]{}
()
[**ABSTRACT**]{}
> We study the constraint structure of the topologically massive theory with one- and two-form fields in the framework of Batalin-Fradkin-Tyutin embedding procedure. Through this analysis we obtain a new type of Wess-Jumino action with novel symmetry, which is originated from the topological coupling term, as well as the Stückelberg action related to the explicit gauge breaking mass terms from the original theory.
>
> 0.5cm
>
> PACS: 11.10.Ef, 11.30.Ly, 11.15.-q\
> Keywords: Topological mass generation; Hamiltonian and Lagrangian embedding; Antisymmetric tensor gauge fields\
Introduction
============
Dirac scheme in the Hamiltonian formalism [@dirac] has been widely used to quantize second-class constraint system in which the Poisson brackets of constraints do not vanish on constraint surface. For second-class constraint system, however, there are under unfavorable circumstances in finding canonically conjugate pairs since the resulting Dirac brackets may be in general field-dependent and/or nonlocal, and have a serious ordering problem between field operators. On the other hand, the quantization of first-class constraint system [@fradvil75; @henneaux85] has been well appreciated in a gauge invariant manner preserving Becci-Rouet-Stora-Tyutin symmetry [@brst]. Therefore, if second-class constraint system can be converted into first-class one using auxiliary degrees of freedom to extend a phase space, we do not actually need to define Dirac brackets and then the remaining quantization program follows the method of Ref. [@fradvil75; @henneaux85; @brst]. This procedure has been established by Batalin, Fradkin, and Tyutin (BFT) [@bf86; @bt91] and extensively studied in the canonical formalism for various models including massive gauge fields [@proca], spontaneously broken gauge theories [@sbgt], $SU(2)$/$SU(3)$ Skyrmion models [@skyrme], non-linear sigma models [@sigma], noncommutative systems [@noncom], and many others [@other]. The BFT embedding formalism has been also widely applied to obtain the Wess-Zumino (WZ) actions [@wz] showing that original theory can be regarded as a gauge-fixed version of extended gauge system, while verifying dual equivalent descriptions [@bft-dual] in particular gauges from the phase space partition function corresponding to the BFT embedded involutive Hamiltonian.
On the other hand, antisymmetric tensor fields appearing first as a mediator of the interaction [@kalb] have been much interested in as an alternative of the Higgs mechanism without residual Higgs scalar [@djt; @alt-higgs]. With the topologically interacting terms of the form $B\wedge F$, this mechanism is considered generic in string phenomenology [@string]. Moreover, various dual descriptions between different models have been widely studied where antisymmetric tensor fields play an important role in realization of dualities [@dual; @duff]. In particular, several years ago Banerjee and Banerjee had studied a master Lagrangian [@banerjee] which is a first order massive spin-one theory involving antisymmetric tensor fields in order to show the dual equivalence of the Proca model and the massive Kalb-Ramond model within a path integral framework. Recently, Harikumar and Sivakumar [@harikumar] have elaborated on the Lagrangian through the Hamiltonian and Lagrangian embedding technique, and have shown that the embedded theory with appropriate gauge fixing is equivalent to the $B\wedge F$ theory on the level of Hamiltonian. The Lagrangian regarded as a topologically massive theory has interesting constraint structure related to the topological coupling term as well as the gauge symmetry breaking mass term. As results, the second-class constraints appears from two different origins: One comes from the explicit gauge symmetry breaking mass term, and the other from the topological coupling term. Since this theory have been studied by a reduction of the set of constraints through Faddeev-Jackiw method [@fj], there still remains to be clarified the role of auxiliary fields, which are originated from the topological interaction term.
In the present paper, we shall fully apply the BFT method to the topologically massive theory with one- and two-form gauge fields. As results, we will explicitly show that auxiliary fields in part for the BFT embedding are nothing but the well-known Stückelberg fields on one hand, and find a new type of WZ action [@nwz] composed of the remaining auxiliary fields, which are originated from symplectic structure of the model on the other hand. We also clarify the underlying constraint structure concerning irreducible and reducible constraints.
In section 2, we newly construct the constraint structure of this model, which is much simpler than that of the previous work [@harikumar] due to the absence of derivatives in their Poisson brackets. In section 3, we carry out the complete BFT embedding of the theory which has local gauge symmetry supplemented with other: Local gauge symmetry is recovered from explicitly gauge symmetry breaking mass terms and the other comes from underlying symplectic structure of topological term. In section 4, by identifying new auxiliary degrees of freedom with Stückelberg vector fields and new type of WZ fields, we obtain simultaneously the Stückelberg Lagrangian related to the explicit gauge breaking mass terms and a new type of WZ action with novel symmetry, which is originated from the symplectic structure of the theory. In section 5, we revisit the original theory making use of gauging technique to show the equivalence of the gauged Lagrangian and the Stückelberg Lagrangian on the constraint surface. Conclusion is devoted in section 6.
Constraint Structure of Topologically Massive Theory
====================================================
In this section, we consider topologically massive theory with one- and two-form fields, described by the following first order Lagrangian [@banerjee; @harikumar]: $$\label{Lag} {\cal L}=
-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}+\frac{1}{2}A_\mu A^\mu
+\frac{1}{2m}\epsilon_{\mu\nu\rho\sigma}B^{\mu\nu}\partial^\rho
A^\sigma.$$ From the symmetrized form of the Lagrangian of $$\label{sym-Lag} {\cal L}=
-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}+\frac{1}{2}A_\mu A^\mu
+\frac{1}{4m}\epsilon_{\mu\nu\rho\sigma}B^{\mu\nu}\partial^\rho
A^\sigma -\frac{1}{4m}\epsilon_{\mu\nu\rho\sigma}\partial^\mu
B^{\nu\rho} A^\sigma,$$ we read the canonical momenta as $$\begin{aligned}
\label{momenta}
&& \pi_0 = 0, ~~~\pi_i = \frac{1}{4m}\epsilon_{ijk}B^{jk}, \nonumber\\
&& \pi_{0i} = 0, ~~~\pi_{ij} = - \frac{1}{2m}\epsilon_{ijk}A^k,\end{aligned}$$ where we denote $\epsilon_{0ijk}=\epsilon_{ijk}$ and $\epsilon^{123}=+1$. Then, the primary Hamiltonian yields $$\label{pri-H} {\cal H}_p={\cal H}_c + \lambda^0 \pi_0 + \lambda^i
\Omega_i +\Sigma^{0i}\pi_{0i}+\Sigma^{ij}\Omega_{ij}$$ with the Lagrange multipliers $\lambda^0$, $\lambda^i$, $\Sigma^{0i}$, and $\Sigma^{ij}$, where the canonical Hamiltonian is given by $$\begin{aligned}
\label{can-H} {\cal
H}_c&=&\frac{1}{4}B_{ij}B^{ij}-\frac{1}{2}A_iA^i
+\frac{1}{2}B_{0i}B^{0i}-\frac{1}{2}A_0A^0\nonumber\\
&-&\frac{1}{m}\epsilon^{ijk}B_{0i}\partial_j
A_k-\frac{1}{2m}A^0\epsilon^{ijk}\partial_iB_{jk},\end{aligned}$$ and the primary constraints are defined as $$\begin{aligned}
\label{pri-con} && \pi_0\approx 0,
~~~\Omega_i\equiv\pi_i-\frac{1}{4m}\epsilon_{ijk}B^{jk}\approx 0,\nonumber\\
&& \pi_{0i}\approx 0,~~~\Omega_{ij}\equiv\pi_{ij}+
\frac{1}{2m}\epsilon_{ijk}A^k\approx 0.\end{aligned}$$ From the time stability conditions of the constraints $\pi_0$ and $\pi_{0i}$, we have obtained two additional secondary constraints as $$\begin{aligned}
\label{sec-con} \Lambda&\equiv&\dot{\pi}_0=\{\pi_0, {\cal
H}_p\}=A_0+\frac{1}{2m}\epsilon^{ijk}\partial_iB_{jk}\approx
0,\nonumber\\
\Lambda_i&\equiv&\dot{\pi}_{0i}=\{\pi_{0i}, {\cal H}_p
\}=-B_{0i}+\frac{1}{m}\epsilon_{ijk}\partial^j A^k\approx 0,\end{aligned}$$ and the constraints $\Omega_i$, $\Omega_{ij}$ fix the Lagrange multipliers $\Sigma^{ij}$, $\lambda^i$, respectively. The other Lagrange multipliers $\lambda^0$, $\Sigma^{0i}$ are also determined by requiring consistency of the secondary constraints $\Lambda$, $\Lambda_i$ with the equations of motion, and thus no further new constraints are generated. As a result, the Poisson brackets of all the constraints (\[pri-con\]) and (\[sec-con\]) are obtained as $$\begin{aligned}
\label{pb}
\{\pi_0, \Lambda\}&=& -\delta(x-y),\nonumber\\
\{\pi_{0i}, \Lambda_j \}&=& \delta_{ij}\delta(x-y),\nonumber\\
\{\Omega_i, \Omega_{jk} \}&=& -\frac{1}{m}\epsilon_{ijk}\delta(x-y),\nonumber\\
\{\Omega_i, \Lambda_j \}&=& \frac{1}{m}\epsilon_{ijk}\partial^k_x\delta(x-y),\nonumber\\
\{\Omega_{ij},\Lambda\}&=&\frac{1}{m}\epsilon_{ijk}\partial^k_x\delta(x-y)\end{aligned}$$ showing that the constraint structure of the topologically massive theory be fully second-class.
Now, instead of reducing the constraints $\Omega_i$, $\Omega_{ij}$ strongly by making use of Faddeev-Jackiw scheme [@fj] and considering only gauge degrees of freedom as done in the previous work [@harikumar], we will keep the full set of the constraints (\[pri-con\]) and (\[sec-con\]). In order for studying the whole constraints efficiently, we further need to modify the constraints (\[sec-con\]) as $$\begin{aligned}
\label{new-con}
\Lambda^\prime&\equiv&\partial^i\Omega_i+\Lambda\nonumber\\
&=&\partial^i\pi_i+A^0+\frac{1}{4m}\epsilon^{ijk}\partial_iB_{jk}\approx
0,\nonumber\\
\Lambda^\prime_i&\equiv&\partial^j\Omega_{ij}-\Lambda_i\nonumber\\
&=&\partial^j\pi_{ij}+B_{0i}-\frac{1}{2m}\epsilon_{ijk}
\partial^jA^k\approx 0,\end{aligned}$$ which are equivalent to the original ones, $\Lambda$, $\Lambda_i$ on the constraint surface. Then, the set of these new constraints makes the constraint algebra (\[pb\]) much simpler and concise as $$\begin{aligned}
\label{con-algebra}
\{\pi_0, \Lambda'\}&=& -\delta(x-y),\nonumber\\
\{\pi_{0i}, \Lambda'_j \}&=& \delta_{ij}\delta(x-y),\nonumber\\
\{\Omega_i, \Omega_{jk} \}&=&
-\frac{1}{m}\epsilon_{ijk}\delta(x-y),\end{aligned}$$ while the others identically vanish[^4].
As results, we have obtained the fully second-class constraints (\[pri-con\]) and (\[new-con\]) for the first order Lagrangian of the topologically massive theory. Notice that this new algebra contains no derivatives unlikely in Eq. (\[pb\]) as well as vanishing brackets between the constraints $\Omega_i$, $\Omega_{ij}$ and $\Lambda$, $\Lambda_i$. In fact, due to the absence of the derivatives in the new Poisson brackets, one can easily convert the second-class constraints (\[pri-con\]) and (\[new-con\]) into corresponding first-class ones making use of the BFT embedding technique, which we will explicitly study in the next section. If we use the constraint algebra as it stands in Eq. (\[pb\]) containing the derivatives, BFT embedded constraints and fields may have non-local expressions which would make the quantization intractable.
On the other hand, making use of the definition of the Dirac brackets as $$\begin{aligned}
\label{def-db} \{A(x), B(y) \}_{DB} &=&\{A(x),B(y)\}_{PB} \nonumber\\
&-&\int dwdz \{A(x), \phi_\alpha (w)\}C^{\alpha\beta}(w,z)
\{\phi_\beta(z), B(y)\}, \nonumber\end{aligned}$$ where the matrix $C^{\alpha\beta}$ is an inverse of $\{\phi_\alpha(x),\phi_\beta(y)\}=C_{\alpha\beta}(x,y)$ along with the constraints denoted by $\phi_\alpha=(\pi_0, \pi_{0i},\Lambda,
\Lambda_i, \Omega_i, \Omega_{ij})$, we have obtained the following non-vanishing Dirac Brackets $$\begin{aligned}
\label{dirac-bra} \{A^0(x), A^i(y)\}_{D} &=&
\partial^i_x \delta(x-y),
\nonumber\\
\{A^0(x), \pi_{ij}(y)\}_{D} &=& -\frac{1}{2m}\epsilon_{ijk}\partial^k_x
\delta(x-y),
\nonumber\\
\{A^i(x), \pi_j(y)\}_{D} &=& \frac{1}{2}\delta^i_j \delta(x-y),
\nonumber\\
\{A^i(x), B^{jk}(y)\}_{D} &=& -m\epsilon^{ijk}\delta(x-y),
\nonumber\\
\{\pi_i(x), B_{0j}(y)\}_{D} &=& \frac{1}{2m}\epsilon_{ijk}\partial^k_x
\delta(x-y),
\nonumber\\
\{\pi_i(x), \pi_{jk}(y)\}_{D} &=& \frac{1}{4m}\epsilon_{ijk} \delta(x-y),
\nonumber\\
\{B^{0i}(x), B^{jk}(y)\}_{D} &=& (\delta^{ij}\partial^k_x - \delta^{ik}
\partial^j_x) \delta(x-y),
\nonumber\\
\{B^{ij}(x), \pi_{kl}(y)\}_{D} &=& \frac{1}{2}(\delta^i_k\delta^j_l-\delta^j_k\delta^i_l)
\delta(x-y)\end{aligned}$$ in order to compare with the results obtained from the BFT embedding which automatically leads to the Dirac brackets at the level of Poisson brackets in the extended phase space.
It seems appropriate to comment on the constraints $\Omega_i$, and $\Omega_{ij}$. These constraints come from the topological term in the Lagrangian (\[Lag\]). In the scheme of Faddeev-Jackiw quantization [@fj] which deals with only the dynamical degrees of freedom, these could be eliminated from the start while modifying other brackets of the fields. Harikumar and Sivakumar [@harikumar] has worked in this direction. However, in the present paper, we will keep these constraints with the others as it stands and embed the whole constraints into much larger phase space than their phase space, which we mean complete BFT embedding. This will give a new type of WZ action with novel symmetry as well as the Stückelberg action with the usual gauge symmetry.
Complete BFT Hamiltonian Embedding
==================================
Since we know how to quantize first-class constraint system very well while second-class system may have serious ordering and/or non-local problems, it is preferred to deal with first-class constraint system. The BFT embedding prescription makes in a systematic way second-class constraint Hamiltonian system into corresponding first-class one. In order for that purpose, we introduce auxiliary fields having involutive relations in which not only modified new constraints in the enlarged space are strongly vanishing with each other but also they have vanishing Poisson brackets, not the Dirac brackets, with physical quantities such as Hamiltonian and fields themselves. Here, ‘physical’ means gauge invariant since resulting modified new quantities would be first-class by construction. Practically, with the aid of auxiliary fields $\Phi^\alpha$, one for each constraint, satisfying with $$\{\Phi^\alpha(x), \Phi^\beta(y)\}= \omega^{\alpha\beta}(x,y),$$ while vanishing with the original fields, we construct the involutive relations as $$\begin{aligned}
\label{inv-const}
\{\tilde{\varphi}_\alpha(x), \tilde\varphi_\beta(y) \} &=& 0, \\
\label{inv-quant} \{\tilde{\varphi}_\alpha(x), \tilde{\cal F}(y)
\} &=& 0,\end{aligned}$$ where the new constraints $\tilde{\varphi}_\alpha$, and physical quantities $\tilde{\cal F}$ are given by $\tilde{\varphi}_\alpha\sim\phi_\alpha
+\sum_{n=1}(\Phi^\alpha)^n$, and $\tilde{\cal F}\sim {\cal F}
+\sum_{n=1}(\Phi^\alpha)^n$, respectively, with appropriate coefficients in front of $(\Phi^\alpha)^n$. In case we set the auxiliary fields $\Phi^\alpha$ to be zero, those quantities are reduced to the original ones, [*i.e.*]{}, $\tilde{\varphi}_\alpha\mid_{\Phi^\alpha=0}=\phi_\alpha$ (or, $\tilde{\cal F}\mid_{\Phi^\alpha=0}={\cal F}$).
Now, let us explicitly solve the involutive relations, Eqs. (\[inv-const\]) and (\[inv-quant\]), by making use of the following Ansatz of $$\label{ansatz} \varphi^{(1)}_\alpha(x)=\int dy~
X_{\alpha\beta}(x,y)\Phi^\beta(y).$$ Inserting the Ansatz to the involutive relations between the constraints (\[inv-const\]), we obtain in the zeroth order of the auxiliary fields $\Phi^\alpha$ as $$\begin{aligned}
\label{sol-x} 0 &=&\label{reln} \{\varphi_\alpha,
\varphi_\beta\}_{\cal O}
+\{\varphi^{(1)}_\alpha, \varphi^{(1)}_\beta\}_{\Phi} \nonumber\\
&=& C_{\alpha\beta}(x,y)+\int dwdz~
X_{\alpha\gamma}(x,w)\omega^{\gamma\delta}(w,z)X_{\delta\beta}(z,y),\end{aligned}$$ and in the first order of $\Phi^\alpha$ as $$\label{first-order} \{\varphi_\alpha, \varphi^{(1)}_\beta\}_{\cal
O} +\{\varphi^{(1)}_\alpha, \varphi_\beta\}_{\cal O}
+\{\varphi^{(1)}_\alpha, \varphi^{(2)}_\beta\}_{\Phi}
+\{\varphi^{(2)}_\alpha, \varphi^{(1)}_\beta\}_{\Phi}=0,$$ and so on. Here, the subscript $\cal O$(or, $\Phi$) denotes the Poisson bracket with respect to the original variables $(q,p)$(or, the auxiliary fields, $\Phi$). Then, by knowing the first order correction $\varphi^{(1)}_\alpha$, we can obtain the next order $\varphi^{(2)}_\alpha$ from Eq. (\[first-order\]), and these procedure continue iteratively until we determine involutive new constraints completely.
Through the similar steps as above, we can also get the first order correction for physical quantity as $$\label{sol-phy} {\cal F}^{(1)}(x)=-\int dy dz dw ~\Phi^\alpha(x)
\omega_{\alpha\beta}(x,y)X^{\beta\gamma}(y,z)\{\varphi_\gamma(z),
A(w)\},$$ in the zeroth order of the auxiliary fields $\Phi^\alpha$. This correction has been obtained explicitly from the ansatz (\[ansatz\]) and the involutive relation (\[inv-quant\]). Also, higher order corrections ${\cal F}^{(n)}$ $(n\ge 2)$ can be obtained iteratively from Eq. (\[inv-quant\]) to find completely involutive quantities out along with the modified new constraints.
It is appropriate to comment on the choice of $\omega^{\alpha\beta}$ and $X_{\alpha\beta}$ which may be the functions of the original variables. Although there are no known criteria to best choice $\omega^{\alpha\beta}$ and $X_{\alpha\beta}$, but final forms of the modified constraints and the physical quantities depend on what the matrix elements of $\omega^{\alpha\beta}$ and $X_{\alpha\beta}$ are. Nevertheless, among different expressions of constraints and physical quantities obtained from choosing any set of $\omega^{\alpha\beta}$ and $X_{\alpha\beta}$, they are related to each other with canonical transformations.
Now return to our model. By following the above BFT prescription, we introduce auxiliary fields paired as $(\theta, \pi_\theta)$, $(Q^i, P_i)$, and $(\Phi^i, \Phi^{jk})$: $$\begin{aligned}
\Phi^\alpha=(\theta, Q^i, \pi_\theta, P^i, \Phi^i, \Phi^{ij}),
\nonumber\end{aligned}$$ which correspond to the constraints $\phi_\alpha=(\pi_0, \pi_{0i},
\Lambda, \Lambda_i, \Omega_i, \Omega_{ij})$, respectively. Without any loss of generality, let us for simplicity make a proper choice for the auxiliary fields canonically conjugated as follows $$\label{omega}
\{\Phi^\alpha(x), \Phi^\beta(y) \}=\omega^{\alpha\beta}(x,y) =
\left(
\begin{array}{cccccc}
0 & 0 & 1 & 0 & 0 & 0\\
0 & 0 & 0 & \delta^{ij} & 0 & 0\\
-1 & 0 & 0 & 0 & 0 & 0 \\
0 & - \delta^{ij} & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 & 0 & \epsilon^{ijk} \\
0 & 0 & 0 & 0 & -\epsilon^{ijk}& 0
\end{array}\right)\delta(x-y).$$ Here, note that the auxiliary fields $\Phi^i$, $\Phi^{ij}$ in the last two columns is explicitly given by $$\begin{aligned}
\label{sym-ext} \{\Phi^i(x), \Phi^{jk}(y)\}=
\epsilon^{ijk}\delta(x-y),\end{aligned}$$ which choice makes it possible to embed the constraints originated from the topological term. With the above choice of $\omega^{\alpha\beta}$, we solve the Eq. (\[sol-x\]) and find a simple solution as $$\label{x} X_{\alpha\beta}(x,y)=\left(
\begin{array}{cccccc}
1 & 0 & 0 & 0 & 0 & 0 \\
0 & \delta^{ij} & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & \delta^{ij}& 0 & 0 \\
0 & 0 & 0 & 0 & I_{3\times 3} & 0 \\
0 & 0 & 0 & 0 & 0 & \frac{1}{m}I_{3\times 3}
\end{array} \right)\delta(x-y),$$ where $I_{3\times 3}$ denotes a $3\times 3$ identity matrix.
As results, we have obtained the strongly involutive new constraints from Eq. (\[ansatz\]) as $$\begin{aligned}
\label{involutive-constraints} \widetilde{\pi}_0 &=& \pi_0
+\theta,~~~
\widetilde{\pi}_{0i} = \pi_{0i} + Q_i, \nonumber\\
\widetilde{\Lambda} &=& \Lambda + \pi_\theta, ~~~
\widetilde{\Lambda}_i = \Lambda_i + P_i, \nonumber\\
\widetilde{\Omega}_i &=& \Omega_i + \Phi_i, ~~~
\widetilde{\Omega}_{ij} = \Omega_{ij}+ \frac{1}{m}\Phi_{ij}\end{aligned}$$ with no higher order contributions of the auxiliary fields due to our proper choice of $\omega^{\alpha\beta}$ and $X_{\alpha\beta}$ while satisfying the involutive relation ([\[inv-const\]]{}). Moreover, from the inverse matrix $\omega_{\alpha\beta}$ and $X^{\alpha\beta}$ and the solution of the physical fields (\[sol-phy\]), we have also gotten the strongly involutive physical fields [@phy-field] as follows $$\begin{aligned}
\label{physfield}
\widetilde{A}^0 &=& A^0 + \pi_\theta, \nonumber\\
\widetilde{A}^i &=& A^i + \partial^i\theta -
\frac{1}{2}\epsilon^{ijk}\Phi_{jk}, \nonumber \\
\widetilde{\pi}_0 &=& \pi_0 + \theta, \nonumber\\
\widetilde{\pi}_i &=& \pi_i - \frac{1}{2m}\epsilon_{ijk}\partial^j
Q^k + \frac{1}{2}\Phi_i, \nonumber \\
\widetilde{B}^{0i} &=& B^{0i}+P^i, \nonumber\\
\widetilde{B}^{ij} &=& B^{ij}-(\partial^iQ^j-\partial^jQ^i)+
m\epsilon^{ijk}\Phi_k, \nonumber\\
\widetilde{\pi}_{0i} &=& \pi_{0i}+Q_i, \nonumber\\
\widetilde{\pi}_{ij} &=& \pi_{ij}-\frac{1}{2m}\epsilon_{ijk}
\partial^k \theta + \frac{1}{2m}\Phi_{ij}\end{aligned}$$ satisfying the involutive relation (\[inv-quant\]). Note that all the physical fields are terminated in the first order of the auxiliary fields of the BFT embedding sequence and there are no higher order contributions. One can easily checked that the Poisson brackets of Eq. (\[physfield\]) in the extended phase space give exactly the same Dirac brackets in the original phase space as given in Eq. (\[dirac-bra\]) as they should be.
On the other hand, canonical Hamiltonian in the extended phase space can be obtained either by solving the strongly involutive relation (\[inv-quant\]) in replacement of $\tilde{\cal F}$ with a Hamiltonian function or by using the physical fields (\[physfield\]) in the canonical Hamiltonian (\[can-H\]) written by the tilde fields. Since these methods of obtaining canonical Hamiltonian are known to be equivalent to each other on the constraint surface, which are weakly equivalent, we will follow the latter approach. Then, the canonical Hamiltonian (\[can-H\]) is simply written in the extended phase space as follows $$\begin{aligned}
\label{ext-cH} \widetilde{\cal H}_c &=&
\frac{1}{4}(B_{ij}-Q_{ij})^2+\frac{1}{2}m\epsilon_{ijk}(B^{ij}-Q^{ij})\Phi^k
-\frac{1}{2}m^2\Phi_i\Phi^i \nonumber\\
&-&\frac{1}{2}(A_i+\partial_i\theta)^2+\frac{1}{2}\epsilon_{ijk}
(A^i+\partial^i\theta)\Phi^{jk}+\frac{1}{4}\Phi_{ij}\Phi^{ij}
\nonumber\\
&+&\frac{1}{2}(B_{0i}+P_i)^2-\frac{1}{m}\epsilon_{ijk}(B^{0i}+P^i)
\partial^jA^k-\frac{1}{m}(B_{0i}+P_i)\partial_j\Phi^{ij}
\nonumber\\
&-&\frac{1}{2}(A_0+\pi_\theta)^2-\frac{1}{2m}(A_0+\pi_\theta)\epsilon_{ijk}
\partial^iB^{jk}+(A_0+\pi_\theta)\partial_i\Phi^i,\end{aligned}$$ where $Q_{ij}$ denote $Q_{ij}=\partial_i Q_j - \partial_j Q_i$. By construction, this canonical Hamiltonian satisfies the strongly involutive relations with the modified constraints $\tilde{\varphi}_\alpha$, [*i.e.*]{}, $\{\tilde{\varphi}_\alpha,
\widetilde {\cal H}_c \}=0$. This ends the BFT embedding for the topologically massive theory with the one- and two-form fields.
Corresponding Lagrangian with new type of WZ term
=================================================
In this section, we find the corresponding Lagrangian of the extended canonical Hamiltonian by making use of the phase space path integral. For this purpose we modify further the canonical Hamiltonian (\[ext-cH\]) to an equivalent one on the constraint surface which generates the Gauss’ constraints naturally. The equivalent Hamiltonian given by $$\label{ext-mH} \widetilde{\cal H}'_c = \widetilde{\cal H}_c +
\pi_\theta \widetilde{\Lambda}+P_i\widetilde{\Lambda}_i$$ yields the Gauss’ constraints as $$\begin{aligned}
\frac{d}{dt}\widetilde{\pi}_0 &=& \{\widetilde{\pi}_0,
\widetilde{\cal H}'_c \} = \widetilde{\Lambda}, \nonumber\\
\frac{d}{dt}\widetilde{\pi}_{0i} &=& \{\widetilde{\pi}_{0i},
\widetilde{\cal H}'_c \} = \widetilde{\Lambda}^i.\end{aligned}$$
Extended Gauge Symmetries
-------------------------
Now, in order for obtaining the corresponding Lagrangian, we write the generating functional of the topologically massive theory in the fully extended phase space as $$\begin{aligned}
\label{path-int} {\cal Z}&=&\int {\cal D}A^\mu {\cal D}\pi_\mu
{\cal D}B^{\mu\nu} {\cal D}\pi_{\mu\nu} {\cal D}\theta {\cal
D}\pi_\theta {\cal D}Q^i {\cal D} P_i {\cal D}\Phi^i {\cal
D}\Phi^{ij}\nonumber\\
&\times&\delta(\widetilde{\varphi}_\alpha)\delta(\Gamma_\beta)~
{\rm det}\mid\{\widetilde{\varphi}_\alpha, \Gamma_\beta \}\mid
e^{iS},\end{aligned}$$ where $$\begin{aligned}
S&=&\int d^4x ~\left[\pi_\mu\dot{A}^\mu+\pi_{0i}\dot{B}^{0i}
+\frac{1}{2}\pi_{ij}\dot{B}^{ij}+\pi_\theta\dot{\theta}+P_i\dot{Q}^i
+\frac{1}{2}\epsilon_{ijk}\Phi^i\dot{\Phi}^{jk}-\widetilde{\cal
H}'_c \right],\nonumber\\\end{aligned}$$ and $\Gamma_\alpha$ are appropriate gauge fixing functions which have non-vanishing Poisson brackets with the modified first-class constraints $\tilde{\varphi}_\alpha$.
First, we can easily integrate the momenta variable, $\pi_0$, $\pi_i$, $\pi_{0i}$, and $\pi_{ij}$ out along with the delta functional, $\delta(\widetilde{\pi}_0)$, $\delta(\widetilde{\Omega}_i)$, $\delta(\widetilde{\pi}_{0i})$, and $\delta(\widetilde{\Omega}_{ij})$, respectively. Then, making use of the Fourier transformations of the constraints $\tilde{\Lambda}$, $\tilde{\Lambda}_i$ as $\delta(\widetilde{\Lambda})=\int {\cal D}\xi \exp(-i\int
d^4x~\xi\widetilde{\Lambda})$, $\delta(\widetilde{\Lambda}_i)=\int
{\cal D}\chi^i \exp(-i\int d^4x~\chi^i\widetilde{\Lambda}_i)$, and transforming $A^0 \rightarrow A^0+\xi$, $B^{0i} \rightarrow
B^{0i}-\chi^i$, and after the Gaussian integrations over $\pi_\theta$ and $P_i$ variables, we finally obtain the generating functional as $$\label{generating-ftn}
{\cal Z}=\int {\cal D}A^\mu {\cal D}B^{\mu\nu} {\cal D}\theta
{\cal D}Q^i {\cal D}\Phi^i {\cal D}\Phi^{ij}{\cal D}Q^0
\delta(Q^0)\delta(\Gamma_\beta)
det\mid\{\widetilde{\varphi}_\alpha, \Gamma_\beta \}\mid e^{iS_T},$$ where $$\begin{aligned}
\label{tot-act}
S_T & = & \int d^4x~\left( {\cal L}_{St} + {\cal
L}_{NWZ}\right) \\
\label{stuckelberg} {\cal L}_{St} &=&
\frac{1}{2}(A_\mu+\partial_\mu\theta)^2 -\frac{1}{4}
(B_{\mu\nu}-Q_{\mu\nu})^2 +
\frac{1}{2m}\epsilon_{\mu\nu\rho\sigma}
(B^{\mu\nu}-Q^{\mu\nu})\partial^\rho(A^\sigma+\partial^\sigma\theta)
\nonumber \\ \\
\label{nwz}{\cal L}_{NWZ} &=& \left[(\partial_i A_0-\partial_0
A_i)- \frac{m}{2}\epsilon_{ijk} (B^{jk}-Q^{jk})+
\frac{m^2}{2}\Phi_i\right]\Phi^i \nonumber\\
&-&\left[\frac{1}{2}\epsilon_{ijk}(A^i+\partial^i\theta)+\frac{1}{m}\partial_k
B_{0j}+\frac{1}{2m}\partial_0
B_{jk}+\frac{1}{4}\Phi_{ij}\right]\Phi^{jk} \nonumber\\
&+&\frac{1}{2}\epsilon_{ijk}\Phi^i\dot{\Phi}^{jk}.\end{aligned}$$
Next, let us construct the gauge transformation generator $G$, following Dirac’s conjecture [@dirac], for the embedded theory in the standard way, $$G=\int d^4x \sum_\alpha \epsilon^\alpha
\widetilde{\varphi}_\alpha,$$ where $\widetilde{\varphi}_\alpha=(\widetilde{\pi}_0,
\widetilde{\Omega}_i, \widetilde{\Lambda}, \widetilde{\pi}_{0i},
\widetilde{\Omega}_{ij}, \widetilde{\Lambda}_i)$ are the first-class constraints in equation (\[involutive-constraints\]) in order, and $\epsilon^\alpha = (\epsilon^{0}_{A},
\epsilon^{i}_{A}, \epsilon_{A}, \epsilon^{0i}_{B},
\epsilon^{ij}_{B}, \epsilon^{i}_{B})$ are, in general, functions of phase space variables. The infinitesimal gauge transformation for a function $F$ of phase space variables is then given by the relation of $\delta F =\{F, G\}_D$, and leads to $$\begin{aligned}
\label{ext-gt} \delta A^0&=&\epsilon^0_A,~~~~~~~~~~~~~~ \delta
B^{0i}=\epsilon^{0i}_B,\nonumber\\
\delta A^i&=&\epsilon^i_A-\partial^i\epsilon_A,~~~~~ \delta
B^{ij}=\partial^i\epsilon^j_B-\partial^j\epsilon^i_B+
\epsilon^{ij}_B-\epsilon^{ji}_B, \nonumber\\
\delta\theta&=&\epsilon_A,~~~~~~~~~~~~~~ \delta Q^i = \epsilon^i_B, \nonumber\\
\delta\Phi^i&=&\frac{1}{m}\epsilon^{ijk}\epsilon^B_{jk},~~~~~~
\delta\Phi^{ij}=-\epsilon^{ijk}\epsilon^A_k.\end{aligned}$$
The above gauge transformation involving the gauge parameters is a symmetry of the Hamiltonian, but not of the Lagrangian. The generator $G$ of the most general local symmetry transformation of a Lagrangian must satisfy the master equation [@rothe99] $$\frac{\partial G}{\partial t}+\{G, H_T\}=0,$$ which, together with (\[ext-gt\]), implies the following restrictions on the gauge parameters and on the Lagrangian multipliers in the primary Hamiltonian: $$\begin{aligned}
\label{mul-rel}
\delta v^\beta &=&
\frac{d\epsilon^\beta}{dt}-\epsilon^P(V^\beta_P+v^\alpha
C^\beta_{\alpha_P}), \nonumber\\
0&=&\frac{d\epsilon^b}{dt}-\epsilon^P(V^b_P+v^\alpha
C^b_{\alpha_P}).\end{aligned}$$ Here the superscripts $\alpha$, $\beta$, ($a$, $b$) denote the primary (secondary) constraints, and $V^P_Q$, $C^P_{QR}$ are the structure functions of the constrained Hamiltonian dynamics defined by $\{H_c, \widetilde{\varphi}_P\}_D =
V^Q_P\widetilde{\varphi}_Q$, $\{\widetilde{\varphi}_P,
\widetilde{\varphi}_Q\}=C^R_{PQ}\widetilde{\varphi}_R$, respectively. From (\[mul-rel\]) we obtain $\epsilon^0_A=-d\epsilon_A/dt$ and $\epsilon^{0i}_B=d\epsilon^i_B/dt$. Thus, the gauge transformations of $A^0$ and $B^{0i}$ in Eq. (\[ext-gt\]) reduce to $$\delta A^0 = -\frac{d\epsilon_A}{dt}, ~~ \delta
B^{0i}=\frac{d\epsilon^i_B}{dt}.$$ Redefining the gauge parameters [@henneaux85] as $\bar{\epsilon}^i_B=\epsilon^i_B-\partial^i\int dt \epsilon^0_B$ along with the variation $\delta Q^0=\epsilon^0_B$, the final gauge transformations[^5] are nicely summarized as $$\begin{aligned}
\label{fin-ext-gt} \delta A^\mu &=&
-\partial^\mu\epsilon_A+\delta^\mu_j \epsilon^j_A,~~ \delta
B^{\mu\nu}=\partial^\mu\epsilon^\nu_B -
\partial^\nu\epsilon^\mu_B+(\epsilon^{kl}_B-\epsilon^{lk}_B)
\delta^\mu_k\delta^\nu_l, \nonumber\\
\delta\theta &=& \epsilon_A, ~~~~~~~~~~~~~~~~~\delta Q^\mu =
\epsilon^\mu_B,
\nonumber\\
\delta\Phi^i&=&\frac{1}{m}\epsilon^{ijk}\epsilon^B_{jk},~~~~~~~~~
\delta\Phi^{ij}=-\epsilon^{ijk}\epsilon^A_k.\end{aligned}$$ Note here that the gauge parameter $\epsilon_A$ is related with the topologically massive one-form fields $A^\mu$ and their WZ field $\theta$ while the gauge parameters $\epsilon^\mu_B$ are connected with the two-form fields $B^{\mu\nu}$ and their WZ fields $Q^\mu$ resulting in the usual symmetry of the Stückelberg Lagrangian. On the other hand, the gauge parameters $\epsilon^i_A$, $\epsilon^{ij}_B$ are obtained from the embedded topological constraints $\tilde{\Omega}_i$, $\tilde{\Omega}_{ij}$ with their WZ fields $\Phi^i$, $\Phi^{ij}$ resulting in the new type of WZ Lagrangian. It is easy to check that the total action (\[tot-act\]) is exactly invariant under those gauge transformation. As results, we have explicitly obtained the usual gauge invariant Stückelberg Lagrangian ${\cal L}_{St}$ for the massive one- and two-form fields, and a new type of the WZ Lagrangian ${\cal L}_{NWZ}$ including the WZ fields, $\Phi^i$, $\Phi^{ij}$.
It is appropriate to comment that in the generating functional (\[generating-ftn\]) there exists the delta functional of a variable $Q^0$ which transforms as $\delta Q^0=\epsilon^0_B$ as shown above. We have introduced this new field to make the final Lagrangian manifestly covariant. Even without this $Q^0$ field, we can show that the resulting Lagrangian successfully reproduces all the BFT embedded constraint structure as in the section 3. However, it fails to have manifest covariance. According to the usage of the Hamiltonian formulation, the constraint structure of the Lagrangian (\[Lag\]) is called irreducible, in other words, the constraints are linearly independent. On the other hand, the constraint structure of the Stückelberg Lagrangian (\[stuckelberg\]) is reducible, [*i.e.,*]{} there is a redundant relation among the constraints. In fact, we have introduced the new variable $Q^0$ in order for keeping the manifest covariance, while giving up the irreducible property between the constraints.
On the other hand, the new type of the WZ action is related to the symplectic constraints, $\Omega_i$, $\Omega_{ij}$, of the Lagrangian (\[Lag\]), which are now converted into the first-class constraints. This seemingly non-covariant form of the Lagrangian ${\cal L}_{NWZ}$ comes from the introduction of the auxiliary fields $\Phi^i$, $\Phi^{ij}$ as given by Eq. (\[sym-ext\]) where the distinction between the fields and the momenta is useless. In other words, they look like another Dirac brackets, and in order for embedding this symplectic structure fully and getting a completely covariant action such as the symplectic structure free theory, we may introduce infinite numbers of auxiliary fields as discussed in Ref. [@kkp].
Various Gauge Fixings
---------------------
Now, let us consider various gauge fixings in the path integral (\[path-int\]). First, unitary gauge fixings: by fixing unitary gauge means that all the auxiliary fields are set to be zero, $\Phi_\alpha=0$, and it is easy task to check that we can recover the original Lagrangian (\[Lag\]) in the unitary gauge as follows $${\cal Z}=\int {\cal D}A^\mu{\cal D}B^{\mu\nu} e^{iS},$$ with the action $S$ in Eq. (\[Lag\]). Furthermore, it is easy to see that the integration over $B^{\mu\nu}$ yields the Proca action, while the integration over $A^\mu$ Kalb-Ramond action. Therefore, we have reconfirmed the well-known result that the action (\[Lag\]) is nothing but a master action leading to a dual description, [*i.e.*]{}, they have a common origin.
Next, let us consider appropriate gauge fixings showing that the topologically massive theory of first order with the one- and two-form fields is equivalent to the $B\wedge F$ theory. From the generating functional (\[path-int\]), we first eliminate the auxiliary fields $\Phi^i=0$, $\Phi^{ij}=0$ by fixing the unitary gauge in part. Then, the resulting Lagrangian is simply the Stückelberg ${\cal L}_{St}$. Next, we consider the following gauge fixings: $$\begin{aligned}
\label{gf}
\partial_iQ^i&=&0, \nonumber\\
\partial_i B^{0i} &=& 0, \nonumber\\
\chi_i &\equiv& A_i -\frac{1}{m}\epsilon_{ijk}\partial^jB^{0k}=0, \nonumber\\
\chi_{ij} &\equiv& B_{ij}-\frac{1}{m}\epsilon_{ijk}\partial^k
A^0=0.\end{aligned}$$ The momenta, $\pi_i$, $\pi_{ij}$, $\pi_0$, $\pi_{0i}$, and $Q^0$ fields are integrated out along with the constraints, $\delta(\widetilde{\Omega}_i)$, $\delta(\widetilde{\Omega}_{ij})$, $\delta(\widetilde{\pi}_0)$, $\delta(\widetilde{\pi}_{0i})$, and $\delta(Q^0)$, respectively. After the momenta $\pi_\theta$, $P_i$, integration along with the constraints $\widetilde{\Lambda}$, $\widetilde{\Lambda}_i$, we obtain the following intermediate generating functional $${\cal Z}=\int {\cal D}A^\mu{\cal D}B^{\mu\nu}{\cal D}\theta {\cal
D}Q^i \delta(\chi_i)\delta(\chi_{ij})~ {\rm
det}\mid\{\widetilde{\varphi}_\alpha, \Gamma_\beta \}\mid e^{iS},$$ where the action $$S=\int d^4x~\left[\frac{1}{2m}\epsilon_{ijk}\dot{A}^iB^{jk} -
\frac{1}{2m}\epsilon_{ijk}\partial^iB^{jk}\dot{\theta} +
\frac{1}{m}\epsilon_{ijk}\dot{Q}^i\partial^jA^k - \widetilde{H}'_c
\right],$$ and $$\widetilde{H}'_c = \frac{1}{4}(B_{ij}-Q_{ij})^2-\frac{1}{2}(A_i+
\partial_i\theta)^2+\frac{1}{4m^2}F_{ij}F^{ij}-\frac{1}{2\cdot3!m^2}
H_{ijk}H^{ijk}.$$ Here, we have used identities $$\begin{aligned}
&&\epsilon_{ijk}\epsilon^{lmn}\partial^iB^{jk}\partial_lB_{mn}=
-\frac{2}{3}H_{ijk}H^{ijk}, \nonumber\\
&&\epsilon_{ijk}\epsilon^{ilm}
\partial^jA^k\partial_lA_m= \frac{1}{2}F_{ij}F^{ij}, \nonumber\end{aligned}$$ with $\partial^iA_i=0$ which comes from the gauge fixing condition $\partial^i\chi_i=0$. We have also denoted $H_{ijk}=\partial_i
B_{jk}+\partial_j B_{ki}+\partial_k B_{ij}$ and $F_{ij}=\partial_i
A_j -\partial_j A_i$. The remaining $\theta$ and $Q^i$ integrations yields the following action $$\begin{aligned}
\label{int-int-Lag} S&=&\int
d^4x~\left[\frac{1}{2m}\epsilon_{ijk}\dot{A}^iB^{jk}-
\frac{1}{4}B_{ij}B^{ij}+\frac{1}{2}A_iA^i-\frac{1}{4m^2}F_{ij}F^{ij}
+\frac{1}{2\cdot3!m^2}H_{ijk}H^{ijk}\right. \nonumber\\
&&~~~~~~~\left.
-\frac{1}{8m^2}\epsilon_{ijk}\epsilon^{ilm}\partial^0B^{jk}\partial_0B_{lm}
-\frac{1}{2m^2}\partial_0A_i\partial^0A^i\right].\end{aligned}$$ Now, making use of the gauge conditions (\[gf\]) and identities $$\begin{aligned}
&&-\frac{1}{8m^2}\epsilon_{ijk}\epsilon^{ilm}\partial^0B^{jk}\partial_0B_{lm}
= \frac{1}{4m^2}H_{0ij}H^{0ij}-\frac{1}{2m^2}B_{0i}\nabla^2B^{0i},
\nonumber \\
&&-\frac{1}{2m^2}\partial_0A_i\partial^0A^i=-\frac{1}{2m^2}F_{0i}F^{0i}
+ \frac{1}{2m^2}A_0\nabla^2A^0, \nonumber\end{aligned}$$ and the equations also obtained from the gauge conditions (\[gf\]) $$\begin{aligned}
&&\nabla^2A^0=\frac{m}{2}\epsilon_{ijk}\partial^iB^{jk}, \nonumber\\
&& \nabla^2B^{0i}=-m\epsilon^{ijk}\partial^jA^k, \nonumber\end{aligned}$$ we can manipulate the last two terms in the action (\[int-int-Lag\]) with the others to obtain the well-known $B\wedge F$ theory of $$\begin{aligned}
{\cal Z} &=& \int {\cal D}A^\mu{\cal D}B^{\mu\nu}{\rm
det}\mid\{\widetilde{\varphi}_\alpha, \Gamma_\beta \}\mid
e^{iS}\nonumber\\
S &=& \int
d^4x~\left[-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2\cdot
3!}H_{\mu\nu\rho}H^{\mu\nu\rho}+\frac{m}{4}\epsilon_{\mu\nu\rho\sigma}
B^{\mu\nu}F^{\rho\sigma} \right],\nonumber\\\end{aligned}$$ where the fields $A^\mu$, $B^{\mu\nu}$ are scaled as $mA^\mu$, $mB^{\mu\nu}$, respectively.
As a result, we have explicitly shown that the equivalence of the topologically massive theory of first order with the one- and two-form fields and the $B\wedge F$ theory on the level of path integral.
Revisit the Gauging of Topologically Massive Theory
===================================================
In this section, we will explicitly show the equivalence of gauged massive theory of first order and the Stückelberg Lagrangian, ${\cal L}_{St}$, in which the Lagrangian is obtained from the BFT embedding.
By gauging the fields $A^\mu$, $B^{\mu\nu}$ means the following transformations $$A^\mu \rightarrow A^\mu+\partial^\mu\theta,
~~~B^{\mu\nu}\rightarrow B^{\mu\nu}-Q^{\mu\nu},$$ where $Q^{\mu\nu}=\partial^\mu Q^\nu-\partial^\nu Q^\mu$. Thus, the gauged massive theory of first order Lagrangian (\[Lag\]) is described by $$\label{gauged-Lg} {\cal L}_G =
-\frac{1}{4}(B_{\mu\nu}-Q_{\mu\nu})^2+\frac{1}{2}
(A_\mu+\partial_\mu\theta)^2
+\frac{1}{2m}\epsilon_{\mu\nu\rho\sigma}(B^{\mu\nu}-Q^{\mu\nu})
\partial^\rho(A^\sigma+\partial^\sigma\theta),$$ which is invariant under the gauge transformations of $$\delta A^\mu = \partial^\mu\epsilon,~~ \delta\theta=-\epsilon,~~
\delta
B^{\mu\nu}=\partial^\mu\epsilon^\nu-\partial^\nu\epsilon^\mu,
~~\delta Q^\mu=\epsilon^\mu.$$ Here, we have redefined the gauge parameters as $\epsilon_A=
-\epsilon$, $\epsilon^\mu_B=\epsilon^\mu$ in which expression is usually seen in the literature and take $\epsilon^i_A
=\epsilon^{ij}_B=0$ in the extended gauge symmetries (\[fin-ext-gt\]) in order to focus mainly on the gauging technique. Note that as you already know through the note in the previous sections by taking $\epsilon^\mu=\partial^\mu\lambda$, $\delta B^{\mu\nu}$ have vanished clearly indicating that the constraints are not all independent, [*i.e.,*]{} reducible.
In order for comparing this action with the Stückelberg Lagrangian ${\cal L}_{St}$ in view of constraint structure, we do partial integration of the terms $G_0\dot\theta$, $H_{0i}\dot{Q}^i$ to $\-\dot{G}_0\theta$, $-\dot{H}_{0i}Q^i$, respectively, in the gauged Lagrangian (\[gauged-Lg\]). Then, the canonical momenta are obtained as $$\begin{aligned}
\pi_0 &=& -\theta,
~~~\pi_i=\frac{1}{4m}\epsilon_{ijk}B^{jk},\nonumber\\
\pi_{0i}&=&-Q^i,~~~\pi_{ij}=-\frac{1}{2m}\epsilon_{ijk}A^k,\nonumber\\
P_i&=&Q_{i0},~~~P_0=0,~~~~~\pi_\theta=\dot\theta,\end{aligned}$$ from which we have the primary constraints as $$\begin{aligned}
&& \Sigma_0 \equiv \pi_0+\theta \approx 0, \nonumber\\
&& \Sigma_i \equiv \pi_{0i}+Q^i\approx 0,\nonumber\\
&& \Omega_i \equiv \pi_i-\frac{1}{4m}\epsilon_{ijk}B^{jk}\approx 0,\nonumber\\
&& \Omega_{ij} \equiv \pi_{ij}+\frac{1}{2m}\epsilon_{ijk}A^k\approx 0,\nonumber\\
&& P_0 \approx 0.\end{aligned}$$ We also get the canonical Hamiltonian as $$\begin{aligned}
{\cal H}_c &=&
\frac{1}{4}(B_{ij}-Q_{ij})^2-\frac{1}{2}(A_i+\partial_i\theta)^2
+\frac{1}{2}B_{0i}B^{0i}-\frac{1}{2}P_iP^i-\frac{1}{2}A_0A^0
\nonumber\\
&+& \frac{1}{2}\pi^2_\theta+ (P_i+B_{0i})\partial^iQ^0-\frac{1}{m}
\epsilon_{ijk}B^{0i}\partial^jA^k-\frac{1}{2m}A^0
\epsilon_{ijk}\partial^iB^{jk}.\end{aligned}$$ From the time stability conditions of the primary constraints with the canonical Hamiltonian, there are additional secondary constraints as follows $$\begin{aligned}
\Delta &\equiv& \frac{d}{dt}P_0 = \partial^i(P_i+B_{0i})\approx 0,
\nonumber\\
\Lambda &\equiv& \frac{d}{dt}\Sigma_0 = A_0 + \frac{1}{2m}
\epsilon_{ijk}\partial^iB^{jk}+\pi_\theta \approx 0, \nonumber\\
\Lambda_i &\equiv& \frac{d}{dt}\Sigma_i = -B_{0i}+
\frac{1}{m}\epsilon_{ijk}\partial^jA^k-P_i \approx 0.\end{aligned}$$ It seems appropriate to comment on the linear independence of the constraints. The combination of the above constraints $\Delta$, $\Lambda_i$ vanishes [*i.e.,*]{} $\Delta+\partial^i \Lambda_i=0$, which indicates again that the whole constraints are not linearly independent, [*i.e.,*]{} reducible.
Finally, making use of the above constraints, we can obtain the following Hamiltonian $$\begin{aligned}
\label{ext-cH-revisit} {\cal H}_c &=& \frac{1}{4}(B_{ij}-Q_{ij})^2
-\frac{1}{2}(A_i+\partial_i\theta)^2 +\frac{1}{2}(B_{0i}+P_i)^2
-\frac{1}{m}\epsilon_{ijk}(B^{0i}+P^i)
\partial^jA^k\nonumber\\
&-&\frac{1}{2}(A_0+\pi_\theta)^2
-\frac{1}{2m}(A_0+\pi_\theta)\epsilon_{ijk}\partial^iB^{jk}
+\pi_\theta\Omega+(P^i+\partial^iQ^0)\Omega_i \nonumber\\
&\approx& \widetilde{\cal H}'_c,\end{aligned}$$ where $\widetilde{\cal H}'_c$ is given in Eq. (\[ext-cH\]). This shows the equivalence of the gauged Lagrangian (\[gauged-Lg\]) and the Stückelberg Lagrangian (\[stuckelberg\]) on the constraint surface.
Conclusion
==========
In this paper, we have applied the complete BFT method to the massive theory with one- and two-form fields. We have newly analyzed the full set of constraint structure of the model having no derivatives in the Poisson brackets, which is much simpler than that of the previous work [@harikumar] and thus makes it possible to apply the BFT embedding further. Then, we have explicitly carried out the complete BFT embedding of the theory including the gauge symmetry breaking terms and the topological term, which was not gauge invariant. Exploiting the complete BFT embedding we have obtained the gauge invariant Lagrangian corresponding to the first class Hamiltonian, and by identifying the auxiliary fields with the Stückelberg vector fields and new type of WZ fields, we have shown simultaneously the Stückelberg Lagrangian related to the explicit gauge breaking mass term and the new type of WZ action to topological term having novel symmetry. Furthermore, by analyzing the gauged version of the theory we have also shown that the usual Stückelberg Lagrangian is exactly equivalent to the BFT embedded one on the constraint space after taking the unitary gauge $\Phi^i=0$, $\Phi^{ij}=0$.
1.0cm The work of YWK was supported by the Korea Research Foundation, Grant No. KRF-2002-075-C00007.
[99]{}
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[^1]: Electronic address: skandjh@empal.com
[^2]: Electronic address: ywkim65@netian.com
[^3]: Electronic address: yjpark@ccs.sogang.ac.kr
[^4]: From now on, we will omit the prime symbols on the constraints $\Lambda$, $\Lambda_i$
[^5]: We have omitted the bar symbol in the final rules of gauge transformations.
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'Rauer, H.'
- 'Queloz, D.'
- '[Csizmadia, Sz.]{}'
- 'Deleuil, M.'
- 'Alonso, R.'
- 'Aigrain, S.'
- 'Almenara, J.M.'
- 'Auvergne, M.'
- 'Baglin, A.'
- 'Barge, P.'
- 'Bord$\acute{e}$, P.'
- 'Bouchy, F.'
- 'Bruntt, H.'
- 'Cabrera, J.'
- 'Carone, L.'
- 'Carpano, S.'
- 'De la Reza, R.'
- 'Deeg, H.J.'
- 'Dvorak, R.'
- 'Erikson, A.'
- 'Fridlund, M.'
- 'Gandolfi, D.'
- 'Gillon, M.'
- 'Guillot, T.'
- 'Guenther, E.'
- 'Hatzes, A.'
- 'H$\acute{e}$brard, G.'
- 'Kabath, P.'
- 'Jorda, L.'
- 'Lammer, H.'
- 'L$\acute{e}$ger, A.'
- 'Llebaria, A.'
- 'Magain, P.'
- 'Mazeh, T'
- 'Moutou, C.'
- 'Ollivier, M.'
- 'P$\ddot{a}$tzold, M.'
- 'Pont, F'
- 'Rabus, M.'
- 'Renner, S.'
- 'Rouan, D.'
- 'Shporer, A.'
- 'Samuel, B.'
- 'Schneider, J.'
- 'Triaud, A.H.M.J.'
- 'Wuchterl, G.'
bibliography:
- '11902references.bib'
date: '00 / 00'
title: |
Transiting exoplanets from the CoRoT space mission\
VII. The “hot-Jupiter"-type planet CoRoT-5b
---
[^1]
Introduction
============
CoRoT started to search for the photometric signal of transiting extrasolar planets in 2007, after its successful launch in December 2006, for details on the satellite see the pre-launch book [@2007baglin; @Boisnard06] and [@auvergne2009]. The satellite monitors about 12000 stars per exoplanet field-of-view in a series of short ($\sim$30 days) and long ($\sim$150 days) observing runs. Its magnitude range is 12 $\le$ m$_v$ $\le$ 16 mag. The resulting stellar lightcurves are searched for periodic signals of transiting extrasolar planets. Radial-velocity follow-up measurements secure the nature of the transiting body and allow us to derive its mass.
The nominal lightcurve analysis for small transiting signals has to await the completion of an observing run and detailed signal analysis. The mission “alarm-mode” [@Quentin06; @surace2008], however, can be used to quickly trigger follow-up measurements during ongoing observations of a target field. The “alarm-mode" is used to increase the transmitted time-sampling for individual stellar lightcurves in the CoRoT exoplanet channel. The sampling is increased from 512 sec to 32 sec if a transit-like signal is detected during the observations. It therefore provides planetary candidates early during an observing run, which are, however, biased towards relatively large planetary candidates because of the limited data set available at this point.
CoRoT-5b is the fifth secured transiting planet detected by CoRoT. As CoRoT-1b to CoRoT-4b [@2008alonso; @2008barge; @deleuil2008; @2008moutou; @2008agrain], it was first detected by the alarm-mode. Here, we present the photometric detection of CoRoT-5b by the satellite based on pre-processed alarm-mode data, the accompanying radial-velocity observations confirming its planetary nature, and the resulting planet parameters.
Observations and data reduction {#observations}
===============================
CoRoT-5b was detected in the LRa01-field, the second long-run field of CoRoT. The field is located near the anti-center direction of the galaxy at RA(2000): 06$^h$46$^m$53$^s$ and DEC(2000): -00$^\circ$12’00” [@2006michel]. The observing sequence started on October 24, 2007 and finished after 112 days duration. CoRoT observations usually have a very high duty cycle since data gaps are mainly caused by the regular crossings of the South Atlantic Anomaly (SAA), which typically last for about 10 min. During the observations of the LRa01 field, however, two longer interruptions occurred. An intermediate interruption of about 12 hours occurred eight days after the beginning of the observing run, and a longer data gap of about 3.5 days started on January 18, 2008, after a DPU reset. Finally, a duty cycle of 93 % was achieved.
![Lightcurve of CoRoT-5 re-sampled to 512 sec time resolution. No corrections for data jumps due to “hot pixels" have been applied in this figure to show the raw data quality. []{data-label="fig1_rauer"}](11902fig1.eps){width="\linewidth" height="6cm"}
The alarm-mode was triggered after 29 days of observations. When seven transit-like signals were detected, the time sampling was switched to 32 sec. The alarm-mode data for CoRoT-5 are based on the analysis of “white light" lightcurves, without using the color information of the CoRoT prism. In total 219,711 data points were obtained, 214,938 of it in oversampling mode. The data pipeline flags data points taken during the SAA crossing or affected by other events decreasing the data quality. When taking only unflagged data into account, the number of data points reduced to 204,092 in total and 199,917 as highly sampled.
The alarm-mode data were processed with a first version of the data reduction pipeline [@auvergne2009]. The pipeline corrects for the CCD zero offsets and gain, the sky background intensity and the telescope jitter. In addition, “hot pixels" [@2008MNRAS.384.1337P] affect the lightcurves, causing sudden jumps in intensity of varying duration. The lightcurve of CoRoT-5 was, however, only moderately affected by such jumps, as can be seen in Fig. \[fig1\_rauer\], which shows the full lightcurve. The oversampled part of the data set was re-binned to display the whole lightcurve with a 512 sec time sampling. The measured intensity decreases during the observing run, as observed for all stars in the fields. Overall, CoRoT-5 only shows a minor level of variability, without clear periodicity.
CoRoT measures stellar intensities by aperture photometry using optimized masks [@Llebaria:2003] that encompass the shape of the stellar point-spread-functions (PSFs). The bi-prism introduced in the light path of the exoplanet channel [@auvergne2009] causes relatively wide PSFs of unusual shapes that vary with e.g. stellar magnitude. Contaminating eclipsing binary stars within the PSF could mimic a planetary transit-like signal. Based on the pre-launch observations of the target field included in the [*Exo-Dat*]{} data base [@deleuil2009], the contamination of the mask of CoRoT-Exo5 is estimated to 8.4 %. Refinement of this value will be performed in a more detailed future analysis using the dedicated windowing mask for this target star. We subtracted this flux level from the lightcurve before normalization to take low level contamination into account.
The overall intensity trend and smaller scale variability of the lightcurve were removed. To do this, we resampled the lightcurve to 512 seconds sampling rate first and convolved this lightcurve with a fourth order Savitzky-Golay filter (similar to the treatment for CoRoT-2b [@2008alonso]). Then median averages were calculated for 24 hour segments of the lightcurve (excluding the transit points and the data jumps), which was fitted by a spline-curve. The original lightcurve was then divided by the spline fit. The filtered lightcurve was used for normalization and further analysis. The out-of-eclipse scatter of CoRoT-5 was determined from the standard deviation of data points in the phase-folded lightcurve. It was found to be 0.0017 mag.
[l l l]{} BJD & RV & Error\
-2400000 & [kms$^{-1}$]{}& [kms$^{-1}$]{}\
\
54463.4939000 & 48.947 & 0.017\
54465.5247100 & 48.816 & 0.028\
54506.3770000 & 48.767 & 0.016\
54525.3478500 & 48.860 & 0.020\
54528.2886100 & 48.933 & 0.031\
54544.3463300 & 48.925 & 0.026\
\
54548.583775 & 48.933 & 0.014\
54550.577783 & 48.792 & 0.021\
54551.584161 & 48.865 & 0.013\
54553.525234 & 48.883 & 0.010\
54554.546158 & 48.819 & 0.012\
54556.554191 & 48.929 & 0.010\
54768.852140 & 48.820 & 0.009\
54769.848137 & 48.900 & 0.008\
54771.850953 & 48.851 & 0.009\
54772.841289 & 48.827 & 0.010\
54773.847921 & 48.900 & 0.008\
54802.777527 & 48.929 & 0.009\
54805.748602 & 48.852 & 0.012\
Photometric follow-up observation
=================================
Photometric follow-up observations with higher spatial resolution than CoRoT’s (of $\approx$ 20x 6) are used to exclude the presence of nearby contaminating eclipsing binaries (Deeg et al., this volume). Such observations of CoRoT-5 were performed at the 80cm telescope at IAC, Tenerife, on the January 12, and March 11, 2008 at a spatial resolution of about 1.5. These data showed only one star bright enough to cause a potential false alarm, about 8 southwest of the target. Observations obtained during and out of a transit (“on/off photometry") showed, however, that this contaminating star varies by less than 0.08 mag. This is far below the variation of about 0.55 mag that is required to explain the observed signal in the CoRoT data.
![Radial velocity measurements and Keplerian fit to the data including the Rossiter effect. Red: SOPHIE, green: HARPS.[]{data-label="fig2_rauer"}](11902fig2.eps){height="8cm"}
![Bisector analysis of CoRoT-5. []{data-label="fig3_rauer"}](11902fig3.eps){width="\linewidth" width="9cm"}
Radial velocity follow-up observations
======================================
In January 2008, after the identification of a transit signal by the alarm-mode, [[CoRoT-5]{}]{} was observed with the SOPHIE spectrograph installed on the 193 cm telescope at the Haute Provence Observatory. Two radial velocity measurements were taken at opposite quadrature phases of the radial velocity variation expected from the transit ephemerides assuming a circular orbit. At this time the data were found to be compatible with a radial velocity amplitude suggesting a Jupiter mass planet. Additional measurements were obtained later in the season to confirm the reality of the signal but not enough to obtain a precise measurement of the orbit eccentricity. One year later, a new series of measurements was obtained with the HARPS spectrograph installed on the 3.6m ESO telescope at La Silla in Chile [@Mayor2003]. Both sets of data (SOPHIE and HARPS) have been processed as in [@bouchy2008]. Radial velocities (RV) were computed by weighted cross-correlation [@baranne1996; @pepe2005] with a numerical G2-spectral template excluding spectral orders below 4200 Å. Radial velocity values are listed in Table \[RVDATA\] and plotted in Fig. \[fig2\_rauer\].
![Stellar abundances of CoRoT-5. Abundances found from neutral lines are marked by circles, for ionized lines box symbols are used. []{data-label="fig4_rauer"}](11902fig4.eps){width="\linewidth" height="3.5cm"}
We analyzed the cross-correlation function computed from the HARPS spectra using the line-bisector technique according to the description in [@queloz2001] to detect possible spectral distortions caused by a faint background eclipsing binary mimicking a small RV amplitude signal. No correlation between the RV data and the bisector span was found at the level of the uncertainty on the data (Fig. \[fig3\_rauer\]).
The stability of the bisector, combined both with the amplitude of the radial velocity and the accuracy of transit lightcurve, is enough to discard an alternate background eclipsing binary scenario. In the case of a hypothetical background eclipsing binary, obtaining a sine-shaped radial-velocity signal would require a superimposed spectrum moving with the same systemic velocity as the brightest component, and on an RV range corresponding to the sum of the width of both CCF line profiles. This prerequisite constrains both on the mass of the potential eclipsing component and its companion. The example of HD41004 provides us with an interesting benchmark [@2002santos]. This system was detected with a similar radial velocity amplitude but with a strong bisector correlation, and could be explained by a superimposed spectrum with 3% flux of the bright star. If one scales down this result to CoRoT-5, which has no bisector correlation, one finds that the contrast ratio between the brightest star and the hypothetical eclipsing binary is such that the eclipse must be very deep and the radius of the eclipsing stars much smaller than CoRoT-5. Considering the quality of the CoRoT lightcurve such a binary scenario does not match the transit ingress and egress timing and the detailed shape of the curve.
parameter value source
--------------------------------------------------- ------------------------------------ -----------------
RA 06$^h$ 45$^m$ 07$^s$ [*Exo-Dat*]{}
DEC 00$^\circ$ 48$\arcmin$ 55$\arcsec$ [*Exo-Dat*]{}
epoch 2000.0
type F9V [*Exo-Dat,*]{}
[*AAOmega*]{}
$V$ 14.0 [*Exo-Dat*]{}
GSC2.3 ID N82O011953
2MASS ID 06450653+0048548
$v \sin i$ \[km s$^{-1}$\] 1$\pm1$ VWA
$\xi_t$ \[km s$^{-1}$\] 0.91$\pm$0.09 VWA
$T_{\rm eff}$ \[K\] 6100$\pm$65 VWA
$\log g$ 4.19$\pm$0.03 VWA
$[M/H]$ $-0.25 \pm 0.06$ VWA
M$_{star}$ \[M$_\odot$\] 1.00$\pm$0.02 Evolut. tracks
R$_{star}$ \[R$_{\odot}$\] 1.186$\pm$0.04 Evolut. tracks
$M^{(1/3)} / R$ \[M$_\odot^{1/3}$ / R$_{\odot}$\] 0.843$\pm$0.024 lightcurve
age \[Gyr\] 5.5 - 8.3 photometry
+Evolut. tracks
: Parameters of the parent star CoRoT-5.[]{data-label="star"}
Properties of the central star
==============================
We determined the fundamental parameters of the host star carrying out a spectral analysis of the set of HARPS spectra acquired for radial velocity measurements. The individual spectra were reduced with the HARPS standard pipeline. The extracted spectra were corrected for cosmics impacts, for the Earth and the stars velocity, and then corrected for the blaze function and normalized, order by order, to increase the signal-to-noise (S/N). The S/N level in the continuum is around 40 in the range 5000-6500Å and it decreases to 15 towards the blue at 4000Å.
Spectroscopic observations of the central star have also been performed in January 2008 with the AAOmega multi-object facility at the Anglo-Australian Observatory. By comparing the low-resolution (R=1300) AAOmega spectrum of the target with a grid of stellar templates, as described in [@2003frasca] and [@2008gandolfi], we derived the spectral type and luminosity class of the star (F9 V).
As for the previous planet host stars, we used different methods to derive Corot-5 atmospheric parameters: line profile fitting with the SME [@1996valenti] and the VWA packages [@bruntt2002; @bruntt08]. We find general agreement and here we quote the results from VWA. The star has a very low projected rotational velocity, $v \sin i = 1\pm1$kms$^{-1}$. More than 600 mostly non-blended lines were selected for analysis in the wavelength range 3990–6810 Å. VWA uses atmosphere models from the grid by [@heiter2002] and atomic parameters from the VALD database [@kupka1999]. The abundance determined for each line is computed relative to the result for the same line in the solar spectrum from [@hinkle2000], following the approach of [@bruntt08]. The results for CoRoT-5 are shown in Table \[star\]. Using these parameters for the atmospheric model, we determined the abundances of 21 individual elements. The uncertainty on the abundances includes a contribution of 0.04 dex due to the uncertainty on the fundamental parameters. The abundance pattern is shown in Fig. \[fig4\_rauer\]. The overall metallicity is found as the mean abundance of the elements with at least 20 lines (Si, Ca, Ti, Cr, Fe, Ni) giving $[M/H]$$=-0.25\pm0.04$. We did not include Mn, as this has a significantly lower abundance. The metallicity and the 1-$\sigma$ error bar is indicated by the horizontal bar in Fig. \[fig4\_rauer\]. There is no evidence of the host star being chemically peculiar, except Mn.
The fundamental parameters of the parent star, its mass and radius were subsequently derived using stellar evolutionary tracks as presented in [@deleuil2008] plotted in a M$^{(1/3)}$/R - $T_{\rm eff}$ HR diagram. The stellar density parameter was derived from the lightcurve fitting (see sect. 7). We determined the mass and radius of the star to: M$_{\rm star}$ = 1.00$\pm$0.02 M$_\odot$ and R$_{star}$ = 1.186$\pm$0.04 R$_\odot$. As a final check, we calculated the corresponding surface gravity $\log g = 4.311\pm$0.033 while the spectroscopic value is $4.19\pm0.03$. These two values of $\log g$ are comparable with each other at the $3\sigma$ level. Based on our photometric analysis, we estimate the age of the star to 5.5 - 8.3 Gyrs. The spectra show no sign of Ca II emission or of a strong Li I absorption line, which is consistent with a relatively evolved star.
![The O-C diagram of the CoRoT-5b system. No clear period variation can be seen. []{data-label="fig5_rauer"}](11902fig5.eps){width="\linewidth"}
Period determination and transit timing variations
==================================================
In total, 27 individual transit events are clearly seen, separated by an orbital period of about 4.03 days. One event was lost in a data gap.
First, we estimated the mid-times of each transit by applying the so-called Kwee-van Woerden method [@kvw56]. This method mirrors the lightcurve around a pre-selected time-point, T, computes the differences of original and mirrored lightcurves and then searches for an optimum T. The $O-C$ diagram of the system was constructed, based on the resulting transit times and an initial guess of the period. A linear fit of this diagram yielded an improved estimate of the period. This period value was then refined with the following procedure. The lightcurve was phase-folded using this previously determined period and then averaged. The size of the bin used was 0.001 in phase (or to 5.81 minutes, using the final period). Then, this lightcurve was fitted (see the next section) by a theoretical transit lightcurve. The transit mid-times were then determined again by cross-correlating the observed and the theoretical lightcurve. This resulted in more precise mid-times of the transit and a new $O-C$ curve. Another linear fit to this $O-C$ diagram yielded a better period value, and the whole procedure was repeated. The final O-C diagram can be seen in Fig. \[fig5\_rauer\]. The resulting ephemeris is given in Table \[planettable\].
There is no obvious period variation present in the $O-C$ diagram. The first part of the lightcurve was obtained with the 512 sec sampling rate, so the first seven minima typically consist of only 20 data points. Thus, they have larger scatter and uncertainties. The next twenty minima were obtained with the high sampling rate (32 sec) and typically consist of a few hundred data points, leading to much higher accuracy. If one takes only these high-resolution minima into account, the constancy of the period is clearer. However, we cannot exclude that small period variations are present in the system. The upper limit of such a period variation was estimated by a quadratic fit to the data, which showed that it should be less than 0.42 seconds/cycle.
Analysis of parameters of [[CoRoT-5b]{}]{}
==========================================
The final phase-folded lightcurve of the transit event is seen in Fig. \[fig6\_rauer\]. The transit signal shows a depth of about 1.4 % and lasts for about 2.7 hours. We derived the planetary parameters by fitting simultaneously the lightcurve of [[CoRoT-5]{}]{} with the SOPHIE and HARPS radial velocities. A planetary model on a Keplerian orbit in the formalism of [@Gimenez2006a] and [@gimenez2006ApJ] was fitted to the data using a Markov Chain Monte-Carlo (MCMC) code described in Triaud et al. (in prep.) but using $e.\cos\omega$ and $e.\sin\omega$ instead of $e$ and $w$ as free parameters for better error estimation. In the fit a quadratic limb-darkening law was assumed at $u_+ =0.616$ and $u_- =0$. In the initial *burn-in* phase of the MCMC adjustment, 15,000 steps were chosen to allow the fit to converge. A further 50,000 steps were used to derive the best parameters and their errors. In the fit, there are eight fitted parameters plus two $\gamma$ velocities and a normalization factor, totalling 11 free parameters. In addition, the fit assumed the presence of a Rossiter-McLaughlin effect with the two fixed parameters $v sin\,i = 1.0$ km s$^{-1}$ and $\lambda = 0$ ($\lambda$: angle between stellar rotation axis and normal vector of the orbital plane). A Bayesian penalty is added to the $\chi^2$ creating a prior for $M_\star = 0.99
\pm 0.02$. The fit to the rv measurements is shown in Fig. \[fig2\_rauer\], and the derived fitting parameters are shown in Table \[RVfittable\].
[l l l]{} Fitted Parameters & Value & Units\
\
${(R_p/R_{star})^2}$& $0.01461^{ +0.00030}_{-0.00032}$&\
$t_T$ &$0.0290 ^{+0.00038} _{ -0.00053}$&\
$b$ & $0.755^{+ 0.017}_ {-0.022}$&\
$K$ &$59.1^{+6.2}_{-3.1}$ & [ms$^{-1}$]{}\
$e \cos\omega$ &$-0.057^{+0.048}_{-0.020}$&\
$e \sin\omega$ &$-0.071^{+0.147} _{-0.130}$&\
$t_T$ denotes the transit duration given in fraction of phase, $b$ the impact parameter and $K$ the RV semi-amplitude.
![Top: Phase-folded lightcurve of CoRoT-5b. Bottom: Residuals of fitted transit curve.[]{data-label="fig6_rauer"}](11902fig6.eps){width="\linewidth"}
[l l l]{} Derived physical parameters & Value & Units\
\
Transit epoch T$_0$ &$2454400.19885\pm0.0002$ & HJD\
Orbital period $P$ & $4.0378962\pm0.0000019$ & days\
Orbital semi-major axis $a$ & $0.04947^{+0.00026}_{-0.00029}$ & AU\
Orbital inclination $ i$ &$ 85.83^{+0.99}_{-1.38}$ & degrees\
Orbital eccentricity $e$ & $ 0.09^{+0.09}_{-0.04} $&\
Argument of periastron $\omega$ &$-2.24^{+5.05}_{-0.84}$ & rad\
Planet radius $R_\textrm{\scriptsize p}$ & $1.388^{+0.046}_{-0.047}$ & $R_\textrm{\scriptsize J}$\
Planet mass $M_\textrm{\scriptsize p}$ &$0.467^{+0.047}_{-0.024}$ & $M_\textrm{\scriptsize J}$\
Mean planet density $\rho_\textrm{\scriptsize p}$ & $0.217^{+0.031}_{-0.025}$ & $g cm^{-3}$\
Planetary surface gravity $\log g_\textrm{\scriptsize p}$¤ & $7.77^{+0.14}_{-0.08}$ & cgs\
Zero albedo equilibrium temperature $T_{eq}$ & **$1438\pm39$** & K\
In addition, a model transit curve [@2002mandel] was fitted to the photometric phase folded transit curve separately. The parameters fitted are the center of transit, the planet radius expressed in stellar radii, the semi-major axis in stellar radii and the orbital inclination. In this fit the limb-darkening coefficients (u$_1$ and u$_2$) were free parameters, assuming a quadratic limb-darkening law. The fitting method follows a Metropolis-Hastings algorithm, which is a kind of Markov Chain Monte-Carlo procedure. The fitting procedure was performed ten times with different starting values to find the global minimum in $\chi^2$. The errors of the fit were estimated from the standard deviations of the points in the chain. In addition to the transit curve, a third light component is included as a free parameter in the fit. In this way, we could check whether another contaminant is present, which remained unresolved in the photometric follow-up. However, no such additional source of light was found. The transformation between contamination factor $c$ and the third light $l_3$ is $c = l_3/(1-l_3)$. We had $c=0.005 \pm 0.024$. Since we already removed the known contaminant factor from the lightcurve (see Section 2), we could therefore conclude that no further observable contaminant is present in the lightcurve of CoRoT-5. The planet parameters derived from this fit agree with the simultaneous fitting within the error bars, so we do not report them again here.
The resulting planetary parameters based on the MCMC approach with fixed limb-darkening coefficients and without any third light are summarized in Table \[planettable\]. The major uncertainties on the planet are, as usual, introduced mainly from the uncertainty of the stellar parameters.
Summary
=======
We report the discovery of a “hot-Jupiter-type" planet, CoRoT-5b, orbiting a type F9V star of 14.0 mag. The planet mass and radius were derived to $0.467^{+0.047}_{-0.024}$ M$_{Jup}$ and $1.388^{+0.046}_{-0.047}$ R$_{Jup}$, respectively. It orbits its central star at $0.04947^{+0.00026}_{-0.00029}$ AU orbital distance. The determined eccentricity is low (see Table \[planettable\]), but further radial velocity measurements would be needed for a more accurate determination.
CoRoT-5b has a density of $ 0.217^{+0.031}_{-0.025}$ g cm$^{-3}$, similar to the planets WASP-12b and WASP-15b [@2009hebb; @west2009], implying that it belongs to the planets with the lowest mean density found so far. As such, it is found to be larger by 20% than standard evolution models [@guillot2006] would predict. Standard recipes that account for missing physics (kinetic energy transport or increased opacities) can explain this large size, and predict that the planet is mostly made of hydrogen-helium, with at most $28\,\rm M_\oplus$ of heavy elements (maximum value obtained in the kinetic energy model, assuming 0.5% of the incoming energy is dissipated at the planet center). Thus, CoRoT-5b supports the proposed link between the metallicity of planets and of their host star.
**Acknowledgements**
HJD and JMA acknowledge support from grant ESP2007-65480-C02-02 of the Spanish Education and Science Ministry. Some of the data published in this article were acquired with the IAC80 telescope operated by the Instituto de Astrofísica de Tenerife at the Observatorio del Teide. The German CoRoT Team (TLS and Univ. Cologne) acknowledges DLR grants 50OW0204, 50OW0603, 50QP07011. RA acknowledges support by grant CNES-COROT-070879. The building of the input [*CoRoT*]{}/Exoplanet catalog was made possible by observations collected for years at the Isaac Newton Telescope (INT), operated on the island of La Palma by the Isaac Newton group in the Spanish Observatorio del Roque de Los Muchachos of the Instituto de Astrofisica de Canarias.
[^1]: Observations made with SOPHIE spectrograph at the Observatoire de Haute Provence (07B.PNP.MOUT), France, and HARPS spectrograph at ESO La Silla Observatory (072.C-0488(E), 082.C-0312(A)), and partly based on observations made at the Anglo-Australian Telescope. The CoRoT space mission, launched on December 27, 2006, was developed and is operated by CNES, with the contribution of Austria, Belgium, Brasil, ESA, Germany, and Spain.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this article we study the impact of the negotiation environment on the performance of several intra-team strategies (team dynamics) for agent-based negotiation teams that negotiate with an opponent. An agent-based negotiation team is a group of agents that joins together as a party because they share common interests in the negotiation at hand. It is experimentally shown how negotiation environment conditions like the deadline of both parties, the concession speed of the opponent, similarity among team members, and team size affect performance metrics like the minimum utility of team members, the average utility of team members, and the number of negotiation rounds. Our goal is identifying which intra-team strategies work better in different environmental conditions in order to provide useful knowledge for team members to select appropriate intra-team strategies according to environmental conditions.'
address: 'Universidad Politécnica de Valencia, Departamento de Sistemas Informáticos y Computación, Camí de Vera s/n, 46022, Valencia, Spain, {sanguix,vinglada,vbotti,agarcia}@dsic.upv.es'
author:
- 'Víctor Sánchez-Anguix'
- Vicente Julián
- Vicente Botti
- 'Ana García-Fornes'
bibliography:
- 'NT-ISFinal.bib'
title: 'Studying the Impact of Negotiation Environments on Negotiation Teams’ Performance'
---
Negotiation Teams ,Agreement Technologies ,Automated Negotiation ,Collective Decision Making ,Multi-agent Systems
Introduction {#sec:introduction}
============
Agreement technologies [@luck08; @sierra11] conform an emergent research area among scholars in artificial intelligence and autonomous agent systems. Autonomous software agents act reactively and proactively with the objective of maximizing their human users’ goals. Nevertheless, as systems tend to be more complex, so do agents’ goals, and agents cannot achieve their goals without the cooperation of other agents. Given the open nature of many multi-agent systems, conflict may be inherent among agents. Hence, distributed mechanisms that allow agents to solve conflict and cooperate are a necessity. Agreement technologies have been actively researched bearing in mind the aforementioned necessity.
Automated negotiation [@kraus97; @jennings01; @lopes08] is one of the core topics in agreement technologies. Basically, agents in conflict engage in an automatic offer exchange process which gradually leads towards a final solution, or agreement, that solves conflict and makes cooperation among agents possible. The most common use for automated negotiation has been electronic commerce [@lomuscio03], but it should be highlighted that the applicability of this technology has been demonstrated in other domains like collaborative design [@klein03], labor management disputes [@sycara93], and mediation between human negotiation parties [@chalamish11].
Despite being widely studied by scholars from different disciplines like artificial intelligence, game theory, and social sciences, studies have largely focused on processes whose parties (bilateral, or multiparty) are formed by single individuals [@faratin98; @ehtamo01; @faratin02; @serrano03; @digiunta06; @fatima06; @behfar08; @halevy08; @ito10]. However, some real world scenarios bring about negotiation parties that are formed by more than a single individual. For instance, when an organization negotiates with another organization the selling of a product line, it is usual for organizations to send a group of representatives to negotiate with the other organization. Another example, probably a more quotidian example, involves a married couple that negotiates the purchase of a house with a seller. In this case, the married couple is actually a negotiation party which is formed by two individuals instead than a single individual party. To conclude with the list of real examples, the reader could also think of a group of friends that want to go on a holiday together. This party, conformed by all the friends, has to negotiate a deal with the travel agency if they want to achieve their desired goal.
This kind of multi-individual party is known in the social sciences as a negotiation team [@thompson96; @thompson01; @behfar08; @halevy08]: *a group of interdependent people that join and act together as a single negotiation party because of their shared interests, related to a negotiation*. The rationale behind negotiation teams is mainly twofold. First, team members may have different expertise and negotiation skills that are needed to tackle the negotiation problem successfully. Second, the multi-individual entity that negotiates may be formed by multiple stakeholders with different sub-goals and preferences regarding the final negotiation outcome. We can imagine how an IT company may send a negotiation team formed by experts (different knowledge and skills) from the sales department, marketing department, and R&D department to successfully negotiate a new project with the local administration, how the wife and the husband may have different opinions with respect to house pricing, location, and facilities, and how each friend may have different interests regarding hotel location, number of days to spend, and pricing regarding their travel.
Electronic applications, and consequently automated negotiation, are not alien to scenarios that may involve agent-based negotiation teams (ABNT). For instance, group travel e-markets, group buying in e-markets, electronic management of farming cooperatives, negotiation support systems for real human teams, and agent-based simulation may be some of the applications where ABNT may be used. From our point of view, we are interested in ABNT whose members may have different preferences regarding the negotiation issues, and, more specifically, we are interested in models for electronic markets.
In this paper, we present four intra-team strategies for an ABNT that negotiates with a single opponent. Intra-team strategies, also known as team dynamics, govern which decisions are taken as a team, and how and when those decisions are taken [@sanchez-anguix10]. The relationship between intra-team strategies and team performance is direct. Hence, it became the focus of our current research. It has been documented that environment conditions such as the deadline, concession speed, and reservation utility may affect the impact of single-individual bilateral strategies [@faratin98]. However, in the team case, new conditions like the number of team members, team preferences’ diversity, and the emergent effect of aggregating team members’ behaviors/actions may also end up affecting team performance. Prior to the negotiation process, negotiation teams face the challenge of selecting which intra-team strategy should be employed. If environmental conditions have an effect on the performance of the different intra-team strategies, the intra-team strategy for the negotiation at hand should be selected accordingly to the current environmental conditions inferred by team members. Our research goal is identifying which intra-team strategies perform better according to different negotiation environments under different team performance measures. The long term goal is employing the results of this article for helping team members to select the proper intra-team strategy.
Hence, four intra-team strategies that guarantee four minimum levels of unanimity regarding team decisions are presented in this article: representative (no unanimity guaranteed), Similarity Simple Voting (plurality/majority guaranteed), Similarity Borda Voting (semi-unanimity guaranteed), and Full Unanimity Mediated (unanimity guaranteed). Due to the large amount of variables that may affect the negotiation, we employ an empirical approach to study the behavior of the four intra-team strategies. We study and identify which are the most appropriate strategies according to different environmental conditions and team performance measures. This article, is partially based on our previous work regarding intra-team strategies for negotiation teams [@sanchez-anguix11; @sanchez-anguix12], where we presented initial results and simulations. In this article, we extend our empirical experiments by incorporating new environmental conditions (i.e., team size, different deadlines), carrying out a more fine-grained analysis of previous environmental conditions (i.e., deadline, concession speeds), and presenting revised versions of the four intra-team strategies.
The article is organized as follows. First, we describe the assumptions of our negotiation model (Section \[sec:assumptions\]). After that, the details of the four intra-team strategies are thoroughly described in Section \[sec:intra\]. Then, in Section \[sec:exp\] the article depicts which negotiation environments and team performance metrics have been studied, and it presents the results and analysis of our experiments. Afterwards, the present work is related to other works in the area of artificial intelligence and automated negotiation (Section \[sec:related\]). Finally, we briefly state the conclusions of our study and point out some future and interesting lines of work in Section \[sec:conclusions\].
General Model Description {#sec:assumptions}
=========================
In this section, we describe the assumptions of our model. We have divided the assumptions in two different categories: general assumptions and opponent assumptions. The general assumptions directly affect the nature of the negotiation at hand and are shared between parties (e.g., protocol, number of parties, attribute types, etc.), whereas opponent assumptions describe the strategy carried out by the opponent.
Negotiation Setting {#sec:general}
-------------------
- The team $A$ is formed by $M$ different agents $a_{i}, 1\leq i \leq M$. It should be stated that team membership is considered static during the negotiation process. Dynamic ABNT are not considered in this work, and they are appointed as future work.
- The common goal of the team $A$ is negotiating a successful deal with the opponent $op$. Thus, in this case we assume an implicit representation of the teams’ goal.
- It is assumed that information is private, even among team members. Therefore, agents do not know other agents’ utility functions, strategies, reservation utilities, or deadlines. On top of that, we also assume agents with bounded computational resources. Thus, we take a heuristic approach which seeks near optimal results while being computationally tractable.
- It is assumed that the team $A$ and the opponent $op$ communicate following an alternating bilateral protocol [@rubinstein82]. One of the two parties acts as the initiator, and it is entitled to propose the first offer. The other party receives the offer and can respond with two different actions: accept the offer (successful negotiation), or propose a counteroffer. If a counteroffer is proposed, the initiator party receives the offer and it can either accept the counteroffer or propose another offer, starting a new negotiation round. Depending on the intra-team strategy, one of the team members or a team mediator is responsible of the communications with the opponent. In this setting, the fact that one of the parties is a team remains unknown to the other party.
- Additionally, it is also assumed that the negotiation is time-bounded, and each party has a private deadline $T_{A}$ (team deadline), $T_{op}$ (opponent deadline). When its deadline is achieved, the party leaves the negotiation and it is considered a failed negotiation. In the case of $T_{A}$, it is considered a joint deadline for all of the team members, who have agreed upon this deadline prior to the negotiation at hand.
- The mediator, if present, is never a perfect mediator that aggregates the utility functions of all the team members. This assumption is taken due to the fact that, depending on the application, some team members may not be completely trustable and may attempt to exaggerate/change their preferences to manipulate the negotiation process. This mischievous behavior is easily carried out when aggregating utility functions.
- The negotiation domain is comprised of $n$ real attributes whose domains can be scaled to $[0,1]$. Thus, the possible number of offers is $[0,1]^{n}$. In this domain, a complete offer is represented as $X=\{x_{1},x_{2},\dots,x_{n}\}$, where $x_{i}$ is a specific instantiation of attribute $i$. Additionally, we use the notation $X^{t}_{i\rightarrow j}$ to denote that offer $X$ was sent at round $t$ from party $i$ to party $j$.
- Every agent $i$ (team member or opponent) has its preferences represented by means of linear additive utility functions in the form: $$U_{i}(X)= w_{i,1}\;V_{i,1}(x_{i,1})+w_{i,2}\;V_{i,2}(x_{2})+...+w_{i,n}\;V_{i,n}(x_{n})$$ where $X$ is a complete offer, $x_{j}$, is the value given to the $j$-th attribute, $V_{i,j}(.)$ is the valuation function for attribute $j$ used by agent $i$ to normalize the attribute value to $[0,1]$, and $w_{i,j}$ is the weight/importance given by agent $i$ to attribute $j$ in the negotiation process. Several observations should be made regarding these utility functions: (i) weights are normalized so that $\sum_{j=1}^{n} w_{i,j}=1$; (ii) attributes are assumed to be independent from each other. Thus, the valuation of one of the attributes does not alter the others attributes’ valuation; (iii) it is assumed that valuation functions are either monotonically increasing or monotonically decreasing. Moreover, we assume that team members share the same type of monotonic function (i.e., increasing or decreasing) for each $V_{i,j}(.)$. As for the opponent, it is assumed that the monotonic function for $V_{i,j}(.)$ is the opposite type to that of team members. It is reasonable to assume this model for valuation functions in e-commerce scenarios. Buyers usually share the same type of valuation function for attributes such as the price (monotonically decreasing), product quality (monotonically increasing), and the dispatch time (monotonically decreasing), whereas sellers usually use the opposite type of monotonic functions (monotonically increasing for price, monotonically decreasing for product quality, and monotonically increasing for dispatch time); (iv) attribute weights $w_{i,j}$ are different for team members. This way, we are able to represent the fact that some team members may be more interested in some attributes whereas other team members may be more interested in other attributes (e.g., some team members prefer price over quality, while others give a higher priority to the product quality). Obviously, the weights of the opponent’s utility function may be different from those of team members; (v) since team members share the same type of monotonic function, if one of the team members increases its utility by increasing/decreasing one of the attribute values, the other team members will stay at the same utility level or they will also increase their utility. Thus, there is potential for cooperation among team members.
- The opponent has a reservation utility $RU_{op}$. Any offer whose utility is lower than $RU_{op}$ will be rejected. Each team member $a_{i}$ has a private reservation utility $RU_{a_{i}}$. This individual reservation utility is not shared among teammates. Therefore, a team member $a_{i}$ will reject any offer whose value is under $RU_{a_{i}}$. In this setting, reservation utilities represent the individual utility of each agent if the negotiation process fails. For the experiments, the reservation utilities are drawn from uniform distributions $RU_{op}=U[0,0.25]$ and $RU_{a_{i}}=U[0,0.25]$.
Opponent Model {#sec:opponent}
--------------
The opponent $op$ is modeled as a single agent for the sake of simplicity. We acknowledge that the other party may be another team (e.g., organization vs organization), but the focus of our study in this article, and the kind of applications in our mind, involve negotiations between one team and one single agent. For the opponent model we used well-known strategies in the agent negotiation literature:
- The opponent uses a time-based tactic during the negotiation process. In multi-attribute, time-bounded negotiations, assuming that agents concede gradually to reach an agreement is usual. This is especially important when agents do not know other agents’ preferences and strategies, since an exploration of the agreement space is required for discovering possible agreements. A concession strategy typically starts by demanding the maximum aspiration and, as the negotiation process advances, the aspiration demanded tends to be lowered. The speed at which the strategy concedes is regulated by the concession speed parameter. In this article, we employed a time-based concession tactic inspired in tactics used by other authors [@faratin98; @lai08]: $$\label{time}
s_{op}(t)=1-(1-RU_{op})(\frac{t}{T_{op}})^{\frac{1}{\beta_{op}}}$$ where $t$ is the current negotiation round and $\beta_{op}$ is a parameter that governs the concession speed. On the one hand, when $\beta_{op}=1$ the concession is linear and each negotiation round the same amount of concession is performed, and when $\beta_{op}<1$ the concession is Boulware and very little is conceded at the start of the negotiation process but the agent concedes faster as the negotiation deadline approaches. On the other hand, when $\beta_{op}>1$ the tactic is conceder and the agents concede fast towards the reservation utility in the first rounds.
- The opponent uses an offer acceptance criterion $ac_{op}(.)$ during the negotiation process. It is formalized as follows: $$ac_{op}(X_{A\rightarrow op}^{t}) = \left\lbrace
\begin{array}{l l}
accept & \mbox{if } s_{op}(t+1) \leq U_{op}(X_{A\rightarrow op}^{t}) \\
reject & \mbox{otherwise}
\end{array}
\right.$$ where $t$ is the current round, $X^{t}_{A \rightarrow op}$ is the offer received from the team, $U_{op}(.)$ is the utility function of the opponent, and $s_{op}(.)$ is the opponent concession strategy. Thus, an offer is accepted by $op$ if it reports a utility that is equal to or greater than the utility of the offer that $op$ would propose in the next round.
- When the opponent has to propose a counteroffer at negotiation round $t$, it proposes an offer $X_{op\rightarrow A}^{t}$ whose utility is $U_{op}(X_{op\rightarrow A}^{t})=s_{op}(t)$. However, depending on the negotiation domain, there may be an infinite number of offers that comply with the previous condition. Similarity heuristics are a largely used family of heuristics in agent-based negotiation literature [@faratin02; @lai08; @sanchez-anguix10]. Thus, we assumed that the opponent attempts to propose the offer that is the most similar to the last offer received from the team, and whose utility is $U_{op}(X_{op\rightarrow A}^{t})=s_{op}(t)$. The Euclidean distance was used as similarity function.
Intra-Team Strategies {#sec:intra}
=====================
An intra-team strategy defines *what* decisions have to be taken by a negotiation team, *how* those decisions are taken, and *when* those decisions are taken. In a bilateral negotiation process between a team and an opponent, the decisions that must be taken (*what*) are which offers are sent to the opponent, and whether or not opponent offers are accepted. Given the fact that a negotiation team is formed by more than a single individual, decisions should take into account the interests of the team members. *How* decisions are taken will determine the satisfaction level of the team with the final decision. In this article, most of the team interactions in intra-team strategies are carried out during the negotiation (*when*).
Next, we describe the four intra-team strategies that we propose: Representative (RE), Similarity Simple Voting (SSV), Similarity Borda Voting (SBV) and Full Unanimity Mediated (FUM). Each strategy is capable of guaranteeing a minimum level of unanimity regarding the offer sent to the opponent, and whether or not to accept the opponent’s offer. Another difference between the four strategies is the presence of a mediator and its level of activity (none, coordination tasks, very active participation). Table \[tab:qualitative\] captures the main qualitative differences between the four intra-team strategies according to the aforementioned criteria.
Representative (RE) {#sec:re}
-------------------
The Representative strategy (RE) is perhaps the simplest intra-team strategy. Basically, one of the team members is selected as representative $a_{re}$ for the team during the negotiation. This agent will act on behalf of the team during the negotiation, making it responsible of selecting which offers are sent to the opponent, and whether or not opponent’s offers are accepted. The only communications are those carried out between the representative agent $a_{re}$ and the opponent $a_{op}$, and, therefore, this strategy is equivalent to a classic bilateral strategy.
The representative agent negotiates according to its own utility function $U_{a_{re}}(.)$ since it does not know the utility function of the other participants. The two decisions that have to be taken during the negotiation are which offers are sent to the opponent, and whether or not the opponent’s offer is accepted.
### Offer proposal
Being a time-bounded negotiation, the representative employs a time-based concession tactic $s_{a_{re}}(.)$ to negotiate with the opponent. It is based on a team deadline $T_{A}$ and a concession speed $\beta_{A}$, which have been agreed upon prior to the negotiation start: $$s_{a_{re}}(t)=1-(1-RU_{a_{re}}) (\frac{t}{T_{A}})^{\frac{1}{\beta_{A}}}$$ The concession strategy defines the aspiration level (utility demanded) by the agent at a specific round $t$. This utility is demanded from the point of view of the representative, and, so, any offer $X_{A\rightarrow op}^{t}$ proposed by $a_{re}$ at round $t$ will obey the following condition: $$U_{a_{re}}(X_{A\rightarrow op}^{t})= s_{a_{re}}(t)$$ Since there is a large number of offers that may obey the equation above, we aimed to satisfy the opponent’s preferences as much as possible. As in the case of the opponent strategy, the representative selects the offer that is the most similar to the previous offer received from the opponent using a similarity heuristic based on the Euclidean distance. $$X_{A\rightarrow op}^{t}=\underset{X | U_{a_{re}}(X)= s_{a_{re}}(t)}{max} Sim(X,X_{op\rightarrow A}^{t-1})$$
### Offer acceptance
A common acceptance criterion in time-bounded negotiations is that an opponent’s offer is accepted if it reports a utility which is higher than or equal to the utility that is to be demanded in the next negotiation step. In the case of the representative, it will accept the opponent’s offer $X_{op\rightarrow A}^{t}$ at round $t$ if it reports a utility $U_{a_{re}}(X_{op\rightarrow A}^{t})$ greater than or equal to $s_{a_{re}}(t+1)$. This can be formalized as follows: $$ac_{a_{re}}(X_{op\rightarrow A}^{t}) = \left\lbrace
\begin{array}{l l}
accept & \mbox{if } s_{a_{re}}(t+1) \leq U_{a_{re}}(X_{op\rightarrow A}^{t}) \\
reject & \mbox{otherwise}
\end{array}
\right.$$
### Unanimity Level
It is clear that since the representative negotiates according to its own utility function and reservation utility, it cannot guarantee any kind of unanimity regarding team decisions. Decisions taken by the representative are acceptable to himself (1 agent), but nothing can be assured about the rest of team members. One could think that if no unanimity can be guaranteed, this strategy is not worth being used. However, when team members tend to be very similar this strategy is expected to yield acceptable results with communication costs equivalent to a bilateral negotiation process.
Similarity Simple Voting (SSV) {#sec:ssv}
------------------------------
The second intra-team strategy relies on a trusted mediator that helps team members to participate in the negotiation process. Its main tasks involve coordination of voting processes and communications with the opponent. It should be highlighted that the mediator communicates team’s decisions to the opponent, and broadcasts opponent’s decisions among team members. Thus, the fact that every team member participates in the negotiation process remains unknown for the opponent. As for intra-team communications, it should be noted that team members do not communicate among them, but they only communicate anonymously with the mediator.
The decision rule used for voting processes is plurality/majority. More specifically, a plurality rule is used in the voting process employed to decide which offer is sent to the opponent, and a majority rule is used in the voting process employed to decide opponent’s offer acceptance. A detailed view of the intra-team strategy can be observed in Algorithm \[alg:ssv\], which describes the whole process from the point of view of the mediator.
### Offer proposal
Whenever a new offer has to be proposed to the opponent at round $t$, the mediator opens a call for proposals among team members. Each team member $a_{i}$ is allowed to communicate anonymously one offer $X_{a_{i}\rightarrow A}^{t}$ to be proposed to the opponent. Once every proposal has been gathered, the mediator opens a voting process where offers proposed $XT^{t}$ are made public among team members. Then, each agent $a_{i}$ anonymously sends a multi-vote $Vote_{a_{i}}$ to the mediator. A multi-vote gathers votes for every offer made public. We use the notation $Vote_{a_{i}}(j)$ to denote the vote given by agent $a_{i}$ to the offer $j$-th from $XT^{t}$, and $XT^{t}(j)$ as the $j$-th offer in $XT^{t}$. The votes can be either positive (1), if the offer $j$-th is acceptable for $a_{i}$ at round $t$, or negative (0), if the offer $j$-th is not acceptable for $a_{i}$ at round $t$. Once all votes have been gathered, the mediator sums up the number of positive votes and the most supported offer $X_{A\rightarrow op}^{t}$ is selected, made public among team members, and sent to the opponent. When a tie is produced, the tie-breaker rule consists in randomly selecting one of the most supported offers. The following Equation describes the selection rule of the previous mechanism: $$X_{A\rightarrow op}^{t}=\underset{X_{j}\in XT^{t}}{\mbox{argmax}}\underset{a_{i}\in A}{\sum}Vote_{a_{i}}(j)
\label{eq:sum}$$ The paragraph above describes the intra-team protocol followed by team members and mediator to determine which offer is sent to the opponent at round $t$. However, team members are faced with two decisions in this intra-team protocol: which offer should be proposed to the mediator during the call for proposals, and the acceptability of each offer proposed during the aforementioned process. We assume that, since the negotiation is time-bounded, team members follow a time-based concession tactic where the concession speed $\beta_{A}$ is common and agreed by teammates prior to the negotiation process: $$s_{a_{i}}(t)=1-(1-RU_{a_{i}})(\frac{t}{T_{A}})^{\frac{1}{\beta_{A}}}
\label{eq:time}$$ For the first decision, proposing an offer to team members, the agent $a_{i}$ proposes an offer $X_{a_{i}\rightarrow A}^{t}$ whose utility is equal to $U_{a_{i}}(X_{a_{i}\rightarrow A}^{t})=s_{a_{i}}(t)$. Since there may be more than a single offer with such utility, the agent has to choose one of those offers. If the agent $a_{i}$ wants its offer $X_{a_{i}\rightarrow A}^{t}$ to be accepted it should maximize the probability of it being the most supported by team members and the probability of it being accepted by the opponent: $$X_{a_{i}\rightarrow A}^{t}= \underset{X|U_{a_{i}}(X)=s_{a_{i}}(t)}{argmax}p_{op}(X) \times p_{A}(X)
\label{eq:prob}$$ where $p_{op}(X)$ is the probability for $X$ to be accepted by the opponent, and $p_{A}(X)$ is the probability for $X$ to be selected by team members. We incorporated agents with a similarity heuristic based on the Euclidean distance over attribute domains scaled to \[0,1\]. It takes into account the last offer proposed by the opponent $X_{op\rightarrow A}^{t-1}$ and the offer sent by team members in the previous negotiation round $X_{A\rightarrow op}^{t-1}$. The most similar an offer is to $X_{op\rightarrow A}^{t-1}$, the more probable it is for the offer to be accepted by the opponent. Analogously, the most similar an offer is to $X_{A\rightarrow op}^{t-1}$, the more probable it is for the offer to be the most supported option in the voting process and, therefore, to be sent to the opponent. Thus, Equation \[eq:prob\] can be approximated by similarity heuristics as follows: $$\begin{array}{l l}
X_{a_{i}\rightarrow A}^{t}= & \underset{X|U_{a_{i}}(X)=s_{a_{i}}(t)}{argmax}p_{op}(X) \times p_{A}(X) \approx\\
& \underset{X|U_{a_{i}}(X)=s_{a_{i}}(t)}{argmax} Sim(X,X_{op\rightarrow A}^{t-1}) \times Sim(X,X_{A\rightarrow op}^{t-1})
\end{array}
\label{eq:similarity}$$ Finally, for determining the acceptability of offers proposed by team members at round $t$, we used a rational criterion so that an agent $a_{i}$ emits a positive vote $Vote_{a_{i}}(j)=1$ for the $j$-th offer if it reports a utility that is greater or equal than the utility marked by the concession strategy $s_{a_{i}}(t)$. Otherwise, the offer is not supported and a negative vote is emitted. This process can be formalized as: $$Vote_{a_{i}}(j) = \left\lbrace
\begin{array}{l l}
1 & \mbox{\;\;if } U_{a_{i}}(XT^{t}(j)) \geq s_{a_{i}}(t) \\
0 & \mbox{\;\;otherwise}
\end{array}
\right.$$
### Offer acceptance
Whenever the mediator receives an offer $X_{op\rightarrow A}^{t}$ from the opponent at round $t$, it broadcasts the offer among team members. Then, the mediator opens up a majority voting process where each agent $a_{i}$ states whether or not the opponent’s offer is acceptable $ac_{a_{i}}(X_{op\rightarrow A}^{t})$ (1 for accept, 0 for reject). The mediator counts the number of acceptances, and if the offers is supported by the majority ($>\frac{|A|}{2}$) then it is accepted by the team. Otherwise, the offer is rejected. If the number of team members is even and a tie has been produced, a random decision is taken by the mediator. This mechanism can be described as follows: $$ac_{A}(X_{op\rightarrow A}^{t}) = \left\lbrace
\begin{array}{l l}
\mbox{accept} & \mbox{\;\;if } \underset{a_{i}\in A}{\sum}ac_{a_{i}}(X_{op\rightarrow A}^{t}) > \frac{|A|}{2}\\
\mbox{reject} & \mbox{\;\;if } \underset{a_{i}\in A}{\sum}ac_{a_{i}}(X_{op\rightarrow A}^{t}) < \frac{|A|}{2} \\
\mbox{random} & \mbox{\;\;otherwise}
\end{array}
\right.$$
How team members $a_{i}$ decide the acceptability of the opponent’s offer $ac_{a_{i}}(X_{op\rightarrow A}^{t})$ follows the rational mechanism that we have employed so far. Basically, the offer is acceptable (1) if it yields a utility which is greater than or equal to the utility demanded by the concession strategy in the next negotiation round $s_{a_{i}}(t+1)$. Otherwise, the offer is not considered acceptable. The following Equation formalizes the acceptance criterion: $$ac_{a_{i}}(X_{op\rightarrow A}^{t}) = \left\lbrace
\begin{array}{l l}
\mbox{1} & \mbox{\;\;if } U_{a_{i}}(X_{op\rightarrow A}^{t}) \geq s_{a_{i}}(t+1)\\
\mbox{0} & \mbox{\;\;otherwise}
\end{array}
\right.$$
$t=0$ Send (Withdraw $\longrightarrow$ $op,A$) Return Failure
### Unanimity Level
The proposed method is capable of guaranteeing team decisions that are supported by a plurality/majority of the participants. More specifically, plurality is assured in case of the offer proposed to the opponent, and majority is assured when deciding opponent’s offer acceptance. Exceptions for this minimum level of team unanimity are ties. For instance, the most extreme case is when team members propose offers to the team, but they only support their own offers. In that case, each proposal sums up exactly 1 positive vote and there is not a clear plurality winner.
Similarity Borda Voting (SBV) {#sec:sbv}
-----------------------------
SSV is capable of assuring majority and plurality decisions within the team. However, some scenarios may need of intra-team strategies that ensure higher levels of unanimity. SBV and FUM (described later) are designed to solve this problem. The basic structure of SBV remains the same than in SSV, but the voting rules employed are different. More specifically, when each team member votes team proposals, borda count is employed to determine the winner, and a unanimity rule is used to determine opponent’s offer acceptance. Next, we briefly describe the aspects which make SBV different to SSV.
### Offer proposal
As in SSV, when the team has to propose an offer to the opponent, the mediator opens a call for proposals where each team member can propose an offer to the mediator. Then, once every offer has been gathered, the mediator makes public the offers proposed to the team members and a voting process starts. The main difference between both intra-team strategies resides in the fact that team members vote according to a Borda count rule [@nurmi10]. Basically, each team member $a_{i}$ ranks the proposals $XT^{t}$ in ascending order according to its own utility function $U_{a_{i}}(.)$. We denote as $rank_{a_{i}}(XT^{t})$ the ascending rank according to $a_{i}$’s utility function, and $Position(X,rank_{a_{i}}(XT^{t}))$ as the position (1 to $|XT^{t}|$) that the offer $X$ occupies in a ranked list. The vote emitted by $a_{i}$ for offer $j$-th in $XT^{t}$ is the position occupied by such offer in the ranked list minus one unit: $$Vote_{a_{i}}(j) = Position(XT^{t}(j),rank_{a_{i}}(XT^{t}))-1$$ Numerical votes for each offer are summed up by the mediator, who finally selects the offer that received the highest sum of scores from the team members (see Equation \[eq:sum\]). It should be highlighted that the similarity heuristic employed by team members is the same than the one employed in SSV.
### Offer acceptance
As for the offer acceptance, the only difference remains in the rule used by the mediator. The opponent’s offer is accepted only if it is acceptable for all the team members. The rationale $ac_{a_{i}}(X_{op\rightarrow A}^{t})$ used by team members to determine if an offer is acceptable at round $t$ is equivalent to the one used in SSV. Thus, the offer acceptance mechanism can be formalized as follows: $$ac_{A}(X_{op\rightarrow A}^{t}) = \left\lbrace
\begin{array}{l l}
\mbox{accept} & \mbox{\;\;if } \underset{a_{i}\in A}{\sum}ac_{a_{i}}(X_{op\rightarrow A}^{t}) = |A| \\
\mbox{reject} & \mbox{\;\;otherwise }\\
\end{array}
\right.$$
### Unanimity Level
When describing the minimum unanimity level guaranteed by SBV, we mentioned the term semi-unanimity. It is clear that if an opponent offer is accepted by the team, it is acceptable for every team member due to the unanimity ruled employed. However, such unanimity is not guaranteed regarding the team decision on which offer is sent to the opponent. Borda count is generally referred as a method that selects broadly accepted options as winners instead of the majority/plurality option (e.g., avoid the tyranny of the majority). In this sense, Borda count entails some degree of unanimity. Nevertheless, the specific degree of unanimity that Borda assures is difficult to determine in our negotiation scenario.
Full Unanimity Mediated (FUM) {#sec:fum}
-----------------------------
The last intra-team strategy, Full Unanimity Mediated (FUM), seeks to reach unanimity regarding all team decisions. In fact, every team decision taken (i.e., offer acceptance, offer proposal) following this intra-team strategy entails unanimity at each round $t$ of the negotiation process. However, the type of mediator required for FUM is more sophisticated than in the rest of strategies presented in this article. It requires that the mediator participates in a pre-negotiation process where team members hand over decision rights over attributes that are not interesting for them. Additionally, the team mediator needs to be able to infer attributes’ importance for the opponent. Finally, it also needs to coordinate unanimity voting processes, and an iterated building process that constructs the offers sent to the opponent. A complete view of the pseudo-algorithm carried out by the mediator can be observed in Algorithm \[alg:fum\].
/\*Pre-negotiation: information sharing\*/ Send (Ask for $NI_{a_{i}}$ $\longrightarrow$ $A$) $t=0$ Send (Withdraw $\longrightarrow$ $op,A$) Return Failure
### Pre-negotiation: information sharing
During the pre-negotiation, team members are allowed to hand over decision rights over some attributes that they do not consider interesting. The iterated offer building process relies on a mechanism which sets attributes’ values one-per-one according to team members’ will. When an agent hands over decision rights on an attribute, it does not participate in the setting of such attribute. All the communications in the pre-negotiation are private with the mediator, who asks each team member regarding the set of attributes which it is willing to hand over. The rationale behind the idea of handing over decision rights is that conflict may be reduced, and, so, the chances to build a more likeable offer for the opponent are increased while maintaining a good quality for one’s own utility function. The fact that some attributes may yield little or no importance at all for some team members is also feasible in a team setting, since some of these attributes may have been introduced to satisfy the interests of a subgroup of team members.
The pre-negotiation protocol goes as follows. First, the mediator opens a call for decision rights, where each team member $a_{i}$ is allowed to send (to the mediator) a set of negotiation attributes $NI_{a_{i}}$, whose decision rights are handed over by $a_{i}$. Once all the responses have been gathered, the mediator keeps track of those attributes that are not interesting for each agent $NI_{a_{i}}$, and those attributes that are not interesting for all team members $\overset{M}{\underset{i=1}{\bigcap}}NI_{a_{i}}$. Once this process has finished, the team and the mediator are ready to start the negotiation process.
Of course, the set of attributes handed over by each team member is not controllable by the mediator. It depends on the behavior of each agent. In our model, the set of attributes handed over by each agent depends on a private parameter $\epsilon_{a_{i}}$. The value of such parameter is related to the weight of the different negotiation attributes in one’s own utility function. More precisely, if $\epsilon_{a_{i}}=0$, then the agent is only willing to hand over the decision rights over those attributes that are not interesting for himself (i.e., weight equal to zero in the utility function). When $\epsilon_{a_{i}}=1$, the agent is willing to hand over decision rights over every attribute in the negotiation. In general, the agent is willing to hand over decision rights over attributes whose sum of weight in the utility function is equal to or lower than $\epsilon_{a_{i}}$: $$\label{eq_team_tolerance}
\underset{j \in NI_{a_{i}}}{\sum}w_{a_{i},j} \leq \epsilon_{a_{i}}$$
Given a certain $\epsilon_{a_{i}}$, a reasonable heuristic is to assume that the agent is willing to concede as many decision rights as possible since this will enhance the possibility of finding an agreement with the opponent. Hence, each team member $a_{i}$ chooses the largest possible set $NI_{a_{i}}$ that fulfills Eq. \[eq\_team\_tolerance\]. A simple algorithm that solves this problem is ordering the negotiation attributes in ascending order by weight in the utility function. The set $NI_{a_{i}}$ starts empty, and, then, the array of ordered attributes is followed. If the attribute weight plus the weights of those attributes already in $NI_{a_{i}}$ exceeds $\epsilon_{a_{i}}$, then the search stops. Otherwise, the attributes is added to $NI_{a_{i}}$ and the algorithm continues with the next attribute. Our initial experiments with FUM [@sanchez-anguix12] suggested that team members should set its private $\epsilon_{a_{i}}$ to 0 and hand over decision rights only over those attributes that are not interesting at all. Therefore, in the experimental setting, we use $\epsilon_{a_{i}}=0$.
### Negotiation: observing opponent’s concessions and building an attribute agenda
Once the negotiation starts, the mediator attempts to guess a ranking of attributes according to the opponent’s preferences. This ranking is used to build an agenda of attributes, which is used in the iterated offer building process. The idea behind the agenda is attempting to satisfy team members as much as possible with those attributes that are less important for the opponent. This way, team members may reach their desired aspiration level with those attributes less interesting for the opponent, and use the rest of attributes to make the offer as satisfactory as possible for the opponent. The only information available for the mediator regarding the opponent’s preferences are the offers received. Thus, the mediator has to infer a ranking of attributes according to that information. A possible heuristic is assuming that agents usually concede less in important attributes and greater concessions are performed in lesser important attributes at the first rounds of the negotiation.
Our proposed heuristic assumes that the mediator observes opponent’s offers for the first $k$ interactions. In our experiments, the value of this parameter was set to $k=\lfloor \frac{T_{A}}{4} \rfloor$ [@sanchez-anguix12]. Then, it calculates the concession performed in each attribute. Since our model assumes that the opponent’s utility function employs the opposite type of valuation function than team members for each attribute, it is relatively easy to calculate the amount of concession performed at each attribute. For instance, if the opponent is a seller, it is reasonable to assume that its valuation functions is monotonically increasing (e.g., higher prices report higher utilities) and, thus, any value below the maximum price can be considered a concession with respect to the maximum price. Therefore, the relative concession can be calculated in each attribute. For each attribute $j$, we calculate the total amount of relative concession $C_{j}$ in the first $k$ offers: $$C_{j}=\overset{k-1}{\underset{t=0}{\sum}} \frac{|X(j)_{op\rightarrow A}^{t}-best\_value(j)|}{max\_value(j)-min\_value(j)}
\label{eq:con}$$ where $X(j)_{op\rightarrow A}^{t}$ it the value of attribute $j$ in the offer $X_{op\rightarrow A}^{t}$, $best\_value(j)$ is the best possible value for the opponent in attribute $j$, and $max\_value(j)$ and $min\_value(j)$ are the maximum and minimum value of the attribute in the negotiation domain. The inner part of the summatory determines the relative concession on attribute $j$ in the offer received at interaction/round $t$. So, the summatory counts the total relative concession for attribute $j$ in the first $k$ offers. The heuristic is that attributes that score lower in Equation \[eq:con\] are usually those more important for the opponent, whilst those attributes scoring higher in Equation \[eq:con\] are those less important for the opponent. Based on the available information (i.e., number of rounds up to $k$), the mediator builds an agenda of attributes according to the scores of $C_{j}$ in descending order. This way, lesser important attributes for the opponent are first in the agenda.
### Negotiation: Offer proposal
In order to determine which offer is sent to opponent, the mediator governs an iterated building process. The aim of this iterated process is building an offer, attribute per attribute, so that the offer sent to the opponent is acceptable for every team member. The order in which the attributes are adjusted is determined by the agenda built by the mediator. The first attribute in the agenda is the one considered less important for the opponent, the second attribute is the next lesser important attribute for the opponent, and so forth. Thus, the first attributes set are those less important for the opponent. The heuristic used by this iterated building process is attempting to satisfy team members’ demands with those attributes that are less important for the opponent, and demand as less as possible from those attributes that are the most important for the opponent. Briefly, the iterated building process goes as follows.
1. The agenda of attributes $agenda$ is built by the mediator according to the available information. The first attribute in the agenda is the one guessed as the less important attribute for the opponent.
2. When the iterated process starts, every team member is considered an active member ($a_{i} \in A'$) in the construction process.
3. The initial partial offer $X_{A\rightarrow op}^{'t}$ starts as an offer whose attributes have not been set.
4. The mediator checks those attributes that are not interesting for every team member $\overset{M}{\underset{i=1}{\bigcap}}NI_{a_{i}}$. These attributes are maximized according to the opponent’s preferences (i.e., if the price was one of these attributes, it would be maximized for the opponent, thus, acquiring its minimum value). The partial offer $X_{A\rightarrow op}^{'t}$ is updated with the new attributes’ values.
5. The next attribute $j$ in the agenda is selected. Those team members active in the construction process ($a_{i} \in A'$) and interested in $j$ ($j \notin NI_{a_{i}}$) are asked by the mediator to submit the value $x_{a_{i},j}$ needed of attribute $j$ to get as close as possible to their aspiration levels.
6. The values $x_{a_{i},j}$ gathered from team members are aggregated. If the assumed valuation function is monotonically increasing, then the $max$ operator is used to aggregate the values and obtain the final value for the attribute $x_{j}$. Otherwise, if the assumed valuation function is monotonically decreasing, then the $min$ operator is used to aggregate the values and obtain $x_{j}$.
7. $x_{j}$ is set in $X_{A\rightarrow op}^{'t}$ and the new partial offer is broadcasted among team members. Every team member that is active in the construction phase is asked if the current partial offer satisfies its current demands.
8. Every response is gathered by the mediator. Those agents that answered positively are removed from the list of active agents. If there are still active agents, the mediator goes back to 5.
9. When every team member has been satisfied by the partial offer $X_{A\rightarrow op}^{'t}$, if there are still attributes that have not been set, those attributes are maximized according to the opponent’s preferences. Then, a final offer $X_{A\rightarrow op}^{t}$ is obtained, made public among team members, and sent to the opponent.
In the protocol described above, team members are asked to submit a value for attributes in which they are interested, and to determine whether or not the partial offer satisfies their needs. In both cases, as in previous strategies, we have assumed that team members follow time-based concession tactics similar to the one described in Equation \[eq:time\], where $\beta_{A}$ has been agreed upon by team members prior to the negotiation process. However, since team members may have handed over some decision rights, it is not possible for agents to demand the maximum utility. The value $\epsilon_{a_{i}}$ has to be subtracted from the maximum utility. Therefore, the concession strategy $s_{a_{i}}(t)$, which determines the level of demand at each negotiation round, can be formalized as: $$s_{a_{i}}(t)=(1-\epsilon_{a_{i}})-(1-\epsilon_{a_{i}}-RU_{a_{i}})(\frac{t}{T_{A}})^{\frac{1}{\beta_{A}}}$$
When team members are asked about a value for $j$, each team member communicates anonymously the value $x_{a_{i},j}$. The value communicated is the one that gets as close as possible to its desired aspiration level $s_{a_{i}}(t)$ at round $t$. Taking the linear additive utility function formula, this can be calculated as: $$\label{eq:bid1}
x_{a_{i},j}= \underset{x\in [0,1]}{\mbox{argmin }} (s_{a_{i}}(t)-U_{a_{i}}(X_{A\rightarrow op}^{'t})-w_{a_{i},j}V_{a_{i},j}(x))$$ where $s_{a_{i}}(t)$ is the utility demanded by the agent $a_{i}$ at round $t$, $U_{a_{i}}(X_{A\rightarrow op}^{'t})$ is the utility reported by the current partial offer, and $w_{a_{i},j}V_{a_{i},j}(x)$ is the weighted utility reported by the value demanded by the agent. Since the value demanded looks to be as close as possible to the utility necessary to get to the current aspiration, the function is minimized. However, the following constraint is fulfilled by team members in order to avoid surpassing the utility demanded: $$\label{eq:bid2}
s_{a_{i}}(t)-U_{a_{i}}(X_{A\rightarrow op}^{'t})-w_{a_{i},j}V_{a_{i},j}(x_{a_{i},j})\geq 0$$
As for determining when a partial offer is acceptable, team members follow a similar criterion to the method proposed in other intra-team strategies. Basically, a partial offer is acceptable for an agent $a_{i}$ if it reports a utility that is greater than or equal to the aspiration level marked by its concession strategy: $$\label{eq:accept}
ac'_{a_{i}}(X_{A\rightarrow op}^{'t}) = \left\lbrace
\begin{array}{l l}
true & \mbox{if } U_{a_{i}}(X_{A\rightarrow op}^{'t}) \geq s_{a_{i}}(t) \\
false & \mbox{otherwise}
\end{array}
\right.$$ where $true$ indicates that the partial offer is acceptable at its current state for agent $a_{i}$, and $false$ indicates the opposite.
### Negotiation: Offer acceptance
Since this strategy looks for unanimity regarding team decisions, we employed the same mechanism employed in SBV for determining whether or not an opponent offer is acceptable. When the mediator receives the opponent’s offer $X_{op\rightarrow A}^{t}$, the offer is publicly announced to all of the team members. Then, the mediator opens a private voting process where each team member $a_{i}$ should specify whether or not it supports acceptance of the opponent’s offer $ac_{a_{i}}(X_{op\rightarrow A}^{t})$. The mediator counts the number of positive votes. The offer is accepted if the number of positive votes is equal to the number of team members. Otherwise, the offer is rejected.
Similarly to SBV, an opponent offer is acceptable for a team member at round $t$ if it reports a utility that is greater than or equal to the aspiration level marked by the concession strategy in the next round: $$\label{op_ac}
ac_{a_{i}}(X_{op\rightarrow A}^{t}) = \left\lbrace
\begin{array}{l l}
true & \mbox{if } s_{a_{i}}(t+1) \leq U_{a_{i}}(X_{op\rightarrow A}^{t}) \\
false & \mbox{otherwise}
\end{array}
\right.$$ where $true$ means that the agent supports the opponent’s offer, $false$ has the opposite meaning, and $s_{a_{i}}(.)$ is the concession strategy employed by agent $a_{i}$ to calculate the aspiration level at each negotiation round $t$.
### Unanimity Level
As stated in the introduction of this section, this strategy is capable of guaranteeing unanimity regarding team decisions. How unanimity is guaranteed in the offer acceptance phase is clear, since a voting process with unanimity rule is employed. In [@sanchez-anguix12] we showed how unanimity is also guaranteed in the offer sent to the opponent. More specifically, the strategy is capable of guaranteeing a strict unanimity: for any team member $a_{i}$, the offer sent to the opponent reports a utility that is greater than or equal to its aspiration level $s_{a_{i}}(t)$. This is possible thanks to the iterated building process and the assumptions in team members’ utility functions. Since team members share the same type of monotonic valuation functions, the use of the max/min operator (max for monotonically increasing valuation functions, min for monotonically decreasing functions) ensures that for each attribute, each team member either gets exactly the value demanded for the attribute or it gets a value that reports a utility greater than or equal to the utility they demanded for the attribute. Hence, when team members demand the exact value needed to get as close as possible to their desired utility level, they will always get the same or greater utility than the one they actually demanded. Thus, in the end, the offer will yield a utility that is equal to or greater than their aspiration levels at round $t$.
Experimental Analysis {#sec:exp}
=====================
In this section we study how the four intra-team strategies presented in this article perform under different environmental conditions. First, we introduce the negotiation case employed for our experiments. Then, the environmental conditions and performance measures studied are introduced and explained to the reader. Finally, we describe the experiments carried out, and we analyze the results provided by each intra-team strategy.
Negotiation Case: Group Booking
-------------------------------
The negotiation case employed for our experiments is based on a group booking negotiation with a hotel, which also illustrates the types of applications that can be built using the intra-team strategies proposed in this article. In this scenario, a group of friends who have decided to spend their holidays together has to book accommodation for their stay. Their destination is Rome, and they want to spend a whole week. Each friend is represented by his/her electronic agent, who acts semi-automatically on behalf of its user. This agent has previously elicited the preferences of its user regarding booking conditions. Each group member has different preferences regarding possible booking conditions. Thus, the final agreement with the hotel should satisfy every friend as much as possible. The group of agents engages in a negotiation with a well-known hotel in their city of destination, which is also represented by an electronic agent. During the pre-negotiation, both parties have decided to negotiate the following issues:
- Price per person ($pp$): The price per person is the amount of money that each friend will pay to the hotel for the accommodation service. The issue domain goes from 210\$, which is the minimum rate (30\$ per night), to 700\$, which is the maximum rate (100\$ per night). A realistic assumption in the group of friends is that friends prefer to pay lower prices to higher prices (i.e., monotonically decreasing valuation function), whereas the seller prefers to charge higher prices to lower prices (i.e., monotonically increasing valuation function).
- Cancellation fee per person ($cf$): When a booking is cancelled, the hotel charges a fee to compensate for losses. The issue domain goes from 0\$ (no cancellation fee) to 150\$. A realistic assumption in the group of friends is that friends prefer to pay lower prices to higher prices (i.e., monotonically decreasing valuation function), whereas the seller prefers to charge higher prices to lower prices (i.e., monotonically increasing valuation function).
- Full payment deadline ($pd$): The full payment deadline indicates when the group of friends has to pay the full price booking in order to confirm their reservation. The domain goes from “Today”=0 days (the date time when the final agreement has been signed) to “Departure Date”=30 days, which indicates that the team should only pay when leaving the hotel. A realistic assumption in the group of friends is that friends prefer to pay as late as possible (i.e., monotonically increasing valuation function), whereas the seller prefers to charge as soon as possible (i.e., monotonically decreasing valuation function).
- Discount in bar ($db$): As a token of respect for good clients, the hotel offers nice discounts at the hotel bar. The issue domain goes from 0% (no discount) to 20%. A realistic assumption in the group of friends is that friends prefer higher discounts to lower discounts (i.e., monotonically increasing valuation function), whereas the seller prefers to offer lower discounts prices to higher discounts (i.e., monotonically decreasing valuation function).
Negotiation Environment Conditions & Team Performance
-----------------------------------------------------
We consider that the negotiation environment plays a very important part in team dynamics. It may not be the same using a representative approach in a setting where all of the team members’ preferences are very similar than a setting where team members’ preferences are exactly the opposite. Since conditions of the negotiation environment highly vary depending on the application domain, we decided to focus on those general conditions that are present in almost every negotiation scenario involving negotiation teams: opponent deadline, team deadline, team members’ preference similarity, opponent concession speed, and team size.
Regarding team performance, it is also acknowledged that there are several well known social welfare measures to assess the quality of decisions in a society. A negotiation team can be considered a small society, and, thus, social welfare measures can also be considered appropriate measures for measuring negotiation teams’ performance. More specifically, we study the impact of the negotiation environment on the minimum utility of team members (i.e., egalitarian social welfare [@chevaleyre06]), and the average utility of team members (i.e., a special case of ordered weighted averaging [@chevaleyre06]). However, we do not only restrain our analysis to social welfare measures. Computational measures like the number of negotiation rounds are also analyzed for all of the intra-team strategies.
### Environment Condition: Opponent Deadline Length
One of the issues that can affect the negotiation process is the number of interactions that the opponent has until he decides that negotiating is no longer worthy, namely opponent deadline $T_{op}$. We partitioned the opponent negotiation deadline in three different classes: short deadline $T_{op}=U[5,10]=S$[^1], medium deadline $T_{op}=U[11,29]=M$, and long deadline $T_{op}=U[30,60]=L$.
### Environment Condition: Team Deadline Length
Similarly, the maximum number of rounds that the team has to negotiate also may impact the performance of the different intra-team strategies. As in the case of the opponent deadline, we partitioned the team deadline in three different classes: short deadline $T_{A}=U[5,10]=S$, medium deadline $T_{A}=U[11,29]=M$, and long deadline $T_{A}=U[30,60]=L$.
### Environment Condition: Team Similarity
25 different linear utility functions were randomly generated. These utility functions represented the preferences of potential team members. 25 linear utility functions were generated to represent the preferences of opponents. These utility functions were generated by taking potential teammates’ utility functions and reversing the type of $V_{i}(.)$.
In order to determine the preference diversity in a team, we decided to compare team members’ utility functions. We introduce a dissimilarity measure based on the utility difference between offers. The dissimilarity between two teammates can be measured as follows: $$D(U_{a_{i}}(.),U_{a_{j}}(.))= \frac{\underset{\forall X \in [0,1]^{n}}{\sum} | U_{a_{i}}(X)-U_{a_{j}}(X) |}{|X \in [0,1]^{n}|}$$ If the dissimilarity between two team members is to be measured exactly, it needs to sample all of the possible offers. However, this is not feasible in the current domain where there is an infinite number of offers. Therefore, we limited the number of sampled offers to 1000 per dissimilarity measure. Due to the fact that a team is composed by more than two members, it is necessary to provide a team dissimilarity measure. We define the team dissimilarity measure as the average of the dissimilarity between all of the possible pairs of teammates.
For all of the teams that had been generated, we measured their dissimilarity and calculated the dissimilarity mean $\bar{dt}$ and standard deviation $\sigma$. We used this information to divide the spectrum of negotiation teams according to their diversity. Our design decision was to consider those teams whose dissimilarity was greater than, or equal to $\bar{dt}+1.5\sigma$ as very dissimilar, and those teams whose dissimilarity was lower than, or equal to $\bar{dt}-1.5\sigma$ as very similar. In each case, 100 random negotiation teams were selected for the tests, that is, 100 teams were selected to represent the very similar team case, and 100 teams were selected to represent the very dissimilar team case. These teams participate in the different environmental scenarios, where they are confronted with one random half of all of the possible individual opponents. Therefore, each environmental scenario (complete instantiation of all the environmental conditions) consists of 100$\times$12$\times$4=4800 different negotiations (each negotiation is repeated 4 times to capture stochastic variations in the different intra-team strategies).
### Environment Condition: Opponent Concession Speed
The concession speed of the opponent during the negotiation process $\beta_{op}$ may determine the final quality of the agreement for team members. For instance, if the opponent concedes very quickly towards its reservation utility, better agreements for the team may come earlier in the negotiation process. In those cases, even intra-team strategies that guarantee less degree of unanimity may achieve good results. We divided the family of concession speeds based on the classic classification of time-tactics: we considered that when $\beta_{op}=U[0.1,0.49]=VB$ the concession speed is very boulware, when $\beta_{op}=U[0.5,0.99]=B$ the concession speed is boulware, when $\beta_{op}=U[1,10]=C$ the concession speed is conceder, when $\beta_{op}=U[11,40]=VC$ the concession speed is very conceder. Similarly, when we refer to $\beta_{A}$ (the team concession speed), we will also employ the same partition in boulware (B), very boulware (VB), conceder (C), and very conceder (VC).
### Environment Condition: Number of Team Members
We think that the number of team members may also influence the performance of the different intra-team strategies. Some of the strategies may become too demanding when the number of team members increases and it may result in more negotiations ending in failure. Therefore, we decided to study the effect of the team size on the performance of the different intra-team strategies. The number of team members $|A|$ ranged from 4 to 8. This number of team members is motivated by the negotiation case employed in our experiments. We consider that groups of friends from 4 to 8 persons are reasonable in practice.
### Team Performance: Number of Negotiation Rounds
The number of negotiation rounds considers the number of interactions between the team and the opponent. It is a measure employed to assess the negotiation time employed by the different negotiation strategies to reach a final agreement. In our study, every pair offer/counter-offer in the negotiation thread is considered as a negotiation round. In equal conditions of utility performance, those intra-team strategies that spend less negotiation rounds are preferred since they employ less negotiation time to reach a final agreement.
### Team Performance: Minimum Utility of Team Members
The minimum utility of team members (Min.) in a negotiation represents the utility of the final agreement for the less benefited team member. If the final agreement is $X$ and the team is composed of M different team members $A=\{a_{1}.a_{2},...,a_{M}\}$, the minimum utility of team members can be calculated as: $$Min.(X)=\underset{1\leq i \leq M}{\mbox{min}}U_{a_{i}}(X)$$ In applications where there is a strong bond among team members (i.e., the group of travelling friends), team members may attempt to maximize the minimum utility of team members in order to avoid extremely unsatisfied team members and a degradation of the relationship among team members. Even if a strong bond is not present among team members, an agent may attempt to maximize the minimum utility of team members if it thinks that its own utility is going to be the less favored utility by the final agreement.
### Team Performance: Average Utility of Team Members
If the final agreement is $X$ and the team is composed of M different team members $A=\{a_{1}.a_{2},...,a_{M}\}$, the average utility of team members can be calculated as: $$Ave.(X)=\frac{1}{M}\underset{1\leq i \leq M}{\sum}U_{a_{i}}(X)$$ A less conservative agent may attempt to maximize the average utility of team members if it thinks that its own utility is not going to be the less favored utility by the final agreement.
Results
-------
### Number of Negotiation Rounds
Although we measured the number of negotiation rounds in each experiment, we found that a general pattern was found in almost every experiment. Thus, instead of commenting the results for the number of negotiation rounds in each experimental section, we decided to present the performance of the four intra-team strategies according to the number of negotiation rounds just once. As a sample for this behavior, we can observe the number of negotiation rounds spent by each intra-team strategy when team and opponent have a long deadline ($T_{op}=L$ and $T_{A}=L$), the number of team $|A|$ members is set to 4, and the opponent uses different concessions speeds $\beta_{op}$ in Table \[rounds\].
As long as the concession speed of the four intra-team strategies can be categorized as the same type, *RE is usually the fastest intra-team strategy in number of negotiation rounds, followed by SSV, then SBV, and finally FUM*. Since less unanimity is guaranteed among team members, it is logical that there may be less conflict with the opponent and, thus, agreements are found faster with low unanimity strategies like RE and SSV. The main exception for this rule is when team members are very similar and the opponent uses either boulware or very boulware concession speeds. In those cases, FUM is able to finalize negotiations successfully in fewer rounds than SBV (and sometimes SSV). The learning heuristic employed by FUM benefits from the fact that the opponent usually concedes more in those attributes that are less important and, thus, it is able to infer a proper agenda and propose better offers to the opponent (ending the negotiation faster). This pattern did not exist when team members are very dissimilar, since in that case, FUM also has to deal with more intra-team conflict. This results in more demanding offers to guarantee unanimity.
Additionally, as expected, *as the concession strategy of team members becomes more conceder, the number of negotiation rounds spent is lower*. Thus, RE using $\beta_{A}=VB$ is slower than RE using $\beta_{A}=B$, which is slower than RE using $\beta_{A}=C$, which is slower than RE using $\beta_{A}=VC$.
The number of negotiation rounds spent by each intra-team strategy is especially interesting to select intra-team strategies when they perform equally in utility terms (minimum or average utility). For instance, if SBV and FUM tie in utility terms, a team is suggested to select SBV most of the times due to the fact that it usually requires less negotiation rounds, if SSV and SBV tie in utility terms, the team should select SSV since it usually requires less rounds than SBV, and so forth.
### Same Type of Deadlines {#parexp1}
The next set of experiments that we conducted consisted in assessing which intra-team strategies work better when both parties have the same type of deadline. More specifically, we chose those scenarios where both parties have short deadlines or long deadlines. Additionally, for each type of deadline, we simulated scenarios where team members were either very dissimilar, or very similar, and gathered information about the minimum and average utility of team members regarding each possible strategy configuration (team concession speeds, intra-team strategies, opponent concession speeds, etc.). The number of team members remained static at $|A|=4$.
The results for this experiment can be found in Table \[exp1\]. It shows the average minimum utility of team members (Min.), the average of the average utility of team members (Ave.), and the average number of rounds (Ro.). It only shows the results for intra-team strategies using a Boulware concession speed since we found that this concession speed worked better than the rest of concession speeds.
When both parties have a short deadline (first and third sub-table in Table \[exp1\]), independently of team similarity, SBV $\beta=B$ and FUM $\beta=B$ are usually the best options for the minimum utility. The unanimity and semi-unanimity rules employed by this strategy make possible for the worst affected team member to ensure that its situation is better than with other strategies. As for the average utility of team members, FUM $\beta=B$ usually is the best option. The only exception for this pattern is when the opponent uses conceder strategies ($\beta_{op}=VC$ or $\beta_{op}=C$). In that case, all of the strategies perform similarly, especially when team members are very similar. For instance, we can observe that RE, SSV, SBV $\beta=B$ are the best option for the average utility of team members when the deadline is short, team members are very similar, and the opponent uses a very conceder strategy. In the same setting, but with the opponent using a conceder strategy, FUM is statistically better but the differences are not very important (less than a 1.8%).
However, *when both parties have a long deadline to negotiate (subtables 2 and 4 in Table \[exp1\]), FUM $\beta=B$ becomes the best choice for the minimum and average utility of team members in almost every scenario*. The only exceptions for this superiority are, again, scenarios where the opponent employs conceder strategies. For instance, when the deadline is long, team members are very dissimilar, and the opponent uses a very conceder strategy, SBV $\beta=B$ is the best intra-team strategy for the minimum and average utility of team members.
We can also observe that *RE and SSV are specially affected by very dissimilar preferences’ scenarios*. When team members are very similar, both strategies are capable of being close to SBV and FUM in the minimum and average utility of team members as long as the opponent plays conceder strategies. However, both intra-team strategies’ results get further from those of SBV and FUM when team members are very dissimilar. These intra-team strategies are not able to tackle situations where team members have very dissimilar preferences due to the type of decision rule applied, and their use in such situations is discouraged.
The reason why several strategies perform similarly in utility terms when the opponent plays conceder strategies is simple: Since the opponent concedes very fast in the first rounds of the negotiation process, as long as the team does not concede very fast (i.e., boulware strategy), all of the strategies are capable of finding a reasonable good agreement in the first rounds by letting the opponent concede and then accepting the opponent’s offer. However, there is an additional reading that explains why strategies like FUM, which guarantees unanimity regarding team decisions, does not perform so well when the opponent uses conceder strategies. FUM relies on the assumption that the opponent concedes very little in those attributes that are important for its interests at the first rounds. However, when the concession strategy carried out by the opponent is conceder or very conceder (a more acute effect) big concessions are usually carried out at the first rounds. Thus, FUM is not able to infer an appropriate agenda. In [@sanchez-anguix12], it was shown that as the agenda gets further from the real ranking of opponent preferences, the more demanding becomes the strategy. This may have a negative effect in the negotiation, since more negotiations may end in failure due to the high demands of the team. In fact a slight effect is observed in the results: when the opponent uses a boulware strategy, the percentage of successful negotiations is $94.6\%$ which is greater than the $92.6\%$ obtained when the opponent uses a conceder strategy and the $93.1\%$ obtained when the opponent uses a very conceder strategy.
Another issue found in the results is the difference between FUM and other strategies when the deadline is long. FUM tends to obtain better results when the deadline is long for both parties. The differences with the other intra-team strategies become greater when compared with the short deadline scenario. The reason for this phenomenon is similar to the reason mentioned in the paragraph above. FUM is a strategy that relies on the information gathered in the negotiation process. Thus, when interactions are lesser, like when deadlines are short, the agenda inferred by the trusted mediator is less close to the ideal agenda. When the agenda deviates from the ideal agenda, offers proposed by the team are more demanding and less probable to be accepted by the opponent. As a matter of fact, the reader can notice that the difference on average between FUM $\beta=B$ in long deadline scenarios (aggregating all of the scenarios where the deadline is long) and the results obtained by FUM $\beta=B$ in short deadline (aggregating the scenarios where the deadline is short) scenarios counterpart is $7.9\%$, whereas it is $5.3\%$ for SBV, $5.4\%$ for SSV and $5.8\%$ for RE. Logically, every intra-team strategy benefits from having a longer deadline, but the results suggest that FUM benefits more than the rest of intra-team strategies due to its learning heuristic, which is based on the amount of information.
### Different Types of Deadlines
The next experiment consisted in studying the behavior of the different intra-team strategies when both parties have different types of deadline. Thus, in this case, one of the two parties has a deadline which is lower than the deadline of the other party. Clearly, the party with a lower deadline is at disadvantage with respect to the other party since it has fewer offers to send before ending the negotiation, and the pressure to accept the opponent’s offers arises earlier.
#### Short Team Deadline and Long Opponent Deadline
First, we start by analyzing the case where the deadline of the team is shorter (short deadline) than the deadline of the opponent party (long deadline). Hence, $T_{A}=U[5,10]$ and $T_{op}=U[30,60]$. The results of this experiment can be found in Table \[exp2\].
In this case, the team has a shorter deadline and, thus, it should be at disadvantage with respect to the opponent. However, we can observe that when the opponent uses a conceder or very conceder strategy, the results are similar to the analogous case where both parties had a short deadline. These results can be explained due to the fact that since the opponent concedes very quickly, a good deal can be found for the team in the first rounds of the negotiation process and the team is not affected by the fact that its deadline is shorter. Nevertheless, as the opponent moves towards Boulware strategies, there is a clear negative effect on the minimum and average utility of team members: *all of the strategies are affected by the fact that the team has a shorter deadline*. In the scenario where both parties have a short deadline (see Table \[exp1\]), the average for the average utility of team members in conceder settings (aggregating those negotiations where $\beta_{op}=C$ or $\beta_{op}=VC$) is 0.67, and the average for the average utility of team members in boulware settings (aggregating those negotiations where $\beta_{op}=B$ or $\beta_{op}=VB$) is 0.45. Thus, the average utility for team members is reduced a 22%. In this experiment (see Table \[exp2\]), the average of the average utility of team members in conceder settings is 0.63, whereas the average of the average utility of team members in boulware settings is 0.10. Therefore, the average utility of team members is reduced a 53%, approximately doubling the difference found in the case where both parties had a short deadline.
When *team members are very similar* (upper sub-table in Table \[exp2\]), it can be observed that, as in the scenario where both parties have a short deadline and team members are very similar, several strategies perform very similarly. The main difference resides in the fact that the only strategy capable of reaching similar results to FUM $\beta=B$ in the minimum and average utility is RE $\beta=B$. Differently to the case when team members are very similar and the deadline for both parties is short, the RE $\beta_{A}=B$ strategy is capable of achieving similar results to the other intra-team strategies even in less conceding settings ($\beta_{op}=C$, $\beta_{op}=B$, and $\beta_{op}=VB$). These results suggest that, *despite not assuring any minimum level of unanimity, employing a representative with a reasonably slow concession (boulware) leads to good results compared with those obtained by other intra-team strategies*. A closer look at the experiments threw some light over these results. For instance, when $\beta_{op}=B$, the number of successful negotiations was 2695 for RE $\beta_{A}=B$, 1925 for FUM $\beta_{A}=B$, 1855 for UBS $\beta_{A}=B$, and 2394 for SSV $\beta_{A}=B$. The average utility for successful negotiations was 0.32 for RE $\beta_{A}=B$, 0.34 for SSV $\beta_{A}=B$, 0.39 for SBV $\beta_{A}=B$, and 0.42 for FUM $\beta_{A}=B$. Hence, despite obtaining less quality results in successful negotiations, the representative approach becomes a good option for these scenarios because it leads to a great number of negotiations ending in success where other intra-team strategies fail to succeed (utility=0). SSV, UBS, and FUM need more interactions to find a satisfactory deal, but when they find it, it is better in utility terms. However, in average, a representative approach may be more adequate for settings where the team has a shorter deadline than the opponent.
As for the scenario where *team members are very dissimilar* (lower sub-table in Table \[exp2\]), we can observe that the negative effect produced by having a shorter deadline is especially acute when the opponent uses boulware or very boulware concessions. The dissimilarities between team members, and the fact that there are very few interactions to find a deal that satisfies both team and opponent, contribute to a strong reduction in the minimum and the average utility of team members. In terms of the minimum utility of team members, $FUM$ and $SBV$ $\beta_{A}=B$ work better when the opponent uses conceder or very conceder concessions. However, *almost every intra-team strategy performs equally bad in terms of the minimum utility of team members when the opponent moves towards boulware concessions* (especially in the very boulware case). In this case, the representative approach can no longer compete with the rest of strategies in terms of utility in most scenarios. Nevertheless, despite team members being very dissimilar and RE not guaranteeing any unanimity regarding team decisions, RE performs slightly better than the rest in terms of the average utility of team members when the opponents concedes using boulware. The explanation to this phenomenon is similar to the case where team members were very similar: a lesser number of negotiations end in failure (26% failures for RE, 33% for SSV, 48% for SBV, and 46% for FUM), which compensates for the dissimilarity between team members’ preferences and the unanimity level guaranteed by RE. In any case, the utility obtained for team members is so low in the average and minimum utility of team members that, in some cases, *it may even be better not to negotiate with such kind of opponent and spend computational resources in looking for another alternative*.
#### Long Team Deadline and Short Opponent Deadline
In this case, the team has an advantage over the opponent since its maximum deadline is longer than the opponent’s deadline. The goal of these experiments is to determine the combination of intra-team strategies and negotiation parameters that maximize the different social welfare measures employed. Thus, if the team has a maximum deadline equal to the uniform distribution $T_{A}=U[30,60]$, the team may decide to play (prior to the negotiation) a different class of deadline like a medium deadline ($T_{A}=U[11,29]$) or a short deadline ($T_{A}=U[5,10]$) if the results of the simulation suggest that better results are obtained by not playing the maximum deadline. Thus, we also show the results for teams that play a medium deadline, and teams that play a short deadline. In this experiment, the opponent plays a short deadline $T_{op}=U[5,10]$. The results of this experiment for the very similar scenario can be observed in Fig. \[exp:more:similar\], whereas the results for the very dissimilar scenario can be observed in Fig. \[exp:more:dissimilar\].
We start by analyzing the results for scenarios where team members are very similar (Fig. \[exp:more:similar\]). We can observe that for situations where the opponent is very conceder, the team benefits from playing strategies with the same deadline. Since the opponent concedes very fast in the first negotiation rounds, the best deals for the team may be proposed in the first negotiation rounds. Playing a longer deadline may be risky since the team may have extremely high aspirations during the whole negotiation, which results in most offers being rejected and ending the negotiation in failure. As a matter of fact, the number of successful negotiations for intra-team strategies playing a short deadline and boulware concession was 95.1%, 68% for medium deadline and boulware concession, and 45% for long deadline and boulware concession, 29% for medium deadline and very boulware concession, and 14% for long deadline and very boulware concession. Other configurations may have a higher number of successful negotiations, but they are not able to retain as much utility as the boulware configuration. As the opponent starts to move towards strategies that concede more slowly, the best intra-team strategies for the team are those played with a medium deadline and boulware strategy (RE, SBV and SSV $\beta=B$). In those cases, the opponent may not propose the best deals for the team until its last negotiation rounds. Thus, playing a slightly longer deadline with a boulware concession comes at an advantage for the team since the team does not fully concede in the whole negotiation and still accepts last opponent’s offers. Some strategies played with a medium deadline like FUM $\beta=B$ are still too demanding, end up in more negotiation failures, and have very little information to learn the opponents’ preferences.
The very dissimilar scenario (Fig. \[exp:more:dissimilar\]) is a little bit different. In this scenario, the team needs to deal with strong divergences in their preferences too. Thus, teams are prone to be more demanding in order to accommodate the preferences of as many team members as possible. We can observe that for cases where the opponent uses conceder strategies, the team should play boulware strategies with the same deadline. Similarly to the very similar scenario, playing a longer deadline is risky since it results in extremely high aspirations and most offers being rejected. However, in the very dissimilar scenario, the transition from selecting short deadline strategies to selecting medium deadline strategies does not appear until the opponent uses boulware strategies. This may be explained precisely due to the dissimilarity among team members, which requires stronger demands that are not met when playing medium deadline. As the opponent starts to concede using boulware strategies, the best intra-team strategies are usually found in the medium deadline, as in the very similar scenario case.
In conclusion, in this experiment we have observed that, generally, *even though the team is able to play a long deadline and the opponent plays a short deadline, the team would benefit more from playing the same type of deadline than the opponent or a slightly longer deadline*.
### Team size effect on intra-team strategies
We also decided to analyze the effect of the team size on the performance of the different intra-team strategies. Thus, we repeated the conditions in \[parexp1\] increasing the number of team members. However, we only analyzed intra-team strategies whose $\beta_{A}=B$ since they were those one that obtained better results in Table \[exp1\]. We excluded the RE strategy from the analysis. Since team members do not interact in RE and no unanimity level is guaranteed, the inclusion of additional team members should not affect the way in which the strategy works. The results of this experiment can be found in Figure \[exp:teamsize\]. It shows the average and minimum utility of team members for teams of size $|A|=\{4,5,6,7,8\}$.
Generally, it can be observed in all of the graphics in Figure \[exp:teamsize\] that, as the number of team members increases, *the quality of the results in terms of the minimum and the average utility is reduced*. This behavior was expected since as the number of agents increases, the set of possible agreements is reduced and the conflict inside the team and with the opponent is increased. However, the reduction in utility terms can be appreciated more easily in the minimum utility of team members. The average for the average utility of team members when $|A|=4$ is 0.70 (aggregating all other factors) and 0.67 for $|A|=8$ (aggregating all other factors), whereas the average for the minimum utility of team members when $|A|=4$ is 0.48 and 0.41 for $|A|=8$. As the number of team members increases, the contribution of each team member to the average utility is lesser, and that is the reason why the negative effect of team size on utility measures can be observed more easily in the minimum utility of team members than in the average utility of team members.
We expected that as the number of team members increased, the performance of unanimity intra-team strategies like FUM would greatly decrease compared to the performance of SSV since more team members would increase the demands of the team and make offers less interesting for the opponent. However, the difference in performance between the three strategies is approximately maintained in almost every graphic as the number of team members increases. Therefore, *team size did not have a different effect on the performance of the three intra-team strategies, affecting all of intra-team strategies equally*. The decision on which intra-team strategy should be chosen is not affected by team size.
The only clear exceptions to this rule are scenarios where the opponent uses conceder strategies ($\beta_{op}=C$ and $\beta_{op}=VC$) and team members’ preferences are very dissimilar (first two graphics in rows 3 and 4, Figure \[exp:teamsize\]). In these scenarios, we can observe that there is a special negative effect of team size on the performance (mininum utility and specially in the average utility) of FUM with respect to the other intra-team strategies, which results in FUM being one of the worst choices when the number of team members in large (e.g., second graphics in rows 3 and 4, Figure \[exp:teamsize\]). As a numeric example of the reduction in the performance of FUM , the difference in the average utility between SBV and FUM goes from approximately a $2\%$ ($|A|=4$) to $10\%$ ($|A|=8$) when $\beta_{op}=VC$ and the deadline is short, from approximately a $0\%$ ($|A|=4$) to $5\%$ ($|A|=8$) when the deadline is short and $\beta_{op}=C$, and from $3\%$ ($|A|=4$) to $8\%$ ($|A|=8$) when the deadline is long and $\beta_{op}=VC$ . This phenomenon has a reasonable explanation. When the opponent uses conceder strategies, FUM has greater difficulties to learn a proper attribute agenda. If the number of team members increases and they are very dissimilar, the demands of team members increase, which summed up to the fact that the agenda does not properly reflect the preferences of the opponent, results in demanding team proposals.
Related Work {#sec:related}
============
Multi-agent systems have gained a growing interest as the infrastructure necessary for the next generation of distributed systems. Due to the inherent conflict among agents, techniques that allow agents to solve their conflicts and cooperate are needed. This need is what has given birth to a group of technologies which have recently been referred to as agreement technologies [@luck08; @sierra11]. Trust and reputation [@sabater05; @such11; @such12], norms [@dignum99; @criado11], agent organizations [@horling04; @esparcia11; @silva12], argumentation [@rahwan03; @pajares11] and automated negotiation [@jennings01; @sanchez-anguix11b] are part of the core that makes up this new family of technologies.
Even though agreement technologies are a novel topic in the community of agent research, some of its core technologies like automated negotiation have been studied by scholars for a few years. In definition, automated negotiation is a process carried out between two or more parties in order to reach an agreement by means of exchange of proposals. Two different research trends can be distinguished in automated negotiation models. The first type of model aims to calculate the optimum strategy given certain information about the opponent and the negotiation environment [@serrano03; @digiunta06; @fatima06]. The second type of model encloses heuristics that do not calculate the optimum strategy but obtain results that aim to be as close to the optimum as possible [@faratin98; @jonker01; @faratin02; @lai08]. These models assume imperfect knowledge about the opponent and the environment, and aim to be computationally tractable while obtaining good results. This present work can be classified into the latter type of models.
In multi-agent systems, most of the research has concentrated on bilateral models where each party is a single individual. The present article studies bilateral negotiations where at least one of the parties is a negotiation team, composed by more than a single individual. It should be noted that the problem of finding an agreement for a negotiation team is inherently complex since it not only requires finding an agreement with the other party but it also entails reaching some type of unanimity within the team. Even though communications with the opponent party may be similar to classical bilateral models, negotiation teams may require an additional level of negotiation that involves team members. Thus, classical bilateral models cannot be applied directly if a certain level of unanimity regarding team decisions is necessary . As far as we know, our previous work [@sanchez-anguix10; @sanchez-anguix11; @sanchez-anguix12; @sanchez-anguix12b] is the only work that focuses on negotiation teams.
In [@sanchez-anguix10] we introduced the topic of negotiation teams in agent research from a descriptive perspective. We analyzed the different phases necessary for an agent-based negotiation team to face such negotiations with success. Apart from the phases that we identified, we also described the current technologies that may be appropriate for the development of such phases. Later, we introduced our first experimental study [@sanchez-anguix11] comparing intra-team strategies in different negotiation environments. That paper should be considered the preliminary basis for our current analysis. We have introduced changes in the intra-team strategies, and our current study applies a more fine-grained analysis of the negotiation environment and its possible scenarios. Additionally, we also studied the properties of the Full Unanimity Mediated intra-team strategy in [@sanchez-anguix12]. However, a thorough analysis of how environmental conditions affect team performance was not carried out. Finally, we should also highlight our work regarding the study of cultural factors in negotiation teams [@sanchez-anguix12b]. The setting is different to the current article. We attempted to propose a computational model for explaining how human cultural factors affect team dynamics in negotiation teams composed by humans. In this present article we do not consider humans but automated agents. Therefore, human factors are not relevant to the present study.
Apart from agent-based negotiation teams, bilateral negotiation is perhaps the most similar topic to our current research. Hence, we describe some of the most important bilateral models that assume imperfect knowledge. Faratin et al. [@faratin98] propose a bilateral negotiation model for service negotiation where agents apply and mix different concession tactics (i.e., time-dependent, imitative and resource-dependent). In their work, they analyze the impact of the model’s parameters and determine which configurations work better in different scenarios by means of experiments. Our proposed work also assumes the use of time-dependent concession tactics for the calculation of agents’ aspirations at each negotiation round. Additionally, we also take an experimental approach to validate the impact of our model’s parameters. Later, the authors proposed a bilateral negotiation model [@faratin02] whose main novelty was the use of trade-offs to improve agreements between two parties. A trade-off consists of reducing the utility obtained from some negotiation issues with the goal of obtaining the same exact utility from other negotiation issues. The rationale behind trade-offs is to make the offer more likable for the opponent while maintaining the same level of satisfaction for the proposing agent. For that purpose, the authors propose a fuzzy similarity heuristic that proposes the most similar offer to the last offer received from the opponent. Some of our intra-team strategies like Similarity Simple Voting and Similarity Borda Voting also employ similarity heuristics to attempt to satisfy team members’ preferences and the opponent’s preferences.
Jonker and Treur propose the Agent-Based Market Place (ABMP) model [@jonker01] where agents, engage in bilateral negotiations. ABMP is a negotiation model where proposed bids are concessions to previous bids. The amount of concession is regulated by the concession factor (i.e., reservation utility), the negotiation speed, the acceptable utility gap (maximal difference between the target utility and the utility of an offer that is acceptable), and the impatience factor (which governs the probability of the agent leaving the negotiation process).
Lai et al. [@lai08] propose a decentralized bilateral negotiation model where agents are allowed to propose up to $k$ different offers at each negotiation round. Offers are proposed from the current iso-utility curve according to a similarity mechanism that selects the most similar offer to the last offer received from the opponent. The selected similarity heuristic is the Euclidean distance since it is general and does not require domain-specific knowledge and information regarding the opponent’s utility function. Results showed that the strategy is capable of reaching agreements that are very close to the Pareto Frontier. Sanchez-Anguix et al. [@sanchez-anguix11b] proposed an enhancement for this strategy in environments where computational resources are very limited and utility functions are complex. It relies on genetic algorithms to sample offers that are interesting for the agent itself and creates new offers during the negotiation process that are interesting for both parties. Results showed that the model is capable of obtaining statistically equivalent results to similar models that had the full iso-utility curve sampled, while being computationally more tractable. As commented above, some of our intra-team strategies use similarity heuristics to satisfy team members’ preferences and the opponent’s preferences.
Another topic that resembles team negotiations are multi-party negotiations. Several works have been proposed in the literature along this line [@ehtamo01; @klein03; @ito10]. For instance, Ehtamo et al. [@ehtamo01] propose a mediated multi-party negotiation protocol which looks for joint gains in an iterated way. The algorithm starts from a tentative agreement and moves in a direction according to what the agents prefer regarding some offers’ comparison. Results showed that the algorithm converges quickly to Pareto optimal points. Klein et al. [@klein03] propose a mediated negotiation model which can be extended to multiple parties. Their main goal is to provide solutions for negotiation processes that use complex utility functions to model agents’ preferences. The negotiation attributes are not independent. Therefore, preference spaces cannot be explored as easily as in the linear case. Later, Ito et al. [@ito10] proposed different types of utility functions (cube and cone constraints) and multiparty negotiation models for such utility functions. The main difference between our work and multi-party negotiations lies in the nature of the conflict and how protocols are devised. Even though each team member could be viewed as a participant in a multi-party negotiation with the opponent, it is natural to think that team members’ preferences are more similar (e.g., a team of buyers, a group of friends, etc.) and they trust other teammates more than the opponent (i.e., they may share more information). Furthermore, multi-party negotiation models may be unfair for agents that are alien to the team if the number of team members exceeds the number of other participants. In that case, multi-party models may be inclined to move the negotiation towards agreements that maximize the preferences of most participants (i.e., team members).
Multi-agent teamwork is also a close research area. Agent teams have been proposed for a variety of tasks such as Robocup [@stone99], rescue tasks [@kitano01], and transportation tasks [@jennings95]. However, as far as we know, there is no published work that considers teams of agents negotiating with an opponent. Most works in agent teamwork consider fully cooperative agents that work to maximize shared goals. The team negotiation setting is different since, even though team members share a common interest related to the negotiation, there may be competition among team members to maximize one’s own preferences.
Conclusions and Future Work {#sec:conclusions}
===========================
An agent-based negotiation team is a group of two or more interdependent agents that join together as a single negotiation party because they share some common interests in the negotiation at hand. Intra-team strategies govern which decisions are taken by the negotiation team, and how and when these decisions are taken. The goal of this article is studying how environmental conditions affect the performance of different intra-team strategies for a team negotiating with an opponent. We studied how the deadline of both parties, the concession speed of the opponent, similarity among team members’ preferences and team size affect the performance of Representative (RE) intra-team strategy, Similarity Simple Voting (SSV) intra-team strategy, Similarity Borda Voting (SBV) intra-team strategy and Full Unanimity Mediated (FUM) intra-team strategy in terms of the minimum utility of team members, the average utility of team members and the number of negotiation rounds. *The results suggest that depending on the environmental conditions and the team performance metric, team members should select different intra-team strategies*, which confirms our initial hypothesis in this article. Next, we summarize some of the most important results found in this paper:
- Generally, when the concession speed is the same for the different intra-team strategies, RE takes less numbers of negotiation rounds than SSV, which takes less number of rounds than SBV, which takes less number of rounds than FUM. The exception for this rule is when team members are very similar and the opponent uses boulware or very boulware strategies, which makes FUM usually faster than SBV.
- FUM tends to clearly outperform the rest of intra-team strategies studied in utility terms (minimum and average utility of team members) when the deadline of both parties is long and the opponent uses either boulware of very boulware concession strategies. When the opponent uses conceder or very conceder strategies, different intra-team strategies tie in terms of the minimum and average utility of team members depending on the rest of environmental conditions.
- When the team deadline is way shorter than the opponent’s deadline, all of the intra-team strategies are negatively affected in the results obtained in the minimum and average utility of team members. Additionally, if team members are very similar, RE becomes one of the best choices for the average utility of team members since it is capable of ending more negotiations successfully where other intra-team strategies fail. If team members are very dissimilar, FUM and SBV tend to work better in terms of utility (minimum and average). However, if the opponent uses boulware or very boulware concession strategies every intra-team strategy performs equally bad and team members are encouraged to look for other negotiation alternatives.
- In situations where the team’s maximum deadline is longer than the opponent’s deadline, the team should not play intra-team strategies with the maximum deadline but intra-team strategies with the same type of deadline than the opponent or a slightly longer type of deadline. Otherwise, the team performance in utility terms is not maximized due to more negotiations ending in failure.
- As the number of team members increases, the performance in utility terms of all of the intra-team strategies is negatively affected. However, in general, all of the intra-team strategies studied are equally affected by the increment in the number of team members. Thus, team size did not have an effect on the intra-team strategy that should be selected by team members to maximize the minimum or the average utility of team members.
The field of negotiation teams is novel in the area of multi-agent systems. Therefore, there is much work to be done in order to advance the state-of-the-art. Current works in agent-based negotiation teams [@sanchez-anguix10; @sanchez-anguix11; @sanchez-anguix12; @sanchez-anguix12b] have focused on negotiation processes where the team has a strong potential for cooperation since team members share the same type of monotonic valuation function for negotiation issues. However, it is possible to assume that in some negotiation scenarios there is more conflict among team members since valuation functions may be of a different type of monotonic function among team members, or the valuation function itself is not predictable in the negotiation domain (e.g., colors, brands, etc.). Our current future work involves designing intra-team strategies that are able to tackle negotiation domains where attribute’s valuation functions may be unpredictable. RE, SSV and SBV are able to handle such types of domains by definition. However, FUM, which is the strategy capable of guaranteeing unanimity regarding team decisions, is not capable of handling domains where attributes are unpredictable (due to the max/min aggregation operator). Hence, our future works consists in proposing an intra-team strategy capable of guaranteeing unanimity for negotiation domains where attributes may not be predictable.
On the other hand, since the results of this present article have shown that environmental conditions do affect the performance of intra-team strategies, we plan to propose a mechanism that allows team members to infer the most probable state of the negotiation environment, and according to that information, advise the use of an appropriate intra-team strategy.
Finally, in our current work we assume that team members have the same knowledge about the negotiation domain and they have the same skills. It may be interesting to study scenarios where team members have different knowledge and skills.
Acknowledgements {#acknowledgements .unnumbered}
================
This work is supported by TIN2011-27652-C03-01, TIN2009-13839-C03-01, CSD2007-00022 of the Spanish government, and FPU grant AP2008-00600 awarded to Víctor Sánchez-Anguix. We would also like to thank anonymous reviewers and assistants of AAMAS 2011 who helped us to improve our previous work, making this present work possible.
[^1]: U\[5,10\] is a uniform distribution from 5 to 10
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The properties of liquid helium have always been a fascinating subject to scientists. The phonon theory of liquids taking into account liquid non-static shear rigidity is employed here for studying internal energy and heat capacity of compressed liquid $^{4}$He. We demonstrate good agreement of calculated and experimental heat capacity of liquid helium at elevated pressures and supercritical temperatures. Unexpectedly helium remains a quantum liquid at elevated pressures for a wide range of temperature supporting both longitudinal and transverse-like phonon excitations. We have found that in the very wide pressure range 5 MPa-500 MPa liquid helium near melting temperature is both solid-like and quantum.'
address:
- '$^1$ School of Physics and Astronomy, Queen Mary University of London, Mile End Road, London, E1 4NS, UK'
- '$^2$ Institute for High Pressure Physics, RAS, 142190, Troitsk, Moscow Region, Russia'
- $^3$ South East Physics Network
author:
- 'D. Bolmatov$^{1}$'
- 'V. V. Brazhkin$^{2}$'
- 'K. Trachenko$^{1,3}$'
title: 'Helium at elevated pressures: Quantum liquid with non-static shear rigidity'
---
Introduction
============
This Letter is concerned with the thermodynamic functions of liquid $^{4}$He and their possible relationships to those of its crystalline counterpart. Compared with crystalline and gas phases the statistical description of the structure and thermal properties of liquids remains relatively incomplete. At low pressures, matter commonly exists either as a dense solid or as a dilute vapour. For each of these states there is a model which is a plausible approximation to reality and which constitutes a basis for detailed theoretical extension. These are the ideal periodic arrangements for crystalline structures and the ideal gas proposition, respectively: in the former, emphasis is placed on structural order modified slightly by zero-point or thermal motion of the atoms while (save perhaps for the ultra-cold gases) the latter model describes the thermal motion of the atoms on the basis of random atomic displacements and associated momenta. But as is well known there is a third state of dense matter, the ubiquitous liquid state [@gund; @coop; @elm; @bryk; @skinner; @pet]. This state occurs over a temperature range that separates the regions occupied by the solid and vapour states. However, the problem of formulating a rigorous mathematical description of the molecular motions in liquids has always been regarded as much more difficult than that of the kinetic theory of gases or collective displacements of crystalline solids [@widom; @bar; @chan; @ashcroft; @cao; @chong; @skir; @anisimov; @bryk1; @bolm]. Approximation, if not judicious, can lead to a description of either high-density gases or of disordered high-temperature solids. Indeed, at one time considerable effort was devoted to the representation of liquids in these terms, but it is now known that liquids do not have a simple interpolated status between gases and solids, although similarities to the properties of both adjacent phases can certainly be observed.
The ability of liquids and solids to form free bounding surfaces obviously distinguish them from gases. The coefficients of self-diffusion of liquids ( $10^{-5}$ $cm^2$ $s^{-1}$ ) and solids ( $10^{-9}$ $cm^2$ $s^{-1}$ ) are orders of magnitude below those of gases. And the viscosities of gases and liquids are some thirteen orders of magnitude lower than those of solids, and this we may easily understand in terms of the molecular processes of momentum exchange. In terms of vibrational states liquids differ from solids because they cannot support static shear stress. However, as will be seen below liquids support shear stress at high frequency. Flow in a solid arises primarily from rupturing of bonds and the propagation of dislocations and imperfections. In a liquid flow is characterized by both configurational and kinetic processes, whilst in a gas the flow is understood purely in terms of kinetic transport. In this very limited sense liquids may have a minor partial interpolated status between gases and solids.
The concept of elasticity and viscosity in liquids merit clarifications. Which property dominates, and what values of the associated parameters are assumed, depends on the stress and duration of application of that stress. If we apply a stress over a very wide spectrum of time, or of frequency, we are able to observe liquid-like properties in solids and solid-like properties in liquids. Frenkel [@frenkel] introduced a relaxation time $\tau$ as the average time between two consecutive local structural rearrangements in a liquid. If $\tau$ is small compared with an observation time it will yield to the process of liquid flow. In this macroscopic hydrodynamic picture, we now have a rather good understanding of most of the fundamental processes operating on such time-scales in liquids, including quantum liquids such as liquid helium.
The investigation of the properties of helium has been one of the most prolific endeavors since its discovery (in 1868) [@temp; @prigogin; @wood]. For most systems the solid phase is the state of lowest energy at one atmosphere. To date liquid helium is the sole exception in this respect: below $1.70^{\circ}$ K liquid helium has a lower free energy than that of solid helium [@blondon; @simon]. Liquid helium is a quantum liquid at low temperatures and quantum effects play a crucial role. Helium is also of considerable practical significance which is related to the rapidly growing industry surrounding the various applications of superconductivity these often relying on liquid helium as a coolant or refrigerant. Considerable interest is still focused on the highly unusual properties of helium especially at low temperatures and the study of the properties of liquid $^{4}$He continues to be an active area of condensed matter research .
Theory has been long drawn to study the condensed isotopes of helium and their mixtures because these liquids are model many-body systems but with fundamental quantum-statistical differences; they are fertile proving grounds for various quantum many-body formalisms. As Landau emphasized [@mandau], these systems are amenable to theoretical attack because, when studied at relatively low temperatures, they are only weakly excited from their ground states. A description in terms of weakly interacting elementary excitations is then appropriate. The heat capacity of liquid helium has been discussed on just such a basis of elementary excitations and to describe it below the critical temperature Landau suggested two classes of elementary excitations, phonons or quanta of longitudinal compressional waves and rotons [@mandau]. The latter are still not completely well understood. The consideration of non-static shear phonon contributions to the energy spectrum of liquid helium has been largely ignored simply because it was not clear whether liquids are actually capable of supporting transverse or shear modes. But according to Frenkel’s proposition a liquid should support transverse modes provided the frequency satisfies (frequency $\omega>\frac{1}{\tau}$, see below).
Nevertheless the idea that at least longitudinal phonons could be excited below $0.5$ K has traditionally been taken as the explanation for the observation that the heat capacity of liquid helium varies as $T^{3}$ [@feynman]. In most cases investigations have been limited to narrow ranges of temperature and pressure these being of immediate concern in early experiments. Therefore, it is some interest to examine the thermodynamic properties of liquid helium both at elevated pressures, but still presenting a liquid phase, and also for wider ranges of temperature.
Accordingly in this paper we present the heat capacity of bosonic liquid helium as determined within in the framework of a phonon theory of liquids. The physical picture of elementary excitations is clarified by means of a study of phonon contributions, both longitudinal-like and transverse-like, to the energy spectrum of liquid helium at elevated pressures. From the analysis of experimental data of viscosity and heat capacity [@NIST] and theoretical calculations we find that liquid helium remains quantum at elevated pressures for a broad temperature range.
The phonon theory of liquids
============================
The fact that the solid-like value of heat capacity in liquids at the melting point may be summarized by $C_{V}=3 N k_{B}$, and the fundamental observation that liquids retain the property of fluidity leaves us with an apparent contradiction. In order to reconcile it, and as noted above, J. Frenkel introduced the average time between two consecutive atomic jumps thus providing a microscopic description of Maxwell’s phenomenological visco-elastic theory of liquid flow [@maxwell]. If $\tau$ is large compared with the period of atomic vibrations, a liquid is characterized by vibrational states as in a solid (solid glass) including shear modes with frequency $\omega>\frac{1}{\tau}$. If $\tau$ is small compared with time during which an external force acts on a liquid, usually liquids flow. The solid-like ability of liquids to sustain high-frequency propagating modes down to wavelengths on the atomic scale, at the temperature around and above melting point, was observed fairly recently [@copley; @pilgrim; @burkel; @morkel; @grimsditch].
As noted, $\tau$ is a fundamental flow property of a liquid, and is directly related to liquid viscosity $\eta$: $\eta=G_{\infty}\tau$ [@frenkel; @maxwell], where $G_{\infty}$ is the instantaneous shear modulus. According to time scale the motion of an atom in a liquid can be viewed as of two types: quasi-harmonic vibrational motion around an equilibrium position as in a solid glass, with Debye vibration period of about $\tau_{D}=$0.1 ps, and diffusive motion between two neighboring positions, where typical diffusion distances exceed vibrational distances by about a factor of ten. When $\tau$ significantly exceeds $\tau_{D}$, the number of diffusing atoms and, therefore, the diffusing energy, becomes small and can be ignored.
The phonon theory of liquids allows us to calculate liquid internal energy in general form which can be compactly presented as $$\label{lenergy}
E=NT\left(1+\frac{\alpha T}{2}\right)\left(3D\left(\frac{\hbar\omega_{\rm D}}{T}\right)-\left(\frac{\omega_{\rm F}}{\omega_{\rm D}}\right)^3D\left(\frac{\hbar\omega_{\rm F}}{T}\right)\right)$$ where $D(x)=\frac{3}{x^3}\int\limits_0^x\frac{z^3{\rm d}z}{\exp(z)-1}$ is the Debye function [@landau], $\alpha$ is the thermal expansion coefficient, $N$ is the number of phonon states, $T$ is the temperature, and $\omega_{D}$ and $\omega_{F}$ are Debye and Frenkel frequencies correspondingly.
Eq.(\[lenergy\]) accounts for longitudinal and also for two high-frequency shear modes with frequency $\omega>\frac{1}{\tau}$. It originates at the same level of approximation as Debye’s phonon theory of solids by using the quadratic density of states. The result for a harmonic solid follows from Eq.(\[lenergy\]) when $\omega_{F}=0$, corresponding to infinite relaxation time, and thermal expansion coefficient $\alpha=0$. This theory of liquids incorporates the effects of anharmonicity and thermal expansion, which is very important not only for classical liquids such as Hg and Rb [@trachenko], but also for liquid helium as we can see further.
The phonon theory of liquids has recently been formulated in a form that predicts the heat capacity of 21 different liquids, among those: noble, metallic, molecular and hydrogen-bonded network liquids. The theory covers both the classical and quantum regimes and agrees with experiment over a wide range of temperatures and pressures [@bolmatov].
Heat capacity of liquid helium
==============================
Accordingly we now take a derivative of energy $E$ (Eq.(\[lenergy\])) with respect to temperature $T$ at constant volume and compare it to experimental data of heat capacity per atom: $c_{V}=\frac{1}{N}\frac{dE}{dT}$. We have used the National Institute of Standards and Technology (NIST) database [@NIST] that contains data for many chemical and physical quantities, including $c_{V}$ for liquid helium. We aimed to check our theoretical predictions in a wider range of temperature, and therefore selected the data at pressures significantly exceeding the critical temperature and pressure of liquid helium ($T_{c}=5.1953$ K and $P_{c}=0.22746$ MPa) where it exists in a liquid form in the broader temperature range. As a result, the temperature range in which we calculate $c_{V}$ is about 40-45 K. Viscosity data was taken from the same database [@NIST], and fitted in order to use in Eq.(\[lenergy\]) to calculate $c_{V}$. We used the Vogel-Fulcher-Tammann (VFT) expression to fit the viscosity data: $\eta=\eta_{0}\exp{A/(T-T_{0})}$. To calculate $c_{V}$ from Eq.(\[lenergy\]), we have taken viscosity data at the same pressures as $c_{V}$ and converted it to $\tau$ using the Maxwell relationship $\tau=\frac{\eta}{G_{\infty}}$, where the Frenkel frequency can be conveniently expressed as $\omega_{F}=\frac{2\pi}{\tau}=\frac{2\pi G_{\infty}}{\eta}$.
Eq.(\[lenergy\]) has no fitting parameters, because the parameters $\omega_{D}$, $\alpha$ and $G_{\infty}$ are fixed by system properties. Values of these parameters used in Eq.(\[lenergy\]) are in a good agreement with typical experimental values. There is a difference between the experimental $\alpha$ and the $\alpha$ used in the calculation. At each pressure the experimental $\alpha$ was estimated from the formula $\alpha=\frac{1}{V}\frac{\Delta V}{\Delta T}$. Experimentally, V$\propto$T only approximately. Consequently, we approximated V$=$V(T) by a linear dependence (an approximation results in somewhat different $\alpha$ used in Eq.(\[lenergy\])). Further, $\tau_{D}=\frac{2\pi}{\omega_{D}}$ used in the calculation (see the caption in Fig. \[fig1\]) is consistent with the known values for low temperature liquid helium under pressure that are typically in the 1-2 ps range [@humprey]. The uncertainty of both experimental heat capacities and viscosities is about 5-10% [@NIST].
Discussions and conclusions
===========================
As noted earlier, there are two basic analytical approaches to calculate liquid energy and heat capacity: from the gas phase and from the solid. The approach from the classical gas phase has two main contributions to liquid energy; kinetic and potential parts and can be presented as $$\label{genergy}
E=K+\int gUdV$$ where K is kinetic energy, g is normalized correlation function and U is the interatomic energy. Generally the expression in Eq.(\[genergy\]) is difficult to evaluate for a many-body systems and it is not clear how to rigorously incorporate quantum effects at elevated temperatures (say tens of Kelvins for He) into Eq.(\[genergy\]). To describe the behaviour of $c_{V}$ on the basis of Bose-Einstein condensation (BEC) in liquid helium at low temperatures was originally initiated by F. London [@london]. The heat capacity of BEC varies as $T^{3/2}$ at low temperatures and must pass through a maximum. This maximum has the character of a cusp which appears at critical temperature $T_{c}$ and it has nothing to do with a subsidiary maximum which liquid helium possesses at elevated pressures and higher temperatures (see Fig. \[fig1\]).
The odd behaviour of the heat capacity of liquid helium at elevated pressures can now be explained in the framework of the phonon theory of liquids. When we calculate liquid energy and heat capacity in this theory the problem of strong interactions is avoided from outset and based on displacive physics associated with phonon contributions. We predict that transverse waves exist in liquid helium at high pressures. This prediction can be verified in a future experimental work. The experimental data for heat capacity and viscosity of liquid helium confirms our hypothesis [@NIST]. Even at elevated pressures liquid helium persists at temperatures which are 3-5 times lower than Debye’s temperature (see Fig.\[fig1\]). As the temperature is raised in liquid helium, more longitudinal and transverse-like phonons become progressively excited and therefore the heat capacity rapidly grows, which is very abnormal for ordinary liquids [@bolmatov]. Thus liquid helium at this P-T region persists as a quantum liquid and also as a solid-like liquid with non-static shear rigidity, similar to classical liquids. When the Debye and the Frenkel temperatures become roughly comparable, $c_{V}$ of liquid helium enters the saturation region (’hump’). Further increase of temperature then leads to the dissipation of transverse-like waves and $c_{V}$ of liquid helium becomes shallow (see Fig.\[fig1\]), implying that at very high temperatures $c_{V}$ reaches its asymptotic value $3/2$.
The results just presented appear to be fairly accurate over the temperature range of experimental importance, despite the fact that the formal expression for liquid internal energy is quite trivial (see Eq.(\[lenergy\])). The agreement is somewhat worse at intermediate part (the slight maximum or ’hump’ region mentioned earlier) of the $c_{V}$ curve, however the maximal difference between the predicted and experimental values is actually comparable to the experimental uncertainty of $c_{V}$, namely of 0.1-0.2 J/K [@NIST]. But this can also be attributed to an oversimplified form of the Debye spectrum of phonon-like states used in our analysis.
Liquid helium in its normal state and at atmospheric pressure is actually above the conditions required for a Frenkel line (see the diagram [@brazhkin]) and barely supports transverse-like elementary excitations. At significant pressures (around 10 GPa) liquid helium quite resembles the other noble-gas liquids; there are transverse-like excitations but at its melting temperature almost all phonons are already excited. This state of liquid helium is rigid but not quantum. Here we are suggesting an interesting intermediate pressure range of 5 MPa-500 MPa where liquid helium near melting temperature is already solid-like but is still significantly quantum.
Acknowledgements
================
D. Bolmatov thanks Myerscough Bequest and K. Trachenko thanks EPSRC for financial support. D. B. acknowledges Thomas Young Centre for Junior Research Fellowship and Cornell University (Neil Ashcroft and Roald Hoffmann) for hospitality. We acknowledge Neil Ashcroft for fruitful discussions.
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| {
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---
abstract: 'We consider the quadratically semilinear wave equation on $( \R^d , \mathfrak{g})$, $d \geq 3$. The metric $\mathfrak{g}$ is non-trapping and approaches the Euclidean metric like $\< x \>^{- \rho}$. Using Mourre estimates and the Kato theory of smoothness, we obtain, for $\rho >0$, a Keel–Smith–Sogge type inequality for the linear equation. Thanks to this estimate, we prove long time existence for the nonlinear problem with small initial data for $\rho \geq 1$. Long time existence means that, for all $n>0$, the life time of the solution is a least $\delta^{-n}$, where $\delta$ is the size of the initial data in some appropriate Sobolev space. Moreover, for $d \geq 4$ and $\rho > 1$, we obtain global existence for small data.'
address: Institut de Mathématiques de Bordeaux UMR 5251 du CNRS Université de Bordeaux I 351 cours de la Libération 33 405 Talence cedex France
author:
- 'Jean-François Bony'
- Dietrich Häfner
title: The semilinear wave equation on asymptotically Euclidean manifolds
---
Introduction {#sec1}
============
This paper is devoted to the study of the quadratically semilinear wave equation on asymptotically Euclidean non-trapping Riemannian manifolds. We show global existence in dimension $d\ge 4$ and long time existence in dimension $d=3$ for small data solutions. In Minkowski space, the semilinear wave equation has been thoroughly studied. Global existence is known in dimension $d\ge 4$ for small initial data (see Klainerman and Ponce [@KlPo83_01] and references therein). Almost global existence in dimension $d=3$ for small data was shown by John and Klainerman in [@JoKl84_01]. Almost global means that the life time of a solution is at least $e^{1/ \delta}$, where $\delta$ is the size of the initial data in some appropriate Sobolev space. Note that, in dimension $d=3$, Sideris [@Si83_01] has proved that global existence does not hold in general (see also John [@Jo81_01]).
In [@KeSmSo02_01], Keel, Smith and Sogge give a new proof of the almost global existence result in dimension 3 using estimates of the form $$\label{KSSE}
( \ln (2+ T ) )^{-1/2} \big\Vert \< x \>^{-1/2} u' \big\Vert_{L^2 ( [0,T] \times \R^3 )} \lesssim \Vert u' (0, \cdot ) \Vert_{L^2 ( \R^3 )} + \int_0^T \Vert G(s, \cdot ) \Vert_{L^2 ( \R^3 )} d s ,$$ and a certain Sobolev type estimate due to Klainerman (see [@Kl85_01]). Here $u$ solves the wave equation $\Box u = G$ in $[0 , + \infty [ \times \R^3$ and $u' = ( \partial_{t} u , \partial_{x} u )$. They also treat the non-trapping obstacle case. In [@KeSmSo04_01], similar results are obtained for the corresponding quasilinear equation. The obstacle case in which the trapped trajectories are of hyperbolic type is treated by Metcalfe and Sogge [@MeSo05_01].
Alinhac shows an estimate similar to on a curved background. In his papers [@Al05_01] and [@Al06_01], the metric is depending on and decaying in time. The results of Metcalfe and Tataru [@MeTa08_01] imply estimates analogous to for a space-time variable coefficients wave equation outside a star shaped obstacle (see also [@MeSo06_01]). Outside the obstacle, their wave operator is a small perturbation of the wave operator in Minkowski space.
The common point of the papers cited so far is that they all use vector field methods. We use in this paper a somewhat different approach. We will show how estimates of type follow from a Mourre estimate [@Mo81_01]. This method will permit us to consider non-trapping Riemannian metrics which are asymptotically Euclidean without requiring that they are everywhere a small perturbation of the Euclidean metric. We will suppose for simplicity that the metric is $C^{\infty}$, but a $C^k$ approach should in principle be possible. Spectral methods for proving dispersive estimates were previously used by Burq. In [@Bu03_01], he obtains global Strichartz estimates for compactly supported non-trapping perturbations of the Euclidean case. In more complicated geometries, conjugate operators are probably not vector fields and it is perhaps worth trying to mix the classical vector field approach with the Mourre theory.
Let us now state our precise results. We consider the asymptotically Euclidean manifold $( \R^d , \mathfrak{g})$ with $d \geq 3$ and $$\mathfrak{g} = \sum_{i,j=1}^{d} g_{i,j} (x) \, d x^i \, d x^j .$$ We suppose $g_{i,j} (x) \in C^{\infty} ( \R^{d} )$ and, for some $\rho >0$, $$\tag{H1} \label{c1}
\forall \alpha \in \N^d \qquad \partial^{\alpha}_x ( g_{i,j} - \delta_{i,j} ) = \CO ( \< x \>^{- \vert \alpha \vert - \rho} ) .$$ We also assume that $$\tag{H2} \label{c15}
\mathfrak{g} \text{ is non-trapping.}$$ Let $g (x) = ( \det ( \mathfrak{g} ) )^{1/4}$. The Laplace–Beltrami operator associated to $\mathfrak{g}$ is given by $$\Delta_{\mathfrak{g}} = \sum_{i,j} \frac{1}{g^2} \partial_i g^{i,j} g^2 \partial_j ,$$ where $g^{i,j} (x)$ denotes the inverse metric. Let us consider the following unitary transform $$\begin{aligned}
{\mathcal V}: \\
{}
\end{aligned}
\left\{ \begin{aligned}
&L^2 ( \R^d , g^{2} \, d x ) &\longrightarrow &&L^2( \R^d , d x ) \\
&v &\longmapsto &&g v .
\end{aligned} \right.$$ The transformation ${\mathcal V}$ sends $- \Delta_{\mathfrak{g}}$ to $$P = - {\mathcal V} \Delta_{\mathfrak{g}} {\mathcal V}^* = - \sum_{i,j} \frac{1}{g} \partial_i g^{i,j} g^2 \partial_j \frac{1}{g} ,$$ which is the operator we are interested in. Let $\widetilde{\partial}_j : = \partial_j g^{-1}$ and $\Omega = \Omega^{k , \ell} : = x_{k} \partial_{\ell} - x_{\ell} \partial_{k}$ be the rotational vector fields. We consider the following semilinear wave equation $$\label{SLW}
\left\{\begin{aligned}
&\Box_{\mathfrak{g}} u = Q (u') , \\
&( u_{\vert_{t=0}} , \partial_t u _{\vert_{t=0}} ) = ( u_0 , u_1 ) .
\end{aligned} \right.$$ Here $\Box_{\mathfrak{g}} = \partial_{t}^{2} + P$ and $Q (u')$ is a quadratic form in $u ' = ( \partial_t u , \widetilde{\partial}_{x} u )$. For $x \in \R$, $\lfloor x \rfloor$ (resp. $\lceil x \rceil$) denotes the largest (resp. smallest) integer such that $\lfloor x \rfloor \leq x \leq \lceil x \rceil$. Our main result is the following theorem.
*\[TSLW\] Assume hypotheses and . Suppose $u_0 , u_1 \in C_0^{\infty} ( \R^d )$ and that, for $M = 2 \left( \left\lceil \frac{d-1}{2} \right\rceil + 1 \right)$, we have $$\label{SW1}
\sum_{\vert \alpha \vert + j \leq M+1} \big\Vert \partial^j_x \Omega^{\alpha} u_0 \big\Vert + \sum_{\vert \alpha \vert + j \leq M} \big\Vert \partial^j_x \Omega^{\alpha} u_1 \big\Vert \leq \delta .$$*
$i)$ Assume $d \geq 3$ and $\rho \geq 1$. For all $n > 0$, there exists a constant $\delta_{n} > 0$ such that, for $\delta \leq \delta_{n}$, the problem has a unique solution $u \in C^{\infty}( [ 0, T ] \times \R^3 )$ with $$T = \delta^{-n} .$$
$ii)$ Assume $d\geq 4$ and $\rho > 1$. For $\delta$ small enough, the problem has a unique global solution $u \in C^{\infty} ( [ 0 , + \infty [ \times \R^d )$.
*One may consider more general nonlinearities. For example, the previous result holds for quadratic nonlinearities of the form $Q (x) ( \< x \>^{- \mu} u , u')$ with $\mu >1$ and $\Vert \partial_{x}^{\alpha} Q (x) \Vert= \CO ( \< x \>^{- \vert \alpha \vert} )$. In particular, one can replace $Q (u')$ by $Q ( \partial_{t} u , \partial_{x} u)$ or work with the wave equation before the transformation by ${\mathcal V}$. To prove this remark, it is enough to combine the proof of Theorem \[TSLW\] with Lemma \[c16\].*
The main ingredient of the proof are estimates of type . Let us therefore consider the corresponding linear equation. Let $u$ be solution of $$\label{LW}
\left\{ \begin{aligned}
&(\partial_t^2+P)u = G(s) , \\
&( u_{\vert_{t=0}} , \partial_t u _{\vert_{t=0}} ) = ( u_0 , u_1 ) .
\end{aligned} \right.$$ With the notation $$F^{\varepsilon}_{\mu} (T) = \left\{ \begin{aligned}
&T^{1-2 \mu +2 \varepsilon} &&\mu \leq 1/2 , \\
&1 &&\mu > 1/2 ,
\end{aligned} \right.$$ we have the following estimate.
*\[TSLW2\] Assume that and hold with $\rho > 0$ and let $0 < \mu \leq 1$. For all $\varepsilon >0$, the solution of satisfies $$\label{EW2}
\big\Vert \< x \>^{-\mu} u' \big\Vert_{L^2 ( [0, T] \times \R^{d} )} \lesssim \< F_{\mu}^{\varepsilon} (T) \>^{1/2} \bigg( \Vert u' ( 0 , \cdot ) \Vert_{L^2 ( \R^{d} )} + \int_0^T \Vert G(s, \cdot ) \Vert_{L^2 ( \R^{d} )} d s \bigg).$$*
To prove the nonlinear theorem, it will be useful to have higher order estimates. To this purpose, let us put $\widetilde{\Omega}^{k, \ell} = x_k \widetilde{\partial}_\ell - x_\ell \widetilde{\partial}_k$, $Z=\{\partial_t , \widetilde{\partial}_x, \widetilde{\Omega}\}$, $Y= \{ \widetilde{\partial}_x, \widetilde{\Omega}\}$, $X = \{ \widetilde{\partial}_x \}$, where $\{\widetilde{\Omega}\}$ (resp. $\{ \widetilde{\partial}_x \}$) are the collections of rotational vector fields (resp. partial derivatives with respect to space variables). Then, we have
*\[TW1\] Assume that and hold with $\rho > 1$ and let $N > 0$ and $1/2 \leq \mu \leq 1$. For all $\varepsilon > 0$, the solution of satisfies $$\begin{aligned}
\sup_{0 \leq t \leq T} \sum_{1 \leq k + j \leq N+1} \big\Vert \partial_{t}^{k} P^{j/2} u (t & , \cdot ) \big\Vert_{L^2 ( \R^{d} )} + \sum_{\vert \alpha \vert \leq N} \< F_{\mu}^{\varepsilon}(T) \>^{-1} \big\Vert \<x\>^{-\mu} Z^{\alpha} u ' \big\Vert_{L^2( [0, T] \times \R^{d} )} \nonumber \\
& \lesssim \sum_{\vert \alpha \vert \leq N} \bigg( \big\Vert (Z^{\alpha}u) ' (0, \cdot ) \big\Vert_{L^2 ( \R^{d} )} + \int_0^T \big\Vert Z^{\alpha}G(s, \cdot ) \big\Vert_{L^2 ( \R^{d} )} d s \bigg). \label{W1}\end{aligned}$$ Moreover, for $\rho = 1$, the same inequality holds with $\< F_{\mu}^{\varepsilon}(T) \>^{-1}$ replaced by $\< T \>^{- \varepsilon}$.*
*\[RW1\] $i)$ Note that, in Theorem \[TSLW\] and Theorem \[TW1\], $\rho \geq 1$ is required whereas Theorem \[TSLW2\] is valid under a general long range condition $\rho > 0$.*
$ii)$ Theorem \[TSLW2\] and Theorem \[TW1\] remain valid if we replace $u '$ by $( \partial_t u , P^{1/2} u )$.
The paper is organized in the following way. In Section \[c20\], we show scattering estimates in a general setting. Section \[secM\] is devoted to the Mourre estimate for the wave equation on our asymptotically Euclidean manifold. Using these results, we prove the estimates for the linear wave equation (Theorem \[TSLW2\] and Theorem \[TW1\]) in Section \[EW\]. From these estimates, we deduce the nonlinear result in Section \[sec6\]. Appendix \[a29\] collects some regularity properties of operators and Appendix \[b56\] contains low frequency resolvent estimates.
The general setting {#c20}
===================
In this section, we obtain some abstract estimates which will be used to prove Theorem \[TSLW2\] and Theorem \[TW1\]. These estimates are not specific to the wave equation and could help to show analogous estimates for other equations. The key ingredients are the limiting absorption principle and the Kato theory of smoothness.
We begin this section with the notion of regularity with respect to an operator. A full presentation of this theory can be found in the book of Amrein, A. Boutet de Monvel and Georgescu [@AmBoGe96_01]. In Appendix \[a29\], we recall the properties which will be used in this paper.
*Let $(A, D(A))$ and $(H,D(H))$ be self-adjoint operators on a separable Hilbert space ${\mathcal H}$. The operator $H$ is of class $C^{k} (A)$ for $k >0$, if there is $z \in \C \setminus \sigma ( H)$ such that $$\R \ni t \longrightarrow e^{i t A} (H-z)^{-1} e^{- i t A},$$ is $C^{k}$ for the strong topology of ${\mathcal L} ( {\mathcal H} )$.*
Let $H \in C^{1} (A)$ and $I\subset \sigma (H)$ be an open interval. We assume that $A$ and $H$ satisfy a Mourre estimate on $I$: $$\label{a3}
\one_{I} (H) i [ H , A ] \one_{I} (H) \geq \delta \one_{I} (H) ,$$ for some $\delta >0$. As usual, we define the multi-commutators $\ad_{A}^{j} B$ inductively by $\ad_{A}^{0} B =B$ and $\ad_{A}^{j+1} B = [ A , \ad_{A}^{j} B ]$.
*\[a4\] Let $H \in C^{2} (A)$ be such that $\ad_{A}^{j} H$, $j=1,2$, are bounded on ${\mathcal H}$. Assume furthermore . Then, for all closed intervals $J \subset I$ and $\mu >1/2$, there exists $C_{J,\mu} >0$ such that $$\label{a5}
\sup_{{\genfrac{}{}{0pt}{}{\scriptstyle \re z \in J}{\scriptstyle \im z \neq 0}}} \big\Vert \< A \>^{- \mu} ( H -z)^{-1} \< A \>^{-\mu} \big\Vert \leq C_{J , \mu} .$$*
If $A$ and $H$ depend on a parameter, the constant in the limiting absorption principle can be specified according to this parameter. In fact, mimicking the proof of [@PeSiSi81_01], we obtain the following estimate.
*\[a6\] Assume that holds uniformly and that $[H ,A]$ is uniformly bounded. Then, for all closed intervals $J \subset I$ and $\mu >1/2$, $$\sup_{{\genfrac{}{}{0pt}{}{\scriptstyle \re z \in J}{\scriptstyle \im z \neq 0}}} \big\Vert \< A \>^{- \mu} ( H -z)^{-1} \< A \>^{-\mu} \big\Vert \leq \widetilde{C}_{J , \mu} \big\< \big\Vert \ad_{A}^{2} H \big\Vert \big\>^{\widetilde{C}_{J , \mu}} ,$$ for some $\widetilde{C}_{J, \mu} >0$.*
We now state a result of Kato [@Ka66_01] which says that, under the conclusions of Theorem \[a4\], $\< A \>^{- \mu} \one_{J} (H)$ is $H$–smooth. For the proof and more details, we refer to Theorem XIII.25 and Theorem XIII.30 of [@ReSi78_01].
*\[a1\] Let $A$ and $H$ be two self-adjoint operators satisfying . Then, for all closed intervals $J \subset I$ and $\mu > 1/2$, $$\int_{\R} \big\Vert \< A \>^{- \mu} e^{-i t H} \one_{J} (H) u \big\Vert^{2} d t \leq 8 C_{J , \mu} \Vert u \Vert^{2} ,$$ for all $u \in {\mathcal H}$.*
In the previous theorem, $C_{J , \mu}$ is the constant appearing in . By interpolation, we get
*\[a2\] Assume . Then, for all closed intervals $J \subset I$ and $0 < \mu \leq 1/2$, $$\int_{0}^{T} \big\Vert \< A \>^{- \mu} e^{-i t H} \one_{J} (H) u \big\Vert^{2} d t \leq M_{J , \mu , \varepsilon} T^{1 - 2 \mu + \varepsilon} \Vert u \Vert^{2} ,$$ for all $0 < \varepsilon < 2 \mu$. Here, $$M_{J , \mu , \varepsilon} = \big( 8 C_{J , \mu / ( 2 \mu - \varepsilon )} \big)^{2 \mu - \varepsilon} .$$*
Since $e^{-i t H}$ is unitary, $$\int_{0}^{T} \big\Vert e^{-i t H} \one_{J} (H) u \big\Vert^{2} d t \leq T \Vert u \Vert^{2} .$$ Combining Theorem \[a1\], the previous estimate and an interpolation argument, we get $$\int_{0}^{T} \big\Vert \< A \>^{- (1- \theta ) \nu} e^{-i t H} \one_{J} (H) u \big\Vert^{2} d t \leq ( 8 C_{J , \nu} )^{1 - \theta} T^{\theta} \Vert u \Vert^{2} .$$ for all $0 \leq \theta \leq 1$ and $\nu >1/2$. Taking $\theta = 1 - 2 \mu + \varepsilon \in [0 ,1]$ (since $\varepsilon < 2 \mu$) and $\nu = \mu / (1 - \theta ) = \mu / ( 2 \mu - \varepsilon ) > 1/2$, the corollary follows.
We now study the non-homogeneous equation using the Fourier transform. Let $G (t) \in L^{1}_{\text{loc}} ( \R_{t}; {\mathcal H} )$ be such that $\supp G \subset [0 , + \infty [$. We consider the solution $u$ of $$\label{a34}
\left\{ \begin{aligned}
&(i \partial_{t} - H )u (t) = \varphi (H) G (t) , \\
&u_{\vert_{t=0}} = 0 ,
\end{aligned} \right.$$ with $\varphi \in L^{\infty} ( \R )$ and $\supp \varphi \subset J$. This means that $$\label{a31}
u (t) = - i \int_{0}^{t} e^{- i (t-s) H} \varphi (H) G (s) \, d s ,$$ and then $u \in C^{0} (\R_{t} ; {\mathcal H} ) \cap {\mathcal S} ' (\R_{t} ; {\mathcal H})$.
*\[a32\] Let $A$ and $H$ be two self-adjoint operators satisfying . Then, for all $\mu >1/2$ and $\varphi \in C^{1} ( \R )$ satisfying $\Vert \varphi \Vert_{\infty} \leq 1$, $\Vert \varphi ' \Vert_{\infty} \leq C_{1}$ and $\supp \varphi \subset J$, we have $$\big\Vert \< A \>^{- \mu} \varphi ( H) ( H-z)^{-1} \< A \>^{- \mu} \big\Vert \leq C_{J , \mu} + C_{1} ,$$ for all $z \in \C \setminus \R$.*
Using Taylor’s expansion formula, we have $$\varphi (x) = \varphi (y) + (x - y) \int_{0}^{1} \varphi ' ( t x + (1 - t) y ) \, d t ,$$ and then $$\varphi (H) (H -z)^{-1} = \varphi ( \re z ) (H - z)^{-1} + \int_{0}^{1} \varphi ' \big( t H + (1-t) \re z \big) \, d t \, (H - \re z) (H -z)^{-1} .$$ Using the spectral theorem, we obtain the following estimates: $$\begin{aligned}
\Big\Vert \int_{0}^{1} \varphi ' \big( t H + (1-t) \re z \big) \, d t \Big\Vert &\leq \int_{0}^{1} \big\Vert \varphi ' \big( t H + (1-t) \re z \big) \big\Vert \, d t \leq C_{1} , \\
\big\Vert (H - \re z ) (H -z)^{-1} \big\Vert &\leq \sup_{x \in \R} \big\vert x ( x - i \im z)^{-1} \big\vert \leq 1 .\end{aligned}$$
Therefore, for $\re z \in J$, we have $$\begin{aligned}
\big\Vert \< A \>^{- \mu} \varphi ( H) ( H-z)^{-1} \< A \>^{- \mu} \big\Vert &\leq \vert \varphi ( \re z) \vert \big\Vert \< A \>^{- \mu} ( H-z)^{-1} \< A \>^{- \mu} \big\Vert + C_{1} \Vert \< A \>^{- \mu} \Vert^{2} \\
&\leq C_{J , \mu} + C_{1} .\end{aligned}$$ On the other hand, for $\re z \notin J$, $\varphi ( \re z ) =0$ and then $$\big\Vert \< A \>^{- \mu} \varphi ( H) ( H-z)^{-1} \< A \>^{- \mu} \big\Vert \leq C_{1} \Vert \< A \>^{- \mu} \Vert^{2} \leq C_{1} .$$ The two last estimates give the lemma.
*\[a33\] Let $A$ and $H$ be two self-adjoint operators satisfying and $\varphi \in C^{1} ( \R )$ as in Lemma \[a32\]. Then, for all $\mu > 1/2$ and $G (t) \in L^{2} ( \R_{t}; D ( \< A \>^{\mu} ) )$ with $\supp G \subset [0 , + \infty [$, the solution $u$ of satisfies $$\int_{0}^{\infty} \big\Vert \< A \>^{- \mu} u (t) \big\Vert^{2} d t \leq ( C_{J , \mu} + C_{1} )^{2} \int_{0}^{\infty} \big\Vert \< A \>^{\mu} G (t) \big\Vert^{2} d t .$$*
Let $u_{\varepsilon} = ( 1+ i \varepsilon H)^{-1} u$. From , $u_{\varepsilon} \in C^{1} (\R_{t} ; {\mathcal H} ) \cap C^{0} ( \R_{t} ; D (H)) \cap {\mathcal S} ' (\R_{t} ; D (H) )$ and $u_{\varepsilon}$ is the solution of the problem $$\label{a30}
\left\{ \begin{aligned}
&(i \partial_{t} - H )u_{\varepsilon} (t) = ( 1+ i \varepsilon H )^{-1} \varphi (H) G (t) , \\
&u_{\varepsilon}{}_{\vert_{t=0}} = 0 .
\end{aligned} \right.$$ Since the support of the temperate distributions $u_{\varepsilon}$ and $G$ is in $[0, + \infty [$, their Fourier transforms are analytic in $\im z < 0$. Then, gives, for $\im z < 0$, $$(z - H) \widehat{u_{\varepsilon}} (z) = (1 + i \varepsilon H )^{-1} \varphi (H) \widehat{G} (z) .$$ Then $$\< A \>^{- \mu} \widehat{u_{\varepsilon}} (z) = \< A \>^{- \mu} ( 1+ i \varepsilon H)^{-1} \varphi ( H) ( z-H)^{-1} \< A \>^{- \mu} \< A \>^{\mu} \widehat{G} (z) .$$ Since $\Vert \varphi ( x) ( 1+ i \varepsilon x )^{-1} \Vert_{\infty} \leq \Vert \varphi \Vert_{\infty}$ and $\Vert \partial_{x} ( \varphi ( x) ( 1+ i \varepsilon x )^{-1} ) \Vert_{\infty} \leq \varepsilon \Vert \varphi \Vert_{\infty} + \Vert \varphi ' \Vert_{\infty}$, Lemma \[a32\] implies $$\big\Vert \< A \>^{- \mu} \widehat{u_{\varepsilon}} (z) \big\Vert \leq ( C_{J , \mu} + C_{1} + \varepsilon ) \big\Vert \< A \>^{\mu} \widehat{G} (z) \big\Vert .$$ Thus, for all $\delta >0$, Plancherel’s theorem gives $$\int_{0}^{+ \infty} e^{- \delta t} \big\Vert \< A \>^{- \mu} u_{\varepsilon} \big\Vert^{2} d t \leq ( C_{J , \mu} + C_{1} + \varepsilon )^{2} \int_{0}^{+ \infty} e^{- \delta t} \big\Vert \< A \>^{\mu} G \big\Vert^{2} d t .$$ Letting $\delta$ and $\varepsilon$ go to $0$, we get the proposition.
By interpolation, we also have
*\[c9\] Assume the hypotheses of Proposition \[a33\]. Then, for all $0 < \mu \leq 1/2$ and $G (t) \in L^{2} ( \R_{t}; D ( \< A \>^{\mu} ) )$ with $\supp G \subset [0 , + \infty [$, $$\int_{0}^{T} \big\Vert \< A \>^{- \mu} u (t) \big\Vert^{2} d t \leq N_{J , \mu , \varepsilon} T^{2 (1 - 2 \mu + \varepsilon )} \int_{0}^{T} \big\Vert \< A \>^{\mu} G (t) \big\Vert^{2} d t ,$$ for all $0 < \varepsilon < 2 \mu$. Here, $$N_{J , \mu , \varepsilon} = \big( C_{J , \mu / ( 2 \mu - \varepsilon )} + C_{1} \big)^{4 \mu - 2 \varepsilon} .$$*
Let $P_{T} : L^{2} ( [0, T] ; {\mathcal H} ) \longrightarrow L^{2} ( [0, T] ; {\mathcal H} )$ be the operator defined by $$( P_{T} G ) (t) = - i \int_{0}^{t} e^{- i (t-s) H} \varphi (H) G (s) \, d s .$$ Proposition \[a33\] gives $$\Vert \< A \>^{- \nu} P_{T} G \Vert_{L^{2} ( [0, T] ; {\mathcal H} )} \leq ( C_{J , \nu} + C_{1} ) \Vert \< A \>^{\nu} G \Vert_{L^{2} ( [0, T] ; {\mathcal H} )} ,$$ for $\nu >1/2$. Moreover, $$\begin{aligned}
\Vert P_{T} G \Vert_{L^{2} ( [0, T] ; {\mathcal H} )} &\leq \sqrt{T} \sup_{t \in [0 , T]} \Big\Vert \int_{0}^{t} e^{- i (t-s) H} \varphi (H) G (s) \, d s \Big\Vert \\
&\leq \sqrt{T} \sup_{t \in [0 , T]} \sqrt{t} \bigg( \int_{0}^{t} \Big\Vert e^{- i (t-s) H} \varphi (H) G (s) \Big\Vert^{2} \, d s \bigg)^{1/2} \\
&\leq T \Vert G \Vert_{L^{2} ( [0, T] ; {\mathcal H} )} .\end{aligned}$$ With these two estimates in mind, one can prove the corollary by mimicking the proof of Corollary \[a2\].
The wave equation and the Mourre estimate {#secM}
=========================================
In this section we will show a Mourre estimate for the wave equation on our asymptotically Euclidean manifold: $$\label{M3}
\left\{ \begin{aligned}
&( \partial_t^2 + P) u = 0 , \\
&u_{\vert_{t=0}} = u_0 , \ \partial_t u_{\vert_{t=0}} = u_1 .
\end{aligned} \right.$$ Recall that $$P = - \sum_{i,j} \frac{1}{g} \partial_i g^{i,j} g^2 \partial_j \frac{1}{g} ,$$ is self-adjoint on $L^2 ( \R^d , d x )$ with domain $D (P) = H^2 ( \R^{d} )$. We define $H^k_{\text{c}} ( \R^{d} )$ as the closure of $H^k ( \R^{d} )$ with respect to the norm $$\Vert u \Vert^2_{H_{\text{c}}^k} = \sum_{j=1}^k \big\Vert P^{j/2} u \big\Vert^2 .$$ Let ${\mathcal E} := H^1_{\text{c}} ( \R^{d} ) \oplus L^{2} ( \R^{d} )$ with $$\Vert ( u_0 , u_1 ) \Vert^{2}_{\mathcal E} = \< P u_0 , u_0 \> + \Vert u_1 \Vert^2 ,$$ be the energy space associated to . The energy of is clearly conserved: $$\big\Vert ( u(t) , \partial_t u(t) ) \big\Vert_{\mathcal E} = \Vert ( u_0 , u_1 ) \Vert_{\mathcal E} .$$ We will rewrite as a first order system $$\label{M4}
\left \{\begin{aligned}
&i \partial_t f = R f , \\
&f_{\vert_{t=0}} = ( u_0 , u_1 ) ,
\end{aligned} \right.$$ with $$R = \left( \begin{array}{cc}
0 & i \\ - i P & 0
\end{array} \right) .$$ The operator $R$ is self-adjoint on ${\mathcal E}$ with domain $D (R) = H_{\text{c}}^2 ( \R^{d} ) \oplus H^{1} ( \R^{d} )$. Let ${\mathcal L} = L^{2} ( \R^{d} ) \oplus L^{2} ( \R^{d} )$. It is useful to introduce the following unitary transform: $$U : {\mathcal E} \longrightarrow {\mathcal L}, \qquad U = \frac{1}{\sqrt{2}}
\left( \begin{array}{cc}
P^{1/2} & i \\ P^{1/2} & -i
\end{array} \right) ,$$ which satisfies $$U^* = U^{-1} = \frac{1}{\sqrt{2}}
\left( \begin{array}{cc}
P^{-1/2} & P^{-1/2} \\ -i & i
\end{array} \right)
\quad \text{ and } \quad L = U R U^* = \frac{1}{\sqrt{2}}
\left( \begin{array}{cc}
P^{1/2} & 0 \\
0 & -P^{1/2}
\end{array} \right) .$$ The operator $( L , D (L) = H^{1} ( \R^{d} ) \oplus H^{1} ( \R^{d} ) )$ is self-adjoint. In order to establish a Mourre estimate for $L$, it is sufficient to establish a Mourre estimate for $P^{1/2}$. We divide this section into the study of the low, the intermediate and the high frequency part.
Low frequency Mourre estimate
------------------------------
${}^{}$
\[secM1\]
For low frequencies, we will make a dyadic decomposition and use a conjugate operator specific to each part of the decomposition. In this section, we will obtain a Mourre estimate for each part. For $\lambda \geq 1$, we set $$\label{b29}
{\mathcal A}_{\lambda} = \varphi ( \lambda P ) A_{0} \varphi ( \lambda P) ,$$ where $$A_{0}= \frac{1}{2} ( x D + D x ) , \quad D ( A_0 ) = \big\{ u \in L^{2} ( \R^{d} ) ; \ A_0 u \in L^{2} ( \R^{d} ) \big\} ,$$ is the generator of dilations and $\varphi \in C^{\infty}_{0} ( ] 0 , + \infty [ ; [0, + \infty [)$ satisfies $\varphi (x) > \delta >0$ on some open bounded interval $I \subset ] 0 , + \infty[$.
For the various estimates that we will establish in this section, the following formula for the square root of an operator will be useful. Making a change of contour and using the Cauchy formula, one can show that $$\label{c8}
\sigma^{-1/2} = \frac{1}{\pi} \int_{0}^{+ \infty} s^{-1/2} (s+ \sigma )^{-1} d s ,$$ for $\sigma \neq 0$. Therefore, the functional calculus gives $$\label{b18}
\varphi ( \lambda P) P^{1/2} = \frac{1}{\pi} \int_{0}^{+ \infty} s^{-1/2} \varphi ( \lambda P) P (s+P)^{-1} d s .$$
It is well known that $P\in C^1(A_0)$. In particular, $\varphi ( \lambda P) : D ( A_0 ) \longrightarrow D ( A_0 )$ and ${\mathcal A}_{\lambda}$ is well defined on $D(A_0)$. Its closure, again denoted ${\mathcal A}_{\lambda}$, is self-adjoint (see [@AmBoGe96_01 Theorem 6.2.5, Lemma 7.2.15]).
*\[PM1\] $i)$ We have $(\lambda P)^{1/2}\in C^2({\mathcal A}_{\lambda})$. The commutators $\ad^j_{{\mathcal A}_{\lambda}}(\lambda P)^{1/2}$, $j=1,2$, can be extended to bounded operators and we have, uniformly in $\lambda$, $$\begin{aligned}
\big\Vert \big[ {\mathcal A}_{\lambda} , ( \lambda P)^{1/2} \big] \big\Vert & \lesssim 1 , \label{Mi1} \\
\big\Vert \ad^2_{{\mathcal A}_{\lambda}} ( \lambda P)^{1/2} \big\Vert & \lesssim
\left\{\begin{aligned}
&1 && \rho > 1 , \\
&\lambda^{\varepsilon} && \rho \leq 1 ,
\end{aligned} \right. \label{Mi2}\end{aligned}$$ where $\varepsilon >0$ can be chosen arbitrary small.*
$ii)$ For $\lambda$ large enough, we have the following Mourre estimate: $$\label{M7}
\one_{I} ( \lambda P) \big[ i ( \lambda P)^{1/2} , {\mathcal A}_{\lambda} \big] \one_{I} ( \lambda P) \geq \frac{\delta^{2} \sqrt{\inf I}}{2} \one_{I} ( \lambda P ) .$$
$iii)$ For $0 \leq \mu \leq 1$ and $\psi \in C^{\infty}_{0} ( ] 0 , + \infty [ )$, we have $$\begin{aligned}
\big\Vert \vert {\mathcal A}_{\lambda} \vert^{\mu} \< x \>^{- \mu} \big\Vert & \lesssim \lambda^{- \mu /2 + \varepsilon} , \\
\big\Vert \< {\mathcal A}_{\lambda} \>^{\mu} \psi ( \lambda P) \< x \>^{- \mu} \big\Vert & \lesssim \lambda^{- \mu /2 + \varepsilon} ,\end{aligned}$$ for all $\varepsilon >0$.
The rest of this section will be devoted to the proof of the above proposition, which will be divided into several lemmas.
*\[LM2\] We have $(\lambda P)^{1/2} \in C^1 ( {\mathcal A}_{\lambda} )$.*
The proof of Lemma \[LM2\] is analogous to the proof of [@Ha01_01 Lemme 3.3]. In this lemma, it is shown that $( ( \lambda P)^{1/2} , {\mathcal A}_{\lambda} )$ fulfill the original conditions of Mourre which imply the $C^1$ regularity. We now have to estimate the commutators. First note that $$\big[ i ( \lambda P)^{1/2} , {\mathcal A}_{\lambda} \big] = i \lambda^{1/2} \varphi ( \lambda P) \big[ P^{1/2} , A_0 \big] \varphi ( \lambda P) .$$ Using formula , we find $$\label{b17}
\varphi ( \lambda P) \big[ P^{1/2} , A_{0} \big] \varphi ( \lambda P ) = \frac{1}{\pi} \int_{0}^{+ \infty} s^{-1/2} \varphi ( \lambda P) \big[ P (s+P)^{-1} , A_{0} \big] \varphi ( \lambda P) \, d s ,$$ with $$\big[ P (s+P)^{-1} , A_{0} \big] = - s \big[ (s+P)^{-1} , A_{0} \big] = s (s+P)^{-1} \big[ P , A_{0} \big] (s+P)^{-1} .$$ A direct calculation gives $$\begin{aligned}
\big[ P , A_{0} \big] =& - i \sum_{j,k} g^{-1} D_{j} \Big( 2 g^{2} g^{j,k} - \sum_{\ell} x_{\ell} \partial_{\ell} ( g^{2} g^{j,k} ) \Big) D_{k} g^{-1} \nonumber \\
&- i \sum_{\ell} g^{-2} x_{\ell} ( \partial_{\ell} g ) \sum_{j,k} D_{j} g^{2} g^{j,k} D_{k} g^{-1} - i \sum_{j,k} g^{-1} D_{j} g^{2} g^{j,k} D_{k} g^{-2} \sum_{\ell} x_{\ell} ( \partial_{\ell} g ) \nonumber \\
=& - 2 i P + i \sum_{j,k} g^{-1} D_{j} \Big( \sum_{\ell} x_{\ell} \partial_{\ell} ( g^{2} g^{j,k} ) - 2 g^{j,k} g \sum_{\ell} x_{\ell} ( \partial_{\ell} g ) \Big) D_{k} g^{-1} \nonumber \\
&+ \sum_{j, k, \ell} g^{-1} \partial_{j} \big( g^{-1} x_{\ell} ( \partial_{\ell} g ) \big) g^{2} g^{j,k} D_{k} g^{-1} - \sum_{j ,k, \ell} g^{-1} D_{j} g^{2} g^{j,k} \partial_{k} \big( g^{-1} x_{\ell} ( \partial_{\ell} g ) \big) g^{-1} \nonumber \\
=& -2 i P - 2 i \sum_{j,k} g^{-1} D_{j} a_{j,k} D_{k} g^{-1} + 2 i \sum_{k} b_{k} D_{k} g^{-1} - 2 i \sum_{j} g^{-1} D_{j} b_{j} , \label{c18}\end{aligned}$$ where $\partial^{\alpha}_{x} a_{j,k} = \CO ( \< x \>^{- \rho - \vert \alpha \vert} )$ and $\partial^{\alpha}_{x} b = \CO ( \< x \>^{- \rho - 1 - \vert \alpha \vert})$ by . In the following, a term $r_{j}$, $j\in \N$, will denote a smooth function such that $$\label{c17}
\partial^{\alpha}_{x} r_{j} (x) = \CO \big( \< x \>^{-\rho - j - \vert \alpha \vert} \big) .$$ Moreover, to clarify the statement, we will not write the sums over $j$, $k$ and $j,k$ and replace the remainder terms in by $\widetilde{\partial}^{*} r_{0} \widetilde{\partial}$, $\widetilde{\partial}^{*} r_{1}$ and $r_{1} \widetilde{\partial}^{*}$. Then, $$\label{c7}
[P , A_{0} ] = - 2 i \big( P + \widetilde{\partial}^{*} r_{0} \widetilde{\partial} + \widetilde{\partial}^{*} r_{1} + r_{1} \widetilde{\partial} \big) .$$ and becomes $$\begin{aligned}
\varphi ( \lambda P) \big[ P^{1/2} , A_{0} \big] \varphi ( \lambda P ) = - \frac{2 i}{\pi} \int_{0}^{+ \infty} & s^{1/2} \varphi ( \lambda P) (s+P)^{-1} \\
& \big( P + \widetilde{\partial}^{*} r_{0} \widetilde{\partial} + \widetilde{\partial}^{*} r_{1} + r_{1} \widetilde{\partial} \big) (s+P)^{-1} \varphi ( \lambda P) d s .\end{aligned}$$ Proceeding as in , one can show that $$\int_{0}^{+ \infty} s^{1/2} \varphi^{2} ( \lambda P ) P (s+P)^{-2} d s = \frac{\pi}{2} \varphi^{2} ( \lambda P ) P^{1/2} .$$ Then, we finally obtain $$\label{M7.1}
\big[ i ( \lambda P)^{1/2} , {\mathcal A}_{\lambda} \big] = ( \lambda P) ^{1/2} \varphi^{2} ( \lambda P ) + R ,$$ with $$\label{b27}
R = \frac{2}{\pi} \lambda^{1/2} \int_{0}^{+ \infty} s^{1/2} \varphi ( \lambda P) (s+P)^{-1} \big( \widetilde{\partial}^{*} r_{0} \widetilde{\partial} + \widetilde{\partial}^{*} r_{1} + r_{1} \widetilde{\partial} \big) (s+P)^{-1} \varphi ( \lambda P) \, d s .$$ The remainder term $R$ can be estimated in the following way.
*\[LM3\] Assume $0 \leq \gamma \leq d/4$. Then, we have $$\Vert R u \Vert \lesssim \lambda^{- \gamma + \varepsilon} \big\Vert \< x \>^{- \min ( \rho , d/2) + 2 \gamma + \varepsilon} u \big\Vert ,$$ for all $\varepsilon >0$ .*
First, we write $$\varphi ( \lambda P) (s+P)^{-1} \widetilde{\partial}^{*} = \lambda^{-1/2} (s+P)^{-1} \varphi ( \lambda P ) ( \lambda P )^{1/2} P^{-1/2} \widetilde{\partial}^{*} .$$ Using Lemma \[b53\] and the functional calculus, this term can be estimated by $$\label{b26}
\big\Vert \varphi ( \lambda P) (s+P)^{-1} \widetilde{\partial}^{*} \big\Vert \lesssim \lambda^{-1/2} \big\Vert (s+P)^{-1} \one_{\supp \varphi} ( \lambda P ) \big\Vert \lesssim \lambda^{-1/2} (s + \lambda^{-1})^{-1} .$$ Moreover, applying Lemma \[b61\] (with $\beta=0$ and $\gamma = 1/2 \leq d/4$), we get $$\begin{gathered}
\big\Vert \varphi ( \lambda P) (s+P)^{-1} \widetilde{\partial}^{*} u \big\Vert \lesssim (s + \lambda^{-1})^{-1} \lambda^{-1 + \varepsilon} \Vert \< x \> u \Vert , \label{b54} \\
\big\Vert \varphi ( \lambda P) (s+P)^{-1} u \big\Vert \lesssim (s + \lambda^{-1})^{-1} \lambda^{-1/2 + \varepsilon} \Vert \< x \> u \Vert . \label{b55}\end{gathered}$$
On the other hand, we write, for $k \in \N$, $$(s+P)^{-1} \varphi ( \lambda P) = \lambda ( \lambda s + \lambda P )^{-1} \psi ( \lambda P ) ( \lambda P + 1)^{-k} ,$$ with $\psi ( \sigma ) = \varphi ( \sigma ) ( \sigma +1 )^{k} \in C^{\infty}_{0} (]0 , + \infty [)$. Using the spectral theorem, we have $$\label{b22}
(s+P)^{-1} \varphi ( \lambda P) = \frac{\lambda}{\pi} \int (\lambda s + z )^{-1} \overline{\partial} \widetilde{\psi} (z) ( \lambda P -z)^{-1} (\lambda P +1)^{-k} L (d z) ,$$ where $\widetilde{\psi} \in C^{\infty}_{0} ( \C )$ is an almost analytic extension of $\psi$. From the form of $\varphi$, one can always assume that $\supp \widetilde{\psi} \subset \{ z \in \C ; \ \re z > \widetilde{\varepsilon} >0 \}$. In particular, $$\label{b23}
\vert ( \lambda s + z )^{-1} \vert \lesssim ( \lambda s + 1)^{-1} ,$$ uniformly for $z \in \supp \widetilde{\psi}$.
Using Proposition \[b45\], Lemma \[b20\] and Remark \[b59\], we obtain $$\begin{aligned}
\big\Vert \< x \>^{- \rho} ( \lambda P -z)^{-1} u \big\Vert &\lesssim \big\Vert \< x \>^{- \rho}( \lambda P -z)^{-1} \< x \>^{\min ( \rho , d/2) - \varepsilon} \< x \>^{- \min ( \rho , d/2) + \varepsilon} u \big\Vert \nonumber \\
&\lesssim \frac{1}{\vert \im z \vert^{C}} \big\Vert \< x \>^{- \min ( \rho , d/2) + \varepsilon} u \big\Vert , \label{b57}\end{aligned}$$ and $$\begin{aligned}
\big\Vert \< x \>^{- \rho} \widetilde{\partial} ( \lambda P -z)^{-1} u \big\Vert &\lesssim \big\Vert \< x \>^{- \rho} \widetilde{\partial} ( \lambda P -z)^{-1} \< x \>^{\min ( \rho , d/2) - \varepsilon} \< x \>^{- \min ( \rho , d/2) + \varepsilon} u \big\Vert \nonumber \\
&\lesssim \frac{\lambda^{-1/2}}{\vert \im z \vert^{C}} \big\Vert \< x \>^{- \min ( \rho , d/2) + \varepsilon} u \big\Vert , \label{b21}\end{aligned}$$ for all $\varepsilon >0$.
Let $\gamma \leq d/4$ and fix $k > \gamma +2$. Applying $k$ times Proposition \[b45\], we get $$\label{b25}
\big\Vert \< x \>^{- \min ( \rho , d/2) + \varepsilon} (\lambda P +1)^{-k} u \big\Vert \lesssim \lambda^{- \gamma + \varepsilon} \big\Vert \< x \>^{- \min ( \rho , d/2) + 2 \gamma + 2 \varepsilon} u \big\Vert ,$$ for all $\varepsilon >0$.
The formula and the estimates , , imply $$\begin{aligned}
\big\Vert \< x \>^{- \rho} ( & s+P)^{-1} \varphi ( \lambda P) u \big\Vert \nonumber \\
&\lesssim \lambda ( \lambda s + 1 )^{-1} \int \vert \overline{\partial} \widetilde{\psi} (z) \vert \big\Vert \< x \>^{- \rho} ( \lambda P -z)^{-1} (\lambda P +1)^{-k} u \big\Vert \, L (d z) \nonumber \\
&\lesssim \lambda ( \lambda s + 1 )^{-1} \bigg( \int \vert \im z \vert^{-C} \vert \overline{\partial} \widetilde{\psi} (z) \vert \, L (d z) \bigg) \big\Vert \< x \>^{- \min ( \rho , d/2) + \varepsilon} (\lambda P +1)^{-k} u \big\Vert \nonumber \\
&\lesssim \lambda^{1 - \gamma + \varepsilon} ( \lambda s + 1 )^{-1} \big\Vert \< x \>^{- \min ( \rho , d/2) + 2 \gamma + 2 \varepsilon} u \big\Vert , \label{b58}\end{aligned}$$ for all $\varepsilon >0$. The same way, using instead of , we obtain $$\big\Vert \< x \>^{- \rho} \widetilde{\partial} (s+P)^{-1} \varphi ( \lambda P) u \big\Vert \lesssim \lambda^{1/2 - \gamma + \varepsilon} ( \lambda s + 1 )^{-1} \big\Vert \< x \>^{- \min ( \rho , d/2) + 2 \gamma + 2 \varepsilon} u \big\Vert , \label{b28}$$ for all $\varepsilon >0$.
Let $R_{1}$ be the term of with $\widetilde{\partial}^{*} r_{0} \widetilde{\partial}$. Using $r_{0} = \CO ( \< x \>^{- \rho})$, and , we get $$\begin{aligned}
\Vert R_{1} u \Vert &\lesssim \lambda^{1/2 - \gamma + \varepsilon} \bigg( \int_{0}^{+ \infty} s^{1/2} (s + \lambda^{-1})^{-1} ( \lambda s + 1 )^{-1} d s \bigg) \big\Vert \< x \>^{- \min ( \rho , d/2) + 2 \gamma + 2 \varepsilon} u \big\Vert \nonumber \\
&\lesssim \lambda^{- \gamma + \varepsilon} \bigg( \int_{0}^{+ \infty} s^{1/2} (s + 1)^{-1} ( s + 1 )^{-1} d s \bigg) \big\Vert \< x \>^{- \min ( \rho , d/2) + 2 \gamma + 2 \varepsilon} u \big\Vert \nonumber \\
&\lesssim \lambda^{- \gamma + \varepsilon} \big\Vert \< x \>^{- \min ( \rho , d/2) + 2 \gamma + 2 \varepsilon} u \big\Vert , \end{aligned}$$ for all $\varepsilon >0$. The same estimate can be proved for the term of with $\widetilde{\partial}^{*} r_{1}$ (resp. $r_{1} \widetilde{\partial}$) from $r_{1} = \CO ( \< x \>^{- \rho -1})$, and (resp. and ).
*\[c11\] For all $\varepsilon >0$, $$\big[ \big[ ( \lambda P )^{1/2} , {\mathcal A}_{\lambda} \big] , {\mathcal A}_{\lambda} \big]= \CO ( \lambda^{\varepsilon} ) .$$*
*If we assume $\rho >1$, Lemma \[c11\] can be proved more simply. In fact, Lemma \[LM3\] and Lemma \[LM4\] give $\Vert R u \Vert \lesssim \Vert \< x \>^{-1} u \Vert$ and $\Vert {\mathcal A}_{\lambda} u \Vert \lesssim \Vert \< x \> u \Vert$. Using $R^{*} = R$, these estimates imply Lemma \[c11\].*
$\bullet$ We start with the commutator between ${\mathcal A}_{\lambda}$ and the first term on the right hand side of . Let $\psi ( \sigma ) = \sigma \varphi^{2} ( \sigma^{2})$ and $\widetilde{\psi}$ be an almost analytic extension of $\psi$. Then, we have $$\begin{aligned}
\big[ ( \lambda P )^{1/2} \varphi^{2} ( \lambda P ) , {\mathcal A}_{\lambda} \big] =& \big[ \psi \big( ( \lambda P )^{1/2} \big) , {\mathcal A}_{\lambda} \big] \\
=& - \frac{1}{\pi} \int \overline{\partial} \widetilde{\psi} (z) \big( ( \lambda P )^{1/2} -z \big)^{-1} \big[ ( \lambda P )^{1/2} , {\mathcal A}_{\lambda} \big] \big( ( \lambda P )^{1/2} -z \big)^{-1} L ( d z ) .\end{aligned}$$ From and Lemma \[LM3\], $[ ( \lambda P )^{1/2} , {\mathcal A}_{\lambda} ]$ is uniformly bounded. Therefore, the commutator $[ ( \lambda P )^{1/2} \varphi^{2} ( \lambda P ) , {\mathcal A}_{\lambda} ]$ is also uniformly bounded.
$\bullet$ We now study the commutator between ${\mathcal A}_{\lambda}$ and $R$ defined in . One can write $$\label{b65}
[ {\mathcal A}_{\lambda} , R ] = \big[ \varphi ( \lambda P ) , R \big] A_{0} \varphi ( \lambda P ) + \varphi ( \lambda P) [ A_{0} , R ] \varphi ( \lambda P ) + \varphi ( \lambda P ) A_{0} \big[ \varphi ( \lambda P ) , R \big],$$ that we note $[ {\mathcal A}_{\lambda} , R ] = S_{1} + S_{2} + S_{3}$. Since $S_{3} = - S_{1}^{*}$, we only study the two first terms.
$\star$ With in mind, the operator $S_{1}$ can be written $$\begin{aligned}
S_{1} = \frac{2}{\pi}\lambda^{1/2} \int_{0}^{+ \infty} s^{1/2} \varphi ( \lambda P) (s+P)^{-1} \big[ \varphi ( \lambda P ) , \widetilde{\partial}^{*} r_{0} \widetilde{\partial} + \widetilde{\partial}^{*} r_{1} + r_{1} \widetilde{\partial} \big] & \nonumber \\
(s+P)^{-1} \varphi ( \lambda P) A_{0} & \varphi ( \lambda P ) \, d s , \label{b66}\end{aligned}$$ where $$\begin{aligned}
\big[ \varphi( \lambda P ) , & \widetilde{\partial}^{*} r_{0} \widetilde{\partial} + \widetilde{\partial}^{*} r_{1} + r_{1} \widetilde{\partial} \big] \nonumber \\
&= - \frac{\lambda}{\pi} \int \overline{\partial} \widetilde{\varphi} (z) ( \lambda P -z)^{-1} \big[ P , \widetilde{\partial}^{*} r_{0} \widetilde{\partial} + \widetilde{\partial}^{*} r_{1} + r_{1} \widetilde{\partial} \big] ( \lambda P -z)^{-1} L ( d z ) , \label{b67}\end{aligned}$$ and $\widetilde{\varphi}$ is an almost analytic extension of $\varphi$. A direct calculation gives $$\big[ P , \widetilde{\partial}^{*} r_{0} \widetilde{\partial} + \widetilde{\partial}^{*} r_{1} + r_{1} \widetilde{\partial} \big] = \widetilde{\partial}^{*} \widetilde{\partial}^{*} r_{1} \widetilde{\partial} + \widetilde{\partial}^{*} r_{2} \widetilde{\partial} + \widetilde{\partial}^{*} r_{3} + r_{3} \widetilde{\partial} ,$$ with the convention of . For the first term in this equality, we write $$\begin{aligned}
\widetilde{\partial}^{*} \widetilde{\partial}^{*} r_{1} \widetilde{\partial} =& \widetilde{\partial}^{*} ( \lambda P +1) ( \lambda P +1)^{-1} \widetilde{\partial}^{*} r_{1} \widetilde{\partial} \\
=& ( \lambda P +1) \widetilde{\partial}^{*} ( \lambda P +1)^{-1} \widetilde{\partial}^{*} r_{1} \widetilde{\partial} - \lambda [ P , \widetilde{\partial}^{*} ] ( \lambda P +1)^{-1} \widetilde{\partial}^{*} r_{1} \widetilde{\partial} .\end{aligned}$$ As before, a direct calculation gives $$[ P , \widetilde{\partial}^{*} ] = \widetilde{\partial}^{*} r_{1} \widetilde{\partial} + \widetilde{\partial}^{*} r_{2} ,$$ with the usual decay on $r_{1}$ and $r_{2}$. Summing up, we get $$\begin{aligned}
\big[ P , \widetilde{\partial}^{*} r_{0} \widetilde{\partial} + \widetilde{\partial}^{*} & r_{1} + r_{1} \widetilde{\partial} \big] = ( \lambda P +1) \widetilde{\partial}^{*} ( \lambda P +1)^{-1} \widetilde{\partial}^{*} r_{1} \widetilde{\partial} \\
&- \lambda \big( \widetilde{\partial}^{*} r_{1} \widetilde{\partial} + \widetilde{\partial}^{*} r_{2} \big) ( \lambda P +1)^{-1} \widetilde{\partial}^{*} r_{1} \widetilde{\partial} + \widetilde{\partial}^{*} r_{2} \widetilde{\partial} + \widetilde{\partial}^{*} r_{3} + r_{3} \widetilde{\partial}^{*} .\end{aligned}$$ Applying Lemma \[b20\] (with $\beta = 1$ and $\gamma = 0$ satisfying $\gamma + \beta /2 \leq d/4$), Lemma \[b20\] (with $\beta =0$ and $\gamma = 1/2$), Remark \[b59\] and Lemma \[b53\], one can show that all the terms (say $\widetilde{r}$) of the last equation, with the exceptions of $\widetilde{\partial}^{*} r_{3}$ and $r_{3} \widetilde{\partial}$, satisfy $$\label{b68}
\big\Vert (\lambda P + 1)^ {-1} (\lambda P -z )^ {-1} \widetilde{r} (\lambda P -z )^ {-1} u \big\Vert \lesssim \frac{\lambda^{- 3/2 + \varepsilon}}{\vert \im z \vert^{C}} \big\Vert \< x \>^{-1} u \big\Vert ,$$ for all $\varepsilon >0$. Writing $$(\lambda P + 1)^ {-1} (\lambda P -z )^ {-1} \widetilde{\partial}^{*} r_{3} (\lambda P -z )^ {-1} = (\lambda P + 1)^ {-1} \< x \>^{-1} \< x \> (\lambda P -z )^ {-1} \widetilde{\partial}^{*} r_{3} (\lambda P -z )^ {-1} ,$$ and using Proposition \[b45\] (with $\beta =0$ and $\gamma = 1/2$), Lemma \[b20\] (with $\beta =1$ and $\gamma = 1/4$) and Proposition \[b45\] (with $\beta =1$ and $\gamma = 1/4$), we get $$\label{b70}
\big\Vert (\lambda P + 1)^ {-1} (\lambda P -z )^ {-1} \widetilde{\partial}^{*} r_{3} (\lambda P -z )^ {-1} u \big\Vert \lesssim \frac{\lambda^{- 3/2 + \varepsilon}}{\vert \im z \vert^{C}} \big\Vert \< x \>^{-1} u \big\Vert .$$ Note that, in the case $d=3$, we have $\gamma + \beta /2 = d /4$. It is why we can not use the additional decay $ \< x \>^{- \rho}$ and loose $\lambda^{\varepsilon}$. In a similar manner, Proposition \[b45\] (with $\beta = 0$ and $\gamma = 3/4$) and Lemma \[b20\] (with $\beta =1$ and $\gamma = 1/4$) imply $$\label{b69}
\big\Vert (\lambda P + 1)^ {-1} (\lambda P -z )^ {-1} r_{3} \widetilde{\partial} (\lambda P -z )^ {-1} u \big\Vert \lesssim \frac{\lambda^{- 3/2 + \varepsilon}}{\vert \im z \vert^{C}} \big\Vert \< x \>^{-1} u \big\Vert .$$ Combining the estimates , and with the identity , we obtain $$\label{b71}
\big\Vert (\lambda P + 1)^ {-1} \big[ \varphi( \lambda P ) , \widetilde{\partial}^{*} r_{0} \widetilde{\partial} + \widetilde{\partial}^{*} r_{1} + r_{1} \widetilde{\partial} \big] u \big\Vert \lesssim \lambda^{- 1/2 + \varepsilon} \big\Vert \< x \>^{-1} u \big\Vert .$$
From the form of $A_{0}$, we have $$A_{0} = - i g^{-1} \partial x g + a,$$ with $\partial_{x}^{\alpha} a (x) = \CO ( \< x \>^{- \vert \alpha \vert} )$. As in , we write $$\label{b72}
(s+P)^{-1} \varphi ( \lambda P) = \frac{\lambda}{\pi} \int (\lambda s + z )^{-1} \overline{\partial} \widetilde{\varphi} (z) ( \lambda P -z)^{-1} L (d z) .$$ The above expression for $A_{0}$, together with Lemma \[b20\] (with $\beta = 1$ and $\gamma =0$), Proposition \[b45\] (with $\beta =0$ and $\gamma = 1/2$) and Remark \[b59\], gives $$\big\Vert \< x \>^{-1} ( \lambda P -z)^{-1} A_{0} \big\Vert \lesssim \frac{\lambda^{- 1/2 + \varepsilon}}{\vert \im z \vert^{C}} .$$ Then, (see also ) implies $$\label{b73}
\big\Vert \< x \>^{-1} (s+P)^{-1} \varphi ( \lambda P ) A_{0} \big\Vert \lesssim \lambda^{1/2 + \varepsilon} (\lambda s +1)^{-1} ,$$ for all $\varepsilon >0$.
Eventually, using the identity , the functional calculus and the estimates and , we obtain $$\begin{aligned}
\Vert S_{1} \Vert \lesssim& \lambda^{1/2} \int_{0}^{+ \infty} s^{1/2} (s + \lambda^{-1} )^{-1} \lambda^{-1/2 + \varepsilon} \lambda^{1/2 + \varepsilon} (\lambda s +1)^{-1} d s \nonumber \\
\lesssim& \lambda^{3/2 + 2 \varepsilon} \int_{0}^{+ \infty} s^{1/2} ( \lambda s +1)^{-2} d s \lesssim \lambda^{2 \varepsilon} \int_{0}^{+ \infty} t^{1/2} ( t +1)^{-2} d t \lesssim \lambda^{2 \varepsilon} , \label{b74}\end{aligned}$$ for all $\varepsilon >0$.
$\star$ We now study $S_{2} = \varphi ( \lambda P) [ A_{0} , R ] \varphi ( \lambda P )$. Using , $S_{2}$ can be decomposed as $$\label{b75}
S_{2} = T_{1} + T_{2} + T_{3} ,$$ with $$\begin{aligned}
T_{1} =& \frac{2}{\pi^{2}} \lambda^{5/2} \iint_{0}^{+ \infty} s^{1/2} \overline{\partial} \widetilde{\varphi} ( z ) ( \lambda s + z)^{-1} \varphi ( \lambda P ) ( \lambda P - z)^{-1} [ P , A_{0} ] ( \lambda P - z)^{-1} \\
&\hspace{160pt} \big( \widetilde{\partial}^{*} r_{0} \widetilde{\partial} + \widetilde{\partial}^{*} r_{1} + r_{1} \widetilde{\partial} \big) (s+P)^{-1} \varphi^{2} ( \lambda P) \, d s \, L (d z ) , \\
T_{2} =& \frac{2}{\pi} \lambda^{1/2} \int_{0}^{+ \infty} s^{1/2} \varphi^{2} ( \lambda P) (s+P)^{-1} \big[ A_{0} , \widetilde{\partial}^{*} r_{0} \widetilde{\partial} + \widetilde{\partial}^{*} r_{1} + r_{1} \widetilde{\partial} \big] (s+P)^{-1} \varphi^{2} ( \lambda P ) \, d s , \\
T_{3} =& \frac{2}{\pi^{2}} \lambda^{5/2} \iint_{0}^{+ \infty} s^{1/2} \overline{\partial} \widetilde{\varphi} ( z ) ( \lambda s +z )^{-1} \varphi^{2} ( \lambda P) (s+P)^{-1} \big( \widetilde{\partial}^{*} r_{0} \widetilde{\partial} + \widetilde{\partial}^{*} r_{1} + r_{1} \widetilde{\partial} \big) \\
&\hspace{175pt} ( \lambda P - z)^{-1} [ P , A_{0} ] ( \lambda P - z)^{-1} \varphi ( \lambda P ) \, d s \, L (d z ) .\end{aligned}$$ Since $T_{3}^{*} = T_{1}$, we only treat $T_{1}$ and $T_{2}$.
From , we know that $$[P , A_{0} ] = - 2 i P + \widetilde{\partial}^{*} r_{0} \widetilde{\partial} + \widetilde{\partial}^{*} r_{1} + r_{1} \widetilde{\partial} .$$ Let $\widetilde{r}$ be a term of the last equation and let $\widehat{r}$ be a term of the sum $$\widetilde{\partial}^{*} r_{0} \widetilde{\partial} + \widetilde{\partial}^{*} r_{1} + r_{1} \widetilde{\partial} .$$ Then the functional calculus, Proposition \[b45\] (with $\beta =0$ and $\gamma =1/2$), Lemma \[b20\] (with $\beta =0$ and $\gamma =1/2$), Remark \[b59\] and Lemma \[b53\] show that $$\big\Vert ( \lambda P -z )^{-1} \widetilde{r} ( \lambda P -z )^{-1} \widehat{r} ( \lambda P +1 )^{-1} \big\Vert \lesssim \frac{\lambda^{-2 + \varepsilon}}{\vert \im z \vert^{C}} ,$$ for all $\varepsilon >0$. Then, $T_{1}$ becomes $$\label{b76}
\Vert T_{1} \Vert \lesssim \lambda^{5/2} \int_{0}^{+ \infty} s^{1/2} (1 + \lambda s)^{-1} \lambda^{-2 + \varepsilon} (s + \lambda^{-1} )^{-1} d s \lesssim \lambda^{\varepsilon} ,$$ for all $\varepsilon >0$.
A direct calculation shows that $$\big[ A_{0} , \widetilde{\partial}^{*} r_{0} \widetilde{\partial} + \widetilde{\partial}^{*} r_{1} + r_{1} \widetilde{\partial} \big] = \widetilde{\partial}^{*} r_{0} \widetilde{\partial} + \widetilde{\partial}^{*} r_{1} + r_{1} \widetilde{\partial} + r_{2} .$$ From Proposition \[b45\] (with $\beta =0$ and $\gamma =1/2$) and Lemma \[b53\], every term (say $\widetilde{r}$) of the previous equation satisfies $$\big\Vert ( \lambda P +1)^{-1} \widetilde{r} ( \lambda P +1)^{-1} \big\Vert \lesssim \lambda^{-1 + \varepsilon} .$$ Then, using the spectral theorem, $T_{2}$ fulfills $$\label{b77}
\Vert T_{2} \Vert \lesssim \lambda^{1/2} \int_{0}^{+ \infty} s^{1/2} ( s + \lambda^{-1})^{-1} \lambda^{-1 + \varepsilon} (s + \lambda^{-1} )^{-1} d s \lesssim \lambda^{\varepsilon} ,$$
Combining with the estimates , and $T_{3}^{*} = T_{1}$, we obtain $$\label{b78}
\Vert S_{2} \Vert \lesssim \lambda^{\varepsilon} .$$
$\star$ The lemma follows from , , and $S_{3} = - S_{1}^{*}$.
*\[LM4\] Let $0 \leq \mu \leq 1$ and $\psi \in C^{\infty}_{0} ( ] 0 , + \infty [ )$. Then, we have $$\begin{gathered}
\big\Vert \vert {\mathcal A}_{\lambda} \vert^{\mu} \< x \>^{- \mu} \big\Vert \lesssim \lambda^{- \mu /2 + \varepsilon} , \label{M11} \\
\big\Vert \< {\mathcal A}_{\lambda} \>^{\mu} \psi ( \lambda P) \< x \>^{- \mu} \big\Vert \lesssim \lambda^{- \mu /2 + \varepsilon} , \label{M12}\end{gathered}$$ for all $\varepsilon >0$.*
From , we have $$\label{b30}
{\mathcal A}_{\lambda} = \varphi ( \lambda P ) \Big( g^{-1} D x g + i \Big( \frac{d}{2} + g^{-1} x ( \partial g ) \Big) \Big) \varphi ( \lambda P) .$$ Lemma \[b53\] gives $$\varphi ( \lambda P ) g^{-1} D = P^{1/2} \varphi ( \lambda P ) P^{-1/2} g^{-1} D = \CO ( \lambda^{-1/2} ).$$ Moreover, Lemma \[b61\] (with $\beta =1$ and $\gamma =0$) implies $$\big\Vert \< x \> \varphi ( \lambda P) u \big\Vert \lesssim \lambda^{\varepsilon} \Vert \< x \> u \Vert ,$$ for all $\varepsilon >0$. Summing up the previous estimates, we get $$\label{b31}
\big\Vert \varphi ( \lambda P ) g^{-1} D x g \varphi ( \lambda P) u \big\Vert \lesssim \lambda^{- 1/2 + \varepsilon} \Vert \< x \> u \Vert ,$$ for all $\varepsilon >0$. Using Lemma \[b61\] (with $\beta =0$ and $\gamma = 1/2$) and that $x ( \partial g )$ is bounded by , we obtain $$\label{b32}
\Big\Vert \varphi ( \lambda P) \Big( \frac{d}{2} + g^{-1} x ( \partial g ) \Big) \varphi ( \lambda P) u \Big\Vert \lesssim \big\Vert \varphi ( \lambda P )u \big\Vert \lesssim \lambda^{- 1/2 + \varepsilon} \Vert \< x \> u \Vert .$$ The inequality follows from , and for $\mu=1$ and for $0\leq \mu \leq 1$ by interpolation. To prove , we write $$\big\Vert \< {\mathcal A}_{\lambda} \>^{\mu} \psi ( \lambda P) \< x \>^{- \mu} \big\Vert \lesssim \big\Vert \psi ( \lambda P) \< x \>^{- \mu} \big\Vert + \big\Vert \vert {\mathcal A}_{\lambda} \vert^{\mu} \psi ( \lambda P) \< x \>^{- \mu} \big\Vert \lesssim \lambda^{- \mu /2 + \varepsilon} ,$$ where we have again used Lemma \[b61\] with $\beta =0$ and $\gamma = 1/2$.
*\[LM7\] For $\lambda$ large enough, we have $$\one_{I} ( \lambda P) \big[ i ( \lambda P)^{1/2} , {\mathcal A}_{\lambda} \big] \one_{I} ( \lambda P) \geq \frac{\delta^{2} \sqrt{\inf I}}{2} \one_{I} ( \lambda P ) .$$*
Recall that gives $$\big[ i ( \lambda P)^{1/2} , {\mathcal A}_{\lambda} \big] = ( \lambda P) ^{1/2} \varphi^{2} ( \lambda P) + R .$$ On the other hand, we know by Lemma \[LM3\] that $\Vert Ru \Vert \lesssim \lambda^{- \widetilde{\varepsilon}} \Vert u \Vert$ for some $\widetilde{\varepsilon} >0$. Using $\varphi (x) > \delta >0$ on $I$ and taking $\lambda$ large enough, we get the lemma.
Intermediate frequency Mourre estimate
---------------------------------------
${}^{}$
\[secM2\]
Here, we obtain a Mourre estimate for frequencies inside the compact $[ 1/C , C ]$. For that, we will use a standard argument in scattering theory. Mimicking Section \[secM1\], we set $${\mathcal A} = \varphi (P) A_{0} \varphi (P) ,$$ where $\varphi \in C^{\infty}_{0} ( ] 0 , + \infty [ ; [0, + \infty [)$ with $\varphi =1$ near $[ 1/C , C ]$. As before, ${\mathcal A}$ is essentially self-adjoint on $D(A_0)$ and we denote again ${\mathcal A}$ its closure.
*\[PM2\] $i)$ We have $P^{1/2}\in C^2({\mathcal A})$. The commutators $\ad^j_{{\mathcal A}} P^{1/2}$, $j=1,2$, can be extended to bounded operators.*
$ii)$ For each $\sigma \in [ 1/C , C ]$, there exists $\delta >0$ such that $$\label{c12}
\one_{[ \sigma - \delta , \sigma + \delta ]} (P) \big[ i P^{1/2} , {\mathcal A} \big] \one_{[ \sigma - \delta , \sigma + \delta ]} (P) \geq \frac{1}{2 \sqrt{C}} \one_{[ \sigma - \delta , \sigma + \delta ]} (P) .$$
$iii)$ For $0 \leq \mu \leq 1$, we have $$\big\Vert \< {\mathcal A} \>^{\mu} \< x \>^{- \mu} \big\Vert \lesssim 1 .$$
The points $i)$ and $iii)$ follow directly from Proposition \[PM1\] with $\lambda =1$. Moreover, using and Lemma \[LM3\], we get $$\big[ i P^{1/2} , {\mathcal A} \big] = P^{1/2} \varphi^{2} (P) + R ,$$ where $\Vert R u \Vert \lesssim \Vert \< x \>^{- \nu} u \Vert$ for some $\nu >0$. Then, $R \varphi (P)$ is a compact operator on $L^{2} ( \R^{d} )$. Let $\sigma \in [ 1/C , C ]$. Since $\sigma$ is not an eigenvalue of $P$ (see [@Do97_01 Corollary 5.4]), we have $$\slim_{\delta \to 0} \one_{[ \sigma - \delta , \sigma + \delta ]} (P) =0 .$$ Thus, we obtain $$\lim_{\delta \to 0} \one_{[ \sigma - \delta , \sigma + \delta ]} (P) R \varphi (P) =0 ,$$ in operator norm. Using $$\begin{aligned}
\one_{[ \sigma - \delta , \sigma + \delta ]} (P) \big[ i P^{1/2} , {\mathcal A} \big] \one_{[ \sigma - \delta , \sigma + \delta ]} (P) \geq & \sqrt{\sigma - \delta} \one_{[ \sigma - \delta , \sigma + \delta ]} (P) \\
&+ \one_{[ \sigma - \delta , \sigma + \delta ]} (P) R \varphi (P) \one_{[ \sigma - \delta , \sigma + \delta ]} (P) ,\end{aligned}$$ part $ii)$ of the proposition follows.
High frequency Mourre estimate
-------------------------------
${}^{}$
In this section, we construct a conjugate operator at high frequencies. We work with the simple $\sigma$–temperate metric $$\gamma = \frac{d x^{2}}{1 + x^{2}} + \frac{d \xi^{2}}{1 + \xi^{2}} .$$ We refer to [@Ho85_01 Section XVIII] for the Weyl calculus of Hörmander. For $m (x , \xi )$ a weight function, let $S (m)$ be the set of functions $f \in C^{\infty} ( \R^{d} \times \R^{d})$ such that $$\vert \partial_{x}^{\alpha} \partial_{\xi}^{\beta} f (x, \xi ) \vert \lesssim m (x , \xi ) \< x \>^{- \vert \alpha \vert} \< \xi \>^{- \vert \beta \vert} ,$$ for all $\alpha , \beta \in \N^{d}$. In fact, $S (m)$ is the space of symbols of weight $m$ for the metric $\gamma$. Let $\Psi (m)$ denote the set of pseudo-differential operators whose symbols are in $S (m)$.
Let $p (x, \xi ) \in S ( \< \xi \>^{2} )$ be the symbol of $P$, and $$p_{0} (x, \xi ) = \sum_{j , k} g^{j , k} (x) \xi_{j} \xi_{k} \in S ( \< \xi \>^{2} ) ,$$ be its principal part. We have $p - p_{0} \in S ( 1)$. Let $${\rm H}_{p_{0}} = \left( \begin{array}{c}
\partial_{\xi} p_{0} \\
- \partial_{x} p_{0}
\end{array} \right) ,$$ be the Hamiltonian of $p_{0}$. Since the metric $\mathfrak{g}$ is non-trapping by assumption, the energy $\{ p_{0} =1 \}$ is non-trapping for the Hamiltonian flow of $p_{0}$. Then, using a result of C. Gérard and Martinez [@GeMa88_01], one can construct a function $b (x, \xi ) \in S ( \< x \> \< \xi \> )$ such that $b = x \cdot \xi$ for $x$ large enough, and $$\label{a9}
H_{p_{0}} b \geq \delta ,$$ for some $\delta >0$ and all $(x, \xi ) \in p_{0}^{-1} ( [ 1 - \varepsilon , 1 + \varepsilon ])$, $\varepsilon >0$. We set $A = \Op (a)$ with $$a (x, \xi ) = b \big( x , (p_{0} +1)^{-1/2} \xi \big) \in S ( \< x \> ) .$$ Let $f \in C^{\infty} ( \R ; \R)$ be such that $f =1$ on $[2 , + \infty [$ and $f=0$ on $] - \infty , 1]$. As conjugate operator at high frequency, we choose $$A_{\infty} = f (P) A f (P) .$$
Let $\varphi \in C^{\infty}_{0} ( \R )$ satisfy $\varphi + f =1$ on $[-1 , + \infty [$. Since $P \geq 0$, we have $f (P) = 1 - \varphi ( P)$. On the other hand, from the functional calculus of pseudo-differential operators, $\varphi (P) \in \Psi ( \< \xi \>^{- \infty} )$ and then $f(P) \in \Psi ( 1 )$. To prove this assertion, one can, for instance, adapt Theorem 8.7 of [@DiSj99_01] or Section D.11 of [@DeGe97_01] to the metric $\gamma$. In particular, $A_{\infty}$ is well defined as a pseudo-differential operator, and we have $$\label{a28}
A_{\infty} = A + \Psi ( \< x \> \< \xi \>^{- \infty} ) \in \Psi ( \< x \> ).$$ The following proposition summarizes the useful properties of $A_{\infty}$.
*\[a26\] $i)$ The operator $A_{\infty}$ is essentially self-adjoint on any core of $\< x \>$ with $D ( \< x \> ) = \{ u \in L^{2} ( \R^{d}) ; \ \< x \> u \in L^{2} ( \R^{d} ) \}$. Moreover, $$\Vert A_{\infty} u \Vert \lesssim \Vert \< x \> u \Vert ,$$ for all $u \in D ( \< x \> )$.*
$ii)$ We have $P^{1/2} \in C^{2} (A_{\infty})$. The commutators $[P^{1/2} ,A_{\infty}]$ and $[ [P^{1/2} ,A_{\infty}] ,A_{\infty} ]$ are in $\Psi (1)$ and can be extended as bounded operators on $L^{2} ( \R^{d} )$.
$iii)$ For $C >0$ large enough, $$\one_{[C , + \infty [} (P) i [ P^{1/2} , A_{\infty} ] \one_{[C , + \infty [} (P) \geq \frac{\delta}{8} \one_{[C , + \infty [} (P) .$$
The rest of this subsection is devoted to the proof of this proposition. It is a direct consequence of the next lemmas. For the first part of the proposition, we will use the following extension of Nelson’s theorem due to C. Gérard and [Ł]{}aba [@GeLa02_01 Lemma 1.2.5] (see also Reed and Simon [@ReSi75_01 Theorem X.36]).
*\[a12\] Let ${\mathcal H}$ be a Hilbert space, $N\geq 1$ a self-adjoint operator on ${\mathcal H}$, $H$ a symmetric operator such that $D(N) \subset D(H)$ and $$\begin{gathered}
\Vert H u \Vert \lesssim \Vert N u \Vert , \\
\vert (H u,N u) - (N u,H u) \vert \lesssim \Vert N^{1/2} u \Vert^{2} ,\end{gathered}$$ for all $u \in D ( N )$. Then, $H$ is essentially self-adjoint on any core of $N$.*
*\[a14\] The operator $A_{\infty}$ is essentially self-adjoint on any core of $(\< x \> , D ( \< x \> ))$ (in particular, on the Schwartz space ${\mathcal S} ( \R^{d} )$) and $$\Vert A_{\infty} u \Vert \lesssim \Vert \< x \> u \Vert ,$$ for all $u \in D ( \< x \> )$.*
The operator $N = \< x \>$ is self-adjoint on $D (N) = D ( \< x\> )$ and essentially self-adjoint on ${\mathcal S} ( \R^{d} )$. Since $A_{\infty} \in \Psi ( \< x \> )$ and $N^{-1} \in \Psi ( \< x \>^{-1} )$, the operator $A_{\infty} N^{-1} \in \Psi (1)$ is bounded on $L^{2} ( \R^{d} )$ by Calderon and Vaillancourt’s theorem. Then, $A_{\infty}$ is defined on $D ( N)$ and $$\Vert A_{\infty} u \Vert \lesssim \Vert N u \Vert ,$$ for all $u \in D(N)$.
By pseudo-differential calculus, $\< x \>^{-1/2} [ A_{\infty} , \< x \> ] \< x \>^{-1/2} \in \Psi ( \< \xi \>^{-1} )$ is bounded on $L^{2} ( \R^{d} )$. Then, working first on ${\mathcal S} ( \R^{d} )$, this gives $$\vert (A u,N u) - (N u,A u) \vert \lesssim \Vert N^{1/2} u \Vert^{2} ,$$ for all $u \in D(N)$. Thus, Theorem \[a12\] implies that $A_{\infty}$ is essentially self-adjoint on any core of $D ( \< x \> )$.
*\[a13\] Let $g \in C^{\infty} ( \R ; [0 , + \infty [ )$ be such that $g = 0$ on $] - \infty , a ]$ and $g =1$ on $[ b , + \infty [$, for some $0 < a < b$. Then, $$g (P) P^{1/2} = \Op \big( ( p_{0} + 1 )^{1/2} \big) + \Psi ( 1 ) \in \Psi ( \< \xi \> ) .$$*
We omit the proof of this classical result. It follows from and the Beals lemma [@Be81_01]. We refer to Section 4.4 of [@HeNi05_01] for similar arguments (see also [@Be81_01 Theorem 4.9]).
*For the subsequent uses, a parametrix will be enough. In fact, since we work with the metric $\gamma$, the remainder terms will decay like $\< ( x , \xi ) \>^{- \infty}$. Therefore, they can “absorb” the pseudo-differential operators of any weight. In particular, this allows to treat the commutators.*
*\[a18\] We have $[P^{1/2} ,A_{\infty}] \in \Psi (1)$ and $[ [P^{1/2} ,A_{\infty}] ,A_{\infty} ] \in \Psi ( \< \xi \>^ {-1})$. These commutators extend as bounded operators on $L^{2} ( \R^{d} )$.*
Let $g \in C^{\infty} ( \R )$ as in Lemma \[a13\] be such that $f g = f$. Then, $$\label{a27}
\big[ P^{1/2} ,A_{\infty} \big] = \big[ g (P) P^{1/2} ,A_{\infty} \big] .$$ Since $g (P) P^{1/2} \in \Psi ( \< \xi \> )$ by Lemma \[a13\] and $A_{\infty} \in \Psi ( \< x \> )$, the pseudo-differential calculus gives $[ g (P) P^{1/2} ,A_{\infty} ] \in \Psi (1)$. The same way, $[ [P^{1/2} ,A_{\infty}] ,A_{\infty} ] \in \Psi ( \< \xi \>^ {-1})$. Using Calderon and Vaillancourt’s theorem and working first on ${\mathcal S} ( \R^{d})$ which is dense in $D ( \< x \> ) \cap H^{1} ( \R^{d} )$, one can prove that these operators extend as bounded operator on $L^{2} ( \R^{d} )$.
*We have $P^{1/2} \in C^{2} ( A_{\infty} )$.*
Let $H = \< D \> = \Op ( \< \xi \> ) \in \Psi ( \< \xi \> )$ be the self-adjoint operator with domain $D (H) = D (P^{1/2} ) = H^{1} ( \R^{d} )$ (see Lemma \[b53\]). We remark that $(H \pm z)^{-1} = \Op ( (\< \xi \> \pm z)^{-1} ) \in \Psi (1)$ is a Fourier multiplier. Thus, $D ( \< x \> )$, which is a core of $A_{\infty}$ from Lemma \[a14\], is stable by $(H \pm z)^{-1}$. On the other hand, $[ H , A_{\infty}] \in \Psi (1)$ can be extend as a bounded operator on $L^{2} ( \R^{d} )$. Then, Theorem \[a15\] implies $H \in C^{1} ( A_{\infty} )$.
Since $H \in C^{1} ( A_{\infty} )$ and $[ H , A_{\infty}]$ is bounded on $L^{2} ( \R^{d})$, Lemma \[a17\] says that $e^{i t A_{\infty}}$ leaves $D (H)$ invariant. Then, $e^{i t A_{\infty}}$ leaves $D (P^{1/2}) = D (H)$ invariant and $[ P^{1/2} , A_{\infty}]$ is bounded from Lemma \[a18\]. Then, Theorem \[a16\] implies that $P^{1/2} \in C^{1} ( A_{\infty} )$.
The lemma follows from Theorem \[a16\], Remark \[a7\] and Lemma \[a18\].
*For $C >0$ large enough, $$\one_{[C , + \infty [} (P) i [ P^{1/2} , A_{\infty} ] \one_{[C , + \infty [} (P) \geq \frac{\delta}{8} \one_{[C , + \infty [} (P) .$$*
Equation and , Lemma \[a13\] and the pseudo-differential calculus give $$\begin{aligned}
i \big[ P^{1/2} , A_{\infty} \big] =& i \big[ g (P) P^{1/2} , A_{\infty} \big] \nonumber \\
=& i \big[ \Op \big( ( p_{0} + 1)^{1/2} \big) , \Op ( a) \big] + \Psi ( \< \xi \>^{-1} ) \nonumber \\
=& \frac{1}{2} \Op \big( ( p_{0} + 1)^{-1/2} {\rm H}_{p_{0}} a \big) + \Psi ( \< \xi \>^{-1} ) \nonumber \\
=& \frac{1}{2} \Op \Big( (p_{0} +1)^{-1/2} (\partial_{\xi} p_{0}) (x, \xi ) \cdot ( \partial_{x} b ) \big( x , (p_{0} +1)^{-1/2} \xi \big) \nonumber \\
&- (p_{0} +1)^{-1} (\partial_{x} p_{0}) (x, \xi ) \cdot ( \partial_{\xi} b ) \big( x , (p_{0} +1)^{-1/2} \xi \big) \Big) + \Psi ( \< \xi \>^{-1} ) \nonumber \\
=& \frac{1}{2} \Op \big( ( {\rm H}_{p_{0}} b ) \big( x , (p_{0} +1)^{-1/2} \xi \big) \big) + \Psi ( \< \xi \>^{-1} ) . \label{a8}\end{aligned}$$ For the last equality, we have used that $p_{0}$ is a homogeneous polynomial of order $2$ in $\xi$.
Note that $$p_{0} \big( x , (p_{0} +1)^{-1/2} \xi \big) = (p_{0} +1)^{-1} p_{0} \in [1 - \varepsilon , 1 + \varepsilon ],$$ for $\xi$ large enough. Then, adding a cut-off function in $\xi$, and imply that $$i \big[ P^{1/2} , A_{\infty} \big] = \Op ( c ) + \Psi ( \< \xi \>^{-1} ) ,$$ with $c \in S (1)$ and $c (x, \xi ) \geq \delta /2$. We write $c (x , \xi ) = \delta / 4 + d^{2} (x, \xi )$ with $d \in S (1)$ real valued. Thus, by the pseudo-differential calculus, $$\begin{aligned}
i \big[ P^{1/2} , A_{\infty} \big] &\geq \delta / 4 + \Op ( d)^{*} \Op ( d) + \Psi ( \< \xi \>^{-1} ) \nonumber \\
&\geq \delta / 4 + \Psi ( \< \xi \>^{-1} ) , \label{a10}\end{aligned}$$ as self-adjoint operators (one can also apply the G[å]{}rding inequality).
Let $R \in \Psi ( \< \xi \>^{-1} )$. Since $P \in \Psi ( \< \xi \>^{2})$, the operator $$R^{*} ( P +1 ) R \in \Psi (1) ,$$ is a bounded operator on $L^{2} ( \R^{d} )$. Then, $( P +1)^{1/2} R$ is also bounded on $L^{2} ( \R^{d} )$. In particular, we have $$\begin{aligned}
\big\Vert \one_{[C , + \infty [} (P) R \big\Vert =& \big\Vert \one_{[C , + \infty [} (P) ( P +1)^{-1/2} ( P +1)^{1/2} R \big\Vert \nonumber \\
\lesssim& \big\Vert \one_{[C , + \infty [} (P) ( P +1)^{-1/2} \big\Vert \big\Vert ( P +1)^{1/2} R \big\Vert \nonumber \\
\lesssim& ( C +1)^{-1/2} . \label{a11}\end{aligned}$$ The lemma follows from and .
Proof of the linear estimates {#EW}
=============================
In this section, we will show the main estimates for the linear wave equation (Theorem \[TSLW2\] and Theorem \[TW1\]). To prove these results, we will make a dyadic decomposition of the low frequencies. We will often consider $\varphi \in C^{\infty}_{0} ( ] 0 , + \infty [ ; [0, + \infty [)$ such that $$\label{c13}
\sum_{\lambda = 2^{n} , \ n \geq 0} \varphi ( \lambda x) = 1 ,$$ for $x \in ] 0 , 1 ]$. To $\varphi$, we will associate $\widetilde{\varphi} \in C^{\infty}_{0} ( ] 0 , + \infty [ ; [0, + \infty [)$ satisfying $\widetilde{\varphi} \varphi = \varphi$.
We begin with a technical lemma which proves Remark \[RW1\] $ii)$.
*\[LW5\] For all $\widetilde{\mu} < \mu \leq 3/2$, we have $$\begin{gathered}
\big\Vert \<x\>^{-\mu} \widetilde{\partial}_{\ell} u \big\Vert \lesssim \big\Vert \<x\>^{- \widetilde{\mu}} P^{1/2} u \big\Vert , \\
\big\Vert \<x\>^{- \mu} P^{1/2} u \big\Vert \lesssim \sum_{\ell =1}^{d} \big\Vert \<x\>^{- \widetilde{\mu}} \widetilde{\partial}_{\ell} u \big\Vert .\end{gathered}$$*
Since the two inequalities can be treated the same way, we only prove the first one. We write $$\big\Vert \<x\>^{-\mu} \widetilde{\partial}_{\ell} u \big\Vert \leq \big\Vert \<x\>^{-\mu} \widetilde{\partial}_{\ell} \Psi (P \leq C) u \big\Vert + \big\Vert \<x\>^{-\mu} \widetilde{\partial}_{\ell} \Psi (P \geq C) u \big\Vert =: I_1 + I_2 .$$
$\bullet$ We first estimate $I_1$. Let $\varphi$ be as in . For $\widetilde{\mu} < \mu$, we have, using Lemma \[b61\], $$\begin{aligned}
I_1 \lesssim& \sum_{\lambda \text{ dyadic}} \big\Vert \< x \>^{- \mu} \widetilde{\partial}_{\ell} \varphi (\lambda P) u \big\Vert \\
=& \sum_{\lambda \text{ dyadic}} \big\Vert \< x \>^{- \mu} ( \lambda^{1/2} \widetilde{\partial}_{\ell} ) \varphi ( \lambda P) ( \lambda P)^{-1/2} \< x \>^{\widetilde{\mu}} \< x \>^{- \widetilde{\mu}} P^{1/2}u \big\Vert \\
\lesssim& \sum_{\lambda \text{ dyadic}} \lambda^{-\frac{\mu-\widetilde{\mu}}{2} +\varepsilon} \big\Vert \< x \>^{-\widetilde{\mu}} P^{1/2} u \big\Vert \\
\lesssim& \big\Vert \< x \>^{-\widetilde{\mu}} P^{1/2} u \big\Vert ,\end{aligned}$$ for $\varepsilon$ small enough.
$\bullet$ We now estimate $I_{2}$. By Lemma \[a13\] and the pseudo-differential calculus, we know that the operator $$\<x\>^{-\mu} \widetilde{\partial}_{\ell} \Psi (P \geq C) P^{-1/2} \<x\>^{\mu} ,$$ is bounded. Therefore, $$I_{2} = \big\Vert \<x\>^{-\mu} \widetilde{\partial}_{\ell} \Psi (P \geq C) P^{-1/2} \< x \>^{\mu} \< x \>^{- \mu} P^{1/2} u \big\Vert \lesssim \big\Vert \< x \>^{- \mu} P^{1/2} u \big\Vert .$$
Using the same type of proof, one can show the following estimate.
*\[c16\] For all $\mu >1$, we have $$\big\Vert \< x \>^{- \mu} u \big\Vert \lesssim \big\Vert P^{1/2} u \big\Vert .$$*
*\[RW2\] Let $\mu>0$ be given. Then, for all $\varepsilon >0$, there exist $0< \widetilde{\mu} < \mu$, $0 < \widetilde{\varepsilon} < \varepsilon$ such that $F^{\widetilde{\varepsilon}}_{\widetilde{\mu}} (T) \leq F^{\varepsilon}_{\mu} (T)$. Then, it is sufficient to bound the different quantities we consider by $F^{\varepsilon}_{\widetilde{\mu}}(T)$ rather than by $F^{\varepsilon}_{\mu} (T)$.*
Theorem \[TSLW2\] will follow from the corresponding result for the group $e^{-i t P^{1/2}}$.
*\[propEW2\] Let $0 < \mu \leq 1$. Then, for all $0 < \varepsilon < \mu$, we have $$\label{EW3}
\int_0^T \big\Vert \<x\>^{- \mu} e^{-i t P^{1/2}} v \big\Vert^2 d t \lesssim F_{\mu}^{\varepsilon} (T) \Vert v \Vert^2 .$$*
We write $$\begin{aligned}
\int_0^T \big\Vert \< x \>^{- \mu} e^{-i t P^{1/2}} v \big\Vert^2 d t \lesssim & \int_0^T \big\Vert \< x \>^{- \mu} e^{-i t P^{1/2}} \Psi (P \leq 1/C) v \big\Vert^2 d t \\
&+ \int_0^T \big\Vert \< x \>^{- \mu} e^{-i t P^{1/2}} \Psi ( 1/C \leq P \leq C) v \big\Vert^2 d t \\
&+ \int_0^T \big\Vert \< x \>^{- \mu} e^{-i t P^{1/2}} \Psi ( P \geq C ) v \big\Vert^2 d t =: I_{1} + I_{2} + I_{3} .\end{aligned}$$
$\bullet$ We first estimate $I_1$. Let $\varphi$, $\widetilde{\varphi}$ be as in . Proposition \[PM1\] gives $$\big\Vert \< x \>^{- \mu} \widetilde{\varphi} ( \lambda P) \< {\mathcal A}_{\lambda} \>^{\mu} \big\Vert^2 \lesssim \lambda^{- \mu + \varepsilon_1} ,$$ for all $\varepsilon_{1} >0$. Then, $$\begin{aligned}
I_1 \lesssim& \sum_{\lambda \text{ dyadic}}\lambda^{-\mu+\varepsilon_1}\int_0^T \big\Vert \< {\mathcal A}_{\lambda}\>^{-\mu}e^{-i t P^{1/2}} \varphi ( \lambda P) v \big\Vert^2 d t \\
=& \sum_{\lambda \text{ dyadic}} \lambda^{- \mu + \varepsilon_1 + 1/2} \int_0^{\lambda^{-1/2} T} \big\Vert \< {\mathcal A}_{\lambda} \>^{- \mu} e^{-i s (\lambda P)^{1/2}}\varphi ( \lambda P) v \big\Vert^2 d s \\
\lesssim& \sum_{\lambda \text{ dyadic}} \lambda^{- \mu + \varepsilon_1 +1/2+ \varepsilon_3} F_{\mu}^{\varepsilon_2} (\lambda^{-1/2} T)\Vert v \Vert^2,\end{aligned}$$ for all $\varepsilon_{2}, \varepsilon_{3} >0$ with $\varepsilon_{2} < \mu$. Here, we have used Proposition \[PM1\], Remark \[a6\], Theorem \[a1\] (for $\mu > 1/2$) and Corollary \[a2\] (for $\mu \leq 1/2$) with $H = ( \lambda P )^{1/2}$.
- If $\mu>1/2$, then, by choosing $\varepsilon_1 , \varepsilon_3$ small enough, the sum is convergent and we find $$I_1 \lesssim \Vert v \Vert^2 .$$
- If $\mu\le1/2$, we find $$I_1 \lesssim \sum_{\lambda \text{ dyadic}} \lambda^{\varepsilon_1 + \varepsilon_3 - \varepsilon_2} T^{1-2\mu+2\varepsilon_2} \Vert v \Vert^2 .$$ Fixing first $\varepsilon_2$ and then $\varepsilon_1 , \varepsilon_3$ small enough makes the sum convergent.
$\bullet$ We now treat $I_{2}$. Since $[1/C , C]$ is a compact interval, Proposition \[PM2\] gives us a finite number of open intervals $I_{k}$, $k =1 , \ldots , K$, satisfying and $$[1/C , C] \subset \bigcup_{k=1}^{K} I_{k} .$$ Then, applying Theorem \[a1\] (for $\mu > 1/2$) and Corollary \[a2\] (for $\mu \leq 1/2$) on each $I_{k}$ (slightly reduced), we obtain $$I_{2} \lesssim F_{\mu}^{\varepsilon} (T) \Vert v \Vert^2 .$$
$\bullet$ Let us finally estimate $I_{3}$. By Proposition \[a26\] and an interpolation argument, we get $$\big\Vert \< x \>^{- \mu} \< {\mathcal A}_{\infty} \>^{\mu} \big\Vert \lesssim 1.$$ Thus, $$\begin{aligned}
\int_0^T \big\Vert \< x \>^{- \mu} e^{-i t P^{1/2}} \Psi (P \geq C) v \big\Vert^2 d t \lesssim& \int_0^T \big\Vert \< {\mathcal A}_{\infty} \>^{- \mu} e^{-i t P^{1/2}} \Psi(P \geq C) v \big\Vert^2 d t \\
\lesssim& F_{\mu}^{\varepsilon}(T) \Vert v \Vert^2 ,\end{aligned}$$ where we have used Theorem \[a1\] (for $\mu > 1/2$) and Corollary \[a2\] (for $\mu \leq 1/2$).
For the proof of Theorem \[TSLW2\], we will need the following theorem of Christ and Kiselev [@ChKi01_01] in a form available in the article of Burq [@Bu03_01].
*Consider a bounded operator ${\mathcal T} : L^p( \R ;B_1) \longrightarrow L^q ( \R ; B_2)$ given by a locally integrable Kernel $K(t,s)$ with value operators from $B_1$ to $B_2$, where $B_{1}$ and $B_{2}$ are Banach spaces. Suppose that $p<q$. Then, the operator $$\widetilde{{\mathcal T}} f (t) = \int_{s<t} K (t,s) f(s) \, d s ,$$ is bounded from $L^p(\R;B_1)$ to $L^q(\R;B_2)$ by $$\big\Vert \widetilde{{\mathcal T}} \big\Vert_{L^p(\R;B_1) \rightarrow L^q(\R;B_2)} \leq \big( 1-2^{-p^{-1}-q^{-1}} \big) \Vert {\mathcal T} \Vert_{L^p( \R ;B_1) \rightarrow L^q ( \R ;B_2)}.$$*
By linearity and uniqueness it is sufficient to consider separately the cases $(u_0,u_1)=0$, $G=0$.
$\bullet$ $G=0$. Thanks to the discussion at the beginning of Section \[secM\], the solution of (\[LW\]) is given by $$\left(\begin{array}{c}
u(t) \\ \partial_t u(t)
\end{array} \right)
= e^{-i t R}
\Big( \begin{array}{c}
u_0 \\ u_1
\end{array} \Big)
\quad \text{ with } \quad R= \left( \begin{array}{cc}
0 & i
\\ - i P & 0
\end{array} \right), \ R = U^* L U .$$ Using Lemma \[LW5\], we see that for $\widetilde{\mu} < \mu$ we have $$\big\Vert \< x \>^{- \mu} u' \big\Vert_{L^2}^2 \lesssim \Big\Vert \< x \>^{- \widetilde{\mu}} M
\left( \begin{array}{c}
u(t) \\
\partial_t u(t)
\end{array} \right)
\bigg\Vert_{L^2\times L^2}^2,$$ with $$\label{c14}
M:= \left( \begin{array}{cc} P^{1/2} & 0 \\ 0 & 1 \end{array} \right) , \ M U^* = \frac{1}{\sqrt{2}} \left( \begin{array}{cc} 1 & 1 \\ -i & i \end{array} \right) .$$ Using Proposition \[propEW2\], we therefore have the following estimate $$\begin{aligned}
\int_0^T \big\Vert \< x \>^{- \mu} u' \big\Vert_{L^2}^2 d t \lesssim&\int_0^T \Big\Vert \< x \>^{- \widetilde{\mu}} M e^{-i t R} \Big( \begin{array}{c} u_0 \\ u_1 \end{array} \Big) \Big\Vert^2_{L^2 \times L^2} d t \\
\lesssim& \int_0^T \Big\Vert \< x \>^{- \widetilde{\mu}} e^{-i t L} U \Big( \begin{array}{c} u_0 \\ u_1 \end{array} \Big) \Big\Vert_{L^2 \times L^2}^2 d t \\
\lesssim& F^{\varepsilon}_{\widetilde{\mu}} (T) \Big\Vert U \Big( \begin{array}{c} u_0 \\ u_1 \end{array} \Big) \Big\Vert_{L^2 \times L^2}^2 = F^{\varepsilon}_{\widetilde{\mu}} (T) \big\Vert (u_0 , u_1 ) \big\Vert^2_{\mathcal E}.\end{aligned}$$
$\bullet$ $(u_0,u_1)=0$. In this case, the solution of (\[LW\]) is given by $$\left( \begin{array}{c} u (t) \\ \partial_t u (t) \end{array} \right) = \int_0^t e^{i(s-t)R} \left( \begin{array}{c} 0 \\ G(s) \end{array} \right) d s .$$ Thus, for all $\widetilde{\mu} < \mu$, $$\label{EW4}
\int_0^T \big\Vert \< x \>^{- \mu} u' \Vert^2_{L^2} \lesssim \int_0^T \left\Vert \int_0^t \< x \>^{-\widetilde{\mu}} e^{i(s-t)L} U \left( \begin{array}{c} 0 \\ G (s) \end{array} \right) d s \right\Vert^2_{L^2 \times L^2} d t .$$ Let $${\mathcal T} f (t) = \int_{\R} \< x \>^{- \widetilde{\mu}} \one_{[ 0,T ]} (s) \one_{[ 0,T ]} (t) e^{i(s-t)L} f (s) \, d s .$$ We estimate $$\begin{aligned}
\Vert {\mathcal T} f \Vert^2_{L^2( \R ; L^2 \times L^2 )} =& \int_0^T \left\Vert \< x \>^{- \widetilde{\mu}} e^{-i t L} \int_0^T e^{i s L} f (s) \, d s \right\Vert^2 d t \\
\lesssim& F^{\varepsilon}_{\widetilde{\mu}} (T) \left\Vert \int_0^T e^{i s L} f (s) \, d s \right\Vert^2 \\
\lesssim& F^{\varepsilon}_{\widetilde{\mu}} (T) \left(\int_0^T \Vert f (s) \Vert \, d s \right)^2.\end{aligned}$$ It follows $$\Vert {\mathcal T} \Vert_{L^1( \R ;L^2 \times L^2) \rightarrow L^2( \R ;L^2 \times L^2)}^{2} \lesssim F^{\varepsilon}_{\widetilde{\mu}} (T).$$ The expression on the right hand side of (\[EW4\]) is $$\big\Vert \widetilde{{\mathcal T}} U (0,G(s)) \big\Vert^2_{L^2( \R ;L^2 \times L^2)} ,$$ with $$\widetilde{{\mathcal T}} f (t) = \int_{s < t} \< x \>^{- \widetilde{\mu}} \one_{[ 0,T ]} (s) \one_{ [ 0 ,T ]} (t) e^{i (s-t) L} f (s) \, d s .$$ We can apply the theorem of Christ and Kiselev to conclude that $$\begin{aligned}
\big\Vert \widetilde{T} U (0,G(s)) \big\Vert^2_{L^2( \R ;L^2 \times L^2)} \lesssim& F^{\varepsilon}_{\widetilde{\mu}} (T) \left( \int_0^T \left\Vert U \left( \begin{array}{c} 0 \\ G(s)\end{array} \right) \right\Vert_{L^2 \times L^2} d s \right)^2 \\
=& F^{\varepsilon}_{\widetilde{\mu}} (T) \left( \int_0^T \left\Vert G (s) \right\Vert_{L^2} d s \right)^2 ,\end{aligned}$$ which finishes the proof.
Theorem \[TW1\] is now proved for $N=0$ using in addition the usual energy estimate $$\label{energy}
\Vert u' \Vert_{L^2 ( \R^{d} )} \lesssim \Vert u'(0, \cdot ) \Vert_{L^2 ( \R^{d} )} + \int_0^T \Vert G(s, \cdot ) \Vert_{L^2 ( \R^{d} )} d s .$$ Note that in the usual energy estimate $u'$ is replaced by $(\partial_t u ,P^{1/2}u)$, but we have $$\sum_k \big\Vert \widetilde{\partial}_k u \big\Vert \lesssim \Vert P^{1/2} u \Vert \lesssim \sum_k \big\Vert \widetilde{\partial}_k u \big\Vert ,$$ by Lemma \[b53\]. It will be useful to have similar estimates to the preceding containing a $L^2( \R^{d+1}, \< x \>^{\mu} d t \, d x )$ norm of $G$ on the right hand side rather than a $L^1_t L^2_x$ norm.
*\[PW4\] Assume $0<\mu\leq 1$.*
$i)$ Let $$\label{W16}
\left\{ \begin{aligned}
&( i \partial_t - P^{1/2} ) v = G , \\
&v_{\vert_{t=0}} = 0 .
\end{aligned} \right.$$ Then we have, for all $0 < \varepsilon < \mu$, $$\label{W17}
\int_0^T \big\Vert \< x \>^{- \mu} v \big\Vert^2 d t \lesssim ( F^{\varepsilon}_{\mu} (T) )^{2} \int_0^T \big\Vert \< x \>^{\mu} G \big\Vert^2 d t .$$
$ii)$ Let $$\label{W18}
\left\{ \begin{aligned}
&( \partial_t^2 + P ) u = G , \\
&( u_{\vert_{t=0}} , \partial_t u_{\vert_{t=0}} ) = 0 .
\end{aligned} \right.$$ Then we have, for all $0 < \varepsilon < \mu$, $$\label{W19}
\int_0^T \big\Vert \< x \>^{- \mu} u' \big\Vert^2 d t \lesssim ( F^{\varepsilon}_{\mu} (T) )^{2} \int_0^T \big\Vert \< x \>^{\mu} G \big\Vert^2 d t .$$
$i)$ We have $$\begin{aligned}
\int_0^T \big\Vert \< x \>^{- \mu} v \big\Vert^2 d t \lesssim& \int_0^T \big\Vert \< x \>^{- \mu} \Psi (P \leq 1/C) v \big\Vert^2 d t + \int_0^T \big\Vert \< x \>^{- \mu} \Psi (1/C \leq P \leq C) v \big\Vert^2 d t \\
&+ \int_0^T \big\Vert \< x \>^{- \mu} \Psi (P \geq C) v \big\Vert^2 d t =: I_1 + I_2 + I_{3} .\end{aligned}$$
$\bullet$ We first estimate $I_1$. Let $\varphi$, $\widetilde{\varphi}$ be as in . By Proposition \[PM1\], we know that $$\big\Vert \< x \>^{- \mu} \varphi ( \lambda P) v \big\Vert^2 = \big\Vert \< x \>^{- \mu} \widetilde{\varphi} ( \lambda P) \< {\mathcal A}_{\lambda} \>^{\mu} \< {\mathcal A}_{\lambda} \>^{- \mu} \varphi ( \lambda P) v \big\Vert^2 \lesssim \lambda^{- \mu + \varepsilon_1} \big\Vert \< {\mathcal A}_{\lambda} \>^{- \mu} \varphi ( \lambda P) v \big\Vert^2 .$$ Therefore, we have $$\begin{aligned}
I_1 \lesssim & \sum_{\lambda \text{ dyadic}} \lambda^{- \mu + \varepsilon_1} \int_0^T \big\Vert\< {\mathcal A}_{\lambda} \>^{- \mu} \varphi ( \lambda P) v (t) \big\Vert^2 d t \\
= & \sum_{\lambda \text{ dyadic}} \lambda^{- \mu + \varepsilon_1 + 1/2} \int_0^{\lambda^{-1/2} T} \big\Vert \< {\mathcal A}_{\lambda} \>^{- \mu} \varphi ( \lambda P) v ( \lambda^{1/2} s ) \big\Vert^2 d s .\end{aligned}$$ Now observe that $\widetilde{v} (s) = v( \lambda^{1/2} s)$ is solution of the equation $$\left\{ \begin{aligned}
&(i \partial_s - ( \lambda P )^{1/2}) \widetilde{v} =& \lambda^{1/2} G ( \lambda^{1/2} s ) , \\
&\widetilde{v}_{\vert_{s=0}} = 0 .
\end{aligned} \right.$$ We now apply Corollary \[c9\] with $H = ( \lambda P )^{1/2}$. Using also again Proposition \[PM1\], we obtain $$\begin{aligned}
\int_0^{\lambda^{-1/2} T} \big\Vert \< {\mathcal A}_{\lambda} \>^{- \mu} & \varphi ( \lambda P) v ( \lambda^{1/2} s) \big\Vert^2 d s \\
\lesssim & ( F_{\mu}^{\varepsilon_2} ( \lambda^{-1/2} T ) )^{2} \lambda \int_0^{\lambda^{-1/2} T} \big\Vert \< {\mathcal A}_{\lambda} \>^{\mu} \varphi ( \lambda P) G ( \lambda^{1/2} s) \big\Vert^2 d s \\
\lesssim & ( F_{\mu}^{\varepsilon_2} ( \lambda^{-1/2} T ) )^{2} \lambda^{1 - \mu + \varepsilon_3} \int_0^{\lambda^{-1/2} T} \big\Vert \< x \>^{\mu} G ( \lambda^{1/2} s ) \big\Vert^2 d s \\
= & ( F_{\mu}^{\varepsilon_2} ( \lambda^{-1/2} T ) )^{2} \lambda^{1/2 - \mu + \varepsilon_3} \int_0^{T} \big\Vert \< x \>^{\mu} G (t) \big\Vert^2 d t .\end{aligned}$$ Thus, $$I_1 \lesssim \sum_{\lambda \text{ dyadic}} \lambda^{1 - 2 \mu + \varepsilon_{1} + \varepsilon_3} ( F_{\mu}^{\varepsilon_2} ( \lambda^{-1/2} T ) )^{2} \int_0^{T} \big\Vert \< x \>^{\mu} G (t) \big\Vert^2 d t .$$ If $\mu\leq 1/2$, then we see that $$I_1 \lesssim \sum_{\lambda \text{ dyadic}} \lambda^{\varepsilon_1 + \varepsilon_3 - 2 \varepsilon_2} T^{2 ( 1-2 \mu + 2 \varepsilon_2 )} \int_0^{T} \big\Vert \< x \>^{\mu} G (t) \big\Vert^2 d t .$$ Once $0 < \varepsilon_2 < \mu$ fixed, it is therefore sufficient to choose $\varepsilon_1 , \varepsilon_3$ small enough such that $\varepsilon_1 + \varepsilon_3 < 2 \varepsilon_2$. If $\mu > 1/2$, we choose $\varepsilon_1 , \varepsilon_3$ small enough such that $\varepsilon_1 + \varepsilon_3 < 2\mu - 1$. Then, $$I_1 \lesssim \int_0^{T} \big\Vert \< x \>^{\mu} G (t) \big\Vert^2 d t .$$
$\bullet$ We now study $I_{2}$. Part $iii)$ of Proposition \[PM2\] implies $$I_{2} \lesssim \int_0^T \big\Vert \< {\mathcal A} \>^{- \mu} \Psi (1/C \leq P \leq C) v \big\Vert^2 d t .$$ As in the proof of Proposition \[propEW2\], Proposition \[PM2\] gives us a finite number of open intervals $I_{k}$, $k =1 , \ldots , K$, satisfying and $$[ 1/C , C] \subset \bigcup_{k=1}^{K} I_{k} .$$ Then, applying Corollary \[c9\] on each $I_{k}$ (slightly reduced) and using Proposition \[PM2\], we obtain $$I_{2} \lesssim ( F^{\varepsilon}_{\mu} (T) )^{2} \int_0^T \big\Vert \< x \>^{\mu} G \big\Vert^2 d t .$$
$\bullet$ We finally estimate $I_{3}$. Proposition \[a26\] and Corollary \[c9\] yield $$\begin{aligned}
I_{3} & \lesssim \int_0^T \big\Vert \< A_{\infty} \>^{- \mu} \Psi (P \geq C) v \big\Vert^2 d t \lesssim ( F^{\varepsilon}_{\mu} (T) )^{2}\int_0^T \big\Vert \< A_{\infty} \>^{\mu} G \big\Vert^2 d t \\
& \lesssim ( F^{\varepsilon}_{\mu} (T) )^{2} \int_0^T \big\Vert \< x \>^{\mu} G \big\Vert^2 d t .\end{aligned}$$
$ii)$ We first write as a first order system $$\begin{aligned}
i \partial_t \left( \begin{array}{c} u \\ \partial_t u \end{array} \right) = R \left( \begin{array}{c} u \\ \partial_t u \end{array} \right) + i \left( \begin{array}{c} 0 \\ G \end{array} \right) , \quad \left( \begin{array}{c} u \\ \partial_t u \end{array} \right) \vert_{t=0} = 0 .\end{aligned}$$ It is sufficient to estimate, for $\widetilde{\mu} < \mu$, $$\int_0^T \left\Vert \< x \>^{-\widetilde{\mu}} M \left( \begin{array}{c} u \\ \partial_t u \end{array} \right) \right\Vert^2_{L^2 \times L^2} d t = \int_0^T \left\Vert \< x \>^{-\widetilde{\mu}} M U^* U \left( \begin{array}{c} u \\ \partial_t u \end{array} \right) \right\Vert^2_{L^2 \times L^2} d t ,$$ with $M$ defined in . But $v = U \left( \begin{array}{c} u \\ \partial_t u \end{array} \right)$ solves $$( i \partial_t - L ) v = i U \left( \begin{array}{c} 0 \\ G \end{array} \right) , \quad v_{\vert_{t=0}} = 0 .$$ By and part $i)$ of the proposition, we find $$\begin{aligned}
\int_0^T \left\Vert \< x \>^{-\widetilde{\mu}} M \left( \begin{array}{c} u \\ \partial_t u \end{array} \right) \right\Vert^2_{L^2 \times L^2} d t \lesssim & \int_0^T \big\Vert \< x \>^{-\widetilde{\mu}} v \big\Vert^2_{L^2 \times L^2} d t \\
\lesssim & ( F^{\varepsilon}_{\widetilde{\mu}} (T) )^{2} \int_0^T \left\Vert \< x \>^{- \widetilde{\mu}} U \left( \begin{array}{c} 0 \\ G \end{array} \right) \right\Vert^2_{L^2 \times L^2} d t \\
= & ( F^{\varepsilon}_{\widetilde{\mu}} (T) )^{2} \int_0^T \big\Vert\< x \>^{-\widetilde{\mu}} G \big\Vert^2_{L^2} d t .\end{aligned}$$ which gives $ii)$ thanks to Remark \[RW2\].
We now want to prove Theorem \[TW1\] for general $N$. In contrast to the Minkowski case, this does not follow directly from the case $N=0$ because the vector fields $\widetilde{\Omega}$, $\widetilde{\partial}_x$ do not commute with the equation. We will therefore need the form of certain commutators. As in , a term $r_{j}$ or $\widetilde{r}_{j}$, $j\in \N$, will denote a smooth function such that $$\begin{aligned}
\partial^{\alpha}_{x} r_{j} (x) &= \CO \big( \< x \>^{-\rho - j - \vert \alpha \vert} \big) , \\
\partial^{\alpha}_{x} \widetilde{r}_{j} (x) &= \CO \big( \< x \>^{- j - \vert \alpha \vert} \big) .\end{aligned}$$ These functions can change from line to line. Direct computations give
*\[LW2\] We have $$\begin{aligned}
&\big[ \widetilde{\partial}_{j} , \widetilde{\partial}_{k} \big] = r_{1} \widetilde{\partial} ,
&& \big[ \widetilde{\Omega}^{j, k} ,P \big] = r_{0} \widetilde{\partial} \widetilde{\partial}+r_{1} \widetilde{\partial} \\
&\big[ \widetilde{\partial}^{*}_{j} , \widetilde{\partial}_{k} \big] = \widetilde{\partial}^{*} r_{1} + r_{2} ,
&&\big[ \widetilde{\Omega}^{j, k} , \widetilde{\partial}_{\ell} \big] = \widetilde{r}_{0} \widetilde{\partial} , \\
&\big[ P , \widetilde{\partial}_{\ell} \big] = \widetilde{\partial}^{*} r_{1} \widetilde{\partial} + r_{2} \widetilde{\partial} .\end{aligned}$$ As before, we have not written the sum over the indexes on the right hand sides.*
We now observe that the vector fields $\widetilde{\partial}_j$ can be replaced by powers of $P$.
*\[LW3\] For $0 < \mu \leq 3/2$ and $n \geq 2$, we have $$\label{W7}
\big\Vert \< x \>^{- \mu} \widetilde{\partial}_{j_1} \cdots \widetilde{\partial}_{j_n} u \big\Vert \lesssim \sum_{j=0}^{\lfloor \frac{n-1}{2} \rfloor} \sum_{q =1}^{d} \big\Vert \< x \>^{- \mu} \widetilde{\partial}_{q} P^j u \big\Vert + \sum_{j=1}^{\lfloor \frac{n}{2} \rfloor} \big\Vert \< x \>^{- \mu} P^j u \big\Vert .$$*
We first show $$\label{W8}
\big\Vert \< x \>^{- \mu} \widetilde{\partial}_k \widetilde{\partial}_{\ell} u \big\Vert \lesssim \Vert \< x \>^{- \mu} P u \Vert + \sum_{q = 1}^{d} \big\Vert \< x \>^{- \mu} \widetilde{\partial}_{q} u \big\Vert .$$ Indeed, we have $$\big\Vert \< x \>^{- \mu} \widetilde{\partial}_k \widetilde{\partial}_{\ell} u \big\Vert \lesssim \big\Vert \< x \>^{- \mu} \widetilde{\partial}_k (P+1)^{-1} \widetilde{\partial}_{\ell} (P+1) u \big\Vert + \big\Vert \< x \>^{- \mu} \widetilde{\partial}_k (P+1)^{-1} \big[ P , \widetilde{\partial}_{\ell} \big] u \big\Vert =: A + B .$$ We estimate $A$. $$A \leq \big\Vert \< x \>^{- \mu} \widetilde{\partial}_k (P+1)^{-1} \widetilde{\partial}_{\ell} P u \big\Vert + \big\Vert \< x \>^{- \mu} \widetilde{\partial}_k (P+1)^{-1} \widetilde{\partial}_{\ell} u \big\Vert .$$ Noting that $\< x \>^{- \mu} \widetilde{\partial}_k (P+1)^{-1} \widetilde{\partial}_{\ell} \< x \>^{\mu}$ and $\< x \>^{- \mu} \widetilde{\partial}_k (P+1)^{-1} \< x \>^{\mu}$ are bounded by Proposition \[b45\] and Lemma \[b20\], we obtain $$\label{W9}
A \lesssim \Vert \< x \>^{- \mu} P u \Vert + \big\Vert \< x \>^{- \mu} \widetilde{\partial}_{\ell} u \big\Vert .$$ Now, recall from Lemma \[LW2\] that $$\big[ P , \widetilde{\partial}_{\ell} \big] = \widetilde{\partial}^{*} r_{1} \widetilde{\partial} + r_{2} \widetilde{\partial} .$$ Thus, as for , we see that $$\label{W10}
B \lesssim \sum_j \big\Vert \< x \>^{- \mu} \widetilde{\partial}_j u \big\Vert .$$ The inequalities , give . We will show by induction over $n\ge2$. For $n=2$ this is exactly . Assume $n \geq 3$. Using , we obtain $$\big\Vert \< x \>^{- \mu} \widetilde{\partial}_{j_1} \widetilde{\partial}_{j_2} \cdots \widetilde{\partial}_{j_n} u \big\Vert \lesssim \big\Vert \< x \>^{- \mu} P \widetilde{\partial}_{j_3} \cdots \widetilde{\partial}_{j_n} u \big\Vert + \sum_{k=1}^{d} \big\Vert \< x \>^{- \mu} \widetilde{\partial}_k \widetilde{\partial}_{j_3} \cdots \widetilde{\partial}_{j_n} u \big\Vert .$$ For the second term, we can use the induction hypothesis. For the first term we commute $P$ through $\widetilde{\partial}_{j_3} \cdots \widetilde{\partial}_{j_n}$. The commutators give terms which can be estimated by terms of the form $\Vert \< x \>^{- \mu} \widetilde{\partial}_{k_m} \cdots \widetilde{\partial}_{k_n} u \Vert$, with $2 \leq m \leq n$, which themselves can be estimated by the induction hypothesis. It remains to consider the term $\Vert\<x\>^{-\mu}\widetilde{\partial}_{j_3} \cdots \widetilde{\partial}_{j_n}Pu\Vert$, which can either be kept ($n=3$) or be estimated applying the induction hypothesis to $P u$ rather than to $u$.
In order to show , it is sufficient to use vector fields in $X$. This is shown in the next lemma.
*\[LW4\] Assume $\rho > 1$. Let $1/2 \leq \mu \leq 1$, $j \in \frac{1}{2} \N$, $\beta$ be a multi-index and $N = 2 j + \vert \beta \vert$. Then, for all $\varepsilon>0$, there exists $\eta_{\varepsilon}> 1/2$ such that $$\begin{aligned}
\< F_{\mu}^{\varepsilon}(T) & \>^{-1} \big\Vert \< x \>^{- \mu} (P^j \widetilde{\Omega}^{\beta}u)' \big\Vert_{L^2( [0 , T] \times \R^{d} )} \nonumber \\
\lesssim & \sum_{\vert\alpha\vert\leq N} \bigg( \big\Vert (Y^{\alpha} u) '(0, \cdot ) \big\Vert_{L^2 ( \R^{d} )} + \int_0^T \Vert Y^{\alpha} G \Vert_{L^2 ( \R^{d} )} d t + \big\Vert \< x \>^{-\eta_{\varepsilon}} (X^{\alpha}u)' \big\Vert_{L^2( [0 , T] \times \R^{d} )} \bigg) . \label{W11}\end{aligned}$$ Moreover, for $\rho = 1$ and $\varepsilon >0$, the same inequality holds with $\< F_{\mu}^{\varepsilon}(T) \>^{-1}$ replaced by $\< T \>^{- \varepsilon}$.*
The inequality will be proven by induction over $\vert \beta \vert$. Assume first $\rho >1$. Since the wave equation commutes with $P$, the case $\vert \beta \vert =0$ follows from Theorem \[TSLW2\] and Lemma \[b53\]. Assume now $\vert \beta \vert \geq 1$ and let $v= P^j \widetilde{\Omega}^{\beta} u$. The function $v$ fulfills the following equation $$\left\{ \begin{aligned}
&( \partial_t^2 + P ) v = P^j \widetilde{\Omega}^{\beta} G + P^j \big[ P , \widetilde{\Omega}^{\beta} \big] u , \\
&( v_{\vert_{t=0}} , \partial_t v_{\vert_{t=0}} ) = ( P^j \widetilde{\Omega}^{\beta} u_0 , P^j \widetilde{\Omega}^{\beta} u_1 ) .
\end{aligned} \right.$$ Let $v_1 , v_2$ be the solutions of $$\label{W13}
\left\{ \begin{aligned}
&( \partial_t^2 + P ) v_1 = P^j \widetilde{\Omega}^{\beta} G , \\
& ( v_{1} {}_{\vert_{t=0}} , \partial_t v_{1} {}_{\vert_{t=0}} ) = ( P^j \widetilde{\Omega}^{\beta} u_0 , P^j \widetilde{\Omega}^{\beta} u_1 ) ,
\end{aligned} \right.$$ $$\label{W14}
\left\{ \begin{aligned}
&( \partial_t^2 + P ) v_2 = P^j [ P , \widetilde{\Omega}^{\beta} ] u , \\
&( v_{2} {}_{\vert_{t=0}} , \partial_t v_{2} {}_{\vert_{t=0}} ) = 0 .
\end{aligned} \right.$$ Clearly $v = v_1 + v_2$. We have, for all $\widetilde{\mu} < \mu$, $$\big\Vert \< x \>^{- \mu} v_1 ' \big\Vert_{L^2( [0 , T] \times \R^{d} )} \lesssim ( F_{\widetilde{\mu}}^{\varepsilon} (T) )^{1/2} \bigg( \big\Vert ( P^j \widetilde{\Omega}^{\beta} u)' (0, \cdot ) \big\Vert_{L^2 ( \R^{d} )} + \int_0^T \big\Vert P^j \widetilde{\Omega}^{\beta} G \big\Vert_{L^2 ( \R^{d} )} d t \bigg) ,$$ where we have used Theorem \[TSLW2\]. If $\mu > 1/2$, we choose $\widetilde{\mu} > 1/2$. We further estimate, by Proposition \[PW4\], $$( F_{\widetilde{\mu}}^{\varepsilon} (T) )^{-1} \Vert \< x \>^{- \mu} v_2 ' \Vert_{L^2( [0 , T] \times \R^{d} )} \lesssim \big\Vert \< x \>^{\widetilde{\mu}} P^j [ P, \widetilde{\Omega}^{\beta} ] u \big\Vert_{L^2( [0 , T] \times \R^{d} )} .$$ Using Lemma \[LW2\], we see that $\< x \>^{\widetilde{\mu}} P^j [ P , \widetilde{\Omega}^{\beta} ] u$ is a sum of terms of the form $$\< x \>^{\widetilde{\mu} - \rho} \widetilde{\partial}_{k_1} \cdots \widetilde{\partial}_{k_{m}} \widetilde{\Omega}^{\gamma} u ,$$ with $1 \leq m \leq 2j+2$ and $\vert \gamma \vert \leq \vert \beta \vert - 1$. Using Lemma \[LW3\], we see that these terms can be estimated in norm by terms of the form $$\big\Vert \< x \>^{\widetilde{\mu} - \rho} \widetilde{\partial}_\ell (P^{q} \widetilde{\Omega}^{\gamma} u ) \big\Vert \quad \text{or} \quad \big\Vert \< x \>^{\widetilde{\mu} - \rho} P^{r} \widetilde{\Omega}^{\gamma} u \big\Vert ,$$ with $q,r \in \N$, $0 \leq q \leq (m-1)/2$ and $1 \leq r \leq m/2$. Applying Lemma \[LW5\], we see that we can replace $P^{1/2}$ in the second term by partial derivatives and apply the induction hypothesis with $\rho - \widetilde{\mu} > 1/2$.
In the case $\rho =1$, it is enough to choose $\widetilde{\mu} = 1/2 - \delta$ with $\delta >0$ small.
The energy term is easily estimated by the observation that $\partial_t$ and $P$ commute with the equation. The same way, note that we can restrict our attention to vector fields in $Y$ for the second term. Also, by Lemma \[LW2\], we can arrange for that the vector fields $\widetilde{\partial}_{x}$ are always on the left of the vector fields $\widetilde{\Omega}$. Using Lemma \[LW2\], we see that we can replace $Y^{\alpha} u '$ by $(Y^{\alpha}u)'$. Using Lemma \[LW5\], Lemma \[LW3\] and Lemma \[LW4\], we see that it is sufficient to estimate $$\< F_{\mu}^{\varepsilon} (T) \>^{-1} \big\Vert \< x \>^{- \mu} P^j v \big\Vert_{L^2( [0 , T] \times \R^{d} )} ,$$ in the case $\rho > 1$ and $$\< T \>^{- \varepsilon} \big\Vert \< x \>^{- \mu} P^j v \big\Vert_{L^2( [0 , T] \times \R^{d} )} ,$$ in the case $\rho = 1$. These terms can be estimated by Theorem \[TSLW2\], because $P$ commutes with the equation.
Proof of the nonlinear result {#sec6}
=============================
In this section we will prove the main theorem, Theorem \[TSLW\]. The proof of the result will follow closely the arguments of Keel, Smith and Sogge in the Minkowski case (see [@KeSmSo02_01]). We start with the now standard Sobolev estimate (see [@Kl85_01]).
*\[LN1\] Suppose that $h \in C^{\infty} ( \R^d )$. Then, for $R>1$, $$\label{N1}
\Vert h \Vert_{L^{\infty} (R/2 \leq \vert x \vert \leq R)} \lesssim R^{\frac{1-d}{2}} \sum_{\vert \alpha \vert \leq \left \lceil \frac{d-1}{2} \right\rceil +1} \Vert Y^{\alpha} h \Vert_{L^2(R/4 \leq \vert x \vert \leq 2R)}.$$*
We now define the bilinear form $\widetilde{Q}$ by $\widetilde{Q}(u',u')=Q(u')$. The following estimate for the nonlinear part will be crucial.
*\[LN2\] Let $\mu_d = \frac{d-1}{4}$. Then, for $L \geq \max \big( 2 \left( \left\lceil \frac{d-1}{2} \right\rceil + 1 \right) , \vert \beta \vert \big)$, we have $$\big\Vert Z^{\beta} \widetilde{Q} (u',v') \big\Vert_{L^2 ( \R^{d} )}^{2} \lesssim \bigg( \sum_{\vert \alpha \vert \leq L} \big\Vert \< x \>^{- \mu_d} Z^{\alpha} u' \big\Vert^2_{L^2 ( \R^{d} )} \bigg) \bigg( \sum_{\vert \alpha \vert \leq L} \big\Vert \< x \>^{- \mu_d} Z^{\alpha} v' \big\Vert^2_{L^2 ( \R^{d} )} \bigg) .$$*
We clearly have the pointwise bound: $$\begin{aligned}
\big\vert Z^{\beta} \widetilde{Q} (u',v') (s,x) \big\vert \lesssim & \bigg( \sum_{\vert\alpha\vert\leq L} \big\vert Z^{\alpha}u'(s,x) \big\vert \bigg) \bigg( \sum_{\vert \alpha \vert \leq \left \lfloor \frac{L}{2} \right \rfloor} \big\vert Z^{\alpha} v' (s,x) \big\vert \bigg) \\
&+ \bigg( \sum_{\vert\alpha\vert\leq L} \big\vert Z^{\alpha} v'(s,x) \big\vert \bigg) \bigg( \sum_{\vert \alpha \vert \leq \left \lfloor \frac{L}{2} \right \rfloor} \big\vert Z^{\alpha} u' (s,x) \big\vert \bigg) .\end{aligned}$$ We only estimate the first term. Using Lemma \[LN1\] for a given $R=2^j$, $j\geq 0$, we get $$\begin{aligned}
\big\Vert Z^{\beta} & \widetilde{Q} (u',v') \big\Vert^{2}_{L^2 ( \{ \vert x \vert \in [ 2^j , 2^{j+1} [ \} )} \\
&\lesssim 2^{j (1-d)} \sum_{\vert \alpha \vert \leq L} \big\Vert Z^{\alpha} u' \big\Vert^{2}_{L^2 ( \{ \vert x \vert \in [ 2^j , 2^{j+1} [ \} )} \sum_{\vert \alpha \vert \leq \left\lfloor \frac{L}{2} \right\rfloor + \left\lceil \frac{d-1}{2} \right\rceil +1} \big\Vert Z^{\alpha} v' \big\Vert^{2}_{L^2 ( \{ \vert x \vert \in [ 2^{j-1} , 2^{j+2} [ \} )} \\
&\lesssim \sum_{\vert \alpha \vert \leq L} \big\Vert \< x \>^{- \mu_d} Z^{\alpha} u' \big\Vert^{2}_{L^2 ( \{ \vert x \vert \in [ 2^{j} , 2^{j+1} [ \} )} \sum_{\vert \alpha \vert \leq L} \big\Vert \< x \>^{- \mu_d} Z^{\alpha} v' \big\Vert^{2}_{L^2 ( \{ \vert x \vert \in [ 2^{j-1} , 2^{j+2} [ \} )} \\
&\lesssim \sum_{\vert \alpha \vert \leq L} \big\Vert \< x \>^{- \mu_d} Z^{\alpha} u' \big\Vert^{2}_{L^2 ( \{ \vert x \vert \in [ 2^{j} , 2^{j+1} [ \} )} \sum_{\vert \alpha \vert \leq L} \big\Vert \< x \>^{- \mu_d} Z^{\alpha} v' \big\Vert^{2}_{L^2 ( \R^{d} )} .\end{aligned}$$ We also have the bound $$\begin{aligned}
\big\Vert Z^{\beta} \widetilde{Q} (u',v') \big\Vert_{L^2 ( \{ \vert x \vert < 1 \} )}^{2} \lesssim \sum_{\vert \alpha \vert \leq L} \big\Vert Z^{\alpha} u' \big\Vert^2_{L^2 ( \{ \vert x \vert <2 \} )} \sum_{\vert \alpha \vert \leq L} \big\Vert Z^{\alpha} v' \big\Vert^2_{L^2 ( \{ \vert x \vert <2 \} )} .\end{aligned}$$ Summing over $j$ gives the lemma.
We follow [@KeSmSo02_01]. Let $u_{-1}=0$. We define $u_k$, $k \in \N$ inductively by letting $u_k$ solve $$\label{N3}
\left\{ \begin{aligned}
&\Box_{\mathfrak{g}} u_k = Q ( u_{k-1} ' ) , \\
&( u_k {}_{\vert_{t=0}} , \partial_t u_k {}_{\vert_{t=0}} ) = ( u_0 , u_1 ) .
\end{aligned} \right.$$ For $T > 0$, we denote $$M_k (T) = \sup_{0 \leq t \leq T} \sum_{1 \leq i + j \leq M+1} \big\Vert \partial_{t}^{i} P^{j/2} u_k \big\Vert_{L^{2} ( \R^{d} )} + \sum_{\vert \alpha \vert \leq M} K_{n} (T)^{-1} \big\Vert \< x \>^{- \mu_d} Z^{\alpha} u_k' \big\Vert_{L^2( [0 , T] \times \R^{d} )} ,$$ with $$K_{n} (T) =
\left\{ \begin{aligned}
&T^{1/n} &&d=3 \text{ or } \rho =1 , \\
&1 &&d\geq 4 \text{ and } \rho > 1 .
\end{aligned} \right.$$ Using Theorem \[TW1\], we see that there exists a constant $C_0$ such that $$M_0 (T) \leq C_0 \delta ,$$ for any $T$. We claim that, for $k \geq 1$, we have $$\label{c10}
M_k ( T_{\delta} ) \leq 2 C_0 \delta ,$$ for $\delta$ sufficiently small and $T_{\delta}$ appropriately chosen. We will prove this inductively. Assume that the bound holds for $k-1$. By Theorem \[TW1\], we have, for $\delta$ small enough, $$\begin{aligned}
M_k ( T_{\delta} ) &\leq C_0 \delta + C \sum_{\vert \alpha \vert \leq M} \int_0^{T_{\delta}} \big\Vert Z^{\alpha} Q ( u_{k-1} ') (s, \cdot ) \big\Vert_{L^2 ( \R^d )} d s \\
& \leq C_0 \delta + C \sum_{\vert \alpha \vert \leq M} \int_0^{T_{\delta}} \big\Vert \< x \>^{- \mu_d} Z^{\alpha} u_{k-1}' \big\Vert^2_{L^2 ( \R^d )} d s \\
& \leq C_0 \delta + C K_{n} ( T_{\delta} ) M_{k-1}^2 ( T_{\delta} ) \\
& \leq C_0 \delta + C K_{n} ( T_{\delta} ) ( 2 C_0 \delta)^2 ,\end{aligned}$$ where we have also used Lemma \[LN2\] and the induction hypothesis. Note that, to estimate the term $\Vert ( Z^{\alpha} u_k ) ' ( 0, \cdot ) \Vert_{L^2}$, we might have to use the equation and Lemma \[LN2\]. We therefore need $\delta$ to be small enough. Then, to prove , it is enough to have $$\label{N4}
C_0 \delta + C K_{n} ( T_{\delta} ) ( 2 C_0 \delta )^2 \leq 2 C_0 \delta \Longleftrightarrow 4 C C_0 K_{n} ( T_{\delta} ) \delta \leq 1 .$$ Therefore, we find:
- If $d=3$ or $\rho =1$, the estimate holds with $T_{\delta} = c_{n} \delta^{- n}$ and $c_{n}$ small enough.
- If $d\geq 4$ and $\rho >1$, is fulfilled if $\delta$ is sufficiently small and we can take $T_{\delta}=\infty$.
To show that the sequence $u_k$ converges, we estimate the quantity $$\begin{aligned}
A_k (T) = \sup_{0 \leq t \leq T} \sum_{1 \leq i + j \leq M+1} \big\Vert & \partial_{t}^{i} P^{j/2} ( u_k - u_{k-1} ) \big\Vert_{L^2 ( \R^d )} \\
& + \sum_{\vert \alpha \vert \leq M} K_{n} (T)^{-1} \big\Vert \< x \>^{- \mu_d} Z^{\alpha} ( u'_k - u'_{k-1} ) \big\Vert_{L^2( [0,T] \times \R^{d} )}.\end{aligned}$$ It is clearly sufficient to show $$\label{N5}
A_k (T) \leq \frac{1}{2} A_{k-1} (T) .$$ Using Lemma \[LN2\] and repeating the above arguments, we obtain $$\begin{aligned}
A_k ( T_{\delta} ) \leq \widetilde{C} \sum_{\vert \alpha \vert \leq M} \int_0^{T_{\delta}} & \big\Vert \< x \>^{- \mu_d} Z^{\alpha} ( u'_{k-1} - u'_{k-2} ) \big\Vert_{L^2 ( \R^d )} \\
& \times \sum_{\vert \alpha \vert \leq M} \Big( \big\Vert \< x \>^{- \mu_d} Z^{\alpha} u'_{k-1} \big\Vert_{L^2 ( \R^d )} + \big\Vert \< x \>^{- \mu_d} Z^{\alpha} u'_{k-2} \Vert_{L^2 ( \R^d )} \Big) d s .\end{aligned}$$ By the Cauchy–Schwarz inequality, we conclude that $$A_k ( T_{\delta} ) \leq \widetilde{C} K_{n} ( T_{\delta} ) ( M_{k-1} ( T_{\delta} ) + M_{k-2} ( T_{\delta} ) ) A_{k-1} (T_{\delta} ) .$$ Using , the above inequality leads to if $\delta$ is small enough. Uniqueness and $C^2$ property of the solution follow from [@Ho97_01 Theorem 6.4.10, Theorem 6.4.11] using that the constructed solution is in $H^{M+1}_{\text{loc}} ( \R^{d+1} ) \subset C^2 ( \R^{d+1} )$. Note also that the solution is bounded in $C^2$ on the interval $[0,T_{\delta} ]$.
Regularity {#a29}
==========
Here, we give some results concerning the regularity with respect to an operator. More details can be found in the book of Amrein, A. Boutet de Monvel and Georgescu [@AmBoGe96_01] and in the paper of C. Gérard and Georgescu [@GeGe99_01]. We start with a useful characterization of the regularity $C^{1} (A)$.
*\[a15\] Let $A$ and $H$ be self-adjoint operators on a Hilbert space ${\mathcal H}$. Then $H$ is of class $C^{1}(A)$ iff the following conditions are satisfied:*
1. there is a constant $c< \infty$ such that for all $u \in D(A)
\cap D(H)$, $$\vert(Au,Hu) - (Hu,Au) \vert \leq c \left( \Vert H u \Vert^{2} + \Vert u\Vert^{2} \right),$$
2. for some $z \in \C\backslash \sigma (H)$, the set $\{ u \in
D(A); \ (H-z)^{-1} u \in D(A) \text{ and } (H-\bar{z})^{-1} u \in D(A)
\}$ is a core for $A$.
If $H$ is of class $C^{1} (A)$, then the following is true:
1. The space $(H-z)^{-1} D(A)$ is independent of $z\in \C \backslash
\sigma (H)$ and contained in $D(A)$. It is a core for $H$ and a dense subspace of $D(A) \cap D(H)$ for the intersection topology (i.e. the topology associated to the norm $ \Vert H u\Vert + \Vert Au\Vert + \Vert u \Vert)$.
2. The space $D(A) \cap D(H)$ is a core for $H$ and the form $[A,H]$ has a unique extension to a continuous sesquilinear form on $D(H)$ (equipped with the graph topology). If this extension is denoted by $[A,H]$, the following identity holds on ${\mathcal H}$ (in the form sense): $$\left[ A, (H-z)^{-1} \right] = - (H-z)^{-1} [A,H] (H-z)^{-1},$$ for $z \in \C \backslash \sigma (H)$.
We also have the following theorem coming from [@AmBoGe96_01 Theorem 6.3.4].
*\[a16\] Let $A$ and $H$ be self-adjoint operators in a Hilbert space ${\mathcal H}$. Assume that the unitary one-parameter group $\{ \exp (i A \tau) \}_{\tau \in \R}$ leaves the domain $D(H)$ of $H$ invariant. Then $H$ is of class $C^{1}(A)$ iff $[H,A]$ is bounded from $D(H)$ to $D(H)^{*}$.*
A criterion for the above assumption to be satisfied is given by the following result of Georgescu and C. Gérard.
*\[a17\] Let $A$ and $H$ be self-adjoint operators in a Hilbert space ${\mathcal H}$. Let $H \in C^{1} (A)$ and suppose that the commutator $[i H , A]$ can be extended to a bounded operator from $D(H)$ to ${\mathcal H}$. Then $e^{i t A}$ preserves $D(H)$.*
In this paper, we will use the following characterization of the regularity $C^{2} (A)$.
*\[a7\] From Section 6.2 of [@AmBoGe96_01], it is known that $H$ if of class $C^{2} (A)$ if the following conditions hold: $i)$ For some $z \in \C \setminus \sigma (H)$, the set $\{ u \in D (A) ; \ (H-z)^{-1} u \in D (A) \text{ and } (H- \overline{z} )^{-1} u \in D (A) \}$ is a core for $A$.*
$ii)$ $[H,A]$ and $[[H,A],A]$ extend as bounded operators on ${\mathcal H}$.
Resolvent estimates at low energies {#b56}
===================================
Estimates for the free Laplacian
---------------------------------
${}^{}$
We begin with some estimates for the free Laplacian $P_{0} = - \Delta$.
*\[b6\] Let $\alpha >0$. Then, for all $\varepsilon >0$, we have $$\big\Vert ( \lambda P_{0} +1)^{- \alpha} u \big\Vert \lesssim \lambda^{- \min ( \alpha - \varepsilon , d/4)} \big\Vert \< x \>^{ \min ( 2 \alpha , d/2 + \varepsilon )} u \big\Vert ,$$ uniformly for $\lambda \geq 1$.*
Here, we denote $$\Vert u \Vert_{p} = \Big( \int_{\R^{d}} \vert u (x) \vert^{p} \, d x \Big)^{1/p} ,$$ the standard norm on $L^{p} ( \R^{d})$. Using the Hölder inequality, we get $$\begin{aligned}
\big\Vert ( \lambda P_{0} +1)^{- \alpha} u \big\Vert =& \Big( \int (\lambda \xi^{2} +1)^{- 2 \alpha} \vert \widehat{u} ( \xi )\vert^{2} d \xi \Big)^{1/2} \\
\leq& \Vert (\lambda \xi^{2} +1)^{- \alpha} \Vert_{2 p} \Vert \widehat{u} \Vert_{2 q} ,\end{aligned}$$ and we choose $p = \max (\frac{d}{4 \alpha} + \mu , 1)$, $\mu >0$, and $p^{-1} + q^{-1} = 1$. In particular, $2 q \geq 2$ and $4 \alpha p >d$. Then, by the Hausdorff–Young inequality, we obtain $$\label{b4}
\big\Vert ( \lambda P_{0} +1)^{- \alpha} u \big\Vert \lesssim \lambda^{- d/ 4p} \Vert u \Vert_{r} ,$$ with $r^{-1} = 1 - (2q)^{-1} = 2^{-1} (1 + p^{-1} )$ satisfying $1 \leq r \leq 2$. Using one more time the Hölder inequality, we have $$\Vert u \Vert_{r} \lesssim \Big( \int \vert u \vert^{r s} \< x \>^{\beta s} d x \Big)^{1/ r s} \Big( \int \< x \>^{- \beta t} d x \Big)^{1/ r t} ,$$ with $s^{-1} +t^{-1} =1$ and $\beta >0$. We choose $s = 2 r^{-1}$ and $\beta = d/t + \nu$, $\nu >0$. Thus, $$\label{b5}
\Vert u \Vert_{r} \lesssim \big\Vert \< x \>^{\beta s /2} u \big\Vert_{2} .$$ The coefficient $\beta s /2$ satisfies $$\begin{aligned}
\frac{\beta s}{2} =& \frac{d s}{2 t} + \frac{\nu s}{2} = \frac{d s}{2} - \frac{d}{2} + \frac{\nu s}{2} = \frac{d}{r} - \frac{d}{2} + \CO (\nu ) = \frac{d}{2 p} + \CO (\nu ) \\
=& \frac{d}{2} \min \Big( \Big( \frac{d}{4 \alpha} + \mu \Big)^{-1} ,1 \Big) + \CO (\nu ) = \frac{d}{2} \min \Big( \frac{4 \alpha}{d} - \frac{16 \alpha^{2} \mu}{d^{2}} + \CO ( \mu^{2} ) ,1 \Big) + \CO (\nu ) \\
=& \min \Big( 2 \alpha - \frac{8 \alpha^{2} \mu}{d} + \CO (\mu^{2}) , \frac{d}{2} \Big) + \CO (\nu ) .\end{aligned}$$ On the other hand, $$\frac{d}{4p} = \frac{d}{4} \min \Big( \Big( \frac{d}{4 \alpha} + \mu \Big)^{-1} ,1 \Big) = \min \Big( \alpha + \CO ( \mu ) , \frac{d}{4} \Big) .$$ Taking first $\mu$ and then $\nu$ small enough, the lemma follows from the estimates and .
*\[b47\] Let $\beta \geq 0$, $0 \leq \gamma \leq \min ( 1 , d/4)$ and $0 \leq \delta \leq d/4$. Then, for all $\varepsilon > 0$, $$\big\Vert \< x \>^{\beta} ( \lambda P_{0} +1)^{- 1} u \big\Vert \lesssim \lambda^{\beta /2 - \delta + \varepsilon} \big\Vert \< x \>^{2 \delta} u \big\Vert + \lambda^{- \gamma + \varepsilon} \big\Vert \< x \>^{\beta + 2 \gamma} u \big\Vert ,$$ uniformly for $\lambda \geq 1$.*
*\[b12\] In the previous lemma, assume $\gamma + \beta /2 \leq d/4$. Then, we can chose $\delta = \gamma + \beta /2$ and we have $$\big\Vert \< x \>^{\beta} ( \lambda P_{0} +1)^{- 1} u \big\Vert \lesssim \lambda^{- \gamma + \varepsilon} \big\Vert \< x \>^{\beta + 2 \gamma} u \big\Vert ,$$ uniformly for $\lambda \geq 1$.*
Assume first that $\beta \in \N$. Using $$\Vert \< x \> u \Vert^{2} = \big\< (x^{2} +1) u , u \big\> = \sum_{j=1}^{d} \Vert x_{j} u \Vert^{2} + \Vert u \Vert^{2} ,$$ it is enough to estimate $\Vert x^{a} ( \lambda P_{0} +1)^{-1} u \Vert$ where $a \in \N^{d}$ is a multi-index of length less or equal to $\beta$. Since $$x_{j} ( \lambda P_{0} +1)^{-1} = ( \lambda P_{0} +1)^{-1} x_{j} - 2 \lambda^{1/2} \big( \lambda^{1/2} \partial_{j} \big) ( \lambda P_{0} +1)^{-2} ,$$ the operator $x^{a} ( \lambda P_{0} +1)^{-1}$ can be written as a finite sum of terms of the form $$T = \lambda^{\frac{\vert a \vert - \vert b \vert}{2}} \big( \lambda^{1/2} \partial \big)^{c} ( \lambda P_{0} +1)^{-1 - \frac{\vert a + c - b \vert}{2}} x^{b} ,$$ where $b , c$ are non-negative multi-indexes such that $b + c \leq a$ and $\vert a + c - b \vert = \vert a \vert + \vert c \vert - \vert b \vert$ is even. Such a term can be written as $$\begin{aligned}
T =& \lambda^{\frac{\vert a \vert - \vert b \vert}{2}} \big( \lambda^{1/2} \partial \big)^{c} ( \lambda P_{0} +1)^{-1 - \frac{\vert a + c - b \vert}{2}} ( \lambda P_{0} +1)^{1 + \frac{\vert a \vert - \vert b \vert}{2}} ( \lambda P_{0} +1)^{- 1 - \frac{\vert a \vert - \vert b \vert}{2}} x^{b} \\
=& \lambda^{\frac{\vert a \vert - \vert b \vert}{2}} B ( \lambda P_{0} +1)^{- 1 - \frac{\vert a \vert - \vert b \vert}{2}} x^{b} ,\end{aligned}$$ where $B$ is a bounded operator on $L^{2} (\R^{d})$ since it is a Fourier multiplier by a uniformly bounded function.
Using Lemma \[b6\] to estimate the powers of the resolvent, we get $$\label{b7}
T = B \lambda^{\frac{\vert a \vert - \vert b \vert}{2} - \min ( \alpha , d/4 ) + \varepsilon} \< x \>^{\vert b \vert + \min (2 \alpha , d/2 + \varepsilon )}$$ where $B$ is an other bounded operator, $0< \varepsilon$ and $0 \leq \alpha \leq 1 + ( \vert a \vert - \vert b \vert ) /2$. We choose $\alpha = \min ( \gamma + ( \vert a \vert - \vert b \vert ) /2 , \delta ) \leq d/4$ and note $b_{0} = \vert a \vert + 2 \gamma - 2 \delta$.
If $\vert b \vert < b_{0}$, then $\alpha = \delta$ and becomes $$\begin{aligned}
T =& B \lambda^{\frac{\vert a \vert - \vert b \vert}{2} - \delta + \varepsilon} \< x \>^{\vert b \vert + 2 \delta} \\
=& \CO \big( \lambda^{\frac{\vert a \vert}{2} - \delta + \varepsilon} \< x \>^{2 \delta} + \lambda^{\frac{\vert a \vert - b_{0}}{2} - \delta + \varepsilon} \< x \>^{b_{0} + 2 \delta} \big) ,\end{aligned}$$ since $y^{\vert b \vert} \leq y^{b_{0}} + y^{0}$ for $0 \leq b \leq b_{0}$ and $y \geq 0$. Using $\vert a \vert \leq \beta$, we get $$\label{b8}
T = \CO \big( \lambda^{\beta /2 - \delta + \varepsilon} \< x \>^{2 \delta} + \lambda^{- \gamma + \varepsilon} \< x \>^{\beta + 2 \gamma} \big) .$$
If $\vert b \vert \geq b_{0}$, then $\alpha = \gamma + ( \vert a \vert - \vert b \vert ) /2$ and gives $$\label{b9}
T = \CO \big( \lambda^{- \gamma + \varepsilon} \< x \>^{\beta + 2 \gamma} \big) .$$ The estimates and imply the lemma for $\beta \in \N$. The case $\beta \in \R^{+}$ follows from an interpolation argument.
Mimicking the previous proofs, one can show the following results
*\[b13\] Let $j \in \{1 , \ldots , d \}$, $\beta \geq 0$ and $0 \leq \gamma \leq 1 /2$ with $\gamma + \beta /2 \leq d/4$. Then, for all $\varepsilon > 0$, we have $$\big\Vert \< x \>^{\beta} (\lambda^{1/2} \partial_{j}) ( \lambda P_{0} +1)^{- 1} u \big\Vert \lesssim \lambda^{- \gamma + \varepsilon} \big\Vert \< x \>^{\beta + 2 \gamma} u \big\Vert ,$$ uniformly for $\lambda \geq 1$.*
*\[b14\] Let $j,k \in \{1 , \ldots , d \}$ and $0 \leq \beta /2 \leq d/4$. Then, for all $\varepsilon > 0$, we have $$\big\Vert \< x \>^{\beta} (\lambda^{1/2} \partial_{j}) ( \lambda P_{0} +1)^{- 1} (\lambda^{1/2} \partial_{k}) u \big\Vert \lesssim \lambda^{\varepsilon} \big\Vert \< x \>^{\beta} u \big\Vert ,$$ uniformly for $\lambda \geq 1$.*
Estimates for an intermediate operator
---------------------------------------
${}^{}$
We now extend these results to the intermediate differential operator $\widetilde{P}$ defined by $$\label{b33}
\widetilde{P} = - \sum_{j,k} \partial_{j} g^{2} g^{j,k} \partial_{k} .$$ Recall from that $g^{2} g^{j,k} - \delta_{j,k} = \CO ( \< x \>^{- \rho })$. The square roots of $\widetilde{P}$ and $P_{0}$ are comparable. More precisely, we have
*\[b16\] For $u \in D (\widetilde{P}^{1/2}) = D (P_{0}^{1/2}) = H^{1} (\R^{d})$, $$\Vert \widetilde{P}^{1/2} u \Vert \lesssim \Vert P_{0}^{1/2} u \Vert \lesssim \Vert \widetilde{P}^{1/2} u \Vert .$$*
For $u \in H^{2} (\R^{d})$, we can write $$( \widetilde{P} u ,u) = \sum_{j,k} ( g^{2} g^{j,k} \partial_{j} u , \partial_{k} u) \quad \text{ and } \quad
(P_{0} u ,u) = \sum_{j} ( \partial_{j} u , \partial_{j} u) .$$ Using the ellipticity of $\widetilde{P}$ and $g^{2} g^{j,k} \in L^{\infty} (\R^{d})$, we get $$( P_{0} u ,u ) \lesssim ( \widetilde{P} u ,u) \lesssim ( P_{0} u ,u ).$$ In particular, we have, for $u \in H^{2} (\R^{d})$, $$\begin{gathered}
\Vert \widetilde{P}^{1/2} u \Vert \lesssim \Vert P_{0}^{1/2} u \Vert \lesssim \Vert \widetilde{P}^{1/2} u \Vert \\
\Vert ( \widetilde{P} +1)^{1/2} u \Vert \lesssim \Vert (P_{0} +1)^{1/2} u \Vert \lesssim \Vert (\widetilde{P}+1)^{1/2} u \Vert .\end{gathered}$$ Then, we obtain $D (\widetilde{P}^{1/2}) = D (P_{0}^{1/2}) = H^{1} (\R^{d})$ and the lemma follows.
*\[b19\] Let $\beta \geq 0$ and $0 \leq \gamma \leq \min ( 1 , d/4)$ with $\gamma + \beta /2 \leq d/4$. Then, for all $\varepsilon > 0$, we have $$\label{b11}
\big\Vert \< x \>^{\beta} ( \lambda \widetilde{P} +1)^{- 1} u \big\Vert \lesssim \lambda^{- \gamma + \varepsilon} \big\Vert \< x \>^{\beta + 2 \gamma} u \big\Vert ,$$ uniformly for $\lambda \geq 1$.*
*\[b51\] Mimicking the proof of Lemma \[b19\], one can show that Lemma \[b13\] (for the operators $(\lambda^{1/2} \partial_{j}) ( \lambda \widetilde{P} +1)^{- 1}$ and $( \lambda \widetilde{P} +1)^{- 1} (\lambda^{1/2} \partial_{j})$) and Lemma \[b14\] hold with $P_{0}$ replaced by $\widetilde{P}$.*
From , we have $$P_{0} - \widetilde{P} = \sum_{j ,k} \partial_{j} r_{j ,k} \partial_{k} ,$$ where $r_{j,k} = \delta_{j ,k} - g^{2} g^{j,k} = \CO (\< x\>^{- \rho})$. In the following, to clarify the statement, we will not write the sum over $j,k$ and simply note $P_{0} - \widetilde{P} = \partial r \partial$. Iterating the resolvent identity, we have $$\begin{aligned}
(\lambda \widetilde{P} +1)^{-1} =& (\lambda P_{0} +1 )^{-1} + (\lambda P_{0} +1 )^{-1} \lambda^{1/2} \partial r \lambda^{1/2} \partial (\lambda P_{0} +1 )^{-1} \nonumber \\
&+ \sum_{j=1}^{2N} (\lambda P_{0} +1 )^{-1} ( \lambda^{1/2} \partial ) \Big( r ( \lambda^{1/2} \partial ) (\lambda P_{0} +1 )^{-1} ( \lambda^{1/2} \partial ) \Big)^{j} r ( \lambda^{1/2} \partial ) (\lambda P_{0} +1 )^{-1} \nonumber \\
&+ (\lambda P_{0} +1 )^{-1} ( \lambda^{1/2} \partial ) \Big( r ( \lambda^{1/2} \partial ) (\lambda P_{0} +1 )^{-1} ( \lambda^{1/2} \partial ) \Big)^{N} \nonumber \\
&\quad \times r ( \lambda^{1/2} \partial ) (\lambda \widetilde{P} +1 )^{-1} ( \lambda^{1/2} \partial ) r \nonumber \\
&\quad \times \Big( ( \lambda^{1/2} \partial ) (\lambda P_{0} +1 )^{-1} ( \lambda^{1/2} \partial ) r \Big)^{N} ( \lambda^{1/2} \partial ) (\lambda P_{0} +1 )^{-1} . \label{b15}\end{aligned}$$
Thanks to Remark \[b12\], the first term of the previous equation satisfies . To treat the second term, we use two times Lemma \[b13\] with a gain equal to $\gamma /2 \leq \max (1/2 , d/4)$.
The sum over $j$ can be studied in a similar way: using Lemma \[b13\], each exterior term $(\lambda^{1/2} \partial_{j}) ( \lambda P_{0} +1)^{- 1}$ gives a factor $\lambda^{- \gamma /2 + \widetilde{\varepsilon}}$, and, using Lemma \[b14\], each interior factor $( \lambda^{1/2} \partial )$ $(\lambda P_{0} +1 )^{-1} ( \lambda^{1/2} \partial )$ gives a factor $\lambda^{\widetilde{\varepsilon}}$. Then, each term in the sum over $j$ can be estimated by $\lambda^{- \gamma + (j +2) \widetilde{\varepsilon}}$. Taking $\widetilde{\varepsilon} = \varepsilon / (2 N + 2)$, each term of the sum over $j$ satisfies .
It remains to study the last term in . As usual, the first term can be estimated by Lemma \[b13\]: $$\big\Vert \< x \>^{\beta} (\lambda P_{0} +1 )^{-1} ( \lambda^{1/2} \partial ) u \big\Vert \lesssim \lambda^{- \gamma /2 + \widetilde{\varepsilon}} \big\Vert \< x \>^{\beta + \gamma} u \big\Vert .$$ Now, using $r = \CO ( \< x \>^{- \rho} )$ together with Lemma \[b14\], we get $$\big\Vert \< x \>^{\mu} r ( \lambda^{1/2} \partial ) (\lambda P_{0} +1 )^{-1} ( \lambda^{1/2} \partial ) u \big\Vert \lesssim \lambda^{\widetilde{\varepsilon}} \big\Vert \< x \>^{\max (\mu - \rho ,0)} u \big\Vert , \label{c2}$$ for $\mu /2 \leq d/4 + \rho /2$. Using $N$ times the last inequality, we obtain $$\begin{aligned}
\Big\Vert \< x \>^{\beta} (\lambda P_{0} +1 )^{-1} ( & \lambda^{1/2} \partial ) \Big( r ( \lambda^{1/2} \partial ) (\lambda P_{0} +1 )^{-1} ( \lambda^{1/2} \partial ) \Big)^{N} u \Big\Vert \nonumber \\
&\lesssim \lambda^{- \gamma /2 + (N +1) \widetilde{\varepsilon}} \big\Vert \< x \>^{\max (\beta + \gamma - \rho N ,0)} u \big\Vert \leq \lambda^{- \gamma /2 + (N +1) \widetilde{\varepsilon}} \Vert u \Vert , \label{c5}\end{aligned}$$ for $N$ large enough. Using two times Lemma \[b16\] and the functional calculus, $$\label{c4}
\big\Vert ( \lambda^{1/2} \partial ) (\lambda \widetilde{P} +1 )^{-1} ( \lambda^{1/2} \partial ) u \big\Vert \lesssim \Vert u \Vert .$$ Finally, applying $N$ times , with $N$ large enough, we get $$\label{c3}
\Big\Vert \< x \>^{\gamma} \Big( r ( \lambda^{1/2} \partial ) (\lambda P_{0} +1 )^{-1} ( \lambda^{1/2} \partial ) \Big)^{N} \Big\Vert \lesssim \lambda^{N \widetilde{\varepsilon}} \Vert u \Vert ,$$ since $\gamma \leq d/4$. Moreover, using $\gamma /2 \leq 1/2$ and taking the adjoint in Lemma \[b13\], we have $$\big\Vert \< x \>^{- \gamma} (\lambda^{1/2} \partial_{j}) ( \lambda P_{0} +1)^{- 1} u \big\Vert \lesssim \lambda^{- \gamma /2 + \widetilde{\varepsilon}} \big\Vert u \big\Vert .$$ Combining the last estimate with the adjoint of , it follows $$\Big\Vert \Big( ( \lambda^{1/2} \partial ) (\lambda P_{0} +1 )^{-1} ( \lambda^{1/2} \partial ) r \Big)^{N} ( \lambda^{1/2} \partial ) (\lambda P_{0} +1 )^{-1} \Big\Vert \lesssim \lambda^{- \gamma /2 + (N +1) \widetilde{\varepsilon}} \Vert u \Vert , \label{c6}$$ for $N$ large enough. Summing up , , and choosing $\widetilde{\varepsilon}$ small enough with respect to $\varepsilon$, the last term in satisfies .
Estimates for the perturbed Laplacian
--------------------------------------
${}^{}$
Here, we extend the previous results to the Laplacian $P$. From , we have $P = g^{-1} \widetilde{P} g^{-1}$. In particular, the resolvent identity gives $$\begin{aligned}
( \lambda P +1)^{-1 } =& g ( \lambda \widetilde{P} + g^{2} )^{-1 } g \nonumber \\
=& g ( \lambda \widetilde{P} + 1 )^{-1 } g + g ( \lambda \widetilde{P} + 1 )^{-1 } (1-g^{2}) g^{-1} ( \lambda P + 1 )^{-1 } \label{b35} \\
=& g ( \lambda \widetilde{P} + 1 )^{-1 } g + ( \lambda P + 1 )^{-1 } g^{-1} (1-g^{2}) ( \lambda \widetilde{P} + 1 )^{-1 } g . \label{b36}\end{aligned}$$ Note that, by , $(1-g^{2}) = \CO ( \< x \>^{- \rho} )$.
*\[b45\] Let $\beta \geq 0$ and $0 \leq \gamma \leq 1$ with $\gamma + \beta /2 \leq d/4$. Then, for all $\varepsilon > 0$, we have $$\big\Vert \< x \>^{\beta} ( \lambda P +1)^{- 1} u \big\Vert \lesssim \lambda^{- \gamma + \varepsilon} \big\Vert \< x \>^{\beta + 2 \gamma} u \big\Vert ,$$ uniformly for $\lambda \geq 1$.*
As in the proof of Lemma \[b19\], we iterate the resolvent identity and obtain $$\begin{aligned}
(\lambda P +1)^{-1} =& g ( \lambda \widetilde{P} + 1 )^{-1 } g + g ( \lambda \widetilde{P} + 1 )^{-1 } (1 -g^{2}) ( \lambda \widetilde{P} + 1 )^{-1 } g \nonumber \\
&+ \sum_{j=1}^{N} g ( \lambda \widetilde{P} + 1 )^{-1 } \Big( (1-g^{2}) ( \lambda \widetilde{P} + 1 )^{-1 } \Big)^{j} (1 -g^{2}) ( \lambda \widetilde{P} + 1 )^{-1 } g \nonumber \\
&+ g ( \lambda \widetilde{P} + 1 )^{-1 } \Big( (1-g^{2}) ( \lambda \widetilde{P} + 1 )^{-1 } \Big)^{N+1} (1-g^{2}) g^{-1} ( \lambda P +1)^{-1} . \label{b37}\end{aligned}$$ The two first terms and the sum over $j$ can be directly estimated by Lemma \[b19\]. For the last term in , we remark that Lemma \[b19\] gives $$\label{b38}
\big\Vert \< x \>^{\beta} g ( \lambda \widetilde{P} + 1 )^{-1 } u \big\Vert \lesssim \lambda^{- \gamma + \widetilde{\varepsilon}} \big\Vert \< x \>^{\beta + 2 \gamma} u \big\Vert ,$$ and $$\label{b39}
\big\Vert \< x \>^{\mu} (1-g^{2}) ( \lambda \widetilde{P} + 1 )^{-1 } u \big\Vert \lesssim \lambda^{\widetilde{\varepsilon}} \big\Vert \< x \>^{\max ( \mu - \rho ,0)} u \big\Vert ,$$ for all $\mu /2 \leq d/4 + \rho /2$. Therefore, applying and $N+1$ times (this can be made since $( \beta + 2 \gamma ) /2 \leq d/4$), we get $$\begin{aligned}
\Big\Vert \< x \>^{\beta} g ( \lambda \widetilde{P} + 1 )^{-1 } \Big( (1-g^{2}) ( \lambda \widetilde{P} + 1 )^{-1 } \Big)^{N+1} u \Big\Vert &\lesssim \lambda^{- \gamma + (N +2 ) \widetilde{\varepsilon}} \big\Vert \< x \>^{\max ( \mu - (N+1) \rho ,0)} u \big\Vert \\
&\lesssim \lambda^{- \gamma + (N +2 ) \widetilde{\varepsilon}} \Vert u \Vert ,\end{aligned}$$ for $N$ large enough. Using $\Vert ( \lambda P + 1 )^{-1 } \Vert \leq 1$ by the spectral theorem and taking $\widetilde{\varepsilon} = \varepsilon / ( N+2)$, this implies $$\Big\Vert \< x \>^{\beta} g ( \lambda \widetilde{P} + 1 )^{-1 } \Big( (1-g^{2}) ( \lambda \widetilde{P} + 1 )^{-1 } \Big)^{N+1} (1-g^{2}) g^{-1} ( \lambda P +1)^{-1} u \Big\Vert \lesssim \lambda^{- \gamma + \varepsilon} \Vert u \Vert ,$$ and the lemma follows.
Mimicking the proof of Proposition \[b45\] and using and Remark \[b51\], one can prove, as for Lemma \[b13\], the following result.
*\[b20\] Let $j \in \{1 , \ldots , d \}$, $\beta \geq 0$ and $0 \leq \gamma \leq 1 /2$ with $\gamma + \beta /2 \leq d/4$. Then, for all $\varepsilon > 0$, we have $$\begin{aligned}
\big\Vert \< x \>^{\beta} ( \lambda P +1)^{- 1} (\lambda^{1/2} \widetilde{\partial}_{j}^{*} ) u \big\Vert & \lesssim \lambda^{- \gamma + \varepsilon} \big\Vert \< x \>^{\beta + 2 \gamma} u \big\Vert \\
\big\Vert \< x \>^{\beta} (\lambda^{1/2} \widetilde{\partial}_{j} ) ( \lambda P +1)^{- 1} u \big\Vert & \lesssim \lambda^{- \gamma + \varepsilon} \big\Vert \< x \>^{\beta + 2 \gamma} u \big\Vert .\end{aligned}$$ uniformly for $\lambda \geq 1$.*
Let $j,k \in \{1 , \ldots , d \}$ and $0 \leq \beta /2 \leq d/4$. Then, for all $\varepsilon > 0$, we have $$\big\Vert \< x \>^{\beta} (\lambda^{1/2} \widetilde{\partial}_{j} ) ( \lambda P +1)^{- 1} (\lambda^{1/2} \widetilde{\partial}_{k}^{*} ) u \big\Vert \lesssim \lambda^{\varepsilon} \big\Vert \< x \>^{\beta} u \big\Vert ,$$ uniformly for $\lambda \geq 1$.
*\[b59\] The results of this section are given for $( \lambda P +1 )^{-1}$, but can be extended to $( \lambda P - z )^{-1}$, with $\im z \neq 0$. In fact, following the previous proofs, one can see that $( \lambda P - z )^{-1}$ satisfies the same results, if we accept a lose of the form $\vert \im z \vert^{-C}$, $C>0$, in the estimates. This is due to $(\lambda P_{0} +1) ( \lambda P_{0} -z)^{-1} = \CO ( \vert \im z \vert^{-1} )$ from the spectral theorem. Note that the constant $C$ does not depend on $\varepsilon \in ]0,1]$, and is uniform with respect to $\alpha , \beta, \gamma , \delta$ in a compact subset.*
For example, Proposition \[b45\] gives the following estimate for $\beta \geq 0$, $\varepsilon >0$ and $0 \leq \gamma \leq 1$ with $\gamma + \beta /2 \leq d/4$: $$\label{b60}
\big\Vert \< x \>^{\beta} ( \lambda P - z )^{- 1} u \big\Vert \lesssim \frac{\lambda^{- \gamma + \varepsilon}}{\vert \im z \vert^{C}} \big\Vert \< x \>^{\beta + 2 \gamma} u \big\Vert ,$$ uniformly for $\lambda \geq 1$ and $z$ in a compact of $\C$.
Using the spectral theorem, this remark implies the following result.
*\[b61\] Let $\chi \in C^{\infty}_{0} ( \R )$, $j,k \in \{1 , \ldots , d \}$ and $\beta , \gamma \geq 0$ with $\gamma + \beta /2 \leq d/4$. Then, for all $\varepsilon > 0$, we have $$\begin{gathered}
\big\Vert \< x \>^{\beta} \chi ( \lambda P ) u \big\Vert \lesssim \lambda^{- \gamma + \varepsilon} \big\Vert \< x \>^{\beta + 2 \gamma} u \big\Vert \\
\big\Vert \< x \>^{\beta} (\lambda^{1/2} \widetilde{\partial}_{j}) \chi ( \lambda P ) u \big\Vert \lesssim \lambda^{- \gamma + \varepsilon} \big\Vert \< x \>^{\beta + 2 \gamma} u \big\Vert \\
\big\Vert \< x \>^{\beta} \chi ( \lambda P ) (\lambda^{1/2} \widetilde{\partial}_{j}^{*} ) u \big\Vert \lesssim \lambda^{- \gamma + \varepsilon} \big\Vert \< x \>^{\beta + 2 \gamma} u \big\Vert \\
\big\Vert \< x \>^{\beta} (\lambda^{1/2} \widetilde{\partial}_{j}) \chi ( \lambda P ) ( \lambda^{1/2} \widetilde{\partial}_{k}^{*} ) u \big\Vert \lesssim \lambda^{- \gamma + \varepsilon} \big\Vert \< x \>^{\beta + 2 \gamma} u \big\Vert ,\end{gathered}$$ uniformly for $\lambda \geq 1$.*
We only prove the first inequality since the others can be treated the same way. Let $k \in \N$ be such that $\gamma / k \leq 1$, $\varphi ( \sigma ) = \chi ( \sigma ) ( \sigma +1 )^{k} \in C^{\infty}_{0} (\R )$ and $\widetilde{\varphi} \in C_{0}^{\infty} ( \C )$ be an almost analytic extension of $\varphi$. From the spectral theorem, we have $$\label{b63}
\< x \>^{\beta} \chi ( \lambda P )= \frac{1}{\pi} \int \overline{\partial} \widetilde{\varphi} (z) \< x \>^{\beta} ( \lambda P -z)^{-1} ( \lambda P +1)^{-k} L (d z) .$$ Estimate with $\gamma = 0$ gives $$\label{b62}
\big\Vert \< x \>^{\beta} (\lambda P -z)^{-1} u \big\Vert \lesssim \frac{\lambda^{\widetilde{\varepsilon}}}{\vert \im z \vert^{C}} \big\Vert \< x \>^{\beta} u \big\Vert .$$ Proposition \[b45\] with $\gamma = \gamma / k \leq 1$ implies $$\big\Vert \< x \>^{\mu} ( \lambda P +1)^{-1} u \big\Vert \lesssim \lambda^{- \gamma / k + \widetilde{\varepsilon}} \big\Vert \< x \>^{\mu + 2 \gamma / k} u \big\Vert ,$$ if $\gamma / k + \mu /2 \leq d/4$. By iteration, we obtain $$\big\Vert \< x \>^{\beta} ( \lambda P +1)^{-k} u \big\Vert \lesssim \lambda^{- \gamma + k \widetilde{\varepsilon}} \big\Vert \< x \>^{\beta + 2 \gamma} u \big\Vert ,$$ since $\gamma + \beta /2 \leq d/4$. Combining this estimate with and taking $\widetilde{\varepsilon} = \varepsilon / (k+1)$, we get $$\big\Vert \< x \>^{\beta} (\lambda P -z)^{-1} (\lambda P +1)^{-k} u \big\Vert \lesssim \frac{\lambda^{- \gamma + \varepsilon}}{\vert \im z \vert^{C}} \big\Vert \< x \>^{\beta + 2 \gamma} u \big\Vert ,$$ and the lemma follows from .
We now state a result which will help us to estimate the square root of $P$. Since this lemma can be proved as Lemma \[b16\], we do not give the proof.
*\[b53\] We have, for $u \in D (P^{1/2}) = H^{1} (\R^{d})$, $$\Vert P^{1/2} u \Vert \lesssim \Vert \nabla g^{-1} u \Vert \lesssim \Vert P^{1/2} u \Vert .$$*
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The development of consistent and stable quasicontinuum models for multi-dimensional crystalline solids remains a challenge. For example, proving stability of the force-based quasicontinuum (QCF) model [@Dobson:2008a] remains an open problem. In 1D and 2D, we show that by [*blending*]{} atomistic and Cauchy–Born continuum forces (instead of a sharp transition as in the QCF method) one obtains positive-definite blended force-based quasicontinuum (B-QCF) models. We establish sharp conditions on the required blending width.'
author:
- 'Xingjie Helen Li, Mitchell Luskin and Christoph Ortner'
bibliography:
- 'BQCFstab.bib'
title: 'Positive-Definiteness of the Blended Force-Based Quasicontinuum Method'
---
[^1]
Introduction
============
Atomistic-to-continuum coupling methods (a/c methods) have been proposed to increase the computational efficiency of atomistic computations involving the interaction between local crystal defects with long-range elastic fields [@curt03; @LinP:2006a; @Miller:2003a; @Shimokawa:2004; @E:2004; @Miller:2008; @Legoll:2005; @bqce11]. Energy-based methods in this class, such as the quasicontinuum model (denoted QCE [@Ortiz:1995a]) exhibit spurious interfacial forces (“ghost forces”) even under uniform strain [@Shenoy:1999a; @Dobson:2008a]. The effect of the ghost force on the error in computing the deformation and the lattice stability by the QCE approximation has been analyzed in [@Dobson:2008a; @Dobson:2008c; @mingyang; @doblusort:qce.stab]. The development of more accurate energy-based a/c methods is an ongoing process [@Shimokawa:2004; @E:2004; @Shapeev2D:2011; @LuskinXingjie.qnl1d; @OrtnerZhang:2011; @XiaoBely:2002]. An alternative approach to a/c coupling is the force-based quasicontinuum (QCF) approximation [@doblusort:qcf.stab; @qcf.stab; @curt03; @Miller:2003a; @Lu.bqcf:2011], but the non-conservative and indefinite equilibrium equations make the iterative solution and the determination of lattice stability more challenging [@qcf.iterative; @qcf.stab; @DobShapOrt:2011]. Indeed, it is an open problem whether the (sharp-interface) QCF method is stable in dimension greater than one.
Many blended a/c coupling methods have been proposed in the literature, e.g., [@xiao:bridgingdomain; @badia:onAtCcouplingbyblending; @bridging; @badia:forcebasedAtCcoupling; @seleson:bridgingmethods; @fish:concurrentAtCcoupling; @prudhomme:modelingerrorArlequin; @bauman:applicationofArlequin; @XiBe:2004]. In the present work, we formulate a blended force-based quasicontinuum (B-QCF) method, similar to the method proposed in [@Lu.bqcf:2011], which smoothly blends the forces of the atomistic and continuum model instead of the sharp transition in the QCF method. In 1D and 2D, we establish sharp conditions under which a linearized B-QCF operator is positive definite.
Our results have three advantages over the stability result proven in [@Lu.bqcf:2011]. Firstly, we establish $H^1$-stability (instead of $H^2$-stability) which opens up the possibility to include defects in the analysis, along the lines of [@OrtnerShapeev:2010; @DobShapOrt:2011]. Secondly, our conditions for the positive definiteness of the linearized B-QCF operator are needed to ensure the convergence of several popular iterative solution methods for the B-QCF equations [@qcf.iterative; @luskin.iter.stat]. We note that the convergence of these popular iterative solution methods for the QCF equations cannot be guaranteed because of its indefinite linearized operator [@qcf.iterative; @luskin.iter.stat]. Thirdly, our results admit much narrower blending regions, which is crucial for the computational efficiency of the method.
The remainder of the paper is split into two sections: In Section \[1DBQCFsection\] we analyze positivity of the B-QCF operator in a 1D model, whereas in Section \[2DBQCFsection\] we analyze a 2D model. Our methods and results are likely more widely applicable to other force-based model couplings.
Analysis of the B-QCF Operator in $1$D {#1DBQCFsection}
======================================
Notation
--------
We denote the scaled reference lattice by ${\epsilon}\mathbb{Z}:=
\{{\epsilon}\ell : \ell\in\mathbb{Z}\}$. We apply a macroscopic strain $F >
0$ to the lattice, which yields $$\mathbf{y}_F := F{\epsilon}\mathbb{Z} = (F {\epsilon}\ell)_{\ell \in \mathbb{Z}}.$$ The space $\mathcal{U}$ of $2N$-periodic zero mean displacements $\mathbf{u}=(u_{\ell})_{\ell \in \mathbb{Z}}$ from $\mathbf{y}_{F}$ is given by $$\mathcal{U}:=\bigg\{\mathbf{u} : u_{\ell+2N}=u_{\ell}
\text{ for }\ell\in \mathbb{Z},
\text{ and }{\textstyle \sum_{\ell=-N+1}^{N}u_{\ell}}=0\bigg\},$$ and we thus admit deformations $\mathbf{y}$ from the space $$\mathcal{Y}_{F}:=\{\mathbf{y}:
\mathbf{y}=\mathbf{y}_{F}+\mathbf{u}\text{ for some }\mathbf{u}\in
\mathcal{U}\}.$$ We set ${\epsilon}=1/N$ throughout so that the reference length of the computational cell remains fixed.
We define the discrete differentiation operator, $D\mathbf{u}$, on periodic displacements by $$(D\mathbf{u})_{\ell}:=\frac{u_{\ell}-u_{\ell-1}}{\epsilon}, \quad
-\infty<\ell<\infty.$$ We note that $\left(D\mathbf{u}\right)_{\ell}$ is also $2N$-periodic in $\ell$ and satisfies the zero mean condition. We will denote $\left(D\mathbf{u}\right)_{\ell}$ by $Du_{\ell}$. We then define $\left(D^{(2)}\mathbf{u}\right)_{\ell}$ and $\left(D^{(3)}\mathbf{u}\right)_{\ell}$ for $-\infty<\ell<\infty$ by $$\left(D^{(2)}\mathbf{u}\right)_{\ell}:=\frac{Du_{\ell+1}-Du_{\ell}}{\epsilon};\quad
\left(D^{(3)}\mathbf{u}\right)_{\ell}:=\frac{Du^{(2)}_{\ell}-Du^{(2)}_{\ell-1}}{\epsilon}.$$ To make the formulas more concise we sometimes denote $Du_{\ell}$ by $u'_{\ell}$, $D^{(2)}u_{\ell}$ by $u''_{\ell}$, etc., when there is no confusion in the expressions.
For a displacement $\mathbf{u}\in \mathcal{U}$ and its discrete derivatives, we employ the weighted discrete $\ell_{\epsilon}^{2}$ and $\ell_{\epsilon}^{\infty}$ norms by $$\begin{aligned}
\|\mathbf{u}\|_{\ell_{\epsilon}^{2}}&:= \left( \epsilon
\sum_{\ell=-N+1}^{N}|u_{\ell}|^{2}\right)^{1/2},\qquad
\|\mathbf{u}\|_{\ell_{\epsilon}^{\infty}}:=\max\limits_{-N+1\le \ell\le N}|u_{\ell}|,\end{aligned}$$ and the weighted inner product $${\langle}\mathbf{u},\mathbf{w}{\rangle}:=\sum\limits_{\ell=-N+1}^{N}\epsilon u_{\ell}w_{\ell}.$$
We will frequently use the following summation by parts identity:
Suppose $\{f_{k}\}_{k = m}^{n+1}$ and $\{g_{k}\}_{k = m}^{n+1}$ are two sequences, then $$\sum\limits_{k=m}^{n}f_{k}\left(g_{k+1}-g_{k}\right)
=\left[f_{n+1}g_{n+1}-f_{m}g_{m}\right]-\sum_{k = n}^{m}g_{k+1}\left(f_{k+1}-f_{k}\right).$$
Also for future reference, we state a discrete Poincar[é]{} inequality [@Ortner:2008a], $$\|\mathbf{v}\|_{\ell_{\epsilon}^{\infty}}\le \|D\mathbf{v}\|_{\ell_{\epsilon}^1}\quad\text{for all}\, \mathbf{v}\in\mathcal{U}.$$
The next-nearest neighbor atomistic model and local QC approximation.
---------------------------------------------------------------------
We consider a one-dimensional ($1$D) atomistic chain with periodicity $2N$, denoted ${\bf y} \in \mathcal{Y}$. The total atomistic energy per period of ${\bf y}$ is given by $\mathcal{E}^{a}(\mathbf{y})-\epsilon
\sum_{\ell=-N+1}^{N}f_{\ell}y_{\ell}$, where $$\label{AtomEnergy1D}
\mathcal{E}^{a}(\mathbf{y})
=\epsilon\sum_{\ell=-N+1}^{N}\left[\phi(y'_{\ell})+\phi(y'_{\ell}+y'_{\ell-1})\right]$$ for a scaled Lennard-Jones type potential [@LennardJones:1924a; @Morse:1929a] $\phi$ and external forces $f_{\ell}$. The equilibrium equations are given by the force balance at each atom: $F_\ell^a + f_\ell = 0$ where $$\begin{aligned}
\label{AtomEquil1D}
F_{\ell}^{a}(\mathbf{y}):=\frac{-1}{\epsilon}\frac{\partial \mathcal{E}^{a}(\mathbf{y})}{\partial y_{\ell}}
=& \frac{1}{\epsilon}\Big\{ \left[\phi'(y'_{\ell+1})+\phi'(y'_{\ell+2}+y'_{\ell+1})\right]
-\left[\phi'(y'_{\ell})+\phi'(y'_{\ell}+y'_{\ell-1})\right]
\Big\}.\end{aligned}$$ We assume that the displacement $\mathbf{u}^a = \mathbf{y}^a -
\mathbf{y}_F$ is “small” and hence linearize the atomistic equilibrium equations about $\mathbf{y}_{F}$ to obtain $$\left(L^{a}\mathbf{u}^a\right)_{\ell}=f_{\ell},\quad\text{for}\quad\ell=-N+1,\dots,N,$$ where $\left(L^a\mathbf{v}\right)$ for a displacement $\mathbf{v}\in \mathcal{U}$ is given by $$\left(L^a\mathbf{v}\right)_{\ell}:=\phi''_{F}\frac{\left(-v_{\ell+1}+2v_{\ell}-v_{\ell-1}\right)}{\epsilon^{2}}+
\phi''_{2F}\frac{\left(-v_{\ell+2}+2v_{\ell}-v_{\ell-2}\right)}{\epsilon^{2}}.$$ Here and throughout we use the notation $\phi''_{F}:=\phi''(F)$ and $\phi''_{2F}:=\phi''(2F)$, where $\phi$ is the potential in . We assume that $\phi''_{F} > 0$, which holds for typical pair potentials such as the Lennard-Jones potential under physically relevant deformations.
We will later require the following characterisation of the stability of $L^a$.
\[th:stab\_atm\] $L^a$ is positive definite, uniformly for $N \in \mathbb{N}$, if and only if $c_0 := \min(\phi_F'', \phi_F'' + 4 \phi_{2F}'') >
0$. Moreover, $${\langle}L^a \mathbf{u},\mathbf{u}{\rangle}\geq c_0
\|D\mathbf{u}\|^2_{\ell_{\epsilon}^2} \qquad \forall \mathbf{u}
\in \mathcal{U}.$$
The case $\phi_{2F}'' \leq 0$ was treated in [@doblusort:qce.stab], hence suppose that $\phi_{2F}'' > 0$. The coercivity estimate is trivial in this case, and it remains to show that it is also sharp. To that end, we note that $${\langle}L^a \mathbf{u},\mathbf{u}{\rangle}= {\epsilon}\sum_{\ell} \phi_F''
(u_\ell')^2 + {\epsilon}\sum_{\ell} \phi_{2F}'' (u_{\ell-1}' + u_\ell')^2.$$ Hence, testing with $u_\ell' = (-1)^\ell$ (this is admissible since there is an even number of atoms per period), the second-neighbor terms drop out and we obtain ${\langle}L^a \mathbf{u},\mathbf{u}{\rangle}=
\phi_F'' \|D\mathbf{u}\|^2_{\ell_{\epsilon}^2}$.
The local QC approximation (QCL) uses the Cauchy–Born extrapolation rule [@Ortiz:1995a; @Shimokawa:2004], that is, approximating $y'_{\ell}+y'_{\ell-1}$ in by $2y'_{\ell}$ in our context. Thus, the QCL energy is given by $$\label{QCLEnergy1D}
\mathcal{E}^{qcl}(\mathbf{y})=\epsilon\sum_{\ell=-N+1}^{N}\left[\phi(y'_{\ell})+\phi(2y'_{\ell})\right].$$ We can similarly obtain the linearized QCL equilibrium equations about the uniform deformation $$\left(L^{qcl}\mathbf{u}^{qcl}\right)_{\ell}=f_{\ell}\quad \text{for}\quad \ell=-N+1,\dots, N,$$ where the expression of $\left(L^{qcl}\mathbf{v}\right)_{\ell}$ with $\mathbf{v}\in \mathcal{U}$ is $$\left(L^{qcl}\mathbf{v}\right)_{\ell}:=
\left(\phi''_{F}+4\phi''_{2F}\right)\frac{\left(-v_{\ell+1}+2v_{\ell}-v_{\ell-1}\right)}{\epsilon^2}.$$
The Blended QCF Operator
------------------------
The blended QCF (B-QCF) operator is obtained through smooth blending of the atomistic and local QC models. Let $\beta : \mathbb{R} \to
\mathbb{R}$ be a “smooth” and $2$-periodic blending function, then we define $$F_\ell^{bqcf}(\mathbf{y}) := \beta_\ell F_\ell^a(\mathbf{y}) +
(1-\beta_\ell) F_\ell^{qcl}(\mathbf{y}),$$ where $F_\ell^{qcl}$ is defined analogously to $F_\ell^a$ and $\beta_\ell := \beta(F\epsilon \ell)$. Linearisation about $\mathbf{y}_F$ yields the linearized B-QCF operator $$(L^{bqcf}\mathbf{v})_{\ell}:=\beta_{\ell} (L^a\mathbf{v})_{\ell}+(1-\beta_{\ell})(L^{qcl}\mathbf{v})_{\ell}.$$ In order to obtain a [*practical*]{} atomistic-to-continuum coupling scheme, we would also need to coarsen the continuum region by choosing a coarser finite element mesh. In the present work we focus exclusively on the stability of the B-QCF operator, which is a necessary ingredient in any subsequent analysis of the B-QCF method.
Positive-Definiteness of the B-QCF Operator
-------------------------------------------
We begin by writing $L^{bqcf}$ in the form $L^{bqcf}=\phi''_{F}L_{1}^{bqcf}+\phi''_{2F}L_{2}^{bqcf}$ where $$\begin{aligned}
\left(L_{1}^{bqcf}\mathbf{v}\right)_{\ell}=&\epsilon^{-2}\left(-v_{\ell+1}+2v_{\ell}-v_{\ell-1}\right),\quad \text{and}\\
\left(L_{2}^{bqcf}\mathbf{v}\right)_{\ell}=&\beta_{\ell}\epsilon^{-2}\left(-v_{\ell+2}+2v_{\ell}-v_{\ell-2}\right)
+(1-\beta_{\ell})4\epsilon^{-2}\left(-v_{\ell+1}+2v_{\ell}-v_{\ell-1}\right).\end{aligned}$$
\[DivformLemma\] For any $\mathbf{u}\in \mathcal{U}$, the nearest neighbor and next-nearest neighbor interaction operator can be written in the form $$\begin{aligned}
\label{Divform}
\begin{split}
{\langle}L^{bqcf}_{1}\mathbf{u},\mathbf{u}{\rangle}=&\|D\mathbf{u}\|_{\ell_{\epsilon}^2}^2,\quad \text{and}\\
{\langle}L^{bqcf}_{2}\mathbf{u},\mathbf{u}{\rangle}=& \big[ 4\|D\mathbf{u}\|_{\ell_{\epsilon}^2}^2
-\epsilon^{2}\|\sqrt{\beta}D^{(2)}\mathbf{u}\|_{\ell_{\epsilon}^2}^2 \big]
+\mathbf{R}+\mathbf{S}+\mathbf{T},
\end{split}\end{aligned}$$ where the terms $\mathbf{R}$ and $\mathbf{S}$ are given by $$\begin{aligned}
\label{SigPart}
\begin{split}
\mathbf{R}=&\sum\limits_{\ell=-N+1}^{N}2\epsilon^3D^{(2)}\beta_{\ell}\left(Du_{\ell}\right)^2 ,
\quad
\mathbf{S}=\sum\limits_{\ell=-N+1}^{N}\epsilon^4 D^{(2)}\beta_{\ell}D^{(2)}u_{\ell}Du_{\ell}\quad \\
&\qquad\qquad\text{and}\quad
\mathbf{T}=\sum\limits_{\ell=-N+1}^{N}\epsilon^3 \left(D^{(3)}\beta_{\ell+1}\right)u_{\ell}Du_{\ell+1}.
\end{split}\end{aligned}$$
Since the proof of the first identity of Lemma \[DivformLemma\] is not difficult, we only prove the identity for $L^{bqcf}_{2}$. The main tool used here is the summation by parts formula. We note that $$\begin{aligned}
{\langle}L^{bqcf}_{2}\mathbf{u},\mathbf{u}{\rangle}=&\sum\limits_{\ell=-N+1}^{N} \epsilon \beta_{\ell} \frac{\left(-u_{\ell+2}+2u_{\ell}-u_{\ell-2}\right)}{\epsilon^2}u_{\ell}
+\epsilon(1-\beta_{\ell})\frac{4\left( -u_{\ell+1}+2u_{\ell}-u_{\ell-1}\right)}{\epsilon^2}u_{\ell}\nonumber\\
=&\sum\limits_{\ell=-N+1}^{N}\epsilon\frac{4\left(-u_{\ell+1}+2u_{\ell}-u_{\ell-1}\right)}{\epsilon^2}u_{\ell}\nonumber\\
&\qquad\qquad+\sum\limits_{\ell=-N+1}^{N}\epsilon \beta_{\ell}\frac{\left(-u_{\ell+2}+4u_{\ell+1}-6u_{\ell}+4u_{\ell-1}-u_{\ell-2}\right)}{\epsilon^2}
u_{\ell}\nonumber\\
=&4\|D\mathbf{u}\|^2_{\ell_{\epsilon}^2}
+\sum\limits_{\ell=-N+1}^{N}\epsilon^2\beta_{\ell} \left(-D^{(3)}u_{\ell+1}+D^{(3)}u_{\ell}\right)u_{\ell}.\label{Divlemeq1}
$$ We then apply the summation by parts formula to the second term of to obtain $$\begin{split}
&\sum\limits_{\ell=-N+1}^{N}\beta_{\ell}\epsilon^2 \left(-D^{(3)}u_{\ell+1}+D^{(3)}u_{\ell}\right)u_{\ell}\\
&\qquad=\sum\limits_{\ell=-N+1}^{N}\epsilon^2 D^{(3)}u_{\ell+1}\left[\beta_{\ell+1}u_{\ell+1}-\beta_{\ell}u_{\ell}\right]
=\sum\limits_{\ell=-N+1}^{N}\epsilon^3 D^{(3)}u_{\ell}\left[\beta_{\ell}Du_{\ell}+u_{\ell-1}D\beta_{\ell}\right].
\end{split}$$ We use the summation by parts formula again and change the index according to the periodicity so that we get $$\begin{aligned}
\sum\limits_{\ell=-N+1}^{N}&\epsilon^3
D^{(3)}u_{\ell}\left[\beta_{\ell}Du_{\ell}+u_{\ell-1}D\beta_{\ell}\right]\nonumber\\
&= \sum_{\ell=-N+1}^{N}\epsilon^2
\left(\beta_{\ell}Du_{\ell}\right)\left(D^{(2)}u_{\ell}-D^{(2)}u_{\ell-1}\right)
+\sum\limits_{\ell=-N+1}^{N}\epsilon^3
\left(D^{(3)}u_{\ell}\right)\,u_{\ell-1}D\beta_{\ell}\nonumber\\
&= \sum\limits_{\ell=-N+1}^{N}\epsilon^2
\left(-D^{(2)}u_{\ell}\right)\left(\beta_{\ell+1}Du_{\ell+1}-\beta_{\ell}Du_{\ell}\right)
+\sum\limits_{\ell=-N+1}^{N}\epsilon^3
\left(D^{(3)}u_{\ell}\right)\,u_{\ell-1}D\beta_{\ell}\nonumber\\
&= \sum\limits_{\ell=-N+1}^{N}\epsilon^2
\left(-D^{(2)}u_{\ell}\right)\left[\beta_{\ell+1}Du_{\ell+1}-\beta_{\ell}Du_{\ell+1}+\beta_{\ell}Du_{\ell+1}-\beta_{\ell}Du_{\ell}\right]
\nonumber\\&\qquad\qquad +\sum\limits_{\ell=-N+1}^{N}\epsilon^3
\left(D^{(3)}u_{\ell}\right)\,u_{\ell-1}D\beta_{\ell}\nonumber\\
&= -\epsilon^2
\|\sqrt{\beta}D^{(2)}\mathbf{u}\|^2_{\ell_{\epsilon}^2} +
\sum\limits_{\ell=-N+1}^{N}\epsilon^3\left[-D^{(2)}u_{\ell-1}D\beta_{\ell}Du_{\ell}+
D^{(3)}u_{\ell}\,u_{\ell-1}D\beta_{\ell}\right].\label{Divlemeq2}
$$ We now focus on the second term of . We repeatedly use the summation by parts formula to obtain $$\begin{aligned}
\sum\limits_{\ell=-N+1}^{N}&\epsilon^3\left[-D^{(2)}u_{\ell-1}D\beta_{\ell}Du_{\ell}+
\left(D^{(3)}u_{\ell}\right)\,u_{\ell-1}D\beta_{\ell}\right]\\
&=
\sum\limits_{\ell=-N+1}^{N}-\epsilon^2D\beta_{\ell}\left[\left(Du_{\ell}\right)^2-\left(Du_{\ell-1}\right)^2\right]\\
&\qquad+\sum\limits_{\ell=-N+1}^{N}\epsilon^2D\beta_{\ell}\left[\left(Du_{\ell}-Du_{\ell-1}\right)Du_{\ell-1}
+\left(D^{(2)}u_{\ell}-D^{(2)}u_{\ell-1}\right)u_{\ell-1}\right]\\
&= \sum\limits_{\ell=-N+1}^{N}\epsilon^3D^{(2)}\beta_{\ell}\left(Du_{\ell}\right)^2
+\sum\limits_{\ell=-N+1}^{N}\epsilon^2D\beta_{\ell}\left[u_{\ell-1}D^{(2)}u_{\ell}-u_{\ell-2}D^{(2)}u_{\ell-1}\right]\\
&=\sum\limits_{\ell=-N+1}^{N}2\epsilon^3 D^{(2)}\beta_{\ell}\left(Du_{\ell}\right)^2
+\sum\limits_{\ell=-N+1}^{N}\epsilon^4 D^{(2)}\beta_{\ell}D^{(2)}u_{\ell}Du_{\ell}
+\sum\limits_{\ell=-N+1}^{N}\epsilon^3 \left(D^{(3)}\beta_{\ell+1}\right)u_{\ell}Du_{\ell+1}\\
&=\mathbf{R}+\mathbf{S}+\mathbf{T},\end{aligned}$$ where $\mathbf{R}$, $\mathbf{S}$ and $\mathbf{T}$ are defined in .
Combining all of the above equalities, we obtain .
We shall see below that the first group in does not negatively affect the stability of the B-QCF operator. By contrast, the three terms ${\bf R}$, ${\bf S}$, ${\bf T}$ should be considered “error terms”. We estimate them in the next lemma.
In order to proceed with the analysis we define $$\mathcal{I}:=\big\{\ell \in \mathbb{Z} : 0 < \beta_{\ell+j} < 1
\text{ for some } j \in \{\pm 1, \pm 2\} \big\},$$ so that $D^{(j)}\beta_\ell = 0$ for all $\ell \in \{-N+1,\dots N\}
\setminus \mathcal{I}$ and $j \in \{1,2,3\}$, and $K:=\sharp\mathcal{I}$.
\[SigEstLemma\] Let $\mathbf{R}$, $\mathbf{S}$ and $\mathbf{T}$ be defined by , then we have the following estimates: $$\label{SigEst}
\begin{split}
|\mathbf{R}|\le~&
\epsilon^{2}\|D^{(2)}\beta\|_{\ell_{\epsilon}^{\infty}}\|D\mathbf{u}\|^2_{\ell_{\epsilon}^2},
\\
|{\bf S}| \leq~& 2\epsilon^2
\|D^{(2)}\beta\|_{\ell_{\epsilon}^{\infty}}\|D\mathbf{u}\|^2_{\ell_{\epsilon}^2},
\quad \text{and} \\
|\mathbf{T}| \le~&
\epsilon^2 \sqrt{2}(K\epsilon)^{1/2} \|D^{(3)}\beta\|_{\ell_{\epsilon}^{\infty}} \,
\|D\mathbf{u}\|^2_{\ell_{\epsilon}^2}.
\end{split}$$
The estimate for ${\bf R}$ follows directly from H[ö]{}lder’s inequality.
To estimate ${\bf S}$ recall that $D^{(2)}u_{\ell}:=\frac{Du_{\ell+1}-Du_{\ell}}{\epsilon}$, which implies $$\|D^{(2)}\mathbf{u}\|^2_{\ell_{\epsilon}^2}\le
\frac{4}{\epsilon^2}\|D\mathbf{u}\|^2_{\ell_{\epsilon}^2}.$$ Therefore, ${\bf S}$ is bounded by $$\begin{aligned}
|{\bf S}| = \left| \sum\limits_{\ell=-N+1}^{N} \epsilon^4
D^{(2)}\beta_{\ell}D^{(2)}u_{\ell}Du_{\ell}\right| \le\epsilon^3
\|D^{(2)}\beta\|_{\ell_{\epsilon}^{\infty}}\|D^{(2)}\mathbf{u}\|_{\ell_{\epsilon}^2}
\|D\mathbf{u}\|_{\ell_{\epsilon}^2}\le 2\epsilon^2
\|D^{(2)}\beta\|_{\ell_{\epsilon}^{\infty}}\|D\mathbf{u}\|^2_{\ell_{\epsilon}^2}.
\end{aligned}$$
Finally, we estimate ${\bf T}$ by $$|{\bf T}| = \left|\sum\limits_{\ell=-N+1}^{N}\epsilon^3D^{(3)}\beta_{\ell+1}Du_{\ell+1}\,u_{\ell}\right|
\le \epsilon^2
\|D^{(3)}\beta\|_{\ell_{\epsilon}^{\infty}}\|\mathbf{u}\|_{\ell_{\epsilon}^2(\mathcal{I})}\|D\mathbf{u}\|_{\ell_{\epsilon}^2},$$ We then apply the H[ö]{}lder inequality, the Poincar[é]{} inequality and Jensen’s inequality successively to $\|\mathbf{u}\|_{\ell_{\epsilon}^2(\mathcal{I})}$ to get $$\|\mathbf{u}\|^2_{\ell_{\epsilon}^2(\mathcal{I})} \le
(K\epsilon)\|\mathbf{u}\|^2_{\ell_{\epsilon}^{\infty}}
\le K\epsilon \|D\mathbf{u}\|^2_{\ell_{\epsilon}^{1}} \le 2K\epsilon
\|D\mathbf{u}\|^2_{\ell_{\epsilon}^{2}}.$$ Therefore, we have $$\left|{\bf T}\right|
\le \epsilon^2
\|D^{(3)}\beta\|_{\ell_{\epsilon}^{\infty}}\|\mathbf{u}\|_{\ell_{\epsilon}^2(\mathcal{I})}\|D\mathbf{u}\|_{\ell_{\epsilon}^2}
\le\sqrt{2} \epsilon^2
\|D^{(3)}\beta\|_{\ell_{\epsilon}^{\infty}}\left(K\epsilon\right)^{1/2}\|D\mathbf{u}\|^2_{\ell_{\epsilon}^2}.$$ Combining the above estimates, we have proven the second inequality in .
We see from the previous result that smoothness of $\beta$ crucially enters the estimates on the error terms ${\bf R}$, ${\bf S}$, ${\bf
T}$. Before we state our main result in 1D we show how quasi-optimal blending functions can be constructed to minimize these terms, which will require us to introduce the [*blending width*]{} into the analysis. The estimate is stated for a single connected interface region, however, an analogous result holds if the interface has connected components with comparable width. A similar result can also be found in [@bqce11].
\[BlendFunEstLemma\]
- Suppose that the blending region is connected, that is $\mathcal{I} = \{ 1, \dots, K\}$ without loss of generality, then $\beta$ can be chosen such that $$\label{eq:BlendFunEst_upper}
\| D^{(j)} \beta \|_{\ell^\infty} \leq C_\beta (K \epsilon)^{-j},
\quad \text{for } j = 1, 2, 3,$$ where $C_\beta$ is independent of $K$ and $\epsilon$.
- This estimate is sharp in sense that, if $\beta_\ell$ attains both the values $0$ and $1$, then $$\label{BlendFunEst}
\|D^{(j)}\beta\|_{\ell^{\infty}}\ge (K\epsilon)^{-j},\quad
\text{for } j=1,2,3.$$
- Suppose that $\mathcal{J} = \{1, \dots, n\} \subset
\mathcal{I}$ such that $\beta(1) = 0$, $\beta(n) = 1$ (or vice-versa), and $0, n+1 \notin \mathcal{I}$, and suppose moreover that is satisfied, then $$\label{eq:blend_interval}
\# \big\{ \ell \in \mathcal{J} : D^{(3)}\beta_\ell \leq
-{{\textstyle \frac{1}{2}}} (\epsilon K)^{-3} \big\}
\geq {{\textstyle \frac{1}{2C_\beta}}} K.$$
[*(i)* ]{} The upper bound follows by fixing a reference blending function $B \in C^3(\mathbb{R})$, $B = 0$ in $(-\infty, 0]$ and $B =
1$ in $[1, +\infty)$, and defining $\beta(x) = B((x-2\epsilon) /
(\epsilon K'))$ for $K' = K-4$. Then $\mathcal{I} = \{1, \dots,
K\}$, and a scaling argument immediately gives .
[*(ii)* ]{} To prove the lower bound, suppose $0 < \beta_\ell < 1$ for $\ell = 1, \dots, K_0-1$, and $\beta_0 = 0$ and $\beta_{K_0} =
1$. Then $\epsilon \sum_{\ell = 1}^{K_0} \beta_\ell' = 1$, from which infer the existence of $K_1 \in \{1, \dots, K_0\}$ such that $\beta_{K_1}' \geq 1 / (\epsilon K_0)$. This establishes the lower bound for $j = 1$. To prove it for $j = 2$ we note that, since $\beta_{K_0} = 1$, $\beta_{K_0+1}' \leq 0$, and hence we obtain $$\epsilon \sum_{\ell = K_1+1}^{K_0} \beta_\ell'' = \beta_{K_0+1}' -
\beta_{K_1}' \leq - 1 / (\epsilon K_0).$$ We deduce that there exists $K_2$ such that $\beta_{K_2}'' \leq - 1 / (\epsilon^2 K_0 (K_0-K_1))\le
-1/(\epsilon K)^2$. This implies for $j =
2$. We can argue similarly to obtain the result for $j = 3$.
[*(iii)* ]{} Finally, to establish , let $m
\in \mathbb{N}$ be chosen minimally such that $\beta_m'' \leq -
(\epsilon K)^{-2}$ and $\beta_0'' = 0$; then $m \leq n$ and we have $$- \frac{1}{(\epsilon K)^2} \geq \beta_m'' - \beta_0'' = \epsilon
\sum_{\ell = 1}^{m} \beta_\ell''' \geq - \frac{\epsilon k
C_\beta}{(\epsilon K)^3} - \frac{\epsilon(m - k)}{2 (\epsilon K)^3},$$ where $k := \#\{ \ell \in \mathcal{J} : \beta_\ell''' \leq - \frac12
(\epsilon K)^{-3} \}$. Rearranging the inequality, we obtain $$- \frac{1}{2 (\epsilon K)^2} \geq
- \frac{1}{(\epsilon K)^2} + \frac{\epsilon(m - k)}{2 (\epsilon
K)^3}
\geq - \frac{\epsilon k
C_\beta}{(\epsilon K)^3} \geq - \frac{k C_\beta}{K (\epsilon K)^2},$$ and we immediately deduce that $k / K \geq 1 / (2C_\beta)$, which concludes the proof of item (iii).
We can summarize the previous estimates and get the following optimal condition for the size $K$ of the blending region provided that $\beta$ is chosen in a quasi-optimal way. Formally, the result states that $L^{bqcf}$ is positive definite if and only if $K \gg
\epsilon^{-1/5}$. In particular, we conclude that the B-QCF operator is positive definite for fairly moderate blending widths.
\[BlendSizeThm\] Let $\mathcal{I}$ and $K$ be defined as in Lemma \[BlendFunEstLemma\], and suppose that $\beta$ is chosen to satisfy the upper bound . Then there exists a constant $C_1 = C_1(C_\beta)$, such that $$\label{eq:1d_coerc_lower}
\langle L^{bqcf} {\bf u}, {\bf u} \rangle \geq \big(c_0 - C_1
|\phi_{2F}''| \big[ K^{-5/2} \epsilon^{-1/2}\big]\big) \|D {\bf
u}\|_{\ell^2_\epsilon}^2
\qquad \forall {\bf u} \in \mathcal{U},$$ where $c_0 = \min(\phi_F'', \phi_F'' + 4 \phi_{2F}'')$ is the atomistic stability constant of Lemma \[th:stab\_atm\].
Moreover, if $\beta_\ell$ takes both the values $0$ and $1$, then there exist constants $C_2, C_3 > 0$, independent of $\mathcal{I}$, $N$, $\phi_F''$ and $\phi_{2F}''$, such that $$\label{eq:1d_coerc_upper}
\inf_{\substack{{\bf u} \in \mathcal{U} \\ \|D{\bf
u}\|_{\ell^2_\epsilon} = 1}} \langle L^{bqcf} {\bf u}, {\bf
u} \rangle \leq \phi_F'' + C_2 |\phi_{2F}''| - C_3 |\phi_{2F}''|
\big[ K^{-5/2} \epsilon^{-1/2} \big].$$
Estimates and establish the asymptotic optimality of the blending width $K \eqsim
\epsilon^{-1/5}$ in the limit as $\epsilon \to 0$: implies that, if $c_0 > 0$ and $K \gg
\epsilon^{-1/5}$, then $L^{bqcf}$ is coercive, while shows that, if $K \ll \epsilon^{-1/5}$ then $L^{bqcf}$ is necessarily indefinite.
We first prove the lower bound. The blended force-based operator satisfies $L^{bqcf}$ $${\langle}L^{bqcf}\mathbf{u},\mathbf{u}{\rangle}=A_F\|D\mathbf{u}\|^2_{\ell_{\epsilon}^2}-\epsilon^2
\phi''_{2F}\|\sqrt{\beta}D^{(2)}\mathbf{u}\|^2_{\ell_{\epsilon}^2}
+\phi''_{2F}(\mathbf{R}+\mathbf{S}+\mathbf{T})$$ where $A_F:=\phi''_{F}+4\phi''_{2F}.$ From Lemma \[SigEstLemma\], we have $$|\mathbf{R}+\mathbf{S}+\mathbf{T}|\le \epsilon^2\left[
4\|D^{(2)}\beta\|_{\ell_{\epsilon}^{\infty}}
+(K\epsilon)^{1/2}\|D^{(3)}\beta\|_{\ell_{\epsilon}^{\infty}}\right]\|D\mathbf{u}\|^2_{\ell_{\epsilon}^2}.$$ Since $\|D^{(j)}\beta\|_{\ell_{\epsilon}^{\infty}}\le
C_{\beta}(K\epsilon)^{-j}$, so we have $$\begin{aligned}
|\mathbf{R}+\mathbf{S}+\mathbf{T}| \le C \epsilon^2\left[ 4(K\epsilon)^{-2}
+(K\epsilon)^{1/2}(K\epsilon)^{-3}\right]\|D\mathbf{u}\|^2_{\ell_{\epsilon}^2}
\le C_3 \left[K^{-5/2}\epsilon^{-1/2}\right]\|D\mathbf{u}\|^2_{\ell_{\epsilon}^2},\end{aligned}$$ where we used the fact that $K^{-2} \leq K^{-5/2} \epsilon^{-1/2}$.
If $\phi''_{2F}\le 0$, then we obtain $${\langle}L^{bqcf}\mathbf{u},\mathbf{u}{\rangle}\ge \left(A_F-C_1|\phi''_{2F}|
\left[ K^{-5/2}\epsilon^{-1/2}\right]\right)\|D\mathbf{u}\|^2_{\ell_{\epsilon}^2}.$$ If $\phi''_{2F} > 0$, then $$\begin{aligned}
{\langle}L^{bqcf}_{2}\mathbf{u},\mathbf{u}{\rangle}=~& A_F\|D\mathbf{u}\|^2_{\ell_{\epsilon}^2}-\epsilon^2
\phi''_{2F}\|\sqrt{\beta}D^{(2)}\mathbf{u}\|^2_{\ell_{\epsilon}^2}+\phi''_{2F}(\mathbf{R}+\mathbf{S}+\mathbf{T})
\\
\geq~& \left(\phi''_{F}-C_3|\phi''_{2F}|
\left[K^{-5/2}\epsilon^{-1/2}\right]\right)\|D\mathbf{u}\|^2_{\ell_{\epsilon}^2},\end{aligned}$$ which is the corresponding result.
To prove the opposite bound, let $\mathcal{J}$ be defined as in Lemma \[BlendFunEstLemma\] (iii). We can assume this without loss of generality upon possibly shifting and inverting the blending function. We define $\mathcal{J}' := \{ \ell \in \mathcal{J} : D^{(3)}
\beta_\ell \leq - {{\textstyle \frac{1}{2}}} (\epsilon K)^{-3} \}$ and $L := \epsilon
\#\mathcal{J}' = \alpha \epsilon K$ for some $\alpha \geq 1 /
(2C_\beta)$, and a test function $\mathbf{v} \in \mathcal{U}$ through $v_0 = {{\textstyle \frac{1}{2}}}$ and $$\label{DvDef1}
\begin{split}
v_{\ell}'=\begin{cases} L^{-1/2}, &\quad \ell\in
\mathcal{J}'\\
0, & \quad \ell\in \mathcal{I}\setminus \mathcal{J}',
\end{cases}
\end{split}$$ and extending $v_\ell'$ outside of $\mathcal{I}$ in such a way that $\|D{\bf v}\|_{\ell^2_\epsilon}$ is bounded uniformly in $\mathcal{I}$ and $N$, and such that ${\bf v}$ is $2N$-periodic (see [@qcf.stab] for details of this construction).
With these definitions we obtain $$\begin{aligned}
{\bf T} =~&
\epsilon^3\sum\limits_{\ell=-N+1}^{N}D^{(3)}\beta_{\ell+1}Dv_{\ell+1}\,v_{\ell}
= \epsilon^3 \sum_{\ell \in \mathcal{J}'} D^{(3)}\beta_{\ell-1}
v_{\ell}' v_{\ell-1} \\
\leq~& - \frac{\epsilon^2 L L^{-1/2}}{4 (\epsilon K)^3} = -
\frac{(\alpha \epsilon K)^{1/2}}{4 \epsilon K^3} = - {{\textstyle \frac{\alpha^{1/2}}{4}}}
K^{-5/2} \epsilon^{-1/2}.\end{aligned}$$ Recall that, by contrast, we have $$|{\bf R} + {\bf S}| \leq C_2 K^{-2} \| D{\bf v} \|_{\ell^2_\epsilon}^2.$$ Combining these estimates, and using the fact that $\|D{\bf
v}\|_{\ell^2_\epsilon}$ is bounded independently of $\mathcal{I}$ and $N$, yields .
Positive-Definiteness of the B-QCF Operator in $2$D {#2DBQCFsection}
===================================================
The triangular lattice
----------------------
For some integer $N\in\mathbb{N}$ and $\epsilon:=1/N$, we define the scaled 2D triangular lattice $$\mathbb{L}:=\mathtt{A}_{6}\mathbb{Z}^2,\quad\text{where}\quad
\mathtt{A}_{6}:=\left[{a}_{1},{a}_{2}\right]
:= \epsilon\left[\begin{array}{cc}
1 & 1/2\\
0 & \sqrt{3}/2
\end{array}\right],$$ where ${a}_{i},\,i=1,2$ are the scaled lattice vectors. Throughout our analysis, we use the following definition of the periodic reference cell $$\Omega:=\mathtt{A}_{6}(-N, N]^2\quad \text{and}\quad \mathcal{L}:=\mathbb{L}\cap\Omega.$$ We furthermore set ${a}_3=(-1/2\epsilon,
\sqrt{3}/2\epsilon)^{\mathtt{T}}$, $a_4 := - a_1, a_5 := -a_2$ and $a_6
:= -a_3$; then the set of *nearest-neighbor directions* is given by $$\mathcal{N}_{1}:=\{\pm{a}_1,\pm{a}_{2},\pm{a}_3\}.$$ The set of *next nearest-neighbor directions* is given by $$\mathcal{N}_{2}:=\{\pm {b}_1,\pm {b}_{2},\pm {b}_3\},
\quad \text{where} \quad b_1:=a_1+a_2,\quad b_2:=a_2+a_3\quad\text{and}\quad b_3=a_3-a_1.$$ We use the notation $\mathcal{N}:=\mathcal{N}_1\cup\mathcal{N}_2$ to denote the directions of the neighboring bonds in the interaction range of each atom (see Figure \[AtomDomainFig\]).
We identify all lattice functions $\mathbf{v} : \mathbb{L} \to
\mathbb{R}^2$ with their continuous, piece affine interpolants with respect to the canonical triangulation $\mathcal{T}$ of $\mathbb{R}^2$ with nodes $\mathbb{L}$.
The atomistic, continuum and blending regions
---------------------------------------------
Let $\mathtt{Hex}(r)$ denote the closed hexagon centered at the origin, with sides aligned with the lattice directions $a_1, a_2, a_3$, and diameter $2r$.
For $R_a< R_b \in \mathbb{N}$, we define the atomistic, blending and continuum regions, respectively, as $$\Omega_a := \mathtt{Hex}(\epsilon R_a), \quad
\Omega_b := \mathtt{Hex}(\epsilon R_b) \setminus \Omega_a,
\quad \text{and} \quad
\Omega_c:={\rm clos}\left(\Omega\setminus\left(\Omega_a\cup\Omega_b\right)\right).$$ We denote the blending width by $K := R_b - R_a$. Moreover, we define the corresponding lattice sites $$\mathcal{L}^a := \mathcal{L}\cap \Omega_a, \qquad
\mathcal{L}^b := \mathcal{L}\cap \Omega_b, \qquad \text{and} \qquad
\mathcal{L}^c := \mathcal{L}\cap \Omega_c.$$ For simplicity, we will again use $\mathcal{L}$ as the finite element nodes, that is, every atom is a repatom.
For a map $\mathbf{v}:\mathbb{L}\rightarrow \mathbb{R}^2$ and bond directions $r,s\in \mathcal{N}$, we define the finite difference operators $$D_{r}v(x):=\frac{v(x+r)-v(x)}{\epsilon}\quad\text{and}\quad
D_{r}D_{s}v(x):=\frac{D_{s}v(x+r)-D_{s}v(x)}{\epsilon}.$$ We define the space of all admissible displacements, $\mathcal{U}$, as all discrete functions $\mathbb{L}\rightarrow \mathbb{R}^2$ which are $\Omega$-periodic and satisfy the mean zero condition on the computational domain: $$\mathcal{U}:=\Big\{\mathbf{u}:\mathbb{L}\rightarrow
\mathbb{R}^2 : \text{$u(x)$ is $\Omega$-periodic and
}{\textstyle \sum_{x\in\mathcal{L}}} u(x)=0 \Big\}.$$ For a given matrix $B \in \mathbb{R}^{2 \times 2}$, $\mathrm{det}(B)>0$, we admit deformations $\mathbf{y}$ from the space $$\mathcal{Y}_{B}:=\big\{ \mathbf{y}
:\mathbb{L}\rightarrow
\mathbb{R}^2: y(x)=Bx+u(x), \,\forall x\in \mathbb{L}\,\text{for some $\mathbf{u}\in\mathcal{U}$}
\big\}.$$
For a displacement $\mathbf{u}\in \mathcal{U}$ and its discrete directional derivatives, we employ the weighted discrete $\ell_{\epsilon}^{2}$ and $\ell_{\epsilon}^{\infty}$ norms given by $$\begin{aligned}
&\|\mathbf{u}\|_{\ell_{\epsilon}^{2}}:= \left( \epsilon^2
\sum_{x\in\mathcal{L}}|u(x)|^{2}\right)^{1/2},\quad
\|\mathbf{u}\|_{\ell_{\epsilon}^{\infty}}:=\max\limits_{x\in\mathcal{L}}|u(x)|,\quad\text{and}\\
&\qquad\quad\|D\mathbf{u}\|_{\ell_{\epsilon}^{2}}:=
\left(\epsilon^2\sum_{x\in\mathcal{L}}\sum_{i=1}^{3}|D_{a_i}u(x)|^2\right)^{1/2}.\end{aligned}$$ The inner product associated with $\ell^2_\epsilon$ is $${\langle}\mathbf{u},\mathbf{w}{\rangle}:=\epsilon^2\sum\limits_{x\in\mathcal{L}}u(x)\cdot w(x).$$
The B-QCF operator
------------------
The total scaled atomistic energy for a periodic computational cell $\Omega$ is $$\begin{aligned}
\mathcal{E}^{a}(\mathbf{y})=&\frac{\epsilon^2}{2}\sum_{x\in\mathcal{L}}\sum_{r\in \mathcal{N}}
\phi(D_{r}y({x}))\label{AtomEnergy2D}
= {\epsilon^2}\sum_{x\in\mathcal{L}}\sum_{i=1}^{3}\big[\phi(D_{a_{i}}y({x}))
+\phi(D_{b_{i}}y({x}))\big],\end{aligned}$$ where $\phi \in C^2(\mathbb{R}^2)$, for the sake of simplicity. Typically, one assumes $\phi(r) = \varphi(|r|)$; the more general form we use gives rise to a simplified notation; see also [@OrtnerShapeev:2010]. We define $\phi'(r) \in \mathbb{R}^2$ and $\phi''(r) \in \mathbb{R}^{2 \times 2}$ to be, respectively, the gradient and hessian of $\phi$.
The equilibrium equations are given by the force balance at each atom, $$\label{AtomEquil2D}
F^{a}(x;y)+f(x;y)=0,\quad \text{for}\quad x\in\mathcal{L},$$ where $f(x;y)$ are the external forces and $F^{a}(x;y)$ are the atomistic forces (per unit volume $\epsilon^2$) $$\begin{aligned}
F^{a}(x;y):=&-\frac{1}{\epsilon^2}\frac{\partial \mathcal{E}^{a}(\mathbf{y})}{\partial y(x)}\\
= &- \frac{1}{\epsilon} \sum_{i=1}^{3}\Big[\phi'\left(D_{a_i}y(x)\right) +\phi'\left(D_{-a_i}y(x)\right)
\Big]
-\frac{1}{\epsilon} \sum_{i=1}^{3}\Big[\phi'\left(D_{b_i}y(x)\right) +\phi'\left(D_{-b_i}y(x)\right)
\Big].\end{aligned}$$ Again, since $\mathbf{u} = \mathbf{y} - \mathbf{y}_B$, where $y_B(x) =
B x$, is assumed to be small we can linearize the atomistic equilibrium equation about $\mathbf{y}_B$: $$\left(L^a\mathbf{u}^a\right)(x)=f(x),\quad \text{for}\quad x\in \mathcal{L},$$ where $\left(L^a\mathbf{v}\right)(x)$, for a displacement $\mathbf{v}$, is given by $$\left(L^a\mathbf{v}\right)(x)=-\sum_{i=1}^{3}\phi''(Ba_{i})D_{a_{i}}D_{a_{i}}v(x-a_{i})
-\sum_{i=1}^{3}\phi''(Bb_{i})D_{b_{i}}D_{b_{i}}v(x-b_{i}),\quad\text{for}\quad x\in\mathcal{L}.$$
The QCL approximation uses the Cauchy–Born extrapolation rule to approximate the nonlocal atomistic model by a local continuum model [@Ortiz:1995a; @Shenoy:1999a; @Miller:2003a]. According to the bond density lemma [@OrtnerShapeev:2010 Lemma 3.2] (see also [@Shapeev2D:2011]), we can write the total QCL energy as a sum of the bond density integrals $$\label{QCLEnergy2D}
\mathcal{E}^{c}(\mathbf{y})= \int_\Omega \sum_{r \in \mathcal{N}}
\phi(\partial_r y) \, dx
= \sum_{x\in \mathcal{L}}\sum_{r\in \mathcal{N}}
\int_{0}^{1}\phi\big(\partial_{r}y(x+tr)\big)dt,$$ where $\partial_{r} y(x) = \frac{d}{dt} y(x+t r)|_{t = 0}$ denotes the directional derivative. We compute the continuum force $F^{c}(x;y) =
-\frac{1}{\epsilon^2} \frac{\partial\mathcal{E}^c}{\partial y(x)}$, and linearize the force equation about the uniform deformation $\mathbf{y}_{B}$ to obtain $$\left(L^{c}\mathbf{u}^{c}\right)(x)=f(x),\quad \text{for}\quad x\in \mathcal{L}.$$
To formulate the B-QCF method, let the blending function $\beta(s):\mathbb{R}^2\rightarrow [0, 1]$ be a “smooth”, $\Omega$-periodic function. We shall suppose throughout that $R_a, R_b$ are chosen in such a way that $$\label{eq:supp_beta}
{\rm supp}(D_{a_{i_1}} D_{a_{i_2}} D_{a_{i_3}} \beta) \subset
\Omega_b \qquad \forall i \in \{1,\dots, 6\}^3.$$ Then, the (nonlinear) B-QCF forces are given by $$F^{bqcf}(x; y) := \beta(x) F^a(x; y) + (1-\beta(x)) F^c(x; y),$$ and linearizing the equilibrium equation $F^{bqcf} + f = 0$ about $y_B$ yields $$\label{eq:2}
\begin{split}
& (L^{bqcf} \mathbf{u}^{bqcf})(x) = f(x), \quad \text{for } x \in
\mathcal{L},\\
&\text{where} \quad (L^{bqcf} \mathbf{v})(x) = \beta(x) (L^a
\mathbf{v})(x) + (1-\beta(x)) (L^c\mathbf{v})(x).
\end{split}$$
Since the nearest neighbor terms in the atomistic and the QCL models are the same, we will focus on the second-neighbor interactions. We rewrite the operator $L^{bqcf}$ in the form $$\begin{aligned}
(L^{bqcf}\mathbf{v})(x) =~& \sum_{r \in \mathcal{N}}
(L^{bqcf}_r \mathbf{v})(x), \\
\text{where} \quad L^{bqcf}_r \mathbf{v}(x) =~& \beta(x) (L^a_r
\mathbf{v})(x) + (1-\beta(x)) (L^c_r \mathbf{v})(x),\end{aligned}$$ where the nearest-neighbor operators are given by $$L^a_{a_j} \mathbf{v}(x) = L^c_{a_j} \mathbf{v}(x) = - \phi''(Ba_j) D_{a_j}
D_{a_j} v(x-a_j),$$ and the second-neighbor operators, stated for convenience only for $b_1 = a_1 + a_2$, by $$\begin{aligned}
\left(L^a_{b_1}\mathbf{u}\right)(x)=&
- \phi''(B b_1) D_{b_{1}}D_{b_{1}}v(x-b_{1}),\quad\text{while}\\
\left(L^{c}_{b_1}\mathbf{u}\right)(x)=&
- \phi''(Bb_1) \big[D_{a_1}D_{a_1}u(x-a_1)+
D_{a_2}D_{a_2}u(x-a_2)\big.\\
&\qquad\qquad\qquad \big.+D_{a_1}D_{a_2}u(x-a_1)+D_{a_1}D_{a_2}u(x-a_2)\big].\end{aligned}$$
Auxiliary results
-----------------
The following is the 2D counterpart of the summation by parts formula. The proof is straightforward.
For any $\mathbf{u} \in \mathcal{U}$ and any direction $r\in
\mathbb{Z}^2$, we have $$\label{SumByPart2D}
\sum_{x\in\mathcal{L}}D_rD_r u(x-r)\cdot u(x)=-\sum_{x\in\mathcal{L}}D_{r}u(x-r)\cdot D_{r}u(x-r).$$
The second auxiliary result we require is a trace- or Poincaré-type inequality to bound $\|\mathbf{u}\|_{\ell^2_\epsilon(\Omega_b)}$ in terms of global norms. As a first step we establish a continuous version of the inequality we are seeking. The key technical ingredient in its proof is a sharp trace inequality, which is stated in Section \[sec:app\_trace\].
\[th:blend\_poincare\_cts\] Let $r_a < r_b \in (0, 1/2]$, and let $H := {\tt Hex}(r_b) \setminus
{\tt Hex}(r_a)$; then there exists a constant $C$ that is independent of $r_a, r_b$ such that $$\label{eq:blend_poincare_cts}
\| u \|_{L^2(H)}^2 \leq C \big[(r_b - r_a) r_b |\log r_b |\big] \| \partial u
\|_{L^2(\Omega)}^2
\qquad \forall u \in H^1(\Omega), \int_\Omega u dx = 0.$$
Let $\Sigma := \partial {\tt Hex}(1)$, and let $dS$ denote the surface measure, then $$\| u \|_{L^2(H)}^2 = \int_{r = r_a}^{r_b} \int_{\Sigma} |u|^2 dS\,dr.$$ Applying with $r_0 = r$ and $r_1 = 1$ to each surface integral, we obtain $$\| u \|_{L^2(H)}^2 \leq (r_b-r_a) \big( C_0 \| u \|_{L^2(\Omega)}^2 + C_1
\| \partial u \|_{L^2(\Omega)}^2 \big),$$ where $C_0 \leq 8 r_b$ and $C_1 = 2 r_b |\log r_b|$. An application of Poincaré’s inequality yields .
In our analysis, we require a result as for discrete norms. We establish this next, using straightforward norm-equivalence arguments.
\[DiscrBlenPoin:lemma\] Suppose that $R_b \leq N/2$, then $$\label{eq:blend_poincare}
\| \mathbf{u} \|_{\ell^2_\epsilon(\mathcal{L}^b)}^2 \leq C\, (\CPab)^2 \| D \mathbf{u}
\|_{\ell^2_\epsilon}^2
\qquad \forall \mathbf{u} \in \mathcal{U}.$$ where $C$ is a generic constant, and $\CPab := \big[ (\epsilon K)
(\epsilon R_b) |\log (\epsilon R_b)| \big]^{1/2}$.
Recall the identification of $\mathbf{u}$ with its corresponding $P_1$-interpolant. Let $T \in \mathcal{T}$ with corners $x_j$, $j =
1, 2, 3$, then $$\int_T u \,dx = \frac{|T|}{3} \sum_{j = 1}^3 u(x_j), \quad
\text{and hence} \quad
\int_\Omega u\, dx = 0 \quad \forall \mathbf{u} \in \mathcal{U}.$$
Let $r_a := \epsilon R_a$ and $r_b := \epsilon R_b$, then $H$ defined in Lemma \[th:blend\_poincare\_cts\] is identical to $\Omega_b$. For any element $T \subset \Omega_b$ it is straightforward to show that $$\| \mathbf{u} \|_{\ell^2_\epsilon(T)} \leq C \| u \|_{L^2(T)}.$$ This immediately implies $$\label{eq:blend_ineq:10}
\| \mathbf{u} \|_{\ell^2_\epsilon(\mathcal{L}^b)} \leq C \| u \|_{L^2(H)},$$ for a constant $C$ that is independent of $\epsilon$, $R_a$, $K$ and $\mathbf{u}$. Applying yields $$\| \mathbf{u} \|_{\ell^2_\epsilon(\mathcal{L}^b)}^2 \leq C \big[
(r_b-r_a)
r_b |\log r_b| \big] \|\partial u \|_{L^2(\Omega)}^2.$$
Fix $T \in \mathcal{T}$ and let $x_j \in T$ such that $x_j + a_j \in
T$ as well. Employing [@OrtnerShapeev:2010 Eq. (2.1)] we obtain $$\sum_{j = 1}^3 \big| D_{a_j} u(x_j) \big|^2 = \sum_{j = 1}^3
\big| (\partial u|_T) a_j \big|^2 = {{\textstyle \frac{3}{2}}} \big| \partial u|_T \big|^2,$$ and summing over $T \in \mathcal{T}, T \subset \bar{\Omega}$ we obtain that $\| \partial u \|_{L^2(\Omega)} \leq C \| D \mathbf{u}
\|_{\ell^2_\epsilon}$. This concludes the proof.
Bounds on $L^{bqcf}_{b_1}$
--------------------------
We focus only on the $b_1$-bonds, however, by symmetry analogous results hold for all second-neighbor bonds. As in the 1D case, we begin by converting the quadratic form ${\langle}L^{bqcf}_{b_1}\mathbf{u},\mathbf{u}{\rangle}$ into divergence form. To that end it is convenient to define the bond-dependent symmetric bilinear forms and quadratic forms (although we write them like a norm they are typically indefinite) $$\begin{aligned}
{\langle}r, s {\rangle}_{b} := r^{\rm T} \phi''(Bb) s, \quad \text{and} \quad
|r|_{b}^2 := {\langle}r, r {\rangle}_{b}, \qquad \text{for } r, s, b \in \mathbb{R}^2.\end{aligned}$$
\[DivformLemma2D\] For any displacement $\mathbf{u}\in \mathcal{U}$, we have $$\label{Divform2D}
{\langle}L^{bqcf}_{b_1}\mathbf{u},\mathbf{u}{\rangle}= {\langle}L^{c}_{b_1}\mathbf{u},\mathbf{u}{\rangle}-\epsilon^4\sum_{x\in\mathcal{L}}\beta(x-a_2)|D_{a_1}D_{a_2}u(x-a_1-a_2)|_{b_1}^2
+\mathbf{R}_{b_1}+\mathbf{S}_{b_1},$$ where $$\begin{aligned}
\label{SigPart2D}
\begin{split}
\mathbf{R}_{b_1}:=&
-\epsilon^4\sum_{x\in\mathcal{L}}\big\{ D_{a_1}\beta(x-2a_1)
\big{\langle}D_{a_1}u(x-2a_1),
D_{a_2}D_{a_2}u(x-a_1-a_2) \big{\rangle}_{b_1}\\
&\qquad\qquad\quad
+ D_{a_2}\beta(x-a_2) \big{\langle}D_{a_1}u(x-a_1),
D_{a_1}D_{a_2}u(x-a_1-a_2) \big{\rangle}_{b_1}
\big\},\quad\text{and}\\
\mathbf{S}_{b_1}:=&-\epsilon^4\sum_{x\in\mathcal{L}}
D_{a_1}D_{a_1}\beta(x-2a_1)
\big{\langle}u(x-a_1), D_{a_2}D_{a_2}u(x-a_1-a_2) \big{\rangle}_{b_1}.
\end{split}\end{aligned}$$
For this purely algebraic proof we may assume without loss of generality that $\phi''(Bb_1) = {\rm I}$. In general, one may simply replace all Euclidean inner products with ${\langle}\cdot, \cdot
{\rangle}_{b_1}$.
Starting from , we have $$\begin{aligned}
{\langle}L^{bqcf}_{b_1}\mathbf{u},\mathbf{u}{\rangle}=& {\langle}L^{c}_{b_1}\mathbf{u},\mathbf{u}{\rangle}+{\langle}L^{a}_{b_1}\mathbf{u}- L^{c}_{b_1}\mathbf{u},\beta\mathbf{u}{\rangle}\\
=& {\langle}L^{c}_{b_1}\mathbf{u},\mathbf{u}{\rangle}-\epsilon^2\sum_{x\in\mathcal{L}}\beta(x)u(x)\cdot\left[D_{b_1}D_{b_1}u(x-b_1)-D_{a_1}D_{a_1}u(x-a_1)
\right.\\
&\qquad\qquad\left. -D_{a_2}D_{a_2}u(x-a_2) -D_{a_1}D_{a_2}u(x-a_1)-D_{a_1}D_{a_2}u(x-a_2)\right].\end{aligned}$$ We will focus our analysis on ${\langle}L^{a}_{b_1}\mathbf{u}- L^{c}_{b_1}\mathbf{u},\beta\mathbf{u}{\rangle}$.
Noting that $b_1=a_1+a_2$, one can recast $D_{b_1}D_{b_1}u(x-b_1)$ as $$\begin{aligned}
D_{b_1}&D_{b_1}u(x-b_1)\\
=&\frac{1}{\epsilon^2}\left[u(x+b_1)-2u(x)+u(x-b_{1})\right]\\
=&D_{a_1}D_{a_2}u(x)+ D_{a_1}D_{a_1}u(x-a_1)
+D_{a_2}D_{a_2}u(x-a_2)+D_{a_1}D_{a_2}u(x-a_1-a_2).\end{aligned}$$ Applying the summation by parts formula to ${\langle}L^{a}_{b_1}\mathbf{u}- L^{c}_{b_1}\mathbf{u},\beta\mathbf{u}{\rangle}$, we get $$\begin{aligned}
{\langle}L^{a}_{b_1}\mathbf{u}- L^{c}_{b_1}\mathbf{u},\beta\mathbf{u}{\rangle}=& -\epsilon^3 \sum_{x\in\mathcal{L}}\beta(x)u(x)\cdot \Big[D_{a_{1}}D_{a_{1}}D_{a_{2}}u(x-a_{1})-D_{a_{1}}D_{a_{1}}D_{a_{2}}u(x-a_1-a_2)\Big]\\
=& -\epsilon^4 \sum_{x\in\mathcal{L}}\beta(x)u(x)\cdot D_{a_{1}}D_{a_{1}}D_{a_{2}}D_{a_2}u(x-a_1-a_2)\\
=& \epsilon^4\sum_{x\in\mathcal{L}}D_{a_1}D_{a_2}D_{a_2}u(x-a_1-a_2)\cdot D_{a_1}\Big(\beta(x-a_1)u(x-a_1)\Big)\\
=& \epsilon^4\sum_{x\in\mathcal{L}}D_{a_1}D_{a_2}D_{a_2}u(x-a_1-a_2)\cdot \Big[\beta(x)D_{a_1}u(x-a_1)+u(x-a_1)D_{a_1}\beta(x-a_1)\Big].\end{aligned}$$ Another application of the summation by parts formula converts ${\langle}L^{a}_{b_1}\mathbf{u}-
L^{c}_{b_1}\mathbf{u},\beta\mathbf{u}{\rangle}$ into $$\begin{aligned}
{\langle}L^{a}_{b_1}\mathbf{u}- L^{c}_{b_1}\mathbf{u},\beta\mathbf{u}{\rangle}=& \epsilon^4\sum_{x\in\mathcal{L}}D_{a_1}D_{a_2}D_{a_2}u(x-a_1-a_2)\cdot \big(u(x-a_1)D_{a_1}\beta(x-a_1)\big)\\
&\quad -\epsilon^4\sum_{x\in\mathcal{L}}D_{a_1}D_{a_2}u(x-a_1-a_2)\cdot \big(D_{a_2}\beta(x-a_2)D_{a_1}u(x-a_1)\big)\\
&\quad\quad- \epsilon^4\sum_{x\in\mathcal{L}}D_{a_1}D_{a_2}u(x-a_1-a_2)\cdot\big( \beta(x-a_2)D_{a_1}D_{a_2}u(x-a_1-a_2)\big).\end{aligned}$$ The first two terms on the right-hand side can be rewritten as $$\begin{aligned}
\epsilon^4&\sum_{x\in\mathcal{L}}\big\{D_{a_1}D_{a_2}D_{a_2}u(x-a_1-a_2)\cdot \big(u(x-a_1)D_{a_1}\beta(x-a_1)\big)\big.\\
&\qquad\Big.-D_{a_1}D_{a_2}u(x-a_1-a_2)\cdot \big(D_{a_2}\beta(x-a_2)D_{a_1}u(x-a_1)\big)\big\}\\
&=-\epsilon^4\sum_{x\in\mathcal{L}}\Big(u(x-a_1)D_{a_1}D_{a_1}\beta(x-2a_1)\Big)\cdot D_{a_2}D_{a_2}u(x-a_1-a_2) \\
&\qquad-\epsilon^4\sum_{x\in\mathcal{L}}\big\{
D_{a_1}\beta(x-2a_1)D_{a_1}u(x-2a_1)\cdot D_{a_2}D_{a_2}u(x-a_1-a_2)\big.\\
&\qquad\qquad\qquad \big.
+D_{a_2}\beta(x-a_2)D_{a_1}u(x-a_1)\cdot D_{a_1}D_{a_2}u(x-a_1-a_2)
\big\}\\
&=\mathbf{S}_{b_1}+\mathbf{R}_{b_1}.\end{aligned}$$ Thus, we obtain and .
Next, we will bound the singular terms $\mathbf{R}_{b_1}$ and $\mathbf{S}_{b_1}$, for which we introduce the notation $$\| D^{(2)} \beta \|_{\ell^\infty_\epsilon} := \max_{1 \leq i, j \leq
6} \| D_{a_i} D_{a_j} \beta \|_{\ell^\infty_\epsilon}, \quad
\text{and} \quad
\| D^{(3)} \beta \|_{\ell^\infty_\epsilon} := \max_{1 \leq i, j, k \leq
6} \| D_{a_i} D_{a_j} D_{a_k} \beta \|_{\ell^\infty_\epsilon}.$$
\[SigEstLemma2D\] The terms $\mathbf{R}_{b_1}$ and $\mathbf{S}_{b_1}$ defined in are bounded by $$\begin{aligned}
\left|\mathbf{R}_{b_1}\right|
\le & 4\epsilon^2|\phi''(Bb_1)| \, \|D\beta\|_{\ell_\epsilon^{\infty}}\|D\mathbf{u}\|_{\ell_\epsilon^{2}}^2, \quad\text{and}\label{SigEst2D}\\
\left|\mathbf{S}_{b_1}\right|
\le&
C\epsilon^{2} |\phi''(Bb_1)| \Big[ \|D^{(2)} \beta\|_{\ell_{\epsilon}^{\infty}}
+\|D^{(3)} \beta\|_{\ell_{\epsilon}^{\infty}}\, \CPab \Big]\|D\mathbf{u}\|_{\ell_{\epsilon}^2}^2,\end{aligned}$$ where $C$ is a generic constant and $\CPab$ is defined in Lemma \[eq:blend\_poincare\].
According to the expression of $\mathbf{R}_{b_1}$ given in and noting that $$\|D_{a_2}D_{a_2}\mathbf{u}\|_{\ell_\epsilon^2}^2\le
\frac{4}{\epsilon^2}\|D\mathbf{u}\|_{\ell_\epsilon^2}^2
\quad \text{and} \quad \|D_{a_1}D_{a_2}\mathbf{u}\|_{\ell_\epsilon^2}^2\le
\frac{4}{\epsilon^2}\|D\mathbf{u}\|_{\ell_\epsilon^2}^2,$$ we immediately obtain the first inequality of .
We first rewrite $\mathbf{S}_{b_1}$ as $$\begin{aligned}
\mathbf{S}_{b_1}=&
-\epsilon^4\sum_{x\in\mathcal{L}}D_{a_1}D_{a_1}\beta(x-2a_1) \big{\langle}D_{a_2}D_{a_2}u(x-a_1-a_2), u(x-a_1) \big{\rangle}_{b_1} \nonumber\\
=&-\epsilon^4\sum_{x\in\mathcal{L}}D_{a_1}D_{a_1}\beta(x-2a_1)
D_{a_2}\big{\langle}D_{a_2}u(x-a_1-a_2), u(x-a_1-a_2)\big{\rangle}_{b_1} \nonumber\\
&\qquad+ \epsilon^4\sum_{x\in\mathcal{L}}D_{a_1}D_{a_1}\beta(x-2a_1)
\big{\langle}D_{a_2}u(x-a_1-a_2), D_{a_2}u(x-a_1-a_2)\big{\rangle}_{b_1}\nonumber\\
=&
\epsilon^4\sum_{x\in\mathcal{L}}D_{a_2}D_{a_1}D_{a_1}\beta(x-2a_1-a_2)
\big{\langle}D_{a_2}u(x-a_1-a_2)\cdot u(x-a_1-a_2) \big{\rangle}_{b_1} \nonumber\\
&\qquad+
\epsilon^4\sum_{x\in\mathcal{L}}D_{a_1}D_{a_1}\beta(x-2a_1)\big|D_{a_2}u(x-a_1-a_2)\big|_{b_1}^2.\label{SigEst2DEq1}
$$ For the second term in , we have $$\Big| \epsilon^4\sum_{x\in\mathcal{L}}D_{a_1}D_{a_1}\beta(x-2a_1)\big|D_{a_2}u(x-a_1-a_2)\big|_{b_1}^2 \Big|
\le
\epsilon^2 | \phi''(Bb_1)| \|D^{(2)} \beta\|_{\ell_{\epsilon}^{\infty}}\|D\mathbf{u}\|_{\ell_{\epsilon}^2}^2.$$ For the first term, we have $$\begin{aligned}
&\Big|\epsilon^4\sum_{x\in\mathcal{L}}D_{a_2}D_{a_1}D_{a_1}\beta(x-2a_1-a_2)
\big{\langle}D_{a_2}u(x-a_1-a_2), u(x-a_1-a_2)\big{\rangle}_{b_1} \Big|\\
&\qquad\le \epsilon^2 \,|\phi''(Bb_1)|\,\|\mathbf{u}\,D_{a_2}D_{a_1}D_{a_1}\beta\|_{\ell_{\epsilon}^2} \|D\mathbf{u}\|_{\ell_{\epsilon}^2}
\le \epsilon^2 \,|\phi''(Bb_1)| \,\|D^{(3)} \beta\|_{\ell_{\epsilon}^{\infty}}
\,\|\mathbf{u}\|_{\ell_{\epsilon}^2(\mathcal{L}^b)} \,\|D\mathbf{u}\|_{\ell_{\epsilon}^2}.\end{aligned}$$ The last inequality comes from the assumption , which ensures that ${\rm supp}(D_{a_2}D_{a_1}D_{a_1}\beta) \subset
\Omega_b$.
Applying Lemma \[DiscrBlenPoin:lemma\] yields the bound for ${\bf
S}_{b_1}$.
To summarize the estimates of this section we define a self-adjoint operator $\tilde{L}$ by $$\label{eq:defn_Ltil}
{\langle}\tilde{L} \mathbf{u}, \mathbf{u} {\rangle}:=
{\langle}L^c \mathbf{u}, \mathbf{u} {\rangle}- \epsilon^4 \sum_{j = 1}^3
\sum_{x \in \mathcal{L}} \beta(x-a_2) \big| D_{a_j}
D_{a_{j+1}} u(x - a_1 - a_2) \big|_{b_1}^2;$$ then, Lemma \[DivformLemma2D\] and Lemma \[SigEstLemma2D\] immediately yield the following result.
\[th:lbqcf\_Ltil\_bound\] Suppose that $R_a$ and $R_b$ are defined such that holds; then, for all $\mathbf{u} \in
\mathcal{U}$, $$\label{eq:Lbqcf_Ltil_bound}
{\langle}L^{bqcf} \mathbf{u}, \mathbf{u} {\rangle}\geq {\langle}\tilde{L}
\mathbf{u}, \mathbf{u} {\rangle}- C \, C'' \big[ \epsilon^2 \| D \beta
\|_{\ell^\infty} + \epsilon^2 \|D^{(2)}\beta\|_{\ell^\infty} +
\epsilon^2 \CPab \| D^{(3)} \beta \|_{\ell^\infty} \big]
\|D\mathbf{u}\|_{\ell^2_\epsilon}^2,$$ where $C$ is a generic constant, $C'' := \max_{j = 1,2,3} |\phi''(B
b_j)|$ and $\CPab$ is defined in Lemma \[eq:blend\_poincare\].
Based on the analysis and numerical experiments in [@OrtnerShapeev:2010] for a similar linearized operator we expect that the region of stability for $\tilde{L}$ is the same as for $L^a$; that is, $\tilde{L}$ is positive definite for a macroscopic strain $B$ if and only if $L^a$ is positive definite. However, we are at this point unable to prove this result. Instead, we have the following weaker result. The proof is elementary.
\[th:stab\_Ltil\] Suppose that $B \in \mathbb{R}^{2 \times 2}$ is such that $L^c$ is positive definite, $${\langle}L^c \mathbf{u}, \mathbf{u} {\rangle}\geq \gamma_c \| D \mathbf{u}
\|_{\ell^2_\epsilon}^2 \qquad \forall \mathbf{u} \in \mathcal{U},$$ and suppose that $\phi''(Bb_j) \leq \delta {\rm I}$ where $\delta <
\gamma_c / 4$, then $\tilde{L}$ is positive definite, $$\label{eq:stab_Ltil}
{\langle}\tilde{L} \mathbf{u}, \mathbf{u} {\rangle}\geq \tilde{\gamma} \| D \mathbf{u}
\|_{\ell^2_\epsilon}^2 \qquad \forall \mathbf{u} \in \mathcal{U},$$ with $\tilde{\gamma} = \gamma_c - 4\delta$.
Positivity of the B-QCF operator in 2D
--------------------------------------
The *blending width* $K$ is again a crucial ingredient in the stability analysis for $L^{bqcf}$. Due to the simple geometry we have chosen it straightforward to generalize Lemma \[BlendFunEstLemma\] to the two-dimensional case, using the same arguments as in 1D.
\[2DBlendFunEstLemma\] It is possible to choose $\beta$ such that $$\label{2Deq:BlendFunEst_upper}
\| D^{(j)} \beta \|_{\ell^\infty} \leq C_\beta (K \epsilon)^{-j}.
\quad \text{for } j = 1, 2, 3,$$
Since we cannot fully characterize the stability of $\tilde{L}$ in terms of the stability of $L^a$ or $L^c$ we will only prove stability of $L^{bqcf}$ subject to the assumption that $\tilde{L}$ is stable. Proposition \[th:stab\_Ltil\] gives sufficient conditions.
\[BlendSizeThm2D\] Suppose that $\beta$ is chosen quasi-optimally so that is attained; then, $${\langle}L^{bqcf} \mathbf{u}, \mathbf{u} {\rangle}\geq \gamma_{bqcf}
\| D\mathbf{u} \|_{\ell^2_\epsilon}^2,$$ where $$\gamma_{bqcf} := \tilde{\gamma} -
C\, C''\, \big[ \epsilon^{-1/2} K^{-5/2} |\epsilon R_b
\log(\epsilon R_b)|^{1/2} \big],$$ where $C$ is a generic constant and $C''$ is defined in Corollary \[th:lbqcf\_Ltil\_bound\].
In particular, if $\tilde{L}$ is positive definite and if $K$ is sufficiently large, then $L^{bqcf}$ is positive definite.
From Corollary \[th:lbqcf\_Ltil\_bound\] and we obtain $$\begin{aligned}
{\langle}L^{bqcf} \mathbf{u}, \mathbf{u} {\rangle}\geq~&
\big\{ \tilde{\gamma} - C\,C''\,\big[ \epsilon^2 (\epsilon K)^{-1}
+ \epsilon^2 (\epsilon K)^{-2} + \epsilon^2 (\epsilon K)^{-5/2}
|\epsilon R_b\log(\epsilon R_b)|^{1/2} \big]\big\}
\| D \mathbf{u} \|_{\ell^2_\epsilon}^2 \\
\geq~& \big\{ \tilde{\gamma} - C\,C''\,\big[ \epsilon^{-1/2} K^{-5/2}
|\epsilon R_b \log(\epsilon R_b)|^{1/2} \big] \big\}
\| D \mathbf{u} \|_{\ell^2_\epsilon}^2. \qedhere
\end{aligned}$$
Suppose that $\tilde{\gamma} > 0$, uniformly as $N \to \infty$ (or, $\epsilon \to 0$). In this limit, we would like to understand how to optimally scale $K$ with $R_a$. (Note that $R_a$ controls the modeling error; cf. Remark \[2DB-QCFremark2\].) We consider three different scalings of $R_a$.
[*Case 1:* ]{} Suppose that $R_a$ is bounded as $\epsilon \to 0$. In that case, we obtain $$\begin{aligned}
\notag
\gamma_{bqcf} - \tilde{\gamma} =~& - C \,C''\,
\epsilon^{-1/2} K^{-5/2} | \epsilon (R_a+K) \log (\epsilon
(R_a+K)) |^{1/2} \\
\notag
=~& - C \,C''\, K^{-2} \big| \big(1 + {{\textstyle \frac{R_a}{K}}}\big)
\big(\log (\epsilon K) + \log(1+{{\textstyle \frac{R_a}{K}}}) \big)\big|^{1/2}
\\
\label{eq:scaling_case2}
\eqsim~& - C\, C''\, K^{-2} |\log(\epsilon K)|^{1/2}.
\end{aligned}$$ From this it is easy to see that $L^{bqcf}$ will be positive definite provided we select $K \gg |\log \epsilon|^{1/4}$.
[*Case 2:* ]{} Suppose that $1 \ll R_a \ll \epsilon^{-1}$; to precise, let $R_a \sim \epsilon^{-\alpha}$ for some $\alpha \in (0,
1)$. Then, a similar computation as yields $$\gamma_{bqcf} - \tilde{\gamma} \eqsim K^{-5/2} \big| (K +
\epsilon^{-\alpha}) (\log\epsilon + \log(K+\epsilon^{-\alpha}))\big|^{1/2},$$ and we deduce that, in this case, $L^{bqcf}$ will positive definite provided we select $K \gg \epsilon^{-\alpha/5} |\log
\epsilon|^{1/5}$.
[*Case 3:* ]{} Finally, the case when the atomistic region is macroscopic, i.e., $R_a = O(\epsilon^{-1})$, can be treated precisely as the 1D case and hence we obtain that, if we select $K
\gg \epsilon^{-1/5}$, then $L^{bqcf}$ is positive.
In summary, we have shown that, in the limit as $\epsilon \to 0$, if $\tilde{L}$ is positive definite, $R_a = O(\epsilon^{-\alpha})$ and if we choose $$\begin{aligned}
\label{BlendSize2D}
K\gg
\begin{cases}
|\log{\epsilon}|^{1/4}, & \quad \alpha = 0, \\
|\log\epsilon|^{1/5}\epsilon^{-\alpha/5},& \quad 0<\alpha<1,\\
\epsilon^{-1/5}, & \quad \alpha = 1,
\end{cases}
\end{aligned}$$ then the B-QCF operator $L^{bqcf}$ is positive definite and $\gamma^{bqcf} \sim \tilde{\gamma}$ as $\epsilon \to 0$. We emphasize that these are very mild restrictions on the blending width.
It remains to show that the sufficient conditions we derived to guarantee positivity of $L^{bqcf}$ are sharp. A result as general as in 1D would be very technical to obtain; instead, we offer a brief formal discussion for a special case.
![\[fig:sharp\_2d\] Visualization of the construction discussed in \[rem:sharp\_2d\]: the white region is the atomistic domain, the light gray region the blending region, the medium gray region and dark gray regions together are the set $\mathcal{J}$ and the dark gray region is the set $\mathcal{J}'$.](sharp_2d_t){width="7cm"}
\[rem:sharp\_2d\] We consider again the limit as $\epsilon \to 0$, and for simplicity restrict ourselves to the case where $0 \ll K \eqsim \epsilon^{-\theta}$ and $0 \ll R_a \eqsim
\epsilon^{-\alpha}$, for $0 < \theta \leq \alpha \leq 1$. In particular, $R_b \eqsim \epsilon^{-\alpha}$ as well.
We assume that $D_{a_3} \beta(x) = 0$ for all $x \in \mathcal{J}
\subset \mathcal{L}^b$, as depicted in Figure \[fig:sharp\_2d\]. The set $\mathcal{J}$ should be chosen so that its size is comparable with that of $\mathcal{L}^b$, but sufficiently small to still allow $\beta$ to satisfy the bound . We can now repeat the 1D argument along atomic layers to obtain that $$D_{a_2} D_{a_1} D_{a_1} \beta(x) \leq - {{\textstyle \frac{1}{2}}} (\epsilon
K)^{-3}
\eqsim - \epsilon^{-3 + 3 \theta}$$ for all $x$ in a subset $\mathcal{J}' \subset \mathcal{J}$ containing entire atomic planes, that has comparable size to $\mathcal{J}$; that is, $\# \mathcal{J}' \eqsim K R_b \eqsim
\epsilon^{-\theta-\alpha}$.
Suppose now that $\phi''(Bb_1)$ has a negative eigenvalue $\lambda$ with corresponding normalized eigenvector $\hat{u} \in
\mathbb{R}^2$, then we seek test functions of the form $u(x) =
\mu(x) \hat{u}$. It is now relatively straightforward, applying the 1D argument in normal direction and using a smooth cut-off in the tangential direction, to construct $\mu$ supported in $\mathcal{J}'$ with $D_{a_2} \mu(x) \eqsim (\epsilon^2 \# \mathcal{J}')^{-1/2}$ so that $\| D \mathbf{u} \|_{\ell^2_\epsilon} \eqsim 1$, and $$\begin{aligned}
&
\epsilon^4\sum_{x\in\mathcal{L}}D_{a_2}D_{a_1}D_{a_1}\beta(x-2a_1-a_2)
\big{\langle}D_{a_2}u(x-a_1-a_2), u(x-a_1-a_2) \big{\rangle}_{b_1} \\
& \hspace{1cm} = \epsilon^4 \lambda_1
\sum_{x\in\mathcal{L}}D_{a_2}D_{a_1}D_{a_1}\beta(x-2a_1-a_2)
D_{a_2}\mu(x-a_1-a_2) \mu(x-a_1-a_2) \\
& \hspace{1cm} \lesssim - \epsilon^4 \lambda_1 (\# \mathcal{J}')
(K \epsilon)^{-3} (\epsilon^2 \# \mathcal{J}')^{-1/2} \eqsim -
\epsilon^{(5\theta-\alpha)/2}.
\end{aligned}$$ This shows that, if $K \ll \epsilon^{-\alpha/5}$, then $L^{bqcf}$ is necessarily indefinite.
In summary, for the specific interface geometry and a particular choice of $\beta$ (which does, however, lead to the quasi-optimal bound ) we have shown that Theorem \[BlendSizeThm2D\] is sharp up to logarithmic terms.
\[2DB-QCFremark2\] In practise, for the computation of different types of defects, we would first choose an appropriate scaling $R_a=\epsilon^{-\alpha}$ for the atomistic region, considering the accuracy of the B-QCF method, and then choose the blending width $K$ in order to ensure stability.
For instance, for a point defect in 2D with zero Burger’s vector it is expected that the displacement field satisfies $u_a(x) = y_a(x) -
Bx \eqsim \epsilon / r$, where $r$ is the distance from the defect [@PointDefect; @OrtnerShapeev:2010]. Without coarse-graining, the local continuum (QCL) model has a modeling error of order $O(\epsilon^2 |\partial^3 u_a|)$ (see [@ortner:qnl1d; @LuskinXingjie.qnl1d; @Dobson:2008b] for proofs in 1D and [@VKOr:blend2] for a proof in arbitrary dimensions); and although we have not established it rigorously, we expect that modeling error for the B-QCF method outside the atomistic region is also of second order; see also [@qcf.stab].
From $u(x) \eqsim \epsilon / r$ we can make the reasonable assumption that $|\partial^3 y_a| \eqsim \epsilon / r^{4}$, from which we obtain (assuming also stability) that the total error is of the order $$\| \partial (y_a - y_{bqcf}) \|_{L^2} \eqsim \epsilon^2
\| \partial^3 y_a \|_{L^2(\Omega \setminus\Omega_a)} \eqsim
\epsilon^3 \Big( \int_{\epsilon R_a}^1 r |r^{-4}|^2 dr \Big)^{1/2}
\eqsim R_a^{-3}.$$ Hence, if we wish to obtain $\| \partial (y_a - y_{bqcf}) \|_{L^2}
\eqsim \epsilon^k$, $0 < k < 3$, then we need to choose $$R_a \eqsim \epsilon^{-k/3}, \quad \text{and
consequently} \quad K \gg \epsilon^{-k/15} |\log
\epsilon|^{1/5}.$$ With this choice we can ensure both the stability and $O(\epsilon^k)$ accuracy of the B-QCF method; provided that our assumption that the B-QCF method has indeed a second-order modelling error is correct.
Conclusion
==========
We have studied the stability a blended force-based quasicontinuum (B-QCF) method. In 1D we were able to identify an asymptotically optimal condition on the width of the blending region to ensure that the linearized B-QCF operator is coercive if and only if the atomistic operator is coercive as well. In the $2$D B-QCF model, we have obtained rigorous sufficient conditions and have presented a heuristic argument suggesting that they are sharp up to logarithmic terms. In 2D our proof of coercivity of $L^{bqcf}$ relies on the coercivity of the auxiliary operator $\tilde{L}$ defined in , for which we cannot give sharp conditions at this point.
The main conclusion of this work is that the required blending width to ensure coercivity of the linearized B-QCF operator is surprisingly small.
Our analysis in this paper is the first step towards a complete a priori error analysis of the B-QCF method, which will require a coercivity analysis of the B-QCF operator linearized about arbitrary states, as well as a consistency analysis in negative Sobolev norms.
Appendix: A Trace Inequality {#sec:app_trace}
============================
In the following trace theorem, $S(1)$ denotes the unit sphere in $\mathbb{R}^d$, $r := |x|$ and $\theta := x / |x|$. Upon taking $\psi
\equiv 1$ and employing standard orthogonal decompositions it is easy to check that the result is sharp. In particular, for $d = 2$, consider the case $u(x) = \log |x|$.
Let $d \geq 2$, $\psi : S(1) \to (0, 1]$ be Lipschitz continuous, and $\Sigma := \{ \psi(\sigma) \sigma : \sigma \in S(1)
\}$. Moreover, let $0 < r_0 < r_1 \leq 1$, and $A := \bigcup_{r_0 <
r < r_1} (r\Sigma)$, then $$\begin{aligned}
\label{eq:general_trace}
& \| u \|_{L^2(r_0 \Sigma)}^2 \leq C_0\| u \|_{L^2(A)}^2 +
C_1 \| \partial u \|_{L^2(A)}^2, \quad \forall u \in H^1(A), \\
& \text{where} \quad
C_0 = \frac{2 d}{r_1 - r_0} \Big(\frac{r_0}{r_1}\Big)^{d-1},
\quad \text{and} \quad
C_1 =
\begin{cases}
2 r_0 |\log r_0|, & d = 2 \\
2 r_0 / (d-2), & d \geq 3.
\end{cases}
\end{aligned}$$
Since $A$ is a Lipschitz domain we may assume, without loss of generality that $u \in C^1(\bar{A})$. The symbol $dS$ denotes the $(d-1)$-dimensional Hausdorff measure in $\mathbb{R}^d$.
Let $r_0 < s < r_1$, then $$\begin{aligned}
\notag
\int_{r_0\Sigma} |u|^2 dS =~& r_0^{d-1} \int_\Sigma
|u(r_0\sigma)|^2 dS_\sigma \\
\notag
=~& r_0^{d-1} \int_\Sigma \bigg| u(s\sigma) - \int_{r = r_0}^s \frac{d}{dr}
u(r \sigma) dr \bigg|^2 dS_\sigma \\
\label{eq:trace:10}
\leq~& 2 r_0^{d-1} \int_\Sigma \big|u(s\sigma)\big|^2 dS_\sigma
+ 2 r_0^{d-1} \int_\Sigma \bigg| \int_{r = r_0}^s \partial u
\cdot \sigma
dr \bigg|^2 dS_\sigma.
\end{aligned}$$ By hypothesis we have $|\sigma| \leq 1$ for all $\sigma \in \Sigma$, hence the second term on the right-hand side can be further estimated, applying also the Cauchy–Schwartz inequality, by $$\begin{aligned}
2 r_0^{d-1} \int_\Sigma \bigg| \int_{r = r_0}^s \partial u
\cdot \sigma
dr \bigg|^2 dS_\sigma
\leq~& 2 r_0^{d-1} \int_\Sigma \int_{r = r_0}^s r^{-d+1} dr \,
\int_{r = r_0}^s r^{d-1} |\partial u(r\sigma)|^2 dr \, dS_\sigma \\
=~& 2 r_0^{d-1} (J(s) - J(r_0)) \int_{r = r_0}^{s}
\int_{r\Sigma} |\partial u|^2 dS \,dr \\
\leq~& 2 r_0^{d-1} (J(s) - J(r_0)) \| \partial u \|_{L^2(A)}^2,
\end{aligned}$$ where $J'(t) = t^{-d+1}$, that is, $J(t) = \log t$ if $d = 2$ and $J(t) = t^{-d+2} / (-d+2)$ if $d \geq 3$. Since $J(s)$ is negative and strictly increasing for $s \leq 1$ we obtain $$\label{eq:trace:20}
2 r_0^{d-1} \int_\Sigma \bigg| \int_{r = r_0}^s \partial u
\cdot \sigma
dr \bigg|^2 dS_\sigma
\leq 2 r_0^{d-1} |J(r_0)| \| \partial u \|_{L^2(A)}^2.$$
Inserting into , multiplying the resulting inequality by $s^{d-1}$ and integrating over $s \in
(r_0, r_1)$ yields $$\begin{aligned}
{{\textstyle \frac{r_1^d - r_0^d}{d}}} \|u\|_{L^2(r_0\Sigma)}^2
=~& \int_{s =
r_0}^{r_1} s^{d-1} \int_{r_0\Sigma} |u|^2 dS\, ds \\
\leq~& 2 r_0^{d-1} \int_{s = r_0}^{r_1} s^{d-1} \int_\Sigma
\big|u(s\sigma)\big|^2 dS_\sigma\, ds
+ 2 r_0^{d-1} J(r_0) {{\textstyle \frac{r_1^d - r_0^d}{d}}} \| \partial u
\|_{L^2(A)}^2.
\end{aligned}$$ Dividing through by $\frac{r_1^d - r_0^d}{d}$ we obtain $$\|u\|_{L^2(r_0\Sigma)}^2 \leq {{\textstyle \frac{2 d r_0^{d-1}}{r_1^d - r_0^d}}}
\| u \|_{L^2(A)}^2 + 2 r_0^{d-1} J(r_0) \| \partial u
\|_{L^2(A)}^2.$$ Finally, estimating $r_0^d - r_1^d \geq (r_1 - r_0) r_1^{d-1}$ yields the stated trace inequality.
Acknowledgments
===============
We appreciate helpful discussions with Brian Van Koten.
[^1]: This work was supported in part by DMS-0757355, DMS-0811039, the PIRE Grant OISE-0967140, and the University of Minnesota Supercomputing Institute. This work was also supported by the Department of Energy under Award Number DE-SC0002085. CO was supported by EPSRC Grant EP/H003096 “Analysis of Atomistic-to-Continuum Coupling Methods”.
| {
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abstract: 'Effect of finite density of nonmagnetic impurities on a coexisting phase of d-density wave (DDW) order and d-wave superconducting (DSC) order is studied using Bogoliubov-de Gennes (BdG) method. The spatial variation of the inhomogeneous DDW order due to impurities has a strong correlation with that of density, which is very different from that of DSC order. The length scale associated with DDW is found to be of the order of a lattice spacing. The nontrivial inhomogeneities are shown to make DDW order much more robust to the impurities, while DSC order becomes very sensitive to them. The effect of disorder on the density of states is also discussed.'
author:
- 'Amit Ghosal$^{1}$ and Hae-Young Kee$^{2}$'
title: ' Robustness of d-Density Wave Order to Nonmagnetic Impurities '
---
#### Introduction
One of the recent proposals in the context of high temperature cuprates is that a true broken symmetry state dubbed as d-density wave state (DDW) is responsible for the pseudogap phenomena.[@sudip] This phase was first suggested in relation to the excitonic insulators[@rice], and it was found as one of the ground states of the t-J type model.[@brad] The DDW is a particle hole condensate with angular momentum 2. The ordered state can be characterized by the circulating current arranged in an alternating pattern on a square lattice, which can be detected as a Bragg scattering signal in neutron scattering measurements.[@brad2; @hykee] But the neutron scattering experiments[@mook; @keimer; @buyers] in cuprates remain controversial. Thus, the definite conclusion on the relevance of the DDW order to the cuprates requires more precise experiments on various doping concentration of cuprates, and further theoretical studies on the properties of this new order. Especially, the effect of the nonmagnetic impurities on DDW order is an important subject to investigate, since any well-prepared cuprate sample contain an intrinsic disorder, minimally from non-stoichiometry.
The simplest possible description of the impurity effect is the self-consistent T-matrix approximation (SCTMA) [@maki2]. This mean field picture excludes not only the freedom of the ordered patterns, but also the interference of the impurities. Within this approximation, the thermodynamics were found to be identical to those of a d-wave BCS superconductor (DSC) in the unitary limit.[@maki] From the density of states, one can see that electrons are localized close to the Fermi energy, and the change in the transition temperature is given by the Abrikosov-Gorkov formula known in BCS superconductors.[@maki] Within the standard non-crossing approximation, the similarity between the DDW and DSC is based on the d-wave symmetry of the gap.
In this paper, we study the effect of impurities on DDW order and for the case where DDW coexists with DSC using Bogoliubov-de Gennes (BdG) technique. This method is the mean field approximation, but it allows spatial inhomogeneity in order parameter. In the case of the disordered DSC with a short coherence length, it was shown that the superfluid stiffness is significantly larger than that predicted by the SCTMA, due to the nontrivial spatial structures of the order parameter[@ghosald].
We found that the DDW order is more robust than the DSC order to the impurities, which cannot be understood within the conventional T-matrix approach. The physical ground for our findings will be discussed later.
#### Model
We model two dimensional disordered DSC and DDW order by the following Hamiltonian. $$\begin{aligned}
{\cal H} &=&
-t\sum_{<ij>,\alpha} (c_{i\alpha}^{\dag} c_{j\alpha} + h.c.)
+ \sum_i \left(V(i)-\mu \right) n_i
\nonumber\\
& & \hspace{-0.5cm}
+ J\sum_{<ij>}\left({\bf S}_i \cdot {\bf S}_j
- n_i n_j /4 \right) + W\sum_{<ij>,\alpha,\beta} n_{i \alpha}
n_{j \beta}.\end{aligned}$$ The first term is the kinetic energy which describes electrons, with spin $\alpha$ at site $i$ created by $c_{i\alpha}^{\dag}$, hopping between nearest-neighbors $<ij>$ on a square lattice. The disorder potential $V(i)$ in the second term is an independent random variable at each site which is either $+ V_0$, with a probability $n_{\rm imp}$ (impurity concentration), or zero, and $\mu$ is the chemical potential. The last, interaction term [@footnote1] is chosen to lead to a coexisting DSC and DDW order ground state in the disorder-free system, where ${\bf S}_i $ and $n_i$ are the spin and density operators, respectively.
The mean field decomposition of the above Hamiltonian leads to following BdG equations [@pgdg; @ghosals]. $$\left(\matrix{\hat\xi & \hat\Delta \cr \hat\Delta^{*} & -\hat\xi^{*}} \right)
\left(\matrix{u_{n} \cr v_{n}} \right) = E_{n}
\left(\matrix{u_{n} \cr v_{n}} \right),
\label {eq:bdg}$$ where $\hat\xi u_{n}(j) = -\sum_{\delta} \{t + \Psi(j;\delta) e^{-i {\bf Q.r_j}} \}
u_{n}(j+\delta) + (V(j)-\tilde{\mu}_j)u_{n}(j)$ and $\hat\Delta u_{n}(j)
= \sum_{\delta}\Delta(j+\delta;\delta) u_{n}(j+\delta)$, and similarly for $v_{n}(j)$. The local DSC pairing ($\Delta$) and DDW ($\chi = {\rm Im} \Psi$) amplitudes on a bond $(j;\delta)$ are defined by $$\begin{aligned}
\Delta(j;\delta) &=& -\frac{J+W}{4}\langle c_{j+\delta \downarrow}
c_{j \uparrow} + c_{j \downarrow}c_{j+\delta \uparrow}\rangle,
\nonumber\\
\Psi(j; \delta) & =&
\frac{ J + 2W}{4}\langle c^{\dag}_{j+\delta \alpha}c_{j \alpha}
- c_{j \alpha} c^{\dag}_{j+\delta \alpha}\rangle e^{-i {\bf Q.r_j}},\end{aligned}$$ where $\delta = \pm{\hat{\bf x}}, \pm{\hat{\bf y}}$. The inhomogeneous Hartee and Fock shifts are given by $\tilde{\mu}_j =
\mu + (\frac{J}{4}+W) \sum_{\delta}\langle n_{j+\delta} \rangle$ and $Re[\Psi(j;\delta)]$ respectively.
We numerically solve for the BdG eigenvalues $E_n \ge 0$ and eigenvectors $\left(u_{n},v_{n}\right)$ on a lattice of $N$ sites with periodic boundary conditions. We then calculate the d-wave pairing amplitude $\Delta(j;\delta) = (J+W)\sum_n\left[u_n(j+\delta)v^*_n(j)
+ u_n(j)v_n^*(j+\delta)\right]/4$ and the DDW order and Fock shift as the imaginary and real parts of $\Psi(j;\delta)
= (J+2W)
\sum_n\left[v^*_n(j)v_n(j+\delta)-u_n(j)u^*_n(j+\delta)\right]/4$ at $T=0$, and the density $\langle n_j \rangle = 2\sum_n |v_n(j)|^2$. These are fed back into the BdG equation, and the process iterated until self consistency [@footnote3] is achieved for [*each*]{} of the (local) variables defined on the sites and bonds of the lattice. The chemical potential $\mu$ is chosen to obtain a given average density $\langle n \rangle = \sum_i \langle n_i \rangle/N$. We define the site dependent order parameters in terms of the bond variables as, $\Delta(j) = \left[\Delta(j;+\hat{x})-\Delta(j;+\hat{y})
+ \Delta(j;-\hat{x}) - \Delta(j;-\hat{y}) \right]/4$ and similarly for $\chi(j)$.
We have studied the model at $T=0$ for a range of parameters and lattice sizes up to $40\times 40$. Here we focus on $J = 1.16$, and $W = 0.6$, in units of $t = 1$, with $\langle n \rangle=0.95$ on systems of typical size $30\times 30$. For these parameters, and $n_{\rm imp} = 0$, the maximum DSC gap is $\Delta_{{\rm max}}=0.16$ and the maximum DDW gap is $\chi_{{\rm max}}=0.31$. In the pure system our calculations reproduce a phase diagram of $\Delta_{{\rm max}}$ and $\chi_{{\rm max}}$ as functions of filling similar to Ref. [@zhu]. For the impurity potential we choose $V_0 = 100$, close to the unitary limit. The results are averaged over 10 different realizations of the random potential.
#### Effect of Impurity on DDW and DSC Orders
We summarize our main results in Fig. (1), where we plot the disorder dependence of different orders (normalized to $n_{\rm imp}=0$ values). Let us first look at the line (a) that represents the behavior of $\chi$ as a function of $n_{\rm imp}$ at half filling ($\langle n \rangle =1$). At this filling DDW is the stable order and DSC order in fact vanishes for the pure system. Comparing the behavior of $\chi$ with the results from SCTMA calculations (represented by (c) curve) we see that the DDW order is more robust to impurities than predicted by SCTMA.
On the other hand, away from half filling when we force $\chi=0$ in BdG equations, DSC becomes the surviving order and the $n_{\rm imp}$ dependence of $\Delta$ is given by the (b) line, which in fact is very similar to $\chi$ in the (a) curve. Such robustness of the DSC order to impurity had been studied before [@ghosald], and it is attributed primarily to the fact that – each impurity affects superconductivity rather inhomogeneously by destroying SC order within a small region (of size determined by coherence length $\xi$) around it. Hence the long range order is not globally affected. We find from our current numerical results that similar picture holds for DDW order as well, and $\chi$ is also affected locally by impurity, keeping long range DDW order robust.
However, similar study for the coexisting phase of DSC + DDW order at $\langle n \rangle=0.95$ reveals surprisingly that, superconducting order is severely affected by disorder (curve (e)) in the coexisting phase, much more so than in the absence of $\chi$. On the contrary, the DDW order (curve (d)) coexisting with DSC order becomes even more robust to impurities. In fact for low $n_{\rm imp}$, $\chi$ even increases with impurity. The rest of the paper is organized towards the detailed understanding of these unexpected results.
From Fig. (1d) we saw that the DDW order increases for small $n_{\rm imp}$. To get a further insight, we study the spatial structures of the order parameter (particularly at large $n_{\rm imp}$) on the lattice for each impurity configuration. In Fig. (2a) we present a Grey-scale plot of $\chi$ on a typical $30\times 30$ lattice at $n_{\rm imp}=0.06$ for a given realization of scatterers. The dark (light) regions represent larger (smaller) values of $\chi$. Comparing this structure with Fig. (2b), that gives the spatial distribution of $|\langle n \rangle -1 |$ for the same $n_{\rm imp}$, we see that $\chi$ is large in space where [local]{} density is close to 1 (half filling). The strong spatial correlation between these two panels is striking, although it is not exact; the scale of modulation of $\chi$ is somewhat larger than that of density. However, the strong tie of local $\langle n \rangle$ and $\chi$ suggest that the length scale of fluctuation of $\chi$ would be governed by that of $\langle n \rangle$, which is rather small (of the order of $k_F^{-1}$). This can be understood as follows.
The length scale associated with the DDW order, $\xi_{\rm DDW} \sim 1/\chi$. When impurity is introduced, the bond current attached to the impurity site is forced to be zero. So does the density. However, the bond current “near” the impurity site is re-constructed to satisfy the current conservation, and one should note that the healing length is of order of a lattice spacing. How the magnitude of the re-constructed bond-current is determined? This magnitude is strongly related to the local density. The electron density depletes close to impurities and increases at locations far from it, to keep the average at the desired value. Since at low disorder, a large number of sites attain $\langle n_i \rangle \sim 1$, $\chi$ increases at those sites; the DDW order is most stable near half filling, where perfect nesting occurs for our model. As a result average $\chi$ increases. At very large $n_{\rm imp}$, local density would be either much larger or smaller than 1, and $\chi$ would decrease everywhere. This argument can be substantiated by looking into our results for each configuration of impurities.
For $\langle n \rangle = 1$, introduction of impurity makes local density only to deviate from half filling. As a result $\chi$ decreases monotonically as found in Fig. (1a). The above argument for the behavior of DDW order with impurity is independent of the coexisting DSC order and we also found similar trend in $\chi$ as in Fig. (1d) for $\langle n \rangle < 1$ even in the absence of DSC order, which is consistent with our picture. This shows that DDW order responds to the density fluctuations due to impurities. On the contrary, the DSC order in the presence of impurities is [*not*]{} related to the local density fluctuations as DDW is, even though the length scale, $\xi_{\rm DSC} \sim 1/\Delta$, which is of the order of a few lattice spacing for high temperature superconductors under considerations. The behavior of $\Delta$ in the presence of impurities is shown to be related to the electron-hole mixing in the real space [@ghosals]; $\Delta$ is large when the density is close to the chemical potential.
Fig. (2c) and (2d) presents the spatial structure of $\Delta$ on lattice with the same $n_{imp}$ configuration in the presence and absence of DDW order. We clearly see that the DSC is strongly suppressed by the impurities when coexisting with DDW order (as also observed in Fig. (1b) and (1e)). The existence of DDW strongly affect the strength of the DSC, because away from the impurities there are regions where the density is near half-filling, hence the DDW becomes strong. Strong DDW allows significant weight of ($\pi$,$\pi$) scattering that mixes the $+$ and $-$ lobes of the DSC order and thereby DSC becomes weak. The regions of small density does not contribute to DSC order as well, due to the absence of enough electrons for pairing! Thus in the inhomogeneous coexistence phase DSC order is suppressed everywhere.
#### Averaged Density of States
Let us now study the (impurity) averaged density of states (DOS) $N(\omega) = {1 \over N}\sum_{n,i} \left[ |u_n(i)|^2\delta(\omega - E_n)
+ |v_n(i)|^2\delta(\omega + E_n) \right]$ (where we broaden the delta functions with a width comparable to average level spacing). To obtain a better statistics for $N(\omega)$, we used “Repeated Zone Scheme", [@ghosalv] that describes a large effective system made out of $10\times 10$ unit cells, each of which is of dimension $30\times 30$.
In Fig. (3) we plot $N(\omega)$ as a function of $\omega$ for different $n_{\rm imp}$ for the case of only DDW order (Panel a) and coexisting DDW + DSC order (Panel b). For the pure system with only DDW order, $N(\omega)$ is the standard d-wave DOS. With increasing $n_{\rm imp}$ we see that the gap-edge singularities get rounded off and a small accumulation of states is produced at the particle side of spectrum close to $\omega=0$. The accumulation of electrons around a single impurity effectively provide impurity screening, which will produce enhanced states at the particle side of the spectrum.[@morr] Such resonances from each impurity contribute to the average $N(\omega)$ and produce a broad band which is reflected as a bump in Fig. (3a). However, the strength of the DDW order is not affected much (given by the relative location of the two coherence peaks). At this point, we should emphasize that the DOS structure for impure DSC state is very different, where coherence peaks get strongly suppressed and a thin gap persists at $\omega=0$ [@ghosald; @peter], so that $N(0)=0$ for all $n_{\rm imp}$. From our results with DDW order, we find that $N(0) \propto n_{\rm imp}$, which is in disagreement with the prediction of T-Matrix result ($N(0) \propto \sqrt{n_{\rm imp}}$)[@maki2; @maki].
In Fig. (3b), for coexisting DSC + DDW, a double-gap DOS is expected at $n_{\rm imp}=0$ [@wonkee]; superconducting gap at $\omega=0$ and d-density wave gap at $\omega=\tilde{\mu}$. With increasing $n_{\rm imp}$, DSC gap gets washed out and by $n_{\rm imp}=0.03$, $N(\omega)$ looks very similar for Fig. (3a) and (3b) (The overall shift for the later case is due to the particle-hole asymmetry). This demonstrates in a different way our main result, that, the DSC order is very sensitive to impurity whereas DDW order is robust in the coexisting phase.
#### Summary and Discussion
We studied the effect of nonmagnetic impurity on DDW ordered state using BdG technique. While the standard SCTMA indicates that the effect of impurity on DDW is similar to that on DSC, we found that the spatial variation of the DDW order has a strong correlation with that of density \[Fig. (2a) and (2b)\], and it is very different from that of DSC order \[Fig. (2a) and (2c)\]. We discussed that this occurs because the length scale associated with the DDW order is of order of a lattice spacing ($\sim 1/k_F$), which suggests that the spatial variation of DDW order is related to the density fluctuation, while the DSC order is related to particle-hole mixing. Therefore, the effect of impurity on the DDW order is very different from that of DSC order, which can not be obtained from the standard SCTMA method.
When DSC and DDW coexist, it turns out that DDW order do not care about the existence of DSC and it still follows the density profile in the presence of impurity. However, DSC order would vanish almost everywhere \[See Fig. (2c)\]. This is because in the region of larger density it is killed by DDW, and in the region of smaller density it is destroyed by disorder. Thus in the inhomogeneous media both DDW and impurity are acting to suppress the DSC order.
Our current picture brings out the unexpected results and their understanding at the mean field level; if the DDW phase exists in cuprates, the Bragg signal would be detected in neutron scattering measurements even in the presence of strong nonmagnetic impurity, while the width of the Bragg peaks depends on strength of impurity. However, the definite answer for its relevance to the cuprates requires the understanding of the role of strong correlation, and interplay between different competing orders, which warrants further studies.
[**Acknowledgments**]{}: We would like to thank Y. B. Kim and A. Vishwanath for illuminating discussions. We acknowledge SHARCNet computational facilities at McMaster University where most of the calculations were carried out. This work is supported by SHARCNet fellowship(AG), NSERC of Canada(HYK), Canada Research Chair(HYK), and Canadian Institute for Advanced Research(HYK).
S. Chakravarty, R. B. Laughlin, D. K. Morr and C. Nayak, Phys. Rev. [**B63**]{}, 94503 (2001).
B. I. Halperin and T. M. Rice, in Solid State Physics, edited by F. Seitz, D. Turnbull and H. Ehrenreich (Academic Press, New York, 1968), VOl. 21, p. 115. I. Affleck and J. B. Marston, Phys. Rev. B [**37**]{}, 3774 (1988); J. B. Marston and I. Affleck, Phys. Rev. B [**39**]{}, 11538 (1989). T. C. Hsu, J. B. Marston, and I. Affleck, Phys. Rev. B [**43**]{}, 2866 (1991). S. Chakravarty, Hae-Yougn Kee, and C. Nayak, Int. J. Mod. Phys. [**15**]{}, 2901 (2001). H. A. Mook, [*et al*]{}, Phys. Rev. B [**64**]{}, 012502 (2001); Phys. Rev. B [**66**]{}, 144513 (2002). Y. Sidis [*et al*]{}, Phys. Rev. Lett. [**86**]{}, 4100 (2001).
C. Stock, [it et al]{}, Phys. Rev. B [**66**]{}, 024505 (2002).
For a review on SCTMA results see K. Maki in “Lectures on the Physics of Highly correlated Electron System", AIP Conf. Proc. [**438**]{}, edited by F. Mancini (AIP, New York, 1998) and references therein.
B. Dora, A. Virosztek, K. Maki, Phys. Rev. B [**66**]{}, 115112 (2002); cond-mat/0302362.
A. Ghosal, M. Randeria and N. Trivedi, Phys. Rev. [**B63**]{}, 20505R (2000).
Note that either of the “spin exchange" term or the “extended Hubbard" term of ${\cal H}_{\rm int}$ could produce both the DSC or the DDW orders. However we keep both terms for stabilizing DDW phase (See Ref. [@zhu]) and also to be able to tune both the orders independently.
J.-X. Zhu, W. Kim and J. P. Carbotte, Phys. Rev. Lett. [**87**]{}, 197001 (2001); C. Wu and V. Liu, Phys. Rev. [**B66**]{}, 20511R (2002). P. G. de Gennes, [*Superconductivity in Metals and Alloys*]{} (Benjamin, New York, 1966).
A. Ghosal, M. Randeria and N. Trivedi, Phys. Rev. Lett. [**81**]{}, 3940 (1998); Phys. Rev. [**B65**]{}, 14501 (2001).
In order to achieve accelerated convergence on multi-variable space we use the Broyden method (see, e.g., W. E. Pickett, Comp. Phys. Rep. [**9**]{}, 115 (1989)). We have checked that the same self-consistent solution is obtained for different initial guesses.
A. Ghosal, C. Kallin and A. J. Berlinsky, Phys. Rev. [**B66**]{}, 214502 (2002).
D. K. Morr, Phys. Rev. Lett. [**89**]{}, 106401 (2002).
W. A. Atkinson, P. J. Hirschfeld, and A. H. MacDonald, Phys. Rev. Lett. [**85**]{}, 3922 (2000).
W. Kim, J.-X. Zhu, J. P. Carbotte, and C. S. Ting, Phys. Rev. [**B65**]{}, 64502 (2002).
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ROTATION AXIS VARIATION DUE TO SPIN ORBIT RESONANCE
Giovanni Gallavotti [^1][A lecture given at the [*$I^o$ Congresso Nazionale di Meccanica Celeste*]{}, L’Aquila, may 22– 26, 1993. This text is archived in $mp\_arc@math.utexas.edu$, \# 93-222; TeXcopies can also be obtained by e-mail from: $gallavotti@vaxrom.infn.it$]{}
Dipartimento di Fisica, Università La Sapienza
P.le Moro 2, 00185, Roma, Italia
=1=110000
Let $\EE$ be a planet modeled by a homogeneous rigid ellipsoid with symmetry axis $NS$, with polar inertia moment $J_3$, equatorial moment $J$ and mechanical flattening coefficient $\h=(J_3-J)/J_3$. [^2][this means that if $R$ is the equatorial radius then the polar radius is $R/(1+2\h)^{1/2}$.]{}
The planet center of mass $T$ is supposed to revolve on a keplerian orbit $t\to\V r_T(t)$ about a focus $S$: the orbit plane will be called the [*ecliptic*]{} plane and $\V{{\bar k}}$ will denote its unit normal vector ([*celestial north*]{}) which sees the planet revolving counterclockwise.
The longitude $\l_T$ of $\V r_T$ on the ecliptic will be reckoned from the aphelion of the ellipse; hence $\l_T=0$ is the [*aphelion*]{} position, where $r_T \equiv |\V r_T|$ is maximal: $r_T(0) = a (1
+e)$, $a$ being the major semiaxis of the keplerian ellipse and $e$ its eccentricity.
With the above conventions, $r_T$ and $\l_T$ are related by: $r_T\=|\V r_T|=p\cdot(1-e\cos\l_T)^{-1},\ p\=a(1-e^2)$ and the Kepler’s laws imply that if $\l$ is the keplerian [*average anomaly*]{} then: $\l\=\l_T+2e\sin\l_T+\fra34e^2\sin2\l_T+\ldots$. and the motion is $\l\to\l+\o_Tt$, where $2\p/\o_T = 2\p a^{3/2}
g_N^{-1/2}$ is the year of the planet, $g_N\=k(m_S+m_T)$ if $k$ is Newton’s constant and $m_T,m_S$ are the masses of the planet and of its star.
The planet is described by means of the following coordinates: 1) The total angular momentum, or [*spin*]{}, $M$ and its projections $K\=M\cos i$ on the axis orthogonal to the ecliptic (“celestial north–south axis”). 2) The angles $\g,\f,\l$ where $\g$ is the angle between an axis fixed on the ecliptic ([*Aries line*]{}) and the spin–ecliptic node (the intersection of the ecliptic and the plane orthogonal to the spin: [*equinox line*]{}); $\f$ is the angle between the spin ecliptic node and the analogous spin–equator node; and $\l$ is the average anomaly of the planet, which rotates uniformly at angular velocity $\o_T$. It is a well known theorem in classical mechanics (Andoyer–Deprit) that the pairs $(M,\f)$ and $(K,\g)$ are canonically conjugate variables (see \[G\], p.318) and, introducing an auxiliary variable $B$ canonically conjugate to $\l$, the energy of the system can be written: $$H=\o_T B+\fra{M^2}{2J}+ V=\o_T B+\fra{M^2}{2J}
-\ig_\EE{k m_T m_S\over|\V r_T+\V x|}{d\V x\over|\EE|}\Eq(1)$$ One could avoid introducing $B$: but one would then have a time dependent hamiltonian, which I do not like.
The complete description of the rigid body configuration would require an extra pair of canonically conjugated varibles, namely $(L,\psi)$ where $\psi$ is the angle between a comoving axis fixed on the equator and the spin–equator node and $L$ is the projection $L=M\cos\th$ of the spin on the $NS$ axis. However, by symmetry, the hamiltonian does not depend on $\psi$; hence $L$ is a constant of motion and one can see that it only gives an additive contribution to the energy, which is dropped in (1).
I shall consider the following approximation for $V$: a) the integral is developed in powers of the ratio $R/a$ and the expansion will be truncated to the order $2$ included neglecting the orders $4$ and higher. [^3][only the 2d order is non trivial among the first three orders.]{} b) the expression thus obtained will be developed in powers of the eccentricity and the expansion will be truncated to 2d order.
The model thus obtained will be called the [*D’ Alembert precession-nutation model*]{}. The $e=0$ model was in fact used by D’Alembert, \[L\], to deduce his celebrated theory of the equinox precession for the Earth and the theory of the lunar plane precession.
It is not interesting to limit ourselves to the D’Alembert’s case $e=0$ because the phenomenon of large variations of the inclination axis are in this approximation not possible.
Define $\o_D\=M/J$, calling it conventionally the [*daily rotation*]{}, and consider the line $\LL$ in the plane of the action variables $\AA=(M,K)$ defined by: $$M=J\o_D\=2 J\o_T=const,\qquad K_0\le K\le K_1\Eq(2)$$ and call $\AA^0$ and $\AA^1$ its extremes. Since $K=M\cos i$ one realizes that proceeding on the above line from $\AA^0$ to $\AA^1$ the spin inclination angle changes between $i_0$ and $i_1$ and the system is locked in a $1:2$ spin orbit resonance.
A motion in which the projection in the $(M,K)$ plane closely follows the line $\LL$ is therefore a motion during which a variation of the inclination axis is observed. Along such line, which will be fixed in the following, the ratio $\cos\th\=L/M$ is a constant which will therefore also be fixed: and which will be supposed different from $\pm1$, (the angle $\th$, [*nutation constant*]{}, between the spin and the $NS$ axis is supposed different from $0$ or $\p$).
The eccentricity will be taken $e=\h^c$ for some $c>1$ so that setting $\h$ small implies that the eccentricity is also (much) smaller.
The question that will be investigated, see \[CG\], is whether there exists a trajectory $X_\eh(t)$ starting at $t=0$ in $\AA_\eh=(M_\eh,K_\eh)$ near $\AA^0$, with phases $(\g_\eh,\f_\eh,\l_\eh)$, and reaching after a long enough time the vicinity of $\AA^1$.
If such a trajectory exists for all $\h$ small enough, [*but different from $0$*]{}, one says that there is a [*Arnold diffusion*]{} phenomenon. [^4][this refers to one possible definition of Arnold’s diffusion, \[A\]: it is a special case of a general definition proposed in \[CG\]. But there are other definitions: different and, often, not very precise.]{} What is remarkable, when the diffusion phenomenon happens, is that it happens in spite of the fact that $K,M$ are adiabatic invariants, : while in the above situation $K$ or $i$ change by a quantity $\sim i_1-i_0$ [*independent on $\eh$*]{}.
The existence of Arnold diffusion says therefore that no averaging approximation can be even approximately correct over very long time scales or for infinite time.
This is not at all a knell for the widely used averaging methods: but it shows the interest of estimating the time scales over which the phenomenon happens: it is becoming well understood that the correctness of the averaging methods give very accurate results over time scales that are often of the order of the age of the universe, \[CeG\].
The result, \[CG\], on the above question, in the case of the described precession nutation model is the following. Therefore in the above precession problem Arnold’s diffusion is possible. The angle of spin inclination can change by an order of magnitude $O(1)$ in a long enough time, no matter how small is the spin orbit coupling ($\h$), provided it is non zero and provided the eccentricity is small enough compared to the flattening. And it becomes interesting to get an idea of the size of the time $T_\h$.
In \[CG\] an explicit estimate for $T_\h$ is derived:
Hence the estimate is a very very long time, compared to the times over which one expects the averaging methods to have some validity, which have scale of order $\sim\exp f\h^{-1/2}$. Also the constants $b,c,d,f$, whose values can actually be computed, see \[CG\], turn out to be very poor for any practical application.
But to get to practical applications a lot of estimates would have to be refined and it is not completely obvious that this is really impossible. One can recall, on this respect, that for a long time it was stated by some people that the KAM estimates were too bad for any practical use: and this turned out to be grossly incorrect, \[CeG\],\[CC\],\[LR\]. In the case of diffusion the situation looks much harder, perhaps desperately so: but we shall see.
This shows that the estimate of the time scale for the diffusion, essentially the time one has to wait to see a violation of the predictions of the averaging approximations, diverges if one tries to bridge gaps in inclination containing $i=0,\fra\p2,\p$.
In particular the above estimate diverges if one tries to find a trajectory in which the spin sign is reversed ($i$ goes through $\fra\p2$). The difficulty in constructing such trajectories might be a manifestation of a physical phenomenon and not just a defect of the techniques of proof; it seems, indeed, that the chaotic motions of the planetary axes are unable, in absence of dissipative effects, to change the spin sign, the sign of $K$, \[LRo\].
Formula (3) is suggestive as it leads to the conjecture that (in general) diffusion along a path $\LL$ in action space might really take place over a time scale dependinng exponentially on some power of the coupling constant [*times a coefficient determined my maximising a suitable functional defined on $\LL$*]{}: therefore I call (3), by analogy, a [*large deviation formula*]{}.
I conclude with a technical comment: the analysis is based on the fact that the hamiltonian system under consideration has three very different time scales: namely the daily time scale ($\o_D^{-1}$, coinciding with the year time scale because the free system is in a $1:2$ resonance), the resonance time scale of order $O(\h^{-1/2}\o_D^{-1})$ (describing the characteristic time of oscillation transversal to the resonance) and the equinox precession scale of order $O(\h^{-1}\o_D^{-1})$ (arising from an application of the method of averaging and describing the precession of the equinox line, when $e=0$). The three time scales become very different from each other when $\h\to0$: this is due to the degeneracy inherent in the problem where the unperturbed hamiltonian [*does not depend on the variable $K$*]{}. And it implies the phenomenon of [*large angles at the homoclinic intersections*]{}, see \[CG\], which in turn implies the existence of diffusion. Deeply different is what happens near resonances of non degenerate systems, where all the action variables appear non linearly in the unperturbed hamiltonian: in such cases there are only two distinct time scales describing the motions near the resonances and the theory of diffusion is much harder (because the homoclinic angles are exponentially small, see \[CG\], \[G2\]).
Thus the above theory is only one more instance of the (well known) fact that the degeneration intrinsically present in all celestial mechanics problems is very often, in fact, making the theory easier rather than harder (as sometimes naively claimed).
For connections between the techniques used in the theory of the above problem and the KAM theorem see also \[G2\],\[G3\].
-200
0.5cm [**References:**]{} 1000010000
[\[A\] ]{} Arnold, V.: [*Instability of dynamical systems with several degrees of freedom*]{}, Sov. Mathematical Dokl., 5, 581-585, 1966.
[\[CC\] ]{} Celletti, A., Chierchia, L.: [*Construction of analytic \*KAM surfaces and effective stability bounds*]{}, Communications in Mathematical Physics, 118, 119– 161, 1988.
[\[CG\] ]{} Chierchia, L., Gallavotti, G.: [*Smooth prime integrals for quasi-integrable Hamiltonian systems*]{} Il Nuovo Cimento, 67 B, 277-295, 1982.
[\[CeG\] ]{} Celletti, A., Giorgilli, A.: [*On the stability of the lagrangian points in the spatial restricted problem of three bodies*]{}, Celestial Mechanics, 50, 31– 58, 1991
[\[G\] ]{} Gallavotti, G.: [*The elements of Mechanics*]{}, Springer, 1983.
[\[G2\] ]{} Gallavotti, G.: [*Twistless KAM tori, quasi flat homoclinic intersections, and other cancellations in the perturbation series of certain completely integrable hamiltonian systems. A review.*]{}, archived in the electronic archive [mp\_arc@math.utexas.edu]{}, \#93-164; e-mail (last version) copies in TeX(plain of course) available also by direct request to the author.
[\[G3\] ]{} Gallavotti, G.: [*Twistless KAM tori*]{}, archived in [mp\_arc@math.utexas.edu]{} \#93-172; e-mail copies (last version) in TeX(plain of course) available also by direct request to the author.
[\[L\] ]{} de la Place, S.: [*Mécanique Céleste*]{}, tome II, book 5, ch. I, 1799, english translation by Bodwitch, E., reprinted by Chelsea, 1966.
[\[LR\] ]{} Llave, R., Rana, D.: [*Accurate strategies for small divisor problems*]{}, Bullettin A.M.S., 22, 85– 90, 1990. See also: [*Accurate strategies for K. A. M. bounds and their implementation*]{} , in “Computer Aided proofs in Analysis”, ed. K. Meyer, D. Schmidt, Springer Verlag, 1991.
[\[LRo\] ]{} Laskar, J. Robutel, P.: [*The chaotic obliquity of the planets*]{}, Nature, 361, 608– 612, 1993. See also the report of J. Laskar at this conference.
[^1]: ${}^*$
[^2]: ${}^1$
[^3]: ${}^2$
[^4]: ${}^3$
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this note, we prove that the kernel of the linearized equation around a positive energy solution in $\rn$, $n\geq 3$, to $-\Delta W-\gamma|x|^{-2}V=|x|^{-s}W^{\crits-1}$ is one-dimensional when $s+\gamma>0$. Here, $s\in [0,2)$, $0\leq\gamma<(n-2)^2/4$ and $\crits=2(n-s)/(n-2)$.'
address: 'Frédéric Robert, Institut Élie Cartan, Université de Lorraine, BP 70239, F-54506 Vand[œ]{}uvre-lès-Nancy, France'
author:
- Frédéric Robert
date: December 29th 2016
title: 'Nondegeneracy of positive solutions to nonlinear Hardy-Sobolev equations'
---
We fix $n\geq 3$, $s\in [0,2)$ and $\gamma<\frac{(n-2)^2}{4}$. We define $\crits=2(n-s)/(n-2)$. We consider a nonnegative solution $W\in C^2(\rnp)\setminus\{0\}$ to $$\label{eq:V}
-\Delta W-\frac{\gamma}{|x|^{2}}W=\frac{W^{\crits-1}}{|x|^{s}}\hbox{ in }\rnp.$$ Due to the abundance of solutions to , we require in addition that $W$ is an energy solution, that is $W\in \dundeux$, where $\dundeux$ is the completion of $C^\infty_c(\rn)$ for the norm $u\mapsto \Vert\nabla u\Vert_2$. Linearizing yields to consider $$\label{def:KV}
K:=\left\{\varphi\in \dundeux/\, -\Delta\varphi-\frac{\gamma}{|x|^2}\varphi=(\crits-1)\frac{W^{\crits-2}}{|x|^s}\varphi\hbox{ in }\dundeux\right\}$$ Equation is conformally invariant in the following sense: for any $r>0$, define $$W_r(x):=r^{\frac{n-2}{2}}W(rx)\hbox{ for all }x\in\rnp,$$ then, as one checks, $W_r\in C^2(\rnp)$ is also a solution to , and, differentiating with respect to $r$ at $r=1$, we get that $$-\Delta Z-\frac{\gamma}{|x|^2}Z=(\crits-1)\frac{W^{\crits-2}}{|x|^s}Z\hbox{ in }\rnp,$$ where $$Z:=\frac{d}{dr}{W_r}_{|r=1}= \sum_ix^i\partial_i W+\frac{n-2}{2}W\in \dundeux.$$ Therefore, $Z\in K$. We prove that this is essentially the only element:
\[th:main\] We assume that $\gamma\geq 0$ and that $\gamma+s>0$. Then $K=\rr Z$. In other words, $K$ is one-dimensional.
Such a result is useful when performing Liapunov-Schmidt’s finite dimensional reduction. When $\gamma=s=0$, the equation is also invariant under the translations $x\mapsto W(x-x_0)$ for any $x_0\in\rn$, and the kernel $K$ is of dimension $n+1$ (see Rey [@Rey] and also Bianchi-Egnell [@BE]). After this note was completed, we learnt that Dancer-Gladiali-Grossi [@dgg] proved Theorem \[th:main\] in the case $s=0$, and that their proof can be extended to our case, see also Gladiali-Grossi-Neves [@ggn].
This note is devoted to the proof of Theorem \[th:main\]. Since $\gamma+s>0$, it follows from Chou-Chu [@ChouChu], that there exists $r>0$ such that $W=\lambda^{\frac{1}{\crits-2}}U_r$, where $$U(x):=\left(|x|^{\frac{2-s}{n-2}\am}+|x|^{\frac{2-s}{n-2}\ap} \right)^{-\frac{n-2}{2-s}}.$$ with $$\eps:=\sqrt{\frac{(n-2)^2}{4}-\gamma}\hbox{ and }\alpha_{\pm}(\gamma):=\frac{n-2}{2}\pm\sqrt{\frac{(n-2)^2}{4}-\gamma}.$$ As one checks, $U\in \dundeux\cap C^\infty(\rnp)$ and $$\label{eq:U}
-\Delta U-\frac{\gamma}{|x|^2}U=\lambda\frac{U^{\crits-1}}{|x|^s}\hbox{ in }\rnp,\hbox{ with }\lambda:=4\frac{n-s}{n-2}\eps^2.$$ Therefore, proving Theorem \[th:main\] reduces to prove that $\tilde{K}$ is one-dimensional, where $$\label{def:tK}
\tilde{K}:=\left\{\varphi\in \dundeux/\, -\Delta\varphi-\frac{\gamma}{|x|^2}\varphi=(\crits-1)\lambda\frac{U^{\crits-2}}{|x|^s}\varphi\hbox{ in }\dundeux\right\}$$
[**I. Conformal transformation.**]{}
We let $\sn:=\{x\in\rn/\, \sum x_i^2=1\}$ be the standard $(n-1)-$dimensional sphere of $\rn$. We endow it with its canonical metric $\can$. We define $$\left\{\begin{array}{cccc}
\Phi: & \rr\times\sn &\mapsto &\rnp\\
&(t,\sigma) & \mapsto & e^{-t}\sigma
\end{array}\right.$$ The map $\Phi$ is a smooth conformal diffeomorphism and $\Phi^\star\eucl=e^{-2t}(dt^2+\can)$. On any Riemannian manifold $(M,g)$, we define the conformal Laplacian as $L_g:=-\Delta_g+\frac{n-2}{4(n-1)}R_g$ where $\Delta_g:=\hbox{div}_g(\nabla)$ and $R_g$ is the scalar curvature. The conformal invariance of the Laplacian reads as follows: for a metric $g'=e^{2\omega}g$ conformal to $g$ ($\omega\in C^\infty(M)$), we have that $L_{g'}u=e^{-\frac{n+2}{2}\omega}L_g(e^{\frac{n-2}{2}\omega}u)$ for all $u\in C^\infty(M)$. It follows from this invariance that for any $u\in C^\infty_c(\rnp)$, we have that $$\label{transfo:delta}
(-\Delta u)\circ \Phi(t,\sigma)=e^{\frac{n+2}{2}t}\left(-\partial_{tt}\hat{u}-\Delta_{\can}\hat{u}+\frac{(n-2)^2}{4}\hat{u}\right)(t,\sigma)$$ for all $(t,\sigma)\in\rr\times \sn$, where $\hat{u}(t,\sigma):=e^{-\frac{n-2}{2}t}u(e^{-t}\sigma)$ for all $(t,\sigma)\in \rr\times\sn$. In addition, as one checks, for any $u,v\in C^\infty_c(\rnp)$, we have that $$\begin{aligned}
\int_{\rn}(\nabla u,\nabla v)\, dx&=& \int_{\rr\times\sn}\left(\partial_t\hat{u}\partial_t\hat{v}+\left(\nabla^\prime\hat{u},\nabla^\prime\hat{v}\right)_{\can}+\frac{(n-2)^2}{4}\hat{u}\hat{v}\right)\, dt\, d\sigma\nonumber\\
&:=&B(\hat{u},\hat{v})\label{def:B}\end{aligned}$$ where we have denoted $\nabla^\prime\hat{u}$ as the gradient on $\sn$ with respect to the $\sigma$ coordinate. We define the space $H$ as the completion of $C_c^\infty(\rr\times\sn)$ for the norm $\Vert\cdot\Vert_H:=\sqrt{B(\cdot,\cdot)}$. As one checks, $u\mapsto \hat{u}$ extends to a bijective isometry $\dundeux\to H$.
The Hardy-Sobolev inequality asserts the existence of $K(n,s,\gamma)>0$ such that $\left(\int_{\rn}\frac{|u|^{\crits}}{|x|^s}\, dx\right)^{\frac{2}{\crits}}\leq K(n,s,\gamma)\int_{\rn}\left(|\nabla u|^2-\frac{\gamma}{|x|^2}u^2\right)\, dx$ for all $u\in C^\infty_c(\rnp)$. Via the isometry $\dundeux\simeq H$, this inequality rewrites $$\left(\int_{\rr\times \sn}|v|^{\crits}\, dt d\sigma\right)^{\frac{2}{\crits}}\leq K(n,s,\gamma)\int_{\rr\times\sn}\left((\partial_t v)^2+|\nabla^\prime v|_{\can}^2+\eps^2v^2\right)\, dtd\sigma,$$ for all $v\in H$. In particular, $v\in L^{\crits}(\rr\times\sn)$ for all $v\in H$.
We define $H_1^2(\rr)$ (resp. $H_1^2(\sn)$) as the completion of $C^\infty_c(\rr)$ (resp. $C^\infty(\sn)$) for the norm $$u\mapsto \sqrt{\int_{\rr}(\dot{u}^2+u^2)\, dx}\; \left(\hbox{resp. }u\mapsto \sqrt{\int_{\sn}(|\nabla^\prime u|^2_{\can}+u^2)\, d\sigma}\right).$$ Each norm arises from a Hilbert inner product. For any $(\varphi,Y)\in C^\infty_c(\rr)\times C^\infty(\sn)$, define $\varphi\star Y\in C^\infty_c(\rr\times\sn)$ by $(\varphi\star Y)(t,\sigma):=\varphi(t)Y(\sigma)$ for all $(t,\sigma)\in\rr\times\sn$. As one checks, there exists $C>0$ such that $$\label{eq:star}
\Vert \varphi\star Y\Vert_H\leq C\Vert \varphi\Vert_{H_1^2(\rr)}\Vert Y\Vert_{H_1^2(\sn)}$$ for all $(\varphi,Y)\in C^\infty_c(\rr)\times C^\infty(\sn)$. Therefore, the operator extends continuously from $H_1^2(\rr)\times H_1^2(\sn)$ to $H$, such that holds for all $(\varphi,Y)\in H_1^2(\rr)\times H_1^2(\sn)$.
\[lem:2\] We fix $u\in C^\infty_c(\rr\times\sn)$ and $Y\in H_1^2(\sn)$. We define $$u_Y(t):=\int_{\sn}u(t,\sigma)Y(\sigma)\, d\sigma=\langle u(t,\cdot),Y\rangle_{L^2(\sn)}\hbox{ for all }t\in\rr.$$ Then $u_Y\in H_1^2(\rr)$. Moreover, this definition extends continuously to $u\in H$ and there exists $C>0$ such that $$\Vert u_Y\Vert_{H_1^2(\rr)}\leq C\Vert u\Vert_H\Vert Y\Vert_{H_1^2(\sn)}\hbox{ for all }(u,Y)\in H\times H_1^2(\sn).$$
[*Proof of Lemma \[lem:2\]:*]{} We let $u\in C^\infty_c(\rr\times\sn)$, $Y\in H_1^2(\sn)$ and $\varphi\in C^\infty_c(\rr)$. Fubini’s theorem yields: $$\int_{\rr}\left(\partial_t u_Y\partial_t\varphi+u_Y\varphi\right)\, dt=\int_{\rr\times\sn}\left(\partial_t u\partial_t(\varphi\star Y)+u\cdot (\varphi\star Y)\right)\, dtd\sigma$$ Taking $\varphi:=u_Y$, the Cauchy-Schwartz inequality yields $$\begin{aligned}
&&\Vert u_Y\Vert_{H_1^2(\rr)}^2\\
&&\leq \sqrt{\int_{\rr\times\sn}\left((\partial_t u)^2+u^2\right)dtd\sigma}
\times \sqrt{\int_{\rr\times\sn}\left((\partial_t (u_Y\star Y))^2+ (u_Y\star Y)^2\right) dtd\sigma}\\
&&\leq C\Vert u\Vert_H\Vert u_Y\star Y\Vert_H\leq C\Vert u\Vert_H\Vert u_Y\Vert_{H_1^2(\rr)}\Vert Y\Vert_{H_1^2(\sn)},\end{aligned}$$ and then $\Vert u_Y\Vert_{H_1^2(\rr)}\leq C\Vert u\Vert_H\Vert Y\Vert_{H_1^2(\sn)}$. The extension follows from density.
[**II. Transformation of the problem.**]{} We let $\varphi\in \tilde{K}$, that is $$-\Delta\varphi-\frac{\gamma}{|x|^2}\varphi=(\crits-1)\lambda\frac{U^{\crits-2}}{|x|^s}\varphi\hbox{ weakly in }\dundeux.$$ Since $U\in C^\infty(\rnp)$, elliptic regularity yields $\varphi\in C^\infty(\rnp)$. Moreover, the correspondance yields $$\label{eq:hphi}
-\partial_{tt}\hphi-\Delta_{\can}\hphi+\eps^2\hphi=(\crits-1)\lambda \hU^{\crits-2}\hphi$$ weakly in $H$. Note that since $\hphi,\hU\in H$ and $H$ is continuously embedded in $L^{\crits}(\rr\times\sn)$, this formulation makes sense. Since $\varphi\in C^\infty(\rnp)$, we get that $\hphi\in C^\infty(\rr\times\sn)\cap H$ and equation makes sense strongly in $\rr\times\sn$. As one checks, we have that $$\hU(t,\sigma)=\left(e^{\frac{2-s}{n-2}\eps t}+e^{-\frac{2-s}{n-2}\eps t}\right)^{-\frac{n-2}{2-s}}\hbox{ for all }(t,\sigma)\in \rr\times\sn.$$ In the sequel, we will write $\hU(t)$ for $\hU(t,\sigma)$ for $(t,\sigma)\in \rr\times\sn$.
The eigenvalues of $-\Delta_{\can}$ on $\sn$ are $$0=\mu_0<n-1=\mu_1<\mu_2<....$$ We let $\mu\geq 0$ be an eigenvalue for $-\Delta_{\can}$ and we let $Y=Y_\mu\in C^\infty(\sn)$ be a corresponding eigenfunction, that is $$-\Delta_{\can}Y=\mu Y\hbox{ in }\sn.$$ We fix $\psi\in C^\infty_c(\rr)$ so that $\psi\star Y\in C^\infty_c(\rr\times\sn)$. Multiplying by $\psi\star Y$, integrating by parts and using Fubini’s theorem yields $$\int_{\rr}\left(\partial_{t}\hphi_Y\partial_t\psi+(\mu+\eps^2)\hphi_Y\psi\right)\, dt=\int_{\rr}(\crits-1)\lambda \hU^{\crits-2}\hphi_Y\psi\, dt,$$ where $\hphi_Y\in H_1^2(\rr)\cap C^\infty(\rr)$. Then $$A_\mu \hphi_Y=0\hbox{ with }A_\mu:=-\partial_{tt}+(\mu+\eps^2-(\crits-1)\lambda \hU^{\crits-2})$$ where this identity holds both in the classical sense and in the weak $H_1^2(\rr)$ sense. We claim that $$\label{eq:phi:0}
\hphi_Y\equiv 0\hbox{ for all eigenfunction }Y\hbox{ of }\mu\geq n-1.$$ We prove the claim by taking inspiration from Chang-Gustafson-Nakanishi ([@gustaf], Lemma 2.1). Differentiating with respect to $i=1,...,n$, we get that $$-\Delta\partial_i U-\frac{\gamma}{|x|^2}\partial_i U-(\crits-1)\lambda\frac{U^{\crits-2}}{|x|^s}\partial_i U=-\left(\frac{2\gamma}{|x|^{4}}U+\frac{s\lambda}{|x|^{s+2}}U^{\crits-1}\right)x_i$$ On $\rr\times\sn$, this equation reads $$-\partial_{tt}\hat{\partial_i U}-\Delta_{\can}\hat{\partial_i U}+\left(\eps^2-(\crits-1)\lambda \hU^{\crits-2}\right)\hat{\partial_i U}=-\sigma_i e^t \left(2\gamma\hU+s\lambda \hU^{\crits-1}\right)$$ Note that $\hat{\partial_i U}=-V\star \sigma_i$, where $\sigma_i:\sn\to \rr$ is the projection on the $x_i$’s and $$V(t):=-e^{-\frac{n-2}{2}t}U^\prime(e^{-t})=e^{(1+\eps)t}\left(\ap +\am e^{2\frac{2-s}{n-2}\eps t}\right)\left(1+e^{2\frac{2-s}{n-2}\eps t}\right)^{-\frac{n-s}{2-s}}>0$$ for all $t\in\rr$. Since $-\Delta_{\can}\sigma_i=(n-1)\sigma_i$ (the $\sigma_i$’s form a basis of the second eigenspace of $-\Delta_{\can}$), we then get that $$A_\mu V\geq A_{n-1}V= e^t\left(2\gamma\hU+s\lambda \hU^{\crits-1}\right)>0\hbox{ for all }\mu\geq n-1\hbox{ and }V>0.$$ Note that for $\gamma>0$, we have that $\am>0$, and that for $\gamma=0$, we have that $\am=0$. As one checks, we have that $$\begin{aligned}
(i)\;\left\{\left(\gamma>0\hbox{ and }\eps>1\right)\hbox{ or }\left(\gamma=0\hbox{ and }s<\frac{n}{2}\right)\right\}&\Rightarrow & V\in H_1^2(\rr)\\
(ii)\; \left\{\left(\gamma>0\hbox{ and }\eps\leq1\right)\hbox{ or }\left(\gamma=0\hbox{ and }s\geq \frac{n}{2}\right)\right\}&\Rightarrow & V\notin L^2((0,+\infty))\end{aligned}$$ [*Assume that case (i) holds:*]{} in this case, $V\in H_1^2(\rr)$ is a distributional solution to $A_\mu V>0$ in $H_1^2(\rr)$. We define $m:=\inf \{\int_{\rr}\varphi A_\mu \varphi\, dt\}$, where the infimum is taken on $\varphi\in H_1^2(\rr)$ such that $\Vert\varphi\Vert_2=1$. We claim that $m>0$. Otherwise, it follows from Lemma \[lem:3\] below that the infimum is achieved, say by $\varphi_0\in H_1^2(\rr)\setminus \{0\}$ that is a weak solution to $A_\mu\varphi_0=m\varphi_0$ in $\rr$. Since $|\varphi_0|$ is also a minimizer, and due to the comparison principle, we can assume that $\varphi_0>0$. Using the self-adjointness of $A_\mu$, we get that $0\geq m\int_{\rr}\varphi_0V\, dt=\int_{\rr}(A_\mu \varphi_0)V\, dt=\int_{\rr}(A_\mu V)\varphi_0\, dt>0$, which is a contradiction. Then $m>0$. Since $A_\mu\varphi_Y=0$, we then get that $\varphi_Y\equiv 0$ as soon as $\mu\geq n-1$. This ends case (i).
[*Assume that case (ii) holds:*]{} we assume that $\varphi_Y\not\equiv 0$. It follows from Lemma \[lem:4\] that $V(t)=o(e^{-\alpha |t|})$ as $t\to -\infty$ for all $0<\alpha<\sqrt{\eps^2+n-1}$. As one checks with the explicit expression of $V$, this is a contradiction when $\eps<\frac{n-2}{2}$, that is when $\gamma>0$. Then we have that $\gamma=0$ and $\eps=\frac{n-2}{2}$. Since $\frac{n}{2}\leq s<2$, we have that $n=3$. As one checks, $(\mu+\eps^2-(\crits-1)\lambda \hU^{\crits-2})>0$ for $\mu\geq n-1$ as soon as $n=3$ and $s\geq 3/2$. Lemma \[lem:4\] yields $\varphi_Y\equiv 0$, a contradiction. So $\varphi_Y\equiv 0$, this ends case (ii).
These steps above prove . Then, for all $t\in\rr$, $\hphi(t,\cdot)$ is orthogonal to the eigenspaces of $\mu_i$, $i\geq 1$, so it is in the eigenspace of $\mu_0=0$ spanned by $1$, and therefore $\hphi=\hphi(t)$ is independent of $\sigma\in\sn$. Then $$-\hphi^{\prime\prime}+(\eps^2-(\crits-1)\lambda \hU^{\crits-2})\hphi=0\hbox{ in }\rr\hbox{ and }\hphi\in H_1^2(\rr).$$ It follows from Lemma \[lem:5\] that the space of such functions is a most one-dimensional. Going back to $\varphi$, we get that $\tilde{K}$ is of dimension at most one, and then so is $K$. Since $Z\in K$, then $K$ is one dimensional and $K=\rr Z$. This proves Theorem \[th:main\].
[**III. Auxiliary lemmas.**]{}
\[lem:5\] Let $q\in C^0(\rr)$. Then $$\hbox{dim}_{\rr}\{\varphi\in C^2(\rr)\cap H_1^2(\rr)\hbox{ such that }-\ddot{\varphi}+q\varphi=0\}\leq 1.$$
[*Proof of Lemma \[lem:5\]:*]{} Let $F$ be this space. Fix $\varphi,\psi\in F\setminus\{0\}$: we prove that they are linearly dependent. Define the Wronskian $W:=\varphi \dot{\psi}-\dot{\varphi}\psi$. As one checks, $\dot{W}=0$, so $W$ is constant. Since $\varphi,\dot{\varphi},\psi,\dot{\psi}\in L^2(\rr)$, then $W\in L^1(\rr)$ and then $W\equiv 0$. Therefore, there exists $\lambda\in\rr$ such that $(\psi(0),\dot{\psi}(0))=\lambda (\varphi(0),\dot{\varphi}(0))$, and then, classical ODE theory yields $\psi=\lambda\varphi$. Then $F$ is of dimension at most one.
\[lem:3\] Let $q\in C^0(\rr)$ be such that there exists $A>0$ such that $\lim_{t\to\pm\infty}q(t)=A$, and define $$m:=\inf_{\varphi\in H_1^2(\rr)\setminus\{0\}}\frac{\int_{\rr}\left(\dot{\varphi}^2+q\varphi^2\right)\, dt}{\int_{\rr}\varphi^2\, dt}.$$ Then either $m>0$, or the infimum is achieved.
Note that in the case $q(t)\equiv A$, $m=A$ and the infimum is not achieved.
[*Proof of Lemma \[lem:3\]:*]{} As one checks, $m\in\rr$ is well-defined. We let $(\varphi_i)_i\in H_1^2(\rr)$ be a minimizing sequence such that $\int_{\rr}\varphi_i^2\, dt=1$ for all $i$, that is $\int_{\rr}\left(\dot{\varphi}_i^2+q\varphi_i^2\right)\, dt=m+o(1)$ as $i\to +\infty$. Then $(\varphi_i)_i$ is bounded in $H_1^2(\rr)$, and, up to a subsequence, there exists $\varphi\in H_1^2(\rr)$ such that $\varphi_i\rightharpoonup \varphi$ weakly in $H_1^2(\rr)$ and $\varphi_i\to \varphi$ strongly in $L^2_{loc}(\rr)$ as $i\to +\infty$. We define $\theta_i:=\varphi_i-\varphi$. Since $\lim_{t\to \pm\infty}(q(t)-A)=0$ and $(\theta_i)_i$ goes to $0$ strongly in $L^2_{loc}$, we get that $\lim_{i\to +\infty}\int_{\rr}(q(t)-A)\theta_i^2\, dt=0$. Using the weak convergence to $0$ and that $(\varphi_i)_i$ is minimizing, we get that $$\int_{\rr}\left(\dot{\varphi}^2+q\varphi^2\right)\, dt+\int_{\rr}\left(\dot{\theta}_i^2+A\theta_i^2\right)\, dt=m+o(1)\hbox{ as }i\to +\infty.$$ Since $1-\Vert\varphi\Vert_2^2=\Vert\theta_i\Vert_2^2+o(1)$ as $i\to +\infty$ and $\int_{\rr}\left(\dot{\varphi}^2+q\varphi^2\right)\, dt\geq m\Vert\varphi\Vert_2^2$, we get $$m\Vert\theta_i\Vert_2^2\geq \int_{\rr}\left(\dot{\theta}_i^2+A\theta_i^2\right)\, dt+o(1)\hbox{ as }i\to +\infty.$$ If $m\leq 0$, then $\theta_i\to 0$ strongly in $H_1^2(\rr)$, and then $(\varphi_i)_i$ goes strongly to $\varphi\not\equiv 0$ in $H_1^2$, and $\varphi$ is a minimizer for $m$. This proves the lemma.
\[lem:4\] Let $q\in C^0(\rr)$ be such that there exists $A>0$ such that $\lim_{t\to\pm\infty}q(t)=A$ and $q$ is even. We let $\varphi\in C^2(\rr)$ be such that $-\ddot{\varphi}+q\varphi=0$ in $\rr$ and $\varphi\in H_1^2(\rr)$.
- If $q\geq 0$, then $\varphi\equiv 0$.
- We assume that there exists $V\in C^2(\rr)$ such that $$-\ddot{V}+qV>0\; ,\; V>0\hbox{ and }V\not\in L^2((0,+\infty)).$$ Then either $\varphi\equiv 0$ or $V(t)=o(e^{-\alpha |t|})$ as $t\to -\infty$ for all $0<\alpha<\sqrt{A}$.
[*Proof of Lemma \[lem:4\]:*]{} We assume that $\varphi\not\equiv 0$. We first assume that $q\geq 0$. By studying the monotonicity of $\varphi$ between two consecutive zeros, we get that $\varphi$ has at most one zero, and then $\ddot{\varphi}$ has constant sign around $\pm\infty$. Therefore, $\varphi$ is monoton around $\pm\infty$ and then has a limit, which is $0$ since $\varphi\in L^2(\rr)$. The contradiction follows from studying the sign of $\ddot{\varphi}$, $\varphi$. Then $\varphi\equiv 0$ and the first part of Lemma \[lem:4\] is proved.
We now deal with the second part and we let $V\in C^2(\rr)$ be as in the statement. We define $\psi:=V^{-1}\varphi$. Then, $-\ddot{\psi}+h \dot{\psi}+Q \psi=0$ in $\rr$ with $h,Q\in C^0(\rr)$ and $Q>0$. Therefore, by studying the zeros, $\dot{\psi}$ vanishes at most once, and then $\psi(t)$ has limits as $t\to\pm\infty$. Since $\varphi=\psi V$, $\varphi\in L^2(\rr)$ and $V\not\in L^2(0,+\infty)$, then $\lim_{t\to +\infty}\psi(t)=0$. We claim that $\lim_{t\to-\infty}\psi(t)\neq 0$. Otherwise, the limit would be $0$. Then $\psi$ would be of constant sign, say $\psi>0$. At the maximum point $t_0$ of $\psi$, the equation would yield $\ddot{\psi}(t_0)>0$, which contradicts the maximum. So the limit of $\psi$ at $-\infty$ is nonzero, and then $V(t)=O(\varphi(t))$ as $t\to-\infty$.
We claim that $\varphi$ is even or odd and $\varphi$ has constant sign around $+\infty$. Since $t\mapsto \varphi(-t)$ is also a solution to the ODE, it follows from Lemma \[lem:5\] that it is a multiple of $\varphi$, and then $\varphi$ is even or odd. Since $\dot{\psi}$ changes sign at most once, then $\psi$ changes sign at most twice. Therefore $\varphi=\psi V$ has constant sign around $+\infty$.
We fix $0<A'<A$ and we let $R_0>0$ such that $q(t)>A'$ for all $t\geq R_0$. Without loss of generality, we also assume that $\varphi(t)>0$ for $t\geq R_0$. We define $b(t):=C_0e^{-\sqrt{A'}t}-\varphi(t)$ for all $t\in\rr$ with $C_0:=2\varphi(R_0)e^{\sqrt{A'}R_0}$. We claim that $b(t)\geq 0$ for all $t\geq R_0$. Otherwise $\inf_{t\geq R_0}b(t)<0$, and since $\lim_{t\to +\infty}b(t)=0$ and $b(R_0)>0$, then there exists $t_1>R_0$ such that $\ddot{b}(t_1)\geq 0$ and $b(t_1)<0$. However, as one checks, the equation yields $\ddot{b}(t_1)<0$, which is a contradiction. Therefore $b(t)\geq 0$ for all $t\geq R_0$, and then $0<\varphi(t)\leq C_0e^{-\sqrt{A'}t}$ for $t\to +\infty$. Lemma \[lem:4\] follows from this inequality, $\varphi$ even or odd, and $V(t)=O(\varphi(t))$ as $t\to-\infty$.
[12]{}
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this paper, we consider generalizations of the Alexander polynomial and signature of 2-bridge knots by considering the Gordon-Litherland bilinear forms associated to essential state surfaces of the 2-bridge knots. We show that the resulting invariants are well-defined and explore properties of these invariants. Finally we realize the boundary slopes of the essential surfaces as a difference of signatures of the knot.'
address:
- 'Department of Mathematics & Statistics, The College of New Jersey, Ewing, NJ 08628'
- 'Department of Mathematics, University of Nebraksa, Lincoln, NE 68588'
author:
- 'Cynthia L. Curtis'
- Vincent Longo
title: 'State invariants of two-bridge knots'
---
[^1]
[^2]
Introduction
============
In [@GL], Gordon and Litherland studied a bilinear form $GL$ associated to a spanning surface of a knot, identifying a correction term enabling them to use this form to compute the signature of the knot. We consider a family of matrices $V$ with $V_{GL} = V+V^T$ the matrix of $GL$ relative to a certain basis. We use these matrices to build invariants generalizing the Alexander polynomial and the signature of the knot.
In order to obtain invariants, we restrict our attention to 2-bridge knots. Further, the spanning surfaces we consider are the essential spanning surfaces of the knot. These surfaces are of particular interest. Any 2-bridge knot has finitely many such surfaces, and these are well-understood up to isotopy due to the work of Hatcher and Thurston in [@HT]. For each such surface we define a [*state polynomial*]{} given by $det(V-tV^T)$ and a [*state signature*]{} $\sigma(V+V^T)$ and show that both quantities are independent of the choices involved in their definitions. We obtain a finite family of polynomials and a finite set of signatures for each 2-bridge knot.
The boundary slopes of these surfaces have become increasingly important computationally, for example in the computation of Culler-Gordon-Luecke-Shalen semi-norms (see [@CGLS], [@BZ01],[@C01] and [@O]), ${{SL(2, {\mathbb C})}}$-Casson invariants (see [@C01], [@BC06], [@BC08], and [@BC12]), and $A$-polynomials (see [@CCGLS], [@BZ01] and [@BC12]). A motivating observation is that the Hatcher-Thurston formula for computing the boundary slope of an essential surface $S$ with boundary a 2-bridge knot is $2(M-M_0)$, where $M$ is a quantity computed from $S$ and $M_0$ is an analogous quantity computed from a Seifert surface for the knot. In fact $M_0$ is just the signature of the knot $K$, and our motivating problem was to realize $M$ as a signature of a matrix defined from $S$. We conclude our paper by realizing the boundary slope of an essential spanning surface of a 2-bridge knot as the difference between the state signature associated to $S$ and the signature of the knot.
We note that our proofs of well-definiton of the invariants rely on the simplicity of the essential spanning surfaces of 2-bridge knots. An interesting question is whether an analogous construction continues to give polynomial invariants for other knots using their essential spanning surfaces.
The paper is outlined as follows: in Section 2.1 we recall the definition of the Gordon-Litherland bilinear form and the definition of the Seifert matrix $V$ which we will generalize. In Section 2.2 we restrict our attention to 2-bridge knots, define the appropriate generalization of Seifert matrices, define the invariants, and show that they are well-defined. We investigate properties of the state polynomials in Section 3.1 and of the state signatures in Section 3.2.
Definitions of the Invariants
=============================
The Gordon-Litherland form for a spanning surface of a knot {#GLsect}
-----------------------------------------------------------
We begin by introducing the bilinear form of Gordon and Litherland associated to a spanning surface following [@GL]. Let $K$ be a knot in $S^3$, and let $S$ be a spanning surface for $K$; that is, a surface with $\partial S = K$. Let $F$ be a closed tubular neighborhood of $S$, and view this as the total space of an $I$-bundle over $S$. Let $\tilde{S}$ be the corresponding $\partial I$-bundle over $S$. Note that $\tilde{S}$ is the orientable double cover of $S$ if $S$ is non-orientable and is the trivial double cover if $S$ is orientable.
Since $S$ and $\partial \tilde{S}$ are disjoint, the linking pairing $lk$ defines a bilinear form $$lk:H_1(S)\times H_1(\tilde{S}) \rightarrow {{\mathbb Z}}.$$ Letting $\tau$ denote the transfer homomorphism $\tau: H_1(S)\rightarrow H_1(\tilde{S})$ and precomposing with $id \times \tau$ we obtain the Gordon Litherland pairing $$GL: H_1(S)\times H_1(S) \rightarrow {{\mathbb Z}}$$ given by $GL(\alpha,\beta) = lk(\alpha, \tau(\beta))$.
If S is isotoped to a single disk with bands attached, then we may choose a set of curves $x_1,x_2,\ldots,x_k$ on $S$ given by the cores of the bands together with arcs in the disk joining the two ends of the core of each band. Orient each curve $x_i$ arbitrarily. The classes of the curves $x_i$ give a basis for $H_1(S)$. Relative to this basis we obtain a symmetric matrix $V_{GL}$ for the Gordon Litherland pairing.
Note that if $S$ is orientable, then the matrix $V_{GL}$ is equal to $V+V^T$ for a certain Seifert matrix $V$ for $K$. The matrix $V$ is obtained directly from the linking pairing on $H_1(S)$ with respect to the chosen basis together with a choice of orientation on $S$. Specifically, denote the positive push-off of $x_i$ by $x^*_i$. We define the entries of $V$ by setting $v_{ij} = lk(x_i,x^*_j)$. It is easy to check that $V_{GL} = V+V^T$.
We remark that any Seifert matrix $V$ for a nontrivial knot is not symmetric. The asymmetry in $V$ arises precisely in the off-diagonal entries $v_{ij}\neq v_{ji}$ whenever the oriented intersection number of $x_i$ and $x_j$ is $\pm 1$. For Seifert surfaces, the intersection paring for the curves $x_i$ remains consistent under band slides, and the Alexander polynomials obtained from the associated Seifert matrices $V$ and $V'$ agree.
This is not quite the case for non-orientable surfaces. To see this, consider the non-orientable surfaces $S$ and $S'$ with boundary the $5_2$ knot shown in Figure \[52surfaces\]. These surfaces are isotopic, as $S'$ is obtained from $S$ by sliding the right foot of the band on the left in $S$ counterclockwise along the right band in $S$. However the curves in a basis coming from the cores of the bands in the first picture intersect, whereas the curves in the basis given by the cores of the second band do not. Defining diagonal terms by $\frac{1}{2}lk(x_i,\tau(x_i))$, we obtain Seifert-like matrices $V =
\left[
\begin{array}{cc}
1 & 0 \\
1 &-3/2
\end{array}
\right]
$ and $V' =
\left[
\begin{array}{cc}
1/2 & 1 \\
1 & -3/2
\end{array}
\right]
$. Both $V_{GL}$ and $V'_{GL}$ are matrices for $GL$ relative to the corresponding bases. However $V$ is asymmetric while $V'$ is symmetric, and the polynomials $det(V-tV^T)$ and $det(V'-tV'^T)$ do not agree.
![The surfaces $S$ and $S'$ corresponding to $[2,3]$[]{data-label="52surfaces"}](2,3surface.pdf "fig:") ![The surfaces $S$ and $S'$ corresponding to $[2,3]$[]{data-label="52surfaces"}](isotoped2,3surface.pdf "fig:")
Thus we must be very careful in generalizing the construction of the Seifert matrix to non-orientable surfaces, so that we obtain a sensible and consistent choice of asymmetry in the resulting matrices. We solve this problem for 2-bridge knots below.
Definitions of the invariants {#refs}
-----------------------------
For the remainder of our paper, we assume $K$ is a 2-bridge knot, and let $M = S^3 - N(K)$, where $N(K)$ is an open tubular neighborhood of $K$. Let $K = K(\alpha , \beta)$ be the standard 2-bridge notation for $K$, where $\alpha$ is the determinant of $K$.
A surface $\Sigma$ in $M$ is said to be *incompressible* if for any disk $D \subset M$ with $D \cap \Sigma = \partial D$, there exists a disk $D' \subset \Sigma$, with $\partial D' = \partial D$. A surface $\Sigma$ is $\partial$[*- incompressible*]{} if for each disk $D \subset M$ with $D \cap \Sigma = \partial_+D$ and $D \cap \partial M = \partial_-D$ there is a disk $D' \subset \Sigma$ with $\partial_+ D' =\partial_+ D$ and $\partial_- D' \subset \partial \Sigma$. A surface $\Sigma \subset M$ is *essential* if it is both incompressible and $\partial$-incompressible.
By [@HT] we know that each essential spanning surface $S$ is isotopic to at least one of the following surfaces obtained by plumbing together half-twisted bands: Specifically, we choose half-twisted bands corresponding to a continued fraction expansion $$\frac{\beta}{\alpha} = r + \frac{1}{n_1 + \frac{1}{n_2 + \ldots +\frac{1}{n_k}}}$$ with $|n_i|\geq 2$ for each $i$. Henceforth we denote this continued fraction expansion by $[n_1,n_2, \ldots, n_k].$ We arrange these bands vertically as shown in Firgure \[HTsurface\], attach parallel untwisted bands as shown, attach the top (level 0) and bottom (level $k$) shaded horizontal rectangular disks shown, and at each level (1,2,…k-1) between the bands we attach either the shaded horizontal rectangular disk shown (the “inner” disk) or its complement in the sphere which is the one-point compactification of the horizontal plane (the “outer” disk).
![The surface corresponding to $[n_1,n_2,...,n_k]$[]{data-label="HTsurface"}](seifert.pdf)
For such a plumbed surface with corresponding continued fraction expansion $[n_1,n_2,...,n_k]$, there is a collection of curves $x_1,x_2,...,x_k$ well-defined up to isotopy of the curve system on the surface, where $x_i$ is essentially the core of the $i^{th}$ “box”, consisting of a vertical arc forming the core of the $i^{th}$ twisted band, a vertical arc following the opposite untwisted band, and arcs joining the ends of the vertical arcs in the level disks (whether inner or outer) in levels $i-1$ and $i$. Note that $x_i \cap x_{i+1}$ can be chosen to be transverse, consisting of a single point in the horizontal disk in level $i$ (either inner or outer), and $x_i \cap x_j = \emptyset $ if $j \notin \{i-1,i, i+1\}$. Fix such a collection of curves, and orient these arbitrarily. We now generalize the construction of the Seifert matrix as follows:
Fix an isotopy of $S$ to a plumbed surface with corresponding continued fraction expansion $[n_1,n_2,\ldots,n_k]$, and (abusing notation) let $x_1,x_2,...,x_k$ also denote the curve system on $S$ corresponding to the chosen curve system $x_1,x_2,...,x_k$ on the plumbed surface. For each $i$ and $j$ such that $j \neq i$ and $x_i\cap x_j \neq \emptyset$ let $D_{ij}$ be a small disk neighborhood of $x_i \cap x_j$ in $S$, chosen so that no two distinct neighborhoods $D_{ij}$ intersect, and let $N_{ij}$ be an arbitrarily chosen normal vector to $D_{ij}$. Denote by $x^{i*}_j$ a curve obtained from $x_j$ by an isotopy supported in a tubular neighborhood of $D_{ij}$ pushing $x_j$ in the direction of $N_{ij}$ if $x_i$ and $x_j$ meet, and let $x^{i*}_j = x_j$ if $x_i$ and $x_j$ are disjoint.
A [*state matrix*]{} $V_S$ for an essential spanning surface $S$ is a matrix $V_S$ with entries $$v_{ij} = \left\{ \begin{array}{ll}
lk(x_i,x^{i*}_j) & \mbox{if } i\neq j \\
\frac{1}{2} lk(x_i, \tau(x_i)) & \mbox{if } i=j. \end{array} \right.$$
Note that if $S$ is nonorientable, then the diagonal entries in $V_S$ will be half-integers rather than integers. Also note that there are several matrices $V_S$ for any surface $S$ due to the choices involved in the definition. As in the orientable case, it is clear that $V_{GL} = V_S + V^T_S$ is a matrix for $GL$ for any state matrix $V_S$.
We now use the state matrices associated to these surfaces to define our invariants. Let $S$ be an essential spanning surface for $K$, and let $V_S$ be an associated state matrix. We define generalizations $\Delta_S(t)$ and $\sigma_S$ of the Alexander polynomial and the signature of the knot using $S$. As for the Alexander polynomial, we define two polynomials to be equivalent if they differ by a unit $\pm t^k$ in the ring of integral Laurent polynomials, where $k \in {{\mathbb Z}}$.
The [*state polynomial*]{} $\Delta_{S}(t)$ is the equivalence class of $det(V_S - t V^{T}_{S})$.
We will frequently abuse notation and identify $\Delta_S(t)$ by a representative of the equivalence class, as is standard for Alexander polynomials.
The [*state signature*]{} $\sigma_{S}$ is the signature of $V_S + V^{T}_{S}$.
\[invariance\] Let $K$ be a 2-bridge knot, and let $S$ be an essential spanning surface of $K$. The state polynomial $\Delta_{S}$ and the state signature $\sigma_{S}$ depend only on the isotopy classes of $K$ and $S$. Thus the collection of state polynomials and the collection of state signatures are well-defined invariants of the knot $K$.
The theorem, and indeed the construction of the polynomials, is somewhat simpler if we view our surfaces as given by a single disk with bands attached rather than as plumbings of twisted bands. Therefore before proving the theorem we first show the following:
\[discwithbands\] Let $K$ be a 2-bridge knot, and let $S$ be an essential spanning surface of $K$ corresponding to the continued fraction expansion $[n_1,n_2,\ldots,n_k]$ which is a plumbing of twisted bands as described above, with corresponding curve system $x_1,x_2,\ldots, x_k$. Then $S$ together with the curve system $x_1,x_2,\ldots x_k$ is isotopic to a single disc with $n$ unknotted bands with curve system given by the cores of the bands, where the $i^{th}$ band has $n_i$ half-twists if $i$ is odd and $-n_i$ half-twists if $i$ is even. The bands are arranged so that the feet of the $i^{th}$ and $(i+1)^{st}$ bands are alternating along the disk, and so that the feet of the $i^{th}$ and $j^{th}$ bands do not alternate along the disk if $i$ and $j$ are not adjacent integers. If all horizontal disks are inner disks, then the $2i^{th}$ band crosses in front of both the $(2i-1)^{st}$ band and the $(2i+1)^{st}$ band. For each level $j$ at which $S$ uses the outer rather than the inner disk, the crossing of bands $j$ and $j+1$ is the reverse of that for the surface with all inner disks.
This realization of $S$ is shown in Figure \[flat73surface\] for the knot $7_3$ where $S$ is the surface corresponding to the continued fraction expansion $[4, -2,2,-2]$ all of whose horizontal disks are inner disks.
![The branched surface corresponding to $[4,-2,2,-2]$, with all inner disks, realized as a single disk with bands[]{data-label="flat73surface"}](actuallycorrect4222.pdf)
View the surface shown in Figure \[HTsurface\] as a disk with bands attached, where the disk is the union of the shaded vertical (untwisted) rectangles together with the horizontal (inner or outer) disks. Flatten this surface into the plane of the even-numbered vertical rectangles.
We now prove Theorem \[invariance\]. We made a number of choices in defining $V_S$, and we must show that both $\Delta_S(t)$ and $\sigma_S$ are is independent of these choices. Specifically, we have we have chosen the curve system $x_1,x_2,\ldots, x_k$ to be the cores of the handles of $S$ after first isotoping $S$ to one of the standard surfaces described above; we must investigate the choice involved in this isotopy. In addition, we have numbered and oriented the curves $x_i$ arbitrarily, and we have chosen normal vectors for the discs $D_{ij}$ arbitrarily.
We first show independence of the numbering and orientation of the curves $x_i$. Renumbering the curves will alter $V_S$ by reordering the rows and corresponding columns of $V_S$ and therefore also the rows and corresponding columns of $V^T$ and $V_S-tV^T_S$. This will leave $\sigma_S$ and $\Delta_S(t)$ unchanged. Reversing the orientation of a curve $x_i$ will alter $V_S$ by multiplying the $i^{th}$ row and $i^{th}$ column by $-1$ to obtain a new state matrix $V'_S$. Then $V_S$ and $V'_S$ differ by congruence via a matrix $U$, where $U$ is a diagonal matrix with $i^{th}$ diagonal entry $-1$ and all other diagonal entries $1$: $V_S' = UV_SU^T$. Therefore $det(V'_S - t V'^{T}_S) = det(V_S - t V^T_S)$, so the state polynomial is unaffected by these choices. Further since $V_S$ and $V'_S$ are conjugate the signs of the eigenvalues of $V_S+V_S^T$ and $V'_S+V'^{T}_S$ agree, so $\sigma_S$ is also independent of these choices.
Next we consider the dependence of $\Delta_S(t)$ and $\sigma_S$ on the choices of normal vectors $N_{ij}$. Note that there is a choice of numbering and orientation for the curves $x_i$ and a choice of normal vectors $N_{ij}$ so that the resulting matrix $V = V_S$ is given by $$\begin{aligned}
\label{standardV}
V=\left[ \begin{array}{cccccl}
\frac{n_1}{2} & 0 & & & & \\
1 & -\frac{n_2}{2} & 0 & & & \text{\huge0} \\
0 & 1 & \frac{n_3}{2} & 0 & & \\
& 0 & 1 & -\frac{n_4}{2} & \ddots & \\
& & \ddots & \ddots & \ddots & \ \ \ \ 0\\
& \text{\huge0} & & 0 & 1 & (-1)^{k+1}\frac{n_k}{2}\\
\end{array}\right].\end{aligned}$$ If we change the orientation of a single normal vector $N_{i,i+1}$ we obtain the matrix $V'$ which is identical to $V$ except at entry $v'_{i,i+1}$, which is 1 rather than 0 and entry $v'_{i+1,i}$, which is 0 rather than 1.
Now $V+V^T$ and $V'+V'^T$ are identical, so $\sigma_S$ is unchanged by the change in $N_{i,i+1}$. To see that $\Delta_S(t)$ is unchanged by the reversal of $N_{i,i+1}$, note that $$\begin{aligned}
\label{V-tVT}
V-tV^T=\left[ \begin{array}{cccccccc}
m_1 & -t & 0 & & &\text{\huge0} \\
1 & -m_2 & -t & 0 & & \\
0 & 1 & m_3 & -t & \ddots & \\
& 0 & 1 & -m_4 & \ddots & 0\\
\text{\huge0}& &\ddots & \ddots & \ddots & -t\\
& & & 0 & 1 & (-1)^{k+1}m_k\\
\end{array}\right],\end{aligned}$$ where $m_j = \frac{n_j}{2} - t \frac{n_j}{2}$, while $$V' -t V'^T=\left[ \begin{array}{ccccccccc}
m_1 & -t & & & & & & &\\
1 & -m_2 & -t & & & \text{\huge0}& & & \\
& 1 & m_3 & \ddots & & & & &\\
& & \ddots & \ddots & -t & & & & \\
& & & 1 & (-1)^{i+1}m_{i} & 1 & & & \\
& & & & -t & (-1)^{i+2}m_{i+1} & -t & & \\
& &\text{\huge0} & & & 1 & \ddots & \ddots &\\
& & & & & & \ddots & \ddots & -t\\
& & & & & & & 1 & (-1)^{k+1}m_k\\
\end{array}\right].$$ We see that $V-tV^T$ is obtained from $V'-tV'^T$ by multiplying each of rows $i+1,i+2,\ldots,k$ of $V'-tV'^T$ by $-1/t$ and each of columns $i+1,i+2,\ldots,k$ of $V'-tV'^T$ by $-t$. Then $det(V - t V^{T}) = det(V'-tV'^T)$, and hence $\Delta_S(t)$ is unchanged by reversing $N_{i,i+1}$.
It remains to be shown that the choice of representation of $S$ as a plumbing of twisted bands given by a continued fraction expansion does not affect the resulting polynomials or signatures. By Theorem 1(b) of [@HT], $S$ must be isotopic to such a surface, and by Theorem 1(d) of [@HT], the continued fraction expansion giving rise to $S$ is unique (as we have required $|n_i| \geq 2$ for each $i$). If $S$ is isotopic to a given surface via two different isotopies, giving rise to distinct curve systems $x_1,x_2,\ldots,x_k$ and $x'_1,x'_2,\ldots,x'_k$ on $S$ each corresponding to the curve system $x_1,x_2,\ldots, x_k$ on the plumbed surface, then the various state matrices defined by these curve systems with will agree up to choices of normal vectors since the linking numbers $lk(x_i,x_j^{i*})$ for and $lk(x'_i,x_j^{'i*})$ must each agree with the linking numbers $lk(x_i,x_j^{i*})$ for the curves on the plumbed surface up to choice of normal vector and similarly $lk(x_i,\tau(x_i))$ and $lk(x'_i,\tau(x'_i))$ on $S$ must both agree with $lk(x_i,\tau(x_i))$ on the plumbed surface.
However by Theorem 1(e) of [@HT] there is any isotopy relation among the various essential spanning surfaces corresponding to a given continued fraction expansion. Specifically, if $n_i = \pm 2$ for one or more $n_i$, then the choice of inner versus outer disk at one or more levels in the representation of $S$ is not uniquely determined by $S$.
We show that this ambiguity in the representation of $S$ does not impact the resulting state polynomial or state signature. In fact, any two essential spanning surfaces for the knot corresponding to the same continued fraction expansion give the same state polynomial and state signature. To see this, recall from the proof of Lemma 2.5 that replacing the $i^{th}$ inner horizontal disk with the $i^{th}$ outer horizontal disk has the effect of reversing the crossing between bands $i$ and $i+1$ in the representation of $S$ as a disk with bands attached given by Lemma 2.5. But reversing the crossing between bands $i$ and $i+1$ has the same effect on the associated state matrix as replacing $N_{i,i+1}$ with its negative and reversing the orientation of $x_{i+1}, x_{i+2},\ldots,x_k$. Therefore by the earlier parts of this proof the state polynomial and the state signature are not affected by a reversal of the crossing between bands. Hence the polynomial and the signature depend only on the isotopy class of $S$.
Properties of the invariants
============================
We continue to let $K$ be a 2-bridge knot with essential spanning surface $S$. Suppose the continued fraction expansion corresponding to $S$ is $[n_1,n_2,\ldots,n_k]$.
Properties of state polynomials and relation to genera
------------------------------------------------------
\[polyprops\] The state polynomial $\Delta_{S}$ has the following properties:
1. The polynomial is symmetric; that is, $\Delta_{S}(t^{-1}) \sim \Delta_{S}(t)$.\
2. $\Delta_{S}(1) = \left\{ \begin{array}{ll}
1 & \mbox{if $k$ is even} \\
0 & \mbox{if $k$ is odd.} \end{array} \right. $\
3. $|\Delta_{S}(-1)| = det(K)$.\
4. When normalized to be a polynomial with lowest term a constant term, $\Delta_{S}$ is a polynomial of degree $k$ with leading coefficient $\frac{1}{2^k} n_1 n_2 \ldots n_k$.
Isotope $S$ to be the disk with bands given by Lemma \[discwithbands\]. Choose the curves $x_i$ and orient the normal vectors $N_{ij}$ so that the resulting matrix $V$ is given by matrix (\[standardV\]) above.
To prove property $i$, note $$\begin{array}{lcl}
\Delta_{S}(t) &=& det(V-tV^T)\\
&=& det[(V-tV^T)^T]\\
&=& det(-tV+V^T)\\
&=& det[-t(V-t^{-1}V^T)]\\
&=& \pm t^k det(V-t^{-1}V^T)\\
&=& \pm t^k \Delta_S(t^{-1}).\\
\end{array}$$ Thus the two polynomials are equivalent.
To prove property $ii$, observe that $$\Delta_S(1) = det(V-V^T) = \left[ \begin{array}{cccccccc}
0 & -1 & 0 & & &\text{\huge0} \\
1 & 0 & -1 & 0 & & \\
0 & 1 & 0 & -1 & \ddots & \\
& 0 & 1 & 0 & \ddots & 0\\
\text{\huge0}& &\ddots & \ddots & \ddots & -1\\
& & & 0 & 1 & 0\\
\end{array}\right].$$ If $k=1$ we have $\Delta_S(1) = det[0] = 0$, and if $k=2$ we have $\Delta_S(1) = det \left[ \begin{array}{cc} 0&-1\\1&0\end{array}\right] =1.$ Further, if $k>1$, it is easy to check by cofactor expansion along the first row and then by cofactor expansion along the first column of the $k-1 \times k-1$ submatrix that $det(V-V^T) = det M$, where $M$ is the $k-2 \times k-2$ matrix of the same form as $V-V^T$. The claim follows by induction.
For property $iii$, note that $\Delta_S(-1) = det(V+V^T)$, and recall that $V+V^T = V_{GL}$. The result is immediate since $V_{GL}$ is a Goeritz matrix by Section 2 of [@GL]. Finally we turn to property $iv$. Note that if $\Delta_S$ is normalized as indicated, then the constant term is given by $\Delta_S(0) = det(V)$ for any state matrix $V$. We have seen that we may take $V$ to be lower triangular, so $det(V) = \pm \frac{1}{2^k} n_1n_2\cdots n_k$. By symmetry this is also the coefficient of $t^k$.
Note that if $S$ is orientable, then $\Delta_{S}$ is the Alexander polynomial of $K$. In this case the properties listed are well-known. Note in particular that if $S$ is orientable then $k$ is even.
We define the *genus* of a non-orientable surface $\Sigma$, like that of an orientable surface, to be $\frac{1 - \chi(\Sigma)}{2}$, or equivalently to be half the number of cross-cap summands in the surface. Note that the genus of $S$ is $g(S) = k/2$, so that in general the degree of the state polynomial $\Delta_S(t)$ is $2g(S)$. We examine the relationship between the various genera and hence the various degrees of the state polynomials for a knot $K$.
Recall that the *genus of K* is the minimum genus among all Seifert surfaces of $K$. We denote this by $g(K)$. In our case this is the genus of the orientable essential surface $S$, corresponding to the unique continued fraction expansion all of whose quotients $n_1,n_2,\ldots,n_k$ are even. We show that $g(K)$ and the genera $g(S)$ for nonorientable essential spanning surfaces $S$ are essentially independent, thereby establishing that the degrees of the Alexander polynomial of $K$ and the other state polynomials of $K$ are independent.
We remark that the *nonorientable genus of K* is defined to be the minimum genus among all nonorientable spanning surfaces of $K$. In Section 3 of [@AK] the authors show that the nonorientable genus of $K$ is equal to the minimum genus among the essential spanning surfaces for $K$ if this genus is realized by a nonorientable surface or is equal to $g(K)+1/2$ otherwise. This is also shown explicitly, and the nonorientable genera are computed explicitly, for 2-bridge knots in [@HiTe]. Note that if the nonorientable genus is realized by an essential spanning surface of $K$ then the nonorientable genus is equal to half the degree of the corresponding state polynomial. On the other hand, if no nonorientable essential spanning surface realizes the nonorientable genus of knot, then the nonorientable genus is realized by a surface obtained by adding a cross-cap to the minimal genus Seifert surface. In this case the nonorientable genus is not equal to half the degree of any state polynomial of $K$.
We prove the following
The degrees of the state polynomials of a 2-bridge knot $K$ are independent of the degree of the Alexander polynomial of $K$. Specifically:
1. There exist 2-bridge knots with linear state polynomials and Alexander polynomials of arbitrarily high degree.\
2. There exist 2-bridge knots with quadratic Alexander polynomials and with state polynomials of arbitrarily high degree.
To prove the theorem, we identify knots with continued fraction expansions of the appropriate lengths. See for example the algorithm of Proposition 3.3 of [@CFLM] for identifying the continued fraction expansions for a knot $K$.
For claim $i$, consider the $(2,m)$ torus knot. This has a continued fraction expansion $[m]$, which corresponds to a surface $S$ with genus 1/2 and linear state polynomial $\Delta_S$. The Seifert surface for the $(2,m)$ torus knot corresponds to the continued fraction expansion $[-2,2,-2,2,\ldots,-2,2]$ of length $m-1$. Then the degree of the Alexander polynomial is $2g(K) = m-1$.
For claim $ii$, consider the 2-bridge knot with continued fraction expansion $[2i,2j]$. This continued fraction expansion yields the Seifert surface of the knot, so the knot is genus one with a quadratic Alexander polynomial. The other two essential spanning surfaces of the knot have continued fraction expansions $[-2,2,-2,\ldots,-2,2j + 1]$ and $[2i + 1,-2,2,-2,\ldots,-2]$ of length $2i$ and $2j$, respectively. These surfaces are of genera $i$ and $j$, respectively, and the corresponding state polynomials have degrees $2i$ and $2j.$
Properties of state signatures and relation to boundary slopes
--------------------------------------------------------------
We next consider properties of the state signatures of the 2-bridge knot $K$. We continue to let $S$ be an essential spanning surface for $K$ corresponding to the continued fraction expansion $[n_1,n_2,\ldots,n_k]$. We begin by explaining how to compute the state signature $\sigma_S$ in terms of the continued fraction expansion. Specifically, let $N^+$ denote the number of entries in the list $n_1,n_2,\ldots n_k$ whose signs agree with the alternating pattern $+,-,+,-,\ldots$, and let $N^-$ denote the number of entries in the list $n_1,n_2,\ldots n_k$ whose signs do not agree with the alternating pattern $+,-,+,-,\ldots$.
The state signature $\sigma_S$ is given by $\sigma_S = N^+ - N^-.$
We are interested in the signature of $V+V^T$. Note that $$\begin{aligned}
V+V^T = \left[ \begin{array}{ccccccc}
n_1 & 1 & 0 & & \text{\huge0}& \\
1 & -n_2 & 1 & 0 & &\\
0 & 1 & n_3 & 1& \ddots &\\
& 0 & 1 & \ddots & \ddots &0\\
\text{\huge0}& & \ddots& \ddots & (-1)^kn_{k-1} &1 \\
& & & 0 & 1 & (-1)^{k+1}n_k \\
\end{array}\right],\end{aligned}$$ which is row equivalent to
$$\begin{aligned}
\left[ \begin{array}{ccccccc}
n_1 & 1 & & & \text{\huge0}\\
0 & -\big(n_2+\frac{1}{n_1}\big) & 1 & & \\
& 0 & \bigg(n_3+\frac{1}{n_2+\frac{1}{n_1}}\bigg) & \ddots& \\
& & \ddots & \ddots & 1 \\
\text{\huge0}& & & 0 & (-1)^{k+1}\bigg(n_k+\frac{1}{n_{k-1}+\ldots+\frac{1}{n_1}}\bigg) \\
\end{array}\right].\end{aligned}$$
Therefore the signs of the eigenvalues of $V+V^T$ agree with the signs of the numbers in the list $n_1, -(n_2+\frac{1}{n_1}), n_3 +\frac{1}{n_2+\frac{1}{n_1}}, \ldots, (-1)^k (n_k+\frac{1}{n_{k-1}+\ldots+\frac{1}{n_1}})$. Since $|n_i| > \left| \frac{1}{n_{i-1}+\ldots+\frac{1}{n_1}}\right|$, we see these signs agree with the signs of the numbers in the list $n_1,- n_2, n_3, \ldots, (-1)^k n_k$. Thus the number of positive eigenvalues is precisely $N^+$, and the number of negative eigenvalues is $N^-$.
This proposition makes the computation of the state signatures simple, and allows us to easily verify some properties of state signatures.
The state signatures $\sigma_S$ for a given knot $K$ have the following properties:
1. $|\sigma_S| \leq 2g(S)$.
2. The width of the set of state signatures for a 2-bridge knot may be arbitrarily large.
3. The numbers $\sigma(K)$ and $\sigma_S$ are independent; that is, either may be arbitrarily large while the other is 0.
Here note that the width of the set of state signatures is the difference between the largest and the smallest state signature.
For the first claim, note that the number of eigenvalues of $V+V^T$ is $2g(S)$, so this is an upper bound for the absolute value of the signature.
For the remaining claims, first consider the knot $K(4m+1,2m)$ with continued fraction expansion $[2,2m]$ giving the Seifert surface and with continued fraction expansion $[3,-2,2,-2,\ldots,-2]$ of length $2m$ giving a second essential spanning surface $S$. We see that for the Seifert surface $N^+ = N^- = 1$, so $\sigma(K) = 0$. However for $S$ we see that $N^+=2m$ and $N^-=0$, so $\sigma_S = 2m$.
In contrast, the knot $K(6\ell+1,2\ell)$ where $\ell$ is any integer has a continued fraction $[4,-2,2,-2,\ldots,-2]$ of length $2\ell$ yielding the Seifert surface and a continued fraction expansion $[3, 2\ell]$ yielding an essential spanning surface $S$. We see that for the Seifert surface $N^+=2 \ell$ and $N^- = 0$, whereas for $S$ we have $N^+=N^-=1$. Hence $\sigma(K) = 2 \ell$ while $\sigma_S = 0$.
Finally, we return to our motivating problem, realizing the boundary slopes of $K$ as a difference of signatures of $K$:
Let $S$ be an essential spanning surface of $K$. Then the boundary slope of $S$ is given by $2(\sigma_S - \sigma(K))$.
By Proposition 2 of [@HT] we know the boundary slope of the essential spanning surface $S$ corresponding to the continued fraction expansion $[n_1,n_2,\ldots,n_k]$ is given by $2(N^+ - N^-) - 2(N^+_0 - N^-_0)$, where $N^+_0$ and $N^-_0$ are $N^+$ and $N^-$ for the all-even continued fraction expansion yielding the essential Seifert surface of $K$. (Here note that our sign conventions do not agree with those of [@HT]; this is their result restated with our sign conventions.) But $2(N^+ - N^-) = 2\sigma_S$, and $2(N^+_0 - N^-_0) = \sigma(K)$.
[*Acknowledgements.*]{} The authors thank William Franczak for contributing Figure \[HTsurface\]. In addition we thank the referee for his or her insightful suggestions.
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[^1]:
[^2]:
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We consider the equilibrium dynamics of Ising spin models with multi-spin interactions on sparse random graphs (Bethe lattices). Such models undergo a mean field glass transition upon increasing the graph connectivity or lowering the temperature. Focusing on the low temperature limit, we identify the large scale rearrangements responsible for the dynamical slowing-down near the transition. We are able to characterize exactly the critical dynamics by analyzing the statistical properties of such rearrangements. We obtain a precise crossover description of the role of activation at the transition. Our approach can be generalized to a large variety of glassy models on sparse random graphs, ranging from satisfiability to kinetically constrained models.'
author:
- 'A. Montanari$^{\,1}$, G. Semerjian$^{\,2}$'
title: From Large Scale Rearrangements to Mode Coupling Phenomenology
---
Understanding the slowing down of relaxational dynamics in glass-forming liquids is an important open problem in statistical physics. Two points of view have been developed in the last years. Mode coupling theory (MCT) [@MCT] is based upon a “self-consistent” closure of dynamical equations, and predicts an ergodicity-breaking transition at temperature $T_{\rm d}$. It was later shown that MCT is exact for a class of fully-connected (FC) mean-field models, the dynamical phase transition (DPT) being related to the proliferation of metastable states [@KiTh; @MCA].
According to an alternative point of view, the dynamical slowing down in supercooled liquids can be traced back to the increasing cooperativity of the dynamics [@BeGaJPB].
It is a recent discovery that MCT implies diverging correlations as the $T_{\rm d}$ is approached [@BB]. This hints at a possible convergence among the above points of view, and may lead to universal predictions. However, the relation between a diverging correlation length and dynamical slowing-down remains qualitative. This paper aims at filling this gap. By analyzing a particular case, we will show that a detailed picture of the critical dynamics can be obtained through the analysis of highly correlated regions whose size diverges at the transition. Several features of MCT are recovered, despite no exact set of MCT equations holds in the system considered here [@SeCuMo].
A further source of motivation comes from the discovery that several [*ensembles*]{} of hard optimization problems, such as satisfiability and coloring, undergo a mean-field DPT [@notrerevue]. An interesting question in this context is: how much time a Monte Carlo (MC) algorithm needs for sampling a low-cost configuration of the problem. Furthermore, a phase transition of the same type, is found in a large variety of other models on random graphs, from kinetically constrained models, to rigidity percolation [@RigidityAndCo].
Finally, there has been a lot of interest in the role of ‘activated processes’ in glasses [@Kob]. In (spherical) FC models free energy barriers vanish above $T_{\rm d}$ while they are extensive (in the system size $N$) below. Activation does not play any role: it is not necessary above $T_{\rm d}$, and it is ineffective below. Schematic MCT does not include activation and is exact for such models, predicting a sharp transition. This picture can be modified in two ways: (A) Requiring barriers to stay finite below $T_{\rm d}$. In finite-dimensional models, this is a consequence of nucleation effects. Activation over such barriers induces a finite time scale for ergodicity restoration, and a smearing of the transition. (B) Introducing finite barriers above $T_{\rm d}$ while keeping extensive ones below. This is the scenario in *diluted* mean field models, such as the one treated below. While the transition remains sharp, the presence of new (activated) relaxation mechanisms may change the slowing down as $T\downarrow T_{\rm d}$, (e.g. the critical exponents).
In this paper we address the problem (B) above. Since in any realistic (finite-dimensional) model crossing over finite-energy barriers exists at any temperature, it is important to understand if this may modify some key predictions of MCT. In particular, we shall consider a regime in which the distinction between activated and non-activated processes has a precise mathematical sense, i.e. in the neighborhood of a $T=0$ bicritical point.
Our approach is to study a class of large scale rearrangements (LSR) which we expect to control the slow dynamics. Let us begin by providing a loose description of the main ideas on the example of a particle system. Consider a low-$T$ [^1] equilibrium configuration and focus on a particular molecule at position $x_i$. Now impose a displacement $\delta x_i$ (a few intermolecular distances) on this molecule and ask what is the [*minimum*]{} number $n_i$ of other molecules which must be moved for this displacement to be possible. The [*minimum*]{} energy barrier $b_i$ to be overcome is a second important property of the displacement $\delta x_i$. A glassy state is characterized by very large sizes $n_i$’s and barriers $b_i$’s (diverging at a sharp DPT) leading in turn to large relaxation times. A third quantity is defined by allowing all the molecules within a distance $\ell$ around $x_i$ to be moved. Let $\ell_i$ be the minimum $\ell$ such that the displacement $\delta x_i$ can be performed. It turns out that $n_i\ll \ell_i^{d}$ (at high enough dimension $d$). Remarkably, molecular dynamics simulations [@rearr] found string-like motions in glass-forming liquids.
Such notions can be completely precised in specific models. In this paper we focus on Ising models with $p$-spin ($p\ge 3$) interactions: $$%
H(\sigma)=\frac{1}{2}
\sum_{(i_1\dots i_p)\in\cG} (1-J_{i_1 \dots i_p} \sigma_{i_1}
\cdots \sigma_{i_p}) \, ,\label{Hamiltonian}
%$$ Here $\sigma_i=\pm 1$ are $N$ Ising spins, $\cG$ is a set of $M=N\gamma$ $p$-uples of indices, and $J_{i_1 \dots i_p}$ are quenched couplings taking values $\pm 1$ with equal probability. The above Hamiltonian counts the number of violated constraints $ \sigma_{i_1} \cdots \sigma_{i_p} =J_{i_1\dots i_p}$. This model is known in computer science as XORSAT [@XOR_CS].
A mean field version of this model is obtained by taking the $M$ interacting $p$-uples of sites $\{(i_1\dots i_p)\}$ to be quenched random variables uniformly distributed in the set of $N \choose p$ possible $p$-uples [@XOR]. In the limit $N\to\infty$, the number of interactions a given spin belongs to (its connectivity) is a Poisson random variable with parameter $p\gamma$. Moreover, the shortest loop through such a spin is (typically) of order $\log_{p\gamma} N$. The phase diagram is sketched in Fig. \[phasediag\]. Two regimes have attracted a particular attention. In the “fully-connected” limit $\gamma\to\infty$, $T\propto \sqrt{\gamma}$, both statics and dynamics can be treated analytically, showing a typical MCT transition. The relaxation time diverges as $|T-T_{\rm d}|^{-\nu}$, while a true thermodynamic transition takes place at $T_{\rm c}$.
![Schematic view of the phase diagram of the diluted $p$-spin model, with its dynamical and static transition line. See the text for details on the two type of divergences encountered at the dynamical transition. The scaling hypothesis of Eq. (\[eq:TemperatureScaling\]) is expected to hold in the shaded region.](sketch_Tgamma.eps){width="7.5cm"}
\[phasediag\]
In the $T\downarrow 0$, finite $\gamma$, limit, probabilistic methods can be used to show that zero-energy ground states with finite entropy density exist for $\gamma<\gamma_{\rm c}$. The set of ground states gets splitted in an exponential number of clusters with extensive Hamming distance (number of spins with different value) separating them for $\gamma_{\rm d}<\gamma<\gamma_{\rm c}$ [@XOR_12]. A finite fraction of the spins does not vary within a particular cluster, while they change when passing from a cluster to the other. At $\gamma_{\rm d}$ the fraction of “frozen” spins jumps from $0$ to a finite value $\phi$. For instance $\gamma_{\rm d}\approx 0.81847$, $\gamma_{\rm c}\approx 0.91793$, and $\phi\approx 0.71533$ when $p=3$. No exact result exists for the dynamics in this regime [@SeCuMo].
In the region $T<T_{\rm d}(\gamma)$, the Gibbs measure splits into an exponential number of pure states separated by barriers of order $N$. However this implies little (if anything) on the correlation time behavior as $T\downarrow T_{\rm d}(\gamma)$. In agreement with our general philosophy, we shall analyze the dynamics in terms of LSR’s as the transition line $T_{\rm d}(\gamma)$ is approached. The discussion below concerns any single-spin-flip Markov dynamics satisfying detailed balance.
Notice that a diverging length cannot be defined through a standard spin-glass correlation function, which remains short-ranged at $T_{\rm d}(\gamma)$. In order to overcome this problem, consider any temperature $T>T_{\rm d}(\gamma)$ and fix a reference thermalized configuration $\us^{(0)}$, a site $i$, and a length $\ell$. Denote by $\<\sigma_i\>_{\ell}$ the Boltzmann average of $\sigma_i$ under the boundary condition $\sigma_j=\sigma^{(0)}_j$ for any site $j$ at a distance larger than $\ell$ from $i$. Define $\ell_i(\ve)$ to be the smallest value of $\ell$ such that $\sigma^{(0)}_i\<\sigma_i\>\le \ve$ ($\ve$ being a small fixed number). A standard recursive calculation yields $\ell_i(\ve)\sim (T-T_{\rm d}(\gamma))^{-1/2}$ as $T\downarrow T_{\rm d}(\gamma)$. A coupling argument from probability theory can be used [@OurFuture; @Proba] to show that this implies a correlation time $\tau\gtrsim (T-T_{\rm d}(\gamma))^{-1/2}$.
The above argument displays clearly the relation between length and time scales. However the estimate for $\tau$ is not tight (the exponent is incorrect). As we shall see next, a highly refined picture can be obtained as $T\to 0$ in the “liquid” phase $\gamma<\gamma_{\rm d}$. In this regime, the system will spend most of its time in quasi-ground states. For the sake of the argument, assume that it is in fact in a ground state and focus on a particular spin $\sigma_i$. The leading mechanism for $\sigma_i$ to relax consists in a trajectory in phase space which brings the system to a new ground state with a reversed value of $\sigma_i$. Let $F_i$ be the set of reversed spins between these two ground states. This set must contain $i$ and, for each interaction $(i_1\dots i_p)\in \cG$, an even number of these $p$ sites. As will be clear from the following, we can restrict in fact to those $F_i$’s which are connected, and such that, for each interaction $(i_1\dots i_p)\in \cG$, either two or none of the sites belong to $F_i$. In the present context, we call $F_i$ a [*rearrangement*]{} for the spin $\sigma_i$.
{width="6.5cm"}
\[fig\_disjoint\]
Each $F_i$ can be assigned a barrier $b(F_i)$, defined as the minimum over all (single-spin-flip) paths in configuration space which flip the spins of $F_i$, of the maximum energy along the path. Assume that each spin in $F_i$ is flipped only once: paths are thus defined by an ordering of the flipped variables. A relaxation time for $\sigma_i$ can be defined by considering the correlation function $C_i(t) = \<\sigma_i(t)\sigma_i(0)\>$ and requiring $C_i(\tau_i) = \ve$ for some fixed $\ve$ (in the following $\ve = 1/2$). At low temperature, Arrhenius law implies $\tau_i \sim \exp[\beta b_i]$, with $b_i = \min_{F_i}b(F_i)$.
Computing $b_i$ requires optimizing both over the choice of the rearrangement $F_i$ (the set of spins to be flipped), and over the paths in configuration space (the flipping order). Consider this second task for a given rearrangement $F_i$. Form the graph with vertices representing the spins of $F_i$, and links between spins belonging to a common interaction in $\cG$. If $F_i$ stays finite in the thermodynamic limit, this graph is a tree rooted at $i$. If one draws this tree with vertices placed on an horizontal axis according to the order in which the spins are flipped, the energy of the system at a certain point of the trajectory is simply the number of links drawn above this point, cf. Fig. \[fig\_disjoint\]. This ordering problem is studied in graph theory as *minimal cutwidth* [@Yanna; @Lengauer].
A simple (and essentially optimal [@OurFuture]) strategy to construct recursively such an ordering is the following. Assume that the site $i=0$ has $l$ neighbors $\{1\dots l\}\equiv[1,l]$, each one being the root of a sub-tree. Then choose a sequential ordering of the sub-trees, i.e. a permutation $P$ of $[1,l]$, and an integer $j \in [0,l]$. Flip all the variables of the sub-tree $P_1$, then do the same on the tree $P_2$, and so on until $P_j$, then flip the $\sigma_0$, and finally flip the spins in the sub-trees $P_{j+1},\dots,P_{l}$. As “proved” in Fig. \[fig\_disjoint\], this construction implies a recursion on the energy barriers $$\begin{aligned}
%
b_0 & = & \min_{P,j} \max [\;\hat{b}_{P_{1}}, 1 +\hat{b}_{P_{2}}, \dots,
j-1 +\hat{b}_{P_{j}}, \label{eq_disjoint_0}
\\ &&\phantom{\min_{P,j} \max[\;} l-j-1 +r+\hat{b}_{P_{j+1}}, \dots,
r+\hat{b}_{P_{l}}\;]\, ,\nonumber
%\end{aligned}$$\
with $r=0$. The $\hat{b}_i$’s are “modified barriers” which obey a similar recursion: $\hat{b}_0$ is given by Eq. (\[eq\_disjoint\_0\]) with $r=1$.
The computation of the barrier $b_i$ for the $p$-spin interaction problem still involves an optimal choice of the rearrangement $F_i$. This step can also be performed recursively: starting from the root $i$, one chooses in each interaction around it the variable $a$ (among the $p-1$ distinct from $i$) which minimizes $\hat{b}_a$. Repeating this step one obtains an admissible rearrangement $F_i$ for $\sigma_i$, with a minimal value of $b(F_i)$. Remarkably, this scheme can be efficiently implemented on a given sample [@OurFuture].
If we consider the ensemble of random hypergraphs ${\cal G}$ described above, the barriers $b_i$ become themselves random variables, and Eq. (\[eq\_disjoint\_0\]) acquires a distributional meaning. The law $Q_b \equiv {\rm Prob} [b_i\ge b]$ can be computed numerically and is plotted in Fig. \[fig\_pih\] for a few values of $\gamma$ approaching $\gamma_{\rm d}$. Notice that $Q_b$ has an immediate physical interpretation in terms of the global correlation function $C(t) = \frac{1}{N}\sum_i \<\sigma_i(0)\sigma_i(t)\>$. At time $t$ only those sites with $b_i\gtrsim T\log t$ contribute to the correlation function. We thus have $\lim_{\beta\to\infty}C(e^{\beta (b-\delta)})= Q_b$ for any $0<\delta<1$.
{width="7.5cm"}
\[fig\_pih\]
The critical behavior of $Q_b$ can be solved analytically. As $\gamma$ approaches $\gamma_{\rm d}$ a plateau develops at height $\phi$: a fraction $\phi$ of the spins have “large” barriers (are freezing), while the other ones have “small” barriers. As the plateau is approached (left) one has $Q_b\simeq \phi + C_- e^{-\omega_ab}$ ($Q_b\simeq \phi - C_+ e^{\omega_bb}$) with $\omega_{a/b}$ the positive solutions of the equations $2e^{\omega_a} - e^{2\omega_a} = 2e^{-\omega_b} - e^{-2\omega_b} = \lambda$, and $\lambda$ a $p$-dependent parameter. Finally the scale of the large barriers diverges as $b_{\rm slow}\simeq - \Upsilon\log(\gamma_{\rm d}-\gamma)$, with $\Upsilon = 1/(2\omega_a) + 1/(2\omega_b)$. For instance, if $p=3$ we get $\omega_a\approx 0.57432$, $\omega_b \approx 1.49574$, and $\Upsilon \approx 1.20488$. The reader will notice the close parallel with the behavior of correlation functions in MCT, with some definite differences: here the divergence is logarithmic rather than power-law; the exponents are no longer fixed by a transcendental relation (see below), but rather through the above quadratic equations for $e^{\omega_{a/b}}$.
Arrhenius law yields a correlation time diverging as $\tau\simeq (\gamma_{\rm d} - \gamma)^{-\beta \Upsilon}$ if the $\gamma\to\gamma_{\rm d}$ limit is taken [*after*]{} $T\to 0$. Inspired by the crossover behavior in diluted ferromagnets [@Henley], we conjecture the following scaling form to hold if $\gamma\to\gamma_{\rm d}$ together with $T\to 0$: $$\begin{aligned}
%
\tau(\beta,\gamma)
\simeq e^{\frac{\Upsilon\beta^2}{2}}f(e^{\beta}(\gamma-\gamma_{\rm d}))\, .
\label{eq:TemperatureScaling}
%\end{aligned}$$ This summarizes the above findings, as well as the low-$T$ behavior of the dynamic transition line [@OurFuture]: $\gamma_{\rm d}(\beta) \simeq \gamma_{\rm d} + x_* e^{-\beta}$ with $x_*\approx 9$ for $p=3$. The scaling function behaves as $f(x)\simeq e^{-\Upsilon(\log|x|)^2/2}$ as $x\to -\infty$, and $f(x)\simeq f_+|x_*-x|^{-\nu(\infty)}$ as $x\to x_*>0$. In Fig. \[fig\_scaling\] we check the scaling hypothesis against numerical simulations. Remarkably, Eq. (\[eq:TemperatureScaling\]) implies a super-Arrhenius behavior at $\gamma_{\rm d}$: $\tau(\beta,\gamma_{\rm d}) \propto e^{\Upsilon\beta^2/2}$.
Several geometrical properties of optimal (minimum barrier) LSR’s can be determined analytically. Their [*size*]{} (number of sites) diverges as $n_{\rm bar}\sim (\gamma_{\rm d}-\gamma)^{-\nu_{\rm bar}}$, with $\nu_{\rm bar} \approx 2.0157$ for $p=3$. They are [*non-compact*]{}: the chemical distance between the root and a random site in an optimal LSR scales as $(\gamma_{\rm d}-\gamma)^{-\zeta}$, with universal exponent $\zeta=1/2$. Further, optimal rearrangements induce [*dynamical correlations*]{}, as can be understood from the following “experiment”. Consider a site $j\neq i$, and ask what is the minimum barrier $b^{(j)}_i$ to be overcome for flipping $\sigma_i$ [*without*]{} flipping $\sigma_j$. Call $Q^{(j)}_b = {\rm Prob}[b^{(j)}_i\ge b]$, and define the “susceptibility” $\chi_b = \sum_j[ Q^{(j)}_b-Q_b]$. As $T\to 0$, $\chi_b$ describes the change in $C_i(t)$ when a pinning field is applied on spin $\sigma_j$ (summed over $j$), and is a geometric analogous of the 4-point susceptibility introduced in [@Chi4]. We show in Fig. \[fig\_pih\] its critical behavior. The main feature is a peak located at $b_{\rm peak}\simeq -\Upsilon\log(\gamma_{\rm d}-\gamma)$ whose height $\chi_{\rm peak}\sim (\gamma_{\rm d}-\gamma)^{-\eta}$ (with universal exponent $\eta=1$) estimates the number of spins $\sigma_j$ which are influential to the dynamics of $\sigma_i$, belonging to all minimal barrier rearrangements for this variable.
![The scaling function of Eq. (\[eq:TemperatureScaling\]) (notice the large corrections to scaling for $\gamma>\gamma_{\rm d}$). Inset: super-Arrhenius behavior at $\gamma_d$. ](plot_scaling.eps){width="8.cm"}
\[fig\_scaling\]
How does our picture generalize to the $\gamma>\gamma_{\rm d}$, $T\downarrow T_{\rm d}(\gamma)$ regime? Summarizing, we presented two types of results: $(i)$ dynamics proceeds by LSR’s of size $n\sim (\gamma_{\rm d}-\gamma)^{-\nu}$; $(ii)$ the energy barriers to be overcome scales as $b\sim z_{\rm act}\log n$, with $z_{\rm act} = \Upsilon/\nu$. The equilibration time was estimated as $\tau\sim e^{\beta b}\sim n^{\beta z_{\rm act}}$. At finite temperature, the Arrhenius argument does not make sense any more, and one cannot understand slowing down in terms of activated processes. However, we still expect that LSR [^2] sizes diverge as $n\sim (T-T_{\rm d}(\gamma))^{-\nu(\gamma)}$, and that a dynamical scaling relation $\tau\sim n^{z}$ holds with an universal exponent $z$. A partial confirmation is provided by the probabilistic argument discussed in the previous pages implying $\tau\gtrsim (T-T_{\rm d}(\gamma))^{-1/2}$.
How is this related to the issue (B) raised in the introduction? The depth and cooperativity of LSR diverge with two universal exponents $\zeta=1/2$ and $\eta=1$. This agrees with MCT calculations [@BB] implying that such universal features of MCT are not modified by activated processes, *even* in the regime $e^{-\beta}\ll (\gamma_{\rm d}-\gamma)$. Other features (e.g. the relation $\Gamma(1-a)^{2}/\Gamma(1-2a)=\Gamma(1+b)^{2}/\Gamma(1+2b)$ between $\alpha$- and $\beta$-relaxation exponents) are indeed modified in a crossover region that can be experimentally relevant. However, the asymptotic $T\downarrow T_{\rm d}(\gamma)$ behavior is governed by usual MCT at any $\gamma>\gamma_{\rm d}$. The crossover between the two regimes is ruled by the ratio $(\gamma-\gamma_{\rm d})^{-1}/e^{\beta}$. In a more general context one should consider the ratio $\xi_{\rm dyn}/\xi_{\rm therm}$, where $\xi_{\rm dyn}$ is a dynamical length scale as measured through 4-points correlations [@BB], and $\xi_{\rm therm}$ is a thermal length (distance between energy defects).
The above ideas can be applied to particle systems. In the particular case of kinetically constrained models on Bethe lattices [@RigidityAndCo], we could show that the same scenario described above holds [@OurFuture]. A challenging direction would be to analyze ensembles of NP-hard decision problems (random $K$-SAT, or the $q$-coloring of random graphs) with a similar phase diagram [@SATColoring]. Finally, we obtained a purely geometrical description of diverging spatial structures at the DPT. This provides a particularly concrete setting for discussing finite-dimensionality effects.
We thank Leticia Cugliandolo for her interest in this work. G.S. has been partially supported by the EU under the EVERGROW project.
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[^2]: Of course we assume that finite-$T$ rearrangements can indeed be properly defined. See [@OurFuture] for a discussion.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
Based on the geometric-optics correspondence between the parameters of a quasinormal mode and the conserved quantities along geodesics, we propose an equation to calculate the typical shadow radius for asymptotically flat and rotating black holes when viewed from the equatorial plane given by $$\notag
\bar{R}_s=\frac{\sqrt{2}}{2}\left(\sqrt{\frac{ r_0^{+}}{f'(r)|_{r_0^{+}}}}+\sqrt{\frac{ r_0^{-}}{f'(r)|_{r_0^{-}}}}\right),$$ with $r_0^{\pm}$ being the radius of circular null geodesics for the corresponding mode. Furthermore we have explicitly related the shadow radius to the real part of QNMs in the eikonal regime corresponding to the prograde and retrograde mode, respectively. As a particular example, we have computed the typical black hole shadow radius for some well known black hole solutions including the Kerr black hole, Kerr-Newman black hole and higher dimensional black hole solutions described by the Myers-Perry black hole.
author:
- Kimet Jusufi
title: Connection Between the Shadow Radius and Quasinormal Modes in Rotating Spacetimes
---
Introduction
============
Einstein’s general theory of relativity revolutionized how we view time, space and gravity. This theory has important astrophysical implications, perhaps the most interesting prediction is the existence of BHs. The strongest evidence to date are the recent experimental announcements the detection of gravitational waves (GWs) [@AbbottBH] by the LIGO and VIRGO observatories and the captured image of the black hole shadow of a supermassive M87 black hole by the Event Horizon Telescope collaboration [Akiyama1,Akiyama4]{}. However there are many other astrophysical phenomena inferring their existence for example the high-energy phenomena such as X-ray emission and jets, and the motions of nearby objects in orbit around the hidden mass.
Gravity waves and BH shadows have opened a new window in our understanding of the Universe. Based on observations, in the near future, we can test many of alternative theories of gravity. In addition, using high-resolution images of BH shadows we can measure the angular diameter the BH or detect tiny effects which are out of reach of the present technology. Thus it remains an open question if future astronomical observations can potentially detect such important effects. Usually there are three stages when we study the evolution of binary black holes. The first stage is the so called inspiral stage and has an inspiral phase signal which encodes valuable information about the masses and the spins of compact objects treated by the post-Newtonian approximation [@Blanchet]. The second stage is the merger phase and describes the rapid collapse, say of two BHs to form a bigger BH [@Pretorius; @Campanelli; @Baker]. Finally there is the ringdown phase is the final stage and describes a perturbed BH that emits GWs in the form of quasinormal radiation [@BertiCardosoWill]. The perturbation theory of Schwarzschild black hole and its stability under small perturbations was studied in Refs. [@Regge; @Zerilli]. Since then the perturbations or also known as the quasinormal modes (QNMs) have been investigated by using other analytic and numerical methods [@1; @2; @3; @4; @5; @6; @7; @8; @9; @10; @11; @12; @13; @14; @15].
On the other hand, the shadow of the Schwarzschild BH was first studied by Synge [@Synge66] and Luminet [@Luminet79] then subsequently the Kerr BH was studied by Bardeen [@DeWitt73]. At first it seems that there is no direct connection between QNMs and shadow radius. However, it turns out that such a connection in fact exists. To understand this connection we fist note that Cardoso et al. [@cardoso] (but see also Refs. [@Hod:2017xkz; @Wei:2019jve]) where the real part of the QNMs is related to the angular velocity of the last circular null geodesic. Later on Stefanov et al. [@Stefanov:2010xz] found a connection between black-hole quasinormal modes in the eikonal limit and lensing in the strong deflection limit. In a recent work )see [@Jusufi:2019ltj]), we have argued that is more suitable to relate the real part of the QNMs with the shadow radius in spherically symmetric and static black holes. As we know the rotation is important in studying the shape of the back hole shadow. In fact, mostly of the BHs in the galactic center are expected to rotate, hence determining the angular momentum is of crucial importance. In this paper, we would like to extend the connection between the shadow radius and the real part of the QNMs for a rotating and asymptotically flat black holes.
This paper is organized as follows. In Section II, we review the shadow of rotating BHs obtained via Newman–Janis algorithm. In Sec. III we will present a method to compute the typical shadow radius based on the correspondence between the shadow radius and the real part of QNMs in the eikonal regime. In Secs. IV-VI, we apply this method to obtain the shadow radius for Kerr BH, KNBH and MP BH. In Sec. VII, we briefly discuss the Teukolsky equation. Finally in Sec. VIII, we comment on our results.
Shadow of rotating black holes
==============================
Consider the rotating black hole spacetime given by the metric [@Azreg-Ainou:2014pra] $$\begin{aligned}
\label{metric}
&ds^2=\left(1-\frac{2\Upsilon(r) r}{\Sigma}\right)dt^2 +2 a\sin^2\theta \frac{2\Upsilon(r) r}{\Sigma}dt d\phi\\
&-\frac{\Sigma}{\Delta}dr^2-\Sigma d\theta^2 - \frac{[(r^2+a^2)^2-a^2\Delta \sin^2\theta] \sin^2\theta}{\Sigma} d\phi^2,\nonumber\end{aligned}$$ where $$\Upsilon(r)=\frac{r (1-f(r))}{2},$$ along with $$\begin{aligned}
\Delta &= & r^{2}f(r)+a^2,\\
\Sigma &=& r^{2}+a^{2}\cos^{2}\theta.\end{aligned}$$
In order to find the contour of a black hole shadow we need to separate the null geodesic equations in the general rotating spacetime metric (1) using the Hamilton-Jacobi equation given by $$\frac{\partial \mathcal{S}}{\partial \sigma}=-\frac{1}{2}g^{\mu\nu}\frac{\partial \mathcal{S}}{\partial x^\mu}\frac{\partial \mathcal{S}}{\partial x^\nu},
\label{eq:HJE}$$ where $\sigma$ is the affine parameter, $\mathcal{S}$ is the Jacobi action. For this purpose we can express the action in terms of known constants of the motion as follows $$\mathcal{S}=-\frac{1}{2}\mu ^2 \sigma + E t - J \phi + \mathcal{S}_{r}(r)+\mathcal{S}_{\theta}(\theta),
\label{eq:action_ansatz}$$ where $\mu$ is the mass of the test particle, $E=p_t$ is the conserved energy and $J=-p_\phi$ is the conserved angular momentum. After we take $\mu=0$ one can obtain the following equations of motion $$\begin{aligned}
\notag
\Sigma\frac{dt}{d\lambda}&=&a(J-aE\sin^{2}\theta)
+\frac{r^{2}+a^{2}}{\Delta}\left[E\,(r^{2}+a^{2})-a\,J\right],\\\notag
\Sigma \frac{dr}{d\lambda}&=& \pm \sqrt{\Re},\\\notag
\Sigma \frac{d\theta}{d\lambda}&=&\pm \sqrt{\Theta},\\
\Sigma\frac{d\phi}{d\lambda}
&=&(J\csc^{2}\theta-a\,E)+\frac{a}{\Delta}\left[E(r^{2}+a^{2})-a\,J\right],\end{aligned}$$ with $\lambda$ the affine parameter, $J$ the angular momentum of the photon, $E$ the energy of the photon (we here use different denotations from the RN case), $\mathcal{K}$ the Carter constant and $$\begin{aligned}
{\Re}&=&\left(a^2 \,E-a\,J+E\, r^2\right)^2-\Delta
\left[\mathcal{K}+(J-a\, E)^2\right],\\
\Theta&=&\mathcal{K}-(J\csc\theta-a\,
E \sin\theta)^2+(J-a\,E)^2.\end{aligned}$$
The motion of the photon can be determined by these two impact parameters. To determine the geometric sharp of the shadow of the black hole, we need to find the critical circular orbit for the photon, which can be derived from the unstable condition $${\label}{condition}
\Re(r)=0,\;\; \frac{d\Re(r)}{dr} =0 ,\;\;\; \frac{d^2 \Re(r)}{dr^2} >0.$$ By using this condition the circular orbit radius $r_{p}$ of the photon can be obtained and the parameters $\xi\equiv J/E$ and $\eta\equiv\mathcal{K}/E^{2}$ can thus be expressed as where $X(r)=(r^2+a^2)$, and $\Delta(r)$ is defined by Eq. (3), while $\mathcal{K}$ is known as the Carter separation constant. From these conditions it follows (see [@100]) $$\xi=\frac{X_{ph}\Delta'_{ph}-2\Delta_{ph}X'_{ph}}{a\Delta'_{ph}},
\label{eq:xi}$$ $$\eta=\frac{4a^2X'^2_{ph}\Delta_{ph}-\left[\left(X_{ph}-a^2\right)\Delta'_{ph}-2X'_{ph}\Delta_{ph} \right]^2}{a^2\Delta'^2_{ph}}.
\label{eq:eta}$$
{width="7.4cm"}
One constraint for the value of the photon’s circular orbit radius is ${\Re(r_{p})>0}$. The shape of the shadow seen by an observer at spatial infinity can be obtained from the geodesics of the photons and described by the celestial coordinates $$\label{celkerrone}
\alpha=-\xi\csc\theta_{0},$$ $$\label{celkerrtwo}
\beta=\pm\sqrt{\eta+a^{2}\cos^{2}\theta_{0}-\xi^{2}\cot^{2}\theta_{0}},$$ where $\theta_{0}$ is the inclination angle of the observer. Working in the case with $\theta_0=\pi/2$, in the present paper, we shall use the definition adopted in Refs. [@Zhang:2019glo; @Feng:2019zzn] where the typical shadow radius is defined in terms of the leftmost and rightmost coordinates $$\bar{R}_S=\frac{1}{2}\left(\alpha^{+}-\alpha^{-}\right),$$ along with the condition $\beta(r=r_A)=\beta(r=r_B)=0$. The typical shadow radius is of major interest since it represents an observable quantity. On the other hand, there is no unique way to define this quantity as a result different definitions have been used (see [@000; @01; @00; @111; @22; @33; @44; @55; @66; @77; @88; @99; @100; @1111; @222; @333]).
{width="7.4cm"} {width="7.4cm"}
Connection between shadow radius and QNMs
=========================================
QNMs are characteristic modes which encode important information about the stability of the black hole under small perturbations. In order to study these characteristic modes one must impose an outgoing boundary condition at infinity and an ingoing boundary condition at the horizon. In general QNMs can be written in terms of the real part and the complex part representing the decaying modes $$\omega_{QNM}=\omega_{\Re}-i \omega_{\Im}.$$
In a seminal paper by Cardoso et al. [@cardoso] it was shown that the real part of the QNMs in the eikonal limit is in fact connected with the angular velocity of the last null geodesic. Furthermore the imaginary part was shown to be related with the Lyapunov exponent [@cardoso] $$\omega_{QNM}=\Omega_c l -i \left(n+\frac{1}{2}\right)|\lambda|.$$
Based on these equation Stefanov et al. [@Stefanov:2010xz] showed a link between the QNMs and the strong deflection limit. In particular they found that $$\Omega_c=\frac{1}{\theta D_{OL}},\,\,\,\lambda=\frac{\ln \tilde{r}}{2 \pi \theta D_{OL}},$$ in which $D_{OL}$ represents the distance between the observer and the lens, $\theta$ gives the angular position of the image that is closest to the BH and finally $\lambda$ is the so-called Lyapunov exponent and determines the instability time scale. In Ref. [@Jusufi:2019ltj] Jusufi showed a connection between the shadow radius and the real part of QNMs. It is straightforward to see this connection by adopting the definition $${\label}{thetas}
\theta=\frac{R_s}{D},$$ hence from (17) and (18) we can explicitly relate the real part of the QNMs with the shadow radius [@Jusufi:2019ltj] $$R_s = \lim_{l \gg 1} \frac{l}{\omega_{\Re}}.$$
We point out again that this relation is accurate only in the eikonal limit having large values of $l$. But one can still use this relation in some cases even for small $l$, for example to investigate the relationship between the QNMs and shadow radius upon different physical quantities (see, [@Liu:2020ola]) due to their inverse relation. Notice that an equivalent expression in terms of QNMs and angular velocity was written in Refs. [@Hod:2017xkz; @Wei:2019jve] but no explicit connection whatsoever between the shadow radius and QNMs was mentioned in those papers. From the physical point of view, it is therefore interesting to use Eq. (20) in studying various problems relating the gravity waves and black hole shadow hole. Equation (20) simply reflects the fact that for a distant observer the gravitational waves can be treated as scalar massless particles propagating along the last null unstable and slowly leaking out to infinity. We point out here that this correspondence is not guaranteed for gravitational fields, for example in the Einstein-Lovelock theory even in the eikonal limit this correspondence may be violated (see, [@Konoplya:2017wot]).
In what follows, we shall argue a connection between QNMs and shadow radius for the rotating black holes metric. Let us start by writing the rotating spacetime $$ds^2=g_{tt}dt^2+2g_{t\phi}dt d\phi+g_{rr}dr^2+g_{\theta \theta}d\theta^2+g_{\phi \phi}d\phi^2,$$ and let us restrict attention to orbits in the equatorial plane $\theta=\pi/2$ for which the appropriate Lagrangian is $$\mathcal{L}=\frac{1}{2}\left(g_{tt}\dot{t}^2+g_{rr} \dot{r}^2+2 g_{t \phi} \dot{t} \dot{\phi}+g_{\phi \phi}\dot{\phi}^2\right).$$
The generalized momenta following from this Lagrangian are $$\begin{aligned}
p_t&=&g_{tt}\dot{t}^2+g_{t \phi} \dot{\phi}=E\\
p_{\phi}&=&g_{t\phi}\dot{t}^2+g_{\phi \phi} \dot{\phi}=-J\\
p_r&=&g_{rr}\dot{r}\end{aligned}$$
From the above equations it is easy to obtain $$\dot{\phi}=\frac{g_{t \phi} E+g_{tt} J}{g_{t \phi}^2-g_{tt}g_{\phi \phi}},$$ and $$\dot{t}=-\frac{g_{\phi \phi} E+g_{t \phi} J}{g_{t \phi}^2-g_{tt}g_{\phi \phi}}.$$
Using the Hamiltonian $$\mathcal{H}=p_t \dot{t}+p_{\phi }\dot{\phi}+p_{r} \dot{r}-\mathcal{L},$$ and considering the null geodesics we obtain $$r^2 \mathcal{V}_r=r^2 E^2+a^2 E^2-J^2+(aE-J)^2(1-f(r)),$$ where we have used $\mathcal{V}_r=\dot{r}^2$. The conditions for the existence of circular geodesics can be written as $$\label{Vreq}
\mathcal{V}_r=\mathcal{V}'_r=0,$$ thus we obtain the following relations $$r^2 E^2+a^2 E^2-J^2+(aE-J)^2(1-f(r))=0,$$ and $$2r E^2-(aE-J)^2f'(r)=0.$$
Now the key point is to use the geometric-optics correspondence between the parameters of a quasinormal mode, and the conserved quantities along geodesics [@Yang:2012he]. In particular, the energy of the particle can be identified with the real part of QNMs, hence we can identify $$E \to \omega_{\Re},$$ and the azimuthal quantum number corresponds to angular momentum $$J \to m.$$
In the eikonal limit in the rotating spacetimes we have $$m=\pm l,$$ corresponding to the prograde and retrograde modes, respectively. With these identifications we can therefore write Eq. (20) as follows $$\omega_{\Re}^{\pm} = \lim_{l \gg 1} \frac{m}{R_S^{\pm}}.$$
Now let us introduce the following quantity $$R_s=\frac{J}{E},$$ combining this with Eq. (32) yields an equation $$2r-(a-R_s)^2\,f'(r)=0,$$ which has the solution $$R_s^{\pm}=a \pm \sqrt{\frac{2 r_0^{\pm}}{f'(r)|_{r_0^{\pm}}}}.$$ In the last equation we have evaluated $r$ at the point $r=r_0$, which gives the radius of circular null geodesics. With this result in hand, from Eq. (31) we obtain $$r_0^2-\frac{2 r_0}{f'(r)|_{r_0^{\pm}}}f(r_0)\mp 2 a \sqrt{\frac{2 r_0}{f'(r)|_{r_0^{\pm}}}}=0.$$
Our aim is to compute the shadow radius, however in general the shape of the shadow depends on the observer’s viewing angle $\theta_0$. In the case with $\theta_0=\pi/2$ we can adopt the definition (15) known as the typical shadow radius which can be written as $$\bar{R}_s=\frac{1}{2}\left(R^+_s|_{r_0^+}-R^-_s|_{r_0^-}\right)$$ where $r_0^{\pm}$ is determined from Eq. (40). This relation simply follows if we combine relation (13) along with the definitions (15) and (37) and choosing the inclination angle $\pi/2$. Now let us use the quantity given by Eq. (41) then, a simple algebra from the last equation, results with an equation for the typical shadow radius $$\bar{R}_s=\frac{\sqrt{2}}{2}\left(\sqrt{\frac{ r_0^{+}}{f'(r)|_{r_0^{+}}}}+\sqrt{\frac{ r_0^{-}}{f'(r)|_{r_0^{-}}}}\right).$$
To the best of our knowledge, this is new equation and has not been reported before. It is important to mention that the roots of the above equation $r_0^{\pm}$ must be chosen that both are outside of the horizon. Finally, sometimes it may be useful to evaluate QNMs numerically and to obtain the shadow radius in terms of the real part of QNMs. For this purpose we can express the typical shadow radius in terms of QNMs as follows $$\bar{R}_s=\lim_{l>>1} \frac{m}{2}\left(\frac{1}{\omega^{+}_{\Re}|_{r_0^+} }- \frac{1}{\omega^{-}_{\Re}|_{r_0^-}}\right),$$ provided $m=l$. Alternatively we can set $m=-l$ and chose an appropriate definition to obtain the same result. As a limiting case we can consider the static spacetime when $a=0$ implying $\omega^{+}_{\Re}=-\omega^{-}_{\Re}=\omega_{\Re}$, thus we obtain Eq. (20).
Kerr black hole
===============
In this section we are going to calculate the typical shadow radius for the Kerr black hole in the equatorial plane having $$\Delta=r^2f(r)+a^2=r^2-2Mr+a^2.$$ Note that $M$ is the mass of the black hole and $a$ is the rotation parameter defined by $a\equiv J/M$ with $J$ the angular momentum of the black hole. Using the relation (42) we find $$\bar{R}_s=\frac{1}{2}\left(r_0^+ \sqrt{\frac{ r_0^{+}}{ M}}+r_0^-\sqrt{\frac{ r_0^{-}}{M}}\right).$$ where $r_0^{\pm}$ corresponds to the prograde and retrograde mode and are determined by solving the equation $$3 r_0 M -r_0^2 \pm 2 a \sqrt{r_0 M}=0.$$
----- -------------------- -------------------- -------------- -- --
$a$ $\omega_{\Re}^{-}$ $\omega_{\Re}^{+}$ $\bar{R}_s$
0.0 -19.24500897 19.24500897 $3 \sqrt{3}$
0.1 -20.86329192 17.87819737 5.193256265
0.2 -22.81414428 16.70659753 5.184452475
0.3 -25.21936783 15.68985638 5.169375540
0.4 -28.27162422 14.79821770 5.147343015
0.5 -32.29695986 14.00922085 5.117211190
0.6 -37.90041288 13.30557023 5.077071505
0.7 -46.36584147 12.67371538 5.023553105
0.8 -61.07624270 12.10287853 4.949897520
0.9 -95.74679312 11.58437233 4.838370314
----- -------------------- -------------------- -------------- -- --
: The Kerr shadow radius against the angular momentum.
{width="8.4cm"}
In Table I we show the values for the typical shadow radius using Eq. (42) for different values of the angular momentum $a$. In addition we have evaluated the real part of QNM for $m=l=100$ using $$\omega_{\Re}^{\pm}=\lim_{l>>1}\frac{m}{a \pm \sqrt{\frac{2 r_0^{\pm}}{f'(r)|_{r_0^{\pm}}}}}.$$
The last equation is nothing but the one obtained in Ref. [@2] in the case of Kerr-Newman black hole. From Fig. 3, we observe that the shadow radius decreases with an increase of the angular momentum. Suppose a spinning black hole has an angular momentum $a=0.9$, this means a decrease of the shadow radius $\Delta \bar{R}_s=0.357782110$. We can estimate now the change in the angular radius $\Delta \theta_s = \Delta R_s M/D$. In the case of M87, for the supermassive black hole M87 mass we have $M = 6.5 \times 10^{9}$M~$\odot$~ and $D =16.8$ Mpc is the distance between the Earth and M87 center black hole. We find $\Delta \theta_s = 9.87098 \times 10^{-6} \Delta R_s(M/$M~$\odot$~)$(1kpc/ D) \mu as= 1.366416092 \mu as$. In the case of Sgr A$^{*}$ black hole we have $M = 4.3 \times 10^{6}$M~$\odot$~ and $D =8.3$ kpc yielding a change in the angular radius of the order $\Delta \theta_s =1.829655207 \mu as$. As a special case we can obtain the shadow radius for the Schwarzschild black hole. Letting $a \to 0$ one has $r_0=r_0^{+}=r_0^{-}=3M$, yielding $$\bar{R}_s=r_0\sqrt{\frac{r_0}{M}}=3 \sqrt{3}M.$$
Furthermore we can use (42) along with $r_0=2 f(r_0)/f'(r)|_{r_0}$, to rewrite the last equation as follows $$\bar{R}_s=\frac{r_0}{\sqrt{f(r_0)}}$$ which is exactly the same result reported for the static spacetime [@Jusufi:2019ltj].
Kerr-Newman black hole
======================
In the case of the Kerr-Newman BH according to Eq. (3) we can write $$\Delta=r^2-2Mr+Q^2+a^2,\,\,\,f(r)=1-\frac{2M}{r}+\frac{Q^2}{r^2}$$ Using Eq. (42) it is not difficult to obtain $$\bar{R}_s=\frac{1}{2}\left(\frac{{(r_0^{+})^{2}}}{\sqrt{Mr_0^+-Q^2}}+\frac{(r_0^{-})^2}{\sqrt{Mr_0^--Q^2}}\right).$$ in which the corresponding radius for the circular geodesics can be found after solving the following equation $$3 r_0 M-2Q^2-r_0^2 \pm 2 a \sqrt{Mr_0-Q^2}=0.$$
For given values of $M$, $Q$ and $a$ we can find the value of $r_0^{\pm}$ corresponding to the prograde and retrograde mode, respectively. In Table II we present our results for the values for the typical shadow radius against the charge for a given value of the angular momentum. We have also evaluated the the real part of QNM for $m=100$. From Fig. 4, we observe that the shadow radius decreases with an increase of the electric charge and this result is verified in Fig. 2 by means of the geodesic approach. But see also Ref. [@444]. Finally as a special case when $a=0$ we obtain the shadow radius for the RN black hole with $$r_0=r_0^+=r_0^-=\frac{1}{2}\left(3M+\sqrt{9 M^2-8 Q^2} \right)$$ and $$\bar{R}_s=\frac{{r_0^{2}}}{\sqrt{Mr_0-Q^2}}.$$
We point out that QNMs in Kerr-Newman spacetime have been studied in Ref. [@1; @2].
----- -------------------- -------------------- ------------- -- --
$Q$ $\omega_{\Re}^{-}$ $\omega_{\Re}^{+}$ $\bar{R}_s$
0.0 -22.86350943 16.72920866 5.175675395
0.1 -22.86350943 16.72920866 5.175675395
0.2 -23.01429916 16.79788649 5.149127295
0.3 -23.27508381 16.91527287 5.104128545
0.4 -23.66197213 17.08616391 5.039439605
0.5 -24.20220158 17.31822837 4.953059103
0.6 -24.94164511 17.62335128 4.841824146
0.7 -25.96181391 18.02021646 4.700566946
0.8 -27.42534331 18.53970579 4.520045660
0.9 -29.74270581 19.23795448 4.280113360
----- -------------------- -------------------- ------------- -- --
: The real parts of QNMs and the shadow radius for different values of the electric charge.
{width="8.4cm"}
Myers-Perry black hole
======================
It is of considerable importance to investigate and extend this method to higher dimensional black hole solutions. Rotating black hole solutions in $d$- dimensions are known as Myers-Perry black holes described by the following metric $$\begin{aligned}
ds^{2}&=&\left(\frac{\Delta-a^2\sin^2\theta}{\rho^{2}}\right)dt^{2}-\frac{\rho^{2}}{\Delta}dr^{2}-\rho^{2}d\theta^{2}\\\notag
&-& \frac{\left[(r^{2}+a^{2})^{2}-\Delta a^{2}\sin^{2}\theta\right]\sin^{2}\theta}{\rho^{2}}d\phi^{2} \\\notag
&+&\frac{2 a (r^2+a^2-\Delta)\sin^2\theta}{\rho^{2}}dtd\phi -r^2 \cos^2\theta d\Omega^2_{d-4},\label{metric}
\end{aligned}$$ where $d\Omega^2_{d-4}$ denotes the standard metric of the unit $d-4$ - sphere and $$\begin{aligned}
\Delta=r^{2}+a^2-\mu r^{5-d},\,\,\,\,\,
\rho^{2}=r^{2}+a^{2}\cos^{2}\theta,\end{aligned}$$
For simplicity, in the present paper we are considering the simplest case having only one angular momentum parameter $a$. Going through the same steps we can obtain the equation for the radius of circular null geodesics given by [@cardoso] $$\frac{d-1}{d-3}r_0^2 \pm 2 a \sqrt{\frac{2 r_0^{d-1}}{(d-3) \mu }}-\frac{2 r_0^{d-1}}{(d-3)\mu}=0.$$
The conditions for the existence of circular geodesics results with the following two equations $$r^2+a^2-R_s^2+\mu r^{3-d}(a-R_s)^2=0,$$ and $$2 r+\mu r^{3-d}(a-R_s)^2=0.$$ evaluated at $r_0$. Furthermore we are going to consider the case of $d=5$ since the equation simplify considerably. The radius of the circular null geodesics (57) reduces to $$r_0^{\pm}=\sqrt{2}\sqrt{\mu \pm a \sqrt{\mu}}.$$
Combining the last result from Eq. (58) we obtain $$R_s^{\pm}=a \pm \frac{2 (\mu \pm a\sqrt{\mu}) }{\sqrt{\mu}}$$
Finally for the typical radius we obtain $$\bar{R}_s=2 \sqrt{\mu}.$$
In other words, we found that the typical shadow radius in the equatorial plane remains unaffected by the rotational parameter $a$. In the case $d=5$ the mass parameter $\mu$ can be written as $\mu=8M/3\pi$. Working in units $\mu=1$, we find $\bar{R}_s=2$. This result is in perfect agreement with Ref. [@Amir:2017slq] where authors studied the black hole shadow using geodesic approach.
Teukolsky equation
==================
An intuitive geometric correspondence between high-frequency QNMs and angular frequencies in the Kerr spacetime has been studied in Ref. [@Yang:2012he]. One can consider the radial Teukolsky equation in Kerr spacetime using the separation of variables (see [@Yang:2012he] for details) $$u(t,r,\theta,\phi) = e^{-i \omega t }e^{i m \phi}u_r(r)u_{\theta}(\theta) \, .$$ Now at the relevant order in $l \gg 1$, the angular equation for $u_\theta(\theta)$ can be stated as follows [@Yang:2012he] $$\begin{aligned}
\frac{1}{\sin \theta}\frac{d}{d\theta}\left[\sin{\theta}
\frac{d u_{\theta}}{d \theta}\right]+
\left[a^2\omega^2\cos^2{\theta}-\frac{m^2}{\sin^2{\theta}}+A_{lm}\right]
u_{\theta}=0 \,,\end{aligned}$$ with $A_{lm}$ being the angular eigenvalue of this equation. The radial equation $u_r(r)$ reduces to [@Yang:2012he] $$\frac{d^2 u_r}{dr_*^2} +V^r u_r=0.$$ in which $$V^r=\frac{[\omega(r^2+a^2)- ma ]^2 -\Delta \left[A_{lm}(a \omega) +a^2\omega^2 -2 m a \omega\right] }{(r^2+a^2)^2}.$$
It was shown in Ref. [@Yang:2012he] that at the leading and next-to-leading order one can find $\omega_R$ by using the condition $$V^r(r_0,\omega_R)=\left.\frac{\partial V^r}{\partial r}\right|_{(r_0,\omega_R)}
=0 \, .$$ yielding $$\lim_{l>>1}\Omega_R^{\pm} =\frac{a(r_{p}-M)\,\mu} {(3M-r_{p})r_{p}^{2}-a^{2}(M+r_{p})}\,,$$ where $r_p$ is the radius of the photon orbits and $$\mu=\frac{m}{L},$$ with $L=l+1/2$ and $$\Omega_{R}=\frac{\omega_{\Re}}{L}.$$
Working in the eikonal limit we can therefore use $\mu=\pm1$ since $\lim_{l>>1}L=l$. making use of the inverse relation between the orbital frequency and shadow radius $\Omega_R=1/R_s$ to finally obtain $$\omega_{\Re}=\lim_{l>>1} \frac{m}{R_s}.$$ This is an alternative way of seeing the problem. One can use the last equation along with (15) and (68) to find the typical shadow radius. This is precisely what was done in Ref. [@Feng:2019zzn]. In the present work, however, we have explicitly related the shadow radius and the real part of QNMs using the geometric-optics correspondence and the conserved quantities along geodesics.\
Conclusion
==========
In this paper we have shown a connection between the shadow radius and the real part of QNMs in rotating BH spacetimes. This connection is obtained using the correspondence between the parameters of a quasinormal mode and the conserved quantities along geodesics. For the typical shadow radius we have obtained an equation given by Eq. (42) provided the observer’s viewing angle is $\theta_0=\pi/2$. Alternatively, we have argued that one can express this result in terms of the real part of QNMs given by Eq. (43). We have applied these equations to explore the typical black hole shadow radius for the the Kerr black hole, Kerr-Newman black hole and five dimensional Myers-Perry black hole. Our results are consistent with the ones obtained via standard methods solving the geodesic equations.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The present work tests the suitability of the narrow transitions $7s \ ^2S_{1/2} \rightarrow 6d \ ^2D_{3/2}$ and $7s \ ^2S_{1/2} \rightarrow 6d \ ^2D_{5/2}$ in Ra$^+$ for optical frequency standard studies. Our calculations of the lifetimes of the metastable $6d$ states using the relativistic coupled-cluster theory suggest that they are sufficiently long for Ra$^+$ to be considered as a potential candidate for an atomic clock. This is further corroborated by our studies of the hyperfine interactions, dipole and quadrupole polarizabilities and quadrupole moments of the appropriate states of this system.'
author:
- |
$^a$B. K. Sahoo , $^b$B. P. Das, $^b$R. K. Chaudhuri, $^c$D. Mukherjee, $^d$R. G. E. Timmermans and $^d$K. Jungmann\
[*$^a$Max Planck Institute for the Physics of Complex Systems, D-01187 Dresden, Germany*]{}\
[*$^b$Non-Accelerator Particle Physics Group, Indian Institute of Astrophysics, Bangalore-34, India*]{}\
[*$^c$Department of Physical Chemistry, IACS, Kolkata-700 032, India*]{}\
[*$^d$KVI, University of Groningen, NL-9747 AA Groningen, The Netherlands*]{}
date: 'Received date; Accepted date'
title: Investigations of Ra$^+$ properties to test possibilities of new optical frequency standards
---
1.0cm
Accurate time and frequency measurement is crucial for the advance of science and technology in many fields. This leads to a number of searches to find candidates for optical frequency standards. The current frequency standard is based on the ground state hyperfine transition in atomic cesium and has a quality factor (Q) of $10^{15}$ [@essen]. Atomic spectral lines with high Q are generally interesting for standards, however good control over systematic line shifts will be essential. As a result of the remarkable advances in the field of ion trapping and laser cooling, single ions like Hg$^+$ [@oskay], In$^+$ [@becker], Ca$^+$ [@champenois], Sr$^+$ [@barwood], Yb$^+$ [@blythe], Cd$^+$ [@tanaka] and Ba$^+$ [@sherman] are particularly interesting as they can be localized using their electric charge rather than light forces, which is necessary for atom trapping. Very accurate measurements have been performed on Hg$^+$ and Sr$^+$, where Q exceeds $10^{17}$. Some of the major systematic errors associated with the clock frequency are the Stark effect, Zeeman effect and quadrupole shifts due to stray electric fields in the ion trap [@itano]. These errors can be estimated from high precision theoretical studies of hyperfine structure constants, polarizabilities and quadrupole moments of the appropriate atomic states. Indeed, studies of these quantities are also essential for parity non-conservation (PNC) studies [@fortson; @bijaya0]. Some of the above mentioned errors can be eliminated by considering the clock transition between suitable hyperfine states [@oskay].
![(color online) Schematic diagram of energy levels of Ra$^+$ with transitions for possible optical frequency standards.[]{data-label="fig1"}](ra+fig.eps){width="6.5cm"}
An experiment is in progress at KVI to search for an suitable optical frequency standard by measuring the frequency of either $7s \ ^2S_{1/2} \rightarrow 6d \ ^2D_{3/2}$ or $7s \ ^2S_{1/2} \rightarrow 6d \ ^2D_{5/2}$ transitions in Ra$^+$. A similar experiment is also being planned at IACS [@manas]. In this paper, we report our theoretical studies on the feasibility of these transitions for the optical frequency studies in Ra$^+$. In the case of Ba$^+$ it has been pointed out that PNC and optical frequency standards experiments share many features in common [@sherman]. The techniques used in the Ba$^+$ experiments can be extended to Ra$^+$ as the electronic structures of the two ions are similar. However, Ra$^+$ has one important advantage: the low lying transition wavelengths (see Fig. \[fig1\]([*a*]{})) of this ion are in the optical regime making them more easily accessible than their counterparts in Ba$^+$. Although, it appears that these wavelengths can be measured very precisely using modern spectroscopic techniques, it is however necessary to find out which transition is the most suitable for optical frequency standard. This can be decided by the experimentalists from the knowledge of different physical quantities of the involved states that can be used to control the sources of error.
First of all, one must determine which of the isotopes of Ra$^+$ merit consideration for optical clock studies. In this context, it is worthwhile to note that only $^{223}$Ra and $^{225}$Ra have half-lives of a few days ($\sim$ 10days) and these isotopes are therefore obvious choices. However, they have different nuclear spins ($I$s); the former has $I=3/2$ whereas the latter has $I=1/2$ and this results in different hyperfine splittings. One has to take into account the various systematic errors while considering both these isotopes. It is possible to eliminate the quadrupole Stark shift by considering the transition between the hyperfine ($F$) states such as $|6s(J=1/2),I=3/2;F=2\rangle \rightarrow |5d(J=3/2),I=3/2;F=0\rangle$ transition (see Fig. \[fig1\]([*b*]{})) for the frequency standard although knowledge of the hyperfine structure constants and the polarizabilities are still required for these experiments. It is necessary to study the hyperfine structure constants, lifetimes and a few other spectroscopic quantities for the $7s$ and $6d$ states of this system in order to assess the suitability of the proposed clock transitions.
Electron correlation and relativistic effects must be treated accurately for Ra$^+$. Relativistic coupled-cluster (RCC) theory; a size-consistent, size-extensive and an all order perturbation method is well suited for this purpose [@szabo]. It has been successfully applied to determine accurately certain ground and excited states properties of Sr$^+$ [@bijaya1] and Ba$^+$ [@bijaya2]. We employ the same method in the present study to obtain accurate results for Ra$^+$. The presence of the non-linear terms in this method makes it challenging to obtain the ground and excited state wave functions for a large system like Ra$^+$. We had observed earlier that these effects are important for accurate studies [@bijaya3] in other heavy systems. In order to obtain the wave functions for Ra$^+$, we solve the RCC equations considering single, double and leading triple excitations (CCSD(T) method). This involves the determination of $10^7$ cluster amplitudes self-consistently. This is one of the largest computations to date for obtaining the wave functions of an atomic system.
The starting point of our work is the relativistic generalization of the valence universal coupled-cluster (CC) theory introduced by Mukherjee et al. [@mukh] which was put later in a more compact form by Lindgren [@lind]. In this approach, the atomic wave function $|\Psi_v \rangle$ for a single valence ($v$) open-shell system is expressed as $$\begin{aligned}
|\Psi_v \rangle = e^T \{1+S_v\} |\Phi_v \rangle ,
\label{eqn1}\end{aligned}$$ where $|\Phi_v \rangle$ is the reference state constructed out of the Dirac-Fock (DF) orbitals of the closed-shell system ($|\Phi_0 \rangle$) by appending the valence electron orbital. Here $T$ and $S_v$ are the excitation operators from the core and valence-core sectors (for example see [@bijaya1; @bijaya3] for the second quantization representations of these operators and equations to obtain their amplitudes). The single particle orbitals in the present calculations are linear combinations of Gaussian type functions [@rajat].
The transition matrix element of a hermitian operator ($O$) corresponding to the initial state $| \Psi_i \rangle$ and the final state $| \Psi_f \rangle$ can be expressed using the RCC method as $$\begin{aligned}
\langle O \rangle_{if} &=& \frac {\langle \Psi_i | O | \Psi_f \rangle} {\sqrt{ \langle \Psi_i | \Psi_i \rangle \langle \Psi_f|\Psi_f \rangle }} \nonumber \\
&=& \frac {\langle \Phi_i |\{1+S_i^{\dagger}\} \overline{O} \{1 +S_f\} | \Phi_f \rangle } {\sqrt{ N_i N_f}} , \ \ \ \ \
\label{eqn2}\end{aligned}$$ where we define $\overline{O}=e^{T^{\dagger}} O e^T$ and $N_v = \langle \Phi_v |e^{T^{\dagger}} e^T + S_v^{\dagger} e^{T^{\dagger}} e^T S_v | \Phi_v \rangle$ for the valence electron $v$. We calculate the above expression using the procedure followed in the earlier works [@bijaya1; @bijaya2; @bijaya3]. The expectation values are determined by considering the special condition $i=f$.
[*Lifetimes of the $6d$ states*]{}: It is necessary to know the lifetimes of the $6d$ metastable states to understand how reliably the proposed experiments can be performed in that time period. The lifetimes (in second (s)) of these states can be determined from the inverse of the total transition probabilities ($A$). The net transition probabilities (in s$^{-1}$) of the $6d$ states are given by $$\begin{aligned}
A_{6 d5/2} &=& A^{\text{E2}}_{6 d5/2 \rightarrow 7 s1/2} + A^{\text{E2}}_{6 d5/2 \rightarrow 6
d3/2} + A^{\text{M1}}_{6 d5/2 \rightarrow 6 d3/2}, \nonumber \\
A_{6 d3/2} &=& A^{\text{E2}}_{6 d3/2 \rightarrow 7 s1/2} + A^{\text{M1}}_{6 d3/2 \rightarrow
7 s1/2} ,
\label{eqn3}\end{aligned}$$ where $$\begin{aligned}
A^{\text{E2}}_{f \rightarrow i} &=& \frac {1.11995 \times 10^{18} }{(2j_f+1) \lambda^5} S^{\text{E2}}_{f \rightarrow i} \\
A^{\text{M1}}_{f \rightarrow i} &=& \frac {2.69735 \times 10^{13} }{(2j_f+1) \lambda^3} S^{\text{M1}}_{f \rightarrow i},
\label{eqn3}\end{aligned}$$ where $S_{f \rightarrow i} = |O_{fi}|^2$ and $\lambda$ (in Å) are the transition line strength for the operator $O$ (in atomic unit (au)) and wavelength, respectively. These quantities depend on both the transition amplitudes and wavelengths, and they can be calculated using a single [*ab initio*]{} method. However, we use experimental wavelengths [@moore] to reduce the errors in the determination of the lifetimes.
-------------------------------------------------- -------------- ----------------------------------- ----------- ---------- --
Transition $O^{\text{M1}}_{f \rightarrow i}$ Lifetime
states (au) (s)
${f \rightarrow i}$ [*Length*]{} [*Velocity*]{}
$|6d_{3/2}\rangle \rightarrow |7s_{1/2}\rangle$ 14.87(7) 14.77(22) 0.0024(2) 0.893(4)
$|6d_{5/2}\rangle \rightarrow |7s_{1/2}\rangle$ 19.04(5) 19.87(1.0) 0.301(3)
$+|6d_{5/2}\rangle \rightarrow |6d_{3/2}\rangle$ 8.80(4) 10.5(2.5) 1.546(1) 0.297(4)
-------------------------------------------------- -------------- ----------------------------------- ----------- ---------- --
: Transition amplitudes (in au) due to M1 and E2 transitions in both length and velocity gauges. Length gauge results of the E2 amplitudes along with M1 amplitudes are considered for the determination of lifetimes.[]{data-label="tab1"}
Since there are no experimental or theoretical predictions of the lifetimes of the $6d$ states, we calculate the E2 transition amplitudes using both the length and velocity gauges in order to assess the numerical accuracies of the results. These results are given in Table \[tab1\] along with the M1 transition amplitudes and the lifetimes of the $6d$ states. We have used the E2 amplitudes in the length gauge as it converges faster than the corresponding values in the velocity gauge. The errors are estimated from the discrepancies of the results obtained with different choices of bases.
Using the RCC method, we find that due to the enhanced role of electron correlation, core polarization effects in particular, the M1 transition amplitude for the $|6d_{3/2}\rangle \rightarrow |7s_{1/2}\rangle$ transition is 0.0024(2)$ea_0$ where the DF value is $\sim 10^{-5}ea_0$. From the calculated E2 amplitudes of this transition, we obtain the lifetime of the $6d_{3/2}$ state as $0.627(4)$s. Inclusion of the above M1 transition probability changes its value to $0.893(4)$s; this change is around 30% of the total result and this finding is different from our earlier studies on similar states of other alkaline earth metal ions [@bijaya4]. However, like the other $d_{5/2}$ states in those systems, the lifetime of the $6d_{5/2}$ state reduces from 0.301s to 0.297s after including the contribution of the M1 transition probability in the $|6d_{5/2}\rangle \rightarrow |6d_{3/2}\rangle$ transition.
[*Quadrupole moments of the $6d$ states:*]{} In order to estimate the error in the frequency of the clock transition arising from quadratic Stark shifts, it is necessary to know the quadrupole moments of the relevant states. The quadrupole moment of a valence state ($v$) is given by $$\begin{aligned}
\Theta(v) = \langle \Psi_v|O^{\text{E2}}|\Psi_v \rangle,
\label{eqn4}\end{aligned}$$ where $O^{\text{E2}}$ is the E2 transition operator. We divide the the above expression into three parts as follows $$\begin{aligned}
\Theta(v) = \Theta_{DF}(v)+ \Theta_{cv}(v) + \Theta_v(v) .
\label{eqn5}\end{aligned}$$ Here $\Theta_{DF}$, $\Theta_{cv}$ and $\Theta_v$ are the DF, core-valence and valence electron correlation effects. In Table \[tab2\], we present these contributions for the $6d_{3/2}$ and $6d_{5/2}$ states. In this table, the difference between the total RCC result and the sum of all the above three contributions is due to the normalization of the wave functions. The quadrupole moment of the $7s$ state is clearly zero as the quadrupole moment operator is of rank two. Therefore, we determine these quantities only for the $6d$ states.
State $\Theta_{DF}$ $\Theta_{cv}$ $\Theta_{v}$ $\Theta$
------------ --------------- --------------- -------------- ----------
$6d_{3/2}$ 3.48 $-$0.01 $-$0.51 2.90(2)
$6d_{5/2}$ 5.19 $-$0.02 $-$0.65 4.45(9)
: Quadrupole moments of atomic states in au.[]{data-label="tab2"}
As given in Table \[tab2\], the dominant contribution comes from $\Theta_{DF}$ followed by $\Theta_v$, which contains core-polarization and pair-correlation effects to all orders, make significant contributions as in Sr$^+$ [@bijaya1] and Ba$^+$ [@bijaya2]. We have followed the same procedure as in the lifetime calculations to estimate errors in these results.
[*Polarizabilities:*]{} We determine the dipole polarizabilities for $7s$ and $6d$ states and quadrupole polarizability of the $7s$ state for our study. The static ($\alpha_0^1(J_v)$) and tensor dipole ($\alpha_2^1(J_v)$) polarizabilities for the valence $v$ state with angular momentum $J_v$ are given by $$\begin{aligned}
\alpha_0^1(v) &=& - 4 \sum_{k \ne v} \frac {|\langle J_v|D|J_k \rangle|^2}{E_v - E_k}
\label{eqn6} \\
\text{and} && \nonumber \\
\alpha_2^1(v) &=& 4\sqrt{ \frac {30 j_v (2j_v -1 )(2j_v + 1)}{(j_v+1)(2j_v+3)}} \sum_{k \ne v} (-1)^{J_v+J_k+1} \nonumber \\
&& \left \{ \matrix { J_v & 1 & J_k \cr 1 & J_v & 2 \cr } \right \} \frac {|\langle J_v|D|J_k \rangle|^2}{E_v - E_k},
\label{eqn7}\end{aligned}$$ respectively, where $D$ is the E1 operator. Similarly, the static quadrupole polarizability ($\alpha_0^2(v)$) is given by $$\begin{aligned}
\alpha_0^2(v) &=& - 4 \sum_{k \ne v} \frac {|\langle J_v| O^{\text{E2}} |J_k \rangle|^2}{E_v - E_k} .
\label{eqn8}\end{aligned}$$ We have used the sum-over-states approach and experimental energies to reduce the errors in the calculations; the calculated energies were used obtained from the RCC method where the experimental energies were not available.
We express generally the polarizabilities as $$\begin{aligned}
\alpha(v) &=& \alpha_{DF}(v)+ \alpha_c(v) + \alpha_{cv}(v) + \alpha_v(v) ,
\label{eqn9}\end{aligned}$$ where each term is defined similar to the corresponding terms of the quadrupole moment expression given in Eq. (\[eqn5\]) except for $\alpha_c$ which is the pure core orbital contribution. We calculate $\alpha_v$ contributions from the calculated intermediate states using the RCC method. However, $\alpha_c$ and $\alpha_{cv}$ are calculated using the second order many-body perturbation theory (MBPT(2)), where the residual Coulomb interaction and E1/E2 operators are treated as perturbation. All these results are tabulated in Table \[tab3\].
State $\alpha_{DF}$ $\alpha_c$ $\alpha_{cv}$ $\alpha_v$ $\alpha_{t}$ $\alpha$s
-------------- ------------ --------------- ------------ --------------- ------------ -------------- -----------
$\alpha_0^1$ $7s_{1/2}$ 99.39 14.57 $-$1.54 $-$4.34 $-$0.02 107.86
$6d_{3/2}$ 926.32 14.57 $-$1.31 39.05 0.03 978.66
$6d_{5/2}$ 1208.13 14.57 $-$7.53 77.18 0.03 1292.38
$\alpha_2^1$ $6d_{3/2}$ $-$220.67 1.21 0.76 65.38 $-$0.01 $-$153.12
$6d_{5/2}$ $-$168.53 1.21 1.27 16.78 $-$0.01 $-$149.28
$\alpha_0^2$ $7s_{1/2}$ 2920.83 56.57 0.34 $-$436.04 5.50 2547.20
: Dipole and quadrupole polarizabilities in au.[]{data-label="tab3"}
We have obtained up to $10s$, $10p$, $10d$, $9f$ and $9g$ low-lying states using the RCC method to calculate the above quantities. Contributions from other higher states are accounted for using MBPT(2). They are just given as tail contributions ($\alpha_t$) in the table. Using the expression $$\begin{aligned}
\alpha_{0,6d}^2(7s) &=& - \sum_{k=6d_{3/2},6d_{5/2}} \frac {|\langle \Psi_{7s}|O^{\text{E2}}|\Psi_k\rangle|^2}{E_{6s} - E_k} ,
\label{eqn10}\end{aligned}$$ we obtain $\alpha_2^0(7s)=1037(7)a_0^5$ along with the corresponding $\alpha_c$ contribution. This is usually necessary for the lifetime measurements of the $6d$ states.
$7s_{1/2}$ $6d_{3/2}$ $6d_{5/2}$
------------ ------------ ------------ ------------
$7p_{1/2}$ 3.28 3.64
$8p_{1/2}$ 0.04 0.07
$7p_{3/2}$ 4.54 1.54 4.92
$8p_{3/2}$ 0.50 0.15 0.40
$5f_{5/2}$ 4.47 1.31
$6f_{5/2}$ 0.86 0.21
$5f_{7/2}$ 6.21
$6f_{7/2}$ 1.08
: Important reduced E1 matrix elements in au used to determine the dipole polarizabilities.[]{data-label="tab3d"}
In Table \[tab3d\], we present the important reduced E1 matrix elements which are used in the determination of dipole polarizabilities. These results are in reasonable agreement with those of Dzuba et al. which are calculated using another many-body approach [@dzuba].
[*Hyperfine structure constants:*]{} Studies of these constants are important to investigate the underlying physics of the wave functions in the nuclear region, especially to estimate the errors of the PNC matrix elements [@bijaya6]. The magnetic dipole ($A_h$) and electric quadrupole ($B_h$) hyperfine structure constants of the valence $v$ state with angular momentum $J_v$ are given by $$\begin{aligned}
&& A_h(v) = \frac{\mu_N g_I}{J_v} \langle \Psi_v |\text{T}^{(1)}| \Psi_v\rangle
\label{eqn11} \\
\text{and} && \nonumber \\
&& B_h(v) = 2e Q_N \langle \Psi_v| \text{T}^{(2)}|\Psi_v\rangle,
\label{eqn12}\end{aligned}$$ respectively. In the above expressions, $\mu_N$, $g_I$ and $Q_N$ are the nuclear magnetic moment, gyromagnetic ratio and quadrupole moment, respectively. Explicit expressions and the single particle matrix elements of $\text{T}^{(1)}$ and $\text{T}^{(2)}$ are given in [@cheng]. We have used $g_I=0.18067$ [@arnold] and $Q_N=1.254$ [@neu] for $^{223}$Ra and $g_I=-1.4676$ [@arnold] for $^{225}$Ra in these calculations.
---------------- ------- -------------- ----------- -------- ---------- --------
$7s_{1/2}$
$A_h$ $A_h$ $B_h$ $A_h$ $B_h$
$^{223}$Ra$^+$
RCC 3567.26 77.08 383.88 $-$23.90 477.09
Expt. 3404.0(1.9)
$^{225}$Ra$^+$
RCC $-$28977.76 $-$626.13 194.15
Expt. $-$27731(13)
---------------- ------- -------------- ----------- -------- ---------- --------
: Hyperfine structure constants in MHz.[]{data-label="tab4"}
The trends of the correlation effects in the hyperfine interactions of the $7s$ and $6d$ states in the present system are similar to the corresponding states in Ba$^+$ [@bijaya2; @bijaya6]. We have found 23%, 31% and 181% correlation contributions with respect to the DF results of $A_h$ in the $7s$, $6d_{3/2}$ and $6d_{5/2}$ states, respectively. The core-polarization (CP) effect in the $6d_{5/2}$ state is very strong and its contribution is larger than the DF result. This gives rise to the unusual behavior of the electron correlation effects.
[*Conclusion:*]{} We have successfully carried out accurate calculations of the lifetimes, polarizabilities, quadrupole moments and hyperfine structure constants in Ra$^+$ using the RCC theory. Our calculated values of the lifetimes of the $6d$ states which are 0.893s and 0.297s, respectively, suggest that Ra$^+$ could be a suitable candidate for an optical frequency standard with an accuracy better than $10^{-18}$. The results of the different properties that we have calculated can serve as benchmarks to guide experimentalists. On the other hand, precise measurements of these quantities can also be used to test our method of calculation.
[*Acknowledgment:*]{} We thank Dr. Manas Mukherjee for fruitful discussions. These computations were performed on C-DAC’s ParamPadma.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We consider the Navier-Stokes equation in a domain with irregular boundaries. The irregularity is modeled by a spatially homogeneous random process, with typical size ${{\varepsilon}}\ll 1$. In the parent paper [@Basson:2007], we derived a homogenized boundary condition of Navier type as ${{\varepsilon}}\rightarrow 0$. We show here that for a large class of boundaries, this Navier condition provides a $O({{\varepsilon}}^{3/2} |\ln {{\varepsilon}}|^{1/2})$ approximation in $L^2$, instead of $O({{\varepsilon}}^{3/2})$ for periodic irregularities. Our result relies on the study of an auxiliary boundary layer system. Decay properties of this boundary layer are deduced from a central limit theorem for dependent variables.'
author:
- 'David Gérard-Varet [^1]'
title: The Navier wall law at a boundary with random roughness
---
[[*Keywords:*]{} Wall laws, rough boundaries, stochastic homogenization, decay of correlations]{}
Introduction
============
The concern of this paper is the effect of a rough boundary on a viscous fluid. In most situations of physical relevance, such effect can not be described in detail: either the precise shape of the roughness is unknown, or its spatial variations are too small for computational grids. Therefore, one may only hope to account for the averaged effect of the irregularities. This is the purpose of [*wall laws*]{}: the irregular boundary is replaced by an artificial smoothed one, and an artificial boundary condition (a wall law) is prescribed there, that should reflect the mean impact of the roughness.
This paper is a mathematical study of wall laws, in the following simple setting: we consider a two-dimensional rough channel $$\Omega^{{\varepsilon}}\: = \: \Omega \cup \Sigma \cup R^{{\varepsilon}}$$ where $\Omega = {{\mathbb R}}\times (0,1)$ is the [*smooth part*]{}, $R^{{\varepsilon}}$ is the rough part, and $\Sigma = {{\mathbb R}}\times \{0\}$ their interface. We assume that the rough part has typical size ${{\varepsilon}}$, that is $$R^{{\varepsilon}}\: = \: \left\{ x, \: x_2 >
{{\varepsilon}}\omega\left(\frac{x_1}{{{\varepsilon}}}\right) \right\}$$ for a $K-$Lipschitz function $\omega : {{\mathbb R}}\mapsto (-1,0)$, $K > 0$. More will be assumed on the boundary function $\omega$ hereafter (see figure for an example of such a rough domain).
We assume that in this channel domain, the viscous fluid obeys to the stationary incompressible Navier-Stokes equations: $$\label{NSeps}
\left\{
\begin{aligned}
& - \Delta u + u \cdot {{\nabla}}u + {{\nabla}}p = 0, \: x \in \Omega^{{\varepsilon}}, \\
& \div u = 0, \: x \in \Omega^{{\varepsilon}}, \\
& \int_{\sigma^{{\varepsilon}}} u_1 = \phi, \\
& u\vert_{{{\partial}}\Omega} = 0,
\end{aligned}
\right.$$ where $\sigma^{{\varepsilon}}$ denotes any vertical cross-section of $\Omega^{{\varepsilon}}$ and $\phi > 0$. The third equation in expresses that a flux $\phi$ is imposed across the channel. Note that this flux does not depend on the cross-section, due to the incompressibility and no-slip condition at the boundary. We also stress that, up to minor changes, we could apply our analysis to many variants of this problem, notably to elliptic type systems or to unstationary Navier-Stokes.
In this simple setting, the search for wall laws resumes to the following problem: to find a boundary operator $B^{{\varepsilon}}(x,D_x)$, regular in ${{\varepsilon}}$, acting at the [*artificial*]{} boundary $\Sigma$, such that the solution of $$\label{NS}
\left\{
\begin{aligned}
& - \Delta u + u \cdot {{\nabla}}u + {{\nabla}}p = 0, \: x \in \Omega, \\
& \div u = 0, \: x \in \Omega, \\
& \int_{\sigma} u_1 = \phi, \quad u\vert_{x_2=1} = 0, \\
& B^{{\varepsilon}}(x,D_x)u\vert_{\Sigma} = 0
\end{aligned}
\right.$$ approximates well the solution $u^{{\varepsilon}}$ of in $\Omega$.
This type of homogenization problems has been considered in many mathematical works. On wall laws for scalar elliptic equations, we refer to [@Achdou:1995]. On wall laws for fluid flows, see [@Achdou:1998a; @Achdou:1998; @Amirat:2001a; @Jager:2001; @Jager:2003; @Bresch:2006]. See also [@Jager:2000] on porous boundaries. These works go along with more formal computations, grounded by empirical arguments ([*cf*]{} for instance [@Bechert:1989; @Luchini:1995]). We finally mention [@Gerard-Varet:2003b; @Bresch:2005] for the study of roughness-induced effects on geophysical systems.
All these studies have been carried under two assumptions:
- compact domains, for instance bounded channels or periodic in the variable $x_1$.
- [*periodic irregularities*]{}, meaning that the boundary function $\omega$ is periodic.
The first restriction is just a small mathematical convenience, that gives direct compactness properties through Rellich type theorems. The second assumption is of course a big simplification, both from the point of view of mathematics and physics. These assumptions were considerably relaxed in the recent article [@Basson:2007] by A. Basson and the author. As the present note extends this article, we now describe shortly its main results and underlying difficulties.
In all papers on wall laws, the starting point is a formal expansion of $u^{{\varepsilon}}$: $$u^{{\varepsilon}}(x) \: \sim \: u^0(x) + 6 \phi {{\varepsilon}}v(x/{{\varepsilon}}) + \dots$$ Formally, the leading term $u^0$ satisfies with the simple no-slip condition $$\label{Dirichlet}
B^{{\varepsilon}}(x,Dx) u \: := \: u = 0 \:\mbox{ at }\: \Sigma$$ The solution of this approximate system is the famous Poiseuille flow : $$u^0(x) \: = \: \left(U(x_2),0\right), \quad U(x_2) \: = \: 6 \phi x_2
(1-x_2)$$ Note that $u^0$ is defined in all ${{\mathbb R}}^2$. This zeroth order asymptotics can be mathematically justified, at least for small fluxes $\phi$: we prove in article [@Basson:2007]
\[Direstimates\] There exists $\phi_0, \: {{\varepsilon}}_0 >0$, such that for all $\phi < \phi_0, \:
{{\varepsilon}}< {{\varepsilon}}_0$, system has a unique solution $u^{{\varepsilon}}$ in $H^1_{uloc}(\Omega^{{\varepsilon}})$. Moreover, $$\| u^{{\varepsilon}}- u^0 \|_{H^1_{uloc}(\Omega^{{\varepsilon}})} \le C \sqrt{{{\varepsilon}}}, \quad \|
u^{{\varepsilon}}- u^0 \|_{L^2_{uloc}(\Omega)} \le C' {{\varepsilon}}.$$
We stress that these estimates hold without further assumption on the boundary: we only assume that $\omega$ has values in $(-1,0)$ and is $K-$Lip. A look at the proof shows that the constants $C$ and $C'$ are only increasing functions of $K$.
Theorem \[Direstimates\] expresses that the wall law provides a $O({{\varepsilon}})$ approximation of $u^{{\varepsilon}}$ in $L^2_{uloc}(\Omega)$. See also [@Jager:2001] for a similar result in a bounded channel. However, this wall law does not account for the behaviour of $u^{{\varepsilon}}$ near the boundary, and can therefore be refined. Indeed, as the Poiseuille flow $u^0$ does not vanish at the lower part of ${{\partial}}\Omega^{{\varepsilon}}$, a boundary layer corrector ${{\varepsilon}}\phi v(x/{{\varepsilon}})$ must be added to the expansion. The (normalized) boundary layer $v = v(y)$ is defined on the rescaled infinite domain $$\Omega^{bl} \: = \: \{ y, \: y_2 > \omega(y_1) \}$$ and formally satisfies the following Stokes problem $$\label{BL}
\left\{
\begin{aligned}
& -\Delta v + {{\nabla}}q = 0, \: x \in \Omega^{bl}, \\
& \div v = 0, \: x \in \Omega^{bl},\\
& v(y_1, \omega(y_1)) \: = \: - (\omega(y_1),0).
\end{aligned}
\right.$$ Note the inhomogeneous Dirichlet condition, that cancels the trace of $u^0$.
. First, the well-posedness is not clear. As the boundary function $\omega$ is not decreasing at infinity, one can expect only local integrability of the solution $v$ in variable $y_1$. The derivation of local bounds is not obvious: the Stokes operator being vectorial, one can not use scalar tools such as the maximum principle or Harnack inequality. Moreover, as $\Omega^{bl}$ is unbounded in all directions, the Poincaré inequality (which allows to get $H^1_{uloc}$ estimates in the channel) is not available. Besides the well-posedness issue, the qualitative properties of $v$ seem also out of reach without further hypothesis.
Under an assumption of periodic irregularities, the analysis of becomes straightforward. If $\omega$ is say $L$ periodic in $y_1$, it is easy to show well-posedness in the space $$\left\{ v \in H^1_{loc}(\Omega_{bl}), \: v \:\:
L-\mbox{periodic in } y_1,
\: \int_0^L \int_{\omega(y_1)}^{+\infty} | {{\nabla}}v |^2 dy_2 dy_1 <
+\infty \right\}.$$ Moreover, a simple Fourier transform in $y_1$ shows that $$\| v(y) - v^\infty \| \le C \, e^{-\delta y_2/L}, \quad v^\infty = (\alpha,
0), \quad \alpha= \frac{1}{L} \int_0^L v_1(s) ds, \quad \delta > 0,$$ that is exponential convergence to a constant field $v^\infty = (\alpha,
0)$ at infinity.
The constant $\alpha$ at infinity is then responsible for a $O({{\varepsilon}})$ tangential slip. Namely, chosing as a wall law the Navier-slip condition $$\label{Navier}
B^{{\varepsilon}}(x, D_x)v \: = \: \left( v_1 - {{\varepsilon}}\alpha {{\partial}}_2 v_1, \:
v_2 \right) = 0 \: \mbox{ at } \Sigma,$$ it can be shown (in this periodic framework) that the solution of provides a $O({{\varepsilon}}^{3/2})$ approximation of $u^{{\varepsilon}}$ in $L^2$. We refer to [@Jager:2001] for all necessary details. The error estimate ${{\varepsilon}}^{3/2}$ comes from the fact that the boundary layer term satisfies $\| {{\varepsilon}}(v(x/{{\varepsilon}})- (\alpha,0)) \|_{L^2} = O({{\varepsilon}}^{3/2})$.
The periodicity hypothesis is a stringent one, and one may wonder if the use of Navier slip condition can be justified in more general configurations. This issue has been adressed rigorously in the recent article [@Basson:2007]. Inspired by the probabilistic modeling of heterogeneous media (see for instance [@Jikov:1994]), we considered irregularities that are not distributed periodically, but randomly, following a stationary stochastic process. Namely, the rough boundary is seen as a realization of a stationary spatial process. Following the well-known construction of Kolmogorov, this amounts to consider the space $$P \: = \: \left\{ \omega : {{\mathbb R}}\mapsto (-1,0), \: \omega \: K-\mbox{Lip}
\right\}$$ of all possible rough boundaries, together with the cylindrical $\sigma-$ field ${\cal C}$ (that is generated by the coordinates $\omega \mapsto
\omega(t)$) and with a stationary measure $\pi$. Stationary means that $\pi$ is invariant by the group of translation $$\tau_h : P \mapsto P, \quad \omega \mapsto \omega(\cdot + h).$$ As a consequence of this modeling, the domains $\Omega^{{\varepsilon}}$, $\Omega^{bl}$, as well as the velocity fields $u^{{\varepsilon}}$ or $v$ depend on the parameter $\omega$. As discussed earlier, the existence result and estimates of theorem \[Direstimates\] are uniform on $P$. Moreover, it was shown in article [@Basson:2007] that the function $\omega \mapsto u^{{\varepsilon}}(\omega, \cdot)$ (extended by $0$ outside $\Omega^{{\varepsilon}}(\omega)$) is measurable as a function from $P$ to $H^1_{loc}({{\mathbb R}}^2)$.
Using this probabilistic structure, we have been able to extend partially the results of the periodic case. Key elements of our analysis are:
- the well-posedness of the boundary layer system, obtained in functional spaces encoding the relation $$v(\tau_h(\omega),y_1,y_2) \: = \: v(\omega, y_1+h, y_2).$$
- the convergence of $v(\omega, y)$ to $(\alpha(\omega), 0)$ as $y_2 \rightarrow + \infty$, both in $L^2(P)$ and almost surely, locally uniformly in $y_1$. Such convergence is deduced from the ergodic theorem.
More on the boundary layer system will be provided in the next sections. As regards the Navier wall law , the main result of [@Basson:2007] resumes to
\[Navestimates\] There exists $\alpha = \alpha(\omega) \in L^2(P)$ such that the solution $u^N$ of , satisfies $$\| u^{{\varepsilon}}- u^N \|_{L^2_{uloc}(P \times \Omega)} = o({{\varepsilon}}).$$
We remind that $ \| w \|_{L^2_{uloc}(P \times \Omega)} \: := \: sup_{x} \left( \int_P
\int_{B(x,1) \cap \Omega} | w |^2 dx dP \right)^{1/2}.$
Theorem \[Navestimates\] shows that a slip condition of Navier type improves the approximation of $u^{{\varepsilon}}$. As in the periodic case, the random variable $\alpha$ in comes from the convergence of the boundary layer $v$. If the measure $\pi$ is ergodic, $\alpha$ does not depend on $\omega$, as pointed out in [@Basson:2007].
A natural concern about this result is the $o({{\varepsilon}})$ bound, which is only a slight improvement of the $O({{\varepsilon}})$ in theorem \[Direstimates\]. A look at article [@Basson:2007] shows that this poor bound is due to the lack of information on the way $v$ converges at infinity. Contrary to the periodic case, where convergence at exponential rate is established, the simple use of the ergodic theorem does not yield any speed rate.
The present paper aims at clarifying this point. Losely, [*we will show that for a large class of boundaries, the Navier wall law provides a $O({{\varepsilon}}^{3/2}
|\ln({{\varepsilon}})|^{1/2})$ approximation of the real solution.*]{} Namely, we will make the two following assumptions on our random roughness:\
[*(H1) The measure $\pi$ is supported by $$P_\alpha = \left\{ \omega : {{\mathbb R}}\mapsto (-1,0), \: \| \omega
\|_{C^{2,\alpha}} \: \le K_\alpha \right\}$$ for some $\alpha > 0$ and some $K_\alpha > 0$.*]{}\
[*(H2) The randon boundary has no correlation at large distances, that is the $\sigma$-fields $$\sigma\left( s \mapsto \omega(s), \: s \le a \right) \: \mbox{ and } \:
\sigma\left( s \mapsto \omega(s), \: s \ge b \right)$$ are independent for $b-a \ge \kappa$, for some $\kappa > 0$.*]{}\
Under these assumptions, the main theorem of the paper reads:
\[refinedestimate\] For small enough $\phi$ and under (H1)-(H2), the following refined estimate holds: $$\| u^{{\varepsilon}}- u^N \|_{L^2_{uloc}(P \times \Omega)} =
O({{\varepsilon}}^{3/2} |\ln({{\varepsilon}})|^{1/2}).$$
Before entering the proof of this theorem, let us give a few hints. Theorem \[refinedestimate\] is deduced from a central limit theorem for the quantity $v(\omega,y)-(\alpha,0)$. Broadly, this theorem comes from good properties of the random variables $$X^n(\omega) \: = \: \int_n^{n+1} v(\omega,y_1, 0) \, dy_1.$$ Due to the elliptic nature of the Stokes operator, such random variables are not independent. However, under assumption (H2), we are able to prove that the correlation terms $E(X_n \, X_0)$ decay fast enough as $n
\rightarrow \infty$. As a result, one can prove a central limit theorem on $X_n$, and then a similar one on $v - (\alpha,0)$. We point out that such type of results for dependent variables with strong decay of correlations is quite classical and has been used in various fields. We refer to [@Baladi:2001] for a review paper related to dynamical systems, and to recent articles [@Varadhan:2007; @Bouard:2007] for applications in a PDE context.
As a consequence of this central limit theorem, we show that the boundary layer converges to a constant as $|y_2^{-1/2}|$. Note that this is in sharp contrast with the periodic case, where exponential convergence holds (we stress that periodic boundaries are highly correlated, thus far from satisfying (H2)).This speed of convergence is resposible for the ${{\varepsilon}}^{3/2} |\ln({{\varepsilon}})|^{1/2}$ in the Navier wall law.
The main difficulty is to obtain the decay of correlations of variables like $X_n$. The proof relies on precise estimates of the Green function for the Stokes operator above a non flat boundary. Such estimates follow from sharp elliptic regularity results, where one must pay attention to the oscillation of the boundary. This is achieved under the regularity assumption (H1), using ideas of Avellaneda and Lin for homogenization of elliptic systems [@Avellaneda:1987; @Avellaneda:1991].
Boundary layer decay and Navier approximation
=============================================
In this section, we explain how Theorem \[Navestimates\] follows from estimates on the solution $v$ of . Such estimates will be established in the following sections. At first, we remind the main features of $v$, as stated in article [@Basson:2007].
The boundary layer system
-------------------------
As emphasized in the introduction, to solve in a deterministic way, that is for each possible boundary $\omega$, is still unclear. Hence, one must take advantage of the probabilistic setting. First, notice that a reasonable solution $v$ should satisfy: $$\label{stationarity}
v(\tau_h(\omega), y_1, y_2) \: = \: v(\omega, y_1+h, y_2).$$ Together with the stationarity assumption, this relation sort of substitutes to the identity $$v(y_1 + L, y_2) \: = \: v(y_1, y_2)$$ used in the treatment of $L-$periodic roughness. It allows to extend the well-posedness result, through an appropriate variational formulation.
This formulation has been described in article [@Basson:2007]. First, one introduces the new unknown $$w(y) \: := \: v(y) \: + \: (y_2,0) \, \mathbf{1}_{\{y_2 < 0\}}(y),$$ and replace system by $$\label{BL2}
\left\{
\begin{aligned}
& -\Delta w + {{\nabla}}q = 0, \: x \in \Omega^{bl}\setminus \{ y_2 = 0\}, \\
& \div w = 0, \: x \in \Omega^{bl},\\
& w\vert_{{{\partial}}\Omega^{bl}} \: = \: 0,\\
& [ w ]\vert_{y_2 = 0} = 0, \quad [{{\partial}}_2 w - (0,q)]\vert_{y_2 = 0} = (-1,0),
\end{aligned}
\right.$$ where $[ \cdot ]\vert_{y_2 = 0}$ denotes the jump at $y_2=0$. Then, one multiplies formally the Stokes equation by a test function $w' = w'(\omega,y)$ that satisfies $\div w' = 0$, $w'\vert_{{{\partial}}\Omega^{bl}} = 0$. Integrating by parts over $\Omega^{bl} \cap \{ |y_1 <
1\}$ yields $$\int_{\Omega^{bl} \cap \{ |y_1 < 1\}} {{\nabla}}w \cdot {{\nabla}}w' \: = \:
\int_{\{ |y_1| < 1, \: y_2 = 0\}} w'_1 \: + \int_{\Omega^{bl} \cap
\{|y_1| = 1\}} \left( {{\partial}}_n w - q n \right) w'.$$ Finally, if $w$, $w'$ satisfy relation , one can integrate with respect to $\omega$, and thanks to the stationarity of $\pi$, get rid of the annoying boundary term at the r.h.s: $${{\mathbb E}}\int_{\Omega^{bl} \cap \{ |y_1 < 1\}} {{\nabla}}w \cdot {{\nabla}}w' \: = \:
{{\mathbb E}}\int_{\{ |y_1 < 1, \: y_2 = 0\}} w'_1$$
Afterwards, this formal variational formulation can be rigorously defined and solved: in short, one can apply the Riesz theorem in a functional space of Sobolev type, made of functions $w$ such that $${{\mathbb E}}\int_{\Omega^{bl} \cap \{ |y_1 < 1\}} | {{\nabla}}w |^2 \: < \: +\infty,$$ and satisfying almost surely , together with $ \div w = 0, \: w\vert_{\Omega^{bl}} = 0$. We refer to [@Basson:2007] for all details. Note that stationarity implies: $$sup_{t, R} \: {{\mathbb E}}\: \frac{1}{R} \int_{\Omega^{bl} \cap
\{ |y_1 -t | < R\}} |
{{\nabla}}w |^2 \: = \: {{\mathbb E}}\int_{\Omega^{bl} \cap \{ |y_1 < 1\}} | {{\nabla}}w |^2
\: < \: +\infty.$$
Back to the original system , this variational solution $w$ provides almost surely a solution $v(\omega, \cdot) \in
H^1_{loc}\left(\Omega^{bl}\right)$ in the sense of distributions. Moreover, the ergodic theorem yields (see [@Basson:2007]) $$\sup_{R} \frac{1}{R} \int_{\Omega^{bl} \cap \{ |y_1| < R\}} | {{\nabla}}v
|^2 \: < \: +\infty, \: \mbox{ almost surely}.$$
In order to understand the origin of the Navier approximation, the next step is to describe the behavior of $v$ as $y_2 \rightarrow +\infty$. For periodic roughness, one can show exponential convergence of $v$ to a constant vector field $v^\infty =
(\alpha, 0)$. However, the rate of convergence goes to zero with the period $L$. When dealing with stationary random boundaries, that broadly speaking contain all periods, the exponential decay does not hold [*a priori*]{}. In other words, there is a problem associated to the Fourier spectrum, that is discrete in the periodic case, and may accumulate to zero in the random case.
Again, this problem has been (partially) overcome in [@Basson:2007]. The first step is to obtain a representation of $v$ in terms of a Stokes double layer potential, [*cf*]{} proposition ??. Almost surely, for any $$\begin{aligned}
v(\omega, y) \: & = \: \int_{{\mathbb R}}G(t,y_2) \, v(\omega,y_1-t,0) \, dt \\
& = \: -\int_{{\mathbb R}}t \, {{\partial}}_t G(y_2) \frac{1}{t} \int_0^t
v(\omega,y_1-s,0) \, ds \, dt
\end{aligned}$$ where $G$ is the Poisson type kernel for the Stokes operator over a half space. Then, the ergodic theorem and a few calculations yield: $$\frac{1}{t} \int_0^t v(\omega,y_1-s,0) \, ds \, \: \rightarrow
v^\infty(\omega) \: = \: (\alpha(\omega), 0), \quad t \rightarrow \pm \infty,$$ where the convergence holds almost surely (locally uniformly in $y_1$), as well as in $L^2(P)$ (uniformly in $y_1$). In the case where the stationary measure $\pi$ is ergodic, the constant $\alpha$ does not depend on $\omega$. Finally, back to the integral representation, and with similar treatment for derivatives of $v$: $$\label{softestimbl}
\forall \beta \in {{\mathbb N}}^2, \: |\beta| \ge 1, \quad
{{\mathbb E}}\left|v(\cdot, 0,,y_2) - \alpha(\cdot)\right|^2 \: + \:
y_2^{2|\beta|} {{\mathbb E}}\left|{{\partial}}^\beta_y \,v(\cdot, 0,y_2)\right|^2 \: \xrightarrow[y_2 \rightarrow
+\infty]{} \: 0.$$ We refer to [@Basson:2007 Proposition 13] for all details.
Refined estimate for Navier wall law
------------------------------------
Most of the analysis of the present paper will be devoted to a refined asymptotic estimate of the boundary layer:
\[theosharp\] Under assumptions (H1), (H2), for all $\beta \in {{\mathbb N}}^2$, $$\label{sharpestimbl}
y_2^{2 |\beta|+1} {{\mathbb E}}\left|{{\partial}}^\beta \left(
v(\cdot, 0,y_2) - \alpha(\cdot)\right) \right|^2 \: \xrightarrow[ y_2
\rightarrow +\infty]{} \sigma_\beta \ge 0.$$
It is of course a much sharper convergence result than . Before tackling its proof, we explain how it implies theorem \[Navestimates\]. Arguments are direct adaptation from section 5 in [@Basson:2007].
On the basis of the boundary layer analysis, one can build an approximation of $u^{{\varepsilon}}$ of boundary layer type. Namely, we introduce $$u^{{\varepsilon}}_{app}(\omega,x) \: = \: u^0(x) \: + \:
\, 6 \, \phi\,{{\varepsilon}}\, v\left(\omega,\frac{x}{{{\varepsilon}}}\right) +
\, 6 \, \phi\,{{\varepsilon}}\, u^1(\omega,x) +
\, 6 \, \phi\,{{\varepsilon}}\,r^{{\varepsilon}}(\omega,x).$$ In this approximation, $u^0$ is the Poiseuille flow and $v(\omega,\cdot)$ is the boundary layer solution of . As $v$ does not converge to zero at infinity, we add a large scale corrector $u^1$ satisfying: $$\label{Couette}
\left\{
\begin{aligned}
& u^0 \cdot {{\nabla}}u^1 + u^1 \cdot {{\nabla}}u^0 -\Delta u^1 + {{\nabla}}p = 0, \: x \in
\Omega,\\
& \div u^1 \: = \: 0, \: x \in \Omega,\\
& \int_0^1 u^1 \cdot e_1 dx_2 = 0, \\
& u^1\vert_{y_2 = 0} = 0, \quad u^1\vert_{y_2 = 1} = -(\alpha, 0).
\end{aligned}
\right.$$ It is just a combination of a Couette and a Poiseuille flow: $u^1 = \alpha x_2 \left( 2 - 3 x_2 \right) e_1$. Still, this approximation does not vanish at the boundary, which explains the addition of another term $r^{{\varepsilon}}(\omega,x)$. It must satisfy $$\left\{
\begin{aligned}
& r^{{\varepsilon}}(x_1,0) \: = \: 0, \\
& r^{{\varepsilon}}(x_1,1) \: = \: v\left(\frac{x_1}{{{\varepsilon}}}, \frac{1}{{{\varepsilon}}}\right) -
(\alpha,0), \\
& \div r^{{\varepsilon}}\: = \: 0, \: x \in \Omega.
\end{aligned}
\right.$$ This remainder can be taken small in the sense of
This problem possesses a (non unique) solution $r^{{\varepsilon}}$ such that $$\sup_x \, {{\mathbb E}}\: \| r^{{\varepsilon}}\|^2_{H^2(B(x,1) \cap \Omega)} \: = \:
O({{\varepsilon}}| \ln {{\varepsilon}}|).$$
[*Proof:*]{} The proof of this result mimics the one of proposition 14 in [@Basson:2007]. The corrector $r^{{\varepsilon}}$ can be chosen in the form $$r^{{\varepsilon}}= {{\nabla}}^\bot \psi, \quad \psi = a(x_1)x_2^3 + b(x_1) x_2^2 +
c(x_1) x_2 + d(x_1).$$ The streamfunction $\psi$ is determined up to a constant and polynomial in $x_2$. Its coefficients have explicit dependence on $v-(\alpha,0)$. Hence, the $H^2$ estimate on $r^{{\varepsilon}}$ follows from the control of various terms involving $v-(\alpha,0)$. For instance, one must bound the $L^2(P \times (-1,1))$ norm of $$\begin{aligned}
\int_0^{x_1} v_2\left(\omega, \frac{t}{{{\varepsilon}}},\frac{1}{{{\varepsilon}}}\right) dt
\: & = \: {{\varepsilon}}\int_0^{x_1/{{\varepsilon}}}
v_2\left(\omega,y_1,\frac{1}{{{\varepsilon}}}\right) dy_1 \\
& = \: {{\varepsilon}}\int_{\omega(x_1/{{\varepsilon}})}^{1/{{\varepsilon}}}
( v_1 - \alpha)\left(\omega,\frac{x_1}{{{\varepsilon}}}, y_2 \right) \, dy_2 \\
& \: - \: {{\varepsilon}}\int_{\omega(0)}^{1/{{\varepsilon}}} (v_1 - \alpha)(\omega,0,y_2) \, dy_2 \: :=
I^{{\varepsilon}}(\omega,x_1) \: - \: I^{{\varepsilon}}(\omega,0)
\end{aligned}$$ where the last equality comes from the Stokes formula. Using stationarity of $\pi$, we get $${{\mathbb E}}\left\|x_1 \mapsto \int_0^{x_1} v_2\left(\cdot, \frac{t}{{{\varepsilon}}},
\frac{1}{{{\varepsilon}}}\right) dt \right\|_{L^2(-1,1)}^2
\: \le \: 4 {{\mathbb E}}| I^{{\varepsilon}}(\cdot,0) |^2.$$ Thanks to the refined estimate , we finally obtain $$\begin{aligned}
{{\mathbb E}}| I^{{\varepsilon}}(\cdot,0) |^2 \: & \le \: C \left(
{{\varepsilon}}^2 \, {{\mathbb E}}\int_{\omega(0)}^{1} |(v_1 -
\alpha)(\cdot,0,y_2)|^2 dy_2 \: + \: {{\varepsilon}}\int_1^{1/{{\varepsilon}}} {{\mathbb E}}\, |(v_1 -
\alpha)(\cdot,0,y_2)|^2 dy_2 \right)\\
& \le \: C' {{\varepsilon}}^2 + C'' {{\varepsilon}}\int_1^{1/{{\varepsilon}}} y_2^{-1} \, dy_2 = O({{\varepsilon}}|\ln {{\varepsilon}}|)\end{aligned}$$ All other terms involve similar computations. These are straightforwardly adapted from the proof of proposition 14 in [@Basson:2007], using instead of .
Once the approximate solution $u^{{\varepsilon}}_{app}$ is built, one can obtain by energy estimates the following bounds, for $\phi$ small enough: $$\begin{aligned}
& \| u^{{\varepsilon}}- u^{{\varepsilon}}_{app} \|_{L^2_{uloc}(P \times \Omega)} \: = \:
O\left({{\varepsilon}}^{3/2} |\ln({{\varepsilon}})|^{1/2}\right), \\
& \| u^{{\varepsilon}}_{app} - u^N \|_{L^2_{uloc}(P \times \Omega)} \: = \:
O\left({{\varepsilon}}^{3/2} |\ln({{\varepsilon}})|^{1/2}\right),\end{aligned}$$ which of course imply theorem \[Navestimates\]. As the proof is very similar to what was done in paper [@Basson:2007], we do not expand more and refer to it for all details.
A central limit theorem
=======================
Up to the end of the paper, we will assume (H1)-(H2), and focus on theorem \[theosharp\]. It is classical that (H2) implies ergodicity of $\pi$, so that the constant $\alpha$ does not depend on $\omega$. We start again from an integral representation $$\label{layer}
\begin{aligned}
{{\partial}}^\beta_y \left( v(\omega, 0,y_2) - (\alpha, 0) \right) \: & = \:
\int_{{\mathbb R}}{{\partial}}^{\beta_1}_t {{\partial}}^{\beta_2}_{y_2} \,
G(t,y_2) \, \left(v(\omega,-t,0) - (\alpha,0) \right) \, dt \\
& = \: -\int_{{\mathbb R}}\, {{\partial}}^{\beta_1+1}_t {{\partial}}^{\beta_2}_{y_2} \,
G(t,y_2) \int_0^t
\left(v(\omega,-s,0) - (\alpha,0) \right) ) \, ds \, dt,
\end{aligned}$$ where the matrix kernel $G$ is given by $$G(y) \: = \: \frac{2 y_2}{\pi (y_1^2 + y_2^2)^2} \begin{pmatrix} y_1^2 &
y_1 y_2 \\ y_1 y_2 & y_2^2 \end{pmatrix}$$ We introduce $$V(\omega,t) \: := \:
\int_{0}^t \left(v(\omega,-s,0) - (\alpha,0) \right) \, ds.$$ A simple change of variable leads to $$y_2^{|\beta|+1/2}
{{\partial}}^\beta_y \left( v(\omega, 0,y_2) - (\alpha, 0) \right) \:
= \: \int_{{\mathbb R}}{{\partial}}^{\beta_1+1}_t {{\partial}}^{\beta_2}_{y_2} G(u,1) \,
y_2^{-1/2} \, V(\omega, y_2 u) \, du.$$ Our first goal is to show that the l.h.s. converges in law to a gaussian distribution, for all $\beta$. We will focus on the case $|\beta| = 0$, the other cases being handled in the exact same way. We state
\[tcl\] The function $V$ satisfies the following properties:
i)
: ${{\mathbb E}}\left|V(\cdot,t)\right|^2 \: \le \: C |t|$
ii)
: The random process $y_2^{-1/2} \, V(\omega, y_2 \, u)$ converges weakly to a gaussian process $B(\omega,u)$ as $y_2$ goes to infinity.
iii)
: The covariance matrices also converge, that is for all indices $i,j$ and for all $s,t$, $${{\mathbb E}}\, y_2^{-1} \, V_i(\cdot, y_2 \, s) \, V_j(\cdot, y_2 \, t)
\xrightarrow[y_2 \rightarrow +\infty]{} {{\mathbb E}}B_i(\cdot, s)
B_j(\cdot, t)$$
We remind that the process $X^n(\omega,t)$ with values in ${{\mathbb R}}^2$ [*converges weakly*]{} to $X(\omega,t)$ if, for all $T > 0$ and all continuous bounded function ${\cal F} : C\left( [-T,T], {{\mathbb R}}^2 \right) \mapsto {{\mathbb R}}$, $${{\mathbb E}}{\cal F}(X^n)
\: \xrightarrow[n \rightarrow +\infty]{} \: {{\mathbb E}}{\cal F}(X).$$ Theorem \[theosharp\] is then a direct consequence of
The random process $y_2^{1/2} \left(v(\omega,0,y_2) -
(\alpha,0)\right)$ converges in law to a gaussian vector with zero average. Moreover, for all $i,j$, $${{\mathbb E}}\left(v_i(\cdot,0,y_2) - (\alpha,0)_i\right) \,
\left(v_j(\cdot,0,y_2) - (\alpha,0)_j\right) \xrightarrow[y_2 \rightarrow
+\infty]{} \sigma_{ij},$$ where $\sigma$ is the covariance matrix of this gaussian vector.
[*Proof of the corollary:*]{} To prove convergence in law to a gaussian vector ${\cal N}_{\sigma}$ of covariance matrix $\sigma$, we need to show that for any $F \in C^\infty_c({{\mathbb R}})$, $${{\mathbb E}}\, F \left( \int_{{\mathbb R}}{{\partial}}_t G(t,1) \,
y_2^{-1/2} \, V(\cdot, y_2 t) \, dt\right) \xrightarrow[y_2 \rightarrow
+\infty]{} {{\mathbb E}}\, F\left({\cal N}_{\sigma}\right).$$ Unsurprisingly, we take $${\cal N}_{\sigma} \: := \: \int_{{\mathbb R}}{{\partial}}_t G(t,1) B(\omega,t) \, dt.$$ We decompose, for any $T > 0$, $$\begin{aligned}
& {{\mathbb E}}\, F \left( \int_{{\mathbb R}}{{\partial}}_t G(t,1) \,
y_2^{-1/2} \, V(\cdot, y_2 t) \, dt\right) - {{\mathbb E}}\, F\left({\cal
N}_{\sigma}\right) \\
\: = &\: {{\mathbb E}}\, F \left( \int_{-T}^T {{\partial}}_t G(t,1) \,
y_2^{-1/2} \, V(\cdot, y_2 t) \, dt\right) -
{{\mathbb E}}\, F \left( \int_{-T}^T {{\partial}}_t G(t,1) \, B(\cdot,t) \, dt\right) \\
\: + & \:{{\mathbb E}}\, F \left( \int_{{\mathbb R}}{{\partial}}_t G(t,1) \,
y_2^{-1/2} \, V(\cdot, y_2 t) \, dt\right) -
{{\mathbb E}}\, F \left( \int_{-T}^T {{\partial}}_t G(t,1) \,
y_2^{-1/2} \, V(\cdot, y_2 t) \, dt\right) \\
\: + \:& {{\mathbb E}}\, F \left( \int_{{\mathbb R}}{{\partial}}_t G(t,1) \,
B(\cdot,t) \, dt\right) -
{{\mathbb E}}\, F \left( \int_{-T}^T {{\partial}}_t G(t,1) \,
B(\cdot,t) \, dt\right) \\
\: := &\: J_1 + J_2 + J_3 \end{aligned}$$ We now show that expressions $J_1$, $J_2$, converge to zero ($J_3$ is similar to $J_2$ and simpler). Let $\delta > 0$. We have $$\begin{aligned}
|J_2| \: & \le \: \max |F'| \: {{\mathbb E}}\int_{|t|>T}
\left|{{\partial}}_t G(t,1)\right| \,
\left|y_2^{-1/2} \, V(\omega, y_2 t) \right| \, dt \\
& \le \: \max |F'| \: \left( \int_{|t|>T} \left|{{\partial}}_t G(t,1)\right| dt
\right)^{1/2} \, \left( \int_{|t|>T} \left|{{\partial}}_t G(t,1)\right|
\, {{\mathbb E}}\left( y_2^{-1} \, |V(\omega, y_2 t)|^2 \,\right)
dt \right)^{1/2} \\
& \le \: C \left( \int_{|t|>T} \left|{{\partial}}_t G(t,1)\right| dt
\right)^{1/2} \left( \int_{|t|>T} \left|{{\partial}}_t G(t,1)\right|
t \, dt \right)^{1/2} \end{aligned}$$ where the last line comes from point ii) of proposition \[tcl\]. Thus, for $T$ large enough, independently of $y_2$, $|J_2| \le \delta/2$. Such $T$ being fixed, for $y_2$ large enough, we get $|J_1| \le \delta/2$ by point i) of proposition \[tcl\]. This yields convergence in law. The convergence of the covariance matrix $$\begin{aligned}
& {{\mathbb E}}\left(v_i(\cdot,0,y_2) - (\alpha,0)_i\right) \,
\left(v_j(\cdot,0,y_2) - (\alpha,0)_j\right) \\
& = \:
\int_{{\mathbb R}}\int_{{\mathbb R}}{{\mathbb E}}\left( {{\partial}}_t G(s,1)
y_2^{-1/2} \, V(\cdot, y_2 \, s) \right)_i \, \left( {{\partial}}_t G(t,1)
y_2^{-1/2} \, V(\cdot, y_2 \, t) \right)_j \, ds \, dt \end{aligned}$$ follows from the dominated convergence theorem, using i) and iii) of proposition \[tcl\]. We get $$\sigma_{ij} \: = \: {{\mathbb E}}\int_{{\mathbb R}}\int_{{\mathbb R}}\left( {{\partial}}_t G(s,1) B(\cdot,s)
\right)_i \, \left( {{\partial}}_t
G(t,1) B(\cdot,t) \right)_j \, ds \, dt.$$ This concludes the proof of the corollary.
It remains to prove theorem \[tcl\]. Note that point ii) is essentially a central limit theorem for the sequence of random variables $$X^n(\omega) = F \circ \tau_n (\omega), \quad F(\omega) = \int_0^1
\left(v(\omega,t,0) - (\alpha(\omega),0) \right) \, dt.$$ The problem is that these random variables are not independent, due to “propagation of information at infinite speed” in the Stokes system. To establish a central limit theorem for such type of sequences is a classical question. The basic idea is that one can extend the central limit theorem to non independent sequences that feature a good decay of correlations as $n$ goes to infinity. We now illustrate this general principle on our problem, using assumption (H2). We follow the presentation of article [@Varadhan:2007], in which a similar question arises for a semilinear heat equation with random source. Let ${\cal C}^n$ the $\sigma$-algebra generated by the applications $y_1 \mapsto \omega(y_1)$, $|y_1| < n$.We state the following lemma:
\[lemmeUn\] Suppose that $\: v^n \: := \: {{\mathbb E}}\left( v(\cdot,0,0) \: | \: {\cal C}^n\right)$ satisfies $${{\mathbb E}}\left| v^n - v(\cdot,0,0) \right|^2 \: \le \:
C \, n^{-\alpha}$$ for some $\alpha >1$. Then, proposition \[tcl\] holds.
[*Proof of the lemma:*]{} We write the decomposition $$v(\cdot,0,0) - (\alpha,0) \: = \:
v^1 - (\alpha,0) \: + \: \sum_{j=1}^{+\infty} \left( v^{2^j}
- v^{2^{j-1}} \right)$$ with the sum converging in $L^2(P)$. The corresponding sum for $V$ is $$V \: = \: \sum_{j=0}^{+\infty} \, V^{j}, \quad V^j(\omega,t) =
\int_{0}^{t} \left( v^{2^j}
- v^{2^{j-1}} \right)\circ \tau_s(\omega) \, ds,$$ where $v^{1/2} := (\alpha,0)$. Then, we have: $ \: \| V(\cdot,t) \|_{L^2(P)} \, \le \,
\sum_{j=0}^{+\infty} \: \| V^{j}(\cdot,t) \|_{L^2(P)}$. By the assumption of independence at large distances, the correlations ${{\mathbb E}}\left( v^{2^j} \circ
\tau_t(\omega) \, v^{2^j} \circ \tau_{t+s}(\omega) \right)$ and ${{\mathbb E}}\left( v^{2^j} \circ
\tau_t(\omega) \, v^{2^{j-1}} \circ \tau_{t+s}(\omega) \right)$ vanish for $|s| \ge \kappa + 2^{j+1}$. We introduce $$n \: := \: \left[ |t|/(\kappa+2^{j+1})\right].$$ If $n=0$, we just write $${{\mathbb E}}\left|V^{j}(\cdot,t)\right|^2 \: \le \: |t|^2 {{\mathbb E}}\left|v^{2^j} - v^{2^{j-1}} \right|^2.$$ If $n \ge 1$, we decompose $$\begin{aligned}
{{\mathbb E}}\left|V^{j}(\cdot,t)\right|^2 \: & = \: {{\mathbb E}}\left| \sum_{k=0}^{n-1}
\int_{kt/n}^{(k+1)t/n} \left( v^{2^j} \circ \tau_s - v^{2^{j-1}}
\circ \tau_s \right) \, ds \right|^2 \\
& \le \: {{\mathbb E}}\left| \int_0^{t/n} \sum_{k=0}^{n-1} \left( v^{2^j} \circ
\tau_{s+kt/n} - v^{2^{j-1}} \circ \tau_{s+kt/n} \right) \, ds
\right|^2 \\
& \le \: 2 \left(\kappa + 2^{j+1} \right)
\, \int_0^{|t|/n} \sum_{k=0}^{n-1} {{\mathbb E}}\left| v^{2^j} -
v^{2^{j-1}} \right|^2
\end{aligned}$$ Using the bound on the conditional expectations, we end up with $$\label{estimateVj}
\| V^{j}(\cdot,t) \|_{L^2(P)}^2 \: \le \: C \, |t| \: \min(|t|, 2^j) \,
2^{-j \alpha}.$$ for some constant $C = C(\kappa)$. We thus get i). To prove ii), we just write the decomposition $$v(\cdot,0,0) - (\alpha, 0) \: = \:
\sum_{j=0}^{+\infty} \left( v^{2^j} \: - \: v^{2^{j-1}} \right), \quad
y_2^{-1/2} \,V(\omega, t y_2) \: = \: y_2^{-1/2} \sum_{j=0}^{+\infty}
\,V^j(\omega, t y_2).$$ It is well-known that each finite sum satisfies a central limit theorem, that is $$\forall k, \quad y_2^{-1/2} \sum_{j=0}^{k} \,V^j(\omega, t y_2)
\xrightarrow[y_2 \rightarrow +\infty]{} B^k(\omega,t)$$ in the sense of weak convergence, to some gaussian process $B^k(\omega,t)$. Moreover, the covariance matrix also converges, that is $$y_2^{-1} {{\mathbb E}}\sum_{j=0}^{k} \,V^j_l(\cdot, s y_2) \,
\sum_{j=0}^{k} \,V^j_m(\cdot, t y_2) \: \xrightarrow[y_2 \rightarrow
+\infty]{}
{{\mathbb E}}B^k_l(\omega,s) \, B^k_m(\omega,t).$$ In short, it is due to the fact that the random variables $$X^{n,j}(\omega) = F^j \circ \tau_n(\omega), \quad
F^j(\omega) = \int_0^1 \left( v^{2^j} - v^{2^{j-1}} \right) \circ
\tau_t(\omega) \, dt, \quad
n \in {{\mathbb Z}},$$ have zero correlations at large distances: see [@Durrett:1996 theorem (7.11) p424] for a similar result and detailed proof. Moreover, thanks to estimate , the remainder $$R^k(\omega,t,y_2) \: = \: \sum_{j=k}^{+\infty} \, y_2^{-1/2}
\,V^j(\omega, t y_2)$$ converges to zero as $k \rightarrow +\infty$, locally uniformly in $t$, uniformly in $y_2$. Hence, points (ii) and (iii) of proposition \[tcl\] hold, which ends the proof of the lemma.
We still have to estimate the variance of $v^n - v(\cdot,0,0)$. Following [@Varadhan:2007], we can turn this question into a question of domain of dependence for solutions of . Precisely, starting from the measure $\pi$ on $P$, we define a new measure $\pi^n$ on the product space $$P^n \: = \: \left\{ (\omega_1, \omega_2) \in P \times P, \quad
\omega_1(t) = \omega_2(t), \: |t| \le n \right\}.$$ endowed with its cylindrical $\sigma-$field. Namely, $\pi^n$ is defined in the following way:
1. $\pi^n(A \times A) \, := \, \pi(A), \: \forall A \in {\cal C}^n$, which determines $\pi^n$ over the sub $\sigma-$field ${\cal D}^n$ generated by the applications $\displaystyle t \mapsto (\omega_1(t), \omega_2(t)), \: |t| \le n$.
2. For all $k \ge 1$, for all $t^1, \dots, t^k$ with $|t^j| > n$, for all borelian subsets $B^1_1, \dots, B^k_1$, $B^1_2, \dots, B^k_2$ of ${{\mathbb R}}$, and $$A_1 \: := \: \cap_{j=1}^k \{ \omega_1, \: \omega_1(t_j) \in
B_1^j \}, \quad A_2 \: := \: \cap_{j=1}^k \{ \omega_2, \: \omega_2(t_j) \in
B_2^j \}$$ $\pi^n(A_1 \times A_2 \, | \, {\cal D}^n)(\omega_1,\omega_2) \, :=
\,\pi(A_1 \, | \, {\cal
C}^n )(\omega_1) \: \pi(A_2 \, | \, {\cal C}^n )(\omega_2)$, which determines $\pi^n$ conditionaly to ${\cal D}^n$.
It is then easy to derive the following identity, see [@Varadhan:2007]: $${{\mathbb E}}\left|v^n - v(\cdot,0,0 ) \right|^2 \: = \: \frac{1}{2} \int_{P^n} \left|
v(\omega_1,0,0) - v(\omega_2,0,0) \right|^2 d\pi_n.$$ Thus, if $\Omega^{bl}(\omega_1)$ and $\Omega^{bl}(\omega_2)$ are two boundary layer domains with boundaries that coincide over $[-n,n]$, we need to estimate the difference of the corresponding boundary layer solutions $v(\omega_1,0,0)$ and $v(\omega_2,0,0)$. This is the purpose of the next section.
Decay of correlations
=====================
[*Throughout the rest of the paper, we will assume (H1).*]{} The main difficuly is to prove the following
\[propdecay\] Under assumption (H1), for all $0 < \tau < 1$, for almost every $\omega_1, \omega_2 \in P$, $$\label{decaybound}
\left| v(\omega_1,0,0) - v(\omega_2,0,0) \right| \: \le \:
\frac{C}{n^{2\tau-1}},$$ where $C$ does not depend on $\omega_1, \omega_2$.
Together with the results of the preceding section, this proposition concludes the proof of theorem \[theosharp\] (take $\tau > 3/4$), and therefore the proof of the main theorem \[Navestimates\]. In fact, the sharper bound $$\left| v(\omega_1,0,0) - v(\omega_2,0,0) \right| \: \le \: \frac{C}{n},$$ that is with $\tau=1$ would still be true. We will discuss this briefly in the last section of the paper. For the sake of brevity, we only prove here the weaker form .
The main difficulty is that the boundary layer solutions $v(\omega_1,y)$ and $v(\omega_2,y)$ of are not defined on the same domain, so that estimates on the difference are not directly available. If the Poisson equation rather than the Stokes system was considered, representation of the solution in terms of Brownian motion would allow to conclude quite easily. Again, this will be explained in the last section of the paper.
In the case of system , we are not aware of such representation, and the bound will come from an accurate description of the (matrix) Green function of the Stokes operator above a humped boundary. We consider for all $\omega \in C^{2,\alpha}$, and for all $z \in
\Omega^{bl}(\omega ) \: = \: \{ y_2 > \omega(y_1)\}$, the system: $$\label{green}
\left\{
\begin{aligned}
-\Delta G_\omega(z,\cdot) + {{\nabla}}P_\omega(z,\cdot) = \delta_z \,
I_2 & \quad \mbox{ in
} \Omega^{bl}(\omega), \\
\div G_\omega(z,\cdot) = 0 & \quad \mbox{ in } \Omega^{bl}(\omega), \\
G_\omega(z, \cdot) = 0 & \quad \mbox{ on } {{\partial}}\Omega^{bl}(\omega).
\end{aligned}
\right.$$ where $\delta_z$ is the Dirac mass at $z$, and $I_2$ is the $2\times2$ identity matrix. Let us remind how to build the matrix Green function $(G_\omega, P_\omega)$. Up to a vertical translation of the domain, we can first assume that $z_2 > 0$. We then introduce the Green function $(G_0, P_0)$ for the Stokes operator in the upper-half plane, see [@Galdi:1994]. Extending $\displaystyle
G_0(z, \cdot), P_0(z, \cdot)$ by $0$ for $y_2 < 0$, the functions $$\displaystyle H(z, \cdot) \: := \; G_\omega(z, \cdot) - G_0(z, \cdot),
\quad Q(z, \cdot)
\: := \: P_\omega(z, \cdot) - P_0(z, \cdot)$$ satisfy formally $$\left\{
\begin{aligned}
-\Delta H(z,\cdot) + {{\nabla}}Q(z,\cdot) = 0 & \quad \mbox{ in
} \Omega^{bl}(\omega), \\
\div H(z,\cdot) = 0 & \quad \mbox{ in } \Omega^{bl}(\omega), \\
H_\omega(z, \cdot) = 0 & \quad \mbox{ on } {{\partial}}\Omega^{bl}(\omega), \\
\left[ H(z,\cdot)\right] = 0, & \quad \bigl[ {{\partial}}_2 H(z,\cdot) - Q(z, \cdot)
\otimes \,e_2 \bigr] = - \left[ {{\partial}}_2 G_0(z,\cdot) - P_0(z, \cdot)\,
\otimes e_2 \right],
\end{aligned}
\right.$$ where $\left[ \: \cdot \: \right]$ is the jump along $\{ y_2 = 0 \}
\cap \Omega^{bl}(\omega)$. The jump on the derivative is explicit, as $${{\partial}}_2 G_0(z,(y_1,0^+)) - P_0(z,(y_1,0^+)) \otimes e_2 \: = \:
\frac{2 z_2}{\pi ((z_1-y_1)^2 + z_2^2)^2} \begin{pmatrix} (z_1-y_1)^2 &
(z_1-y_1) y_2 \\ (z_1-y_1) y_2 & y_2^2 \end{pmatrix}.$$ Standard variational formulation yields existence and uniqueness of a solution $H(z,\cdot)$ with ${{\nabla}}H(z, \cdot)$ in $L^2$. In turn, this provides a unique solution $G_\omega(z, \cdot)$ to . The corresponding pressure field $P_\omega(z, \cdot)$ is determined up to the addition of a constant matrix. Note that uniqueness yields the relation $$\label{stationG}
G_{\tau_h(\omega)}(z,y) \: = \: G_\omega((z_1+h,z_2), (y_1+h,y_2)).$$
Our key estimate is provided by
\[green2\] For all $0 < \tau < 1$, for all $z,y \in \Omega^{bl}(\omega)$ satisfying $|z-y| \ge 1$, we have $$\label{greenestim1}
\sum_{|\beta| \le 2} |{{\partial}}^\beta_y G_\omega(z,y)| \: + \: |{{\nabla}}_y
P_\omega(z,y)| \: \le
\: C \, \frac{ \delta(z)^\tau \, (1+ \delta(y))^\tau}{|z-y|^{2\tau}},$$ where $\delta(\cdot)$ denotes the distance to the boundary ${{\partial}}\Omega^{bl}(\omega)$, and $C$ is a constant depending only on $\tau$ and on $\| \omega \|_{C^{2,\alpha}}$.
Note that by symmetry of $G$, we also have $$\label{greenestim2}
\sum_{|\beta| \le 2} |{{\partial}}^\beta_z G_\omega(z,y)| \: \le \: C \, \frac{
(1+\delta(z))^\tau \, (1+ \delta(y))^\tau}{|z-y|^{2\tau}}.$$ Moreover, in the course of the proof of lemma \[green2\], we will show that for all $y,z \in \Omega^{bl}(\omega)$, $$\label{greenestim3}
| G_\omega(z,y) | \: \le \: C \, \left( \bigl| \ln|z-y| \bigr| + 1\right).$$
. We first need to connect the solution $v(\omega,\cdot)$ of to the Green function $G_\omega$. For this purpose, we rather consider $$w(\omega, y) \: = \: v(\omega,y) + y_2 \, {\bf 1}_{\{y_2 < 0\}}(y).$$ which satisfies . Note that $v$ and $w$ coincide at $y=0$. Formally, $w$ should be equal to $$\label{tildew}
\tilde{w}(\omega,z) \: = \: \int_{\{y_2 =0\}} G_\omega(z,y) \, e_1 \, dy.$$ Using estimates , , it is standard to show that $\tilde{w}$ is a solution of in $H^1_{loc}(\overline{\Omega_{bl}})$. Using bound , one has even $$\int_{\Omega^{bl} \cap \{|z_1| < 1\}} | {{\nabla}}\tilde{w} |^2 \: \le \: C
\: < \: +\infty,$$ for all $\omega \in P_{\alpha}$.
Extending $\tilde{w}(\omega, \cdot)$ by $0$ outside $\Omega^{bl}(\omega)$, one can show that $\omega \mapsto
\tilde{w}(\omega,\cdot)$ is measurable from $P_{\alpha}$ to $H^1_{loc}({{\mathbb R}}^2)$ (see the appendix for details). Moreover, thanks to , $\tilde{w}$ satisfies the stationarity relation $\tilde{w}(\tau_h(\omega),y) = \tilde{w}(\omega,
(y_1+h,y_2))$. Finally, using that $w$ and $\tilde{w}$ both satisfy , a simple energy estimate on the difference leads to $${{\mathbb E}}\int_{\Omega^{bl} \cap \{|z_1| < 1\}} \left| {{\nabla}}\left( \tilde{w} - w
\right) \right|^2 \: = \: 0$$ which shows that $w = \tilde{w}$ almost surely.
It remains to estimate the difference $$v(\omega_1,0,0) - v(\omega_2,0,0) \: = \: \int_{\{y_2 =0\}}
\left( G_{\omega_1}((0,0),y) - G_{\omega_2}((0,0),y) \right) \, e_1,$$ for every $\omega_1$, $\omega_2$ in $P_\alpha$ which coincide over $[-n,n]$. This integral is bounded by $$\begin{aligned}
I_1 + I_2 \: & := \: \int_{y_2=0, |y_1| > n} \left| G_{\omega_1} -
G_{\omega_2}\right|((0,0),y) \, dy \\
& + \: \int_{y_2=0, |y_1| \le n} \left| G_{\omega_1} -
G_{\omega_2}\right|((0,0),y) \, dy\end{aligned}$$ The use of gives $$I_1 \: \le \: C \, \int_{y_2=0, |y_1| > n} \frac{1}{|y_1|^{2\tau}} \,
dy_1 \: \le \: \frac{C}{n^{2\tau-1}}.$$ where $C$, which depends [*a priori*]{} on $\| \omega_i
\|_{C^{2,\alpha}}$, can be chosen uniformly over $P_\alpha$, as all $C^{2,\alpha}$ norms are bounded by $K_{\alpha}$. To bound the second term, we first assume that $\omega_2 >
\omega_1$ for $|y_1| > n$, which is always possible up to introduce an intermediate third boundary. Hence, $\Omega^{bl}(\omega_2) \subset \Omega^{bl}(\omega_1)$. To lighten notations, we introduce $$\Omega^{bl}_{1,2} \: := \:
\Omega^{bl}(\omega_1)\setminus\Omega^{bl}(\omega_2), \quad \Gamma_{1,2} \:
:= \: {{\partial}}\Omega^{bl}(\omega_2)\setminus {{\partial}}\Omega^{bl}(\omega_1),$$ as well as $$\tilde{P}(y) \: := \:
P_{\omega_2}\bigl((0,0),(y_1,\omega_2(y_1))\bigr), \quad y \in \Omega^{bl}_{1,2}$$ which defines a continuous extension of $P_{\omega_2}((0,0), \cdot)$ outside $\Omega^{bl}(\omega_2)$. Finally, we define the vector fields $$\begin{aligned}
& U(y) \: := \: \left(G_{\omega_1} - G_{\omega_2}\right)((0,0),y), \quad
Q(y) \: := \: \left(P_{\omega_1}-P_{\omega_2}\right)((0,0),y), \quad y \in
\Omega^{bl}(\omega_2), \\
& U(y) \: := \: G_{\omega_1}((0,0),y), \hspace{1.9cm} Q(y) \: := \:
P_{\omega_1}((0,0),y) - \tilde{P}(y), \quad y \in
\Omega^{bl}_{1,2}.\end{aligned}$$ They satisfy $$\left\{
\begin{aligned}
-\Delta U + {{\nabla}}Q & = 0, \quad y \in \Omega^{bl}(\omega_2), \\
-\Delta U + {{\nabla}}Q & = -{{\nabla}}\tilde{P}, \quad y \in
\Omega^{bl}_{1,2}, \\
\div U & = 0, \quad y \in \Omega^{bl}(\omega_1), \\
U & = 0, \quad y \in {{\partial}}\Omega^{bl}(\omega_1), \\
\left[ U \right]\vert_{\Gamma_{1,2}}& = 0, \quad \left[ {{\partial}}_n U -
Q \otimes n \right]\vert_{\Gamma_{1,2}} =
-{{\partial}}_n G_{\omega_2}((0,0),y)\vert_{\Gamma_{1,2} }.
\end{aligned}
\right.$$ A direct energy estimate yields $$\begin{aligned}
& \int_{\Omega^{bl}(\omega_1)} \left| {{\nabla}}U \right|^2 \: \le \:
\int_{\Gamma_{1,2}} |{{\partial}}_n G_{\omega_2}((0,0),y)| \, |U| \: + \:
\int_{\Omega^{bl}_{1,2}} |{{\nabla}}\tilde{P}| \, |U| \\
& \: \le \: \Bigl( \int_{\Gamma_{1,2}} |{{\partial}}_n
G_{\omega_2}((0,0),y)|^2 \Bigr)^{1/2} \Bigl( \int_{\Gamma_{1,2}} |U|^2
\Bigr)^{1/2}
\: + \: \Bigl( \int_{\Omega^{bl}_{1,2}} |{{\nabla}}\tilde{P}|^2
\Bigr)^{1/2} \Bigl( \int_{\Omega^{bl}_{1,2}} |U|^2
\Bigr)^{1/2} \\
& \: \le \: C \, \biggl( \Bigl( \int_{\Gamma_{1,2}} |{{\partial}}_n
G_{\omega_2}((0,0),y)|^2 \Bigr)^{1/2} + \Bigl(\int_{\Omega^{bl}_{1,2}} | {{\nabla}}\tilde{P}
|^2 \Bigr)^{1/2} \biggr) \Bigl( \int_{\Omega^{bl}_{1,2}}
| {{\partial}}_{y_2} U |^2 \Bigr)^{1/2}. \end{aligned}$$ Note that all $y$ in both $\Gamma_{1,2}$ and $\Omega^{bl}_{1,2}$ satisfy $|y_1| > n$. Using , we end up with $$\int_{\Omega^{bl}(\omega_1)} \left| {{\nabla}}U \right|^2 \: \le \: C \, n^{1-4\tau}$$ Back to $I_2$, we obtain $$| I_2 | \: \le \: \sqrt{2 n} \, \left( \int_{|y_1| \le n, y_2 = 0} \,
|U|^2 \right)^{1/2} \:
\le \: C \, \sqrt{n} \, \left( \int_{\Omega^{bl}(\omega_1)} \,
|{{\partial}}_{y_2} U|^2 \right)^{1/2} \: \le \: \frac{C}{n^{2\tau-1}}.$$ This ends the proof of proposition .
Green function estimates
========================
[*This section is devoted to the proof of lemma \[green2\]*]{}, that is sharp estimates on the Green function $(G_\omega,P_\omega)$ for the Stokes operator above the humped boundary $y_2 = \omega(y_1)$, where $\omega$ belongs to $C^{2,\alpha}$. A fundamental remark is that the Green function satisfies the scaling $$\label{scaling}
\forall {{\varepsilon}}> 0, \quad G_{\omega^{{\varepsilon}}}({{\varepsilon}}z, {{\varepsilon}}y) \: = \:
G_\omega(z,y), \quad \omega^{{\varepsilon}}(x_1) = {{\varepsilon}}\omega(x_1/{{\varepsilon}}).$$ We want estimates to hold for $|z-y|$ large, that is for ${{\varepsilon}}\: := |z-y|^{-1}$ small. By relation , to establish such estimates amounts to get local estimates for the Green function $G_{\omega^{{\varepsilon}}}$. Thus, this is again a homogenization problem: more precisely, we must show that the oscillations of the boundary at frequency ${{\varepsilon}}^{-1}$ do not affect too much the estimates on $G_{\omega^{{\varepsilon}}}$, so that it behaves as the Green function for a half-plane. A very close problem has been considered in the papers [@Avellaneda:1987; @Avellaneda:1991] by Avellaneda and Lin, namely the derivation of local estimates for elliptic systems $\div\left(A(x/{{\varepsilon}}) {{\nabla}}\, \cdot \, \right)$, in which $A$ is a positive definite matrix with periodic coefficients. Our reasoning follows these papers.
For all $x \in {{\mathbb R}}^2$, $r > 0$, we will denote $D(x,r)$ the disk of center $x$ and radius $r$, and $$D^{{\varepsilon}}(x,r) \: := \:
D(x,r) \cap \{x_2 > \omega^{{\varepsilon}}(x_1)\}, \quad
\Gamma^{{\varepsilon}}(x,r)\: := \: D(x,r) \cap \{x_2 = \omega^{{\varepsilon}}(x_1)\}.$$ An important property is that for all $0 < r < 1$, $$\label{airedeps}
| D^{{\varepsilon}}(x,r) | \: \ge \: \eta \, r^2,$$ for some $\eta > 0$ independent of ${{\varepsilon}}$. More precisely, $\eta$ only involves the Lipschitz norm of $\omega^{{\varepsilon}}$, wich is bounded uniformly in ${{\varepsilon}}$. This will be used implicitly throughout the sequel.
The core of the proof is to derive elliptic estimates uniform with respect to ${{\varepsilon}}$ on the following Stokes problem: $$\label{stokes}
\left\{
\begin{aligned}
-\Delta u + {{\nabla}}p \: = \: \div f, \quad & x \in D^{{\varepsilon}}(x_0,1)\\
\div u = 0, \quad & x \in D^{{\varepsilon}}(x_0,1),\\
u = 0, \quad & x \in \Gamma^{{\varepsilon}}(x_0,1),
\end{aligned}
\right.$$ where $x_0 \in {{\mathbb R}}^2$. More precisely, there are two steps in the proof:
1. We show a ${{\varepsilon}}$-uniform Hölder estimate on $u$: for all $f \in
L^q$, $q>2$ and for $\mu = 1-2/q$, $$\label{holder}
\| u \|_{C^{0,\mu}(D^{{\varepsilon}}(x_0,1/2))} \: \le \: C \left( \| f
\|_{L^q(D^{{\varepsilon}}(x_0,1))} + \| u \|_{L^2(D^{{\varepsilon}}(x_0,1))} \right).$$ where $C$ depends only on $\| \omega \|_{C^{1,\alpha}}$.
2. Thanks to this Hölder estimate, we prove .
The two next paragraphs correspond to these steps.
Hölder estimate
---------------
To obtain a Hölder regularity result, a classical approach is to use a characterization of Hölder spaces due to Campanato (see [@Giaquinta:1983]): for $\Omega$ an open connected bounded set, $\displaystyle v \in C^{0,\mu}(\Omega)$ iff $v \in L^2(\Omega)$ and $$\sup_{x \in \Omega, r > 0} \frac{1}{r^{2+2\mu}}\int_{\Omega(x,r)} |v -
\overline{v}_{x,r}|^2 < \infty, \quad \Omega(x,r) := \Omega \cap D(x,r), \:\:
\overline{v}_{x,r} := \frac{1}{|\Omega(x,r)|} \int_{\Omega(x,r)} v.$$ One then tries to control such local integrals through energy estimates. This approach has been successful in the study of elliptic systems, see the work of Giaquinta and coauthors [@Giaquinta:1983]. It extends to the Stokes type equations, [*cf*]{} article [@Giaquinta:1982]. For us, it amounts to controlling $$I^{{\varepsilon}}_{x,r} \: := \: \frac{1}{r^{2+2\mu}}\int_{D^{{\varepsilon}}(x,r)} |u -
\overline{u}_{x,r}|^2 <
\infty, \quad \overline{u}_{x,r} := \frac{1}{|D^{{\varepsilon}}(x,r)|}
\int_{D^{{\varepsilon}}(x,r)} u$$ where $u$ is solution of . Note that, thanks to (see [@Giaquinta:1983]), $$\| u \|_{C^{0,\mu}(D^{{\varepsilon}}(x_0,1/2))} \: \le \: C_{x_0}
\left( \| u \|_{L^2(D^{{\varepsilon}}(x_0,1/2))} \: +
\sup_{x \in D^{{\varepsilon}}(x_0,1/2), r > 0} I^{{\varepsilon}}(x,r) \right)$$ with $C_{x_0}$ independent of ${{\varepsilon}}$. In our case, the main problem is to keep track of the dependence of $I^{{\varepsilon}}_{x,r}$ on ${{\varepsilon}}$. It involves a discussion of the way ${{\varepsilon}}$ relates to $r$. Broadly speaking, the idea is the following: if $r$ is large compared to ${{\varepsilon}}$, then the oscillations have small enough amplitude to apply the regularity results of the flat case. On the contrary, when $r$ gets as small or even smaller than ${{\varepsilon}}$, one can rescale everything by a factor ${{\varepsilon}}$, so that oscillations of the boundary have frequency $O(1)$, and are no longer annoying. Implementation of this idea is a bit technical, and follows closely the work of Avellaneda and Lin.
We first remind a few elements of regularity theory for Stokes type systems. Let $\Omega$ an open connected bounded set, with [ *Lipschitz boundary*]{}. Then, for any $\varphi \in L^2(\Omega)$ satisfying $\int_\Omega \varphi = 0$, the problem $$\div w = \varphi, \quad w\vert_{{{\partial}}\Omega} = 0$$ has one solution $w$ satisfying $\| w \|_{H^1_0} \: \le \: C \, \| \varphi
\|_{L^2(\Omega)}$, where [*$C$ can be taken as an increasing function of $|\Omega|$ and of the Lipschitz constant $K$ of the boundary*]{}, see [@Galdi:1994]. Thanks to this result, one has quite easily, $$\label{pressure}
\| p - \int_\Omega p \|_{L^2(\Omega)} \: \le \: C \, \| \Delta u +f \|
\|_{H^{-1}(\Omega)}$$ where $(u,p) \in H^1(\Omega) \times L^2(\Omega)$ satisfies (in the distributional sense) $$\label{stokesbis}
-\Delta u + {{\nabla}}p = f, \quad \div u = 0, \quad x \in \Omega.$$ Again, the constant $C$ in depends only on $|\Omega|$ and the Lipschitz constant of the boundary.
We now state the famous Cacciopoli inequality:
For all $0 < r < 1$, any solution $u \in H^1(\Omega)$ of satisfies $$\label{cacciopoli}
\| {{\nabla}}u \|_{L^2(D^{{\varepsilon}}(x,r))} \: \le \: C \left( r^{-1} \, \|
u \|_{L^2(D^{{\varepsilon}}(x,2r))} \: + \: r^\mu \, \| f \|_{L^q(D^{{\varepsilon}}(x,2r))} \right).$$
[*Sketch of proof:*]{} We remind the main elements of proof. Let $\eta$ a smooth function with compact support in $D(x,2r)$, with $\eta=1$ on $D(x,r)$. Note that $|{{\nabla}}\eta| \le C r^{-1}$. Multiplying by the test function $\eta^2 u$, and integrating by parts, one has easily $$\begin{aligned}
\int_{D^{{\varepsilon}}(x,r)}|{{\nabla}}u|^2 \: & \le \: \int_{D^{{\varepsilon}}(x,2r)}\eta^2 |{{\nabla}}u|^2 \: \le \: C \, r^{-2} \, \int_{D^{{\varepsilon}}(x,2r)} |u|^2 \: + \:
C \, \int_{D^{{\varepsilon}}(x,2r)} |f|^2 \\
& \: + \:
\| p - \overline{p}_{x,2r} \|_{L^2(D^{{\varepsilon}}(x,2r))}
\, \| \div (\eta u) \|_{L^2(D^{{\varepsilon}}(x,2r))}.\end{aligned}$$ Using , we get $$\| p - \overline{p}_{x,2r} \|_{L^2(D^{{\varepsilon}}(x,2r))} \le
C \, \| \Delta u + \div f \|_{H^{-1}(D^{{\varepsilon}}(x,2r))} = C \| v
\|_{H^1(D^{{\varepsilon}}(x,2r))}$$ where $v \in H^1_0(D^{{\varepsilon}}(x,2r))$ is the solution of $$-\Delta v + {{\nabla}}p = \Delta u + \div f, \quad \div v = 0,
\quad v\vert_{{{\partial}}D^{{\varepsilon}}(x,2r)} = 0.$$ Note that the previous bound is uniform in ${{\varepsilon}}$, as it only involves the Lipschitz constant of $\omega^{{\varepsilon}}$ which is uniformly bounded. A simple energy estimate on $v$ gives $$\| {{\nabla}}v \|_{L^2(D^{{\varepsilon}}(x,2r))} \: \le \: C \left( \| {{\nabla}}u
\|_{L^2(D^{{\varepsilon}}(x,2r))} \: + \: \| f \|_{L^2(D^{{\varepsilon}}(x,2r))} \right)$$ As $\div(\eta u) = {{\nabla}}\eta \cdot u$, and using Hölder inequality on $f$, we end up with $$\begin{aligned}
\int_{D^{{\varepsilon}}(x,r)}|{{\nabla}}u|^2 \: \le \: \int_{D^{{\varepsilon}}(x,2r)}\eta^2 |{{\nabla}}u|^2 \: & \le \: C \, r^{-2} \, \int_{D^{{\varepsilon}}(x,2r)} |u|^2 \: + \:
C_\delta \, r^{2\mu} \, \|f\|^2_{L^q(D^{{\varepsilon}}(x,2r))} \\
& \: + \:
\delta \| {{\nabla}}u \|^2_{L^2(D^{{\varepsilon}}(x,2r))},\end{aligned}$$ where $\delta > 0$ is arbitrary small. We conclude as in [@Giaquinta:1982 Theorem 1.1, page 180].
Inequality of type has been used by Giaquinta and Modica in the study of elliptic regularity. In the context of Stokes type system, they obtain a local estimate, see [@Giaquinta:1982]:
\[modica\] Let $\Omega$ of class $C^1$, and $(u,p,f) \in H^1(\Omega) \times L^2(\Omega) \times L^q(\Omega)$, $q > 2$, satisfying $$-\Delta u + {{\nabla}}p = \div f, \: \div u = 0, \quad x \in \Omega(x_0,1),
\quad u\vert_{{{\partial}}\Omega \cap D(x_0,1)} = 0.$$ Then, $u \in C^{0,\mu}(\Omega(x_0,1/2))$ for $\mu=1-2/q$, and $$\label{modicaeq}
\| u \|_{C^{0,\mu}(\Omega(x_0,1/2))} \: \le \: C
\left( \| u \|_{L^2(\Omega(x_0,1))} \: + \: \| f \|_{L^q(\Omega(x_0,1))}
\right).$$
Unfortunately, we cannot use this theorem assuch. Indeed, the constant $C$ in the last regularity estimate involves the modulus of continuity of ${{\nabla}}\gamma$, where $x_2 = \gamma(x_1)$ describes the boundary. In our case $\gamma = \omega^{{\varepsilon}}$, such modulus of continuity is not uniformly bounded in ${{\varepsilon}}$. We must proceed in several steps to control the local integrals $I^{{\varepsilon}}_{x,r}$. Note that theorem \[modica\] implies estimate when $D^{{\varepsilon}}(x_0,1)$ is far from the boundary. Thus, we can restrict ourselves to a case in which $x_0$ is close to the oscillating boundary, for instance belongs to the axis $x_2 = 0$.
\[lemme1\] For all $\theta$ small enough, there exists ${{\varepsilon}}_0 > 0$ such that for all ${{\varepsilon}}< {{\varepsilon}}_0$, and for all solutions of satisfying $\displaystyle \quad \| u^{{\varepsilon}}\|_{L^2(D^{{\varepsilon}}(x_0,1/4))} \le 1$, $ \displaystyle \: \| f \|_{L^q(D^{{\varepsilon}}(x_0,1/4))} \le {{\varepsilon}}_0$, one has $$\| u^{{\varepsilon}}\|_{L^2(D^{{\varepsilon}}(x_0,\theta))}
\, \le \, \theta^{\mu+1}.$$
[*Proof of the lemma:*]{} Suppose that the result does not hold. Then one can find $\theta$ arbitrary small, and sequences $u^{{{\varepsilon}}_j}$, $f^j$ satisfying $$\| u^{{{\varepsilon}}_j} \|_{L^2(D^{{{\varepsilon}}_j}(x_0,1/4))} \le 1, \quad
\| f^j \|_{L^2(D^{{{\varepsilon}}_j}(x_0,1/4))} \xrightarrow[j \rightarrow +\infty]{}
0, \quad \| u^{{{\varepsilon}}_j}
\|_{L^2(D^{{{\varepsilon}}_j}(x_0,\theta))} > \theta^{\mu+1}.$$ One can extend all the $u^{{\varepsilon}}_j$, $f^j$ by $0$ outside $D^{{{\varepsilon}}_j}(x_0,1/4)$ so that all these functions are defined on the fixed domain $D(x_0,1/4)$. From the $L^2$ bound on $u^{{{\varepsilon}}_j}$, up to extract a subsequence, we get $$u^{{{\varepsilon}}_j} \mbox{ converges weakly to some } u \mbox{ in }
L^2(D(x_0,1/4))$$ and by Cacciopoli inequality , $$u^{{{\varepsilon}}_j} \mbox{ converges weakly to } u \mbox{ in }
H^1(D(x_0,1/8)), \mbox{ and strongly in }
L^2(D(x_0,1/8)).$$ One can then take the limit in , which yields $$-\Delta u + {{\nabla}}p = 0, \quad \div u = 0, \quad \mbox{in }
D(x_0,1/8) \cap \{x_2 >
0\}, \quad u\vert_{D(x_0,1/8) \cap \{ x_2 = 0 \}} =0.$$ As the upper half plane is a regular domain, we can apply theorem \[modica\], so that for all $\tilde{\mu} > \mu$, for all $\theta$, $$\begin{aligned}
\| u \|_{L^2(D(x_0,\theta) \cap \{x_2 > 0\})} \: & \le \: 2 \pi \, \| u \|_{C^{0,\tilde{\mu}}(D(x_0,\theta) \cap \{x_2 > 0\})}
\, \theta^{\tilde{\mu}+1} \\
& \le \: C \,\| u \|_{L^2(D(x_0,1/8)\cap \{x_2 > 0\})} \,
\theta^{\tilde{\mu}+1} \: \le \: C \, \theta^{\tilde{\mu}+1}. \end{aligned}$$ For $\theta$ small enough, it contradicts the lower bound on $\| u^{{{\varepsilon}}_j} \|_{L^2(D^{{{\varepsilon}}_j}(x_0,\theta))}$.
We fix $\theta$, ${{\varepsilon}}_0$ as in lemma \[lemme1\]. We then state
\[lemme2\] For all ${{\varepsilon}}, k$ satisfying ${{\varepsilon}}/\theta^k \le {{\varepsilon}}_0$ ($k \ge 1$), $$\label{localk}
\int_{D^{{\varepsilon}}(0,\theta^k)} |u^{{\varepsilon}}|^2 \:
\le \: \theta^{2k\mu+2} \, \left( \| u^{{\varepsilon}}\|_{L^2(D^{{\varepsilon}}(0,1/4))} \: + \:
\frac{1}{{{\varepsilon}}_0} \, \| f \|_{L^2(D^{{\varepsilon}}(0,1/4))} \right)^2$$
[*Proof of the lemma:*]{} The lemma is deduced from an induction argument on $k$. For $k=1$, the bound is given by lemma \[lemme1\]. Assume now that this bound holds for $k \ge 1$. Up to a horizontal translation, we can assume that $x_0 =0$. Then, we set $$M := \| u^{{\varepsilon}}\|_{L^2(D^{{\varepsilon}}(0,1/4))} \: + \:
\frac{1}{{{\varepsilon}}_0} \, \| f \|_{L^2(D^{{\varepsilon}}(0,1/4))},$$ and introduce the rescaled functions $$v \: := \: \theta^{-k\mu} M^{-1} u(\theta^k x), \quad g \: = \:
\theta^{k-k\mu} M^{-1} f(\theta^k x).$$ They satisfy $$-\Delta v + {{\nabla}}q = \div g, \quad \div v = 0, \: x \in
D^{{\varepsilon}}(0,\theta^{-k}/4), \quad v\vert_{\Gamma^{{\varepsilon}}(0,\theta^{-k}/4)}.$$ Moreover, one has $\displaystyle \| f \|_{L^q(D^{{\varepsilon}}(0,1/4))} \, \le \,
{{\varepsilon}}_0$, and by the induction assumption $$\displaystyle \| v \|_{L^2(D^{{\varepsilon}}(0,1/4))} \, \le \, 1.$$ Applying lemma \[lemme1\] to $v$ and $g$ yields the result.
We can now finish the proof of estimate . Let $x \in D^{{\varepsilon}}(x_0,1/2)$. We need to bound $I^{{\varepsilon}}(x,r)$, for $r > 0$. There are two cases to handle:
- [*The distance between $x$ and the boundary $\{ x_2 =
\omega^{{\varepsilon}}(x_1)\}$ satisfies $\delta^{{\varepsilon}}(x) \ge \frac{{{\varepsilon}}}{{{\varepsilon}}_0}$.*]{}
Up to take a smaller ${{\varepsilon}}_0$, we can suppose that $\frac{{{\varepsilon}}}{{{\varepsilon}}_0} >
{{\varepsilon}}$, which implies that there exists $x'_0$ on the axis $\{x_2=0\}$ with $|x - x'_0| \le \delta^{{\varepsilon}}(x)$. By lemma \[lemme2\], for all ${{\varepsilon}}/{{\varepsilon}}_0 \le r \le 1/12$, $$\label{deps3r}
\int_{D^{{\varepsilon}}(x'_0, 3r)} | u^{{\varepsilon}}|^2 \: \le \: C \, r^{2\mu+2} \left( \|
u^{{\varepsilon}}\|_{L^2(D^{{\varepsilon}}(x'_0,1/4))} \: + \: \| f \|_{L^q(D^{{\varepsilon}}(x'_0,1/4))}
\right)^2.$$ If $r > \delta^{{\varepsilon}}(x) /2$, $\: D^{{\varepsilon}}(x,r) \subset D^{{\varepsilon}}(x'_0,3r)$, and the previous line implies $$\int_{D^{{\varepsilon}}(x, r)} | u^{{\varepsilon}}|^2 \: \le \: C \, r^{2\mu+2} \left( \|
u^{{\varepsilon}}\|_{L^2(D^{{\varepsilon}}(x'_0,1/4))} \: + \: \| f \|_{L^q(D^{{\varepsilon}}(x'_0,1/4))}
\right)^2.$$ On the contrary, if $r \le \delta^{{\varepsilon}}(x) /2$, then $\: D^{{\varepsilon}}(x,r) \, = \,
D(x,r)$ (it does not intersect the boundary). A simple rescaling of yields $$\| u \|_{C^{0,\mu}(D(x,\delta^{{\varepsilon}}(x)/2))}
\: \le \: C \left( \delta^{{{\varepsilon}}}(x)^{-1-\mu} \| u^{{\varepsilon}}\|_{L^2(D(x,
\delta^{{\varepsilon}}(x)))}
\: + \: \| f \|_{L^q(D(x, \delta^{{\varepsilon}}(x)))} \right).$$ Thus, $$\int_{D^{{\varepsilon}}(x,r)} | u^{{\varepsilon}}|^2 \: \le \: C \, r^{2\mu+2} \, \left(
\delta^{{\varepsilon}}(x)^{-1-\mu} \| u^{{\varepsilon}}\|_{L^2(D(x, \delta^{{\varepsilon}}(x)))} \; + \:
\| f \|_{L^q(D(x, \delta^{{\varepsilon}}(x)))} \right)^2.$$ Now, by lemma \[lemme2\], as $\delta^{{\varepsilon}}(x) \ge {{\varepsilon}}/{{\varepsilon}}_0$, $$\begin{aligned}
\| u^{{\varepsilon}}\|_{L^2(D(x, \delta^{{\varepsilon}}(x)))}
\: & \le \: \| u^{{\varepsilon}}\|_{L^2(D(x'_0, 2 \delta^{{\varepsilon}}(x)))} \\
& \le \:
C \, \delta^{{\varepsilon}}(x)^{\mu+1} \, \left( \| u^{{\varepsilon}}\|_{L^2(D^{{\varepsilon}}(x'_0,1/4))} \: + \: \| f \|_{L^q(D^{{\varepsilon}}(x'_0,1/4))}
\right). \end{aligned}$$ Using the two last inequalities, we end up again with $$\int_{D^{{\varepsilon}}(x, r)} | u^{{\varepsilon}}|^2 \: \le \: C \, r^{2\mu+2} \left( \|
u^{{\varepsilon}}\|_{L^2(D^{{\varepsilon}}(x'_0,1/4))} \: + \: \| f \|_{L^q(D^{{\varepsilon}}(x'_0,1/4))}
\right)^2,$$ which in turn clearly implies $$\int_{D^{{\varepsilon}}(x, r)} | u^{{\varepsilon}}- \overline{u^{{\varepsilon}}}_{x,r} |^2 \: \le \: C
\, r^{2\mu+2}
\left( \| u^{{\varepsilon}}\|_{L^2(D^{{\varepsilon}}(x'_0,1/4))} \: + \: \| f
\|_{L^q(D^{{\varepsilon}}(x'_0,1/4))} \right)^2,$$ As $D^{{\varepsilon}}(x'_0,1/4)) \: \subset \: D^{{\varepsilon}}(x_0,1)$, this gives the required estimate.
- [*The distance between $x$ and the boundary $\{ x_2 =
\omega^{{\varepsilon}}(x_1)\}$ satisfies $\delta^{{\varepsilon}}(x) < \frac{{{\varepsilon}}}{{{\varepsilon}}_0}$.*]{}
This time, there exists $x'_0$ on the axis $\{ x_2 = 0 \}$ such that $|x - x'_0| \le \delta^{{\varepsilon}}(x) + {{\varepsilon}}\, \le \, 2{{\varepsilon}}/{{\varepsilon}}_0$. Again, for all $\: {{\varepsilon}}/{{\varepsilon}}_0 \le r \le 1/12$, $\: D(x,r)\subset D(x'_0,3r) \:$ and implies $$\int_{D^{{\varepsilon}}(x,r)} | u^{{\varepsilon}}|^ 2 \: \le \: C \, r^{2\mu+2} \left( \|
u^{{\varepsilon}}\|_{L^2(D^{{\varepsilon}}(x'_0,1/4))} \: + \: \| f \|_{L^q(D^{{\varepsilon}}(x'_0,1/4))}
\right)^2.$$ It remains to handle the case $r < {{\varepsilon}}/{{\varepsilon}}_0$. Up to a horizontal translation, we can assume that $x'_0=0$. We introduce the rescaled functions $$v \: = \: \left(\frac{{{\varepsilon}}}{{{\varepsilon}}_0}\right)^{-\mu}
u^{{\varepsilon}}\left(\frac{{{\varepsilon}}}{{{\varepsilon}}_0} x\right), \quad g \: = \:
\left(\frac{{{\varepsilon}}}{{{\varepsilon}}_0}\right)^{1-\mu} f\left(\frac{{{\varepsilon}}}{{{\varepsilon}}_0}
x\right).$$ They satisfy in particular $$-\Delta v + {{\nabla}}q = \div g, \quad \div v = 0, \: x \in D^{{{\varepsilon}}_0}(0,1),
\quad v\vert_{\Gamma^{{{\varepsilon}}_0}(0,1)} = 0.$$
It is a Stokes type system set in a domain independent of the small parameter ${{\varepsilon}}$. Hence, we can apply theorem \[modica\], which yields: for all $r < 1$, $$\int_{D^{{{\varepsilon}}_0}(x,r)} | v - \overline{v}_{x,r} |^2 \: \le \: C \, \| v
\|_{C^{0,\mu}(D^{{{\varepsilon}}_0}(x,r))} \, r^{2\mu+2}
\: \le \: C \, r^{2\mu+2} \, \left( \| v \|_{L^2(D^{{{\varepsilon}}_0}(0,2))} \: + \:
\| g \|_{L^2(D^{{{\varepsilon}}_0}(0,2))} \right)^2.$$ Back to the original unknowns $u^{{\varepsilon}}$, $f$, we obtain the control of $I^{{\varepsilon}}_{x,r}$ for $r \le {{\varepsilon}}/{{\varepsilon}}_0$. This ends the proof.
Bounds on the Green function
----------------------------
From the above Hölder estimate, we can deduce the estimate . The arguments are again adapted from article [@Avellaneda:1987 pages 819,829-831]. For the sake of completeness, we describe the ideas at play. The first step is to establish the following bound: for all $x,x'$ in $\{
x_2 > \omega^{{\varepsilon}}(x_1)\}$, $$\label{logG}
| G_{\omega^{{\varepsilon}}}(x,x') | \: \le \: C \, \left( \bigl| \ln |x-x'| \bigr|
\: + \: 1
\right).$$ where $C$ only involves $\| \omega \|_{C^{1,\alpha}}$. Note that it implies . To lighten the notations, we drop the suffix $\omega$, denoting $G^{{\varepsilon}}$, $G$ instead of $G_{\omega^{{\varepsilon}}}, G_\omega$. Let us introduce the Green function $\tilde{G}^{{\varepsilon}}(x,t,x',t')$ for the Stokes operator over $\{
x_2 > \omega^{{\varepsilon}}(x_1)\} \times {{\mathbb T}}$. Namely, it satisfies for all $(x,t)
\in \{ x_2 > \omega^{{\varepsilon}}(x_1)\} \times {{\mathbb T}}$ $$\left\{
\begin{aligned}
-\Delta \tilde{G}^{{\varepsilon}}(x,t,\cdot) + {{\nabla}}\tilde{P}^{{\varepsilon}}(x,t,\cdot) \: = \:
\delta_{x,t} I_3 &
\quad \mbox{ in } \{ x_2 > \omega^{{\varepsilon}}(x_1)\} \times {{\mathbb T}}, \\
\div \tilde{G}^{{\varepsilon}}(x,t,\cdot) = 0 & \quad \mbox{ in } \{ x_2 >
\omega^{{\varepsilon}}(x_1)\} \times {{\mathbb T}}, \\
\tilde{G}^{{\varepsilon}}(x,t, \cdot) = 0 & \quad \mbox{ on } \{ x_2 =
\omega^{{\varepsilon}}(x_1)\} \times {{\mathbb T}}.
\end{aligned}
\right.$$ One has easily that $$G^{{\varepsilon}}(x,x') \: = \: \int_{{{\mathbb T}}} \left( \tilde{G}^{{\varepsilon}}_1(x,0,x',t'),
\tilde{G}^{{\varepsilon}}_2(x,0,x',t') \right) \, dt'.$$ The point is to show that $$|\tilde{G}^{{\varepsilon}}(x,t,x',t')| \: \le \: C \, \frac{1}{ |x-x'| + |t-t'|}.$$ The estimate is then obtained by integration with respect to $t'$, at $t=0$. Such bound on $\tilde{G}^{{\varepsilon}}$ will be deduced from a repeated use of the Hölder estimate . Note that this estimate extends without difficulty to similar Stokes problems in dimension $n \ge 2$, with $q >
n$ and $\mu = 1 - n/q$. In particular, it holds when the domain is $\{
x_2 > \omega^{{\varepsilon}}(x_1)\} \times {{\mathbb T}}$.
Namely, let $\tilde{x} := (x,t)$ and $\tilde{x}' := (x',t')$. Let $r :=
|\tilde{x} - \tilde{x}'|$, and $f \in C^\infty_c(D^{{\varepsilon}}(\tilde{x}',r/3))$. We consider the quantity $$u^{{\varepsilon}}(\tilde{x}) \: = \: \int_{D^{{\varepsilon}}(\tilde{x}',r/3)}
\tilde{G}^{{\varepsilon}}(\tilde{x}, \tilde{z}) \, f(\tilde{z}) \, d\tilde{z}$$ The field $u^{{\varepsilon}}$ satisfies a Stokes equation with source term $f$ over $\{ x_2 > \omega^{{\varepsilon}}(x_1)\} \times {{\mathbb T}}$, with a Dirichlet boundary condition. We therefore apply the estimate to $u^{{\varepsilon}}$. Properly rescaled, it yields $$| u^{{\varepsilon}}(\tilde{x}) | \: \le \: C \, \left( r^{-3}
\int_{D^{{\varepsilon}}(\tilde{x},r/3)} |u^{{\varepsilon}}|^2 \right)^{1/2},$$ where we used the fact that $f$ vanishes over $D^{{\varepsilon}}(\tilde{x},r/3)$. Thus, we get $$\begin{aligned}
& \left| \int_{D^{{\varepsilon}}(\tilde{x}',r/3)} \tilde{G}^{{\varepsilon}}(\tilde{x}, \tilde{z})
\, f(\tilde{z}) \, d\tilde{z} \right| \: \le \: C \, \left( r^{-3}
\int_{D^{{\varepsilon}}(\tilde{x},r/3)} |u^{{\varepsilon}}|^2 \right)^{1/2} \\
& \le \: C' \, \left( r^{-3} \int_{D^{{\varepsilon}}(\tilde{x},r/3)} |u^{{\varepsilon}}|^6
\right)^{1/6} \: \le \:
C' \, r^{-1/2} \, \left( \int_{D^{{\varepsilon}}(\tilde{x},r/3)} |u^{{\varepsilon}}|^6
\right)^{1/6} \\
& \le \: C'' \, r^{-1/2} \left( \int_{\{ x_2 > \omega^{{\varepsilon}}(x_1)\} \times {{\mathbb T}}}
| {{\nabla}}u^{{\varepsilon}}|^2 \right)^{1/2} \: \le \: C'' \, r^{1/2} \| f \|_{L^2}^{1/2} \end{aligned}$$ where the last two inequalities come respectively from the Sobolev imbedding theorem (note that $u^{{\varepsilon}}$ is zero at the boundary so that the imbedding does not involve lower order terms), and from the standard energy estimate on the Stokes system. By duality, we infer that $$\left( r^{-3} \int_{D^{{\varepsilon}}(\tilde{x}', r/3)} |\tilde{G}^{{\varepsilon}}(\tilde{x},
\tilde{z})|^2 \, d\tilde{z} \right)^{1/2} \: \le \: C \, r^{-1}.$$ Using that $\tilde{G}^{{\varepsilon}}(\tilde{x}, \cdot)$ satisfies a homogenenous Stokes system over $D(\tilde{x}',r/3)$, one more application of leads to $$\tilde{G}^{{\varepsilon}}(\tilde{x}, \tilde{y}) \: \le \: C \, \left( r^{-3}
\int_{D^{{\varepsilon}}(\tilde{x}', r/3)} |\tilde{G}^{{\varepsilon}}(\tilde{x}, \tilde{z})|^2 \,
d\tilde{z} \right)^{1/2} \: \le \: C \, r^{-1}.$$
Inequality at hand, we can derive the final estimate on $G$. Let $x, x' \in \{ x_2 > \omega^{{\varepsilon}}(x_1) \}$. Set this time $r := |x - x'|$. For all $\bar{x}$ such that $|\bar{x}-x| < 2r$, implies $$\left| G^{{\varepsilon}}(\bar{x},x') \right| \: \le \: C \left( \bigl|
\ln |\bar{x}-x'| \bigr| +
1 \right)\: \le \: C' \left( \bigl| \ln |x-x'| \bigr| +1 \right).$$ Applying to the function $G^{{\varepsilon}}(\cdot,x')$, we get for any $\tau \in (0,1)$ $$\begin{aligned}
\left| G^{{\varepsilon}}(x,x') \right| \: & \le \: \, \delta^{{\varepsilon}}(x)^\tau
\, \| G^{{\varepsilon}}(\cdot ,x') \|_{C^{0,\tau}(D(x,r/3))} \\
& \le \: C_\tau \,
\delta^{{\varepsilon}}(x)^\tau \, r^{-1-\tau} \, \|
G^{{\varepsilon}}(\cdot ,x') \|_{L^2(D(x,2r/3))} \end{aligned}$$ that leads to $$\left| G^{{\varepsilon}}(x,x') \right| \: \le \: C_\tau \, \left( \bigl|
\ln |x-x'| \bigr| +1 \right) \,
\frac{\delta^{{\varepsilon}}(x)^\tau}{|x-x'|^\tau}.$$ Now reversing the roles of $x$ and $x'$, we obtain $$|G^{{\varepsilon}}(x,x')| \: \le \: C_\tau \, \left( \bigl| \ln |x-x'| \bigr|
+1 \right)
\frac{\delta^{{\varepsilon}}(x)^\tau \, \delta^{{\varepsilon}}(x')^\tau}{|x-x'|^{2\tau}}.$$ Using the scaling relation , we get for all $\: y,z \in
\{ y_2 > \omega(y_1) \}, \:$ for $\: {{\varepsilon}}:= |y-z|^{-1}$, $$G(z,y) = G^{{\varepsilon}}\left({{\varepsilon}}z, {{\varepsilon}}y \right) \: \le \:
C_\tau \left( \bigl| \ln \left({{\varepsilon}}|z-y| \right) \bigr| + 1 \right)
\frac{\delta^{{\varepsilon}}({{\varepsilon}}z)^\tau \, \delta^{{\varepsilon}}({{\varepsilon}}y)^\tau}{|{{\varepsilon}}(y-z)|^{2\tau}} = C_\tau
\frac{\delta(z)^\tau \, \delta(y)^\tau}{|y-z|^{2\tau}}.$$ Using classical local regularity results for the Stokes equation in a $C^{2,\alpha}$ domain (see [@Giaquinta:1982 Theorem 1.3, page 198], which extends theorem \[modica\]): for $|z-y| \ge 1$, $$\begin{aligned}
\sum_{|\beta| \le 2} |{{\partial}}^\beta_y G(z,y)| \: + \: |{{\nabla}}_y
P(z,y)| \: & \le \: C \, \| G(z,\cdot) \|_{L^2(D(y,1/2))} \\
& \le \: C \, \frac{ \delta(z)^\tau \, (1+ \delta(y))^\tau}{|z-y|^{2\tau}}\end{aligned}$$ that is exactly estimate .
Comments
========
Well-posedness of the boundary layer system
-------------------------------------------
As mentioned several times in this paper, the well-posedness of system is not known without a structural assumption on $\omega$, like periodicity or stationarity. We stress however that thanks to our estimates on the Green function $G_\omega$, the representation formula defines a solution of for any $C^{2,\alpha}$ boundary, [*cf*]{} the fourth section. Hence, the open issue is rather to find the appropriate functional space for uniqueness.
Such difficulty does not arise when the Stokes operator is replaced by the Laplacian, or more generally by a scalar elliptic operator. Hence, one can show well-posedness in $L^\infty$ of $$\label{BLbis}
-\Delta v = 0 \: \mbox{ in } \Omega^{bl},
\quad v(y) = \omega(y_1) \: \mbox{ on } {{\partial}}\Omega^{bl}$$ if the function $\omega$ is bounded and Lipschitz. For the existence part, one may consider, for all $n \ge 1$, the solution $v^n$ of $$-\Delta v^n = 0 \mbox{ in } \Omega^{bl} \cap D(0,n),
\quad v^n(y) = \omega(y_1) \: \mbox{ on } {{\partial}}\left( \Omega^{bl} \cap
D(0,n)\right).$$ By the maximum principle, $\| v^n \|_{L^\infty} \le \| \omega
\|_{L^{\infty}}$, so that up to a subsequence it converges to some $v$ in $L^\infty$ weak\*. Straightforwardly, $v$ satisfies .
For the uniqueness part, let $v \in L^\infty$ satisfying $$-\Delta v = 0 \: \mbox{ in } \Omega^{bl},
\quad v(y) = 0 \: \mbox{ on } {{\partial}}\Omega^{bl}$$ Let us show that $v=0$. As we do not know the behavior of $v$ at infinity, it does not follow directly from the maximum principle. In the case of a Lipschitz boundary $\omega$, we can conclude to the uniqueness in the following way. By the change of variables $y := \left(y_1, \, y_2 - \omega(y_1)\right)$, the previous equation becomes $$\div(A(y) {{\nabla}}v) = 0, \: y_2 > 0, \quad v\vert_{y_2 =0} = 0,$$ for some elliptic matrix $A = (a_{ij})$ with bounded coefficients. We extend $A$ and $v$ to $\{y_2 < 0\}$ by the formulas, $\: v(y_1, y_2) := -v(y_1, -y_2), \:$ $$\begin{aligned}
a_{in}(y_1, y_2) & := -a_{in}(y_1, -y_2), \quad
a_{nj}(y_1, y_2) := -a_{nj}(y_1, -y_2), \quad i,j \neq n, \\
a_{ij}(y_1, y_2) & := a_{ij}(y_1, -y_2)\quad \mbox{ otherwise. } \end{aligned}$$ In this way, we get $ \div(A(y) {{\nabla}}v) = 0$ on all ${{\mathbb R}}^2$. Harnack’s inequality for elliptic equations (see [@Gilbarg:2001]) leads to $$\sup_{|y| < R} (M + v) \: \le \: C \, \inf_{|y| < R} (M + v)$$ for any $R > 0$ and $M$ such that $M+u \ge 0$. With $M = max(0,-\inf v)$ and $R$ going to infinity, we obtain that $v = 0$.
Decay of correlations
---------------------
A key element in the paper is the estimate on the Green function for the Stokes operator above an oscillating boundary. This estimate relies itself on the Hölder regularity result . In fact, with some more calculations in the same spirit, one could show a refined bound: for a $C^{1,\alpha}$ boundary $\omega$ and a source term $f\in C^{0,\mu}(D^{{\varepsilon}}(0,1)) \:$ ($\alpha, \mu >0)$, the solution $u^{{{\varepsilon}}}$ of satisfies $$\| {{\nabla}}u^{{\varepsilon}}\|_{L^\infty(D^{{\varepsilon}}(0,1/2))} \: \le \: C \left( \| u
\|_{L^2(D^{{\varepsilon}}(0,1))} \: + \:
\| f \|_{C^{0,\mu}(D^{{\varepsilon}}(0,1))} \right),$$ with $C$ independent of ${{\varepsilon}}$. This yields an optimal bound on the Green function, that is $$\sum_{|\beta| \le 2} |{{\partial}}^\beta_y G_\omega(z,y)| \: + \: |{{\nabla}}_y
P_\omega(z,y)| \: \le
\: C \, \frac{ \delta(z) \, (1+ \delta(y))}{|z-y|^2}.$$ Estimate can in turn be improved as $$|v(\omega_1,0,0) - v(\omega_2,0,0)| \: \le \: C \, n^{-1}.$$ Such bound is far easier to prove when is replaced by the scalar system . In this case, one may use a representation in terms of the standard two-dimensional Brownian motion $ B(m,t) = \left(
B_1(m,t), B_2(m,t) \right)$. If we denote $(M, {\cal M},\mu)$ the probability space on which this Brownian motion is defined, $$v(\omega,0,0) \: = \:
\int_M \left(-\omega,0\right)\bigl(B_1(m, \tau(m)) \bigr) \, d\mu$$ where $\tau$ is the exit time from $\Omega^{bl}(\omega)$ (see [@Varadhan:1968]). We now want to bound $$v(\omega_1,0,0) - v(\omega_2,0,0) = \int_M \, \Bigl(
\omega_1\bigl(B_1(m, \tau_1(m)) \bigr) \, - \omega_2\bigl(B_1(m, \tau_2(m))
\bigr) \, \Bigr) d\mu$$ where $\tau_i$ is the exit time from $\Omega^{bl}(\omega_i)$. We remind that $\omega_1 = \omega_2$ over $[-n,n]$, so the exit times are the same for brownian particles leaving in the region $y_1 \in
[-n,n]$. Hence, $$\begin{aligned}
\left| v(\omega_1,0,0) - v(\omega_2,0,0) \right| \: & \le \: 2 \,
\max_{i=1,2} \: \| \omega_i\|_{L^\infty} \: \mu\left( |B_1(\cdot,
\tau_i)| \ge n \right) \\
& \le \: 2 \, \Bigl(
\mu\left( T_{-n} \le T_{-1} \right) + \mu\left( T_{n} \le
T_{-1} \right) \Bigr)
\end{aligned}$$ where we denote by $T_{\pm n}$ the first time for which $B_1 = \pm n /2$, and $T_{-1}$ the first time for which $ B_2 = -1$. It is well-known that the distributions of these hitting times are $$dT_{\pm n}\left( \mu\right) \: = \: \frac{n}{4 \sqrt{2\pi t^3}}
\exp\left(\frac{-n^2}{8t}\right) {\bf 1}_{t > 0} \, dt, \quad
dT_{-1}\left( \mu \right) \: = \: \frac{1}{ \sqrt{2\pi t^3}}
\exp\left(\frac{-1}{2t}\right) {\bf 1}_{t > 0} \, dt$$ A straightforward calculation provides $$\mu\left( T_{\pm n} \le T_{-1} \right) \: = \: \int_{0 \le t_1 \le
t_2} d T_{\pm n}\left( \mu \right)(t_1) \:\: dT_{-1}\left( \mu
\right)(t_2) \: \le \: C (n^2+1)^{-1/2}$$ which gives the result.
Optimality of the decay rate
----------------------------
Theorem \[theosharp\] shows that the boundary layer solution $v$ converges [*at least as*]{} $y_2^{-1/2}$. One may wonder if this result is optimal, that is if we can find roughness distributions for which the speed of convergence is exactly given by $y_2^{-1/2}$. In other words, is the constant $\sigma_{(0,0)}$ of the theorem positive for some random distribution of roughness ? We have not so far been able to show optimality in this setting, but it can be established for the easier Dirichlet problem $$\left\{
\begin{aligned}
\Delta u & = 0, \quad y_2 > 0 \\
u & = \omega, \quad y_2 = 0
\end{aligned}
\right.$$ where $\omega$ is a given boundary data. Although simpler, this system shares many features with the original system :
- If $\omega = \omega(y_1)$ is say $L$-periodic, the solution $u(y)$ converges exponentially fast to the constant $\alpha :=
L^{-1} \int_0^L \omega(y_1) dy_1$, as $y_2$ goes to infinity.
- If $\omega$ belongs as before to the probability space $(P,{\cal C},
\pi)$, one can show under assumption (H2) that $$y_2 \, {{\mathbb E}}| u(\omega,0,y_2) - \alpha |^2 \: \xrightarrow[y_2
\rightarrow +\infty]{} \: \sigma^2 \: \ge 0, \quad \alpha :=
{{\mathbb E}}(\omega \mapsto \omega(0)).$$
Along the lines of [@Varadhan:2007 pages 21-22], we will exhibit a stationary measure $\pi$ for which $\sigma >
0$. Of course, $\pi$ is the law of the random process $\varphi(\omega, y_1) := \omega(y_1)$, so that we just need to characterize the random initial data. Let $G(\omega,y_1)$ a gaussian random process, of zero mean and covariance $\rho(z_1-y_1)$, where $\rho \ge 0$ is a smooth even function with compact support. Note that such process exists: take $\rho = f * f$, with $f \ge 0$ an even smooth function with compact support. Then, its Fourier transform satisfies $\hat{\rho} = |\hat{f}|^2 \ge 0$, which ensures the required positivity property $$\sum_{z_1,y_1} \, c(z_1) \, c(y_1) \rho(z_1-y_1) \: =\ : \int_{{\mathbb R}}|
\sum_{z_1} c(z_1) e^{i\xi z_1} |^2 \hat{\rho}(\xi) \, d\xi \: \ge \: 0.$$ for any family $c$ with compact support. Note moreover that this process defines almost surely smooth functions of $y_1$: indeed, a simple calculation yields $${{\mathbb E}}\left( \int_{[-R,R]} |{{\partial}}_{y_1}^k X(\cdot,y_1)|^2 \, dy_1 \right)
\: = \: 2R \, (-1)^k \, \rho^{(2k)}(0) \: < \: \infty$$ so that $X(\omega, \cdot)$ is almost surely in the space $H^k_{loc}({{\mathbb R}})$ and therefore smooth. Finally, we introduce $$\varphi(\omega,y_1) \: = \: F(X(\omega,y_1))$$ for a smooth increasing function $F$ with values in $(0,1)$. We stress that $\varphi$ satisfies (H2) as $\rho$ has compact support. We will show that the corresponding $\sigma$ is positive. Suppose [*a contrario*]{} that $\sigma = 0 $. For $y_2 \ge 1$, we introduce the measure $\pi^{y_2}$ associated to the gaussian process with variance $\rho(z_1-y_1)$ but mean $$m(z_1,y_2) \: = \: \int \rho(z_1 - y_1) g(y_1,y_2) \, dy_1$$ where $g$ will be given later. Note that $\pi^{y_2}$ is associated to the random initial data $$\varphi^{y_2}(\omega,y_1) \: := \: F(X(\omega,y_1) + m(y_1,y_2)).$$ Standard computation yields $$\begin{aligned}
R(\omega,y_2) \: := \: \frac{d\pi^{y_2}}{d \pi}(\omega) \: = &
\: \exp\Bigl(\int g(y_1,y_2) \varphi(\omega,y_1) dy_1 \\
& - \, \frac{1}{2}
\, \int\int \rho(z_1 - y_1) g(z_1,y_2) \, g(y_1,y_2) dz_1 \, dy_1
\Bigr). \end{aligned}$$ and $$\int |R(\omega,y_2)|^2 d\pi = e^{H(y_2)}, \quad H(y_2) \: := \: \int\int
\rho(z_1 - y_1) g(z_1,y_2) \, g(y_1,y_2) dz_1 \, dy_1.$$ If $\sigma=0$, then a simple Cauchy-Schwartz inequality $$\int \sqrt{y_2} | u(\omega,0,y_2) - \alpha| d\pi^{y_2} \: \le \:
e^{\frac{1}{2} \, H(y_2)} \, \int y_2 | u(\omega,0,y_2) - \alpha|^2
d\pi$$ and [*goes to zero as $y_2 \rightarrow +\infty$ if $H$ is bounded from above*]{}.
Let $u$ and $v$ solutions associated to the initial data $\varphi(\omega,y_1)$ and $\varphi^{y_2}(\omega,y_1)$. As $m \ge 0$, by monotonicity, $v \ge u$. We can express $u$ and $v$ in terms of the Poisson Kernel, so that $$(v - u)(\omega,0,y_2) \: = \: \frac{2}{\pi} \int_{{\mathbb R}}\frac{y_2}{y_1^2 +
y_2^2} \left( \varphi^{y_2}(\omega,y_1) - \varphi(\omega,y_1) \right)
dy_1.$$ Now, we define for $y_2 \ge 1$ $$g(y_1,y_2) \: = \: \frac{1}{\sqrt{y_2}} \, G\left( \frac{y_1}{y_2}
\right),$$ where $G \ge 0$ has compact support, $G=1$ over $(-1,1)$. On one hand, with this definition of $g$, one can check that $$\sup_{y_2 \ge 1} H(y_2) \: = \: \sup_{y_2 \ge 1} y_2^{-1} \int\int
\rho(z_1 - y_1) G\left( \frac{z_1}{y_2}
\right) \, G\left( \frac{y_1}{y_2}\right) dz_1 dy_1 \: < \: +\infty$$ On the other hand, one has $$\begin{aligned}
& \int \sqrt{y_2} (v - u)(\omega,0,y_2) d\pi \: \ge \: C \int_{{\mathbb R}}\left(
\int F'(X(\omega,0)) d\pi \right) \frac{y_2}{y_1^2 + y_2^2} \,
\sqrt{y_2} \, m(y_1,y_2) \,
dy_1 \\
& \ge \: C' \, \int_{{\mathbb R}}\left( \int F'(X(\omega,0)) d\pi \right) \, \left(
\inf_{y_2 \ge 1} \inf_{|y_1| \le y_2} \int \rho(y_1
- s) G\left( \frac{s}{y_2}\right) ds \, \right)
\left( \int_{-y_2}^{y_2} \frac{y_2}{y_1^2 + y_2^2} dy_1 \right) \\
& \ge \: C'' \, \int_{{\mathbb R}}\rho(s) ds > 0. \end{aligned}$$ This implies that the quantity $ \int \sqrt{y_2} \, (v(\omega,0,y_2) - \alpha)
\, d\pi $ does not go to zero, leading to a contradiction.
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Acknowledgements {#acknowledgements .unnumbered}
================
The author warmly thanks S.R.S Varadhan for pointing to reference [@Varadhan:2007], as well as Luis Silvestre and Thierry Levy for fruitful discussions.
Appendix: Measurability of $\tilde{w}$ {#appendix-measurability-of-tildew .unnumbered}
======================================
We want to show here that $$\tilde{w}(\omega,z) \: := \: \int_{y_2=0} G_\omega(z,y) \, e_1 \, dy,
\quad z \mbox{ in } \Omega^{bl}(\omega),
\quad \tilde{w}(\omega,z) := 0 \quad \mbox{ otherwise}$$ defines a measurable function from $P$ to $H^1_{loc}({{\mathbb R}}^2)$. Let $0 \le \varphi_n \le 1$ a sequence of smooth functions with compact support, $\: \varphi_n\vert_{(-n,n)} =
1$. We define $$w_n \: := \: \int _{y_2=0} G_\omega(z,y) \, (\varphi_n e_1) \, dy,
\quad z \mbox{ in } \Omega^{bl}(\omega),
\quad w_n(\omega,z) := 0 \quad \mbox{ otherwise}.$$ Note that $w_n$ is the (unique) solution of $$\label{BL3}
\left\{
\begin{aligned}
& -\Delta w_n + {{\nabla}}q_n = 0,
\: x \in \Omega^{bl}\setminus \{ y_2 = 0\}, \\
& \div w_n = 0, \: x \in \Omega^{bl},\\
& w_n\vert_{{{\partial}}\Omega^{bl}} \: = \: 0,\\
& [ w_n ]\vert_{y_2 = 0} = 0, \quad [{{\partial}}_2 w_n - (0,q_n)]\vert_{y_2 = 0} =
(-\varphi_n,0),
\end{aligned}
\right.$$ satisfying $ \int_{\Omega^{bl}(\omega)} |{{\nabla}}w_n|^2 < +\infty$. By the dominated convergence theorem applied to the integral formula, we get that $w_n \rightarrow \tilde{w}$ in $L^2_{loc}$. By the Cacciopoli inequality, the convergence is also true in $H^1_{loc}$. Thus, we just have to show measurability of $w_n$.
Let us define $$V \: := \: \left\{ v \in \dot{H}^1({{\mathbb R}}^2), \quad \div v = 0 \right\},
\quad V_\omega \: := \: \left\{ v
\in V, \quad v\vert_{{{\mathbb R}}^2\setminus\Omega^{bl}(\omega)} = 0 \right\}.$$ Following the lines of [@Basson:2007 pages 15-16] it can be shown that the application $\omega \mapsto \pi(\omega)$, where $\pi(\omega) \in {\cal
L}(V,V)$ is the orthogonal projection from $V$ to $V_\omega$, is measurable. Now, $w_n$ is the unique fixed point of the contraction $$w \mapsto \frac{1}{2}\pi(\omega) \left( w - v_n \right)$$ where $v_n$ is the unique function of $H^1({{\mathbb R}}^2)$ satisfying $$\int_{{{\mathbb R}}^2} {{\nabla}}v_n \cdot {{\nabla}}\phi \: = \: 6 \int_{y_2 = 0} \varphi_n \,
\phi_1.$$ The measurability of $w_n$ follows.
[^1]: DMA/CNRS, Ecole Normale Supérieure, 45 rue d’Ulm,75005 Paris, FRANCE
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We evaluate the feasibility of the implementation of two quantum repeater protocols with an existing experimental platform based on a [$^{40}$Ca$^+$]{}-ion in a segmented micro trap, and a third one that requires small changes to the platform. A fiber cavity serves as an ion-light interface. Its small mode volume allows for a large coupling strength of $g_c = 2 \pi \times 20$ MHz despite comparatively large losses $\kappa = 2\pi \times 18.3$ MHz. With a fiber diameter of $125$ m, the cavity is integrated into the microstructured ion trap, which in turn is used to transport single ions in and out of the interaction zone in the fiber cavity. We evaluate the entanglement generation rate for a given fidelity using parameters from the experimental setup. The DLCZ protocol [@Duan2001] and the hybrid protocol [@Loock2006] outperform the EPR protocol [@1367-2630-15-8-085004]. We calculate rates of more than than 100 s$^{-1}$ for non-local Bell state fidelities larger than 0.95 with the existing platform. We identify parameters which mainly limit the attainable rates, and conclude that entanglement generation rates of 750 s$^{-1}$ at fidelities of 0.95 are within reach with current technology.'
author:
- 'A. D. Pfister$^{1\ast}$, M. Salz$^{1}$, M. Hettrich$^{1}$, U. G. Poschinger$^{1}$, F. Schmidt-Kaler$^{1}$'
date: '16.04.2016'
title: 'A Quantum Repeater Node with Trapped Ions: A Realistic Case Example'
---
=1
[example.eps]{} gsave newpath 20 20 moveto 20 220 lineto 220 220 lineto 220 20 lineto closepath 2 setlinewidth gsave .4 setgray fill grestore stroke grestore
Introduction {#sec:intro}
============
One of the most prominent applications in quantum technologies is quantum key distribution (QKD). The fundamental no-cloning theorem of quantum states [@Wooters1982] enables secure communication protocols [@Bennett1984; @Ekert1991]. For distances less than 80 km QKD is commercially available [@IDQuantique; @magiQ] using standard telecom fiber networks, and QKD networks have been set up in multiple locations, e.g. near Tokyo, Vienna and Boston. However, the transmission losses of 0.2 dB/km in fibers lead to restrictions at higher distances, as the secure key rate drops exponentially with the fiber length. Two potential solutions are currently being discussed: Free-space optical links have been established and tested between two of the Canary Islands [@Ma2012a], and between a ground station and a satellite [@Vallone2015]. On the other hand, the abundance and maturity of fiber optical networks appears to be appealing for use in QKD.
The proposal by Briegel, Dür, Cirac and Zoller [@Briegel1998] overcomes the distance limitation by converting the exponential drop in key rates to a polynomial one by using a network of *quantum repeaters* (QR). Here, entanglement is generated at QRs separated by a shorter distance where photonic channels are still efficient (Fig. \[fig:QrepPrinciple\]). By a sequence of entanglement swapping operations, entanglement is generated between the distant endpoints, commonly referred to as *Alice* and *Bob*. These entangled qubits allow Alice and Bob to perform QKD [@Ekert1991].
The building blocks of a QR following Briegel et al. (BDCZ-QR) are (i) an efficient interface between a flying (photonic) qubit and a long-lived stationary quantum memory, (ii) quantum logic operations on the memory, and (iii) error correction protocols [@Bennett1996; @Calderbank1996; @Reichle2006; @Terhal2015]. Practical considerations also suggest (iv) a wavelength-transformer such that the transmitted photons are near telecom wavelengths around 1.5 m, where fiber losses are minimal. An alternative proposal [@Munro2012] relaxes the requirement for long-lived quantum memories, but at the price of a substantially increased number of required qubits. A wide variety of possible implementations is currently being investigated [@Pirandola2015], based either on atomic systems and quantum optical techniques [@Kuzmich2003; @Duan2010; @Ritter2012; @Hucul2014] or on solid state quantum devices [@Press2008; @Gschrey2015]. However, up to date there has been no demonstration of a fully functional QR.
Here, we focus on a trapped ion approach: advantages are the high fidelity gate operations and state readout, and a long coherence time. Many of the requirements for (ii) are met and modern ion trap technology allows for scaling up to modest numbers of qubits required for a QR. Light-atom interfaces (i) in the quantum regime have been demonstrated by placing ions into high-finesse optical cavities [@Mundt2002; @Steiner2014; @Casabone2015]. Furthermore, single photon conversion to telecom wavelengths (iv) has recently been demonstrated [@Ikuta_2011; @Zaske2012]. However, the scalable combination of (i - iv) still appears to be very demanding technologically.
The scope of this work is a detailed investigation of the practical joint implementation of an ion-light interface (i) and quantum logic operations (ii) basic building blocks required for a BDCZ-QR. We do not address wavelength conversion (iv), fiber losses, classical verification and reconciliation protocols [@Brassard1994], or privacy amplification [@Bennett1988; @Deutsch1996]. While (iii) error correction is closely related to the subject of this manuscript, we only focus on the elementary link. We make a detailed comparison between three different entanglement distribution protocols and to identify critical experimental parameters limiting the performance.
After outlining the protocols in general (Sec. \[sec:prot\]), we describe the trapped ion experiment and its key parameters (Sec. \[sec:setup\]), and describe how to implement each of the protocols on the trapped-ion platform (Sec. \[sec:assess\]). Finally, we specify the attainable entanglement generation rates for each protocol. Based on this, we identify the protocols which are most suitable on our specific experimental platform.
![Scheme of long distance QKD using quantum repeaters (QR): a) Entanglement (indicated by colors) is generated between stationary qubits of adjacent repeater nodes via optical channels (black lines). b) A local Bell state measurement allows for an entanglement swapping procedure. c) Consecutive entanglement swapping operations ultimately leave the system with Alice (A) and Bob (B) having one entangled Bell pair. Finally, the Eckert protocol [@Ekert1991] allows for a secret QKD between Alice and Bob. []{data-label="fig:QrepPrinciple"}](1-1_Qrep_principle_v06.pdf){width=".47\textwidth"}
Quantum repeater protocol options {#sec:prot}
=================================
In this work, we will investigate the following entanglement distribution protocols:
- a scheme using *distributed Einstein-Podolski-Rosen-(EPR) states* from a source of entangled photons [@1367-2630-15-8-085004; @Lloyd2001]
- two variations of the *Duan-Lukin-Cirac-Zoller-\
(DLCZ) protocol* [@Duan2001; @Cabrillo1999; @Simon2003]
- a protocol using a combination of discrete and continuous variables dubbed *hybrid protocol* (HP) [@Loock2006]
The key parameters for the assessment of the protocols are the fidelities $F$ and rates of successful entanglement per second $r_e = P / t_d$, where $P$ is the success probability, and $1/t_d$ the repetition rate. The fidelities and rates of entanglement for all three protocols are inferred using experimentally determined or estimated parameters of the apparatus.
All the protocols have in common that they use photonic flying qubits, although the hybrid protocol is unique in that entanglement is distributed using a continuous variable encoded in a coherent light pulse instead of polarization entangled photons. Furthermore, all protocols feature heralding of entanglement creation.
Distributed EPR-states protocol {#EPR}
-------------------------------
One of the most prominent schemes proposed for quantum communication through multiple repeater stations is based on the idea of distributing the constituents of an entangled photon pair, e.g. polarization entangled photons from a spontaneous down conversion source. These are transmitted via fibers to neighboring QR nodes (N) and (N+1), c.f. Fig. \[fig:theo\]a). There, the photon state is mapped onto stationary qubits, giving rise to inter-node entanglement [@Lloyd2001]. Building blocks of this scheme have been realized e.g., in [@Kurz2014; @Schug2014].
For this protocol, the stationary qubits are initially prepared in a superposition state, ${\left| \Psi_i \right\rangle}_q={\left| 0_i \right\rangle} + {\left| 1_i \right\rangle}$. Throughout the paper, we omit the normalization of wavefunctions, unless the normalization factor is of specific interest.
A polarization-entangled photon pair in the state ${\left| \Psi \right\rangle}_p$ interacting with the stationary qubit couples both states ${\left| 0_i \right\rangle}$ and ${\left| 1_i \right\rangle}$, depending on its polarization, to levels of a short lived, excited state, ${\left| 0_e \right\rangle}$ and ${\left| 1_e \right\rangle}$: $$\begin{aligned}
\label{eq:epr:abs}
{\left| \Psi_i \right\rangle}_q^{\otimes 2} \otimes {\left| \Psi \right\rangle}_p =& ({\left| 0_i \right\rangle} + {\left| 1_i \right\rangle})^{\otimes 2} \big( {\left| \sigma^+,\sigma^- \right\rangle} +{\left| \sigma^-,\sigma^+ \right\rangle} \big) \nonumber \\
\xrightarrow{\text{\quad abs.\quad}}& {\left| 0_e,1_e \right\rangle} + {\left| 1_e,0_e \right\rangle} \nonumber \\
+& \big({\left| 0_e,0_i \right\rangle} + {\left| 0_i,0_e \right\rangle} \big) {\left| \sigma^- \right\rangle} \nonumber \\
+& \big({\left| 1_e,1_i \right\rangle} + {\left| 1_i,1_e \right\rangle} \big) {\left| \sigma^+ \right\rangle} \\
+&{\left| 0_i,1_i \right\rangle}{\left| \sigma^-,\sigma^+ \right\rangle} + {\left| 1_i,0_i \right\rangle}{\left| \sigma^-,\sigma^+ \right\rangle} \nonumber\end{aligned}$$ The excited states decay into a long-lived state, ${\left| 0_f \right\rangle}$ and ${\left| 1_f \right\rangle}$, leaving an entangled final state upon two-photon emission: $$\begin{aligned}
\label{eq:epr:em}
{\left| \Psi_f \right\rangle}_q=\big({\left| 0_f,1_f \right\rangle} + {\left| 1_f,0_f \right\rangle}\big).\end{aligned}$$ This decay gives access to herald photons via the spontaneous emission.
Both the initial states and the herald detection basis have to be chosen such that the decay via the distinct channels ${\left| 0_e \right\rangle}\rightarrow{\left| 0_f \right\rangle}$ and ${\left| 1_e \right\rangle}\rightarrow{\left| 1_f \right\rangle}$ is guaranteed. At the same time, the availability of which-path-information has to be prevented for preserving entanglement.
Due to a low absorption efficiency, in most cases either only one or zero photons of an EPR pair interacts at a node. We are interested in the probability of a single-photon-interaction taking place at either node within time $t$, after an initialized stationary qubit is exposed to the EPR source, which is given by $$\label{eq:eprP1}
P_1(t) = 1-e^{-r_1 \cdot t}.$$ The rate of single herald photon emission events $$\label{eq:eprr1}
r_1 = r_\text{EPR} \cdot \eta$$ depends on the brightness of the EPR source, $r_\text{EPR}$, and the probability $\eta$ of a photon from the EPR source to be injected into a cavity at a node and to interact with the stationary qubit.
The probability density for both photons of one EPR pair to interact each with its stationary qubit at time $t$, while *no* single-photon-interaction at neither node has happened before, is $$\label{eq:epr2dens}
p_2(t) = r_2 \, e^{-r_2 \cdot t}\, (1 - P_1(t))^2 = r_2\, e^{-(r_2 + 2 \,r_1) \cdot t}.$$ The rate of *both* photons from one EPR pair interacting with their respective stationary qubit is given by $$\label{eq:eprr2}
r_2 = r_\text{EPR} \eta^2.$$
In Eq. \[eq:epr2dens\] we explicitly exclude a single photon event taking place during time $t$, since such an event would change the state of a stationary qubit and thwart any two-photon event.
Integrating Eq. \[eq:epr2dens\] over $t$, we find the total probability of a successful two-photon mapping within time $t$ after initialization to be $$\label{eq:eprP2}
P_2(t) = \int_0^t{p_2(\tau) d\tau} = \frac{r_2}{r_2+2\, r_1}\big(1-e^{-(r_2+2 r_1)t}\big).$$
The emitted heralds are detected at a probability $P_\text{det}$, leading to a success probability for one experimental run of $$\label{eq:eprheraldprob}
P_e(t) = P_\text{det}^2 P_2(t).$$ Thus, the entanglement generation rate of stationary qubits at neighboring QR nodes is given by $$\label{eq:eprRate}
r_e = \frac{P_e(\tau_W)}{\tau_W+\tau_\text{prep}},$$ where $\tau_W\approx \frac{1}{2r_1}$ is the detection window for coincident herald photons. After expiry of $\tau_W$ without coincident herald detection, the stationary qubits at both nodes are re-initialized, which takes the time $\tau_\text{prep}$. Thus, for fixed $r_1$ and $\tau_\text{prep}$, $\tau_W$ can be chosen to provide an optimum entanglement rate.
![Basic principle for distant entanglement generation for the three protocols investigated. Stationary qubits with shared entanglement are indicated as red balls.\
**a)** Distributed EPR-states protocol. An entangled photon pair (red waves) is generated by an EPR source and sent to two adjacent repeater nodes. The quantum state of the photons is then mapped on a stationary qubit at each node, leaving one qubit at each node entangled with each other after detection of the herald photon (blue)\
**b)** DLCZ protocol. A local operation at each repeater node probabilistically generates a photon in either of the nodes. Interfering the possible photon paths on a beam splitter (BS) before detection entangles the qubits.\
**c)** Continuous variables hybrid protocol, adapted from [@Loock2006]. A coherent light pulse is split into a weak probe pulse (qubus) and a strong local oscillator (LO). The qubus interacts dispersively with the stationary qubit, resulting in a state dependent phase shift of the qubus. After transferring the qubus via the optical channel, the same operation is performed at the second repeater station. Measurement of the qubus phase leaves the system in an non-maximally entangled Bell state, for a correct detection pattern. []{data-label="fig:theo"}](2-1_protocol-principles_v06.pdf){width="48.00000%"}
DLCZ-protocol {#DLCZ}
-------------
This repeater protocol entangles stationary qubits at different repeater nodes by probabilistically creating a stationary qubit-photon pair in either of the nodes (N) and (N+1), see Fig. \[fig:theo\]b). A detection registering the arrival of one photon, but unable to distinguish the source node, projects the stationary qubits at the two nodes into a Bell state.
The stationary qubits are initialized in a state ${\left| q_0 \right\rangle}$. A laser beam then drives a cavity-induced stimulated Raman transition [@Stute2012] via the excited state ${\left| q_e \right\rangle}$ to a stable, final state ${\left| q_1 \right\rangle}$. The quantum state of the stationary qubit and the cavity mode at one node evolves as $$\label{eq:dlczevo}
{\left| q_0 \right\rangle}{\left| 0_c \right\rangle}\xrightarrow{\text{Raman}}{\left| q_0 \right\rangle}{\left| 0_c \right\rangle}+\sqrt{p_1}{\left| q_1 \right\rangle} {\left| 1_c \right\rangle}.$$ Here, ${\left| n_c \right\rangle}$ is the $n$-photon Fock state of the cavity mode.
The entanglement between neighboring nodes is achieved by a *single-photon detection scheme* [@Cabrillo1999]. Driving the Raman transition such that the transition probability fulfills $p_1 \ll 1$, the state evolution in both nodes is $$({\left| q_0 \right\rangle}{\left| 0_c \right\rangle})^{\otimes 2} \rightarrow {\left| q_0, q_1 \right\rangle}{\left| 0_c,1_c \right\rangle} + {\left| q_1,q_0 \right\rangle}{\left| 1_c,0_c \right\rangle},$$ where we have omitted the parts of the final wave function where either no or two photons is emitted. The emitted photons are transmitted to a detection setup, where a 50/50 beamsplitter in front of two detectors erases which-path information, see Fig. \[fig:theo\] b. This scheme results in the two-qubit wave function $$\big( {\left| q_1,q_0 \right\rangle} + {\left| q_0,q_1 \right\rangle} \big) {\left| A \right\rangle}+ \big( {\left| q_1,q_0 \right\rangle} - {\left| q_0,q_1 \right\rangle} \big) {\left| B \right\rangle},$$ where either detector $A$ or $B$ registers a photon.
The success probability of this entanglement creation, with single-photon emission probability $p_1$ and probability to detect the emitted photon $P_\text{det}$, is given by [@Zippilli2008] $$\begin{aligned}
\label{eq:dlcz1suc}
P_{1}=& 2\, P_\text{det}\, p_1\, ( 1- P_\text{det}\, p_1).\end{aligned}$$ The two-photon emission process omitted in Eq. \[eq:dlczevo\] leads to an infidelity of entanglement generation. Therefore, $p_1$ has to be chosen sufficiencly small in order to reach a given threshold fidelity $F_\text{thr}$, $$\label{eq:dlcz:1suc}
p_1 \leq \frac{1-F_\text{thr}}{1-P_\text{det} F_\text{thr}}.$$
The entanglement generation rate $r_e$ with success rate $P_1$ can be found by dividing $P_1$ by the required time per experimental run $\tau_\text{run}$:
$$\label{eq:dlczrate}
r_e = \frac{P_1}{\tau_\text{run}}.$$
Continuous variables hybrid protocol {#sub:Hyb}
------------------------------------
The two previous protocols operate on discrete variables, both for the flying and the stationary qubits. In contrast, the hybrid protocol [@Loock2006; @Ladd2006; @Loock2008] employs continuous variables for encoding photonic quantum information, while retaining the discrete stationary qubit, see Fig. \[fig:theo\]c).
The continuous variable is embodied by a coherent light pulse, termed *qubus*. The stationary qubits are initially prepared in a superposition state ${\left| \Psi_i \right\rangle}={\left| 0 \right\rangle}+{\left| 1 \right\rangle}$, and interact with the qubus, which is injected into a cavity, to enhance the interaction. The cavity field off-resonantly drives the transition ${\left| 1 \right\rangle}\leftrightarrow{\left| e \right\rangle}$ to an auxiliary excited state. The detuning $\Delta$ from this transition is much larger than the vacuum Rabi splitting, $\Delta\gg 2g$, such that the interaction is *dispersive*. The state of the stationary qubit is imprinted into the phase of the qubus state ${\left| \alpha \right\rangle}$. The Hamiltonian pertaining to this regime is given by $$\hat{H}_\text{int} = \hbar \frac{g^2}{\Delta} \hat{\sigma}_z \hat{a}^\dagger \hat{a}.$$ The operator $\hat{a}^\dagger$ ($\hat{a}$) is the creation (annihilation) operator of the field mode, and $\hat{\sigma}_z = {\left| 0 \right\rangle}{\left\langle 0 \right|}-{\left| 1 \right\rangle}{\left\langle 1 \right|}$ is the Pauli z-operator. This Hamiltonian describes an energy shift dependent on the state of the stationary qubit. The evolution operator of this Hamiltonian is $$\hat{U}_\text{int} = \exp{ \Bigg(-i \frac{\theta}{2} \hat{\sigma}_z \hat{a}^\dagger \hat{a}\, \Bigg) } ,$$ with a phase shift $$\theta = \frac{2 \, g^2}{\Delta}\, \tau_\kappa,$$ where $\tau_\kappa$ is the interaction time in the cavity. For a coherent state in the cavity, this leads to the following evolution of a superposition state of the stationary qubit: $$\hat{U}_\text{int}\big( {\left| 0 \right\rangle} + {\left| 1 \right\rangle} \big){\left| \alpha \right\rangle} = {\left| 0 \right\rangle}{\left| \alpha e^{-i\theta/2} \right\rangle}+{\left| 1 \right\rangle}{\left| \alpha e^{i\theta/2} \right\rangle}.$$
Neglecting losses, the interaction of the qubus with two stationary qubits in neighboring QR nodes leads to the state [@Loock2006; @Loock2008] $$\begin{aligned}
{\left| \Psi \right\rangle}={\left| \psi^+ \right\rangle}{\left| \alpha \right\rangle}& + {\left| 00 \right\rangle}{\left| \alpha e^{-i\theta} \right\rangle} + {\left| 11 \right\rangle}{\left| \alpha e^{i\theta} \right\rangle},\end{aligned}$$ A measurement determining the phase of the qubus projects the state into either one of the three components. The Bell state ${\left| \psi^+ \right\rangle}$ is generated in the QR nodes if no phase shift is detected.
We thus require that the *distinguishability* $$\label{eq:hyb:d}
d=\alpha \sin{\theta}$$ of the phase shifted states is sufficiently large to separate the coherent states in phase space [@Loock2006].
While a large amplitude of ${\left| \alpha \right\rangle}$ will guarantee a high distinguishability, it also leads to decoherence when losses in the transmission of the qubus are taken into account.. One thus faces a trade off between fidelity and efficiency.
Taking losses into account, we introduce a total transmission $\eta$ from the cavity in QR (N) to the cavity in QR (N+1), so that on average $(1-\eta)|\alpha|^2$ photons will be lost to the environment while the qubus propagates between neighboring nodes. Following [@Loock2008], we define the coherence parameter $$\label{eq:hybF}
\mu^2 = \frac{1}{2} \big( 1+e^{-(1-\eta) \alpha^2 (1-\cos{\theta})} \big).$$ The initial pure state evolves to a mixed state, after a local operation on each qubit, with a density matrix $$\label{eq:hyb:state}
{\left| \Psi \right\rangle}{\left\langle \Psi \right|} = \mu^2 {\left| \Psi^+ \right\rangle}{\left\langle \Psi^+ \right|}+(1-\mu^2){\left| \Psi^- \right\rangle}{\left\langle \Psi^- \right|},$$ where $$\begin{aligned}
\label{eq:hyb:phip}
{\left| \Psi^{\pm} \right\rangle} =& \frac{1}{\sqrt{2}}{\left| \sqrt{\eta}\alpha \right\rangle}{\left| \psi^{\pm} \right\rangle} \pm \frac{1}{2}e^{-i\eta\xi}{\left| \sqrt{\eta}\alpha e^{i\theta} \right\rangle}{\left| 11 \right\rangle} \\
& + \frac{1}{2} e^{i\eta\xi}{\left| \sqrt{\eta}\alpha e^{-i\theta} \right\rangle}{\left| 00 \right\rangle}. \nonumber \end{aligned}$$
The relative phase $\xi$ is of no further significance for our discussion. Projecting the wave function of the qubus to the non-phase-shifted part at detection selects the maximally entangled Bell states ${\left| \psi^\pm \right\rangle} = {\left| 10 \right\rangle} \pm{\left| 01 \right\rangle} $. A high distinguishability implies reliable identification of the Bell states, but mixes the two pure states ${\left| \Psi^\pm \right\rangle}$, whereas for low distinguishability, a pure state ${\left| \Psi^+ \right\rangle}$ is dominant, at the price of reduced success when identifying the Bell state ${\left| \psi^+ \right\rangle}$.
In order to collapse the wave function to the required part, it is necessary to identify the phase of a coherent state ${\left| \alpha \right\rangle}$. One possibility is *p-homodyne detection*, which requires a setup as depicted in Fig. \[fig:theo\] c. In this setup, the reference signal for homodyning is created by splitting a coherent pulse into the weak qubus signal and a local oscillator (LO) phase reference pulse, which does not interact with the cavities. The p-homodyne measurement amounts to a projection of the qubus state onto the p-quadrature of phase space. Following [@Ladd2006], for an acceptance window $-p_c<p<p_c$, we can assign a probability $P_S$ of a ’non-phase-shifted’ detection event and a fidelity $F$ of the resulting state,
$$\begin{aligned}
\label{eq:Phomo}
P_S =& \frac{1}{4} \Big(2 \operatorname{erf}{\big(\sqrt{2}p_c\big)}+\operatorname{erf}{\big(\sqrt{2}(p_c+\eta d)\big)} \\
& +\operatorname{erf}{\big(\sqrt{2}(p_c - \eta d)\big)}\Big) \nonumber \\
\label{eq:Fhomo}
F =& {\left\langle \psi^+ \right|}\rho{\left| \psi^+ \right\rangle} \nonumber \\
=&\frac{1}{4 P_S}\big(1 + e^{-d^2 (1-\eta^2)/2} \big)\operatorname{erf}{\big(\sqrt{2}p_c\big)}.\end{aligned}$$
In each of the presented protocols, a Bell state is generated first between nodes QR (N) and QR (N+1), and then between QR (N) and QR (N$-$1) in the network (Fig. \[fig:QrepPrinciple\] a). Entanglement swapping [@Bennett1993; @Riebe2008] then creates Bell states in nodes QR (N$-$1) and QR (N+1), which have twice the distance (Fig. \[fig:QrepPrinciple\] b). The repeated application of entanglement swapping finally leads to entanglement between the end nodes Alice and Bob.
Experimental platform {#sec:setup}
=====================
In this section, we describe the relevant components of our ion trap / cavity setup. We explicitly give quantitative parameters of this experimental platform, which are used in Sec. \[sec:assess\] to assess the performance of the different QR protocols. All the following components have been demonstrated to work as described in our labs. A single apparatus with all required parts integrated is not yet operative, the reported values thus pertain to similar trap apparatuses operated in our laboratories [@Schulz2008; @Poschinger2009].
Segmented microtrap {#sub:trap}
-------------------
![Sketch of the microstructured trap bearing the fiber cavity (see inset). The trap design provides access for the fiber cavity in the small processing region. The design of the trap assembly allows the cavity to be flexibly placed with respect to the trap axis.[]{data-label="fig:trap"}](3-1_trap_v04.pdf){width=".48\textwidth"}
Our setup consists of a variation of the segmented, microstructured Paul trap described in [@Schulz2006; @Schulz2008], adapted to accommodate a fiber based cavity similar to [@Hunger2010] as depicted in Fig. \[fig:trap\]. The trap segments are laser-machined out of two microfabricated, gold-sputtered alumina substrates. A spacer separates the two trap layers. The supply range of the dc segments of $\pm10$ V allow for axial trap frequencies of $2 \pi \times 0.2-4 $ MHz. Typically, $160-600$ V$_{pp}$, at drive frequencies in the range $2\pi\times$ 20-40 MHz, are applied to the RF electrodes, resulting in radial trap frequencies of $2\pi\times 2-4$ MHz. The fiber ends are shielded by the trap wafers from laser light, which enters through the trap slit perpendicular to the surface of Fig. \[fig:trap\].
Ion shuttling and separation
----------------------------
A field programmable gate array based arbitrary wave form generator controls the voltages of the dc electrodes [@Walther2012]. It supplies output voltages in the $\pm$10 V range with a resolution of 0.3 mV and analog update rates up to $2.5$ MSamples/s, while having low noise ($\lesssim 10$ nV rms at trap frequencies). Second-order -type low-pass filters for each segment suppress noise arising from voltage updates.
The segmented design allows for performing ion shuttling and ion separation. We have demonstrated fast shuttling over $280$ m in $3.6$ s with an increase in motional quanta of only $0.10(1)$ [@Walther2012; @Bowler2012], and ion crystal separation operations, with an average increase of $\approx 4$ motional quanta per ion in $80$ s for a separation distance of 500 m [@Kaufmann2014; @Ruster2014].
Qubit preparation, manipulation and readout {#sub:qubit}
-------------------------------------------
![Beam geometry of the experimental setup, viewed from the top. Arrows indicate directions for laser beam propagation and the quantization axis defined by the magnetic field $ \vec{B} $. $ R_1 $ and $ R_2 $: Beams used for driving Raman transitions between ${\left| S_{1/2},\pm 1/2 \right\rangle}$. Other lasers are described in the text. A lens in the inverted viewport (bottom) collects scattered 397 nm light (and for the EPR-protocol, also 393 nm light) for detection. The cavity axis is perpendicular to plane of view.[]{data-label="fig:chamber"}](3-2_chamber_v09.pdf){width=".485\textwidth"}
![Level scheme of $^{40}$Ca$^+$ with all relevant transitions. For details see text.[]{data-label="fig:lvl"}](3-3_Ca_termschema_v03.pdf){width=".48\textwidth"}
As the stationary qubit, we employ [$^{40}$Ca$^+$]{}-ions, where all relevant electronic transitions can be driven with commercially available diode lasers. [$^{40}$Ca$^+$]{} allows for encoding a *spin qubit*[@Poschinger2009], where the ground state levels ${\ensuremath{
\ifthenelse{\isempty{-1/2}{}}
{{\left| S_{1/2} \right\rangle}}
{{\left| S_{1/2},-1/2 \right\rangle}}}}$ and ${\ensuremath{
\ifthenelse{\isempty{+1/2}{}}
{{\left| S_{1/2} \right\rangle}}
{{\left| S_{1/2},+1/2 \right\rangle}}}}$ represent the logical states. Alternatively, we can utilize an *optical qubit*[@1367-2630-15-12-123012], where either one or two of the logical states is represented by one of the sublevels of the long-lived metastable D$_{5/2}$ state, see Fig. \[fig:lvl\]. The coherence times for both qubits is in the range of $10-100$ ms, limited by magnetic field fluctuation. As both qubit types are employed for the quantum repeater schemes analyzed in the manuscript, we give a short explanation of how the qubits are implemented.
**Ground state cooling:** For each experimental run, we start with Doppler cooling on the $S_{1/2} \leftrightarrow P_{1/2}$ cycling transition near 397 nm. We obtain a thermal state with a typical average phonon number of $\bar{n} \approx 20$ on the axial mode of vibration. We employ pulsed sideband cooling, by driving a stimulated Raman transition between the Zeeman ground states of the $S_{1/2}$ state, to cool close to the ground state of the axial mode. The repumping is accomplished by employing a circularly polarized laser field, driving the cycling transition. We typically attain average phonon numbers lower than 0.05 in the axial mode.
**Initialization:** The qubit can be initialized in the state ${\left| S_{1/2},-1/2 \right\rangle}$ with high fidelity ($>0.99$) by repetitively transferring population from ${\left| S_{1/2},1/2 \right\rangle}$ to the $D_{5/2}$ state, and quenching the population back into the $S_{1/2}$ state by driving the $D_{5/2} \leftrightarrow P_{3/2}$ transition with a laser field near 854 nm. The population transfer in the first step is done by driving -pulses on a suitable subtransition ${\left| S_{1/2},1/2 \right\rangle} \leftrightarrow {\left| D_{5/2},m_D \right\rangle}$, see Fig. \[fig:lvl\]. The laser pulses are derived from a laser source near 729 nm, stabilized to a linewidth of below 1 kHz. The frequency is controlled using an acousto-optical modulator. The natural linewidth of the ${\left| S_{1/2} \right\rangle}\leftrightarrow{\left| D_{5/2} \right\rangle}$ quadrupole transition of $2 \pi \times 0.14$ Hz and the narrow laser linewidth allow for selectively driving transitions between different Zeeman sublevels.
**Coherent manipulation of single qubits:** Based on the preparation in ${\left| S_{1/2},m_S \right\rangle}$, we can prepare arbitrary superposition states within $S_{1/2}$ and $D_{5/2}$ manifolds. Coherent rotations of the spin qubit are driven by stimulated Raman transitions. At a Raman detuning of about $2\pi\times100$ GHz, we achieve -times of a few s. Coherent rotations on the optical qubit are driven by a laser near 729 nm as explained for the initialization. Note that the coherent dynamics on the quadrupole transition depends on the motional state of the ions, such that we need to keep the ions in the Lamb-Dicke regime to achieve high-fidelity operations. By contrast, the rotations driven by radiofrequency or on the stimulated Raman transition are independent of the motional state. An arbitrary spin qubit state can thus be mapped to the $D_{5/2}$ manifold by using a quadrupole--pulse for each spin state. Note that coherent rotations between Zeeman sublevels of the same manifold can also be driven with radiofrequency pulses.
**State Readout:** Readout of the optical qubit is performed by fluorescence detection on the cycling transition with an EM-CCD camera or a photomulitplier tube (PMT). A bright event corresponds to the $S_{1/2}$ state, and a dark event the $D_{5/2}$ state. For the spin qubit, it is necessary to *shelve* one of the qubit states by transferring population from it with a -pulse on the quadrupole transition. The readout is then analogous to the case of the optical qubit. In both cases, the readout fidelities of $\approx 0.995$ can be obtained.
**Entangling gate:** We entangle two stationary spin qubits by means of the *geometric phase gate* [@Leibfried2003a], where spin-dependent dipole forces are employed to transiently excite motional modes depending on the spin configuration of an entire ion string. For spin configurations where a mode is displaced, a geometric phase is acquired. This conditional phase gives rise to entanglement. The spin-dependent dipole force is created by employing the off-resonant laser beams which are also used for driving stimulated Raman transitions. We achieve Bell-state fidelities of up to 97% (corrected for preparation and measurement errors) for gate durations of about 100 s. The entangled spin qubits can be converted to Bell states of optical qubits by -pulses on the quadrupole transition, as explained for the coherent manipulation.
**Cavity-induced stimulated Raman transition:** For the interaction between flying and stationary qubit, we can employ a cavity-induced stimulated Raman transition between the $S_{1/2}$ and $D_{5/2}$ states, driven by a laser off-resonant to the cycling transition and the cavity field. In this case, the coupling strength is given by $$\Omega^\text{eff}=\frac{G\, g\,\Omega_L}{\Delta},$$ where $\Omega_L$ is the on-resonance Rabi frequency of the laser, $g$ the cavity vacuum coupling rate and $\Delta$ the laser detuning. $G = cg \cdot \mathcal{P_\mathbf{d}}(\mathbf{\boldsymbol\epsilon})$ combines the Clebsch-Gordan coefficient $cg$ for both transitions with the projections $\mathcal{P_\mathbf{d}}(\mathbf{\boldsymbol\epsilon})$ of the polarization $\mathbf{\boldsymbol\epsilon}$ of the laser and the cavity field onto the ionic dipole moment $\mathbf{d}$, see Fig. \[fig:chamber\]. We estimate, based the cavity properties (Sec. \[sub:cavity\]) and the available laser power, that effective transition frequencies of $\Omega^\text{eff} \approx 1$ MHz are within reach.
Fiber based cavity {#sub:cavity}
------------------
In order to achieve a large coupling of the electronic state of an ion with the cavity mode, the mode volume is kept as small as possible. Due to its small size, a fiber based Fabry-Pérot-cavity[@Colombe2007; @Steiner2014; @Brandstaetter2013], where highly reflective dielectric mirrors are sputtered on end facets of optical fibers, can fulfill this requirement. It is suited to be accommodated between the two electrode chips, providing direct coupling into the cavity via one of the fibers.
The cavity drives the $D_{5/2}\leftrightarrow P_{3/2}$ transition near $\lambda=$854 nm. The respective field coupling parameter, i.e., the vacuum Rabi frequency $g_0$, is given by:
$$\label{eq_g0}
g_0= \sqrt{\frac{3 \, c \lambda^2 \gamma_\text{\tiny{PD}}}{\pi^2w_0^2L}}$$
where $ \gamma_\text{\tiny{PD}} = 2 \pi \times 0.67 $ MHz is the radiative field decay rate of the $D_{5/2} \leftrightarrow P_{3/2}$ transition. The cavity mode waist $w_0 $ is set by choosing the radius of curvature (ROC) and the length $ L $ of the cavity. The range of suitable values for $ L $ is predetermined by the trap dimensions: Both the fibers and the mirror surfaces are comprised of insulating materials, which are prone to uncontrolled charging when exposed to UV laser light, leading to uncontrolled electric stray fields in the trap[@Harlander2010; @Herskind2011]. However, the trap dimensions cannot be arbitrarily small, since short distances of the electrode surfaces to the ion increase anomalous heating rates[@Brownnutt2014] and the optical access needs to be ensured. The cavity length $ L $ is thus chosen sufficiently large, such that the fibers are retracted behind the trap electrodes (see Fig. \[fig:trap\]), reducing the electrical feedthrough of the charged insulating surfaces to the trap volume. We utilize a plano-concave cavity setup to reach a high mode matching $ \varepsilon $ between the mode that emanates from the fiber and the cavity mode. The ROC of the concave mirror can be chosen such that the waist is small while cavity stability and high mode matching are ensured.
In our case, the electrodes are separated by $ 250$ m (see Fig. \[fig:trap\]), which demands a large diameter concave mirror structure on the fiber facet to avoid finesse limitations by clipping losses. We developed a novel technique for shaping these facets using a commercial focused ion beam (FIB) device[^1], which allows us to create spherical structures with a large range of possible ROCs[^2]. Our cavity setup has a length of $L= 250~$m and consists of a singlemode fiber with a plane surface and a multimode fiber with $ 350$ m ROC concave facet. The facets are coated with dielectric mirror layers with a target transmission of 50(15) ppm at a wavelength of 854 nm[^3]. The linewidth was determined to $2\,\kappa= 2 \pi \times 36.6(5)~$MHz using frequency modulation as a frequency marker. The field decay rate of the cavity follows as $ \kappa = 2 \pi \times 18.3 (3) $ MHz. A finesse of $\mathcal{F}=1.65(2)\times 10^4$ is deduced. The cavity has a mode waist of $w_0=6.6$ m, i.e., the maximum cavity-ion coupling parameter at the plane mirror is $ g_0= 2\pi\times25.7~\text{MHz} $. In the cavity center, the coupling is reduced to $$\label{eq_gc}
g_c=\frac{w_0}{w(L/2)}g_0$$
The set of cavity parameters thus reads $ (g_c,\kappa,\gamma_\text{\tiny{PD}},\gamma_\text{\tiny{PS}})$ $=2\pi\times(20.1,18.3,0.67,10.7)~$MHz, which means that the cavity operates in the intermediate coupling regime. As the decay rate $P_{3/2}\rightarrow S_{1/2}$ is as strong as the coupling parameter, this system will not display resonant coherent dynamics on the $D_{5/2} \leftrightarrow P_{3/2}$ transition. However, excitation and off-resonant dynamics supported by the cavity can be utilized. Further effective reduction of the cavity coupling $ g_c $ due to geometrical considerations or Clebsch-Gordan coefficients of a particular atomic transition will be taken into account in the discussion of the protocol efficiencies in Sec. \[sec:assess\].
It has been recently shown [@Gallego2015] Mode matching for a fiber based cavity differs from the usual approach for Fabry-Pérot cavities as found, e.g. in [@meschede_book2005]. Most importantly, the minimum of the reflection signal no longer corresponds to the optimal incoupling. However, in our case we find minimal corrections, and will use the latter approach for brevity here. When coupling light into the cavity, two effects reduce the contrast of the reflection dip on resonance $\eta_\text{dip}= \varepsilon\cdot\eta_{\text{imp}}$[@Hood2001; @Gallego2015]: The mode matching $ \varepsilon $ and the impedance matching, described by the coefficient $ \eta_{\text{imp}} $, which depends on the transmission $ T $ of the cavity mirrors and the total losses $ \mathcal{L} $ per round trip. The losses are determined to be $ \mathcal{L}=280(30)~$ppm. For symmetric coating $ T_1=T_2\equiv T $:
$$\label{eq_eta-impedance}
\eta_{\text{imp}}=1-\left(\frac{ \mathcal{L}}{2\,T+ \mathcal{L}}\right)^2= (45.6\pm9.1)\%$$
Comparing this to the experimentally observed contrast of $ \eta_\text{dip}=20.3(1)\% $, one can find
$$\label{eq_epsilon}
\varepsilon=\frac{\eta_\text{dip}}{\eta_{\text{imp}}}=(44.5\pm 8.9)\%$$
The probability that a resonant photon emitted from the ion enters the cavity mode is enhanced by the Purcell effect and given by $$\label{eq:eta_P}
\eta_P = \frac{2 C_c}{2 C_c + 1}=0.97$$ where $C_c$ is the cooperativity in the cavity center, $$\label{eq:C_c}
C_c = \frac{g_c^2}{2 \, \gamma_\text{\tiny{PD}} \, \kappa}=16.5~.$$ Neglecting mode matching, such a photon then has the probability $$\label{eq:eta_CT}
\eta_{out} = \frac{T}{2T + \mathcal{L}} = 0.13$$ of leaving the cavity through the single mode mirror, which is the ratio of this mirror’s transmission loss to the sum of all loss channels of the cavity.
To ensure the frequency stability of the fiber cavity, the fiber cavity can be actively stabilized to a laser before each experimental shot.
The setup so far features only one input/output (I/O) port, in the form of the fiber-based cavity. In order to extend the setup to actual QR chains, the singlemode I/O fiber can also be equipped with a fiber switching device (i.e., before the left fiber of Fig. \[fig:sequence\]).
Assessment of the possible protocol implementations {#sec:assess}
===================================================
{width=".95\textwidth"}
In this section we investigate how the different repeater protocols can be implemented on our hardware platform. We specify the experimenal requirements and derive possible experimental sequences. For each protocol, we quantitatively estimate the attainable fidelities and entanglement generation rates for spatially separated Bell states at QR (N) and QR (N+1). Fig. \[fig:sequence\] depicts the experimental sequence at QR (N) in the chain of repeater nodes (see Fig. \[fig:QrepPrinciple\]). The neighboring QRs have an analogous sequence.
All protocols have similar initialization sequences in the beginning and entanglement swapping sequences in the end, see Sec. \[sub:qubit\]. The duration of the entire protocol run in combination with the success probability determines the entanglement generation rate of the protocol. The time necessary to run one entangling sequence, from initialization to entanglement of a local ion with a distant one, is called $\tau_\text{run}$, and is split into a preparation time $\tau_\text{prep}$, and $\tau_e$. The latter is the time each protocol requires to entangle the two distant ions, once they are initialized and in the cavity. The time $\tau_\text{prep}\approx 210$ s includes the initialization of the ion state $\tau_\text{init} \approx 10$ s, the shuttling of the ion into the cavity $\tau_S \approx 100$ s, as well as the shuttling back to the processor region to be re-initialized if the protocol does not succeed, which again takes the time $\tau_S$. It does not include the post-processing of the ions necessary after the successful entanglement has been heralded. Errors *before* the herald detection decrease the success probability, whereas errors obtained *after* the herald detection reduce the fidelity $F$.
Distributed EPR protocol {#sub:exEPR}
------------------------
To implement the EPR protocol in our apparatus, we finish the initialization process from Sec. \[sub:qubit\] by creating a superposition between the states ${\ensuremath{
\ifthenelse{\isempty{-3/2}{}}
{{\left| D_{5/2} \right\rangle}}
{{\left| D_{5/2},-3/2 \right\rangle}}}}$ and ${\ensuremath{
\ifthenelse{\isempty{+3/2}{}}
{{\left| D_{5/2} \right\rangle}}
{{\left| D_{5/2},+3/2 \right\rangle}}}}$, in the following denoted as ${\left| D_1 \right\rangle}$ and ${\left| D_2 \right\rangle}$, using suitable pulses on the $S \leftrightarrow D$ quadrupole transition (see Fig. \[fig:eprlvl\]).
We thus choose the starting state of each stationary qubit as (Fig. \[fig:eprlvl\](i)) $${\left| \Psi_D(t) \right\rangle}={\left| D_1 \right\rangle}+e^{i\phi_D(t)}{\left| D_2 \right\rangle},$$ with the phase $$\label{eq:epr:phase}
\phi_D(t)=(\Delta m_D g_D - \Delta m_S g_S)\frac{\mu_B}{\hbar}B t + \phi_0.$$ The $\Delta m_i$ are the differences of the magnetic quantum numbers of the respective manifold, the $g_i$ are their Landé factors, $\mu_B$ is the Bohr magneton, and $\phi_0$ the phase offset at initialization.
After transporting the ion into the cavity, an EPR source between QR (N) and QR (N+1) provides the polarization entangled photon pairs as flying qubits for this repeater protocol, producing the state $$\begin{aligned}
{\left| \Psi_\text{EPR} \right\rangle} = {\left| L \right\rangle}{\left| R \right\rangle}+{\left| R \right\rangle} {\left| L \right\rangle}\end{aligned}$$ with left-handed (L) and right-handed (R) circularly polarized light. To ensure that the frequency and linewidth of the photon match the $D_{5/2} \leftrightarrow P_{3/2}$ transition close to $854$ nm, one use a Single Parametric Downconversion (SPDC) source with a filter cavity [@Schuck2010]. The production rate of entangled photon pairs after cavity filtering is denoted $r_\text{EPR}$.
![Level scheme of a $^{40}$Ca$^+$-ion with relevant transitions for the distributed EPR protocol. Encircled numbers indicate protocol steps. (i) After cooling and pumping to the $ {\left| S_2 \right\rangle} $ state, the ion is initialized in a superposition of $ {\left| D_1 \right\rangle} $ and $ {\left| D_2 \right\rangle} $ (dashed red arrows). (ii) In the cavity, the entangled photon induces a transition to one of the $ {\left| P \right\rangle} $-states, depending on its polarization, $ \sigma^+ $ or $ \sigma^- $ (dark red arrows). (iii) The ion quickly decays into one of the $ {\left| S \right\rangle} $-states under emission of a photon at 393 nm as herald. By filtering out circular polarizations (light blue waves), only the events which preserve the information (dark blue wave) are detected.[]{data-label="fig:eprlvl"}](4-1-1_EPR_termschema_v02.pdf){width=".45\textwidth"}
The photons are coupled into the fibers whose ends constitute the cavities of QR (N) and QR (N+1) with an efficiency $\eta_{FC}$, and from there into the cavities with an incoupling efficiency between fiber and cavity $\eta_\text{dip}$. A photon successfully injected into the cavity of the node interacts with the ion (Fig. \[fig:eprlvl\](ii)) with a probability $P_{\text{int}}$, giving rise to the following state evolution:
$$\begin{aligned}
\label{eq:eprtrans}
&{\left| \Psi_D \right\rangle} \times {\left| \Psi_\text{EPR} \right\rangle} & \\
\xrightarrow{abs.}& {\left| P_1 \right\rangle}+ e^{i \phi_D} {\left| P_2 \right\rangle} & \nonumber \\
\xrightarrow{em.}& \big({\left| S_1 \right\rangle} + e^{i \phi_D} {\left| S_2 \right\rangle}\big){\left| \pi_{393} \right\rangle} \nonumber\end{aligned}$$
$\phi_D$ is taken from Eq. \[eq:epr:phase\], at the time $t$ when the ion absorbs the photon and decays into $S$.
In the case of success, the interaction maps the state of one photon onto the two Zeeman sublevels of the ion’s $S_{1/2}$ state, under creation of a -polarized herald photon at a wavelength near $393$ nm (Fig. \[fig:eprlvl\](iii)), detected with probability $P_\text{det}$. A detection event in each node’s herald detector, within a detection window $\tau_c$ of each other, denotes a successful entanglement of ions in QR (N) and QR (N+1), after which the ion is moved out of the fiber cavity, and the second ion is entangled with an ion at QR (N$-$1).
We assume a rate of EPR photon pairs resonant with the $D_{5/2}\leftrightarrow P_{3/2}$ transition is $r_\text{EPR}=7800$ s$^{-1}$ [@Huwer2013], after the filter cavity. Throughout this section, it is assumed that only one photon ever populates the cavity mode at the same mode. This approximation breaks down as soon as the cavity population decay rate $2 \, \kappa$ is no longer considerably greater than the EPR arrival rate at a single cavity, $\kappa\not\gg r_{\text{EPR}}\,\eta_{FC}\,\varepsilon$. With $\kappa = 2 \pi \times 18.3$ MHz, such rates of EPR pair production are beyond current technological proficiency, and our approximation holds.
Solving the Liouville master equation for our setup, including all sublevels of the $S$, $P$ and $D$ states, with cavity parameters as given in Sec. \[sub:cavity\], returns the interaction probability of $P_\text{int} = 0.047$. Required is a Zeeman splitting less than the cavity bandwidth $2 \, \kappa$, thus limiting the magnetic field to $B < 11$ G in order to ensure that the cavity field drives the transitions $D_1\rightarrow P_1$ and $D_2 \rightarrow P_2$ at similar rates.
The $\eta$ in Eqs. \[eq:eprr1\] and \[eq:eprr2\], the probability of one of the photons after the filter cavity to interact with one ion, is given by $$\begin{gathered}
\eta = \eta_{FC}\, \eta_\text{in} \, P_\text{int}\approx 0.005 \\
\eta^2 \approx 3 \times 10^{-5} \,.\nonumber\end{gathered}$$ Here, $\eta_{FC} \approx 0.9$ is the efficiency of coupling the photons into the fiber after the filter cavity, and is the probability of coupling a single photon into the cavity, by time-reversal symmetry to the outcoupling process.
The detection probability $$P_\text{det} = \frac{d\Omega}{4\pi}\, \eta_{QE}^{397} = 0.007$$ depends of the solid angle of our detection lens $d\Omega/4\pi$, and the quantum efficiency $\eta_{QE}^{397}$ of the PMT for UV light (Sec. \[sec:setup\]).
**Entanglement generation Rate:** With these efficiencies, the entanglement generation rate, Eq. \[eq:eprRate\], takes on the shape $$\begin{aligned}
\label{eq:eprre}
r_e(\tau_W) =& \frac{P_e(\tau_W)}{\tau_\text{prep} + \tau_W}\end{aligned}$$ for a given time to wait on an entanglement event $\tau_W$.
We have yet to account for dark counts on the detectors, which both increase the apparent rate of successful events and decrease the fidelity of our final state. A typical dark count rate for UV detectors can be assumed to be $r_\text{dc}=60$ s$^{-1}$ [@Kurz2015]. For a detection window $\tau_c$, during which detector events are counted as concurrent, the probability of registering two simultaneous dark counts is given by $$\label{eq:2dc}
P_{2dc}(t,\tau_c) = (1-e^{-r_\text{dc} t})\cdot(1-e^{-r_\text{dc} \tau_c})\,.$$ Similarly, a photon-qubit interaction in one of the nodes can be concurrent with a dark count in the other, with a probability of $$\label{eq:hdc}
P_{hdc}(t,\tau_c)= P_\text{det} P_1(t)\cdot(1-e^{-r_\text{dc} \tau_c}) \,,$$ with $P_1(t)$ from Eq. \[eq:eprP1\] in Sec. \[EPR\].
The probability $P'_e(t,\tau_c)$ of any two-detector event, true or false positive, happening during time $t$, where the window for coincidenct detection is set to $\tau_c$, is given by $$\begin{gathered}
\label{eq:EPRRate}
P'_e(t,\tau_c) = P_e(t) + P_{2dc}(t,\tau_c) + P_{hdc}(t,\tau_c) \\
\Rightarrow r'_e=\frac{P'_e(t,\tau_c)}{\tau_\text{prep} + \tau_W +\tau_c}\,.\end{gathered}$$
We set $\tau_W=r_1^{-1} \approx 15$ ms (see Eq. \[eq:eprr1\]) to the time by which we can expect one of the photons to have interacted with an ion, making a new initialization necessary. Using $\tau_\text{prep}\approx 210$ s (Sec. \[sec:assess\].0), and $\tau_c =\frac{1}{20 \kappa}$, the repetition rate becomes $r_e(\tau_W) = 6.4 \times 10^{-6}$. Inserting the values of the setup, this culminates in a rate $r'_e \approx 7.4 \times 10^{-6} $ s$^{-1}$, i.e. about one event every 35 hours. We see that a highly efficient detection of herald photons at 397 nm together with an effective in-coupling of EPR photons into the cavity is required to improve this rate.
**Fidelity:** The fidelity of the final state is limited by the ratio of real entanglement events to total heralded events, $F = r_e/r'_e \approx 0.86$. Another fidelity error is the imperfect filtering of -light for detection optics that is not pointlike in extent. For a solid angle of $d\Omega= 4\pi \cdot 0.035$ of the collecting lens, the opening angle is $\theta=21^\circ$. However, integrating the arriving herald wave function over this area shows the amount of -light in the wrong mode to be negligible.
Finally, imperfect initialization and readout (Sec. \[sub:qubit\]) reduce the fidelity to $F \approx 0.86 \cdot 0.995$ for the EPR protocol.
DLCZ protocol {#sub:exDLCZ}
-------------
![Level scheme with relevant transitions for the single-photon DLCZ protocol. After initialization in the $ {\left| S \right\rangle} $-state, a cavity-induced stimulated Raman transition is driven to $ {\left| D \right\rangle} $ via the $ {\left| P \right\rangle} $ state, consisting of a Raman pulse (blue arrow) with detuning $ \Delta $, and the cavity fulfilling the Raman resonance condition with the beam. A single cavity photon is generated for both nodes, and the detection setup eliminates which-way information (see Fig. \[fig:theo\] b) to entangle the nodes in ${\left| S,D \right\rangle}+{\left| D,S \right\rangle}$.[]{data-label="fig:dlczlvl1"}](4-2-2_DLCZ_termschema2_v01.pdf){width=".4\textwidth"}
The implementation of the DLCZ protocol in our apparatus, depicted in Fig. \[fig:dlczlvl1\], starts with initializing the ion in the state
$${\left| S \right\rangle} = {\ensuremath{
\ifthenelse{\isempty{-1/2}{}}
{{\left| S_{1/2} \right\rangle}}
{{\left| S_{1/2},-1/2 \right\rangle}}}}.$$
After transport of the ion into the cavity, a photon is created by a cavity-induced stimulated Raman transition (Sec. \[sub:qubit\]) between the $S_{1/2}$ and the $D_{5/2}$ manifolds, with ${\left| 0 \right\rangle} = {\left| D \right\rangle}$ and ${\left| 1 \right\rangle} = {\left| S \right\rangle}$ chosen as shown in Fig. \[fig:dlczlvl1\].
The photon is emitted into the cavity with a probability $p_1$, which is controlled by the duration of the Raman drive. It is detected with a probability $P_\text{det}$ once emitted. This probability is given by $$\label{eq:dlczdet}
P_\text{det} = \eta_{P} \cdot \eta_\text{out} \cdot \varepsilon \cdot \eta_{QE}^{854}\,,$$ where $\eta_P$ is the probability that the photon is emitted into the cavity mode through Purcell enhancement, $\eta_\text{out}$ is the the cavity outcoupling coefficient, $\varepsilon$ the mode matching efficiency between cavity and fiber and $\eta_{QE}^{854} = 0.5$ the quantum efficiency the photon detectors near $ 854$ nm. All of these parameters are defined and given in Sec. \[sec:setup\], and lead to a detection efficiency of about $P_\text{det}=0.03$.
For a threshold fidelity of $F_\text{thr} = 0.99$, and thus a single photon emission probability $p_1 = 0.01$, these parameters result in a success probability for one experimental run of $P_1 \approx 6 \times 10^{-4}$ according to Eq. \[eq:dlcz1suc\].
**Entanglement generation rate:** Each single experimental run can be done in time $\tau_\text{run}=\tau_\text{prep}+\tau_{R}+\tau_{\kappa}$, which is the sum of the preparation time $\tau_\text{prep}$, the Raman pulse time $\tau_R$, and the cavity decay time $\tau_\kappa$ (negligible in our case). For a threshold fidelity of $F_\text{thr} = 0.99$, the single photon emission probability must fulfill $p_1 \le 0.01$ according to Eq. \[eq:dlcz:1suc\]. A typical duration of the Raman drive pulse is $\tau_R = 2$ s. The preparation time $\tau_\text{prep}\approx 210$ s is substantially longer, leading to $\tau_\text{run} \approx 212$ s. The entanglement generation rate for these parameters results in $r_e \approx 2.8$ s$^{-1}$.
**Fidelity:** The fidelity loss due to events where two photons are generated from driving the stimulated Raman transition is preset by choice of $F_\text{thr}$ and corresponding tuning of the drive pulse area. Additional fidelity loss is due to dark count events. However, choosing a detection window of $\tau_\text{det}=2\tau_\kappa$ after the Raman pulse in order to capture the majority of real events, with typical dark count rates of $20$ Hz for IR detectors, erroneous events are 5 orders of magnitude less frequent than entangling events.
Imperfect initialization and readout (Sec. \[sub:qubit\]) also reduce the fidelity by a factor of $F_\text{init} \approx 0.995$, similar to the EPR protocol, so the final fidelity for $F_\text{thr}=0.99$ is $F_1\approx 0.99 \cdot 0.995$ .
**Two-photon-detection DLCZ:** We also investigate an alternative version of the DLCZ protocol based on deterministic photon emission and coincident two-photon detection [@Simon2003; @Barros2009].
![Level scheme with relevant transitions for the two-photon DLCZ protocol following [@Stute2012]. After initialization in the $ {\left| S \right\rangle} $-state, two Raman transitions are driven simultaneously to $ {\left| D_1 \right\rangle} $ and $ {\left| D_2 \right\rangle} $ via the $ {\left| P \right\rangle} $ state, consisting of a bichromatic Raman pulse (blue arrows) with detunings $ \Delta_1 $, $ \Delta_2 $ and two modes of the cavity ($ V $ and $ H $ with respect to the cavity axis, red arrows). A cavity photon is generated, whose polarization $ V $ or $ H $ is entangled to the electronic state of the ion, $ {\left| D_1 \right\rangle} $ or $ {\left| D_2 \right\rangle} $, respectively.[]{data-label="fig:dlczlvl2"}](4-2-1_DLCZ_termschema_v02.pdf){width=".4\textwidth"}
![Alternative to the DLCZ method introduced in Sec. \[DLCZ\]. Both nodes emit a photon induced by a Raman transition, which is entangled with the ion according to Eq. \[eq:dlczRaman2\]. The photons are brought to interference within a detection setup as pictured. Coincident clicks in detectors {AB, CD, AC, BD} project the two ions onto a Bell state. []{data-label="fig:dlczDet2"}](4-2-3_DLCZ_detection2_v04.pdf){width=".4\textwidth"}
The single Raman beam of Sec. \[DLCZ\] is substituted by a bichromatic Raman beam, so that two cavity-induced stimulated Raman transitions are driven, see Fig. \[fig:dlczlvl2\]: $$\label{eq:dlczRaman2}
{\left| S \right\rangle}{\left| 0 \right\rangle}_p\xrightarrow{\text{Raman}}{\left| D_1 \right\rangle} {\left| V \right\rangle} + {\left| D_2 \right\rangle} {\left| H \right\rangle}\,.$$ Note that ${\left| D_1 \right\rangle}$ and ${\left| D_2 \right\rangle}$ differ from the levels defined in Sec. \[sub:exEPR\]. The photons emitted on the - and -transitions are mapped to the ${\left| V \right\rangle}$ and the ${\left| H \right\rangle}$ mode of the cavity by setting the quantization axis of the ion to right angles with the cavity axis.
The stationary qubits are entangled by two-photon detection. As [@Zippilli2008] elaborates, this requires a 50/50 beamsplitter, with a polarizing beamsplitter (PBS) and two detectors at each output port, as depicted in Fig. \[fig:dlczDet2\]. Coincident clicks in detectors {AB, CD, AC, BD} project the two-qubit state onto the Bell state {$\phi^+$, $\phi^-$, $\psi^+$, $\psi^-$}. As these possibilities represent half of the two-photon detection events possible, this protocol cannot exceed a success probability of $P_2 = \frac{1}{2}$. Including the probability of detection $P_\text{det}$ for each photon, and the probability $p_1$ of successfully inducing the Raman transition from Eq. \[eq:dlczRaman2\], which can be set as close to unity as possible, the success probability of this protocol is $$\label{eq:dlcz2suc}
P_2 = \frac{1}{2} p_1^2 P_\text{det}^2 \approx 4 \times 10^{-4}\,.$$
The two-photon detection eliminates the fidelity’s dependence on photon loss, and is limited by the initialization and readout losses ($F=0.995$), and the width of the cavity compared to the Zeeman splitting of the used transitions. Assuming a cavity centered on the transitions shown in Fig. \[fig:dlczlvl2\], a magnetic field of $10$ G, and taking into account the Clebsch-Gordan-Coefficients for the transitions, the parasitic transition ${\left| S \right\rangle}\rightarrow {\left| D_{5/2},-1/2 \right\rangle}$ has an excitation probability of $\approx0.75$ %. These detrimental effects reduce the fidelity to $F = 0.992$. Due to the coincident detection scheme, typical dark count rates for modern detectors lead to negligible errors.
The single experiment run time of this protocol is slightly lower than in the probabilistic DLCZ case, $\tau_\text{run} \approx 240$ s, as the Raman pulse duration is longer, $\tau_R \approx 30$ s, in order to achieve a -pulse. The entanglement generation rate follows as $r_e \approx 1.6$ s$^{-1}$.
This is slightly higher than the respective rate for the single-photon-detection DLCZ protocol at $F=0.99~.$ Additionally, the two-photon-detection DLCZ rate $r_e$ quickly surpasses that of the single-photon protocol for higher detection quantum efficiencies, a field of active technology development [@Hadfield2009].
Hybrid protocol {#sub:expHyb}
---------------
An interesting alternative to the previous two protocols, the hybrid protocol as introduced in Sec. \[sub:Hyb\] employs a continuous variable qubus. Although the protocol requires an overcoupled cavity, in contrast to the undercoupled one available in our setup, this section will show that upon realization of this condition, the protocol is by far the fastest means of entanglement distribution for medium high finesse. The qubus is encoded in the phase of a coherent light pulse near $854$ nm, to distribute entanglement between QRs, and the optical qubit of the [$^{40}$Ca$^+$]{}-ion (Sec. \[sub:qubit\]) as stationary qubit in the QR nodes. The latter is initialized into the superposition (Fig. \[fig:hqclvl\])
$${\left| \Psi_i \right\rangle} = {\ensuremath{
\ifthenelse{\isempty{+1/2}{}}
{{\left| S_{1/2} \right\rangle}}
{{\left| S_{1/2},+1/2 \right\rangle}}}} + {\ensuremath{
\ifthenelse{\isempty{+5/2}{}}
{{\left| D_{5/2} \right\rangle}}
{{\left| D_{5/2},+5/2 \right\rangle}}}} = {\left| S \right\rangle} + {\left| D \right\rangle}$$
The ion is then shuttled into the cavity for interaction with the qubus. The qubus and the local oscillator (LO) reference pulse (Sec. \[sub:Hyb\]) are created by suitable attenuation and outcoupling of a laser pulse of duration $\tau_q$. For the generation of entanglement the qubus mode successively interacts dispersively and cavity-enhanced with stationary qubits in distant nodes QR (N) and QR (N+1), see Fig. \[fig:theo\]c). The qubus is resonant with the cavity, far detuned by the frequency $\Delta$ from the transition frequency of ${\left| D \right\rangle}\rightarrow{\left| P \right\rangle}$. The detuning must fulfill $\Delta \gg 2 \cdot g_c = 2 \pi \times 40.2$ MHz, in order to realize the dispersive regime.
![Sketch of the continuous variables hybrid protocol. (i) The ion is initialized in a superposition of $ {\left| S \right\rangle} $ and $ {\left| D \right\rangle} $, and then shuttled into the cavity mode. Now, the qubus is coupled into the cavity (ii), where both qubus and cavity are detuned by $ \Delta $ to the $ {\left| P \right\rangle}\leftrightarrow{\left| D \right\rangle} $ transition. The relevant levels of the [$^{40}$Ca$^+$]{}-qubit are shown. (iii) The qubus obtains a phase shift $ \theta $ (see inset) depending on the qubit state, entangling the flying qubus with the stationary qubit in the QR node (iv).[]{data-label="fig:hqclvl"}](4-3-1_Hybrid_termschema_v07.pdf){width=".45\textwidth"}
Unlike the two previous protocols, the implementation of the hybrid protocol suggests the emplyment of an asymmetric cavity, where the mirror on the I/O fiber features a larger transmittance $T_1=200$ ppm, with $T_i$ ($R_i$) the intensity transmittance (reflectance) of mirror $i$. All other cavity parameters are assumed to be the same as in Sec. \[sub:cavity\]. The resulting minor decrease of the cooperativity from $C_c \approx 16$ to $C_c \approx 13$ is of no further relevance in this case.
In order to be able to send the coherent pulse from QR (N) to QR (N+1), we require an optical circulator. This can be realized, e.g., by inserting a $\frac{\lambda}{4}$-waveplate between the PBS and the QR nodes in Fig. \[fig:theo\] c. Our imperfect cavity incoupling efficiency $\eta_\text{dip}$ reduces the quality of our reflection signal. However, this detrimental effect can be eliminated by utilizing pulses split off of the LO for error correction, similar to the correction called tuning displacement in [@Ladd2006]. An LO pulse of correctly chosen amplitude interferes with the qubus at a weak beamsplitter, set after the output port of the optical circulator. The beamsplitter is set up such that the LO pulse is phase shifted by , and the unwanted, directly reflected field is subtracted from the qubus. For a weak beamsplitter, the entangled pulse is almost undisturbed (see [@Ladd2006]). The same error correction must be done after the pulse leaves QR (N+1), before the detection. For an incoming field $E_\text{inc}$, the field $E_\text{ref}$ reflected from the resonant cavity has the form [@meschede_book2005] $$E_\text{ref} = \left(\sqrt{R_1} - \sqrt{\varepsilon} \frac{\sqrt{T_1^2 R_2}}{1-\sqrt{R_1 R_2}}\right) E_\text{inc}\,.$$ The corrective pulse in this scheme is chosen to be by $E_\text{corr} = \sqrt{R_1} E_\text{inc}\, e^{i \pi}$. Combining both pulses results in the following effective reflection efficiency of the coherent pulse, including mode matching: $$\eta_\text{hyb} =\frac{(E_\text{ref} + E_\text{corr})^2}{E_\text{inc}^2} = \varepsilon \frac{T_1^2 R_2}{(1-\sqrt{R_1 R_2})^2} = 0.25$$ for the parameters given above, and the mode matching $\varepsilon = 0.445$.
The qubus pulse duration $\tau_q$ should be suitably long, so that the cavity does not distort the shape of the pulse. We assume a pulse length of $\tau_q=500$ ns $\gg \tau_\kappa$, which satisfies this condition. Due to the length of the qubus, the detection needs to wait the same amount of time, so the entanglement time of the protocol is $\tau_e = 2 \tau_\kappa + \tau_q + \tau_\text{det}\approx 1$ s, plus the time of light travel between the nodes, with $\tau_\kappa$ the cavity decay time, and $\tau_\text{det}$ the time needed to evaluate the detection events. As we aim for comparability between the protocols, we set the travel time to zero here.
The total transmission efficiency is $$\eta = \eta_\text{hyb}^2 \, \eta_{FL} \approx 0.06 \, .$$ The effect of absorption due to fiber length, with a transmission efficiency of $\eta_{FL}$, further reduces $\eta$, but is beyond the scope of this paper, and is set to 1 for the remainder of this section.
The protocol is concluded by a homodyne detection, in order to project the joint state of the qubits in both QR nodes onto the mixed Bell state $\mu {\left| \psi^+ \right\rangle} + \sqrt{1-\mu^2} {\left| \psi^- \right\rangle}$ of Eqs. \[eq:hyb:state\] and \[eq:hyb:phip\]. This homodyne detection has to distinguish the phase, see Eq. \[eq:Fhomo\]. Using the estimated parameters from our setup, we find that the fidelity of Bell states is below 0.5 for any value of the distinguishability, and thus for any combination of $\alpha$ and $\theta$, a result of the relatively large transmission loss $\eta$.
The resulting state is a mixture of the four Bell states, which we cannot distinguish, rendering the homodyne measurement useless for entanglement distribution.
However, we can completely eliminate the *bit-flip* errors that stem from the badly distinguishable phase-rotated and non-phase-rotated parts of Eq. \[eq:hyb:state\], by changing the detection scheme to an unambiguous state discrimination setup, as detailed in the following.
**Unambiguous state discrimination (USD):** USD [@Dusek2000; @Loock2008] is an alternative to homodyne detection, where the measurement is set up so that the possible results are {*definitely entangled, definitely unentangled, unknown*}, ruling out the possibility of bit-flip errors. The scheme introduced here is based on Ref. [@Loock2008], and is derived in detail there.
![Unambiguous state discrimination setup. The incoming qubus ${\left| \beta \right\rangle}$, consisting of the superposition of the three coherent states ${\left| \alpha \right\rangle}$, ${\left| \alpha e^{i \theta} \right\rangle}$, and ${\left| \alpha e^{-i \theta} \right\rangle}$, is transformed by a 50/50 beamsplitter into the two output pulses ${\left| \frac{1}{\sqrt{2}}\beta,\frac{i}{\sqrt{2}}\beta \right\rangle}$. For each output pulse, at another 50/50 beamsplitter a coherent pulse ${\left| \alpha_{\{A/B\}} \right\rangle}$ outcoupled from the LO creates a displacement in phase space. []{data-label="fig:usdDet"}](4-3-2_USD_v03.pdf){width=".4\textwidth"}
We are interested in unambiguously identifying the non-phase-shifted part of Eq. \[eq:hyb:phip\], which projects the qubits onto the mixed Bell state $\mu {\left| \psi^+ \right\rangle} + \sqrt{1-\mu^2} {\left| \psi^- \right\rangle}$. Fig. \[fig:usdDet\] shows the detection setup, with an input port for the qubus (the remaining input port is depicted with the vacuum mode), and two output ports to detectors A and B.
The first beamsplitter changes an incoming coherent state ${\left| \beta \right\rangle}$ to $$\begin{aligned}
\label{eq:hybusd}
{\left| \beta,0 \right\rangle}\rightarrow{\left| \frac{1}{\sqrt{2}}\beta,\frac{i}{\sqrt{2}}\beta \right\rangle}\,.\end{aligned}$$
The other two beamsplitters are used to displace the two resulting coherent pulses in phase space by sending phase-shifted coherent pulses ${\left| \alpha_A \right\rangle}$ and ${\left| \alpha_B \right\rangle}$, outcoupled from the LO, into their respective input ports:
$$\begin{aligned}
\hat{D}_A(\alpha_A) &\otimes \hat{D}_B(\alpha_B) \nonumber \\
= \hat{D}_A(-\frac{1}{\sqrt{2}}\sqrt{\eta}\alpha e^{i\theta}) &\otimes \hat{D}_B(-\frac{1}{\sqrt{2}}\sqrt{\eta}\alpha e^{-i\theta})\,.\end{aligned}$$
The displacements are chosen such that for the two phase shifted parts of the wave function ${\left| \sqrt{\eta} \alpha e^{\pm i \theta} \right\rangle}$ as input pulse ${\left| \beta \right\rangle}$, one of the detection ports is always in the vacuum mode (see Fig. \[fig:usdPhaseShift\]).
This can be seen by applying these transformations to the three different qubus input states from Eq. \[eq:hyb:phip\]:
\[eq:usdDet\] [| ,0 ]{} &[| (1-e\^[i]{}), (1-e\^[-i]{}) ]{},\
[| e\^[i]{},0 ]{} &[| 0, 2 i ]{} ,\
[| e\^[-i]{},0 ]{} &[| - 2 i ,0 ]{}.
Of the 4 possible detector click patterns, only both detectors firing in coincidence definitely identifies the entangled part of Eq. \[eq:hyb:state\].
![Effect of the USD setup on the qubus in phase space. a) The qubus, in the incoming port, is in a superposition of ${\left| \alpha \right\rangle}$ (solid), ${\left| \alpha e^{i \theta} \right\rangle}$ (shaded upper left to lower right), and ${\left| \alpha e^{-i \theta} \right\rangle}$ (shaded lower left to upper right). b) After the first beamsplitter output port $A$ (red) has states unchanged except for the amplitude (not shown), and output port $B$ (green) has states rotated by $\pi$. The displacement pulses $\hat{D}_A$ and $\hat{D}_B$ shift the coherent states in phase space such that in each output port, one of the phase shifted states is displaced onto ${\left| 0 \right\rangle}$. c) A detection event incompatible with ${\left| 0 \right\rangle}$ in both detectors A and B is only possible for ${\left| \alpha \right\rangle}$ []{data-label="fig:usdPhaseShift"}](4-3-3_USD_phasemap_v03.pdf){width=".45\textwidth"}
The success probability for this event is given by $$\begin{aligned}
\label{eq:PUSD}
P_\text{e} = \frac{1}{2}\big( 1 - e^{-\eta\alpha^2 (1-\cos{\theta})} \big)^2\end{aligned}$$ with a fidelity of the final state $$\begin{aligned}
\label{eq:FUSD}
F={\left\langle \psi^+ \right|}\rho{\left| \psi^+ \right\rangle}=\mu^2\,.\end{aligned}$$
Note that $P_e$ already includes the effect of the USD setup, e.g. the signal reduction by the first beamsplitter of Fig. \[fig:usdDet\]. It was assumed that the signal remains strong enough to be clearly discerned from the vacuum state, which should be easily attainable with modern detectors and a moderate pulse amplitude $\alpha$. The final repetition rate of entanglement distribution is given by $$\label{eq:USDrate}
r_e = \frac{P_e}{\tau_\text{run}}\,,$$ for a time per experimental run $\tau_\text{run}$.
**Entanglement generation rates:** Similar to the other two protocols, one has to find a trade off between fidelity and efficiency. Choosing a fidelity $F=0.99$ for good comparison with the DLCZ schemes, Eq. \[eq:PUSD\] gives us a success probability of $P_e \approx 9 \times 10^{-7}$ for optimal values of the distiguishability $d$, Eq. \[eq:hyb:d\]. In order to find the optimal parameters, we require that $F=\mu^2=.99$ (Eq. \[eq:hybF\]), and maximize $P_e$ with this constraint, with respect to $\alpha$ and $\theta$. For, e.g., $\alpha=100$, the optimal phase shift angle is $\theta \approx 2 \times 10^{-3}$. Thus, there are two ways to experimentally achieve the optimal rate: We can either change the qubus intensity $|\alpha|^2$ to optimize $P_e$ for a given interaction strength between qubus and qubit, or we can move the ion within the cavity field, and thus change $\theta$, to optimize $P_e$ for a given qubus intensity $\alpha$.
Since the dispersive interaction does not disturb the ionic state, we can skip ion initializations in between tries until a new cooling cycle is necessary, greatly reducing the time investment required per run as compared to the other two protocols, setting $\tau_\text{prep}=0$. These assumptions lead to a mean time to entanglement of $\tau_e \approx 0.13$ s, and a entanglement generation rate of $r \approx 8$ s$^{-1}$, which includes the ion initialization. Already for a modest drop in fidelity to 0.95, the rate increases to 170 s$^{-1}$, while for $F=0.8$ the rate is $\approx 700$ s$^{-1}$. The initialization and state readout errors once again reduce the fidelities presented in this section by a factor of $0.995$.
Performance comparison of the protocols
---------------------------------------
In the following we discuss key results for all protocols. The entanglement generation rate as a function of the fidelity is plotted for all considered protocol versions in Fig. \[fig:FvR\]. Typically, there is an experimental parameter that can be tuned to trade the fidelity of resulting Bell states for their production rate. Also, the protocols differ considerably in their requirements for inter-node phase coherence of lasers used for state manipulation of the ion, and for the stability of the connecting fiber link.
The tunable parameter for the *EPR protocol* is the detection window $\tau_W$, i.e. the time to wait for re-initialization. A short $\tau_W$ up to a point increases the fidelity by making erroneous dark counts less likely (see Eqs. \[eq:2dc\] and \[eq:hdc\]), while reducing the success probability by stopping the protocol before any EPR photon has interacted with an ion, see Eq. \[eq:EPRRate\]. Choosing this parameter too short, however, reduces the rate quickly as the waiting time becomes shorter than the temporal shape of the herald photon wavepackets leaving the cavities. The interferometric stability of optical frequencies between QR nodes is not required in this protocol: The phase imprinted on the initial ionic state is set by the relative phase of the two transitions near 729 nm at each node (see Fig. \[fig:eprlvl\]), whose frequency difference lies in the RF range. Consequently, phase coherence can be attained by distributing a stable RF reference signal between neighboring nodes. No interferometric stability of the fiber link required, as any phase collected by the entangled photonic state during transmission is a global one. In order to improve the EPR protocol, one would work on the two major inefficiencies: On the one hand, the small photon detection probability of the herald is the largest contributor to the slow entanglement generation rate. Recent work (e.g., [@Kurz2014; @Schug2014]) uses high aperture laser objectives for improved herald collection efficiency. The detection could also be improved by using a high-finesse, dual wavelength cavity for both the detection wavelength near $393$ nm light, and the EPR-pair wavelength near 854 nm. Enhanced emission through the Purcell effect, however, would require a strong coupling regime for UV-cavities and come at the price of technical complexity.
![Entanglement generation rates possible at a given fidelity for each protocol. The EPR protocol (dash-dotted), the single-photon DLCZ protocol (dashed), the two-photon DLCZ protocol (dot), use the undercoupled cavity. The hybrid protocol with USD, for an overcoupled but otherwise identical cavity, is drawn solid. For high fidelities, $F \gtrsim 0.99$, the DLCZ protocol shows the best performance for our apparatus, while at fidelities below 0.99 the hybrid protocol becomes visibly better. For higher quantum efficiencies and an optimized cavity (see text), upper bounds for hybrid and DLCZ protocol rates are also plotted (grey). Results have been obtained numerically from Eqs. \[eq:eprre\], \[eq:dlczrate\] and \[eq:USDrate\][]{data-label="fig:FvR"}](logFvRall_v10_BW.pdf){width=".48\textwidth"}
The *single-photon detection DLCZ* offers as tunable parameter the single photon generation rate, see Eqs. \[eq:dlcz1suc\] and \[eq:dlcz:1suc\]. The rate of Bell pair production rises with a higher rate of single photons, while the fidelity drops caused from a higher probability for simultaneous photon emission in both QR nodes. The absolute phase of the laser field creating the qubit (see Fig. \[fig:dlczlvl1\] ) cannot be controlled, such that phase coherence between neighboring nodes needs to be established at optical frequencies. The relative phase of the photonic state depends on the position of the detector [@Cabrillo1999] which requires that the fiber link between QR nodes must be interferometrically stabilized. For details concerning the interferometric stability of fiber links, see e.g. [@Sangouard2011; @Ma1994; @Minar2008; @Jiang2008]. The rate of entangled pairs would benefit mostly from an improved transfer of the entangled photon to the detection apparatus, which is typically mainly limited by the coupling efficiency between the fiber cavity mode and the fiber mode. We realistically aim for increasing the mode matching $\varepsilon$ from $0.44$ to $\approx 0.54$. Ultimately we are limited by the geometry of our setup to $\varepsilon_\text{max} \approx 0.57$. For small values of the transfer and detection probability $P_\text{det}$, the entanglement rates increase only linearly with mode matching, by at most $\approx 23$ %. Similar gains could be achieved by reducing mirror losses to optimize the impedance matching, Eq. \[eq\_eta-impedance\], or by optimizing the ratio of reflectivity of the two cavity mirrors to improve the cavity outcoupling coefficient $\eta_\text{out}$.
For the *two-photon DLCZ*, decreasing the rate does not improve the fidelity of Bell states. Consequently, its performance is depicted in Fig. \[fig:FvR\] by single points. In this protocol, the ionic state is created by a bichromatic optical field (see Fig. 9), which allows for a phase control by an RF-link. During the photon transmission only the global phase of the photonic state is altered, which means the stability of the fiber link needs to ensure temporal coincidence of both of the photons impinging at the beamsplitter, but interferometric stability is not required [@Sangouard2011]. Any improvement of the mode matching would be most important for this protocol, as this parameter enters quadratically in Eq. \[eq:dlcz2suc\]. A gain in performance (up to a 51 % increase for the optimized mode matching) is expected for a more precisely aligned fiber cavity. Concerning the *hybrid protocol*, the larger the parameters $\theta$ or $\alpha$ are chosen, the more the rate is increased, at the cost of fidelity. This follows from Eqs. \[eq:hybF\] and \[eq:PUSD\] which mirrors the mixing of pure and entangled states described in Sec. \[sub:Hyb\]. The protocol needs optically phase-stable lasers, as a monochromatic field is used to create the initial ionic state (see Fig. \[fig:hqclvl\]). Interferometric stability for the fiber link is not required: The quantum information in the qubus is transmitted together with a local oscillator pulse, and a homodyne-type measurement eliminates all phases collected by both those parts during transmission, also in the case of USD detection. The hybrid protocol would benefit, similar to the DLCZ protocols, from an improved the mode matching between fiber cavity and fiber, as well as an improved impedance matching. Realistically, we could improve the mode matching to $\varepsilon=0.54$ which would yield an almost two orders of magnitude larger entanglement generation rate with $F = 0.99$.
In the hybrid and single-photon DLCZ protocols we might alleviate the requirement for inter-node phase coherence of lasers with an alternative way of creating the superposition of the ${\left| S \right\rangle}$- and ${\left| D \right\rangle}$-states: Creating a coherent superposition between the two $S_{1/2}$-Zeeman states using a Raman laser interaction (see Fig. \[fig:lvl\]) [@Poschinger2009], followed by a coherent population transfer from one of those sublevels to the D-state via rapid adiabatic passage (RAP). After entanglement as been generated, each qubit could be coherently returned to the spin qubit by a second RAP, before the necessary local operations are performed. The phase coherence between QR nodes would then be ensured by a RF reference exchange.
In conclusion and taking above discussion into account, we note that the EPR protocol shows a comparatively small rate of entanglement generation and can hardly be implemented in our ion trap-cavity platform. However, for both the DLCZ protocol and the hybrid protocol utilizing USD, the estimated rates and fidelities for our parameters suggest the possibility of outperforming the state-of-the-art [@Hucul2014], where free-space photon collection is employed. Both protocols would profit from a fiber optical cavity with a high reflectance end mirror on the multimode fiber [@Steiner2014]. If the losses in the mirror coatings and the transmission through the high reflectance mirror could be brought well below 50 ppm, the cavity outcoupling coefficient would improve to a value that we estimate with $\eta_\text{out} \approx 0.26$. This would double $\eta_\text{out}$ and in turn $P_\text{det}$ for both DLCZ protocols, see Eq. \[eq:dlczdet\]. Furthermore, with $\varepsilon \approx 0.54$, these conditions could let the hybrid USD protocol reach a rate of $\approx 750$ s$^{-1}$ for a state fidelity of $F=0.95$ (Fig. \[fig:FvR\]).
Conclusion and outlook {#outlook}
======================
We discussed the implementations of three quantum repeater protocols, namely the distributed EPR protocol, the DLCZ protocol and the hybrid protocol, and various protocol extensions with an ion trap setup. When comparing the protocols, we find that the DLCZ protocol and the hybrid protocol outperform the EPR protocol. The most important limitations are of technical nature. We see that improving the mode matching from the experimentally determined value of 0.44 to 0.54, which can be achieved through an improved cavity alignement, entanglement generation rates of 30 s$^{-1}$ at fidelities of 0.9 with the DLCZ protocol are within reach. Furthermore, the impedance matching of the fiber optical cavity may be improved by coating the mirror of the I/O fiber with a different transmission coefficient as compared to the high reflection end mirror on the multimode fiber, which would allow a rate of up to 60 s$^{-1}$.
The hybrid protocol appears interesting for future investigations, even though we would need to change the fiber cavity: The quick entanglement generation rate at medium-high fidelity makes it an ideal candidate to be used in combination with entanglement distillation schemes [@Bennett1996; @Duan2000; @Pan2001]. That requires a rate of entanglement generation exceeding the decay rate of the stationary quantum memory. The qubit’s coherence time is, in our case, limited by magnetic field fluctuations to 10 ms, which means that for an optimized hybrid rate of up to $750$ s$^{-1}$ at $F=0.95$, already more entanglement per time would be created than lost. Encoding the stationary qubit in a decoherence-free substate [@Haeffner2005] and executing the mapping between qubus and logical qubit comprised of a two-ion Bell state [@Casabone2015] with a coherence time on the order of 10 s would improve the ratio of entanglement distribution time to coherence time to $\approx 7500$.
Yet another option is the use of error correction codes in entanglement distribution. The basic code in [@Calderbank1996; @Schindler2011] requires 2 ancilla qubits per qubit, few enough to be realized in our setup, without the need to build multiple traps per QR node.
Furthermore, combining our platform with a single photon wavelength converter to provide compatibility with telecom fibers opens up a promising perspective of evolving our system from a lab based proof-of-principle experiment to a prototype, which may increase the maximally achievable distance for quantum communication. We thank Peter van Loock, Denis Gonta and Pascal Eich for helpful discussions. We acknowledge financial support by the European commission within the IP SIQS and by the Bundesministerium für Bildung und Forschung via IKT 2020 (Q.com).
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[^1]: *FEI Helios NanoLab*
[^2]: M. Salz, *to be published*
[^3]: *Laser Optik Garbsen*
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We derive the precise late-time asymptotics for solutions to the wave equation on extremal Reissner–Nordström black holes and explicitly express the leading-order coefficients in terms of the initial data. Our method is based on purely physical space techniques. We derive novel weighted energy hierarchies and develop a singular time inversion theory, which allow us to uncover the subtle contribution of both the near-horizon and near-infinity regions to the precise asymptotics. We introduce a new horizon charge and provide applications pertaining to the interior dynamics of extremal black holes. Our work confirms, and in some cases extends, the numerical and heuristic analysis of Lucietti–Murata–Reall–Tanahashi, Ori–Sela and Blaksley–Burko.'
author:
- 'Y. Angelopoulos, S. Aretakis, and D. Gajic'
title: |
Late-time asymptotics for the wave equation\
on extremal Reissner–Nordström backgrounds
---
Introduction {#sec:intro}
============
Introduction and first remarks {#sec:Introduction}
-------------------------------
The existence of *black hole* regions, namely regions of spacetime which are not visible to far away observers, is a celebrated prediction of the *Einstein field equations*. A rigorous understanding of their dynamical properties is of fundamental importance for addressing several conjectures in general relativity such as *the weak and strong cosmic censorship conjectures* as well as for investigating *the propagation of gravitational waves*. Important aspects of the black hole dynamics are captured by the evolution of solutions to the *wave equation* $$\label{eq:waveequation}
\square_g\psi=0$$ on black hole backgrounds. Initial data are prescribed on a Cauchy hypersurface $\Sigma_0$ which intersects *the event horizon* ${\mathcal{H}^{+}}$ and terminates at *the null infinity* ${\mathcal{I}^{+}}$, as in the figure below. A first step towards the non-linear stability of black hole backgrounds is to establish quantitative *dispersive* estimates in the domain of outer communications up to and including the event horizon.
![The initial value problem for the wave equation on black hole backgrounds. []{data-label="fig:p455"}](blackhole-ivp.pdf)
This problem has been extensively studied in both the mathematics and the physics communities.
**Quantitative decay rates** for *scalar fields satisfying and all their higher-order derivatives* have been obtained for the general *sub-extremal Reissner–Nordström* and the general *sub-extremal Kerr* families of black hole spacetimes (see [@part3]). We refer to [@dhr-teukolsky-kerr; @lecturesMD; @MDIR05; @blu1; @Civin2014; @moschidis1; @volker1; @tataru3; @metal; @blukerr; @dssprice; @kro] for additional results in the asymptotically flat setting and to [@peter2; @semyon1; @other1; @gusmu1; @gusmu2] for results in the asymptotically de Sitter and anti de Sitter setting. See also [@HV2016; @Dafermos2016; @klainerman17; @moschidisads] for recent breakthroughs in the understanding of nonlinear stability problems in the context of the Einstein equations.
A definitive proof of the **precise late-time asymptotics** of solutions to the wave equation on the general sub-extremal Reissner–Nordström backgrounds, including the celebrated *Schwarzschild* family of black holes, was obtained in the recent series of papers [@paper1; @paper2; @paper-bifurcate] confirming, in particular, Price’s heuristics (for more details and references see Section \[sec:TheWaveEquationOnBlackHolesBackgrounds\]). In the present paper, we focus on another fundamental class of black holes, namely the *extremal* black holes. These are characterized by the **vanishing** of the *surface gravity* of the event horizon (see also Section \[sec:TheERNManifoldFoliationsAndVectorFields\]). Geometrically, this condition has to do with the fact that the Killing normal to the event horizon coincides with the affine null normal to the event horizon. Extremal black holes play a fundamental role in[^1]
- *astronomy*: according to an abundance of astronomical observations, near-extremal black holes should be ubiquitous in the universe. Such observations concern stellar black holes and supermassive black holes in the centers of galaxies;
- *high energy physics*: they allow for the study of supersymmetric theories of gravity, black hole thermodynamics and of quantum descriptions of gravity;
- *classical general relativity*: they saturate various geometric inequalities concerning the mass, angular-momentum and charge. Furthermore, they have intriguing dynamical properties with no analogue in sub-extremal black holes.
The latter intriguing dynamical properties of extremal black holes are the objects of study in this paper. Specifically, we investigate scalar perturbations of the *extremal Reissner–Nordström* (ERN) one-parameter family of black hole backgrounds. The corresponding metrics take the following form with respect to the so-called ingoing Eddington–Finkelstein coordinates $(v,r,\theta,\varphi)$: $$g_{\text{\tiny{ERN}}}=-\left(1-\frac{M}{r}\right)^2dv^2+dv\otimes dr+dr\otimes dv+r^2(d\theta^2+\sin^2\theta d\varphi^2),
\label{ernmetric}$$ where $M>0$ is the mass parameter.
ERN black hole spacetimes are spherically symmetric, asymptotically flat solutions to the Einstein–Maxwell system $$\begin{split}
R_{\mu\nu}(g)-\frac{1}{2}R(g)\cdot g_{\mu\nu}&=T_{\mu\nu}(F),\\
dF=d\star F&=0.
\end{split}$$ Here, $T_{\mu\nu}(F)$ denote the components of the energy-momentum tensor of the electromagnetic field $F$.
The techniques for establishing decay in time and in fact the precise late-time asymptotics for solutions to the wave equation on sub-extremal backgrounds *break down* in the case of ERN. Indeed, a large portion of the analysis of perturbations of sub-extremal black holes exploits the celebrated *redshift effect* along the event horizon. This *local* version of the redshift effect, which essentially depends on the strict positivity of the surface gravity of the event horizon (see [@redshift; @lecturesMD]), can be illustrated as follows: consider two observers A and B entering the black hole region such that A crosses the event horizon first (see Figure \[fig:p4551ref\]). Suppose A emits a light signal that travels along the event horizon and is intercepted by B. Then the frequency of this signal as measured by B will be “shifted to the red" when compared to the frequency measured by A.
![The local redshift effect for sub-extremal horizons. []{data-label="fig:p4551ref"}](redshift.pdf)
The vanishing of the surface gravity of extremal event horizons means that *the redshift effect along the event horizon degenerates on extremal black holes* and hence cannot be used as a stabilizing mechanism. In fact, it was shown in [@SA10; @aretakis1; @aretakis2] that solutions to the wave equation on ERN satisfy a *conservation law along the event horizon* $\mathcal{H}^{+}$. Consider the translation-invariant vector field $Y=\partial_r$. Note that $Y$ is transversal to the event horizon as in the figure below. Consider the advanced time parameter $\tau$ and denote by $S_{\tau_0}={\mathcal{H}^{+}}\cap \{\tau=\tau_0\}$ the corresponding spherical sections of the event horizon.
\[fig:p4125\]
Then, the surface integrals $$H_0[\psi]:=-\frac{M^2}{4\pi}\int_{S_{\tau}}Y(r\psi) \, d\omega
\label{hcons1}$$ are **independent** of $\tau$. Here $\omega=(\theta,\varphi)$ and $d\omega=\sin\theta d\theta d\varphi$. We will frequently refer to $H_{0}[\psi]$ as a *conserved charge* for $\psi$. This conservation law is certainly an obstruction to decay for generic initial data for which $H_0[\psi]\neq 0$. It can further been shown that higher order derivatives asymptotically **blow up** along ${\mathcal{H}^{+}}$: $$|Y^{k}\psi|_{{\mathcal{H}^{+}}}\sim\ c_k H_0[\psi]\cdot \tau^{k-1}\rightarrow +\infty
\label{instaeq1}$$for $k\geq 2$ as $\tau\rightarrow +\infty$. Here $c_k$ are constants that depend only on $M,k$. The growth along ${\mathcal{H}^{+}}$ of (transversal) derivatives yields a genuine *horizon instability of extremal black holes* which can in fact be measured by local observers who cross the event horizon [@hm2012; @zimmerman2]. On the other hand, it can be shown that *away* from the event horizon $\psi$ and its derivatives $Y^{k}\psi$ **decay** in time. This means that one may regard $H_0$ as a type of *horizon “hair”* associated to the event horizon. We remark that an analogous version of the horizon instability holds also for scalar perturbations of extremal Kerr [@aretakis4; @hj2012] and in fact for many other types of perturbations in various settings (see Sections \[sec:TheHorizonInstabilityOfExtremalBlackHoles\] and \[sec:PreviousWorksOnLateTimeAsymptoticsOnERN\]).
Returning to decay estimates, the following weak decay rate was rigorously established in [@aretakis2] for $\psi$ everywhere in the domain of outer communications up to and including the event horizon: $$|\psi| \lesssim \frac{1}{\tau^{\frac{3}{5}}}.
\label{bound1}$$ Since $\psi$ decays along the event horizon it follows that, in view of the conservation of , the first-order transversal derivative $Y\psi$ of $\psi$ **does not decay** along the horizon for initial data for which $H_0[\psi]\neq 0$.
The above results do *not* provide an insight into the precise asymptotic behavior for $\psi$. There is extensive work in the physics literature regarding late-time asymptotics for scalar fields on extremal Reissner–Nordström via heuristic or numerical methods, see for instance [@other2; @extremal-rn-qnm; @Burko2007; @hm2012; @ori2013; @sela; @sela2; @zimmerman1] and Section \[sec:PreviousWorksOnLateTimeAsymptoticsOnERN\] for more details. However, there has been no mathematically rigorous proof or derivation of these asymptotics. In fact, the heuristic and numerical predictions in the physics literature did not provide the late-time asymptotics in the *full* spacetime, which remained an open problem and is resolved in the present paper. Before we give a more precise statement of this open problem, we introduce the following definition concerning initial data for :
Initial data on the Cauchy hypersurface $\Sigma_0$ are called **horizon-penetrating** if they smoothly extend to the event horizon $\mathcal{H}^{+}$ such that the conserved charge $H_0[\psi]\neq 0$. \[def1intro\]
The following problem had been left completely open
*Obtain the late-time asymptotics of the radiation field along the null infinity $\mathcal{I}^{+}$ for horizon-penetrating compactly supported initial data.*
The physical importance of the above problem lies in the fact that these asymptotics capture the observations made by far-away observers of perturbations of the near-horizon region of extremal black holes. This problem is definitively resolved in the present paper. In fact, in this paper:
*We derive and rigorously prove the **precise** late-time asymptotics for scalar fields on ERN **globally** in the domain of outer communications, for **a general class of initial data**.*
In particular, we derive late-time asymptotics along the event horizon ${\mathcal{H}^{+}}$, along constant $r=r_0$ hypersurfaces and along the null infinity ${\mathcal{I}^{+}}$. The *exact coefficient of the leading-order terms in the asymptotic estimate is obtained in terms of explicit expressions of the initial data*. See Section \[sec:SummaryOfTheMainResults\] for a non-technical summary of the results and Section \[subsec:TheMainTheorems\] for the precise statements of the main theorems. Our results provide, in particular, sharp upper and lower decay rates for the evolution of scalar fields. Our method is based purely on physical space constructions and avoids explicit representations of solutions to the wave equation. We establish a novel elliptic estimate and a new class of hierarchies of weighted estimates adapted to the extremal near-horizon geometry.
Our results provide a rigorous confirmation and proof of the numerics in [@Burko2007; @hm2012] and heuristics in [@ori2013; @sela]. For example, [@hm2012] was the first work to numerically obtain the following late-time asymptotics along the event horizon for horizon-penetrating compactly supported initial data: $$\psi|_{{\mathcal{H}^{+}}}\sim \frac{2}{M}H_0[\psi]\cdot\frac{1}{\tau}.$$ These asymptotics, which were subsequently heuristically derived in [@ori2013; @sela], are indeed rigorously recovered here. Furthermore, as mentioned above, our results extend the works in the physics literature in various directions. Notably, we obtain the asymptotics of the radiation field along the future null infinity ${\mathcal{I}^{+}}$ for horizon-penetrating, compactly supported initial data: $$r\psi|_{{\mathcal{I}^{+}}}\sim \Big(4MH_0[\psi]-I_0^{(1)}[\psi]\Big)\cdot\frac{1}{\tau^2}.$$ Here $I^{(1)}_0[\psi]$ is the Newman–Penrose constant of a *singular* time integral of $\psi$ and depends on the global properties of the initial data (see Sections \[sec:TheNewHorizonHairH01Psi\] and \[sec:GeometricOriginOfTheNewHair\]). We remark that the horizon charge $H_0[\psi]$ of a scalar perturbation that is initially localized near the event horizon in fact appears in the asymptotic behavior along ${\mathcal{I}^{+}}$. In other words, observations along null infinity (that is, arbitrarily far from the event horizon) can in principle be used to measure the charge $H_0[\psi]$ associated to in-falling observers at the horizon. This might be thought of as a “leakage” of horizon information to null infinity and hence could, in principle, be measured by gravitational detectors.
#### Outline of the introduction {#sec:OutlineOfTheIntroduction}
We finish this brief introductory subsection with an outline of the remaining sections in the introduction. In Section \[sec:TheWaveEquationOnBlackHolesBackgrounds\], we review the key mechanisms behind the existence of late-time tails in the asymptotics of scalar fields on sub-extremal black holes. In Section \[sec:PhysicalImportanceOfExtremalBlackHoles\] we list various works which emphasize the importance of the dynamics of extremal black holes and hence serve as a motivation for the work of the present paper. In Section \[sec:TheHorizonInstabilityOfExtremalBlackHoles\] we provide a review of the horizon instability of extremal black holes and in Section \[sec:PreviousWorksOnLateTimeAsymptoticsOnERN\] we discuss the physics literature that is relevant to our problem.
Asymptotics for the wave equation on sub-extremal black holes {#sec:TheWaveEquationOnBlackHolesBackgrounds}
-------------------------------------------------------------
The following late-time *polynomial* tails for solutions to the wave equation with smooth, *compactly supported* initial data on Schwarzschild spacetimes were obtained in a heuristic manner by Price [@Price1972] in 1972 along constant radius $r=r_0$ hypersurfaces away from the event horizon $$\psi|_{r=r_0}(\tau,r=r_0,\omega)\sim \frac{1}{\tau^3}.
\label{eq:1}$$ Subsequent heuristic and numerical works [@leaver; @CGRPJP94b; @LB99] suggested the following asymptotics on the event horizon ${\mathcal{H}^{+}}$: $$\psi|_{\mathcal{H}^{+}}(\tau,r=2M,\omega) \sim \frac{1}{\tau^3},
\label{eq:2}$$ and along the null infinity $\mathcal{I}^{+}$ $$r\psi|_{\mathcal{I}^{+}}(\tau,r=\infty,\omega) \sim \frac{1}{\tau^2}.
\label{eq:3}$$ Here $\tau$ denotes a global time parameter and $\omega\in \mathbb{S}^{2}$.[^2] The following global quantitative estimates which establish rigorously the above asymptotics were obtained for general sub-extremal Reissner–Nordström spacetimes in [@paper2; @paper-bifurcate]:
$$\left|\psi(\tau, r_0,\cdot)+8 I_{0}^{(1)}[\psi]\cdot\frac{1}{\tau^3}\right| \leq C_{r_0} \cdot \sqrt{E_{\Sigma_0}[\psi]}\cdot \frac{1}{\tau^{3+\epsilon}},
\label{our1}$$
$$\left|r\psi|_{\mathcal{I}^{+}}(\tau,\cdot)+2 I_{0}^{(1)}[\psi]\cdot\frac{1}{\tau^2}\right| \leq C \cdot \sqrt{E_{\Sigma_0}[\psi]}\cdot \frac{1}{\tau^{2+\epsilon}},
\label{ourrad1}$$
where $\sqrt{E_{\Sigma_0}[\psi]}$ are weighted norms of the initial data and constant $I_{0}^{(1)}$ is given by the following explicit expression of the initial data on $\Sigma_0$: $$I_{0}^{(1)}[\psi]=\frac{M}{4\pi} \int_{\Sigma_0\cap\mathcal{H}^{+}}\!\!\psi r^2d\omega+\lim_{r_0\rightarrow \infty}\left(\frac{M}{4\pi}\int_{\Sigma_0\cap\{r\leq r_0\}} n_{\Sigma_0}(\psi)d\mu_{\Sigma_0}+\frac{M}{4\pi}\int_{\Sigma_0\cap\{r=r_0\}}\Big(\psi-\frac{2}{M}r\partial_v(r\psi)\Big)r^2d\omega\right),
\label{i011}$$ with $\partial_v$ is an outgoing null derivative and $d\mu_{\Sigma_0}$ denotes the induced volume form on $\Sigma_0$. We note that for compactly supported initial data on the maximal hypersurface $\{t=0\}$, the above expression for the coefficient $I_{0}^{(1)}[\psi]$ reduces to $$I_0^{(1)}[\psi]= \frac{M}{4\pi}\int_{ S_{\text{BF}}}\!\!\psi \, r^2d\omega+\frac{M}{4\pi}\int_{\{t=0\}}\ \frac{1}{1-\frac{2M}{r}}\partial_t\psi\, r^2 dr d\omega,$$ where $S_{\text{BF}}$ denotes the bifurcation sphere.
Generic initial data satisfy $I_{0}^{(1)}[\psi]\neq 0$ and hence give rise to solutions to the wave equation which decay exactly like $\frac{1}{\tau^3}$. This result yielded the first *pointwise lower bounds* for solutions to the wave equation on Schwarzschild backgrounds[^3]. In other words, , and provide a complete characterization of all solutions to which satisfy Price’s law as a lower bound. We remark that the study of precise late-time asymptotic expansions is very important in issues related to black hole interior regions and, in particular, in addressing the strong cosmic censorship conjecture [@MD03; @MD05c; @MD12; @luk2015; @LukSbierski2016; @DafShl2016; @Hintz2015; @Franzen2014; @Luk2016a; @Luk2016b].
It is important to emphasize that the approach of [@paper2; @paper-bifurcate] is based on purely physical space techniques. On the other hand, the heuristic work of Leaver [@leaver] related the late-time power law to the branch cut at $\omega=0$ in the Laplace transform of Green’s function for each fixed angular frequency. This is consistent with the results of [@paper2; @paper-bifurcate], in view of the fact that the geometric origin of the constant $I_{0}^{(1)}[\psi]$ is related to *an obstruction to the invertibility of the time operator* $T=\partial_t$ in a suitable function space (and hence is related to the $\omega=0$ frequency in the Fourier space). Indeed, restricting (strictly) to the future of the bifurcation sphere where $T\neq 0$, we have that **an obstruction to the invertibility of the operator $T$ is the existence of a conservation law along the null infinity** ${\mathcal{I}^{+}}$: For solutions $\psi$ to the wave equation on Reissner–Nordström spacetimes, the limits $$I_{0}[{\psi}](u):=\frac{1}{4\pi}\lim_{r\rightarrow\infty} \int_{\mathbb{S}^{2}}r^2 \partial_r (r{\psi}) (u,r,\omega) \, d\omega$$ are **independent** of the retarded time $u$. Here, we consider the standard outgoing Eddington–Finkelstein coordinates $(u,r,\omega)$ (with $\omega \in\mathbb{S}^2$). The associated constant $$I_{0}[{\psi}]:=I_{0}[{\psi}](u)
\label{np}$$ is called the *Newman–Penrose constant* of ${\psi}$ (see [@NP1; @np2]).
![\[fig:5\]The Newman–Penrose constant on $\mathcal{I}^{+}$.](np_constant.pdf){width="5.5cm"}
The existence of this asymptotic conservation law is an obstruction to inverting the time operator $T$, if the domain of $T$ is taken to be the set of all smooth solutions $\psi$ to the wave equation which satisfy the condition $|r^2\partial_{r}(r\psi)|\in O_{1}(r^0)$ on the initial hypersurface $\Sigma_0$.[^4] Indeed, if there is a regular solution $\psi^{(1)}$ to in the domain of $T$ such that $$T\psi^{(1)}=\psi$$ then we must necessarily have that $$I_{0}[\psi]=I_0[T\psi^{(1)}]=0.$$ Conversely, if we consider a smooth initial data on a Cauchy hypersurface $\Sigma_0$ which crosses the event horizon to the future of the bifurcation sphere (see figure below) such that $I_{0}[\psi]=0$ and $$\lim_{r\rightarrow \infty} \int_{\mathbb{S}^2} r^3 \partial_r(r\psi)|_{\Sigma_0}d\omega<\infty
\label{r3condition}$$ then, by the results in [@paper2], there is a unique smooth spherically symmetric solution $\psi^{(1)}$ to in the domain of $T$ such that $$T\psi^{(1)}=\frac{1}{4\pi}\int_{\mathbb{S}^2} \psi d\omega
\label{timeinvint1}$$in $\mathcal{J}^{+}(\Sigma_0)$.
![\[fig:2432\]Time inversion for the spherical mean $\psi_0$ of $\psi$.](time_inversion1.pdf){width="5.5cm"}
Hence, **$I_0[\psi]$ appears as the unique obstruction to inverting the time operator $T$ on the projection to the spherical mean** of $\psi$. If the Newman–Penrose constant $I_0[\psi]\neq 0$ then has no solution and in this case it is $I_{0}[\psi]$ that appears in the late-time asymptotics of the spherical mean (see [@paper2]): for example, at fixed $r=r_0$ we have that $$\left|\frac{1}{4\pi}\int_{\mathbb{S}^2}\psi(\tau, r_0,\cdot)d\omega-4 I_{0}[\psi]\cdot\frac{1}{\tau^2}\right| \leq C_{r_0} \cdot \sqrt{E_{\Sigma_0}[\psi]}\cdot \frac{1}{\tau^{2+\epsilon}}.
\label{our120}$$ We remark that the restriction to the spherical mean is justified by the fact that the non-spherically symmetric projection : $$\psi_{\ell\geq 1}:=\psi-\frac{1}{4\pi}\int_{\mathbb{S}^2}\psi d\omega,$$ decays at least like $\tau^{-3.5+\epsilon}$ (see [@paper2]), for some small $\epsilon>0$, and hence does **not** contribute to the leading order terms in the late-time asymptotics.
If, on the other hand, holds (and hence $I_0[\psi]=0$) then by the above result $T$ can be inverted to produce the time integral $\psi^{(1)}$. In this case, the Newman–Penrose constant $I_0[\psi^{(1)}]$ of $\psi^{(1)}$ is an obstruction to acting with $T^{-1}$ on $\psi^{(1)}$, or equivalently, an obstruction to acting with $T^{-2}$ on $\frac{1}{4\pi}\int_{\mathbb{S}^{2}}\psi$. This obstruction is precisely the origin of the coefficient $I_{0}^{(1)}$ in and , that is $$I_{0}^{(1)}[\psi]=I_{0}[\psi^{(1)}].$$ Note that $I_{0}^{(1)}[\psi]$ is given in terms of the initial data of $\psi$ by .
Summarizing, we have:
[ l | c ]{} asymptotics for $\psi$ & origin of the coefficient\
$-4I_{0}[\psi]\cdot \frac{1}{\tau^2}$ & $I_{0}[\psi]\neq 0$ unique obstruction to inverting $T$\
$8I_{0}^{(1)}[\psi]\cdot \frac{1}{\tau^3}$ & $I_{0}^{(1)}[\psi]\neq 0$ unique obstruction to inverting $T^2$\
**In the case of ERN there are additional obstructions to inverting the time operator $T$ which cause many subtle difficulties in obtaining the precise late-time asymptotics** (see Sections \[sec:GeometricOriginOfTheNewHair\] and \[sec:OverviewOfTechniques\]).
Physical importance of extremal black holes {#sec:PhysicalImportanceOfExtremalBlackHoles}
-------------------------------------------
As has already been mentioned in Section \[sec:Introduction\], extremal black holes are of fundamental importance in general relativity. Let us emphasize that an understanding of the dynamical properties of “exactly” extremal black holes is relevant also when one is studying the dynamics of “near-extremal” black holes over large (but finite) time intervals. In this section we provide a list of references which underpin the intimate connection of extremal black holes with astronomy/astrophysics, high energy physics and classical general relativity. Results regarding specifically the dynamics of ERN are omitted from this section since they are discussed in detail in the next two sections.
[**Observations of (near-)extremal black holes**]{}
Astronomical evidence suggests that near-extremal black holes are ubiquitous in the universe. Various techniques have been developed to analyze the mechanisms for the formation and distribution of near-extremal black holes [@near-extremal-accretion; @kesden]. It has been suggested that $70\%$ of the stellar black holes, which have been formed from the collapse of massive stars, in the universe are near-extremal [@rees2005].
Using techniques from $X$-ray reflection spectroscopy, it has been shown that many supermassive black holes (whose mass is at least 1 billion times the mass of the sun) are near-extremal [@brenneman-spin; @reynolds-nearextremal]. Such black holes are important for the large scale structure of galaxies and galaxy clusters. Specific near-extremal supermassive black holes are expected to exist in the center of the MCG–06-30-15 galaxy [@xrayextremal] and the NGC 3783 galaxy [@brennemanpaper]. Moreover, the stellar black hole Cygnus X-1 (part of a black hole binary system in the Galaxy) has been shown to have a near-extreme value for the spin parameter [@extremalobservation2]. Another example is the stellar black hole GRS 1915+105 [@extremalobservation1].
[**Observational signatures of extremal black holes**]{}
Many astronomical conclusions are based on calculations for exactly Kerr spacetimes. However, time variability might introduce additional observational signatures of extremal black holes, that is features in the observations that are characteristic to the dynamics of extremal black holes. The near-horizon geometry provides a great background for probing such signatures. Such signatures can be divided in two main categories: gravitational signatures [@gralla2016; @isco-extremal; @ekerr-plunge] and electromagnetic signatures [@extreme-optical1; @extreme-optical2; @grallastrominger]. The asymptotics of the present paper derive a new gravitational signature (see the discussion at the end of Section \[sec:Introduction\]). The physical details will be discussed in upcoming work.
[**Supersymmetry, holography and quantum gravity**]{}
Extremal black holes are often supersymmetric as a consequence of the BPS bound. They have zero entropy and hence play an important role in understanding black hole thermodynamics and the Hawking radiation [@haw95]. Quantum considerations of black hole entropy in five-dimensional extremal black holes and applications in string theory can be found in [@stromingerextremalentropy; @em06]. One can define a near-horizon limit [@gibbonssuper; @nearhorizonrn; @armen] which yields new solutions to the Einstein equations with conformally invariant properties. These limiting geometries have been classified in [@h07; @luciettikund; @k09; @chrus; @hol10]. On the other hand, the conformal properties of the near-horizon geometries allow for a description of quantum gravity via a holographic duality [@strominger-extremal-holography; @conformalnhek; @cft-extremal-cosmo] and the study of bodies orbiting near-extremal black holes [@porfy2014; @porfy2; @porfy3; @porfy4].
[**Uniqueness and classification of extremal black holes**]{}
Extremal event horizon enjoy various rigidity properties [@booth-extremal; @extremalrigidity; @hajicek; @hol09]. Global uniqueness results for extremal black holes in various settings have been obtained in [@pauextremalluci; @nonstaticextrem; @chru; @horow; @meinel08]. We also refer to interesting examples of higher dimensional extremal black holes [@kund16].
[**Extremal black holes as mass minimizers**]{}
Extremal black holes saturate geometric inequalities for the total mass, angular momentum and charge [@dainprl6; @SD08; @chru2008] at higher dimensions [@alaee; @alaeeprl; @alaee17]. They also saturate quasi-local versions of these inequalities for the mass, angular momentum and charge contained in the black hole region [@DJR11; @dain2010; @dainprl11; @reiris2013; @reiris-throat].
[**Quasinormal modes of extremal black holes**]{}
Starobinski [@staro] first investigated the effects of superradiance and extremality. Extensions for quasinormal modes of extremal Kerr were obtained in [@detweiler80] where a sequence of zero damped modes was computed. Subsequent analysis was presented in [@mhighinsta; @glampedakisfull]. The most precise analysis of quasinormal modes in extremal Kerr has been presented in [@zeni13]. As far as other settings are concerned, rapid modes for near extremal Reissner–Nordström–de Sitter spacetimes were discovered in [@hprl18] and slow modes on near-extremal (in fact all sub-extremal) Kerr de Sitter were computed in [@harveyfel18]. Gravitational modes of the near extremal Kerr geometry were studied in [@harvey-modes-2009].
[**Extremality and non-linear effects**]{}
An intriguing aspect of near-extremal black holes is that they exhibit turbulent gravitational behavior [@luis], that is energy is transferred from high frequencies to low frequencies. Non-linear simulations of formation of binary systems of near-extremal black holes were presented in [@extremal-binary-merger]. Furthermore, numerical simulations of the evolutions of the Einstein–Maxwell-scalar field system in a neighborhood of extremal Reissner–Nordström was studied in [@harvey2013]. A general theory of evolution of extremal black holes was developed here [@booth16]. For other non-linear works pertaining to the dynamics of extremal black holes we refer to [@areangel6; @yannis1; @bizon-extremal-nonlinear].
The horizon instability of extremal black holes {#sec:TheHorizonInstabilityOfExtremalBlackHoles}
-----------------------------------------------
The wave equation on ERN in ingoing Eddington–Finkelstein $(v,r,\theta,\varphi)$ coordinates takes the form $$\Box_g\psi= D\partial_r\partial_r\psi+2\partial_v\partial_r\psi+\frac{2}{r}\partial_v\psi+R\partial_r\psi+{\slashed{\Delta}}\psi=0,
\label{weern}$$ where $D(r)=\left(1-\frac{M}{r}\right)^2$ and $R(r)=\frac{dD}{dr}+\frac{2D}{r}$. Here we denote $${\slashed{\Delta}}=\frac{1}{r^2}{\slashed{\Delta}}_{{\mathbb{S}}^2},$$ where ${\slashed{\Delta}}_{{\mathbb{S}}^2}$ is the standard Laplacian on the round unit sphere $\mathbb{S}^{2}$.
We will review here the decay, non-decay and blow-up results for that were established in [@aretakis1; @aretakis2; @SA10; @aag1] and describe the “*horizon instability of extremal black holes*”. We consider smooth initial data on a spherically symmetric Cauchy hypersurface $\Sigma_0$ which crosses the event horizon and terminates at future null infinity. Recall that the event horizon is the hypersurface given by $${\mathcal{H}^{+}}=\{r=M\}.$$ Let $F_{\tau}$ denote the flow of the stationary Killing vector field $T=\partial_v$ and let $\Sigma_{\tau}=F_{\tau}(\Sigma_0)$.
### Conservation laws along the event horizon {#sec:ConservationLawsAlongTheEventHorizon}
Consider the spherical sections $S_{\tau}=\Sigma_{\tau}\cap {\mathcal{H}^{+}}$ of the event horizon. *Restricting to the spherical mean of the wave equation on the event horizon* yields $$\partial_{v}\left(\int_{S_{\tau}}\left(2\partial_r\psi+2M^{-1}\psi\right)\, M^2\, d\omega\right)=0.$$ Since $\partial_v$ is null and normal to the event horizon ${\mathcal{H}^{+}}$, it immediately follows that *the surface integrals* $$H_0[\psi]:=-\frac{M^2}{4\pi}\int_{S_{\tau}}\partial_r(r\psi) \, d\omega
\label{introhorizonH}$$ *are independent of* $\tau$. Here $d\omega=\sin\theta d\theta d\varphi$ is the volume form of the unit round sphere $\mathbb{S}^2$ with $\omega=(\theta,\varphi)$. This gives rise to a conservation law along the event horizon. Surprisingly, an analogous conservation law holds for each projection on the eigenspace of the angular Laplacian. Indeed, it can be shown that if $\psi_{\ell}$ denotes the projection of $\psi$ on the eigenspace $E_{\ell}$ of ${\slashed{\Delta}}$ with eigenvalue $-\frac{\ell (\ell+1)}{r^2}$, then the following derivative $\psi_\ell$ of order $\ell+1$ that is transversal to ${\mathcal{H}^{+}}$, $$\partial_{r}^{\ell} \Big( r\partial_r(r \psi_{\ell})\Big),$$ is *constant along the null generators of the event horizon*.
It is important to emphasize that the derivative $\partial_r$ is translation-invariant (since $[\partial_r,\partial_v]=0$) and hence the above conservation laws provide highly non-trivial *obstructions to decay* for all the geometric quantities associated to a scalar field. Summarizing we have the following:
**Hierarchy of conservation laws on ERN:** *for every fixed angular frequency $\ell$ we have a conservation law along the event horizon involving exactly the first $\ell+1$ translation-invariant, transversal derivatives of the scalar field on the event horizon.*
An analogue of this hierarchy for axisymmetric solutions on extremal Kerr was obtained in [@aretakis4]. Lucietti and Reall [@hj2012] generalized this hierarchy for electromagnetic and gravitational perturbations of extremal Kerr which they used to derive a *gravitational instability of extremal Kerr*. We remark that these conservation laws are a feature characteristic to extremal event horizons. Indeed, it was shown in [@aretakisglue] that non-extremal horizons do **not** admit conservation laws associated to solutions of the wave equation. Further extensions of these conservation laws have recently been provided in [@godazgar17].
### The trapping effect on the event horizon {#sec:TheTrappingEffect}
Let $N$ be a translation-invariant future-directed timelike vector field defined globally in the domain of outer communications up to and including the event horizon. This vector field will be used to measure the energy $E_{\gamma}(s)$ of affinely-parametrized null geodesics $\gamma(s)$: $$E_{\gamma}(s)=g\left(\overset{\cdot}{\gamma}(s),N\right),$$ where $\overset{\cdot}{\gamma}(s)=\frac{d\gamma}{ds}(s)$. A key observation is that for sub-extremal black holes the energy $E_{\gamma}(s)$ of the null generators of the event horizon with positive surface gravity $\kappa>0$ decays exponentially in $s$. On the other hand, the energy $E_{\gamma}(s)$ of the null generators of the event horizon of ERN remains constant for all $s$. This is intimately related to the geometric characterization of extremal horizons, namely that the Killing normal vector field to the event horizon gives rise to an affine foliation of the event horizon. Sbierski [@janpaper] used the Gaussian beam approximation and the above result to show that there are solutions to the wave equation on ERN that are localized in a neighborhood of ${\mathcal{H}^{+}}$ with almost constant energy across $\Sigma_{\tau}$ for arbitrarily large $\tau$. This result immediately yields an obstruction to proving local integrated estimates bounding $$\Gamma_1[\psi]=\int_{0}^{\infty}\left(\int_{\Sigma_{\tau}\cap \{r\leq M+\epsilon\}} |\partial\psi|^2 \right)\, d\tau$$ for some arbitrarily small $\epsilon>0$. Specifically, Sbierski’s result shows that the above integral cannot be bounded purely in terms of the initial energy of $\psi$ on $\Sigma_0$. A Morawetz estimate bounding $\Gamma_1[\psi]$ was established in [@aag1] where it was shown that such an estimate requires
1. *the finiteness of a weighted higher-order norm of the initial data, and*
2. *the vanishing of the conserved charge $H_{0}[\psi]$.*
Furthermore, it was shown that *for smooth and compactly supported initial data, $\Gamma_1[\psi]$ is if and only if $H_0[\psi]\neq 0$.*
The first requirement above is reminiscent to that of the Morawetz estimates on the photon sphere which accounts for the high-frequency solutions localized on the trapped null geodesics. On the other hand, the second requirement is a global (low-frequency) condition on all of the event horizon, that is on all the null generators of the event horizon. This shows that *the event horizon on ERN exhibits a global trapping effect.*
Another characteristic feature of the event horizon on ERN is the following *stable higher-order trapping effect*: *For generic smooth and compactly supported initial data with support away from the event horizon, the following higher-order integral* $$\Gamma_k[\psi]=\int_{0}^{\infty}\left(\int_{\Sigma_{\tau}\cap \{r\leq M+\epsilon\}} |\partial^k\psi|^2 \right)\, d\tau$$ *is infinite, for all* $k\geq 2$.
Bounding the integral in time of the energy flux through $\Sigma_{\tau}$ is further obstructed by the standard photon sphere which is an obstruction present for general black hole spacetimes. We refer to [@lecturesMD; @aretakis2] for the details.
### Energy and pointwise boundedness and weak decay {#sec:EnergyAndPointwiseBoundednessAndDecay}
An important aspect of ERN is that the Killing vector field $T=\partial_v$ is globally causal. That implies that the conserved energy $T$-fluxes are non-negative definite. However, since $T$ is null at the horizon, the $T$-flux $\mathcal{E}^{T}[\psi]$ along $\Sigma_{\tau}$ *degenerates* at the horizon. Schematically, we have $$\mathcal{E}^{T}_{\Sigma_{\tau}}[\psi] \sim \int_{\Sigma_{\tau}} \left(1-\frac{M}{r}\right)^2\cdot |\partial\psi|^2\, d\mu_{\Sigma_{\tau}}.$$ Clearly, we have $$\mathcal{E}^{T}_{\Sigma_{\tau}}[\psi]\leq \mathcal{E}^{T}_{\Sigma_{0}}[\psi].$$ The above estimate was also used in [@dd2012] where various boundedness results where shown for the wave equation on ERN away from the event horizon. One can go beyond such boundedness estimates and derive decay for the $T$-flux (see [@aretakis2]): $$\mathcal{E}^{T}_{\Sigma_{\tau}}[\psi] \leq C\cdot \frac{1}{\tau^2}\cdot E[\psi].
\label{energydecaytintro}$$ where $E[\psi]$ is an appropriate weighted higher-order energy norm of the initial data. Using this type of estimate, it can shown that $\psi$ satisfies the following *pointwise decay estimate* $$|\psi|_{\Sigma_{\tau\cap \{r\geq r_0\}}}\leq C_{r_0}\cdot \frac{1}{\tau}\cdot E[\psi]
\label{eq:awayintro}$$ away from the event horizon $r\geq r_0>M$.
To obtain non-degenerate control of $\psi$ and its derivatives along the event horizon, we consider the energy flux $\mathcal{E}^N[\psi]$ associated to the timelike vector field $N$ which satisfies the *positivity* property $$\mathcal{E}^{N}_{\Sigma_{\tau}}[\psi]\sim \int_{\Sigma_{\tau}} |\partial\psi|^2\, d\mu_{\Sigma_{\tau}}.$$ It turns out that there is a uniform positive constant $C$ such that (see [@aretakis1]) $$\mathcal{E}^{N}_{\Sigma_{\tau}}[\psi]\leq C\cdot \mathcal{E}^{N}_{\Sigma_{0}}[\psi].$$ On the other hand, no decay estimate was known for $J^N$. Nonetheless, via an interpolation argument, it can be shown that $\psi$ does decay along the event horizon: $$|\psi|_{\Sigma_{\tau}\cap {\mathcal{H}^{+}}}\leq C \cdot \frac{1}{\tau^{\frac{3}{5}}}\cdot E[\psi]
\label{eq:onintro}$$ The decay estimates , were the only decay rates that had been proved rigorously for scalar fields $\psi$ on ERN. In this paper, we derive the sharp rates (upper and lower bounds); in fact we derive the precise late-time asymptotics for $\psi$. See Section \[sec:SummaryOfTheMainResults\].
### Energy and pointwise blow-up {#sec:EnergyAndPointwiseBlowUp}
As we shall see, the decay rates in , are not sharp. However, they do suggest that *the decay rate of $\psi$ along the event horizon is slower than the decay rate of $\psi$ away from the horizon*. This statement, which rigorously follows from the main results of the present paper, is a precursor of the horizon instability of ERN. Recall that with respect to the spherical sections $S_{\tau}$ of ${\mathcal{H}^{+}}$, the spherical means $-\frac{1}{4\pi }\int_{S_{\tau}}\left(\partial_r\psi+M^{-1}\psi\right)\, d\omega$ are conserved. On the other hand, for generic initial data on $\Sigma_0$ we have $-\frac{1}{4\pi }\int_{S_{0}}\left(\partial_r\psi+M^{-1}\psi\right)\,d\omega=\frac{1}{M^3}H_0[\psi]\neq 0$. Hence, in view of the estimate , we conclude the following
**Non-decay:** *generically, the spherical mean of the transversal derivative $-\frac{1}{4\pi M^2}\int_{S_{\tau}}\partial_r\psi$ does not decay along the event horizon of ERN. In fact,* $$-\frac{1}{4\pi }\int_{S_{\tau}}\!\partial_r\psi \, d\omega\ \rightarrow \frac{1}{M^3}H_0[\psi], \ \text{ as }\tau\rightarrow \infty.$$ The non-decaying transversal derivative along the event horizon accounts for the different decay rates of $\psi$ on and away from the horizon ${\mathcal{H}^{+}}$. On the other hand, it can be shown that $\partial_r\psi$ decays along the hypersurfaces $\{r=r_0>M\}$ away from the event horizon ${\mathcal{H}^{+}}$. It was observed in [@hm2012] that the above non-decay result implies that *the component ${\mathbf{T}}_{rr}[\psi]$ of the energy-momentum tensor of the scalar field $\psi$ does not decay along* ${\mathcal{H}^{+}}$. In fact, we have $$\frac{1}{4\pi }\int_{S_{\tau}}\!{\mathbf{T}}_{rr}[\psi]\, d\omega\rightarrow \frac{1}{M^6}\left(H_{0}[\psi]\right)^2.$$ Since, ${\mathbf{T}}_{rr}[\psi]$ is related to the energy density measured by an observer crossing ${\mathcal{H}^{+}}$, the authors of [@hm2012] concluded that the conserved charge $H_0[\psi]$ might be thought of as “hair” of the extremal event horizon. It is important to remark that *the results of the present paper yield a new way in potentially measuring this hair from observations along null infinity*. See Section \[sec:SummaryOfTheMainResults\].
By acting with $\partial_r$ on the wave equation , restricting on the event horizon and using the previous results we conclude the following:
**Blow-up:** *the spherical mean of higher-order transversal derivatives $-\frac{1}{4\pi}\int_{S_{\tau}}\partial_r^k\psi\, d\omega$ with $k\geq 2$ generically blows up along the event horizon of ERN. In fact,* $$\frac{1}{4\pi }\int_{S_{\tau}}\!|\partial_r^k\psi| \, d\omega \geq c_k\cdot H_0[\psi]\cdot \tau^{k-1}, \ \text{ as }\tau\rightarrow \infty.$$ Furthermore, the following *higher-order energy blow-up* result generically holds (see [@aretakis2]): $$\mathcal{E}^{N}_{\Sigma_{\tau}}[N^k\psi]\rightarrow \infty$$ for all $k\geq 1$ as $\tau\rightarrow \infty$.
We remark that an extension of the above instabilities to linearized electromagnetic and gravitational perturbations of ERN was presented in [@hm2012] and [@sela2]. Nonlinear extensions have been presented in [@aretakis2013; @harvey2013; @bizon-extremal-nonlinear; @areangel6; @harveyeffective]. For higher-dimensional extensions we refer to [@murata2012; @iapwnes]. For a more detailed discussion of works in the physics literature, see the next section.
Physics literature on the dynamics of extremal Reissner–Nordström {#sec:PreviousWorksOnLateTimeAsymptoticsOnERN}
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In this section we present results in the physics literature which concern the late-time asymptotics for ERN.
### The Blaksley–Burko asymptotic analysis {#sec:TheBlaksleyBurkoAsymptoticAnalysis}
The first work on asymptotics of scalar fields on ERN goes back to 1972 when Bi[č]{}[á]{}k suggested in [@Bicak1972] that scalar fields $\psi_{\ell}$ on ERN with non-vanishing Newman–Penrose constant and with angular frequency $\ell$ decay with the rate $\frac{1}{t^{\ell+2}}$. However, this result was shown to be false in 2007 when Blaksley and Burko [@Burko2007] performed a more accurate heuristic and numerical analysis. Their work considered the following two types of initial data (see Section \[sec:Introduction\] for the relevant definition):
- Type I: horizon-penetrating and null-infinity-extending,
- Type II: Supported away from the horizon and compactly supported.
Define $\mu\in \{0,1\}$ such that $\mu=0$ for data of Type I and $\mu =1$ for data of type II. The authors argued that the *sharp* decay rates for the scalar field are the following:
- Away ${\mathcal{H}^{+}}$ and ${\mathcal{I}^{+}}$: $\ |\psi_{\ell}|_{r=r_0>M} \ \ \text{decays like} \ \ \frac{1}{\tau^{2\ell+2+\mu}}$,
- On ${\mathcal{H}^{+}}$: $\ |\psi_{\ell}|_{{\mathcal{H}^{+}}} \ \ \text{decays like} \ \ \frac{1}{\tau^{\ell+1+\mu}}$,
- On ${\mathcal{I}^{+}}$: $\ |r\psi_{\ell}|_{{\mathcal{I}^{+}}} \ \ \text{decays like} \ \ \frac{1}{\tau^{\ell+1+\mu}}$.
Reference [@Burko2007] did not obtain the precise late-time asymptotics in the above two cases. Moreover, [@Burko2007] did not study other types of initial data, and in particular, did not study horizon-penetrating compactly supported initial data.
### The Lucietti–Murata–Reall–Tanahashi asymptotic analysis {#sec:TheWorkOfReallEtAl}
The asymptotic analysis of Lucietti–Murata–Reall–Tanahashi [@hm2012] was the first work to numerically investigate the precise late-time asymptotics for scalar fields on ERN. The present paper is highly motivated by [@hm2012].
A major result of the numerical analysis of [@hm2012] is the following precise late-time asymptotic behavior of scalar fields with compactly supported initial data $$M\cdot\psi|_{{\mathcal{H}^{+}}} \sim 2H_0[\psi]\cdot \frac{1}{\tau}+4MH_0[\psi]\cdot \frac{\log \tau}{\tau^2}, \ \text{ as } \tau\rightarrow \infty.
\label{harveyhorizon}$$ Furthermore, the authors suggested, using a near-horizon calculation, that the following precise late-time asymptotic behavior off the horizon along $r=r_0>M$ holds: $$\psi|_{\{r=r_0\}} \sim \frac{4M}{r_0-M}H_0[\psi]\cdot\frac{1}{\tau^2}, \ \text{ as } \tau\rightarrow \infty.
\label{harveyaway}$$ Moreover, the authors, extrapolating from numerical simulations for the $\ell=1,2$ angular frequencies, suggested the following sharp rate off the horizon along $r=r_0>M$ $$|\psi_{\ell}|_{\{r=r_0\}} \ \ \text{decays like} \ \ \frac{1}{\tau^{2\ell+2}}.
\label{harveyell}$$ On the other hand, the numerics of [@hm2012] suggested the following asymptotic expression in the case of data with $H_0[\psi]=0$ $$\psi|_{{\mathcal{H}^{+}}}\sim \frac{C_0}{\tau^2}, \ \text{ as } \tau\rightarrow \infty.
\label{harvey0hor}$$ However, the following points were not addressed in [@hm2012]:
- The constant $ C_0$ in was not explicitly computed in terms of the initial data.
- The precise asymptotic estimate was only obtained for compactly supported data, and not, in particular, for data with non-vanishing Newman–Penrose constant.
- The asymptotics of the radiation field $r\psi_{{\mathcal{I}^{+}}}$ along the null infinity ${\mathcal{I}^{+}}$ were not investigated (in the $H_0[\psi]\neq 0$ case).
In the present paper, we address all the above issues (see Section \[sec:SummaryOfTheMainResults\]).
Another important question that was first raised and investigated in [@hm2012] is whether one can trigger the horizon instability using ingoing radiation; that is, using perturbations which are initially supported away from the event horizon and hence necessarily satisfy $H_0[\psi]=0$. The authors found the following stability results $$|\psi|_{{\mathcal{H}^{+}}}\rightarrow 0,\ \ \ |\partial_{r}\psi|_{{\mathcal{H}^{+}}}\rightarrow 0: \ \text{ along }{\mathcal{H}^{+}},$$ and uncovered the following (generic) instability behavior $$|\partial_{r}^2\psi|_{{\mathcal{H}^{+}}}\nrightarrow 0 \ \ \ |\partial_{r}^3\psi|_{{\mathcal{H}^{+}}}\rightarrow \infty: \ \text{ along }{\mathcal{H}^{+}}.$$ This instability behavior, which has also been discussed in [@bizon2012], was subsequently rigorously proved in [@aretakis2012].
Reference [@hm2012] also investigated the late-time behavior of massive scalar fields which solve $\Box_g \psi= m^2\psi$. For such massive fields it is widely believed that the late-time behavior is dominated by the $\omega=\pm m$ frequencies (instead of the $\omega=0$ frequency for massless fields on sub-extremal black holes) which results in a damped oscillatory late-time behavior. In particular, massive fields and all their derivatives are expected to decay like ${\tau^{-\frac{5}{6}}}$ in the domain of outer communications (up to and including the event horizon) of a sub-extremal black hole. The results of [@hm2012] suggest that this remains true on ERN backgrounds off the horizon (a result that had also been seen in [@koyama2001]). On the other hand, [@hm2012] found that the horizon instability persists for a *discrete* set of masses $m^2$. Specifically, if $(mM)^2=n(n+1)$ then the authors argued that $$|\partial_{r}^{n+1}\psi|_{{\mathcal{H}^{+}}}\nrightarrow 0 \ \ \ |\partial_{r}^{n+2}\psi|_{{\mathcal{H}^{+}}}\rightarrow \infty: \ \text{ along }{\mathcal{H}^{+}}.$$ More generally, the numerical analysis of [@harvey2013] suggests the following asymptotic behavior for *general* masses $m^2$: $$\partial_{r}^{k}\psi\ \ \text{ behaves like } \tau^{k-\frac{1}{2}-\sqrt{(mM)^2+\frac{1}{4}}},$$ for all $k\geq 0$. A rigorous proof of the above statements for massive fields remains open.
### The Ori–Sela asymptotic analysis {#sec:TheOriSelaAsymptoticAnalysis}
Ori [@ori2013] and Sela [@sela] used the conservation laws that hold for each fixed angular frequency $\ell$ (see Section \[sec:ConservationLawsAlongTheEventHorizon\]) to heuristically obtain the precise late-time asymptotics of $\psi_{\ell}$ for horizon-penetrating compactly supported initial data. Specifically, they found that along $r=r_0>M$ away from the horizon the following holds: $$\psi_{\ell}|_{\{r=r_0\}}\sim (-4)^{\ell+1}eM^{3\ell+2}\frac{r}{(r-M)^{\ell+1}}\cdot \frac{1}{\tau^{2\ell+2}}, \ \ \text{ as }\tau\rightarrow \infty,$$ where $e$ is an explicit expression of the conserved charge $H_{\ell}[\psi_{\ell}]$ for $\psi_{\ell}$. Hence, the above result improves the statement of [@hm2012].
Furthermore, Ori and Sela derived the precise late-time asymptotics of $\psi_{\ell}$ along the horizon $$\psi_{\ell}|_{{\mathcal{H}^{+}}}\sim e (-M)^{\ell+1}\cdot \frac{1}{\tau^{\ell+1}},\ \ \text{ as }\tau\rightarrow \infty,$$ where $e$ is as above. The recent Fourier based work of Bhattacharjee et al [@ind2018] supported the validity of the above asymptotics.
On the other hand, no asymptotic estimate was derived for the radiation field $r\psi_{\ell}|_{{\mathcal{I}^{+}}}$ along null infinity. Furthermore, the authors did not obtain precise late-time asymptotics in the case where the initial data are supported away from the event horizon and did not provide an explicit expression for the constant $C_0$ that appears in the asymptotic statement .
Sela [@sela2] subsequently used the decay rates obtained in [@ori2013; @sela] in order to obtain decay rates for the coupled electromagnetic and gravitational system for ERN.
### The Murata–Reall–Tanahashi spacetimes {#sec:TheReallSpacetimes}
In a very beautiful work [@harvey2013], *Murata, Reall and Tanahashi studied numerically the fully non-linear evolution of the horizon instability of ERN*. Specifically, the authors of [@harvey2013] investigated perturbations of ERN in the context of the Cauchy problem for the *spherically symmetric Einstein–Maxwell-(massless) scalar field system*. The authors studied various types of perturbations and obtained a great number of results, all of which are consistent with the linear theory described in the previous sections. We will next provide a more detailed summary of their results. It is important to remark that a rigorous treatment of this system remains a (very interesting) open problem.
The initial data on a Cauchy hypersurface $\Sigma_0$ for the spherically symmetric Einstein–Maxwell-scalar field system are completely determined (modulo gauge fixing) by the value of the initial Bondi mass $M$, the conserved charge $e>0$ and the profile of the scalar field $\psi$ on $\Sigma_0$. Note that ERN corresponds to data for which $M=e$ and $\psi$ is trivial on $\Sigma_0$. The authors of [@harvey2013] considered compactly supported scalar fields $\psi$ of size $\epsilon>0$ $$\max_{\Sigma_0}|\psi| =\epsilon.$$ The authors considered the following three types of perturbations of ERN:
**Type I:** *First-order mass perturbation* $M=e+O(\epsilon).$
This is the “largest” of three types of perturbations. An open neighborhood $\mathcal{O}_{\text{trap}}$ of the initial hypersurface $\Sigma_0$ contains trapped surfaces. The evolved spacetime contains a complete null infinity and a well-defined black hole region bounded by a smooth event horizon ${\mathcal{H}^{+}}$. In fact, the spacetime converges asymptotically in time to a sub-extremal RN background with surface gravity $\kappa=O(\sqrt{\epsilon})$, which, in particular, implies that $\psi$ and all higher-order tranvsersal derivatives $\partial^{k}_{r}\psi$ decay along ${\mathcal{H}^{+}}$. On the other hand, the proximity to ERN on the initial hypersurface creates non-trivial effects at initial times, and more specifically at the time scale $\tau\in\left[0,\frac{1}{\sqrt{\epsilon}}\right]$. During this time scale, the third-order transversal derivative $\partial_r^3\psi$ **grows** along the event horizon ${\mathcal{H}^{+}}$, reaches a maximum value and then starts decaying. The crucial observation of Murata, Reall and Tanahashi is that $$\max_{{\mathcal{H}^{+}}}\partial_r^3\psi\ \sim\ \frac{H_0[\psi_0]}{\kappa_1},$$ where $\psi_0$ is the linearization (in $\epsilon$) of the scalar field $\psi$, $H_0$ is the conserved charged on exactly ERN and $\kappa_1$ is the linearization (in $\epsilon$) of the square of the surface gravity $\kappa^2$. The above clearly implies that for these kinds of perturbations $$\max_{{\mathcal{H}^{+}}}\partial_r^3\psi \nrightarrow 0 \ \text{ as }\ \epsilon\rightarrow 0.$$ In other words, *even though the size of the initial perturbation goes to zero (as $\epsilon\rightarrow 0$), the maximum size of higher-order derivatives of the scalar fields does not go to zero*. This may be interpreted as a remnant of the horizon instability in the non-linear theory. We will see below that the situation gets more dramatic when we consider “smaller” perturbations of ERN.
**Type II:** *Second-order mass perturbation* $M=e+O\left(\epsilon^2\right).$
In view of the fact that ERN does not contain trapped surfaces, one would like to consider perturbations which do not contain trapped surfaces on the initial hypersurface. In order to achieve this, one needs to reduce the size of the initial Bondi mass $M$ so that the region $\mathcal{O}_{\text{trap}}$ of trapped surfaces on $\Sigma_0$ reduces to a single surface, namely a marginally trapped surface. This leads to a second-order mass perturbation for which $M=e+O\left( \epsilon^2\right)$. According to [@harvey2013], the evolved spacetime converges asymptotically in time to a sub-extremal RN background with surface gravity $\kappa=O({\epsilon})$, which again implies that $\psi$ and all higher-order transversal derivatives $\partial^{k}_{r}\psi$ decay along ${\mathcal{H}^{+}}$. In this case, the proximity to ERN on the initial hypersurface creates non-trivial effects at the time scale $\tau\in\left[0,\frac{1}{{\epsilon}}\right]$ during which the second-order transversal derivative $\partial_r^2\psi$ **grows** along the event horizon ${\mathcal{H}^{+}}$ reaching a maximum value and then decaying to zero. In fact, the authors calculated $$\max_{{\mathcal{H}^{+}}}\partial_r^2\psi\ \sim\ \frac{H_0[\psi_0]}{\kappa_0},$$ where $\psi_0$ is the linearization (in $\epsilon$) of the scalar field $\psi$, $H_0$ is the conserved charged on exactly ERN and $\kappa_0$ is the linearization (in $\epsilon$) of the surface gravity $\kappa$. The above clearly implies that for these kinds of perturbations $$\max_{{\mathcal{H}^{+}}}\partial_r^2\psi \nrightarrow 0 \ \text{ as }\ \epsilon\rightarrow 0.$$ Once again, we see that the horizon instability is present in the non-linear theory.
**Type III:** *Fine-tuned perturbations* $M=M^{*}(e, \epsilon)$
In the above two cases, the evolved spacetimes converged to sub-extremal RN. In particular, they contained trapped surfaces. The third type of perturbation that was studied by Murata, Reall and Tanahashi treats the case where the evolved spacetime has a regular black hole region but does not have any trapped surfaces and hence has properties which are reminiscent of ERN. For this reason, in fact, the authors called these spacetimes *dynamically extremal*.[^5] In order to numerically construct such spacetimes, the authors considered even smaller fine-tuned values $M^{*}(e, \epsilon)$ of $M$ compared to the case above. We remark that it is conjectured that for initial masses which are less than $M^{*}(e, \epsilon)$ the evolved spacetimes contain naked singularities. Returning the case where the initial mass is exactly equal to $M^{*}(e, \epsilon)$, the corresponding spacetime has a black hole region and converges to ERN **outside** the event horizon. However, on the event horizon, the instability kicks in: $$|\partial_r\psi|\nrightarrow 0\, \ \ \ |\partial_r\partial_r\psi|\rightarrow \infty: \ \text{ along }\ {\mathcal{H}^{+}}$$ for each of these fine-tuned perturbations of ERN. This suggests that *dynamically extremal black holes exhibit a non-linear version of the horizon instability*.
See also Section \[sec:TheInteriorOfBlackHolesAndStrongCosmicCensorship\] for the dynamics of the interior of ERN.
### Addendum: The Casals–Gralla–Zimmerman work on extremal Kerr {#sec:RecentBreakthroughOfCasalsGrallaAndZimmerman}
One of the major open problems in black hole dynamics is to derive the precise late-time asymptotic behavior of general smooth solutions to the wave equation on extremal Kerr (EK) backgrounds. According to [@hj2012; @aretakis3; @aretakis4], axisymmetric fields on EK exhibit exactly the same horizon instability as discussed in Section \[sec:TheHorizonInstabilityOfExtremalBlackHoles\]. For non-axisymmetric scalar fields on EK, however, the horizon instability is significantly amplified. Andersson and Glampedakis [@mhighinsta], following earlier work of Detweiler [@detweiler80], argued that the dominant temporal frequencies $\omega$ for scalar fields $\psi_m$ with *fixed azimuthal frequencies* $m$ occur for $\omega\sim \frac{1}{2M}m$, instead of $\omega\sim 0$ in other settings. Specifically, [@mhighinsta] suggested that away from horizon on $r=r_0>M$ the following sharp rate holds: $$|\psi_{m}|_{\{r=r_0 \}} \ \ \text{decays like} \ \ \frac{1}{\tau}.
\label{glamperate}$$ Important subsequent studies of the distribution of quasinormal modes on EK were presented in [@zeni13; @berti2017] and their findings are consistent with .
Casals, Gralla and Zimmerman [@zimmerman1] were the first to derive the late-time asymptotics along the event horizon for $\psi_m$. Their semi-analytic work, which is based on the mode decomposition method of Leaver [@leaver], yielded the following asymptotic behavior along the horizon $$|\psi_{m}|_{{\mathcal{H}^{+}}} \ \ \text{decays like} \ \ \frac{1}{\sqrt{\tau}}.
\label{zimrate}$$ Reference [@zimmerman1] considered initial data which are compactly supported and *supported away from the event horizon* (and hence they are not horizon-penetrating). Clearly, the rate of is much slower than the sharp decay rates in all other previously discussed settings. Moreover, Casals, Gralla and Zimmerman argued that the instability is further amplified for the first-order transversal to ${\mathcal{H}^{+}}$ derivative $$|\partial_r\psi_{m}|_{{\mathcal{H}^{+}}} \ \ \text{behaves like} \ \ \sqrt{\tau}.
\label{zimrate1}$$ In other words, the results in [@zimmerman1] suggest that for data supported *away* from the horizon the first-order derivative **grows** along the horizon. One would naturally expect that the growth is even more severe in the case where the initial data are horizon-penetrating. However, Hadar and Reall [@harveyeffective] performed a near-horizon analysis which indicates that and (surprisingly!) hold also for scalar fields with horizon-penetrating data. Zimmerman [@zimmerman3] obtained the same rates as and for charged perturbations on ERN. Further extensions have been provided in [@zimmerman2; @zimmerman4; @zimmerman5]. A numerical confirmation of and , as well as stability results for curvature scalars, was presented by Burko and Khanna [@khanna17]. Further extensions to supersymmetric quantum mechanics were presented in [@cardoso-2017].
Proving rigorously , and remains an open problem. We also remark that precise late-time asymptotics for $\psi_m$ on EK are not known. Moreover, even the basic boundedness statement for general solutions $$\psi=\sum_{m=0}^{\infty}\psi_m$$ is completely open.
Outline of the paper {#sec:Outline}
--------------------
The geometry of ERN along is presented in Section \[geometrysection\]. The types of initial data for the wave equation that we will consider are introduced in Section \[sec:TheTypesOfInitialDataABCD\]. A non-technical summary of our results and various applications are presented in Sections \[sec:SummaryOfTheMainResults\] and \[sec:Remarks\], respectively. The main theorems and an overview of the ideas of the proofs can be found in Section \[subsec:TheMainTheorems\]. The weighted hierarchies are derived in Sections \[sec:rweightest\] and \[sec:extendhier\] and pointwise and energy decay results are obtained in Section \[sec:decayest\]. The precise late-time asymptotics are derived in Sections \[sec:asympnonzeroconst\]–\[sec:hoasymp\].
Acknowledgements {#sec:Acknowledgements}
----------------
We would like to thank our mentor Mihalis Dafermos for several insightful discussions. We would also like to thank Harvey Reall, Peter Zimmerman and Samuel Gralla for elucidative conversations. The second author (S.A.) acknowledges support through NSF grant DMS-1265538, NSERC grant 502581, an Alfred P. Sloan Fellowship in Mathematics and the Connaught Fellowship 503071.
The geometry of ERN {#geometrysection}
===================
The ERN metric {#sec:TheERNManifoldFoliationsAndVectorFields}
--------------
The *extremal Reissner–Nordström* spacetimes $(\mathcal{M}_M,g_M)$, $M>0$, are given by the following manifold-with-boundary $$\mathcal{M}_{M}={\mathbb{R}}\times [M ,\infty) \times {\mathbb{S}}^2,$$ equipped with the coordinate chart $(v,r,\theta,\varphi)$, where $v\in {\mathbb{R}}$, $r\in [M,\infty)$ and $(\theta,\varphi)$ is the standard spherical coordinate chart on the round 2-sphere ${\mathbb{S}}^2$. and the following Lorentzian metric $$g_{M}=-D(r)dv^2+2dvdr+r^2(d\theta^2+\sin^2\theta d\varphi^2),$$ where $$D(r)=(1-Mr^{-1})^2.$$ We denote the *future event horizon* as the boundary $\mathcal{H}^+=\partial \mathcal{M}_{M}=\{r=M\}$. We will also denote $$T:=\partial_v,\ \ \ Y:=\partial_r.$$ Note that $Y$ is *transversal* to the event horizon. For ERN we have $$\nabla_{T}T=0$$ on the event horizon. This means that ${\mathcal{H}^{+}}$ has vanishing surface gravity in ERN.
We next introduce the *tortoise* coordinate $r^{*}$ by $$r_*(r)=r-M-M^2(r-M)^{-1}+2M\log\left(\frac{r-M}{M}\right).$$ The *double null coordinate chart* $(u,v,\theta,\varphi)$ in the manifold $\mathring{\mathcal{M}}:=\mathcal{M}\setminus \partial M$, is given by $$u=v-2r_*(r).$$ with $u,v\in {\mathbb{R}}$. In double null coordinates, the extremal Reissner–Nordström metric can be expressed as $$g_{M}=-D(r)dudv+r^2(d\theta^2+\sin^2\theta d\varphi^2).$$ If we consider the vector fields $$L:=\partial_v \ \ \text{ and }\ \ \underline{L}=\partial_u,$$ with respect to the double null coordinates $(u,v,\theta,\varphi)$, then we have the relations $$L=\partial_v+\frac{1}{2}D\partial_r, \ \ \underline{L}=-\frac{1}{2}D\partial_r.$$ Finally, we define $t=(u+v)/2$ and introduce the coordinate system $(t,r^{*},\theta,\varphi)$ with respect to which the metric takes the form $$g_{M}=-D(r)dt^2+D(r)d(r^{*})^2+r^2(d\theta^2+\sin^2\theta d\varphi^2).$$ Note that the vector field $T:=\partial_t$ is Killing and timelike everywhere away from the event horizon.
The null hypersurfaces $C_{\tau} = \{u = \tau \}$ terminate in the future (as $r,v \rightarrow \infty$) at *future null infinity* $\mathcal{I}^{+}$. We will occasionally use the notation $v_{r_0}(u')$, with $r_0>M$, to indicate the value of the $v$ coordinate along the hypersurface $\{r=r_0\}$ at $u=u'$, and similarly, $u_{r_0}(v')$ to indicate the value of the $u$ coordinate along the hypersurface $\{r=r_0\}$ at $v=v'$.
We use the notation $\slashed{\nabla}_{{\mathbb{S}}^2}$ for the covariant derivative with respect to the metric of the unit round 2-sphere and $\slashed{\Delta}_{{\mathbb{S}}^2}$ for the corresponding Laplacian.
We will also use the following “big O notation” with respect to $u$, $v$ and $r$. We use the notation $O_k(r^{-l})$ to indicate functions $f$ on (a subset of) $\mathcal{M}_M$ that satisfy the behavior $|Y^kf|\leq C r^{-\ell-k}$, where $C>0$ is a constant that is independent of $f$ and $k\in {\mathbb{N}}_0$, $l\in {\mathbb{Z}}$. Similarly, we use the notation $O_k(u^{-l})$ and $O_k(v^{-l})$ when $|\underline{L}^kf|\leq C u^{-\ell-k}$ and $|L^kf|\leq C v^{-\ell-k}$, respectively. Finally, we will also employ the notations $O_k((v-u)^{-l})$ and $O_k((u-v)^{-l})$ to group functions $f$ that satisfy, for $k_1+k_2=k$, $k_1,k_2\in {\mathbb{N}}_0 $, $|\underline{L}^{k_1}L^{k_2}f|\leq C |v-u|^{-\ell-k}$ and $|\underline{L}^{k_1}L^{k_2}f|\leq C |u-v|^{-\ell-k}$, respectively. When $k=0$ in the above notation, we will omit the subscript in $O_k$.
The spacelike-null foliation {#sec:TheHyperboloidalFoliation}
----------------------------
Let $\Sigma_0$ be a spherically symmetric hypersurface which crosses the event horizon and terminates at null infinity: $$\Sigma_0:=\{v=v_{\Sigma_0}(r)\},$$ where $v_{\Sigma_0}: [M,\infty)\to {\mathbb{R}}$ is a function defined as follows $$v_{\Sigma_0}(r)=v_{\rm \min}+\int_M^rh(r')\,dr',
\label{definitionh}$$ where we take $v_{\rm \min}\in {\mathbb{R}}_+$ to be a constant and $h: [M,\infty)\to {\mathbb{R}}_{\geq 0}$ is a non-negative function satisfying $$\begin{aligned}
0\leq 2D^{-1}(r)-h(r)&\:=O(r^{-1-\eta}),\end{aligned}$$ for some constant $\eta>0$. We will take $v_{\Sigma_0}(r)$ to be monotonically increasing function. Moreover, $u_{\Sigma_0}(r):=u|_{\Sigma_0}(r)=v_{\Sigma_0}(r)-2r_*(r)$ satisfies $\frac{du_{\Sigma_0}}{dr}=h(r)-2D^{-1}(r)\leq 0$, so $u_{\Sigma_0}(r)$ is a monotonically decreasing function. For convenience, we will assume that $\Sigma_0$ satisfies the following symmetry condition: $$(t,r^{*})\in \Sigma_0 \Longrightarrow (t,-r^{*}) \in \Sigma_0.$$ This condition is here imposed only for convenience because it simplifies the expressions of various new quantities that we introduce in this paper; our results apply for general initial hypersurfaces $\Sigma_0$ as well. An important example of such a hypersurface is defined as follows: Let $M<r_{\mathcal{H}}<2M$ and $r_{\mathcal{I}}>2M$ such that $r^*(r_{\mathcal{H}})=-r^*(r_{\mathcal{I}})$. Then we may further assume that $$\begin{split}
\Sigma_0\cap \{r\leq r_{\mathcal{H}}\}=&{N}_0^{\mathcal{H}}: =\{v=v_0\}\cap \{r\leq r_{\mathcal{H}}\} ,\\
\Sigma_0\cap \{r\geq r_{\mathcal{I}}\}=&{N}_0^{\mathcal{I}}: =\{u=u_0\}\cap \{r\geq r_{\mathcal{I}}\},
\end{split}$$ with $u_0,v_0>0$.
Let $F_{\tau}$ denote the flow of the stationary vector field $T$ where the *time function* $\tau:J^+({\Sigma_0})\to {\mathbb{R}}_{\geq 0}$ is defined as follows $$\begin{aligned}
\tau|_{\Sigma_0}=&\:0,\\
T(\tau)=&\:1.\end{aligned}$$ Note that for all $\tau \geq 1$ we have $$\tau\sim v \text{ for } r\leq r_{\mathcal{H}},\ \ \tau\sim v \sim u \text{ for } r_{\mathcal{H}}\leq r \leq r_{\mathcal{I}}, \ \ \tau \sim u \text{ for } r\geq r_{\mathcal{I}}. $$ We define the following foliation of the future $\mathcal{R}=J^+({\Sigma_0})$ of $\Sigma_0$: $$\mathcal{R}=\cup_{\tau\geq 0}\Sigma_{\tau}=F_{\tau}(\Sigma_0);$$ see Figure \[fig:sigma0\].
We use the notations $d\mu_{\Sigma_{\tau}}$ and $d\mu_{\tau}$ to indicate the natural volume form on $\Sigma_{\tau}$ with respect to the induced metric, where on the null parts $\mathcal{N}^{\mathcal{H}}_{\tau}$ and $\mathcal{N}^{\mathcal{I}}_{\tau}$ we take this volume form to be $r^2d\omega du$ and $r^2d\omega dv$, respectively, where $d\omega=\sin\theta d\theta d\varphi$. Similarly, we denote the normal vector field to $\Sigma_{\tau}$ with $\mathbf{n}_{\Sigma_{\tau}}$ and $\mathbf{n}_{\tau}$, where we take the normal to $\mathcal{N}^{\mathcal{H}}_{\tau}$ and $\mathcal{N}^{\mathcal{I}}_{\tau}$ to be $\underline{L}$ and $L$, respectively.
![The spacelike-null foliation $\Sigma_{\tau}$.[]{data-label="fig:sigma0"}](Sigma0Sigmatau.pdf)
It will be useful to moreover introduce the following corresponding partition of the spacetime region $\mathcal{R}$: $$\mathcal{R}=\mathcal{A}^{\mathcal{H}}\cup\mathcal{B}\cup \mathcal{A}^{\mathcal{I}},$$ where $$\begin{aligned}
\mathcal{A}^{\mathcal{H}}:=\mathcal{R}\cap\{r\geq r_{\mathcal{H}}\}, \ \ \mathcal{B}:=\mathcal{R}\cap\{r_{\mathcal{H}}<r<r_{\mathcal{I}}\},\ \
\mathcal{A}^{\mathcal{I}}:=\mathcal{R}\cap\{r\leq r_{\mathcal{I}}\};\end{aligned}$$ see Figure \[fig:sigmaabc0\].
![The regions $\mathcal{A}^{\mathcal{H}}, \mathcal{B}$ and $\mathcal{A}^{\mathcal{I}}$.[]{data-label="fig:sigmaabc0"}](ABAregions.pdf)
Cauchy data of Type **A**, **B**, **C** and **D**. {#sec:TheTypesOfInitialDataABCD}
--------------------------------------------------
Recall that ERN admits two independent conserved charges: 1) the horizon charge[^6] $H_0[\psi]$ given by , and 2) the Newman–Penrose constant $I_0[\psi]$ at null infinity given by . It is important to emphasize that the values of $H_0[\psi]$ and $I_0[\psi]$ depend **only** on the initial data of $\psi$ at the event horizon ${\mathcal{H}^{+}}$ and at null infinity ${\mathcal{I}^{+}}$, respectively. Hence, compactly supported initial data necessarily satisfy $I_0[\psi]=0$ whereas data for which $I_0[\psi]\neq 0$ are necessarily not compactly supported. Similarly, data supported away from the horizon necessarily satisfy $H_0[\psi]=0$. Recall Definition \[def1intro\] according to which data which satisfy $H_0[\psi]\neq 0$ are called horizon-penetrating. We also introduce the following
Initial data on a Cauchy hypersurface $\Sigma_0$ are called **null-infinity-extending** if the Newman–Penrose constant $I_0[\psi]\neq 0$. \[def1intro2\]
We distinguish the following four types of initial data
**Type A:** *Compactly supported data but horizon-penetrating*.
These data should be thought of as local data in the sense that they reflect perturbations in a neighborhood of the event horizon.
**Type B:** *Compactly supported data that is supported away from the event horizon.*
These data correspond to compact perturbations from afar, that is away from the event horizon.
**Type C:** *Null-infinity-extending and horizon-penetrating data.*
These data correspond to global perturbations with non-trivial support across the whole initial hypersurface $\Sigma_0$. In the physics literature, such data are said to have an “*initial static moment*”.
**Type D:** *Null-infinity-extending but supported away from the horizon data.*
These data correspond to perturbations from afar extending all the way to null infinity.
In summary we have the following table:
[ cV[4]{}c | c ]{} Data & $H_0$ & $I_0$\
Type **A** & $\neq 0$ & $=0$\
Type **B** & $=0$ & $=0$\
Type **C** & $\neq 0$ & $\neq 0$\
Type **D** & $=0$ & $\neq 0$\
As we shall see, each of these types requires a separate treatment and exhibits different asymptotic behavior.
A first version of the main results {#sec:SummaryOfTheMainResults}
===================================
In this section we will present the main theorems of the present paper. We will first present a non-technical version of the results and then in Section \[sec:Remarks\] various applications of our results. Finally we will present the rigorous statements of the main theorems in Section \[subsec:TheMainTheorems\].
We first introduce the notion of a new horizon charge which plays a fundamental role our study of the dynamics of ERN.
The new horizon charge $H_{0}^{(1)}[\psi]$ {#sec:TheNewHorizonHairH01Psi}
-------------------------------------------
We introduce the *dual* scalar field $\widetilde{\psi}$ of $\psi$ as follows $$\widetilde{\psi}(t,r^*,\theta,\phi) = \frac{M}{r-M}\psi(t,-r^*,\theta,\phi).
\label{dual}$$ First observe that duality is self-inverse: $\widetilde{\widetilde{\psi}}=\psi$. Furthermore, $\psi$ satisfies the wave equation if and only if its dual $\widetilde{\psi}$ satisfies . This duality is motivated by the Couch–Torrence conformal symmetry [@couch] of ERN. References [@bizon2012; @hj2012] showed that this duality can be used to relate the horizon charge with the Newman–Penrose constant as follows: $$H_0[\psi] =I_0[\widetilde{\psi}].$$ If the Newman–Penrose constant vanishes $I_0[\psi]=0$ then the following expression $$I_{0}^{(1)}[\psi]=\frac{M}{4\pi} \int_{\Sigma_0\cap\mathcal{H}^{+}}\!\!\psi r^2d\omega+\lim_{r_0\rightarrow \infty}\left(\frac{M}{4\pi}\int_{\Sigma_0\cap\{r\leq r_0\}} n_{\Sigma_0}(\psi)d\mu_{\Sigma_0}+\frac{M}{4\pi}\int_{\Sigma_0\cap\{r=r_0\}}\Big(\psi-\frac{2}{M}r\partial_v(r\psi)\Big)r^2d\omega\right),
\label{timeinvertednp}$$ is finite and conserved[^7]. See the discussion in Section \[sec:TheWaveEquationOnBlackHolesBackgrounds\] and for more details in [@paper-bifurcate]. We refer to $I_{0}^{(1)}[\psi]$ as the time-inverted Newman–Penrose constant. Note that $I_{0}^{(1)}[\psi]$ is only defined for initial data of type **A** and **B**.
In the case where $H_0[\psi]=0$ we introduce the following quantity $$H_{0}^{(1)}[\psi]= I_{0}^{(1)}[\widetilde{\psi}].
\label{h01}$$ We will refer to $H_{0}^{(1)}[\psi]$ as the *time-inverted horizon charge*. Clearly, $H_{0}^{(1)}[\psi]$ is only defined for initial data of Type **B** and **D**.
For a discussion on the geometric importance of the constants $H_{0}^{(1)}[\psi]$ and $I_{0}^{(1)}[\psi]$ and their role in the analysis of the present paper see Section \[sec:GeometricOriginOfTheNewHair\].
The late-time asymptotics {#sec:TheMainTheoremsintrosummary}
-------------------------
We rigorously derive the late-time asymptotics solutions to the wave equation on ERN. We next summarize our results.
### Asymptotics for Type **C** perturbations {#sec:GlobalPerturbationTypeC}
We first consider **global** perturbations of Type **C**. These perturbations satisfy $H_0\neq 0$ and $I_0\neq 0$. Recall from Section \[sec:PreviousWorksOnLateTimeAsymptoticsOnERN\] that the heuristic and numerical work [@Burko2007] argued that the decay *rate* of $r\psi$ is $\tau^{-1}$, $\tau^{-2}$ or $\tau^{-1}$ along ${\mathcal{H}^{+}}$, $\{r=r_0\}$ and ${\mathcal{I}^{+}}$, respectively. However, precise late-time asymptotics for this type of perturbations were not known. The non-vanishing of the conserved constants $H_0$ and $I_0$ might seem to suggest that they appear in a potentially complicated way in the asymptotics for $\psi$. In fact, [@hm2012] conjectured that both $H_0$ and $I_0$ appear in the asymptotics of $\psi$ along the event horizon ${\mathcal{H}^{+}}$. In this paper, we derive and rigorously prove the precise late-time asymptotics for scalar perturbations of Type **C**. We falsify the above conjecture by showing that *the asymptotics along the event horizon are independent of the Newman–Penrose constant $I_0$:* $$r\psi|_{{\mathcal{H}^{+}}}\sim 2H_0[\psi]\cdot\frac{1}{\tau}+4MH_0[\psi]\cdot\frac{\log\tau}{\tau^2} \ \ \text{ as } \ \ \tau\rightarrow \infty.
\label{horasympt}$$ On the other hand, we show that *constants $H_0$ and $I_0$ appear in the leading-order terms for the late-time asymptotics of $\psi|_{\{r=r_0\}}$ along $r=r_0$ hypersurfaces away from the event horizon ($r_0>M$)*: $$\psi|_{\{r=r_0\}} \sim \left(4I_0[\psi]+\frac{4M}{r-M}H_0[\psi]\right)\cdot \frac{1}{\tau^2} \ \ \text{ as } \ \ \tau\rightarrow \infty.
\label{awayasympt}$$ The proof of is particularly subtle since both the horizon region and the null infinity region contribute to the asympotics of $\psi|_{\{r=r_0\}}$ via the constants $H_0$ and $I_0$, respectively. This is in stark contrast with the sub-extremal case (see Section \[sec:TheWaveEquationOnBlackHolesBackgrounds\]) where the dominant terms originate only from the null infinity region. Note that *the term $\frac{4M}{r-M}$ in front of $H_0$ is itself a static solution on ERN*. We remark that in order to show the asymptotics , we need to derive first the asymptotics for the radial derivative[^8] $\partial_{\rho}\psi$ of $\psi$ along $\Sigma_0$: $$\partial_{\rho}\psi|_{\{r=r_0\}}\sim -\frac{4M}{(r-M)^2}H_0[\psi]\cdot \frac{1}{\tau^2} \ \ \text{ as } \ \ \tau\rightarrow \infty.
\label{derivaawayasympt}$$ The crucial insight of is that *the leading-order asymptotics of $\partial_{\rho}\psi|_{\{r=r_0\}}$ are independent of $I_0$ for all values of $r_0>M$!* This is somewhat surprising; it shows that, from the point of view of the derivative $\partial_{\rho}\psi$, the event horizon is, in a sense, more relevant than null infinity. Furthermore, note that the decay rate of $\partial_{\rho}\psi|_{\{r=r_0\}}$ is only $\tau^{-2}$ which is equal to the decay rate of $\psi|_{\{r=r_0\}}$. This is again in stark contrast with the sub-extremal case where $\partial_{\rho}\psi|_{\{r=r_0\}}$ decays like $\tau^{-3}$.
We obtain the following asymptotics along null infinity ${\mathcal{I}^{+}}$: $$r\psi|_{{\mathcal{I}^{+}}}\sim 2I_0[\psi]\cdot\frac{1}{\tau}+4MI_0[\psi]\cdot\frac{\log\tau}{\tau^2} \ \ \text{ as } \ \ \tau\rightarrow \infty.
\label{nullinfasympt}$$ Note that these asymptotics are independent of the horizon charge $H_0$.
### Asymptotics for Type **A** perturbations {#sec:AsymptoticsForTypeAPerturbations}
We next consider **local** horizon-penetrating perturbations of Type **A**. These perturbations, which satisfy $H_0\neq 0$ and $I_0=0$, are the most physically relevant since they represent local perturbations of ERN. In the physics literature, they are said to desribe *outgoing radiation*.
The asymptotics along ${\mathcal{H}^{+}}$, and and along $\{r=r_0\}$ hold in this case as well, where in we have to use that $I_0=0$. On the other hand, the asymptotics along null infinity for the radiation field $r\psi_{{\mathcal{I}^{+}}}$ cannot be read off from . In this case, we derive the following asymptotics $$r\psi|_{\mathcal{I}^+}\sim \left( 4MH_0[\psi]-2I_{0}^{(1)}[\psi] \right)\cdot \frac{1}{\tau^2} \ \ \text{ as } \ \ \tau\rightarrow \infty.
\label{typeanullinf}$$ Here $I_{0}^{(1)}$ is the time-inverted Newman–Penrose constant given by . We observe that *for Type **A** perturbations the dominant term in the asymptotics of the radiation field $r\psi|_{{\mathcal{I}^{+}}}$ contains the horizon charge $H_{0}$*. Such perturbations exhibit the *full* horizon instability $$\partial_{r}\psi|_{{\mathcal{H}^{+}}}\sim -\frac{1}{M^3}H_{0}[\psi], \ \ \ \partial_{r}^2\psi|_{{\mathcal{H}^{+}}}\sim \frac{1}{M^5} H_{0}[\psi]\cdot\tau \ : \ \text{ along }{\mathcal{H}^{+}},
\label{typeainsta}$$ with respect to $(v,r)$ coordinates, the origin of which is the charge $H_0$ (see Section \[sec:TheHorizonInstabilityOfExtremalBlackHoles\] for a review). Therefore, the precise asymptotics yield a way to potentially measure the horizon charge $H_0$ and hence detect the horizon instability of extremal black holes from observations in the far away radiation region.
### Asymptotics for Type **B** perturbations {#sec:AsymptoticsForTypeBPerturbations}
Perturbations of Type **B** consititute another very important class of physically relevant perturbations. Such perturbations are initially compactly supported and supported away from the horizon and hence satisfy $H_0=0$ and $I_0=0$. They represent **local** perturbations **from afar**. In the physics literature, such perturbations are said to describe *ingoing radiation*.
Recall from Section \[sec:TheWorkOfReallEtAl\] that Lucietti–Murata–Reall–Tanahashi [@hm2012] numerically demonstrated that such perturbations exhibit a *weaker* version of the horizon instability, namely $$|\psi|_{{\mathcal{H}^{+}}}\rightarrow 0,\ \ \ |\partial_{r}\psi|_{{\mathcal{H}^{+}}}\rightarrow 0, \ \ \ |\partial_{r}^2\psi|_{{\mathcal{H}^{+}}}\nrightarrow 0 \ \ \ |\partial_{r}^3\psi|_{{\mathcal{H}^{+}}}\rightarrow \infty\ : \ \text{ along }{\mathcal{H}^{+}}.
\label{typebinsta}$$ Perturbations of Type **B** which exhibit the above behavior where rigorously constructed in [@aretakis2012]. However, [@aretakis2012] did not provide a necessary and sufficient condition for perturbations of Type **B** so that holds. In this paper, we provide an answer to this question. In fact, we provide the precise late-time asymptotics for all perturbations of Type **B**.
Recall that the horizon charge $H_{0}^{(1)}[\psi]$ given by which is well-defined for all Type **B** perturbations. We prove that *the weak horizon instability holds if and only if $H_{0}^{(1)}[\psi]\neq 0$.* Specifically, $$\partial_{r}^2\psi \sim \frac{1}{M^5}H_{0}^{(1)}[\psi], \ \ \ \partial_{r}^3\psi \sim -\frac{3}{M^7}H_{0}^{(1)}[\psi]\cdot\tau \ \ \text{ as } \ \ \tau\rightarrow \infty.
\label{tbhorinstabi}$$ Furthermore, we prove that $H_{0}^{(1)}[\psi]$ determines the leading-order asymptotics along the event horizon $$r\psi|_{{\mathcal{H}^{+}}}\sim -2H_{0}^{(1)}[\psi]\cdot \frac{1}{\tau^2}-8MH_0^{(1)}[\psi]\cdot\frac{\log \tau}{\tau^3} \ \ \text{ as } \ \ \tau\rightarrow \infty.
\label{tbhorasympt}$$ On the other hand, the asymptotics of the radiation field depend **only** on the value of the time-inverted Newman–Penrose constant $I_{0}^{(1)}$: $$r\psi|_{{\mathcal{I}^{+}}}\sim -2I_{0}^{(1)}[\psi]\cdot \frac{1}{\tau^2}-8MI_0^{(1)}[\psi]\cdot\frac{\log \tau}{\tau^3} \ \ \text{ as } \ \ \tau\rightarrow \infty.
\label{tbnullasympt}$$ Finally, the asymptotics along $\{r=r_0\}$ depend on the value of both constants $H_{0}^{(1)}[\psi]$ and $I_0^{(1)}[\psi]$: $$\psi|_{\{r=r_0\}}\sim -8\left(I_{0}^{(1)}[\psi]+\frac{M}{r-M}H_{0}^{(1)}[\psi] \right)\cdot \frac{1}{\tau^3} \ \ \text{ as } \ \ \tau\rightarrow \infty.
\label{tbnullasympt2}$$ Note the decay rate of agrees with the decay rate of for Schwarzschild spacetimes. However, in constrast to Schwarzschild, the coefficient of the asymptotic term in depends additionally on the new horizon charge $H_{0}^{(1)}[\psi]$.
### Asymptotics for Type **D** perturbations {#sec:AsymptoticsForTypeDPerturbationsintro}
We conclude our discussion with a brief discussion about the asymptotics of Type **D** perturbations. These are **non-compact** pertrubations supported **away** from the event horizon. They satisfy $H_0=0$ and $I_0\neq 0$. Such perturbations have a well-defined $H_{0}^{(1)}$. Our first result for this case shows that the radiation field $r\psi|_{{\mathcal{I}^{+}}}$ and the scalar field $\psi|_{\{r=r_0\}}$ “see” to leading order **only** $I_0$: $$r\psi|_{{\mathcal{I}^{+}}}\sim 2I_{0}[\psi]\cdot \frac{1}{\tau}, \ \ \ \ \psi|_{\{r=r_0\}}\sim 4I_{0}[\psi]\cdot \frac{1}{\tau^2} \ \ \ \text{ as } \ \ \tau\rightarrow \infty.$$ On the other hand, the asymptotics along ${\mathcal{H}^{+}}$ to leading order to depend on **both** $I_0$ and the horizon charge $H_0^{(1)}$: $$r\psi|_{{\mathcal{H}^{+}}}\sim \left(4MI_{0} -2H_{0}^{(1)}[\psi]\right)\cdot \frac{1}{\tau^2}\ \ \text{ as } \ \ \tau\rightarrow \infty.$$ Type **D** perturbations exhibit the weak version of the horizon instability. In this context, it is important to remark that *even though the asymptotic terms along the event horizon contain both $I_0$ and $H_{0}^{(1)}$, the source of the horizon instability in this case originates purely from the horizon charge $H_{0}^{(1)}$.* In fact, the exact non-decay and blow-up along the event horizon results given by hold for Type **D** perturbations as well. In contrast to Type **A** perturbations, the derivative $\partial_r\psi$ decays faster than $\psi$ away from the horizon: $$\partial_r\psi|_{\{r=r_0\}}\sim\left( \frac{8M}{(r-M)^2}\cdot H_0^{(1)}[\psi]+\frac{8(r^2-M^2)}{(r-M)^2}\cdot I_0[\psi]\right)\cdot \tau^{-3}.$$ The precise asymptotic expression, as above, is derived here for the first time in the literature.
### Summary of the asymptotics {#sec:SummaryOfTheAsymptotics}
We summarize our findings in the table below
[ cV[4]{}c | c |c]{} &\
& $r\psi|_{\mathcal{H}^{+}}$ & $\psi|_{\{r=r_0\}}$ & $r\psi|_{\mathcal{I}^{+}}$\
& $2H_0\cdot \tau^{-1}$ & $ \frac{4M}{r-M}H_0 \cdot \tau^{-2}$ & $\left(4MH_0-2I_0^{(1)}\right) \cdot \tau^{-2}$\
& $-2H_0^{(1)} \cdot \tau^{-2}$ & $-8\left(I_0^{(1)}+\frac{M}{r-M}H_0^{(1)}\right) \cdot \tau^{-3}$ & $-2I_0^{(1)} \cdot \tau^{-2}$\
& $2H_0 \cdot \tau^{-1}$ & $4\left(I_0+\frac{M}{r-M}H_0\right) \cdot \tau^{-2}$ & $2I_0 \cdot \tau^{-1}$\
& $\left(4MI_0-2H_0^{(1)}\right) \cdot \tau^{-2}$ & $4I_0\cdot \tau^{-2}$ & $2I_0\cdot \tau^{-1}$\
We remark that we in fact derive the asymptotics for $T^{k}\psi$ for all $k\geq 0$. The relevant asymptotic expressions can be found by taking the $\frac{\partial^{k}}{\partial \tau^{k}}$ derivative of the expressions in Table \[summarytable\]. Furthermore, we derive the following asymptotics for the transveral derivative $\partial_r\psi$:
[ cV[3]{}c | c ]{} &\
& $\partial_r\psi|_{\mathcal{H}^{+}}$ & $\partial_r\psi|_{\{r=r_0\}}$\
& $-\frac{1}{M^3}\cdot H_0$ & $-\frac{4M}{(r-M)^2}\cdot H_0[\psi]\cdot \tau^{-2}$\
& $\frac{2}{M^2}\cdot H_0^{(1)}\cdot {\tau^{-2}} $ & $\frac{8M}{(r-M)^2}\cdot H_0^{(1)}[\psi]\cdot \tau^{-3}$\
& $-\frac{1}{M^3}\cdot H_0$ & $-\frac{4M}{(r-M)^2}\cdot H_0[\psi]\cdot \tau^{-2}$\
& $\frac{2}{M^2}\cdot H_0^{(1)}\cdot {\tau^{-2}} $ & $\left( \frac{8M}{(r-M)^2}\cdot H_0^{(1)}+\frac{8(r^2-M^2)}{(r-M)^2}\cdot I_0\right)\cdot \tau^{-3}$\
At the horizon, we have the following asymptotics for the higher order transversal derivatives $\partial_r^k\psi$ revealing the strong horizon instability for Type **A** and **C** and the weak horizon instability for Type **B** and **D**.
[ cV[3]{}c | c | c| c ]{} &\
& $\partial_r\psi|_{\mathcal{H}^{+}}$ & $\partial_r^2\psi|_{\mathcal{H}^{+}}$ & $\partial_r^3\psi|_{\mathcal{H}^{+}}$ & $\ \ \partial_r^k\psi|_{\mathcal{H}^{+}},\, k\geq 2$\
& $-\frac{1}{M^3}\cdot H_0$ & $\frac{1}{M^5}\cdot H_0 \cdot\tau$ & $-\frac{3}{2M^7} \cdot H_0 \cdot\tau^2$ & $c_k\cdot H_0\cdot \tau^{k-1}$\
& $\frac{2}{M^2}\cdot H_0^{(1)}\cdot {\tau^{-2}} $ & $\frac{1}{M^5}\cdot H_0^{(1)}$ & $-\frac{3}{M^7}\cdot H_0^{(1)}\cdot \tau$ & $a_k\cdot c_{k-1}\cdot H_0^{(1)}\cdot \tau^{k-2}$\
& $-\frac{1}{M^3}\cdot H_0$ & $\frac{1}{M^5}\cdot H_0 \cdot\tau$ & $-\frac{3}{2M^7} \cdot H_0 \cdot\tau^2$ & $c_k\cdot H_0\cdot \tau^{k-1}$\
& $\frac{2}{M^2}\cdot H_0^{(1)}\cdot {\tau^{-2}} $ & $\frac{1}{M^5}\cdot H_0^{(1)}$ & $-\frac{3}{M^7}\cdot H_0^{(1)}\cdot \tau$ & $a_k\cdot c_{k-1}\cdot H_0^{(1)}\cdot \tau^{k-2}$\
where $$a_k= \frac{(-1)}{M^2}\cdot\binom{k}{2}\ \ \ \text{ and } \ \ \ c_k=(-1)^{k}\cdot\frac{1}{M^3}\cdot \frac{1}{\left(2M^2 \right)^{k-1}}\cdot k!$$
More generally, the late-time asymptotics along ${\mathcal{H}^{+}}$ for $Y^{k}T^{m}\psi$, with $k\geq m+1$ for Type **A** and **C** and $k\geq m+2$ for Type **B** and **D**, can be informally found by taking the $\frac{\partial^{m}}{\partial \tau^{m}}$ derivative of the expressions in Table \[horizontable\] (see also Section \[sec:DecayForScalarInvariants\]).
Applications and additional remarks {#sec:Remarks}
===================================
In this section we present a few applications and remarks about our results.
Singular time inversion and the new horizon charge {#sec:GeometricOriginOfTheNewHair}
--------------------------------------------------
Recall from Section \[sec:TheWaveEquationOnBlackHolesBackgrounds\] that the constants $I_0$ and $I_{0}^{(1)}$ are obstructions to inverting the time operators $T$ and $T^{2}$, respectively. Specifically, $I_0$ and $I_{0}^{(1)}$ are obstructions to defining the operators $T^{-1}$ and $T^{-2}$, respectively, on solutions the wave equation , such that their target functional space consists of solutions the wave equation which decay appropriately in $r$ towards null or spacelike infinity. In sub-extremal black holes, $I_0$ and $I_{0}^{(1)}$ are the only such obstructions. However, for ERN we have an additional obstruction that originates from the geometry of the horizon, namely the conserved charge $H_0$. Indeed, for any smooth solution $\psi$ to the wave equation on ERN we have $$H_{0}[T\psi]=0.\vspace{-0.12cm}
\label{h0t}$$ Hence, *$H_0$ is an obstruction to defining the inverse operator $T^{-1}$ on smooth solutions to such that the image is also a smooth solution to* . On the other hand, if $\psi$ is a smooth solution to with $H_0=0$ then the horizon charge $H_{0}^{(1)}$ is well-defined and satisfies $$H_{0}^{(1)}[T^2\psi]=0. \vspace{-0.12cm}$$ Hence, *$H_0^{(1)}$ is an obstruction to defining the inverse operator $T^{-2}$ on smooth solutions (with $H_0=0$) to such that the image is also a smooth solution to* . The above imply that the horizon associated charges $H_0$ and $H_0^{(1)}$ are related to singularities at time frequencies $\omega \sim 0$. We thus conclude that *the leading order terms in the late-time asymptotic expansion are dominated by the $\omega\sim 0$ frequencies*.
An important aspect of our analysis is that we invert the operators $T$ and $T^2$ even if the images of $T^{-1}$ and $T^{-2}$ do contain smooth function. This is accomplished by developing a **singular time inversion theory**. This theory is needed for Type **A** and Type **D** perturbations. Let’s first consider Type **A** perturbations. Since such perturbations satisfy $H_0\neq 0$ and $I_0=0$, $I_0^{(1)}$ is well-defined whereas $H_{0}^{(1)}$ is undefined. Clearly, there is no smooth solution $T^{-1}\psi$ to . Indeed, if a smooth solution $T^{-1}\psi$ to existed then by replacing $\psi$ with $T^{-1}\psi$ in we would obtain $H_0[\psi]=H_0[T(T^{-1}\psi)]=0$, which is a contradiction. It turns out that we can still *canonically* define a *singular* time inversion $T^{-1}\psi$ such that
- $T^{-1}\psi\rightarrow 0$ as $r\rightarrow \infty$,
- $I_0[T^{-1}\psi]<\infty$,
- $\partial_r(T^{-1}\psi)\sim -2H_{0}[\psi]\cdot \frac{1}{r-M}$ in the region $r\sim M$.
Similar results hold for Type **D** perturbations. For perturbations of Type **A** and **D**, *we develop a low regularity theory which allows us to obtain the precise late-time asymptotics for the singular scalar fields* $T^{-1}\psi$. We remark that for Type **B** perturbations we develop a **regular** time inversion theory, whereas no time inversion is needed for Type **C** perturbations. Summarizing,
[ cV[3]{}c | c| c|cV[3]{}c ]{} &\
& $H_0[\psi]$ & $H_0^{(1)}[\psi]$ & $I_0[\psi]$ & $I_{0}^{(1)}[\psi]$ & [$T^{-1}\psi$]{}\
& $\neq 0$ & $\boldsymbol{=\infty}$ & $=0$ & $<\infty$ & [singular at ${\mathcal{H}^{+}}$]{}\
& $=0$ & $<\infty$ & $\neq 0$ & $\boldsymbol{=\infty}$ & [singular at ${\mathcal{I}^{+}}$]{}\
& $=0$ & $<\infty$ & $=0$ & $<\infty$ & [regular]{}\
Decay for scalar invariants {#sec:DecayForScalarInvariants}
---------------------------
Hadar and Reall [@harveyeffective], assuming the asymptotics on ERN (rigorously established in the present paper), showed that the scalar invariants $|\nabla^k\psi|^2$ decay in time. Similar decay results were presented in [@khanna17] and in [@zimmerman4]. Let’s briefly recall the argument of [@harveyeffective]. First of all, note that the Christoffel symbols $\Gamma^{a}_{bc}$, with $a,b,c\in \{ v,r\}$, vanish on the event horizon and, hence, if $\partial_{i^1},\cdots, \partial_{i^k}\in \{\partial_{v},\partial_{r}\}$ then $\nabla^{k}\psi_{i_1\cdots i_k}=\partial_{i^1}\cdots \partial_{i^k}\psi$ on the event horizon. The following asymptotic decay rates hold along the event horizon for all derivatives: $$\partial_r^k T^m\psi \sim \tau^{k-m-1-\epsilon(k,m)}$$ where $$\epsilon(k,m)=\begin{cases}
0, \ \text{ if }\ k=0 \ \text{ or } \ k\geq m+1, \\
1, \ \text{ if }\ 1\leq k \leq m.
\end{cases}
\label{skipepsilon}$$ Note that the presence of $\epsilon(k,m)$ introduces a *skip* in the decay rates for the derivatives of $\psi$. This skip was also previously observed in [@zimmerman1].
To show that $|\nabla^k\psi|^2$ always decays, it suffices to consider the “slowest” case, namely the case of perturbations of Type **C**. In this case, $$\begin{split}
|\nabla^k\psi|^2 &\sim \sum_{k_1+k_2=k}\partial_{r}^{k_1}T^{k_2}\psi\cdot \partial_{r}^{k_2}T^{k_1}\psi\sim \sum_{k_1+k_2=k}\tau^{k_1-k_2-1-\epsilon(k_1,k_2)}\cdot \tau^{k_2-k_1-1-\epsilon(k_2,k_1)}\\
&\sim \sum_{k_1+k_2=k}\tau^{-2-\epsilon(k_1,k_2)-\epsilon(k_2,k_1)}\sim \tau^{-2},
\end{split}$$ for all $k\geq 1$, since $\epsilon(k_1,k_2), \epsilon(k_2,k_1)\geq 0$ and $\epsilon(k,0)=\epsilon(0,k)=0$. Note that the decay rate for $|\nabla^k\psi|^2$ is independent of $k$.
The interior of black holes and strong cosmic censorship {#sec:TheInteriorOfBlackHolesAndStrongCosmicCensorship}
--------------------------------------------------------
In this paper we have restricted the analysis of the wave equation to the extremal Reissner–Nordström *black hole exterior* (the domain of outer communications). One can also extend the initial data hypersurface $\Sigma_0$ into the black hole *interior* (see Section \[sec:TheHyperboloidalFoliation\] for a precise definition of $\Sigma_0$) and investigate the behavior of solutions to in the restriction of the domain of dependence of the extended initial data hypersurface to the [black hole interior]{}.
![The extended initial value problem that includes the interior region, where $\Sigma$ is the extension of $\Sigma_0$ into the interior.[]{data-label="fig:interiop455"}](fullextrrn.pdf)
An analysis of the behavior of solutions to in the black hole interior of extremal Reissner–Nordström was carried out by the third author in [@gajic] in the setting of a characteristic initial value problem with initial data imposed on a future geodesically complete segment of the future black hole event horizon and initial data imposed on an ingoing null hypersurface intersecting the event horizon to the past. The late-time behaviour of the solution to on extremal Reissner–Nordström along the event horizon was *assumed* to be consistent with the numerical predictions of [@hm2012]. The results of [@gajic] illustrate a remarkably *delicate* dependence of the qualitative behaviour at the inner horizon in the black hole interior on the *precise* late-time behaviour of the solution to along the event horizon of extremal Reissner–Nordström as predicted by numerics and heuristics.
By combining the results stated schematically in Section \[sec:SummaryOfTheMainResults\] and more precisely in Section \[sec:StatementsOfTheTheorems\], that confirm in particular the numerical predictions of [@hm2012], with Theorem 2, 5 and 6 of [@gajic], we conclude that the following theorem holds:
\[thm:interior\] Solutions $\psi$ to on extremal Reissner–Nordström arising from smooth compactly supported data on an extension of $\Sigma_0$ into the black hole interior are extendible across the black hole inner horizon as functions in $C^{0,\alpha}\cap W^{1,2}_{\rm loc}$, with $\alpha<1$. Furthermore, the spherical mean $\frac{1}{4\pi}\int_{{\mathbb{S}}^2}\psi\,d\omega$ can in fact be extended as a $C^2$ function.
It follows from Theorem \[thm:interior\] that for spherically symmetric data one can construct $C^2$ extensions of $\psi$ across the inner horizons that are moreover *classical* solutions to with respect to a smooth extension of the extremal Reissner–Nordström metric across the inner horizon. These extensions of $\psi$, much like the smooth extensions of the metric, are **highly non-unique**!
In order to derive $C^2$ extendibility of the spherical mean across the inner horizon for initial data with $H_0[\psi]\neq 0$, we have to make use of the precise leading order behavior of $\psi$ along the event horizon in .
See also [@gajic2] for extendibility results in the context of in the interior of extremal Kerr–Newman spacetimes.
The extendibility properties in Theorem \[thm:interior\] differ drastically from the extendibility properties of solutions to in the interior of -extremal Reissner–Nordström black holes, which are extendible in $C^0$ across the inner (Cauchy) horizon, but *inextendible* in $W^{1,2}_{\rm loc}$, see [@Franzen2014; @luk2015]. See also [@Hintz2015; @LukSbierski2016; @DafShl2016; @gregjan] for extendibility results in sub-extremal Kerr.
The study of the wave equation in black hole interiors serves as a linear “toy model” for the analysis of dynamical black hole interiors, which is closely related to the *Strong Cosmic Censorship Conjecture* (SCC). As formulated in [@DC09], this conjecture states that “generic” asymptotically flat initial data for the Einstein vacuum equations have maximal globally hyperbolic developments that are inextendible as a Lorentzian manifold with a continuous metric and locally square integrable Christoffel symbols. See for example [@dl-scc] for a more elaborate discussion on SCC.
Building on the pioneering work of Dafermos [@MD03; @MD05c; @MD12] and Dafermos–Rodnianski [@MDIR05], Luk–Oh showed in [@Luk2016a; @Luk2016b] that a $C^2$-version of SCC holds in the spherically symmetric Einstein–Maxwell–scalar field setting: they proved inextendibility of the metric in $C^2$ for generic asymptotically flat two-ended data.
We next consider the case of “dynamical extremal black holes”, i.e. black hole spacetime solutions which approach the extremal Reissner–Nordström suitably rapidly along the event horizon. We remark that in [@harvey2013] dynamical extremal black hole spacetimes are defined as being black hole spacetimes without trapped surfaces and it is shown numerically that the solutions under consideration actually approach an extremal Reissner Nordström solution along the event horizon. Conversely, it can be shown in this setting (massless, uncharged scalar field) that black hole spacetimes that approach extremal Reissner–Nordström along the event horizon will have no radial trapped surfaces in the black hole interior, provided none of the round spheres foliating the event horizon are (marginally) *anti-trapped*; see also Remark 1.7 in [@dejanjon1].
In this dynamical setting, the interior dynamics are significantly different for dynamical extremal black holes compared to the “dynamical sub-extremal black holes” that arise from asymptotically flat two-ended data, as shown in [@Luk2016a; @Luk2016b]. Indeed, in [@dejanjon1] it was shown that the dynamical extremal black holes in consideration are extendible across the inner horizon as *weak* solutions to the spherically symmetric Einstein–Maxwell–(charged) scalar field system of equations (in particular, with Christoffel symbols in $L^2_{\rm loc}$). Hence, *in contrast to the sub-extremal case, dynamical extremal black holes do not conform to the inextendibility properties stated in SCC*. The only way that SCC can therefore still be valid, at least in the spherically symmetric setting under consideration, is if *dynamical extremal black holes do not arise from “generic” initial data*. Analogous numerical results were presented in [@harvey2013] that are moreover compatible with the non-genericity of dynamical extremal black holes (see also Section \[sec:TheReallSpacetimes\] for a discussion on the results of [@harvey2013] in the black hole exterior).
The main theorems and ideas of the proofs {#subsec:TheMainTheorems}
=========================================
Statements of the theorems {#sec:StatementsOfTheTheorems}
--------------------------
We smooth consider solutions $\psi$ to the wave equation on ERN and we derive global late-time asymptotic estimates for $\psi$. We will mostly express our main theorems in terms of the double null coordinates $(u,v)$ (see Section \[sec:TheERNManifoldFoliationsAndVectorFields\] for a review of the geometry of ERN). The constants $H_0$ and $H_0^{(1)}$ are defined in Sections \[sec:ConservationLawsAlongTheEventHorizon\] and \[sec:TheNewHorizonHairH01Psi\], respectively. The constants $I_0$ and $I_0^{(1)}$ are defined in Section \[sec:TheWaveEquationOnBlackHolesBackgrounds\]. The perturbations of Type **A**, **B**, **C** and **D** are introduced in Section \[sec:TheTypesOfInitialDataABCD\]. For simplicity, the initial data norms on the right hand side of the estimates of the main theorems are not presented here and instead are presented in the relevant sections where these theorems are proved. We mention below the exact section where each theorem is proved. Moreover, the quantities $\eta>0$ and $\epsilon>0$ below are suitably small constants, $\beta\in (0,1]$ appears in the initial data norms and $k\in\mathbb{N}_{0}$. Furthermore, $C=C(M,\Sigma_0,r_\mathcal{H},r_\mathcal{I},\eta,\epsilon,\beta, k)>0$ is a universal constant.
\[prop:asympsitheo\] Assume that the initial data of $\psi$ are of Type **C** and that $\psi$ solves the wave equation . The following [global]{} estimate holds: $$\label{eq:asympsiintro}
\begin{split}
\Bigg|T^k\psi_0(u,v)&-4\left(I_0[\psi]+ \frac{M}{r\sqrt{D}}H_0[\psi]\right)T^k\left(\frac{1}{u\cdot v}\right)\Bigg|\\
\leq&\: C\left(\sqrt{E_{0;k+1}^{\epsilon}[\psi]+\sum_{|\alpha|\leq 2}E_{1;k}^{\epsilon}[\Omega^{\alpha}\psi]}+I_0[\psi]+P_{I_0,\beta;k}[\psi]\right)v^{-1}u^{-1-k-\eta}\\
&+C\left(\sqrt{E_{0;k+1}^{\epsilon}[\psi]+\sum_{|\alpha|\leq 2}E_{1;k}^{\epsilon}[\Omega^{\alpha}\psi]}+H_0[\psi]+P_{H_0,1;k}[\psi]\right)D^{-\frac{1}{2}}u^{-1}v^{-1-k-\eta}.
\end{split}$$ \[ctheo\]
Assume that the initial data of $\psi$ are of Type **A** and that $\psi$ solves the wave equation . The following [global]{} estimate holds: $$\begin{split}
\Bigg|T^k\psi_0(u,v)&-4\left[ I_0^{(1)}[\psi]T^{k+1}\left(\frac{1}{u\cdot v}\right)+\frac{M}{r \sqrt{D}}H_0[\psi]T^{k}\left(\frac{1}{u(v+4M-2r)}\right)\right]\Bigg|\\
\leq&\: C\Bigg[\sqrt{E^{\epsilon}_{0, \mathcal{I}; k+1}[\psi]+\sum_{|\alpha|\leq 2}E_{1,\mathcal{I};k}^{\epsilon}[\Omega^{\alpha}\psi]}+P_{I_0,\beta;k+1}[\psi]+P_{H_0,1;k}[\psi]+H_0[\psi]+I_0^{(1)}[\psi] \Bigg]\\
&\cdot \left( v^{-1}u^{-2-k-\eta}+D^{-\frac{1}{2}}u^{-1}v^{-1-k-\eta}\right).
\end{split}$$ \[atheo\]
Assume that the initial data of $\psi$ are of Type **D** and that $\psi$ solves the wave equation . The following [global]{} estimate holds: $$\begin{split}
\Bigg|T^k\psi_0(u,v)&-4\left[ \frac{1}{\sqrt{D}}H_0^{(1)}[\psi]T^{k+1}\left(\frac{1}{u\cdot v}\right)+I_0[\psi]T^{k}\left(\frac{1}{v(u+2M-2M^2(r-M)^{-1})}\right)\right]\Bigg|\\
\leq&\: C\Bigg[\sqrt{E^{\epsilon}_{0, \mathcal{H}; k+1}[\psi]+\sum_{|\alpha|\leq 2}E_{1,\mathcal{H};k}^{\epsilon}[\Omega^{\alpha}\psi]}+P_{I_0,\beta;k}[\psi]+P_{H_0,1;k+1}[\psi]+I_0[\psi]+H_0^{(1)}[\psi] \Bigg]\\
&\cdot \left( v^{-1}u^{-1-k-\eta}+D^{-\frac{1}{2}}u^{-1}v^{-2-k-\eta}\right).
\end{split}$$ \[dtheo\]
Assume that the initial data of $\psi$ are of Type **B** and that $\psi$ solves the wave equation . The following [global]{} estimate holds: $$\label{eq:asympsizeroIHtheo}
\begin{split}
\Bigg|T^k\psi_0(u,v)&-4\left(I_0^{(1)}[\psi]+ \frac{M}{r\sqrt{D}}H_0^{(1)}[\psi]\right)T^{k+1}\left(\frac{1}{v\cdot u}\right)\Bigg|\\
\leq&\: C\left(\sqrt{E_{0, \mathcal{H};k+1}^{\epsilon}[\psi]+E_{0, \mathcal{I};k+1}^{\epsilon}[\psi]+\sum_{|\alpha|\leq 2}E_{2;k+1}^{\epsilon}[\Omega^{\alpha}\psi]}+I_0^{(1)}[\psi]+P_{I_0,\beta;k+1}[\psi]\right)v^{-1}u^{-2-k-\eta}\\
&+C\left(\sqrt{E_{0, \mathcal{H};k+1}^{\epsilon}[\psi]+E_{0, \mathcal{I};k+1}^{\epsilon}[\psi]+\sum_{|\alpha|\leq 2}E_{2;k+1}^{\epsilon}[\Omega^{\alpha}\psi]}+H_0^{(1)}[\psi]+P_{H_0,1;k+1}[\psi]\right)D^{-\frac{1}{2}}u^{-1}v^{-2-k-\eta}.
\end{split}$$ \[btheo\]
Assume that the initial data of $\psi$ are spherically symmetric and of Type **C** and that $\psi$ solves the wave equation . Then, the following estimate holds on ${\mathcal{I}^{+}}$ $$\label{eq:2ndasymphinullinfintro}
\left|r\psi|_{\mathcal{I}^+}(u)-2I_0[\psi]u^{-1}+4MI_0[\psi]u^{-2}\log u\right|\leq C\left(I_0[\psi]+H_0[\psi]+\sqrt{E^{\epsilon}_{0; 1}[\psi]}+P_{\mathcal{H}}[\psi]+P_{\mathcal{I}}[\psi]\right)u^{-2}$$ and the following estimate holds on $\mathcal{H}^{+}$ $$\label{eq:2ndasymphihointro}
\left|r\psi|_{\mathcal{H}^+}(v)-2H_0[\psi]v^{-1}+4MH_0[\psi]v^{-2}\log v\right|\leq C\left(I_0[\psi]+H_0[\psi]+\sqrt{E^{\epsilon}_{0;1}[\psi]}+P_{\mathcal{H}}[\psi]+P_{\mathcal{I}}[\psi]\right)v^{-2}.$$ \[logtheo\]
Assume that the initial data of $\psi$ are spherically symmetric and of Type **B** and that $\psi$ solves the wave equation . Then, the following estimate holds on ${\mathcal{I}^{+}}$ $$\label{eq:2ndasymphinullinftimeintintro}
\begin{split}
\Big|&r\psi|_{\mathcal{I}^+}(u)+2I_0^{(1)}[\psi]u^{-2}-8MI_0^{(1)}[\psi]u^{-3}\log u\Big|\\
\leq&\: C\left(I_0^{(1)}[\psi]+H_0^{(1)}[\psi]+\sqrt{E^{\epsilon}_{0,\mathcal{H}; 1}[\psi]+E^{\epsilon}_{0,\mathcal{I}; 1}[\psi]}+P_{\mathcal{H}, T}[\psi]+P_{\mathcal{I}, T}[\psi]\right)u^{-3},
\end{split}$$ and the following estimate holds on $\mathcal{H}^{+}$ $$\label{eq:2ndasymphihotimintintro}
\begin{split}
\Big|&r\psi|_{\mathcal{H}^+}(v)+2H_0^{(1)}[\psi]v^{-2}-4MH_0^{(1)}[\psi]v^{-3}\log v\Big|\\
\leq&\: C\left(I_0^{(1)}[\psi]+H_0^{(1)}[\psi]+\sqrt{E^{\epsilon}_{0,\mathcal{H}; 1}[\psi]+E^{\epsilon}_{0,\mathcal{I}; 1}[\psi]}+P_{\mathcal{H}, T}[\psi]+P_{\mathcal{I}, T}[\psi]\right)v^{-3}.
\end{split}$$ \[logtheoB\]
We split $\psi=\psi_0+\psi_{\geq 1}$ and prove the appropriate decay estimates for $\psi_{\geq 1}$ in Section \[sec:pdecayest\]. We can then replace $\psi$ with $\psi_0$ in the theorem statements: Theorem \[ctheo\] is proved in Section \[sec:asympnonzeroconst\], Theorem \[atheo\] is proved in Section \[sec:asympzeroconst\] and Theorems \[dtheo\] and \[btheo\] are proved in Section \[sec:AsymptoticsForTypeDPerturbations\]. Finally, Theorem \[logtheo\] and \[logtheoB\] are proved in Section \[sec:hoasymp\].
Overview of techniques {#sec:OverviewOfTechniques}
----------------------
In this section we will give an overview of the main steps and methods involved in proving the theorems stated in Section \[sec:StatementsOfTheTheorems\]. We will moreover highlight the key new ideas that play a role in the proofs.
### The zeroth step {#sec:TheZerothStep}
Deriving the precise late-time asymptotics requires obtaining decay rates for weighted energy fluxes and pointwise norms that are as sharp as possible. Our strategy is based on the integrated $r^{p}$-weighted energy decay approach of Dafermos–Rodnianski [@newmethod] and its extension presented in [@paper1]. The main idea is to derive energy decay by first establishing boundedness for suitable (weighted) *spacetime* integrals. For ERN, the “zeroth” step is the Morawetz estimate of the form (see Appendix \[sec:EnergyBounds\]) $$\int_{\tau_1}^{\tau_2}\int_{\Sigma_{\tau}}(r-M)^{\sigma_1}\cdot \frac{1}{r^{\sigma_2}}\cdot J^{T}[\psi] \, d\tau\ \lesssim \ \int_{\Sigma_{\tau_1} } J^{T}[\psi]+J^{T}[T\psi],
\label{morasche}$$ with $\sigma_1,\sigma_2>2$ sufficient large constants. Here, $J^{T}[\psi]$ denotes the standard $T$-energy current through $\Sigma_{\tau}$. From now on, if the volume form is missing in the integrals, it is implied that we consider the standard volume form with respect to the induced metric on the corresponding hypersurface. The higher-order terms on the right hand side account for *the high-frequency trapping effect on the photon sphere* at $\{r=2M\}$. The $r^{-\sigma_2}$ degenerate coefficient is related to the asymptotic flatness of the spacetime and is present in the analogous estimate for Minkowski spacetime. On the other hand, the degenerate factor $(r-M)^{\sigma_1}$ accounts for *the global trapping effect on the extremal event horizon*, a feature characteristic to ERN (see Section \[sec:TheTrappingEffect\]).
Clearly, one needs to remove the degenerate factors from in order to prove decay for the energy flux $$\mathcal{E}^{T}_{\Sigma_{\tau}}[\psi]:= \int_{\Sigma_{\tau}}\!J^{T}[\psi].
\label{tfluxsigmatau}$$ Dafermos and Rodnianski [@newmethod] and subsequently Moschidis [@moschidis1] showed that the weight at infinity $r^{-\sigma_2}$ can be removed for general asymptotically flat spacetimes by introducing appropriate *growing* $r$ weights on the right hand side yielding *a hierarchy of two $r$-weighted estimates*. In view of the degenerate factors both at the horizon and at infinity in the Morawetz estimate on ERN, one needs to obtain *an analogue of the Dafermos–Rodnianski hierarchy both at the near-infinity region $\mathcal{A}^{\mathcal{I}}$ and at the near-horizon region $\mathcal{A}^{\mathcal{H}}$* (see Section \[sec:TheHyperboloidalFoliation\] for the relevant definitions). This was accomplished in [@aretakis2]. We denote $$N_{\tau}^{\mathcal{I}}= \Sigma_{\tau}\cap \mathcal{A}^{\mathcal{I}}, \ \ \text{ and } \ \ N_{\tau}^{\mathcal{H}}= \Sigma_{\tau}\cap \mathcal{A}^{\mathcal{H}}.$$
![The hypersurfaces $N_{\tau}^{\mathcal{H}}$ and $N_{\tau}^{\mathcal{I}}$.[]{data-label="fig:interiop455ntau"}](Ntau.pdf)
The following ${\mathcal{I}^{+}}-$**localized hierarchy** holds in $\mathcal{A}^{\mathcal{I}}$ for all $0\leq \tau_1<\tau_2$: $$\begin{split}
\int_{\tau_1}^{\tau_2} \left[\int_{{N}^{\mathcal{I}}_{\tau}} J^{T}[\psi]\right] d\tau &\lesssim \int_{{N}^{\mathcal{I}}_{\tau_1}} r\cdot (\partial_v(r\psi))^2\,d\omega dv+\text{l.o.t.},\\
\int_{\tau_1}^{\tau_2} \left[\int_{{N}^{\mathcal{I}}_{\tau}} r\cdot (\partial_v(r\psi))^2\,d\omega dv\right]d\tau &\lesssim \int_{{N}^{\mathcal{I}}_{\tau_1}} r^2\cdot (\partial_v(r\psi))^2\,d\omega dv+\text{l.o.t.},
\label{hierioverview}
\end{split}$$ and the following ${\mathcal{H}^{+}}-$**localized hierarchy** holds in $\mathcal{A}^{\mathcal{H}}$ for all $0\leq \tau_1<\tau_2$: $$\begin{split}
\int_{\tau_1}^{\tau_2} \left[\int_{{N}^{\mathcal{H}}_{\tau}} J^{T}[\psi]\right] d\tau &\lesssim \int_{{N}^{\mathcal{H}}_{\tau_1}} (r-M)^{-1}\cdot (\partial_u(r\psi))^2\,d\omega du+\text{l.o.t.},\\
\int_{\tau_1}^{\tau_2} \left[\int_{{N}^{\mathcal{H}}_{\tau}} (r-M)^{-1}\cdot (\partial_u(r\psi))^2\,d\omega du\right]d\tau &\lesssim \int_{{N}^{\mathcal{H}}_{\tau_1}} (r-M)^{-2}\cdot (\partial_u(r\psi))^2\,d\omega du+\text{l.o.t.}.
\label{hierhoverview}
\end{split}$$ The integral on the right hand side of the second estimate of the ${\mathcal{I}^{+}}-$localized hierarchy corresponds to *the conformal energy near ${\mathcal{I}^{+}}$*. Similarly, the integral on the right hand side of the second estimate of the ${\mathcal{H}^{+}}-$localized hierarchy corresponds to *the conformal energy near ${\mathcal{H}^{+}}$*. We denote $$\hspace{-1.6cm}\text{\textbf{Conformal energy near }}{\mathcal{I}^{+}}: \ \ \ \mathcal{C}_{N_{\tau}^{\mathcal{I}}}[\psi]= \int_{{N}^{\mathcal{I}}_{\tau}} r^2\cdot (\partial_v(r\psi))^2\,d\omega dv
\label{confenergyI}$$ and $$\text{\textbf{Conformal energy near }}{\mathcal{H}^{+}}: \ \ \ \mathcal{C}_{N_{\tau}^{\mathcal{H}}}[\psi]= \int_{{N}^{\mathcal{H}}_{\tau}} (r-M)^{-2}\cdot (\partial_u(r\psi))^2\,d\omega du.
\label{nondegnH}$$ It is important to note that $du=-2\left(1-\frac{M}{r}\right)^{-2}dr$ on $\Sigma_{\tau}$ and $\partial_{u}=-\frac{1}{2}\left(1-\frac{M}{r}\right)^2Y$, where $Y=\partial_r$ is regular vector field at the horizon. Hence, the conformal flux near ${\mathcal{H}^{+}}$ $\mathcal{C}_{N_{\tau}^{\mathcal{H}}}[\psi]\sim \int_{{N}^{\mathcal{H}}_{\tau}}(Y\psi)^2$ is at the level of the *non-degenerate* energy.
If both of the energies and are initially finite, then, by using a standard application of the mean value theorem on dyadic time intervals and the boundedness of the $T$-energy flux, we obtain the decay rate $\tau^{-2}$ for the $T$-energy flux $\mathcal{E}^{T}_{\Sigma_{\tau}}[\psi]$. This decay rate however is quite weak. Faster decay rates for the higher order flux $\mathcal{E}^{T}_{\Sigma_{\tau}}[T\psi]$ were obtained for sub-extremal black holes by Schlue [@volker1] and Moschidis [@moschidis1]. Their method used $\partial_v$, $r\partial_v$ as commutator vector fields in the near-infinity region. Nonetheless, their approach does not yield faster decay for the $T$-flux $\mathcal{E}^{T}_{\Sigma_{\tau}}[\psi]$ itself.
### Commuted hierarchies in the regions $\mathcal{A}^{\mathcal{H}}$ and $\mathcal{A}^{\mathcal{I}}$ {#sec:CommutedHierarchies}
Our strategy for obtaining further decay for $\mathcal{E}^{T}_{\Sigma_{\tau}}[\psi]$ on ERN is to establish *integrated decay estimates for the conformal fluxes*[^9] $\mathcal{C}_{N_{\tau}^{\mathcal{I}}}[\psi]$ and $\mathcal{C}_{N_{\tau}^{\mathcal{H}}}[\psi]$, extending thereby the ${\mathcal{I}^{+}}-$localized and ${\mathcal{H}^{+}}-$localized hierarchies and . However, it is not possible to further extend of and by considering larger powers of $r$ and $(r-M)^{-1}$, respectively. Instead, motivated by the following Hardy inequality (see also Section \[sec:HardyInequalities\] in the Appendix): $$\int_{0}^{\infty}x^2\cdot \left(\partial_{x}f\right)^{2} \,dx \lesssim \int_{0}^{\infty}\Big(\partial_{x}\left(\boldsymbol{x^{2}\partial_{x}}f\right)\Big)^2 \,dx,
\label{basichardy}$$ applied to $f=r\psi$, with $x=r$, $\partial_x=\partial_v$ in $\mathcal{A}^{\mathcal{I}}$, and $x=(r-M)^{-1}$, $\partial_x=\partial_u$ in $\mathcal{A}^{\mathcal{H}}$, we introduce the following $n$-**commuted** quantities: $$\Phi_{(n)}:=(r^2\partial_v)^n(r\psi), \hspace{1.5cm}\underline{\Phi}_{(n)}:=Y^n(r\psi) \sim \Big(-(r-M)^{-2}\partial_{u}\Big)^n(r\psi),$$ where $n\in {\mathbb{N}}_0$. The idea therefore is to derive ${\mathcal{I}^{+}}-$localized and ${\mathcal{H}^{+}}-$localized *commuted* hierarchies which yield decay for weighted fluxes of the commuted functions $\Phi_{(n)}$ and $\underline{\Phi}_{(n)}$, respectively. As we shall see, these hierarchies involve growing $r$ and $(r-M)^{-1}$ weights. Partial results for $\Phi_{(n)}$ and $\underline{\Phi}_{(n)}$ were previously obtained in [@aretakis2; @paper1].
If $\psi$ solves the wave equation on ERN then for all $n\geq 0$ and *for all* $p\in {\mathbb{R}}$ the commuted quantities $\Phi_{(n)}$ and $\underline{\Phi}_{(n)}$ satisfy the following *key identities* in $\mathcal{A}^{\mathcal{I}}$ and $\mathcal{A}^{\mathcal{H}}$ regions, respectively (see Section \[sec:mainest1\]):
**Near-infinity identity:** $$\label{eq:keyid}
\begin{split}
\int_{{\mathbb{S}}^2}& \partial_u\left(r^p(\partial_v\Phi_{(n)})^2\right)+\partial_v\left( r^{p-2}|{\slashed{\nabla}}_{{\mathbb{S}}^2}\Phi_{(n)}|^2-n(n+1)r^{p-2}\Phi_{(n)}^2\right)\,d\omega\\
&+\int_{{\mathbb{S}}^2} (p+4n)r^{p-1} (\partial_v\Phi_{(n)})^2+(2-p)r^{p-3}\left( |{\slashed{\nabla}}_{{\mathbb{S}}^2}\Phi|^2-n(n+1)\Phi_{(n)}^2\right)\,d\omega\\
=&\: n\cdot\sum_{k=0}^{\max\{0,n-1\}} \int_{{\mathbb{S}}^2}O(r^{p-2})\cdot \Phi_{(k)}\cdot \partial_v\Phi_{(n)}\,d\omega+\text{l.o.t.},
\end{split}$$ **Near-horizon identity:** $$\label{eq:keyidhor}
\begin{split}
\int_{\mathbb{S}^{2}}& \partial_v\left((r-M)^{-p}(\partial_u\underline{\Phi}_{(n)})^2\right)+\partial_u\left( (r-M)^{-p+2}|{\slashed{\nabla}}_{{\mathbb{S}}^2}\underline{\Phi}_{(n)}|^2-n(n+1)(r-M)^{-p+2}\underline{\Phi}_{(n)}^2\right)\,d\omega\\
&+\int_{\mathbb{S}^{2}} (p+4n)(r-M)^{-p+1} (\partial_u\underline{\Phi}_{(n)})^2+(2-p)(r-M)^{-p+3}\left( |{\slashed{\nabla}}_{\mathbb{S}^{2}}\underline{\Phi}_{(n)}|^2-n(n+1)\underline{\Phi}_{(n)}^2\right)\,d\omega\\
=&\: n\cdot\sum_{k=0}^{\max\{0,n-1\}} \int_{\mathbb{S}^{2}}O((r-M)^{-p+2})\cdot \underline{\Phi}_{(k)}\cdot \partial_u\underline{\Phi}_{(n)}\,d\omega+\text{l.o.t.}
\end{split}$$ Note that is of the same form as , but with $u$ and $v$ reversed and $r$ replaced by $(r-M)^{-1}$. This is of course related to the existence of the Couch–Torrence conformal inversion of ERN.
After integrating in $u$ and $v$, the “*error*” terms that appear on the right-hand sides of the corresponding spacetime identities can be controlled via *Morawetz and Hardy inequalities* for the following range of weights[^10]: $$-4n<p\leq 2.
\label{pbound}$$ We arrive at the following inequalities (see Section \[sec:mainest\])
**${\mathcal{I}^{+}}-$localized $n-$commuted $p-$inequalities for $\Phi_{(n)}$:** $$\begin{split}
\label{eq:introrweightestI}
&\int_{{N}^{\mathcal{I}}_{\tau_2}} r^p\left(\partial_v\Phi_{(n)}\right)^2\,d\omega dv\\+& \int_{\tau_1}^{\tau_2} \int_{{N}^{\mathcal{I}}_{\tau}} (p+4n)r^{p-1} \left(\partial_v\Phi_{(n)}\right)^2+ (2-p)r^{p-3}\left( |{\slashed{\nabla}}_{{\mathbb{S}}^2}\Phi_{(n)}|^2-n(n+1)\Phi_{(n)}^2\right)\,d\omega dv d\tau\\
&\lesssim_p\: \int_{{N}^{\mathcal{I}}_{\tau_1}} r^p\left(\partial_v\Phi_{(n)}\right)^2\,d\omega dv+\ldots,
\end{split}$$ **${\mathcal{H}^{+}}-$localized $n-$commuted $p-$inequalities for $\underline{\Phi}_{(n)}$:** $$\begin{split}
\label{eq:introrweightestH}
&\int_{{N}^{\mathcal{H}}_{\tau_2}} (r-M)^{-p}\left(\partial_u \underline{\Phi}_{(n)}\right)^2\,d\omega du\\
+& \int_{\tau_1}^{\tau_2} \int_{{N}^{\mathcal{H}}_{\tau}} (p+4n)(r-M)^{-p+1} \left(\partial_u\underline{\Phi}_{(n)}\right)^2+ (2-p)(r-M)^{-p+3}\left({|{\slashed{\nabla}}_{{\mathbb{S}}^2}\underline{\Phi}_{(n)}|^2-n(n+1)\underline{\Phi}_{(n)}^2}\right)\,d\omega du d\tau\\
&\lesssim_p \: \int_{{N}^{\mathcal{H}}_{\tau_1}}(r-M)^{-p}\left(\partial_u \underline{\Phi}_{(n)}\right)^2\,d\omega du+\ldots
\end{split}$$ These inequalities hold for **all** $n$, as long as $p$ satisfies . In order to turn these inequalities into actual estimates we need to guarantee the non-negativity of the terms $|{\slashed{\nabla}}_{{\mathbb{S}}^2}{\Phi}_{(n)}|^2-n(n+1){\Phi}_{(n)}^2$ and $|{\slashed{\nabla}}_{{\mathbb{S}}^2}\underline{\Phi}_{(n)}|^2-n(n+1)\underline{\Phi}_{(n)}^2$. In view of the *Poincaré inequality* on ${\mathbb{S}}^2$ (see Section \[sec:HardyInequalities\] in the Appendix), these terms are non-negative if $\psi$ is supported on angular frequencies $\ell$ such that $$\ell \geq n.
\label{nell}$$ In other words, *we can commute the wave equation $n$ times and obtain two estimates for $\Phi_{(n)}$ and two estimates for $\underline{\Phi}_{(n)}$ for each $n$, as long as $n$ is less or equal than the lowest harmonic mode that is present in a harmonic mode expansion of $\psi$.* The two estimates correspond to the values $p=1$ and $p=2$.
It is worth mentioning that the estimates can be thought of as *degenerate remnants* of the red shift estimates. Note that the degeneracy of the (higher-order) red shift effect is manifested in the additional factor of $(r-M)$ that appears in the spacetime integral of $(\partial_u\underline{\Phi}_{(n)})^2$ on the left-hand side of .
The table below summarizes the number of the ${\mathcal{H}^{+}}-$localized $n-$commuted estimates and the ${\mathcal{I}^{+}}-$localized $n-$commuted estimates for each fixed $n$ as well as the total number of estimates available in the **total hierarchy** over all admissible values of $n$.
: *We define the **length of a hierarchy** to be equal to the number of available and useful integrated estimates in the hierarchy. Useful here means that the exponents $p$ of the weights increase by an integer number or by an almost (modulo $\epsilon>0$) integer number.*
[ cV[5]{}c | c |c]{} &\
& & [**Total hierarchy**]{}\
& & **Length** & **Length**\
& [$0$]{} & $2$ & $2$\
& [$0$]{} & $2$ &\
& [1]{} &2&\
& [$0$]{} & $2$ &\
& [1]{} &2&\
& [2]{} &2&\
### Improved hierarchies for $\ell=0,1$ {#sec:ImprovedHierarchiesForEll01}
The harmonic projections $\psi_{\ell=0}$ and $\psi_{\ell=1}$ of $\psi$ satisfy only two and four estimates in the total hierarchy, respectively, as in Table \[summarytablebasic\]. When dealing with $\ell=0$ (and hence $n=0$) separately, we show in Section \[sec:mainestextended\] that the range of $p$ can actually be *extended* to $0<p<3$ for both the ${\mathcal{H}^{+}}-$localized and the ${\mathcal{I}^{+}}-$localized hierarchies. Note that even though we cannot take $p=3$ exactly in this case, we can take $p=3-\epsilon$ for $\epsilon>0$ arbitrarily small. Additionally, we show that
- if $I_{0}[\psi]=0$ then we can take $0<p<5$ in the ${\mathcal{I}^{+}}-$localized hierarchy, and
- if $H_{0}[\psi]=0$ then we can take $0<p<5$ in the ${\mathcal{H}^{+}}-$localized hierarchy.
Similarly as above, even though we cannot take $p=5$ exactly, we will take $p=5-\epsilon$ for $\epsilon>0$. In this sense, the lengths of the above hierarchies (under the vanishing assumptions) is indeed five. Moreover, these hierarchies are *inextendible* (consistent with the horizon instability results of Section \[sec:TheHorizonInstabilityOfExtremalBlackHoles\]) and hence their length is sharp. It is important to observe that, based on the above result, the lengths of the total hierarchies depend on the type of data. These are summarized in the table below.
By $\mathcal{R}-$**global hierarchy** we mean the hierarchy that arises for weighted fluxed on $\Sigma_{\tau}$ by adding the ${\mathcal{H}^{+}}-$localized hierarchy (in region $\mathcal{A}^{\mathcal{H}}$), the ${\mathcal{I}^{+}}-$localized hierarchy (in region $\mathcal{A}^{\mathcal{I}}$) and the higher-order Morawetz estimates (in region $\mathcal{B}$; see Appendix \[sec:EnergyBounds\]). Recall that $\mathcal{R}=\mathcal{A}^{\mathcal{H}}\cup \mathcal{A}^{\mathcal{I}}\cup \mathcal{B}$.
[ cV[3]{}c | c |c ]{} &\
& ${\mathcal{H}^{+}}-$**localized** & ${\mathcal{I}^{+}}-$**localized** & $\mathcal{R}-$**global**\
& [$3$]{} & [$5$]{} & [$3$]{}\
& [$5$]{} & [$5$]{} & [$5$]{}\
& [$3$]{} & [$3$]{} & [$3$]{}\
& [$5$]{} & [$3$]{} & [$3$]{}\
In order to extend the length of the hierarchies for $\ell=1$ we introduce the following “modified” variants of $\Phi_{(1)}$ and $\underline{\Phi}_{(1)}$ (with $n=1$): $$\widetilde{\Phi}=\widetilde{\Phi}_{(1)}:=r(r-M)\partial_v(r\psi_{\ell=1}), \hspace{1.5cm} \underline{\widetilde{\Phi}}=\underline{\widetilde{\Phi}}_{(1)}:=r\cdot Y(r\psi_{\ell=1}).$$ We obtain the following improved identities for $\psi_{\ell=1}$ (see Section \[sec:mainestextended\]) $$\label{eq:keyidmod}
\begin{split}
\int_{\mathbb{S}^{2}}& \partial_u\left(r^p(\partial_v\widetilde{\Phi})^2\right)\,d\omega+\int_{\mathbb{S}^{2}} (p+4n)r^{p-1} (\partial_v\widetilde{\Phi})^2\,d\omega=\int_{\mathbb{S}^{2}}\boldsymbol{O(r^{p-3})} \cdot r\psi\cdot \partial_v\widetilde{\Phi}\,d\omega+\text{l.o.t}
\end{split}$$ and $$\label{eq:keyidhormod}
\begin{split}
\!\int_{\mathbb{S}^{2}}& \partial_v\left((r-M)^{-p}(\partial_u\widetilde{\underline{\Phi}})^2\right)+\!\int_{\mathbb{S}^{2}} (p+4n)(r-M)^{-p+1} (\partial_u\widetilde{\underline{\Phi}})^2\,d\omega=\!\int_{\mathbb{S}^{2}}\boldsymbol{O((r-M)^{-p+3})}\cdot\! r\psi\cdot \partial_u\widetilde{\underline{\Phi}}\,d\omega+\text{l.o.t}.
\end{split}$$ Note that the error terms (in bold) are now of lower order compared to the error terms in and . This allows us to obtain versions of and with $\Phi_{(1)}$ and $\underline{\Phi}_{(1)}$ replaced by $\widetilde{\Phi}$ and $\underline{\widetilde{\Phi}}$, respectively, where the range of $p$ can be *extended* to either $0<p<3$. We further obtain that:
- the range of the ${\mathcal{I}^{+}}-$localized hierarchy can be further extended to $0<p<4$ if ${\Phi}$ decays sufficiently fast towards ${\mathcal{I}^{+}}$, and
- the range of the ${\mathcal{H}^{+}}-$localized hierarchy can be further extended to $0<p<4$ if $\underline{\Phi}$ decays sufficiently fast towards ${\mathcal{H}^{+}}$.
Again, we cannot take $p=3$ or 4, but we will take $p=3-\epsilon$ or $4-\epsilon$.
The results for $\ell=1$ are summarized in the table below.
[ cV[5]{}c | c |cV[3]{}c|c|cV[3]{}c]{} &\
& & & $\mathcal{R}-$**global**\
& & & & &\
& & **Length** & & $n$& **Length** & &\
& & $2$ & & [$0$]{} & [$2$]{} & &\
& &3& & 1 & 4& &\
& [$0$]{} & $2$ & & [$0$]{} & [$2$]{} & &\
&[1]{} &4& & 1 & 4 & &\
& [$0$]{} & $2$ & & 0 & 2 & &\
& [1]{} &3& & 1 &3 & &\
& [$0$]{} & $2$ & & 0 & 2 & &\
& [1]{} &4& & 1 &3 & &\
[ cV[5]{}cV[2.5]{}c |c|c]{} &\
& **Harmonic mode** & ${\mathcal{H}^{+}}-$**localized** &[${\mathcal{I}^{+}}-$**localized**]{} & $\mathcal{R}-$**global**\
& [$\ell=0$]{} & $3$ & 5 & 3\
&[$\ell=1$]{} &5& 6 & 5\
& [$\ell\geq 2$]{} &6& 6 & 6\
& [$\ell=0$]{} & $5$ & 5 & 5\
& [$\ell=1$]{} &6& 6 & 6\
& [$\ell\geq 2$]{} &6& 6 & 6\
& [$\ell=0$]{} & $3$ & 3 & 3\
& [$\ell=1$]{} &5&5 & 5\
& [$\ell\geq 2$]{} &6& 6 & 6\
& [$\ell=0$]{} & $5$ & 3 & 3\
& [$\ell=1$]{} &6& 5 & 5\
& [$\ell\geq 2$]{} &6& 6 & 6\
\
\
Schlue [@volker1] and Moschidis [@moschidis1] obtained improved energy decay estimates for the time derivative $T\psi$ by considering $r$-weighted estimates for the quantities $\partial_v(r\psi)$ or $r\partial_v(r\psi)$. We generalize in Section \[sec:extendhier\] their approach by establishing estimates for $\partial_v^k\Phi_{(n)}$ in the near-infinity region $\mathcal{A}^{\mathcal{I}}$ and for $\partial_u^k\underline{\Phi}_{(n)}$ in the near-horizon region $\mathcal{A}^{\mathcal{H}}$ (with $n$ as above), where $k\in {\mathbb{N}}$ takes *any* positive value $k\geq 1$. This yields the following: *for each time derivative that we take, we gain two more estimates in the ${\mathcal{I}^{+}}-$localized hierarchy and two more estimates in the ${\mathcal{H}^{+}}-$localized hierarchy.* These improvements play an important role in the subsequent subsections.
### Energy and pointwise decay {#sec:EnergyAndPointwiseDecay}
The total (that is, over all admissible $n$) ${\mathcal{I}^{+}}-$localized and ${\mathcal{H}^{+}}-$localized hierarchies give quantitative decay rates for the conformal fluxes $\mathcal{C}_{N_{\tau}^{\mathcal{I}}}[\psi]$, given by , and $\mathcal{C}_{N_{\tau}^{\mathcal{H}}}[\psi]$, given by , respectively. This is easily obtained via successive application of the mean value theorem in dyadic intervals and the Hardy inequality . The rule is the following: $$\hspace{-0.2cm}\text{\textit{decay rate of the conformal flux }} \mathcal{C}_{N_{\tau}^{\mathcal{I}}}[\psi]=\text{\textit{length}}\left({\mathcal{I}^{+}}\!\!-\!\text{\textit{localized hierarchy}}\right)-2-\epsilon,$$ and $$\text{\textit{decay rate of the conformal flux }} \mathcal{C}_{N_{\tau}^{\mathcal{H}}}[\psi]=\text{\textit{length}}\left({\mathcal{H}^{+}}\!\!-\!\text{\textit{localized hierarchy}}\right)-2-\epsilon$$ for any sufficiently small $\epsilon>0$. The $\epsilon$ loss here has to do with the fact that the maximum value of $p$ in the extended improved hierarchies for $\ell=0$ and $\ell=1$ is not an exact integer.
Having obtained decay rates for the conformal fluxes we can proceed to obtain decay rate for the global $T-$flux $\mathcal{E}^{T}_{\Sigma_{\tau}}[\psi]$. For this we revisit the ${\mathcal{H}^{+}}-$localized and ${\mathcal{I}^{+}}-$localized hierarchies; we add the ${\mathcal{H}^{+}}-$localized hierarchy (in region $\mathcal{A}^{\mathcal{H}}$), the ${\mathcal{I}^{+}}-$localized hierarchy (in region $\mathcal{A}^{\mathcal{I}}$) and the higher-order Morawetz estimates (in region $\mathcal{B}$). Using again successively the mean value theorem in dyadic intervals and appropriate Hardy inequalities we obtain decay estimates for the $T$-energy flux. The rule here is the following: $$\text{\textit{decay rate of the energy flux }} {\mathcal{E}}^{T}_{\Sigma_{\tau}}[\psi]=\text{\textit{decay rate of \underline{slowest} conformal flux }}+2.$$ Unlike the sub-extremal case, in ERN there are **two** independent conformal fluxes that contribute to the decay rate for the energy flux. This feature of ERN creates further complications later in the derivation of the precise asymptotics.
As an illustration of our techniques, let us consider initial data for $\psi$ of Type **A**. As we can see in Table \[summarytablel0\], the length of the total ${\mathcal{I}^{+}}-$localized hierarchy and total ${\mathcal{H}^{+}}-$localized hierarchy is 5 and 3 for $\ell=0$ , respectively. Hence, we obtain schematically the decay estimates for the conformal fluxes (see Section \[sec:decayest\]): $$\begin{aligned}
\mathcal{C}_{N_{\tau}^{\mathcal{H}}}[\psi_{\ell=0}]\ \ \lesssim&\ \ \: E_{\ell=0}\cdot \tau^{-1+\epsilon},\\
\mathcal{C}_{N_{\tau}^{\mathcal{I}}}[\psi_{\ell=0}]\ \ \lesssim&\ \ \: E_{\ell=0}\cdot \tau^{-3+\epsilon}.
\end{aligned}$$ Furthermore, from Tables \[summarytablebasic\] and \[summarytablel1\] we have that the length of the total ${\mathcal{I}^{+}}-$localized hierarchy and total ${\mathcal{H}^{+}}-$localized hierarchy is 6 and 5 for $\ell\geq 1$, respectively. Hence, $$\begin{aligned}
\mathcal{C}_{N_{\tau}^{\mathcal{H}}}[\psi_{\ell\geq 1}]\ \ \lesssim&\ \ \: E_{\ell\geq 1}\cdot \tau^{-3+\epsilon},\\
\mathcal{C}_{N_{\tau}^{\mathcal{I}}}[\psi_{\ell\geq 1}]\ \ \lesssim&\ \ \: E_{\ell\geq 1}\cdot \tau^{-4+\epsilon}.
\end{aligned}$$ We conclude the following decay estimate for the $T-$energy flux: $$\begin{aligned}
\mathcal{E}^{T}_{\Sigma_{\tau}}[\psi_{\ell=0}] \ \ \lesssim&\ \ \: E_{\ell=0}\cdot \tau^{-3+\epsilon},\\
\mathcal{E}^{T}_{\Sigma_{\tau}}[\psi_{\ell\geq 1}] \ \ \lesssim&\ \ \: E_{\ell\geq 1} \cdot \tau^{-5+\epsilon},
\end{aligned}$$ where $E_{\ell=0}$ and $E_{\ell\geq 1}$ denote (higher-order, weighted) initial data energy norms. Furthermore, $$\begin{aligned}
\mathcal{E}^{T}_{\Sigma_{\tau}}[T^k\psi_{\ell=0}] \ \ \lesssim&\ \ \: E_{\ell=0; k}\cdot \tau^{-3-2k+\epsilon},\\
\mathcal{E}^{T}_{\Sigma_{\tau}}[T^k\psi_{\ell\geq 1}] \ \ \lesssim&\ \ \: E_{\ell\geq 1; k} \cdot \tau^{-5-2k+\epsilon},
\end{aligned}$$ for all $k\geq 1$, where $E_{\ell=0;k}$ and $E_{\ell\geq 1;k}$ denote (higher-order, weighted) initial data energy norms.
We next proceed with deriving pointwise decay estimates (see Section \[sec:decayest\]). We will use the following Hardy estimates $$\begin{aligned}
\int_{\mathbb{S}^{2}} (r\psi)^2\,d\omega \ \ &\lesssim \ \ \sqrt{\mathcal{C}_{N_{\tau}^{\mathcal{H}}}[\psi] }\cdot \sqrt{ \mathcal{E}^{T}_{\Sigma_{\tau}}[\psi]} \ \ \ \ \textnormal{ in }\ \ \mathcal{A}^{\mathcal{H}}, \\
\int_{\mathbb{S}^{2}} (r\psi)^2\,d\omega\ \ &\lesssim \ \ \sqrt{\mathcal{C}_{N_{\tau}^{\mathcal{I}}}[\psi] }\cdot \sqrt{ \mathcal{E}^{T}_{\Sigma_{\tau}}[\psi]} \ \ \ \ \textnormal{in }\ \ \mathcal{A}^{\mathcal{I}},\\
\int_{\mathbb{S}^{2}} (r-M)\!\cdot\!\psi^2\,d\omega \ \ &\lesssim \ \ \sqrt{ \mathcal{E}^{T}_{\Sigma_{\tau}}[\psi]} \ \ \ \ \text{ on } \ \Sigma_{\tau}. \end{aligned}$$ For initial data of Type **A**, using the above decay estimates for the conformal energies and the $T$-energy flux, we obtain $$\begin{aligned}
\int_{\mathbb{S}^{2}} (r\psi_{\ell=0})^2\,d\omega \ \ \lesssim& \ \ \: E_{\ell=0}\cdot \tau^{-2+\epsilon}\: \ \ \ \ \textnormal{in }\ \ \mathcal{A}^{\mathcal{H}}, \\
\int_{\mathbb{S}^{2}} (r\psi_{\ell=0})^2\,d\omega \ \ \lesssim&\: \ \ E_{\ell=0} \cdot \tau^{-3+\epsilon} \: \ \ \ \ \textnormal{in }\ \ \mathcal{A}^{\mathcal{I}},\\
\int_{\mathbb{S}^{2}} (r-M)\cdot (\psi_{\ell=0})^2\,d\omega \ \ \lesssim&\: \ \ E_{\ell=0}\cdot \tau^{-3+\epsilon} \: \ \ \ \text{ on } \ \ \Sigma_{\tau}. \end{aligned}$$ Using the standard Sobolev estimates on ${\mathbb{S}}^2$ we immediately obtain $L^{\infty}$ decay estimates for $r\psi_{\ell=0}$ in $\mathcal{A}^H$, $r\psi_{\ell=0}$ in $\mathcal{A}^I$ and $\sqrt{r-M}\cdot\psi_{\ell=0}$ on $\Sigma_{\tau}$, with the decaying factors $\tau^{-1+{\epsilon}}$, $\tau^{-\frac{3}{2}+{\epsilon}}$ and $\tau^{-\frac{3}{2}+{\epsilon}}$, respectively. Similarly, $$\begin{aligned}
\int_{\mathbb{S}^{2}} (r\psi_{\ell\geq 1})^2\,d\omega \ \ \lesssim& \ \ \: E_{\ell=0}\cdot \tau^{-4+\epsilon}\: \ \ \ \ \textnormal{in }\ \ \mathcal{A}^{\mathcal{H}}, \\
\int_{\mathbb{S}^{2}} (r\psi_{\ell\geq 1})^2\,d\omega \ \ \lesssim&\: \ \ E_{\ell=0} \cdot \tau^{-\frac{9}{2}+\epsilon} \: \ \ \ \ \textnormal{in }\ \ \mathcal{A}^{\mathcal{I}},\\
\int_{\mathbb{S}^{2}} (r-M)\cdot (\psi_{\ell\geq 1})^2\,d\omega \ \ \lesssim&\: \ \ E_{\ell=0}\cdot \tau^{-5+\epsilon} \: \ \ \ \text{ on } \ \ \Sigma_{\tau}. \end{aligned}$$ As above, $L^{\infty}$ decay estimates for $r\psi_{\ell\geq 1}$ in $\mathcal{A}^H$, $r\psi_{\ell\geq 1}$ in $\mathcal{A}^I$ and $\sqrt{r-M}\cdot\psi_{\ell\geq 1}$ on $\Sigma_{\tau}$, with the decaying factors $\tau^{-2+{\epsilon}}$, $\tau^{-\frac{9}{4}+{\epsilon}}$ and $\tau^{-\frac{5}{2}+{\epsilon}}$, respectively.
The above estimates illustrate another deviation from the sub-extremal analysis in [@paper1; @paper2]: for Type **A** initial data, the decay rate of $r\psi_{\ell=0}$ in $\mathcal{A}^\mathcal{I}$ *is a power $\frac{1}{2}+\epsilon$ from the sharp decay rate*, whereas in the sub-extremal case, the analogous estimate results in a decay rate that is almost sharp, in other words only $\epsilon$ away from the sharp decay rate. In this case it is the non-vanishing of $H_0$ and hence the slow decay for the conformal energy in the near-horizon region that forms the “bottleneck” for the maximal length of the global hierarchy of weighted estimates for $\psi_{\ell=0}$.
The energy and pointwise decay rates are summarized in the two tables below (see Section \[sec:decayest\]).
[ cV[5]{}c | c |cV[3]{}c|c|c]{} &\
& &\
&$\mathcal{E}^{T}_{\ \Sigma_{\tau}}[\psi] $& $\mathcal{C}_{N_{\tau}^{\mathcal{H}}}[\psi]$ & $\mathcal{C}_{N_{\tau}^{\mathcal{I}}}[\psi]$ &$r\psi|_{\mathcal{H}^{+}}$ & $\psi|_{\{r=r_0\}}$ & $r\psi|_{\mathcal{I}^{+}}$\
& $\tau^{-3+\epsilon}$ & $\tau^{-1+\epsilon}$ & $\tau^{-3+\epsilon}$ &$\tau^{-1+\epsilon}$& $\boldsymbol{\tau^{-\frac{3}{2}+\epsilon}}$ &$\boldsymbol{\tau^{-\frac{3}{2}+\epsilon}}$\
& $\tau^{-5+\epsilon}$ & $\tau^{-3+\epsilon}$ & $\tau^{-3+\epsilon}$ &$\tau^{-2+\epsilon}$&$\boldsymbol{\tau^{-\frac{5}{2}+\epsilon}}$ &$\tau^{-2+\epsilon}$\
& $\tau^{-3+\epsilon}$ & $\tau^{-1+\epsilon}$ & $\tau^{-1+\epsilon}$ &$\tau^{-1+\epsilon}$&$\boldsymbol{\tau^{-\frac{3}{2}+\epsilon}}$ &$\tau^{-1+\epsilon}$\
& $\tau^{-3+\epsilon}$ & $\tau^{-3+\epsilon}$ & $\tau^{-1+\epsilon}$&$\boldsymbol{\tau^{-\frac{3}{2}+\epsilon}}$ &$\boldsymbol{\tau^{-\frac{3}{2}+\epsilon}}$ &$\tau^{-1+\epsilon}$\
[ cV[5]{}c | c |cV[3]{}c|c|c]{} &\
& &\
&$\mathcal{E}^{T}_{\ \Sigma_{\tau}}[\psi] $& $\mathcal{C}_{N_{\tau}^{\mathcal{H}}}[\psi]$ & $\mathcal{C}_{N_{\tau}^{\mathcal{I}}}[\psi]$ &$r\psi|_{\mathcal{H}^{+}}$ & $\psi|_{\{r=r_0\}}$ & $r\psi|_{\mathcal{I}^{+}}$\
& $\tau^{-5+\epsilon}$ & $\tau^{-3+\epsilon}$ & $\tau^{-4+\epsilon}$ &$\tau^{-2+\epsilon}$&$\tau^{-\frac{5}{2}+\epsilon}$ &$\tau^{-\frac{9}{4}+\epsilon}$\
& $\tau^{-6+\epsilon}$ & $\tau^{-4+\epsilon}$ & $\tau^{-4+\epsilon}$ &$\tau^{-\frac{5}{2}+\epsilon}$& $\tau^{-3+\epsilon}$ &$\tau^{-\frac{5}{2}+\epsilon}$\
& $\tau^{-5+\epsilon}$ & $\tau^{-3+\epsilon}$ & $\tau^{-3+\epsilon}$ &$\tau^{-2+\epsilon}$&$\tau^{-\frac{5}{2}+\epsilon}$ &$\tau^{-2+\epsilon}$\
& $\tau^{-5+\epsilon}$ & $\tau^{-4+\epsilon}$ & $\tau^{-3+\epsilon}$&$\tau^{-\frac{5}{2}+\epsilon}$ &$\tau^{-\frac{5}{2}+\epsilon}$ &$\tau^{-2+\epsilon}$\
Note that the decay rates for $\psi_{\{r=r_0\}}$ apply for $\sqrt{r-M}\psi$ for all $r>M$.
### An elliptic estimate for $\ell \geq 1$ {#sec:EllipticEstimates}
The decay rates for $\psi_{\{r=r_0\}}$, in the $\ell=0$ case, in the Table \[summarytablenew\] are $\frac{1}{2}+\epsilon$ away from sharp. Furthermore, the decay rate for $\psi_{\{r=r_0\}}$, in the $\ell\geq 1$ case, as in Table \[summarytablenew1\], is slower than the corresponding expected sharp rate for the $\ell=0$ case. For obtaining late-time asymptotics, the $\ell\geq 1$ rate must be improved.
The desired improvement of the decay rate of $\psi_{\{r=r_0\}}$ will be achieved using an elliptic estimate and the improved decay rates for $T\psi$. The challenge for obtaining the elliptic estimate is that, in contrast to the sub-extremal case, the decaying global energy flux ${\mathcal{E}}^{T}_{\Sigma_{\tau}}$ is highly *degenerate* at the event horizon. Indeed, recall that ${\mathcal{E}}^{T}_{\Sigma_{\tau}}[\psi]\sim \int_{\Sigma_{\tau}}\left(1-\frac{M}{r}\right)^2\cdot |\partial\psi|^{2}$. In other words, we need to obtain a degenerate elliptic estimate on ERN. It turns out that such an estimate is **not** possible for $\ell=0$ and hence we will need to derive the precise asymptotics using the aforementioned weak rates (see the next subsection). On the other hand, we can establish such a degenerate elliptic estimate for $\ell \geq 1$ (see Section \[sec:ellpest\]) which, coupled with a Hardy inequality, schematically gives: $$\int_{\Sigma_{\tau}}\left(1- \frac{M}{r}\right)^4 \cdot \left(\partial_{\rho}\psi_{\ell \geq 1}\right)^2\cdot r^{-2}\, d\mu_{\Sigma_{\tau}}\ \ \lesssim \ \ \int_{\Sigma_{\tau}} \left(1- \frac{M}{r}\right)^2 \cdot \left(\partial_{\rho} T \psi_{\ell \geq 1}\right)^2\, d\mu_{\Sigma_{\tau}},
\label{scheelli}$$ where $\partial_{\rho}$ denotes the radial ($SO(3)-$invariant) vector field tangent to $\Sigma_{\tau}$. Consequently, by denoting $D=\left(1-\frac{M}{r}\right)^2$ and using a standard Hardy inequality and the improved energy decay estimates for $T\psi$ we obtain for Type **B** data: $$\begin{split}
\int_{\mathbb{S}^2} \left(\psi_{\ell\geq 1}\right)^2\,d\omega\ \ \lesssim\ \ &\:\ \ \frac{1}{{D}}\sqrt{ \int_{\Sigma_{\tau}} D^2 \cdot\left(\partial_{\rho} \psi_{\ell\geq 1}\right)^2 \cdot r^{-2}\,d\mu_{\Sigma_{\tau}}}\cdot \sqrt{\int_{\Sigma_{\tau}} \psi_{\ell\geq 1}^2\cdot r^{-2}\,d\mu_{\Sigma_{\tau}}}\\\
\overset{\eqref{scheelli}}{\lesssim} &\:\ \ \frac{1}{{D}}\sqrt{ \int_{\Sigma_{\tau}} D \cdot\left(\partial_{\rho} T \psi_{\ell\geq 1}\right)^2 \,d\mu_{\Sigma_{\tau}}}\cdot \sqrt{\int_{\Sigma_{\tau}} D \cdot\left(\partial_{\rho} \psi_{\ell\geq 1}\right)^2\,d\mu_{\Sigma_{\tau}}}\\
= \ \ &\: \ \ \frac{1}{{D}}\sqrt{\mathcal{E}^{T}_{\Sigma_{\tau}}[T\psi_{\ell\geq 1}]}\cdot \sqrt{\mathcal{E}^{T}_{\Sigma_{\tau}}[\psi_{\ell\geq 1}]}\\
\lesssim\ \ &\: \ \ \frac{1}{{D}} \sqrt{E_{\ell\geq 1;1}}\cdot \sqrt{E_{\ell\geq 1}}\cdot\tau^{-7+\epsilon},
\end{split}$$ where used the decay rates in Table \[summarytablenew1\]. This yields that $\left(1-\frac{M}{r}\right)\cdot \psi_{\ell \geq 1}$ decays with a rate $\tau^{-\frac{7}{2}+\frac{\epsilon}{2}}$. This rate is now indeed sub-dominant (i.e. strictly faster than $\tau^{-3}$). We summarize our findings in the table below:
[ cV[5]{}c | c |cV[3]{}c|c|c]{} &\
& &\
&$\mathcal{E}^{T}_{\ \Sigma_{\tau}}[\psi] $& $\mathcal{C}_{N_{\tau}^{\mathcal{H}}}[\psi]$ & $\mathcal{C}_{N_{\tau}^{\mathcal{I}}}[\psi]$ &$r\psi|_{\mathcal{H}^{+}}$ & $\psi|_{\{r=r_0\}}$ & $r\psi|_{\mathcal{I}^{+}}$\
& $\tau^{-5+\epsilon}$ & $\tau^{-3+\epsilon}$ & $\tau^{-4+\epsilon}$ &$\tau^{-2+\epsilon}$&$\tau^{-\frac{5}{2}+\epsilon}$ &$\tau^{-\frac{9}{4}+\epsilon}$\
& $\tau^{-6+\epsilon}$ & $\tau^{-4+\epsilon}$ & $\tau^{-4+\epsilon}$ &$\tau^{-\frac{5}{2}+\epsilon}$& $\tau^{-\frac{7}{2}+\epsilon}$ &$\tau^{-\frac{5}{2}+\epsilon}$\
& $\tau^{-5+\epsilon}$ & $\tau^{-3+\epsilon}$ & $\tau^{-3+\epsilon}$ &$\tau^{-2+\epsilon}$&$\tau^{-\frac{5}{2}+\epsilon}$ &$\tau^{-2+\epsilon}$\
& $\tau^{-5+\epsilon}$ & $\tau^{-4+\epsilon}$ & $\tau^{-3+\epsilon}$&$\tau^{-\frac{5}{2}+\epsilon}$ &$\tau^{-\frac{5}{2}+\epsilon}$ &$\tau^{-2+\epsilon}$\
### Late-time asymptotics {#sec:LateTimeAsymptotics}
In this section we will provide a summary of the mechanism that gives rise to the precise leading-order asymptotics for $\psi$. Our discussion here complements that of Section \[sec:SummaryOfTheMainResults\]. The complete proofs cover Sections \[sec:asympnonzeroconst\]–\[sec:hoasymp\]. We claim that the decay rates for $\psi_{\ell\geq 1}$ as in Table \[summarytablenewafterelliptic\] are faster than the sharp decay rates for $\psi_{\ell=0}$. Based on this claim, we will derive first the precise late-time asymptotics (and hence the sharp rates) for $\psi_{\ell=0}$. For this reason, we will assume in the rest of this section that $\psi$ is a spherically symmetric (and hence supported only on $\ell=0$) solution to the wave equation on ERN.
We need to overcome the following difficulties
- **Difficulty 1:** Find spacetime regions in which asymptotics can be derived *independently* of their complement in $\mathcal{R}$. An obstruction here is that the decay rates that we have already obtained (as summarized in the previous subsections) are a power $\frac{1}{2}+\epsilon$ from the sharp values in the region $\mathcal{B}=\{r_{\mathcal{H}}\leq r\leq r_{\mathcal{I}}\}$. Compare the rates in Tables \[summarytable\] and \[summarytablenew\].
- **Difficulty 2:** Propagate the above asymptotics *globally* in the region $\mathcal{R}$. The main obstruction here is that for data of Type **A**, **B** and **C** the radial (tangential to $\Sigma_{\tau}$) derivative $\partial_{\rho}\psi$ decays only as fast as $\psi$ itself and hence the corresponding decay estimates cannot be easily integrated to propagate the asymptotics of $\psi$, without first deriving the precise asymptotics of $\partial_{\rho}\psi$. Compare the rates in Tables \[summarytable\] and \[summarytablerho\]. We remark that this is not the case in sub-extremal black holes where radial derivatives decay faster than the scalar field itself.
We consider the timelike hypersurfaces $\gamma^{\mathcal{I}}$ and $\gamma^{\mathcal{H}}$ such that $(v-u) |_{\gamma^{\mathcal{I}}}\sim u^{\alpha}$ and $(u-v) |_{\gamma^{\mathcal{H}}}\sim v^{\alpha}$ where $0<\alpha<1$ is a constant, and we define the following subsets of the near-infinity region $\mathcal{A}^\mathcal{I}$ and the near-horizon region $\mathcal{A}^\mathcal{H}$: $\mathcal{A}^\mathcal{I}_{\gamma^{\mathcal{I}}}:=\mathcal{A}^\mathcal{I}\cap \{r\geq r|_{\gamma^{\mathcal{I}}}\}$ and $\mathcal{A}^\mathcal{H}_{\gamma^{\mathcal{H}}}=\mathcal{A}^\mathcal{H}\cap \{r\leq r|_{\gamma^{\mathcal{H}}}\}$. Note that $(r-M)|_{\gamma^{\mathcal{H}}} \sim r|_{\gamma^{\mathcal{I}}}\sim\tau^{\alpha}$.
![The curves $\gamma^{\mathcal{H}}, \gamma^{\mathcal{I}}$ and the regions $\mathcal{A}^{\mathcal{H}}_{\gamma^{\mathcal{H}}}, \mathcal{A}^{\mathcal{I}}_{\gamma^{\mathcal{I}}}$.[]{data-label="fig:gammaa"}](gammaa.pdf)
We will summarize the resolutions to the above difficulties mainly for initial data of Type **C** and **A** and make a few concluding comments for data of Type **B** and **D**.
**
**Resolution of difficulty 1**
For Type **C** data we derive the leading-order asymptotics of $\psi$ in the near-horizon region $\mathcal{A}^{\mathcal{H}}_{\gamma^{\mathcal{H}}}$ and separately and independently in the near-infinity region $\mathcal{A}^{\mathcal{I}}_{\gamma^{\mathcal{I}}}$. This derivation distinguishes the extremal case from the sub-extremal case treated in [@paper2], where the asymptotics at the near-infinity region can be propagated all the way to the event horizon using that the radial derivative $\partial_{\rho}\psi$ decays faster than $\psi$. The reason we can independently derive the asymptotics in the regions $\mathcal{A}^{\mathcal{H}}_{\gamma^{\mathcal{H}}}$ and $\mathcal{A}^{\mathcal{I}}_{\gamma^{\mathcal{I}}}$ in the extremal case has to do with the existence of the two (independent) conserved charges $H_0$ and $I_0$; moreover, for Type **C** data they are both non-zero, i.e. $H_0\neq 0$ and $I_0\neq 0$. To obtain the precise asymptotics in $\mathcal{A}^{\mathcal{I}}_{\gamma^{\mathcal{I}}}$ and $\mathcal{A}^{\mathcal{H}}_{\gamma^{\mathcal{H}}}$ we propagate the following $v$-asymptotics and $u$-asymptotics of the initial data on $N_{0}^{\mathcal{I}}$ and $N_{0}^{\mathcal{H}}$, respectively, $$\begin{split}
\partial_v(r\psi)|_{N_{0}^{\mathcal{I}}}=&2I_0v^{-2}+O(v^{-2}),\\
\partial_u(r\psi)|_{N_{0}^{\mathcal{H}}}=&2H_0u^{-2}+O(u^{-2})
\end{split}
\label{uvinitialasymptotics}$$ everywhere in $\mathcal{A}^{\mathcal{I}}_{\gamma^{\mathcal{I}}}$ and $\mathcal{A}^{\mathcal{H}}_{\gamma^{\mathcal{H}}}$, respectively. This can be achieved for $\alpha<1$, but sufficiently close to 1. We next integrate the resulting estimates for $\partial_v(r\psi)$ and $\partial_u(r\psi)$ starting from $\gamma^{\mathcal{I}}$ and $\gamma^{\mathcal{H}}$, respectively, to obtain the asymptotics for $r \psi$, and consequently $\psi$, in appropriate sub-regions $\mathcal{A}^{\mathcal{I}}_{{\gamma'}^{\mathcal{I}}}$ and $\mathcal{A}^{\mathcal{H}}_{{\gamma'}^{\mathcal{H}}}$ of $\mathcal{A}^{\mathcal{I}}_{\gamma^{\mathcal{I}}}$ and $\mathcal{A}^{\mathcal{H}}_{\gamma^{\mathcal{H}}}$ obtained by replacing $\alpha$ with appropriate $\alpha'$ such that $\alpha< \alpha'<1$. A crucial observation is that the previously derived decay rates for $\sqrt{r-M}\cdot\psi_{{\gamma'}^{\mathcal{I}}}$ and $\sqrt{r-M}\cdot\psi_{{\gamma'}^{\mathcal{H}}}$ are almost sharp[^11] and hence strong enough to close this argument by showing that, as long as $a<1$, the terms $r\psi|_{\gamma^{\mathcal{I}}}$ and $r\psi|_{\gamma^{\mathcal{H}}}$ decay faster than, say $r\psi|_{{\gamma'}^{\mathcal{I}}}$ and $r\psi|_{{\gamma'}^{\mathcal{H}}}$, and hence are lower order terms.
**Resolution of difficulty 2**
Ideally, we would like to propagate to the left of ${\gamma'}^{\mathcal{I}}$ the asymptotics for $\psi_{{\gamma'}^{\mathcal{I}}}$. In the sub-extremal case this would follow using that $\alpha'<1$ and that the radial derivative $\partial_{\rho}\psi$ decays *faster* that $\psi$. This approach however breaks down in the extremal case in view of the fact that the expected sharp decay rate for $\partial_{\rho}\psi$ is now the *same* as the expected sharp rate for $\psi$.
Instead we obtain first the precise asymptotic behavior of the radial derivative $\partial_{\rho}\psi$. We commute by $T$ and reproduce the above argument to derive the precise late-time asymptotics for $T(r\psi)$ in the near-horizon region $\mathcal{A}^{\mathcal{H}}_{\gamma^{\mathcal{H}}}$. The crucial observation here is that *the asymptotics for $\partial_{\rho}\psi$ in the region $\{M<r\leq r_{{\mathcal{I}}}\}$ depend only on the asymptotics of $T\psi$ along the event horizon, which in turn depend only on $H_0$*! We next derive sharp decay estimates (with growing $r$ weights in the error terms) for $\partial_{\rho}\psi$ up to the curve ${\gamma'}^{\mathcal{I}}$, that is in the region $\{r_{{\mathcal{I}}}\leq r\leq r_{{\gamma'}^{\mathcal{I}}}\}$. The latter step would fail if we were to take $\alpha'=1$. We can next derive the asymptotics for $\psi$ in $\{M<r\leq r_{{\gamma'}^{\mathcal{I}}}\}$ by integrating the estimate for $\partial_{\rho}\psi$ in the same region backwards from ${\gamma'}^{\mathcal{I}}$. In the last step we crucially use again that $\alpha' <1$ and that we have already computed the asymptotics for $\psi|_{{\gamma'}^{\mathcal{I}}}$. Global asymptotics follow using a dual argument from infinity and the asymptotics in $\mathcal{A}^{\mathcal{H}}_{{\gamma'}^{\mathcal{H}}}$. See Section \[sec:asympnonzeroconst\]. Higher order logarithmic corrections are derived in Section \[sec:hoasymp\].
**
**Resolution of difficulty 1**
For Type **A** data we can derive the leading-order asymptotics of $\psi$, and crucially of $T\psi$, in the near-horizon region $\mathcal{A}^{\mathcal{H}}_{\gamma^{\mathcal{H}}}$ as in the Type **C** case, but in contrast to the Type **C** case, [we obtain independently the asymptotics in the near-infinity region]{} $\mathcal{A}^{\mathcal{I}}_{\gamma^{\mathcal{I}}}$ since the first equation of does not provide exact asymptotics anymore, given that $I_0=0$.
**Resolution of difficulty 2**
As in the Type **C** case, we can obtain the precise asymptotics for $\partial_{\rho}\psi$ in the region $\{M<r\leq r_{{\mathcal{I}}}\}$ using the asymptotics of $T\psi$ along the event horizon. However, like before, these asymptotics for $\partial_{\rho}\psi$ do not yield asymptotics for $\psi$ away from $\mathcal{A}^{\mathcal{H}}_{\gamma^{\mathcal{H}}}$. The main idea is that we can, however, derive the precise asymptotics exactly on $\gamma^{\mathcal{I}}$. In other words, equipped with the asymptotics for $\psi$ in $\mathcal{A}^{\mathcal{H}}_{\gamma^{\mathcal{H}}}$ we can next obtain asymptotics only along $\gamma^{\mathcal{I}}$ (and not to the right or to the left of $\gamma^{\mathcal{I}}$ as the asymptotics in these regions will only be derived at a later step). In order to derive asymptotics for $\psi|_{\gamma^{\mathcal{I}}}$ we need to analyze the contributions from the left side (horizon side) and the right side (infinity side) of $\gamma^{\mathcal{I}}$. As we shall see, in order to capture the precise contributions from both sides we will need to make crucial use of $I_0=0$. It turns out that we can only capture the precise contributions at one level of differentiability higher using the following splitting identity $$\Big.\frac{D}{2} r \psi\Big|_{\gamma^{\mathcal{I}}}\hspace{0.5cm}= \Big.\underbrace{r\partial_v (r\psi)\Big|_{\gamma^{\mathcal{I}}}}_{\substack{\text{contribution from} \\ \text{the right side of $\gamma^{\mathcal{I}}$}}}-\Big.\underbrace{r^2 \partial_v\psi\Big|_{\gamma^{\mathcal{I}}}}_{\substack{\text{contribution from} \\ \text{the left side of $\gamma^{\mathcal{I}}$}}}
\label{leftright}$$
*Contribution from the right side of* $\gamma^{\mathcal{I}}$: Recall that we want to show that $r\psi|_{{\gamma^{\mathcal{I}}}}$ decays like $\tau^{-2}$ (see Table \[summarytable\]) and hence all error terms must decay like $\tau^{-2-\epsilon}$. Now propagating in $\mathcal{A}^{\mathcal{I}}_{\gamma^{\mathcal{I}}}$ the first of only yields an $\epsilon$ improvement for $\partial_{v}(r\psi)|_{\gamma^{\mathcal{I}}}$, that is $r\partial_{v}(r\psi)|_{\gamma^{\mathcal{I}}}\sim r\tau^{-2-\epsilon}\sim \tau^{-2-\epsilon +\alpha}$ which is not fast enough since $\alpha$ is close to 1. To circumvent this difficulty, we need to introduce a new technique which we call **the singular time inversion** (see Section \[sec:timeint\]). Specifically, we construct the time integral $\psi^{(1)}$ of $\psi$ that solves the wave equation $\square_g\psi^{(1)}=0$ and satisfies $T\psi^{(1)}=\psi$. Note that if $H_0[\psi]\neq 0$ then $\psi^{(1)}$ is *singular* at the horizon and in fact satisfies $$(r-M)\cdot \partial_{\rho}\psi^{(1)} =-\frac{2}{M}\cdot H_0[\psi]+O(r-M)$$ for $r$ close to $M$, but is smooth away from $r=M$. Using appropriate low regularity estimates we can obtain global-in-time decay estimates for $\psi^{(1)}$ to the right of $\gamma^{\mathcal{I}}$. Moreover, using that for $\psi^{(1)}$ has a well-defined Newman–Penrose constant $I_0[\psi^{(1)}]<\infty$, we can propagate for $\psi^{(1)}$ which yields $\partial_{v}(r\psi^{(1)})|_{\gamma^{\mathcal{I}}}\sim \tau^{-2}$ and hence $r\partial_{v}(r\psi)|_{\gamma^{\mathcal{I}}}\sim r\tau^{-3}\sim \tau^{-3+\alpha}$ which shows that this term does contribute to the asymptotics for $r \psi|_{\gamma^{\mathcal{I}}}$.
*Contribution from the left side of* $\gamma^{\mathcal{I}}$: This is the side that fully contributes to the asymptotics for $r \psi|_{\gamma^{\mathcal{I}}}$ via the term $r^2 \partial_v\psi|_{\gamma^{\mathcal{I}}}$. For we will derive the precise asymptotics for $r^2 \partial_v\psi|_{\gamma^{\mathcal{I}}}$. We make use of the *improved decay rates for the conformal flux* $\mathcal{C}_{N_{\tau}^{\mathcal{I}}}[T\psi]$ (see Table \[summarytablenew\]; Type **A**) which, upon integrating the wave equation on $N_{\tau}^{\mathcal{I}}$, yield that the asymptotics for $r^2\partial_{v}\psi|_{\gamma^{\mathcal{I}}}$ can be derived from the asymptotics of $\partial_{\rho}\psi|_{\{r=r_{\mathcal{I}}\}}$ which we already derived (and recall they depend only on $H_0[\psi]$). Hence, the asymptotics for $r^2\partial_{v}\psi|_{\gamma^{\mathcal{I}}}$ depend only on $H_{0}$ and the respective rate is $\tau^{-2}$.
Concluding, *the precise asymptotics for $r\psi_{\gamma^{\mathcal{I}}}$ depend only on the horizon charge $H_0[\psi]$* and the respective rate is $\tau^{-2}$. The estimate for the conformal flux, as above, in fact yields the asymptotics for $r^2\partial_{v}\psi$ in $\{M< r\leq r_{\gamma^{\mathcal{I}}} \}$ which we can now integrate backwards from $\gamma^{\mathcal{I}}$ (using the asymptotics for $r\psi|_{\gamma^{\mathcal{I}}}$!) to obtain the asymptotics for $r\psi$ in whole region $\{M< r\leq r_{\gamma^{\mathcal{I}}}\}$. It remains to find the asymptotics of $r\psi$ to the right of $\gamma_{\mathcal{\mathcal{I}}}$ all the way up to null infinity. For this, we use the singular time integral $\psi^{(1)}$ once again. Specifically, using the time decay estimates for $\psi^{(1)}$ and that it generically satisfies $I_0[\psi^{(1)}]\neq 0$ we derive the asymptotics of $T(r\psi^{(1)})-T(r\psi^{(1)})|_{\gamma^{\mathcal{I}}}=r\psi-r\psi|_{\gamma^{\mathcal{I}}}$ in $\mathcal{A}^{\mathcal{I}}_{\gamma^{\mathcal{I}}}$ in terms of $I_0[\psi^{(1)}]$. Combined with the above asymptotics for $r\psi|_{\gamma^{\mathcal{I}}}$ we obtain the asymptotics of $r\psi$ in $\mathcal{A}^{\mathcal{I}}_{\gamma^{\mathcal{I}}}$. Note that this shows that both the near-horizon region and the near-infinity region contribute to the asymptotics for the radiation field $r\psi|_{{\mathcal{I}^{+}}}$. This completes the derivation of the asymptotics for $\psi$ everywhere in $\mathcal{R}$. See Section \[sec:asympzeroconst\].
**
In the case of Type **B** initial data the time integral $\psi^{(1)}$ extends smoothly to the horizon, so we can apply the same procedure as for Type **C** data to $\psi^{(1)}$ to derive the global late-time asymptotics of $\psi^{(1)}$ and of $T\psi^{(1)}=\psi$. See Section \[sec:AsymptoticsForTypeDPerturbations\].
**
A modified variant of the methods for Type **A** data can be applied for initial data of Type **D**. In this case $\partial_{\rho}\psi$ decays faster than $\psi$ itself. In order to obtain the asymptotics for $\partial_{\rho}\psi$ we need to first obtain the asymptotics for the weighted derivative $\partial_{\rho}\big((r-M)\psi\big)$, which in fact decays as fast as $\psi$, by starting from null infinity and propagating up to $\gamma^{\mathcal{H}}$. Once we obtain the asymptotics for $\psi$ and its time derivatives then a posteriori we obtain the asymptotics for $\partial_{\rho}\psi$. See Section \[sec:AsymptoticsForTypeDPerturbations\].
The ${\mathcal{H}^{+}}-$localized and ${\mathcal{I}^{+}}-$localized hierarchies {#sec:rweightest}
===============================================================================
In this section we will derive the main hierarchies of commuted $r^p$-weighted estimates near $\mathcal{I}^+$ and the analogous commuted “$(r-M)^{-p}$-weighted” estimates $\mathcal{H}^+$ that lie at the heart of the energy and pointwise decay estimates in the subsequent sections.
The commutator vector fields and basic estimates {#sec:maineq}
------------------------------------------------
We define the main quantities obtained from $\psi$ for which we will derive $r$-weighted estimates.
\[def:horadfields\] We introduce the following *higher-order radiation fields*: let $n\in {\mathbb{N}}_0$ and let $\phi=r\cdot \psi$, with $\psi$ a solution to , then define $$\begin{aligned}
\Phi_{(n)}:=&\:(2D^{-1}r^2L)^n\phi,\\
\widetilde{\Phi}_{(1)}:=&\:2r(r-M)D^{-1}L\phi,\\
\underline{\Phi}_{(n)}:=&\:(2D^{-1}r^2\underline{L})^n\phi,\\
\widetilde{\underline{\Phi}}_{(1)}:=&\:2rD^{-1}\underline{L}\phi.\end{aligned}$$ Denote moreover $\widetilde{\underline{\Phi}}_{(0)}:=\phi$ and $\widetilde{\Phi}_{(0)}:=\phi$.
The lemma below provides the equations for the higher-order radiation fields that are central in deriving the $r$-weighted estimates in a neighbourhood of $\mathcal{H}^+$ and $\mathcal{I}^+$.
\[lm:maineq\] Let $\psi$ be a smooth solution to , then for all $n\in {\mathbb{N}}_0$, we have that $$\begin{aligned}
\label{eq:maincommeqI}
4L\underline{L}\Phi_{(n)}=&\:Dr^{-2}\slashed{\Delta}_{{\mathbb{S}}^2}\Phi_{(n)}+[-4n r^{-1}+O(r^{-2})] L\Phi_{(n)}+ D\left[n(n+1)r^{-2} +O(r^{-3})\right]\Phi_{(n)}\\ \nonumber
&+n\sum_{k=0}^{\max\{0,n-1\}} O(r^{-2}) \Phi_{(k)},\\
\label{eq:maincommeqH}
4\underline{L}L\underline{\Phi}_{(n)}=&\:Dr^{-2}\slashed{\Delta}_{{\mathbb{S}}^2}\underline{\Phi}_{(n)}+[-4M^{-2}n (r-M)+O((r-M)^{2})] \underline{L}\underline{\Phi}_{(n)}\\ \nonumber
&+ \left[n(n+1)M^{-4}(r-M)^{2} +O((r-M)^{3})\right]\underline{\Phi}_{(n)}\\ \nonumber
&+n\sum_{k=0}^{\max\{0,n-1\}} O((r-M)^{2}) \underline{\Phi}_{(k)}.
\end{aligned}$$ Furthermore, $$\begin{aligned}
\label{eq:maincommeqL1I}
4L\underline{L}\widetilde{\Phi}_{(1)}=&\:Dr^{-2}\slashed{\Delta}_{{\mathbb{S}}^2}\widetilde{\Phi}_{(1)}+[-4 r^{-1}+O(r^{-2})] L\widetilde{\Phi}_{(1)}+ D\left[2r^{-2} +O(r^{-3})\right]\widetilde{\Phi}_{(1)}\\ \nonumber
&+[2Mr^{-2}+O(r^{-3})]\phi+[Mr^{-2}+O(r^{-3})] \slashed{\Delta}_{{\mathbb{S}}^2}\phi,\\
\label{eq:maincommeqL1H}
4\underline{L}L\widetilde{\underline{\Phi}}_{(1)}=&\:Dr^{-2}\slashed{\Delta}_{{\mathbb{S}}^2}\widetilde{\underline{\Phi}}_{(1)}+[-4M^{-2} (r-M)+O((r-M)^{2})] \underline{L}\widetilde{\underline{\Phi}}_{(1)}\\ \nonumber
&+ \left[2M^{-4}(r-M)^{2} +O((r-M)^{3})\right]\widetilde{\underline{\Phi}}_{(1)}\\ \nonumber
&+[-2M^{-2}(r-M)^2+O((r-M)^{3})]\phi+[-M^{-2}(r-M)^2+O((r-M)^{3})] \slashed{\Delta}_{{\mathbb{S}}^2}\phi.
\end{aligned}$$
We will first derive and inductively. In all equations, the $n=0$ case follows directly from rewriting as the following equation for $\phi$: $$\label{eq:eqradield}
4\underline{L}L\phi=Dr^2\slashed{\Delta}_{{\mathbb{S}}^2}\phi-\frac{DD'}{r}\phi.$$ See for example [@paper1] for a derivation of in a more general setting. Now, as the inductive step, let us suppose that and hold for some $n=N \in {\mathbb{N}}_0$. We will show that they then must also hold for $n=N+1$.
First, note that $$\begin{split}
2\underline{L} \Phi_{(N+1)}=&\:4\underline{L}(D^{-1}r^2 L\Phi_{(N)})\\
=&\:[-4r+O(r^0)]L\Phi_{(n)}+4D^{-1}r^2 \underline{L}L\underline{\Phi}_{(N)}\\
=&\:[-2Dr^{-1}+O(r^{-2})]\Phi_{(N+1)}+4D^{-1}r^2 \underline{L}L\Phi_{(N)}.
\end{split}$$ We apply with $n=N$ to arrive at $$\begin{split}
2\underline{L} \Phi_{(N+1)}=&\:\slashed{\Delta}_{{\mathbb{S}}^2}\Phi_{(N)}+[-2(N+1) r^{-1}+O(r^{-2})] \Phi_{(N+1)}\\
&+ \left[N(N+1) +O(r^{-1})\right]\Phi_{(N)}+N\sum_{k=0}^{\max\{0,N-1\}} O(r^0) \Phi_{(k)}.
\end{split}$$ Now, we differentiate the above equation in the $L$ direction to obtain: $$\begin{split}
4L\underline{L} \Phi_{(N+1)}=&\:Dr^{-2}\slashed{\Delta}_{{\mathbb{S}}^2}\Phi_{(N+1)}+[-4(N+1) r^{-1}+O(r^{-2})] L\Phi_{(N+1)}\\
&+ D\left[(N(N+1)+2(N+1))r^{-2} +O(r^{-3})\right]\Phi_{(N+1)}+\sum_{k=0}^{N} O(r^{-2}) \Phi_{(k)},
\end{split}$$ which implies that also holds for $n=N+1$.
Note also that $$\begin{split}
2L\underline{\Phi}_{(N+1)}=&\:4L(D^{-1}r^2 \underline{L}\underline{\Phi}_{(N)})\\
=&\:[-2D^{-1}D'r^2+O(r^0)]\underline{L}\underline{\Phi}_{(N)}+4D^{-1}r^2L\underline{L}\underline{\Phi}_{(N)}\\
=&\:[-D'+O((r-M)^2)]\underline{\Phi}_{(N+1)}+4D^{-1}r^2L \underline{L}\underline{\Phi}_{(N)}
\end{split}$$ We apply with $n=N$ to arrive at $$\begin{split}
2L\underline{\Phi}_{(N+1)}=&\:\slashed{\Delta}_{{\mathbb{S}}^2}\underline{\Phi}_{(N)}+[-2M^{-2}(N+1) (r-M)+O((r-M)^{2})]\underline{\Phi}_{(N+1)}\\
&+ \left[N(N+1) +O(r-M)\right]\underline{\Phi}_{(N)}\\
&+N\sum_{k=0}^{\max\{0,N-1\}} O(r^0) \underline{\Phi}_{(k)}.
\end{split}$$ We differentiate the above equation in the $\underline{L}$ direction to obtain: $$\begin{split}
4\underline{L}L\underline{\Phi}_{(N+1)}=&\:Dr^{-2}\slashed{\Delta}_{{\mathbb{S}}^2}\underline{\Phi}_{(N+1)}+[-4M^{-2}(N+1) (r-M)+O((r-M)^{2})] \underline{L}\underline{\Phi}_{(N+1)}\\
&+ \left[(N(N+1)+2(N+1))M^{-4}(r-M)^2 +O((r-M)^3)\right]\underline{\Phi}_{(N+1)}\\
&+\sum_{k=0}^{N} O((r-M)^2) \underline{\Phi}_{(k)},
\end{split}$$ which concludes the proof of .
We are left with deriving and . We will derive these equations using similar arguments to those above. We have that $$\begin{split}
2\underline{L} \widetilde{\Phi}_{(1)}=&\:4\underline{L}(D^{-1}(r^2-Mr) L\phi)\\
=&\:[-4r+O(r^0)]L\phi+4D^{-1}(r^2-Mr) \underline{L}L\phi\\
=&\:[-2Dr^{-1}+O(r^{-2})]\widetilde{\Phi}_{(1)}+4D^{-1}(r^2-Mr) \underline{L}L\phi.
\end{split}$$ By applying we therefore obtain the following equation for $ \widetilde{\Phi}_{(1)}$: $$2\underline{L} \widetilde{\Phi}_{(1)}=[-2Dr^{-1}+O(r^{-2})]\widetilde{\Phi}_{(1)}+(1-Mr^{-1})\slashed{\Delta}_{{\mathbb{S}}^2}\phi-D'(r-M)\phi.$$ By taking the $2L$ derivative of both sides we obtain: $$\begin{split}
4L\underline{L} \widetilde{\Phi}_{(1)}=&\:Dr^2\slashed{\Delta}_{{\mathbb{S}}^2}\widetilde{\Phi}_{(1)}+MDr^{-2}\slashed{\Delta}_{{\mathbb{S}}^2}\phi+[-4Dr^{-1}+O(r^{-2})]L\widetilde{\Phi}_{(1)}+D[2r^{-2}+O(r^{-3})]\widetilde{\Phi}_{(1)}\\
&+[2Mr^{-2}+O(r^{-3})]\phi,
\end{split}$$ which gives .
Similarly, we have that $$\begin{split}
2L \widetilde{\underline{\Phi}}_{(1)}=&\:4L(D^{-1}r^3 \underline{L}\phi)\\
=&\:[-2D^{-1}D'r^3+O(r^0)]\underline{L}\phi+4D^{-1}r^3L\underline{L}\phi\\
=&\:[-D'+O((r-M)^2)]\widetilde{\underline{\Phi}}_{(1)}+4D^{-1}r^3L \underline{L}\phi.
\end{split}$$ We subsequently apply to arrive at $$\begin{split}
2L \widetilde{\underline{\Phi}}_{(1)}=&\:r\slashed{\Delta}_{{\mathbb{S}}^2}\phi+[-D'r^{-1}+O((r-M)^2)]\widetilde{\underline{\Phi}}_{(1)}-D'r^2\phi.
\end{split}$$ and hence, taking the $2\underline{L}$ derivative on both sides leads to: $$\begin{split}
4\underline{L}L \widetilde{\underline{\Phi}}_{(1)}=&\:Dr^2\slashed{\Delta}_{{\mathbb{S}}^2}\widetilde{\underline{\Phi}}_{(1)}-D\slashed{\Delta}_{{\mathbb{S}}^2}\phi+[-4M^{-2}(r-M)+O((r-M)^2)]\underline{L}\widetilde{\underline{\Phi}}_{(1)}\\
&+[2M^{-4}(r-M)^2+O((r-M)^3]\widetilde{\underline{\Phi}}_{(1)}+[DD''r^2+O((r-M)^3)]\phi\\
=&\:Dr^2\slashed{\Delta}_{{\mathbb{S}}^2}\widetilde{\underline{\Phi}}_{(1)}+[-M^{-2}(r-M)^2+O((r-M)^3)]\slashed{\Delta}_{{\mathbb{S}}^2}\phi+[-4M^{-2}(r-M)+O((r-M)^2)]\underline{L}\widetilde{\underline{\Phi}}_{(1)}\\
&+[2M^{-4}(r-M)^2+O((r-M)^3]\widetilde{\underline{\Phi}}_{(1)}+[-2M^{-2}(r-M)^2+O((r-M)^3)]\phi,
\end{split}$$ which gives .
In the following proposition, we establish finiteness of certain limits of the higher-order radiation fields $\Phi_{(n)}$ at $\mathcal{I}^+$.
\[prop:radfieldsinf\] Let $n\in {\mathbb{N}}$ and assume that $$\int_{\Sigma_0}J^T[\psi]\cdot \mathbf{n}_{\Sigma_0}\,d\mu_{\Sigma_0}<\infty.$$
- If we assume that $$\sum_{0\leq k\leq n}\left[\int_{{\mathbb{S}}^2} |\slashed{\Delta}_{{\mathbb{S}}^2}^{n-k} \Phi_{(k)}|^2|_{{N}^{\mathcal{I}}}\,d\omega\right](v)<\infty$$ then for all $u\geq u_0$, we have that there exists a constant $C_u=C_u(M,N,{\Sigma_0},u)>0$, such that $$\label{eq:udepradfieldsinft}
\sum_{0\leq k\leq n}\sup_{u_0\leq u'\leq u}\left[\int_{{\mathbb{S}}^2} |\slashed{\Delta}_{{\mathbb{S}}^2}^{n-k} \Phi_{(k)}|^2|_{{N}^{\mathcal{I}}}\,d\omega\right](v)<C_u\sum_{0\leq k\leq n}\left[\int_{{\mathbb{S}}^2} |\slashed{\Delta}_{{\mathbb{S}}^2}^{n-k} \Phi_{(k)}|^2|_{{N}^{\mathcal{I}}}\,d\omega\right](v).$$
- If we make the stronger assumption that for all $0\leq k \leq n$: $$\lim_{v\to \infty}\left[\int_{{\mathbb{S}}^2} |\slashed{\Delta}_{{\mathbb{S}}^2}^{n-k} \Phi_{(k)}|^2|_{{N}^{\mathcal{I}}}\,d\omega\right](v)<\infty,$$ then, for all $u\geq u_0$ and $0\leq k \leq n$, we have that $$\label{eq:limitradfieldsinft}
\lim_{v\to \infty}\left[\int_{{\mathbb{S}}^2} |\slashed{\Delta}_{{\mathbb{S}}^2}^{n-k} \Phi_{(k)}|^2|_{{N}_{u}^I}\,d\omega\right](v)<\infty.$$
We will prove inductively that holds. The $n=0$ case follows directly from Proposition 3.4 of [@paper1], so we will omit the derivation here. Let $N\in {\mathbb{N}}_0$ and suppose holds for all $0\leq n \leq N$. Then, by applying the fundamental theorem of calculus together with , we have that $$\label{eq:transpest}
\begin{split}
\Phi_{(N+1)}(u,v,\theta,\varphi)=&\:\Phi_{(N+1)}(u_0,v,\theta,\varphi)+\int_{u_0}^u \underline{L}(2r^2D L\Phi_{(N)})(u',v,\theta,\varphi)\,du\\
=&\:\Phi_{(N+1)}(u_0,v,\theta,\varphi)\\
&+\int_{u_0}^u \left[O(r^{-1})\Phi_{(N+1)}+O(r^0)\slashed{\Delta}_{{\mathbb{S}}^2}\Phi_{(N)}+O(r^0)\sum_{k=0}^N \Phi_{(k)}\right](u',v,\theta,\varphi)\,du
\end{split}$$ By applying a Grönwall inequality in $u$, we can therefore estimate $$\sup_{u_0\leq u'\leq u}|\Phi_{(N+1)}|^2(u',v,\theta,\varphi)\leq C(u)\left( |\Phi_{(N+1)}(u_0,v,\theta,\varphi)|^2+ \sup_{u_0\leq u' \leq u}\left[|\slashed{\Delta}_{{\mathbb{S}}^2}\Phi_{(N)}|^2+ \sum_{k=0}^N |\Phi_{(k)}|^2\right](u',v,\theta,\varphi)\right).$$ The above equation integrated over ${\mathbb{S}}^2$, together with , gives the following estimate: $$\begin{split}
\Bigg|&\int_{{\mathbb{S}}^2}|\Phi_{(N+1)}|^2(u,v,\theta,\varphi)d\omega-\int_{{\mathbb{S}}^2}|\Phi_{(N+1)}|^2(u_0,v,\theta,\varphi)d\omega\\
&-\int_{{\mathbb{S}}^2}\left(\int_{u_0}^u \left[O(r^0)\slashed{\Delta}_{{\mathbb{S}}^2}\Phi_{(N)}+O(r^0)\sum_{k=0}^N \Phi_{(k)}\right](u',v,\theta,\varphi)\,du\right)^2\,d\omega\Bigg|\\
\leq& C(u) r^{-1}(u,v) \sup_{u_0\leq u' \leq u}\int_{{\mathbb{S}}^2}|\Phi_{(N+1)}(u_0,v,\theta,\varphi)|^2+ |\slashed{\Delta}_{{\mathbb{S}}^2}\Phi_{(N)}|^2(u',v,\theta,\varphi)+ \sum_{k=0}^N |\Phi_{(k)}|^2(u',v,\theta,\varphi)\,d\omega.
\end{split}$$ By the inductive step, together with the additional fact that the equation for the inductive step immediately holds for $\Phi_{(N)}$ replaced by $\slashed{\Delta}_{{\mathbb{S}}^2}\Phi_{(N)}$, since $[\slashed{\Delta}_{{\mathbb{S}}^2},\square_g]=0$, we can infer that holds and morever, the right-hand side of the equation above goes to zero as $v\to \infty$ and therefore, using once more the inductive step (commuted with $\slashed{\Delta}_{{\mathbb{S}}^2}$), we conclude that $$\lim_{v\to \infty}\int_{{\mathbb{S}}^2}|\Phi_{(N+1)}|^2(u,v,\theta,\varphi)\,d\omega<\infty,$$ which allows us to obtain .
Since $\widetilde{\Phi}_{(1)}=(1+Mr^{-1})\Phi_{(1)}$, and with $n=1$ automatically hold when $\Phi_{(1)}$ is replaced by $\widetilde{\Phi}_{(1)}$.
The key identities {#sec:mainest1}
------------------
In order to establish the $r^p$- and $(r-M)^{-p}$-weighted estimates below, we first derive the following $r^p$- and $(r-M)^{-p}$-weighted *identities* in the following **key lemma**:
\[lm:rweightidentitiess2v1\] Let $p\in {\mathbb{R}}$. Then the following identities hold for all $n\in {\mathbb{N}}$: $$\label{eq:mainidentitysphereinf}
\begin{split}
\int_{{\mathbb{S}}^2}&\underline{L}\left(r^p (L\Phi_{(n)})^2\right)\,d\omega + \frac{1}{2}\int_{{\mathbb{S}}^2}[(p+4n)r^{p-1}+O(r^{p-2})](L\Phi_{(n)})^2\,d\omega\\
&+\frac{1}{8}\int_{{\mathbb{S}}^2} (2-p)r^{p-3}D\left(|\slashed{\nabla}_{{\mathbb{S}}^2}\Phi_{(n)}|^2-n(n+1)\Phi_{(n)}^2\right)\,d\omega\\
=&\: \int_{{\mathbb{S}}^2} \frac{1}{4}L\left(n(n+1)r^{p-2}\Phi_{(n)}^2-r^{p-2}|\slashed{\nabla}_{{\mathbb{S}}^2}\Phi_{(n)}|^2\right)\,d\omega\\
+&\int_{{\mathbb{S}}^2} O(r^{p-3}) \Phi_{(n)}\cdot L\Phi_{(n)}+O(r^{p-3})\slashed{\Delta}_{{\mathbb{S}}^2}\Phi_{(n)}\cdot L\Phi_{(n)}+n\sum_{k=0}^{\max\{0,n-1\}}\int_{{\mathbb{S}}^2} O(r^{p-2}) \Phi_{(k)}\cdot L\Phi_{(n)}\,d\omega,
\end{split}$$ and $$\label{eq:mainidentityspherehor}
\begin{split}
\int_{{\mathbb{S}}^2}&{L}\left((r-M)^{-p} (\underline{L}\underline{\Phi}_{(n)})^2\right)\,d\omega \\
&+ \frac{1}{2}\int_{{\mathbb{S}}^2}(p+4n)M^{-2}[(r-M)^{1-p}+O((r-M)^{2-p})](\underline{L}\underline{\Phi}_{(n)})^2\,d\omega\\
&+\frac{1}{8}\int_{{\mathbb{S}}^2} (2-p)M^{-6}(r-M)^{3-p}\left(|\slashed{\nabla}_{{\mathbb{S}}^2}\underline{\Phi}_{(n)}|^2-n(n+1)\underline{\Phi}_{(n)}^2\right)\,d\omega\\
=&\: \int_{{\mathbb{S}}^2} \frac{1}{4}\underline{L}\left(n(n+1)M^{-4}(r-M)^{2-p}\underline{\Phi}_{(n)}^2-M^{-4}(r-M)^{2-p}|\slashed{\nabla}_{{\mathbb{S}}^2}\underline{\Phi}_{(n)}|^2\right)\,d\omega\\
+&\int_{{\mathbb{S}}^2} O((r-M)^{3-p}) \underline{\Phi}_{(n)}\cdot \underline{L}\underline{\Phi}_{(n)}+O((r-M)^{3-p})\slashed{\Delta}_{{\mathbb{S}}^2}\underline{\Phi}_{(n)}\cdot \underline{L}\underline{\Phi}_{(n)}\,d\omega\\
&+n\int_{{\mathbb{S}}^2}\sum_{k=0}^{\max\{0,n-1\}}\int_{{\mathbb{S}}^2} O((r-M)^{2-p}) \underline{\Phi}_{(k)}\cdot \underline{L}\underline{\Phi}_{(n)}\,d\omega.
\end{split}$$
Let $p\in {\mathbb{N}}$. Then we can use to obtain the following identity: $$\begin{split}
\underline{L}\left(r^p (L\Phi_{(n)})^2\right)=&-\frac{p}{2}Dr^{p-1}(L\Phi_{(n)})^2+2r^p \underline{L}L\Phi_{(n)}\cdot L\Phi_{(n)}\\
=&-\frac{p}{2}Dr^{p-1}(L\Phi_{(n)})^2+\frac{1}{2}Dr^{p-2}\slashed{\Delta}_{{\mathbb{S}}^2}\Phi_{(n)}\cdot L\Phi_{(n)}+[-2n r^{p-1}+O(r^{p-2})] (L\Phi_{(n)})^2\\
&+ \frac{1}{2}D\left[n(n+1)r^{p-2} +O(r^{p-3})\right]\Phi_{(n)}\cdot L\Phi_{(n)}+n\sum_{k=0}^{\max\{0,n-1\}} O(r^{p-2}) \Phi_{(k)}\cdot L\Phi_{(n)}.
\end{split}$$ Note that we can apply the Leibniz rule to rewrite $$\frac{1}{2}n(n+1)r^{p-2}\Phi_{(n)}\cdot L\Phi_{(n)}= L\left(\frac{1}{4}n(n+1)r^{p-2} \Phi_{(n)}^2\right)+\frac{1}{8}(2-p)n(n+1)D \Phi_{(n)}^2.$$ We similarly apply the Leibniz rule with respect to $L$ differentiation, together with integration by parts on ${\mathbb{S}}^2$ to rewrite: $$\int_{{\mathbb{S}}^2} \frac{1}{2}r^{p-2}\slashed{\Delta}_{{\mathbb{S}}^2}\Phi_{(n)}\cdot L\Phi_{(n)} \,d\omega=-\int_{{\mathbb{S}}^2} L\left(\frac{1}{4}r^{p-2}|\slashed{\nabla}_{{\mathbb{S}}^2}\Phi_{(n)}|^2\right) \,d\omega-\int_{{\mathbb{S}}^2}\frac{1}{8}(2-p)r^{p-3} D|\slashed{\nabla}_{{\mathbb{S}}^2}\Phi_{(n)}|^2\,d\omega.$$ By combining the above equations, we arrive at the following equality: $$\begin{split}
\int_{{\mathbb{S}}^2}&\underline{L}\left(r^p (L\Phi_{(n)})^2\right)\,d\omega + \frac{1}{2}\int_{{\mathbb{S}}^2}[(p+4n)r^{p-1}+O(r^{p-2})](L\Phi_{(n)})^2\,d\omega\\
&+\frac{1}{8}\int_{{\mathbb{S}}^2} (2-p)r^{p-3}D\left(|\slashed{\nabla}_{{\mathbb{S}}^2}\Phi_{(n)}|^2-n(n+1)\Phi_{(n)}^2\right)\,d\omega\\
=&\: \int_{{\mathbb{S}}^2} \frac{1}{4}L\left(n(n+1)r^{p-2}\Phi_{(n)}^2-r^{p-2}|\slashed{\nabla}_{{\mathbb{S}}^2}\Phi_{(n)}|^2\right)\,d\omega\\
+&\int_{{\mathbb{S}}^2} O(r^{p-3}) \Phi_{(n)}\cdot L\Phi_{(n)}+O(r^{p-3})\slashed{\Delta}_{{\mathbb{S}}^2}\Phi_{(n)}\cdot L\Phi_{(n)}+n\sum_{k=0}^{\max\{0,n-1\}}\int_{{\mathbb{S}}^2} O(r^{p-2}) \Phi_{(k)}\cdot L\Phi_{(n)}\,d\omega.
\end{split}$$ We conclude that holds.
In order to prove , we proceed in a very similar manner, with the weights $r^p$ replaced by $(r-M)^{-p}$ and $L$ and $\underline{L}$ interchanged. We have that: $$\begin{split}
L\left((r-M)^{-p} (\underline{L}\underline{\Phi}_{(n)})^2\right)=&-\frac{p}{2}r^{-2}(r-M)^{-p+1}(\underline{L}\underline{\Phi}_{(n)})^2+2(r-M)^{-p} \underline{L}L\underline{\Phi}_{(n)}\cdot \underline{L}\underline{\Phi}_{(n)}\\
=&-\frac{p}{2}r^{-2}(r-M)^{-p+1}(\underline{L}\underline{\Phi}_{(n)})^2+\frac{1}{2}r^{-4}(r-M)^{2-p}\slashed{\Delta}_{{\mathbb{S}}^2}\underline{\Phi}_{(n)}\cdot \underline{L}\underline{\Phi}_{(n)}\\
&+[-2M^{-2}n (r-M)^{1-p}+O((r-M)^{2-p})] (\underline{L}\underline{\Phi}_{(n)})^2\\
&+ \frac{1}{2}\left[n(n+1)M^{-4}(r-M)^{2-p} +O((r-M)^{3-p})\right]\underline{\Phi}_{(n)}\cdot \underline{L}\underline{\Phi}_{(n)}\\
&+n\sum_{k=0}^{\max\{0,n-1\}} O((r-M)^{2-p}) \underline{\Phi}_{(k)}\cdot \underline{L}\underline{\Phi}_{(n)}.
\end{split}$$ We apply the Leibniz rule with respect to $\underline{L}$ differentiation and integrate by parts on ${\mathbb{S}}^2$ analogously to above to obtain: $$\begin{split}
\int_{{\mathbb{S}}^2}&{L}\left((r-M)^{-p} (\underline{L}\underline{\Phi}_{(n)})^2\right)\,d\omega+ \frac{1}{2}\int_{{\mathbb{S}}^2}(p+4n)M^{-2}[(r-M)^{1-p}+O((r-M)^{2-p})](\underline{L}\underline{\Phi}_{(n)})^2\,d\omega \\
&+\frac{1}{8}\int_{{\mathbb{S}}^2} (2-p)M^{-6}(r-M)^{3-p}\left(|\slashed{\nabla}_{{\mathbb{S}}^2}\underline{\Phi}_{(n)}|^2-n(n+1)\underline{\Phi}_{(n)}^2\right)\,d\omega\\
=&\: \int_{{\mathbb{S}}^2} \frac{1}{4}\underline{L}\left(n(n+1)M^{-4}(r-M)^{2-p}\underline{\Phi}_{(n)}^2-M^{-4}(r-M)^{2-p}|\slashed{\nabla}_{{\mathbb{S}}^2}\underline{\Phi}_{(n)}|^2\right)\,d\omega\\
+&\int_{{\mathbb{S}}^2} O((r-M)^{3-p}) \underline{\Phi}_{(n)}\cdot \underline{L}\underline{\Phi}_{(n)}+O((r-M)^{3-p})\slashed{\Delta}_{{\mathbb{S}}^2}\underline{\Phi}_{(n)}\cdot \underline{L}\underline{\Phi}_{(n)}\,d\omega\\
&+n\int_{{\mathbb{S}}^2}\sum_{k=0}^{\max\{0,n-1\}}\int_{{\mathbb{S}}^2} O((r-M)^{2-p}) \underline{\Phi}_{(k)}\cdot \underline{L}\underline{\Phi}_{(n)}\,d\omega.
\end{split}$$
We will make use of the following orthogonal projections $$P_{\ell},P_{\leq \ell}, P_{\geq \ell}:L^2({\mathbb{S}}^2)\to L^2({\mathbb{S}}^2),$$ with $\ell\in {\mathbb{N}}_0$, which are defined as follows: let $f\in L^2({\mathbb{S}}^2)$, then $$\begin{aligned}
P_{\ell}f=&\:f_{\ell},\\
P_{\leq \ell}f=&\:\sum_{\ell'=0}^{\ell}f_{\ell'},\\
P_{\geq \ell}f=&\:\sum_{\ell'=\ell}^{\infty}f_{\ell'},\end{aligned}$$ where $f_{\ell'}$ is the $\ell'$-th angular mode. See also Appendix \[sec:HardyInequalities\].
In the lemma below, we prove similar identities for the orthogonal projection $P_1\widetilde{\Phi}_{(1)}$, but we exploit **crucial cancellations** occurring when we apply .
\[lm:rweightidentitiess2v2\] The following identities hold for all $p\in {\mathbb{R}}$: $$\begin{aligned}
\label{eq:rweightidentitiess2v2I}
\int_{{\mathbb{S}}^2}&\underline{L}\left(r^p (LP_1\widetilde{\Phi}_{(1)})^2\right)\,d\omega + \frac{1}{2}\int_{{\mathbb{S}}^2}[(p+4)r^{p-1}+O(r^{p-2})](LP_1\widetilde{\Phi}_{(1)})^2\,d\omega\\ \nonumber
=&\: \int_{{\mathbb{S}}^2} O(r^{p-3}) P_1\widetilde{\Phi}_{(1)}\cdot LP_1\widetilde{\Phi}_{(1)}+\int_{{\mathbb{S}}^2} O(r^{p-3}) P_1\phi\cdot LP_1\widetilde{\Phi}_{(1)}\,d\omega,\\
\label{eq:rweightidentitiess2v2H}
\int_{{\mathbb{S}}^2}&{L}\left((r-M)^{-p} (\underline{L}P_1\widetilde{\underline{\Phi}}_{(1)})^2\right)\,d\omega+ \frac{1}{2}\int_{{\mathbb{S}}^2}(p+4)M^{-2}[(r-M)^{1-p}+O((r-M)^{2-p})](\underline{L}P_1\widetilde{\underline{\Phi}}_{(1)})^2\,d\omega\\ \nonumber
=&\: \int_{{\mathbb{S}}^2} O((r-M)^{3-p}) P_1\widetilde{\underline{\Phi}}_{(1)}\cdot \underline{L}P_1\widetilde{\underline{\Phi}}_{(1)}\,d\omega+\int_{{\mathbb{S}}^2} O((r-M)^{3-p}) P_1\phi\cdot \underline{L}P_1\widetilde{\underline{\Phi}}_{(1)}\,d\omega.\end{aligned}$$
The proof proceeds exactly as the proof of Lemma \[lm:rweightidentitiess2v1\] with $n=1$, but we additionally use the identity together with the fact that $$\slashed{\Delta}_{{\mathbb{S}}^2}P_1\psi+2P_1\psi=0$$ to arrive at additional cancellations resulting in additional factors of $\frac{1}{r}$ and $r-M$ on the right-hand sides of the identities for $P_1\widetilde{\Phi}_{(1)}$ and $P_1\widetilde{\underline{\Phi}}_{(1)}$, respectively.
The main commuted hierarchies {#sec:mainest}
-----------------------------
We have the following
\[prop:generalrpest\] Fix $n\in {\mathbb{N}}_0$ and assume that for all $0\leq k\leq \min\{n-1,0\}$ and $0\leq j \leq n-k$ $$\label{assm:id1}
\lim_{v\to \infty}\left(\int_{{\mathbb{S}}^2}|{\slashed{\nabla}}_{{\mathbb{S}}^2}\slashed{\Delta}_{{\mathbb{S}}^2}^jP_{\geq 1}\Phi_{(k)}|^2\,d\omega\right)(u_0,v)<\infty.$$
Let $\epsilon>0$ be arbitrarily small, then there exists $r_{\mathcal{I}}>0$ sufficiently large, such that for $p\in (-4n,2]$ and for all $0\leq u_1\leq u_2$: $$\label{eq:rpestinf}
\begin{split}
(1-\epsilon)\int_{{N}^{\mathcal{I}}_{u_2}}& r^p(LP_{\geq 1}\Phi_{(n)})^2\,d\omega dv+ \frac{1}{2}(1-\epsilon)\int_{u_1}^{u_2} \int_{{N}^{\mathcal{I}}_u}(p+4n)r^{p-1}(LP_{\geq 1}\Phi_{(n)})^2\,d\omega dv du\\
&+\frac{1}{4}\int_{\mathcal{I}^+(u_1,u_2)}\left[r^{p-2} |{\slashed{\nabla}}_{{\mathbb{S}}^2}P_{\geq 1}\Phi_{(n)}|^2-n(n+1)r^{p-2}(P_{\geq 1}\Phi_{(n)})^2\right]\,d\omega du\\
&+\frac{1}{8}\int_{u_1}^{u_2} \int_{{N}^{\mathcal{I}}_u}(2-p)r^{p-3}D\left(|{\slashed{\nabla}}_{{\mathbb{S}}^2}P_{\geq 1}\Phi_{(n)}|^2-n(n+1)P_{\geq 1}\Phi_{(n)}^2\right)\,d\omega dv du\\
\leq&\: C\int_{{N}^{\mathcal{I}}_{u_1}}r^p(LP_{\geq 1}\Phi_{(n)})^2\,d\omega dv+ C\sum_{k\leq n}\int_{\Sigma_{u_1}} J^T[T^k\psi]\cdot \mathbf{n}_{u_1}\,d\mu_{\Sigma_{u_1}},
\end{split}$$ where $C=C(n,M, {\Sigma_0},r_{\mathcal{H}})>0$ is a constant and we can take $r_{\mathcal{I}}=(p+4n)^{-1}R_0(n,M)>0$.
Furthermore, there exists $r_{\mathcal{H}}>M$, with $r_{\mathcal{H}}-M$ suitably small, such that for $p\in (-4n,2]$ and for all $0\leq u_1\leq u_2$: $$\label{eq:rpesthor}
\begin{split}
(1-\epsilon)\int_{{N}^{\mathcal{H}}_{v_2}}& (r-M)^{-p}(\underline{L}P_{\geq 1}\underline{\Phi}_{(n)})^2\,d\omega du+ \frac{1}{2}(1-\epsilon)\int_{v_1}^{v_2} \int_{{N}^{\mathcal{H}}_u}(p+4n)(r-M)^{1-p}(\underline{L}P_{\geq 1}\underline{\Phi}_{(n)})^2\,d\omega du dv\\
&+\frac{1}{4}M^{-4}\int_{\mathcal{H}^+(v_1,v_2)}\left[(r-M)^{2-p} |{\slashed{\nabla}}_{{\mathbb{S}}^2}\underline{\Phi}_{(n)}|^2-n(n+1)(r-M)^{2-p}\underline{\Phi}_{(n)}^2\right]\,d\omega dv\\
&+\frac{1}{8}\int_{v_1}^{v_2} \int_{{N}^{\mathcal{H}}_v}(2-p)(r-M)^{3-p}M^{-6}\left(|{\slashed{\nabla}}_{{\mathbb{S}}^2}P_{\geq 1}\underline{\Phi}_{(n)}|^2-n(n+1)P_{\geq 1}\underline{\Phi}_{(n)}^2\right)\,d\omega du dv\\
\leq&\: C\int_{{N}^{\mathcal{H}}_{v_1}}r^p(\underline{L}P_{\geq 1}\underline{\Phi}_{(n)})^2\,d\omega du+ C\sum_{k\leq n}\int_{\Sigma_{v_1}} J^T[T^k\psi]\cdot \mathbf{n}_{v_1}\,d\mu_{\Sigma_{v_1}},
\end{split}$$ where $C=C(n,D,r_{\mathcal{H}})>0$ is a constant and we can take $(r_{\mathcal{H}}-M)^{-1}=(p+4n)^{-1}M^{-2}R_0(n,M)>0$.
Observe first of all that the assumption together with the smoothness assumption of the initial data on $\Sigma_0$ imply that $$\int_{\Sigma_0} J^T[\psi_{\geq 1}]\cdot \mathbf{n}_{\Sigma_0}\,d\mu_{\Sigma_0}<\infty.$$ We can therefore appeal to Proposition \[prop:radfieldsinf\] with regards to the limiting behaviour of $P_{\geq 1}\Phi_{(k)}$ at $\mathcal{I}^+$.
We will first derive the estimate . In all the estimates in this proof, we assume for notational convenience that $\int_{{\mathbb{S}}^2}\psi\,d\omega =0$. We introduce a smooth cut-off function $\chi: {\mathbb{R}}\to {\mathbb{R}}$ such that $\chi(r)=0$ for all $r\leq r_{\mathcal{I}}$ and $\chi(r)=1$ for all $r\geq r_{\mathcal{I}}+M$. We will choose $r_{\mathcal{I}}$ appropriately large.
We now integrate both sides of in the $u$ and $v$ directions to obtain $$\label{eq:maineqrpestinf}
\begin{split}
\int_{{N}^{\mathcal{I}}_{u_2}}& r^p(L(\chi\Phi_{(n)}))^2\,d\omega dv+ \frac{1}{2}\int_{u_1}^{u_2} \int_{{N}^{\mathcal{I}}_u}(p+4n)r^{p-1}(L(\chi\Phi_{(n)}))^2\,d\omega dv du\\
&+\frac{1}{4}\int_{\mathcal{I}^+(u_1,u_2)}\left[r^{p-2} |{\slashed{\nabla}}_{{\mathbb{S}}^2}\Phi_{(n)}|^2-n(n+1)r^{p-2}\Phi_{(n)}^2\right]\,d\omega du\\
&+\frac{1}{8}\int_{u_1}^{u_2} \int_{{N}^{\mathcal{I}}_u}(2-p)r^{p-3}D\chi^2\left(|{\slashed{\nabla}}_{{\mathbb{S}}^2}\Phi_{(n)}|^2-n(n+1)\Phi_{(n)}^2\right)\,d\omega dv du\\
=&\: \int_{{N}^{\mathcal{I}}_{u_1}} r^p(L(\chi\Phi_{(n)}))^2\,d\omega dv\\
&+J_1+J_2+J_3+\sum_{|\alpha|\leq 1} \int_{u_1}^{u_2} \int_{{N}^{\mathcal{I}}_u}r^{p-2}\underline{L} (\chi\Phi_{(n)}) \cdot R_{\chi}[\partial^{\alpha}\Phi_{(n)}]\,d\omega dudv,
\end{split}$$ where we use the notation $R_{\chi}[f]$ for terms that are compactly supported in $r_{\mathcal{I}}\leq r\leq r_{\mathcal{I}}+M$ and are linear in the function $f$, and we define $$\begin{aligned}
J_1:=&\: \int_{u_1}^{u_2} \int_{{N}^{\mathcal{I}}_u}O(r^{p-2})(L(\chi \Phi_{(n)}))^2\,d\omega du dv,\\
J_2:=&\: \int_{u_1}^{u_2} \int_{{N}^{\mathcal{I}}_u}O(r^{p-3})\chi \Phi_{(n)}\cdot L(\chi \Phi_{(n)})+ O(r^{p-3})\slashed{\Delta}_{{\mathbb{S}}^2}(\chi\Phi_{(n)})\cdot L(\chi \Phi_{(n)})\,d\omega du dv,\\
J_3:=&\: n\sum_{k=0}^{n-1}\int_{u_1}^{u_2} \int_{{N}^{\mathcal{I}}_u} O(r^{p-2}) \chi \Phi_{(k)}\cdot L(\chi \Phi_{(n)})\,d\omega dv du.\end{aligned}$$ In order to obtain we used that $r^{p-2}\chi^2\Phi_{(n)}^2$ and $r^{p-2}\chi^2\Phi_{(n)}^2$ vanish on $\{r=r_{\mathcal{I}}\}$.
First of all, by and the compactness of the support of $R_{\chi}$ it follows that there exists a constant $C(M,{\Sigma_0},r_{\mathcal{I}})>0$ such that $$\label{est:cptsuppremainder}
\sum_{|\alpha|\leq 1} \int_{u_1}^{u_2} \int_{{N}^{\mathcal{I}}_u}r^{p-2}\underline{L} (\chi\Phi_{(n)}) \cdot R_{\chi}[\partial^{\alpha}\Phi_{(n)}]\,d\omega dudv\leq C\sum_{k\leq n}\int_{\Sigma_{u_1}} J^T[T^k\psi]\cdot \mathbf{n}_{u_1}\,d\mu_{\Sigma_{u_1}}.$$ The strategy for the remainder of the proof will therefore be to absorb $J_1+J_2+J_3$ into the second term on the left-hand side of .\
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We apply Young’s inequality with weights in $\epsilon$ to estimate $$|J_1|\leq \int_{u_1}^{u_2} \int_{{N}^{\mathcal{I}}_u} \epsilon (p+4n) r^{p-1} (L(\chi \Phi_{(n)}))^2+C\epsilon^{-1}(p+4n)^{-1}r^{p-5}\chi^2 \Phi_{(n)}^2 \,d\omega du dv,$$ where we fix $\epsilon>0$ to be suitably small. We absorb the first term into the left-hand side of . We apply to further estimate $$\begin{split}
\epsilon^{-1}(p+4n)^{-1}&\int_{u_1}^{u_2} \int_{{N}^{\mathcal{I}}_u}r^{p-5}\chi^2 \Phi_{(n)}^2\,d\omega dudv \leq C\epsilon^{-1}(p+4n)^{-1} \int_{{N}^{\mathcal{I}}_u}r^{p-3} (L(\chi\Phi_{(n)}))^2\,d\omega dudv\\
\leq &\: C\epsilon^{-1}(p+4n)^{-1}r_{\mathcal{I}}^{-2}\int_{{N}^{\mathcal{I}}_u}r^{p-1} (L(\chi\Phi_{(n)}))^2\,d\omega dudv.
\end{split}$$ For $r_{\mathcal{I}}^2>0$ suitably large (depending linearly on $(p+4n)^{-1}$), we can therefore also absorb the term above in to the second integral on the left-hand side of .\
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To estimate $J_2$, we first consider $O(r^{p-3})\chi \Phi_{(n)}\cdot L(\chi \Phi_{(n)})$ and apply Young’s inequality to obtain: $$\begin{split}
\int_{u_1}^{u_2} \int_{{N}^{\mathcal{I}}_u}O(r^{p-3})\chi \Phi_{(n)}\cdot L(\chi \Phi_{(n)}) \,d\omega du dv\leq&\: \int_{u_1}^{u_2} \int_{{N}^{\mathcal{I}}_u}\epsilon (p+4n)r^{p-1}(L(\chi \Phi_{(n)})^2 \,d\omega du dv\\
&+C {\epsilon}^{-1} (p+4n)^{-1}\int_{u_1}^{u_2} \int_{{N}^{\mathcal{I}}_u} r^{p-5}\chi^2\Phi_{(n)}^2\,d\omega du dv.
\end{split}$$ The first term on the right-hand side can be absorbed into the left-hand side of and the second term on the right-hand side can be absorbed into $J_1$, as above.
In order to estimate $O(r^{p-3})\chi \slashed{\Delta}_{{\mathbb{S}}^2}\Phi_{(n)}\cdot L(\chi \Phi_{(n)})$ we first rearrange the terms in to obtain: $$\label{eq:equationforsphlaplacian}
\frac{1}{2}D \chi \slashed{\Delta}_{{\mathbb{S}}^2} \Phi_{(n)}=2r^2\underline{L}L(\chi \Phi_{(n)})+(2nr+O(r^0))L(\chi \Phi_{(n)})+\sum_{k=0}^n O(r^0) \chi \Phi_{(k)}+\sum_{|\alpha|\leq 1}R_{\chi}[\partial^{\alpha}\Phi_{(n)}],$$ so that $$\begin{split}
\left|\int_{u_1}^{u_2} \int_{{N}^{\mathcal{I}}_u}O(r^{p-3})\chi\slashed{\Delta}_{{\mathbb{S}}^2} \Phi_{(n)}\cdot L(\chi \Phi_{(n)}) \,d\omega du dv\right|\leq&\: \left|\int_{u_1}^{u_2} \int_{{N}^{\mathcal{I}}_u}O(r^{p-1}) \underline{L}L(\chi \Phi_{(n)}) \cdot L(\chi \Phi_{(n)}) \,d\omega du dv\right|\\
&+\left|\int_{u_1}^{u_2} \int_{{N}^{\mathcal{I}}_u}O(r^{p-2}) (L(\chi \Phi_{(n)}))^2 \,d\omega du dv\right|\\
&+\left|\sum_{k=0}^n\int_{u_1}^{u_2} \int_{{N}^{\mathcal{I}}_u}O(r^{p-3}) \chi\Phi_{(k)}\cdot L(\chi \Phi_{(n)}) \,d\omega du dv\right|\\
&+ C\sum_{k\leq n}\int_{\Sigma_{u_1}} J^T[T^k\psi]\cdot \mathbf{n}_{u_1}\,d\mu_{\Sigma_{u_1}}
\end{split}$$ Note that we can absorb the second integral on the right-hand side into $J_1$ and we can group the third integral with the $O(r^{p-3})\chi \Phi_{(n)}\cdot L(\chi \Phi_{(n)})$ term of $J_2$ and with $J_3$ (which we estimate below). It remains to estimate the integral of $O(r^{p-1}) \underline{L}L(\chi \Phi_{(n)}) \cdot L(\chi \Phi_{(n)})$.
We first integrate by parts in the $\underline{L}$ direction: $$\begin{split}
\int_{u_1}^{u_2}& \int_{{N}^{\mathcal{I}}_u}O(r^{p-1}) \underline{L}L(\chi \Phi_{(n)}) \cdot L(\chi \Phi_{(n)}) \,d\omega du dv= \int_{u_1}^{u_2} \int_{{N}^{\mathcal{I}}_u}\underline{L}(O(r^{p-1}) (L(\chi \Phi_{(n)}))^2) \,d\omega du dv\\
&+(p-1)\int_{u_1}^{u_2} \int_{{N}^{\mathcal{I}}_u}O(r^{p-2}) (L(\chi \Phi_{(n)}))^2 \,d\omega du dv\\
=&\: \int_{{N}^{\mathcal{I}}_{u_2}}O(r^{p-1}) (L(\chi \Phi_{(n)}))^2 \,d\omega dv- \int_{{N}^{\mathcal{I}}_{u_1}}O(r^{p-1}) (L(\chi \Phi_{(n)}))^2 \,d\omega dv\\
&+(p-1)\int_{u_1}^{u_2} \int_{{N}^{\mathcal{I}}_u}O(r^{p-2}) (L(\chi \Phi_{(n)}))^2 \,d\omega du dv.
\end{split}$$ We can absorb the third term on the very right-hand side above into $J_1$ and we can absorb the absolute values of the remaining terms into the integrals over ${N}^{\mathcal{I}}_{u_2}$ and ${N}^{\mathcal{I}}_{u_1}$ that appear in (after taking $r_{\mathcal{I}}>0$ suitably large).\
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If $n=0$, there is nothing to estimate. Suppose therefore that $n\geq 1$. **It is only in this step that we will make use of the assumption** $\int_{{\mathbb{S}}^2}\psi\,d\omega=0$. That is to say, using this assumption it follows that there exist functions $f_{(k)}$, with $0\leq k\leq n$, such that $$\slashed{\Delta}_{{\mathbb{S}}^2}f_{(k)}=\Phi_{(k)}.$$ for all $0\leq k\leq n$. We can then estimate $J_3$ as by integrating by parts twice on ${\mathbb{S}}^2$ and then applying once more to obtain: $$\begin{split}
J_3= &\: \sum_{k=0}^{n-1}\int_{u_1}^{u_2} \int_{{N}^{\mathcal{I}}_u} O(r^{p-2}) \chi \slashed{\Delta}_{{\mathbb{S}}^2}\Phi_{(k)}\cdot L(\chi f_{(n)})\,d\omega dv du \\
=&\: \sum_{k=0}^{n-1}\int_{u_1}^{u_2} \int_{{N}^{\mathcal{I}}_u} O(r^{p}) \chi \underline{L}L\Phi_{(k)}\cdot L(\chi f_{(n)})+O(r^{p-1})\chi L\Phi_{(k)}\cdot L(\chi f_{(n)})\\
&+\sum_{m=0}^{k-1} O(r^{p-2})\chi \Phi_{(m)}\cdot L(\chi f_{(n)}) \,d\omega dv du.
\end{split}$$ We integrate by parts in the $\underline{L}$ direction to obtain: $$\begin{split}
\sum_{k=0}^{n-1}\int_{u_1}^{u_2} \int_{{N}^{\mathcal{I}}_u} O(r^{p}) \chi \underline{L}L\Phi_{(k)}\cdot L(\chi f_{(n)})\,d\omega dudv=&\:\sum_{k=0}^{n-1}\int_{u_1}^{u_2} \int_{{N}^{\mathcal{I}}_u} \underline{L}\left(O(r^{p}) \chi L\Phi_{(k)}\cdot L(\chi f_{(n)})\right)\,d\omega dudv\\
&+p\sum_{k=0}^{n-1}\int_{u_1}^{u_2} \int_{{N}^{\mathcal{I}}_u} O(r^{p-1}) \chi L\Phi_{(k)}\cdot L(\chi f_{(n)})\,d\omega dudv+\ldots\\
=&\: \int_{{N}_{u_2}} O(r^{p}) \chi L\Phi_{(k)}\cdot L(\chi f_{(n)})\,d\omega dv\\
&+\int_{{N}_{u_1}} O(r^{p}) \chi L\Phi_{(k)}\cdot L(\chi f_{(n)})\,d\omega dv\\
&+p\sum_{k=0}^{n-1}\int_{u_1}^{u_2} \int_{{N}^{\mathcal{I}}_u} O(r^{p-1}) \chi L\Phi_{(k)}\cdot L(\chi f_{(n)})\,d\omega dudv+\ldots,
\end{split}$$ where for the sake of brevity we employ the schematic notation $\ldots$ to denote all integral terms that are supported in $r_{\mathcal{I}}\leq r\leq r_{\mathcal{I}}+M$.
Note that by applying with $\ell=1$ together with , we can estimate $$\label{eq:poincaref}
\int_{{\mathbb{S}}^2} f_{(n)}^2\,d\omega\leq \frac{1}{2} \int_{{\mathbb{S}}^2} |\slashed{\nabla}_{{\mathbb{S}}^2}f_{(n)}|^2\,d\omega\leq \frac{1}{4}\int_{{\mathbb{S}}^2} \Phi_{(n)}^2\,d\omega$$ and hence, $$\begin{aligned}
\left|\int_{{N}_{u}} O(r^{p}) \chi L\Phi_{(k)}\cdot L(\chi f_{(n)})\,d\omega dv\right|\leq& \:\int_{{N}_{u}} \epsilon r^{p} (L\chi\Phi_{(n)})^2 +C\epsilon^{-1}r^{p-4} \chi^2 (\Phi_{(k+1)})^2\,d\omega dv,\\
\left|\int_{u_1}^{u_2} \int_{{N}^{\mathcal{I}}_u} O(r^{p-1}) \chi L\Phi_{(k)}\cdot L(\chi f_{(n)})\,d\omega dudv \right| \leq& \:\int_{u_1}^{u_2} \int_{{N}_{u}} \epsilon r^{p-1} (L\chi\Phi_{(n)})^2 +C\epsilon^{-1}r^{p-5} \chi^2 (\Phi_{(k+1)})^2\,d\omega dv.\end{aligned}$$
By applying , with $r_1=r_{\mathcal{I}}$ suitably large and $r_2=\infty$, a number $n-k$ times to the second term on the right-hand side above and moreover (which holds by the assumption ) we can conclude that all the boundary terms appearing in vanish, so that we can further estimate for $p<3$: $$\begin{split}
\int_{{N}_{u}} \epsilon r^{p} (L\chi\Phi_{(n)})^2 +C\epsilon^{-1}r^{p-4} \chi^2 (\Phi_{(k+1)})^2\,d\omega dv \leq &\:C\epsilon\int_{{N}_{u}} r^{p} (L\chi\Phi_{(n)})^2\,d\omega dv+C\sum_{m\leq n}\int_{\Sigma_{u}} J^T[T^m\psi]\cdot n_{u}\,d\mu_{\Sigma_{u}},
\end{split}$$ where we take $u=u_1$ or $u=u_2$.
Similarly, $$\int_{u_1}^{u_2} \int_{{N}_{u}} \epsilon r^{p-1} (L\chi\Phi_{(n)})^2 +C\epsilon^{-1}r^{p-5} \chi^2 (\Phi_{(k+1)})^2\,d\omega dv \leq C\epsilon \int_{u_1}^{u_2}\int_{{N}_{u}} r^{p} (L\chi\Phi_{(n)})^2\,d\omega dvdu +\ldots.$$
Putting the above estimates together, we therefore obtain: $$\begin{split}
|J_3|\leq &\: C\int_{u_1}^{u_2} \int_{{N}^{\mathcal{I}}_u}\epsilon r^{p-1}(L\chi\Phi_{(n)})^2+C\epsilon\int_{{N}_{u_2}} r^{p} (L\chi\Phi_{(n)})^2\,d\omega dv+C\epsilon\int_{{N}_{u_1}} r^{p} (L\chi\Phi_{(n)})^2\,d\omega dv\\
&+C\sum_{k\leq n}\int_{\Sigma_{u_1}} J^T[T^k\psi]\cdot \mathbf{n}_{u_1}\,d\mu_{\Sigma_{u_1}}.
\end{split}$$ The first integral on the right-hand side can be absorbed into $J_1$ and the second integral on the right-hand side of the above equation can be absorbed into the left-hand side of .
Hence, we arrive at with $\Phi_{(n)}$ replaced by $\chi \Phi_{(n)}$. In order to remove the cut-off function $\chi$ on the right-hand side of , we estimate: $$\begin{split}
\int_{{N}^{\mathcal{I}}_{u_1}} r^{p} (L\chi\Phi_{(n)})^2\,d\omega dv \leq&\: C\int_{{N}^{\mathcal{I}}_{u_1}} r^{p} (L\Phi_{(n)})^2\,d\omega dv +C \int_{{N}^{\mathcal{I}}_{u_1}\cap\{r_{\mathcal{I}}\leq r\leq r_{\mathcal{I}}+M\}} \Phi_{(n)}^2\,d\omega dv\\
\leq&\: C\int_{{N}^{\mathcal{I}}_{u_1}} r^{p} (L\Phi_{(n)})^2\,d\omega dv +C \sum_{k\leq n-1} \int_{{N}^{\mathcal{I}}_{u_1}\cap\{r_{\mathcal{I}}\leq r\leq r_{\mathcal{I}}+M\}} J^T[T^k\psi]\cdot L \,d\omega dv,
\end{split}$$ where we applied a together with and a standard elliptic estimate on ${N}_{u_1}$ to arrive at the second inequality. We can similarly estimate $$\begin{split}
\int_{{N}^{\mathcal{I}}_{u_2}} r^{p} (L\Phi_{(n)})^2\,d\omega dv \leq&\: C\int_{{N}^{\mathcal{I}}_{u_2}\cap\{r\geq r_{\mathcal{I}}+M\}} r^{p} (L\chi \Phi_{(n)})^2\,d\omega dv +C \int_{{N}^{\mathcal{I}}_{u_2}\cap\{r_{\mathcal{I}}\leq r\leq r_{\mathcal{I}}+M\}} (L\Phi_{(n)})^2\,d\omega dv\\
\leq&\: C\int_{{N}^{\mathcal{I}}_{u_2}} r^{p} (L\chi \Phi_{(n)})^2\,d\omega dv +C \sum_{k\leq n} \int_{{N}^{\mathcal{I}}_{u_2}} J^T[T^k\psi]\cdot L \,d\omega dv\\
\leq&\: C\int_{{N}^{\mathcal{I}}_{u_2}} r^{p} (L\chi \Phi_{(n)})^2\,d\omega dv +C \sum_{k\leq n} \int_{\Sigma_{u_1}} J^T[T^k\psi]\cdot \mathbf{n}_{u_1} \,d\mu_{\Sigma_{u_1}}
\end{split}$$ and, by applying moreover , we also obtain $$\begin{split}
\int_{u_1}^{u_2}\int_{{N}^{\mathcal{I}}_{u}} r^{p-1} (L\Phi_{(n)})^2\,d\omega dvdu \leq&\: C\int_{u_1}^{u_2}\int_{{N}^{\mathcal{I}}_{u}\cap\{r\geq r_{\mathcal{I}}+M\}} r^{p-1} (L\chi \Phi_{(n)})^2\,d\omega dvdu \\
&+C \int_{u_1}^{u_2}\int_{{N}^{\mathcal{I}}_{u}\cap\{r_{\mathcal{I}}\leq r\leq r_{\mathcal{I}}+M\}} (L\Phi_{(n)})^2\,d\omega dvdu\\
\leq&\:C\int_{u_1}^{u_2}\int_{{N}^{\mathcal{I}}_{u}} r^{p-1} (L\chi \Phi_{(n)})^2\,d\omega dvdu +C \sum_{k\leq n} \int_{\Sigma_{u_1}} J^T[T^k\psi]\cdot \mathbf{n}_{u_1} \,d\mu_{\Sigma_{u_1}}.
\end{split}$$
In order to derive we introduce a different smooth cut-off function $\chi: {\mathbb{R}}\to {\mathbb{R}}$ (we use the same notation for this cut-off function for the sake of convenience) such that $\chi(r)=0$ for all $r\geq r_{\mathcal{H}}$ and $\chi(r)=1$ for all $M\leq r\leq M+\frac{1}{2}(r_{\mathcal{H}}-M)$, with $r_{\mathcal{H}}<2M$ and $r_{\mathcal{H}}-M$ appropriately small. We integrate both sides of in the $u$ and $v$ directions to obtain: $$\label{eq:maineqrpesthor}
\begin{split}
\int_{{N}^{\mathcal{H}}_{v_2}}& (r-M)^{-p}(\underline{L}(\chi\underline{\Phi}_{(n)}))^2\,d\omega du+ \frac{1}{2}M^{-2}\int_{v_1}^{v_2} \int_{{N}^{\mathcal{H}}_v}(p+4n)(r-M)^{1-p}(L(\chi\underline{\Phi}_{(n)}))^2\,d\omega du dv\\
&+\frac{1}{4}M^{-4}\int_{\mathcal{H}^+(v_1,v_2)}\left[(r-M)^{2-p} |{\slashed{\nabla}}_{{\mathbb{S}}^2}\underline{\Phi}_{(n)}|^2-n(n+1)(r-M)^{2-p}\underline{\Phi}_{(n)}^2\right]\,d\omega dv\\
&+\frac{1}{8}M^{-6}\int_{v_1}^{v_2} \int_{{N}^{\mathcal{H}}_v}(2-p)(r-M)^{3-p}\chi^2\left(|{\slashed{\nabla}}_{{\mathbb{S}}^2}\underline{\Phi}_{(n)}|^2-n(n+1)\underline{\Phi}_{(n)}^2\right)\,d\omega du dv\\
=&\: \int_{{N}^{\mathcal{H}}_{v_1}} (r-M)^{-p}(L(\chi\underline{\Phi}_{(n)}))^2\,d\omega du\\
&+J_1+J_2+J_3+\sum_{|\alpha|\leq 1} \int_{v_1}^{v_2} \int_{{N}^{\mathcal{H}}_u}r^{p-2}\underline{L} (\chi\underline{\Phi}_{(n)}) \cdot R_{\chi}[\partial^{\alpha}\underline{\Phi}_{(n)}]\,d\omega dudv,
\end{split}$$ where we now use the notation $R_{\chi}[f]$ for terms that are compactly supported in $M+\frac{1}{2}(r_{\mathcal{H}}-M)\leq r\leq r_{\mathcal{H}}$ and are linear in the function $f$, and we define $$\begin{aligned}
\underline{J}_1:=&\: \int_{v_1}^{v_2} \int_{{N}^{\mathcal{H}}_v}O((r-M)^{2-p})(\underline{L}(\chi \underline{\Phi}_{(n)}))^2\,d\omega dv du,\\
\underline{J}_2:=&\: \int_{v_1}^{v_2} \int_{{N}^{\mathcal{H}}_v}O((r-M)^{3-p})\chi \underline{\Phi}_{(n)}\cdot \underline{L}(\chi \underline{\Phi}_{(n)})+ O((r-M)^{3-p})\slashed{\Delta}_{{\mathbb{S}}^2}(\chi\underline{\Phi}_{(n)})\cdot \underline{L}(\chi \underline{\Phi}_{(n)})\,d\omega dv du,\\
\underline{J}_3:=&\: n\sum_{k=0}^{n-1}\int_{v_1}^{v_2} \int_{{N}^{\mathcal{H}}_v} O((r-M)^{2-p}) \chi \underline{\Phi}_{(k)}\cdot \underline{L}(\chi \underline{\Phi}_{(n)})\,d\omega dv du.\end{aligned}$$ We can absorb $\underline{J}_1+\underline{J}_2+\underline{J}_3$ into the left-hand side of by repeating the estimates in $\mathcal{A}^{\mathcal{I}}$ above in the region $\mathcal{A}^{\mathcal{H}}$, using instead of . Note that in the $\mathcal{A}^{\mathcal{H}}$ case there is no need to apply as the analogous estimate at $\mathcal{H}^+$ follows immediately from the smoothness of $\psi$ at $\mathcal{H}^+$ (as we consider smooth initial data).
The third and fourth integrals on the right-hand sides of and have a **positive sign** if we consider $P_{\geq n}\psi$ rather than the more general $P_{\geq 1}\psi$. This follows directly from .
The improved hierarchies for $\ell =0$ and $\ell=1$ {#sec:mainestextended}
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The next proposition yields improved hierarchies for the harmonic mode numbers $\ell=0$ and $\ell=1$.
\[prop:rpestell01\] Fix $n\in \{0,1\}$ and assume that on ${N}^{\mathcal{I}}$: $$\label{asm:radfieldl1}
\lim_{v\to \infty}\left(\int_{{\mathbb{S}}^2}\phi_1^2\,d\omega\right)(u_0,v)<\infty.$$
Then there exists an $r_{\mathcal{I}}>0$, such that for $p\in (-4n,4)$ and for all $0\leq u_1\leq u_2$: $$\label{eq:rpestinftilde}
\begin{split}
\int_{{N}^{\mathcal{I}}_{u_2}}& r^p(LP_{n}\widetilde{\Phi}_{(n)})^2\,d\omega dv+ \frac{1}{2}\int_{u_1}^{u_2} \int_{{N}^{\mathcal{I}}_u}(p+4n)r^{p-1}(LP_{n}\widetilde{\Phi}_{(n)})^2\,d\omega dv du\\
\leq&\: C\int_{{N}^{\mathcal{I}}_{u_1}}r^p(LP_{n}\widetilde{\Phi}_{(n)})^2\,d\omega dv+ C\sum_{k\leq n}\int_{\Sigma_{u_1}} J^T[T^k\psi]\cdot \mathbf{n}_{u_1}\,d\mu_{\Sigma_{u_1}},
\end{split}$$ where $C=C(M,{\Sigma_0},r_{\mathcal{I}})>0$ is a constant and we can take $r_{\mathcal{I}}=(p+4n)^{-1}R_0(n,M)>0$.
Furthermore, there exists an $r_{\mathcal{H}}>M$, such that for $p\in (-4n,4)$ and for all $0\leq u_1\leq u_2$: $$\label{eq:rpeshortilde}
\begin{split}
\int_{{N}^{\mathcal{H}}_{v_2}}& (r-M)^{-p}(\underline{L}P_n\widetilde{\underline{\Phi}}_{(n)})^2\,d\omega du+ \frac{1}{2}\int_{v_1}^{v_2} \int_{{N}^{\mathcal{H}}_u}(p+4n)(r-M)^{1-p}(\underline{L}P_n\widetilde{\underline{\Phi}}_{(n)})^2\,d\omega du dv\\
\leq&\: C\int_{{N}^{\mathcal{H}}_{v_1}}(r-M)^{-p}(\underline{L}P_n\widetilde{\underline{\Phi}}_{(n)})^2\,d\omega du+ C\sum_{k\leq n}\int_{\Sigma_{v_1}} J^T[T^k\psi]\cdot \mathbf{n}_{v_1}\,d\mu_{\Sigma_{v_1}},
\end{split}$$ where $C=C(M,{\Sigma_0},r_{\mathcal{H}})>0$ is a constant and we can take $(r_{\mathcal{H}}-M)^{-1}=(p+4n)^{-1}R_0(n,M)>0$.
If we additionally assume that there exist constants $\eta>0$ and $\mathcal{E}_{\eta}>0$ such that $$\begin{aligned}
\label{addasmforp5I}
\sum_{k\leq n}\int_{{\mathbb{S}}^2}|P_n\widetilde{\Phi}_{(k)}|^2\,d\omega\lesssim \mathcal{E}_{\eta}\cdot (1+u)^{-2-\eta}\quad \textnormal{in $\mathcal{A}^{\mathcal{I}}$},\\
\label{addasmforp5H}
\sum_{k\leq n}\int_{{\mathbb{S}}^2}|P_n\widetilde{\underline{\Phi}}_{(k)}|^2\,d\omega\lesssim \mathcal{E}_{\eta} \cdot (1+v)^{-2-\eta}\quad \textnormal{in $\mathcal{A}^{\mathcal{H}}$},\end{aligned}$$ then we can obtain for $0<p<5$: $$\label{eq:rpestinfp5}
\begin{split}
\int_{{N}^{\mathcal{I}}_{u_2}}& r^p(LP_n\widetilde{\Phi}_{(n)})^2\,d\omega dv+ \frac{1}{2}\int_{u_1}^{u_2} \int_{{N}^{\mathcal{I}}_u}(p+4n)r^{p-1}(LP_n\widetilde{\Phi}_{(n)})^2\,d\omega dv du\\
\leq&\: C\int_{{N}^{\mathcal{I}}_{u_1}}r^p(LP_n\widetilde{\Phi}_{(n)})^2\,d\omega dv+ C\sum_{k\leq n}\int_{\Sigma_{u_1}} J^T[T^k\psi]\cdot \mathbf{n}_{u_1}\,d\mu_{\Sigma_{u_1}}+C(p-5)^{-1}\mathcal{E}_{\eta},
\end{split}$$ and $$\label{eq:rpeshorp5k}
\begin{split}
\int_{{N}^{\mathcal{H}}_{v_2}}& (r-M)^{-p}(\underline{L}P_n\widetilde{\underline{\Phi}}_{(n)})^2\,d\omega du+ \frac{1}{2}\int_{v_1}^{v_2} \int_{{N}^{\mathcal{H}}_u}(p+4n)(r-M)^{1-p}(\underline{L}P_n\widetilde{\underline{\Phi}}_{(n)})^2\,d\omega du dv\\
\leq&\: C\int_{{N}^{\mathcal{H}}_{v_1}}(r-M)^{-p}(\underline{L}P_n\widetilde{\underline{\Phi}}_{(n)})^2\,d\omega du+ C\sum_{k\leq n}\int_{\Sigma_{v_1}} J^T[T^k\psi]\cdot \mathbf{n}_{v_1}\,d\mu_{\Sigma_{v_1}}+C(p-5)^{-1}\mathcal{E}_{\eta}.
\end{split}$$
For $p<4$, the proof proceeds in a similar manner to the proof of Proposition \[prop:generalrpest\], using in the $n=1$ case the identities in Lemma \[lm:rweightidentitiess2v2\] rather than those in Lemma \[lm:rweightidentitiess2v1\]. Note in particular that due to the *lower* powers in $r$ and $(r-M)^{-1}$ appearing in the identities in Lemma \[lm:rweightidentitiess2v2\] and in Lemma \[lm:rweightidentitiess2v1\] if $n=0$ (compared to the general $n$ case), we are able to increase the range of $p$. We omit the details of these steps.
We will now show how we can extend the range of $p$ to $0<p<5$, after invoking the additional assumptions and . We restrict to $\mathcal{A}^{\mathcal{I}}$, because the argument in $\mathcal{A}^{\mathcal{H}}$ proceeds analogously.
By Lemma \[lm:rweightidentitiess2v1\] in the $n=0$ case and Lemma \[lm:rweightidentitiess2v2\] in the $n=1$ case, we only need to estimate $$\int_{u_1}^{u_2} \int_{{N}^{\mathcal{I}}_u} r^{p-3}|\phi| |LP_n \widetilde{\Phi}_{(n)}|+ r^{p-3}|P_n \widetilde{\Phi}_{(n)}||LP_n \widetilde{\Phi}_{(n)}|\,d\omega dv du.$$ We apply Young’s inequality to obtain $$\begin{split}
\int_{u_1}^{u_2}& \int_{{N}^{\mathcal{I}}_u} r^{p-3}|\phi| |LP_n \widetilde{\Phi}_{(n)}|+ r^{p-3}|P_n \widetilde{\Phi}_{(n)}||LP_n \widetilde{\Phi}_{(n)}|\,d\omega dv du\\
\leq&\: \epsilon \int_{u_1}^{u_2} \int_{{N}^{\mathcal{I}}_u} u^{-1-\frac{\eta}{2}}\cdot r^p(LP_n \widetilde{\Phi}_{(n)})^2\,d\omega dv du+ \frac{1}{4\epsilon} \int_{u_1}^{u_2} \int_{{N}^{\mathcal{I}}_u} u^{1+\frac{\eta}{2}}r^{p-6}\left[(\phi)^2+(P_n \widetilde{\Phi}_{(n)})^2 \right]\,d\omega dv du\\
\leq&\: C\epsilon u_0^{-\frac{\eta}{2}} \sup_{u_1\leq u\leq u_2} \int_{{N}^{\mathcal{I}}_u} r^p(LP_n \widetilde{\Phi}_{(n)})^2\,d\omega dv + \frac{C}{\epsilon}(p-5)^{-1} u_0^{-\frac{\eta}{2}} E_{\eta}.
\end{split}$$ For $\epsilon>0$ suitably small, we can absorb the first term on the right-hand side above into the left-hand side of the spacetime integral of the identities (for $n=0$) and (for $n=1$), where we take a supremum in $u$ on the left-hand side.
Extended hierarchies for $T^k\psi$ {#sec:extendhier}
==================================
The preliminary extended identities {#sec:TheLAndUnderlineLCommutedIdenties}
-----------------------------------
In order to obtain *improved* estimates for time-derivatives of $\psi$, which are essential for deriving the late-time asymptotics for $\psi$ itself, we will derive additional hierarchies of $r^p$ and $(r-M)^{-p}$ weighted estimates for $T^k\psi$, with $k\geq 1$. As a first step, we will derive additional hierarchies for $L^k\Phi_{(n)}$ and $L^k\phi_0$ in $\mathcal{A}^{\mathcal{I}}$ and $\underline{L}^k\underline{\Phi}_{(n)}$ and ${\underline{L}}^k\phi_0$ in $\mathcal{A}^{\mathcal{H}}$. We start with the following identities.
\[lm:hoequations\] Let $n\in {\mathbb{N}}_0$ and $k\in {\mathbb{N}}$. Then: $$\label{eq:hoequations1}
\begin{split}
4 L\underline{L}(L^k\Phi_{(n)})=&\:Dr^{-2}\slashed{\Delta}_{{\mathbb{S}}^2}L^k\Phi_{(n)}+[-kDr^{-3}+O(r^{-4})]\slashed{\Delta}_{{\mathbb{S}}^2}L^{k-1}\Phi_{(n)}+[-4nr^{-1}+O(r^{-2})]L^{k+1}\Phi_{(n)}\\
&+D[n(n+1+2k)r^{-2}+O(r^{-3})]L^k\Phi_{(n)}+k(k-1)\sum_{j=\min\{k,2\}}^k O(r^{-2-j})\slashed{\Delta}_{{\mathbb{S}}^2}L^{k-j}\Phi_{(n)}\\
&+\sum_{j=1}^k O(r^{-2-j})L^{k-j}\Phi_{(n)}+n\sum_{m=0}^{\max\{0,n-1\}}\sum_{j=0}^k O(r^{-2-j})L^{k-j}\Phi_{(m)}
\end{split}$$ and $$\label{eq:hoequations2}
\begin{split}
4 L\underline{L}(\underline{L}^k\underline{\Phi}_{(n)})=&\:Dr^{-2}\slashed{\Delta}_{{\mathbb{S}}^2}\underline{L}^k\underline{\Phi}_{(n)}+[-kM^{-6}(r-M)^3+O((r-M)^4)]\slashed{\Delta}_{{\mathbb{S}}^2}{\underline{L}}^{k-1}\underline{\Phi}_{(n)}\\
&+[-4M^{-2}n(r-M)+O((r-M)^2)]{\underline{L}}^{k+1}\underline{\Phi}_{(n)}\\
&+[n(n+1+2k)M^{-4}(r-M)^2+O((r-M)^3)]{\underline{L}}^k\underline{\Phi}_{(n)}\\
&+k(k-1)\sum_{j=\min\{k,2\}}^k O((r-M)^{2+j})\slashed{\Delta}_{{\mathbb{S}}^2}{\underline{L}}^{k-j}\underline{\Phi}_{(n)}\\
&+\sum_{j=1}^k O((r-M)^{2+j}){\underline{L}}^{k-j}\underline{\Phi}_{(n)}+n\sum_{m=0}^{\max\{0,n-1\}}\sum_{j=0}^k O((r-M)^{2+j}){\underline{L}}^{k-j}\underline{\Phi}_{(m)}.
\end{split}$$
The identities and follow from a straightforward induction argument, where we apply Lemma \[lm:maineq\] for the $k=0$ case.
\[lm:hoequationsTpsi\] Let $n\in {\mathbb{N}}_0$. Then $$\label{eq:maincommeqITpsi}
\begin{split}
4LT\Phi_{(n)}=&\: 4L^2\Phi_{(n)}+Dr^{-2}\slashed{\Delta}_{{\mathbb{S}}^2}\Phi_{(n)}+[-4n r^{-1}+O(r^{-2})] L\Phi_{(n)}+ D\left[n(n+1)r^{-2} +O(r^{-3})\right]\Phi_{(n)}\\
&+n\sum_{k=0}^{\max\{0,n-1\}} O(r^{-2}) \Phi_{(k)},\\
\end{split}$$ and $$\label{eq:maincommeqHTpsi}
\begin{split}
4{\underline{L}}T\underline{\Phi}_{(n)}=&\:4{\underline{L}}^2\underline{\Phi}_{(n)}+Dr^{-2}\slashed{\Delta}_{{\mathbb{S}}^2}\underline{\Phi}_{(n)}+[-4M^{-2}n (r-M)+O((r-M)^{2})] \underline{L}\underline{\Phi}_{(n)}\\
&+ \left[n(n+1)M^{-4}(r-M)^{2} +O((r-M)^{3})\right]\underline{\Phi}_{(n)}\\
&+n\sum_{k=0}^{\max\{0,n-1\}} O((r-M)^{2}) \underline{\Phi}_{(k)}.
\end{split}$$
Let $k\geq 2$, then $$\label{eq:hoequations1Tpsi}
\begin{split}
4 L(L^{k-1} T\Phi_{(n)})=&\:4 L^{k+1}\Phi_{(n)}+Dr^{-2}\slashed{\Delta}_{{\mathbb{S}}^2}L^{k-1}\Phi_{(n)}+[-(k-1)Dr^{-3}+O(r^{-4})]\slashed{\Delta}_{{\mathbb{S}}^2}L^{k-2}\Phi_{(n)}\\\
&+[-4nr^{-1}+O(r^{-2})]LL^{k-1}\Phi_{(n)}+D[n(n+1+2(k-1))r^{-2}+O(r^{-3})]L^{k-1}\Phi_{(n)}\\
&+(k-1)(k-2)\sum_{j=\min\{k-1,2\}}^{k-1} O(r^{-2-j})\slashed{\Delta}_{{\mathbb{S}}^2}L^{k-1-j}\Phi_{(n)}\\
&+\sum_{j=0}^{k-2} O(r^{-2-j})L^{k-1-j}\Phi_{(n)}+n\sum_{m=0}^{\max\{0,n-1\}}\sum_{j=0}^{k-1} O(r^{-2-j})L^{k-1-j}\Phi_{(m)}.
\end{split}$$ and $$\label{eq:hoequations2Tpsi}
\begin{split}
4 L(\underline{L}^{k-1}T\underline{\Phi}_{(n)})=&\:4{\underline{L}}{\underline{L}}^k\underline{\Phi}_{(n)}+Dr^{-2}\slashed{\Delta}_{{\mathbb{S}}^2}\underline{L}^{k-1}\underline{\Phi}_{(n)}+[-(k-1)M^{-6}(r-M)^3+O((r-M)^4)]\slashed{\Delta}_{{\mathbb{S}}^2}{\underline{L}}^{k-2}\underline{\Phi}_{(n)}\\
&+[-4M^{-2}n(r-M)+O((r-M)^2)]{\underline{L}}{\underline{L}}^{k-1}\underline{\Phi}_{(n)}\\
&+[n(n+1+2(k-1))M^{-4}(r-M)^2+O((r-M)^3)]{\underline{L}}^{k-1}\underline{\Phi}_{(n)}\\
&+(k-1)(k-2)\sum_{j=\min\{k-1,2\}}^{k-1} O((r-M)^{2+j})\slashed{\Delta}_{{\mathbb{S}}^2}{\underline{L}}^{k-1-j}\underline{\Phi}_{(n)}\\
&+\sum_{j=1}^{k-1} O((r-M)^{2+j}){\underline{L}}^{k-1-j}\underline{\Phi}_{(n)}+n\sum_{m=0}^{\max\{0,n-1\}}\sum_{j=0}^{k-1} O((r-M)^{2+j}){\underline{L}}^{k-1-j}\underline{\Phi}_{(m)}.
\end{split}$$
In order to prove and , we use that $T=\underline{L}+L$ and apply Lemma \[lm:maineq\]. Similarly, and follows immediately from the identities in Lemma \[lm:hoequationsTpsi\].
When we restrict the the spherical mean $\psi_0$, the analogs of Lemma \[lm:hoequations\] and Lemma \[lm:hoequationsTpsi\] (with $n=0$) simplify significantly.
Consider $\psi_0$. Then for all $k\in {\mathbb{N}}$, $$\begin{aligned}
\label{eq:hoequations1l0}
4 L\underline{L}(L^k\phi_0)=&\:O(r^{-2})L^{k+1}\phi_0+O(r^{-3})L^k\phi_0+\sum_{j=1}^k O(r^{-3-j})L^{k-j}\phi_0,\\
\label{eq:hoequations2l0}
4 {\underline{L}}L({\underline{L}}^k\phi_0)=&\:O((r-M)^{2}){\underline{L}}{\underline{L}}^k\phi_0+O((r-M)^{3}){\underline{L}}^k\phi_0+\sum_{j=1}^k O((r-M)^{3+j}){\underline{L}}^{k-j}\phi_0.\end{aligned}$$ and $$\begin{aligned}
\label{eq:hoequations1l0Tpsi}
4 L(L^{k-1}T\phi_0)=&\:4 L(L^k\phi_0)+O(r^{-2})LL^{k-1}\phi_0+O(r^{-3})L^{k-1}\phi_0\\ \nonumber
&+(k-1)\sum_{j=\min\{k-1,1\}}^{k-1} O(r^{-3-j})L^{k-1-j}\phi_0,\\
\label{eq:hoequations2l0Tpsi}
4 {\underline{L}}({\underline{L}}^{k-1}T\phi_0)=&\:4 {\underline{L}}({\underline{L}}^k\phi_0)+O((r-M)^{2}){\underline{L}}{\underline{L}}^{k-1}\phi_0+O((r-M)^{3}){\underline{L}}^{k-1}\phi_0\\ \nonumber
&+(k-1)\sum_{j=\min\{1,k-1\}}^k O((r-M)^{3+j}){\underline{L}}^{k-1-j}\phi_0.\end{aligned}$$
Equations and follow from a standard induction argument, where we apply Lemma \[lm:maineq\] to obtain the $k=0$ case. Equations and then follow by using that $T=L+\underline{L}$.
Before we derive $r$-weighted estimates for $L^k\Phi_{(n)}$ and $\underline{L}^k\underline{\Phi}_{(n)}$, with $k>0$, we need to establish appropriate ($u$-dependent) boundedness estimates near $\mathcal{I}^+$. Recall that in the $k=0$ case these were obtained in Proposition \[prop:radfieldsinf\].
\[prop:radfieldsinfLk\] Let $n\in {\mathbb{N}}$ and $J\in {\mathbb{N}}_0$ and assume that $$\int_{\Sigma_0}J^T[\psi]\cdot \mathbf{n}_{\Sigma_0}\,d\mu_{\Sigma_0}<\infty.$$
- For all $u\geq u_0$, we have that there exists a constant $C_u=C_u(M,J,{\Sigma_0},u)>0$, such that $$\label{eq:udepradfieldsinftLk}
\begin{split}
\sum_{0\leq k\leq n}\sum_{0\leq j\leq J}&\sup_{u_0\leq u'\leq u}\left[\int_{{\mathbb{S}}^2} r^{2j}|\slashed{\Delta}_{{\mathbb{S}}^2}^{n-k} L^j\Phi_{(k)}|^2|_{{N}^{\mathcal{I}}}\,d\omega\right](v)\\
<&\:C_u\sum_{0\leq k\leq n}\sum_{0\leq j\leq J}\left[\int_{{\mathbb{S}}^2}r^{2j} |\slashed{\Delta}_{{\mathbb{S}}^2}^{n-k} L^j\Phi_{(k)}|^2|_{{N}^{\mathcal{I}}}\,d\omega\right](v).
\end{split}$$
- If we assume that for all $0\leq k \leq n$ and $0\leq j\leq J$ $$\lim_{v\to \infty}\left[\int_{{\mathbb{S}}^2}r^{2j} |\slashed{\Delta}_{{\mathbb{S}}^2}^{n-k} L^j\Phi_{(k)}|^2|_{{N}^{\mathcal{I}}}\,d\omega\right](v)<\infty,$$ then, for all $u\geq u_0$, $0\leq k \leq n$ and $0\leq j\leq J$, we have that $$\label{eq:limitradfieldsinftLk}
\lim_{v\to \infty}\left[\int_{{\mathbb{S}}^2}r^{2j} |\slashed{\Delta}_{{\mathbb{S}}^2}^{n-k} L^j\Phi_{(k)}|^2|_{{N}_{u}^I}\,d\omega\right](v)<\infty.$$
The proof proceeds inductively in $J$. The $J=0$ case is proved in Proposition \[prop:radfieldsinf\]. We then suppose that holds for $J$ and prove that it must also hold for $J$ replaced by $J+1$ by applying the same arguments as in the proof of Proposition \[prop:radfieldsinf\], using equation .
We now state the key lemma containing the $r$-weighted *identities* for $L^J\Phi_{(n)}$ and $\underline{L}^j \underline{\Phi}_{(n)}$ with $J>1$. Recall that the $J=0$ case was obtained in Lemma \[lm:rweightidentitiess2v1\].
\[lm:rweightidentitiesho\] Let $p\in {\mathbb{R}}$. Then the following identities hold for all $N\in {\mathbb{N}}_0$ and $J\in {\mathbb{N}}$: $$\label{eq:mainidentitysphereinfho}
\begin{split}
\int_{{\mathbb{S}}^2}&\underline{L}\left(r^p (L^{J+1}\Phi_{(n)})^2\right)\,d\omega+\int_{{\mathbb{S}}^2} L\left(\frac{1}{4}r^{p-2}|\slashed{\nabla}_{{\mathbb{S}}^2}L^J\Phi_{(n)}|^2\right)\,d\omega\\
&+\frac{1}{2}\int_{{\mathbb{S}}^2}L\left(\left[-JDr^{p-3}+O(r^{p-4})\right]\slashed{\nabla}_{{\mathbb{S}}^2}L^J\Phi_{(n)}\cdot {\slashed{\nabla}}_{{\mathbb{S}}^2}L^{J-1}\Phi_{(n)}\right)\,d\omega \\
&+ \frac{1}{2}\int_{{\mathbb{S}}^2}[(p+4N)r^{p-1}+O(r^{p-2})](L^{J+1}\Phi_{(n)})^2\,d\omega\\
&+\frac{1}{8}\int_{{\mathbb{S}}^2} [(2+4J-p)r^{p-3}+O(r^{p-4})]D|\slashed{\nabla}_{{\mathbb{S}}^2}L^J\Phi_{(n)}|^2+(p-2)N(N+1+2J)r^{p-3}D(L^J\Phi_{(n)})^2\,d\omega\\
=&\: \int_{{\mathbb{S}}^2} \frac{1}{4}L\left(N(N+1+2J)r^{p-2}(L^J\Phi_{(n)})^2\right)\,d\omega\\
&+\int_{{\mathbb{S}}^2} O(r^{p-3}) L^J\Phi_{(n)}\cdot L^{J+1}\Phi_{(n)}\,d\omega +\sum_{j=1}^J\int_{{\mathbb{S}}^2}O(r^{p-2-j})L^{J-j}\Phi_{(n)}\cdot L^{J+1}\Phi_{(n)}\\
&+J(J-1)\int_{{\mathbb{S}}^2}\sum_{j=\min\{J,2\}}^J O(r^{p-2-j})\slashed{\Delta}_{{\mathbb{S}}^2}L^{J-j}\Phi_{(n)}\cdot L^{J+1}\Phi_{(n)}\,d\omega\\
&+\int_{{\mathbb{S}}^2} O(r^{p-3})\slashed{\Delta}_{{\mathbb{S}}^2}L^J\Phi_{(n)}\cdot L^{J+1}\Phi_{(n)}\,d\omega+N\sum_{n=0}^{\max\{0,N-1\}}\sum_{j=0}^J\int_{{\mathbb{S}}^2} O(r^{p-2-j}) L^{J-j}\Phi_{(n)}\cdot L^{J+1}\Phi_{(n)}\,d\omega\\
&+\int_{{\mathbb{S}}^2}O(r^{p-4}) \slashed{\nabla}_{{\mathbb{S}}^2}L^{J-1}\Phi_{(n)} \cdot \slashed{\nabla}_{{\mathbb{S}}^2}L^J\Phi_{(n)}\,d\omega,
\end{split}$$ and $$\label{eq:mainidentityspherehorho}
\begin{split}
\int_{{\mathbb{S}}^2}&L\left((r-M)^{-p} ({\underline{L}}^{J+1}\underline{\Phi}_{(n)})^2\right)\,d\omega+\int_{{\mathbb{S}}^2} {\underline{L}}\left(\frac{1}{4}M^{-4}(r-M)^{2-p}|\slashed{\nabla}_{{\mathbb{S}}^2}{\underline{L}}^J\underline{\Phi}_{(n)}|^2\right) \,d\omega\\
&+\frac{1}{2}\int_{{\mathbb{S}}^2}L\left(\left[-JM^{-6}D(r-M)^{3-p}+O((r-M)^{4-p})\right]{\slashed{\nabla}}_{{\mathbb{S}}^2}{\underline{L}}^J\underline{\Phi}_{(n)}\cdot \slashed{\nabla}_{{\mathbb{S}}^2}{\underline{L}}^{J-1}\underline{\Phi}_{(n)}\right)\,d\omega \\
&+ \frac{1}{2}\int_{{\mathbb{S}}^2}[(p+4N)M^{-2}(r-M)^{1-p}+O((r-M)^{2-p})]({\underline{L}}^{J+1}\underline{\Phi}_{(n)})^2\,d\omega\\
&+\frac{1}{8}\int_{{\mathbb{S}}^2}[ (2+4J-p)M^{-6}(r-M)^{3-p}+O((r-M)^{4-p})]|\slashed{\nabla}_{{\mathbb{S}}^2}{\underline{L}}^J\underline{\Phi}_{(n)}|^2\\
&+(p-2)M^{-6}N(N+1+2J)(r-M)^{3-p}({\underline{L}}^J\underline{\Phi}_{(n)})^2\,d\omega\\
=&\: \int_{{\mathbb{S}}^2} \frac{1}{4}M^{-4}{\underline{L}}\left(N(N+1+2J)(r-M)^{2-p}({\underline{L}}^J\underline{\Phi}_{(n)})^2-(r-M)^{2-p}|\slashed{\nabla}_{{\mathbb{S}}^2}{\underline{L}}^J\underline{\Phi}_{(n)}|^2\right)\,d\omega\\
&+\int_{{\mathbb{S}}^2} O((r-M)^{3-p}) {\underline{L}}^J\underline{\Phi}_{(n)}\cdot {\underline{L}}^{J+1}\underline{\Phi}_{(n)}\,d\omega +\sum_{j=1}^J\int_{{\mathbb{S}}^2}O((r-M)^{2+j-p})){\underline{L}}^{J-j}\underline{\Phi}_{(n)}\cdot {\underline{L}}^{J+1}\underline{\Phi}_{(n)}\\
&+J(J-1)\int_{{\mathbb{S}}^2}\sum_{j=\min\{J,2\}}^J O((r-M)^{2+j-p})\slashed{\Delta}_{{\mathbb{S}}^2}{\underline{L}}^{J-j}\underline{\Phi}_{(n)}\cdot {\underline{L}}^{J+1}\underline{\Phi}_{(n)}\,d\omega\\
&+\int_{{\mathbb{S}}^2} O((r-M)^{3-p})\slashed{\Delta}_{{\mathbb{S}}^2}{\underline{L}}^J\underline{\Phi}_{(n)}\cdot {\underline{L}}^J\underline{\Phi}_{(n)}\,d\omega\\
&+N\sum_{n=0}^{\max\{0,N-1\}}\sum_{j=0}^J\int_{{\mathbb{S}}^2} O((r-M)^{2+j-p}) {\underline{L}}^{J-j}\underline{\Phi}_{(n)}\cdot {\underline{L}}^{J+1}\underline{\Phi}_{(n)}\,d\omega\\
&+\int_{{\mathbb{S}}^2}O((r-M)^{4-p}) \slashed{\nabla}_{{\mathbb{S}}^2}{\underline{L}}^{J-1}\underline{\Phi}_{(n)} \cdot \slashed{\nabla}_{{\mathbb{S}}^2}{\underline{L}}^J\underline{\Phi}_{(n)}\,d\omega.
\end{split}$$
The proof of and proceeds in an analogous fashion to the proof of Lemma \[lm:rweightidentitiess2v1\], where we use the more general equations in Lemma \[lm:hoequations\], rather than the equations in Lemma \[lm:maineq\], and we integrate by parts appropriately in the $L$ direction and on ${\mathbb{S}}^2$ and in the ${\underline{L}}$ direction and on ${\mathbb{S}}^2$, respectively.
The preliminary extended hierarchies {#sec:TheExtendedHierarchies}
------------------------------------
We now obtain the *higher-order* (with respect to $L$ or $\underline{L}$ derivation) $r$-weighted estimates.
\[prop:generalrpestLkpsi\] Fix $N\in {\mathbb{N}}_0$ and $J\in {\mathbb{N}}_0$ and assume that there exists a constant $C_0>0$ such that on ${N}^{\mathcal{I}}$: for all $0\leq n\leq \min\{N-1,0\}$ and $0\leq j\leq J$ $$\label{assm:id1b}
\lim_{v\to \infty} \left(\int_{{\mathbb{S}}^2}r^{2j}|{\slashed{\nabla}}_{{\mathbb{S}}^2}\slashed{\Delta}_{{\mathbb{S}}^2}^nL^jP_{\geq 1}\Phi_{(N-n)}|^2\,d\omega\right)(u_0,v)<\infty.$$ Then there exists $r_{\mathcal{I}}>0$ sufficiently large, such that for $p\in (-4N+2J,2+2J]$ and for all $0\leq u_1\leq u_2$: $$\label{eq:rpestinfLkpsi}
\begin{split}
\int_{{N}^{\mathcal{I}}_{u_2}}& r^p(LP_{\geq 1}L^J\Phi_{(N)})^2\,d\omega dv+ \int_{u_1}^{u_2} \int_{{N}^{\mathcal{I}}_u}(p+4N)r^{p-1}(LP_{\geq 1}L^J\Phi_{(N)})^2\,d\omega dv du\\
&+\int_{u_1}^{u_2} \int_{{N}^{\mathcal{I}}_u}r^{p-3}D|{\slashed{\nabla}}_{{\mathbb{S}}^2}P_{\geq 1}L^J\Phi_{(N)}|^2\,d\omega dv du\\
\leq&\: C\sum_{0\leq j\leq J}\int_{{N}^{\mathcal{I}}_{u_1}}r^{p-2j}(LP_{\geq 1}L^{J-j}\Phi_{(N)})^2\,d\omega dv+ C\sum_{m\leq N+J}\int_{\Sigma_{u_1}} J^T[T^m\psi]\cdot \mathbf{n}_{u_1}\,d\mu_{\Sigma_{u_1}}\\
&+C\cdot N \int_{\mathcal{I}^+(u_1,u_2)}r^{p-2-2J}(P_{\leq \max\{N-1,1\}}P_{\geq 1}\Phi_{(n)})^2\,d\omega du\\
&+C\cdot N\int_{u_1}^{u_2} \int_{{N}^{\mathcal{I}}_u} (2+2J-p)r^{p+1-2J}(LP_{\leq \max\{N-1,1\}}P_{\geq 1}\Phi_{(\max\{N-1,0\})})^2\,d\omega dv du,
\end{split}$$ where $C=C(N,J,M,{\Sigma_0},r_{\mathcal{I}})>0$ is a constant and we can take $r_{\mathcal{I}}=(p-2J+4N)^{-1}R_0(N,J,M)>0$.
Furthermore, there exists $r_{\mathcal{H}}>M$, with $r_{\mathcal{H}}-M$ suitably small, such that for $p\in (-4N+2J,2+2J]$ and for all $0\leq u_1\leq u_2$: $$\label{eq:rpesthorLbarkpsi}
\begin{split}
\int_{{N}^{\mathcal{H}}_{v_2}}& (r-M)^{-p}(\underline{L}P_{\geq 1}{\underline{L}}^J\underline{\Phi}_{(N)})^2\,d\omega du+\int_{v_1}^{v_2} \int_{{N}^{\mathcal{H}}_u}(p+4N)(r-M)^{1-p}(\underline{L}P_{\geq 1}{\underline{L}}^J\underline{\Phi}_{(N)})^2\,d\omega du dv\\
&+\int_{v_1}^{v_2} \int_{{N}^{\mathcal{H}}_v}(r-M)^{3-p}D|{\slashed{\nabla}}_{{\mathbb{S}}^2}P_{\geq 1}{\underline{L}}^J\underline{\Phi}_{(N)}|^2\,d\omega du dv\\
\leq&\: C\sum_{0\leq j\leq J}\int_{{N}^{\mathcal{H}}_{v_1}}(r-M)^{-p+2j}(\underline{L}P_{\geq 1}{\underline{L}}^{J-j}\underline{\Phi}_{(N)})^2\,d\omega du+ C\sum_{m\leq N+J}\int_{\Sigma_{v_1}} J^T[T^m\psi]\cdot \mathbf{n}_{v_1}\,d\mu_{\Sigma_{v_1}}\\
&+C\cdot N \int_{\mathcal{H}^+(v_1,v_2)}(r-M)^{-p+2J+2}(P_{\leq \max\{N-1,1\}}P_{\geq 1}\Phi_{(n)})^2\,d\omega dv\\
&+C\cdot N\int_{v_1}^{v_2} \int_{{N}^{\mathcal{H}}_v} (2+2J-p)(r-M)^{-p+2J-1}({\underline{L}}P_{\leq \max\{N-1,1\}}P_{\geq 1}\underline{\Phi}_{(\max\{N-1,0\})})^2\,d\omega du dv,
\end{split}$$ where $C=C(N,J,M,{\Sigma_0},r_{\mathcal{H}})>0$ is a constant and we can take $(r_{\mathcal{H}}-M)^{-1}=(p-2J+4N)^{-1}M^{-2}R_0(N,J,M)>0$.
We prove and inductively in $J$. The $J=0$ case follows immediately from the estimates and . Now suppose and hold for all $0\leq J\leq J'$. Then we need to show that they must also hold with $J$ replaced by $J'+1$. In order to show this, we use the identity , where we either absorb all terms without a sign into the terms with a good sign or we use the induction step to estimate them, applying the Hardy inequalities and where necessary, after introducing a cut-off as in the proof of Proposition \[prop:generalrpest\]. In addition, we integrate by parts the terms with a $\slashed{\Delta}_{{\mathbb{S}}^2}$ derivative that arise on the right-hand side of . We refer to Proposition 4.6 in [@paper1] for additional details in an analogous proof (where $N=2$).
Let us emphasize that for $N\geq 2$ the estimates and applied to $P_{\leq N-1} \psi$ have integrals on the right-hand side that are solely supported on $\Sigma_{u_1}$ or $\Sigma_{v_1}$. We will need *different* $r$-weighted estimates to be able to estimate these terms further. More precisely, we will apply Proposition \[prop:rpestell01\] in the case where $N=2$.
We can similarly extend the estimates for $n=0$ in Proposition \[prop:rpestell01\] to the higher-order quantities $L^k\phi_0$ and $\underline{L}^k\phi_0$ as follows:
\[prop:rpestell01Lkpsi\] Let $k\in {\mathbb{N}}_0$ and consider the following assumptions for all $0\leq j\leq k-1$, $$\begin{aligned}
\label{eq:asmLjphi_01}
\lim_{v\to \infty} r^{j+2} L^{j+1}\phi_0(u_0,v)<&\infty, \\
\label{eq:asmLjphi_02}
\lim_{v\to \infty} r^{j+3} L^{j+1}\phi_0(u_0,v)<&\infty.\end{aligned}$$ Then, if we assume , there exists an $r_{\mathcal{I}}>0$, such that for $p\in (-4n+2k,4+2k)$ and for all $0\leq u_1\leq u_2$: $$\label{eq:rpestinfLkpsi0}
\begin{split}
\int_{{N}^{\mathcal{I}}_{u_2}}& r^p(L^{k+1}\phi_0)^2\,d\omega dv+\int_{u_1}^{u_2} \int_{{N}^{\mathcal{I}}_u}p r^{p-1}(L^{k+1}\phi_0)^2\,d\omega dv du\\
\leq&\: C\int_{{N}^{\mathcal{I}}_{u_1}}r^p(L^{k+1}\phi_0)^2\,d\omega dv+ C\sum_{m\leq k}\int_{\Sigma_{u_1}} J^T[T^m\psi]\cdot \mathbf{n}_{u_1}\,d\mu_{\Sigma_{u_1}},
\end{split}$$ where $C=C(k,M,{\Sigma_0},r_{\mathcal{I}})>0$ is a constant and we can take $r_{\mathcal{I}}=(p-2k)^{-1}R_0(n,M)>0$, and we additionally assume when $p\geq 3+k$.
Furthermore, there exists an $r_{\mathcal{H}}>M$, such that for $p\in (2k-4n,4+2k)$ and for all $0\leq u_1\leq u_2$: $$\label{eq:rpeshorLkpsi0}
\begin{split}
\int_{{N}^{\mathcal{H}}_{v_2}}& (r-M)^{-p}({\underline{L}}^{k+1}\phi_0)^2\,d\omega du+\int_{v_1}^{v_2} \int_{{N}^{\mathcal{H}}_u}p(r-M)^{1-p}({\underline{L}}^{k+1}\phi_0)^2\,d\omega du dv\\
\leq&\: C\int_{{N}^{\mathcal{H}}_{v_1}}(r-M)^{-p}({\underline{L}}^{k+1}\phi_0)^2\,d\omega du+ C\sum_{m\leq k}\int_{\Sigma_{v_1}} J^T[T^m\psi]\cdot \mathbf{n}_{v_1}\,d\mu_{\Sigma_{v_1}},
\end{split}$$ where $C=C(k,M,{\Sigma_0},r_{\mathcal{H}})>0$ is a constant and we can take $(r_{\mathcal{H}}-M)^{-1}=(p+4n)^{-1}R_0(n,M)>0$.
If we additionally assume that there exists constants $\eta>0$ and $\mathcal{E}_{k,\eta}>0$ such that $$\begin{aligned}
\label{eq:hoauxdecayassm1}
\sum_{j\leq k}r^{2j}(L^j\phi_0)^2\lesssim \mathcal{E}_{k,\eta}\cdot (1+u)^{-2-\eta}\quad \textnormal{in $\mathcal{A}^{\mathcal{I}}$},\\
\label{eq:hoauxdecayassm2}
\sum_{j\leq k}(r-M)^{-2j}(\underline{L}^j\phi_0)^2\lesssim \mathcal{E}_{k,\eta} \cdot (1+v)^{-2-\eta}\quad \textnormal{in $\mathcal{A}^{\mathcal{H}}$},\\\end{aligned}$$ and we assume , then we can obtain for $2k<p<5+2k$: $$\label{eq:rpestinfp5k}
\begin{split}
\int_{{N}^{\mathcal{I}}_{u_2}}& r^p(L^{k+1}\phi_0)^2\,d\omega dv+ \int_{u_1}^{u_2} \int_{{N}^{\mathcal{I}}_u}pr^{p-1}(L^{k+1}\phi_0)^2\,d\omega dv du\\
\leq&\: C\int_{{N}^{\mathcal{I}}_{u_1}}r^p(L^{k+1}\phi_0)^2\,d\omega dv+ C\sum_{m\leq k}\int_{\Sigma_{u_1}} J^T[T^m\psi]\cdot \mathbf{n}_{u_1}\,d\mu_{\Sigma_{u_1}}+C(p+2k-5)^{-1}\mathcal{E}_{k,\eta},
\end{split}$$ and $$\label{eq:rpeshorp5}
\begin{split}
\int_{{N}^{\mathcal{H}}_{v_2}}& (r-M)^{-p}({\underline{L}}^{k+1}\phi_0)^2\,d\omega du+\int_{v_1}^{v_2} \int_{{N}^{\mathcal{H}}_u}p(r-M)^{1-p}({\underline{L}}^{k+1}\phi_0)^2\,d\omega du dv\\
\leq&\: C\int_{{N}^{\mathcal{H}}_{v_1}}(r-M)^{-p}({\underline{L}}^{k+1}\phi_0)^2\,d\omega du+ C\sum_{m\leq k}\int_{\Sigma_{v_1}} J^T[T^m\psi]\cdot \mathbf{n}_{v_1}\,d\mu_{\Sigma_{v_1}}+C(p+2k-5)^{-1}\mathcal{E}_{k,\eta}.
\end{split}$$
The estimates follow from an induction argument. The $k=0$ case follows from Proposition \[prop:rpestell01\] with $n=0$ and we use the identities in Lemma \[lm:rweightidentitiesho\] (with $n=0$) in the induction step, as in the proof of Proposition \[prop:generalrpestLkpsi\]. We moreover use that for all $0\leq j \leq k-1$ and $0<p<5$: $$\int_{{N}^{\mathcal{I}}_{u}} r^{p+2j} (L^{j+1}\phi_0)^2\,d\omega dv\lesssim \int_{{N}^{\mathcal{I}}_{u}} r^{p+2j+2} (L^{j+2}\phi_0)^2\,d\omega dv,$$ by together with the assumption $\lim_{v\to \infty}r^{\frac{p+1}{2}+j} L^{j+1}\phi_0(u,v)<\infty$, which follows from the assumption in the $p<3$ case and in the $p<5$ case after a straightforward propagation to $u\geq u_0$ as in the proof of Proposition \[prop:radfieldsinfLk\]. A similar inequality holds along ${N}^{\mathcal{H}}_{v}$, where there is no need for additional initial data assumptions by the assumption of smoothness of the initial data.
The hierarchies for $T^{k}\psi$ {#sec:TheHierarchiesForTKPsi}
-------------------------------
In order to use the extended $r$-weighted hierarchies for $L^k\Phi_{(n)}$ and $\underline{L}^k\underline{\Phi}_{(n)}$ from Proposition \[prop:generalrpestLkpsi\] and \[prop:rpestell01Lkpsi\] to obtain additional $r$-weighted estimates for $T^k\Phi_{(n)}$ and $T^k\Phi_{(n)}$ compared to $\Phi_{(n)}$ and $\underline{\Phi}_{(n)}$, we first *relate* $r$-weighted estimates for $T$-derivatives to $r$-weighted estimates for $L$- or $\underline{L}$-derivatives.
\[lm:intestTpsi\] Let $n\in {\mathbb{N}}_0$, $k\geq1$ and $p\in (2k,2k+2]$, then we can estimate $$\label{eq:intestTpsiinf}
\begin{split}
\int_{u_1}^{u_2}&\int_{{N}^{\mathcal{I}}_{u}} r^{p-1}(L L^{k-1}T\Phi_{(n)})^2\,d\omega dv du\\
\leq &\:C\sum_{|\alpha|=1}\int_{u_1}^{u_2}\int_{{N}^{\mathcal{I}}_{u}} r^{p-1}(LL^{k}\Phi_{(n)})^2+r^{p-3}|{\slashed{\nabla}}_{{\mathbb{S}}^2}L^{k-1}\Omega^{\alpha}\Phi_{(n)}|^2\,d\omega dv du\\
&+ C\sum_{m\leq k+n}\int_{\Sigma_{u_1}} J^T[T^m\psi]\cdot \mathbf{n}_{u_1}\,d\mu_{\Sigma_{u_1}}
\end{split}$$ and $$\label{eq:intestTpsihor}
\begin{split}
\int_{v_1}^{v_2}&\int_{{N}^{\mathcal{H}}_{v}} (r-M)^{1-p}({\underline{L}}{\underline{L}}^{k-1}T\underline{\Phi}_{(n)})^2\,d\omega du dv\\
\leq &\:C\sum_{|\alpha|=1} \int_{v_1}^{v_2}\int_{{N}^{\mathcal{I}}_{v}} (r-M)^{1-p}({\underline{L}}{\underline{L}}^{k}\underline{\Phi}_{(n)})^2+(r-M)^{3-p} |{\slashed{\nabla}}_{{\mathbb{S}}^2} {\underline{L}}^{k-1}\Omega^{\alpha}\underline{\Phi}_{(n)}|^2\,d\omega du dv\\
&+ C\sum_{m\leq k+n}\int_{\Sigma_{v_1}} J^T[T^m\psi]\cdot \mathbf{n}_{v_1}\,d\mu_{\Sigma_{v_1}}.
\end{split}$$ Furthermore, if we let $p\in (2k,2k+5)$, and we assume that for all $0\leq j\leq k$, $$\begin{aligned}
\lim_{v\to \infty} r^{j+2} L^{j+1}\phi_0(u_0,v)<&\infty \quad \textnormal{when $p<3+2k$ and} \\
\lim_{v\to \infty} r^{j+3} L^{j+1}\phi_0(u_0,v)<&\infty\quad \textnormal{when $p<5+2k$,}\end{aligned}$$ then we can estimate $$\label{eq:intestTpsiinfl0}
\begin{split}
\int_{u_1}^{u_2}&\int_{{N}^{\mathcal{I}}_{u}} r^{p-1}(L L^{k-1}T\phi_0)^2\,d\omega dv du\\
\leq &\:C\int_{u_1}^{u_2}\int_{{N}^{\mathcal{I}}_{u}} r^{p-1}(LL^{k}\phi_0)^2\,d\omega dv du+ C\sum_{m\leq k}\int_{\Sigma_{u_1}} J^T[T^m\psi]\cdot \mathbf{n}_{u_1}\,d\mu_{\Sigma_{u_1}}
\end{split}$$ and $$\label{eq:intestTpsihorl0}
\begin{split}
\int_{v_1}^{v_2}&\int_{{N}^{\mathcal{H}}_{v}} (r-M)^{1-p}({\underline{L}}{\underline{L}}^{k-1}T\phi_0)^2\,d\omega du dv\\
\leq &\:C\int_{v_1}^{v_2}\int_{{N}^{\mathcal{I}}_{v}} (r-M)^{1-p}({\underline{L}}{\underline{L}}^{k}\phi_0)^2\,d\omega du dv+ C\sum_{m\leq k}\int_{\Sigma_{v_1}} J^T[T^m\psi]\cdot \mathbf{n}_{v_1}\,d\mu_{\Sigma_{v_1}}.
\end{split}$$
All the estimates in the lemma follow directly from the identities in Lemma \[lm:hoequationsTpsi\] and and and moreover for $\phi_0$. Note that we have made use of to replace $\slashed{\Delta}_{{\mathbb{S}}^2} L^{k-1}\Phi_{(n)}$ appearing in the integral by $\sum_{|\alpha|=1}{\slashed{\nabla}}_{{\mathbb{S}}^2}\Omega^{\alpha}L^{k-1}\Phi_{(n)}$ (and similarly for $\Phi_{(n)}$ replaced with $\underline{\Phi}_{(n)}$).
Time decay estimates {#sec:decayest}
====================
In this section we will derive time decay estimates for $\psi$. First, we will *convert* hierarchies of $r$-weighted estimates from Section \[sec:rweightest\] and \[sec:extendhier\] into time decay estimates for various ($r$-weighted) energy quantities in Section \[sec:edecayest\] and \[sec:hoedecayest\]. Then, we will use these energy decay estimates to obtain *pointwise* time decay estimates in Section \[sec:pdecayest\] by applying in addition certain elliptic estimates that are derived in Section \[sec:ellpest\].
Energy decay estimates {#sec:edecayest}
----------------------
We start by deriving separately energy decay estimates for $\psi_0$ and $\psi_{\geq 1}$.
\[prop:edaypsi0\] For all $\epsilon>0$ there exists a constant $C=C(M,\Sigma_0,\epsilon)>0$ such that $$\begin{aligned}
\label{eq:edecayl01}
\int_{\Sigma_{\tau}} J^T[ \psi_{0}]\cdot \mathbf{n}_{\tau}\,d\mu_{\tau}\leq &\:C\cdot E^{\epsilon}_{0}[\psi] (1+\tau)^{-3+\epsilon},\\
\label{eq:r2edecayl01}
\int_{{N}^{\mathcal{I}}_{\tau}} r^2\cdot (L\phi_0)^2\,d\omega dv+\int_{{N}^{\mathcal{I}}_{\tau}} (r-M)^{-2}\cdot (\underline{L}\phi_0)^2\,d\omega dv\leq &\:C\cdot E^{\epsilon}_{0}[\psi] (1+\tau)^{-1+\epsilon}.\end{aligned}$$ with $$E^{\epsilon}_{0}[\psi]:= \int_{{N}^{\mathcal{I}}} r^{3-\epsilon}(L\phi_0)^2\,d\omega dv+\int_{{N}^{\mathcal{H}}} (r-M)^{-3+\epsilon}(\underline{L}\phi_0)^2\,d\omega du+\int_{\Sigma_0} J^T[\psi_0]\cdot \mathbf{n}_{\Sigma_0}\,d\mu_{\Sigma_0}.$$
Energy decay follows from the hierarchies of $r$-weighted estimates in Proposition \[prop:rpestell01\] with $n=0$ by a repeated use of the *mean value theorem along dyadic time intervals* (sometimes called “the pigeonhole principle”). Indeed, let $\tau_i=2^i$, then we have that $$\begin{split}
\int_{\tau_i}^{\tau_{i+1}}\int_{\Sigma_{\tau}} J^T[\psi_0]\cdot \mathbf{n}_{\tau}\,d\mu_{\tau} d\tau=&\:\int_{\tau_i}^{\tau_{i+1}}\int_{{N}^{\mathcal{I}}_{\tau}} J^T[\psi_0]\cdot L r^2\,d\omega dv d\tau+ \int_{\tau_i}^{\tau_{i+1}}\int_{{N}^{\mathcal{H}}_{\tau}} J^T[\psi_0]\cdot \underline{L} r^2\,d\omega du d\tau\\
&+ \int_{\tau_i}^{\tau_{i+1}}\int_{\Sigma_{\tau} \cap\{r_{\mathcal{H}}\leq r \leq r_{\mathcal{I}}\}}J^T[\psi_0]\cdot \mathbf{n}_{\tau}\,d\mu_{\tau}\\
\lesssim &\: \int_{\tau_i}^{\tau_{i+1}}\int_{{N}^{\mathcal{I}}_{\tau}} (L\phi_0)^2\,d\omega dv d\tau+ \int_{\tau_i}^{\tau_{i+1}}\int_{{N}^{\mathcal{H}}_{\tau}} (\underline{L}\phi_0)^2\,d\omega du d\tau\\
&+ \int_{\Sigma_{\tau_{i}}}J^T[\psi_0]\cdot \mathbf{n}_{\tau_i}\,d\mu_{\tau_i},
\end{split}$$ where we applied the Morawetz inequality together with the Hardy inequalities and . Now we apply and with $n=0$ and $p=1$ to further estimate the right-hand side above and arrive at: $$\begin{split}
\int_{\tau_i}^{\tau_{i+1}}\int_{\Sigma_{\tau}} J^T[\psi_0]\cdot \mathbf{n}_{\tau}\,d\mu_{\tau} d\tau \lesssim &\: \int_{{N}^{\mathcal{I}}_{\tau_i}} r\cdot (L\phi_0)^2\,d\omega dv d\tau_i+ \int_{{N}^{\mathcal{H}}_{\tau_i}} (r-M)^{-1}\cdot (\underline{L}\phi_0)^2\,d\omega du d\tau_i\\
&+ \int_{\Sigma_{\tau_{i}}}J^T[\psi_0]\cdot \mathbf{n}_{\tau_i}\,d\mu_{\tau_i}.
\end{split}$$ After applying the mean-value theorem in $[\tau_i,\tau_{i+1}]$ and using the dyadicity of $\tau_i$: $\tau_{i+1}-\tau_i\sim \tau_i\sim \tau_{i+1}$, we obtain another dyadic sequence $\tau^{(1)}_{i}$ of times along which we have the following decay estimate: $$\begin{split}
\int_{\Sigma_{\tau_i^{(1)}}} J^T[\psi_0]\cdot \mathbf{n}_{\tau_i^{(1)}}\,d\mu_{\tau_i^{(1)}} d\tau \lesssim &\: \frac{1}{1+\tau_i^{(1)}}\cdot \Bigg[\int_{{N}^{\mathcal{I}}_{0}} r\cdot (L\phi_0)^2\,d\omega dv d\tau+ \int_{{N}^{\mathcal{H}}_{0}} (r-M)^{-1}\cdot (\underline{L}\phi_0)^2\,d\omega du d\tau\\
&+ \int_{\Sigma_{0}}J^T[\psi_0]\cdot \mathbf{n}_{0}\,d\mu_{0}\Bigg].
\end{split}$$ By moreover invoking ($T$-energy boundedness) we can replace $\tau_i^{(1)}$ above by *any* $\tau$. Equipped with $\tau^{-1}$ $T$-energy decay, we can now apply once again and with $n=0$ and $p=2-\epsilon$, with $\epsilon>0$ arbitrarily small and the mean-value theorem in the intervals $[\tau_i^{(1)},\tau_{i+1}^{(1)}]$ to show that the energies $$\int_{{N}^{\mathcal{I}}_{\tau}} r\cdot (L\phi_0)^2\,d\omega dv d\tau+ \int_{{N}^{\mathcal{H}}_{\tau}} (r-M)^{-1}\cdot (\underline{L}\phi_0)^2\,d\omega du d\tau$$ decay with a rate $\tau^{-1+\epsilon}$. In this step we moreover need the interpolation estimates from Lemma \[lm:interpolation\] to transfer the $\epsilon$-loss in the $r$-weights into an $\epsilon$-loss in the $\tau$-decay. We can then *improve* the $T$-energy decay rate from $\tau^{-1}$ to $\tau^{-2+\epsilon}$. We now repeat this process, applying with $n=0$ and $p=3-\epsilon$ to arrive at . For additional details, see also the analogous Proposition 7.1 in [@paper1] for the sub-extremal case. Note that follows from applying with $n=0$ and $p=3-\epsilon$, together with and the interpolation estimates from Lemma \[lm:interpolation\].
Before we proceed with proving energy decay for the $\psi_{\geq 1}$ part of the solution, we will show that we can obtain *additional improvements* to the estimates in Proposition \[prop:edaypsi0\] in the cases where $I_0[\psi]=0$ or $H_0[\psi]=0$, after proving the following *auxilliary* decay lemma:
\[lm:auxedaypsi0\] For all $\epsilon>0$ there exists a constant $C=C(M,\Sigma_0,\epsilon)>0$ such that $$\begin{aligned}
\label{eq:auxdecayl01}
\int_{{N}^{\mathcal{I}}_{\tau}} r^2\cdot (L\phi_0)^2\,d\omega dv \leq&\: C\cdot \left[E^{\epsilon}_{0}[\psi]+ \int_{{N}^{\mathcal{I}}_{0}} r^{4-\epsilon}\cdot (L\phi_0)^2\,d\omega dv\right] (1+\tau)^{-2+\epsilon},\\
\label{eq:auxdecayl02}
\int_{{N}^{\mathcal{H}}_{\tau}} (r-M)^{-2}\cdot (\underline{L}\phi_0)^2\,d\omega du&\: C\cdot \left[E^{\epsilon}_{0}[\psi]+ \int_{{N}^{\mathcal{H}}_{0}} (r-M)^{-4+\epsilon}\cdot (\underline{L}\phi_0)^2\,d\omega dv\right] (1+\tau)^{-2+\epsilon}.\end{aligned}$$ Furthermore, $$\begin{aligned}
\label{eq:auxpointdecayl01}
||\phi_0||^2_{L^{\infty}({N}^{\mathcal{I}}_{\tau})}\leq&\: C\cdot \left[E^{\epsilon}_{0}[\psi]+ \int_{{N}^{\mathcal{I}}_{0}} r^{4-\epsilon}\cdot (L\phi_0)^2\,d\omega dv\right] (1+\tau)^{-\frac{5}{2}+\epsilon},\\
\label{eq:auxpointdecayl02}
||\phi_0||^2_{L^{\infty}({N}^{\mathcal{I}}_{\tau})}\leq &\: C\cdot \left[E^{\epsilon}_{0}[\psi]+ \int_{{N}^{\mathcal{H}}_{0}} (r-M)^{-4+\epsilon}\cdot (\underline{L}\phi_0)^2\right] (1+\tau)^{-\frac{5}{2}+\epsilon}.\end{aligned}$$
The estimates and follow after applying, in addition to the estimates in the proof of Proposition \[prop:edaypsi0\], *either* *or* with $n=0$ and $p=4-\epsilon$ and $p=3-\epsilon$ (and applying the mean value theorem twice in an analogous fashion to the the proof of Proposition \[prop:edaypsi0\]. The pointwise decay estimates then follow from a standard application of the fundamental theorem of calculus; see the proof of Proposition \[prop:pointdecay\] for explicit details of this type of computation.
With the $L^{\infty}$ estimates in Lemma \[lm:auxedaypsi0\], we can recover the assumptions *or* and make use of the *full* hierarchy of $r$-weighted estimates from Proposition \[prop:rpestell01\].
\[prop:extraendecay\] For all $\epsilon>0$ there exists a constant $C=C(M,{\Sigma_0},\epsilon)>0$ such that $$\begin{aligned}
\label{eq:optr2decay1}
\int_{{N}_{\tau}^I}r^{2-\epsilon} (L \phi_0)^2\,d\omega dv\leq&\: C \cdot E^{\epsilon}_{0,\mathcal{I}}[\psi] \cdot (1+\tau)^{-3+\epsilon},\\
\label{eq:optr2decay2}
\int_{{N}^{\mathcal{H}}}(r-M)^{-2+\epsilon} (\underline{L}\phi_0)^2\,du\leq&\:C \cdot E^{\epsilon}_{0,\mathcal{H}}[\psi] \cdot (1+\tau)^{-3+\epsilon},\end{aligned}$$ where $$\begin{aligned}
E^{\epsilon}_{0,\mathcal{I}}[\psi]:=&\: E^{\epsilon}_{0}[\psi]+\int_{{N}^{\mathcal{I}}} r^{5-\epsilon} (L \phi_0)^2\,d\omega dv,\\
E^{\epsilon}_{0,\mathcal{H}}[\psi]:=&\: E^{\epsilon}_{0}[\psi]+\int_{{N}^{\mathcal{H}}} (r-M)^{-5+\epsilon} ( \underline{L}\phi_0)^2\,d\omega du.\end{aligned}$$
We repeat the proof of Lemma \[lm:auxedaypsi0\], but we apply either or with $n=0$ and $p=5-\epsilon$, using the $L^{\infty}$ estimates or .
\[prop:edaypsi1\] Assume that for $n=0,1$ and $0\leq k \leq 2-n$: $$\lim_{v\to \infty} |{\slashed{\nabla}}_{{\mathbb{S}}^2}\slashed{\Delta}_{{\mathbb{S}}^2}^kP_{\geq 1}\Phi_{(n)}|^2\,d\omega<\infty.$$ Then, for all $\epsilon>0$ there exists a constant $C=C(M,{\Sigma_0},\epsilon)>0$ such that $$\begin{aligned}
\label{eq:edecayl11}
\int_{\Sigma_{\tau}} J^T[ \psi_{\geq 1}]\cdot \mathbf{n}_{\tau}\,d\mu_{\tau}\leq &\:C\cdot E^{\epsilon}_{1}[\psi] (1+\tau)^{-5+\epsilon},\\
\label{eq:edecaylr211}
\int_{{N}^{\mathcal{I}}_{\tau}} r^2(L \phi_{\geq 1})^2\,d\omega dv+\int_{{N}^{\mathcal{H}}_{\tau}} (r-M)^{-2}(\underline{L} \phi_{\geq 1})^2\,d\omega du \leq &\: C\cdot E^{\epsilon}_{1}[\psi] (1+\tau)^{-3+\epsilon},\end{aligned}$$ with $$\begin{split}
E^{\epsilon}_{1}[\psi]:=&\: \int_{{N}^{\mathcal{I}}} r^{1-\epsilon}(LP_{\geq 1}\Phi_{(2)})^2+\sum_{n=0}^1 \sum_{m=0}^{3-2n} \int_{{N}^{\mathcal{I}}} r^{2}(LT^mP_{\geq 1}\Phi_{(n)})^2+r^{1}(LT^{1+m}P_{\geq 1}\Phi_{(n)})^2\,d\omega dv\\
&+\int_{{N}^{\mathcal{H}}}(r-M)^{-1+\epsilon}(\underline{L}P_{\geq 1}\underline{\Phi}_{(2)})^2 +\sum_{n=0}^1 \sum_{m=0}^{3-2n} \int_{{N}^{\mathcal{H}}} (r-M)^{-2}(\underline{L}T^mP_{\geq 1}\underline{\Phi}_{(n)})^2\\
&+(r-M)^{-1}(\underline{L}T^{1+m}P_{\geq 1}\underline{\Phi}_{(n)})^2\,d\omega du+\sum_{m=0}^{5}\int_{\Sigma_0} J^T[T^m\psi_{\geq 1}]\cdot \mathbf{n}_{\Sigma_0}\,d\mu_{\Sigma_0}.
\end{split}$$
In order to prove we follow a similar strategy to the proof of Proposition \[prop:edaypsi0\]: we apply the mean value theorem on dyadic intervals. However, in this case we appeal to the hierarchies of $r$-weighted estimates in Proposition \[prop:generalrpest\], where we take $n=0,1,2$ and we *relate* the $r$-weighted estimates at different $n$ via the following estimate: for $p<4$, we have that $$\begin{split}
\int_{u_1}^{u_2}\int_{{N}^{\mathcal{I}}_{u}} r^{p-1} (L \chi \Phi_{(n)})^2\,d\omega dvdu\lesssim&\: \int_{u_1}^{u_2}\int_{{N}^{\mathcal{I}}_{u}} r^{p-5} \chi^2 (\Phi_{(n+1)})^2\,d\omega dv+ \sum_{k=0}^n \int_{\Sigma_{u_1}}\int_{\Sigma_{u_1}} J^T[T^k\psi ]\cdot \mathbf{n}_{\Sigma_{u_1}}\,d\mu_{\Sigma_{u_1}}\\
\lesssim&\: \int_{u_1}^{u_2}\int_{{N}^{\mathcal{I}}_{u}} r^{p-3} (L\chi \Phi_{(n+1)})^2\,d\omega dvdu+ \sum_{k=0}^n \int_{\Sigma_{u_1}}\int_{\Sigma_{u_1}} J^T[T^k\psi ]\cdot \mathbf{n}_{\Sigma_{u_1}}\,d\mu_{\Sigma_{u_1}},
\end{split}$$ where $\chi$ is the cut-off function that appears in the estimates in $\mathcal{A}^{\mathcal{I}}$ in the proof of Proposition \[prop:generalrpest\] and the first inequality on the right-hand side above follows from the Morawetz inequality , whereas the second inequality follows from the Hardy inequality . We can similarly estimate for $p<4$: $$\begin{split}
\int_{v_1}^{v_2}\int_{{N}^{\mathcal{H}}_{v}} (r-M)^{-p+1} (\underline{L} \chi \underline{\Phi}_{(n)})^2\,d\omega dudv\lesssim&\: \int_{v_1}^{v_2}\int_{{N}^{\mathcal{H}}_{v}} (r-M)^{-p+3} (\underline{L} \chi \underline{\Phi}_{(n+1)})^2\,d\omega dudv\\
&+ \sum_{k=0}^n \int_{\Sigma_{u_1}}\int_{\Sigma_{u_1}} J^T[T^k\psi ]\cdot \mathbf{n}_{\Sigma_{u_1}}\,d\mu_{\Sigma_{u_1}},
\end{split}$$ where $\chi$ here denotes the cut-off function that appears in the estimates in $\mathcal{A}^{\mathcal{H}}$ in the proof of Proposition \[prop:generalrpest\].
In the $n=2$ case, the left-hand side of the $r$-weighted estimates and are not positive definite due to the presence of the spacetime integrals of the terms $$\begin{aligned}
(2-p)&r^{p-3}\left[|{\slashed{\nabla}}_{{\mathbb{S}}^2}P_{\geq 1} \Phi_{(2)}|^2-6(P_{\geq 1} \Phi_{(2)})^2\right]\quad\textnormal{and}\\
(2-p)&(r-M)^{-p+3}\left[|{\slashed{\nabla}}_{{\mathbb{S}}^2}P_{\geq 1} \underline{\Phi}_{(2)}|^2-6(P_{\geq 1} \underline{\Phi}_{(2)})^2\right].\end{aligned}$$ Note however that, after applying the Poincaré inequality it follows that the above terms *are* positive definite for $P_{\geq 2}\Phi_{(2)}$. To deal with the $\ell=1$ case, we instead use that for $p<2$: $$\begin{split}
\int_{u_1}^{u_2}\int_{{N}^{\mathcal{I}}_{u}} r^{p-3}\chi^2(P_{1}\Phi_{(2)})^2\,d\omega dv du \lesssim&\: \int_{u_1}^{u_2}\int_{{N}^{\mathcal{I}}_{u}} r^{p+1}(L\chi P_{1} \widetilde{\Phi}_{(1)})^2+r^{p-3}\chi^2{\Phi}_{(1)}^2\,d\omega dv du\\
&+\sum_{k=0}^1 \int_{\Sigma_{u_1}}\int_{\Sigma_{u_1}} J^T[T^k\psi ]\cdot \mathbf{n}_{\Sigma_{u_1}}\,d\mu_{\Sigma_{u_1}}\\
\lesssim&\: \int_{u_1}^{u_2}\int_{{N}^{\mathcal{I}}_{u}} r^{p+1}(L\chi P_{1} \widetilde{\Phi}_{(1)})^2+r^{p-1}\chi^2(LP_{1} \Phi_{(1)})^2\,d\omega dv\\
&+\sum_{k=0}^1 \int_{\Sigma_{u_1}}\int_{\Sigma_{u_1}} J^T[T^k\psi ]\cdot \mathbf{n}_{\Sigma_{u_1}}\,d\mu_{\Sigma_{u_1}}\\
\lesssim&\: \int_{u_1}^{u_2}\int_{{N}^{\mathcal{I}}_{u}} r^{p+1}(L\chi P_{1} \widetilde{\Phi}_{(1)})^2+r^{p-5}\chi^2(P_{1}\Phi_{(2)})^2\,d\omega dv du\\
&+\sum_{k=0}^1 \int_{\Sigma_{u_1}}\int_{\Sigma_{u_1}} J^T[T^k\psi ]\cdot \mathbf{n}_{\Sigma_{u_1}}\,d\mu_{\Sigma_{u_1}},
\end{split}$$ where we made use of and . Note that the second term on the very right-hand side above can be absorbed into the left-hand side for sufficiently large $r_{\mathcal{I}}$, similarly, for $r_{\mathcal{H}}-M$ suitably small, we have that for $p<2$: $$\begin{split}
\int_{v_1}^{v_2}\int_{{N}^{\mathcal{H}}_{v}} (r-M)^{-p+3}\chi^2(P_{1}\underline{\Phi}_{(2)})^2\,d\omega du dv
\lesssim&\: \int_{v_1}^{v_2}\int_{{N}^{\mathcal{H}}_{v}} (r-M)^{-p-1}(L\chi P_{1} \widetilde{\underline{\Phi}}_{(1)})^2\,d\omega dudv\\
&+\sum_{k=0}^1 \int_{\Sigma_{v_1}}\int_{\Sigma_{v_1}} J^T[T^k\psi ]\cdot \mathbf{n}_{\Sigma_{v_1}}\,d\mu_{\Sigma_{v_1}}.
\end{split}$$ The above estimate allow therefore allow us to use the $r$-weighted estimates in Proposition \[prop:rpestell01\] with $n=1$ and $p<3$ to estimate the integrals of $r^{p-3}\chi^2(P_{1}\Phi_{(2)})^2$ and $(r-M)^{-p+3}\chi^2(P_{1}\underline{\Phi}_{(2)})^2$ *first* and then combine these with the $n=2$ estimates in Proposition \[prop:generalrpest\] for $p<1$.
Finally, we note that in order to estimate a *global* integrated energy, we moreover apply the Morawetz estimate which has a loss of $T$-derivatives on the right-hand side, and therefore the initial data norms that appear on the right-hand side of the time decay estimates will have *additional* $T$-derivatives.
We will also need the following energy decay estimate that involves higher-order weights in $r$ or $(r-M)^{-1}$ in the initial data norms.
\[prop:edaypsi1b\] Assume that for $n=0,1$ and $0\leq k \leq 2-n$: $$\lim_{v\to \infty} |{\slashed{\nabla}}_{{\mathbb{S}}^2}\slashed{\Delta}_{{\mathbb{S}}^2}^kP_{\geq 1}\Phi_{(n)}|^2\,d\omega<\infty.$$ Then, for all $\epsilon>0$ there exists a constant $C=C(M,{\Sigma_0},\epsilon)>0$ such that $$\begin{aligned}
\label{eq:extrarweightsl1inf}
\int_{{N}^{\mathcal{I}}_{\tau}}& r^2(L \phi_{\geq 1})^2\,d\omega dv \leq C\cdot E^{\epsilon}_{1,\mathcal{I}}[\psi] (1+\tau)^{-4+\epsilon},\\
\label{eq:extrarweightsl1hor}
\int_{{N}^{\mathcal{H}}_{\tau}}& (r-M)^{-2}(\underline{L} \phi_{\geq 1})^2\,d\omega du \leq C\cdot E^{\epsilon}_{1,\mathcal{H}}[\psi](1+\tau)^{-4+\epsilon},\\
\label{eq:edecayl11b}
\int_{\Sigma_{\tau}}& J^T[ \psi_{\geq 1}]\cdot \mathbf{n}_{\tau}\,d\mu_{\tau}\leq C\cdot{E}^{\epsilon}_{2}[\psi] (1+\tau)^{-6+\epsilon}.\end{aligned}$$ with $$\begin{aligned}
E^{\epsilon}_{1,\mathcal{I}}[\psi]:=&\: E^{\epsilon}_{1}[\psi]+ \int_{{N}^{\mathcal{I}}_0} r^{2-\epsilon}(LP_{\geq 1}\Phi_{(2)})^2+r^{4-\epsilon}(LP_1\widetilde{\Phi}_{(1)})^2\,d\omega dv,\\
E^{\epsilon}_{1,\mathcal{H}}[\psi]:=&\: E^{\epsilon}_{1}[\psi]+ \int_{{N}^{\mathcal{H}}_0}(r-M)^{-2+\epsilon}(\underline{L}P_{\geq 1}\underline{\Phi}_{(2)})^2\,d\omega du,\\
{E}^{\epsilon}_{2}[\psi]:=&\: \int_{{N}^{\mathcal{I}}} r^{2-\epsilon}(LP_{\geq 1}\Phi_{(2)})^2+r^{1-\epsilon}(LTP_{\geq 1}\Phi_{(2)})^2+r^{4-\epsilon}(LP_1\widetilde{\Phi}_{(1)})^2+r^{3-\epsilon}(LTP_1\widetilde{\Phi}_{(1)})^2\,d\omega dv\\
&+ \sum_{n=0}^1 \sum_{m=0}^{4-2n} \int_{{N}^{\mathcal{I}}} r^{2}(LT^mP_{\geq 1}\Phi_{(n)})^2+r^{1}(LT^{1+m}P_{\geq 1}\Phi_{(n)})^2\,d\omega dv\\
&+\int_{{N}^{\mathcal{H}}}(r-M)^{-2+\epsilon}(\underline{L}P_{\geq 1}\underline{\Phi}_{(2)})^2+(r-M)^{-1+\epsilon}(\underline{L}TP_{\geq 1}\underline{\Phi}_{(2)})^2+(r-M)^{-4+\epsilon}(LP_1\widetilde{\underline{\Phi}}_{(1)})^2\\
&+(r-M)^{-3+\epsilon}(\underline{L}TP_1\widetilde{\underline{\Phi}}_{(1)})^2\,d\omega du\\
&+ \sum_{n=0}^1 \sum_{m=0}^{4-2n} \int_{{N}^{\mathcal{H}}} (r-M)^{-2}(\underline{L}T^mP_{\geq 1}\underline{\Phi}_{(n)})^2+(r-M)^{-1}(\underline{L}T^{1+m}P_{\geq 1}\underline{\Phi}_{(n)})^2\,d\omega du\\
&+\sum_{m=0}^{6}\int_{\Sigma_0} J^T[T^m\psi_{\geq 1}]\cdot \mathbf{n}_{\Sigma_0}\,d\mu_{\Sigma_0}.\end{aligned}$$
In order to prove we proceed as in the proof of Proposition \[prop:edaypsi1\], but we consider additionally the $r$-weighted estimates in Proposition \[prop:generalrpest\] for $n=2$ with $p=2-\epsilon$ and the $r$-weighted estimates in Proposition \[prop:rpestell01\] for $n=1$ with $p=4-\epsilon$, resulting in an additional power in the energy decay rate.
In the process of deriving , we also obtain and \[eq:extrarweightsl1hor\], but we require a weaker energy norm on the right-hand side. That is to say, the energy norm will contain higher powers in r $(r-M)^{-1}$ on the right hand side (and not both, as in the norm ${E}^{\epsilon}_{2}[\psi]$).
Improved energy decay estimates for time derivatives {#sec:hoedecayest}
----------------------------------------------------
In this section, we obtain improved decay estimates for the time-derivatives $T^J\psi$.
\[prop:hoedaypsi0\] Let $J\in {\mathbb{N}}_0$ and assume that for all $0\leq j\leq J-1$, $$\begin{aligned}
\lim_{v\to \infty} r^{j+2} L^{j+1}\phi_0(u_0,v)<&\infty.\end{aligned}$$ Then, for all $\epsilon>0$, there exists a constant $C=C(M,{\Sigma_0},\epsilon,J)>0$ such that $$\begin{aligned}
\label{eq:hoedecayl01a}
\int_{\Sigma_{\tau}}& J^T[ T^J\psi_{0}]\cdot \mathbf{n}_{\tau}\,d\mu_{\tau}\leq C\cdot E^{\epsilon}_{0; J}[\psi] (1+\tau)^{-3-2J+\epsilon}, \\
\label{eq:hor2edecayl01}
\sum_{\substack{j_1+j_2=J\\ j_1\geq 0, j_2\geq 0}}\int_{{N}^{\mathcal{I}}_{\tau}}& r^{2+2j_1}\cdot (L^{1+j_1}T^{j_2}\phi_0)^2\,d\omega dv+\int_{{N}^{\mathcal{H}}_{\tau}} (r-M)^{-2-2j_2}\cdot (\underline{L}^{j_1+1} T^{j_2}\phi_0)^2\,d\omega du\\ \nonumber
\leq&\: C\cdot E^{\epsilon}_{0;J}[\psi] (1+\tau)^{-1-2j_2+\epsilon}.\end{aligned}$$ with $$E^{\epsilon}_{0;J}[\psi]:= \int_{{N}^{\mathcal{I}}} r^{3+2J-\epsilon}(L^{1+J}\phi_0)^2\,d\omega dv+\int_{{N}^{\mathcal{H}}} (r-M)^{-3-2J+\epsilon}(\underline{L}^{1+J}\phi_0)^2\,d\omega du+\sum_{j=0}^J\int_{\Sigma_0} J^T[T^j\psi_0]\cdot \mathbf{n}_{\Sigma_0}\,d\mu_{\Sigma_0}.$$
We follow the steps in the proof of Proposition \[prop:edaypsi0\], applied to $T^J\psi$ instead of $\psi$ and we *extend* the hierarchy of $r$-weighted estimates in the case $J\geq 1$ by employing the additional $r$-weighted estimates in Proposition \[prop:rpestell01Lkpsi\] and relating them to the original hierarchy of estimates (for $J=0$) via and .
In analogy with Lemma \[lm:auxedaypsi0\], we now obtain auxiliary decay estimates for $J\geq 1$ that will be useful when improving the estimates Proposition \[prop:hoedaypsi0\] in the setting where either $H_0[\psi]=0$ or $I_0[\psi]=0$.
\[lm:hoauxedaypsi0\] Let $J\in {\mathbb{N}}_0$ and assume that for all $0\leq j\leq J-1$, $$\begin{aligned}
\lim_{v\to \infty} r^{j+3} L^{j+1}\phi_0(u_0,v)<&\infty.\end{aligned}$$ Then, for all $\epsilon>0$, there exists a constant $C=C(M,{\Sigma_0},\epsilon,J)>0$ such that $$\begin{aligned}
\label{eq:hoauxdecayl01}
\sum_{\substack{j_1+j_2=J\\ j_1\geq 0, j_2\geq 0}}&\int_{{N}^{\mathcal{I}}_{\tau}}r^{2+2j_1}\cdot (L^{1+j_1}T^{j_2}\phi_0)^2\,d\omega dv\\
\leq&\: C\cdot \left[E^{\epsilon}_{0;J}[\psi]+ \int_{{N}^{\mathcal{I}}_{0}} r^{4+2J-\epsilon}\cdot (L^{1+J}\phi_0)^2\,d\omega dv\right] (1+\tau)^{-2-2j_2+\epsilon},\\
\label{eq:hoauxdecayl02}
\sum_{\substack{j_1+j_2=J\\ j_1\geq 0, j_2\geq 0}}&\int_{{N}^{\mathcal{H}}_{\tau}} (r-M)^{-2-2j_2}\cdot (\underline{L}^{j_1+1} T^{j_2}\phi_0)^2\,d\omega du\\
\leq &\: C\cdot \left[E^{\epsilon}_{0;J}[\psi]+\int_{{N}^{\mathcal{H}}_{0}} (r-M)^{-4-2J+\epsilon}\cdot (\underline{L}^{1+J} \phi_0)^2\,d\omega du\right] (1+\tau)^{-2-2j_2+\epsilon}.\end{aligned}$$ Furthermore, $$\begin{aligned}
\label{eq:hoauxpointdecayl01}
||r^{J}L^J\phi_0||^2_{L^{\infty}({N}^{\mathcal{I}}_{\tau})}\leq&\: C\cdot \left[E^{\epsilon}_{0}[\psi]+\int_{{N}^{\mathcal{I}}_{0}} r^{4+2J-\epsilon}\cdot (L^{1+J}\phi_0)^2\,d\omega dv\right] (1+\tau)^{-\frac{5}{2}+\epsilon},\\
\label{eq:hoauxpointdecayl02}
||r^{J}L^J\phi_0||^2_{L^{\infty}({N}^{\mathcal{I}}_{\tau})}\leq &\: C\cdot \left[E^{\epsilon}_{0}[\psi]+ \int_{{N}^{\mathcal{H}}_{0}} (r-M)^{-4-2J+\epsilon}\cdot (\underline{L}^{1+J} \phi_0)^2\,d\omega du\right] (1+\tau)^{-\frac{5}{2}+\epsilon}.\end{aligned}$$
We obtain and by adding one estimate to the hierarchies of Proposition \[prop:hoedaypsi0\], in analogy to the proof of Lemma \[lm:auxedaypsi0\]. In order to obtain , we need additionally use the estimate $$\int_{{N}^{\mathcal{I}}_{\tau}}r^{1+J}\cdot (L^{1+J}\phi_0)^2\,d\omega dv\leq C\cdot \left[E^{\epsilon}_{0;J}[\psi]+\int_{{N}^{\mathcal{I}}_{0}} r^{4+2J-\epsilon}\cdot (L^{1+J}\phi_0)^2\,d\omega dv\right] (1+\tau)^{-3-2J+\epsilon}.$$ In order to prove , we then apply the fundamental theorem of calculus (together with the Morawetz estimate and the Hardy inequality ) to estimate $r^{2J} \chi^2 (L^J\Phi_0)^2$ using the estimate above together with . We arrive at by applying similar arguments in the region $\mathcal{A}^{\mathcal{H}}$.
The $L^{\infty}$ estimates in Lemma \[lm:hoauxedaypsi0\] allow us to retrieve the assumptions and so that we can use the *full* hierarchy of $r$-weighted estimates in Proposition \[prop:hoedaypsi0\] to obtain the following analog of Proposition \[prop:extraendecay\]:
\[prop:hoextraendecay\] Let $J\in {\mathbb{N}}_0$. Then, for all $\epsilon>0$, there exists a constant $C=C(M,{\Sigma_0},\epsilon,J)>0$ such that $$\begin{aligned}
\label{eq:hooptr2decay1}
\sum_{\substack{j_1+j_2=J\\ j_1\geq 0, j_2\geq 0}}\int_{{N}^{\mathcal{I}}_{\tau}}r^{2+2j_1}\cdot (L^{1+j_1}T^{j_2}\phi_0)^2\,d\omega dv \leq&\: C\cdot E^{\epsilon}_{0,\mathcal{I};J}[\psi] (1+\tau)^{-3-2j_2+\epsilon},\\
\label{eq:hooptr2decay2}
\sum_{\substack{j_1+j_2=J\\ j_1\geq 0, j_2\geq 0}}\int_{{N}^{\mathcal{H}}_{\tau}} (r-M)^{-2-2j_2}\cdot (\underline{L}^{1+j_1} T^{j_2}\phi_0)^2\,d\omega du \leq&\: C\cdot E^{\epsilon}_{0,\mathcal{H};J}[\psi] (1+\tau)^{-3-2j_2+\epsilon},\end{aligned}$$ where $$\begin{aligned}
E^{\epsilon}_{0,\mathcal{I};J}[\psi]=:&\: E^{\epsilon}_{0;J}[\psi]+ \int_{{N}^{\mathcal{I}}_{0}} r^{5+2J-\epsilon}\cdot (L^{1+J}\phi_0)^2\,d\omega dv,\\
E^{\epsilon}_{0,\mathcal{H};J}[\psi]=:&\: E^{\epsilon}_{0;J}[\psi]+ \int_{{N}^{\mathcal{H}}_{0}} (r-M)^{-5-2J+\epsilon}\cdot (\underline{L}^{1+J} \phi_0)^2\,d\omega du,\end{aligned}$$ and we assume for and for .
We now define the following higher-order weighted energy norms for $\psi_{\geq 1}$ and $J\in {\mathbb{N}}_0$. $$\begin{split}
E^{\epsilon}_{1;J}[\psi]:=&\: \sum_{\substack{0\leq j+|\alpha|\leq J\\ |\alpha|\leq J}} \int_{{N}^{\mathcal{I}}} r^{1-\epsilon}|{\slashed{\nabla}}_{{\mathbb{S}}^2}^{\alpha}LP_{\geq 1}T^j\Phi_{(2)}|^2\,d\omega dv\\
&+ \sum_{n=0}^1 \sum_{m=0}^{3-2n+2J} \int_{{N}^{\mathcal{I}}} r^{2}(LT^mP_{\geq 1}\Phi_{(n)})^2+r^{1}(LT^{1+m}P_{\geq 1}\Phi_{(n)})^2\,d\omega dv\\
&+\sum_{\substack{|\alpha|\leq \max\{0,J-1\}\\ i\leq \max\{J-1,0\}\\ 0\leq j+|\alpha|\leq J-2i+\min\{J,1\}}} \int_{{N}^{\mathcal{I}}} r^{1+2i-\epsilon}|{\slashed{\nabla}}_{{\mathbb{S}}^2}^{\alpha}L^{1+i}P_{\geq 1}T^j\Phi_{(2)}|^2\, d\omega dv\\
&+\sum_{\substack{|\alpha|\leq \max\{0,J-1\}\\ i\leq J\\ 0\leq j+|\alpha|\leq 2J-2i+1}} \int_{{N}^{\mathcal{I}}} r^{2i-\epsilon}|{\slashed{\nabla}}_{{\mathbb{S}}^2}^{\alpha}L^{1+i}P_{\geq 1}T^j\Phi_{(2)}|^2\, d\omega dv\\
&+\int_{{N}^{\mathcal{I}}} r^{2+2J-\epsilon}(L^{1+J}P_{\geq 1}\Phi_{(2)})^2\,d\omega dv\\
&+\sum_{\substack{0\leq j+|\alpha|\leq J\\ |\alpha|\leq J}} \int_{{N}^{\mathcal{H}}} (r-M)^{-1+\epsilon}|{\slashed{\nabla}}_{{\mathbb{S}}^2}^{\alpha}\underline{L}P_{\geq 1}T^j\underline{\Phi}_{(2)}|^2\,d\omega du\\
&+ \sum_{n=0}^1 \sum_{m=0}^{3-2n+2J} \int_{{N}^{\mathcal{H}}} (r-M)^{-2}(\underline{L}T^mP_{\geq 1}\underline{\Phi}_{(n)})^2+(r-M)(\underline{L}T^{1+m}P_{\geq 1}\underline{\Phi}_{(n)})^2\,d\omega du\\
&+\sum_{\substack{|\alpha|\leq \max\{0,J-1\}\\ i\leq \max\{J-1,0\}\\ 0\leq j+|\alpha|\leq J-2i+\min\{J,1\}}} \int_{{N}^{\mathcal{H}}} (r-M)^{-1-2i+\epsilon}|{\slashed{\nabla}}_{{\mathbb{S}}^2}^{\alpha}\underline{L}^{1+i}P_{\geq 1}T^j\underline{\Phi}_{(2)}|^2\, d\omega du\\
&+\sum_{\substack{|\alpha|\leq \max\{0,J-1\}\\ i\leq J\\ 0\leq j+|\alpha|\leq 2J-2i+1}} \int_{{N}^{\mathcal{H}}} (r-M)^{-2i+\epsilon}|{\slashed{\nabla}}_{{\mathbb{S}}^2}^{\alpha}\underline{L}^{1+i}P_{\geq 1}T^j\underline{\Phi}_{(2)}|^2\, d\omega du\\
&+\int_{{N}^{\mathcal{H}}} (r-M)^{-1-2J+\epsilon}(\underline{L}^{1+J}P_{\geq 1}\underline{\Phi}_{(2)})^2\,d\omega du\\
&+\sum_{\substack{j+|\alpha|\leq 5+3k\\ |\alpha|\leq k}}\int_{\Sigma_0} J^T[T^j\Omega^{\alpha}\psi_{\geq 1}]\cdot \mathbf{n}_{\Sigma_0}\,d\mu_{\Sigma_0}.
\end{split}$$ and $$\begin{split}
E^{\epsilon}_{2;J}[\psi]:=&\: \sum_{\substack{0\leq j+|\alpha|\leq J\\ |\alpha|\leq J}} \int_{{N}^{\mathcal{I}}} r^{2-\epsilon}|{\slashed{\nabla}}_{{\mathbb{S}}^2}^{\alpha}LP_{\geq 1}T^j\Phi_{(2)}|^2+r^{1-\epsilon}|{\slashed{\nabla}}_{{\mathbb{S}}^2}^{\alpha}LT^{1+j}P_{\geq 1}\Phi_{(2)}|^2+r^{4-\epsilon}(LP_1T^j\widetilde{\Phi}_{(1)})^2\,d\omega dv\\
&+ \sum_{n=0}^1 \sum_{m=0}^{4-2n+2J} \int_{{N}^{\mathcal{I}}} r^{2}(LT^mP_{\geq 1}\Phi_{(n)})^2+r^{1}(LT^{1+m}P_{\geq 1}\Phi_{(n)})^2\,d\omega dv\\
&+\sum_{\substack{|\alpha|\leq \max\{0,J-1\}\\ i\leq \max\{J-1,0\}\\ 0\leq j+|\alpha|\leq J-2i+\min\{J,1\}}} \int_{{N}^{\mathcal{I}}} r^{2+2i-\epsilon}|{\slashed{\nabla}}_{{\mathbb{S}}^2}^{\alpha}L^{1+i}P_{\geq 1}T^j\Phi_{(2)}|^2\, d\omega dv\\
&+\sum_{\substack{|\alpha|\leq \max\{0,J-1\}\\ i\leq J\\ 0\leq j+|\alpha|\leq 2J-2i+1}} \int_{{N}^{\mathcal{I}}} r^{1+2i-\epsilon}|{\slashed{\nabla}}_{{\mathbb{S}}^2}^{\alpha}L^{1+i}P_{\geq 1}T^j\Phi_{(2)}|^2\, d\omega dv\\
&+\int_{{N}^{\mathcal{I}}} r^{2+2J-\epsilon}(L^{1+J}P_{\geq 1}\Phi_{(2)})^2\,d\omega dv\\
&+\sum_{\substack{0\leq j+|\alpha|\leq J\\ |\alpha|\leq J}} \int_{{N}^{\mathcal{H}}} (r-M)^{-2+\epsilon}|{\slashed{\nabla}}_{{\mathbb{S}}^2}^{\alpha}\underline{L}P_{\geq 1}T^j\underline{\Phi}_{(2)}|^2+(r-M)^{-1+\epsilon}|{\slashed{\nabla}}_{{\mathbb{S}}^2}^{\alpha}\underline{L}T^{1+j}P_{\geq 1}\underline{\Phi}_{(2)}|^2\\
&+(r-M)^{-4+\epsilon}(\underline{L}P_1T^j\widetilde{\underline{\Phi}}_{(1)})^2\,d\omega du\\
&+ \sum_{n=0}^1 \sum_{m=0}^{4-2n+2J} \int_{{N}^{\mathcal{H}}} (r-M)^{-2}(\underline{L}T^mP_{\geq 1}\underline{\Phi}_{(n)})^2+(r-M)(\underline{L}T^{1+m}P_{\geq 1}\underline{\Phi}_{(n)})^2\,d\omega du\\
&+\sum_{\substack{|\alpha|\leq \max\{0,J-1\}\\ i\leq \max\{J-1,0\}\\ 0\leq j+|\alpha|\leq J-2i+\min\{J,1\}}} \int_{{N}^{\mathcal{H}}} (r-M)^{-2-2i+\epsilon}|{\slashed{\nabla}}_{{\mathbb{S}}^2}^{\alpha}\underline{L}^{1+i}P_{\geq 1}T^j\underline{\Phi}_{(2)}|^2\, d\omega du\\
&+\sum_{\substack{|\alpha|\leq \max\{0,J-1\}\\ i\leq J\\ 0\leq j+|\alpha|\leq 2J-2i+1}} \int_{{N}^{\mathcal{H}}} (r-M)^{-1-2i+\epsilon}|{\slashed{\nabla}}_{{\mathbb{S}}^2}^{\alpha}\underline{L}^{1+i}P_{\geq 1}T^j\underline{\Phi}_{(2)}|^2\, d\omega du\\
&+\int_{{N}^{\mathcal{H}}} (r-M)^{-2-2J+\epsilon}(\underline{L}^{1+J}P_{\geq 1}\underline{\Phi}_{(2)})^2\,d\omega du\\
&+\sum_{\substack{j+|\alpha|\leq 6+3k\\ |\alpha|\leq k}}\int_{\Sigma_0} J^T[T^j\Omega^{\alpha}\psi_{\geq 1}]\cdot \mathbf{n}_{\Sigma_0}\,d\mu_{\Sigma_0}.
\end{split}$$ We note that the integrals appearing in the energy norm $E^{\epsilon}_{1;J}[\psi]$ that are supported *away* from $\mathcal{H}^+$ are similar to the energy norms defined in Proposition 7.7 of [@paper1] and similarly, the integrals supported away from the horizon in $E^{\epsilon}_{2;J}[\psi]$ appear in the norms defined in Appendix A of [@paper2].
We obtain energy decay estimates for the time derivatives of $\psi_{\geq 1}$.
\[prop:hoedaypsi1\] Let $J\in {\mathbb{N}}_0$. Assume that for $n=0,1$ and for all $0\leq k\leq 2-n$ and $0\leq j\leq J$: $$\lim_{v\to \infty}\int_{{\mathbb{S}}^2}r^{2j}|{\slashed{\nabla}}_{{\mathbb{S}}^2}\slashed{\Delta}_{{\mathbb{S}}^2}^kL^jP_{\geq 1}\Phi_{(n)}|^2\,d\omega|_{u=u_0}<\infty.$$ Then, for all $\epsilon>0$ there exists a constant $C=C(M,{\Sigma_0},\epsilon,J)>0$ such that $$\begin{aligned}
\label{eq:hoedecayl01}
\int_{\Sigma_{\tau}} J^T[ T^J\psi_{\geq 1}]\cdot \mathbf{n}_{\tau}\,d\mu_{\tau}\leq&\: C\cdot E^{\epsilon}_{1;J}[\psi] (1+\tau)^{-5-2J+\epsilon},\\
\label{eq:hoedecaylr211}
\int_{{N}^{\mathcal{I}}_{\tau}} r^2(LT^J\phi_{\geq 1})^2\,d\omega dv+\int_{{N}^{\mathcal{H}}_{\tau}} (r-M)^{-2}(\underline{L}T^J\phi_{\geq 1})^2\,d\omega du \leq&\: C\cdot E^{\epsilon}_{1;J}[\psi] (1+\tau)^{-3-2J+\epsilon}.\end{aligned}$$
The $J=0$ case follows from Proposition \[prop:edaypsi1\]. In order to obtain the estimates for $J\geq 1$ we repeat the steps in the proof of Proposition \[prop:edaypsi1\] applied to $T^J\phi_{\geq 1}$, but we use additionally that for $J\geq 1$ we can extend the hierarchy of $r$-weighted estimates by using the $r$-weighted estimates for $L^J\Phi_{(2)}$ that follow from Proposition \[prop:generalrpestLkpsi\] and combining them with the estimates in Lemma \[lm:intestTpsi\] to extend the hierarchy for $T^J\Phi_{(2)}$. Note that we need to distinguish between $P_{\geq 2}\Phi_{(2)}$ and $P_{1}\Phi_{(2)}$ when estimating $J=0$ terms, so for $J\geq 1$ there is no further need to perform the splitting $P_{\geq 2}L^J\Phi_{(2)}=P_{1}L^J\Phi_{(2)}+P_{\geq 2}L^J\Phi_{(2)}$. We omit further details of the proof and refer to the proof of Proposition 7.7 in [@paper1] for more details on estimates that are analogous to the estimates in $\mathcal{A}^{\mathcal{I}}$ in extremal Reissner–Nordström. The estimates in $\mathcal{A}^{\mathcal{H}}$ follow via very similar arguments.
\[prop:hoedaypsi1b\] Let $J\in {\mathbb{N}}_0$. Assume that $$\lim_{v\to \infty}\sum_{j=0}^J\sum_{n=0}^2 \sum_{k=0}^{2-n}\int_{{\mathbb{S}}^2}r^{2j}|{\slashed{\nabla}}_{{\mathbb{S}}^2}\slashed{\Delta}_{{\mathbb{S}}^2}^kL^jP_{\geq 1}\Phi_{(n)}|^2\,d\omega|_{u=u_0}<\infty.$$ Then, for all $\epsilon>0$ there exists a constant $C=C(M,{\Sigma_0},\epsilon,J)>0$ such that $$\begin{aligned}
\label{eq:hoedecaylr211inf}
\int_{{N}^{\mathcal{I}}_{\tau}} r^2(LT^J\phi_{\geq 1})^2\,d\omega dv\leq&\: C\cdot E^{\epsilon}_{1,\mathcal{I};J}[\psi] (1+\tau)^{-4-2J+\epsilon},\\
\label{eq:hoedecaylr211hor}
\int_{{N}^{\mathcal{H}}_{\tau}} (r-M)^{-2}(\underline{L}T^J\phi_{\geq 1})^2\,d\omega du \leq&\: C\cdot E^{\epsilon}_{1,\mathcal{H};J}[\psi] (1+\tau)^{-4-2J+\epsilon},\\
\label{eq:hoedecayl01b}
\int_{\Sigma_{\tau}} J^T[ T^J\psi_{\geq 1}]\cdot \mathbf{n}_{\tau}\,d\mu_{\tau}\leq&\: C\cdot E^{\epsilon}_{2;J}[\psi] (1+\tau)^{-6-2J+\epsilon},\\\end{aligned}$$ where $$\begin{aligned}
E^{\epsilon}_{1,\mathcal{I};J}[\psi]:=&\: E^{\epsilon}_{1;J}[\psi] + \sum_{\substack{0\leq j+|\alpha|\leq J\\ |\alpha|\leq J}} \int_{{N}^{\mathcal{I}}} r^{2-\epsilon}|{\slashed{\nabla}}_{{\mathbb{S}}^2}^{\alpha}LP_{\geq 1}T^j\Phi_{(2)}|^2+r^{4-\epsilon}(LP_1T^j\widetilde{\Phi}_{(1)})^2\,d\omega dv \\
&+\sum_{\substack{|\alpha|\leq \max\{0,J-1\}\\ i\leq \max\{J-1,0\}\\ 0\leq j+|\alpha|\leq J-2i+\min\{J,1\}}} \int_{{N}^{\mathcal{I}}} r^{2+2i-\epsilon}|{\slashed{\nabla}}_{{\mathbb{S}}^2}^{\alpha}L^{1+i}P_{\geq 1}T^j\Phi_{(2)}|^2\, d\omega dv\\
&+\int_{{N}^{\mathcal{I}}} r^{2+2J-\epsilon}(L^{1+J}P_{\geq 1}\Phi_{(2)})^2\,d\omega dv,\\
E^{\epsilon}_{1,\mathcal{H};J}[\psi]:=&\: E^{\epsilon}_{1;J}[\psi] + \sum_{\substack{0\leq j+|\alpha|\leq J\\ |\alpha|\leq J}} \int_{{N}^{\mathcal{H}}} (r-M)^{-2+\epsilon}|{\slashed{\nabla}}_{{\mathbb{S}}^2}^{\alpha}\underline{L}P_{\geq 1}T^j\underline{\Phi}_{(2)}|^2+(r-M)^{-4+\epsilon}(\underline{L}P_1T^j\widetilde{\underline{\Phi}}_{(1)})^2\,d\omega du \\
&+\sum_{\substack{|\alpha|\leq \max\{0,J-1\}\\ i\leq \max\{J-1,0\}\\ 0\leq j+|\alpha|\leq J-2i+\min\{J,1\}}} \int_{{N}^{\mathcal{H}}} (r-M)^{-2-2i+\epsilon}|{\slashed{\nabla}}_{{\mathbb{S}}^2}^{\alpha}\underline{L}^{1+i}P_{\geq 1}T^j \underline{\Phi}_{(2)}|^2\, d\omega du\\
&+\int_{{N}^{\mathcal{H}}} (r-M)^{-2-2J+\epsilon}(\underline{L}^{1+J}P_{\geq 1}\underline{\Phi}_{(2)})^2\,d\omega du.\\\end{aligned}$$
For $J=0$, the estimates in the proposition are contained in Proposition \[prop:edaypsi1b\]. In order to prove the estimates in the $J\geq 1$ case, we proceed as in Proposition \[prop:hoedaypsi1\] but we moreover apply the *extended* hierarchy in the $J=0$ case in $\mathcal{A}^{\mathcal{I}}$ or $\mathcal{A}^{\mathcal{H}}$ (or in both regions) that is used in the proof of Proposition \[prop:edaypsi1b\].
Degenerate elliptic estimates for $\psi_{\geq 1}$ {#sec:ellpest}
-------------------------------------------------
In this section we derive a degenerate elliptic estimate for solutions $\psi_{\geq 1}$ to that will be used to obtain pointwise decay estimates for $\psi_{\geq 1}$.
\[prop:degenelliptic\] Let $\psi$ be a solution to . Assume moreover that $$\begin{aligned}
\lim_{\rho\to \infty}r^{\frac{1}{2}}T\psi|_{\Sigma_{\tau}}=&0,\\
\lim_{\rho\to \infty}r^{\frac{1}{2}}L\psi|_{\Sigma_{\tau}}=&0.\end{aligned}$$
Then there exists a $C=C(M,\Sigma_0)>0$, such that, with respect to $(\rho,\theta,\varphi)$ coordinates, $$\label{eq:ellipticpsi}
\begin{split}
\int_{M}^{\infty}&\int_{{\mathbb{S}}^2} Dr^2(\partial_{\rho}( D\partial_\rho \psi_{\geq 1}))^2+D^2|{\slashed{\nabla}}_{{\mathbb{S}}^2} \partial_{\rho}\psi_{\geq 1}|^2+Dr^{-2}(\slashed{\Delta}_{{\mathbb{S}}^2}\psi_{\geq 1})^2\,d\omega d\rho\\
\leq&\: C\int_{M}^{\infty}\int_{{\mathbb{S}}^2}Dr^2(\partial_{\rho} T \psi_{\geq 1})^2+Dh(r)^2r^2(T^2\psi_{\geq 1})^2\,d\omega d\rho.
\end{split}$$
For the sake of convenience, we will assume that $\int_{{\mathbb{S}}^2}\psi\,d\omega=0$ so that $\psi=\psi_{\geq 1}$. By Lemma 7.9 in [@paper1] it follows that on (extremal) Reissner–Nordström reduces to the following equation $$\label{eq:waveeqrhocoord}
\partial_{\rho}(Dr^2\partial_{\rho}\psi)+\slashed{\Delta}_{{\mathbb{S}}^2}\psi=-2(1-h\cdot D)r^2 \partial_{\rho}T\psi+(2-h\cdot D)r^2hT^2\psi+((hDr^2)' -2r)T\psi.$$ By squaring both sides of and multiplying the resulting equation with the factor $Dr^{-2}$, we obtain the following estimate: $$\label{eq:waveeqestext}
\begin{split}
\int_{M}^{\infty}\int_{{\mathbb{S}}^2} &r^{-2}D(\partial_{\rho}(Dr^2\partial_{\rho}\psi))^2+r^{-2}D(\slashed{\Delta}_{{\mathbb{S}}^2}\psi)^2+2r^{-2}D\partial_{\rho}(Dr^2\partial_{\rho}\psi)\slashed{\Delta}_{{\mathbb{S}}^2}\psi\,d\omega d\rho\\
\leq&\: C\int_{M}^{\infty}\int_{{\mathbb{S}}^2}r^2D(\partial_{\rho} T \psi)^2+h(r)^2Dr^2(T^2\psi)^2+D(T\psi)^2\,d\omega d\rho.
\end{split}$$
By applying again as follows $$\int_{M}^{\infty}(T\psi)^2\,d\rho\leq 4\int_{M}^{\infty}(r-M)^2(\partial_{\rho} T\psi)^2\,d\rho,$$ so that $$\label{eq:waveeqestextv2}
\begin{split}
\int_{M}^{\infty}\int_{{\mathbb{S}}^2} &r^{-2}D(\partial_{\rho}(Dr^2\partial_{\rho}\psi))^2+r^{-2}D(\slashed{\Delta}_{{\mathbb{S}}^2}\psi)^2+2r^{-2}D\partial_{\rho}(Dr^2\partial_{\rho}\psi)\slashed{\Delta}_{{\mathbb{S}}^2}\psi\,d\omega d\rho\\
\leq&\: C\int_{M}^{\infty}\int_{{\mathbb{S}}^2}Dr^2(\partial_{\rho} T \psi)^2+Dh(r)^2r^2(T^2\psi)^2\,d\omega d\rho.
\end{split}$$ We first consider the mixed derivative term on the left-hand side of . We integrate over ${\mathbb{S}}^2$ and integrate by parts in $\rho$ and the angular variables: $$\begin{split}
\label{eq:ibpangularext}
\int_{M}^{\infty} \int_{{\mathbb{S}}^2}2Dr^{-2}\partial_{\rho}(Dr^2\partial_{\rho}\psi)\slashed{\Delta}_{{\mathbb{S}}^2}\psi\,d\omega d\rho=&\:\int_{M}^{\infty} \int_{{\mathbb{S}}^2}-2Dr^{2}\partial_r(Dr^{-2})\partial_{\rho}\psi\slashed{\Delta}_{{\mathbb{S}}^2}\psi-2D^2\partial_{\rho}\psi\slashed{\Delta}_{{\mathbb{S}}^2}\partial_{\rho}\psi\,d\omega d\rho\\
=&\:\int_{M}^{\infty} \int_{{\mathbb{S}}^2}-2Dr^{2}\partial_r(Dr^{-2})\partial_{\rho}\psi\slashed{\Delta}_{{\mathbb{S}}^2}\psi+2D^2|{\slashed{\nabla}}_{{\mathbb{S}}^2}\partial_{\rho}\psi|^2\,d\omega d\rho,
\end{split}$$ where we used that all resulting boundary terms vanish.
Note that $$\begin{split}
\partial_r(Dr^{-2})&=\partial_r((r^{-1}-Mr^{-2})^2)=-2(r^{-1}-Mr^{-2})(r^{-2}-2Mr^{-3})\\
&=-2r^{-3}\left(1-\frac{M}{r}\right)\left(1-\frac{2M}{r}\right).
\end{split}$$
We now apply Cauchy–Schwarz and to estimate the first term inside the integral on the very right-hand side above: $$\begin{split}
\int_{M}^{\infty}\int_{{\mathbb{S}}^2}\left|2r^{2}D\partial_r(Dr^{-2})\partial_{\rho}\psi\slashed{\Delta}_{{\mathbb{S}}^2}\psi\right|\,d\omega d\rho\leq&\: \int_{M}^{\infty}Dr^{6}(\partial_r(Dr^{-2}))^2(\partial_{\rho}\psi)^2+r^{-2}D(\slashed{\Delta}_{{\mathbb{S}}^2}\psi)^2\,d\omega d\rho\\
= &\: \int_{M}^{\infty}4D^2\left(1-\frac{2M}{r}\right)^2(\partial_{\rho}\psi)^2+r^{-2}D(\slashed{\Delta}_{{\mathbb{S}}^2}\psi)^2\,d\omega d\rho\\
\leq &\:\int_{M}^{\infty}\int_{{\mathbb{S}}^2}2D^2|{\slashed{\nabla}}_{{\mathbb{S}}^2}\partial_{\rho}\psi|^2+r^{-2}D(\slashed{\Delta}_{{\mathbb{S}}^2}\psi)^2\,d\omega d\rho,
\end{split}$$ where we used moreover that $\left|1-\frac{2M}{r}\right|\leq 1$.
We use the above estimates together with to estimate: $$\label{eq:mainineqellipticext}
\begin{split}
\int_{M}^{\infty}\int_{{\mathbb{S}}^2} &Dr^{-2}(\partial_{\rho}(Dr^2\partial_{\rho}\psi))^2+r^{-2}D(1-|D|)(\slashed{\Delta}_{{\mathbb{S}}^2}\psi)^2\,d\omega d\rho\\
\leq&\: C\int_{M}^{\infty}D(\partial_{\rho} T \psi)^2r^2+Dh(r)^2(T^2\psi)^2\,d\omega d\rho.
\end{split}$$
The above estimate allows us to conclude that $$\label{eq:mainineqellipticext3}
\begin{split}
\int_{M}^{\infty}&\int_{{\mathbb{S}}^2} Dr^{-2}(\partial_{\rho}(Dr^2\partial_{\rho}\psi))^2+Dr^{-2}(\slashed{\Delta}_{{\mathbb{S}}^2}\psi)^2+D^2|{\slashed{\nabla}}_{{\mathbb{S}}^2}\partial_{\rho}\psi|^2\,d\omega d\rho\\
\leq&\: C\int_{M}^{\infty}\int_{{\mathbb{S}}^2}D(\partial_{\rho} T \psi)^2r^2+r^2Dh(r)^2(T^2\psi)^2\,d\omega d\rho.
\end{split}$$ Furthermore, we can decompose $$\begin{split}
Dr^{-2}(\partial_{\rho}(Dr^2\partial_{\rho}\psi))^2=&\: Dr^{-2}\left[r^2\partial_{\rho}(D\partial_{\rho}\psi)+2rD\partial_{\rho}\psi\right]^2\\
=&\: r^2D(\partial_{\rho}(D\partial_{\rho}\psi))^2+4D(D\partial_{\rho}\psi)^2+4rD^2\partial_{\rho}\psi\partial_{\rho}(D\partial_{\rho}\psi).
\end{split}$$ By Young’s inequality and we have that $$\begin{split}
\int_{{\mathbb{S}}^2}4rD^2\partial_{\rho}\psi\partial_{\rho}(D\partial_{\rho}\psi)\,d\omega\geq& -\frac{1}{2}\int_{{\mathbb{S}}^2}r^2D(\partial_{\rho}(D\partial_{\rho}\psi))^2\,d\omega -8\int_{{\mathbb{S}}^2}D^3(\partial_{\rho}\psi)^2\,d\omega\\
\geq& -\frac{1}{2}\int_{{\mathbb{S}}^2}r^2D(\partial_{\rho}(D\partial_{\rho}\psi))^2\,d\omega -4\int_{{\mathbb{S}}^2}D^3|{\slashed{\nabla}}_{{\mathbb{S}}^2}\partial_{\rho}\psi|^2\,d\omega.
\end{split}$$ We then apply to conclude that $$\begin{split}
\int_{M}^{\infty}&\int_{{\mathbb{S}}^2} Dr^2(\partial_{\rho}(D\partial_{\rho}\psi))^2+Dr^{-2}(\slashed{\Delta}_{{\mathbb{S}}^2}\psi)^2+D^2|{\slashed{\nabla}}_{{\mathbb{S}}^2}\partial_{\rho}\psi|^2\,d\omega d\rho\\
\leq&\: C\int_{M}^{\infty}\int_{{\mathbb{S}}^2}D(\partial_{\rho} T \psi)^2r^2+r^2Dh(r)^2(T^2\psi)^2\,d\omega d\rho.
\end{split}$$
Pointwise decay estimates {#sec:pdecayest}
-------------------------
In this section we use the energy decay estimates from Section \[sec:edecayest\] and \[sec:hoedecayest\] to derive $L^{\infty}$ estimates.
\[prop:pointdecay\] Let $J\in {\mathbb{N}}_0$ and assume that for $n=0,1$ and for all $0\leq k\leq 2-n$ and $0\leq j\leq J$: $$\begin{aligned}
\lim_{v\to \infty}\int_{{\mathbb{S}}^2}r^{2j}|{\slashed{\nabla}}_{{\mathbb{S}}^2}\slashed{\Delta}_{{\mathbb{S}}^2}^kL^jP_{\geq 1}\Phi_{(n)}|^2\,d\omega|_{u=u_0}<&\:\infty.\end{aligned}$$ and for all $0\leq j\leq J-1$ $$\begin{aligned}
\lim_{v\to \infty} r^{j+2} L^{j+1}\phi_0(u_0,v)<&\infty.\end{aligned}$$ Then, for all $\epsilon>0$, there exists a constant $C=C(M,{\Sigma_0},\epsilon,J)>0$ such that $$\begin{aligned}
\label{eq:pdecayl01v1}
||(r-M)^{\frac{1}{2}}T^J \psi_0||_{L^{\infty}(\Sigma_{\tau})}\leq&\: C\cdot\sqrt{E^{\epsilon}_{0;J}[\psi]} (1+\tau)^{-\frac{3}{2}-J+\frac{\epsilon}{2}},\\
\label{eq:pdecayl01v2}
||rT^J \psi_0||_{L^{\infty}(\Sigma_{\tau})}\leq&\: C\cdot\sqrt{E^{\epsilon}_{0; J}[\psi]} (1+\tau)^{-1-J+\frac{\epsilon}{2}},\\
\label{eq:pdecaylgeq1v1a}
||(r-M)^{\frac{1}{2}} T^J \psi_{\geq 1}||_{L^{\infty}(\Sigma_{\tau})}\leq&\:C\cdot \sqrt{\sum_{|\alpha|\leq 2}E^{\epsilon}_{1;J}[\Omega^{\alpha}\psi]} (1+\tau)^{-\frac{5}{2}-J+\frac{\epsilon}{2}},\\
\label{eq:pdecaylgeq1v1b}
||(r-M)^{\frac{1}{2}}T^J \psi_{\geq 1}||_{L^{\infty}(\Sigma_{\tau})}\leq&\:C\cdot \sqrt{\sum_{|\alpha|\leq 2}E^{\epsilon}_{2;J}[\Omega^{\alpha}\psi]} (1+\tau)^{-3-J+\frac{\epsilon}{2}},\\
\label{eq:pdecaylgeq1v2a}
||rT^J \psi_{\geq 1}||_{L^{\infty}(\Sigma_{\tau})}\leq&\: C\cdot \sqrt{\sum_{|\alpha|\leq 2}E^{\epsilon}_{1;J}[\Omega^{\alpha}\psi]} (1+\tau)^{-2-J+\frac{\epsilon}{2}},\\
\label{eq:pdecaylgeq1v2b}
||rT^J \psi_{\geq 1}||_{L^{\infty}(\Sigma_{\tau})}\leq&\: C\cdot \sqrt{\sum_{|\alpha|\leq 2}E^{\epsilon}_{2;J}[\Omega^{\alpha}\psi]} (1+\tau)^{-\frac{5}{2}-J+\frac{\epsilon}{2}},\\
\label{eq:pdecaylgeq1v3a}
||\sqrt{D} T^J \psi_{\geq 1}||_{L^{\infty}(\Sigma_{\tau})}\leq&\:C\cdot \sqrt{\sum_{|\alpha|\leq 2}E^{\epsilon}_{1;J+1}[\Omega^{\alpha}\psi]} (1+\tau)^{-3-J+\frac{\epsilon}{2}},\\
\label{eq:pdecaylgeq1v3b}
||\sqrt{D}T^J \psi_{\geq 1}||_{L^{\infty}(\Sigma_{\tau})}\leq&\:C\cdot \sqrt{\sum_{|\alpha|\leq 2}E^{\epsilon}_{2;J+1}[\Omega^{\alpha}\psi]} (1+\tau)^{-\frac{7}{2}-J+\frac{\epsilon}{2}},\end{aligned}$$
In order to estimate , and , we apply the fundamental theorem of calculus along the foliation $\Sigma_{\tau}$ as follows: $$\begin{split}
\psi(\tau,\rho,\theta,\varphi)=&- \int_{\rho}^{\infty} \partial_{\rho} \psi (\tau,\rho',\theta,\varphi) \,d\rho'\\
\leq &\:\sqrt{\int_{r}^{\infty} (r'-M)^{-2}\,dr'}\cdot \sqrt{\int_{\rho}^{\infty} Dr^2 (\partial_{\rho}\psi)^2(\tau,\rho',\theta,\varphi) \,d\rho'},
\end{split}$$ where we used that, by the assumptions on the initial data in the proposition and the estimates in Proposition \[prop:radfieldsinfLk\], $\psi$ vanishes as $\rho \to \infty$ and moreover, we applied Cauchy–Schwarz. After applying a standard Sobolev inequality on ${\mathbb{S}}^2$, we therefore have that $$||(r-M)^{\frac{1}{2}}T^J \psi||_{L^{\infty}(\Sigma_{\tau})}\lesssim \sqrt{\sum_{|\alpha|\leq 2} \int_{\Sigma_{\tau}} J^T[\Omega^{\alpha}\psi]\cdot \mathbf{n}_{\tau}\,d\mu_{\tau}}.$$ The estimates , and then follow from the energy decay estimates in Proposition \[prop:hoedaypsi0\], \[prop:hoedaypsi1\] and \[prop:hoedaypsi1b\].
In order to prove the estimates , and we can then restrict to ${N}^{\mathcal{I}}_{\tau}$ and ${N}^{\mathcal{H}}_{\tau}$. Let $\chi(r)$ be a cut-off function that is smooth and compactly supported in $r\geq r_{\mathcal{I}}$ away from $r=r_{\mathcal{I}}$. Then we can apply the fundamental theorem of calculus as follows: $$\begin{split}
(\chi \phi)^2(u',v,\theta,\varphi)=&\:\int_{v_{r_{\mathcal{I}}}(u')}^{v} 2\chi\phi\cdot L(\chi \phi)|_{u=u'}\,dv'\\
\leq &\: 2 \sqrt{\int_{v_{r_{\mathcal{I}}}(u')}^{v} r^{-2}\phi^2 |_{u=u'}\,dv'}\cdot \sqrt{\int_{v_{r_{\mathcal{I}}}(u')}^{v} r^{2}(L(\chi \phi))^2 |_{u=u'}\,dv'}\\
\lesssim &\: \sqrt{\int_{v_{r_{\mathcal{I}}}(u')}^{v} (L\chi \phi)^2 |_{u=u'}\,dv'}\cdot \sqrt{\int_{v_{r_{\mathcal{I}}}(u')}^{v} r^{2}(L(\chi \phi))^2 |_{u=u'}\,dv'},
\end{split}$$ where we applied Cauchy–Schwarz to arrive at the second inequality and to arrive at the third inequality. If we now redefine $\chi$ to be a smooth, compactly supported cut-off function in $r\leq r_{\mathcal{H}}$ away from $r=r_{\mathcal{H}}$, we can similarly apply the fundamental theorem of calculus, Cauchy–Schwarz and to obtain $$\begin{split}
(\chi \phi)^2(u,v',\theta,\varphi) \lesssim &\: \sqrt{\int_{u_{r_{\mathcal{H}}}(v')}^{u} (\underline{L}\chi \phi)^2 |_{v=v'}\,du'}\cdot \sqrt{\int_{u_{r_{\mathcal{H}}}(v')}^{u} (r-M)^{-2}(\underline{L}(\chi \phi))^2 |_{v=v'}\,du'}.
\end{split}$$ It then follows that $$\begin{aligned}
||rT^J \psi||^2_{L^{\infty}({N}^{\mathcal{I}}_{\tau})}\leq &\: \sqrt{ \sum_{|\alpha|\leq 2} \int_{{N}^{\mathcal{I}}_{\tau}} r^2(L\Omega^{\alpha}\phi)^2\,d\omega dv\cdot \int_{\Sigma_{\tau}} J^T[\Omega^{\alpha}\psi]\cdot \mathbf{n}_{\tau}\,d\mu_{\tau}}+ \sum_{|\alpha|\leq 2} \int_{\Sigma_{\tau}} J^T[\Omega^{\alpha}\psi]\cdot \mathbf{n}_{\tau}\,d\mu_{\tau},\\
||rT^J \psi||^2_{L^{\infty}({N}^{\mathcal{H}}_{\tau})}\leq &\: \sqrt{ \sum_{|\alpha|\leq 2} \int_{{N}^{\mathcal{H}}_{\tau}} (r-M)^2(\underline{L}\Omega^{\alpha}\phi)^2\,d\omega du\cdot \int_{\Sigma_{\tau}} J^T[\Omega^{\alpha}\psi]\cdot \mathbf{n}_{\tau}\,d\mu_{\tau}}+ \sum_{|\alpha|\leq 2} \int_{\Sigma_{\tau}} J^T[\Omega^{\alpha}\psi]\cdot \mathbf{n}_{\tau}\,d\mu_{\tau}.\end{aligned}$$ We obtain , and by applying the energy decay estimates in Proposition \[prop:hoedaypsi0\], \[prop:hoedaypsi1\] and \[prop:hoedaypsi1b\].
We are left with proving and . We apply the fundamental theorem of calculus in yet another way: $$\begin{split}
\psi^2_{\geq 1} (\tau,\rho,\theta,\varphi)= &-\int_{\rho}^{\infty} 2 \psi_{\geq 1}\cdot \partial_{\rho}\psi_{\geq 1}(\tau,\rho',\theta,\varphi)\,d\rho'\\
\leq &\: \sqrt{\int_{\rho}^{\infty} D^{-2}\psi_{\geq 1}^2(\tau,\rho',\theta,\varphi)\,d\rho'}\cdot \sqrt{\int_{\rho}^{\infty} D^2(\partial_{\rho}\psi_{\geq 1})^2(\tau,\rho',\theta,\varphi)\,d\rho'}\\
\lesssim &\: D^{-1}(\rho)\sqrt{\int_{\rho}^{\infty} D(\partial_{\rho}\psi_{\geq 1})^2(\tau,\rho',\theta,\varphi)r^2\,d\rho'}\cdot \sqrt{\int_{\rho}^{\infty} D^2(\partial_{\rho}\psi_{\geq 1})^2(\tau,\rho',\theta,
\varphi)\,d\rho'}\\
\lesssim &\: D^{-1}(\rho)\sqrt{\int_{\rho}^{\infty} D(\partial_{\rho}\psi_{\geq 1})^2(\tau,\rho',\theta,\varphi)r^2\,d\rho'}\cdot \sqrt{\int_{\rho}^{\infty} [Dr^2(\partial_{\rho}T\psi_{\geq 1})^2+ Dh^2r^2(T^2\psi_{\geq 1})^2](\tau,\rho',\theta,\varphi)\,d\rho'},
\end{split}$$ where we applied Cauchy–Schwarz to arrive at the first inequality and we applied together with the fact that $D^{-2}(r')\lesssim D^{-2}(r)$ for all $r\leq r'$ to obtain the second inequality. The third inequality then follows from an application of the degenerate elliptic estimate in Proposition \[prop:degenelliptic\], together with the Poincaré inequality from Lemma \[lm:poincare\]. We conclude that $$||\sqrt{D} \psi||^2_{L^{\infty}(\Sigma_{\tau})}\lesssim \sqrt{\sum_{|\alpha|\leq 2} \int_{\Sigma_{\tau}} J^T[\Omega^{\alpha}\psi]\cdot \mathbf{n}_{\tau}\,d\mu_{\tau}\cdot \int_{\Sigma_{\tau}} J^T[T\Omega^{\alpha}\psi]\cdot \mathbf{n}_{\tau}\,d\mu_{\tau}}$$ and we apply the energy decay estimates in Proposition \[prop:hoedaypsi1\] and \[prop:hoedaypsi1b\] to derive and .
It will be necessary to use the following *additional* $L^{\infty}$ estimates with stronger weights in the energy norm *either* in $r$ *or* in $(r-M)^{-1}$ compared to the weights appearing in the norms $E^{\epsilon}_{0; J}[\psi]$ and $E^{\epsilon}_{1; J}[\psi]$:
\[prop:pointdecay2\] Let $J\in {\mathbb{N}}_0$ and assume that for $n=0,1$ and for all $0\leq k\leq 2-n$ and $0\leq j\leq J$: $$\begin{aligned}
\lim_{v\to \infty}\int_{{\mathbb{S}}^2}r^{2j}|{\slashed{\nabla}}_{{\mathbb{S}}^2}\slashed{\Delta}_{{\mathbb{S}}^2}^kL^jP_{\geq 1}\Phi_{(n)}|^2\,d\omega|_{u=u_0}<&\:\infty,\end{aligned}$$ and for all $0\leq j\leq J-1$ $$\begin{aligned}
\lim_{v\to \infty} r^{j+2} L^{j+1}\phi_0(u_0,v)<&\infty.\end{aligned}$$ Then, for all $\epsilon>0$, there exists a constant $C=C(M,{\Sigma_0},\epsilon,J)>0$ such that $$\begin{aligned}
\label{eq:pdecayl01v2impI}
||rT^J \psi_0||_{L^{\infty}({N}^{\mathcal{I}}_{\tau})}\lesssim&\: \sqrt{E^{\epsilon}_{0,\mathcal{I}; J}[\psi]} (1+\tau)^{-\frac{3}{2}-J+\frac{\epsilon}{2}},\\
\label{eq:pdecayl01v2impH}
||rT^J \psi_0||_{L^{\infty}({N}^{\mathcal{H}}_{\tau})}\lesssim&\: \sqrt{E^{\epsilon}_{0,\mathcal{H}; J}[\psi]} (1+\tau)^{-\frac{3}{2}-J+\frac{\epsilon}{2}},\\
\label{eq:pdecaylgeq1v2impI}
||rT^J \psi_{\geq 1}||_{L^{\infty}({N}^{\mathcal{I}}_{\tau})}\leq&\: C\cdot \sqrt{\sum_{|\alpha|\leq 2}E^{\epsilon}_{1,\mathcal{I};J}[\Omega^{\alpha}\psi]} (1+\tau)^{-\frac{9}{4}-J+\frac{\epsilon}{2}},\\
\label{eq:pdecaylgeq1v2impH}
||rT^J \psi_{\geq 1}||_{L^{\infty}({N}^{\mathcal{H}}_{\tau})}\leq&\: C\cdot \sqrt{\sum_{|\alpha|\leq 2}E^{\epsilon}_{1,\mathcal{H};J}[\Omega^{\alpha}\psi]} (1+\tau)^{-\frac{9}{4}-J+\frac{\epsilon}{2}},\end{aligned}$$ where we additionally assumed for .
The estimates follow from the proof of Proposition \[prop:pointdecay\], where we additionally appeal to the stronger weighted energy decay estimates from Proposition \[prop:hoextraendecay\].
Late-time asymptotics for Type **C** perturbations {#sec:asympnonzeroconst}
==================================================
In this section, we will derive the leading-order late-time asymptotics for the spherical mean $\psi_0$ in the case that both $H_0$, the conserved quantity at $\mathcal{H}^+$ and $I_0$, the conserved quantity at $\mathcal{I}^+$ are non-zero. This data is of Type **C**, as defined in Section \[sec:TheTypesOfInitialDataABCD\]. We will make use of the pointwise decay estimates for $\psi_0$ derived in Section \[sec:decayest\].
Late-time asymptotics in the regions $\protect\mathcal{A}^{\mathcal{H}}_{\gamma^{\mathcal{H}}}$ and $\protect\mathcal{A}^{\mathcal{I}}_{\gamma^{\mathcal{I}}}$ {#sec:asympradfieldsnonzeroconst}
--------------------------------------------------------------------------------------------------------------------------------------------------------------
We introduce the following $L^{\infty}$ norms on the derivatives of $\phi_0$ along the initial hypersurfaces ${{N}_0^{\mathcal{H}}}$ and ${{N}_0^{\mathcal{I}}}$: $$\begin{aligned}
P_{H_0,\beta;k}[\psi]:=&\max_{0\leq j \leq k}\left|\left| u^{2+j+\beta}\cdot \underline{L}^j\left(\underline{L}\phi_0|_{{{N}_0^{\mathcal{I}}}}-2\frac{H_0[\psi]}{u^2}\right) \right|\right|_{L^{\infty}}\\
P_{I_0,\beta;k}[\psi]:=&\max_{0\leq j \leq k}\left|\left| v^{2+j+\beta}\cdot L^j\left(L\phi_0|_{{{N}_0^{\mathcal{H}}}}-2\frac{I_0[\psi]}{v^2}\right) \right|\right|_{L^{\infty}},\end{aligned}$$ with $0<\beta\leq 1$.
For the arguments below, we will need to relate decay in terms of the coordinate $r$ to decay in terms of the double null coordinates $u$ and $v$.
\[lm:relationruv\] Let $N\in {\mathbb{N}}$.
- We can estimate in $\{r\geq r_{\mathcal{I}}\}\cap\{u\geq 0\}$: $$\begin{aligned}
r-\frac{v-u}{2}-2M\log (v-u)=&\: O_N((v-u)^0),\\
r^{-1}-\frac{2}{v-u}-8M(v-u)^{-2}\log (v-u)=&\: O_N((v-u)^{-2})\\
r^{-2}-\frac{4}{(v-u)^2}-32M(v-u)^{-3}\log (v-u)=& O_N((v-u)^{-3}),\\
r^{-3}(u,v)=&\:O_N((v-u)^{-3}).\end{aligned}$$ There moreover exists a constant $c_{r_{\mathcal{I}}}>0$ such that $$\begin{array}{ll}
c_{r_{\mathcal{I}}} \cdot v \leq v-u-1 \leq v\quad &\textnormal{if}\quad u_0\leq u\leq \frac{v}{2}+r_*(r_{\mathcal{I}}),\\
c_{r_{\mathcal{I}}}\cdot v\leq u\leq v \quad &\textnormal{if}\quad \frac{v}{2}+r_*(r_{\mathcal{I}})\leq u\leq u_{r_{\mathcal{I}}}(v).
\end{array}$$
- We can estimate in $\{r\geq r_{\mathcal{I}}\}\cap\{u\geq 0\}$: $$\begin{aligned}
M^2(r-M)^{-1}-\frac{u-v}{2}-2M\log (u-v)=&\: O_N((u-v)^0),\\
M^{-2}(r-M)-\frac{2}{u-v}-8M(u-v)^{-2}\log (u-v)=&\: O_N((u-v)^{-2})\\
M^{-4}(r-M)^2-\frac{4}{(u-v)^2}-32M(u-v)^{-3}\log (u-v)=& O_N((u-v)^{-3}),\\
M^{-6}(r-M)^3(u,v)=&\:O_N((u-v)^{-3}).\end{aligned}$$ There moreover exists a constant $c_{r_{\mathcal{H}}}>0$ such that $$\begin{array}{ll}
c_{r_{\mathcal{H}}} \cdot v \leq u-v-1 \leq v\quad &\textnormal{if}\quad v_0\leq v\leq \frac{u}{2}-r_*(r_{\mathcal{H}}),\\
c_{r_{\mathcal{H}}}\cdot u\leq v\leq u \quad &\textnormal{if}\quad \frac{u}{2}-r_*(r_{\mathcal{H}})\leq v\leq v_{r_{\mathcal{H}}}(u).
\end{array}$$
Observe that $$\frac{v-u}{2}=r_*(r)=r-M-M^2(r-M)^{-1}+2M\log\left(\frac{r-M}{M}\right).$$ Hence, we can repeat the proof of Lemma 2.1 in [@logasymptotics], where in the $r\leq r_{\mathcal{H}}$ case, we interchange the roles of $u$ and $v$ and we replace $r$ by $M^{-2}(r-M)^{-1}$.
\[rmk:Pnorminvrcoords\] By applying Lemma \[lm:relationruv\] it follows moreover that $P_{H_0,\beta;k}[\psi]<\infty$ if and only if in $(v,r)$ coordinates: $$\max_{0\leq j\leq k}\left|\left| (r-M)^{j-\beta}\cdot \partial_r^j\left(\partial_r\phi_0|_{{{N}_0^{\mathcal{H}}}}+M^{-2}H_0[\psi]\right) \right|\right|_{L^{\infty}}<\infty$$ Hence, if $\partial_r\phi_0|_{r=M}=-M^{-2}H_0$, we are guaranteed that $P_{H_0,\beta;k}[\psi]<\infty$ for all $k\in {\mathbb{N}}_0$ and $\beta=1$, simply by the smoothness assumption on the initial data for $\psi$ together with Taylor’s theorem.
We moreover introduce the following spacetime subregions contained in either the region $\mathcal{A}^{\mathcal{I}}$ or $\mathcal{A}^{\mathcal{H}}$: for $k\in {\mathbb{N}}_0$ and $\alpha\in (0,1)$ let $$\begin{aligned}
{\mathcal{A}_{\gamma^{\mathcal{H}}_{\alpha}}^{\mathcal{H}}}:=&\:\{r\leq r_{\mathcal{H}}\}\cap \{0\leq v \leq u-v^{\alpha}+2r_*(r_{\mathcal{H}})\}\subset \mathcal{A}^{\mathcal{H}},\\
{\mathcal{A}_{\gamma^{\mathcal{I}}_{\alpha}}^{\mathcal{I}}}:=&\:\{r\geq r_{\mathcal{I}}\}\cap \{0\leq u \leq v-u^{\alpha}+2r_*(r_{\mathcal{I}})\}\subset \mathcal{A}^{\mathcal{I}}.\end{aligned}$$
Note that the boundaries $\partial {\mathcal{A}_{\gamma^{\mathcal{H}}_{\alpha}}^{\mathcal{H}}}$ and $\partial {\mathcal{A}_{\gamma^{\mathcal{I}}_{\alpha}}^{\mathcal{I}}}$ contain subsets of, respectively, the following timelike hypersurfaces: $$\begin{aligned}
\gamma^{\mathcal{H}}_{\alpha}:=&\:\{u-v=v^{\alpha}+2r_*(r_{\mathcal{H}})\},\\
\gamma^{\mathcal{I}}_{\alpha}:=&\:\{v-u=u^{\alpha}+2r_*(r_{\mathcal{I}})\}.\end{aligned}$$
When the value of $\alpha\in (0,1)$ is not relevant, we will occasionally drop the $\alpha$ subscript for convenience and write $\gamma^{\mathcal{H}}$ and $\mathcal{A}_{\gamma^{\mathcal{I}}}^I$.
In the regions ${\mathcal{A}_{\gamma^{\mathcal{H}}_{\alpha}}^{\mathcal{H}}}$ and ${\mathcal{A}_{\gamma^{\mathcal{I}}_{\alpha}}^{\mathcal{I}}}$, we obtain the following additional decay estimates for $r^{-1}$ and $r-M$:
Let $M<r_{\mathcal{H}}<2M$ and $r_{\mathcal{I}}>2M$. Then for all $\eta>0$, we can estimate $$\begin{aligned}
r^{-1}\lesssim &\:v^{-\alpha}\lesssim u^{-\alpha} \quad \textnormal{in ${\mathcal{A}_{\gamma^{\mathcal{I}}_{\alpha}}^{\mathcal{I}}}$},\\
r-M\lesssim &\: u^{-\alpha}\lesssim v^{-\alpha} \quad \textnormal{in ${\mathcal{A}_{\gamma^{\mathcal{H}}_{\alpha}}^{\mathcal{H}}}$}.\end{aligned}$$
As a first step towards obtaining the asymptotics of $\phi_0$, we obtain the asymptotics of $L\phi_0$ and more generally $L^{k+1}\phi_0$, for $k\in {\mathbb{N}}_0$.
\[prop:asympLphi\] Let $k\in {\mathbb{N}}_0$ and $\alpha_k\in (\frac{k+2}{k+3},1)$. Take $\epsilon\in (0,\frac{1}{2}(k+3)\alpha_k-\frac{1}{2}(k+2))$. Assume that $E^{\epsilon}_{0}[\psi]<\infty$ and moreover that there exists a $\beta>0$ such that: $$P_{I_0,\beta;k}[\psi]<\infty.$$ Then, there exists a constant $C=C(M,{\Sigma_0},r_{\mathcal{H}},r_{\mathcal{I}},\alpha_k,\epsilon,k)>0$ such that $$\begin{aligned}
|L^{k+1}\phi_0(u,v)-(-1)^k(k+1)!\cdot 2I_0[\psi]\cdot v^{-2-k}|\leq&\: C \sqrt{E^{\epsilon}_{0}[\psi]}\cdot v^{-3\alpha_k+2\epsilon-k}\\
&+P_{I_0,\beta;k}[\psi]\cdot v^{-2-\beta-k}\quad \textnormal{in ${\mathcal{A}_{\gamma^{\mathcal{I}}_{\alpha_k}}^{\mathcal{I}}}$},\\
|\underline{L}^{k+1}\phi_0(u,v)-(-1)^k (k+1)!\cdot 2H_0[\psi]\cdot u^{-2-k}|\leq&\: C \sqrt{E^{\epsilon}_{0}[\psi]}\cdot u^{-3\alpha_k+2\epsilon-k}\\
&+P_{H_0,1;k}[\psi]\cdot u^{-3-k}\quad \textnormal{in ${\mathcal{A}_{\gamma^{\mathcal{H}}_{\alpha_k}}^{\mathcal{H}}}$}.\end{aligned}$$
The equation for $\psi_0$ on extremal Reissner–Nordström for can be rewritten as follows in double null coordinates: $$\partial_u\partial_v\phi_0=-\frac{1}{4r}DD' \phi_0.$$ From Lemma \[lm:relationruv\] it therefore follows that we can write $$\begin{aligned}
\partial_u\partial_v\phi_0=&\:O_N((v-u)^{-3})\phi_0\quad \textnormal{in $\mathcal{A}^{\mathcal{I}}$},\\
\partial_v\partial_u\phi_0=&\: O_N((u-v)^{-3})\phi_0 \quad \textnormal{in $\mathcal{A}^{\mathcal{H}}$}.\end{aligned}$$ In particular, it follows that the estimates for $L^k\phi_0$ in ${\mathcal{A}_{\gamma^{\mathcal{I}}_{\alpha_k}}^{\mathcal{I}}}$, derived in Proposition 8.3 of [@paper2], apply directly to $\underline{L}^k\phi_0$ in ${\mathcal{A}_{\gamma^{\mathcal{H}}_{\alpha_k}}^{\mathcal{H}}}$, after interchanging the role of $u$ and $v$.
The next step is to apply the estimates for $L^{k+1}\phi_0$ from Proposition \[prop:asympLphi\] in order to obtain asymptotics for $\phi_0$ and, more generally, for $T^k\phi_0$ with $k\in {\mathbb{N}}_0$.
\[prop:asympradfieldnonzeroIH\] Let $k\in {\mathbb{N}}_0$. If we additionally restrict $\alpha_k\in [\frac{2k+5}{2k+7},1)$ and $\epsilon\in (0,\frac{1}{6}(1-\alpha_k))$, we can find a constant $C=C(M,{\Sigma_0},r_{\mathcal{H}},r_{\mathcal{I}},\alpha_k,\epsilon,\beta, k)>0$ such that $$\begin{aligned}
|T^k\phi_0&(u,v)-(-1)^k k!\cdot 2I_0[\psi]\cdot (u^{-1-k}-v^{-1-k})|\\
\leq&\: C \left(\sqrt{E^{\epsilon}_{0;k}[\psi]}+I_0[\psi]\right)\cdot (v+M)^{-\frac{\alpha_k}{2}-\frac{3}{2}+2\epsilon-k}+C\cdot P_{I_0,\beta;k}[\psi]\cdot (u+M)^{-1-\beta-k}\quad \textnormal{in ${\mathcal{A}_{\gamma^{\mathcal{I}}_{\alpha_k}}^{\mathcal{I}}}$},\\
|T^k\phi_0&(u,v)-(-1)^k k!\cdot 2H_0[\psi]\cdot (v^{-1-k}-u^{-1-k})|\\
\leq&\: C \left(\sqrt{E^{\epsilon}_{0;k}[\psi]}+H_0[\psi]\right)\cdot v^{-\frac{\alpha_k}{2}-\frac{3}{2}+2\epsilon-k}+C\cdot P_{H_0,1;k}[\psi]\cdot v^{-2-k}\quad \textnormal{in ${\mathcal{A}_{\gamma^{\mathcal{H}}_{\alpha_k}}^{\mathcal{H}}}$}.\end{aligned}$$ Furthermore, if we impose $\frac{1}{2}(1-\alpha_k)<\beta +2\epsilon$ and assume $H_0[\psi]\neq 0$ and $I_0[\psi]\neq 0$, then the estimates above provide first-order asymptotics for $\phi$ in the regions $\mathcal{A}_{\gamma^{\mathcal{H}}_{\delta}}$ and $\mathcal{A}_{\gamma^{\mathcal{I}}_{\delta}}$, for $1>\delta>\frac{\alpha_k}{2}+\frac{1}{2}+2\epsilon>\alpha_k+2\epsilon$. In particular, $$\begin{aligned}
|T^k\phi_0(u,v)-(-1)^k k!\cdot 2I_0[\psi]\cdot u^{-1-k}|\leq&\: C \left(\sqrt{E^{\epsilon}_{0;k}[\psi]}+I_0[\psi]\right)\cdot u^{-\frac{\alpha_k}{2}-\frac{3}{2}+2\epsilon-k}\\
&+C\cdot P_{I_0,\beta;k}[\psi]\cdot u^{-1-\beta-k},\\
|T^k\phi_0(u,v)-(-1)^k k!\cdot 2H_0[\psi]\cdot v^{-1-k}|\leq&\: C \left(\sqrt{E^{\epsilon}_{0;k}[\psi]}+H_0[\psi]\right)\cdot v^{-\frac{\alpha_k}{2}-\frac{3}{2}+2\epsilon-k}\\
&+C\cdot P_{H_0,1;k}[\psi]\cdot v^{-2-k}.\end{aligned}$$
The proof follows directly from the proof of Proposition 8.4 and 8.5 of [@paper2], where as in case in the proof of Proposition \[prop:asympLphi\], we use that the estimates in the region analogous to ${\mathcal{A}_{\gamma^{\mathcal{I}}_{\alpha_k}}^{\mathcal{I}}}$ in [@paper2] apply directly to ${\mathcal{A}_{\gamma^{\mathcal{H}}_{\alpha_k}}^{\mathcal{H}}}$ after interchanging $u$ and $v$ and $L$ and $\underline{L}$.
Partial asymptotics for $\partial_{\rho}\psi$ away from $\mathcal{H}^+$ up to $\gamma^{\mathcal{I}}$ {#sec:SharpDecayAndAsymptoticsForProtectPartialRhoPsiUpToProtectGammaMathcalI}
----------------------------------------------------------------------------------------------------
Before we discuss the late-time asymptotics of $T^k\psi_0$ for Type **C** data, we will derive the late-time asymptotics for the derivatives $\underline{L}T^k\psi_0$ and $LT^k\psi_0$ in appropriate subsets of $\mathcal{R}$. We will use the asymptotics for $\phi_0$ along $\mathcal{H}^+$ obtained in Proposition \[prop:asympradfieldnonzeroIH\], together with the decay estimates and .
\[prop:partasymdrhopsi\] Let $k\in {\mathbb{N}}_0$. Then there exist $\eta,\epsilon>0$ suitably small, such that in $(v,r)$ coordinates, we have that for all $r\leq r_{\mathcal{I}}$: $$\label{eq:partasymdrhopsi1}
\begin{split}
|-2r^2\underline{L}T^k\psi_0(v,r)&-(-1)^{k+1} (k+1)!\cdot 4MH_0[\psi]\cdot v^{-2-k}|\\
\leq&\: C\left(\sqrt{E^{\epsilon}_{0; k+1}[\psi]}+H_0[\psi]\right)\left[(r-M)^{-\frac{1}{2}}v^{-\frac{5}{2}+\epsilon}+ v^{-2-k-\epsilon'}\right]\\
&+P_{H_0,1;k}[\psi]\cdot v^{-3-k},
\end{split}$$ and in $(u,r)$ coordinates, we can estimate for all $r\geq r_{\mathcal{H}}$: $$\label{eq:partasymdrhopsi2}
\begin{split}
|2LT^k\psi_0(u,r)&-(-1)^{k+1} (k+1)!\cdot 4MH_0[\psi]\cdot D^{-1}r^{-2}u^{-2-k}|\\
\leq &\: C\left(\sqrt{E^{\epsilon}_{0; k+1}[\psi]}+H_0[\psi]\right) r^{-2}u^{-2-k-\epsilon'}\\
&+\left(\sqrt{E^{\epsilon}_{0; k+1}[\psi]}+H_0[\psi]\right) r^{-\frac{1}{2}}u^{-\frac{5}{2}-k+\epsilon}\\
&+P_{H_0,1;k}[\psi]\cdot r^{-2}u^{-3-k},
\end{split}$$ where $C=C(M,{\Sigma_0},r_{\mathcal{H}},r_{\mathcal{I}},\eta,\epsilon,k)>0$ is a constant.
From in follows that $T^k\psi_0$ satisfies the following equation in $(v,r)$ coordinates: $$\label{eq:waveeqvr}
\partial_{r}\left(Dr^2\partial_{r}T^k\psi_0+2rT^{k+1}\phi_0\right)=2T^{k+1}\phi_0.$$ See also with $h=0$ applied to $\psi_0$.
By integrating both sides of along constant $v$ hypersurfaces from $r'=M$ to $r'=r\leq \min\{r_{\mathcal{I}},r_{\Sigma_0}(v)\}$, where $r_{\Sigma_0}(v)$ denotes the value of $r$ along the intersection of the hypersurface of constant $v$ with $\Sigma_0\cap (\mathcal{B} \cup \mathcal{A}^{\mathcal{I}})$ (which is non-empty for $v>v_0$), and using that $Dr^2\partial_rT^k\psi$ vanishes at $\mathcal{H}^+$ for any $T^k\psi$ (using that $\psi$ is smooth), we therefore arrive at: $$Dr^2\partial_{r}T^k\psi_0(v,r)+2rT^{k+1}\phi(v,r)=2MT^{k+1}\phi|_{\mathcal{H}^+}(v)+\int_{M}^r2T^{k+1}\phi(v,r')\,dr'.$$
We first apply Cauchy–Schwarz and , together with and , to estimate $$\begin{split}
\left|\int_{M}^r2T^{k+1}\phi_0(v,r')\,dr'\right|\lesssim&\: \sqrt{\int_{M}^r \int_{{\mathbb{S}}^2}(T^{k+1}\psi_0)^2 \,d\omega dr} \cdot \sqrt{\int_M^{r_{\mathcal{I}}} \,dr'}\\
\lesssim &\: \sqrt{\int_{M}^{ \min\{r_{\mathcal{I}},r_{\Sigma_0}(v)\}}\int_{{\mathbb{S}}^2}(r-M)^2(\partial_r(T^{k+1}\psi_0))^2 \,d\omega dr+ (T^{k+1}\psi_0)^2(v,r_{\mathcal{I}})}\cdot \sqrt{r-M}\\
\lesssim &\:\sqrt{ \int_{\Sigma_{\tau(v, \min\{r_{\mathcal{I}},r_{\Sigma_0}(v)\})}} J^T[T^{k+1}\psi]\cdot \mathbf{n}_{\Sigma_0}\,d\mu_{\Sigma_{\tau(v, \min\{r_{\mathcal{I}},r_{\Sigma_0}(v)\})}}}\cdot \sqrt{r-M}\\
\lesssim &\: \sqrt{E^{\epsilon}_{0; k+1}[\psi]} (r-M)^{\frac{1}{2}}(1+\tau(v,\min\{r_{\mathcal{I}},r_{\Sigma_0}(v)\}))^{-\frac{5}{2}+\epsilon}\\
\lesssim &\: \sqrt{E^{\epsilon}_{0; k+1}[\psi]} (r-M)^{\frac{1}{2}}v^{-\frac{5}{2}+\epsilon}.
\end{split}$$ for $r\leq \min\{r_{\mathcal{I}},r_{\Sigma_0}(v)\}$ and $v\geq v_0$, where in the third inequality we used the conservation property of the $T$-energy flux.
By we moreover have that for all $v\geq 0$: $$\left|2rT^{k+1}\phi_0(v,r)\right|\lesssim \sqrt{E^{\epsilon}_{0;k+ 1}[\psi]} r^2(r-M)^{-\frac{1}{2}}v^{-\frac{5}{2}-k+\epsilon}.$$
Furthermore, by Proposition \[prop:asympradfieldnonzeroIH\], we have that for $\epsilon>0$ suitably small, there exists an $\epsilon'>0$ such that we can estimate $$\begin{split}
\left|T^{k+1}\phi_0|_{\mathcal{H}^+}(v)-(-1)^{k+1} (k+1)!\cdot 2H_0[\psi]\cdot v^{-2-k}\right|\lesssim&\: \left(\sqrt{E^{\epsilon}_{0; k+1}[\psi]}+H_0[\psi]\right)\cdot v^{-2-k-\epsilon'}\\
&+P_{H_0,1;k}[\psi]\cdot v^{-3-k}.
\end{split}$$
Combining all the above decay estimates, we can therefore infer that for all $v\geq v_0$ and $r\leq \min\{r_{\mathcal{I}},r_{\Sigma_0}(v)\}$: $$\label{eq:asymlbarpsidr}
\begin{split}
|Dr^2\partial_rT^k\psi_0(v,r)&-(-1)^{k+1} (k+1)!\cdot 4MH_0[\psi]\cdot v^{-2-k}|\\
\lesssim&\: \left(\sqrt{E^{\epsilon}_{0; k+1}[\psi]}+H_0[\psi]\right)\left[(r-M)^{-\frac{1}{2}}v^{-\frac{5}{2}+\epsilon}+ v^{-2-k-\epsilon'}\right]\\
&+P_{H_0,1;k}[\psi]\cdot v^{-3-k},
\end{split}$$ or equivalently, since we can express $\underline{L}=-\frac{D}{2}\partial_r$ in $(v,r)$ coordinates, we can write $$\label{eq:asymlbarpsi}
\begin{split}
|-2r^2\underline{L}T^k\psi_0(v,r)&-(-1)^{k+1} (k+1)!\cdot 4MH_0[\psi]\cdot v^{-2-k}|\\
\lesssim&\: \left(\sqrt{E^{\epsilon}_{0; k+1}[\psi]}+H_0[\psi]\right)\left[(r-M)^{-\frac{1}{2}}v^{-\frac{5}{2}+\epsilon}+ v^{-2-k-\epsilon'}\right]\\
&+P_{H_0,1;k}[\psi]\cdot v^{-3-k}.
\end{split}$$
By using that $T=\underline{L}+L$ and applying once again , we can rewrite at $r=r_0\geq r_{\mathcal{H}}$ as follows: $$\begin{split}
\label{eq:LpsiestrI}
|2r_{\mathcal{I}}^2LT^k\psi_0|_{r=r_{0}}(u)&-(-1)^{k+1} (k+1)!\cdot 4MH_0[\psi]\cdot u^{-2-k}|\\
\lesssim&\: \left(\sqrt{E^{\epsilon}_{0; k+1}[\psi]}+H_0[\psi]\right) u^{-2-k-\epsilon'}\\
&+P_{H_0,1;k}[\psi]\cdot u^{-3-k}.
\end{split}$$ Let us now switch to $(u,r)$ coordinates in the region $r\geq r_{\mathcal{I}}$. From in follows that $\psi$ satisfies the following equation in $(u,r)$ coordinates: $$\label{eq:waveequr}
\partial_{r}\left(2r^2LT^k\psi_0-2rT^{k+1}\phi_0\right)=-2T^{k+1}\phi_0.$$ We integrate both sides of along constant $u$ hypersurfaces from $r'=r_{\mathcal{I}}$ to $r'=r>r_{\mathcal{I}}$ to arrive at: $$2r^2LT^k\psi_0(u,r)=2r_{\mathcal{I}}^2LT^k\psi_0(u,r_{\mathcal{I}})-2r_{\mathcal{I}}T^{k+1}\phi_0(u,r_{\mathcal{I}})+2rT^{k+1}\phi_0(u,r)-2\int_{r_{\mathcal{I}}}^rT^{k+1}\phi_0(u,r')\,dr'.$$ First of all, we apply to estimate $$|2rT^{k+1}\phi_0(u,r)|\lesssim \sqrt{E^{\epsilon}_{0;k+1}[\psi]} r^{\frac{3}{2}}(1+\tau)^{-\frac{5}{2}-k+\epsilon}$$ for $r\geq r_{\mathcal{I}}$.
We moreover apply Cauchy–Schwarz together with to estimate $$\begin{split}
\left|\int_{r_{\mathcal{I}}}^rT^{k+1}\phi_0(u,r')\,dr'\right| \lesssim&\: \sqrt{\int_{{{N}^{\mathcal{I}}}_u} r^{-2} (T^{k+1} \phi_0)^2\,d\omega dr}\sqrt{\int_{r_{\mathcal{I}}}^r r'^2\,dr'} \\
\lesssim&\: \left[\sqrt{\int_{{{N}^{\mathcal{I}}}_u} (\partial_rT^{k+1}\phi_0)^2\,d\omega dr+(T^{k+1}\phi)^2(u,r_{\mathcal{I}})}\right]\cdot r^{\frac{3}{2}}\\
\lesssim&\: \sqrt{E^{\epsilon'}_{0;k+ 1}[\psi]} r^{\frac{3}{2}}(1+\tau)^{-\frac{5}{2}-k+\epsilon'}.
\end{split}$$ Hence, using that $\partial_r=2D^{-1}L$ in $(u,r)$ coordinates, we have that $$\begin{split}
|\partial_rT^k\psi_0(u,r)&-(-1)^{k+1} (k+1)!\cdot 4MH_0[\psi]\cdot D^{-1}r^{-2}u^{-2-k}|\\
\lesssim&\: \left(\sqrt{E^{\epsilon}_{0; k+1}[\psi]}+H_0[\psi]\right) r^{-2}u^{-2-k-\epsilon'}\\
&+\left(\sqrt{E^{\epsilon}_{0; k+1}[\psi]}+H_0[\psi]\right) r^{-\frac{1}{2}}u^{-\frac{5}{2}-k+\epsilon}\\
&+P_{H_0,1;k}[\psi]\cdot r^{-2}u^{-3-k},
\end{split}$$ for all $r\geq r_{\mathcal{I}}$.
**We note that the estimate provides in particular the late-time asymptotics of $\partial_rT^k\psi_0$ in the region $\{r_{\mathcal{H}}\leq r\leq r_{\mathcal{I}}\}$ ,** so it will be also be relevant when investigating the asymptotics of Type **A** data in Section \[sec:asympzeroconst\] below.
Late-time asymptotics in $\mathcal{R}$ {#sec:latetimeasympsi}
--------------------------------------
In this section, we obtain the asymptotics for $T^k\psi_0$, using fundamentally that both $H_0\neq 0$ and $I_0\neq 0$ in the case of Type **C** data.
\[prop:asympsi\] Let $k\in {\mathbb{N}}_0$. Then there exist $\eta,\epsilon>0$ suitably small, such that we obtain the following *global* estimate: $$\label{eq:asympsi}
\begin{split}
\Bigg|T^k\psi_0(u,v)&-4\left(I_0[\psi]+ \frac{M}{r\sqrt{D}}H_0[\psi]\right)T^k\left(\frac{1}{u\cdot v}\right)\Bigg|\\
\leq&\: C\left(\sqrt{E_{0;k+1}^{\epsilon}[\psi]}+I_0[\psi]+P_{I_0,\beta;k}[\psi]\right)v^{-1}u^{-1-k-\eta}\\
&+C\left(\sqrt{E_{0;k+1}^{\epsilon}[\psi]}+H_0[\psi]+P_{H_0,1;k}[\psi]\right)D^{-\frac{1}{2}}u^{-1}v^{-1-k-\eta},
\end{split}$$ where $C=C(M,{\Sigma_0},r_{\mathcal{H}},r_{\mathcal{I}},\eta,\epsilon, \beta, k)>0$ is a constant.
*Outline of proof:*\
\
For simplicity let use take $k=0$.
- We use the asymptotics for $\phi_0$ along $\mathcal{H}^+$ obtained in Proposition \[prop:asympradfieldnonzeroIH\], together with the decay estimates in and to obtain precise decay estimates for $L\psi_0$ and $\underline{L}\psi_0$ (and hence for the radial derivative). This step has been carried out in Proposition \[prop:partasymdrhopsi\].
- Using Proposition \[prop:asympradfieldnonzeroIH\], we derive the asymptotics for $\psi_0$ in the region $\mathcal{A}^{\mathcal{I}}_{\gamma^{\mathcal{I}}_{\delta}}$, with $\delta<1$ suitably close to 1 and we use the estimates for $L\psi_0$ from **Step 1** to extend the asymptotics of $\psi_0$ to $\mathcal{A}^{\mathcal{I}}$.
- Similarly, we apply Proposition \[prop:asympradfieldnonzeroIH\] to obtain the asymptotics for $\psi_0$ in $\mathcal{A}^{\mathcal{H}}_{\gamma^{\mathcal{H}}_{\alpha}}$. We then integrate $\partial_r\psi_0$ from $r=r_{\mathcal{I}}$ in the direction of decreasing $r$ to obtain moreover the asymptotics for $\psi_0$ in $\mathcal{B}\cup \mathcal{A}^{\mathcal{H}}\setminus \mathcal{A}^{\mathcal{H}}_{\gamma^{\mathcal{H}}_{\alpha}}$.
**Step 2:**\
\
In order to obtain the late-time asymptotics of $\psi_0$ in $\mathcal{A}^{\mathcal{I}}$, we partition the region $\mathcal{A}^{\mathcal{I}}$ into $\mathcal{A}^{\mathcal{I}}_{\gamma^{\mathcal{I}}_{\delta}}=\{r\geq r_{\gamma^{\mathcal{I}}_{\delta}}(u)\}$ and $\mathcal{A}^{\mathcal{I}}\setminus \mathcal{A}^{\mathcal{I}}_{\gamma^{\mathcal{I}}_{\delta}}=\{r< r_{\gamma^{\mathcal{I}}_{\delta}}(u)\}$, with $\delta<1$, where we will choose $1-\delta$ to be suitably small.
We first use the following identity: $$\label{eq:identityuminusv}
u^{-1-k}-v^{-1-k}=\frac{v-u}{v\cdot u^{k+1}}\sum_{j=0}^k\left(\frac{u}{v}\right)^j=(v-u)(-1)^k\frac{1}{k!}T^k\left(\frac{1}{u\cdot v}\right),$$ together with Proposition \[prop:asympradfieldnonzeroIH\] and Lemma \[lm:relationruv\], to find $\eta,\epsilon>0$ suitably small, so that we can estimate: $$\label{eq:asympsiinfnonzeroI0}
\begin{split}
\Bigg|T^k\psi(u,r)-4I_0[\psi]T^k\left(\frac{1}{v\cdot u}\right)\Bigg|\lesssim &\: \left(\sqrt{E^{\epsilon}_{0;k}[\psi]}+I_0[\psi]\right) v^{-1}u^{-1-k-\eta}+P_{I_0,\beta;k}[\psi]\cdot v^{-1}u^{-1-k-\beta}
\end{split}$$ in $\mathcal{A}^{\mathcal{I}}_{\gamma^{\mathcal{I}}_{\delta}}$.
Note that this implies in particular that $$\label{eq:typecasymppsigammaI}
\begin{split}
|T^k\psi_0(u,r_{\gamma^{\mathcal{I}}_{\delta}}(u))-4(-1)^k (k+1)!\cdot I_0[\psi]\cdot u^{-2-k}|\lesssim&\: \left(\sqrt{E^{\epsilon}_{0;k}[\psi]}+I_0[\psi]\right)\cdot u^{-2-k-\eta}\\
&+P_{I_0,\beta;k}[\psi]\cdot u^{-2-\beta-k}.
\end{split}$$
We then integrate in the $-\partial_r$ direction, starting from $r=r_{\gamma^{\mathcal{I}}_{\delta}}(u)$ and we apply from Step 1. By choosing $\eta>0$ and $\epsilon>0$ suitably small, we obtain: $$\label{eq:typecasymppsileftgammaI}
\begin{split}
\Bigg|T^k\psi(u,r)&-T^k\psi(u,r_{\gamma^{\mathcal{I}}_{\delta}}(u))+(-1)^{k+1} (k+1)!\cdot 4MH_0[\psi]u^{-2-k}\int_{r}^{r_{\gamma^{\mathcal{I}}_{\delta}}(u)}\frac{1}{(r'-M)^2}\,dr'\Bigg|\\
\lesssim&\: \left(\sqrt{E^{\epsilon}_{0; k+1}[\psi]}+H_0[\psi]\right) u^{-2-k-\eta}+P_{H_0,1;k}[\psi]\cdot r^{-1}u^{-3-k},
\end{split}$$ with $r_{\mathcal{I}}\leq r\leq r_{\gamma^{\mathcal{I}}_{\delta}}(u)$.
By combining , and , we conclude that in $\mathcal{A}^{\mathcal{I}}$: $$\label{eq:asymppsirI}
\begin{split}
\Bigg|T^k\psi(u,r)&-\left(4MH_0[\psi]\frac{T^k(u^{-2})}{r-M}+4I_0[\psi]T^k\left(\frac{1}{uv}\right)\right)\Bigg|\\
\lesssim&\: \left(\sqrt{E^{\epsilon}_{0; k+1}[\psi]}+H_0[\psi]+I_0[\psi]\right) v^{-1}u^{-1-k-\eta}\\
&+P_{H_0,1;k}[\psi]\cdot r^{-1}u^{-3-k}+P_{I_0,\beta;k}[\psi]\cdot v^{-1}u^{-1-k-\beta}.
\end{split}$$
**Step 3:**\
\
We now turn to the region $\mathcal{A}^{\mathcal{H}}\cup\mathcal{B}=\{r\leq r_{\mathcal{I}}\}$. We will partition this region into the region $\mathcal{A}^{\mathcal{H}}_{\gamma^{\mathcal{H}}_{\alpha}}=\{r\leq r_{\gamma^{\mathcal{H}}_{\alpha}(v)}\}$ and $\{r_{\gamma^{\mathcal{H}}_{\alpha}(v)}\leq r\leq r_{\mathcal{I}}\}$ with $\alpha<1$ and $1-\alpha$ suitably small.
Let us first consider the region $\mathcal{A}^{\mathcal{H}}_{\gamma^{\mathcal{H}}_{\alpha}}$. By using the identity $$\label{eq:identityvminusu}
v^{-1-k}-u^{-1-k}=\frac{u-v}{u\cdot v^{k+1}}\sum_{j=0}^k\left(\frac{v}{u}\right)^j=(u-v)(-1)^k\frac{1}{k!}T^k\left(\frac{1}{u\cdot v}\right)$$ together with Lemma \[lm:relationruv\] and Proposition \[prop:asympradfieldnonzeroIH\], we have that for $1-\alpha$ suitably small we can estimate in $r\leq r_{\gamma^{\mathcal{H}}_{\alpha}}(v)$: $$\label{eq:asympsihornonzeroH0}
\begin{split}
\Bigg|T^k\psi_0(u,v)&-4\frac{M}{r\sqrt{D}}H_0[\psi]T^k\left(\frac{1}{u\cdot v}\right)\Bigg|\\
\leq&\: C\left(\sqrt{E_{0;k+1}^{\epsilon}[\psi]}+H_0[\psi]+P_{H_0,1;k}[\psi]\right)D^{-\frac{1}{2}}u^{-1}v^{-1-k-\eta}.
\end{split}$$
We now consider the region $\{r_{\gamma^{\mathcal{H}}_{\alpha}(v)}\leq r\leq r_{\mathcal{I}}\}$ and use to integrate from $r'=r_{\mathcal{I}}$ to $r'=r\geq r_{\gamma^{\mathcal{H}}_{\alpha}(v)}$ along constant $v$ hypersurfaces with $v\geq v|_{\Sigma_0}(r_{\mathcal{I}})$, for $1-\alpha>0$ suitably small: we have that there exist $\epsilon,\eta>0$ suitably small such that $$\begin{split}
\Bigg|&T^k\psi_0(v,r)- T^k\psi_0(v,r_{\mathcal{I}})+(-1)^{k+1}\frac{(k+1)!4MH_0}{v^{k+2}}\int_{r}^{r_{\mathcal{I}}}(r'-M)^{-2}\,dr'\Bigg|\\
\lesssim&\: \left(\sqrt{E^{\epsilon}_{0; k+1}[\psi]}+H_0[\psi]\right)(r-M)^{-1}v^{-2-k-\eta}\\
&+P_{H_0,1;k}[\psi]\cdot v^{-3-k} .
\end{split}$$ Combined with this implies that for $1-\alpha>0$ suitably small: there exists an $\eta>0$ and $\epsilon>0$ suitably small such that for all $r_{\gamma^{\mathcal{H}}_{\alpha}(v)}\leq r \leq r_{\mathcal{I}}$: $$\label{eq:asymppsirH}
\begin{split}
\Bigg|&T^k\psi_0(v,r)-(-1)^{k}(k+1)!\left(\frac{4MH_0}{r\sqrt{D}}+4I_0[\psi]\right)v^{-k-2}\Bigg|\\
\lesssim&\: \left(\sqrt{E^{\epsilon}_{0; k+1}[\psi]}+H_0[\psi]+I_0[\psi]\right)(r-M)^{-1}v^{-2-k-\eta}\\
&+(r-M)^{-1}P_{H_0,1;k}[\psi]\cdot v^{-3-k}+P_{I_0,\beta;k}[\psi]\cdot v^{-2-\beta-k} .
\end{split}$$
By combining the estimates for $T^k\psi_0$ in the regions $r_{\mathcal{I}}\leq r\leq r_{\gamma^{\mathcal{I}}_{\delta}}(u)$, $r\geq r_{\gamma^{\mathcal{I}}_{\delta}}(u)$, $r_{\gamma^{\mathcal{H}}_{\alpha}(v)}\leq r \leq r_{\mathcal{I}}$ and $r\leq r_{\gamma^{\mathcal{H}}_{\alpha}}(v)$, we arrive at .
We can alternatively consider $\underline{\psi}:=M^{-1}(r-M)\psi=\frac{r}{M}\sqrt{D}\psi$ and reverse the roles of $\mathcal{H}^+$ and $\mathcal{I}^+$ in order to obtain the asymptotics for $\psi_0$ in Proposition \[prop:asympsi\]; see also the proof of Proposition \[prop:explicitexprtimeint2\].
Time inversion theory {#sec:timeint}
=====================
In this section, we will construct an auxilliary “time integral” function ${\psi}_0^{(1)}: \mathcal{R}\setminus \mathcal{H}^+\to {\mathbb{R}}$, which satisfies $T\psi_0^{(1)}=\psi_0$ and $\square_g\psi_0^{(1)}=0$. This construction is fundamental to obtaining asymptotics for $\psi_0$ in Section \[sec:asympzeroconst\], when the initial data is of Type **A**, **B** or **D**; that is to say, when $H_0[\psi]$ or $I_0[\psi]$ vanish.
Regular time inversion in $\protect\mathring{\mathcal{R}}$ {#sec:TheRegularTimeInversionConstructionInOversetOMathcalR}
----------------------------------------------------------
Consider $\mathring{\mathcal{R}}= \mathcal{R}\setminus \partial \mathcal{R}$. We have the following
\[def:timeint\] Let $\psi$ be a smooth, spherically symmetric solution to on extremal Reissner–Nordström with $I_0[\psi]$ a well-defined limit. We then define the *time integral* $\psi^{(1)}$ of $\psi$ to be the function ${\psi}^{(1)}: \mathring{\mathcal{R}}\rightarrow \mathbb{R}$, such that
- $T\psi^{(1)}=\psi_0$,
- $\square_g\psi^{(1)}=0$,
- $\lim_{v\to \infty}\psi^{(1)}(0,v)=0$,
- $\lim_{u\to \infty}\underline{L}\psi_0^{(1)}(u,v_0)=0$.
\[prop:explicitexprtimeint\] The time integral $\psi_0^{(1)}$ of the spherical mean $\psi_0$ of a solution to on extremal Reissner–Nordström satisfies the following identities: $$\begin{aligned}
2r^2\underline{L}\psi_0^{(1)}(u,v_0)=&\: 2\int_{u}^{\infty} r\underline{L}\phi_0(u',v_0)\,du' \quad \textnormal{on ${{N}_0^{\mathcal{H}}}\cap \mathring{\mathcal{R}}$},\\
Dr^2\partial_{\rho}\psi_0^{(1)}(0,\rho)=&\:\int_{r_{\mathcal{H}}}^{\rho}\left[-2(1-h\cdot D)r\partial_{\rho}\phi_0+(2-h\cdot D)rhT\phi_0+r\cdot (hD)' \phi_0\right]|_{\Sigma_0}(\rho')d\rho' \\
&+h\cdot Dr^2\phi_0|_{\Sigma_0}(\rho=r_{\mathcal{H}})-2\int_{u_{r_{\mathcal{H}}}(v_0)}^{\infty} r\underline{L}\phi_0(u',v_0)\,du'\quad \textnormal{on $\Sigma_{\tau}\cap \{r_{\mathcal{H}}\leq r\leq r_{\mathcal{I}}\}$},\\
2r^2L\psi_0^{(1)}(u_0,v)=&\:C_0+2\int_{v_{r_{\mathcal{I}}}(u_0)}^{v} rL\phi_0(u_0,v')\,dv'\quad \textnormal{on ${{N}_0^{\mathcal{I}}}$},\end{aligned}$$ where $h$ is the function of $r$ given by , and we use the shorthand notation $$\begin{split}
4\pi C_0[\psi]:=2\int_{N^{\mathcal{H}}_0} r\underline{L}\phi_0\,d\omega du'+\int_{\Sigma_0\cap \mathcal{B}} n_{\Sigma_0}(\psi)\,d\mu_{0}+4\pi r\phi_0|_{N^{\mathcal{H}}_0} (r=r_{\mathcal{H}})+4\pi r\phi_0|_{N^{\mathcal{H}}_0} (r=r_{\mathcal{I}}).
\end{split}$$ If $\lim_{r\to \infty}r^3\partial_r\phi_0|_{{N^{\mathcal{I}}_0}}<\infty$ (), then we can further express, $$r\psi^{(1)}_0|_{{{N}_0^{\mathcal{I}}}}(r)=-r\left[C_0[\psi]+2\int_{v_{r_{\mathcal{I}}}(u_0)}^{\infty} rL\phi_0(u_0,v')\,dv'\right](r-M)^{-1}+2r\int_{r}^{\infty} (r'-M)^{-2}\int_{r'}^{\infty}r\partial_r\phi_0|_{{{N}_0^{\mathcal{I}}}}(r'')\,dr''dr',$$ and we have that in $(u,r)$ coordinates $$\begin{split}
I_0[\psi^{(1)}]=&MC_0[\psi]+2M\int_{v_{r_{\mathcal{I}}}(u_0)}^{\infty} rL\phi_0(u_0,v')\,dv'- \lim_{r\to \infty}r^3\partial_r\phi_0|_{{{N}_0^{\mathcal{I}}}}\\
=& M^2\phi_0|_{\mathcal{H}^+} (v=v_0)+\frac{M}{4\pi}\int_{N^{\mathcal{H}}_0} \underline{L}\psi_0\,r^2d\omega du'+\frac{M}{4\pi}\int_{\Sigma_0\cap \mathcal{B}} n_{\Sigma_0}(\psi)\,d\mu_{0}\\
&+M r\phi_0|_{N^{\mathcal{H}}_0} (r=r_{\mathcal{I}})+\frac{2M}{4\pi}\int_{N^{\mathcal{I}}_0} rL\phi_0\,d\omega dv'- \lim_{r\to \infty}r^3\partial_r\phi_0|_{{{N}_0^{\mathcal{I}}}}.
\end{split}$$
Note that we can write $$\square_g\psi_0^{(1)}(\tau,\rho)=\square_g\psi_0^{(1)}(0,\rho)+\int_0^{\tau} \square_g\psi_0(\tau',\rho)\,d\tau'=\square_g\psi_0^{(1)}(0,\rho),$$ and therefore $\square_g\psi_0^{(1)}=0$ in $\mathring{\mathcal{R}}$ if and only if $\square_g\psi_0^{(1)}(0,\rho)=0$ for $\rho>M$, which is equivalent to the following equation: $$\underline{L}(r^2\underline{L}\psi_0^{(1)})=r\underline{L}\phi_0.$$ We therefore obtain the following identity on ${{N}_0^{\mathcal{H}}}\cap\mathring{\mathcal{R}}$: $$2r^2\underline{L}\psi_0^{(1)}(u,v_0)= \lim_{u\to \infty}2r^2\underline{L}\psi_0^{(1)}(u,v_0)-2\int_{u}^{\infty} r\underline{L}\phi_0(u',v_0)\,du',$$ where the first term on the right-hand side is zero by definition of $\psi^{(1)}$.
Recall that $\partial_{\rho}=-2D^{-1}\underline{L}+h\cdot T=2D^{-1}L+(h-2D^{-1})\cdot T$, so by the above, we have that $$D(r_{\mathcal{H}})r_{\mathcal{H}}^2\partial_{\rho}\psi_0^{(1)}(u_{r_{\mathcal{H}}}(v_0),v_0)=2\int_{u_{r_{\mathcal{H}}}(v_0)}^{\infty} r\underline{L}\phi_0(u',v_0)\,du'+hD(r_{\mathcal{H}}) r_{\mathcal{H}}\phi_0(u_{r_{\mathcal{H}}}(v_0),v_0).$$ We compute $$\partial_{\rho}(Dr^2\partial_{\rho}\psi_0^{(1)})=-2(1-h\cdot D)r\partial_{\rho}\phi_0+(2-h\cdot D)rhT\phi_0+r\cdot (hD)' \phi_0,$$ so, by using all the above estimates, we can conclude that everywhere on $\Sigma_0\cap \mathcal{B}$: $$\begin{split}
Dr^2\partial_{\rho}\psi_0^{(1)}(0,\rho)=&\:2\int_{u_{r_{\mathcal{H}}}(v_0)}^{\infty} r\underline{L}\phi_0(u',v_0)\,du'+hD(r_{\mathcal{H}}) r_{\mathcal{H}}\phi_0(u_{r_{\mathcal{H}}}(v_0),v_0)\\
&+\int_{r_{\mathcal{H}}}^{\rho}[-2(1-h\cdot D)r\partial_{\rho}\phi_0+(2-h\cdot D)rhT\phi_0+r\cdot (hD)' \phi_0]|_{\Sigma_0}(\rho')d\rho'.
\end{split}$$ By $2L=D\partial_{\rho}+(2-hD)\cdot T$ we also obtain the following expression for $\psi^{(1)}$ on ${{N}_0^{\mathcal{I}}}$: $$\begin{split}
2r_{\mathcal{I}}^2L\psi_0^{(1)}(u_0,v_{r_{\mathcal{I}}}(u_0))=&2\int_{u_{r_{\mathcal{H}}}(v_0)}^{\infty} r\underline{L}\phi_0(u',v_0)\,du'+hD(r_{\mathcal{H}}) r_{\mathcal{H}}\phi_0(u_{r_{\mathcal{H}}}(v_0),v_0)\\
&+\int_{r_{\mathcal{H}}}^{r_{\mathcal{I}}}[-2(1-h\cdot D)r\partial_{\rho}\phi_0+(2-h\cdot D)rhT\phi_0+r\cdot (hD)' \phi_0]|_{\Sigma_0}(\rho')d\rho'\\
&+(2-h(r_{\mathcal{I}})D(r_{\mathcal{I}}))r_{\mathcal{I}}\phi_0(u_0,v_{r_{\mathcal{I}}}(u_0))=:C_0[\psi].
\end{split}$$ By using that the normal $n_{\Sigma_0}$ to $\Sigma_0\cap \mathcal{B}$ can be expressed as follows: $$\sqrt{\det g_{\Sigma_0\cap \mathcal{B}}}n_{\Sigma_0}=r^2\sin \theta\left[(hD-1)\partial_{\rho} +h(2-hD)T\right],$$ we can rewrite $$\begin{split}
[-2&(1-h\cdot D)r\partial_{\rho}\phi_0+(2-h\cdot D)rhT\phi_0+r\cdot (hD)' \phi_0]\sin \theta\\
=&\:2(h D-1)r^2\sin \theta\partial_{\rho}\psi+2(hD-1)\sin \theta \phi_0+(2-hD)h r^2\sin \theta T\psi_0+r(hD)' \sin \theta \phi_0\\
=&\:\partial_{\rho}( (hD-1)r^2 \psi_0)\sin \theta+\sqrt{\det g_{\Sigma_0\cap \mathcal{B}} }n_{\Sigma_0}(\psi).
\end{split}$$ Hence, we obtain $$\begin{split}
4\pi C_0=&2\int_{N^{\mathcal{H}}_0} r\underline{L}\phi_0\,d\omega du'+\int_{\Sigma_0\cap \mathcal{B}} n_{\Sigma_0}(\psi)\,d\mu_{0}+4\pi r\phi_0|_{N^{\mathcal{H}}_0} (r=r_{\mathcal{H}})+4\pi r\phi_0|_{N^{\mathcal{H}}_0} (r=r_{\mathcal{I}})\\
=&4\pi M\phi_0|_{\mathcal{H}^+} (v=v_0)-\int_{N^{\mathcal{H}}_0} \underline{L}\psi_0\,r^2d\omega du'+\int_{\Sigma_0\cap \mathcal{B}} n_{\Sigma_0}(\psi)\,d\mu_{0}+4\pi r\phi_0|_{N^{\mathcal{H}}_0} (r=r_{\mathcal{I}}).
\end{split}$$ Since $\psi^{(1)}$ satisfies $$L(r^2L\psi_0^{(1)})=rL\phi_0,$$ we therefore conclude that everywhere on ${{N}_0^{\mathcal{I}}}$ we can write $$2r^2L\psi_0^{(1)}(u_0,v)=C_0[\psi]+2\int_{v_{r_{\mathcal{I}}}(u_0)}^{v} rL\phi_0(u_0,v')\,dv'.$$
In particular, if $I_0[\psi]=0$, we have that
$$\left|\int_{v_{r_{\mathcal{I}}}(u_0)}^{\infty} rL\phi_0(u_0,v')\,dv'\right|<\infty$$
so we can switch to $(u,r)$ coordinates in order to express: $$\psi^{(1)}_0|_{{{N}_0^{\mathcal{I}}}}(r)=-2\left[C_0[\psi]+2\int_{v_{r_{\mathcal{I}}}(u_0)}^{\infty} rL\phi_0(u_0,v')\,dv'\right](r-M)^{-1}-2\int_{r}^{\infty} (r'-M)^{-1}\int_{r'}^{\infty}r\partial_r\phi_0|_{{{N}_0^{\mathcal{I}}}}(r'')\,dr''$$ The expression for $I_0[\psi_0^{(1)}]$ then follows from multiplying both sides by $r$, using that $\lim_{r\to \infty}r^3\partial_r\phi_0|_{{{N}_0^{\mathcal{I}}}}<\infty$ and taking an $r$ derivative.
\[prop:explicitexprtimeint2\] If $H_0[\psi]=0$, then the time integral $\psi_0^{(1)}$ of $\psi_0$ satisfies moreover $$r\psi^{(1)}_0|_{{{N}_0^{\mathcal{H}}}}(r)=-\tilde{r}\left[\underline{C}_0[\psi]+2\int_{u_{r_{\mathcal{H}}}(v_0)}^{\infty} \tilde{r}\underline{L}\phi_0(u',v_0)\,dv'\right](\tilde{r}-M)^{-1}+2\tilde{r}\int_{\tilde{r
}}^{\infty} (\tilde{r}'-M)^{-2}\int_{\tilde{r}'}^{\infty}\tilde{r}\partial_{\tilde{r}}\phi_0|_{{{N}_0^{\mathcal{H}}}}(\tilde{r}'')\,d\tilde{r}''d\tilde{r}',$$ with $$\begin{aligned}
\tilde{r}=&\:M+M^2(r-M)^{-1},\\
4\pi \underline{C}_0[\psi]:=&\:2\int_{N^{\mathcal{I}}_0}\frac{M}{r-M} \cdot r{L}\phi_0\,d\omega dv+\int_{\Sigma_0\cap \mathcal{B}} n_{\Sigma_0}\left(\frac{M}{r-M}\cdot \psi\right)\,d\mu_{0}\\
&+4\pi \frac{M}{r-M} \cdot r\phi_0|_{N^{\mathcal{H}}_0} (r=r_{\mathcal{H}})+4\pi \frac{M}{r-M} \cdot r\phi_0|_{N^{\mathcal{H}}_0} (r=r_{\mathcal{I}}).\end{aligned}$$ and we have that in $(v,r)$ coordinates $$H_0[\psi^{(1)}]=M\underline{C}_0[\psi]+2M\int_{u_{r_{\mathcal{H}}}(v_0)}^{\infty} \frac{Mr}{r-M}\underline{L}\phi_0(u',v_0)\,du'+M^4\partial_{r}^2\phi_0|_{{{N}_0^{\mathcal{H}}}}(r=M).$$
Consider now the rescaled functions $\underline{\psi}_0=M^{-1}(r-M)\psi_0$ and $\underline{\psi}^{(1)}_0=M^{-1}(r-M)\psi_0^{(1)}$. Note that by Proposition \[prop:explicitexprtimeint\], we have that $$\lim_{u\to \infty}\underline{\psi}^{(1)}_0(u,v_0)=0,\\
\lim_{v\to \infty} L \underline{\psi}^{(1)}_0(u_0,v)=0,$$ so $\underline{\psi}^{(1)}_0$ satisfies analogous boundary conditions to $\psi_0^{(1)}$, but with $u$ and $v$ and $L$ and $\underline{L}$ interchanged. We introduce the notation $\tilde{r}=M+M^2(r-M)^{-1}$ and $\tilde{D}(\tilde{r})=(1-M\tilde{r}^{-1})^2$, we have that $\tilde{r} \underline{\psi}_0=r\psi_0$ and $\tilde{r} \underline{\psi}_0^{(1)}=r\psi_0^{(1)}$, and $\underline{\psi}_0^{(1)}$ satisfies the equations: $$\begin{aligned}
L(\tilde{r}^2L\underline{\psi}^{(1)}_0)=&\tilde{r}L \phi_0,\\
\underline{L}(\tilde{r}^2\underline{L}\underline{\psi}^{(1)}_0)=\:&\tilde{r}\underline{L}\phi_0.\end{aligned}$$
We can therefore repeat the arguments above, starting the integration along ${{N}_0^{\mathcal{I}}}$ rather than ${{N}_0^{\mathcal{H}}}$, to obtain the following expressions: $$\begin{aligned}
2\widetilde{r}^2L\underline{\psi}^{(1)}_0(u_0,v)=&\: 2\int_{v}^{\infty} \widetilde{r}L\phi_0(u_0,v)\,dv' \quad \textnormal{on ${{N}_0^{\mathcal{I}}}$},\\
\tilde{D}(\tilde{r})\tilde{r}^2\partial_{\tilde{\rho}}\underline{\psi}^{(1)}_0(0,\tilde{\rho})=&\:\int^{\tilde{\rho}}_{\tilde{r}(r_{\mathcal{I}})}\left[-2(1-\tilde{h}\cdot \tilde{D})\tilde{r}\partial_{\tilde{\rho}}\phi_0+(2-\tilde{h}\cdot \tilde{D})\tilde{r}\tilde{h}T\phi_0+\tilde{r}\cdot \frac{d(\tilde{h}\tilde{D})}{d\tilde{r}} \phi_0\right]|_{\Sigma_0}(\tilde{\rho}')d\tilde{\rho}' \\
&+\tilde{h}\cdot \tilde{D}(\tilde{r})\tilde{r}^2\phi_0|_{\Sigma_0}(\rho=r_{\mathcal{I}})+2\int_{v_{r_{\mathcal{I}}}(u_0)}^{\infty} \tilde{r}{L}\phi_0(u_0,v')\,dv'\quad \textnormal{on ${S}\cap \{r_{\mathcal{H}}\leq r\leq r_{\mathcal{I}}\}$},\\
2\tilde{r}^2\underline{L}\underline{\psi}^{(1)}_0(u,v_0)=&\:\underline{C}_0[\psi]+2\int_{u_{r_{\mathcal{H}}}(u_0)}^{u} \tilde{r}\underline{L}\phi_0(u',v_0)\,du'\quad \textnormal{on ${{N}_0^{\mathcal{H}}}\cap \mathring{\mathcal{M}}$},\end{aligned}$$ with $$\begin{aligned}
\tilde{h}(\tilde{r})=&\:(2\tilde{D}^{-1}-h M^2(\tilde{r}-M)^{-2})=(2D^{-1}-h) M^2(\tilde{r}-M)^{-2},\\
2-\tilde{h}\tilde{D}=&\:hM^2\tilde{r}^{-2}=hD,\\
\partial_{\tilde{\rho}}=&-M^{-2}(r-M)^{2}\partial_{\rho}=-M^{2}(\tilde{r}-M)^{-2}\partial_{\rho},\\
\underline{C}_0[\psi]:=&\:(2-\tilde{h}\cdot \tilde{D})\tilde{r}\phi_0|_{\Sigma_0}(\rho=r_{\mathcal{H}})\\
&-\int^{\tilde{r}(r_{\mathcal{H}})}_{\tilde{r}(r_{\mathcal{I}})}\left[2(1-\tilde{h}\cdot \tilde{D})\tilde{r}\partial_{\tilde{\rho}}\phi_0-(2-\tilde{h}\cdot \tilde{D})\tilde{r}\tilde{h}T\phi_0-\tilde{r}\cdot \frac{d(\tilde{h}\tilde{D})}{d\tilde{r}} \phi_0\right]|_{\Sigma_0}(\tilde{\rho}')d\tilde{\rho}'\\
&+\tilde{h}\cdot \tilde{D}(\tilde{r})\tilde{r}\phi_0|_{\Sigma_0}(\rho=r_{\mathcal{I}})+2\int_{v_{r_{\mathcal{I}}}(u_0)}^{\infty} \tilde{r}L\phi_0(u_0,v')\,dv'\\
=&\:hD\tilde{r}\phi_0|_{\Sigma_0}(\rho=r_{\mathcal{H}})\\
&-\int_{r_{\mathcal{H}}}^{r_{\mathcal{I}}}\left[2(1-hD)\tilde{r}\partial_{\rho}\phi_0-(2-hD)\tilde{r}hT\phi_0-\tilde{r}\cdot \frac{d(hD)}{dr} \phi_0\right]|_{\Sigma_0}(\rho')d\rho'\\
&+(2-hD)\tilde{r}\phi_0|_{\Sigma_0}(\rho=r_{\mathcal{I}})+2\int_{v_{r_{\mathcal{I}}}(u_0)}^{\infty} \tilde{r}L\phi_0(u_0,v')\,dv'\\
=&\:\:hD\tilde{r}\phi_0|_{\Sigma_0}(\rho=r_{\mathcal{H}})\\
&-\int_{r_{\mathcal{H}}}^{r_{\mathcal{I}}}\frac{M}{r-M}\left[2(1-hD)r\partial_{\rho}\phi_0-(2-hD)rhT\phi_0-r\cdot \frac{d(hD)}{dr} \phi_0\right]|_{\Sigma_0}(\rho')d\rho'\\
&+(2-hD)\tilde{r}\phi_0|_{\Sigma_0}(\rho=r_{\mathcal{I}})+2\int_{v_{r_{\mathcal{I}}}(u_0)}^{\infty} \frac{Mr}{r-M}L\phi_0(u_0,v')\,dv'.\end{aligned}$$
We recall from the proof of Proposition \[prop:explicitexprtimeint\] that $$\begin{split}
\frac{M}{r-M}[-2&(1-h\cdot D)r\partial_{\rho}\phi_0+(2-h\cdot D)rhT\phi_0+r\cdot (hD)' \phi_0]\sin \theta\\
=&\:\frac{M}{r-M}\partial_{\rho}( (hD-1)r^2 \psi_0)\sin \theta+\sqrt{\det g_{\Sigma_0\cap \mathcal{B}} }\frac{M}{r-M}n_{\Sigma_0}(\psi)\\
=&\:\partial_{\rho}\left(\frac{M}{r-M} (hD-1)r^2 \psi_0\right)\sin \theta+\sqrt{\det g_{\Sigma_0\cap \mathcal{B}} } n_{\Sigma_0}\left(\frac{M}{r-M}\psi\right)
\end{split}$$ and hence, $$\begin{split}
4\pi \underline{C}_0[\psi]=&\:2\int_{N^{\mathcal{I}}_0}\frac{M}{r-M} \cdot r{L}\phi_0\,d\omega du'+\int_{\Sigma_0\cap \mathcal{B}} n_{\Sigma_0}\left(\frac{M}{r-M}\cdot \psi\right)\,d\mu_{0}\\
&+4\pi \frac{M}{r-M} \cdot r\phi_0|_{N^{\mathcal{H}}_0} (r=r_{\mathcal{H}})+4\pi \frac{M}{r-M} \cdot r\phi_0|_{N^{\mathcal{H}}_0} (r=r_{\mathcal{I}}).
\end{split}$$ Hence, if $H_0[\psi]=0$, we have that $$\underline{\psi}^{(1)}_0|_{{{N}_0^{\mathcal{H}}}}(\tilde{r})=-\left[\underline{C}_0[\psi]+2\int_{u_{r_{\mathcal{H}}}(v_0)}^{\infty} \tilde{r}\underline{L}\phi_0(u',v_0)\,dv'\right](\tilde{r}-M)^{-1}+2\int_{\tilde{r
}}^{\infty} (\tilde{r}'-M)^{-2}\int_{\tilde{r}'}^{\infty}\tilde{r}\partial_{\tilde{r}}\phi_0|_{{{N}_0^{\mathcal{H}}}}(\tilde{r}'')\,d\tilde{r}''d\tilde{r}',$$ and the expression for $H_0[\psi_0^{(1)}]$ is derived as above.
The above propositions motivative the following definitions:
We define the **time-inverted constants** $I_0^{(1)}[\psi]$ and $H_0^{(1)}[\psi]$ as follows: $$\begin{aligned}
I_0^{(1)}[\psi]:=&\:I_0[\psi_0^{(1)}]\quad \textnormal{if $\lim_{v\to \infty}r^3L\psi_0(u_0,v)<\infty$ (and therefore $I_0[\psi]=0$)},\\
H_0^{(1)}[\psi]:=&\:H_0[\psi_0^{(1)}]\quad \textnormal{if $H_0[\psi]=0$}.\end{aligned}$$
If we assume the qualitative decay statements: $r\psi_0|_{\mathcal{I}^+}(u)\to 0$ as $u\to \infty$ and $r\psi_0|_{\mathcal{H}^+}(v)\to 0$ as $v\to \infty$, we can use the results of Proposition \[prop:explicitexprtimeint\] and \[prop:explicitexprtimeint2\] to obtain the following alternative expressions for $I_0^{(1)}[\psi]$ and $H_0^{(1)}[\psi]$: if $\lim_{r \to \infty} r^3L\phi_0|_{\Sigma_0}<\infty$, then $$\begin{split}
I_0^{(1)}[\psi]=&-M \lim_{r\to \infty} r\psi_0^{(1)}|_{\Sigma_0} (r)-2\lim_{r\to \infty} r^3 L\phi_0|_{\Sigma_0}\\
=&\: M \int_{u_0}^{\infty} r\psi_0|_{\mathcal{I}^+}(u)\,du-2\lim_{r\to \infty} r^3 L\phi_0|_{\Sigma_0},
\end{split}$$ and if $H_0[\psi]=0$, we obtain $$\begin{split}
H_0^{(1)}[\psi]=&-M \lim_{r\downarrow M} r\psi_0^{(1)}|_{\Sigma_0} (r)-M^4Y^2\phi_0|_{\Sigma_0}(r=M)\\
=&\: M \int_{v_0}^{\infty} r\psi_0|_{\mathcal{H}^+}(v)\,dv-M^4Y^2\phi_0|_{\Sigma_0}(r=M).
\end{split}$$ We will recover the above decay assumption for $r\psi_0^{(1)}$ in Proposition \[prop:decayesttimeint\]. See also the discussion in Section 1.6 of [@paper2] for an analogous expression for $I_0^{(1)}[\psi]$ in the sub-extremal setting.
Extension of the time integral $\protect\psi^{(1)}$ in $\protect\mathcal{R}$ {#sec:extendedtimeinversionr}
-----------------------------------------------------------------------------
In this section, we will investigate the regularity properties of the *continuous extensions* of the time integral functions $\psi^{(1)}$ defined in Section \[sec:TheRegularTimeInversionConstructionInOversetOMathcalR\] to the full spacetime region $\mathcal{R}$. We will moreover discuss the singular properties of (derivatives) of the radiation field at $\mathcal{I}^+$.
### Regular extension in $\protect\mathcal{R}$ for Type **B** perturbations {#sec:TheSingularTimeInversionInProtectMathcalRB}
We first consider the case of Type **B** data.
\[eq:smoothextTypeBtimeint\] Let $\psi_0^{(1)}$ be the time integral of a smooth solution $\psi_0$ to corresponding to initial data of Type **B**. Then $\psi_0^{(1)}$ can be extended uniquely as smooth function to $\mathcal{R}$. Furthermore, $I_0[\psi^{(1)}]$ is well-defined.
By Proposition \[prop:explicitexprtimeint\], we have that in $(v,r)$ coordinates $$\label{eq:explicitdrpsi1NH}
\partial_r\psi_0^{(1)} (v_0,r)=\frac{2}{(r-M)^2}\int_M^r r'\partial_r\phi_0(v_0,r')\,dr'$$ along $\mathcal{N}^{\mathcal{H}}_0$.
If we assume that $H_0[\psi]=0$, we can use smoothness of $\phi_0$ together with Taylor’s theorem to obtain the following: for any $N\in {\mathbb{N}}$, we can decompose for $r\leq r_{\mathcal{H}}$: $r\partial_r\phi_0(v_0,r)=\sum_{k=1}^Np_k(r-M)^k+(r-M)^{N+1}f_N(v,r)$, for some smooth function $f_N: [M,r_{\mathcal{H}})\to {\mathbb{R}}$ and coefficients $p_k\in {\mathbb{R}}$, with $k\in 1,\ldots,N$.
Hence, $$\partial_r\psi_0^{(1)} (v_0,r)=\sum_{k=1}^N \frac{2p_k}{(k+1)(r-M)^2}(r-M)^{k+1}+\frac{2}{(r-M)^2}\int_M^r (r'-M)^{N+1}f_N(r')\,dr'.$$ It is clear that $\partial_r^{N}\psi_0$ attains a finite limit at $r=M$. Using that $N$ can be taken to be arbitrarily large and $T\psi_0^{(1)}=\psi_0$, we can conclude that $\psi^{(1)}$ extends smoothly to $r=M$. The second part of the proposition follows immediately from Proposition \[prop:explicitexprtimeint\], using that $\lim_{v\to \infty} r^3L\phi(u_0,v)$ is well-defined for Type **B** data.
**By Proposition \[eq:smoothextTypeBtimeint\], all the estimates in Section \[sec:asympnonzeroconst\] can be applied when $\psi_0$ is replaced by $\psi^{(1)}_0$ in the case of Type **B** data!**
### Singular horizon extension in $\protect\mathcal{R}$ for Type **A** perturbations {#sec:TheSingularTimeInversionInProtectMathcalRA}
We now consider data of Type **A** and show that, due to the non-vanishing of $H_0[\psi]$, the time integral $\psi^{(1)}$ displays singular behaviour at $\mathcal{H}^+$.
\[lm:someregproptimeint\] Let $\psi_0^{(1)}$ be the time integral of a smooth solution $\psi_0$ to corresponding to initial data of Type **A**. Then $\psi_0^{(1)}$ cannot be extended as a continuous function to $\mathcal{R}$. However, $I_0[\psi^{(1)}]$ is well-defined. More precisely, we can decompose $$\begin{aligned}
\label{eq:blowupdrpsi1typeA}
\partial_{r}\psi^{(1)}_0(v,r)=&-2M^{-1}H_0[\psi](r-M)^{-1}+f(v,r),\\
\label{eq:blowupdrpsi1typeA2}
\psi^{(1)}_0(v,r)=& -2M^{-1}H_0[\psi]\log(r-M)+\widetilde{f}(v,r),\end{aligned}$$ for some smooth, spherically symmetric functions $f,\widetilde{f}$ on $\mathcal{R}$.
Furthermore, for $\epsilon>0$ arbitrarily small, we can estimate $$\begin{aligned}
\label{eq:finiteTenergypsi1}
\int_{\Sigma_0}& J^T[\psi^{(1)}]\cdot \mathbf{n}_{\Sigma_0}\,d\mu_{\Sigma_0}+\int_{{{N}_0^{\mathcal{H}}}}(r-M)^{-1+\epsilon}(\underline{L}\phi^{(1)})^2\,du+\int_{{{N}_0^{\mathcal{I}}}}r^{1-\epsilon}({L}\phi^{(1)})^2\,dv<\infty,\\
\label{eq:blowupPenergypsi1}
\int_{\Sigma_0}& \frac{1}{\sqrt{D}}\cdot J^T[\psi^{(1)}]\cdot \mathbf{n}_{\Sigma_0}\,d\mu_{\Sigma_0}=\infty.\end{aligned}$$
We can use smoothness of $\phi_0$ together with Taylor’s theorem and the definition of $H_0[\psi]$ to obtain the following: for any $N\in {\mathbb{N}}$, we can decompose $$\partial_r\phi_0(r,v_0)=-M^{-2}H_0+\sum_{k=1}^Np_k(r-M)^k+(r-M)^{N+1}f_N(v,r).$$ The equation then follows immediately after plugging the above equation into the right-hand side of . Then, and follow directly.
### Singular radiation field $r\protect\protect\psi^{(1)}|_{\mathcal{I}^{+}}$ for Type **D** perturbations {#sec:TheSingularTimeInversionInProtectMathcalRD}
We now consider data of Type **D** and show that the radiation field $r\psi^{(1)}|_{\mathcal{I}^+}$ and Newman–Penrose constant $I_0[\psi^{(1)}]$ are ill-defined in this case.
\[eq:singextTypeDtimeint\] Let $\psi_0^{(1)}$ be the time integral of a smooth solution $\psi_0$ to corresponding to initial data of Type **D** such that moreover $$\partial_r\phi_0(u,r)=I_0[\psi]r^{-2}+O(r^{-3})$$ along $N^{\mathcal{I}}_0$.
Then $\psi_0^{(1)}$ can be extended uniquely as smooth function to $\mathcal{R}$. However, $r\psi^{(1)}$ and $I_0[\psi^{(1)}]$ are ill-defined at $\mathcal{I}^+$; more precisely, $$\begin{aligned}
r\psi_0^{(1)}(u,r)=&\:2I_0[\psi]\log r +O(r^0),\\
r^2\partial_r(r\psi_0^{(1)})(u,r)=&\: 2I_0[\psi] r+O(r^0),\end{aligned}$$ for some constant $p_0\in {\mathbb{R}}$.
By the estimates in the proof of Proposition \[prop:explicitexprtimeint2\], it follows that $$D\tilde{r}^2\partial_r\underline{\psi}_0^{(1)}(u,r)=2\int_{r}^{\infty} \tilde{r}\partial_r\phi_0(u,r')\,dr',$$ where $\tilde{r}-M=M^2(r-M)^{-1}$ and $\underline{\psi}_0=\tilde{r}^{-1}\phi_0$. Hence $$\begin{aligned}
\partial_r\underline{\psi}_0^{(1)}(u,r)=&\:2M^{-1}I_0[\psi]r^{-1}+O(r^{-2}),\\
\underline{\psi}_0^{(1)}(u,r)=&\:2M^{-1}I_0[\psi]\log r+ O(r^0)\end{aligned}$$ so we obtain $$\partial_r\phi_0^{(1)}(u,r)=\partial_r(\tilde{r} \psi_0^{(1)})(u,r)=2I_0[\psi]r^{-1}+ O(r^{-2})$$ and therefore, $$\phi_0^{(1)}=2I_0[\psi]\log r+ O(r^0).$$
### Decay estimates for $\psi^{(1)}$ {#sec:DecayForPsi1}
We now establish some preliminary decay estimates for the time integral $\psi_0^{(1)}$ of $\psi_0$.
\[prop:decayesttimeint\] Let $\psi_0^{(1)}$ be the time integral of $\psi_0$ and let $\epsilon>0$ be arbitrarily small. Then there exists a constant $C=C(M,{\Sigma_0},r_{\mathcal{H}},r_{\mathcal{I}},\epsilon)>0$ such that $$\label{eq:edecaypsi1v1}
\begin{split}
\int_{\Sigma_{\tau}}J^T[\psi_0^{(1)}]\cdot \mathbf{n}_{\tau}\,d\mu_{{\tau}}\leq&\: C(1+\tau)^{-1+\epsilon}\cdot \Bigg[\int_{\Sigma_0} J^T[\psi_0^{(1)}]\cdot \mathbf{n}_{\Sigma_0}\,d\mu_{0}+\int_{{{N}_0^{\mathcal{H}}}}(r-M)^{-1+\epsilon}(\underline{L}\phi_0^{(1)})^2\,d\omega du\\
&+\int_{{{N}_0^{\mathcal{I}}}}r^{1-\epsilon}({L}\phi_0^{(1)})^2\,d\omega dv\Bigg].
\end{split}$$ We can further estimate $$\begin{aligned}
|r\cdot \psi^{(1)}|\leq&\: C\cdot \sqrt{E^{\epsilon}_{0, \mathcal{I}}[\psi]} (1+\tau)^{-\frac{1}{2}+\epsilon}\quad \textnormal{in $\mathcal{A}^{\mathcal{I}}$ if $\lim_{v\to \infty} v^3\partial_v\phi_0|_{N_0^{\mathcal{I}}}<\infty$,} \\
|r\cdot \psi^{(1)}|\leq&\: C\cdot \sqrt{E^{\epsilon}_{0,\mathcal{H}}[\psi]} (1+\tau)^{-\frac{1}{2}+\epsilon}\quad \textnormal{in $\mathcal{A}^{\mathcal{H}}$ if $H_0[\psi]=0$.}\end{aligned}$$
We apply the $r$-weighted estimates from Proposition \[prop:rpestell01\] with $n=0$ and $p=1-\epsilon$, together with the Morawetz estimates (see Appendix \[sec:EnergyBounds\]) to conclude that there exists a sequence of times $(\tau_k)$ along which we can estimate: $$\begin{split}
\int_{N_{\tau_k}^{\mathcal{H}}}& (r-M)^{\epsilon}(\underline{L}\phi^{(1)}_0)^2\,du+\int_{{\Sigma}_{\tau}\cap \mathcal{B}} J^T[\psi_0^{(1)}]\cdot \mathbf{n}_{\tau}\,d\mu_{\tau}+\int_{N_{\tau_k}^{\mathcal{I}}}r^{-\epsilon}(L\phi^{(1)}_0)^2\,dv\\
\lesssim&\: \tau_k^{-1}\left[\int_{\Sigma_0} J^T[\psi^{(1)}]\cdot \mathbf{n}_{\Sigma_0}\,d\mu_{\Sigma_0}+\int_{{{N}_0^{\mathcal{H}}}}(r-M)^{-1+\epsilon}(\underline{L}\phi_0^{(1)})^2\,du+\int_{{{N}_0^{\mathcal{I}}}}r^{1-\epsilon}({L}\phi_0^{(1)})^2\,dv\right].
\end{split}$$ Hence, we estimate in the region $\{M^2\tau_k^{-1}\leq r-M\leq\tau_k\}$: $$\begin{split}
\int_{{\Sigma_0}_{\tau_k}\cap\{M^2\tau_k^{-1}\leq r-M\leq\tau_k\}}& J^T[\psi_0^{(1)}]\cdot \mathbf{n}_{\tau}\,d\mu_{\tau}\\
\lesssim&\:\tau_k^{-1+\epsilon}\left[\int_{\Sigma_0} J^T[\psi_0^{(1)}]\cdot \mathbf{n}_{\Sigma_0}\,d\mu_{\Sigma_0}+\int_{{{N}_0^{\mathcal{H}}}}(r-M)^{-1+\epsilon}(\underline{L}\phi_0^{(1)})^2\,d\omega du+\int_{{{N}_0^{\mathcal{I}}}}r^{1-\epsilon}({L}\phi_0^{(1)})^2\,d\omega dv\right].
\end{split}$$ In the region $\{r-M\leq M^2\tau_k^{-1}\}\cup \{r-M\geq \tau_k\}$ we use again Proposition \[prop:rpestell01\] with $n=0$ and $p=1-\epsilon$ to estimate $$\begin{split}
\int_{N_{\tau_k}^{\mathcal{H}}}& (r-M)^{-1+\epsilon}(\underline{L}\phi^{(1)}_0)^2\,du+\int_{N_{\tau_k}^{\mathcal{I}}} r^{-\epsilon}(L\phi^{(1)}_0)^2\,dv\\
\lesssim&\:\left[\int_{\Sigma_0} J^T[\psi_0^{(1)}]\cdot \mathbf{n}_{\Sigma_0}\,d\mu_{\Sigma_0}+\int_{{{N}_0^{\mathcal{H}}}}(r-M)^{-1+\epsilon}(\underline{L}\phi_0^{(1)})^2\,du+\int_{{{N}_0^{\mathcal{I}}}}r^{1-\epsilon}({L}\phi_0^{(1)})^2\,dv\right]
\end{split}$$ to estimate $$\begin{split}
\int_{{\Sigma}_{\tau_k}\cap\{r-M\leq M^2\tau_k^{-1}\}\cup \{r-M\geq \tau_k\}}& J^T[\psi_0^{(1)}]\cdot \mathbf{n}_{\tau}\,d\mu_{\tau}\\
\lesssim&\:\tau_k^{-1+\epsilon}\left[\int_{\Sigma_0} J^T[\psi_0^{(1)}]\cdot \mathbf{n}_{\Sigma_0}\,d\mu_{\Sigma_0}+\int_{{{N}_0^{\mathcal{H}}}}(r-M)^{-1+\epsilon}(\underline{L}\phi_0^{(1)})^2\,du+\int_{{{N}_0^{\mathcal{I}}}}r^{1-\epsilon}({L}\phi_0^{(1)})^2\,dv\right].
\end{split}$$ Hence, by applying we can conclude that must hold (for all times $\tau\geq 0$). In particular, this implies that $$\lim_{v\to \infty}\psi_0^{(1)}(v,r)=0.$$ The remaining (quantitative) estimates in the proposition then follow immediately from integrating the pointwise decay estimates for $\psi_0=T\psi_0^{(1)}$ obtained in Proposition \[prop:pointdecay\].
We can moreover relate the relevant initial pointwise norms of $\psi^{(1)}$ to analogous pointwise norms of $\psi$.
\[lm:estPnormpsi1\] For all $0\leq \beta\leq 1$ and $k\in {\mathbb{N}}_0$, we can estimate $$\begin{aligned}
P_{H_0,\beta;k}[\psi_0^{(1)}]\lesssim&\: H_0^{(1)}[\psi]+P_{H_0,\beta;k}[\psi]\quad \textnormal{if $H_0[\psi]=0$},\\
P_{I_0,\beta;k}[\psi_0^{(1)}]\lesssim&\: I_0^{(1)}[\psi]+P_{I_0,\beta;k}[\psi]\quad \textnormal{if $\lim_{v\to \infty}r^3L\psi_0(u_0,v)<\infty$},\\
E^{\epsilon}_{0;k+1}[\psi_0^{(1)}]\lesssim&\: E^{\epsilon}_{0, \mathcal{H};k}[\psi]+E^{\epsilon}_{0, \mathcal{I};k}[\psi]\quad \textnormal{if $H_0[\psi]=0$ and $\lim_{v\to \infty}r^3L\psi_0(u_0,v)<\infty$}.\end{aligned}$$
The estimates follow from the expressions for $\psi_0^{(1)}$ in Proposition \[prop:explicitexprtimeint\] and \[prop:explicitexprtimeint2\].
Late-time asymptotics for Type **A** perturbations {#sec:asympzeroconst}
==================================================
In this section we will use the time integral construction from Section to obtain late-time asymptotics for $\psi$ arising from Type **A** data. Recall that Type **A** data includes *generic smooth and compactly supported* data on $\Sigma_0$.
Conditional asymptotics for $r\psi^{(1)}$ in $\protect\mathcal{A}^{\mathcal{I}}_{\gamma^{\mathcal{I}}}$ {#sec:andConditionalAsymptoticsForRPsi}
-------------------------------------------------------------------------------------------------------
We will first obtain estimates in the region . These may be thought of as the analogues of the estimates in Proposition \[prop:asympLphi\] and \[prop:asympradfieldnonzeroIH\] applied to $\phi_0^{(1)}$ rather than $\phi_0$. However, it is important to note the estimates for $\phi_0^{(1)}$ are not as strong as the estimates for $\phi_0$ arising from Type **C** data due to the fact that the upper bound decay estimates that we have for $\phi_0^{(1)}$ (Proposition \[prop:decayesttimeint\]) are *weaker* than the upper bound decay estimates for $\phi_0$ from Proposition \[prop:pointdecay\].
\[prop:improvedasymderphi\] Let $k\in {\mathbb{N}}_0$ and let $\alpha>0$ such that $1-\alpha$ is suitably small. Then, there exists an $\eta>0$ and $\epsilon>0$ suitably small and a constant $C=C(M,\Sigma,r_{\mathcal{H}},r_{\mathcal{I}},\alpha,\epsilon,\beta,\eta)>0$ such that in $\mathcal{A}_{\gamma^{\mathcal{I}}_{\alpha}}^{\mathcal{I}}$: $$\begin{aligned}
|LT^k\phi_0^{(1)}&(u,v)-(-1)^{k}(k+1)! 2I_0^{(1)}[\psi]\cdot v^{-2-k}|\\
\leq&\: C \left[\sqrt{E^{\epsilon}_{0, \mathcal{I}; k}[\psi]}\cdot v^{-2-k-\eta}+\left(P_{I_0,\beta;k}[\psi]+I_0^{(1)}[\psi]\right)\cdot v^{-2-\beta-k}\right]\end{aligned}$$ and moreover, $$\begin{aligned}
|LT^k\phi_0&(u,v)-(-1)^{k+1}(k+2)! 2I_0^{(1)}[\psi]\cdot v^{-3-k}|\\
\leq&\: C \left[\sqrt{E^{\epsilon}_{0, \mathcal{I}; k+1}[\psi]}\cdot v^{-3-k-\eta}+\left(P_{I_0,\beta;k+1}[\psi]+I_0^{(1)}[\psi]\right)\cdot v^{-3-\beta-k}\right].\end{aligned}$$
We apply Proposition \[prop:asympLphi\] to $\phi_0^{(1)}$ instead of $\phi_0$, replacing $k$ with $k+1$, where we only consider the region $\mathcal{A}_{\gamma^{\mathcal{I}}_{\alpha}}^{\mathcal{I}}$. We need the pointwise decay estimates in Proposition \[prop:pointdecay2\] and Proposition \[prop:decayesttimeint\], rather than the decay estimates in Proposition \[prop:pointdecay\]. We also apply Lemma \[lm:estPnormpsi1\] in order to have only norms involving $\psi$ on the right-hand side of our estimates.
\[prop:improvedasymphi\] Let $k\in {\mathbb{N}}_0$ and let $\alpha>0$ such that $1-\alpha$ is suitably small, then there exists a constant $C=C(M,\Sigma,r_{\mathcal{H}},r_{\mathcal{I}},\alpha,\epsilon, \beta, k)>0$ and $\eta>0$ such that in $\mathcal{A}_{\gamma^{\mathcal{I}}_{\alpha}}^{\mathcal{I}}$: $$\begin{aligned}
\Bigg|T^k\phi_0(u,v)&-T^k\phi_0(u,v_{\gamma^{\mathcal{I}}_{\alpha}}(u))- (-1)^{k+1}(k+1)! 2I_0^{(1)}[\psi]\left[u^{-2-k}-v^{-2-k}\right]\Bigg|\\
\leq&\: C\left[\sqrt{E^{\epsilon}_{0, \mathcal{I};k+1}[\psi]}+P_{I_0,\beta;k+1}[\psi]+I_0^{(1)}[\psi]\right]\cdot \frac{v-u}{vu^{2+k+\eta}}.\end{aligned}$$
The estimates in the proposition follow from Proposition \[prop:improvedasymderphi\] in the same way as the estimates in Proposition \[prop:asympradfieldnonzeroIH\] follow from Proposition \[prop:asympLphi\], but we do not estimate $|T^k\phi_0(u,v_{\gamma^{\mathcal{I}}_{\alpha}}(u))|$.
Asymptotics for $\protect\partial_{\rho}\psi$ away from $\protect{\mathcal{H}^{+}}$ up to $\protect\gamma^{\mathcal{I}}$ {#sec:AsymptoticsForPsiAndProtectPartialRhoPsiUpToProtectGammaMathcalIOrProtectGammaMathcalH}
------------------------------------------------------------------------------------------------------------------------
In order to obtain late-time asymptotics from the estimates in Proposition \[prop:improvedasymphi\], we first need to determine, independently, the late-time asymptotics of $T^k\phi_0|_{\gamma^{\mathcal{I}}_{\alpha}}$. This involves a derivation of late-time asymptotics for $L\psi_0$ that are valid all the way up to $\gamma^{\mathcal{I}}_{\alpha}$.[^12]
\[prop:improvedasympLpsiTypeA\] Let $k\in {\mathbb{N}}_0$ and let $\alpha>0$ such that $1-\alpha$ is arbitrarily small. Then, there exists an $\eta>0$ and $\epsilon>0$ suitably small and a constant $C=C(M,\Sigma,r_{\mathcal{H}},r_{\mathcal{I}},\alpha,\epsilon,\eta)>0$, such that in $\mathcal{A}^{\mathcal{I}}\setminus \mathcal{A}_{\gamma^{\mathcal{I}}_{\alpha}}^{\mathcal{I}}$: $$\begin{aligned}
\label{eq:asympLpsigammaTypeA}
\Bigg|2r^2LT^k\psi_0&(u,v)-(-1)^{k+1}(k+1)!4MH_0[\psi]\cdot u^{-2-k}\Bigg|\\ \nonumber
\leq&\: C\left[\sqrt{E^{\epsilon}_{0, \mathcal{I}; k+1}[\psi]}+P_{H_0,1;k}[\psi]+H_0[\psi]\right]\cdot u^{-2-k-\eta}.\end{aligned}$$
We apply the fundamental theorem of calculus in the $L$-direction, together with the equation $L(r^2LT^k\psi_0)=rLT^{k+1}\phi_0$, to obtain: for all $v_{r_{\mathcal{I}}(u)}\leq v\leq v_{\gamma_{\alpha}}(u)$, $$r^2LT^k\psi_0(u,v)=r_{\mathcal{I}}^2LT^k\psi_0(u,v_{r_{\mathcal{I}}}(u))+\int_{v_{r_{\mathcal{I}}}(u)}^{v} rLT^{k+1}\phi_0(u,v')\,dv'.$$ By Cauchy–Schwarz, together with Proposition \[prop:hoextraendecay\], we can estimate $$\begin{split}
\int_{v_{r_{\mathcal{I}}}(u)}^{v_{\gamma_{\alpha}}(u)} r|LT^{k+1}\phi_0|(u,v')\,dv'\lesssim&\: \sqrt{\int_{v_{r_{\mathcal{I}}}(u)}^{v_{\gamma_{\alpha}}(u)} \,dv'}\cdot\sqrt{\int_{v_{r_{\mathcal{I}}}(u)}^{\infty}r^{2} (L T^{k+1}\phi_0)^2\,dv'}\\
\lesssim &\: u^{-\frac{5}{2}+\frac{\epsilon}{2}-k+\frac{\alpha}{2}}\cdot \sqrt{E^{\epsilon}_{0, \mathcal{I}; k+1}[\psi]},
\end{split}$$ and we will take $\alpha+\epsilon<1$.
Now, we appeal to to estimate: $$\begin{split}
\Bigg|2r_{\mathcal{I}}^2LT^k\psi_0|_{r=r_{\mathcal{I}}}(u)&-(-1)^{k+1}(k+1)!\cdot 4MH_0[\psi]\cdot u^{-2-k}\Bigg |\\
\lesssim&\: \left(\sqrt{E^{\epsilon}_{0, \mathcal{I};k+1}[\psi]}+H_0[\psi]+ P_{H_0,1;k}[\psi]\right) u^{-2-k-\eta},
\end{split}$$ for some $\eta>0$. By combining the above estimates, we arrive at .
\[prop:improvedasymphigamma\] Let $k\in {\mathbb{N}}_0$ and let $\alpha>0$ such that $1-\alpha$ is arbitrarily small. Then, there exists an $\eta>0$ and $\epsilon>0$ suitably small and a constant $C=C(M,\Sigma,r_{\mathcal{H}},r_{\mathcal{I}},\alpha,\epsilon,\beta,\eta)>0$, such that $$\label{eq:asymppsigammaIzero}
\begin{split}
\Bigg|T^k\phi_0&(u,v_{\gamma^{\mathcal{I}}_{\alpha}}(u))-(-1)^k(k+1)!4MH_0[\psi]\cdot u^{-2-k}\Bigg|\\ \nonumber
\leq&\: C\Bigg[\sqrt{E^{\epsilon}_{0, \mathcal{I}; k+1}[\psi]}+P_{I_0,\beta;k+1}[\psi]+P_{H_0,1;k}[\psi]+I_0^{(1)}[\psi]+H_0[\psi]\Bigg]\cdot u^{-2-k-\eta}.
\end{split}$$
We split: $$\frac{D}{2}rT^k\psi_0(u,v_{\gamma_{\alpha}}(u))=rLT^k\phi_0(u,v_{\gamma_{\alpha}}(u))-r^2LT^k\psi_0(u,v_{\gamma_{\alpha}}(u)).$$ Now, we apply Proposition \[prop:improvedasymderphi\] together with the estimate $r\lesssim u^{\alpha}$ in $\mathcal{A}_{\gamma^{\mathcal{I}}_{\alpha}}^{\mathcal{I}}$ to estimate for $\epsilon>0$ suitably small: $$\begin{split}
r\cdot& |LT^k\phi_0|(u,v_{\gamma_{\alpha}}(u))\leq C\left[\sqrt{E^{\epsilon}_{0, \mathcal{I}; k+1}[\psi]} +P_{I_0,\beta;k+1}[\psi]+I_0^{(1)}[\psi]\right]\cdot u^{-3-k+\alpha}.
\end{split}$$ Now, we apply to arrive at .
Global asymptotics for $\psi$ in $\protect\mathcal{R}$ {#sec:AsymptoticsInProtectMathcalR}
------------------------------------------------------
In this section, we derive the asymptotics of $\psi$ in the *full* spacetime region, for Type **A** initial data.
\[prop:mainasymptypeA\] Let $k\in {\mathbb{N}}_0$ and assume that $\lim_{v\to \infty}v^3 L\phi_0(u_0,v)<\infty$.
Then there exists an $\eta>0$ and $\epsilon>0$ suitably small, so that we can estimate: $$\begin{split}
\Bigg|T^k\psi_0(u,v)&-4\left[ I_0^{(1)}[\psi]T^{k+1}\left(\frac{1}{u\cdot v}\right)+\frac{M}{r \sqrt{D}}H_0[\psi]T^{k}\left(\frac{1}{u(v+4M-2r)}\right)\right]\Bigg|\\
\leq&\: C\Bigg[\sqrt{E^{\epsilon}_{0, \mathcal{I}; k+1}[\psi]}+P_{I_0,\beta;k+1}[\psi]+P_{H_0,1;k}[\psi]+H_0[\psi]+I_0^{(1)}[\psi] \Bigg]\\
&\cdot \left( v^{-1}u^{-2-k-\eta}+D^{-\frac{1}{2}}u^{-1}v^{-1-k-\eta}\right).
\end{split}$$
By combining the estimates in Proposition \[prop:improvedasymphigamma\] and Proposition \[prop:improvedasymphi\], we arrive at the following estimates for $r\cdot \psi_0(u,v)$: let $\alpha>0$ be sufficiently close to 1, then there exists an $\eta>0$ such that in $\mathcal{A}_{\gamma^{\mathcal{I}}_{\alpha}}^{\mathcal{I}}$: $$\begin{aligned}
\label{eq:asympIzeronearinf}
\Bigg|T^k\phi_0(u,v)&- (-1)^{k+1}(k+1)!\left[ 2I_0^{(1)}[\psi]\left(u^{-2-k}-v^{-2-k}\right)-4MH_0[\psi]u^{-2-k}\right]\Bigg|\\ \nonumber
\leq&\: C\left[\sqrt{E^{\epsilon}_{0, \mathcal{I}; k+1}[\psi]}+P_{I_0,\beta;k+1}[\psi]+P_{H_0,1;k}[\psi]+H_0[\psi]+I_0^{(1)}[\psi] \right]\cdot\frac{v-u}{vu^{2+k+\eta}}.\end{aligned}$$
By applying moreover Lemma \[lm:relationruv\] together with and , we can rewrite as follows: $$\begin{aligned}
\Bigg|T^k\psi_0(u,v)&-\left[ 4I_0^{(1)}[\psi]T^{k+1}\left(\frac{1}{u\cdot v}\right)+4Mr^{-1}H_0[\psi]T^{k}(u^{-2})\right]\Bigg|\\
\leq&\: C\left[\sqrt{E^{\epsilon}_{0, \mathcal{I}; k+1}[\psi]}+P_{I_0,\beta;k+1}[\psi]+P_{H_0,1;k}[\psi]+H_0[\psi]+I_0^{(1)}[\psi] \right]\cdot \frac{1}{vu^{2+k+\eta}}.\end{aligned}$$
To obtain a global estimate for $\psi_0$, we first combine the above estimates with in the region where $r\leq r_{\gamma^{\mathcal{H}}_{\alpha}}(v)$, in the region where $r_{\gamma^{\mathcal{H}}_{\alpha}}(v)\leq r\leq r_{\mathcal{I}}$.
To obtain late-time asymptotics in the remaining region $r_{\mathcal{I}}\leq r\leq r_{\gamma^{\mathcal{I}}_{\alpha}}(u)$, we use and we integrate the estimate from $r=r_{\gamma^{\mathcal{I}}_{\alpha}}(u)$ to any $r\geq r_{\mathcal{I}}$.
We then obtain: $$\begin{split}
\Bigg|T^k\psi_0(u,v)&-4\left[ I_0^{(1)}[\psi]T^{k+1}\left(\frac{1}{u\cdot v}\right)+\frac{M}{r \sqrt{D}}H_0[\psi]T^{k}\left(\frac{1}{u(v+4M-2r)}\right)\right]\Bigg|\\
\leq&\: C\left[\sqrt{E^{\epsilon}_{0, \mathcal{I}; k+1}[\psi]}+P_{I_0,\beta;k+1}[\psi]+P_{H_0,1;k}[\psi]+H_0[\psi]+I_0^{(1)}[\psi] \right]\cdot \left( v^{-1}u^{-2-k-\eta}+D^{-\frac{1}{2}}u^{-1}v^{-1-k-\eta}\right),
\end{split}$$ everywhere in $\mathcal{R}$. Note that $v+4M-2r$ has the property that it approaches $u$ as we increase $v$ and keep $u$ constant, but it remains finite as we approach $r=M$; indeed, we have that everywhere in $\{r\geq 2M\}$, $v-2r+2M\geq u$ and in $\{r\leq 2M\}$, $v\leq v-2r+4M\leq v+2M$.
Asymptotics for Type **B** and **D** perturbations {#sec:AsymptoticsForTypeDPerturbations}
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In this section, we treat the remaining types of initial data: Type **B** and **D**. The late-time asymptotics for Type **B** data follow immediately from Proposition \[prop:asympsi\] applied to $\psi_0^{(1)}$, where we use the regularity properties of $\psi_0^{(1)}$ that follow from Proposition \[eq:smoothextTypeBtimeint\].
\[cor:asympsizeroIH\] Let $k\in {\mathbb{N}}_0$. If $\lim_{v\to \infty} r^3L\phi_0(u_0,v)<\infty$ and $H_0[\psi]=0$, then there exists an $\eta>0$ and $\epsilon>0$ suitably small, such that we obtain the following *global* estimate: $$\label{eq:asympsizeroIH}
\begin{split}
\Bigg|T^k\psi_0(u,v)&-4\left(I_0^{(1)}[\psi]+ \frac{M}{r\sqrt{D}}H_0^{(1)}[\psi]\right)T^{k+1}\left(\frac{1}{v\cdot u}\right)\Bigg|\\
\leq&\: C\left(\sqrt{E_{0, \mathcal{H};k+1}^{\epsilon}[\psi]+E_{0, \mathcal{I};k+1}^{\epsilon}[\psi]}+I_0^{(1)}[\psi]+P_{I_0,\beta;k+1}[\psi]\right)v^{-1}u^{-2-k-\eta}\\
&+C\left(\sqrt{E_{0, \mathcal{H};k+1}^{\epsilon}[\psi]+E_{0, \mathcal{I};k+1}^{\epsilon}[\psi]}+H_0^{(1)}[\psi]+P_{H_0,1;k+1}[\psi]\right)D^{-\frac{1}{2}}u^{-1}v^{-2-k-\eta},
\end{split}$$ where $C=C(M,{\Sigma_0},r_{\mathcal{H}},r_{\mathcal{I}},\eta,\epsilon,\beta,k)>0$ is a constant.
We apply Proposition \[prop:asympsi\] with $k$ replaced by $k+1$ and $\psi_0$ replaced by ${\psi}_0^{(1)}$. We also use Lemma \[lm:estPnormpsi1\].
We are left with Type **D** data. We obtain asymptotics by following arguments analogous to those for Type **A** data in Section \[sec:asympzeroconst\], so we will omit most of the proofs, unless a different argument is needed, compared to the Type **A** data case.
\[prop:improvedasymderphiD\] Let $k\in {\mathbb{N}}_0$ and assume that $H_0[\psi]=0$. Let $\alpha>0$ such that $1-\alpha$ is suitably small. Then, there exists an $\eta>0$ and $\epsilon>0$ suitably small and a constant $C=C(M,\Sigma,r_{\mathcal{H}},r_{\mathcal{I}},\alpha,\epsilon,\eta,k)>0$ such that in $\mathcal{A}_{\gamma^{\mathcal{H}}_{\alpha}}^{\mathcal{H}}$: $$\begin{aligned}
|\underline{L}T^k\phi_0^{(1)}&(u,v)-(-1)^{k}(k+1)! 2H_0^{(1)}[\psi]\cdot u^{-2-k}|\\
\leq&\: C \left[\sqrt{E^{\epsilon}_{0, \mathcal{H}; k}[\psi]}\cdot u^{-2-k-\eta}+\left(P_{H_0,1;k}[\psi]+H_0^{(1)}[\psi]\right)\cdot u^{-2-\beta-k}\right]\end{aligned}$$ and moreover, $$\begin{aligned}
|\underline{L}T^k\phi_0&(u,v)-(-1)^{k+1}(k+2)! 2H_0^{(1)}[\psi]\cdot u^{-3-k}|\\
\leq&\: C \left[\sqrt{E^{\epsilon}_{0, \mathcal{H}; k+1}[\psi]}\cdot u^{-3-k-\eta}+\left(P_{H_0,1;k+1}[\psi]+H_0^{(1)}[\psi]\right)\cdot u^{-3-\beta-k}\right].\end{aligned}$$
We repeat the steps in the proof of Proposition \[prop:improvedasymderphi\] to the region $\mathcal{A}_{\gamma^{\mathcal{H}}_{\alpha}}^{\mathcal{H}}$ instead of $\mathcal{A}_{\gamma^{\mathcal{I}}_{\alpha}}^{\mathcal{I}}$ and interchange the roles of $u$ and $v$.
\[prop:improvedasymphiD\] Let $k\in {\mathbb{N}}_0$ and assume that $H_0[\psi]=0$. Let $\alpha>0$ such that $1-\alpha$ is suitably small. Then, there exists an $\eta>0$ and $\epsilon>0$ suitably small and a constant $C=C(M,\Sigma,r_{\mathcal{H}},r_{\mathcal{I}},\alpha,\epsilon,\eta)>0$ such that in $\mathcal{A}_{\gamma^{\mathcal{H}}_{\alpha}}^{\mathcal{H}}$: $$\begin{aligned}
\Bigg|T^k\phi_0(u,v)&-T^k\phi_0(u_{\gamma^{\mathcal{H}}_{\alpha}}(v),v)- (-1)^{k+1}(k+1)! 2H_0^{(1)}[\psi]\left[v^{-2-k}-u^{-2-k}\right]\Bigg|\\
\leq&\: C \left[\sqrt{E^{\epsilon}_{0, \mathcal{H}; k+1 }[\psi]}+P_{H_0,1;k+1}[\psi]+H_0^{(1)}[\psi]\right] \cdot \frac{u-v}{u v^{2+k+\eta}}.\end{aligned}$$
We repeat the steps in the proof of Proposition \[prop:improvedasymphi\] to the region $\mathcal{A}_{\gamma^{\mathcal{H}}_{\alpha}}^{\mathcal{H}}$ instead of $\mathcal{A}_{\gamma^{\mathcal{I}}_{\alpha}}^{\mathcal{I}}$ and interchange the roles of $u$ and $v$.
In contrast with Lemma \[prop:improvedasympLpsiTypeA\], we cannot yet obtain asymptotics for $\partial_r\psi_0$ in the region $\mathcal{A}^{\mathcal{H}}\setminus \mathcal{A}_{\gamma^{\mathcal{H}}_{\alpha}}^{\mathcal{H}}$ for Type **D** data. Instead, we consider $\partial_r((r-M)\cdot \psi_0)$, which, as we will show, is sufficient for our purposes. See however Proposition \[prop:asympdrpsiTypeD\] at the end of the section, where we do obtain asymptotics for $\partial_r\psi_0$.
\[prop:improvedasympLpsiTypeD\] Let $k\in {\mathbb{N}}_0$ and assume that $H_0[\psi]=0$. Let $\alpha>0$ such that $1-\alpha$ is arbitrarily small. Then, there exists an $\eta>0$ and $\epsilon>0$ suitably small and a constant $C=C(M,\Sigma,r_{\mathcal{H}},r_{\mathcal{I}},\alpha,\epsilon,\eta,\beta,k)>0$, such that in $\mathcal{A}^{\mathcal{H}}\setminus \mathcal{A}_{\gamma^{\mathcal{H}}_{\alpha}}^{\mathcal{H}}$:
$$\label{eq:asympLpsigammaTypeD}
\begin{split}
|M&\partial_r((r-M)\cdot T^k\psi_0)(u,v)-(-1)^{k}(k+1)!4MI_0v^{-2-k}|\\
\leq&\: C\left[\sqrt{E^{\epsilon}_{0,\mathcal{H};k+1}[\psi]}+P_{H_0,1;k+1}[\psi]+P_{I_0,\beta;k}[\psi]+I_0[\psi]\right]v^{-2-k-\eta}.
\end{split}$$
Note that we can rewrite as follows in $(v,r)$ coordinates: $$\partial_r^2((r-M)\cdot T^k\psi_0)=-2(r-M)^{-1}\partial_rT^{k+1}\phi_0.$$ Using the above equation, together with the fundamental theorem of calculus in the $\underline{L}$ direction, we arrive at the following estimate: $$\begin{split}
|&\partial_r((r-M)\cdot T^k\psi_0)(v,r_{\gamma^{\mathcal{H}}_{\alpha}}(v))-\partial_r((r-M)\cdot T^k\psi_0)(v,r_{\mathcal{H}})|\\
\lesssim &\: \int^{u_{\gamma^{\mathcal{H}}_{\alpha}(v)}}_{u(r_{\mathcal{H}})} (r-M)^{-1} |\underline{L} T^{k+1}\phi_0|(u',v)\,du'\\
\lesssim &\: \sqrt{ \int^{u_{\gamma^{\mathcal{H}}_{\alpha}(v)}}_{u(r_{\mathcal{H}})}\,du'}\cdot \sqrt{\int^{u_{\gamma^{\mathcal{H}}_{\alpha}(v)}}_{u(r_{\mathcal{H}})} (r-M)^{-2}(\underline{L} T^{k+1}\phi_0)^2(u',v)\,du'}\\
\lesssim &\: \sqrt{E^{\epsilon}_{0, \mathcal{I}; k+1}[\psi]} \cdot v^{-\frac{5}{2}-k+\frac{\epsilon}{2}+\frac{\alpha}{2}},
\end{split}$$ where we applied Proposition \[prop:extraendecay\] together with the estimate $(r-M)^{-1}\lesssim v^{\alpha}$ to obtain the last inequality.
We moreover have that $$\partial_r((r-M)\cdot T^k\psi_0)(v,r_{\mathcal{H}})=T^k\psi_0(v,r_{\mathcal{H}})+ (r_{\mathcal{H}}-M)\partial_r T^k\psi_0(v,r_{\mathcal{H}}).$$ By it follows that there exists an $\eta>0$ such that $$(r_{\mathcal{H}}-M)|\partial_rT^k\psi_0|(v,r_{\mathcal{H}})\lesssim \left(\sqrt{E^{\epsilon}_{0;k+1}[\psi]} +P_{H_0,1;k}[\psi]\right)v^{-2-k-\eta}.$$ Therefore, we can use at $r=r_{\mathcal{H}}$ to estimate $$\begin{split}
|\partial_r&((r-M)\cdot T^k\psi_0)(v,r_{\mathcal{H}})-(-1)^{-k}(k+1)!4I_0v^{-k-2}|\\
\lesssim&\: \left(\sqrt{E^{\epsilon}_{0;k+1}[\psi] }+P_{H_0,1;k}[\psi]+P_{I_0,\beta;k}+I_0[\psi]\right)v^{-2-k-\eta}.
\end{split}$$ By combining the estimates above, we arrive at .
\[prop:improvedasymphigammaB\] Let $k\in {\mathbb{N}}_0$ and assume $H_0[\psi]=0$. Let $\alpha>0$ such that $1-\alpha$ is arbitrarily small. Then, there exists an $\eta>0$ and $\epsilon>0$ suitably small and a constant $C=C(M,\Sigma,r_{\mathcal{H}},r_{\mathcal{I}},\alpha,\epsilon,\eta,\beta,k)>0$, such that $$\begin{aligned}
\label{eq:asymppsigammaHzero}
\Bigg|T^k\phi_0&(u_{\gamma^{\mathcal{H}}_{\alpha}}(v),v)-(-1)^k(k+1)!4MI_0[\psi]\cdot v^{-2-k}\Bigg|\\ \nonumber
\lesssim&\: \Bigg[\sqrt{E^{\epsilon}_{0, \mathcal{H};k+1}[\psi]}+P_{H_0,1;k+1}[\psi]+P_{I_0,\beta;k}[\psi]+H_0^{(1)}[\psi]+I_0[\psi]\Bigg]\cdot v^{-2-k-\eta}.\end{aligned}$$
We can split in $(v,r)$ coordinates $$\begin{split}
M^2r^{-1}T^k\psi_0(v,r_{\gamma^{\mathcal{H}}_{\alpha}}(v))=&\:M\partial_r(r^{-1}(r-M))\cdot rT^k\psi_0(v,r_{\gamma^{\mathcal{H}}_{\alpha}}(v))\\
=&\: M\partial_r((r-M) \cdot T^k\psi_0)(v,r_{\gamma^{\mathcal{H}}_{\alpha}}(v))-Mr^{-1}(r-M)\partial_rT^k\phi_0(v,r_{\gamma^{\mathcal{H}}_{\alpha}}(v))\\
=&\: M\partial_r((r-M) \cdot T^k\psi_0)(v,r_{\gamma^{\mathcal{H}}_{\alpha}}(v))+2Mr(r-M)^{-1}\underline{L}T^k\phi_0(v,r_{\gamma^{\mathcal{H}}_{\alpha}}(v)).
\end{split}$$
We apply Proposition \[prop:improvedasymderphiD\] together with the estimate $(r-M)^{-1}\lesssim v^{\alpha}$ in $\mathcal{B}^H_{\alpha}$ to estimate $$\begin{split}
2Mr(r-M)^{-1}|\underline{L}T^k\phi_0|(v,r_{\gamma^{\mathcal{H}}_{\alpha}(v)})\lesssim&\: \Bigg[\sqrt{E^{\epsilon}_{0, \mathcal{H}; k+1}}+P_{H_0,1;k+1}[\psi^{(1)}]+H_0^{(1)}[\psi]\Bigg]\cdot v^{-3-k+\alpha}.
\end{split}$$ The estimate then follows by applying .
\[prop:mainasymptypeD\] Let $k\in {\mathbb{N}}_0$ and assume that $H_0[\psi]=0$. Let $\alpha>0$ such that $1-\alpha$ is arbitrarily small. Then, there exists an $\eta>0$ and $\epsilon>0$ suitably small and a constant $C=C(M,\Sigma,r_{\mathcal{H}},r_{\mathcal{I}},\alpha,\epsilon,\eta, \beta, k)>0$, such that $$\label{eq:mainasymptypeD}
\begin{split}
\Bigg|T^k\psi_0(u,v)&-4\left[ \frac{1}{\sqrt{D}}H_0^{(1)}[\psi]T^{k+1}\left(\frac{1}{u\cdot v}\right)+I_0[\psi]T^{k}\left(\frac{1}{v(u+2M-2M^2(r-M)^{-1})}\right)\right]\Bigg|\\
\leq&\: C\Bigg[\sqrt{E^{\epsilon}_{0, \mathcal{H}; k+1}[\psi]}+P_{I_0,\beta;k}[\psi]+P_{H_0,1;k+1}[\psi]+I_0[\psi]+H_0^{(1)}[\psi] \Bigg]\\
&\cdot \left( v^{-1}u^{-1-k-\eta}+D^{-\frac{1}{2}}u^{-1}v^{-2-k-\eta}\right).
\end{split}$$
We apply the previous propositions in this section, together with the asymptotics in derived in Section \[sec:latetimeasympsi\] to arrive at , analogously to what is done in the proof of Proposition \[prop:mainasymptypeA\] (with the roles of $u$ and $v$ reversed).
We moreover used that in $\{r\leq 2M\}$ we can estimate $u+2M-2M^2(r-M)^{-1}\geq v$ and in $\{r\geq 2M\}$, $u+2M-2M^2(r-M)^{-1}\geq u$.
For completeness, we will also derive the precise late-time asymptotics for $\partial_r\psi$ for Type **B** data, *and show that the leading order term decays one power faster compared to the Type **A** and **C** cases*. We will restrict here to a bounded region $\{r\leq r_{\mathcal{I}}\}$ for the sake of convenience, but we note that the estimates providing late-time asymptotics can in principle be extended to the full region $\mathcal{R}$.
\[prop:asympdrpsiTypeD\] Let $k\in {\mathbb{N}}_0$ and assume that $H_0[\psi]=0$. Let $\alpha>0$ such that $1-\alpha$ is arbitrarily small. Then, there exists an $\eta>0$ and $\epsilon>0$ suitably small and a constant $C=C(M,\Sigma,r_{\mathcal{H}},r_{\mathcal{I}},\alpha,\epsilon,\eta,\beta,k)>0$, such that $$\label{eq:asympdrpsihzeroB}
\begin{split}
\Bigg|Dr^2\partial_rT^k\psi_0(v,r)&-8MH_0^{(1)}[\psi]T^k(u^{-3})-8I_0[\psi](r^2-M^2)T^k(v^{-3})\Bigg|\\
\leq&\: C\Bigg[\sqrt{E^{\epsilon}_{0, \mathcal{H}; k+2}[\psi]}+P_{I_0,\beta;k+1}[\psi]+P_{H_0,1;k+2}[\psi]+I_0[\psi]+H_0^{(1)}[\psi] \Bigg]\cdot v^{-3-\eta-k},
\end{split}$$ in $(v,r)$ coordinates, for all $r\leq r_{\mathcal{I}}$.
We apply to obtain in $(v,r)$ coordinates $$\label{eq:maineqestdrpsitypeD}
Dr^2\partial_rT^k\psi_0(v,r)=2M^2T^{k+1}\psi_0(v,M)-2r^2 T^{k+1}\psi_0(v,r)+\int_{M}^r 2r T^{k+1}\psi_0(v,r')\,dr'.$$ We will first estimate $2M^2T^{k+1}\psi_0(v,M)-2r^2 T^{k+1}\psi_0(v,r)$. If $r\leq r_{\gamma^{\mathcal{H}}_{\alpha}}(v)$, we apply Proposition \[prop:improvedasymphiD\] together with Proposition \[prop:improvedasymphigammaB\], with $k$ replaced by $k+1$. If $r\geq r_{\gamma^{\mathcal{H}}_{\alpha}}(v)$, we apply Proposition \[prop:improvedasymphigammaB\] and we integrate the estimate in Lemma \[prop:improvedasympLpsiTypeD\]. We then arrive at the following expressions: $$\begin{aligned}
\Big|&M T^{k+1}\psi_0(v,M)+ 8MI_0[\psi] T^k(v^{-3})- 4H_0^{(1)}[\psi]T^k(v^{-3})\Big| \\
\lesssim&\: \Bigg[\sqrt{E^{\epsilon}_{0, \mathcal{H}; k+2}[\psi]}+P_{I_0,\beta;k+1}[\psi]+P_{H_0,1;k+2}[\psi]+I_0[\psi]+H_0^{(1)}[\psi] \Bigg] v^{-3-k-\eta},\\
\Big|&r^2 T^{k+1}\psi_0(v,r)+ 8r^2I_0[\psi] T^k(v^{-3})- 4H_0^{(1)}[\psi]T^k(v^{-3}-u^{-3})\Big| \\
\lesssim&\: \Bigg[\sqrt{E^{\epsilon}_{0, \mathcal{H}; k+2}[\psi]}+P_{I_0,\beta;k+1}[\psi]+P_{H_0,1;k+2}[\psi]+I_0[\psi]+H_0^{(1)}[\psi] \Bigg] v^{-3-k-\eta}.\end{aligned}$$
Hence, we obtain $$\begin{split}
\Bigg| 2M^2&T^{k+1}\psi_0(v,M)-2r^2T^{k+1}\psi_0(v,r)-8MH_0^{(1)}[\psi]T^k(u^{-3})-16(r^2-M^2)I_0[\psi]T^k(v^{-3})\Bigg|\\
\leq &\: C \Bigg[\sqrt{E^{\epsilon}_{0, \mathcal{H}; k+2}[\psi]}+P_{I_0,\beta;k+1}[\psi]+P_{H_0,1;k+2}[\psi]+I_0[\psi]+H_0^{(1)}[\psi] \Bigg] \cdot v^{-3-k-\eta}.
\end{split}$$
In order to estimate the integral on the right-hand side of , we apply Proposition \[prop:improvedasymphiD\] together with Proposition \[prop:improvedasymphigammaB\] and : $$\Bigg| \int_{M}^r 2r T^{k+1}\psi_0(v,r')+2r'\cdot 8I_0[\psi] T^k(v^{-3})\,dr'\Bigg|\leq C \left[\int_M^{M+v^{-\alpha}}\textnormal{Err}_1\,dr' +\int^r_{M+v^{-\alpha}}\textnormal{Err}_2\,dr'\right],$$ where we take $r>M+v^{-\alpha}$ without loss of generality, and where $$\begin{aligned}
\textnormal{Err}_1:=&\: \left[\sqrt{E^{\epsilon}_{0, \mathcal{H}; k+2}[\psi]}+P_{I_0,\beta;k+1}[\psi]+P_{H_0,1;k+2}[\psi]+I_0[\psi]+H_0^{(1)}[\psi] \right] v^{-3-k-\eta}, \\
\textnormal{Err}_2:=&\: \left[\sqrt{E^{\epsilon}_{0;k+2}[\psi]}+I_0[\psi]+P_{H_0,1;k+1}[\psi]+P_{I_0,\beta;k+1}[\psi]\right] (r-M)^{-1}v^{-3-k-2\eta}.\end{aligned}$$
It follows immediately that (note that the logarithmic term from integrating Err$_2$ can be easily absorbed by the $v$ power) $$\begin{split}
\int_M^{M+v^{-\alpha}}&\textnormal{Err}_1\,dr' +\int^r_{M+v^{-\alpha}}\textnormal{Err}_2\,dr'\\
\leq &\: C \Bigg[\sqrt{E^{\epsilon}_{0, \mathcal{H}; k+2}[\psi]}+P_{I_0,\beta;k+1}[\psi]+P_{H_0,1;k+2}[\psi]+I_0[\psi]+H_0^{(1)}[\psi] \Bigg]\cdot v^{-3-k-\eta}.
\end{split}$$ Finally, we have that $$8I_0[\psi] T^k(v^{-3}) \int_{M}^r 2r'\,dr'= (r^2-M^2)8I_0[\psi] T^k(v^{-3}).$$ When we combine the estimates above, we obtain .
Higher-order asymptotics and logarithmic corrections {#sec:hoasymp}
====================================================
In this section, we derive refined asymptotics along $\mathcal{H}^+$ for data with $H_0[\psi]\neq 0$ and along $\mathcal{I}^+$ for data with $I_0[\psi]\neq 0$. The derivation proceeds in a very similar manner to the arguments in [@logasymptotics].
We first introduce the following additional weighted $L^{\infty}$ norms: we define with respect to $(u,r)$ coordinates, $$\begin{aligned}
P_{\mathcal{I}}[\psi]:=&\:\left| \left| Dr^3\left(\partial_r\phi_0-\frac{I_0[\psi]}{r^2}\right)\right|\right|_{L^{\infty}(\Sigma_0)},\\
P_{\mathcal{I},T}[\psi]:=&\:\left| \left| Dr^4\partial_r\left(D\partial_r\phi_0-D\frac{I_0[\psi]}{r^2}\right)\right|\right|_{L^{\infty}(\Sigma_0)}.\end{aligned}$$ And we define with respect to $(v,r)$ coordinates: $$\begin{aligned}
P_{\mathcal{H}}[\psi]:=&\:\left| \left| D^{-\frac{1}{2}}\left(\partial_r\phi_0+M^2H_0[\psi]\right)\right|\right|_{L^{\infty}(\Sigma_0)},\\
P_{\mathcal{H},T}[\psi]:=&\:\left| \left| \partial_r^2\phi_0\right|\right|_{L^{\infty}(\Sigma_0)}.\end{aligned}$$
\[prop:asymdvphi\] For all $\epsilon>0$, there exists a constant $C=C(M,\Sigma,r_{\mathcal{I}},\epsilon)>0$ such that for all $(u,v)$ in $\mathcal{A}^{\mathcal{I}}$ we can estimate:
- $$\label{eq:2ndoasympdvphiinf}
\begin{split}
&\Bigg|\partial_v(r\psi)(u,v)-2I_0[\psi]v^{-2}-16M I_0[\psi]v^{-3}\log v+8MI_0[\psi]uv^{-3}(v-u)^{-1}\\
&+8MI_0[\psi]v^{-3} \log \left(\frac{vu}{v-u}\right)\Bigg|\\
\leq&\: C(I_0[\psi]+H_0[\psi]+\sqrt{E_{0; 1}^{\epsilon}[\psi]}+P_{\mathcal{I}}[\psi]+P_{\mathcal{H}}[\psi])\textnormal{Err}_{\mathcal{I}}(u,v),
\end{split}$$
where $$\textnormal{Err}_{\mathcal{I}}(u,v):=v^{-3}+v^{-2-\epsilon}\cdot (v-u)^{-1}+v^{-2}\cdot (v-u)^{-2+\eta},$$ with $\eta>0$ arbitrarily small.
- For all $\epsilon>0$, there exists a constant $C=C(M,\Sigma,r_{\mathcal{H}},\epsilon)>0$ such that for all $(u,v)$ in $\mathcal{A}^{\mathcal{H}}$ we can estimate: $$\label{eq:2ndoasympdvphihor}
\begin{split}
&\Bigg|\partial_u(r\psi)(u,v)-2H_0[\psi]u^{-2}-16M H_0[\psi]u^{-3}\log u+8MH_0[\psi]vu^{-3}(u-v)^{-1}\\
&+8MH_0[\psi]u^{-3} \log \left(\frac{vu}{u-v}\right)\Bigg|\\
\leq&\: C(I_0[\psi]+H_0[\psi]+\sqrt{E^{\epsilon}_{0 ;1}[\psi]}+P_{\mathcal{H}}[\psi]+P_{\mathcal{I}}[\psi])\textnormal{Err}_{\mathcal{H}}(u,v),
\end{split}$$ where $$\textnormal{Err}_{\mathcal{H}}(u,v):=u^{-3}+u^{-2-\epsilon}\cdot (u-v)^{-1}+u^{-2}\cdot (u-v)^{-2+\eta},$$ with $\eta>0$ arbitrarily small.
By applying the relations between $r$, $u$ and $v$ from Lemma \[lm:relationruv\], we obtain in $\mathcal{A}^{\mathcal{I}}$: $$\label{eq:maineq2ndasympinf}
\partial_u\partial_v(r\psi)(u,v)=\Big[-2M(v-u)^{-2}+O((v-u)^{-3+\eta})\Big]\cdot \psi\\$$ with $\eta>0$ arbitrarily small, and hence, by Proposition \[prop:asympsi\] we have that there exists an $\epsilon>0$ such that $$\begin{split}
\left|\partial_u\partial_v(r\psi)(u,v)+\frac{8MI_0[\psi]}{vu}(v-u)^{-2}\right|\leq &\:C(I_0[\psi[+H_0[\psi])(v-u)^{-3+\eta}v^{-1}u^{-1}\\
&+ C(I_0[\psi]+H_0[\psi]+\sqrt{E^{\epsilon}_{0; 1}[\psi]}+P_{\mathcal{H}}[\psi]+P_{\mathcal{I}}[\psi])(v-u)^{-2}v^{-1}u^{-1-\epsilon}.
\end{split}$$ The estimate now follows by repeating the arguments in the proof of Proposition 3.1 of [@logasymptotics]
Now, we apply the relations between $r-M$, $u$ and $v$ from Lemma \[lm:relationruv\] in $\mathcal{A}^{\mathcal{H}}$ to obtain: $$\label{eq:maineq2ndasymphor}
\partial_v\partial_u(r\psi)(u,v)=\Big[-2M(v-u)^{-2}+O((v-u)^{-3+\eta})\Big]\cdot \sqrt{D} \psi\\$$ By using together with the estimate for $\sqrt{D}\cdot \psi$ from Proposition \[prop:asympsi\], we can similarly find an $\epsilon>0$ such that for all $(u,v)$ in $\mathcal{A}^{\mathcal{H}}$ $$\begin{split}
\left|\partial_v\partial_u(r\psi)(u,v)+\frac{8MH_0}{uv}(u-v)^{-2}\right|\leq &\:C(I_0+H_0)(u-v)^{-3+\eta}u^{-1}v^{-1}\\
&+ C(I_0+H_0+\sqrt{E^{\epsilon}_{0; 1}[\psi]}+P_{\mathcal{H}}[\psi]+P_{\mathcal{I}}[\psi])(u-v)^{-2}u^{-1}v^{-1-\epsilon}.
\end{split}$$ We obtain by once again repeating the arguments in the proof of Proposition 3.1 of [@logasymptotics] and moreover interchanging the roles of $u$ and $v$ (and $I_0$ and $H_0$).
\[prop:2ndasymphi\] For all $\epsilon>0$, there exists a constant $C=C(M,\Sigma,r_{\mathcal{I}},r_{\mathcal{H}},\epsilon)>0$ such that we can estimate: $$\label{eq:2ndasymphiinf}
\begin{split}
\Bigg|r\psi(u,v)&-2I_0[\psi([u^{-1}-v^{-1})+4MI_0[\psi]u^{-2}\log u-4MI_0[\psi]v^{-2}\log u\\
&+8MI_0[\psi]v^{-2}\log v+4MI_0[\psi](u^{-2}+v^{-2})\log\left(\frac{v-u}{v}\right)\Bigg|\\
\leq&\: C\left(I_0[\psi]+H_0[\psi]+\sqrt{E^{\epsilon}_{0; 1}[\psi]}+P_{\mathcal{H}}[\psi]+P_{\mathcal{I}}[\psi]\right)u^{-2} \quad \textnormal{in $\mathcal{A}^{\mathcal{I}}$}.
\end{split}$$ and $$\label{eq:2ndasymphinearhor}
\begin{split}
\Bigg|r\psi(u,v)&-2H_0[\psi](v^{-1}-u^{-1})+4MH_0[\psi]v^{-2}\log v-4MH_0[\psi]u^{-2}\log v\\
&+8MH_0[\psi]u^{-2}\log u+4MH_0[\psi](v^{-2}+u^{-2})\log\left(\frac{u-v}{u}\right)\Bigg|\\
\leq&\: C\left(I_0[\psi]+H_0[\psi]+\sqrt{E^{\epsilon}_{0; 1}[\psi]}+P_{\mathcal{H}}[\psi]+P_{\mathcal{I}}[\psi]\right)v^{-2} \quad \textnormal{in $\mathcal{A}^{\mathcal{H}}$}.
\end{split}$$ In particular, $$\label{eq:2ndasymphinullinf}
\left|r\psi|_{\mathcal{I}^+}(u)-2I_0[\psi]u^{-1}+4MI_0[\psi]u^{-2}\log u\right|\leq C\left(I_0[\psi]+H_0[\psi]+\sqrt{E^{\epsilon}_{0; 1}[\psi]}+P_{\mathcal{H}}[\psi]+P_{\mathcal{I}}[\psi]\right)u^{-2}$$ and $$\label{eq:2ndasymphihor}
\left|r\psi|_{\mathcal{H}^+}(v)-2H_0[\psi]v^{-1}+4MH_0[\psi]v^{-2}\log v\right|\leq C\left(I_0[\psi]+H_0[\psi]+\sqrt{E^{\epsilon}_{0;1}[\psi]}+P_{\mathcal{H}}[\psi]+P_{\mathcal{I}}[\psi]\right)v^{-2}.$$
In order to prove we integrate $\partial_v(r\psi)$ from $(u,v=u+2r_*(r_{\mathcal{I}}))$ to $(u,v=v')$, estimating the contribution of as in Proposition 3.2 of [@logasymptotics] and moreover using to estimate $$|r\psi|(u,v=u+2r_*(r_{\mathcal{I}}))\lesssim \left(I_0[\psi]+H_0[\psi]+\sqrt{E_{0; 1}[\psi]}+P_{\mathcal{H}}[\psi]+P_{\mathcal{I}}[\psi]\right)u^{-2}.$$ We similarly obtain by integrating $\partial_u(r\psi)$ from $(u=v-2r_*(r_{\mathcal{H}}),v)$ to $(u=u', v)$, estimating the contribution of as in Proposition 3.2 of [@logasymptotics], but with the roles of $u$ and $v$ interchanged, and moreover using to estimate $$|r\psi|(u=v-2r_*(r_{\mathcal{H}}),v)\lesssim \left(I_0[\psi]+H_0[\psi]+\sqrt{E_{0; 1}[\psi]}+P_{\mathcal{H}}[\psi]+P_{\mathcal{I}}[\psi]\right)v^{-2}.$$ The estimates and then follow simply by taking respectively the limit $v\to \infty$ and $u\to \infty$.
We can moreover obtain refined late-time asymptotics along $\mathcal{H}^+$ and $\mathcal{I}^+$ in the case when *both* $I_0[\psi]$ and $H_0[\psi]=0$ are vanishing (i.e. for Type **B** data).
\[prop:2ndasymphitimeint\] Suppose $H_0[\psi]=0$ and $I_0[\psi]=0$. For all $\epsilon>0$, there exists a constant $C=C(M,\Sigma,r_{\mathcal{I}},r_{\mathcal{H}},\epsilon)>0$ such that we can estimate: $$\label{eq:2ndasymphinullinftimeintmain}
\begin{split}
\Big|&r\psi|_{\mathcal{I}^+}(u)+2I_0^{(1)}[\psi]u^{-2}-8MI_0^{(1)}[\psi]u^{-3}\log u\Big|\\
\leq&\: C\left(I_0^{(1)}[\psi]+H_0^{(1)}[\psi]+\sqrt{E^{\epsilon}_{0,\mathcal{H}; 1}[\psi]+E^{\epsilon}_{0,\mathcal{I}; 1}[\psi]}+P_{\mathcal{H}, T}[\psi]+P_{\mathcal{I}, T}[\psi]\right)u^{-3},
\end{split}$$ and the following estimate holds on $\mathcal{H}^{+}$ $$\label{eq:2ndasymphihotimintmain}
\begin{split}
\Big|&r\psi|_{\mathcal{H}^+}(v)+2H_0^{(1)}[\psi]v^{-2}-4MH_0^{(1)}[\psi]v^{-3}\log v\Big|\\
\leq&\: C\left(I_0^{(1)}[\psi]+H_0^{(1)}[\psi]+\sqrt{E^{\epsilon}_{0,\mathcal{H}; 1}[\psi]+E^{\epsilon}_{0,\mathcal{I}; 1}[\psi]}+P_{\mathcal{H}, T}[\psi]+P_{\mathcal{I}, T}[\psi]\right)v^{-3}.
\end{split}$$
By $H_0[\psi]=0$ and $I_0[\psi]=0$, together with the assumption that $P_{\mathcal{I}, T}[\psi]<\infty$, it follows by Proposition \[eq:smoothextTypeBtimeint\] that $\psi_0^{(1)}$ is smooth. We can then apply the arguments in the proof of Proposition 3.3 of [@logasymptotics] in $\mathcal{A}^{\mathcal{I}}$, and similar arguments with the roles of $u$ and $v$ reversed in $\mathcal{A}^{\mathcal{H}}$, to derive the late-time asymptotics of $T\psi_0^{(1)}=\psi_0$. We omit the details of the proof.
Basic inequalities on ERN {#sec:BasicInequalitiesOnERN}
=========================
In this section, we will last some basic inequalities that are used throughout the paper.
Hardy inequalities {#sec:HardyInequalities}
------------------
\[lm:hardy\] Let $q\in {\mathbb{R}}\setminus \{-1\}$ and $r_1>M$. Let $f: \times [v_0,\infty)_v\times [M,\infty)_r \to {\mathbb{R}}$ be a $C^1$ function. Then for all $M<r_1<r_2\leq \infty$ and $u'\geq 0$ $$\label{eq:hardyinf}
\int_{v_{r_1}(u')}^{v_{r_2}(u')} r^qf^2|_{u=u'}\,dv\lesssim (q+1)^{-2} \int_{v_{r_1}(u')}^{v_{r_2}(u')} r^{q+2}(Lf)^2|_{u=u'}\,dv+ 2r_2^{q+1}f^2(u',v_{r_2}(u')),$$ where in the $r_2=\infty$ case, the second term on the right-hand side is interpreted as follows: $$2\lim_{r\to \infty }r^{q+1}f^2(u,v_{r}(u))$$
Furthermore, for all $M\leq r_0<r_1<\infty$ and $v'\geq 0$ $$\label{eq:hardyhor1}
\int_{u_{r_1}(v')}^{u_{r_0}(v')} (r-M)^{-q}f^2|_{v=v'}\,du\lesssim (q+1)^{-2}\int_{u_{r_1}(v')}^{u_{r_0}(v')} (r-M)^{-q-2}(\underline{L}f)^2|_{v=v'}\,du+ 2(r_0-M)^{-q-1}f^2(u_{r_0}(v'),v'),$$ where in the $r_0=M$ case, the second term on the right-hand side is interpreted as follows: $$2\lim_{u\to \infty}(r-M)^{-q-1}f^2|_{v=v'},$$ or equivalently, in $(v,r,\theta,\varphi)$ coordinates, for all $v'\geq 0$ $$\label{eq:hardyhor2}
\int_{r_0}^{r_1} (r-M)^{-2-q}f^2|_{v=v'}\,dr\lesssim (q+1)^{-2}\int_{r_0}^{r_1} (r-M)^{-q}(\partial_rf)^2|_{v=v'}\,du+ 2(r_0-M)^{-q-1}f^2(v',r_0).$$
Poincaré inequalities {#sec:PoincareInequalities}
---------------------
Let $f\in L^2({\mathbb{S}}^2)$. Then we can expand $f$ in terms of spherical harmonics $Y_{\ell,m}$ on ${\mathbb{S}}^2$, which form an orthonormal basis of $L^2({\mathbb{S}}^2)$. We write $$f(\theta,\varphi)=\sum_{\ell=0}^{\infty} f_{\ell}(\theta,\varphi),$$ where the *angular modes* $f_{\ell}$ are defined as follows: $$f_{\ell}(\theta,\varphi):=\sum_{m=-\ell}^{\ell} f_{\ell m}Y_{\ell,m}(\theta,\varphi).$$ We have that $$\slashed{\Delta}_{{\mathbb{S}}^2}f_{\ell}=-\ell(\ell+1)f_{\ell}.$$ Note in particular that $$f_0=\frac{1}{4\pi}\int_{{\mathbb{S}}^2} f(\theta,\varphi)d\omega,$$ where we employed the shorthand notation $d\omega=\sin\theta d\theta d\varphi$.
We will moreover introduce the orthogonal projections $$P_{\ell},P_{\leq \ell}, P_{\geq \ell}:L^2({\mathbb{S}}^2)\to L^2({\mathbb{S}}^2),$$ which are defined as follows $$\begin{aligned}
P_{\ell}f=&\:f_{\ell},\\
P_{\leq \ell}f=&\:\sum_{\ell'=0}^{\ell}f_{\ell'},\\
P_{\geq \ell}f=&\:\sum_{\ell'=\ell}^{\infty}f_{\ell'}.\end{aligned}$$
\[lm:poincare\] Let $f\in H^1({\mathbb{S}}^2)$. Then $$\begin{aligned}
\label{eq:poincareineq}
\int_{{\mathbb{S}}^2} (P_{\geq \ell}f)^2\,d\omega \leq&\: \frac{1}{\ell(\ell+1)} \int_{{\mathbb{S}}^2} |{\slashed{\nabla}}_{{\mathbb{S}}^2}P_{\geq \ell}f|^2\,d\omega,\\
\label{eq:poincareeq}
\int_{{\mathbb{S}}^2} (P_{ \ell}f)^2\,d\omega=&\: \frac{1}{\ell(\ell+1)} \int_{{\mathbb{S}}^2} |{\slashed{\nabla}}_{{\mathbb{S}}^2}P_{\ell}f|^2\,d\omega.\end{aligned}$$ and moreover $$\begin{aligned}
\label{eq:1stordpoincareineq}
\int_{{\mathbb{S}}^2} |{\slashed{\nabla}}_{{\mathbb{S}}^2}f|^2\,d\omega\leq&\: \frac{1}{2} \int_{{\mathbb{S}}^2} (\slashed{\Delta}_{{\mathbb{S}}^2}f)^2\,d\omega,\\
\label{eq:angmomentineq}
\int_{{\mathbb{S}}^2} (\slashed{\Delta}_{{\mathbb{S}}^2}f)^2\,d\omega \leq&\: \sum_{|\alpha|=1}\int_{{\mathbb{S}}^2} |{\slashed{\nabla}}_{{\mathbb{S}}^2}\Omega^{\alpha} f|^2\,d\omega.\end{aligned}$$
Interpolation estimates {#sec:Interpolationestimates}
-----------------------
We will make use of the following interpolation estimates.
\[lm:interpolation\] Let $f: \{(u,v)\in {\mathbb{R}}^2\,|\, u\in [u_0,\infty)\quad v\in [v_{r_{\mathcal{I}} }(u),\infty)\}\to {\mathbb{R}}$ be a function such that the following inequalities hold: there exist $u$-independent constants $\mathcal{E}_1,\mathcal{E}_2>0$, such that $$\begin{aligned}
\int_{v_{r_{\mathcal{I}} }(u)}^{\infty}r^{p-\epsilon}f^2(u,v)\,dv\leq &\:\mathcal{E}_1u^{-q},\\
\int_{v_{r_{\mathcal{I}} }(u)}^{\infty}r^{p+1-\epsilon}f^2(u,v)\,dv\leq&\: \mathcal{E}_2u^{-q+1},\end{aligned}$$ with $q\in {\mathbb{R}}$ and $\epsilon\in (0,1)$.
Then $$\label{eq:interp3I}
\int_{v_{r_{\mathcal{I}} }(u)}^{\infty}r^{p}f^2(u,v)\,dv\leq C \max\{\mathcal{E}_1,\mathcal{E}_2\}u^{-q+\epsilon},$$ with $C>0$ a constant depending only on $M$, $\Sigma_0$ and $r_{\mathcal{I}}$.
Furthermore, let $\underline{f}: \{(u,v)\in {\mathbb{R}}^2\,|\, v\in [v_0,\infty)\quad u\in [u_{r_{\mathcal{H}} }(v),\infty)\}\to {\mathbb{R}}$ be a function such that the following inequalities hold: there exist $v$-independent constants $\mathcal{E}_1,\mathcal{E}_2>0$, such that $$\begin{aligned}
\int_{u_{r_{\mathcal{H}} }(v)}^{\infty}(r-M)^{-p+\epsilon}\underline{f}^2(u,v)\,du\leq &\:\mathcal{E}_1v^{-q},\\
\int_{u_{r_{\mathcal{H}} }(v)}^{\infty}(r-M)^{-p-1+\epsilon}\underline{f}^2(u,v)\,du\leq&\: \mathcal{E}_2v^{-q+1},\end{aligned}$$ with $q\in {\mathbb{R}}$ and $\epsilon\in (0,1)$.
Then $$\label{eq:interp3H}
\int_{u_{r_{\mathcal{H}} }(v)}^{\infty}(r-M)^{-p}\underline{f}^2(u,v)\,du\leq C \max\{\mathcal{E}_1,\mathcal{E}_2\}v^{-q+\epsilon},$$ with $C>0$ a constant depending only on $M$, $\Sigma_0$ and $r_{\mathcal{H}}$.
See the proof of Lemma 2.6 of [@paper1] for the derivation of . The estimate follows after replacing $r$ with $(r-M)^{-1}$ and reversing the roles of $u$ and $v$.
Basic energy estimates {#sec:EnergyBounds}
----------------------
The following energy boundedness estimate holds for all solutions $\psi$ to the wave equation on ERN: $$\label{eq:degenbound}
\int_{\Sigma_{\tau}}J^T[\psi]\cdot \mathbf{n}_{\tau}\,d\mu_{\tau}\leq \int_{\Sigma_{0}}J^T[\psi]\cdot \mathbf{n}_{0}\,d\mu_{0}$$ and it follows straightforwardly from the Killing property of the vector field $T$, together with the non-negativity of the $T$-energy flux through $\mathcal{H}^+$ and $\mathcal{I}^+$ (in a limiting sense).
We next give an overview of the main Morawetz or *integrated local energy decay* estimates that we will make use of throughout the remainder of the paper. A proof of these estimates can be found in [@aretakis1; @aretakis2].
\[thm:ileds\] Let $M<r_1<r_2<2M<r_3<r_4<\infty$. Let $N\in {\mathbb{N}}_0$ and $0\leq \tau_1\leq \tau_2\leq \infty$. Then:
1. There exists a constant $C=C(\Sigma,r_1,r_2,r_3,r_4)>0$ such that $$\label{eq:iledwayps}
\sum_{0\leq k+l+m\leq N+1}\int_{\tau_1}^{\tau_2} \left[\int_{\Sigma_{\tau}\cap (\{r_1\leq r\leq r_2\}\cup \{r_3\leq r\leq r_4\})} |{\slashed{\nabla}}_{{\mathbb{S}}^2}^k\partial_v^{l}\partial_r^m\psi|^2\,d\mu_{\Sigma_{\tau}}\right]\,d\tau\leq C\sum_{k\leq N} \int_{\Sigma_{\tau}} J^T[T^k\psi]\cdot \mathbf{n}_{\tau_1}\,d\mu_{\Sigma_{\tau_1}}.$$
2. There exists a constant $C=C(\Sigma,r_1,r_4)>0$ such that $$\label{eq:iledlossder}
\sum_{0\leq k+l+m\leq N+1}\int_{\tau_1}^{\tau_2} \left[\int_{\Sigma_{\tau}\cap (\{r\geq r_1\}\cap \{r\leq r_4\})} |{\slashed{\nabla}}_{{\mathbb{S}}^2}^k\partial_v^{l}\partial_r^m\psi|^2\,d\mu_{\Sigma_{\tau}}\right]\,d\tau\leq C\sum_{k\leq N+1} \int_{\Sigma_{\tau}} J^T[T^k\psi]\cdot \mathbf{n}_{\tau_1}\,d\mu_{\Sigma_{\tau_1}}.$$
3. There exists a constant $C=C(\Sigma,r_1,r_4)>0$ such that $$\label{eq:iledpsi0}
\sum_{0\leq l+m\leq N+1}\int_{\tau_1}^{\tau_2} \left[\int_{\Sigma_{\tau}\cap (\{r\geq r_1\}\cap \{r\leq r_4\})} |\partial_v^{l}\partial_r^m\psi_0|^2\,d\mu_{\Sigma_{\tau}}\right]\,d\tau\leq C\sum_{k\leq N} \int_{\Sigma_{\tau}} J^T[T^k\psi]\cdot \mathbf{n}_{\tau_1}\,d\mu_{\Sigma_{\tau_1}}.$$
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Department of Mathematics, University of California, Los Angeles, CA 90095, United States, yannis@math.ucla.edu
Princeton University, Department of Mathematics, Fine Hall, Washington Road, Princeton, NJ 08544, United States, aretakis@math.princeton.edu
Department of Mathematics, University of Toronto Scarborough 1265 Military Trail, Toronto, ON, M1C 1A4, Canada, aretakis@math.toronto.edu
Department of Mathematics, University of Toronto, 40 St George Street, Toronto, ON, Canada, aretakis@math.toronto.edu
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, United Kingdom, dg405@cam.ac.uk
[^1]: For more details and an exhaustive list of references for works related to extremal black holes see Section \[sec:PhysicalImportanceOfExtremalBlackHoles\].
[^2]: The time parameter is comparable to the Schwarzschild coordinate time $t$ away from the event horizon and null infinity.
[^3]: Note that the sharpness of the *decay rate* of the time derivative of $\psi$ along the event horizon was first established by Luk and Oh [@luk2015].
[^4]: We refer to Section \[sec:TheERNManifoldFoliationsAndVectorFields\] for a definition of this “big O notation”.
[^5]: See also Section \[sec:TheInteriorOfBlackHolesAndStrongCosmicCensorship\] for a further discussion on the interior of dynamical extremal black holes.
[^6]: The importance of the horizon charge $H_0[\psi]$ for the dynamics of ERN has been discussed in Section \[sec:TheHorizonInstabilityOfExtremalBlackHoles\].
[^7]: that is, it is independent of the choice of the hypersurface $\Sigma_0$.
[^8]: with respect to the coordinate system $(\rho=r,\theta,\varphi)$ on $\Sigma_0$
[^9]: Note that (non-degenerate) integrated decay estimates for the fluxes $\mathcal{C}_{N_{\tau}^{\mathcal{I}}}[\psi]$ and $\mathcal{C}_{N_{\tau}^{\mathcal{H}}}[\psi]$ on ERN are closely related to the *trapping effect* at ${\mathcal{I}^{+}}$ and at ${\mathcal{H}^{+}}$.
[^10]: For spherically symmetric solutions (with harmonic mode number $\ell=0$) we only take $n=0$.
[^11]: Note, however, that the relevant decay rates for $\psi$, without the $\sqrt{r-M}$ weight, are almost sharp; see Table \[summarytablenew\].
[^12]: While the estimate can be used to obtain asymptotics for $L\psi_0$ in spacetime regions of bounded $r$ also in the case of Type **A** data, it fails to provide asymptotics along the curves $\gamma^{\mathcal{I}}_{\alpha}$.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The amount of heat generated by computers is rapidly becoming one of the main problems for developing new generations of information technology. The thermodynamics of computation sets the ultimate physical bounds on heat generation. A lower bound is set by the Landauer Limit, at which computation becomes thermodynamically reversible. For classical computation there is no physical principle which prevents this limit being reached, and approaches to it are already being experimentally tested. In this paper we show that for quantum computation there is an unavoidable excess heat generation that renders it inherently thermodynamically irreversible. The Landauer Limit cannot, in general, be reached by quantum computers. We show the existence of a lower bound to the heat generated by quantum computing that exceeds that given by the Landauer Limit, give the special conditions where this excess cost may be avoided, and show how classical computing falls within these special conditions.'
author:
- 'D. J. Bedingham'
- 'O. J. E. Maroney'
title: The thermodynamic cost of quantum operations
---
Introduction
============
Information processing does not come for free. Physical systems are needed to store, transmit and process information, and these come with physical resource costs, in time, space and energy. The heat generated by information processing is becoming one of the most significant of these costs. Modern integrated circuits have power densities of the order $100 {\rm W/cm}^{2}$ with operating temperatures near to their upper limit [@VASS]. Thermal management is already the main constraint on the performance of modern electronics and advances in technology will increasingly need to test the fundamental limits of energy consumption.
For classical information processing, Landauer’s Principle [@LAN; @BEN1; @LR2003; @Maroney2009] relates the change in information from a computation to a minimum thermodynamic cost in the form of heat generated in the environment. The information content of a source of signals is measured by the Shannon measure $H=-\sum_n p_n \log_2 p_n$, where $p_n$ is the probability of signal $n$ occurring. Landauer’s Principle states that the mean heat generated by an information processing device is given by $$\begin{aligned}
\Delta E_{\text{classical}} \geq - k T \ln 2 \Delta H,
\label{clp}\end{aligned}$$ where $\Delta H$ is the change in the Shannon information.
The limit cost, which corresponds to the condition of thermodynamic reversibility [@OM1; @SAG1], is defined purely in terms of the information processing operation itself: every physical system which performs the task must pay at least this cost [@Pie00; @OM1; @Turgut2006]. Most importantly, it is a tight bound: while there are practical barriers to reaching the limit (finite size and time effects, single shot costs, etc. [@Dahlstein2013; @RW2014]) there is no physical principle that prevents their effects becoming arbitrarily small, with experimental tests increasingly pushing at this boundary [@BER; @JUN; @MARTINI]. This tight relationship between information theory and thermodynamics makes Landauer’s Principle the basis of our understanding of the thermodynamic cost of information processing.
However, both the theoretical studies and experimental tests of the Landauer Limit have almost exclusively considered classical information processing. Quantum computing radically alters the nature of information, leading to many attempts to extend Landauer’s Principle to quantum theory [@VV; @AND; @GOO; @AT; @SAG2; @KIM; @SAG3; @GOO2].
Quantum information processing involves a quantum system ${\rm S}$ with states $\rho^n_{\rm S}$ as signal states, undergoing a quantum operation $Q$, defined by its effect upon the signal states $\rho^n_{\rm S} \rightarrow \rho^{n\prime}_{\rm S} = \mathcal{Q}(\rho^n_{\rm S})$. Given that the input $\rho^n_{\rm S}$ appears with probability $p_n$, the average input is $\rho_{\rm S}=\sum_n p_n \rho^n_{\rm S}$, and the average output is $\rho_{\rm S}'=\sum_n p_n \rho^{n\prime}_{\rm S}$.
The natural quantum generalisation of Shannon information to quantum theory is the Schumacher information measure $S(\rho_{\rm S})=-{{\rm Tr}}[ \rho_{\rm S}\log_2 \rho_{\rm S} ]$ [@Sch95], which agrees with the Shannon information when the signal states are orthogonal. This suggests that Equation (\[clp\]) should be replaced by $$\begin{aligned}
\Delta E_{Q} \geq k T \ln 2 \left[S(\rho_{\rm S})- S(\rho_{\rm S}')\right].
\label{bound}\end{aligned}$$ While it is well known that Equation (\[bound\]) must hold, and that reaching the equality is required for thermodynamic reversibility [@PAV; @Maroney2007a], we show that it fails as a tight bound for quantum information processing. There is an unavoidable excess heat generation in quantum computation that is not present in the classical case.
Summary of the theorem {#summary-of-the-theorem .unnumbered}
----------------------
For any given quantum operation $Q$ there is non negative quantity $\epsilon_Q $, defined independently of how the quantum operation is physically implemented, such that the heat generated by the operation is bounded by: $$\begin{aligned}
\Delta E_Q \geq kT \ln 2\left[ S(\rho_{\rm S}) - S(\rho'_{\rm S})\right] + \epsilon_Q.
\nonumber\end{aligned}$$ In general for quantum operations $\epsilon_Q >0$. Barring exceptional symmetric cases, $\epsilon_Q =0$ if, and only if, all the output signal states $\mathcal{Q}(\rho^n_{\rm S})$ share a common diagonalised basis, and there exists a common left stochastic map from diagonal elements of the input states (in the diagonal basis of $\rho_{\rm S}$) to eigenvalues of the output states. If these conditions do not hold, there is an excess, thermodynamically irreversible cost to any physical process that performs the quantum operation.
This paper will define quantum operations, characterise the thermodynamics of implementing quantum operations, state the theorem, and then give examples of operations which meet, and operations which do not meet, the conditions for $\epsilon_Q=0$.
Quantum Operations
==================
While classical information processing is built up from logical operations such as AND, OR and NOT gates, the fundamental element in quantum information processing is the quantum operation [@NC]. This acts upon one of a number of possible input states, each represented by a different quantum density matrix, $\{\rho^n_{\rm S}\}$, and maps each one to a specific output state $\rho^n_{\rm S} \rightarrow {\rho^{n\prime}_{\rm S}}$.
Not all maps of the form $\rho^n_{\rm S} \rightarrow {\rho^{n\prime}_{\rm S}}$ are physically possible. A quantum operation, $Q$, must be a completely positive, trace preserving linear map: $\mathcal{Q}(\rho^n_{\rm S})=\sum_k Q_k \rho^n_{\rm S} Q^\dag_k$, with $\sum_k Q_k Q_k^\dag=\mathbb{1}_{\rm S}$. When it is not a pure unitary rotation, a quantum operation requires the use of an auxiliary system ${\rm A}$, initially in a standard state, $\rho_{\rm A}$. A joint unitary $V$ acting on $\rho^n_{\rm S}\otimes \rho_{\rm A}\rightarrow \rho_{{\rm SA}}^{n\prime}=V\rho^n_{\rm S}\otimes \rho_{\rm A} V^{\dagger}$ results in an entangled state of system and auxiliary (see Figure \[F0\]). The reduced state of the system, ${\rho^{n\prime}_{\rm S}} = {{\rm Tr}}_{\rm A} [V\rho^n_{\rm S}\otimes \rho_{\rm A} V^{\dagger}]$, is the output state (see [@NC] Chapter 8, for a textbook presentation).
![(i) The basic operation we consider involves a unitary $V$ to convert an input system state $\rho^{n}_{\rm S}$ into an output state $\rho^{n\prime}_{\rm S}$. An auxiliary input $\rho_{\rm A}$ typically becomes entangled with the system and is discarded at the end of the operation. In order to understand the energy cost of the operation we should reset the auxiliary to its standard state $\rho_{\rm A}$ in preparation for further uses. This is done via interaction with a heat bath which starts in the canonical state $\rho_{\rm B}$. (ii) An energetically suboptimal way to do this ignores any correlations between ${\rm S}$ and ${\rm A}$ and directly resets the auxiliary using the unitary $W$ which acts on the joint auxiliary-heat bath state. The output auxiliary state $\bar{\rho}_{\rm A}^n = {{\rm Tr}}_{\rm B}[W\rho^{n\prime}_{\rm A}\otimes\rho_{\rm B}W^{\dagger}]$ should be such that ${\rho}_{\rm A} = \sum_n p_n \bar{\rho}_{\rm A}^n$ where $p_n$ is the probability of input $n$; the auxiliary is reset on average. (iii) The optimal method typically cannot be decomposed in this way and requires a unitary $U$ defined to act on system, auxiliary, and heat bath. $U$ should result in the same output state $\rho^{n\prime}_{\rm S}$ and reset the auxiliary: the output auxiliary state ${\rho}_{\rm A}^n$ should be such that $\rho_{\rm A} = \sum_n p_n {\rho}_{\rm A}^n$. For given average input $\rho_{\rm S}$ and output $\rho'_{\rm S}$, the average energy change which occurs in the heat bath will be lower bounded. []{data-label="F0"}](Fig1.jpg){width="30.00000%"}
Thermodynamics of quantum operations
====================================
It is essential, when quantifying the thermodynamic costs of operations, to keep track of the effects upon auxiliary systems. If an auxiliary is disregarded, then the quantum operation would appear to be performed without generating heat. However, this leaves the auxiliary in a state ${\rho_{\rm A}^{n\prime}}={{\rm Tr}}_{\rm S} [V\rho^n_{\rm S} \otimes \rho_{\rm A} V^\dag]$. Simply discarding the auxiliary would, on average, change the entropy of the environment by $\ln 2 \left[S(\rho_{\rm A}') - S(\rho_{\rm A})\right]$ where $\rho_{\rm A}' = {{\rm Tr}}_{\rm S} [V\rho_{\rm S}\otimes \rho_{\rm A} V^{\dagger}]=\sum_n p_n {\rho_{\rm A}^{n\prime}}$.
The most straightforward way to deal with this is to reset the auxiliary to its original state $\rho_{\rm A}$. To do this a thermal environment is introduced in the form of a heat bath in a canonical state $\rho_{\rm B}$ at temperature $T$. The reset can be performed by some unitary $W$ acting on both the auxiliary and the heat bath such that $\rho_{\rm A}={{\rm Tr}}_{\rm B}[W\rho'_{\rm A}\otimes \rho_{\rm B} W^\dag]$ (see Figure \[F0\]). This operation would transfer at least $kT \ln 2 \left[S(\rho'_{\rm A}) - S(\rho_{\rm A})\right]$ of heat to the heat bath [@PAV]. Either way, the cost to the environment will generally be more than required by Equation (\[bound\]). A quantum operation typically leaves correlations between system and auxiliary, with a mutual information [@VV] of $S(\rho^\prime_{\rm S}\colon\rho^\prime_{\rm A})=\left[S(\rho'_{\rm S}) - S(\rho_{\rm S})\right] + \left[S(\rho_{\rm A}') - S(\rho_{\rm A})\right] \geq 0$. Simply resetting (or discarding) the auxiliary will pay $k T \ln 2 S(\rho_{\rm S}' \colon\rho_{\rm A}')$ as an excess cost unless there is no correlation: $V\rho_{\rm S}\otimes \rho_{\rm A} V^{\dagger}=\rho_{\rm S}' \otimes \rho_{\rm A}'$.
To find the minimum thermodynamic cost of the quantum operation, the auxiliary must be reset more efficiently, exploiting correlations with the system. The unitary $V$ must be embedded in a larger unitary $U$ which includes interactions between system, auxiliary, and heat bath. $U$ must preserve the output of the computation $$\begin{aligned}
{{\rm Tr}}_{\rm AB} [U\rho^n_{\rm S}\otimes\rho_{\rm A}\otimes\rho_{\rm B}U^{\dagger}] &= {\rho^{n\prime}_{\rm S}};
\label{Ereset0}\end{aligned}$$ while resetting the auxiliary $$\begin{aligned}
{{\rm Tr}}_{\rm SB} [U\rho_{\rm S}\otimes\rho_{\rm A}\otimes\rho_{\rm B}U^{\dagger}] &= \rho_{\rm A}
\label{Ereset}\end{aligned}$$ (see Figure \[F0\]). Standard calculations show that this implies Equation (\[bound\]) (see Appendix \[Sqtd\]). If the quantum operation is only defined for a single signal state, then physical implementations are possible which can get arbitrarily close to the equality in (\[bound\]) (Appendix \[ARev\]).
In general, a quantum operation must produce the result ${\rho^{n\prime}_{\rm S}}=\mathcal{Q}(\rho^n_{\rm S})$, for multiple signal states $\{\rho_{\rm S}^n\}$. The same $U$ must satisfy (\[Ereset0\]) for all $n$. Our principal result is to show this additional constraint forces a higher thermodynamic cost than Equation (\[bound\]). An operation which generates a quantity of heat, $\epsilon_Q$, in excess of this bound will be thermodynamically irreversible, as any second operation which restores the original average state $\rho_{\rm S}=\mathcal{Q'}(\rho_{\rm S}')$ must necessarily leave a net heat gain in the heat bath of at least $\epsilon_Q$ over the complete cycle. Quantum computations cannot, in general, be performed in a thermodynamically reversible manner.
The Theorem
===========
Let $\rho^n_{\rm S}\rightarrow \rho^{n\prime}_{\rm S} = {{\rm Tr}}_{\rm AB} [U \rho^n_{\rm S}\otimes \rho_{\rm A}\otimes\rho_{\rm B} U^{\dagger}]$ be a quantum operation $Q$ where $\rho_{\rm B}$ is the state of a canonical heat bath at temperature $T$, and $\rho_{\rm A}$ is a standard state of an auxiliary which should be restored by the operation \[see Equation (\[Ereset\])\].
The average system input state $\rho_{\rm S}$ can be expressed in terms of its diagonal basis vectors $\{|\phi_i\rangle\}$ as $$\begin{aligned}
\rho_{\rm S} = \sum_n p_n \rho^n_{\rm S} =\sum_i \lambda_i |\phi_i\rangle\langle \phi_i |.
\nonumber\end{aligned}$$ Similarly the average system output state can be expressed in terms of its diagonal basis vectors $\{|\phi'_k\rangle\}$ as $$\begin{aligned}
\rho'_{\rm S} = \sum_n p_n \rho^{n\prime}_{\rm S} =\sum_k \lambda'_k |\phi'_k\rangle\langle \phi'_k|.
\nonumber\end{aligned}$$ In terms of these basis vectors the individual inputs and outputs are represented by $$\begin{aligned}
\rho^n_{\rm S} = \sum_{ij} \mu_{ij}^n |\phi_i\rangle\langle \phi_j | \quad \text{and} \quad
\rho^{n\prime}_{\rm S} = \sum_{kl} \mu_{kl}^{n\prime}|\phi'_k\rangle\langle \phi'_l |.
\nonumber\end{aligned}$$ We now state the theorem:
\[th1\] If there exists a stochastic map, $P_Q(k|i)$ with $$\begin{aligned}
\sum_k P_Q(k|i) = 1 \quad \text{and} \quad P_Q(k|i) \geq 0 \; \forall i,k, \nonumber\end{aligned}$$ such that $$\begin{aligned}
\mu^{n\prime}_{kl} = \delta_{kl} \sum_i P_Q(k|i) \mu^n_{ii},
\label{revcon}\end{aligned}$$ for all $n$, then the minimum thermodynamic cost of the operation $$\begin{aligned}
\Delta E_Q \geq kT \ln 2 \left[ S(\rho_{\rm S}) - S(\rho'_{\rm S})\right]+\epsilon_Q,
\nonumber\end{aligned}$$ can approach $\epsilon_Q=0$. Otherwise, provided there are no symmetries of the form $\lambda_i/\lambda_j=\lambda'_k/\lambda'_l$ where $i\neq j$ or $k\neq l$, then necessarily $\epsilon_Q>0$.
Note that the set of properties $\{\{\lambda_i\},\{\lambda'_k\},\{\mu_{ij}^n\},\{{\mu_{kl}^{n\prime}}\}\}$, used to state the theorem are defined solely in terms of the quantum operation, independently of the particular physical process used to implement it.
Outline of proof {#outline-of-proof .unnumbered}
----------------
In Appendix \[NAFS\] we show that the implementation of the operation can be re-written in the form $$\begin{aligned}
\mu^{n\prime}_{kl}=\sum_{ij} q(kl|ij) \mu^n_{ij},
\label{mapp}\end{aligned}$$ where the complex coefficients $q(kl|ij)$ carry the effects of interaction with the environment. They have the properties $q(kk|ii)=q^*(kk|ii)\geq 0$, and $\sum_k q(kk|ij)=\delta_{ij}$. In Appendix \[Sqtd\] we show that for a given physical implementation of the quantum operation where $\rho'= U \rho_{\rm S}\otimes \rho_{\rm A}\otimes \rho_{\rm B} U^{\dagger}$, and $\rho_\star=\rho'_{\rm S} \otimes \rho_{\rm A}\otimes\rho_{\rm B} $, $$\begin{aligned}
\epsilon_Q\geq \frac{1}{2}kT|| \rho'-\rho_\star||_1^2.
\label{newbound}\end{aligned}$$ Finally in Appendix \[SUR\] we show that $$\begin{aligned}
\label{eqbound1}
|| \rho'-\rho_\star||_1 \geq \frac{|\lambda_j\lambda'_k - \lambda_i\lambda'_l|}{\lambda'_k+\lambda'_l} \left| q(kl|ij) \right|.\end{aligned}$$ It follows that if there is a value of $\left| q(kl|ij) \right|>0$ for which $\lambda_i/\lambda_j \neq \lambda'_k/\lambda'_l$, then the implementation has an excess thermodynamic cost.
The coefficients $q(kk|ii)$ imply no bound as $\lambda_i/\lambda_j = \lambda'_k/\lambda'_l$ automatically holds for them. If Equation (\[revcon\]) holds, then $q(kk|ii) = P_Q(k|i)$ gives an implementation with no excess cost (see Appendix \[proto\] for an explicit construction). Otherwise, there must be some $\left|q(kl|ij)\right|>0$ for $i \neq j$ or $k \neq l$. If $\lambda_i/\lambda_j \neq \lambda'_k/\lambda'_l$ for any of these $ijkl$ values, then there is an excess cost for that implementation. We can therefore use numerical optimisation techniques to find the coefficients satisfying (\[mapp\]) which minimise the largest value of (\[eqbound1\]), and Equation (\[newbound\]) shows this gives a lower bound for $\epsilon_Q$.
When the output states do not have a shared diagonalised basis, so that there is some $\mu^{n\prime}_{kl}>0$ for $k \neq l$, there is also an analytical bound: $$\begin{aligned}
\epsilon_Q \geq \frac{1}{2}kT
\max_{n,k,l\neq k}\left\{
\frac
{|\mu^{n\prime}_{kl}|/(\lambda'_k+\lambda'_l)}
{\sum_{ij} |\mu^n_{ij}|/|\lambda_j\lambda'_k - \lambda_i\lambda'_l|}
\right\}^2. \label{eqoffdiagM}\end{aligned}$$ For further details see Appendix \[secPROOF\].
This energy bound and others derived here may not be tight. It is an open problem to demonstrate a protocol which reaches $\epsilon_Q$ - our results only place a non-zero lower bound on the excess energy cost.
{width="80.00000%"}
{width="100.00000%"}
Examples of thermodynamically [reversible]{} operations
=======================================================
There are well known cases of quantum operations where $\epsilon_Q=0$. We now show how they fit into our proof.
*Pure unitary.* In a pure unitary quantum operation, $\rho^{n\prime}_{\rm S}=U\rho^n_{\rm S} U^\dag$. This does not involve an auxiliary or heat bath, and does not change the eigenvalues of any input signal states. Solutions with $q(kl|ij)=\delta_{ik}\delta_{jl}$ satisfy the operation, and the exceptional symmetry condition applies: the only terms which contribute to Equation (\[mapp\]) are ones for which $\lambda_i/\lambda_j=\lambda'_k/\lambda'_l$.
*Single input.* When there is only one possible input $\rho^n_{\rm S} = \rho_{\rm S}$. As $\mu^n_{ij}=\delta_{ij} \lambda_i$ and $\mu^{n\prime}_{kl}=\delta_{kl}\lambda'_k$, choosing $P_Q(k|i) = \lambda'_k$ shows that Equation (\[revcon\]) holds.
*Single output.* Resetting to a standard state is known to satisfy $\epsilon_Q=0$ [@SAG2; @AT]. In this case the output states for all inputs are of the form $\rho'_{\rm S} = \sum_k \lambda'_k {{{\ensuremath{\left| {{\phi_k}} \right\rangle}}{\ensuremath{\left. {\ensuremath{\left\langle {{\phi_k}} \right.}} \right|}}}}$, so $\mu^{n\prime}_{kl}=\delta_{kl}\lambda'_k$. Again, choosing $P_Q(k|i)=\lambda'_k$, shows that Equation (\[revcon\]) holds.
*Classical information processing.* For classical information processing, where the signal states correspond to orthogonal quantum states, $\epsilon_Q=0$ [@Mar05b; @OM1; @SAG1]. $\rho^n_{\rm S} = |\phi_n\rangle\langle \phi_n |$, so $\mu^n_{ij}=\delta_{ij} \delta_{in}$. In general the outputs take the form $\rho^{n\prime}_{\rm S} = \sum_{k} P_Q(k|n) |\phi'_k\rangle\langle \phi'_k |$, representing a distribution of possible outputs $|\phi'_k\rangle$, so $\mu^{n\prime}_{kl}=\delta_{kl} P_Q(k|n)$, and Equation (\[revcon\]) holds.
Case study of a thermodynamically irreversible operation
========================================================
In general it will not be the case that $\epsilon_Q=0$. Even the simplest quantum operations will fail to satisfy Equation (\[revcon\]). As an example, we consider single qubit dephasing operations, which exhibit all the features of the Theorem. Inputs of the form $\rho^n_{\rm S} = |v^n_{\rm S}\rangle\langle v^n_{\rm S}|$ with $|v^n_{\rm S}\rangle = c_0^n|0_{\rm S}\rangle + c_1^n|1_{\rm S}\rangle$ give outputs $\rho^{n\prime}_{\rm S}(r) =r\rho^n_{\rm S}+ (1-r) \left[ |c_0^n|^2|0_{\rm S}\rangle\langle 0_{\rm S}| + |c_1^n|^2|1_{\rm S}\rangle\langle 1_{\rm S}|\right]$. Each value of $r$ defines a different quantum operation, $\rho^{n\prime}_{\rm S}(r)=\mathcal{Q}_r(\rho^n_{\rm S})$, with $Q_1$ the identity, and $Q_0$ completely dephasing the qubit.
The operation $Q_r$ can be implemented straightforwardly by a quantum CNOT gate and an auxiliary, making our Theorem open to experimental investigation [@VYH+2015]. With the auxiliary initially prepared in the state $\alpha|0_{\rm A}\rangle + \beta|1_{\rm A}\rangle$ acting as the target, we have $r=\left(\alpha^* \beta+ \alpha \beta^* \right)$.
*Non-diagonalisable outputs.* When $0<r<1$, the outputs, $\rho^{n\prime}_{\rm S}(r)$, will not typically be simultaneously diagonalisable. Consider $r=\frac{1}{\sqrt{2}}$ with two possible inputs: $|0_{\rm S}\rangle$ occurring with probability $0.3$, and $|+_{\rm S}\rangle = \frac{1}{\sqrt{2}}|0_{\rm S}\rangle + \frac{1}{\sqrt{2}}|1_{\rm S}\rangle$ occurring with probability $0.7$. After tracing away the auxiliary we find that the individual outputs are not simultaneously diagonalisable. There is enough information to calculate the lower bound on $\epsilon_Q$ from Equation (\[eqoffdiagM\]) and this is found to be $0.0007\times kT$. This can be compared with the bound $kT\ln 2\left[S(\rho_{{\rm S}}) - S(\rho'_{{\rm S}})\right] = -0.15 \times kT$. An overall negative value of $\Delta E_Q$ indicates that this operation may still be used to extract energy from the heat bath.
*Diagonalisable outputs.* For $Q_0$, the outputs will be simultaneously diagonalised in the basis $\{{\ensuremath{\left| 0_{\rm S} \right\rangle}},{\ensuremath{\left| 1_{\rm S} \right\rangle}}\}$. We choose two pure state inputs: $\rho^1_{\rm S}$ with probability $p$, and $\rho^2_{\rm S}$ with probability $1-p$. The average input density matrix is therefore $\rho_{\rm S} = p\rho^1_{\rm S} + (1-p)\rho^2_{\rm S}$. Figure \[F1\] shows the minimum energy change in the heat bath with $p$ for four different pairs of inputs. We see that for some combinations of inputs there are no values of $p$ for which the operation can be thermodynamically reversible, and it necessarily requires an excess energy cost. For other inputs there are regions of $p$ where there is no excess cost and regions where there is a minimum non zero excess cost. Figure \[F2\] shows, for a given $\rho^1_{\rm S}$, the regions of $\rho^2_{\rm S}$ on the Bloch sphere where a zero excess cost is possible for various values of $p$. Further details of how to determine the minimum energy change in the heat bath can be found in Appendix \[ADP\].
The qubit dephasing example shows, in particular, that $\epsilon_{Q}=0$ is possible both with $\Delta E_Q<0$ and with non-trivial non-orthogonal input states, and that $\epsilon_{Q}>0$ is possible even when the output states share a common diagonalisation.
Discussion
==========
The excess thermodynamic costs of quantum operations stem from the requirement that the operation should get the computation right for every individual input, and not just on average. For well known cases, including classical information processing, we have shown how thermodynamic reversibility can be reached as a special case. However, for general quantum operations, with non-orthogonal signal states, an excess thermodynamic cost must be paid.
This might seem counter intuitive, since quantum computations can always be represented as unitary operations, which can always be run in reverse. How can thermodynamic irreversibility arise with a reversible unitary operation? Simply running the overall unitary operation in reverse does not just tidy up the auxiliary, returning it to its initial state, but it also undoes the computation, converting the output back to the input. If we wish to retain the result of the computation, we cannot simply reverse the unitary operation. In the classical reversible computing model of Bennett [@BEN2], we save a copy of the output before reversing the computation. For quantum operations this is forbidden by the no-cloning theorem, and no quantum generalisation of Bennett’s procedure is possible [@OM2].
As long as we keep the results of the computation, we are left with the changes in the auxiliaries, representing spent resources and a cost to reset them to their initial states. This is the stage at which thermodynamic costs are incurred. We have shown that in general, for quantum operations this cost is necessarily in excess of the Landauer Limit given in Equation (\[bound\]). Quantum computing requires a thermodynamically irreversible generation of heat.
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank Chris Timpson and Oscar Dahlsten for discussions. This work was funded by the Templeton World Charity Foundation.
Notation and formal set up {#NAFS}
==========================
Given an input $\rho^n_{\rm S}$ and an output ${\cal Q}(\rho^n_{\rm S}) = \rho^{n\prime}_{\rm S}$ of a quantum operation $Q$ on a quantum system ${\rm S}$ we can describe the operation as $$\begin{aligned}
\rho^{n\prime}_{\rm S} = {{\rm Tr}}_{\rm AB} [U \rho_{\rm S}^n\otimes \rho_{\rm A}\otimes \rho_{\rm B} U^{\dagger}],
\label{qop}\end{aligned}$$ where the subscript on the trace indicates which components have been traced over. The unitary $U$ acts on an initial state $\rho_{\rm S}^n \otimes \rho_{\rm A}\otimes \rho_{\rm B}$ which includes, besides the system, a thermal environment in the form of a canonical heat bath, $\rho_{\rm B}$, and an auxiliary to be used as a catalyst $\rho_{\rm A}$. Given that inputs $\rho^n_{\rm S}$ appear with probability $p_n$ we can define an average system input state and an average system output state by $$\begin{aligned}
\rho_{\rm S} = \sum_n p_n \rho^n_{\rm S} \quad \text{and} \quad
\rho'_{\rm S} = \sum_n p_n \rho^{n\prime}_{\rm S},
\nonumber\end{aligned}$$ respectively.
The auxiliary is a resource which is spent by the operation. In order to account for this cost we demand that it must be returned to the same standard state $\rho_{\rm A}$ at the end of the computation $$\begin{aligned}
{{\rm Tr}}_{\rm SB} [U \rho_{\rm S}\otimes \rho_{\rm A}\otimes\rho_{\rm B} U^{\dagger}] = \rho_{\rm A}.
\nonumber\end{aligned}$$ This ensures that the auxiliary can be reused in subsequent computations. We need only demand that the auxiliary is reset on average.
For convenience we define the following notation $$\begin{aligned}
\rho_{\rm AB} &= \rho_{\rm A}\otimes\rho_{\rm B}, \nonumber \\
\rho' &= U (\rho_{\rm S}\otimes\rho_{\rm A}\otimes\rho_{\rm B}) U^{\dagger}, \nonumber \\
\rho_{\rm B}' &={{\rm Tr}}_{\rm SA} [U (\rho_{\rm S}\otimes\rho_{\rm A}\otimes\rho_{\rm B}) U^{\dagger}].
\nonumber\end{aligned}$$
We can choose to work in a basis in which $\rho_{\rm S}$ is diagonal. In general $\rho'_{\rm S}$ will not be diagonal in this basis but we can perform a final rotation to make it so: $$\begin{aligned}
R\rho'_{\rm S} R^{\dagger} = {{\rm Tr}}_{\rm AB} [(RU)\rho_{\rm S} \otimes \rho_{\rm AB} (RU)^{\dagger}] ,
\nonumber\end{aligned}$$ or $$\begin{aligned}
\bar{\rho}'_A = {{\rm Tr}}_{\rm AB} [\bar{U} \rho_{\rm S}\otimes \rho_{\rm AB} \bar{U}^{\dagger}].
\nonumber\end{aligned}$$ Now drop the bars from the notation and assume without loss of generality that the average system inputs and outputs are diagonalised in the same basis. Let us denote these diagonal basis states by $\{|\phi_i\rangle\}$. We can write $$\begin{aligned}
\rho_{\rm S} = \sum_i \lambda_i |\phi_i\rangle\langle \phi_i | \quad \text{and} \quad
\rho'_{\rm S} = \sum_i \lambda'_i |\phi_i\rangle\langle \phi_i |.
\nonumber \end{aligned}$$ In this basis an individual input and output state can in general be expressed as $$\begin{aligned}
\rho^n_{\rm S} = \sum_{ij} \mu^n_{ij} |\phi_i\rangle\langle \phi_j | \quad \text{and} \quad
\rho^{n\prime}_{\rm S} = \sum_{ij} \mu^{n\prime}_{ij }|\phi_i\rangle\langle \phi_j |.
\label{inout}\end{aligned}$$
We define $$\begin{aligned}
A_{ij} = \langle \phi_i |U|\phi_j\rangle.
\nonumber\end{aligned}$$ This is a bounded operator acting on the total environment of auxiliary and heat bath. That it is bounded can be demonstrated as follows: $U$ is a bounded operator since $|U|\psi\rangle| = ||\psi\rangle|$ for a state of system, auxiliary, and heat bath $|\psi\rangle$. This means that the product of the bounded operators $|\phi_i\rangle\langle\phi_i|U|\phi_j\rangle\langle\phi_j|$ is also bounded. In fact $||\phi_i\rangle\langle\phi_i|U|\phi_j\rangle\langle\phi_j|\psi\rangle| \leq ||\psi\rangle|$. If we choose $|\psi\rangle = |\chi\rangle|\phi_j\rangle$ where $|\chi\rangle$ is an arbitrary total environment state we then find that $|A_{ij}|\chi\rangle| \leq ||\chi\rangle|$ so that $A_{ij}$ is a bounded operator.
We can express the unitary operator $U$ as $$\begin{aligned}
U = \sum_{ij} |\phi_i\rangle\langle\phi_j| \otimes A_{ij}.
\label{uphi}\end{aligned}$$ These operators must satisfy $$\begin{aligned}
\langle \phi_i|UU^{\dagger} |\phi_j\rangle &= \sum_k A_{ik}A_{jk}^{\dagger} = \delta_{ij} \mathbb{1}_{\rm AB} ,
\label{g4}\\
\langle \phi_i|U^{\dagger}U |\phi_j\rangle &= \sum_k A_{ki}^{\dagger}A_{kj} = \delta_{ij} \mathbb{1}_{\rm AB}.
\label{g5}\end{aligned}$$ The unitary operation on the complete state results in $$\begin{aligned}
\langle \phi_i|\rho' |\phi_j\rangle
=\sum_k \lambda_k A_{ik} \rho_{\rm AB} A_{jk}^{\dagger}.
\label{g1}\end{aligned}$$ It will also be useful to define the state $\rho_\star = \rho'_{\rm S}\otimes\rho_{\rm AB}$ which we can express as $$\begin{aligned}
\langle \phi_i|\rho_{\star} |\phi_j\rangle
=\lambda'_i \delta_{ij} \rho_{\rm AB},
\label{g2}\end{aligned}$$ and we denote $$\begin{aligned}
\langle \phi_i|(\rho' - \rho_{\star}) |\phi_j\rangle = \Delta_{ij}.
\label{g3}\end{aligned}$$ The result of the quantum operation on individual inputs (\[qop\]) can be written using (\[inout\]) and (\[uphi\]) as $$\begin{aligned}
\mu^{n\prime}_{kl} = \sum_{ij} \mu^n_{ij}{{\rm Tr}}_{} [A_{ki}\rho_{\rm AB} A_{lj}^{\dagger}].
\label{indiv}\end{aligned}$$ We will also use the notation $$\begin{aligned}
q(kl|ij) = {{\rm Tr}}_{} [A_{ki}\rho_{\rm AB} A_{lj}^{\dagger}].
\label{qcoef}\end{aligned}$$
Quantum thermodynamics {#Sqtd}
======================
\[l0\] The energy change in the heat bath as a result of a quantum operation $Q$ on a system ${\rm S}$ implemented by some unitary $U \rho_{\rm S}\otimes \rho_{\rm A}\otimes \rho_{\rm B} U^{\dagger} = \rho'$, involving a resetting of the auxiliary system ${\rm A}$, is given by $$\begin{aligned}
\frac{\Delta E_Q}{kT\ln 2} =\left[ S(\rho_{\rm S}) - S(\rho'_{\rm S})\right] + S(\rho'||\rho_{\star}).
\label{Eresult0}\end{aligned}$$ where $\rho_{\star} = \rho'_{\rm S}\otimes\rho_{\rm A}\otimes\rho_{\rm B}$.
The Schumacher information measure is given by $S(\rho) = -{{\rm Tr}}\left[\rho\log_2\rho\right]$ and we use a definition of the relative entropy using log base 2 $$\begin{aligned}
S(\rho||\sigma) = {{\rm Tr}}\left[\rho\log_2\rho\right]-{{\rm Tr}}\left[\rho\log_2\sigma\right].
\nonumber\end{aligned}$$
The energy change in the bath is given by $\Delta E_Q = {{\rm Tr}}[H_{\rm B}(\rho_{\rm B}'-\rho_{\rm B})]$ where $H_{\rm B}$ is the heat bath Hamiltonian. A standard result is that [@PAV] $$\begin{aligned}
\frac{\Delta E_Q}{kT\ln 2}
=S(\rho'_{\rm B}) - S(\rho_{\rm B}) + S(\rho'_{\rm B}||\rho_{\rm B}).
\nonumber\end{aligned}$$ It follows that $$\begin{aligned}
\frac{\Delta E_Q}{kT\ln 2}
=&S(\rho'_{\rm B}) - \left[S(\rho) - S(\rho_{\rm S}) - S(\rho_{\rm A})\right] + S(\rho'_{\rm B}||\rho_{\rm B}) \nonumber \\
=&S(\rho'_{\rm B}) - \left[S(\rho') - S(\rho_{\rm S}) - S(\rho_{\rm A})\right] + S(\rho'_{\rm B}||\rho_{\rm B}) \nonumber \\
=& \left[ S(\rho_{{\rm S}}) - S(\rho'_{{\rm S}})\right] + S(\rho'_{\rm B}||\rho_{\rm B})\nonumber\\
&\quad+\left[ S(\rho'_{{\rm S}}) + S(\rho_{\rm A}) + S(\rho'_{\rm B}) - S(\rho') \right] .
\label{DEcalc}\end{aligned}$$ Since the last term in square brackets in the last line must be positive by subadditivity and the relative entropy term must be positive as a result of Klein’s inequality, we must have $$\begin{aligned}
\Delta E_Q \geq kT\ln 2\left[ S(\rho_{\rm S}) - S(\rho'_{\rm S})\right] .
\label{LAND}\end{aligned}$$ The operation satisfies thermodynamic reversibility if the inequality is saturated. This can be seen by forming a second operation $\rho_{\rm S} = {\cal Q}'(\rho_{\rm S}')$ which restores the original average state. There must be a net heat gain in the heat bath if, for any part of the cycle, the equality in (\[LAND\]) does not hold.
Write $\rho_\star = \rho_{{\rm S}}'\otimes \rho_{\rm A}\otimes\rho_{\rm B} $. As a result of the fact that $\rho_\star$ factorises we find that $$\begin{aligned}
S(\rho'||\rho_{\star}) = & {{\rm Tr}}[\rho'\log_2\rho'] - {{\rm Tr}}[\rho' \left( \log_2\rho'_{\rm S}\otimes {\mathbb 1}_{\rm A}\otimes{\mathbb 1}_{\rm B}\right)]
\nonumber\\
&- {{\rm Tr}}[\rho' \left( {\mathbb 1}_{\rm S}\otimes \log_2\rho_{\rm A} \otimes{\mathbb 1}_{\rm B}\right)] \nonumber\\
&- {{\rm Tr}}[\rho'\left({\mathbb 1}_{\rm S}\otimes{\mathbb 1}_{\rm A}\otimes\log_2\rho_{\rm B}\right)]\nonumber\\
=& {{\rm Tr}}[\rho'\log_2\rho'] - {{\rm Tr}}[\rho'_{\rm S}\log_2\rho'_{\rm S}]
\nonumber\\ & - {{\rm Tr}}[\rho_{\rm A}\log_2\rho_{\rm A}] - {{\rm Tr}}[\rho'_{\rm B}\log_2\rho_{\rm B}]
\nonumber\\
=&S(\rho_{\rm B}'||\rho_{\rm B})\nonumber\\
&+\left[S(\rho_{\rm S}') + S(\rho_{\rm A}) + S(\rho_{\rm B}') - S(\rho') \right] ,
\label{REcalc}\end{aligned}$$ so that from (\[DEcalc\]) $$\begin{aligned}
\frac{\Delta E_Q}{kT\ln 2} =\left[ S(\rho_{{\rm S}}) - S(\rho'_{{\rm S}})\right] + S(\rho'||\rho_{\star}).
\nonumber\end{aligned}$$ The quantity $S(\rho'||\rho_{\star}) \geq 0$ therefore encodes the extent to which the thermodynamic bound on the energy change (\[LAND\]) is breached.
The demand for $S(\rho'||\rho_{\star})=0$ requires from (\[REcalc\]) that both $S(\rho') = S(\rho_{{\rm S}}') + S(\rho_{\rm A}) + S(\rho_{\rm B}')$ and $S(\rho_{\rm B}'||\rho_{\rm B}) = 0$. In particular the second of these conditions suggests an output state with $\rho_{\rm B}' = \rho_{\rm B}$ and therefore no energy change in the heat bath. In fact, in the limit where the dimension of the heat bath becomes large, it is possible to have $\rho_{\rm B}'$ sufficiently close to $\rho_{\rm B}$ such that $S(\rho'_{\rm B}||\rho_{\rm B})<s$ for some arbitrarily small $s$ whilst at the same time $\Delta E_Q > E$ for some fixed non zero $E$. To see this we write $\rho_{\rm B}' = \rho_{\rm B} + \varepsilon\Delta$ where $\Delta$ is a fixed traceless matrix and $\varepsilon$ is a small parameter. The change in energy of the heat bath is given by $$\begin{aligned}
\Delta E_Q = {{\rm Tr}}[H_{\rm B}(\rho_{\rm B}'-\rho_{\rm B})] = \varepsilon {{\rm Tr}}[ H_{\rm B} \Delta ].
\nonumber\end{aligned}$$ Now suppose that we have $N$ identical independent copies of the same heat bath (this will also be a canonical state). If we further suppose that each copy undergoes the same uncorrelated change $\rho_{\rm B}' = \rho_{\rm B} + \varepsilon\Delta$, then to lowest order, the energy change of the $N$-copy heat bath is $$\begin{aligned}
\Delta E_Q^{(N)} = N\Delta E_Q.
\nonumber\end{aligned}$$ This means that we can take $\varepsilon \propto 1/N$ and as $N$ becomes large, $\Delta E_Q^{(N)}$ remains fixed.
Now consider the relative entropy between $\rho_{\rm B}$ and $\rho_{\rm B}'$ $$\begin{aligned}
S(\rho_{\rm B}'||\rho_{\rm B}) &= \varepsilon\frac{d}{d\varepsilon}S(\rho_{\rm B} + \varepsilon\Delta||\rho_{\rm B})|_{\varepsilon = 0} + {\cal O}(\varepsilon^2) \nonumber\\
&= \varepsilon{{\rm Tr}}\left[\frac{\Delta}{ \ln 2} + \Delta\log_2\rho_{\rm B} - \Delta\log_2\rho_{\rm B}\right]+ {\cal O}(\varepsilon^2) \nonumber\\
&= {\cal O}(\varepsilon^2).
\nonumber\end{aligned}$$ The correction to the relative entropy is at least quadratic in $\varepsilon$. The relative entropy for the $N$-copy heat bath is $$\begin{aligned}
S(\rho_{\rm B}^{\prime\otimes N}||\rho_{\rm B}^{\otimes N}) = N S(\rho_{\rm B}'||\rho_{\rm B}).
\nonumber\end{aligned}$$ Therefore, in the limit that $N$ becomes large and $\varepsilon \propto 1/N$ tends to zero, the relative entropy for the $N$-copy heat bath tends to zero. This simple example demonstrates that it is legitimate for the relative entropy to tend to zero whilst the energy change in the heat bath remains non zero.
\[C0\] The energy change in the heat bath is bounded by $$\begin{aligned}
\Delta E_Q \geq {kT\ln2} \left[ S(\rho_{{\rm S}}) - S(\rho'_{{\rm S}})\right] + \frac{1}{2}kT|| \rho'-\rho_{\star}||_1^2.
\label{Eresult}\end{aligned}$$
We use [@REbound1; @REbound2] $$\begin{aligned}
S(\rho||\sigma) \geq \frac{1}{2\ln 2}|| \rho-\sigma||_1^2.
\nonumber\end{aligned}$$ The result follows from Equation (\[Eresult0\]).
Standard thermodynamically reversible protocol {#ARev}
==============================================
There is a standard protocol for reaching the bound given in Equation (\[LAND\]), appearing in various forms in the literature ([@Maroney2007a; @VV] for examples), when the quantum operation is defined only to act upon an individual density matrix $\rho_{\rm S}'=\mathcal{Q}(\rho_{\rm S})$ of a system ${\rm S}$ (see also [@SH; @AG2013] for alternative protocols to reach the bound). Note that these protocols do not require that either the initial or final states are in thermal equilibrium.
The density matrix is initially in the state $$\begin{aligned}
\rho_{\rm S}=\sum_i \lambda_i {{{\ensuremath{\left| {{\phi_i}} \right\rangle}}{\ensuremath{\left. {\ensuremath{\left\langle {{\phi_i}} \right.}} \right|}}}},
\nonumber\end{aligned}$$ where $\{{\ensuremath{\left| \phi_i \right\rangle}}\}$ are orthonormal eigenstates. The result of the operation $Q$ is a state $$\begin{aligned}
\rho_{\rm S}'=\sum_i \lambda_i' {{{\ensuremath{\left| {{\phi'_i}} \right\rangle}}{\ensuremath{\left. {\ensuremath{\left\langle {{\phi'_i}} \right.}} \right|}}}},
\nonumber\end{aligned}$$ for a possibly different set of orthonormal eigenstates $\{{\ensuremath{\left| \phi'_i \right\rangle}}\}$. We assume that both the initial and final sets of eigenstates are fully degenerate in energy with energy level zero. This avoids the complication of the system being used as a source or sink of energy.
[*Step 1*]{}. Starting with $\rho_{\rm S}$, manipulate the energy levels $E_i$ of each eigenstate ${\ensuremath{\left| \phi_i \right\rangle}}$ until $$\begin{aligned}
\lambda_i = \frac{e^{-\beta E_i}}{\sum_j e^{-\beta E_j}}.
\nonumber\end{aligned}$$ where $\beta = 1/kT$. This has a mean cost $\Delta E_1=\sum_i \lambda_i E_i$. The new density matrix will now be canonically distributed at temperature $T$.
[*Step 2*]{}. Bring the system into contact with a heat bath at temperature $T$.
[*Step 3*]{}. Slowly, isothermally, change the energy eigenstates until they satisfy $$\begin{aligned}
\lambda'_i = \frac{e^{-\beta E'_i}}{\sum_j e^{-\beta E'_j}}.
\nonumber\end{aligned}$$ The energy requirement for this is $\Delta E_2 = kT\ln\sum_j e^{-\beta E_j} - kT\ln\sum_j e^{-\beta E'_j}$. The density matrix is now canonically distributed at temperature $T$, with eigenvalues $\{\lambda'_i\}$.
[*Step 4*]{}. Remove the system from contact with the heat bath.
[*Step 5*]{}. Change the energy levels of the eigenstates back to zero, with a mean cost $\Delta E_3=-\sum_i \lambda_i' E'_i$.
[*Step 6*]{}. Perform a unitary rotation to the final eigenstates ${\ensuremath{\left| \phi_i \right\rangle}} \rightarrow {\ensuremath{\left| \phi_i' \right\rangle}}$.
The operation is complete with a cost of $$\begin{aligned}
\Delta E_1 + \Delta E_2+\Delta E_3 = kT\ln2 \left[S(\rho_{\rm S}) - S(\rho_{\rm S}')\right].
\nonumber\end{aligned}$$ It is worth noting that this protocol works even when the system is a joint system with non-trivial correlations or entanglement between subsystems [@VV].
During the stages in which we manipulate the energy levels of the system we are assuming a time-dependent Hamiltonian $H(t)$ for the system and a standard definition of the mean rate of work given by ${{\rm Tr}}[\rho \partial H/ \partial t]$. This presupposes some idealised work reservoir, capable of doing work on (or taking work from) the system by manipulation of the Hamiltonian.
We will denote this operation $T_{\rho_{\rm S}\rightarrow \rho_{\rm S}'}$. As shown in [@RW2014], the theoretical energy cost can only be acheived in an aymptotic limit. Nevertheless, there is no physical principle which prevents us from getting arbitrarily close this value.
Some useful results {#SUR}
===================
The following relation holds $$\begin{aligned}
&(\lambda_j \lambda'_k - \lambda_i\lambda'_l)q(kl|ij) =
\nonumber\\
& \sum_m\left(\lambda'_k {{\rm Tr}}[A_{ki}A_{mj}^{\dagger}\Delta_{ml}] -
\lambda'_l {{\rm Tr}}[A_{mi}A_{lj}^{\dagger}\Delta_{km}]
\right).
\label{l1}\end{aligned}$$
In order to prove this relation use Equations (\[g1\]), (\[g2\]), and (\[g3\]) to substitute for $\Delta_{ij}$. Then use the identity (\[g5\]) along with the cyclic property of the trace. The coefficients $q(kl|ij)$ are given in (\[qcoef\]).
\[Lem3\] The following bound can be placed on the output state $\rho'$: $$\begin{aligned}
||\rho'-\rho_{\star}||_1 \geq \frac{|\lambda_j\lambda'_k - \lambda_i\lambda'_l|}{\lambda'_k+\lambda'_l}
\left| q(kl|ij) \right| .
\label{lem2EQ}\end{aligned}$$
We start by defining operators acting on the complete state of system, auxiliary, and heat bath $$\begin{aligned}
\nonumber
F_{ijklm} = |\phi_l\rangle\langle\phi_m |\otimes A_{ki}A_{mj}^{\dagger} ,\end{aligned}$$ in terms of which we define $$\begin{aligned}
\nonumber
G^L_{ijkl} = \sum_m F_{ijklm}; \quad
G^R_{ijkl} = \sum_m F_{jilkm}^{\dagger}.\end{aligned}$$ We can then express (\[l1\]) as $$\begin{aligned}
(\lambda_j &\lambda'_k - \lambda_i\lambda'_l)q(kl|ij) = \nonumber\\
&\left(\lambda'_k {{\rm Tr}}[G^L_{ijkl}(\rho'-\rho_{\star})]- \lambda'_l {{\rm Tr}}[G^R_{ijkl}(\rho'-\rho_{\star})]
\right).
\label{el3}\end{aligned}$$ Let us define some arbitrary basis states $\{|\psi_x\rangle\}$ for the complete system, auxiliary, and heat bath. We can write $$\begin{aligned}
G^{L/R}_{ijkl} &= \sum_{xy} g^{L/R}_{xy}|\psi_x\rangle\langle \psi_y | ; \nonumber\\
(\rho' - \rho_{\star}) &= \sum_{xy} r_{xy}|\psi_x\rangle\langle \psi_y |. \nonumber\end{aligned}$$ Then $$\begin{aligned}
\left|{{\rm Tr}}[G^L_{ijkl} (\rho' - \rho_{\star})]\right| &= \left|\sum_{xy} g^L_{xy}r_{yx}\right| \nonumber\\
&\leq \sum_x \left|\sum_y g^L_{xy}r_{yx}\right| \nonumber\\
&\leq \sum_x \left(\sum_y |g^L_{xy}|^2\right)^{\frac{1}{2}}\left(\sum_z |r_{zx}|^2\right)^{\frac{1}{2}} ,
\label{G1eq}\end{aligned}$$ where the second line follows from the triangle inequality and the third line follows from the Cauchy-Schwartz inequality. Similarly $$\begin{aligned}
\left|{{\rm Tr}}[G^R_{ijkl} (\rho' - \rho_{\star})]\right| \leq \sum_x \left(\sum_y |g^R_{yx}|^2\right)^{\frac{1}{2}}\left(\sum_z |r_{xz}|^2\right)^{\frac{1}{2}}.
\label{G2eq}\end{aligned}$$ Next we show that $$\begin{aligned}
\langle \psi_x| G^L_{ijkl}G^{L\dagger}_{ijkl} |\psi_x\rangle = \sum_y |g^L_{xy}|^2 \leq 1,
\label{rdiag}\end{aligned}$$ and $$\begin{aligned}
\langle \psi_x| G^{R\dagger}_{ijkl}G^{R}_{ijkl} |\psi_x\rangle = \sum_y |g^R_{yx}|^2 \leq 1.
\label{ldiag}\end{aligned}$$ The operator products can be written as $$\begin{aligned}
G^L_{ijkl}G^{L\dagger}_{ijkl} &= \sum_m F_{ijklm}F^{\dagger}_{ijklm} ; \nonumber\\
G^{R\dagger}_{ijkl}G^{R}_{ijkl} &= \sum_m F_{jilkm}F^{\dagger}_{jilkm} . \nonumber\end{aligned}$$ We also have $$\begin{aligned}
\nonumber
\delta_{jj'}\delta_{kk'} \mathbb{1} = \sum_{ilm}F_{ijklm} F_{ij'k'lm}^{\dagger},\end{aligned}$$ which follows from (\[g4\]) and (\[g5\]). Since the identity is expressed as a sum of positive operators and since $G^L_{ijkl}G^{L\dagger}_{ijkl}$ is a sum of a subset of these positive operators, it must be the case that the diagonal elements of $G^L_{ijkl}G^{L\dagger}_{ijkl}$ in the arbitrary basis $\{|\psi_x\rangle\}$ must belong to the range $[0,1]$ from which (\[rdiag\]) follows. Similarly for (\[ldiag\]). We can therefore write, from (\[G1eq\]) and (\[G2eq\]), $$\begin{aligned}
\left|{{\rm Tr}}[G^{L}_{ijkl} (\rho' - \rho_{\star})]\right| &\leq \sum_x \left(\sum_z |r_{zx}|^2\right)^{\frac{1}{2}} ,
\nonumber \\
\left|{{\rm Tr}}[G^{R}_{ijkl} (\rho' - \rho_{\star})]\right| &\leq \sum_x \left(\sum_z |r_{xz}|^2\right)^{\frac{1}{2}} .\nonumber\end{aligned}$$ If we choose the basis states $\{|\psi_x\rangle\}$ such that $(\rho' - \rho_{\star})$ is diagonal we have $$\begin{aligned}
\nonumber
\left|{{\rm Tr}}[G^{L/R}_{ijkl} (\rho' - \rho_{\star})]\right| \leq ||\rho' - \rho_{\star}||_1.\end{aligned}$$ Finally from (\[el3\]), using the triangle inequality $$\begin{aligned}
\left|\lambda_j \lambda'_k - \lambda_i\lambda'_l\right| \left| q(kl|ij)\right| & \leq
\nonumber\\
\lambda'_k \left|{{\rm Tr}}[G^{R}_{ijkl}(\rho'-\rho_{\star})]\right| &+
\lambda'_l \left|{{\rm Tr}}[G^{L}_{ijkl}(\rho'-\rho_{\star})]\right|
\nonumber\\
&\leq (\lambda'_k+\lambda'_l)||\rho'-\rho_{\star}||_1, \nonumber\end{aligned}$$ which completes the proof.
\[Lem2\] When the dimension of the system space $d=2$, the following bound can be placed on the output state $\rho'$ for $i\neq j$: $$\begin{aligned}
\nonumber
||\rho'-\rho_{\star}||_1 \geq |\lambda_i - \lambda_j|
\left| q(kk|ij) \right| .\end{aligned}$$
When $k = l$ Equation (\[l1\]) reduces to $$\begin{aligned}
(\lambda_i & - \lambda_j) q(kk|ij) = \nonumber\\
&
\sum_{m\neq k}\left({{\rm Tr}}[A_{mi}A_{kj}^{\dagger}\Delta_{km}] - {{\rm Tr}}[A_{ki}A_{mj}^{\dagger}\Delta_{mk}]
\right).
\label{l2a}\end{aligned}$$ If $i=j$ both sides of this equation are zero. If $i\neq j$ we can define operators acting on the complete system, auxiliary, and heat bath $$\begin{aligned}
\nonumber
H_{ijkm} = |\phi_m\rangle\langle\phi_k |\otimes A_{mi}A_{kj}^{\dagger} - |\phi_k\rangle\langle\phi_m |\otimes A_{ki}A_{mj}^{\dagger},\end{aligned}$$ and then express (\[l2a\]) as $$\begin{aligned}
(\lambda_i - \lambda_j)q(kk|ij) =
\sum_{m\neq k} {{\rm Tr}}[H_{ijkm} (\rho' - \rho_{\star})].
\label{l2b}\end{aligned}$$ Define arbitrary basis states $\{|\psi_x\rangle\}$ for the complete system, auxiliary, and heat bath. We can write $$\begin{aligned}
H_{ijkm} &= \sum_{xy} h_{xy}|\psi_x\rangle\langle \psi_y |; \nonumber\\
(\rho' - \rho_{\star}) &= \sum_{xy} r_{xy}|\psi_x\rangle\langle \psi_y |. \nonumber\end{aligned}$$ Then $$\begin{aligned}
\left|{{\rm Tr}}[H_{ijkm} (\rho' - \rho_{\star})]\right| &= \left|\sum_{xy} h_{xy}r_{yx}\right| \nonumber\\
&\leq \sum_x \left|\sum_y h_{xy}r_{yx}\right| \nonumber\\
&\leq \sum_x \left(\sum_y |h_{xy}|^2\right)^{\frac{1}{2}}\left(\sum_z |r_{zx}|^2\right)^{\frac{1}{2}} .
\label{Heq}\end{aligned}$$ Next we show that $$\begin{aligned}
\langle \psi_x| H_{ijkm} H_{ijkm} ^{\dagger} |\psi_x\rangle = \sum_y |h_{xy}|^2 \leq 1.
\label{hdiag}\end{aligned}$$ The operator product can be written $$\begin{aligned}
H_{ijkm}H^{\dagger}_{ijkm} = F_{ijmmk}F^{\dagger}_{ijmmk} + F_{ijkkm}F^{\dagger}_{ijkkm}.
\label{opprod}\end{aligned}$$ We also have $$\begin{aligned}
\delta_{jj'} \mathbb{1} = \sum_{ikm}F_{ijkkm} F_{ij'kkm}^{\dagger}.
\label{pos2}\end{aligned}$$ Since (\[opprod\]) is the sum of two constituent positive terms of the sum of positive operators (\[pos2\]) then the same argument as that used in Lemma \[Lem3\] can be used to show that (\[hdiag\]) holds. It follows from (\[Heq\]) that $$\begin{aligned}
\nonumber
\left|{{\rm Tr}}[H_{ijkm} (\rho' - \rho_{\star})]\right| \leq \sum_x \left(\sum_z |r_{zx}|^2\right)^{\frac{1}{2}},\end{aligned}$$ and if we choose the basis states $\{|\psi_x\rangle\}$ such that $(\rho' - \rho_{\star})$ is diagonal we have $$\begin{aligned}
\nonumber
\left|{{\rm Tr}}[ H_{ijkm} (\rho' - \rho_{\star})]\right| \leq ||\rho' - \rho_{\star}||_1.\end{aligned}$$ Inserting into (\[l2b\]) we have $$\begin{aligned}
\left|\lambda_i - \lambda_j\right|\left| q(kk|ij)\right| &\leq \sum_{m\neq k}
\left|{{\rm Tr}}[H_{ijkm}(\rho'-\rho_{\star})]\right|
\nonumber\\
&\leq (d-1) ||\rho'-\rho_{\star}||_1,\nonumber\end{aligned}$$ and choosing $d=2$ completes the proof.
Note that Lemma \[Lem3\] can also be applied to the case of $k=l$ and $i\neq j$ for general $d$ with the result $$\begin{aligned}
\nonumber
||\rho'-\rho_{\star}||_1 \geq \frac{|\lambda_i - \lambda_j|}{2} |q(kk|ij)|,\end{aligned}$$ which is half the size of the bound of Lemma \[Lem2\] in the case of $d=2$.
Proof of Theorem \[th1\] {#secPROOF}
========================
Theorem \[th1\] is stated in the main text. We repeat it here for convenience
If there exists a stochastic map, $P_Q(k|i)$ with $$\begin{aligned}
\nonumber
\sum_k P_Q(k|i) = 1 \quad \text{and} \quad P_Q(k|i) \geq 0 \; \forall i,k,\end{aligned}$$ such that $$\begin{aligned}
\mu^{n\prime}_{kl} = \delta_{kl} \sum_i P_Q(k|i) \mu^n_{ii},
\label{revconSI}\end{aligned}$$ for all $n$, then the minimum thermodynamic cost of the operation $$\begin{aligned}
\Delta E_Q \geq kT \ln 2 \left[ S(\rho_{\rm S}) - S(\rho'_{\rm S})\right]+\epsilon_Q,
\nonumber\end{aligned}$$ can approach $\epsilon_Q=0$. Otherwise, provided there are no symmetries of the form $\lambda_i/\lambda_j=\lambda'_k/\lambda'_l$ where $i\neq j$ or $k\neq l$, then necessarily $\epsilon_Q>0$.
In general the output state of the quantum operation is given by (\[indiv\]) $$\begin{aligned}
\mu^{n\prime}_{kl} = \sum_{ij} \mu^n_{ij}q(kl|ij).
\label{EQrewrite1}\end{aligned}$$ By Corollary \[C0\] the thermodynamic bound requires that $||\rho'-\rho_{\star}||_1 = 0$. By Lemma \[Lem3\] this requires that if $|\lambda_j\lambda'_k - \lambda_i\lambda'_l|$ is non zero then $$\begin{aligned}
q(kl|ij) = 0.
\nonumber\end{aligned}$$ It is helpful to separate Equation (\[EQrewrite1\]) into three terms: $$\begin{aligned}
\mu^{n\prime}_{kl}=&\delta_{kl} \sum_i q(kk|ii) \mu^n_{ii} + \delta_{kl} \sum_{i,j\neq i} q(kk|ij) \mu^n_{ij}
\nonumber\\
&+ (1-\delta_{kl})\sum_{ij} q(kl|ij) \mu^n_{ij}.
\label{3split}\end{aligned}$$ The first and second terms only deal with the diagonal elements of the output signal states in the basis of the average output density matrix. The third term deals with the off-diagonal elements.
By Lemma \[Lem3\], the first sum implies only $|| \rho'-\rho_\star||_1\geq 0$, as $\lambda_i\lambda'_k - \lambda_i\lambda'_k=0$ for $q(kk|ii)$. If Equation (\[revconSI\]) holds then $P_Q(k|i)=q(kk|ii)$ gives an implementation in which all other terms are zero, and there is no excess cost. The numbers $q(kk|ii)$ are real, non negative and satisfy $$\begin{aligned}
\sum_k q(kk|ii) = \sum_k {{\rm Tr}}[A_{ki}^{\dagger} A_{ki}\rho_{\rm AB}] = 1,
\nonumber\end{aligned}$$ by Equation (\[g5\]).
If Equation (\[revconSI\]) does not hold, then some values of $q(kl|ij)$ with $i\neq j$ or $k\neq l$ must be non-zero, in any implementation. For a given implementation, the largest value of the lower bound of Equation (\[lem2EQ\]) across all $ijkl$ determines an excess thermodynamic cost. If there are no symmetries of the form $\lambda_i/\lambda_j=\lambda'_k/\lambda'_l$ with $i\neq j$ or $k\neq l$, then this lower bound must be greater than zero. The minimum such value across all possible implementations gives a lower bound for $\epsilon_Q>0$.
It should be noted that the existence of some values of $ijkl$ for which $\lambda_i/\lambda_j=\lambda'_k/\lambda'_l$ holds is not sufficient for $\epsilon_Q=0$. To avoid an excess cost from our Theorem through the exceptional symmetry condition, it must be possible to implement the operation in such a way that $q(kl|ij)=0$ for every $ijkl$ value for which $\lambda_i/\lambda_j \neq \lambda'_k/\lambda'_l$.
When the output signal states do not share a common diagonalisation, we can give a general lower bound for $\epsilon_Q$. The third sum in Equation (\[3split\]) is non-zero whenever there exists $k \neq l$ for which $\mu^{n\prime}_{kl}>0$. For these cases, from Equation (\[EQrewrite1\]), using the triangle inequality $$\begin{aligned}
|\mu^{n\prime}_{kl}| \leq \sum_{ij} |q(kl|ij)|| \mu^n_{ij}|.
\nonumber\end{aligned}$$ From Equation (\[lem2EQ\]) this implies $$\begin{aligned}
|| \rho'-\rho_{\star}||_1 \geq
\max_{n,k,l\neq k}\left\{
\frac
{|\mu^{n\prime}_{kl}|/(\lambda'_k+\lambda'_l)}
{\sum_{ij} |\mu^n_{ij}|/|\lambda_j\lambda'_k - \lambda_i\lambda'_l|}
\right\}>0,
\nonumber\end{aligned}$$ where the maximum is taken across all $k$ and $l$ such that $k\neq l$, and across all inputs $n$. As the right hand side only involves terms independent of the specific implementation, then this bound must hold for all implementations and so by Corollary \[C0\], $$\begin{aligned}
\epsilon_Q \geq \frac{1}{2}kT
\max_{n,k,l\neq k}\left\{
\frac
{|\mu^{n\prime}_{kl}|/(\lambda'_k+\lambda'_l)}
{\sum_{ij} |\mu^n_{ij}|/|\lambda_j\lambda'_k - \lambda_i\lambda'_l|}
\right\}^2. \nonumber\end{aligned}$$ If all $\mu^{n\prime}_{kl}=0$, with $k \neq l$, but Equation (\[revconSI\]) does not hold, then $$\begin{aligned}
\mu^{n\prime}_{kl}=\delta_{kl} \sum_i q(kk|ii) \mu^n_{ii} + \delta_{kl} \sum_{i,j\neq i} q(kk|ij) \mu^n_{ij},
\label{EQdiag}\end{aligned}$$ and any implementation must have values of $|q(kk|ij)| > 0 $ with $i \neq j$. We can find complex coefficients $q(kk|ij)$, which satisfy Equation (\[EQdiag\]), while minimising the largest value of the lower bound of (\[lem2EQ\]), $\frac{1}{2}|\lambda_i - \lambda_j| \left| q(kk|ij) \right|$. This must be done using numerical optimisation techniques on a case-by-case basis. Given the set $\{\{\lambda_i\},\{\lambda'_k\},\{\mu_{ij}^n\},\{\mu_{kl}^{n\prime}\}\}$, this establishes a minimum value for $|| \rho'-\rho_{\star}||_1$ and therefore, by Corollary \[C0\], a minimum excess cost across all possible implementations, so again $\epsilon_Q>0$.
Thermodynamically reversible protocol for special cases {#proto}
=======================================================
We now demonstrate a protocol which can achieve $\epsilon_Q=0$ for the special class of quantum operations which satisfy Equation (\[revconSI\]). For these operations there exists a stochastic map $P_Q(k|i)$ such that the input and output states of a quantum operation are related by $\mu^{n\prime}_{kl} = \delta_{kl} \sum_{i} P_Q(k|i) \mu^n_{ii}$.
Let the initial state of the auxiliary be ${{{\ensuremath{\left| {{0_{\rm A}}} \right\rangle}}{\ensuremath{\left. {\ensuremath{\left\langle {{0_{\rm A}}} \right.}} \right|}}}}$, so the combined system and auxiliary is initially $$\begin{aligned}
\rho_{\rm S}^n\otimes\rho_{\rm A}=\sum_{ij} \mu^n_{ij}{{\ensuremath{\left| {\phi_i} \right\rangle}}{\ensuremath{\left. {\ensuremath{\left\langle {\phi_j} \right.}} \right|}}}\otimes{{{\ensuremath{\left| {{0_{\rm A}}} \right\rangle}}{\ensuremath{\left. {\ensuremath{\left\langle {{0_{\rm A}}} \right.}} \right|}}}}.
\nonumber\end{aligned}$$
[*Step 1*]{}. Use the auxiliary to measure the input system in the basis of the average input density matrix. Correlate the auxiliary to the system using a unitary with ${\ensuremath{\left| \phi_i \right\rangle}}{\ensuremath{\left| 0_{\rm A} \right\rangle}} \rightarrow {\ensuremath{\left| \phi_i \right\rangle}}{\ensuremath{\left| i_{\rm A} \right\rangle}}$. The joint state is now $$\begin{aligned}
\sum_{ij} \mu^n_{ij} {{\ensuremath{\left| {\phi_i} \right\rangle}}{\ensuremath{\left. {\ensuremath{\left\langle {\phi_j} \right.}} \right|}}}\otimes{{\ensuremath{\left| {i_{\rm A}} \right\rangle}}{\ensuremath{\left. {\ensuremath{\left\langle {j_{\rm A}} \right.}} \right|}}}.
\nonumber\end{aligned}$$
[*Step 2*]{}. The operation $T_{{{{\ensuremath{\left| {{\phi_i}} \right\rangle}}{\ensuremath{\left. {\ensuremath{\left\langle {{\phi_i}} \right.}} \right|}}}}\rightarrow \rho^{(i)}_{\rm S}}$ (see Appendix \[proto\]) performs a thermodynamically optimal conversion of ${{{\ensuremath{\left| {{\phi_i}} \right\rangle}}{\ensuremath{\left. {\ensuremath{\left\langle {{\phi_i}} \right.}} \right|}}}}$ to $\rho^{(i)}_{\rm S}=\sum_k P_Q(k|i) {{{\ensuremath{\left| {{\phi_k}} \right\rangle}}{\ensuremath{\left. {\ensuremath{\left\langle {{\phi_k}} \right.}} \right|}}}}$. The conditional operation $\sum_i T_{{{{\ensuremath{\left| {{\phi_i}} \right\rangle}}{\ensuremath{\left. {\ensuremath{\left\langle {{\phi_i}} \right.}} \right|}}}} \rightarrow \rho^{(i)}_{\rm S}} \otimes {{{\ensuremath{\left| {{i_{\rm A}}} \right\rangle}}{\ensuremath{\left. {\ensuremath{\left\langle {{i_{\rm A}}} \right.}} \right|}}}} $ puts the joint system in the state $$\begin{aligned}
\sum_{ki}P_Q(k|i)\mu^n_{ii} {{{\ensuremath{\left| {{\phi_k}} \right\rangle}}{\ensuremath{\left. {\ensuremath{\left\langle {{\phi_k}} \right.}} \right|}}}} \otimes {{{\ensuremath{\left| {{i_{\rm A}}} \right\rangle}}{\ensuremath{\left. {\ensuremath{\left\langle {{i_{\rm A}}} \right.}} \right|}}}},
\nonumber\end{aligned}$$ transferring mean heat to the heat bath $\Delta E_{1}=-k T \ln 2 \sum_{ni} p_n \mu^n_{ii} S(\rho^{(i)}_{\rm S})$. The average state takes the form $$\begin{aligned}
\sum_{ki}P_Q(k|i) \lambda_i {{{\ensuremath{\left| {{\phi_k}} \right\rangle}}{\ensuremath{\left. {\ensuremath{\left\langle {{\phi_k}} \right.}} \right|}}}} \otimes {{{\ensuremath{\left| {{i_{\rm A}}} \right\rangle}}{\ensuremath{\left. {\ensuremath{\left\langle {{i_{\rm A}}} \right.}} \right|}}}}.
\nonumber\end{aligned}$$
[*Step 3*]{}. We now exploit the correlations between system and auxiliary to optimally reset the auxiliary. The operation $T_{\rho^{(k)}_{\rm A}\rightarrow {{{\ensuremath{\left| {{0_{\rm A}}} \right\rangle}}{\ensuremath{\left. {\ensuremath{\left\langle {{0_{\rm A}}} \right.}} \right|}}}}}$ performs an optimal conversion of $\rho^{(k)}_{\rm A}=\sum_{i} P_Q(k|i)\lambda_i/\left(\sum_j P_Q(k|j)\lambda_j\right) {{{\ensuremath{\left| {{i_{\rm A}}} \right\rangle}}{\ensuremath{\left. {\ensuremath{\left\langle {{i_{\rm A}}} \right.}} \right|}}}}$ to ${{{\ensuremath{\left| {{0_{\rm A}}} \right\rangle}}{\ensuremath{\left. {\ensuremath{\left\langle {{0_{\rm A}}} \right.}} \right|}}}}$, where $P_Q(k|i)\lambda_i/\left(\sum_j P_Q(k|j)\lambda_j\right)$ is the mean conditional probability of the auxiliary state being $|i_{\rm A}\rangle$ given that the system state is $|\phi_k\rangle$. The conditional operation $\sum_k {{{\ensuremath{\left| {{\phi_k}} \right\rangle}}{\ensuremath{\left. {\ensuremath{\left\langle {{\phi_k}} \right.}} \right|}}}} \otimes T_{\rho^{(k)}_{\rm A}\rightarrow {{{\ensuremath{\left| {{0_{\rm A}}} \right\rangle}}{\ensuremath{\left. {\ensuremath{\left\langle {{0_{\rm A}}} \right.}} \right|}}}}}$ resets the auxiliary to ${{{\ensuremath{\left| {{0_{\rm A}}} \right\rangle}}{\ensuremath{\left. {\ensuremath{\left\langle {{0_{\rm A}}} \right.}} \right|}}}}$ while transferring mean heat $\Delta E_{2}=k T \ln 2 \sum_{ki} P_Q(k|i)\lambda_i S(\rho^{(k)}_{\rm A})$.
The joint state is now $$\begin{aligned}
\sum_{ki}P_Q(k|i)\mu^n_{ii} {{{\ensuremath{\left| {{\phi_k}} \right\rangle}}{\ensuremath{\left. {\ensuremath{\left\langle {{\phi_k}} \right.}} \right|}}}} \otimes {{{\ensuremath{\left| {{0_{\rm A}}} \right\rangle}}{\ensuremath{\left. {\ensuremath{\left\langle {{0_{\rm A}}} \right.}} \right|}}}},
\nonumber\end{aligned}$$ and provided the conditions of the theorem hold, the system is in the correct output state, up to a unitary. The average heat production is readily confirmed to be $\Delta E_Q =\Delta E_1 +\Delta E_2 = k T \ln 2 [S(\rho_{\rm S})-S(\rho_{\rm S}')]$ thus proving that $\epsilon_Q = 0$. It is important to note that, if the conditions of the theorem do not apply, this protocol not only fails to be optimal, but fails to correctly implement the operation at all, i.e. $\rho_{\rm S}^n$ is not mapped to the correct $\rho^{n\prime}_{\rm S}$.
Qubit dephasing example {#ADP}
=======================
We analyse the specific example of a qubit dephasing operation. This can be implemented using a CNOT. For this operation an input of the form $\rho^n_{\rm S} = |v^n_{\rm S}\rangle\langle v^n_{\rm S}|$ with $|v^n_{\rm S}\rangle = c^n_0|0\rangle + c^n_1|1\rangle$ gives the output $\rho^{n\prime}_{\rm S} = |c^n_0|^2|0\rangle\langle 0 | +|c^n_1|^2|1\rangle\langle 1|$. All outputs are simultaneously diagonalisable in the $\{|0\rangle,|1\rangle\}$ basis. We choose two pure state inputs $\rho^1_{\rm S}$ and $\rho^2_{\rm S}$ such that the average input state is $\rho_{\rm S} = p\rho^1_{\rm S} +(1-p) \rho^2_{\rm S}$ for some probability $p$.
All of the elements $\mu^1_{ij},\mu^2_{ij},\mu^{1\prime}_{ij},\mu^{2\prime}_{ij}$ are specified by the operation together with the form and relative frequency of the inputs. To find $\mu^1_{ij},\mu^2_{ij}$ we determine the diagonal basis of $\rho_{\rm S}$ and express the individual inputs in this basis $$\begin{aligned}
\rho^1_{\rm S} =& \mu^1_{11}|\phi_1\rangle\langle \phi_1|+\mu^1_{12}|\phi_1\rangle\langle \phi_2| \nonumber\\
&\quad+\mu^1_{21}|\phi_2\rangle\langle \phi_1|+\mu^1_{22}|\phi_2\rangle\langle \phi_2|, \nonumber \\
\rho^2_{\rm S} =& \mu^2_{11}|\phi_1\rangle\langle \phi_1|+\mu^2_{12}|\phi_1\rangle\langle \phi_2| \nonumber \\
&\quad+\mu^2_{21}|\phi_2\rangle\langle \phi_1|+\mu^2_{22}|\phi_2\rangle\langle \phi_2|.
\nonumber\end{aligned}$$ The output elements are $$\begin{aligned}
\mu^{n\prime}_{11} &= |c^n_0|^2, \nonumber\\
\mu^{n\prime}_{22} &= |c^n_1|^2, \nonumber\\
\mu^{n\prime}_{kl} & = 0 \text{ for } k\neq l.\nonumber\end{aligned}$$ The elements must satisfy Equation (\[EQrewrite1\]) for some set of coefficients $q(kl|ij)$. The off-diagonal elements of the outputs can be solved by choosing $q(kl|ij) = 0$ for $k\neq l$, implying no constraint on excess heat cost by Lemma \[Lem3\]. For the diagonal elements of the outputs we denote $q(kk|ii) = q_{ki}$, and $q(11|12) =w = -q(22|12)$, $q(11|21) = w^* = -q(22|21)$, so that from Equation (\[EQrewrite1\]) we can write $$\begin{aligned}
\mu^{1\prime}_{11} &= |c^1_0|^2 = \mu^1_{11} q_{11} + \mu^1_{12}w + \mu^1_{21}w^* + \mu^1_{22}q_{12}, \nonumber \\
\mu^{2\prime}_{11} &= |c^2_0|^2 = \mu^2_{11} q_{11} + \mu^2_{12}w + \mu^2_{21}w^* + \mu^2_{22}q_{12} , \nonumber\end{aligned}$$ and $$\begin{aligned}
\mu^{1\prime}_{22} &= |c^1_1|^2 = \mu^1_{11} q_{21} - \mu^1_{12}w - \mu^1_{21}w^* + \mu^1_{22}q_{22}, \nonumber\\
\mu^{2\prime}_{22} &= |c^2_1|^2 = \mu^2_{11} q_{21} - \mu^2_{12}w - \mu^2_{21}w^* + \mu^2_{22}q_{22} .
\nonumber\end{aligned}$$ Since $|c^n_0|^2+|c^n_1|^2 = 1$, $\mu^n_{11}+\mu^n_{22} = 1$, and $q_{1k} +q_{2k}=1$, these pairs of equations are equivalent, so we focus on the first pair. The remaining numerical problem is to find the $w$ with the smallest magnitude which can solve this pair of equations with $0\leq q_{11}, q_{12} \leq 1$. This is done by scanning complex values of $w$ of increasing magnitude and solving for $q_{11}$ and $q_{12}$ until they both fit within the required range. Once we have found $|w|_{min}$ we can use Lemma \[Lem2\] to bound $||\rho'-\rho_{\star}||_1$: $$\begin{aligned}
||\rho'-\rho_{\star}||_1\geq |\lambda_1-\lambda_2| |w|_{min},
\nonumber\end{aligned}$$ where $\lambda_i$ are the eigenvalues of $\rho_{\rm S}$. Finally we can use Equation (\[Eresult\]) to bound the energy cost: $$\begin{aligned}
\Delta E_{Q_0} \geq kT\ln 2\left[ S(\rho_{\rm S}) - S(\rho'_{\rm S})\right] + \frac{1}{2} kT (\lambda_1-\lambda_2)^2 |w|^2_{min}.
\nonumber\end{aligned}$$ Examples are given in Figure \[F1\].
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abstract: |
In this paper, we study the existence of minimizers to the following functional related to the nonlinear Choquard equation: $$E(u)=\frac{1}{2}{\displaystyle}\int_{{{\mathbb R}}^N}|\nabla u|^2+\frac{1}{2}{\displaystyle}\int_{{{\mathbb R}}^N}V(x)|u|^2-\frac{1}{2p}{\displaystyle}\int_{{{\mathbb R}}^N}(I_{\alpha}*|u|^p)|u|^p$$ on $\widetilde{S}(c)=\{u\in H^1({{\mathbb R}}^N)|\ \int_{{{\mathbb R}}^N}V(x)|u|^2<+\infty,\ |u|_2=c,c>0\},$ where $N\geq1$ ${\alpha}\in(0,N)$, $\frac{N+\alpha}{N}\leq p<\frac{N+\alpha}{(N-2)_+}$ and $I_{\alpha}:{{\mathbb R}}^N\rightarrow{{\mathbb R}}$ is the Riesz potential. We present sharp existence results for $E(u)$ constrained on $\widetilde{S}(c)$ when $V(x)\equiv0$ for all $\frac{N+\alpha}{N}\leq p<\frac{N+\alpha}{(N-2)_+}$. For the mass critical case $p=\frac{N+\alpha+2}{N}$, we show that if $0\leq V(x)\in L_{loc}^{\infty}({{\mathbb R}}^N)$ and $\lim\limits_{|x|\rightarrow+\infty}V(x)=+\infty$, then mass minimizers exist only if $0<c<c_*=|Q|_2$ and concentrate at the flattest minimum of $V$ as $c$ approaches $c_*$ from below, where $Q$ is a groundstate solution of $-\Delta u+u=(I_\alpha*|u|^{\frac{N+\alpha+2}{N}})|u|^{\frac{N+\alpha+2}{N}-2}u$ in ${{\mathbb R}}^N$.\
[**Keywords:**]{} Choquard equation; Mass concentration; Normalized solutions; Sharp existence.\
[**Mathematics Subject Classification(2010):**]{} 35J60, 35Q40, 46N50\
address: '$^*$ H. Y. Ye, College of Science, Wuhan University of Science and Technology, Wuhan, 430065, P. R. China'
author:
- Hong yu Ye
title: 'Mass minimizers and concentration for nonlinear Choquard equations in ${{\mathbb R}}^N$'
---
[^1]
Introduction
============
In this paper, we consider the following semilinear Choquard problem $$\label{1.1}
-\Delta u-\mu u=(I_{\alpha}*|u|^{p})|u|^{p-2}u,\ \ \ x\in{{\mathbb R}}^N,\ \mu\in{{\mathbb R}}$$ where $N\geq 1,$ ${\alpha}\in(0,N)$, $\frac{N+\alpha}{N}\leq p<\frac{N+\alpha}{(N-2)_+}$, here $\frac{N+\alpha}{(N-2)_+}=\frac{N+\alpha}{N-2}$ if $N\geq3$ and $\frac{N+\alpha}{(N-2)_+}=+\infty$ if $N=1,2$. $I_{\alpha}:{{\mathbb R}}^N\rightarrow{{\mathbb R}}$ is the Riesz potential [@r] defined as $$I_{\alpha}(x)=\frac{\Gamma(\frac{N-{\alpha}}{2})}{\Gamma(\frac{\alpha}{2})\pi^{\frac{N}{2}}2^{\alpha}}\frac{1}{|x|^{N-{\alpha}}},\ \ \ \forall\ x\in{{\mathbb R}}^N\backslash\{0\}.$$
Problem is a nonlocal one due to the existence of the nonlocal nonlinearity. It arises in various fields of mathematical physics, such as quantum mechanics, physics of laser beams, the physics of multiple-particle systems, etc. When $N=3$, $\mu=-1$ and ${\alpha}=p=2$, turns to be the well-known Choquard-Pekar equation: $$\label{1.2}
-\Delta u+u=(I_2*|u|^2)u,\ \ \ \ x\in{{\mathbb R}}^3,$$ which was proposed as early as in 1954 by Pekar [@pekar], and by a work of Choquard 1976 in a certain approximation to Hartree-Fock theory for one-component plasma, see [@lieb; @ls]. is also known as the nonlinear stationary Hartree equation since if $u$ solves then $\psi(t,x)=e^{it}u(x)$ is a solitary wave of the following time-dependent Hartree equation $$i\psi_t=-\Delta \psi-(I_\alpha*|\psi|^p)|\psi|^{p-2}\psi\ \ \ \hbox{in}\ {{\mathbb R}}^+\times{{\mathbb R}}^N,$$ see [@gv; @mpt].
In the past years, there are several approaches to construct nontrivial solutions of , see e.g. [@csv; @lieb; @lions; @l; @m; @mpt; @tm] for $p=2$ and [@ms1; @ms2]. One of them is to look for a constrained critical point of the functional $$\label{1.3}
I_p(u)=\frac12{\displaystyle}\int_{{{\mathbb R}}^N}|\nabla u|^2-\frac{1}{2p}{\displaystyle}\int_{{{\mathbb R}}^N}(I_\alpha*|u|^p)|u|^p$$ on the constrained $L^2$-spheres in $H^1({{\mathbb R}}^N)$: $$S(c)=\{u\in H^1({{\mathbb R}}^N)|\ |u|_2=c,c>0\}.$$ In this way, the parameter $\mu\in{{\mathbb R}}$ will appear as a Lagrange multiplier and such solution is called a normalized solution. By the following well known Hardy-Littlewood-Sobolev inequality: For $1<r,s<+\infty$, if $f\in L^r({{\mathbb R}}^N),$ $g\in L^s({{\mathbb R}}^N)$, $\lambda\in(0,N)$ and $\frac{1}{r}+\frac1s+\frac{{\lambda}}{N}=2$, then $$\label{1.20}
{\displaystyle}\int_{{{\mathbb R}}^N}{\displaystyle}\int_{{{\mathbb R}}^N}\frac{f(x)g(y)}{|x-y|^{\lambda}}\leq C_{r,{\lambda},N}|f|_r|g|_s,$$ we see that $I_p(u)$ is well defined and a $C^1$ functional. Set $$\label{1.4}
I_p(c^2)=\inf\limits_{u\in S(c)}I_p(u),$$ then minimizers of $I_p(c^2)$ are exactly critical points of $I_p(u)$ constrained on $S(c)$.
Normalized solutions for equation have been studied in [@lieb; @lions]. In this paper, one of our purposes is to get a general and sharp result for the existence of minimizers for the minimization problem .
To state our main result, we first prove the following interpolation inequality with the best constant: For $\frac{N+\alpha}{N}<p<\frac{N+\alpha}{(N-2)_+}$, $$\label{1.5}
{\displaystyle}\int_{{{\mathbb R}}^N}(I_\alpha*|u|^p)|u|^p \leq \frac{p}{|Q_p|_2^{2p-2}}\left({\displaystyle}\int_{{{\mathbb R}}^N}|\nabla u|^2\right)^{\frac{Np-(N+\alpha)}{2}}\left({\displaystyle}\int_{{{\mathbb R}}^N}|u|^2\right)^{\frac{N+\alpha-(N-2)p}{2}},$$ where equality holds for $u=Q_p$, where $Q_p$ is a nontrivial solution of $$\label{1.6}
-\frac{Np-(N+\alpha)}{2}\Delta Q_p+\frac{N+\alpha-(N-2)p}{2}Q_p=(I_\alpha*|Q_p|^p)|Q_p|^{p-2}Q_p,\ \ \ x\in\ {{\mathbb R}}^N.$$ In particular, $Q_{\frac{N+\alpha+2}{N}}$ is a groundstate solution, i.e. the least energy solution among all nontrivial solutions of . Moreover, when $p=\frac{N+\alpha+2}{N}$, all groundstate solutions of have the same $L^2$-norm (see Lemma \[lem3.2\] below).
Recall in [@lieb2] that for $p=\frac{N+\alpha}{N}$, the following Hardy-Littlewood-Sobolev inequality with the best constant: $$\label{1.12}
{\displaystyle}\int_{{{\mathbb R}}^N}(I_\alpha*|u|^{\frac{N+\alpha}{N}})|u|^{\frac{N+\alpha}{N}} \leq \frac{1}{|Q_{\frac{N+{\alpha}}{N}}|_2^{\frac{2(N+\alpha)}{N}}}\left({\displaystyle}\int_{{{\mathbb R}}^N}|u|^2\right)^{\frac{N+\alpha}{N}}$$ with equality if and only if $u=Q_{\frac{N+{\alpha}}{N}}$, where $Q_{\frac{N+{\alpha}}{N}}=C\left(\frac{\eta}{\eta^2+|x-a|^2}\right)^{\frac{N}{2}},$ $C>0$ is a fixed constant, $a\in{{\mathbb R}}^N$ and $\eta\in(0,+\infty)$ are parameters.\
Then our first result is as follows:
\[th1.1\] Assume that $N\geq1$, $\alpha\in(0,N)$ and $\frac{N+\alpha}{N}\leq p<\frac{N+\alpha}{(N-2)_+}$.
$(1)$ If $p=\frac{N+{\alpha}}{N}$, for any $c>0$,$$I_{\frac{N+{\alpha}}{N}}(c^2)=-\frac{N}{2(N+{\alpha})}(\frac{c}{|Q_{\frac{N+{\alpha}}{N}}|_2})^{\frac{2(N+{\alpha})}{N}}$$ and $I_{\frac{N+{\alpha}}{N}}(c^2)$ has no minimizer.
$(2)$ If $\frac{N+\alpha}{N}<p<\frac{N+\alpha+2}{N}$, then $I_p(c^2)<0$ for all $c>0$, moreover, $I_p(c^2)$ has at least one minimizer for each $c>0$.
$(3)$ If $p=\frac{N+\alpha+2}{N}$, let $c_*:=|Q_{\frac{N+\alpha+2}{N}}|_2$, then
(i) $I_{\frac{N+\alpha+2}{N}}(c^2)=\left\{\begin{array}{ll}
0, & \hbox{if~$0<c\leq c_*$}, \\
-\infty, & \hbox{if~$c>c_*$};\\
\end{array}\right.$
(ii) $I_{\frac{N+\alpha+2}{N}}(c^2)$ has no minimizer if $c\neq c_*$;
(iii) each groundstate of is a minimizer of $I_{\frac{N+\alpha+2}{N}}(c_*^2)$.
(iv) there is no critical point for $I_{\frac{N+\alpha+2}{N}}(u)$ constrained on $S(c)$ for each $0<c<c_*$.
$(4)$ If $\frac{N+\alpha+2}{N}<p<\frac{N+\alpha}{(N-2)_+}$, then $I_p(c^2)$ has no minimizer for each $c>0$ and $I_p(c^2)=-\infty$.
\[rem1.2\] Theorem \[th1.1\] can be seemed as a consequence of the results in Theorem 9 of [@lieb] for $p=2$ and in Theorem 1 of [@ms1]. However, we still state and prove Theorem \[th1.1\] here by using an alternative method since our result is delicate and it provides a framework to our subsequent main considerations.
\[rem1.3\]
$(1)$ $c_*$ is unique.
(2) Since the positive solution of with $\alpha=p=2$ is uniquely determined up to translations see e.g. [@cg; @klr; @ll], it follows that if $N=4$ and $\alpha=2$, then up to translations, **the minimizer of $I_{\frac{N+\alpha+2}{N}}(c_*^2)$ is unique and there exists no critical point for $I_{\frac{N+\alpha+2}{N}}(u)$ constrained on ${S(c)}$ for each $c\neq c_*$.**
(3) For $N\geq3$ and $\frac{N+\alpha+2}{N}<p<\frac{N+\alpha}{N-2}$, it has been proved in [@ly] that for each $c>0$, $I_p(u)$ has a mountain pass geometry on $S(c)$ and there exits a couple $(u_c,\mu_c)\in S(c)\times{{\mathbb R}}^-$ solution of with $I_p(u_c)=\gamma(c)$, where $\gamma(c)$ denotes the mountain pass level on $S(c)$.
By Theorem 1.1, $p=\frac{N+\alpha+2}{N}$ is called $L^2$-critical exponent for . In order to get critical points under the mass constraint for such $L^2$-critical case, we add a nonnegative perturbation term to the right hand side of , i.e. considering the following functional: $$\label{1.7}
E(u)=\frac{1}{2}{\displaystyle}\int_{{{\mathbb R}}^N}|\nabla u|^2+\frac{1}{2}{\displaystyle}\int_{{{\mathbb R}}^N}V(x)|u|^2-\frac{N}{2(N+\alpha+2)}{\displaystyle}\int_{{{\mathbb R}}^N}(I_\alpha*|u|^{\frac{N+\alpha+2}{N}})|u|^{\frac{N+\alpha+2}{N}},$$ where $$V(x)\in L^{\infty}_{loc}({{\mathbb R}}^N),\ \ \inf\limits_{x\in{{\mathbb R}}^N}V(x)=0\ \ \hbox{and}\ \ \lim\limits_{|x|\rightarrow+\infty}V(x)=+\infty.\eqno{(V_0)}$$ Based on $(V_0)$, we introduce a Sobolev space $\mathcal{H}=\{u\in H^1({{\mathbb R}}^N)|\ \int_{{{\mathbb R}}^N}V(x)|u^2<+\infty\}$ with its associated norm $\|u\|_{\mathcal{H}}=(\int_{{{\mathbb R}}^N}(|\nabla u|^2+|u|^2+V(x)|u|^2))^{\frac12}.$
\[th1.2\] Assume that $N\geq1$, $\alpha\in(0,N)$ and $(V_0)$ holds. Set $$\label{1.8}
e_c=\inf\limits_{u\in \widetilde{S}(c)}E(u),$$ where $\widetilde{S}(c)=\{u\in \mathcal{H}|\ |u|_2=c\}.$ Let $c_*$ be given in Theorem \[th1.1\].
(1) If $0<c<c_*$, then $e_c$ has at least one minimizer and $e_c>0$;
(2) Let $N-2\leq \alpha<N$ if $N\geq3$ and $0<{\alpha}<N$ if $N=1,2$, then for each $c\geq c_*$, $e_c$ has no minimizer; Moreover, $e_c=\left\{\begin{array}{ll}
0, & \hbox{if~$c=c_*$} \\
-\infty, & \hbox{if~$c>c_*$}\\
\end{array}\right.$ and $\lim\limits_{c\rightarrow (c_*)^-}e_c=e_{c_*}.$
We also concern the concentration phenomena of minimizers of $e_c$ as $c$ converges to $c_*$ from below. Let $u_c$ be a minimizer of $e_c$ for each $0<c<c_*$, then by and Theorem \[th1.2\], we see that $\int_{{{\mathbb R}}^N}V(x)|u_c|^2\rightarrow0$ as $c\rightarrow (c_*)^-$, i.e. $u_c$ can be expected to concentrate at the minimum of $V(x)$. To show this fact, besides condition $(V_0)$, we assume that there exist $m\geq1$ distinct points $x_i\in{{\mathbb R}}^N$ and $q_i>0$ $(1\leq i\leq m)$ such that $$\mu_i:=\lim\limits_{x\rightarrow x_i}\frac{V(x)}{|x-x_i|^{q_i}}\in (0,+\infty).\eqno{(V_1)}$$ Set $$q:=\max\{q_1,q_2,\cdots,q_m\}.$$ Let $\{c_k\}\subset(0,c_*)$ be a sequence such that $c_k\rightarrow c_*$ as $k\rightarrow+\infty$. Then Our main result is as follows:
\[th1.3\] Suppose that $N\geq1$, $\alpha\in[N-2,N)$ if $N\geq3$ and ${\alpha}\in(0,N)$ if $N=1,2$ and $(V_0)(V_1)$ hold. Then there exists a sequence $\{x_k\}\subset{{\mathbb R}}^N$ and a groundstate solution $W_0$ of the following equation $$\label{1.9}
-\Delta W_0+W_0=(I_\alpha*|W_0|^{\frac{N+\alpha+2}{N}})|W_0|^{\frac{N+\alpha+2}{N}-2}W_0,\ \ \ x\in{{\mathbb R}}^N$$ and $$\lambda:=\min\limits_{1\leq i\leq m}\left\{\lambda_i|\ \lambda_i=\left(\frac{q_i}{2c_*^2}\mu_i{\displaystyle}\int_{{{\mathbb R}}^N}|x|^{q_i}|W_0(x)|^2\right)^{\frac{1}{q_i+2}}\right\}$$ such that up to a subsequence, $$\label{1.10} [1-(\frac{c_k}{c_*})^{\frac{2(\alpha+2)}{N}}]^{\frac{1}{q+2}\frac{N}{2}}u_{c_k}([1-(\frac{c_k}{c_*})^{\frac{2(\alpha+2)}{N}}]^{\frac{1}{q+2}} x+x_k)\rightarrow[(\frac{\alpha+2}{N})^{\frac{1}{q+2}}\lambda]^{\frac{N}{2}} W_0((\frac{\alpha+2}{N})^{\frac{1}{q+2}}\lambda x)$$ in $L^{\frac{2Ns}{N+\alpha}}({{\mathbb R}}^N)$ for $\frac{N+\alpha}{N}\leq s<\frac{N+\alpha}{(N-2)+}$ as $k\rightarrow+\infty$. Moreover, there exists $x_{j_0}\in \{x_i|~\lambda_i=\lambda,~1\leq i\leq m\}$ such that $x_k\rightarrow x_{j_0}$ as $k\rightarrow+\infty$.
\[rem1.1\] It has been proved in [@ms1] that for $\alpha\in[N-2,N)$ if $N\geq3$ and ${\alpha}\in(0,N)$ if $N=1,2$, then each groundstate solution $u$ of satisfies that $\lim\limits_{|x|\rightarrow+\infty} |u(x)||x|^{\frac{N-1}{2}}e^{|x|}\in(0,+\infty)$. Hence $\lambda_i\in(0,+\infty)$.
The result in Theorem \[th1.3\] is different from that in [@me] studying the case $p<\frac{N+\alpha+2}{N}$, where one considered the concentration behavior of minimizers as $c\rightarrow+\infty$. The concentration phenomena have also been studied in [@ms3] and [@css] by considering semiclassical limit of the Choquard equation $$-\varepsilon^2\Delta u+Vu=\varepsilon^{-\alpha}(I_\alpha*|u|^p)|u|^{p-2}u~~~~\ \ \ \hbox{in}~~{{\mathbb R}}^N.$$ However, since the parameter is different, we need a different technique to obtain our result.
The main proof of Theorem \[th1.3\] is based on optimal energy estimates of $e_c$ and $\int_{{{\mathbb R}}^N}|\nabla u_c|^2$ for each minimizer $u_c$. The main idea to prove Theorem \[th1.3\] comes from [@gs], which was restricted to the case of local nonlinearities. But due to the fact that our nonlinearity is nonlocal and that the assumption imposed on $(V)$ is more general than that in [@gs], the method used in [@gs] can not be directly applied here. It needs some improvements and careful analysis. First, by choosing a suitable test function, we get that $0<e_c\leq C_1[1-(\frac{c}{c_*})^{\frac{2(\alpha+2)}{N}}]^{\frac{q}{q+2}}$ as $c\rightarrow (c_*)^-$ for some constant $C_1>0$ independent of $c$. The lower bound now is not optimal. The method in [@gs] by using the perturbation term $\int_{{{\mathbb R}}^N}V(x)u^2$ to remove the local nonlinearity term does not work in our cases. To obtain an optimal lower bound, we notice that $\int_{{{\mathbb R}}^N}|\nabla u_c|^2\rightarrow+\infty$ as $c\rightarrow (c_*)^-$, moreover, $$\lim\limits_{c\rightarrow (c_*)^-}\frac{\frac{N}{N+\alpha+2}\int_{{{\mathbb R}}^N}(I_\alpha*|u_c|^{\frac{N+\alpha+2}{N}})|u_c|^{\frac{N+\alpha+2}{N}}}{\int_{{{\mathbb R}}^N}|\nabla u_c|^2}=1.$$ Then by taking a special $L^2$-preserving scaling as: $$\label{1.11}
w_c(x)=\varepsilon_c^{\frac{N}{2}}u_c(\varepsilon_c x+\varepsilon_cy_c),$$ where $$\varepsilon^2_c=\frac{2(N+\alpha+2)}{N\int_{{{\mathbb R}}^N}(I_\alpha*|u_c|^{\frac{N+\alpha+2}{N}})|u_c|^{\frac{N+\alpha+2}{N}}}\rightarrow0\ \ \ \hbox{as}\ c\rightarrow (c_*)^-$$ and the sequence $\{y_c\}$ is derived from the vanishing lemma, we succeeded in proving that there is a constant $C_2>0$ independent of $c$ such that $${\displaystyle}\int_{{{\mathbb R}}^N}V(\varepsilon_cx+\varepsilon_cy_c)|w_c(x)|^2\geq C_2\varepsilon_c^q\ \ \ \hbox{as}\ c\rightarrow (c_*)^-,$$ which and implies that $e_c\geq C_3[1-(\frac{c}{c_*})^{\frac{2(\alpha+2)}{N}}]^{\frac{q}{q+2}}$ for some constant $C_3>0$ independent of $c$. In succession, there exist two constants $0<C_4<C_5$ independent of $c$ such that $C_4[1-(\frac{c}{c_*})^{\frac{2(\alpha+2)}{N}}]^{\frac{-2}{q+2}}\leq \int_{{{\mathbb R}}^N}|\nabla u_c|^2\leq C_5[1-(\frac{c}{c_*})^{\frac{2(\alpha+2)}{N}}]^{\frac{-2}{q+2}}$. Finally, by using the Euler-Lagrange equation $u_c$ satisfied and the scaling again with $\varepsilon_c=[1-(\frac{c}{c_*})^{\frac{2(\alpha+2)}{N}}]^{\frac{1}{q+2}}$, we show that $$e_c\approx[1-(\frac{c}{c_*})^{\frac{2(\alpha+2)}{N}}]^{\frac{q}{q+2}}\frac{q+2}{q}\frac{\lambda^2c_*^2}{2}\left(\frac{N}{\alpha+2}\right)^{\frac{q}{q+2}}\ \ \hbox{as}\ c\rightarrow (c_*)^-,$$ which implies .
Throughout this paper, we use standard notations. For simplicity, we write $\int_{\Omega} h$ to mean the Lebesgue integral of $h(x)$ over a domain $\Omega\subset{{\mathbb R}}^N$. $L^{p}:= L^{p}({{\mathbb R}}^{N})~(1\leq
p<+\infty)$ is the usual Lebesgue space with the standard norm $|\cdot|_{p}.$ We use “ $\rightarrow"$ and “ $\rightharpoonup"$ to denote the strong and weak convergence in the related function space respectively. $C$ will denote a positive constant unless specified. We use “ $:="$ to denote definitions. We denote a subsequence of a sequence $\{u_n\}$ as $\{u_n\}$ to simplify the notation unless specified.
The paper is organized as follows. In Section 2, we will determine the best constant for the interpolation estimate and give the proof of Theorem \[th1.1\]. In section 3, we prove Theorems \[th1.2\] and \[th1.3\].
Proof of Theorem \[th1.1\]
===========================
In this section, we first prove the interpolation estimate . It is enough to consider the following minimization problem: $$S_p=\inf\limits_{u\in H^1({{\mathbb R}}^N)\backslash\{0\}}W_p(u),$$ where $$W_p(u)=\frac{\left(\int_{{{\mathbb R}}^N}|\nabla u|^2\right)^{\frac{Np-(N+\alpha)}{2}}\left(\int_{{{\mathbb R}}^N}|u|^2\right)^{\frac{N+\alpha-(N-2)p}{2}}}{\int_{{{\mathbb R}}^N}(I_\alpha*|u|^p)|u|^p}.$$
\[lem2.1\]([@ms1], Lemma 2.4) Let $N\geq1$, $\alpha\in (0,N)$, $p\in [1,\frac{2N}{N+\alpha})$ and $\{u_n\}$ be a bounded sequence in $L^{\frac{2Np}{N+\alpha}}({{\mathbb R}}^N)$. If $u_n\rightarrow u$ a.e. in ${{\mathbb R}}^N$ as $n\rightarrow+\infty$, then $$\lim\limits_{n\rightarrow+\infty}\left({\displaystyle}\int_{{{\mathbb R}}^N}(I_\alpha*|u_n|^p)|u_n|^p-{\displaystyle}\int_{{{\mathbb R}}^N}(I_\alpha*|u_n-u|^p)|u_n-u|^p\right)
={\displaystyle}\int_{{{\mathbb R}}^N}(I_\alpha*|u|^p)|u|^p.$$
\[lem2.6\] ([@wi], Vanishing Lemma) Let $r>0$ and $2\leq q<2^*$. If $\{u_n\}$ is bounded in $H^1({{\mathbb R}}^N)$ and $$\sup\limits_{y\in{{\mathbb R}}^N}{\displaystyle}\int_{B_r(y)}|u_n|^q\rightarrow0,~~n\rightarrow+\infty,$$ then $u_n\rightarrow0$ in $L^s({{\mathbb R}}^N)$ for $2<s<2^*$.
\[lem2.2\] Let $N\geq1$, $\alpha\in(0,N)$ and $\frac{N+\alpha}{N}<p<\frac{N+\alpha}{(N-2)_+}$, then $S_{p}$ is achieved by a function $Q_p\in H^1({{\mathbb R}}^N)\backslash\{0\}$, where $Q_p$ is a nontrivial solution of equation and $$S_{p}=\frac{|Q_p|_2^{2p-2}}{p}.$$
The lemma can be viewed as a consequence of Proposition 2.1 in [@ms1] and Theorem 9 in [@lieb], but we give an alternative proof here. The idea of the proof comes from [@we], but some details are delicate.
Since $W_{p}(u)\geq0$ for any $u\in H^1({{\mathbb R}}^N)\backslash\{0\}$, $S_{p}$ is well defined. Let $\{u_n\}\subset H^1({{\mathbb R}}^N)\backslash\{0\}$ be a minimizing sequence for $S_p$, i.e. $W_{p}(u_n)\rightarrow S_{p}$ as $n\rightarrow+\infty$. Set $$\lambda_n:=\frac{(\int_{{{\mathbb R}}^N}|u_n|^2)^{\frac{N-2}{4}}}{(\int_{{{\mathbb R}}^N}|\nabla u_n|^2)^{\frac{N}{4}}},\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \mu_n:=\frac{(\int_{{{\mathbb R}}^N}|u_n|^2)^{\frac12}}{(\int_{{{\mathbb R}}^N}|\nabla u_n|^2)^{\frac12}}$$ and $$v_n(x):=\lambda_nu_n(\mu_nx).$$ Then $\int_{{{\mathbb R}}^N}|v_n|^2=\int_{{{\mathbb R}}^N}|\nabla v_n|^2=1$ and $$\label{2.1}
W_{p}(v_n)=W_{p}(u_n)\rightarrow S_{p}\ \ \ \hbox{as}\ \ n\rightarrow+\infty,$$ i.e. $\{v_n\}$ is a bounded minimizing sequence for $S_{p}$.
Let $\delta:=\lim\limits_{n\rightarrow+\infty}\sup\limits_{y\in{{\mathbb R}}^N}\int_{B_1(y)}|v_n|^2.$ If $\delta=0$, then by Lemma \[lem2.6\], $v_n\rightarrow 0$ in $L^s({{\mathbb R}}^N)$, $2<s<2^*$. Hence by the Hardy-Littlewood-Sobolev inequality , $$W_{p}(v_n)=\frac{1}{\int_{{{\mathbb R}}^N}(I_\alpha*|v_n|^p)|v_n|^p}\rightarrow+\infty,$$ which contradicts . Therefore, $\delta>0$ and there exists a sequence $\{y_n\}\subset{{\mathbb R}}^N$ such that $$\label{2.2}
\int_{B_1(y_n)}|v_n|^2\geq\frac{\delta}{2}>0.$$ Up to translations, we may assume that $y_n=0$. Since $\{v_n\}$ is bounded in $H^1({{\mathbb R}}^N)$ and by , there exists $v_{p}\in H^1({{\mathbb R}}^N)\backslash\{0\}$ such that $v_n\rightharpoonup v_{p}$ in $H^1({{\mathbb R}}^N)$. Then by the Brezis Lemma and Lemma \[lem2.1\], we have $$\begin{array}{ll}
S_{p} &\leq W_{p}(v_{p})\\[5mm]
&\leq\lim\limits_{n\rightarrow+\infty}{\displaystyle}\left[W_{p}(v_n)\frac{\int_{{{\mathbb R}}^N}(I_\alpha*|v_n|^p)|v_n|^p}{\int_{{{\mathbb R}}^N}(I_\alpha*|v_{p}|^p)|v_{p}|^p}-
W_{p}(v_n-v_{p})\frac{\int_{{{\mathbb R}}^N}(I_\alpha*|v_n-v_{p}|^p)|v_n-v_{p}|^p}{\int_{{{\mathbb R}}^N}(I_\alpha*|v_p|^p)|v_{p}|^p}\right]\\[5mm]
&\leq S_{p} \lim\limits_{n\rightarrow+\infty}{\displaystyle}\left(\frac{\int_{{{\mathbb R}}^N}(I_\alpha*|v_n|^p)|v_n|^p-\int_{{{\mathbb R}}^N}(I_\alpha*|v_n-v_p|^p)|v_n-v_{p}|^p}{\int_{{{\mathbb R}}^N}(I_\alpha*|v_p|^p)|v_{p}|^p}\right)\\[5mm]
&=S_{p},
\end{array}$$ i.e. $W_{p}(v_{p})=S_{p}.$ Moreover, $|\nabla v_{p}|_2=|v_{p}|_2=1$ and $S_p=\frac{1}{\int_{{{\mathbb R}}^N}(I_\alpha*|v_{p}|^p)|v_{p}|^p}.$
Therefore, for any $h\in H^1({{\mathbb R}}^N)$, $\frac{d}{dt}\Big{|}_{t=0}W_{p}(v_p+th)=0$, i.e. $v_p$ satisfies the following equation $$-[Np-(N+\alpha)]\Delta v_p+[N+\alpha-(N-2)p]v_p=2pS_{p}(I_\alpha*|v|^p)|v_p|^{p-2}v_p,~~~\hbox{in}~~{{\mathbb R}}^N.$$ Let $v_p=(\frac{1}{pS_{p}})^{\frac{1}{2p-2}}Q_p$, then $Q_p$ is a nontrivial solution of and $S_{p}=\frac{|Q_p|_2^{2p-2}}{p}$.\
Next we give the proof of Theorem \[th1.1\]. For any $u\in S(c)$, set $$A(u):={\displaystyle}\int_{{{\mathbb R}}^N}|\nabla u|^2,\, \, \, \, \, \, \, \, \, \, B(u):={\displaystyle}\int_{{{\mathbb R}}^N}(I_\alpha*|u|^p)|u|^p,$$ then $I_p(u)=\frac{1}{2}A(u)-\frac{1}{2p}B(u).$ It follows from that for $\frac{N+{\alpha}}{N}<p<\frac{N+{\alpha}}{(N-2)_+}$, $$\label{2.4}
B(u)\leq \frac{p}{|Q_p|_2^{2p-2}}A(u)^{\frac{Np-(N+\alpha)}{2}}c^{N+\alpha-(N-2)p}$$ with equality for $u=Q_p$ given in , moreover, $$\label{2.5}
A(Q_p)=\frac{1}{p}B(Q_p)=|Q_p|_2^2.$$
\[lem2.3\] Let $N\geq1$ and ${\alpha}\in(0,N)$.
$(1)$ If $\frac{N+\alpha}{N}<p<\frac{N+\alpha+2}{N}$, then $I_p(u)$ is bounded from below and coercive on $S(c)$ for all $c>0$, moreover, $I_{p}(c^2)<0$.
$(2)$ If $p=\frac{N+\alpha+2}{N}$, then $I_{\frac{N+\alpha+2}{N}}(c^2)=\left\{\begin{array}{ll}
0, & 0<c\leq c_*:=|Q_{\frac{N+\alpha+2}{N}}|_2, \\
-\infty, & c>c_*,\\
\end{array}\right.$
$(3)$ If $\frac{N+\alpha+2}{N}<p<\frac{N+\alpha}{(N-2)_+}$, then $I_{p}(c^2)=-\infty$ for all $c>0$.
$(1)$ For any $c>0$ and $u\in S(c)$, by , there exists $C:=\frac{c^{N+\alpha-(N-2)p}}{|Q_p|_2^{2p-2}}$ such that $$\label{2.6}
I_p(u)\geq \frac{A(u)-
CA(u)^{\frac{Np-(N+\alpha)}{2}}}{2}.$$ Since $\frac{N+\alpha}{N}<p<\frac{N+\alpha+2}{N}$, $0<Np-(N+\alpha)<2$. Then implies that $I_p(u)$ is bounded from below and coercive on $S(c)$ for any $c>0$.
Set $u^t(x):=t^{\frac{N}{2}}u(tx)$ with $t>0$, then $u^t\in S(c)$ and $$\label{2.7}
I_p(u^t)=\frac{t^2}{2}A(u)-\frac{t^{Np-(N+\alpha)}}{2p}B(u)<0\ \ \ \ \ \hbox{for}\ t>0\ \hbox{small~enough}$$ since $0<Np-(N+\alpha)<2$, which implies that $I_{p}(c^2)<0$ for each $c>0$.
$(2)$ When $p=\frac{N+\alpha+2}{N}$, $Np-(N+\alpha)=2$, similarly to and , we have $$I_{\frac{N+\alpha+2}{N}}(u)\geq \frac{A(u)}{2}\left[1-\left(\frac{c}{c_*}\right)^{\frac{2(\alpha+2)}{N}}\right]\geq0\ \ \ \ \hbox{if}\ \ \ 0<c\leq c_*$$ and $I_{\frac{N+\alpha+2}{N}}(c^2)\leq I_{\frac{N+\alpha+2}{N}}(u^t)\rightarrow 0$ as $t\rightarrow 0^+$ for all $c$. Then $I_{\frac{N+\alpha+2}{N}}(c^2)=0$ if $0<c\leq c_*$.
If $c>c_*$, set $Q^t(x):=\frac{ct^{\frac{N}{2}}}{c_*}Q_{\frac{N+\alpha+2}{N}}(tx)$, then by , $$I_{\frac{N+\alpha+2}{N}}(Q^t)=\frac{c^2t^2}{2c_*^2}\left[1-\left(\frac{c}{c_*}\right)^{\frac{2(\alpha+2)}{N}}\right]\rightarrow -\infty\ \ \hbox{as}\ \ t\rightarrow+\infty,$$ then $I_{\frac{N+\alpha+2}{N}}(c^2)=-\infty$ for $c>c_*$.
$(3)$ If $\frac{N+\alpha+2}{N}<p<\frac{N+\alpha}{(N-2)_+}$, then $Np-(N+\alpha)>2$, hence by , we have $I_p(u^t)\rightarrow -\infty$ as $t\rightarrow+\infty$, so $I_{p}(c^2)=-\infty$ for all $c>0$.
\[lem2.4\] If $\frac{N+\alpha}{N}<p<\frac{N+\alpha+2}{N}$, then
$(1)$ the function $c\mapsto I_p(c^2)$ is continuous on $(0,+\infty)$;
$(2)$ $$\label{2.8}
I_p(c^2)<I_p(\alpha^2)+I_p(c^2-\alpha^2),\ \ \ \ \ \ \forall\ 0<\alpha<c<+\infty.$$
The proof of $(1)$ follows from Lemma \[lem2.3\] and is similar to that of Theorem 2.1 in [@bs], so we omit it.
$(2)$ For any $c>0$, let $\{u_n\}\subset S(c)$ be a minimizing sequence for $I_p(c^2)<0$, then by Lemma \[lem2.3\], $\{u_n\}$ is bounded in $H^1({{\mathbb R}}^N)$ and there exists a constant $K_1>0$ independent of $n$ such that $B(u_n)\geq K_1$. Set $u_n^\theta=\theta u_n$ with $\theta>1$, then $u_n^\theta\in S(\theta c)$ and $$I_p(u_n^\theta)-\theta^2 I(u_n)=\frac{\theta^2-\theta^{2p}}{2p}B(u_n)\leq \frac{\theta^2-\theta^{2p}}{2p}K_1<0.$$ Letting $n\rightarrow+\infty$, we have $I_p(\theta^2c^2)<I_p(c^2)$, $\theta>1$, which easily implies by using Lemma \[lem2.3\] (1).
\[lem2.5\] Let $N\geq1$, ${\alpha}\in(0,N)$ and $\frac{N+\alpha}{N}<p<\frac{N+\alpha}{(N-2)_+}$. If $u$ is a critical point of $I_p(u)$ constrained on $S(c)$, then there exists $\mu_c<0$ such that $I_p'(u)-\mu_c u=0$ in $H^{-1}({{\mathbb R}}^N)$ and $$A(u)-\frac{Np-(N+\alpha)}{2p}B(u)=0.$$
Since $(I_p|_{S(c)})'(u)=0$, there exists $\mu_c\in{{\mathbb R}}$ such that $I_p'(u)-\mu_cu=0$ in $H^{-1}({{\mathbb R}}^N)$. Then $$A(u)-B(u)=\mu_cc^2.$$ By Proposition 3.5 in [@ms2], $u$ satisfies the following Pohozaev identity, $$\frac{N-2}{2}A(u)-\frac{N+\alpha}{2p}B(u)=\frac{N}{2}\mu_cc^2.$$ Hence $A(u)=\frac{Np-(N+\alpha)}{2p}B(u)$ and $$\mu_c=\frac{(N-2)p-(N+\alpha)}{2pc^2}B(u)<0.$$
$\textbf{Proof of Theorem 1.1}$\
(1) If $p=\frac{N+{\alpha}}{N}$, for any $c>0$ and $u\in S(c)$, by we have $$I_{\frac{N+\alpha}{N}}(u)\geq-\frac{N}{2(N+{\alpha})}\left(\frac{c}{|Q_{\frac{N+\alpha}{N}}|_2}\right)^{\frac{2(N+{\alpha})}{N}}.$$ Set $Q_{\frac{N+\alpha}{N}}^t(x):=\frac{ct^{\frac N2}}{|Q_{\frac{N+\alpha}{N}}|_2}Q_{\frac{N+\alpha}{N}}(tx)$, then by again, we see that $$I_{\frac{N+\alpha}{N}}(Q_{\frac{N+\alpha}{N}}^t)=\frac{c^2t^2}{2|Q_{\frac{N+\alpha}{N}}|_2^2}A(Q_{\frac{N+\alpha}{N}})
-\frac{N}{2(N+{\alpha})}\left(\frac{c}{|Q_{\frac{N+\alpha}{N}}|_2}\right)^{\frac{2(N+{\alpha})}{N}},$$ letting $t\rightarrow 0^+$, then $I_{\frac{N+\alpha}{N}}(c^2)=-\frac{N}{2(N+{\alpha})}(\frac{c}{|Q_{\frac{N+\alpha}{N}}|_2})^{\frac{2(N+{\alpha})}{N}}.$
By contradiction, if for some $c>0$, there is $u\in S(c)$ such that $I_{\frac{N+\alpha}{N}}(u)=I_{\frac{N+\alpha}{N}}(c^2)$, then shows that $$0\leq\frac{1}{2}A(u)=\frac{N}{2(N+{\alpha})}\left[B(u)-\left(\frac{c}{|Q_{\frac{N+\alpha}{N}}|_2}\right)^{\frac{2(N+{\alpha})}{N}}\right]\leq0,$$ which implies that $u=0$. It is a contradiction. So $I_{\frac{N+\alpha}{N}}(c^2)$ has no minimizer for all $c>0$.
(2) If $\frac{N+\alpha}{N}<p<\frac{N+{\alpha}+2}{N}$, for any $c>0$, by Lemma \[lem2.3\], $I_p(c^2)<0$. Let $\{u_n\}\subset S(c)$ be a minimizing sequence for $I_p(c^2)$, then Lemma \[lem2.3\] (1) implies that $\{u_n\}$ is bounded in $H^1({{\mathbb R}}^N)$ and for some constant $C>0$ independent of $n$, $B(u_n)\geq C$. Hence there exists $u\in H^1({{\mathbb R}}^N)$ such that $$\label{2.9}
u_n\rightharpoonup u\ \ \hbox{in}\ H^1({{\mathbb R}}^N),\ \ \ \ \ \ \ u_n(x)\rightarrow u(x)\ \ \hbox{a.e.}\ \hbox{in}\ {{\mathbb R}}^N.$$ Moreover, by the Vanishing Lemma \[lem2.6\], up to translations, we may assume that $u\neq0$. Then $0<|u|_2:=\alpha\leq c$. We just suppose that $\alpha<c$, then $u\in S(\alpha)$. By and the Brezis lemma, we have $$\lim\limits_{n\rightarrow+\infty}|u_n-u|_2^2=\lim\limits_{n\rightarrow+\infty}|u_n|_2^2-|u|_2^2=c^2-\alpha^2.$$ Then by Lemma \[lem2.1\] and Lemma \[lem2.4\] (1), we have $$I_p(c^2)=\lim\limits_{n\rightarrow+\infty}I_p(u_n)
=\lim\limits_{n\rightarrow+\infty}I_p(u_n-u)+I_p(u)
\geq I_p(c^2-\alpha^2)+I_p(\alpha^2),$$ which contradicts . So $|u|_2=c$, i.e. $u_n\rightarrow u$ in $L^2({{\mathbb R}}^N)$. By , we have $B(u_n)\rightarrow B(u)$. Then $$I_p(c^2)\leq I_p(u)\leq \lim\limits_{n\rightarrow+\infty}I_p(u_n)=I_p(c^2),$$ i.e. $u$ is minimizer for $I_p(c^2)$.
$(3)$ (i) has been proved in Lemma \[lem2.4\] (2). To prove (ii), by contradiction, if there exists $c_0\in (0,c_*)$ such that $I_{\frac{N+\alpha+2}{N}}(c_0^2)$ has a minimizer $u_0\in S(c_0)$, i.e. $I_{\frac{N+\alpha+2}{N}}(u_0)=I_{\frac{N+\alpha+2}{N}}(c_0^2)=0$, then by , $$A(u_0)=\frac{N}{N+\alpha+2}B(u_0)\leq \left(\frac{c_0}{c_*}\right)^{\frac{2(\alpha+2)}{N}}A(u_0)<A(u_0),$$ which is impossible. So combining (i), we see that $I_{\frac{N+\alpha+2}{N}}(c^2)$ has no minimizer for all $c\neq c_*$.
By , we see that $I_{\frac{N+\alpha+2}{N}}(Q_{\frac{N+\alpha+2}{N}})=0=I_{\frac{N+\alpha+2}{N}}(c_*^2),$ i.e. $Q_{\frac{N+\alpha+2}{N}}$ is a minimizer for $I_{\frac{N+\alpha+2}{N}}(c_*^2)$. Moreover, by Lemmas \[lem3.1\] (2) and \[lem3.2\] below, each groundstate solution of is a minimizer of $I_{\frac{N+\alpha+2}{N}}(c_*^2)$. So we proved (iii).
For any $c>0$, suppose that $u$ is a critical point of $I_{\frac{N+\alpha+2}{N}}(u)$ constrained on $S(c)$, then by and Lemma \[lem2.5\], we have $$A(u)=\frac{N}{N+\alpha+2}B(u)\leq \left(\frac{c}{c_*}\right)^{\frac{2(\alpha+2)}{N}}A(u),$$ which implies that $c_*\leq c$. Therefore, there exists no critical point for $I_{\frac{N+\alpha+2}{N}}(u)$ constrained on $S(c)$ if $0<c<c_*$. So (iv) is proved.
(4) By Lemma \[lem2.3\] (3), $I_{p}(c^2)$ has no minimizer for all $c>0$ if $\frac{N+{\alpha}+2}{N}<p<\frac{N+{\alpha}}{(N-2)_+}$.
Proof of Theorems \[th1.2\] and \[th1.3\]
==========================================
For $p=\frac{N+\alpha+2}{N}$, turns to be $$\label{3.4}
B(u)\leq \frac{N+\alpha+2}{N}\left(\frac{1}{c_*}\right)^{\frac{2(\alpha+2)}{N}}A(u)|u|_2^{\frac{2(\alpha+2)}{N}},$$ with equality for $u=Q_{\frac{N+\alpha+2}{N}}$ and $c_*:=|Q_{\frac{N+\alpha+2}{N}}|_2$, where $Q_{\frac{N+\alpha+2}{N}}$ is a nontrivial solution of $$-\Delta Q_{\frac{N+\alpha+2}{N}}+\frac{\alpha+2}{N}Q_{\frac{N+\alpha+2}{N}}=(I_\alpha*|Q_{\frac{N+\alpha+2}{N}}|^{\frac{N+\alpha+2}{N}})|Q_{\frac{N+\alpha+2}{N}}|^{\frac{N+\alpha+2}{N}-2}Q_{\frac{N+\alpha+2}{N}},\ \ \hbox{in}\ {{\mathbb R}}^N.$$ Set $Q_{\frac{N+\alpha+2}{N}}(x)=\left(\sqrt{\frac{\alpha+2}{N}}\right)^{\frac{N}{2}}\widetilde{Q}_{\frac{N+\alpha+2}{N}}(\sqrt{\frac{\alpha+2}{N}}x)$, then $\widetilde{Q}_{\frac{N+\alpha+2}{N}}$ satisfies the equation $$\label{3.1}
-\Delta \widetilde{Q}_{\frac{N+\alpha+2}{N}}+\widetilde{Q}_{\frac{N+\alpha+2}{N}}
=(I_\alpha*|\widetilde{Q}_{\frac{N+\alpha+2}{N}}|^{\frac{N+\alpha+2}{N}})|\widetilde{Q}_{\frac{N+\alpha+2}{N}}|^{\frac{N+\alpha+2}{N}-2}\widetilde{Q}_{\frac{N+\alpha+2}{N}}, \ \ \hbox{in}\ {{\mathbb R}}^N.$$
The following Lemma is a direct conclusion of Theorems 1-4 in [@ms1].
\[lem3.1\] Assume that $N\geq1$ and $\alpha\in(0,N)$.
$(1)$ There is at least one groundstate solution $u\in H^1({{\mathbb R}}^N)$ to with $$F(u)=d:=\inf\{F(v)|~v\in H^1({{\mathbb R}}^N)\backslash\{0\}~\hbox{is~a~weak~solution~of}~\eqref{3.1}\},$$ where $F(v)={\displaystyle}\frac{1}{2}\int_{{{\mathbb R}}^N}(|\nabla v|^2+|v|^2)-\frac{N}{2(N+\alpha+2)}{\displaystyle}\int_{{{\mathbb R}}^N}(I_\alpha*|v|^{\frac{N+\alpha+2}{N}})|v|^{\frac{N+\alpha+2}{N}}.$
$(2)$ If $u\in H^1({{\mathbb R}}^N)$ is a nontrivial solution of , then $u\in L^1({{\mathbb R}}^N)\cap C^2({{\mathbb R}}^N),$ $u\in W^{2,s}({{\mathbb R}}^N)$ for every $s>1$ and $u\in C^\infty({{\mathbb R}}^N\backslash u^{-1}(\{0\})$. Moreover, $$\label{3.2}
\frac{N+\alpha+2}{N}A(u)=\frac{N+\alpha+2}{\alpha+2}
{\displaystyle}\int_{{{\mathbb R}}^N}|u|^2
=B(u).$$
$(3)$ If $u$ is a groundstate solution of , then $u$ is either positive or negative and there exists $x_0\in{{\mathbb R}}^N$ and a monotone function $v\in C^{\infty}(0,+\infty)$ such that $$u(x)=v(|x-x_0|),~~~~~~\forall~x\in{{\mathbb R}}^N.$$
$(4)$ Let $N-2\leq \alpha<N$ if $N\geq3$ and $0<{\alpha}<N$ if $N=1,2$. If $u$ is a groundstate solution of , then $$\lim\limits_{|x|\rightarrow+\infty}|u(x)||x|^{\frac{N-1}{2}}e^{|x|}\in (0,+\infty).$$ Moreover, $|\nabla u(x)|=O(|x|^{-\frac{N-1}{2}}e^{-|x|})$ as $|x|\rightarrow+\infty.$
\[lem3.2\] (1) $d=\frac{c_*^2}{2}$.
(2) $u$ is a nontrivial solution of with $|u|_2=c_*$ if and only if $u$ is a groundstate solution.
For any nontrivial solution $u$ of , then by Lemma \[lem3.1\] (1)(2) and , we have $$c_*\leq |u|_2$$ and $$d\leq F(u)=\frac{1}{2}\int_{{{\mathbb R}}^N}|u|^2$$ where equality holds only if $u$ is a groundstate solution. In particular, since $\widetilde{Q}_{\frac{N+\alpha+2}{N}}$ is a nontrivial solution of , $$d\leq F(\widetilde{Q}_{\frac{N+\alpha+2}{N}})= \frac{|\widetilde{Q}_{\frac{N+\alpha+2}{N}}|_2^2}{2}=\frac{c_*^2}{2}.$$
Therefore, if $u$ is a groundstate solution of , then by Lemma \[lem3.1\] (3), $u$ is nontrivial and $$\frac{c_*^2}{2}\leq \frac{|u|_2^2}{2}=F(u)=d\leq \frac{c_*^2}{2},$$ which shows that $d=\frac{c_*^2}{2}$ and $|u|_2=c_*$.
On the other hand, if $u$ is a nontrivial solution of with $|u|_2=c_*$, then $$\frac{ c_*^2}{2}=d\leq F(u)=\frac{1}{2}\int_{{{\mathbb R}}^N}|u|^2=\frac{ c_*^2}{2},$$ which implies that $F(u)=d$, i.e. $u$ is a groundstate solution.
\[rem3.3\] $\widetilde{Q}_{\frac{N+\alpha+2}{N}}$ is a groundstate solution of .
\[lem3.4\]([@a]) Suppose that $V\in L^{\infty}_{loc}({{\mathbb R}}^N)$ and $\lim\limits_{|x|\rightarrow +\infty}V(x)=+\infty$, then the embedding $\mathcal{H}\hookrightarrow L^s({{\mathbb R}}^N)$, $2\leq s<2^*$ is compact.
$\textbf{Proof of Theorem \ref{th1.2}}$\
Set $$C(u):={\displaystyle}\int_{{{\mathbb R}}^N}V(x)|u|^2\geq0,\ \ \ \forall~u\in H^1({{\mathbb R}}^N),$$ then $$E(u)=\frac{A(u)}{2}+\frac{C(u)}{2}-\frac{N}{2(N+\alpha+2)}B(u).$$
(1) By , for any $0<c\leq c_*$ and $u\in \widetilde{S}(c)$, $$\label{3.5}
E(u)\geq \frac{1}{2}\left[1-\left(\frac{c}{c_*}\right)^{\frac{2(\alpha+2)}{N}}\right]A(u)+\frac{1}{2}C(u)\geq0,$$ then $e_c=\inf\limits_{u\in\widetilde{S}(c)}E(u)\geq0$ is well defined for $0<c\leq c_*$.
For each $0<c<c_*$, let $\{u_n\}\subset \widetilde{S}(c)$ be a minimizing sequence for $e_c$, then by , $\{u_n\}$ is bounded in $\mathcal{H}$. Hence there exists $u_c\in \mathcal{H}$ such that $u_n\rightharpoonup u_c$ in $\mathcal{H}$. By Lemma \[lem3.4\], $u_n\rightarrow u_c$ in $L^s({{\mathbb R}}^N)$, $2\leq s<2^*$, which implies that $|u_c|_2=c$ and $B(u_n)\rightarrow B(u_c)$. So $e_c\leq E(u_c)\leq \lim\limits_{n\rightarrow+\infty}E(u_n)=e_c$, i.e. $u_c\in \widetilde{S}(c)$ is a minimizer of $e_c$. Moreover, by , $e_c>0$. So $e_c>0$ has at least one minimizer for all $0<c<c_*$.
(2) Let $N-2\leq \alpha<N$ if $N\geq3$ and $0<{\alpha}<N$ if $N=1,2$. For any $c>0$, let $\varphi\in C_0^{\infty}({{\mathbb R}}^N)$ such that $0\leq \varphi(x)\leq1$, $\varphi(x)\equiv1$ for $|x|\leq1$, $\varphi(x)\equiv0$ for $|x|\geq2$ and $|\nabla \varphi|\leq 2$. For any $x_0\in{{\mathbb R}}^N$ and any $t>0$, set $$\label{3.6}
\widetilde{Q}^t(x)=\frac{cA_t t^{\frac{N}{2}}}{c_*}\varphi(x-x_0)\widetilde{Q}_{\frac{N+\alpha+2}{N}}(t(x-x_0)),$$ where $A_t>0$ is chosen to satisfy that $|\widetilde{Q}^t|_2=c$. By the exponential decay of $\widetilde{Q}_{\frac{N+\alpha+2}{N}}$, we see that $$\frac{1}{A_t^2}=1+\frac{1}{c_*^2}{\displaystyle}\int_{{{\mathbb R}}^N}\left(\varphi^2(\frac{x}{t})-1\right)|\widetilde{Q}_{\frac{N+\alpha+2}{N}}(x)|^2
\rightarrow1$$ as $t\rightarrow +\infty$. Then $A_t$ depends only on $t$ and $\lim\limits_{t\rightarrow+\infty}A_t=1$. Since $V(x)\varphi^2(x-x_0)$ is bounded and has compact support, $C(\widetilde{Q}^t)\rightarrow \frac{c^2}{c_*^2}V(x_0)$. $$\begin{array}{ll}
B(\widetilde{Q}^t)&={\displaystyle}(\frac{c A_t}{c_*})^{\frac{2(N+\alpha+2)}{N}}t^2{\displaystyle}\left\{B(\widetilde{Q}_{\frac{N+\alpha+2}{N}})\right.\\[5mm]
& \left.+{\displaystyle}\int_{{{\mathbb R}}^N}\{I_\alpha*[(|\varphi(\frac{y}{t})|^{\frac{N+\alpha+2}{N}}-1)|\widetilde{Q}_{\frac{N+\alpha+2}{N}}(y)|^{\frac{N+\alpha+2}{N}}]\}
(|\varphi(\frac{x}{t})|^{\frac{N+\alpha+2}{N}}+1)|\widetilde{Q}_{\frac{N+\alpha+2}{N}}(x)|^{\frac{N+\alpha+2}{N}}\right\}\\[5mm]
&:={\displaystyle}(\frac{c A_t}{c_*})^{\frac{2(N+\alpha+2)}{N}}t^2\left[B(\widetilde{Q}_{\frac{N+\alpha+2}{N}})+f_1(t)\right].
\end{array}$$ By the Hardy-Littlewood-Sobolev inequality and the exponential decay of $\widetilde{Q}_{\frac{N+\alpha+2}{N}}$, we have there exists a constant $C>0$ such that $$\begin{array}{ll}
|f_1(t)|&\leq C\left({\displaystyle}\int_{{{\mathbb R}}^N}|[\varphi(\frac{x}{t})]^{\frac{N+\alpha+2}{N}}-1|^{\frac{2N}{N+\alpha}}|\widetilde{Q}_{\frac{N+\alpha+2}{N}}(x)|^{\frac{2(N+\alpha+2)}{N+\alpha}}\right)^{\frac{N+\alpha}{2N}}\\[5mm]
&\leq C\left({\displaystyle}\int_{|x|\geq t}|\widetilde{Q}_{\frac{N+\alpha+2}{N}}(x)|^{\frac{2(N+\alpha+2)}{N+\alpha}}\right)^{\frac{N+\alpha}{2N}}\\[5mm]
&\leq C\left({\displaystyle}\int_{t}^{+\infty}r^{-\frac{2(N-1)}{N+\alpha}}e^{-\frac{2(N+\alpha+2)}{N+\alpha}r}\right)^{\frac{N+\alpha}{2N}}\leq Ct^{-\frac{2(N-1)}{2N}}e^{-\frac{N+\alpha+2}{N}t}\ \ \ \ \ \ \ \hbox{as}\ \ t\rightarrow+\infty.
\end{array}$$ Then by the exponential decay of $\widetilde{Q}_{\frac{N+\alpha+2}{N}}$ and $|\nabla \widetilde{Q}_{\frac{N+\alpha+2}{N}}|$, we have $$\label{3.7}
E(\widetilde{Q}^t)=\frac{c^2}{2c_*^2}t^2A(\widetilde{Q}_{\frac{N+\alpha+2}{N}})\left[1-\left(\frac{c}{c_*}\right)^{\frac{2(\alpha+2)}{N}}\right]+t^2f_2(t)+\frac{c^2}{2c_*^2}V(x_0)\ \ \hbox{as}\ \ t\rightarrow+\infty,$$ where $f_2(t)$ denotes a function satisfying that $\lim\limits_{t\rightarrow+\infty}|f_2(t)|t^r=0$ for all $r>0$.
If $c>c^*$, then by , $e_c\leq \lim\limits_{t\rightarrow+\infty}E(\widetilde{Q}^t)=-\infty$, hence $e_c=-\infty$ and there exists no minimizer for $e_c$.
If $c=c^*$, then by and , $0\leq e_{c_*}\leq \frac{V(x_0)}{2}.$ Taking the infimum over $x_0$, $e_{c_*}=0$. We just suppose that there exists $u\in \widetilde{S}(c_*)$ such that $E(u)=e_{c_*}$, then it follows from that $$\label{3.9}
C(u)=0,$$ which and the condition $(V_0)$ imply that $u$ must have compact support. On the other hand, $(E|_{\widetilde{S}(c_*)})'(u)=0$. Then there exists $\mu_{c_*}\in{{\mathbb R}}$ such that $E'(u)-\mu_{c_*}u=0$, i.e. for any $h\in C_0^{\infty}({{\mathbb R}}^N)$, $$\label{3.10}\begin{array}{ll}
0&=\langle E'(u)-\mu_{c_*}u,h\rangle\\[5mm]
&={\displaystyle}\int_{{{\mathbb R}}^N}(\nabla u\nabla h-\mu_{c_*}uh)-{\displaystyle}\int_{{{\mathbb R}}^N}(I_\alpha*|u|^{\frac{N+\alpha+2}{N}})|u|^{\frac{N+\alpha+2}{N}-2}uh\\[5mm]
&=\langle I_{\frac{N+\alpha+2}{N}}'(u)-\mu_{c_*}u,h\rangle,
\end{array}$$ where we have used the fact that $\int_{{{\mathbb R}}^N}V(x)uh=0$ due to the Hölder inequality and . Then by Lemma \[lem2.5\], we see that $\mu_{c_*}<0$. Set $u(x):=(\sqrt{-\mu_{c_*}})^{\frac{N}{2}}w(\sqrt{-\mu_{c_*}}x)$, then by , $w$ is a nontrivial solution of with $|w|_2=c_*$, hence by Lemma \[lem3.2\] $w$ is a groundstate solution. So by Lemma \[lem3.1\] (4), $\lim\limits_{|x|\rightarrow +\infty}|u(x)||x|^{\frac{N-1}{2}}e^{|x|}\in(0,+\infty),$ which contradicts .
Moreover, we conclude from and that $\limsup\limits_{c\rightarrow (c_*)^-}e_c\leq \frac{V(x_0)}{2}$ as $t\rightarrow+\infty$. By the arbitrary of $x_0$, we have $\lim\limits_{c\rightarrow (c_*)^-}e_c=0=e_{c_*}.$
In the following, we consider the concentration behavior of minimizers as $c$ approaches $c_*$ from below when $N-2\leq \alpha<N$ if $N\geq3$ and $0<{\alpha}<N$ if $N=1,2$ and the potential $V(x)$ satisfies conditions $(V_0)(V_1)$.
\[lem3.6\] Suppose that $(V_0)(V_1)$ hold, then there exist two positive constants $M_1<M_2$ independent of $c$ such that $$M_1\left[1-\left(\frac{c}{c_*}\right)^{\frac{2(\alpha+2)}{N}}\right]^{\frac{q}{q+2}}\leq e_c\leq M_2\left[1-\left(\frac{c}{c_*}\right)^{\frac{2(\alpha+2)}{N}}\right]^{\frac{q}{q+2}}\ \ \ \ \hbox{as}\ \ c\rightarrow (c_*)^-,$$ where $q=\max\{q_1,q_2,\cdots,q_m\}$.
The proof consists of two steps.
**Step 1.** Without loss of generality, we may assume that $q=q_{i_0}$ for some $1\leq i_0\leq m$. By $(V_1)$, there exists $R>0$ small such that $V(x)\leq 2\mu_{i_0}|x-x_{i_0}|^{q_{i_0}}$ for $|x-x_{i_0}|\leq R$. Similarly to , let $$u(x):=\frac{cA_{R,t} t^{\frac{N}{2}}}{c_*}\varphi\left(\frac{2(x-x_{i_0})}{R}\right)\widetilde{Q}_{\frac{N+\alpha+2}{N}}(t(x-x_{i_0}))\in \widetilde{S}(c),$$ where $A_{R,t}>0$ and $A_{R,t}\rightarrow1$ as $t\rightarrow+\infty$. Then $$C(u)\leq \frac{2\mu_{i_0}c^2A_{R,t}^2}{c_*^2}t^{-q_{i_0}}{\displaystyle}\int_{{{\mathbb R}}^N}|x|^{q_{i_0}}|\widetilde{Q}_{\frac{N+\alpha+2}{N}}|^2.$$ Hence similarly to , for large $t$, $$e_c\leq \frac{A(\widetilde{Q}_{\frac{N+\alpha+2}{N}})}{2}t^2\left[1-\left(\frac{c}{c_*}\right)^{\frac{2(\alpha+2)}{N}}\right]
+2\mu_{i_0}t^{-q_{i_0}}{\displaystyle}\int_{{{\mathbb R}}^N}|x|^{q_{i_0}}|\widetilde{Q}_{\frac{N+\alpha+2}{N}}(x)|^2+t^2h(t),$$ where $\lim\limits_{t\rightarrow+\infty}|h(t)|t^2=0$. By taking $t=[1-(\frac{c}{c_*})^{\frac{2(\alpha+2)}{N}}]^{-\frac{1}{q_{i_0}+2}},$ then there exists a constant $M_2>0$ independent of $c$ such that $$e_c\leq M_2 \left[1-\left(\frac{c}{c_*}\right)^{\frac{2(\alpha+2)}{N}}\right]^{\frac{q}{q+2}}.$$
**Step 2.** For any $0<c<c_*$, there exists $u_c\in \widetilde{S}(c)$ such that $E(u_c)=e_c$. By and Theorem \[th1.2\], we see that $$\label{3.11}
C(u_c)\leq e_c\rightarrow 0 \ \ \ \hbox{as}\ \ c\rightarrow (c_*)^-.$$ We claim that $$\label{3.12}
A(u_c)\rightarrow+\infty\ \ \ \hbox{as}\ \ c\rightarrow(c_*)^-.$$ In fact, by contradiction, if there exists a sequence $\{c_k\}\subset(0,c_*)$ with $c_k\rightarrow c_*$ as $k\rightarrow+\infty$ such that the sequence of minimizers $\{u_{c_k}\}\subset \widetilde{S}(c_k)$ is uniformly bounded in $\mathcal{H}$, then we may assume that for some $u\in\mathcal{H}$, $u_{c_k}\rightharpoonup u$ in $\mathcal{H}$ and by Lemma \[lem3.4\] and , $$u_{c_k}\rightarrow u\ \ \ \hbox{in} \ L^2({{\mathbb R}}^N)\ \ \ \hbox{and} \ \ \ B(u_{c_k})\rightarrow B(u).$$ Hence $u\in \widetilde{S}(c_*)$ and $0\leq e_{c_*}\leq E(u)\leq \lim\limits_{k\rightarrow+\infty}E(u_{c_k})=\lim\limits_{k\rightarrow+\infty}e_{c_k}=0,$ i.e. $u$ is a minimizer of $e_{c_*}$, which contradicts Theorem \[th1.2\].
Since $$0\leq \frac{1}{2}A(u_c)-\frac{N}{2(N+\alpha+2)}B(u_c)\leq e_c,$$ we see that $$\lim\limits_{c\rightarrow (c_*)^-}\frac{\frac{N}{N+\alpha+2}B(u_c)}{A(u_c)}=1.$$ Then by , set $$\label{3.13}
\varepsilon_c^{-2}:=\frac{N}{2(N+\alpha+2)}B(u_c)\rightarrow+\infty\ \ \ \hbox{as}\ \ c\rightarrow(c_*)^-$$ and $\tilde{w}_c(x):=\varepsilon_c^{\frac{N}{2}}u_c(\varepsilon_cx).$ Then $|\tilde{w}_c|_2=c$ and $$\label{3.14}
\frac{N}{2(N+\alpha+2)}B(\tilde{w}_c)=1, \ \ \ \ \ \ \ 2\leq A(\tilde{w}_c)\leq 2+2\varepsilon_c^2e_c.$$ Let $\delta:=\lim\limits_{c\rightarrow(c_*)^-}\sup\limits_{y\in{{\mathbb R}}^N}\int_{B_1(y)}|\tilde{w}_c|^2.$ If $\delta=0$, then $\tilde{w}_c\rightarrow 0$ in $L^s({{\mathbb R}}^N)$ as $c\rightarrow(c_*)^-$, $2< s<2^*$, hence by , $B(\tilde{w}_c)\rightarrow0$, which contradicts . So $\delta>0$ and there exists $\{y_c\}\subset {{\mathbb R}}^N$ such that $\int_{B_1(y_c)}|\tilde{w}_c|^2\geq\frac{\delta}{2}>0.$ Set $$w_c(x):=\tilde{w}_c(x+y_c)=\varepsilon_c^{\frac{N}{2}}u_c(\varepsilon_cx+\varepsilon_cy_c),$$ then $$\label{3.15}
\int_{B_1(0)}|w_c|^2\geq \frac{\delta}{2}>0.$$
We claim that $\{\varepsilon_cy_c\}$ is uniformly bounded as $c\rightarrow (c_*)^-$. Indeed, if there exists a sequence $\{c_k\}\subset(0,c_*)$ with $c_k\rightarrow c_*$ as $k\rightarrow+\infty$ such that $|\varepsilon_{c_k}y_{c_k}|\rightarrow +\infty$ as $k\rightarrow+\infty$, then by $(V_0)$, and and the Fatou’s Lemma, we have $$\begin{array}{ll}
0=\liminf\limits_{k\rightarrow+\infty}{\displaystyle}\int_{{{\mathbb R}}^N}V(x)|u_{c_k}|^2&=\liminf\limits_{k\rightarrow+\infty}{\displaystyle}\int_{{{\mathbb R}}^N}V(\varepsilon_{c_k}x+\varepsilon_{c_k}y_{c_k})|w_{c_k}(x)|^2\\[5mm]
&\geq{\displaystyle}\int_{{{\mathbb R}}^N}\liminf\limits_{k\rightarrow+\infty}[V(\varepsilon_{c_k}x+\varepsilon_{c_k}y_{c_k})|w_{c_k}(x)|^2]\\[5mm]
&\geq{\displaystyle}\int_{B_1(0)}\liminf\limits_{k\rightarrow+\infty}[V(\varepsilon_{c_k}x+\varepsilon_{c_k}y_{c_k})|w_{c_k}(x)|^2]\\[5mm]
&\geq {\displaystyle}(+\infty)\cdot\frac{\delta}{2}=+\infty,
\end{array}$$ which is impossible. So $\{\varepsilon_cy_c\}$ is uniformly bounded as $c\rightarrow (c_*)^-$. Moreover, there exists $x_{j_0}\in\{x_1,\cdots,x_m\}$ such that $$\label{3.20}
\left\{\frac{\varepsilon_cy_c-x_{j_0}}{\varepsilon_c}\right\}\ \hbox{is~uniformly~bounded~as}~c\rightarrow(c_*)^-.$$
Indeed, by contradiction, we just suppose that for any $x_i\in\{x_1,\cdots,x_m\}$, there exists $c_k\rightarrow(c_*)^-$ as $k\rightarrow+\infty$ such that $|\frac{\varepsilon_{c_k}y_{c_k}-x_{i}}{\varepsilon_{c_k}}|\rightarrow+\infty$ as $k\rightarrow+\infty$. By $(V_1)$, and the Fatou’s Lemma, for any positive constant $C$, $$\begin{array}{ll}
\liminf\limits_{k\rightarrow+\infty}\varepsilon_{c_k}^{-q_i}{\displaystyle}\int_{{{\mathbb R}}^N}V(\varepsilon_{c_k}x+\varepsilon_{c_k}y_{c_k})|w_{c_k}(x)|^2
&\geq{\displaystyle}\int_{{{\mathbb R}}^N}\liminf\limits_{k\rightarrow+\infty}\frac{V(\varepsilon_{c_k}x+\varepsilon_{c_k}y_{c_k})}{\varepsilon_{c_k}^{q_i}}|w_{c_k}(x)|^2\\[5mm]
&\geq{\displaystyle}\int_{{{\mathbb R}}^N}\liminf\limits_{k\rightarrow+\infty}\frac{V(\varepsilon_{c_k}x+x_{i})}{\varepsilon_{c_k}^{q_i}}|w_{c_k}(x+\frac{x_i-\varepsilon_{c_k}y_{c_k}}{\varepsilon_{c_k}})|^2\\[5mm]
&\geq{\displaystyle}\mu_i{\displaystyle}\int_{{{\mathbb R}}^N}\liminf\limits_{k\rightarrow+\infty}|x|^{q_i}|w_{c_k}(x+\frac{x_i-\varepsilon_{c_k}y_{c_k}}{\varepsilon_{c_k}})|^2\\[5mm]
&\geq{\displaystyle}\mu_i{\displaystyle}\int_{B_{1}(0)}\liminf\limits_{k\rightarrow+\infty}|x+\frac{\varepsilon_{c_k}y_{c_k}-x_i}{\varepsilon_{c_k}}|^{q_i}|w_{c_k}(x)|^2\\[5mm]
&\geq {\displaystyle}\frac{\mu_i\delta}{2}C.
\end{array}$$ Hence by and , $$\label{3.21}
\begin{array}{ll}
e_{c_k}&={\displaystyle}\frac{1}{\varepsilon_{c_k}^2}\left(\frac{A(w_{c_k})}{2}-\frac{N B(w_{c_k})}{2(N+\alpha+2)}\right)
+\frac{1}{2}{\displaystyle}\int_{{{\mathbb R}}^N}V(\varepsilon_{c_k}x+\varepsilon_{c_k}y_{c_k})|w_{c_k}|^2\\[5mm]
&\geq {\displaystyle}\frac{1}{\varepsilon_{c_k}^2}\left[1-\left(\frac{c}{c_*}\right)^{\frac{2(\alpha+2)}{N}}\right]+\frac{\mu_i \delta C}{4}\varepsilon_{c_k}^{q_i}\\[5mm]
&\geq{\displaystyle}(1+\frac{2}{q_i})\left(\frac{q_i\delta\mu_i}{8}\right)^{\frac{2}{q_i+2}}\left[1-\left(\frac{c_k}{c_*}\right)^{\frac{2(\alpha+2)}{N}}\right]^{\frac{q_i}{q_i+2}}C^{\frac{2}{q_i+2}}\\[5mm]
&\geq{\displaystyle}(1+\frac{2}{q_i})\left(\frac{q_i\delta\mu_i}{8}\right)^{\frac{2}{q_i+2}}C^{\frac{2}{q_i+2}}\left[1-\left(\frac{c_k}{c_*}\right)^{\frac{2(\alpha+2)}{N}}\right]^{\frac{q}{q+2}}\ \ \ \hbox{as}\ \ k\rightarrow+\infty,
\end{array}$$ which contradicts the upper bound obtained in **Step 1** since $C>0$ is arbitrary. Then holds. So for some $y_0\in{{\mathbb R}}^N,$ $$\frac{\varepsilon_cy_c-x_{j_0}}{\varepsilon_c}\rightarrow y_0\ \ \ \hbox{and}\ \ \ \varepsilon_cy_c\rightarrow x_{j_0} \ \ \ \hbox{as}\ c\rightarrow(c_*)^-.$$ By the definition of $\{w_c\}$ and , $\{w_c\}$ is uniformly bounded in $H^1({{\mathbb R}}^N)$. Then up to a subsequence, we may assume that for some $w_0\in H^1({{\mathbb R}}^N)$, $$w_c\rightharpoonup w_0\ \ \hbox{in}\ H^1({{\mathbb R}}^N),\ \ \ \ w_c\rightarrow w_0\ \ \hbox{in}\ L^s_{loc}({{\mathbb R}}^N),\ 1\leq s<2^*$$ and $$w_c(x)\rightarrow w_0(x)\ \ \hbox{a.e.}\ \hbox{in}\ {{\mathbb R}}^N.$$ Then by $(V_1)$ and the Fatou’s Lemma again, there exists a constant $C_2>0$ independent of $c$ such that $$\label{3.16}
\begin{array}{ll}
&\ \ \ \liminf\limits_{c\rightarrow(c_*)^-}\varepsilon_{c}^{-q_{j_0}}{\displaystyle}\int_{{{\mathbb R}}^N}V(\varepsilon_{c}x+\varepsilon_{c}y_{c})|w_{c}(x)|^2\\[5mm]
&\geq {\displaystyle}\int_{{{\mathbb R}}^N}\liminf\limits_{c\rightarrow(c_*)^-}\frac{V(\varepsilon_{c}x+\varepsilon_{c}y_{c})}{|\varepsilon_{c}|^{q_{j_0}}}|w_{c}(x)|^2\\[5mm]
&\geq {\displaystyle}\int_{{{\mathbb R}}^N}\liminf\limits_{c\rightarrow(c_*)^-}\frac{V(\varepsilon_{c}x+\varepsilon_{c}y_{c})}{|\varepsilon_{c}x+\varepsilon_{c}y_{c}-x_{j_0}|^{q_{j_0}}}|x+\frac{\varepsilon_{c}y_{c}-x_{j_0}}{\varepsilon_c}|^{q_{j_0}}|w_{c}(x)|^2\\[5mm]
&\geq \mu_{j_0}{\displaystyle}\int_{B_1(0)}|x+y_0|^{q_{j_0}}|w_{0}(x)|^2:=C_2>0.
\end{array}$$ Similarly to , we have $$e_c\geq(1+\frac{2}{q_{j_0}})\left(\frac{q_{j_0}C_2}{2}\right)^{\frac{2}{q_{j_0}+2}}\left[1-\left(\frac{c}{c_*}\right)^{\frac{2(\alpha+2)}{N}}\right]^{\frac{q}{q+2}}:=M_1\left[1-\left(\frac{c}{c_*}\right)^{\frac{2(\alpha+2)}{N}}\right]^{\frac{q}{q+2}}$$ as $c\rightarrow(c_*)^-.$
\[lem3.7\] Suppose that $u_c$ is a minimizer of $e_c$ and $V(x)$ satisfies $(V_0)(V_1)$, then there exist two positive constants $K_1<K_2$ independent of $c$ such that $$K_1\left[1-\left(\frac{c}{c_*}\right)^{\frac{2(\alpha+2)}{N}}\right]^{-\frac{2}{q+2}}\leq A(u_c)\leq K_2\left[1-\left(\frac{c}{c_*}\right)^{\frac{2(\alpha+2)}{N}}\right]^{-\frac{2}{q+2}}\ \ \ \hbox{as}\ c\rightarrow(c_*)^-.$$
The idea of the proof comes from that of Lemma 4 in [@gs], but it needs more careful analysis.
By , we see that $$e_c=E(u_c)\geq\frac{1}{2}\left[1-(\frac{c}{c_*})^{\frac{2(\alpha+2)}{N}}\right]A(u_c),$$ then by Lemma \[lem3.6\], $$A(u_c)\leq 2M_2\left[1-\left(\frac{c}{c_*}\right)^{\frac{2(\alpha+2)}{N}}\right]^{-\frac{2}{q+2}},$$ where $M_2$ is given in Lemma \[lem3.6\].
For any fixed $b\in(0,c)$, there exist two functions $u_b\in \widetilde{S}(b)$, $u_c\in \widetilde{S}(c)$ such that $e_b=E(u_b)$ and $e_c=E(u_c)$ respectively. Then by , we see that $$e_b\leq E\left(\frac{b}{c}u_c\right)< e_c+\frac{1}{2}\left[1-\left(\frac{b}{c}\right)^{\frac{2(\alpha+2)}{N}}\right]A(u_c).$$ Let $\eta:=\frac{c-b}{c_*-c}>0$, then $\eta\rightarrow +\infty$ as $c\rightarrow(c_*)^-$. Then by Lemma \[3.6\], we have $$\begin{array}{ll}
{\displaystyle}\frac{1}{2}A(u_c)&>{\displaystyle}\frac{e_b-e_c}{1-(\frac{b}{c})^{\frac{2(\alpha+2)}{N}}}\\[5mm]
&\geq{\displaystyle}\frac{M_1(1-(\frac{b}{c_*})^{\frac{2(\alpha+2)}{N}})^{\frac{q}{q+2}}-M_2(1-(\frac{c}{c_*})^{\frac{2(\alpha+2)}{N}})^{\frac{q}{q+2}}}
{1-(\frac{b}{c})^{\frac{2(\alpha+2)}{N}}}\\[5mm]
&\geq {\displaystyle}\left[1-\left(\frac{c}{c_*}\right)^{\frac{2(\alpha+2)}{N}}\right]^{-\frac{2}{q+2}}
{\displaystyle}\frac{M_1{\displaystyle}[\frac{1-(\frac{b}{c_*})^{\frac{2(\alpha+2)}{N}}}{1-(\frac{c}{c_*})^{\frac{2(\alpha+2)}{N}}}]^{\frac{q}{q+2}}-M_2}{(1-(\frac{b}{c})^{\frac{2(\alpha+2)}{N}})[1-(\frac{c}{c_*})^{\frac{2(\alpha+2)}{N}}]^{-1}}\\[5mm]
&\geq{\displaystyle}\left[1-\left(\frac{c}{c_*}\right)^{\frac{2(\alpha+2)}{N}}\right]^{-\frac{2}{q+2}}
{\displaystyle}\frac{M_1(\frac{N}{2(\alpha+2)})^{\frac{q}{q+2}}(1+\eta)^{\frac{q}{q+2}}-M_2}{\eta},
\end{array}$$ which gives the desired positive lower bound as $c\rightarrow(c_*)^-.$
$\textbf{Proof of Theorem \ref{th1.3}}$\
Let $\{c_k\}\subset(0,c_*)$ be a sequence satisfying $c_k\rightarrow (c_*)^-$ as $k\rightarrow+\infty$ and denote $\{u_{c_k}\}\subset \widetilde{S}(c_k)$ to be a sequence of minimizers for $e_{c_k}$. Set $$\label{3.22}
\varepsilon_k:=[1-(\frac{c_k}{c_*})^{\frac{2(\alpha+2)}{N}}]^{\frac{1}{q+2}}>0.$$ By , Lemmas \[lem3.6\] and \[lem3.7\], we see that $$K_1\varepsilon_k^{-2}\leq A(u_{c_k})\leq K_2\varepsilon_k^{-2},\ \ \ 0\leq C(u_{c_k})\leq 2M_2\varepsilon_k^q$$ Let $$\tilde{w}_{c_k}(x):=\varepsilon_k^{\frac{N}{2}}u_{c_k}(\varepsilon_k x),$$ then $|\tilde{w}_c|_2=c$ and $$\label{3.23}
K_1\leq A(\tilde{w}_{c_k})\leq K_2, \ \ \ \ \ B(\tilde{w}_{c_k})\leq \frac{N+\alpha+2}{N}K_2$$
Let $\delta:=\lim\limits_{k\rightarrow+\infty}\sup\limits_{y\in{{\mathbb R}}^N}\int_{B_1(0)}|\tilde{w}_{c_k}|^2.$ If $\delta=0$, then by the Vanishing Lemma \[lem2.6\], $\tilde{w}_{c_k}\rightarrow0$ in $L^s({{\mathbb R}}^N)$ as $k\rightarrow+\infty$, $2<s<2^*$. Hence by , $B(\tilde{w}_{c_k})\rightarrow0$. So $$0<\frac{K_1}{2}\leq \frac{A(\tilde{w}_{c_k})}{2}\leq e_{c_k}\varepsilon_k^2+\frac{N}{2(N+\alpha+2)}B(\tilde{w}_{c_k})\rightarrow0\ \ \hbox{as}\ k\rightarrow+\infty,$$ which is a contradiction. Then $\delta>0$ and there exists $\{y_k\}\subset{{\mathbb R}}^N$ such that $\int_{B_1(y_k)}|w_{c_k}|^2\geq\frac{\delta}{2}>0.$ Set $$w_{c_k}(x):=\tilde{w}_{c_k}(x+y_{k})=\varepsilon_k^{\frac{N}{2}}u_{c_k}(\varepsilon_k x+\varepsilon y_k),$$ then $$\label{3.24}
\int_{B_1(0)}|w_{c_k}|^2\geq\frac{\delta}{2}>0$$ and $$\label{3.25}
{\displaystyle}\int_{{{\mathbb R}}^N}V(\varepsilon_k x+\varepsilon_k y_k)|w_{c_k}(x)|^2=C(u_{c_k})\leq 2M_2\varepsilon_k^q.$$ Similar to the proof in Lemma \[lem3.6\], one can show that $\{\varepsilon_k y_k\}$ is uniformly bounded as $k\rightarrow+\infty$.
Since $u_{c_k}\in \widetilde{S}(c_k)$ is a minimizer of $e_{c_k}$, $(E|_{\widetilde{S}(c_k)})'(u_{c_k})=0$, i.e. there exists a sequence $\{\lambda_k\}\subset{{\mathbb R}}$ such that $E'(u_{c_k})-\lambda_ku_{c_k}=0$ in $\mathcal{H}^{-1}$, where $\mathcal{H}^{-1}$ denotes the dual space of $\mathcal{H}$. Then $$\varepsilon_k^2\lambda_k
=\frac{2\frac{N+\alpha+2}{N}\varepsilon_k^2e_{c_k}-\frac{\alpha+2}{N}\varepsilon_k^2 C(u_{c_k})-\frac{\alpha+2}{N}A(w_{c_k})}{c^2_k},$$ which and imply that there exists $\beta>0$ such that $$\varepsilon_k^2\lambda_k\rightarrow-\beta^2\ \ \hbox{as}\ k\rightarrow+\infty.$$ By the definition of $w_{c_k}$, we see that $w_{c_k}$ satisfies the following equation $$\label{3.26}
-\Delta w_{c_k}+\varepsilon_k^2V(\varepsilon_k x+\varepsilon_k y_k)w_{c_k}-(I_\alpha*|w_{c_k}|^{\frac{N+\alpha+2}{N}})|w_{c_k}|^{\frac{N+\alpha+2}{N}-2}w_{c_k}
=\lambda_k\varepsilon_k^2 w_{c_k}\ \ \hbox{in}\ {{\mathbb R}}^N.$$ Since $\{w_{c_k}\}$ is uniformly bounded in $H^1({{\mathbb R}}^N)$, there exists $w_0\in H^1({{\mathbb R}}^N)$ such that $$w_{c_k}\rightharpoonup w_0\ \ \hbox{in}\ H^1({{\mathbb R}}^N),\ \ \ \ \ \ \ w_{c_k}\rightarrow w_0\ \ \hbox{in}\ L^s_{loc}({{\mathbb R}}^N),\ 1\leq s<2^*$$ and $$w_{c_k}(x)\rightarrow w_0(x)\ \ \ \hbox{a.e.~in}\ {{\mathbb R}}^N.$$ Moreover, implies that $w_0\neq0$. Then $w_0$ is a nontrivial solution of $-\Delta w_0+\beta^2 w_0=(I_\alpha*|w_0|^{\frac{N+\alpha+2}{N}})|w_0|^{\frac{N+\alpha+2}{N}-2}w_0$ in ${{\mathbb R}}^N.$ Set $$w_0(x):=\beta^{\frac{N}{2}}W_0(\beta x),$$ then $W_0$ is a nontrivial solution of $$\label{3.27}
-\Delta W_0+W_0=(I_\alpha*|W_0|^{\frac{N+\alpha+2}{N}})|W_0|^{\frac{N+\alpha+2}{N}-2}W_0,\ \ \ x\in{{\mathbb R}}^N.$$ Hence by Lemma \[lem3.1\] (2), we have $A(W_0)=\frac{N}{N+\alpha+2}B(W_0).$ So it follows from that $$c_*^{\frac{2(\alpha+2)}{N}}\leq \frac{\frac{N+\alpha+2}{N}A(W_0)|W_0|_2^{\frac{2(\alpha+2)}{N}}}{B(W_0)}=|W_0|_2^{\frac{2(\alpha+2)}{N}}=|w_0|_2^{\frac{2(\alpha+2)}{N}}\leq \lim\limits_{k\rightarrow+\infty}|w_{c_k}|_2^{\frac{2(\alpha+2)}{N}}=c_*^{\frac{2(\alpha+2)}{N}},$$ i.e. $|w_0|_2=|W_0|_2=c_*$. Hence $w_{c_k}\rightarrow w_0$ in $L^2({{\mathbb R}}^N)$ and then by the interpolation inequality, $$w_{c_k}\rightarrow w_0\ \ \hbox{in}\ L^s({{\mathbb R}}^N)\ \ \hbox{for\ all}\ \ 2\leq s<2^*.$$ Moreover, Lemma \[lem3.2\] shows that $W_0$ is a groundstate solution of . So by Lemma \[lem3.1\] (3)(4), $W_0(x)=O(|x|^{-\frac{N-1}{2}}e^{-|x|})$ as $|x|\rightarrow+\infty$ and we may assume that up to translations, $W_0(x)$ is radially symmetric about the origin.
By , we see that for any $q_i\in\{q_1,\cdots,q_m\}$, $$\frac{1}{\varepsilon^{q_i}}{\displaystyle}\int_{{{\mathbb R}}^N}V(\varepsilon_k x+\varepsilon_ky_k)|w_{c_k}(x)|^2\leq 2M_2.$$ Similarly to the proof of , there exists $x_{j_0}\in\{x_1,\cdots,x_m\}$ and $y_0\in{{\mathbb R}}^N$ such that $\varepsilon_k y_k\rightarrow x_{j_0}$ and $\frac{\varepsilon_k y_k-x_{j_0}}{\varepsilon_k}\rightarrow y_0$ as $k\rightarrow +\infty$. Then similarly to , we see that $$\label{3.28}\begin{array}{ll}
{\displaystyle}\liminf\limits_{k\rightarrow +\infty}\frac{1}{\varepsilon_k^{q}}{\displaystyle}\int_{{{\mathbb R}}^N}V(\varepsilon_kx+\varepsilon_ky_k)|w_{c_k}(x)|^2
&\geq{\displaystyle}\int_{{{\mathbb R}}^N}{\displaystyle}\liminf\limits_{k\rightarrow +\infty}\frac{ V(\varepsilon_kx+\varepsilon_ky_k)}{\varepsilon_k^{q_{j_0}}}|w_{c_k}(x)|^2\\[5mm]
&\geq\mu_{j_0}{\displaystyle}\int_{{{\mathbb R}}^N}|x+y_0|^{q_{j_0}}|w_0(x)|^2\\[5mm]
&={\displaystyle}\frac{\mu_{j_0}}{\beta^{q_{j_0}}}{\displaystyle}\int_{{{\mathbb R}}^N}|x+\beta y_0|^{q_{j_0}}|W_0(|x|)|^2\\[5mm]
&\geq {\displaystyle}\frac{\mu_{j_0}}{\beta^{q_{j_0}}}{\displaystyle}\int_{{{\mathbb R}}^N}|x|^{q_{j_0}}|W_0(x)|^2,
\end{array}$$ where the last inequality is strict if and only if $y_0\neq0$. Hence similarly to , $$\label{3.29}\begin{array}{ll}
{\displaystyle}\liminf\limits_{k\rightarrow +\infty}\frac{e_{c_k}}{\varepsilon_k^{q}}
&\geq {\displaystyle}\frac{1}{2}A(w_0)+\liminf\limits_{k\rightarrow +\infty}{\displaystyle}\frac{1}{2\varepsilon_k^q}{\displaystyle}\int_{{{\mathbb R}}^N}V(\varepsilon_k x+\varepsilon_ky_k)|w_{c_k}(x)|^2\\[5mm]
&\geq{\displaystyle}\frac{1}{2}\left(\beta^2c_*^2\frac{N}{\alpha+2}+{\displaystyle}\frac{\mu_{j_0}}{\beta^{q_{j_0}}}{\displaystyle}\int_{{{\mathbb R}}^N}|x|^{q_{j_0}}|W_0(x)|^2\right)\\[5mm]
&\geq {\displaystyle}c_*^2\left(\frac{\beta^2}{2}\frac{N}{\alpha+2}+\frac{\lambda_{j_0}^{q_{j_0}+2}}{q_{j_0}\beta^{q_{j_0}}}\right)\\[5mm]
&\geq{\displaystyle}\frac{\lambda_{j_0}^2c_*^2}{2}\left(\frac{ N}{\alpha+2}\right)^{\frac{q_{j_0}}{q_{j_0}+2}}\frac{q_{j_0}+2}{q_{j_0}}\\[5mm]
&\geq{\displaystyle}\frac{\lambda^2c_*^2}{2}\left(\frac{N}{\alpha+2}\right)^{\frac{q}{q+2}}\frac{q+2}{q},
\end{array}$$ where $\lambda=\min\limits_{1\leq i\leq m}\lambda_i$ and $q=\max\limits_{1\leq i\leq m}q_i$.
On the other hand, for any $x_i\in \{x_1,\cdots,x_m\}$ and $t>0$, let $v_k(x)=A_k\frac{c_k}{c_*}\left(\frac{t}{\varepsilon_k}\right)^{\frac{N}{2}}\varphi(x-x_i)W_0(\frac{t(x-x_i)}{\varepsilon_k})$, where $\varphi$ is a cut-off function given as in and $A_k>0$ is chosen to satisfy that $v_k\in \widetilde{S}(c_k)$. Then $A_k\rightarrow1$ as $k\rightarrow+\infty$. Similarly to , by the Dominated Convergence theorem, we see that $$\label{3.30}\begin{array}{ll}
{\displaystyle}\lim\limits_{k\rightarrow +\infty}\frac{E(v_k)}{\varepsilon_k^{q}}&={\displaystyle}\frac{t^2}{2}
A(W_0)+\lim\limits_{k\rightarrow +\infty}\frac{t^N}{2\varepsilon_k^{N+q}}{\displaystyle}\int_{{{\mathbb R}}^N}V(x)|\varphi(x-x_i)W_0(\frac{t(x-x_i)}{\varepsilon_k})|^2\\[5mm]
&={\displaystyle}\frac12\left(\frac{t^2c_*^2 N}{\alpha+2}
+\frac{\overline{\mu}_i}{t^{q}}{\displaystyle}\int_{{{\mathbb R}}^N}|x|^{q}|W_0(x)|^2\right)\\[5mm]
&\leq{\displaystyle}c_*^2\left(\frac{t^2}{2}\frac{N}{\alpha+2}
+\frac{\overline{\lambda}_i^{q+2}}{qt^{q}}\right),\end{array}$$ where $$\overline{\mu}_i=\lim\limits_{x\rightarrow x_i}\frac{V(x)}{|x-x_i|^q}=\left\{\begin{array}{ll}
\mu_i, & \hbox{if~$q=q_i$}, \\
+\infty, & \hbox{if~$q\neq q_i$}\\
\end{array}\right.$$ and $$\overline{\lambda}_i=\left(\frac{\overline{\mu}_i q}{2c_*^2}\int_{{{\mathbb R}}^N}|x|^{q}|W_0(x)|^2\right)^{\frac{1}{q+2}}=\left\{\begin{array}{ll}
\lambda_i, & \hbox{if~$q=q_i$}, \\
+\infty, & \hbox{if~$q\neq q_i$}\\
\end{array}\right..$$ So, since $t>0$ is arbitrary, by taking the infimum over $\{\overline{\lambda}_i\}_{i=1}^{m}$ in and combining , we see that $${\displaystyle}\lim\limits_{k\rightarrow+\infty}\frac{e_{c_k}}{\varepsilon_k^{q}}= {\displaystyle}\frac{\lambda^2c_*^2}{2}\left(\frac{N}{\alpha+2}\right)^{\frac{q}{q+2}}\frac{q+2}{q}.$$ Then - must be equalities, which imply that $$y_0=0,\ \ \ \beta=\left(\frac{\alpha+2}{N}\right)^{\frac{1}{q+2}}\lambda$$ and $\varepsilon_k y_k\rightarrow x_{j_0}\in\{x_i|~\lambda_i=\lambda,~1\leq i\leq m\}$. Therefore, $$\varepsilon_k^{\frac{N}{2}}u_{c_k}(\varepsilon_k x+\varepsilon_k y_k)=w_{c_k}(x)\rightarrow w_0(x)= \left((\frac{\alpha+2}{N})^{\frac{1}{q+2}}\lambda\right)^{\frac{N}{2}} W_0((\frac{\alpha+2}{N})^{\frac{1}{q+2}}\lambda x)$$ in $L^{\frac{2Ns}{N+\alpha}}({{\mathbb R}}^N)$ for all $\frac{N+\alpha}{N}\leq s<\frac{N+\alpha}{(N-2)_+}$.
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[^1]: a: Partially supported by NSFC No: 11371159
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We present the first study of the spatial distribution of star formation in $z\sim0.5$ cluster galaxies. The analysis is based on data taken with the Wide Field Camera 3 as part of the Grism Lens-Amplified Survey from Space (GLASS). We illustrate the methodology by focusing on two clusters (MACS0717.5+3745 and MACS1423.8+2404) with different morphologies (one relaxed and one merging) and use foreground and background galaxies as field control sample. The cluster+field sample consists of 42 galaxies with stellar masses in the range 10$^8$-10$^{11}$ M$_\odot$, and star formation rates in the range 1-20 M$_\odot \, yr^{-1}$. Both in clusters and in the field, [H$\alpha$]{} is more extended than the rest-frame UV continuum in 60% of the cases, consistent with diffuse star formation and inside out growth. In $\sim$ 20% of the cases, the [H$\alpha$]{}emission appears more extended in cluster galaxies than in the field, pointing perhaps to ionized gas being stripped and/or star formation being enhanced at large radii. The peak of the [H$\alpha$]{} emission and that of the continuum are offset by less than 1 kpc. We investigate trends with the hot gas density as traced by the X-ray emission, and with the surface mass density as inferred from gravitational lens models and find no conclusive results. The diversity of morphologies and sizes observed in [H$\alpha$]{} illustrates the complexity of the environmental process that regulate star formation. Upcoming analysis of the full GLASS dataset will increase our sample size by almost an order of magnitude, verifying and strengthening the inference from this initial dataset.'
author:
- Benedetta Vulcani
- Tommaso Treu
- 'Kasper B. Schmidt'
- 'Bianca M. Poggianti'
- Alan Dressler
- Adriano Fontana
- Marusa Bradač
- 'Gabriel B. Brammer'
- Austin Hoag
- 'Kuan-Han Huang'
- Matthew Malkan
- Laura Pentericci
- Michele Trenti
- Anja von der Linden
- Louis Abramson
- Julie He
- Glenn Morris
bibliography:
- 'biblio\_SFR.bib'
title: 'THE GRISM LENS-AMPLIFIED SURVEY FROM SPACE (GLASS). V. Extent and spatial distribution of star formation in $z\approx0.5$ cluster galaxies'
---
Introduction
============
Over the past decade, observations have shown that the star formation activity in galaxies has strongly declined since $z\sim 2$ [see, e.g., @hopkins06; @madau14], with a large number of star-forming galaxies evolving into passive galaxies at later times, and the star formation rate at fixed mass progressively decreasing [@bell04; @bell07; @noeske07; @daddi07; @karim11]. The evolution of the star formation activity is coupled to the evolution of galaxy morphologies [@poggianti09], with a significant fraction of today’s early-type galaxies having evolved from late types at relatively recent epochs. Even though transformations occur both in galaxy clusters [@dressler97; @fasano00] and in the field [@oesch10; @capak07], the strength of the decline has been found to depend on environment: galaxies in clusters experience a stronger evolution in star formation activity compared to galaxies in the field [e.g., @poggianti06; @cooper06; @guglielmo15].
Central for our progress in understanding galaxy evolution is identifying the cause of the decline of star formation and of the emergence of the different galaxy types. The mass of galaxies and the environment where they reside are generally believed to play a role for quenching the star formation [e.g., @peng10], but the specific physical mechanisms involved remain obscure. There is no consensus on whether there is one process that dominates quenching across all environments or whether some processes play a larger role in driving galaxy evolution in dense environments than they do in the field [@butcher84; @poggianti99; @dressler99; @treu03; @dressler13].
Each of the processes that have been proposed to quench star formation in galaxies should leave a different signature on the spatial distribution of the star formation activity within the galaxy. For example, ram-pressure stripping from the disk due to the interaction between the galaxy interstellar medium (ISM) and the intergalactic medium [IGM, @gunngott72] is expected to partially or completely remove the ISM, leaving a recognizable pattern of star formation with truncated [H$\alpha$]{}disks smaller than the undisturbed stellar disk [e.g., @yagi15]. Strangulation, which is the removal of the hot gas halo surrounding the galaxy either via ram-pressure or via tidal stripping by the halo potential [@larson80; @balogh00], should deprive the galaxy of its gas reservoir, and leave the existing interstellar medium in the disk to be consumed by star formation. Strong tidal interactions and mergers, tidal effects of the cluster as a whole and harassment, that is the cumulative effect of several weak and fast tidal encounters [@moore96], thermal evaporation [@cowie77] and turbulent/viscous stripping [@nulsen82] can also deplete the gas in a non homogeneous way.
In order to address how star formation is suppressed in the different regions of the galaxy, a key ingredient is the spatial distributions of both the past star formation, as traced by the existing stellar population, and of the instantaneous star formation. The latter can be traced by the [H$\alpha$]{}line emission as it scales with the quantity of ionizing photons produced by hot young stars [@kennicutt98].
In the local universe, a few studies have focused on the analysis of [H$\alpha$]{}spatial distribution of a limited number of systems in clusters [e.g. @merluzzi13; @fumagalli14], detecting debris of material that appear to be stripped from the main body of the galaxy, and whose morphology is suggestive of gas-only removal mechanisms, such as ram pressure stripping.
However, our current understanding is that much of the activity in cluster galaxies happens beyond the local universe at $z=0-1$ and it is therefore essential to gather information in this redshift range. A number of [H$\alpha$]{}surveys up to $z\sim$1 have been undertaken in the field using narrow-band imaging [e.g., @sobral13] and with WFC3 grism observations [e.g., @atek10; @straughn11]. In clusters, narrow-band [H$\alpha$]{}studies are available for just a few systems at $z = 0.3-1$ [@kodama04; @finn05; @koyama11] and a few other higher-$z$ overdense regions (@kurk04a 2004a; @kurk05 2004b; @geach08 [@hatch11; @koyama13]). These ground-based studies provide integrated [H$\alpha$]{}fluxes, and no spatial information.
Recently, spatially resolved star formation maps at $z\sim1$ have been obtained for field galaxies using both the ACS I band and the G141 grism on the Wide Field Camera 3 (WFC3) on board the Hubble Space Telescope (HST) as part of the 3D-HST Survey (@vandokkum11 [@brammer12; @schmidt13]; Momcheva et al. in prep). @nelson12 [@nelson13] mapped the [H$\alpha$]{}and stellar continuum with high resolution for $\sim$ 60 galaxies and showed that star formation broadly follows the rest-frame optical light, but is slightly more extended. By stacking the [H$\alpha$]{}emission, they measured structural parameters of stellar continuum emission and star formation, finding that star formation occurred in approximately exponential distributions. They concluded that star formation at $z\sim$1 generally occurred in disks.
[@wuyts13] expanded the sample analyzed by @nelson12 [@nelson13] and characterized the resolved stellar populations of $\sim$500 massive star-forming galaxies, with multi-wavelength broad-band imaging from CANDELS [@wuyts12] and [H$\alpha$]{}surface brightness profiles. They found the [H$\alpha$]{}morphologies to resemble more closely those observed in the ACS I band than in the WFC3 H band, especially for the larger systems. They also found that the rate of ongoing star formation per unit area tracks the amount of stellar mass assembled over the same area. Off-center clumps are characterized by enhanced [H$\alpha$]{}equivalent widths, bluer broad-band colors and correspondingly higher specific star formation rates (SFRs) than the underlying disk, implying they are a star formation phenomenon.
More recently, [@nelson15], exploiting a much larger sample, studied the behavior of the [H$\alpha$]{}profiles above and below the main sequence and showed that star formation is enhanced at all radii above the main sequence, and suppressed at all radii below the main sequence.
In this paper we present a pilot study characterizing the spatial distribution of the [H$\alpha$]{}emission in cluster galaxies beyond the local universe based on WFC3-IR data drawn from the Grism Lens-Amplified Survey from Space (GLASS; GO-13459; PI: Treu,[^1] @schmidt14 [@treu15]). The GLASS G102 data yield spatially resolved [H$\alpha$]{}fluxes for all star-forming galaxies in the core ($<1$ Mpc) of 10 clusters at $z = 0.31 - 0.69$, with an order of magnitude improvement in sensitivity compared to previous studies [@sobral13]. Each cluster is observed at two different position angles. These two orientations allow us to mitigate the impact of contamination from overlapping spectra, and reliably measure for the first time the relative position of the [H$\alpha$]{} emission with respect to the continuum.
We illustrate the methodology and first results of this approach by analyzing two of the ten clusters in the GLASS sample. Among the first clusters that have been observed by GLASS we selected MACS0717.5+3745 (hereafter [MACS0717]{}) and MACS1423.8+2404 (hereafter [MACS1423]{}) based on the following criteria. First, we required similar redshift, so as to minimize evolutionary effects and differences in the sensitivity/selection function ($z=0.55$). Second, we selected them to be in very different dynamical states (MACS1423 is relaxed, MACS0717 is an active merger), so as to span the range of expected environments. A homogeneous control field sample is obtained by selecting objects in the immediate foreground and background of the two clusters. In total the sample presented in this pilot paper consists of 42 objects, evenly distributed between the two clusters and the field (15 and 10 cluster galaxies and 9 and 8 field galaxies in [MACS0717]{}and [MACS1423]{}, respectively). In a forthcoming paper, to appear after the complete GLASS data have been processed, we will present an analysis of the entire sample.
We assume $H_{0}=70 \, \rm km \, s^{-1} \, Mpc^{-1}$, $\Omega_{0}=0.3$, and $\Omega_{\Lambda} =0.7$. The adopted initial mass function (IMF) is that of [@kr01] in the mass range 0.1–100 $\textrm{M}_{\odot}$.
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -- -- -- -- -- -- -- -- --
**[cluster]{} & **[RA]{} (J2000) & **[DEC]{} (J2000) & **[z]{} & **[HST imaging]{} & **[L$_{\rm X}$]{} (10$^{44}$erg s$^{-1}$) & **[M$_{\rm 500}$]{} (10$^{14}$M$_\odot$) & **[r$_{\rm 500}$]{} (Mpc) & [**PA1**]{} & [**PA2**]{}\
MACS0717.5+3745 & 07:17:31.6 & +37:45:18 & 0.548 & CLASH/HFF2 & 24.99$\pm$0.92& 24.9$\pm$2.7 & 1.69$\pm$0.06 & 020& 280\
MACS1423.8+2404 & 14:23:47.8 & +24:04:40 & 0.545 & CLASH & 13.96$\pm$0.52& 6.64$\pm$0.88 & 1.09$\pm$0.05 & 008& 088\
****************
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ -- -- -- -- -- -- -- -- --
The Grism Lens-Amplified Survey from Space data set {#sec:glass}
===================================================
GLASS is a 140 orbit slitless spectroscopic survey with [*HST*]{} in cycle 21. It has observed the cores of 10 massive galaxy clusters with the WFC3 NIR grisms G102 and G141 providing an uninterrupted wavelength coverage from 0.8$\mu$m to 1.7$\mu$m. Observations for GLASS were completed in January 2015. Amongst the 10 GLASS clusters, 6 are targeted by the Hubble Frontier Fields (HFF; P.I. Lotz) and 8 by the Cluster Lensing And Supernova survey with Hubble (CLASH; P.I. Postman, @postman12). Prior to each grism exposure, imaging through either F105W or F140W is obtained to assist the extraction of the spectra and the modeling of contamination from nearby objects on the sky. The total exposure time per cluster is 10 orbits in G102 (with either F105W or F140W) and 4 orbits in G141 with F140W. Each cluster is observed at two position angles (PAs) approximately 90 degrees apart to facilitate clean extraction of the spectra for objects in crowded cluster fields.
Data reduction {#sec:reduction}
--------------
The GLASS observations are designed after the 3D-HST observing strategy and were processed with an updated version of the 3D- HST reduction pipeline[^2] described by [@brammer12]. The updated pipeline combines the individual exposures into mosaics using AstroDrizzle [@gonzaga12], replacing the MultiDrizzle package [@koekemoer03] used in earlier versions of the pipeline.
The direct images were sky subtracted by fitting a 2nd order polynomial to each of the source-subtracted exposures. Each exposure is then interlaced to a final G102(G141) grism mosaic. Before sky-subtraction and interlacing, each individual exposure was checked and corrected for elevated backgrounds due to the He Earth-glow described by [@brammer14]. From the final mosaics, the spectra of individual objects are extracted by predicting the position and extent of each two-dimensional spectrum based on the SExtractor [@bertin96] segmentation map combined with deep mosaic of the direct NIR GLASS and CLASH images. As this is done for each object, the contamination, i.e., the dispersed light from neighboring objects in the direct image field-of-view, can be estimated and accounted for. Full details on the sample selection, data observations and data reduction are given in [@treu15], while a complete description of the 3D-HST image preparation pipeline, spectral extractions, and spectral fitting, is provided by Momcheva et al. (in prep.).
The spectra analyzed in this study were all visually inspected with the publicly available GLASS inspection GUI, GiG[^3] [@treu15], in order to identify and flag erroneous models from the reduction, assess the degree of contamination in the spectra and flag and identify strong emission lines and the presence of a continuum.
Redshift determinations
-----------------------
In order to determine redshifts, templates are compared to each of the four available grism spectra independently (G102 and G141 at two PAs each) to compute a posterior distribution function for the redshift. If available, photometric redshift distributions can be used as input priors to the grism fits in order to reduce computational time. Then, with the help of the publicly available GLASS inspection GUI for redshifts (GIGz, @treu15), we flag which grism fits are reliable or alternatively enter a redshift by hand if the redshift is misidentified by the automatic procedure. Using GIGz we assign a quality flag to the redshift (4=secure; 3=probable; 2=possible; 1=tentative, but likely an artifact; 0=no-$z$). These quality criteria take into account the signal to noise ratio of the detection, the possibility that the line is a contaminant, and the identification of the feature with a specific emission line. This procedure is carried out independently by at least two inspectors per cluster and then their outputs are combined [see @treu15 for details].
We note that for [MACS0717]{}, one of the clusters analyzed in this paper (see §\[sec:sample\]), a redshift catalog was already published by [@ebeling14]. Considering only galaxies with quality flag $>$2.5, four objects are in common between the two catalogs (cross match within 1") and the reported redshift agree at the 10$^{-3}$ level, consistent with the resolution of the grism.
The sample {#sec:sample}
----------
Even though all GLASS data have been obtained and reduced, their inspection and quality control is still underway, and is expected to be completed and released by Winter 2016 [@treu15]. Among the clusters for which quality control is sufficiently advanced for this work, we select two with similar redshift, [MACS0717]{}and [MACS1423]{}, whose properties are listed in Table \[tab:clus\].
From the redshift catalogs, we extract galaxies with secure redshift (flag$\geq$2.5) and consider as cluster members galaxies with redshift within $\pm$0.03 the cluster redshift. Then, we select galaxies with visually detected [H$\alpha$]{}in emission. Given the cluster redshifts, [H$\alpha$]{}is found at an observed wavelength of $\sim$10,100 Å, and we therefore only exploit the G102 grism data in our analysis.
 
We assemble a control sample which includes all galaxies with secure redshift, [H$\alpha$]{}in emission detected in the G102 grism and redshift outside the cluster redshift intervals ($z<z_{cl}-0.03$ or $z>z_{cl}+0.03$). The field sample includes galaxies in the redshift range 0.2$<z<$0.7, and differences among these galaxies are therefore potentially due to evolutionary effects, although the average redshift is very similar between the two samples. We note that we do not have additional information on the environments in which these galaxies reside, therefore they might be located in some groups.
Overall, our sample includes 15 cluster members and 9 field galaxies from the [MACS0717]{}field, and 10 cluster members and 8 field galaxies from the [MACS1423]{}field.
Stellar masses have been estimated using the broad-band CLASH photometry [@postman12] and a set of templates, computed with standard spectral synthesis models [@bc03], and fixing the redshift at the spectroscopic one. As it was done in previous papers [e.g., @fontana04; @fontana06; @santini15 for details], we have used a range of exponential time scales ranging from $0.1$ to $\infty$ Gyr. A [@salpeter55] IMF, ranging over a set of metallicities (from Z = 0.02Z$_\odot$ to Z=2.5Z$_\odot$) and dust extinction (0<E(B-V)<1.1, with a Calzetti extinction curve) has been initially chosen, and then converted to a [@kr01] IMF. We have also added emission lines in a self-consistent way, as described by [@castellano14], that provide an important contribution to our H$\alpha$-emitting galaxies. Uncertainties on the estimated masses have been derived by scanning (for each galaxy) all the templates and retaining only masses corresponding to models with $P(\chi^2)>0.1$ [@santini15].
Figure \[fig:z\_m\_distr\] shows the redshift and mass distribution for cluster and field samples separately. We note that, while the mass range spanned in the two environments is similar, going from 10$^8$ to 10$^{11}$ M$_\odot$, the field galaxies are systematically less massive than cluster galaxies. Therefore, in the following, when comparing cluster and field populations, differences might be due to the different mass distribution, and not only to purely environmental effects.
Methodology
===========
[H$\alpha$]{}maps
-----------------
The combined spatial resolution on the WFC3 and of the grism yield a spectrum that can be seen as images of a galaxy taken at $\sim$24 Å increments ($\sim$12 Å after interlacing) and placed next to each other (offset by one pixel) on the detector. Thus, an emission feature in a high spatial resolution slitless spectrum is essentially an image of a galaxy at that wavelength.
 
Figure \[fig:methods\] shows two examples of the procedure we followed to create [H$\alpha$]{}emission line maps and therefore SFR maps. In the first steps, we treat the spectra coming from the different exposures (one per PA) of each galaxy independently, and only in the last step we combine them. Both panels $1a$ and $1b$ show the flux-calibrated galaxy 2D continuum spectra, after the sky background and the contamination have been subtracted. From two regions contiguous to the [H$\alpha$]{}emission we determine the $y$-position of the peak of the continuum. This position will be needed to measure the offset in the $y$-direction of the [H$\alpha$]{}emission with respect to the galaxy center in the light of the continuum. Subsequently, we subtract the 2D stellar continuum model obtained by convolving the best-fit 1D SED without emission lines with the actual 2D data, ensuring that all model flux pixels are non-negative (panels $2a$ and $2b$). If the sky, the contamination and the continuum were fit perfectly, we should be left only with the flux coming from the emission lines. We find that counts around the lines are slightly negative, suggesting that the continuum subtraction is somewhat too aggressive. Therefore, we select a box just above and one just below the emission line and measure the median flux which is further subtracted from the entire spectrum. The residual is a map of the galaxy in the light of the [H$\alpha$]{}line (panels $3a$ and $3b$). As a last step, we superimpose the [H$\alpha$]{}map onto an image of the galaxy taken with the F475W filter (rest-frame UV). We use F475W to map relatively recent ($\sim$100 Myr) star formation, as opposed to ongoing ($\sim$10Myr) star formation traced by [H$\alpha$]{}. To do so, we align each map to the image of the galaxy in the light of the continuum, rotating each map by the angle of its PA, keeping the $y$-offset unaltered with the respect to the continuum. On the $x$-axis, there is a degeneracy between the spatial dimension and the wavelength uncertainty, it is therefore not possible to determine very accurately the central position of the [H$\alpha$]{}map for each PA separately. Nonetheless, for the cases in which spectra from both PAs are reliable (28/42), we use the fact that the 2 PAs differ by almost $90\degree$, therefore the $x$-direction of one spectrum roughly corresponds to the $y$-direction of the second spectrum and vice-versa. We can therefore shift the two spectra independently along their $x$-direction to force the center of the emission of the two maps to coincide. The results are shown in Figure \[fig:methods\], panels $4$. For the galaxies with reliable spectra in both PAs, we can also measure the real distance between the peak of the [H$\alpha$]{}emission and the continuum emission, obtained as the quadratic sum of the two offsets.
Finally, for cluster galaxies, we also measure the magnitude of the offset between the [H$\alpha$]{}and the continuum as projected along the cluster radial direction, determined by the line connecting the cluster-center and the galaxy center in the continuum light. We assign a positive sign to the projected offset when the peak of the [H$\alpha$]{}is between the cluster center and the peak of the continuum.
[H$\alpha$]{}map sizes
----------------------
Since one of our aims is to compare the extent of [H$\alpha$]{}light to the extent of the continuum light, we estimate galaxy sizes at different wavelengths by measuring the second order moment of the light distribution, which gives us the width of the distribution and therefore the extension of the galaxy: $$\sigma=\sqrt{\frac{\sum_{i=1}^{N} \left[I(x_i) \cdot x_i^2\right]}{\sum_{i=1}^{N} I(x_i)}-\left[\frac{\sum_{i=1}^{N} \left[I(x_i) \cdot x_i\right]}{\sum_{i=1}^{N} I(x_i)}\right]^2}$$ with $x_i$ along the spectrum, and $I(x_i)$ flux at the corresponding position. We measure sizes both along the $x-$ and $y-$direction. The average size is obtained by taking the mean of the two and summing errors in quadrature. This adopted size definition is independent on the galaxy’s center. When spectra from both PAs are reliable, we take the average size and sum the errors in quadrature, after having checked that the measurements from the two PAs are consistent within the uncertainty.
Besides on the [H$\alpha$]{}light, we compute sizes both on the F475W filter, to map the star formation occurred in roughly the last 100 Myr, and on the F110W filter, which probes the rest-frame optical continuum and therefore maps the older stellar population. We correct our size estimates for the point spread function (PSF) of our observations. We estimate the mean full width half maximum (FWHM) in each band by taking the average of the FWHM of 5 stars. We then subtract in quadrature the PSF (=FWHM/2.355) from the sizes. The values we obtain are $\sim$0.03$^{\prime\prime}$ in the F475W, and $\sim$0.055$^{\prime\prime}$ in the F110W and G102 filters. We note that the PSF correction is generally much smaller than the sizes we observe therefore the impact of the correction is negligible.
We note that more robust measurements are currently underway for the entire GLASS sample and will be presented in a forthcoming paper.
---------------- ------------- ------------- -------- ---------------------- -------------- -------------- --------------- ------- ------
MACS0717-00173 07:17:35.64 +37:45:59.2 0.556 9.5$^{+0.4}_{-0.5}$ 23$\pm$2 3.4$\pm$0.5 0.20$\pm$0.03 0.276 466
MACS0717-00234 07:17:35.14 +37:45:52.9 0.549 10.5$^{+0.1}_{-0.3}$ 14.1$\pm$0.3 8.6$\pm$0.8 0.16$\pm$0.01 0.247 418
MACS0717-00431 07:17:36.59 +37:45:40.1 0.5495 8.9$^{+0.2}_{-0.3}$ 29$\pm$3 2.9$\pm$0.4 0.30$\pm$0.06 0.226 382
MACS0717-00596 07:17:37.76 +37:45:30.1 0.5475 9.9$^{+0.2}_{-0.4}$ 65$\pm$2 14.0$\pm$0.6 0.54$\pm$0.03 0.230 388
MACS0717-00624 07:17:33.44 +37:45:28.9 0.5725 9.0$^{+0.1}_{-0.3}$ 35$\pm$4 2.3$\pm$0.5 0.15$\pm$0.04 0.153 259
MACS0717-00674 07:17:34.98 +37:45:27.4 0.574 9.3$^{+0.2}_{-0.2}$ 24$\pm$5 0.8$\pm$0.4 0.08$\pm$0.06 0.152 257
MACS0717-00977 07:17:38.86 +37:45:20.0 0.567 9.3$^{+0.2}_{-0.3}$ 18.5$\pm$0.6 4.8$\pm$0.7 0.10$\pm$0.02 0.248 419
MACS0717-01208 07:17:32.79 +37:44:41.3 0.5585 9.5$^{+0.3}_{-0.7}$ 63$\pm$5 5.1$\pm$0.4 0.53$\pm$0.06 0.062 105
MACS0717-01305 07:17:35.55 +37:44:41.8 0.5285 10.7$^{+0.2}_{-0.2}$ 18.1$\pm$0.5 18$\pm$1 0.14$\pm$0.01 0.074 1256
MACS0717-02181 07:17:31.51 +37:44:13.1 0.564 9.0$^{+0.1}_{-0.4}$ 11$\pm$4 1.3$\pm$0.4 0.16$\pm$0.07 0.176 298
MACS0717-02189 07:17:33.76 +37:44:08.4 0.5275 10.0$^{+0.3}_{-0.4}$ 6.2$\pm$0.2 3.0$\pm$0.6 0.17$\pm$0.06 0.154 260
MACS0717-02297 07:17:30.17 +37:44:04.1 0.5485 9.2$^{+0.3}_{-0.3}$ 19$\pm$1 3.1$\pm$0.4 0.28$\pm$0.04 0.242 408
MACS0717-02334 07:17:32.35 +37:43:59.4 0.534 9.9$^{+0.2}_{-0.2}$ 13.8$\pm$0.3 7.4$\pm$0.5 0.35$\pm$0.03 0.202 341
MACS0717-02432 07:17:31.45 +37:43:50.7 0.548 9.4$^{+0.4}_{-0.3}$ 55$\pm$9 2.9$\pm$0.5 0.17$\pm$0.03 0.249 420
MACS0717-02574 07:17:31.76 +37:43:33.8 0.5375 10.2$^{+0.6}_{-0.4}$ 29$\pm$1 9.1$\pm$0.7 0.25$\pm$0.02 0.302 511
MACS1423-00152 14:23:49.65 +24:05:43.0 0.53 9.0$^{+0.2}_{-0.3}$ 21$\pm$2 2.0$\pm$0.4 0.15$\pm$0.03 0.345 376
MACS1423-00229 14:23:46.25 +24:05:32.6 0.563 8.3$^{+0.1}_{-0.1}$ 110$\pm$30 4.1$\pm$0.5 0.28$\pm$0.05 0.313 342
MACS1423-00310 14:23:45.62 +24:05:27.3 0.53 9.9$^{+0.2}_{-0.4}$ 26$\pm$1 3.7$\pm$0.5 0.26$\pm$0.05 0.319 348
MACS1423-00319 14:23:48.24 +24:05:20.7 0.536 10.4$^{+0.2}_{-0.5}$ 36$\pm$4 3.0$\pm$0.7 0.12$\pm$0.03 0.198 215
MACS1423-00446 14:23:45.18 +24:05:16.4 0.548 9.9$^{+0.3}_{-0.4}$ 57$\pm$2 8.7$\pm$0.6 0.40$\pm$0.04 0.304 331
MACS1423-00487 14:23:47.81 +24:05:13.6 0.53 9.7$^{+0.3}_{-0.4}$ 61$\pm$5 7.1$\pm$0.6 0.38$\pm$0.03 0.161 175
MACS1423-00831 14:23:49.24 +24:04:52.3 0.575 8.5$^{+0.9}_{-0.5}$ 41$\pm$4 2.6$\pm$0.5 0.11$\pm$0.03 0.081 89
MACS1423-01516 14:23:48.56 +24:04:14.6 0.54 10.6$^{+0.2}_{-0.3}$ 27.1$\pm$0.3 14.8$\pm$0.7 0.48$\pm$0.03 0.190 208
MACS1423-01253 14:23:53.13 +24:04:29.4 0.556 9.1$^{+0.1}_{-0.2}$ 13$\pm$1 2.4$\pm$0.5 0.12$\pm$0.02 0.400 437
MACS1423-01910 14:23:49.20 +24:03:42.7 0.532 8.5$^{+0.2}_{-0.3}$ 100$\pm$10 3.2$\pm$0.4 0.35$\pm$0.06 0.383 418
---------------- ------------- ------------- -------- ---------------------- -------------- -------------- --------------- ------- ------
---------------- ---------------- ---------------- --------------- ------------- ------ ------ ------ ------ ------ ------ -------
MACS0717-00173 - 0.4$\pm$0.1 4.1$\pm$0.3 1.0$\pm$0.2 1.63 0.82 3.77 1.24 3.25 1.62 -47.6
MACS0717-00234 - -0.4$\pm$0.1 3.7$\pm$0.5 2.0$\pm$0.8 3.37 2.44 3.17 2.13 5.32 3.2 -16.3
MACS0717-00431 -0.43$\pm$0.1 0.18$\pm$0.07 1.38$\pm$0.09 1.0$\pm$0.2 1.09 0.98 1.62 1.45 1.81 1.61 17.4
MACS0717-00596 0.11$\pm$0.05 -0.16$\pm$0.03 3.2$\pm$0.4 2.6$\pm$0.3 2.84 1.82 3.39 2.22 3.5 2.39 -61.8
MACS0717-00624 -0.4$\pm$0.2 -0.5$\pm$0.3 3.7$\pm$0.8 4$\pm$1 1.88 1.20 0.80 0.54 2.58 1.83 -81.5
MACS0717-00674 0.2$\pm$0.4 0.7$\pm$0.3 4.3$\pm$0.3 3$\pm$1 2.28 2.11 5.69 6.28 1.96 1.51 -58.4
MACS0717-00977 - 0.1$\pm$0.2 3.3$\pm$0.7 3.7$\pm$0.6 3.06 2.71 3.67 3.45 4.28 3.55 -80.3
MACS0717-01208 -0.19$\pm$0.07 0.31$\pm$0.04 3.9$\pm$0.6 3.2$\pm$0.5 3.20 3.11 1.64 1.95 1.95 1.57 -44.7
MACS0717-01305 -1.6$\pm$0.1 -1.0$\pm$0.2 3.1$\pm$0.3 2.9$\pm$0.4 3.71 3.73 3.57 3.12 7.06 5.4 23.5
MACS0717-02181 -0.2$\pm$0.2 0.1$\pm$0.2 4$\pm$1 3.7$\pm$0.9 1.07 0.91 - 4.61 1.68 1.52 -47.0
MACS0717-02189 -1.9$\pm$0.3 -0.1$\pm$0.1 3.2$\pm$0.3 2.7$\pm$0.4 2.98 3.06 3.05 1.40 3.5 1.5 -55.1
MACS0717-02297 -0.06$\pm$0.09 - 2$\pm$1 1.5$\pm$0.6 0.94 0.97 2.89 1.40 2.01 1.7 -14.1
MACS0717-02334 - 0.62$\pm$0.05 1$\pm$1.0 1.7$\pm$0.6 2.77 2.14 2.23 1.52 3.35 2.02 -74.7
MACS0717-02432 0.4$\pm$0.2 - 1$\pm$1 0.8$\pm$0.1 2.68 0.59 2.89 0.73 3.69 1.48 -80.5
MACS0717-02574 0.11$\pm$0.09 - 3.6$\pm$0.3 3.5$\pm$0.3 - - 3.93 1.25 7.34 1.57 -74.0
MACS1423-00152 -0.2$\pm$0.2 - 6$\pm$2 5$\pm$2 1.16 1.12 0.98 1.03 0.71 0.61 3.1
MACS1423-00229 0.23$\pm$0.08 -0.1$\pm$0.1 4.4$\pm$0.4 3$\pm$1 3.09 1.68 3.01 0.61 1.28 1.05 -6.0
MACS1423-00310 0.16$\pm$0.07 0.5$\pm$0.2 4.2$\pm$0.5 2.8$\pm$0.7 2.02 2.10 1.85 2.49 2.17 1.96 16.7
MACS1423-00319 - 0.11$\pm$0.09 4.5$\pm$0.5 5.0$\pm$0.5 2.92 2.14 1.11 0.84 3.65 3.18 4.0
MACS1423-00446 -0.28$\pm$0.05 -0.18$\pm$0.06 3.2$\pm$0.3 3.3$\pm$0.3 2.60 1.95 2.92 2.09 6.71 2.56 86.4
MACS1423-00487 - 0.42$\pm$0.07 3.1$\pm$0.3 1.4$\pm$0.1 3.03 1.72 1.90 0.89 0.81 0.44 -55.4
MACS1423-00831 -0.1$\pm$0.2 -0.2$\pm$0.3 4.1$\pm$0.6 4.9$\pm$0.5 3.06 1.75 1.88 1.13 1.0 0.8 -12.0
MACS1423-01253 -0.0$\pm$0.2 - 4.0$\pm$0.8 3.2$\pm$0.9 2.09 1.75 1.25 1.09 2.59 1.73 70.7
MACS1423-01516 -0.29$\pm$0.04 -0.76$\pm$0.06 3.8$\pm$0.2 3.7$\pm$0.2 2.50 2.33 2.83 2.90 1.57 1.37 75.6
MACS1423-01910 -0.15$\pm$0.05 0.5$\pm$0.1 2$\pm$1 4$\pm$1 1.62 1.16 1.45 1.90 0.68 0.53 22.1
---------------- ---------------- ---------------- --------------- ------------- ------ ------ ------ ------ ------ ------ -------
---------------- ------------- ------------- -------- ---------------------- -------------- -------------- ---------------
MACS0717-00236 07:17:34.48 +37:45:52.1 0.39 10.2$^{+0.3}_{-0.3}$ 11.3$\pm$0.2 15$\pm$1 0.29$\pm$0.03
MACS0717-00450 07:17:37.39 +37:45:34.9 0.5965 8.5$^{+0.2}_{-0.5}$ 90$\pm$40 2.4$\pm$0.5 0.13$\pm$0.04
MACS0717-01234 07:17:37.56 +37:45:09.3 0.2295 8.1$^{+0.3}_{-0.3}$ 110$\pm$10 7.2$\pm$1.6 0.6$\pm$0.2
MACS0717-01416 07:17:39.72 +37:44:50.7 0.5095 9.7$^{+0.3}_{-0.3}$ 32$\pm$1 6.0$\pm$0.4 0.51$\pm$0.03
MACS0717-01477 07:17:29.74 +37:44:46.2 0.45 9.5$^{+0.1}_{-0.3}$ 9.6$\pm$0.6 9$\pm$1 0.13$\pm$0.02
MACS0717-01589 07:17:32.33 +37:44:37.6 0.385 8.0$^{+0.9}_{-0.1}$ 40$\pm$2 10$\pm$1 0.24$\pm$0.02
MACS0717-02371 07:17:31.95 +37:44:01.7 0.263 8.8$^{+0.3}_{-0.5}$ 17$\pm$3 2.7$\pm$0.8 0.4$\pm$0.2
MACS0717-02390 07:17:34.63 +37:43:53.8 0.473 8.7$^{+0.1}_{-0.2}$ 35$\pm$2 6$\pm$1 0.17$\pm$0.05
MACS0717-02445 07:17:32.38 +37:43:51.0 0.49 8.7$^{+0.1}_{-0.2}$ 90$\pm$30 2.8$\pm$0.5 0.20$\pm$0.06
MACS1423-00246 14:23:49.68 +24:05:33.6 0.66 7.0$^{+0.9}_{-0.7}$ 70$\pm$50 0.4$\pm$0.2 0.12$\pm$0.06
MACS1423-00256 14:23:45.35 +24:05:31.4 0.71 7.8$^{+0.8}_{-0.5}$ - 0.2$\pm$0.2 0.1$\pm$0.1
MACS1423-00463 14:23:46.44 +24:05:15.4 0.62 8.5$^{+0.3}_{-0.1}$ 77$\pm$9 1.8$\pm$0.4 0.13$\pm$0.03
MACS1423-00610 14:23:49.10 +24:05:02.6 0.655 10.3$^{+0.2}_{-0.3}$ 24.3$\pm$0.5 9.0$\pm$0.7 0.22$\pm$0.02
MACS1423-00677 14:23:45.68 +24:04:55.2 0.665 8.9$^{+0.2}_{-0.2}$ 49$\pm$5 4.3$\pm$0.4 0.33$\pm$0.05
MACS1423-01729 14:23:46.13 +24:04:00.2 0.455 8.7$^{+0.4}_{-0.4}$ 42$\pm$4 3.4$\pm$0.5 0.28$\pm$0.06
MACS1423-01771 14:23:44.43 +24:03:56.1 0.65 10.1$^{+0.2}_{-0.2}$ 38$\pm$1 11.8$\pm$0.7 0.34$\pm$0.02
MACS1423-01972 14:23:48.07 +24:03:34.6 0.278 8.3$^{+0.5}_{-0.3}$ 140$\pm$50 3.4$\pm$0.6 0.9$\pm$0.2
---------------- ------------- ------------- -------- ---------------------- -------------- -------------- ---------------
---------------- --------------- ---------------- ------------- ------------- ------ ------ ------ ------ ------ ------ ------- -- -- -- -- -- --
MACS0717-00236 0.1$\pm$0.1 0.6$\pm$0.1 3.8$\pm$0.1 3.3$\pm$0.2 3.09 2.62 3.39 2.85 4.85 3.59 47.9
MACS0717-00450 -2.2$\pm$0.5 0.8$\pm$0.2 3.2$\pm$0.5 0.8$\pm$0.2 3.76 0.83 3.72 3.82 3.14 1.85 4.8
MACS0717-01234 0$\pm$1 0.3$\pm$0.2 2$\pm$2 0.7$\pm$0.2 1.27 1.13 1.12 1.01 2.88 1.31 76.2
MACS0717-01416 - -0.24$\pm$0.05 5$\pm$1 5$\pm$1 - - 2.25 2.56 1.94 1.82 -0.9
MACS0717-01477 -1.3$\pm$0.3 -0.2$\pm$0.2 3.5$\pm$0.4 3.7$\pm$0.3 3.29 2.90 3.50 3.29 5.29 4.08 -19.3
MACS0717-01589 - -0.1$\pm$0.1 2.4$\pm$0.7 1$\pm$1 3.29 3.37 3.40 3.59 3.91 3.5 -29.1
MACS0717-02371 0.4$\pm$0.4 -0.1$\pm$0.7 1.4$\pm$0.6 1.5$\pm$0.9 0.63 1.04 0.69 0.89 1.34 1.15 -27.2
MACS0717-02390 0.1$\pm$0.1 0.1$\pm$0.1 1.5$\pm$0.8 0.9$\pm$0.1 2.03 1.63 3.90 2.72 6.32 3.36 -86.5
MACS0717-02445 0.1$\pm$0.1 -0.1$\pm$0.3 4$\pm$1 3$\pm$1 1.61 0.48 2.88 0.96 2.43 1.25 -77.7
MACS1423-00246 -0.2$\pm$0.2 - - 5$\pm$1 - - 5.67 7.03 0.7 0.44 -77.6
MACS1423-00256 - -1.2$\pm$0.7 - 0.9$\pm$0.3 1.41 4.48 4.35 2.51 2.63 2.27 -11.5
MACS1423-00463 0.3$\pm$0.2 - 34$\pm$1 3$\pm$2 2.43 2.36 2.14 1.46 1.58 0.86 32.7
MACS1423-00610 -1.2$\pm$0.1 0.11$\pm$0.09 4.0$\pm$0.3 3.2$\pm$0.3 3.59 2.53 3.79 2.32 1.72 0.71 74.3
MACS1423-00677 0.01$\pm$0.05 -0.3$\pm$0.1 4.0$\pm$0.3 3.6$\pm$0.4 2.17 1.73 2.16 3.52 1.34 1.17 20.5
MACS1423-01729 -0.1$\pm$0.1 -0.7$\pm$0.2 3$\pm$1 3.4$\pm$0.6 1.86 1.65 1.88 1.81 1.84 1.43 47.5
MACS1423-01771 - 0.07$\pm$0.07 3.7$\pm$0.3 3.4$\pm$0.3 3.12 2.84 3.35 3.26 4.22 2.49 34.3
MACS1423-01972 0.2$\pm$0.1 -0.1$\pm$0.1 2.3$\pm$0.8 0.9$\pm$0.3 0.78 1.19 1.21 1.25 4.05 1.67 -40.8
---------------- --------------- ---------------- ------------- ------------- ------ ------ ------ ------ ------ ------ ------- -- -- -- -- -- --
SFRs and EW([H$\alpha$]{})s
---------------------------
From the [H$\alpha$]{}maps we also derive SFRs. We use the conversion factor derived by [@kennicutt94] and [@madau98]: $$\textrm{SFR} [\textrm{M}_\odot \, \textrm{yr}^{-1}] = 5.5 \times 10^{-42}\textrm{L}(H\alpha)[\textrm{erg}\, \textrm{s}^{-1}]$$ valid for a [@kr01] IMF. We compute both the surface SFR density ($\Sigma$, $\textrm{M}_\odot \, \textrm{yr}^{-1}\, \textrm{kpc}^{-2}$) and the total SFRs ($\textrm{M}_\odot \, \textrm{yr}^{-1}$), separately for the spectra coming from the two PAs and then we combine them taking the mean values. Errors are summed in quadrature. The total SFRs are obtained summing the surface SFR density within the Kron radius[^4] of the galaxy.
There are two major limitations when using [H$\alpha$]{}as SFR estimator: the contamination by the \[NII\] line doublet, and uncertainties in the extinction corrections to be applied to each galaxy.
To correct for the scatter due to the \[NII\] contamination, we apply the locally calibrated correction factor given by [@james05]. As opposed to previous works which considered only central regions, these authors developed a method which takes into account the variation of the [H$\alpha$]{}/\[NII\] with radial distance from the galaxy center, finding an average value of [H$\alpha$]{}/([H$\alpha$]{}+ \[NII\])= 0.823. This approach is appropriate given our goal to investigate extended emission.
![$\Sigma$SFR-SFR for cluster (blue) and field (red) galaxies in our sample. \[fig:SFR\_SFRd\]](SFR_SFR_dens_475_PSFcorr.png)
The second major problem when deriving SFR([H$\alpha$]{}) is the effect of dust extinction. Star formation normally takes place in dense and dusty molecular clouds, so a significant fraction of the emitted light from young stars is absorbed by the dust and re-emitted at rest-frame IR wavelengths. [@hopkins01] modeled a SFR-dependent attenuation by dust, characterized by the Calzetti reddening curve of the form $$\begin{aligned}
\log(\textrm{SFR}_i) = \log(\textrm{SFR}_o(H\alpha)) + 2.614 \times \\
\log \left[ \frac{0.797\times \log(\textrm{SFR}_i ) + 3.834}{2.88} \right]\end{aligned}$$ which allows us to estimate attenuation and intrinsic SFR, even for observations of a single [H$\alpha$]{}emission line. We use this correction to obtain the intrinsic SFRs.
Figure \[fig:SFR\_SFRd\] correlates the total $\Sigma$SFRs to the SFR and shows that our $\Sigma$SFR limit is around $10^{-1}$ M$_\odot$ yr$^{-1}$ kpc$^{-2}$ for SFR$\sim1 M_\odot yr^{-1}$.
Finally, we also compute [H$\alpha$]{}equivalent widths (EW([H$\alpha$]{})) from the collapsed 2D spectra. We define the line profile by adopting a fixed rest frame wavelength range, centered on the theoretical wavelength, 6480-6650 Å, and then obtain the line flux, $f_{\textrm{line}}$, by summing the flux within the line. The continuum is defined by two regions of 100 Å located at the two extremes of the line profile. We fit a straight line to the average continuum in the two regions and sum the flux below the line, to obtain $f_{\textrm{cont}}$. The rest-frame EW([H$\alpha$]{}) is therefore defined by $$\textrm{EW}(H_\alpha) = \frac{f_{\textrm{line}}}{f_{\textrm{cont}}\times(1+z)}$$ We note that our approach ignores underlying [H$\alpha$]{}absorption. As usual, when two spectra for the same galaxy are reliable, the final value is given by the average of the two EW estimates, and the error is obtained by summing in quadrature the individual errors. The measurements from the two PAs are consistent within the uncertainty. Otherwise we just use a single spectrum.
Results {#sec:results}
=======
 
Tables \[tab:clu\_gal\] to \[tab:fie\_gal\_2\] summarize the properties of the galaxies in our cluster and field samples, respectively. They include galaxy positions, redshifts, stellar masses, EW([H$\alpha$]{})s, SFRs, $\Sigma$SFRs, sizes in different bands (F475W, F110W and [H$\alpha$]{}) along both the $x-$ and $y-$ direction, the offset between the peak of the light in [H$\alpha$]{}and in the rest-frame UV continuum, and, for clusters, the cluster-centric distances (both in kpc and in units of $r_{500}$).
Figure \[fig:size\] shows the distribution of the ratio of the size as measured from the [H$\alpha$]{}light (r([H$\alpha$]{})) to the size as measured from the rest-frame UV continuum (r(F745W)) and rest-frame optical continuum (r(F110W)), both for the two directions separately and for the mean sizes, obtained as average between the two directions. We therefore compare the currently star forming regions to the younger stellar population (traced by the observed F475W continuum) and to the older one (traced by the observed F110W continuum). In a forthcoming analysis we will also compare [H$\alpha$]{}maps to maps of the even older stellar populations, as traced by the rest-frame Infrared.
In both environments, there is no preferential axis for the [H$\alpha$]{}emission. Ratios obtained using the F110W and the F475W filters agrees within the errors, in both clusters and field. Looking at the continuum sizes, we do not detect strong trends with the wavelengths, most likely because of the our currently small number statistics. Distributions peak around 1, showing that many galaxies have comparable sizes in the line and continuum. However, distributions are slightly skewed toward values $>1$. In cluster galaxies mean size ratios are systematically slightly larger than field galaxies when the F475W is used, but not when the F110W filter is considered (clusters: $(r(H\alpha)/r(F475W))_x=1.8\pm0.3$, $(r(H\alpha)/r(F475W))_y=2.2\pm0.3$, $\langle r(H\alpha)\rangle/ \langle r(F745W)\rangle=1.9\pm0.3$; $(r(H\alpha)/r(F110W))_x=1.6\pm0.2$, $(r(H\alpha)/r(F110W))_y=1.8\pm0.2$, $\langle r(H\alpha)\rangle/ \langle r(F1110W)\rangle=1.7\pm0.2$; field: ($r(H\alpha)/r(F475W))_x=1.3\pm0.2$, $(r(H\alpha)/r(F475W))_y=1.2\pm0.2$, $\langle r(H\alpha)\rangle/ \langle r(F745W)\rangle=1.3\pm0.1$; $(r(H\alpha)/r(F110W))_x=1.6\pm0.3$, $(r(H\alpha)/r(F110W))_y=1.6\pm0.2$, $\langle r(H\alpha)\rangle/ \langle r(F110W)\rangle=1.4\pm0.1$). Mean values for cluster galaxies are driven by a subpopulation of galaxies ($\sim$ 20%) which present [H$\alpha$]{}emission at least two-three time as extended as the light in the rest-frame UV continuum. No such examples are present in the field. This might suggest that in all environments star formation is probably occurring over a larger area than that of the recent star formation ($\sim100$ Myr), but in clusters there might be some additional mechanisms that are stripping the gas and star formation is continuing in the stripped material. A Kolomgorov-Smirnov (K-S) test can not reject the hypothesis that the distributions of the two samples are the same, giving probabilities lower than 80% in all the three cases. We note that both samples are quite small, which might explain why the K-S test is inconclusive.
We use the information in the top right panel of Fig. \[fig:size\] to group galaxies into different classes, as described in the next subsection.
Maps of [H$\alpha$]{}and continuum emission
-------------------------------------------
      \
  
              
         
   
   
We group galaxies according to 1) the ratio of the average [H$\alpha$]{}size to the average size of the rest-frame UV continuum shown in the top right panel of Fig. \[fig:size\], and 2) the axis ratio of the continuum. The first classification scheme states whether the current star formation is occurring at larger or smaller radii than the recent star formation. We note that similar results are obtained when we consider the rest-frame optical continuum, which traces the older stars in the galaxy. The second is a rough attempt to describe the galaxy morphology in the light of the continuum. However, we note that all these classes of objects contain very heterogeneous cases with a variety of different features. It is therefore quite hard to perform a strict classification.
Figures 5-10 show the [H$\alpha$]{}maps obtained as described in Section 3, for all galaxies in our sample. For each galaxy, also a color composite image of the galaxy based on the CLASH [@postman12] HST data is shown. The blue channel is composed by the F435W, F475W, F555W, F606W, and F625W (the last one only for [MACS0717]{}) filters, the green by the F775W, F814W, F850lp, F105W, F110W filters, and the red by the F125W, F140W, F160W filters.
Figures \[fig:c\_size\_Hec\] and \[fig:f\_size\_Hec\] show the 6/25 cluster and 3/17 field galaxies with similar sizes in [H$\alpha$]{}and in the rest-frame UV continuum ($0.8< \langle r($[H$\alpha$]{}$)\rangle/ \langle r(F745W)\rangle <1.2$). 4 cluster galaxies show elongated sizes in the light of the rest-frame UV continuum (axis ratio >1.2), while all field galaxies show symmetric shapes. Nonetheless, 3 galaxies in the field have clearly spiral morphologies.
Figures \[fig:c\_size\_Hgc\] and \[fig:f\_size\_Hgc\] show all galaxies with size of the [H$\alpha$]{}light larger than the size measured from the continuum ($ \langle r($[H$\alpha$]{}$)\rangle/ \langle r(F745W)\rangle >1.2$), in clusters and in the field respectively. The great majority of cluster and field galaxies fall into this class (15/25 and 10/17 respectively). Of these, 11 in clusters and 6 in the field show an elongated shape. Though being star forming, most of these galaxies show an early-type morphology in the color composite images. In clusters, this might be a sign of ongoing stripping.
Few galaxies have [H$\alpha$]{}sizes smaller than continuum sizes ($ \langle r($[H$\alpha$]{}$)\rangle/ \langle r(F745W)\rangle <0.8$) and are shown in Figures \[fig:c\_size\_Hsc\] and \[fig:f\_size\_Hsc\]. In both environments, 2 out of 4 galaxies show symmetric profiles.
In general, our sample includes galaxies with a variety of morphologies and we find that there is no clear correlation between the extent of the [H$\alpha$]{}emission and the galaxy color or morphology in the color images. This might suggest that there is no a unique mechanism responsible for extension of the [H$\alpha$]{}, but that different processes might be at work in galaxies of different types.
Overall, both in clusters and in the field 60% of galaxies show [H$\alpha$]{}emission more extended than the emission in the rest-frame UV continuum. Half of the galaxies in the field show a symmetric shape, 35% in clusters.
When comparing the maps at different wavelengths, we also observe that the peak of the [H$\alpha$]{}emission is displaced with respect to the F475W continuum emission. Figure \[fig:offset\] shows the distribution of the absolute value of the offsets in the two directions (obtained from the two different PAs) and, for the galaxies with both PAs, the real distance between the two peaks, obtained by combining the offsets. In both environments, the displacement is always smaller than 1 kpc. There is no a preferential direction of the offset. There are hints that galaxies in clusters are characterized by a marginally larger offset than field galaxies, but a larger number statistics will be needed to confirm the trends. The existence of the offset suggests that in most galaxies the bulk of the star formation is not occurring in the galaxy cores. Unsurprisingly given the small sample statistics, cluster and field means are compatible within the errors and a K-S test can not reject the hypothesis that the two distributions are drawn from the same parent distribution.
We note that in our analysis we have made the assumption that there is no spatial variation in extinction across the galaxy. Nonetheless, high-resolution imaging in multiple HST bands [@wuyts12] as well as analysis of such data in combination with [H$\alpha$]{}maps extracted from grism spectroscopy [@wuyts13] indicate that such an assumption may be over-simplistic, particularly in the more massive galaxies where the largest spatial color variations are seen. It is hard to anticipate how corrections for non-uniform extinction might affect our conclusions, since the correction to the sizes will depend on the actual distribution of dust. For example, if dust is mostly in the centers (like a dust lane), it would make us overestimate the F475W sizes more than the [H$\alpha$]{}sizes. Conversely, if dust is mostly in the outskirts the correction could lead us in the opposite direction. Reaching a firm conclusion would require obtaining Hb maps to trace the Balmer decrement. Unfortunately H$_\beta$ is too blue for the G102 grism for these clusters.
Maps of [H$\alpha$]{}and position within the clusters
-----------------------------------------------------
   
For cluster galaxies, we can correlate their morphology with their location in the cluster, the surface mass density distribution and the X-ray emission, as shown in Fig. \[fig:image\].
The mass maps were produced using SWUnited reconstruction code described in detail in @bradac04b and @bradac09. The method uses both strong and weak lensing mass reconstruction on a non-uniform adapted grid. From the set of potential values we determine all observables (and mass distribution) using derivatives. The potential is reconstructed by maximizing a log likelihood which uses image positions of multiply imaged sources, weak lensing ellipticities, and regularization as constraints. For both MACS1423 and MACS0717 we use CLASH data. In addition, for MACS0717 we make use of the arcs identified in HST archival imaging prior to Hubble Frontier Fields [@zitrin09; @limousin10] and spectroscopic redshifts obtained by @ma09 [@limousin12; @ebeling14]. For MACS1423 we use spectroscopic redshifts and images identified in @limousin10; and we add additional multiply imaged systems discovered by our team.
The X-ray images are based on Chandra data, and are described in [@mantz10] and [@vonderlinden14a]. For the contours shown in Fig.\[fig:image\], the images have been adaptively smoothed, after removing point sources identified in [@ehlert13].
In both clusters, galaxies are located within $\sim$0.4r$_{500}$ and do not seem to avoid the cluster cores, even though there might be possible projection effects. The two clusters present very different surface mass density distribution and X-ray emission: while [MACS0717]{}extends along the north-west - south-east direction and has more than one main peak in the emissions, [MACS1423]{}shows a symmetric surface mass density distribution and X-ray emission. We note that [MACS1423]{}passes the very strict requirements on relaxedness defined in [@mantz14]. In [MACS0717]{}we find galaxies with both truncated or extended [H$\alpha$]{}with respect to the rest-frame UV continuum, in [MACS1423]{}all galaxies have [H$\alpha$]{}light more extended than the continuum light. Despite the small number statistics, it appears that the truncated objects are only found between the merging clusters, suggesting that the spatial distribution of [H$\alpha$]{}is indeed related to cluster dynamics: the most extreme cases of stripping are expected to take place in interacting systems [e.g., @owen06; @smith10; @owers12].
 
Figure \[fig:dist\_offset\] quantifies the relation between the projected offset along the cluster radial direction and the distance of the galaxy from the cluster center, for cluster members. While most of the galaxies have a projected offset within $\pm$0.2 kpc, there are some galaxies with a larger offset. Almost half of the galaxies (55%) have a positive offset, the other half have a negative offset. No trends with distance are detected, indicating that the cluster center is not affecting the position of the peak of the [H$\alpha$]{}emission. Galaxies with different [H$\alpha$]{}extension are not clustered in particular regions of the clusters.
![Galaxy properties - cluster properties correlations. Upper row: X-ray emission, bottom row: Surface mass density. Left panels: projected offset along the line connecting the peak of the emission in the continuum and the cluster center. Right panels: ratio of r([H$\alpha$]{}) to r(F745W). Shocks and strong gradients in the X-ray IGM might alter the relative position between the peak of the [H$\alpha$]{}emission and the peak of the light of the of the young stellar population ($\sim$100 Myr), even though there is not statistically significant evidence to support this conclusion. Shocks and gradients do not alter the relative extension of the [H$\alpha$]{}with respect to the continuum light. \[fig:corr\]](clusters_xray_projoffset_475_PSFcorr.png "fig:") ![Galaxy properties - cluster properties correlations. Upper row: X-ray emission, bottom row: Surface mass density. Left panels: projected offset along the line connecting the peak of the emission in the continuum and the cluster center. Right panels: ratio of r([H$\alpha$]{}) to r(F745W). Shocks and strong gradients in the X-ray IGM might alter the relative position between the peak of the [H$\alpha$]{}emission and the peak of the light of the of the young stellar population ($\sim$100 Myr), even though there is not statistically significant evidence to support this conclusion. Shocks and gradients do not alter the relative extension of the [H$\alpha$]{}with respect to the continuum light. \[fig:corr\]](clusters_xray_sizeratio_475_PSFcorr.png "fig:") ![Galaxy properties - cluster properties correlations. Upper row: X-ray emission, bottom row: Surface mass density. Left panels: projected offset along the line connecting the peak of the emission in the continuum and the cluster center. Right panels: ratio of r([H$\alpha$]{}) to r(F745W). Shocks and strong gradients in the X-ray IGM might alter the relative position between the peak of the [H$\alpha$]{}emission and the peak of the light of the of the young stellar population ($\sim$100 Myr), even though there is not statistically significant evidence to support this conclusion. Shocks and gradients do not alter the relative extension of the [H$\alpha$]{}with respect to the continuum light. \[fig:corr\]](clusters_surf_mass_projoffset_475_PSFcorr.png "fig:") ![Galaxy properties - cluster properties correlations. Upper row: X-ray emission, bottom row: Surface mass density. Left panels: projected offset along the line connecting the peak of the emission in the continuum and the cluster center. Right panels: ratio of r([H$\alpha$]{}) to r(F745W). Shocks and strong gradients in the X-ray IGM might alter the relative position between the peak of the [H$\alpha$]{}emission and the peak of the light of the of the young stellar population ($\sim$100 Myr), even though there is not statistically significant evidence to support this conclusion. Shocks and gradients do not alter the relative extension of the [H$\alpha$]{}with respect to the continuum light. \[fig:corr\]](clusters_surf_mass_sizeratio_475_PSFcorr.png "fig:")
There is increasing evidence for a correlation between the efficiency of the stripping phenomenon and the presence of shocks and strong gradients in the X-ray IGM [e.g., @owers12; @vijayaraghavan13]. Indeed, Figure \[fig:corr\] hints at potential correlations between the offset and X-ray emission or surface mass density. Similar results are obtained if we project the offset along the line that connects the galaxy to the peak of the X-ray emission. Nonetheless, Spearman rank-order correlation tests show that these trends are not statistically significant. Likewise, it seems that the extent of the [H$\alpha$]{}size with respect to the continuum size does not strongly correlate with the X-ray emission nor the surface mass density, as confirmed by a Spearman rank-order correlation test.
Overall, it seems that shocks and strong gradients in the X-ray IGM might alter the relative position between the peak of the [H$\alpha$]{}emission and the peak of the light of the recent star formation, even though we do not detect clear signs of gas compression and/or stripping.
We note that the lack of strong correlations does not allow us to identify a unique strong environmental effect that originates from the cluster center. We hypothesize that local effects, uncorrelated to the cluster-centric radius, play a larger role. Such effects weaken potential radial trends.
Notes on remarkable objects
---------------------------
In the following we describe some interesting objects presented in Figures 5-10.
Among galaxies with [H$\alpha$]{}size similar to the rest-frame UV continuum size (Figures \[fig:c\_size\_Hec\] and \[fig:f\_size\_Hec\]), we note that [MACS0717]{}-00236 is a spiral galaxy with three main peaks of [H$\alpha$]{}emission. Indeed, the strongest emission comes from the two spiral arms, while the flux in the core of the galaxy is less important. A complex [H$\alpha$]{}structure extends throughout the entire galaxy. Only one PA covers the entire galaxy, while the other misses one of the arms. In contrast, [MACS0717]{}-01477, even though showing a similar appearance in the color image to [MACS0717]{}-00236, is characterized by a much weaker and clumpy [H$\alpha$]{}emission. [MACS1423]{}-01771 shows extended features both in the continuum and in the [H$\alpha$]{}light.
Among galaxies with [H$\alpha$]{}size larger than the continuum size (Figures \[fig:c\_size\_Hgc\] and \[fig:f\_size\_Hgc\]), [MACS0717]{}-02189 and [MACS0717]{}-02574 show very elongated shapes in the continuum, but quite regular [H$\alpha$]{}emission extended in both sizes. [MACS1423]{}-00229, [MACS1423]{}-01253 and [MACS1423]{}-01516 show very regular shapes in the continuum and very extended [H$\alpha$]{}emission, in the case of [MACS1423]{}-01253 the emission is only in one direction. In [MACS1423]{}-00319 the [H$\alpha$]{}emission is orthogonal to the continuum emission.
Finally, among galaxies with [H$\alpha$]{}size smaller than the continuum size (Figures \[fig:c\_size\_Hsc\] and \[fig:f\_size\_Hsc\]), [MACS0717]{}-02334 shows an [H$\alpha$]{}emission which is bent with respect to the continuum. In this case, the truncated [H$\alpha$]{}disk might be an example of ram pressure stripping, which removed the ISM. The orientation of the tail does not point away from the cluster center, but the bending might suggests that the galaxy formed from an infalling population experiencing the cluster environment for the first time [see, e.g., @smith10].
It is also worth noting that [MACS0717]{}-01305 (Fig. \[fig:c\_size\_Hec\]) is very close to the cluster center, is located on a surface mass density distribution peak and is quite close to a peak in the X-ray emission. The shape of the [H$\alpha$]{}emission seems to be unaffected by this peculiar position within the cluster, but the galaxy shows the largest projected offset along the line of sight.
Star Formation Rates
--------------------
![GLASS cluster SFR-mass relation over plotted to the field relation at similar redshift [from @noeske07] and the cluster relation at similar redshift [from @vulcani10]. Red filled circles: GLASS clusters; red filled stars: GLASS field. Blue squares: EDisCS clusters, green triangles: AEGIS field. Empty symbols: red galaxies with detected emission lines, filled symbols: blue galaxies with detected emission lines and galaxies detected at 24$\mu m$ [refer to @noeske07; @vulcani10 for details on the sample selection]. \[fig:sfr\_mass\]](sfr_mass_v2.png)
Figure \[fig:sfr\_mass\] shows the SFR-mass plane for our cluster and field galaxies, together with that found by [@noeske07] for field galaxies at $z\sim0.5$ and by [@vulcani10] for cluster galaxies also at $z\sim0.5$. Our galaxies lay on the field SFR-mass relation of blue galaxies with emission lines or detected in the Infrared [see @noeske07; @vulcani10 for details on their sample selection], and even trace the upper limit. To some extent, this was expected having selected star forming cluster galaxies. This results, however, shows that at these redshifts cluster galaxies can be as star forming as field galaxies of similar mass. The location of our galaxies on the plane is also in line with what was found by [@poggianti15] for the local universe, who showed that galaxies with signs of ongoing stripping tend to be located above the best fit to the relation, indicating a SFR excess with respect to galaxies of the same mass but that are not being stripped. Recall that the field galaxies span a wide redshift range (0.2<$z$<0.7), therefore they lay on different regions of the SFR-mass plane simply due to the evolution of the SFR-M$_\ast$ relation with $z$ [e.g., @noeske07].
Both in clusters and in the field, the $\Sigma$SFR ranges from $\sim$0.1 to 1 $M_\odot \, \textrm{yr}^{-1} \, \textrm{kpc}^{-2}$, suggesting that the physical conditions in star forming galaxies do not strongly depend on environment.
Discussion and Conclusions
==========================
In this pilot study we have carried out an exploration of the spatial distribution of star formation in galaxies beyond the local universe, as traced by the [H$\alpha$]{} emission in two of the GLASS clusters, [MACS0717]{}and [MACS1423]{}. For this purpose, we have developed a new methodology to produce [H$\alpha$]{}maps taking advantage of the WFC3-G102 data at two orthogonal PAs. We then visually selected galaxies with [H$\alpha$]{}emission and, based on their redshift, assigned their membership to the cluster. We have used galaxies in the foreground and background of the two clusters to compile a field control sample. Both for field and cluster galaxies, we computed the extent of the emission and its position within the galaxy and compared these quantities to the younger stellar population as traced by the rest frame UV continuum (obtained by images in the F475W filter) and the older stellar population as traced by the rest-frame optical continuum (obtained by images in the F110W filter). We correlated galaxy properties to global and local cluster properties, in order to look for signs of cluster specific processes.
The main results of this analysis can be summarized as follows:
- Both in clusters and the field $\sim$60% of the galaxies are more extended in [H$\alpha$]{}than in the rest-frame UV continuum. The emission appears larger in the cluster than in field galaxies. Trends are driven by a subpopulation ($\sim$20%) of cluster galaxies with [H$\alpha$]{}emission at least three time as extended the continuum emission.
- Both in clusters and in the field there is an offset between the peak of the [H$\alpha$]{}emission and that in the rest-frame UV continuum. The displacement can reach $\sim$1 kpc. In clusters the offset appears to be marginally larger.
- Comparing the extent of the offset and the cluster properties, we find a tentative correlation between the projected offset and both the X-ray emission and the mass surface density: the larger the emission, the bigger the offset between the emission in the rest-frame UV continuum and the emission in [H$\alpha$]{}. This offset seems to also to point at a cluster-specific mechanism.
- [MACS0717]{}and [MACS1423]{}present very different surface mass density distribution and X ray emission, indicating that [MACS1423]{}is much more relaxed than [MACS0717]{}, which in contrast presents a double peak in the distribution. In [MACS1423]{}all galaxies have [H$\alpha$]{}disk larger than the rest-frame UV continuum, while in [MACS0717]{}galaxies with both extend and truncated [H$\alpha$]{}are observed. This finding suggests that gradients in the X-ray IGM might alter the relative position between the peak of the [H$\alpha$]{}emission and the peak of the light of the young stellar population, even though at this stage correlations are not supported by statistical tests.
From our analysis a complex picture emerges and a simple explanation can not describe our observations. Even though galaxies in clusters and in the field present similar [H$\alpha$]{}properties, the variety of their morphology suggests that they are at different stages of their evolution, therefore there can not be a unique mechanism acting on galaxies in the different environments. For example, the larger extension of [H$\alpha$]{} with respect to the continuum seems to be a generic indication of inside-out growth [see also @nelson12]. Specific examples of this case include [MACS1423]{}-01253 and [MACS1423]{}-01910 in clusters (Fig.\[fig:c\_size\_Hgc\]) and [MACS1423]{}-01972 in the field (Fig.\[fig:f\_size\_Hgc\]). Investigating whether this growth is localized in a disk component will require careful bulge to disk decompositions which is planned for a future work. However, given the variety of morphologies, well-ordered disks do not appear to be the only site of star formation.
Furthermore, the larger average sizes in the cluster point to an additional cluster-specific mechanism responsible for stripping the ionized gas or perhaps for triggering additional star formation in the outskirts of the galaxies. The mechanism could be ram pressure stripping of the ionized gas or perhaps tidal compression of the outskirts or the galaxies, or both. Galaxies [MACS0717]{}-02189 and [MACS1423]{}-00446 (Fig.\[fig:c\_size\_Hgc\]) are examples of this possible mechanism.
Finally, in some cases, galaxies have been already deprived from their gas and are left with a smaller [H$\alpha$]{}disk than the recent star formation. [MACS0717]{}-02334 (Fig.\[fig:f\_size\_Hsc\]) is a clear example of galaxies in such stage.
We note that the observed differences between cluster and field galaxies might be also due to the different mass and redshift distributions of the two samples, and not only to purely environmental effects.
The results from this pilot study illustrate the power and feasibility of space-based grism data to learn qualitatively new information about the mechanisms that regulate star formation in different environments during the second half of the life of the universe. Having developed the methods we are now in a position to carry out a larger scale investigation on the full GLASS cluster sample, when visual inspections and quality controls have been completed. The extended analysis will allow us not only to further distinguish and classify the processes acting in clusters from those acting on the field, but also to better correlate galaxy properties with the cluster global properties, investigating in detail the role of the environment in shutting down star formation.
Acknowledgments {#acknowledgments .unnumbered}
===============
Support for GLASS (HST-GO-13459) was provided by NASA through a grant from the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. We are very grateful to the staff of the Space Telescope for their assistance in planning, scheduling and executing the observations. B.V. acknowledges the support from the World Premier International Research Center Initiative (WPI), MEXT, Japan and the Kakenhi Grant-in-Aid for Young Scientists (B)(26870140) from the Japan Society for the Promotion of Science (JSPS).
\[lastpage\]
[^1]: <http://glass.physics.ucsb.edu>
[^2]: <http://code.google.com/p/threedhst/>
[^3]: <https://github.com/kasperschmidt/GLASSinspectionGUIs>
[^4]: Kron radii are measured by Sextractor from a combined NIR image.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We study fairness in collaborative-filtering recommender systems, which are sensitive to discrimination that exists in historical data. Biased data can lead collaborative-filtering methods to make unfair predictions for users from minority groups. We identify the insufficiency of existing fairness metrics and propose four new metrics that address different forms of unfairness. These fairness metrics can be optimized by adding fairness terms to the learning objective. Experiments on synthetic and real data show that our new metrics can better measure fairness than the baseline, and that the fairness objectives effectively help reduce unfairness.'
author:
- |
Sirui Yao\
Department of Computer Science\
Virginia Tech\
Blacksburg, VA 24061\
`ysirui@vt.edu`\
Bert Huang\
Department of Computer Science\
Virginia Tech\
Blacksburg, VA 24061\
`bhuang@vt.edu`\
bibliography:
- 'yao-nips17.bib'
title: |
Beyond Parity:\
Fairness Objectives for Collaborative Filtering
---
Introduction
============
This paper introduces new measures of unfairness in algorithmic recommendation and demonstrates how to optimize these metrics to reduce different forms of unfairness. Recommender systems study user behavior and make recommendations to support decision making. They have been widely applied in various fields to recommend items such as movies, products, jobs, and courses. However, since recommender systems make predictions based on observed data, they can easily inherit bias that may already exist. To address this issue, we first formalize the problem of unfairness in recommender systems and identify the insufficiency of demographic parity for this setting. We then propose four new unfairness metrics that address different forms of unfairness. We compare our fairness measures with non-parity on biased, synthetic training data and prove that our metrics can better measure unfairness. To improve model fairness, we provide five fairness objectives that can be optimized, each adding unfairness penalties as regularizers. Experimenting on real and synthetic data, we demonstrate that each fairness metric can be optimized without much degradation in prediction accuracy, but that trade-offs exist among the different forms of unfairness.
We focus on a frequently practiced approach for recommendation called collaborative filtering, which makes recommendations based on the ratings or behavior of other users in the system. The fundamental assumption behind collaborative filtering is that other users’ opinions can be selected and aggregated in such a way as to provide a reasonable prediction of the active user’s preference [@ekstrand2011collaborative]. For example, if a user likes item A, and many other users who like item A also like item B, then it is reasonable to expect that the user will also like item B. Collaborative filtering methods would predict that the user will give item B a high rating.
With this approach, predictions are made based on co-occurrence statistics, and most methods assume that the missing ratings are missing at random. Unfortunately, researchers have shown that sampled ratings have markedly different properties from the users’ true preferences [@marlin2012collaborative; @marlin2009collaborative]. Sampling is heavily influenced by social bias, which results in more missing ratings in some cases than others. This non-random pattern of missing and observed rating data is a potential source of unfairness. For the purpose of improving recommendation accuracy, there are collaborative filtering models [@marlin2012collaborative; @beutel2017beyond; @sahebi2015takes] that use side information to address the problem of imbalanced data, but in this work, to test the properties and effectiveness of our metrics, we focus on the basic matrix-factorization algorithm first. Investigating how these other models could reduce unfairness is one direction for future research.
Throughout the paper, we consider a running example of unfair recommendation. We consider recommendation in education, and unfairness that may occur in areas with current gender imbalance, such as science, technology, engineering, and mathematics (STEM) topics. Due to societal and cultural influences, fewer female students currently choose careers in STEM. For example, in 2010, women accounted for only 18% of the bachelor’s degrees awarded in computer science [@broad2014recruiting]. The underrepresentation of women causes historical rating data of computer-science courses to be dominated by men. Consequently, the learned model may underestimate women’s preferences and be biased toward men. We consider the setting in which, even if the ratings provided by students accurately reflect their true preferences, the bias in which ratings are reported leads to unfairness.
The remainder of the paper is organized as follows. First, we review previous relevant work in \[sec:related\]. In \[sec:approach\], we formalize the recommendation problem, and we introduce four new unfairness metrics and give justifications and examples. In \[sec:experiments\], we show that unfairness occurs as data gets more imbalanced, and we present results that successfully minimize each form of unfairness. Finally, \[sec:conclusion\] concludes the paper and proposes possible future work.
Related Work {#sec:related}
============
As machine learning is being more widely applied in modern society, researchers have begun identifying the criticality of algorithmic fairness. Various studies have considered algorithmic fairness in problems such as supervised classification [@pedreshi2008discrimination; @lum2016statistical; @zafar2017fairness]. When aiming to protect algorithms from treating people differently for prejudicial reasons, removing sensitive features (e.g., gender, race, or age) can help alleviate unfairness but is often insufficient. Features are often correlated, so other unprotected attributes can be related to the sensitive features and therefore still cause the model to be biased [@kamishima2011fairness; @zemel2013learning]. Moreover, in problems such as collaborative filtering, algorithms do not directly consider measured features and instead infer latent user attributes from their behavior.
Another frequently practiced strategy for encouraging fairness is to enforce *demographic parity*, which is to achieve statistical parity among groups. The goal is to ensure that the overall proportion of members in the protected group receiving positive (or negative) classifications is identical to the proportion of the population as a whole [@zemel2013learning]. For example, in the case of a binary decision $\hat{Y} \in \{0, 1\}$ and a binary protected attribute $A \in \{0, 1\}$, this constraint can be formalized as [@hardt2016equality] $$\Pr\{\hat{Y} =1 | A = 0\} = \Pr\{\hat{Y} =1 | A = 1\}~.
\label{eq:parity}$$
Kamishima et al. [@kamishima2011fairness; @kamishima2012enhancement; @kamishima2013efficiency; @kamishima2014correcting; @kamishima2016model] evaluate model fairness based on this non-parity unfairness concept, or try to solve the unfairness issue in recommender systems by adding a regularization term that enforces demographic parity. The objective penalizes the differences among the average predicted ratings of user groups. However, demographic parity is only appropriate when preferences are unrelated to the sensitive features. In tasks such as recommendation, user preferences are indeed influenced by sensitive features such as gender, race, and age [@chausson2010watches; @daymont1984job]. Therefore, enforcing demographic parity may significantly damage the quality of recommendations.
To address the issue of demographic parity, Hardt et al. [@hardt2016equality] propose to measure unfairness with the true positive rate and true negative rate. This idea encourages what they refer to as *equal opportunity* and no longer relies on the implicit assumption of demographic parity that the target variable is independent of sensitive features. They propose that, in a binary setting, given a decision $\hat{Y} \in \{0, 1\}$, a protected attribute $A \in \{0, 1\}$, and the true label $Y \in \{0, 1\}$, the constraints are equivalent to [@hardt2016equality] $$\Pr\{\hat{Y} =1 | A = 0, Y = y\} = \Pr\{\hat{Y} =1 | A = 1, Y = y\}, y \in \{0, 1\}~.
\label{eq:equalopportunity}$$ This constraint upholds fairness and simultaneously respects group differences. It penalizes models that only perform well on the majority groups. This idea is also the basis of the unfairness metrics we propose for recommendation.
Our running example of recommendation in education is inspired by the recent interest in using algorithms in this domain [@sacin2009recommendation; @thai2010recommender; @dascalu2016educational]. Student decisions about which courses to study can have significant impacts on their lives, so the usage of algorithmic recommendation in this setting has consequences that will affect society for generations. Coupling the importance of this application with the issue of gender imbalance in STEM [@beede2011women] and challenges in retention of students with backgrounds underrepresented in STEM [@smith2011women; @griffith2010persistence], we find this setting a serious motivation to advance scientific understanding of unfairness—and methods to reduce unfairness—in recommendation.
Fairness Objectives for Collaborative Filtering {#sec:approach}
===============================================
This section introduces fairness objectives for collaborative filtering. We begin by reviewing the matrix factorization method. We then describe the various fairness objectives we consider, providing formal definitions and discussion of their motivations.
Matrix Factorization for Recommendation
---------------------------------------
We consider the task of collaborative filtering using matrix factorization [@koren2009matrix]. We have a set of users indexed from 1 to $\numusers$ and a set of items indexed from 1 to $\numitems$. For the $i$th user, let $\group_i$ be a variable indicating which group the $i$th user belongs to. For example, it may indicate whether user $i$ identifies as a woman, a man, or with a non-binary gender identity. For the $j$th item, let $\itemgroup_j$ indicate the item group that it belongs to. For example, $\itemgroup_j$ may represent a genre of a movie or topic of a course. Let $\rating_{ij}$ be the preference score of the $i$th user for the $j$th item. The ratings can be viewed as entries in a rating matrix $\ratingmat$.
The matrix-factorization formulation builds on the assumption that each rating can be represented as the product of vectors representing the user and item. With additional bias terms for users and items, this assumption can be summarized as follows: $$\rating_{ij} \approx \uservec_i ^\top \itemvec_j + \userbias_i + \itembias_j ~ ,
\label{eq:reconstruction}$$ where $\uservec_i$ is a $d$-dimensional vector representing the $i$th user, $\itemvec_j$ is a $d$-dimensional vector representing the $j$th item, and $\userbias_i$ and $\itembias_j$ are scalar bias terms for the user and item, respectively. The matrix-factorization learning algorithm seeks to learn these parameters from observed ratings $\trainingdata$, typically by minimizing a regularized, squared reconstruction error: $$J(\usermat, \itemmat, \userbiasvec, \itembiasvec) = \frac{\lambda}{2} \left( ||\usermat||^2_{\mathrm{F}} + ||\itemmat||^2_{\mathrm{F}} \right) + \frac{1}{|\trainingdata|} \sum_{(i, j) \in \trainingdata} \left( \prediction_{ij} - \rating_{ij} \right)^2 ~,
\label{eq:MF-objective}$$ where $\userbiasvec$ and $\itembiasvec$ are the vectors of bias terms, $|| \cdot ||_{\mathrm{F}}$ represents the Frobenius norm, and $$\prediction_{ij} = \uservec_i ^\top \itemvec_j + \userbias_i + \itembias_j.$$ Strategies for minimizing this non-convex objective are well studied, and a general approach is to compute the gradient and use a gradient-based optimizer. In our experiments, we use the Adam algorithm [@kingma2014adam], which combines adaptive learning rates with momentum.
Unfair Recommendations from Underrepresentation {#sec:example}
-----------------------------------------------
In this section, we describe a process through which matrix factorization leads to unfair recommendations, even when rating data accurately reflects users’ true preferences. Such unfairness can occur with imbalanced data. We identify two forms of underrepresentation: *population imbalance* and *observation bias*. We later demonstrate that either leads to unfair recommendation, and both forms together lead to worse unfairness. In our discussion, we use a running example of course recommendation, highlighting effects of underrepresentation in STEM education.
Population imbalance occurs when different types of users occur in the dataset with varied frequencies. For example, we consider four types of users defined by two aspects. First, each individual identifies with a gender. For simplicity, we only consider binary gender identities, though in this example, it would also be appropriate to consider men as one gender group and women and all non-binary gender identities as the second group. Second, each individual is either someone who enjoys and would excel in STEM topics or someone who does and would not. Population imbalance occurs in STEM education when, because of systemic bias or other societal problems, there may be significantly fewer women who succeed in STEM (WS) than those who do not (W), and because of converse societal unfairness, there may be more men who succeed in STEM (MS) than those who do not (M). This four-way separation of user groups is not available to the recommender system, which instead may only know the gender group of each user, but not their proclivity for STEM.
Observation bias is a related but distinct form of data imbalance, in which certain types of users may have different tendencies to rate different types of items. This bias is often part of a feedback loop involving existing methods of recommendation, whether by algorithms or by humans. If an individual is never recommended a particular item, they will likely never provide rating data for that item. Therefore, algorithms will never be able to directly learn about this preference relationship. In the education example, if women are rarely recommended to take STEM courses, there may be significantly less training data about women in STEM courses.
We simulate these two types of data bias with two stochastic block models [@holland1976local]. We create one block model that determines the probability that an individual in a particular user group likes an item in a particular item group. The group ratios may be non-uniform, leading to population imbalance. We then use a second block model to determine the probability that an individual in a user group rates an item in an item group. Non-uniformity in the second block model will lead to observation bias.
Formally, let matrix $\rateblockmat \in [0,1]^{|\group| \times |\itemgroup|}$ be the block-model parameters for rating probability. For the $i$th user and the $j$th item, the probability of $\rating_{ij}=+1$ is $\rateblock_{(\group_i, \itemgroup_j)}$, and otherwise $\rating_{ij} = -1$. Morever, let $\takeblockmat \in [0,1]^{|\group| \times |\itemgroup|}$ be such that the probability of observing $\rating_{ij}$ is $\takeblock_{(\group_i, \itemgroup_j)}$.
Fairness Metrics
----------------
In this section, we present four new unfairness metrics for preference prediction, all measuring a discrepancy between the prediction behavior for disadvantaged users and advantaged users. Each metric captures a different type of unfairness that may have different consequences. We describe the mathematical formulation of each metric, its justification, and examples of consequences the metric may indicate. We consider a binary group feature and refer to disadvantaged and advantaged groups, which may represent women and men in our education example.
The first metric is *value unfairness*, which measures inconsistency in signed estimation error across the user types, computed as $$\metric_\val = \frac{1}{\numitems} \sum_{j = 1}^\numitems \left| \left( \avgpredF_j - \avgrateF_j \right) - \left( \avgpredM_j - \avgrateM_j \right) \right| ~ ,
\label{eq:value-unfairness}$$ where $\avgpredF_j$ is the average predicted score for the $j$th item from disadvantaged users, $\avgpredM_j$ is the average predicted score for advantaged users, and $\avgrateF_j$ and $\avgrateM_j$ are the average ratings for the disadvantaged and advantaged users, respectively. Precisely, the quantity $\avgpredF_j$ is computed as $$\avgpredF_j := \frac{1}{|\{ i: \left( (i, j) \in \trainingdata \right) \wedge \group_i \}|} \sum_{i: \left( (i, j) \in \trainingdata \right) \wedge \group_i } \prediction_{ij} ~,$$ and the other averages are computed analogously.
Value unfairness occurs when one class of user is consistently given higher or lower predictions than their true preferences. If the errors in prediction are evenly balanced between overestimation and underestimation or if both classes of users have the same direction and magnitude of error, the value unfairness becomes small. Value unfairness becomes large when predictions for one class are consistently overestimated and predictions for the other class are consistently underestimated. For example, in a course recommender, value unfairness may manifest in male students being recommended STEM courses even when they are not interested in STEM topics and female students not being recommended STEM courses even if they are interested in STEM topics.
The second metric is *absolute unfairness*, which measures inconsistency in absolute estimation error across user types, computed as $$\metric_\absolute = \frac{1}{\numitems} \sum_{j = 1}^\numitems \left| \left| \avgpredF_j - \avgrateF_j \right| - \left| \avgpredM_j - \avgrateM_j \right| \right| ~.
\label{eq:abs-unfairness}$$ Absolute unfairness is unsigned, so it captures a single statistic representing the quality of prediction for each user type. If one user type has small reconstruction error and the other user type has large reconstruction error, one type of user has the unfair advantage of good recommendation, while the other user type has poor recommendation. In contrast to value unfairness, absolute unfairness does not consider the direction of error. For example, if female students are given predictions 0.5 points below their true preferences and male students are given predictions 0.5 points above their true preferences, there is no absolute unfairness. Conversely, if female students are given ratings that are off by 2 points in either direction while male students are rated within 1 point of their true preferences, absolute unfairness is high, while value unfairness may be low.
The third metric is *underestimation unfairness*, which measures inconsistency in how much the predictions underestimate the true ratings: $$\metric_\underest = \frac{1}{\numitems} \sum_{j = 1}^\numitems \left| \max \{ 0, \avgrateF_j - \avgpredF_j \} - \max \{ 0, \avgrateM_j - \avgpredM_j \} \right| ~.
\label{eq:under-unfairness}$$ Underestimation unfairness is important in settings where missing recommendations are more critical than extra recommendations. For example, underestimation could lead to a top student not being recommended to explore a topic they would excel in.
Conversely, the fourth new metric is *overestimation unfairness*, which measures inconsistency in how much the predictions overestimate the true ratings: $$\metric_\overest = \frac{1}{\numitems} \sum_{j = 1}^\numitems \left| \max \{ 0, \avgpredF_j - \avgrateF_j \} - \max \{ 0, \avgpredM_j - \avgrateM_j \} \right| ~.
\label{eq:over-unfairness}$$ Overestimation unfairness may be important in settings where users may be overwhelmed by recommendations, so providing too many recommendations would be especially detrimental. For example, if users must invest large amounts of time to evaluate each recommended item, overestimating essentially costs the user time. Thus, uneven amounts of overestimation could cost one type of user more time than the other.
Finally, a *non-parity* unfairness measure based on the regularization term introduced by Kamishima et al. [@kamishima2011fairness] can be computed as the absolute difference between the overall average ratings of disadvantaged users and those of advantaged users: $$\metric_\parity = \left| \avgpredF - \avgpredM \right|~.$$
Each of these metrics has a straightforward subgradient and can be optimized by various subgradient optimization techniques. We augment the learning objective by adding a smoothed variation of a fairness metric based on the Huber loss [@huber1964robust], where the outer absolute value is replaced with the squared difference if it is less than 1. We solve for a local minimum, i.e, $$\min_{\usermat, \itemmat, \userbiasvec, \itembiasvec}
~ J(\usermat, \itemmat, \userbiasvec, \itembiasvec) +
\metric ~.
\label{eq:fullobj}$$ The smoothed penalty helps reduce discontinuities in the objective, making optimization more efficient. It is also straightforward to add a scalar trade-off term to weight the fairness against the loss. In our experiments, we use equal weighting, so we omit the term from \[eq:fullobj\].
Experiments {#sec:experiments}
===========
We run experiments on synthetic data based on the simulated course-recommendation scenario and real movie rating data [@harper2016movielens]. For each experiment, we investigate whether the learning objectives augmented with unfairness penalties successfully reduce unfairness.
Synthetic Data
--------------
In our synthetic experiments, we generate simulated course-recommendation data from a block model as described in \[sec:example\]. We consider four user groups $\group \in \{\textrm{W}, \textrm{WS}, \textrm{M}, \textrm{MS}\}$ and three item groups $\itemgroup \in \{ \textrm{Fem}, \textrm{STEM}, \textrm{Masc} \}$. The user groups can be thought of as women who do not enjoy STEM topics (W), women who do enjoy STEM topics (WS), men who do not enjoy STEM topics (M), and men who do (MS). The item groups can be thought of as courses that tend to appeal to most women (Fem), STEM courses, and courses that tend to appeal to most men (Masc). Based on these groups, we consider the rating block model $$\rateblockmat = \left[
\begin{tabular}{c|ccc}
& Fem & STEM & Masc\\
\midrule
W & 0.8 & 0.2 & 0.2 \\
WS & 0.8 & 0.8 & 0.2 \\
MS & 0.2 & 0.8 & 0.8 \\
M & 0.2 & 0.2 & 0.8
\end{tabular}
\right].$$
We also consider two observation block models: one with uniform observation probability across all groups $\takeblockmat^{\textrm{uni}} = [0.4]^{4 \times 3}$ and one with unbalanced observation probability inspired by how students are often encouraged to take certain courses $$\takeblockmat^{\textrm{bias}} = \left[
\begin{tabular}{c|ccc}
& Fem & STEM & Masc\\
\midrule
W & 0.6 & 0.2 & 0.1 \\
WS & 0.3 & 0.4 & 0.2 \\
MS & 0.1 & 0.3 & 0.5 \\
M & 0.05 & 0.5 & 0.35
\end{tabular}
\right]~.$$
We define two different user group distributions: one in which each of the four groups is exactly a quarter of the population, and an imbalanced setting where 0.4 of the population is in W, 0.1 in WS, 0.4 in MS, and 0.1 in M. This heavy imbalance is inspired by some of the severe gender imbalances in certain STEM areas today.
For each experiment, we select an observation matrix and user group distribution, generate 400 users and 300 items, and sample preferences and observations of those preferences from the block models. Training on these ratings, we evaluate on the remaining entries of the rating matrix, comparing the predicted rating against the true expected rating, $2\rateblock_{(g_i, h_j)} - 1$.
### Unfairness from different types of underrepresentation
Using standard matrix factorization, we measure the various unfairness metrics under the different sampling conditions. We average over five random trials and plot the average score in \[fig:bars\]. We label the settings as follows: uniform user groups and uniform observation probabilities (U), uniform groups and biased observation probabilities (O), biased user group populations and uniform observations (P), and biased populations and biased observations (P+O).
![Average unfairness scores for standard matrix factorization on synthetic data generated from different underrepresentation schemes. For each metric, the four sampling schemes are uniform (U), biased observations (O), biased populations (P), and both biases (O+P). The reconstruction error and the first four unfairness metrics follow the same trend, while non-parity exhibits different behavior.[]{data-label="fig:bars"}](unfairness_bar_chart_Error.eps "fig:"){width="32.00000%"} ![Average unfairness scores for standard matrix factorization on synthetic data generated from different underrepresentation schemes. For each metric, the four sampling schemes are uniform (U), biased observations (O), biased populations (P), and both biases (O+P). The reconstruction error and the first four unfairness metrics follow the same trend, while non-parity exhibits different behavior.[]{data-label="fig:bars"}](unfairness_bar_chart_Value.eps "fig:"){width="32.00000%"} ![Average unfairness scores for standard matrix factorization on synthetic data generated from different underrepresentation schemes. For each metric, the four sampling schemes are uniform (U), biased observations (O), biased populations (P), and both biases (O+P). The reconstruction error and the first four unfairness metrics follow the same trend, while non-parity exhibits different behavior.[]{data-label="fig:bars"}](unfairness_bar_chart_Absolute.eps "fig:"){width="32.00000%"} ![Average unfairness scores for standard matrix factorization on synthetic data generated from different underrepresentation schemes. For each metric, the four sampling schemes are uniform (U), biased observations (O), biased populations (P), and both biases (O+P). The reconstruction error and the first four unfairness metrics follow the same trend, while non-parity exhibits different behavior.[]{data-label="fig:bars"}](unfairness_bar_chart_Under.eps "fig:"){width="32.00000%"} ![Average unfairness scores for standard matrix factorization on synthetic data generated from different underrepresentation schemes. For each metric, the four sampling schemes are uniform (U), biased observations (O), biased populations (P), and both biases (O+P). The reconstruction error and the first four unfairness metrics follow the same trend, while non-parity exhibits different behavior.[]{data-label="fig:bars"}](unfairness_bar_chart_Over.eps "fig:"){width="32.00000%"} ![Average unfairness scores for standard matrix factorization on synthetic data generated from different underrepresentation schemes. For each metric, the four sampling schemes are uniform (U), biased observations (O), biased populations (P), and both biases (O+P). The reconstruction error and the first four unfairness metrics follow the same trend, while non-parity exhibits different behavior.[]{data-label="fig:bars"}](unfairness_bar_chart_Parity.eps "fig:"){width="32.00000%"}
The statistics demonstrate that each type of underrepresentation contributes to various forms of unfairness. For all metrics except parity, there is a strict order of unfairness: uniform data is the most fair; biased observations is the next most fair; biased populations is worse; and biasing the populations and observations causes the most unfairness. The squared rating error also follows this same trend. In contrast, non-parity behaves differently, in that it is heavily amplified by biased observations but seems unaffected by biased populations. Note that though non-parity is high when the observations are imbalanced, because of the imbalance in the observations, one should actually expect non-parity in the labeled ratings, so it a high non-parity score does not necessarily indicate an unfair situation. The other unfairness metrics, on the other hand, describe examples of unfair behavior by the rating predictor. These tests verify that unfairness can occur with imbalanced populations or observations, even when the measured ratings accurately represent user preferences.
### Optimization of unfairness metrics
As before, we generate rating data using the block model under the most imbalanced setting: The user populations are imbalanced, and the sampling rate is skewed. We provide the sampled ratings to the matrix factorization algorithms and evaluate on the remaining entries of the expected rating matrix. We again use two-dimensional vectors to represent the users and items, a regularization term of $\lambda = 10^{-3}$, and optimize for 250 iterations using the full gradient. We generate three datasets each and measure squared reconstruction error and the six unfairness metrics.
The results are listed in \[tab:synthetic\]. For each metric, we print in bold the best average score and any scores that are not statistically significantly distinct according to paired t-tests with threshold 0.05. The results indicate that the learning algorithm successfully minimizes the unfairness penalties, generalizing to unseen, held-out user-item pairs. And reducing any unfairness metric does not lead to a significant increase in reconstruction error.
The complexity of computing the unfairness metrics is similar to that of the error computation, which is linear in the number of ratings, so adding the fairness term approximately doubles the training time. In our implementation, learning with fairness terms takes longer because loops and backpropagation introduce extra overhead. For example, with synthetic data of $400$ users and $300$ items, it takes $13.46$ seconds to train a matrix factorization model without any unfairness term and $43.71$ seconds for one with value unfairness.
While optimizing each metric leads to improved performance on itself (see the highlighted entries in \[tab:synthetic\]), a few trends are worth noting. Optimizing any of our new unfairness metrics almost always reduces the other forms of unfairness. An exception is that optimizing absolute unfairness leads to an increase in underestimation. Value unfairness is closely related to underestimation and overestimation, since optimizing value unfairness is even more effective at reducing underestimation and overestimation than directly optimizing them. Also, optimizing value and overestimation are more effective in reducing absolute unfairness than directly optimizing it. Finally, optimizing parity unfairness leads to increases in all unfairness metrics except absolute unfairness and parity itself. These relationships among the metrics suggest a need for practitioners to decide which types of fairness are most important for their applications.
Real Data
---------
We use the Movielens Million Dataset [@harper2016movielens], which contains ratings (from 1 to 5) by 6,040 users of 3,883 movies. The users are annotated with demographic variables including gender, and the movies are each annotated with a set of genres. We manually selected genres that feature different forms of gender imbalance and only consider movies that list these genres. Then we filter the users to only consider those who rated at least 50 of the selected movies.
The genres we selected are *action*, *crime*, *musical*, *romance*, and *sci-fi*. We selected these genres because they each have a noticeable gender effect in the data. Women rate musical and romance films higher and more frequently than men. Women and men both score action, crime, and sci-fi films about equally, but men rate these film much more frequently. lists these statistics in detail. After filtering by genre and rating frequency, we have 2,953 users and 1,006 movies in the dataset.
We run five trials in which we randomly split the ratings into training and testing sets, train each objective function on the training set, and evaluate each metric on the testing set. The average scores are listed in \[tab:movielens\], where bold scores again indicate being statistically indistinguishable from the best average score. On real data, the results show that optimizing each unfairness metric leads to the best performance on that metric without a significant change in the reconstruction error. As in the synthetic data, optimizing value unfairness leads to the most decrease on under- and overestimation. Optimizing non-parity again causes an increase or no change in almost all the other unfairness metrics.
Conclusion {#sec:conclusion}
==========
In this paper, we discussed various types of unfairness that can occur in collaborative filtering. We demonstrate that these forms of unfairness can occur even when the observed rating data is correct, in the sense that it accurately reflects the preferences of the users. We identify two forms of data bias that can lead to such unfairness. We then demonstrate that augmenting matrix-factorization objectives with these unfairness metrics as penalty functions enables a learning algorithm to minimize each of them. Our experiments on synthetic and real data show that minimization of these forms of unfairness is possible with no significant increase in reconstruction error.
We also demonstrate a combined objective that penalizes both overestimation and underestimation. Minimizing this objective leads to small unfairness penalties for the other forms of unfairness. Using this combined objective may be a good approach for practitioners. However, no single objective was the best for all unfairness metrics, so it remains necessary for practitioners to consider precisely which form of fairness is most important in their application and optimize that specific objective.
#### Future Work
While our work in this paper focused on improving fairness among users so that the model treats different groups of users fairly, we did not address fair treatment of different item groups. The model could be biased toward certain items, e.g., performing better at prediction for some items than others in terms of accuracy or over- and underestimation. Achieving fairness for both users and items may be important when considering that the items may also suffer from discrimination or bias, for example, when courses are taught by instructors with different demographics.
Our experiments demonstrate that minimizing empirical unfairness generalizes, but this generalization is dependent on data density. When ratings are especially sparse, the empirical fairness does not always generalize well to held-out predictions. We are investigating methods that are more robust to data sparsity in future work.
Moreover, our fairness metrics assume that users rate items according to their true preferences. This assumption is likely to be violated in real data, since ratings can also be influenced by various environmental factors. E.g., in education, a student’s rating for a course also depends on whether the course has an inclusive and welcoming learning environment. However, addressing this type of bias may require additional information or external interventions beyond the provided rating data.
Finally, we are investigating methods to reduce unfairness by directly modeling the two-stage sampling process we used to generate synthetic, biased data. We hypothesize that by explicitly modeling the rating and observation probabilities as separate variables, we may be able to derive a principled, probabilistic approach to address these forms of data imbalance.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Heckman et al. (2005) used the Galaxy Evolution Explorer (GALEX) UV imaging survey to show that there exists a rare population of nearby compact UV-luminous galaxies (UVLGs) that closely resembles high redshift Lyman break galaxies (LBGs). We present HST images in the UV, optical, and H$\alpha$, and resimulate them at the depth and resolution of the GOODS/UDF fields to show that the morphologies of UVLGs are also similar to those of LBGs. Our sample of 8 LBG analogs thus provides detailed insight into the connection between star formation and LBG morphology. Faint tidal features or companions can be seen in all of the rest-frame optical images, suggesting that the starbursts are the result of a merger or interaction. The UV/optical light is dominated by unresolved ($\sim$100-300 pc) super starburst regions (SSBs). A detailed comparison with the galaxies Haro 11 and VV 114 at $z=0.02$ indicates that the SSBs themselves consist of diffuse stars and (super) star clusters. The structural features revealed by the new HST images occur on very small physical scales and are thus not detectable in images of high redshift LBGs, except in a few cases where they are magnified by gravitational lensing. We propose, therefore, that LBGs are mergers of gas-rich, relatively low-mass ($M_*\sim10^{10}$ $M_\odot$) systems, and that the mergers trigger the formation of SSBs. If galaxies at high redshifts are dominated by SSBs, then the faint end slope of the luminosity function is predicted to have slope $\alpha\sim2$. Our results are the most direct confirmation to date of models that predict that the main mode of star formation in the early universe was highly collisional.'
author:
- 'Roderik A. Overzier, Timothy M. Heckman, Guinevere Kauffmann, Mark Seibert, R. Michael Rich, Antara Basu-Zych, Jennifer Lotz, Alessandra Aloisi, Stéphane Charlot, C. Hoopes, D. Christopher Martin, David Schiminovich, Barry Madore'
title: |
HST morphologies of local Lyman break galaxy analogs I:\
Evidence for starbursts triggered by merging
---
Introduction {#sec:intro}
============
How did galaxies form? Ultimately, this simple question captures most, if not all of the most widely pursued research topics in cosmology. Surveys of galaxies at different redshifts are being used to study which galaxies contain most of the stellar mass at a given epoch, and which galaxies are undergoing the strongest evolution. These measurements serve to constrain models of structure formation. The main formative processes associated with typical galaxies (e.g. star formation, merging, and feedback) were largely completed within the first half of the Hubble time, making their study a very challenging one. One of the best probes for studying star formation in the early universe is provided by galaxies that are luminous in the rest-frame UV due to intense star formation. The most luminous of these, the Lyman break galaxies (LBGs), are easily detected at $z=2-6$ in deep pencil beam surveys from the ground and with the [*Hubble Space Telescope*]{} (HST) [e.g. @steidel93; @steidel99; @shapley01; @giavalisco04; @adelberger04; @bouwens06; @yoshida06]. The cosmic star formation rate (SFR) reached a maximum at $z\sim1-3$ and decreased dramatically toward $z=0$ [e.g. @lilly96; @schiminovich05]. There is only a modest decrease in the UV luminosity density from $z\sim3$ to $z\sim6$ [e.g. @ouchi04; @bouwens06], indicating that LBGs represent a major phase in the early stages of galaxy formation and evolution. Based on their clustering and number statistics, LBGs are the precursors of present day massive galaxies undergoing a phase of intense star formation, and a large fraction of LBGs may merge to form elliptical galaxies that are situated in present-day groups and clusters [@governato01; @giavalisco01; @moustakas02; @ouchi04b; @adelberger05].
Studies of the structure and sizes of LBGs indicate that high redshift galaxies are compact [$\sim$0.1–0.3, or $\sim$1–2.5 kpc; @giavalisco96a; @lowenthal97; @ferguson04; @bouwens04], and that large ($\gtrsim$0.4) low surface brightness galaxies are rare. The morphologies are best characterized as being dominated by one to several UV-bright knots embedded in diffuse emission, with little difference between the rest-frame UV and optical light distributions [@papovich01; @papovich05]. The morphologies of LBGs are unlike those of nearby galaxies on the Hubble sequence. This is true even if the images of the nearby objects are degraded so that they have similar resolution and depth to those of LBGs. LBG morphologies are more similar to those of ultraluminous infrared galaxies (ULIRGs), which occupy a region of the concentration/asymmetry plane that lies between double-nucleated mergers, spheroids and disks [e.g. @abraham96; @conselice04; @lotz04; @lotz06; @law07a and references therein]. However, the interpretation of the UV morphologies of high redshift populations remains problematic [see @law07a], because it is unclear whether the irregular morphologies are a consequence of the merging of gas-rich galaxies, or whether one is just seeing patchily distributed star formation within a single system. The answer to this question and how it evolves with redshift is essential for understanding how galaxies formed.
A detailed study of LBGs is limited by two factors. First, their distances render them small and faint, and, second, star forming galaxies at high redshift are systematically different from starburst galaxies in the nearby universe. Local starbursts (SBs) show strong correlations among their basic properties that are different from LBGs. In particular, there is a systematic increase in the amount of dust obscuration ($L_{FIR}/L_{FUV}$) with increasing SFR, corresponding to a systematic increase in metallicity with mass. LBGs are substantially less obscured than local SBs of similar SFR or mass [e.g. @reddy06]. The relatively low extinction of LBGs is also a characteristic of blue compact dwarfs (BCDs, alsoor galaxies), but the latter have SFRs that are typically two orders of magnitude smaller than LBGs and they have lower masses [e.g. @fanelli88; @telles97; @hunter99; @gildepaz03]. Although BCDs may resemble the faintest LBGs [e.g. @meurer95; @noeske06], and LIRGS and ULIRGS may resemble the dustiest LBGs [e.g. @adelberger00; @goldader02; @daddi05; @chapman05; @huang05; @vandokkum06], typical local SBs are different from typical LBGs. LBGs have systematically lower metallicities and higher gas-mass fractions than local SBs with the same SFRs. The typical mechanisms that trigger starbursts might also be very different at high redshift compared to low redshift.
Clearly, if we could find relatively nearby starburst galaxies whose properties are the best match to LBGs, we could study for the first time at a much higher physical resolution how the vigorous star formation and morphologies of typical high redshift galaxies are related. However, local LBG analogs are currently very rare; as indicated by the Galaxy Evolution Explorer [GALEX; @martin05] imaging survey in the near (NUV, $\lambda$$\sim$2250Å) and far (FUV, $\lambda$$\sim$1550Å) UV, the co-moving number density of galaxies with FUV luminosities similar to LBGs has declined by a factor of several hundred between $z=3$ and $z=0$. @heckman05 and @hoopes07 (henceforward called “H05” & “H07”) identified a rare population of low-redshift ($z<0.3$) galaxies with properties remarkably similar to those of LBGs by matching sources in the GALEX all-sky survey with the Sloan Digital Sky Survey [SDSS; @york00] spectroscopic sample, and selecting according to 2 criteria specifically designed to match the typical UV properties of LBGs [e.g. @steidel96; @shapley01]: $L_{FUV} \geq 10^{10.3} L_{\odot}$ $\wedge$ $I_{FUV} \geq 10^9
L_{\odot} \mathrm{kpc}^{-2}$, where $L_{FUV}$ is the FUV luminosity, $\lambda P_\lambda$, and $I_{FUV}$ is the mean FUV surface brightness interior to the SDSS [*u*]{}-band half-light radius ($I_{FUV} =
\frac{1}{2} L_{FUV}/\pi r_{e,u}^2$). By further limiting the sample to $z<0.3$ and excluding broad line active galactic nuclei (AGN), this resulted in a sample of 31 “supercompact UV-luminous galaxies” (UVLGs). Interestingly, once selected according to these criteria, the UVLG sample proved to be similar to typical LBGs in all other measurable properties (see H05 and H07). Further details on the sample are given by @basuzych07, who investigated the radio to UV continuum properties of the sample and found that they follow the radio-far infrared correlation of normal star-forming galaxies, but generally have less dust than other star forming galaxies with such high specific SFRs. In this respect, they are similar to LBGs. This suggests that the process that causes star formation in the supercompact UVLGs differs from other local star forming galaxies, but may be similar to LBGs. Our sample of UVLGs (“local LBG analogs”) is therefore highly valuable for understanding the nature of Lyman break galaxies at high redshifts[^1].
In this paper (Paper I) we present the first set of high resolution HST images of 8 UVLGs and discuss their morphologies. In a subsequent paper (Paper II) we will carry out a more detailed study using a large HST data set to be observed in Cycle 16. The structure of this paper is as follows. In Sect. 2 we describe the new observations with HST. In Sections 3 and 4 we present the data, investigate UVLG morphologies both qualitatively and quantitatively, resimulate our data at higher redshift, and compare with morphologies of star-forming galaxies and LBGs as well as two of the most nearby LBG-like galaxies known (Haro 11 and VV 114). We discuss the implications of our results for understanding the nature of LBGs at high redshifts in Section 5, followed by a summary of the results (Section 6). We use a cosmology \[$\Omega_M$, $\Omega_\Lambda$, $h$\]$=$\[0.27,0.73,0.73\] with $H_0=100h$ km s$^{-1}$ Mpc$^{-1}$.
Data
====
HST observations
----------------
We have observed 8 of the nearest ($0.091<z<0.204$) and brightest local LBG analogs of H05 & H07 with HST in Cycle 15. To date, 7 UVLGs have been observed with the Advanced Camera for Surveys (ACS) High Resolution Camera (HRC) in the filter F330W, and with the Wide Field Channel (WFC) through the filter F850LP. In addition, one UVLG has been observed with the Solar Blind Channel (SBC) in the filter F150LP, and with the Wide Field and Planetary Camera 2 (WFPC2) through F606W, given the new constraints following the failure of ACS during the course of 2007. Ramp filter images with a central wavelength equal to that of redshifted H$\alpha$ were also obtained for all but the latter source. Each filter probes star formation and morphology at a different characteristic timescale. The F330W filter (central wavelength $\lambda_c$$\approx$3334Å with an effective width $\Delta\lambda\sim548\AA$) is the most sensitive to star formation over the past $\sim$100 Myr [@leitherer95] and provide the closest match to the GALEX NUV filter ($\lambda_c$$\approx$2315Å, $\Delta\lambda\sim730\AA$). The F150LP filter ($\lambda_c$$\approx$1614Å, $\Delta\lambda\sim$234Å), available for one of the UVLGs, provides an excellent match to the GALEX FUV filter ($\lambda_c$$\approx$1530Å, $\Delta\lambda\sim255\AA$). The F606W ($\lambda_c$$\approx$6001Å) and F850LP ($\lambda_c$$\approx$9170Å) images are well-matched to the SDSS images in the [*r-*]{} and [ *i*]{}-band and probe older ($>$Gyr) stellar populations. The ACS ramp filters (FR647M with a width of 207Å or FR782N with a width of 52Å, depending on the redshift) probe redshifted H$\alpha$ from regions tracing the youngest stellar population (O stars with lifetimes $<$10 Myr), and allow us to search for evidence of galaxy-wide outflows of ionized gas.
The targets were observed for one orbit per filter with ACS, and two orbits per filter with WFPC2. The F330W image of SDSS J092600.41+442736.1 failed to execute due to a guide star problem. The F150LP, F330W, ramp filter, F606W, and F850LP images consisted of 3, 3, 2, 6 and 3 exposures, respectively, to facilitate the removal of cosmic rays. The images were combined using [*Multidrizzle*]{} [@koekemoer02], producing registered, cosmic-ray free, geometrically corrected images. The SBC and HRC images have a plate scale of 0025 pixel$^{-1}$ and a resolution of $\sim$$0\farcs07$ (FWHM). The WFPC2/PC images have a plate scale of 0046 pixel$^{-1}$ with a resolution of $\approx$$0\farcs11$ (FWHM). The WFC images have a scale of 005 pixel$^{-1}$ with a resolution of $\approx$$0\farcs12$ (FWHM). Magnitudes in the AB system were measured using SExtractor [@bertin96] from $m_{AB}=-2.5\mathrm{log}_{10}(\mathrm{counts}/T_{exp})+\mathrm{ZPT}$, where the zeropoints (ZPT) are 22.448, 24.085, 23.004, and 24.862 mag for F150LP, F330W, F606W and F850LP, resp. All magnitudes were corrected for Galactic extinction using the dust maps of @schlegel98. The reader is referred to Table \[tab:log\] for a log of the observations, Table \[tab:sfrs\] for SFRs and stellar masses of the objects in our sample, and Table \[tab:phot\] for the main photometric data from GALEX and ACS.
Continuum subtracted [H$\alpha$]{} images
-----------------------------------------
The [H$\alpha$]{} ramp filter images were continuum-subtracted as follows. First we scaled the ACS [$z_{850}$]{} image to an artificial [$i_{775}$]{} continuum image, by determining the ratio of the flux of the [$i_{775}$]{} continuum at the wavelength of redshifted [H$\alpha$]{} (measured from the SDSS fiber spectrum on opposite sides of the [H$\alpha$]{} line) to the total flux in the [$z_{850}$]{} image. The total flux was measured within an aperture similar in size to the 3 diameter SDSS fiber aperture. We then created a “continuum-free” [H$\alpha$]{} image by subtracting the artificial [$i_{775}$]{}continuum image from the ACS ramp filter image. The resulting [H$\alpha$]{}image should be a very good approximation, under the (very reasonable) assumption that the morphology of the continuum does not change between [$i_{775}$]{} and [$z_{850}$]{}. In the case of object SDSS J080844.26+394852.4 the subtraction was problematic due to the fact that it relied on subtraction of two bright point sources in [$z_{850}$]{} and [H$\alpha$]{}.
Spitzer observations
--------------------
Infrared photometry was obtained with the Infrared Array Camera (IRAC) and the Multi-band Imaging Photometer for Spitzer (MIPS) aboard the [*Spitzer Space Telescope*]{} (PI: C. Hoopes, \#20390). The total integration time with IRAC at 3.6, 4.5, 5.8 and 8.0 $\mu$m was 60 s for each channel. The total integration time with MIPS at 24 and 70 $\mu$m was 42 and 60 s, resp. We used a minimum 5 step dither pattern for removal of cosmic rays. The post-basic calibrated data (BCD) delivered by the pipeline were used to measure the integrated flux densities. The infrared photometry is given in Table \[tab:ir\].
Comparison data
---------------
The UVLGs in our sample, which have a median redshift of $z\sim0.15$, lie at redshifts that are intermediate between local starburst galaxies ($\sim0.02$) and LBGs at high redshift ($z>1.5$). Throughout the paper, we therefore find it instructive to present our results with respect to the following two sets of comparison data:
### Archival imaging data of [*Haro 11*]{} and [*VV 114*]{} {#sec:archival}
As shown in H07 (see their Fig. 12), a number of very nearby blue compact starburst galaxies and (U)LIRGS fall very near to the boundary of the UVLG selection window as defined in the $L_{FUV}$ vs. $I_{FUV}$ plane. Because there is quite some leverage in the definition of a typical ‘LBG’, we will compare our data to two of these objects, which have been found to possess some properties that are similar to high redshift LBGs. The two local comparison objects are Haro 11 and VV 114, both of which lie at $z=0.02$ [for details see @knop94; @scoville00; @goldader02; @bergvall02; @kunth03; @grimes06; @grimes07 and references therein]. Haro 11 is a blue compact dwarf galaxy consisting of a number of UV bright knots believed to be in the process of merging. VV 114 is a merging system consisting of a UV-luminous Western component (VV 114W), and an IR-luminous Eastern component (VV 114E) with very little associated UV emission.
In this paper, we make use of an ACS/HRC image of Haro 11 taken through the filter F220W ($\lambda_c$$\approx$2255Å) obtained from the HST archive (Program 10575, PI: Göran Östlin). For VV 114(W), we use a Space Telescope Imaging Spectrograph (STIS) image in the NUV ($\lambda_c$$\approx$2365Å) from Program 8201 (PI: Gerhardt Meurer).
### High redshift samples from @lotz06
Our high redshift comparison samples consist of 55 starburst galaxies at $z\sim1.5$ selected from GOODS, and two (largely non-overlapping) samples of in total 82 $z\sim4$ LBGs in GOODS and the UDF from @lotz06. The galaxy sample at $z\sim1.5$ is based on a large spectroscopic sample of strong emission line galaxies. We used the publicly available GOODS and UDF data, and extracted postage stamps of the objects in [$B_{435}$]{} for the $z\sim1.5$ sample, and in [$V_{606}$]{}$+$[$i_{775}$]{} for the $z\sim4$ sample. A detailed description of the data and sample selection can be found in @lotz06.
Results
=======
Images and notes on individual galaxies
---------------------------------------
In Fig. \[fig:uv\] we show the GALEX FUV and NUV images. The UV, H$\alpha$, and optical images taken with HST are shown in Fig. \[fig:stamps\], which have a resolution of $\sim50$ times that of the GALEX images. Qualitative remarks on the morphologies of the UVLGs in Fig. \[fig:stamps\] are as follows.\
[**SDSS J005527.46–002148.7**]{}. This object consists of a strong point source surrounded by diffuse emission that can be seen in [$U_{330}$]{}, [$z_{850}$]{} and [H$\alpha$]{}. The diffuse emission has a high surface brightness region extending to the NE with respect to the nucleus, which suggests either the infall of a small diffuse galaxy or an additional off-center star forming region. The continuum-subtracted [H$\alpha$]{}emission is peculiar with an ‘arm’ of emission extending to about 1 South of the nucleus. The morphology of the [H$\alpha$]{} is very different from the [$z_{850}$]{} morphology, and at its outer extremity, the [H$\alpha$]{} is not associated with any UV emission. We suspect that the extended [H$\alpha$]{} is due to an outflow over the larger region probed by the [$z_{850}$]{} continuum.
[**SDSS J032845.99+011150.8**]{}. The [$z_{850}$]{} image suggests a merger of two low surface brightness galaxies, as evidenced by the tidal tails extending symmetrically from the center to the Northwest and Northeast. The starburst dominates the nuclear region, in which five isolated point sources can be seen. The easternmost knot dominates in [H$\alpha$]{}.
[**SDSS J040208.86–050642.0**]{}. The [$z_{850}$]{} image suggests a merger of two diffuse systems as the contours of faint [$z_{850}$]{} continuum emission show a sharp bend in position angle. The image center is dominated by three point sources, and several fainter ones lie further out on either side of the center. One of these faint, compact regions (about 1 South of the nucleus) is more extended in [H$\alpha$]{}.
[**SDSS J080844.26+394852.4**]{}. There is a bright, unresolved source seen in the [$U_{330}$]{} image that is located in the centre of an extended, highly diffuse galaxy seen in [$z_{850}$]{}. The unresolved source is remarkably bright as both [$U_{330}$]{} and [H$\alpha$]{} show the ACS diffraction spikes. The [H$\alpha$]{} image suffers from poor point source subtraction. The object seen $\sim2\arcsec$ to the South lies at a similar redshift, and may be interacting. The nucleus of this companion is also detected in [$U_{330}$]{}, but its flux is $\sim75\times$ lower.
[**SDSS J092600.41+442736.1**]{}. The faint contours of the diffuse emission in [$z_{850}$]{} suggest that at least two diffuse galaxies may be merging. A bright, compact component dominates in [$z_{850}$]{} and [H$\alpha$]{}. The faint extension about 1 East of the nucleus in [$z_{850}$]{} has little associated [H$\alpha$]{} emission. The [$U_{330}$]{} image failed to execute due to a guide star problem.
[**SDSS J102613.97+484458.9**]{}. This galaxy has a ring-shaped morphology with most of the starburst occurring along the Eastern rim. There is also an isolated knot to the Southwest. The star-forming region contains several bright, unresolved components and diffuse emission. A very faint diffuse companion can be seen in [$z_{850}$]{}, about 2 to the South, and very faint extended emission, possibly tidal debris, is seen directly to the East along the entire extent of the galaxy. The overall morphology is very similar to that of the “drop-through” ring galaxy NGC922 studied by @wong06.
[**SDSS J135355.90+664800.5**]{}. This UVLG is highly irregular in all bands, showing many star-forming knots and diffuse emission. The object is interacting or merging with a warped, edge-on or filamentary galaxy located just to the East, which is much redder and only just detected in [$U_{330}$]{}. A longslit spectrum along the major axis of the system shows that the two objects are likely to be a counter-rotating merger (Overzier et al., in prep.).
[**SDSS J214500.25+011157.6**]{}. This is the only object in our sample for which the ACS image provides a direct match at the wavelength of the GALEX FUV image. The F150LP image shows a very compact, but elongated starburst region in which we can discern three different knots. Fainter FUV emission comes from a slightly more extended region of $\sim$1. The optical morphology in [$V_{606}$]{} is strikingly different. The image is still dominated by the emission surrounding the starburst region, and a faint spiral structure may be present. A companion object appears to have gone straight through the star forming nucleus as evidenced by its tadpole-like morphology with a faint tail pointing towards the main galaxy and a trail of tidal debris that can be traced back to a location that is about equidistant from the nucleus on the opposite side of the main galaxy.
Emission lines
--------------
Although the sample of UVLGs does not contain any broad line AGN, we want to make sure that we are studying the morphologies of starburst galaxies rather than those of narrow line AGN. The main optical emission line diagnostic diagrams involving the lines \[OIII\]$\lambda5007$Å, H$\beta$, \[OI\]$\lambda6300$Å, H$\alpha$, \[NII\]$\lambda6584$Å, and the \[SII\]$\lambda\lambda$6713,6731Ådoublet (measured from the SDSS spectra) are shown in Fig. \[fig:bpt\] (see also H07). The 8 objects observed by HST are indicated by the large filled circles and their IDs, while small filled circles indicate objects in our supercompact UVLG sample without HST data. Most of the UVLGs lie along the main star forming sequence (points) in log(\[OIII\]/H$\beta$) vs. log(\[NII\]/H$\alpha$) (left panel of Fig. \[fig:bpt\]), albeit offset towards higher values of log(\[OIII\]/H$\beta$ probably due to a more intense ionizing radiation field. It is interesting to note that a similar offset from the main star-forming sequence has been observed for high-redshift LBGs as well [@shapley05; @erb06a].
About one quarter of the sample falls in the region between the relations of @kauffmann03 (solid line) and @kewley06 (dashed line) that is typically populated by objects having a composite spectrum consisting of a metal rich stellar population and an AGN. When we plot log(\[OI\]/H$\alpha$) vs. log(\[OIII\]/H$\beta$) (middle panel) or log(\[SII\]/H$\alpha$) vs. log(\[OIII\]/H$\beta$) (right panel), the objects that were in the AGN/starburst composite region move to the far, opposite side of the star forming track (see object SDSS J080844.26+394852.4 that is most relevant to this paper). It is not exactly clear what causes this behavior of the line ratios, but it is consistent with starburst model predictions having very high ionization parameters [@kewley01] as might be expected for our sample. A more detailed analysis of the ionization parameters and the possibly remaining contribution from AGN will be given elsewhere (Overzier et al., in prep.). In any case, @kauffmann03 have shown that the continuum emission from local narrow-line AGN is almost always dominated by its stellar population; this assures us that the HST images accurately reflect the morphology of the stellar light.
HST-based size measurements {#sec:sizes}
---------------------------
### UV sizes
Our sample of UVLGs is selected to have high surface brightness, which is estimated from the FUV flux and a half-light radius measured from fitting a seeing-convolved exponential profile to the SDSS [ *u*]{}-band images (see Section 1, H05 & H07). The resulting seeing-deconvolved sizes were on the order of 1–2 kpc, but were somewhat uncertain because most of the UVLGs are only barely resolved in the [*u*]{}-band SDSS images. The F330W images are a factor 10 higher in resolution and thus provide a reliable and independent test for the true sizes of these UVLGs. We have used SExtractor’s circular aperture method to estimate the radii; the 50 and 90% radii were computed using the light enclosed within 2.5 times the elliptical Kron radius also determined using SExtractor (Table \[tab:sizes\]). The sizes measured from the HST images are somewhat smaller than our previous estimates based on SDSS. In the case of SDSS J080844, approximately half the light in F330W is coming from a region that is still unresolved even at the resolution of the HRC ($\sim0\farcs07$, or $\sim$110 pc at $z=0.091$). Excluding this extreme case, the radii range from 0.4 to 1.9 kpc.
One could question (see Appendix) whether the sizes measured in [$U_{330}$]{}or [*u*]{}-band (used to calculate the UV surface brightness) are representative for the sizes of the regions that emit in the FUV and NUV as seen by GALEX at the much lower resolution of $\sim4\arcsec$ (FWHM). We can test this simply by measuring the slope, $\beta$, of the UV continuum (with $f_\lambda\propto\lambda^\beta$) across the GALEX bands, and comparing the flux measured in the HST image to the extrapolated value at [$U_{330}$]{}. The UV slopes[^2] are given in Table \[tab:phot\], and we note that they are very similar to that of high-redshift LBGs [e.g. @ouchi07]. The measured flux in [$U_{330}$]{}is lower than the predicted values by a factor of $\sim1.3$ on average. The measured flux in [*u*]{}-band is $\sim1.2\times$ lower than the value extrapolated from the UV. The generally good correspondence between the predicted and measured continuum flux at 1500–3500Å indicates that the emission structure seen in the HST and SDSS images is representative of the light distribution in the FUV. A further test is provided by the case of SDSS J214500 for which we have an ACS/SBC image taken through F150LP, a filter that is almost identical to the GALEX FUV filter [see @teplitz06]. The corresponding FUV surface brightness calculated from the ACS data is $\textrm{log}_{10}I_{FUV,ACS}=9.53$ $L_\odot$ kpc$^{-2}$, in agreement with H07.
### H$\alpha$ and Optical sizes
The distribution of the optical and [H$\alpha$]{} light need not be the same. We find, however, that the sources are compact at all observed wavelengths (Table \[tab:sizes\]), and the starbursts dominate the structures. The half-light radii measured from the continuum subtracted [H$\alpha$]{} images are slightly larger than those measured from [$U_{330}$]{} ($\sim$0.5–2 kpc). In some cases, the [H$\alpha$]{} morphology suggests an outflow (SDSS J005527, and perhaps J032845 and J092600; see Fig. \[fig:stamps\]), but this does not affect the measured sizes. The half-light radii in F850LP are typically 1–2 kpc, with the exception of SDSS J135355 which has $r_{50,850}\approx4$ kpc (note, however, that it has a nearby companion). Both the 50 and 90% radii are typically twice as large as the UV size ($\sim$2–5 kpc), indicating that these systems have relatively faint, underlying structures that were already in place prior to the current episode of star formation.
UV-optical Colors
-----------------
We have used SExtractor to derive (circular) [$U_{330}$]{}–[$z_{850}$]{} radial color profiles out to the 90% flux radius. The results are shown in Fig. \[fig:radial\]. Most objects have large gradients in their UV-optical color profile, with objects typically being bluer near the center, and redder further out. Because the profiles were calculated with respect to the centroid of the objects in the optical image, for some objects (032845, 040208 & 102613) the bluest regions are sometimes slightly offset from the nominal center of the galaxy (see Fig. \[fig:stamps\]). We have defined an “inner” color measured within the [$U_{330}$]{} half-lightradius and an “outer” color measured over the region between the 50 and 90% radii in [$z_{850}$]{} (see Table \[tab:sizes\]). In all cases the inner color is bluer than the outer color, and the steepest color gradients are seen for the most compact objects (in [$U_{330}$]{}). This indicates that these objects are composites of recent, central starbursts within older (or more dusty) extended structures.
Extinction {#sec:slopes}
----------
For starburst galaxies there exists a good correlation between the amount of UV flux that is absorbed by dust (usually taken to be a foreground “screen”), and the amount of flux (re-)emitted in the far-IR [@meurer97; @calzetti01; @kong04; @seibert05]. To estimate the internal extinction for our sample we have calculated the attenuation in the FUV, $A_{FUV}$, using the ratio of the bolometric dust luminosity, $L_{TIR}$, to bolometric FUV luminosity, $L_{FUV}$, and the fitting formula of @burgarella05: $$A_{FUV}=a_1x^3+a_2x^2+a_3x+a_4,$$ with $[a_1,a_2,a_3,a_4]=[-0.028,0.392,1.094,0.546]$ and $x=\mathrm{log_{10}}(L_{TIR}/L_{FUV})$. The bolometric dust luminosity is given by [@dale02]:\
$$\L_{TIR}=\xi_1\nu_{24}L_{24}+\xi_2\nu_{70}L_{70}+\xi_3\nu_{160}L_{160},$$ with $[\xi_1,\xi_2,\xi_3]=[1.559,0.7686,1.347]$. Although we do not have observed flux densities at 160 $\mu$m, we can estimate 160 $\mu$m using the strong empirical correlation between $(f_8/f_{24})_{\mathrm{dust}}$ and $(f_{70}/f_{160})$ by employing the synthetic models of @dale02. The correlation arises because more intense radiation fields have larger $(f_{70}/f_{160})$ due to hotter large grains, and smaller $(f_8/f_{24})_{\mathrm{dust}}$ due to larger emission by small grains at 24 $\mu$m. The dust emission at 8.0 and 24 $\mu$m was calculated from the observed flux densities using $(f_8)_{\mathrm{dust}}=f_8-0.232f_{3.6}$ and $(f_{24})_{\mathrm{dust}}=f_{24}-0.032f_{3.6}$, which includes a correction for the contribution due to stars based on the 3.6 $\mu$m flux [@dale05]. Finally, we calculate the extinction using the formula $E(B-V)_{\mathrm{stars}}= A_{FUV}/k(\lambda)$ with $k(\lambda)\approx10.77$ [@calzetti01]. The extinction ranges from $E(B-V)\approx0.01$ to 0.14. The IR flux densities, bolometric luminosities, $A_{FUV}$ and extinction are given in Table \[tab:ir\]. The extinction will be needed in Section 4 when we estimate masses and ages from the data using a set of model star formation histories. We refer the reader to @basuzych07 for a more detailed discussion and analysis of the extinction properties of this sample using alternative estimators.
Morphologies
------------
### Gini coefficient, $M_{20}$, and Concentration {#sec:morphologies}
In order to compare the morphologies of the UVLGs to those of nearby galaxies and of LBGs at high redshift, we calculated the Gini coefficient ($G$; a measure of the equality with which the flux is distributed across a galaxy), $M_{20}$ (the log of the ratio of the second order moment of the pixels containing the 20% brightest flux to the total second order moment), and concentration ($C$; five times the log of the ratio of the circular radii containing 80 and 20% of the flux). To calculate the structural parameters we follow the procedures described in full detail in @lotz04 [@lotz06]. First, we use SExtractor to make an object segmentation map and mask out neighboring objects. The image is background subtracted, and we calculate an initial Petrosian radius ($r_P$ with $\eta\equiv0.2$) using the object center and (elliptical) shape information from SExtractor. We then smooth the image by $\sigma=r_P/5$ and create a new segmentation map by selecting those pixels that have a surface brightness higher than the mean surface brightness at the Petrosian radius. We recalculate the object center by minimizing the second order moment of the flux, and then recalculate the Petrosian radius in the original image using this center. The total flux is defined as the flux within a radius of $1.5\times r_P$ and $C$ is then calculated in circular apertures. The individual measurements are listed in Table \[tab:morphologies\], and the results are shown in Fig. \[fig:morphologies\]. For each object, the [$U_{330}$]{} measurements are plotted as stars and the [$z_{850}$]{} as circles, and are connected by a dotted line. For 080844, $G$ and $C$ values in [$U_{330}$]{} must be considered lower limits, and its $M_{20}$ value is an upper limit because the central pixel of the unresolved component contains $\approx$20% of the total flux.
Although our sample is small, we will attempt to make a general comparison with the morphologies of nearby galaxies. As shown in Fig. \[fig:morphologies\], the UVLGs populate the region roughly defined by $M_{20}\lesssim-1.3$ and $0.53\lesssim G\lesssim0.7$ (left panel), and $3\lesssim C\lesssim5$ (right panel). We can compare these measurements to the various divisions of morphological parameter space discussed by @lotz06. Our UVLGs have smaller $M_{20}$ than merging galaxies with two clearly separated nuclei ($M_{20}\gtrsim-1.1$, left hatched region), larger $G$ than the expectations for exponential disks (dotted box), $M_{20}$ similar or larger than those expected for bulge-dominated objects (right hatched region), and $G$ and $C$ that are either similar or smaller than those expected for bulge-dominated galaxies seen face-on (solid box). We do not find any systematic differences between the morphologies measured in [$U_{330}$]{} and [$z_{850}$]{} for the sample as a whole. The morphological parameters in the UV and optical are thus dominated by the same (bright) regions, even though some of the objects possess very faint, extended structures in [$z_{850}$]{}.
Fig. \[fig:morphologies\] shows that two UVLGs qualify as bulge-dominated in [$U_{330}$]{}, while four objects qualify as bulge-dominated in [$z_{850}$]{}. None of the objects qualifies as a pronounced merger, and only one object lies just on the border of the exponential disk region in [$z_{850}$]{}, with the remainder populating the regions in between. The overall distribution of morphologies is very similar to that of star forming galaxies at $z\sim$1.5 and LBGs at $z\sim4$ selected from GOODS and the UDF. @lotz06 find median values of $G\sim0.55$, $M_{20}\sim-1.5$ and $C\sim3.3$ at $z\sim1.5$, and $G\sim0.58$, $M_{20}\sim-1.6$ and $C\sim3.8$ at $z\sim4$ (encircled crosses in Fig. \[fig:morphologies\]). However, a fair comparison of the morphologies of UVLGs with the higher redshift comparison samples should be carried out at the same $S/N$ and resolution. In the next section, we will carry out such a comparison.
### Redshift simulations {#sec:sims}
We now investigate whether the conclusions of the previous subsection are still upheld if the analysis is carried out using images that are simulated to have the same depth and resolution as LBGs observed at high redshift.
We follow common practice [e.g., see @giavalisco96b; @hibbard97; @bouwens98; @papovich03; @conselice03; @lotz04], and apply corrections for cosmological surface brightness dimming and for changes in physical resolutions to the [$U_{330}$]{} images, in order to match the instrumental conditions under which our objects would be observed in typical surveys when placed at $z=1-4$.
The first step of the procedure is to rebin the [$U_{330}$]{} images by a factor $b=(\theta_1/\theta_2)(s_2/s_1)$, where $\theta_i$ is the angle on the sky of an object of fixed size $d$ at $z=z_i$, and $s_i$ is the instrumental pixel scale (in arcsec pixel$^{-1}$). The rebinning factor can be expressed in terms of either the angular diameter distance, $D_{A_i}=d/\theta_i$, or the luminosity distance, $D_{L_i}=(1+z_i)^2D_{A_i}$, at low and at high redshift ($z_2>z_1$): $$b=\frac{D_{A_2}}{D_{A_1}}\frac{s_2}{s_1}=\left(\frac{1+z_1}{1+z_2}\right)^2\frac{D_{L_2}}{D_{L_1}}\frac{s_2}{s_1},$$ The second step is to reduce the surface brightness of each (rebinned) pixel according to the relative amount of cosmological dimming of a galaxy at $z_2$ with respect to that at $z_1$. We calculate the scaling by making use of the fact that the absolute rest-frame magnitude (or luminosity) of the object before and after redshifting will be conserved ($M_{\lambda_2/(1+z_2)}=M_{\lambda_1/(1+z_1)}$ with matched filters so that $\lambda_2=\lambda_1(1+z_2)/(1+z_1)$).
In order to compare our results to the high redshift samples of @lotz06, objects were simulated at $z=1.5$ in [$B_{435}$]{}, and at $z=3.0$ and 4.0 in both [$V_{606}$]{} and [$i_{775}$]{}. Objects were simulated at the depths of GOODS (3, 2.5 and 2.5 orbits in [$B_{435}$]{}, [$V_{606}$]{} and [$i_{775}$]{}, resp.) and the UDF (56, 56, 150 orbits in [$B_{435}$]{}, [$V_{606}$]{} and [$i_{775}$]{}, resp.). For completeness and future reference, we also simulated the sample at the shallower COSMOS survey in the [$I_{814}$]{} filter (1 orbit). For each filter and for each redshift, we used the HST images that correspond most closely in terms of rest-frame central wavelength. Where necessary, we rescaled the flux of the input images assuming the UV slopes determined in Section \[sec:slopes\]. The final steps of the simulation consist of applying Poissonian noise to the simulated profiles based on the typical exposure times of COSMOS/GOODS/UDF, convolving the images with a Gaussian to match the desired output PSF size ($\sim0\farcs12$), and placing the simulated object inside an empty region in a COSMOS, GOODS or UDF image to obtain a realistic background. The simulated pixel scale was $0\farcs03$ pixel$^{-1}$ for GOODS and the UDF, and $0\farcs05$ pixel$^{-1}$ for COSMOS.
The rest-frame UV images of Haro 11 and VV 114 (see top panels of Fig. \[fig:haro11vv114sims\]) were artificially redshifted in an identical manner using a two-step process. First, we simulated their F330W images at a redshift ($z=0.15$), depth (2500 s), plate scale ($0\farcs025$ pixel$^{-1}$), and seeing ($\sim0\farcs075$) comparable to our observations of the UVLGs. These images are shown in the bottom panels of Fig. \[fig:haro11vv114sims\]. Both Haro 11 and VV 114 possess multiple bright nuclei ($\sim$1 apart) when observed in [$U_{330}$]{} at $z=0.15$. VV 114 also shows a significant amount of diffuse emission in between the knots. Next, these images were used to simulate how these objects would appear at high redshift, analogous to the UVLG simulations described above. To match the desired rest-frame wavelengths of the output filters, small flux extrapolations were performed using $\beta=-1.4$ for both objects as measured by @goldader02 and @bergvall06.
### Results
Figs. \[fig:sims\_goods\_uvlgs\], \[fig:sims\_udf\_uvlgs\], and \[fig:sims\_cosmos\_uvlgs\] compare postage stamps of the local LBG analogs (including Haro 11 and VV 114) that are simulated at the depths of the GOODS, UDF, and COSMOS surveys. Results are shown at the three different redshifts discussed above ($z=1.5$, 3.0 and 4.0). The images demonstrate that as redshift increases, components that are well separated at low redshift blend due to the lower spatial resolution, and that low surface brightness features are lost due to surface brightness dimming. The latter effect is most severe in the shallow COSMOS data, and least severe in the deep UDF data.
Fig. \[fig:morphologies\_redshift\] shows how the measured morphological parameters $G$ (top panel), $M_{20}$ (middle panel) and $C$ (bottom panel) change as a function of redshift and survey depth. There is a systematic drop of $\sim$0.05–0.10 in $G$ from $z\sim0.15$ to $z\sim1.5$, followed by a further less significant decrease of a few hundredths out to $z=4$. The first decrease is caused by the strong drop in resolution, as well as the loss of faint extended features that cause the flux to be more evenly distributed over the galaxy profile. Simulations by @lotz06 show that the second decrease is mainly caused by the loss of low surface brightness pixels lowering $G$ somewhat further at $z>1.5$. The loss of low surface brightness features with redshift also tends to increase $M_{20}$ (the total second order moment is lowered while the second order moment of the 20% brightest flux stays roughly constant), and tends to lower the concentration, because the 80% flux radii are systematically underestimated. The measurements at the COSMOS depth show quite a large scatter with respect to GOODS and the UDF, indicating that the lower $S/N$ and its larger pixel scale result in relatively unstable morphology measurements for LBGs.
The main results of our morphological comparison are presented in Fig. \[fig:morphologies\_lotz\]. We compare the UVLG morphologies (filled stars) with those of the low redshift LBG analogs (open stars) and the high redshift comparison sample (plusses) of @lotz06. In order to make sure that the morphologies are measured in a consistent manner, we used our own code to recalculate the morphologies of the 55 $z\sim1.5$ starburst galaxies in GOODS (bottom panels), and the 82 $z\sim4$ LBGs in GOODS (middle panels) and the UDF (top panels) analyzed by @lotz06. @lotz06 showed that the morphologies of the $z\sim1.5$ sample are very similar to that of the $z\sim4$ sample. The distribution of the UVLGs is very similar to that of the high redshift objects as well, although the latter has a larger scatter due to the much larger sample size.
### Comparison with Haro 11 and VV 114
Haro 11 and VV 114 (open stars in Fig. \[fig:morphologies\_lotz\]) have higher $M_{20}$ and lower $G$ and $C$ when compared to the UVLGs. We can understand this given that both objects possess several bright nuclei with a separation that enables them to be significantly resolved even when simulated at $z=0.15-4.0$. Their qualitative morphology perhaps most closely resembles that of the high redshift “clump-cluster galaxies” [e.g., see @elmegreen04]. Although objects with similar morphological characteristics are certainly present in high redshift samples, they do not make up the majority of LBGs as can be seen from the distribution in morphological types of LBGs [e.g. @lotz06; @ravindranath06; @elmegreen07 and Fig. \[fig:morphologies\_lotz\] in this paper].
It is important to note that our GALEX/SDSS selection of local LBG analogs at $z\approx0.1-0.3$ is likely biased against finding objects having such widely separated nuclei as Haro 11 and VV 114. Because we selected objects principally on having a high FUV luminosity, as well as a high UV surface brightness based on the half-light radius measured in the [*u*]{}-band, we are most sensitive to objects that are not or barely resolved in SDSS. The surface brightness requirement is necessary in order to obtain a relatively clean separation between the LBG analogs on one hand, and large, UV-luminous (predominantly spiral) galaxies on the other. However, we will further investigate whether we can find more LBG analogs of the type of Haro 11/VV 114 by looking closer at the objects that straddle the boundary of the LBG analog selection criteria in a future paper (Overzier et al., in prep.).
### Summary
We conclude that the morphologies of the UVLGs cannot be distinguished from the morphologies of high redshift LBGs when measured from the redshifted images simulated at the same depth as GOODS and the UDF. This further strengthens our conclusion that the relatively nearby supercompact UVLGs and high redshift LBGs are very similar.
In the following sections, we will turn our attention again to the undegraded HST images of Fig. \[fig:stamps\]. In Sect. \[sec:sscs\] we will study in detail the nature of the starburst regions, and in Sect. \[sec:disc\] we discuss the connection between morphology and the mechanisms responsible for triggering the vigorous star formation observed in these local LBG analogs.
Super starburst regions {#sec:sscs}
=======================
As discussed in Section 3, in all our sources the UV light is distributed in a series of bright, unresolved knots embedded in a region of more diffuse emission. In this section we will show that these regions can be interpreted as being (super) starburst regions (SSBs). An understanding of the physical nature of these regions is crucial for interpreting the UV emission from LBGs.
Identification and photometry
-----------------------------
In Fig. \[fig:ssc\_apertures\] we indicate all the starburst regions identifiable by eye from the [$U_{330}$]{}-band images (small circles). Although our identification by eye is somewhat subjective, most regions are bright and isolated and thus easily recognised. We identify a total of 41 regions in our sample of 8 galaxies. Some galaxies contain only one knot, while others contain as many as ten. Next, we measured the fluxes and colors of each region using circular aperture photometry in matched [$U_{330}$]{} and [$z_{850}$]{} images (resolution of $\sim0\farcs12$, FWHM). We were able to isolate all of the starburst regions identified in Fig. \[fig:ssc\_apertures\] using circular apertures of 03 in diameter (corresponding to physical radii of $\sim$240-480 pc). The encircled energy (EE) measurements of [@sirianni05] show that circular apertures of this diameter enclose $\sim80$% of the total light of a point source observed with ACS. We did not apply the EE correction. Magnitudes were corrected for Galactic extinction, and the magnitude distribution of the 41 starburst regions is shown in Fig. \[fig:ssc\_maghist\]. Absolute magnitudes were calculated from $M=m-5\mathrm{log}_{10}(D_L\mathrm{(pc)})+5+2.5\mathrm{log}_{10}(1+z)$. We did not apply a $K$-correction as the central rest-frame wavelengths at which the absolute magnitudes are determined differ only by a small amount ($\lesssim300$Å). Fig. \[fig:ssc\_maghist\] shows that our ‘by eye’ selection is relatively complete at $m_{330}\lesssim23$ mag ($M_{330/(1+z)}\lesssim-16.5$). We also determined the flux and color of the large, diffuse region that generally surrounds the compact starburst knots (large circles in Fig. \[fig:ssc\_apertures\]). These regions measure $\sim$2-5 kpc in radius. The fluxes measured in the small apertures were subtracted from the larger aperture.
We find that the UV light is typically dominated by emission from the compact, luminous regions: the combined flux of the unresolved regions contributes an average of $\sim$50% to the total flux in [$U_{330}$]{}. Object 102613 has the lowest contribution from SSBs ($\sim34$%) and object 080844 the highest ($\sim80$%). Although we do not have a [$U_{330}$]{} image for 092600, we can infer that its structure is very similar to the other objects given that the unresolved knots contribute $\sim$42% to the total flux in [$z_{850}$]{}, and its morphology in [H$\alpha$]{} (assumed to be a good proxy for its [$U_{330}$]{} morphology, see Fig. \[fig:stamps\]), shows a similar knotty structure.
Color-magnitude diagram
-----------------------
In Fig. \[fig:ssc\_cm\] we show the color-magnitude diagram of all regions identified in Fig. \[fig:ssc\_apertures\]. Objects 092600 and 214500 were omitted. Starburst regions from each galaxy are plotted using a different filled symbol, with their corresponding ID number from Fig. \[fig:ssc\_apertures\] indicated for reference. Open symbols of corresponding shape indicate the diffuse regions that surround the compact regions. In order to compare the measured colors and magnitudes to model starburst tracks, we ‘dereddened’ the measurements using the global reddening value for the stellar continuum that was derived from the bolometric dust to FUV luminosity ratio (see Sect. \[sec:slopes\]). We note that this correction is in general quite small. The compact regions span quite a large range both in color and in absolute magnitude (filled symbols in Fig. \[fig:ssc\_cm\]). The diffuse regions, defined by the larger areas in Fig. \[fig:ssc\_apertures\] but with the circular regions around all the identified point sources removed, lie at $M_U\sim-20$ and ([$U_{330}$]{}–[$z_{850}$]{})$\sim0.5$ (open symbols in Fig. \[fig:ssc\_cm\]).
To interpret these colors, we have used STARBURST99 [@leitherer95; @leitherer99] to predict the color and magnitude evolution of a starburst as observed through our filters. The tracks in the left panel of Fig. \[fig:ssc\_cm\] represent an [ *instantaneous burst model*]{} with burst masses of $M=10^7$ $M_\odot$ and $M=10^8$ $M_\odot$, while tracks in the right panel indicate a [*continuous star formation model*]{} having SFRs of 0.1, 1 and 5 $M_\odot$ yr$^{-1}$. We use a Kroupa initial mass function (IMF) with a slope of $\alpha=1.3$ between 0.1 and 0.5 $M_\odot$ and $\alpha=2.3$ between 0.5 and 100 $M_\odot$. We also use the Padova 1994 models with thermally pulsating AGB stars added [@vazquez05]. The tracks shown were obtained after redshifting the rest-frame spectra to $z=0.15$, the average redshift of our sample, and measuring the magnitudes in [$U_{330}$]{} and [$z_{850}$]{}. The blue lines correspond to a metallicity equal to that of the Large Magellanic Cloud ($Z=0.008$), while the red lines are for solar metallicity. Ages (in Myr) have been indicated along the tracks. Dashed lines indicate the contribution from the nebular continuum (assuming no reddening). The general behaviour of all tracks is to become redder with age. After about 10 Myr, the colors become steadily redder, because the most massive O stars have become supernovae and lower mass O stars have become red supergiants. The red supergiant dominated phase at 10–20 Myr is highly dependent on metallicity, as their number is lower and their temperature is warmer in lower metallicity starbursts [@vazquez05]. After 20 Myr, O stars have disappeared leading to a gradual reddening with age. Continuous models become brighter, while instantaneous models fade with time.
Results
-------
The instantaneous burst and continuous star formation models have been chosen to bracket the range of possible star formation histories of our compact starburst regions. Most of the regions are clearly detected in [H$\alpha$]{} indicating that they must at least have had very recent star formation. In Fig. \[fig:ew\] we plot the SSBs in the plane of color vs. the rest-frame equivalent width (EW) of [H$\alpha$]{}measured for each SSB. The measured equivalent widths are 30–500Å, and are seen to correlate with the [$U_{330}$]{}-[$z_{850}$]{} color, suggesting that the [H$\alpha$]{} EW is a reasonably good age indicator. We have overplotted the predicted EW of [H$\alpha$]{} based on the STARBURST99 models shown in Fig \[fig:ssc\_cm\]. For instantaneous models, the [H$\alpha$]{} EW is less than $\sim$100Å after 6 Myr, and less than 10Å after only 10 Myr (dashed lines). For continuous star forming models, a high [H$\alpha$]{} EW is maintained over a much longer period of time, with $EW_{H\alpha}\gtrsim$100Å for ages $\lesssim$1 Gyr.
Comparison of the instantaneous burst and continuous star formation tracks with the observed compact and diffuse regions lead us to the following conclusions:
[*(i)*]{} The $M\simeq10^{7-8}$ $M_\odot$ instantaneous starburst tracks shown in the left panel of Fig. \[fig:ssc\_cm\] provide a generally good match to the majority of the compact regions, but the ages of $\sim$10-100 Myr inferred are inconsistent with their large [H$\alpha$]{} EWs, as shown in Fig. \[fig:ew\]. A better match to both the color-magnitude diagram and the [H$\alpha$]{} EWs is provided by the continuous star formation models shown in the right panel of Fig. \[fig:ssc\_cm\]. The exact ages are uncertain, because of small-scale differences between the [$U_{330}$]{} and [H$\alpha$]{} morphologies (note that the apertures were defined on the [$U_{330}$]{} image), uncertainties in the small-scale dust distribution (now assumed to be a single foreground screen), and because of the unknown fraction of the light that is due to the background population inside each of the small apertures (not subtracted). Our best estimates for the ages of the starburst regions range from a few tens of Myr to a few hundreds of Myr based on a comparison with the tracks in Fig \[fig:ssc\_cm\]. These ages should be regarded as upper limits, because we may have underestimated the contribution from dust and older stars. The scatter in $M_U$ magnitude can be interpreted as a sequence in SFR ranging from $\sim$0.1 to $\sim$5 $M_\odot$ yr$^{-1}$ (or total burst mass ranging from $\sim10^7$ $M_\odot$ to several $10^8$ $M_\odot$). The scatter in ([$U_{330}$]{}–[$z_{850}$]{}) color can be interpreted as a sequence in age (or burst age in the case of instantaneous bursts). Note that our results are almost independent of metallicity.
[*(ii)*]{} For the brightest starburst regions, there is a strong tendency for neighbouring regions to have a very similar color and magnitude, e.g. regions (2,4,7) in object 135355, regions (1,4,5) in object 040208, and regions (4,5) in object 032845. This suggests that the starburst in these objects tends to occur in the form of several co-eval and equally massive bursts.
[*(iii)*]{} The two compact regions that are outliers (in the sense that they appear to be both extremely young, $\sim$6 Myr, as well as very massive, $M_*\sim10^8$ $M_\odot$) each correspond to objects where the UV emission is dominated by a [*single*]{}, luminous knot (objects 005527 & 080844). Both objects have a very large [H$\alpha$]{}EW, indicating that they are massive, young starbursts. The SDSS spectrum[^3] of 005527 shows a broad feature around a bright and narrow HeII 4686Å line that is characteristic of Wolf-Rayet (WR) galaxies [e.g. @conti91; @sargent91; @izotov97]. The WR phase is a relatively short phase in the life of the most massive stars during which they experience a large mass loss. The presence of WR stars is a unique indicator of young starbursts, as WR stars disappear after $\sim6-10$ Myr depending on the metallicity of the burst, with less mass loss occurring at lower metallicity [@leitherer95]. Further confirmation of this young age comes from the large equivalent width of H$\beta$ ($\sim$60Å), consistent with an age of $\sim$6 Myr [@leitherer95].
[*(iv)*]{} Some other regions with very blue colors and high EW in [H$\alpha$]{} (e.g. region 2 in 032845 and region 1 in 102613) appear to be as young as the previous two, but are $\sim$2-3 magnitudes fainter.
[*(v)*]{} The more diffuse, inner regions in which the SSBs are embedded (open symbols in Fig. \[fig:ssc\_cm\]) are typically a few tenths of a magnitude redder than the average color of the compact SSBs they surround, and their total magnitudes are very similar to the combined flux of their SSBs. These regions contain an older stellar population. Given their large physical extent, the instantaneous burst model is unlikely to be appropriate. We find that the colors and luminosities of these regions are in good agreement with continuous star formation models with SFRs of $\simeq1-5$ $M_\odot$ yr$^{-1}$ and ages of $\simeq0.2-1.0$ Gyr.
[*(vi)*]{} The outer annuli (defined as the region between $r_{50,850}$ and $r_{90,850}$, see Table \[tab:sizes\]) are consistent with older populations having ages between a few Gyr to a Hubble time, depending on the star formation history.
Summary
-------
The starburst regions observed in the local LBG analogs are highly compact (radii of 100-300 pc), and characterized by bright, unresolved knots of emission within a larger region of diffuse star formation that extends out to a radius of a few kpc. The total UV emission from these compact starburst regions is substantial, indicating that 30–80% of the total SFR is generated in these knots. Comparison with STARBURST99 evolutionary tracks indicates that some of the regions are due to very young ($\sim$6 Myr) bursts, while other regions may have been forming stars in a more continuous manner for several tens to several hundreds of Myr as indicated by their relatively red colors and large [H$\alpha$]{} equivalent widths. There is a tendency for neighboring knots to have similar colors and luminosities. This implies that they may be co-eval and of similar mass. Most of the SSBs are still unresolved in the unbinned [$U_{330}$]{} images ($\approx0\farcs075$, FWHM). The stellar mass densities implied are $\sim10^{2-3}$ $M_\odot$ pc$^{-2}$ estimated from their masses of $\sim10^{7-8}$ $M_\odot$ and assuming an effective radius of $\sim$100 pc.
The masses of the super starburst regions are one to two orders of magnitude larger than the masses of the most massive clusters found in the local Universe. We may therefore conclude that the (unresolved) super starburst regions seen in Fig. \[fig:ssc\_apertures\] are likely composed of smaller units. @meurer95 found that typical starburst regions in local starburst galaxies consist of diffuse stars as well as clusters in the ratio of respectively 80 to 20%. The most prominent of the clusters, the so-called super star clusters (SSCs), have masses of $\approx10^{5-6}$ $M_\odot$ and sizes of a few to $\sim$10 pc and likely correspond to globular clusters in the process of formation (provided the stellar IMF extends down to 0.1 $M_\odot$).
We can get an even better idea of the sub-resolution structure of the super starburst regions by comparing the starburst regions in Haro 11 and VV 114 shown in the top panels of Fig. \[fig:haro11vv114sims\] with the same regions simulated at $z=0.15$ shown in the bottom panels of the same figure. The majority of the bright nuclei seen in the images simulated at $z=0.15$ are a blend of numerous smaller super starclusters that can clearly be seen in their unredshifted images. However, the most Northern and brightest knot in the redshifted image of Haro 11 is still largely unresolved in its unredshifted image. This indicates that single, highly massive star clusters are capable of dominating the rest-frame UV/optical morphology of LBG-like galaxies at virtually any redshift. A detailed determination of the sizes, masses and ages of the star clusters in Haro 11 and VV 114 is beyond the scope of this paper, and can be found elsewhere [@scoville00; @hayes07 Adamo et al., in prep.].
Discussion {#sec:disc}
==========
Triggering mechanism
--------------------
Although the number of objects in our HST sample is currently quite small, the results presented in this paper can nevertheless guide us in the interpretation of the morphologies of these local starburst galaxies, and by extension, those of LBGs at higher redshifts.\
[*(i) Are the morphologies evidence for merging?*]{}\
Although some of the SDSS images showed evidence for close companions or faint extended emission, the galaxies were extremely compact and it was not at all clear from the SDSS images that they were highly disturbed. The biggest surprise from the HST images was that most of the UVLGs show disturbed morphologies at scales well below the SDSS seeing or sensitivity. Each UVLG shows at least one of the typical signs of merging or interaction, such as multiplicity or twists in the faint, outer isophotes (e.g., 005527, 032845, 040208, 092600), tidal debris or tails (e.g., 032845, 102613, 214500), and close companions (e.g., 080844, 102613, 135355, 214500).
While none of the UVLGs appear as fully-formed Hubble sequence galaxies, in one case, 214500, a small spiral structure seems to be present both in the FUV and in [$V_{606}$]{}. Nonetheless, even in this object there is circumstantial evidence that the starburst was triggered by a merging event, as evidenced by its companion that appears to have gone straight through its nucleus leaving behind a trail of tidal debris. The projected distance between the centroid of the starburst region and the companion is about 5 kpc. The companion galaxy would traverse this distance in $\sim$10-100 Myr assuming a velocity of a few hundred km s$^{-1}$ and a straight trajectory in the plane of the sky. If we asume that the starburst is triggered by the interaction (as suggested by the [$V_{606}$]{} morphology), this timescale constrains the maximum possible age of the burst. The results are consistent with 30-100 Myr ages derived from the colors and magnitudes of these systems. In many of the other cases, the region that dominates the UV morphology often lies at the interface of two diffuse merging structures, as identified by the contours of their outer isophotes in [$z_{850}$]{}. In the remaining cases, a companion is always seen at a distance of less than $\lesssim$5 kpc from the UVLG. It is well-known that close neighbours can affect the SFR in galaxies, with close pairs having separations of $\lesssim50$ kpc showing enhanced star formation [e.g. @larson78; @lambas03; @nikolic04; @woods06; @owers07 Li et al. in prep.] This supports the idea that the starbursts are linked to mergers. The merging galaxies must be very gas-rich in order to trigger the very high level of enhanced star formation and the luminous super starburst regions that we observe.
It has been proposed that starburst galaxies at high redshift, such as Lyman break galaxies, are triggered by mergers of relatively gas-rich disk galaxies [@somerville01]. Simulations indicate that mergers of gas-rich galaxies can trigger massive starbursts. The conditions of the starburst depends on a number of key parameters of the merging system [e.g. see @mihos94a; @mihos94b; @springel05; @dimatteo07 and references therein]. During a merger of disk galaxies, bar-like patterns develop that efficiently funnel gas into the central region. The gas flow to the center is generally not disrupted or depleted due to star formation during the early stages of the merger, although the inflow of gas can be limited due to resonances if a bulge component is present, reducing the central gas densities and overall SFR and duration of the central starburst. The presence of gas is not always a sufficient condition for generating a starburst. The star formation efficiency of a merger depends largely on the galaxy spin direction (retrogade encounters have higher efficiency), and on the amount of gas that is being expelled due to the tidal interaction at first passage [@springel05; @dimatteo07].
We remind the reader that typical ($L$$\sim$$L^*_{z=3}$) LBGs are not dwarf galaxies, but have masses of $\sim$10$^{10}$ $M\odot$ and SFRs that are much higher than those of typical (unobscured) local starbursts [@papovich01; @shapley01]. They do not show evidence for spiral structure [@papovich05]. As such, they are very similar to our local LBG analogs. We conclude that the main mechanism responsible for triggering star formation in both high redshift LBGs and their local analogs is likely to be mergers.\
[*(ii) Are the compact super-starburst regions triggered by the mergers?*]{}\
Luminous star clusters (e.g. SSCs) occur in many irregular and dwarf galaxies (e.g. 30 Dor in the LMC). The triggering of compact SSBs as massive as the ones in our sample of UVLGs probably requires a highly efficient inflow and compression of the gas that is typically only seen in galaxy mergers or interactions. The inflow ensures that the pressure in the interstellar medium becomes larger than the internal pressure of giant molecular clouds, and that the clouds can collapse and form SSCs before they are disrupted by supernova explosions [@bekki04]. SPH simulations of galaxy mergers suggest that the properties of the starburst are determined by the magnitude, timescale and geometry of this inflow [see @mihos94a; @bekki01; @bekki04; @dimatteo07]. @mihos94a found that the central starbursts resulting from gas-rich mergers can have total sizes of $\sim350$ pc, and contain most of the total mass in new stars. @bekki01 predict that (i) SSCs have a narrow age distribution ($\sim200$ Myr) because they form most efficiently during the stage when the gas density is the highest, (ii) the total number and intrinsic masses of SSCs are larger in major mergers due to larger tidal effects and higher peak SFRs, and (iii) SSC production is more efficient in merging galaxies [e.g. ULIRGS, @sanders96] than in tidally interacting galaxies [e.g. M82, @kennicutt98].
Observations show that starbursts indeed occur in regions where the gas densities are the highest, typically in the cores [e.g. Arp 220, @shaya94] and in circum-nuclear starburst regions [e.g. VV114, @scoville00]. Stars may form in large complexes having total stellar masses of $\sim10^{7-9}$ $M_\odot$, and containing several tens to hundreds of SSCs as well as smaller star clusters. This is true both in ULIRGS and in compact blue galaxies [e.g. @meurer95; @ostlin98; @whitmore99; @zepf99; @goldader02 and therein].
We conclude that the characteristic morphologies, ages, sizes and masses of the starburst regions we have observed are consistent with those predicted by simulations of gas-rich mergers. They are also analogous to the large star-forming complexes observed in very nearby merging galaxies.
Low redshift lessons for high redshift LBGs
-------------------------------------------
Because of their relative proximity, studies of local star-forming galaxies can help us understand the nature of starburst galaxies at high redshifts. As explained in Section 1, up to now these comparisons have been of limited usefulness, because typical local starburst galaxies are systematically different from LBGs at high redshift. The sample of local LBG analogs first studied by H05 was specifically designed to bridge this gap. In this paper, we have studied the detailed morphologies of these systems and we now discuss how our results impact on the study of star-forming galaxies at high redshifts.
### Morphologies and sizes
Our analysis of the HST data has established that UVLGs have a light distribution dominated by very compact starbursting regions (ranging in number from 1 to more than 10 per galaxy) that are triggered by mergers or interactions. In most cases, an older, diffuse, stellar component is also present. Our redshift simulations showed that the sizes and morphologies at $z=1.5-4$ are similar to that of LBGs [e.g. @giavalisco96a; @conselice06; @lotz06; @overzier07; @younger07]. The morphological parameters measured at rest-frame UV and optical wavelengths are very similar. This is unlike typical local galaxies, but very similar to LBGs as both are dominated by young stars [e.g. @papovich05].
The redshift simulations presented in Section 3 yield a number of important conclusions: (i) at high redshift, the compact starburst regions can blend giving rise to a galaxy image that appears to have the one or two bright ‘knots’ with a total half-light radius of 1–2 kpc, (ii) the diffuse emission is partly lost in the background noise or may blend with the knots, and (iii) the [*undegraded*]{} optical images were in most cases required to establish a definite connection between the (UV) starbursts and merging (Section 5.1).
If ‘knots’ are present in LBGs they are typically not resolved in HST observations, ground-based spectroscopy with adaptive optics [resolution $\sim0\farcs1$ or $\sim$1 kpc; @genzel06; @law07b], or standard spectroscopy [resolution $\sim0\farcs5$ or $\sim$4 kpc; @forster06; @erb06b]. Because imaging observations of high redshift LBGs lack the sensitivity and resolution to see the (often subtle) features associated with merging, 2D or 3D spectroscopy is the most promising method of studying the connection between star formation and morphology in these objects. Small samples studied to date show complex kinematics and high nuclear gas fractions that are consistent with the merger hypothesis [@erb03; @erb06b; @forster06; @law07b]. However, some star-forming objects at $z\sim2$ may have more ordered underlying structures [@forster06; @genzel06; @wright07].
Coming back to the important, long-standing question of whether the irregular morphologies of LBGs are a sign of merging or of patchy star formation within a single (forming) disk [e.g. @law07a], it is safe to say that in [*virtually every single case*]{} presented in this paper we do not see any direct evidence for merging based on the HST UV images alone. However, in [*virtually every single case*]{} evidence for merging is readily apparent from the rest-frame optical image, and we reiterate that most of the features suggestive of merging would be too faint or too small to be seen at $z$$\sim$2–4. In some of the local LBG analogs the burst occurs predominantly in only one of the members of a merging or interacting pair. In others the merger seems in such an advanced state that one cannot speak about separate systems any more, making the distinction between ‘merging’ and ‘patchy disk’ essentially irrelevant. We conclude that the UV morphologies of LBGs are likely [*patchy as the result of merging*]{}.
Although both the UV and optical morphologies are dominated by compact starbursts, some of our objects show evidence for an older, extended component, which is generally not seen in high redshift LBGs [e.g. @papovich05]. In this respect, the UVLGs may be more evolved than LBGs at high redshift. However, it is important to remember that the age of the universe at $z\sim3$ is only $\sim2$ Gyr. The spectral energy distributions of LBGs are consistent with the presence of such a “maximally old” stellar component @papovich01. The limiting surface brightness of rest-frame optical images of LBGs in the NICMOS HDFN indicates that these features are likely often too faint to be detected. Also, the rest-frame wavelength of the reddest NICMOS images of LBGs ($H_{160}$-band) corresponds to only about half that of our [ *z-*]{}band observations of the UVLGs and so are less sensitive to detection of an older population. Resolution effects could play a role as well. In paper II we will carry out redshift simulations of a large set of rest-frame optical images of UVLGs to investigate these optical structures in detail.
Because high surface brightness regions in LBGs are very likely to be blends of super starburst regions, we stress that one must be cautious in interpreting size/morphology measurements in terms of disk- or bulge-like components. The half-light or effective radius may be more directly related to the typical radius over which the compact, super starburst regions are distributed, rather than to the scale size of a (forming) bulge or disk [see also @noguchi99; @immeli04; @law07b], although the two are likely closely related if stars predominantly form inside these burst regions and relaxation or tidal disruption spreads them out over time. Although we do not rule out the possibility that some fraction of LBGs has bulges at redshifts as high as $z\sim4$, the high physical resolution data presented here suggests that this is possibly far less common than currently believed based on morphological parameters (e.g. Lotz et al. 2006 estimate $\sim$30%).
Evidence that the small-scale structure of LBG knots is indeed of the nature advocated in this paper is found in a few rare cases where LBGs are observed at high magnification due to gravitational lensing. By studying a lensed galaxy at $z=4.92$, @franx97 were the first to show that high surface brightness super starburst regions having sizes of a few hundred pc form an essential contribution ($\sim$75%) to the UV flux in LBGs. Similar features can be seen in other lensed systems that have been discovered at high redshift [@bunker00; @ellis01; @smail07; @swinbank07]. We note that some fraction of LBGs with very knotty structures may be discarded in current large surveys, because their light profiles are similar to those of stars. Our sample indicates that this number could be on the order of 13-25% (1–2 out of 8 objects). If we compare the substructure of the super starburst regions in Haro 11 and VV 114 at $z=0.02$ with the same regions simulated at $z=0.15$ (Fig. \[fig:haro11vv114sims\]), we see that majority of the bright nuclei seen in the images simulated at $z=0.15$ are a blend of smaller super star clusters identified in their unredshifted images. However, the most Northern and brightest knot in the redshifted image of Haro 11 is still largely unresolved in its unredshifted image, indicating that single, luminous star clusters are capable of dominating the rest-frame UV/optical morphology of LBG-like galaxies at virtually any redshift.
As shown by @dahlen07, the fraction of star-forming galaxies having a bulge-like morphology at rest-frame 2800Å decreases from $\sim$30% at $z\approx2$ to $\sim$10% at $z\approx0.5$, illustrating the relative importance of concentrated star formation at high redshift compared to low redshift. @zirm07 have pointed out the existence of a population of compact, massive galaxies at $z\sim2.5$ having effective radii and high stellar mass surface densities. It is possible that these are the result of massive and concentrated starbursts in highly dissipative, gas-rich mergers at high redshift ($z\sim3-6$). Extrapolation of our results to higher redshifts suggests that very dense stellar ‘cores’ are indeed actively being formed in the LBG population.
### Prediction for the faint end slope of the LBG luminosity function
The UV luminosity function (LF) has now been evaluated over the entire redshift range from $z=0-6$. Interestingly, the faint end slope of the LF has been found to flatten with decreasing redshift from $\alpha\sim$1.74 at $z=6$ to $\sim1.2$ at $z=0$ [@yan04a; @yan04b; @wyder05; @bouwens07; @ryan07]. Various simulations and models have been tried to reproduce the LF and explain its flattening at faint magnitudes [e.g. @night06; @finlator07; @khochfar07]. As we will show, the existence of compact starbursts in LBGs may offer a natural explanation for the steep slopes that are observed at high redshifts. The LF of star-forming regions is well-known to have a slope of $\alpha\approx2$ over a wide range in physical scales from regions to SSCs and beyond [e.g. @kennicutt89; @meurer95; @elmegreen97; @zepf99; @alonso02; @bradley06 and references therein].
We consider the following, very simple, toy model. Let $N(L,z)$ be the total number density of galaxies with UV luminosity between $L$ and $L+dL$ and redshift between $z$ and $z+dz$. We will also assume that the stellar IMF, as well as the star formation efficiency in galaxies, does not change with redshift. The UV luminosity of an individual galaxy at a given time $t$ is the sum over the luminosity of all its previous star-formation events, $L_i$, at that time: $$\label{eq:lf}
L_{\mathrm{LBG}}^{UV}(t)=\sum_i L_i (t).$$ The luminosity arising from an ‘event’ can take any arbitrary form (e.g. burst, constant, declining, etc.), and it is not important what caused the event. For LBGs, it is instructive to make a distinction between the contribution to the total luminosity from a series of $N$ (semi-)discrete starbursts, $L_{\mathrm{burst}}^i$, and that due to a diffuse star-forming component that is slowly evolving with time, $L_{\mathrm{diffuse}}$, so that $$L_{\mathrm{LBG}}^{UV}(t) = L_{\mathrm{diffuse}}(t) + \sum_{i=1}^N L_{\mathrm{burst}}^i(t).$$ We have learned that the UV luminosity from an LBG is typically dominated by a series of $N_{\mathrm{knot}}$ ‘knots’ ($\sim$1–2 kpc in size), and that each knot can be further resolved into a series of $N_{\mathrm{SSB}}$ super starburst regions ($\sim$100-300 pc in size), and thus $$L_{\mathrm{LBG}}^{UV}(t)=L_{\mathrm{diffuse}}(t) + \sum_i^{N_{\mathrm{knot}}} \sum_j^{N_{\mathrm{SSB}}} L_{\mathrm{SSB}}^{i,j} (t),$$ where $L_{\mathrm{SSB}}^{i,j}(t)$ is the luminosity of a SSB region $j$ in knot $i$ at time $t$. For (proto-)galaxies that are just undergoing their first episode of massive star formation the above expression will simplify enormously, as we can neglect the diffuse component, $L_{\mathrm{diffuse}}(t)$, and both $N_{\mathrm{knot}}$ and $N_{\mathrm{SSB}}$ will be near unity. The most simple case is represented by $N_{\mathrm{knot}}=N_{\mathrm{SSB}}=1$, as observed in some of our local LBG analogs, and (at least in this extreme case) the total UV LF will be purely determined by a random sampling of the LF of the SSBs over the galaxy population in a given redshift interval. As shown by @meurer95, the ‘anatomy of starbursts’ typically consists of irregularly shaped cloud(s) (SSBs) with embedded compact sources (SSCs), the brightest of which are massive forming (globular) clusters. Star formation thus occurs in hierarchically clustered systems ranging from sub-pc to multi-kpc scales, and is believed to be generated at all scales by self-gravity and turbulence within an underlying hierarchical gas mass distribution $n(M)dM\propto
M^{-\alpha} dM$ [@elmegreen96; @elmegreen97 and references therein]. Because the UV luminosity of starbursts is dominated by its OB stars at all scales, on average $M\propto L$ with $\alpha\sim1.7-2$ for the references given above. [*Therefore, in the regime where the LF is dominated by young galaxies generating their first significant amount of stellar mass, i.e. at relatively faint magnitudes and high redshift, our simple model predicts that the galaxy LF observed will essentially be a random subsampling of the LF of starburst regions thereby maintaining its intrinsic slope.*]{} It is interesting to note that, for example, the faintest [$B_{435}$]{}, [$V_{606}$]{}, or [$i_{775}$]{}-dropout galaxies observed at $z\sim4-6$ have stellar masses very similar to those of our starburst regions ($\sim10^{8-9}$ $M_\odot$).
Generally, it will be impossible to model the full UV LF quantitatively in this way (e.g. Eq. \[eq:lf\]) without resorting to semi-analytic models or simulations because it involves the convolution of large numbers of galaxies with a range of star formation histories including effects such as merging, feedback and dust. However, qualitatively one expects that, when all processes are considered, the slope will flatten from its initial value. Thus, we expect that the faint end slope of the LF at least steepens with redshift and will deviate less and less from the ‘pure’ star formation LF with a slope of $\alpha\approx2$, exactly as observed.
Conclusions
===========
The study of galaxies in the early Universe can benefit enormously from the study of objects that are relatively nearby, provided that suitable samples of local analogs can be found. H05 and H07 selected “supercompact”, UV-luminous galaxies in a large GALEX-SDSS cross-matched sample (see §1). Because these objects match LBGs in terms of size, SFR, surface brightness, mass, metallicity, kinematics, dust, and UV–optical color, this sample of local LBG analogs is well suited for investigating the connection between star formation and morphology at a level of detail and precision that is impossible for high redshift galaxies. Emission line diagnostics indicate that they are dominated by starbursts, and that the line ratios show small offsets with respect to the local star-forming population analogous to those observed for high redshift LBGs (§3.2).
In this paper we present HST imaging data in [$U_{330}$]{}, [$z_{850}$]{} and [H$\alpha$]{} of a subsample of 8 local LBG analogs in order to investigate their morphologies. The effective radii estimated from the [$U_{330}$]{} images range from unresolved ($\lesssim$100 pc) to $\sim2$ kpc, confirming that the objects are highly compact. The objects are slightly more extended in [$z_{850}$]{} ($\sim$1–2 kpc), and they generally have color gradients with relatively blue inner colors and redder outer colors. These gradients are due to the fact that the starburst regions are very compact and are embedded in an older, more extended population (§§3.3–3.4). The internal extinctions amount to $E(B-V)\approx0.01-0.14$ mag based on the bolometric dust to FUV luminosity ratios observed with Spitzer and GALEX (§3.5). We have calculated the morphological parameters $G$, $M_{20}$ and $C$ and find that local LBG analogs have irregular morphologies that, on average, lie in between those expected for major mergers, disks and bulges. We have simulated our data at the depth and resolution of LBGs at $z=[1.5,3.0,4.0]$ in the COSMOS, GOODS, and UDF surveys, and we have carried out direct comparisons with the morphologies of $z\sim1.5$ star-forming galaxies and $z\sim4$ LBGs extracted from these surveys. The morphological parameters measured for the artificially redshifted sample are very similar to those obtained for the distant LBGs. This establishes that local LBG analogs and high-redshift LBGs are likely to be in a similar phase of their evolution (§3.6). The UVLGs are more typical of LBGs in comparison with two very nearby LBG analogs, Haro 11 and VV 114 at $z=0.02$. The latter two are more similar to the “clump cluster galaxies” due to their multiple nuclei at relatively large separation [@elmegreen07]. Although more local LBG analogs of this type may be found as well, we note that our current sample is biased against finding such objects (§3.6.4).
Although a large fraction of LBGs at $z\sim3-4$ is consistent with having exponential or $r^{1/4}$ light profiles, their visual morphologies include multiple, double and perturbed systems. Morphological parameters indicate that the majority of LBGs are likely to be in a range of stages associated with minor and major galaxy merging [e.g. @giavalisco96a; @lotz06; @ravindranath06; @elmegreen07]. In contrast, @burgarella06 find that the majority of luminous, UV-selected star-forming galaxies at $z\sim1$ has a disk or spiral morphology. At a similar redshift, but at much lower UV luminosities, @demello06 find a mix of morphological types that include early-type galaxies as well as starbursts. We note that this is a direct consequence of the selection of galaxies purely based on their UV emission. As discussed here and by H05, at relatively low redshifts ($z\sim0-1$) the construction of a sample of galaxies that most closely resembles high redshift LBGs in terms of their physical properties requires additional selection criteria (e.g. UV surface brightness) rather than selection on UV luminosity or the presence of a ‘Lyman break’ alone.
We have carried out a detailed investigation of the small-scale structure of the local LBG analogs in terms of their star forming regions (§4). The starburst regions observed in the local LBG analogs are highly compact (radii of 100-300 pc), and characterized by bright, unresolved knots of emission within a larger region of diffuse star formation that extend up to a few kpc in size. The total UV emission from these compact super starburst regions (SSBs) is substantial, indicating that 30–80% of the total SFR is generated in these knots. Comparison with STARBURST99 evolutionary tracks indicates that some of the regions are due to very young ($\sim$6 Myr) and massive ($\sim10^{8}$ $M_\odot$) bursts, while other regions may have been forming stars in a more continuous manner for several tens to several hundreds of Myr as indicated by their relatively red colors and high [H$\alpha$]{} equivalent widths. The super starburst regions are likely to be a blend of diffuse star forming regions, stellar associations and super or globular star clusters, as demonstrated by comparing the images of Haro 11 and VV 114 at $z=0.02$ to the same images redshifted to $z=0.15-4.0$. The SSBs are generally embedded in a diffuse, older component having ages ranging from a few Gyr to a Hubble time, depending on the star formation history. In this respect, the precursors of the UVLGs may have been more evolved than those of LBGs at high redshift (although the presence of a “maximally old” stellar component in LBGs is consistent with the observations).
Although some of the SDSS images showed limited evidence for close companions or faint extended emission, the optical HST images clearly reveal disturbed morphologies at scales well below the SDSS seeing or sensitivity. Each object shows evidence of merging or interactions, such as multiplicity of position angles, tidal debris or tails, or close companions. Most morphological information about high redshift galaxies has been derived from rest-frame UV images. In our sample, the [$U_{330}$]{}-band does not establish a definite case for a merger in any single one of our galaxies. The disturbed optical ([*z*]{}-band) morphologies, together with the luminous, compact super starburt regions, suggests that mergers of relatively gas-rich objects trigger vigorous episodes of star formation, with general properties reminiscent of simulations of collisional starbursts (§5.1). Our results on local LBGs and their similarity to their high-$z$ LBGs constitute the most direct evidence to date that the onset of star formation and morphological structures of high redshift LBGs are largely driven by highly dissipational merging.
We discuss several implications of our results for the interpretation of lower resolution data on high redshift LBGs. Resolution effects will cause the compact starburst regions to blend into typically one or two bright LBG-like ‘knots’. Although these knots will appear relatively featureless with total half-light radii of 1–2 kpc, the substructure of our LBG analogs indicates that they should generally not be interpreted as being evidence of bulges or disks. Evidence that the substructure of high redshift LBGs is indeed dominated by SSBs is found in a few cases where LBGs are highly magnified by gravitational lensing (§5.2.1). However, the strong relation we find between merging and the triggering of very compact, massive super starburst regions indicates that these events may be closely linked to the formation of stellar bulges. Furthermore, we suggest that the prominence of luminous, unresolved (super) starburst regions in forming galaxies may provide a natural explanation for the value of the faint end slope of the UV luminosity function at high redshift by an extension of the local star-forming region luminosity function which is well-known to have a power law slope of $\alpha\approx2$ (§5.2.2).
[*Future work.–*]{} In Paper II we will study a significantly larger sample of local LBG analogs to be observed with HST in Cycle 16 in the FUV/optical using the ACS/SBC and WFPC2. Follow-up spectroscopy is being used to study emission line ratios and kinematics, measure outflows, and identify companion objects. The ongoing surveys with GALEX will provide larger samples of local LBGs that will be used to derive better sample statistics.
This paper has benefited from discussions and helpful comments from numerous friends and colleagues. We thank Casey Papovich, Masami Ouchi and Isa Oliveira for carefully reading through the manuscript. We further thank Rychard Bouwens, Nick Cross, Ricardo Demarco, Marijn Franx, Lisa Kewley, Cheng Li, Crystal Martin, Alessandro Rettura, Samir Salim, Christi Tremonti, Arjen van der Wel and Andrew Zirm for discussion of various parts of this paper. RAO thanks Gabrelle Saurage for her excellent support during observations at APO.
Based on observations made with the NASA/ESA Hubble Space Telescope, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. These observations are associated with program \# 10920. Based on observations obtained with the Apache Point Observatory (APO) 3.5-meter telescope, which is owned and operated by the Astrophysical Research Consortium.
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-------------------------- ----------------------- ------------------------------------ ------- ---------- ------ ---------------- ------ ------ ------ ------
$E(B-V)_{gal}$
(J2000) (J2000)
SDSS J005527.46–002148.7 00$^h$55$^m$27.46$^s$ –00$\degr$21$\arcmin$48.7$\arcsec$ 0.167 11/01/06 – 2514 2302 – 2238 0.03
SDSS J032845.99+011150.8 03$^h$28$^m$45.99$^s$ +01$\degr$11$\arcmin$50.8$\arcsec$ 0.142 10/07/06 – 2514 2302 – 2238 0.11
SDSS J040208.86–050642.0 04$^h$02$^m$08.86$^s$ –05$\degr$06$\arcmin$42.0$\arcsec$ 0.139 10/31/06 – 2514 2302 – 2238 0.10
SDSS J080844.26+394852.4 08$^h$08$^m$44.26$^s$ +39$\degr$48$\arcmin$52.3$\arcsec$ 0.091 10/30/06 – 2541 2356 – 2211 0.05
SDSS J092600.41+442736.1 09$^h$26$^m$00.40$^s$ +44$\degr$27$\arcmin$36.1$\arcsec$ 0.181 11/06/06 – – 2340 – 2274 0.02
SDSS J102613.97+484458.9 10$^h$26$^m$13.97$^s$ +48$\degr$44$\arcmin$58.9$\arcsec$ 0.160 11/22/06 – 2565 2354 – 2289 0.01
SDSS J135355.90+664800.5 13$^h$53$^m$55.90$^s$ +66$\degr$48$\arcmin$00.5$\arcsec$ 0.198 01/04/07 – 2661 2468 – 2334 0.02
SDSS J214500.25+011157.6 21$^h$45$^m$00.25$^s$ +01$\degr$11$\arcmin$57.3$\arcsec$ 0.204 07/10/07 2514 – – 3600 – 0.06
-------------------------- ----------------------- ------------------------------------ ------- ---------- ------ ---------------- ------ ------ ------ ------
-------------------------- ------------- -------- -------------- ------
SDSS J005527.46–002148.7 $4.2\pm0.8$ 28 14.9$\pm$2.0 1.17
SDSS J032845.99+011150.8 $3.2\pm1.1$ $<$3.4 $<$4.6 0.55
SDSS J040208.86–050642.0 $3.7\pm0.6$ $<$2.8 $<$4.1 0.25
SDSS J080844.26+394852.4 $3.1\pm0.4$ 7.3 $<$3.7 2.14
SDSS J092600.41+442736.1 $6.8\pm0.4$ 3.7 7.8$\pm$1.4 0.16
SDSS J102613.97+484458.9 $3.4\pm0.1$ 6.6 9.2$\pm$1.3 0.81
SDSS J135355.90+664800.5 $7.1\pm0.9$ 14.6 22.5$\pm$2.7 0.98
SDSS J214500.25+011157.6 $3.4\pm0.8$ 7.5 14.5$\pm$2.9 1.32
-------------------------- ------------- -------- -------------- ------
-------------------------- ------------------------- ------------------------- ----------- ---------------- ----------- ----------------
(mag) (mag) (mag) (mag) (mag)
SDSS J005527.46–002148.7 19.23$^{+0.15}_{-0.17}$ 18.85$^{+0.08}_{-0.08}$ 18.78 $18.65\pm0.02$ 18.25 $-1.13\pm0.22$
SDSS J032845.99+011150.8 19.42$^{+0.24}_{-0.31}$ 19.23$^{+0.14}_{-0.16}$ 18.97 $18.68\pm0.04$ 18.15 $-1.56\pm0.39$
SDSS J040208.86–050642.0 18.90$^{+0.06}_{-0.06}$ 18.70$^{+0.03}_{-0.03}$ 18.81 $18.67\pm0.03$ 18.08 $-1.54\pm0.09$
SDSS J080844.26+394852.4 18.22$^{+0.10}_{-0.11}$ 17.59$^{+0.05}_{-0.05}$ 17.62 $17.48\pm0.01$ 17.05 $-0.54\pm0.15$
SDSS J092600.41+442736.1 18.83$^{+0.11}_{-0.13}$ 18.89$^{+0.07}_{-0.08}$ –$^a$ $19.04\pm0.03$ 19.28 $-2.12\pm0.18$
SDSS J102613.97+484458.9 19.27$^{+0.03}_{-0.03}$ 19.00$^{+0.02}_{-0.02}$ 18.86 $18.85\pm0.02$ 18.23 $-1.38\pm0.05$
SDSS J135355.90+664800.5 18.99$^{+0.04}_{-0.04}$ 18.57$^{+0.02}_{-0.02}$ 18.42 $18.35\pm0.02$ 17.67 $-1.03\pm0.06$
SDSS J214500.25+011157.6 19.94$^{+0.25}_{-0.32}$ 19.22$^{+0.12}_{-0.13}$ 19.61$^b$ $19.24\pm0.04$ 18.62$^c$ $-0.35\pm0.41$
-------------------------- ------------------------- ------------------------- ----------- ---------------- ----------- ----------------
-------------------------- ------- ---------- ---------- ---------- ---------- ------- ------- ------- -------
(kpc) (kpc) (kpc) (kpc) (kpc) (kpc) (kpc) (mag) (mag)
SDSS J005527.46–002148.7 0.77 0.37 1.26 0.78 2.13 0.50 1.69 0.03 0.89
SDSS J032845.99+011150.8 1.66 0.87 1.96 1.79 4.93 0.93 2.49 0.24 1.32
SDSS J040208.86–050642.0 1.16 0.84 1.94 1.44 4.26 1.23 2.81 0.48 0.84
SDSS J080844.26+394852.4 0.47 0.16 1.41 0.57 2.77 0.47 2.94 –0.58 1.52
SDSS J092600.41+442736.1 0.97 –$^a$ –$^a$ 1.10 3.33 0.87 2.67 –$^a$ –$^a$
SDSS J102613.97+484458.9 2.07 1.89 3.86 1.99 4.48 2.13 4.22 0.53 0.67
SDSS J135355.90+664800.5 1.81 1.46 3.19 3.56 9.06 2.74 7.23 0.48 0.81
SDSS J214500.25+011157.6 0.74 1.10$^b$ 3.55$^b$ 1.30$^c$ 4.64$^c$ –$^d$ –$^d$ 0.93 1.00
-------------------------- ------- ---------- ---------- ---------- ---------- ------- ------- ------- -------
-------------------------- ----------------- ----------------- -------------- --------------- ------- -------- ----- ------ ------
($\mu$Jy) ($\mu$Jy) (mJy) (mJy) (mJy)
SDSS J005527.46–002148.7 725.8$\pm$33.2 4280.5$\pm$80.8 33.2$\pm$0.4 163.4$\pm$6.4 115.4 16 3.2 1.48 0.14
SDSS J032845.99+011150.8 218.3$\pm$39.0 972.0$\pm$71.9 3.4$\pm$0.4 $<$23.4$^f$ 25.9 $<$1.9 1.8 0.56 0.05
SDSS J040208.86–050642.0 131.6$\pm$34.8 415.3$\pm$66.6 1.1$\pm$0.1 $<$19.8$^f$ 25.5 $<$1.6 2.8 0.31 0.03
SDSS J080844.26+394852.4 1048.6$\pm$39.8 5786.5$\pm$66.2 39.9$\pm$0.6 156.0$\pm$6.8 117.0 4.1 2.0 0.91 0.08
SDSS J092600.41+442736.1 90.7$\pm$4.2 229.4$\pm$24.0 3.4$\pm$0.4 20.7$\pm$3.7 10.2 2.1 5.5 0.16 0.01
SDSS J102613.97+484458.9 176.6$\pm$36.2 614.5$\pm$56.8 4.2$\pm$0.3 42.5$\pm$4.2 31.6 3.7 2.7 0.70 0.06
SDSS J135355.90+664800.5 369.2$\pm$32.4 1660.3$\pm$53.1 44.4$\pm$0.4 70.1$\pm$4.3 27.2 8.2 6.0 0.71 0.07
SDSS J214500.25+011157.6 200.4$\pm$31.5 1617.5$\pm$64.2 5.0$\pm$0.4 23.0$\pm$5.5 27.7 4.2 2.6 0.79 0.07
-------------------------- ----------------- ----------------- -------------- --------------- ------- -------- ----- ------ ------
-------------------------- ------------- ----------- -------------- ----------- ------------- ----------- -------------- -----------
$r_{P,330}$ $G_{330}$ $M_{20,330}$ $C_{330}$ $r_{P,850}$ $G_{850}$ $M_{20,850}$ $C_{850}$
() ()
SDSS J005527.46–002148.7 0.55 0.63 -2.41 4.73 0.83 0.58 -1.98 3.53
SDSS J032845.99+011150.8 1.66 0.66 -1.48 3.97 2.62 0.58 -1.55 3.34
SDSS J040208.86–050642.0 1.02 0.56 -1.60 3.29 1.92 0.60 -1.98 3.92
SDSS J080844.26+394852.4 0.13 $>$0.57 $<$-2.55 $>$3.01 2.97 0.66 -3.23 7.11
SDSS J092600.41+442736.1 –$^a$ –$^a$ –$^a$ –$^a$ 1.29 0.73 -1.60 3.27
SDSS J102613.97+484458.9 2.21 0.60 -1.31 3.28 1.92 0.58 -1.48 2.93
SDSS J135355.90+664800.5 1.26 0.60 -1.27 2.88 1.29 0.53 -1.57 2.88
SDSS J214500.25+011157.6 0.75$^b$ 0.56$^b$ -1.47$^b$ 2.95$^b$ 1.40$^c$ 0.61$^c$ -1.80$^c$ 3.56$^c$
-------------------------- ------------- ----------- -------------- ----------- ------------- ----------- -------------- -----------
{width="75.00000%"}
{width="\textwidth"} {width="\textwidth"} {width="\textwidth"} {width="\textwidth"}
{width="\textwidth"} {width="\textwidth"} {width="\textwidth"} {width="\textwidth"} Fig. 1.– Cont.
![\[fig:radial\]UV-optical radial color profiles of the UVLGs shown in Fig. \[fig:stamps\]. Lines indicate the (cumulative) color determined in circular apertures measured from the centroids of the object detected in the optical image out to the 90% flux radius (dashed colored lines), and with respect to the object centroids defined in the UV image (dotted colored lines). The UVLGs are generally very blue within approximately the half light radius, and become redder at larger radii. The objects often have steeper inner color gradients when then UV image is used for object detection (dotted lines), compared to when the optical image is used (dashed lines) due to the different positions of the object centroids in the UV and optical. For 214500, the UV-optical color corresponds to $FUV_{150}$–[$V_{606}$]{}, whereas for the other objects the colors measured are [$U_{330}$]{}–[$z_{850}$]{}. Errors are 3$\sigma$.](f4.ps){width="\columnwidth"}
{width="50.00000%"}{width="50.00000%"}
![\[fig:haro11vv114sims\][*Top panels:*]{} NUV archival images of the local starburst galaxies Haro 11 and VV 114, both at $z=0.02$. The image of Haro 11 was taken with the ACS/HRC (Program 10575, PI: Göran Östlin). The image of VV 114 was taken with STIS (Program 8201, PI: Gerhardt Meurer). See Sect. \[sec:archival\] for details. [*Bottom panels:*]{} Simulated F330W images of Haro 11 (left) and VV 114 (right) at $z=0.15$ for comparison with our UVLG sample. The images measure 3$\times$3. See Sect. \[sec:sims\] for details.](f6a.ps "fig:"){width="0.4\columnwidth"}![\[fig:haro11vv114sims\][*Top panels:*]{} NUV archival images of the local starburst galaxies Haro 11 and VV 114, both at $z=0.02$. The image of Haro 11 was taken with the ACS/HRC (Program 10575, PI: Göran Östlin). The image of VV 114 was taken with STIS (Program 8201, PI: Gerhardt Meurer). See Sect. \[sec:archival\] for details. [*Bottom panels:*]{} Simulated F330W images of Haro 11 (left) and VV 114 (right) at $z=0.15$ for comparison with our UVLG sample. The images measure 3$\times$3. See Sect. \[sec:sims\] for details.](f6b.ps "fig:"){width="0.4\columnwidth"}\
![\[fig:haro11vv114sims\][*Top panels:*]{} NUV archival images of the local starburst galaxies Haro 11 and VV 114, both at $z=0.02$. The image of Haro 11 was taken with the ACS/HRC (Program 10575, PI: Göran Östlin). The image of VV 114 was taken with STIS (Program 8201, PI: Gerhardt Meurer). See Sect. \[sec:archival\] for details. [*Bottom panels:*]{} Simulated F330W images of Haro 11 (left) and VV 114 (right) at $z=0.15$ for comparison with our UVLG sample. The images measure 3$\times$3. See Sect. \[sec:sims\] for details.](f6c.ps "fig:"){width="0.4\columnwidth"}![\[fig:haro11vv114sims\][*Top panels:*]{} NUV archival images of the local starburst galaxies Haro 11 and VV 114, both at $z=0.02$. The image of Haro 11 was taken with the ACS/HRC (Program 10575, PI: Göran Östlin). The image of VV 114 was taken with STIS (Program 8201, PI: Gerhardt Meurer). See Sect. \[sec:archival\] for details. [*Bottom panels:*]{} Simulated F330W images of Haro 11 (left) and VV 114 (right) at $z=0.15$ for comparison with our UVLG sample. The images measure 3$\times$3. See Sect. \[sec:sims\] for details.](f6d.ps "fig:"){width="0.4\columnwidth"}
![\[fig:sims\_goods\_uvlgs\]Redshift simulations of the UVLG sample at the depth of GOODS (Sect. \[sec:sims\]). Panels ($3\arcsec\times3\arcsec$) show the original, unredshifted HRC/F330W or SBC/F150LP images (left), and GOODS [$B_{435}$]{} or [$V_{606}$]{}+[$i_{775}$]{} simulations at $z=1.5$, $z=3.0$ and $z=4.0$. The last two rows show the redshift simulations of Haro 11 and VV 114 (Sect. \[sec:sims\]) for comparison.](f7a.ps "fig:"){width="0.5\columnwidth"}\
![\[fig:sims\_goods\_uvlgs\]Redshift simulations of the UVLG sample at the depth of GOODS (Sect. \[sec:sims\]). Panels ($3\arcsec\times3\arcsec$) show the original, unredshifted HRC/F330W or SBC/F150LP images (left), and GOODS [$B_{435}$]{} or [$V_{606}$]{}+[$i_{775}$]{} simulations at $z=1.5$, $z=3.0$ and $z=4.0$. The last two rows show the redshift simulations of Haro 11 and VV 114 (Sect. \[sec:sims\]) for comparison.](f7b.ps "fig:"){width="0.5\columnwidth"}
![\[fig:sims\_udf\_uvlgs\]Redshift simulations at the depth of the UDF. See Sect. \[sec:sims\] and the caption of Fig. \[fig:sims\_goods\_uvlgs\] for details.](f8a.ps "fig:"){width="0.5\columnwidth"}\
![\[fig:sims\_udf\_uvlgs\]Redshift simulations at the depth of the UDF. See Sect. \[sec:sims\] and the caption of Fig. \[fig:sims\_goods\_uvlgs\] for details.](f8b.ps "fig:"){width="0.5\columnwidth"}
![\[fig:sims\_cosmos\_uvlgs\]Redshift simulations at the depth of COSMOS. See Sect. \[sec:sims\] and the caption of Fig. \[fig:sims\_goods\_uvlgs\] for details.](f9a.ps "fig:"){width="0.5\columnwidth"}\
![\[fig:sims\_cosmos\_uvlgs\]Redshift simulations at the depth of COSMOS. See Sect. \[sec:sims\] and the caption of Fig. \[fig:sims\_goods\_uvlgs\] for details.](f9b.ps "fig:"){width="0.5\columnwidth"}
![\[fig:morphologies\_redshift\]Morphological parameters of the UVLGs measured from the simulated images at different redshift. Objects were redshifted to $z=1.5$, $z=3$ and $z=4$, and simulated at the depths of COSMOS (red dashed lines), GOODS (green dotted lines) and the UDF (blue solid lines).](f10.ps){width="70.00000%"}
![\[fig:morphologies\_lotz\]Comparison of the morphologies of Lyman break galaxies at $\sim4$ and starburst galaxies at $z\sim1.5$ from the samples of @lotz06 (plusses) with the morphologies of the UVLGs (stars). The UVLGs were artificially redshifted to the mean redshift of the comparison samples, and simulated at a similar depth. Top panels show the results for LBGs and UVLGs at the UDF depth at $z\sim4$, middle panels show LBGs and UVLGs at the GOODS depth at $z\sim4$, and bottom panels show star forming galaxies and UVLGs at the GOODS depth at $z\sim1.5$. Haro 11 (’H’) and VV 114 (’V’) were simulated at the same redshift and depth as the UVLGs. For comparison, we have indicated the following (all adapted from @lotz06): regions populated by likely mergers having multiple nuclei (left hatched area) and by bulge-dominated objects (right hatched area), and the typical errors for the high redshift samples (large crosses). In all panels, the UVLGs span a very similar range in parameter space compared to the comparison samples, indicating that UVLGs and LBGs have very similar quantitative morphologies.](f11a.ps "fig:"){width="70.00000%"}\
![\[fig:morphologies\_lotz\]Comparison of the morphologies of Lyman break galaxies at $\sim4$ and starburst galaxies at $z\sim1.5$ from the samples of @lotz06 (plusses) with the morphologies of the UVLGs (stars). The UVLGs were artificially redshifted to the mean redshift of the comparison samples, and simulated at a similar depth. Top panels show the results for LBGs and UVLGs at the UDF depth at $z\sim4$, middle panels show LBGs and UVLGs at the GOODS depth at $z\sim4$, and bottom panels show star forming galaxies and UVLGs at the GOODS depth at $z\sim1.5$. Haro 11 (’H’) and VV 114 (’V’) were simulated at the same redshift and depth as the UVLGs. For comparison, we have indicated the following (all adapted from @lotz06): regions populated by likely mergers having multiple nuclei (left hatched area) and by bulge-dominated objects (right hatched area), and the typical errors for the high redshift samples (large crosses). In all panels, the UVLGs span a very similar range in parameter space compared to the comparison samples, indicating that UVLGs and LBGs have very similar quantitative morphologies.](f11b.ps "fig:"){width="70.00000%"}\
![\[fig:morphologies\_lotz\]Comparison of the morphologies of Lyman break galaxies at $\sim4$ and starburst galaxies at $z\sim1.5$ from the samples of @lotz06 (plusses) with the morphologies of the UVLGs (stars). The UVLGs were artificially redshifted to the mean redshift of the comparison samples, and simulated at a similar depth. Top panels show the results for LBGs and UVLGs at the UDF depth at $z\sim4$, middle panels show LBGs and UVLGs at the GOODS depth at $z\sim4$, and bottom panels show star forming galaxies and UVLGs at the GOODS depth at $z\sim1.5$. Haro 11 (’H’) and VV 114 (’V’) were simulated at the same redshift and depth as the UVLGs. For comparison, we have indicated the following (all adapted from @lotz06): regions populated by likely mergers having multiple nuclei (left hatched area) and by bulge-dominated objects (right hatched area), and the typical errors for the high redshift samples (large crosses). In all panels, the UVLGs span a very similar range in parameter space compared to the comparison samples, indicating that UVLGs and LBGs have very similar quantitative morphologies.](f11c.ps "fig:"){width="70.00000%"}
{width="\textwidth"}
{width="45.00000%"} {width="45.00000%"}
![\[fig:ew\]Rest-frame equivalent width [H$\alpha$]{} vs. [$U_{330}$]{}–[$z_{850}$]{} color for the super starburst regions. Tracks indicate the age evolution of the [H$\alpha$]{} EW for instantaneous burst ([*dashed lines*]{}, with ages marked at 5,6,7,8,9,10 Myr) and continuous star formation ([*solid lines*]{}, with ages marged at 5,6,7,8,9,10,20,30,40,50,60,70,80,90,100,200,500,1000 Myr) models from STARBURST99. See text and the caption of Fig. \[fig:ssc\_cm\] for further details.](f15.ps){width="0.5\columnwidth"}
Scarpa et al. (2007) revisited {#sec:scarpa}
==============================
In a recent paper, @scarpa07 have tried to argue that the sample of compact UVLGs selected by H05 and H07 are not true analogs of LBGs. @scarpa07 claim that the sizes estimated from the SDSS [*u*]{}-band images grossly underestimate the actual size in the UV, and that the UVLGs therefore are much larger and lower in surface brightness than LBGs. Here, we will briefly show that this claim is based on the combination of a faulty analysis and incorrect assumptions, that are easily refuted by the HST data presented in this paper.\
$\bullet$ @scarpa07 claim that it is only possible to detect central bulges in the SDSS [*u*]{}-band images, and that fainter, extended disks would be missed. They reach this conclusion by fitting the SDSS [*r*]{}-band images with a two component model (point source plus disk) and then using the half-light radius of [*only the disk*]{} component to recompute the size of the UVLGs. In doing this, the authors made a serious mistake, because they ignore the fact that on average, half the light in their model fits comes from the central [*“point source”*]{} (see their Table 1).\
$\bullet$ The claim in the @scarpa07 paper that most of the FUV flux in compact UVLGs should come from a large disk component relies on the observation that there exists a population of galaxies with relatively red bulges not visible in the FUV and very blue outer rings or disks that are only apparent in the UV [e.g. @thilker05]. It should be clear both from the results presented in H05 and H07, and from the analysis of HST data presented here, that UVLGs are an entirely different class of objects. In fact, the UVLGs are clearly more extended in the optical than in the UV due to an underlying (older) component. The UVLGs have very blue inner colors, and become redder outwards (see Fig. \[fig:radial\] and Table \[tab:sizes\]). This is opposite to the assumption of @scarpa07.\
$\bullet$ The HST [$U_{330}$]{} images confirm that the UVLGs are indeed very compact objects. In fact, in most cases, the half light radius is even smaller than the seeing deconvolved SDSS [*u*]{}-band size (Table \[tab:sizes\]). In the case of both J005527.46–002148.7 and J080844.26+394852.3 the UV flux is dominated by an unresolved component even at the HST resolution. Furthermore, when we extrapolate the slope of the UV continuum measured from the FUV–NUV color and predict the [$U_{330}$]{} flux, we find good agreement with the measured [$U_{330}$]{}flux (Sect. \[sec:sizes\]), indicating that the [$U_{330}$]{} (and SDSS [ *u*]{}-band) image is a good tracer of the FUV morphology. In the case of J214500.25+011157.3 we have an ACS/SBC image taken through F150LP, a filter that is almost identical to the GALEX FUV filter [see @teplitz06]. The corresponding FUV surface brightness calculated solely from the ACS data, $\textrm{log}_{10}I_{FUV,ACS}=9.53$ $L_\odot$ kpc$^{-2}$, is slightly lower than the value of H07 based on GALEX and SDSS ($\textrm{log}_{10}I_{FUV}=9.82$ $L_\odot$ kpc$^{-2}$), but still well above our LBG selection threshold of $\textrm{log}_{10}I_{FUV}=9$ $L_\odot$ kpc$^{-2}$.\
$\bullet$ As we have discussed above, the galaxy sizes defined by @scarpa07 pertain only to the disk component and thus cannot be compared with the half-light radii quoted by H05 and H07. @scarpa07 also did not apply their profile fitting methodology to the full sample in a uniform way. They only applied their analysis to the objects that were marked as “supercompact” by H07 and not to the other UVLGs. As a result, their concluding Figure 5 is misleading. It appears to show that the LBG analogs have a similar FUV surface brightness as the main sample. Had they applied their technique to all the objects in the sample, the compact UVLGs would still form an upper envelope to the $I_{FUV}$ distribution, with the overall $I_{FUV}$ distribution shifted downwards toward lower surface brightness due to the fact that the “point source” flux in their fits was ignored.\
$\bullet$ Finally, we would like to remind the reader that the real test of whether or not a given sample of nearby galaxies are good analogs of LBGs is not the UV selection per sé (note that the LBG luminosity function extends over many orders of magnitude [e.g. @bouwens07]), but whether their main physical properties are indeed similar. We firmly believe that our sample of LBG analogs have passed this test.
[^1]: See the Appendix for a detailed analysis that demonstrates that the interpretation and conclusions presented in the recent paper by Scarpa et al. (2007) are in fact in error.
[^2]: Calculated from $\beta_{GALEX}=2.32(m_{FUV}-m_{NUV})_{AB}-2$.
[^3]: http://cas.sdss.org/dr5/en/tools/explore/obj.asp?id=588015508738539596
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this letter we communicate the identification of a new Galactic O2If\* star (MTT 68) isolated at a projected linear distance of 3 pc from the centre of the star-burst cluster NGC 3603. From its optical photometry I computed a bolometric luminosity M$_{Bol}$ = -10.7, which corresponds to a total stellar luminosity of 1.5$\times$10$^{6}$ L$_\odot$. It was found an interesting similarity between MTT 68 and the well known multiple system HD 93129. From Hubble Space Telescope F656N images of the NGC 3603 field, it was found that MTT 68 is actually a visual binary system with an angular separation of 0.38$\arcsec$, which corresponds to a projected (minimum) linear distance of *r*$_{A-B}$ = 1.4$\times$10$^{-2}$ pc. This value is similar to that for the HD 93129A (O2If\*) and HD 93129B (O3.5) pair, *r*$_{A-B}$ = 3.0$\times$10$^{-2}$ pc. On the other hand, HD93129A has a third closer companion named HD 93129Ab (O3.5) at only 0.053$\arcsec$, and taking into account that the X-ray to total stellar luminosity ratio for the MTT 68 system (L$_X$/L$_{Bol}$ $\sim$1$\times$10$^{-5}$) is about two orders of magnitude above the canonical value expected for single stars, I suspect that the MTT 68 system probably hosts another massive companion possibly to close to be properly resolved by the HST archive images.'
author:
- |
A. Roman-Lopes$^{1}$[^1]\
$^{1}$Department of Physics - Universidad de La Serena - Cisternas, 1200 - La Serena - Chile\
title: 'An O2If\* star found in isolation in the backyard of NGC 3603[^2]'
---
\[firstpage\]
Stars: Wolf-Rayet; Infrared: Stars: Individual: HD93129A, MTT 68; Galaxy: open clusters and associations: individual: NGC 3603
Introduction
============
NGC 3603 is the closest star-burst like cluster, being an invaluable template for the modern theory of formation and evolution of *very massive stars*. Indeed, with at least four exemplars in its core (three WN6ha + one O2If\*/WN6 - @b7), plus two O2If\*/WN6 stars (WR42e and MTT58) recently identified at only a few arcminutes from its centre [@b8; @b9], this cluster may be the host of the larger concentration of extremely massive *hydrogen core burning* stars in the Galaxy [@b10; @b3; @b7].
@b6 detected a large number of X-ray sources toward the cluster centre, concluding that the majority of them should be probably compound by pre-main sequence stars. However, the detection of two O2If\*/WN6 stars in the NGC 3603’s periphery may indicate that some other very massive stars could be present in the NGC 3603 cluster field. In this context and based on the presence of very strong X-Ray counterparts in the BMW-Chandra catalogue [@b13], we performed NIR follow-up spectroscopic observations of a very interesting X-ray point source previously cataloged by @b2 as MTT 68. It is now confirmed to be an O2If\* star isolated in the periphery of NGC 3603, at about 1.4$\arcmin$ from its core.
Near-Infrared spectroscopic observations and data reduction
===========================================================
The NIR spectroscopic observations were performed with the Ohio State Infrared Imager and Spectrometer (OSIRIS) at the Southern Astrophysics Research (SOAR) telescope. The J-, H- and K-band data were acquired in 31th January 2012 with the night presenting good atmospheric conditions.
In Table 1 it is shown a summary of the NIR observations. The raw frames were reduced following well known NIR reduction procedures. The two-dimensional frames were subtracted for each pair of images taken at the two shifted positions, with the resultant images being divided by a master normalized flat. For each processed frame, the J-, H- and K-band spectra were extracted using the task APALL within IRAF, with the wavelength calibration made through the IRAF tasks IDENTIFY and DISPCOR applied to a set of OH sky line spectra in the range 12400Å -23000Å . The corrections of the telluric atmospheric features on the science data, were performed using J-, H- and K-band spectra of A type stars obtained before and after the science targets. The photospheric absorption lines present in the high signal-to-noise telluric spectra, were subtracted from a careful modeling (through the use of Voigt and Lorentz profiles) of the hydrogen absorption lines using the corresponding adjacent continuum. At the end, the J-, H- and K-band spectra were combined by using the IRAF task SCOMBINE with the mean signal-to noise ratio of the resulting spectra well above 100.
----------------------- ------------
Date 31/01/2012
Telescope SOAR
Instrument OSIRIS
Mode XD
Camera f/3
Slit 1“ x 27”
Resolution 1000
Coverage ($\mu$m) 1.25-2.35
Seeing (") & 0.8-1.3\
----------------------- ------------
: Summary of the SOAR/OSIRIS spectroscopic observations.[]{data-label="catalog"}
![The three color image of the north and west regions towards the NGC 3603 cluster’s centre, constructed from the 3.6$\mu$m (blue), 4.5$\mu$m (green) and 5.8$\mu$m (red) Spitzer IRAC data taken from the NASA/IPAC Infrared Science Archive. Notice that the new star is found in isolation at about 1.4 arcmin from the cluster’s centre, which for an estimated heliocentric distance of 7.6 kpc corresponds to a projected distance of about 3 parsecs.[]{data-label="FigVibStab"}](figure2c.eps){width="8.7"}
![The J- H- and K-band continuum normalized SOAR-OSIRIS spectra of MTT 68, with the main H, He and N emission lines identified by labels.[]{data-label="FigVibStab"}](Figure1c.eps){width="8.7"}
Results
=======
Coordinates and photometric parameters of MTT 68 are shown in Table 2. The V-, R- and I-band optical magnitudes were taken from the work of @b20, while the near-infrared values were obtained from the Two-Micron All Sky Survey [@b21], with the absorption-corrected 0.5-10keV Chandra X-ray flux taken from the work of @b13.
The OSIRIS NIR spectra of MTT 68: An O2If\* star isolated in the backyard of NGC 3603
-------------------------------------------------------------------------------------
As mentioned before, the star subject of this letter was previously cataloged as MTT 68 by @b2. In Figure 1 it is shown a three colour image of part of the NGC 3603 field, made from the 3.6$\mu$m (blue), 4.5$\mu$m (green) and 5.8$\mu$m (red) Spitzer IRAC data taken from the NASA/IPAC Infrared Science Archive[^3]. From that figure, we can see that the new star appears isolated at about 1.4 arcmin from the cluster’s centre, which for an estimated heliocentric distance of 7.6 kpc [@b3] corresponds to a projected radial distance of about 3 parsecs. From a search in the CADC HST/HLA/WFPC2B Science Archive[^4] for optical images of the NGC 3603 field, it was found a non-saturated F656N image (IB6WA1070 - P.I. O’Connell) in which we see that MTT 68 has a visual companion (MTT 68B - please see the inset in Figure 1) at an angular separation of 0.38$\arcsec$. More on this subject will be discussed in Section 3.2.
Figure 2 shows the telluric corrected (continuum normalized) J-, H- and K-band SOAR-OSIRIS spectra of MTT 68, in which we can see the Paschen beta and Bracket hydrogen emission lines at 1.283$\mu$m, 1.736$\mu$m and 2.166$\mu$m, as well as the N[iii]{} and He[ii]{} lines (also in emission) at 2.116$\mu$m and 2.189$\mu$m, respectively. In Figure 3 it is presented the MTT 68’s J-, H- and K-band SOAR-OSIRIS spectra, together with those taken for HD93129A (O2If\*), WR20a (O3If\*/WN6 + O) and WR25 (O2.5If\*/WN6). The spectral type O2If\* was introduced in 2002 [@b32], with HD93129A being probably the only known Galactic template of the class. It is considered the earliest, hottest, most massive and luminous O star in the Galaxy, showing an extremely powerful wind with a terminal velocity above 3000 km s$^{-1}$, and a mass-loss rate above 10$^{-5}$ M$_{\odot}$ yr$^{-1}$ [@b31]. From the comparison of the MTT 68’s NIR spectra with those for the templates, we can see that the MTT 68’s spectrograms resembles well those for HD93129A, indicating that it is probably a new Galactic exemplar of the rare O2If\* type.
{width="10"}
{width="5"}
------------- ------------- ------- ------- ------- ------- ------ ------ ------ ------------------------ --
RA Dec B V R I J H Ks X-Ray
(J2000) (J2000) (Wm$^{-2}$)
11:14:59.48 -61:14:33.9 16.31 14.72 13.64 12.05 9.98 9.17 8.74 12.9$\times$10$^{-16}$
------------- ------------- ------- ------- ------- ------- ------ ------ ------ ------------------------ --
Mass, luminosity and binary status of MTT 68
--------------------------------------------
We can estimate the mass of the MTT 68 components taking into account that the star is actually a visual binary system presenting an angular separation of 0.38$\arcsec$, and by comparing their combined V-band magnitude and V-I color with those of other NGC 3603 cluster members, presented in Figure 7 of @b20. Also, from the comparison of the instrumental magnitudes of the two (non-saturated) point sources in the HST F656N image, we estimate as 1.2 mag the $\Delta_m$ difference for the MTT 68A and MTT 68B. Finally, and in order to simplify the process, we will assume that both stars have the same bolometric corrections, which is a reasonable assumption considering their probable very early spectral types. In Figure 4 we show an adapted version of the V $\times$ (V-I) diagram for NGC 3603 [@b20], with MTT 68A, MTT 68B and WR42e (estimated mass of $\sim$ 130 M$_\odot$ - @b8 [@b24] represented by black circles and gray triangle, respectively. WR42e is also shown there because it is a star of the O2If\*/WN6 spectral type and as the MTT 68 system, possibly belongs to the same complex. From this diagram we can see that MTT 68A and MTT 68B probably presented initial masses well above 100 M$_\odot$ and 40 M$_\odot$, respectively.
To estimate the total luminosity of the MTT 68 binary system, it is necessary to compute the associated visual extinction considering that the interstellar reddening law for NGC 3603 is possibly abnormal, with a ratio of total to selective extinction value R$_V$=3.55$\pm$0.12 [@b20]. From Table 2, we can see that MTT 68 presents (B-V) color $\sim$ 1.6 mag, which for an assumed mean intrinsic (B-V)$_0$ value of -0.3 mag (typical for the hottest early-type stars), corresponds to a color excess E(B-V) $\sim$ 1.9 mag or A$_V$ $\sim$ 6.7$\pm$0.3 mag. From the computed color excess and visual extinction, we can estimate the MTT 68’s absolute magnitude using the distance modulus equation, assuming that the star is part of the NGC 3603 complex at an heliocentric distance of 7.6$\pm$0.4 kpc [@b3]. We computed M$_V$=-6.4 mag for the binary system (or individually M$_V$=-6.1 and M$_V$=-4.9 for components *A* and *B*, respectively), which for an assumed mean bolometric correction BC $\sim$ -4.3 mag [@b3; @b7], results in a bolometric magnitude M$_{Bol}$ $\sim$ -10.7 and in a total stellar luminosity of 1.5$\times$10$^{6}$ L$_\odot$. This total luminosity is similar to that derived by @b6, which found L$\sim$1.3$\times$10$^{6}$ L$_\odot$ with an X-ray to total stellar luminosity ratio L$_X$/L$_{Bol}$ $\sim$1$\times$10$^{-5}$, a value two orders of magnitude greater than the canonical value expected for single stars, e.g. L$_X$/L$_{Bol}$ $\sim$ 10$^{-7}$ [@b31].
Similarities with the HD93129 system
------------------------------------
From the measured angular separation (0.38$\arcsec$) and the assumed heliocentric distance of 7.6 kpc, it is possible to compute the linear projected (minimum) radial distance of the MTT 68 binary components as *r*$_{A-B}$ = 1.4$\times$10$^{-2}$ pc. HD93129A is known to be an extremely powerful X-ray source, being part of the Trumpler 14 cluster in the Carina Nebula, at an heliocentric distance of 2.3 kpc (for more on it please see the work of @b31 and references therein). As the new O2If\* star, it has a visual companion named HD93129B (O3.5) at an angular separation of 2.7$\arcsec$, which for the quoted distance corresponds to a linear projected radial distance of 3$\times$10$^{-2}$ pc, a value similar to that for the MTT 68 components. On the other hand, HD93129A has also another closer companion (HD 93129Ab) at only 0.053$\arcsec$ that is supposed to be also of the same spectral type of HD 93129B [@b31]. Considering the observed X-ray to total stellar luminosity ratio of the MTT 68 system (L$_X$/L$_{Bol}$ $\sim$1$\times$10$^{-5}$), it is reasonable to speculate that this might be also the case for MTT 68, e.g., the probable existence of a very close companion not resolved in the F656N HST images.
Certainly, new further spectro-photometric studies of the MTT 68 binary system are needed. In one hand, its present evolutionary stage can help us to better understand how such kind of system can be found (and build) in relative isolation (at about 3.1 pc from the NGC 3603 cluster centre). On the other hand (and may be even more important), its presence in the periphery of a star burst like cluster may represent a challenge to the present theory of formation and evolution of very massive stellar systems, in the sense that may be the unique conditions found in such kind of environment could produce very massive stars well beyond the core of stellar clusters like NGC 3603.
Summary
=======
In this work we communicate the identification of a new Galactic O2If\* star (MTT 68) that is found in isolation at about 1.4$\arcmin$ (about 3 pc considering a quoted heliocentric distance of 7.6 kpc) from the core of the star-burst cluster NGC 3603. Our main conclusions and results are:
1- By comparing its NIR spectra with those for HD 93129A (O2If\*), WR20a (O3If\*/WN6), and WR25 (O2.5If\*/WN6), we conclude that the MTT 68’s spectrograms resembles well those of the former, indicating that it is probably a new Galactic exemplar of the rare O2If\* type.
2- From the inspection of F656N HST images of NGC 3603, it was determined that MTT 68 is actually a visual binary presenting an angular separation of 0.38$\arcsec$. Also, from the instrumental photometry of the non-saturated MTT 68 point sources, it was possible to determine a magnitude difference $\Delta_m$ = 1.2 magnitudes, which combined with the B-, V- and I-band photometry taken from the literature, resulted in a combined bolometric luminosity M$_{Bol}$ = -10.7 (M$_{Bol}$ = -10.4 and -9.2 for MTT 68A and MTT 68B, respectively) equivalent to a total stellar combined luminosity of 1.5$\times$10$^{6}$ L$_\odot$.
3- The total luminosity derived by us is similar to that obtained by @b6, which found L$\sim$1.3$\times$10$^{6}$ L$_\odot$ with an X-ray to total stellar luminosity ratio L$_X$/L$_{Bol}$ $\sim$1$\times$10$^{-5}$, a value two orders of magnitude above the canonical value for single stars, e.g. L$_X$/L$_{Bol}$ $\sim$ 10$^{-7}$ [@b31].
4- From the associated V- and I-band photometry, the magnitude differences obtained from the HST F656N image and stellar models previously published for the NGC 3603 massive stellar population, it was possible to conclude that MTT 68A and MTT 68B probably presented initial masses above 100 M$_\odot$ and 40 M$_\odot$, respectively.
5- We found some interesting similarities with the well known multiple system HD 93129. In one hand, the measured angular separation of the MTT 68 binary components (0.38$\arcsec$) corresponds to a projected (minimum) linear distance of *r*$_{A-B}$ = 1.4$\times$10$^{-2}$ pc, a value similar to that for HD 93129A (O2If\*) and HD 93129B (O3.5), e.g., *r*$_{A-B}$ = 3.0$\times$10$^{-2}$ pc. On the other hand, HD93129A has another closer companion named HD 93129Ab (O3.5) at only 0.053$\arcsec$ [@b31]. Considering the observed X-ray to total stellar luminosity ratio for the MTT 68 system (L$_X$/L$_{Bol}$ $\sim$1$\times$10$^{-5}$), it is possible that some another very close massive companion, not resolved in the F656N HST images, is still to be detected in the MTT 68’s binary system.
Acknowledgments {#acknowledgments .unnumbered}
===============
I would like to thank the anonymous referee by the careful reading of the manuscript. Her/his comments and criticism were welcome. This research has made use of the NASA/ IPAC Infrared Science Archive, which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration. This publication makes use of data products from the Two Micron All Sky Survey, which is a joint project of the University of Massachusetts and the Infrared Processing and Analysis Center/California Institute of Technology, funded by the National Aeronautics and Space Administration and the National Science Foundation. Based on observations obtained at the Southern Astrophysical Research (SOAR) telescope, which is a joint project of the Ministério da Ciência, Tecnologia, e Inovação (MCTI) da República Federativa do Brasil, the U.S. National Optical Astronomy Observatory (NOAO), the University of North Carolina at Chapel Hill (UNC), and Michigan State University (MSU). Based (in part) on observations made with the NASA/ESA Hubble Space Telescope, and obtained from the Hubble Legacy Archive, which is a collaboration between the Space Telescope Science Institute (STScI/NASA), the Space Telescope European Coordinating Facility (ST-ECF/ESA) and the Canadian Astronomy Data Centre (CADC/NRC/CSA). This work was partially supported by the Department of Physics of the Universidad de La Serena. ARL thanks financial support from Diretoria de Investigación - Universidad de La Serena through Project DIULS REGULAR PR13144.
Cohen, David H., Gagné, Marc, Leutenegger, Maurice A., MacArthur, James P., Wollman, Emma E., Sundqvist, Jon O., Fullerton, Alex W., Owocki, Stanley P. 2011, MNRAS, 415, 3354 Crowther, P. A., Schnurr, O., Hirschi, R., Yusof, N., Parker, R. J., Goodwin, S. P., Kassim, H. A. 2010, MNRAS, 408, 731 Crowther, P. A. & Walborn, N. R. 2011, MNRAS, 416, 1311 Cutri, R. M., Skrutskie, M. F., van Dyk, S., Beichman, C. A., Carpenter, and 20 more authors 2003, Gvaramadze, V. V., Kniazev, A. Y., Chené, A.-N.; Schnurr, O. 2013, MNRAS, 430, L20 Melnick, J., Tapia, M., Terlevich, R. 1989,A&A, 213, 89 Moffat, A. F. J., Corcoran, M. F., Stevens, I. R., Skalkowski, G., Marchenko, S. V., Mücke, A., Ptak, A., Koribalski, B. S., Brenneman, L., Mushotzky, R., Pittard, J. M., Pollock, A. M. T., Brandner, W. 2002, ApJ, 573, 191 Roman-Lopes, A. 2012, MNRAS, 427, 65 Roman-Lopes, A. 2013, MNRAS, 433, 712 Romano, P., Campana, S., Mignani, R. P., Moretti, A., Mottini, M., Panzera, M. R., Tagliaferri, G. 2008, A&A, 488, 1221 Smith, N. & Conti, P. S. 2008, ApJ, 679, 1467 Sung, H., Bessell, M. S. 2004, AJ, 127, 1014 Walborn, Nolan R., Howarth, Ian D., Lennon, Daniel J., Massey, Philip, Oey, M. S., Moffat, Anthony F. J., Skalkowski, Gwen, Morrell, Nidia I., Drissen, Laurent, Parker, Joel Wm 2002, AJ, 123, 2754
[^1]: roman@dfuls.cl
[^2]: Based on observations obtained at the Southern Astrophysical Research (SOAR) telescope
[^3]: http://irsa.ipac.caltech.edu/data/SPITZER/docs/spitzerdataarchives/
[^4]: http://www3.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/hst/new/
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Wireless Sensor Network (WSN) applications range from domestic Internet of Things systems like temperature monitoring of homes to the monitoring and control of large-scale critical infrastructures. The greatest risk with the use of WSNs in critical infrastructure is their vulnerability to malicious network level attacks. Their radio communication network can be disrupted, causing them to lose or delay data which will compromise system functionality. This paper presents Antilizer, a lightweight, fully-distributed solution to enable WSNs to detect and recover from common network level attack scenarios. In Antilizer each sensor node builds a self-referenced trust model of its neighbourhood using network overhearing. The node uses the trust model to autonomously adapt its communication decisions. In the case of a network attack, a node can make neighbour collaboration routing decisions to avoid affected regions of the network. Mobile agents further bound the damage caused by attacks. These agents enable a simple notification scheme which propagates collaborative decisions from the nodes to the base station. A filtering mechanism at the base station further validates the authenticity of the information shared by mobile agents. We evaluate Antilizer in simulation against several routing attacks. Our results show that Antilizer reduces data loss down to 1% (4% on average), with operational overheads of less than 1% and provides fast network-wide convergence.'
author:
- 'Ivana Tomić, Po-Yu Chen, Michael J. Breza and Julie A. McCann'
bibliography:
- 'references.bib'
title: 'Antilizer: Run Time Self-Healing Security for Wireless Sensor Networks'
---
<ccs2012> <concept> <concept\_id>10002978.10003001.10003003</concept\_id> <concept\_desc>Security and privacy Embedded systems security</concept\_desc> <concept\_significance>500</concept\_significance> </concept> <concept> <concept\_id>10010520.10010553.10003238</concept\_id> <concept\_desc>Computer systems organization Sensor networks</concept\_desc> <concept\_significance>300</concept\_significance> </concept> <concept> <concept\_id>10002978.10002986.10002987</concept\_id> <concept\_desc>Security and privacy Trust frameworks</concept\_desc> <concept\_significance>300</concept\_significance> </concept> </ccs2012>
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We characterize the heterogeneous character of the dynamics of liquids approaching the glass transition through an experimental determination of the number of dynamically correlated molecules $N_{corr}$ as obtained from dynamical susceptibilities. To do so, we have obtained a new set of dielectric spectroscopy data for liquid dibutyl-phtalate on a fine and extended temperature and pressure grid, and we have used it in conjunction with high-pressure data from the literature. We have been able to evaluate the contributions to $N_{corr}$ that are due to fluctuations associated with density and with temperature separately, thereby improving the estimate of $N_{corr}$. We find that $N_{corr}$ increases along the glass transition line, and more generally along any isochronic line, as pressure increases (up to $1\,GPa$), a result which is at odds with recent reports and theoretical predictions.'
author:
- 'Christiane Alba-Simionesco'
- 'C[é]{}cile Dalle-Ferrier'
- Gilles Tarjus
title: Effect of pressure on the number of dynamically correlated molecules when approaching the glass transition
---
Introduction
============
A tempting explanation of the dramatic slowing down of the dynamics observed in liquids approaching their glass transition is the existence of a growing underlying length scale. The relevance of such a length, characterizing the cooperative nature of the structural relaxation, was already suggested by Adam and Gibbs in the sixties [@Adam65]. The search for length scales went in two directions [@Tarjus11]: on the one hand, studies of nontrivial spatial correlations in the structure, associated with some hidden order parameter or in the form of point-to-set correlations describable through the influence of amorphous boundary conditions [@PTS] (both being essentially inaccessible in laboratory experiments); on the other hand, based on the mounting evidence in experiments and simulations of the increasingly heterogeneous character of the dynamics [@DHexpts; @Hurley95; @DHbook], investigations of a “dynamic” length describing the spatial extent of the dynamic heterogeneities. However, direct experimental measurement of the latter in molecular liquids is hard to obtain. In consequence, the corresponding results are scarce [@DHexpts; @Tracht98; @Reinsberg], moreover providing no information on the evolution with temperature.
A few years ago, on the basis of theoretical arguments, a new method for estimating the typical length scale of the dynamical heterogeneities and access its temperature dependence was suggested [@Berthier05]. It relies on the introduction and the study of multi-point dynamical susceptibilities. No information on spatial correlations in the dynamics can indeed be derived from the mere consideration of standard 2-time correlation functions, $F(t)=<\delta O(0)\delta O(t)>$ (with $O(t)$ some observable depending on the probe), which describe the “average” dynamics of a liquid. Higher-order space-time correlation functions, or their volume integrals that represent susceptibilities and may therefore be more easily accessible, are required to probe the fluctuations around the average dynamics, whose manifestation is precisely the dynamic heterogeneities [@Franz99]. From the multi-point dynamic susceptibilities, one can extract an estimate of the “number of dynamically correlated molecules” [@Berthier05; @Dalle07], which is the focus of the present work.
Specifically, the four-point dynamical susceptibility $\chi_4(t)=N<\delta C(0,t)^2>$, where $N$ is the number of molecules in the system and $\delta C(0,t) = \delta O(0)\delta O(t) - F(t)$ characterizes deviations from the average dynamics, quantifies the overall amount of spatial correlations between *spontaneous* fluctuations taking place between times 0 and $t$. This quantity is still, unfortunately, very difficult to directly measure in experiments, especially in the case of molecular liquids. Interestingly, however, it is related through fluctuation-dissipation relations [@Berthier05; @Berthier07] to other dynamical susceptibilities associated with fluctuations that are *induced* by an external control parameter $x$ (such as temperature, pressure or, for colloids, volume fraction): these are three-point susceptibilities, defined as $\chi_x(t)=\frac{\partial F(t)}{\partial x}$, which describe the response of the two-time correlation function $F(t)$ to a change in an external parameter. Such quantities are much easier to experimentally access than $\chi_4(t)$, as the dependence of $F(t)$ on temperature, pressure, or any external field, may be measured by several well-developed methods such as dielectric or photon correlation spectroscopy for example [@Berthier05]. It was moreover shown by computer simulation studies [@Berthier07] that a good estimate for $\chi_4(t)$ is obtained from the $\chi_x(t)$’s in the viscous liquid regime. (Note that another 3-point dynamical susceptibility has also been directly measured by means of an investigation of the nonlinear dielectric response [@Ladieu].)
Thanks to these high-order dynamical susceptibilities, direct experimental evidence of the increase of the number of dynamically correlated molecules $N_{corr}(T)$ while approaching the glass transition by cooling was obtained for several molecular liquids and polymers [@Dalle07; @Dalle08; @Capaccioli08; @Fragiadakis09; @Fragiadakis11; @Ladieu]. The growth of $N_{corr}(T)$ was found to be moderate compared to the dramatic change in relaxation time in the same temperature range, with $N_{corr}$ reaching a value of the order of $10^2$ at the glass transition temperature $T_g$ at atmospheric pressure. No clear conclusions could be drawn concerning the influence of the chemical nature of the system on the variation of $N_{corr}$. In particular, no correlation was observed between the growth of $N_{corr}(T)$ and the fragility of the system, *i.e.* the deviation of the temperature dependence of relaxation time from an Arrhenius behavior.
When cooling a liquid at constant pressure, its density increases. It is therefore useful to disentangle the effect coming from the densification of the system from that due to the decrease of thermal energy. This provides clearer insight into the physics of the slowing down of relaxation [@Ferrer98]. In addition, when considering estimates of the four-point dynamical susceptibility $\chi_4(t)$, it may prove important because a better description of the associated fluctuations in the $NPT$ ensemble is provided by summing the separate contributions of the density-induced and the temperature-induced fluctuations. However, except for partial results at atmospheric pressure [@Dalle07], this latter procedure was never attempted. Previous pressure studies along several isobars considered the crudest bound on $\chi_4(t)$ only [@Fragiadakis09; @Fragiadakis11].
In this work, we consider the effect of density and temperature on the number of dynamically correlated molecules in a fragile glass-forming liquid, dibutylphtalate. To do so, we have measured the (linear) dielectric susceptibility along both isobars and isotherms, covering thermodynamic points on a fine grid in temperature and pressure, and combined these results with existing high-pressure data [@Cook93; @Paluch03]. We have calculated the responses of the linear dielectric susceptibility to temperature along isobars and to density along isotherms to provide estimates of the variation of the number of dynamically correlated molecules. To go beyond this and, as explained above, improve the estimate of $N_{corr}$ by considering separately the contributions associated with density and temperature at any state point, we have used the density-temperature scaling already shown to describe relaxation data in supercooled liquids in a large domain of temperature and density [@Alba02; @Alba04; @Roland04; @Dreyfus04; @Roland05]. With the help of this scaling description, one can more easily compute the various dynamic quantities entering in the expression of $N_{corr}$. We have taken special care to minimize and to evaluate the systematic errors that come with the calculation of $N_{corr}$. Our main finding is that $N_{corr}$ significantly increases (by a factor of almost 3) along the glass transition line as one increases the pressure up to $1\,GPa$ (the density changes by $25\,\%$). The same trend is found for other isochronic lines (*i.e.* lines of equal relaxation time). Contrary to previous reports [@Fragiadakis09; @Fragiadakis11] and to theoretical predictions [@Xia00; @Lubchenko07], the extent of the spatial correlations in the dynamics is therefore not uniquely determined by the relaxation time.
Experimental details
====================
Sample and experimental setup
-----------------------------
The dibutylphtalate (DBP) sample ($99\%$ purity) was acquired from Sigma Aldrich and used as purchased. The experimental pressure dielectric setup for the dielectric measurements was described in a previous work [@Niss07]. The main characteristic of this setup is that it insures a completely isotropic compression by the use of an external pressure fluid. For the present study, we measured a new set of data with very small pressure or temperature steps (see Fig. \[fig1\] for an illustration). This data set is fully compatible with that reported in Ref. \[\] but is more complete in order to perform the detailed analysis involved in estimating $N_{corr}$. The reproducibility was always checked very carefully, especially under pressure where the experiments were carried out alternatively by compression and decompression for comparison.
![\[fig1\] Imaginary part of the dielectric spectrum of liquid DPB measured at $T=216\,K$ from atmospheric pressure (leftmost curve) to $P=250\,MPa$ (rightmost curve) obtained by both compression and decompression. The average pressure step between 50 and 250 $MPa$ is about $10\,MPa$.](fig1-eps-converted-to){width="\linewidth"}
The dielectric spectra were fitted by Havriliak-Negami expressions [@Havriliak67]: $$\label{eq:havriliak}
\epsilon(\omega)=\epsilon_{\infty} + \frac{\Delta \epsilon}{[1+(i\omega \tau)^{\alpha}]^{\gamma}}\;,$$ with $\Delta \epsilon$, $\tau$, $\alpha$ and $\gamma$ temperature and pressure dependent adjustable parameters. This provided a convenient and faithful parametrization of all data, allowing for instance to conveniently extract the relaxation function through Fourier transformation and to take numerical derivatives with respect to the control parameters via the dependence of the adjustable parameters.
Thermodynamic data
------------------
Information on the thermodynamics is needed for the computation of the number of dynamically correlated molecules and it must be considered with care in order to minimize the sources of uncertainty. The PVT data was obtained from the literature and included very high pressure results [@Cook94; @Cook93; @Bridgman32]. (This has been already described in Ref. \[\].) The density and temperature dependent isothermal compressibility $\kappa_T(T,\rho)$ and the thermal expansion coefficient $\alpha_P(T,\rho)$ were estimated from this data set.
Some results for the temperature dependence of the heat capacity $C_P$ of DBP are available in the literature [@Mizukami95; @Krueger99], but only at atmospheric pressure. We estimated from thermodynamic arguments and comparisons to other molecular liquids that the pressure variation of $C_P$ is negligible for DBP. This is in part due to the fact that the absolute value of $C_P$ is very high for DBP (about $440\,JK^{-1}mol^{-1}$ at $T_g$ and atmospheric pressure [@Mizukami95]). The variation for the constant volume heat capacity $C_V$ can in turn be estimated from the relation $C_V = C_P - T\alpha_P^2\,V/\kappa_T$ and is also found to be small (typically less than $10\,\%$). The jump of $C_P$ at $T_g$ is known at atmospheric pressure [@Mizukami95] and is about $150\,JK^{-1}mol^{-1}$. Unfortunately, no data are available for the jump of $C_V$ nor for that of $\kappa_T$ at the glass transition. (No data either are available for the pressure dependence of these jumps.) For other molecular glass-forming liquids, one observes that the heat capacity $C_P$ of the liquid at $T_g$ slightly increases with pressure: a few $\%$ for 3-methylpentane and 1-propanol up to $200\,MPa$ [@Takahara94], a few $\%$ for m-fluoroaniline up to $400\,MPa$ [@Alba91], and $10-12\,\%$ for toluene up to $400\,MPa$ [@Ter88]. However, the $C_P$ of the glass at $T_g$ has been found to also increase in 3-methylpentane and 1-propanol where it has been measured [@Takahara94], so that the heat capacity jump $\Delta C_{P,g}( P)$ barely changes with pressure in these systems.
Number of dynamically correlated molecules $N_{corr}$ along isobars and isotherms
=================================================================================
We are interested in evaluating the four-point dynamical susceptibility $\chi_4$ and the associated number of dynamically correlated molecules in the $NPT$ ensemble. From considerations about fluctuation-dissipation relations [@Berthier05; @Berthier07], two expressions may be used. The first one, $$\label{eq:eqNcorrP}
\chi_4^{NPT}(t)=\frac{k_BT^2}{c_P}[\chi_T^{NPT}(t)]^2+\chi_4^{NPH}(t)\, ,$$ where $c_P$ is the isobaric heat capacity per molecule, relates $\chi_4^{NPT}(t)$ to the three-point-dynamical susceptibility $\chi_T^{NPT}(t)=\left(\frac{\partial F(t)}{\partial T}\right)_P$, with a strictly positive residual term describing the fluctuations in the $NPH$ ensemble (where $H$ is the enthalpy). The second expression takes into account the contributions associated with temperature (at constant volume) and density (at constant temperature) separately: $$\begin{split}
\chi_4^{NPT}(t)=\frac{k_BT^2}{c_V}[\chi_T^{NVT}(t)]^2
+\rho^3 k_B T\kappa_T [\chi_\rho^{NPT}(t)]^2+\chi_4^{NVE}(t)\, ,
\end{split}
\label{eq:eqNcorrrhoT}$$ where $c_V$ is the isochoric heat capacity per molecule, $\chi_T^{NVT}(t)=\left(\frac{\partial F(t)}{\partial T}\right)_\rho$, $\chi_{\rho}^{NVT}(t)=\left(\frac{\partial F(t)}{\partial \rho}\right)_T$, and $\chi_4^{NVE}$ is the (strictly positive) four-point-dynamical susceptibility in the $NVE$ ensemble. For a more detailed discussion concerning these different quantities, see Refs. \[\].
In simulation studies on model glass-formers [@Berthier07], $\chi_4^{NPH}$ and $\chi_4^{NVE}$ were found to be small compared to the other terms in the viscous liquid regime at low temperature, becoming even negligible at the lowest temperatures. This was also indirectly confirmed by a comparison between results obtained from $(k_B T^2/c_P)[\chi_T^{NPT}(t)]^2$ and from the direct experimental measurement of a nonlinear dielectric susceptibility [@Ladieu]. It should nonetheless be stressed that, since $\chi_4^{NVE} \leq \chi_4^{NPH}$, considering the separate contributions associated with temperature and density, $(k_BT^2/c_V)[\chi_T^{NVT}(t)]^2+\rho^3k_BT\kappa_T [\chi_\rho^{NPT}(t)]^2$, provides a better approximation to the dynamical four-point susceptibility than $(k_B T^2/c_P)[\chi_T^{NPT}(t)]^2$.
In this work we focus on the numbers of dynamically correlated molecules $N_{corr}$ which may be defined as the maximum over time of the relevant time-dependent susceptibilities (or their absolute value if negative). This maximum takes place for a time of the order of the average relaxation time of the system $\tau_\alpha$. One therefore considers $N_{corr,4}^{NPT}=max_t[\chi_4^{NPT}(t)]$, $N_{corr,T}^{NPT}=\sqrt{k_B/c_P}\,T max_t\{\vert \chi_T^{NPT}(t)\vert \}$, $N_{corr,T}^{NVT}=\sqrt{k_B/c_V}\,T max_t\{\vert \chi_T^{NVT}(t)\vert \}$, and $N_{corr,\rho}^{NPT}=\sqrt{\rho^3 k_B T \kappa_T} max_t\{\chi_{\rho}^{NPT}(t)\}$. From Eqs. (\[eq:eqNcorrP\]) and (\[eq:eqNcorrrhoT\]), $N_{corr,4}^{NPT}$ may then be estimated either from $$\label{eq:eqNcorrP_N}
N_{corr,4}^{NPT} \gtrsim (N_{corr,T}^{NPT})^2$$ or from $$\label{eq:eqNcorrrhoT_N}
N_{corr,4}^{NPT} \gtrsim (N_{corr,T}^{NVT})^2+ (N_{corr,\rho}^{NPT})^2,$$ with the latter expression providing a better approximation than the former one.
We have first evaluated the four-point dynamical susceptibility along isobars through the simpler formula in Eq. (\[eq:eqNcorrP\]). To do so we have computed $\chi_T^{NPT}(t)$ from a numerical derivative of the Fourier transform of the Havriliak-Negami fits of the experimental dielectric data, as explained in detail in Ref. \[\]. We display the resulting estimate for $N_{corr,4}^{NPT}$ versus temperature $T$ on different isobars in Fig. \[fig2ab\] (a). The results are similar to those obtained at atmospheric pressure in previous papers [@Dalle07; @Dalle08; @Capaccioli08] and in Ref. \[\]: the number of dynamically correlated molecules is found to increase when the temperature is decreased along any isobar, indicating growing spatial correlations in the dynamics as one approaches the glass transition.
![ \[fig2ab\] **-a** Temperature dependence of the estimate of $N_{corr,4}^{NPT}$ from Eq. (\[eq:eqNcorrP\_N\]) along several isobars. **-b** Density dependence of $N_{corr,\rho}^{NPT}$ along several isotherms : closed squares $T=216\,K$, closed triangles $T=220\,K$, open squares $T=227\,K$, crosses $T=295\,K$ (the data at $T=295\,K$ are from Ref. \[\]). Both quantities have been computed by a numerical differentiation of the Havriliak-Negami fits of the dielectric spectra. ](fig2ab-eps-converted-to){width="\linewidth"}
We have also considered the contribution coming from the density-triggered fluctuations along isotherms, *i.e.* $N_{corr,\rho}^{NPT}(\rho)$. To compute this term, we have numerically differentiated with respect to density the Havrilliak-Negami fits (after converting the pressure data to density data) and included the necessary thermodynamic factors. The results are shown in Fig.\[fig2ab\] (b).
We observe that $N_{corr,\rho}^{NPT}$ increases with increasing density along any isotherm. The other contribution involving $\chi_T^{NVT}(t)=\left(\frac{\partial F(t)}{\partial \rho}\right)_T$ is virtually impossible to get directly from the dielectric data with a good accuracy, due to the fact that data are not collected along isochores. A different method will then be considered in the following section. Note finally that the span of most curves in Figs. 2 (a) and (b) is quite restricted. In spite of the large number of state points studied and the fine $T,P$ grid covered, the range of relaxation times that are directly accessible experimentally on given isobars and isotherms remains limited, especially at low temperature and high pressure.
$N_{corr}$ through the scaling description of the relaxation time
=================================================================
We have already stressed that considering separately the fluctuation contributions due to density and temperature provides a crisper estimate of the number of dynamically correlated molecules $N_{corr,4}^{NPT}$. However, to compute the various terms required \[see Eqs. (\[eq:eqNcorrrhoT\]) and (\[eq:eqNcorrrhoT\_N\])\], an alternative procedure to the direct differentiation of the experimental spectra must be employed. We have therefore relied on the, by now, well established scaling description of the density and temperature dependences of the relaxation time.
Density-temperature scaling law for relaxation time
---------------------------------------------------
The relaxation times $\tau_\alpha$ obtained from the Havriliak-Negami fits of the dielectric spectra \[$\tau_{\alpha}$ is equivalent to $\tau$ in Eq. (\[eq:havriliak\])\] are plotted versus pressure at different temperatures in the inset of Fig. \[fig3ab\] (a). The main panel of this figure shows that the same points, combined with literature data [@Cook93; @Paluch03], can be collapsed onto a master curve when plotted versus a scaling parameter $e(\rho)/T$: the function $e(\rho)$ is obtained from the best collapse of the whole data set and has the physical meaning (up to an overall constant factor) of a bare or noncooperative activation energy characterizing the dynamics at high temperature [@Alba02]. The fact that all the $\tau_\alpha$’s can be plotted on a master curve, $$\label{eq_scaling_tau}
\log \left(\frac{\tau_\alpha(\rho,T)}{\tau_0}\right)=\mathcal F(\frac{e(\rho)}{T})\, ,$$ with $\mathcal F$ a species-dependent scaling function, has been already proposed and verified for a great variety of glass-forming liquids and polymers [@Alba02; @Alba04; @Roland04; @Dreyfus04; @Roland05]. In many cases, the function $e(\rho)$ can be approximated by a power law, $\rho^x$, [@Alba04; @Roland04; @Dreyfus04]. Often, however, the density range under study is too limited to allow a discrimination between different functional forms for $e(\rho)$. Interestingly, in the case of DBP, data are available up to $1\,GPa$, and it was found in Ref. \[\] that a simple power law does not provide a good collapse of the data over the whole density range and should be replaced by a more general function, *e.g.* a polynomial function. This is illustrated in Fig.\[fig3ab\] (b). In such a case, the exponent $x$ can be generalized to a density dependent parameter $$\label{eq_scaling_x}
x(\rho)= \frac{\partial \log e(\rho)}{\partial \log (\rho)} \,,$$ which varies between 1.5 and 4 for DBP over the range of density covered. As discussed elsewhere[@Alba06] (and recently confirmed in the case of “strongly correlating” liquids [@Dyre]), there are no physical reasons for why the density-temperature scaling law of the relaxation time should always be expressible in terms of a simple power law. In this respect, the study presented in this paper is more general than previous ones [@Fragiadakis09; @Fragiadakis11] that have only considered liquids and ranges of pressure for which the exponent $x$ can roughly be taken as constant. We will see that this feature has some important consequences on the variation of the number of dynamically correlated molecules.
![ \[fig3ab\] **-a** Logarithm (base 10) of the (dielectric) relaxation time $\tau_\alpha$ (in units of $\tau_0=1 \rm{sec}$) of DBP measured at different pressures and temperatures versus the scaling parameter $e(\rho)/T$. The latter is adjusted to obtain the best collapse of all data points. The blue symbols denote our new set of data, the black ones our previous data from Ref. \[\], and the violet ones are for literature data from Refs. \[\]. Inset: All our data points plotted versus pressure. **-b** Bare activation energy $e(\rho)$ obtained from the collapse of relaxation time data (dotted line). For DPB the function cannot be simply described by a power-law dependence $\rho^x$: the full line represents the best fit with a constant $x=2.5$. ](fig3ab-eps-converted-to){width="\columnwidth"}
Density and temperature contributions to $N_{corr,4}^{NPT}$
-----------------------------------------------------------
The density-temperature scaling of the relaxation time provides a continuous parametrization of the latter over the whole domain of density and temperature and more specifically allows us to compute derivatives with respect to the temperature at constant density, an otherwise horrendous task experimentally. To further facilitate the computation of the various contributions to $N_{corr,4}^{NPT}$, we have simplified the form of the relaxation function $F(t)$ by fitting the result of the Havriliak-Negami parametrization to a stretched exponential, $F(t)= \exp [-(t/\tau_{\alpha})^{\beta}]$, with $\beta$ a state dependent parameter. This enables us to rewrite $N_{corr,T}^{NVT}$, $N_{corr,\rho}^{NPT}$, and $N_{corr,T}^{NPT}$ as $$N_{corr,T}^{NVT}=\sqrt{\frac{k_B}{c_V}}\, \beta\, e^{-1} \bigg [\frac{e(\rho)}{T} \mathcal F'(\frac{e(\rho)}{T}) \bigg ] \,
\label{eq:eqscaling_NcorrNVT}$$ $$N_{corr,\rho}^{NPT}=\sqrt{\rho k_B T \kappa_T}\, \beta\, e^{-1}
x(\rho) \bigg [\frac{e(\rho)}{T} \mathcal F'(\frac{e(\rho)}{T}) \bigg ] \, ,
\label{eq:eqscaling_Ncorrrho}$$ and $$N_{corr,T}^{NPT}=\sqrt{\frac{k_B}{c_P}}\, \beta\, e^{-1} \bigg [1+x(\rho) T \alpha_P\bigg]
\bigg [\frac{e(\rho)}{T}\mathcal F'(\frac{e(\rho)}{T}) \bigg ],
\label{eq:eqscaling_NcorrNPT}$$ where $\alpha_P$ is the thermal expansion coefficient, $\beta$ is the stretching parameter, $e^{-1}$ is the inverse of Euler’s mathematical constant, which should not be confused with the activation energy $e(\rho)$, $x(\rho)$ is defined in eq. (\[eq\_scaling\_x\]), and $\mathcal F'$ is the first derivative of the scaling function in Eq. (\[eq\_scaling\_tau\]). To derive the above equations, we have used the fact that the maximum over time of the various susceptibilities calculated from a stretched exponential description occurs for $t=\tau_{\alpha}$.
Finally, $N_{corr,4}^{NPT}$ is estimated from Eq. (\[eq:eqNcorrP\_N\]) as $$N_{corr,4}^{NPT}\simeq (N_{corr,T}^{NVT})^2 (1+R),
\label{eq:eqNcorr_full}$$ where $N_{corr,T}^{NVT}$ is given in Eq. (\[eq:eqscaling\_NcorrNVT\]) and $R$ is defined as [@Dalle07] $$R=\rho T \kappa_T c_V \,x(\rho)^2 \,.
\label{eq:eqR}$$ $R$ is therefore a measure of the relative importance of the density-induced contribution to the number of dynamically correlated molecules at constant pressure.
Before discussing the results obtained from Eq. (\[eq:eqNcorr\_full\]), it is worth assessing various sources of uncertainty in the calculation. We have thus compared the values of $N_{corr,\rho}^{NPT}$ and of $N_{corr,T}^{NPT}$ computed from Eqs. (\[eq:eqscaling\_Ncorrrho\]) and (\[eq:eqscaling\_NcorrNPT\]), respectively, to their counterparts obtained through a numerical derivative of the Havriliak-Negami parametrization of the dielectric data (see previous section). We find that the difference between the two sets are at most of the order of $10\, \%$ \[which amounts to at most $20\,\%$ in the square values entering in $N_{corr,4}^{NPT}$: see Eqs. (\[eq:eqNcorrP\_N\],\[eq:eqNcorrrhoT\_N\])\], with no systematic trend with density or temperature.
Additional errors come with the account of the thermodynamic input. First, there can be a change of an overall factor for the absolute value of the $N_{corr}$’s according to whether one considers the total fluctuations of energy, enthalpy or density (as rigorously enters in the lower bound for $N_{corr,4}^{NPT}$) or only that part of the fluctuations which is associated with the structural relaxation and can be ascribed to the values in excess to the glass ones (as physically motivated) [@Dalle07]. For instance, as described in section II-B, $\Delta C_P$ is about $1/3$ of $C_P$ for DBP at the glass transition, which, when replacing $C_P$ by $\Delta C_P$ in the estimate of the number of dynamically correlated molecules, introduces an overall multiplying factor of $3$. However, we are not interested in *absolute* magnitudes but in *relative* variations with $T$, $\rho$, or $P$. Concerning the latter, the situation is much more favorable, as one can estimate that the relative change in the various thermodynamic factors is never more than $10\,\%$ over the whole range of state points under study. To summarize this discussion, we conservatively conclude that a variation of the estimate of $N_{corr,4}^{NPT}$ that is beyond $20-30\, \%$ is physically significant. Smaller ones need to be interpreted with caution.
![ \[fig4\] 3D-plot for the number of dynamically correlated molecules $N_{corr,4}^{NPT}$ of DBP, estimated via Eqs. (\[eq\_scaling\_tau\]), (\[eq:eqNcorr\_full\]) and (\[eq:eqR\]), for a wide range of $T$ and $\rho$. $N_{corr,4}^{NPT}$ increases with increasing density and with decreasing temperature. The changes of color schematically indicate isochronic (constant relaxation time) conditions.](fig4-eps-converted-to){width="\columnwidth"}
Fig. \[fig4\] shows the variation with temperature and density of the number of correlated molecules $N_{corr,4}^{NPT}$ in DBP, as obtained from Eq. (\[eq:eqNcorr\_full\]) in conjunction with Eqs. (\[eq\_scaling\_tau\],\[eq\_scaling\_x\], \[eq:eqscaling\_NcorrNVT\],\[eq:eqR\]). $N_{corr,4}^{NPT}$ is found to increase when the temperature decreases or when density increases, as anticipated. The number of dynamically correlated molecules obtained through this method is furthermore always slightly larger than those calculated through the other, less efficient, bound in Eq. (\[eq:eqNcorrP\_N\]) or (\[eq:eqscaling\_NcorrNPT\]).
$N_{corr,4}$ along the glass transition line and other isochrones
=================================================================
From the 3-dimensional plot displayed in Fig. 4, one can follow the evolution of the number of dynamically correlated molecules along any chosen path in the $(T,\rho)$ plane. An interesting choice of paths is provided by the isochronic, *i.e.* equal relaxation time, lines. The most commonly considered among such lines is the glass transition line, where $T_g$ is defined by a given value of the relaxation time, say $\tau_{\alpha}=100\,sec$. It was reported in Ref. \[\] for 4 glass-forming liquids and polymers that the number of dynamically correlated molecules is uniquely determined by the relaxation time. If this were true in general, $N_{corr,4}^{NPT}$ should be constant along isochronic lines. As shown in Fig. \[fig5ab\] (a), this is however not valid in the case of DBP. We find in particular that $N_{corr,4}^{NPT}$ increases with pressure (and temperature) along the $T_g$ line, at least up to the highest pressure available (the evolution appears to saturate or reach a maximum for the highest pressures or temperatures considered). The same systematic trend is observed for all isochrones, although the increase in $N_{corr,4}^{NPT}$ becomes smaller as the relaxation time decreases \[see Fig. \[fig5ab\] (b)\]. We stress that the pressure range covered in the present study on DBP is significantly larger than in Ref. \[\]: here, we consider $P$ from the atmospheric value up to $1\,GPa$ (with a change in density of almost $25\,\%$) whereas $P$ varies from the atmospheric value to a few tens to a few hundreds $MPa$ in \[\].
![ \[fig5ab\] **-a** Evolution of the number of dynamically correlated molecules $N_{corr,4}^{NPT}$ along the glass transition line for DBP ($T_g$ is defined for a dielectric relaxation of $100\, sec$). **-b** Same as (a) but for various isochronic lines (the value of the $\log_{10}$ of the dielectric relaxation time in $sec$ is given in the caption)](fig5ab-eps-converted-to){width="\columnwidth"}
Two remarks are worth making in connection with the above result. First, the stretching parameter $\beta$ of the relaxation functions was shown to be constant along any isochronic line, and in particular along the $T_g$ line, in several glass-forming liquids [@Roland05; @Niss07]. The same observation applies here to liquid DBP. Secondly, the isochoric fragility $m_\rho$, defined as $m_\rho=\left(\frac{\partial \log(\tau_\alpha/\tau_0)}{T\partial (1/T)}\right)_\rho$ is also found to be constant along any isochronic line, as a result of the density-temperature scaling of the relaxation time [@Alba02; @Alba06]. The important consequence is that neither the stretching of the relaxation nor the dynamic fragility are correlated with the increase in the number of dynamically correlated molecules observed along the glass transition line and other isochrones.
As the stretching parameter $\beta$ and the isochoric fragility $m_{\rho}$ are essentially constant, whereas $c_V$ varies by $10\,\%$ along the glass transition line, it follows from Eqs. (\[eq:eqNcorr\_full\]) and (\[eq:eqscaling\_NcorrNVT\]) that the variation of $N_{corr,4}^{NPT}$ with pressure should be mostly due to that of the parameter $R$. This is indeed what is observed, as illustrated in Fig. \[fig6\]: $1+R$ increases by a factor of almost 3 (and $R$ by a factor of almost 4), which roughly corresponds to the increase in $N_{corr,4}^{NPT}$.
![ \[fig6\] Variation along the $T_g$ line of $1+R$, where the parameter $R$ quantifies the relative importance of density-induced versus temperature-induced contributions to $N_{corr,4}^{NPT}$. As for the number of dynamically correlated molecules, $1+R$ appears to saturate or reach a maximum at the highest accessible pressures (and therefore temperatures)](fig6-eps-converted-to){width="\columnwidth"}
The physical significance of this increase in $R$ is that the fluctuation effects triggered by density become increasingly dominant over those associated with temperature as pressure and temperature increase. We stress that this should not be confused with the relative importance of density effects (versus temperature effects) in the slowing down of the relaxation. While $R$ concerns the *fluctuations* around the average dynamics, the latter deals with the *average* dynamics (see the Introduction) and, in the scaling description of the relaxation time, it is measured by the factor $T \alpha_{P}(T,\rho)x(\rho)$ [@Alba06]. For DBP along the $T_g( P)$ line, $T \alpha_{P}$ is equal to $0.12$ at atmospheric pressure and decreases by about $25\,\%$ at high pressure, whereas, as already quoted, $x$ varies from 2.5 to 4. As a result, the factor $T \alpha_{P}x$ changes at the glass transition from $0.30$ to $0.36$ over the same pressure range. At $T_g( P)$, the influence of density on the average dynamics at constant pressure is thus significantly less than that of temperature and increases by only $20\,\%$ under pressure, a result very different from that found for $R$ in connection with the dynamical heterogeneities.
As the thermodynamic factor $\rho T \kappa_T c_V$ varies weakly with pressure along the glass transition line, the increase in $R$ can be assigned to the change in the activation energy parameter $x(\rho)$ \[compare with Eq. (\[eq:eqR\])\]. The latter indeed increases from $2.5$ to $4$ along the $T_g( P)$ line, which, once squared, explains the growth of $R$ and of $N_{corr,4}^{NPT}$. The present study therefore highlights the role of the deviation from simple power-law behavior of the bare activation energy $e(\rho)$ in the variation of the number of dynamically correlated molecules along the glass transition line. As a check, we have repeated for another molecular glass-forming liquid, ortho-terphenyl, the very same procedure as done here. This leads to a value of the number of dynamically correlated molecules that increases along the glass transition line with increasing pressure up to $400\,MPa$, but only by less than $20\,\%$. Accordingly, it was found that the parameter $x(\rho)$ can be taken as essentially constant over the same domain of pressure [@Dreyfus04; @Alba02].
Finally, we note that our finding (for DBP up to $1\,GPa$) of an increase of the number of dynamically correlated molecules, and as a consequence of the associated correlation length, along the $T_g( P)$ line is inconsistent with the theoretical predictions made on the basis of the Random First-Order Transition (RFOT) theory [@Xia00; @Lubchenko07]. In the RFOT theory, the dynamic length at $T_g$ is predicted to be $\xi =5.8\,a$ [@Xia00; @Lubchenko07], where $a$ is the diameter of an elementary component of the molecule or the polymer, referred to as a “bead”. This prediction, which only depends on the relative value of the relaxation time compared to a microscopic time, is claimed to be valid for all glass-formers under any thermodynamic conditions. Some uncertainty do exist as to how beads should be defined and operational procedures have been devised to compare different systems [@Xia00; @Lubchenko07]. However, when looking at the same liquid under different pressures, the bead should keep unchanged, which sidesteps the fuzziness of its definition. The fact that $N_{corr,4}$ increases along the glass transition line then directly contradicts the claim of the RFOT theory (except if one thinks that the relation between length scale and number of molecules should change with pressure, an *a priori* groundless assumption). One may of course take the variation by a factor of slightly less than 3 which is observed observed for DBP as “within the noise” of the theoretical predictions and the experimental procedures. Nevertheless, considering the care with which the various sources of uncertainty have been estimated in our calculation, it rather seems as a fact to be seriously taken into account in further theoretical and experimental investigations.
Conclusion
==========
In this work, we have characterized the spatial extent of the dynamical heterogeneities in a glass-forming liquid through an experimental determination of the number of dynamically correlated molecules obtained from dynamical susceptibilities. We have focused on the fragile liquid dibutylphatalate (DBP), for which, from our own measurements and from existing ones, a very large domain of temperature and pressure is experimentally covered for both dielectric spectroscopy and thermodynamics. We have calculated different estimates of the number of dynamically correlated molecules $N_{corr,4}^{NPT}$, including a crisp one making use of the separate contributions due to fluctuations associated with density and with temperature. We have shown that the range of the spatial correlations in the dynamics of DBP grow when approaching the glass transition along any thermodynamic path. In addition, we have found that $N_{corr,4}^{NPT}$ varies along the glass transition line (by a factor of about 3), and more generally along any isochronic line, as pressure (or equivalently temperature) increases, a result which is at odds with recent reports and theoretical predictions. The spatial extent of the correlations in the dynamics is further found to be uncorrelated with both the (isochoric) fragility of the system and the stretching of the relaxation function. This contradicts the naive view of the dynamics of glass-forming liquids that ties cooperativity, fragility, stretching and growing heterogeneity altogether.\
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We propose a supervised machine learning approach for boosting existing signal and image recovery methods and demonstrate its efficacy on example of image reconstruction in computed tomography. Our technique is based on a local nonlinear fusion of several image estimates, all obtained by applying a chosen reconstruction algorithm with different values of its control parameters. Usually such output images have different bias/variance trade-off. The fusion of the images is performed by feed-forward neural network trained on a set of known examples. Numerical experiments show an improvement in reconstruction quality relatively to existing direct and iterative reconstruction methods.'
author:
- 'Joseph Shtok, Michael Zibulevsky and Michael Elad, Fellow, IEEE. [^1]'
bibliography:
- 'ct\_neural\_net\_arxive.bib'
title: 'Spatially-Adaptive Reconstruction in Computed Tomography using Neural Networks'
---
=1
Computed Tomography, Low-Dose Reconstruction, Neural Networks, Supervised Learning, Filtered-Back-Projection (FBP).
Introduction
============
Computed tomography (CT) imaging produces an attenuation map of the scanned object, by sequentially irradiating it with X-rays from several directions. The integral attenuation of the X-rays, measured by comparing the radiation intensity entering and leaving the body, forms the raw data for the CT imaging. In practice, these photon count measurements are degraded by stochastic noise, typically modeled as instances of Poisson random variables. There are also other degradation effects due to a number of physical phenomena – see [*e.g.*]{} [@RiBi06] for a detailed account.
Given the projection data, known as the sinogram, a reconstruction process can be performed in order to recover the attenuation map. Various such algorithms exist, ranging from the simple and still very popular Filtered-Back-Projection (FBP) [@ShKr78], and all the way to the more advanced Bayesian-inspired iterative algorithms (see e.g., [@ElFe02; @WaLi06b]) that take the statistical nature of the measurements and the unknown image into account. Since CT relies on X-ray, which is an ionizing radiation known to be dangerous to living tissues, there is a dire and constant need to improve the reconstruction algorithms in an attempt to enable reduction of radiation dose.
In this work we are concerned with the question of image post-processing, following the CT reconstruction, for the purpose of getting better quality CT image, thereby permitting an eventual radiation-dose reduction. The proposed method does not focus on a specific CT reconstruction algorithm, nor the properties of the images it produces. Instead, we take a generic approach which adapts, in an off-line learning process, to any such given algorithm. The only requirement is the access to design parameters of the reconstruction procedure which influence the nature of the output image, such as the resolution-variance trade-off.
We aim to exploit the fact that any reconstruction algorithm can provide more image information if instead of one fixed value of a parameter (or a vector of them) controlling the reconstruction, few different values are used (leading to different versions of the image). In order to extract this information from a collection of image versions, we use an Artificial Neural Network (ANN) [@Hayk94]. The proposed method can also use other techniques for computing a non-linear multivariate regression function.
Neural networks have been used extensively in medical imaging, particularly for the purpose of CT reconstruction (see Section \[sect:bkgnd-ANN\] for an overview). Here we propose a new constellation, which consists in a local fusion of the different image versions, aimed at an improved reconstruction quality. We use a set of intensity values from a neighborhood of a pixel $q$, taken from the different versions, as inputs to the network, and train it to compute a (smaller) neighborhood of $q$ which values are as close as possible (in Mean-Squared-Error or other sense) to those found in the reference image. As we show in this paper, the proposed approach enables an improvement of the variance-resolution trade-off of a given reconstruction algorithm, i.e. producing images with a reduced amount of noise without compromising the spatial resolution and without introducing artifacts.
This paper is organized as follows: Sections \[sect:CT\] and \[sect:bkgnd-ANN\] are devoted to a brief and general discussion on CT scan/reconstruction and artificial neural networks. Readers familiar with these topics may skip and start reading at Section \[sect:RVTF-concept\], where the core concept of this work is detailed. This section also contains an illustration on one-dimensional piece-wise constant signals, where it is easy to appreciate the action of the proposed algorithm and the effect of local fusion performed by a neural network. In the sequel, the proposed method is implemented on two tomographic reconstruction methods: boosting the Filtered Back-Projection (FBP) is presented in Section \[sect:RVTF-FBP\] and the same for Penalized Weighted Least-Squares (PWLS) method is described in Section \[sect:RVTF-PWLS-it\]. We conclude this work by discussing the computational complexity of the proposed algorithms in Section \[sect:complex\], and a summary of this work and its potential implications in Section \[sect:summary\].
Background on Computed Tomography {#sect:CT}
=================================
Mathematical Model of CT Scan {#sect:model-scan}
-----------------------------
In the process of a CT scan, the object is radiated with X-rays. In this work we consider a reconstruction in a plane from rays incident only to this plane (the two-dimensional tomography). From the mathematical point of view, the considered object is a function $f(x)$ in the plane, which values are the attenuation coefficients of composing materials (i.e., tissues). When the measured photon counts are perfect, the measurements are directly related to the X-ray transform of the function $f(x)$ as a collection of all the straight lines passing through the object, and the value associated with each such line is the integral of $f(x)$ along it. In two dimensions, and under the assumption of a full rotated and parallel beam scan, this coincides with the Radon transform ${\mathbf}{R}f$.
Let $\ell$ be a straight line from an X-ray source to a detector. The ideal photon count $\lambda_{\ell}$, measured by the detector is related to ${\mathbf}{R}f$ via the function $$\label{eq:model1}
\lambda_{\ell} = \lambda_0e^{-[{\mathbf}{R}f]_{\ell}},$$ where $\lambda_0$ is the blank scan count. The scanned data is stored in a matrix which columns correspond to the sampled angle $\theta$; each such column is referred to as a “view” or a “projection”, and is acquired, schematically, by a parallel array of X-rays passing through the object at the corresponding angle. The rows of the matrix, corresponding to the sampled values of the distance $s$, are called the “bins” of each projection. According to the Equation (\[eq:model1\]), for reconstruction purposes the measurements data undergoes the log transform $$\label{eq:g-y-connection}
g_{\ell}=-log(\dfrac{\lambda_{\ell}}{\lambda_0}).$$ We refer to $g$ as the *sinogram*. The name indicates that every point in the image space traces a sine curve in this domain. Since the sinogram matrix is the (sampled) Radon transform of the original image $f(x)$, a discrete version of the image can be reconstructed by applying the inverse Radon transform (see Section \[sect:BK-recon\]).
Each measured photon count $y_{\ell}$ is typically interpreted as an instance of the random variable $Y_{\ell}$ following a Poisson distribution $ Y_{\ell} \sim Poisson(\lambda_{\ell}) $[@RiBi06; @ThBo06; @Bors09]. This reflects the photon count statistics at the detectors [@Hans81]. For a random variable $X\sim Poisson(\lambda)$, the standard deviation $\sigma_X$ satisfies $\sigma_X=\sqrt{{\mathbb}{E}(X)}$, and therefore the signal-to-noise ratio of $X$, $SNR(X)={\mathbb}{E}(X)/\sigma_X = \sqrt{{\mathbb}{E}(X)}$ monotonously increases with its expected value.
In the sinogram domain, the standard deviation of the error between the ideal sinogram and the one computed from the measurements, $\hat{g}-\hat{\bar{g}}$, is $\lambda^{-1/2}$ [@Maco83], and this is well approximated by $\hat{y}^{-1/2}$ [@LiLi04]. In Figure \[fig:BK-demo1\] we display a sinogram matrix and the corresponding Poisson noise image. Below, one can observe the resulting reconstruction artifacts produced by the standard FBP algorithm (see next sub-section). The sinogram error image has a high-energy regions where the sinogram values are relatively high; this corresponds to the predicted behavior of the noise variance. The reconstruction from the noisy sinogram is contaminated with anisotropic noise, mainly in the form of streaks. Their appearance is related to large errors in sinogram values.
![Upper row, left to right: Exact sinogram, and the absolute-valued difference between this and the corrupted sinogram (darker shade corresponds to higher error). Lower left: true (reference) image, corresponding to the true sinogram. Lower right: reconstruction from the noisy data with FBP algorithm. Images are displayed in HU window $[-220,350]$.[]{data-label="fig:BK-demo1"}](Figures/BK-noise_demo1.pdf "fig:"){width="1\columnwidth"}\
![Upper row, left to right: Exact sinogram, and the absolute-valued difference between this and the corrupted sinogram (darker shade corresponds to higher error). Lower left: true (reference) image, corresponding to the true sinogram. Lower right: reconstruction from the noisy data with FBP algorithm. Images are displayed in HU window $[-220,350]$.[]{data-label="fig:BK-demo1"}](Figures/BK-noise_demo2.pdf "fig:"){width="1\columnwidth"}
Reconstruction Algorithms for Computed Tomography {#sect:BK-recon}
-------------------------------------------------
There are various reconstruction algorithms that aim at computing the attenuation map of the scanned object from its projections. In this paper we shall refer and work with two such algorithms: (i) the Filtered Back-Projection (FBP) ([@ShKr78], which is a direct Radon inversion approach. This is a popular technique despite its known flaws; and (ii) an iterative reconstruction algorithm that takes the statistical nature of the unknown and the noise into account (e.g. [@ElFe02]). Bayesian methods achieve better image quality than the direct Radon inversion, at the expense of longer processing time. We now describe these two methods is somewhat more details.
[**Filtered-Back-Projection Method:**]{} Mathematically, FBP is the linear operator of the form $$\label{eq:FBP-formula}
\mathbf{T}_{\text{FBP}}=\mathbf{R}^*\mathbf{F}_{low}\mathbf{F}_{RL}.$$ Here ${\mathbf}{R}^*$ is the adjoint of the Radon transform, known in the literature as “back-projection”. $\mathbf{F}_{RL}$ is a 1-D convolution filter, applied to each individual projection (column in the sinogram matrix). It uses the Ram-Lak kernel $k$ [@RaLa71], defined in the Fourier domain by $\hat{k}(\omega)=|\omega|$, and $\mathbf{F}_{low}$ is a low-pass filter which prevents the noise amplification at high frequencies, typical for the Ram-Lak action. In clinical CT scanners, the parameters of $\mathbf{F}_{low}$ are tuned for specific needs of the radiologist: different preset values are chosen to view bones, soft tissues, high contrast/smooth images, specific anatomical regions, etc.
Without the low-pass filter, the FBP is an exact inverse of the Radon transform in the continuous domain [@NaWu01] for the noiseless case. Moving from theory to practice, the FBP algorithm does not perform very well. The low-pass 1-D convolution filter in the sinogram domain is not an effective remedy for the projections noise. The problem of photon starvation manifests through outlier values in the sinogram, which propagate to the output image in the form of streak artifacts. They corrupt the image contents and jeopardize its diagnostic value. Those artifacts can be explained as follows: each measured line integral is effectively smeared back over that line through the image by the back-projection; an incorrect measurement results in a (partial) line of wrong intensity in the image. Typically, the streaks radiate from bone regions or metal implants.
[**Statistically-Based Method:**]{} The relation between $f$, the sought CT image, and the vector of measured counts $y$ can be described as $$\label{eq:PL-meas}
\log(y)={\mathbf}{A}f+e,$$ where ${\mathbf}{A}$ approximates the Radon transform and models the scan process in reality. The additive error $e$ (which also depends of $f$) stems from the statistical noise. In the Bayesian framework, the reconstruction is performed by computing the Maximum a-Posteriory (MAP) estimate of the image $$\label{eq:MAP}
\tilde{f} = \arg\max_f{\mathbf}{P}(f|y) =\arg\max_f\dfrac{{\mathbf}{P}(y|f){\mathbf}{P}(f)}{{\mathbf}{P}(y)}.$$ For CT, an accurate statistical model for the data is quite complicated and is often replaced by a Gaussian approximation with a suitable diagonal weighting term whose components are inversely proportional to the measurement variances. This leads to a penalized weighted least-squares (PWLS) formulation, see e.g.[@RamaniFessler2012] $$\label{eq:PWLS}
\tilde{f} = \arg\min_f \| \log(y) - {\mathbf}{A}f \|_D +\beta R(f),$$ where $ \| u\|_D=u^T {\mathbf}{D}u$, ${\mathbf}{D}$ is a diagonal matrix of weights, which in simplistic model are proportional to photon counts $y$; The penalty term $R(f)$ also referred to as the *prior*, expresses assumptions on the behavior of the clean CT image. In [@Fess06] this expression is chosen as $$\label{eq:BK-penalty}
R(f)= \sum_q\sum_{k\in {\mathcal}{N}(q)}\psi_\delta(f_q-f_k),$$ where for each image location $q$, a scalar function $\psi_\delta(x)$ is the convex edge-preserving Huber penalty $$\label{eq:FR-huber}
\psi_\delta(x)=\left\{ \begin{array}{ccc}
\dfrac{x^{2}}{2}, & |x|<\delta \\ \nonumber
\delta|x|-\dfrac{\delta^{2}}{2},& |x|\geq\delta
\end{array}\right\},$$ In order to minimize , we have used the L-BFGS optimization method [@NoWr06]. The Matlab/C implementation of the algorithm is the courtesy of Mark Schmidt.
Artificial Neural Networks (ANN) {#sect:bkgnd-ANN}
================================
For completeness of this paper, we provide here a brief background on ANN, and in particular their role in CT and medical imaging. ANN, mimicking after the biological networks of neurons which comprise the nervous system, are intensively used in many domains of Computer Science. In this work we focus on the multi-layer feed-forward ANN with no cycles. This is best represented by a directed, weighted graph which has an array of input nodes (data inputs), inner nodes (neurons) implementing specific (linear or non-linear) scalar functions, and another array of output nodes. The input argument of each neuron is the weighted sum of all its inputs, where the weights are associated with the edges. Those weights are learned during the network training and, effectively, define the regression function produced by the ANN.
More specifically, the first layer consists of $m$ inputs, coming from the outside world; then $N_l$ neurons are situated in the $l$-th layer ($l>1$), and the last one contains $n$ output nodes. Each input $x_i$ is connected to each neuron $j$ in the second (hidden) layer by a weighted edge with weight $w^1_{i,j}$. The output of each neuron is connected to the input of every $k$-th neuron in the second layer by the weight $w^2_{j,k}$, and so on. Finally, each neuron of the last layer is connected to the output $y_s$ with a weight $v_{s,j}$. We denote by $\sigma$ the function implemented in each neuron. There is a number of popular choices for this function, for instance $\sigma(x)=\tanh(x)$.
For example, here is the explicit definition of a network with one hidden layer: $$\label{eq:BK-ANN}
y(x;w,v,b) = \sum_j v_j\sigma\left(\sum_i w_{i,j}x_j+b_j\right).$$ The weights $\{w,v,b\}$ define the multi-variable regression function $y=y(x)$ which approximates any continuous function implied by the set of training examples[^2]. A training set for the network comprises of a collection of examples $(X^k,Y^k)$, where $X^k$ is the vector of inputs and $Y^k$ is the true output related to this vector. Training the network consists of optimizing the weights $\{w,v,b\}$ for a minimal error, $$\label{eq:nn_train}
(w,v,b) = \arg\min_{w,v,b}\sum_{k=1...K} E\left(y(X^k; w, v,b), Y^k\right),$$ where the sum is over the training set, and E(a,b) is an error measure of some sort (e.g. $E(a,b)=(a-b)^2$). The popular method for solution of this problem is the iterative backpropagation method [@RuHi86]. A scheme of such network is depicted in Figure \[fig:ANNsc\].
![A scheme of a multi-layer feed-forward ANN.[]{data-label="fig:ANNsc"}](Figures/BK-ANN_multi.pdf){width="\columnwidth"}
Since the development of the back-propagation algorithm for ANN in mid-eighties, the image processing community (among others) has attained a powerful tool to attack virtually any regression or discrimination task. Among the wealth of applications neural networks found in this area (see [@EgRi02] for a broad and comprehensive overview), some were designed for medical imaging. As such, Hopfield ANN were used for computer-aided screening for cervical cancer [@CeNa00], breast tumors [@BrKa00] and segmentation [@ChLi96]. ANN are also used for compression and classification in cardiac studies [@HiMa94] and ECG beat recognition [@OsHo01]. Tasks of filtering, segmentation and edge detection in medical images are addressed with cellular ANN in [@AiAi01]. Our group has used neural networks for optimal photon detection in scintillation crystals in PET [@nnpet].
As for reconstruction problems, a series of works has appeared in which the ANN replaces the overall reconstruction chain by learning the net contribution of all detector readings to each pixel in the image. For Electron Magnetic Resonance (EMR), such an algorithm is proposed in [@DuKr07]. Floyd et. al. have used this approach for SPECT reconstruction [@Floy91] with feed-forward networks and also for lesion detection in this modality [@FlTo92]. We remark that such naive application of the ANN for reconstruction is limited to low-resolution $n\times n$ images, since the network must have ${\mathcal}{O}(n^2)$ inputs and outputs. For instance, in [@DuKr07], a $64\times 64$ image is reconstructed. Application of ANN for SPECT reconstruction was also studied by J. P. Kerr and E. B. Bartlett [@KeBa95a; @KeBa95b].
Imaging modalities like PET and SPECT, where low-resolution images are produced, are a natural domain for ANN application. However, some works tackle also the problem of CT reconstruction where the image size is larger. Ref. [@Cier08] proposes using a neural network structure with training based on a minimization of a maximum entropy energy function. Reconstruction in Electrical Impedance Tomography was treated with ANN in [@AdGu94]. Another variety, an Electrical Capacitance Tomography and an ANN-based reconstruction method for it, are described in [@NoHo97].
Despite the abundance of applications, there is still place for innovation in the domain of ANN application for medical imaging. First, the CT reconstruction problem is rarely attacked with this tool due to the high dimensions of raw data and the resulting images, which render the naive application of ANN as the black box converting measurements to image unfeasible. Indeed, in our work we do not propose such a scheme per se – rather, our ANN is employed to perform a locally-adaptive fusion of a number of image versions, produced by a given reconstruction algorithm upon using different configurations. This brings us naturally to the next section where we describe our algorithm.
The Proposed Scheme {#sect:RVTF-concept}
===================
Local Fusion with a Regression Function
---------------------------------------
We consider the general setup of the non-linear inverse problem. Assume we are given the measurements vector $y$ of the form $$\label{eq:RV-problem}
y = {\mathbf}{H}x+\xi,$$ where ${\mathbf}{H}$ is some transformation, $\xi$ represents the noise, and $x$ is the signal to be recovered. Assume further that ${\mathbf}{T}_{{\mathbf}{p}}$ is some restoration algorithm designed to recover $x$ from this type of measurements, i.e., $$\bar{x}_{{\mathbf}{p}} = {\mathbf}{T}_{{\mathbf}{p}}(y)$$ The scalar parameter ${\mathbf}{p}$ controls the behavior of ${\mathbf}{T}$ and therefore influences certain characteristics of the estimate $\bar{x}$. For example, when ${\mathbf}{p}$ is responsible for variance-resolution tradeoff of the algorithm, the estimate $\bar{x}_{{\mathbf}{p}}$ may be obtained with different noise levels and corresponding spatial resolution characteristics.
The described situation is common in many signal/image processing scenarios. As a basic example, we consider a simple image denoising algorithm, which recovers the signal $x$ from noisy measurements $y= x+\xi$ by a shift-invariant low-pass filter, realized as a 2-D convolution with prescribed kernel. For some fixed shape of this kernel (say, a simple boxcar function or a 2-D Gaussian rotation-invariant kernel), its width (spread) can be parameterized by a scalar variable ${\mathbf}{p}$. A wider such kernel will perform a more aggressive noise reduction, by averaging the noisy signal over a larger area, at the cost of reducing the spatial resolution.
A second, and more relevant example to this work, is from the domain of CT reconstruction. Recovery of the attenuation map is classically performed by the Filtered Back-Projection algorithm. The latter involves a 1-D low-pass filter, applied to the individual projections. As in the above example, the cut-off frequency of this filter controls the variance-resolution properties of the reconstructed image. In these examples, and also in a general such situation, no single value for the parameter ${\mathbf}{p}$ makes the best of the processing algorithm. For different signals, different values may be optimal in the sense of MSE or other quality measure. Indeed, in the same image, computed with two different values of ${\mathbf}{p}$, different regions will get the best treatment by different values of ${\mathbf}{p}$. For each specific case, ad-hoc considerations for tuning this scalar parameter are applied.
In the domain of non-parametric statistics, there is a noise reduction algorithm with proven near-optimality that devises a switch rule for selecting at each location of the signal an appropriate local filter [@GoNe97]. In effect, the signal is processed by a low-pass filter adaptive to the local signal smoothness. In the context of our discussion, one can say that this algorithm performs a fusion of a number of filtered versions of a signal with varying filter parameter. The switch rule, developed for this adaptive signal smoothing, is based on the balance of the stochastic and structural noise components and model assumptions, and as such, it is very difficult to devise. Moreover, better output may be obtained if we allow to use some combination of the given image versions in each pixel, rather than selecting one of them alone. To our knowledge, no mathematical theory offering a descriptive rule for such local fusion is available for signal estimators, used for denoising or CT reconstruction.
Borrowing from the above switch-rule idea between filters, the solution we propose for the problem described above is a local fusion of a sequence of estimates $\bar{x}_{{\mathbf}{p}_1},...,\bar{x}_{{\mathbf}{p}_N}$ with a specific regression function, learned on a training dataset consisting of similar cases. Among known regression methods, we choose to work with ANN, due to their strong adaptivity and generalization properties [@Hayk94]. The supervised learning is done with a training set of examples: For each location in the processed signals, the features (input vector) are sample values extracted from the corresponding location in the sequence of reconstructed versions for this signal. The output is a small region of sample in the desired destination signal. Contemporary training algorithms employ error back-propagation to optimize the objective function, measuring the discrepancy between the correct output values and those predicted by the ANN [@RuHi86]. In our work we employ the Matlab Neural Network toolbox; the training was performed with the Levenberg-Marquardt algorithm [@Marq63; @Gavi11]. Our networks consist of two hidden layers. We use the function $\sigma(x)=x/(1+|x|)$, which has similar properties to the classical sigmoid and is computationally cheaper and is more robust to saturation caused by large arguments.
In this work, the outlined general concept is specialized to reconstruction algorithms for CT. Specifically, we consider representatives of the two types of those algorithms: the direct FBP and the iterative PWLS (Section \[sect:BK-recon\]) methods. For FBP, we propose making a sweep over the cut-off frequency of its low-pass filter in the sinogram domain. This parameter controls the noise-resolution tradeoff and has a major influence on the visual impression of the resulting images. For the iterative PWLS algorithm, a sequence of images is extracted along its execution by saving a version of the CT result every few iterations. In following sections we illustrate this approach on a simple 1-D denoising problem and work out a number of applications for CT reconstruction algorithms, as detailed above. Along the way, we discuss the choice of training set and design of features extracted for the ANN.
An Example: ANN Fusion for 1-D Signal Denoising {#sect:RVTF-PWC}
-----------------------------------------------
To illustrate the proposed concept, we begin with the simple signal denoising algorithm as mentioned above. We assume that the original signal is 1-D piece-wise constant (PWC). This choice is beneficial for the test we are about to present, since random PWC signals can easily be generated for training/testing purposes, and the effect of low-pass filter denoising is easily observed. We generate such a signal $x$ of length $n$ by choosing $n/30$ step locations uniformly in random, and choosing the intensity value for each step uniformly at random as well, in the \[0,1\] segment.
Assume that such a signal $x$ has been created and is contaminated with i.i.d. Gaussian noise $\xi \sim{\mathcal}{N}(0,\sigma_n)$ with $\sigma_n=0.06$. For the noise reduction, we perform a convolution of $y=x+\xi$ with a Gaussian kernel $G(p)={\mathcal}{N}(0,p)$. For some chosen values of the standard deviation $p=p_1,...,p_{8}$ we obtain the sequence of estimates $$\hat{x}_i = y*G(p_i),\;\;\;i=1,...,8.$$ In Figure \[fig:RVTF-PWC-1\] we display an instance of such a signal, the corresponding noisy version, and a number of signal estimates obtained with convolution filters of different widths.
![Left to right, top to bottom: clean train signal $x$, noisy signal $y$, and several restored signals $\hat{x}_i$ obtained by convolution with a Gaussian kernel of increasing width.[]{data-label="fig:RVTF-PWC-1"}](Figures/RVTF_PWC_1.pdf){width="1\columnwidth"}
In this setup we train the ANN for a better signal restoration. For each location $q$ in $y$, we extract a set of small neighborhoods of the same location $q$ from each of the signals $\hat{x}_1,...,\hat{x}_K$. Those are concatenated into one vector which serves as the ANN input. Specifically, we take a $11$-samples window from each processed signal in the sequence of $K=8$ signals. Thus, overall the feature vector for each location is of size $8\cdot11=88$ samples. In the training stage, every such vector is matched with a label – the correct value $x(q)$, which is provided to the ANN as the desired output.
For the training procedure we generate a signal $x$ of size $n=2\cdot 10^4$ (=number of training samples) and extract the training data as described above. The obtained ANN is tested on another signal of length $300$, randomly generated with the same engine. In Figure \[fig:RVTF-PWC-3\] such test results are presented. The neural network has improved the SNR of the best linear estimate from $19.85$dB to $26.18$dB, and this difference is observed in the fact that the ANN estimate fits the original signal much closer. The SNR values are calculated over an interval of $200$ samples in the center of the test signal, so as to avoid boundary problems.
![Test results: a zoom-in on a portion of the clean signal $x$, the noisy one $y$ ($17.67$dB), the SNR-best linearly restored signal $\hat{x}_{i}$ ($19.85$ dB), and the restoration by the ANN ($26.18$dB).[]{data-label="fig:RVTF-PWC-3"}](Figures/SP_PWC_2b_ver2.pdf){width="0.8\columnwidth"}
The presented algorithm has various design variables: the number, shape and width of the applied filters, the size and structure of the neural network, the structure of a input vector for each example (set of features). The questions of algorithm design will be pursued in the following sections, where CT reconstruction algorithms, relevant to our study, are invoked in the similar setup of performance boosting by local ANN fusion.
Error Measures
--------------
Just before we conclude this section and move to present the specific details of boosting CT reconstruction algorithms, we should discuss the choice of the error function to use in the learning process, and the error measure to use when evaluating the quality of the reconstruction.
[**C.2 Training Risk**]{}
In the supervised learning procedure, we design the ANN weights so as to minimize the regression error between the ANN output and the desired labels (training output data). In many cases, the natural choice for this function would be the Mean-Squared-Error (MSE). However, in CT, we should contemplate whether MSE is the proper choice to use. Consider a homogeneous region in a CT image (corresponding to some tissue) with a small detail of a different yet similar intensity (a cavity or a lesion). The MSE penalty paid by an over-smoothing reconstruction filter that blurs this lesion is small, and therefore such faint details may be lost while leading to better MSE. The remedy for this problem could be to penalize not only for the difference in intensity values between the reference image $f_0$ and the reconstruction $\tilde{f}$ , but also for the difference in the derivatives of these two images. Alternatively, We can weight the training examples so as to boost the importance of such faint edge regions, at the expense of more pronounced parts of the image, where the edges are sufficiently strong. In this spirit, building on the general error term written in Equation (\[eq:nn\_train\]), we propose to use $$\begin{aligned}
\label{eq:grad-penalty}
\theta^* & = & \arg\min_{\theta}\sum_{k=1...K} E\left(y(X^k,\theta),Y^k\right) \\
& = & \arg\min_{\theta}\sum_{k=1...K} \rho_k \cdot \left(y(X^k,\theta)-Y^k\right)^2. \nonumber\end{aligned}$$ In this expression $(X^k,Y^k)_k$ is the training data consisting of pairs of feature (input) vectors and their desired label (output), and the function $y(X^k,\theta)$ is the output of the ANN, governed by its control parameters $\theta$. This is a simple weighted MSE, and the idea mentioned above is encompassed in the choice of $\rho_k$, the scalar weights assigned to the training examples. In our work we have chosen $\rho_k$ to be zero for examples having a very low variance in the input image, which correspond to air regions. Specifically, the threshold is set to $10^{-6}$ times the maximal variance of $X^k$. A zero weight is also assigned to all the examples where the accumulated gradient over the input patch (in the idea image) is above $2\%$ of its maximal value. The later pruning is introduced in order to avoid the bias of the very strong bone-flesh, flesh-air edges in the training process. As for the remaining examples, we assign their weight to be proportional to the accumulated gradient of the patch (again, in the ideal image). This way, the remaining informative edges get a more pronounced effect in the learning procedure.
[**C.2 Quality Assessment**]{}
The quality measures of CT images used in this study, are the following:
- **Signal-to-Noise Ratio** (SNR), defined for the ideal signal $f$ and a deteriorated version $\hat{f}$ by SNR$(f,\hat{f})=-20 log_{10}(\|f-\hat{f}\|_2/\|f\|_2)$. In practice, we consider the signal $\hat{f}$ up to a multiplicative constant and compute $$SNR(f,\hat{f})=\max_{\alpha}-20
log_{10}(\|f-\alpha\hat{f}\|_2/\|f\|_2).$$ To make the error measurement more meaningful, the SNR is only computed in the image region where the screened object resides, ignoring the background area. We have used an active contour technique to find the object region in the image; specifically we have used the Chan-Vese method [@ChVe01].
- **Windowed Signal-to Noise Ratio**. The dynamic range of the HU values in a CT image is very large, from $-1000$ for air to $1500-2000$ for bones. Often, the diagnostic interest lies in the soft tissues, the HU values of which are near zero (HU of water). For axial sections of thighs, we chose (by a criterion of best visibility ) the window of $[b_1=-220, b_2=350]$ HU; our algorithms are tuned for best reconstruction in this HU range. Therefore, an appropriate SNR measurement considers only the regions in the image that fall in this range. Technically, the reference image $f$ and the noisy image $\hat{f}$ are pre-processed before the standard SNR is computed by projecting values lower or higher than $b_1$ and $b_2$ respectively to these values.
- **Structured Similarity** (SSIM) measure [@WaBo04]. This measure of similarity between two images comes to replace the standard Mean Squared Error (the expression $\|f-\hat{f}\|_2$ appearing in the SNR formula), which is known for its problamatic correlation with the human visual perception system (see [@WaBo04] and the references 1-9 therein). SSIM compares small corresponding patches in the two images, after a normalization of the intensity and contrast. The explicit formula involves first and second moments of the local image statistics and the correlation between the two compared images. In our numerical experiments, we use the Matlab code provided by the authors of [@WaBo04], which is available at $\sim$ .
- **Spatial resolution measure**: the spatial resolution properties of a non-shift-invariant reconstruction method should be characterized using a local impulse response (LIR) function, which replaces the standard point-spread function [@FeRo96]. We compute the LIRs by placing sharp impulses (single pixel) in many random locations in the reference image and by taking the difference between the reconstructed images, scanned with or without the spikes. For each LIR, the Full-Width Half-Maximum (FWHM) value is computed as follows: first, the 2-D image matrix of the response function is resized into an image larger by $\times 16$ in each axis, in order to reduce the discretization effect. Then, the number of pixels with intensity higher than half-maximum is counted and divided by the refinement factor of $256$. This is the FWHM resolution measure at the specific location.
FBP Boost – Algorithm Design {#sect:RVTF-FBP}
============================
The Low-Pass FBP Filter Parameters
----------------------------------
The method of local fusion, advocated in the previous section, is now applied to the standard Filtered Back-Projection (FBP) algorithm for CT reconstruction. The fusion is performed over the parameters of the low-pass sinogram filter, applied before the Back-Projection. This one-dimensional low-pass filter is realized as a multiplication with the Butterworth window $H$ in the Fourier domain, defined by $$\label{eq:SP-Butterworth}
|\hat{H}(\omega)| = \left(1+\left(\dfrac{\omega}{\phi_0}\right)^{2p}\right)^{-1/2}.$$ We sweep through the range of the parameter $\phi_0$ (expressing the cut-off frequency of the filter), thus changing the resolution-variance tradeoff of the FBP. We also change the parameter $p$, which controls the steepness of the window roll-off. While $\phi_0$ controls the amount of blur introduced during the reconstruction, the parameter $p$ influences the texture of reconstructed image.
In Figure \[fig:RV-FBP1b\] we show the reconstruction for a fixed value of $p=3$ and an increasing cut-off frequency $\phi_0$. Visually, the strong low-pass filter produces a cleaner image (which also have a higher SNR), but looses in the spatial resolution. The displayed sequence corresponds to values $\phi_0=[0.4, 0.8, 1.15, 2.0, 120 ,\infty]$ (the last corresponds to no filter).
![FBP reconstruction with different cut-off frequency value. Upper to lower, Left to right: $\phi_0= [0.4, 0.8, 1.15, 2.0, 120,\infty]$ (the last image is compute without the low-pass filter).[]{data-label="fig:RV-FBP1b"}](Figures/SP-FBP-filter-tune2.pdf){width="1\columnwidth"}
After testing various combinations, we chose to use only three FBP images with cut-off frequencies $\phi_0= [0.4, 1.15, \infty]$ and $p=3$. Those were selected from eight images – three with the frequencies $\phi_0= [0.4, 0.8, 1.15]$ and $p=1$, another three images with the same frequencies and $p=3$, and the last two are obtained with $\phi_0= [2.0, 120]$ and $p=3$. The reason for the restriction to three images is the smaller ANN required.
Design of The ANN Fusion and Training Setup {#sect:SP-ANN-design}
-------------------------------------------
Let $\tilde{f}_1,..., \tilde{f}_K$ be a given set of versions of a CT image, reconstructed by FBP with different low-pass filters in the sinogram domain.[^3] We describe the fusion procedure used to compute the output image $\hat{f}$ of the algorithm:
- For each location $q$ in the image matrix, extract its disk-shaped neighborhood from each of the $K$ images $\tilde{f}_i$, $i=1,...,K$. The radius of the disk is set to $3$ pixels (containing $29$ pixels).
- Compose a set of inputs for the ANN by stacking the pixel intensities from the $K$ neighborhoods into one vector. Normalize this vector in the training stage (discussed below).
- Apply the ANN to produce a set of output values, which are the intensity values in the disk-shaped neighborhood of $q$ in the image $\hat{f}$. This disk has the same radius of $3$ pixels.
- By this design, each pixel in the output image is covered by $29$ disk-shaped patches; its final value is computed by averaging all those contributions.
We detail now on the several of the steps in the list above. In the training stage, the neural network is tuned to minimize the discrepancy between true values in each output vector and those produced by the network from the set of noisy inputs. A vector of inputs is built, as described above, for a location $q$ in a reference image $f$ from a training set, using data from noisy reconstructions. The corresponding vector of outputs is the disk-shaped neighborhood of $q$ in the reference image. Thus, for each image $f$ we produce the set $\tilde{f}_1, ...,\tilde{f}_K$ using pre-defined FBP filters and sample them to build the training dataset. The image is sampled on a cartesian grid, choosing every third pixel $q$ both in horizontal and vertical directions. The pair of input and output vectors for the neural networks is an example used in the training process. Examples from all the training images $f$ are put in one pool. A portion of this pool, having a very low variance in the inputs vector, is discarded (specifically, the threshold is set to $10^{-6}$ times the maximal variance). Those examples correspond to regions of air, since no constant patch in any kind of tissue can be observed in the noisy FBP images. This step leads to an empirical improvement in the performance of the ANN.
It is generally acknowledged, that data normalization improves performance of neural networks [@lecun1998efficient]. Our data matrix $A$, which columns are the individual example vectors, is normalized by $$A \Leftarrow A - \min_i(A(i))\mbox{ and then } A \Leftarrow A/\max_i(A(i)).$$ The two constants $\alpha_1 = \min_i(A(i))$ (the minimum value of the matrix $A$) and $\alpha_2 = 1/\max_i(A(i))$ are stored along with the weights of the neural network, and the new data matrix in the test stage is transformed with those pre-computed constants. Given intensity values in the neighborhood of a pixel $q$ in several noisy images, the network should predict a single value in this pixel for the fusion image. However, as a step of regularization, we design the ANN to produce a vector output which is interpreted as a small neighborhood of $q$. the fusion image is then built from such disk-shaped overlapping patches, which are averaged to produce the final result. This is done to avoid possible artifacts, which can be produced by the network: in the training stage, if the ANN produces a single outlier intensity value, its penalty will be smaller than of a vector of such incorrect intensities. Such regularization reduces the performance the ANN can achieve on the training set, since more equations are imposed, but its performance on test images is expected to be more stable.
FBP Boost – Empirical Study
===========================
Evaluating the Algorithm Performance
------------------------------------
In the experiments we have used sets of clinical CT images, axial body slices extracted from a 3D CT scan of a male head, abdomen and thighs. The images are courtesy of Visible Human Project. The intensity levels of those grayscale images correspond to Hounsfield Units. The training set comprises of $461\times 461$ male thighs sections. The image set for ANN training consists of $12$ images, from which $30,000$ examples are extracted. This number, in our experience, suffices to avoid an over-fitting for the chosen size of neural network ($40$ neurons in the hidden layer, $90$ network inputs, overall $3720$ weights). The vector of features for each example is built from the pixel neighborhoods of radius $3$ pixels, coming from the three corresponding FBP reconstructions. These images are a subset of the $8$ FBP reconstruction images mentioned before, seeking (manually) for the subgroup that would perform the best. The size of the input vector is $3\times 29 = 87$ entries.
In Figure \[fig:SPF1\] we present a reconstruction of a test image. This test image is taken $10$cm away from the region where the training data was taken from. The middle upper image is the result of a fusion of the number of FBP versions, performed with the trained ANN. By the visual impression, the noise-resolution balance in the fused image $\hat{f}$ is better than in any of the FBP versions forming it. The texture of tissues is closer to the original (observed in the reference image, upper left). The level of streaks and general noise are lower than in the central and right FBP images, and the image sharpness is higher than in the left and the central images. Thus, the fusion image enjoys the good properties offered by each of the FBP versions and is superior than any of them.
Recall that the training was done with a set of weights, corresponding to our penalty component from Equation \[eq:grad-penalty\]. The quantitative error measures we compute for this comparison include plain SNR, SNR weighted by those weights, the training risk and the SSIM measures. These values are given in Table \[tbl:SPF-parade1\]. As observed from the table, the weighted SNR of the fusion image is $1.8$dB higher than the highest attainable value in FBP images. For plain SNR this increment is $1.5$dB. Values of the training risk measure behave expectedly: the weights of ANN training were designed to implicitly reduce this measure for the fusion image. Indeed, it is by $20\%$ lower than that of the optimal FBP image. Finally, the SSIM measure supports the claim the fusion image has the best visual appearance, since it admits the larger value for this measure.
---------------- -------------- --------------- ----------------- ---------
Image FBP FBP FBP Fusion
$\phi_0=0.4$ $\phi_0=1.15$ $\phi_0=\infty$ result
SNR (uniform) 25.3059 22.1515 19.2833 26.8692
SNR (weighted) 24.3437 22.3835 19.4414 26.1060
Training-Risk 40.6049 25.3577 55.7256 20.5624
SSIM 0.8839 0.8939 0.6892 0.9298
---------------- -------------- --------------- ----------------- ---------
: Quantitative measures for the FBP reconstructions and the fusion result.[]{data-label="tbl:SPF-parade1"}
![Upper left: reference image. Upper middle: the ANN fusion result. Other: FBP images participating in the fusion, produced with different low-pass filters.[]{data-label="fig:SPF1"}](Figures/SP-parade1b.pdf){width="1\columnwidth"}
Size of Local Neighborhood
--------------------------
We study the algorithm performance with different amounts of local data provided for the ANN fusion. A sequence of test image reconstructions is produced, where the radius $r$ of the pixel neighborhood, extracted for the fusion, is increased from $r=0$ (single pixel) to $r=4$ ($49$ pixels). The input vector for the ANN is built from three such neighborhoods, coming from FBP reconstructions corresponding to cut-off frequencies $\phi_0 = [0.4, 1.15, \infty]$ of the low-pass filter. We remark that in the special case of $r=0$, the regression function learned by the network incorporates only the relations between the pixel values in the different image versions, while with larger neighborhood sizes there is also a possibility to perform some local filtering in each image.
In Figure \[fig:SP-r1-seq\] we display graphs of SNR values[^4] computed for the test image. Observably, the quality increment with the neighborhood radius is exhausted around $r=4$. Our choice is to use $r=3$, which requires a smaller number of variables (comparing to $r=4$) without almost no loss in quality. We also notice in these graphs that the fusion using only the central pixel $p$ has a performance very close to that of the best FBP version (but slightly higher, which testifies to the necessity to provide a larger neighborhood of each pixel for a successful fusion. We should note that large neighborhood allows the network to perform a kind of directionally anisotropic filtering matched to the direction of edges.
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![Graphs of the SNR values corresponding to reconstructions with ANN fusion using input pixel neighborhoods of radius $r=0,1,2,3,4$ (x-axis). []{data-label="fig:SP-r1-seq"}](Figures/seq_r1a.pdf "fig:"){width="0.75\columnwidth"}
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We also compare two cases of output vectors produced by the ANN. In the lower row of Figure \[fig:SP-ym1\], the image on the right is produced by the fusion process where a single pixel is recovered by the ANN for each input vector. The image on the left is produced by computing $5$-pixel neighborhoods of each pixel and averaging the overlapping regions. The visual difference between the image is negligible, and the difference in SNR is $0.2$ dB in favor of the averaging approach. Judging from this (and other similar) tests, we conclude that forcing the neural networks to evaluate a number of pixels in the neighborhood of the one being recovered does not reduce its performance. We don’t have an empirical evidence that such a step is truly necessary, since no artifacts in single-pixel-estimation case were observed in this test.
![Upper left: reference image. Upper right: best-SNR FBP reconstruction. Lower left: fusion result where ANN output size is $5$ pixel. Lower right: a fusion result where the ANN produces a single pixel value.[]{data-label="fig:SP-ym1"}](Figures/SP-ym1-compare.pdf){width="1\columnwidth"}
Single-Image “Fusion”
---------------------
A special case of the proposed method is to perform local processing with the ANN using only one FBP image. This, in fact, is a post-processing algorithm based on a regression function, which implements some non-linear local filter. In the following experiment we compare the performance of two fusion methods, one using three FBP images (sharp, normal and blurred) and another using only one FBP image produced with no low-pass filter. The results are displayed in Figure \[fig:SPF-single\]. Visually, in the single-image fusion case some artificial streaks are observed, which do not appear in the multi-image fusion (where also a lower MSE is achieved). On the other hand, the single-image fusion produces sharper images.
![Two test images (corresponding to the upper and lower rows) of thighs sections. Left column: reference images. Middle column: ANN fusion of a single FBP image with no sinogram filter ($\phi_0 = \infty$). Quality of the upper middle image: SNR = 26.13 dB; lower middle: SNR = 27.02 dB. Right column: ANN fusion of three FBP versions, corresponding to filter cut-off frequency of $\phi_0 = [0.4, 1.15, \infty]$. Quality of the upper right image: SNR = 27.53 dB; lower right: SNR = 27.51 dB.[]{data-label="fig:SPF-single"}](Figures/SP-single-img.pdf){width="1\columnwidth"}
PWLS Boost - Algorithm Design and Empirical Study {#sect:RVTF-PWLS-it}
=================================================
Algorithm Description
---------------------
The iterative PWLS algorithm (see Section \[sect:BK-recon\]) can be boosted by gathering intermediate versions of the image at different numbers of iterations. The idea is to capture the gradual transformation of the image from the initial to the final state. If the initial image is a blurred one, it gradually changes along the iterations towards a sharper version; the intermediate stages contain important information that can contribute to further improve the algorithm output.
The method is very similar to the one proposed in the previous section. At the training stage, a CT reconstruction is performed with a high-quality reference at hand. The examples for ANN training are produced in the following manner: the vector of inputs, corresponding to a location $q$ in the image, is assembled using neighborhoods of $q$ in the different versions of the image, gathered along the PWLS iterations. Specifically, we take a small neighborhood of pixels from each image in this sequence (see details below). The “correct answer”, corresponding to this vector of ANN inputs, is the value of the pixel $q$ in the reference image. As was done previously, the objective function for ANN training is augmented with weights which determine the importance of the individual examples.
PWLS Boost - Empirical Study
----------------------------
We conducted numerical experiments to demonstrate the proposed method using the same setup as in the FBP experiment. Training data for the ANN was obtained using a data-set of $12$ axial male thighs section images. For each, an initial image $\tilde{f}$ is computed with the FBP algorithm using a sinogram filter with cut-off frequency value of $2.0$ (see Figure \[fig:RV-FBP1b\]). The PWLS algorithm is implemented as described in Section \[sect:BK-recon\], with parameters $\delta =0.02$, $\lambda=8\cdot10^{-5}$. We have performed $90$ iterations, saving an image version every $10$ iterations - overall we have a sequence of $10$ images. In practice, we use three images out of this sequence, namely those from iterations 20, 60 and 80. From the first and the third images, neighborhoods of radius $4$ ($49$ pixels) were taken for the estimation of the pixel value, and the second image contributed a neighborhood of radius $1$ ($5$ pixels). Overall, the ANN has $2*49+5=103$ inputs. It is set to be a network with $30$ neurons in the (single) hidden layer. It has a single output, set to produce only the central pixel of the provided neighborhood. These specific settings were obtained with a manual tuning of the design parameters.
In Figure \[fig:SPI-1\] we display the fusion result along with individual PWLS reconstructions, used in the fusion process. The lower part of the figure contains the absolute-valued error images. The fusion result has a higher visual quality than any of the three underlying images. Comparing to those images, the noise level in the fusion image is the lowest, and the tissue texture is closer to the original. The sharpness is the same as in the lower middle PWLS image. The SNR values (stated in the Figure) also point to the improvement in quality. The SSIM of the fusion image is $0.95$, while the sequence of PWLS results have the SSIM values of $0.93, 0.92, 0.86$ (corresponding to the lower row of Figure \[fig:SPI-1\], left to right). A reconstruction of an additional test image is displayed in Figure \[fig:SPI-2\]. The effect of the fusion observed here is similar to the one in the previous reconstruction. We conclude that the ANN-based fusion can contribute also to the iterative reconstruction, without requiring any additional iterations; the computational cost of the fusion, exercised after the reconstruction, is lower by an order of magnitude than that of the iterative process.
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![Images of thighs section. Upper row, left to right: reference image, ANN-fused PWLS (SNR=28.18 dB). Middle rows: three PWLS versions (20 iterations, SNR=26.05, 60 iterations,SNR=26.86 dB, 80 iterations, SNR=24.77 dB). Lower row: absolute-valued error images for the fusion image(left) and best-SNR PWLS (right). Darker shade corresponds to a larger error.[]{data-label="fig:SPI-1"}](Figures/SPI-2a.pdf "fig:"){width="1\columnwidth"}
![Images of thighs section. Upper row, left to right: reference image, ANN-fused PWLS (SNR=28.18 dB). Middle rows: three PWLS versions (20 iterations, SNR=26.05, 60 iterations,SNR=26.86 dB, 80 iterations, SNR=24.77 dB). Lower row: absolute-valued error images for the fusion image(left) and best-SNR PWLS (right). Darker shade corresponds to a larger error.[]{data-label="fig:SPI-1"}](Figures/SPI-2ae.pdf "fig:"){width="1\columnwidth"}
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![A different test image. Left to right: reference image, ANN-fused PWLS (SNR=27.44 dB), three PWLS versions (20 iters., SNR=25.19, 60 iters.,SNR=26.02 dB, 80 iters., SNR=24.67 dB). []{data-label="fig:SPI-2"}](Figures/SPI-2b.pdf){width="1\columnwidth"}
To summarize the fusion effect on the outcome of standard reconstruction algorithms, we display in Figure \[fig:SPF-SPI\] images produced by both FBP and PWLS methods, before and after applying the proposed method of the ANN-based fusion; these images were previously given in Figures \[fig:SPF1\],\[fig:SPI-1\].
![FBP and PWLS reconstruction before (middle column) and after (right column) the ANN fusion. Upper left: reference image. Upper row: FBP (central), FBP+ANN (right). Lower row: PWLS (central), PWLS+ANN (right). []{data-label="fig:SPF-SPI"}](Figures/SP_FI-2a.pdf){width="1\columnwidth"}
In order to test the robustness of the training results, we apply the ANN trained with the thigh sections, for a reconstruction of images of other body parts – sections of the head and the abdomen. Reconstruction results are presented in Figure \[fig:SPI-3\] in the same order as in the previous comparison: middle image in the upper row is the result of fusion, which components are presented in the lower row. The head reconstruction is improved substantially by the fusion process, as visual observation shows. However, the SNR values (given in Table \[table:SPI-anatomic\]) point to the favor of the PWLS image corresponding to $60$ iterations (lower middle image). The highest SSIM value does belong to the fusion result, though. In the case of the abdomen section, the fusion image is similar to the $40$-iterations version but contains less noise; its quantitative measures are somewhat better than those of the individual PWLS images.
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Upper part: head reconstruction. Left to right, top to bottom: reference image, ANN-fused image, three PWLS versions (20, 60, 80 iterations). Lower part: abdomen reconstruction with the same arrangement.[]{data-label="fig:SPI-3"}](Figures/SPI-2ah.pdf "fig:"){width="1\columnwidth"}
![Upper part: head reconstruction. Left to right, top to bottom: reference image, ANN-fused image, three PWLS versions (20, 60, 80 iterations). Lower part: abdomen reconstruction with the same arrangement.[]{data-label="fig:SPI-3"}](Figures/SPI-2aa.pdf "fig:"){width="1\columnwidth"}
---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
----------------- ------------- ------------- ------------- --------
Image PWLS PWLS PWLS Fusion
$40$ iters. $60$ iters. $20$ iters. result
Head section
SNR (plain) 24.91 28.67 28.12 28.09
SSIM 0.878 0.873 0.858 0.881
Abdomen section
SNR (plain) 26.68 27.15 25.24 27.94
SSIM 0.813 0.800 0.761 0.821
----------------- ------------- ------------- ------------- --------
: Quantitative measures for the head and abdomen reconstructions.[]{data-label="table:SPI-anatomic"}
As a last experiment, we consider the special case where the ANN only performs a local filtering of the single version of the image, without a reference to the other versions. A neighborhood of radius $r=4$ ($49$ pixels) was extracted for each location in the PWLS image, corresponding to iteration number $60$. The fusion result is visually compared in Figure \[fig:SPI-single\] versus the image produced from $10$ PWLS versions, as before. It can be observed that the processing by ANN reduces the noise appearing in the PWLS image, but it is slightly inferior to the fusion image produced from several PWLS versions.
![Left to right: PWLS image (60 iterations, SNR=26.02 dB), single-image fusion (SNR = 27.25 dB), multi-image fusion (SNR = 27.44 dB). []{data-label="fig:SPI-single"}](Figures/SPI-single.pdf){width="1\columnwidth"}
Computational Complexity of The Method {#sect:complex}
======================================
We analyze the number of computations required for the proposed method in the cases of FBP and PWLS reconstruction. First, we consider the complexity of applying the ANN to perform the pixel-wise fusion from a number of image versions produced at the reconstruction stage. For an $n\times n$ image, $n^2$ activations of the ANN is required. Typically, the dimension of the input vector of the ANN is of the order of $100$ samples, the output dimension has up to $5$ elements, and a single hidden layer of up to $40$ neurons is used. Thus, the network contains $40\cdot100+5\cdot 40 = 4200$ weights[^5]. Each neuron also implements a sigmoid function thus requiring $40$ sigmoid calculations to produce the ANN output values. Therefore, the cost of performing a local fusion by the ANN is $4200\cdot n^3$ operations.
When the method is used for the FBP reconstruction, a number of FBP versions must be produced; in our experiments, three reconstructions suffice. Each FBP reconstruction is of computational complexity of ${\mathcal}{O}(n^3)$. Therefore, if no hardware changes in an existing scanner are made, producing the fusion image will require roughly four times the extent of a single reconstruction (three FBP processes and the fusion step). Of course, the regular FBP image will be available for the radiologist after the usual time of a single FBP reconstruction.
As for the iterative PWLS algorithm, no changes in the reconstruction process are needed, since we only sample images along the standard iterations. We do not have an accurate estimate for the time complexity of the PWLS, since it depends on the optimization method and its efficient implementation. However, the iterative process necessarily involves an application of the system matrix (${\mathcal}{O}(n^3)$ operations) in each iteration, and therefore it is by order of magnitude slower than the FBP. Adding the fusion step in the end of this process will only marginally increase the total reconstruction time.
Summary {#sect:summary}
=======
We have introduced a method for quality improvement for a general parametric signal estimator. The concept is to use a regression function for a local fusion of a number of estimator’s outputs, corresponding to different parameter settings. The regression proposed is realized with feed-forward artificial neural networks. The fusion process consists of two components: first, the behavior of the signal in its different versions is gathered; second, the ANN performs its own non-linear filtering of the signal versions in small neighborhoods of the estimated pixel.
The proposed method is very general and CT reconstruction is only one possible application for it. The local fusion can be used to solve any linear on non-linear inverse problem where an algorithm, producing a solution estimate, exists. The proposed method will enable to incorporate the algorithm outputs, produced with different values of a core parameter, to a single improved result, thus removing the need for tuning this parameter.
In this work this concept was illustrated for the case of CT reconstruction, done with two basic algorithms – the FBP and the PWLS. Empirical results suggest that the local fusion can improve on the resolution variance trade-off of the given reconstruction algorithm, thus adding to the visual quality of the CT images. The post-processing method is not very time-consuming, and the cost of the local fusion can be well below the extent of one FBP reconstruction.
[^1]: All authors are with the Computer Science Department, Technion - Israel Institute of Technology, Israel.
[^2]: The Universal Approximation Theorem states that a network with just one hidden layer, where each neuron is realized as a monotonically-increasing continuous function, can uniformly approximate any given multivariate continuous function up to an arbitrary small error bound [@Cybe89]. In practice, adding hidden layers shows an improvement in the ANN performance.
[^3]: Note that all these images are produced from the very same raw sinogram, which means that the patient is exposed to radiation only once.
[^4]: Very similar effect was observed with SSIM.
[^5]: This number can be reduced if a parallel implementation of the ANN is available, since each neuron output can be calculated separately
| {
"pile_set_name": "ArXiv"
} |
---
---
\
[ **M. Pieroni$^{ab}$** ]{}\
\
\
[*${}^a$ Laboratoire AstroParticule et Cosmologie, Université Paris Diderot* ]{}\
[*${}^b$ Paris Centre for Cosmological Physics, F75205 Paris Cedex 13*]{}\
[ **Abstract**]{}
[ We discuss the introduction of a non minimal coupling between the inflaton and gravity in terms of our recently proposed $\beta$-function formalism for inflation. Via a field redefinition we reduce to the case of minimally coupled theories. The universal attractor at strong coupling has a simple explanation in terms of the new field. Generalizations are discussed and the possibility of evading the universal attractor is shown. ]{}
Introduction
============
Inflation is the most suitable extension of standard cosmology to solve the horizon, monopoles and flatness problems. The Planck mission [@Ade:2015lrj] and other cosmological observations help to fix several constraints on the general mechanism driving this phenomenon. The chaotic model [@Linde:1983gd] with potential $V(\phi) = \lambda \phi^4$ with a non-minimal coupling of the scalar field with gravity $\frac{\xi \phi^2}{2} R$ has been recently proposed by Bezrukov and Shaposhnikov [@Bezrukov:2007ep] as a natural extension of the Standard Model in order to include inflation. For a large number $N$ of e-folding this model gives predictions for the scalar spectral index and the tensor to scalar ratio: $$\label{ns-r-attractor}
n_s = 1 - \frac{2}{N}, \qquad \qquad \qquad r = \frac{12}{N^2}.$$ Assuming that $N\sim50-60$ we find numerical values in good agreement with Planck data. As Starobinsky model [@Starobinsky:1980te] and many other inflationary models are also predicting similar values for $n_s$ and $r$ it is important to define a systematic classification in order to avoid this degeneracy. Some proposals to explain this degeneracy have been formulated by Mukhanov [@Mukhanov:2013tua] and Roest [@Roest:2013fha]. In this spirit we recently proposed a $\beta-$function formalism for inflation [@Binetruy:2014zya]. This new approach is based on the idea of providing universality classes of models of inflation by relying on the approximate scale invariance during the inflationary epoch. This suggestion has a deep connection with the idea proposed by McFadden and Skenderis [@McFadden:2010na] of applying the holographic principle to describe the inflationary Universe. In the language of the well known (A)dS/CFT correspondence of Maldacena [@Maldacena:1997re], the asymptotic de Sitter spacetime is dual to a (pseudo) Conformal Field Theory. In this framework the equations describing the cosmological evolution are thus interpreted as holographic Renormalization Group (RG) equations for the corresponding QFT [@Kiritsis:2013gia]. This correspondence suggests that universality classes for inflationary models should be defined in terms of the Wilsonian picture of fixed points (exact deSitter solutions), scaling regions (inflationary epochs), and critical exponents (scaling exponents of the power spectra related with the slow roll parameters). It is important to stress that, in analogy with statistical mechanics, these universality classes should be considered as sets of theories that share a common scale invariant limit. As results obtained in this framework are not only valid for particular models but for whole sets of theories, it should be clear that they should be conceived as more general than the ones obtained using the standard methods.\
In this paper we discuss inflationary models where a scalar field is non-minimally coupled with gravity. A discussion of this topic has been recently proposed by Linde, Kallosh and Roest [@Kallosh:2013tua] in terms of the standard picture of defining inflationary models by identifying the inflationary potentials and they proved the existence of a universal attractor at strong coupling. Both to have a deeper comprehension of the inflationary regime and to produce a further generalization of the results presented in [@Kallosh:2013tua], it is interesting to treat theories for scalar field with a non-minimal coupling with gravity in terms of the $\beta-$function formalism. In Sec.\[sec:model\_definition\], we present a model of a scalar field with a non-minimal coupling with gravity. In Sec.\[sec:beta\_function\] we formulate the problem in terms of the $\beta-$function formalism and we present the weak and strong limits. In Sec.\[sec:general\_case\], we consider a more general class of models by relaxing an assumption on the expression for the potential. In this context we prove that it is possible to evade the universal attractor and that other attractors can be reached. In Sec.\[sec:conclusions\], we finally present our conclusions.
Setting up the model {#sec:model_definition}
====================
The simplest action to describe the inflating universe consists of a the standard Einstein-Hilbert term to describe gravity plus the action for a homogeneous scalar field in curved spacetime[^1]: $$\label{minimal-action}
S =\int\mathrm{d}^4x\sqrt{-g}\left( - \frac{1}{2\kappa^2}R + X - V_J(\phi) \right),$$ where $X\equiv g^{\mu \nu} \partial_\mu \phi \partial_\nu \phi /2 = \dot{\phi}^2 /2 $ is the standard kinetic term for a homogeneous scalar field. Let us consider a generalization of this action to include a non-minimal coupling between the scalar field and gravity. In this paper, we follow the proposal of [@Kallosh:2013tua], and we consider the action: $$\label{non-minimal-action2}
S =\int\mathrm{d}^4x\sqrt{-g}\left( - \frac{\Omega(\phi)}{2\kappa^2}R + X - V_J(\phi) \right).$$ As gravity is not described by a standard Einstein-Hilbert term, this should be considered as the Jordan frame formulation of the model. Notice that we have not imposed any constraint on the explicit expression of $V_J(\phi)$. Let us consider: $$\label{omega}
\Omega(\phi) = 1 + \xi f(\phi),$$ where $\xi$ is the coupling constant and $f(\phi)$ is a function of $\phi$. It is again interesting to stress that this parametrization is quite general as we are not imposing any constraint on the explicit expression for $f(\phi)$. It should also be stressed that $\xi = 0$ corresponds to the standard case of a scalar field minimally coupled with gravity.\
It is well known that by means of a conformal transformation i.e. $$\label{metric}
g_{\mu\nu} \rightarrow \Omega(\phi)^{ -1} g_{\mu\nu},$$ we can recover the standard Einstein-Hilbert term for gravity. The action in terms of the new metric can be expressed as: $$\label{non-minimal-action}
S=\int\mathrm{d}^4x\sqrt{-g}\left( - \frac{1}{2\kappa^2}R + F(\phi)X - \bar{V}(\phi) \right),$$ where $F(\phi)$ and $\bar{V}(\phi)$ are defined by: $$\label{generalF}
F(\phi) \equiv \Omega^{-1} + \frac{3}{2} \left( \frac{\mathrm{d} \ln \Omega}{\mathrm{d} \phi}\right)^2, \qquad \qquad \qquad \bar{V}(\phi) \equiv \frac{V_J (\phi)}{\Omega(\phi)^2}.$$ This is usually known as the Einstein frame formulation for the theory. From now on, we impose $\kappa^2 = 1 $ to simplify the notation. As discussed in [@Kallosh:2013tua], it is interesting to consider the particular expression for $V_J(\phi)$: $$V_J(\phi) = \lambda^2 f^2(\phi).$$ This parametrization is motivated by the possibility of defining a natural supergravity embedding [@Kallosh:2010ug] for this class of models. It is important to stress that in the limit of small coupling $\xi \ll 1 $, both $\Omega(\phi)$ and $F(\phi)$ are close to one. In this limit, fixing an explicit parametrization for $f(\phi)$ we are directly fixing the potential for the theory. At this point it should be clear that in this regime different choices for $f(\phi)$ correspond to different predictions for $n_s$, scalar spectral index, and $r$, tensor to scalar ratio. As discussed in [@Kallosh:2013tua], it is interesting to consider the limit of a strong coupling $1 \ll \xi$. It possible to show that in this regime the expression for $N$, number of e-foldings, simply reads: $$\label{e-foldings}
N(\phi) \simeq \frac{3}{4} \xi f(\phi).$$ It is also possible to show that in this limit, the expressions for $n_s$ and $r$ are simply given by eq.. It is important to stress that this result is independent on the explicit choice for $f(\phi)$. As different theories share the same asymptotic behavior in the limit of $1 \ll \xi$, this proves the existence of a universal attractor at strong coupling. In the rest of this work we will focus both on the interpretation of this attractor in terms of the $\beta$ function formalism of [@Binetruy:2014zya], and on the possibility of extending these results for more general classes of models. In particular, in Sec.\[sec:general\_case\], we will discuss the consequences of chosing a different parametrizetion for $V_J(\phi)$ i.e. $$\Omega(\phi) = 1 + \xi f(\phi) ,\qquad \qquad \qquad V_J(\phi) = \lambda^2 g^2(\phi) ,$$ with $f(\phi) \neq g(\phi)$.
$\beta$-function formalism {#sec:beta_function}
==========================
Let us consider the model described in Sec.\[sec:model\_definition\]. By means of a field redefinition it is possible to reduce to the problem of a scalar field with a canonically normalized kinetic terms. In particular let us define[^2] the new field $\varphi$ as: $$\label{def_varphi}
\left( \frac{\mathrm{d} \varphi}{ \mathrm{d} \phi} \right)^2 = F(\phi).$$ By definition the kinetic term of $\varphi$ is canonically normalized and thus we can directly follow the procedure discussed in [@Binetruy:2014zya]. Assuming that the time evolution of the scalar field $\varphi(t)$ is *piecewise monotonic* we can invert to get $t(\varphi)$ and use the field as a clock. Under this assumption we can thus describe the dynamics of the system in terms of the Hamilton-Jacobi approach of Salopek and Bond [@Salopek:1990jq]. In this framework we define $W(\varphi) \equiv -2H(\varphi)$, that satisfies $\dot{\varphi} = W_{,\varphi} (\varphi)$ and also $$\label{superpotential}
2 V(\varphi) = \frac{3}{2} W^2(\varphi) - \left[W_{,\varphi} (\varphi)\right]^2.$$ The latter expression leads to call the function $W(\varphi)$ *superpotential* because of a similar parametrisation in the context of supersymmetry. In analogy with QFT we define: $$\label{beta_varphi}
\beta(\varphi) \equiv \frac{\mathrm{d} \varphi}{\mathrm{d} \ln a} = - 2\frac{\mathrm{d} \ln W(\varphi)}{\mathrm{d} \varphi}.$$ It is important to notice that the equation of state for the scalar field in terms of $\beta$ reads: $$\label{eq_of_state}
\frac{p+ \rho}{\rho} = \frac{\beta^2 (\varphi)}{3}.$$ This expression for the equation of state implies that an inflationary epoch is associated with the neighborhood of a zero of $\beta(\varphi)$. In fact, by specifying a parametrization for $\beta(\varphi)$, we are fixing the evolution of the system (or equivalently the RG flow) close to a fixed point. As a single asymptotic behavior can be reached by several models, the parametrization of $\beta(\varphi)$ is not simply specifying a single inflationary model but rather a set of theories sharing a scale invariant limit. In particular, using the language of statistical mechanics, we are specifying a *universality class* for inflationary models. It is important to stress that in this framework all the informations on the inflationary phase are thus enclosed in the parametrization of $\beta(\varphi)$ in terms of the critical exponents. Substituting eq. into eq. we express the potential $V(\varphi)$ as: $$\label{potential_1}
V(\varphi)=\frac{3 W^{2}(\varphi)}{4}\left[1-\frac{\beta^{2}(\varphi)}{6}\right].$$ During an inflationary epoch $\beta(\varphi)$ must be close to zero and thus at the lowest order, we can approximate[^3] eq. with: $V(\varphi) \sim \frac{3}{4} W(\varphi)^2$. From eq. , we can notice that $\beta^2 /2 $ is equal to the first slow-roll parameter $\epsilon = - \dot{H}/H^2$. In the slow rolling regime the $\beta$-function formalism is thus equivalent to the horizon-flow approach of Hoffman and Turner [@Hoffman:2000ue; @Kinney:2002qn; @Liddle:2003py; @Vennin:2014xta]. In this limit we can thus express $\beta(\varphi)$ as: $$\label{beta_approx_varphi}
\beta(\varphi ) \sim - \frac{\mathrm{d} \ln V(\varphi)}{\mathrm{d} \varphi} = - 2 \frac{\tilde{f}_{,\varphi}(\varphi)}{\tilde{f}(\varphi) \left[1 + \xi \tilde{f}(\varphi) \right]} ,$$ where $\tilde{f}(\varphi) \equiv f(\phi(\varphi))$. Characterizing the system in terms of $\varphi$ helps to have a deeper comprehension of this model and leads to an interpretation of the attractor at strong coupling. In the rest of this section we discuss the limits of large and small $\xi$ and we present an explicit example to understand the interpolation between these two regimes.
Strong and weak coupling limits. {#sec:strong}
--------------------------------
In the strong coupling limit we have $1 \ll \xi $ and thus the lowest order approximation for simply reads: $$\label{beta_strong}
\beta(\varphi ) \simeq - \frac{2}{\xi} \frac{\tilde{f}_{,\varphi} (\varphi )}{ \tilde{f}^2 (\varphi)} .$$ Using eq. we can get the lowest order expression for $F(\phi)$ i.e. $$\label{F_strong}
F(\phi) \simeq \frac{3}{2}\left( \frac{f_{,\phi} (\phi)}{ f(\phi)} \right)^2.$$ We can substitute eq. into eq. and integrate to get: $$\label{def_varphi_strong}
f(\phi(\varphi)) = \tilde{f}(\varphi) = f_{\mathrm{f}}\exp \left[ \sqrt{\frac{2}{3}} (\varphi - \varphi_{\mathrm{f}} ) \right],$$ where we defined $f_{\mathrm{f}} \equiv \tilde{f}(\varphi_{\mathrm{f}})$. It is crucial to notice that in the limit $1 \ll \xi$ the expression for $f(\varphi)$ does not depend on the explicit choice for $f(\phi)$! As shown in eq., the expression of $\beta(\varphi)$ in the limit of a strong coupling is only depending on $\tilde{f}(\varphi)$. We can thus substitute eq. into eq. to get: $$\label{explicit_beta_strong}
\beta(\varphi) = -\sqrt{\frac{8}{3}} \frac{1}{\xi f_{\mathrm{f}}} \exp \left[ - \sqrt{\frac{2}{3}} (\varphi - \varphi_{\mathrm{f}} ) \right].$$ It is important to stress that at the end of inflation $1 + p/\rho $ is close to one and thus eq. implies that $\left| \beta(\varphi_{\mathrm{f}}) \right| \sim 1 $. Eq. then leads to $f_{\mathrm{f}} \sim \sqrt{8/3} / \xi$ that can be substituted into eq. to conclude that: $$\label{beta_strong_ff}
\beta(\varphi) = -\exp \left[ - \sqrt{\frac{2}{3}} (\varphi - \varphi_{\mathrm{f}} ) \right].$$ It is then clear that the expression for $\beta(\varphi)$, in the limit of big $\xi$, is independent on the explicit choice for $f(\phi)$. As the dynamics of the system during the inflationary phase is completely specified by $\beta(\varphi)$, this directly leads to the universality. In particular, we notice that $\beta(\varphi)$ approaches the exponential class of [@Binetruy:2014zya]. This universality class is entirely determined by a single critical exponent, denoted with $\gamma$, that in this case is equal to the $\sqrt{2 /3 }$ factor in the exponential of . As discussed in [@Binetruy:2014zya], the scalar spectral index and the tensor to scalar ratio are given by: $$\begin{aligned}
\label{ns_strong}
n_{s} - 1 &\simeq& - \frac{2}{N}, \\
\label{r_strong}
r &\simeq& \frac{8}{\gamma^2 N^2} = \frac{12}{N^2} .
\end{aligned}$$ As expected, these results are in perfect agreement with the ones discussed in [@Kallosh:2013tua]. In this framework, the independence of $\beta(\varphi)$ on $f(\phi)$ directly leads to the universality for the values of $n_s$ and $r$. In particular, in terms of the $\beta$-function formalism, the appearence of a universal attractor at strong coupling corresponds to the flow of the system into a particular universality class.\
For completeness we can also express $N(\varphi)$ as: $$\label{efold_ff}
N(\varphi) = - \int_{\varphi_f}^{\varphi} \frac{1}{\beta(\hat{\varphi})} d\hat{\varphi} = \sqrt{ \frac{3}{2}} \left\{ \exp \left[ \sqrt{\frac{2}{3}} \left( \varphi - \varphi_{\mathrm{f}} \right) \right] - 1 \right\} .$$ Substituting eq. into eq. we find that choosing values for $N$ in the range $ [50,60]$ we get $\beta \in [-0.024,-0.02]$. Notice that in the limit of strong coupling, the dynamics in terms of $\varphi$ does not depend on $\xi$. It is also interesting to point out that in this case the asymptotic fixed point is reached for $N \rightarrow \infty$ that corresponds to $\varphi \rightarrow \infty$.\
It is interesting to point out that $\xi = 0$ corresponds to a minimal coupling between the inflaton and gravity. In this case we can again use the equations derived in Sec.\[sec:beta\_function\], but the expression for $\tilde{f}(\varphi)$ given by eq. does not hold. As a consequence we are not expecting to obtain a model independent expression for $\beta(\varphi)$ and thus results will be model dependent. In particular, by choosing particular parametrizations for $\beta(\varphi)$, we can reproduce the universality classes introduced in [@Binetruy:2014zya]. As in the limit of a weak coupling we are just introducing a small variation with respect to the case of $\xi = 0$, we will only obtain a little departure from the standard results. In particular it is possible to prove that in the limit of a weak coupling the lowest order expressions for $n_s$ and $r$ correspond to the ones presented in [@Kallosh:2013tua].
An explicit example. {#sec:example}
--------------------
In this section we present an example to be have a better understanding of the transition from the weak to the strong coupling limits. In particular, we consider some particular models by specifying an explicit expression for $f(\phi)$. From eq. and eq., it should be clear that $\beta(\varphi) \sim - 2\alpha/\varphi$ simply gives: $$W(\varphi) = W_\textit{f} \left (\frac{\varphi}{\varphi_\textit{f}} \right)^{\alpha}, \qquad \qquad V(\varphi) = \frac{3 W_\textit{f}^2}{4} \left (\frac{\varphi}{\varphi_\textit{f}} \right)^{2\alpha} = V_\textit{f} \left (\frac{\varphi}{\varphi_\textit{f}} \right)^{2\alpha}.$$ This clearly corresponds to the well known case of chaotic inflation [@Linde:1983gd]. As in the limit of small $\xi$ we have $\varphi \sim \phi$ and $\beta(\varphi) \sim - 2 \tilde{f}_{,\varphi}(\varphi)/\tilde{f}(\varphi)$, to obtain this expression for $\beta(\varphi)$ we simply choose $f(\phi) = \phi^\alpha$. It is well known that in this case the lowest order predictions for the $n_s$, scalar spectral index, and $r$, tensor to scalar ratio, are given by: $$\label{chaotic_predictions}
n_{s} \simeq 1 - \frac{1+\alpha}{N}, \qquad \qquad r \simeq \frac{8\alpha}{N}.$$ On the contrary, the strong limit predictions have been discussed in Sec \[sec:strong\], and these are given by eq. and eq.. Variating the value of $\xi$, we expect to shift from the model dependent regime to the universal attractor at strong coupling. Numerical results for our choice for $f(\phi)$ are shown in Fig. \[figure1\] and Fig. \[figure2\]. In this particular case the parametrization of $\beta(\varphi)$ is completely specified by the value of the critical exponent $\alpha$. Once this constant is fixed, we can compute numerical predictions as a function of $N$, number of e-foldings. in Fig. \[figure1\] and Fig. \[figure2\] we use different colors to plot models associated with a different values for $\alpha$. The solid black lines in the plot of Fig. \[figure2\] are used to follow the variation of $\xi$ while the values of $\alpha$ and $N$ are fixed. The thick line corresponds to $ N = 60 $ and the thin one corresponds to $N = 50$. Numerical results are compared with the ones obtained for the chaotic class i.e. $\beta(\varphi) = - \alpha / \varphi$ with some values for $\alpha$ in the range $[0.1,3]$ and for the exponential class i.e. $\beta(\varphi) = - \exp \left[ - \gamma \varphi \right]$ with $\gamma = \sqrt{2/3}$. In limit of a weak coupling numerical predictions match with the chaotic class while in the strong coupling limit we approach the predicted exponential class attractor. In the intermediate region we have a whole set of valid inflationary models that actually interpolate between the two fundamental classes.\
\
\
This behavior is quite similar to the mechanism of interpolation discussed in [@Binetruy:2014zya]. In these paper we have shown that, by introducing a new scale $f$, it is possible to construct a $\beta$-function that approaches different universality classes as we consider different values for $f$. In particular we have shown that a small and a large field regime can be reached. For example let us consider a model with a scalar field $\chi$ and let us assume that $\beta(\chi) = g(\chi)$ is the $\beta$-function for this model. As we are interested in studying an inflationary stage, the system is close to zero of the $\beta$-function. Without loss of generality we assume that $\beta(\chi = 0) = 0$. Finally we consider a model with $\beta$-function defined by $\beta(\chi) = \epsilon g(\chi)$ where $\epsilon \ll 1$ is a constant. It should be clear that this system inflates for all the values for $\chi$ such that $\beta(\chi) = \epsilon g(\chi) \ll 1$ and thus we can have inflation even for $g(\chi) \sim 1$. As a matter of fact the small field regime is stretched and it can be reached even for bigger values for $\chi$. In the case of a scalar field with a non-minimally coupling with gravity the role of the scale $f$ appears to be played by the coupling $\xi$. In particular this can be shown expressing the $\beta$-function in terms of $\phi$: $$\label{beta_phi}
\bar{\beta}(\phi) = \beta(\varphi(\phi)) = - \left( \frac{\mathrm{d} \phi}{\mathrm{d} \varphi} \right) \frac{\mathrm{d} \ln \bar{V}(\phi)}{\mathrm{d} \phi}.$$ Usign eq. and eq. we express the $\beta$-function in the limit of strong coupling as: $$\label{final_beta_phi}
\bar{\beta}(\phi) = - \sqrt{\frac{8}{3}} \frac{\phi^{-\alpha}}{\xi}.$$ It should be clear that a zero of this function is reached for $1 \ll \phi^\alpha$. However, by choosing a large value for $\xi$, it is still possible to have inflation in the limit of $\phi^\alpha \ll 1$. In particular, consistently with [@Kallosh:2013tua], this mechanism allows the production of cosmological perturbations in the regime $ \phi^\alpha \ll 1$. By choosing a large value of $\xi$ we have thus stretched the asymptotic large field regime so that it can be obtained even for small values of $\phi$.
A more general discussion on non-minimal coupling {#sec:general_case}
=================================================
In this section we are interested in discussing a more general parametrization for the model for a scalar field with a non minimal coupling with gravity. In particular we follow the proposal of [@Kallosh:2013tua] and we start by considering the same lagrangian of eq. i.e. $$\label{non_minimal_action_2}
S =\int\mathrm{d}^4x\sqrt{-g}\left( - \frac{\Omega(\phi)}{2\kappa^2}R + X - V_J(\phi) \right),$$ but we introduce an additional functional freedom in the model i.e. $$\Omega(\phi) = 1 + \xi f(\phi) ,\qquad \qquad \qquad V_J(\phi) = \lambda^2 g^2(\phi) ,$$ where both $f(\phi)$ and $g(\phi)$ are generic functions of $\phi$. As argued in [@Kallosh:2013tua], by studying the system in terms of $\phi$, it seems reasonable to assume that small variations of the potential should not affect the occurrence of the attractor. In the following we will prove that in general this conclusion does not appear to hold.\
By means of a conformal transformation we recover the Einstein frame action of eq.. In this case the expressions for $\Omega(\phi)$ and $V(\phi)$ are given by: $$\label{omega_V}
\Omega(\phi) = 1 + \xi f(\phi) , \qquad \qquad \bar{V}(\phi) = \frac{\lambda^2 g^2(\phi) }{\Omega^2}.$$ Following the procedure defined in Sec.\[sec:beta\_function\], we describe the system in terms of a new field $\varphi$ with a canonically normalized kinetic term. Substituting eq. into eq. we get: $$\label{general_vaphi}
\left( \frac{\mathrm{d} \varphi}{\mathrm{d} \phi} \right)^2 = F(\phi) = \frac{1 + \xi f + \frac{3}{2} \xi^2 f^2_{,\phi} }{(1 + \xi f(\phi))^2}.$$ The expression for the $\beta$-function associated with the system reads: $$\label{beta_general}
\beta(\varphi) \sim - \frac{\mathrm{d} \ln V(\varphi)}{\mathrm{d} \varphi} = - 2 \left( \frac{\tilde{g}_{,\varphi}(\varphi)}{\tilde{g}(\varphi)} - \frac{\xi \tilde{f}_{,\varphi}(\varphi)}{1 + \xi \tilde{f}(\varphi)} \right),$$ where in analogy with $\tilde{f}(\varphi)$, we defined $\tilde{g}(\varphi) = g(\phi(\varphi))$. Without loss of generality we can parametrizations $\tilde{g}(\phi)$ as: $$\label{eq_parametization}
\tilde{g}(\varphi) = \tilde{f}(\varphi) \tilde{h}(\varphi),$$ where $\tilde{h}(\varphi)$ is a generic function of $\varphi$. It is important to stress that we are not specifying an explicit expression for $\tilde{h}(\varphi)$ and thus we can produce a quite general description of the problem. We can substitute eq. into eq. to get: $$\label{beta_general_2}
\beta(\varphi) = - 2 \left\{ \frac{\tilde{h}_{,\varphi}(\varphi)}{\tilde{h}(\varphi)} + \frac{ \tilde{f}_{,\varphi}(\varphi)}{\tilde{f}(\varphi) \left[ 1 + \xi \tilde{f}(\varphi) \right]} \right\}.$$ It is easy to notice that in the case of $\tilde{h}_{,\varphi}(\varphi)/\tilde{h}(\varphi) = 0$, this equation is exactly equal to eq.. In particular, in the limit of strong coupling $\xi$ eq. simply reads: $$\label{beta_general_3}
\beta(\varphi) = - 2 \left[ \frac{\tilde{h}_{,\varphi}(\varphi)}{\tilde{h}(\varphi)} + \frac{ \tilde{f}_{,\varphi}(\varphi)}{ \xi \tilde{f}^2(\varphi) } \right].$$ It is interesting to point out that choosing $\tilde{f}(\varphi) = \tilde{g}(\varphi)$ or equivalently $\tilde{h}(\varphi) = 1$, the zero order term in $1/\xi$ is set equal to zero. Under this assumption the expression for $\beta(\varphi)$ is thus dominated by the first order term $1/\xi$. In particular the $\beta$-function is simply given by eq. and thus we reduce to the case discussed in Sec. \[sec:strong\]. Relaxing the assumption of $\tilde{h}(\varphi) = 1$, we can consider the case of a zero order term different from zero. As an inflationary stage corresponds to $\beta(\varphi) \rightarrow 0$ any choice of $\tilde{h}(\varphi)$ that satisfies: $$\label{eq_evasion}
\frac{ \tilde{f}_{,\varphi}(\varphi)}{ \xi \tilde{f}^2(\varphi) } \ll \frac{\tilde{h}_{,\varphi}(\varphi)}{\tilde{h}(\varphi)} \rightarrow 0,$$ describing a viable inflationary model. As no other restriction has been imposed on the choice for $\tilde{h}(\varphi)$, we can immediately conclude that in general the attractor at strong coupling can be evaded. In the Sec. \[sec:General\_example\] we present an explicit example to discuss the conditions to preserve the attractor at strong coupling. In Sec. \[sec:Further\_generalizations\] we show that models defined via further generalizations of the action eq. are still included in this class and we investigate the characterization of the $\alpha$-attractors of Kallosh and Linde [@Kallosh:2013hoa; @Kallosh:2013daa; @Kallosh:2013yoa; @Kallosh:2015lwa] in terms of our formalism. Some other examples of the parametrization for $\tilde{h}(\varphi)$ are discussed in Appendix \[Appendix\].
Polynomial expansion. {#sec:General_example}
---------------------
Let us assume that $f(\phi)$ and $g(\phi)$ admit a Taylor expansion in terms of $\phi$: $$\label{Taylor}
f(\phi) = \sum_{i = 0}^\infty f_i \phi^i, \qquad \qquad g(\phi) = \sum_{i = 0}^\infty g_i \phi^i .$$ Let us restrict to the case of both $f(\phi)$ and $g(\phi)$ vanishing for a certain value of $\phi$. By means of a field redefinition we can fix $f_0 = g_0 = 0$. Without loss of generality we can also rescale $\lambda $ and $\xi$ to impose $f_1 = g_1 = 1$. The case $f(\phi) = g(\phi)$ has been discussed in Sec. \[sec:strong\], and in particular we have shown that under this assumption it is possible to choose $\xi$ such that $\phi \ll 1$. As the first order terms of eq. are imposed to be equal, and high order orders in terms of $\phi$ are expected to be negligible, it would be reasonable to conclude that the attractor at strong coupling should be preserved. Surprisingly, expressing the dynamics in terms of $\varphi$, it is possible to show that the attractor at strong coupling may be evaded! Let us fix a particular expression for $f(\phi)$ and $g(\phi)$, in particular we choose: $$\label{explicit_expansion}
f(\phi) = \phi, \qquad \qquad g(\phi) =\phi + g_{n+1} \phi^{n+1} = \phi (1 +g_{n+1} \phi^{n} ) .$$ In the strong coupling limit, the lowest order approximation for eq. simply reads: $$\label{example_beta_strong}
\left( \frac{ \textit{d} \varphi}{ \textit{d} \phi} \right)^2 = F(\phi) \simeq \frac{3}{2}\left( \frac{f_{,\phi} (\phi)}{ f(\phi)} \right)^2.$$ We can integrate eq. to get an explicit expression for $\tilde{f}(\varphi)$: $$\label{example_varphi}
\tilde{f}(\varphi) = \tilde{f}_{\mathrm{f}}\exp \left[ \sqrt{\frac{2}{3}} (\varphi - \varphi_{\mathrm{f}} ) \right].$$ Finally we substitute into eq. to get: $$\begin{aligned}
\label{example_phi_varphi}
\phi &=& f(\phi) = \tilde{f}(\varphi) = \tilde{f}_{\mathrm{f}}\exp \left[ \sqrt{\frac{2}{3}} (\varphi - \varphi_{\mathrm{f}} ) \right], \\
\label{explicit_g}
\tilde{g} (\varphi) &=& \tilde{f}_{\mathrm{f}}\exp \left[ \sqrt{\frac{2}{3}} (\varphi - \varphi_{\mathrm{f}} ) \right] \left\{ 1 + g_{n+1} \tilde{f}_{\mathrm{f}}^{\ n} \exp \left[ \sqrt{\frac{2}{3}} n (\varphi - \varphi_{\mathrm{f}} ) \right] \right\},\end{aligned}$$ where $\tilde{g} (\varphi) = g(\phi(\varphi))$. It should be clear that this corresponds to: $$\label{example_h}
\tilde{h}(\varphi) = 1 + g_{n+1} \tilde{f}_{\mathrm{f}}^{\ n} \exp \left[ \sqrt{\frac{2}{3}} n (\varphi - \varphi_{\mathrm{f}} ) \right].$$ Using eq. we can then compute the explicit expression for $\beta(\varphi)$: $$\label{explicit_beta}
\beta(\varphi) = -\sqrt{\frac{8}{3}} \left\{ \frac{n g_{n+1} \tilde{f}^{\ n}_{\mathrm{f}} \exp \left[ \sqrt{\frac{2}{3}} n (\varphi - \varphi_{\mathrm{f}} ) \right] } {1 + g_{n+1} \tilde{f}^{\ n}_{\mathrm{f}}\exp \left[ \sqrt{\frac{2}{3}} n (\varphi - \varphi_{\mathrm{f}} ) \right] } + \frac{1 }{1 + \xi \tilde{f}_{\mathrm{f}}\exp \left[ \sqrt{\frac{2}{3}} (\varphi - \varphi_{\mathrm{f}} ) \right] } \right\}.$$ Notice that the first term on the right hand side of eq. gives the zero order contribution in $1/\xi$ while the second term on the right hand side of eq. is a first order term in $1/\xi$. It is important to stress that imposing $ g_{n+1} = 0$, is equivalent to fix $f(\phi) = g(\phi)$. As discussed in Sec.\[sec:strong\], in this case the inflationary regime is reached for large positive values for $\varphi$ and $\beta(\varphi)$ is approximated by eq.. On the contrary when $ g_{n+1} \neq 0$, the second term on the right hand side of eq. is negligible with respect to the first one[^4]. Under this assumption $\beta(\varphi)$ can be approximated as: $$\label{explicit_beta_2}
\beta(\varphi) \sim -\sqrt{\frac{8}{3}} \left\{ \frac{ n g_{n+1} \tilde{f}^{\ n}_{\mathrm{f}} \exp \left[ \sqrt{\frac{2}{3}} n (\varphi - \varphi_{\mathrm{f}} ) \right] } {1 + g_{n+1} \tilde{f}^{\ n}_{\mathrm{f}}\exp \left[ \sqrt{\frac{2}{3}} n (\varphi - \varphi_{\mathrm{f}} ) \right] } \right\}.$$ In this case the zero of $\beta(\varphi)$ that corresponds to the inflationary phase is thus reached for large negative values for $\varphi$. In this regime the expressions for $\beta(\varphi)$ and $N(\varphi)$ are: $$\begin{aligned}
\label{explicit_beta_3}
\beta(\varphi) &\sim& -\sqrt{\frac{8}{3}} n g_{n+1} \tilde{f}^{\ n}_{\mathrm{f}} \exp \left[ \sqrt{\frac{2}{3}} n (\varphi - \varphi_{\mathrm{f}} ) \right] , \\
\label{efold_explicit}
N(\varphi) &=& - \frac{3}{4} \frac{1}{n^2 g_{n+1} \tilde{f}^{\ n}_{\mathrm{f}}} \left\{ \exp \left[ - \sqrt{\frac{2}{3}} n \left( \varphi - \varphi_{\mathrm{f}} \right) \right] - 1 \right\}.\end{aligned}$$ To ensure $0 \leq N(\varphi)$, we impose $0 < - f^n_{\mathrm{f}} g_{n+1}$. Following the same procedure of Sec.\[sec:strong\], we also impose the condition $\left| \beta(\varphi_{\mathrm{f}}) \right| \sim 1 $ to fix the value of $\beta(\varphi)$ at the end of inflation: $$\label{condition}
\left| \beta(\varphi_{\mathrm{f}}) \right| = \left| \sqrt{\frac{8}{3}} n g_{n+1} \tilde{f}^{\ n}_{\mathrm{f}} \right| \sim 1.$$ As $n$ and $g_{n+1}$ are expected to be of order one, we can conclude that $ f_{\mathrm{f}}$ is expected to be of order one too. Finally, using eq. we express $\beta(\varphi)$ and $N(\varphi)$ as: $$\begin{aligned}
\label{explicit_beta_final}
\beta(\varphi) &\sim& \exp \left[ \sqrt{\frac{2}{3}} n (\varphi - \varphi_{\mathrm{f}} ) \right], \\
\label{explicit_efold_final}
N(\varphi) &=& \sqrt{\frac{3}{2 n^4}} \left\{ \exp \left[ - \sqrt{\frac{2}{3}} n \left( \varphi - \varphi_{\mathrm{f}} \right) \right] - 1 \right\}.\end{aligned}$$ The expression for $\beta(\varphi)$, in the limit of big $\xi$, thus depends on $n$ and this leads to the evasion from the univerality. In particular, $\beta(\varphi)$ approaches the exponential class of [@Binetruy:2014zya] with $\gamma = n \sqrt{2 /3 }$. The corrisponding expression for the scalar spectral index and for the tensor to scalar ratio are given by: $$\label{ns_r_evaded_attractor}
n_s = 1 - \frac{2}{N}, \qquad \qquad \qquad r = \frac{12}{n^2 N^2}.$$ It is interesting to notice that the attractor at strong coupling of [@Kallosh:2013tua] can be reproduced by imposing $n = 1$. Actually it is possible to go further and prove that the attractor can be recovered under some more general condition. As argued during this section, the inflationary phase is reached for $\varphi \ll -1$ and this corresponds to $\phi \ll 1$. In this regime higher order corrections to the expression of $g(\phi)$: $$\label{generalized_expansion}
f(\phi) = \phi, \qquad \qquad g(\phi) =\phi + g_{n+1} \phi^{n+1} + \sum_{i = n+2 }^\infty g_i \phi^i ,$$ are not producing significat changes in the lowest order expressions for $\beta(\varphi)$ and $N(\varphi)$ given in eqs.,. In particular this implies that assuming $g_2 \neq 0$, the dominant contribution to the expression of $\beta(\varphi)$ is fixed by the term with $n=1$. This condition is thus sufficient to preserve the attractor[^5]. Conversely, other attractors are find for different values of $1<n$.\
To conclude this section we discuss the consistency of the assumption that the second term on the right hand side of eq. is negligible with respect to the first one. To be sure that this term is subdominant from the end of inflation up to the production of cosmological perturbations we need: $$\label{attractor_condition}
\frac{1 }{1 + \xi \tilde{f}_{\mathrm{f}}\exp \left[ \sqrt{\frac{2}{3}} (\varphi_{\mathrm{H}} - \varphi_{\mathrm{f}} ) \right] } \ll \exp \left[ \sqrt{\frac{2}{3}} n (\varphi_{\mathrm{H}} - \varphi_{\mathrm{f}} ) \right] \ll 1,$$ where $\varphi_{\mathrm{H}}$ is the value of $\varphi$ at the production of cosmological perturbation. Using the expression for $N(\varphi)$ given by eq. it is clear that eq. satisied if $N_{\mathrm{H}}^2/\xi \ll 1$.\
$\alpha$-attractors. {#sec:Further_generalizations}
--------------------
It is interesting to notice that the class of models described in this section also includes further generalizations of the lagrangian of eq.. In particular some of these generalizations have been presented in [@Galante:2014ifa] and [@Kallosh:2014laa]. Following the proposal of [@Galante:2014ifa] we consider the general Jordan frame action[^6] to describe a homogeneous scalar field with a non-minimal coupling with gravity non-minimally: $$\label{general_action_jordan}
S=\int\mathrm{d}^4x\sqrt{-g}\left( - \Omega(\phi)\frac{R}{2} + K_J(\phi) X - V_J(\phi) \right).$$ As usual we perform a conformal transformation: $$\label{conformal}
g_{\mu\nu} \rightarrow \Omega(\phi)^{ -1} g_{\mu\nu},$$ to get the Einstein frame formulation of the theory: $$\label{general_action_einstein}
\mathcal{L}_E = - \frac{R}{2} + F(\phi) X - V(\phi) ,$$ where we defined $F(\phi)$ and $V(\phi)$ as: $$F(\phi) \equiv \left[ \frac{K_J (\phi)}{\Omega(\phi)} + \frac{3}{2} \left( \frac{\textrm{d} \ln \Omega(\phi)}{\textrm{d}\phi}\right)^2 \right] \qquad \qquad V(\phi) = \frac{V_J(\phi)}{\Omega^2(\phi)}.$$ It is clear that the cases discussed in the previous sections can be recovered simply imposing $K_J (\phi) = 1$. Again we can define a new field $\varphi$: $$\label{general_varphi_def}
\left( \frac{\textrm{d} \varphi}{\textrm{d}\phi}\right)^2 \equiv F(\phi) = \left[ \frac{K_J (\phi)}{\Omega(\phi)} + \frac{3}{2} \left( \frac{\textrm{d} \ln \Omega(\phi)}{\textrm{d}\phi}\right)^2 \right] ,$$ that has a canonically normalized standard kinetic term. In particular the lagrangian for this field simply reads: $$\label{general_varphi_einstein}
\mathcal{L}_E = - \frac{R}{2} + \frac{(\partial \varphi)^2}{2} - \tilde{V}(\varphi) ,$$ where $\tilde{V}(\varphi) $ is defined as $\tilde{V}(\varphi) = V(\phi(\varphi))$. As in terms of the canonically normalized field $\varphi$ the three functional dependece are merged into $\tilde{V}(\varphi)$, the model construction reduces to fixing a particular parametrization for this function. Usign eq., and the lowest order approximation $V(\varphi) \sim \frac{3}{4} W^2(\varphi)$ we can finally express the $\beta$-function as: $$\label{beta_varphi_def}
\beta(\varphi) \sim - \frac{\textrm{d} \ln \tilde{V}(\varphi)}{\textrm{d}\varphi} .$$ Again the whole dynamics of the model is thus fixed by the parametrization of the $\beta$-function. As different choices for $\Omega(\phi), K_J(\phi)$ and $V_J(\phi)$ lead to the same expression for $\beta$, this explains the possibility for degeneracies to arise.\
Several models described by the action of eq. has been presented in [@Galante:2014ifa] and [@Kallosh:2014laa]. In this paper we consider the $\alpha$-attractors of [@Kallosh:2015lwa] as an interesting example for this class of models. In particular let us consider the case of *T-models* [@Kallosh:2013hoa]. *T-models* can be described in terms of the action of eq. by fixing: $$\label{alpha_attrators}
\left( \frac{\textrm{d} \varphi}{\textrm{d}\phi}\right)^2 = F(\phi) = \left( 1 - \frac{\phi^2}{6 \alpha}\right)^{-2} \qquad \qquad \qquad V(\phi) = \frac{m^2}{2} \phi^2.$$ Using eq. we define the canonically normalized field and using eq. and eq. we can compute the explicit expression for $h(\phi)$. In particular we get: $$\phi = \sqrt{6\alpha} \tanh\left(\frac{\varphi}{\sqrt{6\alpha}} \right), \qquad \qquad \qquad h(\phi) \sim \phi.$$ Finally we can use eq. to compute the expplicit expression for the $\beta$-function: $$\label{beta_alpha_attractors}
\beta(\varphi) = - \sqrt{\frac{2}{3 \alpha}} \left[ \frac{1 - \tanh^2 \left( \frac{\varphi}{\sqrt{6 \alpha}} \right)}{\tanh \left( \frac{\varphi}{\sqrt{6 \alpha}} \right)} \right] \sim - \exp \left[ - \sqrt{\frac{2}{3 \alpha}} ( \varphi-\varphi_f) \right].$$ Eq. imples that the $\beta$ function for *T-models* falls in the exponential class of [@Binetruy:2014zya]. As already discussed in this paper, the predictions for $n_s$ and $r$ are thus given by: $$\label{ns_r_alpha_attractor}
n_s = 1 - \frac{2}{N}, \qquad \qquad \qquad r = \frac{8}{\gamma^2 N^2} = \frac{12 \alpha}{N^2}.$$
Similar conclusions can be draw for the other models for $\alpha$-attractors presented in [@Kallosh:2013hoa].
Conclusions {#sec:conclusions}
===========
In the analysis of this paper, by means of a conformal transformation and of a field redefinition, we have discussed the problem of inflationary models with a non-minimal coupling with gravity in terms of a single field with a canonically normalized kinetic term. In particular we have shown, the application of the $\beta-$function formalism, helps to understand the asymptotic behavior of the system during the inflationary phase. In this framework, the fall of the system into the attractor is interpreted as the approach of a universality class. In this sense, the formulation of the problem in this framework, should not be seen as a simple rewriting of the results obtained with standard methods, but on the contrary it should be considered as a further generalization.\
The $\beta-$function formalism appears to be extremely useful when we investigate the stability of the attractor at strong coupling under generalizations of the theory. In particular, once we have defined the $\beta-$function associated with our system, it has been easy to identify the dominant contribution to characterize inflation. Specifically, in Sec.\[sec:general\_case\], we have discussed the possibility of introducing an additional functional freedom in the model. In this case the behavior of the system is dominated by the zeroth order term that conversely was set equal to zero in the treatment followed in Sec.\[sec:beta\_function\]. As in general this term can be chosen arbitrarily, it leads to the possibility of evading the attractor at strong coupling. A critical review of the conditions required to preserve the attractor at strong coupling has been presented and the existence of different attractors has been shown.\
The further generalization discussed in [@Galante:2014ifa] and [@Kallosh:2014laa] have been presented. In these works it was shown that a slight modification of the theory may lead to the existence of other attractors. Indeed for these models an analogous of the treatment presented in this paper can be carried out and it leads to similar conclusions. In particular we have presented the application of our formalism to the case of the $\alpha$-attractors of [@Kallosh:2015lwa]. A further generalization of the formalism proposed in [@Binetruy:2014zya] can be also useful to have a deeper understanding of more general models with a non-standard kinetic term or with more scalar fields[@Kaiser:2013sna]. It seems reasonable to suppose that in analogy with the case of the non-minimal coupling, the $\beta$-function formalism can be coherently applied to these models as well.
Acknowledgements {#acknowledgements .unnumbered}
=================
I would like to thank Nathalie Deruelle, David Kaiser, Andrei Linde and Diderik Roest for their suggestions. In particular I would like to thank Pierre Binétruy for all the useful discussions that lead to the production of this work. I acknowledge the financial support of the UnivEarthS Labex program at Sorbonne Paris Cité (ANR-10-LABX-0023 and ANR-11-IDEX-0005-02) and the Paris Centre for Cosmological Physics.
Appendix {#appendix .unnumbered}
========
Some explicit examples. {#Appendix}
=======================
In Sec.\[sec:general\_case\], we discussed the consequences of introducing a further functional freedom in the Jordan frame forumalation of the model. In particular we considered: $$\label{non_minimal_action_3}
S =\int\mathrm{d}^4x\sqrt{-g}\left( - \frac{\Omega(\phi)}{2\kappa^2}R + X - V_J(\phi) \right),$$ where $\Omega(\phi)$ and $V_J (\phi)$ have been defined as: $$\Omega(\phi) = 1 + \xi f(\phi) ,\qquad \qquad \qquad V_J(\phi) = \lambda^2 g^2(\phi).$$ After the usual conformal transformation we recover the Einstein frame formulation of the theory. By means of a field redefinition we are finally able to describe the system in terms of a field with a canonically normalized kinetic term. The strong coupling expression for $\tilde{f}(\varphi)$ is fixed by eq. and thus the model definition reduces to fixing an explicit expression for $g(\phi)$. In this appendix we consider some parametrizations for $\tilde{g}(\varphi)$ to study the possibility of preserving and evading the attractor. In particular we show that different universality classes can be reached.
- **Exponential.** Let us consider: $$\tilde{g}(\varphi) = \exp\left[ \sqrt{\frac{2}{3}} \left( \varphi - \varphi_{\mathrm{f}} \right) - \frac{e^{-\alpha (\varphi - \varphi_{\mathrm{f}})}}{\alpha} \right].$$ It is straightforward to derive the lowest order expression for $\beta(\varphi)$: $$\beta(\varphi) \sim - \frac{2 e^{-\alpha (\varphi - \varphi_{\mathrm{f}})}}{\alpha}.$$ This expression for $\beta(\varphi)$ corresponds to the exponential class presented in [@Binetruy:2014zya]. In this case the predictions for $n_s$ and $r$ are: $$\begin{aligned}
n_{s} &\simeq& 1 - \frac{2}{N}, \\
r & \simeq& \frac{8}{\alpha^2 N^2}.
\end{aligned}$$ It may be interesting to notice that the attractor at strong coupling of [@Kallosh:2013tua] can only be reproduced for $\alpha = \sqrt{2/3}$. For any other value for $\alpha$ the attractor is evaded.
- **Chaotic.** Let us consider: $$\tilde{g}(\varphi) = \left( \varphi - \varphi_{\mathrm{f}} \right)^\alpha \exp\left[ \sqrt{\frac{2}{3}} \left( \varphi - \varphi_{\mathrm{f}} \right) \right]$$ clearly $ \tilde{g}_{,\varphi}(\varphi) / \tilde{g}(\varphi) = \sqrt{2/3} + \alpha /( \varphi - \varphi_{\mathrm{f}})$ that gives the lowest order expression: $$\label{beta_general_ex}
\beta(\varphi) = \frac{- 2\alpha}{ \varphi - \varphi_{\mathrm{f}}}.$$ This case corresponds to the Chaotic class discussed in [@Binetruy:2014zya] and gives: $$\begin{aligned}
n_{s} &\simeq &1 - \frac{2+a}{2N}, \\
r & \simeq & \frac{4a}{N^2}.
\end{aligned}$$ In this case the attractor at strong coupling is clearly evaded.
- **Polynomial.** Let us consider: $$\tilde{g}(\varphi) = \left\{ 1 + \frac{P_1(\varphi)}{\exp \left[ \sqrt{\frac{2}{3}} \left( \varphi - \varphi_{\mathrm{f}} \right) \right]} \right\} \exp\left[ \sqrt{\frac{2}{3}} \left( \varphi - \varphi_{\mathrm{f}} \right) \right]$$
where $P_1(\varphi)$ is a polynomial in $\varphi$. It is possible to prove that in this case the lowest order expression for $\beta(\varphi)$ reads: $$\beta(\varphi) \sim \frac{P_2(\varphi)}{ \tilde{f}(\varphi)},$$ where $P_2(\varphi)$ is a polynomial in $\varphi$. It is possible to show that at the lowest order the expressions for $n_s$ and $r$ are: $$\begin{aligned}
n_{s} - 1 &\simeq& - \frac{2}{N}, \\
r & \simeq& \frac{12}{N^2} .
\end{aligned}$$ In this case the attractor is always preserved independently on the explicit expression for $P_1(\varphi)$.
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[^1]: We use the convention $ds^2 = dt^2 - a(t)^2 (dr^2 + r^2 d\Omega^2)$
[^2]: Notice that this definition may drive to an ambiguity as it implies $d\varphi / d \phi = \pm \sqrt{F(\phi)}$. To solve this problem we simply have to select a solution and be consistent with our choice. In this paper we will consider the $+$ solution. Clearly an equivalent treatment can be achieved in terms of the $-$ solution.
[^3]: As eq. implies: $$W(\varphi) = W_\textrm{f} \exp \left[ - \int_{\varphi_f}^{\varphi} \frac{\beta(\hat{\varphi})}{2} d\hat{\varphi} \right],$$ to be consistent with this approximation we need: $$\left| \beta(\varphi)\right|^2 \ll \left| \int_{\varphi_f}^{\varphi} \beta(\hat{\varphi}) d\hat{\varphi} \right|.$$ In the rest of this paper we will consider explicit expressions for $\beta(\varphi)$ that satisfy this requirement.
[^4]: The consistency of this assumption is discussed in the following.
[^5]: Notice that this is a specific feature of the parametrization of eq. . As discussed in Sec. \[sec:general\_case\], the attractor can be evaded under the quite general condition of eq.. Some explicit examples of the evasion are presented in the appendix \[Appendix\].
[^6]: $\kappa^2$ is set equal to 1.
| {
"pile_set_name": "ArXiv"
} |
[**Symmetric Powers of Galois Modules on Dedekind Schemes**]{}\
by\
> [. We prove a certain Riemann-Roch type formula for symmetric powers of Galois modules on Dedekind schemes which, in the number field or function field case, specializes to a formula of Burns and Chinburg for Cassou-Noguès-Taylor operations.]{}
Introduction {#introduction .unnumbered}
============
Let $G$ be a finite group and $E$ a number field. Let ${{\cal O}}_E$ denote the ring of integers in $E$, $Y:= {{\rm Spec}}({{\cal O}}_E)$, and $${{\rm Cl}}({{\cal O}}_YG) := \textrm{ker}({{\rm rank}}: K_0({{\cal O}}_EG) {\rightarrow}{{\mathbb Z}})$$ the locally free classgroup associated with $E$ and $G$. For any $k\ge 1$, Cassou-Noguès and Taylor have constructed a certain endomorphism $\psi_k^{{\rm CNT}}$ of ${{\rm Cl}}({{\cal O}}_YG)$ which, via Fröhlich’s Hom-description of ${{\rm Cl}}({{\cal O}}_YG)$, is dual to the $k$-th Adams operation on the classical ring of virtual characters of $G$ (see [@CNT]). Now, let $\gcd(k, {{\rm ord}}(G)) =1$ and $k'\in {{\mathbb N}}$ an inverse of $k$ modulo ${{\rm ord}}(G)$. In the paper [@KoCl], we have shown that then the endomorphism $\psi_{k'}^{{\rm CNT}}$ is a simply definable symmetric power operation $\sigma^k$.
Now, let $F/E$ be a finite tame Galois extension with Galois group $G$. Let $f: X:= {{\rm Spec}}({{\cal O}}_F) {\rightarrow}Y$ denote the corresponding $G$-morphism and $f_*$ the homomorphism $$f_*: K_0(G,X) {\rightarrow}{{\rm Cl}}({{\cal O}}_YG), \quad [{{\cal E}}] \mapsto
[f_*({{\cal E}})] - {{\rm rank}}({{\cal E}}) \cdot [{{\cal O}}_YG],$$ from the Grothendieck group $K_0(G,X)$ of all locally free ${{\cal O}}_X$-modules with (semilinear) $G$-action to ${{\rm Cl}}({{\cal O}}_YG)$. Furthermore, let ${{\cal D}}$ denote the different of $F/E$ and $\psi^k$ the $k$-th Adams operation on $K_0(G,X)$. The paper [@BC] by Burns and Chinburg together with the identification of $\psi_{k'}^{{\rm CNT}}$ with $\sigma^k$ mentioned above then implies the following Riemann-Roch type formula for all $x \in K_0(G,X)$: $$\sigma^k(f_*(x)) = f_*\left(\sum_{i=0}^{k'-1} [{{\cal D}}^{-ik}] \cdot \psi^k(x)\right)
\quad \textrm{in} \quad {{\rm Cl}}({{\cal O}}_YG)/{{\rm Ind}}_1^G{{\rm Cl}}({{\cal O}}_Y)$$ (see Theorem 5.6 and Theorem 3.7 in [@KoCl]).
We now assume that $Y$ is an arbitrary Dedekind scheme and that $X$ is the normalization of $Y$ in a finite Galois extension $F$ of the function field $E$ of $Y$ with Galois group $G$. We again assume that the corresponding $G$-morphism $f:X{\rightarrow}Y$ is tamely ramified. Similarly to the number field case, we define the locally free classgroup ${{\rm Cl}}({{\cal O}}_YG)$ (see section 2 or [@ABGr]), the symmetric power operation $\sigma^k$ on ${{\rm Cl}}({{\cal O}}_YG)$ (see sections 1 and 2), and the homomorphism $f_*: K_0(G,X) {\rightarrow}{{\rm Cl}}({{\cal O}}_YG)$ (see section 3). The object of this paper is to study the following natural question. Does the formula () still hold in this more general situation?
First of all, we mention that the paper [@BC] also implies that the formula () holds if $Y$ is a projective smooth curve over a finite field $L$ and the characteristic of $L$ does not divide the order of $G$ (see Theorem 3.5(b)). In this semisimple function field case, a Hom-description of ${{\rm Cl}}({{\cal O}}_YG)$ again exists and the operation $\sigma^k$ is dual to the Adams operation $\psi^{k'}$ as in the number field case (see Theorem 2.10). In particular, Fröhlich’s techniques can be applied as in the number field case (see [@BC]).
In this paper, we moreover obtain the following results whose proof however requires completely different methods since there is no Hom-description of ${{\rm Cl}}({{\cal O}}_YG)$ available in general.
[**Theorem A**]{}. The formula () holds if one of the following assumptions is satisfied:\
(a) $k=1$.\
(b) The group $G$ is Abelian and $f:X{\rightarrow}Y$ is unramified.
[**Theorem B**]{}. The formula () holds after passing from ${{\rm Cl}}({{\cal O}}_YG)/{{\rm Ind}}_1^G{{\rm Cl}}({{\cal O}}_Y)$ to $\hat{K}_0(G,Y)[k^{-1}]/({{\rm Ind}}_1^G K_0(Y))
\hat{K}_0(G,Y)[k^{-1}]$ via the Cartan homomorphism.\
Here, $K_0(G,Y)$ denotes the Grothendieck group of all locally free ${{\cal O}}_Y$-modules with $G$-action and $\hat{K}_0(G,Y)[k^{-1}]$ denotes the $I$-adic completion of $K_0(G,Y)[k^{-1}]$ where $I$ is the augmentation ideal of $K_0(G,Y)[k^{-1}]$.
The proof of Theorem A in the case $k=1$ relies on the results of the paper [@Ch] by Chase (see Proposition 3.2). Note that, despite the fact $\sigma^k = {{\rm id}}$ for $k=1$, the formula () is non-trivial since $k'$ may be an arbitrary natural number in the coset $1 + {{\rm ord}}(G){{\mathbb Z}}$. If $G$ is Abelian and $f:X{\rightarrow}Y$ is unramified, the proof of Theorem A relies on the following two facts (see Theorem 3.5). Firstly, applying the operation $\sigma^k$ to the element $[{{\cal Q}}]-[{{\cal P}}]$ in ${{\rm Cl}}({{\cal O}}_YG)$ is the same as pulling back the $G$-action on ${{\cal P}}$ and ${{\cal Q}}$ along the automorphism $G{\rightarrow}G$, $g \mapsto g^k$ (see Theorem 2.7). Secondly, the map $H^1(Y,G) {\rightarrow}{{\rm Cl}}({{\cal O}}_YG)$ which maps a principal $G$-bundle $f:X{\rightarrow}Y$ to the class $[f_*({{\cal O}}_X)] - [{{\cal O}}_YG]$ is a homomorphism (by Theorem 5 in the paper [@Wa] by Waterhouse). Theorem $B$ follows from the equivariant Adams-Riemann-Roch theorem (see [@KoGRR]) and the case $k=1$ of Theorem A (see Theorem 3.3). Moreover, in the semisimple function field case mentioned above, the formula () modulo torsion can be deduced from Theorem B if the order of $G$ is a power of a prime (see Remark 3.6).
[**Acknowledgments**]{}. I would like to thank David Burns for many very helpful discussions and for his hospitality during my stay at the King’s College in London. In particular, he has drawn my attention to the paper [@Ch] which is fundamental for the first case in Theorem A and also for Theorem B.
§1 Symmetric Power Operations on $K_0$-, $K_1$-, and Relative Grothendieck Groups {#symmetric-power-operations-on-k_0--k_1--and-relative-grothendieck-groups .unnumbered}
=================================================================================
Let $X$ be a Noetherian scheme and $G$ a finite group.
First, we introduce the category of locally projective modules over the group ring ${{\cal O}}_XG$. Then, we (purely algebraically) construct symmetric power operations on the Grothendieck group $K_0({{\cal O}}_XG)$ and the Bass group $K_1^{\det}({{\cal O}}_XG)$ associated with this category. While these constructions are more or less obvious generalizations of the constructions in section 1 of [@KoCl] (for $K_0$ and $K_1$), the subsequent construction of symmetric power operations on relative Grothendieck groups (in the sense of [@Ba]) is new. We furthermore show that these operations are compatible with the maps in the localization sequence. Finally, we present some cases in which the relative Grothendieck groups can be identified with Grothendieck groups of certain torsion modules.
By a (quasi-)coherent ${{\cal O}}_XG$-module we mean a (quasi-)coherent ${{\cal O}}_X$-module ${{\cal P}}$ together with an action of $G$ on ${{\cal P}}$ by ${{\cal O}}_X$-homomorphisms. Homomorphisms and exact sequences of quasi-coherent ${{\cal O}}_XG$-modules are defined in the obvious way. We call a coherent ${{\cal O}}_XG$-module ${{\cal P}}$ [*locally projective*]{} iff the stalk ${{\cal P}}_x$ is a projective ${{\cal O}}_{X,x}G$-module for all $x\in X$. Let $K_0({{\cal O}}_XG)$ denote the Grothendieck group of all locally projective ${{\cal O}}_XG$-modules.
[**Remark 1.1**]{}. Let $X= {{\rm Spec}}(A)$ be affine. Then, a finitely generated module $P$ over the group ring $AG$ is projective if and only if the corresponding coherent ${{\cal O}}_XG$-module ${{\cal P}}= \tilde{P}$ is locally projective; indeed, the exactness of the Hom-functor ${{\rm Hom}}_{AG}(P,-)$ is equivalent with the exactness of the Hom-functors ${{\rm Hom}}_{{{\cal O}}_{X,x}G}({{\cal P}}_x, -)$, $x\in X$. In particular, the Grothendieck group $K_0({{\cal O}}_XG)$ coincides with the usual Grothendieck group $K_0(AG)$ of all f. g. projective $AG$-modules.
We are now going to construct the above-mentioned symmetric power operations. As in section 1 of [@KoCl], it is convenient to introduce the following categories. For any $i\ge 1$, let ${{\cal M}}_i$ denote the smallest full subcategory of the Abelian category of all coherent ${{\cal O}}_XG$-modules which is closed under extensions and kernels of ${{\cal O}}_XG$-epimorphisms and which contains all the modules of the form ${{\rm Sym}}_{{{\cal O}}_X}^{i_1}({{\cal P}}_1) \otimes_{{{\cal O}}_X} \ldots
\otimes _{{{\cal O}}_X} {{\rm Sym}}_{{{\cal O}}_X}^{i_r}({{\cal P}}_r)$ where ${{\cal P}}_1, \ldots, {{\cal P}}_r$ are locally projective coherent ${{\cal O}}_XG$-modules, $i_1, \ldots, i_r$ are natural numbers with $i_1 + \ldots + i_r = i$, and $G$ acts diagonally. So, ${{\cal M}}_1$ is the category of all locally projective coherent ${{\cal O}}_XG$-modules. By Proposition 1.1 in [@KoCl], the category ${{\cal M}}_i$ is contained in ${{\cal M}}_1$ if $\gcd(i, {{\rm ord}}(G))$ is invertible on $X$. It is easy to see that, for all $i,j
\ge 1$, the functor $${{\cal M}}_i \times {{\cal M}}_j {\rightarrow}{{\cal M}}_{i+j}, \quad ({{\cal P}}, {{\cal Q}}) \mapsto {{\cal P}}\otimes_{{{\cal O}}_X}{{\cal Q}},$$ is well-defined and bi-exact (cf. Lemma 1.2 in [@KoCl]). In particular, we obtain products $K_0({{\cal M}}_i) \times K_0({{\cal M}}_j) {\rightarrow}K_0({{\cal M}}_{i+j})$, $i,j \ge 1$, and the set $1+\prod_{i\ge 1}K_0({{\cal M}}_i)t^i$ consisting of all power series $1+\sum_{i\ge 1}a_i t^i$ with $a_i \in K_0({{\cal M}}_i)$ forms an Abelian group with respect to multiplication of power series. As usual, one shows that the association $[{{\cal P}}] \mapsto \sum_{i\ge 0} [{{\rm Sym}}^i_{{{\cal O}}_X}({{\cal P}})]t^i$ can be extended to a well-defined homomorphism $$\sigma: K_0({{\cal O}}_XG) {\rightarrow}1+\prod_{i\ge 1}K_0({{\cal M}}_i) t^i$$ (see §1 of Chapter V in [@FL] and Lemma 1.3 in [@KoCl]). The $i$-th component of this homomorphism is denoted by $\sigma^i$. We have for all $x,y \in K_0({{\cal O}}_XG)$: $$\begin{aligned}
\sigma^i(x-y) &=& \sum_{{a\ge 0, b_1, \ldots, b_u \ge 1} \atop {a+b_1+\ldots+b_u =i}}
(-1)^u \sigma^a(x) \sigma^{b_1}(y) \cdots \sigma^{b_u}(y) \\
&=& \sum_{{a,b_1, \ldots, b_u \ge 1} \atop {a+b_1+\ldots+b_u =i}}
(-1)^u (\sigma^a(x)-\sigma^a(y))\sigma^{b_1}(y) \cdots \sigma^{b_u}(y)\end{aligned}$$ in $K_0({{\cal M}}_i)$ (cf. section 2 in [@Gr2]). If $\gcd(i,{{\rm ord}}(G))$ is invertible on $X$, let $\sigma^i$ also denote the composition $$K_0({{\cal O}}_XG) \,\, \stackrel{\sigma^i}{\longrightarrow}\,\,
K_0({{\cal M}}_i) \,\, \stackrel{{\rm can}}{\longrightarrow}\,\, K_0({{\cal O}}_XG).$$ The map $\sigma^i$ is called [*$i$-th symmetric power operation*]{}.
Now, let $K_0({{\mathbb Z}},{{\cal M}}_i)$ denote the Grothendieck group of all pairs $({{\cal P}},\alpha)$ where ${{\cal P}}$ is an object of ${{\cal M}}_i$ and $\alpha$ is an ${{\cal O}}_XG$-automorphism of ${{\cal P}}$. We put $K_0({{\mathbb Z}},{{\cal O}}_XG):=K_0({{\mathbb Z}},{{\cal M}}_1)$. As above, the association $(({{\cal P}},\alpha), ({{\cal Q}},\beta)) \mapsto ({{\cal P}}\otimes_{{{\cal O}}_X}
{{\cal Q}}, \alpha \otimes_{{{\cal O}}_X} \beta)$ induces a multiplication map $$K_0({{\mathbb Z}}, {{\cal M}}_i) \times K_0({{\mathbb Z}}, {{\cal M}}_j) {\rightarrow}K_0({{\mathbb Z}},{{\cal M}}_{i+j})$$ (for all $i, j \ge 1$) and the association $({{\cal P}},\alpha) \mapsto
\sum_{i\ge 0}({{\rm Sym}}^i_{{{\cal O}}_X}({{\cal P}}), {{\rm Sym}}^i_{{{\cal O}}_X}(\alpha))t^i$ induces a homomorphism $$\sigma: K_0({{\mathbb Z}},{{\cal O}}_XG) {\rightarrow}1+\prod_{i\ge 1}K_0({{\mathbb Z}}, {{\cal M}}_i)t^i.$$ By restricting, we obtain symmetric power operations $$\sigma^i: \tilde{K}_0({{\mathbb Z}},{{\cal O}}_XG)=\tilde{K}_0({{\mathbb Z}}, {{\cal M}}_1) {\rightarrow}\tilde{K}_0({{\mathbb Z}},{{\cal M}}_i),\quad i\ge1,$$ between the reduced Grothendieck groups $$\tilde{K}_0({{\mathbb Z}},{{\cal M}}_i) := \textrm{ker}(K_0({{\mathbb Z}},{{\cal M}}_i) {\rightarrow}K_0({{\cal M}}_i), \;
[{{\cal P}},\alpha] \mapsto [{{\cal P}}]), \quad i\ge1.$$ We denote the factor group of $K_0({{\mathbb Z}},{{\cal M}}_i)$ modulo the subgroup generated by the relations of the form $[{{\cal P}},\alpha\beta]-[{{\cal P}},\alpha]-[{{\cal P}},\beta]$ by $K_1^{\det}({{\cal M}}_i)$. If $X={{\rm Spec}}(A)$ is affine, the group $K_1^{\det}({{\cal O}}_XG) =
K_1^{\det}({{\cal M}}_1)$ coincides with the usual Bass-Whitehead group $K_1(AG)$ of the group ring $AG$ (by Remark 1.1). In the sequel, we consider $K_1^{\det}({{\cal M}}_i)$ as the factor group of $\tilde{K}_0({{\mathbb Z}},{{\cal M}}_i)$ modulo the subgroup $I_i$ generated by the relations of the form $[{{\cal P}},\alpha\beta] - [{{\cal P}},\alpha] - [{{\cal P}},\beta] + [{{\cal P}},{{\rm id}}]$. Since $$\begin{aligned}
\lefteqn{([{{\cal P}},\alpha\beta] - [{{\cal P}},\alpha] - [{{\cal P}},\beta] + [{{\cal P}},{{\rm id}}]) \cdot
[{{\cal Q}},\gamma]}\\
&=& \left([{{\cal P}}\otimes{{\cal Q}}, \alpha\beta\otimes\gamma] - [{{\cal P}}\otimes{{\cal Q}}, \alpha\otimes\gamma]
- [{{\cal P}}\otimes {{\cal Q}}, \beta \otimes {{\rm id}}] + [{{\cal P}}\otimes{{\cal Q}}, {{\rm id}}\otimes {{\rm id}}]\right) \\
&& - \left([{{\cal P}}\otimes{{\cal Q}}, \beta\otimes\gamma] - [{{\cal P}}\otimes{{\cal Q}},{{\rm id}}\otimes\gamma]
-[{{\cal P}}\otimes{{\cal Q}}, \beta\otimes{{\rm id}}] + [{{\cal P}}\otimes{{\cal Q}}, {{\rm id}}\otimes {{\rm id}}] \right),\end{aligned}$$ the group $I_iK_0({{\mathbb Z}},{{\cal M}}_j)$ is contained in $I_{i+j}$ and we obtain a multiplication map $$K_1^{\det}({{\cal M}}_i) \times K_1^{\det}({{\cal M}}_j) = \tilde{K}_0({{\mathbb Z}},{{\cal M}}_i)/I_i \times
\tilde{K}_0({{\mathbb Z}},{{\cal M}}_j)/I_j {\rightarrow}K_1^{\det}({{\cal M}}_{i+j})$$ (for all $i,j \ge 1$) which is obviously trivial, i.e., the product of any two power series $\sum_{i\ge 0}x_it^i$, $\sum_{i\ge 0} y_it^i$ in $1 + \prod_{i\ge 1}K_1^{\det}({{\cal M}}_i)t^i$ is $1+ \sum_{i\ge 1}(x_i + y_i)t_i$.
[**Lemma 1.2**]{}. The homomorphism $\sigma: \tilde{K}_0({{\mathbb Z}}, {{\cal O}}_XG) {\rightarrow}1+\prod_{i\ge 1}\tilde{K}_0({{\mathbb Z}},{{\cal M}}_i)t^i$ induces a homomorphism $\sigma: K_1^{\det}({{\cal O}}_XG) {\rightarrow}1 + \prod_{i\ge 1} K_1^{\det}({{\cal M}}_i)t^i$. Each component $\sigma^i: K_1^{\det}({{\cal O}}_XG) {\rightarrow}K_1^{\det}({{\cal M}}_i)$ of $\sigma$ is a homomorphism.
[**Proof**]{}. Let ${{\cal P}}\in {{\cal M}}_1$ and $\alpha, \beta \in {{\rm Aut}}_{{{\cal O}}_XG}({{\cal P}})$. We write ${{\rm S}}$ for ${{\rm Sym}}$. Then, for all $a \ge 1$, the element $$\begin{aligned}
\lefteqn{[{{\rm S}}^a({{\cal P}}\oplus{{\cal P}}), {{\rm S}}^a(\alpha\beta \oplus {{\rm id}})] -
[{{\rm S}}^a({{\cal P}}\oplus{{\cal P}}), {{\rm S}}^a(\alpha \oplus \beta)] }\\
&=&\sum_{c=0}^a \Big([{{\rm S}}^c({{\cal P}})\otimes {{\rm S}}^{a-c}({{\cal P}}),
{{\rm S}}^c(\alpha\beta) \otimes {{\rm S}}^{a-c}({{\rm id}})]
- [{{\rm S}}^c({{\cal P}}) \otimes {{\rm S}}^{a-c}({{\cal P}}), {{\rm S}}^c(\alpha) \otimes {{\rm id}}]\\
&&-[{{\rm S}}^c({{\cal P}})\otimes {{\rm S}}^{a-c}({{\cal P}}), {{\rm S}}^c(\beta) \otimes {{\rm id}}] +
[{{\rm S}}^c({{\cal P}})\otimes{{\rm S}}^{a-c}({{\cal P}}), {{\rm id}}\otimes {{\rm id}}]\Big)\\
&& - \sum_{c=0}^a \Big([{{\rm S}}^c({{\cal P}}) \otimes {{\rm S}}^{a-c}({{\cal P}}), {{\rm S}}^c(\alpha) \otimes
{{\rm S}}^{a-c}(\beta)]
- [{{\rm S}}^c({{\cal P}}) \otimes {{\rm S}}^{a-c}({{\cal P}}), {{\rm S}}^c(\alpha) \otimes {{\rm id}}]\\
&&-[{{\rm S}}^c({{\cal P}}) \otimes {{\rm S}}^{a-c}({{\cal P}}), {{\rm id}}\otimes {{\rm S}}^{a-c}(\beta)]
+[{{\rm S}}^c({{\cal P}}) \otimes {{\rm S}}^{a-c}({{\cal P}}), {{\rm id}}\otimes {{\rm id}}]\Big)\end{aligned}$$ is contained in $I_a$. Since $$\sigma^i(x-y) = \sum_{{a,b_1, \ldots, b_u \ge 1} \atop {a+b_1+\ldots+b_u =i}}
(-1)^u(\sigma^a(x) - \sigma^a(y)) \sigma^{b_1}(y) \cdots \sigma^{b_u}(y)$$ (for all $x,y \in K_0({{\mathbb Z}}, {{\cal M}}_1)$), this implies that the element $$\sigma^i\left([{{\cal P}},\alpha\beta]-[{{\cal P}},\alpha]-[{{\cal P}},\beta]+[{{\cal P}},{{\rm id}}]\right)
=\sigma^i\left([{{\cal P}}\oplus {{\cal P}},\alpha \beta \oplus {{\rm id}}] - [{{\cal P}}\oplus {{\cal P}},\alpha \oplus \beta]\right)$$ is contained in $I_i$, as was to be shown. For all $x,y \in K_1^{\det}({{\cal M}}_1)$, we have $$\sigma(x+y) = \sigma(x) \cdot \sigma(y) = 1+\sum_{i\ge 1}
(\sigma^i(x) + \sigma^i(y))t^i \quad \textrm{in} \quad 1+\prod_{i\ge 1}
K_1^{\det}({{\cal M}}_i)t^i;$$ thus, $\sigma^i$ is a homomorphism for all $i\ge 1$.
Now, let $j: U {\rightarrow}X$ be a morphism between Noetherian schemes. Similarly to §5 of Chapter VII in [@Ba], let $K_0({{\rm co}}(j_i^*))$ denote the Grothendieck group of all triples $({{\cal P}}, \alpha, {{\cal Q}})$ where ${{\cal P}}$ and ${{\cal Q}}$ are objects in ${{\cal M}}_i$ and $\alpha: j^*({{\cal P}}) {\rightarrow}j^*({{\cal Q}})$ is an ${{\cal O}}_UG$-isomorphism. As above, the association $$(({{\cal P}},\alpha, {{\cal Q}}),({{\cal P}}',\alpha',{{\cal Q}}')) \mapsto
({{\cal P}}\otimes_{{{\cal O}}_X}{{\cal P}}', \alpha\otimes_{{{\cal O}}_U}\alpha', {{\cal Q}}\otimes_{{{\cal O}}_X}{{\cal Q}}')$$ induces, for all $i,i'\ge 1$, a multiplication map $$K_0({{\rm co}}(j_i^*)) \times K_0({{\rm co}}(j_{i'}^*)) {\rightarrow}K_0({{\rm co}}(j_{i+i'}^*))$$ and the association $({{\cal P}}, \alpha, {{\cal Q}}) \mapsto \sum_{i\ge 0}
({{\rm Sym}}^i_{{{\cal O}}_X}({{\cal P}}), {{\rm Sym}}^i_{{{\cal O}}_U}(\alpha), {{\rm Sym}}^i_{{{\cal O}}_X}({{\cal Q}}))t^i$ induces a homomorphism $$\sigma: K_0({{\rm co}}(j_1^*)) {\rightarrow}1+ \prod_{i\ge 1} K_0({{\rm co}}(j_i^*))t^i.$$ By restricting, we obtain symmetric power operations $$\sigma^i: \tilde{K}_0({{\rm co}}(j_1^*)) {\rightarrow}\tilde{K}_0({{\rm co}}(j_i^*)), \quad i\ge 1,$$ between the reduced Grothendieck groups $$\tilde{K}_0({{\rm co}}(j_i^*)):= \textrm{ker}(K_0({{\rm co}}(j_i^*)) {\rightarrow}K_0({{\cal M}}_i), \;
[{{\cal P}},\alpha,{{\cal Q}}] \mapsto [{{\cal P}}]).$$ Let $K_0(j_i^*)$ denote the factor group of $K_0({{\rm co}}(j_i^*))$ modulo the subgroup generated by the relations of the form $[{{\cal P}}, \beta \alpha, {{\cal R}}]
-[{{\cal P}}, \alpha, {{\cal Q}}] - [{{\cal Q}}, \beta, {{\cal R}}]$ (see also Proposition (5.1) on p. 370 in [@Ba]). In the sequel, we consider $K_0(j_i^*)$ as the factor group of $\tilde{K}_0({{\rm co}}(j_i^*))$ modulo the subgroup $I_i$ generated by the elements of the form $[{{\cal P}},\beta \alpha, {{\cal R}}] - [{{\cal P}}, \alpha, {{\cal Q}}]
- [{{\cal Q}}, \beta, {{\cal R}}] + [{{\cal Q}}, {{\rm id}}, {{\cal Q}}]$. As above, one easily sees that $I_i K_0({{\rm co}}(j_{i'}^*))$ is contained in $I_{i+i'}$ and we obtain a multiplication map $$K_0(j_i^*) \times K_0(j_{i'}^*) {\rightarrow}K_0(j_{i+i'}^*)$$ for all $i,i' \ge 1$ which however (in contrast to $K_1^{\det}$) seems not to be trivial in general.
[**Lemma 1.3**]{}. The homomorphism $\sigma: \tilde{K}_0({{\rm co}}(j_1^*)) {\rightarrow}1+ \prod _{i\ge 1} \tilde{K}_0({{\rm co}}(j_i^*))t^i$ induces a homomorphism $\sigma: K_0(j_1^*) {\rightarrow}1+ \prod_{i\ge 1} K_0(j_i^*) t^i$.
[**Proof**]{}. Similarly to Lemma 1.2.
The association $[{{\cal P}}, \alpha, {{\cal Q}}] \mapsto [{{\cal Q}}] - [{{\cal P}}]$ obviously defines a homomorphism $$\nu_i: K_0(j_i^*) {\rightarrow}K_0({{\cal M}}_i)$$ for all $i\ge 1$.
[**Lemma 1.4**]{}. The multiplication maps are compatible with the homomorphisms $\nu_i$, $i\ge 1$. The same holds for the symmetric power operations $\sigma^i$, $i\ge 1$; i.e., the following diagram commutes for all $i\ge 1$: $$\xymatrix{
K_0(j_1^*) \ar[r]^{\nu_1} \ar[d]^{\sigma^i}& K_0({{\cal M}}_1) \ar[d]^{\sigma^i}\\
K_0(j_i^*) \ar[r]^{\nu_i} & K_0({{\cal M}}_i). }$$
[**Proof**]{}. We only prove the assertion for $\sigma^i$. Let ${{\cal P}}, {{\cal Q}}, {{\cal R}}\in {{\cal M}}_1$ and $\alpha: j^*({{\cal P}}) \,\, \tilde{{\rightarrow}}\,\, j^*({{\cal Q}})$, $\beta: j^*({{\cal Q}}) \,\,
\tilde{{\rightarrow}} \,\, j^*({{\cal R}})$ ${{\cal O}}_UG$-isomorphisms. We again write ${{\rm S}}$ for ${{\rm Sym}}$. Then we have in $K_0({{\cal M}}_i)$: $$\begin{aligned}
\lefteqn{\nu_i\sigma^i({{\cal P}},\alpha,{{\cal Q}}) = \nu_i\sigma^i(({{\cal P}},\alpha,{{\cal Q}})-({{\cal P}},{{\rm id}},{{\cal P}}))}\\
&=& \nu_i\Bigg(\sum_{{a \ge 0, b_1, \ldots, b_u \ge 1} \atop {a+b_1+\ldots+b_u = i}}
(-1)^u \Big({{\rm S}}^a({{\cal P}}) \otimes {{\rm S}}^{b_1}({{\cal P}})\otimes \ldots \otimes {{\rm S}}^{b_u}({{\cal P}}),
{{\rm S}}^a(\alpha) \otimes {{\rm id}}\otimes \ldots \otimes {{\rm id}}, \\
&& \hspace*{43ex}{{\rm S}}^a({{\cal Q}}) \otimes {{\rm S}}^{b_1}({{\cal P}}) \otimes
\ldots \otimes {{\rm S}}^{b_u}({{\cal P}})\Big)\Bigg)\\
&=& \sum_{{a,b_1, \ldots, b_u \ge 1}\atop {a+b_1+\ldots+ b_u=i}}(-1)^u
\left([{{\rm S}}^a({{\cal Q}})] - [{{\rm S}}^a({{\cal P}})]\right) \cdot [{{\rm S}}^{b_1}({{\cal P}})\otimes \ldots
\otimes {{\rm S}}^{b_u}({{\cal P}})]\\
&=&\sigma^i([{{\cal Q}}]-[{{\cal P}}]) = \sigma^i\nu_1({{\cal P}},\alpha,{{\cal Q}}).\end{aligned}$$
We now assume that $U = {{\rm Spec}}(F)$ is affine. Then, by Proposition (2.1) on p. 393 in [@Ba], the association $({\mathop{\oplus}\limits}^m FG,\alpha) \mapsto
({\mathop{\oplus}\limits}^m{{\cal O}}_XG,\alpha,{\mathop{\oplus}\limits}^m {{\cal O}}_XG)$ induces a [*connecting homomorphism*]{} $$\partial: K_1(FG){\rightarrow}K_0(j_1^*)$$ with $\nu_1 \circ \partial=0$.
[**Lemma 1.5**]{}. Let $\gcd(i, {{\rm ord}}(G))$ be invertible on $X$. Then we have: $$\sigma^i \circ \partial = \partial \circ \sigma^i \quad \textrm{in} \quad
{{\rm Hom}}(K_1(FG),K_0(j_1^*)).$$ The multiplication maps are compatible with $\partial$ (in the obvious sense), too. In particular, the multiplication on $\textrm{Image}(\partial)$ is trivial and the operation $\sigma^i$ is a homomorphism on $\textrm{Image}(\partial)$.
[**Proof**]{}. Easy.
[**Proposition 1.6**]{}. The following sequence is exact: $$K_1(FG) \,\, \stackrel{\partial}{{\rightarrow}} \,\, K_0(j_1^*) \,\,
\stackrel{\nu_1}{{\rightarrow}} \,\, K_0({{\cal O}}_XG)\,\, \stackrel{j^*}{{\rightarrow}}\,\, K_0(FG).$$
[**Proof**]{}. Apply Theorem (2.2)(b) on p. 396 in [@Ba].
Now, let ${{\cal H}}$ denote the category of all coherent ${{\cal O}}_XG$-modules ${{\cal V}}$ which allow a resolution by locally projective coherent ${{\cal O}}_XG$-modules of length $\le 1$ and for which $j^*({{\cal V}})=0$ holds. Furthermore, let $K_0T({{\cal O}}_XG)$ denote the Grothendieck group of ${{\cal H}}$. By mapping the class $[{{\cal V}}]$ of a coherent ${{\cal O}}_XG$-module ${{\cal V}}$ with the resolution $0 {\rightarrow}{{\cal P}}\,\, \stackrel{\alpha}{{\rightarrow}}\,\,
{{\cal Q}}{\rightarrow}{{\cal V}}{\rightarrow}0$ and with $j^*({{\cal V}}) =0$ to the element $({{\cal P}}, j^*(\alpha), {{\cal Q}})$ in $K_0(j_1^*)$, we obviously obtain a homomorphism $$\psi: K_0T({{\cal O}}_XG) {\rightarrow}K_0(j_1^*).$$
[**Proposition 1.7**]{}. The homomorphism $\psi$ is bijective in the following cases:\
(a) $X = {{\rm Spec}}(A)$ is affine, $F$ is the localization $A_S$ of $A$ by a multiplicative set $S$ of non-zero-divisors in $A$, and $j: U={{\rm Spec}}(F) {\rightarrow}X={{\rm Spec}}(A)$ is the canonical morphism.\
(b) The morphism $j:U={{\rm Spec}}(F) {\rightarrow}X$ is an open immersion and the ideal ${{\cal I}}$ of the complement $Y:=X\backslash U$ is locally generated by a non-zero-divisor.\
(c) $X$ is a Dedekind scheme (i.e., Noetherian, regular, irreducible, and $\dim(X) =1$), $F$ is the function field of $X$ and $j:U={{\rm Spec}}(F) {\rightarrow}X$ is the canonical morphism.
[**Proof**]{}. The assertion (a) follows from (the proof of) Theorem (5.8) on p. 429 in [@Ba]. In the case (b), we construct an inverse map as follows: Let $({{\cal P}},\alpha,{{\cal Q}})$ be a generator of $K_0(j_1^*)$. Then, the image of the composition $$\tilde{\alpha}: {{\cal P}}\,\,\stackrel{{\rm can}}{\longrightarrow}\,\, j_*j^*({{\cal P}}) \,\,
\stackrel{j_*(\alpha)}{\longrightarrow} \,\, j_*j^*({{\cal Q}}) =
\cup_{n\ge 0} {{\cal I}}^{-n}{{\cal Q}}$$ (see Lemma 2 on p. 231 in [@Gr1] for the last equality) is contained in ${{\cal I}}^{-n}{{\cal Q}}$ for some $n\ge 0$. We put $$\phi({{\cal P}},\alpha,{{\cal Q}}):= [{{\rm coker}}({{\cal P}}\,\,\stackrel{\tilde{\alpha}}{\hookrightarrow}\,\,
{{\cal I}}^{-n}{{\cal Q}})]-[{{\rm coker}}({{\cal Q}}\,\,\stackrel{{\rm can}}{\hookrightarrow}\,\,
{{\cal I}}^{-n}{{\cal Q}})] \in K_0T({{\cal O}}_XG).$$ As in loc. cit., one easily checks that the association $({{\cal P}},\alpha,{{\cal Q}}) \mapsto
\phi({{\cal P}},\alpha,{{\cal Q}})$ induces a well-defined map $\phi: K_0(j_1^*){\rightarrow}K_0T({{\cal O}}_XG)$ which is an inverse of $\psi$. In the case (c), we construct an inverse map as follows. Let $({{\cal P}},\alpha,{{\cal Q}})$ be a generator of $K_0(j_1^*)$. The isomorphism $\alpha: j^*({{\cal P}}) \,\, \tilde{{\rightarrow}}\,\, j^*({{\cal Q}})$ can be extended to an isomorphism ${{\cal P}}|_U \,\, \tilde{{\rightarrow}}\,\, {{\cal Q}}|_U$ where $U$ is an open subset of $X$. The ideal ${{\cal I}}$ of the complement $Y:= X\backslash U$ is then locally generated by a non-zero-divisor. We now define $\phi({{\cal P}},\alpha,{{\cal Q}})$ as in the case (b). As in loc. cit., one again easily checks that the association $({{\cal P}}, \alpha, {{\cal Q}})
\mapsto \phi({{\cal P}},\alpha,{{\cal Q}})$ induces a well-defined map $\phi: K_0(j_1^*) {\rightarrow}K_0T({{\cal O}}_XG)$ which is an inverse of $\psi$.
[**Remark 1.8**]{}. We assume that one of the conditions (a), (b), (c) of Proposition 1.7 holds.\
(a) The $K$-theory space of the exact category ${{\cal H}}$ is homotopy equivalent to the homotopy fibre of the canonical continuous map from the $K$-theory space of ${{\cal M}}_1$ to the $K$-theory space of the exact category consisting of all f. g. projective $FG$-modules (see [@Gr1] and [@ABGr]). Hence, we have a long exact (localization) sequence $$\ldots {\rightarrow}K_1(FG) {\rightarrow}K_0T({{\cal O}}_XG) {\rightarrow}K_0({{\cal O}}_XG) {\rightarrow}K_0(FG).$$ The end of this sequence can be identified with the exact sequence in Proposition 1.6 by virtue of Proposition 1.7.\
(b) If $\gcd(i,{{\rm ord}}(G))$ is invertible on $X$, we obtain a symmetric power operation $\sigma^i:K_0T({{\cal O}}_XG) {\rightarrow}K_0T({{\cal O}}_XG)$ by virtue of the isomorphism $\psi$. It maps the class $[{{\cal V}}]$ of a coherent ${{\cal O}}_XG$-module ${{\cal V}}$ in ${{\cal H}}$ with the resolution $0{\rightarrow}{{\cal P}}\,\, \stackrel{\alpha}{{\rightarrow}} \,\, {{\cal Q}}{\rightarrow}{{\cal V}}{\rightarrow}0$ to the element $$\begin{aligned}
\lefteqn{
{\mathop{\sum}\limits}_{{a, b_1, \ldots, b_u\ge 1} \atop {a+b_1+\ldots + b_u=i}} (-1)^u
\Big[{{\rm coker}}\Big({{\rm Sym}}^a({{\cal P}}) \otimes
{{\rm Sym}}^{b_1}({{\cal P}}) \otimes \ldots \otimes {{\rm Sym}}^{b_u}({{\cal P}})}\\
&&\xymatrix{ {}\ar[rrr]^{{{\rm Sym}}^a(\alpha)\otimes {{\rm id}}\otimes \ldots \otimes {{\rm id}}} &&&{}}
{{\rm Sym}}^a({{\cal Q}}) \otimes {{\rm Sym}}^{b_1}({{\cal P}}) \otimes \ldots \otimes {{\rm Sym}}^{b_u}({{\cal P}})\Big)\Big]. \end{aligned}$$ Alternatively, the operation $\sigma^i$ on $K_0T({{\cal O}}_XG)$ can also be constructed as follows. Let ${{\cal E}}$ denote the exact category of all short exact sequences $0{\rightarrow}{{\cal P}}{\rightarrow}{{\cal Q}}{\rightarrow}{{\cal V}}{\rightarrow}0$ with ${{\cal P}}, {{\cal Q}}\in {{\cal M}}_1$ and ${{\cal V}}\in {{\cal H}}$. Then, we have a canonical isomorphism $$K_0T({{\cal O}}_XG) = K_0({{\cal H}}) \cong \textrm{ker}(K_0({{\cal E}}) {\rightarrow}K_0({{\cal O}}_XG), \;
[0{\rightarrow}{{\cal P}}{\rightarrow}{{\cal Q}}{\rightarrow}{{\cal V}}{\rightarrow}0] \mapsto [{{\cal P}}]).$$ The association $$[0{\rightarrow}{{\cal P}}\,\, \stackrel{\alpha}{{\rightarrow}}\,\, {{\cal Q}}{\rightarrow}{{\cal V}}{\rightarrow}0]
\mapsto [0 {\rightarrow}{{\rm Sym}}^i({{\cal P}}) \,\, \stackrel{{{\rm Sym}}^i(\alpha)}{\longrightarrow}\,\,
{{\rm Sym}}^i({{\cal Q}}) {\rightarrow}{{\rm coker}}({{\rm Sym}}^i(\alpha)){\rightarrow}0]$$ induces an operation $\sigma^i$ on $K_0({{\cal E}})$ as usual. It is then easy to check that its restriction to $K_0T({{\cal O}}_XG)$ coincides with the operation $\sigma^i$ constructed above. Moreover, the latter construction can be extended to all higher $K$-groups $K_q({{\cal H}})$, $q\ge 0$, by using the methods of [@Gr2]. On the other hand, we have a symmetric power operation $\sigma^i$ on the $K$-theory space of ${{\cal M}}_1$ and on the $K$-theory space of the category consisting of all f. g. projective modules (see section 1 in [@KoCl]), hence also on the homotopy fibre mentioned in (a) and finally on $K_q({{\cal H}})$, $q\ge 0$. It seems to be plausible that these two constructions of $\sigma^i$ on $K_q({{\cal H}})$, $q \ge 0$, coincide. I hope to say more on this in a future paper.
§2 Symmetric Power Operations on Locally Free Classgroups of Dedekind Schemes {#symmetric-power-operations-on-locally-free-classgroups-of-dedekind-schemes .unnumbered}
=============================================================================
Let $X$ be a Dedekind scheme (i.e., Noetherian, regular, irreducible and $\dim(X) \le 1$) with function field $F$, and let $G$ be a finite group.
First, we recall the definition of the locally free classgroup ${{\rm Cl}}({{\cal O}}_XG)$ (see [@ABGr] or [@BC]). Using the tools developed in section 1 and Hattori’s theorem, we then show that the locally free classgroup coincides with the analogously defined locally projective classgroup and that the operations $\sigma^i$, $i\ge 1$, constructed in section 1 are homomorphisms on ${{\rm Cl}}({{\cal O}}_XG)$. Furthermore, we prove the following concrete interpretations of the operations $\sigma^i$, $i\ge 1$, on ${{\rm Cl}}({{\cal O}}_XG)$. Firstly, if $G$ is Abelian and $\gcd(i,{{\rm ord}}(G))=1$, then pulling back the action of $G$ on locally free ${{\cal O}}_XG$-modules along the automorphism $G{\rightarrow}G, \; g\mapsto g^i$, induces the operation $\sigma^i$ on ${{\rm Cl}}({{\cal O}}_XG)$. Secondly, if $X$ is a smooth curve over an (algebraically closed or) finite field $L$ such that the characteristic of $L$ does not divide the order of $G$, then the identification of the locally free with the locally projective classgroup allows us a simple module theoretic description of the isomorphism between ${{\rm Cl}}({{\cal O}}_XG)$ and ${{\rm Hom}}_{{\rm Galois}}
(K_0(\bar{L}G),{{\rm Cl}}(\bar{X}))$ (developed in [@ABGr]), and the operation $\sigma^i$ on ${{\rm Cl}}({{\cal O}}_XG)$ is dual to the adjoint Adams operation $\hat{\psi}^i$ on $K_0(\bar{L}G)$ with respect to this isomorphism. The proof of the latter result presented here can also be applied in the number field case and then simplifies the proof of Theorem 3.7 in [@KoCl].
A coherent ${{\cal O}}_XG$-module ${{\cal P}}$ is called [*locally free over ${{\cal O}}_XG$*]{} iff the stalk ${{\cal P}}_x$ is a free ${{\cal O}}_{X,x}G$-module for all $x \in X$. By Proposition (30.17) on p. 627 in [@CR], this is equivalent to the condition that ${{\cal P}}_x \otimes_{{{\cal O}}_{X,x}}\hat{{{\cal O}}}_{X,x}$ is a free $\hat{{{\cal O}}}_{X,x}G$-module for all closed points $x\in X$. (Here, $\hat{{{\cal O}}}_{X,x}$ denotes the ${\mathfrak m}_x$-adic completion of ${{\cal O}}_{X,x}$ and ${\mathfrak m}_x$ the maximal ideal in ${{\cal O}}_{X,x}$.) Let $K_0^{{\rm lf}}({{\cal O}}_XG)$ denote the Grothendieck group of all coherent ${{\cal O}}_XG$-modules which are locally free over ${{\cal O}}_XG$.
[**Remark 2.1**]{}. Let $X={{\rm Spec}}(A)$ be affine. Then we also write $K_0^{{\rm lf}}(AG)$ for $K_0^{{\rm lf}}({{\cal O}}_XG)$. This is the Grothendieck group considered for instance in [@F1]. If $A$ is a local Dedekind domain, then the rank (over $AG$) induces an isomorphism $K_0^{{\rm lf}}(AG) \,\, \tilde{{\rightarrow}}\,\, {{\mathbb Z}}$. If ${\rm char}(A) =0$ and and no prime divisor of ${{\rm ord}}(G)$ is a unit in $A$, then any f. g.projective $AG$-module is already locally free by Swan’s theorem (see Theorem (32.11) on p. 676 in [@CR]). The same holds if $p={\rm char}(A) > 0$ and $G$ is a $p$-group since then the group rings ${{\cal O}}_{X,x}G$, $x\in X$, are local rings. We will prove in Proposition 2.4 that the locally free classgroup defined below always coincides with the analogously defined locally projective classgroup.
[**Definition 2.2**]{}. The group $${{\rm Cl}}({{\cal O}}_XG) := \textrm{ker}(K_0^{{\rm lf}}({{\cal O}}_XG) \,\, \stackrel{{\rm can}}{\longrightarrow}
\,\, K_0^{{\rm lf}}(FG) \cong {{\mathbb Z}})$$ is called the [*locally free classgroup associated with $X$ and $G$*]{}.
Let $K_0T({{\cal O}}_XG)$ (resp., $K_0^{{\rm lf}}T({{\cal O}}_XG)$) denote the Grothendieck group of all coherent ${{\cal O}}_XG$-modules which are ${{\cal O}}_X$-torsion modules and which allow a resolution of length $\le 1$ by locally projective (resp., locally free) ${{\cal O}}_XG$-modules. The notation $K_0T({{\cal O}}_XG)$ obviously agrees with the notation introduced in section 1 (if $j:U={{\rm Spec}}(F) {\rightarrow}X$ is the canonical morphism).
[**Lemma 2.3**]{}. The canonical homomorphisms $$K_0T({{\cal O}}_XG) {\rightarrow}{\mathop{\oplus}\limits}_{x \in X \;{\rm closed}} K_0T({{\cal O}}_{X,x}G) \quad
\textrm{and} \quad K_0^{{\rm lf}}T({{\cal O}}_XG) {\rightarrow}{\mathop{\oplus}\limits}_{x\in X \;{\rm closed}}
K_0^{{\rm lf}}T({{\cal O}}_{X,x}G)$$ are bijective.
[**Proof**]{}. Let $x$ be a closed point of $X$ and $V$ a f. g. ${{\cal O}}_{X,x}G$-module which is ${{\cal O}}_{X,x}$-torsion and which allows an ${{\cal O}}_{X,x}G$-projective (resp., ${{\cal O}}_{X,x}G$-free) resolution $0{\rightarrow}P {\rightarrow}Q \,\, \stackrel{\varepsilon}{{\rightarrow}}
V {\rightarrow}0$. Let $i: {{\rm Spec}}({{\cal O}}_{X,x}) \hookrightarrow X$ denote the inclusion. It suffices to show that $i_*(V)$ has a (global) locally projective (resp., locally free) resolution of length $\le 1$. If $P$ and $Q$ are ${{\cal O}}_{X,x}G$-free, i.e., if they are isomorphic to ${\mathop{\oplus}\limits}^m {{\cal O}}_{X,x}G$ for some $m \ge 0$, then the composition $\tilde{\varepsilon}: {\mathop{\oplus}\limits}^m {{\cal O}}_XG \,\,
\stackrel{{\rm can}}{\longrightarrow} \,\, i_*({\mathop{\oplus}\limits}^m({{\cal O}}_{X,x}G)) \,\,
\stackrel{i_*(\varepsilon)}{\longrightarrow} \,\, i_*(V)$ is surjective and $\textrm{ker}(\varepsilon)$ is a locally free ${{\cal O}}_XG$-module, i.e., $i_*(V)$ has a locally free resolution of length $1$. If $P$ and $Q$ are only projective over ${{\cal O}}_{X,x}G$, we choose a (non-equivariant) surjective homomorphism ${{\cal E}}{\rightarrow}i_*(V)$ with a locally free ${{\cal O}}_X$-module ${{\cal E}}$. Then, the induced homomorphism $\tilde{\varepsilon}: {{\cal O}}_XG \otimes_{{{\cal O}}_X} {{\cal E}}{\rightarrow}i_*(V)$ is an equivariant surjection and the coherent ${{\cal O}}_XG$-module $\textrm{ker}(\varepsilon)$ is locally projective by Schanuel’s Lemma, i.e., $i_*(V)$ has a locally projective resolution of length $1$.
[**Proposition 2.4**]{}. The canonical homomorphism $K_0^{{\rm lf}}({{\cal O}}_XG) {\rightarrow}K_0({{\cal O}}_XG)$ induces an isomorphism $${{\rm Cl}}({{\cal O}}_XG) \,\, \tilde{{\rightarrow}}\,\, \textrm{ker}(K_0({{\cal O}}_XG) \,\,
\stackrel{{\rm can}}{\longrightarrow} \,\, K_0(FG)).$$
[**Proof**]{}. We have a natural commutative diagram of groups $$\xymatrix{
K_1(FG) \ar[r] \ar@{=}[d] & K_0^{{\rm lf}}T({{\cal O}}_XG) \ar[r] \ar[d] &
K_0^{{\rm lf}}({{\cal O}}_XG) \ar[r] \ar[d] & K_0^{{\rm lf}}(FG) \ar[d] \\
K_1(FG) \ar[r] & K_0T({{\cal O}}_XG) \ar[r] & K_0({{\cal O}}_XG) \ar[r] & K_0(FG); }$$ here, the lower row is the exact localization sequence constructed in Proposition 1.6 and Proposition 1.7; the maps in the upper row are defined as in the lower row; one can prove as in section 1 or as in Theorem 1(ii) on p. 3 in [@F2] that also the upper sequence is exact. Thus, it suffices to prove that the map $K_0^{{\rm lf}}T({{\cal O}}_XG) {\rightarrow}K_0T({{\cal O}}_XG)$ is bijective. By Lemma 2.3, it furthermore suffices to prove that the map $K_0^{{\rm lf}}T({{\cal O}}_{X,x}G)
{\rightarrow}K_0T({{\cal O}}_{X,x}G)$ is bijective for all closed points $x\in X$. We have a natural commutative diagram of groups $$\xymatrix{
K_1({{\cal O}}_{X,x}G) \ar[r] \ar@{=}[d] & K_1(FG) \ar[r] \ar@{=}[d] &
K_0^{{\rm lf}}T({{\cal O}}_{X,x}G) \ar[r] \ar[d] & 0 \\
K_1({{\cal O}}_{X,x}G) \ar[r] & K_1(FG) \ar[r]& K_0T({{\cal O}}_{X,x}G) \ar[r] &
K_0({{\cal O}}_{X,x}G) \ar[r] & K_0(FG) }$$ with exact rows (e.g., see Theorem 1(ii) on p. 3 in [@F2]). Furthermore, the map $K_0({{\cal O}}_{X,x}G) {\rightarrow}K_0(FG)$ is injective by Hattori’s Theorem (see Theorem (32.1) on p. 671 in [@CR]). This proves Proposition 2.4.
Let $K_0(G,X)$ denote the Grothendieck group of all coherent ${{\cal O}}_XG$-modules which are locally free as ${{\cal O}}_X$-modules.
[**Corollary 2.5**]{}. If ${{\rm ord}}(G)$ is invertible on $X$, the Cartan homomorphism $K_0^{{\rm lf}}({{\cal O}}_XG) {\rightarrow}K_0(G,{{\cal O}}_X)$ induces an isomorphism $${{\rm Cl}}({{\cal O}}_XG) \,\, \tilde{{\rightarrow}} \,\, \textrm{ker}\left(K_0(G,X) \,\,
\stackrel{{\rm can}}{\longrightarrow} \,\, K_0(G,F) \cong K_0(FG)\right).$$
[**Proof**]{}. This immediately follows from Proposition 2.4 and the fact that a f. g. ${{\cal O}}_{X,x}G$-module is projective over ${{\cal O}}_{X,x}G$ if and only if it is projective over ${{\cal O}}_{X,x}$.
Now, we fix $i\in {{\mathbb N}}$ such that $\gcd(i,{{\rm ord}}(G))$ is invertible on $X$. By section 1, we have a symmetric power operation $\sigma^i:
K_0({{\cal O}}_XG) {\rightarrow}K_0({{\cal O}}_XG)$. By restricting, we obtain an operation $\sigma^i$ on $\textrm{ker}(K_0({{\cal O}}_XG) {\rightarrow}K_0(FG)) \cong {{\rm Cl}}({{\cal O}}_XG)$. In the same way, we obtain a multiplication map on ${{\rm Cl}}({{\cal O}}_XG)$.
[**Proposition 2.6**]{}. The multiplication on ${{\rm Cl}}({{\cal O}}_XG)$ is trivial and the operation $\sigma^i$ on ${{\rm Cl}}({{\cal O}}_XG)$ is a homomorphism.
[**Proof**]{}. Since the canonical homomorphism $K_0T({{\cal O}}_XG) {\rightarrow}{{\rm Cl}}({{\cal O}}_XG)$ is surjective, it suffices to show the corresponding assertions for $K_0T({{\cal O}}_XG)$ (by Lemma 1.4). By Lemma 2.3, we may furthermore assume that $X={{\rm Spec}}(A)$ where $A$ is a local Dedekind domain. Then, the connecting homomorphism $\partial: K_1(FG) {\rightarrow}K_0T({{\cal O}}_XG)$ is surjective (see the proof of Proposition 2.4), and Proposition 2.6 follows from Lemma 1.5.
[**Theorem 2.7**]{}. Let $G$ be Abelian and $\gcd(i, {{\rm ord}}(G))=1$. We fix $i' \in
{{\mathbb N}}$ such that $ii' \equiv 1 \textrm{ mod } e(G)$ where $e(G)$ denotes the exponent of $G$. Let $\phi_{i'}$ denote both the ${{\cal O}}_X$-algebra automorphism ${{\cal O}}_XG {\rightarrow}{{\cal O}}_XG$ given by $[g] \mapsto [g^{i'}]$ and the automorphism of $K_0({{\cal O}}_XG)$ or ${{\rm Cl}}({{\cal O}}_XG)$ induced by the association $[{{\cal P}}] \mapsto
[{{\cal O}}_XG \otimes_{{{\cal O}}_XG} {{\cal P}}]$ (where ${{\cal O}}_XG$ is considered as an ${{\cal O}}_XG$-algebra via $\phi_{i'}$). Then we have: $$\sigma^i=\phi_{i'} \quad \textrm{on} \quad {{\rm Cl}}({{\cal O}}_XG).$$
[**Proof**]{}. As in Proposition 2.6, it suffices to show the corresponding assertion for $K_1(FG)$ where $\phi_{i'}$ on $K_1(FG)$ is defined analogously. Since $FG$ is semilocal and commutative, the canonical homomorphism $(FG)^\times {\rightarrow}K_1(FG)$ is bijective (see Corollary (9.2) on p. 267 in [@Ba]). Under this isomorphism, the automorphism $\phi_{i'}$ corresponds to the restriction of the (analogously defined) automorphism $\phi_{i'}$ of $FG$. Thus it suffices to show that the following diagram commutes: $$\xymatrix{
(FG)^\times \ar[r]^{\sim} \ar[d]^{\phi_{i'}} & K_1(FG) \ar[d]^{\sigma^i} \\
(FG)^\times \ar[r]^{\sim} & K_1(FG). }$$ Now, let $W$ be a local domain of characteristic $0$ whose residue class field is isomorphic to $F$. (If $\textrm{char}(F) =0$, we may choose $F$ itself for $W$. If $p =\textrm{char}(F) > 0$, the ring of infinite Witt vectors over $F$ associated with the prime $p$ is such a ring.) Since the group ring $WG$ is semilocal and commutative, the canonical map $(WG)^\times {\rightarrow}K_1(FG)$ is bijective (see loc. cit.) and the canonical homomorphism $(WG)^\times {\rightarrow}(FG)^\times$ is surjective. Thus it suffices to show that the following diagram commutes: $$\xymatrix{
(WG)^\times \ar[r]^{\sim} \ar[d]^{\phi_{i'}} & K_1(WG) \ar[d]^{\sigma^i}\\
(WG)^\times \ar[r]^\sim & K_1(WG).}$$ In a similar way, we conclude that it suffices to show that the corresponding diagram commutes if $W$ is replaced by the quotient field $Q$ of $W$ and finally by the algebraic closure $\bar{Q}$ of $Q$. In the latter case, the commutativity follows from Theorem 1.6(d) in [@KoCl], Theorem 3.3 in [@KoAdHi], and Lemma 3.6(b) in [@KoCl]. This ends the proof of Theorem 2.7.
[**Remark 2.8**]{}. Let $\gcd(i,{{\rm ord}}(G))=1$. Theorem 2.7 implies in particular that $\sigma^{i + e(G)} = \sigma^i$ on ${{\rm Cl}}({{\cal O}}_XG)$ if $G$ is Abelian. This also holds if $X= {{\rm Spec}}({{\cal O}}_F)$ where ${{\cal O}}_F$ is the ring of integers in a number field $F$ (see Corollary 3.8 in [@KoCl]) or if $X$ is a smooth curve over a finite field (this follows from Theorem 2.10). It is not clear to me whether this is true in general.
Now, let $L$ be an algebraically closed field such that $\textrm{char}(L)$ does not divide ${{\rm ord}}(G)$, and let $p: X{\rightarrow}{{\rm Spec}}(L)$ be an irreducible smooth curve over $L$. Then, for any f. g. $LG$-module $V$, the pull-back $p^*(V)$ is a locally projective coherent ${{\cal O}}_XG$-module. Furthermore, for any locally projective coherent ${{\cal O}}_XG$-module ${{\cal P}}$, ${{\cal P}}'$, the ${{\cal O}}_X$-module ${\cal H}{\rm om}_{{{\cal O}}_X}({{\cal P}},{{\cal P}}') \cong {{\cal P}}^\vee \otimes_{{{\cal O}}_X} {{\cal P}}$ is again a locally projective ${{\cal O}}_XG$-module. Finally, for any locally projective ${{\cal O}}_XG$-module ${{\cal P}}$, the ${{\cal O}}_X$-module ${{\cal P}}^G$ of $G$-fixed elements is locally free since ${{\rm ord}}(G)$ is invertible on $X$. Thus, we obtain a well-defined homomorphism $$\xymatrix@R=1ex{
K_0({{\cal O}}_XG) \ar[r] & {{\rm Hom}}(K_0(LG), K_0(X))\\
[{{\cal P}}] \ar@{|->}[r]& ([V] \mapsto [{{\rm Hom}}_{{{\cal O}}_XG}(p^*(V), {{\cal P}})]). }$$ This homomorphism is bijective (see the proof of Proposition (2.2) on p. 133 in [@Se]) and induces an isomorphism $${{\rm Cl}}({{\cal O}}_XG) \,\, \tilde{{\rightarrow}} \,\, {{\rm Hom}}(K_0(LG), {{\rm Cl}}(X))$$ by Proposition 2.4.
Let $\psi^i$ denote the $i$-th Adams operation on $K_0(LG)$. In the sequel, we will identify $K_0(LG)$ with the ring of virtual characters of $G$. Then $\psi^i$ maps a character $\chi$ to the character $G {\rightarrow}L$, $g \mapsto
\chi(g^i)$. Let $\hat{\psi}^i$ denote the adjoint operation (with respect to the usual character pairing). Note that the assumption $\textrm{char}(L) {{\,\not{\kern-0.075em|}\,}}{{\rm ord}}(G)$ implies that $\gcd(i,{{\rm ord}}(G))$ is invertible on $G$ for all $i\in {{\mathbb N}}$.
[**Theorem 2.9**]{}. Under the isomorphism (), the operation $\sigma^i$ on ${{\rm Cl}}({{\cal O}}_XG)$ corresponds to the endomorphism ${{\rm Hom}}(\hat{\psi}^i, {{\rm Cl}}(X))$ of ${{\rm Hom}}(K_0(LG), {{\rm Cl}}(X))$.
[**Proof**]{}. By Theorem 3.3 on p. 145 in [@KoAdHi] and Theorem 1.6(d)(ii) in [@KoCl], the operation $\sigma^i$ on $K_1(FG)$ (constructed e.g. in section 1) corresponds to the endomorphism ${{\rm Hom}}(\hat{\psi}^i, K_1(F))$ of ${{\rm Hom}}(K_0(LG), K_1(F))$ under the isomorphism $$\xymatrix@R=1ex{
K_1(FG) \ar[r]^\sim & {{\rm Hom}}(K_0(LG), K_1(F)) \\
(P,\alpha) \ar@{|->}[r] &
\Big([V] \mapsto ({{\rm Hom}}_{FG}(F\otimes_L V,P), {{\rm Hom}}_{FG}(F\otimes_L V, \alpha))\Big).}$$ For any closed point $x\in X$, the association $[M] \mapsto ([V] \mapsto
[{{\rm Hom}}_{{{\cal O}}_{X,x}G}({{\cal O}}_{X,x} \otimes_L V, M)])$ induces an isomorphism $K_0T({{\cal O}}_{X,x}G) \,\, \tilde{{\rightarrow}}\,\, {{\rm Hom}}(K_0(LG), K_0T({{\cal O}}_{X,x}))$ (both sides are isomorphic to $K_0(LG)$!) such that the following diagram commutes: $$\xymatrix{
K_1(FG) \ar@{->>}[rrrr]^\partial \ar[d]^\wr &&&& K_0T({{\cal O}}_{X,x}G) \ar[d]^\wr\\
{{\rm Hom}}(K_0(LG), K_1(F)) \ar@{->>}[rrrr]^{{{\rm Hom}}(K_0(LG), \partial)} &&&&
{{\rm Hom}}(K_0(LG),K_0T({{\cal O}}_{X,x})).}$$ Hence, by Lemma 1.5, the operation $\sigma^i$ on $K_0T({{\cal O}}_{X,x}G)$ corresponds to the endomorphism ${{\rm Hom}}(\hat{\psi}^i, K_0T({{\cal O}}_{X,x}))$ of ${{\rm Hom}}(K_0(LG),K_0T({{\cal O}}_{X,x}))$. Under the isomorphism of Lemma 2.3, the operation $\sigma^i$ on $K_0T({{\cal O}}_{X}G)$ obviously corresponds to the endomorphism ${\mathop{\oplus}\limits}_{x\in X \;{\rm closed}} \sigma^i$ of ${\mathop{\oplus}\limits}_{x\in X \;{\rm closed}} K_0T({{\cal O}}_{X,x}G)$. Thus, under the isomorphism $K_0T({{\cal O}}_XG) \cong {{\rm Hom}}(K_0(LG), K_0T({{\cal O}}_X))$, $[{{\cal M}}] \mapsto
([V] \mapsto [{{\rm Hom}}_{{{\cal O}}_XG}(p^*(V),{{\cal M}})])$, the operation $\sigma^i$ on $K_0T({{\cal O}}_XG)$ corresponds to the endomorphism ${{\rm Hom}}(\hat{\psi}^i, K_0T({{\cal O}}_X))$ of ${{\rm Hom}}(K_0(LG),K_0T({{\cal O}}_X))$. Furthermore, the following diagram obviously commutes: $$\xymatrix{
K_0T({{\cal O}}_XG) \ar[r]^{{\rm can}} \ar[d]^\wr & K_0({{\cal O}}_XG) \ar[d]^\wr \\
{{\rm Hom}}(K_0(LG), K_0T({{\cal O}}_X)) \ar[r]^{{\rm can}} & {{\rm Hom}}(K_0(LG), K_0(X)).}$$ Now, Theorem 2.9 follows from Lemma 1.4 and Proposition 1.6.
Now, let $L$ be a finite field with $\textrm{char}(L) {{\,\not{\kern-0.075em|}\,}}{{\rm ord}}(G)$ and $p: X{\rightarrow}{{\rm Spec}}(L)$ an irreducible smooth curve over $L$. Let $\bar{L}$ denote an algebraic closure of $L$ and $\bar{p}: \bar{X}
:= X\times_L \bar{L} {\rightarrow}{{\rm Spec}}(\bar{L})$ the corresponding curve over $\bar{L}$. Then, the composition of the canonical map $K_0({{\cal O}}_XG) {\rightarrow}K_0({{\cal O}}_{\bar{X}}G)$ with the isomorphism $K_0({{\cal O}}_{\bar{X}}G) \cong {{\rm Hom}}(K_0(\bar{L}G), K_0(\bar{X}))$ constructed above obviously induces a homomorphism $$K_0({{\cal O}}_XG) {\rightarrow}{{\rm Hom}}_{{\rm Gal}(\bar{L}/L)}(K_0(\bar{L}G), K_0(\bar{X})).$$
[**Theorem 2.10**]{}. This homomorphism is bijective. In particular, we obtain an isomorphism $${{\rm Cl}}({{\cal O}}_XG) \,\, \tilde{{\rightarrow}} \,\, {{\rm Hom}}_{{\rm Gal}(\bar{L}/L)}
(K_0(\bar{L}G), {{\rm Cl}}(\bar{X})).$$ Under this isomorphism, the operation $\sigma^i$ on ${{\rm Cl}}({{\cal O}}_XG)$ corresponds to the endomorphism ${{\rm Hom}}_{{\rm Gal}(\bar{L}/L)}(\hat{\psi}^i, {{\rm Cl}}(\bar{X}))$ of ${{\rm Hom}}_{{\rm Gal}(\bar{L}/L}(K_0(\bar{L}G), {{\rm Cl}}(\bar{X}))$.
[**Proof**]{}. The bijectivity can be shown as in section 6 of [@ABGr] using Morita equivalence and the Galois descent property $K_0(X \times_L L') \cong
K_0(\bar{X})^{{\rm Gal}(\bar{L}/L')}$ (for any finite extension $L \subseteq L'
\subset \bar{L}$ of $L$). Proposition 2.4 then yields the Hom-description of the classgroup. The last assertion immediately follows from Theorem 2.9.
§3 Equivariant Riemann-Roch Type Formulas for Tame Extensions of Dedekind Schemes {#equivariant-riemann-roch-type-formulas-for-tame-extensions-of-dedekind-schemes .unnumbered}
=================================================================================
The aim of this section is to prove Theorem A and Theorem B presented in the introduction.
Let $Y$ be a Dedekind scheme and $G$ a finite group of order $n$. Let ${{\rm Ind}}_1^G: {{\rm Cl}}({{\cal O}}_Y) {\rightarrow}{{\rm Cl}}({{\cal O}}_YG)$ and ${{\rm Ind}}_1^G: K_0T({{\cal O}}_Y) {\rightarrow}K_0^{{\rm lf}}T({{\cal O}}_YG)$ denote the induction maps. The following lemma generalizes Lemma 2.6 on p. 933 in [@BC].
[**Lemma 3.1**]{}. The image of the natural multiplication maps $$K_0T({{\cal O}}_Y) \times K_0^{{\rm lf}}({{\cal O}}_YG) {\rightarrow}K_0^{{\rm lf}}T({{\cal O}}_YG) \quad
\textrm{and} \quad {{\rm Cl}}({{\cal O}}_Y) \times K_0^{{\rm lf}}({{\cal O}}_YG) {\rightarrow}{{\rm Cl}}({{\cal O}}_YG)$$ is contained in ${{\rm Ind}}_1^GK_0T({{\cal O}}_Y)$ resp. ${{\rm Ind}}_1^G{{\rm Cl}}({{\cal O}}_Y)$.
[**Proof**]{}. The assertion for the first map is clear. The assertion for the second map follows from this since the natural map $K_0T({{\cal O}}_Y) {\rightarrow}{{\rm Cl}}({{\cal O}}_Y)$ is surjective.
Now, let $F/E$ be a finite Galois extension of the function field $E$ of $Y$ with Galois group $G$. Let $X$ denote the normalization of $Y$ in $F$. Then $X$ is a Dedekind scheme endowed with a natural $G$-action and the corresponding $G$-morphism $f:X{\rightarrow}Y$ is finite (see the proof of Theorem (8.1) on p. 47 in [@N]). We assume that $f$ is tamely ramified. As in Lemma 5.5 in [@KoCl], one easily shows that then, for any locally free coherent ${{\cal O}}_X$-module ${{\cal E}}$ with (semilinear) $G$-action, the direct image $f_*({{\cal E}})$ is a locally free coherent ${{\cal O}}_YG$-module in the sense of section 2. Let $K_0(G,X)$ denote the Grothendieck group of all such modules ${{\cal E}}$. Thus, we have a homomorphism $$f_*: K_0(G,X) {\rightarrow}K_0^{{\rm lf}}({{\cal O}}_YG), \quad [{{\cal E}}] \mapsto [f_*({{\cal E}})].$$ The different ${{\cal D}}:= {{\cal D}}_{X/Y} := {\cal A}{\rm nn}_{{{\cal O}}_X}(\Omega^1_{X/Y})$ is a $G$-stable ideal in ${{\cal O}}_X$, hence a module ${{\cal E}}$ as above. The following proposition generalizes formula (2.8) on p. 933 in [@BC].
[**Proposition 3.2**]{}. For all $x\in K_0(G,X)$ we have: $$f_*\left(x \cdot \sum_{i=0}^{n-1}[{{\cal D}}^{-i}]\right) =0 \quad \textrm{in} \quad
K_0^{{\rm lf}}({{\cal O}}_YG)/({{\rm Ind}}_1^G{{\rm Cl}}({{\cal O}}_Y) \oplus n {{\mathbb Z}}[{{\cal O}}_YG]).$$
[**Proof**]{}. We may assume that $x=[{{\cal E}}]$ where ${{\cal E}}$ is a module as above. Let $r:= {{\rm rank}}_{{{\cal O}}_X}({{\cal E}})$. Then we have: $$\begin{aligned}
\lefteqn{\sum_{i=0}^{n-1}\Big([f_*({{\cal E}}\otimes {{\cal D}}^{-i})] - r [{{\cal O}}_YG]\Big) }\\
&=&n\Big([f_*({{\cal E}})] - r[{{\cal O}}_YG]\Big) + \sum_{i=1}^{n-1}\Big([f_*({{\cal E}}\otimes {{\cal D}}^{-i})]
- [f_*({{\cal E}})]\Big) \quad \textrm{in} \quad {{\rm Cl}}({{\cal O}}_YG).\end{aligned}$$ In the sequel, let ${{\cal M}}\mapsto {{\cal M}}^t$ denote the forgetful functor from the category of ${{\cal O}}_YG$-modules to the category of ${{\cal O}}_Y$-modules. (We will consider ${{\cal M}}^t$ also as an ${{\cal O}}_YG$-module with trivial $G$-action.) Then, the elements $[f_*({{\cal O}}_X)^t]-n$ and $[f_*({{\cal E}})^t] -nr$ are contained in ${{\rm Cl}}({{\cal O}}_Y)$. Hence, we have by Lemma 3.1: $$n([f_*({{\cal E}})] -r [{{\cal O}}_YG]) = [f_*({{\cal O}}_X)^t \otimes f_*({{\cal E}})] -
[f_*({{\cal E}})^t \otimes {{\cal O}}_YG] \quad \textrm{in} \quad
{{\rm Cl}}({{\cal O}}_YG)/{{\rm Ind}}_1^G{{\rm Cl}}({{\cal O}}_Y).$$ The homomorphism $$f_*({{\cal O}}_X)^t \otimes f_*({{\cal E}}) {\rightarrow}f_*({{\cal E}})^t \otimes {{\cal O}}_YG,
\quad a\otimes b \mapsto \sum_{g\in G} a g(b) \otimes [g^{-1}],$$ of ${{\cal O}}_YG$-modules is generically bijective since $F/E$ is a Galois extension and any f. g. module over the twisted group ring $F\#G$ is isomorphic to ${\mathop{\oplus}\limits}^m F$ for some $m\ge 0$. In particular, this map is a monomorphism and the cokernel ${{\cal R}}_{X/Y}({{\cal E}})$ is an ${{\cal O}}_YG$-torsion module. Hence, it suffices to show that we have: $$[{{\cal R}}_{X/Y}({{\cal E}})] = \sum_{i=1}^{n-1}[f_*({{\cal E}}\otimes {{\cal D}}^{-i}/{{\cal O}}_X)] \quad
\textrm{in} \quad K_0^{{\rm lf}}T({{\cal O}}_YG)/{{\rm Ind}}_1^GK_0T({{\cal O}}_Y).$$ By Lemma 2.3, it furthermore suffices to show that we have $$[{{\cal R}}_{X/Y}({{\cal E}})_y] = \sum_{i=1}^{n-1}[f_*({{\cal E}}\otimes {{\cal D}}^{-i}/{{\cal O}}_X)_y] \quad
\textrm{in} \quad K_0^{{\rm lf}}T({{\cal O}}_{Y,y}G)/{{\rm Ind}}_1^GK_0T({{\cal O}}_{Y,y})$$ for all closed points $y \in Y$.\
We now fix $y\in Y$ and $x\in X$ with $f(x) =y$. Let $G_x := \{g\in G: xg =x\}$ denote the decomposition group of $x$. Furthermore, let $f':X':= {{\rm Spec}}(\hat{{{\cal O}}}_{X,x})
{\rightarrow}{{\rm Spec}}(\hat{{{\cal O}}}_{Y,y}) =: Y'$ denote the induced $G_x$-morphism where $\hat{}$ denotes completion. We identify the category of coherent torsion modules on $Y'$ with the category of coherent torsion modules on $Y$ supported in $y$. An easy generalization of Corollary 3.11(b) on p. 239 in [@Ch] shows that ${{\cal R}}_{X/Y}({{\cal E}})_y$ is isomorphic to the direct sum of $[G:G_x]$ copies of ${{\rm Ind}}_{G_x}^G {{\cal R}}_{X'/Y'}(\hat{{{\cal E}}}_x)$. Furthermore, it is clear that $f_*({{\cal E}}\otimes {{\cal D}}^{-i}_{X/Y}/{{\cal O}}_X)_y$ is isomorphic to ${{\rm Ind}}_{G_x}^G f'_*(\hat{{{\cal E}}}_x \otimes {{\cal D}}^{-i}_{X'/Y'}/{{\cal O}}_{X'})$ for all $i \ge 0$. For $i \equiv j \textrm{ mod } {{\rm ord}}(G_x)$, we finally have $$[f'_*(\hat{{{\cal E}}}_x \otimes {{\cal D}}_{X'/Y'}^{-i}/{{\cal O}}_{X'})] =
[f'_*(\hat{{{\cal E}}}_x \otimes {{\cal D}}^{-j}_{X'/Y'}/{{\cal O}}_{X'})] \quad \textrm{in} \quad
K_0^{{\rm lf}}T({{\cal O}}_{Y'}G_{x})/{{\rm Ind}}_1^{G_x} K_0T({{\cal O}}_{Y'})$$ since the ideal ${{\cal D}}^{{{\rm ord}}(G_x)}_{X'/Y'}$ of ${{\cal O}}_{X'}$ can be written as $(f')^*({\mathfrak a})$ with some ideal ${\mathfrak a}$ in ${{\cal O}}_{Y'}$ and since, for any locally free coherent ${{\cal O}}_{Y'}G$-module ${{\cal P}}$, we have $$[{{\cal P}}/{\mathfrak a} {{\cal P}}] = [{{\cal O}}/{\mathfrak a} \otimes {{\cal P}}] = 0 \quad
\textrm{in} \quad K_0^{{\rm lf}}({{\cal O}}_{Y'}G_{x})/{{\rm Ind}}_1^{G_x} K_0T({{\cal O}}_{Y'})$$ by Lemma 3.1. Thus it suffices to prove that $$[{{\cal R}}_{X'/Y'}(\hat{{{\cal E}}}_x)] = \sum_{i=1}^{{{\rm ord}}(G_x)-1}
[f'_*(\hat{{{\cal E}}}_x \otimes {{\cal D}}^{-i}_{X'/Y'} / {{\cal O}}_{X'})] \quad \textrm{in}
\quad K_0T({{\cal O}}_{Y'}G_x)/{{\rm Ind}}_1^{G_x}K_0T({{\cal O}}_{Y'}).$$ We now write $G$ for $G_x$, $X$ for $X'$, ${{\cal E}}$ for $\hat{{{\cal E}}}_x$, and so on. Let $\Delta \subseteq G$ denote the inertia group, $e$ the order of $\Delta$, ${\mathfrak P}$ the ideal in ${{\cal O}}_X$ which corresponds to the closed point in $X$, and $\chi$ the $\Delta$-module ${\mathfrak P}/{\mathfrak P}^2$. We decompose $f: X{\rightarrow}Y$ into $X\,\, \stackrel{g}{{\rightarrow}}\,\, Z \,\, \stackrel{h}{{\rightarrow}} \,\,Y$ where $Z:= {{\rm Spec}}(\Gamma(X,{{\cal O}}_X)^\Delta)$; i.e., the function field of $Z$ is the inertia field of $F/E$. Since $K_0(G,X)$ is generated by the classes of fractional $G$-stable ideals in ${{\cal O}}_X$ (see Lemma 5.5(c) in [@KoCl]), we may assume that ${{\cal E}}= {\mathfrak P}^j$ for some $j\in {{\mathbb Z}}$. An easy generalization of Corollary 3.8 on p. 236 and Theorem 2.8 on p. 222 in [@Ch] shows that we have the following isomorphisms: $$\begin{aligned}
\lefteqn{{{\cal R}}_{X/Y}({{\mathfrak P}}^j) \cong {{\rm Ind}}_\Delta^G h_*({{\cal R}}_{X/Z}({{\mathfrak P}}^j))}\\
&\cong&{{\rm Ind}}_\Delta^Gh_*\left({\mathop{\oplus}\limits}_{i=1}^{e-1} g_*\Big(({{\mathfrak P}}^j/{{\mathfrak P}}^{j+i})^t \otimes
\chi^{j+i}\Big)\right) \\
&\cong& {{\rm Ind}}_\Delta^G f_*\left({\mathop{\oplus}\limits}_{i=1}^{e-1}({{\mathfrak P}}^j/{{\mathfrak P}}^{j+i})^t \otimes
\chi^{j+i} \right).\end{aligned}$$ Thus we have: $$[{{\cal R}}_{X/Y}({{\mathfrak P}}^j)] = \sum_{i=1}^{e-1} i [{{\rm Ind}}_\Delta^G f_*(\chi^{j+i})] \quad
\textrm{in} \quad K_0T({{\cal O}}_YG).$$ Since ${{\cal D}}= {{\mathfrak P}}^{e-1}$ and ${{\mathfrak P}}^e = f^*({{\mathfrak p}})$ (where ${{\mathfrak p}}$ is the ideal in ${{\cal O}}_Y$ which corresponds to the closed point in $Y$), we can conclude as above using Lemma 3.1: $$\begin{aligned}
\lefteqn{\sum_{i=1}^{n-1}[f_*({{\mathfrak P}}^j\otimes {{\cal D}}^{-i}/{{\cal O}}_X)] =
\frac{n}{e} \sum_{i=1}^{e-1}[f_*({{\mathfrak P}}^j \otimes {{\cal D}}^{-i}/{{\cal O}}_X)] =
\frac{n}{e} \sum_{i=1}^{e-1} [f_*({{\mathfrak P}}^{j+i}/{{\mathfrak P}}^{j+e})]}\\
&=& \frac{n}{e} \sum_{i=1}^{e-1} i [f_*({{\mathfrak P}}^{j+i}/{{\mathfrak P}}^{j+i+1})] \quad
\textrm{in} \quad K_0T({{\cal O}}_YG)/{{\rm Ind}}_1^GK_0T({{\cal O}}_Y). \hspace*{10ex}\end{aligned}$$ Thus it suffices to prove that the ${{\cal O}}_YG$-modules ${{\rm Ind}}_\Delta^G f_*(\chi^i)$ and ${\mathop{\oplus}\limits}^{n/e}f_*({{\mathfrak P}}^i/{{\mathfrak P}}^{i+1})$ are isomorphic for all $i\in {{\mathbb Z}}$. For this, we consider the ${{\cal O}}_YG$-homomorphism $$\xymatrix@R=1ex{
h_*({{\cal O}}_Z)^t \otimes f_*({{\mathfrak P}}^i/{{\mathfrak P}}^{i+1})
\ar[r]& {\rm Maps}_\Delta(G, f_*({{\mathfrak P}}^i/{{\mathfrak P}}^{i+1})) \\
a\otimes b \ar@{|->}[r] & (g \mapsto ag(b)).}$$ This homomorphism is bijective since $h$ is unramified (e.g., see pp. 214-215 in [@Ch]). Furthermore, the left hand side is obviously isomorphic to ${\mathop{\oplus}\limits}^{n/e}f_*({{\mathfrak P}}^i/{{\mathfrak P}}^{i+1})$ and the right hand side is isomorphic to ${{\rm Ind}}_\Delta^G f_*(\chi^i)$. So, Proposition 3.2 is proved.
Now, let $k\in {{\mathbb N}}$ with $\gcd(k,n)=1$ and $k'\in {{\mathbb N}}$ with $kk' \equiv 1
\textrm{ mod } n$. Let $\sigma^k$ denote the $k$-th symmetric power operation on $K_0(G,Y)$ and $\psi^k$ the $k$-th Adams operation on $K_0(G,Y)$ or $K_0(G,X)$ (e.g., see section 1 in [@KoCl]). The composition of the map $f_*:
K_0(G,X) {\rightarrow}K_0^{{\rm lf}}({{\cal O}}_YG)$ with the Cartan homomorphism $K_0^{{\rm lf}}({{\cal O}}_YG)
{\rightarrow}K_0(G,Y)$ is denoted by $f_*$ again. Finally, let $\hat{K}_0(G,Y)[k^{-1}]$ denote the $J$-adic completion of $K_0(G,Y)[k^{-1}]$ where $J:={\rm ker}(K_0(G,Y) \,\, \stackrel{{\rm rank}}{\longrightarrow}\,\, {{\mathbb Z}})[k^{-1}]$ is the augmentation ideal in $K_0(G,Y)[k^{-1}]$.
[**Theorem 3.3**]{}. For all $x \in K_0(G,X)$ we have: $$\sigma^k(f_*(x)- {{\rm rank}}(x)\cdot [{{\cal O}}_YG]) =
f_*\left(\sum_{i=0}^{k'-1}[{{\cal D}}^{-ik}] \cdot \psi^k(x)\right)$$ in $\hat{K}_0(G,Y)[k^{-1}]/({{\rm Ind}}^G_1 K_0(Y)) \hat{K}_0(G,Y)[k^{-1}]$.
[**Proof**]{}. Let $$\hat{f}_*: \hat{K}_0(G,X)[k^{-1}] := K_0(G,X) \otimes_{K_0(G,Y)}
\hat{K}_0(G,Y)[k^{-1}] {\rightarrow}\hat{K}_0(G,Y)[k^{-1}]$$ denote the homomorphism which is induced by $f_*: K_0(G,X) {\rightarrow}K_0(G,Y)$, and let $\theta^k({{\cal D}}^{-1}):= 1+[{{\cal D}}^{-1}] + \ldots + [{{\cal D}}^{-(k-1)}] \in K_0(G,X)$ denote the Bott element. As in Theorem 5.4 in [@KoCl], one easily deduces the following assertion from the equivariant Adams-Riemann-Roch theorem (see Theorem (4.5) in [@KoGRR]): The element $\theta^k({{\cal D}}^{-1})$ is invertible in $\hat{K}_0(G,X)[k^{-1}]$ and we have $$\psi^k(f_*(x)) = \hat{f}_*(k \cdot \theta^k({{\cal D}}^{-1})^{-1} \cdot \psi^k(x))
\quad {\rm in} \quad \hat{K}_0(G,Y)[k^{-1}]$$ for all $x\in K_0(G,X)$. Furthermore, we have: $$\theta^k({{\cal D}}^{-1}) \cdot \left(\sum_{i=0}^{k'-1}[{{\cal D}}^{-ik}]\right) =
\sum_{j=0}^{k-1}\sum_{i=0}^{k'-1}[{{\cal D}}^{-(j+ik)}] =
\sum_{i=0}^{kk'-1} [{{\cal D}}^{-i}] = [{{\cal O}}_X] + \sum_{i=1}^{kk'-1}[{{\cal D}}^{-i}]$$ in $K_0(G,X)$. Thus, we have: $$\theta^k({{\cal D}}^{-1})^{-1} = \sum_{i=0}^{k'-1}[{{\cal D}}^{-ik}] -
\theta^k({{\cal D}}^{-1})^{-1}\sum_{i=1}^{kk'-1} [{{\cal D}}^{-i}] \quad
\textrm{in} \quad \hat{K}_0(G,X)[k^{-1}].$$ Hence, we obtain the equality $$\begin{aligned}
\lefteqn{\psi^k(f_*(x)) = k \cdot \hat{f}_*\left(\left( \sum_{i=0}^{k'-1}[{{\cal D}}^{-ik}] -
\theta^k({{\cal D}}^{-1})^{-1}\cdot \sum_{i=1}^{kk'-1}[{{\cal D}}^{-i}]\right) \cdot
\psi^k(x)\right)}\\
&=& k \cdot f_*\left(\sum_{i=0}^{k'-1}[{{\cal D}}^{-ik}]\cdot \psi^k(x)\right) \quad
\textrm{in} \quad \hat{K}_0(G,Y)[k^{-1}]/({{\rm Ind}}_1^GK_0(Y)) \hat{K}_0(G,Y)[k^{-1}]\end{aligned}$$ by Proposition 3.2. Since we have $\psi^k = k \cdot \sigma^k$ on ${{\rm Cl}}({{\cal O}}_YG)$ (by Proposition 2.6) and $\psi^k([{{\cal O}}_YG]) = [{{\cal O}}_YG]$ (by Theorem 1.6(e) in [@KoCl]), this implies Theorem 3.3.
Note that the formula of Theorem 3.3 lives within the somewhat complicated group $\hat{K}_0(G,Y)[k^{-1}]/({{\rm Ind}}_1^GK_0(Y)) \hat{K}_0(G,Y)[k^{-1}]$. The next proposition computes this group in a special case.
[**Proposition 3.4**]{}. Let $L$ be an algebraically closed field, $Y$ a projective smooth irreducible curve over $L$, and $n= {{\rm ord}}(G)$ a power of a prime $l\not=
{\rm char}(L)$. Let $I$ denote the augmentation ideal in $K_0(LG)$. Then we have: $$\hat{K}_0(G,Y)[k^{-1}] \cong K_0(Y)[k^{-1}] \oplus I\otimes {{\mathbb Z}}_l \oplus
I\otimes {{\mathbb Z}}_l;$$ under this isomorphism, the extended ideal $({{\rm Ind}}_1^G K_0(Y))\hat{K}_0(G,Y)[k^{-1}]$ corresponds to the subgroup $\{(ny,([{{\mathbb Z}}G]-n) \otimes {{\rm rank}}(y), ([{{\mathbb Z}}G] -n) \otimes
\deg \det(y)): y \in K_0(Y)[k^{-1}]\}$ of $K_0(Y)[k^{-1}] \oplus
I \otimes {{\mathbb Z}}_l \oplus I \otimes {{\mathbb Z}}_l$.
[**Proof**]{}. The canonical map $K_0(LG) \otimes K_0(Y) {\rightarrow}K_0(G,Y)$ is an isomorphism by Proposition (2.2) on p. 133 in [@Se]. Since the augmentation ideal of $K_0(Y)$ is nilpotent (e.g., by Proposition 2.6) and the $I$-adic topology on $I$ coincides with the $l$-adic topology (see Proposition 1.1 on p. 277 in [@AT]), the completion $\hat{K}_0(G,Y)[k^{-1}]$ is isomorphic to the direct sum of $K_0(Y)[k^{-1}]$ and the $l$-adic completion of $I\otimes K_0(Y)[k^{-1}]$. Furthermore, we have $K_0(Y) \cong {{\mathbb Z}}\oplus {{\mathbb Z}}\oplus {{\rm Pic}}^0(Y)$ where ${{\rm Pic}}^0(Y)$ denotes the group of line bundles on $Y$ of degree $0$. Since ${{\rm Pic}}^0(Y)$ is an $l$-divisible group (see item (iv) on p. 42 in [@Mu]), the $l$-adic completion of $I\otimes K_0(Y)[k^{-1}]$ is isomorphic to $I\otimes {{\mathbb Z}}_l \oplus I\otimes {{\mathbb Z}}_l$. Thus, we have $$\hat{K}_0(G,Y)[k^{-1}] \cong K_0(Y)[k^{-1}] \oplus I\otimes {{\mathbb Z}}_l \oplus
I\otimes {{\mathbb Z}}_l.$$ Under the isomorphism $K_0(G,Y) \cong K_0(LG) \otimes K_0(Y)$, the ideal ${{\rm Ind}}_1^G K_0(Y)$ of the ring $K_0(G,Y)$ corresponds to the ideal ${{\rm Ind}}_1^G K_0(L) \otimes
K_0(Y)$ ($\cong K_0(Y)$) of $K_0(LG) \otimes K_0(Y)$ which is generated by the element $[{{\mathbb Z}}G] \otimes 1 = n\otimes 1 + ([{{\mathbb Z}}G]-n) \otimes 1$. One easily deduces the second assertion of Proposition 3.4 from this. (Note that $[{{\mathbb Z}}G] \cdot x =0$ for all $x \in I$.)
Now, let $f_*: K_0(G,X) {\rightarrow}{{\rm Cl}}({{\cal O}}_YG)$ denote the composition of $f_*:K_0(G,X) {\rightarrow}K_0^{{\rm lf}}({{\cal O}}_YG)$ with the canonical projection $K_0^{{\rm lf}}({{\cal O}}_YG) \cong {{\rm Cl}}({{\cal O}}_Y G) \oplus {{\mathbb Z}}[{{\cal O}}_Y G] {\rightarrow}{{\rm Cl}}({{\cal O}}_Y G)$.
[**Theorem 3.5**]{}. Suppose that one of the following conditions holds:\
(a) $Y= {{\rm Spec}}({{\cal O}}_E)$ where ${{\cal O}}_E$ is the ring of integers in a number field $E$.\
(b) $Y$ is an irreducible projective smooth curve over a finite field $L$ and\
$\gcd({\rm char}(L), n) =1$.\
(c) The group $G$ is Abelian and $f:X{\rightarrow}Y$ is unramified.\
(d) $k=1$.\
Then we have for all $x\in K_0(G,X)$: $$\sigma^k(f_*(x)) = f_*\left(\sum_{i=0}^{k'-1} [{{\cal D}}^{-ik}] \cdot \psi^k(x)\right)
\quad \textrm{in} \quad {{\rm Cl}}({{\cal O}}_YG)/{{\rm Ind}}_1^G{{\rm Cl}}({{\cal O}}_Y).$$
[**Proof**]{}. In the case (a), Theorem 3.5 can be deduced from Corollary 2.7 on p. 933 in [@BC] using Theorem 3.7 and Lemma 5.5 in [@KoCl] (see also the proof of Theorem 5.6 in [@KoCl]). The same can be done in the case (b) by using Lemma 3.6(a) in [@KoCl] and Theorem 2.9 (in place of Theorem 3.7 in [@KoCl]) and an obvious generalization of Lemma 5.5(c) in [@KoCl]. (For completeness sake, we mention that it is easy to check that the additional assumptions in Theorem 2.1 on p. 932 in [@BC] about the absolute discriminant or the characteristic of $E$ are not necessary for Corollary 2.7 on p. 933 in [@BC].) We now prove Theorem 3.5 in the case (c), i.e., we want to show the formula $$\sigma^k(f_*(x)) = k'\cdot f_*(\psi^k(x)) \quad \textrm{in} \quad
{{\rm Cl}}({{\cal O}}_YG)/{{\rm Ind}}_1^G{{\rm Cl}}({{\cal O}}_Y)$$ for all $x\in K_0(G,X)$. First, we show that it suffices to prove the formula () for $x=1=[{{\cal O}}_X]$. Indeed, for an arbitrary $x \in K_0(G,X)$, there is a $y \in K_0(Y) \subseteq K_0(G,Y)$ such that $x = f^*(y)$ (e.g., see Theorem 1(B) on p. 112 in [@Mu]). Furthermore, we have $\sigma^k({{\rm Ind}}_1^G{{\rm Cl}}({{\cal O}}_Y)) \subseteq {{\rm Ind}}_1^G{{\rm Cl}}({{\cal O}}_Y)$. This follows from Proposition 1.1 in [@KoCl] as there is a polynomial $Q_k \in
{{\mathbb Z}}[X_1, \ldots, X_k; Y_1, \ldots, Y_k]$ which is homogeneous of weight $k$ in both sets of variables such that $$\sigma^k(z \cdot [{{\cal O}}_Y G]) = Q_k\left(\sigma^1(z), \ldots, \sigma^k(z);
[{{\rm Sym}}^1({{\cal O}}_YG)], \ldots, [{{\rm Sym}}^k({{\cal O}}_YG)]\right)\;\; \textrm{in} \;\;
{{\rm Cl}}({{\cal O}}_YG)$$ for all $z \in Cl({{\cal O}}_Y)$ (by Theorem 2.2 in [@KoCl]). Thus we have in ${{\rm Cl}}({{\cal O}}_YG)/{{\rm Ind}}_1^G{{\rm Cl}}({{\cal O}}_Y)$: $$\begin{aligned}
\lefteqn{\sigma^k(f_*(x)) = \sigma^k(f_*(f^*(y))) = \sigma^k(y \cdot f_*(1))
\quad \textrm{(Projection formula)}}\\
&=& \sigma^k({{\rm rank}}(y) \cdot f_*(1)) \quad \textrm{(Lemma 3.1)}\\
&=& {{\rm rank}}(y) \cdot \sigma^k(f_*(1)) \quad \textrm{(Proposition 2.6)}\\
&=& {{\rm rank}}(y) \cdot k' \cdot f_*(1) \quad \textrm{(by assumption)}\\
&=& k' \cdot \psi^k(y) \cdot f_*(1) \quad \textrm{(Lemma 3.1)}\\
&=& k' \cdot f_*(\psi^k(f^*(y))) = k' \cdot f_*(\psi^k(x)) \quad
\textrm{(Projection formula).}\end{aligned}$$ We now prove formula () for $x=1$. Since $f$ is unramified, the scheme $X$ is a principal $G$-bundle over $Y$ (see Proposition 2.6 on p.115 in [@SGA1]). There is a well-known natural bijection between the set of all principal $G$-bundles over $Y$ and the cohomology group $H^1(Y,G)$. We write $[X]$ for the corresponding element in $H^1(Y,G)$. We define a new principal $G$-bundle $X_{k'}$ over $Y$ as follows: $X_{k'} =X$ as $Y$-schemes and the new action $*$ of $G$ on $X_{k'}$ is given by $x*g:=xg^k$ for “$x \in X$” and $g\in G$. Then, it is easy to check that the association $X \mapsto X_{k'}$ corresponds to the multiplication with $k'$ on $H^1(Y,G)$. Let ${\rm cl}: H^1(Y,G) {\rightarrow}{{\rm Cl}}({{\cal O}}_Y G)$ denote the map which maps a principal $G$-bundle $f: X {\rightarrow}Y$ to the class $[f_*({{\cal O}}_X)] - [{{\cal O}}_YG]$. This map is a homomorphism by Theorem 5 and the subsequent remarks on p. 189 in [@Wa] and by Proposition 3.9 in [@ABGr]. Thus we have: $$\begin{aligned}
\lefteqn{\sigma^k(f_*([{{\cal O}}_X])) = \phi_{k'}({\rm cl}([X])) \quad
\textrm{(Theorem 2.7)}}\\
&=& {\rm cl}([X_{k'}]) = {\rm cl}(k' \cdot [X])
= k' \cdot {\rm cl}([X]) = k' \cdot f_*([{{\cal O}}_X]).\end{aligned}$$ in ${{\rm Cl}}({{\cal O}}_Y G)$, as was to be shown. In the case (d), Theorem 3.5 immediately follows from Proposition 3.2.
[**Remark 3.6**]{}. If one of the conditions (a), (b), (c), (d) of Theorem 3.5 is satisfied, then Theorem 3.3 follows from Theorem 3.5 by passing from ${{\rm Cl}}({{\cal O}}_YG) \subset K_0({{\cal O}}_YG)$ to $K_0(G,Y)$ via Cartan homomorphism and finally by passing from $K_0(G,Y)$ to the completion $\hat{K}_0(G,Y)[k^{-1}]$ of $K_0(G,Y)[k^{-1}]$. In particular, in the case (a), the formula of Theorem 3.3 is substantially weaker than the formula in Theorem 3.5, as already the passage from $K_0({{\cal O}}_YG)$ to $K_0(G,Y)$ loses much information. On the other hand, in the case (b), the formula of Theorem 3.5 modulo torsion follows from the formula in Theorem 3.3 if $n$ is a power of a prime. This can be proved as follows. The Cartan homomorphism $K_0({{\cal O}}_YG) {\rightarrow}K_0(G,Y)$ is bijective since $n$ is invertible on $Y$. Furthermore, the canonical map ${{\rm Cl}}({{\cal O}}_YG)/{{\rm Ind}}_1^G{{\rm Cl}}({{\cal O}}_Y) \subseteq K_0({{\cal O}}_YG)/{{\rm Ind}}_1^G K_0(Y) {\rightarrow}K_0({{\cal O}}_{\bar{Y}}G)/{{\rm Ind}}_1^GK_0(\bar{Y})$ is injective by Theorem 2.10. (Here, $\bar{Y}$ denotes the curve $Y \times_L \bar{L}$ over the algebraic closure $\bar{L}$ of $L$.) Hence, it suffices to prove the formula $$\sigma^k(\bar{f}_*(x)) = \bar{f}_*\left(\sum_{i=0}^{k'-1}[{{\cal D}}_{\bar{X}/\bar{Y}}^{-ik}]
\cdot \psi^k(x)\right) \quad \textrm{in}\quad K_0(G,\bar{Y})_{{\mathbb Q}}/
{{\rm Ind}}_1^GK_0(\bar{Y})_{{\mathbb Q}}$$ for all $x\in K_0(G,\bar{X})$. Furthermore, we have $K_0(G,\bar{Y})_{{\mathbb Q}}\cong
K_0(\bar{L}G)_{{\mathbb Q}}\otimes K_0(\bar{Y})_{{\mathbb Q}}$ and $K_0(G,\bar{Y})_{{\mathbb Q}}/{{\rm Ind}}_1^GK_0(\bar{Y})_{{\mathbb Q}}\cong I\otimes K_0(\bar{Y})_{{\mathbb Q}}\cong I_{{\mathbb Q}}\oplus I_{{\mathbb Q}}$ (see the proof of Proposition 3.4). On the other hand, $\left(\hat{K}_0(G,\bar{Y})[k^{-1}]/
({{\rm Ind}}_1^GK_0(\bar{Y}))\hat{K}_0(G,\bar{Y})[k^{-1}]\right)_{{\mathbb Q}}$ is isomorphic to $I\otimes {{\mathbb Q}}_l \oplus I\otimes {{\mathbb Q}}_l$ by Proposition 3.4. Hence, the canonical map $K_0(G,\bar{Y})_{{\mathbb Q}}/{{\rm Ind}}_1^GK_0(\bar{Y})_{{\mathbb Q}}$ ${\rightarrow}\left(\hat{K}_0(G,\bar{Y})[k^{-1}]/
({{\rm Ind}}_1^GK_0(\bar{Y}))\hat{K}_0(G,\bar{Y})[k^{-1}]\right)_{{\mathbb Q}}$ is injective, and formula () follows from Theorem 3.3.
[**Remark 3.7**]{}.\
(a) If one of the conditions (a), (b), or (c) holds, Theorem 3.5 can be slightly strengthened: It suffices to assume that $k'$ is an inverse modulo the exponent of $G$ (see [@BC] and Theorem 2.7, respectively). It is not clear to me whether this is true also in the case (d).\
(b) Let $Y$ be an irreducible smooth projective curve over a finite field $L$. Then, the case (c) is particularly interesting as complementary case of the semisimple case which is assumed in the case (b). Indeed, if $G$ is an (Abelian) ${\rm char}(L)$-group, then the tameness condition already implies that $f$ is unramified.
[SGA1]{}
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
Let $\zz[1/p]$ denote the ring of integers with the prime $p$ inverted. There is a canonical homomorphism $\Psi\co \hskip-.07in \oplus \Theta^3_{\zz[1/p]} \to \Theta^3_{\qq}$, where $\Theta^3_{R}$ denotes the three-dimensional smooth $R$–homology cobordism group of $R$–homology spheres and the direct sum is over all prime integers. Gauge theoretic methods prove the kernel is infinitely generated. Here we prove that $\Psi$ is not surjective, with cokernel infinitely generated. As a basic example we show that for $p$ and $q$ distinct primes, there is no rational homology cobordism from the lens space $L(pq,1) $ to any $M_p \cs M_q$, where $H_1(M_p) = \zz_{p}$ and $H_1(M_q) = \zz_{q}$. More subtle examples include cases in which a cobordism to such a connected sum exists topologically but not smoothly. (Conjecturally, such a splitting always exists topologically.) Further examples can be chosen to represent 2–torsion in $\Theta^3_{\qq}$.
Let $\calk$ denote the kernel of $\Theta^3_\qq \to \widehat{\Theta}^{3}_\qq$, where $\widehat{\Theta}^{3}_\qq$ denotes the topological homology cobordism group. Freedman proved that $\Theta^3_\zz \subset \calk$. A corollary of results here is that $\calk / \Theta^3_\zz $ is infinitely generated. We also demonstrate the failure in dimension three of splitting theorems that apply to higher dimensional knot concordance groups.
address:
- 'Se-Goo Kim: Department of Mathematics and Research Institute for Basic Sciences, Kyung Hee University, Seoul 130–701, Korea '
- 'Charles Livingston: Department of Mathematics, Indiana University, Bloomington, IN 47405 '
author:
- 'Se-Goo Kim'
- Charles Livingston
title: 'Non-splittability of the rational homology cobordism group of 3–manifolds'
---
[^1]
Introduction. {#sectionintroduction}
=============
In [@furuta], Furata applied instanton theory to reveal unexpectedly deep structure in the homology cobordism group of smooth homology 3–spheres, $\Theta^3_\zz$. Here we will use the added algebraic structures associated to Heegaard–Floer theory to identify further complications in the rational cobordism group, $\Theta^3_\qq$.
As a simple example, an application of Lisca’s rational homology cobordism classification of lens spaces [@lisca] implies that for $p$ and $q$ relatively prime, the lens space $L(pq,1)$ is not $\qq$–homology cobordant to any connected sum $L(p,a) \# L(q,b)$. A simple consequence of the work here is that $L(pq,1)$ is not $\qq$–homology cobordant to any connected sum $M_p \#M_q$ where $H_1(M_p) = \zz_p$ and $H_1(M_q) = \zz_q$.
We let $\Theta^3_R$ denote the $R$–homology cobordism group of three-dimensional $R$–homology spheres. Note that $\Theta^3_{\zz[1/p]}$ is generated by three-manifolds $M$ with $H_1(M)$ $p$–torsion. There is a canonical map $$\Phi \co \oplus_{p \in \calp} \Theta^3_{\zz[1/p]} \to \Theta^3_{\qq}.$$ Rochlin’s Theorem and Furuta’s result imply that the kernel of $\Phi$ is infinitely generated. Our main result is the following:.05in
[**Proposition.**]{} [*The cokernel of $\Phi$, $ \Theta^3_{\qq}/ \Phi(\oplus_{p \in \calp} \Theta^3_{\zz[1/p]}) $, contains an infinite free subgroup generated by lens spaces of the form $L(pq,1)$ and infinite two-torsion, generated by lens spaces of the form $L(4n^2+1, 2n)$. An infinite subgroup is also generated by three-manifolds that bound $\qq$–homology balls topologically.* ]{} .05in
We also present applications to the study of knot concordance and present families of elements in the kernel $\Theta^3_\qq / \Theta^3_\zz \to \widehat{\Theta}^3_\qq$, where $\widehat{\Theta}^3_\qq$ denotes the topological cobordism group. Similar examples were presented in [@hlr], with the additional condition that bordisms were assumed to be Spin.
An important perspective is provided by considering the torsion linking form of three-manifolds, which yields a homomorphism $\Theta^3_\qq \to W(\qq /\zz)$, the Witt group of nonsingular $\qq/\zz$–valued linking forms on finite abelian groups. According to [@kk] this homomorphism is surjective. Again by Rochlin’s theorem and Furuta’s result, it has infinitely generated kernel (in the topological category it is conjecturally an isomorphism). A basic result of Witt theory is that $W(\qq/\zz)$ splits into primary components, $\oplus_{p \in \calp} W(\ff_p) \xrightarrow{\cong} W(\qq/\zz)$, where $W(\ff_p)$ is the Witt group of linking forms of $\ff_p$–vector spaces and $\calp$ is the set of prime integers. The conjecture that topological cobordism is determined by the linking form implies that $\widehat{\Theta}^3_\qq $ has a corresponding primary decomposition. One thrust of our work here is to display the extent of the failure of the existence of such a primary decomposition in the smooth setting.
The following commutative diagram organizes the groups of interest. In the diagram, hats denote the topological category and $\calk$ denotes the kernel of the canonical homomorphism from the smooth to the topological $\qq$–homology cobordism group. With the exception of the inclusion of the kernel, all horizontal arrows are surjective. Conjecturally, the right square consists of isomorphisms. .1in
$\begin{diagram}
\dgARROWLENGTH=1.4em
\node{ } \node{ \oplus_{p \in \calp} \Theta^3_{\zz[1/p]}} \arrow{e}\arrow{s,r}{\Phi}\node{ \oplus_{p \in \calp} \widehat{\Theta}^3_{\zz[1/p]}}\arrow{e}\arrow{s,r}{\widehat{\Phi}}\node{ \oplus_{p \in \calp} W(\ff_p)}\arrow{s,r}{\cong} \\
\node{\calk}\arrow{e} \node{ \Theta^3_{\qq}}\arrow{e} \node{ \widehat{\Theta}^3_{\qq}}\arrow{e} \node{ W(\qq/\zz)}
\end{diagram}
$
.1in
The proposition above states that $\Theta^3_\qq / \text{Image} (\Phi)$ is infinitely generated containing an infinite free subgroup and infinite two-torsion and that furthermore, the image of $\calk$ in $\Theta^3_\qq / \text{Image} (\Phi)$ similarly contains an infinite subgroup.
.05in
[**Definition.**]{} A three manifold $M$ is said to [*split*]{} if it represents a class in the image of $\Phi$. That is, a manifold does not split if it is nontrivial in the cokernel of $\Phi$. .05in [**Outline**]{} In Sections \[secdefinitions\], \[secmetabolizers\] and \[spincsection\] we present some of the basic definitions used throughout the paper, isolate a basic result concerning metabolizers of linking forms, and discuss [Spin$^c$]{}–structures. Section \[secobstructions\] presents one of our main results, describing an obstruction based on Heegaard–Floer $d$–invariants to a class in $\Theta^3_\qq$ being in the image of $\oplus_{p \in \calp} \Theta^3_{\zz[1/p]}$..05in Following this we provide a series of examples:
- Section \[basiclensspace\] demonstrates that lens spaces $L(pq, 1)$ with $p$ and $q$ square free and relatively prime do not split, and extends this to finite connected sums of such lens spaces, with all $p$ and $q$ distinct, thus proving that $ \Theta^3_{\qq}/ \Phi(\oplus_{p \in \calp} \Theta^3_{\zz[1/p]}) $ is infinite. Section \[infiniteorderlens\] further extends this, demonstrating that the set of lens spaces of the form $L(pq,1)$ (with $p$ and $q$ now required to be prime) generate an infinite free subgroup of infinite rank contained in $ \Theta^3_{\qq}/ \Phi(\oplus_{p \in \calp} \Theta^3_{\zz[1/p]}) $. .05in
- Section \[order2lensspacesec\] considers specific lens spaces of the form $L(4n^2 +1, 2n)$ to provide elements of order 2 in $\Theta^3_\qq$ that do not split, in particular showing that $ \Theta^3_{\qq}/ \Phi(\oplus_{p \in \calp} \Theta^3_{\zz[1/p]}) $ contains 2–torsion. Section \[inftwotor\] expands on this, providing an infinite family of independent elements of order 2..05in
- Section \[secbasicexample\] begins the examination of the failure of splittings among manifolds that do split topologically; that is, we consider manifolds representing classes in $\calk$. The main example is built from surgery on the connected sum of the torus knot $T_{3,5}$ and the untwisted Whitehead double of the trefoil knot, $Wh(T_{2,3}) = D$. We show that $S^3_{15}(T_{3,5} \cs D)$ splits topologically but not smoothly. Section \[secexamplestopsplit1\] generalizes that example to an infinite family, using $(p, p+2)$ torus knots, with $p$ odd. .05in
- Section \[concordance\] applies the results of Section \[basiclensspace\] to demonstrate the failure of a splitting theorem for knot concordance which, by a result of Stoltzfus [@stoltzfus], applies algebraically and in dimensions greater than 3..05in
- According to the Freedman’s work [@fr; @freedman-quinn], all homology spheres bound contractible 4–manifolds topologically, so $\Theta^3_\zz \subset \calk$. In Section \[toptrivialbordism\] we outline the proof that the quotient $\calk / \Theta^3_\zz$ contains an infinitely generated free subgroup. This was proved in [@hlr] with the added constraint that one restricts the cobordism groups by considering only manifolds that are $\zz_2$–homology spheres or by requiring that all spaces have Spin–structures. We briefly indicate how results here permit one to remove those restrictions in the argument in [@hlr].
.05in
[*Acknowledgements.*]{} We are grateful for Matt Hedden’s help in better understanding Heegaard–Floer homology. His results regarding the Heegaard–Floer theory of doubled knots is central here, and our specific examples are inspired by those that Matt pointed us toward in our collaborations with him.
Definitions {#secdefinitions}
===========
We will consider $\qq$–homology 3–spheres: these are closed 3–manifolds $M^3$ with $H_n(M^3, \qq) \cong H_n(S^3, \qq)$ for all $n$. For each such $M$ there is a symmetric linking form $\beta \co H_1(M) \times H_1(M) \to \qq/\zz $ which is nonsingular in the sense that the induced map $\beta^* \co H_1(M) \to \text{Hom}(H_1(M), \qq/\zz)$ is an isomorphism. If $M = \partial X^4$ where $X$ is a compact 4–manifold and $H_n(X, \qq) = H_n(B^4, \qq)$ for all $n$, then the kernel $\calm$ of the map $H_1(M) \to H_1(X)$ is a metabolizer for $\beta$ (see [@CG1]). That is, $\calm^\perp = \calm$, and in particular $|\calm|^2 = |H_1(M)|$. The Witt group $W(\qq/\zz)$ is built from the set of all pairs $(G, \beta)$ where $G$ is a finite abelian group and $\beta$ is a non-degenerate symmetric bilinear form taking values in $\qq/\zz$. There is an equivalence relation on this set: $(G, \beta)\sim (G', \beta')$ if $(G \oplus G', \beta \oplus -\beta')$ has a metabolizer, and under this relation it becomes an abelian group under direct sum, denoted $W(\qq/\zz)$. It can be proved (e.g. [@ahv]) that a pair $(G, \beta)$ is Witt trivial if and only if it has a metabolizer. The proof of this fact includes the following, which we will be using.
\[metasplitprop\] If $(G_1, \beta_1) \oplus (G_2, \beta_2)$ has metabolizer $\calm$ and $ (G_2, \beta_2)$ has metabolizer $\calm_2$, then $\calm_1 = \{ g \in G_1\ |\ (g,h) \in \calm \text{ for some } h \in \calm_2 \}$ is a metabolizer for $(G_1, \beta_1)$.
The Witt groups $W(\qq / \zz, \left<p\right>)$ are defined as is $W(\qq / \zz)$, considering only $p$–torsion abelian groups, and the decomposition $W(\qq/\zz)\cong \oplus_{p \in \calp } W(\qq / \zz, \left< p \right>) $ is easily proved. The Witt group of non-degenerate symmetric forms on $\ff_p$–vector spaces is denoted $W(\ff_p)$. The inclusion $W(\ff_p) \to W(\qq / \zz, \left<p\right>)$ is an isomorphism. In the proof of this, the inclusion is clearly injective, and an inverse map $ W(\qq / \zz, \left<p\right>) \to W(\ff_p) $ is explicitly constructed via “divessage” [@ahv; @mh].
Let $R$ be a commutative ring. Two closed 3–manifolds, $M_1$ and $M_2$, are called *$R$–homology cobordant* if there is a compact smooth 4–manifold $X$ with boundary the disjoint union $ M_1 \cup -M_2$ such that the inclusions $H_*(M_i, R) \to H_*(X, R)$ are isomorphisms. Equivalently, they are $R$–cobordant, written $M_1 \sim_R M_2$, if $M_1 \cs - M_2$ bounds an $R$–homology 4–ball. The set of $R$–cobordism classes of $R$–homology spheres forms an abelian group with operation induced by connected sum. This group is denoted $\Theta^3_R$.
The ring $\zz[1/p]$ is the ring of integers with $p$ inverted, consisting of all rational numbers with denominators a power of $p$. A closed $M$ is a $\zz[1/p]$–homology sphere if and only if $H_1(M)$ is $p$–torsion. The linking form provides well-defined homomorphisms $\Theta^3_\qq \to W(\qq / \zz)$ and $\Theta^3_{\zz[1/p]} \to W(\ff_p)$ for which the following diagram commutes.
$\begin{diagram}
\dgARROWLENGTH=1.5em
\node{ \oplus_{p \in \calp} {\Theta}^3_{\zz[1/p]}} \arrow{e,t}{ } \arrow{s,r}{\Phi}\node{ \oplus_{p \in \calp} W(\ff_p)}\arrow{s,r}{\cong} \\
\node{ {\Theta}^3_{\qq}} \arrow{e,t}{}\node{ W(\qq/\zz)}\\
\end{diagram}
$ -.3in $\ $
If we switch to the topological category, all these maps are conjecturally isomorphisms.
Metabolizers for connected sums {#secmetabolizers}
===============================
Metabolizers
------------
If a connected sum of 3–manifolds bounds a rational homology ball, the associated metabolizer of the linking form does not necessarily split relative to the connected sum. However, the existence of the connected sum decomposition does place constraints on the metabolizer.
\[metabolizerthm2\] If $p$ is prime, $G$ is a finite abelian group, and a given nonsingular linking form $\beta_1 \oplus \beta_2$ on $\zz_p \oplus G$ has metabolizer $\calm$, then for some $a \in G$, $(1,a) \in \calm$.
Let $G_p$ denote the $p$–torsion in $G$. There is a metabolizer $\calm_p$ for the form restricted to $\zz_p \oplus G_p$. If $\calm_p \subset G_p$, then it would represent a metabolizer for the linking form restricted to $G_p$, implying that the order of $G_p$ is an even power of $p$. But since the form on $\zz_p \oplus G_p$ is metabolic, the order of $G_p$ must be an odd power of $p$. It follows that there is an element $(a',a'') \in \calm_p$ with $a' \ne 0$. Multiplying by $(a')^{-1} \mod p$, we see that $ (1,a) \in \calm_p \subset \calm$ for some $a \in G_p$.
In the following corollary, for each integer $k$, $G_k$ denotes a finite abelian group of order dividing a power of $k$.
\[productthm\] If $m$ is a square free integer, $G_m\oplus G_n$ is a finite abelian group with $\gcd(m,n)=1$, and a given linking form $\beta_1 \oplus \beta_2\oplus \beta_3$ on $\zz_m \oplus G_m\oplus G_n$ has metabolizer $\calm$, then for some $a \in G_m$, $(1,a,0) \in \calm$.
Write $\zz_m = \zz_{p_1} \oplus \cdots \oplus \zz_{p_k}$. By Theorem \[metabolizerthm2\], the projection of $\calm$ to each $\zz_{p_i}$ summand is surjective. Since the $p_i$ are relatively prime, the projection to $\zz_m$ is similarly surjective.
In order to construct elements of infinite order, we will need to consider multiples of linking forms. Without loss of generality, we will be able to assume that the multiplicative factors are divisible by 4.
\[thmmultiplemetab\] Suppose that $p$ is prime and the nonsingular form $4 k(\beta_1 \oplus \beta_2)$ on $(\zz_p \oplus G)^{4k}$ has a metabolizer $\calm$. Then $\calm$ contains an element of the form $(1, 1, \ldots , 1, \alpha_{2k+1}, \cdots , \alpha_{4k}) \oplus b$ for some set of $\alpha_i \in \zz_p$ and some $b \in G^{4k}$.
The Witt group $W(\qq/\zz)$ is 4–torsion [@mh], and thus $4 k \beta_2$ has a metabolizer $\calm'$. By Proposition \[metasplitprop\], the set of elements $x$ such that $(x,y) \in \calm$ for some $y \in \calm'$ is a metabolizer, denoted $\caln$, for $4k \beta_1$, and thus is $2k$–dimensional. As argued in [@LN1], a simple application of the Gauss–Jordan algorithm applied to a generating set for $\caln$ yields a generating set consisting of vectors of the form $(1,0 ,0 ,0 , \ldots , 0, *, * \ldots )$, $(0,1 ,0 ,0 , \ldots , 0, *, * \ldots )$, $(0,0, 1 ,0 , \ldots , 0, *, * \ldots )$, $\ldots$, where each initial sequence of a 1 and 0s is of length $2k$.
By adding these vectors together, we find that the metabolizer $\caln$ contains an element of the form $ (1,1,\cdots, 1, \alpha_{2k+1} , \cdots, \alpha_{4k}) \in \zz_p^{4k}$. Finally, since each element in $\caln$ pairs with an element in the metabolizer $\calm'$ to give an element in $\calm$, we get the desired element $b$.
[Spin$^c$]{}–structures {#spincsection}
=======================
We need the following facts about [Spin$^c$]{}($Y)$, the set of [Spin$^c$]{}–structures on an arbitrary space $Y$.
- The first Chern class is a map $c_1 \co \text{{Spin$^c$}}( Y ) \to H^2(Y).$.05in
- There is a transitive action $H^2(Y) \times \text{{Spin$^c$}}(Y) \to \text{{Spin$^c$}}(Y) $ denoted $(\alpha,{\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}) \to \alpha \cdot {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}$..05in
- For $Y \subset W$, the restriction map $r$ is functorial: If ${\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}\in \text{{Spin$^c$}}(W)$, $\alpha \in H^2(W)$ then $$r(\alpha \cdot {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}) = r(\alpha) \cdot r({\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}).$$.05in
- For all $\alpha \in H^2(Y) $ and ${\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}\in \text{{Spin$^c$}}(Y)$, $c_1(\alpha \cdot {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}) - c_1({\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}) = 2 \alpha $..05in
- As a corollary, if $|H^2(Y)|$ is finite and odd, then $c_1 \co \text{{Spin$^c$}}( Y ) \to H^2(Y) $ is a bijection. .05in
- There is a canonical bijection: [Spin$^c$]{}($Y \cs W) \to$ [Spin$^c$]{}($Y) \times$ [Spin$^c$]{}($W$). .05in
For every smooth 4–manifold $X$, the set [Spin$^c$]{}$(X)$ is nonempty. (See [@gs] for a proof.) As a consequence, we have the following.
\[extendthm\] Let $ N = \partial X$ and let $ {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}\in \text{{Spin$^c$}}(N) $ be the restriction of a [Spin$^c$]{}–structure on $X$. Then the set of [Spin$^c$]{}–structures on $N$ which extends to $X$ are those of the form $ \alpha \cdot {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}$ for $ \alpha $ in the image of the restriction map $r\co H^2(X) \to H^2(N)$.
Identifying $H_1(N)$ and $H^2(N)$.
----------------------------------
Suppose that $N$ is a rational homology 3–sphere bounding a rational homology ball $X$. Then by Poincaré duality, $H_1(N) \cong H^2(N)$. We have denoted kernel($H_1(N) \to H_1(X))$ by $\calm$. Via duality, it corresponds to the image of $H^2(X)$ in $H^2(N)$. Thus, we will use $\calm$ to denote this subgroup of $H^2(N)$.
Spin–structures
---------------
If the order $|H_1(M)|$ is odd, then there is a unique Spin–structure on $M$ that lifts to a canonical [Spin$^c$]{}–structure that we will denote ${\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_0$. With this, there is a natural identification of $H^2(M)$ with [Spin$^c$]{}$(M)$. However, we face the complication that in assuming that $M$ bounds a rational homology 4–ball $X$, we cannot assume that $X$ has a Spin–structure. The following result permits us to adapt to this possibility. (In addition to playing a role in considering splittings of classes in $\Theta^3_\qq$, in Section \[toptrivialbordism\] we will use this result to extend a theorem from [@hlr] in which an added hypothesis was needed to ensure the existence of a Spin–structure on $X$.)
\[lemmaextendspin\] Suppose that $ N_1 \cs N_2 = \partial X$ for some smooth rational homology 4–ball $X$ and that the order of $H_1(N_1)$ is odd. Then the image of the restriction map [Spin$^c$]{}$(X) \to $ [Spin$^c$]{}$(N_1)$ contains the Spin–structure ${\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_0 \in $ [Spin$^c$]{}$(N_1)$. In particular, every element in the image of this restriction map is of the form $\alpha \cdot {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_0$ for $\alpha \in \text{Image}(H^2(X) \to H^2(N_1))$.
Let $H = \text{Image}(H^2(X) \to H^2(N_1))$ and $S =\text{Image}($[Spin$^c$]{}$(X) \to$ [Spin$^c$]{}$(N_1))$. As usual, the choice of an element ${\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}\in S$ determines a bijection between $H$ and $S$. In particular, the number of elements in $S$ is the same as in $H$, which is odd. Conjugation defines an involution on $S$ which commutes with restriction. Thus, since $S$ is odd, conjugation has a fixed point in $S$. But the only fixed element under conjugation is the Spin–structure, since $ c_1(\bar{{\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}}) = -c_1({\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi})$.
Basic obstructions from $d$–invariants {#secobstructions}
======================================
To each rational homology 3–sphere $M$ and ${\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}\in \text{{Spin$^c$}}(M)$ there is associated an invariant $d(M,{\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}) \in \qq$, defined in [@os2]. It is additive under connected sum: $d(M\cs N, ({\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_1 , {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_2)) = d(M\ , {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_1 ) + d( N, {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_2) $. A key result relating the $d$–invariant and bordism is the following from [@os2].
If $M = \partial X$ with $H_*(X,\qq) \cong H_*(B^4,\qq)$, and $ {\ifmmode{{\mathfrak t}}\else{${\mathfrak t}$\ }\fi}\in \text{{Spin$^c$}}(X)$, then $d(M,{\ifmmode{{\mathfrak t}}\else{${\mathfrak t}$\ }\fi}|_{M}) = 0$.
Obstruction theorem
-------------------
Suppose that $|H_1(M)|$ is odd and ${\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_0$ is the unique Spin–structure on $M$. For $\alpha \in H^2(M)$, we abbreviate $d(M, \alpha \cdot {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_0) $ by $d(M, \alpha)$.
$\bar{d}(M, \alpha) = d(M, \alpha) -d(M,0)$.
The following result will be sufficient to prove that $\Theta^3_\qq / \Phi(\oplus \Theta^3_{\zz[1/p]})$ is infinite.
\[obstructthm\]Suppose $\{M_i\}$ is a collection of 3–manifolds for which $H_1(M_i) = \zz_{m_i} \oplus \zz_{n_i}$, where $m_i$ and $n_i$ are square free and odd, and the full set $\{m_i, n_i\}$ is pairwise relatively prime. If a finite connected sum $\cs_{k=1}^N \pm M_{i_k}$ represents a class in $\Theta^3_{\qq}$ that is in the image of $\oplus_{p} \Theta^3({\zz[1/p]})$, then for all $i=i_k, 1 \le k \le N$, and for all $(a,b) \in \zz_{m_i} \oplus \zz_{n_i}$, $$\bar{ d} (M_{i}, (a,b) ) = \bar{d}(M_{i}, (a,0) ) + \bar{d}(M_{i}, (0,b)) .$$
Suppose that $Y = \cs_k \pm M_{i_k} \in \Phi(\oplus_{p} \Theta^3({\zz[1/p]}))$. We consider $k= 1$, abbreviating $M_{i_1} = M$ and $H_1(M) \cong \zz_m \oplus \zz_n$. Suppose that $Y$ is in the image. Then $Y \cs \oplus Y_{p_i} = \partial X$ for some collection of $Y_{p_i}$ which are $\zz[p_{i}^{-1}]$–homology spheres and $X$ is a rational homology ball. Collecting summands, we can write $M \cs N_m \cs N_n \cs N = \partial X$, where the prime factors of $|H_1(N_m)|$ all divide $m$, the prime factors of $|H_1(N_n)| $ all divide $n$, and $|H_1(N)|$ is relatively prime to $mn$. Let $({\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_0, {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_1, {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_2, {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_*) \in \text{Image({Spin$^c$}}(X))$. (By Theorem \[lemmaextendspin\] we can assume that the structure ${\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_0 \in \text{{Spin$^c$}}(M)$ is the Spin–structure.) Then by Corollary \[productthm\], for all $a \in \zz_m$ and $b \in \zz_n$, there are elements $a' \in H_1(N_m)$ and $b' \in H_1(N_n)$ such that: .05in
- $((a,0) \cdot {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_0, a' \cdot {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_1, {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_2, {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_*) \in \text{Image({Spin$^c$}}(X))$. .05in
- $( (0,b) \cdot {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_0, {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_1,b' \cdot {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_2, {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_*) \in \text{Image({Spin$^c$}}(X))$. .05in
- $( (a,b) \cdot {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_0, a' \cdot {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_1,b' \cdot {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_2, {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_*) \in \text{Image({Spin$^c$}}(X))$. .05in
Thus, we have the following vanishing conditions on the $d$–invariants: .05in
- $d(M , {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_0) + d(N_m, {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_1) + d(N_n, {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_2) + d(N, {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_*) = 0$..05in
- $d(M , (a,0)\cdot {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_0) + d(N_m, a' \cdot {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_1) + d(N_n, {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_2) + d(N, {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_*) = 0$..05in
- $d(M ,(0,b) \cdot {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_0) + d(N_m, {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_1) + d(N_n, b' \cdot {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_2) + d(N, {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_*) = 0$..05in
- $d(M , (a,b) \cdot {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_0) + d(N_m, a' \cdot {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_1) + d(N_n,b' \cdot {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_2) + d(N, {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_*) = 0$..05in
Subtracting the second and third equality from the sum of the first and fourth yields: $$d(M , (a,b) \cdot {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_0) - d(M , (a,0)\cdot {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_0) - d(M , (0,b) \cdot{\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_0) +d(M , {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_0) = 0.$$ Recalling that $\bar{d}(M, \alpha )$ denotes $d(M, \alpha \cdot {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_0) - d(M, {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_0)$, this can be rewritten as $$\bar{d}(M, (a,b)) - \bar{d}(M, (a,0 ))- \bar{d}(M, (0,b)) =0.$$ Repeating for each $M_i$ completes the proof of the theorem.
Lens Space Examples: $L(pq,1)$. {#basiclensspace}
===============================
Let $\{p_i,q_i\}$ be a set of pairs of odd integers such that the union of all pairs are pairwise relatively prime. We prove:
\[lensspacethm\] No finite linear combination $\cs_k \pm L(p_{i_k}q_{i_k} ,1)$ represents an element in the image $\Phi(\oplus_{p \in \calp} \Theta^3_{\zz[1/p]}) \subset \Theta^3_\qq$.
We consider the first term $L(p_1q_1,1)$ and simplify notation by writing $p = p_1$ and $q = q_1$. By Theorem \[obstructthm\] we would have for all $(a,b) \in \zz_{p} \oplus \zz_{q}$, $$\bar{ d} (L(pq,1), (a,b) ) = \bar{d}(L(pq,1), (a,0) ) + \bar{d}(L(pq,1), (0,b)) .$$ According to [@os2], for some enumeration of [Spin$^c$]{}–structures on $L(m,n)$, denoted ${\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_i$, $0\le i <m$, if we let $D(m,n,i) = d(-L(m,n), {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_i)$, there is the recursive formula: $$D(m,n,i) = \frac{ mn - (2i +1 - m-n)^2}{4mn} - D(n,m',i'),$$ where the primes denote reductions modulo $n$, $0<n <m$, and $0 \le i < m$. The base case in the recursion is by definition $D(1,0,0) = 0$. For every [Spin$^c$]{}–structure ${\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}$ there is a conjugate structure $\bar{{\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}}$ for which $d(M, {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}) = d(M, \bar{{\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}})$ and ${\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}\ne \bar{{\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}}$ unless ${\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}$ is the Spin–structure. We claim that for $L(pq,1)$, the [Spin$^c$]{}–structure ${\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_0$ does correspond to the Spin–structure. To see this, observe that an algebraic compuation shows $4pqD(pq,1,i) = -4i^2 +4pq i +pq(1-pq)$ and in particular, $pqD(pq,1,0) = pq(1-pq)$. The difference $4pqD(pq,1,i) - 4pqD(pq,1,0) = 4i(pq -i)$, does not take on the value 0 for any $0<i<pq$. Since the value of $D(pq,1,0)$ is unique among the $d$–invariants, it must correspond to the Spin–structure. In applying Theorem \[obstructthm\], we identify $\zz_p \oplus \zz_q \cong \zz_{pq}$, so that the pair $(a,b) \in \zz_p \oplus \zz_q $ corresponds to $aq+bp \in \zz_{pq}$. In this case, the criteria becomes $$D(pq,1, ap+bq) - D(pq,1,ap) - D(pq,1,bq)+ D(pq,1,0) = 0.$$ Certainly $p+q <pq$, so we can apply the formula for $D$ with $a = b =1$ . However, in this case the sum is immediately calculated to equal $-2 \ne 0$.
Infinite order examples {#infiniteorderlens}
=======================
The examples of the previous section are sufficient to demonstrate that the quotient $\Theta^3_\qq / \Phi(\oplus \Theta^3_{\zz[1/p]})$ is infinite. We now present an argument to show it contains an infinite free subgroup. To carry out this argument we need to make the additional assumption of primeness for the relevant $p$ and $q$. Let $\{p_i , q_i\}$ be a set of distinct odd prime pairs with all elements distinct. This section is devoted to the proof of the following theorem.
\[infiniteorderthm\] The lens spaces $L(p_i q_i, 1)$ are linearly independent in the quotient $\Theta^3_\qq / \Phi(\oplus \Theta^3_{\zz[1/p]})$.
Notation
--------
Suppose that $\sum_i b_i L(p_iq_i ,1) \subset \text{Image\ } (\Phi)$. We can assume that $b_1 \ne 0$. We simplify notation, writing $p $ and $q $ for $p_1$ and $q_1$, respectively. There is no loss of generality in assuming that for all $i$, $b_i = 4k_i$ for some $k_i$, and write $k= k_1$. At times we also abbreviate $L(pq,1) = L_{pq}$.
Following our earlier approach, we will show that a contradiction arises from the assumption that $N = 4k L(pq,1) \cs M_p \cs M_q \cs M_*= \partial X$ for some rational homology 4–ball $X$, where the orders of $H_1(M_p)$ and $H_1(M_q)$ are powers of $p$ and $q$, respectively, and the order of $H_1(M_*)$ is relatively prime to $pq$.
According to Theorem \[thmmultiplemetab\], the $p$–primary part of the associated metabolizer, $\calm_p$, includes a vector $A = ((1, \ldots, 1, \alpha_{2k+1} , \ldots , \alpha_{4k}), g) \in (\zz_p)^{4k} \oplus H_1(M_p)$. Similarly, the $q$–primary part of the associated metabolizer, $\calm_q$, includes a vector $B = ((1, \ldots, 1, \beta_{2k+1} , \ldots , \beta_{4k}), h) \in (\zz_q)^{4k} \oplus H_1(M_q)$.
Constraints on the $d$–invariants
----------------------------------
We let the Spin–structures on $L(pq,1)$, $M_p$, and $M_q$ be ${\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_0, {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_0'$ and ${\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_0''$, respectively. Consider now the vectors 0, $aA$, $bB$, and $aA + bB \in \calm$. Computing the $d$-invariant associated to each, we find that each of the following sums is 0.
- $2k d(L_{pq}, s_0) + \sum_{i = 2k+1}^{ 4k} d(L_{pq} , {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_0) + d(M_p, {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_0' ) + d(M_q, {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_0'') +d(M_*,{\ifmmode{{\mathfrak t}}\else{${\mathfrak t}$\ }\fi}) $..05in
- $2k d(L_{pq}, aq \cdot s_0) + \sum_{i = 2k+1}^{4k} d(L_{pq}, aq \alpha_i \cdot {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_0) + d(M_p, a g \cdot {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_0' ) + d(M_q, {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_0'') +d(M_*,{\ifmmode{{\mathfrak t}}\else{${\mathfrak t}$\ }\fi})$..05in
- $2k d(L_{pq}, bp\cdot s_0) + \sum_{i = 2k+1}^{ 4k} d(L_{pq}, bp \beta_i \cdot {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_0) + d(M_p, {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_0' ) + d(M_q, bh\cdot {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_0'') +d(M_*,{\ifmmode{{\mathfrak t}}\else{${\mathfrak t}$\ }\fi})$..05in
- $2k d(L_{pq}, (aq + bp) \cdot s_0) + \sum_{i = 2k+1}^{ 4k} d(L_{pq} , (aq \alpha_i + bp \beta_i) \cdot {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_0) + d(M_p, a g \cdot {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_0') )+ d(M_q, bh\cdot {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_0'')+d(M_*,{\ifmmode{{\mathfrak t}}\else{${\mathfrak t}$\ }\fi}) $..05in
[**Note.**]{} We have again used that the inclusion $\zz_p \subset \zz_{pq}$ takes $\alpha$ to $\alpha q$, and similarly for $\zz_q$ and $\beta$. We now take the sum of the first and last equation, and subtract the sum of the middle two. The result is that for some set of $a_i$ and $b_i$: $$\hskip-.5in 2k\left( { d} (L_{pq}, aq + bp) - { d} (L_{pq}, aq )- { d} (L_{pq}, bp) + {d} (L_{pq}, 0) \right)+$$ $$\hskip.5in \sum_{i = 2k+1}^{4k} \left( { d} (L_{pq}, a_i q + b_i p) - { d} (L_{pq}, a_i q )- { d} (L_{pq}, b_i p) + { d} (L_{pq}, 0)\right)
= 0.$$ We now introduce further notation: let $$\delta (L_{pq} , a,b) = {d} (L_{pq}, aq + bp) - {d} (L_{pq}, aq)- {d} (L_{pq}, bp) + {d} (L_{pq}, 0) .$$ With this, we have proved the following lemma.
\[deltaboundlemma\] If the lens spaces $L_{p_i q_i}$ are linearly dependent in $\Theta^3_\qq / \Phi(\oplus \Theta^3_{p})$ and, for $p=p_1$ and $q=q_1$, $L_{pq}$ has nonzero coefficient in some linear relation, then for all $a$ and $b$ there are $k$, $a_i$ and $b_i$ such that, $$2k {\delta} (L_{pq},a,b) + \sum_{i = 2k+1}^{ 4k} {\delta} (L_{pq},a_i ,b_i ) = 0.$$
Computation of bounds on $\delta (L_{pq},a,b)$
----------------------------------------------
Note that $\delta(L_{pq},a,b)=0$ if $a=0$ or $b=0$. Given Lemma \[deltaboundlemma\], the proof of Theorem \[infiniteorderthm\] is completed with the following result.
For all $a \ne 0 \mod p$ and $b \ne 0 \mod q$, $\delta( L_{pq}, a,b) < 0 $.
All [Spin$^c$]{}–structures are included by considering the range $-\frac{p-1}{2} \le a \le \frac{p-1}{2}$ and $-\frac{q-1}{2} \le b \le \frac{q-1}{2}$. By symmetry we can exclude the case $a<0$. Since the formula for the $d$–invariant $d(L(pq,1), i)$ assumes $i \ge 0$, there are three cases to consider.
1. $a > 0 , b>0$..05in
2. $a>0 , -\frac{aq}{p} < b <0$. .05in
3. $a>0 , b< -\frac{aq}{p}$. .05in
The formula for the $d$–invariant in the current case is $$4 n(d(L(n,1), i)) = { n- (2i + 1 -n -1)^2} = n - n^2 + 4n i - 4i^2,$$ for $0 \le i < n$. We now compute $4pq \delta(L_{pq}, aq+bp)$ in each of the three cases. First note that $\delta(L_{pq},aq + bp) = d(L_{pq}, aq+bp) - d(L_{pq}, aq) - d(L_{pq},bp) + d(L_{pq},0).$ In places we write $pq = n$ to simplify the appearance of the formula.
1. Since all entries are now positive we find $$\begin{aligned}
4n \delta(L_{pq},a,b) &= \left( n - n^2 + 4n(aq +bp) - 4(aq+bp)^2 \right)\\
&- \left( n - n^2 + 4n(aq ) - 4(aq)^2 \right)\\
&- \left( n - n^2 + 4n(bp ) - 4(bp)^2 \right)\\
&+ \left( n - n^2 + 4n(0 ) - 4(0)^2 \right).\end{aligned}$$ This simplifies to $-8abpq$, which is negative. .05in
2. In this case $bp<0$, so we replace $d(L_{pq}, bp) $ with $d(L_{pq}, -bp)$ in the computation. $$\begin{aligned}
4n \delta(L_{pq},a,b) & = \left( n - n^2 + 4n(aq +bp) - 4(aq+bp)^2 \right)\\
&- \left( n - n^2 + 4n(aq ) - 4(aq)^2 \right)\\
&- \left( n - n^2 + 4n(-bp ) - 4(-bp)^2 \right)\\
&+ \left( n - n^2 + 4n(0 ) - 4(0)^2 \right).\end{aligned}$$ This simplifies to give $-8b(a-p)pq$. Since $b<0$ and $a < \frac{p-1}{2}$, this is negative..05in
3. In this case, both $bp$ and $aq + bp <0$. Thus, we compute $$\begin{aligned}
4n \delta( L_{pq},a,b ) & = \left( n - n^2 + 4n(-aq -bp) - 4(-aq-bp)^2 \right)\\
&- \left( n - n^2 + 4n(aq ) - 4(aq)^2 \right)\\
&- \left( n - n^2 + 4n(-bp ) - 4(-bp)^2 \right)\\
&+ \left( n - n^2 + 4n(0 ) - 4(0)^2 \right).\end{aligned}$$ This simplifies to give $-8apq(b+q)$. Since $b > -\frac{q-1}{2}$, this is again negative.
An Order 2 lens space that does not split {#order2lensspacesec}
=========================================
We now consider a lens space that represents 2–torsion in $\Theta^3_\qq$. Let $M= L(65,8)$; since $8^2 = -1 \mod 65$, $M= -M$ and $2M = 0 \in \Theta^3_\qq$. We show that $M$ does not split. It follows quickly from the fact that $ L(65,8) $ is of finite order in $\Theta^3_\qq$ that for the Spin-structure ${\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}^*$, $d(L(65,8), {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}^*) = 0$. On can compute directly from the formula for $D$ given above that the value 0 is realized only by ${\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_{36}$. Thus, in applying Theorem \[obstructthm\] we identify the homology class $x \in H_1(L(65,8))$ with the [Spin$^c$]{}–structure ${\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_{36+x}$, where the index is taken modulo 65. The matrix in Figure \[fig1a\] presents the values of $d(L(65,8), 13a + 5b)$ (multiplied by 65 to clear denominators). Rows correspond to the values of $a$ and columns to $b$. The central row and left column correspond to $a=0$ and $b=0$ respectively. Symmetry permits us to list only the values with $b \ge 0$. In Figure \[fig2a\] we list the differences, $d(L(65,8), 13a + 5b) - d(L(65,8), 13a ) -d(L(65,8), 5b)$, with the nonzero entries demonstrating the failure of additivity.
$$\begin{array}{c|c|c|c|c|c|c|c|}
& b=0 & b= 1 & b= 2 & b= 3 & b= 4 & b=5 & b= 6\\
\hline
a=2 & -52 & 18 & -32 & 58 &28 & 8 & 128 \\
\hline
a=1 & 52 & -8 & 72 & 32 &2 &112 & -28\\
\hline
a=0 &0 & 70 & 20 & -20 & 80 &-70& -80 \\
\hline
a=-1 &52 &-8 & -58 & 32 &-128 & 18 & -28 \\
\hline
a =-2 & -52 &-112 &-32 & -72 & 28 &8 & -2 \\
\hline
\end{array}$$
$$\begin{array}{c|c|c|c|c|c|c|c|}
& b=0 & b= 1 & b= 2 & b= 3 & b= 4& b=5 & b= 6\\
\hline
a=2 & 0 & 0 & 0 & 2 & 0 & 2 & 4 \\
\hline
a=1 & 0 & -2 & 0 & 0 & -2 & 2 & 0 \\
\hline
a=0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
\hline
a=-1 & 0 &-2 & -2 & 0& -4 &0 & 0 \\
\hline
a =-2 & 0 & -2 & 0 & 0 & 0 & 2 & 2 \\
\hline
\end{array}$$
Infinite 2–torsion {#inftwotor}
==================
We now generalize the previous example to describe an infinite subgroup of $\Theta^3_\qq$ consisting of 2–torsion that injects into the quotient $\Theta^3_\qq / \Phi(\oplus_{p \in \calp} \Theta^3_{\zz[1/p]})$. Consider the family $N_n = L( 4(5n +1)^2 +1, 2(5n+1))$; for $n=-1$ we have $-L(65, 8)$ as in the previous section, but we simplify the computations by restricting to $n>0$. Expanding, we have $N_n = L(5 ( 20n^2 +8n +1), 2(5n+1))$. If $n \ne 3 \mod 5$, then $20n^2 + 8n +1$ is not divisible by 5. By Appendix \[appendpi\] we can further assume that the $n$ are selected so that $n$ is divisible by 5 and the set of integers $20n^2 + 8n +1$ are pairwise relatively prime and square free. We enumerate the set of such $n$ as $n_i$ and abbreviate the corresponding lens spaces as $L(5p_i, q_i) = N_{n_i}$. The remainder of this section is devoted to proving the following.
\[twotorthm\]The set $\{N_{n_i}$} generates an infinite subgroup consisting of elements of order 2 in $\Theta^3_\qq / \Phi(\oplus_{p \in \calp} \Theta^3_{\zz[1/p]})$.
To begin, we need to identify the Spin–structure. We use the recursion formula $$D(m,n,i) = \frac{ mn - (2i +1 - m-n)^2}{4mn} - D(n,m',i')$$ to compute relevant $d$–invariants. We are interested in the lens spaces $L(4r^2 +1, 2r)$. One step of the recursion reduces this to $L(2r, 1)$, and another step reduces it to $S^3$. Since we need to reduce modulo $2r$, for $0\le i<4r^2+1$, let $y$ be the remainder of $i$ modulo $2r$ and $x$ the quotient so that $2rx+y=i$. So we write [Spin$^c$]{}–structures as ${\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_{2rx + y}$ for $0 \le y < 2r$ and $0\le 2rx+y<4r^2+1$. Carrying out the arithmetic yields:
\[spinlemma\] For any $r>0$, $x$ and $y$ with $0\le y<2r $ and $0\le 2rx+y<4r^2+ 1$,
1. $d(L(4r^2 +1, 2r), {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_{2rx+y}) =\frac{2\left(rx^2+(y-r(2r+1))x-r(y^2-(2r-1)y-r)\right)} {4r^2+1}$..05in
2. The discriminant of the numerator, viewed as a quadratic polynomial in the variable $x$, is $4(y-r)^2(4r^2+1)$. Moreover, it is the square of an integer if and only if $y=r$..05in
3. $d(L(4r^2 +1, 2r), {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_{2rx+y})=0$ if and only if $x=r$ and $y=r$..05in
4. The Spin–structure on $L(4r^2 +1, 2r)$ is ${\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_{2r^2 +r}$..05in
In our case $r = 5n +1$ and the Spin–structure is ${\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_{50n^2+25n+3}$.
For each $n$, we write $N_n = L(5p_n , q_n)$ and assume that some linear combination $\sum N_{n_i} = 0 \in \Theta^3_\qq / \Phi(\oplus_{p \in \calp} \Theta^3_{\zz[1/p]})$. We write the first term in the sum as $N = L(5p,q)$ where $p = 20n^2 + 8n +1$.
Since the sum splits, for some collection of primes $r_j$ and manifolds $M_{r_j}$ with $H_1(M_{r_j}) $ being $r_j$–torsion, we have $$N \#_{i >1} N_{n_i} \#_j M_{r_j} = \partial X,$$ where $X$ is a rational homology ball. We can collect terms as $N \# M_p \# M_m = \partial X$ where $M_p$ includes all the $M_{r_j}$ for which $r_j$ divides $p$, and $M_m$ contains all the other summands, including all the $N_{n_i}$ with $i>1$.
The homology of this connected sum of three manifolds splits into the direct sum of three groups: $(\zz_5 \oplus \zz_p) \oplus G_p \oplus G_m$, where the order of $G_p$ is a product of prime factors of $p$, 5 does not divide the order of $G_p$, and the orders of $G_p$ and $G_m$ are relatively prime. It follows that the $5$–torsion in the metabolizer, $\calm_5$, is contained in $ (\zz_5,0) \oplus 0 \oplus G_m$. The direct sum of all primary parts of the metabolizer for primes that divide $p$, $\calm_p$, is contained in $\calm_p = (0,\zz_p ) \oplus G_p \oplus 0$.
Now, as in our previous arguments, $\calm_5$ contains an element of the form $(1,0) \oplus 0 \oplus a''$ and $\calm_p$ contains an element $(0,1) \oplus b'' \oplus 0$. Continuing as in the early proofs, we find that for all $a$ and $b$, $$\bar{d}(L(5p,q), (a,b)) = \bar{d}(L(5p,q), (a,0)) + \bar{d}(L(5p,q), (0,b) ).$$ Or, writing $\zz_5 \oplus \zz_p$ as $\zz_{5p}$, $$\bar{d}(L(5p,q), pa + 5b) = \bar{d}(L(5p,q), pa ) + \bar{d}(L(5p,q),5b ).$$ Since $L(5p,q)$ is of order two, for the Spin–structure the $d$–invariant vanishes, so the $\bar{d}$–invariant is the same as the $d$–invariant. We let $a = 1$ and $b=-1$, and arrive at a contradiction by showing the following equality does not hold: $${d}(L(5p,q), p-5) = {d}(L(5p,q), p ) + {d}(L(5p,q), -5 ).$$ To apply Lemma \[spinlemma\] we need to express each of $(50n^2+25n+3)+p-5$, $(50n^2+25n+3)+p $, and $(50n^2+25n+3)-5 $, as $2(5n+1)x +y$. Simple algebra yields the following pairs $(x,y)$ for these three respective [Spin$^c$]{}–structures:
- $a=1, b= -1 \longrightarrow (x,y) = ( 7n+1, 9n-3)$..05in
- $a=1, b= 0 \longrightarrow (x,y) = ( 7n+1, 9n+2)$..05in
- $a=0, b= -1 \longrightarrow (x,y) = ( 5n+1, 5n-4)$..05in
Finally, one uses these expressions to determine that for all $n$, $${d}(L(5p,q), p-5) - {d}(L(5p,q), p ) - {d}(L(5p,q), -5 )= 4.$$ Since the difference is not zero, no splitting exists and the proof of Theorem \[twotorthm\] is complete.
Topologically split examples {#secbasicexample}
============================
In this section, we apply Theorem \[obstructthm\] to find examples of manifolds that split topologically but not smoothly. We begin by carefully examining an example in which the splitting exists smoothly, focusing on the computation of the $d$–invariants, and next illustrate the modifications which do not change its topological cobordism class, but alter it smoothly. The deepest aspect of the work is in the determination of the $d$–invariants. In brief, the manifold we look at is 15–surgery on the $(3,5)$–torus knot, $T_{3,5}$, denoted $ S^3_{15}(T_{3,5}) $. This is homeomorphic to the connected sum $L(3,5) \cs - L(5,3)$. Next, letting $D $ denote the untwisted double of the trefoil knot ($D = Wh(T_{2,3})$), which is topologically slice, we consider $ S^3_{15}(T_{3,5} \cs D ) $, and prove that it does not split in the cobordism group.
In this section and the next, and also Appendix \[torusknotpoly\], we develop properties of the Heegaard-Floer complex of specific torus knots as well as tensor products of certain of these complexes. Related and more extensive computations appear in [@hhn].
$\bf \bar{d} (S^3_{15}(T_{3,5}),i ))$
--------------------------------------
We now determine the doubly filtered Heegaard-Floer complex $CFK^\infty(S^3, T_{3,5})$. This complex is by definition a doubly filtered, graded chain complex over $\ff_2$. Thus a set of filtered generators can be illustrated on a grid with the coordinates representing the filtration levels and the grading marked. There is an action of $\zz$ on the complex, and if we let $U$ be the generator, this makes the complex a $\ff_2[U, U^{-1}]$–module. The action of $U$ on the complex lowers filtration levels by 1 and gradings by 2.
We now show that $CFK^\infty(S^3, T_{3,5})$ is as illustrated in Figure \[hf35-4.pdf\]. In order to find this decomposition, we start by focusing on the central column (for which the top-most generator is at filtration level $j = 4$ and is labeled with its grading 0). The vertical column, $i = 0$, represents the sub-quotient complex $\widehat{CFK}(S^3 ,T_{3,5})$. We begin by explaining why it appears as it does in the illustration. According to [@os3 Theorem 1.2], since for torus knots there is an integer surgery that yields a lens space, $\widehat{HFK}(S^3, T_{3,5}, j)$, the quotients of the $j$-filtration level by the $(j-1)$–filtration level is completely determined by the Alexander polynomial, $$\Delta_{T_{3,5}}(t) = 1 -(t^{-1}+t) +(t^{-3} + t^3) - (t^{-4} + t^4).$$ This explains the location of the generators of $\widehat{CFK}(S^3 ,T_{3,5})$. Similarly, [@os3] determines the grading of the generators. The fact the complex $\widehat{CFK}(S^3 ,T_{3,5})$ is a filtration of the complex $\widehat{CF}(S^3 )$ which has homology $\ff_2$ with its generator at grading level 0, forces the vertical arrows, presenting the boundary maps, to be as illustrated. To build the $ {CFK}^\infty$ diagram from the $\widehat{CFK}$ diagram, we first apply the action of $U$ to fill in the generators as well as the all the vertical arrows. We next note that the homology groups $\widehat{HFK}(T_{3,5},i)$ can be computed using the horizontal slice $j=0$ instead of the vertical slice, and this forces the existence of the horizontal arrows as drawn. With this much of the diagram drawn, and the action of $U$ lowering grading by $2$, the gradings of all the elements in the diagram are determined. Finally, we note that the fact that the boundary map lowers gradings by 1 rules out the possibility of any other arrows.
[ ![[]{data-label="hf35-4.pdf"}](hf35-4.pdf "fig:") ]{}
According to [@os4], the complex $CFK^+(S^3_{15}(T_{3,5}),s)$, for $-7 \le s \le 7$ is given by the quotient $$CFK^\infty(S^3 , T_{3,5} ) / CFK^\infty(S^3, T_{3,5} )_{i <0, j<s}[- \eta],$$ where the quotienting subgroup is shaded in the diagram for $s=-4$. Here $\eta$ is a grading shift: $$\eta = \frac{-(2s-15)^2 +15}{60}.$$ By definition, the $d$–invariant is the minimal grading among all classes in the group $HFK^+(S^3_{15}(T_{3,5}),s )$ which are in the image of $U^n$ for all $n$. From the diagram, without shifting the gradings, we see this minimum for $HFK^+(S^3_{15}(T_{3,5}),-4 )$ is $-8$: one generator of grading level $-10$ has been killed, and all such generators are homologous. The values for all [Spin$^c$]{}–structures, $s = -7, -6, \ldots , 6,7$ are given in order as $$\{-14, -12,-10,-8,-8,-6,-4,-4,-2,-2,-2,0,0,0,0\}.$$ After the grading shift, the values are all of the form $a_i/30$, where, in order, the $a_i$ are: $$\{-7,-3,5,17,-27,-7,17,-15,17,-7,-27,17,5,-3,-7\}.$$.05in Finally, to compute $\bar{d}$, we subtract $-15/30$ (the value for the Spin structure) to each entry, and find that the values of $\bar{d}$ are given by $b_i/30$ for the following values of $b_i$ in order. $$\{8,12,20,32,-12,8,32,0,32,8,-12,32,20,12,8\}.$$ We have listed these values in the chart of Figure \[dt35\], in which we write each value of $s$ as $5a+3b \mod 15$ for $-1\le a \le 1$ and $-2\le b \le 2$. .1in
$$\begin{array}{c|c|c|c|c|c|}
& b= -2 &b= -1 & b=0 & b= 1&b= 2 \\
\hline
a= 1 & 32 &8 & \bf 20 & 8& 32 \\
\hline
a=0 &\bf 12&\bf -12&\bf 0& \bf -12 &\bf 12\\
\hline
a= -1 & 32 & 8& \bf 20& 8& 32\\
\hline
\end{array}$$
.1in
Since $S^3_{15}(T_{3,5})$ is the connected sum of lens spaces, Theorem \[obstructthm\] predicts a pattern in the chart: each element should be the sum of the entries of its projection on the the main axes. This is the case. Notice for instance that the top right entry 32 in position $(a,b) = (1,2) \in \zz_3 \oplus \zz_5$ (which represents $1(5) +2(3) = 11 \in \zz_{15}$), is the sum of the entries in positions $(2,0)$ and $(0,1)$, 12 and 20, respectively.
$\bf \bar{d}(S^3_{15}(T_{3,5} \cs D), {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi})$.
---------------------------------------------------------------------------------------------
In order to compute the $\bar{d}$–invariants that are associated to surgery on the connect sum, we first must compute $CFK^\infty$ for the connected sum of knots. The complex $CFK^\infty(T_{2,3})$ is illustrated in Figure \[hf23figure\], and it follows from [@hedden] that, modulo acyclic subcomplexes, the homology of the double $D(T_{2,3})$ is the same.
[ ![[]{data-label="hf23figure"}](hf23.pdf "fig:") ]{}
At this point we need to analyze the tensor product, $$C = CFK^\infty(T_{3,5}) \otimes_{\ff[U,U^{-1}]} CFK^\infty(T_{2,3}).$$ This complex is fairly complicated, containing 21 generators, but it is easily seen that it contains a subcomplex $C'$ as illustrated in Figure \[hf35doublefigure\]. This subcomplex carries the homology of the overall complex, but does not contain all generators of a given grading. However, it has the following property.
The complex $C_{i<m, j<n}$ contains a generator of grading 0 if and only if $C'_{i<m, j<n}$ contains a generator of grading 0. In particular, $d$–invariants for $C$ can be computed using $C'$.
[ ![[]{data-label="hf35doublefigure"}](hf35double.pdf "fig:") ]{}
Using this diagram to compute the minimal gradings of classes in $$CFK^\infty(T_{3,5} \cs D ) / CFK^\infty(T_{3,5}\cs D)_{i <0, j<s}$$ for $-7 \le s \le 7$ we get the following: $$\{ -14, -12, -10,-10,-8, -6, -6, -4, -4, -2, -2, -2, 0,0,0\}.$$ After shifting gradings by $-\eta$, the values are of the form $a_i/30$, where the $a_i$ are, in order, $$\{ -7, -3, 5, -43, -27, -7, -43, -15, -43, -7, -27, -43, 5, -3, -7\}.$$ To compute $\bar{d}$, we add $15/30$ to each term, yielding the values $b_i/30$, where the $b_i$ are: $$\{ 8,12,20,-28,-12,8,-28,0,-28,8,-12,-28,20,12,8\}.$$ We can arrange these in a chart shown in Figure \[dt35D\].
$$\begin{array}{c|c|c|c|c|c|}
& b= -2 &b= -1 & b=0 & b= 1&b= 2 \\
\hline
a= 1 & \underline{-28} &8 & \bf 20 & 8& \underline{-28} \\
\hline
a=0 &\bf 12&\bf -12&\bf 0& \bf -12 &\bf 12\\
\hline
a= -1 & \underline{-28} & 8& \bf 20& 8& \underline{-28} \\
\hline
\end{array}$$
Notice that the entries on the axes are unchanged, but the underlined entries are no longer the sum of the values of the projections; that is, $-28 \ne 12 + 20$. Thus, according to Theorem \[obstructthm\], this manifold is not $\qq$–homology cobordant to any manifold of the form $M_3 \cs M_5 \cs M_q$.
Second Example
--------------
As a second example we consider the case of $S^3_{35}(T_{5,7})$ and $S^3_{35}(T_{5,7}\cs D)$ and illustrate the analogous charts as above (this time multiplied by 70 to clear denominators). The first chart, Figure \[dt57\] necessarily demonstrates additivity, the second, in Figure \[dt57D\], upon examination does not. This becomes more apparent by considering the third chart, in Figure \[dt57Ddiff\], formed as the difference of the first two, but not multiplied by 70. The underlined entries illustrate the failure of additivity. Considering this difference is a simplifying approach of the general proof in the next section.
$$\begin{array}{c|c|c|c|c|c|c|c|}
& b= -3 &b= -2 & b=-1 & b= 0&b= 1& b= 2&b= 3 \\
\hline
a= 2 & -68 & -108 & -48 &\bf -28 & -48 & -108 & -68 \\
\hline
a=1 &-12 & -52 & 8 & \bf 28 & 8 & -52 & -12 \\
\hline
a= 0 &\bf -40 &\bf -80 &\bf -20 &\bf 0 &\bf -20 &\bf -80 &\bf -40 \\
\hline
a= -1 &-12 & -52 & 8 &\bf 28 & 8 & -52 & -12 \\
\hline
a= -2 & -68 & -108 & -48 &\bf -28 & -48 & -108 & -68\\
\hline
\end{array}$$
$$\begin{array}{c|c|c|c|c|c|c|c|}
& b= -3 &b= -2 & b=-1 & b= 0&b= 1& b= 2&b= 3 \\
\hline
a= 2 &\underline{72} & \underline{32} & 92 &\bf 112 & 92 & \underline{32} & \underline{72} \\
\hline
a=1 &128 & 88 & 8 &\bf 28 & 8 & 88 & 128 \\
\hline
a= 0 &\bf100 &\bf 60 &\bf -20 &\bf 0 &\bf -20 &\bf 60 &\bf 100\\
\hline
a= -1 &128 & 88 & 8 &\bf 28 & 8 & 88 & 128 \\
\hline
a= -2 &\underline{72} & \underline{32} & 92 &\bf 112 & 92 & \underline{32} & \underline{72}\\
\hline
\end{array}$$
$$\begin{array}{c|c|c|c|c|c|c|c|}
& b= -3 &b= -2 & b=-1 & b= 0&b= 1& b= 2&b= 3 \\
\hline
a= 2 &\underline{2} & \underline{2} &2 & 2 & 2 & \underline{2} & \underline{2} \\
\hline
a=1 &2 & 2 &0 & 0 & 0 &2& 2 \\
\hline
a= 0 & 2 & 2 & 0 & 0 & 0 & 2 & 2\\
\hline
a= -1 &2 & 2&0 & 0 &0 & 2 & 2 \\
\hline
a= -2 &\underline{2} &\underline{2}&2& 2 & 2 & \underline{2} & \underline{2}\\
\hline
\end{array}$$
Topologically split examples, general case. {#secexamplestopsplit1}
===========================================
We now wish to generalize the examples of the previous section. To do so, we begin by choosing an infinite set of integers $\{p_i\}$ with the following properties: (1) all $p_i$ are odd; (2) the full set of integers $\{p_i, p_i +2\}$ is pairwise relatively prime; and, (3) each $p_i$ and $p_i +2$ is square free. The existence of such a set is demonstrated in Appendix \[appendpi\], and throughout this section we assume all $p$ are selected from this set. In the previous example we needed to track grading shifts. It will simplify our discussion if we avoid dealing the grading shifts as follows: define $\tilde{d}(S^3_n(K), s) = d(S^3_n(K),s) +\eta$. That is, $\tilde{d}$ is computed as is the $d$–invariant, except without the grading shift, the induced grading on $$CFK^+(S^3_N(K),s)=CFK^\infty(S^3,K)/CFK^\infty(S^3,K)_{\{i<0,j<s\}}$$ Since $p$ is odd, we can write $ p= 2n+1$ and let $q = p+2 = 2n +3$. Our manifolds of interest are $S^3_{pq}(T_{p,q})$ and $S^3_{pq}(T_{p,q} \cs D)$. We collect here the results of a few elementary calculations.
\[pqcalcs\] $ \ $
1. The surgery coefficient is $$pq = 4n^2 + 8n +3.$$ .05in
2. The three-genus satisfies $$g(T_{p,q}) = 2n(n+1) = 2n^2+2n \hskip.2in \text{and} \hskip.2in g(T_{p,q} \cs D) = 2n^2+2n +1.$$.05in
3. [Spin$^c$]{}–structures are parameterized by $s$, with $$-(2n^2 +4n+1) \le s \le (2n^2 +4n+1).$$.05in
4. Generators of $\widehat{CFK}(T_{p,q})$ have filtration level $j$, where $$-2n(n+1) \le j \le 2n(n+1).$$.05in
The main result of this section is the following.
\[pqtorus\] $\bar{d}(S^3_{pq}(T_{p,q} \cs D), s)$ does not satisfy additivity as given in Theorem \[obstructthm\].
The space $S^3_{pq}(T_{p,q})$ satisfies the additive property as in Theorem \[obstructthm\]. Suppose that $S^3_{pq}(T_{p,q} \cs D)$ also satisfies additivity property. Then the difference $\bar{d}(S^3_{pq}(T_{p,q}),(a,b))-\bar{d}(S^3_{pq}(T_{p,q} \cs D),(a,b))$ also satisfies the additivity property. We denote this difference by $\bar{d}'(a,b)$ or $\bar{d}'(aq+bp)$. Note that it is unnecessary to add the grading shift $\eta$ to the amount we get from the diagram when computing either of the values $\bar{d}(S^3_{pq}(T_{p,q}),(a,b))$ or $\bar{d}(S^3_{pq}(T_{p,q} \cs D),(a,b))$ since they have the same grading shift. Namely, $$\begin{aligned}
\bar{d}'(a,b) &= \tilde{d}(S^3_{pq}(T(p,q)),(a,b)) -\tilde{d}(S^3_{pq}(T_{p,q} \cs D),(a,b))\\
&-\tilde{d}(S^3_{pq}(T_{p,q}),0) +\tilde{d}(S^3_{pq}(T_{p,q} \cs D),0).\end{aligned}$$ From our choice of $p$ and $q$, we have $(n+1)p+(-n)q=1$. Thus, the additivity property implies the equality $$\bar{d}'(1)=\bar{d}'((n+1)p)+\bar{d}'(-nq),$$ or equivalently, $$\begin{aligned}
\tilde{d}(S^3_{pq}(T_{p,q}),1) &-\tilde{d}(S^3_{pq}(T_{p,q} \cs D),1) \nonumber \\
& =\tilde{d}(S^3_{pq}(T_{p,q}),(n+1)p) -\tilde{d}(S^3_{pq}(T_{p,q} \cs D),(n+1)p) \label{eqn:tilded}\\
&+\tilde{d}(S^3_{pq}(T_{p,q}),-nq) -\tilde{d}(S^3_{pq}(T_{p,q} \cs D),-nq) \nonumber \\
&-\tilde{d}(S^3_{pq}(T_{p,q}),0) +\tilde{d}(S^3_{pq}(T_{p,q} \cs D),0). \nonumber\end{aligned}$$ Since $(n+1)p=2n^2+3n+1$ lies between the genus of $T(p,q)$ (and of $T_{p,q} \cs D$) and the upper bound on the parameters for the [Spin$^c$]{}–structures: $$2n^2 +2n +1 < 2n^2+3n +1 < 2n^2 +4n +1,$$ the values of the $\tilde{d}$–invariants are easily seen to be 0. On the other hand, the number $-nq$ is greater than the lower bound on the parameters for the [Spin$^c$]{}–structures and less than the negative of the genus: $$-(2n^2 + 4n +1) < -(2n^2 + 3n) < -(2n^2 + 2n + 1 )$$ and thus one sees that the $\tilde{d}$–invariants take the same value $-2s = 2(2n^2+3n)$ for both $ T_{p,q}$ and $T_{p,q} \cs D$.
Thus, in contradicting additivity, it remains to show that the equality $$\tilde{d}(S^3_{pq}(T_{p,q}),1) -\tilde{d}(S^3_{pq}(T_{p,q} \cs D),1) =
-\tilde{d}(S^3_{pq}(T_{p,q}),0) +\tilde{d}(S^3_{pq}(T_{p,q} \cs D),0)$$ does not hold.
Now we will compute $\tilde{d}$ of both spaces for [Spin$^c$]{}–structures $0$ and $1$. Observe that within width 1 from the diagonal $j=i$, the complex $CFK^\infty(S^3,T_{p,q})$ looks like $CFK^\infty(S^3,T_{2,3})$ if $n$ is odd, or $CFK^\infty(S^3,T_{2,5})$ if $n$ is even. This depends on the fact that near the origin the complex $CFK^\infty(S^3,K) $ looks like that of the $(2,k)$–torus knots. In Appendix \[torusknotpoly\] we prove that the Alexander polynomial of $T_{p,p+2}$ is of the form 1+ $\sum_{i>0} a_i(t^{-i} + t^{i})$ where $a_i = \pm 1$ for $i\le (p-1)/2$. As in the example of the previous section, this determines the “zig-zag” feature of the $CFK^\infty$ complex near the origin. Tensoring with the trefoil complex does not alter this pattern.
The generators of the same grading $2l$ of $[x,-1,0]$ if $n$ is odd (or, $[x,0,0]$ if $n$ is even) lies above the anti-diagonal $i+j=-1$ (or, $i+j=0$). So, in order to compute $\tilde{d}(S^3_{pq}(T(p,q)),s)$ for $s=0,1$, we may assume in the computations that the complex we are considering is one of $$\begin{cases}
CFK^\infty(S^3,T_{2,3}) & \text{if $n$ is odd}, \\
CFK^\infty(S^3,T_{2,5}) & \text{if $n$ is even}.
\end{cases}$$ It is now easy to compute $$\tilde{d}(S^3_{pq}(T_{p,q}),s)=
\begin{array}{|c|c|c|} \hline
s & n \text{ odd} & n \text{ even} \\ \hline
1 & 2l+2 & 2l \\ \hline
0 & 2l & 2l \\ \hline
\end{array}.$$ Near the diagonal $j=i$, the complex $CFK^\infty(S^3,T_{p,q} \cs D)$ looks like: $$\begin{cases}
CFK^\infty(S^3,T_{2,5}) & \text{if $n$ is odd}, \\
CFK^\infty(S^3,T_{2,3})[-2] & \text{if $n$ is even}.
\end{cases}$$ The grading of $[x,-1,0]$ is $2l-2$ if $n$ is even and the grading of $[x,0,0]$ is $2l$ if $n$ is odd. Thus, we have $$\tilde{d}(S^3_{pq}(T_{p,q} \cs D),s)=
\begin{array}{|c|c|c|} \hline
s & n \text{ odd} & n \text{ even} \\ \hline
1 & 2l & 2l \\ \hline
0 & 2l & 2l-2 \\ \hline
\end{array}.$$ We see that $$\tilde{d}(S^3_{pq}(T_{p,q}),s)-\tilde{d}(S^3_{pq}(T_{p,q} \cs D),s)=
\begin{array}{|c|c|c|} \hline
s & n \text{ odd} & n \text{ even} \\ \hline
1 & 2 & 0 \\ \hline
0 & 0 & 2 \\ \hline
\end{array}.$$ This shows that (\[eqn:tilded\]) cannot be satisfied. We conclude that the space $S^3_{pq}(T_{p,q} \cs D)$ does not satisfy the additive property of Theorem \[obstructthm\].
The image of $\calk$ in $\Theta^3_\qq / \Phi(\oplus_{p \in \calp} \Theta^3_{\zz[1/p]})$ is infinite.
----------------------------------------------------------------------------------------------------
This follows from the following result.
The spaces $N_{p,q} = S^3_{pq}(T_{p,q} \cs D) \cs - S^3_{pq}(T_{p,q} ) \in \calk$ are distinct in the quotient $\Theta^3_\qq / \Phi(\oplus_{p \in \calp} \Theta^3_{\zz[1/p]})$.
Observe that $S^3_{pq}(T_{p,q} \cs D) \cs - S^3_{pq}(T_{p,q} ) \in \calk$, since the knots are topologically concordant. We next observe that these manifolds have the property that no linear combination with all coefficients $\pm 1$ is trivial in the quotient. Suppose that some such linear combination was trivial. Then focusing on any particular pair $(p,q)$, we would have that $S^3_{pq}(T_{p,q} \cs D) \# M_p \#M_q \# M_m = \partial X$ for a rational homology ball $X$, where the order of $M_p$ is a product of prime factors of $p$, the order of $M_q$ is a product of prime factors of $q$, and the order of $M_m$ is relatively prime to $pq$. (This uses the fact that $S^3_{pq}(T_{p,q} ) $ does split as a connected sum.)
The existence of this connect sum decomposition implies the additivity for $d$–invariants of $S^3_{pq}(T_{p,q} \cs D)$ in a way that contradicts Theorem \[pqtorus\].
Knot concordance {#concordance}
================
We denote by $\calc$ the classical smooth knot concordance group. Levine [@levine] defined the algebraic concordance group $\calg$ and the rational algebraic concordance group, $\calg^\qq$. He also defined a surjective homomorphism $\calc \to \calg$, proved that natural map $\calg \to \calg^{\qq}$ is injective, and proved that $\calg^{\qq}$ is isomorphic to an infinite direct sum of groups isomorphic to $\zz, \zz_2$ and $\zz_4$. He also proved that the image of $\calg$ in $\calg^\qq$ is isomorphic to a similar infinite direct sum. In [@levine] it is observed that $\calg^\qq$ has a natural decomposition as a direct sum $\oplus \calg^\qq_{p(t)}$, where the $p(t)$ are symmetric irreducible rational polynomials. We will not present the details here, but note that if the Alexander polynomial of $K$, $\Delta_K(t)$, is irreducible, then the image of $K$ in $\calg^\qq$ is in the $\calg^\qq_{\Delta(t)}$ summand. Stoltzfus [@stoltzfus] observed that the algebraic concordance group $\calg$ does not have a similar splitting. Thus, there is not an immediate analog in concordance for the decompositions we have been studying for homology cobordism. However, he did prove that in some cases such a splitting exists. The following, Corollary 6.5 from [@stoltzfus], is stated in terms of knot concordance, but given the isomorphism of higher dimensional concordance and $\calg^\zz$, the same splitting theorem holds in the algebraic concordance group.
If $\Delta_K(t)$ factors as $p(t)q(t)$ with $p(t)$ and $q(t)$ symmetric and the resultant *Res*$(p(t), q(t)) = 1$, then $K$ is concordant to a connected sum $K_1 \cs K_2$, with $\Delta_{K_1}(t) = p(t)$ and $\Delta_{K_2}(t) = q(t)$.
Here we observe that this result does not hold in dimension 3..05in [**Example.**]{} Consider the ten crossing knot $K = 10_{5}$. It has Alexander polynomial $$\Delta = (1 - t + t^2) (1 - 2 t + 2 t^2 - t^3 + 2 t^4 - 2 t^5 + t^6).$$ These two factors are irreducible and have resultant 1.
The knot $10_5$ is not concordant to any connected sum $K_1 \cs K_2$ where $\Delta_{K_1} = 1 - t + t^2 $ and $\Delta_{K_2} = 1 - 2 t + 2 t^2 - t^3 + 2 t^4 - 2 t^5 + t^6$.
The 2-fold branched cover of $K$ is the lens space $L(33,13)$. If the desired concordance existed, then $L(33,13)$ would split in rational cobordism as a connected sum $M_3 \cs M_{11}$, with $H_1(M_3) = \zz_{3}$ and $H_1(M_{11}) = \zz_{11}$. In order to compute the relevant $d$–invariants, one first identifies ${\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_6$ as the Spin–structure ${\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_*$ by computing that the value of $d(L(33,13), {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_6)= 33$, a value that is not attained by any other [Spin$^c$]{}–structure. The values of the $d$–invariants, $d(L(33,13), (a,b)\cdot {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_*) - d(L(33,13), {\ifmmode{{\mathfrak s}}\else{${\mathfrak s}$\ }\fi}_*)$ for $(a,b) \in \zz_3 \oplus \zz_{11}$ are given in the chart in Figure \[dl33\] (multiplied by 33 to clear denominators).
$$\begin{array}{c|c|c|c|c|c|c|c|}
& b= 0 &b= 1 & b=2 & b= 3&b= 4& b= 5 \\
\hline
a=2 &\bf 22 &10 &40 & -20 & 28 & -14 \\
\hline
a= 0 &\bf 0 &\bf 54 &\bf 18 &\bf 24 &\bf 6 &\bf 30 \\
\hline
a= -1 & \bf 22 & 10 & 40 & 46 & 28 & 52 \\
\hline
\end{array}$$
The next chart, in Figure \[dl33diff\], presents the values $$\begin{aligned}
\delta(L(33,13),(a,b)) &= d(L(33,13),(a,b)) - d(L(33,13),(a,0)) \\
&- d(L(33,13),(0,b)) + d(L(33,13),(0,0)).\end{aligned}$$
$$\begin{array}{c|c|c|c|c|c|c|c|}
& b= 0 &b= 1 & b=2 & b= 3&b= 4& b= 5 \\
\hline
a=2 & \bf 0 & 2 & 0 & 2 & 0 & 2 \\
\hline
a= 0 &\bf 0 &\bf 0 &\bf 0 &\bf 0 &\bf 0 &\bf 0 \\
\hline
a= -1 &\bf 0 & 2 & 0 &0 & 0 & 0 \\
\hline
\end{array}$$
The presence of the nonzero entries implies the nonsplittability of the manifold, as desired.
[**Note.**]{} In unpublished work [@livingst] the second author constructed similar but much more complicated examples in the topological category.
Topologically trivial bordism {#toptrivialbordism}
=============================
In [@hlr] the quotient $\Theta^T_{\qq, spin} / \Theta^I_{\qq, spin}$ was studied. Here, the cobordism group has been restricted to spin 3–manifolds and spin bordisms which have the rational homology of $S^3$. The notation $\Theta^T_{\qq, spin}$ denotes the subgroup generated by representatives which bound topological homology balls and $ \Theta^I_{\qq, spin}$ is generated by those that are cobordant to $\zz$–homology spheres. (Note we have changed the notation from that of [@hlr] to be consistent with the results of the current paper. There is a similar result in [@hlr] replacing $(\qq, spin)$, with $\zz_2$. (Recall that every $\zz_2$ homology sphere is spin.)
Here we observe that Theorem \[lemmaextendspin\] permits us to generalize this result, eliminating the need to constrain the cobordism group to being spin or to use $\zz_2$ coefficients. Let $\Theta^T_\qq$ denote the subgroup of $\Theta^3_\qq$ generated by rational homology spheres that are trivial in the topological rational cobordism group, that is, the kernel of $\calk$.
The quotient group $\Theta^T_\qq / \Theta^3_\zz$ is infinitely generated.
We outline how the argument in [@hlr] can be generalized.
In [@hlr] there is a family of rational homology spheres constructed, $M_{p^2}$, for an infinite set of primes $p$. These are constructed so that they bound topological balls. The proof of the theorem consists of showing that no linear combination $N = \cs_i a_iM_{p_i^2} \cs M_0$ bounds a spin rational homology ball (or $\zz_2$ homology ball) $W$, where $M_0$ is a $\zz$–homology sphere. The existence of a unique Spin–structure was used to identify [Spin$^c$]{} of the relevant manifolds with the second homology.
If all $p$ are odd, then there is a unique [Spin$^c$]{}–structure on $N$ and according to Theorem \[lemmaextendspin\], it is the restriction of a [Spin$^c$]{}–structure on $W$. Given this, Proposition 2.1 of [@hlr], which required that $W$ be spin, continues to apply to identify the [Spin$^c$]{}–structures on $N$ which extend to $W$ with a metabolizer of the linking form on $H_1(N)$. That identification is what is used to obstruct the existence of $W$ via $d$–invariants, as described in Thoerem 3.2 of [@hlr]. Thus, the remainder of the proof goes through as in that paper.
Finding the $p_i$ {#appendpi}
==================
The proof of Theorem \[pqtorus\] requires a sequence of odd pairs $\{ p_i, p_i +2\}$ so that the elements of the full set of $\{p_i\} \cup \{p_i +2\}$ are pairwise relatively prime and square free. Since $p_i $ and $p_i +2$ are relatively prime, we need to choose the $p_i$ so that the set of all elements of $\{ p_i(p_i +2)\}$ are pairwise relatively prime and each element is square free. If we let $p_i = n_i -1$, then $p_i(p_i+2) = n_i^2 -1$, and so we are seeking an infinite sequence of positive integers $\{n_i\}$ such that:
1. $n_i$ is even for all $i$..05in
2. All elements of $\{n_i^2 -1\}$ are relatively prime..05in
3. Each $n_i^2 -1 $ is square free.
In Section \[inftwotor\] we need a sequence of integers $n_i$ such that $n_i = 0 \mod5$ with the property that the integers $20n_i^2 +8n_i +1$ are relatively prime and square free. Here is a theorem that covers both cases.
Let $f(x)\in \zz[t]$ be an quadratic polynomial with constant term 1 that is not the square of a linear polynomial. Let $\alpha$ be a fixed integer and $s_n= \alpha n$ be an arithmetic sequence. There exists an infinite set of $s_i$ such that values of $f(s_i)$ are pairwise relatively prime and square free.
It is known that if $g(n)$ is a quadratic polynomial that is not a square of a linear polynomial and which has the property that its coefficients have greatest common divisor one, then $g(n)$ is square free for an infinite set of $n$ (see, for example, [@erdos]). We wish to construct the sequence of $s_i$ inductively. To find $s_1$, let $f_1(n) = f(\alpha n)$, which is irreducible with constant term one. Choose $n_1$ so that $f_1(n_1)$ is square free. Let $s_1 = \alpha n_1$. Assume that $s_i$ has been defined for $i < k$. We find $s_{k} $ with the desired properties as follows. Let $P = \prod_{i=1}^{k-1} f(s_i)$. Consider the function $f_k(n) = f(\alpha P n)$. Again, this polynomial is irreducible with constant term one, so there exists an $n_k$ for which $f_k(n_k)$ is square free. Since $f_k(n_k) = f(\alpha P n_k)$, we let $s_k = \alpha P n_k$. Notice that for each prime divisor $p$ of $P$, $f(\alpha P n) = 1 \mod p$, since evaluating $f$ at $\alpha P n$ gives a quadratic polynomial in $n$, with the quadratic term and linear term divisible by $P$ and the constant term one. It follows that $f(s_k)$ is relatively prime to all $f(s_i), i < k$.
The Alexander polynomial of $T_{p,p+2}$. {#torusknotpoly}
========================================
Normalized to be symmetric, the Alexander polynomial of a knot can be written in the form $\Delta_K(t) = a_0 +\sum_{i=1}^{n} a_i(t^{-i} + t^i)$, where $a_0 + 2\sum a_i = \pm 1$. In Section \[secexamplestopsplit1\] we use the following fact.
If $K = T_{p,p+2}$ with $p$ odd then $$\Delta_{T_{p,p+2}}(t) = a_0 +\sum_{i=1}^{(p^2-1)/2} a_i(t^{-i} + t^i),$$ where $a_i = \pm 1$ for $i \le (p-1)/2.$
[**Note.**]{} With more care, all the coefficients or $ \Delta_{T_{p,p+2}}(t)$ can be described in closed form.
As a polynomial (as opposed to the normalized Laurent polynomial) with nonzero constant term, the Alexander polynomial of $T_{p,q}$ is $(1-t^{pq})(1-t)/(1-t^p)(1-t^q)$. Expanding each term of the denominator in a power series and noting that multiplying by the $t^{pq}$ term in the numerators does not affect terms of the product of degree less than $2g = (p-1)(q-1)$, the degree of the Alexander polynomial, we can focus on the expression: $$(1-t) ( 1 + t^p + t^{2p} + t^{3p}\cdots)(1+t^q + t^{2q} +\cdots),$$ which we write as the product $$(1-t)\sum_{i=0}^{\infty} b_i t^i.$$ Here $b_i $ is the number of solutions to $xp + y q = i$, with $x, y \ge 0$. In the case of interest, $q = p+2$ and the genus $g = (p^2-1)/2$. We will now show that for $i$ in the range $g- A \le i \le g$, the values $b_i$ are alternately 0 and 1, where $A$ is a constant to be determined. Thus, using the fact that the Alexander polynomial is symmetric, upon multiplying by $(1-t)$ we have the coefficients of the Alexander polynomial are all $\pm 1$ near $t^g$. To show that the coefficients $b_i$ alternate between 0 and 1 for $g - A \le i \le g$, we first observe that in a given range of $i$, all $b_i \ge 1$ for $i$ even. To see this, write $p= 2n+1$ and $q= 2n+3$; thus $g = 2n^2 + 2n$. Consider the sum $$\frac{n+j}{2}p +\frac{n-j}{2}q = 2n^2 +2n -j,$$ where $j$ is selected to have the same parity as $n$. (We require here that $j \le n$, that is, we need $A \le \frac{p-1}{2}$.) To complete the argument, we next observe that the difference $|b_{i} - b_{j}| \le 1$ if $|i-j| \le 1$. Suppose otherwise. That is, suppose that there are [*distinct*]{} nonnegative solutions to equations: $$x p + y q = i$$ and $$x' p +y'q = j$$ with $x, y, x' , y' \ge 0$, $|i-j|\le 1$, and $i, j \le g$. The conditions that $i\le g$ and $y \ge 0$ imply that $xp \le g = (pq - p -q -1)/2$, which imply that $x < (q-1)/2$. We first consider the case that $i \ne j$. After possibly rordering, the difference would give $$(x- x') p + (y-y')q =1.$$ One solution to this equation is $$\frac{q-1}{2} p -\frac{p-1}{2}q = 1.$$ Every other solution is given by adding a multiple of $(-q,p)$ to the coefficient vector (note that $-q(p) + p(q) = 0$ is a primitive solution since $p$ and $q$ are relatively prime). Thus, the solutions with the smallest absolute values of the $x$–coordinate to the unital equation are the one above and $$-\frac{q+1}{2} p +\frac{p+1}{2} q =1.$$ That is, the smallest possible value for $(x-x')$ is $x-x' = \frac{q-1}{2}$. But, since $x$ and $x'$ both are nonnegative and less than $\frac{q-1}{2}$, this is impossible. As an example, if $p = 21$ and $q=23$, (so $g = 220$) we have the solutions $$11(21) - 10(23)=1$$ and $$-12(21) + 11(23) =1.$$ with $g = 220$. We also have $x(21) + y(23) \le 220$ which imply that $x \le 220 /21$, so $0 \le x \le 10$. Similarly for $x'$, so it is not possible for $|x - x'| = 11$. Finally, we consider the case $i = j$. Thus, our coefficients would satisfy $$(x-x')p + (y-y')q =0.$$ This implies that $x - x' $ is a multiple of $q$. But this would imply that they are equal, since under our assumptions, both are nonnegative and also $xp \le pq - p -q +1 \le pq$, so $x < q$ and $x' <q$. .1in In summary, if we write the Alexander polynomial of the $T_{p,q}$ torus knot, with $q-p=2$ as $\pm 1$ as $a_0 + \sum_{i=1}^g a_i(t^i +t^{-i})$, then for $i \le \frac{p-1}{2}$, we have shown that $a_i = (-1)^i$.
.1in
[BBVB]{}
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[^1]: This work was supported in part by the National Science Foundation under Grant 1007196 and by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MEST) No. 2011–0012893.\
| {
"pile_set_name": "ArXiv"
} |
---
bibliography:
- 'main.bib'
- 'LHCb-PAPER.bib'
- 'LHCb-CONF.bib'
- 'LHCb-DP.bib'
- 'LHCb-TDR.bib'
- 'local.bib'
---
=1
| {
"pile_set_name": "ArXiv"
} |
---
address: 'Department of Physics, Kent State University, Kent, OH 44242, USA'
author:
- Pieter Maris
title: Calculating the critical exponents of the chiral phase transition
---
Introduction
============
It is anticipated that the restoration of chiral symmetry, which accompanies the formation of a quark-gluon plasma at nonzero temperature $T$, is a second-order phase transition in QCD with 2 light flavours. Such transitions are characterised by two critical exponents: $(\beta,\delta)$, which describe the response of the chiral order parameters, ${\cal X}$, to changes in $T$ and in the current-quark mass, $m$. Denoting the critical temperature by $T_c$, and introducing the reduced-temperature $t:= T/T_c-1$ and reduced mass $h:=m/T$, then $$\begin{aligned}
\label{aa} {\cal X} \propto (-t)^\beta\,,\;\;\; && t\to 0^-\,,\;h=0\,,\\
\label{ab} {\cal X} \propto h^{1/\delta}\,,\;\;\;&& h\to 0^+\,,\;t=0\,.\end{aligned}$$ Calculating the critical exponents is an important goal because of the notion of [*universality*]{}, which states that their values depend only on the symmetries and dimensions, but not on the microscopic details of the theory.
The success of the nonlinear $\sigma$-model in describing long-wavelength pion dynamics underlies a conjecture[@pisarski] that chiral symmetry restoration at finite $T$ in 2-flavour QCD is in the same universality class as the 3-dimensional, $N=4$ Heisenberg magnet ($O(4)$ model), with critical exponents:[@ofour] $\beta^H=0.38 \pm
.01$, $\delta^H=4.82 \pm .05$. However, recently it was argued that the compositeness of QCD’s mesons affects the nature of the phase transition and the Gross–Neveu model was presented as a counterexample to universality.[@kogut] Subsequent studies[@stephanov] indicated that nontrivial $1/N$ corrections are important and that this model has the same critical exponents as the Ising model, as was argued on the notion of universality, but only in a scaling region of width $1/N$.
Calculating the exponents $\beta$ and $\delta$ directly[@prl] from Eqs. (\[aa\]) and (\[ab\]) is often difficult because of numerical noise near the critical temperature. Another method is to consider[@arnea] the chiral and thermal susceptibilities: $$\label{defchih}
\chi_h(t,h) :=
\left.\frac{\partial\, {\cal X}(t,h)}
{\!\!\!\!\!\!\partial h}\right|_{t}\,,\;\;\;
\chi_t(t,h) :=
\left.\frac{\partial\, {\cal X}(t,h)}
{\!\!\!\!\!\!\partial t}\right|_{h}\,.$$ At each $h$, $\chi_i(t,h)$, $i=h,t$, are smooth functions of $t$ with maxima $\chi_i^{\rm pc}$ at the pseudocritical points $t_{\rm pc}^i$. Near the critical point $t = 0 = h$ we have $$\begin{aligned}
\label{pchpct}
&& t_{\rm pc}^h \propto\, h^{1/(\beta \delta)}\,
\propto\, t_{\rm pc}^t\,,\\
\label{deltaslope}
&& \chi_h^{\rm pc}\; = \; \chi_h(t_{\rm pc}^h,h) \;\propto\;
h^{-z_h}\,,\;\;\;
z_h := 1 - {\mbox{\footnotesize $\displaystyle \frac{1}{\delta}$}} \,,\\
\label{betaslope}
&& \chi_t^{\rm pc} \; = \; \chi_t(t_{\rm pc}^t,h) \; \propto \;
h^{-z_t}\,,\;\;\;
z_t:= {\mbox{\footnotesize $\displaystyle \frac{1}{\beta\delta}$}}\,(1-\beta)\,.\end{aligned}$$ Therefore, by calculating the chiral and thermal susceptibilities and locating the pseudocritical points, one can determine $T_c$ and the critical exponents.[@arnea; @hmr98]
Quark Dyson–Schwinger Equation
==============================
We have analysed[@hmr98] $\chi_h(t,h)$ and $\chi_t(t,h)$ in a class of confining DSE models that underlies many successful phenomenological applications[@pct] at both zero and finite-$(T,\mu)$.[@rs99] The foundation of our study is the renormalised quark DSE $$\begin{aligned}
S^{-1}(\vec{p},\omega_k) & :=&
i\vec{\gamma}\cdot \vec{p} \,A(p^2,\omega_k)
+ i\gamma_4\,\omega_k \,C(p^2,\omega_k) + B(p^2,\omega_k) \\
&= &Z_2^A \,i\vec{\gamma}\cdot \vec{p}
+ Z_2^C \, i\gamma_4\,\omega_k
+ Z_4 \,m_R(\zeta) + \Sigma^\prime(\vec{p},\omega_k)\,.
\label{qDSE} \end{aligned}$$ Here $\omega_k= (2 k + 1)\,\pi T$ is the fermion Matsubara frequency and $m_R(\zeta)$ is the current quark mass at the renormalisation point $\zeta$. The self-energy is $$\begin{aligned}
\Sigma^\prime(\vec{p},\omega_k) & =&
T\sum_{l=-\infty}^\infty \!\int\! \frac{d^3q}{(2\pi)^3}\,
{\mbox{\footnotesize $\displaystyle \frac{4}{3}$}}\,g^2\,D_{\mu\nu}(\vec{p}-\vec{q},\Omega_{k-l})\,
\gamma_\mu S(\vec{q},\omega_l)\Gamma_\nu \,,
\label{regself}\end{aligned}$$ with $D_{\mu\nu}(\vec{k},\Omega_j)$ the renormalised dressed-gluon propagator and $\Gamma_\nu$ the renormalised dressed-quark-gluon vertex. In renormalising the DSE we require $$\label{subren}
\left.S^{-1}(\vec{p},\omega_0)\right|_{p^2+\omega_0^2=\zeta^2} =
i\vec{\gamma}\cdot \vec{p} + i\gamma_4\,\omega_0 + m_R\;.$$ Equations (\[qDSE\])-(\[subren\]) define the exact QCD [*gap equation*]{}.
We use the rainbow trunctation for the vertex, $\Gamma_\nu =
\gamma_\nu$, which is the leading term in a $1/N_c$-expansion of the vertex, and consider three models in which the long-range part of the interaction is an integrable infrared singularity,[@mn83] motivated by $T=0$ studies of the gluon DSE:[@pennington] $$\begin{aligned}
g^2 D_{\mu\nu}(\vec{k},\Omega_j) &=&
P_{\mu\nu}^L(\vec{k},\Omega_j) {\cal D}(\vec{k},\Omega_j;m_g) +
P_{\mu\nu}^T(\vec{k},\Omega_j) {\cal D}(\vec{k},\Omega_j;0) \,,
\nonumber\\
\label{delta}
{\cal D}(\vec{k},\Omega_j;m_g) &:=&
2\pi^2 D\,{\mbox{\footnotesize $\displaystyle \frac{2\pi}{T}$}}\delta_{0\,j} \,\delta^3(\vec{k})
+ {\cal D}_{\rm M}(k^2+\Omega_j^2+m^2_g)\,,\end{aligned}$$ where $P_{44}^T=P_{4i}^T=0$, $P_{ij}^T=\delta_{ij}-k_i k_j/k^2$, $P_{\mu\nu}^L=\delta_{\mu\nu}-k_\mu k_\nu/(k^2+\Omega^2)-P_{\mu\nu}^T$, $m_g$ is a Debye mass, and $D$ is a mass-parameter fitted to $m_\pi$ and $f_\pi$ at $T=0$. We compare the results for 3 different models, denoted by ${\cal D}_{\rm M}$, ${\rm M} = {\rm A, B, C}$.
One order parameter for dynamical chiral symmetry breaking is the quark condensate[@mr97] $\langle \bar q q\rangle_\zeta^0$. There are other, equivalent order parameters and in calculating the chiral and thermal susceptibilities we employ $${\cal X}:= B(p^2=0,\omega_0), \; \; \;
{\cal X}_C:= \frac{B(p^2=0,\omega_0)}{C(p^2=0,\omega_0)}.$$ They should be equivalent and, as we will see, the onset of that equivalence is a good way to determine the $h$-domain on which Eqs. (\[pchpct\])-(\[betaslope\]) are valid. Further, we have verified numerically that in the chiral limit ($m=0$) and for $t \sim
0$: $f_\pi \propto \langle\bar q q\rangle \propto {\cal X}(t,0)$; i.e., that these quantities are all equivalent, [*bona fide*]{} order parameters. It thus follows from the pseudoscalar mass formula:[@mr97] $f_\pi^2\,m_\pi^2 = 2\,m_R(\zeta)\langle\bar q q
\rangle_\zeta^0$, that $m_\pi$ increases with temperature.[@prl]
Results
=======
The first model we consider is an infrared dominant model with ${\cal
D}_{\rm A}(s) \equiv 0$, and the mass-scale $D=0.56\,$GeV$^2$ fixed[@mn83] by fitting $\pi$- and $\rho$-meson masses at $T=0$. A current-quark mass of $m=12\,$MeV yields . The quark DSE obtained with ${\cal D}_A$ is an algebraic equation. Chiral symmetry restoration is therefore easy to analyse and either directly, via Eqs. (\[aa\]) and (\[ab\]), or using the susceptibilities and Eqs. (\[pchpct\])-(\[betaslope\]), it is straightforward to establish[@arnea] that this model has mean field critical exponents and to determine the critical temperature in Table \[taba\]. The exponents are unchanged[@hmr98] and $T_c$ reduced by $<\,$2% upon the inclusion of some higher-order $1/N_c$-corrections to the dressed-quark-gluon vertex.[@brs96]
Model B: QED-like tail
----------------------
To improve the ultraviolet behaviour, we consider a model[@prl] with $$\label{modelfr}
{\cal D}_{\rm B}(s) = {\mbox{\footnotesize $\displaystyle \frac{16}{9}$}}\pi^2\,
\frac{1-{\rm e}^{- s /(4m_t^2)}}{s}\,,$$ and $D=(8/9) \, m_t^2 $. The mass-scale $m_t=0.69\,{\rm
GeV}=1/0.29\,{\rm fm}$ marks the boundary between the perturbative and nonperturbative domains, and was fixed[@fr] by requiring a good description of $\pi$- and $\rho$-meson properties at $T=0$.
The quark DSE obtained with this model can be solved numerically and $\chi_h^{\rm pc}(h)$ and $\chi_t^{\rm pc}(h)$ are depicted in Fig. \[frchi\](a).
Following Eqs. (\[deltaslope\]) and (\[betaslope\]), the critical exponents can be determined by defining a local critical exponent as a function of the reduced mass $h$ for each of the equivalent order parameters: $$\label{zi}
z_i(h):= \,-\,h\,\frac{ \partial \ln \chi^{\rm pc}_i}{\partial h}\,,$$ see Fig. \[frchi\](b). $h$ lies in the scaling region when $z_i$ is independent of the order parameter. This shows that the scaling relations are not valid until $$\label{mfr}
\log_{10} (h/h_u)< -7\,,$$ where $h_u = m_R/T_c$ corresponds to the current-quark mass that gives in this model. The values of $z_h$ and $z_t$ in Table \[taba\] are obtained by a Padé fit to the five smallest
mean field A B C
------------- ------------ ------- ----------------- -------------------- --
$T_c$ (MeV) 169 174 120
$z_h$ 2/3 0.666 0.67 $\pm$ 0.01 0.667 $\pm $ 0.001
$z_t$ 1/3 0.335 0.33 $\pm$ 0.02 0.333 $\pm $ 0.001
: Critical temperature for chiral symmetry restoration and critical exponents characterising the second-order transition in the three exemplary models.\[taba\]
$h$-values in Fig. \[frchi\](b), and extrapolating to $h \rightarrow
0^+$. The critical temperature is obtained using Eq. (\[pchpct\]); its value is insensitive to whether $t_{\rm pc}^h$ or $t_{\rm pc}^t$ is used and to which of the equivalent order parameters is used.
Model C: Logarithmic tail
-------------------------
Finally, we consider the finite-$T$ extension[@hmr98] of a model which further improves the ultraviolet behaviour, via the inclusion of the one-loop $\ln$-suppression at $s \gg \Lambda_{\rm QCD}^2$. Again, the parameters are fixed at $T=0$ by requiring a good fit to a range of $\pi$-, $K$-meson properties;[@mr97] recent calculations show that the vector mesons are also described well in this model.[@mt99] The study of chiral symmetry restoration in this model is very similar to the previous study, with the additional $\ln[s]$-suppression in the ultraviolet making the numerical analysis easier. The critical temperature and exponents are presented in Table \[taba\]. Also in this case the scaling relations are only valid for very small current-quark masses: $\log_{10} (h/h_u) < -5$. These results are qualitatively, and for the critical exponents quantitatively, independent of the parameters in this model. Direct calculation of the critical exponents using Eqs. (\[aa\]) and (\[ab\]) are in good agreement with the critical exponents found using the susceptibilities.
Conclusions
===========
It is clear from Table \[taba\] that each of these models is mean field in nature. In hindsight that may be not surprising because the long-range part of the interaction is identical. However, the models differ by the manner in which the interactions approach their long-range limits, and our numerical demonstration of their equivalence required extremely small values of the current-quark mass, Eq. (\[mfr\]). This might also be true in QCD; i.e., while $T_c$ is relatively easy to determine, very small current-quark masses may be necessary to accurately calculate the critical exponents from the susceptibilities. In that case, calculation of $\beta$ and $\delta$ via lattice-QCD will not be easy. The discrepancies found in recent lattice calculations[@el98] could be a signal of this difficulty.
The class of models we have considered can describe the long-wavelength dynamics of QCD very well[@pct; @rs99] in terms of mesons that are quark-antiquark [*composites*]{}. The characteristic feature is the behaviour of the confining interaction. It provides a driving term in the quark DSE proportional to the dressed-quark propagator, which means that boson Matsubara zero-modes do not influence the critical behaviour determined from the gap equation. The class of Coulomb gauge models[@reinhard] also describes mesons as composite particles and it too exhibits mean field critical exponents. The long-range part of the interaction in that class of models corresponds to the regularised Fourier amplitude of a linearly rising potential. Hence it is not equivalent to ours in any simple way, except insofar as zero modes do not influence the gap equation.
The quark DSE is the QCD gap equation and the many equivalent chiral order parameters are directly related to properties of its solution. We have observed that several classes of models exhibit the same (mean field) critical exponents. Only in our simplest confining model did we consider the effect of $1/N_c$-corrections to the quark-gluon vertex, and in that case the critical exponents were unchanged. These results suggest that mean field exponents are a feature of the essential fermion substructure in the gap equation. It can likely only be false if nonperturbative corrections to the vertex are large in the vicinity of the transition. In this context the role of mesonic bound states, which can appear as nonperturbative contributions in the dressed-quark-gluon vertex, has to be studied in more detail. This might also give a nontrivial dependence on the number of fermion flavours, as is anticipated on universality arguments.
Acknowledgments {#acknowledgments .unnumbered}
===============
I would like to thank ECT\* and the organizers for providing such a stimulating environment for this workshop. This work was done in collaboration with A. Höll and C.D. Roberts and funded in part by the National Science Foundation under grants no. INT-9603385 and PHY97-22429, and benefited from the resources of the National Energy Research Scientific Computing Center.
[99]{} R. Pisarski and F. Wilczek, Phys. Rev. D[**29**]{}, 338 (1984). G. Baker, B. Nickel and D. Meiron, Phys. Rev. B[**17**]{}, 1365 (1978). A. Kocić and J. Kogut, Phys. Rev. Lett. [**74**]{}, 3109 (1995). J. Kogut, M. Stephanov and C. Strouthos, Phys. Rev. D[**58**]{}, 96001 (1998). A. Bender [*et al.*]{}, Phys. Rev. Lett. [**77**]{}, 3724 (1996). D. Blaschke [*et al.*]{}, Phys. Rev. C[**58**]{}, 1758 (1998). A. Höll, P. Maris and C.D. Roberts, Phys. Rev. C[**59**]{}, 1751 (1999). P.C. Tandy, Prog. Part. Nucl. Phys. [**39**]{}, 117 (1997). C.D. Roberts and S. Schmidt, nucl-th/9903075, these proceedings. H.J. Munczek and A.M. Nemirovsky, Phys. Rev. D[**28**]{}, 3081 (1983). N. Brown and M.R. Pennington, Phys. Rev. D[**39**]{}, 2723 (1989). P. Maris and C.D. Roberts, Phys. Rev. C[**56**]{}, 3369 (1997). A. Bender, C.D. Roberts and L. v. Smekal, Phys. Lett. B [**380**]{}, 7 (1996). M.R. Frank and C.D. Roberts, Phys. Rev. C[**53**]{}, 390 (1996). P. Maris and P.C. Tandy, nucl-th/9905056. E. Laermann, Nucl. Phys. B [**63**]{} (Proc. Suppl.), 114 (1998). R. Alkofer, P.A. Amundsen and K. Langfeld, Z. Phys. C [**42**]{}, 199 (1989).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We present a multi-contact walking pattern generator based on preview-control of the 3D acceleration of the center of mass (COM). A key point in the design of our algorithm is the calculation of contact-stability constraints. Thanks to a mathematical observation on the algebraic nature of the frictional wrench cone, we show that the 3D volume of feasible COM accelerations is always an upward-pointing cone. We reduce its computation to a convex hull of (dual) 2D points, for which optimal $\calO(n \log n)$ algorithms are readily available. This reformulation brings a significant speedup compared to previous methods, which allows us to compute time-varying contact-stability criteria fast enough for the control loop. Next, we propose a conservative *trajectory-wide* contact-stability criterion, which can be derived from COM-acceleration volumes at marginal cost and directly applied in a model-predictive controller. We finally implement this pipeline and exemplify it with the HRP-4 humanoid model in multi-contact dynamically walking scenarios.'
author:
- 'Stéphane Caron$^{1}$ and Abderrahmane Kheddar$^{1,2}$ [^1] [^2] [^3]'
bibliography:
- 'refs.bib'
title: |
**Multi-contact Walking Pattern Generation based on\
Model Preview Control of 3D COM Accelerations**
---
Introduction
============
Years ago, humanoid robots were considered as research platforms with vague perspectives in terms of concrete applications. Without much conviction, they were envisioned for entertainment, as receptionists, or as a high-tech show-case for other businesses. Some projects are challenging humanoids to be a daily companion or an assistant for frail persons[^4]. The DARPA robotics challenge boosted the idea that humanoid robots can operate in disaster interventions. The challenge exhibited interesting developments while highlighting the road ahead. Nowadays, Airbus Group seriously envisions humanoids as manufacturing robots to act in large-scale airliner assembly lines. What makes humanoid robots a plausible solution in these applications is their physical ability to move in confined spaces, on non-flat floors, using stairs, etc. In such environments, there are large parts where the robot has to walk robustly.
![ HRP-4 walking on a circular staircase with tilted stepping stones using a preview controller based on 3D COM accelerations. By bounding future COM positions (preview trajectory in yellow) into a polytope $\calT$ (red box), we derive a *trajectory-wide* contact-stability condition, the COM acceleration cone $I_\calT$ (in green), that is efficiently computed using a 2D convex hull algorithm. (Scale and position of this acceleration cone were chosen arbitrarily for depiction purposes.) See the accompanying video [@code] for demonstrations of the controller in various multi-contact scenarios. []{data-label="fig:staircase"}](figures/staircase.pdf){width="0.98\columnwidth"}
Walking robustly on uneven floors is still an open problem in humanoid research. Recently, Boston Dynamics released an impressive video showing robust humanoid walks on various terrains[^5]. This demonstration proves that the goal can be achieved. One key difficulty in locomotion is that the viability (the ability to avoid falling) of the states traversed while walking depends on future contacts. This problem can be addressed geometrically, as done in [@englsberger2015tro] using the generalized 3D capture point, or using dynamic programming as done in [@zhao2016rss] walking in the phase-space of the center of mass (COM), the latter being constrained on predefined surfaces that conform to terrain shape.
Model Predictive Control (MPC) is another widely applied framework that gives controllers the required hindsight to tackle this question. Following the design introduced by Hirukawa et al. [@hirukawa2007icra], one active line of research [@audren2014iros; @herzog2015humanoids; @carpentier2015icra] uses contact forces as control variables, which produces optimization problems with simple inequality constraints but a high number of control variables. Another line of research reduces the problem to the center of mass motion [@brasseur2015humanoids; @naveau2017ral; @vanheerden2017ral; @caron2016tro]. Optimization problems are then much smaller, but their inequality constraints become quadratic (and non-positive-semidefinite [@vanheerden2017ral]). So far, this problem has only been addressed for walking on parallel horizontal surfaces: in [@brasseur2015humanoids], by bounding vertical COM accelerations to keep the formulation linear, and in [@naveau2017ral; @vanheerden2017ral], where Sequential Quadratic Programming was used to cope with quadratic inequalities.
In this paper, we introduce a method that decouples the quadratic inequalities of the general multi-contact problem into pairs of linear constraints, thus opening the way for resolution with classical MPC solvers.
Background
==========
Screw algebra
-------------
Humanoid robots are commonly modeled as a set of rigid bodies and joints whose motion can be described by *screws* [@featherstone2014]. A screw $\bfs_O = (\bfr, \bfm_O)$ is a vector field generated by two vectors: its *resultant* $\bfr$ and its *moment* $\bfm_O$ at a given point $O$. From $\bfm_O$ and $\bfr$, the moment at any other point $P$ results from the Varignon formula: $$\begin{aligned}
\label{varignon}
\bfm_P & = & \bfm_O + \overrightarrow{PO} \times \bfr.\end{aligned}$$
The generalized velocity of a rigid body, called *twist*, is the screw $\bft_O = (\bfv_O, \bfomega)$ with resultant $\bfomega$, the angular velocity, and moment $\bfv_O$, its velocity at given point $O$. A generalized force acting on the body, called *wrench*, is a screw $\bfw_O = (\bff, \bftau_O)$ with resultant net force $\bff$, and net moment $\bftau_O$, at a reference point $O$. Although the coordinate vector $\bfs_P$ of a screw depends on the point $P$ where it is taken, the screw itself does not depend on the choice of $P$ as a consequence of the Varignon formula . We denote screws with hats $(\hat\bft$, $\hat\bfw)$ and their coordinates with point subscripts $(\bft_O$, $\bfw_O)$.
Twists and wrenches live in two dual spaces [@featherstone2014]: the motion space $\mathsf{M}^6$ and the force space $\mathsf{F}^6$. The scalar product between a twist $\hat\bft \in \mathsf{M}^6$ and a wrench $\hat\bfw \in
\mathsf{F}^6$ is given by: $$\hat\bft \cdot \hat\bfw \ \defeq \ \bft_O \cdot \bfw_O \ = \ \bfv_O \cdot
\bff + \bfomega \cdot \bftau_O.$$ From , this number does not depend on the point $O$ where it is computed. When $\hat\bft$ and $\hat\bfw$ are acting on a single rigid body, their product is the instantaneous power of the motion.
Newton-Euler equations
----------------------
Let $m$ denote the total mass of the robot and $G$ its center of mass (COM). We write $\bfp_A$ the coordinate vector of a point $A$ in the inertial frame and denote by $O$ the origin of this frame (so that $\bfp_O = \bm{0}$). Suppose that contacts between the robot and its environment are described by $K$ contact points. (This formulation includes surface contacts; see *e.g.* [@caron2015icra].) The Newton-Euler equations of motion of the whole robot are then given by: $$\label{newton-euler}
\left[ \begin{array}{c}
m \pdd_G \\
\dot{\bfL}_G
\end{array} \right]
\ = \
\left[ \begin{array}{c}
m \bfg \\
{\bm{0}}
\end{array}\right]
\, + \,
\sum_{i=1}^K
\colvec{\bff_{i} \\ {\overrightarrow{GC}_{i}} \times \bff_{i}}$$ where $\bfL_G$ is the angular momentum of the robot around $G$, $\bfg = [0\ 0\
{-g}]^\top$ is the gravity vector defined from the gravity constant $g \approx
9.81\, \textrm{m}\,\textrm{s}^{-1}$ and $\bff_i$ is the force exerted onto the robot at the $\th{i}$ contact point $C_i$. We say that a contact force $\bff_i$ is *feasible* when it lies in the friction cone $\calC_i$ directed by the contact normal $\bfn_i$, $$\begin{aligned}
\label{fric-cones}
\| \bff_i - (\bff_i \cdot \bfn_i) \bfn_i \|_2 & \leq & \mu_i (\bff_i \cdot
\bfn_i)\end{aligned}$$ where $\mu_i$ is the static friction coefficient. The problem of *contact stability* (also called *force balance*) is to find whole-body motions for which admits solutions with feasible contact forces $\{\bff_i\}$.
The Newton-Euler equations describe the components of motion that are independent from the actuation power of the robot, and play a critical role in locomotion. Most of today’s trajectory generators [@kajita2003icra; @hirukawa2007icra; @mordatch2010tog; @brasseur2015humanoids; @herzog2015humanoids; @englsberger2015tro; @vanheerden2017ral] focus on solving these equations and rely on whole-body controllers to take actuation limits into account at a later stage of the motion generation process.
Wrench cones
------------
Equations - include a large number of force variables. Although some walking pattern generators chose to work directly on this representation [@hirukawa2007icra], another line of research [@qiu2011isdhm; @caron2015rss] found that these force variables can be eliminated by propagating their inequality constraints into inequalities on the target rate of change $(m \pdd_G, \dot{\bfL}_G)$ of the whole-body momentum.
Define the net *contact wrench* by: $$\label{cwc-def}
\bfw_O \ =\ \colvec{\bff \\ \bftau_O}
\ \defeq\ \sum_{i=1}^K \colvec{\bff_{i} \\ {\overrightarrow{OC}_{i}}
\times \bff_{i}}$$ In matrix form, $\bfw_O = \bfG_O \bff_\all$ where $\bff_\all$ is the stacked vector of contact forces and $\bfG_O$ is the *grasp matrix*. This wrench can be directly computed from whole-body motions, as it only differs from the whole-body momentum by a constant $\hat\bfw^g$ due to gravity.
Next, one can linearize regular friction cones $\calC_i$ into polyhedral convex cones $\widetilde{\calC}_i$, so that becomes in matrix form (see *e.g.* [@caron2015rss] for details): $$\label{lin-fric-cones}
\bff_i \in \widetilde{\calC}_i\ \Leftrightarrow\ \bfF_i \bff_i \leq \bm0$$ This form is known as the *halfspace representation* of a polyhedral cone. From the Weyl-Minkowski theorem, any polyhedron thus described can be equivalently written as: $$\widetilde{\calC}_i \ = \ \conv(\{\bfv_i\}) + \rays(\{\bfr_j\}),$$ where $\conv(\{\bfv_i\}) = \{ \sum_i \alpha_i \bfv_i, \forall i\,\alpha_i \geq 0
\sum_i \alpha_i = 1\}$ is the *convex hull* of a set of vertices, and similarly $\rays(\{\bfr_j\}) = \{ \sum_i \lambda_i \bfr_i, \forall i\, \lambda_i
\geq 0\}$ denotes positive combinations of a set of rays. This form is known as the *vertex representation* of a polyhedron.
Using suitable conversions between these two representations [@qiu2011isdhm; @caron2015rss], one can finally compute the Contact Wrench Cone (CWC) described in halfspace representation by: $$\label{cwc-hrepr}
\bfA_O \bfw_O \leq \bm0$$ By construction, a net contact wrench $\bfw_O$ belongs to the CWC if and only if there exists a set of contact forces $\{\bff_i\}$ satisfying both - and . Hence, the CWC provides a necessary and sufficient condition for the contact stability of whole-body motions.
Friction Cones are Dual Twists
==============================
Let us consider a row $\bfa = [\bfa_1^\top \bfa_2^\top]^\top$ of the CWC matrix $\bfA_O$. It defines an inequality constraint of the form $$\label{fc1}
\bfa_1 \cdot \bff + \bfa_2 \cdot \bftau_O \ \leq \ 0$$ Applying the Varignon formula , the cone for the same wrench $\bfw_G$ taken at a different point $G$ is subject to: $$\bfa_1 \cdot \bff + \bfa_2 \cdot (\bftau_G + \overrightarrow{OG} \times
\bff) \ \leq \ 0$$ Using the invariance of the mixed product under circular shift, we can rewrite the left-hand side as: $$\label{fc2}
(\bfa_1 + \overrightarrow{GO} \times \bfa_2) \cdot \bff + \bfa_2 \cdot
\bftau_G \ \leq \ 0$$ Let us now defined the *dual twist* $\hat{\bfa} \in \mathsf{M}^6$ by: $$\colvec{\bfa_O \\ \bfa} \ \defeq \ \colvec{\bfa_1 \\ \bfa_2}$$ Equations and rewrite to: $$\begin{aligned}
\label{fcs1} \bfa_O \cdot \bff + \bfa \cdot \bftau_O & \leq & 0 \\
\label{fcs2} \bfa_G \cdot \bff + \bfa \cdot \bftau_G & \leq & 0\end{aligned}$$ where $\bfa_G = \bfa_1 + \overrightarrow{GO} \times \bfa_2$. In concise form: $$\hat\bfa \cdot \hat\bfw \ \leq \ 0$$ This inequality is independent from $O$ where $\bfA_O$ is computed. Therefore, the CWC can be interpreted as a set of dual twists, the coordinates $\bfA_O$ of which one can compute at a fixed reference point using known techniques [@qiu2011isdhm; @caron2015rss].
This shift in the way of considering the cone has an important implication: using the Varignon formula, we can now calculate [analytically]{} the cone $\bfA_G$ at a mobile point $G$ using a fixed solution $\bfA_O$ and the vector coordinates $\bfp_G$. Our following contributions build upon this property.
Contact Stability Areas and Volumes {#areas-and-volumes}
===================================
Static-equilibrium COM polygon {#sep-poly}
------------------------------
Bretl and Lall [@bretl2008tro] showed how static equilibrium can be sustained by feasible contact forces if and only if the (horizontal projection of the) center of mass lies inside a specific polygon, henceforth called the *static-equilibrium polygon*. In fact, the static-equilibrium polygon is embedded in the CWC. Suppose that its matrix $\bfA_O$ was computed at a given point $O$, and let $\hat\bfa$ denote a twist of the CWC corresponding to the inequality . In static equilibrium, the whole-body momentum is zero, so that the net contact wrench $\bfw_G$ at the center of mass $G$ is simply opposed to gravity: $$\bfw_G \ = \ \colvec{\bff \\ \bftau_G} \ = \ \colvec{-m \bfg \\ \bm0}$$ Then, expressing at $G$, yields: $$\bfa_G \cdot (-m \bfg) + \bfa \cdot \bm0 \ \leq \ 0$$ which also writes, since $m>0$: $$-(\bfa_O + \bfa \times \bfp_G) \cdot \bfg \ \leq \ 0$$ Expanding this scalar product yields: $$\label{fsc2d}
a_{Oz} - a_y x_G + a_x y_G\ \leq\ 0$$ where $\bfa_O = [a_{Ox}\ a_{Oy}\ a_{Oz}]^\top$ and $\bfa = [a_x\ a_y\
a_z]^\top$. The set of inequalities over all twists $\hat\bfa$ of the CWC provides the half-plane representation of the static-equilibrium polygon. Note that the static-equilibrium polygon does not depend on the mass, which was not observed in previous works [@bretl2008tro; @delprete2016icra; @caron2016tro; @zhang2016ijhr].
In what follows, we will use the following equivalent formulation. Let us define the *slackness* of by: $$\sigma_{\hat\bfa}(x_G, y_G) \ \defeq \ -a_{Oz} + a_y x_G - a_x y_G$$ it is the signed distance between $(x_G, y_G)$ and the supporting line $-a_y x + a_x y + a_{Oz} = 0$ of the corresponding static-equilibrium polygon’s edge. A point $(x_G, y_G)$ is then inside the polygon if and only if $\sigmabfa(x_G, y_G) \geq 0$ for all the CWC twists $\hat\bfa$.
Vertex enumeration for polygons {#vertex-enum}
-------------------------------
The half-plane representation is best-suited for COM feasibility tests. Meanwhile, the vertex representation is best-suited for planning. Converting from halfspace to vertex representation is known as the *vertex enumeration* problem, for which the *double description method* [@fukuda1996double] has been applied in previous works [@bouyarmane2009icra; @qiu2011isdhm; @escande2013ras; @caron2015rss].
For general $d$-dimensional polyhedra, vertex enumeration has polynomial, yet super-linear time complexity. For example, the Avis-Fukuda algorithm [@avis1992dcg] runs in $\calO(d h v)$, with $h$ and $v$ the numbers of hyperplanes and vertices, while the original double-description method by Motzkin has a worst-case time complexity of $\calO(h^2
v^3)$ [@fukuda1996double]. Yet, for $d=2$, the problem boils into computing the *convex hull* of a set of points, for which optimal algorithms (for instance [@kirkpatrick1986siam]) are known that match the theoretical lower-bound of $\Omega(h \log v)$.
Our formulation allows us to enumerate vertices in 2D. Let us assume for now that the origin $(x_G, y_G) = (0, 0)$ lies in the interior of the polygon, and divide each inequality by $a_{Oz}$ to put the overall inequality system in polar form: $$\label{polar}
\bfB \colvec{x_G \\ y_G} \ \leq \ \bm1$$ We run a convex hull algorithm on the rows of the matrix $\bfB$. By duality, the cyclic order of extreme points thus computed corresponds to a cyclic order of adjacent edges for the primal problem. Intersecting pairs of adjacent lines in this order yields the vertices of the initial polygon. The conversion of inequalities to being $\calO(n)$, computing the output polygon is done overall in $\calO(h \log v)$. See the Appendix for a comparison with existing approaches.
To construct , we assumed that the origin lies inside the polygon. When this is not the case, one can simply compute the Chebyshev center $(x_C,
y_C)$ by solving a single Linear Program (LP) as detailed *e.g.* in [@boyd2004convex] p. 148. From there, a translation $(x_G', y_G') = (x_G -
x_C, y_G - y_C)$ brings the origin inside the polygon.
Pendular ZMP support areas
--------------------------
Let us revisit the derivation of the pendular ZMP support area [@caron2016tro] using our new approach. To achieve linear-pendulum mode of the Newton-Euler equations of the system, the following four equality constraints are applied to the contact wrench: $$\begin{aligned}
\label{lp1} \bfn \cdot \bff & = & m (\bfn \cdot \bfg) \\
\label{lp2} \bftau_G & = & \bm0\end{aligned}$$ where $\bfn$ denotes the unit vector normal to the plane in which the ZMP is taken. In what follows, we suppose that $\bfn$ is opposite to gravity, so that $\bfn \cdot \bfg = -g$. Equation is used to linearize the pendulum dynamics, the ZMP being defined in general by the non-linear formula $\bfp_Z
\defeq \frac{\bfn \times \bftau_O}{\bfn \cdot \bff} + \bfp_O$. Under Equations -, the resultant force can be computed from COM and ZMP positions by [@caron2016tro]: $$\label{forcelin}
\bff \ = \ \frac{mg}{h} \colvec{x_Z - x_G \\ y_Z - y_G \\ 1}$$ where $h = z_Z - z_G$ is the constant difference between ZMP and COM altitudes. This value can be positive or negative: for the sake of exposition, we will take $h > 0$. Injecting Equations - into an inequality constraint of the CWC yields: $$\label{fgba}
(\bfa_O + \bfa \times \bfp_G) \cdot \bff + \bfa \cdot \bm0 \ \leq \ 0$$ Substituting into yields: $$a_i (x_Z - x_G) + b_i (y_Z - y_G)
\ \leq \ h \sigma_{\hat\bfa}(\bfp_G)
\label{link-sep-0}$$ where $$\colvec{a_i \\ b_i} \ = \
\colvec{a_{Ox} \\ a_{Oy}} + \colvec{
a_y z_G - a_z y_G \\
-a_x z_G + a_z x_G}$$ Assuming that the COM lies inside the static-equilibrium polygon,[^6] the right-hand side of this expression is positive from . The inequality is then expressed in polar form as $$\label{zmp-polar}
\bfB_{\mathrm{ZMP}}(\bfp_G) \colvec{x_Z - x_G \\ y_Z - y_G} \ \leq \ \bm1$$ where the origin $(x_Z, y_Z) = (x_G, y_G)$ lies inside the polygon by construction. As in , a convex hull algorithm can finally be applied to compute the vertices of the pendular ZMP support area.
3D Volume of COM accelerations {#3dvol}
------------------------------
Equation is a limitation of the linear-pendulum mode in that the COM trajectory needs to lie in a plane pre-defined by the vector $\bfn$. This limitation is all the less grounded that, from Equation , controlling the ZMP in this mode is equivalent to controlling the resultant contact force $\bff$. We therefore propose to directly control this force, or equivalently, to directly control the three-dimensional COM acceleration $\pdd_G = \frac1m \bff + \bfg$.
Substituting this acceleration into , one gets: $$\label{fcs3}
(\bfa_O + \bfa \times \bfp_G) \cdot \pdd_G
\ \leq \ (\bfa_O + \bfa \times \bfp_G) \cdot \bfg
%\ = \ g \sigma_{\hat\bfa}(\bfp_G)$$ Expanding scalar products, this inequality rewrites to: $$\label{link-with-sep}
a_i \xdd_G + b_i \ydd_G - \sigma_{\hat\bfa} \zdd_G \ \leq \ g
\sigma_{\hat\bfa}$$ (We dropped the argument $\bfp_G$ of $\sigma_{\hat\bfa}$ to alleviate notations.) Assuming that the COM lies in the interior of the polygon$^3$ ($\sigma_{\hat\bfa} > 0$) and that $\zdd_G > -g$, $$\label{force-polar}
\left(\frac{a_i}{\sigmabfa}\right) \cdot \frac{\xdd_G}{g + \zdd_G} +
\left(\frac{b_i}{\sigmabfa}\right) \cdot \frac{\ydd_G}{g + \zdd_G} \ \leq \
1$$ This expression is in polar form $\bfB_{\mathrm{3D}}(\bfp_G) [\xt\ \yt]^\top
\leq \bm1$ for the new coordinates: $$\colvec{\xt \\ \yt} \ = \ \frac{1}{g + \zdd_G} \colvec{\xdd_G \\ \ydd_G}$$ We can enumerate the vertices $\{(\xt_i, \yt_i)\}$ of the corresponding polygon using a convex hull again. For a given vertical acceleration $\zdd_{G,i} > -g$, the COM acceleration coordinates $(\xdd_{G,i}, \ydd_{G,i})$ corresponding to a vertex $(\xt_i, \yt_i)$ are: $$\colvec{\xdd_{G,i} \\ \ydd_{G,i}} \ = \ (g + \zdd_{G,i}) \colvec{\xt_i \\
\yt_i}$$ We recognize the equation of a 3D polyhedral convex cone, pointing upward, with apex located at $(\xdd_G, \ydd_G, \zdd_G) = (0, 0, -g)$ and rays defined by $\bfr_i = [\xt_i\ \yt_i\ 1]^\top$. Figure \[fig:config3\] shows the cone in a sample contact configuration.
![ Static-equilibrium polygon (in green at the altitude of the center of mass) and cone of 3D COM accelerations (in red, with the zero acceleration centered on the COM) in a double-support configuration. For the latter, the scaling from accelerations to positions is $0.08\
\textrm{s}^2$, and the cone was cut at $\zdd_G = g$ to show its cross-section. []{data-label="fig:config3"}](images/config3.png){width="0.98\columnwidth"}
Overall, we have thus both (1) a geometric characterization and (2) an algorithm to compute the cone of feasible COM accelerations when the angular momentum is regulated to zero.[^7] This construction generalizes the pendular support area defined in [@caron2016tro]. Furthermore, as a consequence of Equations -, we have the following property:
The COM is in the interior of the static-equilibrium polygon if and only if the set of feasible COM accelerations (equivalently, whole-body ZMPs) under zero angular momentum contains a neighborhood of the origin.
In other words, the static-equilibrium polygon is not only related to static equilibrium: it is also the set of positions from which the robot can accelerate its center of mass in any direction (with zero angular momentum). When the COM reaches the edge of this polygon, the zero acceleration touches a facet of the acceleration cone. Denoting by $\bfd$ the facet normal, all feasible accelerations $\pdd_G$ are then such that $\bfd \cdot \pdd_G \geq 0$, $\bfd$ is an “irresistible” direction of motion.
Preview control of COM accelerations {#mpc101}
====================================
Let the control variable $\bfu$ be the COM acceleration $\bfu :=
\pdd_G$. The discretized COM dynamics with sampling $\Delta T$ are: $$\begin{aligned}
\label{disdyn}
\bfx(k + 1) & = & \bfA \bfx(k) + \bfB \bfu(k)\end{aligned}$$ where $\bfx(k) = [\bfp_G(k\,\Delta T)^\top \ \pd_G(k\,\Delta T)^\top]^\top$ and, denoting by $\bfE_3$ the $3 \times 3$ identity matrix, $$\bfA \: = \: \left[
\begin{array}{rr}
\bfE_3 & \Delta T \bfE_3 \\
\bm{0}_{3 \times 3} & \bfE_3
\end{array} \right]
\quad
\bfB \: = \: \left[
\begin{array}{r}
\frac12 \Delta T^2 \bfE_3 \\
\Delta T \bfE_3
\end{array} \right]$$ At each control step, a preview controller receives the current state $\bfx_0 = (\bfp_0, \pd_0)$ and computes a sequence of controls $\bfu(0), \ldots, \bfu(N)$ driving the system from $\bfx_0$ to $\bfx(N)$ at the end of the time horizon $T = N \Delta T$ of the preview window. By recursively applying , $\bfx(k)$ can be written as a function of $\bfx_0$ and of $\bfu(0), \ldots,
\bfu(k-1)$ (see [@audren2014iros]): $$\begin{aligned}
\label{iter-disdyn}
\bfx(k) & = & \bfPhi_k \bfx_0 + \bfPsi_k \bfU(k-1) \\
\bfU(k) & = & [\bfu(0)^\top \ \cdots \ \bfu(k)^\top]^\top\end{aligned}$$ A necessary condition for contact-stability throughout the trajectory is that all accelerations $\bfu(k)$ lie in the COM-acceleration cone $\calC(\bfp_G(k \Delta T))$. Expanding its inequalities for an arbitrary twist $\hat\bfa \in \CWC$ yields: $$\label{csfun1}
\bfp_G(k\,\Delta T)^\top [-\bfa \times] (\bfu(k) - \bfg) + \bfa_O^\top (\bfu(k) - \bfg) \leq 0$$ Let $L$ denote the number of twists in the CWC. By stacking up the $N$ inequalities , we get in more concise form: $$\label{tensor-form}
\bfp_G(k\,\Delta T)^\top \bbA_\times (\bfu(k) - \bfg) + \bfA'_O (\bfu(k) - \bfg) \leq \bm0$$ where $\bbA_\times$ is a $3 \times L \times 3$ tensor, and $\bfA'_O$ consists of the first three columns of $\bfA_O$. Combining and , this condition yields a set of quadratic inequality constraints of the form: $$\label{quad-ineq}
\forall k < N,\ \bfU(k)^\top \bbP_k \bfU(k) + \bfQ_k \bfU(k) + \bfl_k \leq
\bm0$$ where $\bbP_k$ is a $3(k+1) \times L \times 3(k+1)$ tensor, $\bfQ_k$ is a $L
\times 3(k+1)$ matrix $\bfl_k$ is an $L$-dimensional vector. One can thus formulate the preview control problem as a Quadratically Constrained Quadratic Program (QCQP). Although a QCQP formulation was successfully applied for walking on even terrains with variable COM height [@vanheerden2017ral], we chose not to do so for the following reasons:
- QCQP is a harder class of problems than Quadratic Programming (QP), especially when inequality constraints are not positive-semidefinite [@vanheerden2017ral] so that the problem non-convex. Real-timeness implies that only a small number of SQP iterations can be run in the control loop (two in [@vanheerden2017ral]), thus with no convergence guarantee.
- Constraints are given without any redundancy elimination. In practice, eliminating redundancy (in our case, by applying a convex hull algorithm) significantly reduces the number of inequality constraints (Table \[table:dimrec\]).
Instead, we propose a trajectory-wide contact-stability criterion that yields linear inequality constraints, and for which we can apply the convex-hull reduction from Section \[areas-and-volumes\].
Robust trajectory-wide contact-stability criterion {#sec:tubes}
--------------------------------------------------
We want to drive the COM from $\bfx_0 = (\bfp_0, \pd_0)$, its current state, to a goal position $\bfx_T$ through a trajectory $t \in [0, T] \mapsto \bfp_G(t)$. Due to the initial velocity $\pd_G(0)$ and real-world uncertainties, the trajectory will not be exactly a line segment $[\bfp_0, \bfp_T]$. Yet, we assume that it lies within a polyhedral “tube” $\calT = \conv(\{\bfnu_1,
\ldots, \bfnu_q\})$ containing $[\bfp_0, \bfp_T]$. A point $\bfp_G \in \calT$ can be written as a convex combination $\bfp_G = \sum_{i=1}^q \alpha_i
\bfnu_i$, where the $\alpha_i$’s are positive and sum-up to one, so that the inequality becomes: $$\label{csfun2}
\sum_{i=1}^q \alpha_i (\bfnu_i^\top \bbA_\times + \bfA_O') (\pdd_G - \bfg) \ \leq \ \bm0$$ A particular way to enforce the negativity of a convex combination is to ensure that all of its terms are negative. Thus, a sufficient condition for the satisfaction of is $$\label{csint1}
\bfC_\calT (\pdd_G - \bfg) \leq \bm0, \quad
\bfC_\calT := \colvec{\bfnu_1^\top \bbA_\times + \bfA_O' \\ \vdots \\
\bfnu_q^\top \bbA_\times + \bfA_O'}$$ which is the halfspace representation of the intersection $$I_\calT \ := \ \calC(\bfnu_1) \cap \cdots \cap \calC(\bfnu_q)
\ \subseteq \ \calC(\bfp_G)$$
$I_\calT$ is the set of COM accelerations that are feasible everywhere in $\calT$, $$\label{I-bigcap}
I_\calT = \bigcap_{\bfp \in \calT} \calC(\bfp)$$
We showed that $\pdd_G \in I_\calT$ is feasible at any position $\bfp_G \in \calT$ (by construction, $\Rightarrow$ $\Rightarrow$ ). Conversely, suppose that $\pdd_G$ is feasible for all $\bfp_G \in \calT = \conv(\{\bfnu_1, \ldots,
\bfnu_k\})$. In particular, holds when $\bfp_G$ is equal to any vertex $\bfnu_j$. Thus, $\pdd_G \in \calC(\bfnu_j)$, and since the index $j$ can be chosen arbitrarily, $\pdd_G \in \cap_j \calC(\bfnu_j) =
I_\calT$.
A trajectory $\bfp_G(t)$ is *contact-stable within $\calT$* if it satisfies: $$\forall t \in [0, T],\ \bfp_G(t) \in \calT\ \textrm{and}\ \pdd_G(t) \in I_\calT$$
This condition can be compared with the previous trajectory-wide stability criterion from [@caron2015rss], where trajectories were computed such that $$\forall t \in [0, T],\ \pdd_G(t) \in \calC(\bfp_G(t))$$ The implicit assumption behind such a strategy is that either trajectory-tracking is perfect, or small deviations $\delta \bfp_G$ from the reference $\bfp_G(t)$ can be coped with if $\pdd_G$ lies “inside enough” of its cone. By taking $\calT$ around the trajectory, we explicitly model how far $\pdd_G$ needs to be inside the cone to cope with a set of deviations $\delta
\bfp_G$. In this sense, our criterion is a *robust* condition for trajectory-wide contact stability, with explicit modelling of the robustness margin.
Although the tube-wise intersection formalized by Equation seems like a severely restrictive condition, we noticed (to our surprise) that the acceleration cones left after intersection are still sufficient for locomotion in challenging scenarios (see Section \[sec:xp\]).
$|\bfC_\calT|$ $|\bfC'_\calT|$
---------------- ---------------- -----------------
Single support $376 \pm 226$ $6 \pm 2$
Double support $484 \pm 221$ $6 \pm 2$
: Effect of the dual convex-hull reduction on the size of the inequality-constraint matrix $\bfC_\calT$. Averages are given on $26$ matrices (one for each step of the circular staircase). []{data-label="table:dimrec"}
Integration with preview control
--------------------------------
At each new control step, the current state $\bfx_0 = (\bfp_0, \pd_0)$ of the center of mass feeds the preview controller. We define the target state at the end of the time horizon by $\bfx_T = (\bfp_T, \bfv_T)$, where $\bfp_T$ is determined from the current stance in the step sequence, $\bfv_T$ is a reference velocity of $0.4$ m.s$^{-1}$ oriented in the direction of motion, and $T$ is calculated from the time remaining in the current gait phase. From there, our method goes as follows:
1. Define $\calT$ containing $[\bfp_0, \bfp_T]$ and compute its halfspace representation $\bfP_\calT$ using the double-description. In practice, we defined $\calT$ by a polyhedral cylinder centered on $[\bfp_0, \bfp_T]$ with square cross-sections and a robustness radius of 5 cm.
2. Compute the halfspace representation $\bfC_\calT$ of $I_\calT$ from the contact wrench cone $\bfA_O$ (which is computed only once per support phase).
3. Reduce $\bfC_\calT$ to polar form $\bfB_\calT {[x\ y]^\top} \leq
\bm1$ and compute its vertex representation $\bfg + \rays(\{\bfr_i\})$ using a convex hull algorithm[^8] (Section \[3dvol\]).
4. Compute its non-redundant halfspace-representation $\bfC'_\calT$ using again the double-description.
Next, we formulate the preview control problem as a quadratic program: $$\begin{aligned}
\label{qp-obj}
\min_{\bfU} & : & \| \bfx(N) - \bfx_T \|^2 + \epsilon \|\bfU\|^2 \\
\textrm{s.t.} & : & \forall k,\ \bfC'_\calT \bfu(k) \leq \bfC'_\calT \bfg \label{qp-lin-ineq} \\
& & \forall k,\ \bfP_\calT \bfx(k) \leq \bm1
\label{qp-poly-ineq}\end{aligned}$$ where $\bfU \defeq \bfU(N-1)$ is the stacked vector of controls from which all $\bfx(k)$ and $\bfu(k)$ derive by Equation . The inequality constraints and are a linear decoupling of via polyhedral bounds $\calT$.
As a matter of fact, using polyhedral bounds can be thought of as a general linearization technique. In [@brasseur2015humanoids], a linear decoupling was also obtained by bounding vertical COM accelerations. Similarly, polytopes of robust COM positions $\bfp_G$ were obtained in [@caron2015rss; @delprete2016icra] by defining polyhedral bounds on disturbances $\bfepsilon$, thus eliminating the bilinear coupling between $\bfp_G$ and $\bfepsilon$.
Eliminating redundancy in the pipeline above is a significant computational step. From one of our experiments (Table \[table:dimrec\]), the number of lines $|\bfC'_\calT|$ in $\bfC'_\calT$ is two orders of magnitude smaller than that of $\bfC_\calT$. Given that the number of inequalities in the above QP is $N |\bfC'_\calT|$, this makes the difference in practice between solving a problem of size $100$ versus $10,000$ (we use $N=10$ steps).
Function Output Time (ms)
---------------------------------------- --------------- ---------------
Double description of $\calT$ (ß and ) $\bfP_\calT$ $0.3 \pm 0.1$
H-representation of $I_\calT$ (ß and ) $\bfC_\calT$ $0.2 \pm 0.1$
Convex hull of $I_\calT$ (ß and ) $\bfC'_\calT$ $2.4 \pm 1.2$
Other matrix operations – $0.4 \pm 0.4$
Solving final QP $(N=10)$ $\bfU$ $0.2 \pm 0.1$
Total $3.5 \pm 2.1$
: Breakdown of computation times inside the predictive controller, averaged over 2000 calls in Experiment VI.B. []{data-label="table:times-stair"}
Locomotion state machine
------------------------
Our preview controller is applied to the HRP-4 humanoid in various environments. The inputs to the controller are the current COM position $\bfp_0$ and velocity $\pd_0$, the preview time horizon $T$, a target COM position $\bfp_T$, $\calT$ and its acceleration cone $I_\calT$. These computation of these inputs is supervised by a Finite State Machine (FSM) that cycles between four phases $\varphi \in \{\SSL, \DSR, \SSR, \DSL\}$, where $\textsf{\textsc{ss}}$ (resp. ) stands for single-support (resp. double-support), while and indicate that the phase ends on the left and right foot, respectively. Each phase is thus associated with a unique foot contact. The target COM position $\bfp_G^*(\varphi)$ of a phase is then taken $0.8$ m above this contact.
Phase durations are set to $T_\ss = 1$ s for single-support and $T_\ds = 0.5$ s for double-support. At each iteration of the control loop, the input to the preview controller is decided based on the time $T_\rem$ remaining until the next phase transition. Let us denote by $\varphi$ the current phase in the FSM and $\varphi'$ the phase after $\varphi$. We define preview targets by:
1. if $\varphi$ is single-support and $T_\rem < \frac12
T_\ss$:
- $T \leftarrow T_\rem + T_\ds + \frac12 T_\ss$
- $\bfp_G \leftarrow \bfp_G^*(\varphi'')$
2. otherwise, if $\varphi$ is double-support:
- $T \leftarrow T_\rem + \frac12 T_\ss$
- $\bfp_G \leftarrow \bfp_G^*(\varphi)$
3. otherwise ($\varphi$ is single-support and $T_\rem > \frac12 T_\ss$):
- $T \leftarrow T_\rem$
- $\bfp_G \leftarrow \bfp_G^*(\varphi)$
Case 1) switches the target of the preview controller to the next staircase step in the middle of single-support phases[^9], which forces the robot to start its next step while allowing it to re-use the kinetic momentum in the direction of motion.
Note that cases 1) and 2) imply contact switches in the middle of preview trajectories, which different cones $\bfC'_\calT$ depending on the step $k$ in the preview problem. To take this into account, we compute the switching step $k_\rem = T_\rem / \Delta T$ along with two tubes $\calT_\ss \subset \calT_\ds$ and their dual cones (computations between these two overlapping tubes can be factored; see [@code] for details). The corresponding matrices $(\bfC'_{\calT_\ss}, \bfP_{\calT_\ss})$ and $(\bfC'_{\calT_\ds},
\bfP_{\calT_\ds})$ are then respectively used in Equations - for $k \leq k_\rem$ and $k >
k_\rem$.
A breakdown of computation times inside the overall predictive controller is reported in Table \[table:times-stair\]. All computations were run on an average laptop computer (Intel<span style="font-variant:small-caps;">(r)</span> Core<span style="font-variant:small-caps;">(tm)</span> i7-6500U CPU @ 2.50 Ghz).
Whole-body controller
---------------------
The last step of the pipeline is to convert task objectives, such as COM or foot positions, into joint commands sent to motor controllers. For this, we used our own solver implemented in the *pymanoid* library.[^10] It solves a single quadratic program on five weighted tasks (see [@caron2016tro] for details), by decreasing priority: support foot, swing foot, COM tracking, constant angular-momentum, and posture tracking for regularization. The corresponding task weights were respectively set to $(10^4, 10^2, 10, 1,
10^{-1})$. Each QP solution provides joint-angle velocities $\qd_\textrm{ref}$, which is then sent to the robot. See Figure \[fig:pipeline\] for a summary of our pipeline.
![ Overview of our control pipeline. The COM trajectory polyhedron $\bfP_\calT$ and its corresponding contact-stability cone $\bfC_\calT$ are computed using the method described in Section \[mpc101\]. []{data-label="fig:pipeline"}](figures/pipeline.pdf){width="0.98\columnwidth"}
Experiments {#sec:xp}
===========
The ground truth for contact stability is the existence of feasible contact forces $\bff_\all$ summing up to the net wrench $\hat{\bfw}$ of the motion. In all experiments, we validate our trajectories by checking the existence of such $\bff_\all$ at each time instant. In both experiments, friction coefficients were set to $\mu=0.7$.
Regular staircase and walking into an aircraft
----------------------------------------------
Our first scenario, provided by Airbus Group, takes place in a 3D realistic model of scale 1:1 for a section of the A350 airplane. This mock-up will be used for experiments with the real robot when the research matures to an integrated software, and gets approval from Airbus Group. In order to access a predefined spot in the A350, the robot needs to climb stairs (size of those available in the factory), walk on a platform (flat ground) to finally reach the mockup floor composed of removable tiles that can be uneven and disposed in various locations, see Figure \[fig:airbus\]. In this simulation, the footprints were given together with the timing for the and phases (respectively 0.5 s and 1 s).
Slanted circular staircase with tilted steps
--------------------------------------------
The slanted circular staircase depicted in Figure \[fig:staircase\] has 26 steps randomly rolled, pitched and yawed by angles $(\theta_r, \theta_p,
\theta_y) \in [-0.5, +0.5]^3$ rad. The average radius of the staircase is $1.4$ m, and the altitude difference between the highest and lower steps is also $1.4$ m.
For this scenario, the reference durations of single and double support phases were set to $T_\ss = 1$ s and $T_\ds = 0.5$ s, respectively. We concur with [@audren2014iros] that the question of finding proper timings becomes crucial in multi-contact. Having constant durations overlooks the fact that some steps are harder to take than others (due to their respective tilting, altitude difference, etc.).[^11] Being unable to find a single pair of constants $(T_\ss, T_\ds)$ suited to the whole staircase, we opted for the following workaround condition:
- At the end of a double-support phase, wait for the COM to be above the static-equilibrium polygon of the next single-support before activating the phase transition.
This choice is motivated by the link between the 3D cone of COM accelerations and the position of the COM in the static-equilibrium polygon. In practice, it allows the use of “optimistic” values of $(T_\ss,
T_\ds)$ while only extending $T_\ds$ when necessary.
![ HRP-4 accessing the floor-shop of an A350 through stairs and reach the working areas by walking on tiles. Foot trajectories (dashed lines) step over two staircase steps at a time, as done in natural stair climbing. The COM trajectory (blue line) illustrates the progression of the robot until its target configuration inside the aircraft. Contact stability of the whole motion was cross-validated by checking the existence of groundtruth contact forces $\bff_\all$ at each timestep, as shown in the accompanying video [@code]. []{data-label="fig:airbus"}](images/airbus.png){width="0.98\columnwidth"}
Conclusion
==========
We presented a multi-contact walking pattern generator based on preview-control of the 3D acceleration of the center of mass. Our development builds upon algebraic manipulations of friction cones as dual twists, thanks to which we can recompute 3D cones of feasible COM accelerations in real-time. We then showed how to intersect these cones over the preview window of a model-preview controller to construct a conservative trajectory-wide contact-stability criterion. We implemented this pipeline and illustrated it with the HRP-4 humanoid model in multi-contact dynamically walking scenarios. All our source code is released at [@code].
Acknowledgment {#acknowledgment .unnumbered}
==============
The authors would like to warmly thank Komei Fukuda for his helpful feedback and Patrick Wensing for pointing out a calculation mistake in a preliminary version of the paper. This paper has also benefitted from rich discussions with Hervé Audren and Adrien Escande.
We compare the convex hull reduction to the original calculation [@bretl2008tro] of the static-equilibrium polygon. Computation times for randomly sampled contact configurations are reported in Table \[table:times2\] for four algorithms:
- *cdd + hull*: the method described in Section \[vertex-enum\], where *cdd* [@fukuda1996double] is used to compute the CWC while convex hulls are computed with *Qhull* [@barber1996quickhull].
- *Parma + hull*: same approach, using the Parma Polyhedra Library[^12] rather than *cdd* to compute the CWC.
- *cdd only*: as described in [@zhang2016ijhr], *cdd* can also be used to compute the static-equilibrium polygon directly.
- *Bretl & Lall*: the algorithm from [@bretl2008tro], in the implementation from [@pham2015tm] but using GLPK as LP solver.[^13]
The *Parma + hull* solution is the slowest but most numerically stable, while *cdd only* is only competitive in single-support. Neck to neck are *Bretl & Lall* and *cdd + hull*, with the latter faster in single- and double-support. But the real benefit of our approach comes with the computation of time-varying criteria: in double- and triple-support, we see that executing *hull only* is more than ten times faster than applying any other algorithm from scratch. We also highlight it as the fastest solution for single-support, as in this case the CWC is known analytically [@caron2015icra] and there is no need for the *cdd* step.
--------------------------------------------------------------------------------------------------------------
Algorithm Single support Double support Triple support
--------------------------------------------------- ----------------- -------------------- -------------------
Parma + hull $6.02 \pm 0.20$ $21.0 \pm 4.2$ $42 \pm 11$
cdd only $0.38 \pm 0.01$ $7.0 \pm 2.7$ $> 500$
Brel & Lall [@bretl2008tro] $1.00 \pm 0.02$ $3.1 \pm 0.8$ $\bf 5.9 \pm 1.6$
cdd + hull $0.60 \pm 0.01$ $\bf 2.7 \pm 0.6$ $7.1 \pm 1.9$
*hull only & $\bf 0.17 \pm 0.003$ & $\it 0.28 \pm
0.04$ & $\it 0.38 \pm 0.09$\
*
--------------------------------------------------------------------------------------------------------------
: Time (in ms) to compute the static-equilibrium polygon, averaged over 100 random contact configurations. []{data-label="table:times2"}
[^1]: \*This work is supported in part by H2020 EU project COMANOID <http://www.comanoid.eu/>, RIA No 645097.
[^2]: $^{1}$CNRS-UM2 LIRMM, IDH group, UMR5506, Montpellier, France.
[^3]: $^{2}$CNRS-AIST Joint Robotics Laboratory (JRL), UMI3218/RL. Corresponding author: [stephane.caron@normalesup.org]{}
[^4]: <http://projetromeo.com/>
[^5]: <https://www.youtube.com/watch?v=rVlhMGQgDkY>
[^6]: Otherwise, center the polygon on its Chebyshev center as previously.
[^7]: The same derivation can be applied with non-zero angular-momentum references; however, the question of finding such references is still open.
[^8]: We used *Qhull* [@barber1996quickhull], available from <http://www.qhull.org/>
[^9]: The same behavior is present in [@kajita2003icra]: if the control from Figure 5 of this paper was followed to the end, the COM velocity would go to zero between each step, which is not the behavior observed in Figures 7 and 8.
[^10]: <https://github.com/stephane-caron/pymanoid>
[^11]: This question is less critical for walking on horizontal or well structured floors, where all steps are similar.
[^12]: <https://github.com/haudren/pyparma>
[^13]: Using an efficient LP solver is crucial here: in a preliminary version of this paper, we used the more general CVXOPT, which resulted in computations around $10 \times$ slower than those we now report using GLPK.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We consider the current induced in a quantum wire by external electromagnetic radiation. The photocurrent is caused by the interplay of spin-orbit interaction (Rashba and Dresselhaus terms) and external in-plane magnetic field. We calculate this current using a Wigner functions approach taking into account radiation-induced transitions between transverse subbands. The magnitude and the direction of the current depend on the Dresselhaus and Rashba constants, strength of magnetic field, radiation frequency and intensity. The current can be controlled by changing some of these parameters.'
author:
- Arkady Fedorov
- 'Yuriy V. Pershin'
- Carlo Piermarocchi
title: 'Spin-photovoltaic effect in quantum wires due to inter-subband transitions'
---
Introduction
============
The energy spectrum of free electrons in a perfect quantum wire without spin-orbit interaction consists of spin-degenerate subbands due to the transverse confinement in two directions. In each subband, the energy depends quadratically on the one-dimensional momentum. In the presence of an external radiation intersubband excitation probabilities are equal for states with opposite momentum. Therefore, there is no change in current associated with external radiation.
During the last 15 years there has been a great interest in theoretical and experimental investigations of photovoltaic effects and photoconductance in quantum wires (see Refs. - and references therein). Mechanisms for pure spin current generation in 2D and 1D systems with spin-orbit interaction have also been discussed [@sipe; @ivchenko]. It is known that a photovoltaic effect is possible in quantum wires without inversion symmetry. For example, a photovoltaic effect in quantum wires with spatially dependent lateral confinement was predicted in Ref. . In the present paper we consider a different [*spin*]{}-based mechanism for photovoltaic effect, which is very interesting in the context of the fast growing field of spintronics. [@book; @review; @ourrev] The most important component of our scheme is the spin-orbit interaction. However, the spin-orbit interaction alone is not sufficient to generate a charge photocurrent if the quantum wire is spatially homogeneous. Therefore, we consider a wire in an in-plane magnetic field which breaks the inversion symmetry.
We can identify the following groups of inter-subband transitions that lead to a photovoltaic effect in quantum wires:
1. Transitions between spin-splitted subbands with the same confinement quantum numbers.
2. Transitions between subbands with different confinement quantum numbers.
The main difference between these two groups is that the first is generated by the magnetic field component of electromagnetic radiation, while transitions from the second group are due to the electric field component. In a recent paper [@perpier] a spin-photovoltaic effect in quantum wires due to transitions of the first type was considered. It was found that a special role in the photovoltaic effect is played by transitions in which the direction of electron velocity changes. The importance of this velocity inversion was outlined earlier, see e.g. Ref. , in studies of photoconductance. Here we consider intersubband transitions of the second type. An important feature in semiconductor-based quantum wires is that the spin-orbit interaction constants are different for subbands with different confinement quantum numbers. This peculiarity is essential in our scheme for the generation of photocurrent.
The goal of this paper is to calculate the current in a quantum wire at zero bias voltage due to external radiation. The current as a function of radiation frequency is found numerically from coupled equations involving Wigner functions. The Wigner function formalism has many advantages for investigating transport problems [@Kluksdahl; @Frensley]. Among them we mention the phase-space nature of Wigner functions which are similar to the classical Boltzmann distribution functions. This feature makes possible to separate the incoming and outgoing components of the electron distribution at the boundaries which, in turn, facilitates the modeling of an ideal contact. The commonly used assumptions are: the distribution of electrons emitted in the quantum wire can be described by the equilibrium distribution function of the leads reservoirs, and all electron are collected by the leads reservoirs without reflection. In this work we extend the description of the transport dynamics to include inter-subband transitions due to electro-magnetic wave excitation.
We show that the current is sensitive to many control parameters like, e.g., the spin-orbit coupling and external magnetic field. Therefore, the current can be used to determine materials parameters. The calculated current strength for a realistic set of parameters is of the order of $0.1$ nA and consequently can be measured using standard experimental techniques.
This paper is organized as follows. The single particle energy spectrum and wave functions are introduced in Sec. \[sec2\]. A set of coupled equations for Wigner functions is derived in Sec. \[sec3\]. The discrete model used for numerical solutions is presented in Sec. \[discrete\]. Numerical results are given and discussed in Sec. \[sec4\] and concluding remarks are in Sec. \[sec5\].
System {#sec2}
======
Fig. \[fig1\] shows a possible experimental realization of the system under investigation. The two-dimensional electron gas is split into two parts by a potential applied to the gate electrodes. The narrow channel between the gates then forms a quantum wire. Let us define a coordinate system such that the direction of the electron transport through the wire is in the $x$-direction and the lateral confinement is in the $y$-direction. We assume that an external magnetic field is applied in the ($x,y$) plane. Previously, several interesting investigations of quantum wires with spin-orbit interaction in the presence of an in-plane magnetic field were reported. [@QWSOMF1; @QWSOMF2; @QWSOMF3; @QWSOMF5; @QWSOMF6]
![(Color online) Quantum wire with an applied magnetic field in ($x,y$) plane, and irradiated by a electro-magnetic wave linearly polarized in the $y$-direction.[]{data-label="fig1"}](fig1){width="8.5cm"}
In the quantum wire, the Hamiltonian for the conduction electrons can be written in the form $$H=H_0+H_1,\label{ham}~,$$ where $$H_0=\frac{\mathbf{p}^2}{2m^*}-\frac{\alpha}{\hbar}p_x\sigma_y-\frac{\beta}{\hbar}p_x\sigma_x
+V(y)+U(z)+\frac{g^*\mu_B}{2}\boldsymbol{\sigma B}, \label{ham0}$$ and $$H_1= -\frac{e}{m^*}\mathbf{A}\mathbf{p}=-\frac{eE_y
p_y}{m^*\omega}\cos(\omega t ). \label{ham1}$$ Here, $H_0$ is the time-independent part of the Hamiltonian, $H_1$ describes the interaction with the electro-magnetic field, $\mathbf{p}$ is the momentum of the electron, $m^*$ is the effective mass, $V(y)$ is the lateral confinement potential due to the gates, $U(z)$ is the confinement potential in $z$-direction, $\mu_B$ and $g^*$ are the Bohr magneton and effective g-factor, $E_y$ and $\omega$ are the amplitude and frequency of electric field of the polarized electro-magnetic wave, and $\boldsymbol{\sigma}$ is the vector of the Pauli matrices. The effect of the external field $\mathbf{B}$ on the spatial motion is neglected, assuming strong confinement in the $z$-direction. The second and third terms in (\[ham\]) represent the Rashba and Dresselhaus spin-orbit interaction [@rashba; @dressel] for an electron moving in the $x$-direction, $\alpha$ and $\beta$ are the corresponding coupling constants.
The spin-orbit interactions included into the Hamiltonian (\[ham0\]) originate from bulk inversion asymmetry (giving rise to a Dresselhaus interaction [@dressel]) and structural inversion asymmetry (giving rise to a Rashba interaction [@rashba]). It is well known that the spin-orbit interaction constants are different for electrons in different transverse subbands [@sipe; @pramanik]. In our model we assume that the Rashba spin-orbit interaction constant $\alpha$ depends on the index $m$ and the Dresselhaus spin-orbit interaction constant $\beta$ depends both on $n$ and $m$, where $n=0,1,...$ and $m=0,1,..$ are subband indices due to confinement in $y$ and $z$ directions respectively. [@sipe; @pramanik] In the model of rigid quantum wire walls $\beta=\gamma\left( \left(\pi n /
W_z\right)^2-\left(\pi m / W_y\right)^2\right)$, where $\gamma$ is a constant.
![(Color online) Dispersion relations (two lowest spin-splitted subbands) calculated for (a) $B_x=2$T, $B_y=2$T and (b) $B_x=2$T, $B_y=-2$T. These plots were obtained using the parameters values: $m^*=0.067m_e$, $g^*=-0.44$, $\alpha_0=0.5\times10^{-11}$eV m, $\beta_{0,0}=0.84\times10^{-11}$eV m, $\beta_{1,0}=0.36\times10^{-11}$eV m, $\epsilon_1-\epsilon_0=
3.64$meV.[]{data-label="fig2"}](fig2a "fig:"){width="8.5cm"} ![(Color online) Dispersion relations (two lowest spin-splitted subbands) calculated for (a) $B_x=2$T, $B_y=2$T and (b) $B_x=2$T, $B_y=-2$T. These plots were obtained using the parameters values: $m^*=0.067m_e$, $g^*=-0.44$, $\alpha_0=0.5\times10^{-11}$eV m, $\beta_{0,0}=0.84\times10^{-11}$eV m, $\beta_{1,0}=0.36\times10^{-11}$eV m, $\epsilon_1-\epsilon_0=
3.64$meV.[]{data-label="fig2"}](fig2b "fig:"){width="8.5cm"}
At $E_y = 0$, the solutions of the Schrödinger equation can be written in the form
$$\Psi_{m,n,\pm}(k)=\frac{e^{ikx}}{\sqrt{2}}\binom{\pm
e^{i\varphi_{n,m}}}{1}\phi_m(y)\eta_n(z), \label{wf}$$
where $$\begin{aligned}
\varphi_{n,m}=\pi\theta(\beta_{n,m} k-\frac{g^* \mu_B}{2} B_x)+
\nonumber \\ \textnormal{arctan}\left[-\frac{-\alpha_n k+\frac{g^*
\mu_B}{2} B_y}{-\beta_{m,n} k+\frac{g^* \mu_B}{2} B_x}\right],
\label{phi}\end{aligned}$$ $\theta(..)$ is the step function, $\phi_m(y)$ and $\eta_n(z)$ are the wave function of the transverse modes (due to the confinement potentials $V(y)$ and $U(z)$). The eigenvalue problem can be solved to obtain
$$\begin{aligned}
E_{m,n,\pm}(k)=\frac{\hbar^2 k^2}{2m^*}+\epsilon_m+E_n\pm
\nonumber
\\ \sqrt{\left(-\alpha_n k+\frac{g^* \mu_B}{2}
B_y\right)^2+\left(-\beta_{n,m} k+\frac{g^* \mu_B}{2}
B_x\right)^2}. \label{spectr}\end{aligned}$$
In this expression, $\epsilon_m$ and $E_n$ are the eigenvalues of decoupled Scrödinger equations in $y$ and $z$ directions. In the experimental setup depicted in Fig. \[fig1\] the confinement in $z$-direction is stronger than the confinement in $y$-direction, thus we will consider $E_1-E_0\gg\epsilon_1-\epsilon_0$.
In what follows we assume that in the absence of radiation the chemical potential in the wire is located between the ground ($0,0$) and first ($1,0$) transverse subbands so that only the ground subband is occupied by electrons, and we focus our attention on radiation-induced transitions between these two transverse subbands. The energy spectrum in Eq. (\[spectr\]) is illustrated in Fig. \[fig2\] for various directions of $\boldsymbol{B}$. Notice that the energy dispersion is different (not simply shifted) for the ground and the first transverse subbands. Fig. \[fig2\] shows that the energy spectrum is strongly asymmetric and significantly dependent on the magnetic field direction. We also note that the gaps between spin-splitted subbands are due to the magnetic field.
The electron velocity is defined by the slope of $E_{m,n,\pm}(k)$ and is given by
$$\begin{aligned}
v_{n,m,\pm}(k)=\frac{1}{\hbar}\frac{\partial
E_{n,m,\pm}(k)}{\partial k}=\frac{\hbar k}{m^*}\pm \nonumber \\
\frac{\alpha_n \left(\alpha_n k-\frac{g^* \mu_B}{2}
B_y\right)+\beta_{n,m} \left(\beta_{n,m} k-\frac{g^* \mu_B}{2}
B_x\right)}{\hbar \sqrt{\left(-\alpha_n k+\frac{g^* \mu_B}{2}
B_y\right)^2+\left(-\beta_{n,m} k+\frac{g^* \mu_B}{2}
B_x\right)^2}}. \label{velocity}\end{aligned}$$
We emphasize that the direction of velocity changes in local extrema points of the spectrum. The radiation-induced transitions that we will consider conserve $k$. Since the positions of local extrema are different in the ground ($0,0$) and first ($1,0$) transverse subbands, transitions reversing the velocity direction are possible. The spectrum asymmetry results in asymmetric transition rates and as a result in a finite current at zero bias voltage.
Wigner functions {#sec3}
================
Interaction Hamiltonian
-----------------------
Assuming a parabolic confinement in $y$ direction we write the ground and the first excited wave functions of the corresponding transverse mode explicitly as $$\begin{aligned}
\label{harmonic}
\phi_0 (y) &=& \left( \frac{2}{\pi\bar y^2 } \right)^{ - 1/4} \exp
(\frac{-y^2}{\bar y^2}),\nonumber\\
\phi_1 (y) &=&2 \left( \frac{2}{\pi\bar y^6 } \right)^{ - 1/4} y
\exp (\frac{-y^2}{\bar y^2}),\end{aligned}$$ where $\bar y$ is the characteristic width of the quantum wire in the $y$ direction. The energy gap between the ground and the first excited states can be estimated as $\epsilon_1-\epsilon_0=(2\hbar^2)/(m^* {\bar y}^2)$. Taking the form of the solution for the Schrödinger equation (\[phi\]) and (\[harmonic\]) we obtain the matrix form of the interaction Hamiltonian (\[ham1\]) $$\label{ham1t}
H_1=i\hbar g \left( {e^{i\omega t} + e^{ - i\omega t} }
\right)\left( {\sigma ^ + s^{01} - h.c. } \right),$$ where $g=(\hbar eE_y) /(2m^* \bar y \omega)$ is the coupling constant, $\sigma^{\pm}$ is the ladder operator acting on the $y$ component of the wave function: $\sigma^{\pm}\phi_{m}(y)=\phi_{m\pm1}(y)$ and $s^{01}$ is an operator in the space of spin degree of freedom, $$\label{s12}
s^{01}=\left(\begin{array}{cc}
s_+ & s_- \\
s_- & s_+ \\
\end{array}\right).$$ Here $s_{\pm}=\left(1/2\right)\left(1 \pm \exp(i\Delta
\varphi_n)\right)$ and $\Delta
\varphi_n=\varphi_{n,1}-\varphi_{n,0}$, and $\varphi_{n,m}$ is defined in Eq. (\[phi\]). Neglecting high-oscillatory terms we can write the Hamiltonian (\[ham1t\]) in the rotating wave approximation [@Louisell] as $$\label{ham1tt}
H_1=i\hbar g \left( e^{-i\omega t}\sigma ^ + s^{01} - h.c.\right).$$
Equations for the Wigner Functions
----------------------------------
The Liouville-von Neumann equation for the density operator of the electron $\rho_{mm',ss'}(x,x',t)$ is given by $$\label{Liouville}
i\hbar \dot \rho=[H,\rho],$$ where $s,s'=\pm$ are variables associated with spin degree of freedom. Henceforth we omit the indices $n,n'$ due to confinement in $z$ direction since we are interested only in the transitions between the ground $m=0$ and $m=1$ transverse $y$ subbands. The Wigner function can be obtained by integrating [@Wigner] $$\label{Wignerf}
W_{mm',ss'}(R,k,t)=\int \rho_{mm',ss'}(R,\Delta r,t) \exp(-i k \Delta
r) d\Delta r,$$ where the density operator is written in the new spatial variables $R=(r+r')/2$ and $\Delta r=r-r'$ and $k$ is the electron wave vector.
Following the standard procedure (see, e.g., [@Saikin]) we derive a set of transport equations by neglecting non-local correlations in the diagonal components of the Wigner functions
$$\begin{aligned}
\label{Wignersystemdiag}
\dot W_{11,++} &+& v_{1,+} \frac{{\partial W_{11,++} }}{{\partial
x}} + \frac{{eE_x}}{\hbar }\frac{{\partial W_{11, ++ } }}{{\partial
k}} = g\left[ \left( s_+ W_{21, ++ } + c.c. \right) + \left( s_-
W_{21, -+ } + c.c. \right) \right], \\ \dot W_{11, - - } &+& v_{1,
- } \frac{{\partial W_{11, - - } }}{{\partial x}} +
\frac{{eE_x}}{\hbar }\frac{{\partial W_{11, - - } }}{{\partial k}}
= g\left[ \left( s_ - W_{21, + - } + c.c. \right) + \left(
s_ + W_{21, - - } + c.c. \right) \right],\\
\dot W_{22,--}&+&v_{2,-} \frac{\partial W_{22,--} }{\partial x} + \frac{eE_x}{\hbar
}\frac{\partial W_{22,--} }{\partial k} = -g\left[ \left( s_ -
W_{21, -+} + c.c. \right) + \left( s_ + W_{21,-- } + c.c.
\right) \right],\\ \dot W_{22,++}&+& v_{2,+} \frac{\partial
W_{22,++} }{\partial x} + \frac{eE_x}{\hbar }\frac{\partial
W_{22,++}}{\partial k} = -g\left[ \left( s_ + W_{21, ++} + c.c.
\right) + \left( s_ - W_{21,+- } + c.c. \right) \right].\end{aligned}$$
For the amplitudes of the off-diagonal components we define $W'_{mm',ss'}(t)=W_{mm',ss'}(t)\exp(-i\omega t)$, and we obtain
$$\begin{aligned}
\dot W'_{12, ++} + \frac{v_{1,+}+v_{2,+}}{2} \frac{\partial
W'_{12, + + } }{\partial x} + \frac{{eE_x}}{\hbar
}\frac{{\partial W'_{12, + + } }}{{\partial k}} &-& i(\omega _{12,
+ + } - \omega )W'_{12, + + }\\
& =& g\left[ {s_ + \left( {W_{22, + + } - W_{11, + + } } \right) + s_ - \left( {W_{22, - + } - W_{11, + - } } \right)} \right],
\nonumber\\
\dot W'_{12, + - } + \frac{v_{1,+}+v_{2,-}}{2} \frac{{\partial
W'_{12, + - } }}{{\partial x}} + \frac{{eE_x}}{\hbar
}\frac{{\partial W'_{12, + - } }}{{\partial k}} &-& i(\omega _{12,
+ - } - \omega )W'_{12, + -}\\ &=& g\left[ {s_ + \left(
{W'_{22, + - } - W'_{11, + - } } \right) + s_ - \left( {W_{22,
- - } - W_{11, + + } } \right)} \right], \nonumber\\ \dot
W'_{12, - - } + \frac{v_{1,-}+v_{2,-}}{2} \frac{{\partial W'_{12,
- - } }}{{\partial x}} + \frac{{eE_x}}{\hbar }\frac{{\partial
W_{12, - - } }}{{\partial k}} &-& i(\omega _{12, - - } - \omega
)W_{12, - -}\\ &=& g\left[ {s_ + \left( {W_{22, - - } - W_{11,
- - } } \right) + s_ - \left( {W'_{22, + - } - W'_{11, - + } }
\right)} \right], \nonumber\\
\dot W'_{12, - + } + \frac{v_{1,-}+v_{2,+}}{2} \frac{{\partial W'_{12, - + } }}{{\partial x}} + \frac{{eE_x}}{\hbar }\frac{{\partial W'_{12, - + } }}{{\partial k}} &-& i(\omega _{12, - + } - \omega )W'_{12, - +}\\
&=& g\left[ {s_ + \left( {W'_{22, - + } - W'_{11, - + } } \right) + s_ - \left( {W_{22, + + } - W_{11, - - } } \right)} \right], \nonumber\\
\dot W'_{11, - + } + \frac{v_{1,-}+v_{1,+}}{2} \frac{{\partial W'_{11, - + } }}{{\partial x}} + \frac{{eE_x}}{\hbar }\frac{{\partial W'_{11, - + } }}{{\partial k}} &-& i(\omega _{11, - + } - \omega )W'_{11, - + }\\
&=& g\left[ {\left( {s_ - W'_{21, + + } + s_ + W'_{21, - + } } \right) + \left( {s_ + ^* W'_{12, - + } + s_ - ^* W'_{12, - - } } \right)} \right], \nonumber\\
\dot W'_{22, - + } + \frac{v_{2,-}+v_{2,+}}{2} \frac{{\partial W'_{22, - + } }}{{\partial x}} + \frac{{eE_x}}{\hbar }\frac{{\partial W'_{22, - + } }}{{\partial k}} &-& i(\omega _{22, - + } - \omega )W'_{22, - + }
\label{Wignersystemoffdiag}\\
&=& - g\left[ {\left( {s_ - ^* W'_{12, + + } + s_ + ^* W'_{12, - + } } \right) + \left( {s_ + W'_{21, - + } + s_ - W'_{21, - - } } \right)}
\right].
\nonumber\end{aligned}$$
For completeness we also included a static electric field $E_x$ along the $x$ direction, for example, due to a bias voltage. $\omega_{mm',ss'}=\hbar^{-1}(E_{m',s'}-E_{m,s})$. The left part of the equations (\[Wignersystemdiag\]-\[Wignersystemoffdiag\]) describes the ballistic transport of the electron in the quantum wire, the right part is responsible for excitations induced by the radiation. We consider the “ideal” contact boundary conditions for a wire of length $L$ $$\label{boundary1}
W_{mm,ss} (0,k)|_{v_{m,s}(k)>0}=f(k,\mu_l),$$ $$\label{boundary2}
W_{mm,ss} (L,k)|_{v_{m,s}(k)<0}=f(k,\mu_r).$$ where $f(k,\mu)=1/(1+\exp[(E_{m,s}(k)-\mu)/(k_B T)])$ is the Fermi function, $k_B$ is the Boltzmann constant, $T$ is the electron temperature and $\mu_{l/r}$ are chemical potentials of the left/right lead, respectively. We also assume that only internal part of the quantum wire is irradiated, that is $E_y=0$ for $x<0$ and $x>L$.
The electron charge density, electric charge and spin currents can be obtained from Wigner functions as $$\label{density}
n(x)= \frac{e}{2\pi} \sum_{m,s}\int_{-\infty}^{\infty} W_{mm,ss}(x,k)
dk,$$ $$\label{current}
I(x)= \frac{e}{2\pi} \sum_{m,s}\int_{-\infty}^{\infty}v_{m,s}(k) W_{mm,ss}(x,k)
dk,$$ and $$\begin{aligned}
\label{spincurrent}
I_{\gamma}^{s}(x)= \frac{1}{2\pi} \sum_{m,s}\int_{-\infty}^{\infty}
\langle \Psi_{m,s}| \sigma_\gamma | \Psi_{m,s}\rangle v_{m,s}(k) \nonumber \\W_{mm,ss}(x,k)
dk,\end{aligned}$$ respectively. Here $\gamma=(x,y,z)$ and different matrix elements can be found in accordance with (\[phi\]) as $\langle
\Psi_{m,\pm}| \sigma_x | \Psi_{m,\pm}\rangle=\pm \cos(\varphi_m)$, $\langle \Psi_{m,\pm}| \sigma_y | \Psi_{m,\pm}\rangle=\mp \sin
(\varphi_m)$. The details of the numerical model and solution for the derived system of equations are given in the next sections.
Discrete Model {#discrete}
==============
The form of Eqs.(\[Wignersystemdiag\]-\[Wignersystemoffdiag\]) does not allow us to solve the problem analytically even for the stationary case $\partial W/\partial t=0$ and unbiased channel $E_x=0$. The solution is complicated by different inter-subband transitions with the change of spin state $s$ originating from the subband asymmetry in $k$ domain. This effect plays a central role in the electric current generation and should be taken into account. Thus, the system of equations (\[Wignersystemdiag\]-\[Wignersystemoffdiag\]) was solved numerically for $\bar y=25~\rm{nm}$, $\mu=0.0001~\rm{eV}$ and $T=0.1K$. We model the domain $x\in [0,L]$ and $k\in[-k_{max},
k_{max}]$ with the mesh sizes of $\Delta x=L/(N_x-1)$ and $\Delta
k=2 k_{max}/(N_k-1)$, respectively. In the calculation we used $N_x=25$ and $N_k=80$ and the length of the quantum wire $L=2.5\;\mu\rm{m}$. The value $k_{max}=3.37\cdot10^7 \rm{m^{-1}}$ was chosen to ensure that all filled states in $k$ space are taken into account. We fixed the values of diagonal components $W_{mm,ss}$ on the boundaries at $x=0$ for $k$ with $v_{m,s}(k)>0$ and at $x=L$ for $k$ with $v_{m,s}(k)<0$, accordingly to (\[boundary1\],\[boundary2\]). Similarly, for off-diagonal components we fixed the values $W_{m\neq m',s\neq s'}=0$ at $x=0$ for $k$ with $(v_{m,s}(k)+v_{m',s'}(k))>0$ and at $x=L$ for $k$ with $(v_{m,s}(k)+v_{m',s'}(k))<0$."
The first order upwind difference scheme was used for the propagation of the Wigner functions in $\{x,k\}$ domain and second order two-step Lax-Wendroff scheme [@Anderson] to describe the time-dependent inter-subband transfer due to due to interaction with the electro-magnetic field. The discretized Liouville equation for the Wigner function can be written as
$$\begin{aligned}
\label{discretLiouvill}
&& W(x_i,k_j,t_{l+1/2})=W(x_i,k_j,t_{l})-\frac{\Delta
t}{2}\left[v(k_j)\frac{\Delta W(x_i,k_j,t_l)}{\Delta
x}-F(W(x_i,k_j,t_l))\right],\\
&&W(x_i,k_j,t_{l+1})=W(x_i,k_j,t_{l})-\Delta t\left[v(k_j)\frac{\Delta W(x_i,k_j,t_l)}{\Delta
x}-F(W(x_i,k_j,t_{l+1/2}))\right]\nonumber
.\nonumber\end{aligned}$$
where we consider the case $E_x=0$ and do not show subband indices for brevity. The upwind/downwind difference is chosen in accordance with $$\label{difference}
\Delta W(x_i,k_j,t_l)=\left\{
\begin{array}{c}
W(x_i,k_j,t_l)- W(x_{i-1},k_j,t_l) \\
\\
{\rm if} \; v(k_j)>0, \\
\\
W(x_{i+1},k_j,t_l)- W(x_i,k_j,t_l)\\
\\
{\rm if} \; v(k_j)<0.\\
\end{array}
\right.$$ The function $F(W(x_i,k_j,t_l))$ embodies the remaining part of Eqs. (\[Wignersystemdiag\]-\[Wignersystemoffdiag\]) which depends only on the Wigner function $W(x_i,k_j,t_l)$ itself and does not contain partial derivatives. The upwind differencing is stable [@Anderson] provided the time step is small enough: $\Delta x/\Delta t\leq v_{max}$, where $v_{max}$ is the maximum possible absolute value of the velocity. Additionally, the time step must be much smaller than the highest frequency of the solution. This condition is satisfied by $\Omega_R^{max} \Delta
t\ll1$ where $\Omega_R^{max}=\max\left\{\sqrt
{(\omega_{mm',ss'}-\omega)^2+4g^2}\right\}$ is the maximum possible Rabi frequency involved in the problem. The calculation proved to be stable if these two conditions are met.
![(Color online) The different subband transition due to the interaction with electro-magnetic wave are sketched. The interaction causes some electrons to change the direction of the velocity.[]{data-label="fly"}](fly){width="8.5cm"}
The investigation of the effects of the external bias and charge redistribution in the quantum wire is not in the scope of this paper. However, the way to include these effects is straightforward. To consider the effect of voltage applied to the quantum wire it suffices to add the $x$ component of the electric field $E_x(x,V(t))$ in the discrete model (\[discretLiouvill\]). This electric field is a function of applied potential difference $V(t)=(\mu_r-\mu_l)/e $ which can be time-dependent and $x$-dependent. The latter is defined by the leads geometry. Additional components of electric field $E_x^q(n(x))$ can be calculated self -consistently at each time step to incorporate the effect of charge redistribution.
As was mentioned earlier, transitions between subbands can force the electron to change the direction of the velocity. As a result the flows of electrons moving in the opposite directions inside the quantum wire intermix as shown in Fig. \[fly\]. Without reflections in the wire, the steady-state solution can be obtained by simply imposing the condition $\partial W/\partial t=0$ and advancing from the given values at left/right boundaries for $v(k)\gtrless k$. However, in the presence of radiation, the electron distribution at the boundary acquires an additional component, due to the electrons whose velocity direction is changed, and the described above scheme fails. In order to achieve the steady-state solution we considered the temporal dynamics of the system evolving from some initial state until it reached stationary conditions: $\partial W/\partial t\backsimeq0$ and $I(x)\backsimeq I(x')$ for all $x,x'\in [0,L]$. As a initial state we took the values of Wigner function at equilibrium $W_{mm',ss'}
(x,k)=\delta_{mm'}\delta_{ss'}f(k,\mu)$. This corresponds to the uniformly distributed electron density along the channel and is a solution of Eqs. (\[Wignersystemdiag\]-\[Wignersystemoffdiag\]) in the absence of radiation and external bias: $E_x=E_y=0$. The chosen method of obtaining the steady-state solution provides us also with the transient behavior and, thus, gives more insight into the problem. The drawback is a serious computational effort. The electron distribution reaches the stationary state within the effective time of flight through the quantum wire. Electrons constantly change the direction of velocity due to the interaction with electro-magnetic wave, therefore this time can be very long compared to the time step $\Delta t$. Fortunately, the slower electrons give the smaller contribution to the current and a steady-state solution can be always found within certain accuracy. The number of steps in time can reach values as much as $N_t\sim10^5$. The results of the numerical simulation are presented in the following section \[sec4\].
Results and Discussion {#sec4}
======================
![(Color online) Current through the wire as a function of photon energy $\hbar \omega$ for (a) $B_x=2$T, $B_y=2$T and (b) $B_x=2$T, $B_y=-2$T. The parameters values are as in Fig. \[fig2\], $E_y=200$V/m.[]{data-label="fig3"}](fig3a "fig:"){width="8.5cm"} ![(Color online) Current through the wire as a function of photon energy $\hbar \omega$ for (a) $B_x=2$T, $B_y=2$T and (b) $B_x=2$T, $B_y=-2$T. The parameters values are as in Fig. \[fig2\], $E_y=200$V/m.[]{data-label="fig3"}](fig3b "fig:"){width="8.5cm"}
The photo-induced current through the wire is shown in Fig. \[fig3\] as a function of the photon energy. The directions and strength of the magnetic field in this graph are the same as in Fig. \[fig2\]. The amplitude of the electric field $E_y=200$V/m used in our calculations was selected close to the electric field amplitude used in recent experiments [@mani]. Fig. \[fig3\] clearly shows a number of current peaks corresponding to different transitions. These peaks depend on the magnetic field strength and direction, as a consequence of the magnetic field dependence of the energy spectrum.
![(Color online) Transitions in the regions of photocurrent peaks for $B_x=2$T, $B_y=-2$T. The energy spectrum and difference energies are shown in the top and bottom panels, respectively. Horizontal lines in the bottom panel correspond to excitation energies in peak regions, vertical arrows in the top panel denote transitions when electron velocity direction changes.[]{data-label="fig4"}](fig4){width="8.5cm"}
In order to understand transitions leading to a specific peak formation, we consider in detail the current dependence on photon energy for $B_x=2$T, $B_y=-2$T (Fig. \[fig3\](b)). It follows from Fig. \[fig3\](b) that the current as a function of photon energy has a well pronounced positive peak at $\hbar \omega \simeq
3.64$meV, a double negative peak with a minimum at $\hbar
\omega\simeq 3.67$meV, a small negative peak at $\hbar
\omega\simeq 3.565$meV, and a broad positive peak of small amplitude at $\hbar \omega\simeq 3.75$meV. Fig. \[fig4\] represents a graphical determination of relevant transitions.
The energy spectrum of two lowest spin-splitted subbands is plotted in the top panel of Fig. \[fig4\]. The bottom panel of Fig. \[fig4\] shows the energy difference between different transverse subbands. By plotting horizontal lines corresponding to the peak energies, in the bottom panel and, by drawing vertical lines through the intersection points of those horizontal lines with energy difference, we can finallyidentify the points in the top panel corresponding to the peak formation. As it was mentioned above, the important transitions are those that lead to a change of the electron velocity direction. These transitions are shown by arrows in the top panel of Fig. \[fig4\].
In particular, let us consider the large negative peak in the current at $\hbar \omega\simeq 3.67$meV (Fig. \[fig3\](b)). Fig. \[fig4\] shows that the horizontal line $3.67$meV intersects only $E_{1,0,-}-E_{0,0,-}$ curve in two points. We remind that the electron velocity is determined by the slope of $E_{m,n,\pm}(k)$ according to Eq. (\[velocity\]). The left intersection point gives $k$-vector of transition in which the electron velocity direction is conserved), because the slopes of $E_{1,0,-}$ and $E_{0,0,-}$ at this value of $k$ are in the same direction. The right intersection point of $3.67$meV line with $E_{1,0,-}-E_{0,0,-}$ curve gives a transition with a change of electron velocity direction, specifically, with a back-scattering of left-moving electrons. Consequently, the electron flux from the right to the left decreases and, because of the negative electron charge, a negative current appears.
----------------------------------------------------------------------------------
Excitation energy, meV Transitions
------------------------ ---------------------------------------------------------
3.565 $E_{0,0,+}\rightarrow E_{1,0,-}$
3.64 $E_{0,0,-}\rightarrow E_{1,0,-}$; $E_{0,0,+}\rightarrow
E_{1,0,+}$
3.67 $E_{0,0,-}\rightarrow E_{1,0,-}$
3.75 $E_{0,0,-} \rightarrow
E_{1,0,+}$
----------------------------------------------------------------------------------
: Transitions giving contribution to photocurrent at selected radiation frequencies for $B_x=2$T, $B_y=-2$T.[]{data-label="table1"}
Similarly, one can consider transitions at other radiation frequencies. An interesting situation occurs for $\hbar \omega \simeq
3.64$meV excitation, since at this particular frequency three out of four transitions are characterized by the reverse of electron velocity direction. We summarize transitions contributing to the photocurrent at selected radiation frequencies in Table \[table1\]. The same analysis can also be applied to the result presented in Fig. \[fig3\](a), but, because of a more distorted spectrum, the roles of different possible transitions are more difficult to interpret.. Moreover, we would like to note that, generally, transition probabilities from $E_{0,0,-}$ to $E_{1,0,+}$ and from $E_{0,0,+}$ to $E_{1,0,-}$ subbands are smaller than transition probabilities from $E_{0,0,-}$ to $E_{1,0,-}$ and from $E_{0,0,+}$ to $E_{1,0,+}$ subbands because of the different spin direction in initial and final states. This results in a smaller current peaks at $\hbar \omega\simeq 3.565$meV and $\hbar \omega\simeq 3.75$meV in Fig. \[fig3\](b).
![(Color online) Spin current components for $B_x=2$T, $B_y=-2$T at $x=0.$[]{data-label="fig5"}](fig5){width="8.5cm"}
As electrons carry spin as well as charge, the external radiation also changes the spin current through the wire. Notice that even without the radiation, the spin current is not zero, due to the spin-orbit interaction. Fig. \[fig5\] shows spin current components at $x=0$ for $B_x=2$T, $B_y=-2$T. We note that the spin current dependence on the radiation frequency has features similar to the charge current (Fig. \[fig3\](b)). However, we found that radiation-induced changes in spin current are much less than the equilibrium spin current in the wire. From an experimental point of view, the spin currents are not so easy detectable. Therefore, the observation of this photovoltaic effect through spin current seems unpractical.
Conclusions {#sec5}
===========
In summary, in this paper we have investigated the photovoltaic effect in quantum wires with spin-orbit interaction and in-plane magnetic field. We have found that the peculiarities of the energy spectrum lead to a photocurrent generation. The dependence of the photoinduced current on the excitation frequency was calculated numerically using the Wigner functions formalism. A system of coupled equations for the Wigner functions was derived and solved numerically for “ideal” contact boundary conditions. We used the first order upwind differencing for the propagation in the spatial domain and the second order two-step Lax-Wendroff differencing for time-dependent inter-subband transitions due to electro-magnetic wave excitation. Stable numerical solutions were found under appropriate choices of the time-step $\Delta t$. The calculations can be extended to introduce the effects of an external bias and self-consistent potentials due to charge density redistributions which can be a topic for a future investigation. The frequency dependence of the photoinduced current consists of a set of peaks related to transitions between different points of the spectrum. Therefore, the energy spectrum can be reconstructed from photocurrent measurements. Material parameters, such as spin-orbit coupling constants, can be obtained from the analysis of the photocurrent.
Acknowledgments {#acknowledgments .unnumbered}
===============
We gratefully acknowledge useful discussions with M. Cheng and L. Fedichkin. This research was supported by the National Science Foundation, Grants DMR-0121146 and DMR-0312491.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The mass shifts of the $P$-wave $D_s$ and $B_s$ mesons due to coupling to $DK$ and $BK$ channels are calculated in the coupling channel model without fitting parameters. The strong mass shifts down for $0^+$ and ${1^+}''$ states have been obtained, while ${1^+}''''$ and $2^+$ states remain almost in situ. The masses of $0^+$ and ${1^+}''$ states of $B_s$ mesons have been predicted.'
address: 'ITEP, Moscow, Russia'
author:
- 'A.M.Badalian, Yu.A.Simonov, M.A.Trusov'
title: 'Chiral shifts in heavy-light mesons'
---
After the experimental discovery of the $D_s(2317)$ and $D_s(2460)$ mesons [@1-2], a necessity to study the chiral dynamics in heavy-light mesons became quite clear. The masses of these states proved to be much lower than expected values in ordinary quark models while their widths were surprisingly small. The problem was studied in different approaches: in relativistic quark model calculations [@3I-4I]–[@6I], on the lattice [@7I], in QCD Sum Rules [@8I; @9I], in chiral models [@10I; @11I-12I] (for reviews see also [@13I; @14I]). The masses of $D_s(0^+)$ and $D_s(1^{+'})$ in closed-channel approximation typically exceed by $\sim$ 140 and 90 MeV their experimental numbers. The main theoretical goal seems for us to understand dynamical mechanism responsible for such large mass shifts of the $0^+$ and $1^{+'}$ levels and explain why the position of other two levels remains practically unchanged. The importance of second fact has been underlined by S.Godfrey in [@5I].
The mass shifts of the $D_s(0^+,1^{+'})$ mesons have already been considered in a number of papers with the use of unitarized coupled-channel model [@15I], in nonrelativistic Cornell model [@16I], in semi-relativistic model with inverse heavy quark mass expansion [@Matsuki], and in different chiral models [@17I]–[@19I]. Here we address again this problem with the aim to calculate also the mass shifts of the $D_s(1^{+'})$ and $B_s(0^+,1^{+'})$ states and the widths of the $2^+$ and $1^+$ states, following the approach developed in [@18I], for which strong coupling to the S-wave decay channel, containing a pseudoscalar ($P$) Nambu-Goldstone (NG) meson, is crucially important. Therefore in this approach principal difference exists between vector-vector ($VV$) and $VP$ (or $PP$) channels. This analysis of two-channel system is performed with the use of the chiral quark-pion Lagrangian which has been derived directly from the QCD Lagrangian [@20I] in the frame of the Field Correlator Method (FCM) and does not contain fitting parameters, so that the shift of the $D^*_s(0^+)$ state $\sim$ 140 MeV is only determined by the conventional decay constant $f_K$.
From the common point of view, due to spin-orbit and tensor interactions the $P$-wave multiplet of a HL meson is split into four levels with $J^P =0^+, 1^+_L, 1^+_H, 2^+$ [@29I]. Here we use the notation H(L) for the higher (lower) $1^+$ state because a priori one cannot say which of them mostly consists of the light quark $j=1/2$ contribution. In fact, starting with the Dirac’s $P$-wave levels, one has the states with $j=1/2$ and $j=3/2$. And $1^+_{L,H}$ eigenstates can be parameterized by introducing the mixing angle $\phi$: $$\begin{gathered} |1^+_H{\rangle}= \cos \phi |j=\frac12{\rangle}+ \sin
\phi|j=\frac32{\rangle},\\ |1^+_L{\rangle}= -\sin\phi |j=\frac12{\rangle}+
\cos\phi |j=\frac32{\rangle}.
\end{gathered}\label{5-6}$$ In the heavy-quark (HQ) limit the states with $j=\frac32 $ and $j=\frac12$ are not mixed, but for finite $m_Q$ they can be mixed even in closed-channel approximation [@10I; @29I].
Taking the meson emission to the lowest order, one obtains the effective quark-pion Lagrangian in the form $$\Delta L_{FCM} =- \int \psi_i^+ (x) \sigma
|{\mbox{\boldmath${\rm x}$}}|\gamma_5 \frac{\varphi_a\lambda_a}{f_\pi}
\psi_k(x)d^4x.\label{17}$$ Writing the equation (\[17\]) as $\Delta L_{FCM}=- \int V_{if} dt$, one obtains the operator matrix element for the transition from the light quark state $i$ (i.e. the initial state $i$ of a HL meson) to the continuum state $f$ with the emission of a NG meson $(\varphi_a\lambda_a)$. Thus we are now able to write the coupled channel equations, connecting any state of a HL meson to a decay channel which contains another HL meson plus a NG meson.
Consider a complete set of the states $|f{\rangle}$ in the decay channel 2 and the set of unperturbed states $|i{\rangle}$ in channel 1. One arrives at the nonlinear equation for the shifted mass: $$m[i]=m^{(0)}[i]-\sum\limits_f \dfrac{|<i|\Hat V|f>|^2}{E_f-m[i]},
\label{21}$$ where $m^{(0)}[i]$ is the initial mass, calculated in the single-channel approximation (assumed to be known), $m[i]$ – is the final one, $E_f$ is the energy of the final state, and the operator $\hat V$ provides the transitions between the channels. Note, that in our approximation we do not take into account the final state interaction in the $DK$ system and neglect the $D$-meson motion. Also, in the w.f. we neglect possible (very small) mixing between the $D(1^-_{1/2})$, $D(1^-_{3/2})$ states and between $D_s(2^+_{3/2})$, $D_s(2^+_{5/2})$ states; physical $D_s(1^+)$ states can be mixed, though.
In subsequent analysis it is convenient to define the masses with respect to nearby threshold: $m_{\text{thr}}= m_K+m_D$. So, we introduce the following notations: $$E_0=m^{(0)}[D_s]-m_D-m_K,\quad \delta m=m[D_s]-m^{(0)}[D_s],$$ $$\Delta = E_0+\delta m=m[D_s]-m_D-m_K,$$ where $\Delta$ determines the deviation of the $D_s$ meson mass from the threshold, and can be complex if a decay to $DK$ pair is allowed. In what follows we consider unperturbed masses $m_0(J^P)$ of the ($Q\bar q$) levels as given (our results do not change if we slightly vary their position, in this way the analysis is actually model-independent).
For further calculations we should insert the explicit meson w.f. to the matrix element in (\[21\]). In our approximation for a HL meson we consider a light $q$ (or strange $s$) quark with current (pole) mass $m_{q,s}$ moving in the static field of a heavy antiquark $\bar Q$, and take its w.f. as a 4-spinor obeying the Dirac equation with the linear scalar potential and the vector Coulomb potential with frozen $\alpha_s=\text{const}$: $$U=\sigma r,\quad V_C=-\dfrac{\beta}{r},\quad
\beta=\dfrac{4}{3}\alpha_s.$$
Finally, after long cumbersome calculations which are omitted here, we arrive at the the following equations to determine meson masses and widths: $$\begin{array}{l} D_s(0^+): \quad
E_0[0^+]-\Delta=\tilde{\mathcal{F}}_0(\Delta),
\\[3mm]
D_s(1^+_L): \\
E_0[1^+_L]-\Delta=
\cos^2\phi\cdot\tilde{\mathcal{F}}_0(\Delta)+\sin^2\phi\cdot\tilde{\mathcal{F}}_2(\Delta),
\\[3mm] D_s(1^+_H):
\\
E_0[1^+_H]-\Delta=
\sin^2\phi\cdot\tilde{\mathcal{F}}_0(\Delta)+\cos^2\phi\cdot\tilde{\mathcal{F}}_2(\Delta),
\\ \vphantom{\bigg|}
\Gamma[1^+_H]=\sin^2\phi\cdot\tilde\Gamma_0(\Delta)
+\cos^2\phi\cdot\tilde\Gamma_2(\Delta),
\\[3mm] D_s(2^+_{3/2}):
\\
E_0[2^+_{3/2}]-\Delta= \dfrac{3}{5}\cdot\tilde{\mathcal{F}}_2(\Delta),
\quad \Gamma[2^+_{3/2}]=\dfrac{3}{5}\cdot\tilde\Gamma_2(\Delta),
\end{array}
\label{misha_table_4}$$ where ${\mathcal{F}}_{0,2}$ and $\Gamma_{0,2}$ are some universal functions; definition of those, together with calculation details, can be found in [@BST].
In our analysis the 4-component (Dirac) structure of the light quark w.f. is crucially important. Specifically, the emission of a NG meson is accompanied with the $\gamma_5$ factor which permutes higher and lower components of the Dirac bispinors. For the $j=1/2,P$ -wave and the $j=1/2,S$ -wave states it is exactly the case that this “permuted overlap” of the w.f. is maximal because the lower component of the first state is similar to the higher component of the second state and vice-versa, while for the analogous overlap between $j=3/2,P$ -wave and the $j=1/2,S$ -wave states the situation is opposite. In the end, it leads to the functions ${\mathcal{F}}_0$, $\Gamma_0$ being much larger than ${\mathcal{F}}_2$, $\Gamma_2$ for almost all reasonable values of $\Delta$. Thus the large shift of the ${1^+}'$ state with a concurrent small one for ${1^+}''$ state reveals a natural explanation (see below).
Now we turn directly to the mass computations. We will take into account the following pairs of mesons in coupled channels ($i$ refers to first (initial) channel, while $f$ refers to second (decay) one):
$$\begin{tabular}{cc} $i$ & $f$ \\ \hline \hline $D_s(0^+)$
& $D(0^-)+K(0^-)$ \\ \hline
$D_s(1^+)$ & $D^*(1^-)+K(0^-)$ \\ \hline $D_s(2^+)$ &
$D^*(1^-)+K(0^-)$ \\ \hline
\end{tabular}$$
and analogously for $B$-meson case, with corresponding masses and threshold values (in MeV): $$\begin{gathered}
m_{D^+}=1869, \quad m_{D^+}+m_{K^-}=2363, \\
m_{D^{*+}}=2010, \quad m_{D^{*+}}+m_{K^-}=2504,\\
m_{B^+}=5279, \quad m_{B^+}+m_{K^-}=5772, \\
m_{B^*}=5325, \quad m_{B^*}+m_{K^-}=5819. \\
\end{gathered}
$$
The light quark eigenfunction is calculated numerically via Dirac equation with the following set of parameters: (the same as in our previous papers [@dirac]): $$\begin{gathered} \sigma=0.18~\text{GeV}^2,\quad \alpha_s=0.39,
\\ m_s=210~\text{MeV},\quad m_q\sim 0~\text{MeV}, \end{gathered}$$ The choice of $\sigma$ and $\alpha_s$ is a common one in the frame of the FCM approach, and the value of the light quark mass really does not influence here on any physical results because of its smallness in comparison with the natural mass scale $\sqrt{\sigma}$. The strange quark mass is taken from [@ms], where it was found from the ratio of experimentally measured decay constants $f(D_s)/f(D)$; the same value can be obtained by a renormalization group evolution starting from the conventional value $m_s(\text{2~GeV}) = 90\pm 15\text{~GeV}$.
state $m^{(0)}$ $m^{\text{(theor)}}$ $m^{\text{(exp)}}$ $\delta m$
------------ ----------- ---------------------- -------------------- ------------
$D_s(0^+)$ 2475 (30) 2330(20) 2317 -145
$B_s(0^+)$ 5814(15) 5709 (15) [not seen]{} -105
: $D_s(0^+)$-meson mass shift due to the $DK$ decay channel and $B_s(0^+)$-meson mass shift due to the $BK$ decay channel (all in MeV)[]{data-label="misha_table_11"}
state $m^{(0)}$ $m^{\text{(theor)}}$ $m^{\text{(exp)}}$ $\Gamma^{\text{(theor)}}_{(D^*K)}$ $\Gamma^{\text{(exp)}}_{(D^*K)}$ $\delta m$
------------------ ----------- ---------------------- -------------------- ------------------------------------ ---------------------------------- ------------
$D_s(1^+_H)$ 2568(15) 2458(15) 2460 $\times$ $\times$ -110
$D_s(1^+_L)$ 2537 2535 2535(1) 1.1 $<1.3$ -2
$D_s(2^+_{3/2})$ 2575 2573 2573(2) 0.03 [not seen]{} -2
state $m^{(0)}$ $m^{\text{(theor)}}$ $m^{\text{(exp)}}$ $\Gamma^{\text{(theor)}}_{(B^*K)}$ $\Gamma^{\text{(exp)}}_{(B^*K)}$ $\delta m$
------------------ ----------- ---------------------- -------------------- ------------------------------------ ---------------------------------- ------------
$B_s(1^+_H)$ 5835(15) 5727(15) [not seen]{} $\times$ $\times$ -108
$B_s(1^+_L)$ 5830 5828 5829 (1) 0.8 $<2.3$ -2
$B_s(2^+_{3/2})$ 5840 5838 5839(1) $<10^{-3}$ [not seen]{} -2
The ultimate results of our calculations are presented in Tables \[misha\_table\_11\]–\[misha\_table\_13\]. A priori one cannot say whether the $|j=\frac12{\rangle}$ and $|j=\frac32{\rangle}$ states are mixed or not. If there is no mixing at all, in this case the width $\Gamma(D_{s1}(2536))= 0.3$ MeV is obtained in [@35], while the experimental limit is $\Gamma<2.3$ MeV [@26I] and recently in [@36] the width $\Gamma=1.0\pm 0.17$ MeV has been measured. Therefore small mixing is not excluded and here we take the mixing angle $\phi$ slightly deviated from $\phi=0^{\circ}$ ( no mixing case). Then we define those angles $\phi$ which are compatible with experimental data for the masses and widths of both $1^+$ states.
The large value $\cos^2\phi$ for the $1_{H}^+(j=1/2)$ state provides large mass shift ($\sim 100$ MeV) of this level and at the same time does not produce the mass shift of the $1^{+}_L$ level, which is almost pure $j=\frac32$ state. We would like to stress here that the mass shifts weakly differ for $D_s$ and $B_s$, or, in other words, weakly depend on the heavy quark mass.
Thus we have obtained the shifted masses $M(B_s,0^+)=5710(15)$ MeV and $M(B_s,1^{+'})=5730(15)$ MeV, which are in agreement with the predictions in [@14I] and of S.Narison [@9I] and by $\sim
100$ MeV lower than in [@3I-4I],[@10I]. The masses of the $2^+$ and $1^+$ states precisely agree with experiment.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors would like to acknowledge support from the President Grant No. 4961.2008.2 for scientific schools. One of the authors (M.A.T.) acknowledges partial support from the President Grant No. MK-2130.2008.2 and the RFBR for partial support via Grant No. 06-02-17120.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We are conducting COLD GASS, a legacy survey for molecular gas in nearby galaxies. Using the IRAM 30m telescope, we measure the CO(1-0) line in a sample of $\sim$ 350 nearby ($D_L\simeq100-200$ Mpc), massive galaxies ($\log(M_{\ast}/M_{\odot})>10.0$). The sample is selected purely according to stellar mass, and therefore provides an unbiased view of molecular gas in these systems. By combining the IRAM data with SDSS photometry and spectroscopy, GALEX imaging and high-quality Arecibo HI data, we investigate the partition of condensed baryons between stars, atomic gas and molecular gas in $0.1-10L^{\ast}$ galaxies. In this paper, we present CO luminosities and molecular hydrogen masses for the first [222]{} galaxies. The overall CO detection rate is 54%, but our survey also uncovers the existence of sharp thresholds in galaxy structural parameters such as stellar mass surface density and concentration index, below which all galaxies have a measurable cold gas component but above which the detection rate of the CO line drops suddenly. The mean molecular gas fraction $M_{H2}/M_{\ast}$ of the CO detections is $0.066\pm0.039$, and this fraction does not depend on stellar mass, but is a strong function of [NUV$-r$]{} colour. Through stacking, we set a firm upper limit of $M_{H2}/M_{\ast}=0.0016\pm0.0005$ for red galaxies with [NUV$-r$]{}$>5.0$. The average molecular-to-atomic hydrogen ratio in present-day galaxies is 0.3, with significant scatter from one galaxy to the next. The existence of strong detection thresholds in both the HI and CO lines suggests that “quenching” processes have occurred in these systems. Intriguingly, atomic gas strongly dominates in the minority of galaxies with significant cold gas that lie above these thresholds. This suggests that some re-accretion of gas may still be possible following the quenching event.'
author:
- |
Amélie Saintonge$^{1,2}$[^1], Guinevere Kauffmann$^{1}$, Carsten Kramer$^{3}$, Linda J. Tacconi$^{2}$, Christof Buchbender$^{3}$, Barbara Catinella$^{1}$, Silvia Fabello$^{1}$, Javier Graciá-Carpio$^{2}$, Jing Wang$^{1,4}$, Luca Cortese$^{5}$, Jian Fu$^{6,1}$, Reinhard Genzel$^{2}$, Riccardo Giovanelli$^{7}$, Qi Guo$^{8,9}$, Martha P. Haynes$^{7}$, Timothy M. Heckman$^{10}$, Mark R. Krumholz$^{11}$, Jenna Lemonias$^{12}$, Cheng Li$^{6,13}$, Sean Moran$^{10}$, Nemesio Rodriguez-Fernandez$^{14}$, David Schiminovich$^{12}$, Karl Schuster$^{14}$ and Albrecht Sievers$^{3}$\
$^{1}$Max-Planck Institut für Astrophysik, 85741 Garching, Germany\
$^{2}$Max-Planck Institut für extraterrestrische Physik, 85741 Garching, Germany\
$^{3}$Instituto Radioastronomía Milimétrica, Av. Divina Pastora 7, Nucleo Central, 18012 Granada, Spain\
$^{4}$Center for Astrophysics, University of Science and Technology of China, 230026 Hefei, China\
$^{5}$European Southern Observatory, Karl-Schwarzschild-Str. 2, 85748 Garching, Germany\
$^{6}$Key Laboratory for Research in Galaxies and Cosmology, Shanghai Astronomical Observatory, Chinese Academy of Sciences,\
Nandan Road 80, Shanghai 200030, China\
$^{7}$Center for Radiophysics and Space Research, Cornell University, Ithaca, NY 14853, USA\
$^{8}$National Astronomical Observatories, Chinese Academy of Sciences, Beijing 100012, China\
$^{9}$Institute for Computational Cosmology, Department of Physics, Durham University, South Road, Durham DH1 3LE, UK\
$^{10}$Johns Hopkins University, Baltimore, Maryland 21218, USA\
$^{11}$Department of Astronomy and Astrophysics, University of California, Santa Cruz, CA 95064, USA\
$^{12}$Department of Astronomy, Columbia University, New York, NY 10027, USA\
$^{13}$Max-Planck-Institut Partner Group, Shanghai Astronomical Observatory\
$^{14}$Institut de Radioastronomie Millimétrique, 300 Rue de la piscine, 38406 St Martin d’Hères, France
title: 'COLD GASS, an IRAM Legacy Survey of Molecular Gas in Massive Galaxies: I. Relations between H$_2$, HI, Stellar Content and Structural Properties'
---
galaxies: fundamental parameters – galaxies: evolution – galaxies: ISM – radio lines: galaxies – surveys
Introduction {#intro}
============
Perhaps the most fascinating aspect of nearby galaxies is the intricately interwoven system of correlations between their global properties. These correlations form the basis of the so-called “scaling laws", which are fundamental because they provide a quantitative means of characterizing how the physical properties of galaxies relate to each other. Galaxy scaling relations also provide the route to understanding the internal physics of galaxies, as well as their formation and evolutionary histories.
We currently enjoy a diverse array of scaling laws that describe the stellar components of galaxies, for example the Tully-Fisher relation for spiral galaxies [@tf77], and the fundamental plane for ellipticals [@jorgensen96]. Both relations provide important constraints on how these systems have assembled. However, few well-established scaling laws exist describing how the cold gas is correlated with the other global physical properties of galaxies. The only well-studied relation is the Schmidt-Kennicutt star formation law [@kennicutt98a], relating the formation rate of new stars and the surface density of cold gas in disks.
The reason why so few scaling laws involving cold gas and global galaxy properties, such as masses, sizes and bulge-to-disk ratios, exist in the literature, is the difficulty in acquiring suitable data. There are four general requirements on the data if the derived scaling laws are to be reliable: (1) homogeneous and accurate measurements of all the physical properties under consideration, (2) unbiased measurement of every property with respect to every other property, (3) sample selection that ensures large dynamic range of the various physical properties under consideration, (4) a large enough sample to define both the mean relation and the scatter about the mean. As we will describe, existing data sets do not, in general, meet all of these conditions.
Line emission from the CO molecule was first detected in the central parts and disks of nearby galaxies 35 years ago [@rickard75; @solomon75; @combes77]. A decade later, CO measurements existed for $\sim100$ galaxies [see the compilation of @verter85], and in the following years, several larger systematic studies of molecular gas in nearby galaxies were performed. The largest effort was the FCRAO Extragalactic CO Survey [@young95], which measured the CO $J=1\rightarrow0$ line (hereafter, CO(1-0)) in 300 nearby galaxies. Because of a strong correlation between infrared luminosity and CO luminosity [e.g. @sanders91; @sanders96], a significant part of this early work was done targeting luminous infrared galaxies [e.g. @radford91; @solomon97]. Even the FCRAO Survey, still considered as the reference for CO measurements in the nearby universe, targeted galaxies selected on infrared or $B-$band luminosity.
Recognizing this bias toward “exceptional" galaxies (e.g. starbursts and interacting systems), @braine93 observed the CO $1\rightarrow0$ and $2\rightarrow1$ rotational transition lines with the IRAM 30m telescope for a magnitude-limited sample of 81 normal spiral galaxies. Other attempts at measuring molecular gas in normal galaxies include the work of @kenney88 for Virgo cluster spirals, of @sage93 for nearby non-interacting spirals, and of @boselli97 for Coma cluster spirals.
These pioneering studies constrained molecular gas properties in nearby galaxies as a function of morphology [e.g. @wiklind89; @thronson89], star formation rate or infrared luminosity [e.g. @sanders85; @gao04], atomic gas contents [e.g. @young89a], environment [e.g. @kenney89; @casoli91; @boselli97], and for resolved studies, position within galaxy disks. Highlights from these studies include the observations that molecular gas distributions decline monotonically with galaxy-centric radius unlike the atomic gas distributions, that IR-luminous galaxies are also CO-bright, with molecular gas concentrated within the inner kpc of these mostly interacting systems, and that the total gas mass fraction as well as the molecular-to-atomic ratio are functions of Hubble type.
Nevertheless, most of the samples did not meet all of the criteria listed above that would allow for accurate scaling laws to be derived; some samples were biased towards a particular galaxy type (e.g. infrared-bright objects), some of the more unbiased samples did not cover enough parameter space (e.g. targeting only spiral galaxies), some samples suffered from aperture problems, some samples were too small, and attempts to combine different samples to remedy these problems led to inhomogeneous datasets (see also §\[archive\]).
Recently, much effort has been put into obtaining homogeneous and relatively deep high spatial resolution molecular gas maps covering the optical disks of nearby galaxies [@regan01; @kuno07; @leroy09]. These samples are excellent for studying star formation laws within galaxies [e.g. @bigiel08], but the number of objects is too small to adequately define global scaling relations.
With reliable measurements of molecular gas for a large, unbiased sample of galaxies, it is possible not only to quantify scaling relations, but also to construct an accurate molecular gas mass function. Current estimates are based on inherently inhomogeneous samples [@keres03; @obreschkow09]. We can also investigate the molecular gas properties of galaxy samples for which dedicated surveys do exist, but where the number of objects studied has been very small, for example early-type galaxies [e.g. @combes07; @krips10], and galaxies with active nuclei [e.g. @helfer93; @sakamoto99; @garcia03]. A large unbiased sample of galaxies which can serve as a reference for such particular objects would also be very valuable.
In this paper, we introduce COLD GASS, a new survey for molecular gas in nearby galaxies. Upon completion, it will have measured fluxes in the CO(1-0) line for a purely mass-selected sample of at least 350 galaxies. The sample contains galaxies with a wide range of Hubble types from star-forming spirals to “red and dead" ellipticals. With its new, large-bandwidth receivers, the IRAM 30m telescope is the instrument of choice to conduct a new large molecular gas survey, allowing the community to move from dedicated studies of particular types of galaxies, to larger systematic efforts. COLD GASS will provide a definitive, unbiased census of the partition of condensed baryons in the local Universe into stars, atomic and molecular gas in galaxies covering over two orders of magnitude in luminosity.
In §\[sample\]-\[iramdata\], we present an overview of the survey and of the sample selection, and describe the CO measurements and ancillary datasets. In §\[scalrel\] and \[compHI\] we present the first COLD GASS scaling relations, correlating molecular gas masses with global galaxy parameters including stellar mass and atomic gas mass. Throughout the paper, distance-dependent quantities are calculated for a standard flat $\Lambda$CDM cosmology with $H_0=70$[km s$^{-1}$]{} Mpc$^{-1}$, and we adopt a conversion factor from CO luminosity to $H_2$ mass of $\alpha_{CO}=3.2$[$M_{\odot}$]{} (K [km s$^{-1}$]{} pc$^2$)$^{-1}$ (which does not account for the presence of Helium), unless otherwise specified.
Survey description and Sample selection {#sample}
=======================================
The conditions required to obtain reliable scaling laws and listed in §\[intro\] are routinely met by optically-selected samples of galaxies at low redshift. The Sloan Digital Sky Survey [SDSS; @sdss] with its 5-band optical imaging campaign over a quarter of the sky and its follow-up spectroscopy of close to a million galaxies has facilitated the study of galaxy scaling relations studies at an unprecedented level of detail.
At radio wavelengths, a series of large blind HI surveys have become possible thanks to a number of new multi-feed arrays. The most advanced of these, the Arecibo Legacy Fast ALFA Survey [ALFALFA; @ALFALFA1], will have detected upon completion $\sim 30,000$ galaxies out to distances of $\sim 200$Mpc. Although ALFALFA measurements are accurate, homogeneous, and unbiased, the survey is shallow, with the result that it does not probe a large dynamic range in HI-to-stellar mass ratio for all but the very nearest galaxies. For example, in the redshift range $0.025<z<0.05$, the median value of $M_{HI}/M_{\ast}$ for ALFALFA detections with $M_{\ast}>10^{10}$[$M_{\odot}$]{} is $\sim25\%$.
GASS
----
To overcome this issue, the [*GALEX*]{} Arecibo SDSS Survey [GASS; @GASS1] was designed to measure the neutral hydrogen content for a large, unbiased sample of $\sim1000$ massive galaxies ($M_{\ast}>10^{10}$ [$M_{\odot}$]{}), via longer pointed observations. GASS is a large program currently under way at the Arecibo 305m telescope, and is producing some of the first unbiased atomic gas scaling relations in the nearby universe [@GASS1; @GASS2; @fabello10].
Details about the GASS survey design, target selection, and observing procedures are given in @GASS1. In short, the galaxies observed as part of GASS are selected at random out of a larger parent sample of galaxies that meet the following criteria:
1. They are located within the area of overlap of the SDSS spectroscopic survey, the ALFALFA survey, and the projected footprint of the [*GALEX*]{} Medium Imaging Survey (MIS).
2. They lie in the redshift range $0.025<z<0.05$.
3. They have a stellar mass in the range $10^{10}<M_{\ast}/M_{\odot}<10^{11.5}$.
The GASS sample is selected out of this parent sample to produce a flat $\log M_{\ast}$ distribution. No other selection criteria on colour, morphology, or spectral properties for example are applied. This sample therefore provides us with a complete picture of how the cold atomic gas relates to other properties such as stellar mass, luminosity, stellar surface mass density and colour.
@GASS1 present the first GASS data release, which includes $\sim20\%$ of the final sample. They show that there exist strong anti-correlations between the atomic gas mass fraction and stellar mass, stellar mass surface density and [NUV$-r$]{} colour. GASS also aims at studying the galaxies that are transitioning between a blue, star-forming state and a red passive state (and vice-versa). These are identified as outliers from the mean scaling relations. The ultimate goal is to understand the physical processes that affect the gas content of these galaxies (e.g. accretion or quenching) and in turn the star formation process.
COLD GASS {#sampleselect}
---------
![Distribution of sources observed as of 25 Oct. 2010 (filled blue histograms), compared to the proposed final COLD GASS sample (filled gray histograms). The solid line in each panel shows the distribution of objects in the superset of $\sim$12000 galaxies out of which the GASS sample is extracted, scaled down to the number of objects in the COLD GASS master list (350). The GASS sample is selected as to produce a flat distribution in $\log M_{\ast}$. \[distribs\]](fig1.eps){width="84mm"}
We are in the process of constructing a CO Legacy Database for the GASS survey (COLD GASS), measuring the molecular gas content of a significant subsample of the GASS galaxies. We will then be able to quantify the link between atomic gas, molecular gas and stars in these systems.
COLD GASS is designed to meet all the requirements to produce reliable gas scaling relations:
1. Galaxies in our redshift range ($0.025<z<0.05$) have angular diameters that are small enough to enable accurate measurement of the total CO line flux with a single IRAM 30m pointing for the majority (80%) of the galaxies. For the remaining objects, we recover the total flux by adding a single offset pointing (see §\[offsets\] for details).
2. In the mass range we study ($10^{10}<M_{\ast}/M_{\odot}<10^{11.5}$), the metallicity of the galaxies is around solar [@tremonti04]. The CO line flux therefore provides a reasonably accurate measurement of the total molecular gas content using a single conversion factor, $\alpha_{CO}$.
3. The $\sim 350$ targets are selected at random from the GASS survey, the sample is therefore unbiased.
4. We integrate until the CO line is detected, or until we reach an upper limit in molecular gas mass to stellar mass ratio ($f_{H_2}\equiv M_{H_2}/M_{\ast}$) of $\sim1.5\%$.
5. Upon completion of the survey, the sample size of at least 350 galaxies will be large enough to determine accurately a set of scaling laws involving three parameters and to measure the scatter around these relations.
Distributions of some basic parameters of the COLD GASS sample are shown in Figure \[distribs\] and compared to a purely volume-limited superset of galaxies. The COLD GASS targets are selected randomly from the GASS survey and therefore share the flat $\log M_{\ast}$ distribution designed for that survey to ensure even sampling of the stellar mass parameter space. As seen in Figure \[distribs\], a purely volume-limited sample is richer in low mass galaxies, and poorer in high mass systems. The uniform mass distribution also has the effect of flattening the colour distribution and reducing the number of galaxies with low stellar mass surface densities. We note that when deriving scaling relations and calculating sample averages, we statistically correct for this “mass bias", as done by @GASS1 for the GASS sample (see §\[scalrel\]) .
SDSS, [*GALEX*]{} and Arecibo Observations {#data}
==========================================
Optical and UV measurements
---------------------------
Parameters such as redshifts, sizes, magnitudes, and Galactic extinction factors are retrieved from the database of SDSS DR7 [@DR7]. The UV data are taken from the [*GALEX*]{} All-sky and Medium Imaging surveys [AIS and MIS, respectively, see @martin05].
The SDSS and [*GALEX*]{} images are reprocessed following @wang10, in order to obtain accurate aperture photometry. The process includes registering the images and smoothing them to a common PSF. The SDSS $r-$band images are convolved to the resolution of the UV imaging before Sextractor is used to calculate magnitudes in consistent apertures, therefore ensuring that measurements in different bands represent similar physical regions of the galaxies. The derived [NUV$-r$]{} colours are corrected for Galactic extinction using the prescription of @wyder07 [see also @GASS1].
Stellar masses are calculated from the SDSS photometry using the SED-fitting technique of @salim07, assuming a Chabrier IMF [@chabrier03]. A variety of model SEDs from the @bc03 library are fitted to each galaxy, building a probability distribution for its stellar mass. The stellar mass assigned to a galaxy is then the mean of this distribution, while the measurement error is estimated from its width. The systematic uncertainty between different technique to derive photometric stellar masses from SDSS measurements is $<0.1$dex, as estimated by @dutton11.
The main optical- and UV-derived parameters used throughout this paper are presented in Table \[params\], for all galaxies within the present COLD GASS data release. Column 1 and 2 give the GASS and SDSS ID numbers, respectively, column 3 gives the optical redshift from SDSS spectroscopy, while column 4 lists the stellar mass and column 5 the stellar mass surface density, which we calculate as: $$\mu_{\ast}=\frac{M_{\ast}}{2\pi R_{50,z}^2},$$ where $R_{50,z}$ is the $z-$band 50% flux intensity petrosian radius, in kiloparsecs. In column 6, we give the $g-$band optical diameter ($D_{25}$), and in column 7 the concentration index ($C\equiv R_{90}/R_{50}$, where $R_{50 }$ and $R_{90}$ are from $r-$band photometry). Finally, columns 8 and 9 present the [NUV$-r$]{} colour and the $r-$band model magnitude, both corrected for Galactic extinction.
--------- --------------------- ------------ -------------------- ---------------------------------- --------------- ----------------- --------- ---------
GASS ID SDSS ID $z_{SDSS}$ $M_{\ast}$ $\mu_{\ast}$ $D_{25}$ $R_{90}/R_{50}$ NUV$-r$ $r$
$[\log M_{\odot}]$ $[\log M_{\odot} \rm{kpc}^{-2}]$ \[$\arcsec$\] \[mag\] \[mag\]
11956 J000820.76+150921.6 0.0395 10.09 8.48 22.5 2.15 3.04 16.28
12025 J001934.54+161215.0 0.0366 10.84 9.13 34.3 3.03 5.93 14.73
12002 J002504.00+145815.2 0.0367 10.48 9.41 24.2 3.17 6.25 15.46
11989 J002558.89+135545.8 0.0419 10.69 9.18 23.7 3.02 5.79 15.13
27167 J003921.66+142811.5 0.0380 10.37 9.14 21.1 2.77 4.48 15.49
3189 J004023.48+143649.4 0.0384 10.05 7.92 37.7 1.96 2.77 15.65
3261 J005532.61+154632.9 0.0375 10.08 8.57 22.8 2.54 2.63 15.48
3318 J010238.29+151006.6 0.0397 10.53 8.98 26.4 3.05 5.73 15.21
3439 J010905.96+144520.8 0.0386 10.35 8.78 32.5 2.90 3.05 15.48
3465 J011221.82+150039.0 0.0292 10.19 8.93 28.7 2.89 3.63 15.33
3645 J011501.75+152448.6 0.0307 10.33 8.93 28.1 2.71 3.97 15.11
3509 J011711.65+132027.3 0.0484 10.81 9.18 31.1 3.11 4.14 15.27
3519 J011728.11+144215.9 0.0427 10.74 8.64 34.2 2.20 3.68 14.94
3505 J011746.76+131924.5 0.0479 10.21 8.83 17.7 3.30 4.92 16.35
3504 J011823.44+133728.4 0.0380 10.16 7.91 37.7 1.84 2.85 15.34
--------- --------------------- ------------ -------------------- ---------------------------------- --------------- ----------------- --------- ---------
$^a$ Table 1 is published in its entirety in the electronic edition of the journal. A portion is shown here as example of its format and contents.
HI masses
---------
Details of the HI observations are described in @GASS1, so we only provide a brief overview here. The survey builds upon existing HI databases: the Cornell digital HI archive [@springob05] and the ALFALFA survey [@ALFALFA1]. HI data for about 20% of the GASS sample (the most gas-rich objects), can be found in either of these sources. For the rest of the sample, observations are carried out at the Arecibo Observatory. Integration times are set such as to detect HI gas mass fractions ($f_{HI}=M_{HI}/M_{\ast}$) of 1.5% or more. Observations are carried out using the $L-$band Wide receiver and the interim correlator, providing coverage of the full frequency interval of the GASS targets at a velocity resolution of 1.4 [km s$^{-1}$]{} before smoothing. Data reduction includes Hanning smoothing, bandpass subtraction, radio frequency interference (RFI) excision, flux calibration and weighted combination of individual spectra. Total HI-line fluxes, velocity widths and recessional velocities are then measured using linear fitting of the edges of the HI profiles [e.g. @springob05; @catinella07].
IRAM Observations {#iramdata}
=================
Observing procedure \[obs\_setup\]
----------------------------------
Observations are carried out at the IRAM 30m telescope. We use the Eight Mixer Receiver (EMIR) to observe the CO(1-0) line (rest frequency, 115.271 GHz) . The CO $1\rightarrow0$ transition traces well the entire molecular gas contents of the galaxies at $n(H_2)>10^2$ cm$^{-3}$ [see e.g. the Appendix in @tacconi08]. In the 3mm band (E090), EMIR offers two sidebands with 8 GHz instantaneous bandwidth per sideband and per polarisation. With a single tuning of the receiver at a frequency of 111.4081 GHz, we are able to detect the redshifted CO(1-0) line for all the galaxies in our sample ($0.025<z<0.05$), within the 4 GHz bandwidth covered by the correlators. This single tuning procedure results in enormous time savings of 15 minutes per source, and in an improved relative calibration accuracy. Also, the frequency range covered benefits from a considerably improved atmospheric transmission as compared to the CO(1-0) rest frequency. The second band is tuned to a frequency of 222.8118 GHz (E230 band), to cover the redshifted CO(2-1) line which falls within the available 4 GHz bandwidth for about 75% of our sample. We postpone the presentation and analysis of the CO(2-1) data until the survey is completed, to maximize the sample size.
The wobbler-switching mode is used for all the observations with a frequency of 1Hz and a throw of 180 . The Wideband Line Multiple Autocorrelator (WILMA) is used as the backend, covering 4 GHz in each linear polarisation, for each band. WILMA gives a resolution of 2 MHz ($\sim 5$ [km s$^{-1}$]{} for the 3mm band). We also simultaneously record the data with the 4MHz Filterbank, as a backup.
Observations for this first data release were conducted between 2009 December and 2010 October. Atmospheric conditions varied greatly, with an average of 6mm of precipitable water vapor (PWV). We also fold into this catalog 15 galaxies observed in 2009 June, as part of a pilot program designed to test the feasibility of the survey. These galaxies were selected to be HI-rich ($M_{HI}/M_{\ast}>0.1$), but this selection bias does not affect the overall sample.
Observing strategy {#obsstrat}
------------------
![Fraction of galaxies with a detection in the CO(1-0) line as a function of (a) stellar mass, (b) stellar mass surface density, (c) concentration index (defined as $R_{90}/R_{50}$, the ratio between the $r-$band radii encompassing 90% and 50% of the light), and (d) [NUV$-r$]{} colour. The results are shown in equally populated bins, each containing 37 galaxies. The gray shaded region shows the overall detection rate of $53.6\%$, down to $47.3\%$ if all tentative detections are excluded. The downward error bars show the effect of excluding tentative detections in each individual bin. In panels (b-d), the vertical dotted line indicates the critical value where the detection rate suddenly drops below $\sim80\%$. \[detfrac\]](fig2.eps){width="84mm"}
Observations are carried out in fixed observing blocks and as poor-weather backups for higher frequency programs. We accommodate to the changing weather conditions by making “real time" decisions on targets. We observe the bluer galaxies (generally CO-luminous) under poorer weather conditions. These galaxies require on average an rms sensitivity of 1.7 mK per 20 [km s$^{-1}$]{}-wide channel to achieve a reliable detection of the CO(1-0) line with $S/N>5$. As seen in Figure \[detfrac\]d, galaxies with colour [NUV$-r$]{}$<4.5$ have a detection rate greater than 80%. When the atmospheric water vapor level is low, we preferentially observe the redder galaxies, which have a very low detection rate (Fig. \[detfrac\]d). In order to set firm upper limits for these galaxies, we require low noise values to reach our integration limit of $M_{H2}/M_{\ast}=1.5\%$. Under good observing conditions, sensitivity to this minimum gas fraction, or an absolute minimum rms of 1.1mK (per 20[km s$^{-1}$]{}-wide channel), is reached within 1-1.5 hour. This absolute minimum rms is imposed in order to keep the integration time per galaxy $<2$ hours, and translates in a detection limit that is higher than the nominal value of 1.5% for galaxies with $\log M_{\star}/M_{\odot}<10.6$.
The efficiency of the observations is also maximized by our single tuning approach (see §\[obs\_setup\]), and by the fact that the galaxies are mostly concentrated in a declination strip with $0^{\circ} < \delta < +15^{\circ}$, as shown in Figure \[sky\]. Not only does this allow us to move quickly from one source to the next without repeating pointing correction measurements, it makes it possible to almost always observe at elevations larger than 45$^{\circ}$, minimizing atmospheric opacity.
{width="165mm"}
Data reduction {#linemes}
--------------
The data are reduced with the CLASS software. All scans are visually examined, and those with distorted baselines, increased noise due to poor atmospheric conditions, or anomalous features are discarded. The individual scans for a single galaxy are baseline-subtracted (first order fit) and then combined. This averaged spectrum is finally binned to a resolution of $\sim 20$[km s$^{-1}$]{}, and the standard deviation of the noise per such channel is recorded ($\sigma_{rms}$).
Flux in the CO(1-0) line is measured by adding the signal within an appropriately defined windowing function. If the line is detected, the window is set by hand to match the observed line profile. If the CO line is undetected or very weak, the window is set either to the full width of the HI line ($W50_{HI}$) or to a width of 300 [km s$^{-1}$]{} in case of an HI non-detection. For the non-detection, an upper limit for the flux of $5\epsilon_{obs}$ (see Eq. \[errmes\]) is set.
The central velocity and total width of the detected CO lines are then measured using a custom-made IDL interactive script. The peaks of the signal are identified, and a linear fit is applied to each side of the profile between the 20% and 80% peak flux level. The width of the line, $W50_{CO}$, is then measured as the distance between the points on each of the fits corresponding to 50% of the peak intensity. The recession velocity is taken as the midpoint of this line. This method is described in @springob05 and @catinella07, and is also used to measure the HI line-widths of the GASS sample.
Aperture Corrections {#offsets}
---------------------
![Aperture corrections for the IRAM 30m observations of the COLD GASS sample. (a): Ratio between the flux recovered by a central pointing with the IRAM 22 beam to the total flux of the galaxy ($S_{CO,obs}/S_{tot}$), as a function of optical diameter ($D_{25}$). Small gray points indicate the results from the simulated observations of a sample of 40 nearby spiral galaxies with high quality CO(1-0) maps [@kuno07] when they are placed at various redshifts in the range 0.01-0.05 (each set of points connected by a broken line corresponds to one of the galaxies, placed at different redshifts). A single central pointing recovers at least 60% of the flux in galaxies with $D_{25}<40\arcsec$. The histogram indicates the size distribution of the COLD GASS sample (with the distribution of blue galaxies, [NUV$-r$]{}$<$4.3, indicated in blue). (b): With an additional offset observation 0.75 beam (i.e. $16\arcsec$) from the centre of the galaxy along the major axis ($S_{off}$), it is possible to derive the fraction of CO(1-0) line flux detected at the center and from that estimate the total molecular gas mass in galaxies with $D_{25}>40\arcsec$. The large red symbols in panel (a) indicate the aperture corrections estimated using this method for the [25]{} COLD GASS galaxies for which we performed offset pointings to date, and the dashed line is the size threshold (40) for a galaxy to require an offset pointing. \[figapcor\]](fig4.eps){width="84mm"}
Because the galaxies targeted are at a distance of at least 100 Mpc ($z>0.025$), most of them can be observed with a single pointing of the IRAM 30m, which has a beam with a FWHM of 22 at a wavelength of 3mm. However some of the galaxies have optical diameters in excess of this, and an aperture correction needs to be applied.
We derive aperture corrections using a set of nearby galaxies with accurate CO maps [@kuno07]. We simulate the impact of observing galaxies with the IRAM beam by taking each of the maps, placing it at different redshifts in the range $0.025<z<0.05$, and computing the ratio between the flux as would be measured by a 22 Gaussian beam to the total flux in the map ($S_{CO,obs}/S_{tot}$). We find that a single central observation recovers most $(>60\%)$ of the CO line flux in galaxies with $D_{25}<40\arcsec$. Results are shown in Figure \[figapcor\]a. Based on the best fit to these data, we apply the following aperture correction: $$S_{CO,cor} = S_{CO,obs} / (1.094 - 0.008 D_{25} + 2.0\times10^{-5} D_{25}^2)
\label{Icor1}$$ where $S_{CO,obs}$ is the observed flux in the central pointing and $S_{CO,cor}$ the extrapolated total flux.
For galaxies with optical diameters larger than 40, however, there is a significant scatter in the $S_{CO,obs}/S_{tot}$ ratio, and additional information is required to recover the total CO line flux. In the right panel of Figure \[figapcor\], we show that with a single offset pointing at three-quarter beam from the central position ($0.75\times$22) along the major axis, the total flux can be recovered with much better accuracy[^2]. An offset pointing at a full beam also does well, but the mean ratio $S_{off}/S_{CO,obs}$ then drops from 33% to 15%. We adopted the three-quarter beam offset as a compromise between the requirements for independent flux measures and a modest fraction of our total observing time going into off-center pointings.
Our requirement to perform an offset pointing is that a galaxy has (1) a large angular size ($D_{25}>40$) and (2) a bright CO line in the central pointing, such that a detection in the offset pointing can be made with an hour of integration or less. For these galaxies, the flux in the central pointing is corrected based on the ratio between the flux in the offset pointing to that in the center ($f_{off}\equiv S_{off}/S_{CO,obs}$), based on the best fit to data in Figure \[figapcor\]b: $$S_{CO,cor}=S_{CO,obs} / (1.587 - 3.361 f_{off} + 2.107 f_{off}^2).
\label{Icor2}$$ If a galaxy is larger than 40 but the CO line in the central pointing is weak or undetected, then the measured central flux is corrected according to Eq. \[Icor1\].
So far, we have performed offset pointings for [25]{}galaxies, that met the requirements listed above (see Figs. \[spectra\_off\] and \[spectra\_off2\]). We used Eq. \[Icor2\] and Fig. \[figapcor\]b to infer the fraction of the total flux that was measured in the central pointing ($S_{CO,obs}/S_{tot}$), and we show in Fig. \[figapcor\]a how this ratio depends on $D_{25}$ and how it compares to the range of values predicted by our simulated observations.
$M_{H2}$ and associated error budget
------------------------------------
After correcting the CO(1-0) line fluxes for aperture effects using either Eq. \[Icor1\] or Eq. \[Icor2\], whichever case applies to each galaxy, we compute the total CO luminosities following @solomon97: $$L'_{CO}=3.25\times10^7 S_{CO,cor} \nu_{obs}^{-2} D_{L}^2 (1+z)^{-3},
\label{Ico}$$ where $S_{CO,cor}$ in units of Jy [km s$^{-1}$]{}is the integrated line flux[^3], $\nu_{obs}$ is the observed frequency of the CO(1-0) line in GHz, $D_L$ is the luminosity distance in units of Mpc, and $L'_{CO}$ is the CO luminosity in \[K [km s$^{-1}$]{} pc$^2$\].
The total molecular hydrogen masses are then calculated as $M_{H2}=L'_{CO}\alpha_{CO}$. We adopt a constant Galactic conversion factor of $\alpha_{CO}=3.2$ M$_{\odot}$(K [km s$^{-1}$]{} pc$^2)^{-1}$, which does not include a correction for the presence of Helium. Our choice of $\alpha_{CO}$ is roughly the mean of values estimated in the Milky Way and in nearby galaxies [e.g. @strong96; @dame01; @blitz07; @draine07; @heyer09; @abdo10]. The virial method used to measure $\alpha_{CO}$ has been validated by other independent techniques such as $\gamma-$ray observations, and shown to also hold for the ensemble average of the virialized clouds of entire galaxies, as long as the factor $n({\rm H}_2)^{0.5}/T$ is constant throughout the galaxy and the CO line is optically thick [@dickman86; @tacconi10].
This constant value for $\alpha_{CO}$ has been shown to hold for galaxies in the Local Group, when the metallicity goes from solar down to SMC values [@bolatto08]. It may be that the conversion factor is instead a function of a parameter such as gas surface density or metallicity [@tacconi08; @obreschkow09]. Based on the well-known metallicity-luminosity relation, @boselli02 for example proposed a luminosity-dependent conversion factor. However, such a prescription has yet to be observationally or theoretically validated. Furthermore, it has since been shown that the mass-metallicity relation is even more fundamental [@tremonti04]. At stellar masses above $\sim 10^{10.5}$[$M_{\odot}$]{}, the relation flattens out, and therefore the metallicity of our COLD GASS galaxies is expected to be $\sim$solar with little variations across the sample. This expectation is confirmed by our long-slit spectroscopy measurements [@moran10; @COLDGASS2]. Based on this fact, adopting a constant Galactic conversion factor $\alpha_{CO}=3.2$ M$_{\odot}$(K [km s$^{-1}$]{} pc$^2)^{-1}$ is the simplest yet most justified assumption we can make at this point. We however consider the variations in $\alpha_{CO}$ by a factor of $\sim$2 at fixed metallicity [@bolatto08] as the systematic uncertainty on our values of $M_{H_2}$.
We calculate the formal measurement error on the observed line flux, $S_{CO,obs}$, as: $$\epsilon_{obs}=\frac{\sigma_{rms} W50_{CO}}{\sqrt{W50_{CO} \Delta w_{ch}^{-1}}},
\label{errmes}$$ where $\sigma_{rms}$ is the rms noise per spectral channel of width $\Delta w_{ch}=21.57$[km s$^{-1}$]{}, and $W50_{CO}$ is the FWHM of the CO(1-0) line. This definition takes into account that for a given total flux, the $S/N$ per channel is largest when the emission line is narrower. Considering all detections, the mean fractional error ($\epsilon_{obs}/S_{CO,obs}$) is $11\%$.
Other contributions to the error budget on $M_{H2}$ include a flux calibration error ($10\%$ at a wavelength of 3mm, under average atmospheric conditions) and the uncertainty on the aperture correction (which we estimate to be $15\%$ based on Fig.\[figapcor\] for the median galaxy in our survey). The average pointing rms error is $\Delta_{point}=2\arcsec$, and also contributes to the uncertainty on the measured flux. Using the resolved CO(1-0) maps of @kuno07, we simulate the observation of COLD GASS galaxies both with a beam perfectly centered on the object, and with a beam offset by $\Delta_{point}$ from the center. We find that a $2\arcsec$ positional error generates a 2.1% uncertainty on the measured line flux. Given that the redshift error is negligible compared to these other contributions, the fractional error on $L'_{CO}$ is obtained by adding in quadrature all error contributions to $S_{CO,cor}$. We therefore find a mean error of $<\epsilon_{Lco}>\simeq20\%$. The random measurement error on our quoted values of $M_{H2}$ is then $\sim20\%$, to which we add the systematic error coming from the uncertainty in the value of $\alpha_{CO}$ bringing the total error on $\log M_{H_2}$ to 0.3 dex.
Stacking
--------
To extract information from the CO non-detections, we also perform a stacking analysis. We stack the spectra after converting them in “gas fraction" units following @fabello10, who provide an extensive discussion of the merits of this technique. We convert each spectrum $S$ into a scaled spectrum $S_{gf}$: $$S_{gf}=\frac{S D_L^2}{M_{\ast} (1+z)^3},$$ where the factor $D_L^2 (1+z)^{-3}$ is a consequence of Eq. \[Ico\]. This gives us scaled spectra in units of \[mJy Mpc$^2$ [$M_{\odot}$]{}$^{-1}$\].
For any subsample of $N$ galaxies with CO non-detection, we calculate a mean stacked spectrum: $$S_{stack}=\frac{\sum_{i=1}^N w_i S_{gf,i}}{\sum_{i=1}^N w_i},$$ where $w$ is the weight given to each galaxy in the stack, based on its stellar mass, in order to compensate for the fact that the $\log M_{\ast}$ distribution of the COLD GASS sample is by design flatter than in a complete volume-limited sample (Figure \[distribs\]a).
To obtain $<M_{H2}/M_{\ast}>$, the mean molecular gas mass fraction for the family of $N$ galaxies stacked, we measure a line flux or set an upper limit as described in §\[linemes\], and multiply by the constant factors $3.25\times10^7 \nu_{obs}^{-2} \alpha_{CO}$ to obtain the dimensionless mass fraction.
Catalog presentation {#catalog}
--------------------
In table \[COtab\], we summarize the results of the IRAM observations of the first [222]{} COLD GASS galaxies ([119]{} CO detections and [103]{} non-detections). The contents of the catalog are as follows:\
\
Column (1): GASS ID. Galaxies are in the same order as in Table \[params\] to ease cross-referencing.\
\
Column (2): rms noise per channel in mK, after binning the spectra to a resolution of $\Delta w_{ch}=21$ [km s$^{-1}$]{}.\
\
Column (3): Signal-to-noise ratio of the detected CO line, calculated as $S_{CO,obs}/\epsilon_{obs}$, where $\epsilon_{obs}$ is calculated according to Eq. \[errmes\]. In our analysis, we consider sources with $S/N>5$ as secure, and the detections with $S/N<5$ as tentative.\
\
Column (4): Integrated CO line flux in Jy [km s$^{-1}$]{}. The measured flux in antenna temperature units is converted to these units using the ratio $S/T_a^{\ast}=6.0$ Jy/K for the IRAM 30m telescope at our observing frequency of 111GHz.\
\
Column (5): Corrected total line flux in Jy [km s$^{-1}$]{}, computed following Equations \[Icor1\] and \[Icor2\].\
\
Column (6): When available, the measured ratio between the flux in the offset pointing to that in the central pointing, as described in §\[offsets\].\
\
Column (7): Total molecular hydrogen gas mass. We adopt a constant Galactic conversion factor of $\alpha_{CO}=3.2$ M$_{\odot}$(K [km s$^{-1}$]{} pc$^2)^{-1}$, which does not include a correction for the presence of Helium. The numbers quoted are either measured masses in the case of detections, or $5\sigma$ upper limits for the non-detections (see Column 9).\
\
Column (8): Molecular gas mass fraction, $f_{H2} \equiv M_{H2}/M_{\ast}$, and upper limits for non-detections.\
\
Column (9): CO emission line flag, set to 1 for detections and 2 for non-detections.
--------- ---------- ----------- ----------------------- ----------------------- ----------- -------------------- ------------------- ------ --
GASS ID $\sigma$ $S/N$ $S_{CO,obs}$ $S_{CO,cor}$ $f_{off}$ $M_{H2}$ $M_{H2}/M_{\ast}$ Flag
\[mK\] $[\rm{Jy~km~s}^{-1}]$ $[\rm{Jy~km~s}^{-1}]$ $[\log M_{\odot}]$
11956 1.07 2.35 1.16 1.25 [$...$]{} 8.46 0.024 1
12025 1.06 [$...$]{} [$...$]{} [$...$]{} [$...$]{} 8.78 0.009 2
12002 1.18 [$...$]{} [$...$]{} [$...$]{} [$...$]{} 8.79 0.021 2
11989 1.07 [$...$]{} [$...$]{} [$...$]{} [$...$]{} 8.79 0.013 2
27167 1.17 [$...$]{} [$...$]{} [$...$]{} [$...$]{} 8.74 0.023 2
3189 1.24 6.69 3.19 3.88 [$...$]{} 8.93 0.076 1
3261 1.96 8.57 4.27 4.62 [$...$]{} 8.98 0.080 1
3318 1.03 [$...$]{} [$...$]{} [$...$]{} [$...$]{} 8.81 0.019 2
3439 1.03 [$...$]{} [$...$]{} [$...$]{} [$...$]{} 8.79 0.028 2
3465 1.17 4.28 2.89 3.28 [$...$]{} 8.62 0.027 1
3645 1.08 [$...$]{} [$...$]{} [$...$]{} [$...$]{} 8.62 0.020 2
3509 1.15 7.24 5.02 5.80 [$...$]{} 9.30 0.031 1
3519 1.53 7.72 5.51 6.52 [$...$]{} 9.24 0.032 1
3505 1.30 [$...$]{} [$...$]{} [$...$]{} [$...$]{} 8.93 0.052 2
3504 1.53 9.19 2.83 3.44 [$...$]{} 8.87 0.051 1
--------- ---------- ----------- ----------------------- ----------------------- ----------- -------------------- ------------------- ------ --
$^a$ Table 1 is published in its entirety in the electronic edition of the journal. A portion is shown here as example of its format and contents.
Molecular gas fraction scaling relations {#scalrel}
========================================
Molecular gas fraction and global galaxy parameters {#fH2_s1}
---------------------------------------------------
{width="150mm"}
We first investigate how the molecular gas mass fraction[^4] ($f_{H2}\equiv M_{H2}/M_{\ast}$) varies across the COLD GASS sample. The values of $f_{H2}$ range from 0.009 (G7286), at the very edge of the survey’s detection limit, up to 0.20 for the most $H_2-$rich galaxy in the sample observed so far (G41969). Overall, the molecular gas contents are more modest, averaging at $<f_{H2}>=0.066\pm0.039$ for the CO detections, and at $0.043\pm0.022$ when non-detections are included in the mean at the value of their upper limits. These sample averages are measured by weighting each galaxy differently according to its stellar mass, in order to compensate for the flat $\log M_{\ast}$ distribution (see Section \[fH2\_s2\] for more details).
In Figure \[fH2plots\], we investigate how the molecular gas mass fraction depends on stellar mass, stellar mass surface density, concentration index and [NUV$-r$]{} colour. For all four relations, we quantify the strength of the dependence of $f_{H2}$ on the x-axis parameter in two ways. Firstly, we compute for all secure detections the Pearson correlation coefficient of the relation, $r$. Secondly we perform an ordinary least squares linear regression of $f_{H2}$ on the x-axis parameter for the secure detections, and report the standard deviation of the residuals about this best fitting relation, $\sigma$, in $\log M_{H2}/M_{\ast}$ units.
There is a weak dependency of $f_{H2}$ on stellar mass, $M_{\ast}$, as seen in Fig. \[fH2plots\]a, with a correlation coefficient $r=-0.40$, and a scatter of $\sigma=0.292$ dex, but some of this correlation is caused by the detection limit. The strength of the correlation of $f_{H2}$ on stellar mass surface density, $\mu_{\ast}$, is similar with $r=-0.33$ and $\sigma=0.301$ dex (Fig. \[fH2plots\]b). There is however a striking difference in the behaviour of the non-detections; while the detection rate is roughly constant as a function of [$M_{\ast}$]{}($\sim50\%$, see Fig. \[detfrac\]a), it is a strong function of [$\mu_{\ast}$]{}. There is a surface density threshold of $\mu_{\ast}\simeq10^{8.7}$ [$M_{\odot}$]{} kpc$^{-2}$, below which the detection rate is almost $100\%$ and above which it quickly drops. A similar threshold is observed by @GASS1 in the relation between the atomic gas mass fraction ($f_{HI}\equiv M_{HI}/M_{\ast}$) and [$\mu_{\ast}$]{} (see §\[compHI\] for more details).
To test the impact of galaxy morphology on the molecular mass fraction, we plot in Fig. \[fH2plots\]c $f_{H2}$ as a function of the concentration index, defined as $C\equiv R_{90}/R_{50}$ where $R_{90}$ and $R_{50}$ are the radii enclosing 90% and 50% of the $r-$band flux, respectively. The concentration index is a good proxy from the bulge-to-total ratio, as would be recovered by full two-dimentional bulge/disk decompositions [@weinmann09]. The galaxies with a CO line detection show no correlation, with a correlation coefficient of $-0.14$, and a scatter about the best-fit relation (0.317 dex) as large as the scatter in $f_{H2}$ itself (also 0.317 dex), suggesting that the presence of a bulge does not affect strongly how much of the mass in these galaxies is in the form of molecular gas. A similar result was found by @GASS1 and @fabello10 for the HI contents of similar galaxies. The only effect seen is again in the detection fraction, with the success rate of measuring the CO line dropping sharply from $100\%$ for the most disk-dominated systems, to $0\%$ for the most bulge-dominated ones.
![Stacked spectrum of all galaxies with [NUV$-r$]{}$>5.0$. Individually, the galaxies included are all non-detections in CO, and even their stacked average does not yield a detection, setting instead a stringent upper limit of $<M_{H2}/M_{\ast}>=0.0016\pm0.0005$. The vertical dotted line show an expected line width of $\sim 500$[km s$^{-1}$]{} for such a stacked signal. \[figstack\]](fig6.eps){width="84mm"}
The only parameter upon which the measured molecular mass fractions does depend significantly is colour ($r=-0.57$, $\sigma=0.260$ dex, see Fig. \[fH2plots\]d). Since [NUV$-r$]{} colour is a proxy for specific star formation rate, this dependency of $f_{H2}$ is expected, because star formation and molecular gas are known to be strongly correlated. It is interesting to note that not a single red sequence galaxy ([NUV$-r$]{} $>5.0$) has a measurable molecular gas component ($f_{H2}<0.015$ in all cases). To test this further, we stack all non-detections with [NUV$-r$]{}$>5$ using the technique described in §\[stacking\]. The result is shown in Figure \[figstack\]. Even in the stack, the red galaxies lead to a non-detection of the CO line, thus setting an even more restrictive upper limit on the molecular gas mass fraction of $0.0016\pm0.0005$, which at their median stellar mass corresponds to an upper limit on the molecular gas mass of $\sim1.1\times10^{8} M_{\odot}$. Molecular gas has been detected in early-type galaxies [e.g. in the SAURON sample, @combes07], but at levels even below this limit. Our results are therefore not in contradiction with these studies.
Mean scaling relations {#fH2_s2}
----------------------
{width="150mm"}
After having looked separately at the properties of CO detections and non-detections as a function of several global galaxy parameters (Fig. \[fH2plots\]), we combine all measurements into mean molecular gas mass fraction scaling relations.
As described in §\[sampleselect\], the COLD GASS sample has been generated to have a stellar mass distribution that is flatter than it is in a purely volume-limited sample (see Fig. \[distribs\]). When building the scaling relations, we correct for this by weighing each point according to its stellar mass. Following @GASS1, the galaxies are placed in bins of stellar mass of width 0.2 dex and assigned as a weight the ratio between the total number of galaxies in the unbiased volume-limited parent sample and the total number of COLD GASS galaxies within that same mass bin. In other words, low mass galaxies are given a higher weight in the computation of the mean scaling relations, because these galaxies are underrepresented in the COLD GASS sample compared to a volume-limited sample.
------------------- ------- ------------------ ------------------ -- ------------------ ------------------ -- ------------------ ------------------
$x$ $x_0$ $m$ $b$ $m$ $b$ $m$ $b$
$\log M_{\ast}$ 10.70 $-0.202\pm0.040$ $-1.300\pm0.592$ $-0.455\pm0.069$ $-1.607\pm1.034$ $-0.346\pm0.203$ $-1.552\pm3.038$
$\log \mu_{\ast}$ 8.70 $-0.283\pm0.019$ $-1.265\pm0.237$ $-0.595\pm0.055$ $-1.441\pm0.682$ $-0.816\pm0.127$ $-1.454\pm1.587$
$R_{90}/R_{50}$ 2.50 $-0.078\pm0.048$ $-1.258\pm0.176$ $-0.447\pm0.049$ $-1.420\pm0.185$ $-0.644\pm0.145$ $-1.388\pm0.529$
NUV$-r$ 3.50 $-0.197\pm0.012$ $-1.275\pm0.066$ $-0.219\pm0.013$ $-1.363\pm0.077$ $-0.293\pm0.013$ $-1.349\pm0.069$
------------------- ------- ------------------ ------------------ -- ------------------ ------------------ -- ------------------ ------------------
$^a$ The relations are parametrized as $\log t_{dep}({\rm H}_2)[{\rm yr}^{-1}]=m(x-x_0)+b$.
Each scaling relation is computed and plotted using three different subsamples: (1) only the galaxies with CO detections are considered, (2) both the detections and non-detections are used, and the upper limit on $f_{H2}$ is used for the non-detections, and (3) both the detections and non-detections are used, but this time a value of $f_{H2}=0.0$ is assigned to the non-detections. In all cases, the weighted mean of $\log f_{H2}$ is then calculated in equally populated bins of either [$M_{\ast}$]{}, [$\mu_{\ast}$]{}, concentration index or [NUV$-r$]{} colour, and the relations are fitted, weighting galaxies according to their stellar mass. The resulting mean relations are plotted in Figure \[fH2scalrel\], with the error bars representing the uncertainty on the position of $<\log f_{H2}>$ in equally populated bins, as determined by bootstrapping: the error is the standard deviation in the value of $<\log f_{H2}>$ for 1000 resamples of the original data in each bin, with repetitions. The best-fit linear relations are also plotted and summarized in Table \[ALLtab\]. Note that in case (3), because of the null values of $f_{H2}$ for the non-detections, we cannot directly average and fit the values of $\log f_{H2}$ as we do for cases (1) and (2). Instead, we measure the logarithm of the mean value of $f_{H2}$ in each bin, and fit these average values. This is the relation plotted in Figure \[fH2scalrel\] and given in Table \[ALLtab\] for case (3).
The choice between setting the non-detections to their upper limits or to a constant value of zero only significantly affects the scaling relations at large values of [$\mu_{\ast}$]{}, $C$ and [NUV$-r$]{} where few galaxies have a detected CO line. Both sets of scaling relations are included in Table \[ALLtab\], labeled as $<\log M_{H2}/M_{\odot}>_{lim}$ and $<\log M_{H2}/M_{\odot}>_{0}$ for the upper limit and zero value cases, respectively.
The mean molecular gas mass fraction is a roughly constant function of stellar mass, both for detections alone and when including non-detections, as shown in Fig. \[fH2scalrel\]a. This is a consequence of the flat detection rate of the CO line as a function of [$M_{\ast}$]{}. Adding non-detections turns the mostly-flat relations between $f_{H2}$ and both [$\mu_{\ast}$]{} and $C$ into monotonically decreasing functions. The strongest correlation is still with [NUV$-r$]{} colour, both before and after including non-detections. Because [NUV$-r$]{} colour is a proxy for specific star formation rate, its correlation with $f_{H2}$ is not surprising, given e.g. the Kennicutt-Schmidt relation. The correlation of $f_{H2}$ with [NUV$-r$]{} could therefore be seen as a consequence rather than a cause.
For the other parameters describing the underlying properties of the galaxies, independently of the current star formation rate ([$M_{\ast}$]{}, [$\mu_{\ast}$]{}, concentration index), there is very little dependence of the measured values of $f_{H2}$. In §\[fH2\_s1\], we showed that no matter their stellar mass, about half of our sample has detectable CO line emission, which converts into a molecular mass fraction of $\sim 6\%$ across the stellar mass interval sampled. On the other hand, while the molecular mass fraction of $detected$ galaxies is mostly independent of [$\mu_{\ast}$]{}, the detection fraction is a strong function of that quantity. These observations trace a picture where: (1) the conditions required for the formation/consumption of molecular gas are a strong function of [$\mu_{\ast}$]{} (and concentration index) but not of [$M_{\ast}$]{}, but (2) when these conditions are met, a roughly constant fraction of the stellar mass is found in the form of molecular gas.
The relationship between HI and $H_2$ {#compHI}
=====================================
![Comparison between atomic and molecular hydrogen gas masses. COLD GASS galaxies with detections of both the HI and CO lines are plotted as filled and open blue circles, for secure and tentative CO detections, respectively. The arrows shows limits in the cases of non-detecion of either or both the HI and CO lines. The best bisector linear fit to the detections is show as a solid line, and the $\pm 1.5 \sigma$ region around this fit marked with dashed lines. For comparison, we overplot as filled squares the integrated measurements for the HERACLES galaxies with $M_{\ast}>10^{10}M_{\odot}$, taken from @leroy08. \[HIH2\]](fig8.eps){width="84mm"}
Under the assumption that molecular gas forms out of lower density clouds of atomic gas, one might naïvely expect a tight correlation between $M_{HI}$ and $M_{H2}$. The actual situation is, however, quite different, as seen in Figure \[HIH2\]. Within the subsample of galaxies detected both in HI and CO, the fraction $M_{H2}/M_{HI}$ varies greatly, from 0.037 (G13775) up to 4.09 (G38462). For galaxies with both CO and HI-detections, the correlation coefficient between $\log M_{HI}$ and $\log M_{H2}$ is $r=0.37$, indicating that the two quantities are only weakly correlated. A bisector linear fit to the same subsample reveals that on average, $M_{H2}$ is 0.295 times the value of $M_{HI}$, with a large scatter of 0.41 dex in $\log(M_{H2}/M_{HI})$ (see Fig. \[HIH2\]).
To further investigate the relationship between $M_{H2}$ and $M_{HI}$, we look into how the ratio between these two quantities varies as a function of different physical parameters. We define the molecular fraction as the ratio between the molecular hydrogen gas mass and the atomic gas mass of the system: $$R_{mol}=\log \left( \frac{M_{H2}}{M_{HI}} \right).
\label{fmol}$$ In Figure \[MH2MHI\], we plot [$R_{mol}$]{} as a function of the same four global parameters as in the previous figures. Galaxies with non-detections in both HI and $H_2$ are not plotted, as $R_{mol}$ is completely unconstrained in these cases. he best fit linear relations, taking into account the weights correcting for the flat $\log M_{\ast}$ distribution, are measured in two ways: (1) including only the secure detections in HI and [$H_2$]{}, and (2) including the non-detections in either of these quantities as lower or upper limits, respectively. In Figure \[MH2MHI\], we also report the Pearson correlation coefficients and the scatter around these relations. The best-fit scaling relations are given the form $R_{mol}=m(x-x_0)+b$, with the parameters presented in Table \[Rmoltab\].
{width="150mm"}
------------------- ------- ------------------ ------------------ -- ------------------ ------------------
$x$ $x_0$ $m$ $b$ $m$ $b$
$\log M_{\ast}$ 10.70 $ 0.425\pm0.097$ $-0.387\pm1.464$ $ 0.303\pm0.084$ $-0.489\pm1.253$
$\log \mu_{\ast}$ 8.70 $ 0.533\pm0.128$ $-0.451\pm1.562$ $ 0.346\pm0.142$ $-0.562\pm1.772$
$R_{90}/R_{50}$ 2.50 $ 0.152\pm0.031$ $-0.475\pm0.113$ $-0.106\pm0.098$ $-0.571\pm0.367$
NUV$-r$ 3.50 $ 0.206\pm0.057$ $-0.436\pm0.289$ $ 0.075\pm0.032$ $-0.575\pm0.169$
------------------- ------- ------------------ ------------------ -- ------------------ ------------------
$^a$ The relations are parametrized as $\log M_{H_2}/M_{HI}=m(x-x_0)+b$.
The first panel shows that [$R_{mol}$]{} is a very weakly increasing function of [$M_{\ast}$]{} ($r=0.23$). Both with and without the non-detections included, the slope of the relation is positive at the 3$\sigma$ significance level (see Table \[Rmoltab\]). As demonstrated previously, $f_{H2}$ does not appear to depend significantly on [$M_{\ast}$]{}(Fig. \[fH2plots\]a), while $f_{HI}$ is a fairly strongly declining function of [$M_{\ast}$]{} [@GASS1]. The combination of these two trends produces the weak increase in $R_{mol}$ as a function of stellar mass.
@GASS1 also reported a strong anti-correlation of $f_{HI}$ with the stellar mass surface density, [$\mu_{\ast}$]{}. For galaxies with CO detections, the values of $f_{H2}$ show a similar, but considerably weaker trend as a function of [$\mu_{\ast}$]{}. In their resolved study, @leroy08 show that the molecular-to-atomic ratio is a strong function of local properties within the disks of spiral galaxies. In particular, they find a dependence on stellar mass surface density, with the molecular fraction steadily increasing from surface mass densities of $10^{7.5}$ to $10^9$ [$M_{\odot}$]{} kpc$^{-2}$. Our molecular fractions are smaller because our measurements are integrated over entire galaxies, but the same qualitative trend is observed for our global measurements.
Even though $f_{HI}$ and $f_{H2}$ show different dependencies on [$\mu_{\ast}$]{}, the fraction of galaxies with non-detections as a function of [$\mu_{\ast}$]{} in the HI and CO samples exhibits very similar behaviour. As shown in Figure \[fH2plots\]b and in @GASS1, there is a critical mass surface density of $\mu_{\ast}=10^{8.7}$ [$M_{\odot}$]{} kpc$^{-2}$ below which all galaxies have a sizeable HI and $H_2$ component, and above which cold gas seems to have mostly disappeared. There are similar thresholds in concentration index and [NUV$-r$]{} colour (see e.g. our detection fractions in Fig. \[detfrac\]). However, while the mean value of $f_{HI}$ never falls below $\sim2\%$, even at high [$\mu_{\ast}$]{} or [NUV$-r$]{} colour [@fabello10], $f_{H2}$ drops sharply below that level, as shown in Fig. \[fH2scalrel\] and evidenced by the results of our stacking experiment (Fig. \[figstack\]). Furthermore, even though @GASS1 found some red sequence galaxies with a surprisingly large HI component, none of these galaxies have a sizeable molecular gas mass; none of the galaxies with [NUV$-r$]{}$>5$ are securely detected in CO. It therefore seems that above our empirical thresholds in [$\mu_{\ast}$]{}, concentration index and [NUV$-r$]{} colour, an increasing fraction of galaxies with any form of cold gas at all [*appears to be dominated by atomic gas*]{}. This result is also striking in Fig. \[MH2MHI\]c, where we see the same population of atomic-gas dominated galaxies (i.e. the upper limits in $R_{mol}$) almost exclusively at $C>2.6$. We note that this critical value of concentration index (C=2.6) corresponds to the observed transition between late- and early-type galaxies [e.g. @shimasaku01; @nakamura03; @weinmann09].
Finally, Figure \[MH2MHI\]d shows that [$R_{mol}$]{} is an increasing function ($r=0.25$) of [NUV$-r$]{} colour for galaxies with detections of both HI and CO. @GASS1 reported a strong anti-correlation between HI mass fraction and colour. Figure \[fH2plots\]d shows that there is also a fairly strong anti-correlation bewteen $f_{H2}$ and [NUV$-r$]{} colour. The fact that the anti-correlation of $f_{HI}$ with [NUV$-r$]{} appears to “win” probably reflects the fact that HI is dominant in regions such as outer galaxy disks where dust content is low and most of the starlight emitted by forming populations is emitted at UV wavelengths, whereas molecular gas tends to occur in the inner regions of galaxies where dust content is high and much of the light from young stars may be emitted in the infrared. In future work, we plan to look more carefully at these issues.
Comparison with previous work {#archive}
=============================
{width="165mm"}
To put the new COLD GASS results in context, we assembled CO data from the literature for 263 nearby galaxies in the SDSS survey. They are taken from the compilations of @bettoni03, @yao03, @casasola04, @albrecht07, @komugi08 and @obreschkow09. When multiple measurements are found for the same galaxy, the newest is assumed to supersede previous values. The values of [$M_{H_2}$]{} were then homogenized to the best of our ability using a common conversion factor ($X_{CO}=2.3\times10^{20}$ cm$^{-2}$ \[K [km s$^{-1}$]{}\]$^{-1}$) and cosmology ($H_0=70$ [km s$^{-1}$]{} Mpc$^{-1}$). In addition, the SDSS photometry was reprocessed using the same technique used for the COLD GASS galaxies (see §\[data\]), and used to measure reliable and homogeneous stellar masses.
In Fig. \[figarch\]a, we show the relation between $M_{H2}/M_{\ast}$ and $M_{\ast}$ in this reference sample. The molecular gas mass fraction has a spread of more than two orders of magnitude, and does not appear to correlate with stellar mass. A significant fraction of this observed scatter can be attributed to measurement errors and inhomogeneities in the sample. The vast majority of these galaxies have $z<0.02$, and therefore tend to be significantly larger than the typical observing beam, resulting in important aperture problems. Additional contributions to the artificially large scatter include different telescope calibrations, low $S/N$ detections and selection on IR luminosity, which tends to bias $f_{H2}$ high.
As a comparison, we also show as contours in Figure \[figarch\]a the relations produced by @fu10 through semi-analytic modeling of galaxy formation including detailed prescriptions for the break-up of gas between the atomic and molecular phases. The models predict a significantly smaller range in $M_{H2}/M_{\ast}$ than seen in the literature compilation. But it is also clear that a systematically measured set of galaxies will produce more consistent results, for example the THINGS/HERACLES sample [@THINGS; @leroy09], which is also shown for comparison in Figure \[figarch\].
The equivalent relation from COLD GASS is plotted in Figure \[figarch\]b, showcasing the significantly reduced observational scatter compared to the literature compilation, but the increased dynamic range due to the rigorously measured upper limits for the CO non-detections. We note that the COLD GASS and HERACLES galaxies span a similar region in the plots. The HERACLES galaxies provide resolved CO maps for a much smaller sample of galaxies, so the two approaches are highly complementary.
The main results presented in this paper are overall qualitatively similar to some earlier observations. For example, @sage93 found that $M_{H2}/M_{dyn}$ is independent of morphology, just like we find $M_{H2}/M_{\ast}$ to be independent of concentration index. However, they found that $M_{H2}/M_{HI}$ is a strong function of Hubble type [see also @young89], which we do not. Based on these studies, we would have expected to find a significant population of early-type galaxies with large values of $M_{H2}/M_{HI}$. We see no evidence for such systems in our sample.
Much of the earlier work regarding CO in nearby galaxies focussed on the trends between $M_{H2}$, $M_{HI}$ and morphology, interaction state [e.g. @braine93b], or far infrared properties of the systems [e.g. @sanders85]. Using a set of galaxies still mostly based on the infrared-based FCRAO sample, @bothwell09 however derive relations similar to ours, between gas fraction (atomic and molecular) and $B-$band luminosity. Their findings are qualitatively similar to ours; they see that $M_{HI}/M_{\ast}$ decreases with luminosity, but $M_{H2}/M_{\ast}$ does not. The breakthrough here is that COLD GASS allows us to [*quantitatively*]{} describe, in an unbiased sample, how the molecular gas component varies with several key physical parameters which are at the basis of the theoretical effort towards understanding the star formation process.
Summary
=======
We are conducting COLD GASS, a legacy survey for molecular gas in nearby galaxies. We target at least 350 massive galaxies ($M_{\ast}>10^{10}$[$M_{\odot}$]{}) in the CO(1-0) emission line with the IRAM 30m telescope. Because the survey is unbiased, it will provide us with a complete view of the molecular gas properties of massive galaxies in the local universe, as well as the relations between molecular gas and other global galaxy properties. The stellar mass and redshift ranges also ensure that we recover the total CO line flux of the galaxies with a single pointing of the IRAM 30m telescope, and that a single CO luminosity to $M_{H2}$ conversion factor is likely adequate. Finally, our observations provide stringent upper limits on molecular gas fraction $M_{H2}/M_{\ast}<0.015$ in the case of CO non-detections.
In this paper, we present a catalog of CO(1-0) fluxes and $H_2$ masses (or upper-limits) for the first [222]{} galaxies observed as part of COLD GASS, and report on their properties:
1. The detected molecular gas mass fractions ($M_{H2}/M_{\ast}$) are in the range of 0.9% to 20%, with a mean value $<M_{H2}/M_{\ast}>=0.066\pm0.039$. The mean gas mass fraction among the [*detected*]{} galaxies does not vary strongly with any global galaxy property except colour.
2. We detect the CO(1-0) line in $\sim50\%$ of COLD GASS galaxies, and while the detection rate is independent of stellar mass, it is a strongly decreasing function of stellar mass surface density, concentration index and [NUV$-r$]{} colour. None of the [68]{} galaxies redder than [NUV$-r$]{}=5 were detected, and stacking them leads to a non-detection and a stringent upper limit of $<M_{H2}/M_{\ast}>=0.0016\pm0.0005$.
3. The mean molecular gas mass fraction (averaged over galaxies with detections and non-detections of the CO line) is a roughly constant function of stellar mass, but a decreasing function of stellar mass surface density and concentration index. The observed trends are weaker than those observed by @GASS1 for the atomic gas mass fraction in a similar sample of galaxies. Of all parameters investigated here, the molecular gas correlates most strongly with [NUV$-r$]{} colour, which is a tracer of specific star formation.
4. The molecular-to-atomic mass ratio, $R_{mol}$, has a mean value of $30\%$ over the entire sample. It is a weakly increasing function of $M_{\ast}$ ($r=0.23$) and $\mu_{\ast}$ ($r=0.37$).
One result that we wish to highlight is the existence of sharp thresholds in galaxy structural parameters such as stellar surface mass density, below which most galaxies have measurable atomic and molecular gas components, but above which the detection rate of both the HI and CO lines drops drastically. This result was discussed previously for the HI in @GASS1. The fact that [*the same sharp thresholds also apply to the CO*]{} strongly suggests that onset of “quenching” processes in galaxies was associated with a change in their structure. We note that the same sharp drop in cold gas content is not seen as a function of stellar mass. Intriguingly, atomic gas dominates in the minority of galaxies that are above threshold and that have significant cold gas content. One possible interpretation is that re-accretion of gas may still be possible following the quenching event. In future work, we will examine galaxies on either side of the “quenching threshold” in more detail. We will also look more closely at the relationhip between molecular gas content and star formation.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work is based on observations carried out with the IRAM 30 m telescope. IRAM is supported by INSU/CNRS (France), MPG (Germany), and IGN (Spain).
We are grateful to Nario Kuno for providing us with total fluxes for his sample of nearby galaxies. We wish to thank the staff of the IRAM observatory for their tremendous help in conducting our observations. We thank the anonymous referee for a constructive and helpful report.
RG and MPH are supported by NSF grant AST-0607007 and by a grant from the Brinson Foundation.
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Spectral Gallery {#atlas}
================
Galaxies with CO line detections
--------------------------------
In Figures \[spectra\_det\]-\[lastspec\_det\], we present SDSS imaging and IRAM spectra for the COLD GASS galaxies present in this data release for which we have securely detected the CO(1-0) line. These are defined as those with $S/N>5$ in the CO line. The tentative detections, those with $S/N<5$, are shown in Figures \[spectra\_tent\] and \[spectra\_tent\_last\]. All galaxies with a secure CO line detection have colours bluer that [NUV$-r$]{}=5.0, but cover uniformly the stellar mass range probed by COLD GASS.
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Galaxies with non-detection of the CO line
------------------------------------------
For completeness, we show in Figures \[spectra\_nondet\]-\[spectra\_nondet\_last\] the SDSS three-colour images of the CO non-detections. These objects tend to be red ([NUV$-r$]{}$>5$), early-type looking ($C>2.6$) galaxies.
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Offset pointings
----------------
As described in §\[offsets\], we perform additional off-center pointings for a small fraction of the COLD GASS sources. They are the larger galaxies ($D_{25}>40\arcsec$), with a strong detection in the central pointing. These offsets are taken 16(or three-quarters of the main beam) away from the centers. In Figures \[spectra\_off\] and \[spectra\_off2\], we show the SDSS images and IRAM spectra for the offset pointings. The data reduction process is identical to that used for the central pointings and described in Section \[linemes\]. The emission lines are identified by examining the spectra at the expected positions based on the redshift on each galaxy, and the width of CO the line in the central pointing is used as additional information. Fluxes are measured by integrating over the region identified though this process, and the flux ratio between offset and central pointings is used to determine an appropriate aperture correction for each galaxy using the technique described in Section \[offsets\].
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[^1]: E-mail: amelie@mpe.mpg.de
[^2]: During the pilot observations of 2009 June, we took offset pointings one full beam from the central position. These galaxies are identified clearly by asterisks in Col. 6 of Tab. \[COtab\], and aperture corrections are performed using a version of Eq. \[Icor2\] appropriate for these larger offsets.
[^3]: Calculated from antenna temperature units using the conversion $S/T_a^{\ast}=6.0 {\rm Jy/K}$, specific for the IRAM 30m at our observing frequency of 111GHz.
[^4]: We warn the reader that our definition of $f_{H2}$ differs from some previous studies, where $f_{H2}$ is also defined as $M_{H2}/M_{dyn}$ or $M_{H2}/(M_{\ast}+M_{HI}+M_{H2})$.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We consider a generalized two-species population dynamic model and analytically solve it for the amensalism and commensalism ecological interactions. These two-species models can be simplified to a one-species model with a time dependent extrinsic growth factor. With a one-species model with an effective carrying capacity one is able to retrieve the steady state solutions of the previous one-species model. The equivalence obtained between the effective carrying capacity and the extrinsic growth factor is complete only for a particular case, the Gompertz model. Here we unveil important aspects of sigmoid growth curves, which are relevant to growth processes and population dynamics.'
address:
- |
Faculdade de Filosofia, Ciências e Letras de Ribeirão Preto (FFCLRP),\
Universidade de São Paulo (USP)\
Avenida Bandeirantes, 3900\
14040-901, Ribeirão Preto, São Paulo, Brazil.
- |
Departamento de Ciências Exatas (DEX),\
Universidade Federal de Lavras (UFLA)\
Caixa Postal 3037\
37200-000 Lavras, MG, Brazil.
- Instituto Nacional de Ciência e Tecnologia em Sistemas Complexos
author:
- Brenno Caetano Troca Cabella
- Fabiano Ribeiro
- Alexandre Souto Martinez
title: 'Effective carrying capacity and analytical solution of a particular case of the Richards-like two-species population dynamics model'
---
,
and
,
population dynamics (ecology) , generalized logarithmic and exponential functions , growth models , Richards model ,two-species model 89.75.-k , 87.23.-n , 87.23.Cc ,
Introduction {#intro}
============
Population dynamics models are useful when one tries to understand, describe or predict the behavior of a wide range of time dependent processes in several disciplines. To easily formulate and write the solutions of the population dynamics model, we present a one-parameter generalization of the logarithmic and exponential functions. This generalization has been first introduced in the context of non-extensive statistical mechanics [@tsallis_1988; @tsallis_qm; @arruda_2008; @martinez:2008b; @Martinez:2009p1410]. The generalized logarithm or the so-called $\tilde{q}$-*logarithm function* is defined as: $$\ln_{\tilde{q}}(x) = \lim_{\tilde{q}' \rightarrow \tilde{q}}\frac{x^{\tilde{q}'} -1 }{\tilde{q}'} = \int_1^x \frac{dt}{t^{1-\tilde{q}}} \; .
\label{eq-ln-q}$$ The natural logarithm function is retrieved for $\tilde{q} \rightarrow 0$. The inverse of the $\tilde{q}$-logarithm function is the $\tilde{q}$-*exponential function* $$e_{\tilde{q}}(x) =
\left\{ \begin{array}{ll}
\lim_{\tilde{q}^{'} \to \tilde{q}} (1+ \tilde{q}^{'} x)^{1/\tilde{q}' } & ,\textrm{ if $\tilde{q}x > -1$} \\
0 & , \textrm{ otherwise}
\end{array} \right. \; ,
\label{def-eq}$$ so that for $\tilde{q} =0$, one retrieves the usual exponential function. The use of these functions is convenient since it has allowed us to find and simplify (using their properties) the solution of the models with time-dependent intrinsic and extrinsic growth rates.
The simplest way to deal with population growth is to consider *one-species models*. In these models, individuals do not explicitly interact with the external ones and, at time $t$, the number of individuals is $N(t) \ge 0$, with initial condition $N_0 \equiv N(0) > 0$. The parameters are the intrinsic growth rate $\kappa > 0$ and the environment carrying capacity $K = N(\infty) > 0$, which takes into account all possible interactions among individuals and resources [@blanco_1993]. If $N(t)/K \ll 1$, a general model is the *von Foerster et al.* model [@vonfoerster], where the per capita growth rate is $d \ln N/ dt = \kappa N^{\tilde{q}}$. Considering $\tau \equiv \kappa t \ge 0$ as a dimensionless time unity, the model solution is: $N(\tau) = N_0 e_{-\tilde{q}}[\tau/(\tilde{q} T)]$, with $N_0 = (\tilde{q} T)^{-1/\tilde{q}}$ and produces a population size divergence at a finite time $T$, obtained from the $\tilde{q}$-exponential $\tilde{q} \tau /(\tilde{q} T) > 1$ [@Strzalka:2009p2964]. As $\tilde{q} \to 0$, the exponential population growth (*Malthus* model) $N(t) = N_0 e^{\kappa t}$, with divergence at an infinite time, is obtained. The population size divergence can be dismantled considering a finite carrying capacity.
The richness ecological community cannot be properly described only by the one-species models. The interactions between species becomes better formalized with the (Malthus-like) Lotka-Volterra equations [@murray], which only explains prey-predator behavior in its original formulation and presents stability problems. Besides the predation interaction, there are many other different kinds of interaction taking place between two biological species. For instance, if species negatively affects each other, when they occupy the same ecological niche and use the same resources, there is *competition*. If species favor each other, there is *mutualism* or *symbiosis*. These ecological richness is partially appreciated in the competitive (Verhulst-like) Lotka-Volterra model studied in Ref. [@ribeiro_2011_2]. This model presents complete analytical solutions. It has a non-trivial phase diagram and solves the stability problem of the Malthus-like two species model.
Here, we consider a generalized two-species population dynamic model and analytically solve it for particular ecological interactions. These two-species models can be simplified to a one-species model, with a time dependent extrinsic growth factor. With a one-species model with an effective carrying capacity, one is able to retrieve the steady state solutions of the previous one-species model. The equivalence obtained between the effective carrying capacity and the extrinsic growth factor is complete only for the Gompertz model.
Our presentation is organized as follows: In the Sec. \[gen1\], the Richards model is presented in terms of the generalized functions, which has the Verhulst and Gompertz models as particular cases. We show that the steady state solution of the Richards model with extrinsic growth rate is the same for the Richards model without extrinsic growth rate, but with a modified carrying capacity. Also, for a particular case of the Richards model ($\tilde{q} \rightarrow 0$), that is the Gompertz model, not only the steady state is the same but also the whole system evolution (transient). In the Sec. \[gen2\], we present a generalized model of two interacting species. We show that the interaction between species can also be interpreted as an extrinsic growth factor. This allows one to represent a two-species system by a one-species model with a modified carrying capacity. In Sec. \[conclusion\], we present our final remarks.
Generalized one-species model {#gen1}
=============================
The growth of individual organisms [@laird65], tumors [@bajzer96] and other biological systems [@zwietering90] are well described by mathematical models considering a finite carrying capacity $K$, which lead to sigmoid growth curves [@boyce_diprima; @murray; @Keshet]. In this context, it is convenient to express the population size $N$ with respect to its equilibrium value, i.e. $p=N/K$. The *Richards* model [@richards_1959] $$\frac{d }{d \tau} \ln p = \ln_{\tilde{q}} p,$$ binds the *Gompertz* ($\tilde{q} = 0$, $d \ln p/d \tau = \ln p$) and the *Verhulst* ($\tilde{q} = 1$, $d \ln p/d \tau = 1 - p$) models through the $\tilde{q}$ parameter, which microscopic interpretation is given in Refs. [@martinez:2008b; @Mombach:2002p1065; @donofrio]. This parameter is related to the range of interaction between the individuals that compose the population and the fractal pattern of the population structure. This generalization is corroborated by several different approaches in terms of Tsallis statistics [@Strzalka:2008p2511] and logistic equations of arbitrary order [@grabowski:2010p3081] for the classical Verhulst and Malthus models. The solution of the Richards model is: $$p(\tau)=\frac{1}{e_{\tilde{q}}\{\ln_{\tilde{q}}[1/p_0]e^{-\tau}\}},$$ whose initial condition is $p_0 \equiv p(0)$ and the steady state solution is $p^* = p(\infty) = 1$ [@brenno_2011].
Adding an extrinsic growth rate $\epsilon$ to the Richards’ model, one has the so called *Richards-Schaefer* model and respective solution: $$\begin{aligned}
\frac{d \ln p}{d\tau} & = & - \ln_{\tilde{q}}p + \epsilon
\label{steady_state_ii} \\
p(\tau) & = & \frac{e_{\tilde{q}}(\epsilon)}{e_{\tilde{q}}\{\ln_{\tilde{q}}[e_{\tilde{q}}(\epsilon)/p_0]e^{-[1+\tilde{q}\epsilon]\tau}\}} \; ,
\label{transient_ii}\end{aligned}$$ with steady state $p(\infty) \equiv p^*=e_{\tilde{q}}(\epsilon)$, which behaves as an order parameter. The Richards’ model solution is retrieved for $\epsilon = 0$. The parameter $\epsilon$ plays the role of an external factor that removes or insert individuals in the population. For instance, it can be an interaction factor between different species, or the effect of the cancer treatment when one deals with tumor cells population. Survival of the species is obtained when $\tilde{q} \epsilon > -1$, and extinction occurs otherwise. The critical value $\epsilon^{(c)} = -1/\tilde{q}$ separates the extinction from the survival phase, which behaves as $p^* \sim [\epsilon - \epsilon^{(c)}]^{1/\tilde{q}}$ near the critical point. This transition occurs for $\tilde{q} \ne 0$, so that only the Gompertz model does not present the extinction-survival transition.
From the analytical solution of Eq. (\[transient\_ii\]), we show that the insertion ($\epsilon > 0$) or removal ($\epsilon < 0$) of individuals from the population modifies the steady state solution. Instead of considering the population size with respect to the medium (bare) carrying capacity, we propose to consider the population size with respect to a rescaled carrying capacity $K'= e_{\tilde{q}}(\epsilon)K$. It means that the $N$ population individuals live in an environmental with an effective carrying capacity $K'$, but now without insertion or removal of individuals. The population size with respect to the new equilibrium value is $p'= N/K'= p/e_{\tilde{q}}(\epsilon)$ and one has the Richards’ model and its respective solution: $$\begin{aligned}
\frac{d \ln p' }{d\tau} & = & -\ln_{\tilde{q}}\left[ p'\right] \; ,
\label{steady_state_iii} \\
p'(\tau) & = & \frac{p(\tau)}{e_{\tilde{q}}(\epsilon)} =\frac{1}{e_{\tilde{q}}[\ln_{\tilde{q}}(1/p_0')e^{-\tau}]} \ ;
\label{transient3}\end{aligned}$$ with steady state: $p'(\infty) = (p')^* = 1$, which of course leads to: $p^*=e_{\tilde{q}}(\epsilon)$ in the original variable. Observe that the Eq. (\[steady\_state\_iii\]) is valid only for the survival phase, i.e. $\tilde{q} \epsilon > -1$, otherwise $K'$ vanishes.
Comparing the solutions of Eqs. (\[transient\_ii\]) and (\[transient3\]), one sees that the considered models present the same steady state solutions. Nevertheless, they have different transient behavior, except for Gompertz model ($\tilde{q}\rightarrow 0$), where the equivalence between the original and rescaled models is complete. The plots of Fig. \[fig:equivalence\] illustrate this behavior. In fact, this property comes from the argument of the exponential function: while in Eq, (\[transient\_ii\]) this argument is $-\tau$ , in Eq. (\[transient3\]) the argument is $-(1+\tilde{q}\epsilon)\tau$. Thus the evolution of the two systems become identical only if $\tilde{q} \epsilon =0$. As we are dealing with $\epsilon \ne 0$, the evolution of the two model is the same only when $\tilde{q} \rightarrow 0$ i.e. the Gompertz model.
![Plot of population $p(\tau)$ \[Eqs. \[transient\_ii\] and \[transient3\]\] as a function of $\tau$, with: $p_0=0.1$, $\epsilon=-0.3$, $\tilde{q}=-1$, $\tilde{q} \rightarrow 0$ (Gompertz model) and $\tilde{q}=1$ (Verhulst model). The full equivalence of the models with extrinsic growth factor and modified carrying capacity is only obtained for the steady state solutions. Nevertheless, for $\tilde{q} \rightarrow 0$, the transient solution is also equivalent.[]{data-label="fig:equivalence"}](equivalence.eps){width=".95\columnwidth"}
Generalized two-species model {#gen2}
=============================
Here, we introduce a generalization of the Verhulst-like two species model, considered in Ref. [@ribeiro_2011_2], which we call the *Richards-like two-species model*. We consider the Richards term instead of the Malthus one in the original Lotka-Volterra equations. In this type of model, the two species interact according to the Lotka-Volterra equation, however the species also suffer from inter-species competition. As in Ref. [@ribeiro_2011_2], we let the interaction parameter vary from negative to positive values to cover all possible ecological regimes. The proposed model is written as: $$\begin{aligned}
\frac{d p_1}{d\tau} &=& p_1[ \ln_{\tilde{q}_1} p_1 + \epsilon_1 p_2] \label{generalized-model_1} \\
\frac{d p_2}{d\tau} &=& \rho p_2[ \ln_{\tilde{q}_2} p_2 + \epsilon_2 p_1] \label{generalized-model_2} \; , \end{aligned}$$ where $p_i = N_i/K_i \ge 0$, for $i=1,2$, where $N_i \ge 0$, $\kappa_i$ and $K_i > 0$ are the number of individuals (size), net reproductive rate, and the carrying capacity of species $i$, respectively. The carrying capacity $K_1$ represents the restriction on resources that comes from any kind of external factors that do not have to do with species 2 and similarly for $K_2$. Time is measured with respect to the net reproductive rate of species 1, $\tau = \kappa_1 t \ge 0$. The scaled time is positive since we take the initial condition at $t_0 = 0$. Moreover, the two net reproductive rates form a single parameter $\rho = \kappa_2/\kappa_1 > 0$, fixing a second time scale: $\tau' \equiv \rho \tau = \kappa_2 t$. In the right side of Eq. (\[generalized-model\_1\]), the term $p_1 \ln_{\tilde{q}_1} p_1$ represents the competition between individuals of the same species (intraspecific competition) and $\epsilon_1 p_1 p_2$ represents the interaction between individuals of different species (interspecific interaction) [@murray; @Keshet]. The right side of Eq. (\[generalized-model\_2\]) behaves similarly. The non-dimensional population interaction parameters are $\epsilon_1 $ and $\epsilon_2$, which are not restricted and may represent different ecological interactions. Contrary to $\rho$, which has no major relevance to this model (since we consider only $\rho>0$), the product $\epsilon_1\epsilon_2$ plays an important role, so that $\epsilon_1 \epsilon_2 < 0$ means predation; $\epsilon_1\epsilon_2 = 0$ means commensalism, amensalism, or neutralism; and $\epsilon_1 \epsilon_2 > 0$ means either mutualism or competition. For $\tilde{q}_1 = \tilde{q}_2 = 1$, one retrieves the model of Ref. [@ribeiro_2011_2].
Here, we restricted ourselves to the particular case when $\epsilon_1 \epsilon_2 = 0$. Thus, only one species is affected by the other or they do not interact at all. Closed analytical solutions can be found in this particular case. Consider that the individuals of species $1$ is unaffected by species $2$. In this case, species $1$ is described by Eqs. \[steady\_state\_ii\] and \[transient\_ii\], so that: $$\begin{aligned}
\frac{d \ln p_1}{d\tau} & = & - \ln_{\tilde{q}_1}p_1
\label{steady_state_i_a} \\
p_1(\tau) & = & \frac{1}{e_{\tilde{q}_1}[\ln_{\tilde{q}_1}(p_{1,0}^{-1})e^{-\tau}]} \; ,
\label{transient_i_a}\end{aligned}$$ where the initial condition is $p_{1,0} = p_1(0)$ and steady state: $p_1(\infty) = p_1^*=1$. The individuals of species $2$ are affected by species $1$ through an extrinsic growth factor $\epsilon'_2(\tau)$, and one writes the equation for species $2$ as: $$\begin{aligned}
\frac{d \ln p_2}{d\tau} & = & -\rho \left[ \ln_{\tilde{q}_2}p_2-\epsilon_2'(\tau) \right] \;
\label{steady_state_i}\end{aligned}$$ where, $$\epsilon'_2(\tau)=\epsilon_2 p_1(\tau)=\frac{\epsilon_2}{e_{\tilde{q}_1}[\ln_{\tilde{q}_1}\left(p_{10}^{-1}\right) e^{-\tau}]} \; .
\label{epsilon2l}$$ The solution of Eq. (\[steady\_state\_i\]) has been obtained in Ref. [@brenno_2011]: $$\begin{aligned}
p_2(\tau) & = & \frac{e_{\tilde{q}_2}[\epsilon_2'(\tau)]}
{e_{\tilde{q}_2}\left\{
\ln_{\tilde{q}_2} \left\{\frac{e_{\tilde{q}_2} [\epsilon_2'(0)]}{p_{2,0}} \right\}
\frac{e_{\tilde{q}_2}[\epsilon_2'(0)]}
{e_{\tilde{q}_2}[\epsilon_2'(\tau)]}
e^{- \left[1 + \tilde{q}_2 \overline{\epsilon}_2'(\tau) \right] \rho \tau}
\right\}} \; ,
\label{transient_i}\end{aligned}$$ where the initial condition is $p_{2,0} = p_2(0) $ and steady state: $p_2(\infty) = p_2^*=e_{\tilde{q}_2}(\epsilon_2)$. In Eq. (\[transient\_i\]), $\overline{\epsilon}_2'(\tau) = (1/\tau) \int_0^{\tau} d \tau' \epsilon_2'(\tau')$ is the mean value of $\epsilon_2'(\tau)$ up to $\tau$.
For $\epsilon_2 < 0$ ($\epsilon_2 > 0$), one has amensalism (commensalism), so that species $1$ adversely (positively) affects species $2$. The bread mold penicillium is a common example of amensalism. The analytical solution for the amensalism and commensalism regimes is obtained considering the time dependent extrinsic growth factor as a function of species $1$, according to Eq. (\[epsilon2l\]). Penicillium secretes penicillin, a chemical that kills bacteria. Consider the two-species model of Eqs. (\[steady\_state\_i\_a\]) and (\[steady\_state\_i\]), with $\epsilon_2<0$, the interaction term $\epsilon_2'$ can be interpreted as an extrinsic growth rate of a one-species model. As we have seen, an extrinsic factor can be incorporated into the carrying capacity. In this way, a two species model can be interpretated as a one-species model, with modified carrying capacity, as depicted in Fig \[fig:penicillium\].
![Representation of the amensalism ecological interaction between the bread mold and bacteria. A two-species model can be simplified to an one species model with a modified carrying capacity (gray area). [**(a)**]{} Bacteria occupying all the available space with carrying capacity $K$. [**(b)**]{} Restricted bacteria growth due to the interaction with bread mold, carrying capacity $K'<K$.[]{data-label="fig:penicillium"}](bact.eps){width=".95\columnwidth"}
Species 2 evolves as a time dependent one-species (Richards-Schaefer) model, whose solution has been obtained in Ref. [@brenno_2011] and extinction occurs for $\epsilon_2'(\tau) \tilde{q}_2 < -1$. The evolution of both species is presented in Fig. (\[amensa-out-eq\]) for two different values of $\tilde{q}_1$. The time transient depends on species 1, through the effective extrinsic factor $\epsilon_2'(\tau)$. For $\tilde{q}\ne0$, there is a dependence on the mean value $\overline{\epsilon}_2'(\tau)$, see Eq. (\[transient\_i\]). Considering the Gompertz model ($\tilde{q}_2 \rightarrow 0$), the solution can be simplified to: $$\begin{aligned}
p_2(\tau) = e^{\epsilon_2'(\tau)}\left[\frac{p_{2,0}}{e^{\epsilon_2'\left(0\right)}}\right]^{e^{[\epsilon_2'(0) - \epsilon'_2(\tau)-\rho\tau]}}.
\label{p2taugom}\end{aligned}$$
![For several values of $\tilde{q}_2$, plots of $p_1(\tau)$ (thick lines), given by Eq. \[transient\_i\_a\] and of $p_2(\tau)$ (thin lines), given by Eq. \[transient\_i\], where the integral of Eq. \[transient\_i\] has been calculated numerically. In both cases $\rho = 1$, $\epsilon_1 = 0$ and $\epsilon_2 = 1/2$ : [**(a)**]{} $\tilde{q}_1 = 1/2$ and [**(b)**]{} $\tilde{q}_1 = 2$. Notice that if $\epsilon_2 \tilde{q}_2 < -1$, $p_2^* = 0$.[]{data-label="amensa-out-eq"}](cc1.eps "fig:"){width=".95\columnwidth"} [**(a)**]{}\
![For several values of $\tilde{q}_2$, plots of $p_1(\tau)$ (thick lines), given by Eq. \[transient\_i\_a\] and of $p_2(\tau)$ (thin lines), given by Eq. \[transient\_i\], where the integral of Eq. \[transient\_i\] has been calculated numerically. In both cases $\rho = 1$, $\epsilon_1 = 0$ and $\epsilon_2 = 1/2$ : [**(a)**]{} $\tilde{q}_1 = 1/2$ and [**(b)**]{} $\tilde{q}_1 = 2$. Notice that if $\epsilon_2 \tilde{q}_2 < -1$, $p_2^* = 0$.[]{data-label="amensa-out-eq"}](cc2.eps "fig:"){width=".95\columnwidth"} [**(b)**]{}
Conclusion
==========
We have proposed a generalized two-species population model and have found its analytical solution for the commensalism and amensalism regimes. From this result, we are able to show that the solution of the two-species model is equivalent when one considers one-species model with a time dependent extrinsic growth factor. Also, the extrinsic growth factor can be incorporated into the carrying capacity and the full equivalence, i.e. steady state and transient solution, is retrieved for the Gompertz model. Otherwise, the equivalence persists only for the steady state solutions. This feature reveals an important aspect of sigmoid growth curves. Considering two well known growth models, Verhulst and Gompertz , besides presenting a similar sigmoid behavior, they are differently affected by an extrinsic growth factor. For the Gompertz model, the extrinsic growth factor can be fully incorporated into the carrying capacity; i.e. removing individuals from the population has the same effect if one considers that this population grows in a more limited environment. Any kind of external influence can be seen as a modification in the environment limitations. Although the extrinsic factor can also be incorporated in the Verhulst model, it correctly describe only the steady state solution, the equivalence is lost when one considers the transient solution.
Acknowledgements {#acknowledgements .unnumbered}
================
B. C. T. C. acknowledges support from CAPES. F. R. acknowledges support from CNPq (151057/2009-5). A. S. M. acknowledges the Brazilian agency CNPq (305738/2010-0 and 476722/2010-1) for support.
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| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'M. Axelsson[^1]'
- 'H. T. Ihle'
- 'S. Scodeller'
- 'F. K. Hansen'
bibliography:
- 'prefs.bib'
title: 'Testing for foreground residuals in the Planck foreground cleaned maps: A new method for designing confidence masks'
---
Introduction
============
The recent results from ESA’s [@planck:data] experiment have significantly improved cosmological parameter estimates, and today we understand many of the processes that have formed our universe. A plethora of phenomena are explained by the best fit $\Lambda$CDM model, which complies with the cosmological principles of homogeneity and isotropy. For over a decade it has withstood serious challenges brought forth by confrontation with high precision data delivered by the WMAP satellite [@Bennett:wmap1; @Hinshaw:wmap3; @Hinshaw:wmap5; @Jarosik:wmap7; @Bennett:wmap9], not to mention other numerous experiments such as BOOMERanG [@boom:2003], MAXIMA [@Maxi:2001], DASI [@Dasi:2002] ACBAR [@acbar:2007], and others.
Notably, the BICEP2 experiment [@BICEP2:exp] might have further cemented one of the most crucial hypotheses put forth by the standard model, the inflation theory, through direct detection of B-mode polarization at a significance $>5\sigma$. The signal peaks at the correct angular scales, and could be an indirect observation of gravitational waves which are, according to inflation theory, produced by quantum fluctuations in the gravity field. As they travel towards our detectors their wavelengths become stretched, generating a faint B-mode signal. There is some apparent tension between the tensor-scalar ratio found in and the corresponding value predicted from the BICEP 2 experiment which is not fully resolved yet, but it could well be a statistical fluke.
There is still debate whether the BICEP2 results are valid, as the foreground subtraction procedure is up for scrutiny, and it seems the results are also consistent with a model without gravitational waves, but with a significant dust polarization signal [@flauger:2014; @Adam:2014bub]. Further analysis of polarization data, soon to be released, will hopefully shed more light on the subject.
There are, however, still issues to be resolved, regardless of the BICEP2 results. It seems, the most intriguing discrepancies between observed CMB data and the best fit model occurs at the very largest angular scales. The so-called hemispherical power asymmetry first reported by [@Eriksen:2004; @Hansen2004], and subsequently re-analyzed in a number of papers, see [@Hansen2009] and also observed in [@planck:isotropy], has been shown to be statistically significant at least at the $3.3\sigma$ level. The fact that this curious effect persists in several experiments, argues against an explanation in terms of systematic effects, and may pose a challenge to the standard model.
It is of utmost importance, that any cosmological analysis is performed on maps where foreground contaminations are at a minimum, hence, consistency checks should always be performed whenever possible. In this paper, we aim to shine a bright light on the publicly available data maps, and especially examine the level of any residuals, if there are any. Foregrounds have been subtracted from raw data using four separate cleaning algorithms: (Spectral Matching Independent Component Analysis), (Needlet Internal Linear Combination), (Spectral Expectation Via Maximization-Expectation), and `Commander-Ruler`. Common to all methods, is the use of observations at multiple frequencies in order to reduce foregrounds. The method has been dubbed the main product in the first release.
The [@smica:2003] method consists of three basic steps. In the first step, spectral statistics are derived from a matrix computed from correlations between observations in harmonic space, where each observation is assumed to be a superposition of individual components. Subsequently, a component model is fitted to the result which is then used to estimate a Wiener filter in harmonic space. The filtered spectral components are then transformed back into pixel space, using the inverse spherical harmonic transform.
The [@sevem:2003] method treats all components, except the CMB signal, as generalized noise. Internal templates [@wifit] are fitted and subtracted from the frequency maps.
The [@nilc:2011] is a generalization of the WMAP ILC method, which constructs multidimensional filters that are used to estimate the emission from complex components, spawned by multiple correlated emissions. Hence, from a given map, which can be thought of as a superposition of components, the CMB is removed, as opposed to the usual procedure of removing the non-cosmological signals. It is generalized, in the sense that the number of foreground components is not assumed fixed. The method performs local estimation of the foregrounds, in order to suppress the instrumental noise levels.
The `Commander-Ruler` method [@Commander:2008] (henceforth referred to as ) implements Bayesian component separation in pixel space, fitting a parametric model to the data by sampling the posterior distribution. Gibbs sampling is used to fit foreground amplitude and spectral parameters at low resolution (typically $N_{\mathrm{side}}=256$), and subsequently, the amplitudes are converted to high resolution by solving a least squares system of equations in each pixel, with the spectral parameters fixed to their values from the low-resolution run, while at the same time taking pixelization effects into account in order to avoid sharp boundaries in the high-resolution map.
In the first release, each method provided its own mask based on the properties of each cleaned CMB map. The available sky fraction in these masks varies from $75\%$ to $93\%$. In most cosmological analyses the so-called U73 mask, the product of all these individual masks, is applied. The aim of this paper is to investigate (1) if the cleaned maps are sufficiently clean outside the U73 mask and (2) if some areas of the sky inside the masked pixels of the U73 maps may be safe for cosmological analysis. Both the galactic mask as well as the point source mask will be investigated. In order to assess these questions, we will (1) study the local power spectra around the galactic plane, (2) study the mean, variance and skewness of needlet coefficients in bands around the U73 cut, both in the fully foreground separated maps as well as in the difference maps between the different methods, and (3) investigate the presence of residual unmasked point sources in the difference maps based on the approach described in [@scodeller].
A large part of the analysis undertaken in this paper is based on needlets. Their localization properties both in pixel- as well as in harmonic space make them particularly suited to locate foreground residuals. Wavelets (and in particular needlets) have previously been applied to several aspects of statistical CMB analysis, such as tests for non-Gaussianity and asymmetries [@Vielva:2004; @Cabella:2004; @Wiaux:2006; @MceWen:2008; @Wiaux:2008; @Mar2008; @Piet2008; @Rud2009], polarization analysis [@Cabella:2007], foreground component separation and reduction [@wifit], point source detection in CMB data [@Scodeller2012], power spectrum estimation [@Basak2012]. Also, the cold spot was first detected through wavelet analysis [@Cruz:2005]. For a general introduction to needlets and their properties, see [@Baldi:2006; @Mari:2011].
The approach which we develop and apply to temperature data in this paper is a methodology which allows the construction of a common mask based on data cleaned with many different methods. We will here show the importance of applying such a procedure in order to obtain a consistency test of component separation methods as well as in designing a fiducial mask. For the coming release of polarization data were the foreground properties are less known, such an approach may become even more important.
In section \[data\] we discuss the data products used in this paper. In section \[method\] we discuss the details of our methodology and define several tests applied to the cleaned maps. In section \[analysis1\] we analyze individual maps, whereas in section \[analysis2\] the analysis is repeated, but this time on difference maps in order to perform consistency checks. In section \[mask\] we use difference maps in needlet space to manipulate the mask in order to explore how statistics is affected by either adding, or subtracting, parts of the sky close to the galactic plane. The point source mask is investigated in section \[ps\] and we discuss our findings in section \[conclusion\].
Data
====
In this paper we use the publicly available , , and foreground cleaned maps as well as their beam functions and accompanying FFP6 simulation sets. The sixth round full focal plane (FFP6) simulations have been passed through the component separation pipeline and therfore have beam and noise properties similar to the foreground cleaned maps. The method specific masks as well as the common mask based on the product of these are used. We also create jack-knife maps based on the difference between half-ring maps of the data. The advantage of jack-knife maps is that they have noise properties very close to the noise properties of the actual data. As described in detail below, they are used to adjust the noise level in the simulations, in order to obtain best possible agreement with the noise properties in the data.
Method
======
In order to study the variation of possible foreground residuals with distance from the galactic plane, we construct 7 bands in each hemisphere starting from the borders of the U73 mask proceeding out toward the polar caps. We will number these bands from 1 to 7, each “band” consists of the sum of the corresponding bands in both hemispheres. The northern and southern bands are combined in order to increase statistics. Band 1 consists of the two bands closest to the U73 mask, band 7 consists of the polar caps (see \[bands\]). The bands are constructed by smoothing the U73 mask with a large beam, then including all pixels below a certain threshold. This process is repeated for each band. The sky fractions covered by bands 1 to 7 are 0.145, 0.138, 0.126, 0.11, 0.096, 0.078 and 0.041 respectively. A further division of the first bands will be necessary as detailed below.
{width="0.65\linewidth"}
![ Constructed bands in the interior of the U73 mask, prior to data reduction. []{data-label="inside"}](figs/CandidateBands.eps){width="0.56\linewidth"}
Furthermore, using the same approach as described above to construct bands outside U73, we have also constructed five bands inside the U73 mask. This in order to test whether some of these areas appear sufficiently clean for cosmological analysis. In \[inside\] we show these inside bands. The LFI and HFI point source masks [@planck:ps] are used to ensure that no pixels in the inside bands are contaminated by point sources.
We have estimated a set of quantities in each of these bands and compared to the corresponding quantities within the same band on simulated maps. The indicators of foreground residuals which will be used are the following:
1. We have estimated the power spectrum within each band using the MASTER approach [@hivon2002]. Due to the small sky fraction available to each band, we needed to bin the resulting spectra in bins of 10 multipoles.
2. We have calculated the mean, variance and skewness of needlet coefficients for each band. In this process each needlet coefficient is weighted by the inverse of its CMB+noise variance. We use standard needlets with needlet base $B=1.8393$ and scales $j=[2,11]$ which correspond to multipoles in the range $\ell=[2,1500]$.
These indicators have been calculated on two sets of maps:
1. The officially released , , and cleaned maps.
2. On difference maps between pairs of cleaned maps. For each difference map, we smooth the maps to a common resolution and subtract. In the difference maps, the CMB cancels out and only noise as well as differences in foreground residuals are present. We found that the noise properties of the data difference maps deviate significantly from the simulated difference maps. We used the jack-knife difference maps for the data to fit and adjust an amplitude correction factor to the noise levels in the simulated maps, scale by scale and band by band (although the variation with band is very small). After this correction we found a very good agreement between the noise level in the simulated maps and in the jack-knife maps. The correction factors for some difference maps are shown in \[corfacsU73\].
![Plot of bias correction factors in each band outside the U73 mask for selected difference maps, see \[bands\]. *Top*: Correction factors in pixel space for -. *Middle*: Correction factors for -, *Bottom*: Correction factors applied to -. \[corfacsU73\]](figs/biascorfactors_ch0_withfullsky.eps "fig:"){width="\linewidth"} ![Plot of bias correction factors in each band outside the U73 mask for selected difference maps, see \[bands\]. *Top*: Correction factors in pixel space for -. *Middle*: Correction factors for -, *Bottom*: Correction factors applied to -. \[corfacsU73\]](figs/biascorfactors_ch1_withfullsky.eps "fig:"){width="\linewidth"} ![Plot of bias correction factors in each band outside the U73 mask for selected difference maps, see \[bands\]. *Top*: Correction factors in pixel space for -. *Middle*: Correction factors for -, *Bottom*: Correction factors applied to -. \[corfacsU73\]](figs/biascorfactors_ch2_withfullsky.eps "fig:"){width="\linewidth"}
We further applied the approach in [@scodeller] to amplify point sources in the difference maps. We found that the needlet scales $B=1.5$ and $j=17$ gave the largest increase in point source amplitudes.
We will assess significance of our results by studying deviations by plots of $(x-\langle x\rangle)/\sigma$ where $x$ is any of the aforementioned indicators, $\langle x\rangle$ and $\sigma$ are their corresponding mean and standard deviation from simulations.
Single map analysis {#analysis1}
===================
![ Plot of $(C_\ell-\langle C_\ell\rangle)/\sigma_\ell$ obtained from the map. The data have been binned in $\Delta \ell = 10$ sized bins in order to avoid singular matrices. The legend label “BX” refers to band number “X”, as defined above. The corresponding plots for the other methods are very similar and not shown. \[clchis\]](figs/chi_ch1_forpaper_save.eps){width="\linewidth"}
{width="\linewidth"}
{width="\linewidth"}
{width="\linewidth"}
\
{width="\linewidth"}
{width="\linewidth"}
{width="\linewidth"}
{width="\linewidth"}
{width="\linewidth"}
{width="\linewidth"}
{width="\linewidth"}
{width="\linewidth"}
{width="\linewidth"}
![Same as \[needchis\] for variance only but now plotted with the band number on the x-axis and with color codes indicating needlet scales. \[needchis\_j\]](figs/sigma_chi_single_v2_ch0.eps){width="\linewidth"}
![Same as \[needchis\] for variance only but now plotted with the band number on the x-axis and with color codes indicating needlet scales. \[needchis\_j\]](figs/sigma_chi_single_v2_ch1.eps){width="\linewidth"}
![Same as \[needchis\] for variance only but now plotted with the band number on the x-axis and with color codes indicating needlet scales. \[needchis\_j\]](figs/sigma_chi_single_v2_ch2.eps){width="\linewidth"}
![Same as \[needchis\] for variance only but now plotted with the band number on the x-axis and with color codes indicating needlet scales. \[needchis\_j\]](figs/sigma_chi_single_v2_ch3.eps){width="\linewidth"}
In this section we present the results for each individual foreground cleaned map. These maps have both CMB and noise present, although the noise is sub-dominant on most scales. In \[clchis\], \[needchis\] and \[needchis\_j\] we show results on the power spectrum as well as needlet mean, variance and skewness, band by band and scale by scale. While \[needchis\] shows the results as a function of needlet scale, \[needchis\_j\] shows the same but as a function of band on the x-axis. The purpose of the former is to show the scale dependence, the purpose of the latter is to show whether there is an increase towards band 1 (galactic plane) which could indicate foreground residuals.
We find very good agreement between data and simulations, although from \[needchis\] one can clearly see, in particular for the variance of the needlet coefficients, the effect of unresolved point sources on small scales (high $j$). This effect is seen even clearer in the difference maps presented in the next section. Note also that both the mean and skewness of band 2 appears systematically below zero over most scales. In the simulated data we found that in $30\%$ of the cases, the mean lies below zero on all scales in at least one band. For skewness this occurred in $12\%$ of the simulations. Therefore we conclude that the behavior of band 2 can be well explained as a statistical fluctuation. Note further in \[needchis\_j\] that for the three largest scales there is a clear increase towards the galactic plane in all methods. This is particularly seen in bands 1-3, the ones closest to the galactic equator. This is only seen in the variance of the needlet coefficients. The variance can only increase with foregrounds (while the mean and skewness can increase or decrease), as foreground residuals would generally not subtract power from the map. This increase in variance towards the galactic plane, although the increase is towards the expected variance, can therefore be interpreted as an increasing level of foreground residuals. This is further supported by the fact that this increase disappears with an extended mask as we will show later. We do not show mean and skewness in this figure as no signs of residuals were seen in those cases.
Difference map analysis {#analysis2}
=======================
In this section we analyze inter-method consistency of the component separation by looking at difference maps between six pairs of the four available foreground cleaned maps. These difference maps consist only of noise and differences in foreground residuals between the methods. Since the CMB has been eliminated the difference maps are much more sensitive to foreground residuals and we will use these maps to quantify to which degree the foreground cleaned maps are reliable. Then in the next section we will use these results in order to suggest an improved common mask.
Due to the higher sensitivity of the difference maps to foreground residuals, we find residuals in most bands and scales for most of the computed quantities. Knowledge of whether these residuals may bias cosmological results is of very high interest. We therefore plot $(x-\langle x\rangle)/\sigma_\mathrm{CMB}$ instead of $(x-\langle x\rangle)/\sigma$ where $\sigma_\mathrm{CMB}$ is the standard deviation derived from maps with both CMB and noise in them. On the other hand $\sigma$ is the expected noise standard deviation of the difference maps. In this way we measure the residuals in units of fraction of the standard deviation of CMB fluctuations. If the residuals are larger than $0.2\sigma_\mathrm{CMB}$ it means that they may bias cosmological results by the order of $0.2\sigma_\mathrm{CMB}$. For needlet skewness the residuals are still small, so in this case we show $(x-\langle x\rangle)/\sigma$ as previously.
![$(C_\ell-\langle C_\ell\rangle)/\sigma_\mathrm{CMB}$ for -, -, -and -derived from MASTER estimated power spectra. \[chival\]](figs/CL_diff_highbchi_ch0.eps "fig:"){width="0.8\linewidth"} ![$(C_\ell-\langle C_\ell\rangle)/\sigma_\mathrm{CMB}$ for -, -, -and -derived from MASTER estimated power spectra. \[chival\]](figs/CL_diff_highbchi_ch1.eps "fig:"){width="0.8\linewidth"} ![$(C_\ell-\langle C_\ell\rangle)/\sigma_\mathrm{CMB}$ for -, -, -and -derived from MASTER estimated power spectra. \[chival\]](figs/CL_diff_highbchi_ch2.eps "fig:"){width="0.8\linewidth"} ![$(C_\ell-\langle C_\ell\rangle)/\sigma_\mathrm{CMB}$ for -, -, -and -derived from MASTER estimated power spectra. \[chival\]](figs/CL_diff_highbchi_ch3.eps "fig:"){width="0.8\linewidth"}
The results for the power spectrum are shown in \[chival\], for the multipole interval $\ell \in [500,1500]$. For values $[0,500]$ the agreement between simulations and data is perfect, and hence not shown. We will first consider the , and maps: First note the general increase towards smaller scales from unresolved point sources visible in all bands. As we approach $\ell = 1500$ we notice that the difference increases, especially for the difference maps including . This increase is particularly large in the two bands close to the galactic plane where most foreground residuals are expected. The difference -is generally much smaller than the differences including suggesting residuals which are either present only in or common for both and . In order to obtain further information, we continue with wavelet space analysis.
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![Same as \[diffchivals\] for variance only but now plotted with the band number on the x-axis and with color codes indicating needlet scales. \[diffchivals\_j\] ](figs/sigma_jj_chival_ch0.eps){width="\linewidth"}
![Same as \[diffchivals\] for variance only but now plotted with the band number on the x-axis and with color codes indicating needlet scales. \[diffchivals\_j\] ](figs/sigma_jj_chival_ch1.eps){width="\linewidth"}
![Same as \[diffchivals\] for variance only but now plotted with the band number on the x-axis and with color codes indicating needlet scales. \[diffchivals\_j\] ](figs/sigma_jj_chival_ch2.eps){width="\linewidth"}
![Same as \[diffchivals\] for variance only but now plotted with the band number on the x-axis and with color codes indicating needlet scales. \[diffchivals\_j\] ](figs/sigma_jj_chival_ruler_ch1.eps){width="\linewidth"}
In \[diffchivals\] and \[diffchivals\_j\] we show the results for the moments of the needlet coefficients. Looking at the variance measure for larger scales, one important observation in variance is that while for , and combinations, the residuals are $<0.2\sigma_\mathrm{CMB}$ for bands $>2$, -(and all other combinations, not shown) have residuals $>0.2\sigma_\mathrm{CMB}$ for all bands. In fact, even when extending the mask as described in the next section, we are unable to improve results with combinations significantly. We conclude that the map has larger differences compared to the other three maps, than the other three maps have between themselves. We are therefore, as detailed in the next section, capable of creating an extended mask with improved results using , and only. The map however is too different to allow for construction of a common mask which brings all four maps in full agreement. We therefore decided to exclude the map from the work in the next section.
However, we want to point out (1) the fact that while the map is different from the other three maps, these differences are still so tiny that they did not show up when single channel analysis including CMB was performed in the previous section. Furthermore (2) we cannot conclude from this that is the map with the highest foreground residuals. The approach used in the construction of the map takes better into account variation of foreground properties across the sky compared to the other three methods. It can therefore not be excluded that there are common residuals in the other three maps which give rise to the larger difference between and other methods.
In the following we will consider only combinations with , and . Looking at the mean of needlet coefficients we find again that and are very similar with differences $<0.1\sigma_{\mathrm{CMB}}$ for all bands, while shows differences $>0.1\sigma_{\mathrm{CMB}}$ for band 1 (close to the galactic plane) compared to the other two maps.
We find that the variance measure is the measure most sensitive to foreground residuals. First of all the strong increase at the last 2-3 needlet scales due to unresolved point sources is now very visible. We observe that band 1 again shows strong deviation ($>0.2\sigma_{\mathrm{CMB}}$) between methods, now also visible in the difference -. The skewness measure also supports the fact that there are large differences between methods in band 1. Band 4 shows a very strong outlier in skewness only for the -difference map. We have not been able to identify the source of this latter difference, however, with the new mask which is derived in the following sections, we find that the skewness outliers previously present on band 1 now disappear. Looking at \[diffchivals\_j\] we can clearly see the increase towards the galactic plane for the large scales, in particular for bands 1 and 2.
These results provide an incentive to further study the bands closest to the galactic center, bands 1 and 2. It is already clear from the inferences made so far, that these bands are not consistent between foreground reduction algorithms, however, we may not infer from the obtained data, which of the methods, if not all, have residuals causing these inconsistencies. The strategy now, is to examine the needlet coefficients belonging to difference maps -, -, and -and use these to construct a new confidence mask.
Improving the mask {#mask}
==================
The U73 mask is defined to be the union of all individual foreground method masks, meaning that if one of the method masks excludes a given pixel, then the combined mask excludes it as well, even if the other masks include it. One might contemplate if it can be made smaller, or given the results from the previous section, be extended. The current galactic mask, including masking of point sources, allows a fraction $f_{\mathrm{sky}} = 73.7\%$ of the sky to be used for cosmological analysis, deeming $26.3 \%$ of the sky improper. This stands in stark contrast to, for example, the mask which has an $f_{\mathrm{sky}} \sim 88\%$. To our knowledge, no analysis using all three foreground method maps simultaneously has been done in order to construct a joint confidence mask. Such an analysis is the topic of this section.
Our methodology is to examine the needlet coefficients scale by scale in wavelet space. This will allow construction of “scale masks”, $M_{j}(i)$, where $j$ is the needlet scale, and $i$ is a pixel index. The advantage we have over methods that use pixel maps is that we can examine each scale individually and thus be more flexible. From these masks we can then define the complete mask as the product of scale masks over all relevant scales: M(i) = \_j M\_j(i) where $M(i)$ is the total mask for a given difference map and the product runs over all relevant scales $j$.
We obtain the scale mask for a given pixel $i$ from the needlet coefficients of the difference map divided by their standard deviation obtained from simulations: M\_j(i) = {
[cl]{} 1 & \
\
0 &
. where $\beta_j(i)$ is the needlet coefficent from a difference map and $\sigma_{j}^\mathrm{CMB}(i)$ is the corresponding standard deviation including the expected standard deviation from CMB. We use difference maps for $j \in [3,11]$. In order to minimize the influence of foreground residuals, we require these to have values less than $0.1\sigma_{j}^\mathrm{CMB}$ for $j\le 7$. For $j>7$ the noise level is higher than $0.1\sigma_{j}^\mathrm{CMB}$ in some pixels and we therefore use the maximum value in the jack-knife difference map as a threshold. For the pixels exceeding the threshold we zero all pixels within a disc with scale dependent radius ranging between $24^\circ$ and $0.18^\circ$ at $j=3$ and $j=11$ respectively. The disk radius is calculated according to the recommended procedure in [@Scodeller2012]. In this way we obtain a new and more conservative mask.
As it turns out that large regions of band 1 are removed in the extended mask, we found that these bands needed a further subdivision into bands 1a, 1b and 1c, where band 1a lies closest to the galactic plane, and band 1c lies farthest away from it. We use these smaller bands in order to test the results with the extended mask close to the borders of the U73 mask. In \[chivals2\] and \[chivals2\_j\] we show results on the variance of needlet coefficients from the analysis with the extended mask. Included in the plots are results from analysis on the first band inside the U73 mask, defined in figure \[inside\], and labeled iB1. Pixels analyzed on the inside bands have undergone the same mask extension procedure, as the bands on the outside of U73. After this mask extension only a sky fraction of $2.2\%$ of the original $4.1\%$ remains in band iB1. Still, we clearly see from the figure that this band is unsuitable for cosmological analysis, the same conclusion is valid for all five inside bands. Note that in \[chivals2\] we show the full band 1 and 2 analyzed with the U73 mask whereas bands 1a,b,c and iB1 were analysed with the extended mask. We find that band 1a has to be fully discarded in order to achieve residuals $<0.2\sigma_\mathrm{CMB}$ for the large scales whereas bands 1b and 1c can be kept with this new extended mask. From \[chivals2\_j\] we can see how the new extended mask has removed the increase in variance towards the galactic plane. In fact, only in band 1b are there signs of an increase, but it is well below $<0.2\sigma_\mathrm{CMB}$. Also notice that band 2 has been subdivided into bands 2a and 2b, as done previously with band 1, in order to examine if band 2 may be fully used with the new mask. Band 2a lies closest to the galactic plane while band 2b lies farthest away. From the plots shown we conclude that entire band 2 may be kept.
We have thus arrived at a further extended mask which equals the mask obtained above but with the further extension that all pixels in band 1a are set to zero. This new mask gives satisfactory results for all measures used in this paper allowing $f_{\mathrm{sky}} = 65.9\%$ of the sky for cosmological analysis.
![*From top*: $(x-\langle x\rangle)/\sigma_\mathrm{CMB}$ for -, -, and -for variance after applying the extended mask. We show results on the full bands 1 and 2 using the old U73 mask. Band 1 has been divided into B1a, B1b and B1c, We show results for these smaller bands as well as for the first band (iB1) inside the U73 mask after the mask extension described in the text has been applied. \[chivals2\]](figs/chiplot_newbands_ch0.eps "fig:"){width="\linewidth"} ![*From top*: $(x-\langle x\rangle)/\sigma_\mathrm{CMB}$ for -, -, and -for variance after applying the extended mask. We show results on the full bands 1 and 2 using the old U73 mask. Band 1 has been divided into B1a, B1b and B1c, We show results for these smaller bands as well as for the first band (iB1) inside the U73 mask after the mask extension described in the text has been applied. \[chivals2\]](figs/chiplot_newbands_ch1.eps "fig:"){width="\linewidth"} ![*From top*: $(x-\langle x\rangle)/\sigma_\mathrm{CMB}$ for -, -, and -for variance after applying the extended mask. We show results on the full bands 1 and 2 using the old U73 mask. Band 1 has been divided into B1a, B1b and B1c, We show results for these smaller bands as well as for the first band (iB1) inside the U73 mask after the mask extension described in the text has been applied. \[chivals2\]](figs/chiplot_newbands_ch2.eps "fig:"){width="\linewidth"}
![Same as \[chivals2\] but now plotted with the band number on the x-axis and with color codes indicating needlet scales. Band 2 has been divided into B2a and B2b. In this figure we only show results based on the new extended mask. \[chivals2\_j\]](figs/jplot_ch0.eps "fig:"){width="\linewidth"} ![Same as \[chivals2\] but now plotted with the band number on the x-axis and with color codes indicating needlet scales. Band 2 has been divided into B2a and B2b. In this figure we only show results based on the new extended mask. \[chivals2\_j\]](figs/jplot_ch1.eps "fig:"){width="\linewidth"} ![Same as \[chivals2\] but now plotted with the band number on the x-axis and with color codes indicating needlet scales. Band 2 has been divided into B2a and B2b. In this figure we only show results based on the new extended mask. \[chivals2\_j\]](figs/jplot_ch2.eps "fig:"){width="\linewidth"}
Point source extensions {#ps}
=======================
![The U73 mask with the 276 new point sources indicated by large discs for illustration, the actual holes are much smaller. \[src\]](figs/srcmask.ps){width="0.65\linewidth"}
![New extended U66 mask (yellow) with the U73 mask (blue). \[finalmask\]](figs/newmask.ps){width="0.65\linewidth"}
We have seen in the previous plots that unresolved point sources give rise to large discrepancies between the methods on smaller angular scales. We cannot do much to remove the unresolved sources, but we will check if there are sources left in the difference maps which can be resolved and therefore masked.
Following the approach in [@scodeller], we find 276 point sources at $>5\sigma$ in the difference maps. These point sources include only those which are not already masked by the above described extended U73 mask. Many of these are common to several difference map combinations, others are detected only in one combination but is present but slightly below the detection limit in others. We include point source holes with radius 0.1 degrees for all these sources in our extended U73 mask. The final extended mask now has a usable sky fraction of $65.9\%$. In \[src\] we show the position of these 276 sources and in \[finalmask\] we show the final extended U66 mask which we have made publicly available[^2].
Conclusions {#conclusion}
===========
In this work, the , , and foreground cleaned data maps have been compared to simulated data. It is known that the current maps are recommended for joint cosmological analysis up to $\ell_{\mathrm{max}}=1500$, but for smaller scales, the complex foregrounds and noise properties of the maps are not yet fully understood. We have therefore limited our study to $\ell<1500$.
The aim of this work was to test for foreground residuals in the cleaned maps outside and inside the U73 mask, checking whether the U73 mask needs extension or if it can be made smaller and still be suitable for cosmological analysis. We divided the sky outside U73 into 7 bands north and south of the galactic equator and tested for foreground residuals in these bands by analyzing their local power spectra as well as mean, variance and skewness of needlet coefficients at several scales. We performed this test, both on the individual foreground cleaned maps as well as on difference maps constructed from pairs of these maps. We found that in particular the variance of needlet coefficients on difference maps was highly sensitive to residuals.
Based on the needlet variance test, we found that all the difference maps where the foreground cleaned map was present, the differences to the other maps were so large that we decided to exclude the map from further analysis. Even with a highly extended mask we were unable to make the map agree with the other maps at a satisfactory level. Note that this difference was not seen in the full maps including CMB, only in the difference maps, and only at a level of 0.3 to 0.4 CMB standard deviations. This may influence some cosmological analyses, but is too small to significantly influence the power spectrum. Note however that it is not clear whether this difference comes from large residuals in the map or in the other three maps.
The other three methods however were found to agree with differences less than $20\%$ of the standard deviation of the CMB over most scales after an extended U73 mask was applied. This extended U73 map was constructed by removing pixels where the needlet coefficients were found to be higher than a certain scale dependent threshold. Analysis of bands inside the U73 mask revealed such high levels of foreground contamination that we can confirm that areas which are currently masked by U73 cannot be reliably used for cosmologial analysis. Our extended mask was finally further extended by point source holes for point sources detected in the difference maps. 276 point sources which are not masked in U73 were detected in the difference maps. Our final extended U66 mask, including point source holes for the additional sources has a usable sky fraction of $65.9\%$ and is publicly available (see previously specified url). We recommend the use of this mask rather than the U73 mask for cosmological analysis of the foreground cleaned maps, in particular for analyses which are performed on smaller patches on the sky rather than on the full sky. The effect of the foreground residuals which we detected outside the U73 mask, mostly close to the mask borders, will probably be small for any CMB analysis using the full sky. We also point out that we did not detect these residuals in the individual foreground cleaned maps, only in the differences between these.
We further note that the method presented here can easily be extended to polarization. We have seen that simply taking the product of the individual method specific masks does not necessarily yield a common mask which masks all residuals. By using the differences between the cleaned maps we can extend this simple common mask according to the desired acceptance level of foreground residuals. This may be of even higher importance for the soon-to-be-released polarization maps as the properties of polarized foregrounds are much less known than for temperature.
Maps and results have been derived using the Healpix[^3] software package developed by [@Gorski:2004by]. This work was performed on the Abel Cluster, owned by the University of Oslo and the Norwegian metacenter for High Performance Computing (NOTUR), and operated by the Department for Research Computing at USIT, the University of Oslo IT-department, http://www.hpc.uio.no/. This work is based on observations obtained with (http://esa.int/Planck), an ESA science mission with instruments and contributions directly funded by ESA Member States, NASA, and Canada. The development of has been supported by: ESA; CNES and CNRS/INSU-IN2P3-INP (France); ASI, CNR, and INAF (Italy); NASA and DoE (USA); STFC and UKSA (UK); CSIC, MICINN and JA (Spain); Tekes, AoF and CSC (Finland); DLR and MPG (Germany); CSA (Canada); DTU Space (Denmark); SER/SSO (Switzerland); RCN (Norway); SFI (Ireland); FCT/MCTES (Portugal); and PRACE (EU). We acknowledge the use of the Planck Legacy Archive.
[^1]: e-mail: `magnusax@astro.uio.no`
[^2]: http://folk.uio.no/frodekh/PS\_catalogue/\
planck\_extended\_mask.fits
[^3]: http://healpix.jpl.nasa.gov
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The angular distribution of the Cosmic Microwave Background Radiation (CMBR) in sky shows a dipole asymmetry, ascribed to the observer’s motion (peculiar velocity of the solar system!), relative to the local comoving coordinates. The peculiar velocity thus determined turns out to be $370$ km s$^{-1}$ in the direction RA$=168^{\circ}$, Dec$=-7^{\circ}$. On the other hand, a dipole asymmetry in the sky distribution of radio sources in the NRAO VLA Sky Survey (NVSS) catalog, comprising 1.8 million sources, yielded a value for the observer’s velocity to be $\sim 4$ times larger than the CMBR value, though the direction turned out to be in agreement with that of the CMBR dipole. This large difference in observer’s speeds with respect to the reference frames of NVSS radio sources and of CMBR, confirmed since by many independent groups, is rather disconcerting as the observer’s motion with respect to local comoving coordinates should be independent of the technique used to determine it. A genuine difference in relative speeds of two cosmic reference frames could jeopardize the cosmological principle, thence it is crucial to confirm such discrepancies using independent samples of radio sources. We here investigate the dipole in the sky distribution of radio sources in the recent TIFR GMRT Sky Survey (TGSS) dataset, comprising 0.62 million sources, to determine observer’s motion. We find a significant disparity in observer’s speeds relative to all three reference frames, determined from the radio source datasets and the CMBR, which does not fit with the cosmological principle, a starting point for the standard modern cosmology.'
author:
- 'Ashok K. Singal'
title: A large disparity in cosmic reference frames determined from the sky distributions of radio sources and the microwave background radiation
---
INTRODUCTION
============
Due to the assumed isotropy of the Universe – à la cosmological principle – an observer stationary with respect to the comoving coordinates of the cosmic fluid, should find the number counts of distant radio sources as well as the sky brightness therefrom (i.e., an integrated emission from discrete sources per unit solid angle), to be uniform over the sky. However, an observer moving with a velocity $v$ relative to the cosmic fluid will find, as a combined effect of aberration and Doppler boosting, the number counts and the sky brightness to vary as $\propto \delta^{2+x(1+\alpha)}$, where $\delta$ ($=1+(v/c)\cos\theta$, for a non-relativistic case) is the Doppler factor, $c$ is the velocity of light, $\alpha$ ($\approx 0.8$) is the spectral index, defined by $S \propto \nu^{-\alpha}$, and $x$ is the index of the integral source counts of extragalactic radio source population, which follows a power law $N(>S)\propto S^{-x}$ ($x \sim 1$) [@6; @20; @7]. The angular variation of the number counts as well as of the sky brightness can be expressed as $1+{\cal D}\cos\theta$, implying a dipole anisotropy over the sky with an amplitude [@6; @20; @3; @4; @7; @10] $$\label{eq:1}
{\cal D}=\left[2+{x(1+\alpha)}\right]\frac{v}{c}\;.$$ By observing such angular variation over the sky for a sufficiently large dataset of distant radio sources, one can compute the dipole ${\cal D}$ and thereby velocity $v$ of the observer with respect to the comoving coordinates.
Let $ \bf{\hat{r}}_i$ be the angular position of $i^{th}$ source of observed flux density $S_i$ with respect to the stationary observer, who should find $\Sigma S_i \bf{\hat{r}}_i=0$. However, for a moving observer, due to the dipole anisotropy over the sky, it would yield a finite vector along the direction of the dipole. Let $\bf{\hat{d}}$ be a unit vector in the direction of the dipole, then writing $\Delta {\cal F}=\Sigma S_i\: \bf{\hat{d}}\cdot\bf{\hat{r}}_i$ and ${\cal F}=\Sigma S_i\: |\bf{\hat{d}}\cdot\bf{\hat{r}}_i|$, a summation over the whole sky determines magnitude of the dipole in the sky brightness as [@20; @7; @10] $$\label{eq:2}
{\cal D}=\frac {{\cal D}_{\rm o}}{k}=\frac {3}{2k}\frac{\Delta {\cal F}}{ {\cal F}}
=\frac {3}{2k}\frac{\Sigma S_i\: \cos \theta_i}{\Sigma S_i\: |\cos \theta_i|}\:,$$ where $\theta_i$ is the polar angle of the $i^{th}$ source with respect to the dipole direction. Here ${\cal D}_{\rm o}$ is the dipole determined from observational data and might be affected by any gaps in the sky coverage and some other factors as discussed later, and $k$ is the correction factor, of the order of unity, and as such would need to be determined numerically for individual samples.
A study of the angular variation in the temperature distribution of the CMBR has given quite accurate measurements of a dipole anisotropy, supposedly arising from the observer’s motion (peculiar velocity of the solar system!) of $370$ km s$^{-1}$, in the direction $l=264^{\circ}, b=48^{\circ}$ or equivalently, RA$=168^{\circ}$, Dec$=-7^{\circ}$ [@1; @2; @16]. Some earlier attempts to determine the dipole from the observed angular asymmetry in the sky distribution of distant radio sources claimed the radio source dipole to match the CMBR dipole within statistical uncertainties [@20; @3]. However, Singal [@7], from a study of the anisotropy in the number counts as well as in the sky brightness from discrete radio sources in the NVSS catalog [@5], covering whole sky north of declination $-40^{\circ}$ and containing $\sim 1.8$ million sources with a flux-density limit $S>3$ mJy at 1.4 GHz, found the solar peculiar motion $\sim 4$ times the CMBR value at a statistically significant ($\sim 3\sigma$) level. At the same time, the direction of the velocity vector, though, was surprisingly found to be in agreement with the CMBR value. These unexpected findings of the NVSS dipole being many times larger than the CMBR dipole, have since been confirmed in a number of publications [@8; @9; @18; @19; @12]. Such a difference between two dipoles would imply a relative motion between two cosmic reference frames which will be against the cosmological principle on which the whole modern cosmology is based upon. Therefore it is imperative that an investigation of the radio source dipole be made employing some independent radio source samples. A recent estimate from the TGSS data [@11], has yielded an even larger amplitude for the radio dipole [@12], which is all the more disturbing. Here we investigate this radio dipole in further details, by choosing different flux-density levels, to examine the self-consistency of the dipole in the TGSS data and relate the results to those from NVSS catalog by using the spectral index information between the two datasets [@17; @15].
TGSS Dataset
============
TIFR GMRT Sky Survey (TGSS) is a 150 MHz, continuum survey, carried out between 2010 and 2012 using the Giant Metrewave Radio Telescope (GMRT) [@14], and the raw data are available at the GMRT archive. A First Alternative Data Release of the TGSS (TGSS-ADR1) [@11], that includes direction-dependent calibration and imaging, is available online in the public domain. The TGSS-ADR1, henceforth called TGSS, dataset covers whole sky north of declination $-53^{\circ}$, a total of $3.6\pi$ sr, amounting to $90\%$ of the celestial sphere, with an rms noise below 5 mJy/beam and an approximate resolution of $25''\times25''$. Using a detection limit of 7-sigma, the TGSS catalog comprises 0.62 Million radio sources with an accuracy of about $2''$ or better in RA and Dec. From the spectral index data of common sources in the TGSS and NVSS catalogs [@12; @17; @15], it has been demonstrated that the number counts in the two datasets could be compared above the flux-density limits of 100 mJy and 20 mJy respectively, using a relation $S_{\rm NVSS}\approx 0.2\: S_{\rm TGSS}$, and that the two catalogs are essentially complete above these respective flux-density limits.
As the TGSS catalog [@11] has a gap of sources for Dec $<-53^{\circ}$, in that case our assumption of $\Sigma S_i \bf{\hat{r}}_i=0$ for a stationary observer does not hold good. However if we drop all sources with Dec $> 53 ^{\circ}$ as well, then with equal and opposite gaps on two opposite sides, $\Sigma S_i \bf{\hat{r}}_i=0$ is valid for a stationary observer [@7]. Exclusion of such sky-strips, which affect the forward and backward measurements identically, to a first order do not have systematic effects on the results [@6]. We also exclude all sources from our sample which lie in the galactic plane ($|b|<10^{\circ}$), otherwise a large number of galactic sources in the galactic plane is likely to have an unwanted influence on our determination of the radio source dipole. To ascertain effects of some systematics like local clustering (mainly the Virgo super-cluster), we also examined any alterations in our results by restricting our dataset to regions outside the super-galactic plane by rejecting sources with low super-galactic latitude ($|{\rm SGB}| < 10^{\circ}$).
We used Monte–Carlo simulations to create an artificial radio sky with a similar number density of sources as in the TGSS catalog, distributed at random positions in the sky. However, for the flux-density distribution we took the observed TGSS sample, so that the source counts remain unchanged. On this we superimposed Doppler boosting and aberration effects of an assumed motion, choosing a different velocity vector for each simulation. This artificial sky was then used to retrieve the velocity vector under conditions similar as in our actual TGSS sample (e.g., with $|{\rm Dec}|> 53 ^{\circ}, |b|<10^{\circ}$ gaps in the sky), and compared with the input value in that particular realization. This not only validated our procedure as well as the computer routine, but also helped us make an estimate of errors in the dipole co-ordinates from one hundred simulations we made, each time with randomly chosen radio source sky positions and a different velocity vector assumed for the solar peculiar motion. The error in ${\Delta {\cal F}}/{\cal F}$ is given by $(\Sigma S_i^2)^{1/2}/{\sqrt 3}{\cal F}$ [@7], which allows us to compute error in the dipole magnitude $\cal D$ from Eq. (\[eq:2\]).
Results and Discussion
======================
Sky brightness
--------------
As a relatively small number of strong sources at high flux-density levels could introduce large statistical fluctuations in the sky brightness, we have restricted our sample here to below 5,000 mJy level. At the other end we chose 100 mJy as the lowest cut-off limit since the TGSS catalog is essentially complete only above that flux-density level [@11; @17].
Results for the dipole, determined from the anisotropy in sky brightness for the TGSS dataset, for sources in various flux-density bins, are presented in Table I. Here $N$ is the total number of sources in the corresponding flux-density bin, RA and Dec give the dipole direction in sky, $\cal D_{\rm o}$ is the ’raw’ dipole value computed from $3\Delta {\cal F}/ 2{\cal F}$ that needs to be corrected for various effects as explained below, $k$ being the correction factor.
The numerical factor $k$ includes the effect of gaps in sky coverage of the data as well as the effects, if any, of the variations of power index $x$ and the spectral index $\alpha$ with flux density, and is therefore determined separately for each flux-density bin. Now, from Table I, our estimate of the direction of the dipole (RA$=172^{\circ} \pm 10^{\circ}$, Dec$=-05^{\circ} \pm 09^{\circ}$), is quite in agreement with that determined from the CMBR (RA$=168^{\circ}$, Dec$=-7^{\circ}$, with errors less than a degree) [@1; @2; @16]. Taking this into account we can make our estimates of $k$ better by using this information in our simulations. Accordingly, these simulations differ from the ones above in the sense that the dipole used now is in the direction of the CMBR dipole direction, superimposed on randomly assigned source positions in the sky. It should be noted that this procedure allows us to make a more realistic assessment of the influence of sky-gaps on our determination of the dipole magnitude and does not introduce bias of any kind. A comparison of the dipole derived from each simulation with the input dipole magnitude yielded the correction for that simulation. A set of 100 different simulations was used to determine an average value of the correction factor $k$, given in Table I. The corrected dipole $\cal D$ is obtained by dividing the ’raw’ dipole strength $\cal D_{\rm o}$ by $k$ for each flux-density bin.
From Table I we see a trend that the dipole strength ${\cal D}$ falls systematically with a decrease in the lower flux-density cut-off. From 250 mJy to 100 mJy, ${\cal D}$ gradually falls as much as by $\sim 10 \%$, though it comes nowhere near an order of magnitude weaker CMBR dipole. For a given peculiar velocity $v$, the dipole $\cal D$ could get affected by two quantities, the power law index $x$ and the spectral index $\alpha$ (Eq. (\[eq:1\])). To estimate effects of the power index $x$ on $\cal D$, we have plotted in Fig. (1) the integrated source counts, $N(>S)$ for different $S$ for the TGSS and NVSS samples. The index $x$ in the power law relation, $N(>S)\propto S^{-x}$, can be estimated from the slope of the $\log-\log S$ plot in Fig. (1), where we find that the index $x$ steepens from low to high flux-density levels for both samples. From piece-wise straight line fits to the $\log N-\log S$ data, we find that $x$ steepens from $-0.85$ ($-0.95$) at 100 (20) mJy to $-1.7$ ($-1.65$) at 5000 (1000) mJy, with a value $-1.05$ ($-1.1$) at 250 (50) mJy in the TGSS (NVSS) data. It should be noted that the value of $x$ that enters into Eq. (\[eq:1\]) is the one at the lower cut-off flux density of the bin. On the other hand variation in $\alpha$ is much smaller. From a comparison of TGSS and NVSS samples, the spectral index for the flux-density range $100<S_{\rm TGSS}<200$ mJy was found to be $0.758\pm 0.245$ while for $S_{\rm TGSS}>200$ mJy it turned out to be $0.802\pm 0.225$ [@15]. This slight steepening of $\alpha$ with flux density would affect the dipole value only by $\stackrel{<}{_{\sim}} 1 \%$. For definiteness, we use $\alpha=0.76$ for the $100$ ($20$) mJy bin and $\alpha=0.8$ for the $200$ ($40$) mJy bin in the TGSS (NVSS) data. For other flux-density bins we use interpolated or extrapolated values for $x$ and $\alpha$. The peculiar velocity $v$, calculated accordingly from Eq. (\[eq:1\]), is listed for each flux-density bin in Table I, where no significant trend with changing flux-density cut-off levels is seen. The underlying assumption throughout is that $v$ represents motion of the observer (solar system!), with respect to the corresponding reference frame of radio sources, in the direction given by RA and Dec of the dipole, and its value should not vary from one flux-density bin to another.
As we mentioned above, the direction of the dipole from TGSS data is quite in agreement with that of the CMBR dipole. However, the strength of the dipole (${\cal D}\simeq 5.2 \times 10^{-2}$, the best estimate in Table I) appears an order of magnitude larger than the CMBR dipole ($\simeq 0.47 \times 10^{-2}$) [@7], even though it is smaller by a factor of $\sim 1.4$ than an earlier estimate ($ 7 \times 10^{-2}$) [@12]. When compared with the dipole determined from the sky brightness in the NVSS dataset [@7], the TGSS dipole is a factor of $\sim 2.5$ stronger than the NVSS dipole ($\simeq 2.1 \times 10^{-2}$).
{width="\columnwidth"}
To rule out the possibility that the excessive dipole strength in Table I might be the result of some local clustering (e.g., the Virgo super-cluster), we determined the dipole from the sky brightness from radio sources outside the super-galactic plane by dropping sources with low super-galactic latitude, $|{\rm SGB}|<10^\circ$. We see that the computed dipole (Table II) is still an order of magnitude larger than the CMBR dipole, but lies in the same direction, and that this anomalous result is not due to a local clustering.
Number counts
-------------
We have determined the dipole and the solar peculiar velocity from the number counts as well. First the direction of the dipole was determined from $\Sigma\bf{\hat{r}}_i$. With $\bf{\hat{d}}$ as a unit vector in the direction of the dipole, we define the fractional difference as $$\begin{aligned}
\label{eq:3}
\frac {\Delta {\cal N}}{{\cal N}}&=& \frac{\Sigma \bf{\hat{d}}\cdot\bf{\hat{r}}_i}{\Sigma |\bf{\hat{d}}\cdot\bf{\hat{r}}_i|}= \frac{\Sigma \cos \theta_i}{\Sigma |\cos \theta_i|},\end{aligned}$$ where $\theta_i$ is the polar angle of the $i^{th}$ source with respect to the dipole direction. The dipole magnitude is then calculated from the fractional difference $$\begin{aligned}
\label{eq:4}
{\cal D}=\frac {{\cal D}_{\rm o}}{k}= \frac{3}{2k}\frac {\Delta {\cal N}}{{\cal N}}\:,\end{aligned}$$ similar to that from ${\Delta {\cal F}}/{\cal F}$ in the case of sky brightness (Eq. (\[eq:2\])). Since, unlike in the case of sky brightness, a small number of bright sources do not adversely affect the number counts, in the latter case we have relaxed the upper limit of 5000 mJy on the flux density.
Like in the case of sky brightness, here too we created an artificial radio sky with sources distributed at random positions in the sky, but with a flux-density distribution as of the TGSS sample, so that the source counts remain unchanged. On this was superimposed a mock dipole oriented in a random direction and of random magnitude. One hundred such independent simulations were made to estimate the expected errors in the dipole co-ordinates. The error in ${\Delta {\cal N}}/{\cal N}$ is given by $2/\sqrt {3N}$, then from Eq. (\[eq:4\]), error in $\cal D_{\rm o}$ is $\sqrt {3/N}$. For estimating $k$, another set of 100 simulations were made by choosing dipoles oriented in the direction of the CMBR dipole direction, superimposed on randomly assigned source positions in the sky. Such derived correction factor $k$ was used to divide $\cal D_{\rm o}$ to get $\cal D$ for each flux-density bin. The peculiar velocity $v$, was then calculated using appropriate $x$ nd $\alpha$ values for each flux-density bin.
Results from the number count are summarized in Table III. Comparing with Table I we notice that the directions of the dipoles determined both from the sky brightness as well as from the number counts, are consistent with that of the CMBR. However, the number counts yield a magnitude of the dipole (and the thereby inferred solar peculiar velocity) to be somewhat smaller ($\sim 15 \%$) than that from the radio source sky brightness. It should be noted that in the sky brightness case, stronger sources get more weight, while in the number counts method, weaker source, being more numerous, dominate the dipole determination, so the results could differ somewhat. At a first look it may appear that since in the case of sky brightness, the number counts are being weighted by the flux density, for a power law integrated counts, even with a constant $x$, this may result in a dipole estimate to be higher by a factor of $\sim 1.4$ [@8]. However, it has been shown [@10] that since the flux-density cut-offs for all directions, including the forward and backward directions, are selected in the moving observer’s frame and not in the stationary observer’s frame, the above argument does not hold good. In any case, the dipole strength in number counts still remains an order of magnitude larger than the CMBR value.
If we now compare the dipole determined from number counts for the TGSS dataset with that determined from the NVSS dataset [@7; @18; @8; @9; @19; @12], we find that the directions of the dipole from both these datasets match well with the CMBR measurements, implying that the cause of the dipoles is common and a peculiar motion of the solar system seems to be the only reasonable interpretation for that. However such a statistically significant disparity in their magnitudes, with the TGSS dipole being an order of magnitude (a factor of $\sim 10$) larger than the CMBR dipole, while the NVSS dipole being $\sim 4$ times larger than the CMBR dipole, is rather unsettling.
Radio survey dipoles with respect to the CMBR dipole direction
--------------------------------------------------------------
The fact that the directions of the dipole from the radio source data and the CMBR measurements are matching well, suggests that the direction of the CMBR dipole, known with high accuracy, could be taken to be the direction for the radio source dipoles too. However, we need to first explicitly examine for both TGSS and NVSS datasets if there exist indeed dipoles in the radio source sky distribution with respect to the CMBR dipole direction. For this we compute the dipole strength and the inferred velocity for both radio source datasets, but now with respect to the CMBR dipole direction, viz. RA$=168^{\circ}$, Dec$=-7^{\circ}$. For this we employ an alternate procedure which is more transparent, simpler in nature and more easily visualized.
Using the great circle at $90^\circ$ from the CMBR dipole direction, we divide the sky in two equal hemispheres, $\Sigma_1$ and $\Sigma_2$, with $\Sigma_1$ containing the CMBR dipole, and $\Sigma_2$ containing the direction opposite to the CMBR dipole. Then if there is indeed a motion of the observer along the CMBR dipole direction, due to a combined effect of the aberration and Doppler boosting, the number counts will have a dipole anisotropy, $1+{\cal D}\cos\theta$, over the sky with an amplitude ${\cal D}=[2+x(1+\alpha)]\:v/c$ (Eq. (\[eq:1\])), $\theta$ being the angle measured from the CMBR dipole direction. Then the number of sources in the hemisphere $\Sigma_1$, should be larger than the number of sources in the hemisphere $\Sigma_2$. Let $\phi$ be a complementary angle to $\theta$, i.e. $\phi=\pi/2 - \theta$, with $\phi$ measured towards the CMBR dipole direction, starting from the great circle that divides the sky into hemispheres $\Sigma_1$ and $\Sigma_2$. Now, counting from the great circle, if we denote by $N_1$ the number of sources between 0 and $\phi$ in $\Sigma_1$, then we can write $$\label{eq5}
N_1=2\pi N_0{\int^{\phi}_{0}(1+ {\cal D}\sin\phi) \cos \phi\:{\rm d}\phi},$$ where $N_0$ is the number density per unit solid angle for an isotropic distribution, in the absence of any peculiar motion. Similarly we can write the number of sources in the opposite hemisphere $\Sigma_2$ as $$\label{eq:6}
N_2=2\pi N_0{\int^{0}_{-\phi}(1+ {\cal D}\sin\phi) \cos \phi\:{\rm d}\phi},$$
Then the fractional excess in number of sources in sky region between 0 and $\phi$ in $\Sigma_1$ over the corresponding, symmetrically placed, opposite region in $\Sigma_2$ will be $$\label{eq:7}
\frac {\delta N}{N}=\frac {N_1 - N_2} {N_1 + N_2}= \frac{{\cal D}\sin\phi}{2}=\frac{{\cal D}\cos\theta}{2}.$$ Thus the dipole ${\cal D}$ could then be determined from ${\delta N}/N$ computed for the whole sky ($\phi=\pi/2$). $$\label{eq:8}
{\cal D}=\frac {{\cal D}_{\rm o}}{k}=\frac {2}{k}\:\frac {\delta N}{N}.$$ where $k$ is a constant, of the order of unity, to be determined numerically for individual samples.
For estimating $k$, Monte–Carlo simulations were made by choosing mock dipoles, oriented in the direction of the CMBR dipole direction, superimposed on randomly assigned source positions in the sky but with a flux-density distribution as of the TGSS sample, so that the source counts remain unaffected. From $\sim 100$ such computer simulations, we estimated $k$ for our TGSS sample for different flux-density cut-offs.
Our results are presented in Table IV, which is almost self-explanatory. Dipole ${\cal D}$ was estimated for samples containing all sources with flux-density levels $>~\!\!\!S$, starting from $S=250$ mJy and going down to $S=100$ mJy levels. Of course the accuracy in our estimate improves as we go to lower flux-density limits since the number of sources increases as $N(>S) \propto S^{-x}$ (with $x\sim 1$).
This method is less prone to statistical errors in the radio source data. For one thing, the errors in the CMBR dipole direction themselves are negligible [@1; @2; @16], secondly errors in the sky positions of individual sources do not affect the count of total numbers in each hemisphere. Only a very small number of sources in the two small strips of widths $\sim 2$ arcsec (which is the typical error in source positions in the TGSS catalog) on either side of the great circle at the boundary between the two hemispheres could add to the error in dipole magnitude. However the solid angle covered by this strip of width $\sim 2\times2/ (2 \times 10^5) =2\times 10^{-5}$ radian is $\sim 2 \pi\times 2\times 10^{-5}$ sr or $10^{-5}$ fraction of the sky, which for the $N$ values in Table IV contains only one or two sources, with negligible contribution to $\delta N$ (or even to $\sigma _{\rm N}$) at any flux-density level.
Our estimate of the magnitude of the velocity vector ($v \simeq 3770\pm 340$ km s$^{-1}$) from Table IV appears order of magnitude higher than the CMBR value ($370$ km s$^{-1}$). The quoted errors for $v$ in Table IV are from the expected uncertainty $\sigma_N(=\sqrt N)$ in $\delta N(=N_1-N_2)$, the uncertainty here being that of a binomial distribution, similar to that of the random-walk problem (see, e.g. [@22]). The corresponding $1\sigma$ uncertainty in $\delta N/N$ is $1/\sqrt N$, then from Eq. (\[eq:8\]), error in $\cal D_{\rm o}$ is $2/\sqrt {N}$.
For a comparison, we also employed the same technique to estimate the magnitude of the velocity vector from the NVSS data, with respect to the CMBR dipole direction. The results for NVSS dataset are also summarized in Table-IV, where $v$ turns out to be $\simeq 1430\pm 290$ km s$^{-1}$. In Table IV, the four rows of TGSS dataset at various flux-density levels could be compared to the corresponding four rows of the NVSS dataset. We notice that while the total number $N$ of sources at each flux-density level do match reasonably well (within $1\%$ to $\stackrel{<}{_{\sim}} 10 \%$), the difference $\delta N$ between two hemispheres in respective flux-density bins, and the thereby derived magnitudes of dipole $\cal D$ and velocity $v$, differ as much as by a factor of $\sim 2.5$. Of course all estimates of dipole $\cal D$ and velocity $v$ in either dataset are way above the values expected from the CMBR.
From the rms errors ($\sim 300$ to 400 km s$^{-1}$, Table 1V), determined basically by the radio source density field, for the peculiar velocity estimates for the two samples, detection of a peculiar velocity like the CMBR value ($\sim 370$ km s$^{-1}$) would not have been possible as it would be within $1\sigma$ level. However, because of the much larger amplitude of the peculiar velocity, by a factor of $\sim 4$ for the NVSS and a factor of $\sim 10$ for the TGSS data, a positive detection became possible at statistically significant, $\sim 4 \sigma$ and $\sim 10 \sigma$ levels, respectively.
{width="\columnwidth"}
Here we have explored the radio source dipole by studying any excess of radio source density with respect to the CMBR direction, the latter itself incidentally did not use any information from the radio survey datasets. Moreover, as we move to lower flux-density levels, $\delta N$ seems to steadily increase, specifically, in none of the flux-density bin, for either dataset, we find $N_2$ to be larger than $N_1$. Now if an excess in the sources due to some local clustering in certain regions of the sky were indeed masquerading as a radio source dipole, only in a very contrived situation would one expect to get $N_1 > N_2$ [*at all flux-density levels*]{} and that too [*for both radio catalogs*]{}.
From Eq. (\[eq:7\]), we expect that the fractional excess, ${\delta N}/{N}$, should have a $\sin\phi$ or $\cos\theta$ dependence. We can verify this $\sin\phi$ dependence of ${\delta N}/{N}$ by making cumulative counts of $N_1$ and $N_2$ as a function of $\phi$. We should expect large departures from the expected $\sin\phi$ behavior for small $\phi$, not just because of larger statistical fluctuations in a Binomial distribution for small numbers, but also because dipole strength builds up only at larger $\phi$ (number distribution having a dipole ${\cal D}\cos\theta={\cal D}\sin\phi$), i.e., when one approaches the dipole direction at smaller $\theta$. Figure (2) shows the fractional excess, ${\delta N}/{N}$, for both TGSS and NVSS data as a function of $\phi$ or $\theta$. We have used the number counts for the bins $S_{\rm TGSS}>100$ and $S_{\rm NVSS}>20$ as these have the largest number of sources in our samples. As expected, in Fig. (2), we see large fluctuations for smaller $\phi$. While the fractional excess in the NVSS data does stabilize at $\phi \approx 30^\circ$, in the TGSS data it seems to stabilizes only around $\phi \approx 60^\circ$. However, from Fig. (2) it is clear that not only in both TGSS and NVSS cases is the fractional excess, ${\delta N}/{N}$, way above that expected from the CMBR value of peculiar velocity, viz. 370 km s$^{-1}$, but also that TGSS dipole is much stronger than the NVSS dipole.
The evidence seems irrefutable that the peculiar velocity of the solar system estimated from the distant radio source distributions in sky is indeed much larger than that inferred from the CMBR sky distribution. Such a statistically significant difference in the estimates of the magnitude of the peculiar velocity is puzzling and one cannot escape the conclusion that there is a genuine disparity in the three reference frames defined by the radio source populations selected at different frequencies and the CMBR.
From the clustering properties of radio sources in the TGSS angular spectrum on large angular scales, corresponding to multipoles $ 2 \le l \le 30$, the amplitude of the TGSS angular power spectrum is found to be significantly larger than that of the NVSS, and from that questions have been raised [@13] that some unknown systematic errors may be present in the TGSS dataset. At the same time, while the amplitude of the dipole ($l=1$) too is significantly larger than that of the NVSS, the self-consistency of the TGSS dipole in different flux-density bins and the fact that the direction of the dipole coincides with that of the CMBR dipole, indicates that the TGSS dataset may not be affected to such a great extent by systematics.
Now, unless one wants to disregard radio dipoles, derived from TGSS and NVSS datasets, altogether, at this stage one may be left mainly with only two alternatives. One of them is to say that there may be something amiss in the interpretation of the observed dipoles (including the CMBR) as reflecting observer’s motion (peculiar velocity of the solar system!) and that the strength of a dipole may not be representing an observer’s peculiar speed. In this line of thinking, one will then have to explain the existence of a common direction of all the dipoles and that what is so peculiar about this direction and whether it represents some sort of an “axis” of the universe. The other alternative would be to still follow the conventional wisdom that these dipoles are arising as a result of observer’s motion and that these dipole magnitudes differing by as much as an order of magnitude, indicates that there may be a large relative motion of the various cosmic reference frames. Either alternative does not fit with the cosmological principle, which is the starting point for the standard modern cosmology. Perhaps it points out to the need for a fresh look at the role of the cosmological principle in the cosmological models.
Conclusions
===========
From the dipole anisotropy computed for the TGSS dataset, it was found that the dipole strength and the thereby inferred peculiar motion of the solar system is an order of magnitude larger than that inferred from the CMBR dipole. The TGSS dipole is also larger than the NVSS dipole by a factor of $\sim 2.5$. But the direction of the dipole in all these cases turns out to be the same within errors. An obvious inference is that the reference frames determined from some of the most distant observables, viz. the CMBR, the NVSS 1400 MHz dataset of radio sources and the TGSS 150 MHz dataset of radio sources, somehow do not coincide with each other, which raises uncomfortable questions about the cosmological principle, the basis of the modern cosmology.
Acknowledgements {#acknowledgements .unnumbered}
================
Thanks are due to the staff of the GMRT that made the TGSS observations possible. GMRT is run by the National Centre for Radio Astrophysics of the Tata Institute of Fundamental Research. I thank an anonymous referee for his/her comments and suggestions that helped improve the paper.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
We consider the scenario in which neutrino data are explained by the interplay of type I and II seesaw terms in the Majorana neutrino mass matrix ${\cal M}_\nu = {\cal M}_L - {\cal M}_D {\cal M}_R^{-1} {\cal M}_D^T$. We construct a predictive model with ${\cal M}_L$ proportional to the unit matrix, 3 diagonal texture zeros in ${\cal M}_R$, and ${\cal M}_D$ diagonal. We show how this pattern can be maintained by the non-Abelian discrete symmetry $A_4$, and discuss its phenomenological consequences. It turns out that the two types of seesaw give contributions of the same order to ${\cal M}_\nu$. In the CP conserving case, we find $\sin\theta_{13}\approx 2/(\tan 2\theta_{23}
\tan 2\theta_{12})$ and we predict inverted (normal) ordering of the mass spectrum for $\tan^2\theta_{12} < 0.5 ~(>0.5)$.
---
UCRHEP-T387\
hep-ph/0504081\
April 2005
[**Hybrid Seesaw Neutrino Masses\
with $A_4$ Family Symmetry\
**]{}
PACS: 14.60.Pq, 11.30.Hv, 14.60.St
In many well-motivated extensions of the Standard Model of particle interactions, the Majorana neutrino mass matrix is in general given by $${\cal M}_\nu = {\cal M}_L - {\cal M}_D {\cal M}_R^{-1} {\cal M}_D^T ~,
\label{master}$$ where the first term comes from the coupling of two left-handed neutrinos to a heavy Higgs triplet with a naturally small vacuum expectation value (type II seesaw [@seesawII]) and the second term comes from the canonical seesaw mechanism [@seesaw] assuming the existence of heavy singlet right-handed neutrinos. In the past, perhaps for reasons of simplicity or economy, the common practice was to assume the dominance of one or the other of these two terms. After all, if both terms were considered, predictability would be largely lost. However, if there exists a symmetry which limits the forms of both terms, a simple and realistic Hybrid Seesaw model may still emerge.
An early discussion of the role of the two contributions in the generation of a large mixing angle can be found in [@wetterich]. One common strategy (see e.g. [@ak05] and references therein) was to assume a symmetry such as $SO(3)$ which requires ${\cal M}_L$ to be proportional to the unit matrix, but allow it to be broken arbitrarily in the second term. Another recent paper [@RX] applies an $S_3\times S_3$ symmetry to both terms, but a number of symmetry-breaking parameters are needed to fit data. Here we propose that the structure of both terms is fixed by the same family symmetry and thus obtain the first example of a predictive Hybrid Seesaw model with a well-defined symmetry (the discrete group $A_4$) for the complete Lagrangian.
Consider first the type I contribution. If ${\cal M}_D$ is diagonal (which may be maintained by the $A_4$ symmetry), then the texture zeros of ${\cal M}_R$ are reflected in ${\cal M}_\nu$ as zero sub-determinants [@m05]. In fact, in the perspective of the type I seesaw formula, instead of the texture zeros of ${\cal M}_\nu$ [@fgm02], those of ${\cal M}_R$ [@l04] are expected to have a deeper theoretical meaning (the two types of zeros may be related [@KKST]). In particular, consider the case $${\cal M}_R = \pmatrix{0 & \times & \times \cr \times & 0 & \times \cr \times
& \times & 0},
\label{MR}$$ where $\times$ denotes a nonzero entry. This structure is rather unique, as it is the only possibility to have more than 2 zeros in ${\cal M}_R$ (and therefore less than 4 free parameters) without inducing 2 or more zeros in ${\cal M}_R^{-1}$. Assuming ${\cal M}_D$ diagonal, one then obtains $${\cal M}^I_\nu = \frac 1a \pmatrix{a^2 & ab & ac \cr ab & b^2 & -bc \cr
ac & -bc & c^2},
\label{typeI}$$ which has no texture zero but 3 zero sub-determinants. This three-parameter structure, as we will show, cannot reproduce all present neutrino data [@data]. On the other hand, it is possible that a significant contribution comes from ${\cal M}_L$ and, if it is proportional to the unit matrix, Eq. (\[master\]) becomes $${\cal M}_\nu = \pmatrix{d+a & b & c \cr b & d+b^2/a & -bc/a \cr
c & -bc/a & d+c^2/a},
\label{genM}$$ which (i) turns out to fit all present data and (ii) may be stabilized by a simple family symmetry, as shown below. This model of Hybrid Seesaw, depending on 4 parameters, is the most minimal constructed so far.
To maintain the pattern of ${\cal M}_\nu$ in Eq. (4), a suitable family symmetry is $A_4$, the discrete group of the even permutation of four objects. It is also the symmetry group of the regular tetrahedron (Plato’s “fire” [@fire]), and has been applied to the neutrino mass matrix in a number of ways [@mr01; @bmv03]. The irreducible representations of $A_4$ are ${\bf1}, {\bf 1'}, {\bf 1''}, {\bf 3}$. The group multiplication rule [@mr01] is $${\bf 3} \times {\bf 3} = {\bf 1} + {\bf 1'} +
{\bf 1''} + {\bf 3}_1 + {\bf 3}_2 ~,$$ where $\psi_i,~\varphi_j \sim {\bf 3}$ implies $$\begin{aligned}
{\bf 1} &=& \psi_1 \varphi_1 + \psi_2 \varphi_2 + \psi_3 \varphi_3~, \\
{\bf 1'} &=& \psi_1 \varphi_1 + \omega^2 \psi_2 \varphi_2 +
\omega \psi_3 \varphi_3~, \\
{\bf 1''} &=& \psi_1 \varphi_1 + \omega \psi_2 \varphi_2 +
\omega^2 \psi_3 \varphi_3~, \\
{\bf 3}_1 &=& (\psi_2 \varphi_3, \psi_3 \varphi_1, \psi_1 \varphi_2)~, \\
{\bf 3}_2 &=& (\psi_3 \varphi_2, \psi_1 \varphi_3, \psi_2 \varphi_1)~, \end{aligned}$$ with $\omega = e^{2 \pi i/3}$.
Here we make the following assignment: the 3 families of leptons transform as a triplet, $$(\nu_i,l_i), ~ l^c_i, ~ \nu^c_i \sim {\bf 3}~,$$ and the scalar sector consists of three Higgs doublets $\Phi_i \sim {\bf 1}, {\bf 1'},
{\bf 1''}$, one Higgs triplet $\xi \sim {\bf 1}$, and three Higgs singlets $\Sigma_i \sim {\bf 3}$. This implies that the Dirac mass matrices linking $l_i$ to $l^c_j$ (${\cal M}_l$) as well as $\nu_i$ to $\nu^c_j$ (${\cal M}_D$) are both diagonal, with 3 independent entries each. Explicitly, $$\left(\begin{array}{c} m_e \\ m_\mu \\ m_\tau \end{array}\right) =
\frac{1}{\sqrt{3}}\left(\begin{array}{ccc}
1 & 1 & 1 \\
1 & \omega & \omega^2 \\
1 & \omega^2 & \omega \\
\end{array}\right)
\left(\begin{array}{c} y_{l1} \langle\Phi_1\rangle \\
y_{l2} \langle\Phi_2\rangle \\ y_{l3} \langle\Phi_3\rangle \\
\end{array}\right)~,$$ where the Yukawa couplings $y_{li}$ should be tuned to fit the charged lepton masses, as in the Standard Model. Our assignment also implies that ${\cal M}_L$ in Eq.(\[master\]), which is generated by $\langle \xi \rangle$, is proportional to the unit matrix and ${\cal M}_R$ has nonzero off-diagonal entries, as in Eq.(\[MR\]): $({\cal M}_R)_{ij}=f_R \langle \Sigma_k \rangle$ with $i\ne j\ne k$. Notice that, even if $\langle \Sigma_k \rangle$ were related among each other by the symmetry of the scalar potential, the parameters $a,b,c$ in Eq.(\[typeI\]) would be completely independent, since they are determined by the 3 unknown diagonal entries of ${\cal M}_D$. This is what is needed to obtain Eq.(\[genM\]).
However, the bare Majorana mass term $\nu^c_i \nu^c_i$ is invariant under $A_4$ and it cannot be removed by hand. This is a generic issue in models with texture zeros in the mass matrix of gauge singlets. Thus one is naturally led to consider a left-right gauge extension of the Standard Model with $(l^c,\nu^c)$ transforming as a doublet under $SU(2)_R$. In that case, these bare mass terms are forbidden by the gauge symmetry and $\Sigma_i$ should now be considered as triplets under $SU(2)_R$, i.e. the counterpart of $\xi$ which is a triplet under $SU(2)_L$. In this way our initial assumption in Eq.(\[MR\]) is justified and the pattern of our proposed Hybrid Seesaw model in Eq.(\[genM\]) is completely stabilized.
Let us briefly consider the phenomenology associated with the three $SU(2)_L$ doublets $\Phi_i$. Since charged lepton Yukawa couplings are diagonal, Lepton Flavor Violation processes are suppressed by the smallness of neutrino masses and therefore negligible. The Standard Model like Higgs doublet is given by $\Phi = (v_1\Phi_1+v_2\Phi_2+v_3\Phi_3)/v$, where $v^2=v_1^2+v_2^2+v_3^2=(174$ GeV$)^2$. The orthogonal combinations $\Phi'$ and $\Phi''$ decay into $e^+ e^-$, $\mu^+ \mu^-$ and $\tau^+ \tau^-$ with similar rates (the couplings are of the order $m_\tau/v$, the exact values depending on the scalar potential parameters). One loop contributions to the anomalous magnetic moment of the muon, $g_\mu-2$, are induced, but their size is generically negligible even for Higgs masses as light as 100 GeV (for a precise estimation in a similar model, see [@Z22]).
Notice also that, if the 3 families of quarks transform as an $A_4$ triplet in the same way as leptons, up and down quark mass matrices are both diagonal, thus describing in first approximation the smallness of CKM mixing angles. Then, since all fermions transform in the same way under $A_4$, they may be embedded in multiplets of a Grand Unified gauge group. However, the construction of an appropriate scalar sector is highly non-trivial and beyond the purposes of the present paper.
Let us study the phenomenological implications for neutrino masses and mixing angles. Data on neutrino oscillations [@data] indicate that $\theta_{23}$ is close to maximal and $\theta_{13}$ is small. One can check that the matrix structure (\[genM\]) may accommodate $\theta_{23}=\pi/4$ and $\theta_{13}=0$ if and only if $b^2 = c^2$. It is useful to discuss first this limiting case and, in the following, possible deviations from it.
The matrix ${\cal M}_\nu$ has a form [@m02] such that $\theta_{13} = 0$ and $\theta_{23} = \pi/4$. In the basis spanning $\nu_e$, $(\nu_\mu +
\nu_\tau)/\sqrt 2$, and $(\nu_\tau - \nu_\mu)/\sqrt 2$, it becomes $${\cal M}_\nu = \pmatrix{d+a & \sqrt 2 b & 0 \cr \sqrt 2 b & d & 0 \cr
0 & 0 & d+2b^2/a}.
\label{2b2}$$ The parameters $a,b,d$ are in general complex. Diagonalizing ${\cal M}_\nu
{\cal M}_\nu^\dagger$, we obtain $$\tan 2 \theta_{12} = {2|B| \over |d|^2 - |d+a|^2}~,
\label{t12}$$ where $B = \sqrt 2 [2 Re(bd^*) + ab^*]$. Notice that $\theta_{12}<\pi/4$ implies $|a|^2 + 2 Re(ad^*) < 0$. The mass squared differences are given by $$\Delta m^2_{sol} \equiv |m_2|^2 - |m_1|^2 =
\sqrt{(|d|^2-|d+a|^2)^2 + 4|B|^2} =
{|d|^2 - |d+a|^2 \over \cos 2 \theta_{12}}~,$$ $$\pm \Delta m^2_{atm} \equiv |m_3|^2 - {1 \over 2} (|m_2|^2+|m_1|^2) =
4 \left| {b^2 \over a} \right|^2 + 4 Re \left( {d^* b^2 \over a} \right)
- 2|b|^2 + \frac 12 (|d|^2-|d+a|^2) ~.
\label{DMA}$$
![The mass eigenvalue $m_3$ as a function of $\tan^2\theta_{12}$, in the limit $\theta_{13}=0$, $\theta_{23}=\pi/4$ and no complex phases (subcase (1)). The displayed interval is the $99\%$ C.L. allowed range for $\tan^2\theta_{12}$ [@SV]. The solid line corresponds to the best fit values for $\Delta m^2_{sol}$ and $\Delta m^2_{atm}$, whereas the shaded region accounts for the $99\%$ C.L. allowed ranges of the mass squared differences. The star indicates the best fit. The ordering of the mass spectrum is inverted for $\tan^2\theta_{12}<0.5$ (left branch) and normal for $\tan^2\theta_{12}>0.5$ (right branch).[]{data-label="fig1"}](fig1.eps){width="270pt" height="220pt"}
[**Subcase (1):**]{} If the parameters $a,~b$ and $d$ are real, they are uniquely determined by the experimental values of $\theta_{12}$, $\Delta m^2_{sol}$ and $\Delta m^2_{atm}$. In particular $\Delta m^2_{sol}
\ll \Delta m^2_{atm}$ implies $2d\approx -a$, so that $d=0$ (pure type I seesaw) is not a solution, as already mentioned. Since $$|m_3|^2 - {1 \over 2} (|m_2|^2+|m_1|^2) = - \frac{a^2 \tan^2
2\theta_{12}}{2} \left(1-\frac 18 \tan^2 2\theta_{12}\right) + {1 \over 2}
\Delta m^2_{sol} \left( 1 - {1 \over 2} \tan^2 2 \theta_{12} \right)~,
\label{phyDMA}$$ the ordering of the mass spectrum is inverted for $\tan^2\theta_{12}<0.5$, as favored (but only at about $1\sigma$ level) by present data. For the best fit values of oscillation parameters ($\tan^2 \theta_{12} = 0.45$, $\Delta m^2_{sol} = 8.0 \times 10^{-5}$ eV$^2$, $\Delta m^2_{atm} = 2.5
\times 10^{-3}$ eV$^2$ [@SV]), one finds $a = \pm 0.057$ eV, $|b| = 0.049$ eV, $d = \mp 0.029$ eV and $|m_{1,2,3}|$ are respectively 0.0748, 0.0753, 0.0560 eV, i.e. a mild inverted ordering. In this case the effective mass parameter relevant for neutrinoless $2\beta$-decay takes the value $m_{ee}\equiv |d+a| = 0.028$ eV. However, for values of $\tan^2 \theta_{12}$ closer to $0.5$, the absolute mass scale increases and the spectrum becomes quasi-degenerate, as shown in Fig.\[fig1\]. Correspondingly, $m_{ee}
\approx \cos2\theta_{12} |m_1|$ becomes larger.
[**Subcase (2):**]{} If the parameters $a$ and $d$ are real and $b=i|b|$ is imaginary, then Eq. (\[phyDMA\]) is replaced by $$|m_3|^2 - {1 \over 2}(|m_2|^2+|m_1|^2) = {(\Delta m^2_{sol})^4 \sin^4 2
\theta_{12} \over 16 a^6} + {(\Delta m^2_{sol})^3 \sin^2 2 \theta_{12}
\cos 2 \theta_{12} \over 4 a^4} + {1 \over 2} \Delta m^2_{sol} \cos 2
\theta_{12}.$$ This is a solution with normal ordering and again the 3 experimental conditions (best fit values) determine $a$, $|b|$, and $d$, i.e. $\pm0.0032$ eV, 0.0084 eV, and $\mp 0.0064$ eV, with $|m_{1,2,3}| = 0.011,
0.014, 0.052$ eV respectively. Differently from subcase (1), the absolute mass scale depends weakly on $\tan^2 \theta_{12}$ within the experimental range.
![The dependence of the neutrino mass eigenvalues $m_i$ on complex phases, in the limit $\theta_{13}=0$ and $\theta_{23}=\pi/4$ (Eq.(\[2b2\])). In the left panel we assume $d$ real and $a,b$ having the same phase $\phi$ (subcase $(1')$). In the right panel we assume $d$ real, $a=|a|e^{i\phi}$ and $b=i|b|$ purely imaginary (subcase $(2')$). We take the best fit values of mass squared differences and $\tan^2\theta_{12}=0.45$.[]{data-label="fig2"}](fig2inv.eps "fig:"){width="230pt"} ![The dependence of the neutrino mass eigenvalues $m_i$ on complex phases, in the limit $\theta_{13}=0$ and $\theta_{23}=\pi/4$ (Eq.(\[2b2\])). In the left panel we assume $d$ real and $a,b$ having the same phase $\phi$ (subcase $(1')$). In the right panel we assume $d$ real, $a=|a|e^{i\phi}$ and $b=i|b|$ purely imaginary (subcase $(2')$). We take the best fit values of mass squared differences and $\tan^2\theta_{12}=0.45$.[]{data-label="fig2"}](fig2nor.eps "fig:"){width="230pt"}
More in general, when $b=c$ there are two complex free parameters given by the relative phases among $a$, $b$ and $d$. For illustration, let us consider the following extensions of subcases (1) and (2), with only one additional degree of freedom. [**Subcase $(1')$:**]{} Let $d$ be real with $a=|a|e^{i\phi}$ and $b=|b|e^{i\phi}$, i.e. $a$ and $b$ have the same phase. Then the 3 relations for $\tan 2 \theta_{12}$, $\Delta m^2_{sol}$, and $\Delta m^2_{atm}$ are exactly the same as in subcase (1), with the replacements $a\rightarrow |a|$, $b \rightarrow |b|$ and $d \rightarrow d \cos \phi$. This means that we again have an inverted ordering for $\tan^2 \theta_{12} < 0.5$. Since only the combination $d \cos \phi$ is determined by phenomenology and $$|m_3|^2 = \left| d + {2 b^2 \over a} \right|^2 = \left( d \cos \phi +
2 \left| {b^2 \over a} \right| \right)^2 + (d \cos \phi)^2 \tan^2\phi ~,$$ the overall scale of neutrino masses increases with increasing values of $\tan^2 \phi$. This means that the mass spectrum can be quasi-degenerate independently from the value of $\tan^2\theta_{12}$. [**Subcase $(2')$:**]{} Let $d$ be real with $a=|a|e^{i\phi}$ and $b=i|b|$. The 3 conditions are the same as in subcase (2), with $a$ replaced by $|a|$ and $d$ by $d \cos \phi$. We now have $$|m_3|^2 = \left( d \cos \phi - 2 \left| {b^2 \over a} \right| \right)^2
+ (d \cos \phi)^2 \tan^2\phi ~,$$ so that, as in subcase $(1')$, the overall mass scale increases with $\tan^2 \phi$. The dependence of neutrino masses on $\phi$ is shown in Fig.\[fig2\], for both subcases $(1')$ and $(2')$.
In this general case $\theta_{13}$ may be non-zero and $\theta_{23}$ may deviate from the maximal value $\pi/4$. If one neglects complex phases, the type II term ${\cal M}_L = d {\mathbb I}$, being proportional to the identity, does not affect the mixing angles but only the mass spectrum: calling $\lambda_i$ the eigenvalues of ${\cal M}^I_\nu$ in Eq.(\[typeI\]), one has simply $m_i = d +\lambda_i$. In order to extract the constraints on the mixing angles and $\lambda_i$, one should notice that $({\cal M}^I_\nu)^{-1}$ has 3 texture zeros on the diagonal, by construction. It then follows that $$\begin{array}{l}
{\displaystyle 0=\frac{1}{\lambda_3}+\frac{1}{\lambda_2}+\frac{1}{\lambda_1}}
~,\\ \\
{\displaystyle \tan^2\theta_{13} = \frac{\lambda_2\cos^2\theta_{12}+
\lambda_1\sin^2\theta_{12}}{\lambda_1+\lambda_2}} ~,\\ \\
{\displaystyle \tan2\theta_{23}=
\frac{\lambda_2\sin^2\theta_{12}+\lambda_1\cos^2\theta_{12}}
{(\lambda_1-\lambda_2)\cos\theta_{12}\sin\theta_{12}} ~
\frac{1}{\sin\theta_{13}}}~.\\
\end{array}
\label{tris}$$ Therefore, given the values of $\theta_{12}$ and $\theta_{23}$ ($\theta_{13}$), the ratio $\lambda_1/\lambda_2$ and $\theta_{13}$ ($\theta_{23}$) are predicted. In particular, taking into account that $\tan^2\theta_{13}<0.05\ll 1$, one finds $$\sin\theta_{13} \approx \frac{1}{\tan 2\theta_{23}}\frac{2}{\tan 2\theta_{12}}
~,$$ so that the size of the $1-3$ mixing angle is proportional to the deviation from maximal atmospheric mixing. This result is illustrated in Fig.\[fig3\], which shows that the present upper bound $\sin\theta_{13}< 0.2$ can be saturated, given the experimental uncertainty on $\theta_{23}$ and $\theta_{12}$.
![The correlation between $\theta_{13}$ and $\theta_{23}$ in the CP conserving case (no complex phases). The displayed interval is the $99\%$ C.L. allowed range for $\tan^2\theta_{23}$ [@SV]. The curves depend only on the value of the solar mixing angle $\theta_{12}$ (they are independent of the neutrino mass spectrum). []{data-label="fig3"}](fig3.eps){width="270pt" height="200pt"}
![The correlation between $m_3$ and $\theta_{13}$, for different values of the solar mixing angle $\theta_{12}$. The value of $\theta_{12}$ determines if the ordering of the mass spectrum is inverted (left panel) or normal (right panel). The lines correspond to the best fit values for $\Delta m^2_{sol}$ and $\Delta m^2_{atm}$, whereas the shaded regions account for the $99\%$ C.L. allowed ranges of the mass squared differences. Complex phases are put to zero.[]{data-label="fig4"}](fig4inv.eps "fig:"){width="230pt" height="220pt"} ![The correlation between $m_3$ and $\theta_{13}$, for different values of the solar mixing angle $\theta_{12}$. The value of $\theta_{12}$ determines if the ordering of the mass spectrum is inverted (left panel) or normal (right panel). The lines correspond to the best fit values for $\Delta m^2_{sol}$ and $\Delta m^2_{atm}$, whereas the shaded regions account for the $99\%$ C.L. allowed ranges of the mass squared differences. Complex phases are put to zero.[]{data-label="fig4"}](fig4nor.eps "fig:"){width="230pt" height="220pt"}
Since $m_i = d + \lambda_i$, the parameters $d$ and $\lambda_{1,2}$ are uniquely determined once $\Delta m^2_{sol}$ and $\Delta m^2_{atm}$ are given, so that the mass spectrum is predicted too. After some algebra, one finds that the ordering is inverted for $$\tan^2\theta_{12} < \frac {1+\tan^2\theta_{13}}{2-\tan^2\theta_{13}}
= 0.5 \div 0.54 ~
\label{perf}$$ and normal for $\tan^2\theta_{12}$ larger. The absolute neutrino mass scale $|m_3|$ is shown, as a function of $\theta_{13}$ ($\theta_{23}$), in Fig.\[fig4\] (Fig.\[fig5\]). The dependence of $|m_3|$ on $\theta_{12}$ is strong, analogously to the case $b=c$: quasi-degeneracy of the spectrum (and accordingly sizable $m_{ee}$) is obtained for $\tan^2\theta_{12}$ close to the right-hand side of Eq.(\[perf\]).
![The same as Fig.\[fig4\] but as a function of $\theta_{23}$. The displayed interval is the $99\%$ C.L. allowed range for $\tan^2\theta_{23}$. []{data-label="fig5"}](fig5inv.eps "fig:"){width="230pt" height="220pt"} ![The same as Fig.\[fig4\] but as a function of $\theta_{23}$. The displayed interval is the $99\%$ C.L. allowed range for $\tan^2\theta_{23}$. []{data-label="fig5"}](fig5nor.eps "fig:"){width="230pt" height="220pt"}
If complex phases are introduced, the type II contribution will affect not only the mass spectrum but also the mixing angles. In general there will be more freedom to fit data and we do not elaborate further in this direction. Just notice that $\theta_{13}\ne 0$ can be possibly associated with Dirac type CP violation.
In conclusion, we have considered the Hybrid Seesaw scenario, where light neutrino masses receive comparable contributions from super-heavy right-handed neutrinos and scalar isotriplets. We have shown that a family symmetry (the discrete group $A_4$) is able (i) to control the structure of type I and type II seesaw terms at the same time and (ii) to restrict the number of free parameters so that predictions are possible and experimentally testable.
[**Acknowledgments:**]{} This work was supported in part by the U. S. Department of Energy under Grant No. DE-FG03-94ER40837.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Let $(X, T)$ be a topological dynamical system and let $\Phi: X^r \to \mathbb{R}$ be a continuous function on the product space $X^r= X\times \cdots \times X$ ($r\ge 1$). We are interested in the limit of V-statistics taking $\Phi$ as kernel: $$\lim_{n\to \infty} n^{-r}\sum_{1\le i_1, \cdots, i_r\le n} \Phi(T^{i_1}x, \cdots, T^{i_r} x).$$ The multifractal spectrum of topological entropy of the above limit is expressed by a variational principle when the system satisfies the specification property. Unlike the classical case ($r=1$) where the spectrum is an analytic function when $\Phi$ is Hölder continuous, the spectrum of the limit of higher order V-statistics ($r\ge 2$) may be discontinuous even for very nice kernel $\Phi$.'
address:
- 'LAMFA, UMR 7352 CNRS, University of Picardie, 33 rue Saint Leu, 80039 Amiens, France'
- 'MCMS, Lund Institute of Technology, Lund University Box 118 SE-221 00 Lund, Sweden'
- 'LAMFA, UMR 7352 CNRS, University of Picardie, 33 rue Saint Leu, 80039 Amiens, France'
author:
- Aihua Fan
- Jörg Schmeling
- Meng Wu
title: 'The multifractal spectra of V-statistics'
---
Introduction
============
Consider a topological dynamical system $(X, T)$, where $T: X \to X$ is a continuous transformation on a compact metric space $X$ with metric $d$. For $r\ge 1$, let $X^r = X\times \cdots \times X$ (product of $r$ copies of $X$) and let $C(X^r)$ be the space of continuous functions $\Phi: X^r \to \mathbb{R}$.
For $\Phi \in C(X^r)$ and $n\ge 1$, let $$V_\Phi(n, x) = n^{-r}\sum_{1\le i_1, \cdots, i_r\le n} \Phi(T^{i_1}x, \cdots, T^{i_r} x)$$ and $
V_\Phi(x) = \lim_{n\to \infty } V_\Phi(n, x)
$ if the limit exists. For $\alpha \in \mathbb{R} $, define $$E_\Phi(\alpha) =\left\{x \in X: \lim_{n\to \infty } V_\Phi(n, x) = \alpha\right\}.$$
The problem treated in the present paper is to measure the sizes of the sets $E_\Phi(\alpha)$. To measure the sizes of the sets $E_\Phi(\alpha)$, we adopt the notion of topological entropy introduced by Bowen ([@Bowen]), denoted by $h_{\rm
top}$. We denote by $\mathcal{M}_{\rm inv}$ the set of all $T$-invariant probability Borel measures on $X$ and by $\mathcal{M}_{\rm erg}$ its subset of all ergodic measures. The measure-theoretic entropy of $\mu$ in $\mathcal{M}_{\rm inv} $ is denoted by $h_\mu$.
For $\mu \in \mathcal{M}_{\rm inv}$, the set $G_{\mu}$ of [*$\mu$-generic points*]{} is defined by $$G_{\mu}:=\left\{x \in X: \frac{1}{n}\sum_{j=0}^{n-1} \delta_{T^j x}
\stackrel{w^*}{\longrightarrow} \mu \right\},$$ where $\stackrel{w^*}{\longrightarrow}$ stands for the weak star convergence of measures. Bowen ([@Bowen]) proved that on any dynamical system, we have $h_{\rm top} (G_\mu)\le h_\mu$ for any $\mu \in
\mathcal{M}_{\rm inv}$. For ergodic measure measure $\mu$, we get equality. But in general, the equality doesn’t hold. A dynamical system $(X, T)$ is said to be [*saturated*]{} if for any $\mu \in \mathcal{M}_{\rm inv}$, we have $h_{\rm
top}(G_{\mu})=h_{\mu}$. It is proved in [@FLP] that systems of specification are saturated.
In this paper, we shall prove a variational principle which relates the topological entropy $h_{\rm
top}(E_\Phi(\alpha))$ to the measure theoretic entropies of invariant measures in the following set, called $(\Phi, \alpha)$-fiber, $$\mathcal{M}_\Phi(\alpha) =\left\{\mu \in \mathcal{M}_{\rm inv} : \int_{X^r} \Phi d \mu^{\otimes r}=\alpha \right\}$$ where $\mu^{\otimes r} =\mu \times \cdots \times \mu$ is the product of $r$ copies of $\mu$.
\[VP\] Suppose that the dynamical system $(X, T)$ is saturated. Let $\Phi \in C(X^r)$ ($r\ge 1$). If $
\mathcal{M}_\Phi(\alpha)= \emptyset$, we have $E_\Phi(\alpha)=\emptyset$. If $ \mathcal{M}_\Phi(\alpha)\not=
\emptyset$, we have $$\label{variational-principle}
h_{\rm top} (E_\Phi(\alpha)) =
\sup_{\mu \in \mathcal{M}_\Phi(\alpha)} h_\mu.$$
Theorem \[VP\] is well known when $r=1$ (see e.g. [@FFW; @FLP; @BSS; @Barreira]). In particular, it is known that for regular potential $\Phi$, $\alpha \mapsto h_{\rm top} (E_\Phi(\alpha))$ is an analytic function (see e.g. [@Fan1994; @Ruelle]). But as we shall see, when $r\geq 2$, this function can admit discontinuity even for “very regular” potentials.
The above consideration was motivated by the following problem. Recently the multiple ergodic limit $$\label{MEA}
M_{\Phi}(x):=
\lim_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}\Phi(\sigma^ix,\sigma^{2i}x,\cdots,T^{ri}x)$$ have been studied by Furstenberg ([@Furstenberg]), Bergelson ([@Bergelson]), Bourgain ([@Bourgain]), Assani ([@Assani]), Host and Kra ([@HK]), and others. Fan, Liao and Ma proposed in [@FLM] to give a multifractal analysis of the multiple ergodic average $M_{\Phi}$, in other words, to determine the Hausdorff dimensions of the level sets $$L_{\Phi}(\alpha)=\{x\in X: M_{\Phi}(x)=\alpha \}.$$ This problem in its generality remains open.
However, there are two results for the shift dynamics on symbolic space and for some special potentials $\Phi$. The first one concerns the case where $X=\{-1,1\}^\N$, $T$ is the shift and $\Phi(x_1,\cdots,x_r)=x_1^{(1)}\cdots x_r^{(1)}$ ($x_i^{(1)}$ being the first coordinate of $x_i$). By using Riesz products, the authors in [@FLM] proved that for $\alpha\in [-1,1]$ we have $$\dim L_{\Phi}(\alpha)=1-\frac{1}{r}-\frac{1}{r}\left(\frac{1-\alpha}{2}\log_2\frac{1-\alpha}{2}+\frac{1+\alpha}{2}\log_2\frac{1+\alpha}{2}\right).$$ The second one concerns the case where $X=\{0,1\}^\N$, $T$ is the shift and $\Phi(x_1,x_2)=F(x_1^{(1)},x_2^{(1)})$ is a function depending only on the first coordinates $x_1^{(1)}$ and $x_2^{(1)}$ of $x_1$ and $x_2$. The multifractal analysis of these double ergodic average was determined in [@FSW]. A related work was done in [@KPS] answering a question in [@FLM] about the Hausdorff dimension of a subset of $L_{\Phi}(\alpha)$ for extremal values of $\alpha$.
As shown in [@FSW], the dimension of the “mixing part” of $L_{\Phi}(\alpha)$ which is defined by $$\sup \left\{\dim \mu : \mu(L_{\Phi}(\alpha))=1,\ \mu \ {\rm is\ mixing}\right\}$$ is equal to $$\sup\left\{\dim \mu : \int \Phi d\mu^{\otimes r}=\alpha,\ \mu\ {\rm is \ mixing}\right\}.$$ This equality is very similar to the variational principal stated in Theorem \[VP\].
In Section \[V-stat\], we recall some facts about V-statistics. In Section \[Top Ent\], we recall some notions like topological entropy, generic points and specification property. The main theorem, Theorem \[VP\], is proved in Section \[proof\]. In Section \[examples\], we examine the special case of full shift together with some examples. We will see that, even for very regular function $\Phi$, the function $\alpha\to h_{\rm top} (L_\Phi(\alpha))$ may admit discontinuity.
To finish this introduction, we emphasise that the problem of multifractal analysis of multiple ergodic limits remains largely open.
V-statistics {#V-stat}
============
V-statistics are tightly related to U-statistics which are well known in statistics. Let $\mu$ be a probability law on $\mathbb{R}$. A U-parameter of $\mu$ is defined through a function called kernel $h: \mathbb{R}^d \to \mathbb{R}$ by $$\theta(\mu) = \theta_h(\mu) = \int_{\mathbb{R}^d} h d \mu^{\otimes d}$$ where $\mu^{\otimes d}$ is the product measure $\mu \times \cdots \times \mu$ ($d$ times) on $\mathbb{R}^d$. This $U$-statistics is well defined for all $\mu$ such that the integral exists.
In statistics, U-parameters are also called estimable parameters and they constitute the set of all parameters that can be estimated in an unbiased fashion. A fundamental problem in statistics is the estimation of a parameter $\theta(\mu)$ for an unknown probability law $\mu$. To estimate a U-parameter $\theta_h$, people employ the U-statistics for $\theta_h$: $$U_h(X_1, \cdots, X_n) =\frac{(n-d)!}{n!} \sum h(X_{i_1}, \cdots, X_{i_d})$$ where the sum is taken over all $(i_1, \cdots, i_d)$ with $i_j$’s distinct and $1\le i_j\le n$, where $X_1, \cdots, X_d$ is a sequence of observations of $\mu$. Closely related to U-statistics is the V-statistics (von Mises statistics): $$V_h(X_1, \cdots, X_n) = n^{-d}\sum_{1\le i_1, \cdots, i_d \le n} h(X_{i_1}, \cdots, X_{i_d}).$$
People expect that $U_h(X_1, \cdots, X_n)$ converges almost surely to $\theta_h$. This fact, if it holds, allows one to estimate $\theta_h$ using observations. If it is the case, we say the U-parameter strong law of large numbers (SLLN) holds. The U-statistics SLLN had been well studied for independent observations. In [@ABDGHW], the authors have studied the U-statistics SLLN for ergodic stationary process $(X_n)$, i.e. $X_n = f\circ T^n$ where $T$ is ergodic measure-preserving transformation on a probability space $(\Omega, \mathcal{A}, \mathbb{P})$, $f: \Omega \to \mathbb{R}$ is a measurable function and $X_1$ admits $\mu$ as probability law.
If $h$ is a kernel bounded by a integrable function and if $(X_n)$ is ergodic, it can be proved (see [@ABDGHW]) that almost surely $$\lim_{n\to \infty}|U_h(X_1, \cdots, X_n)- V_h(X_1, \cdots, X_n)|=0.$$ It is also proved in [@ABDGHW] that the U-statistics SLLN holds if the kernel $h$ is continuous. In the following, we consider only V-statistics.
Topological entropy {#Top Ent}
===================
For any integer $n\ge 1$, the Bowen metric $d_n$ on $X$ is defined by $$d_n(x, y) = \max_{0\le j <n} d(T^jx, T^j y).$$ For any $\epsilon >0$, we will denote by $B_n(x, \epsilon)$ the open $d_n$-ball centered at $x$ of radius $\epsilon$.
Let $ Z \subset X$ be a subset of $X$. Let $\epsilon >0$. A cover is a collection of Bowen balls (at most countable) $R=\{B_{n_i}(x_i,
\epsilon)\}$ such that $Z \subset \bigcup_i B_{n_i}(x_i,
\epsilon)$. For such a cover $R$, we put $n(R) = \min_i
n_i$. Let $s\ge 0$. Define $$H^s_n (Z, \epsilon) = \inf_R \sum_i \exp(-s n_i),$$ where the infimum is taken over all covers $R$ of $Z$ with $n(R)
\ge n$. The quantity $H^s_n (Z, \epsilon)$ being a non-decreasing function of $n$, the following limit exists $$H^s (Z, \epsilon) = \lim_{n \to \infty} H^s_n (Z, \epsilon).$$ Consider the quantity $H^s (Z, \epsilon)$ as a function of $s$, there exists a critical value, which we denote by $h_{\rm
top} (Z, \epsilon)$, such that $$H^s (Z, \epsilon) =\left\{ \begin{array} {ll} +\infty, & s <
h_{\rm top} (Z, \epsilon) \\ 0 , & s> h_{\rm
top} (Z, \epsilon). \end{array} \right.$$ The following limit exists $$h_{\rm top} (Z) = \lim_{\epsilon \to 0} h_{\rm top} (Z,
\epsilon).$$ The limit $h_{\rm top} (Z)$ is called the [*topological entropy*]{} of $Z$ ([@Bowen]).
For $x \in X$, we denote by $V(x)$ the set of all weak limits of the sequence of probability measures $n^{-1}\sum_{j=0}^{n-1}
\delta_{T^j x}$. Recall that $X$ is compact. It is clear then that for any $x$ we have $$\emptyset \not= V(x)
\subset \mathcal{M}_{\rm inv}.$$ The following lemma is due to Bowen ([@Bowen]).
\[Bowen\] For $t\ge 0$, we have $h_{\rm top} (B^{(t)})\le t$ where $$B^{(t)} =\left\{x\in X: \exists\ \mu \in V(x) \
\mbox{\rm satisfying}\ h_\mu \le t\right\}.$$
The set $G_\mu$ of $\mu$-generic points is the set of all $x$ such that $V(x)=\{\mu\}$. The Bowen lemma implies that $$h_{\rm
top}(G_\mu)\le h_\mu$$ for any invariant measure $\mu$. It is simply because $x \in G_\mu $ implies $\mu \in V(x)$. Bowen also proved that the inequality becomes equality when $\mu$ is ergodic. However, in general, we do not have the equality and it is even possible that $G_\mu =
\emptyset$. In fact, $\mu(G_\mu) = 1 \ {\rm or}\ 0$ according to whether $\mu$ is ergodic or not (see [@DGS]).
The equality $h_{\rm
top}(G_\mu)= h_\mu$ does hold for any invariant probability measure in any dynamical system with specification ([@FLP]).
\[FLP\] Any dynamical system with specification $(X, T)$ is saturated. In other words, $h_{\rm
top}(G_\mu)= h_\mu$ for any $\mu \in \mathcal{M}_{\rm inv}$.
A dynamical system $(X,T)$ is said to satisfy the [*specification property*]{} if for any $\epsilon >0$ there exists an integer $m(\epsilon)\ge 1$ having the property that for any integer $k\ge2$, for any $k$ points $x_1,\ldots,x_k$ in $X$, and for any integers $$a_1 \le b_1 < a_2 \le b_2 < \cdots < a_k \le b_k$$ with $
a_i - b_{i-1} \ge m(\epsilon) \quad (\forall 2 \le i \le k),
$ there exists a point $y\in X$ such that $$d(T^{a_i+n} y,T^n x_i) < \epsilon
\qquad (\forall
\ 0 \le n \le b_i-a_i, \quad \forall 1 \le i \le k).$$
The specification property implies the topological mixing. Blokh ([@Blokh]) proved that these two properties are equivalent for continuous interval transformations. Mixing subshifts of finite type satisfy the specification property. In general, a subshift satisfies the specification if for any admissible words $u$ and $v$ there exists a word $w$ with $|w|\le k$ (some constant $k$) such that $uwv$ is admissible. For $\beta$-shifts defined by $T_\beta x = \beta x (\!\!\mod 1)$, there is only a countable number of $\beta$’s such that the $\beta$ shifts admit Markov partition (i.e. subshifts of finite type), but an uncountable number of $\beta$’s such that the $\beta$-shifts satisfy the specification property ([@Schmel97]).
We finish this section by mentioning that continuous functions on $X^r$ can be uniformly approximated by tensor functions. It is a consequence of the Stone-Weierstrass theorem.
\[SW\] Let $F \in C(X^r)$. For any $\epsilon>0$, there exists a function of the form $$\widetilde{F}(x_1, \cdots, x_r) = \sum_{j=1}^n f_j^{(1)}(x_1)f_j^{(2)}(x_2) \cdots f_j^{(r)}(x_r)$$ where $f_j^{(i)} \in C(X)$, such that $\|F-\widetilde{F}\|_\infty<\epsilon$.
We will write $$\widetilde{F} = \sum_{j=1}^n f_j^{(1)}\otimes f_j^{(2)}\otimes \cdots \otimes f_j^{(r)}.$$
Proof of Theorem \[VP\] {#proof}
=======================
We can actually consider Banach-valued $V$-statistics. More than Theorem \[VP\] can be proved.
Let $\mathbb{B}$ be a real Banach space and $\mathbb{B}^*$ its dual space. The duality will be denoted by $\langle y, x\rangle$ ($x \in \mathbb{B}, y\in \mathbb{B}^*$). We consider $\mathbb{B}^*$ as a locally convex topological space with the weak star topology $\sigma(\mathbb{B}^*, \mathbb{B})$. For any $\mathbb{B}^*$-valued continuous function $\Phi: X \to \mathbb{B}^*$, we consider its $V$-statistics $V_\Phi(n, x)$ as before, formally in the same way.
Fix a subset $W \subset \mathbb{B}$. For a sequence $\{\xi_n\} \subset
\mathbb{B}^*$ and a point $\xi \in \mathbb{B}^*$, we denote by $\limsup_{n \to \infty} \xi_n \stackrel{W}{\le} \xi$ the fact $$\limsup_{n \to \infty} \langle \xi_n , w\rangle \le \langle \xi,
w\rangle\ {\rm \ for \ all\ } w \in W.$$ It is clear that $\limsup_{n \to \infty} \xi_n \stackrel{\mathbb{B}}{\le} \xi$ means $\xi_n$ converges to $\xi$ in the weak star topology $\sigma(\mathbb{B}^*, \mathbb{B})$.
Given $\alpha \in \mathbb{B}^*$ and $W \subset \mathbb{B}$. We define $$E_\Phi(\alpha, W) = \left\{ x \in X: \limsup_{n \to \infty}
V_\Phi(n, x) \stackrel{W}{\le}
\alpha
\right\}$$ $$\mathcal{M}_\Phi(\alpha, W) = \left\{\mu\in \mathcal{M}_{\rm
inv}: \int \Phi d\mu \stackrel{W}{\le} \alpha \right\}$$ where $\int \Phi d \mu$ denotes the vector-valued integral in Pettis’ sense (see [@Rudin]) and the inequality “$\stackrel{W}{\le}$" means $
\int \langle \Phi, w \rangle d\mu \le
\langle \alpha, w \rangle
\ \ \mbox{\rm for \ all }\ w \in W.
$
\[VP2\] Suppose that the dynamical system $(X, T)$ is saturated. If $
\mathcal{M}_\Phi(\alpha, W)= \emptyset$, we have $E_\Phi(\alpha, W)=\emptyset$. If $ \mathcal{M}_\Phi(\alpha, W)\not=
\emptyset$, we have $$\label{variational-principle}
h_{\rm top} (E_\Phi(\alpha, W)) =
\sup_{\mu \in \mathcal{M}_\Phi(\alpha, W)} h_\mu.$$
We prove the first assertion by showing that $E_\Phi(\alpha, W)\not=\emptyset$ implies $\mathcal{M}_\Phi(\alpha, W)\not= \emptyset$. Let $x\in E_\Phi(\alpha, W)$. There exists a measure $\mu \in V(x)\subset \mathcal{M}_{\rm inv}$ and a sequence of integers $(n_k)$ such that $$\label{limit_mu}
\mu = w^*\!-\!\lim_{k \to \infty} \frac{1}{n_k}\sum_{j=1}^{n_k} \delta_{T^j x}.$$ We are going to show that $\mu \in \mathcal{M}_\Phi(\alpha, W)$.
Let $w\in W$. Then $\langle \Phi, w \rangle$ is a continuous function on $X$. For an arbitrarily small $\epsilon >0$, by the Stone-Weierstrass theorem (See Lemma\[SW\]) there exists a function $\widetilde{\Phi}$ of the form $$\widetilde{\Phi}= \sum_j f_j^{(1)} \otimes f_j^{(2)} \otimes\cdots \otimes f_j^{(r)}$$ (finite sum of tensor products) such that $$\|\langle \Phi, w \rangle - \widetilde{\Phi}\|_\infty\le \epsilon.$$ Notice that $$V_{\widetilde{\Phi}}(n, x) = \sum_j \prod_{i=1}^r \frac{S_n f_j^{(i)}(x)}{n}$$ where $$S_n f(x)= \sum_{k=1}^n f(T^k x)$$ denotes the ergodic sum for a given function $f$. According to (\[limit\_mu\]), we have $$\label{limit_Tphi}
\lim_{k \to \infty} V_{\widetilde{\Phi}}(n_k, x)=
\sum_j \prod_{i=1}^r \int_X f_j^{(i)}d \mu = \int_{X^r} \widetilde{\Phi} d\mu^{\otimes d}.$$ On the other hand, we write $$\int \langle \Phi, w\rangle d \mu^{\otimes d} - \langle \alpha, w\rangle
= \sigma_1+ \sigma_2 + \sigma_3 + \sigma_4$$ where $$\begin{aligned}
\sigma_1 & = & \int (\langle \Phi, w\rangle-\widetilde{\Phi}) d\mu^{\otimes d} \\
\sigma_2 & =& \int \widetilde{\Phi} d\mu^{\otimes d}- V_{\widetilde{\Phi}}(n_k, x)\\
\sigma_3 & = & V_{ \widetilde{\Phi}}(n_k, x)- V_{\langle\Phi, w\rangle}(n_k, x)\\
\sigma_4 &=& V_{\langle\Phi, w\rangle} (n_k, x)-\langle \alpha, w\rangle.\end{aligned}$$ We have $$|\sigma_1| \le \epsilon, \quad |\sigma_3| \le \epsilon, \quad
\lim \sigma_2=0, \quad \limsup \sigma_4 \le 0.$$ So, we get $$\int \langle \Phi, w\rangle d \mu^{\otimes d} \le \langle \alpha, w\rangle+2 \epsilon.$$ Since $\epsilon$ is arbitrary, we have thus proved that $\mu \in \mathcal{M}_{\rm inv}(\alpha, W)$. The first assertion is then proved.
Prove now the second assertion. What we have just proved also implies $$E_\Phi(\alpha, W) \subset B^{(t)}=\{x \in X: \exists \mu \in V(x) \ {\rm such \ that} \ h_\mu \le t\}$$ where $t = \sup_{\mu \in \mathcal{M}_\Phi(\alpha, W)} h_\mu$. By the Bowen lemma (Lemma \[Bowen\]), we get $$\label{upperbound}
h_{\rm top}(E_\Phi(\alpha))
\le
\sup_{\mu \in \mathcal{M}_\Phi(\alpha)} h_\mu.$$ To finish the proof of the second assertion, it suffices to prove the reverse inequality of (\[upperbound\]). Let $\mu \in \mathcal{M}_\Phi(\alpha)$. Let $x\in G_\mu$. For any $\epsilon >0$ and any $w\in W$, consider $\widetilde{\Phi}$ as above. We have $$\lim_{n \to \infty} V_{\widetilde{\Phi}}(n, x) = \int_{X^r} \widetilde{\Phi} d \mu^{\otimes r}.$$ It follows that $$\begin{aligned}
\limsup_{n \to \infty} V_{\langle \Phi, w\rangle }(n, x)
& \le & \lim_{n \to \infty} V_{\widetilde{\Phi}}(n, x) +\epsilon\\
& = & \int_{X^r} \widetilde{\Phi} d \mu^{\otimes r} +\epsilon \\
& \le & \int_{X^r} \langle \Phi, w\rangle d \mu^{\otimes r} +2\epsilon\le \langle \alpha, w\rangle
+2\epsilon.\end{aligned}$$ Letting $\epsilon \to 0$ we get $$\limsup_{n \to \infty} \langle V_{{\Phi}}(n, x), w\rangle \le \langle \alpha, w\rangle.$$ In other words, we have proved $G_\mu \subset E_\Phi(\alpha, W)$ for all $\mu \in \mathcal{M}_{\rm inv}(\alpha, W)$. So, $$h_{\rm top}(E_\Phi(\alpha))\ge h_{\rm top} (G_\mu).$$ By Lemma \[FLP\], $h_{\rm top} (G_\mu)=h_\mu$. Taking the supremum over $\mu \in \mathcal{M}_{\rm inv}(\alpha, W)$ leads to the reverse inequality of (\[upperbound\]).
Example: Shift dynamics {#examples}
=======================
Let $(X, T) = (\Sigma_m, \sigma)$ with $m\ge 2$, where $\sigma\colon\Sigma_m\to \Sigma_m$ is the shift on the space $\Sigma_m=\{0,1, \cdots, m-1\}^\N$.
Let $$L_{(\Phi, W)}=\{ \alpha \in \mathbb{B}^*: E_\Phi(\alpha, W)\not= \emptyset\}.$$ If $W=\mathbb{B}$, we write $L_{\Phi}=L_{(\Phi, W)}$. Define $f_{(\Phi, W)}: L_{(\Phi, W)}\to \mathbb{R}$ by $$f_{(\Phi, W)}(\alpha)= h_{\rm top} (E_\Phi(\alpha, W)).$$
\[USC\] $f_{(\Phi, W)}: L_{(\Phi, W)}\to \mathbb{R}$ is upper semi-continuous.
Let $\alpha_n, \alpha \in L_{(\Phi, W)}$. Suppose $\alpha_n \to \alpha$ in the weak star topology. We have to show that $$\limsup_n f_{(\Phi, W)}(\alpha_n)\le f_{(\Phi, W)}(\alpha).$$ Since each fiber like $\mathcal{M}_{\rm inv}(\alpha, W)$ is compact, there are maximizing measures $\mu_{\alpha_n}\in \mathcal{M}_{\rm inv}(\alpha_n, W)$ and $\mu_\alpha\in \mathcal{M}_{\rm inv}(\alpha, W)$ such that $$\label{USC1}
f_{(\Phi, W)}(\alpha_n)= h_{\alpha_n}, \qquad f_{(\Phi, W)}(\alpha)= h_{\alpha}.$$ Without loss of generality, we can assume that $\mu_{\alpha_n}$ converge weakly, say to $\mu^*$. Since $$\forall w\in W, \quad \int \langle\Phi, w \rangle d \mu_n \le \langle \alpha_n, w \rangle,$$ taking limit shows that $\mu^* \in \mathcal{M}_{\rm inv}(\alpha, W)$. It follows that $$\label{USC2}
h_{\mu^*}\le h_{\mu_\alpha}.$$ On the other hand, recall that for the shift dynamics, the entropy function $\mu \mapsto h_\mu$ is upper semi-continuous. So, $$\label{USC3}
\limsup_n h_{\alpha_n} \le h_{\mu^*}.$$ We combine (\[USC1\]),(\[USC2\]) and (\[USC3\]) to finish the proof.
\[USC\] Assume that $\Phi$ is a function defined on $\Sigma_m^r$ ($r\ge 1$) which depends only on the first $k$ coordinates of each of its variables ($k\ge 1$). Then the suppremum in the variational principle (\[variational-principle\]) is attained by a $(k-1)$-Markov measure.
This is just because the integral $\int \Phi d\mu^{\otimes r}$ depends only on the values $\mu([a_1, \cdots,a_k])$ of the measure $\mu$ on cylinders $[a_1, \cdots, a_k]$ and there exists a $(k-1)$-Markov measure $\nu$ such that $$\mu([a_1, \cdots,a_k]) = \nu([a_1, \cdots,a_k])$$ for all cylinders $[a_1, \cdots, a_k]$ and such that $h_\nu \ge h_\mu$.
In particular, if $k=1$, maximizing measures are Bernoulli measures. For the Bernoulli measure $\mu_p$ determined by a probability vector $p=(p_0, \cdots, p_{m-1})$, we have $h_{\mu_p}=H_1(p)$ where $$H_1(p) = -\sum_{j=0}^{m-1} p_j \log p_j.$$ Suppose that the function $\Phi$ is a product of $r$ functions and each of its factor depends only on the first coordinate, i.e. $$\Phi(x^{(1)}, \cdots, x^{(r)})
= \phi_1(x^{(1)}_1)\cdots \phi_r(x^{(r)}_1).$$ Let $$A(p)= \int_{\Sigma_m^r} \Phi(x^{(1)}, \cdots, x^{(r)}) d\mu_p(x^{(1)})\cdots d\mu_p(x^{(r)}).$$ Notice that $E_\Phi(\alpha)\not=\emptyset$ iff $\alpha=A(p)$ for some probability vector $p=(p_0, \cdots, p_{m-1})$. The following result is a direct consequence of the last theorem.
\[k=1\] Let $\Phi(x^{(1)}, \cdots, x^{(r)})
= \phi_1(x^{(1)}_1)\cdots \phi_r(x^{(r)}_1)$. We have $$A(p) = \prod_{k=1}^r \sum_{j=0}^{m-1} \phi_k(j) p_j.$$ For any $\alpha$ satisfying $E_\Phi(\alpha)\not=\emptyset$, we have $$h_{\rm top}(E_\Phi(\alpha)) = \max_{A(p)=\alpha} H_1(p)$$ where the maximum is taken over all probability vectors $p$ satisfying $A(p)=\alpha$.
If $k=2$, maximizing measures are Markov measures. A Markov measure $\mu_{p, P}$ is determined by a probability vector $p$ and a transition matrix $P$. Its entropy is equal to $$H_2(p, P) = -\sum_{i=0}^{m-1} p_i \sum_{j=0}^{m-1} p_{i,j} \log p_{i,j}.$$ Suppose $\Phi(x^{(1)}, \cdots, x^{(r)})$ is of the form $
\phi_1(x^{(1)}_1, x^{(1)}_2)\cdots \phi_r(x^{(r)}_1, x^{(r)}_2).
$ Let $$A(p, P)= \int_{\Sigma_m^r} \Phi(x^{(1)}, \cdots, x^{(r)}) d\mu_{p,P}(x^{(1)})\cdots d\mu_{p,P}(x^{(r)}).$$
\[k=2\] Let $\Phi(x^{(1)}, \cdots, x^{(r)})
= \phi_1(x^{(1)}_1, x^{(1)}_2)\cdots \phi_r(x^{(r)}_1, x^{(r)}_2)$. We have $$A(p,P) = \prod_{k=1}^r \sum_{i, j=0}^{m-1} \phi_k(i,j) p_ip_{i,j}.$$ For any $\alpha$ satisfying $E_\Phi(\alpha)\not=\emptyset$, we have $$h_{\rm top}(E_\Phi(\alpha)) = \max_{A(p, P)=\alpha} H_2(p, P)$$ where the maximum is taken over all couples $(p, P)$ satisfying $A(p, P)=\alpha$.
Let us consider two examples. We will use the following trivial property of the entropy function $H(x)=-x\log x-(1-x)\log(1-x)$.
Given two numbers $p_1, p_2 \in [0,1]$. We have $$H(p_1)<H(p_2)\ \ \mbox{\rm iff} \ \ |p_1-1/2|> |p_2-1/2|.$$ We have $
H(p_1)= H(p_2)$ iff $ |p_1-1/2|= |p_2-1/2|$.
[*Example 1.*]{} Consider the case $m=2$, $k=1$ and $r=2$. Let $x=p_1$. Then $p_0=1-x$ and we have $$A(p) = [\phi_1(0)(1-x) +\phi_1(1)x][\phi_2(0)(1-x) +\phi_2(1)x].$$ For simplicity, we write $A(x)$ for $A(p)$. Suppose that $\phi_1(0)\neq \phi_1(1)$ and $\phi_2(0)\neq
\phi_2(1)$. Otherwise, the question is trivial. By multiplying $\phi$ by a constant we can suppose that $A(x)$ is of the form $$A(x)=(x-a)(x-b).$$ Let $x=x^*$ be the critical point of the quadratic function $A$ (i.e., $x^*=\frac{a+b}{2}$).
Using the last lemma, it is easy to find the unique point $x_\alpha$ such that $$A(x_\alpha)=\alpha, \ \ h_{\rm top
}(E_{\Phi}(\alpha))=H(x_\alpha).$$ The point $x_{\alpha}$ is the closest to $1/2$ among those $x$ such that $A(x)=\alpha$.
We distinguish three cases.
[*Case I.*]{} $x^* \le 0$ or $x^*\ge 1$ (see Figure \[figure1\]).\
1. $A(x)$ is strictly monotonic in the interval $[0, 1]$.\
2. $L_\Phi$ is the interval with end points $ab$ and $(1-a)(1-b)$.\
3. For any $\alpha\in L_\Phi$ , $A(x_\alpha)=\alpha$ admits a unique solution $x_\alpha$ in $[0, 1]$.
[*Case II.*]{} $0<x^* \le 1/2$ (see Figure \[figure2\]).\
1. $A(x)$ is strictly monotonic in the intervals $[x^*, 1]$.\
2. $L_\Phi$ is the interval with end points $A(x^*)$ and $(1-a)(1-b)$.\
3. For any $\alpha\in L_\Phi$, $A(x_\alpha)=\alpha$ admits a unique solution $x_\alpha$ in $[x^*, 1]$.
[*Case III.*]{} $1/2\le x^* <1$ (see Figure \[figure3\]).\
1. $A(x)$ is strictly increasing in the interval $[0, x^*]$.\
2. $L_\Phi$ is the interval with end points $ab$ and $A(x^*)$.\
3. For any $\alpha\in L_\Phi$, $A(x_\alpha)=\alpha$ admits a unique solution $x_\alpha$ in $[0, x^*]$.
![[Case $x^*=1$ (with $a=0.5$, $b=1.5$)]{}[]{data-label="figure1"}](1.eps){width="6.8cm"}
![[Case $x^*=1$ (with $a=0.5$, $b=1.5$)]{}[]{data-label="figure1"}](2.eps){width="6.8cm"}
![Case $0<x^*<1/2$ (with $a=0.2, b=0.4$)[]{data-label="figure2"}](3.eps){width="6.8cm"}
![Case $0<x^*<1/2$ (with $a=0.2, b=0.4$)[]{data-label="figure2"}](4.eps){width="6.8cm"}
![Case $1/2<x^*<1$ (with $a=0.6, b=0.9$)[]{data-label="figure3"}](5.eps){width="6.8cm"}
![Case $1/2<x^*<1$ (with $a=0.6, b=0.9$)[]{data-label="figure3"}](6.eps){width="6.8cm"}
We can see in the case $m=2$, $k=1$ and $r=2$ the spectrums are always continuous (in fact, they are differentiable in the interior of $L_{\Phi}$). In the following examples we will see that this is no longer the case when $m=2$, $k=1$ and $r=3$.
[*Example 2.*]{} Consider the case $m=2$, $k=1$ and $r=3$. We have $$A(x) = [\phi_1(0)(1-x) +\phi_1(1)x][\phi_2(0)(1-x) +\phi_2(1)x][\phi_3(0)(1-x)+\phi_3(1)x].$$
By multiplying $\phi$ by a constant, we can always suppose that $A$ is of the form $$A(x)=(x-a)(x-b)(x-c).$$
This cubic polynomial function is either increasing or admit a local maximal point $x_{\max}$ and a local minimal point $x_{\min}$ and then we must have $x_{\max}<x_{\min}$. As we will see, the continuity of the spectrum depends on the location of $x_{\max}$ and $x_{\min}$.
When $A$ is increasing or when $x_{\max},x_{\min}\notin (0,1)$, $L_\Phi$ is the interval with $-abc$ and $(1-a)(1-b)(1-c)$ as end points. For any $\alpha$ in the interval, $A(x_\alpha)=\alpha$ admits a unique solution $x_\alpha$ in $[0,1]$ and $h_{\rm top}(E_{\Phi}(\alpha))=H(x_\alpha)$. In this case the spectrum is continuous (and differentiable).
Suppose now that $A(x)$ admits a local maximal point $x_{\max}$ and a local minimal point $x_{\min}$ (with $x_{\max}<x_{\min}$). Then there exist a unique $x'>x_{\min}$ and a unique $x''<x_{\max}$ such that $$A(x')=A(x_{\max}),\ \ A(x'')=A(x_{\min}).$$ We point out that there are three possible situations: the spectrum is continuous, admits one discontinuous point or admits two discontinuous points. Before present in detail these three situations we prove the following lemma which will be useful for our discussion.
Let $P$ be a polynomial of degree 3 with positive leading coefficient. Suppose that $P$ admits a local maximal point $x_{\max}$ and a local minimal point $x_{\min}$. Then $x_{\max}<x_{\min}$ and $$x_1<x_{\max}<x_2,|x_1-x_{\max}|=|x_2-x_{\max}| \Rightarrow P(x_1)<P(x_2)$$ $$y_1<x_{\min}<y_2,|y_1-x_{\min}|=|y_2-x_{\min}| \Rightarrow P(y_1)<P(y_2)$$
The fact $x_{\max}<x_{\min}$ follows from $P(-\infty)=-\infty$ and $P(+\infty)=+\infty$. By the existence of the extremal points, we can write $$P'(x)=\lambda(x-x_{\max})(x-x_{\min})$$ with $\lambda>0$. It follows that $$u<x_{\max}<v, x_{\max}-u=v-x_{\max} \Rightarrow \frac{|P'(u)|}{|P'(v)|}=\frac{|u-x_{\min}|}{|v-x_{\min}|}>1.$$ This means that for two equidistant points from $x_{\max}$, the left point climbs quicker than the right point descents. By integration, we get $$P(x_1)=P(x_{\max})+\int_{x_{\max}}^{x_1}P'(u)du,\
P(x_2)=P(x_{\max})+\int_{x_{\max}}^{x_2}P'(u)du.$$ Making the change of variable $v-x_{\max}=x_{\max}-u$, we obtain $$\int_{x_{\max}}^{x_1}P'(u)du=-\int_{x_1}^{x_{\max}}|P'(u)|du<-\int_{x_{\max}}^{x_2}|P'(v)|dv\le
P(x_2)-P(x_{\max}).$$ The first equality holds since $P'$ is positive in $(x_{1},x_{\max})$. Hence $P(x_1)<P(x_2)$. We prove $P(y_1)<P(y_2)$ in the same way.
In the following we present three situations. We use the last two lemmas. In each situation, there is a unique point $x_\alpha$ such that $$A(x_\alpha)=\alpha,\
h_{\rm top}(E_\Phi(\alpha))=H(x_\alpha).$$ We call $x_\alpha$ the maximizing point. For every $\alpha\in L_{\Phi}$, there could be one, two or three points $x$ such that $A(x)=\alpha$. The maximizing point $x_{\alpha}$ is the one which is the nearest to 1/2. In Figures \[figure4\], \[figure5\] and \[figure6\], those parts of graph of $A$ corresponding to the maximizing points will be traced by solid lines, other parts will be traced by dotted lines.
[*Situation I.*]{} $1/2\le x_{\max} <1<x_{\min}$ (see Figure \[figure4\]).\
Let $a=0.4$, $b=1$, and $c=2$. Then $x_{\max}=2/3$ and $x_{\min}=1.6$. The spectrum is continuous. The following hold:\
1. $L_\Phi=[A(0),A(x_{\max})]$.\
2. The maximizing points lie in $[0,x_{\max}]$.\
3. $A(x)$ is strictly monotonic in $[0,x_{\max}]$.
![Situation $1/2< x_{\max} <1<x_{\min}$ ($a=0.4$, $b=1$, and $c=2$)[]{data-label="figure4"}](7.eps "fig:"){width="6.8cm"} \[figure 1.1\]
![Situation $1/2< x_{\max} <1<x_{\min}$ ($a=0.4$, $b=1$, and $c=2$)[]{data-label="figure4"}](8.eps){width="6.8cm"}
[*Situation II.*]{} $1/2\le x_{\max} <x_{\min}<1$ (see Figure \[figure5\]).\
Let $a=0.4$, $b=0.7$, and $c=0.8$. Then $x_{\max}=0.5131$, $x_{\min}=0.7375$ and $x'=0.8737$. The spectrum admits one discontinuous point. The following hold:\
1. $L_\Phi=[A(0),A(1)]$.\
2. The maximizing points lie in $[0,x_{\max}]\cup (x',1]$.\
3. $A(x)$ is strictly monotonic in each of above two intervals.\
4. The spectrum has one discontinuous point at $A(x_{\max})(=A(x'))$, the entropy jumps from $H(x_{\max})$ to $H(x')$.
![Situation $1/2< x_{\max} <x_{\min}<1$ ($a=0.4$, $b=0.7$, and $c=0.8$)[]{data-label="figure5"}](9.eps){width="6.8cm"}
![Situation $1/2< x_{\max} <x_{\min}<1$ ($a=0.4$, $b=0.7$, and $c=0.8$)[]{data-label="figure5"}](10.eps){width="6.8cm"}
[*Situation III.*]{} $0<x_{\max} <1/2< x_{\min}<1$ (see Figure \[figure6\]).\
Let $a=0.15$, $b=0.7$, and $c=0.8$. Then $x_{\max}=0.3479$, $x_{\min}=0.7520$, $x'=0.9541$ and $x''=0.1458$. The spectrum admits two discontinuous points. The following hold:\
1. $L_\Phi=[A(0),A(1)]$.\
2. The maximizing points lie in the intervals $[0,x'')\cup [x_{\max},x_{\min}]\cup (x',1]$.\
3. $A(x)$ is strictly monotonic in each of above three intervals.\
4. The spectrum has two discontinuity points. One is $A(x'')(=A(x_{\min}))$, where the entropy jumps from $H(x'')$ to $H(x_{\min})$, the other is $A(x_{\max})(=A(x'))$, where the entropy jumps from $H(x_{\max})$ to $H(x')$.
![Situation $0< x_{\max}<1/2 <x_{\min}<1$ ($a=0.15$, $b=0.7$, and $c=0.8$)[]{data-label="figure6"}](11.eps){width="6.8cm"}
![Situation $0< x_{\max}<1/2 <x_{\min}<1$ ($a=0.15$, $b=0.7$, and $c=0.8$)[]{data-label="figure6"}](12.eps){width="6.8cm"}
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We show that asymptotically the first Betti number $b_1$ of a Shimura curve satisfies the Gauss–Bonnet equality $2\pi(b_1 - 2) = \vol$ where $\vol$ is hyperbolic volume; equivalently $2g - 2 = (1+o(1))\vol$ where $g$ is the arithmetic genus. We also show that the first Betti number of a congruence hyperbolic 3–orbifolds asymptotically vanishes relatively to hyperbolic volume, that is $b_1/\vol \to 0$. This generalises results from [@7samurai] and [@fraczyk] and we rely on results and techniques from these works, most importantly the notion of Benjamini–Schramm convergence of locally symmetric spaces.'
address:
- 'Alfréd Rényi Institute of Mathematics, Reáltanoda utca 13-15, H-1053, Budapest, Hungary'
- |
Institut de Mathématiques de Toulouse ; UMR5219\
Université de Toulouse ; CNRS\
UPS IMT, F-31062 Toulouse Cedex 9, France
author:
- Mikołaj Frczyk and Jean Raimbault
bibliography:
- 'bib.bib'
title: 'Betti numbers of Shimura curves and arithmetic three–orbifolds'
---
Introduction
============
Benjamini–Schramm convergence
-----------------------------
Let $G$ be a semisimple Lie group, $K \subset G$ a maximal compact subgroup and $X = G/K$ the associated symmetric space. Benjamini–Schramm convergence of locally symmetric orbifolds $\Gamma \bs X$ of finite volume was introduced in [@7samurai]. The Benjamini–Schramm convergence of a sequence of finite volume locally symmetric spaces $(\Gamma_i\bs X)_{i\in\mathbb N}$ to the symmetric space $X$ is equivalent to the following simple geometric condition: $$\label{BS_symmetric}
\forall R>0,\, \lim_{i\to\infty}\frac{\vol((\Gamma_i \bs X)_{<R})}{\vol(\Gamma_i \bs X)}=0,$$ where $M_{<R}$ denotes the $R$-thin part of a Riemannian orbifold $M$ (which we take to include the full singular set, see below).
In addition to $X$ there are other possible limits in the Benjamini-Schramm topology. In order to describe them it is convenient to pass to the language of invariant random subgroups (IRS) of the group $G$. These are the Borel probability measures on the Chabauty space $\sub_G$ of closed subgroups which are invariant under conjugation by elements of $G$. For every lattice $\Gamma$ of $G$ there is a unique $G$-invariant probability measure on $G/\Gamma$ and its pushforward by the map $g\Gamma \mapsto g\Gamma g^{-1}$ gives an IRS denoted $\mu_\Gamma$. It was observed in [@7samurai] that $(\Gamma_i\bs X)$ converges to $X$ if and only if $\mu_{\Gamma_i}$ converge weakly-\* to the trivial IRS $\delta_{\{1\}}.$ In general a sequence $(\Gamma_i\bs X)$ converges Benjamini-Schramm if and only if $\mu_\Gamma$ converges weakly-\* to some IRS $\nu$. The limit IRS $\nu$ is always supported on discrete subgroups and the Benjamini-Schramm limit is the random locally symmetric space $X/\Lambda$ where $\Lambda$ is a $\nu$-random subgroup of $G$.
It was proven in [@7samurai], as a consequence of the Nevo–Stück–Zimmer theorem, that if $G$ is semisimple of higher rank, with all factors having property (T) then any sequence of irreducible locally symmetric spaces converges in the Benjamini–Schramm sense to $X$. This was extended to all nontrivial products in [@levit] (see also [@Matz] for more precise results in a very specific case).
This statement is known to be false when $G = \SO(n, 1)$ or $\SU(n, 1)$, because in those cases there are lattices $\Gamma\subset G$ such that $H^1(\Gamma, \mathbb R)\neq 0$ (see [@Millson], [@Li_Millson], [@Kazhdan1977]). On the other hand restricting attention to the family of [*arithmetic congruence lattices*]{} in $G$ (see \[lattices\] below for a short description) the first author proved in [@fraczyk] that for $G=\SO(2,1),\SO(3,1)$ the symmetric space $X = \HH^2, \HH^3$ is the only possible limit in the Benjamini-Schramm topology for a sequence of congruence lattices. Previously the second author [@raimbault] had proven a similar result for the family of non-uniform, not necessarily torsion-free lattices (nonuniformity makes them much easier to deal with algebraically). In this paper we remove the torsion-free hypothesis in general.
\[Main\] If $G = \PGL_2(\RR)$ or $\PGL_2(\CC)$ and $\Gamma_n$ is a sequence of irreducible arithmetic lattices in $G$, which are either all congruence and pairwise distinct, or pairwise non-commensurable, then the sequence of locally symmetric spaces $\Gamma_n \bs X$ converges in the Benjamini–Schramm sense to $X$.
In [@fraczyk] the torsion free assumption was necessary because the methods only allowed to control the volume of the subset of thin part consiting of the collars of short geodesics. For a sequence of general arithmetic congruence orbifolds $(\Gamma_n\bs X)_{n\in\N}$ it could *a priori* happen that the vast majority of the thin part comes from the cusps or the conical singularities so the sequence does not converge to $X$. Theorem \[Main\] excludes this possibility. For the proof we use the estimates developped in [@fraczyk] to show that any weak-\* limit of the sequence $\mu_{\Gamma_n}$ is supported on elementary subgroups. By [@Osin] the only IRS supported on this set is the trivial IRS, hence the theorem. We carry out the second step of this scheme of proof in detail in Proposition \[main\_tech\], which is valid for all sequences of lattices in proper Gromov-hyperbolic spaces.
We note that because we are using a soft method our approach does not indicate the rate of decay of $\vol((\Gamma_n\bs X)_{<R})/\vol(\Gamma_n\bs X)$ as opposed to [@fraczyk].
Genus of Shimura curves
-----------------------
One application of Theorem \[Main\] is to determine the asymptotic genus of congruence surfaces of large volume. For compact surfaces without singularities the genus and volume are essentially linearly related by the Gauss-Bonnet formula. However for 2-orbifolds terms coming from cone points and cusps appear in the formula, and it is easy to see that there exists sequences of hyperbolic orbifolds with underlying space a sphere and volume going to infinity. This also has an algebraic interpretation: if $S$ is isomorphic as a Riemann surface to the $\CC$-points of an algebraic variety defined over a number field, which is the case for orbifolds obtained from congruence groups (so-called Shimura curves, see [@Shimura]), then its geometric genus is given by the Riemann–Hurwitz formula and essentially proportional to the volume while its arithmetic genus equals the topological genus of the underlying surface and can be arbitrarily smaller than the former.
It is known that this phenomenon cannot occur for congruence orbifolds: using the uniform spectral gap for congruence quotients (see [@Clozel_tau]) and a theorem of P. Zograf [@Zograf] it follows that there is a lower bound of the form $g \ge c \vol$ for congruence subgroups (see also [@LMR_genus_zero]). As a consequence of Theorem \[Main\] we obtain the following asymptotically more precise result (we note that it was known for congruence covers of the modular surface by a result of J. G. Thompson [@Thompson]).
\[genus\] Let $\Gamma_n$ be a sequence of congruence lattices in $\PSL_2(\mathbb R)$, and let $g_n$ be the topological genus of the orbifold $O_n = \Gamma_n \bs \HH^2$. Then, assuming the $\Gamma_n$ are not pairwise conjugated, we have $$\lim_{n\to +\infty} \frac{g_n}{\vol O_n} = \frac 1{4\pi}.$$
Betti numbers of 3–orbifolds
----------------------------
Theorem \[genus\] implies the weaker result that $b_1(\Gamma_n)/\vol(\Gamma_n \bs \HH^2)$ converges to $1/2\pi$ for a sequence of congruence lattices. Indeed, the rank of abelianisation is essentially equal to twice the genus in a BS-convergent sequence. This can be proven more directly by analytical means, as $1/2\pi$ is the first $L^2$-Betti number of the hyperbolic plane. While more complicated, the analytic approach generalizes to the dimension 3 and where obtain the following result.
\[Betti\] Let $\Gamma_n$ be a sequence of congruence lattices in $\PSL_2(\mathbb C)$. Then $$\lim_{n \to +\infty} \frac{b_1(\Gamma_n)}{\vol(\Gamma_n \bs \HH^3)} = 0.$$
This was proven in [@raimbault] for non-uniform lattices, and in [@fraczyk] in the case of all torsion-free lattices. Our proof is very similar to the proof for hyperbolic 3–manifolds appearing in [@7samurai].
Congruence lattices {#lattices}
-------------------
For completeness we give an explicit description of the congruence arithmetic latices in $G={\rm PGL}(2,\RR),{\rm PGL}(2,\CC)$, though we will not directly use this structure theory in the rest of the paper. Let $\mathbb K=\RR,\CC$. We start by choosing a number field $k$ with Archimedean places $\nu_1,\ldots,\nu_d$ such that $k_{\nu_1}\simeq \mathbb K$ and $k_{\nu_i}\simeq \mathbb R$ for $i\geq 2$. In what follows $\mathbb A,\mathbb A_f$ stand for the ring of adèles, respectively finite adèles of $k$. We will write $k\ni x\mapsto (x)_\nu\in k_\nu$ for the embedding of $k$ in its completion $k_\nu$. Let $a,b\in k^\times$ be such that $(a)_{\nu_i},(b)_{\nu_i}$ are positive for $i\geq 2$ and $(a)_{\nu_1}$ or $(b)_{\nu_1}$ is negative if $\mathbb K\simeq \RR$. We define the quaternion algebra $A$ as $$A = k+ {\mathbf i}k+{\mathbf j}k+ \mathbf{ij}k,$$ subject to the relations ${\mathbf i}^2=-a,{\mathbf j}^2=-b,\mathbf{ij}=-\mathbf{ji}$. By our choice of $a,b$ we have $A\otimes_k k_{\nu_1}\simeq M(2,\mathbb K)$ and for $i\geq 2$ the algebra $A\otimes_k k_{\nu_i}$ is isomorphic to the Hamilton’s quaternions. We form an algebraic group ${\mathrm{PA}^\times}=A^\times/k^\times$. It is an adjoint simple group of type $A_1$ defined over $k$. Note that $\mathrm{PA}^\times(\mathbb A)=\mathrm{PA}^\times(k\otimes_\QQ \RR)\times \mathrm{PA}^\times(\mathbb A_f)$ and $$\mathrm{PA}^\times(k\otimes_\mathbb{Q} \mathbb R)=\prod_{i=1}^d\mathrm{PA}^\times(k_{\nu_i})\simeq {\rm PGL}(2,\mathbb K)\times {\rm PO}(3)^{d-1}.$$ Choose an open compact subgroup $U$ of $\mathrm{PA}^\times(\mathbb A_f).$ Let $\Gamma_U=\mathrm{PA}^\times(k)\cap (\mathrm{PA}^\times(k\otimes_\QQ \RR)\times \mathrm{PA}^\times(\mathbb A_f))$. By a classical result of Borel-Harish-Chandra [@BHC] the group $\Gamma_U$ is a lattice in $\mathrm{PA}^\times(k\otimes_\QQ \RR)\times \mathrm{PA}^\times(\mathbb A_f)\simeq {\rm PGL}(2,\mathbb K)\times {\rm PO}(3)^{d-1}\times U$. The projection of $\Gamma_U$ to the factor $\PGL(2,\mathbb K)$ is a [*congruence arithmetic lattice*]{} in ${\rm PGL}(2,\mathbb K)$. Every congruence arithmetic lattice of ${\rm PGL}(2,\mathbb K)$ arises in this way.
Outline of the paper
--------------------
In Section \[sec\_BSconv\] we describe a general criterion for the Benjamini–Schramm convergence of lattices in the isometry group of a proper Gromov-hyperbolic spaces and we apply it, together with the estimates from [@fraczyk], to deduce Theorem \[Main\]. Next, in section \[sec\_sing\] we give a precise metric description of the singular locus of hyperbolic 2- and 3-orbifolds, and a way to smooth the boundary of the thick part while keeping control of the geometry (the technical details of which are left to Appendix \[appendix\_smooth\]). We use the description of singularities and Theorem \[Main\] to deduce Theorem \[genus\] in section \[sec\_genus\]. In section \[sec\_betti\] we use heat kernel methods (for which we need the precise description of the smoothed thick part) to deduce Theorem \[Betti\] from Theorem \[Main\].
Benjamini–Schramm convergence of quotients of hyperbolic spaces {#sec_BSconv}
===============================================================
Orbital integrals on hyperbolic spaces
--------------------------------------
Let $X$ be a proper Gromov-hyperbolic space and $G = \isom(X)$. With the compact-open topology $G$ is a locally compact second countable topological group. For $\gamma \in G$ we denote by $G_\gamma$ its centraliser. The following lemma is a slight generalisation of [@Bridson_Haefliger Corollary 3.10(2) on p. 463]—the latter dealing only with discrete groups. It might be possible to straightforwrdly adapt the arguments in loc. cit. to our case, but we give a different, mostly self-contained proof.
\[cocompact\_centr\] Let $\gamma \in G$ be an hyperbolic isometry. Then $G_\gamma / \langle \gamma \rangle$ is compact.
For the proof we use the following lemma, which should be standard but we could not find in the literature. The proof is a bit long and technical and we put it in Appendix \[proof\_lemma\].
\[dist\_trans\] Let $\gamma$ be an hyperbolic isometry of $X$. For any $x \in X$ there exists constants $C = C(x, \gamma, \delta)$ and $A = A(x, \gamma, \delta)$ such that for any $y \in X$ and any $k$ sufficiently large (depending on $\gamma, x, \delta$) we have $$d(y, \gamma^k y) \ge Ck + 2d(y, \langle \gamma\rangle x) - A.$$
Let $\tau = d(\gamma) := \inf\{d(y,\gamma y)|y\in X\}$ be the minimal displacement of $\gamma$. Fix $x \in X$, let $k, A, C$ as given by Lemma \[dist\_trans\] and define: $$D = \{y\in X| d(y,\gamma^k y)\leq k\tau + 1\}.$$ It is a non-empty (by definition of $\tau$) closed $G_\gamma$-invariant subset of $X$. Given that the action of $G_\gamma$ on $D$ is proper, the Lemma will follow once we prove that $\langle \gamma\rangle \bs D$ is compact. The previous Lemma implies that $$D \subset \{ y \in X :\: d(y, \langle \gamma\rangle x) \le (\tau - C)k + A + 1 \}$$ so that $D \subset \gamma^\ZZ B(x, R)$ for some sufficiently large $R$, and as $X$ is proper this in turn implies that $\langle \gamma\rangle \bs D$ is compact.
Let $dg$ be a fixed Haar measure on $G$. According to the lemma above the subgroup $G_\gamma$ admits a lattice so it is unimodular and we have a decomposition $dg = dx dh$ where $dx$ is a $G$-invariant measure on $G / G_\gamma$ and $dh$ a Haar measure on $G_\gamma$, both depending only on the original choice of $dg$. For a function $f \in C_0(G)$ we can then define the [*orbital integral*]{} associated to $\gamma$ by: $$\label{orb_int}
\mathcal O_f(\gamma) = \int_{G/G_\gamma} f(\gamma^{-1} x \gamma) dx$$ which depends only on the $G$-conjugacy class $[\gamma]_G$.
General criterion for Benjamini–Schramm convergence
---------------------------------------------------
Here again $X$ is always a proper Gromov-hyperbolic space and $G = \isom(X)$. We assume that the action of $G$ on $X$ is non-elementary. The [*elliptic radical*]{} of $G$ can then be defined as its unique maximal normal compact subgroup (see [@Osin Proposition 3.4]; in our context, by properness of $X$ means that bounded elements are the same as compact ones). The following lemma is a special case of [@Osin Theorem 1.5].
\[full\_limit\_set\] Let $\mu$ be an invariant random subgroup of $G$. Then either $\mu$ is supported on the elliptic radical or it has full limit set.
Recall from [@Gelander_ICM Section 3] that there is a “Benjamini–Schramm topology” on the set of Borel probability measures on the Gromov–Hausdorff space of pointed proper metric spaces (up to isometry). The set of measures supported on spaces locally isometric to $X$ is precompact in this topology. Moreover, if $X$ is a locally symmetric space then is equivalent to $\Gamma_i \bs X$ converging in the Benjamini–Schramm topology to $X$.
There is a continuous injective map from the space of invariant random subgroups of $G$ to the Benjamini–Schramm space. If $\Gamma_i$ are lattices in $G$ then the sequence of uniformly pointed spaces $\Gamma_i \bs X$ converges to $X$ if and only if the IRSs $\mu_{\Gamma_i}$ converge to the trivial IRS. We will use this to prove the following criterion for convergence.
\[main\_tech\] Let $U$ the set of hyperbolic isometries in $G$. Assume that the elliptic radical of $G$ is trivial. If $\Gamma_n$ is a sequence of lattices in $G$ which satisfies: $$\label{regular_sums}
\lim_{n \to +\infty} \frac{\sum_{[\gamma]_{\Gamma_n} \subset U} \vol((\Gamma_n)_\gamma\bs G_\gamma)\mathcal O_f(\gamma)}{\vol(\Gamma_n \bs G)} = 0$$ then the sequence of metric spaces $\Gamma_n \bs X$ converges to $X$ in the Benjamini–Schramm topology.
Let $\mu_n$ be the invariant random subgroup of $G$ supported on the conjugacy class of $\Gamma_n$. We want to prove that any weak limit $\mu$ of a subsequence of $(\mu_n)$ is equal to the trivial IRS $\delta_e$. By Lemma \[full\_limit\_set\], and the fact that a subgroup of $G$ containing no hyperbolic isometries has at most one limit point (cf. [@Gromov_hyp Section 8.2]) it suffices to prove that any such $\mu$ contains no hyperbolic isometries.
To prove this choose a covering $U = \bigcup_{C \in \mathcal C} C$ of $U$ where $\mathcal C$ is countable and every $C \in \mathcal C$ is compact. We can do this since $\sub_G$ is metrizable [@dlHarpe1 Proposition 2]. Let $W_C = \Lambda : \Lambda \cap C \not= \emptyset$ ; this is a Chabauty-closed subset of $\sub_G$. If $\nu$ is a nontrivial IRS then by Lemma \[full\_limit\_set\] and previous paragraph it almost surely contains a hyperbolic element. Hence, there is $C \in \mathcal C$ such that $\nu(W_C) > 0$. We need to prove the opposite for $\mu$, which amounts to the following : for every $C$ there exists a non-negative Borel function $F$ on $\sub_G$ which is positive on $W_C$ and such that $\int_{\sub_G} F(\Lambda) d\mu(\Lambda) = 0$.
Let us fix $C \in \mathcal C$ and prove this. There exists an open relatively compact subset $V$ with $C \subset V$ and $\ovl V\subset U$. Choose any $f \in C^\infty(G)$ such that $f > 0$ on $C$ and $f = 0$ on $G \setminus V$ and define : $$F(\Lambda) = \begin{cases} \sum_{\lambda\in\Lambda} f(\lambda) &\text{if } \Lambda \text{ is discrete; } \\
1 & \text{ if } \Lambda \text{ is not discrete and intersects } C;\\
0 & \text{otherwise.} \end{cases}$$ Then $F$ is lower semicontinuous on $\sub_G$, non-negative and positive on $W_C$. On the other hand we have : $$\begin{aligned}
\int_{\sub_G} F(\Lambda) d\mu_n(\Lambda) &= \frac 1 {\vol(\Gamma_n \bs G)} \int_{G/ \Gamma_n} \sum_{\gamma \in g\Gamma_n g^{-1}} f(\gamma) dg \\
&= \frac 1 {\vol(\Gamma_n \bs G)} \sum_{[\gamma]_{\Gamma_n} \subset U} \vol((\Gamma_n)_\gamma\bs G_\gamma)\mathcal O_\gamma(f).
\end{aligned}$$ By the so-called “Portemanteau theorem” [@klenke Theorem 13.16] the limit inferior of the left-hand side is larger or equal to $\int_{\sub_G} F(\Lambda) d\mu(\Lambda)$. By we have that the right-hand side converges to 0. It follows that $$\int_{\sub_G} F(\Lambda) d\mu(\Lambda) = 0$$ which finishes the proof.
Proof of Theorem \[Main\]
-------------------------
If $X$ is a rank-one irreducible symmetric space such as $\HH^2$ or $\HH^3$ and $G = \isom(X)$ then $G$ is a simple Lie group of non-compact type and its elliptic radical is trivial. Theorem \[Main\] thus follows immediately from Proposition \[main\_tech\] and the following result extracted from [@fraczyk].
\[reg\_conv\] Let $G = \PGL_2(\RR)$ or $\PGL_2(\CC)$ and let $U$ be the set of hyperbolic elements of $G$. Let $\Gamma_n$ a sequence of arithmetic congruence lattices in $G$, such that $\vol(\Gamma_n \bs G) \to +\infty$ or any sequence of pairwise non-commensurable arithmetic lattices. Then for any $f \in C_0^\infty(G)$ we have : $$\label{sum_hyp_integrals}
\frac 1 {\vol(\Gamma_n \bs G)} \left| \sum_{[\gamma] \subset U} \vol((\Gamma_n)_\gamma\bs G_\gamma)\mathcal O_f(\gamma) \right| \xrightarrow{n \to +\infty}{} 0.$$
If $\Gamma$ is an arithmetic lattice in $\PGL_2(\RR)$ or $\PGL_2(\CC)$ then an element $\gamma \in \Gamma$ is hyperbolic if and only if it is semisimple and of infinite order. In the proof of [@fraczyk Theorem 1.8], starting form the lines (10.7-10.9) the author bounds the sum $$\label{eqUpperBound}\sum_{\substack{[\gamma]_{\Gamma}\\ \textrm{non torsion}}}\vol(\Gamma_\gamma\bs G_\gamma)\mathcal O_\gamma(f)$$ for congruence arithmetic lattices. The line (10.7) of [@fraczyk p. 67] is the adèlic version of the last sum where we group together the classes conjugate over $\mathrm{PA}^\times(k)$, where $\mathrm{PA}^\times$ is the group used to construct the lattice $\Gamma$ as explained in Section \[lattices\]. The passage between the adèlic and classical trace formula is explained in [@fraczyk Theorem 4.21]. Proceeding as in [@fraczyk p. 67-69] we obtain the bound $$\sum_{\substack{[\gamma]_{\Gamma}\\ \textrm{non torsion}}}\vol(\Gamma_\gamma\bs G_\gamma)\mathcal O_\gamma(f)\ll \vol(\Gamma\bs G)^{0.986}.$$ Any hyperbolic conjugacy class $[\gamma]_\Gamma$ is non-torsion so we can deduce the that the sum converges to 0 as $\vol(\Gamma\bs X)\to\infty$ and $\Gamma$ is a congruence arithmetic lattice. In order to establish the convergence for sequences of pairwise non-commensurable arithmetic lattices $(\Gamma_n)_{n\in\NN}$ we choose for each $n$ a maximal arithmetic lattice $\Lambda_n$ containing $\Gamma_n$. It is always a congruence arithmetic lattice. We have $$\begin{aligned}
\frac{1}{\vol(\Gamma_n\bs X)}\left| \sum_{[\gamma]_{\Gamma_n}\in U}\vol((\Gamma_n)_\gamma\bs G_\gamma)\mathcal O_f(\gamma) \right|&\leq \\
\frac{1}{\vol(\Gamma_n\bs X)}\sum_{[\gamma]_{\Gamma_n}\in U}\vol((\Gamma_n)_\gamma\bs G_\gamma)\mathcal O_{|f|}(\gamma) &\leq \\
\frac{1}{\vol(\Lambda_n\bs X)}\sum_{[\gamma]_{\Lambda_n}\in U}\vol((\Lambda)_\gamma\bs G_\gamma)\mathcal O_{|f|}(\gamma)&=o(1).\end{aligned}$$
Structure of the singular locus of closed hyperbolic orbifolds {#sec_sing}
==============================================================
To be able to deduce from the sole Benjamini–Schramm convergence of a sequence of orbifolds further asymptotic results on topological invariants we need a fine metric description of the singular locus. The results in this section provide it; they are not really original but precise statements such as we need are not found in the litterature. As usual our main tool is the Margulis lemma.
\[margulis\] For every $n \ge 2$ there exists $\eps = \eps(n) > 0$ such that the following holds. Let $\Gamma$ be a discrete subgroup of isometries of $\HH^n$, then for any $x \in \HH^n$ the subgroup $$\Gamma_\eps := \langle \gamma \in \Gamma : d(x, \gamma x) \le \eps \rangle$$ is virtually abelian.
In the sequel we will only work in 2 or 3-dimensional hyperbolic space, and we let $\eps$ denote a Margulis constant which is valid for both cases. Recall that $O_{\le \eps}$ stands for the $\eps$-thin part of an orbifold $O$, for which we use the following definition: if $O = \Gamma \bs X$ where $X$ is the orbifold universal cover and we assume $X$ to be CAT(0) then $$\label{def_thinpart}
O_{\le \eps} = \Gamma \bs \{ \tilde x \in X :\: \exists \gamma \in \Gamma \setminus\{\mathrm{Id}\},\, d(\tilde x, \gamma\tilde x) \le \eps \}$$ which includes the singular locus of $O$—note that in the litterature, e.g. in [@BMP], a different convention is often used where only points with large stabilisers are included. The closure of the complement of $O_{\le \eps}$ (the $\eps$-thick part) will be denoted by $O_{\ge \eps}$.
In fact we need to tweak a bit the definition of the thin part around that part of the singular locus where the cone angle is $\pi$: around these vertices or geodesics we put a collar whose width is $\eps/6$ (instead of $\eps/2)$.
2-dimensional orbifolds
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In $\PGL_2(\RR)^+$ all the virtually abelian discrete subgroups are given by the following list:
1. An infinite cyclic group generated by an hyperbolic or parabolic isometry;
2. \[isolated\_cone\_pt\] A finite cyclic group generated by an elliptic isometry;
3. \[pair\_half\_turns\] An infinite dihedral group generated by two elliptic isometries of order 2.
As a first consequence we see that the singular locus of an orientable hyperbolic 2-orbifold consists only of [*cone points*]{}, that is all non-manifold points have a neighbourhood which is isometric to the quotient of a disc by a finite cyclic group.
In addition we can deduce from this classification a metric description of the singular locus. We need the following notation: given an elliptic isometry $\gamma$ with fixed point $x$ and rotation angle $\theta$, let $\ell(\theta, \eps)$ be the smallest $\ell$ such that $d(y, \gamma y) \ge \eps$ for $d(x, y) = \ell$. Similarly, given a hyperbolic isometry $\gamma$ of minimal displacement $\ell$ we define $r(\ell, \eps)$ to be the minimal distance from its axis at which an hyperbolic isometry translates of at least $\eps$.
\[2dim\_sing\] Let $O = \Gamma \bs \HH^2$ be an orientable hyperbolic 2-orbifold and $x$ a point in its singular locus. Then $x$ is an isolated cone point and one of the following possibilities hold:
1. If its angle is $2\pi/m$ with $m \ge 3$ then there is no other singular point in the ball $B_O(x, \ell)$ where $\ell = \ell(2\pi/m, \eps)$.
2. If the angle is equal to $\pi$ then either there is no other singular point within distance $\ell(\pi, \eps)$, or there is one (and its cone angle is also $\pi$) at distance $\ell_x < \ell(\pi, \eps)$ but no other within distance $r(\ell_x, \eps)$ of $x$.
Let $\Gamma x\in O$ be as in the statement, with $x\in\HH^2$. Then $x$ is a fixed point of a non-trivial element of $\Gamma$, and it follows that the subgroup $$\Gamma_x^\varepsilon = \{ \gamma \in \Gamma :\: d(x, \gamma x) \le \eps \}$$ must be one of those described in (\[isolated\_cone\_pt\]) or (\[pair\_half\_turns\]) at the beginning of this section; let $\gamma_0$ be a generator (with minimal rotation angle) of the cyclic subgroup fixing $x$ and $m > 1$ its order.
In any case $x$ lies above a conical point in $O$. Assume now that $m \ge 3$; then $\Gamma_x = \langle \gamma_0 \rangle$ and by the Margulis lemma there is no other fixed point of a non-trivial element in $\Gamma$ within the set $$C = \{ y \in \HH^2 :\: d(y, \gamma_0 y) \le \eps) \}.$$ By definition the ball $B_{\HH^2}(x, \ell(2\pi/m, \eps))$ is contained in $C$, so it contains no other singular point.
If $m = 2$ and there is another elliptic fixed point $x' \in \HH^2$ with $d(x, x') \le \ell(\pi, \eps)$ then we might assume that $x'$ is the closest such point. By the previous paragraph any nontrivial $\gamma_0' \in \Gamma$ fixing $x'$ must be of order $2$. Let $\eta = \gamma_0\gamma_0'$. It is a hyperbolic isometry with axis containing the geodesic $\alpha$ joining $x$ to $x'$ and translation distance $2d(x, x')$. Write $\Gamma_\alpha$ for the setwise stabilizer of $\alpha$ in $\Gamma$. For every $\gamma\in\Gamma_\alpha$ not fixing $x$ we will have $d(x,\gamma x)\geq 2d(x,x')$ as otherwise $\gamma_0\gamma$ would have a fixed point closer to $x$ than $x'$. We deduce that $\Gamma_\alpha=\langle\gamma_0, \gamma_0'\rangle$. The former is a maximal virtually abelian subgroup of $\Gamma$ (it is an intersection of $\Gamma$ with the normaliser of a split torus). The Margulis lemma now implies that within the ball $B_{\HH^2}(x, \ell(\pi, \eps))$ (resp. $B_{\HH^2}(x, r(\ell_x, \eps))$) any other elliptic fixed point must be a translate of either $x$ or $x'$ by a power of $\eta$, as any such point is moved by at most $\eps$ by $\gamma_0$ (resp. $\eta$) and hence its stabiliser in $\Gamma$ must belong to $\Gamma_\alpha$.
3-dimensional orbifolds
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### Description of the singular locus
The list of discrete virtually abelian subgroups of $\PGL_2(\CC)$ is long enough to make us avoid giving a complete description. Rather, we will assume that $\Gamma$ is a cocompact lattice in $\PGL_2(\CC)$ and $\Lambda$ a maximal virtually abelian subgroup of $\Gamma$ which contains torsion elements (which is all we need to prove Theorem \[Betti\]). If $\Lambda$ contains a hyperbolic element $\gamma$ then it must normalise $\langle\gamma\rangle$, so it is contained in the normalizer of a maximal torus. Any such normalizer is isomorphic to $\CC^\times \rtimes \ZZ/2$. Otherwise $\Lambda$ contains only elements if finite order and so by Burnside’s theorem it must be a finite subgroup of the maximal compact $\PU(2)$. It follows that $\Lambda$ is one of the following groups:
1. $\langle \gamma, \eta \rangle \cong \ZZ \times \ZZ/m$ where $\gamma, \eta$ are respectively hyperbolic and elliptic isometries sharing the same axis;
2. $\langle \gamma, \eta, \rho \rangle \cong (\ZZ \times \ZZ/m) \rtimes \ZZ/2$ where $\eta, \gamma$ are as above (with $\eta$ possibly trivial) and $\rho$ is an elliptic of order 2 with axis orthogonal to that of $\gamma$ or $\eta$;
3. One of the finitely many non-dihedral finite subgroups of $\PU(2)$.
We see from this description that the singular locus of an hyperbolic 3–orbifold consists of closed geodesics (which we’ll call [*singular geodesics*]{}), which can intersect each other. A singular point not on the intersection of two singular geodesics has a neighbourhood isometric to the quotient of a ball by a rotation; the angle of the latter we will call the cone angle of the singular geodesic. We will call a vertex which is at the intersection of two or more singular geodesics a [*vertex*]{} of the singular locus.
Together with the Margulis lemma the list above allows us to give the following metric description of the singular locus (see also [@BMP Corollary 6.3] for a more geometric description, and loc. cit., Fig. 5 on p. 33 for illustrations). This description is analogous to the situation from Lemma \[2dim\_sing\].
\[3dim\_sing\] Let $O$ be a compact orientable 3-dimensional hyperbolic orbifold and $\Sigma$ its singular locus. Let $x \in \Sigma$ be a vertex. Then one of the two following possibilities hold.
1. \[exception\] The $\eps/2$-neighbourhood of $x$ is isometric to one of a finite list of orbifolds, whose singular locus has only one vertex and all singular geodesics go through $x$.
2. \[dihedral\] There is at most one other singular vertex $x'$ within distance $\eps/2$ of $x$; $x$ and $x'$ are joined by a singular geodesic $c$ of length $\ell$ and cone angle $2\pi/m$, there are two singular geodesics with cone angle $\pi$ and orthogonal to $c$ each going through one of $x$ or $x'$. There are no further components of the singular locus within distance $\max(\ell(2\pi/m, \eps), \ell(r, \eps))$ of $x$ and $x'$.
Moreover if two non-intersecting singular geodesics of $O$ are within distance $\eps/2$ of each other then both have angle $\pi$.
Let $O = \Gamma \bs \HH^3$ a closed hyperbolic 3–orbifold. Let $x$ be a vertex in the singular locus of $O$ and $\Pi$ the subgroup of $\Gamma$ fixing a lift $\tilde x$ of $x$ to $\HH^3$. Then $\Pi$ is either a dihedral group $\ZZ/m \rtimes \ZZ/2$ or one of finitely many finite non-dihedral subgroups of $\PU(2)$, according to the list of virtually abelian subgroups of $\Gamma$ above. We note that under the condition in .
If the vertex is as in and $\gamma \in \Gamma$, $\gamma \not\in \Pi$ is an elliptic isometry of order $m$ then as (by the Margulis Lemma) $\Pi$ contains all isometries moving $\tilde x$ by more than $\eps$ any fixed point of $\gamma$ must be at distance at least $\ell(2\pi/m, \eps) \ge \ell(\pi, \eps) = \eps/2$ of $\tilde x$. Similarly any hyperbolic isometry in $\Gamma$ must move $\tilde x$ by at least $\eps$. Hence the quotient $\Pi \bs B(\tilde x, \eps/2)$ embeds into $O$.
If the vertex has a dihedral stabiliser as in let $\gamma$ be a generator of the $\ZZ/m$-subgroup and $\eta$ a generator of the $\ZZ$-subgroup commuting with $\gamma$. Then we might assume that either $\ell < \eps/2$ or $m > 5$ (otherwise we can add its neighbourhood to the finite list in ). Then any elliptic element of $\Gamma$ which does not normalise $\langle\gamma\rangle$ cannot fix a point in $B(\tilde x, \eps)$ (otherwise it and $\gamma$ would generate a subgroup moving a point by less than $\eps$ but not in the list given above, which is not possible by the Margulis Lemma). Similarly it cannot fix a point within $\ell(\ell, \eps)$ of the axis of $\eta$.
### Smoothing the thick part
Let $C = (C_0, C_1, \ldots) \in [0, +\infty[^\NN$. As (a slight variation of) the definition in [@Lueck_Schick] we say that a Riemannian manifold has [*$C$-bounded geometry*]{} if its injectivity radius is at least $C_0$, the normal geodesic flow up to $C_0$ gives coordinates for a collar neighbourhood of the boundary, and the $k$th derivatives of the metric tensor and its inverse (in normal coordinates) are bounded in sup norm by $C_k$. In this section we prove the following lemma.
\[smoothing\] There exists $C$ such that for any hyperbolic 3–orbifold $O$ there exists a smooth submanifold $O'$ such that:
- $O_{\ge \eps} \subset O'$ and this is an homotopy equivalence;
- $O'$ is of $C$-bounded geometry.
We will deduce the lemma from the description of the singular locus and the following general proposition, the proof of which we give in appendix \[appendix\_smooth\].
\[general\_smoothing\] Let $X$ be a Riemannian $d$-manifold and $H_1, H_2$ two open subsets whose closures have smooth boundary. Assume the following holds:
- they intersect transversally in a compact subset; let $\alpha_0$ such that the dihedral angles at the intersection stay within the interval $]\alpha_0, \pi-\alpha_0[$.
- Both manifolds $X \setminus H_i$ are of bounded geometry.
Then for any $\delta>0$ there exists an open subset $H$ of $X$ such that:
1. \[contain\] $H \supset H_1 \cup H_2$ and they are equal outside of the $\delta$-neighbourhood of $H_1 \cap H_2$;
2. \[smooth\_bd\] the closure of $H$ has a smooth boundary;
3. $X \setminus H$ is of bounded geometry; the bounds depend only on $\delta$, on the bounds on the geometry of $X$ and $X \setminus H_i$ and on $\alpha_0$.
Observe first that the boundary of the thin part is smooth away from the geodesics with cone angle $\pi$ and the vertices of the singular locus, as follows from the third part of Lemma \[3dim\_sing\]. Thus the non-smooth part of $\pl O_{\ge \eps}$ comes from intersecting tubular neighbourhoods of singular geodesics and short geodesics. There are finitely many possible configurations where the geodesics are not orthogonal to each other (corresponding to case of Lemma \[3dim\_sing\]); we do not need to deal in detail with these, so the only problem left to deal with is the following: at all points in the intersection of the tubular neighbourhood $N_1$ (with varying radius) of one geodesic, and the $\eps/6$-tubular neigbourhood $N_2$ of another geodesic orthogonal to the first, the dihedral angle between $\pl N_1$ and $\pl N_2$ stays bounded away from 0 and from $\pi$[^1].
To prove this note that the maximum and minimum values for these angles both are continuous functions of the radius $0 \le r < +\infty$ of $N_1$. It can be continuously extended to $r = +\infty$, the values then being those of the angle (in a conformal model of $\HH^3$) between $\pl N_1$ and the boundary at infinity of $\HH^3$. As $N_1$ and $N_2$ are never tangent to each other we see by compactness that the maximal and minimal values stay bounded away from $0$ and $\pi$.
The genus of congruence orbifolds {#sec_genus}
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In this section we prove Theorem \[genus\]. Let $O$ be an hyperbolic orbifold of dimension 2, which is a quotient of the hyperbolic plane $\HH^2$ by a lattice of $\mathrm{PSL}_2(\RR)$. Then the underlying topological space $|O|$ is a surface of finite type, that is it is homeomorphic to a compact surface $S$ with a finite number of points removed. The [*genus*]{} of $O$ is defined to be the genus of $S$.
Suppose that $O$ has genus $g$, that it has $k$ punctures and $r$ conical singularities with angles $2\pi/m_1, \ldots, 2\pi/m_r$ (the tuple $(g, k, m_1, \ldots, m_r)$ is then called the [*signature*]{} of $O$). Then, computing the volume of a well-chosen fundamental polygon we get the following equality (see [@Beardon Theorem 10.4.2]): $$\label{GB}
\vol O = 2\pi\left( 2g - 2 + k + \sum_{i=1}^r \left( 1 - \frac 1{m_i} \right) \right).$$ From this equation we obtain the bound: $$\left| g - \frac{\vol(O)}{4\pi} \right| \le \frac {k + r + 2}{4\pi}.$$ We now see that Theorem \[genus\] follows from Theorem \[Main\] together with the following proposition.
Let $O_n$ be a sequence of hyperbolic 2–orbifolds which is Benjamini–Schramm convergent to $\HH^2$. Let $k_n, r_n$ be the number of cusps and conical points of $O_n$, respectively. Then $k_n + r_n = o(\vol O_n)$.
To prove that $r_n = o(\vol O_n)$ we associate to each conical point $x$ with angle $\theta$ the region $$\Omega_x = B(x, \ell(\theta, \eps))$$ if there is no other singular point within distance $\ell(\theta, \eps)$. Otherwise let $\ell_x$ be the distance to the nearest singular point and put $$\Omega_x = B(x, r(\ell_x, \eps)).$$ We will check below the following facts:
1. \[lowbd\_vol\] there exists $c > 0$ such that $\vol \Omega_x > c$ for all $n$ and $x \in O_n$;
2. \[almost\_disjoint\] if $x \in O_n$ is a conical point then there is at most one conical point $x' \not= x$ such that $x \in \Omega_{x'}$;
3. \[thin\_part\] for all conical points $x \in O_n$ we have $\Omega_x \subset (O_n)_{\le \eps}$.
It follows from these that: $$r_n \le \frac 1 c\sum_{x \in \Sigma_{O_n}} \vol \Omega_x \le \frac 2 c \vol\left( \bigcup_{x \in \Sigma_{O_n}} \Omega_x \right) \le \frac 2 c \vol(O_n)_{\le\eps}$$ and as the right-hand side is $o(\vol O_n)$ in a BS-convergent sequence we get that $r_n = o(\vol O_n)$.
That \[thin\_part\] holds follows immediately from the definitions of $\ell(\theta, \eps)$ and $r(\ell, \eps)$. Point \[almost\_disjoint\] follows from Lemma \[2dim\_sing\].
It remains to prove \[lowbd\_vol\]. Let $x \in O_n$ be a singularity with cone angle $2\pi/m$ with $m > 2$, let $\tilde x$ be a lift of $x$ to $\HH^2$ and $\ell = \ell(2\pi/m, \eps)$. Then we have $$\vol(B_{O_n}(x, \ell)) = \frac 1 m B_{\HH^2}(\tilde x, \ell) \gg \frac{e^\ell}m$$ so we need to prove that $e^\ell \gg m$. This follows easily from distance computations in the disk model: by definition of $\ell(\theta, \eps)$ we have that $\ell(\theta, \eps) = \log((1+r)/(1-r))$ where $0 < r < 1$ is such that $d(r, re^{i\theta}) = \eps$. It follows that $$\cosh(\eps) = 1 + \frac{2r^2|1 - e^{i\theta}|^2}{(1 - r^2)^2}$$ and by standard computations we get that $$r = 1 - \frac \theta{\sqrt 2\sinh(\eps)} + O(\theta^2)$$ whence it follows that $$\ell(\theta, \eps) = -\log(\theta) - c + O(\theta)$$ for some constant $c$ depending on $\eps$. We finally get that $\ell \gg e^{\log(m/2\pi)} \gg m$.
Assume now that $m = 2$ and that there is another singular point $x'$ within $\ell(2, \eps)$ of $x$. In this case the volume of $\Omega_x$ is half that of a collar around a closed geodesic of length $r(\ell_x, \eps) \ll \eps$; as the latter is bounded below (see [@Halpern]) so is that of $\Omega_x$.
The proof that $k_n = o(\vol O_n)$ is similar: by the Margulis lemma the regions of the $\eps$-thin part where a given conjugacy class of parabolic isometries realises the injectivity radius are pairwise disjoint, and an easy hyperbolic area computation shows that the volume of such a region is bounded below.
Betti numbers of arithmetic 3–orbifolds {#sec_betti}
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Recall that $\eps$ is a Margulis constant for $\HH^3$. Let $O$ be a 3–orbifold, then we will write $O'$ for the manifold with boundary obtained by Lemma \[smoothing\]. We write $\Delta_\abs^1$ for the maximal self-adjoint extension of the Hodge–Laplace operator on $O'$ with absolute boundary condition. The goal of this section is to prove the following proposition, which we prove by extending the analysis at the end of section 7 in [@7samurai] to the orbifold case.
\[limit\_hk\] Let $O_n$ be a sequence of closed hyperbolic 3–orbifolds which BS-converge to $\HH^3$, and let $O_n'$ be the smoothings described in Lemma \[smoothing\]. Then for all $t > 0$ we have that $$\limsup_{t \to +\infty} \lim_{n \to +\infty} \frac{ \otr(e^{-t\Delta_\abs^1[O_n']}) }{ \vol O_n } = 0.$$
Before giving the proof we explain how this implies Theorem \[Betti\]: let $O_n = \Gamma_n \bs \HH^3$. By Hodge theory we have $b_1(O_n') \le \otr(e^{-t\Delta_\abs^1[O_n']})$ for all $t$, and so Proposition \[limit\_hk\] implies that $$\lim_{n \to +\infty} \frac{b_1(O_n')}{\vol O_n} = 0.$$ On the other hand we have that the orbifold fundamental group $\Gamma_n$ is a quotient of $\pi_1(O_n')$. Indeed, the universal cover of $(O_n)_{\ge \eps}$ is a cover of the connected subset $(\widehat O_n)_\eps$ of those $x \in \HH^3$ which are not displaced by less than $\eps$ by some non-trivial element of $\Gamma_n$, and $(O_n)_{\ge \eps}$ is homotopy equivalent to $O_n'$. Moreover $O_n'$ is aspherical (as the cover $(\widehat O_n)_\eps$ constructed above is) so that $H_1(O_n')$ is the abelianisation of $\pi_1(O_n')$. From these two facts it follows that $b_1(\Gamma_n) \le b_1(O_n')$, so that $b_1(\Gamma_n) = o(\vol O_n)$ as well.
The proof of Proposition \[limit\_hk\] is done in three steps: first we observe convergence of the part of the trace formula for $O_n$ coming from the $\eps$-thick part: see . The two next steps together imply that the trace of the heat kernel on $O_n'$ is asymptotically the same as that computed in : first we analyse the integral of the difference on the $R$-thick part and show that it limit superior is $o(R)$ (see , then we prove that the integral on the $R$-thin part of $O_n'$ asymptotically vanishes (see ). Altogether these three steps imply that $$\lim_{n \to +\infty} \frac{ \otr(e^{-t\Delta_\abs^1[O_n']}) }{ \vol O_n } = \tr e^{-t\Delta^1[\HH^3]}$$ where we denoted $\tr e^{-t\Delta^1[\HH^3]} = \tr e^{-t\Delta^1[\HH^3]}(\tilde x, \tilde x)$ for any $\tilde x \in \HH^3$. The proposition now follows from the vanishing of the first $L^2$-Betti number of $\HH^3$, which means that $\lim_{t \to +\infty} \tr e^{-t\Delta^1[\HH^3]} = 0$ (see [@Lueck_book]).
Trace formula on the thick part
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Let $O_n$ be a sequence as in Proposition \[limit\_hk\]. We prove here that $$\label{lim_hk_thick}
\int_{(O_n)_{\ge\eps}} \tr e^{-t\Delta^1[O_n]}(x, x) dx - \tr e^{-t\Delta^1[\HH^3]} \cdot \vol O_n = o(\vol O_n).$$ Let $\mathcal C_{n, e}$ and $\mathcal C_{n, h}$ be the sets of conjugacy classes of respectively elliptic and hyperbolic elements in $\Gamma_n$. For $\gamma \in \Gamma$ let $\mathcal F_\gamma$ be a fundamental domain for the centraliser $Z_\gamma$ of $\gamma$ in $\Gamma$ and $\mathcal F_\Gamma^{\ge \eps}$ the part of it on which no non-trivial element of $\Gamma$ displaces by less than $\eps$. The proof of the Selberg trace formula then gives that $$\begin{gathered}
\label{trace_formula}
\int_{(O_n)_{\ge\eps}} \tr e^{-t\Delta^1[O_n]}(x, x) dx = \vol(O_n)_{\ge\eps}\tr e^{-t\Delta^1[\HH^3]} \\ + \sum_{[\gamma] \in \mathcal C_{n, e} \cup \mathcal C_{n, h}} \int_{\mathcal F_\gamma^{\ge\eps}} \tr(\gamma^* e^{-t\Delta^1[\HH^3]} (x, \gamma x)) dx.\end{gathered}$$ Because of Benjamini–Schramm convergence we have $\vol O_n - \vol(O_n)_{\ge\eps} = o(\vol O_n)$. Then will follow from together with the following limit: $$\label{vanishing_contrib}
\sum_{[\gamma] \in \mathcal C_{n, e} \cup \mathcal C_{n, h}} \int_{\mathcal F_\gamma^{\ge\eps}} \tr(\gamma^* e^{-t\Delta^1[\HH^3]} (x, \gamma x)) dx = o(\vol O_n).$$ We proceed to prove (\[vanishing\_contrib\]). The proof for the hyperbolic part is exactly the same as in [@7samurai Section 7].
We deal now with the elliptic part; to simplify notation we cheat slightly by integrating over the part of $\mathcal F_\gamma$ where elliptic elements in $Z_\gamma$ translate by at least $\eps$, which we will continue to denote by $\mathcal F_\gamma^{\ge\eps}$ (note that it is larger than what we denoted by $\mathcal F_\gamma^{\ge \eps}$ above). If $[\gamma]$ is an elliptic conjugacy class we let $\theta_\gamma$ be its rotation angle and $\ell_\gamma$ the minimal translation length of an hyperbolic isometry in $Z_\gamma$. Then we have by integrating in polar coordinates around the axis of $\gamma$ that $$\int_{\mathcal F_\gamma^{\ge\eps}} \tr(\gamma^* e^{-t\Delta^1[\HH^3]} (x, \gamma x)) dx = \theta_\gamma\ell_\gamma \int_{\max(\ell(\theta_\gamma, \eps), r(\ell_\gamma, \eps))}^{+\infty} f_\theta(r) dr$$ where $f_\theta(r) = \sinh(r)\cosh(r)\tr(\gamma^*e^{-t\Delta[\HH^3]}(x, \gamma x))$ for a point $x$ at distance $r$ from the axis. This is a consequence of desintegration of hyperbolic volume in cylindrical coordinates [@fenchel1 p. 205]. Let $\Sigma_n$ be the set of singular geodesics in $O_n$ (so each is the image of an axis of an elliptic conjugacy class in $\Gamma_n$). If $\gamma$ is an elliptic isometry of order $m$, primitive in $\Gamma$, there are $m-1$ elliptic elements in $Z_\gamma$ sharing the same axis. So we get that $$\sum_{[\gamma] \in \mathcal C_{n, e}} \int_{\mathcal F_\gamma^{\ge\eps}} \tr(\gamma^* e^{-t\Delta^1[\HH^3]} (x, \gamma x)) dx = \sum_{c \in \Sigma} 2\pi \ell_c \frac{o_c - 1}{o_c} \int_{\max(\ell(\theta_\gamma, \eps), r(\ell_\gamma, \eps))}^{+\infty} f_{2\pi/o_c}(r) dr$$ where $\ell_c$ is the length of $c$ and $2\pi/o_c$ its cone angle. By the Gaussian estimate of the heat kernel of $\HH^3$ we have that $$f_{2\pi/o_c}(r) \ll C(t)e^{-c(t)r^2}$$ uniformly for $r \ge \ell(\theta_\gamma, \eps)$ and it follows that $$\sum_{[\gamma] \in \mathcal C_{n, e}} \int_{\mathcal F_\gamma^{\ge\eps}} \tr(\gamma^* e^{-t\Delta^1[\HH^3]} (x, \gamma x)) dx \ll \sum_{c \in \Sigma_n} \ell_c$$ and the right-hand side is an $o(\vol O_n)$ by Benjamini–Schramm convergence.
Comparison between heat kernels
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We prove here that $$\label{comphk_thick}
\lim_{R \to +\infty} \limsup_{n \to +\infty} \frac 1 {\vol O_n} \int_{(O_n)_{\ge R}} \tr (e^{-t\Delta^1[O_n]} - e^{-t\Delta^1[O_n']})(x, x) dx = 0.$$ To do this we let $U_n$ be the subset of $\HH^3$ covering $O_n'$ and choose a fundamental domain $D_n$ for $\Gamma$ acting in the subset of $U_n$ covering $(O_n)_{\ge R}$ (we assume $R$ is large enough so that $(O_n)_{\ge R} \subset O_n'$). Then we can write $$\begin{aligned}
\int_{(O_n)_{\ge R}} \tr (e^{-t\Delta^1[O_n]} - e^{-t\Delta^1[O_n']})(x, x) dx &= \int_{D_n} \sum_{\gamma\in \Gamma} \tr\gamma^*(e^{-t\Delta^1[\HH^3]} - e^{-t\Delta[U_n]})(x, \gamma x) dx \\
&\ll e^{-\frac{R^2}{Ct}} \int_{D_n} \sum_{\gamma\in \Gamma} e^{-\frac{d(x, \gamma x)^2}{Ct}} dx\end{aligned}$$ where the second line follows from [@Lueck_Schick Theorem 2.26]. By the same arguments as used above to demonstrate the integral is $O(\vol O_n)$ (with a constant independent of $R$ as the domain of integration shrinks when we take $R$ to infinity). In the end we get that $$\limsup_{n \to +\infty} \frac 1 {\vol O_n} \int_{(O_n)_{\ge R}} \tr (e^{-t\Delta^1[O_n]} - e^{-t\Delta^1[O_n']})(x, x) dx \ll e^{-\frac{R^2}{Ct}}$$ from which follows immediately.
Heat kernel near the boundary
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Here we prove the final ingredient for the proof of Proposition \[limit\_hk\]: for all $R > 0$ we have $$\label{near_bd}
\int_{O_n' \setminus (O_n)_{\ge R}} \tr e^{-t\Delta^1[O_n']}(x, x) dx = o(\vol O_n).$$ By Benjamini–Schramm convergence we have that $\vol(O_n' \setminus (O_n)_{\ge R}) = o(\vol O_n)$. So to prove it suffices to see that $\tr e^{-t\Delta^1[O_n']}(x, x) = O_t(1)$ for $x \in O_n'$. As in [@7samurai (7.19.4)] this follows from [@Lueck_Schick Theorem 2.35]; the latter is applicable with a uniform constant in our context by Lemma \[smoothing\].
Proof of Lemma \[dist\_trans\] {#proof_lemma}
==============================
Let $x, y \in X$. As $\gamma$ is hyperbolic there exists $a, c$ such that $L = \langle \gamma\rangle x$ is a $(c, a)$-quasi-geodesic. Regarding the conclusion of the proposition it does not change anything if we assume that $x$ is the approximate projection of $y$ on $L$, meaning that any point $x'$ of $L$ within distance $d(y, L)$ of $y$, satisfies $d(x', x) \le K$ (where $K$ depends only on the hyperbolicity constant $\delta$).
Let $\ell = d(x, \gamma x)$. Note first that if $k$ is large enough so that $$\label{k_large}
k > 100c\ell^{-1}K\log(k) + ac$$ holds, and $y$ is close enough to $L$ so that $$\label{y_away}
d(y, x) > c^2\ell^{-1}\log(k) + cK(2+\log(2+k)) + ca$$ does not then the conclusion is immediate by the triangle inequality. Thus from now on we will assume that both hold.
Let $x_i = \gamma^i x$, $y_i = \gamma^i y$ for $0 \le i \le k$. Let $F$ be the finite set $$F = \{x_0, x_1, \ldots, x_k \} \cup \{ y_0, y_k \};$$ by [@Bowditch Proposition 7.3.1] there exists a choice of a “spanning tree” on $F$ (that is, a tree whose edges are a subset of all pairs of geodesics segment between points of $F$) such that $$\label{spanning_tree}
\forall p, q \in F :\: d(p, q) \ge d_{T_F}(p, q) - (1 + \log(2+k))K$$ where $K$ depends only on $\delta$ (so we take it equal to the $K$ introduced above to simplify notation). One of $y_0, y_k$ must be connected to one of the $x_i$ in $T_F$; we may assume that $[y_0, x_i]$ is an edge in $T_F$ for some $i$. We claim that this $i$ must be unique, and we must have $$\label{i_small}
i < c\ell^{-1}\left( (\log(k+2) + 2)K +a \right).$$ Indeed, let $i$ be the smallest integer such that $[x_i, y_0] \subset T_F$. Then, because $$d_{T_F}(x_0, y_0) \le d(x_0, y_0) + (\log(k+2) + 1)K$$ and $$d_{T_F}(x_0, y_0) \ge d(x_0, x_i) + d(x_i, y_0) \ge \frac{i \ell} c - a + d(x_0, y_0) - K$$ we see that $i$ must verify . Now assume that there is a $j > i$ such that $[x_j, y_0] \subset T_F$, and take it to be the smallest such; we want to reach a contradiction. Consider $i \le l < j$ to be maximal such that the path in $T_F$ from $x_l$ to $x_i$ does not go through $y_0$. Then the path in $T_F$ from $x_l$ to $x_{l+1}$ must go through $y_0$ (otherwise we would have a path from $x_{l+1}$ to $x_i$ via $x_l$ avoiding $y_0$). We have thus $d_{T_F}(x_l, x_{l+1}) \ge d(x_0, y_0) - K$ which together with and contradicts the fact that $d(x_l, x_{l+1}) = \ell$.
We now want to prove that $[y_0, y_k]$ is not an edge in $T_F$. To do so we must consider two possibilities. Assume first that $[y_k, x_j] \subset T_F$ for some $j$. Then reasoning as above we see that $j$ is the only such index, and $j > k - c\ell^{-1}\left( (\log(k+2) + 2)K +a \right) > i$. In this case we reach a contradiction in the same way as in the previous paragraph: considering a maximal $i \le l < j$ such that the path from $x_l$ to $x_i$ does not go through $y_0$ we see that $d_{T_F}(x_l, x_{l+1})$ is too large.
If there is no edge $[y_k, x_j]$ in $T_F$ then the path from $x_k$ to $y_k$ must go first to $x_i$, then to $y_0$ and finally to $y_k$. But as $d(x_k, x_i) > (\log(k+2) + 1)K$ by and we see that this contradicts $d(x_0, y_0) = d(x_k, y_k)$.
So we get that there must be a unique edge $[y_k, x_j]$ in $T_F$, and the path in $T_F$ from $y_0$ to $y_k$ must go through $x_j$ and $x_i$. As before we must have $$j > k - c\ell^{-1}\left( (\log(k+2) + 2)K +a \right)$$ and we finally get using first , then the fact that $(x_0, \ldots, x_k)$ is a quasi-geodesic, and finally the above together with that: $$\begin{aligned}
d(y_0, y_k) &\ge d(y_0, x_i) + d(x_i, x_j) + d(x_j, y_k) - K - K\log(2 + k)\\
&\ge 2d(x_0, y_0) + c^{-1}(j-i)\ell - a - 3K - K\log(2 + k) \\
&\ge 2d(x_0, y_0) + c^{-1}\ell k - B - b\log(k)
\end{aligned}$$ where $B, b$ depend only on $x, \gamma, \delta$. From the last inequality and the conclusion is immediate.
Smoothing corners {#appendix_smooth}
=================
In this appendix we prove Proposition \[general\_smoothing\]; as the argument is technical but has no subtleties we will be quite sketchy in presenting it.
Recall that we have the following situation: $X$ is a manifold with bounded geometry, $H_1, H_2 \subset X$ such that $X \setminus H_i$ both have bounded geometry, meet transversally and the dihedral angle between them is bounded away from 0 and $\pi$. We remark that constructing a smoothing of $Y = X \setminus (H_1 \cup H_2)$ satisfying the conclusions of Proposition \[general\_smoothing\] is immediate in the case where the intersection $I = H_1 \cap H_2$ has a neighbourhood in $Y$ which is isometric to the product $[0, \delta[^2 \times I$. In general we will prove the following statement: there exists a diffeomorphism $\varphi$ from $[0, \delta[^2 \times I$ to a neighbourhood of $I$ in $Y$ such that $\varphi$ and $\varphi^{-1}$ have all their derivatives uniformly bounded. In view of the preceding remark this proves the proposition.
To define $\varphi$ we need some more auxiliary notation: for a vector field $V$ and $t \ge 0$ we let $\Phi_V^t$ be its flow at time $t$; if $H \subset Z$ is open with smooth boundary we denote by $N_H^Z$ the normal field of $H$ in $Z$. We put: $$\varphi_1(x, t, s) = \Phi_{N_{H_1}^X}^t(\Phi_{N_I^{H_1}}^s(x))$$ and $$\varphi_2(x, t, s) = \Phi_{N_{H_2}^X}^s(\Phi_{N_I^{H_2}}^t(x))$$ We fix a smooth non-decreasing function $h : \RR \to [0, 1[$ such that $h$ is zero on negative numbers, and at infinity it tends to 1 and all its derivatives vanish at all orders. Let $0 < a <1$ such that the convex hull of all $\varphi_1(x, t, s)$ and $\varphi_2(x, t, s)$ for $as \le t \le a^{-1}s$ is contained in $Y$. For $x, y \in X$ and $u \in [0, 1]$ let $ux + (1-u)y$ denote the barycenter of $x, y$ on the geodesic segment between them[^2]. With this notation we define: $$\varphi(x, t, s) = h\left( \frac{at - s}{as - t}\right) \varphi_1(x, t, s) + \left(1 - h\left( \frac{at - s}{as - t}\right) \right) \varphi_2(x, t, s)$$ and we claim that $\varphi$ has the desired properties. It is smooth as a composition of smooth maps. To deduce the remaining properties we will use the following lemma.
\[angle\] For $i=1, 2$ there is $c$ depending only on the bounds on the geometry of $H_i$ such that the following properties hold.
1. \[angle2\] Let $z \in \pl H_i$ and $0 \le t \le \delta$. The linear map $D_z\Phi_{N_{H_i}^X}^t$ is $c$-Lipschitz on angles. The same holds for $x \in I$ and $D_x\Phi_{N_I^{H_i}}^t$.
2. \[angle1\] For all $x \in I$ and all $0 \le s, t < \delta$, let $y = \Phi_{N_{H_i}^X}^t(\Phi_{N_I^{H_i}}^s(x))$. Let $\gamma$ be the geodesic (in $X$) from $x$ to $y$, $u_i$ the parallel transport along $\gamma$ of the outward normal vector to $H_i$ at $x$ and $v_i = \left. \frac{\pl}{\pl\tau} \right|_{\tau=t} \Phi_{N_{H_i}^X}^{\tau}(\Phi_{N_I^{H_i}}^s(x))$. Then the angle between $u_i$ and $v_i$ is at most $c\delta$.
follows from the boundedness of coefficients of the metric tensor and its inverse in normal exponential coordinates (in both $I \subset H_i$ and $\pl H_i \subset X$). follows from , together with the fact that parallel transport along a closed curve stays close to the identity within the $\delta$-neighbourhood.
Let $V_i$ be the vector fields given by the vectors $v_i$ defined in the lemma. As for any $x \in I$ we have that the angle between $V_1(x)$ and $V_2(x)$ lies in $[\alpha_0, \pi - \alpha_0]$ it follows from that if we choose $\delta < c^{-1}\alpha_0/2$ we have that the angle between $V_1$ and $V_2$ at any point $x$ in the $\delta$-neighbourhood of $I$ lies in $[\alpha_0/2, \pi - \alpha_0/2]$. In particular $V_1, V_2$ define a plane field, and we define $J$ to be its orthogonal.
Let $\pi_J$ be orthogonal projection on $J$. The block decomposition of $D\varphi$ according to $TX = J \oplus (V_1+V_2)$ is: $$D_{(x, t, s)}\varphi = \begin{pmatrix} \pi_JD_x\varphi & C \\(1-\pi_J)D_x\varphi & B \end{pmatrix}.$$ We need to prove that:
1. $D_x\varphi, B$ and $C$ have bounded coefficients (in terms of the bounds on the geometry);
2. $\pi_J D_x\varphi$ and $B$ are everywhere invertible and their inverses are bounded.
Indeed, this shows that the map $\varphi$ has a derivative which everywhere invertible. In particular, it is a local diffeomorphism and as it is the identity on $I$ it is also a global diffeomorphism. This also implies that its derivative is uniformly bounded in terms of the geometry of $H_i$ and $\alpha_0$, and so is its inverse.
We deal first with $D_x\varphi$. We note that $$(D_x \varphi)_{(x, t, s)} = h\left( \frac{at - s}{as - t}\right) D_x\varphi_1(x, t, s) + \left(1 - h\left( \frac{at - s}{as - t}\right) \right) D_x\varphi_2(x, t, s) + O(\delta)$$ because of bounded geometry and the fact that to obtain $\varphi$ we move $\varphi_1$ and $\varphi_2$ by at most $\delta$. It follows that $D_x\varphi$ is bounded. By point of the Lemma we have that at all points the angle between the image of $D_x\varphi$ and $V_i$ is at most $c\delta$; it follows that $\|(1-\pi_J)D_x\varphi\| \ll \delta$. Moreover $D_x\varphi$ is everywhere invertible with bounded inverse, because both $A_1 = D_x\varphi_1$ and $A_2 = D_x\varphi_2$ are, and for $w \in T_xI$ the vectors $A_1(w), A_2(w)$ have an angle $\le c\delta$ between them by .
We also have $$D_t\varphi = h\left( \frac{at - s}{as - t}\right) D_t\varphi_1(x, t, s) + \left(1 - h\left( \frac{at - s}{as - t}\right) \right) D_t\varphi_2(x, t, s) + O(\delta)$$ and similarly for $D_s\varphi$, so the coefficients of $B, C$ are bounded.
It remains to prove that $B$ is invertible and $\det(B)$ is bounded away from zero. At a point $x \in I$ we have $D_t\varphi$ and $D_s\varphi$ belong to two disjoint open convex cones in $T_xX/J_x$; by and this remains true in the $\delta$-neighbourhood and the angle between the cones remains bounded away from zero, hence the matrix $B$ is invertible with uniformly bounded inverse.
[^1]: Note that the neighbourhoods corresponding to two geodesics orthogonal to a third one cannot intersect each other, because we took their radius to be $\eps/3$ and the distance between the geodesics outside the $\eps$-thin part is at least $\eps/2$
[^2]: This is well-defined for those pairs of points in $X$ that we consider, as long as we take $\delta \ll \mathrm{inj}(X)$
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The propagation of infectious diseases and its impact on individuals play a major role in disease dynamics, and it is important to incorporate population heterogeneity into efforts to study diseases. As a simplistic but illustrative example, we examine interactions between urban and rural populations in the dynamics of disease spreading. Using a compartmental framework of susceptible–infected–susceptible ($\mathrm{SI\widetilde{S}}$) dynamics with some level of immunity, we formulate a model that allows nonlinear reinfection. We investigate the effects of population movement in the simplest scenario: a case with two patches, which allows us to model movement between urban and rural areas. To study the dynamics of the system, we compute a basic reproduction number for each population (urban and rural). We also compute steady states, determine the local stability of the disease-free steady state, and identify conditions for the existence of endemic steady states. From our analysis and computational experiments, we illustrate that population movement plays an important role in disease dynamics. In some cases, it can be rather beneficial, as it can enlarge the region of stability of a disease-free steady state.'
author:
- 'Juan G. Calvo[^1]'
- '[Alberto Hernández]{}[^2]'
- '[Mason A. Porter]{}[^3]'
- '[Fabio Sanchez]{}[^4]'
date: |
Received: xx-xx-xx; Revised: xx-xx-xx;\
Accepted: xx-xx-xx
title: |
A two-patch epidemic model with nonlinear reinfection\
Un modelo epidémico de dos poblaciones con reinfección no lineal
---
Dynamical systems; population dynamics; mathematical modeling; biological contagions; population movement.
La propagación de enfermedades infecciosas y su impacto en individuos juega un gran rol en la dinámica de la enfermedad, y es importante incorporar heterogeneidad en la población en los esfuerzos por estudiar enfermedades. De manera simplística pero ilustrativa, se examina interacciones entre una población urbana y una rural en la dinámica de la propagación de una enfermedad. Utilizando un sistema compartimental de dinámicas entres susceptibles–infectados–susceptibles ($\mathrm{SI\widetilde{S}}$) con cierto nivel de inmunidad, se formula un modelo que permite reinfecciones no lineales. Se investiga los efectos de movimiento de poblaciones en el escenario más simple: un caso con dos poblaciones, que permite modelar movimiento entre un área urbana y otra rural. Con el fin de estudiar la dinámica del sistema, se calcula el número básico reproductivo para cada comunidad (rural y urbana). Se calculan también puntos de equilibrio, la estabilidad local del estado libre de enfermedad, y se identifican condiciones para la existencia de estados de equilibrio endémicos. Del análisis y experimentos computacionales, se ilustra que el movimiento en la población jueva un rol importante en la dinámica del sistema. En algunos casos, puede ser beneficioso, pues incrementa la región de estabilidad del punto de equilibrio del estado libre de infección.
Sistemas dinamicos, dinámica de poblaciones; modelado matemático; contagios biológicos; movimiento de poblaciones.
92D25, 92D30
Introduction
============
It is relatively easy for individuals to move between towns, cities, countries, and even continents; and incorporating movement between populations has become increasingly prevalent in the modeling and analysis of disease spreading [@ccc; @pastor2015]. It is also important to consider movement in which individuals travel to a distinct location from their place of origin and then return to their original location in a relative short time. Such movement can lead to rapid spreading of infectious diseases, and examination of connected environments can give clues about the types of strategies that are needed for control of disease propagation [@bichara2016; @martens2000].
In 2003, the Severe Acute Respiratory Syndrome (SARS) epidemic was a major concern among public health officials worldwide [@cdc2018a]. This new infectious disease spread rapidly, and scientists and researchers scrambled to try to discern how to contain its spread (e.g., by reducing the spreading rate) and to seek treatments and a vaccine. The best control measure that was found at the time was to isolate individuals who had been in contact with infected individuals. The rapid spread of the disease was associated with movement of a doctor who was identified as “Patient 0” for SARS [@chowell2003; @sanchez2005]. Population movement has also played an important role in subsequent events, such as the spread of ebola to the western hemisphere [@cdc2018b], the spread of measles in some parts of the world by travelers [@cdc2018c], and the resurgence of malaria through the massive migration of Nicaraguans to the northern part of Costa Rica [@malariacr].
The use of compartmental models to describe the spreading of diseases has been explored thoroughly in numerous scenarios [@ccc; @pastor2015]. For example, when there is nonlinear reinfection, an individual who was infected previously can become infected again through contact with an infectious individual after losing immunity [@treno2007]. Several models that allow individuals to lose immunity and become infectious again also exhibit a backward bifurcation in which a stable endemic equilibrium coexists with a stable disease-free equilibrium when the associated basic reproduction number is smaller than $1$ [@feng2000; @sanchez2007; @song2006; @song2013; @sanchez2018]. Moreover, in social contagion processes (e.g., spread of use of drugs, adoption of products, and so on), after an initial “contagion”, some models include a backward bifurcation that can arise via social inputs [@song2006].
In the present paper, we generalize the compartmental model from [@sanchez2007], who studied a continuous dynamical system (in the form of coupled ordinary differential equations) that describes interactions between susceptible and infected individuals with the possibility of reinfection after loss of immunity, by incorporating population movement between urban and rural environments. Taking population movement into account is important for studies of disease dynamics in practice, and it changes the qualitative dynamics of disease spreading. We explore the simplest case, in which a population has two patches, and yield insights that will be useful for later explorations of disease spreading in a population that includes a network of patches.
Our paper proceeds as follows. In , we describe our compartment model of disease spreading (including nonlinear reinfection) between urban and rural environments. In , we give a formula for the model’s basic reproduction number $\mathcal{R}_0$, analyze the existence and local stability of the disease-free state, and study the existence of endemic equilibria. In , we illustrate several example scenarios with numerical computations. Finally, in , we conclude and discuss the biological insights of our model.
A two-patch compartmental model {#sec:model}
===============================
We present a two-patch model of disease spreading in humans that incorporates nonlinear reinfection and population movement. Specifically, we generalize the model by Sanchez et al. [@sanchez2007] by incorporating the idea of urban versus rural environments. We use the subscript $u$ for urban variables and parameters and the subscript $r$ for rural variables and parameters. For $j\in\lbrace u,r\rbrace$, let $S_j$, $I_j$, and $\widetilde{S}_j$ denote the numbers of susceptible, infected, and post-recovery susceptible individuals, respectively. New susceptible individuals enter the system in proportion to the total population $N_j$, where $N_j=S_j+I_j+\widetilde{S}_j$. Let $\mu_j$ denote the rate of both births and deaths (for simplicity, we assume that they are the same.) Susceptible individuals become infected at a rate of $\beta_j$, infected individuals transition to a state of post-recovery susceptibility at a rate of $\gamma_j$, and post-recovery susceptible individuals can be reinfected at a rate of $\rho_j$. Such reinfection corresponds to infectious diseases (such as tuberculosis, malaria, and others [@glynn2008; @grun1983]) in which subsequent infections are possible after loss of immunity. Typically, when there exists the possibility of reinfection, an initial infection tends to produce stronger symptoms [@crutcher1996]. We model movement between patches using the functions $\delta_{ij}(t)$, which denote the fraction of individuals who travel from patch $i \in\lbrace u,r\rbrace$ to patch $j \in\lbrace u,r\rbrace$ (with $i \neq j$) at time $t$.
Our model consists of the following coupled system of ordinary differential equations: $$\label{eq:sys}
\begin{array}{rcl}
\dfrac{d S_u}{dt} &=& \mu_u N_u - \beta_u \dfrac{I_u}{N_u} S_u -\mu_u S_u + {\delta_{ru}}S_r - {\delta_{ur}}S_u\,,\\
\dfrac{d I_u}{dt} &=& \beta_u \dfrac{I_u}{N_u} S_u - (\mu_u+\gamma_u)I_u + \rho_u \dfrac{I_u}{N_u} \widetilde{S}_u +{\delta_{ru}}I_r - {\delta_{ur}}I_u\,,\\
\dfrac{d \widetilde{S}_u}{dt} &=& \gamma_u I_u - \rho_u \dfrac{I_u}{N_u} \widetilde{S}_u - \mu_u \widetilde{S}_u +{\delta_{ru}}\widetilde{S}_r - {\delta_{ur}}\widetilde{S}_u\,,\\
\dfrac{d S_r}{dt} &=& \mu_r N_r - \beta_r \dfrac{I_r}{N_r} S_r -\mu_r S_r + {\delta_{ur}}S_u - {\delta_{ru}}S_r\,,\\
\dfrac{d I_r}{dt} &=& \beta_r \dfrac{I_r}{N_r} S_r - (\mu_r+\gamma_r)I_r + \rho_r \dfrac{I_r}{N_r} \widetilde{S}_r +{\delta_{ur}}I_u - {\delta_{ru}}I_r\,,\\
\dfrac{d \widetilde{S}_r}{dt} &=& \gamma_r I_r - \rho_r \dfrac{I_r}{N_r} \widetilde{S}_r - \mu_r \widetilde{S}_r +{\delta_{ur}}\widetilde{S}_u - {\delta_{ru}}\widetilde{S}_r\,,\\
\end{array}$$ with initial conditions $S_j(0)=S_{j0}$, $I_j(0)=I_{j0}$, and $\widetilde{S}_j(0)=\widetilde{S}_{j0}$ (for $j\in\lbrace u,r\rbrace$).
By adding the first three and last three equations in , we see that $N_u$ and $N_r$ satisfy the linear dynamical system $$\dfrac{d}{dt}\left[
\begin{array}{c}
N_u \\
N_r
\end{array} \right] =
\left[
\begin{array}{rr}
-{\delta_{ur}}& {\delta_{ru}}\\
{\delta_{ur}}& -{\delta_{ru}}\end{array}
\right]
\left[
\begin{array}{c}
N_u \\
N_r
\end{array} \right]\,.$$ Its solution is $$\begin{aligned}
N_u(t) &= e^{-\int_0^t \delta(s) ds}\left[ N_{u,0}\ + (N_{u,0}+N_{r,0})\ \int_0^t {\delta_{ru}}(s) e^{\int_0^s \delta(h) dh}\ ds\right]\,,\\
N_r(t) &= e^{-\int_0^t \delta(s) ds}\left[ N_{r,0}\ + (N_{u,0}+N_{r,0})\ \int_0^t {\delta_{ur}}(s) e^{\int_0^s \delta(h) dh}\ ds\right]\,,\end{aligned}$$ where $\delta(t) = {\delta_{ur}}(t)+{\delta_{ru}}(t)$ is the net movement of individuals at time $t$. The initial conditions are $N_{u,0} = N_u(0)$ and $N_{r,0} = N_r(0)$.
In our analysis, we assume at first that ${\delta_{ru}}$ and ${\delta_{ur}}$ are constant, such that $N_u$ and $N_r$ simplify to $$\begin{aligned}
N_u(t) &= N_{u,0} e^{-t ({\delta_{ur}}+{\delta_{ru}})} + (N_{u,0}+N_{r,0})\dfrac{{\delta_{ru}}}{{\delta_{ru}}+{\delta_{ur}}} \left(1-e^{-t ({\delta_{ur}}+{\delta_{ru}})}\right)\,, \\
N_r(t) &= N_{r,0} e^{-t ({\delta_{ur}}+{\delta_{ru}})} + (N_{u,0}+N_{r,0})\dfrac{{\delta_{ur}}}{{\delta_{ru}}+{\delta_{ur}}} \left(1-e^{-t ({\delta_{ur}}+{\delta_{ru}})}\right)\,. \end{aligned}$$ There are three cases:
1. There is no movement (i.e., ${\delta_{ur}}= {\delta_{ru}}= 0$), which corresponds to the case of independent patches that was studied in [@sanchez2007]. In this case, each patch has a backward bifurcation, and the steady state can depend on the number of initially infected individuals.
2. Movement occurs exclusively from one patch to the other. For example, suppose that ${\delta_{ur}}= 0$ and ${\delta_{ru}}>0$. In this case, $\displaystyle\lim_{t\to\infty} N_r(t) = 0$ (i.e., eventually, the entire population is in $u$). There is a backward bifurcation in this case as well, but now there is a total population of $N_{u,0}+N_{r,0}$ in the urban patch. We explore this case in Example \[ex:1\] in Section \[sec:sims\].
3. There is movement in both directions between the two patches (i.e., ${\delta_{ur}}>0$ and ${\delta_{ru}}>0$). This is the primary scenario (and the principal novel contribution) of the present paper.
Analysis of our model {#sec:analysis}
=====================
Disease-free steady state and basic reproduction number {#sec:dfe}
-------------------------------------------------------
Because the total population is constant, we can eliminate $\widetilde{S}_r$ from the dynamical system . We then compute the Jacobian matrix of and evaluate it at the disease-free steady state $$\label{free}
(S_u^*,I_u^*,\widetilde{S}_u^*,S_r^*,I_r^*,\widetilde{S}_r^*)=(S_u^*,0,0,S_r^*,0,0)$$ to obtain the matrix $$\label{blah}
\left[
\begin{array}{rrrrr}
-{\delta_{ur}}& {\mu_u}-{\beta_u}&{\mu_u}& {\delta_{ru}}& 0\\
0& \eta_{ur} & 0 & 0 & {\delta_{ru}}\\
-{\delta_{ru}}& {\gamma_u}-{\delta_{ru}}& -{\delta_{ru}}-{\delta_{ur}}-{\mu_u}&-{\delta_{ru}}& -{\delta_{ru}}\\
{\delta_{ur}}-{\mu_r}& -{\mu_r}& -{\mu_r}&-{\mu_r}-{\delta_{ru}}&-{\beta_r}\\
0& {\delta_{ur}}&0 &0 & \eta_{ru} \\
\end{array}
\right]\,,$$ where $\eta_{ur} = {\beta_u}- {\delta_{ur}}- {\gamma_u}- {\mu_u}$ and $\eta_{ru} = {\beta_r}- {\delta_{ru}}- {\gamma_r}- {\mu_r}$. For the dynamical system , equilibria with $I_u=I_r=0$ necessarily also satisfy $\widetilde{S}_u=\widetilde{S}_r=0$.
The five eigenvalues of the matrix are $$\begin{aligned}
\lambda_1 &= -({\delta_{ru}}+{\delta_{ur}})\,,\\
\lambda_{2\pm} &= \frac{1}{2} \left[-({\delta_{ru}}+ {\delta_{ur}}+ {\mu_r}+ {\mu_u}) \pm \sqrt{({\delta_{ru}}- {\delta_{ur}}+ {\mu_r}- {\mu_u})^2 + 4 {\delta_{ru}}{\delta_{ur}}} \right]\,,\\
\lambda_{3\pm} &= \frac{1}{2}\left[ \eta_{ur}+\eta_{ru} \pm \sqrt{\left( \eta_{ur}-\eta_{ru} \right)^2 +4{\delta_{ru}}{\delta_{ur}}} \right]\,.\\\end{aligned}$$ All eigenvalues are real, and $\lambda_1$ and $\lambda_{2\pm}$ are negative. The two remaining eigenvalues $\lambda_{3\pm}$ are negative as long as $$\begin{aligned}
\label{eq:condEigNeg}
{\delta_{ru}}{\delta_{ur}}&< \eta_{ur}\eta_{ru}\,, \quad \eta_{ur}+ \eta_{ru} < 0 \,.\end{aligned}$$
When there is only one population (i.e., one patch), the basic reproduction number is $\mathcal{R}_0=\frac{\beta}{\mu+\gamma}$. For our multiple-patch case, we define a “local basic reproduction number” for each patch: $$\mathcal{R}_{0u} = \dfrac{{\beta_u}}{{\gamma_u}+{\mu_u}}\,, \quad \mathcal{R}_{0r} = \dfrac{{\beta_r}}{{\gamma_r}+{\mu_r}}\,.$$ We can then express the conditions in as
\[eq:condR0\] $$\begin{aligned}
\mathcal{R}_{0u} &< 1 + \dfrac{{\delta_{ur}}}{{\mu_u}+{\gamma_u}}\,, \label{eq:condR0a}\\
\mathcal{R}_{0r} &< 1 + \dfrac{{\delta_{ru}}}{{\mu_r}+{\gamma_r}},\label{eq:condR0b}\\ \dfrac{{\delta_{ur}}}{{\mu_u}+{\gamma_u}}\dfrac{{\delta_{ru}}}{{\mu_r}+{\gamma_r}} &< \left(\mathcal{R}_{0u}-1-\dfrac{{\delta_{ur}}}{{\mu_u}+{\gamma_u}}\right)\left(\mathcal{R}_{0r}-1-\dfrac{{\delta_{ru}}}{{\mu_r}+{\gamma_r}}\right)\,. \label{eq:condR0c}\end{aligned}$$
See Figure \[fig:stab\_reg\] for an illustration of a typical region in which all eigenvalues are negative. For progressively smaller ${\delta_{ur}}$ and ${\delta_{ru}}$, the shaded region approaches the unit square.
![A typical region of local asymptotic stability of the disease-free steady state of the dynamical system from the conditions as a function of the local basic reproduction numbers $\mathcal{R}_{0u}$ and $\mathcal{R}_{0r}$. \[fig:stab\_reg\]](stab_reg-eps-converted-to.pdf){width="0.7\linewidth"}
We have thus established the following lemma.
\[lem:diseaseFree\] Assume that Eqs. hold. It then follows that the disease-free steady state of the dynamical system is locally asymptotically stable.
[The conditions in are satisfied when $\mathcal{R}_{0u}<1$ and $\mathcal{R}_{0r}<1$. We then have local stability in the rural and urban patches if we treat them as independent. Additionally, from Lemma \[lem:diseaseFree\], we see that it is possible to obtain local asymptotic stability for the disease-free steady state even when one or both local basic reproduction numbers are larger than $1$. In such a scenario, movement is beneficial, as it leads to local asymptotic stability of the disease-free steady state in situations that would not be the case for independent patches. We illustrate such a scenario in Example \[ex:2\] in Section \[sec:sims\]. ]{}
Technical tool: The Poincaré–Miranda theorem {#sec:ee}
--------------------------------------------
As preparation for analyzing the existence of steady states in our model, we briefly recall some classical results by Poincaré and others. See [@frankowska2018; @kulpa1997; @szyman2015; @turzanski2012] for detailed accounts of the relevant theory. In 1817, Bolzano proved the following well-known theorem:
Let $f:[a,b]\longrightarrow \mathbb{R}$ be a continuous function such that $f(a) \cdot f(b)< 0$. There then exists $c \in (a,b)$ such that $f(c) = 0$.
Let $I^n = [0,1]^n$, and let $\partial I^n$ denote its boundary. For each $i \in \{1, \dots, n\}$, let $$\begin{aligned}
I_i^- = \{x \in I^n | x_i = 0\} \,, \qquad
I_i^+ = \{x\in I^n | x_i = 1 \}\,,\end{aligned}$$ be the opposite $i$-th faces of the boundary $\partial I^n$.
In 1883–1884, Poincaré announced a generalization of Bolzano’s theorem without providing a proof.
Let $F: I^n \longrightarrow \mathbb{R}^n$, with $F = (F_1,\cdots, F_n)$, be a continuous map such that $$F_i(I_{i}^-) \subseteq (-\infty, 0]$$ and $$F_i(I_{i}^+) \subseteq [0, +\infty)$$ for every $i \in \{1, \dots, n\}$. It then follows that there exists $c\in I^n$ such that $F(c) = 0$.
In the 1940s, Miranda rediscovered Poincaré’s theorem and showed that it is logically equivalent to Brower’s fixed-point theorem. Since then, this result has often been called the Poincaré–Miranda theorem. We require a modified version of the Poincaré–Miranda theorem in two dimensions.
\[prop:miranda\] Let $F: [0,1]^2 \longrightarrow \mathbb{R}^2$ such that $F(x,y)= (F_1(x,y),F_2(x,y))^T$ is continuous and $F(0,0) = 0$. Assume that $$\begin{array}{cc}
F_1(x,0) > 0 \quad \mathrm{for\,\, all}\,\,\, x\in (0,1]\,, & F_1(x,1) <0\ \quad \mathrm{for\,\, all}\,\,\, x\in [0,1]\,,\\
F_2(0,y) > 0 \quad \mathrm{for\,\, all}\,\,\, y\in (0,1]\,, & F_2(1,y) <0\ \quad \mathrm{for\,\, all}\,\,\, y\in [0,1]\,.
\end{array}$$ Assume additionally that $ \partial{F_1} / \partial{y}$ and $\partial{F_2} / \partial{x}$ are both continuous from the right at $(0,0)$, with $\partial{F_1} / \partial{y}(0,0)>0$ and $\partial{F_2} / \partial{x}(0,0)>0$. It then follows that there exists $(x_0,y_0) \in (0,1)^2$ such that $F(x_0,y_0) = (0,0)^T$.
Because $F$ is continuous, both $F_1$ and $F_2$ are also continuous. Using the facts that $F_1(x,0) >0$ for all $x \in (0,1]$ and that $\partial {F_1} / \partial {x}$ is continuous from the right, it follows that there exists $\epsilon_0 > 0$ such that $\partial {F_1} /\partial x (x,0) > 0$ for all $x \in (0, \epsilon_0)$. By the implicit function theorem, for $x \in (0,\epsilon_0)$, one can write $y = g_1(x)$, where the function $g_1$ is differentiable. Therefore, $$\label{inverse thm}
g_1'(0) = -\frac{ \frac{\partial F_1}{\partial x}(0,0) }{\frac{\partial F_1}{\partial y}(0,0)} \,.$$ By the continuity of $F_1$ in the compact set $[0,1]^2$ and using Equation , it follows that there exists $\epsilon_1 > 0$ such that $F_1(x,\epsilon_1) > 0$ for all $x \in [0,1]$. By the same argument, there also exists $\epsilon_2 \in (0,1)$ such that $F_2(\epsilon_2,y)> 0$ for all $y \in [0,1]$. By the Poincaré–Miranda theorem, there must exist $(x_0,y_0) \in [\epsilon_1, 1]\times [\epsilon_2,1]$ such that $F(x_0,y_0) = (0,0)$.
Existence of multiple-population endemic steady states
------------------------------------------------------
We now examine endemic steady states, in which there are infected individuals at steady state in both the urban and rural patches.
We start by proving the following lemma.
Suppose that $\mathcal{R}_{0u} > 1+\dfrac{{\delta_{ur}}}{{\mu_u}+{\gamma_u}}$ and $\mathcal{R}_{0r} > 1+\dfrac{{\delta_{ru}}}{{\mu_r}+{\gamma_r}}$. It then follows that exists at least one endemic state, for which both $I_u^*>0$ and $I_r^*>0$. This is, there are infected individuals at steady state in both the urban and the rural patches.
We deduce the existence of a solution of the following nonlinear system of algebraic equations: $$\label{eq:systemSS}
\begin{array}{rcl}
0 &=& \mu_u N_u - \beta_u \dfrac{I_u}{N_u} S_u -\mu_u S_u + {\delta_{ru}}S_r - {\delta_{ur}}S_u\,,\\
0 &=& \beta_u \dfrac{I_u}{N_u} S_u - (\mu_u+\gamma_u)I_u + \rho_u \dfrac{I_u}{N_u} \widetilde{S}_u +{\delta_{ru}}I_r - {\delta_{ur}}I_u\,,\\
0 &=& \gamma_u I_u - \rho_u \dfrac{I_u}{N_u} \widetilde{S}_u - \mu_u \widetilde{S}_u +{\delta_{ru}}\widetilde{S}_r - {\delta_{ur}}\widetilde{S}_u\,,\\
0 &=& \mu_r N_r - \beta_r \dfrac{I_r}{N_r} S_r -\mu_r S_r + {\delta_{ur}}S_u - {\delta_{ru}}S_r\,,\\
0 &=& \beta_r \dfrac{I_r}{N_r} S_r - (\mu_r+\gamma_r)I_r + \rho_r \dfrac{I_r}{N_r} \widetilde{S}_r +{\delta_{ur}}I_u - {\delta_{ru}}I_r\,,\\
0 &=& \gamma_r I_r - \rho_r \dfrac{I_r}{N_r} \widetilde{S}_r - \mu_r \widetilde{S}_r +{\delta_{ur}}\widetilde{S}_u - {\delta_{ru}}\widetilde{S}_r\,.\\
\end{array}$$
Given two parameters $(B_u,B_r) \in [0,1]^2$, we consider the auxiliary linear system $$\label{eq:systemSS2}
\begin{array}{rcl}
0 &=& \mu_u N_u - \beta_u B_u S_u -\mu_u S_u + {\delta_{ru}}S_r - {\delta_{ur}}S_u\,,\\
0 &=& \beta_u B_u S_u - (\mu_u+\gamma_u)I_u + \rho_u B_u \widetilde{S}_u +{\delta_{ru}}I_r - {\delta_{ur}}I_u\,,\\
0 &=& \gamma_u I_u - \rho_u B_u \widetilde{S}_u - \mu_u \widetilde{S}_u +{\delta_{ru}}\widetilde{S}_r - {\delta_{ur}}\widetilde{S}_u\,,\\
0 &=& \mu_r N_r - \beta_r B_r S_r -\mu_r S_r + {\delta_{ur}}S_u - {\delta_{ru}}S_r\,,\\
0 &=& \beta_r B_r S_r - (\mu_r+\gamma_r)I_r + \rho_r B_r \widetilde{S}_r +{\delta_{ur}}I_u - {\delta_{ru}}I_r\,,\\
0 &=& \gamma_r I_r - \rho_r B_r \widetilde{S}_r - \mu_r \widetilde{S}_r +{\delta_{ur}}\widetilde{S}_u - {\delta_{ru}}\widetilde{S}_r\,.\\
\end{array}$$
Suppose that $(S_u,I_u,\widetilde{S}_u,S_r,I_r,\widetilde{S}_r)$ is a solution of the linear system . This solution also satisfies the nonlinear system if $I_u/N_u = B_u$ and $I_r/N_r=B_r$. By adding all of the equations in , we see that ${\delta_{ur}}N_u = {\delta_{ru}}N_r$. We then rescale variables in by substituting $$\begin{aligned}
s_u &=\dfrac{S_u}{N_u}\,,\quad s_r =\dfrac{S_r}{N_r}\,,\\
i_u &=\dfrac{I_u}{N_u}\,, \quad i_r =\dfrac{I_r}{N_r}\,,\\
\widetilde{s}_u &=\dfrac{\widetilde{S}_u}{N_u}\,,\quad \widetilde{s}_r =\dfrac{\widetilde{S}_r}{N_r}\end{aligned}$$ to obtain a linear system of algebraic equations with parameters $B_u$ and $B_r$. This system is
\[eq:linearSystem\] $$\begin{aligned}
\mu_u -\left(\beta_u B_u +\mu_u+{\delta_{ur}}\right ) s_u + {\delta_{ur}}s_r &= 0\,, \label{eq:su}\\
\beta_u B_u s_u - (\mu_u+\gamma_u + {\delta_{ur}}) i_u + \rho_u B_u \widetilde{s}_u + {\delta_{ur}}i_r &= 0\,,\label{eq:iu}\\
\gamma_u i_u - \rho_u B_u \widetilde{s}_u - \left(\mu_u + {\delta_{ur}}\right)\widetilde{s}_u + {\delta_{ur}}\widetilde{s}_r &= 0\,,\label{eq:ru}\\
\mu_r - \left(\beta_r B_r +\mu_r+{\delta_{ru}}\right ) s_r + {\delta_{ru}}s_u &= 0\,,\label{eq:sr}\\
\beta_r B_r s_r - (\mu_r+\gamma_r + {\delta_{ru}}) i_r +\rho_r B_r \widetilde{s}_r + {\delta_{ru}}i_u &= 0\,,\label{eq:ir}\\
\gamma_r i_r - \rho_r B_r \widetilde{s}_r - \left( \mu_r + {\delta_{ru}}\right)\widetilde{s}_r + {\delta_{ru}}\widetilde{s}_u &= 0\label{eq:rr}\,.\end{aligned}$$
We now solve Eqs. and to obtain $$\begin{aligned}
s_u(B_u,B_r) &= \dfrac{\beta_r \mu_u B_r+{\delta_{ur}}\mu_r+{\delta_{ru}}\mu_u+\mu_r\mu_u}{\left(\beta_u B_u +\mu_u+{\delta_{ur}}\right )\left(\beta_r B_r +\mu_r+{\delta_{ru}}\right )-{\delta_{ru}}{\delta_{ur}}}\,,\\
s_r(B_u,B_r) &= \dfrac{\beta_u \mu_r B_u+{\delta_{ur}}\mu_r+{\delta_{ru}}\mu_u+\mu_r\mu_u}{\left(\beta_u B_u +\mu_u+{\delta_{ur}}\right )\left(\beta_r B_r +\mu_r+{\delta_{ru}}\right )-{\delta_{ru}}{\delta_{ur}}}\,.\end{aligned}$$ Using Eqs. , , , , we solve for $i_u$ and $i_r$ to obtain $$\begin{aligned}
i_u(B_u,B_r) &= \dfrac{1}{\Delta}\left( s_r {\beta_r}{\delta_{ur}}B_r \Delta_{i_u}^{(1)} + s_u {\beta_u}B_u \Delta_{i_u}^{(2)} \right)\,,\\
i_r(B_u,B_r) &= \dfrac{1}{\Delta}\left( s_u {\beta_u}{\delta_{ru}}B_u \Delta_{i_r}^{(1)} + s_r {\beta_r}B_r \Delta_{i_r}^{(2)} \right)\,,\end{aligned}$$ where $$\begin{aligned}
\Delta_{i_u}^{(1)} &= {\delta_{ur}}{\mu_r}+ {\delta_{ru}}{\mu_u}+ {\mu_u}{\mu_r}+ {\rho_r}({\mu_u}+ {\delta_{ur}}) B_r + {\rho_u}({\gamma_r}+ {\mu_r}+ {\delta_{ru}}) B_u + {\rho_u}{\rho_r}B_u B_r\,,\\
\Delta_{i_u}^{(2)} &= ({\delta_{ru}}+ {\gamma_r}+ {\mu_r}) ({\delta_{ur}}{\mu_r}+ {\delta_{ru}}{\mu_u}+ {\mu_r}\mu) + ({\delta_{ru}}+ {\mu_r}) ({\delta_{ur}}+ {\mu_u}) {\rho_r}B_r + \\
& \hspace{1cm} + ({\delta_{ru}}+ {\mu_r}) ({\delta_{ru}}+ {\gamma_r}+ {\mu_r}) {\rho_u}B_u + ({\delta_{ru}}+ {\mu_r}) {\rho_u}{\rho_r}B_u B_r\,,\\
\Delta_{i_r}^{(1)} &= {\delta_{ur}}{\mu_r}+ {\delta_{ru}}{\mu_u}+ {\mu_u}{\mu_r}+ {\rho_u}({\mu_r}+ {\delta_{ru}}) B_u + {\rho_r}({\gamma_u}+ {\mu_u}+ {\delta_{ur}}) B_r + {\rho_u}{\rho_r}B_u B_r\,,\\
\Delta_{i_r}^{(2)} &= ({\delta_{ur}}+ {\gamma_u}+ \mu) ({\delta_{ur}}{\mu_r}+ {\delta_{ru}}{\mu_u}+ {\mu_r}\mu) + ({\delta_{ur}}+ {\mu_u}) ({\delta_{ru}}+ {\mu_r}) {\rho_u}B_u + \\
& \hspace{1cm} + ({\delta_{ur}}+ {\mu_u}) ({\delta_{ur}}+ {\gamma_u}+ {\mu_u}) {\rho_r}B_r + ({\delta_{ur}}+ {\mu_u}) {\rho_u}{\rho_r}B_u B_r\,,\\
\Delta &= ({\delta_{ur}}{\mu_r}+ {\delta_{ru}}{\mu_u}+{\mu_r}{\mu_u}) ( ({\delta_{ru}}+ {\gamma_r}+ {\mu_r}+ B_r {\rho_r}) ({\delta_{ur}}+ {\gamma_u}+ {\mu_u}+ B_u {\rho_u}) -{\delta_{ur}}{\delta_{ru}})\,.\end{aligned}$$ One can write similar expressions for $\widetilde{s}_u(B_u,B_r)$ and $\widetilde{s}_r(B_u,B_r)$. The solution $(s_u,i_u,\widetilde{s}_u,s_r,i_r,\widetilde{s}_r)$ of satisfies the nonlinear algebraic system if and only if $i_u(B_u,B_r)=B_u$ and $i_r(B_u,B_r)=B_r$. We then define $$\begin{aligned}
G_1(B_u,B_r) &= i_u(B_u,B_r)-B_u\,,\\
G_2(B_u,B_r) &= i_r(B_u,B_r)-B_r\,.\end{aligned}$$ We will show that the system $$\label{eq_systemG1G2}
G_1(B_u,B_r) = G_2(B_u,B_r)=0$$ has at least one solution $(B_u,B_r)\in (0,1)^2$. This implies the existence of a solution of the nonlinear algebraic system with $i_u>0$ and $i_r>0$. This solution corresponds to a steady state with a nonzero infected population in both the urban and the rural patches.
Consider the surface $z_1 = G_1(B_u,B_r)$. We have that $G_1(0,B_r) = i_u(0,B_r) \geq 0$ for $B_r\in [0,1]$, with equality when $B_r=0$. Because $G_1(0,0)=0$, we need to cut the origin to guarantee the existence of a positive solution of Eq. . A straightforward calculation yields $$\begin{aligned}
\dfrac{\partial G_1}{\partial B_u}(0,0) &= \dfrac{-{\delta_{ur}}({\gamma_r}+ {\mu_r}) + ({\beta_u}-{\gamma_u}- {\mu_u}) ({\delta_{ru}}+ {\gamma_r}+ {\mu_r}) }{{\delta_{ur}}({\gamma_r}+ {\mu_r}) + ({\delta_{ru}}+ {\gamma_r}+ {\mu_r}) ({\gamma_u}+ {\mu_u})}\\
&> \dfrac{{\delta_{ur}}{\delta_{ru}}}{{\delta_{ur}}({\gamma_r}+ {\mu_r}) + ({\delta_{ru}}+ {\gamma_r}+ {\mu_r}) ({\gamma_u}+ {\mu_u})} >0\,,\end{aligned}$$ because ${\beta_u}>{\gamma_u}+{\mu_u}+{\delta_{ur}}$ by hypothesis. Analogously, we compute that $$\dfrac{\partial G_2}{\partial B_r}(0,0)>0\,.$$
We also compute that $G_1(1,B_r) = i_u(1,B_r) - 1<0$ for $B_r\in [0,1]$ and that $G_2(B_u,1)<0$ for $B_u\in [0,1]$. By Proposition \[prop:miranda\], we conclude that there exists $(B_u,B_r) \in (\epsilon_1,1)\times (\epsilon_2,1)$ such that $G_1(B_u,B_r) = G_2(B_u,B_r)=0$. Therefore, the nonlinear algebraic system has a solution that corresponds to a steady state with positive values for both $I_u^*$ and $I_r^*$.
[\[rem:severalPoints\] Using numerical computations, we can observe the existence of two or more distinct endemic steady states. However, we need to explore them further to characterize them; see Figure \[fig:remarkEnd\] and Examples \[ex:3\] and \[eq:ninePts\] in Section \[sec:sims\]. ]{}
Examples {#sec:sims}
========
We now present some numerical simulations of the dynamical system for a variety of parameter values. Using these examples, we illustrate that population movement strongly influences how a disease can propagate.
\[ex:1\] We first explore the behavior of our model when ${\delta_{ur}}=0$ and ${\delta_{ru}}\neq 0$, which describes movement in one direction (specifically, from the rural patch to the urban one). In some countries, it is common for individuals in rural areas to move to urban areas for work [@coffee2015]. This is a type of short-term mobility. We consider the following parameter values: $$\begin{array}{cccc}
{\mu_u}= 1/(365\cdot 80)\,, & {\rho_u}= 0.08\,, & {\gamma_u}= 0.01\,, & {\beta_u}= 0.03\,,\\
{\mu_r}= 1/(365\cdot 70)\,, & {\rho_r}= 0.04\,, & {\gamma_r}= 0.01\,, & {\beta_r}= 0.02\,,
\end{array}$$ with initial conditions $$\begin{array}{ccc}
S_{u0} = 999\,, & I_{u0} = 1\,, & \widetilde{S}_{u0} = 0\,,\\
S_{r0} = 300\,, & I_{r0} = 0\,, & \widetilde{S}_{r0} = 0\,.
\end{array}$$ In this case, $$\begin{aligned}
\mathcal{R}_{0u} > 1 \quad \text{and} \quad \mathcal{R}_{0r} \approx 2 \,.\end{aligned}$$ In Figure \[fig:ex1\_dur\_change\], we show $I_u(t)$ as we vary ${\delta_{ru}}$. We show the solutions for both the urban and the rural patches when ${\delta_{ru}}= 0$ (i.e., when there is no movement) and ${\delta_{ru}}= 0.01$ in Figure \[fig:ex1\]. We observe that movement in one direction increases the number of infected individuals and that larger values of ${\delta_{ru}}$ affects only the approach speed to the steady state.
![The number $I_u$ of infected individuals in the urban patch as a function of time (in days) for ${\delta_{ur}}= 0$ and different values of ${\delta_{ru}}$; see Example \[ex:1\]. \[fig:ex1\_dur\_change\]](example1_iu_all_fn_deltaru-eps-converted-to.pdf){width=".8\linewidth"}
\[ex:2\]
We now compare the effects of conditions and to the standard condition $\mathcal{R}_0<1$ for local asymptotic stability of the disease-free steady state . We use the parameter values $$\begin{array}{cccc}
{\mu_u}= 1/(365\cdot 80)\,, & {\rho_u}= 0.08\,, & {\gamma_u}= 0.01\,, & {\beta_u}= 3\cdot 10^{-2}\,,\\
{\mu_r}= 1/(365\cdot 70)\,, & {\rho_r}= 0.40\,, & {\gamma_r}= 0.10\,, & {\beta_r}= 2\cdot 10^{-5}
\end{array}$$ and initial conditions $$\begin{array}{ccc}
S_{u0} = 999, & I_{u0} = 1\,, & \widetilde{S}_{u0} = 0\,,\\
S_{r0} = 300, & I_{r0} = 0\,, & \widetilde{S}_{r0} = 0\,.
\end{array}$$ We calculate that $\mathcal{R}_{0u} \approx 2.9$ and $\mathcal{R}_{0r} \approx 2\cdot 10^{-4}$. Therefore, in the absence of movement, disease persists in the urban patch but dies out in the rural patch; see Figures \[fig:ex2\_u1\] and \[fig:ex2\_r1\]. For ${\delta_{ur}}={\delta_{ru}}=0.05$, we calculate that ${\delta_{ur}}/({\gamma_u}+\mu) \approx 5.98$, so conditions are satisfied. Therefore, the disease-free steady state is locally asymptotically stable; see Figures \[fig:ex2\_u2\] and \[fig:ex2\_r2\].
In this example, there exists one endemic state, with $(I_u^*/N_u^*, I_r^*/N_r^*) \approx (0.82, 0.76)$. We study two variations:
1. When the initial number of infected individuals increases, we reach the endemic steady state. See Figure \[fig:ex2c1\] for an illustration with initial conditions $S_{u0} = 900$ and $I_{u0} = 100$.
2. When we slightly increase ${\beta_u}$ to ${\beta_u}= 5.3\cdot 10^{-2}$, condition is no longer satisfied, and the system has an endemic state with just one initial infected individual in the urban patch; see Figure \[fig:ex2c2\].
\[ex:3\]
We now give an example with two endemic states. We use the parameter values $$\begin{array}{ccccc}
{\mu_u}= 1/(365\cdot 70)\,, & {\rho_u}= 0.80\,, & {\gamma_u}= 0.15\,, & {\beta_u}= 3\cdot 10^{-4}\,, & {\delta_{ur}}= 0.04\,,\\
{\mu_r}= 1/(365\cdot 70)\,, & {\rho_r}= 0.40\,, & {\gamma_r}= 0.10\,, & {\beta_r}= 2\cdot 10^{-5}\,, & {\delta_{ru}}= 0.05\,.
\end{array}$$ We fix the initial conditions $$\begin{array}{ccc}
N_{u0} = 1000\,, & \widetilde{S}_{u0} = 0\,,\\
N_{r0} = 300\,, & I_{r0} = 0 \,, & \widetilde{S}_{r0} = 0
\end{array}$$ and vary the initial number of infected individuals in the urban patch from $0$ to $N_u$. In this case, there are two endemic states, with $(I_u^*/N_u^*, I_r^*/N_r^*) \approx (0.09, 0.07)$ and $(I_u^*/N_u^*, I_r^*/N_r^*) \approx (0.49, 0.45)$, which we illustrate in Figure \[fig:remarkEnd\]. We show $I_u^*$ and $I_r^*$ as a function of $I_{0u}$ in Figure \[fig:ex3a\]. We observe that the steady state $(I_u^*/N_u^*, I_r^*/N_r^*) \approx (0.09, 0.07)$ is unstable.
![The number of infected individuals in the urban and rural patches in an endemic steady state. We observe convergence of $I_u^*$ to a persistent number of infections as a function of $I_{0u}$; see Example \[ex:3\]. \[fig:ex3a\]](example2b_istar_asfn_su0-eps-converted-to.pdf){width=".7\linewidth"}
\[ex:6\]
In this example, we explore the dependence on $\mathcal{R}_{0u}$ and $\mathcal{R}_{0r}$ in our model to illustrate the region of local asymptotic stability from Eqs. . We consider the parameter values $$\begin{array}{ccccc}
{\mu_u}= 1/(365\cdot 70)\,, & {\rho_u}= 0.80\,, & {\gamma_u}= 0.01\,, & {\delta_{ur}}= 0.005\,,\\
{\mu_r}= 1/(365\cdot 70)\,, & {\rho_r}= 0.40\,, & {\gamma_r}= 0.05\,, & {\delta_{ru}}= 0.010\,.
\end{array}$$ We fix the initial conditions $$\begin{array}{ccc}
N_{u0} = 999\,, & I_{u0} = 1 \,,&\widetilde{S}_{u0} = 0\,,\\
N_{r0} = 300\,, & I_{r0} = 0 \,, &\widetilde{S}_{r0} = 0
\end{array}$$ and vary ${\beta_u}$ and ${\beta_r}$; see our results in Figure \[fig:ex6\]. We observe that the conditions in Eqs. precisely describe the disease-free region (in gray). Therefore, $\mathcal{R}_{0u}$ or $\mathcal{R}_{0r}$ can have a value that is slightly larger than $1$ in situations with a disease-free steady state. This is not the case when we consider just one patch, so movement can be beneficial.
\[ex:6b\]
In this example, we study how a disease spreads through the two populations as we vary ${\delta_{ur}}$ and ${\delta_{ru}}$ when initially one infected person arrives at one patch. We consider the parameter values $$\begin{array}{ccccc}
{\mu_u}= 1/(365\cdot 70)\,, & {\rho_u}= 0.80\,, & {\gamma_u}= 0.15\,, & {\beta_u}= 3\cdot{10}^{-1}\,,\\
{\mu_r}= 1/(365\cdot 70)\,, & {\rho_r}= 0.40\,, & {\gamma_r}= 0.10\,, & {\beta_r}= 2\cdot{10}^{-3}\,.
\end{array}$$ We fix the initial conditions $$\begin{array}{ccc}
N_{u0} = 99999\,, & I_{u0} = 1\,, &\widetilde{S}_{u0} = 0\,,\\
N_{r0} = 30000\,, & I_{r0} = 0\,, &\widetilde{S}_{r0} = 0
\end{array}$$ and vary ${\delta_{ur}}$ and ${\delta_{ru}}$. We show our results in Figure \[fig:ex6B\].
\[ex:varDeltas\]
To model different rates of movement between patches on weekdays and weekends, we now take ${\delta_{ur}}$ and ${\delta_{ru}}$ to be piecewise constant and periodic. Specifically, we take ${\delta_{ur}}(t)$ and ${\delta_{ru}}(t)$ as in Figure \[fig:deltas\]. In Figure \[fig:ex3\_delta\_var\], we show the results of our numerical computations using the parameter values $$\begin{array}{ccccc}
{\mu_u}= 1/(365\cdot 70)\,, & {\rho_u}= 0.80\,, & {\gamma_u}= 0.15\,, & {\beta_u}= 3\cdot{10}^{-4}\,,\\
{\mu_r}= 1/(365\cdot 70)\,, & {\rho_r}= 0.40\,, & {\gamma_r}= 0.10\,, & {\beta_r}= 2\cdot{10}^{-5}
\end{array}$$ with initial conditions $$\begin{array}{ccc}
N_{u0} = 999\,, & I_{u0} = 1\,, &\widetilde{S}_{u0} = 0\,,\\
N_{r0} = 300\,, & I_{r0} = 0\,, &\widetilde{S}_{r0} = 0\,.
\end{array}$$
\[eq:ninePts\]
We now revisit Remark \[rem:severalPoints\], where we stated that we can observe numerically the existence of several distinct endemic steady states. Equation is equivalent to a polynomial equation (as a function of $B_u$ or $B_r)$ of degree $9$, for which there can exist $9$ solutions (for $i_u^*$ or $i_r^*$). For the parameter values $$\begin{array}{ccccc}
{\mu_u}= 1/(365\cdot 70)\,, & {\rho_u}= 1\,, & {\gamma_u}= 0.15\,, & {\beta_u}= 3\cdot{10}^{-4}\,, & {\delta_{ur}}= 4\cdot 10^{-6}\,,\\
{\mu_r}= 1/(365\cdot 70)\,, & {\rho_r}= 2\,, & {\gamma_r}= 1.00\,, & {\beta_r}= 2\cdot{10}^{-3}\,, & {\delta_{ru}}= 5\cdot 10^{-5}\,,
\end{array}$$ we see that all $9$ solutions are in the interval $[0,1]$. They correspond to feasible steady states $(i_u^*,i_r^*)$; see Figure \[fig:ex\_9pts\]. We see numerically that $4$ of these points are locally stable.
Conclusions and Discussion {#sec:disc}
==========================
The dynamics of spreading diseases are influenced significantly by spatial heterogeneity and population movement. In this paper, we illustrated the importance of incorporating movement into models of disease dynamics using a simple but biologically meaningful model. Specifically, we constructed a two-patch compartmental model that incorporates movement between urban and rural populations, as well as the possibility of reinfection after recovery.
When there is a lot of population movement, there are regions of local asymptotic stability of the disease-free steady-state even when the basic reproduction number $\mathcal{R}>1$. This arises predominantly from the numerous individuals who move between patches.
The exploration of interacting populations plays an important role in the understanding of disease dynamics. Many models of disease spreading focus on a single population [@ccc], but populations do not exist in isolation. Using our two-patch model with urban and rural environments, we illustrated several examples of plausible real-world scenarios in which movement yields insightful information about disease spreading and epidemics. We expect that such dynamics will be relevant for studies of disease spreading on networks, such as when many people commute daily between their homes in rural areas and work in urban centers (as is the case in many countries in South and Central America).
Acknowledgements
================
We thank the Research Center in Pure and Applied Mathematics and the Mathematics Department at Universidad de Costa Rica for their support during the preparation of this manuscript. The authors gratefully acknowledge institutional support for project B8747 from an UCREA grant from the Vice Rectory for Research at Universidad de Costa Rica. We also acknowledge helpful discussions with Prof. Luis Barboza, Prof. Carlos Castillo-Chavez, and Prof. Esteban Segura.
[99]{} “Costa Rica Once Again Under Malaria Alert”, in <https://news.co.cr/costa-rica-once-again-under-malaria-alert/73681>, accessed 25/04/2019.
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[^1]: CIMPA-Escuela de Matemática, Universidad de Costa Rica, San José, Costa Rica. E-Mail: [juan.calvo@ucr.ac.cr](mailto: juan.calvo@ucr.ac.cr)
[^2]: CIMPA-Escuela de Matemática, Universidad de Costa Rica, San José, Costa Rica. E-Mail: [albertojose.hernandez@ucr.ac.cr](mailto: albertojose.hernandez@ucr.ac.cr)
[^3]: Department of Mathematics, University of California Los Angeles, USA. E-Mail: [mason@math.ucla.edu](mailto: mason@math.ucla.edu)
[^4]: CIMPA-Escuela de Matemática, Universidad de Costa Rica, San José, Costa Rica. E-Mail: [fabio.sanchez@ucr.ac.cr](mailto: fabio.sanchez@ucr.ac.cr)
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Energy conversion via reconnecting current sheets is common in space and astrophysical plasmas. Frequently, current sheets disrupt at multiple reconnection sites, leading to the formation of plasmoid structures between sites, which might affect energy conversion. We present in situ evidence of the firehose instability in multiple reconnection in the Earth’s magnetotail. The observed proton beams accelerated in the direction parallel to magnetic field and ion-scale fluctuations of whistler type imply the development of firehose instability between two active reconnection sites. The linear wave dispersion relation, estimated for the measured plasma parameters, indicates a positive growth rate of firehose-related electromagnetic fluctuations. Simulations of temporal evolution of the observed multiple reconnection by using a 2.5D implicit particle-in-cell code show that, as the plasmoid formed between two reconnection sites evolves, the plasma at its edge becomes anisotropic and overcomes the firehose marginal stability threshold, leading to the generation of magnetic field fluctuations. The combined results of observations and simulations suggest that the firehose instability, operating between reconnection sites, converts plasma kinetic energy into energy of magnetic field fluctuations, counteracting the conversion of magnetic energy into plasma energy occurring at reconnection sites. This suggests that magnetic energy conversion in multiple reconnection can be less efficient than in the case of the single-site reconnection.'
author:
- Alexandra Alexandrova
- Alessandro Retinò
- Andrey Divin
- Lorenzo Matteini
- Olivier Le Contel
- Hugo Breuillard
- Filomena Catapano
- Giulia Cozzani
- Ivan Zaitsev
- Jan Deca
bibliography:
- 'Bibliography.bib'
title: In situ evidence of firehose instability in multiple reconnection
---
Introduction
============
The dynamics of magnetized, collisionless astrophysical plasmas implies the formation of current sheets accompanied by the accumulation of magnetic energy [@parker1994], and the fast release of such energy through magnetic reconnection [@priestforbes2000; @yamada2010]. At the reconnection site, or X-line, nonlinear kinetic-scale processes mediate the large magnetohydrodynamical-scale rearrangement of the magnetic field, leading to plasma bulk acceleration, heating and non-thermal acceleration of particles. Depending on the global conditions, the current sheet disruption may develop into single or multiple reconnection sites. During multiple reconnection, looped magnetic field structures (magnetic islands, plasmoids or flux ropes) tend to form between the adjacent X-lines. Though such plasmoid chains are considered to be an inevitable primary stage of the current sheet disruption [@bhattacharjee2009; @fulviapucci2018; @uzdensky2016 and references therein] as well as an important stage of the single X-line evolution [@daughton2006], the effect of plasmoid dynamics on the large-scale energy redistribution is not yet fully understood.
Numerical studies showed that plasmoid contraction leads to [the acceleration]{} of trapped electrons [@drake2006] and ions [@drake2010] parallel [to the magnetic field]{}[(further referred to as parallel acceleration)]{} by first-order Fermi mechanism. Accordingly, the multi-layered current sheets disrupted [into]{} plasmoids, can be responsible for non-thermal acceleration of particles, e.g., at the heliopause [@drake2010] and in the eruptive solar flares [@guidoni2016]. Ion parallel acceleration is limited by the firehose instability, which manifests a[t]{} later stage[s]{} of plasmoids contraction in 2D particle-in-cell [@drake2010] and 3D hybrid @burgess2016 simulations.
In [space]{}[ plasmas]{}, statistical analysis of plasma properties showed that [the]{} firehose instability limits parallel acceleration of particles in the solar wind [@hellinger2006; @bale2009] and in magnetotail reconnection jets [@voros2011; @wu2013], except for an extreme case of long-duration [reconnection]{} exhaust [@hietala2015], where [plasma]{} acceleration by reconnection appeared to [prevail over the effect of the]{} instability. The plasmoid dynamics in relation to the firehose instability, [however, was not studied ]{} from observations.
In situ observations of multiple reconnection, [showing a]{} plasmoid as well as neighboring reconnection sites, [were]{} provided in the Earth’s magnetotail [@hwang2013; @alexandrova2015]. A case study of [a]{} passage of two- X-lines by the spacecraft [@alexandrova2016] revealed [the]{} highly variable magnetic field topology between the X-lines, representing complex stages of the plasmoid [evolution]{}.
[Here]{} we [present]{} in situ observations of multiple reconnection in the Earth’s magnetotail [@alexandrova2016], with a focus on the [ion]{} temperature anisotropy between [two]{} reconnection sites. The analysis of particle distribution [functions]{}, electromagnetic fluctuations and plasma stability conditions associated with the parallel temperature anisotropy of ions reveal typical conditions for the development of the firehose instability. The observations are supported by 2.5D implicit particle-in-cell (PIC) simulations [which allow to follow the]{} space-time dynamics of the plasmoid in relation to [development of]{} the instability.
Overview of the multiple reconnection event in the Earth’s magnetotail
======================================================================
On 2002 August 18, between 17:07:00-17:13:00 UT, [the]{} Earth’s magnetosphere was quiet. Ground based observations of the ionosphere showed no signatures of a substorm. Cluster four-probe spacecraft [@escoubet2001] was moving from northern to southern hemisphere [across]{} the magnetotail current sheet, at about 17.7 Earth radii ($R_E$) tailward and 5 $R_E$ dawnward in the Geocentric Solar Magnetic coordinate system (GSM). The spacecraft detected [the]{} typical signature[s]{} of [a]{} consecutive passage of two reconnection sites and [of]{} the region in-between[,]{} where counterstreaming reconnection jets interact [@alexandrova2016]. The signature[s]{} involve [three consecutive reversals in the following characteristics: (i) plasma bulk velocity; (ii) magnetic field component in the normal to the current sheet direction (reconnected field) (iii) Hall magnetic field component]{} (see Figure 1 in [@alexandrova2016]). A large value of the magnetic field [component]{} parallel to the current sheet ($\approx 16$ [nT]{}) and the jets’ speed [($\approx 200$ km/s)]{} being smaller than the characteristic Alfvén speed [($\approx 800$ km/s)]{} indicate that the structures were detected [at the ed[g]{}e]{} of the current sheet. Between the reconnection sites, two Cluster probes observed two different stages of the jets[’]{} collision process, showing [the]{} formation and compression of a[n]{} ion[-]{}scale boundary separating [the]{} two counter-streaming jets, which is accompanied [by]{} strong wave activity (see Figures 2 and 3 in [@alexandrova2016]). [All t]{}his indicates that [the observed]{} plasmoid was not steady during Cluster observations, but [was]{} rather evolving in time. [Here]{} we focus on the [region]{} between [the two]{} reconnection sites where [one of the Cluster probes, C1,]{} observed [enhanced]{} parallel temperature anisotropy of [ions]{}.
In situ observations of firehose instability
=============================================
Figure \[f.overview\] presents Cluster C1 probe observations of magnetic field ($22.4$ Hz resolution \[Balogh et al., 2001\]) and plasma ($H^{+}$ ion, $4$ s resolution plasma moments and distribution function[s]{} \[Rème et al., 2001\]) in the region [between the two reconnection sites]{}. [Measurements of $He^{+}$ and $O^{+}$ ions show densities of more than an order smaller than the $H^{+}$ [density]{}, thus we [analyze]{} $H^{+}$ ions [only]{}.]{} Data are represented in the current sheet conventional coordinates LMN, where **L** is parallel to the current sheet and perpendicular to the reconnection line, **M** is parallel to the reconnection line, **N** is perpendicular to the current sheet. LMN was calculated [through]{} [M]{}inimum [V]{}ariance [A]{}nalysis [(MVA)]{} [of magnetic field]{} \[Sonnerup and Cahill, 1967; Sonnerup et al., 2006\] to the current sheet crossing prior to the reconnection activity at 16:40 - 17:00 UT for C1 probe [@alexandrova2016]. In GSM [coordinates]{}, $\mathbf{L} = (0.99, 0.03, −0.09)$, $\mathbf{M} = (−0.00, 0.95, 0.31)$, and $\mathbf{N} = (0.10, −0.31, 0.95)$. [In t]{}he time interval 17:08:30-17:11:00 [the]{} [spacecraft were located]{} below the magnetotail current sheet ($B_L<0$, Figure \[f.overview\]a). After the first X-line detection at $\sim$ 17:08:30, the probe C1 entered the earthward reconnection [jet]{} ($V_L>0$, $B_N>0$, $B_M<0$, Figure \[f.overview\]a-c) and at $\sim$ 17:09:45, it detected the tailward [jet]{} ($V_L<0$, $B_N<0$, $B_M>0$, Figure \[f.overview\]a-c) [coming]{} [from]{} the second X-line observed later at $\sim$ 17:11:00. The velocity perpendicular to the magnetic field, $V_{\perp_{L}}$, is negligible, supporting that the spacecraft is located at the current sheet edge where ions propagate mostly in the parallel direction. The magnetic field configuration [is consistent with the observations of magnetic field and velocity, see]{} Figure \[f.overview\] [sketch]{}. According to the [timing analysis applied to the reversals in the component of magnetic field normal to the current sheet, $B_N$,]{} [@alexandrova2015; @alexandrova2016], the plasmoid [observed in between the two reconnection sites]{} is moving tailward with [a]{} speed $U_L \approx 130$ km/s [and]{} has a scale in the **L** direction of $\Delta L_o \approx 19500$ km $\approx 3$ $R_E \approx 3.4 $ $di$, where $di = 580$ km is the ion inertial length. At the time 17:09:04-17:09:26, associated with the region between the first X-line and the jets[’ collision site]{} [(marked by vertical lines in Figure \[f.overview\])]{}, [C1 measurements showed parallel temperature anisotropy of [ions]{}, $T_{||}>T_\perp$]{}, where $T_{||}$ [and]{} $T_\perp$ are ion temperatures parallel [and perpendicular]{} to the local magnetic field, respectively (Figure \[f.overview\]d). [The average temperature ratio is]{} $T_\perp/T_{||} = 0.67$. According to linear analysis in the framework of [MHD theory]{}, the parallel temperature anisotropy [can]{} cause the firehose instability when $\alpha = (\beta_\parallel - \beta_\perp)/2 > 1$, where $\beta_\parallel$ and $\beta_\perp$ are the ratios of the parallel and perpendicular plasma pressures, respectively, to the magnetic field pressure [@gary1998]. At the time corresponding to the temperature anisotropy, the observations show an increase of $\beta_\parallel$ (Figure \[f.overview\]e) as well as $\alpha$ (Figure \[f.overview\]f). However the peak value is lower than the predicted firehose threshold. [As for the other Cluster probes, C2 and C3 were far from the region of interest in M and in N direction, respectively, [therefore they are not used in this analysis]{}. The probe C4 was observing the region $\sim 10$ s later and showed parallel temperature anisotropy only in one measurement point, at 09:14.]{}
[During the time of interest $\approx 20$ s]{} associated with the ion parallel temperature anisotropy [([in ]{}between the vertical lines in Figure \[f.overview\])]{}, the [ion velocity distributions]{} show [the]{} presence of parallel and anti-parallel beams superposed to the bulk velocity of earthward propagating [ions]{} (anti-parallel to the background field), [see]{} Figure \[f.overview\]g, 17:09:14.7 and 17:09:22.7 [times]{}. These beams are not observed [either]{} before ([Figure \[f.overview\]g, time]{} 17:08:54.7) [or]{} after ([Figure \[f.overview\]g, time]{} 17:09:26.7) the region of anisotropy.
We study [the]{} magnetic field fluctuations associated with the [observed]{} parallel temperature anisotropy. Figure \[f.waves\]a [shows]{} the wavelet spectrum of the magnetic field $B_N$ component, which is the reconnected magnetic field [component]{}. The fluctuations which directly correspond to the time associated with the ion temperature anisotropy [and the $\beta_\parallel$ increase]{} (17:09:04-17:09:20), are seen in the frequency range of $f_0 \approx 0.08-0.11$ Hz. These frequencies are nearly twice lower than the corresponding [ion]{} cyclotron frequency $f_{ci}=0.22$ Hz. We apply a [bandpass]{} filter for the observed fluctuations (17:09:04-17:09:20, $f = 0.08-0.11$ Hz ) and the Minimum Variance Analysis (MVA) [@sonnerupcahill1967] to calculate the direction of propagation and the polarization of these fluctuations [@thorne1973; @smithtsurutani1976]. The wave components in the MVA coordinate system ($lmn_{wave}$) are [shown]{} in Figure \[f.waves\]b. The orientation [of the wave coordinate system ]{}in [the ]{}LMN system is $\mathbf{l_{wave}} = (0.02, 0.64, 0.77)$, $\mathbf{m_{wave}} = (0.74, -0.53, 0.42)$, $\mathbf{n_{wave}} = (0.68, 0.56, 0.49)$. The medium to minimum eigenvalue ratio is $188$, which indicates that the normal direction is well defined. The ellipticity, defined as the square root of the medium to maximum eigenvalue ratio is $e=0.57$. [With respect to]{} the background magnetic field, calculated as an average over the spacecraft spin, $\mathbf{B_{db}}$, the wave is propagating with an angle $\Theta = \angle (\mathbf{n_{wave}}, \mathbf{B_{bg}}) \sim 23^{o}$. The MVA analysis contains a $180^{o}$ ambiguity in the normal vector $n_{wave}$ direction. Under [the]{} assumption of wave propagation preferentially parallel to the background field, ${n_{wave_{L}}>0}$, the elliptically polarized wave exhibit right-hand rotation around the magnetic field, see the wave hodograph in Figure \[f.waves\]c. The observed waves characteristics and frequency ranges [are consistent with]{} the low branch whistler waves, which are related to the linear firehose instability [@gary1998]. According to the plasmoid tailward speed $U_L \approx -130$ km/s [@alexandrova2016], the ion bulk speed $V_L \approx 160$ km/s (Figure \[f.overview\]c between the dashed lines), the Alfvén speed $V_A\approx 800$ km/s, and the period of the ion gyration $\tau_i \approx 4$ s, we can roughly estimate the Doppler shift to be about $f_D = (V-U_L)/V_A T_i \approx 0.09 $ Hz. As long as background plasma propagates in the [direction ]{}opposite to the wave [propagation ]{}direction, the expected real wave frequency is [therefore]{} about $f \approx 0.2$ Hz, [which]{} is almost equal to the ion cyclotron frequency. The time interval of $16$ s corresponding to the temperature anisotropy [and $\beta_\parallel$ increase]{} is associated with one wave period. According to the plasmoid speed, the wave has a scale of $\Delta L \approx 1950$ km $\approx 0.3$ $R_E \approx 3.4$ $d_i$. The amplitude of the observed fluctuation is $\delta \mathbf{|B|}/ |B_{bg}| \approx 0.03$.
In order to verify [whether ]{}the firehose instability [was operating]{}, we perform [a]{} plasma stability analysis. Figure \[f.instability\]a shows comparison between $T_\perp/T_{||}$ and $\beta_\parallel$, measured in [between the two reconnection sites]{} (17:08:30-17:11:00), with the predictions of Vlasov linear theory for the marginal stability thresholds of typical plasma instabilities (mirror, ion cyclotron, oblique and parallel firehose) calculated for the maximum growth rate $\gamma\approx 10^{-3}$[,]{} according to the fitting parameters from @hellinger2006, Section 2 and Table 1. The five data points, related to the parallel temperature anisotropy observed at 17:09:04-17:09:20, lie close to the parallel firehose threshold (marked with rectangle in Figure \[f.instability\]a). To investigate the growth rate of the possible firehose instability, we solve the Vlasov-Maxwell equations by using WHAMP solver [@roennmark1982], assuming [a]{} bi-Maxwellian gyrotropic [ion]{} distribution and different values of the wave vector direction. It should be noted that the observed [ion]{} distribution function is neither bi-Maxwellian, nor gyrotropic. In such a case a [recent ]{} method developed in @astfalkjenko2017 might give more correct estimates, however it requires better resolution than Cluster provides. Therefore, we use WHAMP calculations to obtain [indicative]{} estimates. [T]{}he solution with positive growth rate is presented in Figure \[f.instability\]b, solid line. For the observed plasma parameters: the average magnetic field $\langle B\rangle = 15$ nT, electron temperature $\langle T_e\rangle = 1$ keV, [ion]{} density $\langle n\rangle = 0.16$, [ion]{} temperature $\langle T_{\parallel} \rangle = 5$ keV, and temperature anisotropy $ \langle T_{\perp}/T_{\parallel} \rangle = 0.67$. The maximum growth rate is about $\gamma = 10^{-5}$ $\Omega_p$, where $\Omega_p$ is the [ion]{} frequency. The dashed line in Figure \[f.instability\]c represents the growth rate calculated for [an]{} [ion]{} temperature enhanced by $30\%$ and anisotropy enhanced by $10\%$, [to include]{} possible temperature underestimation [due to]{} instrument[al errors]{} [Figure 6.7 of @paschmanndaly2000]. For the enhanced parameters, the growth rate reaches the marginal stability threshold of $\gamma = 10^{-3}$ $\Omega_p$. For the observed parameters, the solution with the maximum positive growth rate corresponds to [a]{} wave with frequency $\omega = 0.48$ $\Omega_p$, which is about $0.07$ Hz, see Figure \[f.instability\]c, propagating parallel to magnetic field and have right-hand polarization. The resulting electromagnetic fluctuations [are consistent]{} with the firehose instability.
Reconstruction of observations with 2.5D PIC simulations
========================================================
The [presented]{} [Cluster]{} magnetotail observations represent [only]{} single-spacecraft measurements of anisotropic plasma in [the]{} localized region of plasmoid between two X-lines, [leaving a]{} detailed investigation of the [overall]{} large-scale plasmoid [spatio-temporal evolution unresolved]{}. Thus, we employ a 2.5D numerical simulation which reproduces the formation of a plasmoid, followed by its compression by reconnected plasma flows and [by the]{} development of firehose instability at later stages (Figure \[f.3dpic\]a-\[f.3dpic\]c). The iPIC3D implicit PIC code [@markidis2010] is used. The system of coordinates is as follows: the $\mathbf{x}$ axis is directed parallel to the reconnecting magnetic field [(corresponds to the **L** direction)]{}; the $\mathbf{y}$ axis is normal to the current sheet at time t=0 [(corresponds to the **N** direction)]{}; the $\mathbf{z}$ axis complements the right-hand triple [(corresponds to the **M** direction)]{}. The simulation is performed in 2D rectangular domain with the dimensions $(L_x,L_y)$. The model is translationally invariant in [the]{} third [direction, **z**, which is the direction of current]{}. The simulation is initialized with a pair of conventional Harris current sheets located at $y=L_y/4$ (active) and $y=3L_y/4$ (remains quiet in the present study, not shown). A uniform background population of density $n_b=0.1 n_0$ is added, with $n_0$ being the peak density of the Harris current sheet. [The length unit (${d_i \, \approx \, 509}$ km) is computed [from]{} the plasmoid edge density (${\approx 0.35 n_0 = 0.2 }$ cm$^{-3}$) to ease comparison with observations. For such a normalization, the computational box dimensions amount to ${(L_x,L_y)=(60d_i,15d_i)}$, and the number of grid points in each dimension is ${(N_x,N_y)=(2304, 576)}$. The magnetic field $B_0$ (asymptotic magnetic field outside of [the]{} Harris sheets) is normalized to $16$ nT, which is the largest magnetic field observed by the [Cluster]{} spacecraft [in]{} the parallel temperature anisotropy region (see Figure \[f.overview\]a).]{} Derived units are the Alfvén speed of $ 780 $ km/s and the ion cyclotron frequency $\Omega_{ci0} \sim 1.5$ $s^{-1}$. The initial ion-to-electron temperature ratio is similar to that [of the [Cluster]{} observations described above]{} ($T_i/T_e=5$). The ion-to-electron mass ratio is $m_i/m_e=256$, the ratio of the speed of light to the characteristic Alfvén speed is 256.
A localized X-point perturbation [[see, e.g., @divin2012]]{} ignites reconnection at $(0, L_y/4)$. Ion jets are formed at early stages once reconnection at the main X-line has reached the steady state. These jets propagate nearly unperturbed in the $\mathbf{x}$ and $-\mathbf{x}$ directions up to $t \approx 30 $ $\Omega_{ci0}^{-1}$. To mimic the dynamical stage of the plasmoid evolution, we impose periodic boundary conditions and allow plasma jets to run head-to-head producing the domain-large plasmoid (Figures \[f.3dpic\]a-\[f.3dpic\]c). In essence, such periodic configuration can be viewed as interaction of two X-lines. As reconnection progresses, ion flows compress the plasmoid, producing regions with different kinds of anisotropy: (i) perpendicular anisotropy (${T_{\perp}/T_{||}>1}$) is found in the plasmoid core and at reconnection fronts, ${x \approx 15.6 d_i}$ and ${x \approx 44 d_i}$ for the discussed times); (ii) parallel anisotropy (${T_{\perp}/T_{||}<1}$) is found typically in the low-$\beta$ regions [close to the edges of the plasmoid]{} (shown with deep blue color in Figures \[f.3dpic\]a-\[f.3dpic\]c). In Figure \[f.3dpic\]a-\[f.3dpic\]c, we highlight a few field lines of constant magnetic flux to trace their evolution in time. At the beginning of strong interaction between the counterstreaming jets, [approximately at]{} ${t=30.6} \Omega_{ci0}^{-1}$, the plasmoid scale is $32 d_i \approx 16000$ km. This estimate is rather close to the scale of the [observed]{} magnetotail plasmoid described above. In course of [simulation]{} time, the plasmoid shrinks and becomes about ${24} d_i \approx 12200$ km by ${t=36.7} \Omega_{ci0}^{-1}$. [Further shrinking of the plasmoid, proceeding between ${t=36.7} \Omega_{ci0}^{-1}$ and ${t=45.5} \Omega_{ci0}^{-1}$ leads to a strong bending of the magnetic field lines [at the edges]{} of the plasmoid, while inside the plasmoid the magnetic field structure becomes strongly inhomogeneous following the nonlinear pressure growth.]{}
In Figure \[f.3dpic\]d we visualize the growth of magnetic field fluctuations at the [edge]{} of the plasmoid by plotting the time stack plots of $B_y$ component along a cut through ${y=2.06} d_i$. Very weak $B_y$ at earlier times ($t<30 \Omega_{ci0}^{-1}$, ${15 d_i < x < 45 d_i}$) are most likely attributed to the weak tearing instability present at earlier stage [[see, e.g., @pritchett1991 Section 3]]{}. Colliding jet fronts (peak $B_y$ locations are marked with the squares of corresponding colors) host large $B_y$ variations at $10 d_i<x<20 d_i$ and $40 d_i<x<50 d_i$. Note that the colorbar is compressed to reveal weaker $B_y$ perturbations in $20 d_i<x<40 d_i$. Compression of the plasmoid produces conditions favorable for the excitation of the firehose instability. Gray crosses indicate regions where the temperature anisotropy overcomes the firehose marginal stability threshold, calculated according to @hellinger2006. Notably, fluctuations are strongly amplified at these times ($30 \Omega_{ci0}^{-1}<t<40 \Omega_{ci0}^{-1}$), before saturating at the de-compression stage. Black lines in Figure \[f.3dpic\]d are virtual streamlines of fluid elements located at $y=2.06 d_i$ (presuming that the velocity component along the $\mathbf{y}$ direction is unimportant). The streamlines trace well the jet front locations. Although not exactly, the $B_y$ fluctuations follow the streamlines as expected for the firehose instability.
We focus on the region, marked by the white rectangle in Figures \[f.3dpic\]a-\[f.3dpic\]c: $22.25 d_i <x<23.56 d_i$ and ${2.0 d_i <y<2.12 d_i}$, which closely reproduces the spacecraft observations by a combination of field and plasma parameters. Figure \[f.3dpic\][d]{} shows temporal changes in the plasma distribution of $T_{\perp}/T_{||}$ plotted against $\beta_{||}$ for the selected region. The thresholds for the plasma marginal stability are shown according to @hellinger2006. The ion temperature anisotropy in relation to $\beta_{||}$ indicate isotropic and stable plasma at the early simulation time ($\Delta t_1$, cyan). [A]{}t later stages[, the]{} plasma becomes more anisotropic and [the anisotropy]{} exceeds the parallel firehose threshold ($\Delta t_2$, pink and $\Delta t_3$, dark pink). [Then,]{} at the time corresponding to the active jets collision ($\Delta t_4$, red)[, the]{} plasma becomes more isotropic and close to the marginal stability threshold, and at the end of simulations ($\Delta t_5$, dark red)[, the]{} plasma becomes stable. [[In order to get a clear visualization]{} in Figure \[f.3dpic\]d, [we plot selected points which represent the overall behavior.]{}]{} Figures \[f.3dpic\]a, \[f.3dpic\]b and \[f.3dpic\]c reflect plasmoid development stages for three selected times from the periods $\Delta t_2$, $\Delta t_3$ and $\Delta t_4$, respectively. The magnetic field temporal changes taken in the middle of the selected region, ${{x=22.9} ~d_i}$, ${{y=2.06} ~ d_i}$ represent one wave period (Figure \[f.3dpic\]f). In two locations in $\mathbf{x}$ direction separated by about ${1.2 ~ d_i \approx 600}$ km the fluctuations are almost similar and shifted in time by $0.2$ $\Omega_{ci0}^{-1}$ $\approx 0.13$ s, indicating the wave phase speed [to be]{} about ${500} $ km/s, which is [smaller [than]{}]{} the corresponding Alfvén speed (${780} $ km/s). The MVA analysis [@sonnerupcahill1967] gives the orientation of the normal $\mathbf{n_{wave}}=(0.92,0.11,0.3)$, the maximum and medium variance components are $\mathbf{l_{wave}}=(-0.25, 0.9, 0.34)$ and $\mathbf{m_{wave}}=(0.28,0.41,-0.87)$, respectively, in the PIC $(x,y,z)$ coordinates. Figure \[f.3dpic\]e shows the wave in the $lmn_{wave}$ system and the rotation of magnetic field in the plane perpendicular to the normal direction, which indicate right-hand elliptical polarization. Fluctuations show characteristics typical for the firehose instability, with the magnitude of about $\delta \mathbf{|B|} / B \approx 0.15$. Note, that the amplitude of the magnetic field fluctuations changes in time revealing the nonlinear evolution.
Discussion
==========
At the [edge]{} of the plasmoid forming between two X-lines and having a scale of approximately $35$ ion inertial length[s]{} (about three Earth’s radii), [Cluster]{} observations show distinctive signatures of the firehose instability, including
\(1) [a]{} parallel temperature anisotropy of [ions]{} of about $T_{\perp}/T_{\parallel}{\approx}$ $0.7$ (Figures \[f.overview\]d,e and \[f.instability\]a);
\(2) parallel and anti-parallel ion beams, superimposed to the bulk motion of plasma in the reconnection jet (Figure \[f.overview\]g);
\(3) magnetic field fluctuations [at frequencies around]{} the ion cyclotron [frequency]{}, right-hand polarized and quasi-parallel propagating (Figure \[f.waves\]);
\(4) plasma condition[s]{} corresponding to the marginal firehose state (Figure \[f.instability\]a), with positive growth rate of [the ]{}instability (Figure \[f.instability\]b) for the firehose-like fluctuations (Figure \[f.instability\]c) [in]{} approximately [the]{} ion cyclotron frequency range.
2.5D iPIC simulations of the plasmoid formation between two reconnection jets running head-to-head (see Figures \[f.3dpic\]a-\[f.3dpic\]c)[,]{} [with]{} plasma and magnetic field parameters, [similar to the ones observed,]{} show [that]{}
\(5) in course of time the plasmoid is compressed[,]{} producing regions at [its]{} [edge]{} with [a]{} large parallel temperature anisotropy of the maximum value $T_{\perp}/T_{||} \sim 0.3$ (Figure \[f.3dpic\]d, \[f.3dpic\]e)
\(6) firehose-like electromagnetic fluctuations with relatively large amplitude $\delta B / B \approx 0.15$ (Figure \[f.3dpic\]f, \[f.3dpic\]g) arise when the [maximum]{} temperature anisotropy is reached (Figure \[f.3dpic\]d, $t>30$ ${\Omega_{ci0}^{-1}}$).
\(7) at later stages, the anisotropy relaxes though the fluctuations are still present (Figure \[f.3dpic\]d, $t>44$ ${\Omega_{ci0}^{-1}}$).
\(8) the fluctuations [at the edge]{} of the plasmoid are affected by the inhomogeneous plasma pressure growth inside the plasmoid and become strongly nonlinear at later stages (Figure \[f.3dpic\]c).
[The combination of Cluster observations and PIC simulations leads to]{} the following interpretation. During multiple reconnection, [ions]{}, accelerated by reconnection, propagate mostly parallel to the magnetic field at the [edge]{} of the plasmoid forming between [two]{} X-lines. As reconnection proceeds, the island contracts which leads to additional acceleration of [ions]{} in the magnetic trap between the X-lines. As the temperature anisotropy reaches the firehose marginal stability threshold, the firehose instability [excites]{} magnetic field fluctuations, which are [confined]{} in the looped magnetic field. Further compression of magnetic field by ongoing reconnection may lead to the nonlinear evolution of [these ]{}fluctuations and their transformation [in]{}to bent field line[s]{} [leading to the formation of a]{} thin boundary, similar to the one observed between the X-lines [after]{} [the temperature anisotropy was observed]{} [@alexandrova2016]. [Note ]{}[that both active X-lines are observed, thus reconnection does not cease with the development of instability, unlikely to what was [shown in simulations by]{} @drake2006 [@drake2010; @burgess2016]]{}. [In such periodic-boundary simulations]{}, the development of the firehose-induced fluctuations[,]{} together with the inhomogeneous pressure growth in the plasmoid center, constrains the plasmoid shrinking at later stages and initiate the phase of jet fronts [repulsion]{} with [consequent]{} decrease of the reconnection rate [at]{} the X-lines. Simulations with open [boundary conditions]{}[,]{} as well as observations covering larger temporal and [spatial]{} scale[s,]{} might be helpful in describing particular conditions for [different ways of the plasmoid-chain evolution]{}.
[It is]{} important to note that the performed stability analysis was based on the theoretical assumptions of a stable background magnetic field and [of]{} bi-Maxwellian particle distribution[s]{}, while none of these assumptions are rigorously valid in the observations. However, the consistency of the magnetic field fluctuations with the temperature anisotropy of plasma indicates that[,]{} despite [of these issues]{}, the firehose instability is [identified]{}.
Previous studies of plasma stability in the context of temperature anisotropies in the solar wind [@hellinger2006; @bale2009; @matteini2013] and in magnetotail reconnection jets [@voros2011; @hietala2015] performed statistical analysis of the average characteristics of magnetic field fluctuations and typical marginal stability thresholds. In the present study, we [analyzed in details]{} the development of the firehose instability and [of]{} the associated waves excitation, for the [specific]{} magnetic configuration of [multiple reconnection with a]{} plasmoid forming between two reconnection sites. The analysis showed that[,]{} differently to the average solar wind conditions, multiple reconnection results in highly inhomogeneous anisotropic plasma and nonlinear development of firehose-related fluctuations. Our simulations show that the anisotropy grows [far]{} beyond the marginal stability threshold before the firehose-related fluctuations arise. However, fluctuations are present even after plasma reaches the marginal stability. This [supports [the fact]{}]{} that the observations of relatively small growth rate, but quite intensive fluctuations, indicate that the plasmoid in the magnetotail was detected [during]{} instability [decaying]{} and plasma isotropisation.
An important aspect is that the instability [may]{} affect the global energy conversion in the multiple reconnection configuration. The magnetic energy conversion in the neighboring X-lines leads to particle acceleration, which in turn [leads]{} to the development of the firehose instability [at the edge of the plasmoid between the X-lines]{}. [A]{}s a consequence, energy of accelerated particles [is converted]{} back to magnetic field fluctuations. Understanding the ratio between the energy [converted]{} to plasma by reconnection and the energy withdrawn by the [firehose]{} instability might help [to]{} better quantify the [impact of]{} current sheet disruption [to multiple reconnection for]{} space plasma dynamics and would be a valuable direction for a future research.
Conclusions
===========
We studied [the]{} [dynamics of]{} [a]{} plasmoid between two active X-lines observed in situ by [the]{} Cluster spacecraft in the Earth’s magnetotail current sheet. [At th[e]{} edge]{} of the plasmoid having a scale of about $35$ ion inertial length[s]{} (about three Earth’s radii), [a]{} parallel temperature anisotropy of [ions]{} [due to]{} parallel and anti-parallel ion beams [was observed]{}. The plasma conditions corresponded to [a]{} firehose marginal stability [state]{}, [as]{} also supported by the excitation of the whistler low-frequency branch waves (at [about]{} half of the ion cyclotron frequency). Reconstruction of the magnetotail observations by using PIC simulations [allowed us to reproduce the evolution of]{} the firehose instability during multiple reconnection. [Simulations]{} support the scenario [in which]{} the looped magnetic field [of the plasmoid]{} between [two]{} reconnection sites [undergoes]{} fluctuations due to the [firehose instability caused by the]{} excess of parallel [ion]{} acceleration [at the edges]{} of the plasmoid. In such a way, some part of the plasma energy gained in reconnection might be converted back to magnetic field. [The present study indicates that the firehose instability in multiple reconnection can play an important role for energy partition not only in the terrestrial magnetotail, but also in solar and astrophysics plasmas where multiple reconnection is expected to be ubiquitous.]{}
Authors acknowledge the Cluster Science Archive for use of the Cluster spacecraft data. The research was supported by the project of Sorbonne Université/Ecole Polytechnique, Convention 2800, and the LABEX PLAS@PAR project with the financial state aid managed by the Agence Nationale de la Recherche, as a part of the Programme ’Investissements d’Avenir’ under the reference ANR-11-IDEX-0004-02. J.D. acknowledges support from NASA’s Solar System Exploration Research Virtual Institute (SSERVI): Institute for Modeling Plasmas, Atmosphere, and Cosmic Dust (IMPACT), and the NASA High-End Computing (HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames Research Center.
![Overview of Cluster C1 observations in the Earth’s magnetotail on 2002 August 18 at 17:08:30-17:11:00 of (a) magnetic field components, (c) plasma bulk velocity in the parallel to the current sheet direction, $V_{L}$, (d) [ion ]{} temperature ratio, (e) parallel plasma beta, (f) firehose instability threshold according to linear theory, (g) [ion]{} distribution functions for the selected times before, during and after the parallel temperature anisotropy observations. Sketch above the panels [represents]{} an interpretation of the observations.[]{data-label="f.overview"}](PaperFigure_Overview.pdf){width="0.5\linewidth"}
![(a) wavelet spectrum of the normal to the current sheet magnetic field component $B_N$, (b) waveform of the filtered fluctuations at $0.08-0.11$ Hz, (c) hodograph of the filtered fluctuations in the plane perpendicular to the wave normal.[]{data-label="f.waves"}](PaperFigure_Waves.pdf){width="0.6\linewidth"}
![(a) parallel temperature anisotropy of protons in relation to the parallel plasma beta in comparison to the marginal stability thresholds of the plasma instabilities, (b) instability growth rate and (c) real frequency of the corresponding fluctuations according to the stability analysis by WHAMP solver for the observed plasma parameters (solid lines) and presumed plasma parameters within the $30\%$ of inaccuracy of measurements (dashed lines).[]{data-label="f.instability"}](PaperFigure_Instability.pdf){width="1\linewidth"}
![$2.5$D iPIC simulations representing the dynamics of plasmoid forming between two reconnection sites. (a)-(c) parallel temperature anisotropy of [ions]{} (in color) and magnetic field configuration (red lines) for three stages of plasmoid development, (d) temporal evolution of the normal to the current sheet magnetic field (in color) and the regions where temperature anisotropy overcomes the marginal stability threshold of the firehose instability (gray crosses) for a cut in $y=2.06 d_i$. (e) temporal evolution of the temperature anisotropy in the selected region (white rectangle in panels (a)-(c)) in comparison to the marginal stability thresholds of plasma instabilities, (f) magnetic field fluctuations in the center of the selected region, (g) magnetic field fluctuations in the minimum variance coordinate system and the hodograph in the plane perpendicular to the wave normal direction.[]{data-label="f.3dpic"}](PaperFigure_PIC.png){width="0.7\linewidth"}
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In his article [@V] on generating functions Viterbo constructed a bi-invariant metric on the group of compactly supported Hamiltonian symplectomorphisms of $\mathbb{R}^{2n}$. Using the set-up of [@mio] we extend the Viterbo metric to the group $\text{Cont}_0^{\phantom{0}c}\,(\mathbb{R}^{2n}\times S^1)$ of compactly supported contactomorphisms of $\mathbb{R}^{2n}\times S^{1}$ isotopic to the identity. We also prove that $\text{Cont}_0^{\phantom{0}c}\,(\mathbb{R}^{2n}\times S^1)$ is unbounded with respect to this metric.'
address: 'Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal'
author:
- Sheila Sandon
title: 'An integer valued bi-invariant metric on the group of contactomorphisms of $\mathbb{R}^{2n}\times S^{1}$'
---
Introduction
============
Gromov’s non-squeezing theorem [@Gr] marked the beginning of the study of symplectic capacities, which are global symplectic invariants that measure the size of symplectic manifolds. Some years later Hofer [@H] discovered a way of measuring the size (or *energy*) of Hamiltonian symplectomorphisms, by looking at the total variation of their generating Hamiltonians. This notion gave rise in fact to the definition of a bi-invariant metric (the Hofer metric) on the group of Hamiltonian symplectomorphisms [@H; @P; @LM]. The Hofer metric is deeply related to symplectic capacities. It was proved by Hofer that for any domain $\mathcal{U}$ of $\mathbb{R}^{2n}$ the Hofer-Zehnder capacity $c_{HZ}(\mathcal{U})$ is related to the displacement energy $E(\mathcal{U})$ via the energy-capacity inequality $c_{HZ}(\mathcal{U})\leq E(\mathcal{U})$. Lalonde and McDuff [@LM] gave moreover a direct geometric construction relating non-degeneracy of the Hofer metric to Gromov’s non-squeezing theorem.\
\
Using the theory of generating functions, Viterbo constructed in [@V] a new symplectic capacity for domains of $\mathbb{R}^{2n}$ and a new bi-invariant metric[^1] $d_V$ on the group $\text{Ham}^c\,(\mathbb{R}^{2n})$ of compactly supported Hamiltonian symplectomorphisms, and proved an energy-capacity inequality relating these two notions. Moreover he defined a partial order $\leq_V$ on $\text{Ham}^c\,(\mathbb{R}^{2n})$, giving $\big(\,\text{Ham}^c\,(\mathbb{R}^{2n}),d_V\,\big)$ the structure of a partially ordered metric space [^2].\
\
The generalization of the constructions in [@V] to the contact case [@B; @mio] has been motivated by the theory of *orderability* of contact manifolds introduced by Eliashberg and Polterovich [@EP], and by the related contact rigidity phenomena studied in [@EKP]. Bhupal [@B] extended the Viterbo partial order to the group $\text{Cont}_0^{\phantom{0}c}\,(\mathbb{R}^{2n+1})$ of compactly supported and isotopic to the identity contactomorphisms of $\mathbb{R}^{2n+1}$, thereby proving orderability of $\mathbb{R}^{2n+1}$. In [@mio] we obtained a new proof of the contact non-squeezing theorem of Eliashberg, Kim and Polterovich [@EKP] by extending the Viterbo capacity to the contact manifold $\mathbb{R}^{2n}\times S^1$.\
\
In introducing the notion of orderability Eliashberg and Polterovich were in fact motivated by the question of finding some geometric structure on the group of contactomorphisms. They showed in [@EP] that the concept of *relative growth*, which is available in any partially ordered group, can be applied to the contactomorphism group of an orderable contact manifold $(M,\xi)$ in order to associate to it a metric space $\big(Z(M,\xi),\delta\big)$. This can be done by defining, in terms of the relative growth, a pseudo-distance $\delta$ on the group of those contactomorphisms of $(M,\xi)$ that are generated by a positive Hamiltonian, and then by considering the quotient of this group by the equivalence classes of elements which are at zero distance from each other.\
\
In the present article we show that in the case of the contact manifold $\mathbb{R}^{2n}\times S^{1}$ it is possible to define a bi-invariant metric directly on the group of all (not necessarily positive) compactly supported contactomorphisms isotopic to the identity. This metric is a generalization of the Viterbo metric and can be easily constructed by using the set-up developed in [@mio]. However a crucial difference is that, in contrast with the symplectic case, our metric only takes values in $\mathbb{Z}$. We refer to the introduction of [@mio] for an explanation, in terms of generating functions, of the special role played by the integers in the study of rigidity phenomena for the contact manifold $\mathbb{R}^{2n}\times S^{1}$.\
\
Our results can be summarized in the following theorem.
\[main\] The group $\text{Cont}_0^{\phantom{0}c}\,(\mathbb{R}^{2n}\times S^1)$ admits an unbounded integer valued bi-invariant metric $d$. This metric is compatible with the Bhupal partial order $\leq_B$ in the sense that $\leq_B$ turns $\big(\,\text{Cont}_0^{\phantom{0}c}\,(\mathbb{R}^{2n}\times S^1),d\,\big)$ into a partially ordered metric space.
The proof of Theorem \[main\] is based on results of [@mio].\
\
The article is organized as follows. In the first two sections we recall the set-up of [@mio]. In particular, in Section \[prel\] we give some preliminaries on generating functions and in Section \[inv\] we discuss the generalization to $\text{Cont}_0^{\phantom{0}c}\,(\mathbb{R}^{2n}\times S^1)$ of the invariants $c^+$ and $c^-$ that were constructed by Viterbo in [@V]. In Section \[metric\] we define the metric $d$ on $\text{Cont}_0^{\phantom{0}c}\,(\mathbb{R}^{2n}\times S^1)$, and discuss an energy-capacity inequality relating it to the contact capacity for domains of $\mathbb{R}^{2n}\times S^1$ that was constructed in [@mio]. In Section \[order\] we recall the definition of the Bhupal partial order $\leq_B$ on $\text{Cont}_0^{\phantom{0}c}\,(\mathbb{R}^{2n}\times S^1)$ and show that $d$ and $\leq_B$ are compatible. We also show how this implies that the energy (i.e. the distance to the identity) does not decrease along contact isotopies that are generated by a non-negative Hamiltonian. In the last section we prove that $d$ is unbounded.
Acknowledgements {#acknowledgements .unnumbered}
----------------
I thank my supervisor Miguel Abreu for his support and mathematical guidance, and Leonid Polterovich for feedback on preliminary versions of this article. My research was supported by an FCT graduate fellowship, program POCTI-Research Units Pluriannual Funding Program through the Center for Mathematical Analysis Geometry and Dynamical Systems and Portugal/Spain cooperation grant FCT/CSIC-14/CSIC/08.
Preliminaries on generating functions {#prel}
=====================================
We start by presenting some preliminaries on generating functions, referring to [@mio] and the bibliography therein for more details and background information.\
\
Let $B$ be a closed manifold, and consider a function $S:E\rightarrow\mathbb{R}$ defined on the total space of a fiber bundle $p:E \longrightarrow B$. We will assume that $dS: E\longrightarrow T^{\ast}E$ is transverse to $N_E:=\{\,(e,\eta)\in T^{\ast}E \;|\; \eta = 0 \;\text{on} \;\text{ker}\,dp\,(e)\,\}$, so that the set $\Sigma_S$ of fiber critical points is a submanifold of $E$, of dimension equal to the dimension of $B$. To any $e$ in $\Sigma_S$ we associate an element $v^{\ast}(e)$ of $T^{\phantom{p}\ast}_{p(e)}B$ by defining $v^{\ast}(e)\,(X):=dS\,(\widehat{X})$ for $X \in T_{p(e)}B$, where $\widehat{X}$ is any vector in $T_eE$ with $p_{\ast}(\widehat{X})=X$. Then $i_S:\Sigma_S\longrightarrow T^{\ast}B$, $e\mapsto \big(p(e),v^{\ast}(e)\big)$ is an exact Lagrangian immersion, with $i_S^{\phantom{S}\ast}\, \lambda_{\text{can}}=d\,(S_{|\Sigma_S})$. Its lift to $\big(J^1B=T^{\ast}B\times\mathbb{R}\,,\,\text{ker}(dz-\lambda_{\text{can}})\big)$ is the Legendrian immersion $j_S:\Sigma_S\rightarrow J^1B$, $e\mapsto\big(p(e), v^{\ast}(e),S(e)\big)$. The function $S:E\rightarrow\mathbb{R}$ is called a *generating function* for $L_S:=i_S\,(\Sigma_S)\subset T^{\ast}B$ and for its lift $\widetilde{L_S}:=j_S(\Sigma_S)\subset J^1B$. A fundamental property of $S$ is that its critical points correspond under $i_S$ to intersections of $L_S$ with the 0-section, and under $j_S$ to intersections of $\widetilde{L_S}$ with the 0-wall. Note also that if $e$ is a critical point of $S$ then its critical value is given by the $\mathbb{R}$-coordinate of the point $j_S(e)$.\
\
A generating function $S:E \longrightarrow \mathbb{R}$ is said to be *quadratic at infinity* if $p:E \longrightarrow B $ is a vector bundle and if there exists a non-degenerate quadratic form $Q_{\infty}: E \longrightarrow \mathbb{R}$ such that $dS-\partial_vQ_{\infty}: E \longrightarrow E^{\ast}$ is bounded, where $\partial_v$ denotes the fiber derivative. Existence of generating functions quadratic at infinity for all Legendrian submanifolds of $J^1B$ contact isotopic to the 0-section was proved by Chaperon [@Ch] and Théret [@Th], and independently by Chekanov [@C]. Their theorem is a generalization of the analogous result for Lagrangian submanifolds of $T^{\ast}B$ Hamiltonian isotopic to the 0-section, that was proved by Sikorav [@S; @S2] using ideas of [@LS] and [@Ch1]. A second fundamental result is the uniqueness theorem for generating functions quadratic at infinity, that is due to Viterbo [@V] and Théret [@Th; @Th2].\
\
Relying on the Uniqueness Theorem, Viterbo [@V] applied Morse theoretical methods to generating functions in order to define invariants for Lagrangian submanifolds of $T^{\ast}B$ Hamiltonian isotopic to the 0-section. As observed by Bhupal [@B], Viterbo’s invariants can also be defined in the more general class $\mathcal{L}$ of Legendrian submanifolds of $J^1B$ contact isotopic to the 0-section. The construction goes as follows. Let $L$ be an element of $\mathcal{L}$ with generating function $S:E\rightarrow \mathbb{R}$. Denote by $E^{a}$, for $a\in \mathbb{R}\cup\infty$, the sublevel set of $S$ at $a$, and by $E^{-\infty}$ the set $E^{-a}$ for $a$ big. We consider the inclusion $i_a: (E^a,E^{-\infty})\hookrightarrow (E,E^{-\infty})$, and the induced map on cohomology $$i_a^{\phantom{a}\ast}: H^{\ast}(B)\equiv H^{\ast}(E,E^{-\infty})\longrightarrow H^{\ast}(E^a,E^{-\infty})$$ where $H^{\ast}(B)$ is identified with $H^{\ast}(E,E^{-\infty})$ via the Thom isomorphism. For any $u\neq 0$ in $H^{\ast}(B)$ we define $$c(u,L):=\text{inf}\,\{\,a\in \mathbb{R} \;|\;i_a^{\phantom{a}\ast}(u)\neq 0\,\}.$$ Note that $c(u,L)$ is a critical value of $S$.
\[inv\_leg\] Let $\mu\in H^n(B)$ denote the orientation class of $B$, and $0_B$ the 0-section in $J^1B$. The map $H^{\ast}(B)\times \mathcal{L}\longrightarrow\mathbb{R}$, $(u,L)\longmapsto c(u,L)$ satisfies the following properties:
1. $$c\big(u\cup v, L_1+L_2\big)\geq c(u,L_1)+c(v,L_2)$$ where $L_1+L_2$ is defined [^3] by $$L_1+L_2:=\{\;(q,p,z)\in J^1B \;|\; p=p_1+p_2, \; z=z_1+z_2, \;(q,p_1,z_1)\in L_1, \; (q,p_2,z_2)\in L_2 \;\}.$$
2. $$c(\mu,\bar{L})=-c(1,L),$$ where $\bar{L}$ denotes the image of $L$ under the map $J^1B\rightarrow J^1B$, $(q,p,z)\mapsto(q,-p,-z)$.
3. Assume that $L\cap0_B\neq\emptyset$. Then $c(\mu, L)=c(1,L)$ if and only if $L$ is the $0$-section. In this case we have $c(\mu,L)=c(1,L)=0$.
The proof of this lemma is purely algebraic topological, and does not require any argument of symplectic or contact topology. It was originally given by Viterbo in the setting of Lagrangian submanifolds of $T^{\ast}B$ Hamiltonian isotopic to the 0-section, but its extension to the contact case is immediate (see [@B] or [@mio]). In the symplectic case, the symplectic character of $c$ is given by the fact that $$\label{V}
c\big(u,\Psi(L)\big)=c\big(u,L-\Psi^{-1}(0_B)\big)$$ for every Hamiltonian symplectomorphism $\Psi$ of $T^{\ast}B$ (see [@V] or [@mio]). The analogue of this result does not hold in the contact case. However Bhupal proved that the following weaker statement is still true.
\[weaker\] For any contactomorphism $\Psi$ of $J^1B$ contact isotopic to the identity, $0\neq u \in H^{\ast}(B)$ and $L\in \mathcal{L}$ it holds $$c\big(u,\Psi(L)\big)=0 \quad\Leftrightarrow \quad c\big(u,L-\Psi^{-1}(0_B)\big)=0.$$
The idea of the proof is to study the bifurcation diagram of a 1-parameter family $S_t$ of generating functions of $\Psi_t^{\phantom{t}-1}\Psi(L)-\Psi_t^{\phantom{t}-1}(0_B)$, where $\Psi_t$ is a contact isotopy connecting $\Psi$ to the identity, and to show that there cannot be a 1-parameter family $c_t$ of critical values of $S_t$ crossing the critical value $0$. The key reason why the critical value $0$ plays a special role is that critical points with critical value $0$ correspond to intersections of the generated Legendrian submanifold with the 0-section. As we observed in [@mio], if $\Psi$ is 1-periodic in the $\mathbb{R}$-coordinate of $J^1B=T^{\ast}B\times\mathbb{R}$ then the argument of Bhupal can also be applied if we replace $0$ by any other integer, to get the following result. We denote by $\lceil\cdot\rceil$ (respectively $\lfloor\cdot\rfloor$) the smallest (respectively largest) integer that is greater or equal (respectively smaller or equal) to the given number.
\[1per\] Let $\Psi$ be a contactomorphism of $J^1B$ which is 1-periodic in the $\mathbb{R}$-coordinate of $J^1B=T^{\ast}B\times\mathbb{R}$, and isotopic to the identity through 1-periodic contactomorphisms. Then for every $u\neq 0$ in $H^{\ast}(B)$ and $L\in\mathcal{L}$ it holds $$\lceil c\big(u,\Psi(L)\big)\rceil=\lceil c\big(u,L-\Psi^{-1}(0_B)\big)\rceil \quad \text{and} \quad \lfloor c\big(u,\Psi(L)\big)\rfloor=\lfloor c\big(u,L-\Psi^{-1}(0_B)\big)\rfloor.$$
The invariants $c^+$ and $c^-$ {#inv}
==============================
The invariants for Lagrangian submanifolds of $T^{\ast}B$ discussed in the previous section where applied by Viterbo [@V] to the special case of a compactly supported Hamiltonian symplectomorphism $\phi$ of $\big(\,\mathbb{R}^{2n}\,,\,\omega=dx\wedge dy\,\big)$, by regarding its compactified graph as a Lagrangian submanifold of $T^{\ast}S^{2n}$. Viterbo obtained in this way two invariants $c^+(\phi)$ and $c^-(\phi)$, defined by using respectively the orientation and unit cohomology classes of $S^{2n}$. The invariants $c^+$ and $c^-$ were generalized in [@B] and [@mio] respectively to the case of compactly supported contactomorphisms of $\mathbb{R}^{2n+1}$ and $\mathbb{R}^{2n}\times
S^1$. We will review in this section the construction and properties of $c^+$ and $c^-$, discussing directly the contact case[^4].\
\
Let $\phi$ be a contactomorphism of $\big(\mathbb{R}^{2n+1},\xi_0=\text{ker}\,(dz-ydx)\big)$, with $\phi^{\ast}(dz-ydx)=e^g(dz-ydx)$. Following Bhupal [@B], we define a Legendrian embedding $\Gamma_{\phi}:\mathbb{R}^{2n+1}\longrightarrow J^1\mathbb{R}^{2n+1}$ to be the composition $\tau \circ \text{gr}_{\phi}$, where $\text{gr}_{\phi}:\mathbb{R}^{2n+1}\longrightarrow \mathbb{R}^{2(2n+1)+1}$ is the map $q\mapsto (q,\phi(q),g(q))$ and $\tau:\mathbb{R}^{2(2n+1)+1}\longrightarrow J^1\mathbb{R}^{2n+1}$ the contact embedding $(x,y,z,X,Y,Z,\theta)\mapsto \big(x,Y,z, Y-e^{\theta}y, x-X, e^{\theta}-1, xY-XY+Z-z\big)$. Note that $\Gamma_{\phi}$ can also be written as $\Gamma_{\phi}=\Psi_{\phi}\,(\text{0-section})$ where $\Psi_{\phi}$ is the local contactomorphism of $J^1\mathbb{R}^{2n}$ defined by the diagram $$\xymatrix{
\quad \mathbb{R}^{2(2n+1)+1} \quad \ar[r]^{\overline{\phi}} \ar[d]_{\tau} &
\quad \mathbb{R}^{2(2n+1)+1} \quad \ar[d]^{\tau} \\
\quad J^1\mathbb{R}^{2n+1} \quad \ar[r]_{\Psi_{\phi}} & \quad J^1\mathbb{R}^{2n+1}}\quad$$ with $\overline{\phi}$ the contactomorphism $(p,P,\theta)\mapsto (p,\phi(P), g(P)+\theta)$. This shows in particular that $\Gamma_{\phi}$ is contact isotopic to the 0-section. Notice also that the diagram above behaves well with respect to composition: for all contactomorphisms $\phi$, $\phi_1$ and $\phi_2$ we have namely that $\Psi_{\phi_1}\circ\Psi_{\phi_2}=\Psi_{\phi_1\phi_2} $ (in particular $\Gamma_{\phi_1\,\circ\,\phi_2}=\Psi_{\phi_1}\,(\Gamma_{\phi_2})$) and $\Psi_{\phi}^{\phantom{\phi}-1}=\Psi_{\phi^{-1}}$.\
\
If $\phi$ is compactly supported then we can see $\Gamma_{\phi}$ as a Legendrian submanifold of $J^1S^{2n+1}$, thus we can associate to it a generating function. The same is true if $\phi$ is a contactomorphism of $\mathbb{R}^{2n+1}$ which is 1-periodic in the $z$-coordinate and compactly supported in the $(x,y)$-plane (i.e. a compactly supported contactomorphism of $\mathbb{R}^{2n}\times S^1$) because then we can see $\Gamma_{\phi}$ as a Legendrian submanifold of $J^1\big(S^{2n}\times S^1\big)$. We denote respectively by $\text{Cont}_0^{\phantom{0}c}\,(\mathbb{R}^{2n+1})$ and $\text{Cont}_0^{\phantom{0}c}\,(\mathbb{R}^{2n}\times S^1)$ the groups of compactly supported contactomorphisms of $\mathbb{R}^{2n+1}$ and $\mathbb{R}^{2n}\times S^1$ that are isotopic to the identity. In the following we will always regard contactomorphisms of $\mathbb{R}^{2n}\times S^1$ as 1-periodic contactomorphisms of $\mathbb{R}^{2n+1}$.\
\
For $\phi$ in $\text{Cont}_0^{\phantom{0}c}\,(\mathbb{R}^{2n+1})$ or in $\text{Cont}_0^{\phantom{0}c}\,(\mathbb{R}^{2n}\times S^1)$ we define $$c^+(\phi):=c(\mu,\Gamma_{\phi})$$ $$c^-(\phi):=c(1,\Gamma_{\phi})$$ where $\mu$ and $1$ are respectively the orientation and the unit cohomology class either of $S^{2n+1}$ or of $S^{2n}\times S^1$. Note that $c^+(\phi)$ and $c^-(\phi)$ are critical values for any generating function of $\Gamma_{\phi}$. Moreover, exactly as in the symplectic case, they satisfy the following properties.
\[inv\_sympl\] For all $\phi$, $\psi$ in $\text{Cont}_0^{\phantom{0}c}\,(\mathbb{R}^{2n+1})$ or $\text{Cont}_0^{\phantom{0}c}\,(\mathbb{R}^{2n}\times S^1)$ it holds:
1. $c^+(\phi)\geq 0$ and $c^-(\phi)\leq 0$.
2. $c^+(\phi)=c^-(\phi)= 0$ if and only if $\phi$ is the identity.
Point (ii) is an immediate consequence of Lemma \[inv\_leg\](iii). As for (i), it can be seen as follows [@V]. We will prove that $c(1,\Gamma_{\phi})\leq 0$ and $c(1,\overline{\Gamma_{\phi}})\leq 0$ for any $\phi$, so that $c^-(\phi)\leq 0$ and, using Lemma \[inv\_leg\](ii), $ c^+(\phi)=c(\mu,\Gamma_{\phi})=-c(1,\overline{\Gamma_{\phi}}) \geq 0$. Since $c(1,\Gamma_{\phi})=\text{inf}\,\{\,a\in \mathbb{R} \;|\;i_{a}^{\phantom{a}\ast}(1)\neq 0\,\}$, we need to prove that $i_0^{\phantom{0}\ast}(1)\neq 0$. Let $S: E\rightarrow\mathbb{R}$ be a g.f.q.i. for $\Gamma_{\phi}$ (respectively $\overline{\Gamma_{\phi}}$) and take a point $P$ in $B$, where $B$ denotes either $S^{2n+1}$ or $S^{2n}\times S^1$, outside the support of $\phi$. Consider the commutative diagram $$\xymatrix{
H^{\ast}(E^0,E^{-\infty}) \ar[r] &
H^{\ast}(E_P^{\phantom{P}0},E_P^{\phantom{P}-\infty}) \\
H^{\ast}(B) \ar[r] \ar[u]_{(i_0)^{\ast}} &
H^{\ast}(\{P\}) \ar[u]_{\cong}}$$ where the horizontal maps are induced by the inclusions $\{P\}\hookrightarrow B$ and $E_P\hookrightarrow E$. Since $P$ is outside the support of $\phi$ we have that $\Gamma_{\phi}$ and $\overline{\Gamma_{\phi}}$ coincide with the 0-section on a neighborhood of $P$, and so $S_{|E_P}: E_P \rightarrow \mathbb{R}$ is a quadratic form. It follows that the vertical map on the right hand side is an isomorphism. Since the horizontal map on the bottom sends 1 to 1, we see that $i_0^{\phantom{0}\ast}(1)\neq 0$ as we wanted.
In the symplectic case the relation (\[V\]), together with the properties in Lemma \[inv\_leg\], implies that for every $\phi$, $\psi$ in $\text{Ham}^c\,(\mathbb{R}^{2n})$ we have $c^-(\phi)=-c^+(\phi^{-1})$, $c^+(\phi\psi)\leq c^+(\phi) + c^+(\psi)$ and $c^-(\phi\psi)\geq c^-(\phi) + c^-(\psi)$ (see [@V] or [@mio]). In the contact 1-periodic case we only get the following weaker statement, using Lemma \[1per\].
\[inv\_per\] For all $\phi$, $\psi$ in $\text{Cont}_0^{\phantom{0}c}\,(\mathbb{R}^{2n}\times S^1)$ it holds:
1. $\lfloor c^-(\phi)\rfloor=-\lceil c^+(\phi^{-1})\rceil$.
2. $\lceil c^+(\phi\psi)\rceil\leq\lceil c^+(\phi)\rceil+\lceil c^+(\psi)\rceil$ and $\lfloor c^-(\phi\psi)\rfloor\geq\lfloor c^-(\phi)\rfloor+\lfloor c^-(\psi)\rfloor$.
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1. Note first that $\lceil c(u,\Gamma_{\phi^{-1}})\rceil=\lceil c(u,\overline{\Gamma_{\phi}})\rceil$ for all $u$ (apply Lemma \[1per\] to $L=0_B$ and $\Psi=\Psi_{\phi^{-1}}$). Using this and Lemma \[inv\_leg\](ii) we have $$\lfloor c^-(\phi)\rfloor =\lfloor c(1,\Gamma_{\phi}) \rfloor=-\lceil c(\mu,\overline{\Gamma_{\phi}})\rceil=-\lceil c(\mu,\Gamma_{\phi^{-1}})\rceil=-\lceil c^+(\phi^{-1})\rceil.$$
2. We have $ c^+(\psi)=c(\mu,\Gamma_{\psi})=c\big(\mu, \Psi_{\phi^{-1}}(\Gamma_{\phi\psi})\big)$ thus by Lemma \[1per\] it holds $\lceil c^+(\psi)\rceil=\lceil c\big(\mu,\Gamma_{\phi\psi}-\Psi_{\phi}(0_B)\big)\rceil$. But, by Lemma \[inv\_leg\](i)-(ii) $$c\big(\mu,\Gamma_{\phi\psi}-\Psi_{\phi}(0_B)\big)\geq
c\big(\mu,\Gamma_{\phi\psi}\big)+c\big(1,\overline{\Gamma_{\phi}}\big)=
c^+(\phi\psi)-c^+(\phi).$$ Thus $$\lceil c^+(\psi) \rceil \geq \lceil c^+(\phi\psi)-c^+(\phi) \rceil \geq \lceil c^+(\phi\psi)\rceil-\lceil c^+(\phi) \rceil$$ as we wanted. The statement about $c^-$ follows now from (i).
Similarly, in the case of $\mathbb{R}^{2n+1}$ we can use Lemma \[weaker\] to show that $c^-(\phi)=0$ if and only if $c^+(\phi^{-1})=0$ and that if $c^{\pm}(\phi)=c^{\pm}(\psi)= 0$ then $c^{\pm}(\phi\psi)= 0$ (see [@B]).
A fundamental property of $c^+$ and $c^-$ in the symplectic case is that they are invariant by conjugation, i.e. $c^{\pm}(\phi)=c^{\pm}(\psi\phi\psi^{-1})$ for all $\phi$, $\psi$ in $\text{Ham}^c\,(\mathbb{R}^{2n})$ (see [@V] or [@mio]). This is a consequence of the fact that the set of critical values of the generating function of a Hamiltonian symplectomorphism $\phi$ of $\mathbb{R}^{2n}$ coincides with the action spectrum of $\phi$, which is invariant by conjugation: if $q$ is a fixed point of $\phi$ then $\psi(q)$ is a fixed point of $\psi\phi\psi^{-1}$ with the same symplectic action. This crucial fact does not hold in the contact case. Given a contactomorphism $\phi=(\phi_1,\phi_2,\phi_3)$ of $\mathbb{R}^{2n+1}$ with $\phi^{\ast}(dz-ydx)=e^g(dz-ydx)$, the critical points of a generating function of $\phi$ coincide with the *translated points* of $\phi$, i.e. the points $q=(x,y,z)$ such that $\phi_1(q)=x$, $\phi_2(q)=y$ and $g(q)=0$. Moreover, the critical value is given by the *contact action* of the corresponding translated point, i.e. the value $\phi_3(q)-z$ (see [@B] or [@mio]). Note that the contact action is not invariant by conjugation. In fact, not even the property of being a translated point is invariant by conjugation: if $q$ is a translated point of $\phi$ then in general $\psi(q)$ is not a translated point of $\psi\phi\psi^{-1}$. However this is the case if the contact action is $0$, because translated points with contact action $0$ are fixed points of $\phi$. This fact has been used by Bhupal to prove that, for all $\phi$, $\psi$ in $\text{Cont}_0^{\phantom{0}c}\,(\mathbb{R}^{2n+1})$, $c^{\pm}(\phi)=0$ if and only if $c^{\pm}(\psi\phi\psi^{-1})=0$. As for Lemma \[weaker\], the idea of the proof is to study the bifurcation diagram of a 1-parameter family $S_t$ of generating functions for $\psi_t\phi\psi_t^{\phantom{t}-1}$, where $\psi_t$ is a contact isotopy connecting $\psi$ to the identity, and to show that there can be no path $c_t$ of critical values for $S_t$ crossing the critical value $0$. As observed in [@mio], in the 1-periodic case the same argument can also be applied if we replace $0$ by any other integer, to show that the integer part of $c^+$ and $c^-$ is invariant by conjugation.
\[conj\_mio\] For all $\phi$, $\psi$ in $\text{Cont}_0^{\phantom{0}c}\,(\mathbb{R}^{2n}\times S^1)$ it holds that $\lceil c^{\pm}(\phi)\rceil=\lceil c^{\pm}(\psi\phi\psi^{-1})\rceil$ and $\lfloor c^{\pm}(\phi)\rfloor=\lfloor c^{\pm}(\psi\phi\psi^{-1})\rfloor$.
\[lift\] Every Hamiltonian symplectomorphism $\varphi$ of $\mathbb{R}^{2n}$ can be lifted to a contactomorphism $\widetilde{\varphi}$ of $\mathbb{R}^{2n+1}$ or $\mathbb{R}^{2n}\times S^1$ by defining $\widetilde{\varphi}(x,y,z)=\big(\varphi_1(x,y),\varphi_2(x,y), z+F(x,y)\big)$ where $\varphi=(\varphi_1,\varphi_2)$ and $F$ is the compactly supported function satisfying $\varphi^{\ast}(ydx)-ydx=dF$. It can be proved (see [@mio]) that $c^+(\widetilde{\varphi})=c^+(\varphi)$ and $c^-(\widetilde{\varphi})=c^-(\varphi)$.
The bi-invariant metric $d$ on $\text{Cont}_0^{\;c}\,(\mathbb{R}^{2n}\times S^1)$ {#metric}
=================================================================================
In [@V] Viterbo used the invariants $c^+$ and $c^-$ to construct a bi-invariant partial order $\leq_V$ and a bi-invariant metric $d_V$ on $\text{Ham}^c\,(\mathbb{R}^{2n})$, and a symplectic capacity for domains in $\mathbb{R}^{2n}$. Bhupal showed in [@B] that the weaker properties of $c^+$ and $c^-$ that are still satisfied in the case of $\text{Cont}_0^{\phantom{0}c}\,(\mathbb{R}^{2n+1})$ are in fact enough to extend the Viterbo partial order to that group. Note that, by Lemma \[inv\_per\] and Lemma \[conj\_mio\], these same properties are also satisfied by elements of $\text{Cont}_0^{\phantom{0}c}\,(\mathbb{R}^{2n}\times S^1)$ so that Bhupal’s contruction can be applied to the case of $\mathbb{R}^{2n}\times S^1$ as well (see [@mio]). However, we will now show that Lemma \[inv\_per\] and Lemma \[conj\_mio\], that are only available in the 1-periodic case, allow us to extend also the Viterbo metric to $\text{Cont}_0^{\phantom{0}c}\,(\mathbb{R}^{2n}\times S^1)$.\
\
Recall that the Viterbo metric on $\text{Ham}^c\,(\mathbb{R}^{2n})$ is defined by $d_V\,(\phi,\psi):=c^+(\phi\psi^{-1})-c^-(\phi\psi^{-1})$. Similarly, our metric $d$ on $\text{Cont}_0^{\phantom{0}c} (\mathbb{R}^{2n}\times S^1)$ is defined by $$d\,(\phi,\psi):=\lceil c^+(\phi\psi^{-1})\rceil-\lfloor c^-(\phi\psi^{-1})\rfloor.$$
\[metric\_c\] $d$ is a bi-invariant metric on $\text{Cont}_0^{\phantom{0}c} (\mathbb{R}^{2n}\times S^1)$, i.e.
1. (positivity) $d\,(\phi,\psi)\geq 0$ for all $\phi$, $\psi$.
2. (non-degeneracy) $d\,(\phi,\psi)=0$ if and only if $\phi=\psi$.
3. (symmetry) $d\,(\phi,\psi)=d\,(\psi,\phi)$.
4. (triangle inequality) $d\,(\phi,\varphi)\leq d\,(\phi,\psi) + d\,(\psi,\varphi)$
5. (bi-invariance) $d\,(\phi\varphi,\psi\varphi)=d\,(\varphi\phi,\varphi\psi)=d\,(\phi,\psi)$.
Positivity and symmetry follow from Lemma \[inv\_sympl\](i) and Lemma \[inv\_per\](i) respectively. Using Lemma \[inv\_per\](ii) we have $$\begin{aligned}
d(\phi,\varphi)&=&\lceil c^+(\phi\varphi^{-1})\rceil-\lfloor c^-(\phi\varphi^{-1})\rfloor=
\lceil c^+(\phi\psi^{-1}\psi\varphi^{-1})\rceil-\lfloor c^-(\phi\psi^{-1}\psi\varphi^{-1})\rfloor\\
&\leq & \lceil c^+(\phi\psi^{-1})\rceil+\lceil c^+(\psi\varphi^{-1})\rceil-\lfloor c^-(\phi\psi^{-1})\rfloor-\lfloor c^-(\psi\varphi^{-1})\rfloor=
d\,(\phi,\psi) + d\,(\psi,\varphi)\end{aligned}$$ proving the triangle inequality. By Lemma \[inv\_sympl\](ii) we have $c^+(\text{id})=c^-(\text{id})=0$, thus $d\,(\phi,\phi)=0$. Suppose now that $d\,(\phi,\psi)=0$. Then, because of Lemma \[inv\_sympl\](i), we must have $c^+(\phi\psi^{-1})=c^-(\phi\psi^{-1})=0$ and so $\phi=\psi$ by Lemma \[inv\_sympl\](ii). This proves non-degeneracy. As for bi-invariance, we have $$d\,(\phi\varphi,\psi\varphi)=
\lceil c^+(\phi\varphi\varphi^{-1}\psi^{-1})\rceil -\lfloor c^-(\phi\varphi\varphi^{-1}\psi^{-1})\rfloor=
\lceil c^+(\phi\psi^{-1})\rceil-\lfloor c^-(\phi\psi^{-1})\rfloor=d\,(\phi,\psi)$$ and, by Lemma \[conj\_mio\], $$d\,(\varphi\phi,\varphi\psi)=
\lceil c^+(\varphi\phi\psi^{-1}\varphi^{-1})\rceil -\lfloor c^-(\varphi\phi\psi^{-1}\varphi^{-1})\rfloor=
\lceil c^+(\phi\psi^{-1})\rceil -\lfloor c^-(\phi\psi^{-1})\rfloor=d\,(\phi,\psi).$$
The **energy** of an element $\phi$ of $\text{Cont}_0^{\phantom{0}c} (\mathbb{R}^{2n}\times S^1)$ is defined to be its distance to the identity, i.e. $$E(\phi):=\lceil c^+(\phi)\rceil-\lfloor c^-(\phi)\rfloor.$$ Given an open and bounded domain $\mathcal{V}$ of $\mathbb{R}^{2n}\times S^1$, its **displacement energy** is defined by $$E(\mathcal{V}):=\text{inf}\;\{\; E(\psi) \;|\; \psi(\mathcal{V})\cap\mathcal{V}= \emptyset\;\}.$$ This definition can be extended to arbitrary domains of $\mathbb{R}^{2n}\times S^1$ by setting $E(\mathcal{U})=\text{sup}\,\{\,E(\mathcal{V}) \;|\; \mathcal{V}\subset\mathcal{U}, \:\mathcal{V} \:\text{bounded}\,\}$ if $\mathcal{U}$ is open, and $ E(A)=\text{inf}\,\{\,E(\mathcal{U}) \;|\; \mathcal{U} \:\text{open,}\;A\subset\mathcal{U}\,\}$ for an arbitrary domain $A$. In [@mio] we extended the Viterbo capacity to domains of $\mathbb{R}^{2n}\times S^1$ by defining $c(\mathcal{V})=\text{sup}\,\{\,\lceil c^+(\phi)\rceil \:|\: \phi\in\text{Cont}\,(\mathcal{V})\,\}$ where $\text{Cont}\,(\mathcal{V})$ denotes the set of time-1 maps of contact Hamiltonians supported in $\mathcal{V}$. The **energy-capacity inequality** $$c(\mathcal{V})\leq E(\mathcal{V})$$ follows from [@mio 3.6.1].
Relation with the Bhupal partial order {#order}
======================================
Recall from [@B] and [@mio] that, similarly to the symplectic case, the **Bhupal partial order** $\leq_B$ on $\text{Cont}_0^{\phantom{0}c}\,(\mathbb{R}^{2n+1})$ and $\text{Cont}_0^{\phantom{0}c}\,(\mathbb{R}^{2n}\times S^1)$ is defined by $$\phi_1 \leq_B \phi_2 \quad \text{if} \quad c^+(\phi_1\phi_2^{\phantom{2}-1})=0.$$ We will now show that the metric $d$ and the partial order $\leq_B$ are compatible.
\[c\_part\_ord\_metric\] $\big(\,\text{Cont}_0^{\phantom{0}c} (\mathbb{R}^{2n}\times S^1),d,\leq_B\big)$ is a partially ordered metric space.
Suppose that $\phi_1\leq_B\phi_2\leq_B\phi_3$. Then, since $c^+(\phi_1\phi_2^{\phantom{2}-1})=c^+(\phi_2\phi_3^{\phantom{3}-1})=c^+(\phi_1\phi_3^{\phantom{3}-1})=0$, using Lemma \[inv\_per\](i) and (ii) we get $$\begin{aligned}
d(\phi_1,\phi_2)&=&\lceil c^+(\phi_1\phi_2^{\phantom{2}-1})\rceil-\lfloor c^-(\phi_1\phi_2^{\phantom{2}-1})\rfloor\\
&=&-\lfloor c^-(\phi_1\phi_2^{\phantom{2}-1})\rfloor=\lceil c^+(\phi_2\phi_1^{\phantom{1}-1})\rceil \leq \lceil c^+(\phi_2\phi_3^{\phantom{3}-1})\rceil+\lceil c^+(\phi_3\phi_1^{\phantom{1}-1})\rceil=\lceil c^+(\phi_3\phi_1^{\phantom{1}-1})\rceil\\
&=&
\lceil c^+(\phi_1\phi_3^{\phantom{3}-1})\rceil+\lceil c^+(\phi_3\phi_1^{\phantom{1}-1})\rceil=\lceil c^+(\phi_1\phi_3^{\phantom{3}-1})\rceil-\lfloor c^-(\phi_1\phi_3^{\phantom{3}-1})\rfloor=d(\phi_1,\phi_3)\end{aligned}$$ i.e. $d(\phi_1,\phi_2)\leq d(\phi_1,\phi_3)$.
Consider now the relation $\preceq$ on $\text{Cont}_0^{\phantom{0}c} (\mathbb{R}^{2n}\times S^1)$ or $\text{Cont}_0^{\phantom{0}c} (\mathbb{R}^{2n+1})$ defined by setting $\phi_1\preceq\phi_2$ if $\phi_2\phi_1^{-1}$ can be written as the time-1 flow of a non-negative Hamiltonian. This relation is clearly reflexive and transitive. The deep fact that $\preceq$ is also anti-symmetric (hence a partial order) follows from antisymmetry of $\leq_B$ and the implication $$\label{e}
\phi_1\preceq\phi_2 \quad \Rightarrow \quad \phi_1\leq_B\phi_2.$$ In the language of Eliashberg and Polterovich [@EP], antisymmetry of $\preceq$ proves that $\mathbb{R}^{2n}\times S^1$ and $\mathbb{R}^{2n}\times S^1$ are orderable contact manifolds. The implication (\[e\]) can easily be proved using the fact that $c^+$ and $c^-$ are monotone with respect to $\preceq$ : if $\phi_1\preceq\phi_2$ then $c^+(\phi_1)\leq c^+(\phi_2)$ and $c^-(\phi_1)\leq c^-(\phi_2)$ (see [@B] or [@mio]).\
\
Notice that (\[e\]) and Proposition \[c\_part\_ord\_metric\] immediately imply that $\big(\,\text{Cont}_0^{\phantom{0}c} (\mathbb{R}^{2n}\times S^1),d,\preceq\big)$ is a also partially ordered metric space, so that in particular the energy does not decrease along non-negative contact isotopies.
Unboundedness of $d$
====================
We will now show that the diameter of $\text{Cont}_0^{\phantom{0}c} (\mathbb{R}^{2n}\times S^1)$ with respect to our metric is infinite, or in other words that $d$ is unbounded. As we will see, this fact follows immediately from the analogous result in the symplectic case and Remark \[lift\].\
\
Unboundedness of the Viterbo metric on $\text{Ham}^c\,(\mathbb{R}^{2n})$ is well known. It can be seen for instance by considering the sequence of Hamiltonian symplectomorphisms of $\mathbb{R}^{2n}$ supported in $B^{2n}(R)= \{\,\pi \sum_{i=1}^n x_i^{\,2} + y_i^{\,2} <R\,\}$ that was constructed by Traynor in [@T] to calculate the symplectic homology of $B^{2n}(R)$, and by noticing that the energy of the elements of this sequence tends to $R$ (that can be chosen to be arbitrarily big). Traynor’s sequence $\phi^{\rho_1}\preceq\phi^{\rho_2}\preceq\phi^{\rho_3}\preceq\cdots$ is constructed as follows. Let $H:\mathbb{R}^{2n}\rightarrow\mathbb{R}$ be the function $H(x_1,y_1,\cdots,x_n,y_n)=\sum_{i=1}^{n} \frac{\pi}{R}(x_i^{\phantom{i}2}+y_i^{\phantom{i}2})$ and consider $H_{\rho}=\rho\circ H$, where $\rho:[0,\infty)\rightarrow [0,\infty)$ is a function supported in $[0,1]$ with $\rho''>0$. Take then a sequence $\rho_1$, $\rho_2$, $\rho_3$, $\cdots$ of functions of this form, with $\lim_{i \to \infty} \rho_i(0) = \infty$, $\lim_{i \to \infty} \rho'_i(0) = -\infty$, and such that $H_{\rho_1}\leq H_{\rho_2}\leq H_{\rho_3}\leq\cdots$ with $H_{\rho_i}$ getting pointwise arbitrarily big on $B^{2n}(R)$. Since the $H_{\rho_i}$ are positive, by monotonicity of $c^-$ we have that $c^-(\phi^{\rho_i})=0$ and thus $E(\phi^{\rho_i})=c^+(\phi^{\rho_i})$. Moreover, it was proved by Traynor [@T] that $c^+(\phi^{\rho_i})$ tends to $R$ for $i\rightarrow \infty$. If we now lift the sequence $\phi^{\rho_1}\leq\phi^{\rho_2}\leq\phi^{\rho_3}\leq\cdots$ to $\mathbb{R}^{2n}\times S^1$ as explained in Remark \[lift\], we get a sequence of contactomorphisms whose energy tends to the integer part of $R$, which can be chosen arbitrarily big. It follows thus that $\text{Cont}_0^{\phantom{0}c} (\mathbb{R}^{2n}\times S^1)$ is also not bounded.\
\
The following terminology is taken from [@BIP]. Two norms on a group $G$ are said to be *equivalent* if their ratio is bounded away from $0$ and $\infty$. In particular, a norm $ \nu$ on $G$ is equivalent to the trivial one (i.e. the norm that is everywhere $1$ except at the identity) if and only if it is bounded and not *fine*. A norm $\nu$ on $G$ is called fine if $0$ is a limit point of $\nu(G)$. Since the metric $d$ on $\text{Cont}_0^{\phantom{0}c} (\mathbb{R}^{2n}\times S^1)$ takes values in $\mathbb{Z}$, it is not fine. However, being unbounded, it is not equivalent to the trivial one. An unbounded norm $\nu$ on a group $G$ is called *stably unbounded* if $\lim_{n \to \infty} \frac{\nu(f^n)}{n}\neq 0$ for some $f$ in $G$. Note that, by definition of the capacity $c$, for every $\phi$ in $\text{Cont}_0^{\phantom{0}c} (\mathbb{R}^{2n}\times S^1)$ which is the time-1 flow of a Hamiltonian supported in $\mathcal{V}$ we have that $E(\phi)=\lceil c^+(\phi)\rceil + \lceil c^+(\phi^{-1})\rceil\leq 2\,c(\mathcal{V})$. If $\phi$ is generated by a Hamiltonian supported in $\mathcal{V}$ then so is $\phi^n$ as well, thus $\lim_{n \to \infty} \frac{E(\phi^n)}{n}= 0$ for all $\phi$. Hence $\text{Cont}_0^{\phantom{0}c} (\mathbb{R}^{2n}\times S^1)$ is not stably unbounded. Similarly, $\text{Ham}^c\,(\mathbb{R}^{2n})$ is unbounded but not stably unbounded with respect to the Viterbo metric[^5].\
\
We conclude by discussing the case of the contact manifold $\big(S^1,\text{ker}(dz)\big)$, that can be seen as $\big(\mathbb{R}^{2n}\times S^1,\text{ker}(dz-ydx)\big)$ for $n=0$. It was proved in [@BIP] that the diffeomorphism group of $S^1$ does not admit any non-trivial (up to equivalence) bi-invariant metric. We will now explain why the construction of our metric on $\mathbb{R}^{2n}\times S^1$ does not contradict this result. Namely, we will see that although the definition of $d$ given in Section \[metric\] also applies to $S^1$, the outcome is not metric in this case. Note that for a diffeomorphism $\phi$ of $S^1$ it holds that $\phi^{\ast}dz=\phi'dz$, thus $\phi$ is a contactomorphism if and only if $\phi'>0$ i.e. if and only if $\phi$ is orientation preserving. Let $\phi$ be an orientation preserving diffeomorphism of $S^1$, regarded as a 1-periodic diffeomorphism of $\mathbb{R}$. We can then associate to $\phi$ a Legendrian submanifold $\Gamma_{\phi}$ of $J^1\mathbb{R}$ by defining $\Gamma_{\phi}=\tau\circ\text{gr}_{\phi}$, where $\tau:\mathbb{R}^3\rightarrow J^1\mathbb{R}$ is the contact embedding $$\tau(z,Z,\theta)=\big(\,z,e^{\theta}-1,Z-z\,\big).$$ Thus $\Gamma_{\phi}:\mathbb{R}\rightarrow J^1\mathbb{R}$ is given by $\Gamma_{\phi}(q)=\big(\,q,\phi'(q)-1,\phi(q)-q\,\big)$. We have that $\Gamma_{\phi}$ is the 1-jet of the function $S:\mathbb{R}\rightarrow\mathbb{R}$, $S(q)=\phi(q)-q$ thus in other words $S$ is a generating function for $\phi$. Notice that $c^+(\phi)=\text{max}(S)$ is not necessarily non-negative, and $c^-(\phi)=\text{min}(S)$ not necessarily non-positive. Hence if we define $d$ as in Section \[metric\] we do not get a metric in this case, because positivity fails. Notice that the proof of non-negativity of $c^+(\phi)$ and non-positivity of $c^-(\phi)$ in Lemma \[inv\_sympl\](i) used in a crucial way the possibility of choosing a point outside the support of $\phi$. This cannot be done in general in the $S^1$ case.
[99]{}
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[^1]: It was proved by Bialy and Polterovich [@BP] that the Viterbo and the Hofer metric coincide on a neighborhood of the identity in $\text{Ham}^c\,(\mathbb{R}^{2n})$, but as far as I know it is still an open question whether or not they coincide on the whole $\text{Ham}^c\,(\mathbb{R}^{2n})$.
[^2]: Recall that a *partially ordered metric space* is a metric space $(Z,d)$ endowed with a partial order $\leq$ such that for every $a$, $b$, $c$ in $Z$ with $a\leq b\leq c$ it holds $d(a,b)\leq d(a,c)$.
[^3]: Note that $L_1+L_2$ is not necessarily a smooth submanifold. However it still has a generating function, given in terms of the generating functions of $L_1$ and $L_2$ (see [@V] or [@B]).
[^4]: Although $c^-$ did not appear in [@mio] it can be treated exactly as $c^+$, which is what in [@mio] we just called $c$.
[^5]: The same also holds for $\text{Ham}^c\,(\mathbb{R}^{2n})$ with respect to the Hofer metric [@S3].
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We study the magnetization and the spin dynamics of the Cr$_7$Ni ring-shaped magnetic cluster. Measurements of the magnetization at high pulsed fields and low temperature are compared to calculations and show that the spin Hamiltonian approach provides a good description of Cr$_7$Ni magnetic molecule. In addition, the phonon-induced relaxation dynamics of molecular observables has been investigated. By assuming the spin-phonon coupling to take place through the modulation of the local crystal fields, it is possible to evaluate the decay of fluctuations of two generic molecular observables. The nuclear spin-lattice relaxation rate $1/T_1$ directly probes such fluctuations, and allows to determine the magnetoelastic coupling strength.'
address:
- '$^1$Dipartimento di Fisica, Università di Parma, Viale Usberti 7/A, I-43100 Parma, Italy; and S$^3$-CNR-INFM, I-41100 Modena, Italy'
- '$^2$Division of Physics, Hokkaido University, Sapporo 060-0810, Japan'
- '$^3$Department of Chemistry, Kyoto University, Kyoto 606-8502, Japan; and CREST Japan Science and Technology (JST)'
- '$^4$The Institute for Solid State Physics, The University of Tokyo, Kashiwa, Chiba 277-8581, Japan'
- '$^5$Dipartimento di Fisica A. Volta, Università di Pavia; and CNR-INFM, Via Bassi 6, I-27100 Pavia, Italy'
- '$^6$Istituto di Fisologia Generale e Chimica Biologica, Università di Milano, I-20134 Milano, Italy; CNR-INFM, I-27100 Pavia, Italy; and S$^3$-CNR-INFM, I-41100 Modena, Italy'
- '$^7$School of Chemistry, University of Manchester, Oxford Road, Manchester, M13 9PL United Kingdom'
author:
- 'A. Bianchi$^1$, S. Carretta$^1$, P. Santini$^1$, G. Amoretti$^1$, Y. Furukawa$^2$, K. Kiuchi$^2$, Y. Ajiro$^3$, Y. Narumi$^4$, K. Kindo$^4$, J. Lago$^5$, E. Micotti$^5$, P. Arosio$^6$, A. Lascialfari$^{5,6}$, F. Borsa$^5$, G. Timco$^7$ and R. E. P. Winpenny$^7$'
title: 'Magnetization and spin dynamics of a Cr-based magnetic cluster: Cr$_{7}$Ni'
---
Introduction {#Intro}
============
The great efforts devoted to the synthesis and investigation of nanosize magnetic molecules are motivated both by interests in fundamental physics and by the envisaged technological applications. For instance, some of these systems have shown phenomena such as quantum tunneling of magnetization between quasi-degenerate levels, slow relaxation at low $T$, and revealed to be promising for high density information storage and quantum computing [@Gatteschi; @Sessoli; @Leuenberger]. The magnetic core of molecular magnets is constituted by transition metal ions sorrounded by an organic shell which prevents intramolecular magnetic interactions. As a result, the microscopic properties of these nanoscale clusters can be investigated by means of bulk samples. Among these systems, there are homonuclear antiferromagnetic (AF) ring-shaped molecules formed by $n$ transition metal ions in an almost planar ring. In particular, in even membered rings the dominant AF exchange interactions lead to a singlet $S_T=0$ ground state and the energy spectrum is characterized by rotational bands, with the lowest-lying levels approximately following the so-called Landé’s rule [@Gatteschi]. In this paper we study the magnetization and the phonon-induced relaxation in the heterometallic ring Cr$_7$Ni. This compound derives from the even membered AF ring Cr$_8$ and thus provides a opportunity of a deeper insight in the role of topology in the static and dynamical quantum properties of magnetic wheels [@CarrettaTopol; @Furukawa]. Cr$_7$Ni compound is obtained by the chemical substitution of a Cr$^{3+}$ ion with a Ni$^{2+}$ ion in the structure of the Cr$_8$ ring. This leads to a new molecular system formed by an odd number of unpaired electrons with dominant AF nearest neighbour interactions as inferred by susceptibility measurements [@Larsen]. The resulting $S_T=1/2$ ground state has been shown to be suitable to encode a qubit [@Troiani].
Magnetization
=============
The magnetic molecule has been theoretically analyzed within a spin Hamiltonian approach, with the Hamiltonian given by: $$\begin{aligned}
H=\sum_{i>j}{J_{ij}\textbf{s}(i)\cdot\textbf{s}(j)}+\sum_i{d_i\left[s^{2}_{z}(i)-s_i(s_i+1)/3\right]}\nonumber\\
+\sum_{i>j}\textbf{s}(i)\cdot\textbf{D}_{ij}\cdot\textbf{s}(j)-\mu_B\sum_i{g_{i}\textbf{H}\cdot\textbf{s}(i)},\end{aligned}$$ where $\textbf{s}(i)$ is the spin operator of the the $i$th ion in the molecule ($s(i)$=3/2 for Cr$^{3+}$ ions, and $s(i)$=1 for the Ni$^{2+}$ ion). The first term of the above equation is the dominant nearest neighbour Heisenberg exchange interaction. The second and the third terms describe the uniaxial local crystal fields and anisotropic intracluster spin-spin interactions respectively (with the $z$ axis assumed perpendicular to the plane of the ring). The last term represents the Zeeman coupling with an external field $\textbf{H}$. The parameters of the above Hamiltonian were determined by inelastic neutron scattering (INS) experiments [@Caciuffo; @CarrettaPRL]. In order to corroborate the microscopic description of the Cr$_7$Ni from INS data, a detailed study of high field magnetization is very powerful. In fact, with high pulsed fields up to almost 60T, spin multiplets not accessible to a standard INS experiment can be explored. In Fig.\[fig:Cr7NiM-article\]a the magnetization curve as a function of the magnetic field $\textbf{H}$ is reported. A clear staircase structure with plateaus at $\approx 1\mu_B$, $\approx 3\mu_B$ and odd multiples of $\mu_B$ reflects the change in the ground state due to the external field at the level anticrossing fields $H_n$. An hysteresis of the measured magnetization curves has been observed. The effect arises from the non-equilibrium condition due to the high pulsed magnetic field with a few millisecond duration [@Furukawa] and has been discussed in terms of phonon bottle-neck effects and magnetic Foehn effects [@Nakano]. There is a very good agreement between the measured and calculated magnetization curves. This is clearly visible in Fig.\[fig:Cr7NiM-article\]b where the positions of the main peaks of the calculated and measured $dM/dH$ matches correctly. The smaller peaks in the experimental $dM/dH$ are due to level anticrossings between excited energy levels. The effects are caused by the non-equilibrium exeperimental conditions and are not included in equilibrium calculations reported in Fig.\[fig:Cr7NiM-article\] [@Furukawa]. These results confirm that the microscopic picture derived from INS experiments [@Caciuffo; @CarrettaPRL] perfectly holds even at very high applied magnetic fields.
![(Color online) Magnetization curve of Cr$_7$Ni (top) and derivative $dM/dH$ (bottom) at T=1.3K. Red (dark gray) and black lines represent the down and up experimental magnetic field processes respectively. The dashed blue lines represent the theoretical calculation with the following parameters: $J_{Cr-Cr}$=16.9K, $J_{Cr-Ni}$=19.6K, $d_{Cr}$=-0.3K, $d_{Ni}$=-4K, $g_{Cr}$=1.98, $g_{Ni}$=2.2.[]{data-label="fig:Cr7NiM-article"}](Cr7NiM-article.eps){width="45.00000%"}
Spin dynamics
=============
A major obstacle to the proposed technological applications of magnetic molecules is constituted by phonon-induced relaxation. In fact, molecular observables, e.g. the magnetization, are deeply affected by the interaction of the spins with other degrees of freedom such as phonons [@SantiniNMR]. Here we investigate the molecular spin-spin correlations through an approach based on a density matrix theory [@SantiniNMR]. The irreversible evolution of the density matrix $\hat{\rho}(t)$ can be determined through the secular approximation and focusing on time scales detectable by low-frequency techiques such as NMR. Within this theoretical framework, a general expression for the quasi-elastic part of the Fourier transform of cross correlation functions is given by [@SantiniNMR; @Bertaina]: $$\begin{aligned}
\label{eq:S}
S_{AB}(\omega)=\sum_{m,n=1,N}p^{(eq)}_{m}(B_{mm}-\left\langle \hat{B}\right\rangle_{eq})\nonumber\\ \times(A_{nn}-\left\langle\hat{A}\right\rangle_{eq})
Re\left\{\left(\frac{1}{i\omega-W}\right)_{nm}\right\},\end{aligned}$$ where $N$ is the dimension of the Hilbert spin space of the molecule, $p^{(eq)}_{m}$ is the equilibrium population of the $m$th level and $B_{mm}=\left\langle m\right|\hat{B}\left|m\right\rangle$, $\left|m\right\rangle$ being the $m$th eigenstate of the spin Hamiltonian, while $W$ is the so called rate-matrix. The $mn$ element $W_{mn}$ of $W$ represents the probability per unit time of a transition between eigenstates $\left|m\right\rangle$ and $\left|n\right\rangle$ induced by the interaction of the spins with phonons. By assuming that spin-bath interaction takes place mostly through modulation of local crystal fields, the rate matrix can be calculated on the basis of the eigenstates of molecular spin Hamiltonian by first-order perturbation theory. With the choice of a spherical magnetoelastic (ME) coupling [@CarrettaNi10] the transition rates $W_{mn}$ are given by: $$\begin{aligned}
\label{eq:W}
W_{mn}=\gamma\pi^{2}\Delta^{3}_{mn}n(\Delta_{mn})\sum_{\stackrel{i,j=1,N}{q_1,q_2=x,y,z}}\left\langle m\right|O_{q_1,q_2}(\textbf{s}_i)\left|n\right\rangle
\nonumber\\
\times\left\langle n\right|O_{q_1,q_2}(\textbf{s}_j)\left|m\right\rangle,\end{aligned}$$ with $n(x)=(e^{\beta\hbar x}-1)^{-1}$, $\Delta_{mn}=(E_m-E_n)/\hbar$ the gap between the eigenstates $\left|m\right\rangle$ and $\left|n\right\rangle$ of the molecule. In the last equation $O_{q_1,q_2}(\textbf{s}_i)=(s_{q1,i}s_{q2,i}+s_{q2,i}s_{q1,i})/2$ are quadrupolar operators [@CarrettaNi10]. Finally, $\gamma$ represents the spin-phonon coupling strength, which can be determined by comparing the theoretical results with experimental data. In fact, the nuclear spin-lattice relaxation rate $1/T_1$ probes the fluctuations of molecular observables, thus giving information on the relaxation dynamics [@SantiniNMR]. Exploiting the Moriya formula [@Moriya], the proton NMR $1/T_1$ can be evaluated in absolute units using as inputs the positions of the Cr and Ni ions and of the hydrogens of the molecule: $${\label{eq:T1}}
\frac{1}{T_1}=\sum_{\stackrel{i,j=1,N}{q,q'=x,y,z}}\alpha^{qq'}_{ij}
\left(S_{s^{q}_{i},s^{q'}_{j}}(\omega_L)+S_{s^{q}_{i},s^{q'}_{j}}(-\omega_L)\right),$$ where the $S_{s^{q}_{i},s^{q'}_{j}}(\omega_L)$ are the Fourier transforms of the cross correlation functions from Eq. (\[eq:S\]) calculated at the Larmor angular frequency $\omega_L$, while the $\alpha^{qq'}_{ij}$ are geometric coefficients of the hyperfine dipolar interaction between magnetic ions and protons probed by NMR.
![(Color online) Experimental data (scatters) and calculations (lines) of reduced proton NMR $1/(T_1\chi T)$ for different values of the applied field along z (parallel to the ring axis).[]{data-label="fig:Cr7Ni-T1-Hz"}](Cr7Ni-T1-Hz.eps){width="45.00000%"}
The occurence of a peak in the proton NMR $1/T_1$ has been clearly explained in homonuclear ring-shaped molecules with small anisotropy such as Cr$_8$ [@SantiniNMR]. In fact, in this case $1/T_1\propto S_{S_z,S_z}(\omega_L)$, where $S_{S_z,S_z}(\omega,\textbf{H},T)$ is the Fourier transform of the autocorrelation function of $M$ [@SantiniNMR]: $S_{S_{z},S_{z}}(\omega,T,\textbf{H})=\sum_{i=1,N}A(\lambda_{i},T,\textbf{H})\lambda_{i}(T,\textbf{H})/
[\lambda_{i}(T,\textbf{H})^{2}+\omega^{2}]$. This equation shows that the spectrum of fluctuations of $M$ is given by a sum of $N$ Lorentzians, each with characteristic frequency $\lambda_i$, given by the eigenvalues of $-W$. For a wide range of $\textbf{H}$ and $T$ in these systems only a single relaxation frequency $\lambda_0$ significantly contribute to $S_{S_{z},S_{z}}(\omega,T,\textbf{H})$. As a result, if the dominant frequency $\lambda_0$ intersects the Larmor angular frequency, i.e. when $\lambda_0(T_0)=\omega_L$, at the temperature $T_0$ the proton NMR $1/T_1$ shows a sharp peak [@SantiniNMR; @Baek]. Being an heterometallic ring, this explanation does not hold for Cr$_7$Ni and Eq.(\[eq:T1\]) has to be used. Nevertheless, our calculations show that a peak in the reduced $1/(T_{1}\chi T)$ occurs in agreement with experimental data (see Fig.\[fig:Cr7Ni-T1-Hz\]). By fitting the observed peak position we have obtained $\gamma=0.8\times10^{-7}$THz$^{-2}$.
Conclusions
===========
A magnetization study of the heteronuclear antiferromagnetic ring-shaped nanomagnet Cr$_7$Ni has been performed. A clear step-wise increase of magnetization with increasing field is observed. The very good agreement of high field magnetization measurements up to almost 60T with calculation shows the spin Hamiltonian approach to be suitable even at very high fields. The relaxation dynamics of the compound has been investigated by the proton nuclear-spin relaxation rate $1/T_1$. Our calculations are in very good quantitative agreement with experimental data.
D. Gatteschi, R. Sessoli, and J. Villain, *Molecular Nanomagnets*, Oxford University Press, Oxford (2006). R. Sessoli, D. Gatteschi, A. Caneschi, and M. A. Novak, Nature (London) **365**, 141 (1993). M. N. Leuenberger and D. Loss, Nature (London) **410**, 789 (2001). F. K. Larsen, E. J. L. McInnes, H. El Mkami, J. Overgaard, S. Piligkos, G. Rajaraman, E. Rentschler, A. A. Smith, G. M. Smith, V. Boote, M. Jennings, G. A. Timco, and R. E. P. Winpenny, Ang. Chemie **42**, 101 (2003). F. Troiani, A. Ghirri, M. Affronte, S. Carretta, P. Santini, G. Amoretti, S. Piligkos, G. Timco, and R. E. P. Winpenny, Phys. Rev. Lett. **94**, 207208 (2005). S. Carretta, P. Santini, G. Amoretti, M. Affronte, A. Ghirri, I. Sheikin, S. Piligkos, G. Timco, and R. E. P. Winpenny, Phys. Rev. B **72**, 060403 (2005). Y. Furukawa et al., to be published. R. Caciuffo, T. Guidi, G. Amoretti, S. Carretta, E. Liviotti, P. Santini, C. Mondelli, G. Timco, C. A. Muryn, and R. E. P. Winpenny, Phys. Rev. B **71**, 174407 (2005). S. Carretta, P. Santini, G. Amoretti, T. Guidi, J. R. D. Copley, Y. Qiu, R. Caciuffo, G. Timco, and R. E. P. Winpenny, Phys. Rev. Lett. **98**, 167401 (2007). P. Santini, S. Carretta, E. Liviotti, G. Amoretti, P. Carretta, M. Filibian, A. Lascialfari, and E. Micotti, Phys. Rev. Lett. **94**, 077203 (2005). S.-H. Baek, M. Luban, A. Lascialfari, E. Micotti, Y. Furukawa, F. Borsa, J. van Slageren, and A. Cornia, Phys. Rev. B **70**, 134434 (2004). H. Nakano and S. Miyashita, J. Phys. Soc. Jpn. **71**, 2580 (2002). S. Bertaina, B. Barbara, R. Giraud, B. Z. Malkin, M. V. Vanuynin, A. I. Pominov, A. L. Stolov, and A. M. Tkachuk, Phys. Rev. B **74**, 184421 (2006). S. Carretta, P. Santini, G. Amoretti, M. Affronte, A. Candini, A. Ghirri, I. S. Tidmarsh, R. H. Laye, R. Shaw, and E. J. L. McInnes, Phys. Rev. Lett. **97**, 207201 (2006). T. Moriya, Progr. Theor. Phys. **16**, 23 (1956).
| {
"pile_set_name": "ArXiv"
} |
---
author:
- 'Takuto <span style="font-variant:small-caps;">Kawakami</span>[^1], Yasumasa <span style="font-variant:small-caps;">Tsutsumi</span>, and Kazushige <span style="font-variant:small-caps;">Machida</span>'
title: 'Singular and Half-Quantum Vortices and Associated Majorana Particles in Superfluid $^3$He-A between Parallel Plates'
---
Introduction
============
There has been much attention on topological orders and the associated Fermionic quasi-particles with low energies [@fu; @ghaemi; @nilsson; @bargman]. This is particularly true when the quasi-particles are Majorana [@majorana] character, namely its creation operator $\gamma^{\dagger}$ is equal to its annihilation operator $\gamma$; $\gamma^{\dagger}=\gamma$. This unusual and intriguing Fermion, quite different from the usual Dirac particle, is thought to be useful for fault-tolerant quantum computations because it obeys non-Abelian statistics [@ivanov] and its existence is protected topologically to avoid decoherence. These situations are ideal for quantum computation [@dassarma]. The candidate systems, which support the Majorana particle, are quite rare; chiral spinless $p_x\pm ip_y$ superconductors, $p$-wave Feshbach resonanced superfluid [@mizushima], and the fractional quantum Hall state with the 5/2 filling. The former superconductors have not been identified yet in nature. It has often been argued that Sr$_2$RuO$_4$ may be a prime candidate [@maki; @sarma; @chung; @vakaryuk], but strong doubt has been cast on this possibility of Sr$_2$RuO$_4$ of its triplet pairing [@machida2; @lebed; @mazin]. Note that the first discovered triplet superconductor UPt$_3$ is an $f$-wave pairing, not chiral $p$-wave [@machida_a; @machida_b; @sauls_b]. It is proven theoretically that the half-quantum vortex (HQV) in the chiral superconductors with the $p$-wave pairing possesses the Majorana particle with zero energy localized at the vortex core [@tewari; @volovik].
Superfluid $^3$He-A phase is characterized by a chiral $p$-wave pairing. There is no doubt on this identification [@leggett; @Vollhardt]. In fact, Volovik and Mineev [@mineev] are the first to point out the possibility to the realization of HQVs in 1976. Since then, there have been several general arguments on the stability of a HQV in connection with $^3$He-A phase [@cross; @salomaa; @salomaaRMP; @volovikbook]. However, there are no serious calculations which consider it in a realistic situation in superfluid $^3$He-A phase on how to stabilize it and on what boundary conditions are needed for it.
Recently, Yamashita, $et$ $al.$ [@yamashita] have performed an experiment intended to observe HQVs in superfluid $^3$He-A in parallel plate geometry. The superfluid is confined in a cylindrical region with the radius $R=1.5$ mm and the height 12.5 $\mu$m sandwiched by parallel plates. A magnetic field $H=26.7$ mT ($\parallel$$\bm{z}$) is applied perpendicular to the parallel plates under pressure $P$=3.05 MPa. Since the gap 12.5 $\mu$m between plates is narrow compared to the dipole coherence length $\xi_d\sim 10 \mu$m, the $l$-vector, which signifies the direction of orbital angular momentum of Cooper pairs, is always perpendicular to the plates. Also the $d$-vector is confined within the plane by an applied field $\bm{H} \parallel \bm{z} $ because the dipole magnetic field $H_d\sim 2.0 $ mT [@leggett; @Vollhardt], where $\bm{H}$ tends to align the $d$-vector perpendicular to the field direction. They investigate to seek out various parameter spaces, such as temperature $T$, or the rotation speed $\Omega$ up to $\Omega=6.28$ rad/sec by using the rotating cryostat in ISSP, Univ. Tokyo, capable for the maximum rotation speed $\sim$12 rad/sec, but there is no evidence for the existence of the HQV [@yamashita].
The aims in this paper are to investigate the possible vortex structures which can accommodate the Majorana particle in the core under the above realistic experimental situations at ISSP for superfluid $^3$He-A phase confined in the parallel plates. The candidate vortex structures in this situation are either the singular vortex (SV) with odd winging number or HQVs mentioned above. Thus after examining the sufficient condition for the Majorana zero energy particle to exist in the SV, we determine the phase diagram in the system size with the radius $R$ and the external rotation frequency $\Omega$ under applied fields with arbitrary angle relative to the plates. Note that the external field is necessary for performing NMR detection [@yamashita].
The arrangement of the paper is as follows: In \[General\] we examine the possible order parameters and its spatial structures, or textures. By utilizing general symmetry properties of the superfluid $^3$He-A phase, we demonstrate that the SV with odd winding number even in the spinful situation can accommodate the Majorana zero mode in the core. In \[SV\] we investigate the stable vortices and textures within the Ginzburg-Landau (GL) formalism when the magnetic field is applied to an arbitrary angle relative to the plates, in order to find the stable SV region for various system parameters, external rotation $\Omega$, the system size $R$ and the field orientation $\theta _H$. In \[HQV\] the stability problem of the single HQV and a pair of HQVs are analyzed within the same framework. We devote to summary and conclusions in the final section.
A part of the \[HQV\] is published in ref. \[kawakamihqv\].
General Considerations {#General}
======================
Order parameters and textures
-----------------------------
Generally, in $p$-wave superfluids, the order parameter (OP) is described by $$\begin{aligned}
\hat{\Delta}(\bm{r},\hat{\bm{p}})=
\left(
\begin{array}{cc}
\Delta _{\uparrow \uparrow}(\bm{r},\hat{\bm{p}}) & \Delta _{\uparrow \downarrow} (\bm{r},\hat{\bm{p}}) \\
\Delta _{\downarrow \uparrow}(\bm{r},\hat{\bm{p}}) & \Delta _{\downarrow \downarrow}(\bm{r},\hat{\bm{p}})
\end{array}
\right).\end{aligned}$$ The matrix element is described by $$\begin{aligned}
{\nonumber}\Delta _{\sigma \sigma '}(\bm{r},\hat{\bm{p}}) &=& A_{\sigma \sigma',+}(\bm{r})\hat{p}_+
+ A_{\sigma \sigma',-}(\bm{r})\hat{p}_- \\
& &+ A_{\sigma \sigma',z}(\bm{r})\hat{p}_z,\end{aligned}$$ where $\sigma$ is spin index $\uparrow$ or $\downarrow$ , $\hat{p} _{\pm}=\mp (\hat{p}_x \pm i\hat{p}_y)/\sqrt{2}$, and $\hat{\bm{p}}$ is the unit vector in the momentum space. In the parallel plate geometry, the momentum $\hat{p}_z$ component is suppressed. Here each component $A _{\sigma \sigma',\pm }(\bm{r})$ $(\sigma = \uparrow,\downarrow)$ can have its own phase winding whose winding number is denoted by $w _{\sigma \sigma', \pm}$. There are three possible textures at rest and under rotation as shown in Table \[winding\]: $(w _{\sigma \sigma',+},w _{\sigma \sigma',-})=(0,2)$; A-phase texture (AT), (1,3); SV, and $(w _{\uparrow \uparrow,+},w _{\downarrow \downarrow,+},w _{\uparrow \uparrow,-},w _{\downarrow \downarrow,-})=(0,1,2,3)$; HQV. These textures and vortices are allowed in axisymmetric situation (see \[Hpara\]). We investigate the detailed configuration of these textures, the relative energetics, and the associated Majorana quasi-particle in the last two textures.
$w _{\uparrow \uparrow, +}$ $w _{\uparrow \downarrow, +}$ $w _{\downarrow \downarrow, +}$ $w _{\uparrow \uparrow, -}$ $w _{\uparrow \downarrow, -}$ $w _{\downarrow \downarrow, -}$
----- --------------------------------- --------------------------------- --------------------------------- --------------------------------- --------------------------------- ---------------------------------
AT 0 0 0 2 2 2
SV 1 1 1 3 3 3
HQV 0 $\times$ 1 2 $\times$ 3
: The combination of phase windings of each OP component for the A-phase texture (AT), the singular vortex (SV) and the half-quantum vortex (HQV). []{data-label="winding"}
Majorana bound state -symmetry considerations- {#Majorana}
----------------------------------------------
In this subsection, we examine the existence of the Majorana particle in the SV and the HQV in a most general situation. By assuming that the configurations of the OP are the bulk A-phase, the SV texture is described as $$\begin{aligned}
{\nonumber}\Delta _{\sigma \sigma '}(\bm{r},\hat{\bm{p}}) &=& A_{\sigma \sigma',+}(\bm{r})\hat{p}_+ \\
&=& d_{\sigma \sigma'}(\bm{r})A_+(r)\exp(i\phi)\hat{p}_+,\end{aligned}$$ where $\phi$ is the azimuthal angle of the center-of-mass coordinate $\bm{r}$, $$\begin{aligned}
{\nonumber}d_{\uparrow \uparrow(\downarrow \downarrow)}(\bm{r}) &=& \mp \frac{1}{\sqrt{2}}(d_x(\bm{r}) \mp id_y(\bm{r})) , \\
{\nonumber}d_{\uparrow \downarrow }(\bm{r}) &=& d_z(\bm{r}), \end{aligned}$$ and $d$-vector $\bm{d}=(d_x,d_y,d_z)$ is real. Under this assumption, the direction of $d$-vector is determined by the dipole interaction and the interaction with the external field. The polar and azimuthal angle of direction of $d$-vector are determined as $$\begin{aligned}
\theta _d ^{(bulk)} &=& \frac{1}{2}\tan ^{-1} \left [\frac{\sin 2\theta _H}{(H_d/H)^2-\cos 2\theta _H}\right],
\label{bulkdt}\\
\phi _d ^{(bulk)} &=& 0. \label{bulkdp}\end{aligned}$$ The angle $\theta _d$ and $\theta _H$ are defined as shown in Fig. \[def\_theta1\]. The angle $\phi _d$ is the azimuthal angle of the direction of $d$-vector. In other words, when the direction of the spin quantization axis is in the angle of $(\theta _d ^{(bulk)}+\pi/2,\phi _d ^{(bulk)})$, $d_{\uparrow \downarrow }(\bm{r})=0$ throughout the system.
![ (Color online) Schematic diagram of the parallel plate system. The external magnetic field $\bm{H}$ tilts by $\theta _H$ from the $z$-axis. $\theta_d$ is the polar angle of the $d$-vector. The external field tends to align the $d$-vector perpendicular to $\bm{H}$ and the dipole interaction tends to aligned the $d$-vector parallel to the direction $\bm{z}$. Then $0\leq \theta_d\leq \pi/2-\theta _H$. []{data-label="def_theta1"}](fig_def_theta1.eps){width="60mm"}
When $\Delta _{\uparrow \downarrow }(\bm{r_1},\bm{r}_2)=\Delta _{\downarrow \uparrow }(\bm{r},\hat{\bm{p}})=0$, $p$-wave mean field Hamiltonian is written as $$\begin{aligned}
\mathcal{H}=\int d \bm{r}_1 d \bm{r}_2 \bm{\Psi }^\dagger(\bm{r_1})
\left(
\begin{array}{cc}
\hat{\mathcal{K}} _{\uparrow \uparrow } &0 \\
0 & \hat{\mathcal{K}} _{\downarrow \downarrow }
\end{array}
\right)
\bm{\Psi}(\bm{r_2}),\end{aligned}$$ where $$\begin{aligned}
{\nonumber}\bm{\Psi}(\bm{r})=[\psi _\uparrow (\bm{r}), \psi _\uparrow ^\dagger(\bm{r})
,\psi _\downarrow(\bm{r}), \psi _\downarrow^\dagger(\bm{r})]^T, \end{aligned}$$ $$\begin{aligned}
{\nonumber}\hat{\mathcal{K}}_{\sigma \sigma }=
\left[
\begin{array}{cc}
H _0 ^\sigma (\bm{r}_1,\bm{r}_2) & \Delta _{\sigma \sigma}(\bm{r}_1,\bm{r}_2) \\
-\Delta _{\sigma \sigma } ^*(\bm{r}_1,\bm{r}_2) & -H_0 ^{\sigma*} (\bm{r}_1,\bm{r}_2) \\
\end{array}
\right], \end{aligned}$$ $$\begin{aligned}
{\nonumber}H_0^{(\sigma)}(\bm{r}_1,\bm{r}_2)&=&
\left[
-\frac{\nabla _1 ^2}{2m_3}+V(\bm{r}_1)-\mu_\sigma \right.\\
{\nonumber}& & \left. +i\Omega \left(x_1 \frac{\partial }{\partial y_1}-y_1\frac{\partial }{\partial x_1} \right)
\right]
\delta(\bm{r}_1-\bm{r}_2),\end{aligned}$$ where $m_3$, $V$, and $\mu _\sigma$ are the mass of the $^3$He atom, the single particle potential, and the chemical potential of the particle whose spin $\sigma$. We carry out the Bogoliubov transformation to quasi particles whose creation and annihilation operators $$\begin{aligned}
{\nonumber}\eta _{\nu \uparrow } &=&
\int d \bm{r}
\left[
u_{\nu 1}^*(\bm{r})\psi _\uparrow (\bm{r})
+ u_{\nu 2}^*(\bm{r})\psi _\downarrow (\bm{r})
\right.\\
& & \left.
+ v_{\nu 1}^*(\bm{r})\psi _\uparrow ^\dagger(\bm{r})
+ v_{\nu 2}^*(\bm{r})\psi _\downarrow ^\dagger(\bm{r})
\right], \\
{\nonumber}\eta _{\nu \downarrow } &=&
\int d \bm{r}
\left[
v_{\nu 1}(\bm{r})\psi _\uparrow (\bm{r})
+ v_{\nu 2}(\bm{r})\psi _\downarrow (\bm{r})
\right.\\
& & \left.
+ u_{\nu 1}(\bm{r})\psi _\uparrow ^\dagger(\bm{r})
+ u_{\nu 2}(\bm{r})\psi _\downarrow ^\dagger(\bm{r})
\right].\end{aligned}$$ Obtained Bogoliubov-de Gennes (BdG) equation is $$\begin{aligned}
\label{BdG}
{\nonumber}\int d\bm{r}_2
\left[
\begin{array}{cc}
\hat{{\mathcal K}}_{\uparrow \uparrow }(\bm{r}_1,\bm{r}_2) &0 \\
0 & \hat{{\mathcal K}}_{\downarrow \downarrow }(\bm{r}_1,\bm{r}_2)
\end{array}
\right]
\underline{u}_\nu(\bm{r_2})
= \\
\underline{u}_\nu (\bm{r}_1)
\left[
\begin{array}{cc}
\hat{E}_\nu^\uparrow & 0 \\
0 & \hat{E}_\nu^\downarrow
\end{array}
\right],\end{aligned}$$ where $\hat{E}_\nu ^\sigma = \sigma _3 E_\nu^\sigma$. This $\sigma _3$ is the Pauli matrix. We can reduce the BdG equation (\[BdG\]) to four eigenvalue equations, $$\begin{aligned}
{\nonumber}\int d\bm{r}_2
\hat{{\cal K}}_{\uparrow \uparrow }(\bm{r}_1,\bm{r}_2)
\left(
\begin{array}{c}
u_{\nu 1}(\bm{r}_2) \\
v_{\nu 1}(\bm{r}_2)
\end{array} \right)
= \\
\sgn(\sigma)E_\nu^{\sigma }
\left(
\begin{array}{c}
u_{\nu 1}(\bm{r}_1) \\
v_{\nu 1}(\bm{r}_1)
\end{array} \right), \end{aligned}$$ $$\begin{aligned}
{\nonumber}\int d\bm{r}_2
\hat{{\cal K}}_{\downarrow \downarrow }(\bm{r}_1,\bm{r}_2)
\left(
\begin{array}{c}
u_{\nu 2}(\bm{r}_2) \\
v_{\nu 2}(\bm{r}_2)
\end{array} \right)
= \\
\sgn(\sigma)E_\nu^{\sigma }
\left(
\begin{array}{c}
u_{\nu 2}(\bm{r}_1) \\
v_{\nu 2}(\bm{r}_1)
\end{array} \right). \end{aligned}$$ These eigenvalue equations are solved numerically [@tsutsumi]. If $\Delta_{\sigma \sigma }(\bm{r},\hat{\bm{p}})$ has odd phase winding, we can obtain the zero energy state ($E ^\sigma _\nu=0$) and $\left(u_{\nu,i}(\bm{r}),v_{\nu,i}(\bm{r})\right)=\left(v_{\nu,i} ^*(\bm{r}),u_{\nu,i}^*(\bm{r})\right)$ so that $\eta _{\nu \sigma } = \eta _{\nu\sigma } ^{ \dagger }$. That is, the Bogoliubov quasi-particles exhibit the Majorana character.
In the SV texture all components of the OP have the odd winding number. Thus the SV has the Majorana quasi-particle with the external field toward any direction. In the HQV texture, either $\Delta _{\uparrow \uparrow } (\bm{r},\hat{\bm{p}})$ or $\Delta _{\downarrow \downarrow } (\bm{r},\hat{\bm{p}})$ has the odd phase winding. The component that has no phase windings does not have the lower excitation. That is, the HQV has the lower excitation that is induced by the odd phase winding and these excitations contain the Majorana quasi-particle. Consequently, both the SV and the HQV have the Majorana quasi-particle at these vortex cores.
However, in the above analysis, we do not consider the influence of the vortex core and the edge of system. In these situation, the configuration of the OP is different from the bulk A-phase due to variance of the OP, mixture of minor components, and so on. Furthermore, the Majorana quasi-particle exists at the vortex core and the edge of system. Then we have to clarify OP textures by more realistic and serious calculation using GL theory.
Ginzburg-Landau Functional and Phase Diagram {#SV}
============================================
Ginzburg-Landau functional
--------------------------
The GL free-energy functional invariant under gauge transformation, spin, and orbital space rotations is well established [@leggett; @Vollhardt; @salomaaRMP; @sauls_a; @greywall; @kita; @fetter; @thuneberg; @wheatley] and given by a standard form $$\begin{aligned}
\label{GLfun}
f_{total}=f_{grad}+f_{bulk}+f_{dipole}+f_{field},\end{aligned}$$ $$\begin{aligned}
{\nonumber}f_{grad} = K\left[(\partial _{i}^*A_{\mu j}^*)(\partial _{i}A_{\mu j}) + (\partial _{i}^*A_{\mu j}^*)(\partial _{j}A_{\mu i})\right. \\
{\nonumber}+ \left.(\partial _{i}^*A_{\mu i}^*)(\partial _{j}A_{\mu i})\right], \end{aligned}$$ $$\begin{aligned}
{\nonumber}f_{bulk} & = & -\alpha _{\uparrow \uparrow } A_{\uparrow \uparrow ,i }^*A_{\uparrow \uparrow ,i}
-\alpha _{\uparrow \downarrow } A_{\uparrow \downarrow,i }^*A_{\uparrow \downarrow,i} \\
{\nonumber}& & -\alpha _{\downarrow \downarrow } A_{\downarrow \downarrow,i }^*A_{\downarrow \downarrow,i} \\
{\nonumber}& & + \beta _1 A_{\mu i}^*A_{\mu i}^*A_{\nu j}A_{\nu j}
+ \beta _2 A_{\mu i}^*A_{\nu j}^*A_{\mu i}A_{\nu j} \\
{\nonumber}& & + \beta _3 A_{\mu i}^*A_{\nu i}^*A_{\mu j}A_{\nu j}
+ \beta _4 A_{\mu i}^*A_{\nu j}^*A_{\mu j}A_{\nu i} \\
{\nonumber}& & + \beta _5 A_{\mu i}^*A_{\mu j}^*A_{\nu i}A_{\nu j}, \end{aligned}$$ $$\begin{aligned}
{\nonumber}f_{dipole}=g_d(A_{\mu \mu }^*A_{\nu \nu } + A_{\mu \nu }^*A_{\nu \mu } - \frac{2}{3} A_{\mu \nu }^*A_{\mu \nu }), \end{aligned}$$ $$\begin{aligned}
{\nonumber}f_{field}=g_m H_\mu A_{\mu i}^*H_\nu A_{\nu i}, \end{aligned}$$ where $\mu,i=x,y,z$ and $\partial_i=\nabla_i-(2im_3/\hbar)(\bm\Omega \times \bm r)_i$ $(\bm \Omega \parallel \bm{z})$ with $m_3$ being the mass of the $^3$He atom. The coefficients of quadratic terms $$\begin{aligned}
\label{alpha}
\alpha _{\sigma \sigma'}=\alpha_0(1-T/T_c+\sgn(\sigma +\sigma ')\Delta T/T_c),\end{aligned}$$ where $\alpha_0=N(0)/3$. The Zeeman effect splits the transition temperature $T_{c \uparrow }$ of the $\left| \uparrow \uparrow \right>$ pair and $T_{c \downarrow }$ of the $\left| \downarrow \downarrow \right>$ pair [@ambegaokar]. The split width $\Delta T \equiv (T_{c \uparrow } - T_{c \downarrow })/2$ in eq. (\[alpha\]). However, we consider the region that $\Delta T \ll T_c-T$ so-called weak field region until \[HQV\]. The coefficient of fourth order terms $\beta _i$ [@GLparameters] are obtained in ref. \[sauls\] appropriate for the experiment at $P=3.05$ MPa. We use the gradient coupling constant with weak coupling limit $K=7\zeta(3)N(0)(\hbar v_F)^2/240(\pi k_B T_c)$. The $g_d$ [@thuneberg] and the $g_m$ are the coupling constant of the dipole interaction and the interaction with external field respectively, $$\begin{aligned}
g_d=\frac{\mu _0}{40}\left[ \gamma \hbar N(0) \ln \frac {1.1339 \times 0.45 T_F}{T_c}\right],\end{aligned}$$ $$\begin{aligned}
g_m=\frac{7\zeta(3)N(0)(\gamma \hbar)^2}{48[(1+F^a_0)\pi k_B T_c]}.\end{aligned}$$ The transition temperature under no magnetic field $T_c$, the density of states $N(0)$, Fermi velocity $v_F$, the permeability of vacuum $\mu _0$, the gyromagnetic ratio $\gamma$, Landau parameter $F_0 ^a$, and Fermi temperature $T_F$ are given by experiments [@wheatley; @greywall]. In order to carry out realistic calculations taking account of the configuration of the vortex core and the edge of the system, our formulation includes strong coupling corrections in the bulk terms where the A-phase is stable over the B-phase. Thus the gradient terms are within the weak coupling limit $(\rho _{sp}/\rho _s=1)$ because there is no established method to properly take into account the Fermi liquid correction for the OP form generalized beyond the A-phase (see for standard method ref. \[vollhardt\]). We minimize the GL functional eq. (\[GLfun\]) where the external field is applied to arbitrary direction relative to the plates and absolute value. First, we consider two limits where the direction of the external magnetic field is parallel and perpendicular to the plates.
Parallel field $H_{||}$ {#Hpara}
-----------------------
In the situation where the external field applies parallel to the plates, both the dipole interaction and the external field suppresses the components of the OP $\Delta _{\uparrow \uparrow }(\bm{r})$ and $\Delta _{\downarrow \downarrow }(\bm{r})$. In the bulk A-phase system, the two components $A _{\uparrow \downarrow, \pm }(\bm{r})$ survive. Since the phase winding of the OP components increases the kinetic energy of Cooper pairs, it is energetically favorable at rest that large component of the OP has small winding number. Furthermore, in the cylindrical symmetric system, $w_{\uparrow \downarrow,+}=w_{\uparrow \downarrow -}-2$ due to orbital coupling effect [@kawakami]. Under low rotations the stable texture is the AT whose phase winding $(w_{\uparrow \downarrow,+},w_{\uparrow \downarrow,-})=(0,2)$ and the dominant component of the OP is $A_{\uparrow \downarrow,+}(\bm{r})$.
Under the anti-clock wise rotation the component that has the positive phase winding gains the rotating energy through a term $-w_{\uparrow \downarrow \pm}|A_{\uparrow \downarrow \pm}|^2$ in the GL functional eq. (\[GLfun\]). As the rotating speed $\Omega$ is higher, the free energy of the SV whose dominant component of the OP has winding number $w=1$ becomes lower than that of the AT. According to the above analysis, the candidates of stable SVs are $(w_{\uparrow \downarrow,+},w_{\uparrow \downarrow,-})=(-1,1)$ and (1,3). The $(-1,1)$ texture has smaller winding number than the $(1,3)$ texture, from the kinetic energy view point the $(-1,1)$ texture seems to be more stable than the$(1,3)$. However, under rotation the minor component $A_{\uparrow \downarrow -}(\bm{r})$ of $(1,3)$ texture gains more rotating energy than the minor component $A_{\uparrow \downarrow +}(\bm{r})$ of $(-1,1)$ texture. Under rotation the stable texture is determined by competition of those two factors. The results of our calculation show that when the rotating speed is large enough to stabilize the SV not AT, $(1,3)$ texture is always more stable than $(-1,1)$ texture.
In Fig. \[tex\_para\], the cross section of the amplitude of the OP component of the AT and the SV when the direction of the external field is parallel to the plates and the quantization axis of spin is chosen perpendicular to the plates. Choosing the quantization axis parallel to the plates same as the direction of the field, we can obtain the SV texture whose OP component $\Delta _{\uparrow \downarrow }(\bm{r}) = 0$ exactly. The SV texture has Majorana quasi-particle state at the vortex core.
![ (Color online) Order parameter amplitude $\left| A_{\sigma \sigma', \pm }\right|$ normalized by $\pi k_B T_c$. Left (right) figure shows the cross section of the OP components along the radial direction $\bm{r}$ of the AT (SV) texture for $R=5$ $\mu$m, $T/T_c=0.95$, $H=10$ mT, $\theta _H=\pi/2$. We choose the spin quantization axis perpendicular to the plates; the $\bm{z}$ direction. The spin component $\Delta _{\sigma \sigma }$ vanishes. []{data-label="tex_para"}](fig_tex_para.eps){width="80mm"}
![ System size $R$ dependence of the critical angular velocity $\Omega _c$ from the AT to the SV $T/T_c=0.95$, $\bm{H} \parallel \mathrm{plates}$ ($\theta _H =\pi /2$). An extrapolated value of $\Omega _c = 0.06$ rad/sec is found at $R=1.5$ mm. The inset shows the energy differences $\Delta f$ between the two textures as a function of $\Omega$ for $R = 100$ $\mu$m. []{data-label="phase_diagram"}](fig_phase_diagram.eps){width="75mm"}
The phase diagram of textures in the space consisting of the rotating speed $\Omega$ and system size $R$ is shown in Fig. \[phase\_diagram\] in a fixed temperature $T/T_c=0.95$. We calculate in various temperatures and absolute values of external field, showing that the temperature and the absolute value of external field change $\Omega _c$ only slightly, so that $\Omega _c$ is determined by almost only $R$. We obtain the critical rotating speed $\Omega _c$ quantitatively as a function of system size $R$ smaller than 100 $\mu$m. Therefore, we can extrapolate $\Omega _c = 0.06$ rad/sec in $R = 1.5$ mm, that is the system size of the sample using the experiment in ISSP. This rotating speed can be well controlled by the present experimental technique.
Perpendicular field $H_\perp$
-----------------------------
In this subsection we consider the situation where the external field apply parallel to the plates. When the external field larger than the dipole field $H_d$, the $d$-vectors lie parallel to the plates. The result of numerical calculation shows that $\Delta _{\uparrow \downarrow }(\bm{r},\hat{\bm{p}})=0$ exactly throughout the system as shown in Fig. \[tex\_perp1\]. These SV textures have the Majorana quasi-particle state at the vortex core. When the external field is smaller than the dipole field $H_d$, in the bulk A-phase the $d$-vector aligns perpendicular to the plates. The OP texture is same as in the case that the external field is parallel to the plates shown in Fig. \[tex\_para\]. Choosing the quantization axis parallel to the plates, we can obtain the OP component $\Delta_{\uparrow \downarrow }(\bm{r})=0$ exactly. However, when $d\mbox{-vector} \parallel \bm{H}$ taking account of Zeeman effect in BdG equation (\[BdG\]), the symmetry of the equation arrows the solution $\left(u_{\nu,1}(\bm{r}),v_{\nu,1}(\bm{r})\right)=\left(v_{\nu,2} ^*(\bm{r}),u_{\nu,2}^*(\bm{r})\right)$. Thus the core bound state of the SV cannot satisfy Majorana condition $\left(u_{\nu,i}(\bm{r}),v_{\nu,i}(\bm{r})\right)=\left(v_{\nu,i} ^*(\bm{r}),u_{\nu,i}^*(\bm{r})\right)$. Therefore only in the case where the external field is larger than the dipole field $H_d$, the stabilized SV has the Majorana quasi-particle.
The phase diagram of textures are same as that in \[Hpara\] since in the result of our calculation, the external magnetic field change critical rotating speed $\Omega _c$ of the texture transition from the AT to the SV only slightly as shown in \[Harbitral\].
In the case that $\bm{H}\perp \mathrm{plates}$ where the external field is larger than the dipole field $H_d$, the $d$-vector is perpendicular to the $l$-vector. Then the dipole energy of the bulk system can be neglected. Thus this case is favorable for realization of the HQV texture. We discuss the HQV texture in \[HQV\] in detail.
![ (Color online) Order parameter amplitude $\left| A_{\sigma \sigma', \pm }\right|$ normalized by $\pi k_B T_c$ for $R=5$ $\mu$m, $T/T_c=0.95$, $H=10$ mT, $\theta _H=0$. Left (right) figure shows the cross section of the OP components along the radial direction $\bm{r}$ of the AT (SV) texture. We choose the spin quantization axis perpendicular to the plates; $\bm{z}$ direction. The spin component $\Delta _{\uparrow \downarrow }(\bm{r},\hat{\bm{p}})=0$ exactly. []{data-label="tex_perp1"}](fig_tex_perp.eps){width="80mm"}
Arbitrary oriented field {#Harbitral}
------------------------
In the bulk A-phase system, the direction of the $d$-vector is determined by the competition between the dipole interaction favoring that the $d$-vector is parallel to $l$-vector and the external field favoring that the $d$-vector is perpendicular to the $\bm{H}$. The direction of the $d$-vector is given by eqs. (\[bulkdt\]) and (\[bulkdp\]). When the quantization axis is perpendicular to the plates, all components $\Delta _{\sigma \sigma'}(\bm{r},\hat{\bm{p}})$ are non-vanishing as shown in Fig. \[tex\_10\].
![ (Color online) Order parameter amplitude $\left| A_{\sigma \sigma', \pm }\right|$ normalized by $\pi k_B T_c$ for $R=5$ $\mu$m, $T/T_c=0.95$, $H = 2$ mT, $\theta _H = \pi/18$. Left (right) figure shows the cross section of the OP components along the radial direction $r$ of the AT (SV) texture. We choose the spin quantization axis perpendicular to the plates; $\bm{z}$ direction. All spin components $\Delta_{\sigma \sigma'}(\bm{r},\hat{\bm{p}})$ are non-vanishing. When we choose the spin quantization axis to be the direction $\theta=\theta_d^{(bulk)}+\pi/2$ and $\phi=0$, $A_{\uparrow \downarrow, \pm }\ll A_{\sigma \sigma, \pm }$. []{data-label="tex_10"}](fig_tex_10.eps){width="80mm"}
We consider the energetics of these textures by calculating the stable textures under various external magnetic fields; $H$ = 1, 2, 3, and 10 mT, for various polar angles $\theta _H=n\pi/18$ $n=1,2, \cdots$. The result of the calculation in the system $R=5$ $\mu$m which is shown in Fig. \[omega\_th\]. Under these external fields, $\Omega _c$ coincides with each other for the precision in second order after the decimal point. We see that the external magnetic field scarcely changes the phase diagram shown in Fig. \[phase\_diagram\]. Their reasons are as follows. The critical rotation $\Omega _c$ is determined by the competition of the kinetic energy and the energy gain from angular momentum, namely, the gradient term. On the other hands the influence of the interaction with the external field is no more than $10^{-6}$ order of the gradient energy.
![ (Color online) The critical angular velocity $\Omega _c$ dependence of $\theta _H$ from the AT to the SV for $T/T_c=0.95$, $R = 5$ $\mu$m, $H= 1$, $2$, $3$, and $10$ mT. Under the various direction and absolute value of the external fields, $\Omega _c$ coincides with each other for the precision in second order after the decimal point. []{data-label="omega_th"}](fig_omg_th.eps){width="80mm"}
We discuss the possibility that the Majorana quasi-particle exists in these textures. As shown in \[Majorana\], the SV has the Majorana quasi-particle when their OP component $\Delta _{\uparrow \downarrow}(\bm{r},\hat{\bm{p}})=0$ for appropriate quantization axis throughout the system. Thus we turn the quantization axis to be their polar and azimuthal angle $(0,0) \mapsto (\theta _d ^{(bulk)}+\pi/2, \phi _d^{(bulk)})$ in order to change the expression to that suppressing the component of the OP $\Delta _{\uparrow \downarrow}(\bm{r},\hat{\bm{p}})$. Then we obtain changed OP textures as shown in Fig. \[amp\_minor\]. In these textures, the components of the OP $A_{\uparrow \downarrow,\pm}(\bm{r})\neq 0$, but finite amplitude with the order $10^{-3}$ against the dominant component $A_{\sigma \sigma \pm}(\bm{r})$. These amplitudes vary spatially, and their phase cannot be specified by the accurate A-phase OP. In addition, when the $d$-vector is not perpendicular to the magnetic field $\bm{H}$, we should note the influence of Zeeman effect for the Majorana condition. Therefore, we cannot conclude whether or not those SVs have the Majorana quasi-particle. In order to clarify these points, we have to solve spinful BdG equation include off-diagonal terms of eq. (\[BdG\]). These remain as a future problem.
{width="160mm"}
We notice these induced components. In Fig. \[amp\_H\], we show the maximum value of amplitude of induced component $A_{\uparrow \downarrow,+}(\bm{r})$ as a function of $\theta _H$ in various external field values. We show that when the direction of the external field are perpendicular to the plates ($\theta _H=0$) and parallel to the plates ($\theta _H=\pi/2$), the component $A_{\uparrow \downarrow,+}(\bm{r}) = 0$ exactly. Moreover, when the external field comparable to the dipole field $H_d$ ($\simeq 2$ mT), the induced component is enhanced maximally under such a magnitude of the field.
The system size dependence of them is shown in Fig. \[amp\_size\]. When the system size becomes large, the amplitude of the component $A_{\uparrow \downarrow,+}(\bm{r})$ tend to be large. Therefore, even if the large sample such as that used in experiment $R=1.5$ mm, these components are finite.
We suggest that the reason why the induced component is non-zero is due to the existence of induced $A_{\sigma \sigma',-}$ component. Taking account of the $A_{\sigma \sigma',-}$ component, the direction of $d$-vector determined by minimizing $f_{dipole} + f_{field}$ has the component parallel to the spin quantization axis except for the case that $\bm{H}\parallel$ plates and $\bm{H}\perp$ plates. In addition, these components varies spatially according to spatial modulation of $A_{\sigma \sigma',-}$ component.
![ (Color online) The maximum value of $\left| A_{\uparrow \downarrow +} \right|$ as a function of $\theta_H$ for $H = 2$, $3$, and $27$ mT, $R = 5$ $\mu$m, $T/T_c=0.95$. When $\theta _H =0$ or $\pi/2$, the amplitude of the component $\Delta_{\uparrow \downarrow }(\bm{r},\hat{\bm{p}}) =0$ exactly When the external field is comparable to dipole field ($\simeq 2$) mT, the amplitude of these components is enhanced. []{data-label="amp_H"}](fig_amp_H.eps){width="75mm"}
![ The maximum value of $\left| A_{\uparrow \downarrow +} \right|$ as a function of $R$ for $H = 1$, $2$, and $3$ mT, $\theta _H = \pi/6$ and $\pi/3$ rad, $\mu$m, $T/T_c=0.95$. When the system size becomes larger, the amplitude of these components becomes larger. In the system using experiment, $R = 1.5$ mm, $\left| A_{\uparrow \downarrow \pm} \right|$ are non-vanishing. []{data-label="amp_size"}](fig_amp_size.eps){width="75mm"}
Half-Quantum Vortex {#HQV}
===================
Single half-quantum vortex
--------------------------
When the direction of the external field is perpendicular to the plates, it is a favorable situation for the HQV to realize. Then in this section, we consider possible stable HQV texture and their energetics. First, we consider the case that there is a single HQV in the system. The external magnetic field splits $T_c$ of the $\left| \uparrow \uparrow \right>$ and $\left| \downarrow \downarrow \right>$ pairs by the Zeeman effect [@ambegaokar]. The splitting is defined as $\Delta T = (T_{c\uparrow }-T_{c\downarrow })/2$. When $\Delta T > 0$ ($\Delta T < 0$), the amplitude of the OP $\Delta _{\uparrow \uparrow }$ ($\Delta _{\downarrow \downarrow }$) becomes larger than $\Delta _{\downarrow \downarrow }$ ($\Delta _{\uparrow \uparrow }$). The splitting $\Delta T$ is generally much smaller than the amplitude of the OP in the so-called weak field region $H\simeq$ 10 mT. In this region, the texture of the HQV are shown in Fig. \[tex\_HQV\]. The rotation speed $\Omega _{c1}$ ($\Omega _{c2}$) at which the free energy of the HQV (SV) and the AT (HQV) are intersect. In Fig. \[HQV\_df\_omg\], we show that the HQV never becomes the absolute stable texture. The reasons are the strong coupling effect in the bulk fourth terms of GL functional eq. (\[GLfun\]). In the weak coupling limit, under some critical rotation, the AT, the SV, and the HQV degenerate each other. The strong coupling effect stabilize the A phase, then the structure of the central region of the AT are more favorable than the others. Therefore, the critical rotation $\Omega _{c1}$ are larger than $\Omega _{c2}$.
![ (Color online) Order parameter amplitude of (a) $A_{\uparrow \uparrow, \pm }$ and (b) $A_{\downarrow \downarrow, \pm }$ components normalized by $\pi k_B T_c$ for $R=5$ $\mu$m, $T/T_c=0.95$, $H = 10$ mT, $\theta _H = 0$. We choose the spin quantization axis perpendicular to the plates; $\bm{z}$ direction. At the vortex core, $A_{\uparrow \uparrow, \pm }$ component is only non-vanishing, so that the HQV is A$_1$-core vortex. []{data-label="tex_HQV"}](fig_tex_HQV.eps){width="80mm"}
![ (Color online) Free-energy comparison for the AT, the HQV, the SV, the two HQVs system (2HQV), and the two SVs system (2SV) as a function of $\Omega$ for $R = 10$ $\mu$m and $T/T_c=0.97$ at the weak field region. The $\Delta f$ is the relative free energy to the AT. In the weak field region, $\Omega _{c2} < \Omega _{c1}$. []{data-label="HQV_df_omg"}](fig_HQV_df_omg2.eps){width="80mm"}
When the temperature becomes larger, the amplitude of the OP becomes smaller. Then the split $\Delta T$ cannot be neglected. When the split $\Delta T>0$, the $\left| \uparrow \uparrow \right>$ pair becomes dominant. Thus, the HQV texture and the energy difference $\Delta f$ in Fig. \[HQV\_df\_omg\] become close to the AT. In this process, $\Omega _{c1}$ become smaller than $\Omega _{c2}$. Furthermore, when the temperature becomes larger, the stable region of the HQV becomes larger as shown in Fig. \[HQV\_omg\_t\]. Consequently, the phase diagram is shown in Fig. \[HQV\_omg\_r\]. We can extrapolate that the stability region of the HQV is $0.05 < \Omega < 0.06$ rad/sec in the sample whose size $R = 1.5$ mm. We also notice that by changing the radius $R$ of the system one can control the width of the stability region. For example, in $R=100$ $\mu$m, the stability region of the HQV is $7 < \Omega < 8$ rad/sec.
![ (Color online) Stability region of the HQV in $\Omega$ versus $T/T_c$ ($R = 10$ $\mu$m and $\Delta T/T_c = 0.05$). A$_1$T denotes the A$_1$ phase texture where only $\left| \uparrow \uparrow \right>$ pair exist. The stability region of the HQV becomes wide at the vicinity of the transition temperature from the A-phase to the A$_1$-phase. []{data-label="HQV_omg_t"}](fig_HQV_omg_t.eps){width="80mm"}
![Stability region of HQV sandwiched between $\Omega _{c1}$ and $\Omega _{c2}$ as a function of $R$ for $T/T_c = 0.97$ and $\Delta T/T_c = 0.05$ ($H\simeq 100$ mT). The critical rotation $\Omega _{c1} = 0.05$ rad/sec is the extrapolated value for $R = 1.5$ mm. Inset shows the free-energy comparison for $R = 20$ $\mu$m, displaying the successive transitions from the AT to the HQV at $\Omega _{c1}$ and from the HQV to the SV at $\Omega _{c2}$. []{data-label="HQV_omg_r"}](fig_HQV_omg_r.eps){width="80mm"}
A pair of half-quantum vortices
-------------------------------
We examine the case that there are more than two half-quantum vortices in the system. There are two possibilities to enter the two HQVs in the system. Namely, the one case is that there are the HQVs whose winding number $(w_{\uparrow \uparrow +}, w_{\downarrow \downarrow +}, w_{\uparrow \uparrow -}, w_{\downarrow \downarrow -}) = (0,1,0,3)$ and $(1,0,3,0)$ in the system, the other case is that there are two (0,1,2,3)-HQVs. We calculate these two cases and conclude that the former is not solution of our calculation. The center of phase winding of the component of the OP does not coincide with the center of the system, which is not the energetically advantageous form for the angular momentum. Furthermore, there is no repulsion between HQVs because we assume that the gradient energy are the weak coupling limit form.
We investigate the case that there are two (0,1,2,3)-HQVs in the system. In the Fig. \[2HQV\_amp\] we show the texture of the two HQVs system (2HQV). In this texture, the component that has the phase winding are the same spin state. Thus the distance between them is determined by the competition of the kinetic energy loss and the energy gain from the angular momentum. However, we conclude these are just metastable texture from our calculation.
First of all, we consider the case that $\Delta T<0$ where the $\left| \downarrow \downarrow \right>$ pair is dominant. In this case, the texture of the 2HQV is close to two SVs system (2SV). From the analogy with the single HQV case, it can be guessed that critical rotation $\Omega _{(\mathrm{AT} \rightarrow \mathrm{2HQV})}$ where the free energy of the 2HQV cross to that of the AT is larger than the critical rotation $\Omega _{(\mathrm{2HQV} \rightarrow \mathrm{2SV})}$ where the free energy of the 2SV crosses to that of the AT. Namely, the 2HQV cannot be stable in this course.
On the other hands, we consider the case that $\Delta T > 0$ where the $\left| \uparrow \uparrow \right>$ pair is dominant. In this case the texture of the 2HQV are energetically close to the AT. From the analogy with the single HQV case, it is possible that $\Omega _{(\mathrm{AT} \rightarrow \mathrm{2HQV})}<\Omega _{(\mathrm{2HQV} \rightarrow \mathrm{2SV})}$. However, in these regions, the SV is more stable than the AT. From our calculation, we show that the critical rotation speed $\Omega _{(\mathrm{HQV} \rightarrow \mathrm{2HQV})}$ where the free energy of the 2HQV crosses to that of the HQV cannot smaller than the critical rotation where the free energy of the SV crosses to that of the HQV as shown in Fig. \[2HQV\_omg\_t\]. Namely the 2HQV cannot also be stable in this course.
![ (Color online) Order parameter amplitude (a) $A_{\uparrow \uparrow , \pm }$ and (b) $A_{\downarrow \downarrow, \pm }$ components normalized by $\pi k_B T_c$ for the metastable 2HQV system, $R=5$ $\mu$m, $T/T_c=0.95$, $\theta _H = 0$. We choose the spin quantization axis perpendicular to the plates; $\bm{z}$ direction. There are two vortices whose winding number 1 in the $\left| \downarrow \downarrow \right>$ space. The sense of the phase winding and its number are shown. The low figures are cross sections of each component across the vortex cores. []{data-label="2HQV_amp"}](fig_2HQV_amp.eps){width="80mm"}
![ Critical angular velocity $\Omega _{(\mathrm{HQV} \rightarrow \mathrm{2HQV})}$ from the HQV to the 2HQV for $R=5$ $\mu$m and $\Delta T/T_{c}$=0.05. Even if the temperature becomes higher, $\Omega _{(\mathrm{HQV} \rightarrow \mathrm{2HQV})}$ does not become lower than $\Omega _{c2}$. []{data-label="2HQV_omg_t"}](fig_2HQV_omg_t.eps){width="80mm"}
Summary an conclusions
======================
We have studied the relative stability of the three kinds of the order parameter textures; the A-phase texture, the singular vortex, and the half-quantum vortex for superfluid $^3$He-A phase confined by narrow parallel plates with the radius $R$. After showing in \[General\] that the spinful singular vortex can accommodate the Majorana quasi-particle localized at the core, we examine the condition that the Majorana quasi-particle exists, namely, the stability of the singular vortex and also half-quantum vortex which is known to accommodate the Majorana quasi-particle.
We have found that under the external field applied to either exactly parallel or perpendicular to the plates, the singular vortex state becomes stable above the critical rotation speed $\Omega _c$ which is evaluated. Thus the Majorana quasi-particles can be observed in those situation. The on-going experiment using the rotating cryostat at ISSP, Univ. Tokyo is particularly suited for this task because $\Omega _c \simeq 0.06$ rad/sec is within well controlled rotation speed. Moreover since singular vortex carries the Majorana quasi-particle localized at the core, we can design the braiding experiment by manipulating two or more Majorana quasi-particles.
Our calculations on the relative stability of the relevant textures are based on the standard Ginzburg-Landau functional which is firmly established through the cross checking between experiments and theories over several decades. Thus we believe that our results should be reliable not only qualitatively, but also quantitatively.
There are a few question remained to be clarified; (1) When the field is applied to exactly neither $\theta _H \neq 0$, or $\theta _H \neq \pi /2$, there appears the unwanted OP component $\Delta _{\uparrow \downarrow }$ whose magnitude is an order of $10^{-3}$ compared with the main component. At present we do not know whether or not this component is really harmful for the existence of the Majorana quasi-particle for the singular vortex. (2) We do not know how to take into account the so-called strong coupling effect in the gradient term in eq. (\[GLfun\]) for general order parameter state. This remains a future problem.
In conclusion, independent of the above two reservations (1) and (2) we emphasize the fact that the parallel plate geometry whose gap is an order of 10 $\mu$m for superfluid $^3$He-A phase is one of the best systems for detecting the Majorana quasi-particle.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank T. Mizushima and M. Ichioka for useful discussions.
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The following GL parameters are used: $\alpha_0=3.81\times10^{50}[J^{-1}m^{-3}]$, and $\beta_1=-3.75,
\beta_2=6.65,\beta_3=6.56,\beta_4=5.99, \beta_5=-8.53$ in units of $10^{99}[J^{-3}m^{-3}].$
[^1]: E-mail address: kawakami@mp.okayama-u.ac.jp
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The objective of this work is to study the applicability of various Machine Learning algorithms for prediction of some rock properties which geoscientists usually define due to special lab analysis. We demonstrate that these special properties can be predicted only basing on routine core analysis (RCA) data. To validate the approach core samples from the reservoir with soluble rock matrix components (salts) were tested within 100+ laboratory experiments. The challenge of the experiments was to characterize the rate of salts in cores and alteration of porosity and permeability after reservoir desalination due to drilling mud or water injection. For these three measured characteristics, we developed the relevant predictive models, which were based on the results of RCA and data on coring depth and top and bottom depths of productive horizons. To select the most accurate Machine Learning algorithm a comparative analysis has been performed. It was shown that different algorithms work better in different models. However, two hidden layers Neural network has demonstrated the best predictive ability and generalizability for all three rock characteristics jointly. The other algorithms, such as Support Vector Machine and Linear Regression, also worked well on the dataset, but in particular cases. Overall, the applied approach allows predicting the alteration of porosity and permeability during desalination in porous rocks and also evaluating salt concentration without direct measurements in a laboratory. This work also shows that developed approaches could be applied for prediction of other rock properties (residual brine and oil saturations, relative permeability, capillary pressure, and others), which laboratory measurements are time-consuming and expensive.'
author:
- |
Andrei Erofeev\
Skolkovo Institute of Science and Technology\
Skolkovo Innovation Center\
Building 3, Moscow, 143026, Russia\
`Andrei.Erofeev@skoltech.ru` Denis Orlov\
Skolkovo Institute of Science and Technology\
Skolkovo Innovation Center\
Building 3, Moscow, 143026, Russia\
`D.Orlov@skoltech.ru` Alexey Ryzhov\
Gazprom VNIIGAZ LLC\
box 130, Moscow, 115583, Russia\
`ARyzhov@vniigaz.gazprom.ru` Dmitry Koroteev\
Skolkovo Institute of Science and Technology\
Skolkovo Innovation Center\
Building 3, Moscow, 143026, Russia\
`D.Koroteev@skoltech.ru`
bibliography:
- 'sourses.bib'
title: Prediction of Porosity and Permeability Alteration based on Machine Learning Algorithms
---
Introduction {#intro}
============
Laboratory study of reservoir rock samples of a geologic formation (Core Analysis) is the direct way to determine reservoir properties and to provide accurate input data for geological models [@Andersen2013CoreTruth]. Geoscientists have developed a variety of approaches for measuring properties of reservoir rocks, such as porosity, permeability, residual oil saturation, and many others. The information obtained from core analysis aids in formation evaluation, reservoir development, and reservoir engineering [@mcphee2015core; @mahzari2018co]. It can be used to calibrate log and seismic measurements and to help with well placement, completion design, and other aspects of reservoir production.
Common applications of Core Analysis include [@Gaafar2015]:
- definitions of porosity and permeability, residual fluid saturations, lithology and prediction of possible production of gas, condensate, oil or water;
- definition of spatial distributions of porosity, permeability and lithology to characterize a reservoir in macro scale;
- definition of fluids distribution in a reservoir (estimation of fluids contacts, transition zones);
- performing special core analysis tests to define the most effective field development plan to maximize oil recovery and profitability.
Unfortunately, Core Analysis is expensive and tedious. Laboratory study requires careful planning to obtain data with minimum uncertainties [@BardOttesen2008Core]. Proper results of basic laboratory tests, provides the reservoir management team with a vital information for further development and production strategy.
Core analysis is generally categorized into two groups: conventional or routine core analysis (RCA) and special core analysis (SCAL) [@dandekar2006petroleum]. RCA generally refers to the measurements for defining porosity, grain density, absolute permeability, fluid saturations, and a lithologic description of the core. Samples for conventional core analysis are usually collected three to four times per meter [@monicard1980properties]. Fine stratification features and spatial variations in lithology may require more frequent sampling.
Probably the most prominent SCAL tests are two-phase or three-phase fluid flow experiments in the rock samples for defining relative permeability, wettability, and capillary pressure. In addition, SCAL tests may also include measurements of electrical and mechanical properties, petrographic studies and formation damage tests [@orlov2018self]. Petrographic and mineralogical studies include imaging of the formation rock samples through thin-section analysis, X-ray diffraction, scanning electron microscopy (SEM), and computed tomography (CT) scanning in order to obtain better visualization of the pore space [@dandekar2006petroleum; @liu2017pore; @soulaine2016micro]. SCAL is a detailed study of rock characteristics, but it is time-consuming and expensive. As a result, a number of SCAL measurements is much less than a number of RCA measurements (5-30% of RCA tests). In this way, SCAL data space requires correct expansion or extrapolation to the data space covered by RCA. To provide the expansion, core samples set used in SCAL tests should be highly representative and contain all the rock types and cover a wide range of permeability and porosity [@stewart2011well]. Even then, sometimes it is difficult to estimate correlations between conventional and special core analysis results and expand SCAL data to the available RCA dataset. There are few common approaches on stretching the SCAL data to RCA data space:
- typification (defining rock types with typical SCAL characteristics in certain ranges of RCA parameters);
- petrophysical models (SCAL characteristics included as parameters in functional dependencies between RCA characteristics);
- prediction models based on machine learning (RCA parameters used as features to predict SCAL characteristics).
The first approach leads to a significant simplification of reservoir characterization and is based on subjective conclusions. Petrophysical models allow predicting only a few of SCAL characteristics (basically capillary curves and residual saturations). The last approach looks more promising as it accounts for all the available features (measurements) and builds implicit correlations among the features [@meshalkin2018robotized; @tahmasebi2018rapid].
The purpose of this research is to demonstrate the performance of Machine Learning (ML) at maximizing the effect of RCA and SCAL data treatment. Machine Learning is a subarea of artificial intelligence based on the idea that an intelligent algorithm can learn from data, identify patterns and make decisions with minimal human intervention [@kotsiantis2007supervised].
Commonly spread feature of fields in Eastern Siberia is salts (the ionic compound that can be formed by the neutralization reaction of an acid and a base) presented in the pore volume of the deposits. Salts distribution in the reservoirs depends on a complex of sedimentation processes. Thus, the key challenge of this work is to develop prediction models, which can characterize the quantity of soluble rock matrix components (sodium chloride and other ionic compounds) and an increase of porosity and permeability after reservoir desalination due to drilling mud or water injection (ablation).
One of the main challenges for geoscientists is forecasting salts distribution in productive horizons together with porosity and permeability alteration due to the salts ablation. It is very important for:
- estimations of original porosity and permeability in wells as water-based drilling muds can change pore structure during wellbore drilling and coring,
- RCA and SCAL data validation and correction due to pore structure alteration during core sample preparation and consequent measurements,
- reservoir engineering (IOR&EOR based on water injection).
In this work salts content and alteration of porosity and permeability after desalination could be considered as a SCAL measurements because it is expensive and time-consuming. The procedure of alteration estimation includes porosity and permeability measurements before and after water injection in core samples and its desalinization during long-term one phase water filtration. Porosity and permeability before desalination, sample density and lithology and texture description are the RCA input data for our predictive models.
The significant benefit of ML predictive models is that one may not have to perform SCAL measurements for all the core samples, but can conduct prediction of the results [@unsal2005genetic]. Once a predictive model of any SCAL results is trained it could be effectively used for future forecasting. There are a lot of ML algorithms to build a predictive model [@hastie2001elements]. In our work we used the following algorithms: linear regression (with and without regularization) [@boyd2004convex; @freedman2009statistical], decision tree [@Quinlan1986], random forest [@ho1995random], gradient boosting [@Friedman2000Greedy], neural network [@haykin1994neural] and support vector machines [@Cortes1995SVN]. The choice of algorithm strongly depends on the considered problem, data quality and size of the dataset. For example, it would be unnecessary to build convolutional or recursive NN in our problem due to the small dataset size and its structure. However, more simple algorithms (mentioned above) could be adopted for discussed cases.
Accordingly, we have two goals in our research. First is to develop a predictive model of salts concentration using information of RCA and some additional data about coring depth and top and bottom depths of productive horizons. Second is to develop relevant predictive models of porosity and permeability.
The main innovation elements of the research are:
- Special experimental investigations of porosity and permeability increasing in core waterflooding tests,
- Validation of predictive algorithms to define the best predictive model,
- Accounting 10 features of core samples to predict porosity and permeability after rock desalination and 9 features to predict salts content;
- The high quality of models for prediction of porosity and permeability after rock desalination and rather good quality of model for evaluation of rock salinity.
Materials and Methods {#sec:2}
=====================
Hydrocarbon reservoir characterization {#sec:2.1}
--------------------------------------
The Chayandinskoye oil and gas condensate field is located in the Lensk district of Sakha (Yakutia) Republic in Russia and hosted towards the south of the Siberian platform within the Nepa arch. The field belongs to the Nepa-Botuobinsky oil and gas area, which contains rich hydrocarbons reserves. The main gas and oil resources are associated with the Vendian terrigenous deposits (Talakh, Khamakin and Botuobinsk horizons) which are overlapped by a thick series of the salt-bearing sediments.
Chayandinskoye field is characterized by a complex geological structure and special thermobaric formation conditions (reservoir pressure of 36-38 MPa, overburden stress of 50 MPa, temperature of 11-17$^{\circ}$C). The Vendian deposits consist predominantly of quartz sandstones and aleurolits with a low level of cementation and development of indentation and incorporation of grains. Another essential feature of the field is salts presented in the pore volume of the deposits. Salts distribution in the reservoir is exceptionally irregular due to various sedimentation processes: change in thermobaric condition during regional uplifts and erosional destruction of deposits, paleoclimate cooling and glaciation, in addition to filtration of brines through rock faults and fractured zones [@Ryzhov2014filtration]. Usually, the most common salt in rock matrix is sodium chloride (NaCl), but many other salts occur in varying smaller quantities. The same conclusion based on TDS analysis (measurement of the total ionic concentration of dissolved minerals in water) is correct for brine composition. Rocks analysis demonstrates that highly salinized formations are coarse-grain poorly sorted rocks with mass salts concentration – ranging from 4 to 30%. The porosity of the salted rocks is 1 - 8% (seldom $\geq$ 10%). After core desalination permeability could be increased up to 60 times and porosity - up to 2.5 times.
Dataset {#sec:2.2}
-------
We included the following features to the dataset for our prediction models:
- measurements of salts mass concentrations for core samples with various values of initial porosity and permeability, lithology, depth, horizons’ ID and wells’ ID;
- measurements of porosity and absolute permeability before and after desalination.
All tests are performed on 102 cylindrical core samples with 30 mm radius and 30 mm length. Sample preparation included delicate extraction in the alcohol-benzene mixture at room temperature (to avoid premature desalination) and drying up to constant weight. Absolute permeability was measured at ambient conditions in the steady-state regime of nitrogen flow. Porosity was determined by a gas-volumetric method based on Boyle’s Law [@rp401998recommended].
For salts ablation, we injected in each core sample more than 10 pore volumes of brine with low salinity (30 mg/cc). To enhance ablation, we also performed additional extraction in the alcohol-benzene mixtureto remove oil films preventing salts dissolution. After that, the samples were dried up to a constant weight to measure porosity and permeability after desalination. Results of porosity and permeability measurements (before and after salts ablation) are presented in figures \[fig1:a\], \[fig1:b\].
Measurements of salts mass concentrations for core samples were based on the data of sample weighting before and after desalination. The resulting expression for salts concentration could be defined as: $$\label{eq1:saltC}
C_{salt} = \frac{\Delta m}{m_0},$$ where $\Delta m$ - is the sample mass difference before and after ablation (g); $m_0$ - sample mass before desalination (g). All the measurements were made on oven-dry samples. Results of salts mass concentrations measurements presented in figure \[fig1:c\].
For salts concentration predictive model, we have used 9 features: formation top depth, formation bottom depth, initial (before desalination) porosity and permeability, sample depth adjusted to log depth, sample density (before desalination), average grain size (by lithology and texture description), sample colour and horizon ID. Average grain size was quantified from textual lithology description in the following way:
Gravel –- 1 mm;
Coarse sand -– 0.5 mm;
Medium sand –- 0.25 mm;
Fine sand -– 0.1 mm;
Coarse silt –- 0.05 mm;
Fine silt –- 0.01 mm;
Clay –- 0.005 mm.
For sample colour and horizon type, we have used the classification scheme containing 6 colour types and 3 horizon types. If the sample has any of 6 colours and any of 3 horizons we mark “1”, otherwise, we mark “0”.
For porosity and permeability predictive models we have used 9 previously described features plus salts concentration. All 10 features accounting in machine learning algorithms are presented in table \[tab:1\].
[lll]{} No. & Feature & Unit\
1 & salts concentration & g/g\
2 & formation top depth & m\
3 & formation bottom depth & m\
4 & porosity before desalination & %\
5 & absolute permeability before desalination & mD\
6 & sample depth & m\
7 & sample density & g/cc\
8 & average grain size & mm\
9 & color & -\*\
10 & depth horizon & -\*\
Prediction models {#sec:2.3}
-----------------
We have used 9 models: linear regression (simple, with L1 and L2 regularization), decision tree, random forest, gradient boosting (two different implementations with and without regularization) and neural network, support vector machines to compare their predictive power.
**Linear regression** [@hastie2001elements; @freedman2009statistical]. Simple linear regression expresses predicting value as linear combination of the features: $$\label{eq2:LR}
y = w_0x_0 + w_1x_1 + ... + w_px_p + b$$ where $y$ is predicting parameter; $x$ is a vector of features; $\mathbf{w}$ is a vector of optimizing coefficients.
The optimization problem for regression is given by the expression: $$\label{eq3:OP_LR}
\min_{\textbf{w}, b} F(\textbf{w}, b) = \frac{1}{m} \sum_{i=1}^{m} (\textbf{wx} + b - y_i)^2$$ Coefficients $\mathbf{w}$, $b$ are defined from a training set of data ($y_i$ is the actual value of the predicting parameter; $m$ is the size of the training set). Further, the linear regression model with predefined coefficients could be effectively applied to fit new data. This algorithm is implemented in LinearRegression() method of Python scikit-learn library [@Pedregosa2012Scikit-learn].
Sometimes regression with regularization works better than simple regression. In a case when we have many features, linear regression procedure leads to overfitting: enormous weights $\textbf{w}$ that fit the training data very well, but poorly predicts future data. “Regularization” means modifying the optimization problem to prefer small weights. To avoid the numerical instability of the Least Squares procedure regression with L2 and L1 regularizations are often applied [@hastie2001elements].
**Linear regression with L2 regularization (Ridge)** [@boyd2004convex]. This approach is based on Tikhonov regularization, which addresses the numerical instability of the matrix inversion and subsequently produces lower variance models: $$\label{eq4:LR_L2}
\min_{\textbf{w}, b} F(\textbf{w}, b) = \frac{1}{m} \sum_{i=1}^{m} (\textbf{wx} + b - y_i)^2 + \lambda ||\textbf{w}||^2_2$$ All variables have the same meanings as in linear regression case. Optimal regularization parameter $\lambda$ is chosen in a way to get the best model fitting while weights $\textbf{w}$ are small. This algorithm is implemented in Ridge() method of Python scikit-learn library [@Pedregosa2012Scikit-learn].
**Linear regression with L1 regularization (Lasso)** [@boyd2004convex]. While L2 regularization is an effective approach of achieving numerical stability and increasing predictive performance, it does not address another problem with Least Squares estimates, parsimony of the model and interpretability of the coefficient values [@Tibshirani2011Regression]. Another trend has been to replace the L2-norm with an L1-norm: $$\label{eq5:LR_L1}
\min_{\textbf{w}, b} F(\textbf{w}, b) = \frac{1}{m} \sum_{i=1}^{m} (\textbf{wx} + b - y_i)^2 + \lambda ||\textbf{w}||_1$$
This L1 regularization has many of the beneficial properties of L2 regularization, but yields sparse models that are more easily interpreted [@hastie2001elements]. L1 regularization algorithm is implemented in Lasso() method of Python scikit-learn library [@Pedregosa2012Scikit-learn].
**Decision Tree** [@Quinlan1986]. Decision tree is a tree representation of a partition of feature space. There are numbers of different types of tree algorithms, but here we will consider only CART (Classification and Regression Trees) [@breiman2017classification] approach. A classification tree is a decision tree which returns a categorical answer (class, text, color and other) while a regression tree is a decision tree which responses with an exact number. Figure \[fig2:a\] demonstrates a simple example of the decision tree in a case of two-dimensional space based on two features X1 and X2.
A decision tree consists of sets of leaves and nodes. One may builts very detailed deep tree with many nodes however such tree will suffer from overfitting. Usually, the maximum depth of the tree is restricted. To build a tree, the recursive partition is applied until a sufficient size of a tree would be obtained. The criteria for tree splitting is often given by Gini index or information gain criteria in classification case and mean squared error or mean absolute error in the regression case. These functions are used to measure the quality of a split and choose the optimal point of partition. Tree construction could involve not all input variables, so tree managed to demonstrate which variables are relatively important, but it could not rank the input variables. Decision tree algorithm exploited in this work is implemented in DecisionTreeRegressor() method of Python scikit-learn library [@Pedregosa2012Scikit-learn].
**Random Forest** [@ho1995random], [@Breiman2001]. The main idea of this method is to build many independent decision trees (ensemble of trees), train them on data subset and receive predictions. The algorithm uses bootstrap re-sampling to prevent overfitting. Bootstrapping is a re-sampling with replacement: bootstrap sets are built on initial data, where several samples are replaced with other repeating samples. Each tree is built on individual bootstrap set (so, for N tree estimators, we need N different bootstrap representations). Consequently, all trees are different as they are built on different datasets and hold different predictions. Then all trees are aggregated together after training and the final prediction is obtained by averaging (in the case of regression) predictions of each tree. One useful feature of Random Forest algorithms is that it could rank input features. It is implemented in RandomForestRegressor() method of Python scikit-learn library [@Pedregosa2012Scikit-learn].
**Gradient Boosting** [@Friedman2000Greedy]. This method uses “boosting” of the ensemble of weak learners (often decision trees). Boosting algorithm combines trees sequentially in such a way that the next estimator (tree) learns from the error of previous one: this method is iterative, and each next tree is built as a regression on pseudo-remainders. Similar to any other ML algorithm, Gradient Boosting uses loss function to minimize. Also, gradient descent is applied to minimize error (loss function) associated with adding a new estimator. The final model is obtained by combining the initial estimation with all subsequent estimations with appropriate weights. Gradient boosting method used in this study implemented in the scikit-learn library in GradientBoostingRegressor() method [@Pedregosa2012Scikit-learn]. Also, XGBoost library [@chen2016xgboost] was considered because it allows adding regularization to the model.
**Support Vector Machine (SVM)** [@Cortes1995SVN]. The idea of Support vector machine (in case of regression) is to find a function that approximates data in the best possible way. This function has at most $\epsilon$ deviation out from real train values $y_i$ and as flat as possible. Such linear function $f(\textbf{X})$ could be expressed as:
$$\label{eq11:SVM_linfun}
f(\textbf{X}) = w\textbf{X} + b$$
The optimization problem could be formulated in the following form:
$$\label{eq12:SVM_OP}
\begin{split}
&\min_{W} \frac{1}{2}||w||^2 \\
subject & \: to:
\begin{cases}
y_i - w \textbf{X} - b \leq \epsilon\\
w \textbf{X} - b - y_i \leq \epsilon
\end{cases}
\end{split}$$
Often, some errors beyond $\epsilon$ are allowed, which requires introducing of slack variables $\xi_i, \xi_i^*$ into the problem: $$\label{eq12b:SVM_OP}
\begin{split}
&\min_{W} \frac{1}{2}||w||^2 + C\sum_{i = 1}^{m}(\xi_i + \xi_i^*) \\
subject & \: to:
\begin{cases}
y_i - w \textbf{X} - b \leq \epsilon + \xi_i\\
w \textbf{X} - b - y_i \leq \epsilon + \xi_i^*\\
\xi_i, \xi_i^* \geq 0
\end{cases}
\end{split}$$
The solution of SVR problem is usually given in a dual form[@hsieh2008dual], which includes calculation of Lagrange multipliers $\alpha_i, \alpha'_i$. In this formulation solution looks as follows:
$$\label{eq13:SVM_DP}
f(x) = \sum_{i = 1}^{m}(\alpha'_i - \alpha_i)K(x_i, x) + b$$
where $K(x_i, x)$ is a kernel [@crammer2001algorithmic].
In this work, the standard Gaussian kernel has been applied:
$$\label{eq15:SVM_GaussKernel}
K(x, x') = exp\left(-\frac{||x - x'||^2}{2 \sigma ^ 2}\right)$$
SVM algorithm is implemented in SVR() method of Python scikit-learn library [@Pedregosa2012Scikit-learn].
**Neural Network** [@haykin1994neural; @Schmidhuber2014Deep]. Artificial neural network (ANN) is a mathematical representation of biological neural network (figure \[fig2:b\]). It consists of several layers with units that are connected by links [@mcculloch1943logical]. Each link has associated weight and activation level. Each node has an input value, an activation function and an output. In ANN information propagates (forward pass) from first (not hidden) layer with inputs to next hidden layer and then to further hidden layers until the output layer would be achieved. The value in each node of the first hidden layer obtained after calculation of activation function for the dot product of inputs and weights. Next hidden layer receives the output of the previous one and puts its dot product with weights to the activation an so on. Initially, all weights for all nodes are assigned randomly. ANN calculates first output with random weights. Then compare it to real value, calculate the error and adopts weights to obtain smaller error on the next iteration via backpropagation (ANN training algorithm). After training all weights are tuned, and one may make a prediction for a new data by passing them into an ANN which will calculate output via forward propagation throw all activations. [@rumelhart1986learning]. Scikit-learn library provides ANN representation in MLPRegressor() method [@Pedregosa2012Scikit-learn]. This implementation allows to indicate the number of hidden layers, number of nodes in each layer, activation function, learning rate and some other parameters.
Methodology of using machine learning algorithms {#sec:2.4}
------------------------------------------------
**Metrics.** To evaluate the accuracy of applied methods and compare them between each other, the following metrics have been exploited.
The coefficient of determination R2 is the proportion of the total (corrected) sum of squares of the dependent variable “explained” by the independent variables in the model. R2 score is a part of dispersion of dependent variable that is predictable by independent variables:
$$\label{eq16:R2}
R^2 = 1 - \frac{\sum_{i}(y_i - \hat y_i)^2}{\sum_{i}(y_i - \bar y)^2}\\$$
where $y_i$ are real values, $\hat y_i$ are predicted values, $\bar y$ is a mean value.
Mean squared error corresponds to quadratic errors:
$$\label{eq17:MSE}
MSE(y, \hat y) = \frac{1}{n_{samples}} \sum_{i = 0}^{n_{samples} - 1} (y_i - \hat y_i)^2$$
Mean absolute error corresponds to absolute errors: $$\label{eq18:MAE}
MAE(y, \hat y) = \frac{1}{n_{samples}} \sum_{i = 0}^{n_{samples} - 1} |y_i - \hat y_i|$$
**Cross-validation.** Since this study is limited in the amount of data, cross-validation (CV) has been applied. There are several cross-validation techniques. For example, k-fold, when we split whole data into k parts, then use first part for testing the performance of ML model after training it on the other k-1 parts. Next, we could take the second part for testing and the rest parts for training and so on k times. In the end we will have k different values of metrics - R2, MAE, MSE (eqs. (\[eq16:R2\]) –- (\[eq18:MAE\])) calculated for each fold. So, via cross-validation, we could obtain mean values of metrics and their standard deviation. Another cross-validation technique called random permutation supposes random division of data into train and test set, then data shuffle and we can obtain new division on test and train set. This process repeats n times and each time metrics are calculated. Similarly, in the end, we can evaluate the mean values of metrics and standard deviations. So, cross-validation allows not only calculate R2 (or MAE and MSE) for the test set but do it several times by independent data splitting into test and train sets. Since in our task we are restricted in the amount of data, cross-validation was used several times (for hyperparameters tuning, model’s performance estimation and making predictions to plot predicted results versus real data). Models building and estimation was done in 3 steps:
1. Hyperparameters tuning was done with the help of exhaustive grid search [@bergstra2012random]. This process allows to search through given ranges of each hyperparameter and define optimal values which led to the best R2 (or MAE, MSE etc.) scores. This process implemented in scikit-learn Python library [@Pedregosa2012Scikit-learn] in method GridSearchCV(). The method simply calculates CV score for each combination of hyperparameters in a given range. Random permutation approach with 10 repeats was chosen as CV iterator. GridSearchCV() allows not only find the best hyperparameters but also calculate metrics in the optimal point. However, just 10 repeats are not enough for very accurate final estimation of mean value and deviation, while taking more repeats require more computational time. So, final estimation with known hyperparameters would be done next.
2. Evaluation of the ML model with optimal hyperparameters (defined on the first step) was also done via CV with random permutation approach. However, here, we take 100 repeats, and it is enough for a fair result.
3. To plot predictions versus real values, we applied k-fold CV. In our particular case, we took k equals the number of samples. So, first of all, we train our model on all data without one point, then predict at this point (testing) by trained model. Next, we take the second point - remove it from the dataset and train model from scratch, then obtain prediction in point. This process was made for all data points (102 times). It allows to obtain predictions for all points and compare them to initial data visually.
Some researchers [@choubineh2019estimation] suggest the other way to validate the quality of ML models. The data records of the dataset are divided into training, validation and testing subsets, respectively. Where validation set is used for hyperparametres tuning while training on training set and test set is used for final model evaluation. However, unfortunately, a single random partition of the data on subsets could not be enough for correct model estimation due to non-uniformity of the dataset. Another random partition will give another value of metrics (R2, MAE, MSE and others). Single partition of the data is reasonable only in case of the big size of the dataset.
**Normalization of data.** Some machine learning methods require normalized data to proceed correctly (SVM and Neural Network), so in these cases, the data has been normalized by using the mean and standard deviation of the training set:
$$\label{eq19:normalize}
X_{scaled_i} = \frac{X_i - \bar X}{\sigma_X}$$
where $X_i$ is the feature vector, $\bar X$ is the mean value of feature vector, $\sigma_X$ is a variance.
Results & Discussion {#sec:3}
====================
In this section results of porosity, permeability and salts concentration predictions are presented, analyzed and compared. We made predictions of these reservoir properties on the basis of features described in section 2.2 by algorithms described in section 2.3. However, only Linear regression with L1 regularization was taken into account out of 3 Linear regression algorithms (because algorithms are very similar, implemented within the same library and results are very close, and regression with L1 regularization performs slightly better in our task). Also, two different libraries for Gradient boosting calculation was applied (they reported separately) because XGBoost library allows regularization while scikit-learn not. So, we reported results for 7 different algorithms. Each algorithm was adopted to the experimental data to get the best performance due to the cross-validation procedure.
Since our dataset is not such big from Big Data point of view and contains different measurements errors, it turns out that the proportion of its splitting via cross-validation procedure is important. We found out via grid-search that for our small dataset would be optimal to left 35% of data for testing on each cross-validation pass to obtain the estimation of algorithm performance.
Porosity prediction {#sec:3.1}
-------------------
Porosity model was the first ML model we built. Actually, the material balance equation defines specific dependence between porosity and salts concentration:
$$\label{eq20:salt_conc}
\phi = \phi_0 + \frac{\rho_{r}^{0}}{\rho_{ha}}C_{salt}$$
where $\phi$ is porosity after desalinization; $\phi_0$ is porosity before desalinization; $C_{salt}$ is the mass salts concentration; $\rho_{ha}$ is salt density; $\rho_r^0$ is core sample density before desalinization. Equation \[eq20:salt\_conc\] allows to estimate performance of ML models. In this case, data-driven algorithms should find hidden correlations within parameters and demonstrate appropriate predictive abilities. The appearance of the physical model gives an opportunity widely to test ML instruments.
For predictive models of porosity we took into account influence of all 10 characteristics of rock samples (Table \[tab:1\]). Almost all surveyed methods (except Decision tree) has demonstrated promising results and high values of determination coefficient in the case of porosity prediction. Table \[tab:2\] demonstrates results of models evaluation via cross-validation process: mean value and standard deviation. Here and further we reported linear regression only with L1 regularization, because of other implementations (without regularization and with L2 regularization) show very close results. However, this algorithm works slightly better. In general, the highest value of R2-metric corresponds to the lowest values of MSE and MAE metrics. In porosity case SVM, Neural network, Gradient boosting and Linear regressions have the best scores. The best two models are Support Vector Machines with $\textrm{R2} = 0.86 \pm 0.14, \: \textrm{MAE} = 0.82 \pm 0.19, \: \textrm{MSE} = 1.63 \pm 1.47$ and Neural network with $\textrm{R2} = 0.84 \pm 0.1, \: \textrm{MAE} = 0.94 \pm 0.16, \: \textrm{MSE} = 1.79 \pm 0.94$.
[llllllll]{} No. & Model & $R2$ & $\sigma_{R2}$ & MAE & $\sigma_{MAE}$ & MSE & $\sigma_{MSE}$\
1 & Linear regression with L1 regularization & 0.792 & 0.116 & 1.035 & 0.178 & 2.361 & 1.110\
2 & Decision tree & 0.490 & 0.226 & 1.749 & 0.322 & 5.882 & 2.244\
3 & Random forest & 0.675 & 0.090 & 1.427 & 0.213 & 3.815 & 1.073\
4 & Gradient boosting & 0.763 & 0.078 & 1.173 & 0.192 & 2.791 & 0.943\
5 & Gradient boosting (XGBoost) & 0.782 & 0.081 & 1.112 & 0.186 & 2.526 & 0.840\
6 & Support Vector Machines & 0.855 & 0.144 & 0.816 & 0.194 & 1.634 & 1.472\
7 & Neural Network & 0.841 & 0.098 & 0.935 & 0.164 & 1.793 & 0.935\
Grid search calculated optimal regularization value of L1 linear regression which equals 0.001. In a similar search process, optimal depth of decision tree was obtained - 7 and the optimal number of estimators (trees) for random forest - 150 along with the maximum depth of each tree - 8. For Gradient Boosting model the following parameters were selected: 16000 estimators (trees), maximum depth of each tree - 2, subsample - 0.7 (it means that each tree takes only 70% of initial data to fit, the next tree takes another 70% randomly etc., this idea helps to prevent overfitting), max-features - 0.9 (the concept is similar to subsample, the only difference is that instead of using part of the samples, algorithm takes part of features to fit each tree), regularization - 0.001. Neural Network architecture was also defined with the help of grid search since we have the small dataset and can calculate several architectures fast. It has 2 hidden layers with 2 and 4 nodes in each layer. SVM was built with Gaussian kernel which has two parameters to tune: C and gamma. The exhaustive search showed optimal gamma - 0.0001 and optimal C - 40000.
Performance of the SVR and MLPRegressor models could be demonstrated by plotting predicted values (via cross-validation) of porosity after ablation versus the actual values (figure \[fig3\]). One can see that the data points located along the mean line (bisectrix).
Python’s XGBoost allows arranging features concerning a degree of their influence on the prediction model [@Friedman2000Greedy], see figure \[fig4\]. The importance provides a score (referred as F score) that indicates how useful each feature was in the construction of the boosted decision trees within the model. This metric shows how many times the feature was used to split tree on [@freedman2009statistical]. The feature importance is then averaged across all of the decision trees within the model.
![Feature Selection in porosity model with XGBoost[]{data-label="fig4"}](08_XGB_poro.png){width="80.00000%"}
In figure \[fig4\] one can see that porosity and permeability before desalination and salts concentration have the most significant influence on the porosity prediction results. What is in a good correspondence with the geometric correlation between porosity and salts concentration (equation \[eq20:salt\_conc\]). Core sample density before desalinization ($\rho_r^0$) also among the five influential features in ML algorithms (Figure \[fig4\]). These observations demonstrate that predictive ML models simulate the same correlations as the physical model. A strong influence of the permeability before desalinization ($K_0$) on the porosity after desalinization ($\phi$) can be explained by a strong correlation between $K_0$ and $\phi_0$ (as, for example, in Kozeny-Carman equation form [@carman1956flow]). The next few features on Figure \[fig4\], which also significantly influence on porosity increase, are depths of sample, formation top and formation bottom. This fact confirms that the salts are distributed in formation non-uniformly and the distribution strongly depends on the geological condition of the reservoir. The colour features and the horizon types have the lowest influence on the prediction models.
Permeability prediction {#sec:3.2}
-----------------------
For permeability prediction all the methods also look promising and demonstrate high values of R2-metrics (table \[tab:3\]). In this case Support Vector Machines, Neural network and linear regressions show the best performance. The highest scores were obtained for SVR with $\textrm{R2} = 0.86 \pm 0.08, \: \textrm{MAE} = 105 \pm 20, \: \textrm{MSE} = 39957 \pm 17906$ and Linear regression with $\textrm{R2} = 0.85 \pm 0.07, \: \textrm{MAE} = 118 \pm 20, \: \textrm{MSE} = 40864 \pm 16434$
[llllllll]{} No. & Model & R2 & $\sigma_{R2}$ & MAE & $\sigma_{MAE}$ & MSE & $\sigma_{MSE}$\
1 & Linear regression with L1 regularization & 0.852 & 0.074 & 118 & 20 & 40864 & 16434\
2 & Decision tree & 0.677 & 0.196 & 167 & 42 & 90988 & 47737\
3 & Random forest & 0.775 & 0.103 & 139 & 36 & 68091 & 40814\
4 & Gradient boosting & 0.806 & 0.085 & 139 & 33 & 57696 & 29888\
5 & Gradient boosting (XGBoost) & 0.809 & 0.093 & 137 & 36 & 57345 & 36071\
6 & Support Vector Machines & 0.856 & 0.078 & 105 & 20 & 39957 & 17906\
7 & Neural Network & 0.850 & 0.123 & 108 & 34 & 40256 & 32030\
Similar to the previous section, one may define optimal hyperparameters of algorithms for permeability prediction via grid search [@bergstra2012random]. So, for Linear regression model, we have used L1 regularization with regularization parameter equals to 10. The decision tree was built with maximum depth - 10. 25 trees with the maximum depth of 8 were defined for Random Forest algorithm. In Gradient boosting model we obtained the following optimal values: 300 estimators with a maximum depth of each tree equals 2. Only 80% of the samples and 90% of the features have been used for each tree to fit the model and regularization parameter - 0.1. Neural network had 2 hidden layers with 77 and 102 nodes in each layer. SVM parameters included the Gaussian kernel with gamma equals to 0.0001 and C equals to 50000.
Performance of the SVR and Linear regression models is demonstrated on the mean line plots in figure \[fig5\]. Similarly, the prediction was made via cross-validation for all points. Data points are mainly located along the bisectrix, but generally matching between observed and predicted permeability is weaker than in porosity case.
The XGBoost method was also used to arrange features concerning their influence on the predictive model. In figure \[fig6\] one can see that porosity and permeability before desalination and salts concentration have the most influence on the permeability prediction results. It is very similar to the results of Feature Selection in porosity model. We also obtained that the next features, which significantly influence permeability increase, are connected with the geological condition of the reservoir (sample depth, formation top and bottom depths). The colour features and the horizon type also occurred in the lowest influencers.
![Feature Selection in permeability model with XGBoost[]{data-label="fig6"}](11_XGB_perm.png){width="80.00000%"}
Salts concentration prediction {#sec:3.3}
------------------------------
The last part of the research is devoted to the prediction of salts concentration. The models work worse and demonstrate rather weak performance with R2-metric hardly reaching 0.6. Only a few methods look promising and demonstrate reasonable values of R2-metrics (Table \[tab4\]). The best algorithms are Neural network, Gradient boosting and Random forest. Linear regression and Decision tree models are unacceptable with very small R2-metrics. R2 for Support Vector Regression reached almost 0.5. The best two models with the highest scores are Neural network (from MLPRegressor model of Scikit-learn) with $\textrm{R2} =0.66 \pm 0.25, \: \textrm{MAE} =0.77 \pm 0.18, \: \textrm{MSE} = 1.69 \pm 0.96$ and Gradient boosting with $\textrm{R2} =0.59 \pm 0.27, \: \textrm{MAE} =0.93 \pm 0.19, \: \textrm{MSE} = 2.23 \pm 1.13$. Also, in this case, very high standard deviation (up to 100%) in defining of R2, MSE and MAE metrics are obtained. This could be explained by non-uniformity of the experimental data.
[llllllll]{} No. & Model & R2 & $\sigma_{R2}$ & MAE & $\sigma_{MAE}$ & MSE & $\sigma_{MSE}$\
1 & Linear regression with L1 regularization & 0.014 & 0.385 & 1.579 & 0.174 & 5.747 & 1.996\
2 & Decision tree & 0.253 & 0.566 & 1.092 & 0.279 & 4.084 & 2.602\
3 & Random forest & 0.528 & 0.287 & 1.021 & 0.177 & 2.610 & 1.213\
4 & Gradient boosting & 0.593 & 0.265 & 0.929 & 0.191 & 2.227 & 1.131\
5 & Gradient boosting (XGBoost) & 0.565 & 0.299 & 0.950 & 0.189 & 2.276 & 1.275\
6 & Support Vector Machines & 0.484 & 0.411 & 0.951 & 0.1909 & 2.608 & 1.221\
7 & Neural Network & 0.664 & 0.251 & 0.774 & 0.175 & 1.686 & 0.959\
By analogy with porosity and permeability, we defined optimal hyperparameters of algorithms via grid search process [@bergstra2012random]. Regularization parameter for L1 regression is equal to 1.0. The optimal decision tree has a depth of 9. Random Forest performs better with 10 estimators of the depth of 1. Gradient Boosting model runs with 300 estimators of the depth of 10. 95% of samples and 50% of features have been used for training of each tree. Regularization parameter is small and equals to 0.00001. The neural network contains 3 layers and 55, 10, 86 nodes in each layer respectively. SVM was performed with gamma equals to 0.1 and C equals to 25.
Performance of the Gradient boosting and Neural network models are demonstrated on the mean line plots (figure \[fig7\]). Data points partially located along the mean line. Accordingly, the correlation between observed and predicted values is much weaker than in porosity and permeability cases.
Results of XGBoost features arrangement is in figure \[fig8\]. As one can see, porosity before desalination has the most substantial influence on the salts concentration prediction results. The next two features affecting the prediction results are sample depth and permeability before desalination. Results of this Feature Rating differs from the results obtained for porosity and permeability models. We can state that the prediction of porosity and permeability alteration is primarily controlled by its initial values and amount of salts in the pore volume. Salts concentration, in its turn, strongly depends not only on the initial porosity and permeability but also on the formation pattern characteristics, which are linked with post-sedimentation processes. Therefore, the prediction model attempts to learn from training dataset where and how strong these processes are developed in the certain reservoir beds with various depth and location in the oilfield (through the formation top and bottom depths).
![Feature Selection in salts concentration model with XGBoost[]{data-label="fig8"}](14_XGB_salt.png){width="80.00000%"}
Comparison of the predictive models with traditional approaches {#sec:3.4}
---------------------------------------------------------------
All obtained R2 scores with its variances for all algorithms are represented in figure \[fig9\]. The worst results could be associated with Decision tree method where we obtained not only the lowest values for R2 metric but the largest standard deviation of R2. Support Vector Machines and Linear regression demonstrate good results only for porosity and permeability prediction, but these methods are inappropriate for salts concentration prediction. The best machine learning method for prediction of all three petrophysical characteristics is Neural network in MLPRegression implementation. This algorithm demonstrates the most significant values of R2 metrics and the smallest standard deviation. Gradient boosting and Random forest could also be recommended as effective methods for prediction of salts concentration and permeability and porosity alteration due to salts ablation.
![R2 scores for all models[]{data-label="fig9"}](15_R2_results.png){width="100.00000%"}
The benefits of using machine learning models to estimate rock properties in comparison with standard one-feature approximation are obvious. When we talk about standard one-feature approximation, we assume the next approach. In case we do not know the law (or physical model) controlling the correlation between core sample characteristics and the target rock property the simplest and the fastest way to build the petrophysical model of the property is consistent single variable function analysis - finding consistently the target rock property functional dependence on each variable (characteristic of rock). The best single variable correlation in this approach could be considered as a one-feature approximation model. Instead of this expert approach ML algorithms allow building multi-feature approximations, which are more relevant to real rock properties correlations. Using one-feature approximation analysis, we found that porosity and permeability after ablation have the strongest correlation with salts concentration and corresponding dependencies showed in figures \[fig10\] and \[fig11\]. However, it does not mean that other core sample characteristics are useless. Over against, ML algorithms accounting all 10 characteristics should demonstrate better results. In our case, the one-feature approximation is the cubic polynomial which accounts for the dependency of porosity (permeability) alteration on salt content.
![Dependence of real and predicted porosity alteration on salts content. $\Delta\phi = \phi - \phi_0$, where $\phi_0$ is porosity before salts ablation and $\phi$ is porosity after salts ablation.[]{data-label="fig10"}](16_poro_comparison.png){width="100.00000%"}
![Dependence of real and predicted permeability alteration on salts content. $\Delta K = K - K_0$, where $K_0$ is permeability before salts ablation and $K$ is permeability after salts ablation.[]{data-label="fig11"}](17_perm_comparison.png){width="100.00000%"}
To compare ML methods with one-feature approximation approach predictions of porosity and permeability alterations were performed in three ways:
- using one-feature approximations for porosity and permeability dependencies on salts content (here we take into account only one feature, solid lines in figures \[fig10\] and \[fig11\]);
- using prediction models based on Neural network with all 10 predetermined features (dataset with salts concentration measurements, green dots in figures \[fig10\] and \[fig11\]);
- using prediction models based on Neural network with only 9 predetermined features (dataset without salts concentration measurements, red dots in figures \[fig10\] and \[fig11\]).
The last approach includes a two-step procedure. First, we estimate salts concentrations with corresponding prediction model and second, use this predicted values in porosity and permeability predictions. This approach is applicable in the case when we do not have experimental measurements of salts concentrations.
The more detailed comparison of machine learning models and standard one-feature approximation presented in figures \[fig12\], \[fig13\]. Here, the experimentally measured porosity (figure \[fig12\]) and permeability (figure \[fig13\]) were compared with the predicted values from ML (with and without salt content measurements) and one-feature approximation. These plots demonstrate the difference between three applied approaches for estimation of porosity and permeability after desalinization. Originally one feature approximation were obtained from dependency of porosity (permeability) alteration on salt content (figures \[fig10\], \[fig11\]). Than these alterations were used to obtain values of porosity and permeability after desalinization. Blue triangles in figures \[fig12\], \[fig13\] relate to ML model with known salt content. These points are located near the plot diagonal (ideal case). Black points relate to one-feature approximation, and these points are the most distant from diagonal. Yellow squares are approximation by ML with salt content preliminary predicted by ML. These predictions were made by using the approach described in section 2.4 in cross-validation as a third step (k-fold cross-validation with k equals to the number of samples). This method helps to compare approaches of permeability evaluation between each other and depict them on the plot. It does not evaluate the performance of ML algorithms well (methods evaluation was given in tables 2,3,4), because we have only one value of R2 (equation \[eq16:R2\]) and do not have confidence intervals. From figures \[fig12\], \[fig13\] one can see, that machine learning models work better.
![Comparison of prediction by ML model and one-feature approximation for porosity[]{data-label="fig12"}](porosity_full_comparison.png){width="70.00000%"}
![Comparison of prediction by ML model and one-feature approximation for permeability[]{data-label="fig13"}](permeability_full_comparison.png){width="70.00000%"}
Quantitative comparison of models is presented in Table \[tab5\]. Negative value of R2 for the one-feature approximation of permeability was obtained because of several points, which after recalculation (from normalized permeability to absolute values) lay very far from experimental data and add huge error in R2 calculation (equation \[eq16:R2\]). One may remove these points and obtain much better estimation, but ML models work with satisfying accuracy at these points, so, we’ve left results as it is to demonstrate the superiority of ML over old method. Table \[tab5\] confirm that simple polynomial regression taking into account only one feature at the same time works not so well as machine learning models considering many different features. We can also see that restriction of the dataset (case without salts concentration measurements) does not strongly affect prediction quality. However, it makes it possible to predict porosity and permeability alterations using only formation and core sample depths, initial porosity and permeability, rock density and lithology description. Feature ranking for salts concentration, permeability and porosity alterations models with Python’s XGBoost method demonstrate that sample colour and horizon have a feeble influence on the predictive models and could be excluded from feature list for further applications.
No. Metric ML (salts is known) ML (salts is unknown) one-feature approx.
----- -------------- --------------------- ----------------------- ---------------------
Porocity
1 R2 0.90623 0.73156 0.69427
2 MAE 0.69695 1.20640 1.41261
3 MSE 1.1197 3.20539 3.65065
Permeability
1 R2 0.87924 0.80541 -0.22504
2 MAE 95.923 140.877 188.569
3 MSE 37042.88 59687.87 375778.98
: Performance of the prediction models[]{data-label="tab5"}
Conclusion {#sec:4}
==========
In this paper applicability of various Machine Learning algorithms for prediction of some rock properties were tested. We demonstrated that three special properties of salted reservoirs of Chayandinskoye field could be predicted only basing on routine core analysis data. The target properties were:
- alteration of open porosity,
- alteration of absolute permeability,
- salts mass concentration.
After core desalination permeability could be increased up to 60 times and porosity - up to 2.5 times. Usually these characteristics are out of RCA scope because it is time-consuming and occasional analysis. It is very useful for reservoir development planning to have the predictive models in case of lack of this type of data. Porosity and permeability before desalination, sample density, lithology and texture description are the RCA input data for our predictive models.
To build relevant predictive models the dataset with results of 100+ laboratory experiments was formed. The main 9 features were: formation top depth, formation bottom depth, initial (before desalination) porosity and permeability, sample depth adjusted to log depth, sample density (before desalination), average grain size (by lithology and texture description), sample colour and horizon ID. These features were used to build the salts concentration predictive model. For porosity and permeability alteration prediction we additionally used the 10th feature – the salts concentration. From a technical point of view, there is no matter these concentrations measured or predicted with other ML model. We reported 7 algorithms:
- Linear regression with L1 regularization;
- Decision tree;
- Random Forest;
- Gradient boosting;
- XGBoost;
- Support vector machine;
- Artificial neural network.
The best two algorithms for porosity and permeability alteration prediction were Support Vector Machines with and Neural network. For permeability the Linear regression with regularization also showed good results. The best models demonstrate the determination coefficient R2 of 0.85+ for porosity and permeability. High precision of developed models looks to be helpful in decreasing of geological uncertainties in modelling of salted reservoirs. It was shown, that porosity and permeability before water intrusion along with the matrix density, sample depth and salts content are the most influencing features on permeability and porosity alteration.
The predictive model of salts concentration has been developed using the results of routine core analysis and data on core depth and top and bottom depths of productive horizons. The best algorithms here were Gradient boosting and Neural network. The highest coefficient of determination R2 for salts concentration in rocks equals 0.66. The precision of salts model is lower than the precision of porosity and permeability models. Nevertheless, the developed models allows to estimate the salts content in rocks without special experiments.
Combining all three models, it is also possible to make precise porosity and permeability alterations predictions using only a minimal volume of routine core analysis data: formation and core sample depths, initial porosity and permeability, rock density and lithology description. Accordingly, with these instruments geocientists and reservoir engineers can estimate the porosity and permeability alteration at waterflooding conditions having RCA measurements only.
It was shown that different algorithms work better in different models. However, the best machine learning method for prediction of all three parameters was two hidden layer Neural network in MLPRegression implementation. This algorithm gave the highest values of R2 metric and the smallest standard deviation. Gradient boosting and Random forest could also be recommended as alternative methods for predictions but with lower precision.
Finally, this work showed that machine learning methods could be applied for the prediction of rock properties, which laboratory measurements are time-consuming and expensive.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The complex oxide Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$ was synthesized and investigated by means of X-ray powder diffraction, electron diffraction, magnetic susceptibility measurements and band structure calculations. Its structure is similar to that of MV$_2$O$_5$ compounds (M = Na, Ca) giving rise to a spin system of coupled $S=1/2$ two-leg ladders. Magnetic susceptibility measurements reveal a spin gap-like behavior with $\Delta\approx 270$ K and a spin singlet ground state. Band structure calculations suggest Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$ to be a system of weakly coupled dimers in perfect agreement with the experimental data. Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$ provides an example of the modification of the spin system in layered vanadium oxides by cation substitution. Simple correlations between the cation size, geometrical parameters and exchange integrals for the MV$_2$O$_5$-type oxides are established and discussed.'
author:
- 'Alexander A. Tsirlin'
- 'Roman V. Shpanchenko'
- 'Evgeny V. Antipov'
- Catherine Bougerol
- Joke Hadermann
- Gustaaf
- Walter Schnelle
- Helge Rosner
title: 'Spin ladder compound Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$: synthesis and investigation'
---
Introduction
============
Complex vanadium oxides often present attractive examples of low-dimensional magnetic systems and exhibit unusual phenomena like spin gap formation or a spin liquid ground state.[@ueda; @vasiliev] These phenomena are not fully understood yet, therefore the search and investigation of novel compounds revealing low-dimensional spin systems is still a challenging task for solid state physics. Basically, there are two possible ways to search for new spin systems. The first way deals with the study of new structural types that are completely different from those of well-known and thoroughly investigated compounds. The second way suggests a systematic investigation of a number of related compounds. Both ways have been successfully realized during the last decade.[@na2v3o7; @li2vosio4; @ueda] However, the search for new structural types requires substantial intuition and luck. Systematic studies are more predictable if one can understand the relationship between the composition, crystal structure of the solid and the magnetic interactions in it.
 The projection of the [Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$]{} crystal structure along the $c$ axis (upper panel) and the system of coupled two-leg spin ladders (lower panel). In the upper panel circles denote the mixed Pb/Cd position, white arrows show three different nearest-neighbor exchange interactions. In the lower panel solid lines correspond to the legs of the ladders, dashed lines show the rungs and dots indicate inter-ladder coupling. The inter-layer hopping $t_{\perp}$ is not shown.](fig1)
MV$_2$O$_5$ compounds (M is an alkaline or alkaline-earth metal) are promising candidates for a systematic study of magnetic interactions in vanadium oxides. All of them (except for that with M = Cs) have layered structures with layers formed by edge- and corner-sharing VO$_5$ square pyramids (see Fig. \[structure\]). The M cations are situated between the layers. If M is a monovalent cation the average oxidation state of vanadium is +4.5 and both charge ordered (LiV$_2$O$_5$) or disordered (NaV$_2$O$_5$ above 35 K) states may be realized along with 1D magnetic behavior.[@ueda; @liv2o5_elstructure; @liv2o5; @charge; @liv2o5_nmr] Actually, the properties of NaV$_2$O$_5$ are even more complicated since different patterns of charge ordering have been recently found for this compound at low temperature.[@nav2o5_order; @nav2o5_devil]
The physics of MV$_2$O$_5$ compounds with divalent M cations (Ca, Mg) is simpler due to the absence of charge degrees of freedom. In both oxides vanadium atoms are tetravalent and a system of coupled $S=1/2$ two-leg spin ladders is realized. However, exchange interactions in CaV$_2$O$_5$ and MgV$_2$O$_5$ are quite different. In CaV$_2$O$_5$ the strongest interaction runs along the rung of the ladder ($J_1\approx 600$ K, see Fig. \[structure\]) while other interactions ($J_2,\ J_3,\ J_4$) do not exceed 100 – 150 K.[@korotin; @korotin_prl; @cav2o5_qmc; @cav2o5_nmr] Thus, weakly coupled spin dimers are formed and CaV$_2$O$_5$ reveals a spin gap $\Delta\approx J_1$ \[experimental values of $\Delta$ range from 560 to 660 K (Refs. )\]. In contrast, MgV$_2$O$_5$ reveals comparable nearest-neighbor interactions $J_1,\ J_2,\ J_3$ of about 100 K.[@korotin; @korotin_prl] The presence of a spin gap in MgV$_2$O$_5$ is still argued and the suggested values of $\Delta$ are quite small ($15-20$ K).[@mgv2o5_prb; @mgv2o5_japan; @onoda-mg; @raman] The change of V–O–V angles has been claimed in Ref. as the main reason for the differences between CaV$_2$O$_5$ and MgV$_2$O$_5$.
A general phase diagram for the coupled $S=1/2$ two-leg spin ladders is rather complicated revealing unusual ground states (namely, spiral order) and a number of quantum critical points.[@diagram] However, most of the regions of this diagram still lack experimental study. The remarkable difference between the magnetic properties of CaV$_2$O$_5$ and MgV$_2$O$_5$ suggests that an appropriate substitution in the M cation position of MV$_2$O$_5$ enables to modify the spin system and to shift between different regions of the phase diagram. Here we present the results of synthesis and investigation of [Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$]{} – an example of MV$_2$O$_5$-type compounds combining two different cations in the M position. The joint experimental and computational study of [Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$]{} allows us to carry out a comparison between several MV$_2$O$_5$-type compounds and to establish correlations between the cation size, geometrical parameters and exchange integrals.
Methods
=======
A powder sample of [Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$]{} was obtained by heating a mixture of Pb$_2$V$_2$O$_7$, CdO, V$_2$O$_3$ and V$_2$O$_5$ (ratio 2:4:1:1) in an evacuated and sealed silica tube at 700$^0$C for 24 hours. A change in the Pb:Cd ratio always resulted in the appearence of trace amounts of impurity phases in the samples. A change of heating conditions also led to the formation of impurities.
X-ray powder diffraction (XPD) data for the structure refinement were collected on a STOE diffractometer (transmission mode, CuK$\alpha_1$-radiation, Ge-monochromator, linear-PSD). The JANA2000 program [@jana] was used for Rietveld refinement.
Transmission electron microscopy (electron diffraction (ED) and EDXS) was performed using a Philips CM20 microscope with an Oxford Instruments Inca EDXS analyser.
Magnetic susceptibility measurements were carried out using a Quantum Design MPMS SQUID magnetometer in the range between 2 and 400 K at fields $\mu_0H$ of 0.01, 0.2 and 1 T.
Scalar relativistic band-structure calculations were performed using the full-potential non-orthogonal local-orbital minimum-basis scheme[@fplo] and the parametrization of Perdew and Wang for the exchange and correlation potential.[@perdew] V$(3s,3p,3d,4s,4p)$, Pb$(5s,5p,5d,6s,6p,6d)$, Cd$(4s,4p,4d,5s,5p)$ and O$(2s,2p,3d)$ states, respectively, were chosen as the basis set. All lower-lying states were treated as core states. The inclusion of V$(3s,3p)$, Pb$(5s,5p)$ and Cd$(4s,4p)$ states in the basis set was necessary to account for non-negligible core-core overlaps due to the relatively large extension of the corresponding wave functions. The O $3d$ and Pb $6d$ states were taken into account to get a more complete basis set. A $k$ mesh of 768 points in the Brillouin zone (189 in the irreducible part) was used. The spatial extension of the basis orbitals, controlled by a confining potential[@eschrig] $(r/r_0)^4$, was optimized to minimize the total energy. Convergence with respect to the $k$ mesh was carefully checked. A four-band tight-binding (TB) model was fitted to the resulting band structure to find the values of the relevant hopping parameters. The latter values were used to estimate the exchange integrals.
Results
=======
Crystal structure
-----------------
[Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$]{} has an orthorhombic unit cell with lattice parameters $a=11.3565(2)$ A, $b=3.6672(1)$ A, $c=4.9017(1)$ A. Single crystals of [Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$]{} can not be obtained since above 700$^0$C the compound gradually decomposes without melting. Therefore, the XPD data were used for the structural study.
The lattice parameters for [Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$]{} and CaV$_2$O$_5$ ($a=11.351$ A, $b=3.604$ A, $c=4.893$ A, space group $Pmmn$ [@onoda]) are rather close, therefore we used the crystal structure of CaV$_2$O$_5$ as an initial model for the Rietveld refinement. Indexing of XPD and ED (see below) patterns revealed only the reflection conditions $hk0,\ h+k=2n$ indicating two possible space groups: $Pmmn$ and $P2_1mn$, a subgroup of $Pmmn$. The lead and cadmium atoms were placed into $2b$ position (corresponding to calcium in CaV$_2$O$_5$) and occupancy factors $g_{\text{Pb}}=g_{\text{Cd}}=0.5$ were set.
------ ------ ----------- ----------- ------ ----------- -------------------
Atom Site Occupancy $x$ $y$ $z$ $U_{iso}$ (A$^2$)
Pb $2b$ 0.55(1) 0.75 0.25 0.1557(1) 0.0122(4)
Cd $2b$ 0.45(1) 0.75 0.25 0.1557(1) 0.0122(4)
V $4f$ 1 0.4063(1) 0.25 0.3883(4) 0.0076(7)
O(1) $4f$ 1 0.3884(3) 0.25 0.062(1) 0.008(1)
O(2) $4f$ 1 0.5733(4) 0.25 0.4732(8) 0.008(1)
O(3) $2a$ 1 0.25 0.25 0.549(1) 0.008(1)
------ ------ ----------- ----------- ------ ----------- -------------------
: \[coordinates\]Atomic coordinates and isotropic thermal displacement parameters for [Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$]{}
The refinement in both $Pmmn$ and $P2_1mn$ space groups gave similar results, therefore the highest possible symmetry ($Pmmn$) was chosen for the final refinement. The atomic displacement parameters of all oxygen atoms were constrained. Anisotropic strain broadening for the profile function was refined to achieve a proper fit for all reflections in the XPD pattern.[@foot1] Finally, occupancy factors of lead and cadmium atoms were also refined with a constraint $g_{\text{Pb}}+g_{\text{Cd}}=1$ yielding $g_{\text{Pb}}=0.55(1),\ g_{\text{Cd}}=0.45(1)$ and the [Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$]{} composition. This composition is in a good agreement with the EDXS result Pb:Cd:V = 1.2(1):1.0(1):3.8(1) but slightly differs from the initial Pb$_{0.5}$Cd$_{0.5}$V$_2$O$_5$ composition. The clear halo in the XPD pattern at $2\theta=15-25^0$ indicates the presence of an amorphous component in the prepared sample. One may roughly estimate the amount of this component by comparing integrated intensities for amorphous (halo) and crystalline (sharp peaks) phases and assuming their scattering powers to be similar. Thus, we find $\sim 25$% of the amorphous component.
The crystal structure of [Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$]{} is shown in Fig. \[structure\]. The experimental, calculated and difference Rietveld plots are shown in Fig. \[rietveld\]. Table \[coordinates\] presents atomic coordinates and thermal displacement parameters, while Table \[distances\] summarizes the main interatomic distances and angles in the crystal structure. The final residuals of the refinement are $R_P=0.022,\ R_F=0.020$, and $\chi^2=1.29$.
 Experimental, calculated and difference XRD patterns for [Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$]{}. The halo at $2\theta=15-25^0$ indicates the presence of an amorphous impurity in the sample under investigation.](fig2)
The crystal structure of [Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$]{} is very similar to that of CaV$_2$O$_5$. Vanadium atoms form VO$_5$ square pyramids with one short vanadyl bond and four longer equatorial bonds typical for tetravalent vanadium. The V$^{+4}$O$_5$ square pyramids with opposite orientation of vanadyl bonds are linked into zigzag chains via common edges. The pyramids in neighboring chains share their corners and form \[V$_2$O$_5$\] layers. Lead and cadmium atoms randomly occupy one crystallographic position between the vanadium-oxygen layers. The M–O distances are rather short for lead due to the presence of the smaller cadmium cation in the same position.
------------------ -------------------- ---------- --------------------
M–O(1) $4\times 2.639(3)$ V–O(1) 1.614(5)
M–O(2) $2\times 2.540(4)$ V–O(2) 1.942(5)
M–O(3) $2\times 2.336(4)$ V–O(2) $2\times 1.969(2)$
V–O(3) 1.942(3)
V–O(2)–V $(J_2)$ 137.3(2) V–O(2)–V 100.88(14)
V–O(3)–V $(J_1)$ 132.1(3)
------------------ -------------------- ---------- --------------------
: \[distances\]Selected interatomic distances (in A) and angles (in deg). M denotes the mixed Pb/Cd position
Lead and cadmium cations are fairly different in size (ionic radii 1.45 A and 1.21 A, respectively), therefore one would expect cation ordering in [Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$]{} resulting in a decrease of symmetry and superstructure formation. However, all ED patterns (Fig. \[ed\]) could be indexed in the $Pmmn$ space group with the cell parameters found from the XPD data. The appearance of the forbidden reflections $h00:h\neq 2n$ in the $[0\bar 11]$ zone is due to double diffraction since these reflections disappear after rotating the crystal away from the perfect orientation. Thus, one may conclude that lead and cadmium atoms in [Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$]{} are randomly distributed.
![\[ed\]Electron diffraction patterns along the \[001\] and $[0\bar 11]$ directions.](fig3)
We also prepared Pb$_{1-x}$Cd$_x$V$_2$O$_5$ samples with . The MV$_2$O$_5$-type phase was detected for $0.3\leq x\leq 0.7$, but single phase samples were obtained only for $x=0.5$. The lattice parameters for the CaV$_2$O$_5$-type phase were almost identical to those refined for [Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$]{}. Thus, the [Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$]{} phase has no considerable homogeneity range.
Magnetic susceptibility
-----------------------
The $\chi(T)$ curve of [Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$]{} (Fig. \[suscept\]) reveals a broad maximum, typical for low-dimensional spin systems. The fast decrease of the susceptibility below the maximum may be indicative of a spin gap while the upturn below 30 K is usually attributed to the paramagnetic impact of impurities and defects. In general, the $\chi(T)$ curve of [Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$]{} is similar to that for CaV$_2$O$_5$.[@onoda] Nevertheless, in CaV$_2$O$_5$ the susceptibility maximum is at $T_{\max}\approx 350$ K while in the case of [Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$]{} $T_{\max}$ is half this value (about 170 K). The position of the susceptibility maximum is usually a characteristic of the strongest exchange interaction in the system. Therefore one may suppose that the spin systems of Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$ and CaV$_2$O$_5$ are qualitatively similar, but a slight change of the crystal structure results in a weakening of the magnetic interactions.
 Magnetic susceptibility ($\chi_{\text{exp}}$) *vs.* temperature for [Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$]{} measured at $\mu_0H=0.2$ T. Circles show experimental points while the solid line is the fit with the isolated dimer model. The inset shows the intrinsic susceptibility $\chi_{int}$ of [Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$]{}: $\chi_{int}=\chi_{exp}-\chi_0-C/T$ (see equation \[eq\]).](fig4)
CaV$_2$O$_5$ is considered to be a system of weakly coupled dimers, therefore we follow Ref. and fit the experimental curve in Fig. \[suscept\] with the expression $$\chi=\chi_0+C/T+\dfrac{Ng^2\mu_B^2}{k_BT}\dfrac{1}{e^{\Delta/k_BT}+3}
\label{eq}$$ where $\chi_0$ is a temperature-independent term, $C/T$ corresponds to the paramagnetic signal of defects (impurities) while the last term corresponds to a system of isolated dimers. $\Delta$ is the spin gap value and $N=2N_{\text{A}}$ since there are two vanadium atoms per formula unit. We found a good fit with $\chi_0=4.0(1)\cdot 10^{-4}$ emu/mol, $C=3.1(1)\cdot 10^{-3}$ emu$\cdot$K/mol, $g=1.55(2),\ \Delta=270.7(3)$ K. The $g$ value is somewhat lower than one can expect for V$^{+4}$ ($g=1.93-1.96$, see, for example, Ref. ). The underestimate of $g$ may be caused by the presence of the amorphous impurity in the samples under investigation. According to XRD (see above) the samples contain an appreciable ($\sim 25$%) amount of the amorphous component that may really lead to an error in the sample weight and result in the decrease of $g$.[@foot2] The deviation of the spin system of [Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$]{} from the simple dimer model may be another reason for the decrease of $g$. However, the influence of this factor is much weaker than that of the amorphous component. We are convinced that a spin dimer model is the best choice for the reliable fitting of the present experimental data, since the use of the improved model (including four exchange integrals, see \[section-band\]) makes the fit very unstable.
The subtraction of $\chi_0$ and of the paramagnetic contribution $C/T$ shows that the intrinsic susceptibility of [Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$]{} drops down to zero at low temperatures (see the inset of Fig. \[suscept\]) indicating a spin singlet ground state.
Band structure and exchange interactions {#section-band}
----------------------------------------
Fitting of experimental data provides a phenomenological way to study magnetic interactions. However, the crystal structure of [Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$]{} results in a rather complicated spin system with several different types of nearest-neighbor interactions (see Fig. \[structure\]), therefore a mere phenomenological description of the system may appear unreliable. A microscopic picture will provide additional information about exchange interactions in the system.
 Total and atomic resolved density of states for the hypothetical PbV$_2$O$_5$ compound with the structure of [Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$]{}. The Fermi level is at zero energy.](fig5)
A direct estimate of the exchange interactions can be derived on the basis of band structure data. The latter one is usually obtained by density-functional calculations. However, there is an additional difficulty in case of [Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$]{} since lead and cadmium atoms randomly occupy one crystallographic position. Sophisticated techniques (like CPA, VCA) or supercell calculations are required to treat correctly such type of disorder. Nevertheless, one may try a more simple way in order to study exchange interactions in [Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$]{}. According to the recent study of CaV$_2$O$_5$ and MgV$_2$O$_5$ [@korotin] the exchange integrals in these compounds are not sensitive to the type of metal cation (Ca or Mg) but depend on the geometry of the \[V$_2$O$_5$\] layer (namely, V–O–V angles). Therefore, we used the structural parameters of [Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$]{} (Table \[coordinates\]) but set the occupancy factors $g_{\text{Pb}}=1,\ g_{\text{Cd}}=0$, or vice versa. Thus, the band structure for the two hypothetical compounds (PbV$_2$O$_5$ and CdV$_2$O$_5$) was calculated.
The density of states (DOS) plots for PbV$_2$O$_5$ and CdV$_2$O$_5$ are very similar, therefore we show only one of them in Fig. \[dos\]. The states below $-2$ eV are mainly formed by oxygen orbitals and provide the bonding between vanadium, lead or cadmium and oxygen. The highest occupied states have predominantly vanadium character with an admixture of oxygen. Note that gapless energy spectra of insulating transition metal compounds are a typical failure of LDA due to an underestimate of strong electron-electron correlations in the V $3d$ shell. The energy gap is readily reproduced by means of LSDA+$U$ (see, for example, Ref. ).
 Band structure of PbV$_2$O$_5$ (upper panel) and CdV$_2$O$_5$ (lower panel) near the Fermi level. Black dots indicate the contribution of the V $3d_{xy}$ states. Thick solid lines show the fit of the tight-binding model to the LDA band structure.](fig6)
According to simple crystal field considerations, the lowest (and hence occupied) vanadium $3d$ orbital in the V$^{+4}$O$_5$ square pyramid is the $d_{xy}$ one. There are four vanadium atoms in the unit cell of [Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$]{}, therefore we find near the Fermi level four bands formed by V $d_{xy}$ orbitals (Fig. \[bands\]). These bands are suitable for the tight-binding fit.
Similar four-band tight-binding models were constructed for both PbV$_2$O$_5$ and CdV$_2$O$_5$. The leading interactions of the models are shown in Fig. \[structure\]. A number of long-range interactions were also included in the model in order to provide a proper fit. Hopping parameters $t$ corresponding to the main interactions are listed in Table \[hopping\]. Other $t$’s were found to be less than 0.020 eV, therefore one can neglect them while considering the overall magnetic behavior of the spin system.
(meV) $t_1$ $t_2$ $t_3$ $t_4$ $t_{\perp}$
-------------- ------- ------- ------- ------- -------------
PbV$_2$O$_5$ 159 63 72 57 $-17$
CdV$_2$O$_5$ 147 26 52 98 $-14$
Averaged 153 44.5 62 77.5 $-15.5$
(K) $J_1$ $J_2$ $J_3$ $J_4$ $J_{\perp}$
303 26 50 78 3
: \[hopping\]Tight-binding hopping parameters $t$ (in units of meV) and derived exchange integrals $J$ (in units of K)
The hopping parameters of PbV$_2$O$_5$ and CdV$_2$O$_5$ are quite similar. If one considers the spin lattice in terms of coupled ladders, then the largest hopping ($t_1$) corresponds to the rungs of the ladders. Other nearest-neighbor hoppings are about half this value. The significant difference between the values of $t_2$ and $t_4$ for PbV$_2$O$_5$ and CdV$_2$O$_5$ may be caused by the influence of the metal cation on the corresponding interactions. Nevertheless, the basic feature of the magnetic interactions in both compounds is the same: strong coupling along the rungs of the ladder. Now, the $t$ values can be averaged and antiferromagnetic contributions to the exchange integrals are estimated as $J_i^{{\text{AFM}}}=4\bar t_i^2/U$ where $U$ is the effective on-site Coulomb repulsion.
In general, one should also take into account ferromagnetic contributions since $J_i=J_i^{{\text{AFM}}}+J_i^{{\text{FM}}}$. Unfortunately, the estimation of $J_i^{{\text{FM}}}$ implies LSDA+U calculation, and the latter one is a difficult task for a partially disordered structure of [Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$]{}. According to Refs ferromagnetic interactions in layered vanadium oxides are relatively weak. For that reason we neglect $J_i^{{\text{FM}}}$ and assume that $J_i=J_i^{{\text{AFM}}}$. Remarkable agreement between experimental and calculated values of $J_1$ (see below) *a posteriori* justifies this approach.
We set $U=3.6$ eV according to Ref. and find $J_1^{{\text{AFM}}}=303$ K. This value may be compared with the spin gap $\Delta\approx 270$ K found by our fit of the susceptibility curve. In case of isolated dimers the spin gap is equal to the intradimer interaction and $J_1\approx\Delta\approx 270$ K in perfect agreement with the computational result. Other nearest-neighbor interactions are at least four times weaker than $J_1$, hence [Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$]{} may be considered as a system of weakly coupled dimers. The interlayer coupling is very weak ($J_{\perp}\approx 3$ K) as one can expect for the layered structure with the magnetically active $d_{xy}$ orbitals parallel to the layers.
Discussion
==========
Experimental and computational data clearly show that [Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$]{} is similar to CaV$_2$O$_5$ and reveals a spin system of weakly coupled dimers. However, the intradimer interactions in these compounds differ by a factor of two. Now we will try to uncover a structural evidence of this change. The comparison of [Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$]{} with two known MV$_2$O$_5$ compounds (M = Ca, Mg) allows us to establish reliable correlations between exchange integrals and geometrical parameters.
The various magnetic interactions in the MV$_2$O$_5$-type compounds have different origin. $J_1$ and $J_2$ run via superexchange paths, $J_3$ is a superposition of direct V–V exchange and $90^0$ superexchange. $J_4$ corresponds to a more complicated superexchange path that involves two oxygen atoms and probably the M cation as well. Further we will focus on $J_1$ and $J_2$ since only single V–O–V paths may provide reasonably simple correlations between exchange integrals and geometrical parameters. In general, such paths are characterized by three parameters: V–O–V angle and two corresponding V–O distances. In MV$_2$O$_5$ compounds the parameters are related, therefore two parameters are sufficient: the angle and one distance (V–O or V–V). The geometrical parameters corresponding to $J_1$ and $J_2$ are listed in Table \[exchange\].
-------------------------------------- ----------- -------------- ---------------------- ----------- -------------- ----------------------
$J_1$ (K) $d$(V–V) (A) $\angle$VO(3)V (deg) $J_2$ (K) $d$(V–V) (A) $\angle$VO(2)V (deg)
[Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$]{} 303 3.550(5) 132.1(3) 26 3.667(7) 137.3(2)
CaV$_2$O$_5$ 608 3.493 132.9 122 3.604 135.3
MgV$_2$O$_5$ 92 3.372 117.6 144 3.692 141.1
-------------------------------------- ----------- -------------- ---------------------- ----------- -------------- ----------------------
The values in Table \[exchange\] are easily understood if one considers the change of the crystal structure caused by the decrease of the cation size. In CaV$_2$O$_5$ calcium (ionic radius $r=1.26$ A) has eight-fold coordination and the \[V$_2$O$_5$\] layers are flat. Smaller cations like magnesium ($r=1.03$ A) require a decrease of the coordination number (six in case of Mg) that is achieved by the corrugation of the layers (see Fig. \[layers\]). Edge-sharing connections of the square pyramids are rigid in contrast to corner-sharing ones, therefore the corrugation of the layers results in a significant change of the V–O(3)–V angle. The averaged ionic radius of the effective Pb$_{0.55}$Cd$_{0.45}$ cation (1.33 A) is even larger than that of calcium and the layers remain flat. For that reason, angles remain almost constant but V–V distances (and hence V–O distances) increase.
 Comparison of the MgV$_2$O$_5$ (upper panel) and CaV$_2$O$_5$ (lower panel) structures. The small size of the Mg cation results in a corrugation of the \[V$_2$O$_5$\] layers.](fig7)
Now we turn to the exchange integrals. According to Ref. the dramatic decrease of $J_1$ in MgV$_2$O$_5$ compared to that in CaV$_2$O$_5$ is caused by the decrease of the angle. In case of [Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$]{} the decrease of $J_1$ is less pronounced since interatomic distances are changed instead of the angle. This result is quite reasonable as magnetic interactions are known to be far more sensitive to the variation of angles than to interatomic distances.
The values of $J_2$ are more difficult to explain. The relevant bonding angles in the two compounds are quite similar. The V–V distance in MgV$_2$O$_5$ and [Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$]{} is higher than in CaV$_2$O$_5$ but $J_2\text{(MgV}_2\text{O}_5)\approx J_2\text{(CaV}_2\text{O}_5)\gg J_2$([Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$]{}). One may suggest two reasons for such changes of $J_2$. First, tight-binding model reveals significant next-nearest-neighbor diagonal interaction ($J_4\approx 80$ K) in [Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$]{} while lower estimates ($J_4\approx 20$ K) were given for CaV$_2$O$_5$ and MgV$_2$O$_5$.[@korotin; @korotin_prl] This explanation seems to be reasonable (at least qualitatively) since lead and cadmium have different relevant orbitals compared to calcium or magnesium. Therefore, Pb and Cd may mediate superexchange interactions better. Note however that we discuss antiferromagnetic contributions to the exchange integrals only (see Band structure section), and ferromagnetic contributions may slightly change the situation. Second, the consideration of the geometrical parameters corresponding to V–O–V path only may be an oversimplification as other factors (e.g. local environment of vanadium: all the V–O distances and O–V–O angles) are also important.
Thus, in case of complex superexchange paths it is difficult to impose a simple correlation between exchange integrals and geometrical parameters as the spin system in the MV$_2$O$_5$-type compounds is rather complicated. Nevertheless, we succeed in the explanation of the basic difference between [Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$]{} and CaV$_2$O$_5$. We found that the CaV$_2$O$_5$ structure is conserved rather well and the increase of the effective cation size in [Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$]{} results in a mere increase of interatomic distances. Both $J_1$ and $J_2$ are decreased and the type of the spin system (weakly coupled dimers) remains unchanged.
The corrugation of the \[V$_2$O$_5$\] layer (alike the one that happens in MgV$_2$O$_5$) would be necessary to induce a really significant change of the spin system. The layers in CaV$_2$O$_5$ are almost flat and the further increase of the cation size can not give rise to their distortion. It means that one has to introduce a smaller (namely, smaller than calcium) cation between the \[V$_2$O$_5$\] layers in order to achieve an appreciable change of the spin system. The number of the appropriate cations is rather limited, therefore one may try to use a pair of different cations instead. The present study demonstrates that even two cations with fairly different size may randomly accommodate interstices between the vanadium-oxygen layers. This approach seems to be very promising for the search of new layered vanadium oxides since combinations of two different cations provide an easy way to adjust the cation size and modify the spin system in a controlled manner.
In conclusion, we prepared and investigated the complex vanadium oxide [Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$]{}. Its crystal structure is similar to that of CaV$_2$O$_5$. The new compound reveals a system of weakly antiferromagnetically coupled dimers with an intradimer interaction of about 270 K. In our approach using microscopic model with averaged hopping parameters we were able to describe the influence of structural changes on the magnetic interactions. Nearest-neighbor exchange interactions in the \[V$_2$O$_5$\] layers are decreased with the increase of the interatomic distances from CaV$_2$O$_5$ to [Pb$_{0.55}$Cd$_{0.45}$V$_2$O$_5$]{} while the type of spin system remains unchanged.
The authors acknowledge the financial support of RFBR (grant 07-03-00890), ICDD (GiA APS91-05), GIF (grant No. I-811-257.14/03) and the Emmy Noether Program. ZIH Dresden is acknowledged for computational facilities. A.Ts. is also grateful to the support of Alfred Toepfer Foundation and to MPI CPfS for hospitality.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'For $x\in {\mathbb{R}}^d- \{0\}$, in dimension $d=3$, we study the asymptotic behavior of the local time $L_t^x$ of super-Brownian motion $X$ starting from $\delta_0$ as $x \to 0$. Let $\psi(x)=((1/2\pi^2) \log (1/|x|))^{1/2}$ be a normalization, Theorem 1 implies that $(L_t^x-(1/2\pi|x|))/\psi(x)$ converges in distribution to a standard normal distributed random variable as $x\to 0$. For dimension $d=2$, Theorem 2 implies that $L^x_t-(1/\pi)\log(1/|x|)$ is $L^1$ bounded as $x\to 0$. To do this, we prove a Tanaka formula for the local time which refines a result in Barlow, Evans and Perkins \[1\].'
author:
- Jieliang Hong
title: Local behavior of local times of super Brownian motion
---
Introduction and main results
=============================
Introduction
------------
Super Brownian Motion arises as a scaling limit of critical branching random walk. Let $M_F=M_F({\mathbb{R}}^d)$ be the space of finite measures on ${\mathbb{R}}^d$ equipped with Borel $\sigma$- algebra $\mathfrak{B}({\mathbb{R}}^d)$ and $(\Omega,{\mathcal F},{\mathcal F}_t,P)$ be a filtered probability space. The $\mathit{Super}$-$\mathit{Brownian}$ $\mathit{ Motion}$ $X$ starting at $\mu\in M_F({\mathbb{R}}^d)$ is a continuous $M_F({\mathbb{R}}^d)$-valued adapted strong Markov process defined on $(\Omega,{\mathcal F},{\mathcal F}_t,P)$ with $X_0=\mu$ a.s. which is the unique in law solution of a martingale problem (see (1) below).\
For $0\leq t<\infty$, the weighted occupation time process is defined to be $$Y_t(A):=\int_0^t X_s(A) ds, \ A\in \mathfrak{B}({\mathbb{R}}^d).$$ If $\mu$ is a measure on ${\mathbb{R}}^d$ and $\psi$ is a real-valued function on ${\mathbb{R}}^d$, we write $\mu(\psi)$ for $\int_{{\mathbb{R}}^d} \psi(y) d\mu(y)$.\
Local times of superprocesses have been studied by many authors. Sugitani \[7\] has proved that given the joint continuity of $\mu q_t(x)=\int \mu(dy) \int_0^t p_s(x-y) ds$ in $(t,s)$, the local time $L_t^x$ has a jointly continuous version which satisfies that for any $\phi \in C_b({\mathbb{R}}^d)$, $$\int_0^t X_s(\phi) ds =\int_{{\mathbb{R}}^d} L_t^x \phi(x) dx.$$ $L_t^x$ is called the local time of $X$ at point $x\in {\mathbb{R}}^d$ and time $t>0$ and it also can be defined as $$L_t^x:=\lim_{\epsilon \to 0} \int_0^t X_s(p_\epsilon^x) ds,$$ where $p_\epsilon^x(y)=p_\epsilon(y-x)$ is the transition density of Brownian motion. In general, for any fixed $\epsilon>0$, $L_t^x-L_\epsilon^x$ is jointly continuous in $t\geq \epsilon$ and $x\in {\mathbb{R}}^d$.\
However, the condition of continuity of $\mu q_t(x)$ fails in $x=0$ when $\mu=\delta_0$ in $d=2$ and $d=3$ (joint continuity still holds for $L_t^x-L_\epsilon^x$). Our main result Theorem 1 gives precise information about the local behavior of local times of super-Brownian motion in dimension $d=3$. Let $x\in {\mathbb{R}}^d -\{0\}$ and $X$ be a super-Brownian motion initially in $\delta_0$, and $L_t^x$ be the local time of $X$ at time $t$ and point $x$. Theorem 1 tells us that as $x \to 0$ $L_t^x$ blows up like $1/|x|$ and has a variation like $\sqrt[]{\log 1/|x|}$. We can view this as an analogue to the classical Central Limit Theorem. For $d=2$, we derive a refined Tanaka formula in Proposition 3 compared to the one in \[1\] and Theorem 2 tells us that $\big|L_t^x-\frac{1}{\pi} \log 1/|x|\big|$ is $L^1$ bounded.
Notations and Properties of super-Brownian motion
-------------------------------------------------
We denote by $p_t(x)=(2\pi t)^{-d/2} e^{-|x|^2/2t}, t>0, x\in {\mathbb{R}}^d$ the transition density of d-dimensional Brownian motion $B_t$. Let $P_t$ be the corresponding Markov semigroup, then for any function $\phi$, $$P_t \phi(x)=\int p_t(y) \phi(x-y) dy.$$ Let $C_b^2({\mathbb{R}}^d)$ denotes the set of all twice continuously differentiable functions on ${\mathbb{R}}^d$ with bounded derivatives of order less than 2. It is known that super-Brownian motion $X$ solves a martingale problem (Perkins \[5\], II.5): For any $ \phi \in C_b^2({\mathbb{R}}^d)$, $$X_t(\phi)=X_0(\phi)+M_t(\phi)+\int_0^t X_s(\frac{\Delta}{2}\phi) ds,$$ where $M_t(\phi)$ is an ${\mathcal F}_t$ martingale such that $M_0(\phi)=0$ and the quadratic variation of $M(\phi)$ is $$[M(\phi)]_t=\int_0^t X_s(\phi^2) ds.$$
For the first two moments of Super-Brownian motion, Konno and Shiga \[4\] gives us $$E_{X_0} X_t(\phi)=X_0(P_t\phi),$$ and $$E_{X_0} \Big(X_t(\phi)^2\Big)=\Big(X_0(P_t\phi)\Big)^2+\int_0^t X_0\Big(P_s\big((P_{t-s}\phi)^2\big)\Big) ds.$$ We drop the subscript $X_0$ when there is no confusion.\
\
$\mathbf{Notations.}$ $c_3=1/2\pi$, $c_{3.1}=2c_3^2=1/2\pi^2$, $c_2=1/\pi$. The weird order here is to emphasize the dimension the constant is for.
Main result
-----------
(d=3) Let $\psi(|x|)=(c_{3.1} \log 1/|x|)^{1/2}$, and $X$ be a super-Brownian motion in $\mathbb{R}^3$ with initial value $\delta_0$. Then for each $0<t\leq \infty$ as $x\to 0$, we have
$$\Big(X,\frac{L_t^x-c_3 \frac{1}{|x|}}{\psi(|x|)}\Big) \xrightarrow[]{d} \Big(X,Z\Big)$$
where $Z$ is a random variable with standard normal distribution and independent of $X$. Moreover, convergence in probability fails.
(d=2) Let $X$ be a super-Brownian motion in $\mathbb{R}^2$ with initial value $\delta_0$. Then we have $$\limsup_{x \to 0} E\Big|L_t^x-c_2 \log\frac{1}{|x|}\Big | <\infty.$$
Proof of Theorem 1
==================
Fix $x\in {\mathbb{R}}^3-{\{0\}}$, we will use the Tanaka formula for local times of super-Brownian motion (see \[1\], Theorem 6.1). Let $\phi_x(y)=c_3/|y-x|$, under the assumption $X_0(\phi_x)=\delta_0(\phi_x)=c_3/|x|<\infty,$ we have $P_{\delta_0}-$ almost surely that
$$L_t^x=c_3 \frac{1}{|x|}+M_t(\phi_x)-X_t(\phi_x),$$
where $M_t(\phi_x)$ is an $\mathcal{F}_t$ martingale, with $M_0(\phi_x)=0$ and quadratic variation $$[M(\phi_x)]_t=\int_0^t X_s(\phi_x^2) ds=\int_0^t \int \frac{c_3^2}{|y-x|^2} X_s(dy) ds.$$ To prove Theorem 1, we need several propositions which are stated below and proofs of them will be shown in Section 2.2 after finishing the proof of Theorem 1.\
\
$\mathbf{Notations.}$ We define $g_x(y):=\log|y-x|$ for $x,y \in {\mathbb{R}}^3$.
For $d=3$, we have almost surely that $$X_t(g_x)=\delta_0(g_x)+ M_t(g_x)+\frac{1}{2} \int_0^t \int \frac{1}{|y-x|^2} X_s(dy) ds.$$
For $d>1$, we have $$\int_0^t \int \frac{1}{|y-x|} p_s(y) dy ds \leq \frac{2}{d-1} E|B_t|, \ \ \ \forall x.$$
Proof of Theorem 1
------------------
Before proceeding to the proof, we state some lemmas which will be used in proving Theorem 1.
For any $u,v \in {\mathbb{R}}^d-{\{0\}}$, we have $$\Big|\log \frac{|u+v|}{|v|}\Big| \leq \sqrt[]{\frac{|u|}{|v|}}+\sqrt[]{\frac{|u|}{|u+v|}}.$$
Let $f(u)=\sqrt[]{u}-\log(1+u)$ for $u\geq 0$. Observe that $f(0)=0$ and $$f'(u)=\frac{1}{2\sqrt[]{u}}-\frac{1}{1+u}=\frac{(\sqrt[]{u}-1)^2}{2\sqrt[]{u} (1+u)} \geq 0,$$ therefore $f(u)\geq 0$ and $\log(1+u) \leq \sqrt[]{u}$ for all $u\geq 0$.\
If $|u+v|\geq |v|$, then $$\Big|\log \frac{|u+v|}{|v|}\Big|=\log \frac{|u+v|}{|v|} \leq \log \frac{|u|+|v|}{|v|}\leq \sqrt[]{\frac{|u|}{|v|}}\leq \sqrt[]{\frac{|u|}{|v|}}+\sqrt[]{\frac{|u|}{|u+v|}}.$$ If $|u+v|\leq |v|$, then $$\Big|\log \frac{|u+v|}{|v|}\Big|=\log \frac{|v|}{|u+v|} \leq \log \frac{|v+u|+|u|}{|u+v|}\leq \sqrt[]{\frac{|u|}{|u+v|}}\leq \sqrt[]{\frac{|u|}{|v|}}+\sqrt[]{\frac{|u|}{|u+v|}}.$$ So Lemma 1 follows.
For any $t>0$, we have $$\limsup_{x\to 0} E\Big[ \Big(\int \frac{1}{|y-x|} X_t(dy)\Big)^2 \Big]<\infty.$$
$$\begin{aligned}
E\Big[ \Big(\int \frac{1}{|y-x|} X_t(dy)\Big)^2 \Big] &=& \Big[\int p_t(y) \frac{1}{|y-x|} dy \Big]^2\\
&+& \int_0^t \ ds \int p_s(y) \ dy \Big( \int p_{t-s}(y-z) \frac{1}{|z-x|} \ dz \Big)^2 .\end{aligned}$$
For the first term, $$\begin{aligned}
& &\int p_t(y) \frac{1}{|y-x|} dy\\
&\leq & 1+\int_{|y-x|<1} (\frac{1}{\sqrt[]{2\pi t}})^3 e^{\frac{-|y|^2}{2t}} \frac{1}{|y-x|} dy\\
&\leq & 1+ (\frac{1}{\sqrt[]{2\pi t}})^3 \int_{{\mathbb{R}}^3} \frac{1}{|y-x|} 1_{\{|y-x|<1\}} dy\\
&=& 1+ (\frac{1}{\sqrt[]{2\pi t}})^3 \ 4\pi \int_0^1 r^2 \ dr \ \frac{1}{r} <\infty. \end{aligned}$$ For the second term, we use Cauchy Schwarz to get $$\begin{aligned}
& &(\int p_{t-s} (y-z) \frac{1}{|z-x|} dz)^2\\
&\leq & \int p_{t-s}(y-z) dz \cdot \int p_{t-s}(y-z) \frac{1}{|z-x|^2} dz\\
&=& \int p_{t-s}(y-z) \frac{1}{|z-x|^2} dz,\end{aligned}$$ and by Chapman-Kolmogorov $$\begin{aligned}
& & \int_0^t \ ds \int p_s(y) \ dy \Big( \int p_{t-s}(y-z) \frac{1}{|z-x|} \ dz \Big)^2\\
&\leq & \int_0^t ds \int p_s(y) dy \int p_{t-s}(y-z) \frac{1}{|z-x|^2} dz\\
&=& \int_0^t ds \int \frac{1}{|z-x|^2} dz \int p_s(y) p_{t-s} (y-z) dy\\
&=& \int_0^t ds \int \frac{1}{|z-x|^2} \ dz \cdot p_t(z) =t \int \frac{1}{|z-x|^2} p_t(z) dz.\end{aligned}$$ Using the same trick in the first term, we get$$\int \frac{1}{|z-x|^2} p_t(z) dz \leq 1+ (\frac{1}{\sqrt[]{2\pi t}})^3 4\pi<\infty.$$ Therefore we get $$\limsup_{x\to 0} E\Big[ \Big(\int \frac{1}{|y-x|} X_t(dy)\Big)^2 \Big]<\infty.$$
For any $t>0$, $$\text{(i) } \limsup_{x\to 0} E\Big(X_t^2(g_{x})\Big) <\infty$$ and $$\text{(ii) } \limsup_{x\to 0} E\Big(M_t^2(g_{x})\Big) <\infty.$$
[ ]{} (i) For $|y-x|<1$, we bound $|g_{x}(y)|=\log 1/|y-x|$ by $1/|y-x|$, so $$\begin{aligned}
& & \limsup_{x\to 0} E \Big[ \Big(\int_{|y-x|<1} \log |y-x| X_t(dy)\Big)^2\Big] \\
& & \leq \limsup_{x\to 0} E\Big[\Big(\int \frac{1}{|y-x|} X_t(dy) \Big)^2\Big]<\infty\end{aligned}$$ according to Lemma 2.\
For $|y-x|\geq 1$, we bound $|g_{x}(y)|=\log |y-x|$ by $|y-x|$, so $$\begin{aligned}
& & E \Big[ \Big(\int_{|y-x|\geq 1} \log |y-x| X_t(dy)\Big)^2\Big] \leq E\Big[\Big(\int |y-x| \ X_t(dy) \Big)^2\Big]\\
&=& \Big(\int p_t(y) |y-x| \ dy\Big)^2+\int_0^t ds \int p_s(z) dz \Big(\int |y-x| \ p_{t-s} (z-y) dy\Big)^2.\end{aligned}$$ It is clear that the first term is finite for any $x$ and for the second term, $$\begin{aligned}
& & \int_0^t ds \int p_s(z) dz \Big(\int p_{t-s} (z-y) |y-x| dy\Big)^2\\
&\leq &\int_0^t ds \int p_s(z) dz \int p_{t-s} (z-y) |y-x|^2 dy\\
&=& \int_0^t ds \int p_{t} (y) |y-x|^2 dy<\infty.\end{aligned}$$ So $$\begin{aligned}
& & \limsup_{x\to 0} E\Big[\Big(X_t(g_{x})\Big)^2\Big]\\
&=&\limsup_{x\to 0} E\Big[\Big(\int_{|y-x|<1} \log |y-x| \ X_t(dy) +\int_{|y-x|\geq 1} \log |y-x| \ X_t(dy)\Big)^2\Big]\\
&\leq &2 \ \limsup_{x\to 0} E\Big[\Big(\int_{|y-x|<1} \log |y-x| \ X_t(dy)\Big)^2\Big] \\
& \ \ \ & +2 \ \limsup_{x\to 0} E\Big[\Big(\int_{|y-x|\geq 1} \log |y-x| \ X_t(dy)\Big)^2\Big]<\infty.\end{aligned}$$\
(ii) Since $M_t(g_x)$ is a martingale with quadratic variation $[M(g_x)]_t=\int_0^t X_s(g_x^2) ds$, we get $$\begin{aligned}
& & E\Big(M_t^2(g_x)\Big)=E \int_0^t X_s(g_x^2) \ ds= \int_0^t ds \int p_s(y) \Big(\log |y-x|\Big)^2 dy\\
&\leq & \int_0^t ds \int p_s(y) \frac{1}{|y-x|} 1_{\{|y-x|<1\}}dy+\int_0^t ds \int p_s(y) |y-x| 1_{\{|y-x|\geq 1\}}dy\\
&\leq & \int_0^t ds \int p_s(y) \frac{1}{|y-x|} dy+\int_0^t ds \int p_s(y) |y-x| dy. \ \ \ \ \ \
(\star)\end{aligned}$$ We use the fact that $\log u\leq \log(1+u) \leq \sqrt{u}$ for $u \geq 1$ by Lemma 1.\
By Proposition 2 in $d=3$, we get $$\int_0^t ds \int p_s(y) \frac{1}{|y-x|} dy \leq E |B_t|<\infty.$$ As it is obvious that the latter term in $(\star)$ above is finite, we get $$\limsup_{x\to 0} E\Big(M_t^2(g_{x})\Big) <\infty.$$
### Convergence in distribution
[ ]{} Observe that combining (3) and (4), we obtain $$[M(\phi_x)]_t=2 c_3^2 \Big(X_t(g_x)-\delta_0(g_x)- M_t(g_x)\Big).$$ Note that $\delta_0(g_x)= \log |x|=-\log 1/|x|$, so $$E\Big[\Big([M(\phi_x)]_t-c_{3.1} \log \frac{1}{|x|}\Big)^2\Big]= c_{3.1}^2 E\Big[\Big(X_t(g_x)-M_t(g_x)\Big)^2\Big]$$ where $c_{3.1}=2 c_3^2$. $$\begin{aligned}
& & E\Big[\Big(\frac{[M(\phi_x)]_t-c_{3.1} \log \frac{1}{|x|}}{c_{3.1} \log \frac{1}{|x|}}\Big)^2\Big]= \frac{c_{3.1}^2}{(c_{3.1} \log \frac{1}{|x|})^2} E\Big[\Big(X_t(g_x)-M_t(g_x)\Big)^2 \Big]\\
&\leq &\frac{2}{(\log \frac{1}{|x|})^2} \Big[E\Big(X_t^2(g_x)\Big)+E\Big(M_t^2(g_x)\Big)\Big] \to 0 \text{ as } x \to 0,\end{aligned}$$ by Lemma 3. Hence we have shown that $$\frac{[M(\phi_x)]_t}{c_{3.1} \log \frac{1}{|x|}} \xrightarrow[]{L^2} 1 \text{ as } x\to 0.$$ Since $\frac{[M(\phi_x)]_t}{c_{3.1} \log \frac{1}{|x|}}$ is the quadratic variation of martingale $\frac{M_t(\phi_x)}{\sqrt[]{c_{3.1} \log \frac{1}{|x|}}}$, using the Dubins-Schwarz theorem (see \[6\], Theorem V.1.6), we can find some Brownian motion $B^{x} (t)$ in dimension 1 depending on $x$ such that $$\frac{M_t(\phi_x)}{\sqrt[]{c_{3.1} \log \frac{1}{|x|}}}=B^{x} \Big(\frac{[M(\phi_x)]_t}{c_{3.1} \log \frac{1}{|x|}}\Big).$$ For any sequence ${\{x_n\}}$ that goes to 0, (6) implies that $$\tau_n:=\frac{[M(\phi_{x_n})]_t}{c_{3.1} \log \frac{1}{|x_n|}} \xrightarrow[]{\text{ P}} 1 \text{ as } n \to \infty,$$ and we claim that $$B_{\tau_n}^{x_n}=B^{x_n} \Big(\frac{[M(\phi_{x_n})]_t}{c_{3.1} \log \frac{1}{|x_n|}}\Big) \xrightarrow[]{d} Z,$$ where $Z\sim N(0,1)$ in dimension 1.\
In fact for any bounded uniformly continuous function $h(x)$, $\forall \ \epsilon>0, \exists \ \delta>0$ such that $|h(x)-h(y)|<\epsilon$ holds for any $x,y \in {\mathbb{R}}$ with $|x-y|<\delta$. So $$E|h(B_{\tau_n}^{x_n})-h(B_1^{x_n})| \leq \epsilon+ 2 \|h\|_{\infty} \cdot P(|B_{\tau_n}^{x_n}-B_1^{x_n}|>\delta),\\$$ and for any $\gamma>0$, we have $$\begin{aligned}
& & P(|B_{\tau_n}^{x_n}-B_1^{x_n}|>\delta) \\
&\leq& P(|B_{\tau_n}^{x_n}-B_1^{x_n}|>\delta,|\tau_n-1|<\gamma)+P(|\tau_n-1|>\gamma)\\
&\leq& P(\sup_{|s-1| \leq \gamma} |B_s^{x_n}-B_1^{x_n}|> \delta)+P(|\tau_n-1|>\gamma)\\
&=& P(\sup_{|s-1| \leq \gamma} |B_s-B_1|> \delta)+P(|\tau_n-1|>\gamma)\\
&<& \epsilon+P(|\tau_n-1|>\gamma), \text{ if we pick } \gamma \text{ small enough.}\end{aligned}$$ Since $\tau_n$ converge in probability to 1, for $n$ large enough, we have $P(|\tau_n-1|>\gamma)<\epsilon$ and so $$E|h(B_{\tau_n}^{x_n})-h(B_1^{x_n})|\leq \epsilon+ 2 \|h\|_{\infty} 2 \epsilon$$ and hence $$\frac{M_t(\phi_{x_n})}{\sqrt[]{c_{3.1} \log \frac{1}{|x_n|}}}=B_{\tau_n}^{x_n} \xrightarrow[]{d} Z,$$ where $Z\sim N(0,1)$. Recall that $\phi_{x_n}(y)=c_3 /|y-x_n|$ and by Lemma 2 $$\lim_{n\to \infty} E\Big[\Big(\frac{X_t(\phi_{x_n})}{c_{3.1} \log \frac{1}{|x_n|}}\Big)^2\Big]=0,$$ hence $$\frac{X_t(\phi_{x_n})}{\sqrt[]{c_{3.1} \log \frac{1}{|x_n|}}} \xrightarrow[]{\text{ p} } 0.$$ Combining (7) and (8), by Theorem 25.4 in Billingsley \[2\] , we have $$\frac{L_t^{x_n}-\frac{1}{|x_n|}}{\sqrt[]{c_{3.1} \log \frac{1}{|x_n|}}}=
\frac{M_t(\phi_{x_n})}{\sqrt[]{c_{3.1} \log \frac{1}{|x_n|}}}- \frac{X_t(\phi_{x_n})}{\sqrt[]{c_{3.1} \log \frac{1}{|x_n|}}} \xrightarrow[]{d} Z.$$ So any sequence that approaches $0$ converges in distribution to $Z$ as above, which implies that $$\frac{L_t^{x}-\frac{1}{|x|}}{\sqrt[]{c_{3.1} \log \frac{1}{|x|}}} \xrightarrow[]{d} Z \text{ as } x\to 0.$$ For $t=\infty$, let $\rho$ be the life time of super Brownian motion $X$, then $L_{\infty}^x=L_\rho^x$. Chp II.5 in Perkins \[5\] tells us that $\rho<\infty$ a.s.. Sugitani \[7\] gives us $$L_t^x-L_\epsilon^x \text{ is continuous in } x \text{ for any } 0<\epsilon<t,$$ with the initial condition being $\delta_0$.\
Fix $\epsilon$ small, we define $L_\rho^x-L_\epsilon^x=0$ if $\rho<\epsilon$. As $x\to 0$, we get $$\frac{L_\rho^{x}-L_\epsilon^{x}}{(c_{3.1} \log \frac{1}{|x|})^{1/2}} \to 0\text{ a.s.},$$ and by Theorem 25.4 in Billingsley \[2\] again we get$$\frac{L_\rho^{x}-\frac{c_3}{|x|}}{(c_{3.1} \log \frac{1}{|x|})^{1/2}} =\frac{L_\rho^{x}-L_\epsilon^{x}}{(c_{3.1} \log \frac{1}{|x|})^{1/2}} +\frac{L_\epsilon^{x}-\frac{c_3}{|x|}}{(c_{3.1} \log \frac{1}{|x|})^{1/2}} \xrightarrow[]{d} Z.$$ $\hfill\square$
### Remaining Part of Theorem 1
\(i) Fix $0<t\leq \infty$, let $Z_t^{x_n}$ denotes $(L_t^{x_n}-c_3/|x_n|)/(c_{3.1}\log 1/|x_n|)^{1/2}$. By tightness of each component in $(X,Z^{x_n}_t)$, we clearly have tightness of $(X,Z^{x_n}_t)$ as $x_n\to 0$, so it suffices to show all weak limit points coincide. Assume $(X,Z^{x_n}_t)$ converges weakly to $(X,Z)$ for some sequence $x_n\to 0$. Let $(X,Z)$ be defined on $(\tilde{\Omega},\tilde{{\mathcal F}}_t,\tilde{P})$ where $X$ is super-Brownian motion and $Z$ is standard normal under $\tilde{P}$.\
For any $0<t_1<t_2<\cdots<t_m$, let $\phi_0: {\mathbb{R}}\to {\mathbb{R}}$ and $\psi_i: M_F \to {\mathbb{R}}$, $1\leq i\leq m$ be bounded continuous, we have $$\lim_{n\to\infty} E\Big[\psi_1(X_{t_1})\cdots \psi_m(X_{t_m}) \phi_0(Z^{x_n}_t)\Big]=\tilde{E}\Big[\psi_1(X_{t_1})\cdots \psi_m(X_{t_m}) \phi_0(Z)\Big]$$ since we assume that $(X,Z^{x_n}_t)$ converge weakly to $(X,Z)$.\
Pick $\epsilon>0$ such that $\epsilon<t_1$ and $\epsilon<t$, by Sugitani \[7\], $$L_t^x-L_\epsilon^x \text{ is continuous in } x \text{ for any } 0<\epsilon<t$$ with the initial condition being $\delta_0$, when $n\to \infty$ we get $$Z_t^{x_n}-Z_\epsilon^{x_n}=\frac{L_t^{x_n}-L_\epsilon^{x_n}}{(c_{3.1} \log \frac{1}{|x_n|})^{1/2}} \to 0\text{ a.s.}.$$ and hence $$(0,Z_t^{x_n}-Z_\epsilon^{x_n}) \to (0,0) \text{ a.s.. }$$ By Theorem 25.4 in Billingsley \[2\] again $$(X,Z_\epsilon^{x_n})=(X,Z_t^{x_n})-(0,Z_t^{x_n}-Z_\epsilon^{x_n}) \text{ converge weakly to } (X,Z).$$ Therefore since $Z_\epsilon^{x_n} \in {\mathcal F}_\epsilon^{X}$, $$\begin{aligned}
I &=& \tilde{E} \Big[\psi_1(X_{t_1})\cdots \psi_m(X_{t_m}) \cdot \phi_0(Z)\Big]\\
&=& \lim_{n\to \infty} E \Big[\psi_1(X_{t_1})\cdots \psi_m(X_{t_m}) \cdot \phi_0(Z_\epsilon^{x_n})\Big]\\
&=& \lim_{n\to \infty} E \Big[E \Big( \psi_1(X_{t_1})\cdots \psi_m(X_{t_m}) \big| {\mathcal F}_\epsilon^X \Big) \cdot \phi_0(Z_\epsilon^{x_n})\Big]\\
&=&\lim_{n\to \infty} E \Big[E_{X_\epsilon} \Big( \prod_{i=1}^m \psi_i(X_{t_i-\epsilon})\Big) \cdot \phi_0(Z_\epsilon^{x_n})\Big]\end{aligned}$$ Define $$F_\epsilon(\mu)=E_\mu\Big(\prod_{i=1}^m \psi_i(X_{t_i-\epsilon})\Big)$$ for $\mu\in M_F$ and we prove by induction that $F_\epsilon \in C_b(M_F)$. For $m=1$ we have $$F_\epsilon(\mu)=E_\mu \Big( \psi_1(X_{t_1-\epsilon})\Big)=P_{t_1-\epsilon} \psi_1(\mu).$$ By Theorem II.5.1 in Perkins \[5\], if $P_t F(\mu) = E_\mu F(X_t)$, then $P_t: C_b(M_F) \to C_b(M_F)$ so $F_\epsilon = P_{t_1-\epsilon} \psi_1 \in C_b(M_F)$ since $\psi_1 \in C_b(M_F)$. Suppose it holds for $m-1$, then $$\begin{aligned}
F_\epsilon(\mu)&=&E_\mu \Big( \prod_{i=1}^m \psi_i(X_{t_i-\epsilon}) \Big)\\
&=& E_\mu \Big[ \prod_{i=1}^{m-2} \psi_i(X_{t_i-\epsilon}) \cdot E_\mu \Big( \psi_{m-1}(X_{t_{m-1}-\epsilon}) \psi_m(X_{t_m-\epsilon}) \big| {\mathcal F}_{t_{m-1}-\epsilon}^X \Big)\Big]\\
&=& E_\mu \Big[ \prod_{i=1}^{m-2} \psi_i(X_{t_i-\epsilon}) \cdot \psi_{m-1}(X_{t_{m-1}-\epsilon}) P_{t_m-t_{m-1}} \psi_m(X_{t_{m-1}-\epsilon}) \Big]\\
&=& E_\mu \Big[ \prod_{i=1}^{m-2} \psi_i(X_{t_i-\epsilon}) \cdot \tilde{\psi}_{m-1}(X_{t_{m-1}-\epsilon}) \Big]\end{aligned}$$ where $\tilde{\psi}_{m-1} $ defined to be $\psi_{m-1} P_{t_m-t_{m-1}} \psi_m$ is in $C_b(M_F)$. It is reduced to the case $m-1$ where we already have $F_\epsilon \in C_b(M_F)$, so it holds for case $m$.\
Therefore by the weak convergence of $(X,Z_\epsilon^{x_n})$ to $(X,Z)$, we have $$\lim_{n\to \infty} E \Big[F_\epsilon(X_\epsilon) \cdot \phi_0(Z_\epsilon^{x_n})\Big]=\tilde{E} \Big[F_\epsilon(X_\epsilon) \cdot \phi_0(Z)\Big]$$ and hence $$\begin{aligned}
I &=& \lim_{n\to \infty} E \Big[E_{X_\epsilon} \Big( \prod_{i=1}^m \psi_1(X_{t_i-\epsilon})\Big) \cdot \phi_0(Z_\epsilon^{x_n})\Big]\\
&=& \lim_{n\to \infty} E \Big[F_\epsilon(X_\epsilon) \cdot \phi_0(Z_\epsilon^{x_n})\Big]=\tilde{E} \Big[F_\epsilon(X_\epsilon) \cdot \phi_0(Z)\Big]\\
&=&\tilde{E} \Big[\tilde{E}_{X_\epsilon} \Big( \prod_{i=1}^m \psi_1(X_{t_i-\epsilon})\Big) \cdot \phi_0(Z)\Big]\\
&=& \tilde{E} \Big[\tilde{E} \Big( \psi_1(X_{t_1})\cdots \psi_m(X_{t_m}) \big| \tilde{{\mathcal F}}_\epsilon^X \Big) \cdot \phi_0(Z)\Big]\end{aligned}$$ Let $\epsilon \to 0$, by martingale convergence we have $$\tilde{E} \Big( \psi_1(X_{t_1})\cdots \psi_m(X_{t_m}) \big| \tilde{{\mathcal F}}_\epsilon^X \Big) \xrightarrow[]{L^1 } \tilde{E} \Big( \psi_1(X_{t_1})\cdots \psi_m(X_{t_m}) \big| \tilde{{\mathcal F}}_{0+}^X\Big)=\tilde{E} \Big( \psi_1(X_{t_1})\cdots \psi_m(X_{t_m}) \Big).$$ The equality follows from Blumental 0-1 law that $\tilde{{\mathcal F}}_{0+}^X$ is trivial. Therefore $$\begin{aligned}
I=& \tilde{E} \Big[\psi_1(X_{t_1})\cdots \psi_m(X_{t_m}) \cdot \phi_0(Z)\Big]\\
=& \lim_{\epsilon \to 0 }\tilde{E} \Big[ \tilde{E} \Big( \psi_1(X_{t_1})\cdots \psi_m(X_{t_m}) \big| \tilde{{\mathcal F}}_\epsilon^X \Big) \cdot \phi_0(Z)\Big]\\
=& \tilde{E} \Big[\tilde{E} \Big( \psi_1(X_{t_1})\cdots \psi_m(X_{t_m}) \Big) \cdot \phi_0(Z)\Big]\\
=& \tilde{E} \Big( \psi_1(X_{t_1})\cdots \psi_m(X_{t_m}) \Big) \cdot \tilde{E} \phi_0(Z)\end{aligned}$$
The above functionals are a determining class on $C([0,\infty),M_F) \times R$ and so we get weak convergence of $(X,Z^x_t) \to (X,Z)$ where the latter are independent.\
(ii) Suppose we find convergence in probability for $0<t\leq \infty$, $$\frac{L_t^{x_n}-\frac{1}{|x_n|}}{(c_{3.1} \log \frac{1}{|x_n|})^{1/2}} \xrightarrow[]{\text{ P} } Z$$ for some random variable $Z$, then it must converge in distribution to $Z$ as well, so $Z$ is a standard normal distributed random variable. By taking a further subsequence we may assume a.s. convergence holds:$$\frac{L_t^{x_n}-\frac{1}{|x_n|}}{(c_{3.1} \log \frac{1}{|x_n|})^{1/2}} \xrightarrow[]{\text{ a.s. } } Z.$$ By Sugitani \[7\], $$L_t^x-L_\epsilon^x \text{ is continuous in } x \text{ for any } 0<\epsilon<t$$ with the initial condition being $\delta_0$, we get $$\frac{L_t^{x_n}-L_\epsilon^{x_n}}{(c_{3.1} \log \frac{1}{|x_n|})^{1/2}} \to 0\text{ a.s.}.$$ Therefore $$\frac{L_\epsilon^{x_n}-\frac{1}{|x_n|}}{(c_{3.1} \log \frac{1}{|x_n|})^{1/2}} =\frac{L_t^{x_n}-\frac{1}{|x_n|}}{(c_{3.1} \log \frac{1}{|x_n|})^{1/2}} -\frac{L_t^{x_n}-L_\epsilon^{x_n}}{(c_{3.1} \log \frac{1}{|x_n|})^{1/2}} \to Z \text{ a.s.}$$ Because (9) holds for any $\epsilon>0$, we get $$Z \in \bigcap_{t>0} {\mathcal F}_t^X={\mathcal F}_{0+}^X,$$ and Blumenthal 0-1 law tells us that any event in ${\mathcal F}_{0+}^X$ is an event of probability $0$ or $1$, hence $Z$ is a.s. constant. This contradicts the fact that $Z$ is standard normal. So we get a contradiction by assuming that $(L_t^{x_n}-c_3/|x_n|)/(c_{3.1} \log \frac{1}{|x_n|})^{1/2}$ converges in probability. $\hfill\square$
Proof of Proposition 1 and 2
----------------------------
### Some useful lemmas
For any $0<\alpha<3$, there exists a constant $C=C(\alpha)$ such that for any $x\neq 0$ and $t>0$, $$\int_{{\mathbb{R}}^3} p_t(y) \frac{1}{|y-x|^\alpha} dy<C \frac{1}{|x|^\alpha}.$$
Fix $\delta=|x|/2$, $$\begin{aligned}
& & \int_{{\mathbb{R}}^3} p_t(y) \frac{1}{|y-x|^\alpha} dy \\
&\leq& \frac{1}{\delta^\alpha}+ \int_{|y-x|<\delta} p_t(y) \frac{1}{|y-x|^\alpha} dy.\end{aligned}$$ For $|y-x|<\delta$, we have $|y|\geq |x|-|y-x|>|x|-\delta=\delta$, therefore $$\begin{aligned}
& & \int_{|y-x|<\delta} (\frac{1}{2\pi t})^{3/2} e^{-\frac{|y|^2}{2t}} \frac{1}{|y-x|^\alpha} dy\\
&\leq& \int_{|y-x|<\delta} (\frac{1}{2\pi t})^{3/2} e^{-\frac{\delta^2}{2t}} \frac{1}{|y-x|^\alpha} dy\\
&=& (\frac{1}{2\pi t})^{3/2} e^{-\frac{\delta^2}{2t}} \int_0^\delta \frac{1}{r^\alpha} r^2 dr \cdot 4\pi\\
&=& 4\pi M(\delta) \cdot \frac{1}{3-\alpha} \delta^{3-\alpha}\end{aligned}$$ where $$M(\delta):=\sup_{t> 0} (\frac{1}{2\pi t})^{3/2} e^{-\frac{\delta^2}{2t}}
\overset{\underset{ u=\frac{\delta^2}{t} }{}}{=}\frac{1}{\delta^3} \sup_{u\geq 0} (\frac{u}{2\pi})^{3/2} e^{-u/2}:=C_0 \frac{1}{\delta^3}.$$ Therefore $$\int_{{\mathbb{R}}^3} p_t(y) \frac{1}{|y-x|^\alpha} dy < \frac{1}{\delta^\alpha}+4 \pi \cdot C_0 \frac{1}{\delta^3} \cdot \frac{1}{3-\alpha} \delta^{3-\alpha} =C(\alpha) \frac{1}{|x|^\alpha}.$$
For any $0<\alpha<3$, there exists a constant $C=C(\alpha)$ such that for any $x\neq 0$ and $t>0$, $$E\int_0^t \int \frac{1}{|y-x|^\alpha} X_s(dy) ds =\int_0^t ds \int_{{\mathbb{R}}^3} p_s(y) \frac{1}{|y-x|^\alpha} dy<C \frac{1}{|x|^\alpha} t.$$
It directly follows from Lemma 4.
In ${\mathbb{R}}^3$, for any fixed $s>0$ and $y\neq x$, we have $$\Delta_y P_s g_x(y)= \int p_s(y-z) \frac{1}{|z-x|^2}dz.$$
Idea of this proof is from Evans \[3\]. For any fixed $s>0$, $p_{s}(y)=(2\pi s)^{-3/2} e^{-|y|^2/2s} \in C_0^\infty({\mathbb{R}}^3)$, we have $$\|Dp_s\|_{L^\infty({\mathbb{R}}^3)}<\infty \text{ and }\|\Delta p_s\|_{L^\infty({\mathbb{R}}^3)}<\infty.$$ Here $Du=D_x u=(u_{x_1},u_{x_2}, u_{x_3})$ denotes the gradient of $u$ with respect to $x=(x_1,x_2,x_3).$\
For any $\delta \in (0,1)$, $$\begin{aligned}
& & \Delta_y \int_{{\mathbb{R}}^3} p_{s}(y-z) g_x(z)dz\\
&=& \int_{B(x,\delta)} \Delta_y p_{s}(y-z) g_x(z)dz+\int_{{\mathbb{R}}^3-B(x,\delta)} \Delta_y p_{s}(y-z) g_x(z)dz\\
&=:& I_\delta+J_\delta.\end{aligned}$$\
Now $$|I_\delta|\leq \|\Delta p_s\|_{L^\infty({\mathbb{R}}^3)} \int_{B(x,\delta)} |g_x(z)| dz\leq C \delta^3 |\log \delta| \to 0.$$ Note that $\Delta_y p_{s}(y-z)=\Delta_z p_{s}(y-z)$. Integration by parts yields $$\begin{aligned}
J_\delta &=& \int_{{\mathbb{R}}^3-B(x,\delta)} \Delta_z p_{s}(y-z) g_x(z)dz\\
&=& \int_{\partial B(x,\delta)} g_x(z) \frac{\partial p_{s}}{\partial \nu}(y-z) dz-\int_{{\mathbb{R}}^3-B(x,\delta)} D_z p_{s}(y-z) D_z g_x(z)dz\\
&=:& K_\delta+L_\delta,\end{aligned}$$ $\nu$ denoting the inward pointing unit normal along $\partial B(x,\delta).$ So $$|K_\delta|\leq \|Dp_s\|_{L^\infty({\mathbb{R}}^3)} \int_{\partial B(x,\delta)} |g_x(z)|dz \leq C \delta^2 |\log \delta| \to 0.$$ We continue by integrating by parts again in the term $L_\delta$ to find $$\begin{aligned}
L_\delta &=& \int_{{\mathbb{R}}^3-B(x,\delta)} p_{s}(y-z) \Delta_z g_x(z)dz-\int_{\partial B(x,\delta)} p_{s}(y-z) \frac{\partial g_x}{\partial \nu}(z) dz\\
&=:& M_\delta+N_\delta.\end{aligned}$$ Now $Dg_x(z)=\frac{z-x}{|z-x|^2}(z\neq x)$ and $\nu=\frac{-(z-x)}{|z-x|}=\frac{-(z-x)}{\delta}$ on $\partial B(x,\delta)$. Hence $\frac{\partial g_x}{\partial \nu}(z)=\nu \cdot Dg_x(z)=-\frac{1}{\delta} $ on $\partial B(x,\delta)$. Since $4\pi \delta^2$ is the surface area of the sphere $\partial B(x,\delta)$ in ${\mathbb{R}}^3$, we have $$N_\delta=4\pi \delta \cdot \frac{1}{4\pi \delta^2} \int_{\partial B(x,\delta)} p_{s} (y-z)dz \to 0\cdot p_{s}(y-x) = 0 \text{ as } \delta \to 0.$$ By direct calculation, we have $\Delta_z g_x(z)=\frac{1}{|x-z|^2}$ when $z \in {\mathbb{R}}^3-B(x,\delta)$, therefore $$M_\delta=\int_{{\mathbb{R}}^3-B(x,\delta)} p_{s}(y-z) \frac{1}{|x-z|^2} dz.$$ Lemma 4 gives $$\int p_{s}(y-z) \frac{1}{|x-z|^2} dz<\infty,$$ by Dominated Convergence Theorem, we have $$M_\delta=\int p_{s}(y-z) \frac{1}{|x-z|^2} 1_{\{|z-x|\geq \delta\}} dz \to \int p_{s}(y-z) \frac{1}{|x-z|^2} dz$$ as $\delta \to 0$.
### Proof of Proposition 1
Define $\eta\in C^\infty({\mathbb{R}}^d)$ by $$\eta(x):=C \exp\Big(\frac{1}{|x|^2-1}\Big)
1_{\{|x|<1\} },$$ the constant $C$ selected such that $\int_{{\mathbb{R}}^d} \eta dx=1$.\
Let $\chi_{n}$ be the convolution of $\eta$ and the indicator function of the ball $B_{n}=\{x: |x|<n\}$, we get $$\chi_{n} (x)=\int_{{\mathbb{R}}^d} 1_{\{|x-y|<n\}} \eta (y) dy=\int_{B_{1}} 1_{\{|x-y|<n\}} \eta(y) dy.$$ It is known that $\chi_{n}$ is a $C^{\infty}$ function with support in $B_{n+1}$ and for $x\in B_{n-1}$, we have $|x-y|<n$ since $|x|<n-1$ and $|y|<1$, so $$\chi_{n} (x)=\int_{B_{1}} 1_{\{|x-y|<n\}} \eta(y) dy=\int_{B_{1}} \eta(y) dy=1.$$ It’s easy to see that $\chi_n$ increases to $1$ as $n$ goes to infinity.
Recall that $g_x(y)=\log|y-x|$ and let $g_{n,x}(y)=g_x(y) \cdot \chi_n(y-x)$, then $$\begin{aligned}
P_{\epsilon} g_{n,x}(z)=&\int_{|y-x|<n-1} p_\epsilon(z-y) \log |y-x| dy\\
&+ \int_{n-1<|y-x|<n+1} p_\epsilon(z-y) \log |y-x| \chi_n(y-x) dy \in C_b^{2},\end{aligned}$$ and $$\begin{aligned}
\Delta_z P_{\epsilon} g_{n,x}(z)=&\int_{|y-x|<n-1} \Delta_z p_\epsilon(z-y) \log |y-x| dy\\
&+ \int_{n-1<|y-x|<n+1} \Delta_z p_\epsilon(z-y) \log |y-x| \chi_n(y-x) dy.\end{aligned}$$ It is easy to see that $P_{\epsilon} g_{n,x}(z)$ and $\Delta_z P_{\epsilon} g_{n,x}(z)$ increases to $P_{\epsilon} g_{x}(z)$ and $\Delta_z P_{\epsilon} g_{x}(z)$ respectively.\
For $P_{\epsilon} g_{n,x} \in C_b^2({\mathbb{R}}^3)$, we have following equation hold a.s., $$X_t(P_{\epsilon} g_{n,x} )=\delta_0(P_{\epsilon} g_{n,x})+ M_t(P_{\epsilon} g_{n,x})+\int_0^t X_s(\frac{\Delta}{2} P_{\epsilon} g_{n,x}) ds,$$ where $M_t(P_{\epsilon} g_{n,x})$ is a martingale with quadratic variation $$[M(P_{\epsilon} g_{n,x})]_t=\int_0^t X_s\Big((P_{\epsilon} g_{n,x})^2\Big) ds.$$ As $n$ goes to infinity, by monotone convergence, we have $$X_t(P_{\epsilon} g_{n,x} ) \to X_t(P_{\epsilon} g_{x} ),\ \delta_0(P_{\epsilon} g_{n,x})\to \delta_0(P_{\epsilon} g_{n,x}),$$and $$\int_0^t X_s(\frac{\Delta}{2} P_{\epsilon} g_{n,x}) ds \to \int_0^t X_s(\frac{\Delta}{2} P_{\epsilon} g_{x}) ds.$$ Note that $$\begin{aligned}
& & E \int_0^t X_s\Big((P_{\epsilon} g_x)^2\Big) ds\\
&=& \int_0^t ds \int p_s(y) dy \Big(\int p_\epsilon(y-z) \log |z-x| dz \Big)^2\\
&\leq& \int_0^t ds \int p_s(y) dy \int p_\epsilon(y-z) \Big(\log |z-x| \Big)^2 dz \\
&=& \int_0^t ds \int p_{s+\epsilon}(z) \Big(\log |z-x| \Big)^2 dz \\
&\leq& \int_0^{t+\epsilon} ds \int p_{s}(z) \Big(\log |z-x| \Big)^2 dz <\infty.
\end{aligned}$$ The last is by $(\star)$ in Lemma 3 when calculating $E(M_t^2(g_x))$. So we conclude that $$E\Big[\Big(M_t(P_{\epsilon} g_{n,x})-M_t(P_{\epsilon} g_x)\Big)^2\Big]=E\int_0^t X_s\Big((P_{\epsilon}g_{n,x}-P_{\epsilon}g_x)^2\Big)ds \to 0$$ by Dominated Convergence Theorem since $$(P_{\epsilon}g_{n,x}-P_{\epsilon}g_x)^2 \to 0 \text{ and } (P_{\epsilon}g_{n,x}-P_{\epsilon}g_x)^2 \leq 4 (P_{\epsilon} g_x)^2.$$ So the $L^2$ convergence of a martingale $M_t(P_{\epsilon} g_{n,x})$to $M_t(P_{\epsilon} g_{x})$ follows, which makes $M_t(P_{\epsilon} g_{x})$ a martingale as well. By taking a subsequence we have the following equation holds a.s. $$X_t(P_{\epsilon} g_x)=\delta_0(P_{\epsilon} g_x)+ M_t(P_{\epsilon} g_x)+\int_0^t X_s(\frac{\Delta}{2} P_{\epsilon} g_x) ds,$$ where $M_t(P_{\epsilon} g_x)$ is a martingale with integrable quadratic variation $$[M(P_{\epsilon} g_x)]_t=\int_0^t X_s\Big((P_{\epsilon} g_x)^2\Big) ds.$$ Let $\epsilon$ goes to $0$, we will show in (i)-(iv) the $L^1$ convergence of each term in (10) to the corresponding term in Proposition 1, i.e. $$X_t(g_x)=\delta_0(g_x)+ M_t(g_x)+\frac{1}{2} \int_0^t \int \frac{1}{|y-x|^2} X_s(dy) ds.$$ (i)
First we have $$\begin{aligned}
& & \bigg|\delta_0(P_\epsilon g_x)-\delta_0(g_x)\bigg|=\bigg|\int_{{\mathbb{R}}^2} p_\epsilon(y) \log|y-x| dy-\log|x| \bigg|\\
&\leq & \int_{{\mathbb{R}}^3} p_\epsilon(y) \Big|\log|y-x| -\log|x|\Big| dy=E\Big[\Big|\log|B_\epsilon-x|-\log|x|\Big|\Big]\\
&=& E\Big[\Big|\log\frac{|B_\epsilon-x|}{|x|}\Big|\Big].\end{aligned}$$
As a result, $$\begin{aligned}
& & \bigg|\delta_0(P_\epsilon g_x)-\delta_0(g_x)\bigg| \leq E\Big[\Big|\log\frac{|B_\epsilon-x|}{|x|}\Big|\Big]\\
&\leq& E \Big[\sqrt[]{\frac{|B_\epsilon|}{|x|}}\Big]+E\Big[\sqrt[]{\frac{|B_\epsilon|}{|B_\epsilon-x|}}\Big] \text{ by Lemma 1}\\
&\leq& \frac{1}{|x|^{\frac{1}{2}}} E|B_\epsilon|^{\frac{1}{2}} + \Big(E|B_\epsilon|\Big)^{\frac{1}{2}}\cdot \Big(E\frac{1}{|B_\epsilon-x|}\Big)^{\frac{1}{2}} \\
&\leq & \frac{1}{|x|^{\frac{1}{2}}} E|B_\epsilon|^{\frac{1}{2}} + \Big(E|B_\epsilon|\Big)^{\frac{1}{2}}\cdot \Big(C \frac{1}{|x|}\Big)^{\frac{1}{2}} \text{ by Lemma 4}\\
&\to& 0 \text{ as } \epsilon \to 0.\end{aligned}$$ (ii)
Let $B_t$ and $B_t^{'}$ be two independent standard Brownian motion in ${\mathbb{R}}^3$, $$\begin{aligned}
& & E\Big[\Big|X_t(P_\epsilon g_x)-X_t(g_x)\Big|\Big] \leq E \Big[X_t\Big(|P_\epsilon g_x-g_x|\Big)\Big]\\
&=& \int p_t(y) dy \bigg|\int p_\epsilon(z) \log|z-(y-x)|dz-\log|y-x|\bigg|\\
&\leq& \int p_t(y) dy \int p_\epsilon(z) \bigg|\log|z-(y-x)|-\log|y-x|\bigg| dz\\
&=& E\Bigg(\Big|\log \frac{|B'_\epsilon-(B_t-x)|}{|B_t-x|}\Big|\Bigg)\\
&\leq& E \Big[\sqrt{\frac{|B'_\epsilon| }{|B_t-x|}}\Big]+ E \Big[\sqrt{\frac{|B'_\epsilon|}{|B'_\epsilon+B_t-x|}}\Big] .\end{aligned}$$ Since $E\sqrt[]{|B'_\epsilon|} \to 0$ and by Lemma 4 $$E \sqrt{\frac{|B'_\epsilon| }{|B_t-x|}}= E \sqrt[]{|B'_\epsilon|} \cdot E\sqrt[]{\frac{1}{|B_t-x|}} \leq E \sqrt[]{|B'_\epsilon|} \cdot C\frac{1}{|x|^{\frac{1}{2}}}\to 0.$$ For the second term, we use Cauchy Schwarz Inequality, $$\bigg(E \sqrt{\frac{|B'_\epsilon|}{|B'_\epsilon+B_t-x|}}\bigg)^2 \leq E|B'_\epsilon| \cdot E\frac{1}{|B'_\epsilon+B_t-x|}= E|B'_\epsilon| \cdot E\frac{1}{|B_{t+\epsilon}-x|}.$$ So again by Lemma 4$$E \sqrt{\frac{|B'_\epsilon|}{|B'_\epsilon+B_t-x|}} \leq \bigg(E|B'_\epsilon|\bigg)^{1/2} \cdot \bigg(C \frac{1}{|x|^{\frac{1}{2}}}\bigg)^{1/2} \to 0 \text{ as } \epsilon \to 0.$$ and the $L^1$ convergence of $X_t(P_\epsilon g_x)$ to $X_t(g_x)$ follows.\
\
(iii)
Next we deal with $M_t(P_\epsilon g_x)-M_t(g_x)$ and we use its quadratic variation to compute its second moment. $$\begin{aligned}
& & \Big(E|M_t(P_\epsilon g_x)-M_t(g_x)|\Big)^2 \leq E \Big[\Big(M_t(P_\epsilon g_x)-M_t(g_x)\Big)^2\Big]\\
&=& E \int_0^t X_s\Big((P_\epsilon g_x-g_x)^2\Big) ds\\
&=& \int_0^t ds \int p_s(y) dy \bigg( \int p_\epsilon(z) \Big(\log|z+y-x| -\log|y-x|\Big) dz\bigg)^2 \\
&\leq& \int_0^t ds \int p_s(y) dy \int p_\epsilon(z) \Big(\log|
z+y-x| -\log|y-x|\Big)^2 dz \\
&=& \int_0^t E \Big[\Big(\log |B_\epsilon^{'}+B_s-x|-\log |B_s-x|\Big)^2\Big] ds.\end{aligned}$$ By Lemma 1 we get $$\begin{aligned}
& & \int_0^t E \Big[\Big(\log \frac{|B_\epsilon^{'}+B_s-x|}{|B_s-x|}\Big)^2\Big] ds\\
&\leq& \int_0^t E \Big[\Big(\sqrt{\frac{|B_\epsilon^{'}|}{|B_s-x|}}+\sqrt{\frac{|B_\epsilon^{'}|}{|B'_\epsilon+B_s-x|}}\Big)^2\Big] ds \\
&\leq& 2 \int_0^t E \Big({\frac{|B_\epsilon^{'}|}{|B_s-x|}}\Big)+E\Big({\frac{|B_\epsilon^{'}|}{|B'_\epsilon+B_s-x|}}\Big)ds\\
&:=& 2I.\end{aligned}$$ For the first term in $I$, $$\begin{aligned}
& & \int_0^t E {\frac{|B_\epsilon^{'}|}{|B_s-x|}}ds =E|B_\epsilon^{'}|
\cdot \int_0^t E {\frac{1}{|B_s-x|}}ds\\
&\leq & C \epsilon^{1/2} \int_0^t ds \int p_s(y) \frac{1}{|y-x|} dy \to 0 \text{ as } \epsilon \to 0 \ \text{ by Corollary 1} .\end{aligned}$$ For the second term in $I$, note that $B'_\epsilon+B_s=^d B_{s+\epsilon}$ as they are independent Brownian motion, so $$\begin{aligned}
& & \int_0^t E {\frac{|B_\epsilon^{'}|}{|B'_\epsilon+B_s-x|}}ds \leq \int_0^t \Big(E\big(|B'_\epsilon|^2\big)\Big)^{1/2} \cdot \Big(E\frac{1}{|B'_\epsilon+B_s-x|^2}\Big)^{1/2} ds\\
&=& C\epsilon^{1/2} \int_0^t\Big(E\frac{1}{|B_{s+\epsilon}-x|^2}\Big)^{1/2} ds\leq C\epsilon^{1/2} \Big(\int_0^t E\frac{1}{|B_{s+\epsilon}-x|^2}ds\Big)^{1/2} \cdot \Big(\int_0^t 1^2 ds\Big)^{1/2}\\
&\leq & C\epsilon^{1/2} \cdot t^{1/2} \Big(\int_0^{t+1} ds \int p_s(y) \frac{1}{|y-x|^2} dy\Big)^{1/2} \to 0 \text{ as } \epsilon \to 0 \ \text{ by Corollary 1}.\end{aligned}$$ The $L^1$ convergence of $M_t(P_\epsilon g_x)$ to $M_t(g_x)$ follows.\
\
(iv)
For the convergence of the last term in (10), by Lemma 5 we get $$\begin{aligned}
& & E \Big|\int_0^t X_s\big(\frac{\Delta}{2} P_{\epsilon} g_x) ds-\frac{1}{2}\int_0^t ds \int \frac{1}{|y-x|^2} X_s(dy) \ \Big|\\
&=& \frac{1}{2} \ E \Big|\int_0^t ds \int X_s(dy) \int p_{\epsilon}(y-z) \frac{1}{|z-x|^2} dz-\int_0^t ds \int X_s(dy) \frac{1}{|y-x|^2} \Big|\\
&\leq& \frac{1}{2} \ E \int_0^t ds \int X_s(dy) \ \Big|\int p_{\epsilon}(y-z) \frac{1}{|z-x|^2} dz- \frac{1}{|y-x|^2} \Big|\\
&=& \frac{1}{2} \int_0^t ds \int_{{\mathbb{R}}^3} \Big|\int p_\epsilon(y-z) \frac{1}{|z-x|^2}dz-\frac{1}{|y-x|^2}\Big| p_s(y) dy.\end{aligned}$$ $\mathit{Claim:}$ $$\Big|\int p_\epsilon(y-z) \frac{1}{|z-x|^2}dz-\frac{1}{|y-x|^2}\Big| \to 0 \text{ as } \epsilon\to 0\text{ for } y\neq x .$$
For $\xi=y-x \neq 0$, $$\begin{aligned}
& & \Big|\int p_\epsilon(y-z) \frac{1}{|z-x|^2}dz-\frac{1}{|y-x|^2}\Big|\\
&=& \Big|\int p_\epsilon(z) \frac{1}{|z-(y-x)|^2}dz-\frac{1}{|y-x|^2}\Big|\\
&\leq& \int p_\epsilon(z) \Big|\frac{1}{|z-\xi|^2}-\frac{1}{|\xi|^2}\Big| dz\\
&=& E\bigg(\Big|\frac{1}{|B_\epsilon-\xi|^2}-\frac{1}{|\xi|^2}\Big|\bigg)\\
&=& E \bigg(\Big||B_\epsilon-\xi|-|\xi|\Big| \cdot \Big(\frac{|B_\epsilon-\xi|+|\xi|}{|B_\epsilon-\xi|^2 |\xi|^2}\Big)\bigg)\\
&\leq& E \bigg(|B_\epsilon| \cdot \Big(\frac{|B_\epsilon-\xi|+|\xi|}{|B_\epsilon-\xi|^2 |\xi|^2}\Big)\bigg)\\
&=& E\bigg(|B_\epsilon|\cdot \frac{1}{|B_\epsilon-\xi|^2 |\xi|}\bigg)+E\bigg( |B_\epsilon| \cdot \frac{1}{|B_\epsilon-\xi|\ |\xi|^2}\bigg).\end{aligned}$$ For the first term, we use Holder’s inequality with $1/p=1/5$ and $1/q=4/5$ to get $$\begin{aligned}
& & E\Big(|B_\epsilon|\cdot \frac{1}{|B_\epsilon-\xi|^2 |\xi|}\Big)\leq \frac{1}{|\xi|} \cdot \Big(E(|B_\epsilon|^5)\Big)^{1/5} \cdot \Big(E\big((\frac{1}{|B_\epsilon-\xi|^2 })^{5/4}\big)\Big)^{4/5}.\end{aligned}$$ By Lemma 4, we have $$E\frac{1}{|B_\epsilon-\xi|^{5/2}} \leq C\cdot |\xi|^{-\frac{5}{2}}<\infty,$$ so $$E\Big(|B_\epsilon|\cdot \frac{1}{|B_\epsilon-\xi|^2 |\xi|}\Big) \leq \frac{1}{|\xi|} \Big(C |\xi|^{-\frac{5}{2}}\Big)^{4/5} \Big(E|B_\epsilon|^5\Big)^{1/5} \to 0 \text{ as } \epsilon \to 0.$$ Similarly $$E \Big(|B_\epsilon| \cdot \frac{1}{|B_\epsilon-\xi| |\xi|^2}\Big) \to 0.$$
Note that we have just proved that $$|\int p_\epsilon(y-z) \frac{1}{|z-x|^2}dz-\frac{1}{|y-x|^2}| \to 0$$ almost everywhere ($y\neq x$) as $\epsilon \to 0$. Corollary 1 gives us $$\int_0^t ds \int_{{\mathbb{R}}^3} p_s(y) \frac{1}{|y-x|^2} dy<\infty,$$ and by Lemma 4 $$\Big|\int p_\epsilon(y-z) \frac{1}{|z-x|^2}dz-\frac{1}{|y-x|^2}\Big| \leq (C+1) \frac{1}{|y-x|^2} \text{ for all } \epsilon,$$ by Dominated Convergence Theorem, $$\int_0^t ds \int_{{\mathbb{R}}^3} \Big|\int p_\epsilon(y-z) \frac{1}{|z-x|^2}dz-\frac{1}{|y-x|^2}\Big| p_s(y) dy \to 0,$$ and we proved that $$\int_0^t X_s(\frac{\Delta}{2} P_{\epsilon} g_x)ds \xrightarrow[]{L^1} \int_0^t \int \frac{1}{|y-x|^2} X_s(dy) ds.$$\
(v) Combining (i)-(iv), we build the $L^1$ convergence of each term in (10) to the corresponding term in (4), therefore (4) holds a.s. and the proof of Proposition 1 is done.$\hfill\square$
### Proof of Proposition 2
Let $h_{\epsilon,x} (y)=\sqrt[]{|y-x|^2+\epsilon}$, then $$\nabla h_{\epsilon,x} (y)=\frac{y-x}{\sqrt[]{|y-x|^2+\epsilon}}$$ and $$\Delta h_{\epsilon,x} (y)=\frac{(d-1)|y-x|^2+d \epsilon}{(|y-x|^2+\epsilon)^{3/2}}.$$ By Ito’s Lemma, we have $$\begin{aligned}
\sqrt[]{|B_t-x|^2+\epsilon}=\sqrt[]{|x|^2+\epsilon}&+&\int_0^t \frac{B_s-x}{\sqrt[]{|B_s-x|^2+\epsilon}}\cdot dB_s\\
&+& \frac{1}{2} \int_0^t \frac{(d-1)|B_s-x|^2+d \epsilon}{(|B_s-x|^2+\epsilon)^{3/2}} ds\end{aligned}$$ Let $H_s=\frac{B_s-x}{\sqrt[]{|B_s-x|^2+\epsilon}}$, then $$M_t^\epsilon:=\int_0^t \frac{B_s-x}{\sqrt[]{|B_s-x|^2+\epsilon}}\cdot dB_s \in cM_{0,\text{loc}}.$$ Since $$E[M^\epsilon]_t=E \int_0^t \frac{|B_s-x|^2}{|B_s-x|^2+\epsilon}ds \leq E \int_0^t 1 ds=t<\infty,$$ then $M^\epsilon$ is a martingale and hence by taking expectation $$E \sqrt[]{|B_t-x|^2+\epsilon}=\sqrt[]{|x|^2+\epsilon}+\frac{1}{2} \int_0^t E \frac{(d-1)|B_s-x|^2+d \epsilon}{(|B_s-x|^2+\epsilon)^{3/2}} ds.$$ By Fatou’s Lemma, $$\begin{aligned}
& & \frac{1}{2}\int_0^t E \frac{d-1}{|B_s-x|} ds= \frac{1}{2} \int_0^t E \ \lim_{\epsilon \to 0} \frac{(d-1)|B_s-x|^2+d \epsilon}{(|B_s-x|^2+\epsilon)^{3/2}} ds\\
&\leq& \frac{1}{2}\int_0^t \ \liminf_{\epsilon \to 0} E\frac{(d-1)|B_s-x|^2+d \epsilon}{(|B_s-x|^2+\epsilon)^{3/2}} ds\\
&\leq& \liminf_{\epsilon \to 0} \frac{1}{2} \int_0^t \ E\frac{(d-1)|B_s-x|^2+d \epsilon}{(|B_s-x|^2+\epsilon)^{3/2}} ds\\
&=& \liminf_{\epsilon \to 0} \Big[E \sqrt[]{|B_t-x|^2+\epsilon}- \sqrt[]{|x|^2+\epsilon}\Big] \text{ by (11)}\\
&=& E |B_t-x|-|x|\leq E|B_t|.\end{aligned}$$ The last equality is from $$0\leq \sqrt[]{|x|^2+\epsilon} -|x| \leq \sqrt[]{\epsilon} \to 0,$$ and $$\begin{aligned}
0 &\leq& E \sqrt[]{|B_t-x|^2+\epsilon}-E|B_t-x|\\
&\leq& E\Big[|B_t-x|+\sqrt[]{\epsilon}\Big]-E|B_t-x|=\sqrt[]{\epsilon} \to 0.\end{aligned}$$ So $$\begin{aligned}
&& \int_0^t \int \frac{1}{|y-x|} p_s(y) dy ds=\int_0^t E \frac{1}{|B_s-x|} ds \leq\frac{2}{d-1} E|B_t|<\infty.\\\end{aligned}$$ $\hfill\square$
Proof of Theorem 2
==================
To prove Theorem 2, we need the Tanaka formula for $d=2$, which are stated below and the proof will follow after the proof of Theorem 2.
(Tanaka formula for d=2) Let $c_2=1/\pi$ and $g_x(y)= \log|y-x|$, where $x\neq 0$. Then we have a.s. that $$L_t^x=c_2\Big[X_t(g_x)-\delta_0 (g_x)-M_t(g_x)\Big].$$
$\mathbf{Remark.}$ Barlow, Evans and Perkins \[1\] gives a Tanaka formula for local time of Super-Brownian Motion in $d=2$, which is $$X_t(g_{\alpha,x})=X_0(g_{\alpha,x})+M_t(g_{\alpha,x})+\alpha \int_0^t X_s(g_{\alpha,x}) ds-L_t^x,$$ for all $t\geq 0$ a.s.. Here $g_{\alpha,x}(y)$ is defined to be $\int_0^\infty e^{-\alpha t} p_t(x-y) dt$. We can see that $g_{\alpha,x}$ is not well defined for $\alpha=0$ and our result effectively extends the Tanaka formula in \[1\] to the $\alpha=0$ case.
Proof of Theorem 2
------------------
By (12), note that $\delta_0(g_x)=\log|x|=-\log 1/|x|$, $$L_t^x-c_2\log \frac{1}{|x|}=c_2\Big[ X_t(g_x)-M_t(g_x) \Big],$$ therefore $$E\Big|L_t^x-c_2\log \frac{1}{|x|}\Big| \leq c_2 E\Big|X_t(g_x)\Big|+ c_2 E\Big|M_t(g_x)\Big|.$$ For the first term, $$\begin{aligned}
& & E\Big|X_t(g_x)\Big|\leq E X_t(|g_x|)= \int p_t(y) \Big|\log |y-x|\Big| dy\\
&\leq & \int p_t(y) \frac{1}{|y-x|} 1_{\{|y-x|<1\}} dy+\int p_t(y) |y-x| 1_{\{|y-x|\geq 1\}} dy\\
&\leq & \frac{1}{2\pi t} \int \frac{1}{|y-x|} 1_{\{|y-x|<1\}} dy+\int p_t(y) (|y|+|x|) dy\\
&=& \frac{1}{2\pi t} \cdot 2\pi+|x| +E|B_t| \to \frac{1}{2\pi t} \cdot 2\pi +E|B_t|.\end{aligned}$$ For the second term, $$\Big(E|M_t(g_x)|\Big)^2\leq E\Big(M_t^2(g_x)\Big)=E\int_0^t X_s(g_x^2) ds,$$ and by Lemma 1 $$\begin{aligned}
& & E\int_0^t X_s(g_x^2) ds=\int_0^t ds \int p_s(y) (\log|y-x|)^2 dy\\
&\leq& \int_0^t ds \int p_s(y) \frac{1}{|y-x|} 1_{\{|y-x|<1\}} dy+\int_0^t ds \int p_s(y) |y-x| 1_{\{|y-x|\geq 1\}} dy\\
&\leq& \int_0^t ds \int p_s(y) \frac{1}{|y-x|} dy+\int_0^t ds \int p_s(y) (|y|+|x|) dy.\end{aligned}$$ By Proposition 2 in $d=2$, $$\int_0^t ds \int p_s(y) \frac{1}{|y-x|} dy\leq 2 E|B_t|<\infty,$$ and $$\int_0^t ds \int p_s(y) (|y|+|x|) dy=|x| t+\int_0^t E |B_s| ds \to \int_0^t E |B_s| ds <\infty.$$ Therefore $$\limsup_{x\to 0} E\Big|L_t^x-c_2\log \frac{1}{|x|}\Big|<\infty.$$
Proof of Proposition 3
----------------------
### Some useful lemmas
In ${\mathbb{R}}^2$, for $0<\alpha<2$, there exists a constant $C=C(\alpha)$ such that for any $x\neq 0$ and $t>0$, $$\int_{{\mathbb{R}}^2} p_t(y) \frac{1}{|y-x|^\alpha} dy<C \frac{1}{|x|^\alpha}.$$
The proof follows from the proof of Lemma 4 after some modification.
In ${\mathbb{R}}^2$, for any $0<\alpha<2$, there exists a constant $C=C(\alpha)$ such that for any $x\neq 0$ and $t>0$ $$E\int_0^t \int \frac{1}{|y-x|^\alpha} X_s(dy) \ ds =\int_0^t ds \int_{{\mathbb{R}}^2} p_s(y) \frac{1}{|y-x|^\alpha} dy<C \frac{1}{|x|^\alpha} t.$$
It follows from Lemma 6.
Let $g_x(y)= \log|y-x|$, where $x, y\in {\mathbb{R}}^2$, then for any $s>0$ and $y\neq x$, we have $$\frac{\Delta_y}{2} P_s g_x(y)= \pi p_s(y-x).$$
Idea of this proof is from Evans \[3\]. For any fixed $s>0$, $p_{s}(y)=(2\pi s)^{-1} e^{-|y|^2/{2s}} \in C_0^\infty({\mathbb{R}}^2)$, we have $$\|Dp_s\|_{L^\infty({\mathbb{R}}^2)}<\infty \text{ and } \|\Delta p_s\|_{L^\infty({\mathbb{R}}^2)}<\infty.$$ For any $\delta \in (0,1)$, $$\begin{aligned}
& & \Delta_y \int_{{\mathbb{R}}^2} p_{s}(y-z) g_x(z)dz\\
&=& \int_{B(x,\delta)} \Delta_y p_{s}(y-z) g_x(z)dz+\int_{{\mathbb{R}}^2-B(x,\delta)} \Delta_y p_{s}(y-z) g_x(z)dz\\
&=:& I_\delta+J_\delta.\end{aligned}$$ Now $$|I_\delta|\leq \|\Delta p_s\|_{L^\infty({\mathbb{R}}^2)} \int_{B(x,\delta)} |g_x(z)| dz\leq C \delta^2 |\log \delta| \to 0.$$ Note that $\Delta_y p_{s}(y-z)=\Delta_z p_{s}(y-z)$. Integration by parts yields $$\begin{aligned}
J_\delta &=& \int_{{\mathbb{R}}^2-B(x,\delta)} \Delta_z p_{s}(y-z) g_x(z)dz\\
&=& \int_{\partial B(x,\delta)} g_x(z) \frac{\partial p_{s}}{\partial \nu}(y-z) dz-\int_{{\mathbb{R}}^2-B(x,\delta)} D_z p_{s}(y-z) D_z g_x(z)dz\\
&=:& K_\delta+L_\delta,\end{aligned}$$ $\nu$ denoting the inward pointing unit normal along $\partial B(x,\delta).$ So $$|K_\delta|\leq \|Dp_s\|_{L^\infty({\mathbb{R}}^2)} \int_{\partial B(x,\delta)} |g_x(z)|dz \leq C \delta |\log \delta| \to 0.$$ We continue by integrating by parts again in the term $L_\delta$ to find $$\begin{aligned}
L_\delta &=& \int_{{\mathbb{R}}^2-B(x,\delta)} p_{s}(y-z) \Delta_z g_x(z)dz-\int_{\partial B(x,\delta)} p_{s}(y-z) \frac{\partial g_x}{\partial \nu}(z) dz\\
&=& -\int_{\partial B(x,\delta)} p_{s}(y-z) \frac{\partial g_x}{\partial \nu}(z) dz\\
&=:& M_\delta.\end{aligned}$$ since $\Delta_z g_x(z)=0$ when $z$ is away from $x$.\
\
Now $Dg_x(z)=\frac{z-x}{|z-x|^2}(z\neq x)$ and $\nu=\frac{-(z-x)}{|z-x|}=\frac{-(z-x)}{\delta}$ on $\partial B(x,\delta)$. Hence $\frac{\partial g_x}{\partial \nu}(z)=\nu \cdot Dg_x(z)=-\frac{1}{\delta} $ on $\partial B(x,\delta)$. Since $2\pi \delta$ is the surface area of the sphere $\partial B(x,\delta)$ in ${\mathbb{R}}^2$, we have $$M_\delta=2\pi \cdot \frac{1}{2\pi \delta} \int_{\partial B(x,\delta)} p_{s} (y-z)dz \to 2 \pi p_{s}(y-x) \text{ as } \delta \to 0.$$ Therefore we proved $$\frac{\Delta_y}{2} P_s g_x(y)= \pi p_s(y-x)= \pi p_s^x(y).$$
### Proof of Proposition 3
Using the same argument in proving Proposition 1 in Section 2.2.2, by a smooth cutoff $\chi_n$ of $\log$ and let $n$ goes to infinity, we have following equation hold a.s., $$X_t(P_{\epsilon} g_x)=\delta_0(P_{\epsilon} g_x)+ M_t(P_{\epsilon} g_x)+\int_0^t X_s\Big(\frac{\Delta}{2} P_{\epsilon} g_x\Big) ds,$$ where $M_t(P_{\epsilon} g_x)$ is a martingale with quadratic variation being $$[M(P_{\epsilon} g_x)]_t=\int_0^t X_s((P_{\epsilon} g_x)^2) ds,$$ and $M_t^2(P_{\epsilon} g_x)-[M(P_{\epsilon} g_x)]_t$ is also a martingale.\
Let $\epsilon$ goes to $0$, we will show the a.s. convergence of each term in (13) to the corresponding term in Proposition 3, which is equivalent to $$X_t(g_x)=\delta_0(g_x)+M_t(g_x)+\pi L_t^x.$$ By Lemma 7, we have $$\int_0^t X_s(\frac{\Delta}{2} P_\epsilon g_x) ds= \pi \int_0^t X_s(p_\epsilon^x) ds \xrightarrow[]{\text{a.s. }} \pi L_t^x \text{ as } \epsilon \to 0.$$ Then in (i)-(iii) we will build the $L^1$ convergence of the rest three terms in (13) to the corresponding term in (14) and we can take a subsequence along which all four terms converge a.s. and therefore (14) holds a.s..\
(i)
Let $B_t$ and $B_t^{'}$ be two independent standard Brownian motion in ${\mathbb{R}}^2$, $$\begin{aligned}
& & E\Big|X_t(P_\epsilon g_x)-X_t(g_x)\Big| \leq E \Big[X_t\Big(|P_\epsilon g_x-g_x|\Big)\Big]\\
&=& \int p_t(y) dy \Big|\int p_\epsilon(z) \log|z-(y-x)|dz-\log|y-x|\Big|\\
&\leq& \int p_t(y) dy \int p_\epsilon(z) \Big|\log|z-(y-x)|-\log|y-x|\Big| dz\\
&=& E\Big[\Big|\log \frac{|B'_\epsilon-(B_t-x)|}{|B_t-x|}\Big|\Big]\\
&\leq& E \sqrt{\frac{|B'_\epsilon| }{|B_t-x|}}+ E \sqrt{\frac{|B'_\epsilon|}{|B'_\epsilon+B_t-x|}} \text{ by Lemma 1. }\end{aligned}$$ Since $E\sqrt[]{|B_\epsilon|} \to 0$ and $E\sqrt[]{\frac{1}{|B_t-x|}}<\infty$ by Lemma 6, $$E \sqrt{\frac{|B'_\epsilon| }{|B_t-x|}}= E \sqrt[]{|B'_\epsilon|} \cdot E\sqrt[]{\frac{1}{|B_t-x|}} \to 0.$$ For the second term, we use Cauchy Schwarz Inequality, $$\bigg(E \sqrt{\frac{|B'_\epsilon|}{|B'_\epsilon+B_t-x|}}\bigg)^2 \leq E|B'_\epsilon| \cdot E\frac{1}{|B'_\epsilon+B_t-x|}= E|B'_\epsilon| \cdot E\frac{1}{|B_{t+\epsilon}-x|}.$$ So by Lemma 6 $$E \sqrt{\frac{|B'_\epsilon|}{|B'_\epsilon+B_t-x|}} \leq \bigg(E|B'_\epsilon|\bigg)^{1/2} \cdot \bigg(E\frac{1}{|B_{t+\epsilon}-x|}\bigg)^{1/2} \to 0 \text{ as } \epsilon \to 0.$$ and the $L^1$ convergence of $X_t(P_\epsilon g_x)$ to $X_t(g_x)$ follows.\
(ii) Similarly we have $$\begin{aligned}
& & \bigg|\delta_0(P_\epsilon g_x)-\delta_0(g_x)\bigg|=\bigg|\int_{{\mathbb{R}}^2} p_\epsilon(y) \log|y-x| dy-\log|x| \bigg|\\
&\leq & \int_{{\mathbb{R}}^2} p_\epsilon(y) \Big|\log|y-x| -\log|x|\Big| dy=E\Big|\log|B_\epsilon-x|-\log|x|\Big|\\
&\leq& E \sqrt[]{\frac{|B_\epsilon|}{|x|}}+E\sqrt[]{\frac{|B_\epsilon|}{|B_\epsilon-x|}} \to 0.\end{aligned}$$ (iii) For the convergence of the martingale term $M_t(P_\epsilon g_x)$ to $M_t(g_x)$, $$\begin{aligned}
& & \Big(E|M_t(P_\epsilon g_x)-M_t(g_x)|\Big)^2 \leq E\Big[\Big(M_t(P_\epsilon g_x)-M_t(g_x)\Big)^2\Big]\\
&=& E \int_0^t X_s\Big((P_\epsilon g_x-g_x)^2\Big) ds\\
&=& \int_0^t ds \int p_s(y) dy \Big(\int p_\epsilon(z) (\log|z-(y-x)|-\log|y-x|) dz\Big)^2\\
&\leq& \int_0^t ds \int p_s(y) dy \int p_\epsilon(z) \Big(\log|z-(y-x)|-\log|y-x|\Big)^2 dz\\
&=& \int_0^t E \Bigg[\Bigg(\log \frac{|B'_\epsilon-(B_s-x)|}{|B_s-x|}\Bigg)^2\Bigg] ds\\
&\leq& \int_0^t E \Bigg[\Bigg(\sqrt[]{\frac{|B'_\epsilon|}{|B_s-x|}}+\sqrt[]{\frac{|B'_\epsilon|}{|B'_\epsilon-(B_s-x)|}}\Bigg)^2\Bigg] ds \hfill{\text{ (by Lemma 1)}}\\
&\leq& 2 \int_0^t E \Bigg({\frac{|B'_\epsilon|}{|B_s-x|}}+{\frac{|B'_\epsilon|}{|B'_\epsilon-(B_s-x)|}}\Bigg) ds\\
&:=& 2J.\end{aligned}$$ For the first term in $J$, by Corollary 2, $$\begin{aligned}
\int_0^t E \frac{|B'_\epsilon| }{|B_s-x|} ds=E |B'_\epsilon| \cdot \int_0^t E\frac{1}{|B_s-x|} ds\to 0.\end{aligned}$$ and for the second term in $J$, using Holder’s inequality with $1/p=1/3$ and $1/q=2/3$ twice, we get $$\begin{aligned}
& & \int_0^t E \frac{|B'_\epsilon| }{|B'_\epsilon+B_s-x|} ds\leq \int_0^t
\Big(E (|B'_\epsilon|^3)\Big)^{1/3}\cdot \Big(E\frac{1}{|B'_\epsilon+B_s-x|^{3/2}}\Big)^{2/3} ds \\
&=& \Big(E (|B'_\epsilon|^3)\Big)^{1/3} \cdot \int_0^t \Big(E\frac{1}{|B_{s+\epsilon}-x|^{3/2}}\Big)^{2/3} ds \\
&\leq& \Big(E (|B'_\epsilon|^3)\Big)^{1/3} \cdot \Big(\int_0^t E\frac{1}{|B_{s+\epsilon}-x|^{3/2}} ds\Big)^{2/3} \cdot \Big(\int_0^t 1^3 ds\Big)^{1/3} \\
&\leq& \Big(E (|B'_\epsilon|^3)\Big)^{1/3} \cdot \Big(\int_0^{t+1} \int E \frac{1}{|B_{s}-x|^{3/2}} ds\Big)^{2/3} \cdot t^{1/3} \to 0\end{aligned}$$ as $\epsilon \to 0$ by Corollary 2.\
This completes the proof of Proposition 3.$\hfill\square$
[5]{}
M. Barlow, S. Evans and E. Perkins (1991). Collision local times and measure-valued diffusions. Can. J. Math. 43, 897-938.
P. Billingsley (1995). Probability and Measure. Wiley Series in Probab. and Mathematical Stat.
L. Evans (2010). Partial Differential Equations. American Mathematical Soc., 2010.
N. Konno and T. Shiga (1988). Stochastic partial differential equations for some measure-valued diffusions. Probab. Theory Related Fields 79 (1988), 201-225.
E. Perkins (1999). Dawson?Watanabe superprocesses and measure-valued diffusions, in: Lectures on Probability Theory and Statistics, Ecole d?été de Probabilités de Saint-Flour XXIX, in: Lecture Notes in Math., vol. 1781, Springer, 1999.
D. Revuz and M. Yor (1994). Continuous Martingales and Brownian Motion, Springer, Berlin, 1994.
S. Sugitani (1987). Some properties for the measure-valued diffusion process. J. Math. Soc. Japan 41, 437-462.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The transport of quantum electrons through hierarchical lattices is of interest because such lattices have some properties of both regular lattices and random systems. We calculate the electron transmission as a function of energy in the tight binding approximation for two related Hanoi networks. HN3 is a Hanoi network with every site having three bonds. HN5 has additional bonds added to HN3 to make the average number of bonds per site equal to five. We present a renormalization group approach to solve the matrix equation involved in this quantum transport calculation. We observe band gaps in HN3, while no such band gaps are observed in linear networks or in HN5.'
author:
- 'S. Boettcher'
- 'C. Varghese'
- 'M. A. Novotny'
title: Quantum Transport through Hierarchical Structures
---
Introduction
============
Understanding and controlling the transport of electrons is central to the operation of all electrical and electronic devices [@switch]. In many cases of interest in nanomaterials, the electron transport is coherent, and therefore must be analyzed using the Schr[ö]{}dinger equation. Interference effects can then lead to a metal-insulator transition and Anderson localization [@anderson]. Even more than fifty years after Anderson’s publication of his celebrated paper, the effect remains an active area of study [@Lage2009; @monthus; @montusRG; @caliskan; @islam; @Liu2007; @Ostr2006]. The main goal is to calculate the transmission probability as a function of energy, $E$, for an incoming electron, i.e. the probability that an electron that comes in from $x=-\infty$ can be observed at $x=+\infty$.
The starting point to quantum calculations of (spinless) electrons through a material is often a tight-binding model [@datta]. In such a model each node can be considered an atom, the on-site energy at a node is associated with a potential energy at the site, and there is a hopping term which comes from discretization of the kinetic energy term of the Schr[ö]{}dinger equation [@datta]. The electron transport calculation via the Schr[ö]{}dinger equation thus reduces to the solution of an infinite matrix equation. The solution of the matrix equation is often accomplished using a Green’s function method [@andrade; @caliskan; @datta; @datta3; @young; @pouthier; @ryndyk]. In this paper we instead use an ansatz approach introduced by Daboul, Chang, and Aharony [@daboul], which is simpler to describe at an undergraduate level [@Solomon2010]. The ansatz reduces the size of the matrix equation to that of the number of tight-binding cites in the scattering volume, plus one for both the incoming lead and the outgoing lead. This approach has been used by other authors [@islam; @cuansing]. We find that the ansatz approach is particularly well suited to our calculations of transport through hierarchical structures. To perform the calculation we construct a decimation Renormalization Group (RG) procedure to reduce the number of sites of the hierarchical structure. Our RG is related to that utilized by others for tight-binding models [@aoki; @Banav1983; @Niu1986; @Mama2003]. However, our RG has been explicitly constructed for the calculation of the transmission probabilities. We find that our RG procedure greatly simplifies the calculation, albeit only for certain select networks that have hierarchical structures. Although our RG does not significantly simplify the calculation of associated wavefunctions, we nevertheless give a recipe for the RG calculation of the wavefunctions.
The specific models we solve here for quantum transport are motivated by four considerations. One is to understand how hierarchical models, in particular the Hanoi networks [@SWPRL], affect quantum transport. The second is that such hierarchical models provide an intermediate between regular lattices and ones that have a small-world property [@Watts], and would be of interest to understand quantum transport of nanomaterials that have a small-world property [@caliskan; @Novotny2004; @Novotny2005; @Zhang2006; @yancey2009]. The third is that often phase transitions such as the metal-insulator transition or a ferromagnetic transition have some universal quantities that depend only on the dimension. Hierarchical models then sometimes provide insight into how these universal quantities behave as a function of the dimension [@Gefen1980; @Banav1983; @LeGuillou1987; @Novotny1992; @Novotny1993]. Lastly, experimental realizations of hierarchical materials may possess novel physical properties [@Lopes2001; @Boker2004; @Lin2005; @Ravind2005; @Chen2008].
The hierarchical models we study are the Hanoi networks HN3 and HN5. These networks have particularly interesting properties. First, they are both planar, and consequently could be experimentally constructed on a surface. Second, both networks have typical paths’ (defined precisely in Sec. \[sec:Graph-Structure\]) that grow more slowly with system size than do paths in regular lattices. Finally, Anderson localization is associated with randomness in the system, while randomness in the Hanoi networks depends on the scale. For example, for HN3 locally every site has three bonds connecting it with other sites, choosing sites from HN3 at random the connections to other sites seem random, but these connections are actually from a hierarchical arrangement and hence at the larger scale there is regularity to the network. Therefore studying electron transport and Anderson localization in these lattices is of interest.
In Section \[sec:Graph-Structure\] we provide a brief description of the Hanoi networks. For transport properties, these networks can be connected to the leads in many ways, but we choose to present results only for symmetric linear and symmetric ring lead attachments. In Section \[MatrixRG\] we develop the RG equations for calculating the transmission through these networks, with details of the RG presented in Appendix \[Sec:AppA\]. In Section \[analysis\] we analyze these RG equations for these networks. This involves iterating the RG until the system is comprised only of a few lattice points, and these small lattice solutions are presented in Appendix \[Sec:AppB\]. Section \[conclusion\] contains our conclusions and a discussion of our results. The appendices contain the basic matrix algebra used to develop the RG equations.
Network Structure\[sec:Graph-Structure\]
========================================
Each of the networks considered in this paper possess a simple geometric backbone, a one-dimensional line of $N=2^{k}$ sites formed into a ring as depicted in Fig. \[fig:3hanoi\] for HN3 and $k=5$. Alternatively, we can connect the network to the incoming and outgoing leads in a linear arrangement with $2^k+1$ sites, as depicted in Fig. \[fig:5hanoi\] and \[fig:HN3scatterKoch\]. Each site is at least connected to its nearest neighbor left and right on the backbone. For consistency, we call the ordinary one-dimensional ring HN2 (for Hanoi Network of degree 2). For example, HN2 is the linear lattice in Fig. \[fig:HN3scatterKoch\] formed by only the black bonds.
To generate the small-world hierarchy in these networks, consider parameterizing any number $n<N$ (except for 0) *uniquely* in terms of two other integers $(i,j)$, $i\geq1$ and $1\leq j\leq2^{k-i}$, via $$\begin{aligned}
n & = & 2^{i-1}\left(2j-1\right).
\label{eq:numbering}\end{aligned}$$ Here, $i$ denotes the level in the hierarchy whereas $j$ labels consecutive sites within each hierarchy. To generate the network HN3, we connect each site $n=2^{i-1}(4j-3)$ also with a long-distance neighbor $n'=2^{i-1}(4j-1)$ for $1\leq
j\leq 2^{k-i-1}$. (In the ring, if an index $n$ equals or exceeds the system size $N$, we assume that the site $n$ mod $N$ is implied.)
For the linear arrangement, the sites zero and $2^k+1$ are connected to the input/output leads, and site $2^{k-1}$ will not be connected to any other site or to the input/output leads \[see, for instance, HN5 in Fig. \[fig:5hanoi\]\]. For the ring arrangement, the sites with number zero and $2^{k-1}$ are connected to the input/output leads \[see Fig. \[fig:3hanoi\]\], and site zero is connected to site $2^k-1$ to form the ring.
![\[fig:3hanoi\] Depiction of the 3-regular network HN3 with a one-dimensional periodic backbone forming a ring, here with $k=5$. The top and bottom sites $n=0$ and $n=2^{k-1}$ require special treatment and are connected to external leads. With these connections, the entire network becomes 3-regular. Note that the graph is planar.](./SmallWorld)
Previously [@SWPRL], it was found that the average chemical path” between sites on HN3 scales as $$d^{HN3}\sim\sqrt{l}
\label{eq:3dia}$$ with the distance $l$ along the backbone. In some ways, this property is reminiscent of a square-lattice consisting of $N$ lattice sites with diagonal $\sim\sqrt{N}$.
While the preceding networks are of a fixed, finite degree, we can extend HN3 in the following manner to obtain a new planar network of average degree 5, hence called HN5, at the price of a distribution in the degrees that is exponentially falling. In addition to the bonds in HN3, in HN5 we also connect all even sites to both of its nearest neighboring sites *within* the same level of the hierarchy $i>1$ in Eq. (\[eq:numbering\]). The resulting network remains planar but now sites have a hierarchy-dependent degree, as shown in Fig. \[fig:5hanoi\]. To obtain the average degree, we observe that 1/2 of all sites have degree 3, 1/4 have degree 5, 1/8 have degree 7, and so on, leading to an exponentially falling degree distribution of ${\cal P}\left\{ \alpha=2i+1\right\}
\propto2^{-i}$ for $i=1,\>2,\>3,\cdots$. Then, the total number of bonds $L$ in the (linear) system of size $N=2^{k}+1$ is $$\begin{aligned}
2L & =2(2k-1)+ & \sum_{i=1}^{k-1}\left(2i+1\right)2^{k-i}=5\times 2^{k}-4,
\label{eq:TotalLinksHN5}\end{aligned}$$ thus, the average degree is $$\begin{aligned}
\left\langle \alpha\right\rangle & = & \frac{2L}{N}\sim5.
\label{eq:averageDegreeHN5}\end{aligned}$$
In HN5, the end-to-end distance is trivially 1, see Fig. \[fig:5hanoi\]. Therefore, we define as the diameter the largest of the shortest paths possible between any two sites, which are typically odd-index sites furthest away from long-distance bonds. For the $N=32$ site network depicted in Fig. \[fig:5hanoi\], for instance, that diameter is 5 as measured between site 3 and 19 (0 is the left-most site), although there are many other such pairs. It is easy to show recursively that this diameter grows strictly as $$\begin{aligned}
d^{HN5} & = & 2\left\lfloor k/2\right\rfloor +1\sim\log_{2}N
\label{eq:5dia}\end{aligned}$$ with $\lfloor x\rfloor$ the integer portion of $x$. We have checked numerically that the *average* shortest path between any two sites also increases logarithmically with system size $N$.
![\[fig:HN3scatterKoch\] (Color online) Scattering a quantum electron off a linear version of HN3, here drawn as a branched Koch curve. The incoming electron on the left gets scattered into an outgoing transmitted portion (right) and reflected portion (left) on the attached external leads (blue-shaded). In this form of HN3, the one-dimensional backbone is marked by black links while the small-world links are shaded in red. Note, for instance, that the shortest end-to-end path here is the baseline of the Koch curve.](./HN3scatterKoch.pdf)
Matrix RG for Hanoi Networks\[MatrixRG\]
========================================
![(Color online) Decimation of a block in the RG. The two sites with on-site energy $\kappa_1$ with a connecting bond of strength $\tau_1$ are decimated.[]{data-label="BuildingBlock"}](./BuildingBlock.pdf "fig:"){width=".5\textwidth"}\
At each RG step, we decimate all the odd sites. Take site 0 to be at level $i =k+1$ in a linear geometry and at level $i =k$ in a ring geometry. As the odd sites each have only one small-world-type bond, we can divide the network into blocks containing 5 sites and decimate the pairs of two odd sites block by block. Let us start with the first block which contain sites 0,1,2,3 and 4. This decimation process for a linear geometry is shown in Fig. \[BuildingBlock\]. Thus, here (see Appendix \[Sec:AppA\]) we have $${\bf A} =
\begin{pmatrix}
\kappa_{k+1} & \lambda_{1} & \lambda_{2} \\
\lambda_{1} & \kappa_{2} & \lambda_{1} \\
\lambda_{2} & \lambda_{1} & \kappa_{3} \\
\end{pmatrix}$$ and $$\begin{aligned}
{\bf D} =
\begin{pmatrix}
\kappa_{1} & \tau_{1} \\
\tau_{1} & \kappa_{1} \\
\end{pmatrix}
; \qquad {\bf B} =
\begin{pmatrix}
\tau_{0} & 0 \\
\tau_{0} & \tau_{0} \\
0 & \tau_{0} \\
\end{pmatrix} = \tau_{0}\begin{pmatrix}
1 & 0 \\
1 & 1 \\
0 & 1 \\
\end{pmatrix} \label{AdB}
.\end{aligned}$$
After decimation, $$\begin{aligned}
{\bf A'} =
\begin{pmatrix}
\kappa'_{k} & \tau'_{0} & \lambda'_{1} \\
\tau'_{0} & \kappa'_{1} & \tau'_{0} \\
\lambda'_{1} & \tau'_{0} & \tilde{\kappa}_{2} \\
\end{pmatrix}
= {\bf A} -{\bf B}{\bf D}^{-1}{\bf B}^{\rm T}
.\end{aligned}$$ After simplification, we find that $$\begin{aligned}
\kappa'_{k} &=& \kappa_{k+1} -
\frac{\tau_{0}^{2}\kappa_{1}}{\kappa_{1}^{2}-\tau_{1}^{2}} \label{endsite}\\
\kappa'_{1} &=& \kappa_{2} - \frac{2\tau_{0}^{2}}{\kappa_{1} + \tau_{1}} \\
\tilde{\kappa}_{2} &=& \kappa_{3} - \frac{\tau_{0}^{2}\kappa_{1}}{\kappa_{1}^{2}-\tau_{1}^{2}} \\
\tau'_{0} &=& \lambda_{1} - \frac{\tau_{0}^{2}}{\kappa_{1} + \tau_{1}} \\
\lambda'_{1} &=& \lambda_{2} + \frac{\tau_{0}^{2}\tau_{1}}{\kappa_{1}^{2}-\tau_{1}^{2}}
\label{nexttonear}
.\end{aligned}$$ Here, the primed and unprimed quantities represent the $1$-st and the $0$-th level respectively in the RG recursion. Notice that the decimation of sites connected to site 4 \[the right-most site in Fig. \[BuildingBlock\](a)\] is not complete yet and therefore its on-site energy will be modified further when we decimate the odd sites of the next block which contain sites 4,5,6,7 and 8.
After the decimation of the next block we get $$\begin{aligned}
\kappa'_{2} &=& \tilde{\kappa}_{2} - \frac{\tau_{0}^{2}\kappa_{1}}{\kappa_{1}^{2}-\tau_{1}^{2}} =
\kappa_{3} - \frac{2\tau_{0}^{2}\kappa_{1}}{\kappa_{1}^{2}-\tau_{1}^{2}}
.\end{aligned}$$ Continuing the decimation block by block in this way, we find that $$\begin{aligned}
\kappa'_{i} &=& \kappa_{i+1} - \frac{2\tau_{0}^{2}\kappa_{1}}{\kappa_{1}^{2}-\tau_{1}^{2}}
\quad \textrm{for} \quad \forall i \in \{2,\ldots,k\} \label{evensite}\\
\tau'_{i} &=& \tau_{i+1} \quad \forall i\geq1 \label{taumatrix}\\
\lambda'_{i} &=& \lambda_{i+1} \quad \forall i\geq2 \label{lambdamatrix}
.\end{aligned}$$ At first it appears that there are a lot of RG variables to worry about. However most of these RG variables are interdependent. It can be deduced from Eqs. (\[endsite\],\[evensite\], \[taumatrix\],\[lambdamatrix\]) that $$\begin{aligned}
\tau_{1}^{(m)} &=& \tau_{m+1} \\
\lambda_{2}^{(m)} &=& \lambda_{m+2}
,\end{aligned}$$ and that the on-site energy parameter of the even sites is related to those of the odd sites as
$$\begin{aligned}
\kappa_{i}^{(m)} &=&
\kappa_{1}^{(m)} + \kappa_{m+i} - \kappa_{m+1} - 2(\lambda_{1}^{(m)} - \lambda_{m+1})
\quad \textrm{for} \quad \forall i \in \{2,\ldots,k-m\} \label{aries}
\end{aligned}$$
and that specifically for a linear geometry, the on-site energy parameter of the end sites $$\begin{aligned}
\kappa_{k+1-m}^{(m)} &=& \kappa_{k+1} + (\kappa_{2}^{(m)} - \kappa_{m+2})/2 \quad \textrm{for}
\quad \forall m \in \{0,\ldots,k-2\} \label{cancer}
.\end{aligned}$$ Thus we are left with just three independent RG variables which are $\kappa_{1}^{(m)}$, $\tau_{0}^{(m)}$ and $\lambda_{1}^{(m)}$ governed by the RG equations $$\begin{aligned}
\kappa_{1}^{(m+1)} \>=\> \kappa_{1}^{(m)} + \kappa_{m+2} - \kappa_{m+1} - 2(\lambda_{1}^{(m)} -
\lambda_{m+1}) - \frac{2[\tau_{0}^{(m)}]^{2}}{\kappa_{1}^{(m)}+ \tau_{m+1}} \quad
& \forall m \in \{0,\ldots,k-2\} \label{kitten}\\
\tau_{0}^{(m+1)} \>=\> \lambda_{1}^{(m)} -
\frac{[\tau_{0}^{(m)}]^{2}}{\kappa_{1}^{(m)}+ \tau_{m+1}}
\quad & \forall m \in \{0,\ldots,k-1\} \label{calf}\\
\lambda_{1}^{(m+1)} \>=\> \lambda_{m+2} +
\frac{[\tau_{0}^{(m)}]^{2}\tau_{m+1}}{[\kappa_{1}^{(m)}]^{2}
- \tau_{m+1}^{2}} \quad & \forall m \in \{0,\ldots,k-1\} \label{puppy}
.\end{aligned}$$
Analysis of the RG Equations \[analysis\]
=========================================
One-dimensional Lattice (HN2)\[sub:HN2trans\]
---------------------------------------------
It will prove helpful to demonstrate the general set of recursions \[Eqs. (\[kitten\], \[calf\], \[puppy\])\] by way of the one-dimensional ($d=1$) ring of $N=2^{k}$ sites. For consistency we call this the HN2 network, or Hanoi network with 2 bonds per site (with additional bonds for the input/output leads in the ring geometry). We can employ the recursions to explore the transmission through a $d=1$ ring of $N=2^k$ sites. The energy scale chosen throughout is such that the (uniform) transmissivity for each bond has a unit weight. With that, we obtain the initial conditions $$\begin{aligned}
\kappa_{i}^{(0)} & = & E,\qquad(i\geq1),\nonumber \\
\tau_{0}^{(0)} & = & -1,\label{eq:transIC_HN2}\\
\tau_{i}^{(0)} & = & 0,\qquad(i\geq1),\nonumber \\
\lambda_{i}^{(0)} & = & 0,\qquad(i\geq1)\>.\nonumber\end{aligned}$$ Eqs. (\[kitten\], \[calf\], \[puppy\]) simplify to $$\begin{aligned}
\kappa_{m+1} & = &
\kappa_{m}-\frac{2\tau_{m}^{2}}{\kappa_{m}},
\label{eq:RG-HN2}\\
\tau_{m+1} & = & -\frac{\tau_{m}^{2}}{\kappa_{m}},
\nonumber\end{aligned}$$ where $\kappa_m\equiv \kappa_{i}^{(m)}$ and $\tau_{i}=\lambda_{i}\equiv0$ for all $i\geq1$ and $\tau_m\equiv \tau_{0}^{(m)}$. These nonlinear recursions are easily solved by defining $s_{m}=-\kappa_{m}/\tau_{m}$ for which $s_{m+1}=s_{m}^{2}-2$, obtained by dividing the $2^{\rm nd}$ by the $3^{\rm rd}$ line in Eqs. (\[eq:RG-HN2\]). Formally, the solution is $$s_{m}=2\cos\left[2^{m}\arccos\left(\frac{\kappa^{(0)}}{2\tau^{(0)}}\right)\right]=
2T_{2^{m}}\left(\frac{\kappa^{(0)}}{2\tau^{(0)}}\right),
\label{eq:sn}$$ where $T_{n}(x)$ refers to the $n$-th Chebyshev polynomial of the first kind [@abramowitz:64]. Inserting into Eqs. (\[eq:RG-HN2\]) and applying the initial conditions in Eqs. (\[eq:transIC\_HN2\]), generates the results $$\begin{aligned}
\tau^{(m)} & = & -\prod_{i=0}^{m-1}\frac{1}{s_{i}},\nonumber \\
\kappa^{(m)} & = & s_{m}\prod_{i=0}^{m-1}\frac{1}{s_{i}},
\label{eq:HN2solutions}\end{aligned}$$ where the last equality emerges under reordering factors in the products.
Eq. (\[linear1tT\]) in Appendix \[Sec:AppB\] shows that the transmission amplitude $t$ is directly proportional to $\tau^{(k)}$. Clearly, if there is no transmission on any bond, [*i.e.*]{} $\tau^{(k)}=0$, for a given input energy $E$, there can be no transmission through the network itself, no matter what happens on the sites. But instead of plotting $\tau^{(k)}$, it will prove more instructive to plot $\kappa^{(k)}$. It is easy to see from Eqs. (\[eq:HN2solutions\]) that $\kappa^{(k)}$ varies rapidly whenever $\tau^{(k)}$ does, but that $\kappa^{(k)}$ varies smoothly whenever $\tau^{(k)}$ vanishes. In the following, we will see that this behavior remains true for HN3 and HN5, in which case the variation of $\kappa^{(k)}$ with $\kappa^{(0)}=E$ provides more information beyond the mere vanishing of $\tau^{(k)}$.
We have evolved the RG-recursion in (\[eq:RG-HN2\]) for the initial conditions in (\[eq:transIC\_HN2\]) and plotted $\kappa^{(k=10)}$ as a function of $\kappa^{(0)}=E$ in Fig. \[fig:q10HN2\]. Even at that system size, $N=2^{k}=1024$, delocalized states completely cover the domain $-2\leq E\leq2$.
![\[fig:q10HN2\] (Color online) Plot of $\kappa^{(10)}$ in Eq. (\[eq:RG-HN2\]) as a function of its initial condition $\kappa^{(0)}=E$. Even at this small order, the function varies extremely rapidly, such that its (green-shaded) line completely covers the shown domain. Thus, for $-2\leq E\leq2$, $\kappa^{(m)}$ for any sufficiently large $m$ is a random function. Correspondingly, the transmission spectrum is dense, with full transmission close to any input energy $E$ such that particles do not localize.](./q10_HN2.pdf)
Case HN3\[sub:Case-HN3transm\]
------------------------------
Again, we can employ the RG Eqs. (\[kitten\], \[calf\], \[puppy\]) for transmission through HN3 consisting of a ring of $N=2^{k}$ sites, as in Fig. \[fig:3hanoi\]. Since all initial diagonal entries are identical, the hierarchy for the $\kappa_{i}$ collapses and we retain only two nontrivial relations, one for $\kappa_{1}$ and one for all other $\kappa_{i}\equiv \kappa_{2}$ for all $i\geq2$. Here, all $\tau_{i}$ are non-zero, encompassing the backbone links ($i=0$) and all levels of long-range links ($i\geq1$). But it remains $\tau_{i}\equiv-1$ for $i\geq1$ at any step $m$ of the RG, in particular, $\tau_{1}^{(m)}\equiv-1$ throughout; only the backbone $\tau_{0}$ renormalizes non-trivially. Although all links of type $\lambda_{i}$ are initially absent in this network, the details of the RG calculation shows that under renormalization terms of type $\lambda_{1}$ emerge while those for $\lambda_{i}$ for $i\geq2$ remain zero at any step. Thus, we obtain far more elaborate RG recursion equations compared to those of HN2. Abbreviating $\kappa_m\equiv \kappa_{1}^{(m)}$, $\tau_m\equiv \tau_{0}^{(m)}$, and $\lambda_m=\lambda_{1}^{(m)}$, Eqs. (\[kitten\], \[calf\], \[puppy\]) and their initial conditions reduce to $$\begin{matrix}
\kappa_{m+1} & = &
\kappa_{m}-2\lambda_{m}-\frac{2\tau_{m}^{2}}{\kappa_{m}-1},
& \qquad & (\kappa_{0}=E), \\
\tau_{m+1} & = &
\lambda_{m}-\frac{\tau_{m}^{2}}{\kappa_{m}-1},
& \qquad & (\tau_{0}=-1), \\
\lambda_{m+1} & = &-\frac{\tau_{m}^{2}}{\kappa_{m}^{2}-1},
& \qquad & (\lambda_{0}=0). \\
\end{matrix}
\label{eq:transmRG-HN3_redux}$$ We have evolved the RG-recursion in (\[eq:transmRG-HN3\_redux\]) and plotted $\kappa_{k=200}$ as a function of $\kappa_{0}=E$ in Fig. \[fig:q200HN3\]. Even at that enormous (and definitely asymptotic) system size, $N=2^{200}\approx10^{70}$, domains of localized states remain asymptotically inside the physically relevant domain of $-2\leq
E\leq2$. For comparison, the radius of the visible universe is only about $10^{40}$ fm.
In the next subsection, we will explore the asymptotic properties of these recursions for large $m$. We will find domains in $E$ of stationary solutions, which are particular for HN3, and show that the special points where this analysis fails correspond to transitions between localized and delocalized behavior.
![\[fig:q200HN3\] (Color online) Plot of $\kappa_{200}$ in Eq. (\[eq:transmRG-HN3\_redux\]) as a function of its initial condition $\kappa_{0}=E$. Bands of localized and delocalized states intermix. Correspondingly, there are localization-delocalization transitions already before the addition of any additional randomness in HN3, merely as a function of the input energy $E$. On the horizontal axis, we have marked the solutions $E_{s}^{(i)}$ of Eq. (\[eq:qcritical\]) for $s=1$ (red dots), $s=2$ (blue dots), and $s=6$ (small black dots). The accumulation of the latter demonstrates (even for such a small value of $s$) that the band gaps are associated with the absence of such solutions. While the solutions for $s=1$ happen to be interior to the bands, some of those for $s=2$ appear to mark the band edges, in particular the one at $E=E_{2}^{(2)}=-0.637875$.](./q200_HN3.pdf)
### Analysis of the Steady State:\[sub:SteadyStateAnalysis\]
We can analyze the absorbing steady state, which is the unique feature of HN3 (in contrast to HN2 and HN5) leading to band gaps, as follows. Numerical trails show that the RG recursions in Eq. (\[eq:transmRG-HN3\_redux\]) reach a steady state for certain initial conditions $E$ when for all $m$ larger than some $m_{0}$ it is $$\begin{aligned}
1 & \gg &
\lambda_{m}\gg\frac{\tau_{m}^{2}}{\kappa_{m}-1},\qquad(m\to\infty).
\label{eq:Asymp}\end{aligned}$$ The leading contribution for $\kappa_{m}$ in Eq. (\[eq:transmRG-HN3\_redux\]) then suggests $$\begin{aligned}
\kappa_{m+1} & \sim & \kappa_{m}\sim \kappa_{\infty},
\label{eq:qinfty}\end{aligned}$$ which is a constant that is difficult to derive from the initial conditions, $\kappa_{0}=E$, unfortunately.
From the recursion for $\tau_{m}$ in Eq. (\[eq:transmRG-HN3\_redux\]), we further obtain $$\begin{aligned}
\tau_{m+1} & \sim & \lambda_{m} \sim -\frac{\tau_{m-1}^{2}}{\kappa_{\infty}^{2}-1}.
\label{eq:pasymp}\end{aligned}$$ This 2nd order difference equation has two solutions, of which we discard the oscillatory one, to get for large $m>m_{0}$: $$\begin{aligned}
\tau_{m} & \sim & -\left(\kappa_{\infty}^{2}-1\right)\exp\left\{ -C\sqrt{2^{m}}\right\} ,\\
\lambda_{m} & \sim & -\left(\kappa_{\infty}^{2}-1\right)\exp\left\{
-C\sqrt{2^{m+1}}\right\} ,\end{aligned}$$ where $C>0$ is another undetermined constant that depends on $E$. Using the recursion for $\kappa_{m}$ in Eq. (\[eq:transmRG-HN3\_redux\]) to next-to-leading order yields $$\begin{aligned}
\kappa_{m+1} & \sim & \kappa_{m}-2\lambda_{m},
\label{eq:q_correction}\\
& \sim & \kappa_{m}+2\left(\kappa_{\infty}^{2}-1\right)\exp\left\{
-C\sqrt{2^{m+1}}\right\} ,\end{aligned}$$ which, when summed from a $m>m_{0}$ to $\infty$ results in $$\begin{aligned}
\kappa_{m} & \sim &
\kappa_{\infty}-2\left(\kappa_{\infty}^{2}-1\right)\exp\left\{
-C\sqrt{2^{m+1}}\right\} ,
\label{eq:q_asymp}\end{aligned}$$ where we have kept only the first term in the sum, as the summand is exponentially decaying.
The consequences of this analysis are quite dramatic. If such a steady-state solution is reached, both transmission rates $\tau$ and $\lambda$ vanish, only leaving a finite on-site energy $\kappa_{\infty}$. Hence, there can not be any transmission through the network when such a state is reached, and gaps emerge in the transmission spectrum. Since the system size is given by $N=2^{m}$, this result implies that finite size corrections scale with $\exp\left\{ -C\sqrt{2N}\right\} $, [*i.e.*]{} finite-size corrections decay rapidly with a stretched exponential. Note that there are no steady-state solutions that cross $\kappa_{\infty}=\pm1$ (dashed lines in Fig. \[fig:q200HN3\]), where the correction in Eq. (\[eq:q\_asymp\]) would break down. Instead, the approach of $\kappa_{\infty}\to\pm1$ frequently appears to be associated with the emergence of a band edge between localized and delocalized states.
### Band-Edge Analysis:\[sub:BandEdgeAnalysis\]
The non-trivial band structure warrants some further investigation. In particular, we can associate such band edges with initial conditions $E=E_{s}^{(i)}$ for which there exists a $m=s$ such that $$\begin{aligned}
\kappa_{s}\left(E_{s}^{(i)}\right) & = & -1,
\label{eq:qcritical}\end{aligned}$$ a singular point in the recursion for $\lambda_{s+1}$. [\[]{}Interestingly, any singularity at $\kappa_{m}=+1$ appears to be benign in that it does *not* affect the continuity in $\kappa_{m}$ as a function of $E$ for $m\to\infty$; it afflicts each quantity in Eqs. (\[eq:transmRG-HN3\_redux\]) *simultaneously*, leading to a divergence in $\kappa_{m+1}$, $\tau_{m+1}$, and $\lambda_{m+1}$ just so that $\kappa_{m+2}\approx \kappa_{m-1}$, $\tau_{m+2}\approx
\tau_{m-1}$, $\lambda_{m+2}\approx \lambda_{m-1}$.[\]]{} In Fig. \[fig:q200HN3\], we have also marked the real solutions $E_{s}^{(i)}$ of Eq. (\[eq:qcritical\]) for $s=1$, 2, and 6. Clearly, those solutions strongly correlate with the bands, and there appear to be none within the gaps (although we have not been able to prove this conjecture). But while those solutions for $s=1$, $E_{1}^{(1,2)}=\pm\sqrt{3}$, are located well within some band (as are those for $s=6$), the four real solutions of Eq. (\[eq:qcritical\]) for $s=2$ satisfy the quartic equation $$\begin{aligned}
0 & = & 1-2E-6E^{2}+E^{4}
\label{eq:quarticQ}\end{aligned}$$ and appear to be all associated with some more or less significant band edge, see blue dots in Fig. \[fig:q200HN3\]. We can speculate that there is a whole hierarchy of transitions, each associated with one of the solutions $E_{m}^{(i)}$, which may become dense on certain intervals. While we don’t know what determines those intervals precisely, we can analyze the behavior in the neighborhood of Eq. (\[eq:qcritical\]). We observed that the recursions in Eq. (\[eq:transmRG-HN3\_redux\]) possess stable steady-state solutions for large $m$ characterized by $\tau_{m}\sim
\lambda_{m}\to0$, [*i.e.*]{} vanishing bond-strength between input and output. These solutions prevail in the observed band gaps, which accordingly correspond to localized states. It seems that the reason for the persistence of gaps derives from that stability: band gaps emerge whenever the steady state is reached *before* Eq. (\[eq:qcritical\]) can be satisfied. For instance, in the case of HN5 below, any putative steady-state solution proves unstable for sufficiently large $m$ such that any band gaps are transitory only, see Fig. \[fig:q10HN5\].
For the analysis of the recursion Eq. (\[eq:transmRG-HN3\_redux\]), we assume that for some $m=s$, we reach $$\begin{aligned}
\kappa_{s} & \sim & -1+\epsilon,\qquad(\epsilon\ll1),
\label{eq:qAnsatz}\end{aligned}$$ where $\epsilon=\epsilon(E)$ may be of either sign, depending on $\Delta E=E-E_{s}^{(i)}$. Generically, $\epsilon\propto\Delta E$, see Sec. \[sub:Scaling-Relation:\] below. Assuming that $\tau_{s},\lambda_{s}\ll1/\epsilon$ leads to $$\begin{aligned}
\kappa_{s+1} & \sim & -1-2\lambda_{s}+\tau_{s}^{2}+O(\epsilon),\nonumber \\
\tau_{s+1} & \sim & \lambda_{s}+\frac{\tau_{s}^{2}}{2}+O(\epsilon),
\label{eq:s1}\\
\lambda_{s+1} & \sim & \frac{\tau_{s}^{2}}{2\epsilon}+\frac{\tau_{s}^{2}}{4}+O(\epsilon),
\nonumber\end{aligned}$$ which leaves only $\lambda_{s+1}$ singular. After one more recursion step, we get instead $$\begin{aligned}
\kappa_{s+2} & \sim &
-\frac{\tau_{s}^{2}}{\epsilon}+\frac{2+6\lambda_{s}+2\lambda_{s}^{2}-2\tau_{s}^{2}-5\lambda_{s}
\tau_{s}^{2}}{-2-2\lambda_{s}+\tau_{s}^{2}}+O(\epsilon),
\nonumber \\
\tau_{s+2} & \sim &
\frac{\tau_{s}^{2}}{2\epsilon}+
\frac{2\lambda_{s}^{2}+\tau_{s}^{2}+3\lambda_{s}\tau_{s}^{2}}
{4+4\lambda_{s}-2\tau_{s}^{2}}+O(\epsilon),
\label{eq:s2}\\
\lambda_{s+2} & \sim & O(1).
\nonumber\end{aligned}$$ At this point, Eq. (\[eq:transmRG-HN3\_redux\]) decouple to leading order, as $\lambda_{s+2+i+1}\sim
-\tau_{s+2+i}^{2}/\kappa_{s+2+i}^{2}\sim -1/4$ remains of order $O(1)$ while both $\kappa_{s+2+i}$ and $\tau_{s+2+i}$ are of order $O(1/\epsilon)$, and we get for some $i\geq0$ $$\begin{aligned}
\kappa_{s+2+i+1} & \sim & \kappa_{s+2+i}-\frac{2\tau_{s+2+i}^{2}}{\kappa_{s+2+i}},
\nonumber \\
\tau_{s+2+i+1} & \sim & -\frac{\tau_{s+2+i}^{2}}{\kappa_{s+2+i}}.
\label{eq:asymp_qp}\end{aligned}$$ These are *exactly* the same recursions we obtained in Eq. (\[eq:RG-HN2\]) for HN2, with the solution in Eq. (\[eq:sn\]): $$\begin{aligned}
\frac{\kappa_{s+2+i}}{2\tau_{s+2+i}} & \sim &
-T_{2^{i}}\left(-\frac{\kappa_{s+2}}{2\tau_{s+2}}\right),
\label{eq:asymp_T}\end{aligned}$$ where from Eq. (\[eq:s2\]) we have $$\begin{aligned}
\label{eq:asymp_IC}
\frac{\kappa_{s+2}}{2\tau_{s+2}} & \sim &
-1+A\frac{\epsilon}{\tau_{s}^{2}}, \\
A & = & \frac{2+6\lambda_{s}
-3\tau_{s}^{2}-8\lambda_{s}\tau_{s}^{2}}{2+2\lambda_{s}-\tau_{s}^{2}}
\>.\end{aligned}$$ With that inserted into Eq. (\[eq:asymp\_T\]), we can deduce $$\begin{aligned}
\frac{\kappa_{s+2+i}}{2\tau_{s+2+i}} & \sim &-T_{2^{i}}\left(1-A\frac{\epsilon}{\tau_{s}^{2}}\right),
\nonumber \\
& \sim & 1-A\frac{\epsilon}{\tau_{s}^{2}}T_{2^{I}}'\left(1\right),
\nonumber \\
& \sim & 1-2^{2i}A\frac{\epsilon}{\tau_{s}^{2}},
\label{eq:asymp_correction}\end{aligned}$$ since $T'_{n}(x)=nU_{n-1}(x)$ and $U_{n-1}(1)=n$, referring to the Chebyshev polynomial of the 2nd kind, $U_{n}(x)$ [@abramowitz:64]. With the exponential growth in $i$ of the correction amplitude in the asymptotic expansion in Eq. (\[eq:asymp\_correction\]), the expansion breaks down at some $i\sim i_{0}$ such that the correction itself becomes of $O(1)$, [*i.e.*]{} $$\begin{aligned}
i_{0} & \sim & \frac{1}{2}\log_{2}\left(\frac{\tau_{s}^{2}}{\left|A\epsilon\right|}\right).
\label{eq:i0}\end{aligned}$$ For $i>i_{0}$, according to the first line of Eq. (\[eq:asymp\_correction\]) the ratio $\kappa_{s+2+i}/\tau_{s+2+i}$ either rises or falls exponentially, depending on whether $A\epsilon<0$ or $A\epsilon>0$, respectively. In the latter case, $\kappa$ becomes less relevant and the bonds $\tau$ and $\lambda$ determine the future evolution in $m$, leading again to the chaotic behavior in $\kappa_{m}$ observed within the bands in Fig. \[fig:q200HN3\]. On the other hand, if $A\epsilon>0$, the on-site energies $\kappa$ dominate exponentially over the couplings $\tau$ and $\lambda$, evolving towards an absorbing steady state on the band-gap side of the transition.
### Scaling Relation for $\kappa_{\infty}(E)$:\[sub:Scaling-Relation:\]
As mentioned in Sec. \[sub:SteadyStateAnalysis\], we can not generally predict the dependence of the asymptotic behavior on the initial condition $E$. But we can use the analysis of that section at least to determine the behavior of $\kappa_{\infty}(E)$ on the approach to those band edges where it diverges.
It is easy to show, using just the singular terms for $\kappa$ and $\tau$ in Eq. (\[eq:s2\]) inserted into the recursions in Eq. (\[eq:asymp\_qp\]), that both simultaneously decay exponentially with $i$ at least while $i\lesssim i_{0}$ from Eq. (\[eq:i0\]), $$\begin{aligned}
\kappa_{s+2+i} & \sim & -\frac{\tau_{s}^{2}}{2^{i}\epsilon}+O(1),
\label{eq:qs_asymp}\\
\tau_{s+2+i} & \sim & \frac{\tau_{s}^{2}}{2^{i+1}\epsilon}+O(1).
\nonumber\end{aligned}$$ At $i\sim i_{0}$ there is a cross-over beyond which for all $i>i_{0}$ it is $\tau_{s+2+i}\to0$ and a saturated value of $\kappa_{s+2+i}\to \kappa_{\infty}$ is reached. We can obtain the dominant asymptotic behavior at the cross-over from $$\begin{aligned}
\kappa_{\infty} & \sim & \kappa_{s+2+i_{0}} \sim
-\frac{\tau_{s}^{2}}{\epsilon\sqrt{\frac{\tau_{s}^{2}}{\left|A\epsilon\right|}}},
\nonumber \\
& \sim &
-\sqrt{\left|A\right|\tau_{s}^{2}}\frac{\mathrm{sgn}\left(\epsilon\right)}
{\left|\epsilon\right|^{\frac{1}{2}}}.
\label{eq:qinfty_eps}\end{aligned}$$ We can further establish a (generic) relation between $\epsilon$ and $$\begin{aligned}
\Delta E & \sim & E-E_{s}^{(i)}\end{aligned}$$ by extending the discussion of Eq. (\[eq:qcritical\]) in Sec. \[sub:BandEdgeAnalysis\]. We set $$\begin{aligned}
-1+\epsilon\left(\Delta E\right) & \sim & \kappa_{s}\left(E_{s}^{(i)}+\Delta E\right),\\
& \sim & \kappa_{s}\left(E_{s}^{(i)}\right)+\kappa'_{s}\left(E_{s}^{(i)}\right)\Delta E,\\
& \sim & -1+\kappa'_{s}\left(E_{s}^{(i)}\right)\Delta E,\end{aligned}$$ since $\kappa_{s}\to-1$ is a regular limit for $\Delta E\to0$, as Eq. (\[eq:quarticQ\]), for example, suggests. Hence, $$\begin{aligned}
\epsilon\left(\Delta E\right) & \sim &
\kappa'_{s}\left(E_{s}^{(i)}\right)\Delta E,\end{aligned}$$ and we conclude $$\begin{aligned}
\kappa_{\infty} & \sim &
-\sqrt{\frac{\left|A\right|\tau_{s}^{2}}
{\left|\kappa'_{s}\left(E_{s}^{(i)}\right)\right|}}
\frac{\mathrm{sgn}\left(\kappa'_{s}\left(E_{s}^{(i)}\right)\Delta
E\right)}{\left|\Delta E\right|^{\frac{1}{2}}}
\label{eq:qinftyDQ}\end{aligned}$$ with $\Delta E\to0$. In Fig. \[fig:qinftyDQ\] we have tested the asymptotic relation $1/\kappa_{\infty}\sim\sqrt{\left|\Delta E\right|}$ in the band gap near $E_{s}^{(2)}=-0.637875\ldots$, a solution of Eq. (\[eq:quarticQ\]) marked blue in Fig. \[fig:q200HN3\].
![\[fig:qinftyDQ\] (Color online) Plot of $1/\kappa_{\infty}$ as a function of $\sqrt{E-E_{s}^{(i)}}$ for $E\to E_{s}^{(i)}$ to test Eq. (\[eq:qinftyDQ\]). Here, $E_{2}^{(2)}=-0.637875$, marked as the 2nd blue dot from the left in Fig. \[fig:q200HN3\].](./qDE_HN3.pdf)
Interpolation between HN2 and HN3\[sub:Interpolation-between-HN2\]
------------------------------------------------------------------
It proves fruitful to consider an interpolation between the case of HN2 in Sec. \[sub:HN2trans\] and HN3 in Sec. \[sub:Case-HN3transm\] in terms of a one-parameter family of models. To wit, we can accomplish such an interpolation by weighting the transmission along the small-world links (see red-shaded links in Fig. \[fig:HN3scatterKoch\]) by a factor of $y$ relative to that of the backbone links (see black links in Fig. \[fig:HN3scatterKoch\]). Clearly, more generally, hierarchy and/or distance-dependent weights could be introduced as well. For $y=0$, small-world links are non-existent, and we have the linear lattice HN2. Although still mostly delocalized, the states of the systems immediately change behavior when $y>0$, and we find localized states which expand their domain towards $y=1$, corresponding to HN3, and continue to do so until at about $y=4$ no transmission is possible any longer: The more we weight small-world links here, which classically would *expedite* transport [@TASEP09], the less quantum transport is possible! In the next section we will see that even more small-world links, as in HN5, can lead to more transmission again. Hence, the detailed structure of the links matter.
To explore this $y$-family of models, we have to generalize Eq. (\[eq:transmRG-HN3\_redux\]) appropriately: $$\label{eq:RG-HN2y}
\begin{matrix}
\kappa_{m+1} & = &
\kappa_{m}-2\lambda_{m}-\frac{2\tau_{m}^{2}}{\kappa_{m}-y},
& \qquad & (\kappa_{0}=E), \\
\tau_{m+1} & = &
\lambda_{m}-\frac{\tau_{m}^{2}}{\kappa_{m}-y},
& \qquad & (\tau_{0}=-1), \\
\lambda_{m+1} & = & -\frac{y\,\tau_{m}^{2}}{\kappa_{m}^{2}-y^{2}},
& \qquad & (\lambda_{0}=0),
\end{matrix}$$ since for all non-backbone links in Eqs. (\[kitten\], \[calf\], \[puppy\]) it is $\tau_{i\geq1}^{(0)}=y\, \tau_{0}^{(0)}$, [*i.e.*]{} $\tau_{i\geq1}^{(m)}=-y$ at every RG-step. Note that these equations reduce to Eqs. (\[eq:RG-HN2\]) for $y=0$ (with all $\lambda_{i}\equiv0$) and to Eq. (\[eq:transmRG-HN3\_redux\]) for $y=1$.
In Fig. \[fig:HN2yplot\] we map out the state of Eq. (\[eq:RG-HN2y\]) after the $1000^{\rm th}$ iteration based on whether a steady state has been reached or not, depending on the incoming energy $E$ and the relative weight $y$. For any $y>0$, the ability to transmit has a strong chaotic dependence on these parameters, and ceases completely for $y>4$. Even within domains of apparent transmission there are often sub-domains where no transmission is possible, and it is not clear whether true conduction bands exist. Since in this model the long-range links are not connected to each other except through the backbone, one may speculate that even at high weight these links merely lead to localized resonances that interfere with transport along the backbone instead of conveying it. (A similar confinement effect was observed for the RG applied to random walks on HN3 in Ref. [@SWN].)
It is straightforward to generalize the discussion for HN3 in Sec. \[sub:Case-HN3transm\] to this model. In particular, for the band-edge analysis we have to generalize Eq. (\[eq:qcritical\]) to read $$\begin{aligned}
\kappa_{s}\left(E_{s}^{(i)}\right) & = & -y.
\label{eq:y_qcritical}\end{aligned}$$ For $s=0,1,2$, and 3, the numerical solutions $E_{s}^{(i)}(y)$ are also plotted as lines in Fig. \[fig:HN2yplot\]. The result underlines the contention made before for HN3 that the transitions between transmission and localization are closely associated with these singular points of the Eq. (\[eq:RG-HN2y\]). For instance, the big blue-shaded dots in Fig. \[fig:q200HN3\] correspond here to the intersection of the simple-dashed line for $s=2$ with the dotted horizontal line along $y=1$ (i. e. HN3). Dominant features emerge, such as the line $\kappa_{0}=E=-y$. Other interesting points become apparent, for instance, the one at $y=-E=1/\sqrt{2}$. While there is otherwise no apparent relation to solutions of $\kappa_{s}=+y$, it should be noted that its $s=0$ case *does* produce a distinct feature in the line $E=y$.
Case HN5\[sub:Case-HN5transm\]
------------------------------
In close correspondence with the treatment in Sec. \[sub:Case-HN3transm\], we can employ the RG in Eqs. (\[kitten\], \[calf\], \[puppy\]) for transmission through HN5 consisting of a ring of $N=2^{k}$ sites. The sole difference with Sec. \[sub:Case-HN3transm\] is that all links of type $\lambda_{i}$ are initially present in this network. Yet, the details of the RG calculation in Sec. \[MatrixRG\] show that under renormalization only links of type $\lambda_{1}$ renormalize while those $\lambda_{i}$ for $i\geq2$ remain unrenormalized at any step. The diagonal elements are again hierarchy-independent, $\kappa_{i}^{(0)}\equiv E$, while the recursion for $\lambda$ changes. Abbreviating $\kappa\equiv \kappa_{1}$, $\tau\equiv
\tau_{0}$, and $\lambda=\lambda_{1}$, Eqs. (\[kitten\], \[calf\], \[puppy\]) and their initial conditions reduce to $$\label{eq:transmRG-HN5_redux}
\begin{matrix}
\kappa_{m+1} & = &
\kappa_{m}-2\lambda_{m}-2-2\frac{\tau_{m}^{2}}{\kappa_{m}-1},
& \qquad & (\kappa_{0}=E), \\
\tau_{m+1} & = & \lambda_{m}-\frac{\tau_{m}^{2}}{\kappa_{m}-1},
& \qquad & (\tau_{0}=-1), \\
\lambda_{m+1} & = &
1-\frac{\tau_{m}^{2}}{\kappa_{m}^{2}-1},
& \qquad & (\lambda_{0}=-1).
\end{matrix}$$
We have evolved the RG-recursion in Eq. (\[eq:transmRG-HN5\_redux\]) and plotted $\kappa^{(k=10)}$ as a function of $\kappa^{(0)}=E$ in Fig. \[fig:q10HN5\]. At that (definitely not asymptotic) system size, $N=2^{10}\approx10^{3}$, domains of localized states remain which will disappear asymptotically, as for the case of HN2 in Fig. \[fig:q10HN2\]. Unlike for HN3, there are no steady-state solutions for Eq. (\[eq:transmRG-HN5\_redux\]) that could signal localization. It is interesting to analyze the cause of this behavior. Since the $\lambda$-links \[green-shaded in Fig. \[fig:5hanoi\]\] are an original feature of the network, the RG recursion in Eq. (\[eq:transmRG-HN5\_redux\]) for $\lambda_{m}$ obtain a constant offset preventing $\lambda_{m}$, and hence $\tau_{m}$, from vanishing. Unlike for HN3, these small-world links allow perpetual transmission within a given level of the hierarchy, instead of interfering with other paths, and transmission is enhanced. In fact, we have studied also an interpolation between HN3 and HN5 by attaching a relative weight $y$ to these $\lambda$-links in HN5, relative to the otherwise uniform links present in HN3. Thus, for $y=0$ HN3 is obtained, and for $y=1$ we get HN5. Yet, we find that for any $y>0$, this model eventually behaves like HN5, with unfettered transmission throughout the energy spectrum.
![\[fig:q10HN5\] (Color online) Plot of $\kappa^{(10)}$ in Eqs. (\[eq:transmRG-HN5\_redux\]) for HN5 as a function of its initial condition $\kappa^{(0)}=E$. Bands of localized and delocalized states intermix, but those localized intervals disappear asymptotically. ](./q10_HN5.pdf)
Summary and Discussion \[conclusion\]
=====================================
We have devised a decimation RG procedure within the matrix methodology of Ref. [@daboul] to obtain transmission of quantum electrons through networks within the tight-binding model approximation. This decimation RG procedure of Appendix \[Sec:AppA\] can in principle be implemented for any network, in that Eq. (\[Eq:AppA01\]) with $n+m$ sites is reduced to Eq. (\[Eq:AppA07\]) with $n$ sites. For general networks the bookkeeping required could become prohibitive. However, we find the RG equations well suited to the networks we have chosen to analyze, namely the hierarchical networks called HN3 (Hanoi Network with 3 bonds per site) and HN5 (Hanoi Network with an average of 5 bonds per site). This is because the RG can proceed block-wise, as depicted in Fig. \[BuildingBlock\]. We have analyzed both the ring geometry (Fig. \[fig:3hanoi\]) and the linear geometry (Figs. \[fig:3hanoi\] and \[fig:HN3scatterKoch\]) of these networks, with the only difference being the last steps of the RG (Appendix \[Sec:AppB\]). We have also analyzed how the transmission for the linear lattice (which we label HN2) changes with the strength of the small-world-type bonds added to form HN3 (Fig. \[fig:HN2yplot\]).
The Hanoi networks are hierarchical models that provide an intermediary between regular lattices and lattices that have random small-world bonds placed on regular lattices. Since the small-world-type bonds in the Hanoi networks provide short-cuts between sites, one might expect intuitively that they should provide extra paths for transmission. However, because of the hierarchical nature of the networks, the networks no longer possess translational symmetry. This broken symmetry is seen by the incoming quantum electrons, and can lead to Anderson-type localization. For the HN3 network with variable strength $y$ for the small-world-type bonds, we find that the more we weight the small-world links, which classically would [*expedite*]{} transport [@TASEP09], the less quantum transport is possible (Fig. \[fig:HN2yplot\]). Furthermore, we find that HN3 has band edges at particular energies $E$ of the incoming electrons, between band gaps with near zero transmission and regions of extended wavefunctions and transmission near unity. The network HN5 adds still more small-world type bonds to HN3, but we find that for any non-zero strength of these additional bonds the band edges seen in HN3 disappear and approximately unfettered transmission is seen for large enough lattices for any energy of the incoming electrons. Thus the hierarchical nature of these lattices lead to very interesting transmission properties. Thus for these hierarchical lattices, the metal-insulator transitions depend on quantities other than just the embedding dimension. Similar effects have been seen for critical phenomena in hierarchical lattices, but only where translational symmetry is broken [@Gefen1980; @Banav1983; @LeGuillou1987; @Novotny1992; @Novotny1993].
Since the HN3 and HN5 networks are planar (see for example Fig. \[fig:HN3scatterKoch\]), experimental realizations of these networks should be possible to construct using etching techniques. These experiments would lead to very interesting device physics, in particular at the energy-dependent band edges we have analyzed for HN3.
The authors thank Lazarus Solomon for helpful discussions. MAN thanks Emory University for hospitality during a one-month sabbatical stay. SB acknowledges support from the U.S. National Science Foundation through grant number DMR-0812204.
General Matrix Formulation of the RG \[Sec:AppA\]
=================================================
As in Ref. [@daboul], a blob’ of atoms in the tight-binding approximation is considered to be connected to two semi-infinite leads. Each semi-infinite lead is considered to be a linear arrangement of tight-binding sites with on-site energy $0$ and a hopping parameter of $-1$ (setting the energy scale). The incoming (outgoing) lead is connected to the blob’ sites by a vector of hopping parameters ${\vec w}$, (respectively, ${\vec u}$). The Schr[ö]{}dinger equation for the infinite system, ${\cal H}_\infty {\vec\Psi} = E {\vec\Psi}$ must be solved with the appropriate boundary conditions. The [*ansatz*]{} made [@daboul] is that the wavefunction at every site in the in-coming lead has the form $\psi_{m-1}=e^{i m q}+ r e^{-i m q}$ with $m=-\infty,\cdots,-2,-1,0$, and the outgoing lead has the wavefunction $\psi_{m+1} = t e^{i m q}$ with $m=0,1,2,\cdots,\infty$. The wavevector $q$ is related to the energy of the incoming electron by $E=2\cos(q)$. The reflection probability is $R=|r|^2$ and the transmission probability is $T=|t|^2$. With this ansatz, the required solution of the infinite matrix Sch[ö]{}dinger equation reduces to the solution of a finite matrix equation of dimension two larger than then number of sites in the blob’. Unlike Ref. [@daboul], we assume no direct hopping between the leads, [*i.e.*]{} no short-cut path around the blob’.
Consider the case with $n+m$ sites in the blob. We specialize to the case where all hopping parameters ($\tau$ or $\lambda$) and on-site energies (convoluted with $E$ to give $\kappa$) are real. The $(n+m+2)\times(n+m+2)$ matrix to solve for the transmission $T=|t|^2$ is [@daboul] $$\label{Eq:AppA01}
\begin{pmatrix}
\xi & {\vec w}^{\rm T} & {\vec w}_{d}^{\rm T} & 0 \\
{\vec w} & {\bf A} & {\bf B} & {\vec u} \\
{\vec w}_{d} & {\bf B}^{\rm T} & {\bf D} & {\vec u}_{d} \\
0 & {\vec u}^{\rm T} & {\vec u}_{d}^{\rm T} & \xi \\
\end{pmatrix}
\begin{pmatrix}
1+r \\
{\vec \psi} \\
{\vec \psi}_d \\
t \\
\end{pmatrix}
\>=\>
\begin{pmatrix}
2 i \Im (\xi) \\
{\vec 0}_n \\
{\vec 0}_m \\
0 \\
\end{pmatrix}$$ with $\Im(\xi)$ the imaginary part of the complex function $\xi$, and the definition $$\label{Eq:AppA02}
\xi = {\rm e}^{iq} - E = -\frac{E}{2} + i \frac{\sqrt{4-E^2}}{2}
.$$ The matrix ${\bf A}$ is of size $n\times n$ and includes all interactions between the $n$ sites that will remain after the RG. The matrix ${\bf D}$ is of size $m\times m$ and includes all interactions between the $m$ sites that will be decimated by the RG. The matrix ${\bf B}$ is of size $n\times m$ and includes all interactions between the the $n$ sites that will remain and the $m$ sites that will be decimated. The matrices ${\bf A}$ and ${\bf D}$ are both symmetric matrices, while in general ${\bf B}$ is not symmetric. The vectors ${\vec w}$, ${\vec u}$, ${\vec \psi}$, and ${\vec 0}_n$ are all of length $n$, while the vectors ${\vec w}_{d}$, ${\vec u}_{d}$, ${\vec
\psi}_d$, and ${\vec 0}_m$ are all of length $m$.
Multiplying out the two middle rows gives the equations $$\label{Eq:AppA03}
(1+r){\vec w} +{\bf A}{\vec\psi}+{\bf B}{\vec\psi}_d + t{\vec u} = {\vec 0}_n$$ and $$\label{Eq:AppA04}
(1+r){\vec w}_{d} +{\bf B}^{\rm T}{\vec\psi}+{\bf D}{\vec\psi}_d +
t{\vec u}_{d} = {\vec 0}_m
.$$ Solve Eq. (\[Eq:AppA04\]) for ${\psi}_d$ to give $$\label{Eq:AppA05}
{\vec \psi}_d =
-{\bf D}^{-1}
\left[
(1+r){\vec w}_{d} +{\bf B}^{\rm T}{\vec\psi}+ t{\vec u}_{d}
\right]
.$$
Substituting ${\vec\psi}_d$ into Eq. (\[Eq:AppA03\]) and collecting terms allows the equation to be rewritten as $$\label{Eq:AppA06}
\left[{\vec w} - {\bf B}{\bf D}^{-1}{\vec w}_{d}\right] (1+r) +
\left[{\bf A}-{\bf B}{\bf D}^{-1}{\bf B}^{\rm T}\right]{\vec\psi} +
\left[{\vec u}-{\bf B}{\bf D}^{-1}{\vec u}_{d}\right] t
= {\vec 0}_n
.$$ Note that since ${\bf D}$ is symmetric, so is ${\bf D}^{-1}$. We can also substitute ${\vec\psi}_d$ from Eq. (\[Eq:AppA03\]) in for expressions obtained from multiplying out the top and bottom rows of Eq. (\[Eq:AppA01\]). This gives that the matrix equation $$\label{Eq:AppA07}
\begin{pmatrix}
\xi -{\vec w}_{d}^{\rm T}{\bf D}^{-1}{\vec w}_{d} &
{\vec w}^{\rm T} -{\vec w}_{d}^{\rm T}{\bf D}^{-1}{\bf B}^{\rm T} &
-{\vec w}_{d}^{\rm T}{\bf D}^{-1}{\vec u}_{d} \\
{\vec w} -{\bf B}{\bf D}^{-1}{\vec w}_{d} &
{\bf A} -{\bf B}{\bf D}^{-1}{\bf B}^{\rm T} &
{\vec u} -{\bf B}{\bf D}^{-1}{\vec u}_{d} \\
-{\vec w}_{d}^{\rm T}{\bf D}^{-1}{\vec u}_{d} &
{\vec u}^{\rm T} -{\vec u}_{d}^{\rm T}{\bf D}^{-1}{\bf B}^{\rm T} &
\xi -{\vec u}_{d}^{\rm T}{\bf D}^{-1}{\vec u}_{d} \\
\end{pmatrix}
\begin{pmatrix}
1+r \\
{\vec \psi} \\
t \\
\end{pmatrix}
\>=\>
\begin{pmatrix}
2 i \Im (\xi) \\
{\vec 0}_n \\
0 \\
\end{pmatrix}$$ has the same solutions for $r$, $t$, and ${\vec\psi}$ as does Eq. (\[Eq:AppA01\]). For the cases in this paper we will not have interactions between the input site and output site and the $m$ sites to be decimated, so both ${\vec w}_{d}$ and ${\vec u}_{d}$ will be zero and the $(n+2)\times(n+2)$ matrix equation to solve for $t$ becomes $$\label{Eq:AppA08}
\begin{pmatrix}
\xi \>\> & {\vec w}^{\rm T} & 0 \\
{\vec w} & {\bf A} -{\bf B}{\bf D}^{-1}{\bf B}^{\rm T} & {\vec u} \\
0 & {\vec u}^{\rm T} & \xi \\
\end{pmatrix}
\begin{pmatrix}
1+r \\
{\vec \psi} \\
t \\
\end{pmatrix}
\>=\>
\begin{pmatrix}
2 i \Im (\xi) \\
{\vec 0}_n \\
0 \\
\end{pmatrix}
.$$ This completes the decimation RG of the $m$ sites.
![(Color online) The last steps in the RG for the linear (left) and the ring (right) geometries. In both cases one is left with only two sites connected to the external leads. Note for clarity we have dropped the superscripts that denote the RG number on all variables.[]{data-label="EndGame"}](./EndGame.pdf "fig:"){width="\textwidth"}\
Transmission from Small Renormalized Lattices \[Sec:AppB\]
==========================================================
Equations(\[aries\] – \[puppy\]) can be used to construct the $m=k-2$ state shown at the top of Fig. \[EndGame\]. The subsequent RG steps require use of these equations with care.
Linear geometry
---------------
For the linear geometry (left side of Fig. \[EndGame\]), for the $m=k-1$ RG step we proceed as follows. Using the RHS of Eq. (\[aries\]) with $i=2$ we calculate $$\label{}
\tilde{\kappa}_{2}^{(k-1)} = \kappa_{1}^{(k-1)} + \kappa_{k+1} - \kappa_{k}
- 2(\lambda_{1}^{(k-1)} - \lambda_{k})$$ and then substitute $\kappa_{2}^{(k-1)}$ by $\tilde{\kappa}_{2}^{(k-1)}$ in the RHS of Eq. (\[cancer\]) to get $$\label{}
\kappa_{2}^{(k-1)} = \kappa_{k+1} + \lambda_{k} - \lambda_{1}^{(k-1)}
+ \frac{1}{2}[\kappa_{1}^{(k-1)} - \kappa_{k}]
.$$
This completes the $m=k-1$ RG step. Now for $m=k$, we proceed in a similar way. Using the RHS of Eq. (\[kitten\]) with $m=k-1$ and $\tau_k=0$, we calculate $$\label{}
\tilde{\kappa}_{1}^{(k)} = \kappa_{1}^{(k-1)} + \kappa_{k+1} - \kappa_{k}
- 2(\lambda_{1}^{(k-1)} - \lambda_{k}) - \frac{2[\tau_{0}^{(k-1)}]^{2}}{\kappa_{1}^{(k-1)}}$$ and then with $m=k$, $i=2$, $\lambda_{k+1} =0$ and substituting $\kappa_{1}^{(k)}$ by $\tilde{\kappa}_{1}^{(k)}$ in the RHS of Eq. (\[aries\]), we calculate $$\label{}
\tilde{\kappa}_{2}^{(k)}= \tilde{\kappa}_{1}^{(k)} + \kappa_{k+2} -
\kappa_{k+1} - 2\lambda_{1}^{(k)}
\>.$$ Substituting $m=k-1$ in Eq. (\[puppy\]) gives $\lambda_{1}^{(k)} =0$. Finally, with $m=k$ and substituting $\kappa_{2}^{(k)}$ by $\tilde{\kappa}_{2}^{(k)}$ in the RHS of Eq. (\[cancer\]), we get $$\begin{aligned}
\label{}
\kappa_{1}^{(k)} &=& \kappa_{k+1} + \lambda_{k} - \lambda_{1}^{(k-1)}
- \frac{[\tau_{0}^{(k-1)}]^{2}}{\kappa_{1}^{(k-1)}} + \frac{1}{2}[\kappa_{1}^{(k-1)}
- \kappa_{k}] \nonumber \\
&=& \kappa_{k+1} + \lambda_{k} + \tau_{0}^{(k)}- 2\lambda_{1}^{(k-1)}
+ \frac{1}{2}[\kappa_{1}^{(k-1)} - \kappa_{k}] \> .\end{aligned}$$
After performing $k$ RG steps, we are left with just $1 +
2^{k-k} =2$ sites with an on-site energy corresponding to $\kappa_{1}^{(k)}$ and an interaction of $\tau_{0}^{(k)}$ between them. One site is connected to the input by $w$ while the other is connected to the output by $u$. In order to decimate these two sites, we have $$\begin{aligned}
{\bf A} =
\begin{pmatrix}
\xi & 0 \\
0 & \xi
\end{pmatrix}
; \quad\quad {\bf D} =
\begin{pmatrix}
\kappa_{1}^{(k)} & \tau_{0}^{(k)} \\
\tau_{0}^{(k)} & \kappa_{1}^{(k)}
\end{pmatrix}
; \quad\quad {\bf B} =
\begin{pmatrix}
w & 0 \\
0 & u
\end{pmatrix}
\\
{\bf A'} = {\bf A} -{\bf B}{\bf D}^{-1}{\bf B}^{\rm T} =
\begin{pmatrix}
\xi -\kappa_{1}^{(k)}\beta/\gamma & \beta \tau_{0}^{(k)} \\
\beta \tau_{0}^{(k)} & \xi -\kappa_{1}^{(k)}\beta\gamma
\end{pmatrix}\end{aligned}$$ where $\beta = wu/([\kappa_{1}^{(k)}]^{2} - [\tau_{0}^{(k)}]^{2})$ and $\gamma = u/w$. Thus after decimating all $N = 1+ 2^{k}$ sites, we have $$\begin{aligned}
\begin{pmatrix}
\xi -\kappa_{1}^{(k)}\beta/\gamma & \beta \tau_{0}^{(k)} \\
\beta \tau_{0}^{(k)} & \xi -\kappa_{1}^{(k)}\beta\gamma
\end{pmatrix}
\begin{pmatrix}
1+r \\
t
\end{pmatrix}
=
\begin{pmatrix}
2i\Im(\xi)\\
0
\end{pmatrix}
.\end{aligned}$$
From the above matrix equation by taking the inverse of the $2\times 2$ matrix, $$\begin{pmatrix}
1+r\\t
\end{pmatrix}
=
\frac{2i\Im(\xi)}{(\xi -\kappa_{1}^{(k)}\beta/\gamma)(\xi - \kappa_{1}^{(k)}\beta\gamma) -
\beta^{2}[\tau_{0}^{(k)}]^{2}}\begin{pmatrix}
\xi -\kappa_{1}^{(k)}\beta\gamma \\ -\beta \tau_{0}^{(k)}
\end{pmatrix} \label{linear1tT}
.$$
Thus we have found the transmission $T = |t|^2$.
Now using Eq. (\[Eq:AppA05\]) we can calculate the wavefunctions associated with the various sites. For simplicity we take $w=u$. To find the wavefunction associated with the highest level sites ($i=k+1$), we start with the $m=k$ RG step given in Fig. \[EndGame\]. Here, for Eq. (\[Eq:AppA05\]), $\boldsymbol{\psi} =0$, $
\mathbf{w_{d}} =
\begin{pmatrix}
w \\
0 \\
\end{pmatrix}
$, $
\mathbf{u_{d}} =
\begin{pmatrix}
0 \\
w \\
\end{pmatrix}
$ whereas the ${\bf D}$ and ${\bf B}$ matrices are given by Eq. (\[AdB\]). Making these substitutions, we get $$\label{HighWave}
\boldsymbol{\psi}_{k+1} = \frac{-w}{[\kappa_{1}^{(k)}]^{2} - [\tau_{0}^{(k)}]^{2}}
\begin{pmatrix}
(1+r)\kappa_{1}^{(k)} - t\tau_{0}^{(k)} \\
t\kappa_{1}^{(k)} - (1+r)\tau_{0}^{(k)} \\
\end{pmatrix}
.$$ Next we can find the wavefunction associated with the immediate lower level site ($i=k$). Here, for Eq. (\[Eq:AppA05\]), $\boldsymbol{\psi} = \boldsymbol{\psi}_{k+1}$, $\mathbf{w_{d}} = 0 = \mathbf{u_{d}}$, ${\bf D} =
\kappa_{1}^{(k-1)}$ and ${\bf B} = \tau_{0}^{(k-1)}
\begin{pmatrix}
1 \\
1 \\
\end{pmatrix}
$. Making these substitutions, we get $$\label{NextHighWave}
\boldsymbol{\psi}_{k} = \frac{\tau_{0}^{(k-1)}w(1+r+t)}
{\kappa_{1}^{(k-1)}[\kappa_{1}^{(k)}+\tau_{0}^{(k)}]}
\> .$$ We can continue in this manner, in principle, to find the wavefunction of all the levels below. However the calculations get very tedious and the expressions very complicated below $i=k$ and therefore are not given explicitly.
Ring geometry
-------------
Next consider the ring geometry (right side of Fig. \[EndGame\]). We perform $k-1$ RG steps to be left with $2^{k-(k-1)} =2$ sites. Site 1 is the nearest neighbor of site 0 and site 2. This makes the interaction between the two sites to be $2\tau_{0}^{(k-1)}$. This adds a factor of $2\lambda_{1}^{(k-1)}$ to the on-site energy of site 0. Thus the on-site energy of the even site 0 corresponds to $(\kappa_{k-1} - 2l_{k-1}) + 2l_{k-1} = \kappa_{k-1}$. The decimation of these two sites is similar to the case of the linear geometry, except that we need to replace $\kappa_{1}^{(k)}$ by $\kappa_{1}^{(k-1)}$ and $\tau_{0}^{(k)}$ by $2\tau_{0}^{(k-1)}$. So $\beta = wu/([\kappa_{1}^{(k-1)}]^{2} - 4[\tau_{0}^{(k-1)}]^{2})$ and
$$\begin{pmatrix}
1+r\\t
\end{pmatrix} = \frac{2i\Im(\xi)}{(\xi -\kappa_{1}^{(k-1)}\beta/\gamma)(\xi
- \kappa_{1}^{(k-1)}\beta\gamma) - 4\beta^{2}[\tau_{0}^{(k-1)}]^{2}}\begin{pmatrix}
\xi -\kappa_{1}^{(k-1)}\beta\gamma \\ -2\beta \tau_{0}^{(k-1)}
\end{pmatrix} \label{linear2tT}
.$$
This completes the RG, giving the transmission $T =|t|^2$.
To find the wave functions here, we start at the $m=k-1$ RG step and proceed as we did in the case of the linear geometry. We find that $$\begin{aligned}
\boldsymbol{\psi}_{k} &=& \frac{-w}{[\kappa_{1}^{(k-1)}]^{2} - 4[\tau_{0}^{(k-1)}]^{2}}
\>
\begin{pmatrix}
(1+r)\kappa_{1}^{(k-1)} - 2t\tau_{0}^{(k-1)} \\
t\kappa_{1}^{(k-1)} - 2(1+r)\tau_{0}^{(k-1)} \\
\end{pmatrix}\end{aligned}$$ and $$\begin{aligned}
\boldsymbol{\psi}_{k-1} &=& \frac{\tau_{0}^{(k-2)}w(1+r+t)}
{[\kappa_{1}^{(k-2)}+\tau_{1}^{(k-2)}][\kappa_{1}^{(k-1)}+2\tau_{0}^{(k-1)}]}
\>
\begin{pmatrix}
1 \\
1 \\
\end{pmatrix}
.\end{aligned}$$
Notice that the two $i =k-1$ sites are symmetric. Therefore the wavefunctions obtained above for the two sites are identical, as expected. Again the wavefunction expressions for the lower $i$ sites become complicated upon further iteration of this methodology.
[99]{} natexlab\#1[\#1]{}bibnamefont \#1[\#1]{}bibfnamefont \#1[\#1]{}citenamefont \#1[\#1]{}url \#1[`#1`]{}urlprefix\[2\][\#2]{} \[2\]\[\][[\#2](#2)]{}
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'The standard approach to compressive sampling considers recovering an unknown deterministic signal with certain known structure, and designing the sub-sampling pattern and recovery algorithm based on the known structure. This approach requires looking for a good representation that reveals the signal structure, and solving a non-smooth convex minimization problem (e.g., basis pursuit). In this paper, another approach is considered: We learn a good sub-sampling pattern based on available training signals, without knowing the signal structure in advance, and reconstruct an accordingly sub-sampled signal by computationally much cheaper linear reconstruction. We provide a theoretical guarantee on the recovery error, and show via experiments on real-world MRI data the effectiveness of the proposed compressive MRI scheme.'
address: |
Laboratory for Information Inference Systems\
École Polytechnique Fédérale de Lausanne
bibliography:
- 'list.bib'
title: 'Learning Data Triage: Linear Decoding Works for Compressive MRI'
---
Compressive sampling, magnetic resonance imaging (MRI), learning, least squares estimation, sub-modular minimization
Introduction {#sec_intro}
============
The standard theory of compressive sampling (CS) considers recovering an unknown deterministic signal with certain *known* structure, and designing sampling and recovery schemes based on the known structure [@Foucart2013]. For example, if the unknown signal is known to be sparse, one can measure it by a sub-sampling matrix satisfying the restricted isometry property (RIP), and apply basis pursuit to obtain an estimate of the signal [@Candes2008; @Candes2006a]. Similar ideas can be extended to low-rank matrix recovery [@Candes2009b], and in general signal recovery problems where the signal structures can be encoded by atomic norms or other convex functions [@Bach2013a; @Chandrasekaran2012; @ElHalabi2015].
Despite its success in many applications, we note that there are some undesired features of the standard CS theory:
1. The signal structure must be known *in advance*. This usually requires seeking for a good signal representation to reveal the signal structure, a non-trivial task called dictionary learning [@Tosic2011].
2. The recovery scheme is *computationally expensive*. Typical examples are basis pursuit and the Lasso; both are non-smooth convex optimization problems.
While those features seem to be necessary according to existing literature on CS, in some applications, the real-world setting can deviate from the standard setting of CS. This creates an opportunity of getting rid of those undesired features.
We focus on one important observation which the standard CS theory does not take into consideration—we usually have training signals, i.e., signals that are given and similar to the unknown signal in some sense.
In fact, practitioners are indeed applying this learning-based approach in a naïve way. For example, it is by examining a large amount of real-world images that we discovered sparsity or more sophisticated structures, under proper representations [@Baraniuk2010; @Cevher2009; @Mallat2009]. Although this naïve learning procedure can be made rigorous and automated by dictionary learning, training signals are still required.
In this paper, we propose alternative to compressive sampling, and apply it to compressive magnetic resonance imaging (MRI). The proposed scheme *automatically* adapts to the given training signals, without any *a priori* knowledge on the signal structure. We highlight the following contributions:
1. We propose a novel statistical learning view point to the compressive sampling problem, which allows us to study the effect of training signals.
2. Our compressive MRI scheme is computationally efficient: The learning procedure can be cast as a combinatorial optimization program, which can be exactly solved by an efficient algorithm; the recovery algorithm we consider is simply least-squares (LS) reconstruction.
3. In contrast to the standard approach using random sub-sampling patterns [@Candes2006a; @Lustig2007; @Roman2014], our sub-sampling scheme is fixed given the training signals, and hence simpler for implementation.
4. We provide a theoretical guarantee on the reconstruction error, and characterize its dependence on the number of training signals.
5. We show via experiments on real MRI images that the reconstruction error performance of the proposed scheme is comparable to the performance using a finely-tuned sub-sampling pattern given in [@Lustig2007].
Review of Existing Approaches
=============================
Compressive MRI is essentially a linear inverse problem. The goal is to recover an unknown signal ${x^\natural}\in \mathbb{C}^p$, given a a sub-sampling pattern $\Omega \subset {\left\{ 1, \ldots, p \right\}}$ with ${\left\vert \Omega \right\vert} = n$ for some $n < p$, and the outcome of compressive sampling: $$y := P_{\Omega} {\mathcal{F}}{x^\natural}\notag$$ where ${\mathcal{F}}: \mathbb{C}^p \to \mathbb{C}^p$ is the Fourier transform matrix, and $P_{\Omega}: \mathbb{C}^p \to \mathbb{C}^n$ is a linear operator that only keeps entries of ${\mathcal{F}}{x^\natural}$ indexed by $\Omega$. In practice, ${x^\natural}$ is usually a 2D or 3D object, and ${\mathcal{F}}$ should be replaced by the corresponding multidimensional Fourier transform.
Existing approaches to compressive MRI can be briefly summarized as follows:
1. Find a wavelet transform matrix $\Psi : \mathbb{C}^p \to \mathbb{C}^p$, such that ${x^\natural}= \Psi^{-1} {z^\natural}$ and ${z^\natural}$ possesses certain *structure*. For example, the sparsity of ${z^\natural}$ and smoothness of ${x^\natural}$ were exploited in [@Lustig2007], the tree sparsity of ${z^\natural}$ was considered in [@Chen2012a], and the multi-level sparsity of ${z^\natural}$ was considered in [@Roman2014].
2. Choose a *random* sub-sampling pattern $\Omega$ and sample ${\mathcal{F}}{x^\natural}$ accordingly; the probability distribution might be dependent on the knowledge about the structure of ${z^\natural}$ [@Lustig2007; @Roman2014].
3. Finally, apply *non-linear* decoding algorithms to reconstruct ${x^\natural}$. The standard basis pursuit estimator was considered in [@Roman2014]. A basis pursuit like estimator minimizing a linear combination of the $\ell_1$-norm and the total variation semi-norm was proposed in [@Lustig2007]. A closely-related Lasso like estimator with the $\ell_1$-norm and total variation semi-norm penalization was considered in [@Yang2010]. A similar Lasso like estimator with one additional penalization term for tree sparsity was introduced by [@Chen2012a].
We note that existing approaches essentially follow the standard theory of compressive sampling, and hence inherit the two undesired features which we mentioned in the introduction.
Learning Data Triage
====================
The standard approach to compressive MRI models ${x^\natural}$ as a deterministic unknown signal. Here we adopt another modeling philosophy: We assume that ${x^\natural}$ is a random vector following some *unknown* probability distribution $\mathbb{P}$, and we have access to $m$ training signals $x_1, \ldots, x_m \in \mathbb{C}^p$, which are independent and identically distributed random vectors also following $\mathbb{P}$, and are independent of ${x^\natural}$. Note that this is different from Bayesian compressive sampling [@Ji2008], as $\mathbb{P}$ is unknown in our model.
We consider LS reconstruction. For any given sub-sampling pattern $\Omega$, the estimator has an explicit form: $$\begin{aligned}
{\hat{x}}_{\Omega} &= \arg \min_{x} {\left\{ {\left\Vert y - P_{\Omega} {\mathcal{F}}x \right\Vert}_2^2 : x \in \mathbb{R}^p \right\}} \notag \\
&= {\mathcal{F}}^H P_{\Omega}^T y. \notag\end{aligned}$$ Once the reconstruction scheme is fixed, the only issue is to choose $\Omega$ that optimizes the resulting estimation performance.
We show in Section \[sec\_prof\_eqR\] that for any given $\Omega$, the expected normalized reconstruction error satisfies $$\mathbb{E}\, \frac{{\left\Vert {\hat{x}}_{\Omega} - {x^\natural}\right\Vert}_2^2}{{\left\Vert {x^\natural}\right\Vert}_2^2} = 1 - \mathbb{E}\, f_{\Omega} ( x ), \label{eq_R}$$ where the expectations are with respect to ${x^\natural}\sim \mathbb{P}$ and $x \sim \mathbb{P}$, respectively, and we define $$f_{\Omega} ( x ) := \frac{{\left\Vert P_{\Omega} {\mathcal{F}}x \right\Vert}_2^2}{{\left\Vert x \right\Vert}_2^2} \notag$$ for convenience. This implies that the optimal sub-sampling pattern $\Omega$, denoted by $\Omega_{\text{opt.}}$, is given by any solution of the following combinatorial optimization program: $$\Omega_{\text{opt}} \in \arg \max_{\Omega} {\left\{ \mathbb{E}\, f_{\Omega} ( x ) : \Omega \subset {\left\{ 1, \ldots, p \right\}}, {\left\vert \Omega \right\vert} = n \right\}}. \label{eq_RM}$$ However, since $\mathbb{P}$ is assumed unknown, the optimization program is not tractable.
Motivated by the idea of empirical risk minimization in statistical learning theory [@Vapnik1999], we make use of the training signals and approximate $\Omega_{\text{opt.}}$ via any solution of the optimization program: $$\Omega_m \in \arg \max_{\Omega} {\left\{ \hat{\mathbb{E}}_m \, f_{\Omega} ( x ) : \Omega \subset {\left\{ 1, \ldots, p \right\}}, {\left\vert \Omega \right\vert} = n \right\}} \label{eq_ERM}$$ where $\hat{\mathbb{E}}_m$ denotes the expectation with respect to the empirical measure, i.e., $$\hat{\mathbb{E}}_m\, f_{\Omega} ( x ) := \frac{1}{m} \sum_{i = 1}^m \frac{{\left\Vert P_{\Omega} {\mathcal{F}}x_i \right\Vert}_2^2}{{\left\Vert x_i \right\Vert}_2^2}. \notag$$ This optimization program is tractable, because we only need to solve it for *any realization of* the training signals $x_1, \ldots, x_m$. Note that then ${\hat{\mathbb{E}}_m}\, f_{\Omega} ( x )$ depends on $x_1, \ldots, x_m$ and is random, and so does $\Omega_m$. The overall systems is summarized as follows:
1. Find a sub-sampling pattern $\Omega_m$ by (\[eq\_ERM\]).
2. Sub-sample ${x^\natural}$ using $\Omega_m$ and obtain the measurement outcome $$y := P_{\Omega_m} {\mathcal{F}}{x^\natural}. \notag$$
3. Recover ${x^\natural}$ by $${\hat{x}}:= {\mathcal{F}}^H P_{\Omega_m}^T y. \notag$$
**On Computing $\Omega_m$:** The optimization program (\[eq\_ERM\]) is modular, and hence can be exactly solved by a simple greedy algorithm [@Fujishige2005]: Let $\phi_i^T$ be the $i$-th row of $\Phi$. Compute the values $$v_i := \frac{1}{m} \sum_{i = 1}^m \left( \phi_i^T x_i \right)^2, \notag$$ and set $\Omega_m$ as the set of indices corresponding to the largest $n$ $v_i$’s. The computational complexity is dominated by computation of $v_i$’s, which behaves as $\mathcal{O} ( m p ^2 )$ in general, and $\mathcal{O} ( m p \log p )$ if $\Phi$ is suitably structured, such as the Fourier and Hadamard matrices.
Performance Analysis
====================
We analyze the reconstruction error of the proposed learning-based compressive sampling system.
If we could solve the optimization program (\[eq\_RM\]), the estimation performance would be given by $$\mathbb{E}\, \frac{{\left\Vert {\hat{x}}_{\Omega_{\text{opt}}} - {x^\natural}\right\Vert}_2^2}{{\left\Vert {x^\natural}\right\Vert}_2^2} = 1 - \varepsilon_{\mathbb{P}}, \notag$$ where we define $$\varepsilon_{\mathbb{P}} := \max_{\Omega} {\left\{ \mathbb{E}\, f_{\Omega} ( x ) : \Omega \subset {\left\{ 1, \ldots, p \right\}}, {\left\vert \Omega \right\vert} = n \right\}}. \notag$$ Note that $\varepsilon_{\mathbb{P}}$ is a constant for any given $\mathbb{P}$, independent of the training signals.
Since now the optimization program (\[eq\_RM\]) is replaced by its empirical version (\[eq\_ERM\]), a reasonable guess is that the estimation performance would behave as $$\mathbb{E}\, \frac{{\left\Vert {\hat{x}}_{\Omega_m} - {x^\natural}\right\Vert}_2^2}{{\left\Vert {x^\natural}\right\Vert}_2^2} \leq 1 - \varepsilon_{\mathbb{P}} + \varepsilon_m, \notag$$ for some $\varepsilon_m > 0$ with high probability (with respect to the training signals $x_1, \ldots, x_m$), and $\varepsilon_m$ should converge to $0$ as $m \to \infty$. The following proposition verifies this guess.
\[prop\_main\] For any $\beta \in ( 0, 1 )$, we have $$\varepsilon_m \leq \sqrt{ \frac{2}{m} \left[ \log { p \choose n } + \log \frac{2}{\beta} \right] } \notag$$ with probability at least $1 - \beta$.
This means a size of training signals of the order $\mathcal{O} ( n \log p )$ suffices to have small enough $\varepsilon_m$, with high probability. We note that this is a worst-case guarantee, as it is *distribution-independent*. In practice, $m$ can be much smaller.
Discussions
===========
The essential idea of the learning-based approach can be summarized as follows: Fix a decoder, and find the optimal sub-sampling pattern that minimizes the corresponding expected recovery error, which can be approximated by empirical risk minimization. The performance is essentially determined by the distribution of signal ensemble.
In this paper, we consider the linear decoder for computational efficiency, and it works well on the ensemble of MRI images. For other signal ensembles, it is possible to have a better recovery error performance by a non-linear decoder, such as basis pursuit or the Lasso, and realize a trade-off between computational complexity and recovery performance. Note that the idea of the learning-based approach still applies, while the empirical risk minimization formulation for choosing the sub-sampling pattern should be modified accordingly given the decoder. We are currently working in this research direction.
Acknowledgement
===============
The authors would like to thank Baran Gözcü and Luca Baldassarre for providing numerical results, and Jonathan Scarlett for the complexity discussion.
Proofs
======
Proof of ([\[eq\_R\]]{}) {#sec_prof_eqR}
------------------------
In fact, the equality holds deterministically, as $$\begin{aligned}
&{\left\Vert \hat{x}_{\Omega} - {x^\natural}\right\Vert}_2^2 \notag \\
&\quad = {\left\Vert \hat{x}_{\Omega} \right\Vert}_2^2 - 2 \left\langle \hat{x}_{\Omega}, {x^\natural}\right\rangle + {\left\Vert {x^\natural}\right\Vert}_2^2 \notag \\
&\quad = {\left\Vert \mathcal{F}^H P_{\Omega}^T P_{\Omega} \mathcal{F} {x^\natural}\right\Vert}_2^2 - 2 \left\langle F^H P_{\Omega}^T P_{\Omega} \mathcal{F} {x^\natural}, {x^\natural}\right\rangle + {\left\Vert {x^\natural}\right\Vert}_2^2 \notag \\
&\quad = {\left\Vert P_{\Omega} \mathcal{F} {x^\natural}\right\Vert}_2^2 - 2 {\left\Vert P_{\Omega} \mathcal{F} {x^\natural}\right\Vert}_2^2 + {\left\Vert {x^\natural}\right\Vert}_2^2. \notag\end{aligned}$$ In the third equality, we used the fact that $A A^\dagger A = A$ for any matrix $A$ and its Moore-Penrose generalized inverse $A^\dagger$, by setting $A := P_{\Omega} \mathcal{F}$.
Proof of Proposition [\[prop\_main\]]{}
---------------------------------------
It suffices to choose $\varepsilon_m$ such that with probability at least $1 - \beta$, $$\Delta_m := \mathbb{E}\, f_{\Omega_{\text{opt}}} ( x ) - \mathbb{E}\, f_{\Omega_m} ( x ) \leq \varepsilon_m. \notag$$
We note that $$\begin{aligned}
\Delta_m = \, \ & \left( \mathbb{E}\, f_{\Omega_{\text{opt}}} ( x ) - {\hat{\mathbb{E}}_m}\, f_{\Omega_{\text{opt}}} ( x ) \right) + \notag \\
& \left( {\hat{\mathbb{E}}_m}\, f_{\Omega_{\text{opt}}} ( x ) - {\hat{\mathbb{E}}_m}\, f_{ \Omega_m } ( x ) \right) + \notag \\
& \left( {\hat{\mathbb{E}}_m}\, f_{ \Omega_m } ( x ) - \mathbb{E}\, f_{\Omega_m} ( x ) \right). \notag\end{aligned}$$ The second summand on the right-hand side cannot be positive by definition. Then we have $$\Delta_m \leq 2 \max_{\Omega} {\left\{ {\left\vert {\hat{\mathbb{E}}_m}\, f_{\Omega} ( x ) - \mathbb{E}\, f_{\Omega} ( x ) \right\vert} : \Omega \in \mathcal{A} \right\}} \notag$$ where $\mathcal{A} := {\left\{ \Omega : \Omega \subset {\left\{ 1, \ldots, p \right\}}, {\left\vert \Omega \right\vert} = n \right\}}$.
As the random variables $f_{\Omega} ( x )$ are bounded ($0 \leq f_{\Omega} ( x ) \leq 1$), we can use Hoeffding’s inequality and the union bound to obtain an upper bound of $\Delta_m$ that holds with high probability, as in [@Audibert2007 B.3].
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'A phenomenological pion-nucleon interaction is used to obtain pionic mass modification in presence of constant homogeneous magnetic field background at finite temperature and chemical potential in the real time formalism of thermal field theory. The magnetically modified propagator in its complete form is used to obtain the one loop self-energy for pions. For charged pions we find that the effective mass increases with the magnetic field at given temperature and chemical potential. Since the transverse momentum of charged pion is quantized and its contribution to Dyson-Schwinger Equation is large compared to the loop correction, the charged pion mass remains constant with both temperature and chemical potential for a given landau level. In order to unveil the role of the real part of the self-energy, we also calculate the effective mass neglecting the trivial shift. The effective mass for charged pions shows an oscillatory behaviour which is attributed to the thermal contribution of the self-energy. It is argued that the magnetic field dependent vacuum contribution to the self-energy influences the behaviour of the effective mass both qualitatively and quantitatively. We also find that very large field is necessary for neutral pions to condense.'
author:
- 'Arghya Mukherjee$^{a,c}$'
- 'Snigdha Ghosh$^{b,c}$'
- 'Mahatsab Mandal$^{a,d}$'
- 'Pradip Roy$^{a,c}$'
- 'Sourav Sarkar$^{b,c}~$'
---
Introduction
============
Understanding quantum chromodynamics(QCD) in presence of magnetic background has gained a lot of contemporary research interest [@ln871]. It is important to study QCD in presence of external magnetic field not only for its relevance to astrophysical phenomena [@prep442; @prl95; @prd76; @prl100; @prdN76; @prl105; @prd82] but also due to the possibility of strong magnetic field production in non-central heavy-ion collisions [@ijmp] which sets the stage for investigation of these magnetic modifications. Although the background fields produced in RHIC and LHC are much smaller in comparison with the field strengths that prevailed during the cosmological electro-weak phase transition which may reach up to $eB\approx200m_\pi^2$ [@plb265], they are strong enough to cast significant influence on the hadronic properties which bear the information of the chiral phase transition. Moreover, among the hadrons, mesons possess direct connection to the chiral phase transition[@prdW86] that make them more important candidates than baryons for the understanding of the phenomena. At vanishing chemical potential, modification due to the presence of magnetic background can be obtained from first principle using lattice QCD simulations [@prd82M; @prd83] which shows monotonic increase in critical temperature with the increasing magnetic field, also known as magnetic catalysis(MC). The effects of external magnetic field on the chiral phase transition has been studied using different effective models in recent years [@tmph90; @prd81; @prd84; @prc83; @prd83M; @prc83A; @prd82R; @prd83K; @prd85; @jhep08; @prd82A; @prd85S; @prd84D; @prd86E]. QCD being a confining theory at low energies, effective theories are employed to describe the low energy behaviour of the strong interaction. These effective field theoretic models in general contain a few parameters which can be fixed from experimental inputs. Although most of the model calculations are in support of MC, different lattice results had shown inverse magnetic catalysis(IMC) where critical temperature follows the opposite trend [@jhep1202; @prdB86; @jhep04; @prd90; @prc90; @prd91; @plb759]. It was pointed out in [@jhep1304] that IMC is attributed to the dominance of the sea contribution over the valence contribution of the quark condensate. The sea effect has not been incorporated even in the Polyakov loop extended versions of Nambu–Jona-Lasinio (PNJL) model and Quark-Meson(PQM) model which might be a possible reason of the disagreement. To investigate the apparent contradiction, significant works have been done [@rmp88] in quest of proper modifications of the effective models, most of which are focused on the magnetic field dependency of the coupling constants or other magnetic field dependent parameters in the model. Very recently, IMC has been observed in NJL model, with Pauli-Villars regularization scheme [@plb758] which gives markedly different behaviour in comparison with the usual soft-cutoff approach.
It has been argued in [@prd79] that the study of pion-nucleon interaction plays an important role in the behavioural description of the deconfinement critical temperature in terms of pion mass and isospin. Pion mass modification in presence of magnetic background has been calculated employing chiral perturbation theory in Ref.[@prd86]. It has been shown that in presence of magnetic field, the charged pions are no longer the Goldstone modes and the critical temperature of chiral phase transition shows magnetic catalysis. In NJL approach [@epjc76], it has been found that there exists a sudden leap in the effective masses of charged pions near the same critical temperature from where the $\sigma$ and $\pi^0$ meson become nearly degenerate. Here also, the pseudo-critical temperature is found to increase with the increasing magnetic field. In Ref.[@colucci] pion effective mass variation with $eB$ has been observed for large magnetic fields where Lowest Landau Level(LLL) approximation is reasonable. Also in Ref.[@aguirre], the medium modification of pion effective mass has been obtained in a self-consistent way with LLL approximation. However, it is a common trend to ignore the magnetic field dependent [*vacuum*]{} contribution of the self-energy function in case of mass and dispersion calculations. It has been shown [@spa; @me] (also see the references therein) that it can have significant influence on mesonic properties like effective mass, dispersion relation, decay width and spectral function. Merely on grounds of simplicity, it appears unreasonable to neglect this $eB$ dependent vacuum contribution apriori unless one compares the dependencies on other external parameters with it. It is also interesting to observe the interplay between the medium and the vacuum effect of the external magnetic field to find the complete $eB$ dependence of the physical properties. In this article we revisit the mass modification of pions in presence of finite temperature and chemical potential in a homogeneous magnetic background with a well known pion-nucleon interaction in isospin-symmetric nuclear matter. Unlike NJL model, here pionic fields are treated as elementary. However, the non-trivial mass correction in presence of magnetic field occurs due to the modification of the nucleon propagators. The influence of the magnetic field dependent vacuum contribution in case of the pseudo-vector pion-nucleon interaction has been studied in detail. In the Dyson-Schwinger formalism, instead of restricting ourselves to the strong/weak field region, the full propagator is used to obtain the pion self-energy. Since the charged pion transverse momentum is quantized in magnetic field, we obtain Landau Level(LL) dependent self-energy and the corresponding Dyson-Schwinger Equation (DSE) is modified because of the presence of the transverse momentum. Thus we obtain LL dependent effective mass for charged pions. To see the importance of $eB$ dependent vacuum contribution we neglect the trivial shift. In case of neutral pions, although the evaluation is restricted to the special case where the external pion momentum is parallel to the field direction, this restriction does not put any constraint in case of mass calculation. More specifically, pion effective mass has been obtained with full magnetic field dependence up to one loop order. Effective mass variation with external magnetic field due to $eB$ dependent vacuum is compared with the $eB$ dependent thermal contribution. It is argued that neglecting vacuum contribution may even influence the qualitative predictions of the effective mass dependences for different pion species.
The article is organized as follows. In Sec.II we discuss the formalism for calculating the one loop pion self-energy function for phenomenological pion-nucleon interaction in presence of constant external magnetic field in dense thermal medium. The section comprises two subsections, one for the charged pions and the other for the neutral pion where the magnetic field dependent vacuum contribution and thermal contribution of the self-energy of the corresponding species are calculated. The Dyson-Schwinger equation that relates the effective mass with the real part of the self-energy is also obtained. Pion mass variation with respect to the independent parameters are presented in Sec.III. The effect of incorporation of magnetic field dependent vacuum part is also discussed. Finally we summarize our work in Sec.IV.
Formalism
=========
The Dyson-Schwinger Equation(DSE) for the effective propagator of pion is given by D\^[-1]{}(q)&=&D\_0\^[-1]{}(q)-(q) where $D_0^{-1}(q)=q^2-m_\pi^2+i\epsilon$ and $\Pi(q)$ is the pion self-energy. One can obtain the effective mass by finding the pole of $D(q)$. Here, we are interested in finding the thermal modification of pion mass in presence of constant external magnetic field along with finite baryon density due to the effective pion-nucleon interaction, given by [@prc62] \^[int]{} \_&=&-|5\^(\_). where $\psi$ is the two component nucleon field and $\vec{\tau}=(\sigma_x,\sigma_y,\sigma_z)$, with $\sigma_a$ denoting the $a$th Pauli spin matrix. The pionic fields are represented by the isovector $\vec{\pi}$. Expanding the interaction Lagrangian, one finds the Feynman diagrams for the one loop self-energy of pions as given in Fig.\[feyn\].
[![One loop Feynman diagrams for pion-nucleon interaction.[]{data-label="feyn"}](feynman_diagram.eps "fig:"){width="6.5in"}]{}
In the real time formalism of thermal field theory, the propagators as well as the one loop self-energy function assume a $2\times2$ matrix structure. The 11-components of the matrices for neutral and charged pions are given by
\^[11]{}\_0(q,\_N,T)&=&-ig\^2\_-ig\^2\_\
\^[11]{}\_+(q,\_N,T)&=&-2ig\^2\_\
\^[11]{}\_-(q,\_N,T)&=&-2ig\^2\_\[mother\]
where $S^{11}_p(k)$ and $S^{11}_n(k)$ are the 11-components of the thermal propagators for proton and neutron respectively. The real time thermal propagators can be decomposed into two parts as [@olivo] S\^[11]{}\_p(k)&=&S\_p(k)-2i(k) S\_p(k)\
S\^[11]{}\_n(k)&=&S\_n(k)-2i(k) S\_n(k)\
(k)&=&(k\^0)n\_k\^++(-k\^0)n\_k\^-\
n\^\_k&=&. Here, $S_p(k)$ is the momentum space representation of the fermionic propagator in presence of magnetic field, $\theta$ denoting the unit step function and $\beta=\frac{1}{T}$ is the inverse temperature in natural unit. Fermionic propagators in presence of magnetic field possess a phase factor which can not be taken as translationally invariant in general. However, in the current context, the phase factor can be removed by suitable gauge transformation [@prd92] and we can work with the momentum space representation of the translationally invariant part which is given by [@nuphyb462] S\_p(k)&=&-\_[n=0]{}\^[n=]{}(-1)\^n e\^[-]{} where $\a=-\kpr^2/eB$ and $L_n\equiv L_n^0$ with $L_n^\a$ representing the generalized Laguerre polynomials. The $\epsilon$ in the denominator is an infinitesimal positive parameter. It should be mentioned here that in this article we use $g^{\mu\nu}_{||}={\rm diag}(1,0,0,-1)$ and $g^{\mu\nu}_{\perp}={\rm diag}(0,-1,-1,0)$ with metric defined as $g^{\mu\nu}=g^{\mu\nu}_{||}+g^{\mu\nu}_{\perp}$. A general four vector can be decomposed as $a^\mu=a^\mu_{||}+a^\mu_{\perp}$ with $a_{||}^2=a_0^2-a_3^2$ and $a_{\perp}^2=-a_1^2-a_2^2$. The imaginary part of the propagator is S\_p(k)&=&\_[n=0]{}\^[n=]{} (-1)\^n e\^[-]{}\
&&(\^2-m\^2-2n eB). However, the neutron propagator $S_n(k)$ is not influenced by the presence of the magnetic field and is given by S\_n(k)&=&-\
S\_n(k)&=&(+m)(k\^2-m\^2). Now, the propagators have two distinct parts, one with the thermal distribution function and another without it. On this basis the self-energy function can be expressed as a sum of three different portions given by \^[11]{}\_[0,]{}&=&(\_[0,]{})\_+(\^\_[0,]{})\_+(\^\_[0,]{})\_[\^2]{}. The term with quadratic dependence on the distribution function is purely imaginary. As we are only interested in the real part of the self-energy, we have (\^[11]{}\_[0,]{})(\_[0,]{}) &=&(\^\_[0,]{})\_+(\^\_[0,]{})\_where $\overline{\Pi}$ represents the 11-component of the diagonal self-energy matrix (see e.g [@sirbook]). Let us now consider the explicit forms of the real part of the self-energy for charged and neutral mesons separately.
Charged pions
-------------
The medium independent vacuum self-energy will be same for the charged pions $\pi^+$ and $\pi^-$ and can be obtained as follows. (\^\_[+]{})\_&=&-2ig\^2\_\
&=&-2ig\^2\_\_[n=0]{}\^\
&=&i\_[n=0]{}\^(n,\^2,\^2,\^2)\[pi+\]\
(n,\^2,\^2,\^2)&=&\
\_n&=&-2g\_\^2(-1)\^ne\^[-\_p]{}\
&=&-8g\_\^2(-1)\^ne\^[-\_p]{}. \[i\] Here $\a _p=-p_{\perp}^2/eB$ and $p=q+k$ with $q$ being the external momentum of pions. The 2-dimensional $\kp$ integration can be performed using standard Feynman parametrization and dimensional regularization technique to obtain I(n,,\^2,\^2)&=&-8g\_\^2(-1)\^n e\^[-\_p]{}\_0\^1dx\
A&=&q\^2\^2+2x\^2\^2+{(1+2x)\^2-\^2-2()}()\
B\_n&=&-2 m\^2 q\^2-x(1-x)(\^4-\^2\^2-\^2 q\^2)+2(1-x)\^2()+(1-x)q\^2\^2-2 n x eB q\^2\
&=&B\_0-2 n x eB q\^2\
\_[n]{}&=&m\^2-x(1-x)\^2-i-(1-x)\^2+2 n x eB\
&=&-(1-x)\^2+2n x eB\[ifinal\]. where $\mu_0$ is the scale which appears in the process of dimensional regularization. Now, with this $I(n,\kpr,\qp^2,\qpr^2)$, the summation in Eq.(\[pi+\]) can be taken inside the $\kpr$ integral which gives (\_+)\_&=&g\^2\_\_0\^1dxe\^[-\_p]{}\
\_1&=&\_n\^(-1)\^nL\_[n-1]{}\^1(2\_p)\
\_2&=&\_n\^(-1)\^n(L\_n(2\_p)-L\_[n-1]{}(2\_p))\
\_3&=&\_n\^(-1)\^n(L\_n(2\_p)-L\_[n-1]{}(2\_p))\[sum\]. It is possible to find compact expressions for these summations by casting them into known series of Laguerre polynomials as discussed in detail in the appendix. The final expression of the vacuum contribution of the charged pion self-energy is given by (\_+)\_&=&\_0\^1\_0\^1dxz\^[-y(1-x)-1]{}\
&=&(1-x)+(xz)\
y&=&(xz)\[finalpi+\]. One can observe here that instead of a 2-dimensional $\kpr$ integral and an infinite series summation, now we have one integration over the parameter $z$ which is more convenient for numerical evaluation. Another important feature is that the expression is not in the form of any polynomial of $eB$ which signifies its non-perturbative character. It should be pointed out that the scale $\mu$ present here appears in the process of parametrization with $z$ and is different from the scale $\mu_0$ that appeared from dimensional regularization of $\kp$. It can be shown from Eq.(\[finalpi+\]) that at $eB=0$ the self energy becomes (\_+)\_(eB=0)&=&g\^2\_\_0\^1 dx () which is exactly twice the contribution of $nn$ loop in case of neutral pions as will be seen later. Thus the $(\Pi_+)_{\mbox{vac}}$ can be decomposed as (\_+)\_&=&(\_+)\_(eB0)+(\_+)\_(eB=0) where $(\Pi_+)_{\mbox{vac}}(eB\ne0)$ represents the external $eB$ dependent part of the self-energy and will be used in the DSE to obtain the effective mass.
The procedure to obtain the thermal part of the self-energy is relatively simpler and only the final expressions are presented. The real part of the thermal contribution of the self-energy for $\pi^+$ is given by (\_+)\_&=&\_[l=0]{}\^ \[nl\] where the expression for $\mathcal{N}_l$ is given in Eq.(\[i\]) for the dummy index $l=n$. One can observe from Eq.(\[mother\]) that \^[11]{}\_+(-q,\_N,T)&=&\^[11]{}\_-(q,\_N,T). \[pi+pi-\] Thus the expression of $\mathcal{R}\mbox{e}(\Pi_-)_\eta$ can be easily obtained from $\mathcal{R}\mbox{e}(\Pi_+)_\eta$ by successively changing $k\rightarrow k-q$ and $q\rightarrow-q$. It should be mentioned here that unlike the vacuum case, the infinite sum could not be performed analytically and as a result the Laguerre polynomials remain in the numerator within the single sum structure of the thermal contribution.
As in this case the external particles are charged, the external transverse momentum suffers Landau quantization in presence of $eB$. Thus to obtain the effective mass of $\pi^{\pm}$ as a function of temperature, chemical potential and external magnetic field, we now solve the DSE of charged pions given by m\_[\^]{}\^[\*2]{}-m\_[\^]{}\^[2]{}+\_(m\_[\^]{}\^[\*2]{},\^2=(2n+1)eB,eB)-(2n+1)eB=0 \[dsepm\] where $\mathcal{R}\mbox{e}\Pi_{\pm}$ contains the sum of explicit $eB$ dependent vacuum and thermal contributions. Here, $m_{\pi^{\pm}}^{}$ and $m_{\pi^{\pm}}^{*}$ denotes the renormalized mass and the effective mass of the charged pions respectively.
Neutral pions
-------------
At first we consider the magnetic field dependent vacuum contributions from $pp$ and $nn$ loops. Here, the vacuum contribution refers to the part of the self-energy which is independent of the thermal distribution functions and can be written as (\^\_0)\_&=&(\^\_0)\^[pp]{}\_+(\^\_0)\_\^[nn]{}. For the $pp$ loop (\^\_0)\^[pp]{}\_&=&-ig\^2\_\
&=&-ig\^2\_\_[n,l=0]{}\^\
D\_n(k)&=&.
At this point, one can observe that with $q_x=q_y=0$ the $\kpr$ integration can be done analytically and standard Feynman parametrization technique can be applied to obtain (\^\_0)\^[pp]{}\_&=&-i\_[n,l=0]{}\^\_0\^1 dx. Dropping the odd terms after the momentum shift $\kp\rightarrow\kp-x\qp$, we get \^[n,l]{}\_[||]{}()&=&4g\^2\_eB\
\_[nl]{}&=&m\_n\^2-x(1-x)\^2+x(m\_l\^2-m\_n\^2)\
m\_n\^2&=&m\^2+2neB-i. After the momentum integration we obtain (\^\_0)\^[pp]{}\_&=&g\^2\_\^2\_0\^1dx\_[n,l=0]{}\^\
&=&\_[n,l=0]{}\^\
&=&\_[n,l=1]{}\^ -\
&&\_[00]{}==m\^2-x(1-x)\^2-i\
&=&-\_[n=1]{}\^-\
&=&-- () =.
It should be noted that $\d_{n,-1}=0$ here as Laguerre polynomial with negative index is taken to be zero. Finally, after regularization in the $\overline{MS}$ scheme, the vacuum part of self-energy for the $pp$ loop with $\vec{\,\,\,\,\qpr}=0$ becomes (\^\_0)\^[pp]{}\_&=&g\^2\_\_0\^1 dx. \[pp\_selfenergy\] Note that it contains the pure vacuum part i.e one without explicit $eB$ dependence as well as the explicit magnetic field dependent vacuum contribution. The vacuum contribution from $nn$ loop is given by (\^\_0)\^[nn]{}\_&=&-ig\^2\_\
&=&-i\
N(k)&=&4g\^2\_. This is a divergent integral and the momentum integration can be performed after standard Feynman parametrization. After regularization in $\overline{MS}$ scheme the finite part of the vacuum self-energy for the $nn$ loop can be obtained as (\^\_0)\^[nn]{}\_&=&g\^2\_\_0\^1 dx () =0 \[nn\_selfenergy\] and $\mu$ is the scale of the theory having dimension of square mass. At zero magnetic field, the self-energy contribution of the $pp$ loop must coincide with the contribution from the $nn$ loop in isospin symmetric matter. Now since the complete form of the propagators is used in order to derive Eq.(\[pp\_selfenergy\]), we obtain a non-perturbative result as long as expansion in terms of $eB$ is concerned. For that reason, we can not simply put $eB=0$ there to obtain the zero field contribution. Instead, we obtain a perturbative expansion of the $pp$ result around $eB=0$. The $eB\rightarrow 0$ expansion of Eq.(\[pp\_selfenergy\]) neglecting $\mathcal{O}((eB)^2)$ term is given by (\^\_0)\^[pp]{}\_&=&g\^2\_\_0\^1 dx ()\
&=&g\^2\_\_0\^1 dx .\
\[ppeB0\] The first term in the square bracket exactly matches with Eq.(\[nn\_selfenergy\]) whereas the second term diverges at $eB=0$. Hence, to match the two expressions identically i.e irrespective of the value of external momentum we must modify Eq.(\[pp\_selfenergy\]) as, (\^\_0)\^[pp]{}\_&=&g\^2\_\_0\^1 dx. By demanding identical contributions from $pp$ and $nn$ loop at vanishing magnetic field, one in fact imposes here a physical condition to extract out the finite part of the self energy.
We now turn to the thermal contribution. The real part of the thermal contribution for proton-proton($pp$) loop can be obtained following a similar procedure employed in case of charged pions and is given by (\^[pp]{}\_0)\_&=&\_[n,l=0]{}\^[where]{}\
\^[n,l]{}\_[||]{}(k)&=&4g\^2\_eB\[double\_sum\] with $\omega_k^n=\sqrt{k_z^2+m^2+2neB}$ and $\omega_p^l=\sqrt{p_z^2+m^2+2leB}=\sqrt{(q_z+k_z)^2+m^2+2leB}$. Here $\mathcal{P}$ represents the principle value of the argument. In case of neutron-neutron($nn$) loop (\^[nn]{}\_0)\_&=&\
(k)&=&4 g\^2\_with $\omega_k=\sqrt{\vec{k}^{2}+m^2}$ and $\omega_p=\sqrt{(\vec{q}+\vec{k})^2+m^2}$. It should be mentioned here that in case of $pp$ loop, the expression is obtained with the simplifying assumption that $\vec{\,\,\,\,\qpr}=0$ so that the $\kpr$ integral can be performed exactly using the orthogonality condition of generalized Laguerre polynomials which renders the products of Laguerre polynomials into simple Kronecker Deltas and in turn makes it trivial to convert the double summation structure of the self-energy into a single sum over Landau Levels. Obviously, the same assumption does not provide any such simplification for $nn$ loop. In this case also $\vec{\,\,\,\,\qpr}=0$ is taken for consistency. Thus, the one loop vacuum self-energy of neutral pion becomes (\^\_0)\_&=&(\^\_0)\^[pp]{}\_+(\^\_0)\^[nn]{}\_\
&=&g\^2\_\_0\^1 dx\
&=&g\^2\_\_0\^1 dx+2g\^2\_ \_0\^1 dx\
&=&(\^\_0)\_(eB0)+(\^\_0)\_(eB=0).
Unlike the charged pion case, the external transverse momentum of the neutral pions is continuous. To obtain the effective mass of the neutral pions as a function of $T$, $\mu_N$ and $eB$, we solve the DSE given by m\^[2]{}\_[\^0]{}-m\_[\^0]{}\^2+\_[0]{}(m\^[2]{}\_[\^0]{},=0,eB)=0 \[dse\] where $\mathcal{R}\mbox{e}\Pi_{0}$ contains the sum of explicit $eB$ dependent vacuum and thermal contribution and the renormalized pion mass in vacuum $m_{\pi^0}$ is taken to be same as $m_{\pi^{\pm}}^{}$ and will be denoted as $m_{\pi}$ in subsequent sections.
Results and discussions
=======================
\
In this section we present the numerical results obtained by solving the DSE given in Eq.(\[dse\]). We have taken $f^2_{\pnn}/{4\pi}=0.0778$ and $m_\pi$=0.14 GeV. The nucleon mass is taken as 0.938 GeV. In case of evaluating the effective masses, we have summed up to 300 Landau levels of the loop particles. For stronger magnetic fields i.e $eB>0.1$ GeV$^2$,the results are found to converge at much lower value of the maximum Landau Level used. However, for $eB\rightarrow 0$, more than 200 landau levels are needed to produce a convergent numerical result. In case of charged pions we have taken the scale $\mu$ as the square of the neucleon mass. At first we solve the DSE for the charged pions given by m\_\^[\*2]{}-m\_\^[2]{}+\_(m\^[\*2]{},\^2=(2n+1)eB,eB)-(2n+1)eB=0 \[dsepm1\] where $m_{}^{*}=m_{\pi^+}^{*}(m_{\pi^-}^{*})$ when $\mathcal{R}\mbox{e}\Pi_{+}(\mathcal{R}\mbox{e}\Pi_{-})$ is used. The variation of $m_{}^{*}$ with $eB$, $T$ and $\mu_N$ for $\pi^+$ is shown in Fig.\[mstarpm\]. It can be seen from Fig.\[mstarpm\](a) that the plots are completely dominated by the trivial Landau quantization of the external pion. In other words, for each value of Landau Level, the linear $(2n+1)eB$ term present in Eq.(\[dsepm\]) affects the effective mass much more compared to the one loop self-energy correction. Moreover, as the temperature and $\mu_N$ dependence of the effective mass comes only through the self-energy, one can expect that in the $eB$ dominated scale, they will be insignificantly small. Accordingly $m^\ast$ seems to be independent of the variation of $T$ and $\mu_N$ as shown in (b) and (c) part of Fig.\[mstarpm\]. It should be mentioned here that as the DSE of $\pi^-$ is different from that of $\pi^+$ only in the expression of $\mathcal{R}\mbox{e}\Pi$, the $m^\ast$ plots for $\pi^-$ will be exactly superimposed on those of $\pi^+$. However, differences between the two charged species can be observed in Fig.(\[reprtpm\]) where the variation of the real part of the self-energy with the invariant mass is shown for two different values of $eB$ with first three Landau Levels. Although both of the self-energies decrease with invariant mass and remain negative throughout, the small oscillatory behaviour present in case of $\mathcal{R}\mbox{e}\Pi_{+}$ can not be observed for $\mathcal{R}\mbox{e}\Pi_{-}$. To unveil the contribution of the real part of the self-energy correction, now we neglect the trivial shift and solve m\^[2]{}\_[\^]{}-m\_\^2+\_(m\^[2]{}\_[\^]{},=0,eB)=0. \[dse\_subtracted\] In other words, the effective mass is measured with respect to the trivial Landau shift. This kind of situation may occur in principle when different species of charged particles are present in the system and all of which undergo equivalent trivial Landau shifts. In that case the real part of the self-energies will play the deciding role in the characterization of the effective masses. Moreover, comparing Eq.(\[dse\]) and Eq.(\[dse\_subtracted\]), one can observe that Eq.(\[dse\_subtracted\]) brings down the charged pions in equal footing with the neutral pions only in the difference of the self-energy.
In Fig.\[drv\], the effective mass variation as a function of external magnetic field has been shown at a given temperature of 160 MeV and chemical potential of 200 MeV. Considering magnetic field dependent vacuum contribution and the thermal contribution of the self-energy separately, one can compare with the total contribution as shown. When only the thermal contribution is taken into account, it can be noticed that with the increase of external magnetic field, the effective mass of the charged pions develops smooth oscillations. However, no such oscillations have been observed for neutral pion which instead, shows marginal increase in the thermal effective mass with $eB$. On the other hand, the effective mass due to the magnetic field dependent vacuum part decreases monotonically with $eB$ for $\pi_0$ as well as for $\pi_{\pm}$. However, the decreasing nature is more pronounced in case of neutral pion. It is clear from the figure that the field dependent vacuum part of the self-energy can influence the $eB$ dependence of the effective mass significantly. One can notice that, for neutral pions, even the qualitative nature of the $eB$ dependence of the effective mass changes i.e from a slowly increasing nature it becomes a decreasing function of $eB$ due to the incorporation of the vacuum part.
To analyze the markedly different behaviour of the thermal contribution in the effective masses of $\pi^+$ and $\pi^-$, the real part of the self-energy is shown in Fig.\[reprt\]. Real part of the thermal contribution is plotted as a function of $q^0$, keeping the magnitude of the $z$-component of the external momentum fixed at 200 MeV. With $eB$=0.1 GeV$^2$ and T=160 MeV, the oscillatory behaviour in case of charged pions can be observed for $\mu_N$ =0 as well as for $\mu_N$ =200 MeV. At vanishing $\mu_N$, the self-energies of $\pi^+$ and $\pi^-$ are almost superimposed. The slight difference is present because of the non-vanishing $q_z$. However, the introduction of large $\mu_N$ reduces the oscillations for $\pi^-$ only in the positive values of $q^0$ while they get enhanced in the domain of negative $q^0$. Exactly the opposite behaviour is seen in case of $\pi^+$. This follows from the fact that $\Pi^+_\eta(-q,\mu_N,T)=\Pi^-_\eta(q,\mu_N,T)$ as mentioned earlier in Eq.(\[pi+pi-\]). Thus the difference in the behaviour of the effective mass in case of charged pions is attributed to the asymmetric behaviour of the real part of the thermal self-energy in presence of non-zero chemical potential. Fig.\[massv\] describes the external parameter dependencies of the effective mass of pions. At a given temperature T=160 MeV, variation of effective mass as a function of $eB$ is shown in Fig.4(a) and 4(b) which correspond to two different values of chemical potentials, $\mu_N$ =50 MeV and 200 MeV respectively. Clearly, the effective mass shows non-trivial oscillatory behaviour in case of charged pions as expected from earlier discussions. Moreover, one should notice that with the increase of $\mu_N$, the charged pions behave differently. The oscillations in effective mass of $\pi^-$ gets reduced with the increase in $\mu_N$ while it enhances the oscillations of $\pi^+$. It is also worth mentioning that apart from the anticipated mass splitting, there exist small windows within the given range of the magnetic field values, in which the mass hierarchy of different pion species gets altered. Moreover, the possibility of a large window exists due to the fact that, for $eB>0.1$ GeV$^2$ the decreasing rate of the effective mass for $\pi^+$ is higher in comparison with that of $\pi^-$ i.e at even higher values of magnetic field, the negatively charged pions become more massive than positively charged pions. However, in extremely large magnetic background, the chiral power counting does not hold anymore [@colucci] which imposes serious restrictions on the validity of the model calculation.
Effective mass as a function of chemical potential is presented in Fig.4(c) and 4(d) for two different values of magnetic field, $eB$=0.1 GeV$^2$ and 0.2 GeV$^2$ keeping the temperature fixed at T=160 MeV. In a similar fashion, the temperature dependence is plotted in Fig.4(e) and 4(f) with constant $\mu_N$=200 MeV and with $eB$=0.1 GeV$^2$ and 0.2 GeV$^2$ as before. In case of $\pi^+$ and $\pi^0$, the effective mass slightly increases with T whereas for $\pi^-$, it almost remain independent of temperature variation which is analogous to the $\mu_N$ dependence. However, unlike the temperature dependence, careful observation suggests that for $eB=0.2$ GeV$^2$, $m^\ast _{\pi^-}$ follows a decreasing trend with $\mu_N$. Moreover, in plots with higher $eB$ value, there exists a noticeable initial mass difference between neutral and charged pions. One can also observe that the rate of increase of effective mass as a function of T as well as $\mu_N$ for all the pion species gets reduced for higher magnetic fields.
\
\[a\] \[b\]\
\[c\] \[d\]\
\[e\] \[f\]\
summary and conclusions
=======================
We have evaluated the one loop pion self-energy in presence of constant homogeneous magnetic field for finite temperature and chemical potential. As far as the strength of the external magnetic field is concerned, we have not made any approximation and used the complete form of the fermionic propagator represented in terms of a sum over infinite landau levels. We have used the real time formalism in the evaluation of the thermal part of pion self-energy. We have solved LL dependent DSE to obtain the effective masses as a function of different external parameters. It is shown that by taking the trivial Landau shift term, the effective mass increases with $eB$ for the charged pions. Although the real part of the self-energy depends on $T$, $\mu$ and $eB$ it is sub-leading in comparison to the trivial Landau shift. Thus the effective mass of the charged pions remain constant as a function of both $T$ and $\mu$ for a given $eB$. To extract the contribution of the real part of the self-energy, we also solve the DSE by neglecting the trivial shift. It is shown that the effective masses of the charged pions possess oscillatory behaviour. However, the same oscillatory behaviour is not seen in case of the neutral pions. We have also shown that the oscillatory behaviour with finite chemical potential is not similar for $\pi^+$ and $\pi^-$. With increasing chemical potential, the oscillation in the effective mass of positive pion is found to be enhanced while that of $\pi^-$ gets reduced. Along with the thermal contribution, the magnetic field dependent vacuum contribution is also taken into account. Our results suggest that the external magnetic field dependent vacuum part of the self-energy significantly influences not only the quantitative behaviour but also the qualitative behaviour of the effective mass.
acknowledgement
===============
SG acknowledges Centre for Nuclear Theory, VECC for support.
Evaluation of the summation
===========================
In Eq.(\[sum\]) we find that the vacuum contribution for the charged pion self-energy possess the sum of three different infinite series $\mathcal{S}_1$ , $\mathcal{S}_2$ and $\mathcal{S}_3$. Here we discuss the procedure to obtain the compact expressions for these summations one by one. These compact expressions will be useful for evaluating the subsequent $\kpr$ integral as will be seen below. \_1&=&\_n\^(-1)\^nL\_[n-1]{}\^1(2\_p)\
&=&\_n(-1)\^nL\_[n-1]{}\^1(2\_p)\_0\^1z\^tz\^[2nxeB/]{}t=-(1-x)-1\
&=&\_0\^1z\^t(-z\^[2xeB/]{})(1+z\^[2xeB/]{})\^[-2]{}where we have used the identity \_[n=0]{}\^L\_n\^(x)z\^n=(1-z)\^[--1]{}|z|1. In a similar way with $\theta=z^{2xeB/\mu}$ we find \_2&=&\_0\^1z\^t()\
\_[n=0]{}\^nz\^n&=&|z|1 is used. $\mathcal{S}_3$ is a divergent series and to extract the momentum dependent finite part we use derivative regularization as follows. \_3&=&\_n\^(-1)\^n(L\_n(2\_p)-L\_[n-1]{}(2\_p))\
&=&\_n\^(-1)\^n(L\_n(2\_p)-L\_[n-1]{}(2\_p))\
&=&\_0\^1z\^t x(x-1)()\
\_3&=&\_0\^1z\^t()\
\_[n=0]{}\^z\^n&=&|z|1. It might seem that the scale is absent here but in fact is hidden in $\theta=z^{2xeB/\mu}$. Moreover, one can observe that the $\mathcal{S}_3$ is obtained after an indefinite integral over $\qp^2$ which must contain an integration constant independent of $\qp^2$. In fact this constant must be infinity as the series we started with is divergent in nature. However this procedure extracts out the finite momentum dependent part that we require and the infinite contribution can be taken care by redefining the scale $\mu$ in such a way that it renormalizes the bare mass to the physical one. The vacuum self-energy now becomes (\_+)\_&=&g\^2\_\_0\^1dxe\^[-\_p]{}\_0\^1 e\^[tz]{}()\
&=&g\^2\_\_0\^1dx\_0\^1z\^[-1]{}z\^[-y(1-x)\^2/]{}\
&&\
&=&(1-x)+(xz)\
y&=&(xz). Now, we shift the $\kpr\rightarrow\kpr-y\qpr$. Droping the odd terms we get (\_+)\_&=&g\^2\_\_0\^1dx\_0\^1z\^[-y(1-x)-1]{}e\^[-\^2]{}\
&&\
&=&q\^2(\^2+y\^2\^2)+2x\^2\^2-y\^2{(1+2x)\^2-\^2}-2{()\^2+y\^2\^4}\
&=&-2m\^2q\^2+2(x-y)(1-x)\^2\^2+(1-x)q\^2(\^2+y\^2\^2). Now, the 2 dimensional $\kpr$ integral can be easily evaluated using the standard Gaussian integral identities given by d\^2e\^[-\^2]{}&=&-\
d\^2\^2 e\^[-\^2]{}&=&-\
d\^2()\^2e\^[-\^2]{}&=&-\^2e()<0.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We study the heat conduction of a cold, thermal cloud in a highly asymmetric trap. The cloud is axially hydrodynamic, but due to the asymmetric trap radially collisionless. By locally heating the cloud we excite a thermal dipole mode and measure its oscillation frequency and damping rate. We find an unexpectedly large heat conduction compared to the homogeneous case. The enhanced heat conduction in this regime is partially caused by atoms with a high angular momentum spiraling in trajectories around the core of the cloud. Since atoms in these trajectories are almost collisionless they strongly contribute to the heat transfer. We observe a second, oscillating hydrodynamic mode, which we identify as a standing wave sound mode.'
author:
- 'R. Meppelink, R. van Rooij, J. M. Vogels and P. van der Straten'
title: 'Enhanced heat flow in the hydrodynamic-collisionless regime'
---
The field of Bose-Einstein condensation in dilute atomic gases provides a fruitful playground to test well-developed theories of quantum fluids. Research using Bose-Einstein condensates (BECs) can address open questions relating to the many-body aspects of two-component quantum liquids, namely the interaction between the hydrodynamic normal and the superfluid component at finite temperatures [@JLowTempPhys.116.277]. After the first realization of BEC some pilot experiments have been carried out, but detailed experiments are missing [@PhysRevLett.78.764; @PhysRevLett.81.500]. This has to be compared to the case of liquid helium below the $\lambda$ point, where many experiments since the 1950s have added to our understanding of novel phenomena in quantum liquids, like collective excitations, first and second sound, and others. One of the drawbacks of liquid helium is that the interactions are so strong that a clear distinction between the two components is difficult.
The reason for the lack of detailed experiments in BECs to study quantum liquids and in particular the hydrodynamical aspects of it, is the limited number of atoms (typically 1–10 million) in the experiments leaving the thermal atoms virtually collisionless. Efforts to decrease the mean free path by increasing the confinement limits the lifetime of the sample, since the density is limited by three-body decay. This makes the observation of sound propagation in a BEC a challenge.
As to theory, hydrodynamical damping of trapped Bose gases has been described above and below the transition temperature $T_\text{c}$ [@JLowTempPhys.111.793; @JLowTempPhys.116.277]. These theories yield the oscillation frequencies and damping rates of several low-lying modes, where it is assumed that the sample is fully hydrodynamic in all directions. Experiments on dilute clouds of cold atoms are generally conducted in highly asymmetric traps. In these elongated, cigar-shaped geometries the mean free path of the atoms can become much shorter than the size of the cloud in the long, axial direction, but at the same time exceeds the size in the other, radial directions. In this so-called hydrodynamic-collisionless regime the system is axially hydrodynamic and radially collisionless. In our setup, described in detail in Ref. [@RevSciInstr.78.013102], we have created BECs containing up to 3 $\times$ 10$^8$ sodium atoms by evaporation of atoms in an axially strongly decompressed trap with an aspect ratio of 1:65. Hot atoms created in three-body collisions are able to leave the sample in this highly asymmetric trap, before they can heat other atoms in an avalanche [@PhysRevA.75.031602]. The sample is axially hydrodynamic, but due to the large aspect ratio collisionless in the radial direction. Such samples seem ideal for the observation of sound propagation in the axial direction, since the axial length of the condensates exceeds a few mm. However, neither experiments nor theoretical descriptions exist to determine if the collisions in the radial direction will affect the damping rates to a degree that the observation of sound remains elusive.
From a practical point of view, the hydrodynamic-collisionless regime is of relevance for the realization of a continuous atom laser by evaporatively cooling a magnetically guided atom beam [@PhysRevLett.93.093003]. Here the efficiency of the cooling process is expected to be limited by the heat transfer between the hot, upstream and cold, downstream parts of the beam.
In this Letter we report the experimental determination of the heat conduction in a cold, thermal gas above the transition temperature $T_\text{c}$, which is hydrodynamic in the axial direction, but collisionless in the radial directions. The heat conduction is determined by locally heating the cloud and subsequently observing the equilibration of the temperature distribution. Two previously unobserved hydrodynamic modes are reported; a thermal dipole mode and a standing wave sound mode. Furthermore, by reducing the number of atoms the transition in the axial direction from the hydrodynamic regime to the collisionless regime is observed. We find that the heat conduction is five times stronger than calculations for the homogeneous case predict.
{width="65.00000%"}
We measure the heat flow by locally heating the thermal cloud, after which the equilibration is studied. The heat is induced by exciting the thermal cloud using Bragg scattering with a laser beam aligned perpendicular to the axial axis of the cloud, which is aimed at its tail and retro-reflected [@PhysRevLett.82.871]. The ($1/{\ensuremath{\text{e}}}$)-width of the intensity of this beam is $0.8$ mm, which is close to half of the axial ($1/{\ensuremath{\text{e}}}$)-size of the cloud. The asymmetric excitation is chosen since it yields the maximum separation between the cold, unperturbed part and the heated part of the cloud, resulting in a long observation time. The laser beam is detuned 2 nm below of the ${\ensuremath{^{23}}\text{Na}} \, \text{D}_2$ transition in order to reduce resonant scattering and prevent superradiant scattering. The excited particles will locally redistribute their momentum and energy through collisions with the other particles, resulting after a few collisions in a *local thermal equilibrium. The experiments are conducted on a cloud containing up to $1.3 \times 10^9$ atoms confined in a clover leaf type magnetic trap (MT) characterized by the radial trap frequencies $\omega_\text{rad}=2 \pi\times 95.6$ Hz and the axial trap frequency $\omega_\text{ax}=2 \pi \times 1.46$ Hz at a temperature between 1.2 and 2 $\mu$K, which is above $T_\text{c} \approx 1$ $\mu$K. Once a cloud is excited, it is allowed to rethermalize in the MT during an adjustable hold time $\tau$, after which the confinement is turned off and an absorption image is taken after time-of-flight (TOF). The TOF duration is chosen in such a way that the optical density will not exceed 3.5, resulting in a time-of-flight of 40 ms for the highest number of atoms.*
We introduce a measure for the hydrodynamicity in the axial direction $\bar{\gamma}\equiv \gamma_\text{col}/\omega_\text{ax}$, where the collision rate $\gamma_\text{col} = n_\text{eff} \,\sigma\, v_\text{rel}$ is the average number of collisions. Here, the relative velocity $v_\text{rel}=\sqrt{2} \bar{v}_\text{th}$, where $\bar{v}_\text{th}=\sqrt{8{\ensuremath{k_\text{B}}}T/m \pi}$ is the thermal velocity at temperature $T$ and $m$ is the mass and $\sigma =8\pi a^{2}$ is the isotropic cross section of two bosons with s-wave scattering length $a$. Furthermore, $n_\text{eff}=\int n^2\left (\vec{r}\right ){\ensuremath{\text{d}}}V/\int n\left(\vec{r}\right ){\ensuremath{\text{d}}}V = n_0/\sqrt{8}$ for an equilibrium distribution in a harmonic potential, where $n_0$ is the peak density. Written in terms of the number of atoms $N=n_0 \left ( 2 \pi {\ensuremath{k_\text{B}}}T/ \left ( m \bar{\omega}^2 \right ) \right )^{3/2}$ and the geometric mean of the angular trap frequencies $\bar{\omega}^3 \equiv \omega^2_\text{rad} \omega_\text{ax}$, this results in $\gamma_\text{col}= N m \sigma \bar{\omega}^3/(2 \pi^2 {\ensuremath{k_\text{B}}}T) \approx 90 \text{ s}^{-1}$ for the highest number of atoms and corresponds to a hydrodynamicity of $\bar{\gamma}\lesssim$ 10 in the axial direction. Even at the highest hydrodynamicity the lifetime of the cloud, limited by 3-body decay, is more than 60 s, which is more than sufficient for the experiments described below. Note that the hydrodynamicity parameter in the radial direction is due to the anisotropic trap potential $\bar{\gamma}_\text{rad}\lesssim 10/65$ and the cloud is in the radially collisionless regime. By reducing the number of atoms the heat flow through the thermal cloud can also be measured in the axially collisionless regime.
The images are analyzed using a least square fit to a 2D Gaussian distribution. In this distribution the radial size as a function of the axial position is modeled by a hyperbolic tangent function which adds a gradient to the width that resembles the asymmetric distribution. The fit to the 2D distribution yields the temperature gradient, axial and radial cloud sizes, and the optical density. In the following we will focus on the temperature gradient, which is a measure of the imbalance of the temperature in the cloud. In this analysis we assume that a local temperature equilibrium is established at all times. Since this is not a valid assumption in the first few tenths of milliseconds after excitation especially for lower collision rates, we cannot accurately describe the data at these times.
Heating the thermal cloud will also cause a quadrupole motion of the atoms, since the cloud is excited non-adiabatically to a higher temperature. Since the resulting compression and decompression is homogeneous over the cloud it does not influence the temperature gradient. The quadrupole mode, induced by perturbing the magnetic confinement, has been studied experimentally by Buggle and coworkers [@PhysRevA.72.043610]. They found the damping rate and oscillation frequency of this mode to be in good agreement with a theoretical model [@PhysRevA.60.4851]. Since the quadrupole mode damps slower than the thermal dipole mode considered in this paper, the maximum hold time for most of our measurement series turns out to be insufficient to accurately determine the damping rate of the quadrupole mode. Although our results are less accurate, we have confirmed that the frequency and damping rate of the quadrupole mode are in agreement with the results reported in Refs. [@PhysRevA.72.043610; @PhysRevA.60.4851].
A series of measurements consists of about 100 shots at various hold time $\tau$ from which the temperature gradient is determined. The number of atoms and temperature is determined from an average over all shots, which yields the hydrodynamicity $\bar{\gamma}$. We plot the temperature gradient as a function of $\tau$ for three values of $\bar{\gamma}$ in [Fig. \[fig:thermcond-coll\]]{}.
[Fig. \[fig:thermcond-coll\]]{}(a) is the result of a measurement at small $\bar{\gamma}$ and shows a slowly damped oscillation, where the temperature gradient after half a trap oscillation has changed sign. The heated atoms are then at the opposite side of the cloud with respect to the excitation side, oscillating a frequency $\omega_\text{d}/ \omega_\text{ax}=1$. We refer to this mode as the thermal dipole mode, which has not been observed previously. The oscillation frequency $\omega_\text{d}$ and damping rate $\Gamma_\text{d}$ are determined by fitting the data to a damped sinusoidal and are shown in [Fig. \[fig:thermal-dipole\]]{} as a function of $\bar{\gamma}$. For small collision rates ($\bar{\gamma}\approx1$) the damping rate is proportional to the collision rate. As a consequence the frequency of the mode will decrease for increasing $\bar{\gamma}$ until it reaches zero, when the system is critically damped. In the experiment we observe oscillatory behavior for $\bar{\gamma} \lesssim 2.5$.
For higher values of $\bar{\gamma}$ the temperature gradient as a function of $\tau$ becomes critically damped, as can be seen in [Fig. \[fig:thermcond-coll\]]{}(b). For even higher values of $\bar{\gamma}$ the system becomes hydrodynamic and atoms cannot move through the cloud without colliding frequently. The heat transport will become diffusive, which is a slower process than the harmonic oscillation. As a consequence we expect the damping of the temperature gradient to decrease, but remain non-oscillatory. A measurement for high $\bar{\gamma}$ is shown in [Fig. \[fig:thermcond-coll\]]{}(c). For $\tau<0.1$ s, a double exponential decay can be seen, where the fast decay due to higher order modes and the slow decay of the lowest thermal dipole mode can be discriminated from each other due to the strong inequality of the damping rates. The reduced chi-squared for all fits are of the order of unity, as can also be seen from [Fig. \[fig:thermcond-coll\]]{}, since the curves go smoothly through the data points.
![The measured normalized damping rate $\Gamma_\text{d}/\omega_\text{ax}$ (a) and normalized frequency $\omega_\text{d}/\omega_\text{ax}$ (b) of the thermal dipole mode as a function of the hydrodynamicity $\bar{\gamma}$. The solid line in (a) is a fit of the data points with $\bar{\gamma}>5$ to the solution of [Eq. (\[eqn:heatdiffusion\])]{} with $\kappa_0=6.4$. The dashed lines are a guide to the eye. The vertical error bars show only statistical errors; the main contribution to the uncertainty in $\bar{\gamma}$ is the uncertainty in the number of atoms.[]{data-label="fig:thermal-dipole"}](thermal_dipole.pdf){width="40.00000%"}
The measurements in the hydrodynamic regime can be analyzed by numerically solving the heat diffusion equation in the axial direction $$c_p n(z) {\frac{\partial}{\partial t}} T(z,t) = {\frac{\partial}{\partial z}}\left[\kappa {\frac{\partial}{\partial z}} T(z,t)\right].
\label{eqn:heatdiffusion}$$ Here the specific heat capacity $c_p=7/2$, the heat conductivity $\kappa \equiv \kappa_0 \pi v_\text{th} \Sigma/(\sqrt{8}\sigma)$ with $\kappa_0$ the dimensionless heat conductivity coefficient and $\Sigma=2 \pi {\ensuremath{k_\text{B}}}T/(m \omega_\text{rad}^2)$ is the effective cloud surface. The damping rate of the lowest order solution in this regime is found to be $\Gamma_\text{d}=0.542 \kappa_0/\bar{\gamma}$. Fitting the measurements for $\Gamma_\text{d}$ with $\bar{\gamma}>5$ yields $\kappa_0=$[6.4]{}$\pm$[0.4]{}. This value is a factor of five larger than the Chapman-Enskog value $\kappa_0 =75/64 \approx 1.17$ for a homogeneous hydrodynamic system [@Chapman1916; @Enskog1917].
The measurements in the hydrodynamic regime also show a damped oscillation of the decay of the temperature gradient for $\tau > 0.1$ s ([Fig. \[fig:thermcond-coll\]]{}(c)). As this oscillation is only seen in the hydrodynamic regime, where the collisionless oscillation is completely damped out, we conclude that it is the result of another hydrodynamic mode, which we identify as a standing wave sound mode. In our experiments this sound mode can only be seen for values of $\bar{\gamma}$ exceeding 5. Using a least-square fit we determine both the oscillation frequency $\omega_\text{s}$ and damping rate $\Gamma_\text{s}$ of the sound mode for all values of $\bar{\gamma}>5$ as a function of $\bar{\gamma}$ (see [Fig. \[fig:thermal-sound\]]{}). The measured normalized frequency $\omega_\text{s}/\omega_\text{ax} \approx 2.1$ confirms that the mode differs from a center-of-mass motion $\omega_\text{ax}$ and the quadrupole mode $\omega_\text{q}/\omega_\text{ax} \approx \sqrt{12/5}$ [@PhysRevA.60.4851].
$\begin{tabular}{cc}
\includegraphics[width=0.22\textwidth]{sound1.pdf} & \includegraphics[width=0.22\textwidth]{sound2.pdf} \\
\end{tabular}$
This sound mode can be found theoretically by solving the hydrodynamic equations [@JLowTempPhys.111.793] in the limit of no damping and we find $\omega=\sqrt{19/5}\omega_\text{ax}\approx 1.95\omega_\text{ax}$. This mode resembles the quadrupole mode, although the standing wave sound mode is even in the axial velocity $v_z$ and has two nodes in the velocity profile instead of one. A schematic representation of this mode is given in [Fig. \[fig:sound\]]{}. As a consequence, this sound mode contributes to the temperature gradient and can be observed in [Fig. \[fig:thermcond-coll\]]{}(c). This is the first direct experimental observation of a thermal sound mode in a cold gas.
A rigorous theoretical model to calculate the oscillation frequency and damping rate of the modes for arbitrary hydrodynamicity $\bar{\gamma}$ will be presented in Ref. [@danismartien10]. The analysis yields $\kappa_0 = 5.98$ in the hydrodynamic-collisionless regime, which confirms the experiment. Furthermore, for a linear confinement as is used for magnetically guided atomic beams the enhancement of the heat flow is even stronger; up to two orders of magnitude. This result implies that the efficiency of evaporatively cooling of a linearly guided atomic beam is strongly diminished due to the large heat flow and questions the feasibility of realizing a continuous atom laser. The increase of $\kappa_0$ is found to be caused by two effects. First, the shape of the density of states is altered, which results in a relatively lower collision rate and causes the presence of more atoms with a high transverse energy. The second effect lies in the presence of atoms with a high angular momentum. These atoms are in trajectories spiraling around the core of the cloud. These trajectories are almost collisionless and cause a very strong contribution to the heat flow.
![The measured normalized damping rate $\Gamma_\text{s}/\omega_\text{ax}$ (a) and normalized frequency $\omega_\text{s}/\omega_\text{ax}$ (b) of the hydrodynamic sound mode as a function of the hydrodynamicity $\bar{\gamma}$. The dashed line shows the frequency of this mode in the limit of no damping, $\omega=\sqrt{19/5}$.[]{data-label="fig:thermal-sound"}](standing_sound.pdf){width="40.00000%"}
In conclusion, we have successfully excited and measured two previously unobserved modes; a thermal dipole mode and a standing wave sound mode. Observation of the latter demonstrates both the hydrodynamic behavior of the cloud and the presence of sound propagation in a dilute thermal gas. In the hydrodynamic regime we have measured the heat conduction coefficient $\kappa_0=$[6.4]{}$\pm$[0.4]{}, which is five times higher than calculations for the homogeneous case predict. This effect is expected to be even stronger for a linear confined cloud in the axially hydrodynamic-radially collisionless regime. This result implies that the efficiency of evaporative cooling in a continuous atom laser, where the confinement is linear, is strongly reduced by the large heat transfer between the hot and cold parts of the beam.
This work is supported by the Stichting voor Fundamenteel Onderzoek der Materie “FOM” and by the Nederlandse Organisatie voor Wetenschaplijk Onderzoek “NWO”. We are grateful to W. C. Germs for contributing in the early stages to the theoretical description and to D. Guéry-Odelin and J. Dalibard for stimulating discussions.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
In this paper we prove a rate of escape theorem and a central limit theorem for isotropic random walks on Fuchsian buildings, giving formulae for the speed and asymptotic variance. In particular, these results apply to random walks induced by bi-invariant measures on Fuchsian Kac-Moody groups, however they also apply to the case where the building is not associated to any reasonable group structure. Our primary strategy is to construct a renewal structure of the random walk. For this purpose we define cones and cone types for buildings and prove that the corresponding automata in the building and the underlying Coxeter group are strongly connected. The limit theorems are then proven by adapting the techniques in [@HMM:13]. The moments of the renewal times are controlled via the retraction of the walks onto an apartment of the building. \
random walk, limit theorems, Fuchsian building, Kac-Moody group, Cannon automaton <span style="font-variant:small-caps;">2010 Mathematics Subject Classification: 60G50, 60F05, 60B15, 20E42, 51E24</span>
author:
- 'L. Gilch'
- 'S. Müller[^1]'
- 'J. Parkinson[^2]'
title: 'Limit theorems for random walks on Fuchsian buildings and Kac-Moody groups'
---
Introduction
============
Let $(X_{n})_{n\geq 1}$ be i.i.d. random variables taking values in $\ZZ^{d}$. Under a second moment condition the classical central limit theorem gives $$\frac{\sum_{i=1}^{n} X_{i}-nv}{\sqrt n } \stackrel{\mathcal{D}}{\longrightarrow} \mathcal{N}(0,\sigma^{2}),$$ where $v=\EE[X_{1}]$ is the rate of escape (or drift) and $\sigma^{2}$ the asymptotic variance. A natural and influential question, dating back to Bellman [@bellman] and Furstenberg and Kesten [@FK], is to what extent this phenomenon generalizes to the situation where $(X_{n})_{n\geq 1}$ takes values in a group, or more generally, the situation where $(X_n)_{n\geq 1}$ is a random walk on a graph.
There are various settings in which central limit theorems have been established, with key results in the contexts of Lie groups and hyperbolic groups. In the hyperbolic setting, Sawyer and Steger [@ST] studied the case of the free group $F_{d}$ with $d$ standard generators and the corresponding word distance $d(\cdot,\cdot)$. Under a technical moment condition they show, using analytic extensions of Green functions, that $(d(e,X_n)-nv)/\sqrt{n}$ converges in law to some non-degenerate Gaussian distribution, where $e$ is the group identity in $F_{d}$. Another proof was given by Lalley [@La:93] using algebraic function theory and Perron-Frobenius theory, and a geometric proof was later presented by Ledrappier [@Le:01]. A generalization to trees with finitely many cone types can be found in Nagnibeda and Woess [@NW]. Another generalization to free products of graphs was given by Gilch [@gilch:PhD].
More recently Björklund [@bjorklund] proved a central limit theorem for hyperbolic groups with respect to the Green metric, and this was pushed forward by Benoist and Quint [@BeQu:14] for random walks on hyperbolic groups with respect to the word metric under the optimal second moment condition.
Another approach to the central limit theorem for surface groups has been developed by Haissinski, Mathieu, and Müller [@HMM:13], where the planarity and hyperbolicity of the Cayley graph are employed to develop a renewal theory for random walks on these groups. The resulting central limit theorem comes complete with formulae for the speed and variance of the walk.
Central limit theorems for semisimple real Lie groups were established by Wehn [@wehn:62], Tutubalin [@Tutubalin:65], Virtser [@virtser], Stroock and Varadhan [@Stroock:73], and Guivarc’h [@Gui:80] in a variety of contexts, using a wide range of techniques. There is an extensive literature on this subject, with further limit theorems for real Lie groups given in [@BeQu:13; @bougerol; @GoGu:96; @guivarc'h2; @GLeP:04; @kaimanovich; @LePage].
The case of $p$-adic Lie groups is also rather well understood, with central limit theorems established by Lindlbauer and Voit [@LV], Cartwright and Woess [@CW], and Parkinson [@P3]. Further limit theorems for $p$-adic Lie groups are given in [@BS; @PS; @tolli]. Many of these papers employ a remarkable geometric object called the *affine building* associated to the $p$-adic group, and utilize the rich representation theory available in the $p$-adic setting.
The current paper lies at the confluent of the hyperbolic and Lie theoretic settings. Here we prove limit theorems for random walks on *Fuchsian buildings* and the *Kac-Moody groups* associated to them (see below for some descriptions). From the point of view of Lie theory, this is a natural next step in the progression from ‘spherical-type’ Lie groups (the semisimple real Lie groups) and ‘affine-type’ Lie groups (the $p$-adic case) to a theory for random walks on buildings and Kac-Moody groups of arbitrary type. From the hyperbolic point of view, the buildings that we consider contain many copies of the hyperbolic disc tessellated using a ‘Fuchsian Coxeter group’, and thus while the buildings are certainly not planar, some of the renewal theory techniques from the planar surface group case [@HMM:13] can be pushed through.
Before stating our main results, let us give a brief description of the objects involved in this paper. Buildings are geometric/combinatorial objects that can be defined axiomatically. Initial data required to define a building includes a Coxeter system $(W,S)$, and then a *building $(\Delta,\delta)$ of type $(W,S)$* consists of a set $\Delta$ (whose elements are the *chambers* of the building) along with a “generalised distance function” $\delta:\Delta\times\Delta\to W$ satisfying various axioms (see Definition \[defn:building\]). Thus the “distance” between chambers $x,y\in\Delta$ is an element $\delta(x,y)$ of the Coxeter group $W$, and by taking word length in $W$ this gives rise to a metric $d(\cdot,\cdot)$ on the building. We fix a base chamber $o\in\Delta$. The ‘spherical’ buildings are those with $|W|<\infty$, and the ‘affine’ buildings are those where $W$ is a Euclidean reflection group.
The theory of spherical and affine buildings has been extensively studied. However there are many Coxeter systems which are neither finite nor affine. Examples of buildings of these more ‘exotic types’ arise naturally in Kac-Moody theory. These groups can be seen as generalisations of the classical ‘groups of Lie type’, since they admit presentations reminiscent of the Chevalley presentations of the classical groups based around an associated Coxeter system $(W,S)$ (see [@titskac]). To each Kac-Moody group of type $(W,S)$ there is naturally associated a building of type $(W,S)$, and the Kac-Moody group acts highly transitively on this building.
While the above construction produces a lot of very interesting buildings, it is certainly not true that all buildings arise in this way (see [@ronanconstruction], for example). Thus in this paper we consider the building as the primary object of interest. Results concerning groups may then be deduced as corollaries, although it is important to note that our results apply equally well to the situation where there is no underlying group. (Indeed the building may have trivial automorphism group!).
In this paper we consider the natural class of *isotropic random walks* $(X_n)_{n\geq 0}$ on buildings, where the transition probabilities $p(x,y)$ of the random walk depend only on the generalised distance $\delta(x,y)$. If the building comes from a Kac-Moody group, then isotropic random walks are induced by measures on the group which are bi-invariant with respect to the ‘Borel subgroup’ $B$. The results of much of the preliminary sections are valid for buildings of any type, however our main results concern the class of *Fuchsian buildings*. These are buildings whose Coxeter groups are discrete subgroups of $PGL_2(\mathbb{R})$, and since they are neither spherical nor affine they are an interesting “non-classical” class of buildings.
We use a mixture of algebraic, geometric and probabilistic techniques. We observe that the transition operator of an isotropic random walk can naturally be regarded as an element of a *Hecke algebra*, and we use this result to show that our buildings are nonamenable. Next we develop the theory of cones, cone types, and automata for buildings, and we show that the Cannon automaton of a Fuchsian building is *strongly connected*. Connectivity properties of automata have various applications and are interesting in their own right. To our knowledge our results on the strong connectivity of the Cannon automaton are the first besides the trivial cases of free groups and surface groups. There are also some interesting features of cones in buildings that are in contrast to the theory of cones in groups. For example cones of the same type in the building are not necessarily isomorphic as graphs (see Remark \[rem:noniso\]).
We use our the theory of cones in buildings to develop a *renewal theory* for the isotropic random walks on Fuchsian buildings. The idea here is to find a decomposition of the trajectory of the walk into aligned pieces in such a way that these pieces are identically and independently distributed. To do this we define renewal times $(R_n)_{n\geq 1}$ for the walk as follows. Fix a (recurrent) cone type ${\mathbf{T}}$ and let $R_1$ be the first time that the walk visits a cone of type ${\mathbf{T}}$ and never leaves this cone again. Inductively define $R_{n+1}$ to be the first time after $R_n$ that the walk visits a cone of type ${\mathbf{T}}$ and never leaves it again (see (\[eq:renewal\]) for a more formal definition). Our main results are as follows.
\[thm:LLNbuilding\] Let $(\Delta, \delta)$ be a regular Fuchsian building and let $(X_n)_{n\geq 0}$ be an isotropic random walk on $\Delta$ with bounded range. Then, $$\begin{aligned}
\label{eq:v}
\frac{1}{n} d(o,X_{n}) \stackrel{a.s.}{\longrightarrow} v= \frac{\EE[ d(X_{ R_{2}}, X_{ R_{1}})]}{\EE[ R_{2}- R_{1}]}>0~\mbox{ as } n\to\infty.\end{aligned}$$
\[thm:CLTbuilding\] Let $(\Delta, \delta)$ be a regular Fuchsian building and let $(X_n)_{n\geq 0}$ be an isotropic random walk on $\Delta$ with bounded range. Then, $$\frac{d(o,X_{n}) -nv }{\sqrt{n}} \stackrel{\cD}{\longrightarrow} \cN (0, \sigma^2),$$ with $v$ as in (\[eq:v\]) and $$\begin{aligned}
\sigma^{2}=\frac{\EE[(d(X_{R_{2}}, X_{R_{1}}) - (R_{2}- R_{1})v)^{2}]}{\EE[R_{2}-R_{1}]}.\end{aligned}$$
When a group acts suitably transitively on a regular Fuchsian building the above theorems give limit theorems for random walks associated to these groups. For example, suppose that $G$ is a Kac-Moody group over a finite field with Coxeter system $(W,S)$ (see [@titskac]). Let $B$ be the positive root subgroup of $G$. Then $\Delta=G/B$ is the set of chambers of a locally finite regular building of type $(W,S)$, where $\delta(gB,hB)=w$ if and only if $g^{-1}hB\subseteq BwB$. Then we have the following corollary.
\[cor:Kac\] Let $G$ be a Kac-Moody group over a finite field with Fuchsian Coxeter system $(W,S)$, and let $(\Delta,\delta)$ be the associated Fuchsian building. Let $\varphi$ be the density function of a $B$-bi-invariant probability measure on $G$, and assume that $\varphi$ is supported on finitely many $BwB$ double cosets. Then the assignment $$p(go,ho)=\varphi(g^{-1}h)$$ defines an isotropic random walk on $(\Delta,\delta)$, and Theorems \[thm:LLNbuilding\] and \[thm:CLTbuilding\] provide a rate of escape theorem and a central limit theorem for this random walk.
To conclude this introduction, let us outline the structure of this paper. Section \[sect:2\] gives definitions and examples of Coxeter groups and buildings. In Section \[sect:3\] we develop the theory of automata for buildings (and Coxeter groups). In Section \[sect:4\] we introduce isotropic random walks on buildings, and use algebraic techniques to prove general results on irreducibility and the spectral radius. We also introduce the retracted walk in this section, which is a main tool in our investigations. In Section \[sect:renewal\] we restrict our attention to Fuchsian buildings, and develop renewal theory for isotropic random walks on these buildings. We prove our main theorems in this section, following the general proof strategy of [@HMM:13]. Finally, in Appendix \[app:A\] we explicitly construct the automaton for each Fuchsian Coxeter system, and deduce that these automata are *strongly connected* (a property that was useful in the work of Section \[sect:renewal\]).
Coxeter groups and buildings {#sect:2}
============================
Coxeter systems
---------------
A *Coxeter system* $(W,S)$ is a group $W$ generated by a finite set $S$ with relations $$s^2=1\quad\textrm{and}\quad (st)^{m_{st}}=1\quad\textrm{for all $s,t\in S$ with $s\neq t$},$$ where $m_{st}=m_{ts}\in\ZZ_{\geq 2}\cup\{\infty\}$ for all $s\neq t$ (if $m_{st}=\infty$ then it is understood that there is no relation between $s$ and $t$). The *rank* of $(W,S)$ is $|S|$. The *length* of $w\in W$ is $$\ell(w)=\min\{n\geq 0\mid w=s_1\cdots s_n\textrm{ with }s_1,\ldots,s_n\in S\},$$ and an expression $w=s_1\cdots s_n$ with $n=\ell(w)$ is called a *reduced expression* for $w$. If $w\in W$ and $s\in S$ then $\ell(ws)\in\{\ell(w)-1,\ell(w)+1\}$. In particular, $\ell(ws)=\ell(w)$ is impossible. The *distance* between elements $u\in W$ and $v\in W$ is $$d(u,v)=\ell(u^{-1}v).$$ The *ball* of radius $R\geq 0$ with centre $u\in W$ is $\cB(u,R)=\{v\in W\mid d(u,v)\leq R\}$ and the *sphere* of radius $R\geq 0$ with centre $u\in W$ is $\cS(u,R)=\{v\in W\mid d(u,v)=R\}$.
If $I\subseteq S$ let $W_I$ be the subgroup of $W$ generated by $I$. Then $(W_I,I)$ is a Coxeter system. The subgroup $W_I$ is called the *standard parabolic subgroup of type $I$*. A Coxeter system $(W,S)$ is *irreducible* if there is no partition of the generating set $S$ into disjoint nonempty sets $S_1$ and $S_2$ such that $s_1s_2=s_2s_1$ for all $s_1\in S_1$ and all $s_2\in S_2$. We will always assume that $(W,S)$ is irreducible.
Fuchsian Coxeter groups
-----------------------
We now define a special class of Coxeter groups that are discrete subgroups of $PGL_2(\mathbb{R})$, called *Fuchsian Coxeter groups*. Let $n\geq 3$ be an integer, and let $k_1,\ldots,k_n\geq 2$ be integers satisfying $$\begin{aligned}
\label{eq:hyperbolic}
\sum_{i=1}^n\frac{1}{k_i}<n-2.\end{aligned}$$ Assign the angles $\pi/k_i$ to the vertices of a combinatorial $n$-gon $F$. There is a convex realisation of $F$ (which we also call $F$) in the hyperbolic disc $\mathbb{H}^2$, and the subgroup of $PGL_2(\mathbb{R})$ generated by the reflections in the sides of $F$ is a Coxeter group $(W,S)$ (see [@davis Example 6.5.3]). If $s_1,\ldots,s_{n}$ are the reflections in the sides of $F$ (arranged cyclically), then the order of $s_is_j$ is $$\begin{aligned}
\label{eq:hyperbolic2}
\begin{aligned}
m_{ij}=\begin{cases}k_i&\textrm{if $j=i+1$}\\
\infty&\textrm{if $|i-j|>1$},
\end{cases}
\end{aligned}\end{aligned}$$ where the indices are read cyclically with $n+1\equiv 1$.
A Coxeter system $(W,S)$ given by data (\[eq:hyperbolic\]) and (\[eq:hyperbolic2\]) is called a *Fuchsian Coxeter system*. Observe that these systems are always infinite. The group $W$ acts on $\mathbb{H}^2$ with fundamental domain $F$. Note that this action does not preserve orientation, however the index $2$ subgroup $W'$ generated by the even length elements of $W$ is orientation preserving. Thus $W'$ is a discrete subgroup of $PSL_2(\mathbb{R})$, and so is a ‘Fuchsian group’ in the strictest sense of the expression.
The Fuchsian Coxeter system $(W,S)$ induces a tessellation of $\mathbb{H}^2$ by isometric polygons $wF$, $w\in W$. The polygons $wF$ are called *chambers*, and we usually identify the set of chambers with $W$ by $wF\leftrightarrow w$. We call this the *hyperbolic realisation* of the Coxeter system $(W,S)$ (it is closely related to the *Davis complex* from [@davis], see the discussion in [@AB Example 12.43]).
\[ex:21\] (a) Let $a,b,c\geq 2$ be integers, and let $W_{abc}$ be the group generated by $S=\{s,t,u\}$ subject to the relations $$s^2=t^2=u^2=1\quad\textrm{and}\quad (st)^a=(tu)^b=(us)^c=1.$$ These Coxeter groups are called *triangle groups*, for they can be realised as groups generated by the reflections in the sides of a triangle on the sphere $\SS^2$ (when $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}>1$), the Euclidean plane $\RR^2$ (when $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$), or the hyperbolic disc $\HH^2$ (when $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}<1$). In the latter case the Coxeter group is Fuchsian. Up to permutation of the triple $(a,b,c)$, the irreducible spherical triangle groups are given by $(a,b,c)=(3,3,2),(4,3,2),(5,3,2)$, and the Euclidean triangle groups are given by $(a,b,c)=(3,3,3),(4,4,2),(6,3,2)$. (b) Let $k_i=2$ for each $1\leq i\leq n$ in (\[eq:hyperbolic\]). Thus each internal angle of the $n$-gon $F$ is a right angle, and the corresponding Coxeter group is called a *right angled polygon group* (by (\[eq:hyperbolic\]) this group is Fuchsian if and only if $n\geq 5$).
Definition of buildings
-----------------------
We now give an axiomatic definition of buildings, following [@AB].
\[defn:building\] Let $(W,S)$ be a Coxeter system. A *building of type $(W,S)$* is a pair $(\Delta,\delta)$ where $\Delta$ is a nonempty set (whose elements are called *chambers*) and $\delta:\Delta\times\Delta\to W$ is a function (called the *Weyl distance function*) such that if $x,y\in\Delta$ then the following conditions hold:
1. $\delta(x,y)=1$ if and only if $x=y$.
2. If $\delta(x,y)=w$ and $z\in\Delta$ satisfies $\delta(y,z)=s$ with $s\in S$, then $\delta(x,z)\in\{w,ws\}$. If, in addition, $\ell(ws)=\ell(w)+1$, then $\delta(x,z)=ws$.
3. If $\delta(x,y)=w$ and $s\in S$, then there is a chamber $z\in\Delta$ with $\delta(y,z)=s$ and $\delta(x,z)=ws$.
Let $(\Delta,\delta)$ be a building of type $(W,S)$ and let $s\in S$. Chambers $x,y\in\Delta$ are *$s$-adjacent* (written $x\sim_s y$) if $\delta(x,y)=s$. One useful way to visualise a building is to imagine an $|S|$-gon with edges labelled by the generators $s\in S$ (think of the edges as being coloured by $|S|$ different colours). Call this $|S|$-gon the *base chamber* which we denote by $o$. Now take one copy of the base chamber for each element $x\in\Delta$, and glue these chambers together along edges so that $x\sim_s y$ if and only if the chambers are glued together along their $s$-edges.
A *gallery of type $(s_1,\ldots,s_n)$* joining $x\in\Delta$ to $y\in\Delta$ is a sequence $x_0,x_1,\ldots,x_n$ of chambers with $$x=x_0\sim_{s_1}x_1\sim_{s_2}\cdots\sim_{s_n}x_n=y. $$ This gallery has *length $n$*.
The Weyl distance function $\delta$ has a useful description in terms of minimal length galleries in the building: If $s_1\cdots s_n$ is a reduced expression in $W$ then $\delta(x,y)=s_1\cdots s_n$ if and only if there is a minimal length gallery in $\Delta$ from $x$ to $y$ of type $(s_1,\ldots,s_n)$. The *(numerical) distance* between chambers $x,y\in\Delta$ is $$d(x,y)=(\textrm{length of a minimal length gallery joining $x$ to $y$})=\ell(\delta(x,y)),$$ Note that we use the same notation $d(\cdot,\cdot)$ for distance in both the Coxeter system and the building.
A building $(\Delta,\delta)$ is called *thick* if $|\{y\in\Delta\mid x\sim_s y\}|\geq 2$ for all chambers $x\in\Delta$, and *thin* if $|\{y\in\Delta\mid x\sim_s y\}|=1$ for all chambers $x\in\Delta$. A building $(\Delta,\delta)$ is *regular* if $$q_s:=|\{y\in\Delta\mid x\sim_s y\}|\quad\text{is finite and does not depend on $x\in\Delta$}.$$ For the remainder of this paper we will assume that $(\Delta,\delta)$ is regular. The numbers $(q_s)_{s\in S}$ are called the *thickness parameters* of the building.
If $I\subseteq S$ and $x\in\Delta$ then the set $R_I(x)=\{y\in\Delta\mid \delta(x,y)\in W_I\}$ (called the *$I$-residue of $x$*) is a building of type $(W_I,I)$ with thickness parameters $(q_s)_{s\in I}$ (see [@AB Corollary 5.30]).
For each $x\in\Delta$ and each $w\in W$, let $$\Delta_w(x)=\{y\in\Delta\mid \delta(x,y)=w\}\quad\text{be the ``sphere of radius $w$'' centred at $x$.}$$ By [@P1 Proposition 2.1] the cardinality $q_w=|\Delta_w(x)|$ does not depend on $x\in \Delta$, and is given by $$q_w=q_{s_1}\cdots q_{s_{\ell}}\quad\textrm{whenever $w=s_1\cdots s_{\ell}$ is a reduced expression.}$$
We call $(\Delta,\delta)$ a *Fuchsian building* if $(W,S)$ is a Fuchsian Coxeter system, and we call $(\Delta,\delta)$ a *triangle building* if $W$ is an infinite triangle group.
Finally, a word about notation. Typically the letters $u,v,w$ will be used for elements of a Coxeter group $W$, and the letters $x,y,z$ will be used for chambers of a building $(\Delta,\delta)$.
Examples of buildings
---------------------
We now give some examples of buildings that are relevant to this paper. We also show that the class of locally finite thick Fuchsian buildings is sufficiently rich by proving existence of many such buildings.
Let $(W,S)$ be a Coxeter system. Let $\Delta=W$, and define $\delta:\Delta\times \Delta\to W$ by $\delta(u,v)=u^{-1}v$. It is immediate that $(\Delta,\delta)$ is a building of type $(W,S)$. This rather simple example is called the *Coxeter complex* of $(W,S)$. It is a thin building, because $\{v\in W\mid u\sim_s v\}=\{us\}$ for each $u\in W$. Conversely it is not difficult to see that every thin building is isomorphic to a Coxeter complex.
If $(W,S)$ is a dihedral group of order $2m$ (that is, $S=\{s,t\}$ with $s^2=t^2=(st)^m=1$) then buildings of type $(W,S)$ are called *generalised $m$-gons*. These ‘basic building blocks’ play an important role in the theory (see the monograph [@HVM] which is devoted to the study of generalised $m$-gons). The Feit-Higman Theorem [@feithigman] implies that locally finite thick generalised $m$-gons only exist for $m\in\{2,3,4,6,8,\infty\}$.
If $(\Delta,\delta)$ is a locally finite thick regular building of general type $(W,S)$, then the “rank $2$” residues $R_{st}(x)=R_{\{s,t\}}(x)$ are generalised $m_{st}$-gons, and so necessarily $$\begin{aligned}
\label{eq:feithigman}
m_{st}\in\{2,3,4,6,8,\infty\}\quad\text{for all $s,t\in S$ with $s\neq t$}.\end{aligned}$$ A sufficient condition for the existence of a locally finite thick regular building of type $(W,S)$ is that $m_{st}\in\{2,3,4,6,\infty\}$ for all $s,t\in S$ (see Example \[ex:kac\]). Allowing $m_{st}=8$ introduces some complications, see Proposition \[prop:existence\]).
\[ex:kac\] Let $G$ be a group with a $BN$-pair $(B,N)$ and Coxeter system $(W,S)$ (see [@AB § 6.2.6] for the definition of $BN$-pairs). An instructive example is $G=GL_n(\mathbb{F})$ where $\mathbb{F}$ is a field, with $B$ the upper triangular invertible matrices, $N$ the monomial matrices (matrices with exactly one nonzero entry in each row and column) and $W=N/(N\cap B)$ the symmetric group on $n$ letters (represented as permutation matrices) with $S$ being the elementary transpositions.
The group $G$ admits a *Bruhat decomposition* $G=\bigsqcup_{w\in W}BwB$. Let $\Delta=G/B$, and define $\delta:\Delta\times\Delta\to W$ by $$\begin{aligned}
\delta(gB,hB)=w\quad\textrm{if and only if}\quad g^{-1}hB\subseteq BwB.\end{aligned}$$ Then $(\Delta,\delta)$ is a thick building of type $(W,S)$ (see [@AB Theorem 6.56]).
All *groups of Lie type* (classical groups, Chevalley groups, Steinberg groups, Suzuki-Ree groups) admit a $BN$-pair. More generally, every “Kac-Moody group” admits a $BN$-pair. A *Kac-Moody algebra* (cf. [@kac]) is a generalisation of the more familiar semisimple Lie algebras. These algebras share many properties with their finite dimensional counterparts, for example, Cartan subalgebras, root space decompositions, and Weyl groups. However in contrast to the semisimple Lie algebra case, the root systems and Weyl groups for infinite dimensional Kac-Moody algebras are infinite. There are Kac-Moody algebras associated to each *crystallographic* Coxeter system (that is, $m_{st}\in\{2,3,4,6,\infty\}$ for all $s,t\in S$). To each such algebra, and for each choice of ground field $\mathbb{F}$, one can define a *Kac-Moody group* $G=G(\mathbb{F})$ by generators and relations in an analogous way to the construction of Chevalley groups in the finite dimensional setting (see [@steinberg] for the finite dimensional theory, and [@titskac] for the Kac-Moody case). The group $G$ has a $BN$-pair, with Coxeter system $(W,S)$. The associated building $(G/B,\delta)$ has uniform thickness parameter $|\mathbb{F}|$, and so taking $\mathbb{F}=\mathbb{F}_q$ to be the finite field with $q$ elements yields a regular building of type $(W,S)$ with thickness $q$.
\[ex:ronan\] Suppose that $(W,S)$ is a Coxeter system such that every irreducible rank $3$ parabolic subgroup is infinite. Suppose that $(q_s)_{s\in S}$ is a sequence of integers, and that for each pair $s,t\in S$ with $s\neq t$ there exists a generalised $m_{st}$-gon $\Gamma_{st}$ with parameters $(q_s,q_t)$. Then Ronan’s free construction [@ronanconstruction] implies that there exists a locally finite thick regular building $(\Delta,\delta)$ of type $(W,S)$ with thickness parameters $(q_s)_{s\in S}$.
It is obvious that every irreducible rank $3$ parabolic subgroup of a Fuchsian Coxeter system $(W,S)$ is infinite, and thus Ronan’s free construction applies to Fuchsian buildings. Thus to exhibit the existence of a thick regular Fuchsian building $(\Delta,\delta)$ of type $(W,S)$ with thickness parameters $(q_s)_{s\in S}$ it is sufficient to exhibit the existence of a family $\{\Gamma_{st}\mid s,t\in S, s\neq t\}$ of generalised $m_{st}$-gons $\Gamma_{st}$ with thickness parameters $(q_s,q_t)$. In the following proposition we use this idea to classify those infinite triangle Coxeter systems admitting locally finite thick regular buildings. This is elementary, although we have been unable to find a reference in the literature.
\[prop:existence\] Let $(W,S)$ be an infinite triangle Coxeter system, with the generators $s,t,u$ arranged so that $m_{st}\geq m_{tu}\geq m_{us}$. A locally finite thick triangle building of type $(W,S)$ exists if and only if $$(m_{st},m_{tu},m_{us})\in\{(a,b,c)\mid a,b,c\in\{2,3,4,6,8\}\}\setminus \{(8,3,3),(8,6,3),(8,6,6),(8,8,8)\}$$ Thus there are precisely $24$ infinite triangle Coxeter systems (up to permuting the generators) admitting locally finite thick triangle buildings. Moreover, for each of these infinite triangle Coxeter systems $(W,S)$ there are infinitely many pairwise nonisomorphic buildings of type $(W,S)$.
Suppose that a locally finite thick regular building $(\Delta,\delta)$ of type $(W,S)$ exists. By (\[eq:feithigman\]) we have $m_{st}\in\{2,3,4,6,8,\infty\}$, and the case $m_{st}=\infty$ is excluded for triangle groups by definition. Since infinite triangle groups have $m_{st}^{-1}+m_{tu}^{-1}+m_{us}^{-1}\leq 1$ this leaves precisely $28$ infinite triangle groups with $m_{st}\geq m_{tu}\geq m_{us}$ and $m_{st},m_{tu},m_{us}\in\{2,3,4,6,8\}$. We now show that the four cases $(m_{st},m_{tu},m_{us})=(8,3,3)$, $(8,6,3)$, $(8,6,6)$, or $(8,8,8)$ do not admit locally finite thick buildings. We recall from [@HVM §1.7] that in a finite thick generalised $m$-gon with parameters $(q,q')$ we necessarily have that $q=q'$ if $m=3$, $\sqrt{qq'}\in\mathbb{Z}$ if $m=6$, and $\sqrt{2qq'}\in\mathbb{Z}$ if $m=8$. For example, consider the $(m_{st},m_{tu},m_{us})=(8,6,6)$ case. If a locally finite thick building with parameters $q_s,q_t,q_u$ exists, then $R_{st}(o)$ is a generalised $8$-gon with parameters $(q_s,q_t)$, and $R_{tu}(o)$ and $R_{us}(o)$ are generalised $6$-gons with parameters $(q_t,q_u)$ and $(q_u,q_s)$ (respectively). This implies that $\sqrt{2q_sq_t}\in\mathbb{Z}$, and $\sqrt{q_tq_u},\sqrt{q_uq_s}\in\mathbb{Z}$, a contradiction. The remaining cases are similar.
We now show that there exist locally finite thick regular buildings for each of the remaining $24$ infinite triangle Coxeter systems, and moreover, that for each of these triangle Coxeter systems there are infinitely many buildings. For this we recall some known examples of generalised $m$-gons (see [@HVM] for details). If $m=2,3,4$ or $6$ then there is a generalised $m$-gon with parameters $(q,q)$ for each prime power $q$. Thus if $m_{st},m_{tu},m_{us}\in\{2,3,4,6\}$ we can take generalised $m$-gons with parameters $(q,q)$ as the basic building blocks, verifying the claim in this case. The cases where at least one of the $m$’s is $8$ require a little more care. We recall that there are generalised $4$-gons with parameters $(q,q^2)$ and $(q^2,q)$ for each prime power $q$, and that for each $r=2^{2k+1}$ there are generalised $8$-gons with parameters $(r,r^2)$ and $(r^2,r)$ (in fact, these are the only known examples of finite thick generalised $8$-gons). For example, consider the $(8,6,4)$ triangle group. For each $r=2^{2k+1}$ there exists a generalised $8$-gon with parameters $(r^2,r)$, a generalised $6$-gon with parameters $(r,r)$, and a generalised $4$-gon with parameters $(r,r^2)$, and so there is a thick regular triangle building of type $(W,S)$ with parameters $(r^2,r,r)$. By varying $k$ we obtain infinitely many buildings (pairwise non-isomorphic because they have different thicknesses). The remaining examples are similar.
Similar ideas show that there are infinitely many Fuchsian Coxeter systems $(W,S)$ with $|S|\geq 4$ for which locally finite thick regular buildings of type $(W,S)$ exist, and therefore the class of Fuchsian buildings is reassuringly rather large.
Apartments and retractions {#sect:aptret}
--------------------------
Let $(\Delta,\delta)$ be a building of type $(W,S)$. The thin sub-buildings of $(\Delta,\delta)$ of type $(W,S)$ are called the *apartments* of $(\Delta,\delta)$. Thus each apartment is isomorphic to the Coxeter complex of $(W,S)$. Two key facts concerning apartments are as follows:
1. If $x,y\in\Delta$ then there is an apartment $A$ containing both $x$ and $y$.
2. If $A$ and $A'$ are apartments containing a common chamber $x$ then there is a unique isomorphism $\theta:A'\to A$ fixing each chamber of the intersection $A\cap A'$.
In fact conditions (A1) and (A2) can be taken as an alternative, equivalent definition of buildings (see [@AB Definition 4.1] for the precise statement, and [@AB Theorem 5.91] for the equivalence of the two axiomatic systems).
Given chambers $x,y\in\Delta$, the *convex hull* $[x,y]$ of $x$ and $y$ is the union of all chambers on minimal length galleries from $x$ to $y$. That is, $
[x,y]=\{z\in\Delta\mid d(x,y)=d(x,z)+d(z,y)\}.
$ Another useful fact about apartments is:
1. If $A$ is an apartment containing $x$ and $y$ then $[x,y]\subseteq A$.
In fact, if $\Delta$ is thick then $[x,y]$ is the intersection of all apartments $A$ containing $x$ and $y$.
The hyperbolic realisation of each apartment of a Fuchsian building is a tesselation of the hyperbolic disc, as in Figure \[fig:triangle\](b) and (c). Roughly speaking, the properties (A1) and (A2) ensure that the hyperbolic metric on each apartment can be coherently ‘glued together’ to make $(\Delta,\delta)$ a $\mathrm{CAT}(-1)$ space (see [@davis Theorem 18.3.9] for details).
*Retractions* play an important role in building theory, and indeed in this current work. Let $A$ be an apartment, and let $x$ be a chamber of $A$. The *retraction $\rho_{A,x}$ of $\Delta$ onto $A$ with centre $x$* is defined as follows: For each chamber $y\in\Delta$, $$\rho_{A,x}(y)=z,\quad\textrm{where $z$ is the unique chamber of $A$ with $\delta(x,z)=\delta(x,y)$}.$$ Alternatively, let $A'$ be any apartment containing $x$ and $y$ (using (A1)) and let $\theta:A'\to A$ be the isomorphism from (A2) fixing $A\cap A'$. Then $$\rho_{A,x}(y)=\theta(y).$$ Thus $\rho_{A,x}:\Delta\to A$ “radially flattens” the building onto $A$, with centre $x\in A$.
Fix, once and for all, an apartment $A_0$ and a chamber $o\in A_0$. Canonically identify $A_0$ with the Coxeter complex of $(W,S)$ such that $o$ is identified with $1$, the neutral element of $W$. Thus we regard $W=A_0$ as a “base apartment” of $\Delta$. To simplify notation, we write $
\rho=\rho_{W,o}
$ for the retraction of $\Delta$ onto the apartment $W$ with centre $o$. Thus $$\begin{aligned}
\label{eq:canonicalretraction}
\rho:\Delta\to W\quad\textrm{is given by}\quad \rho(x)=\delta(o,x).\end{aligned}$$ We also note that in the apartment $A_0=W$ the Weyl distance function is given by $$\delta(u,v)=u^{-1}v\quad\text{for all $u,v$ in the base apartment~$W$}.$$
Automata for Coxeter groups and buildings {#sect:3}
=========================================
The notions of *cones*, *cone types*, and *automata* are well established for finitely generated groups, with [@cannon] being a standard reference. Let us briefly recall these notions in the specific context of Coxeter groups, and then extend the ideas into the (non-group) realm of buildings.
Let $(W,S)$ be a Coxeter system. Let $w\in W$. The *cone of $(W,S)$ with root $w$* is the set $$C_W(w)=\{v\in W\mid d(1,v)=d(1,w)+d(w,v)\}.$$ Thus $C_W(w)$ is the set of all elements $v\in W$ such that there exists a geodesic from $1$ to $v$ passing through $w$. The *cone type* of the cone $C_W(w)$ is $$T_W(w)=\{w^{-1}v\mid v\in C(w)\}=w^{-1}C_W(w).$$
Let $\mathcal{T}(W,S)$ be the set of cone types of $(W,S)$. By [@howlett Theorem 2.8] there are only finitely many cone types in a Coxeter system $(W,S)$, and so $|\mathcal{T}(W,S)|<\infty$.
\[defn:cannon\] The *Cannon automaton* of the Coxeter system $(W,S)$ is the directed graph $\mathcal{A}(W,S)$ with vertex set $\mathcal{T}(W,S)$ and with labelled edges defined as follows. There is a directed edge with label $s\in S$ from cone type ${\mathbf{T}}$ to cone type ${\mathbf{T}}'$ if and only if there exists $w\in W$ such that ${\mathbf{T}}=T_W(w)$ and ${\mathbf{T}}'=T_W(ws)$ and $d(1,ws)=d(1,w)+1$.
A cone type ${\mathbf{T}}'$ is *accessible* from the cone type ${\mathbf{T}}$ if there is a path from ${\mathbf{T}}$ to ${\mathbf{T}}'$ in the (directed) graph $\cA(W,S)$. In this case we write ${\mathbf{T}}\to {\mathbf{T}}'$. A cone type ${\mathbf{T}}$ is called *recurrent* if ${\mathbf{T}}\to{\mathbf{T}}$, and otherwise it is called *transient*. The set of recurrent vertices induces a (directed) subgraph $\cA_R(W,S)$ of $\cA(W,S)$. We call the automaton $\cA(W,S)$ *strongly connected* if each recurrent cone type is accessible from any other recurrent cone type in the subgraph $\cA_R(W,S)$.
\[scale=1.3\] (0,0) node \[shape=rectangle,draw\] (1,-1) node \[shape=rectangle,draw\] (2,-2) node \[shape=rectangle,draw\] (1,-3) node \[shape=rectangle,draw\] (0,-3) node \[shape=rectangle,draw\] (-1,-3) node \[shape=rectangle,draw\] (-2,-2) node \[shape=rectangle,draw\] (-1,-1) node \[shape=rectangle,draw\] (2,0) node \[shape=rectangle,draw\] (2,1) node \[shape=rectangle,draw\] (1,2) node \[shape=rectangle,draw\] (3,2) node \[shape=rectangle,draw\] (0,1) node \[shape=rectangle,draw\] (-1,2) node \[shape=rectangle,draw\] (-2,1) node \[shape=rectangle,draw\] (-2,0) node \[shape=rectangle,draw\] (-3,2) node \[shape=rectangle,draw\] (0,-4) node \[shape=rectangle,draw\] ; (0,-3.2) – (0,-3.8); (-1.2,1.8) – (-1.8,1.2); (1.2,1.8) – (1.8,1.2); (-1.2,-0.8) – (-1.8,-0.2); (1.2,-0.8) – (1.8,-0.2); (-1.8,-1.8) – (1.7,-0.1); (1.8,-1.8) – (-1.7,-0.1); (-0.9,-2.8) – (-2,-0.2); (0.9,-2.8) – (2,-0.2); (0,0.2) – (0,0.8); (-2.8,1.8) – (1.7,1.1); (2.8,1.8) – (-1.7,1.1); (0.2,-0.2) – (0.8,-0.8); (-1.2,-1.2) – (-1.9,-1.8); (-0.7,-3) – (-0.35,-3); (1.8,-2.2) – (1.2,-2.8); (0.2,1.2) – (0.8,1.8); (2,0.2) – (2,0.8); (-2.2,1.2) – (-3,1.8); (-1,1.8) – (-2,-1.8); (3.1,1.8) – (1.1,-2.8); (0.4,-4) – (2,-3.5) – (3,-1) – (2.2,0.8); (-1.7,0.1) – (0.75,2); (-0.2,-0.2) – (-0.8,-0.8); (1.2,-1.2) – (1.9,-1.8); (0.7,-3) – (0.35,-3); (-1.8,-2.2) – (-1.2,-2.8); (-0.2,1.2) – (-0.8,1.8); (-2,0.2) – (-2,0.8); (2.2,1.2) – (3,1.8); (1,1.8) – (2,-1.8); (-3.1,1.8) – (-1.1,-2.8); (-0.4,-4) – (-2,-3.5) – (-3,-1) – (-2.2,0.8); (1.7,0.1) – (-0.75,2);
Figure \[fig:automata\] shows the Cannon automaton for the (Fuchsian) triangle group $W_{(3,3,4)}$ (see Appendix \[app:A\] for details). The generators are labelled $1$, $2$, and $3$, and the labels on the edges are indicated by colours (green, blue, red respectively). The cone types are given by the base element of a representative cone of that type. Thus the vertex $131$ is the cone type $T(131)$. All cone types, except for $\emptyset$, $1$, $2$, and $3$, are recurrent. This automaton is strongly connected. For example, the sequence $121\to 1212\to 12123\to 232\to 2321\to 212\to 23$ shows that $121\to 23$.
The existence of a strongly connected Cannon automaton is important for our renewal theory arguments in Section \[sect:renewal\], thus in Appendix \[app:A\] we prove:
\[thm:stronglyconnected\] The Cannon automaton of a Fuchsian Coxeter system is strongly connected.
It does not appear to be known in the literature which Coxeter systems have strongly connected automata. For example, our direct calculations in Appendix \[app:A\] show that affine triangle groups do not have strongly connected Cannon automata, and we suspect that no affine Coxeter group has a strongly connected Cannon automaton.
We now extend the above concepts to buildings. Let $(\Delta,\delta)$ be a building of type $(W,S)$ with fixed base chamber $o$. Let $x\in \Delta$ be a chamber. The *cone of $(\Delta,\delta)$ with root $x$* is the set $$C_{\Delta}(x)=\{y\in \Delta\mid d(o,y)=d(o,x)+d(x,y)\}.$$ Thus $C_{\Delta}(x)$ is the set of all chambers $y\in \Delta$ such that there exists a geodesic from $o$ to $y$ passing through $x$. The *cone type* of the cone $C_{\Delta}(x)$ is $$T_{\Delta}(x)=\{\delta(x,y)\mid y\in C_{\Delta}(x)\}.$$
If $A$ is an apartment of $\Delta$ containing $o$ and $x\in A$ we write $$C_A(x)=\{y\in A\mid d(o,y)=d(o,x)+d(x,y)\}.$$ We collect together some useful facts about cones and cone types in buildings, and the connection with cones and cone types in Coxeter systems. Recall the definition of the canonical retraction $\rho:\Delta\to W$ from (\[eq:canonicalretraction\]).
\[prop:conetypes\] Let $(\Delta,\delta)$ be a building of type $(W,S)$.
1. If $A$ is an apartment containing the chambers $o$ and $x$ then the isomorphism $\rho|_A:A\to W$ maps $C_A(x)$ onto $C_W(\rho(x))$.
2. $\rho(C_{\Delta}(x))=C_W(\rho(x))$ for all $x\in \Delta$.
3. $T_{\Delta}(x)=T_W(\rho(x))$ for all $x\in\Delta$.
4. $\rho^{-1}(C_W(w))=\bigsqcup_{x\in\Delta_w(o)}C_{\Delta}(x)$ for all $w\in W$.
If $A$ is an apartment containing $o$ and $x$ then the restriction $\rho|_A:A\to W$ is an isomorphism. Thus $\rho|_A$ and $\rho|_A^{-1}$ map minimal galleries to minimal galleries, and hence part $1$ follows.
Next we claim that $C_{\Delta}(x)=\bigcup_{A}C_A(x)$ where the union is over all apartments $A$ containing $o$ and $x$. It is clear that $C_A(x)\subseteq C_{\Delta}(x)$ for each apartment $A$ containing $o$ and $x$, and thus $\bigcup_{A}C_A(x)\subseteq C_{\Delta}(x)$. On the other hand, suppose that $y\in C_{\Delta}(x)$. Let $A$ be an apartment containing $o$ and $y$. Then $A$ contains $x$ by (A3), and so $y\in C_A(x)$, completing the proof of the claim. Part $2$ follows using part $1$, since $
\rho(C_{\Delta}(x))=\bigcup_{A}\rho(C_A(x))=C_W(\rho(x)).
$
To prove part $3$, note that by part $2$, $$\begin{aligned}
T_W(\rho(x))&=\rho(x)^{-1}C_W(\rho(x))=\rho(x)^{-1}\rho(C_{\Delta}(x))=\rho(x)^{-1}\{\rho(y)\mid y\in C_{\Delta}(x)\}.\end{aligned}$$ If $y\in C_{\Delta}(x)$ then $\delta(o,y)=\delta(o,x)\delta(x,y)$. Thus $\rho(y)=\rho(x)\delta(x,y)$, and so $
T_W(\rho(x))=T_{\Delta}(x)
$.
From part $2$ it is immediate that $\rho^{-1}(C_W(w))=\bigcup_{x\in\Delta_w(o)}C_{\Delta}(x)$ for all $w\in W$. To see that the union is disjoint, suppose that $y\in C_{\Delta}(x)\cap C_{\Delta}(x')$ with $x,x'\in\Delta_w(o)$. Let $A$ be an apartment of $\Delta$ containing $o$ and $y$. Since $x$ and $x'$ are both on minimal galleries from $o$ to $y$, (A3) implies that $x,x'\in A$. Since $\rho|_A:A\to W$ is an isomorphism, and since $\rho(x)=w=\rho(x')$, we have $x=x'$.
We make a completely analogous definition to Definition \[defn:cannon\] for the Cannon automaton $\cA(\Delta,\delta)$ of a building $(\Delta,\delta)$.
\[defn:cannon2\] Let $(\Delta,\delta)$ be a building of type $(W,S)$. The *Cannon automaton* of $(\Delta,\delta)$ is the directed graph $\mathcal{A}(\Delta,\delta)$ with vertex set $\mathcal{T}(\Delta,\delta)$ and with labelled edges defined as follows. There is a directed edge with label $s\in S$ from cone type ${\mathbf{T}}$ to cone type ${\mathbf{T}}'$ if and only if there exists $x\in \Delta$ and $y\in\Delta_s(x)$ such that ${\mathbf{T}}=T_{\Delta}(x)$ and ${\mathbf{T}}'=T_{\Delta}(y)$ and $d(o,y)=d(o,x)+1$.
Let $(\Delta,\delta)$ be a building of type $(W,S)$. Then $\cA(\Delta,\delta)\cong\cA(W,S)$.
By Proposition \[prop:conetypes\] there is a bijection between the vertex sets of $\cA(\Delta,\delta)$ and $\cA(W,S)$, and it is elementary to check that this bijection preserves labelled oriented edges.
For the remainder of this paper, when it is clear from context we will typically write $C(\cdot)$ and $T(\cdot)$ for cones and cone types in either Coxeter groups or buildings.
The *boundary* of a cone $C$ of $(\Delta,\delta)$ is $$\partial C=\{y\in C\mid \text{ there exists $z\in \Delta\setminus C$ with $d(y,z)=1$}\}.$$ If $L\geq 1$, the *$L$-boundary* of a cone $C$ of $(\Delta,\delta)$ is defined to be $$\begin{aligned}
\label{eq:boundaryofcone}
\partial_L C=\{y\in C\mid \text{there exists $z\in \Delta\setminus C$ with $d(y,z)\leq L$}\}.\end{aligned}$$ In particular, $\partial_1C=\partial C$. We call $\mathrm{Int}_{L} C=C\setminus \partial_{L} C$ the *$L$-interior* of $C$. We make analogous definitions for the boundary, $L$-boundary and $L$-interior of a cone $C$ of $(W,S)$.
The *$L$-boundary of a cone type* ${\mathbf{T}}$ (of $\Delta$ or $W$) is defined by $$\partial_L{\mathbf{T}}=\{w\in{\mathbf{T}}\mid\text{ there exists $v\in W\setminus{\mathbf{T}}$ with $d(w,v)\leq L$}\},$$ and the *$L$-interior of the cone type* ${\mathbf{T}}$ is $\mathrm{Int}_L {\mathbf{T}}={\mathbf{T}}\setminus \partial_L{\mathbf{T}}$.
\[lem:onestep\] Let $x\in\Delta$ and $y\in C_{\Delta}(x)$. If there is a chamber $z\in\Delta$ with $d(y,z)=1$ and $z\notin C_{\Delta}(x)$ then there is a chamber $z'\in\Delta$ with $d(y,z')=1$ and $\rho(z')\notin C_W(\rho(x))$.
Since $d(y,z)=1$ we have $\delta(y,z)=s$ for some $s\in S$. If $\ell(\delta(o,y)s)=\ell(\delta(o,y))+1$ then every minimal gallery from $o$ to $y$ can be extended to a minimal gallery from $o$ to $z$, and thus since $y\in C_{\Delta}(x)$ there is a minimal gallery from $o$ to $z$ passing through $x$, a contradiction.
Thus $\ell(\delta(o,y)s)=\ell(\delta(o,y))-1$. Let $A$ be an apartment containing $o$ and $y$, and hence $A$ contains $x$ by (A3). Let $z'\in A$ be the unique chamber of $A$ with $\delta(y,z')=s$. We claim that $\rho(z')\notin C_W(\rho(x))$. Suppose, for a contradiction, that $\rho(z')\in C_W(\rho(x))$. Then part 1 of Proposition \[prop:conetypes\] gives $z'\in C_A(x)$, and hence $z'\in C_{\Delta}(x)$. In particular $z'\neq z$ and so since both $z$ and $z'$ are $s$-adjacent to $y$ we have $\delta(z',z)=s$. Since $z'\in C_{\Delta}(x)$ there is a minimal gallery from $o$ to $z'$ passing through $x$, and since $\delta(o,z')=\delta(o,y)s$ we have $\ell(\delta(o,z')s)=\ell(\delta(o,z'))+1$ and so we can extend this minimal gallery to give a minimal gallery from $o$ to $z$ passing through $x$. Thus $z\in C_{\Delta}(x)$, a contradiction, and so $\rho(z')\notin C_W(\rho(x))$.
\[prop:Lboundaries\] Let $(\Delta,\delta)$ be a building of type $(W,S)$. Then for each $x\in \Delta$ and each $L\geq 1$ we have $$\rho(\partial_LC_{\Delta}(x))=\partial_LC_{W}(\rho(x)).$$
Suppose that $v\in \partial_LC_W(\rho(x))$. Thus there is an element $v'\in W$ with $d(v,v')\leq L$ and $v'\notin C_W(\rho(x))$. Choose any apartment $A$ containing $o$ and $x$. Let $y$ be the unique chamber of $A$ with $\delta(o,y)=v$, and let $y'$ be the unique chamber of $A$ with $\delta(o,y')=v'$. Since $\rho(y)=v$ and $\rho(y')=v'$ and since $\rho|_A:A\to W$ is an isomorphism we have $d(y,y')=d(v,v')$. Moreover, from part $1$ of Proposition \[prop:conetypes\] we have $y'\notin C_A(x)$ and it follows, using (A3), that $y'\notin C_{\Delta}(x)$. Thus $y\in\partial_L C_{\Delta}(x)$ and so $v=\rho(y)\in\rho(\partial_LC_{\Delta}(x))$, giving $\partial_L C_W(\rho(x))\subseteq \rho(\partial_LC_{\Delta}(x))$.
Suppose that $y\in \partial_LC_{\Delta}(x)$, and so there is a chamber $z$ with $d(y,z)\leq L$ such that $z\notin C_{\Delta}(x)$. Choose this chamber $z$ with $d(y,z)$ minimal, and let $y=y_0\sim y_1\sim\cdots\sim y_{k-1}\sim y_k=z$ be a minimal length gallery from $y$ to $z$. By minimality of $d(y,z)$ we have that $y_{k-1}\in C_{\Delta}(x)$. Since $z\notin C_{\Delta}(x)$ Lemma \[lem:onestep\] implies that there is a chamber $z'$ adjacent to $y_{k-1}$ such that $\rho(z')\notin C_W(\rho(x))$. Since $d(\rho(y),\rho(z'))\leq d(y,z')=d(y,z)\leq L$ we have $\rho(y)\in\partial_LC_W(\rho(x))$, and hence $\rho(\partial_LC_{\Delta}(x))\subseteq \partial_LC_W(\rho(x))$.
\[rem:noniso\] In the traditional setup of cones in groups, two cones with the same cone type are necessarily isomorphic since there is a group element taking one cone to the other. In the context of buildings the situation is quite different, for it follows from Ronan’s free construction [@ronanconstruction] of buildings with no rank $3$ residues of spherical type that two cones in $\Delta$ of the same type are not necessarily isomorphic as graphs. In fact one can construct buildings in which there are *infinitely many* pairwise non-isomorphic cones of a fixed type. However we note that Proposition \[prop:conetypes\] still guarantees that there will be only finitely many distinct cone types for the building.
Isotropic random walks on regular buildings {#sect:4}
===========================================
In this section we investigate the structure of isotropic random walks in the general context of a regular building (not necessarily Fuchsian).
Definitions and transition operators {#subsec:prelim}
------------------------------------
We will henceforth write $(\Delta,\delta)$ for a thick regular building of type $(W,S)$. A random walk $(X_n)_{n\geq 0}$ on the set $\Delta$ of chambers of the building $(\Delta,\delta)$ is *isotropic* if the transition probabilities of the walk satisfy $$p(x,y)=p(x',y')\quad\textrm{whenever $\delta(x,y)=\delta(x',y')$}.$$ In other words, the probability of jumping from $x$ to $y$ in one step depends only on the Weyl distance $\delta(x,y)$. Thus an isotropic random walk is determined by the probabilities $$\begin{aligned}
\label{eq:transitionprob}
p_w=\mathbb{P}[X_1\in \Delta_w(x)\mid X_0=x],\quad\textrm{so that}\quad p(x,y)=p_w/q_{w}\quad\textrm{if $\delta(x,y)=w$},\end{aligned}$$ and the transition operator of an isotropic random walk $(X_n)_{n\geq 0}$ on $\Delta$ with governing probabilities (\[eq:transitionprob\]) is given by $$\begin{aligned}
\label{eq:transitionop}
P=\sum_{w\in W}p_wP_w,\end{aligned}$$ where for each $w\in W$, the operator $P_w$ acts on the space of all functions $f:\Delta\to\mathbb{C}$ by $$P_wf(x)=\frac{1}{q_w}\sum_{y\in\Delta_w(x)}f(y).$$
For each $n\geq 0$ let $$p^{(n)}(x,y)=\mathbb{P}[X_n=y\mid X_0=x].$$ Then $P^n=\sum_{w\in W}p_w^{(n)}P_w$, where $p^{(n)}(x,y)=p^{(n)}_w/q_w$ whenever $\delta(x,y)=w$.
The random walk $(X_n)_{n\geq 0}$ is *irreducible* if for every pair $x,y\in\Delta$ there is an integer $n\geq 1$ such that $p^{(n)}(x,y)>0$. The spectral radius of an irreducible random walk $(X_n)_{n\geq 0}$ with transition operator $P$ is $$\varrho(P)=\limsup_{n\to\infty}p^{(n)}(x,y)$$ (by irreducibility this value does not depend on the pair $x,y\in\Delta$).
We will assume that the random walk has bounded range (although most of this section only requires a finite first moment assumption). Let $L_0=\max\{\ell(w)\mid p_w>0\}$, and so the largest possible jump of the random walk has length $L_0$.
There is a beautiful algebraic structure underlying isotropic random walks. In particular the geometry of the building implies that (see [@P1 Theorem 3.4]) $$\begin{aligned}
\label{eq:algebra}
\begin{aligned}
P_wP_s=\begin{cases}P_{ws}&\textrm{if $\ell(ws)=\ell(w)+1$}\\
q_s^{-1}P_{ws}+(1-q_s^{-1})P_w&\textrm{if $\ell(ws)=\ell(w)-1$},
\end{cases}
\end{aligned}\end{aligned}$$ from which it immediately follows that the vector space $\mathscr{A}$ over $\mathbb{C}$ with basis $\{P_w\mid w\in W\}$ is an algebra under composition (called the *Hecke algebra* of the building, cf. [@P1]). The transition operator $P$ of a bounded range isotropic random walk is an element of the Hecke algebera $\mathscr{A}$.
The following interpretation of the structure constants in the Hecke algebra, and the “distance regularity” statement (\[eq:distreg\]) that follows from this interpretation, will be crucial to our investigations.
\[prop:distanceregular\] Let $(\Delta,\delta)$ be a regular locally finite building of type $(W,S)$ and let $u,v\in W$. Then $$P_uP_v=\sum_{w\in W}\alpha_{u,v}^wP_w,\quad\textrm{where}\quad \alpha_{u,v}^w=\frac{q_w}{q_uq_v}|\Delta_u(x)\cap \Delta_{v^{-1}}(y)|$$ for any pair of chambers $x,y\in\Delta$ with $\delta(x,y)=w$. In particular, the numbers $$\begin{aligned}
\label{eq:distreg}
a_{u,v}^w=|\Delta_u(x)\cap \Delta_{v}(y)|\quad\textrm{with $\delta(x,y)=w$}\end{aligned}$$ do not depend on the particular pair $x,y\in\Delta$ with $\delta(x,y)=w$.
Since $\mathscr{A}$ is an algebra, we have $P_uP_v=\sum_w \alpha_{u,v}^wP_w$ for some numbers $\alpha_{u,v}^w\in\mathbb{C}$. Let $y\in\Delta$, and let $\delta_y:\Delta\to\mathbb{C}$ be the Kronecker delta function. Then $P_w\delta_y(x)=q_w^{-1}$ if $y\in\Delta_w(x)$ and $0$ otherwise, and a direct calculation shows that $P_uP_v\delta_y(x)=q_u^{-1}q_v^{-1}|\Delta_u(x)\cap\Delta_{v^{-1}}(y)|$, completing the proof (see also [@P1 Proposition 3.9]).
\[lem:add1\] If $\alpha_{u,v}^w\neq 0$ then $w=uv'$ for some $v'\in W$ with $\ell(v')\leq \ell(v)$.
We prove the lemma by induction on $\ell(v)$, with the base case $\ell(v)=0$ being trivial. Suppose that the result is true for $\ell(v)=k$, and let $s\in S$ with $\ell(vs)=\ell(v)+1$. Then by (\[eq:algebra\]) and the induction hypothesis we have $$P_uP_{vs}=P_uP_vP_s=(P_uP_v)P_s=\sum_{z\in W\,:\,\ell(z)\leq \ell(v)}\alpha_{u,v}^{uz}P_{uz}P_s.$$ By (\[eq:algebra\]) we have either $P_{uz}P_s=P_{uzs}$ (in the case that $\ell(uzs)=\ell(uz)+1$), or $P_{uz}P_s=q_s^{-1}P_{uzs}+(1-q_s^{-1})P_{uz}$ (in the case that $\ell(uzs)=\ell(uz)-1$). Since $\ell(z)\leq \ell(v)<\ell(vs)$ and $\ell(zs)\leq\ell(z)+1\leq \ell(v)+1=\ell(vs)$ we see that $P_uP_{vs}$ is a linear combination of the operators $P_{uz'}$ with $\ell(z')\leq \ell(vs)$, hence the result.
Irreducibility and aperiodicity
-------------------------------
Let $P=\sum_{w\in W}p_wP_w$ be the transition operator of an isotropic random walk $(X_n)_{n\geq 0}$ on $\Delta$. The *support* of $P$ is $\mathrm{supp}(P)=\{w\in W\mid p_w>0\}$.
\[lem:irreducible\] Let $(X_n)_{n\geq 0}$ be an isotropic random walk on a thick regular building with transition operator $P$ as in (\[eq:transitionop\]), and write $P^n=\sum_wp_w^{(n)}P_w$.
1. If the support of $P$ generates $W$ then $(X_n)_{n\geq 0}$ is irreducible.
2. If $(X_n)_{n\geq 0}$ is irreducible then for each $k>0$ there is $M_k>0$ such that $p_w^{(M_k)}>0$ for all $w\in W$ with $\ell(w)\leq k$.
3. If $(X_n)_{n\geq 0}$ is irreducible, then $(X_n)_{n\geq 0}$ is aperiodic.
1\. Let $x,y\in \Delta$, and let $A$ be an apartment containing $x$ and $y$. Since the support of $P$ generates $W$ there are elements $w_1,\ldots,w_n\in
\mathrm{supp}(P)$ such that $\delta(x,y)=w_1w_2\cdots w_n$. Let $x_0=x$ and let $x_1,\ldots,x_n\in A$ be the unique chambers of the apartment $A$ with $\delta(x,x_k)=w_1\cdots w_k$ for $k=1,\ldots,n$. In particular, $x_n=y$. Then, since $x_0,x_1,\ldots,x_k$ all lie in the apartment $A$, we have $\delta(x_{k-1},x_k)=\delta(x,x_{k-1})^{-1}\delta(x,x_k)=w_k$. Thus $p(x_{k-1},x_k)=p_{w_k}/q_{w_k}>0$, and so $$p^{(n)}(x,y)\geq p(x,x_1)p(x_1,x_2)\cdots p(x_{n-1},y)>0,$$ showing that $P$ is irreducible.
2\. Suppose that $(X_n)_{n\geq 0}$ is irreducible. Thus for each $s\in S$ there is $N_s\geq 1$ such that $p_{s}^{(N_s)}>0$. The formula $P_{s}^2=q_{s}^{-1}I+(1-q_s^{-1})P_{s}$ from (\[eq:algebra\]) implies that $p_1^{(2N_s)}>0$ and $p_{s}^{(2N_s)}>0$. Thus setting $N=2\sum_{s\in S} N_s$ we have $p_1^{(N)}>0$ and $p_s^{(N)}>0$ for all $s\in S$. Thus taking $M_k=kN$ gives $p_w^{(M_k)}>0$ for all $w\in W$ with $\ell(w)\leq k$.
3\. Suppose that $p_w>0$, and let $k=\ell(w)$. By the previous part we have $p_{w^{-1}}^{(M_k)}>0$. If $w=s_1\cdots s_{k}$ is reduced, then using (\[eq:algebra\]) we have $$\begin{aligned}
P_wP_{w^{-1}}&=P_{s_1}\cdots P_{s_{k}}P_{s_{k}}\cdots P_{s_1}=q_w^{-1}I+\cdots,\end{aligned}$$ where “$+\cdots$” is a nonnegative linear combination of the $P_v$ with $v\in W$. Therefore $$P^{M_k+1}=PP^{M_k}=p_wp_{w^{-1}}^{(M_k)}P_wP_{w^{-1}}+\cdots=q_w^{-1}p_wp_{w^{-1}}^{(M_k)}I+\cdots,$$ and so $p_1^{(M_k+1)}>0$. Since we also have $p_1^{(M_k)}>0$ the walk is aperiodic.
If $(X_n)_{n\geq 0}$ is irreducible then it is not necessarily true that $\{w\in W\mid p_w>0\}$ generates $W$. For example if $p_w>0$ if and only if $\ell(w)=2$ then the random walk $(X_n)_{n\geq 0}$ is irreducible, yet $\{w\in
W \mid \ell(w)=2\}$ only generates the index $2$ subgroup of all even length elements of $W$.
The retracted walk
------------------
An indispensable technique in our analysis of isotropic random walks $(X_n)_{n\geq 0}$ on $(\Delta,\delta)$ is to look at the image $\overline{X}_n=\rho(X_n)$ of the random walk under the canonical retraction $\rho:\Delta\to W$. In Proposition \[prop:project\] below we show that the stochastic process $(\overline{X}_n)_{n\geq 0}$ on $W$ is in fact a random walk on $W$, which we call the *retracted walk*. However we note in advance that the retracted walk is not $W$-invariant. That is, $\overline{p}(wu,wv)\neq \overline{p}(u,v)$ in general. However we we will prove a more delicate invariance property in Proposition \[prop:invariance\] later in this section.
\[prop:project\] The isotropic random walk $(X_n)_{n\geq 0}$ is factorisable over $W$ with respect to the partition of $\Delta$ into sets $\Delta_w(o)$ with $w\in W$. Moreover, the transition probabilities $\overline{p}(u,v)$ of the factor walk $(\overline{X}_n)_{n\geq 0}$ (where $\overline{X}_n=\rho(X_n)$) on $W$ are given by $$\overline{p}(u,v)=\sum_{w\in W}a_{v,w}^uq_w^{-1}p_w=q_u^{-1}q_v\sum_{w\in W}\alpha_{v,w^{-1}}^up_w,$$ where $a_{v,w}^u\geq 0$ and $\alpha_{v,w^{-1}}^u\geq 0$ are the numbers appearing in Proposition \[prop:distanceregular\].
Let $u,v\in W$, and let $x\in\Delta_u(o)$. Then by Proposition \[prop:distanceregular\], $$\begin{aligned}
\sum_{y\in\Delta_v(o)}p(x,y)&=\sum_{w\in W}\sum_{y\in\Delta_v(o)\cap \Delta_w(x)}p(x,y)=\sum_{w\in W}|\Delta_v(o)\cap \Delta_w(x)|q_w^{-1}p_w=\sum_{w\in W}a_{v,w}^uq_w^{-1}p_w.\end{aligned}$$ This proves the first equality, and the final equality follows from the definitions of the numbers $a_{v,w}^u$ and $\alpha_{v,w^{-1}}^u$.
The following proposition tells us that the return probabilities for the random walk $(X_n)_{n\geq 0}$ can be obtained from the return probabilities for the retracted walk $(\overline{X}_n)_{n\geq 0}$.
\[prop:retractedwalkspectralradius\] Let $P$ be an irreducible isotropic random walk on a regular building $(\Delta,\delta)$ of type $(W,S)$, and let $\overline{P}$ be the transition operator of the retracted walk on $(W,S)$. Then $$p^{(n)}(o,o)=\overline{p}^{(n)}(1,1)\qquad\textrm{for all $n\geq 1$},$$ and thus $\vrho(\overline{P})=\vrho(P)$.
From Proposition \[prop:project\] (applied to $P^n$) we have $$\overline{p}^{(n)}(1,1)=\sum_{w\in W}a_{1,w}^1q_w^{-1}p_w^{(n)},$$ and since $a_{1,w}^1=|\Delta_1(o)\cap\Delta_w(o)|=\delta_{w,1}$ we have $\overline{p}^{(n)}(1,1)=p_1^{(n)}=p^{(n)}(o,o)$. Since $P$ and $\overline{P}$ are irreducible it follows that $\varrho(P)=\limsup_{n\to\infty}p^{(n)}(o,o)=\limsup_{n\to\infty}\overline{p}^{(n)}(1,1)=\varrho(\overline{P})$.
The retracted walk is not $W$-invariant, however we have the following weaker invariance property which roughly says that the transition probabilities of the retracted walk in two cones of the same type are the same.
\[prop:invariance\] Let ${\mathbf{T}}$ be a cone type of $(W,S)$ and $w_{1},w_{2}\in W$ with $T(w_1)=T(w_2)={\mathbf{T}}$. Then $$\overline{p}(w_1u,w_1v)=\overline{p}(w_2u,w_2v)\quad\textrm{for all $u\in {\mathbf{T}}$ and all $v\in \mathrm{Int}_{L_0} {\mathbf{T}}={\mathbf{T}}\setminus\partial_{L_0}{\mathbf{T}}$}.$$
By the formula for $\overline{p}(\cdot,\cdot)$ in Proposition \[prop:project\] it is sufficient to show that $$q_{w_1u}^{-1}q_{w_1v}\alpha_{w_1v,w^{-1}}^{w_1u}=q_{w_2u}^{-1}q_{w_2v}\alpha_{w_2v,w^{-1}}^{w_2u}\quad\textrm{whenever $w\in W$ is such that $p_w>0$}.$$ First note that since $u,v\in {\mathbf{T}}$ we have $\ell(w_iu)=\ell(w_i)+\ell(u)$ and $\ell(w_iv)=\ell(w_i)+\ell(v)$ for each $i=1,2$, and therefore $q_{w_iu}=q_{w_i}q_u$ and $q_{w_iv}=q_{w_i}q_v$ for each $i=1,2$. Therefore it is sufficient to show that $$\begin{aligned}
\label{eq:alphaeq}
\alpha_{w_1v,w}^{w_1u}=\alpha_{w_2v,w}^{w_2u}\quad\text{whenever $\ell(w)\leq L_0$}\end{aligned}$$ (we have replaced $w$ by $w^{-1}$, and noted that $p_{w^{-1}}>0$ implies that $\ell(w)\leq L_0$).
Since $\ell(w_1v)=\ell(w_1)+\ell(v)$ it follows from (\[eq:algebra\]) and Proposition \[prop:distanceregular\] that $$\begin{aligned}
P_{w_1v}P_{w}&=P_{w_1}P_vP_{w}=P_{w_1}\sum_{w'\in W}\alpha_{v,w}^{w'}P_{w'}=\sum_{w'\in W}\alpha_{v,w}^{w'}P_{w_1}P_{w'}.\end{aligned}$$ By Lemma \[lem:add1\] we see that if $\alpha_{v,w}^{w'}\neq 0$ then $w'=v\tilde{w}$ for some $\tilde{w}$ with $\ell(\tilde{w})\leq \ell(w)$, and therefore $$d(v,w')=\ell(v^{-1}w')=\ell(\tilde{w})\leq\ell(w)\leq L_0.$$ Thus $w'\in {\mathbf{T}}$ (since $v\in {\mathbf{T}}\setminus\partial_{L_0} {\mathbf{T}}$), and therefore $\ell(w_1w')=\ell(w_1)+\ell(w')$, giving $P_{w_1}P_{w'}=P_{w_1w'}$. Thus $$\begin{aligned}
\label{eq:psum1}
P_{w_1v}P_{w}=\sum_{w'\in W}\alpha_{v,w}^{w'}P_{w_1w'}.\end{aligned}$$ On the other hand we have $$\begin{aligned}
\label{eq:psum2}
P_{w_1v}P_{w}=\sum_{w''\in W}\alpha_{w_1v,w}^{w''}P_{w''}=\sum_{w'\in W}\alpha_{w_1v,w}^{w_1w'}P_{w_1w'}.\end{aligned}$$ Comparing (\[eq:psum1\]) and (\[eq:psum2\]) and using the linear independence of the operators gives $$\alpha_{w_1v,w}^{w_1w'}=\alpha_{v,w}^{w'}\qquad\textrm{for all $w,w'\in W$ with $\ell(w)\leq L_0$}.$$ The same formula holds with $w_1$ replaced by $w_2$, and (\[eq:alphaeq\]) follows by taking $w'=u$.
The spectral radius
-------------------
In the following theorem we give a sufficient condition for the spectral radius of an isotropic random walk on a regular building to have spectral radius strictly less than $1$.
Let $(W,S)$ be a Coxeter system, and let $\cF=\{I\subseteq S\mid \text{$W_I$ is finite}\}$. For each $w\in W$, let $R(w)=\{s\in S\mid \ell(ws)=\ell(w)-1\}$ be the *right descent set of $w$*. By [@AB Corollary 2.18] we have that $R(w)\in\cF$ for all $w\in W$.
\[thm:spectralradius\] Let $(W,S)$ be a Coxeter system with $W$ infinite and let $(\Delta,\delta)$ be a regular building of type $(W,S)$. Let $P$ be the transition operator of an irreducible isotropic random walk on $(\Delta,\delta)$. If $$\begin{aligned}
\label{eq:Iineq}
\sum_{s\in S\setminus I}q_s\geq |I|\quad\text{for all $I\in \cF$}\end{aligned}$$ then the spectral radius $\vrho(P)$ is strictly less than $1$. In particular, if $q_s\geq |S|-1$ for all $s\in S$ then $\vrho(P)<1$.
Suppose first that $P$ is the simple random walk on $\Delta$. Furthermore, suppose first that strict inequality holds in (\[eq:Iineq\]) for all $I\in \cF$, and let $C=\min_{I\in \cF}(\sum_{s\in S\setminus I}q_s-|I|)/Q>0$ where $Q=\sum_{s\in S}q_s$ is the total number of chambers adjacent to any given chamber. Let $x\in\Delta$ and $w=\rho(x)$. Let $I=R(w)\in\cF$. Let $Y_n=d(o,X_n)$. Then $$\begin{aligned}
\mathbb{E}[Y_{n+1}-Y_n\mid X_n=x]&=\mathbb{P}[Y_{n+1}-Y_n=1\mid X_n=x]-\mathbb{P}[Y_{n+1}-Y_n=-1\mid X_n=x]\\
&=\frac{\sum_{s\in S\setminus I}q_s-|I|}{Q}.\end{aligned}$$ Thus $\mathbb{E}[Y_{n+1}-Y_n\mid X_n]\geq C$, and so the sequence $Z_n=Y_n-Cn$ is a submartingale with respect to $(X_n)_{n\geq 0}$. We have $$p^{(n)}(o,o)=\mathbb{P}[Y_n=0\mid X_0=o]\leq \mathbb{P}[Z_n\leq -Cn\mid X_0=o]\leq e^{-C^2n/2}$$ where the last inequality is Azuma’s Inequality. Thus $\vrho(P)\leq e^{-C^2/2}<1$.
We now briefly sketch the proof in the more general case where we do not assume strict inequality in (\[eq:Iineq\]), with $P$ still the simple random walk on $\Delta$. Note that the singleton $I=\{s'\}$ is in $\cF$, and that $\sum_{s\neq s'}q_s>|I|$. It can be seen that there is a number $K>0$ such that for each chamber $x\in\Delta$ there is an element $x'$ with $d(x,x')\leq K$ such that $R(\rho(x'))=\{s'\}$. Using this fact, and looking at the $(K+1)$-step walk $P^{K+1}$, an argument analogous to the above, using a telescoping sum, shows that $\mathbb{E}[Y_{n+K+1}-Y_n\mid X_n]\geq C(1/Q)^{K+1}$, where $C=\sum_{s\neq s'}q_s-1>0$. The result now follows as above.
Now let $P=\sum_{w\in W} p_wP_w$ be an arbitrary isotropic random walk on $\Delta$. By Lemma \[lem:irreducible\] there is $N>0$ such that $p_s^{(N)}>0$ for all $s\in S$. Thus, writing $\tilde{P}$ for the simple random walk operator on $\Delta$, we have $$P^N=b\tilde{P}+\sum_{w\in W}b_wP_w\qquad\textrm{where $b>0$ and $b_w\geq 0$ for all $w\in W$}.$$ The condition $\sum_{w\in W} p_w^{(N)}=1$ gives $b+\sum b_w=1$. Since $\tilde{P}$ is symmetric we have $\|\tilde{P}\|=\vrho(\tilde{P})<1$ by the above argument (where $\|P\|$ is the operator norm of $P:\ell^2(\Delta)\to\ell^2(\Delta)$). Thus, since $\|P_w\|\leq 1$ for all $w$, we see that$$\begin{aligned}
\vrho(P)^N=\vrho(P^N)&\leq\|P^N\|\leq b\|\tilde{P}\|+\sum_{w\in W}b_w\|P_w\|< b+\sum_{w\in W}b_w=1.\end{aligned}$$ (The first equality holds since $P$ is irreducible and aperiodic, see [@woessbook Exercise 1.10]).
\[cor:spectralradiusbuilding\] An isotropic random walk on any regular thick Fuchsian building has spectral radius strictly less than $1$.
Let $(W,S)$ be a Fuchsian Coxeter system. Any three distinct elements of $S$ generate an infinite group, and so $|I|\leq 2$ for all $I\in \cF=\{I\subseteq S\mid \text{$W_I$ is finite}\}$. Thus if $|S|\geq 4$ we have $$\sum_{s\in S\setminus I}q_s\geq 2|S\setminus I|=2(|S|-|I|)\geq 2(4-2)=4> |I|\quad\textrm{for all $I\in \cF$},$$ and so the result follows from Theorem \[thm:spectralradius\]. If $|S|=3$ and $|I|=1$ then $\sum_{s\in S\setminus I}q_s\geq 4>|I|$, and if $|S|=3$ and $|I|=2$ then $\sum_{s\in S\setminus I}q_s\geq 2=|I|$, completing the proof.
We do not think that Theorem \[thm:spectralradius\] is optimal. In fact, we believe that every thick regular building $(\Delta,\delta)$ of type $(W,S)$ with $W$ infinite has $\vrho(P)<1$ for all irreducible isotropic random walks. However the conclusion of Corollary \[cor:spectralradiusbuilding\] is sufficient for our purposes.
The path space
--------------
Let $\cT_{x}\subset \Delta^{\mathbb{N}}$ denote the space of all paths in $\Delta$ starting at $x\in\Delta$ (with jumps of any length allowed). More formally, the path space is defined as the inverse limit $$\cT_x=\varprojlim\cT_x^n=\bigg\{\gamma\in\prod_{n\geq 0}\cT_x^n\,\big|\,\gamma_i=\pi_{ij}(\gamma_j)\text{ for all $i\leq j$}\bigg\}$$ where $\cT_x^n=\{x\}\times \Delta^{n-1}\subset\Delta^n$ is the space of all paths in $\Delta$ of length $n-1$ starting at $x\in\Delta$, and $\pi_{ij}:\cT_x^j\to \cT_x^i$ are the natural projections. From this description we see (from Tychonoff’s Theorem) that $\cT_x$ is a compact Hausdorff topological space.
In the case of a Cayley graph of a group, there is naturally an automorphism of the graph taking any given vertex $x$ to any other vertex $y$, and thus there is a bijection $\psi_{xy}:\cT_x\to\cT_y$ mapping paths based at $x$ to “isomorphic” paths based at $y$. In effect, this gives the intuition that random walks starting at $x$ “behave the same as” random walks starting at $y$. In our context there is typically not an automorphism of $\Delta$ taking $x$ to $y$, and so we need to work a little harder to construct a suitable bijection $\psi_{xy}:\cT_x\to\cT_y$. The distance regularity of Proposition \[prop:distanceregular\] plays a crucial role here.
\[prop:pathbijection\] For each $x,y\in \Delta$ there is a bijection $\psi_{xy}:\cT_x\to\cT_y$ such that:
1. If $\gamma=(x_0,x_1,x_2,\ldots)\in\cT_x$ and $\psi_{xy}(\gamma)=(y_0,y_1,y_2,\ldots)$, then $\delta(x_{i},x_{i+1})=\delta(y_{i},y_{i+1})$ and $\delta(x,x_i)=\delta(y,y_i)$ for all $i\geq 0$.
2. For all $L\geq 0$ and for all $x,y$ of same cone type, if $\gamma=(x=x_0,x_1,x_2,\ldots)$ and $\psi_{xy}(\gamma)=(y=y_0,y_1,y_2,\ldots)$ and if $x_j\in\mathrm{Int}_{L} C(x)$ for some $j\geq 0$, then $y_j \in \mathrm{Int}_{L} C(y)$.
3. We have $\mathbb{P}_x[ (X_{n})_{n\geq 0}\in A]=\mathbb{P}_y[ (X_{n})_{n\geq 0}\in \psi_{xy}(A)]$ for all measurable $A\subseteq \cT_x$, where $\PP_{x}$ denotes the distribution of the isotropic random walk $(X_{n})_{n\geq 0}$ started at $X_0=x\in\Delta$.
We inductively build bijections $\psi_{xy}^n:\cT_x^n\to\cT_y^n$ satisfying part 1 of the proposition (for $0\leq i\leq n$). The case $n=0$ is trivial. Suppose that $\psi_{xy}^n:\cT_x^n\to\cT_y^n$ has been constructed. For any finite path $\gamma=(x_{1},\ldots, x_{n})$ and any point $x_{n+1}$ we define $\gamma\circ x_{n+1}=(x_{1},\ldots, x_{n}, x_{n+1})$.
Write $$\begin{aligned}
\cT_x^{n+1}&=\{\gamma_n\circ x_{n+1}\mid \gamma_n\in\cT_x^n,\,x_{n+1}\in\Delta\}\\
&=\bigsqcup_{u,v\in W}\{\gamma_n\circ x_{n+1}\mid\gamma_n\in\cT_x^n,\,x_{n+1}\in\Delta_u(x)\cap\Delta_v(x_n)\},\end{aligned}$$ where $\gamma_n=(x_0,\ldots,x_n)$. For each $\gamma_n\in\cT_x^n$, the set $\gamma_n\circ \{x_{n+1}\mid x_{n+1}\in\Delta_u(x)\cap\Delta_v(x_n)\}$ has cardinality $a_{uv}^w$, where $w=\delta(x,x_n)$ (see Proposition \[prop:distanceregular\]). We also have $$\cT_y^{n+1}=\bigsqcup_{u,v\in W}\{\psi_{xy}^n(\gamma_n)\cdot y_{n+1}\mid \gamma_n\in\cT_x^n,\,y_{n+1}\in \Delta_u(y)\cap \Delta_v(y_n)\}$$ where $\psi_{xy}^n(\gamma_n)=(y_0,\ldots,y_n)$. For each $\gamma_n\in\cT_x^n$ the set $\psi_{xy}^n(\gamma_n)\circ\{ y_{n+1}\mid y_{n+1}\in \Delta_u(y)\cap \Delta_v(y_n)\}$ also has cardinality $a_{uv}^w$ since $\delta(y,y_n)=\delta(x,x_n)=w$ (by the induction hypothesis). Thus for each fixed $\gamma_n\in\cT_x^n$ and each $u,v\in W$ we can choose a bijection $$\theta_{xy}^{uv}[\gamma_n]:\{x_{n+1}\mid x_{n+1}\in\Delta_u(x)\cap\Delta_v(x_n)\}\to \{y_{n+1}\mid y_{n+1}\in \Delta_u(y)\cap \Delta_v(y_n)\}.$$ Thus for each $\gamma_n\in\cT_x^n$ we obtain a bijection $\theta_{xy}[\gamma_n]:\Delta\to\Delta$ (depending on $\gamma_n$) by the rule $$\theta_{xy}[\gamma_n](x_{n+1})=\theta_{xy}^{uv}[\gamma_n](x_{n+1})\quad\text{if $x_{n+1}\in\Delta_u(x)\cap\Delta_v(x_n)$}.$$ Then define $\psi_{xy}^{n+1}:\cT_x^{n+1}\to\cT_y^{n+1}$ by $$\begin{aligned}
\label{eq:project}
\psi_{xy}^{n+1}(\gamma_{n+1})=\psi_{xy}^n(\gamma_n)\circ \theta_{xy}[\gamma_n](x_{n+1}).\end{aligned}$$ By construction this bijection satisfies the conditions in part $1$ of the proposition for $0\leq i\leq n+1$.
We now construct a bijection $\psi_{xy}:\cT_x\to\cT_y$ satisfying part 1 of the proposition. For $\gamma=(\gamma_0,\gamma_1,\ldots)\in\cT_x$ let $$\psi_{xy}(\gamma)=(\psi_{xy}^0(\gamma_0),\psi_{xy}^1(\gamma_1),\ldots).$$ By (\[eq:project\]) we see that $\psi_{xy}(\gamma)\in\cT_y$ for all $\gamma\in\cT_x$, and hence $\psi_{xy}:\cT_x\to\cT_y$. If $\psi_{xy}(\gamma)=\psi_{xy}(\gamma')$ then $\gamma_i=\gamma_i'$ for all $i\geq 0$, and hence $\gamma=\gamma'$ and so $\gamma$ is injective. To check surjectivity, if $\gamma=(\gamma_0,\gamma_1,\ldots)\in\cT_y$ then let $\gamma'=((\psi_{xy}^0)^{-1}(\gamma_0),(\psi_{xy}^1)^{-1}(\gamma_1),\ldots)$. Then $\gamma'\in \cT_x$ (using (\[eq:project\])), and hence $\psi_{xy}$ is surjective.
It follows that $\psi_{xy}:\cT_x\to\cT_y$ is a bijection satisfying the conditions in part 1 of the proposition. Then from Proposition \[prop:conetypes\] and Proposition \[prop:Lboundaries\] we have $$\partial_{L}C(x)=\{z\in C(x)\mid \delta(x,z)\in \partial_{L}T(\rho(x))\},$$ and part 2 of the proposition follows from this description. Since $(X_n)_{n\geq 0}$ is an isotropic random walk part 1 of the proposition implies part 3.
On occasion we will consider $\psi_{xy}$ as a bijection $\psi_{xy}:\cT_x^n\to\cT_y^n$ for each fixed $n\geq 0$ (that is, we write $\psi_{xy}$ in place of $\psi_{xy}^n$).
Isotropic random walks and groups {#sect:6}
---------------------------------
The following proposition (cf. [@CW Lemma 8.1]) illustrates how isotropic random walks naturally arise from bi-invariant probability measures on groups acting on buildings.
\[prop:Kac\] Let $G$ be a locally compact group acting transitively on a regular building $(\Delta,\delta)$, and let $B$ be the stabiliser in $G$ of a fixed base chamber $o$. Normalise the Haar measure on $G$ so that $B$ has measure $1$. Let $\varphi$ be the density function of a $B$-bi-invariant probability measure on $G$. If the group $B$ acts transitively on each set $\Delta_w(o)$ with $w\in W$, then the assignment $$p(go,ho)=\varphi(g^{-1}h)$$ for $g,h\in G$ defines an isotropic random walk on $(\Delta,\delta)$.
To check that $p(\cdot,\cdot)$ is well defined, suppose that $g_1o=go$ and $h_1o=ho$. Then $g_1^{-1}g\in B$ and $h^{-1}h_1\in B$, and thus $g_1^{-1}h_1\in Bg^{-1}hB$, and so $\varphi(g_1^{-1}h_1)=\varphi(g^{-1}h)$.
For each $x\in\Delta$ use transitivity to fix an element $g_x\in G$ with $g_xo=x$. Then $G$ is the disjoint union of cosets $g_xB$, $x\in\Delta$, and thus $$\sum_{y\in\Delta}p(x,y)=\sum_{y\in\Delta}\varphi(g_x^{-1}g_y)=\sum_{y\in\Delta}\int_{g_x^{-1}g_yB}\varphi(g)\,dg=\int_G\varphi(g)\,dg=1.$$
Clearly $p(gx,gy)=p(x,y)$ for all $g\in G$ and all $x,y\in\Delta$, and since $B$ is transitive on each set $\Delta_w(o)$ it follows that $p(\cdot,\cdot)$ is isotropic.
Thus Theorems \[thm:LLNbuilding\] and \[thm:CLTbuilding\] give a rate of escape theorem and a central limit theorem (with formulas for the speed and variance) for random walks induced by $B$-bi-invariant measures on groups acting, as in Proposition \[prop:Kac\], on Fuchsian buildings, where $B$ is the stabiliser of a chamber. The finite range assumption amounts to assuming that the density function of the $B$-bi-invariant measure is supported on finitely many $B$ double cosets. An important example is the case where $G=G(\mathbb{F}_q)$ is a Fuchsian Kac-Moody group over a finite field $\mathbb{F}_q$, acting on its natural building $G/B$ (as in Example \[ex:kac\]), and thus Corollary \[cor:Kac\] follows from Proposition \[prop:Kac\] and Theorems \[thm:LLNbuilding\] and \[thm:CLTbuilding\].
Isotropic random walks on regular Fuchsian buildings {#sect:renewal}
====================================================
We now restrict our attention to irreducible isotropic random walks on a thick regular Fuchsian building. Thus in this section $(W,S)$ denotes a Fuchsian Coxeter system, $(\Delta,\delta)$ is a thick regular Fuchsian building of type $(W,S)$, and $P=\sum_{w\in W}p_wP_w$ is the transition operator of an isotropic random walk $(X_n)_{n\geq 0}$ on $\Delta$. For the remainder of this section we fix a recurrent cone type ${\mathbf{T}}$.
We will assume that $(X_n)_{n\geq 0}$ has bounded range. Thus there is a minimal number $L_0\geq 0$ such that $$\begin{aligned}
\label{eq:L0}\text{$p_w\neq 0$ implies that $\ell(w)\leq L_0$}.\end{aligned}$$
It is sufficient to prove Theorems \[thm:LLNbuilding\] and \[thm:CLTbuilding\] under the assumption that $p_s>0$ for all $s\in S$, and so there is an $\varepsilon>0$ such that $$\begin{aligned}
\label{eq:support}\text{$p(x,y)>\varepsilon$ whenever $d(x,y)=1$}.\end{aligned}$$ To see this, note that by Lemma \[lem:irreducible\].2 there is an $M\geq 1$ such that the $M$-step walk $(X_{nM})_{n\geq 0}$ satisfies $p_s^{(M)}>0$ for all $s\in S$, and by the bounded range assumption proving Theorems \[thm:LLNbuilding\] and \[thm:CLTbuilding\] for the $M$-step walk implies the theorems for the $1$-step walk $(X_n)_{n\geq 0}$. Thus, without loss of generality we will assume (\[eq:support\]) throughout this section.
Renewal times and the proofs of Theorems \[thm:LLNbuilding\] and \[thm:CLTbuilding\] {#sect:yT}
------------------------------------------------------------------------------------
In this section we setup a renewal structure for isotropic random walks on Fuchsian buildings. The main result is Theorem \[thm:reg\], which is the key ingredient in the proofs of our rate of escape and central limit theorems. The proof of Theorem \[thm:reg\] will occupy Sections \[sec:proofreg1\] and \[sec:proofreg2\].
We note that Theorems \[thm:LLNbuilding\] and \[thm:CLTbuilding\] can be proven by only developing a renewal structure for the retracted random walk $(\overline{X}_n)_{n\geq 0}$ on $W$. However here we will develop a more satisfying picture by proving a renewal structure for the walk $(X_n)_{n\geq 0}$ on the building. This only requires a small amount more work, and in our opinion is more natural.
We start by recalling the crucial fact that geodesics in a hyperbolic group either stay within bounded distance of each other or diverge exponentially. More precisely, there exists some exponential divergence function $e: \NN_0 \rightarrow \RR$ such that the following holds: for all $u\in W$ and all geodesics $\gamma_{1}$ from $u$ to any $v_{1}\in W$ and $\gamma_{2}$ from $u$ to any $v_{2}\in W$ and all $r,R\in \NN_0$ with $R+r\leq \min\{d(u,v_{1}), d(u,v_{2})\}$ and $d(\gamma_{1}(R), \gamma_{2}(R))\geq e(0)$, all paths starting in $\gamma_{1}(R+r)$, visiting only vertices in $W\setminus
\cB(u,R+r)$ and ending in $\gamma_{2}(R+r)$ have length of at least $e(r)$. Here $\gamma_i(n)$ is the point on $\gamma_i$ at distance $n\in\NN_0$ to $u$. In particular, two geodesics that have been at least $e(0)$ apart can never intersect again.
*(c.f. [@HMM:13 Lemma 2.4])*\[lem:boundary-rays\] Let be $u\in W\setminus \{e\}$. Then the boundary $\partial_1 C_W(u)$ is contained in the union of two geodesic rays starting at $u$ in the Coxeter complex.
Since the Coxeter complex is homeomorphic to the hyperbolic disc, it can be endowed with an orientation. Let $r_1, r_2: \mathbb{N}_0\to W$ be two infinite geodesic rays in the Coxeter complex going through $u$ which coincide up to $u$. Let $c_1$ and $c_2$ be the geodesic rays extracted from $r_1,r_2$ starting at $u$. Let $V$ be a component of $W\setminus \{c_1 \cup c_2\}$ which does not contain $e$; here we identify $c_1$ and $c_2$ as the sets of vertices which lie on the geodesics. Let us prove that $V$ is contained in $C_{W}(u)$: let $v\in V$, and let us consider a geodesic segment $c_v$ joining $e$ to $v$. Since the Coxeter complex is planar, JordanÕs Theorem implies that $c_v$ intersects $\partial V=\{ w_1\in V \mid \exists w_2\in W\setminus V: d(w_1,w_2)=1\}$ at some point $w$, hence $c_1$ or $c_2$. Let us assume that it intersects $c_1$. Since $c_1$ is geodesic, we may replace the portion of $c_v$ before $w$ by $c_1$: it follows that the concatenation of $c_1$ up to $w$ and $c_v$ from $w$ to $v$ is geodesic; this implies that $v\in C_W(u)$.
By Arzela-AscoliÕs theorem and the planarity of the Coxeter complex, we may find two rays $c_\ell$ and $c_r$ going through $u$ such that $C_W(u)$ is the union of those rays with all the components of their complement which do not contain $e$.
Recall that we have fixed a recurrent cone type ${\mathbf{T}}$, and we now fix some $x_{{\mathbf{T}}}\in \Delta$ with $T(x_{{\mathbf{T}}})={\mathbf{T}}$. Set $L_1=\max \{L_0,e(0)\}$. In the following we will construct some $y_{{\mathbf{T}}}\in \mathrm{Int}_{L_1} C_{\Delta}(x_{{\mathbf{T}}})$ such that $C_{\Delta}(y_{{\mathbf{T}}})\subset \mathrm{Int}_{L_1} C_{\Delta}(x_{{\mathbf{T}}})$. Let us remark that $\mathrm{Int}_{L_1} C_{\Delta}(x_{{\mathbf{T}}})$ always contains at least one infinite connected component.
\[lem:equal-distance\] Let be $L\geq 1$ and $y\in \mathrm{Int}_{L} C_{\Delta}(x_{{\mathbf{T}}})$. Then: $$d(y,\partial_{L} C_{\Delta}(x_{{\mathbf{T}}}) )= d(\rho(y), \partial_{L} C_{W}(\rho(x_{{\mathbf{T}}}))).$$
Since retractions decrease the distance, and since $\rho(\partial_{L}C_{\Delta}(x_{{\mathbf{T}}}))=\partial_{L}C_W(\rho(x_{{\mathbf{T}}}))$ (see Proposition \[prop:Lboundaries\]) we have $d(y,\partial_{L} C_{\Delta}(x_{{\mathbf{T}}}) )\geq d(\rho(y), \partial_{L} C_{W}(\rho(x_{{\mathbf{T}}})))$. It remains to show the other inequality to finish the proof. For this purpose, take a path of length $K=d(\rho(y), \partial_{L} C_{W}(\rho(x_{{\mathbf{T}}})))$ from $\rho(y)$ to some $v\in \partial_{L} C_{W}(\rho(x_{{\mathbf{T}}}))$, say the path $(w_0=\rho(y),w_1,\dots,w_K=v)$. We now want to construct a path of length $K$ from $y$ to $\partial_{L} C_{\Delta}(x_{{\mathbf{T}}})$. Let $A$ be an apartment which contains $o$ and $y$ (and thus $x_{{\mathbf{T}}}$ by (A3)). By Proposition \[prop:conetypes\] the retraction $\rho$ maps $C_A(x_{{\mathbf{T}}})$ isometrically onto $C_W(\rho(x_{{\mathbf{T}}}))$. Therefore, $$\pi=((\rho|_A)^{-1}(w_0)=y,(\rho|_A)^{-1}(w_1),\dots,(\rho|_A)^{-1}(v))$$ is a path of length $K$ from $y$ to $z=(\rho|_A)^{-1}(v) \in \partial_{L} C_A(x_{{\mathbf{T}}})$. Now choose any $z'\in A$ with $d(z,z')=L$ and $z'\notin C_A(x_{{\mathbf{T}}})$. Then $\rho(z')\notin \partial_{L} C_W(\rho(x_{{\mathbf{T}}}))=\rho (\partial_{L} C_{\Delta}({\mathbf{x}}_{{\mathbf{T}}}))$, and hence $z'\notin \partial_{L} C_{\Delta}({\mathbf{x}}_{{\mathbf{T}}})$. That is, $\pi$ is a path of length $K$ in $A\subset \Delta$ which connects $y$ with $\partial_{L} C_{\Delta}(x_{{\mathbf{T}}})$. This finishes the proof.
The following lemma and its corollary will be used to construct $y_{{\mathbf{T}}}$.
Let $L\geq e(0)$ and let $u\in W\setminus \{e\}$ be such that $T(u)={\mathbf{T}}$. Then there is some $v\in \mathrm{Int}_{L} C_W(u)$ such that $T(v)={\mathbf{T}}$ and $C_W(v)\subset \mathrm{Int}_{1} C_W(u)$.
Let $u\in W\setminus \{e\}$ with $T(u)={\mathbf{T}}$. By Lemma \[lem:boundary-rays\] there are two geodesic rays $\gamma_1,\gamma_2$ starting from $e$ whose union contains $\partial_1 C_W(u)$ and which coincide up to $u$. Since $T(u)$ is recurrent we can choose any end $\xi\in \partial_\infty C_W(u)$ which is different from the ends described by $\gamma_1$ and $\gamma_2$. Let $\pi$ be any geodesic ray which starts at $e$, follows $\gamma_1$ up to $u$ and describes $\xi$. It follows that, for every $i\in\{1,2\}$, the distance $d(\pi(t),\gamma_i(t))$ cannot be bounded for $t\geq 0$. Hence, there are $t_1,t_2\in\mathbb{N}$ such that $d(\pi(t_i),\gamma_i(t_i))\geq L+1\geq e(0)+1$ implying that $d(\pi(t),\gamma_i(t))\geq e(0)+1$ for all $t\geq \max\{t_1,t_2\}$ and all $i\in\{1,2\}$. That is, $\pi$ and $\gamma_i$, $i\in\{1,2\}$, diverge exponentially. In particular, there must be some $t_0\geq \max\{t_1,t_2\}$ such that $v'=\pi(t_0)\in \mathrm{Int}_{L} C_W(u)$ with $T(v')$ being recurrent. Denote by $\gamma_1'$ and $\gamma_2'$ the geodesic rays starting at $e$ whose union contains $\partial_1 C_W(v')$ and pass through $v'$. Due to exponential divergence of $\gamma_i$ and $\gamma_j'$, where $i,j\in \{1,2\}$, we have that $\gamma_i(t)\neq \gamma_j'(t)$ for all $t\geq t_0$. This yields $C_W(v')\subset \mathrm{Int}_1(u)$
The cone $C_W(v')$ contains an element $v$ with $T(v)={\mathbf{T}}$, and since $C_W(v)\subset C_W(v')$ the result follows.
We can iterate the last step by replacing the role of $u$ by $v$. This leads then to the following corollary:
\[cor:find-v\] Let be $u\in W\setminus \{e\}$ such that $T(u)={\mathbf{T}}$. Then there is some $v\in \mathrm{Int}_{L_1} C_W(u)$ such that $T(v)={\mathbf{T}}$ and $C_W(v)\subset \mathrm{Int}_{L_1}C_W(u)$.
We now show how to construct $y_{{\mathbf{T}}}$: take an apartment $A$ which contains $o$ and $x_{{\mathbf{T}}}$ and recall that $\rho|_A$ denotes the restriction of $\rho$ to $A$ which becomes an isomorphism mapping $A$ onto $W$. We apply Corollary \[cor:find-v\] on $u=\rho|_A(x_{{\mathbf{T}}})$ and find some $v\in W$ such that $C_W(v)\subset \mathrm{Int}_{L_1} C_W(u)$. Due to Proposition \[prop:Lboundaries\] and Lemma \[lem:equal-distance\] we then must have $C_{\Delta}((\rho|_A)^{-1}(v))\subset \mathrm{Int}_{L_1} C_{\Delta}(x_{{\mathbf{T}}})$. Fix now for the rest of this section such a chamber $y_{{\mathbf{T}}}=(\rho|_A)^{-1}(v)$ in dependence of $x_{{\mathbf{T}}}$, $v$ and $A$. Furthermore, fix a shortest path $\pi_{{\mathbf{T}}}=[x_{{\mathbf{T}}}, x_{{\mathbf{T}}}^{(1)},\dots,x_{{\mathbf{T}}}^{(k-1)},y_{{\mathbf{T}}}]$ from $x_{{\mathbf{T}}}$ to $y_{{\mathbf{T}}}$ contained in $C(x_{{\mathbf{T}}})$. Note that for $x\in\Delta$ with $T(x)={\mathbf{T}}$ the bijection $\psi_{x_{{\mathbf{T}}}x}$ maps $\pi_{{\mathbf{T}}}$ onto a path from $x$ to some $y\in
\mathrm{Int}_{L_1} C(x)$ contained in $C_{\Delta}(x)$ (see Proposition \[prop:pathbijection\].2).
We now give the definition of *renewal times*. For each $x\in\Delta$ with $T(x)={\mathbf{T}}$ let $\widehat{\mathcal{T}}_{x}$ be the set of all paths which start at $x$, initially follow $$\pi_x=\left( \psi_{x_{{\mathbf{T}}}x}(x_{{\mathbf{T}}}^{(1)}),\psi_{x_{{\mathbf{T}}}x}(x_{{\mathbf{T}}}^{(2)}), \dots,\psi_{x_{{\mathbf{T}}}x}(y_{{\mathbf{T}}})\right)$$ and stay in $\mathrm{Int}_{L_1} C(x)$ afterwards forever. We define $ R_{0}=0$ and let $$R_{1}=\inf\{{k\geq 0}\mid (X_{i})_{i\geq k} \in \widehat{\mathcal{T}}_{X_k},~T( X_{k})={\mathbf{T}}\}$$ be the first time $k\in\mathbb{N}$ that the random walk hits the root of a cone of type ${\mathbf{T}}$, visits consecutively $\psi_{x_{{\mathbf{T}}}X_k}(x_{{\mathbf{T}}}^{(1)}),\psi_{x_{{\mathbf{T}}}X_k}(x_{{\mathbf{T}}}^{(2)}),\dots,\psi_{x_{{\mathbf{T}}}X_k}(y_{{\mathbf{T}}})$ and stays in $\mathrm{Int}_{L_1} C(X_k)$ afterwards forever. Inductively, $$\begin{aligned}
\label{eq:renewal}
R_{n}=\inf\{k> R_{n-1}\mid (X_{i})_{i\geq k} \in \widehat{\mathcal{T}}_{X_k},~T( X_{k})={\mathbf{T}}\}.
\end{aligned}$$
Recall the notion of random variables with exponential moments. A real valued random variable $Y$ has *exponential moments* if $\EE[\exp(\lambda Y)]<\infty$ for some $\lambda>0$, or equivalently, if there are positive constants $C>0$ and $c<1$ such that $\PP[Y=n]\leq C c^{n} $ for all $n\in \NN_0$.
\[thm:reg\] Let $(X_n)_{n\geq 0}$ be an isotropic random walk on a thick regular Fuchsian building $(\Delta, \delta)$ with bounded range.
1. The renewal times $R_{n}$ are almost surely finite, $d(o,X_{R_{n}})=\sum_{i=1}^{n} d( X_{R_{i-1}}, X_{R_{i}})$, and $(d( X_{R_{i-1}}, X_{R_{i}}))_{i\geq 2}$ are i.i.d. random variables.
2. The renewal time $ R_{1}$ and the increments $(R_{i+1}- R_{i})$ for $i\geq 1$ have exponential moments. The same holds true for $d(o,X_{ R_{1}})$ and $d( X_{ R_{i}}, X_{ R_{i+1}})$ for $i \geq 1$.
The proof of Theorem \[thm:reg\] will be given in Sections \[sec:proofreg1\] and \[sec:proofreg2\]. Assuming Theorem \[thm:reg\] for the moment, one can now argue as in [@HMM:13] (verbatim modulo some notations) to prove our law of large numbers and central limit theorem.
We content ourselves with giving the main idea and refer to [@HMM:13] for the technical details. The role of the cones in the definition of the renewal times was that the trajectory of the walk observed at renewal times is the “aligned” sum of i.i.d. pieces, Theorem \[thm:reg\]; that is, $$d(o,X_{R_{n}})=\sum_{i=1}^{n} d(R_{i-1}, R_{i}).$$ Now, the law of large numbers and central limit theorem for real-valued random variables apply and the statements in Theorems \[thm:LLNbuilding\] and \[thm:CLTbuilding\] follow for the process $d(o,X_{R_{n}})$. It remains therefore to control the distance or “error” between $X_{n}$ and the position of the last renewal before time $n$. More precisely, we define the last renewal time before time $n$: $$k(n)=\sup\{k\mid R_{k}\le n\}.$$ We have that $$\frac{n}{k(n)}= \frac{n}{R_{k(n)}}\frac{R_{k(n)}}{k(n)}.$$ By the strong law of large numbers the second factor tends a.s. to $\EE[R_2-R_1]$. For the first factor we observe that $R_{k(n)}\le n \le R_{k(n)+1}$, hence $$\limsup_{n\to\infty} \frac{R_{k(n)}}{n}\le 1.$$ On the other hand, since $n\ge k(n)$ and $(R_{k(n)}-R_{k(n)+1})$ have finite first moments, $$\lim_{n\to\infty} \frac{R_{k(n)}-R_{k(n)+1}}{n}=0\quad \hbox{a.s.}$$ and hence $$\liminf_{n\to\infty}\frac{R_{k(n)}}{n}\ge \liminf_{n\to\infty}\left( \frac{R_{k(n)}-R_{k(n)+1}}{n}\right)+ \frac{R_{k(n)+1}}{n}\ge 1.$$ Eventually, we have that $$\frac{n}{k(n)} \xrightarrow[n\to \infty]{a.s.} \EE[R_{2}-R_{1}]<\infty.$$ Denote $$M_{k}=\sup\{d(Z_{n},Z_{R_{k}})\mid R_{k}\le n \leq R_{k+1}\}, ~k\geq 1.$$ The random variables $(M_{k})_{k\geq 1}$ form an i.i.d. sequence of random variables with exponential moments. This is a consequence of the fact that $d( X_{ R_{i+1}}, X_{ R_{i}})$ have exponential moments, see [@HMM:13 Corollary 4.2]. As a consequence we have that $$\lim_{n\to\infty}\frac{d(Z_{n},e)-d(Z_{R_{k(n)}},e)}n \le \lim_{k\to\infty}\frac{M_k}k=0 \quad\hbox{a.s.}.$$ Since the strong law of large numbers guarantees that $$\frac{d(Z_{R_{k(n)}},e)}{k(n)} \xrightarrow[n\to \infty]{a.s.} \EE[ d(Z_{ R_{2}}, Z_{ R_{1}})]$$ we can conclude the proof of Theorem \[thm:LLNbuilding\]: $$\begin{aligned}
\frac{d(Z_{n},e)}n& =& \frac{d(Z_{n},e)-d(Z_{R_{k(n)}},e)}n + \frac{d(Z_{R_{k(n)}},e)}{k(n)} \frac{k(n)}{n}\cr
& \xrightarrow[n\to \infty]{a.s.} & 0 + \frac{\EE[ d(Z_{ R_{2}}, Z_{ R_{1}})]}{\EE[ R_{2}- R_{1}]}.\end{aligned}$$ The proof of Theorem \[thm:CLTbuilding\] is more involved; we refer [@HMM:13 Section 4.2] for the remaining details.
Proof of Theorem \[thm:reg\].1 {#sec:proofreg1}
------------------------------
Recall that $\overline{X}_{n}=\rho(X_n)$ denotes the retracted walk on $W$ and its transition probabilities are given by $\overline{p}(u,v)$ and its transition operator is $\overline{P}$. This retracted walk is necessarily irreducible and aperiodic ($P$ is irreducible and thus aperiodic by Lemma \[lem:irreducible\]). Proposition \[prop:retractedwalkspectralradius\] and Corollary \[cor:spectralradiusbuilding\] give $\vrho(\overline{P})=\vrho( P)<1$
The retraction induces a probability measure on the space of trajectories $\cT$ in the underlying Coxeter group, we also denote this by $\PP$. Recall that by (\[eq:support\]) we have a uniform bound on the next neighbour one-step probabilities: we have $\overline{p}(u,v)>\varepsilon$ for all $u,v\in W$ with $d(u,v)=1$.
For each cone $C(u)$ in $(W,S)$ let $\partial_{\infty} C(u)$ denote the closure of $C(u)$ at infinity, that is, in the Gromov hyperbolic compactification. If $u\in W$ has cone type ${\mathbf{T}}$ let $\widehat{\mathcal{T}}_u$ be the set of all paths starting at $u$, initially following the path $$\begin{aligned}
\label{eq:piu}
\pi_u=\left(\rho(\psi_{x_{{\mathbf{T}}}u}(x_{{\mathbf{T}}}^{(1)})),
\rho(\psi_{x_{{\mathbf{T}}}u}(x_{{\mathbf{T}}}^{(2)})),\dots,\rho(\psi_{x_{{\mathbf{T}}}u}(y_{{\mathbf{T}}}))\right),\end{aligned}$$ and staying in $\mathrm{Int}_{L_1} C(u)$ afterwards.
The invariance properties given in Propositions \[prop:invariance\] and \[prop:pathbijection\] induce the following invariance property for the retracted walk.
\[lem:coneinvariance\] For all $u,v\in W$ with $T(u)=T(v)={\mathbf{T}}$ and all measurable sets $A\subseteq\widehat{\mathcal{T}}_u$ we have $$\PP_{u}[(\overline{X}_{n})_{n\geq 0}\in A ]=\PP_{v}[(\overline{X}_{n})_{n\geq 0}\in vu^{-1} A ].$$
Since $(\overline X_{n})_{n\geq 0}$ is an irreducible Markov chain on a hyperbolic graph with bounded range and spectral radius $\vrho(\overline{P})<1$ the Markov chain $(\overline X_{n})_{\geq 0}$ converges almost surely to a random point $\overline{X}_{\infty}$ of the hyperbolic boundary $\partial W$; since the detailed structure of Gromov hyperbolic boundary is not needed for our purposes, we refer e.g. to [@woessbook Theorem 22.19] for further details. The harmonic measure $\nu$ is defined as the law of $\overline{X}_{\infty}$. More precisely, it is the probability measure on the hyperbolic boundary $\partial W$ such that $\nu(A)=\PP[\overline{X}_{\infty}\in A]$ for each $A\subset\partial W$.
\[lem:nonatomic\] The harmonic measure $\nu$ of $(\overline{X}_{n})_{n\geq 0}$ is not concentrated on a finite number of atoms.
Let us assume that $\nu$ is concentrated on the finite set $\{\xi_{1}, \xi_{2},\ldots,
\xi_{k}\}\subset \partial W$. Let $u\in W$ be such that $T(u)={\mathbf{T}}$ and that $\xi_1\in\mathrm{Int}(\partial_{\infty} C(u))$. Then by the definition of harmonic measure, $$\PP_{1}[\overline{X}_{\infty}=\xi_{1 }, \overline{X}_{n}\in C(u) \mbox{ for all but finitely many } n]= \nu(\xi_{1})>0.$$ Consequently there exists some $v\in C(u)$ such that $$\PP_{v}[\overline{X}_{\infty}=\xi_{1}, \overline{X}_{n}\in C(u) \mbox{ for all } n]>0.$$ Since there exists a path of positive probability inside $C(u)$ from $u$ to $v$, we have $$\PP_{u}[\overline{X}_{\infty}=\xi_{1}, \overline{X}_{n}\in C(u) \mbox{ for all } n]>0.$$ As there are only a finite number of atoms and the automaton $\mathcal{A}(W,S)$ is strongly connected (Theorem \[thm:stronglyconnected\]), there exists some $w\in W$ with cone type $T(u)={\mathbf{T}}$ such that $\partial_{\infty}C(w)$ does not contain any of the atoms $\xi_1,\ldots,\xi_k$. However by Lemma \[lem:coneinvariance\] we have that there exists some $\xi_{k+1}\in \partial_{\infty}C(w)$ such that $$\PP_{w}[\overline{X}_{\infty}=\xi_{k+1},\overline{X}_{n}\in C(w) \mbox{ for all } n]>0,$$ and so $\xi_{k+1}$ is an atom, a contradiction.
The next lemma will be crucial for our proofs.
\[lem:stayincone\] There exists a constant $\overline{p}_{esc}>0$ such that for all $u\in W$ with $T(u)={\mathbf{T}}$ we have that $$\PP_{u}[(\overline{X}_{n})_{n\geq 0}\in \widehat{\mathcal{T}}_u]\ge\overline{p}_{esc}.$$
First we claim that $$\label{lem:stayincone:eq:prelim}
\PP_{w}[\overline{X}_{n}\in C(w) \text{ for all } n]>0\quad\textrm{for all $w\in W$ with $T(w)={\mathbf{T}}$}.$$ Due to strongly connectedness of the automaton $\mathcal{A}(W,S)$ (see Theorem \[thm:stronglyconnected\]) and by the definition of recurrent cone types there exists some $R\geq 0$ such that the sphere $\cS(1,R)$ contains only elements whose cone types are recurrent. Furthermore, by definition we have $W\setminus \cB(1,R-1)=\bigcup_{w'\in
\cS(1,R)} C(w')$. By Lemma \[lem:nonatomic\] the support of $\nu$ cannot be contained in the set of Gromov boundary points determined by the finitely many geodesics (from Lemma \[lem:boundary-rays\]) describing the boundaries of the cones $C(w')$ with $w'\in \cS(1,R)$. Thus there exists some $v\in \cS(1,R)$ and some open set $O\subset \partial_{\infty} C(v)$ such that $\PP[\overline{X}_{\infty}\in O]>0$. On the event that $\overline{X}_{\infty}\in O$, at some moment the random walk $(\overline{X}_n)$ enters $C(v)$ and never leaves it afterwards. If $w\in W$ with $T(w)={\mathbf{T}}$, then the cone $C(w)$ contains an element $v_1$ with $T(v_1)=T(v)$ (since ${\mathbf{T}}$ is recurrent and $\mathcal{A}(W,S)$ is strongly connected). By (\[eq:support\]) there is positive probability of walking from $w$ to $v_1$ via a shortest path, and necessarily this path is contained in $C(w)$. Hence (\[lem:stayincone:eq:prelim\]) is established.
Let $u\in W$ with $T(u)={\mathbf{T}}$, and let $w=u\delta(x_{{\mathbf{T}}},y_{{\mathbf{T}}})$. By the construction in Section \[sect:yT\] we have $T(w)={\mathbf{T}}$, and $C(w)\subset C(u)$. By (\[eq:support\]) there is positive probability that the retracted random walk with $X_0=u$ follows the path $\pi_u$ from (\[eq:piu\]) initially, and so (\[lem:stayincone:eq:prelim\]) implies the Lemma.
Recall from Proposition \[prop:pathbijection\].3 that for all $x,y\in\Delta$ with $T(x)=T(y)$, and all subsets $A\subset \widehat{\mathcal{T}}_{x}$, we have $$\label{eq:coneinvariantbuilding}
\PP_{x}[(X_{n})_{n\geq 0}\in A ] = \PP_{y}[(X_{n})_{n\geq 0}\in \psi_{xy}(A) ].$$
\[lem:stayinconebuilding\] There exists some constant $p_{esc}>0$ such that for all $x\in \Delta$ $$\PP_{x}[(X_{n})_{n\geq 0}\in \widehat{\mathcal{T}}_{x}]\geq p_{esc}.$$
This is a consequence of Proposition \[prop:Lboundaries\] and Lemma \[lem:stayincone\].
This is an adaption of [@HMM:13 Theorem 3.1]. We sketch the proof and refer to [@HMM:13] for the details. The fact that the Cannon automaton $\mathcal{A}(W,S)$ is strongly connected implies that there exists some $R\in\mathbb{N}$ such that for all $x\in \Delta$ the ball $\cB(x,R)$ contains at least one chamber of cone type ${\mathbf{T}}$. This can be extended to prove the following fact. Denote by $\widehat{\mathcal{T}}^{(n)}_{x}$ the set of $y\in\Delta$ such that there exists a path $(x_{0}, x_{1},x_{2},\ldots) \in \widehat{\mathcal{T}}_{x}$ such that $x_{n}=y$. For $x\in\Delta$, denote the first exit time of $\widehat{\mathcal{T}}_{x}$ by $D_{x}=\inf\{n\geq 1 \mid X_{n}\not\in \widehat{\mathcal{T}}^{(n)}_{x}\}$. Then there exists some constant $p_{h}$ and some $K$ such that for all choices of $x\in\Delta$ and all $y\in \mathrm{Int}_{L_1} C(x)$ we have $$\PP_{x}[ \{T(X_{n})\}_{n=1}^{K}\ni {\mathbf{T}}]\ge p_{h} \quad\text{and}\quad \PP_{y}[ \{T(X_{n})\}_{n=1}^{K}\ni {\mathbf{T}}\mid D_{x}=\infty]\ge p_{h}$$ Thus wherever the walk is it will reach a chamber $x\in\Delta$ of cone type ${\mathbf{T}}$ after at most $K$ steps with some positive probability of at least $p_{h}$. By Lemma \[lem:stayinconebuilding\], each time the walk is at a chamber of cone type ${\mathbf{T}}$ it has a positive probability of at least $p_{esc}$ to follow the walk $\psi_{x_{{\mathbf{T}}}x}(\pi_{{\mathbf{T}}})$ and to stay in the $L_1$-interior of this cone forever. If it does, this means that a renewal step was performed, and otherwise, the walk exits this last cone. Now, again the walk will hit a chamber of cone type ${\mathbf{T}}$ in at most $K$ steps with probability at least $p_{h}$ and we continue as above until we eventually performed one renewal step. Hence by induction, the random times $R_{n}$ are almost surely finite.
It is clear that $d(o,X_{ R_{n}})=\sum_{i=1}^{n} d( X_{ R_{i-1}}, X_{ R_{i}})$, because the chambers $(X_{R_n})_{n\geq 0}$ lie in a sequence of nested cones, and so there is a geodesic from $o=X_0$ to $X_{R_n}$ passing through $X_{R_1},X_{R_2},\ldots,X_{R_{n-1}}$. The fact that what happens between two subsequent renewal times is independent is a consequence of the following crucial property. For any $x,y\in\Delta$ with cone type ${\mathbf{T}}$ and any $A\subset \widehat{\mathcal{T}}_{x}$, (\[eq:coneinvariantbuilding\]) implies that $$\PP_x[ (X_{n})_{n\geq 0}\in A \mid D_{x}=\infty]=\PP_{y} [(X_{n})_{n\geq 0}\in \psi_{xy}(A) \mid D_{y}=\infty].$$ Thus we may introduce a new probability measure: for $A\subset \psi_{xo}(\widehat{\mathcal{T}}_x)$ let $$\QQ_{{\mathbf{T}}}[ (X_{n})_{n\geq 0}\in A]= \PP_{x}[ (X_{n})_{n\geq0 }\in \psi_{ox}(A) \mid D_{x}=\infty],$$ where $x$ is of cone type ${\mathbf{T}}$. Define the $\sigma$-algebras $$\cG_{n}=\sigma(R_{1},\ldots, R_{n}, X_{0}, \ldots, X_{ R_{n}}),~n\geq 1.$$ Although the $ R_{n}$’s are not stopping times, a check of the definition of conditional probability yields the following “Markov property”: for any measurable set $A\subset\psi_{xo}(\widehat{\cT}_{x})$ and any $x\in
\Delta$ of cone type ${\mathbf{T}}$, $$\PP_{x}[ (X_{ R_{n}+k})_{k\geq 0}\in \psi_{oX_{R_{n}}}(A)\mid \cG_{n}] = \QQ_{{\mathbf{T}}} [ (X_{k})_{k\geq 0}\in A]$$ (see [@HMM:13 Lemma 3.3] for details). Thus $d(( X_{ R_{i-1}}, X_{ R_{i}}))_{i\geq 2}$ are i.i.d. random variables.
Proof of Theorem \[thm:reg\].2 {#sec:proofreg2}
------------------------------
As the Cayley graph of $(W,S)$ is planar, its Cayley $2$-complex is such that the one skeleton is given by the Cayley graph and the $2$-cells are bounded by loops. These loops are described by the relations $$s^2=1\quad\textrm{and}\quad (st)^{m_{st}}=1\quad\textrm{for all $s,t\in S$ with $s\neq t$},$$ where $m_{st}=m_{ts}\in\ZZ_{\geq 2}\cup\{\infty\}$ for all $s\neq t$. Denote by $k$ the maximal length of all finite relations. Then, every loop in the Cayley $2$-complex has length of at most $k$. At various places, we will use the fact that the $2$-complex is homeomorphic to the hyperbolic disc and can be endowed with an orientation.
We make use of the following type of connectedness of spheres in Cayley graphs. We give an adaption of the results in [@BaBe:99] and [@Gournay] to our setting. Define the annulus $$\cS^{(k)}(w,K)=\{w'\in W\mid K-k/2\leq d(w,w')\leq K+k/2\}, \quad k,K\in\NN_0, w\in W.$$
\[lem:connected\_sphere\] Let $K>k/2$. Then there is a simple cycle in $\cS^{(k)}(w,K)$ that forms a simple closed curve around $w$ in the Cayley $2$-complex.
By planarity we can order the elements in the sphere $\cS(w,K)$ in clockwise order and say that two elements are neighbors on the sphere if they are neighbors in the ordering. Pick two neighbors $u,v\in \cS(w,K)$. Since the Cayley graph is one-ended and planar there exists a loop in the Cayley $2$-complex that contains $u,v$. Hence, $d(u,v)\leq k/2$ and thus there exists a path in $\cS^{(k)}(w,K)$ connecting $u$ and $v$. The concatenation of all paths connecting the neighbors is a cycle $C$ that is contained in $\cS^{(k)}(w,K)$. Since $K> k/2$ every infinite path starting from $w$ intersects the cycle $C$. It is straightforward to see that the cycle $C$ contains a simple cycle as a subset that forms a simple closed curve around $w$ in the Cayley $2$-complex.
The following Proposition is a key ingredient in the proof of Theorem \[thm:reg\].2.
\[lem:expdecay\] There exist $C<\infty$ and $\lambda<1$ such that for any $u \in W\setminus \{e\}$ and all $v\in C(u)$ $$F(u,v)=\PP_{u}[\overline{X}_{n}=v\text{ for some } n]\leq C \lambda^{d(u,v)}.$$
Fix $K>k/2$ and let $w\in W$ be such that $d(1,w)>K+k/2$. Lemma \[lem:connected\_sphere\] and the Jordan curve theorem yields now that $\cS^{(k)}(1,d(1,w))\cap \cS^{(k)}(w,K)\neq \emptyset$. Let $\pi^{+}$ be a geodesic ray from $1$ passing through $w$ and $\pi^{-}$ be a geodesic ray starting from $1$ and not passing through $\cS^{(k)}(w,K)$. Denote by $\overline\pi$ the bi-infinite path consisting of geodesics $\pi^{+}$ and $\pi^{-}$. There are at least two points $v_{1},v_{2}\in \cS^{(k)}(1,d(1,w))\cap \cS^{(k)}(w,K)$ on different sides of $\overline\pi$, i.e. each path between $v_{1}$ and $v_{2}$ has to cross $\pi^{-}$ or $\pi^{+}$. We will make use of this fact later in the proof.
Let $\gamma$ be a geodesic from $1$ to $v$ passing through $u$, and let $d=d(u,v)$. Let $D\in\mathbb{N}$ be such that $D>2e(0)+2k$ and $e(D)>4e(0)+4k$. For $i\in\{1,\ldots,\lfloor d/D\rfloor\}$ denote by $u_{i}=\gamma(d(1,u)+i\cdot D)$ and let $\cB(u_{i},2e(0)+2k)$ the ball of radius $2e(0)+2k$ around $u_{i}$. We define $$B_{i}=\bigcup_{w\in \cB(u_{i},2e(0)+2k) } C(w) $$ The boundary of $B_i$ is given by $\partial B_i=\{w\in B_i \mid \exists w'\in
W\setminus B_i: w'\sim w\}$.
The boundaries $\partial B_i$’s are disjoint.
*Proof of the claim.* The choice of $D > 4e(0)+4k$ implies that the balls $\cB(u_{i},2e(0)+2k)$ are disjoint. Let us assume that there exists some $w\in \partial
B_i\cap \partial B_{i+1}$ for some $i$ and show that this yields a contradiction. Denote by $\gamma^{+}$ a geodesic continuation of $\gamma$ that is contained in all $B_{i}$’s. Let $\gamma^{-}$ be a geodesic ray emanating from $1$ and not intersecting the $\partial B_{i}$’s, and let $\overline \gamma$ the bi-infinite path consisting of the elements of $\gamma^{+}$ and $\gamma^{-}$. Let $\gamma_{1}$ be the geodesic from $1$ to $w$ that maximizes $d(\gamma_{1}(t),\gamma^{+}(t))$ for all $t\leq d(1,w)$ among all geodesics from $1$ to $w$; this construction is well-defined since geodesics can only cross in vertices.
Our next step is to show that $\gamma_{1}$ is sufficiently “far away” from $u_{i}$. Due to planarity, there exists a geodesic ray $\gamma_{1}'$ starting from $1$, that passes through $\cS^{(k)}(1,d(1,u_{i}))\cap \cS^{(k)}(u_{i},2 e(0)+3k/2)$, and is between $\gamma^{+}$ and $\gamma_{1}$. To see this, take any $w_{1}'\in \cS^{(k)}(1,d(1,u_{i}))\cap \cS^{(k)}(u_{i},2 e(0)+3k/2)$ on the same side of $\overline \gamma$ as $\gamma_{1}$. By maximality of $\gamma_{1}$ the vertex $w_{1}'$ has to lie between $\gamma_{1}$ and $\gamma^{+}$. First, choose a geodesic from $1$ to $w_{1}$ that lies in between $\gamma^{+}$ and $\gamma_{1}$. Now, augment this geodesic step by step (following some way in the automaton) till forever or until one hits $\gamma_{1}$ in which case we follow $\gamma_{1}$ afterwards as long as we can, and then continue to follow some path in the automaton. Since $d(1,u_{1})-k/2\leq d(1,w'_{1})\leq d(1,u_{1})+k/2$ and $ d(u_{1},w'_{1})\geq 2e(0)+k$ we have that $ d( \gamma_{1}'(d(1,u_{i})),u_{i})\geq 2 e(0)+k/2 $. The maximality of $\gamma_{1}$ implies now that $$\label{eq:lem:expdecay}
d(\gamma_{1}(d(1,u_{i})),u_{i})\geq 2 e(0)+k/2.$$
Since $w\in \partial B_{i+1}$ there exists a geodesic $\gamma_{2}$ from $1$ to $w$ that passes through $\cB(u_{i},2e(0)+2k)$. Denote by $v$ a point in the intersection of $\gamma_{2}$ and $\cB(u_{i+1},2e(0)+2k)$. We have $$d(1,u_{i+1})-2e(0)-2k \leq d(1, v)\leq d(1,u_{i+1})+2e(0)+2k \mbox{ and } d(v,u_{i+1})\leq 2e(0)+2k.$$ Therefore, $$d(\gamma_{2}(d(1,u_{i+1}), \gamma(d(1,u_{i+1})))\leq 4e(0)+4k.$$ Eventually, by the exponential divergence of geodesics and since $e(D)>4e(0)+4k$ we have that $d(\gamma(d(1,u_{i}), \gamma_{2}(d(1,u_{i})))<e(0)$. Inequality (\[eq:lem:expdecay\]) implies now that $$d(\gamma_{1}(d(1,u_{i}), \gamma_{2}(d(1,u_{i})))>e(0),$$ and hence $\gamma_{1}$ and $\gamma_{2}$ diverge which contradicts the fact that they intersect in $w$. This proves the claim.
We define the stopping times $$\tau_{i}=\inf\{n\geq 0: \overline{X}_{n}\in B_{i}\}.$$ In order to walk from $u$ to $v$ the walk has to enter each $B_{i}$, and so we find that $$F(u,v)\leq \PP_{u}[\tau_{1}<\infty, \tau_2<\infty,\ldots, \tau_{\lfloor d/D\rfloor}<\infty].$$ Our proof strategy is now to prove that $$\label{eq:lem:decay}
\PP[\tau_{i+1}=\infty \mid \tau_{i}<\infty]\geq c\quad \textrm{for all } i\in\{1,\ldots,\lfloor d/D\rfloor\}$$ for some constant $c>0$. An application of the strong Markov property yields then that $$F(u,v)\leq \PP_{u}[\tau_{1}<\infty, \tau_2<\infty,\ldots, \tau_{\lfloor d/D\rfloor}<\infty]\leq (1-c)^{\lfloor d/D\rfloor},$$ which proves the claim. Therefore it remains to prove (\[eq:lem:decay\]). Assume $\tau_{i}<\infty$ and denote $w=\overline{X}_{\tau_{i}}$. Due to the connectivity of spheres there exists some $w_{1}$ such that $e(0)+k/2\leq d(w,w_{1})\leq e(0)+3k/2$, $d(1,w)-{k/2}\leq d(1, w_{1})\leq d(1, w)+k/2$ and $w_{1}\notin B_{i}$. Now, due to the fact that geodesics either stay at bounded distance at most $e(0)$ or diverge, we find that $C(w_{1})$ does not intersect $B_{i+1}$. Hence, a walk started in $w_{1}$ will stay with positive probability of at least $\overline{p}_{esc}$ in $C(w_{1})$ and therefore will never visit $B_{i+1}$. As the probability that a walk started in $w$ will visit $w_{1}$ is bounded below by $\varepsilon^{e(0)+3k/2}$ we obtain (\[eq:lem:decay\]) with $c=\varepsilon^{e(0)+3k/2}\overline{p}_{esc}$.
For each $u\in W$ let $$D_{u}=\inf\{n\geq 1\mid \overline{X}_{n}\not\in \rho(\widehat{\mathcal{T}}^{(n)}_{x})\}.$$ It is crucial to bound the moments of $ D_u$. In the following $\EE_v$ denotes the expectation given that $\overline{X}_0= v$, $v\in W$.
\[lem:expmomentsDretract\] There exist constants $\lambda_{ D},K_{ D}$ such that for all $u\in W$ with $T(u)={\mathbf{T}}$ we have $$\EE_{v}[\exp(\lambda_{ D} D_{u}) {\mathbf{1}}_{\{ D_{u}<\infty\}}]\leq K_{ D}\quad\text{for all $v\in C(u)$}.$$
Since $p(v,z)\geq \varepsilon$ whenever $d(v, z)=1$, [@woessbook Lemma 8.1] guarantees the existence of a constant $A$ such that for all $v,z\in W$ we have $$\label{eq:Arho}
\bar p^{(n)}(v,z)\leq A^{d(v,z)} \vrho(\bar P)^{n}.$$ We proceed with the tails of $\PP_{v}[ D_{u}=n]$ for $u$ such that $T(u)={\mathbf{T}}$ and $v\in C(u)$. Let $\delta>0$ to be chosen later, then $$\label{eq:lem:momentbound:1}
\PP_{v}[ D_{u}=n+1]\leq \PP_{v}[d(v,\overline{X}_{n})\leq \delta n, D_{u}=n+1]+ \PP_{v}[d(v,\overline{X}_{n})\geq \delta n, D_{u}=n+1].$$ We will make use of the fact that $ D_{u}=n+1$ implies $\overline X_{n} \in \partial_{2L_1} C(u)$ for $n$ sufficiently large. Due to the planarity of $(W, S)$ we have that $ c(n)=|\partial_{2L_1} C(u)\cap \cB(v,\delta n)|$ grows linearly in $n$. The first summand in inequality (\[eq:lem:momentbound:1\]) can be bounded as follows: $$\begin{aligned}
\PP_{v}\left[d(v,\overline{X}_{n}) \leq \delta n, D_{u}=n+1\right]
&\leq & \PP_{v}[ d(v,\overline{X}_{n}) \leq \delta n, \overline{X}_{n}\in \partial_{2L_1} C(u)] \cr
&\leq & \sum_{z\in \partial_{2L_1} C(u)\cap \cB(v,\delta n)} \bar p^{(n)}(v,z)\cr
&\leq & \sum_{z\in \partial_{2L_1} C(u)\cap \cB(v,\delta n)} A^{d(v,z)} \vrho(\bar P)^{n}\cr
& \leq& c(n) \max\{1,A^{\delta n}\} \vrho(\bar P)^{n}.
\end{aligned}$$ Choose $\delta>0$ sufficiently small so that the latter sum decays exponentially in $n$.
For the second summand in (\[eq:lem:momentbound:1\]) we have $$\PP_{v}[d(v,\overline{X}_{n})\geq \delta n, D_{u}=n+1] \leq \sum_{z\in
\partial_{2L_1} C(u)\setminus {\cB(v,\delta n)}} F(v,z)
\leq \sum_{k=\lceil \delta n \rceil}^{\infty} c(k) C \lambda^{k},$$ which decays exponentially in $n$. The result follows.
It turns out that in order to give a good estimate on the length of the time intervals between renewal times, it suffices to control the tails of $D_{x}=\inf\{n\geq 1 \mid X_{n}\not\in \widehat{\mathcal{T}}^{(n)}_{x}\}$ on the event that $D_{x}$ is finite. However, this can be achieved by comparison with the retracted walk.
\[lem:expmomentsD\] There exists constants $\lambda_{D}',K_{D}'$ such that for all $x$ of type ${\mathbf{T}}$ we have $$\EE_{y}[\exp(\lambda_{D}' D_{x}) {\mathbf{1}}_{\{D_{x}<\infty\}}]\leq K_{D}'\quad\text{for all $y\in C(x)$}.$$
Let $u=\rho(x)$, then due to Proposition \[prop:Lboundaries\] and Lemma \[lem:equal-distance\] we have $$\begin{aligned}
\{ D_{x}=k\}&=& \{X_{k}\notin \widehat{\mathcal{T}}^{(k)}_{x}, \forall m<k: X_{m}\in \widehat{\mathcal{T}}^{(m)}_{x}) \}\cr
&\subseteq& \{\overline{X}_{k}\notin \rho(\widehat{\mathcal{T}}^{(k)}_{x}), \forall m<k: \overline{X}_{m}\in \rho(\widehat{\mathcal{T}}^{(m)}_{x}) \}\cr
&=& \{D_{u}=k\}.\end{aligned}$$ Hence, for all $y\in C(x)$ and $v=\rho(y)$ we have that $\PP_{y}[
D_{x}=k]\leq \PP_{v}[ D_{u}=k]$ and the claim follows with Lemma \[lem:expmomentsDretract\].
The essential ingredients are Lemmata \[lem:stayinconebuilding\] and \[lem:expmomentsD\] and the fact that Cannon automaton is strongly connected. Since the proof is analogous to the proof of [@HMM:13 Lemma 4.1] we only give a sketch of the arguments here. In fact, the proof is a more quantitative analysis of the arguments given in the proof of Theorem \[thm:reg\].1. Recall that wherever the walk is, it will reach a chamber of cone type ${\mathbf{T}}$ after at most $K$ steps with probability of at least $p_{h}$. By Lemma \[lem:stayinconebuilding\], each time the walk is at a chamber (in the $L_{1}$-interior) of cone type ${\mathbf{T}}$ it has a positive probability of at least $p_{esc}$ to perform a renewal step. If it does not perform a renewal step, it takes the walk a random time $D$ to exit $\widehat{\mathcal{T}}$. Now, again the walk will hit a chamber of cone type ${\mathbf{T}}$ in at most $K$ steps with probability of at least $p_{h}$ and we continue as above until we eventually performed one renewal step. The time until the walk does a renewal step can therefore be bounded by $\sum_{i=1}^{G} D_{i}$ where the $D_{i}$ are i.i.d. copies of $D$ and $G$ is a geometric random variable (independent of the $D_{i}$’s) with success probability $p_{h}p_{esc}$. Since the $G$ and the $D_{i}$’s have exponential moments one proves that $\sum_{i=1}^{G} D_{i}$ has exponential moments, too.
Automata and the proof of Theorem \[thm:stronglyconnected\] {#app:A}
===========================================================
The automatic structure of Coxeter groups was first proven by Brink and Howlett [@howlett] (see also [@bjorner Chapter 4]). In this section we explicitly construct the Cannon automaton for each Fuchsian Coxeter system, and deduce that these automata are strongly connected (hence proving Theorem \[thm:stronglyconnected\]). We also envisage that our explicit description of the automata will be useful for future work where precise information regarding the automata is required.
It is convenient to divide the set of all Fuchsian Coxeter systems into $4$ classes. First consider triangle groups. Let $(W,S)$ be a triangle group generated by $s,t,u$. Let $a=m_{st}$, $b=m_{tu}$, and $c=m_{us}$, and rename the generators if necessary so that $a\geq b\geq c\geq 2$. Then $W$ is infinite if and only if (see Example \[ex:21\]) $$\begin{aligned}
(a,b,c)\in\{(k_1,k_2,k_3),(k_4,k_5,2),(k_6,3,2)\mid k_1\geq k_2\geq k_3\geq 3,\, k_4\geq k_5\geq 4,\, k_6\geq 6\}.\end{aligned}$$ We partition the infinite triangle groups into $3$ classes:
- Class I consists of those groups with $a\geq b\geq c\geq 3$.
- Class II consists of those groups with $a\geq b\geq 4$ and $c=2$.
- Class III consists of those groups with $a\geq 6$, $b=3$, and $c=2$.
Note that the “root” of each class (that is, the group with $a+b+c$ minimal) is an affine triangle group: $(a,b,c)=(3,3,3),(4,4,2),(6,3,2)$. All other infinite triangle groups are Fuchsian. For no extra work we will include the affine triangle groups in our considerations.
The remaining Fuchsian Coxeter systems are those with $|S|\geq 4$. We call these Fuchsian Coxeter systems of Class IV.
Preliminaries
-------------
Before constructing the automata, we give some general background (see [@humphreys] for details). Let $(W,S)$ be a Fuchsian Coxeter complex (or an affine triangle group). The conjugates of the generators $S$ are called *reflections*. Thus the set of all reflections is $
R=\{wsw^{-1}\mid s\in S,w\in W\}.
$ Each reflection $r\in R$ determines a *wall* (also called a *hyperplane*) $H_r=\{\zeta\in\mathbb{H}^2\mid r\zeta=\zeta\}$ in $\mathbb{H}^2$ (or in $\mathbb{R}^2$ for Euclidean triangle groups).
Let $\mathcal{H}=\{H_r\mid r\in R\}$ be the set of all walls. Given a wall $H\in \mathcal{H}$ we write $s_H$ for the reflection in the wall $H$. Thus $s_H=r$ if $H=H_r$. More generally, if $H$ is the wall separating $w$ from $ws$ then $s_H=wsw^{-1}$.
Each wall $H\in\mathcal{H}$ determines two (closed) *half-spaces* of the hyperbolic disc $\HH^2$. The *positive side* of $H$ is the half-space $H^+$ which contains the identity chamber $1$, and the *negative side* of $H$ is the half space $H^-$ which does not contain $1$. Alternatively, we have $$\begin{aligned}
\label{eq:halfspaces}
H^+=\{w\in W\mid \ell(s_Hw)>\ell(w)\}\quad\textrm{and}\quad H^-=\{w\in W\mid \ell(s_Hw)<\ell(w)\}.\end{aligned}$$
If $w=s_1\cdots s_{\ell}$ is a reduced expression then for each $1\leq k\leq \ell$ the element $w$ is on the negative side of each wall $H_{r_k}$, where $r_k$ is the reflection $r_k=s_1\cdots s_{k-1}s_ks_{k-1}\cdots s_1$, and $\{H_{r_k}\mid k=1,\ldots,\ell\}$ is precisely the set of all walls separating $1$ from $w$.
\[lem:intersect\] Walls $H,H'$ of a Fuchsian Coxeter system (or affine triangle group) intersect if and only if the corresponding reflections $s_H$ and $s_{H'}$ generate a finite group.
Suppose that the walls $H$ and $H'$ intersect. If $H=H'$ then $s_H$ generates a group of order $2$, and if $H\neq H'$ then $H$ and $H'$ intersect at a single point $x\in \mathbb{H}^2$ (or $x\in\mathbb{R}^2$). By construction of the realisation there are only finitely many walls through $x$, and the group generated by the reflections in these walls is a conjugate of a finite standard parabolic subgroup. The group generated by $s_H$ and $s_{H'}$ is a subgroup of this finite group, and is thus finite.
On the other hand, if the reflections $s_H$ and $s_{H'}$ generate a finite group then by [@AB Proposition 2.87] the reflections $s_H$ and $s_{H'}$ both lie in a conjugate of a finite parabolic subgroup. Therefore the walls $H$ and $H'$ intersect.
The *left descent set* of $w\in W$ is $$L(w)=\{s\in S\mid \ell(sw)=\ell(w)-1\}.$$ Equivalently, $L(w)$ is the set of generators $s\in S$ for which there is a reduced expression for $w$ starting with the letter $s$. By [@AB Corollary 2.18] the subgroup of $W$ generated by $L(w)$ is finite for each fixed $w\in W$. Moreover, if $v$ is an element of the group generated by $L(w)$ then by [@AB Proposition 2.17] there exists an expression $$\begin{aligned}
\label{eq:startexpression}
w=vw'\quad\textrm{with $\ell(w)=\ell(v)+\ell(w')$}.\end{aligned}$$
Class I
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\[lem:reduction\] Let $(W,S)$ be a Coxeter system and let $s,t,u\in S$ be distinct generators. If $m_{st},m_{tu},m_{us}\geq 3$ then the subgroup of $W$ generated by $u$ and $tst$ is infinite.
Consider the word $w_n=(tstu)^n=tstutstutstu\cdots tstu$. If $m_{st},m_{tu}>3$ then $w_n$ has no available Coxeter moves, and is thus reduced, and so the subgroup generated by $u$ and $tst$ is infinite. In the cases that one or both of $m_{st}$ and $m_{tu}$ are $3$ then there are Coxeter moves available, although it is not hard to see that no reduction in the length of $w_n$ is possible, and so the subgroup is finite in these cases too.
\[lem:automaton1\] Let $(W,S)$ be a triangle group with $S=\{s,t,u\}$ and $3\leq
m_{st},m_{tu},m_{su}<\infty$. Let $x$ be the longest element of $W_{st}$. Let $v\in W_{st}$ with $v\notin \{x,1\}$, and let $v=s_{\ell}\cdots s_1$ be the unique reduced expression for $v$. Then
1. $T(vu)=T(s_1u)$,
2. $T(xus)=T(sus)$ and $T(xut)=T(tut)$.
1\. By symmetry we may suppose that $s_1=s$. We are required to show that for each fixed $w\in W$, $$\textrm{$vuw$ is reduced}\quad\textrm{if and only if}\quad \textrm{$suw$ is reduced}.$$ It is clear that if $vuw$ is reduced then $suw$ is also reduced (since truncations of reduced expressions are reduced). Suppose, for a contradiction, that $suw$ is reduced and that $vuw$ is not reduced. Let $2\leq k\leq \ell$ be minimal subject to $s_{k-1}\cdots s_1uw$ is reduced and $s_k\cdots s_1uw$ is not reduced. Since $\{s_{k-1},s_k\}=\{s,t\}$ we see that $s,t\in L(s_{k-1}\cdots s_1uw)$. Thus by (\[eq:startexpression\]) there is a reduced expression for $s_{k-1}\cdots s_1uw$ starting with any chosen reduced word in $W_{st}$. Since $\ell(s_{k-1}\cdots s_1)\leq m_{st}-2$ the word $s_{k-1}\cdots s_1ts\in W_{st}$ is reduced, and thus $s_{k-1}\cdots s_1uw=s_{k-1}\cdots s_1tsv'$ for some $v'\in W$ (with the expressions on both sides being reduced) and so $uw=tsv'$ (again with both expressions reduced). Therefore the element $uw=tsv'$ lies on the negative side of the wall $H$ separating $1$ from $u$, and on the negative side of the wall $H'$ separating $t$ from $ts$, and so $H^-\cup(H')^-\neq\emptyset$. The reflection in the hyperplane $H$ is $s_H=u$, and the reflection in the hyperplane $H'$ is $s_{H'}=tut$. Either $H^-\subseteq (H')^-$, or $(H')^-\subseteq H^-$, or $H$ and $H'$ intersect. If $H^-\subseteq (H')^-$ then $u\in (H')^-$, and so $\ell(s_{H'}u)<\ell(u)=1$. But $s_{H'}u=tstu$ has length $4$. Similarly, if $(H')^-\subseteq H^-$ then $ts\in H^-$ and so $\ell(s_Hts)<\ell(ts)=2$, but $s_Hts=uts$ has length $3$. Therefore $H$ and $H'$ intersect, and so by Lemma \[lem:intersect\] the group generated by $s_H=u$ and $s_{H'}=tst$ is finite, contradicting Lemma \[lem:reduction\].
2\. We show that $T(xus)=T(sus)$. We are required to show that for each fixed $w\in W$, $$\textrm{$xusw$ is reduced}\quad\textrm{if and only if}\quad \textrm{$susw$ is reduced}.$$ Choose the reduced expression $x=s_{\ell}\cdots s_1$ with $s_1=s$. Thus the expression $susw$ can be regarded as a truncation of the expression $xusw$, and so it follows that if the latter is reduced then the former is also reduced. Suppose, for a contradiction, that $susw$ is reduced and that $xusw$ is not reduced. Let $2\leq k\leq \ell$ be minimal subject to $s_{k-1}\cdots s_1usw$ is reduced and $s_k\cdots s_1usw$ is not reduced. Then $\{s,t\}\in L(s_{k-1}\cdots s_1usw)$ and so there is a reduced expression $s_{k-1}\cdots s_1usw=s_{k-1}\cdots s_1tv'$ for some $v'\in W$, and so $usw=tv'$ with both sides reduced. Hence, as above, the group generated by $usu$ and $t$ is finite, contradicting Lemma \[lem:reduction\].
Lemma \[lem:automaton1\] completely determined the Cannon automaton for all triangle groups in Class I. An illustration is given in Figure \[fig:I\].
\[scale=6\] (0,0) – (0,0.25); (0,0) – ([0.25\*cos(-30)]{},[0.25\*sin(-30)]{}); (0,0) – ([0.25\*cos(210)]{},[0.25\*sin(210)]{}); plot ([0.3\*cos()]{},[0.3\*sin()-0.335]{}); plot ([0.3\*cos()]{},[0.3\*sin()-0.335]{}); plot ([0.3\*cos()]{},[0.3\*sin()-0.335]{}); plot ([0.3\*cos()]{},[0.3\*sin()-0.335]{}); plot ([0.3\*cos()]{},[0.3\*sin()-0.335]{}); plot ([0.3\*cos()]{},[0.3\*sin()-0.335]{}); plot ([0.3\*cos()]{},[0.3\*sin()-0.335]{}); plot ([0.3\*cos()]{},[0.3\*sin()-0.335]{}); ([0.3\*cos(-90)]{},[0.3\*sin(-90)-0.335]{}) – ([0.3\*cos(-90)]{},[0.3\*sin(-90)-0.335-0.16]{}); ([0.3\*cos(-90)]{},[0.3\*sin(-90)-0.335-0.16]{}) – ([0.3\*cos(-90)-0.16]{},[0.3\*sin(-90)-0.335-0.16]{}); ([0.3\*cos(-90)]{},[0.3\*sin(-90)-0.335-0.16]{}) – ([0.3\*cos(-90)+0.16]{},[0.3\*sin(-90)-0.335-0.16]{}); ([0.3\*cos(170)]{},[0.3\*sin(170)-0.335]{}) – ([0.2\*cos(170)]{},[0.2\*sin(170)-0.335]{}); ([0.3\*cos(205)]{},[0.3\*sin(205)-0.335]{}) – ([0.2\*cos(205)]{},[0.2\*sin(205)-0.335]{}); ([0.3\*cos(240)]{},[0.3\*sin(240)-0.335]{}) – ([0.2\*cos(240)]{},[0.2\*sin(240)-0.335]{}); ([0.3\*cos(10)]{},[0.3\*sin(10)-0.335]{}) – ([0.2\*cos(10)]{},[0.2\*sin(10)-0.335]{}); ([0.3\*cos(-25)]{},[0.3\*sin(-25)-0.335]{}) – ([0.2\*cos(-25)]{},[0.2\*sin(-25)-0.335]{}); ([0.3\*cos(-60)]{},[0.3\*sin(-60)-0.335]{}) – ([0.2\*cos(-60)]{},[0.2\*sin(-60)-0.335]{}); (0,-0.05) node ([0.25\*cos(-30)-0.02]{},[0.25\*sin(-30)-0.04]{}) node ([-0.25\*cos(-30)+0.02]{},[0.25\*sin(-30)-0.04]{}) node (-0.04,0.23) node ([0.3\*cos(10)+0.05]{},[0.3\*sin(10)-0.335]{}) node ([0.3\*cos(-25)+0.06]{},[0.3\*sin(-25)-0.335]{}) node ([0.3\*cos(-60)+0.03]{},[0.3\*sin(-60)-0.335-0.06]{}) node ([-0.3\*cos(10)-0.05]{},[0.3\*sin(10)-0.335]{}) node ([-0.3\*cos(-25)-0.06]{},[0.3\*sin(-25)-0.335]{}) node ([-0.3\*cos(-60)-0.03]{},[0.3\*sin(-60)-0.335-0.06]{}) node ([-0.3\*cos(-90)]{},[0.3\*sin(-90)-0.335+0.05]{}) node ([-0.3\*cos(-90)]{},[0.3\*sin(-90)-0.335+0.05-0.25]{}) node ([-0.3\*cos(-90)+0.21]{},[0.3\*sin(-90)-0.335-0.16]{}) node \[gray\] ([-0.3\*cos(-90)-0.21]{},[0.3\*sin(-90)-0.335-0.16]{}) node \[gray\] ([0.2\*cos(10)-0.04]{},[0.2\*sin(10)-0.335]{}) node \[gray\] ([0.2\*cos(-25)-0.04]{},[0.2\*sin(-25)-0.335]{}) node \[gray\] ([0.2\*cos(-60)-0.04+0.01]{},[0.2\*sin(-60)-0.335+0.025]{}) node \[gray\] ([-0.2\*cos(10)+0.04]{},[0.2\*sin(10)-0.335]{}) node \[gray\] ([-0.2\*cos(-25)+0.04]{},[0.2\*sin(-25)-0.335]{}) node \[gray\] ([-0.2\*cos(-60)+0.04-0.01]{},[0.2\*sin(-60)-0.335+0.025]{}) node \[gray\] ;
plot ([0.3\*cos()]{},[0.3\*sin()-0.335]{}); plot ([0.3\*cos()]{},[0.3\*sin()-0.335]{}); plot ([0.3\*cos()]{},[0.3\*sin()-0.335]{}); plot ([0.3\*cos()]{},[0.3\*sin()-0.335]{}); plot ([0.3\*cos()]{},[0.3\*sin()-0.335]{}); plot ([0.3\*cos()]{},[0.3\*sin()-0.335]{}); plot ([0.3\*cos()]{},[0.3\*sin()-0.335]{}); plot ([0.3\*cos()]{},[0.3\*sin()-0.335]{}); ([0.3\*cos(-90)]{},[0.3\*sin(-90)-0.335]{}) – ([0.3\*cos(-90)]{},[0.3\*sin(-90)-0.335-0.16]{}); ([0.3\*cos(-90)]{},[0.3\*sin(-90)-0.335-0.16]{}) – ([0.3\*cos(-90)-0.16]{},[0.3\*sin(-90)-0.335-0.16]{}); ([0.3\*cos(-90)]{},[0.3\*sin(-90)-0.335-0.16]{}) – ([0.3\*cos(-90)+0.16]{},[0.3\*sin(-90)-0.335-0.16]{}); ([0.3\*cos(170)]{},[0.3\*sin(170)-0.335]{}) – ([0.2\*cos(170)]{},[0.2\*sin(170)-0.335]{}); ([0.3\*cos(205)]{},[0.3\*sin(205)-0.335]{}) – ([0.2\*cos(205)]{},[0.2\*sin(205)-0.335]{}); ([0.3\*cos(240)]{},[0.3\*sin(240)-0.335]{}) – ([0.2\*cos(240)]{},[0.2\*sin(240)-0.335]{}); ([0.3\*cos(10)]{},[0.3\*sin(10)-0.335]{}) – ([0.2\*cos(10)]{},[0.2\*sin(10)-0.335]{}); ([0.3\*cos(-25)]{},[0.3\*sin(-25)-0.335]{}) – ([0.2\*cos(-25)]{},[0.2\*sin(-25)-0.335]{}); ([0.3\*cos(-60)]{},[0.3\*sin(-60)-0.335]{}) – ([0.2\*cos(-60)]{},[0.2\*sin(-60)-0.335]{}); ([0.3\*cos(10)+0.05]{},[0.3\*sin(10)-0.335]{}) node ([0.3\*cos(-25)+0.06]{},[0.3\*sin(-25)-0.335]{}) node ([0.3\*cos(-60)+0.03]{},[0.3\*sin(-60)-0.335-0.06]{}) node ([-0.3\*cos(10)-0.05]{},[0.3\*sin(10)-0.335]{}) node ([-0.3\*cos(-25)-0.06]{},[0.3\*sin(-25)-0.335]{}) node ([-0.3\*cos(-60)-0.03]{},[0.3\*sin(-60)-0.335-0.06]{}) node ([-0.3\*cos(-90)]{},[0.3\*sin(-90)-0.335+0.05]{}) node ([-0.3\*cos(-90)]{},[0.3\*sin(-90)-0.335+0.05-0.25]{}) node ([-0.3\*cos(-90)+0.21]{},[0.3\*sin(-90)-0.335-0.16]{}) node \[gray\] ([-0.3\*cos(-90)-0.21]{},[0.3\*sin(-90)-0.335-0.16]{}) node \[gray\] ([0.2\*cos(10)-0.04]{},[0.2\*sin(10)-0.335]{}) node \[gray\] ([0.2\*cos(-25)-0.04]{},[0.2\*sin(-25)-0.335]{}) node \[gray\] ([0.2\*cos(-60)-0.04+0.01]{},[0.2\*sin(-60)-0.335+0.025]{}) node \[gray\] ([-0.2\*cos(10)+0.04]{},[0.2\*sin(10)-0.335]{}) node \[gray\] ([-0.2\*cos(-25)+0.04]{},[0.2\*sin(-25)-0.335]{}) node \[gray\] ([-0.2\*cos(-60)+0.04-0.01]{},[0.2\*sin(-60)-0.335+0.025]{}) node \[gray\] ;
plot ([0.3\*cos()]{},[0.3\*sin()-0.335]{}); plot ([0.3\*cos()]{},[0.3\*sin()-0.335]{}); plot ([0.3\*cos()]{},[0.3\*sin()-0.335]{}); plot ([0.3\*cos()]{},[0.3\*sin()-0.335]{}); plot ([0.3\*cos()]{},[0.3\*sin()-0.335]{}); plot ([0.3\*cos()]{},[0.3\*sin()-0.335]{}); plot ([0.3\*cos()]{},[0.3\*sin()-0.335]{}); plot ([0.3\*cos()]{},[0.3\*sin()-0.335]{}); ([0.3\*cos(-90)]{},[0.3\*sin(-90)-0.335]{}) – ([0.3\*cos(-90)]{},[0.3\*sin(-90)-0.335-0.16]{}); ([0.3\*cos(-90)]{},[0.3\*sin(-90)-0.335-0.16]{}) – ([0.3\*cos(-90)-0.16]{},[0.3\*sin(-90)-0.335-0.16]{}); ([0.3\*cos(-90)]{},[0.3\*sin(-90)-0.335-0.16]{}) – ([0.3\*cos(-90)+0.16]{},[0.3\*sin(-90)-0.335-0.16]{}); ([0.3\*cos(170)]{},[0.3\*sin(170)-0.335]{}) – ([0.2\*cos(170)]{},[0.2\*sin(170)-0.335]{}); ([0.3\*cos(205)]{},[0.3\*sin(205)-0.335]{}) – ([0.2\*cos(205)]{},[0.2\*sin(205)-0.335]{}); ([0.3\*cos(240)]{},[0.3\*sin(240)-0.335]{}) – ([0.2\*cos(240)]{},[0.2\*sin(240)-0.335]{}); ([0.3\*cos(10)]{},[0.3\*sin(10)-0.335]{}) – ([0.2\*cos(10)]{},[0.2\*sin(10)-0.335]{}); ([0.3\*cos(-25)]{},[0.3\*sin(-25)-0.335]{}) – ([0.2\*cos(-25)]{},[0.2\*sin(-25)-0.335]{}); ([0.3\*cos(-60)]{},[0.3\*sin(-60)-0.335]{}) – ([0.2\*cos(-60)]{},[0.2\*sin(-60)-0.335]{}); ([0.3\*cos(10)+0.05]{},[0.3\*sin(10)-0.335]{}) node ([0.3\*cos(-25)+0.06]{},[0.3\*sin(-25)-0.335]{}) node ([0.3\*cos(-60)+0.03]{},[0.3\*sin(-60)-0.335-0.06]{}) node ([-0.3\*cos(10)-0.05]{},[0.3\*sin(10)-0.335]{}) node ([-0.3\*cos(-25)-0.06]{},[0.3\*sin(-25)-0.335]{}) node ([-0.3\*cos(-60)-0.03]{},[0.3\*sin(-60)-0.335-0.06]{}) node ([-0.3\*cos(-90)]{},[0.3\*sin(-90)-0.335+0.05]{}) node ([-0.3\*cos(-90)]{},[0.3\*sin(-90)-0.335+0.05-0.25]{}) node ([-0.3\*cos(-90)+0.21]{},[0.3\*sin(-90)-0.335-0.16]{}) node \[gray\] ([-0.3\*cos(-90)-0.21]{},[0.3\*sin(-90)-0.335-0.16]{}) node \[gray\] ([0.2\*cos(10)-0.04]{},[0.2\*sin(10)-0.335]{}) node \[gray\] ([0.2\*cos(-25)-0.04]{},[0.2\*sin(-25)-0.335]{}) node \[gray\] ([0.2\*cos(-60)-0.04+0.01]{},[0.2\*sin(-60)-0.335+0.025]{}) node \[gray\] ([-0.2\*cos(10)+0.04]{},[0.2\*sin(10)-0.335]{}) node \[gray\] ([-0.2\*cos(-25)+0.04]{},[0.2\*sin(-25)-0.335]{}) node \[gray\] ([-0.2\*cos(-60)+0.04-0.01]{},[0.2\*sin(-60)-0.335+0.025]{}) node \[gray\] ;
Figure \[fig:I\] shows the Cannon automaton for Class I triangle groups. The generators are labelled $1$, $2$, and $3$, and the labels on the edges are indicated by colours, with $1=\text{green}$, $2=\text{blue}$, and $3=\text{red}$. The cone types are given by the base element of a representative cone of that type. The cone types in grey are duplicates, and the reader should instead imagine the arrow pointing to the corresponding cone type in black. Finally, $x=121\cdots$, $y=232\cdots$ and $z=131\cdots$ are the longest elements of the parabolic subgroups $W_{12}$, $W_{23}$, and $W_{13}$ respectively.
Class II
--------
The proof of the following lemma, which determines the Cannon automata for the triangle groups in Class II, is similar to the proof of Lemma \[lem:automaton1\] and the details are omitted.
\[lem:lem:automaton2\] Let $(W,S)$ be a triangle group with $S=\{s,t,u\}$. Suppose that $m_{st},m_{tu}\geq 4$ and that $m_{su}=2$. Let $x$ be the longest element of $W_{st}$. Let $v\in W_{st}$ with $v\notin \{x,xs\}$, and write $v=s_{\ell}\cdots s_1$ for the unique reduced expression for $v$. Then $$\begin{aligned}
T(vu)&=T(s_1u)&T((xs)ut)&=T(tut)&T(xutu)&=T(tutu)&T(xutsu)&=T(tutsu)\\
T(xutst)&=T(stst)&T(suts)&=T(sts)&T(sutu)&=T(utu).\end{aligned}$$
The resulting automaton has $a+b+c+4$ vertices, given by $T(w)$ with $w\in{\mathbf{W}}$, where $${\mathbf{W}}=W_{st}\cup W_{tu}\cup W_{us}\cup \{sut\}\cup x\cdot\{su,u,ut,uts\}\cup y\cdot\{us,s,st,stu\},$$ where $x=sts\cdots$ is the longest element of $W_{st}$ and $y=tut\cdots$ is the longest element of $W_{tu}$. See Figure \[fig:II\] for an illustration.
Class III
---------
The proof of the following lemma, which determines the Cannon automaton for the triangle groups in Class III, is similar to the proof of Lemma \[lem:automaton1\] and the details are omitted.
\[lem:automaton3\] Let $(W,S)$ be a triangle group with $S=\{s,t,u\}$. Suppose that $m_{st}\geq 6$, $m_{tu}=3$, and $m_{su}=2$. Let $x$ be the longest element element of $W_{st}$. Let $v\in W_{st}$ with $v\notin \{1,x,xs,xt,xts\}$, and let $v=s_{\ell}\cdots s_1$ be the unique reduced expression for $v$. Then $$\begin{aligned}
T(vu)&=T(s_1u)&T(xutstst)&=T(ststst)&T((xs)ut)&=T(tut)\\
T(xutstst)&=T(ststst)&T(xutststut)&=T(stststut)&T(xutststuts)&=T(stststuts)\\
T(xutststutstu)&=T(stststutstu)&T((xt)utstutststs)&=T(tststs)&T((xts)ut)&=T(tut)\\
T((xt)utsts)&=T(ststs)&T((xt)utstsu)&=T(ststsu)&T(tutstst)&=T(tstst)\\
T(utstuts)&=T(tuts)&T(utstut)&=T(tut)&T(utst)&=T(tst)\\
T(stut)&=T(tut)&T(ustst)&=T(stst)&T(stuts)&=T(tuts)\end{aligned}$$
An illustration of the resulting automaton is given in Figure \[fig:III\].
Class IV
--------
The proof of the following lemma, which determines the Cannon automata for the triangle groups in Class IV, is similar to the proof of Lemma \[lem:automaton1\] and the details are omitted.
\[lem:automaton4\] Let $(W,S)$ be a Fuchsian Coxeter system, and let $s,t,u\in S$ be pairwise distinct generators.
1. If $2\leq m_{st},m_{tu}<\infty$ and $m_{us}=\infty$ then for all $v\in W_{st}$ we have $$T(vu)=\begin{cases}
T(u)&\textrm{if $\ell(vt)=\ell(v)+1$}\\
T(tu)&\textrm{if $\ell(vt)=\ell(v)-1$}.
\end{cases}$$
2. If $2\leq m_{st}<\infty$ and $m_{tu}=m_{us}=\infty$ then for all $v\in W_{st}$ we have $
T(vu)=T(u)
$
The resulting automaton has vertices given by $\{T(w)\mid w\in{\mathbf{W}}\}$, where ${\mathbf{W}}$ is the union of all finite parabolic subgroups: $${\mathbf{W}}=W_{1,2}\cup W_{2,3}\cup \cdots \cup W_{n-1,n}\cup W_{n,1},$$ where the generators are labelled $1,2,\ldots,n$.
Proof of Theorem \[thm:stronglyconnected\]
------------------------------------------
Consider Class I first. It is clear that the cone types $\emptyset$, $1$, $2$ and $3$ are not recurrent (in the notation of Figure \[fig:I\], we will simply write $w$ for the cone type $T(w)$).
Suppose, without loss of generality, that $m_{12}>3$. We first note that there is a cycle $$c=(12\to23\to31\to12\to 121\to13\to32\to 21\to 212\to 23\to31\to 12)$$ containing all cone types $w$ with $\ell(w)=2$ (it is important here that $121$ and $212$ are not the longest words of $W_{12}$).
Now let $w$ be a cone type other than $\emptyset$, $1$, $2$, or $3$. From any such $w$ there is a path $\gamma_1$ in the automaton to some cone type $w_{ij}k$ where $(i,j,k)$ is some permutation of the generating set $(1,2,3)$ and $w_{ij}=iji\cdots$ is the longest element of $W_{ij}$. Then $$\begin{aligned}
w_{ij}k\to jkj\to\begin{cases}
ji&\text{if $m_{jk}>3$}\\
jkji\to jij\to jk&\text{if $m_{jk}=3$ and $m_{ij}>3$}\\
jkji\to jij\to ijik\to kik\to kj&\text{if $m_{jk}=m_{ij}=3$ and $m_{ik}>3$},
\end{cases}\end{aligned}$$ and so in all cases there is a path $\gamma_2$ from $w_{ij}k$ to a cone type $i'j'$ of length $2$, and so there is a path $\gamma_1\gamma_2$ from $w$ to $i'j'$ in the automaton. Furthermore, there is a path $\gamma_3$ from some cone type $i''j''$ to $w$ (because every reduced path in $W$ must pass through a word of length $2$). Thus, using $c$, there is a loop $\gamma$ from $12$ to $12$ passing through $w$. This shows that $w$ is recurrent, and readily implies that the automaton is strongly connected.
Now consider Class II. The set $\mathbf{R}$ of recurrent cone types is as follows: $$\begin{aligned}
\mathbf{R}=\begin{cases}
\{T(w)\mid w\in{\mathbf{W}}\text{ with }\ell(w)\geq 2\}\backslash\{st,ut\}&\text{if $m_{st},m_{tu}>4$}\\
\{T(w)\mid w\in{\mathbf{W}}\text{ with }\ell(w)\geq 2\}\backslash\{st,ut,tu\}&\text{if $m_{st}=4$ and $m_{tu}>4$}\\
\{T(w)\mid w\in{\mathbf{W}}\text{ with }\ell(w)\geq 2\}\backslash\{st,ut,ts\}&\text{if $m_{st}>4$ and $m_{tu}=4$}\\
\{T(w)\mid w\in{\mathbf{W}}\text{ with }\ell(w)\geq 2\}\backslash\{st,ut,ts,tu\}&\text{if $m_{st}=m_{tu}=4$}.
\end{cases}\end{aligned}$$ Direct inspection (see Figure \[fig:II\]) shows that the automaton is strongly connected as long as either $m_{st}>4$ or $m_{tu}>4$. We omit the full details, however for example consider the concrete case $m_{st}=4$ and $m_{tu}=5$ (thus in Figure \[fig:II\] we have $x=1212=2121$ and $y=23232=32323$). We have the following paths (in the notation of Figure \[fig:II\]) $$\begin{aligned}
\gamma_1&=(13\to132\to 121\to 1212=x\to x3\to x32)\\
\gamma_2&=(x32\to x321\to 1212=x\to x3\to x32\to 2323\to 13)\\
\gamma_3&=(13\to132\to 323\to 3232=y3\to y31\to 212=x1\to x13\to 232\to 2323\to 13)\\
\gamma_4&=(x32\to 2323\to 23232=y\to y1\to y12\to y123\to 2323\to 13),\end{aligned}$$ and the concatenation $\gamma_1\gamma_2\gamma_3\gamma_1\gamma_4$ gives a loop containing all recurrent cone types. Thus the automaton is strongly connected.
We omit the details of Class III, and refer the reader to Lemma \[lem:automaton3\] and Figure \[fig:III\].
Finally, consider the groups in Class IV. Let $1,2,\ldots,n$ be the generators of $W$, arranged cyclically around the fundamental chamber. If $n\geq 5$, then for each pair $(i,i+1)$ there is a generator $j$ with $m_{i,j}=\infty$ and $m_{{i+1},j}=\infty$, and thus $i\to j \to {i+1}$ in the automaton. Moreover, for any $w\in W_{i,i+1}$ we have $w\to j$. It follows that every node other than $\emptyset$ is recurrent, and moreover that the automaton is strongly connected. If $n=4$ then we may assume that $m_{12}\geq 3$ (for if $m_{ij}=2$ for all $i,j$ then $W$ is affine type $\tilde{A}_1\times\tilde{A}_1$, where $\tilde{A}_1$ is the infinite dihedral group). Then $21\to 3$ and $12\to 4$. Thus $1\to 12\to 4\to 2\to21\to3\to1$. Again it follows that every node other than $\emptyset$ is recurrent, and that the automaton is strongly connected.
The Cannon automata for the affine triangle groups $(3,3,3)$, $(4,4,2)$ and $(6,3,2)$ are not strongly connected. For example, for the $(4,4,2)$ group, note that the cone types $13$ and $1212$ are recurrent, yet there is no path from $1212$ to $13$ in the automaton.
\[scale=2.7\] ;
\[scale=4\] ;
[100]{}
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Lorenz GilchInstitut für Mathematische StrukturtheorieTechnische Universität GrazSteyrergasse 308010 Graz, Austria`gilch@TUGraz.at`
Sebastian MüllerAix Marseille UniversitéCNRS Centrale Marseille Institut de Mathématiques de Marseille (I2M)UMR 737313453 Marseille France`mueller@cmi.univ-mrs.fr`
James ParkinsonSchool of Mathematics and StatisticsUniversity of SydneyCarslaw Building, F07NSW, 2006, Australia`jamesp@maths.usyd.edu.au`
[^1]: Research partly supported under CIRM Research in Pairs program
[^2]: Research partly supported under the Australian Research Council (ARC) discovery grant DP110103205, Austrian Science Fund W1230 and P24028, and CIRM Research in Pairs program
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'In this paper, we investigate a commonsense inference task that unifies natural language understanding and commonsense reasoning. We describe our attempt at SemEval-2020 Task 4 competition: Commonsense Validation and Explanation (ComVE) challenge. We discuss several state-of-the-art deep learning architectures for this challenge. Our system uses prepared labeled textual datasets that were manually curated for three different natural language inference tasks. The goal of the first subtask is to test whether a model can distinguish between natural language statements that make sense and those that do not make sense. We compare the performance of several language models and fine-tuned classifiers. Then, we propose a method inspired by question/answering tasks to treat a classification problem as a multiple choice question task to boost the performance of our experimental results (96.06%), which is significantly better than the baseline. For the second subtask, which is to select the reason why a statement does not make sense, we stand within the first six teams (93.7%) among 27 participants with very competitive results[^1].'
author:
- |
Sirwe Saeedi\
Western Michigan University, USA\
[sirwe.saeedi@wmich.edu]{}\
Aliakbar Panahi\
Virginia Commonwealth University, USA\
[panahia@vcu.edu]{}\
Seyran Saeedi\
Virginia Commonwealth University, USA\
[saeedis@vcu.edu]{}\
Alvis C Fong\
Western Michigan University, US\
[alvis.fong@wmich.edu]{}
bibliography:
- 'semeval2020.bib'
title: 'CS-NLP team at SemEval-2020 Task 4: Evaluation of State-of-the-art NLP Deep Learning Architectures on Commonsense Reasoning Task'
---
[**Keywords**]{}: Artificial Intelligence, Natural Language Processing, Commonsense Reasoning and Knowledge, Language Models, Transformers, Self-Attention.
Introduction
============
Commonsense is unstated background knowledge that is used to perceive, infer, and understand the physical world, human emotions, reactions, and knowledge of the common facts that most people agree with. Ordinary commonsense helps us to differentiate between simple false and true statements or answer questions, such as “can an elephant fit into the fridge” quickly, but they can be difficult for automatic systems [@c206796aef5c4a1e8fda075d6fd94673]. Recent advances in machine learning emphasize the importance of commonsense reasoning in natural language processing (NLP) and as a critical component of artificial intelligence (AI). In the fifty-year history of AI research, progress was extremely slow [@4aba22e1f5b0492bab5811af4028ff48] in automated commonsense reasoning. However, when transfer learning [@Yosinski2014HowTA; @Goodfellow-et-al-2016] and then transformers were introduced to the NLP world [@vaswani1706attention], great breakthroughs and developments have occurred at an unprecedented pace [@pan2009survey; @Tan2018ASO]. Advances in machine learning and deep learning methods have been achieved in numerous studies and wide range of disciplines [@Panahi2019word2ketSW; @article; @rs12091361; @arodz2019quantum].
This paper describes a system participating in the SemEval-2020 “Commonsense Validation and Explanation (ComVE) Challenge”, multiple tasks of commonsense reasoning and Natural Language Understanding (NLU) designed by [@wang-etal-2020-semeval]. The competition is divided into three subtasks, which involve testing commonsense reasoning in automatic systems, multiple choice questions, and text generation. In these tasks, participants are asked to improve the performance of previous efforts [@wang2019does]. We apply transfer learning to reuse a pretrained model on different data distribution and feature space as the starting point of our target tasks. Applying Transfer Learning to NLP significantly improves the learning process in terms of time and computation through the transfer of knowledge from a related task that has already been learned [@10.5555/1803899].
Recently, there have been some excellent advancements towards transfer learning, and its success was illuminated by OpenAI transformer (GPT-2) [@radford2019language], Universal Language Model Fine-tuning for Text Classification (ULMFiT) by fast.ai founder Jeremy Howard [@howard2018universal], ELMo [@Peters:2018], and BERT [@devlin2018bert], and other new waves of cutting-edge methods and architectures like XLNet [@yang2019xlnet], Facebook AI RoBERTa: A Robustly Optimized BERT Pretraining Approach [@liu2019roberta], ALBERT: A Lite BERT for Self-supervised Learning of Language Representations [@lan2019albert], T5 team google [@raffel2019exploring], and CTRL [@keskar2019ctrl]. For this work, we employ and fine-tune some of these suitable models.
The paper “Attention Is All You Need” [@vaswani1706attention] describes a sequence-to-sequence architecture called transformers relying entirely on the self-attention mechanism and does not rely on any recurrent network such as GRU [@chung2014empirical] and LSTM [@hochreiter1997long]. Transformers consist of Encoders and Decoders. The encoder takes the input sequence and then decides which other parts of the sequence are important by attributing different weights to them. Decoder turns the encoded sentence into another sequence corresponding to the target task.
A huge variety of downstream tasks have been devised to test a model’s understanding of a specific aspect of language. The General Language Understanding Evaluation (GLUE) [@wang2018glue] and The Natural Language Decathlon (decaNLP) benchmarks [@mccann2018natural] consist of difficult and diverse natural language task datasets. These benchmarks span complex tasks, such as question answering, machine translation, textual entailment, natural language inference, and commonsense pronoun resolution. The majority of state-of-the-art transformers models publish their results for all tasks on the GLUE benchmark. For example, models like different modified versions of BERT, RoBERTa, and T5 outperform the human baseline benchmark [@zhu2019freelb; @wang2019structbert]. For the evaluation phase, GLUE follows the same approach as SemEval.
Our attempts at SemEval-2020 Task4 competition, boost performance on two subtasks significantly. Our idea in reframing the first subtask helps to outperform results of state-of-the-art architecture and language models like BERT, AlBERT, and ULMFiT. We were ranked 11 with a very competitive result on the first subtask and achieved rank 6 for the second subtask amongst 40 and 27 teams, respectively.
This paper is organized as follows. In Section 2, we introduce the three subtasks and their datasets. In Section 3, we describe our different applied models and various strategies that were used to fine tune the models for each individual subtask. In Section 4, we present the performance of our system. Finally, we conclude the paper in Section 5.
Task Definition and Datasets
============================
As discussed, SemEval-2020 Task 4 consists of three subtasks, each designed for a different natural language inference task. Figure \[fig:Figure1\] shows a sample for each subtask and the corresponding answer of the model.
- [**SubtaskA (Commonsense Validation)**]{}: Given two English statements with similar wordings, decide which one does not make sense. We had access to 10,000, and 2021 human-labeled pairs of sentences for training and trial models, respectively. After releasing the dev set, we combined these two datasets for training phase and used the dev set to test our models.
- [**SubtaskB (Explanation)**]{}: Given the nonsense statement, select the correct option among three to reason why the statement conflicts with human knowledge. The number of samples in the datasets for this task is similar to subtaskA. Each sample contains the incorrect statement from subtaskA and three candidate reasons to explain why this is against commonsense knowledge.
- [**SubtaskC (Reason Generating)**]{}: Given the nonsense statement, generate an understandable reason in the form of a sequence of words to verify why the statement is against human knowledge. Training samples of datasets for this subtask are all of the false sentences in subtaskA as well as for trial and dev set.
![Sample of training data for each subtask[]{data-label="fig:Figure1"}](figure){width="9cm" height="9cm"}
Model Description
=================
Large pretrained language models are definitely the main trend of the latest NLP breakthroughs. As transformers occupy the NLP leaderboards, we choose several state-of-the-art architectures to outperform the baseline of all subtasks significantly. For each subtask, we describe our system separately below.
SubtaskA (Commonsense Validation)
---------------------------------
We consider two approaches to address this task: the first method is based on language models, and the second approach uses classifiers. Our experimental process begins with language models which the key idea behind the first approach is to find the probability of appearing each word in statements and then select one with higher multiplication of probabilities.
Our first try involves fine-tuning pretrained model on AWD-LSTM, which performs as poor as a random guess. We also try two other different language models: ‘BERT’ the Masked Language Model (MLM) that attempts to predict the original value of the masked words, based on the non-masked words in the sequence of words, and then transformer.
Original BERT uniformly selects 15% of the input tokens for possible replacement. Of the selected tokens, 80% are replaced with ‘\[MASK\]’, 10% are left unchanged, and 10% are replaced by a randomly selected vocabulary token [@devlin2018bert] . However, our way of using MLM follows these steps:
1. Add special tokens to the beginning and end of each sentence.
2. Replace each token from left to right by ‘\[MASK\]’ each time.
\[‘\[MASK\]’, ‘He’, ‘drinks’, ‘apple’, ‘\[SEP\]’\],
\[‘\[CLS\]’, ‘\[MASK\]’, ‘drinks’, ‘apple’, ‘\[SEP\]’\],
\[‘\[CLS\]’, ‘He’, ‘\[MASK\]’, ‘apple’, ‘\[SEP\]’\],
\[‘\[CLS\]’, ‘He’, ‘drinks’, ‘\[MASK\]’, ‘\[SEP\]’\],
\[‘\[CLS\]’, ‘He’, ‘drinks’, ‘apple’, ‘\[MASK\]’\]
3. Feed them to MLM for predicting the probabilities of the original masked tokens.
4. Normalize predicted probabilities using softmax activation function in the output layer.
5. Multiply predicted probabilities of masked tokens for each pair of statement. The correct sentence has a higher probability.
During the consideration of dataset homogeneity, we observe some samples are ended by periods and some others are not. The most frequent reason for using periods is to mark the end of sentences that are not questions or exclamations. By adding a period at the end of all statements, we increase the accuracy by 4%, which is remarkable. We also try normalization and padding to boost performance of the model to minimize the impact of sequence length. Surprisingly, during normalization in step 5 we observe that normalizing by the length of sequence root of multiplied probabilities do not improve the performance of model. Similarly, normalization using perplexity does not increase the accuracy of model. We observe that padding does not make any differences in terms of accuracy. Therefore, the result of BERT MLM is almost the same with the baseline, which is achieved by fine-tuned ELMo as reported by [@wang2019does]. As a result, our observation shows BERT MLM model is more suitable for long document understanding; however, the maximum length (27) of our samples is too short.
As mentioned, we consider classifiers as the second approach to deal with this task. We show that the classification-based approach is more efficient in recognizing nonsense statements except Universal Language Model Fine-tuning (ULMFiT) for text classification.
General-Domain of ULMFiT is to predict the next word in a sequence by a widely used pretrained AWD-LSTM network [@merity2017regularizing] on the WikiText-103 dataset. ULMFiT could outperform many text classification tasks like emotion detection, early detection of depression, and medical images analysis [@lundervold2019overview; @xiao2019figure; @trotzek2018utilizing]. However, applying ULMFiT for this task is similar to choosing between any two statements, randomly. On the other hand, our results show the fine-tuned classifier on the pretrained AWD-LSTM, transformer, and random guess yielded results with almost close to 50% accuracy. As shown in Table 1, these models can not differentiate sentences that make sense from those that do not make sense, properly.
In addition, we apply the ubiquitous architecture of transformers for classification, such as BERT, Albert, and RoBERTa. All these models allow us to pretrain a model on a large corpus of data, such as all Wikipedia articles and English book corpus, and then fine-tune them on downstream tasks. Looking at Table 1, we see RoBERTa outperforms all other models. We find out a significant difference when using fine-tuned Albert and BERT classification. Table \[table:kysymys\] summarizes the performance of these systems on dev in terms of accuracy.\
------------------------------------------- -- -- --
**Models & **Accuracy\
AWD-LSTM & 52.45\
Transformer & 53.8\
ULMFiT & 59.8\
BERT MLM & 74.29\
BERT classification & 88\
Albert classification & 92\
RoBERTa classification & 95\
RoBERTa multiple choice question & 96.08\
****
------------------------------------------- -- -- --
: Experimental results for subtaskA on dev set. []{data-label="table:kysymys"}
Our idea to boost the performance of all these applied models is reframing the input of subtaskA as a binary classification task to the input of another downstream task, multiple choice questions. As a result, we show fine-tuned RoBERTa for multiple choice questions task gives better results than RoBERTa for classification problem on both dev and test set (See Table \[table:kysymys\]).
The difference between these two models is paying attention to the statements. In the self-attention layer, the encoder looks at other words in the input sentence as it encodes a specific word. For binary classification models like BERT, RoBERTa, and Albert, we concatenate two statements and then self-attention layer attends to each position in the input sequence, including both statements. However, for RoBERTa multiple choice questions task, we feed each statement to the network separately. Therefore, the attention layer attends to the sequence of words for each individual statement for gathering information that can lead to better encoding for each word.
Question answering task usually provides a paragraph of context and a question. The goal is to answer the question based on the information in the context. For subtaskA, we do not have the context and question; all we have is two options corresponding to the statements which are fed to the network, separately. Our goal is to select the correct statement (answer) from the two options.
As expected, determining optimal hyper-parameters has a significant impact on the accuracy on the performance of the model, and their optimization needs careful evaluation of many key hyper-parameters. We primarily follow the default hyper-parameters of RoBERTa, except for the maximum sequence length, weight decay, and learning rate $\in\{1e-5, 2e-5, 3e-5\}$ which is warmed up over 320 steps with a maximum of 5336 numbers of step to a peak value and then linearly decayed. The other hyper-parameters remained as defaults during the training process for 5 epochs. Fine-tuned hyper-parameters achieve 96.08% and 94.7% accuracy on dev and test set, respectively. Our result is a big jump from 74.1% baseline accuracy and competes with 99.1% accuracy of human performance.
SubtaskB (Explanation)
----------------------
As described earlier, subtaskB requires world knowledge and targets commonsense reasoning to answer why nonsense statements do not make sense. This type of task seems trivial for humans with a basic knowledge but is still one of the most challenging tasks in the NLP world. However, the baseline for human performance, 97.8% shows how it is difficult to reason even with a comprehensive commonsense knowledge.
Our goal is to investigate whether transformers like RoBERTa (which its performance was confirmed on subtaskA) can learn commonsense inference given a nonsense statement. The architecture of RoBERTa-large is comprised of of 24-layer, 1024-hidden dimension, 16-self attention heads, 355M parameters and pretrained on book corpus plus English Wikipedia, English CommonCrawl News, and WebText corpus.
SubtaskB is a multiple choice question task and we fine-tune hyper-parameters of RoBERTa model to answer questions. In this setting, we concatenate the nonsense statement (context) with each option (endings) and then use three statements as the input of model. For example, ‘He drinks apple.’ is the context and \[‘Apple juice are very tasty and milk too.’, ‘Apple can not be drunk.’, ‘Apple cannot eat a human.’\] is the list of endings. We want to select the ending from three options that is entailed by the context:
- “He drinks apple. Apple juice are very tasty and milk too.”
- “He drinks apple. Apple can not be drunk.”
- “He drinks apple. Apple cannot eat a human.”
The set of concatenated examples is fed into the model to predict the answer of questions that require reasoning. We considered a few hyper-parameter settings and figured out the model with hyper-parameters in Table \[Table 2\] yields the surprising results 93.7%, compared to the baseline accuracy of 45.6%.
------------------------------- -- -- --
**hyper-parameters & **value\
batch size & 16\
learning rate & $1e-5$\
weight decay & 0.1\
adam epsilon & $1e-8$\
num\_train\_epochs & 5\
max\_steps & 5336\
warmup\_steps & 320\
****
------------------------------- -- -- --
: Tuned hyper-parameters of RoBERTa for subtaskB.[]{data-label="Table 2"}
SubtaskC (Reason Generating)
----------------------------
Based on the subtaskC definition, we can frame subtaskC as a conditional text generation problem. Given a nonsense statement, we expect that the language model will generate commonsense reasons to explain why statement conflicts with our knowledge. We applied the full version of OpenAI GPT2 (Generative Pre-Training), a large-scale unsupervised language model with billions of parameters, trained on a very large corpus of text data. The goal of this model is to automatically generate text, given a sequence of natural language words. The performance of GPT-2 in a zero-shot setting is competitive on many language modeling datasets and various tasks like reading comprehension, translation, and question answering.
GPT-2 architecture claims that the model performs well in generating coherent samples depending on the context, which are fairly represented during the training process. However, we observed that employing GPT-2 for generating texts against the given nonsense statements is poor in performance with unnatural topic switching and 6.1732 BLEU score.
Notably, we submitted the original test set for the evaluation phase on SemEval-2020 portal and surprisingly, we stand among the first four teams. The competitive BLEU score of 17.2 with the top team shows that subtaskC is challenging enough to receive more research attentions. We believe that our simple efforts indicate significant opportunities for future research to utilize reasoning on commonsense knowledge.
Conclusion
==========
We evaluated architectures for three commonsense reasoning tasks. First, we found that RoBERTa-large performs better substantially in differentiating sentences that make sense from those that do not make sense compared to other cutting-edge architectures (e.g. Albert, BERT, and ULMFiT). We reframe this classification task to a question answering task to enhance the performance of the fine-tuned RoBERTa to 96.08%. Second, we achieved significant results on reasoning why false statements do not make sense. We showed that RoBERTa performs well in selecting the correct option among three to infer the commonsense reason and it yields significant result with 93.7% accuracy compare to baseline using BERT, 45.6%. With a little effort on generating reasons to explain why false statement conflicts with commonsense knowledge, we observe that the original test set produces 17.2 BLEU score which ranked us among first four teams in the competition with a very competitive results. Our experimental result showed that GPT-2 performs as poor as random generating of a sequence of words for this task. We believe this task has many potentials and challenges for upcoming NLP researches. As another future work, we believe that ensemble learning can reduce the variance of predictions and also improve prediction performance.
[^1]: <https://github.com/Sirwe-Saeedi/Commonsense-NLP>
| {
"pile_set_name": "ArXiv"
} |
---
abstract: '$\phi$ and K$^-$ mesons from Ni+Ni collisions at the beam energy of 1.91A GeV have been measured by the FOPI spectrometer, with a trigger selecting central and semi-central events amounting to 51% of the total cross section. The phase space distributions, and the total yield of K$^-$, as well as the kinetic energy distribution and the total yield of $\phi$ mesons are presented. The $\phi$/K$^-$ ratio is found to be $0.44 \pm 0.07(\text{stat}) ^{+0.18}_{-0.12} (\text{syst})$, meaning that about 22% of K$^-$ mesons originate from the decays of $\phi$ mesons, occurring mostly in vacuum. The inverse slopes of direct kaons are up to about 15 MeV larger than the ones extracted within the one-source model, signalling that a considerable share of gap between the slopes of K$^+$ and K$^-$ could be explained by the contribution of $\phi$ mesons to negative kaons.'
author:
- 'K. Piasecki'
- 'N. Herrmann'
- 'R. Averbeck'
- 'A. Andronic'
- 'V. Barret'
- 'Z. Basrak'
- 'N. Bastid'
- 'M.L. Benabderrahmane'
- 'M. Berger'
- 'P. Buehler'
- 'M. Cargnelli'
- 'R. Čaplar'
- 'P. Crochet'
- 'O. Czerwiakowa'
- 'I. Deppner'
- 'P. Dupieux'
- 'M. Dželalija'
- 'L. Fabbietti'
- 'Z. Fodor'
- 'P. Gasik'
- 'I. Gašparić'
- 'Y. Grishkin'
- 'O.N. Hartmann'
- 'K.D. Hildenbrand'
- 'B. Hong'
- 'T.I. Kang'
- 'J. Kecskemeti'
- 'Y.J. Kim'
- 'M. Kirejczyk'
- 'M. Kiš'
- 'P. Koczon'
- 'R. Kotte'
- 'A. Lebedev'
- 'Y. Leifels'
- 'A. Le Fèvre'
- 'J.L. Liu'
- 'X. Lopez'
- 'V. Manko'
- 'J. Marton'
- 'T. Matulewicz'
- 'R. Münzer'
- 'M. Petrovici'
- 'F. Rami'
- 'A. Reischl'
- 'W. Reisdorf'
- 'M.S. Ryu'
- 'P. Schmidt'
- 'A. Schüttauf'
- 'Z. Seres'
- 'B. Sikora'
- 'K.S. Sim'
- 'V. Simion'
- 'K. Siwek-Wilczyńska'
- 'V. Smolyankin'
- 'K. Suzuki'
- 'Z. Tymiński'
- 'P. Wagner'
- 'I. Weber'
- 'E. Widmann'
- 'K. Wiśniewski'
- 'Z.G. Xiao'
- 'I. Yushmanov'
- 'Y. Zhang'
- 'A. Zhilin'
- 'V. Zinyuk'
- 'J. Zmeskal'
title: 'Influence of $\phi$ mesons on negative kaons in Ni+Ni collisions at 1.91A GeV beam energy'
---
Introduction
============
Nucleus-nucleus collisions at the beam kinetic energies of 1-2A GeV offer the unique possibility to study the onset of the strangeness production. The emergence of strangeness at beam energies below the thresholds in free nucleon-nucleon (NN) collisions is facilitated by the appearance of resonances and mesons in the heated (T $\approx 100$ MeV) and compressed (2-3 times above the normal nuclear density $\rho_0$) collision zone, where the basic properties of particles like effective mass and decay constant are modified [@Fuch06; @Lutz04; @Hart11; @Hong97; @Wisn00; @Fors07; @Lop07].
The knowledge about the emission yields and kinematical properties of $\phi$ mesons produced in nucleus-nucleus collisions is very limited at the beam energy below 10A GeV as the experimental data is scarce in this beam energy region [@Mang02; @Agak09; @Gasi10; @Back04]. Possible channels of in-medium $\phi$ meson production include BB $\rightarrow$ BB$\phi$, MB $\rightarrow$ N$\phi$, $\rho\phi \rightarrow \phi$ (B = \[N, $\Delta$\], M = \[$\rho$, $\pi$\]).
Calculations within the BUU transport model for the central Ni+Ni collisions at the beam energy of 1.93A GeV favour the dominance of the MB channels [@Barz02]. This system was measured in a previous work by the FOPI Collaboration, but the statistics (23$\pm$7 events attributed to $phi$ mesons) was insufficient for a quantitative comparison with the calculations [@Mang02].
{width="95.00000%"}
{width="95.00000%"}
Since the mean decay path of $\phi$ is 46 fm, most of these particles decay outside the collision zone. As its dominant decay channel is $\phi \rightarrow K^+K^-$ (BR = 48.9%) [@PDG], and the freeze-out yields of $\phi$ and K$^-$ are found to be of comparable order [@Mang02; @Agak09; @Gasi10], $\phi$ decays are the source of K$^-$ mesons that are mostly unaffected by presence inside the medium, in contrast to those negative kaons produced directly in the collision zone. Therefore, evaluation of the $\phi$/K$^-$ ratio is of importance for the studies of the modifications of K$^-$ properties in-medium. In the discussed energy range this ratio was reported for the Ar+KCl collisions at 1.756A GeV [@Agak09].
In this paper we present the yield and kinetic energy distribution of $\phi$, as well as the $\phi$/K$^-$ ratio for the collisions of Ni+Ni at the beam kinetic energy of 1.91A GeV, covering the most central 51% of the geometrical cross section.
The experiment
==============
The experiment was carried out with the FOPI spectrometer, installed at the heavy-ion synchrotron SIS-18 in GSI, Darmstadt. The innermost detector is the azimuthally symmetric Central Drift Chamber (CDC), covering the wide range of polar angles ($27^\circ < \theta_{\text{lab}} < 113^\circ$). CDC is surrounded by two detectors in the barrel geometry, dedicated for the Time-of-Flight (ToF) measurements: Multi-strip Multi-gap Resistive Plate Counter [@Kis11], spanning $30^\circ < \theta_{\text{lab}} < 53^\circ$, and the Plastic Scintillation Barrel (PSB), covering $55^\circ < \theta_{\text{lab}} < 110^\circ$. These devices are encircled by the magnet solenoid, delivering the magnetic field of B = 0.617 T, and covered at front by the Plastic scintillation Wall (PlaWa). More details on characteristics and performance of the FOPI apparatus can be found in [@FOPI].
The $^{58}$Ni ions, accelerated to the kinetic energy of 1.91A GeV, were incident on the 405 $\mu$m-thick $^{58}$Ni target (corresponding to 1% interaction probability). By requiring the multiplicities of charged hits in PlaWa (PSB) to be $\geq$ 5 ($\geq$ 1), the trigger selected the sample of $7.6\times 10^7$ central and semi-central events amounting to 51% of the total geometrical cross section. Assuming the simple geometrical model of interpenetrating spheres, and the sharp cut-off approximation between the maximum impact parameter and the total reaction cross section, the mean number of participant nucleons averaged over the impact parameter was estimated to be $\langle A_{\text{part}} \rangle_{\text{b}} = 50$.
Data analysis
=============
Particles traversing the CDC detector activate sense wires along their flight path, leaving series of [*hits*]{}, which are collected into [*tracks*]{} by the off-line procedure. While hitting the MMRPC (PSB) detector, particles activate one or a few neighbouring strips, merged off-line into hits. Subsequently, tracks in the CDC and hits in the ToF detectors are matched. A collection of tracks is used to calculate the position $\vec v$ of the event vertex. To reject the collisions occurred outside the target, a cut was applied on the component of the vertex position in the beam direction: $\left| v_{\text{z}} \right| < 15$ cm.
In order to select a sample of good-quality tracks, a set of cuts was applied. To suppress the contribution from the discontinuous tracks, the track was required to be constructed of at least 36 (32) hits for K$^-$ (K$^+$). The asymmetry in the number of hits is due to the non-radial profile of sense wires of the CDC, so the negative particles generate more hits on average than the positive ones. As both charged kaons are produced in the target, two more conditions were imposed on transverse and longitudinal distance of closest approach between the track and the collision vertex: $\left| d_0 \right| < 1.5$ cm and $\left| z_0 \right| < 30$ cm, accordingly. Particles with the absolute values of charge higher than the elementary charge were rejected. Since some anisotropy was found in the azimuthal distribution of tracks in the backward zone of the CDC, the region of $\theta_{\text{lab}} > 90^\circ$ was excluded from the analysis. The requirement of CDC-ToF matching imposes a lower limit on the transverse momentum of a particle spiralling in the magnetic field: $p_{\text{t}} \geq 0.1$ GeV/c. To ensure the good matching between the track in the CDC device and the hit in a ToF detector, two additional conditions were imposed on the azimuthal and longitudinal distance between the extrapolation of the CDC track to the relevant ToF module, and the hit therein: $\left| \Delta \varphi \right| < 1.5^\circ$, $\left| \Delta z \right| < 30$ cm for the PSB, and $\left| \Delta \varphi \right| < 0.6^\circ$, $\left| \Delta z \right| < 25$ cm for the MMRPC.
Charged particle identification based on information from the CDC and ToF detectors is performed by tracing the correlation between momentum and velocity of a particle, as shown in Fig. \[fig:pvplot\]. Events on this plane can be projected onto the mass parameter, using the relativistic formula $p = m\gamma v$. The identification capability of charged kaons, limited to modest momenta for the PSB, has been largely enhanced in the MMRPC, due to the excellent timing performance and better granularity of the latter [@Kis11].
Negative kaons
--------------
For the analysis of K$^-$, basing on the observation of signal and background ratio the high $p_{\text{t}}$ limits were imposed of 0.57 GeV/$c$ (MMRPC) and 0.35 GeV/$c$ (PSB). To reconstruct the phase space population of K$^-$, the experimental mass distribution was analysed for every $p_{\text{t}}$-$y_{\text{lab}}$ cell, where $y_{\text{lab}}$ denotes rapidity in the laboratory frame, and the kaonic signal was separated from the background composed of $\pi^-$ mesons. A total of 9870 negative kaons were found in the CDC-ToF acceptance region. Influences of the choice of the binning of $p_{\text{t}}$, $y_{\text{lab}}$ and mass histograms, and of the choice of the minimum number of hits in a CDC track on the spectra presented in this paper were included in the relevant systematic errors.
 Invariant mass plot of (solid line) true, and (scattered points) mixed K$^+$K$^-$ pairs. (b) $\phi$ meson signal obtained after background subtraction.](s325_phi_minv.eps){width="8.6cm"}
$\phi$ mesons
-------------
For the reconstruction of the $\phi$ meson via the K$^+$K$^-$ decay, the maximum momentum of K$^+$ (K$^-$) was set to 1.2 (0.85) GeV/$c$ for the tracks matched with the MMRPC, and 0.72 (0.60) GeV/$c$ for the PSB. To minimize the side effects on the edges of ToF detectors, the observation region of the $\phi$ meson phase space was trimmed to $95^\circ < \theta_{\text{NN}} < 150^\circ$, where $\theta_{\text{NN}}$ is the polar emission angle in the nucleon-nucleon centre of mass frame.
The $\phi$ mesons were identified via the invariant mass analysis of K$^+$K$^-$ pairs (see Fig. \[fig:phiminv\]), a dominant $\phi$ meson decay channel. The uncorrelated background was obtained with the mixed events method and normalized to the true pair distribution in the region $1.05 < M_{\text{inv}} < 1.18$ GeV/$c^2$. After subtraction of the background, about 170 $\phi$ mesons were found under the peak. Within the range of $\pm$2 standard deviations of the fitted Gaussian distribution the signal to background ratio was found to be 1.1, and significance 9.3. The values of cut parameters leading to the identification of $\phi$ mesons were further varied, and the propagation of these changes on the results were included in their systematic uncertainties.
Efficiency evaluation
=====================
Standard efficiency correction
------------------------------
### Negative kaons
The GEANT [@GEANT] package for detector simulation was employed to obtain the efficiency correction. Negative kaons were generated according to the homogeneous distribution, and embedded in the events of products of the Ni+Ni collisions at the beam energy of 1.91A GeV, simulated by the IQMD code [@IQMD]. For both K$^-$ and $\phi$ mesons, the simulated events were treated using the same off-line analysis package, as for the experimental data. The efficiency distribution of K$^-$ for the regions of phase space covered by MMRPC and PSB detectors is shown in Fig. \[fig:efficiencies\]a. For the momenta higher than 0.5 GeV/$c$ it reaches about 50%. A drop of efficiency at lower $p_{\text{t}}$ is caused by the higher probability for the kaon to decay on its path to either of two ToF detectors.
### $\phi$ mesons
For the efficiency evaluation, the mass of the $\phi$ mesons was sampled from the Breit-Wigner distribution, and the phase space was populated by pulling from the Boltzmann distribution scaled by the anisotropy term $$\label{EQ:phigeantmodel}
\frac{d^2N}{dEd\vartheta} \sim ~pE~ \exp (-E\slash T_{\text{s}}) \cdot \left(1 + \alpha \cos^2 \vartheta\right) ,$$ where $T_{\text{s}}$ is the temperature of the source, and $\alpha$ is the anisotropy parameter. Sampled mesons were subsequently boosted to the laboratory frame. The parameters in the Eq. \[EQ:phigeantmodel\] were varied in the range of $T_{\text{s}} \in$ , $\alpha \in$ . Differences of the obtained efficiency corrections due to variations of these parameters were further included in the systematic uncertainties of the investigated physics variables, as discussed below. $\phi$ mesons were subsequently embedded in the Ni+Ni collisions, simulated with the IQMD code. Fig. \[fig:efficiencies\]b shows the phase space efficiency distribution for $T_{\text{s}} = 100$ MeV, and $\alpha = 0$. As the $\phi$ mesons have been analysed both inclusively and by studying the kinetic energy distribution, the appropriate efficiencies were obtained for either case separately.
Internal efficiency of ToF detectors
------------------------------------
Following the reported inhomogeneity in the longitudinal position response of the MMRPC detector, shown in Fig. 12 of Ref. [@Kis11], the phenomenological study of the internal efficiency of this detector was performed. While the effects of geometry and matching are the regular part of the GEANT-based efficiency determination procedure, possible internal MMRPC inefficiencies were not included so far. This additional efficiency factor was pursued by constructing first the ratio of CDC tracks with associated hit in a ToF (MMRPC, PSB) detector to all the reconstructed CDC tracks. This ratio was obtained independently for the experimental and simulated data. Next, both results were divided, to yield the internal efficiency factor $f$, according to:
$$f^{ToF} \left( \vartheta, p_{\text{t}} \right)
= \frac{N^{\text{ToF}}_{\text{exp}}}{N^{\text{CDC}}_{\text{exp}}}
\slash
\frac{N^{\text{ToF}}_{\text{sim}}}{N^{\text{CDC}}_{\text{sim}}}$$
where ToF $\in$ (MMRPC, PSB). The resulting f($\vartheta$, $p_{\text{t}}$) maps for both PSB and MMRPC detectors are shown in Fig. \[fig:efficiencies\]. Particularly noticeable is the heap structure around the centre of length of the MMRPC detector ($\vartheta \approx 38^\circ$), reported also in [@Kis11]. A correction of the distributions of K mesons was done on an event-by-event basis by weighting every kaon event with the factor . In order to minimize the possible sensitivity of $f$ to particle’s charge, corrections for negatively charged kaons were performed using the map obtained from $\pi^-$ mesons. Differences between maps obtained from protons, and $\pi^-$ were found to be small (in the order of 5-7%) and were manifested mainly in the global normalization.
Results
=======
Negative kaons
--------------
The transverse momentum spectra for consecutive slices of rapidity within the $0.34 < y_{\text{lab}} < 0.89$ range, covered by the MMRPC, are shown in Fig. \[fig:kmphase\]a. They were fitted according to the Boltzmann-like function:
$$\label{EQ:ptfit}
\frac{d^2 N}{dp_{\text{t}} dy_{\text{NN}}} = N ~p_{\text{t}}
E ~\exp (-m_{\text{T}} \slash T_{\text{B}} )$$
where $m_{\text{T}} = \sqrt{p_{\text{T}}^2 + m^2}$ is the transverse mass, and for every slice $y_{\text{NN}}$ denotes the average value of rapidity in the NN frame, $N$ is the normalization factor, and $T_{\text{B}}$ is the inverse slope (also called the apparent temperature). The $p_{\text{t}}$ spectra measured in the PSB region ($0 < y_{\text{lab}} < 0.3$) had too poor statistics for the two-parameter fitting. Therefore, the normalization parameters were extracted only, by fitting the formula \[EQ:ptfit\] with one of two fixed values of $T_{\text{B}}$: 45 and 60 MeV (differences in the results were accounted for in the evaluation of the systematic errors). The points of the rapidity distribution were obtained by an analytic integration of the formula \[EQ:ptfit\] from 0 to $\infty$ for every slice,
$$\frac{dN}{dy_{\text{NN}}} = N \cdot \cosh (y_{\text{NN}}) \cdot T_{\text{B}}^3
\left( \frac{m^2}{T_{\text{B}}^2} + 2\frac{m}{T_{\text{B}}} + 2 \right)
\exp \left( -\frac{m}{T_{\text{B}}} \right) \ ,$$
where $m$ is the particle’s mass, and substituting the parameters obtained in the fit above. The obtained distribution is shown in Fig. \[fig:kmphase\]b. Note, that the N and T$_{\text{B}}$ fit parameters are typically characterized by the strong anticorrelation term in the covariance matrix, which was included in the error evaluation of the rapidity distribution.The magenta boxes on the abovementioned plot correspond to the systematic errors arising from variations of all the applied cuts and binnings, except for the contribution from the choice of binning of the rapidity axis. Benefitting from the symmetry of the colliding system, the measured data points were reflected with respect to the midrapidity (y$^{\text{CM}}_{\text{NN}} = 0.89$), which allowed for a wide coverage of the K$^-$ rapidity spectrum.
In order to obtain the K$^-$ yield, the tails of the rapidity distribution were extrapolated. To assess the systematic error of this procedure, the data was fitted using the Gaussian and linear functions, each one in two ranges: $y_{\text{lab}} < 0.3$, and 0.5. The variant resulting in the largest (smallest) reconstructed yield is shown as dashed (dotted) line in Fig. \[fig:kmphase\]b. The total yield of K$^-$ was found to be:
$$\rm{P(K^-) = (9.84 ~\pm~ 0.21~(stat) ~^{+0.63}_{-0.57}~(syst)) \times 10^{-4}}$$
per triggered event. Note, that the relatively wide coverage of phase space allowed to minimize the assumptions on the overall shape profile of the rapidity distribution, and in consequence on the total yield of K$^-$.
$\phi$ mesons
-------------
The limitation of the sample of $\phi$ mesons to about 170 events does not permit for a full-fledged analysis of its phase space, and thus for the reconstruction of the yield in a model-independent fashion. As a first step the yield was reconstructed directly by dividing the total number of events within the phase space region reported above by the inclusive efficiency for this region. However, the result revealed a clear correlation with the input temperature $T_{\text{s}}$ of the source simulated in the efficiency evaluation procedure (cf. Eq. \[EQ:phigeantmodel\]). It ranged between $3.2 \times 10^{-4}$ for $T_{\text{s}} = 80$ MeV, and $6.2 \times 10^{-4}$ for $T_{\text{s}} = 130$ MeV, not accounting for any statistical or systematic errors. This uncertainty was a motivation to study the kinetic energy spectrum of $\phi$ mesons, in hope that, apart from carrying the kinematical information itself, it may provide constraints on the temperature, and thus limit the uncertainty of the total yield. The reconstructed kinetic energy spectrum, was fitted with the Boltzmann-like function of the form
 Kinetic energy distribution of $\phi$ mesons fitted with Boltzmann-like function, assuming the following input parameters to the efficiency calculation: $T_{\text{s}} = 100$ MeV, $\alpha = 0$ (see text for details).](s325_phi_ekin.eps){width="8.6cm"}
$$\label{EQ:ekfit}
\frac{dN}{dE_{\text{k}}} = N ~pE~ \exp (-E\slash T_{\text{eff}}) ,$$
where $N$ is the normalization parameter, total energy $E = E_{\text{k}} + m$, momentum $p = \sqrt{E^2 - m^2}$, and $T_{\text{eff}}$ is called the inverse slope. An exemplary case of fitting the kinetic energy spectrum at $T_{\text{s}} = 100$ MeV, $\alpha = 0$, and events grouped in four bins, is shown in Fig. \[fig:phiekin\]. The extracted value of $T_{\text{eff}}$ was found to depend a little on a choice of binning of the $E_{\text{k}}$ spectrum, $T_{\text{s}}$ and $\alpha$ efficiency input parameters, particle identification cuts, and fit range imposed. However, only the fits obtained with the input $T_{\text{s}}$ around the middle of the probed range of $~\mbox{[80, 130]~MeV}$ resulted in values of $T_{\text{eff}}$ fit parameter consistent with $T_{\text{s}}$. Following this finding, a self-consistency condition was imposed: $\left| T_{\text{s}} - T_{\text{eff}} \right| < 15$ MeV, where 15 MeV is the typical statistical error of the fitted $T_{\text{eff}}$. Note, that this condition not only narrows down the ranges of input $T_{\text{s}}$ parameter, and the systematic error of $T_{\text{eff}}$, but the selection of the $T_{\text{s}}$ region also limits the systematic error of the total $\phi$-meson yield discussed above. Within the imposed condition, the inverse slope of the kinetic energy distribution of $\phi$ mesons was found to be $T_{\text{eff}} = 105 \pm 18 \text{(stat)} ^{+19}_{-13} \text{(syst)}$. The total yield was extracted by extrapolation of the data points with the fitted curve (Eq. \[EQ:ekfit\]), and found to be
$$\rm{P(\phi) = (4.4 ~\pm~ 0.7 ~\text{(stat)} ~^{+1.7}_{-1.4} ~(syst))
\times 10^{-4}}$$
per triggered event. We found, that the impact of non-zero anisotropy parameter $\alpha$ on the final results is on the level of 5% of this value.
Influence of $\phi$ production on K$^-$ yields
----------------------------------------------
The nearly 50% branching ratio of the $\phi \rightarrow \text{K}^+ \text{K}^-$ decay channel, and the comparable yields of $\phi$ and K$^-$ mesons suggest that their emission should be considerably correlated. However, while direct K$^-$ are emitted from the hot collision zone, most decay products of $\phi$ mesons are created outside this region. Therefore, one may expect that negative kaons emitted from both these sources could have different kinematical characteristics. Hence, the question arises, what fraction of the observed K$^-$ originate from the decays of $\phi$ mesons. For this analysis we find the following ratio of yields:
$$\rm{\frac{P(\phi)}{P(K^-)} = 0.44 \pm 0.07 (stat) ^{+0.18}_{-0.12} (syst)
\quad ,}$$
where the systematic errors were obtained by combining all the variations of both components due to their own systematic uncertainties, and rejecting 5% of values on the tails of the resulting distribution. Taking into account that 48.9% $\phi$ mesons decay into K$^+$K$^-$ pairs, it translates into $22 \pm 3 ^{+9}_{-6} ~\%$ of observed K$^-$ originating from decays of $\phi$ mesons. It is worthwhile noting that the same values within errors have been obtained for the Ar+KCl system colliding at even more subthreshold beam energy of 1.756A GeV [@Agak09]. Also, for the central Al+Al collisions at 1.9A GeV, a similar value of $= 0.30 \pm 0.08 \text{(stat)} ^{+0.04}_{-0.06} \text{(syst)}$ was found [@Gasi10; @GasiTh]. However, for the elementary pp collisions at 2.7 GeV, this ratio was found to be slightly above 1 [@Maed08]. One possible explanation of this discrepancy arises in the context of the statistical model where the volume of open strangeness production is assumed to be limited and parametrized by the canonical radius $R_{\text{C}}$. According to the calculations reported in [@Agak09], the $\phi$/ K$^-$ ratio measured for Ar+KCl is consistent with $R_{\text{C}}$ between 2.2 and 3.2 fm, while that for pp is reproduced for $R_{\text{C}} \approx 1.2$ fm, although the latter prediction disagrees with the experimental data at higher beam energies. Unfortunately, performing the statistical model fit to the data was not possible for the present experiment, due to a meager amount of reconstructed particle yields available at centralities corresponding to $\langle A_{\text{part}} \rangle_{\text{b}} = 50$. Much wider array of particle yields is available for the central collisions of the studied system (c.f. Fig. 4b in [@Pias09]). A measurement of ratio for this centrality range would give an opportunity to extract the canonical radius parameter for the Ni+Ni system and compare with that for the Ar+KCl collisions.
 Rapidity distribution of $T_{\text{B}}$ (apparent temperatures) for negative kaons within the one-source hypothesis (open squares), and two-source approach (K$^-$ from $\phi$ decays - full triangles, direct kaons - open circles). Inverse slope of K$^+$ (K$^-$) obtained by the KaoS Collaboration is marked by full circle (square). See text for details. ](s325_kminus_temps-sources.eps){width="8.6cm"}
In order to estimate the influence of $\phi$ mesons on the kinematic properties of the observed K$^-$, PLUTO code [@PLUTO] was first employed to generate the thermal, and isotropic $\phi$ mesons with temperature of 105 MeV, which subsequently decayed, giving rise to K$^-$ production. The inverse slopes of the $p_{\text{t}}$ spectra of such kaons, shown in Fig. \[fig:kaon2src\] in full triangles, were found to be clearly lower than those of experimentally measured K$^-$, depicted with open squares. Following this finding, the emission of negative kaons was assumed to arise from two sources: the collision zone emitting kaons directly, and $\phi$ mesons decaying in vacuum. Their contributions were weighted according to the experimentally found ratio, and the phase space distributions of both were assumed to be Boltzmann-like. For every slice of rapidity, the temperature parameter (inverse slope) of kaons from the $\phi$-meson decays was fixed at a value obtained from the simulation described above, while the slope for the direct kaons was extracted by fitting the two-source model to the experimental data, giving the results shown in open circles in Fig. \[fig:kaon2src\]. Keeping in mind the limited statistics, and simplicity of the model, the results suggest that removing 22% contribution from $\phi$ mesons could systematically raise the inverse slope of K$^-$ by up to about 15 MeV.
In contrast, as the K$^-$/K$^+$ yield ratio is about 3% [@Fors07], the influence of $\phi$ mesons on positively charged kaons is negligible. The inverse slope parameters of the kinetic energy distributions of K$^+$ and K$^-$, obtained by the KaoS Collaboration at midrapidity for the same colliding system, and similar $\langle A_{\text{part}} \rangle_{\text{b}}$, were found to exhibit a gap of about 25 MeV. Our finding suggests, that the contribution from $\phi$ mesons to K$^-$ emission significantly ,,cools down" the overall spectrum of negative kaons. Thus, this effect may account for a sizeable share of this gap, possibly competing with modifications of kaonic properties in the nuclear medium.
Summary
=======
Production of $\phi$ and K$^-$ mesons was investigated in Ni+Ni collisions at the beam kinetic energy of 1.91A GeV. The trigger selected a sample of central and semi-central collisions amounting to 51% of the geometrical cross section. The $p_{\text{t}}$ and $y_{\text{lab}}$ distribution of K$^-$ were analysed in a wide region of phase space. The total yield of K$^-$ was found to be $(9.84 \pm 0.21~(\text{stat}) ^{+0.63}_{-0.57}~
(\text{syst})) \times 10^{-4}$ per triggered event. About 170 $\phi$ mesons were reconstructed. The inverse slope of kinetic energy distribution was found to be $T = 105 \pm 18 (\text{stat}) ^{+19}_{-13} (\text{syst})$ MeV, and the total yield $(4.4 \pm 0.7 (\text{stat}) ^{+1.7}_{-1.4}
~(\text{syst})) \times 10^{-4}$ per triggered event.
The found $\phi$/K$^-$ ratio of $0.44 \pm 0.07 (\text{stat})
^{+0.18}_{-0.12} (\text{syst})$ means that $22 \pm 3 \, ^{+9}_{-6} ~\%$ of K$^-$ originate from decays of $\phi$ mesons, occurring mostly in vacuum. The influence of this additional source of negative kaons on the transverse momentum spectra was studied within a two-sources approach, where the contribution from $\phi$-meson decays was modelled by an isotropic Boltzmann-like distribution. The inverse slopes of K$^-$ produced directly in the collision zone seem to be up to about 15 MeV higher than the values extracted within the one-source hypothesis. This effect, compared to the 25 MeV gap between the inverse slopes of K$^+$ and K$^-$, signals that a considerable share of the gap could be explained by feeding of negative kaons by the $\phi$ meson decays. Thus, it seems crucial to account for the contribution of $\phi$-originating negative kaons in the studies of the in-medium modifications of these particles via comparisons of K$^-$/K$^+$ or flow (e.g. $v_{1,2}$) distributions to the predictions of the transport codes.
This work was supported by the German BMBF Contract No. 05P12VHFC7, the Korea Science and Engineering Foundation (KOSEF) under Grant No. F01-2006-000-10035-0, by the German BMBF Contract No. 05P12RFFCQ, by the Polish Ministry of Science and Higher Education (DFG/34/2007), the agreement between GSI and IN2P3/CEA, the HIC for FAIR, the Hungarian OTKA Grant No. 71989, by NSFC (Project No. 11079025), by DAAD (PPP D/03/44611), by DFG (Projekt 446-KOR-113/76/04) and by the EU, 7th Framework Program, Integrated Infrastructure: Strongly Interacting Matter (Hadron Physics), Contract No. RII3-CT-2004-506078.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We present for the first time a master formula for $\epe$, the ratio probing direct CP violation in $K \to \pi\pi$ decays, valid in [*any*]{} ultraviolet extension of the Standard Model (BSM). The formula makes use of hadronic matrix elements of BSM operators calculated recently in the Dual QCD approach and the ones of the SM operators from lattice QCD. We emphasize the large impact of several scalar and tensor BSM operators in the context of the emerging $\epe$ anomaly. We have implemented the results in the open source code flavio.'
author:
- Jason Aebischer
- Christoph Bobeth
- 'Andrzej J. Buras'
- 'Jean-Marc G[é]{}rard'
- 'David M. Straub'
bibliography:
- 'refs.bib'
title: 'Master formula for epsilon’/epsilon beyond the Standard Model'
---
The non-conservation of the product of parity (P) and charge-conjugation (C) symmetries in nature, known under the name of CP violation, was established experimentally for the first time in 1964 via $K\to \pi\pi$ decays [@Christenson:1964fg]. Since then, this fundamental phenomenon has been confirmed also in other processes in the quark sector and is rather consistently described by the so-called Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix [@Cabibbo:1963yz; @Kobayashi:1973fv] within the Standard Model (SM) of elementary particle physics. Currently there are experimental efforts to establish analogous CP violation in the lepton sector.
CP violation proves to be a prerequisite [@Sakharov:1967dj] for our present understanding of matter dominance over anti-matter in the universe. However, the CP-violating contribution from the CKM matrix in the SM fails to account for this observation and it remains to be seen whether the CP-violating contributions in the lepton sector will be able to do so. As direct collider searches have not yet revealed any presence of new physics, rare processes in the quark sector remain a good territory to search for new sources of CP violation. This is especially the case for the kaon physics observables $\varepsilon$ and $\varepsilon^\prime$, which measure indirect and direct CP violation in $K^0$-$\bar K^0$ mixing and $K^0$ decay into $\pi\pi$, respectively.
Recently, there has been a renewed interest in the ratio $\epe$ [@Buras:2014sba; @Buras:2015yca; @Blanke:2015wba; @Buras:2015kwd; @Buras:2016dxz; @Buras:2015jaq; @Kitahara:2016otd; @Endo:2016aws; @Endo:2016tnu; @Cirigliano:2016yhc; @Alioli:2017ces; @Bobeth:2016llm; @Bobeth:2017xry; @Crivellin:2017gks; @Bobeth:2017ecx; @Endo:2017ums; @Haba:2018byj; @Chen:2018ytc; @Chen:2018vog; @Matsuzaki:2018jui; @Haba:2018rzf], due to hints for a significant tension between measurements and the SM prediction from the RBC-UKQCD lattice collaboration [@Bai:2015nea; @Blum:2015ywa] and the Dual QCD approach (DQCD) [@Buras:2015xba; @Buras:2016fys]. While on the experimental side the world average from the NA48 [@Batley:2002gn] and KTeV [@AlaviHarati:2002ye; @Abouzaid:2010ny] collaborations reads $$\begin{aligned}
\label{EXP}
(\epe)_\text{exp} &
= (16.6 \pm 2.3) \times 10^{-4} \,,\end{aligned}$$ the lattice collaboration [@Bai:2015nea; @Blum:2015ywa] and the NLO analyses in [@Buras:2015yba; @Kitahara:2016nld] based on their results find $(\epe)_{\text{SM}}$ in the ballpark of $(1-2) \times 10^{-4}$, that is by one order of magnitude below the data, but with an error in the ballpark of $5\times 10^{-4}$. An independent analysis based on hadronic matrix elements from DQCD [@Buras:2015xba; @Buras:2016fys] gives a strong support to these values and moreover provides an *upper bound* on $(\epe)_{\text{SM}}$ in the ballpark of $6\times 10^{-4}$. A different view has been expressed in [@Gisbert:2017vvj] where, using ideas from chiral perturbation theory but going beyond it, the authors find $(\epe)_{\text{SM}} = (15 \pm 7) \times 10^{-4}$ in agreement with the data, albeit with a large uncertainty.
The results from RBC-UKQCD and DQCD motivated several authors to look for various extensions of the SM which could bring the theory to agree with data. For a recent review see [@Buras:2018wmb]. In all the models studied to date, the rescue comes from the modification of the Wilson coefficient of the dominant electroweak left-right (LR) penguin operator $Q_8$, but also solutions through a modified contribution of the dominant QCD LR penguin operator $Q_6$ could be considered [@Buras:2015jaq]. However, in generic BSM scenarios, also operators not present in the SM could play an important role. The very recent calculation of the $K\to\pi\pi$ hadronic matrix elements of all BSM four-quark operators, in particular scalar and tensor operators in DQCD [@Aebischer:2018rrz] and the one of the chromo-magnetic operator by the ETM lattice collaboration [@Constantinou:2017sgv] and in DQCD [@Buras:2018evv], allow for the first time the study of $\epe$ in an arbitrary extension of the SM. While the matrix element of the chromo-magnetic operator has been found to be much smaller than previously expected, the values of the BSM matrix elements of scalar and tensor operators are found to be in the ballpark of the ones of $Q_8$, the dominant electroweak penguin operator in the SM. Consequently, these operators could help in the explanation of the emerging $\epe$ anomaly.
As far as short-distance contributions encoded in the Wilson coefficients are concerned, they have been known for the SM operators already for 25 years at the NLO level [@Buras:1991jm; @Buras:1992tc; @Buras:1992zv; @Ciuchini:1992tj; @Buras:1993dy; @Ciuchini:1993vr] and for the BSM operators two-loop anomalous dimensions have been known [@Ciuchini:1997bw; @Buras:2000if] for almost two decades. First steps towards the NNLO predictions for $\epe$ have been made in [@Bobeth:1999mk; @Buras:1999st; @Gorbahn:2004my; @Brod:2010mj] and the complete NNLO result should be available soon [@Cerda-Sevilla:2016yzo].
Having all these ingredients from long-distance and short-distance contributions at hand, we are in the position to present for the first time a master formula for $\epe$ that can be applied to any ultraviolet extension of the SM. Neglecting isospin breaking corrections, $\epe$ can be written as $$\begin{aligned}
\label{eq:epe-formula}
\left(\frac{\varepsilon'}{\varepsilon}\right)_\text{th} &
= -\frac{\omega}{\sqrt{2}|\epsK|}
\left[ \frac{\text{Im}A_0}{\text{Re}A_0}
- \frac{\text{Im}A_2}{\text{Re}A_2} \right]\,,\end{aligned}$$ where $\omega = {\text{Re}A_2}/{\text{Re}A_0}$ and $A_{0,2}$ are the $K\to\pi\pi$ isospin amplitudes, $$\begin{aligned}
A_{0,2} &
= \big\langle (\pi\pi)_{I=0,2}\, \big|\; \mathcal{H}_{\Delta S = 1}^{(3)}
\;\big|\, K^0 \big\rangle \,.\end{aligned}$$ Isospin breaking corrections have been considered in [@Cirigliano:2003gt; @Cirigliano:2003nn]. These corrections will affect only the $A_0$ contributions that are suppressed by $\omega \sim 1/22$. They can only be relevant in NP scenarios in which, similar to the case of the SM, the Wilson coefficients of the operators contributing to $A_0$ are by more than one order of magnitude larger than those relevant for the $A_2$ amplitude. Here $\mathcal{H}_{\Delta S = 1}^{(3)}$ denotes the $\Delta S=1$ effective Hamiltonian with only the three lightest quarks ($q=u,d,s$) being dynamical, obtained by decoupling the heavy $W^\pm$, $Z^0$, and $h^0$ bosons and the top quark at the electroweak scale ${{\mu_\mathrm{ew}}}\sim m_W$ and the bottom and charm quarks at their respective mass thresholds [@Buchalla:1995vs].
Assuming that no particles beyond the SM ones with mass below the electroweak scale exist, any BSM effect is encoded in the Wilson coefficients of the most general $\Delta S = 1$ dimension-six effective Hamiltonian. The values of the Wilson coefficients $C_i({{\mu_\mathrm{ew}}})$ in this effective Hamiltonian at the electroweak scale with $N_f = 5$ active quark flavours, $$\begin{aligned}
\label{eq:DS1-Hamiltonian}
\mathcal{H}_{\Delta S = 1}^{(5)} &
= - \mathcal{N}_{\Delta S = 1} \sum_i C_i \, O_i \,,\end{aligned}$$ are connected to those of $\mathcal{H}_{\Delta S = 1}^{(3)}$, entering $\epe$, by the usual QCD and QED renormalization group (RG) evolution. In full generality, three classes of operators can contribute, directly or via RG mixing, to $K\to\pi\pi$ decays:
#### four-quark operators:
$$\begin{aligned}
\label{eq:DS1-psi4-col1}
O_{XAB}^q &
= (\bar s^i \Gamma_X P_A d^i) (\bar q^j \Gamma_X P_B q^j) \,,
\\
\label{eq:DS1-psi4-col8}
\widetilde{O}_{XAB}^q &
= (\bar s^i \Gamma_X P_A d^j) (\bar q^j \Gamma_X P_B q^i) \,,\end{aligned}$$
#### electro- and chromo-magnetic dipole operators:
$$\begin{aligned}
\label{eq:DS1-dipole-QED}
O_{7\gamma}^{(\prime)} &
= m_s(\bar s \sigma^{\mu\nu} P_{L(R)} d) F_{\mu\nu} \,,
\\
\label{eq:DS1-dipole-QCD}
O_{8g}^{(\prime)} &
= m_s(\bar s \sigma^{\mu\nu} T^a P_{L(R)} d) G^a_{\mu\nu} \,,\end{aligned}$$
#### semi-leptonic operators:
$$\begin{aligned}
O_{XAB}^\ell &
= (\bar s\, \Gamma_X P_A d) (\bar \ell\, \Gamma_X P_B \ell) \,.\end{aligned}$$
Here $i,j$ are colour indices, $A,B=L,R$, and $X=S,V,T$ with $\Gamma_S=1$, $\Gamma_V=\gamma^\mu$, $\Gamma_T=\sigma^{\mu\nu}$ [^1]. Throughout it is sufficient to consider the case $A = L$, whereas results for the case $A = R$ follow analogously due to parity conservation of QCD and QED. We will choose the overall normalization factor $\mathcal{N}_{\Delta S = 1}$ below such that the coefficients $C_i$ are dimensionless.
In the following, we will neglect the electro-magnetic dipole and semi-leptonic operators, which only enter through small QED effects. This leaves 40 four-quark operators for $N_f = 5$ and one chromo-magnetic dipole operator of a given chirality which have to be considered at the electroweak scale. A detailed renormalization group analysis of these operators, model independently and in the context of the Standard Model effective field theory (SMEFT), is performed in [@Aebischer:2018csl]. The goal of the present letter is to provide the central result of [@Aebischer:2018csl] and [@Aebischer:2018rrz], the master formula for $\epe$, in a form that could be used by any model builder or phenomenologist right away without getting involved with the technical intricacies of these analyses.
Writing $$\begin{aligned}
\label{eq:mastertotal}
\left(\frac{\varepsilon'}{\varepsilon}\right) &
= \left(\frac{\varepsilon'}{\varepsilon}\right)_\text{SM}
+ \left(\frac{\varepsilon'}{\varepsilon}\right)_\text{BSM},\end{aligned}$$ our formula allows to calculate automatically $(\epe)_\text{BSM}$ once the Wilson coefficients of all contributing operators are known at the electroweak scale ${{\mu_\mathrm{ew}}}$. It reads as follows: $$\begin{aligned}
\label{eq:master}
\left(\frac{\varepsilon'}{\varepsilon}\right)_\text{BSM} &
= \sum_i P_i({{\mu_\mathrm{ew}}}) ~\text{Im}\left[ C_i({{\mu_\mathrm{ew}}})-C^\prime_i({{\mu_\mathrm{ew}}})\right],\end{aligned}$$ where $$\label{eq:master2}
P_i({{\mu_\mathrm{ew}}}) = \sum_{j} \sum_{I=0,2} p_{ij}^{(I)}({{\mu_\mathrm{ew}}}, {{\mu}})
\,\left[\frac{\langle Q_j ({{\mu}})\rangle_I}{\text{GeV}^3}\right]\,.$$
class $O_i$ $P_i$ $\frac{\Lambda}{\text{TeV}}$ <span style="font-variant:small-caps;">smeft</span>
----------- -------------------------------------------------------------------------------------------------- ------------------ ------------------------------ -----------------------------------------------------
$O_{VLL}^u = (\bar s^i \gamma_\mu P_L d^i)(\bar u^j \gamma^\mu P_L u^j)$ $-4.3 \pm 1.0$ 65
$O_{VLR}^u = (\bar s^i \gamma_\mu P_L d^i)(\bar u^j \gamma^\mu P_R u^j)$ $-126 \pm 10$ 354
${\widetilde{O}}_{VLL}^u = (\bar s^i \gamma_\mu P_L d^j)(\bar u^j \gamma^\mu P_L u^i)$ $1.5 \pm 1.7$ 38
${\widetilde{O}}_{VLR}^u = (\bar s^i \gamma_\mu P_L d^j)(\bar u^j \gamma^\mu P_R u^i)$ $-436 \pm 35$ 659
\[0.1cm\] $O_{VLL}^d = (\bar s^i \gamma_\mu P_L d^i)(\bar d^j \gamma^\mu P_L d^j)$ $2.3 \pm 0.5$ 48
$O_{VLR}^d = (\bar s^i \gamma_\mu P_L d^i)(\bar d^j \gamma^\mu P_R d^j)$ $123 \pm 10$ 350
$O_{SLR}^d = (\bar s^i P_L d^i)(\bar d^j P_R d^j)$ $214 \pm 19$ 462
\[0.1cm\] $O_{VLL}^s = (\bar s^i \gamma_\mu P_L d^i)(\bar s^j \gamma^\mu P_L s^j)$ $-0.4 \pm 0.1$ 18
$O_{VLR}^s = (\bar s^i \gamma_\mu P_L d^i)(\bar s^j \gamma^\mu P_R s^j)$ $-0.32 \pm 0.05$ 17
$O_{SLR}^s = (\bar s^i P_L d^i)(\bar s^j P_R s^j)$ $0.0 \pm 0.1$ 6
\[0.1cm\] $O_{VLL}^c = (\bar s^i \gamma_\mu P_L d^i)(\bar c^j \gamma^\mu P_L c^j)$ $0.7 \pm 0.1$ 25
$O_{VLR}^c = (\bar s^i \gamma_\mu P_L d^i)(\bar c^j \gamma^\mu P_R c^j)$ $0.7 \pm 0.1$ 26
${\widetilde{O}}_{VLL}^c = (\bar s^i \gamma_\mu P_L d^j)(\bar c^j \gamma^\mu P_L c^i)$ $0.2 \pm 0.2$ 13
${\widetilde{O}}_{VLR}^c = (\bar s^i \gamma_\mu P_L d^j)(\bar c^j \gamma^\mu P_R c^i)$ $0.4 \pm 0.2$ 20
\[0.1cm\] $O_{VLL}^b = (\bar s^i \gamma_\mu P_L d^i)(\bar b^j \gamma^\mu P_L b^j)$ $-0.30 \pm 0.03$ 17
$O_{VLR}^b = (\bar s^i \gamma_\mu P_L d^i)(\bar b^j \gamma^\mu P_R b^j)$ $-0.28 \pm 0.03$ 16
${\widetilde{O}}_{VLL}^b = (\bar s^i \gamma_\mu P_L d^j)(\bar b^j \gamma^\mu P_L b^i)$ $0.0 \pm 0.1$ 4
${\widetilde{O}}_{VLR}^b = (\bar s^i \gamma_\mu P_L d^j)(\bar b^j \gamma^\mu P_R b^i)$ $-0.1 \pm 0.1$ 8
$O_{8g} \;\;\, = m_s (\bar s \sigma^{\mu\nu} T^a P_L d)\, G^{a}_{\mu\nu}$ $-0.35 \pm 0.12$ 18
\[0.1cm\] $O_{SLL}^s = (\bar s^i P_L d^i)(\bar s^j P_L s^j)$ $0.05 \pm 0.02$ 7
$O_{TLL}^s = (\bar s^i \sigma_{\mu\nu} P_L d^i)(\bar s^j \sigma^{\mu\nu} P_L s^j)$ $-0.14 \pm 0.05$ 12
\[0.1cm\] $O_{SLL}^c = (\bar s^i P_L d^i)(\bar c^j P_L c^j)$ $-0.26 \pm 0.09$ 16
$O_{TLL}^c = (\bar s^i \sigma_{\mu\nu} P_L d^i)(\bar c^j \sigma^{\mu\nu} P_L c^j)$ $-0.15 \pm 0.05$ 12
${\widetilde{O}}_{SLL}^c = (\bar s^i P_L d^j)(\bar c^j P_L c^i)$ $-0.23 \pm 0.07$ 15
${\widetilde{O}}_{TLL}^c = (\bar s^i \sigma_{\mu\nu} P_L d^j)(\bar c^j \sigma^{\mu\nu} P_L c^i)$ $-5.9 \pm 1.9$ 76
\[0.1cm\] $O_{SLL}^b = (\bar s^i P_L d^i)(\bar b^j P_L b^j)$ $-0.35 \pm 0.12$ 18
$O_{TLL}^b = (\bar s^i \sigma_{\mu\nu} P_L d^i)(\bar b^j \sigma^{\mu\nu} P_L b^j)$ $-0.11 \pm 0.03$ 10
${\widetilde{O}}_{SLL}^b = (\bar s^i P_L d^j)(\bar b^j P_L b^i)$ $-0.34 \pm 0.11$ 18
${\widetilde{O}}_{TLL}^b = (\bar s^i \sigma_{\mu\nu} P_L d^j)(\bar b^j \sigma^{\mu\nu} P_L b^i)$ $-13.4 \pm 4.5$ 115
$O_{SLL}^u = (\bar s^i P_L d^i)(\bar u^j P_L u^j)$ $74 \pm 16$ 272
$O_{TLL}^u = (\bar s^i \sigma_{\mu\nu} P_L d^i)(\bar u^j \sigma^{\mu\nu} P_L u^j)$ $-162 \pm 36$ 402
${\widetilde{O}}_{SLL}^u = (\bar s^i P_L d^j)(\bar u^j P_L u^i)$ $-15.6 \pm 3.3$ 124
${\widetilde{O}}_{TLL}^u = (\bar s^i \sigma_{\mu\nu} P_L d^j)(\bar u^j \sigma^{\mu\nu} P_L u^i)$ $-509 \pm 108$ 713
$O_{SLL}^d = (\bar s^i P_L d^i)(\bar d^j P_L d^j)$ $-87 \pm 16$ 295
$O_{TLL}^d = (\bar s^i \sigma_{\mu\nu} P_L d^i)(\bar d^j \sigma^{\mu\nu} P_L d^j)$ $191 \pm 35$ 436
$O_{SLR}^u = (\bar s^i P_L d^i)(\bar u^j P_R u^j)$ $-266 \pm 21$ 515
${\widetilde{O}}_{SLR}^u = (\bar s^i P_L d^j)(\bar u^j P_R u^i)$ $-60 \pm 5$ 244
: \[tab:P\_i\] Table of $\Delta S = 1$ operators contributing to $(\epe)_\text{BSM}$ with coefficients $P_i({{\mu_\mathrm{ew}}})$ for ${{\mu_\mathrm{ew}}}=160$GeV, and corresponding suppression scales. The Hamiltonian is normalized as $\mathcal{H}_{\Delta S = 1}^{(5)} = - \sum_i \frac{C_i({{\mu_\mathrm{ew}}})}{(1\,\text{TeV})^2} \, O_i \,$.
The sum in extends over the Wilson coefficients $C_i$ of the linearly independent four-quark and chromo-magnetic dipole operators listed in . The $C_i'$ are the Wilson coefficients of the corresponding chirality-flipped operators obtained by replacing $P_L\leftrightarrow P_R$. The relative minus sign accounts for the fact that their $K\to\pi\pi$ matrix elements differ by a sign. Among the contributing operators are also operators present already in the SM but their Wilson coefficients in include only BSM contributions.
The dimensionless coefficients $p_{ij}^{(I)}({{\mu_\mathrm{ew}}},{{\mu}})$ include the QCD and QED RG evolution from ${{\mu_\mathrm{ew}}}$ to ${{\mu}}\sim {\mathcal{O}}(1\,\text{GeV})$ for each Wilson coefficient as well as the relative suppression of the contributions to the $I=0$ amplitude due to ${\text{Re}A_2} / {\text{Re}A_0}\ll 1$ for the matrix elements $\langle Q_j ({{\mu}}) \rangle_I$ of all the operators $Q_j$ present at the low-energy scale. The index $j$ includes also $i$ so that the effect of self-mixing is included. We refer the reader to [@Aebischer:2018csl] for the numerical values of the $p_{ij}^{(I)}({{\mu_\mathrm{ew}}},{{\mu}})$ and $\langle Q_j
({{\mu}}) \rangle_I$ for our choice of the set of $Q_j$. The details given their allow to easily account for future updates of the matrix elements. The $P_i({{\mu_\mathrm{ew}}})$ do not depend on ${{\mu}}$ to the considered order, because the ${{\mu}}$-dependence cancels between matrix elements and the RG evolution operator. Moreover, it should be emphasized that their values are [*model-independent*]{} and depend only on the SM dynamics below the electroweak scale, which includes short distance contributions down to ${{\mu}}$ and the long distance contributions represented by the hadronic matrix elements. The BSM dependence enters our master formula in (\[eq:master\]) [ *only*]{} through the Wilson coefficients $C_i({{\mu_\mathrm{ew}}})$ and $C^\prime_i({{\mu_\mathrm{ew}}})$. That is, even if a given $P_i$ is non-zero, the fate of its contribution depends on the difference of these two coefficients. In particular, in models with exact left-right symmetry this contribution vanishes as first pointed out in [@Branco:1982wp].
The numerical values of the $P_i({{\mu_\mathrm{ew}}})$ are collected in for $$\begin{aligned}
{{\mu_\mathrm{ew}}}&=160\,\text{GeV} \,,&
\mathcal{N}_{\Delta S = 1}&=(1\,\text{TeV})^{-2} \,.\end{aligned}$$ They have been calculated with the flavio package [@Straub:2018kue], where we have implemented general BSM contributions to $\epe$. As seen in (\[eq:master2\]), the $P_i$ depend on the hadronic matrix elements $\langle Q_j ({{\mu}}) \rangle_I$ and the RG evolution factors $p_{ij}^{(I)}({{\mu_\mathrm{ew}}}, {{\mu}})$. The numerical values of the hadronic matrix elements rely on lattice QCD in the case of SM operators [@Bai:2015nea; @Blum:2015ywa] and on results for scalar and tensor operators obtained in DQCD [@Aebischer:2018rrz]. The matrix element of the chromo-magnetic dipole operator is from [@Constantinou:2017sgv] and [@Buras:2018evv] which agree with each other.
The operators in have been grouped into five distinct classes.
[**Class A:**]{} All hadronic matrix elements can be expressed in terms of the ones of SM operators calculated by lattice QCD [@Bai:2015nea; @Blum:2015ywa].
[**Class B:**]{} All operators contribute only through RG mixing into the chromo-magnetic operator $O_{8g}$ so that only one hadronic matrix element is involved and taken from [@Constantinou:2017sgv; @Buras:2018evv].
[**Class C:**]{} RLRL type operators with flavour $(\bar sd)(\bar uu)$ that contribute via BSM matrix elements [@Aebischer:2018rrz] or by generating the chromo-magnetic dipole matrix element [@Constantinou:2017sgv; @Buras:2018evv] through mixing.
[**Class D:**]{} RLRL type operators with flavour $(\bar sd)(\bar dd)$ that contribute via BSM matrix elements [@Aebischer:2018rrz] or the chromo-magnetic dipole matrix element [@Constantinou:2017sgv; @Buras:2018evv].
[**Class E:**]{} RLLR type operators with flavour $(\bar sd)(\bar uu)$ that contribute exclusively via BSM matrix elements [@Aebischer:2018rrz].
Besides the $P_i$, we provide in the next-to-last column of the suppression scale $\Lambda$ that would generate $(\epe)_\text{BSM}=10^{-3}$ for $C_i=(1\,\text{TeV})^2/\Lambda^2$. It gives an indication of the maximal scale probed by $\epe$ for any given operator.
Among the 40 four-quark operators present in $\mathcal{H}_{\Delta S = 1}^{(5)}$, four have been omitted in , namely $O_{SLR}^{b,c}$ and $\widetilde{O}_{SLR}^{b,c}$, since they neither contribute directly nor via RG mixing at the level considered, i.e. they have $P_i=0$.
In models with a mass gap above the electroweak scale, $v\ll\Lambda$, where $v$ is the Higgs vacuum expectation value and $\Lambda$ the BSM scale, some of the operators in are not generated at leading order in an expansion in $v/\Lambda$. As discussed in more detail in [@Aebischer:2018csl], these operators violate hypercharge, that is conserved in the SMEFT above the electroweak scale [^2]. In the rightmost column of , we have indicated whether the operator can arise from a tree-level matching of SMEFT at dimension six onto the $\Delta S=1$ effective Hamiltonian (cf. [@Aebischer:2015fzz; @Jenkins:2017jig]).
Inspecting the results in , the following comments are in order.
- The largest $P_i$ values in Class A can be traced back to the large values of the matrix elements $\langle Q_{7,8}\rangle_2$, the dominant electroweak penguin operators in the SM, and the enhancement by $1/\omega\approx 22$ of the $I=2$ contributions.
- The small $P_i$ values in Class B are the consequence of the fact that each one is proportional to $\langle O_{8g} \rangle_0$, which has recently been found to be much smaller than previously expected [@Constantinou:2017sgv; @Buras:2018evv]. Moreover, as $\langle O_{8g}\rangle_2=0$, all contributions in this class are suppressed by the factor $1/\omega$ relative to contributions from other classes.
- The large $P_i$ values in Classes C and D can be traced back to the large hadronic matrix elements of scalar and tensor operators calculated recently in [@Aebischer:2018rrz]. Due to the smallness of $\langle O_{8g} \rangle_0$, the contribution of the chromo-magnetic dipole operator is negligible.
- While the operators in Classes D have sizable $P_i$, they violate hypercharge as discussed above, so they do not arise in a tree-level matching from SMEFT at dimension six.
- While the $I=0$ matrix elements of the operators in Class E cannot be expressed in terms of SM ones, the $I=2$ matrix elements can, and the large $P_i$ values can be traced back to the large SM matrix elements $\langle Q_{7,8}\rangle_2$.
Almost all existing BSM analyses of $\epe$ in the literature are based on the contributions of operators from Class A or the chromo-magnetic dipole operator. shows that also other operators, in particular the ones in Class C, could be promising to explain the emerging $\epe$ anomaly and can play an important role in constraining BSM scenarios.
However, in a concrete BSM scenario, the Wilson coefficients with the highest values of $P_i$ could vanish or be suppressed by small couplings. Moreover, additional constraints on Wilson coefficients can come in SMEFT and from correlations with other observables as discussed in more detail in [@Aebischer:2018csl].
Next, we would like to comment on the accuracy of the values of the $P_i$ listed in . As far as short distance contributions to the $P_i$ are concerned, they have been calculated in the leading logarithmic approximation to RG improved perturbation theory using the results of [@Aebischer:2017gaw; @Jenkins:2017dyc; @Aebischer:2017ugx; @flavio; @Aebischer:2018bkb]. Although the inclusion of next-to-leading corrections is possible already now, such contributions are renormalization scheme dependent and can only be cancelled by the one of the hadronic matrix elements. While in the case of SM operators this dependence has been included in the DQCD calculations in [@Buras:2014maa], much more work still has to be done in the case of BSM operators.
The uncertainties from the matrix elements depend on the operator classes in . In the $P_i$ column, we have given the uncertainties obtained from varying the individual matrix elements, assuming them to be uncorrelated. In Class A, they stem from the lattice matrix elements. Here we point out that due to the enhancement of the $I=2$ contributions by the factor $1/\omega\approx22$, the largest $P_i$ are dominated by the $I=2$ matrix elements, which are known to 5–7% accuracy from lattice QCD [@Blum:2015ywa]. For the small $P_i$ in Class A, in some cases there are cancellations between contributions from different matrix elements, leading to larger relative uncertainties. The matrix elements of four-quark BSM operators entering Classes C–E have only been calculated recently in DQCD approach [@Aebischer:2018rrz] and it will still take some time before lattice QCD will be able to provide results for them. Previous results of DQCD imply that it is a successful approximation of low-energy QCD and that the uncertainties in the largest $P_i$ are at most at the level of $20\%$. While not as precise as ultimate lattice QCD calculations, DQCD offered over many years an insight in the lattice results and often, like was the case of the $\Delta I = 1/2$ rule [@Bardeen:1986vz] and the parameter $\hat B_K$ [@Bardeen:1987vg], provided results almost three decades before this was possible with lattice QCD. The agreement between results from DQCD and lattice QCD is remarkable, in particular considering the simplicity of the former approach compared to the sophisticated and computationally demanding numerical lattice QCD one. The most recent example of this agreement was an explanation by DQCD of the pattern of values of $B_6^{(1/2)}$ and $B_8^{(3/2)}$ entering $\epe$ obtained by lattice QCD [@Buras:2015xba; @Buras:2016fys] and of the pattern of lattice values for BSM parameters $B_i$ in $K^0$-$\bar K^0$ mixing [@Buras:2018lgu]. This should be sufficient for the exploration of new phenomena responsible for the hinted $\epe$ anomaly. Similar comments apply to the hadronic matrix element of the chromo-magnetic dipole operator, entering mainly the $P_i$ in Class B, that was recently calculated in DQCD in [@Buras:2018evv] and found to be in agreement with the lattice QCD result from [@Constantinou:2017sgv]. Since the $P_i$ in Class B only receive a single contribution, their relative uncertainties mirror the relative uncertainty of the chromo-magnetic matrix element, that was estimated at 30% in [@Buras:2018evv].
The usefulness of our master formula is twofold. First, it opens the road to an efficient search for BSM scenarios behind the $\epe$ anomaly and through the values of $P_i$ in indicates which routes could be more successful than others. This will play an important role if the $\epe$ anomaly will be confirmed by more precise lattice QCD calculations. Second, it allows to put strong constraints on models with new sources of CP violation, in many cases probing scales up to hundreds of TeV, as shown in . Thus, even if future lattice QCD calculations within the SM will confirm the data on $\epe$, our master formula will be instrumental in putting strong constraints on the parameters of a multitude of BSM scenarios.
#### Acknowledgments. {#acknowledgments. .unnumbered}
The work of J. A., C. B., A. J. B., and D. M. S. is supported by the DFG cluster of excellence “Origin and Structure of the Universe”. We thank Andreas Crivellin for pointing out misprints in .
[^1]: For $\Gamma_T$ there is only $P_A = P_B$ in four dimensions but not $P_A \neq P_B$.
[^2]: As an exception, the hypercharge constraint can be avoided for the operator $\tilde{O}_{SLR}^u$, if in the intermediate SMEFT the dimension-six operator with right-handed modified $W^\pm$ couplings ($\mathcal{O}_{Hud}$ in the basis of [@Grzadkowski:2010es]) is generated, as for example in a left-right symmetric UV completion of the SM due to tree-level $W_L$–$W_R$ mixing. The $\tilde{O}_{SLR}^u$ is then generated at the electroweak scale by the tree-level $W^\pm$ exchange of a single insertion of $\mathcal{O}_{Hud}$ with a dimension-four SM coupling of $W^\pm$ and quarks.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'A transfer matrix function representation of the fundamental solution of the general-type discrete Dirac system, corresponding to rectangular Schur coefficients and Weyl functions, is obtained. Connections with Szegö recurrence, Schur coefficients and structured matrices are treated. Borg-Marchenko-type uniqueness theorem is derived. Inverse problems on the interval and semiaxis are solved.'
author:
- 'B. Fritzsche, B. Kirstein, I. Roitberg, A.L. Sakhnovich'
title: 'Discrete Dirac system: rectangular Weyl functions, direct and inverse problems'
---
MSC 2010: 34B20, 34L40, 39A12, 47A57.
[*Keywords: Discrete Dirac system, Szegö recurrence, Weyl function, inverse problem, $j$-theory, Schur coefficient.*]{}
Introduction {#intro}
============
In this paper we deal with a discrete Dirac-type (or simply Dirac) system: $$\label{0.1}
y_{k+1}(z)=(I_m+ {\mathrm{i}}z
j C_k)
y_k(z) \quad \left( k \in \BN_0
\right),$$ where $\BN_0$ stands for the set of non-negative integer numbers, $I_m$ is the $m \times m$ identity matrix, $"{\mathrm{i}}"$ is the imaginary unit (${\mathrm{i}}^2=-1$) and the $m \times m$ matrices $\{C_k\}$ are positive and $j$-unitary: $$\label{0.2}
C_k>0, \quad C_k j C_k=j, \quad j: = \left[
\begin{array}{cc}
I_{m_1} & 0 \\ 0 & -I_{m_2}
\end{array}
\right] \quad (m_1+m_2=m, \, \, m_1, \, m_2 \not= 0).$$ Discrete systems are of great interest and their study is sometimes more complicated than the study of the corresponding continuous systems (see, e.g., [@AbC1; @AbC2; @AG0; @BoSu; @FaMoL] and references therein). The subcase $m_1=m_2$ of system (satisfying ) corresponds to the self-adjoint Dirac-type systems, which were studied in [@FKRS08] (and the subcase $j=I_m$ of system corresponds to the skew-self-adjoint Dirac-type systems, an important subclass of which was investigated in [@KaSa; @SaA8]). The analogies between system and continuous Dirac-type systems are also discussed in [@FKRS08; @KaSa; @SaA8] in detail. Here we follow the paper [@FKRS12] on the continuous case, where $m_1$ does not necessarily equals $m_2$ and the $m_2 \times m_1$ Weyl matrix functions are, correspondingly, rectangular.
It is essential that Dirac system , is equivalent to the very well-known Szegö recurrence (see, e.g., [@DFK; @Si2]). This connection is discussed in detail in Section \[SzRc\]. Inverse problems for the subcase of the scalar Schur (or Verblunsky) coefficients were studied, for instance, in [@AG2; @Si2] (see also various references therein), and here we deal with the rectangular matrix Schur coefficients.
In this paper $\im$ denotes image of a matrix (or operator), $\s(A)$ stands for the spectrum of $A$ and “span” stands for the linear span.
Dirac system and Szegö recurrence {#SzRc}
=================================
The next simple proposition is essential for our future research and could be of independent interest in the theory of functions $($and powers, in particular$)$ of matrices, which is developed in a series of works $($see, e.g., [@BiI; @VeS] and references therein$)$.
\[PnC\] Let an $m\times m$ matrix $C$ satisfy relations $$\label{0.2'}
C>0, \quad C j C=j \quad (j=j^*=j^{-1}).$$ Then the following relations hold for all $s\in \BR$: $$\label{0.2!}
C^s>0, \quad C^s j C^s=j.$$
. Since $C>0$, it admits a representation $$\begin{aligned}
\label{C2}&
C=u^*D u,\end{aligned}$$ where $D$ is a diagonal matrix and $$\begin{aligned}
\label{C3}&
D>0, \quad u^* u=u u^*=I_m .\end{aligned}$$ We substitute into the second equality in to derive $$\begin{aligned}
\nn &
u^*D u j u^*D u=j,\end{aligned}$$ or, equivalently, $$\begin{aligned}
\label{C4}&
D J D=J, \quad J=J^*=J^{-1}:=u j u^*.\end{aligned}$$ Formula yields $D^{-1}=J D J$ and, taking power $s$ of the both parts of this equality, we obtain $$\begin{aligned}
\label{C5}&
D^{-s}=J D^{s} J, \qquad D^{s}J D^{s}= J.\end{aligned}$$ Finally, using – we have $$\begin{aligned}
\label{C6}&
u^*D^{s}u j u^*D^{s}u=j.\end{aligned}$$
We substitute $s=1/2$ and apply Proposition \[PnC\] to matrices $C_k$ in order to obtain the next proposition.
\[Rkgb\] Let matrices $C_k$ satisfy . Then they admit representations $$\begin{aligned}
\label{C1'}&
C_k=2\bt(k)^*\bt(k)-j, \quad \bt(k)j \bt(k)^*=I_{m_1}, \\
\label{C1}&
C_k=j+2\g(k)^*\g(k), \quad \g(k) j \g(k)^*=-I_{m_2},\end{aligned}$$ where $\bt(k)$ and $\g(k)$ are $m_1 \times m$ and $m_2 \times m$ matrices given by and , respectively.
. We note that matrices $C_k$ satisfy conditions of Proposition \[PnC\] and so holds for $C=C_k$. Next we put $$\begin{aligned}
\label{C7'}&
\bt(k):=\begin{bmatrix}
I_{m_1} & 0
\end{bmatrix}C_k^{1/2}\end{aligned}$$ and take into account the equality $$C_k=C_k^{1/2}\Big(2
\begin{bmatrix}
I_{m_1} & 0
\end{bmatrix} ^*
\begin{bmatrix}
I_{m_1} & 0
\end{bmatrix} -j \Big)C_k^{1/2}.$$ Now, representation is apparent from taken with $s=1/2$. In a similar way, formula and equality $I_m=j +2
\begin{bmatrix}
0 &
I_{m_2}
\end{bmatrix} ^*
\begin{bmatrix}
0 &
I_{m_2}
\end{bmatrix}
$ imply representation for $$\begin{aligned}
\label{C7}&
\g(k)=\begin{bmatrix}
0 &
I_{m_2}
\end{bmatrix}C_k^{1/2}.\end{aligned}$$
Now, we will consider interrelations between Dirac system , and Szegö recurrence, which is given by the formula $$\label{C8}
X_{k+1}(\la)={\cal D}_k H_k \left[
\begin{array}{cc}
\la I_{m_1} & 0 \\ 0 & I_{m_2}
\end{array}
\right] X_k(\la),$$ where $$\label{C9}
H_k= \left[
\begin{array}{cc}
I_{m_1} & \rho_k \\ \rho_k^* & I_{m_2}
\end{array}
\right], \quad {\mathcal D}_k= {\mathrm{diag}}\Big\{
\big(
I_{m_1}-\rho_k \rho_k^* \big)^{-\frac{1}{2}}, \, \,
\big(I_{m_2}-\rho_k^*\rho_k\big)^{-\frac{1}{2}}\Big\},$$ and the $m_1 \times m_2$ matrices $\rho_k$ are strictly contractive, that is, $$\begin{aligned}
&\label{C9'}
\|\rho_k\|<1.\end{aligned}$$
\[SzRec\] When $m_1=m_2=1$, one easily removes the factor $(1-|\rho_k|^2)^{-1/2}$ in to obtain systems as in [@AG1; @AG2], where direct and inverse problems for the case of scalar strictly pseudo-exponential potentials have been treated. The square matrix version $($i.e., the version where $m_1=m_2)$ of Szegö recurrence, its connections with Schur coefficients and applications are discussed in [@DGK1; @DGK2] $($see also references therein$)$. For the rectangular matrices $\rho_k$ see, for instance, [@DFK]. We note that $\cld_k H_k $ is the so called Halmos extension of $\rho_k$ $($see [@DFK p. 167]$)$, and that the matrices $\cld_k$ and $H_k $ commute $($which easily follows, e.g., from [@DFK Lemma 1.1.12]$)$. The matrix ${\cal D}_k H_k$ is $j$-unitary and positive, that is, $$\begin{aligned}
& \label{C10}
{\cal D}_k H_k j H_k{\cal D}_k=H_k{\cal D}_k j {\cal
D}_k H_k=j, \\
\label{nov0} & {\cal D}_k H_k>0.\end{aligned}$$
According to [@Dy0 Theorem 1.2], any $j$-unitary matrix $C$ admits a representation, which is close to Halmos extension. More precisely, partitioning $C$ into blocks $C=\{c_{ik}\}_{i,k=1}^2$ we see that the $m_1 \times m_1$ block $c_{11}$ and the $m_2\times m_2$ block $c_{22}$ are invertible. Then, putting $$\begin{aligned}
& \nn
\rho=c_{12}c_{22}^{-1}=(c_{11}^{-1})^*c_{21}^*, \quad u_1=\big(I_{m_1}-\rho \rho^*\big)^{1/2}c_{11}, \quad u_2=\big(I_{m_2}- \rho^* \rho\big)^{{1}/{2}}c_{22},\end{aligned}$$ we have the respresentation: $$\begin{aligned}
\label{nov1}&
C= {\cal D} H\begin{bmatrix} u_1 & 0\\ 0 & u_2 \end{bmatrix}, \quad u_i^*u_i=u_iu_i^*=I_{m_i}; \quad H= \left[
\begin{array}{cc}
I_{m_1} & \rho \\ \rho^* & I_{m_2}
\end{array}
\right],
\\ \label{nov2}&
{\mathcal D}= {\mathrm{diag}}\Big\{
\big(
I_{m_1}- \rho \rho^* \big)^{-\frac{1}{2}}, \, \,
\big(I_{m_2}- \rho^* \rho\big)^{-\frac{1}{2}}\Big\}, \quad \rho^*\rho <I_{m_2}.\end{aligned}$$ Although relations - are well-known, we could not find in the literature a statement, which is converse to , . Hence, we prove it below.
\[novHE\] Let an $m\times m$ matrix $C$ be $j$-unitary and positive. Then it admits a representation $$\begin{aligned}
\label{nov3}&
C= {\cal D} H,\end{aligned}$$ where $ H$ and $ {\cal D}$ are of the form and $($i.e., the last factor on the right-hand side of the first equality in is removed$)$.
. Recall that $C$ admits representation . We fix a unitary matrix $\wt U$ such that $ \cld H= \wt U \wt D \wt U^*$, where $ \wt D$ is a diagonal matrix, $ \wt D>0$. Then, relations $C=C^*$ and yield the equality $$\wt U \wt D \wt U^* \begin{bmatrix} u_1 & 0\\ 0 & u_2 \end{bmatrix} =\begin{bmatrix} u_1^* & 0\\ 0 & u_2^* \end{bmatrix} \wt U \wt D \wt U^*,$$ which we rewrite in the form $$\begin{aligned}
& \label{nov3'}
\wt D \wh U=\wh U^* \wt D, \quad \wh U:=\wt U^*\begin{bmatrix} u_1 & 0\\ 0 & u_2 \end{bmatrix}\wt U.\end{aligned}$$ According to , $\wt D \wh U$ is a selfadjoint matrix, and so $\wt D^{1/2}\wh U \wt D^{-1/2}$ is a selfadjoint matrix too, that is, there is a representation $$\begin{aligned}
& \label{nov4}
\wt D^{1/2}\wh U \wt D^{-1/2}= \breve U D_1 \breve U^*,\end{aligned}$$ where $\breve U$ and $D_1=D_1^*$ are unitary and diagonal matrices, respectively. The definition of $\wh U$ in implies that $\wh U$ is unitary. Therefore, in view of , $D_1$ is linear similar to a unitary matrix, that is, its entries are $\pm 1$. Moreover $D_1>0$, since $C>0$ and formulas , and yield $$\begin{aligned}
& \label{nov5}
C=\wt U \wt D \wt U^* \begin{bmatrix} u_1 & 0\\ 0 & u_2 \end{bmatrix}=\wt U \wt D \wh U \wt U^*=
\wt U \wt D^{1/2}\breve U D_1 \breve U^* \wt D^{1/2}\wt U^*.\end{aligned}$$ From the inequality $D_1>0$ and the fact that the entries of $D_1$ equal either $1$ or $-1$, we have $D_1=I_m$. Thus, the last equality in implies $C=\wt U \wt D \wt U^* $, that is, holds.
Proposition \[novHE\] completes Propositions \[PnC\] and \[Rkgb\] on representations and properties of $C_k$. Taking into account , and Proposition \[novHE\], we rewrite Szegö recurrence in an equivalent form $$\begin{aligned}
\label{C8'} &
X_{k+1}(\la)=\wt C_k \left[
\begin{array}{cc}
\la I_{m_1} & 0 \\ 0 & I_{m_2}
\end{array}
\right] X_k(\la), \quad k \in \BN_0,
\\ \label{nov6} &
\wt C_k>0, \quad \wt C_k j\wt C_k=j.\end{aligned}$$ Using we see that the matrix functions $U_k$, which are given by the equalities $$\label{C11}
U_0:=I_m, \quad U_{k+1}:={\mathrm{i}}U_k\wt C_k j=\prod_{r=0}^k({\mathrm{i}}\wt C_r j)
\quad (k \geq 0),$$ are also $j$-unitary. From and we have $$\begin{aligned}
\nn &
({\mathrm{i}}+z)U_{k+1}(I_m+{\mathrm{i}}z j)\wt C_k \begin{bmatrix}
\frac{z-{\mathrm{i}}}{z+ {\mathrm{i}}} I_{m_1} & 0 \\ 0 & I_{m_2}
\end{bmatrix}(I_m+{\mathrm{i}}z j)^{-1}U_k^{-1}
\\ \label{C12} &
=I_m+ {\mathrm{i}}z U_{k+1} j U_{k+1}^{-1} .\end{aligned}$$ In view of , the function $y_k$ of the form $$\label{C13}
y_k(z)=({\mathrm{i}}+z)^k U_k (I_m
+ {\mathrm{i}}z j)
X_k\left(\frac{z-{\mathrm{i}}}{z+ {\mathrm{i}}}\right)$$ satisfies , where $y_0(z)=(I_m+{\mathrm{i}}z j)X_0(z)$ and $C_k=jU_{k+1}jU_{k+1}^{-1}$. Since $U_{k+1}$ is $j$-unitary, we rewrite $C_k$ as $$\begin{aligned}
\label{C14} &
C_k=jU_{k+1}U_{k+1}^*j,\end{aligned}$$ and so holds. Because of , and $j$-unitarity of $U_k$, we have $jU_k^*C_kU_k j=\wt C_k^2$, that is, $$\begin{aligned}
\label{C15} &
\wt C_k=(jU_k^*C_kU_k j)^{1/2}.\end{aligned}$$ The following theorem describes interconnections between systems and .
\[TmSzRc\] Dirac systems , and Szegö recurrences , are equivalent. The transformation ${\mathfrak M}: \, \{\wt C_k\} \rightarrow \{C_k\}$ of Szegö recurrence into Dirac system, and the transformation of their solutions, are given, respectively, by formulas and , where matrices $\{U_k\}$ are defined in . The mapping ${\mathfrak M}$ is bijective, and the inverse mapping is obtained by applying $($and substitution of the result into $)$ for the successive values of $k$.
. It is proved already above that the formulas and describe a mapping of Szegö recurrence and its solution into Dirac system and its solution, respectively. Moreover, the mapping ${\mathfrak M}$ is injective, since we can successively and uniquely recover $\wt C_k$ and $U_{k+1}$ from $C_k$ and $U_k$ using formulas and , respectively.
Next, we prove that ${\mathfrak M}$ is surjective. Indeed, given an arbitrary sequence $\{C_k\}$ satisfying , let us apply to the matrices from this sequence relation (and substitute the result into ) for the successive values of $k$. In this way we construct a sequence $\{\wt C_k\}$. Since the matrices $jU_k^*C_kU_k j$ are positive and $j$-unitary, we see, from and Proposition \[PnC\], that the matrices $\wt C_k$ are also positive and $j$-unitary. Next, we apply to $\{\wt C_k\}$ the mapping ${\mathfrak M}$. Taking into account and , we derive $$\begin{aligned}
\label{nov7} &
jU_{k+1}U_{k+1}^*j=jU_k\wt C_k^2U_k^*j=jU_k(jU_k^*C_kU_k j)U_k^*j=C_k,\end{aligned}$$ that is, ${\mathfrak M}$ maps the constructed sequence $\{\wt C_k\}$ into the initial sequence $\{C_k\}$. Recall that we started from an arbitrary $\{C_k\}$ satisfying . Hence, ${\mathfrak M}$ is surjective.
Weyl theory: direct problems {#gcdp}
=============================
In this section we introduce Weyl functions for matricial discrete Dirac systems (\[0.1\]). Next we prove the Weyl function’s existence and, moreover, give a procedure to construct it (direct problems). Finally, we construct the $S$-node, which corresponds to system , and the transfer matrix function representation of the fundamental solution $W_k$. (See, e.g., [@SaL1; @SaL2; @SaL3] on the $S$-nodes and the transfer matrix functions in Lev Sakhnovich sense.)
The fundamental $m \times m$ solution $\{W_k \}$ of we normalize by the condition $$\begin{aligned}
\label{1.1}&
W_0(z)=I_m.\end{aligned}$$ Similar to the continuous analog of in [@FKRS12; @FKRSp12] (see also canonical system case [@SaL3 p. 7]), the Weyl functions of system on the interval $[0, \, r]$ (i.e., system considered for $0 \leq k \leq r$) are defined by the Möbius (linear-fractional) transformation: $$\begin{aligned}
\label{1.6}&
\vp_r(z, \clp)=\begin{bmatrix}
0 &I_{m_2}
\end{bmatrix}W_{r+1}(z)^{-1}\clp(z)\Big(\begin{bmatrix}
I_{m_1} & 0
\end{bmatrix}W_{r+1}(z)^{-1}\clp(z)\Big)^{-1},\end{aligned}$$ where $\clp(z)$ are nonsingular $m \times m_1$ matrix functions with property-$j$. That is, $\clp(z)$ are meromorphic in $\BC_+$ matrix functions such that $$\begin{aligned}
\label{1.7}&
\clp(z)^*\clp(z)>0, \quad \clp(z)^*j\clp(z) \geq 0\end{aligned}$$ for all points in $\BC_+$ (excluding, possibly, a discrete set). The first inequality in means non-singularity (non-degeneracy) of $\clp$ and the second inequality is called property-$j$. Since $\clp$ is meromorphic, property-$j$ almost everywhere in $\BC_+$ and the first inequality in at some $z_0 \in \BC_+$ suffice for the conditions on $\clp$ to hold.
It is apparent from and that $$\begin{aligned}
\label{Z3}&
W_{r+1}(z)=\prod_{k=0}^r(I_m+{\mathrm{i}}z jC_k ).\end{aligned}$$ In view of and we obtain $$\begin{aligned}
\label{Z9}&
W_{r+1}({\mathrm{i}})=(-2)^{r+1}\prod_{k=0}^r \big(j\g(k)^*\g(k)\big).\end{aligned}$$ Hence, $\det W_{r+1}({\mathrm{i}})=0$, and we don’t consider $z={\mathrm{i}}$ in this section.
\[zi\] We note that the behavior of Weyl functions in the neighborhood of $z= {\mathrm{i}}$ is essential for the inverse problems that are dealt with in the next section. Therefore, unlike the Weyl disc case $($see Notation \[cln\]$)$, in the definition of the Weyl functions on the interval we assume that $\clp$ is not only nonsingular with property-$j$ but has also an additional property. Namely, it is well-defined and nonsingular at $z={\mathrm{i}}$. We don’t use this additional property in this section, though, in important cases, it could be obtained via multiplication by a scalar function.
The lemma below shows that transformations $\vp_r(z, \clp)$ are well-defined.
\[det\] Fix any $z\in \BC_+$ such that the inequalities $\det W_r(z) \not=0$ and hold. Then we have the inequality $$\begin{aligned}
\label{Z1}&
\det\Big(\begin{bmatrix}
I_{m_1} & 0
\end{bmatrix}W_{r+1}(z)^{-1}\clp(z)\Big)\not=0.\end{aligned}$$
. Using and we obtain $$\begin{aligned}
\nn&
(I_m+{\mathrm{i}}z jC_k )^*j (I_m+{\mathrm{i}}z jC_k )=(1+{\mathrm{i}}(z- \ov{z})+|z|^2)j+2{\mathrm{i}}(z- \ov{z})\g(k)^*\g(k)
\\ & \label{Z2}
\leq (1-2\Im(z)+|z|^2)j, \qquad (1-2\Im(z)+|z|^2) >0 \quad {\mathrm{for}} \quad z\not= {\mathrm{i}}.\end{aligned}$$ Since the equality holds, formula implies that $$\begin{aligned}
\label{Z4}&
\big(W_{r+1}(z)^{-1}\big)^*jW_{r+1}(z)^{-1}\geq (1-2\Im(z)+|z|^2)^{-r-1}j \quad (z\in \BC_+, \,\, z\not={\mathrm{i}}).\end{aligned}$$ Because of and , we see that $\wt \clp:= W_{r+1}(z)^{-1}\clp(z)$ satisfies the inequality $\wt \clp^* j \wt \clp \geq 0$. It is apparent that the same inequality holds for the matrix $\begin{bmatrix}
I_{m_1} & 0
\end{bmatrix}^*$. In other words, $\im W_{r+1}(z)^{-1}\clp(z)$ and $\im \begin{bmatrix}
I_{m_1} & 0
\end{bmatrix}^*$ are maximal $j$-nonnegative subspaces. Therefore, the inequality follows in a standard way from $j$-theoretic considerations (see, e.g., the proof of or the proof of [@FKRS08 inequality (5.6)] for such considerations).
\[CyW-1\] The following inequalities hold for the fundamental solution $W_{r+1}$ of $($where $\{C_k\}$ satisfy $):$ $$\begin{aligned}
&\label{Z5}
\det W_{r+1}(z)\not=0, \quad W_{r+1}(z)^{-1}=(1+z^2)^{-r-1}jW_{r+1}(\ov{z})^*j \qquad (z\not= \pm {\mathrm{i}}).\end{aligned}$$
. Relations and imply that $$W_{r+1}(z)^*jW_{r+1}(z)=(1+z^2)^{r+1}j, \qquad z=\ov{z}.$$ Hence, using analyticity considerations, we obtain $$\begin{aligned}
\label{Z6}&
W_{r+1}(\ov{z})^*jW_{r+1}(z)\equiv (1+z^2)^{r+1}j,\end{aligned}$$ and is apparent.
\[cln\] The set of values of matrices $\vp_r(z, \clp)$, which are given by the transformation where parameter matrices $\clp(z)$ satisfy , is denoted by $\cln(r,z)$ $($or, sometimes, simply $\cln(r))$.
Usually, $\cln(r,z)$ is called the Weyl disk.
\[Cyj\] The sets $\cln(r,z)$ are embedded $($i.e., $\cln(r,z)\subseteq \cln(r-1,z))$ for all $r >0 $ and $z \in \BC_+$, $\, z \not= {\mathrm{i}}$. Moreover, for all $\vp_k$ $(k \geq 0)$ we have $$\begin{aligned}
\label{Z8}&
\vp_k(z)^*\vp_k(z)\leq I_{m_1}.\end{aligned}$$
. It follows from Corollary \[CyW-1\] that the matrices $W_{r+1}(z)$, $W_r(z)$ and $(I_m+{\mathrm{i}}z jC_r )$ are invertible. Hence formulas and imply that $\wt \clp:=(I_m+{\mathrm{i}}z jC_r )^{-1}\clp(z)$ satisfies . Therefore, we rewrite in the form $$\begin{aligned}
\label{Z7}&
\vp_r(z, \clp)=\begin{bmatrix}
0 &I_{m_2}
\end{bmatrix}W_{r}(z)^{-1}\wt \clp(z)\Big(\begin{bmatrix}
I_{m_1} & 0
\end{bmatrix}W_{r}(z)^{-1}\wt \clp(z)\Big)^{-1},\end{aligned}$$ and see that $\vp_r(z)\in \cln(r-1,z)$ ($r>0$). Inequality is obtained for the matrices from $\cln(0,z)$ via substitution of $r=0$ into .
Weyl functions of system on the semiaxis $\BN_0$ of non-negative integers are defined in a different and more traditional way (in terms of summability), see definition below. We will show also that the definitions of Weyl functions on the interval and semiaxis are interrelated.
\[defWeyl\] The Weyl-Titchmarsh $($or simply Weyl$)$ function of Dirac system $($which is given on the semiaxis $0\leq k < \infty$ and satisfies $)$ is an $m_2\times m_1$ matrix function $\vp(z)$ $\, (z \in \BC_+)$, such that the following inequality holds: $$\begin{aligned}
\label{1.3}&
\sum_{k=0}^\infty q(z)^k
\begin{bmatrix}
I_{m_1} & \vp(z)^*
\end{bmatrix}
W_k(z)^*C_k W_k (z)
\begin{bmatrix}
I_{m_1} \\ \vp(z)
\end{bmatrix}<\infty,
\\ & \label{1.5}
q(z):=(1+|z|^2)^{-1}.\end{aligned}$$
\[inta\] If $\vp_r(z)\in \cln(r,z)$, we have the inequality $$\begin{aligned}
\label{1.3'}
\sum_{k=0}^r q(z)^k
\begin{bmatrix}
I_{m_1} & \vp_r(z)^*
\end{bmatrix}
W_k(z)^*C_k W_k (z)
\begin{bmatrix}
I_{m_1} \\ \vp_r(z)
\end{bmatrix} \leq & \frac{1+|z|^2}{{\mathrm{i}}( \ov{z}-z)}
\\ \nn & \times
\big(I_m-\vp_r(z)^*\vp_r(z)\big).\end{aligned}$$
. Because of (\[0.1\]) and (\[0.2\]) we have $$\begin{aligned}
\nn
W_{k+1}(z)^*jW_{k+1}(z)&=W_{k}(z)^*\Big(
I_m -
{\mathrm{i}}{\ov z} C_k j \Big)j\Big( I_m
+{\mathrm{i}}z j
C_k \Big)W_{k}(z)
\\
\label{1.2}&
=q(z)^{-1}W_k(z)^*jW_k(z)+{\mathrm{i}}(z-\ov{z})W_k(z)^*C_k W_k(z).\end{aligned}$$ Using and , we derive a summation formula, which is similar to the formula for the case that $m_1=m_2$, see [@FKRS08 formula (4.2)]: $$\label{1.4}
\sum_{k=0}^r
q(z)^k W_k(z)^*C_k W_k (z)=\frac{1+|z|^2}{{\mathrm{i}}( \ov{z}-z)}
\Big(j- q(z)^{r+1}W_{r+1}(z)^*jW_{r+1}(z)\Big).$$ On the other hand, it follows from that $$\label{1.4!}
\begin{bmatrix}
I_{m_1} \\ \vp_r(z)
\end{bmatrix} =W_{r+1}(z)^{-1}\clp(z)\Big(\begin{bmatrix}
I_{m_1} & 0
\end{bmatrix}W_{r+1}(z)^{-1}\clp(z)\Big)^{-1},$$ and so formula yields $$\label{1.4'}
\begin{bmatrix}
I_{m_1} & \vp_r(z)^*
\end{bmatrix}
W_{r+1}(z)^*j W_{r+1}(z)
\begin{bmatrix}
I_{m_1} \\ \vp_r(z)
\end{bmatrix} \geq 0.$$ Formulas and imply .
Now, we are ready to prove the main direct theorem.
\[Tm3.8\] There is a unique Weyl function of the discrete Dirac system , which is given on the semi-axis $ 0\leq k < \infty$ and satisfies . This Weyl function $\vp$ is analytic and non-expansive $($i.e., $\vp^*\vp \leq I_{m_1})$ in $\BC_+$.
. The proof consists of 3 steps. First, we show that there is an analytic and non-expansive function $$\begin{aligned}
\label{Z10}&
\vp_{\infty}(z)\in \bigcap_{r \geq 0} \cln(r,z).\end{aligned}$$ Next, we show that $\vp_{\infty}(z)$ is a Weyl function. Finally, we prove the uniqueness.
[**Step 1.**]{} This step is similar to the corresponding part of the proof of [@FKRSp12 Proposition 2.2]. Indeed, from Corollary \[Cyj\] we see that the set of functions $\vp_r(z,\clp)$ of the form is uniformly bounded in $\BC_+$. So, Montel’s theorem is applicable and there is an analytic matrix function, which we denote by $\vp_{\infty}(z)$ and which is a uniform limit of some sequence $$\begin{aligned}
& \label{Z11}
\vp_{\infty}(z)=\lim_{i \to \infty} \vp_{r_i}(z,\clp_i) \quad (i \in \BN, \quad r_i \uparrow, \quad \lim_{i \to \infty}r_i=\infty)\end{aligned}$$ on all the bounded and closed subsets of $\BC_+$. Clearly, $\vp_{\infty}$ is non-expansive. Since $r_i \uparrow$, the sets $\cln(r,z)$ are embedded and equality is valid, it follows that the matrix functions $$\begin{aligned}
& \nn
\clp_{ij}(z):=W_{r_i+1}(z)\begin{bmatrix}
I_{m_1} \\ \vp_{r_j}(z,\clp_j)
\end{bmatrix} \quad (j \geq i)\end{aligned}$$ satisfy relations . Therefore, using we derive that holds for $$\begin{aligned}
& \label{pizphi}
\clp_{i, \infty}(z):=W_{r_i+1}(z)\begin{bmatrix}
I_{m_1} \\ \vp_{\infty}(z)
\end{bmatrix},\end{aligned}$$ which implies that we can substitute $\clp=\clp_{i, \infty}$ and $r=r_i$ into to obtain $$\begin{aligned}
& \label{Z12}
\vp_{\infty}(z)\in\cln(r_i,z).\end{aligned}$$ Since holds for all $i\in \BN$, we see that is fulfilled.
[**Step 2.**]{} Because of , the function $\vp_{\infty}$ satisfies condition of Lemma \[inta\]. Hence, holds for any $r \geq 0$ and $\vp_r=\vp_{\infty}$, which implies . Therefore, $\vp_{\infty}$ is a Weyl function.
[**Step 3.**]{} It is apparent from that $$\label{Z13}
W_k(z)^*C_k W_k(z)\geq W_k(z)^*(-j) W_k(z).$$ Using we derive also $$\label{Z14}
q(z)^k W_k(z)^*(-j) W_k(z)\geq q(z)^{k-1}W_{k-1}(z)^*(-j) W_{k-1}(z).$$ Formulas , and yield the basic for Step 3 inequality $$\label{Z15}
q(z)^k W_k(z)^*C_k W_k(z)\geq -j.$$ Therefore, the following equality is immediate for any $g\in
\BC^{m_2}$: $$\label{Z16}
\sum_{k=0}^\infty g^*[0 \quad I_{m_2}]
q(z)^kW_k(z)^*C_k W_k(z)\left[\begin{array}{c}
0 \\ I_{m_2}
\end{array}
\right]g =\infty .$$ It was shown in Step 2 that $\vp = \vp_{\infty}$ satisfies . According to (\[1.3\]) and (\[Z16\]), the dimension of the subspace $L \in \BC^m$ of vectors $h$ such that $$\label{Z17}
\sum_{k=0}^\infty
h^*q(z)^kW_k(z)^*C_k W_k(z)h <\infty$$ equals $m_1$. Now, suppose that there is a Weyl function $\wt \vp
\not= \vp_{\infty}$. Then we have $$\im \left[\begin{array}{c} I_{m_1} \\ \vp_{\infty}(z)
\end{array}
\right] \subseteq L, \quad
\im \left[\begin{array}{c}
I_{m_1} \\ \wt
\vp(z)
\end{array}
\right] \subseteq L .$$ Therefore, $\dim L>m_1$ (for those $z$, where $\wt \vp(z) \not=
\vp_{\infty}(z)$) and we arrive at a contradiction.
Finally, let us construct representations of $W_{r+1}$ $\, (r \geq 0)$ via $S$-nodes. First, recall that matrices $\{C_k\}$ generate via formula a set $\{\g(k)\}$ of the $m_2 \times m$ matrices $\g(k)$. Using $\{\g(k)\}$, we introduce $m_2(r+1) \times m$ matrices $\G_r$ and $m_2(r+1) \times m_2(r+1)$ matrices $K_r$ $\,(0 \leq r < \infty)$: $$\begin{aligned}
\label{1.8}&
\G_r:=\begin{bmatrix}\g (0)\\ \g(1) \\ \ldots \\ \g(r) \end{bmatrix}; \quad
K_r:=\begin{bmatrix}\vk_r (0)\\ \vk_r(1) \\ \ldots \\ \vk_r(r) \end{bmatrix},
\\ \label{1.9}&
\vk_r(k):={\mathrm{i}}\g(k)j
\begin{bmatrix}\g(0)^* & \ldots & \g(k-1)^* & \g(k)^*/2 & 0 & \ldots & 0 \end{bmatrix}.\end{aligned}$$ It is apparent from and that the identity $$\begin{aligned}
\label{1.10}&
K_r-K_r^*={\mathrm{i}}\G_r j \G_r^*\end{aligned}$$ holds. The $m_2(r+1) \times m_2(r+1)$ matrices $A_r$ are introduced by the equalities: $$\begin{aligned}
\label{1.11}&
A_r=\{a_{p-k}\}_{k,p=0}^r, \quad a_n=-\left\{\begin{array}{l}
0 \quad {\mathrm{for}} \quad n>0, \\
({\mathrm{i}}/2) I_{m_2} \quad {\mathrm{for}} \quad n=0, \\
{\mathrm{i}}I_{m_2} \quad {\mathrm{for}} \quad n<0.
\end{array}
\right.\end{aligned}$$
\[PnSym\] Matrices $K_r$ and $A_r$ are linear similar$:$ $$\begin{aligned}
\label{1.12}&
K_r=E_rA_rE_r^{-1}.\end{aligned}$$ Moreover, the similarity transformations $E_r$ can be constructed so that $$\begin{aligned}
\label{1.13}&
E_r=\begin{bmatrix} E_{r-1} &0 \\
X_r & e^{-}_r
\end{bmatrix} \quad (r>0), \quad E_r^{-1}\G_{r,2}=\Phi_{r,2}, \quad \Phi_{r,2}:=
\begin{bmatrix}I_{m_2} \\ \ldots \\ I_{m_2} \end{bmatrix},
\\ \label{1.14}& E_0=e^-_0=\g_2(0),\end{aligned}$$ where $\G_{r,p}$ are $m_2(r+1) \times m_p$ blocks of $\G_r=
\begin{bmatrix}\G_{r,1} & \G_{r,2} \end{bmatrix}$ and $\g_{p}(k)$ are $m_2 \times m_p$ blocks of $\g(k)=
\begin{bmatrix}\g_{1}(k) & \g_{2}(k) \end{bmatrix}$.
. It follows from , , and that $$\begin{aligned}
\label{1.15}&
K_0=A_0=-({\mathrm{i}}/2)I_{m_2}, \quad \det \g_2(0)\not= 0, \\ \label{1.16}&
\vk_r(r)={\mathrm{i}}\begin{bmatrix}\g(r) j \g(0)^* & \ldots & \g(r) j \g(r-1)^* & -I_{m_2}/2 \end{bmatrix}.\end{aligned}$$ We see that and imply for $r=0$. Next, we prove by induction. Assume that $K_{r-1}=E_{r-1}A_{r-1}E_{r-1}^{-1}$ and let $E_r$ have the form , where $\det e^-_r \not=0$. Then we obtain $$\begin{aligned}
\label{1.17}&
E_r^{-1}=\begin{bmatrix} E_{r-1}^{-1} &0 \\
- (e^{-}_r)^{-1}X_r E_{r-1}^{-1} & (e^{-}_r)^{-1}
\end{bmatrix},\end{aligned}$$ and, in view of , , , , it is necessary and sufficient (for to hold) that $$\begin{aligned}
\nn &
\Big(\begin{bmatrix} X_rA_{r-1} & -({\mathrm{i}}/2)e^-_r
\end{bmatrix}-{\mathrm{i}}e^-_r
\begin{bmatrix}
I_{m_2} & \ldots & I_{m_2} &0
\end{bmatrix}\Big)
\begin{bmatrix} I_{rm_2} \\ -(e^-_r)^{-1}X_r
\end{bmatrix}E_{r-1}^{-1}
\\& \label{1.18}
=
{\mathrm{i}}\g(r) j \begin{bmatrix} \g(0)^* & \ldots & \g(r-1)^*
\end{bmatrix}.\end{aligned}$$ We can rewrite in the form $$\begin{aligned}
\nn
X_r\big(A_{r-1}+({\mathrm{i}}/2)I_{rm_2}\big)=&{\mathrm{i}}\g(r)j
\begin{bmatrix}\g(0)^* & \ldots & \g(r-1)^*
\end{bmatrix}E_{r-1}
\\\label{1.19}&
+{\mathrm{i}}e^-_r
\begin{bmatrix} I_{m_2} & \ldots & I_{m_2}
\end{bmatrix}.\end{aligned}$$ We partition $X_r$ ($r>1$) into two $m_2\times m_2$ and $m_2\times (r-1)m_2$, respectively, blocks $$\begin{aligned}
\label{1.19'}&
X_r=\begin{bmatrix} x_{r}^- & \wt X_r
\end{bmatrix},\end{aligned}$$ and we will need also partitions of the matrices $A_{r-1}+({\mathrm{i}}/2)I_{rm_2}$ and $E_{r-1}$, which follow (for $r>1$) from and : $$\begin{aligned}
\label{1.20}&
\big(A_{r-1}+({\mathrm{i}}/2)I_{rm_2}\big)=
\begin{bmatrix} 0 & 0 \\
\big(A_{r-2}-({\mathrm{i}}/2)I_{(r-1)m_2}\big) & 0
\end{bmatrix}, \quad
E_{r-1}
\begin{bmatrix} 0 \\ I_{m_2}
\end{bmatrix}=\begin{bmatrix} 0 \\ e_{r-1}^-
\end{bmatrix}.\end{aligned}$$ Using and we see that is equivalent to the relations $$\begin{aligned}
\label{1.21}
e_r^-=&-\g(r)j
\g(r-1)^*
e_{r-1}^-
\quad \mathrm{for} \quad r \geq 1;
\\ \nn
\wt X_r=&
{\mathrm{i}}\Big( \g(r)j
\begin{bmatrix}\g(0)^* & \ldots & \g(r-1)^*
\end{bmatrix}E_{r-1}
+ e^-_r
\begin{bmatrix} I_{m_2} & \ldots & I_{m_2}
\end{bmatrix}\Big) \\ \label{1.23}&
\times \begin{bmatrix} \big(A_{r-2}-({\mathrm{i}}/2)I_{(r-1)m_2}\big)^{-1} \\ 0
\end{bmatrix} \quad \mathrm{for} \quad r>1.\end{aligned}$$ Hence, if $e_r^-$ and $X_r$ satisfy and , respectively, and $\det e_r^- \not =0$, the similarity relation holds. The inequalities $\det e_r^- \not =0$ are apparent (by induction) from , and the inequalities $$\begin{aligned}
\label{1.24}&
\det (\g(r)j
\g(r-1)^*)\not=0,
\end{aligned}$$ and it remains to prove . Indeed, let $\g(r)j
\g(r-1)^*g=0$, $\, g \not=0$. Then, the subspaces $\im \g(r)^*$ and $\spa \g(r-1)^*g$ are $j$-orthogonal. The second equality in (taken for $k=r$ and $k=r-1$) implies that these subspaces are also $j$-negative, have zero intersection and have dimensions $m_2$ and $1$, respectively. Thus, $\spa \big( \g(r-1)^*g \cup \im \g(r)^*\big)$ is an $m_2+1$-dimensional $j$-negative subspace, which does not exist. Therefore, the relation , and so also equality , is proved.
Formula shows that the second equality in holds for $r=0$. Now, we choose $X_r$ (for $r=1$) and $x_r^-$ (for $r>1$) so that the second equality in holds in the case that $r>0$. Taking into account , and using induction, we see that this equality is valid when $$\begin{aligned}
\label{1.25}&
X_1=\g_2(1)-e_1^{-}, \qquad x_r^{-}=\g_2(r)-e_r^{-}-\wt X_r\Phi_{r-2,2} \quad (r>1).\end{aligned}$$
We note that inequalities, which are similar to and , are often required in the study of completion problems and Weyl theory. Therefore, the next proposition, which is easily proved using the same considerations as in the proof of , could be of more general interest.
\[Pn!\] Let the $m \times m$ matrix $J$ satisfy equalities $J=J^*=J^{-1}$ and have $m_1>0$ positive eigenvalues. Let $m \times m_1$ matrices $\vt$ and $\wt \vt$ satisfy inequalities $$\begin{aligned}
\label{prop!} &
\vt^* \vt >0, \quad \vt^* J \vt>0, \quad \wt \vt^* \wt \vt>0, \quad \wt \vt ^*J \wt \vt\geq 0.\end{aligned}$$ Then we have $$\begin{aligned}
\label{ineq!} &
\det \vt^* J \wt \vt \not= 0. \end{aligned}$$
Let us substitute into to derive $$\begin{aligned}
\label{1.26}&
E_rA_rE_r^{-1}-\big(E_r^*\big)^{-1}A_r^*E_r^*
={\mathrm{i}}\G_r j \G_r^*.\end{aligned}$$ Multiplying both sides of by $E_r^{-1}$ and $\big(E_r^*\big)^{-1}$ from the left and right, respectively, we obtain the operator identity $$\begin{aligned}
\label{1.27}&
A_rS_r-S_rA_r^*={\mathrm{i}}\Pi_r j \Pi_r^*={\mathrm{i}}(\Phi_{r,1}\Phi_{r,1}^*-\Phi_{r,2}\Phi_{r,2}^*),\end{aligned}$$ where $$\begin{aligned}
\label{1.28}&
S_r:=E_r^{-1}\big(E_r^*\big)^{-1}, \quad \Pi_r:=E_r^{-1} \G_r=\begin{bmatrix} \Phi_{r,1} & \Phi_{r,2} \end{bmatrix}.\end{aligned}$$
\[DnSnd\] The triple of matrices $\{A_r,\, S_r, \, \Pi_r\}$ forms a symmetric $S$-node if the operator (matrix) identity holds, $S_r=S_r^*$ and $\det S_r\not=0$.
The transfer matrix function $($in Lev Sakhnovich form$)$, which corresponds to the $S$-node, is given by the formula $$\label{1.37}
w_A(r, \lambda)=I_{m}-{\mathrm{i}}j \Pi_r^*S_r^{-1}\big(A_r-
\lambda
I_{(r+1)m_2} \big)^{-1} \Pi_r.$$
\[RkSn\] A symmetric $S$-node corresponding to Dirac system $($which satisfies $)$ on the interval $0 \leq k \leq r$ is constructed using formulas and , where $\G_r$ is given in .
Recall that $S$-nodes, transfer matrix functions $w_A$ and the method of operator identities are introduced and studied in [@SaLopid1; @SaL1; @SaL2; @SaL3] (see also references therein).
For $r>0$ introduce projectors: $$\begin{aligned}
\label{1.29}
& P_1:=\begin{bmatrix}I_{r m_2} & 0 \end{bmatrix},
\quad P_2= P:=\begin{bmatrix}0 &
\ldots & 0 & I_{m_2} \end{bmatrix}.\end{aligned}$$ Since $E_r^{-1}$ is a block lower triangular matrix, we easily derive from and that $$\begin{aligned}
\label{1.30}
& P_1S_r P_1^*=E_{r-1}^{-1}\big(E_{r-1}^*\big)^{-1}=S_{r-1}, \quad P_1\Pi_r=\Pi_{r-1}.\end{aligned}$$ It is apparent that $$\begin{aligned}
&\label{1.31}
\det S_{r-1}\not=0, \quad P_1 A_rP_1^*=A_{r-1}.\end{aligned}$$ In view of and , the factorization Theorem 4 from [@SaL1] (see also [@SaL3 p. 188]) yields $$\begin{aligned}
\nn
w_A(r, \lambda)=& \Big(I_{m} -{\mathrm{i}}j
\Pi_r^*S_r^{-1}P^*\big(PA_rP^*- \lambda I_{m_2}
\big)^{-1}\big(PS_r^{-1}P^*\big)^{-1}P S_r^{-1}\Pi_r \Big)
\\ \label{1.38}
&\times w_A(r-1, \lambda).\end{aligned}$$
\[FundSol\] The fundamental solution $W$ of the system , where $W$ is normalized by the condition and the [potential]{} $\{C_k\}$ satisfies , admits reprezentation $$\label{1.36}
W_{r+1}(z)=(1+ {\mathrm{i}}z)^{r+1}
w_A \big(r, (2z)^{-1}\big).$$
. Formulas and imply the following equalities $$\begin{aligned}
\label{1.32}
& W_{r+1}(z)=(1 + {\mathrm{i}}z)\big(I_m+2{\mathrm{i}}z(1 + {\mathrm{i}}z)^{-1}j\g(r)^*\g(r)\big) W_r(z) \quad (r \geq 0).\end{aligned}$$ On the other hand, we easily derive from , , and that $$\begin{aligned}
\label{1.33}&
\big(PA_rP^*- \lambda I_{m_2}
\big)^{-1}=-\big( \lambda + {\mathrm{i}}/2\big)^{-1} I_{m_2}, \quad S_r^{-1}=E_r^*E_r, \\
\label{1.34} &
PS_r^{-1}P^*=(e_r^-)^*e_r^-,
\quad PS_r^{-1}\Pi_r=PE_r^*\G_r=(e_r^-)^*\g(r).\end{aligned}$$ We substitute and into to obtain $$\begin{aligned}
&\label{1.35}
w_A(r, \lambda)= \Big(I_{m} +\frac{2 {\mathrm{i}}}{ 2 \la+{\mathrm{i}}} j\g(r)^*\g(r)\Big)
w_A(r-1, \lambda) \quad (r \geq 1).\end{aligned}$$ In a similar way, we rewrite (for the case that $r=0$) in the form $$\begin{aligned}
&\label{1.35'}
w_A(0, \lambda)= I_{m} +\frac{2 {\mathrm{i}}}{2 \la+{\mathrm{i}}} j\g(0)^*\g(0).\end{aligned}$$ Finally, we compare with and (and take into account ) to see that $W_1(z)=(1+ {\mathrm{i}}z)
w_A \big(0, (2z)^{-1}\big)$ and iterative relations for the left- and right-hand sides of ) coincide.
Weyl theory: inverse problems {#gcip}
==============================
The values of $\vp$ and its derivatives at $z={\mathrm{i}}$ will be of interest in this section. Therefore, using we rewrite in the form $$\begin{aligned}
\label{nc-1}&
\vp_r(z, \clp)=-\begin{bmatrix}
0 &I_{m_2}
\end{bmatrix}W_{r+1}(\ov{z})^{*}\clp(z)\Big(\begin{bmatrix}
I_{m_1} & 0
\end{bmatrix}W_{r+1}(\ov{z})^{*}\clp(z)\Big)^{-1},\end{aligned}$$ where $\clp$ in differs from $\clp$ in by the factor $j$ (and so this $\clp$ is also a nonsingular matrix function with property-$j$).
\[defWeyl2\] Weyl functions of Dirac system $($which is given on the interval $0 \leq k \leq r$ and satisfies $)$ are $m_2\times m_1$ matrix functions $\vp(z)$ of the form , where $\clp$ are nonsingular matrix functions with property-$j$ such that $\clp({\mathrm{i}})$ are well-defined and nonsingular.
It is apparent that is equivalent to $$\begin{aligned}
\label{nc-1'}&
\begin{bmatrix}I_{m_1} \\ \vp_r(z, \clp)\end{bmatrix}=jW_{r+1}(\ov{z})^{*}\clp(z)\Big(\begin{bmatrix}
I_{m_1} & 0
\end{bmatrix}W_{r+1}(\ov{z})^{*}\clp(z)\Big)^{-1}.\end{aligned}$$
\[LaI\] Let $\clp$ satisfy conditions from Definition \[defWeyl2\]. Then we have the inequality $$\begin{aligned}
\label{nc0}&
\det\Big(\begin{bmatrix}
I_{m_1} & 0
\end{bmatrix}W_{r+1}(-{\mathrm{i}})^{*}\clp({\mathrm{i}})\Big) \not= 0.\end{aligned}$$
. First note that in view of we obtain $$\begin{aligned}
\label{nc}&
I_m+C_kj=2\bt(k)^*\bt(k)j.\end{aligned}$$ Formulas and imply $$\begin{aligned}
\nn
\begin{bmatrix}
I_{m_1} & 0
\end{bmatrix}W_{r+1}(-{\mathrm{i}})^{*}\clp({\mathrm{i}})=&2^{r+1}\big(\begin{bmatrix}
I_{m_1} & 0
\end{bmatrix}\bt(0)^*\big)(\bt(0)j \bt(1)^*)\ldots
\\ \label{nc'}& \times
(\bt(r-1)j \bt(r)^*)(\bt(r)j \clp({\mathrm{i}})).\end{aligned}$$ Using Proposition \[Pn!\] (and the second equality in ) and putting, correspondingly, $\vt = \bt(k)^*$ and $\wt \vt=\bt(k+1)^*$ or $\wt \vt=\clp({\mathrm{i}})$, we derive inequalities $$\begin{aligned}
&\label{btp}
\det (\bt(k)j \bt(k+1)^*)\not=0 \quad {\mathrm{and}} \quad\det (\bt(r)j \clp({\mathrm{i}}))\not=0,
\end{aligned}$$ respectively. In the same way we obtain $\det\big(\begin{bmatrix}
I_{m_1} & 0
\end{bmatrix}\bt(0)^*\big)\not=0.$ Now, inequality follows from .
Our next proposition is proved similar to Corollary \[Cyj\].
\[Pnwtr\] Suppose $\vp$ is a Weyl function of Dirac system on the interval $0 \leq k \leq r$, where the potential $\{C_k\}$ satisfies . Then $\vp$ is a Weyl function of the same system on all the intervals $0 \leq k \leq \wt r$ $(\wt r \leq r)$.
. Clearly, it suffices to show that the statement of the proposition holds for $\wt r=r-1$ (if $r>0$). That is, in view of Definition \[defWeyl2\], we should prove that $\wt \clp(z):= (I_m-{\mathrm{i}}z C_r j)\clp(z)$ has property-$j$, that $\wt \clp({\mathrm{i}})$ is well-defined and that the first inequality in written for $ \wt \clp$ at $z={\mathrm{i}}$ always holds (i.e., $\wt \clp({\mathrm{i}})$ is nonsingular), if only $\clp$ has these properties.
Indeed, since we have $$\begin{aligned}
\label{vst1}&
(I_m-{\mathrm{i}}z C_r j)^*j(I_m-{\mathrm{i}}z C_r j)=(1+|z|^2)j+{\mathrm{i}}(\ov{z}-z)jC_rj \geq (1+|z|^2)j,\end{aligned}$$ the matrix function $ \wt \clp$ has property-$j$. The non-singularity of $\wt \clp({\mathrm{i}})= (I_m+ C_r j)\clp({\mathrm{i}})$ is apparent from and .
\[Tm2.2\] Suppose $\vp$ is a Weyl function of Dirac system on the interval $0 \leq k \leq r$, where the potential $\{C_k\}$ satisfies . Then $\{C_k\}_{k=0}^r$ is uniquely recovered from the first $r+1$ Taylor coefficients of $\vp\left({\mathrm{i}}\frac{1-z}{1+z} \right)$ at $z=0$.
If $\vp\left({\mathrm{i}}\frac{1-z}{1+z} \right)=\sum_{k=0}^r\phi_k z^k+ O(z^{r+1})$, then matrices $\Phi_{k,1}$ are recovered via the formula $$\begin{aligned}
\label{ncip1}&
\Phi_{k,1}=-\begin{bmatrix} \phi_0 \\ \phi_0+\phi_1 \\ \ldots \\ \phi_0+\phi_1+ \ldots + \phi_k \end{bmatrix}.\end{aligned}$$ Using $\Phi_{k,1}$ we easily recover consecutively $\Pi_{k}=\begin{bmatrix} \Phi_{k,1} & \Phi_{k,2} \end{bmatrix}$ $($where $\Phi_{k,2} $ is given in $)$ and $S_k$, which is the unique solution of the matrix identity $A_kS_k-S_kA_k^*={\mathrm{i}}\Pi_k j \Pi_k^*$. Next, we construct $$\begin{aligned}
\label{ncip2}&
\g(k)^*\g(k)= \Pi_k^*S_k^{-1}P^*(PS_k^{-1}P^*)^{-1}PS_k^{-1}\Pi_k, \quad P=\begin{bmatrix} 0 & \ldots & 0 & I_{m_2}\end{bmatrix}.\end{aligned}$$ Finally, we use $\g(k)^*\g(k)$ to recover $C_k$ via .
. Put $$\label{nc1}
{\cal A}(z):=|1+z^2|^{-2(r+1)}\begin{bmatrix}I_{m_1} & \vp(z)^* \end{bmatrix} W_{r+1}(z)^*jW_{r+1}( z)\left[
\begin{array}{c}
I_{m_1} \\ \vp(z)
\end{array}
\right].$$ According to and we have $$\begin{aligned}
\nn
{\cal A}(z)=& \Big(\Big(\begin{bmatrix}
I_{m_1} & 0
\end{bmatrix}W_{r+1}(\ov{z})^{*}\clp(z)\Big)^{-1}\Big)^*\clp(z)^*j\clp(z)
\\ \label{nc2} &
\times \Big(\begin{bmatrix}
I_{m_1} & 0
\end{bmatrix}W_{r+1}(\ov{z})^{*}\clp(z)\Big)^{-1}.\end{aligned}$$ From (\[nc0\]) and (\[nc2\]) we see that ${\cal A}$ is bounded in the neighbourhood of $z={\mathrm{i}}$: $$\label{nc3}
\|{\cal A}(z)\|=O(1) \quad {\mathrm{for}} \quad
z \to
{\mathrm{i}}.$$ Let us include into considerations the $S$-node (corresponding to Dirac system), which is constructed in accordance with Remark \[RkSn\]. Substitute (\[1.36\]) into (\[nc1\]) to obtain $$\begin{aligned}
\label{nc4}
{\cal A}(z)=&\big((1-{\mathrm{i}}z)(1+{\mathrm{i}}\ov{z})\big)^{-r-1}\begin{bmatrix}I_{m_1} & \vp(z)^* \end{bmatrix}
\\
\nn &
\times
\Big(j- \frac{\Im(z)}{|z|^2}\Pi_r^*\Big(A_r^*-\frac{1}{2\ov{z}}I\Big)^{-1}S_r^{-1}\Big(A_r-\frac{1}{2z} I\Big)^{-1}\Pi_r\Big)\begin{bmatrix}I_{m_1} \\ \vp(z) \end{bmatrix},\end{aligned}$$ where $I=I_{(r+1)m_2}$. Here we used the important equality $$\begin{aligned}
\label{nc5}
w_A(r,\la)^*jw_A(r, \wt \la)=j-{\mathrm{i}}(\wt \la - \ov{\la})\Pi_r^*(A_r^*-\ov{\la}I)^{-1}S_r^{-1}(A_r-\wt \la I)^{-1}\Pi_r,\end{aligned}$$ which follows from and (see, e.g., [@SaAEx; @SaL1]).
Notice that $S_r>0$. Hence, formulas , (\[nc3\]) and (\[nc4\]) imply that $$\label{nc6}
\left\|
\Big(A_r-\frac{1}{2z} I\Big)^{-1}\Pi_r\begin{bmatrix}I_{m_1} \\ \vp(z) \end{bmatrix}
\right\|=O(1) \quad {\mathrm{for}} \quad
z \to {\mathrm{i}}.$$ Using the block representation $\Pi_r=\begin{bmatrix}\Phi_{r,1} & \Phi_{r,2} \end{bmatrix}$ from and multiplying both sides of by $\left\| \Big(\Phi_{r,2}^*\Big(A_r-\frac{1}{2z} I\Big)^{-1}\Phi_{r,2}\Big)^{-1}\Phi_{r,2}^*\right\|$ we rewrite the result: $$\begin{aligned}
\nn &
\left\|
\vp(z)+\Big(\Phi_{r,2}^*\Big(A_r-\frac{1}{2z} I\Big)^{-1}\Phi_{r,2}\Big)^{-1}\Phi_{r,2}^*\Big(A_r-\frac{1}{2z} I\Big)^{-1}\Phi_{r,1}\Big)^{-1}
\right\|
\\ \label{nc7} &
=O\left(\left\| \Big(\Phi_{r,2}^*\Big(A_r-\frac{1}{2z} I\Big)^{-1}\Phi_{r,2}\Big)^{-1}\right\|\right) \quad {\mathrm{for}} \quad
z \to {\mathrm{i}}.\end{aligned}$$ In order to obtain we applied also the matrix (operator) norm inequality $\|X_1X_2\| \leq \|X_1\| \|X_2\|$.
The resolvent $(A - \la I)^{-1}$ is easily constructed explicitly (see, for instance, formula (1.10) in [@SaAtepl]). In particular, we derive $$\label{nc8}
\Phi_{r,2}^*\Big(A_r-\frac{1}{2z} I\Big)^{-1}=
-\frac{2z}{1+{\mathrm{i}}z}\begin{bmatrix}\wh q(z)^r & \wh q(z)^{r-1} & \ldots & I_{m_2}\end{bmatrix}
, \quad \wh
q:=\frac{1
-{\mathrm{i}}z}{1 +{\mathrm{i}}z}I_{m_2}.$$ From we see that $$\label{nc9}
\Phi_{r,2}^*\Big(A_r-\frac{1}{2z} I\Big)^{-1}\Phi_{r,2}={\mathrm{i}}\Big(1-\Big( \frac{1
-{\mathrm{i}}z}{1 +{\mathrm{i}}z}\Big)^{r+1}\Big)I_{m_2}.$$ Partitioning $\Phi_{r,1}$ into $m_2 \times m_1$ blocks $\Phi_{r,1}(k)$ and using - we obtain $$\begin{aligned}
\nn &
\vp\left({\mathrm{i}}\frac{1-z}{1+z} \right)+\frac{1-z}{1-z^{r+1}}\sum_{k=0}^r z^k\Phi_{r,1}(k)=O(z^{r+1}) \quad {\mathrm{for}} \quad
z \to 0,\end{aligned}$$ which can be easily transformed into $$\begin{aligned}
\label{nc10} &
\vp\left({\mathrm{i}}\frac{1-z}{1+z} \right)+(1-z)\sum_{k=0}^r z^k\Phi_{r,1}(k)=O(z^{r+1}) \quad {\mathrm{for}} \quad
z \to 0,\end{aligned}$$ and follows for $k=r$. Since $\s(A_r) \cap \s(A_r^*)=\emptyset$ the matrix $S_r$ is uniquely recovered from the matrix identity . Finally, for the case, where $k=r$, is apparent from . From Proposition \[Pnwtr\], we see that $\vp$ is a Weyl function of our Dirac system on all the intervals $0 \leq k \leq \wt r$ $(\wt r \leq r)$ and so all $C_{\wt r}$ are recovered in the same way as $C_r$.
The next corollary is a discrete version of Borg-Marchenko-type uniqueness theorems. The active study of such theorems was triggered by the seminal papers by F. Gesztesy and B. Simon [@GS; @GeSi].
\[CyBM\] Suppose $\vp$ and $\wt \vp$ are Weyl functions of two Dirac systems with potentials $\{C_k\}$ and $\{\wt C_k\}$, which are given on the intervals $0 \leq k \leq r$ and $0 \leq k \leq \wt r$, respectively. We suppose that matrices $\{C_k\}$ and $\{\wt C_k\}$ are positive and $j$-unitary. Moreover, we assume that $$\begin{aligned}
\label{bm1} &
\vp\left({\mathrm{i}}\frac{1-z}{1+z} \right)- \wt \vp\left({\mathrm{i}}\frac{1-z}{1+z} \right)= O(z^{p+1}), \quad z \to \infty, \quad p\in \BN_0, \quad p \leq \min(r, \, \wt r).\end{aligned}$$ Then we have $C_k=\wt C_k$ for all $0\leq k \leq p$.
. According to Proposition \[Pnwtr\] both functions $\vp$ and $\wt \vp$ are Weyl functions of the corresponding Dirac systems on the same interval $[0, \, p]$. From we see that the first $p+1$ Taylor coefficients of $\vp\left({\mathrm{i}}\frac{1-z}{1+z} \right)$ and $\wt \vp\left({\mathrm{i}}\frac{1-z}{1+z} \right)$ coincide. Hence, the uniqueness of the recovery of the potential from Taylor coefficients in Theorem \[Tm2.2\] yields $C_k=\wt C_k$ $(0\leq k \leq p)$.
Taking into account , we derive that the first $r+1$ Taylor coefficients of $\vp_r\left({\mathrm{i}}\frac{1-z}{1+z} \right)$ at $z=0$ (for any Weyl function $\vp_r$ of a fixed Dirac system) can be uniquely and in the same way recovered from the matrix $\Phi_{r,1}$, which, in turn, can be constructed as proposed in Remark \[RkSn\]. Therefore, the next theorem is apparent.
\[TmStr\] Let Dirac system , where matrices $C_k$ satisfy , be given on the interval $0 \leq k \leq r$.Then all the functions $\vp_d(z)=\vp_r\Big({\mathrm{i}}\frac{1-z}{1+z}, \clp\Big)$, where $\vp_r$ are Weyl functions of this Dirac system, are non-expansive in the unit disk and have the same first $r+1$ Taylor coefficients $\{\phi_k\}_0^r$ at $z=0$.
Step 1 in the proof of Theorem \[Tm3.8\] shows that the Weyl function $\vp_{\infty}$ of Dirac system on the semi-axis can be constructed as a uniform limit of Weyl functions $\vp_r$ on increasing intervals. Hence, using Theorem \[TmStr\] we obtain the following corollary.
\[Wfonax\] Let $\vp(z)$ be the Weyl function of some Dirac system , which is given on the semi-axis and satisfies . Assume that $\vp_r$ is a Weyl function of the same system on the finite interval $0 \leq k \leq r$. Then the first $r+1$ Taylor coefficients of $\vp\left({\mathrm{i}}\frac{1-z}{1+z} \right)$ and $ \vp_r\left({\mathrm{i}}\frac{1-z}{1+z} \right)$ coincide. Therefore, the system can be uniquely recovered from $\vp$ via procedure from Theorem \[Tm2.2\].
Operator identities and interpolation problems {#OpIC}
==============================================
One can easily derive (see, e.g, [@FKSELA p. 474]) that the equality $$\begin{aligned}
\label{id1} &
s_{k+1, p+1}-s_{kp}=Q_{kp}+Q_{k+1,p+1}-Q_{k+1,p}-Q_{k,p+1}, \quad -1 \leq k,p \leq r-1\end{aligned}$$ holds for the blocks $s_{kp}$ and $Q_{kp}$ of the block matrices $S_r=\{s_{kp}\}_{k,p=0}^r$ and $Q_r=\{Q_{kp}\}_{k,p=0}^r$, respectively, which satisfy the operator identity $$\begin{aligned}
\label{id2} &
A_rS_r-S_rA_r^*+{\mathrm{i}}Q=0,\end{aligned}$$ where $A_r$ is given by . Here we add sometimes commas between the indices of blocks and put also $$\begin{aligned}
\label{id3} &
s_{-1, p}=s_{k,-1}=Q_{-1,p}=Q_{k,-1}=0.\end{aligned}$$ For the case that $S_r$ corresponds to Dirac system, we rewrite below in an equivalent form and obtain the structure of $S_r$.
\[PnStr\] Let $S_r$ satisfy , where $A_r$, $ \Phi_{r,1}$ and $\Phi_{r,2}$ are given by , and the last equality in , respectively. Then $S_r$ has the following structure: $$\begin{aligned}
\label{id4} &
s_{00}= I_{m_2}-\phi_0\phi_0^* \quad {\mathrm{and}} \quad s_{k+1, p+1}-s_{kp}=\phi_{k+1}\phi_{p+1}^* \end{aligned}$$ for $ -1 \leq k,p \leq r-1, \quad k+p+2>0$.
The following statement is immediate from Theorem \[TmStr\] and Proposition \[PnStr\].
\[TmStr1\] Let Dirac system , where matrices $C_k$ satisfy , be given on the interval $0 \leq k \leq r$.Then all the functions $\vp_d(z)=\vp_r\Big({\mathrm{i}}\frac{1-z}{1+z}, \clp\Big)$, where $\vp_r$ are given by , matrix functions $\clp(z)$ in have property-$j$ and matrices $\clp({\mathrm{i}})$ are non-singular, are non-expansive in the unit disk and have the same first $r+1$ Taylor coefficients $\{\phi_k\}_0^r$ at $z=0$. The matrix $S_r$ determined by these coefficients via is positive.
On the other hand, if we assume only that the coefficients $\{\phi_k\}_0^r$ are fixed and $S_r$ given is positive, two related interpolation problems appear.
[**Interpolation problem I.**]{} Describe all the analytic and non-expansive in the unit disk matrix functions $\vp_d$ such that the coefficients $\{\phi_k\}_0^r$ are their first $r+1$ Taylor coefficients.
[**Interpolation problem II.**]{} Describe all the positive continuations of $S_r$, which preserve the structure given by .
[**Acknowledgements.**]{} The research of I. Roitberg was supported by the German Research Foundation (DFG) under grant No. KI 760/3-1. The research of A.L. Sakhnovich was supported (at different periods) by the German Research Foundation (DFG) under grant No. KI 760/3-1 and by the Austrian Science Fund (FWF) under Grant No. P24301.
[AGKS]{}
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*B. Fritzsche,\
Fakultät für Mathematik und Informatik,\
Mathematisches Institut, Universität Leipzig,\
Johannisgasse 26, D-04103 Leipzig, Germany,\
e-mail: [fritzsche@math.uni-leipzig.de ]{}\
$ $\
B. Kirstein,\
Fakultät für Mathematik und Informatik,\
Mathematisches Institut, Universität Leipzig,\
Johannisgasse 26, D-04103 Leipzig, Germany,\
e-mai: [kirstein@math.uni-leipzig.de ]{}\
$ $\
I. Roitberg,\
Fakultät für Mathematik und Informatik,\
Mathematisches Institut, Universität Leipzig,\
Johannisgasse 26, D-04103 Leipzig, Germany,\
e-mail: [i$_-$roitberg@yahoo.com ]{}\
$ $\
A.L. Sakhnovich,\
Fakultät für Mathematik, Universität Wien,\
Nordbergstrasse 15, A-1090 Wien, Austria\
e-mail: [al$_-$sakhnov@yahoo.com ]{}*
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Studying the coherence of an optical field is typically compartmentalized with respect to its different optical degrees of freedom (DoFs) – spatial, temporal, and polarization. Although this traditional approach succeeds when the DoFs are uncoupled, it fails at capturing key features of the field’s coherence if the DOFs are indeed correlated – a situation that arises often. By viewing coherence as a ‘resource’ that can be shared among the DoFs, it becomes possible to convert the entropy associated with the fluctuations in one DoF to another DoF that is initially fluctuation-free. Here, we verify experimentally that coherence can indeed be reversibly exchanged – without loss of energy – between polarization and the spatial DoF of a partially coherent field. Starting from a linearly polarized spatially incoherent field – one that produces no spatial interference fringes – we obtain a spatially coherent field that is unpolarized. By reallocating the entropy to polarization, the field becomes invariant with regards to the action of a polarization scrambler, thus suggesting a strategy for avoiding the deleterious effects of a randomizing system on a DoF of the optical field.'
author:
- 'Chukwuemeka O. Okoro'
- 'H. Esat Kondakci'
- 'Ayman F. Abouraddy'
- 'Kimani C. Toussaint, Jr.'
bibliography:
- 'sample.bib'
title: 'Demonstration of an optical-coherence converter'
---
[^1]
Optical coherence is evaluated by assessing the correlations between field fluctuations at different points in space and time [@Mandel95Book]. When multiple degrees of freedom (DoFs) of an optical field – spatial, temporal, and polarization – are relevant, the coherence of each DoF is typically studied separately. For example, spatial coherence is evaluated via double-slit interference [@Zernike38P], temporal coherence through two-path (e.g., Michelson) interference [@Born99Book], and polarization coherence by measuring the Stokes parameters [@Brosseau98Book]. Although this traditional approach succeeds when the DoFs are uncoupled, it fails at capturing key features of the field coherence if they are correlated [@Qian11OL; @Kagalwala13NP].
Here we show that coherence can be viewed as a ‘resource’ that can be reversibly converted from one DoF of the field to another. We demonstrate experimentally the reversible and energy-conserving (unitary) conversion of coherence between the spatial and polarization DoFs of an optical field. Starting from a linearly polarized field having *no spatial coherence* (a complete lack of double-slit interference visibility), we convert the field without filtering or loss of energy into one that displays spatial coherence (high-visibility interference fringes) – but is *unpolarized*. The optical arrangement we describe engenders an internal reorganization of the field energy that leads to a migration of the entropy associated with the statistical fluctuations from one DoF (spatial) to another (polarization). This coherency conversion is confirmed by measuring the full $4\times4$ coherency matrix that provides a complete description of two-point vector-field coherence via ‘optical coherency matrix tomography’ (OCmT) [@Abouraddy14OL; @Kagalwala15SR]. The tomographic measurement of coherence is carried out at different stages in the experimental setup to confirm the transformations involved in the coherence conversion process. As an application to highlight the usefulness of reallocating the entropy from one DoF of the field to another, we show that the field can be reconfigured to be invariant under the impact of a depolarizer or polarization scrambler that transforms any input polarization to unpolarized light. By transferring all the field entropy into polarization, the polarization scrambler cannot further increase the polarization entropy, which thus emerges unchanged. Since the coherence conversion procedure is reversible and no energy is lost, the field may be reversed to its original fully polarized configuration after traversing the polarization scrambler.
The paper first briefly reviews the matrix approach to quantifying optical coherence for a single DoF and multiple DoFs. Second, we describe the concept of an optical-coherence converter, a system that reversibly transforms coherence – viewed as a resource – from one DoF of a field to another without loss of energy. Starting from a field with one coherent DoF and another incoherent DoF, we reversibly convert the field such that the former DoF becomes incoherent and the latter coherent. Third, we present the experimental arrangement used in confirming these predictions and the measurement scheme to identify multi-DoF beam coherence. Finally, we demonstrate the invariance of a field with respect to a polarization scrambler before presenting our conclusions.
Multi-DoF coherence
===================
Polarization coherence and spatial coherence
--------------------------------------------
Partial polarization is described by a polarization coherency matrix $\mathbf{G}_{\mathrm{p}}=\left(\small{\begin{array}{cc}G^{\mathrm{HH}}&G^{\mathrm{HV}}\\G^{\mathrm{VH}}&G^{\mathrm{VV}}\end{array}}\right)$, where $\mathrm{H}$ and $\mathrm{V}$ identify the horizontal and vertical polarization components, respectively, $G^{ij}=\langle E^{i}(E^{j})^{*}\rangle$, $i,j=\mathrm{H},\mathrm{V}$, and $\langle\cdot\rangle$ is an ensemble average [@Wolf07Book]. Here $\mathbf{G}_{\mathrm{p}}$ is Hermitian ($\mathbf{G}_{\mathrm{p}}^{\dag}=\mathbf{G}_{\mathrm{p}}$), semi-positive, normalized such that $G^{\mathrm{HH}}+G^{\mathrm{VV}}=1$, $G^{\mathrm{HH}}$ and $G^{\mathrm{VV}}$ are the contributions of H and V to the total power, respectively, and $G^{\mathrm{HV}}$ is their normalized correlation. The degree of polarization is $D_{\mathrm{p}}=|\lambda_{\mathrm{H}}-\lambda_{\mathrm{V}}|$, where $\lambda_{\mathrm{H}}$ and $\lambda_{\mathrm{V}}$ are the eigenvalues of $\mathbf{G}_{\mathrm{p}}$ [@Al-Qasimi07OL]. Spatial coherence can be similarly described via a spatial coherency matrix for points $\vec{a}$ and $\vec{b}$, $\mathbf{G}_{\mathrm{s}}=\left(\small{\begin{array}{cc}G_{aa}&G_{ab}\\G_{ba}&G_{bb}\end{array}}\right)$, where $G_{kl}=\langle E_{k}E_{l}^{*}\rangle$, $k,l=a,b$. The properties of $\mathbf{G}_{\mathrm{s}}$ are similar to those of $\mathbf{G}_{\mathrm{p}}$. The visibility of the interference fringes observed by superposing the fields from $\vec{a}$ and $\vec{b}$ is $V=2|G_{ab}|$. Alternatively, the degree of spatial coherence $D_{\mathrm{s}}=|\lambda_{a}-\lambda_{b}|$ represents the *maximum* visibility obtained after equalizing the amplitudes at $\vec{a}$ and $\vec{b}$, where $\lambda_{a}$ and $\lambda_{b}$ are the eigenvalues of $\mathbf{G}_{\mathrm{s}}$ [@Zernike38P; @Abouraddy17OE].
A DoF represented by a $2\times2$ coherency matrix carries up to 1 bit of entropy; e.g., the polarization entropy is $S_{\mathrm{p}}=-\lambda_{\mathrm{H}}\log_{2}\lambda_{\mathrm{H}}-\lambda_{\mathrm{V}}\log_{2}\lambda_{\mathrm{V}}$, where $0\leq S_{\mathrm{p}}\leq1$. The zero-entropy state $S_{\mathrm{p}}=0$ corresponds to a fully polarized field (no statistical fluctuations), whereas the maximal-entropy state $S_{\mathrm{p}}=1$ corresponds to an unpolarized field (maximal fluctuations) [@Brosseau06PO]; similarly for the spatial DoF based on $\mathbf{G}_{\mathrm{s}}$. Entropy so defined is a *unitary invariant* of the field DoF: it cannot be changed by applying lossless deterministic optical transformations.
Joint polarization and spatial coherence formalism
--------------------------------------------------
Evaluating $\mathbf{G}_{\mathrm{p}}$ and $\mathbf{G}_{\mathrm{s}}$ is *not* sufficient to completely identify the coherence of a vector field in which the polarization and spatial DoFs are potentially correlated. A $4\times4$ coherency matrix $\mathbf{G}$ is necessary to capture the full vector-field coherence [@Gori06OL; @Kagalwala13NP], $$\mathbf{G}=\left(\begin{array}{cccc}
G_{aa}^{\mathrm{HH}}&G_{aa}^{\mathrm{HV}}&G_{ab}^{\mathrm{HH}}&G_{ab}^{\mathrm{HV}}\\[.4em]
G_{aa}^{\mathrm{VH}}&G_{aa}^{\mathrm{VV}}&G_{ab}^{\mathrm{VH}}&G_{ab}^{\mathrm{VV}}\\[.4em]
G_{ba}^{\mathrm{HH}}&G_{ba}^{\mathrm{HV}}&G_{bb}^{\mathrm{HH}}&G_{bb}^{\mathrm{HV}}\\[.4em]
G_{ba}^{\mathrm{VH}}&G_{ba}^{\mathrm{VV}}&G_{bb}^{\mathrm{VH}}&G_{bb}^{\mathrm{VV}}
\end{array}\right), \nonumber$$ where $G_{kl}^{ij}=\langle E_{k}^{i}(E_{l}^{j})^{*}\rangle$, $i,j=\mathrm{H},\mathrm{V}$, and $k,l=a,b$. The matrix $\mathbf{G}$ is Hermitian positive semi-definite and normalized such that $\mathrm{Tr}\{\mathbf{G}\}=1$ (‘$\mathrm{Tr}$’ is the matrix trace). The diagonal elements are the power-fractions from the mutually exclusive contributions: $G_{aa}^{\mathrm{HH}}$ and $G_{aa}^{\mathrm{VV}}$ are the H and V components at $\vec{a}$, respectively, and $G_{bb}^{\mathrm{HH}}$ and $G_{bb}^{\mathrm{VV}}$ are those at $\vec{b}$. The off-diagonal elements are normalized correlations between field components. The double-slit visibility observed when overlapping the fields from $\vec{a}$ and $\vec{b}$ is $V=2|G_{ab}^{\mathrm{HH}}+G_{ab}^{\mathrm{VV}}|$ [@Wolf03PLA]. Crucially, $V$ is *not a unitary invariant* of the field [@Tervo03OE; @Setala04OE], and reversible optical transformations that span the spatial and polarization DoFs can increase $V$ [@Gori07OL; @Herrero07OL].
Each *physically independent* DoF (spatial and polarization) carries one bit of entropy, so the vector field now carries 2 bits of entropy: $S=-\lambda_{a\mathrm{H}}\log_{2}\lambda_{a\mathrm{H}}-\lambda_{a\mathrm{V}}\log_{2}\lambda_{a\mathrm{V}}-\lambda_{b\mathrm{H}}\log_{2}\lambda_{b\mathrm{H}}-\lambda_{b\mathrm{V}}\log_{2}\lambda_{b\mathrm{V}}$, where $0\leq S\leq2$ and $\{\lambda\}=\{\lambda_{a\mathrm{H}},\lambda_{a\mathrm{V}},\lambda_{b\mathrm{H}},\lambda_{b\mathrm{V}}\}$ are the real positive eigenvalues of $\mathbf{G}$. The zero-entropy state $S=0$ corresponds to a fully polarized *and* spatially coherent field (no statistical fluctuations in either DoF and $\{\lambda\}=\{1,0,0,0\}$), whereas the maximal-entropy state $S=2$ corresponds to an unpolarized spatially incoherent field (maximal fluctuations in both DoFs and $\{\lambda\}=\tfrac{1}{4}\{1,1,1,1\}$).
Concept of optical coherency conversion
=======================================
In general $S\leq S_{\mathrm{p}}+S_{\mathrm{s}}$, with equality achieved only when the two DoFs are independent, in which case $\mathbf{G}$ can be written in separable form $\mathbf{G}=\mathbf{G}_{\mathrm{s}}\otimes\mathbf{G}_{\mathrm{p}}$. In general, $S_{\mathrm{s}}$ and $S_{\mathrm{p}}$ are obtained from the $2\times2$ ‘reduced’ spatial and polarization coherency matrices $$\begin{aligned}
\mathbf{G}_{\mathrm{s}}^{(\mathrm{r})}=\left(\begin{array}{cc}
G_{aa}^{\mathrm{HH}}+G_{aa}^{\mathrm{VV}}
&G_{ab}^{\mathrm{HH}}+G_{ab}^{\mathrm{VV}}\\ [0.4em]
G_{ba}^{\mathrm{HH}}+G_{ba}^{\mathrm{VV}}
&G_{bb}^{\mathrm{HH}}+G_{bb}^{\mathrm{VV}}
\end{array}\right), \nonumber \\
\mathbf{G}_{\mathrm{p}}^{(\mathrm{r})}=\left(\begin{array}{cc}
G_{aa}^{\mathrm{HH}}+G_{bb}^{\mathrm{HH}}
&G_{aa}^{\mathrm{HV}}+G_{bb}^{\mathrm{HV}}\\ [0.4em]
G_{aa}^{\mathrm{VH}}+G_{aa}^{\mathrm{VH}}
&G_{aa}^{\mathrm{VV}}+G_{bb}^{\mathrm{VV}}
\end{array}\right), \nonumber \end{aligned}$$ which are obtained from $\mathbf{G}$ by a ‘partial trace’ [@Peres95Book], that is, by tracing over one DoF [@Kagalwala13NP; @Abouraddy14OL]
The concept of an optical-coherence converter is illustrated in Fig. \[fig:GeneralConcept\](a). Consider the case when the field carries *one bit* of entropy ($S=1$) and the DoFs are independent ($S=S_{\mathrm{p}}+S_{\mathrm{s}}$), in which case a single DoF can accommodate this entropy. The field may be maximally *incoherent* but polarized ($S_{\mathrm{s}}=1$ and $S_{\mathrm{p}}=0$), whereupon no interference fringes can be observed \[Fig. \[fig:GeneralConcept\](b)\]. Alternatively, the field may be spatially coherent but *un*polarized ($S_{\mathrm{s}}=0$ and $S_{\mathrm{p}}=1$), in which case full-visibility fringes can be observed \[Fig. \[fig:GeneralConcept\](c)\]. We demonstrate here that an optical field can be *reversibly* transformed from the former configuration to the latter without loss of energy, thus *converting* coherence from one DoF (polarization) to the other (spatial). Throughout the procedure $S$ remains constant; that is, no uncertainty is added or removed from the field, only an internal reorganization of the field engendered by a unitary transformation confines the statistical fluctuations to one DoF while freeing the other from uncertainty. We call such a system a ‘coherence converter’.
The optical arrangement we propose to convert coherency between the spatial and polarization DoFs is depicted in Fig. \[fig:SetupExp1and2\]. We start from two points $\vec{a}'$ and $\vec{b}'$ of equal intensity in a spatially *in*coherent H-polarized field (the fields are mutually incoherent or statistically independent), which thus produce no interference fringes. The polarization at $\vec{b}'$ is rotated to become orthogonal to that at $\vec{a}'$ $(\mathrm{H}\rightarrow\mathrm{V})$ before combining the fields at a polarizing beam splitter (PBS), which yields an *un*polarized field. We then split the field into two points $\vec{a}$ and $\vec{b}$ using a *non*polarizing beam splitter, which creates two copies of the field that can demonstrate high-visibility interference fringes. We proceed now to present the measurements at each step of this coherency-conversion process.
![Concept of an optical-coherence converter. (a) Starting with a polarized but spatially incoherent field ($S_{\mathrm{p}}=0$ and $S_{\mathrm{s}}=1$, $S=S_{\mathrm{p}}+S_{\mathrm{s}}=1$), coherence is converted from polarization to the spatial DoF, thereby yielding an unpolarized but spatially coherent field ($S_{\mathrm{p}}=1$ and $S_{\mathrm{s}}=0$) but without introducing further fluctuations (fixed total entropy $S=1$). The device thus converts the statistical fluctuations (and the attendant entropy) from one DoF to the other. (b) When a polarized but spatially incoherent field is incident on a double-slit, no interference fringes are observed. (c) After converting coherence from polarization to the spatial DoF, high-visibility (but unpolarized) interference fringes appear.[]{data-label="fig:GeneralConcept"}](Figure1)
Source characterization
=======================
The optical field we study is extracted from a broadband, unpolarized LED (center wavelength 850 nm, 30-nm-FWHM bandwidth; Thorlabs M850L3 IR). The field is spectrally filtered (10-nm-FWHM), polarized along H, and spatially filtered through a 100-$\mu$m-wide slit placed at a distance of 180 mm from the source. The ‘input’ plane that includes the points $\vec{a}'$ and $\vec{b}'$ (each defined by a 100-$\mu$m-wide slit) is located 420 mm away from the slit \[the source in Fig. \[fig:SetupExp1and2\](a)\]. We first confirm that the field is spatially coherent within $\vec{a}'$ and $\vec{b}'$ separately (i.e., the spatial coherence width of the field, estimated to be $\sim$1 mm, is larger than the slit width). This is accomplished using a narrow pair of slits (50-$\mu$m-wide separated by $\Delta=150$ $\mu$m) at either $\vec{a}'$ or $\vec{b}'$ and observing the double-slit interference on a CCD camera (Hamamatsu 1394) at a distance of $d=200$ mm away. High-visibility fringes ($V=0.98$) are observed separated by $\lambda d/\Delta\approx1170$ $\mu$m. Next, we superpose the fields from $\vec{a}'$ and $\vec{b}'$ \[’measurement’ in Fig. \[fig:SetupExp1and2\](a)\] and observe no interference fringes \[Fig. \[fig:Figure3\](a)\], confirming that the two points are separated by more than the field coherence width. We have thus confirmed the relationship between the two length scales involved: the size of the locations at $\vec{a}'$ and $\vec{b}'$ (100 $\mu$m) is smaller than the coherence width, and the separation between them (10 mm) is larger. The field that we start from is linearly polarized (scalar) but spatially incoherent, thus $\mathbf{G}$ has the form $$\begin{aligned}
\label{G_1}
\mathbf{G}_{1}&=\tfrac{1}{2}\left(\begin{array}{cccc}
1&0&0&0\\
0&0&0&0\\
0&0&1&0\\
0&0&0&0
\end{array}\right)=\tfrac{1}{2}\mathrm{diag}\{1,0,1,0\} \nonumber \\
&=\tfrac{1}{2}\left(\begin{array}{cc}1&0\\0&1\end{array}\right)_{\mathrm{s}}\otimes\left(\begin{array}{cc}1&0\\0&0\end{array}\right)_{\mathrm{p}},\end{aligned}$$ where the subscripts ‘s’ and ‘p’ refer to the spatial and polarization DoFs, respectively, and the notation $\mathrm{diag}\{\dot\}$ identifies a diagonal matrix with the entries along the diagonal listed between the curly brackets.
![(a) Schematic depicting the input field preparation (source) and characterization (measurement). The field at points $\vec{a}'$ and $\vec{b}'$ is spatially incoherent but fully polarized (scalar). F: Filter; P: polarizer; L: lens; PA: polarization analyzer; CCD: charge-coupled device camera. (b) A coherency converter maps the spatially incoherent but polarized field at $\vec{a}'$ and $\vec{b}'$ to a spatially coherent but unpolarized field at $\vec{a}$ and $\vec{b}$. (c) Schematic of the optical setup for the coherence-converter. A bi-convex lens (L: $f=20$ cm) images $\vec{a}'$ and $\vec{b}'$ to $\vec{a}$ and $\vec{b}$, respectively, with $2\times$ magnification. The delay lines enable matching pairs of paths within the source temporal coherence length. HWP: Half-wave plate; PBS: polarizing beam splitter; BS: beam splitter. The planes at which the coherency matrices $\mathbf{G}_{1}$, $\mathbf{G}_{2}$, and $\mathbf{G}_{3}$ are reconstructed are marked.[]{data-label="fig:SetupExp1and2"}](Figure2)
![(a) The four measurements required to reconstruct the spatial coherence matrix $\mathbf{G}_{\mathrm{S}}$ for a scalar field at $\vec{a}$ and $\vec{b}$. The intensity pattern is recorded with both slits open (left), and two measurements are made: the intensity on the optical axis (red dot) and at the location mid-way along the first expected fringe location calculated from the slit separation (green dot). No fringes are observed here since the field is spatially incoherent. Next, the intensity on the optical axis is recorded when $\vec{a}$ (left) and $\vec{b}$ (right) are blocked (the red dots; see Refs. [@Abouraddy14OL; @Kagalwala15SR] for details). (b) Plot depicting graphically the real parts of the elements of the spatial-polarization coherency matrix $\mathbf{G}_{1}$ for the source plane as reconstructed from OCmT that utilizes the measurements in (a) when carried out in conjunction with polarization measurements. (c) Plot depicting graphically the elements of the theoretically expected coherency matrix $\mathbf{G}=\tfrac{1}{2}\mathrm{diag}\{1,0,1,0\}$, corresponding to a scalar H-polarized field that is spatially incoherent (Eq. \[G\_1\]).[]{data-label="fig:Figure3"}](Figure3)
To fully characterize the field coherence across the spatial and polarization DoFs, we measure $\mathbf{G}_{1}$ via OCmT [@Abouraddy14OL; @Kagalwala15SR], which requires 16 measurements to reconstruct $\mathbf{G}_{1}$. Since 4 polarization projections are required to identify $\mathbf{G}_{\mathrm{p}}$ and 4 spatial projections are required to determine $\mathbf{G}_{\mathrm{s}}$ for a scalar field, $4\times4$ linearly independent combinations of these spatial and polarization projections are necessary to reconstruct $\mathbf{G}$ subject to the constraints of Hermiticity, semi-positiveness, and unity-trace. These measurements are in one-to-one correspondence to those required to reconstruct a two-qubit density matrix in quantum mechanics, a process known as ‘quantum state tomography’ [@Wooters90CEPI; @James01PRA1; @Abouraddy02OptComm]. Carrying out these optical measurements (see [@Kagalwala15SR] for details), $\mathbf{G}_{1}$ is reconstructed \[Fig. \[fig:Figure3\](b)\] and is found to be in good agreement with the theoretical expectation \[Fig. \[fig:Figure3\](c)\], with the remaining slight deviations attributable to unequal powers at $\vec{a}'$ and $\vec{b}'$.
The measured coherency matrix $\mathbf{G}_{1}$ in the $(\vec{a}',\vec{b}')$-plane yields $S=1.001$, and the reduced spatial and polarization coherency matrices $\mathbf{G}_{\mathrm{s}}^{(\mathrm{r})}$ and $\mathbf{G}_{\mathrm{p}}^{(\mathrm{r})}$ obtained from $\mathbf{G}$ yield $S_{\mathrm{s}}=0.991$, $S_{\mathrm{p}}=0.037$, respectively. The field entropy is thus associated with the spatial DoF and *not* polarization, resulting in an absence of interference fringes \[Fig. \[fig:Figure3\](a)\]. The lack of interference fringes is consistent with the fact that all measures of spatial coherence or double-slit interference fringes for a *vector field* rely on the cross-correlation matrix [@Wolf03PLA] or beam-coherence matrix [@Gori98OL], $\mathbf{G}_{ab}=\footnotesize{\left(\begin{array}{cc}G_{ab}^{\mathrm{HH}}&G_{ab}^{\mathrm{HV}}\\G_{ab}^{\mathrm{VH}}&G_{ab}^{\mathrm{VV}}\end{array}\right)}$, which is the top right $2\times2$ block of the coherency matrix $\mathbf{G}$. For example, the degree of coherence proposed by E. Wolf [@Wolf03PLA], the degree of electromagnetic coherence [@Tervo03OE; @Setala04OE], the complex degree of mutual polarization [@Ellis04OL], the visibility predicted through a generalized form of the Fresnel-Arago law [@Mujat04JOSAA], or the maximal visibility obtainable through local unitary transformations [@Gori07OL; @Herrero07OL] all predict that the observed visibility will be zero if all the entries in $\mathbf{G}_{ab}$ are zero. Because the measured and theoretically expected $\mathbf{G}_{1}$ has all zero entries in the $\mathbf{G}_{ab}$ block, it follows that interference fringes do not form. We proceed to show that interference fringes nevertheless appear once the coherence has been *reversibly converted* from polarization to the spatial DoF.
![(a) Schematic for the setup to combine two linearly polarized fields from $\vec{a}'$ and $\vec{b}'$ that are statistically independent or spatially incoherent ($S_{\mathrm{p}}=0$ and $S_{\mathrm{s}}=1$) into $\vec{a}''$ and $\vec{b}''$ whereupon the field becomes unpolarized but spatially coherent ($S_{\mathrm{p}}=1$ and $S_{\mathrm{s}}=0$), without loss of power or increase in total entropy $S=1$. HWP: Half-wave plate rotated to implement the transformation H$\rightarrow$V; PBS: polarizing beam splitter. (b) Malus curves for fields at the two input ports of the PBS highlight the linear polarization (one orthogonal to the other) and that for the field at the output port highlights its random polarization. The dashed and continuous lines are the flat and sinusoidal curves associated with unpolarized and V-polarized light, respectively. (c) Graphical depiction of the elements of the full coherency matrix $\mathbf{G}_{2}$ is obtained experimentally and (d) expected theoretically.[]{data-label="Figure4"}](Figure4)
Converting coherence from polarization to space
===============================================
Converting the coherence reversibly from polarization to the spatial DoF entails maximizing the polarization entropy $S_{\mathrm{p}}$ and minimizing the spatial entropy $S_{\mathrm{s}}$ at *fixed* total entropy $S$. A half-wave plate (HWP) placed after $\vec{b}'$ rotates the polarization H$\rightarrow$V, resulting in a new coherency matrix $\mathbf{G}=\tfrac{1}{2}\mathrm{diag}\{1,0,0,1\}$. The fields from $\vec{a}'$ and $\vec{b}'$ are then directed by mirrors to the two input ports of a PBS (Thorlabs CM1-PBS252), where the V component is transmitted and H is reflected \[Fig. \[Figure4\](a)\]. Consequently, the H and V components overlap in the same output port at $\vec{a}''$ (minimal power in the other port $\vec{b}''$). Note that a PBS is a reversible device: when the two output fields reverse their direction and return to the PBS, the input fields are reconstituted. The field is now unpolarized, which we confirm by registering a flat Malus curve and comparing it to the sinusoidal Malus curve produced by the linearly polarized input fields \[Fig. \[Figure4\](b)\]. That is, each incident field on the PBS is linearly polarized, whereas their superposition at the output – with the initial optical power now concentrated in a single path – is randomly polarized. The coherency matrix $\mathbf{G}_{2}$ at $\vec{a}''$ and $\vec{b}''$ is reconstructed via OCmT \[Fig. \[Figure4\](c)\], and is found to be in good agreement with the expected form \[Fig. \[Figure4\](d)\]: $$\begin{aligned}
\mathbf{G}_{2}&=\tfrac{1}{2}\left(\begin{array}{cccc}
1&0&0&0\\
0&1&0&0\\
0&0&0&0\\
0&0&0&0
\end{array}\right)=\tfrac{1}{2}\mathrm{diag}\{1,1,0,0\} \nonumber \\
&=\left(\begin{array}{cc}1&0\\0&0\end{array}\right)_{\mathrm{s}}\otimes\tfrac{1}{2}\left(\begin{array}{cc}1&0\\0&1\end{array}\right)_{\mathrm{p}}.\end{aligned}$$
From the reconstructed $\mathbf{G}_{2}$ in the $(\vec{a}'',\vec{b}'')$-plane, the total entropy is $S=0.997$, but the new values of entropy for the spatial and polarization DoFs are $S_{\mathrm{s}}=0.001$, and $S_{\mathrm{p}}=0.996$, respectively. *The entropy has now been converted between the two DoFs*. To observe interference fringes, the randomly polarized field at $\vec{a}''$ is split symmetrically into two halves by a 50:50 *non*-polarizing beam splitter to points $\vec{a}$ and $\vec{b}$ \[Fig. \[Figure5\](a)\], which can then be overlapped to produce high-visibility fringes. This step does not change the values of $S$, $S_{\mathrm{s}}$, or $S_{\mathrm{p}}$. The $(\vec{a},\vec{b})$-plane is the image plane relayed from the $(\vec{a}',\vec{b}')$-plane by a lens \[Fig. \[fig:SetupExp1and2\](c)\]. The coherency matrix $\mathbf{G}_{3}$ in this plane: $$\label{G_3}
\mathbf{G}_{3}=\frac{1}{4}\left(\begin{array}{cccc}
1&0&1&0\\
0&1&0&1\\
1&0&1&0\\
0&1&0&1
\end{array}\right)=\frac{1}{2}\left(\begin{array}{cc}1&1\\1&1\end{array}\right)_{\mathrm{s}}\otimes\frac{1}{2}\left(\begin{array}{cc}1&0\\0&1\end{array}\right)_{\mathrm{p}}.$$ The measured $\mathbf{G}_{3}$ reconstructed via OCmT \[Fig. \[Figure5\](b)\] is in good agreement with the theoretical expectation \[Fig. \[Figure5\](c)\].
Now that we have $G_{ab}^{\mathrm{HH}}+G_{ab}^{\mathrm{VV}}=\tfrac{1}{2}$, the predicted visibility is $V=1$ [@Wolf03PLA]. Furthermore, because the two DoFs are independent *and* the field is unpolarized (as is clear from the separable form of $\mathbf{G}_{3}$ in Eq. \[G\_3\]), projecting the polarization on any direction will not affect the high visibility \[Fig. \[Figure5\](d)\]. Note that the coherence time of the field is determined by its spectral bandwidth, and observing the fringes \[Fig. \[Figure5\](d)\] requires introducing a relative delay in the path of the fields from $\vec{a}$ or $\vec{b}$ before overlapping them \[delay line 2 in Fig. \[fig:SetupExp1and2\](c)\]. The variation in the measured visibility with introduced relative time delay corresponds to the expected field coherence time $\approx100$ fs (coherence length $\approx30$ $\mu$m) \[Fig. \[Figure5\](e)\].
Surviving randomization through entropy reallocation
====================================================
The ability to redistribute or reallocate the field entropy $S$ between the DoFs can be exploited in protecting a DoF from the deleterious impact of a randomizing medium. Consider a depolarizing medium represented by a Mueller matrix $\hat{M}=\mathrm{diag}\{1,0,0,0\}$ that converts *any* state of polarization into a completely unpolarized state. The initial field $\mathbf{G}_{1}$ \[Fig. \[fig:Figure3\](b,c)\] would be converted into the incoherent unpolarized field $\mathbf{G}'_{1}=\tfrac{1}{4}\mathrm{diag}\{1,1,1,1\}$ upon traversing this medium with $S_{\mathrm{p}}\rightarrow1$, such that $S\rightarrow2$. If, however, coherence is first reversibly converted from the spatial DoF to polarization ($S_{\mathrm{p}}\rightarrow1$ and $S_{\mathrm{s}}\rightarrow0$), then traversing a depolarizing medium *cannot increase* $S_{\mathrm{p}}$, and the field is thus left unchanged. Subsequently, the coherence-conversion can be reversed and a polarized field retrieved after emerging from the depolarizing medium without loss of energy.
We have carried out the proof-of-concept experiment depicted in Fig. \[Figure6\](a) where we place a depolarizer or polarization scrambler in the path of the field $\mathbf{G}_{2}$ in the $(\vec{a}'',\vec{b}'')$-plane. The polarization scrambler is implemented by rotating a HWP. The CCD camera exposure time is increased to 10 s, corresponding to the rotation time of the waveplate, to capture the averaged interference pattern in a single shot. The Mueller matrix associated with a polarization scrambler is $\hat{M}=\mathrm{diag}\{1,0,0,0\}$. The visibility observed after the $(\vec{a},\vec{b})$-plane remains high. This result can be modeled theoretically by first noting that a unitary transformation $\hat{U}$ transforms the field according to $\mathbf{G}_{2}\rightarrow\mathbf{G}'_{2}=\hat{U}\mathbf{G}_{2}\hat{U}^{\dag}$, where $\hat{U}$ is a $4\times4$ unitary transformation that spans the spatial and polarization DoFs. If the device in question impacts the two DoFs independently, then $\hat{U}=\hat{U}_{\mathrm{s}}\otimes\hat{U}_{\mathrm{p}}$, where $\hat{U}_{\mathrm{s}}$ and $\hat{U}_{\mathrm{p}}$ are $2\times2$ unitary transformations for the spatial and polarization DoFs, respectively. The impact of a randomizing but energy-conserving transformation can be modeled as a statistical ensemble of unitary transformations [@Kim87JOSAA; @Gil00JOSAA; @Gamel11OL]. The transformation of $\mathbf{G}_{2}$ upon traversing a polarization scrambler can be expressed as $$\label{G2AfterScrambler}
\mathbf{G}_{2}'=\sum_{j=1}^{4}p_{j}\{\hat{\mathbb{I}}\otimes\hat{U}_{\mathrm{p}}^{(j)}\}\mathbf{G}_{2}\{\hat{\mathbb{I}}\otimes\hat{U}_{\mathrm{p}}^{(j)\dag}\},$$ where $\hat{\mathbb{I}}$ is the $2\times2$ identity matrix and the ensemble $\{U_{\mathrm{p}}^{(j)}\}$ comprises with equal probabilities $p_{j}=\tfrac{1}{4}$ the Jones matrices: $\tfrac{1}{\sqrt{2}}\footnotesize{\left(\begin{array}{cc}1&-1\\1&1\end{array}\right)}$, $\tfrac{1}{\sqrt{2}}\footnotesize{\left(\begin{array}{cc}1&1\\-1&1\end{array}\right)}$, $\footnotesize{\left(\begin{array}{cc}1&0\\0&-1\end{array}\right)}$, $\footnotesize{\left(\begin{array}{cc}0&1\\1&0\end{array}\right)}$, which correspond to polarization rotations of $\pm45^{\circ}$ in the H-V basis, and a HWP in the H-V basis and rotated by $45^{\circ}$. Substituting the ensemble $\{\hat{U}_{\mathrm{p}}^{(j)}\}$ into Eq. \[G2AfterScrambler\] yields $\mathbf{G}_{2}'=\mathbf{G}_{2}$, which entails that the high-visibility seen in Fig. \[Figure5\](d) should be retained, as confirmed in Fig. \[Figure6\](a). In other words, $\mathbf{G}_{2}$ is *invariant with regards to any polarization randomization*.
.[]{data-label="Figure6"}](Figure6)
If the polarization scrambler is position-dependent, the coherency matrix $\mathbf{G}_{2}$ will no longer be invariant under randomization (because the spatial and polarization DoFs become coupled). In the experiment illustrated in Fig. \[Figure6\](b), the polarization scrambler is placed at $\vec{a}$ in the plane of $\mathbf{G}_{3}$. The spatial-polarization transformation of $\mathbf{G}_{3}$ takes the form $$\begin{aligned}
\label{G3AfterScrambler}
\mathbf{G}_{3}'=&\{\hat{\Lambda}_{b}\otimes\mathbb{I}\}\mathbf{G}_{3}\{\hat{\Lambda}_{b}^{\dag}\otimes\mathbb{I}\} \nonumber \\
&+\sum_{j=1}^{4}p_{j}
\left\{\hat{\Lambda}_{a}\otimes\hat{U}_{\mathrm{p}}^{(j)}\right\}\mathbf{G}_{3}\left\{\hat{\Lambda}_{a}^{\dag}\otimes\hat{U}_{\mathrm{p}}^{(j)\dag}\right\},\end{aligned}$$ where we have $\hat{\Lambda}_{a}=\footnotesize{\left(\begin{array}{cc}1&0\\0&0\end{array}\right)}$ and $\hat{\Lambda}_{b}=\footnotesize{\left(\begin{array}{cc}0&0\\0&1\end{array}\right)}$, resulting in $\mathbf{G}_{3}'=\tfrac{1}{2}\hat{\mathbb{I}}\otimes\tfrac{1}{2}\hat{\mathbb{I}}=\tfrac{1}{4}\mathrm{diag}\{1,1,1,1\}$; that is, an unpolarized and spatially incoherent field with maximal entropy $S=2$ is produced. No fringes will appear in this case \[Fig. \[Figure6\](b)\].
Discussion and conclusion
=========================
To facilitate the analysis of the coherence of optical fields encompassing multiple DoFs, it is becoming increasingly clear that the mathematical formalism of multi-partite quantum mechanical states is most useful [@Simon10PRL; @Qian11OL; @Kagalwala13NP]. The underlying foundation for this utility is the analogy between the mathematical description used in these domains. The Hilbert space of a multi-partite system is the tensor product of the Hilbert spaces of the single-particle subsystems. In classical optics, the multiple DoFs of a beam are described in a linear vector space having formally the structure of a tensor product of the linear vector spaces of the individual DoFs. In the quantum setting, pure multi-partite states that cannot be separated into products of single-particle states are said to be entangled; whereas in the classical setting, fields in which the DoFs cannot be separated are now being called ‘classically entangled’ [@Kagalwala13NP], coherence can be viewed as a ‘resource’ that may be reversibly converted from one DoF of the beam to another, just as entanglement is a resource shared among multiple quantum particles. There is a critical difference though between the quantum and classical settings. Entanglement between initially separable particles requires nonlocal operations of particle-particle interactions (which are particularly challenging for photons [@Gaeta13NP]); on the other hand, entangling operations that couple different DoFs of a beam are readily available in classical optics. Adopting this information-driven standpoint has led to a host of novel opportunities and applications. For example, Bell’s measure, originally developed for ascertaining nonlocality, becomes a useful in quantifying the coherence of a multi-DoF beam and assessing the resources required to synthesize a beam of given characteristics [@Kagalwala13NP]; beams in which spatial modes and polarization are classically entangled have been shown to enable fast particle tracking [@Berg15Optica] and full Mueller characterization of a sample [@Toppel14NJP; @Tripathi09OE]; and introducing spatio-temporal spectral correlations leads to propagation-invariant pulsed optical beams [@Kondakci16OE; @Parker16OE].
We have demonstrated – for the first time to the best of our knowledge – the ‘conversion’ or transformation of coherence from one DoF of an optical field to another; namely, from polarization to the spatial DoF. Starting from a field that is spatially incoherent but polarized, we redirect the statistical fluctuations from space towards polarization, resulting in an unpolarized field that is spatially coherent – reversibly, without losing optical energy along the way. Specifically, the 1 bit of entropy characterizing the spatial DoF was removed and added instead to the initially zero-entropy polarization. Entropy-engineering of partially coherent optical fields can open the path to a variety of possible future extensions and applications with regards to optimizing the interaction of optical fields with disordered media.\
**Funding Information**. Office of Naval Research (ONR) contract N00014-14-1-0260.\
**Acknowledgments**. The authors thank A. Dogariu and D. N. Christodoulides for useful discussions.
[^1]: These authors contributed equally.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We demonstrate a mesoscopic spin polarizer/analyzer system that allows the spin polarization of current from a quantum point contact in an in-plane magnetic field to be measured. A transverse focusing geometry is used to couple current from an emitter point contact into a collector point contact. At large in-plane fields, with the point contacts biased to transmit only a single spin ($g < e^2/h$), the voltage across the collector depends on the spin polarization of the current incident on it. Spin polarizations of $> 80 \%$ are found for both emitter and collector at $300~mK$ and $7~T$ in-plane field.'
author:
- 'R. M. Potok'
- 'J. A. Folk'
- 'C. M. Marcus'
- 'V. Umansky'
date: '6/12/02'
title: 'Detecting Spin-Polarized Currents in Ballistic Nanostructures '
---
The detection of single electron spins has been the aim of extensive experimental efforts for many years. In addition to providing a new tool to investigate the physics of mesoscopic devices, there is hope that the ability to manipulate and measure electron spins in a solid state system may open the way for quantum information processing [@Loss; @Golovach]. However, the long coherence times [@Kikkawa] that make electron spins such a promising system for quantum manipulation result fundamentally from their weak coupling to the environment, and this makes the task of measuring spin difficult.
In this Letter we demonstrate a technique to measure spin by converting the problem into the easier one of measuring charge. At low field and low temperature, a narrow constriction in a 2D electron gas (2DEG), known as a quantum point contact (QPC) \[see Fig. 1(a)\], transmits through two spin-degenerate channels, producing conductance plateaus at integer multiples of $2e^2/h$. When a large in-plane magnetic field is applied, the degeneracy is lifted and conductance becomes quantized in multiples of $1e^2/h$ \[Fig. 1(b)\] [@Wharam; @vanWees]. While it is widely believed that the $e^2/h$ plateau is associated with spin-polarized transmission, this has not been established experimentally to our knowledge. One key result of this Letter is the demonstration that point contacts do operate as spin emitters and detectors, and therefore allow the detection of spin polarization to be accomplished by simply measuring electrical resistance.
\[fig1\] {width="3.25in"}
Our experiment is based on a technique known as transverse electron focusing [@vanHouten], which has been used previously to study phenomena ranging from anisotropy in the band structure of metals [@Sharvin; @Tsoi] and semiconductors [@Goldoni; @Ohtsuka] to composite fermions in the fractional quantum Hall regime [@Goldman]. This device geometry \[Fig. 1(a)\] allows electrons from a spin-polarizing emitter—in this case a QPC—to be coupled into a second QPC serving as a spin-sensitive collector. A magnetic field, $B_\perp$, applied perpendicular to the 2DEG plane, bends and focuses ballistic electron trajectories from the emitter to the collector, resulting in peaks in the base-collector voltage \[Figs. 1(c) and 1(d)\] whenever the spacing between point contacts is an integer multiple of the cyclotron diameter, $2m^*v_F/eB_\perp$, where $m^*$ is the effective electron mass and $v_F$ the Fermi velocity.
The coupling efficiency between emitter and collector can be quite high in clean 2DEG materials, allowing the two QPCs to be separated by several microns. This separation is useful for investigating spin physics in mesoscopic structures because it allows spin measurements to be decoupled from the device under test, simplifying the interpretation of results. A further advantage of a focusing geometry is that only ballistic trajectories contribute to the signal, so spin detection occurs very quickly ($ < 10~ps$) after the polarized electrons are emitted, leaving little time for spin relaxation.
In the present experiment, the focusing signal is measured as a voltage between collector and base regions, with fixed current applied between emitter and base \[Fig. 1(a)\]. With the collector configured as a voltage probe, current injected ballistically into the collector region at the focusing condition must flow back into the base region, giving rise to a voltage $V_{c}=I_{c}/g_{c}$ between collector and base, where $I_{c}$ is the current injected into the collector and $g_{c}$ is the conductance of the collector point contact. For this experiment both point contacts are kept at or below one channel of conductance; therefore the collector voltage may be written in terms of the transmission of the collector point contact, $T_c$ ($\le 1$), as $V_c=(2e^2/h)^{-1} I_c/T_c$.
To analyze how spin polarization affects the base-collector voltage, we assume $I_{c}\propto I_{e} T_{c}$, where $I_e$ is the emitter current, and the constant of proportionality does not depend on the transmissions of either of the point contacts. In the absence of spin effects, one then expects $V_{c}$ to be independent of $g_{c}$. Because $I_e$ is fixed, $V_{c}$ would also be independent of the emitter conductance, $g_{e}$.
Taking into account different transmissions for the two spin channels, however, one expects the voltage on the collector to double if both emitter and collector pass the same spin, or drop to zero if the two pass opposite spins. This conclusion assumes that a spin polarized current injected into the collector region will lose all polarization before flowing out again. Under these conditions, the collector voltage generally depends on the polarization of the emitter current $P_e=(I_{\uparrow} - I_{\downarrow})/(I_{\uparrow} + I_{\downarrow})$ and the spin selectivity of the collector $P_c=(T_{\uparrow}-T_{\downarrow})/(T_{\uparrow}+T_{\downarrow})$ in the following simple way [@me]: $$V_{c}\propto\frac{h}{2e^2}I_{e}(1+P_eP_c).$$ Note from Eq. 1 that colinear and complete spin polarization ($P_e = 1$) and spin selectivity ($P_c =
1$) gives a collector voltage twice as large as when [*either*]{} emitter or collector is not spin polarized.
The focusing device was fabricated on a high-mobility two-dimensional electron gas (2DEG) formed at the interface of a $\rm GaAs/Al_{0.36}Ga_{0.64}As$ heterostructure, defined using Cr/Au surface depletion gates patterned by electron-beam lithography, and contacted with nonmagnetic (PtAuGe) ohmic contacts. The 2DEG was $68~nm$ from the Si delta-doped layer ($n_{Si}= 2.5 \times 10^{12}
~cm^{-2}$) and $102~nm$ below the wafer surface. Mobility of the unpatterned 2DEG was $5.5\times10^6~cm^2/Vs$ in the dark, limited mostly by remote impurity scattering in the relatively shallow structure, with an estimated background impurity level $<
5\times10^{13}~cm^{-3}$. With an electron density of $\sim 1.3 \times 10^{11}~cm^{-2}$, the transport mean free path was $\sim 45~\mu m$, much greater than the distance ($1.5~\mu m$) between emitter and collector point contacts. The Fermi velocity associated with this density is $v_F=2\times
10^7~cm/s$, consistent with the observed $\sim 80~mT$ spacing between focusing peaks.
Measurements were performed in a $^3He$ cryostat with a base temperature of $300~ mK$. A conventional superconducting solenoid was used to generate in-plane fields, $B_\parallel$, and a smaller superconducting coil wound on the refrigerator vacuum can allowed fine control of the perpendicular field, $B_\perp$ [@Folk01]. $B_\parallel$ was oriented along the axis between the two point contacts, as shown in Fig. 1(a).
Independent ac current biases of $1~nA$ were applied between base and emitter ($17~Hz$), and base and collector ($43~Hz$), allowing simultaneous lock-in measurement of the emitter conductance (base-emitter voltage at $17~Hz$), collector conductance (base-collector voltage at $43~Hz$), and the focusing signal (base-collector voltage at $17~Hz$). The base-collector current bias was found to have no effect on the focusing signal. Additionally, the focusing signal was found to be linear in base-emitter current for the small currents used in this measurement.
Measurements were taken over several thermal cycles of the device. While details of focusing peak shapes and point contact conductance traces changed somewhat upon thermal cycling, their qualitative behavior did not change. Although all of the data presented in this paper comes from a single device, the results were confirmed in a similar device on the same heterostructure.
Spin polarization and spin selectivity of the point contacts were detected by comparing the focusing signal (the collector voltage at the top of a focusing peak) for various conductances of the emitter and collector point contacts. At $B_\parallel=0$, where no static spin polarization is expected, the focusing signal was found to be nearly independent of the conductances of both emitter and collector point contacts, as shown in Fig. 1(c). In contrast, at $B_\parallel=7~T$, the focusing signal observed when both the emitter and collector point contacts were set well below $2e^2/h$ was larger by a factor of $\sim 2$ compared to the signal when either emitter or collector was set to $2e^2/h$, as seen in Fig. 1(d). A factor-of-two enhancement is consistent with Eq. (1) for fully spin polarized emission and aligned, fully spin-selective detection.
To normalize for overall variations in transmission through the bulk from the emitter to the collector (for instance upon thermal cycling), the focusing signal at any emitter or collector setting can be normalized by the value when both the emitter and collector are set to $2e^2/h$. We denote the point contact settings as $(x:y)$ where $x$ is the conductance of the emitter and $y$ is the conductance of the collector, both in units of $e^2/h$. For instance, $(2:2)$ indicates both emitter and collector set to $2e^2/h$ (expected to be unpolarized in any field), while $(0.5:0.5)$ indicates both point contacts set to $0.5e^2/h$ (expected to be polarized in a sizable in-plane field). Ratios are then denoted $(x: y)/(2:2)$.
\[fig2\] {width="3.25in"}
Figures 2 and 3 show the focusing signal ratios for the third focusing peak ($B_\perp \sim
230-250~mT$), chosen because its height and structure in the $(2:2)$ condition were less sensitive to $B_\parallel$ and small variations in point contact tuning compared to the first and second peaks. However, all peaks showed qualitatively similar behavior.
Figure 2(a) shows that only the ratio $(0.5:0.5)/(2:2)$ grows with $B_\parallel$, reaching a value $\sim 2$ at $7T$, while the other ratios, $(2:0.5)/(2:2)$ and $(0.5:2)/(2:2)$, are essentially independent of in-plane field, as expected from Eq. (1) if no spin selectivity exists when the conductance is $2~e^2/h$. At $B_\parallel = 0$, we find $(0.5:0.5)/(2:2) \sim 1.4$, rather than the expected $1.0$, for this particular cooldown. As discussed below, these ratios fluctuate somewhat between thermal cycles.
Temperature dependences of the $(0.5:0.5)/(2:2)$ ratio are shown in Fig. 2(b) for a different cooldown. At $B_\parallel = 7T$, the ratio $(0.5:0.5)/(2:2)$ decreases from $\sim2.2$ at $T=300~mK$ to the zero-field value of $1.4$ above $2K$. Note that $2K$ is roughly the temperature at which $g
\mu B_\parallel/kT \sim 1$, using the GaAs g-factor $g =-0.44$. At $B_\parallel = 0$, the ratio $(0.5:0.5)/(2:2)$ remains near $1.4$, with only a weak temperature dependence up to $6K$.
The inset of Fig. 2(b) shows that focusing data at different values of $B_{\parallel}$ scale to a single curve when plotted as a function of $kT/g\mu B_\parallel$, suggesting that both spin-polarized emission and spin-selective detection arise from an energy splitting that is linear in $B_\parallel$. A simple model that accounts roughly for the observed scaling of the focusing signal assumes that the point contact transmission, $T(E)$, is $0$ for $E< E_0$, and $1$ for $E> E_0$, where $E$ is the electron kinetic energy and $E_0$ is a gate-voltage-dependent threshold. Spin selectivity then results from the Zeeman splitting of the two spin sub-bands, and is reduced by thermal broadening. Except for a vertical offset of $\sim 0.4$, this simple model agrees reasonably well with the data \[Fig. 2(b), inset\].
Fig. 3(a) shows the evolution of spin selectivity in the collector point contact as a function of its conductance. At $B_\parallel=6~T$, with the emitter point contact set to $0.5 e^2/h$, the collector point contact is swept from $2
e^2/h$ to 0. The focusing signal increases as the collector point contact conductance is reduced below $2 e^2/h$, saturating as the collector conductance goes below the $e^2/h$ spin-split plateau. The polarization saturates completely only well into the tunneling regime, below $\sim0.5 e^2/h$. Similar to the effect seen in Fig. 2(b), spin selectivity decreases with increasing temperature, approaching the zero field curve at $1.3~K$.
\[fig3\] {width="3.25in"}
Fig. 3(b) shows the same measurement taken at $B_\parallel=0$. The focusing peak rises slightly when both point contacts are set below one spin degenerate channel. Unlike at high field, however, the increase of the focusing signal is very gradual as the point contact is pinched off. In addition, temperature has only a weak effect.
As mentioned above, both the low and high field ratios $(0.5 : 0.5)/(2:2)$ were measured to be larger than their ideal theoretical values of 1 and 2 respectively. Sampled over multiple thermal cycles, several gate voltage settings (shifting the point contact centers by $\sim 100~nm$), and different focusing peaks, the ratio at $B_{\parallel}=0$ varied between 1.0 and 1.6, with an average value of 1.25 and a standard deviation $\sigma= 0.2$. The average value of the ratio at $B_\parallel=7~T$ was 2.1, with $\sigma=0.1$.
Both point contacts display a modest amount of zero-field 0.7 structure [@Thomas; @Cronenwett], as seen in Figs. 1(b) and 3(b). Although a static spin polarization associated with 0.7 structure would be consistent with our larger-than-one ratio $(0.5:0.5)/(2:2)$ at zero field, this does not explain the enhanced ratio found [*both*]{} at zero field and high field. Rather, we believe the enhancement is due to a slight increase in the efficiency of focusing for $(T_c, T_e) < 1$. For example, more of the emitted current may be focused into the collector as the point contacts are pinched off, causing deviations from the assumption $I_c\propto I_e T_c$. This explanation is also consistent with the weak temperature dependence of the zero-field ratio up to $4~K$, which would not occur if the enhancement were due a static polarization at zero field.
An unexplained feature of our data is the relative suppression of the lower-index focusing peaks—particularly the first peak—in a large in-plane field, as seen in Figs. 1(c) and 1(d). This effect was observed over multiple thermal cycles and for all point contact positions. The effect is not readily explained as a field-dependent change in the scattering rate, as neither the bulk mobility, nor the width of the focusing peak is affected. Also, the effect is not obviously related to spin, as it occurred for both polarized and unpolarized point contacts.
In conclusion, we have developed a new method for creating and remotely detecting spin currents using quantum point contacts. The technique has allowed a first demonstration of what was widely expected, namely that a point contact in an in-plane field can act as a spin polarized emitter and a spin sensitive detector. From our perspective, however, this result also has a larger significance: it is the first demonstration of a wholly new technique to measure spin-current from a mesoscopic device using a remote electrical spin detector. In future work, this technique can be applied to more subtle mesoscopic spin systems such as measuring spin currents from open or Coulomb-blockaded quantum dots, or directly measuring spin precession due to a spin-orbit interaction.
We acknowledge valuable discussions with H. Bruus, S. Cronenwett, A. Johnson, and H. Lynch. This work was supported in part by ARO-MURI (DAAD 19-99-1-0215) and DARPA-SpinS (DAAD 19-01-1-0659). JAF acknowledges partial support from the Stanford Graduate Fellowship; RMP acknowledges support as an ARO Graduate Research Fellow.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'As multi-agent networks grow in size and scale, they become increasingly difficult to synchronize, though agents must work together even when generating and sharing different information at different times. Targeting such cases, this paper presents an asynchronous optimization framework in which the time between successive communications and computations is unknown and unspecified for each agent. Agents’ updates are carried out in blocks, with each agent updating only a small subset of all decision variables. To provide robustness to asynchrony, each agent uses an independently chosen Tikhonov regularization. Convergence is measured with respect to a weighted block-maximum norm in which convergence of agents’ blocks can be measured in different p-norms and weighted differently to heterogeneously normalize problems. Asymptotic convergence is shown and convergence rates are derived explicitly in terms of a problem’s parameters, with only mild restrictions imposed upon them. Simulation results are provided to verify the theoretical developments made.'
author:
- 'Stefan Hochhaus$^{\star,\dagger}$ and Matthew T. Hale$^{\star}$[^1] [^2]'
bibliography:
- 'CDC.bib'
title: |
Asynchronous Distributed Optimization with\
Heterogeneous Regularizations and Normalizations
---
Introduction
============
Distributed optimization techniques have been applied in many areas ranging from sensor networks [@Khan2009; @Cortes2004; @Rabbat2004] and communications [@Mitra1994; @Chiang2007], to robotics [@Soltero2014] and smart power grids [@Caron2010]. With this diversity in applications, there have emerged correspondingly diverse problem formulations which address a wide variety of practical considerations. As multi-agent systems become increasingly complex, a key practical consideration is the ability to tightly couple agents and the timing of their behaviors. Often, perfect synchrony among agents’ communications and computations is difficult or impossible because closely coupling all agents in large networks is also difficult or impossible. Instead, one must sometimes utilize information that is asynchronously generated and shared. This paper examines how to do so in a distributed optimization setting.
There is a significant existing literature on distributed optimization, including a large corpus of work on asynchronous optimization. One common approach is to assume that delays in communications and computations are bounded, and this approach is used for example in [@Chen2012; @Jadbabaie2003; @Moreau2005; @Nedic2007; @Nedic2009; @Nedic2010; @Olshevsky2009; @Ren2005; @Touri2009], and the delay bound parameter explicitly appears in convergence rates in [@Chen2012; @Nedic2007; @Nedic2009; @Olshevsky2009; @Touri2009]. However, in some cases, delay bounds cannot be enforced. For example, agents with mutually interfering communications may be unable to ensure that delay lengths stay below a certain threshold because delays are outside their control. Similarly, agents facing anti-access/area-denial (A2AD) measures may be unable to predict when transmissions will be received or even measure delay lengths at all. As a result, some works have addressed asynchronous optimization with unbounded delays. Early work in this area includes [@Bertsekas1989], as well as [@bertsekas1989parallell], which gives a textbook-level treatment and simplified proof of the main results in [@Bertsekas1989].
Work in [@Bertsekas1989] was expanded upon in [@Hale2017], where it was shown that a fixed Tikhonov regularization implies the existence of the nested sets required in [@Bertsekas1989] for asymptotic convergence. However, developments in [@Hale2017] require every agent to apply the same regularization, which can be difficult to enforce and verify in practice, especially in large decentralized networks. Moreover, convergence in [@Hale2017] is measured with respect to the same un-weighted norm for all agents. There is a wide variety of statistical and machine learning problems which must be normalized due to disparate numerical scales across potentially many orders of magnitude [@bishop1995neural], and which may require measuring convergence of different components in different norms. While such problems are commonly solved using distributed optimization techniques, they are not accounted for by the work in [@Hale2017]. Therefore, a fundamentally new approach is required to account for heterogeneous regularizations and normalizations in the setting of distributed optimization.
In this paper we develop an asynchronous optimization framework to address this gap. In particular, we examine set-constrained optimization problems with potentially non-separable cost functions, and we allow agents’ communications and computations to be arbitrarily asynchronous, subject only to mild assumptions. Agents are permitted to independently choose regularization parameters with no restrictions on the disparity between them. Under these conditions, agents’ convergence is measured with respect to a weighted block-maximum norm which allows for heterogeneous normalizations of agents’ distance to an optimum in order to accommodate problems with different numerical scales. Convergence rates are developed in terms of agents’ communications and computations without specifying when they must occur. The framework developed in this paper uses a block-based update scheme in which each agent updates only a subset of all decision variables in a problem in order to provide a scalable update law for large convex programs. The contributions of this paper therefore consist of a scalable optimization framework that accommodates heterogeneous regularizations and normalizations, together with its convergence rate.
The rest of the paper is organized as follows. Section II defines the optimization problems to be solved and regularizations used. Next, Section III defines the block-based multi-agent update law, and Section IV proves its convergence and derives its convergence rate. After that, Section V presents simulation results and Section VI provides concluding remarks.
Tikhonov Regularization and Problem Statement
=============================================
In this section we describe the class of problems to be solved and the assumptions imposed upon problem data. We then introduce heterogeneous regularizations and the need for heterogeneous normalizations. Then we give a formal problem statement that is the focus of the remainder of the paper.
We consider convex optimization problems spread across teams of agents. In particular, we consider teams comprised of $N$ agents, where agents are indexed over $i\in\left[N\right]\coloneqq\left\{ 1,\ldots,N\right\}$. Agent $i$ has a decision variable $x_{i}\in\mathbb{R}^{n_{i}}$, $n_{i}\in\mathbb{N}$, which we refer to as its state, and we allow for $n_{i}\neq n_{j}$ when $i\neq j$. The state $x_{i}$ is subject to the set constraint $x_{i}\in X_{i}\subset\mathbb{R}^{n_{i}}$, which can represent, e.g., that a mobile robot must stay in a given area. We make the following assumption about each $X_{i}$.
For all $i\in\left[N\right]$, the set $X_{i}\subset\mathbb{R}^{n_{i}}$ is non-empty, compact, and convex. $\triangle$
Towards making a formal problem statement, we aggregate agents’ set constraints by defining $X\coloneqq X_1\times\dots\times X_N$, and Assumption 1 ensures that $X$ is also non-empty, compact, and convex. We further define the ensemble state as $x\coloneqq\left(x_{1}^{T},\ldots,x_{N}^{T}\right)^{T}\in X\subset\mathbb{R}^{n}$, where $n=\underset{i\in\left[N\right]}{\sum}n_{i}$. We consider problems in which each agent has a local objective function $f_{i}$ to minimize, which can represent, e.g., a mobile robot’s desire to minimize its distance to a target location; only agent $i$ needs to know $f_{i}$. The agents are also collectively subject to a coupling cost $c$, which can represent the cost of communication congestion in a network, and we allow for $c$ to be non-separable. We then make the following assumption about the functions $f_{i}$ and $c$.
The functions $f_{i}$, $i\in\left[N\right]$, and $c$ are convex and $C^{2}$ (twice continuously differentiable) in $x_{i}$ and $x$, respectively. $\triangle$
In particular, $\nabla{f}$ is Lipschitz and we denote its Lipschitz constant by $L$. The sum of these costs then gives the aggregate cost function $$f\left(x\right)\coloneqq c\left(x\right)+\underset{i\in[N]}{\sum}f_{i}\left(x_{i}\right),$$ and the agents will jointly minimize $f$. For simplicity of the forthcoming analysis, we assume that $f$ has a unique minimizer. To endow $f$ with an inherent robustness to asynchrony, we will regularize it before agents start optimizing. In particular, we regularize $f$ on a per-agent basis, where agent $i$ uses the regularization parameter $\alpha_{i}>0$ and where we allow $\alpha_{i}\neq\alpha_{j}$ when $i\neq j$. Regularizing $f$ makes it strongly convex, and this will be shown to provide robustness to asynchrony below. The regularized form of $f$ is denoted $f_{A}$, and is defined as $$f_{A}\left(x\right)\coloneqq f\left(x\right)+\frac{1}{2}x^{T}Ax,$$ where $A=\textrm{diag}\left(\alpha_{1}I_{n_{1}},\ldots,\alpha_{N}I_{n_{N}}\right),$ and where $I_{n_{i}}$ is the $n_{i}\times n_{i}$ identity matrix.
In some optimization settings, some decision variables evolve at drastically different numerical scales [@bishop1995neural]. To more meaningfully evaluate the convergence of agents with respect to one another, it would be useful to normalize each agent’s distance to an optimum to prevent the error of one agent dominating the convergence analysis. Allowing heterogeneous normalizations would therefore give a more useful estimate of the distance to an optimum, and this should be accounted for by our framework. Moreover, each agent may wish to evaluate the convergence of its own state using a particular $p$-norm. Therefore, our framework should accommodate agents measuring the distance to an optimum in different norms. Bearing these criteria in mind, we now state the problem that is the focus of the rest of the paper.
For a team of $N$ agents, $$\underset{x\in X}{\textrm{minimize }}f_{A}\left(x\right)$$ while measuring convergence with heterogeneous normalization constants and norms across the agents. $\lozenge$
Section III specifies the structure of the asynchronous communications and computations used to solve Problem 1.
Block-Based Multi-Agent Update Law
==================================
To define the exact update law for each agent’s state, we must describe what information is stored and how agents communicate. Each agent will store a vector containing its own states and those of agents it communicates with. Each agent only updates its own states within the vector it stores onboard. States stored onboard agent $i$ which correspond to other agents’ states are only updated when those agents send their states to agent $i$. This type of block-based update can be used to capture, for example, when an agent does not have the information required to update other agents’ states, or when it is desirable to parallelize updates to reduce each agent’s computational burden.
Formally, we will denote agent $i$’s full vector of states by $x^{i}$. Agent $i$’s own states in this vector are then denoted by $x_{i}^{i}$. The current values stored onboard agent $i$ for agent $j$ are denoted by $x_{j}^{i}$. At timestep $k$, agent $i$’s full state vector is denoted $x^{i}\left({k}\right)$, with its own states denoted $x_{i}^{i}\left({k}\right)$ and those of agent $j$ denoted $x_{j}^{i}\left({k}\right)$. At any single timestep, agent $i$ may or may not update its states due to asynchrony in agents’ computations, and the times of these updates must be accounted for. We define the set $K^{i}$ to be the collection of time indices $k$ at which agent $i$ updates $x_{i}^{i}$; agent $i$ does not compute an update for time indices $k\notin K^{i}$. Using this notation, agent $i$’s update law can be written as $$x_{i}^{i}\left(k+1\right)=\left\{ \begin{array}{cc}
x_{i}^{i}\left(k\right)-\gamma\nabla_{i}f_{A}\left(x^{i}\left(k\right)\right) & k\in K^{i}\\
x_{i}^{i}\left(k\right) & k\notin K^{i}
\end{array}\right.,$$ where agent $i$ uses stepsize $\gamma >0$, which will be bounded below. Here $\nabla_{i}f_{A}\coloneqq\frac{\partial f_{A}}{\partial x_{i}}$ is the gradient of the regularized cost function with respect to $x_{i}$. The significance of agent $i$’s choice of regularization parameter can be seen by expanding $\nabla_{i}f_{A}\left(x^{i}\left(k\right)\right)$ as $\nabla_{i}f_{A}\left(x^{i}\left(k\right)\right)=\nabla_{i}f\left(x^{i}\left(k\right)\right)+\alpha_{i}x_{i}^{i}\left(k\right)$, where $\alpha_{i}>0$ is set by agent $i$ alone.
In order to account for communication delays we use $\tau_{j}^{i}\left(k\right)$ to denote the time at which the value of $x_{j}^{i}\left(k\right)$ was originally computed by agent $j$. For example, if agent $j$ computes a state update at time $k_{a}$ and immediately transmits it to agent $i$, then agent $i$ may receive this state update at time $k_{b}>k_{a}$ due to communication delays. Then $\tau_{j}^{i}$ is defined so that $\tau_{j}^{i}\left(k_{b}\right)=k_{a}$, the time at which agent $j$ originally computed the update just received by agent $i$. Concerning $K^{i}$ and $\tau_{j}^{i}\left(k\right)$, we have the following assumption.
For all $i\in\left[N\right]$, the set $K^{i}$ is infinite. Moreover, for all $i\in\left[N\right]$ and $j\in\left[N\right]\backslash\left\{i\right\}$, if $\left\{k_{d}\right\}_{d\in\mathbb{N}}$ is a sequence in $K^{i}$ tending to infinity, then $$\lim_{d\rightarrow\infty} \tau_{j}^{i}\left(k_{d}\right)=\infty.$$ $\triangle$
Assumption 3 is quite mild in that it simply requires that no agent ever permanently stop updating and sharing information. For $i\neq j$, the sets $K^{i}$ and $K^{j}$ need not have any relationship because agents’ updates are asynchronous. The entire update law for all agents can then be written as follows.
For all $i\in\left[N\right]$ and $j\in\left[N\right]\backslash\left\{i\right\}$, execute $$\begin{aligned}x_{i}^{i}\!\left(k\!+\!1\right) & \!=\!\left\{ \!\!\!\begin{array}{cc}
x_{i}^{i}\left(k\right)-\gamma\nabla_{i}f_{A}\left(x^{i}\left(k\right)\right) & k\in K^{i}\\
x_{i}^{i}\left(k\right) & k\notin K^{i}
\end{array}\right.\\
x_{j}^{i}\!\left(k\!+\!1\right) & \! =\!\left\{ \!\!\!\begin{array}{cc}
x_{j}^{j}\left(\tau_{j}^{i}\left(k+1\right)\right) & \!\!\!\!\textrm{i receives j's state at k+1}\\
x_{j}^{i}\left(k\right) & \textrm{otherwise}
\end{array}\right.\!\!\!\!\!.
\end{aligned}$$ $\diamond$
In Algorithm 1 we see that $x_{j}^{i}$ changes only when agent $i$ receives a transmission from agent $j$; otherwise it remains constant. Agent $i$ can therefore reuse old values of agents $j$’s state many times and can reuse different agents’ states different numbers of times. Showing convergence of this update law must take these delays into account, and that is the subject of the next section.
Convergence of Asynchronous Optimization
========================================
In this section we prove the convergence of the multi-agent block update law in Algorithm 1. We first define the block-maximum norm used to measure convergence and then define a collection of nested sets that will be used to show asymptotic convergence of all agents adapted from the approach in [@Hale2017]. Then a convergence rate is developed using parameters from these sets.
Block-Maximum Norms
-------------------
We begin by analyzing the convergence of the optimization algorithm using block maximum norms similar to those defined in [@Bertsekas1989], [@bertsekas1989parallell], and [@Hale2017], and we do so to accommodate the need for heterogeneus normalizations and norms in Problem 1. Due to asynchrony in agents’ communications, we will generally have $x^{i}\left(k\right)\neq x^{j}\left(k\right)$ for all agents $i$ and $j$ and all timesteps $k$. We will refer to $x_{i}^{i}$ as the $i^{th}$ block of $x^{i}$ and $x_{j}^{i}$ as the $j^{th}$ block of $x^{i}$. With these blocks defined we next define the block-maximum norm that will be used to measure convergence below.
Let $x\in\mathbb{R}^{n}$ consist of $N$ blocks, with $x_{i}\in\mathbb{R}^{n_{i}}$ being the $i^{th}$ block. The $i^{th}$ block is weighted by some normalization constant $w_{i}\geq1$ and is measured in the $p_{i}$-norm for some $p_{i}\in\left[1,\infty\right]$. The norm of the full vector $x$ is defined as the maximum norm of any single block, i.e., $$\left\Vert x\right\Vert _{\max}\coloneqq\underset{i\in\left[N\right]}{\max}\frac{\left\Vert x_{i}\right\Vert _{p_{i}}}{w_{i}}.$$$\triangle$
The following lemma allows us to upper-bound the induced block-maximum matrix norm by the Euclidian matrix norm, which will be used below in our convergence analysis. In this lemma, we use the notion of a block of an $n\times n$ matrix. Given a matrix $B\in\mathbb{R}^{n\times n}$, where $n=\sum_{i=1}^{N}n_{i}$, the $i^{th}$ block of $B$, denoted $B^{\left[i\right]}$, is the $n_{i}\times n$ matrix formed by rows of $B$ with indices $\sum_{k=1}^{i-1}n_{k}+1$ through $\sum_{k=1}^{i}n_{k}$. We then have the following result.
Let $p_{\min}\coloneqq\min_{i\in\left[{N}\right]} p_{i}$ and let $w_{\min}=\min_{i\in\left[{N}\right]} w_{i}$. Then for all $B\in\mathbb{R}^{n\times n}$, $$\left\Vert B\right\Vert _{\max}\leq\left\{ \begin{array}{cc}
n^{\left(p_{\min}^{-1}-\frac{1}{2}\right)}w_{\min}^{-1}\left\Vert B\right\Vert _{2} & p_{\min}<2\\
\frac{1}{w_{\min}}\left\Vert B\right\Vert _{2} & p_{\min}\geq2
\end{array}\right..$$
For $B^{\left[i\right]}$ the $i^{th}$ block of $B$ and any $x\in\mathbb{R}^{n}$, by definition we have $$\begin{aligned}\frac{\left\Vert B^{\left[i\right]}x\right\Vert _{p_{i}}}{w_{i}} & =\frac{1}{w_{i}}\left(\sum_{k=1}^{n_{i}}\left|\sum_{j=1}^{n}B_{k,j}^{\left[i\right]}x_{j}\right|^{p_{i}}\right)^{\frac{1}{p_{i}}}.\end{aligned}
\label{eq:BlockNorm}$$ From the definition of a $p$-norm, the right side of Equation will always be non-negative. Thus summing the right-hand side over every block results in $$\begin{aligned}\frac{\left\Vert B^{\left[i\right]}x\right\Vert _{p_{i}}}{w_{i}} & \leq\sum_{i=1}^{N}\left(\frac{1}{w_{i}}\left(\sum_{k=1}^{n_{i}}\left|\sum_{j=1}^{n}B_{k,j}^{\left[i\right]}x_{j}\right|^{p_{i}}\right)^{\frac{1}{p_{i}}}\right)\end{aligned}.$$ Next, recalling that $\left\Vert{x}\right\Vert_{q}\leq\left\Vert{x}\right\Vert_{r}$ for all vectors $x\in\mathbb R^{n}$ and all $q\geq r>0$, we find that $$\begin{aligned}\frac{\left\Vert B^{\left[i\right]}x\right\Vert _{p_{i}}}{w_{i}} & \leq\frac{1}{w_{min}}\sum_{i=1}^{N}\left(\sum_{k=1}^{n_{i}}\left|\sum_{j=1}^{n}B_{k,j}^{\left[i\right]}x_{j}\right|^{p_{i}}\right)^{\frac{1}{p_{i}}}\\
& \leq\frac{1}{w_{\min}}\sum_{i=1}^{N}\left(\sum_{k=1}^{n_{i}}\left|\sum_{j=1}^{n}B_{k,j}^{\left[i\right]}x_{j}\right|^{p_{\min}}\right)^{\frac{1}{p_{\min}}}.
\end{aligned}$$ This then allows us to express the sum over all rows of $B$ via $$\frac{\left\Vert B^{\left[i\right]}x\right\Vert _{p_{i}}}{w_{i}}\leq\frac{1}{w_{min}}\left(\sum_{l=1}^{n}\left|\sum_{j=1}^{n}B_{l,j}x_{j}\right|^{p_{\min}}\right)^{\frac{1}{p_{\min}}}.$$ If $p_{min}\geq2$, then $\left\Vert B^{\left[i\right]}x\right\Vert _{p_{i}}\leq\left\Vert B^{\left[i\right]}x\right\Vert _{2}$ for all $p_i$. If $p_{min}<2$, we recall that $\left\Vert x\right\Vert _{l}\leq\left\Vert x\right\Vert _{p_{\min}}\leq n^{\left(p_{\min}^{-1}-l^{-1}\right)}\left\Vert x\right\Vert _{l}$, which follows from H[ö]{}lder’s inequality for $0<p_{min}<l$, and observe that $\left\Vert B^{\left[i\right]}x\right\Vert _{p_{i}}\leq\left\Vert B^{\left[i\right]}x\right\Vert _{p_{\min}}\leq n^{\left(p_{\min}^{-1}-\frac{1}{2}\right)}\left\Vert Bx\right\Vert _{2}$. Combining these inequalities we find that $$\frac{\left\Vert B^{\left[i\right]}x\right\Vert _{p_{i}}}{w_{i}}\leq\left\{ \begin{array}{cc}
n^{\left(p_{\min}^{-1}-\frac{1}{2}\right)}w_{\min}^{-1}\left\Vert Bx\right\Vert _{2} & p_{\min}<2\\
\frac{1}{w_{\min}}\left\Vert Bx\right\Vert _{2} & p_{\min}\geq2
\end{array}\right.$$ for all $i$. Thus the weighted block maximum norm of $Bx$ for any $x\in\mathbb{R}^{n}$ can be bounded as $$\begin{aligned}\left\Vert Bx\right\Vert _{\max} & =\underset{i\in\left[N\right]}{\max}\frac{\left\Vert B^{\left[i\right]}x\right\Vert _{p_{i}}}{w_{i}}\\
& \leq\left\{ \begin{array}{cc}
n^{\left(p_{\min}^{-1}-\frac{1}{2}\right)}w_{\min}^{-1}\left\Vert Bx\right\Vert _{2} & p_{\min}<2\\
\frac{1}{w_{\min}}\left\Vert Bx\right\Vert _{2} & p_{\min}\geq2
\end{array}\right.,
\end{aligned}
\label{eq:BlockNormBound}$$ and the lemma follows by taking the supremum over all unit vectors $x$.
Convergence Via Nested Sets
---------------------------
We now begin the convergence analysis for the block-based update law in Algorithm 1 where agents are asynchronously optimizing. In order for this system to converge using the communications described in the previous section, we construct a sequence of sets, $\left\{ X\left(s\right)\right\} _{s\in\mathbb{N}}$, based on work in [@Bertsekas1989] and [@bertsekas1989parallell]. Below we use the notation $\hat{x}_{A}\coloneqq\arg\min_{x\in X}f_{A}\left(x\right)$ to specify the minimizer of the regularized cost function $f_{A}$. We state the conditions imposed upon these sets as an assumption, and this assumption will be shown below to be satisfied using the heterogeneous regularization applied by $A$.
The sets $\left\{ X\left(s\right)\right\} _{s\in\mathbb{N}}$ satisfy:
1. $\cdots\subset X\left(s+1\right)\subset X\left(s\right)\subset\cdots\subset X$
2. $\underset{s\rightarrow\infty}{\lim}X\left(s\right)=\left\{ \hat{x}_{A}\right\} $
3. $X_{i}\left(s\right)\subset X_{i}$ for all $i\in\left[N\right]$ and $s\in\mathbb{N}$ such that $X\left(s\right)=X_{1}\left(s\right)\times\cdots\times X_{N}\left(s\right)$
4. $\theta_{i}\left(y\right)\in X_{i}\left(s+1\right)$, where $\theta_{i}\left(y\right)\coloneqq y_{i}-\gamma\nabla_{i}f_{A}\left(y\right)$ for all $y\in X\left(s\right)$ and $i\in\left[N\right]$.$\triangle$
Assumptions 4.1 and 4.2 together show that these sets are nested as they converge to the minimum $\hat{x}_{A}$. Assumption 4.3 allows for the blocks to be updated independently by the agents, and Assumption 4.4 ensures that state updates always progress down the chain of nested sets such that only forward progress toward $\hat{x}_{A}$ is made. It is shown in [@Bertsekas1989] and [@bertsekas1989parallell] that the existence of such a sequence of sets implies asymptotic convergence of the asynchronous update law in Algorithm 1, and we therefore use this construction to show asymptotic convergence in this paper. Defining the Lipschitz constant of $\nabla_{i}f_{A}$ as $L_{i}$, we further define $L_{\max}\coloneqq\underset{i\in\left[N\right]}{\max}\ L_{i}$, and then define the constant $$q=\max\left\{ \underset{i\in\left[N\right]}{\max}\left|1-\gamma\alpha_{i}\right|,\underset{i\in\left[N\right]}{\max}\left|1-\gamma L_{i}\right|\right\} .$$ Letting $\gamma\in\left(0,\frac{2}{L_{\max}}\right)$ and $\alpha\in\left(0,L_{\max}\right)$, we find $q\in\left(0,1\right)$; a proof for this can be seen in [@polyak1987]. We then proceed to define $D_{o}$ as $$D_{o}\coloneqq\underset{i\in\left[N\right]}{\max}\left\Vert x^{i}\left(0\right)-\hat{x}_{A}\right\Vert _{\max},$$ which is the worst-performing block onboard any agent with respect to distance to $\hat{x}_{A}$ at timestep $0$. We then define the sequence of sets $\left\{ X\left(s\right)\right\} _{s\in\mathbb{N}}$ as $$\label{eq:Xsets}
X\left(s\right)=\left\{ y\in X:\left\Vert y-\hat{x}_{A}\right\Vert _{\max}\leq q^{s}D_{o}\right\},$$ and this construction is shown in the following theorem to satisfy Assumption 4, thereby ensuring asymptotic convergence of Algorithm 1.
The collection of sets $\left\{ X\left(s\right)\right\} _{s\in\mathbb{N}}$ as defined in Equation satisfies Assumption 4.
For Assumption 4.1 we see that $$X\left(s+1\right)=\left\{ y\in X:\left\Vert y-\hat{x}_{A}\right\Vert _{\max}\leq q^{s+1}D_{o}\right\} .$$ Since $q\in\left(0,1\right)$, we have $q^{s+1}<q^{s}$, which results in $\left\Vert y-\hat{x}_{A}\right\Vert_{\max} \leq q^{s+1}D_{o}<q^{s}D_{o}$. Then $y\in{X}\left(s+1\right)$ implies $y\in{X}\left(s\right)$ and $X\left(s+1\right)\subset X\left(s\right)\subset X$, as desired.
From Assumption 4.2 we find $$\begin{aligned}\underset{s\rightarrow\infty}{\lim}X\left(s\right) & =\underset{s\rightarrow\infty}{\lim}\left\{ y\in X:\left\Vert y-\hat{x}_{A}\right\Vert _{\max}\leq q^{s}D_{o}\right\} \\
& =\left\{ y\in X:\left\Vert y-\hat{x}_{A}\right\Vert _{\max}\leq0\right\} \\
& =\left\{ \hat{x}_{A}\right\} ,
\end{aligned}$$ and Assumption 4.2 is therefore satisfied. The structure of the weighted block-maximum norm then allows us to see that $\left\Vert y-\hat{x}_{A}\right\Vert _{\max}\leq q^{s}D_{o}$ if and only if $\frac{1}{w_{i}}\left\Vert y_{i}-\hat{x}_{A,i}\right\Vert _{p_{i}}\leq q^{s}D_{o}$ for all $i\in\left[N\right]$. It then follows that $$X_{i}\left(s\right)=\left\{ y_{i}\in X_{i}:\frac{1}{w_{i}}\left\Vert y_{i}-\hat{x}_{A,i}\right\Vert _{p_{i}}\leq q^{s}D_{o}\right\} ,$$ which shows $X\left(s\right)=X_{1}\left(s\right)\times\cdots\times X_{N}\left(s\right)$, thus satisfying Assumption 4.3.
In order to show Assumption 4.4 is satisfied we recall the following exact expansion of $\nabla f_{A}$:
$$\begin{aligned} \nabla f_{A}\left(y\right)-\nabla f_{A}\left(\hat{x}_{A}\right)
& \ =\int_{0}^{1}\nabla^{2}f_{A}\left(\hat{x}_{A}+\tau\left(y-\hat{x}_{A}\right)\right)\left(y-\hat{x}_{A}\right)d\tau\\
& \ =\left(\int_{0}^{1}\nabla^{2}f_{A}\left(\hat{x}_{A}+\tau\left(y-\hat{x}_{A}\right)\right)d\tau\right)\cdot\left(y-\hat{x}_{A}\right)\\
& \ \eqqcolon H\left(y\right)\left(y-\hat{x}_{A}\right),
\end{aligned}
\label{eq:heshian}$$
where we have defined $$H\left(y\right)=\int_{0}^{1}\nabla^{2}f_{A}\left(\hat{x}_{A}+\tau\left(y-\hat{x}_{A}\right)\right)d\tau.$$
We then see that for $y\in X\left(s\right)$,
$$\begin{aligned} \frac{\left\Vert \theta_{i}\left(y\right)-\hat{x}_{A,i}\right\Vert _{p_{i}}}{w_{i}}
& =\frac{1}{w_{i}}\left\Vert y_{i}-\gamma\nabla_{i}f_{A}\left(y\right)-\hat{x}_{A,i}+\gamma\nabla_{i}f_{A}\left(\hat{x}_{A}\right)\right\Vert _{p_{i}}\\
& \leq\underset{i\in\left[N\right]}{\max}\frac{1}{w_{i}}\left\Vert y_{i}-\gamma\nabla_{i}f_{A}\left(y\right)-\hat{x}_{A,i}+\gamma\nabla_{i}f_{A}\left(\hat{x}_{A}\right)\right\Vert _{p_{i}}\\
& =\left\Vert y-\hat{x}_{A}-\gamma\nabla f_{A}\left(y\right)+\gamma\nabla f_{A}\left(\hat{x}_{A}\right)\right\Vert _{\max}\\
& =\left\Vert y-\hat{x}_{A}-\gamma\left(\nabla f_{A}\left(y\right)-\nabla f_{A}\left(\hat{x}_{A}\right)\right)\right\Vert _{\max}\\
& =\left\Vert y-\hat{x}_{A}-\gamma H\left(y\right)\left(y-\hat{x}_{A}\right)\right\Vert _{\max}\\
& \leq\left\Vert I-\gamma H\left(y\right)\right\Vert _{\max}\left\Vert {y-\hat{x}_{A}}\right\Vert _{\max}\\
& \leq\left\{ \begin{array}{cc}
\frac{n^{\left(p_{\min}-\frac{1}{2}\right)}}{w_{\min}}\left\Vert I\!\!-\!\gamma H\!\left(y\right)\right\Vert _{2}\left\Vert {y-\hat{x}_{A}}\right\Vert _{\max} & p_{\min}<2\\[5pt]
\frac{1}{w_{\min}}\left\Vert I-\gamma H\left(y\right)\right\Vert _{2}\left\Vert{y-\hat{x}_{A}}\right\Vert _{\max} & p_{\min}\geq2
\end{array}\right.,
\end{aligned}$$
where we have used Equation in the fourth equality and Lemma 1 in the third inequality. We then define the vector $\nabla f_{A}=\left(\nabla_{1}f_{A},\ldots,\nabla_{N}f_{A}\right)^{T}$ which has a Lipschitz constant of $M=\sqrt{\sum_{i=1}^{N}L_{i}^{2}}$. It then follows from the definition of $f_{A}$ that $A\preceq H\left(\cdot\right)\preceq MI$, which implies that the eigenvalues of $H\left(\cdot\right)$ are bounded below by the smallest diagonal entry of $A$ and above by $M$. Since $H\left(y\right)$ is a symmetric matrix it follows that
$$\begin{aligned} \left\Vert I-\gamma H\left(y\right)\right\Vert _{2}
& =\max\left\{ \left|\lambda_{\min}\left(I-\gamma H\left(y\right)\right)\right|,\left|\lambda_{\max}\left(I-\gamma H\left(y\right)\right)\right|\right\} \\
& =\max\left\{ \underset{i\in\left[N\right]}{\max}\left|1-\gamma\alpha_{i}\right|,\underset{i\in\left[N\right]}{\max}\left|1-\gamma L_{i}\right|\right\} \\
& =q,
\end{aligned}$$
where $\lambda_{\min}\left(\cdot\right)$ and $\lambda_{\max}\left(\cdot\right)$ are the minimum and maximum eigenvalues of a matrix, respectively. Using the hypothesis that $y\in{X}\left(s\right)$, we find $$\begin{aligned} \frac{\left\Vert \theta_{i}\left(y\right)-\hat{x}_{A,i}\right\Vert _{p_{i}}}{w_{i}}
& \leq\left\{ \begin{array}{cc}
n^{\left(p_{\min}^{-1}-\frac{1}{2}\right)}w_{\min}^{-1}q\left\Vert y-\hat{x}_{A}\right\Vert _{\max} & p_{\min}<2\\
\frac{1}{w_{\min}}q\left\Vert y-\hat{x}_{A}\right\Vert _{\max} & p_{\min}\geq2
\end{array}\right.\\
& \leq\left\{ \begin{array}{cc}
n^{\left(p_{\min}^{-1}-\frac{1}{2}\right)}w_{\min}^{-1}q^{s+1}D_{o} & p_{\min}<2\\
\frac{1}{w_{\min}}q^{s+1}D_{o} & p_{\min}\geq2
\end{array}\right.\\
& \leq\left\{ \begin{array}{c}
q^{s+1}D_{o}\\
q^{s+1}D_{o}
\end{array}\right.,
\end{aligned}$$ where the bottom case follows from $w_{\min}\geq {1}$ and the top case follows from $w_{\min}\geq {1}$ and $p_{\min}^{-1}-\frac{1}{2}<1$. Then $\theta_{i}\left(y\right)\in X_{i}\left(s+1\right)$ and Assumption 4.4 is satisfied.
As noted above, the fact that the construction in Equation satisfies Assumption 4 implies asymptotic convergence of Algorithm 1 for all $i\in\left[N\right]$ from [@Bertsekas1989] and [@bertsekas1989parallell]. With this in mind, we next derive a rate of convergence for Algorithm 1.
Convergence Rate
----------------
The structure of the sets $\left\{ X\left(s\right)\right\} _{s\in\mathbb{N}}$ allows us to determine a convergence rate. However, to do so we must first define the notion of a *communication cycle*. Starting at time $k=0$, one cycle occurs when all agents have calculated a state update and this updated state has been sent to and received by each other agent. It is only then that each agents’ copy of the ensemble state is moved from $X\left(0\right)$ to $X\left(1\right)$. Once another cycle is completed the ensemble state is moved from $X\left(1\right)$ to $X\left(2\right)$. This process repeats indefinitely, and coupled with Assumption 4, means the convergence rate is geometric in the number of cycles completed, which we show now.
Let Assumptions 1-4 hold and let $\gamma\in\left(0,\frac{2}{L_{\max}}\right)$. At time $k$, if $c\left(k\right)$ cycles have been completed, then $$\left\Vert x^{i}\left(k\right)-\hat{x}_{A}\right\Vert _{\max}\leq q^{c\left(k\right)}D_{o}$$ for all $i\in\left[N\right]$.
From the definition of $D_{o}$, for all $i\in\left[N\right]$ we have $x^{i}\left(0\right)\in X\left(0\right)$. If agent $i$ computes a state update, then $\theta_{i}\left(x^{i}\left(0\right)\right)\in X_{i}\left(1\right)$ and after one cycle is completed, say at time $k$, we have $x^{i}\left(k\right)\in X\left(1\right)$ for all $i$. Iterating this process, after $c\left(\bar{k}\right)$ cycles have been completed by some time $\bar{k}$, $x^{i}\left(\bar{k}\right)\in X\left(c\left(\bar{k}\right)\right)$. The result follows by expanding the definition of $\{X\left(s\right)\}_{s\in\mathbb{N}}$.
Theorem 3 can be used by a network operator to bound agents’ convergence by simply observing them and without specifying when or how often agents should generate or share information. Having shown convergence of Algorithm 1, we next demonstrate its performance in practice.
Simulation
==========
In this section we present a problem to be solved using Algorithm 1. The simulation uses a network consisting of 8 nodes and 9 edges, where we define the set $\varepsilon\coloneqq\left[9\right]$ as the set of indices of the edges. There are $N=8$ agents that are users of this network and they are each tasked with routing a flow between two nodes. The network itself is shown in Figure \[fig:network\]; we emphasize that the nodes in the network are not the agents themselves, but instead are simply source/destination pairs for users to route flows between. The starting and ending nodes as well as the edges traversed for each agents’ flow are listed in Table \[tab:Edges-Traversed-by\].
![The network through which eight agents must route a flow between two nodes[]{data-label="fig:network"}](network.pdf){width="2.6in"}
Agent Number Start Node$\rightarrow$End Node Edges Traversed
-------------- --------------------------------- ---------------------------------
1 $1\rightarrow7$ $e_{1},e_{3},e_{6}$
2 $2\rightarrow8$ $e_{4},e_{7},e_{8}$
3 $3\rightarrow4$ $e_{2},e_{4},e_{7},e_{5}$
4 $5\rightarrow6$ $e_{3},e_{4},e_{7}$
5 $1\rightarrow4$ $e_{1},e_{3},e_{6},e_{7},e_{5}$
6 $3\rightarrow8$ $e_{2},e_{4},e_{9}$
7 $4\rightarrow5$ $e_{5},e_{8},e_{9},e_{6}$
8 $6\rightarrow2$ $e_{7},e_{4}$
: Edges traversed by each agent’s flow \[tab:Edges-Traversed-by\]
The cost function of agent $i$ is $f_{i}\left(x_{i}\right)=-100\log\left(1+x_{i}\right)$, and the coupling cost is $c\left(x\right)=\frac{1}{20}x^{T}C^{T}Cx$, where the network connection matrix is defined as $$C_{k,i}=\left\{ \begin{array}{cc}
1 & \textrm{if flow }i\textrm{ traverses edge }k\\
0 & \textrm{otherwise}
\end{array}\right..$$ This problem was then implemented such that agent $i$ had its own regularization parameter $\alpha_{i}>0$, normalization constant $w_{i}\geq 1$, and $p_{i}$ norm with $p_{i}\in\left[ 1,\infty\right]$. In particular, these parameters were chosen using $w = [12,8,6,7,6,10,9,10]$ and $p = [\infty,20,3,90,6,12,2,9]$, where $w_{i}$ is the $i^{th}$ element in $w$ and $p_{i}$ is defined analogously. All agents’ behaviors were randomized to give each agent a $10\%$ chance of computing an update at any timestep and to give each pair of agents a $10\%$ chance of communicating at each timestep. Three total simulation runs were executed using the three different choices of $A$ listed to demonstrate its effects upon convergence, with
$$\begin{aligned}
A_1 & = \textnormal{diag}[3\!\!\times \!\!10^{-4},1\!\!\times \!\!10^{-4},9\!\!\times \!\!10^{-4},2\!\!\times \!\!10^{-4},0.001,0.001,5\!\! \times \!\! 10^{-4},4\!\! \times \!\! 10^{-4}] \\
A_2 & = \textnormal{diag}[0.01,0.01,0.003,0.005,0.002,0.01,0.005,0.002] \\
A_3 & = \textnormal{diag}[0.08,0.1,0.1,0.09,0.009,0.1,0.08,0.04].
\end{aligned}$$
A plot of error versus iteration count for a run with $A_1$ is shown in Figure \[fig:error0.001\], which shows that the regularization provided by $A_1$ can provide robustness to asynchrony without significantly impacting the final point obtained by Algorithm 1. In addition, close convergence to a minimizer is attained in a reasonable number of iterations, even when agents infrequently generate and share information.
![Regularized and unregularized error for agent 1 where $\left\Vert{A_{1}}\right\Vert =0.001$. Here, the regularized error is shown as a line and the unregularized error is shown by the circles. As expected, both errors converge to small final values, indicating close convergence to both $\hat x$ and $\hat{x}_{A}$ when $\left\Vert{A}\right\Vert$ is small.[]{data-label="fig:error0.001"}](Reg3.eps){width="3.3in"}
To demonstrate the impact of larger regularizations, a simulation was run with $A_{2}$, and an error plot for this run is shown in Figure \[fig:error0.01\].
![Regularized and unregularized error for agent 1 where $\left\Vert{A_{2}}\right\Vert =0.01$. The regularized error is shown as a line and the unregularized error is shown by the circles. Because $\left\Vert{A}\right\Vert$ is larger, the agents converge to a minimum faster, though there is a larger discrepancy between $\hat x$ and $\hat{x}_{A}$, as evidenced by the asymptotic disagreement between the two curves shown here.[]{data-label="fig:error0.01"}](Reg2.eps){width="3.3in"}
To further illustrate the effects of regularizing, a third and final simulation was run with $A_3$, and a plot of error in this case is shown in Figure \[fig:error0.1\].
![Regularized and unregularized error for agent 1 where $\left\Vert{A_{3}}\right\Vert =0.1$. The regularized error is shown as a line and the unregularized error is shown by the circles. As expected, this run converges faster (because its value of $q$ smaller), but with the largest error in the final solution obtained, indicating that a significant acceleration in convergence comes in exchange for a less accurate solution.[]{data-label="fig:error0.1"}](Reg1.eps){width="3.3in"}
To enable numerical comparisons of these convergence results, final error values for all three runs are shown in Table \[tab:Errors\], where we see that larger values of $\Vert{A}\Vert$ do indeed lead to larger errors.
$\left\Vert{A}\right\Vert$ Final Regularized Error Final Unregularized Error
---------------------------- ------------------------- ---------------------------
0.001 $2.2575\times 10^{-8}$ $2.9558\times 10^{-4}$
0.01 $2.1837\times 10^{-8}$ $8.4922\times 10^{-4}$
0.1 $7.9827\times 10^{-10}$ $0.0848$
: Errors for agent 1\[tab:Errors\]
Conclusion
==========
This work presented an asynchronous optimization framework which allows for arbitrarily delayed communications and computations. Future extensions to this work include incorporating constraints in order to accommodate broader classes of problems [@Hale2014], and using time-varying regularizations to always reach exact solutions. Future applications include use in robotic swarms where communications are unreliable and asynchrony is unavoidable.
[^1]: $^{\star}$The authors are with the Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL, USA. Email: `smhochhaus@ufl.edu, matthewhale@ufl.edu`.
[^2]: $^\dagger$ Corresponding author.
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
A new method of cooling positronium down is proposed to realize Bose-Einstein condensation of positronium. We perform detail studies about three processes (1) thermalization processes between positronium and silica walls of a cavity, (2) Ps-Ps scatterings and (3) Laser cooling. The thermalization process is shown to be not sufficient for BEC. Ps-Ps collision is also shown to make a big effect on the cooling performance. We combine both methods and establish an efficient cooling for BEC. We also propose a new optical laser system for the cooling.\
\
\
PACS numbers: 36.10.Dr, 67.85.Jk
address:
- '$^1$Department of Physics, Graduate School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan'
- '$^2$International Center for Elementary Particle Physics (ICEPP), The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan'
- '$^*$Corresponding authors'
author:
- |
K. Shu$^{1,*}$, X. Fan$^{1,*}$, T. Yamazaki$^2$, T. Namba$^2$, S. Asai$^1$,\
K. Yoshioka$^1$, M. Kuwata-Gonokami$^1$
bibliography:
- 'ref.bib'
title: 'Study on Cooling of Positronium for Bose-Einstein Condensation'
---
[*Keywords:* positronium, Bose-Einstein condensation, laser cooling]{}
Introduction
============
Positronium (Ps), a bound state of an electron and a positron, is the lightest atom, whose mass $m_{\mathrm{Ps}}=$1022keV. The two ground states of Ps, the triplet state ($1^{3}S_{1}$) and the singlet state ($1^{1}S_{0}$), are known as *ortho*-positronium (*o*-Ps) and *para*-positronium (*p*-Ps) respectively. *Ortho*-positronium decays slowly into three photons with a lifetime of 142ns[@life]. On the other hand, *p*-Ps quickly decays into two photons. The two states can be mixed with a weak magnetic field. Ps is therefore a good source of 511keV $\gamma$ ray. Furthermore Ps is a good tool to probe the gravity of anti-particle, since Ps is a purely particle and anti-particle system.
Bose-Einstein condensation (BEC) is one of the most interesting phenomena of the quantum physics. Behavior of quantum particles can be magnified into a macroscopic level to be directly observed in BEC state, and BEC provides various applications. The first observation of BEC of weakly interacting bosonic atomic gas was found using $^{87} \mathrm{Rb}$ gas in 1995[@RbBEC] and opened a new era of studying the macroscopic behavior of quantum gas. Ps BEC is very attractive since it would provide a 511keV $\gamma$ ray laser[@Vanyashin1994; @gammarayLaser1; @gammarayLaser2] and macroscopic behavior of anti-particle gravity could be observed.
The de Broglie wave length, $\lambda_D$, and the density, $n$, play an important role for BEC. The critical temperature, $T_\mathrm{C}$, is determined by the following formula[@pethick2002bose]: $$n \lambda_D^3 = n \left(\frac{2 \pi \hbar ^2}{m k_B T_C}\right)^{3/2} = 2.612,$$
$\hbar$: The reduced Planck constant, $m$: A mass of an atom, $k_B$: The Boltzmann constant.
\
Figure \[fg:BECTc\] shows a relation between $T_\mathrm{C}$ and $n$, for $^{87}\mathrm{Rb}$, $^1$H and Ps. Since Ps is very light, BEC can be achieved at a few hundreds millikelvins for $n \sim 10^{15}\,\mathrm{/cm^3}$. This $T_\mathrm{C}$ is much higher than that for $^{87}\mathrm{Rb}$ and $^1$H.
![Relation between density and critical temperature of BEC transition. Left side of each line is a region where the atoms are in BEC phase.[]{data-label="fg:BECTc"}](./Figure/BECTc.pdf){width="7.5cm"}
It is essential to rapidly cool dense Ps atoms down in order to achieve a high phase-space density. The conventional way to cool Ps down is creating Ps atoms in a cold material with voids into which Ps can escape. The most recent experiment could cool them down to 150K by using cold silicon nanochannels (less than 10nm in diameters)[@Mariazzi2010] in which Ps atoms were created and thermalized via many collisions with the channel’s walls. This minimum temperature was limited by the sizes of voids. If a material with smaller voids is used, momentum exchanges between Ps and the walls will be more efficient. However, a quantum confinement effect which is remarkable due to the large de Broglie wave length of Ps prevents Ps atoms from being cold inside a too small void[@Mariazzi2008]. Cooling with a laser has also been proposed to achieve less than 10K of Ps atoms[@Liang1988; @Kumita2002; @Crivelli2014]. As for density, $2\times 10^{15}\,\mathrm{/cm^3}$ was achieved recently[@Cassidy2007] by a positron accumulator with $\mathrm{^{22}Na}$ radioisotope. Much effort is still necessary for both cooling and accumulation to achieve the phase-space density for BEC phase transition.
In order to realize BEC of Ps atoms, using a cold silicon cavity whose dimension was around 1$\mathrm{\mu m}$ was proposed for trapping and cooling down Ps by collisions between the walls and Ps[@Platzman1994]. However, we will later show that the cooling with the large cold cavity is insufficient. In this article, we propose a new cooling method using both the cold cavity and the laser cooling. The efficiency of the cooling is estimated taking into account effects of dense Ps atoms.
Setup {#sc:Setup}
=====
A conceptional view of our experimental setup is shown in Fig. \[fg:Setup\].
![The schematic diagram of the experimental setup. The two laser beams which are perpendicular to the sheet are not written.[]{data-label="fg:Setup"}](./Figure/Setup.pdf){width="13cm"}
Positrons are stored in a small trapping cavity made by silica. A 5keV positron beam of $10^7$ positrons per bunch will be used. This number of positrons is already possible elsewhere[@Cassidy2006]. The positrons are focused and injected into the cavity which is cooled at 1K. The cavity has an internal void whose dimension is assumed to be a cube of $100\,\mathrm{nm} \times 100\,\mathrm{nm} \times 100\,\mathrm{nm}$. About $4\times 10^3$ fully spin-polarized Ps atoms are assumed to be left inside the void, which in turn means $n=4\times 10^{18}\,\mathrm{/cm^3}$, while the focusing technique and overall efficiency of the conversion from the positrons to Ps have to be studied in future. Ps atoms have an initial kinetic energy of around 0.8eV[@Nagashima1995] and are confined inside the cavity. The cavity is irradiated by UV laser beams which are configured as optical molasses: by three orthogonal pairs of counter-propagating beams. The laser photons can go through into the cavity because silica is transparent to this light. The laser system is described in section \[sc:LaserImplementaion\].
In this setup, Ps atoms have the following interactions:
- Thermalization by interactions with silica walls of the cold cavity,
- Ps-Ps two-body interactions,
- Cooling by optical transitions and momentum recoils by photons.
As discussed in the following sections, the cooling through the thermalization process is efficient for Ps atoms with high energy while the opposite for laser cooling. These two cooling processes are complementary, so we propose a new cooling scheme in two stages: initially by Ps-silica interactions and then by laser fields after the former becomes inefficient. We evaluate cooling efficiency of each process and see whether it is enough to achieve BEC transition.
Thermalization in the silica cavity {#sc:Thermalization}
===================================
The thermalization process is evaluated by using the classical interaction model[@Nagashima1995]. This model presumes that thermalization will proceed through classical elastic collisions between Ps and grains of a surrounding material. According to the model, an average kinetic energy of Ps, $E$, evolves as follows: $$\frac{\mathrm{d}E}{\mathrm{d}t} = -\frac{2}{LM}\sqrt{2m_{\mathrm{Ps}}E}\left(E-\frac{3}{2}k_BT\right).
\label{eq:Thermalization}$$ Here $M$ is an effective mass of surrounding grains, $L$ is the mean free path of the collisions with grains and $T$ is temperature of a surrounding material. This model can well describe the thermalization process which were measured by various techniques[@Chang1987; @Takada2000; @Shibuya2013].
Experimental results are used in order to determine $M$. Various results of the measurements[@Nagashima1995; @Shibuya2013; @Chang1987; @Nagashima1998] are shown in Fig. \[fg:EffectiveM\].
$M$ is shown as a function of Ps kinetic energy in which it was measured, because the effective mass could depend on kinetic energy of interacting Ps as suggested[@Nagashima1998]. This is because the number of phonon modes which Ps can excite at collisions decreases as kinetic energy of Ps does. We estimate energy dependence of $M$ in a.m.u as $M=21+308\,\exp{\left(-\frac{E}{0.16\,\mathrm{eV}}\right)}$ in order to reproduce the experimental results as shown in Fig. \[fg:EffectiveM\] by the solid line which is named as “Best fit”. The uncertainty is large because precision of the measurements is still limited. The range of the predictions is also shown in Fig.\[fg:EffectiveM\] as “Fast” and “Slow”.
In addition to the interaction with the silica walls, the Ps-Ps interactions must be taken into account in the case of high density. The main process is the elastic $s$-wave scattering of spin-polarized Ps atoms. This process leads Ps atoms to quasi thermal equilibrium: the energy distribution of atoms becomes the Maxwell-Boltzmann distribution with an approximation of classical scattering. The total cross section ($\sigma$) of the scattering is given by $\sigma=4\pi a^2$ where $a$ is a scattering length, $a=0.16\,\mathrm{nm}$[@Oda2001; @Ivanov2002]. The mean free time ($\tau$) of scatterings depends on a number density of Ps ($n$), an average velocity ($\bar{v}$), and $\sigma$ as $\tau = 1/n \sigma \bar{v}$. This interval with $n=10^{18}\,\mathrm{/cm^3}$ is less than 100ps even at 10K. The quasi thermalization among Ps atoms is quite fast compared to the thermalization process between Ps atoms and the silica walls.
The interactions with the silica wall and Ps-Ps elastic scatterings are simulated by a Monte Carlo method, which is explained in an appendix in detail. Time evolutions of temperature are shown in Fig. \[fg:Thermalization\].
As for “Best fit” estimation of $M$, Ps can be cooled to 300K in 100ns from the initial energy of 0.8eV. The further cooling is so slow that it takes 500ns to reach 100K even though the cavity is at 1K. Our estimation is consistent with previous studies in which Ps would not thermalize at low temperatures[@Kiefl1983; @Saito1999; @SurkoSaito2001].
However the thermalization process strongly depends on the effective mass of a silica grain as shown in Fig. \[fg:Thermalization\]. It is necessary to perform additional measurements precisely in order to determine $M$.
There is another interaction model with silica[@Mariazzi2008; @Morandi2014]. In this model, Ps atoms emit or absorb acoustic phonons on the cavity walls through deformation potential scatterings. The interaction rates can be deduced by the first-order perturbation theory. We also evaluate the thermalization process in this model by fitting to reproduce the experimental results[@Chang1987; @Takada2000; @Shibuya2013]. There is no observable difference between the two models for Ps atoms of higher than around 300K. At lower energy, the thermalization process in this phonon model is more insufficient than in the classical model because of less stimulated emission at low temperatures in the phonon model. In our cooling scheme, the thermalization process is crucial only at the initial stage of cooling to around 300K. The difference to the final result is therefore negligible.
Laser Cooling {#sc:LaserCooling}
=============
The evaluation of laser cooling
-------------------------------
It is necessary to accelerate the cooling down from a few hundreds of Kelvins. It is efficient to use $1s\leftrightarrow 2p$ transition for laser cooling of Ps because of a large recoil momentum by photons. The difference between these two energy levels corresponds to 243nm UV light.
Optical transitions induced by the laser can be modeled with the rate equation approach[@Iijima2001] because the time scale of cooling down, more than 100ns, is much longer than the time constant 3.2ns of the spontaneous emission from $2p$ to $1s$. The stimulated transition rate, which is called as “Einstein $B$ coefficient”, can be calculated by the flux of photons and the cross section of the interaction between Ps and resonant photons. This coefficient can be calculated as follows: $$B(t, \vec{x}, \vec{v}) = \int \mathrm{d}\omega\, \frac{I(t,\vec{x},\omega)}{\hbar \omega} \cdot \frac{4}{3}\pi^2 \alpha \omega_0 |X_{12}|^2 \cdot \frac{1}{2\pi} \frac{\Gamma/2}{(\omega(1-\vec{\hat{k}}\cdot \vec{v}/c)-\omega_0)^2+(\Gamma/2)^2},
\label{eq:B}$$
$t$: time, $\vec{x}$: Position of Ps, $\vec{v}$: Velocity of Ps, $\vec{\hat{k}}$: Direction of laser photons,
$I(t,\vec{x},\omega)$: Intensity per frequency of the laser,
$X_{12}$: The matrix element of $1s$-$2p$ transition,
$\omega$: The frequency of interacting photons in the laser
$\omega_0$: The resonant frequency of $1s$-$2p$ transition.
\
Gaussian profiles are assumed for frequency/timing/space domains of the intensity. $\Gamma=313~\mathrm{MHz}$ is the natural line width so the last term in the equation (\[eq:B\]) represents the Breit-Wigner line shape including the first-order Doppler effect. The internal state evolves according to this stimulated transition rate and the spontaneous emission rate from $2p$ state. A Ps is recoiled by $\hbar\omega/m_{\mathrm{Ps}}c\simeq 1.5\times 10^3~\mathrm{m/s}$ when it emits or absorbs a photon. The direction of the recoil is random for spontaneous emission while for stimulated absorption/emission it is the same as photons in the laser. An important feature is that the annihilation to gamma-rays from $2p$ state is very slow (10$\mathrm{s^{-1}}$) compared to that from $1s$ state. This means that the maximum excitation effectively increases the lifetime of Ps by twice to 284ns.
Table \[tb:LaserParameters\] shows a summary of laser parameters.
Parameter Name Value
------------------------------ ------------------------
Pulse energy 40$\mathrm{\mu J}$
Center frequency 1.23PHz$-\Delta(t)$
Frequency detune $\Delta(t)$ $\Delta$(0ns)=300GHz
$\Delta$(300ns)=240GHz
Bandwidth(2$\sigma$) 140GHz
Time duration(2$\sigma$) 300ns
Beam size(2$\sigma$) 200$\mathrm{\mu m}$
: The summary of laser parameters of the 243nm laser system.
\[tb:LaserParameters\]
The laser is a pulsed laser and its energy is 40$\mu$J, which is divided into the six beams and focused into 200$\mu$m at the cavity. The timing of the peak intensity is delayed by 200ns from the creation of Ps atoms as shown in the upper part of Fig. \[fg:LaserCooledTemperature\]. The time duration is 300ns in order to cover the long duration necessary for cooling. The center frequency of the laser field is detuned from 1.23PHz which corresponds to 243nm wavelength. This detune, $\Delta(t)$, is 300GHz at the beginning and then up-chirped to 240GHz in order to compensate the decrease of Ps velocities. The bandwidth around the center frequency is 140GHz in order to excite Ps atoms with a wide range of velocities. The required frequency chirp and bandwidth are quite large compared to standard systems for cooling other atoms. It is because of the large Doppler shift of Ps due to its light mass compared to any other atoms. The laser system with these features is a challenge.
The cooling effect by the laser is evaluated by another Monte Carlo simulation, which includes laser effects with the thermalization and the Ps-Ps two-body scatterings. The details are also given in the appendix. “Best fit” estimation of $M$ in Fig. \[fg:EffectiveM\] is used for the thermalization process. Distributions of Ps velocity at different times are shown in Fig. \[fg:VelocityDistribution\].
The distributions quickly become Maxwell-Boltzmann distributions by Ps-Ps scatterings. This means that Ps atoms are always in quasi thermal equilibrium and have well-defined temperature which can be calculated by a width of a velocity distribution. The number of remained atoms is increased with the laser because of the longer lifetime as described before. The time evolution of temperature and $T_\mathrm{C}$ are shown in the lower part of Fig. \[fg:LaserCooledTemperature\].
![The upper part: Irradiated laser intensity in an arbitrary unit versus time. The lower part: Time evolutions of temperature and $T_\mathrm{C}$. $T_\mathrm{C}$ decreases as the density of $1s$ Ps does due to the annihilation.[]{data-label="fg:LaserCooledTemperature"}](./Figure/LaserCooledTemperature_fixed.pdf){width="10cm"}
$T_\mathrm{C}$ is calculated from the density of Ps in $1s$ state. The cooling effect by the laser becomes dominant below several hundreds of Kelvins. After around 400ns, the temperature becomes less than $T_\mathrm{C}$. This means that the phase transition to BEC can be achieved by our cooling scheme. Figure \[fg:CondensateFraction\] shows condensate fractions over remained atoms, $R_C=1-\left( T/T_C \right)^{\frac{3}{2}}$[@pethick2002bose], which are calculated with an assumption that Ps atoms are non-interacting bosonic systems.
More than 30% of the remained atoms will be in the condensate with the laser system described above.
A gain of the chirp is demonstrated in Fig. \[fg:CondensateFraction\], in which $R_C$ are calculated with $\Delta(0\,\mathrm{ns})$ being changed to 270GHz or 240GHz while $\Delta(300\,\mathrm{ns})$ are fixed at 240GHz. $R_C$ without chirp is only around 0.1 and the time interval of condensation is shortened to less than 100ns. Even 30GHz chirp can increase the fraction by twice than that without chirp. Therefore, 30-60GHz chirp is enough for the efficient cooling.
Implementation of the laser system {#sc:LaserImplementaion}
----------------------------------
Though the specific parameters listed in Table \[tb:LaserParameters\] are technically challenging, it can be achievable using various techniques which already exist. The technically challenging points are the large and fast frequency chirp with the optical amplification for the long time duration of more than 100ns. The basic idea is to use the third harmonics generation of 729nm light, whose frequency and pulse shape are precisely controlled. Figure \[fg:laser\] shows a conceptual diagram of our designed laser system.
![The conceptual diagram of our designed laser system. The 729 nm single mode CW light is frequency-controlled at EOMs. Pulse shaping and amplification are conducted at the following injection seeded Ti:Sapphire laser. Ti:sapphire crystal is pumped by Q-switched 532nm pulsed laser. The light is converted to 243 nm at the last SHG and THG part. The pictures on the right side show the frequency information of the 729nm light: (a) before entering EOMs, (b) ejected from EOM 1 which generates up-shift to 20GHz, (c) ejected from EOM 2 which generates sidebands of about 20GHz range. The frequency shift and broadening are multiplied by three at the last THG and the desired parameters in Table \[tb:LaserParameters\] can be obtained.[]{data-label="fg:laser"}](./Figure/laser.pdf){width="15cm"}
A single mode 729nm CW laser is used as a master laser. The CW light is modulated at the electro-optic modulator 1, EOM 1, to generate a 20GHz sideband. This generated sideband is used as a center frequency of the frequency chirp. The advantage to use the sideband as a center frequency is that the chirp can be electrically controlled by changing modulation frequency applied to EOM 1. At EOM 2, 20GHz broad sidebands are generated around the first sideband. 1GHz modulation frequency to EOM with 20 sidebands generation is enough to cover all the Ps Doppler broadening. After the sideband generation, the seed light is injected into injection seeded Ti:sapphire laser. The gain of the laser can be controlled by adjusting the waveform of the pump laser and reflective indexes of an output coupler so the desired amplification along 300ns can be obtained. The pulse energy is about 10mJ. Now the amplified pulse is well controlled in both time domain and frequency domain, and enters the following Second Harmonics Generation, SHG and Third Harmonics Generation, THG, to achieve desired 243nm light. The 10mJ injected pulse energy is high enough to obtain 40$\mu$J energy. Furthermore, the 20GHz frequency shift and 20GHz broadening are also multiplied by three at THG, so the frequency parameters in Table \[tb:LaserParameters\] can be obtained. The 243nm light will be sent to Ps generation cavity and used as cooling light.
The repetition rate of the positron beam will be less than 10 Hz, so the assumed pulse energy is not so hard to achieve. Coincidence between the positron source and the laser system will also be easy by synchronising the positron system and the pulsed pump laser and EOMs. The most challenging part will be the large and fast frequency modulation at the two parts of EOM. In order to achieve this large frequency modulation, broadband traveling-wave type optical modulator will be used as a frequency shifter. Though this type of modulators have been mainly used in optical communication wavelength, modulation up to 100GHz also exists in 1064nm[@EOM]. We are developing this broadband device compatible with 729nm.
The Roadmap for Ps BEC
======================
There are three steps to achieve Ps BEC. At first, the cooling process with silica should be confirmed. We are now measuring the thermalization process precisely. The second step is to develop the laser system. Some studies are ongoing with basic technology already developed. The last step is to develop the focus system of the slow positron beam. The beam should be focused into 100nm while it can currently be focused only into 25$\mu$m[@PositronMicroBeam] by different technique.
The detection of the transition to BEC will also be possible by the same technique as the precise measurement of the Ps thermalization. Another method could be using a characteristic spectrum of annihilation gamma-rays from the condensate. It is also under study.
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to express sincere gratitude to Prof. Y. Nagashima, Prof. H. Saito, Dr. J. Omachi, and Mr. Y. Morita for helpful discussions. We warmly thank Dr. A. Ishida for his useful advice.
Appendix {#appendix .unnumbered}
========
Details of the two Monte Carlo simulations are given in this appendix. For both simulations, $10^4$ atoms are created initially with monochromatic (0.8eV) and isotropic velocities. A velocity and an internal state of each atom are traced at the same time as a brute-force method. Time evolution of those quantities are divided into short time steps. Random numbers are generated at each step to compare with rates of interactions. Durations of the steps, which are typically 0.1ps, are determined so that all possible interactions happen with low probability ($<1\%$). Results shown in the text are obtained by one execution without any averaging. Some notes are as following:
- Positions of the atoms are not traced because the trapping region of Ps, the 100nm cube, is narrow enough compared with the laser beam size.
- The number of simulated atoms ($10^4$) is more than the assumed initial number, $4\times 10^3$, in order to decrease a statistical uncertainty of the simulation, but the rate of the Ps-Ps scatterings is scaled to represent the assumed condition in section \[sc:Setup\].
Following interactions are coded in the simulations.
Thermalization through collisions with the silica cavity wall
: \
A change of an average Ps kinetic energy is calculated by differential Eq. (2) with a linear approximation. For the parameters, $L=100\,\mathrm{nm}$ assuming $L=\sqrt[3]{V}$[@SurkoSaito2001] where $V$ is a volume of the trapping cavity and $M$ is determined by the curves in Fig. \[fg:EffectiveM\]. A kinetic energy of each Ps atom is then evolved according to the result of the calculation for each time step whose duration is determined to suppress energy changes to be less than a percent for this interaction.
Ps-Ps two-body interactions
: \
Pairs of Ps atoms are made randomly and then momenta of final states are calculated as a result of the elastic $s$-wave scatterings for each pair. Rates of scatterings are calculated as $1/\tau(=n\sigma \bar{v})$, here $n$ is scaled to match the initial density ($4\times 10^{18}\,\mathrm{/cm^3}$).
The annihilations and the spontaneous de-excitation of Ps
: \
The annihilation of $1s$ Ps (142ns lifetime) is included, while the annihilation of $2p$ Ps is ignored because a time duration of the simulations, 600ns, is much shorter than its lifetime (100ms). After a Ps atom annihilates, the atom is not included in the simulation. The spontaneous de-excitation of $2p$ Ps into $1s$ Ps with 3.2ns time constant is also included.
The interactions between Ps and laser photons
: \
Rates of stimulated emissions/absorptions are calculated by Eq. (\[eq:B\]) for each Ps atom. A laser intensity is approximated as uniform by using the peak value because Ps atoms are assumed to be confined in the small region.
References {#references .unnumbered}
==========
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We consider a dark energy model with a relation between the equation of state parameter $w$ and the energy density parameter $\Omega_\phi$ derived from thawing scalar field models. Assuming the variation of the fine structure constant is caused by dark energy, we use the observational data of the variation of the fine structure constant to constrain the current value of $w_0$ and $\Omega_{\phi 0}$ for the dark energy model. At the $1\sigma$ level, the observational data excluded some areas around $w_0=-1$, which explains the positive detection of the variation of the fine structure constant at the $1\sigma$ level, but $\Lambda$CDM model is consistent with the data at the $2\sigma$ level.'
address: |
$^1$MOE Key Laboratory of Fundamental Quantities Measurement, School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China\
$^2$Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
author:
- 'Qing Gao$^1$[^1] and Yungui Gong$^{1,2}$[^2]'
title: Constraints on thawing scalar field models from fundamental constants
---
Introduction
============
The observations of type Ia supernovae in 1998 [@acc1; @acc2] found that the universe is experiencing an accelerating expansion. To explain the cosmic acceleration, an exotic energy with negative pressure which contributes approximately 70% of the energy density in the universe was introduced to the matter component. This exotic energy component is called dark energy. For a recent review of dark energy, please see Refs. . We usually use a minimally coupled scalar field with positive kinetic energy term to model the quintessence [@wetterich88; @peebles88; @quintessence], and a minimally coupled scalar field with negative kinetic energy term to model the phantom [@phantom]. For a scalar field with a nearly flat potential, approximate relations between the equation of state parameter $w=p/\rho$ and the energy density parameter $\Omega_\phi$ can be obtained [@Robert2008; @Robert; @Sourish; @Crittenden:2007yy]. By using this generic relation, more general dark energy models can be derived from known models with $\Omega_\phi$, and the derived dark energy models can be approximated by the Chevallier-Polarski-Linder (CPL) parametrization [@cpl1; @cpl2; @Gong:2013bn]. In this paper, we focus on the thawing scalar field models with nearly flat potentials with a starting value of $w$ that is close to, but not exactly equal to -1 at early times [@Caldwell:2005tm].
Fundamental constants like the fine structure constant $\alpha=e^2/\hbar c$ play important roles in physics. The variation of the fine structure constant $\alpha$ with time produce shifts in the molecular spectra [@Rodger]. So observations of molecular spectra in high redshift objects can track the value of the fundamental constant in the early universe. It is possible that the variation of the fine structure constant $\alpha$ is caused by the non-minimally coupling between dark energy and the electromagnetic field strength [@Bekenstein]. The measurements of the values of the fundamental constant throughout the history of the universe provide strong constraints on the property of dark energy [@Martins]. A lot of efforts have been made to measure the variation of the fundamental constants [@King2009; @Wendt; @Thompson; @Malec; @King2011; @Murphy2008; @Muller2011; @Julian; @Murphy:2003mi; @Murphy2004; @Chand2004; @Webb]. These observations can be used to test cosmological model [@zhangtj]. On the other hand, the variations of the fundamental constants may caused by the interaction of the vacuum with the matter [@Fritzsch:2012qc]. In particular, it is possible that the vacuum energy is time-varying with $w_\Lambda=-1$ due to the renormalization group running of the cosmological constant in the framework of quantum field theory in curved space-time [@Shapiro:1999zt; @Shapiro:2009dh; @Basilakos:2009wi; @Grande:2011xf].
In this paper, we use the observational data on the variation of the fine structure constant $\Delta \alpha/\alpha$ to constrain the property of dark energy. The paper is organized as follows. In section 2, we review the generic relationship between $w$ and $\Omega_\phi$ for both quintessence and phantom fields satisfying the slow-roll conditions. In section 3, we relate the variation of the fine structure constant $\Delta \alpha/\alpha$ with the thawing scalar field. In section 4, we derive the thawing dark energy model and apply the observational data to constrain the model. The conclusions are drawn in section 5.
Slow-roll scalar fields
=======================
Taking $w$ to be the ratio of pressure to energy density of the dark energy, $$\begin{aligned}
\label{w}
w=p_{DE}/\rho_{DE},\end{aligned}$$ models for which $w>-1$ are called quintessence [@wetterich88; @peebles88; @quintessence] and models with $w<-1$ are called phantom [@phantom]. Freezing models start with the equation of state parameter $w$ different from -1 at early times and approaching -1 at the present time while thawing scalar field models start with $w$ close to -1 at early times and deviate from -1 at the present epoch [@Caldwell:2005tm]. In this paper, we use a scalar field to model dark energy.
Quintessence
------------
For the quintessence models, we assume that dark energy is provided by a minimally coupled scalar field $\phi$. The pressure and energy density of the scalar field are given by $$\label{p}
p_\phi=\frac{\dot{\phi}^2}{2}-V(\phi),$$ and $$\label{rho}
\rho_\phi=\frac{\dot{\phi}^2}{2}+V(\phi).$$ The equation of motion of the scalr field is given by $$\label{field}
\ddot{\phi} + 3 H \dot{\phi} + \frac{dV}{d\phi}=0,$$ where the Hubble parameter $H$ is given by $$\label{h}
H=\frac{\dot a}{a}=\kappa\,\sqrt{\rho/3}.$$ Here $a$ is the scale factor, $\rho$ is the total energy density and $\kappa^2=8\pi G$. We consider the spatially flat model only throughout the paper.
If the scalar field has a nearly flat potential $V(\phi)$ with initial value $\phi_0$, i.e., at $\phi=\phi_0$, the scalar field satisfies the slow-roll conditions: $$\label{slow1}
(\frac{1}{V}\frac{dV}{d\phi})^2\ll 1, \quad
\frac{1}{V}\frac{d^2V}{d{\phi}^2}\ll 1,$$ then over the region in which the above conditions (\[slow1\]) apply, a generic relationship between $w$ and the energy density $\Omega_\phi=\kappa^2\rho_\phi/3H^2$ for the quintessence was found [@Robert2008; @Sourish], $$\label{eq13}
1+w=\frac{\lambda_0 ^2}{3}\left[\frac{1}{\sqrt{\Omega_\phi}}-\left(\frac{1}{\Omega_\phi}-1\right)(\tanh^{-1}\sqrt{\Omega_\phi}+C)\right]^2,$$ where $\lambda_0$ is the value of $\lambda=V^{-1}dV(\phi)/d\phi$ at the initial value of the scalar field $\phi_0$ before it begins to roll down the potential, and the integration constant $C$ is determined by the initial values $w_i$ and $\Omega_{\phi i}\ll 1$ [@Sourish] $$\label{eq14}
C=\pm\frac{\sqrt{3(1+\omega_i)}\,\Omega_{\phi i}}{\lambda_0}.$$ For simplicity, we neglect the early dark energy and take $C=0$ which corresponds to the thawing scalar field models. As shown in Figures 2-4 with two explicit potentials $V(\phi)=\phi^2$ and $V(\phi)=\phi^{-2}$ in Ref. , the generic relation (\[eq13\]) between $w$ and $\Omega_\phi$ is a good approximation for arbitrary potentials once the slow-roll conditions (\[slow1\]) are satisfied. In terms of the current values $w_0$ and $\Omega_{\phi 0}$, we get $$\begin{aligned}
\label{lambda0}
\lambda_0=\frac{\sqrt{3(1+w_0)}}{\Omega_{\phi0}^{-1/2}-(\Omega_{\phi0}^{-1}-1)\tanh^{-1}(\sqrt{\Omega_{\phi0}})}.\end{aligned}$$ Substituting Eq. (\[lambda0\]) for $\lambda_0$ into Eq. (\[eq13\]), we get $$\begin{aligned}
\label{eq15}
1+w=(1+w_0)\left[\frac{1}{\sqrt{\Omega_{\phi0}}}-(\Omega_{\phi0}^{-1}-1)\tanh^{-1}
\sqrt{\Omega_{\phi0}}\right]^{-2}\nonumber\\
\times\left[\frac{1}{\sqrt{\Omega_\phi}}-\left(\frac{1}{\Omega_\phi}-1\right)\tanh^{-1}(\sqrt{\Omega_\phi})\right]^2.\end{aligned}$$ Note that the above result does not depend on the specific form of the potential $V(\phi)$ and holds for general potentials satisfying the slow-roll conditions (\[slow1\]). Once a potential $V(\phi)$ is given, we can solve the coupled cosmological equations to find the equation of state parameter $w$ for the scalar field. However, with the result (\[eq15\]), we can get the function $w(z)$ from $\Omega_\phi$ without specifying the potential $V(\phi)$. In other words, instead of specifying $V(\phi)$, we use $\Omega_\phi$ to obtain the equation of state $w(z)$ for the scalar field $\phi$, and it provides us with a particular model for $w(a)$ by this way.
Phantom
-------
For the phantom model, we consider a scalar field $\phi$ with negative kinetic term and potential $V(\phi)$, the energy density and pressure of the phantom are given by $$\begin{aligned}
\label{rho1}
\rho_\phi=-\frac{\dot{\phi}^2}{2}+V(\phi),\end{aligned}$$ and $$\begin{aligned}
\label{p1}
p_\phi=-\frac{\dot{\phi}^2}{2}-V(\phi),\end{aligned}$$ so that the equation of state parameter is $$\begin{aligned}
\label{w1}
w=\frac{\dot{\phi}^2+2V(\phi)}{\dot{\phi}^2-2V(\phi)}.\end{aligned}$$ The evolution of $\phi$ is given by $$\begin{aligned}
\label{field1}
\ddot{\phi} + 3 H \dot{\phi} - \frac{dV}{d\phi}=0.\end{aligned}$$ Similarly, for the phantom field satisfying the slow-roll conditions, we get [@Robert] $$\begin{aligned}
\label{w2}
1+w=-\frac{\lambda_0 ^2}{3}\left[\frac{1}{\sqrt{\Omega_\phi}}-\left(\frac{1}{\Omega_\phi}-1\right)(\tanh^{-1}\sqrt{\Omega_\phi}+C)\right]^2.\end{aligned}$$ Again, we consider the case $C=0$ for simplicity. Expressing $\lambda_0$ in terms of $w_0$ and $\Omega_{\phi 0}$, the result between $w$ and $\Omega_\phi$ for the phantom is the same as that for quintessence given by equation (\[eq15\]).
The variation of fundamental constants
======================================
For realistic dark energy models, the scalar field should couple to other matter components in the universe. Here we consider the coupling between the scalar field and photon with the interaction [@Bekenstein], $$\label{intereq1}
L_{\phi F}=-\frac{1}{4}B_F(\phi)F_{\mu\nu}F^{\mu\nu},$$ where $B_F(\phi)=1-\zeta_\alpha\kappa(\phi-\phi_0)$ and the coupling constant $\zeta_\alpha$ is constrained to be $|\zeta_\alpha|\sim10^{-4}-10^{-7}$ [@copeland04; @Nunes]. Due to the coupling, the fine structure $\alpha$ will evolve with the scalar field, $$\label{alphaeq1}
\frac{\Delta \alpha}{\alpha}=\frac{\alpha-\alpha_0}{\alpha_0}=\zeta_\alpha\kappa(\phi-\phi_0).$$ So $$\begin{aligned}
\label{ophi}
\frac{\alpha'}{\alpha}=\zeta_\alpha\kappa \phi',\end{aligned}$$ where $\alpha'=d\alpha/d \ln a={\dot\alpha}/H$. On the other hand, $$\begin{aligned}
\label{phid}
|w+1|=\frac{\dot{\phi}^2}{\rho_\phi}=\frac{(\kappa\phi')^2}{3\Omega_\phi},\end{aligned}$$ where $\phi'=d\phi/d \ln a={\dot\phi}/H$. Substituting equation (\[ophi\]) into equation (\[phid\]), we get [@Rodger] $$\begin{aligned}
\label{a1}
(\alpha'/\alpha)^2 =3\zeta_\alpha^2\, |w+1|\, \Omega_\phi.\end{aligned}$$ Therefore, $$\begin{aligned}
\label{dverk}
\left|\frac{\Delta\alpha}{\alpha}\right|&=&\int_1^a \sqrt{3\zeta_\alpha^2\Omega_\phi(x)|w(x)+1|}d\ln x \nonumber\\
&=&\int_0^z (1+z)^{-1}\,\sqrt{3\zeta_\alpha^2\,\Omega_\phi(z)|w(z)+1)|}\, dz.\end{aligned}$$ For a model with known $w(z)$, we can solve the cosmological equations to get $\Omega_\phi$ and then calculate the variation of the fine structure constant $\Delta\alpha/\alpha$.
If $\Omega_\phi$ is given, substituting equation (\[eq15\]) with the $\Omega_\phi$ into the above equation (\[dverk\]), we get [@Rodger] $$\begin{aligned}
\label{varalpha1}
\left|\frac{\Delta\alpha}{\alpha}\right|=\sqrt{3\zeta_\alpha^2|1+w_0|}\left|\frac{1}{\sqrt{\Omega_{\phi0}}}-(\Omega_{\phi0}^{-1}-1)\tanh^{-1}\sqrt{\Omega_{\phi0}}\right|^{-1}\nonumber\\
\times\int_0^z (1+z)^{-1} \left|1-\left(\Omega_\phi^{-1/2}-\sqrt{\Omega_\phi}\right)\tanh^{-1}(\sqrt{\Omega_\phi})\right|dz.\end{aligned}$$
Cosmological constraints
========================
Now we apply the observational data on $\Delta\alpha/\alpha$ to constrain the property of dark energy. The data sample consists of 151 absorbers of quasar absorption-line spectra obtained using the Ultraviolet and Visual Echelle Spectrograph on the Very Large Telescope in Chile [@Julian], and 140 absorbers obtained with the Keck telescope at the Keck Observatory in Hawaii [@Murphy:2003mi]. We apply the $\chi^2$ statistics to the 291 data [@Julian; @Murphy:2003mi], $$\begin{aligned}
\label{chi}
\chi^2=\sum_i\left[\frac{(\Delta\alpha/\alpha)_{th,i}-(\Delta\alpha/\alpha)_{obs,i}}{\sigma_i}\right]^2,\end{aligned}$$ where the subscripts $"th"$ and $"obs"$ stand for the theoretically predicted value and observed ones respectively. The theoretical value of $\Delta\alpha/\alpha$ is calculated by equation (\[dverk\]) or (\[varalpha1\]). Since $\Delta\alpha/\alpha$ is proportional to $\sqrt{\zeta_\alpha^2|1+w_0|}$, so the positive detection of the variation of the fine structure constant means that dark energy is not a cosmological constant if the variation of the fine structure is caused by dark energy with the interaction (\[intereq1\]). However, if the variation of $\alpha$ is not due to dark energy, then the above statement does not hold. By neglecting the recent dark energy domination, an analytic expression for the behavior of $\alpha$ was proposed [@Vielzeuf], $$\begin{aligned}
\label{new}
\frac{\Delta\alpha}{\alpha}=-4\epsilon\ln(1+z),\end{aligned}$$ where $\epsilon$ gives the magnitude of the variation. In order to get the best-fit result of the parameter $\epsilon$ in equation (\[new\]), we apply the $\chi^2$ statistics to the 291 $\Delta\alpha/\alpha$ observational data. The $1\sigma$ constraint on $\epsilon$ is $\epsilon=(4.2\pm 2.3)\times10^{-7}$ with $\chi^2=296.48$. By using the best-fit value of $\epsilon$, we plot the variation of $\Delta\alpha/\alpha$ in figure \[fig1\]. If we choose $\epsilon=0$ which corresponds to the model with $\Delta\alpha/\alpha=0$, we get $\chi^2=299.82$. The model with no variation of the fine structure is consistent with the data at the $2\sigma$ level. It is clear that $\alpha$ varies with redshift at the $1\sigma$ level. In the following, we apply the data to constrain dark energy models.
![The 291 $\Delta\alpha/\alpha$ observational data [@Julian; @Murphy:2003mi]. The solid line is the evolution of $\Delta\alpha/\alpha$ as a function of $z$ with equation (\[new\]) for the best fitting value $\epsilon=4.2\times10^{-7}$. The dashed line is the horizonal line indicating no variation of $\Delta\alpha/\alpha$.[]{data-label="fig1"}](deltaalpha.eps){width="84mm"}
If $w$ is near $-1$, then $\Omega_\phi$ can be approximated with $\Lambda$CDM model [@Robert2008; @Crittenden:2007yy], $$\begin{aligned}
\label{phi}
\Omega_\phi=\frac{1}{1+(\Omega_{\phi 0}^{-1}-1)a^{-3}}.\end{aligned}$$ Substituting equation (\[phi\]) into equation (\[eq15\]), we get [@Robert2008; @Robert] $$\begin{aligned}
\label{wzeq1}
w(a)=-1+(1+w_0)\left[\frac{1}{\sqrt{\Omega_{\phi0}}}-(\Omega_{\phi0}^{-1}-1)\tanh^{-1}
\sqrt{\Omega_{\phi0}}\right]^{-2}\nonumber\\
\times\left[\sqrt{1+(\Omega_{\phi 0}^{-1}-1)a^{-3}}-(\Omega_{\phi 0}^{-1}-1)a^{-3}\tanh^{-1}[1+(\Omega_{\phi 0}^{-1}-1)a^{-3}]^{-1/2}\right]^2.\end{aligned}$$ In other words, we consider the dark energy model [@Robert2008; @Robert] with $w(a)$ given by equation (\[wzeq1\]) and the model is different from $\Lambda$CDM model because $w(z)\neq -1$. Taking Taylor expansion of $w(a)$ around $a=1$, the model can be approximated as the CPL parametrization at low redshift with the degeneracy relationship $$\label{waeq1}
w_a=6(1+w_0)\frac{\Omega_{\phi0}^{-1/2}-\sqrt{\Omega_{\phi 0}}-(\Omega_{\phi0}^{-1}-1)\tanh^{-1}(\sqrt{\Omega_{\phi0}})}
{\Omega_{\phi 0}^{-1/2}-(\Omega_{\phi0}^{-1}-1)\tanh^{-1}(\sqrt{\Omega_{\phi0}})}.$$ If we take $\Omega_{\phi 0}=0.7$, then we get $w_a=-1.42(1+w_0)$ which is consistent with the numerical result $w_a\approx -1.5(1+w_0)$ obtained in Refs. . Note that we derived the analytical expression for $w_a$ in terms of both $\Omega_{\phi 0}$ and $w_0$. With this explicit degeneracy relation, the CPL parametrization can be used to give tighter constraints on $\Omega_{\phi 0}$ and $w_0$. Following Ref. , we consider the model (\[waeq1\]) and substitute equation (\[phi\]) into equation (\[varalpha1\]) to get the change of the fine structure constant. By using the observational data, we get constraints on $\log_{10}( \sqrt{\zeta_\alpha^2 |1+w_0|})$ and $\Omega_{\phi0}$ and the results are shown in Figure \[fig2\]. The best fitting results are $\log_{10}(\sqrt{\zeta_\alpha^2 |1+w_0|})=-4.77$ and $\Omega_{\phi0}=0.05$ with $\chi^2=293.97$. At the $1\sigma$ level, $\Omega_{\phi 0}$ reaches the physical boundaries 0 and 1, and $\log_{10}(\sqrt{\zeta_\alpha^2 |1+w_0|})=-4.77^{+0.21}_{-0.43}$. If we take the observational value $\Omega_{\phi0}=0.72$ [@gong13], then the $1\sigma$ constraint is $\log_{10}(\sqrt{\zeta_\alpha^2|1+w_0|})=-5.5^{+0.1}_{-0.3}$ with $\chi^2=294.32$. It is clear that the data is not sensitive to $\Omega_{\phi 0}$ and bigger value of $\zeta_\alpha^2|1+w_0|$ is needed to compensate smaller $\Omega_{\phi 0}$. Taking the observational value $|1+w_0|=0.02$ [@gong13], the best fitting $\zeta_\alpha$ is roughly equal to $1\times10^{-5}$. By using the best fitting value $|\zeta_\alpha|=1\times10^{-5}$, and the priors $-2.0\leqslant w_0 \leqslant -0.3$ and $0.05\leqslant \Omega_{\phi0} \leqslant 0.99$, we get constraints on $w_0$ and $\Omega_{\phi0}$, and the best fitting values are $\Omega_{\phi0}=0.13$ and $w_0=-2.0$ with $\chi^2=293.99$. Figure \[fig2\] shows the $1\sigma$ and $2\sigma$ contours on $\Omega_{\phi0}$ and $w_0$.
![ The $1\sigma$ and $2\sigma$ contours on the dark energy model with $w(z)$ given by equation (\[wzeq1\]). Dark gray and light gray areas show the $1\sigma$ and $2\sigma$ contours, respectively. The contours between $\Omega_{\phi 0}$ and $w_0$ in the right panel were obtained by fixing $|\zeta_\alpha|=1\times 10^{-5}$.[]{data-label="fig2"}](lcdm.eps){width="84mm"}
Conclusions
===========
Starting with a generic relationship (\[eq15\]) between $w$ and $\Omega_\phi$ for thawing scalar field models, we discussed the dark energy model with varying $w(z)$ by using $\Omega_\phi$ from $\Lambda$CDM. Since at the $1\sigma$ level, the fine structure constant is found to vary with redshift according to $-4\epsilon \ln (1+z)$ as shown in Figure \[fig1\], so we apply the data of $\Delta \alpha/\alpha$ to constrain the property of thawing scalar field models with the assumption that the variation of the fine structure constant is caused by dark energy.
The variation of the fine structure constant $\Delta\alpha/\alpha$ is proportional to the coupling constant $\zeta_\alpha$ by the factor $\sqrt{\zeta_\alpha^2|1+w|}$, so there are three parameters $\zeta_\alpha$, $w_0$ and $\Omega_{\phi 0}$ to be fitted. However, the data only constrains $\Omega_{\phi 0}$ and $\sqrt{\zeta_\alpha^2|1+w_0|}$. If we can determine the values of $w_0$ and $\Omega_{\phi 0}$ from other astronomical observations, then the 291 data of $\Delta\alpha/\alpha$ can be used to constrain the coupling constant $\zeta_\alpha$. If we take the observational values $\Omega_{\phi 0}=0.72$ and $|1+w_0|=0.02$, we get the best fitting value $|\zeta_\alpha|=1\times 10^{-5}$. Because $\Delta\alpha/\alpha$ is proportional to $\sqrt{\zeta_\alpha^2|1+w_0|}$, so $\sqrt{\zeta_\alpha^2|1+w_0|}$ determines the magnitude of $\Delta\alpha/\alpha$ and is independent of the variation of $\Delta\alpha/\alpha$ with redshift, we need to find other ways to break the the degeneracy between $\zeta_\alpha$ and $w_0$. A bigger value of $|1+w_0|$ is needed if smaller value of $\zeta_\alpha$ is chosen, that is the reason why the best-fit value of $w_0$ approaches the cut-off value $w_0=-2$ and the best-fit value of $\Omega_{\phi 0}$ is small. However, the value of $\chi^2$ does not change much when we choose the observational values $|1+w_0|=0.02$ and $\Omega_{\phi 0}=0.72$ instead of the best-fit values obtained here. Due to the big uncertainties of the measurements, the observational constraints are not tight. The 291 data of $\Delta\alpha/\alpha$ excluded some areas around $w_0=-1$ at the $1\sigma$ level, $w_0$ is unbounded from below for phantom fields and unbounded from above for quintessence fields. Because $w_0=-1$ gives $\Delta\alpha/\alpha=0$, so positive detection of the variation of the fine structure at the $1\sigma$ level means $w_0\neq -1$ at the $1\sigma$ level if the variation of the fine structure is caused by dark energy with the interaction (\[intereq1\]). That is why we get an excluded area around $w_0=-1$. However, $\Lambda$CDM model is consistent with the data at $2\sigma$ level.
Acknowledgement {#acknowledgement .unnumbered}
===============
The authors would like to thank Tong-Jie Zhang for helpful discussions. This work was partially supported by the National Basic Science Program (Project 973) of China under grant No. 2010CB833004, the NNSF of China under grant Nos. 10935013 and 11175270, the Program for New Century Excellent Talents in University and the Fundamental Research Funds for the Central Universities.
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[^1]: Email: gaoqing01good@163.com
[^2]: Email: yggong@mail.hust.edu.cn
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Second-order nonlinear optical response allows to detect different properties of the system associated with the inversion symmetry breaking. Here, we use a second harmonic generation effect to investigate the alignment of a graphene/hexagonal Boron Nitride heterostructure. To achieve that, we activate a commensurate-incommensurate phase transition by a thermal annealing of the sample. We find that this structural change in the system can be directly observed through a strong modification of a nonlinear optical signal. This result reveals the potential of a second harmonic generation technique for probing structural properties of layered systems.'
author:
- 'E. A. Stepanov'
- 'S. V. Semin'
- 'C. R. Woods'
- 'M. Vandelli'
- 'A. V. Kimel'
- 'K. S. Novoselov'
- 'M. I. Katsnelson'
bibliography:
- 'Ref.bib'
title: ' Nonlinear optical study of commensurability effects in graphene-hBN heterostructures '
---
[^1]
[^2]
Two-dimensional (2D) materials attract a lot of attention due to their remarkable characteristics. These systems have a rich variety of structural modifications and chemical compositions, which results in a high tunability of their physical properties. Moreover, collective many-body effects that are essentially strong in two dimensions give rise to nontrivial phases of matter, which appear promising for various electronic and opto-electronic applications. For example, transition metal dichalcogenides (TMDs) reveal competing charge [@PhysRevB.82.075130; @C5CP01326G] and spin-ordered [@PhysRevB.93.054429; @PhysRevB.94.035120], as well as Mott insulating [@cho2016nanoscale; @ma2016metallic] states in the mono- and multilayered phases. Also, 2D systems show a superconducting behaviour [@Uchihashi_2016] that was theoretically predicted for black phosphorus [@Shao_2014; @Ge_2015] and antimony [@PhysRevB.99.064513], and experimentally observed in black phosphorus [@zhang2017intercalant] and various TMDs [@Ye1193; @Lu1353; @xi2016ising; @Lu3551].
In a multilayered phase, characteristics of quasi-2D systems depend not only on internal properties of individual layers. Here, the number of layers and, what is even more important, the form of their stacking also plays a significant role. A famous example is a twisted bilayer graphene that at a magic misorientation angle of $1.1^{\circ}$ between two graphene lattices drastically changes electronic properties and becomes superconducting. This remarkable effect has been first theoretically predicted [@Bistritzer12233; @PhysRevB.82.121407] and recently confirmed experimentally [@cao2018unconventional; @cao2018correlated; @Yankowitz1059]. Also, a strong correlation between the interlayer stacking and optical properties has been reported for MoS$_2$ [@hsu2014second; @Yin488] and hexagonal Boron Nitride (hBN) [@kim2013stacking; @yang2019stacking].
Recently, it became possible to experimentally produce hybrid systems combining different two-dimensional materials in a layered heterostructure [@2dmat]. This gives even more freedom in designing materials with special characteristics in a controlled way. For heterostructures, the effect of different stacking is much less explored than in the case of multilayered homostructures. However, this issue should also be of a crucial importance as it involves different types of materials. For instance, a change of a misorientation angle between graphene and hBN lattices results in a commensurate-incommensurate phase transition, which strongly affects a crystal symmetry and electronic properties of the system [@woods2014commensurate]. Therefore, it is very important to have a simple yet sensitive tool that provides the information about the structural properties of a material, because even a small mismatch of different layers dramatically changes characteristics of the system.
![\[fig:Scheme\] Sketch of the experiment. Green and red hexagonal tiles represent hBN and graphene, respectively. Red arrows depict the incident 800nm light. Blue arrows indicate the collected SHG response at 400nm from different parts of the sample. In the incommensurate phase (a), the signal of the SHG is uniform for the entire sample. The strong modification of the SHG response is expected after the structural phase transition (b) from the aligned graphene area. ](experiment_idea_small.pdf){width="0.95\linewidth"}
Conventional experimental techniques that allow to probe various electronic and symmetry properties of nanostructures are Raman spectroscopy [@woods2014commensurate] and transport measurements [@Gorbachev448]. Raman technique is by no means a powerful method that, however, requires an accurate interpretation of results, which might not be straightforward even for a simple case of graphene [@Malard2009; @FERRARI200747]. Also, Raman signals are integrated over rather thick layer of material ($>$500nm) which may result in additional difficulties during data analysis, as relevant signals might originate from buried layers or interfaces [@Malard2009]. Transport experiments are much more difficult to perform as they require additional sample preparationtion steps and corresponding facilities. In this regard, an optical second harmonic generation (SHG) is a very promising tool for investigation of different structural properties of 2D materials, being sensitive to the inversion symmetry breaking in the system and, therefore, crucially depends on the number of layers, stacking, alignment and etc. Thus, the SHG has already been used for description of non-centrosymmetric 2D systems, such as MoS$_2$ [@li2013probing; @hsu2014second; @ultrastrong; @PhysRevB.87.201401; @PhysRevB.87.161403; @Yin488] and WS$_2$ [@janisch2014extraordinary; @seyler2015electrical; @expTMDs]. Also, the optical SHG response has been observed from graphene, where the inversion symmetry was broken by a presence of an electric field [@PhysRevB.85.121413; @PhysRevB.91.205405].
Experimental SHG studies of quasi-2D homostructures are almost entirely dedicated to mono-, or few-layered systems. An exception is a very recent study where a nonzero SHG response from rather thick hBN (ca. hundred layers) flake has been reported [@2019arXiv190209060K]. SHG from layered heterostructures is much less studied, both theoretically [@marg; @PhysRevB.99.165432] and experimentally [@doi:10.1063/1.3275740; @PhysRevB.82.125411; @an2013enhanced], and is mostly focused on graphene/graphite films, where the inversion symmetry is broken by the interface with a substrate.
An effect of the stacking on the nonlinear optical response in layered heterostructures has not been investigated experimentally so far. Here, we address this important question for the cases of mono- and multilayer graphene disposed on an insulating hBN substrate. We show that the change of the SHG response clearly indicates the commensurate-incommensurate phase transition for both considered heterostructures. The change of the alignment in the case of the monolayer graphene has been additionally confirmed by Raman measurements. At the same time, Raman spectroscopy was not able to characterize the structural change in the multilayer graphene/hBN heterostructure. This result suggest that the SHG can be used as a simple and efficient method for probing structural properties and alignment of layered heterostructures.
![\[fig:Sample\] Optical image of the graphene/hBN sample. We use a green filter to enhance contrast. Monolayer graphene regions are shaded red. Gray square area corresponds to a multilayer graphene. The substrate hBN crystal is outlined in light green. Orange lines indicate cross sections shown in Fig. \[fig:cut\]. Scale bar is 20um.](Sample1_small.pdf){width="0.95\linewidth"}
We investigate a nonlinear optical response from graphene flakes placed on top of a hexagonal Boron Nitride (hBN) substrate. In general, a small mismatch between lattice constants of these two materials leads to hexagonal moiré patterns formed in the system [@xue2011scanning; @yankowitz2012emergence]. If a misorientation angle between graphene and hBN lattices is less than $1^{\circ}$, a commensurate stacking is energetically more preferable [@woods2014commensurate]. In this case, electronic properties of the system can be effectively described by Dirac electrons with a nonzero mass [@PhysRevB.87.245408; @PhysRevB.84.195414; @PhysRevLett.115.186801]. The latter appears as a result of an inversion symmetry breaking in a graphene/hBN heterostructure. Importantly, the value of the mass varies in space with a period of moiré pattern, but the average value of the mass stays nonzero [@PhysRevB.89.201404; @PhysRevB.84.195414; @PhysRevLett.115.186801]. Apart from the theoretical prediction, the nonzero average mass in the commensurate phase was also observed in transport experiments [@Gorbachev448]. At larger values of the misorientation angle the system undergoes a structural phase transition towards an incommensurate phase [@woods2014commensurate]. In this phase, the inversion symmetry is still broken locally, but the average value of the mass becomes zero [@PhysRevB.89.201404]. Since a typical focused laser spot is much larger than an interatomic distance, only the average value of the effective mass can be probed in optical experiments. The SHG is known to be very sensitive to the lack of inversion symmetry in the point group of the medium. Therefore, such a nonlinear optical technique a very promising tool for detection of the alignment of such layered heterostructures.
![ TPL (a, c) and SHG (b, d) signals from incommensurate (a, b) and commensurate (c, d) stacking of graphene/hBN heterostructure. Scans were performed on the same sample before and after the structural phase transition. Color bar depicts the intensity of the nonlinear response in arbitrary units. A lighter color indicates a larger value of a signal. Green line outlines the hBN sample area.[]{data-label="fig:2x2"}](Fig_2x2_small.pdf){width="0.95\linewidth"}
Schematic representation of the experiment is shown in Fig. \[fig:Scheme\]. The optical image of the considered sample obtained from optical microscope through a $\times100$ objective is shown in Fig. \[fig:Sample\]. Here, red areas highlight single-layer graphene flakes placed on top of the hBN substrate depicted by a light green color. The incommensurate phase for both flakes is confirmed by Raman spectroscopy through the broadening of the 2D-peak in the Raman spectrum [@eckmann2013raman]. A central gray square area corresponds to a multilayer graphene, which alignment with respect to the hBN was not possible to define by Raman measurements. Details of the sample preparation and Raman experiments can be found in the Section “Methods”. A combined optical response has been measured from the sample excited by a 800 nm femtosecond laser. A two-photon luminescence (TPL) signal that was collected at $390-650$ nm is shown in Fig. \[fig:2x2\]a. A narrow bandpass filter centered at 400 nm ($\pm$ 20 nm) was used to detect the SHG signal only (see Fig. \[fig:2x2\]b). Description of the experimental set-up can also be found in the Section “Methods”.
Since the TPL is an incoherent process, it is not sensitive to the point group of the medium. Thus, we observe the TPL response only from graphene areas. Here, the strongest signal comes from a multilayer graphene as it has more complex band structure than its single-layer realization. The hBN does not contribute to the TPL, because the corresponding excitation energy $\hbar\omega\sim 1.9-3.2$eV is lower than the band gap in this material, which is about 6eV [@PhysRevB.51.6868; @cassabois2016hexagonal]. Thus, a nonzero TPL signal only confirms the presence of graphene flakes on the hBN substrate. As expected, this incoherent optical process is not sensitive to the alignment of the considered heterostructure.
On contrary, the SHG is a coherent process, whose efficiency must obey the symmetry principle [@nye1985physical]. Hence SHG can serve as a probe of symmetries of the point group in the studied medium. For considered materials, the latter is mediated by the value of the average mass (band gap) [@PhysRevB.99.165432]. We observe that the intensity of the SHG response shown in Fig. \[fig:2x2\]b is uniform for the entire sample. This fact suggests that there is no inversion symmetry breaking associated with the interaction of graphene with the hBN substrate. Thus, we find that single-layer graphene flakes are not aligned with the hBN, which is consistent with Raman measurements. This result also confirms that the average mass of graphene, which is effectively probed by the SHG, is zero in the incommensurate phase. Importantly, we also do not observe any change of the SHG signal from the multilayer graphene area. This indicates that the multilayer graphene flake is not aligned with hBN substrate as well. As a consequence, the SHG response from the sample is related only to the hBN.
We note that the SHG from a thick hBN is strong. This can be explained by a simple phenomenological model. The SHG arises from the nonlinear polarization ${\bf P}(2\omega)$ induced by an incident laser field ${\bf E}(\omega)$. To the leading orders in ${\bf E}(\omega)$, this polarization can be written as [@HEINZ1991353]: $$\begin{aligned}
P_{i}(2\omega) = \chi^{d}_{ijk}E_{j}(\omega)E_{k}(\omega) + \chi^{q}_{ijkl}E_{j}(\omega)\nabla_{k}E_{l}(\omega),\end{aligned}$$ where the first and second terms describe the electric-dipole and quadrupole contributions to the SHG, respectively. The total SHG response from the system can be divided into the surface and bulk parts. According to Neumann’s principle [@nye1985physical], the electron-dipole SHG process is allowed only for a non-centrosymmetric medium. This results in a strong second-order optical response from the surface of hBN [@PhysRevB.99.165432]. On contrary, the bulk of hBN obeys the inversion symmetry and contributes to the SHG only through the quadrupole polarization. Although the latter is much smaller than the electric-dipole one, in birefringent materials the total quadrupole contribution from the bulk can be of the same order of magnitude as the surface SHG due to the phase-matching [@PhysRevLett.51.1983; @HEINZ1991353]. Moreover, in some cases these two processes can hardly be separated from each other [@PhysRevB.38.7985; @HEINZ1991353]. However, the strength and the relative phase of the quadrupole contribution to the SHG strongly depends on the particular structure of the bulk.
![\[fig:cut\] Vertical (a) and horizontal (b, c) cross-sections of the SHG signal depicted in Fig. \[fig:Sample\]. Blue and green lines correspond to the SHG response before (Fig. \[fig:2x2\]b) and after (Fig. \[fig:2x2\]d) the incommensurate-to-commensurate phase transition, respectively. Light red areas highlight the reduction of the SHG signal due to a presence of the aligned graphene. The enhanced response at gray area indicates the commensurate multilayer graphene. The value of a signal is given in arbitrary units.](Hor_Ver_small.pdf){width="0.9\linewidth"}
To modify the alignment of the sample, we exploit the finding of the work [@woods2016macroscopic], where thermal annealing was used to activate the incommensurate-to-commensurate phase transition. For this aim, we induce a local laser heating of the graphene/hBN heterostructure via a long irradiation of the sample. This was done in such a way that the system was under the irradiation for tens of seconds. The estimated energy of the irradiation is $\sim$0.06 J/cm$^2$ (the pulse energy is 1.88 nJ, the spot size area is $\sim$ 1 $\mu$m$^2$). Heated by a laser, the structure relaxed from the incommensurate state to a more favorable commensurate one. After that, a Raman spectroscopy measurement was performed to confirm the phase transition. It has been observed that single-layer graphene flakes became aligned with the underlying hBN forming moiré patterns with the periodicity of 12.5 and 14.0 nm. The alignment of a multilayer graphene has not been determined by Raman spectroscopy. For more details see the SI [@SI].
Now, we repeat TPL and SHG measurements. Corresponding results are shown in Fig. \[fig:2x2\]c and d, respectively. As expected, the TPL signal remains unchanged after the phase transition. This follows from the fact that the electronic band structure of the system at a characteristic energy of nonlinear optical excitations does not change under small rotations [@PhysRevB.89.201404]. On contrary, the SHG response is drastically modified after graphene becomes aligned with the hBN substrate. Indeed, now the SHG intensity picture explicitly shows the position of graphene flakes. According to a theoretical prediction [@PhysRevB.99.165432], this is a clear signature of the inversion symmetry breaking that occurs due to the interaction of the aligned graphene with the underlying hBN layer. As a consequence, electrons in the commensurate phase of the graphene/hBN heterostructure gain a nonzero average mass [@PhysRevB.87.245408; @PhysRevB.84.195414; @PhysRevLett.115.186801; @Gorbachev448], which is precisely captured by the SHG experiment. Remarkably, we find that the SHG response from the area of the multilayer graphene has also been modified indicating the phase transition towards the commensurate phase.
A precise comparison of the normalized intensity of the SHG for different alignments is shown in Fig. \[fig:cut\]. Results are obtained for cross-sections along vertical (a) and horizontal (b and c) directions depicted by orange lines in Fig. \[fig:Sample\]. Blue and green lines in Fig. \[fig:cut\] indicate the SHG signal in the incommensurate and commensurate phases, respectively. Remarkably, we observe a completely different change of the SHG response from single- and multilayer graphene/hBN heterostructures after the phase transition. Indeed, the SHG is suppressed at light red ares that correspond to single-layer graphene flakes. On contrary, the SHG from the aligned multilayer graphene depicted by gray color is enhanced.
It is worth noting that the electric-dipole polarization of the hBN surface has only the imaginary component, because the considered frequency of the light is below the band gap of this material [@PhysRevB.99.165432]. This leads to a phase shift of the corresponding SHG signal with respect to the second-order optical response from the graphene/hBN interface, for which the electric-dipole polarization can have both, real and imaginary components [@PhysRevB.99.165432]. Therefore, the interference between commensurate graphene and underlying hBN can be either destructive or constructive, depending on a particular structure of the system. This fact makes the SHG very appealing for investigation of the inversion symmetry breaking in layered heterostructures, because already [*the change*]{} of the SHG clearly indicates a structural phase transition. In this context, the Raman technique is a less direct method, since it requires additional data processing steps, such as the calculation of the broadening of the 2D peak in the spectrum [@eckmann2013raman].
For additional confirmation of obtained results, we have performed the TPL and SHG measurements of another aligned single-layer graphene encapsulated between two hBN flakes. In this case, we also observe a strong modification of the SHG signal associated with the presence of the aligned graphene flake between hBN layers. For more details of this experiment please see the SI [@SI].
In conclusion, we have observed that structural changes in graphene/hBN heterostructures can be explicitly captured using the optical SHG technique. We realized that the incommensurate-to-commensurate phase transition in considered systems can be activated by a local laser heating of the sample. We have found that the SHG method is able to detect the alignment not only of the single-, but also of the multilayer graphene flake disposed on the hBN surface, contrary to Raman spectroscopy. In addition to transport measurements, our nonlinear optical study confirmed that the average mass of electrons in graphene is zero in the incommensurate phase, and is nonzero for the commensurate structure. Our results suggest that the proposed method for detection of the alignment is more direct and, thus, has an advantage over a standard Raman technique.
Methods {#sec:Methods .unnumbered}
=======
Samples for these measurements were fabricated by dry-peel methodology described elsewhere [@woods2014commensurate; @kretinin2014]. Control of the alignment is achieved by positioning long, straight edges of the crystals (which tend to be one crystallographic axis, zig-zag or arm-chair) parallel to each other. We use single- and multilayer graphene flakes placed on top of thick hBN ($\sim150$nm).
Raman spectroscopy measurements were performed using a Horiba Raman spectrometer (grating 1200 GPI) operating with an incident laser at a wavelength of 532nm and $\sim0.5$mW power. A confocal microscope was used to focus on the sample through a $\times100$ objective. For more details please see the SI [@SI].
For nonlinear characterization of the samples WITec alpha300 S confocal microscope was used in reflection geometry. Samples were irradiated by Ti:sapphire oscillator at 800nm and $\sim100$fs pulse width. Typical laser power was $\sim220$mW before the microscope where about 70% of power reached and were focused on a sample with $\times20$ Zeiss objective. Detected nonlinear response was separated from fundamental wavelength by use of two types of filters. SCHOTT BG39 filter (390-650nm transmission) was used for experiments when SHG/TPL combined response was detected. Thorlabs FB400-40 filter was used in experiments where only SHG signal was of interest.
Authors thank Chris Berkhout for technical support. Authors also thank Clement Dutreix and Vladimir Kukushkin for fruitful discussions and useful comments. The work of E.A.S. was supported by the Russian Science Foundation, Grant No. 17-72-20041. K.S.N acknowledges support from EU Graphene Flagship Program (contract CNECTICT-604391), European Research Council Synergy Grant Hetero2D, the Royal Society, EPSRC grant EP/N010345/1. The work of M.I.K. was supported by European Research Council via Synergy Grant 854843 - FASTCORR.
\
Raman characterisation of the sample {#raman-characterisation-of-the-sample .unnumbered}
====================================
![Optical picture of the sample (a) and Raman spectroscopy for single-layer graphene flakes (b and c) after the phase transition. Labels (1) and (2) on the panel c indicate regions with different moiré periodicity specified in the text.[]{data-label="fig:periodicity"}](SI_Fig2_small.pdf){width="0.7\linewidth"}
We confirm the presence or absence of a commensurate phase at the interface between the crystals by Raman spectroscopy. This is because the commensurate phase for graphene on hBN is characterised by the appearance of a strain distribution when the crystals near to alignment [@woods2014commensurate]. The Raman spectrum of graphene is sensitive to slight changes in uniaxial/biaxial strain. In particular, the 2D-peak responds to the commensurate phases’ strain distribution by broadening [@eckmann2013raman]. Here, we observe that the full-width half-maximum of the 2D-peak (FWHM(2D)) before our experiments is 23 cm$^{-1}$, which is consistent with an unaligned flake ($>1.5^{\circ}$ misalignment) [@eckmann2013raman], or graphene on a rough substrate (SiO$_2$, polymers etc.) [@ferrari2013raman]. This indicates that the graphene and hBN crystals cannot be in the commensurate phase.
After the phase transition, the FWHM(2D) shows significant change. The width of the peak has broadened to $\sim36$cm$^{-1}$, which is consistent with the most aligned case of graphene on hBN ($0.3^{\circ}$ alignment, moiré period is $L=13.5$nm). Corresponding results are shown in Fig. \[fig:periodicity\]. Here, insets b and c depict two single-layer graphene flakes. The moiré period obtained for the flake b is $L=12.5$nm (FWHM(2D) is 33cm$^{-1}$). Moiré periods for areas (1) and (2) of the flake c are $L=14.0$nm (FWHM(2D) is 40cm$^{-1}$) and $L=12.5$ nm (FWHM(2D) is 33cm$^{-1}$), respectively. This result demonstrates unambiguously that the graphene is now in a commensurate phase with the hBN crystal.
Encapsulated graphene {#encapsulated-graphene .unnumbered}
=====================
![Optical picture of the encapsulated and aligned graphene sample. A single-layer graphene flake is shown in red. The substrate hBN crystal is outlined in light green. Orange line indicates the cross sections shown in Fig. \[fig:enc2\]c. Scale bar is 20um. The inset is a Raman spectroscopy result.[]{data-label="fig:enc"}](Sample_col17_small.pdf){width="0.7\linewidth"}
For additional confirmation of our findings, we repeat the nonlinear optical study on another sample. The optical picture of the sample is shown in Fig. \[fig:enc\]. Here, a red area highlights an aligned monolayer graphene encapsulated between two hBN layers depicted by a light green color. The alignment of the graphene is confirmed by Raman spectroscopy. The moiré period is found to be $L=13.5$nm (FWHM(2D) is 37cm$^{-1}$).
![\[fig:enc2\] TPL (a) and SHG (b) signals from the aligned and encapsulated graphene/hBN heterostructure. Color bar depicts the intensity of the nonlinear response. A lighter color indicates a larger value of a signal. Panel (c) shows a cross-sections of the SHG signal depicted in Fig. \[fig:enc\]. Light red area highlights the reduction of the SHG signal due to a presence of the aligned graphene. The value of a signal is given in arbitrary units. ](Sample_col17_SHGandSection_small.pdf){width="0.7\linewidth"}
The TPL and SHG intensity pictures are shown in Fig. \[fig:enc2\]a and b, respectively. Here, both, the TPL and SHG responses clearly shows the presence of a graphene flake. The change of the SHG signal from graphene with respect to the hBN environment confirms that the monolayer graphene flake is in the commensurate phase. The strong suppression of the SHG can be explicitly seen in Fig. \[fig:enc2\]c that shows the cross-section of the SHG signal depicted by the orange line in Fig. \[fig:enc\]. Here, the light red area corresponds to the position of the aligned single-layer graphene flake.
[^1]: These authors contributed equally
[^2]: These authors contributed equally
| {
"pile_set_name": "ArXiv"
} |
---
author:
- |
Ashwin S. Pande[^1],\
Department of Mathematics,\
Harish-Chandra Research Institute,\
Jhusi, Allahabad, India.
title: 'Topological T-duality for Stacks using a Gysin Sequence'
---
Introduction {#secIntro}
============
Topological T-duality is a recent theory inspired by the theory of T-duality in String Theory. Principal circle (and torus) bundles with a class in $H^3$ of the total space of the bundle possess an unusual symmetry, namely, from this data it is possible to naturally construct [*T-dual*]{} bundles which are also principal circle bundles over the same base with a class in $H^3$ of the total space of the T-dual bundle (See Ref. [@MRCMP; @Bunke] for examples).
In Refs. [@Bunke1; @Bunke2], the authors generalize Topological T-duality to principal bundles of topological stacks (see Refs. [@Bunke1; @Noohi1; @Noohi2; @Heinloth] for an introduction) ${{\mathcal E}}\to {{\mathcal Y}}$ with an $S^1$-gerbe ${{\mathcal G}}$ on the stack ${{\mathcal E}}$. The authors show that in such a situation, the T-dual exists and is also a principal bundle of stacks together with a gerbe on it. Since this result applies to stacks, it applies to spaces with circle group actions which are not necessarily free and, in particular, to semi-free $S^1$-spaces.
A brief outline of the paper is as follows: In Ref. [@Pande] the T-duals of some semi-free $S^1$-spaces were derived using $C^{\ast}$-algebraic techniques. In Ref. [@MaWu] a general formalism using the Borel construction was used to derive the T-dual of any semi-free $S^1$-space. Neither of these constructions used stack theory, and it would be interesting to compare the T-duals obtained using all three theories. In Sec. (\[secComp\]) below, we attempt to do this in a simple example.
We calculate Topological T-duals of semi-free spaces for the examples of Ref. [@Pande] (these are the Kaluza-Klein monopole backgrounds of string theory) using the methods of Ref. [@Bunke1] in Sec. (\[secTDKK\]). We also comment on the results obtained.
A phenomenon seen in backgrounds with Kaluza-Klein monopoles is the dyonic coordinate (See Refs. [@Sen; @HaJen] for details). A model for this using $C^{\ast}$-algebraic methods was developed in Ref. [@Pande]. We point out in Sec. (\[secDyon\]) that this phenomenon may also be obtained completely independently of the $C^{\ast}$-algebraic formalism in the stack theory formalism of Ref. [@Bunke1] using the results from Ref. [@TTDA].
In Ref. [@Ginot], the authors give a Gysin sequence for a $S^1$-stack. Very roughly, this is an exact sequence of stack cohomology groups which is derived by taking classifying spaces for ${{\mathcal E}}$ and ${{\mathcal Y}},$ obtaining an ordinary principal bundle, and then using the ordinary (homology) Gysin sequence. It is clear that a cohomology Gysin sequence can also be obtained from this argument. Can this be used to obtain the Topological T-dual of a semi-free space in analogy to the argument in ordinary Topological T-duality? In Sec. (\[secGysin\]) we develop this theory and calculate a few T-duals. We prove that the T-dual of a semi-free space obtained using this method is the one obtained by Mathai and Wu in Ref. [@MaWu].
In this paper we restrict ourselves to $S^1$-actions on stacks and stacks which are principal $S^1$-bundles over a stack.
Comparision of the Three Formalisms {#secComp}
===================================
We use the convention of Ref. [@Heinloth]: If $X$ is a topological space then $\underline{X}$ is $X$ viewed as a [*stack*]{} using the Yoneda Lemma. Consider a space which is a point ${\protect{\underline{\mathrm{pt}}}}$ with an $S^1$-action which fixes that point. It is interesting to compare the T-duals obtained for this space using the formalisms of Refs. [@MRCMP; @Bunke1; @MaWu]. We have the following theorem:
Consider a point ${\protect{\underline{\mathrm{pt}}}}$ with an $S^1$-action which fixes the point. Then,
1. The $C^{\ast}$-algebraic T-dual is ${{\mathbb R}}$ with quotient space the point.
2. The Topological T-dual in the formalism of Bunke et al. (see Refs. [@Bunke; @Bunke1; @Bunke2]) is the principal bundle of stacks $[{\protect{\mathrm{pt}}}/S^1] \times S^1 \to [{\protect{\mathrm{pt}}}/S^1]$ with a gerbe on the total space corresponding to $H$-flux.
3. The Topological T-dual in the formalism of Mathai-Wu (see Ref. [@MaWu]) is $ BS^1 \times S^1 \times {\protect{\mathrm{pt}}}$ with $H$-flux.
\[ThmComp\]
Suppose one considers a point with a $S^1$-action which fixes the point. The quotient is still a point.
1. In the $C^{\ast}$-algebraic formalism, to this geometry we would assign the $C^{\ast}$-algebra of compact operators ${{\mathcal K}}$ (since the spectrum of ${{\mathcal K}}$ is the point). The trivial action of $S^1$ on ${{\mathcal K}}$ lifts to a trivial action $\alpha$ of ${{\mathbb R}}$ on ${{\mathcal K}}$. Also, any other action of ${{\mathbb R}}$ on ${{\mathcal K}}$ is exterior equivalent to this trivial one. The T-dual would be the spectrum of the crossed product ${{\mathcal K}}\underset{\alpha}{\rtimes} {{\mathbb R}}\simeq C_0({{\mathbb R}})$ i. e. ${{\mathbb R}}$ where $\simeq$ denotes Morita equivalence (See Ref. [@RaeRos]). The group action on the T-dual $C^{\ast}$-algebra $C_0({{\mathbb R}})$ is induced from the translation action of ${{\mathbb R}}$ on itself (See Ref. [@RaeRos] for details). The quotient would be the point as expected.
2. In Ref. [@Bunke1], Sec. (4.2, 4.3), Prop. (4.3), the T-dual of a stack is obtained by the following procedure: One passes to the geometric realization of the simplicial space of the groupoid associated with that stack. This gives an ordinary principal bundle with $H$-flux from the principal bundle of stacks one one began with. This principal bundle may be T-dualized in the normal way [@Bunke1].
Consider the principal bundle of stacks $ {\protect{\underline{\mathrm{pt}}}}\to
[{\protect{\mathrm{pt}}}/ S^1] $ and the atlas $\mbox{pt} \to [{\protect{\mathrm{pt}}}/S^1].$ Let $Y = \mbox{pt} \underset{[ {\protect{\underline{\mathrm{pt}}}}/S^1]}{\times} {\protect{\underline{\mathrm{pt}}}}$ be an atlas for ${\protect{\underline{\mathrm{pt}}}}$ (See Ref. [@Bunke1] Sec. (4.2)). We have a commutative square $$\begin{CD}
Y @>>> {\protect{\underline{\mathrm{pt}}}}\\
@VVV @VVV \\
\mbox{pt} @>>> [{\protect{\mathrm{pt}}}/ S^1 ].
\end{CD}$$
The groupoid associated to $ {\protect{\underline{\mathrm{pt}}}}\to [ {\protect{\mathrm{pt}}}/ S^1 ]$ is $ \mbox{pt} \times S^1 \rightrightarrows \mbox{pt} $ as there is a canonical isomorphism $ Y \simeq ( \mbox{pt} \times S^1 )$ since $( \mbox{pt} \times S^1 )$ is the canonical bundle over $ \mbox{pt} $ (See Heinloth Ex. 2.5 and following). Similarly the groupoid associated to the atlas $Y= \left( \mbox{pt} \underset{[ {\protect{\mathrm{pt}}}/ S^1 ]}{\times}
{\protect{\underline{\mathrm{pt}}}}\right) \to {\protect{\underline{\mathrm{pt}}}}$ is $Y \underset{{\protect{\underline{\mathrm{pt}}}}}{\times}{Y} \rightrightarrows {\protect{\mathrm{pt}}}$. Since the fiber product of $Y$ with itself over ${\protect{\mathrm{pt}}}$ is $ \mbox{pt} \times ( S^1 )^2$ the associated groupoid would be $\mbox{pt} \times ( S^1 )^2 \to \mbox{pt} $.
It is clear that the iterated fiber product of $Y$ with itself $n$ times would be isomorphic to $\mbox{pt} \times ( S^1 )^n$. The total space of the associated simplicial bundle would then (by definition of $EG$) be $ES^1$ and the base would (by the construction above) be $BS^1 $. Therefore the T-dual of the simplicial bundle would be $BS^1 \times S^1 $ with a gerbe on total space (See Refs. [@Bunke; @Bunke1]). This corresponds to the T-dual bundle $[{\protect{\mathrm{pt}}}/S^1] \times S^1 \to [{\protect{\mathrm{pt}}}/S^1]$ with a gerbe on the total space of the bundle corresponding to the $H$-flux.
3. In the formalism of Mathai and Wu (See Ref. [@MaWu]), the original space would be replaced by $ES^1 \times {\protect{\mathrm{pt}}}$ as a principal circle bundle over $BS^1 \times {\protect{\mathrm{pt}}}$ and the T-dual would be $BS^1 \times S^1 \times {\protect{\mathrm{pt}}}$ as a principal circle bundle over $BS^1 \times {\protect{\mathrm{pt}}}$ with $H$-flux. The T-dual obtained here namely $BS^1 \times S^1 \times {\protect{\mathrm{pt}}}$ should be compared with the T-dual $[{\protect{\mathrm{pt}}}/S^1] \times S^1$ obtained in Part (2).
Thus, the formalisms of Bunke-Schick and Mathai-Wu give similar answers here for this example and the $C^{\ast}$-algebraic formalism gives a different one. This difference is probably due to the fact that in Ref. [@MRCMP], an $S^1$-action on a principal bundle lifts to an ${{\mathbb R}}$-action (with ${{\mathbb Z}}$-stabilizers) on the $C^{\ast}$-dynamical system associated to that space while in Ref. [@MaWu], the $S^1$-action remains an $S^1$-action. That is, in the $C^{\ast}$-algebraic formalism a circle action is viewed as an ${{\mathbb R}}$-action with ${{\mathbb Z}}$-stabilizers while in the other formalisms an $S^1$-action is viewed only as an $S^1$-action.
In Topological T-duality, it is expected that the original and T-dual spaces have the same $K$-theory up to a degree shift. It is clear, from the Connes-Thom isomorphism, that the $C^{\ast}$-algebraic T-dual will have this property. Similarly, the answers obtained by the other two formalisms will also have this property, one would have to use $K$-theory twisted by the $H$-flux on the T-dual to perform the above calculation.
T-dual of Kaluza-Klein monopole Backgrounds {#secTDKK}
===========================================
We now restrict our attention to spaces with semi-free circle actions. We may further restrict ourselves to spaces which contain Kaluza-Klein monopoles ($KK$-monopoles)[^2]. Away from fixed points such spaces are equivariantly $S^1$-homeomorphic to the total space of a principal circle bundle.
We begin this section by making a remark about the existence of T-dual stacks. In this paper, we will use the method of Ref. [@Bunke1] to calculate the T-dual of a stack associated to a semi-free space. In the method of Ref. [@Bunke1], if we restrict ourselves to $U(1)$-bundles over such stacks, the associated simplicial bundles, being circle bundles, may always be T-dualized. There is always a ‘T-duality diamond’ of Ref. [@Bunke] (see Diagram (2.14) in Lemma (2.13) of that reference) for the associated simplicial bundles, since these are only circle bundles. This gives a diagram of the form Diagram (4.1.2) in Ref. [@Bunke2] for the associated [*stacks*]{}. Hence, any $U(1)$-bundle over such a stack may be T-dualized in the sense of Def. (4.1.4) of Ref. [@Bunke2], even if the base is not an orbispace.
In the following, the stacks we dualize are not orbispaces in the sense of Ref. [@Bunke1], but, due to the above, the principal bundles $p:{{\mathcal E}}\to {{\mathcal B}}$ may be completed into a diagram of the form of Diagram (4,1.2) of Ref. [@Bunke2], and hence these stacks may be T-dualized (by Ref. [@Bunke2], Def. (4.1.4) ).
In a neighbourhood of a fixed point such spaces are equivariantly $S^1$-homeomorphic to ${{\mathbb R}}^{4}$ with an orthogonal $S^1$-action. The associated topological stacks are the stacks $[CS^3/{{\mathbb Z}}_k].$ In Ref. [@Pande], the $C^{\ast}$-algebraic approach of Ref. [@MRCMP] was used to compute the T-duals of some semi-free spaces.
First, we consider the T-dual of spaces (See Thm. (\[ThmKK\]) below) which are the total space of $KK$-monopoles of charge $l \in {{\mathbb N}}, l > 0.$ As argued above, these correspond to stacks of the form $E=[CS^3/{{\mathbb Z}}_l]$ with $E/S^1 = CS^2$ for $l \geq 2$, and $E=CS^3$ and $E/S^1=CS^2$ for $l=1$. For $l=1$ the associated principal bundle of stacks would be $\underline{CS^3} \to [CS^3/S^1]$. For $l \geq 2$, the associated principal bundle of stacks would be $[CS^3/{{\mathbb Z}}_l] \to [CS^3/S^1]$. We consider the T-dual of spaces containing multiple $KK$-monopoles in Thm. (\[ThmTDGen\]) and Cor. (\[CorKKMulti\]) below.
Consider a spacetime which is a $KK$-monopole spacetime with charge $1$. Physically, the T-dual would have a source of $H$-flux over the set in the base corresponding to the image of the singular fiber and the $H$-flux would be undefined at the location of the source[^3]. In the $C^{\ast}$-formalism of Topological T-duality the $C^{\ast}$-algebra describing the background loses the continuous-trace property exactly on this locus. In Ref. [@Pande], it was argued that this is a model for a space with a [*source*]{} of $H$-flux.
In the formalisms of Refs. [@Bunke1; @MaWu], however, the T-dual $H$-flux would be everywhere defined, that is, there would be no [*source*]{} of $H$-flux present. In particular, for these two theories, the following would hold: The T-dual of a single NS5-brane with a background of $k$-units of (sourceless) $H$-flux would be indistinguishable from the T-dual of a space with $(k+1)$-units of sourceless $H$-flux.
Let ${{\mathcal E}}\to {{\mathcal Y}}$ be a principal $S^1$-bundle of stacks over ${{\mathcal Y}}$. It is clear from the axioms of a principal $S^1$-bundle (see Ref. [@Heinloth] after Remark (2.14)) that the topological stack ${{\mathcal E}}$ is a space with a left $S^1$-action (in the sense of Ref. [@Ginot], Sec. (3)).
1. Let $E$ be a space with a circle action. Then, $\underline{E} \to [E/S^1]$ is a principal bundle of topological stacks.
2. Let $p:{{\mathcal E}}\to {{\mathcal Y}}$ be a principal $S^1$-bundle of topological stacks. Then ${{\mathcal E}}$ is a stack with a left $S^1$-action in the sense of Ref. [@Ginot] and ${{\mathcal Y}}\simeq \left[ S^1 \backslash {{\mathcal E}}\right].$
3. For a space $E$ with a circle action, $[S^1 \backslash \underline{E}] \simeq [E/S^1]$.
\[StLem1\]
1. $E$ is a space with a $S^1$-action and satisfies the conditions for a principal $S^1$-bundle described after Remark (2.14) in Ref. [@Heinloth]. This is equivalent to the definition of a principal bundle of stacks using atlases[^4] by the Claim before Example (2.15) in Ref. [@Heinloth].
2. That ${{\mathcal E}}$ is a stack with a left $S^1$-action follows from the definition of a principal bundle of stacks (See Ref. [@Heinloth]).
From Ginot and Noohi (Ref. [@Ginot]), Prop. (4.8), $\left[S^1 \backslash {{\mathcal E}}\right]$ is a stack. Let $\tilde{p}:{{\mathcal E}}\to \left[S^1 \backslash {{\mathcal E}}\right]$ be the natural map defined in Sec. (4.1) of Ref. [@Ginot].
Let $T \to {{\mathcal Y}}$ be an atlas for ${{\mathcal Y}}$. By definition of a principal bundle of stacks, (see Ref. [@Heinloth]), we have a commutative square $$\begin{CD}
P @>>> {{\mathcal E}}\\
@VVV @VVV \\
T @>>> {{\mathcal Y}}\label{CDYB}
\end{CD}$$ where $P$ is a principal bundle over $T$ and an atlas for ${{\mathcal E}}$. As noted above, we have a natural map $\tilde{p}:{{\mathcal E}}\to [S^1\backslash {{\mathcal E}}]$. By the second part of the proof of Prop. (4.8) of Ref. [@Ginot], $P$ is also an atlas for $[S^1\backslash {{\mathcal E}}]$. By Ref. [@Ginot], Sec. (4.1) the stack $[S^1 \backslash {{\mathcal E}}]$ is the stackification of the prestack $\lfloor S^1 \backslash {{\mathcal E}}\rfloor$. Then, we have
$$\begin{gathered}
\lfloor S^1 \backslash {{\mathcal E}}\rfloor(P) \simeq S^1 \backslash ({{\mathcal E}}(P)),
(\mbox{by definition of $\lfloor S^1 \backslash {{\mathcal E}}\rfloor$, see
Sec.\ (4.1) of Ref.\ \cite{Ginot}}),\nonumber \\
\simeq {{\mathcal Y}}(P),(\mbox{By definition, see Sec.\ (4.1) of Ref.\ \cite{Ginot} }),
\nonumber \\
\simeq {{\mathcal Y}}(P/S^1) \simeq {{\mathcal Y}}(T),(\mbox{because the Diagram (\ref{CDYB})
commutes}). \nonumber \end{gathered}$$
Hence, the stackification of ${{\mathcal Y}}$ is isomorphic to the stackification of $\lfloor S^1 \backslash {{\mathcal E}}\rfloor$. Since ${{\mathcal Y}}$ is a stack, this implies that ${{\mathcal Y}}\simeq [S^1 \backslash {{\mathcal E}}]$.
3. This follows from Parts (1) and (2) above.
Let ${{\mathcal X}}$ be a stack with a $S^1$-action in the sense of Ginot et. al (See Ref. [@Ginot], Def. (3.1)). Let $q:{{\mathcal X}}\to [S^1 \backslash {{\mathcal X}}]$ be the quotient map of Ref. [@Ginot], Sec. (3.2). Then $[S^1 \backslash {{\mathcal X}}]$ is a topological stack and $q:{{\mathcal X}}\to [S^1 \backslash {{\mathcal X}}]$ is a principal bundle of stacks in the sense of Ref. [@Heinloth]. \[StLem2\]
By definition, we have an action $\mu$ of $S^1$ on the stack $X$. By Prop. (4.8) of Ref. [@Ginot], $[S^1 \backslash {{\mathcal X}}]$ is also a topological stack. By Prop. (4.7) of the same reference, the map $q$ is representable. It can be checked that the conditions for a principal bundle of stacks given after Remark (2.14) in Ref. [@Heinloth] are satisfied with $act = \mu,p=q.$
$p:{{\mathcal E}}\to {{\mathcal Y}}$ is a principal bundle of stacks iff ${{\mathcal E}}$ is a stack with a left $S^1$-action in the sense of Ref. [@Ginot] and ${{\mathcal Y}}\simeq [S^1 \backslash {{\mathcal E}}].$ \[CorStLem\]
This follows from Lemma (\[StLem1\]), Part (2), and Lemma (\[StLem2\]) above.
In Thm. (\[ThmComp\]) (2) we determined the Topological T-dual of the principal bundle of stacks ${\protect{\underline{\mathrm{pt}}}}\to [{\protect{\mathrm{pt}}}/S^1]$. The calculation for the Topological T-dual of $CS^3$ is nearly similar to that for the Topological T-dual of ${\protect{\underline{\mathrm{pt}}}}\to [{\protect{\mathrm{pt}}}/S^1]$ for the following two reasons: Firstly, the space $CS^3$ is equivariantly homotopy equivalent to its vertex ${\protect{\mathrm{pt}}}$. Secondly, $CS^3$ and $CS^2$ are contractible and are homeomorphic to ${{\mathbb R}}^4$ and ${{\mathbb R}}^3$ respectively. As are result, any principal bundle over $CS^3$ is trivial. Due to this, we can take the above proof and replace ${\protect{\underline{\mathrm{pt}}}}$ with $CS^3$ and $[ {\protect{\mathrm{pt}}}/S^1 ]$ with $CS^2$ and obtain a working proof.
1. The Topological T-dual of the principal bundle of stacks $\underline{CS^3} \to [CS^3/S^1]$ (associated to a $KK$-monopole) in the formalism of [@Bunke1] is the principal bundle of stacks $[CS^3/S^1 ]\times S^1 \to [CS^3/S^1] $ with a gerbe on the stack $[CS^3/S^1] \times S^1$.
2. Consider the $S^1$-action on a point which fixes the point. This gives a principal bundle of stacks ${\protect{\underline{\mathrm{pt}}}}\to [{\protect{\mathrm{pt}}}/S^1]$. Consider the subgroup ${{\mathbb Z}}_k \hookrightarrow S^1$ for any natural number $k > 1$. Then, $[{\protect{\mathrm{pt}}}/{{\mathbb Z}}_k] \to [{\protect{\mathrm{pt}}}/S^1]$ is also a principal bundle of stacks. This bundle with no $H$-flux has as T-dual the principal bundle of stacks $[{\protect{\mathrm{pt}}}/S^1] \times S^1 \to [{\protect{\mathrm{pt}}}/S^1]$ with $H$-flux of $k$ units.
3. For $k$ any natural number larger than $1$, there is a principal bundle of stacks $[CS^3/{{\mathbb Z}}_k] \to [CS^3/S^1]$ (corresponding to a $KK$-monopole of charge $k$). The Topological T-dual of this bundle with no $H$-flux, in the formalism of [@Bunke1], is the principal bundle of stacks $([CS^3/S^1] \times S^1) \to [CS^3/S^1]$ with $k$ units of $H$-flux.
\[ThmKK\]
1. Consider the principal bundle of stacks $\underline{CS^3} \to \left[CS^3/S^1 \right]$. The space ${{\mathbb R}}^4 \simeq CS^3$ is an atlas for the stack $[CS^3/S^1]$. Let $Y = CS^3 \underset{[CS^3/S^1]}{\times} \underline{CS^3}$ be an atlas for $\underline{CS^3}$ (See Ref. [@Bunke1] Sec. (4.2)) induced by the atlas for the stack $[CS^3/S^1]$. We have a commutative square $$\begin{CD}
Y @>>> \underline{CS^3} \\
@VVV @VVV \\
CS^3 @>>> [CS^3/S^1].
\end{CD}$$
The atlas associated to $\underline{CS^3}$ is $ CS^3 \times S^1 \rightarrow \underline{CS^3} $: There is a canonical isomorphism $Y \simeq ( CS^3 \times S^1 )$, since, due to the contractibility of $CS^3 \simeq {{\mathbb R}}^4$, $(CS^3 \times S^1 )$ is the canonical bundle over $ CS^3 $ (See Heinloth Ex. 2.5 and following).
The groupoid associated to the atlas $Y= \left( CS^3 \underset{[CS^3/S^1]}{\times} \underline{CS^3} \right)
\to \underline{CS^3}$ is $Y \underset{\underline{CS^3}}{\times}Y \rightrightarrows CS^3$. The fiber product $Y \underset{\underline{CS^3}}{\times}Y$ is $ CS^3 \times ( S^1 )^2$ by definition. Therefore, the groupoid is $CS^3 \times (S^1)^2 \rightrightarrows CS^3$. It is clear from the definition that the iterated fiber product of $Y$ with itself $n$ times is $CS^3 \times (S^1)^n$.
The associated simplicial space in each degree would be $CS^3 \times (S^1)^n$. The simplicial space is thus the fiber product of $CS^3 \to CS^2 \sim *$ with $* \times (S^1)^n \to *$. Therefore by Ref. [@May], Cor. (11.6), the simplicial space is the fiber product $(A \underset{*}{\times} ES^1)$ where $A$ is the geometric realisation of the simplicial space which is $CS^3$ in each degree. Since $CS^3$ is contractible, the space $(A \underset{*}{\times} ES^1)$ would be homotopic to $ES^1$.
We had noted above that the total space of the bundle is the simplicial space which in each degree is $CS^3 \times (S^1)^n$. Hence, the base of the simplical bundle would be the simplicial space which in each degree $>1$ is $CS^2 \times (S^1)^{n-1}$ and at degree $1$ is $CS^2 \times \mbox{pt}$. Let $B$ be the simplicial space which is $CS^2$ in each degree. By the above argument the base is the fiber product $(B \underset{*}{\times} BS^1)$. Since $B$ is contractible, the base has the homotopy type of $BS^1$. Thus, the simplicial bundle associated to the bundle of stacks $\underline{CS^3} \to [CS^3/S^1]$ is $(A \underset{*}{\times} ES^1) \to (B \underset{*}{\times} BS^1)$.
Therefore the T-dual of the simplicial bundle would be $(B \underset{*}{\times}BS^1) \times S^1 $ with a gerbe on total space (See Refs. [@Bunke; @Bunke1]). The class in $H^3$ of the total space associated to the gerbe would correspond to the generator of $H^3((B \underset{*}{\times} BS^1) \times S^1, {{\mathbb Z}}) \simeq
H^3(BS^1 \times S^1, {{\mathbb Z}})$. It is clear from the above that this is the simplicial bundle associated to $[CS^2/S^1] \times S^1$. Also, the $H$-flux on the total space of the bundle is $1$. Therefore the T-dual stack has a gerbe on it.
2. Consider the $S^1$-action on a point which fixes the point. This gives a $S ^1$-bundle of stacks ${\protect{\underline{\mathrm{pt}}}}\to [{\protect{\mathrm{pt}}}/S^1]$. Then ${{\mathbb Z}}_k \hookrightarrow S^1$ acts on ${\protect{\underline{\mathrm{pt}}}}$ as well. The $S^1$-action on ${\protect{\underline{\mathrm{pt}}}}$ gives an action of $S^1$ on $[{\protect{\mathrm{pt}}}/{{\mathbb Z}}_k]$. The quotient is still $[{\protect{\mathrm{pt}}}/S^1].$ Therefore, $[{\protect{\mathrm{pt}}}/{{\mathbb Z}}_k] \to [{\protect{\mathrm{pt}}}/S^1]$ is a principal bundle of stacks by Lemmas (\[StLem1\],\[StLem2\]) above.
The groupoid associated to the stack $[{\protect{\mathrm{pt}}}/{{\mathbb Z}}_k]$ is ${{\mathbb Z}}_k \rightrightarrows {\protect{\underline{\mathrm{pt}}}}$ and the associated simplicial space is $B{{\mathbb Z}}_k$ (See Ref. [@Bunke1] Sec. (5.1) ). The simplicial space associated to $[{\protect{\mathrm{pt}}}/S^1]$ is $BS^1$. The associated simplicial principal bundle is $B{{\mathbb Z}}_k \to BS^1$. This is $ES^1/{{\mathbb Z}}_k \to BS^1$. Thus the T-dual simplicial bundle is $BS^1 \times S^1$ with $H$-flux $k$. This is the simplicial bundle associated to the stack $[{\protect{\mathrm{pt}}}/S^1] \times S^1$ with a gerbe on the stack.
3. For every $k > 1,$ there is an $S^1$-action on $CS^3/{{\mathbb Z}}_k$ induced by the $S^1$-action on the $S^1$-space $CS^3$ with quotient $CS^3/S^1.$ Thus there is an $S^1$-action on $[CS^3/{{\mathbb Z}}_k]$ such that the quotient under the $S^1$-action is $[CS^3/S^1].$ Therefore, by Lemmas (\[StLem1\],\[StLem2\]), there is a principal bundle of stacks $[CS^3/{{\mathbb Z}}_k] \to [CS^3/S^1]$.
The simplicial space associated to $[CS^3/{{\mathbb Z}}_k]$ is determined in a manner similar to Part (1). Consider the principal bundle of stacks $\underline{CS^3} \to [CS^3/{{\mathbb Z}}_k]$.
Let $W$ be an atlas for the stack $\underline{CS^3}$ induced by the the atlas $CS^3$ for the stack $[CS^3/{{\mathbb Z}}_k]$. Then, $W$ is a principal ${{\mathbb Z}}_k$-bundle over $CS^3$. We have a commutative square $$\begin{CD}
W @>>> \underline{CS^3} \\
@VVV @VVV \\
CS^3 @>>> [\underline{CS^3}/{{\mathbb Z}}_k].
\end{CD}$$ There is a canonical isomorphism $W \simeq (CS^3 \times {{\mathbb Z}}_k)$ by the same argument as in Part (1) with $S^1$ replaced by ${{\mathbb Z}}_k$. The groupoid associated to the atlas $W$ is $W \underset{\underline{CS^3}}{\times} W \rightrightarrows CS^3$. We have that $W \underset{\underline{CS^3}}{\times} W \simeq CS^3
\times ({{\mathbb Z}}_k)^2$. Also, $W \underset{\underline{CS^3}}{\times} \ldots
\underset{\underline{CS^3}}{\times} W \simeq CS^3 \times ({{\mathbb Z}}_k)^n$. Let $A$ be the simplicial space which is $CS^3$ in each degree. By an argument similar to that in Part (1), the simplicial space associated to $\underline{CS^3}$ by this atlas is then $(A \underset{*}{\times} E{{\mathbb Z}}_k)$. This has a natural action of ${{\mathbb Z}}_k$ on each factor. It is a principal ${{\mathbb Z}}_k$-bundle over the simplicial space associated to $[CS^3/{{\mathbb Z}}_k]$. (By the above diagram, $W$ is a principal ${{\mathbb Z}}_k$ bundle over $CS^3,$ the result follows from the definition of the simplicial space associated to a groupoid.) Hence, the simplicial space associated to $[CS^3/{{\mathbb Z}}_k]$ by this atlas is $(A/{{\mathbb Z}}_k \underset{\ast}{\times} B{{\mathbb Z}}_k).$ It is also a principal circle bundle over the simplicial space associated to $[CS^3/S^1]$.
By Part (1), the simplicial space associated to $[CS^3/S^1]$ is $(B \underset{*}{\times} BS^1)$. Therefore the simplicial circle bundle associated to the principal bundle of stacks $[CS^3/{{\mathbb Z}}_k] \to [CS^3/S^1]$ is $(A/{{\mathbb Z}}_k \underset{*}{\times} B{{\mathbb Z}}_k) \to (B \underset{*}{\times} BS^1)$.
Thus, the T-dual would be $(B \underset{*}{\times} BS^1) \times S^1 \to (B \underset{*}{\times} BS^1)$ with $H$-flux. This is the principal bundle associated to $([CS^3/S^1] \times S^1) \to [CS^3/S^1]$ with a gerbe on it.
It is interesting to observe that if a property holds for T-duality pairs in the sense of Ref. [@Bunke], it can be applied to the pair consisting of the simplicial bundle and $H$-flux. Sometimes, this has interesting consequences for spaces with a non-free $S^1$-action. We present two examples below using this property: In the first we calculate the T-dual of a semi-free space with countably many isolated fixed sets. In the second example done in the next Section, we develop a model for the Dyonic coordinate of Ref. [@Pande] using the stack-theoretic approach.
Note the following property of T-dual principal $S^1$-bundles: Let $p:E \to B$ be a principal $S^1$-bundle. Let $h \in H^3(E,{{\mathbb Z}})$ be an $H$-flux. Let $W_1,\ldots,W_k$ be open subsets of $B$ such that $W_1 \cup \ldots \cup W_k = B$. Let $E_i = p^{-1}(W_i)$ be the induced open cover of $E$. Let $h_i$ be the $H$-flux restricted to $E_i$. Let $q:E^{\#} \to B$ be the T-dual of the principal bundle $E \to B$ with $H$-flux $h^{\#}.$ Let $E^{\#}_i=q^{-1}(W_i)$ and let $h^{\#}_i = h^{\#}|_{E_i}$. Then we claim that $(E^{\#}_i,h^{\#}_i)$ is the T-dual of $(E_i,h_i).$ This follows from the existence of a classifying space for a pair and the properties of the T-duality map on it in Ref. [@Bunke]: If $R$ is the classifying space of pairs of Ref. [@Bunke], $T:R \to R$ the T-duality map and $\phi:B \to R$ the classifying map for the pair $(E,h),$ we have, on restriction to $W_i,$ $(T\circ \phi)|_{W_i}=T\circ (\phi|_{W_i}).$ However, by definition, $(T \circ \phi)|_{W_i}$ classifies $(E^{\#}_i,h^{\#}_i)$ while $\phi|_{W_i}$ classifies $(E_i,h_i)=(E|_{W_i},h|_{E_i}).$
A similar result holds for semi-free spaces:
1. Let $E$ be a semi-free $S^1$-space with at most countably many isolated fixed sets of the $S^1$-action ${F_1,\ldots,F_k,\ldots}$. Suppose we are given disjoint neighbourhoods $U_i$ of $F_i$. Let $\underline{V_i} = [U_i/S^1].$ Then these data determine the T-dual of $E$.
2. The T-dual of a semi-free space $E$ with at most countably isolated fixed sets is the principal bundle of stacks $[E/S^1] \times S^1 \to [E/S^1]$ with $H$-flux. There will be $H$-flux on the T-dual coming from an NS5-brane if the T-duals of any of the $U_i$ (see previous part) possess $H$-flux.
\[ThmTDGen\]
1. Let $E$ be a semi-free $S^1$-space with at most countably many isolated fixed sets of the $S^1$-action ${F_1,\ldots,F_k,\ldots}$. Let $W = E/S^1$ and let $U_l,l=1,\ldots,k,\ldots$ be open disjoint subsets of $E$, such that each $F_l \subsetneq U_l$. Since the fixed sets are isolated, we may always assume that $U_l \cap U_k = \phi$ for every $l \neq k$. Then, by the classification theorem for spaces with finitely many orbit types (see proof of Cor. (\[CorKKMulti\])), $P=(E - \bigcup^{k}_{l=1} U_l)$ is a principal circle bundle over $V=(W - \bigcup^{k}_{l=1}V_l)$ where $V_l=(U_l/S^1).$ Also, $E$ is determined by $P$ and the gluing data for the $U_l$. Since there is no $H$-flux on $E,$ these data determine the T-dual of $E.$
2. Given the data of the previous part, for every $i,$ suppose we are given atlases $V_i$ for $\underline{V_i}$ and induced atlases $Q_i$ for $\underline{U_i}$ in the sense of Ref. [@Bunke1]. Let $SV$ be the simplical space associated to $V$, and, for every $i,$ $SV_i$ the simplical space associated to $V_i$ by the above atlases. Similarly, let $SP$ be the simplicial space associated to $P$ and, for every $i,$ $SU_i$ the simplicial space associated to $U_i$ by the above atlases.
Consider the principal bundle of stacks $\underline{E} \to \underline{W}$. Consider the atlas $X=V \cup_{i} V_i$ for $W$. Then $V_i \cap V_j = \phi$ for all $i\neq j$. Also, $V_i \cap V$ need not be empty, but, $V_i \cap V \subseteq V_i.$ Now $X \underset{\underline{W}}{\times} X \simeq V \cup V_i \cup (V \cap V_i).$ Similarly, for the same reason, the $n$-fold fiber product $X \underset{\underline{W}}{\times} \ldots
\underset{\underline{W}}{\times}X \simeq V \cup_{i}V_i \cup (V \cap V_i).$ However, since $(V \cap V_i) \subseteq V_i,$ $X \underset{\underline{W}}{\times} X$ may always be written as $V \cup_{i} V_i.$ Also, in the associated simplicial space, $V$ is always glued to each $V_i$ while $V_i$ glue to themselves. Then, the simplicial space associated to $X$ is $SV \cup_f SV_1 \cup_{g_1} \ldots \cup_{g_k} SV_k \cup \ldots $ for some gluing maps $f,g_i$.
Consider the atlas $Y=P \cup_{i} Q_i$ for $\underline{E}$. Here also, we have that $Q_i \cap Q_j = \phi$ for all $i\neq j$. Also, $Q \cap Q_i \subseteq Q_i$ for every $i.$ This implies that $Y \underset{\underline{E}}{\times} Y$ may always be written as $P \cup_{i} Q_i$ by the intersection property of $P$ and $Q_i$ described above. Similarly the $n$-fold fiber product $Y \underset{\underline{E}}{\times} \ldots
\underset{\underline{E}}{\times}Y$ may always be written as $P \cup_{i} Q_i$ by the intersection property described above. Also, in the associated simplicial space, $P$ is always glued to each $P_i$ while the $P_i$ glue to themselves. Then, the simplicial space associated to $Y$ is $SP \cup_{f'} SQ_1 \cup_{g_1'} \ldots \cup_{g_k'} SQ_k \ldots $ for some gluing maps $f',g_i'$.
Therefore we have the associated principal bundle $(SP \cup_{f'} SQ_1 \cup_{g_1'} \ldots \cup_{g_k'} SQ_k \ldots)
\to (SV \cup_f SV_1 \cup_{g_1} \ldots \cup_{g_k} SV_k \ldots)$ where $f',f,
g_i', g$ are defined above.
By the remark before this theorem, the T-dual will be $$E^{\#}= (\left( SP \cup_f SQ_1 \cup_{g_1}
\ldots \cup_{g_k} SQ_k \ldots \right) \times S^1)$$ as a principal bundle over $$B^{\#} = \left( SV \cup_f SV_1 \cup_{g_1} \ldots \cup_{g_k} SV_k \ldots
\right).$$
Note that this is the principal simplicial bundle associated to $([E/S^1] \times S^1) \to [E/S^1].$ There will be nonzero $H$-flux on $E^{\#}$ due to the fact that the original bundle $E$ had nontrivial topology. There will be additional $H$-flux on $E^{\#}$ due to NS5-branes if there is nonzero $H$-flux on the T-dual of any of the bundles $SQ_i$ when T-dualized by themselves: By the remark before this Theorem, the $H$-flux on the total space of the simplicial bundle associated to $E^{\#}$ must restrict to this $H$-flux on the subspace $SV_i \times S^1$.
This Theorem lets us determine the T-dual of any semi-free space with countably many isolated fixed sets. This covers most of the semi-free spaces that would occur in a physical context. In particular, we may now determine the T-dual of a space with at most countably many isolated $KK$-monopoles.
Let $E$ be a semi-free $S^1$-space with at most countably many Kaluza-Klein monopoles ${p_1,\ldots,p_k,\ldots}$. Then, the T-dual is a trivial principal bundle glued to spaces of the form $([CS^3/S^1] \times S^1)$. There is $H$-flux present on the T-dual. \[CorKKMulti\]
This is an elementary application of Thm. (\[ThmTDGen\]). In $E$, since the $KK$-monopoles which are the fixed points of the $S^1$-action are isolated, it is possible to enclose each one in an open set homeomorphic to a ball $CS^3$. Thus, as topological spaces, each $U_i$ is equivariantly homeomorphic to $CS^3$. The atlases $U_i$ and $V_i$ may be chosen as in Thm. (\[ThmKK\]). This construction is always possible by the classification theorem for spaces with finitely many orbit types since there are only two orbit types (fixed points and free orbits) and the fixed points are at most countably many and isolated (See Ref. [@Bredon] Chap. V Sec. (5)).
Given this, the T-dual may be found. Note that $SV_i$ are simplicial bundles associated to spaces of the form $[CS^3/S^1] \times S^1$ with $H$-flux. Thus, the T-dual of $E$ is a stack which is a trivial principal bundle glued to stacks of the form $([CS^3/S^1] \times S^1)$.
There is $H$-flux present on the T-dual. There will be nonzero $H$-flux on the T-dual bundle due to the fact that the original bundle had nontrivial topology. There will be additional $H$-flux on this bundle due to NS5-branes if there is nonzero $H$-flux on the T-dual of any of the $U_i$. This is because these will then contribute to a nonzero $H$-flux on the associated simplicial bundle: By Thm. (\[ThmKK\]), there is nonzero $H$-flux on the T-dual of any of the $SU_i \subseteq E$, i. e. there is a $H$-flux on $(SV_i \times S^1) \subseteq E^{\#}$. By the remark before Thm. (\[ThmTDGen\]), the $H$-flux on $(SV_i \times S^1)$ is the restriction of the $H$-flux on the T-dual to $SV_i \times S^1$. However, by the above, this restriction is nonzero hence the T-dual $H$-flux cannot be zero.
Note that unlike the $C^{\ast}$-algebraic case ([@Pande]) there is no [*source*]{} of $H$-flux on the T-dual. However, the T-dual does possess $H$-flux. We make this precise in the following:
1. The T-dual $E^{\#}$ of a semi-free $S^1$-space $E$ with at most countably many $KK$-monopoles is the principal bundle of stacks $[E/S^1] \times S^1 \to [E/S^1]$ with $H$-flux.
2. $E^{\#}$ is a topological stack which is not equivalent to a topological space if and only if the $S^1$-action on $E$ has fixed sets.
3. The natural map $\phi_{mod}:\underline{E^{\#}} \to E^{\#}_{mod}$ is a homeomorphism iff the $S^1$-action on $\underline{E}$ has no fixed sets.
\[CorESt\]
1. It follows from the proof of Thm. (\[ThmTDGen\]) that the simplicial bundle associated to the T-dual stack is the trivial bundle over the base with $H$-flux. This is the simplicial bundle associated to the principal bundle of stacks $[\underline{E}/S^1] \times S^1 \to [\underline{E}/S^1]$ with $H$-flux.
2. First note the following: If $\underline{X}$ is a stack equivalent to a topological space $X,$ then, restricting the equivalence to a substack shows that every substack of $\underline{X}$ is equivalent to a topological space. Suppose the action had no fixed sets, i. e. none of the $U_i$ was present in $E$, then, from the proof of Thm. (\[ThmTDGen\]) $E$ would be a topological space and so would $E^{\#}$. Now suppose the $S^1$-action on $E$ had fixed sets. Then one of the $U_i$ would be present in $E$, then, from the same proof, the T-dual would contain substacks of the form $[U_i/S^1] \times S^1$. Here, by the classification theorem for spaces with finitely many orbit types (see proof of Cor. (\[CorKKMulti\]) and by the proof of Thms. (\[ThmKK\],\[ThmTDGen\])), each of these would be equivalent to stacks of the form $[CS^3/S^1] \times S^1$. These are not equivalent to topological spaces[^5]. As a result, the T-dual could not be equivalent to a topological space.
3. Suppose the $S^1$-action on $E$ had fixed points and the map $\overline{\phi_{mod}}:\underline{E^{\#}} \to \underline{E^{\#}_{mod}}$ induced by $\phi_{mod}$ was an equivalence of stacks. Then, by the proof of the previous part choosing suitable neighbourhoods of the fixed points will give an inclusion of stacks $[CS^3/S^1] \to \underline{E^{\#}}.$ The above would imply that the map $\overline{\phi'_{mod}}:[CS^3/S^1] \to \underline{CS^2 \times S^1}.$ would be an equivalence of stacks. Since the stack cohomology groups of $CS^2 \times S^1$ are different from the stack cohomology groups of $[CS^3/S^1] \times S^1,$ this is impossible.
Conversely, suppose the $S^1$-action on $E$ had no fixed points. Then, by the previous part of the theorem, the T-dual stack would be equivalent to a space and so $\phi_{mod}$ would give an equivalence $\overline{\phi_{mod}}:\underline{E^{\#}} \to \underline{E^{\#}_{mod}}.$
Consider the T-dual of $\underline{CS^3}$: The coarse moduli space of $[CS^3/S^1]$ is $CS^2$ (See Ref. [@Noohi1] Example (4.13), $[CS^3/S^1]$ is the quotient stack of the transformation groupoid $((CS^3 \times S^1)\rightrightarrows CS^3)$). However, $H^3(CS^2 \times S^1,{{\mathbb Z}})=0$, so there can be no $H$-flux on the topological space $CS^2 \times S^1$. By the above, however, the [*stack*]{} $[CS^3/S^1] \times S^1$ possesses $H$-flux. This is because the simplicial bundle associated to this stack (see proof of Thm. (\[ThmKK\])) is nontrivial, and so the [*stack*]{} cohomology groups of $[CS^3/S^1]$ are nontrivial.
Since the $H$-flux on the T-dual stack would vanish (see Cor. \[CorKKMulti\]) if there were no fixed points of the $S^1$-action on the original space, presumably this $H$-flux is the flux generated by the T-dual NS5-brane. Note that this also happens for the T-dual of $[CS^3/{{\mathbb Z}}_k]$ for $k>1$ since the T-duals are the same as the case above only the $H$-flux changes.
This should also happen in the example in Cor. (\[CorKKMulti\]) above: The T-dual is a principal bundle $P$ glued to copies of $([CS^3/S^1] \times S^1)$. By the proof of Ref. ([@Bunke]), the T-dual is a topological stack. As a space, the coarse moduli space of the T-dual will be $P \times S^1$ glued to $CS^2 \times S^1$. Also, by Cor. (\[CorESt\]) the T-dual coarse moduli space will be a trivial principal circle bundle. The $CS^2$ factor is contractible and the resulting space cannot have nonzero $H$-flux coming from an NS5-brane. (The space will have $H$-flux only due to the $H$-flux on $P^{\#}$). However, the T-dual [*stack*]{} does have $H$-flux coming from this source.
Note that in all these T-duals (see also Cor. (\[CorESt\])) above, the reason the T-dual has a nontrivial $H$-flux is due to the fact that the stack cohomology groups of $[CS^3/S^1]$ are [*different*]{} from those of the coarse moduli space $\underline{CS^2}.$
The Dyonic Coordinate
=====================
In String Theory backgrounds which contain $KK$-monopoles possess a dyonic coordinate. (See Ref. [@Sen] for details. See also Ref. [@HaJen]). Roughly speaking, a large gauge transformation of the $B$-field on a $KK$-monopole background under T-duality corresponds to a rotation of the T-dual NS5-brane around its circle fiber. A model for this was constructed for $KK$-monopole backgrounds using $C^{\ast}$-algebraic methods in Ref. [@Pande].
Large gauge transformations of a gerbe on a space $X$ are given by a class in $H^2(X,{{\mathbb Z}})$ (See Ref. [@TTDA]). We would like to understand the behaviour of these classes under Topological T-duality for [*semi-free*]{} spaces. As we have argued earlier, the $S^1$-spaces underlying $KK$-monopole spacetimes are semi-free spaces.
Using the results of Ref. [@TTDA], we show below that for these semi-free spaces $X$, an automorphism of a trivial gerbe on $\underline{X}$ gives a class in $H^2(X^{\#},{{\mathbb Z}})$ under Topological T-duality.
\[secDyon\]
1. Consider the principal bundle of stacks $[{\protect{\mathrm{pt}}}/{{\mathbb Z}}_k] \to [{\protect{\mathrm{pt}}}/S^1]$ for $k=2$. Consider a trivial gerbe on this stack. Each cyclic subgroup of the group of automorphisms of the gerbe on $[{\protect{\mathrm{pt}}}/{{\mathbb Z}}_k]$ gives rise to a cyclic subgroup of $H^2([{\protect{\mathrm{pt}}}/S^1]\times S^1,{{\mathbb Z}})$. For $k=2$, this may be calculated explicitly.
2. Consider the principal bundle of stacks corresponding to a $KK$-monopole of charge $k$. Consider a trivial gerbe on the total space of the principal bundle. Each cyclic subgroup of automorphisms of the trivial gerbe on the $KK$-monopole of charge $k>1$ gives rise to a cyclic subgroup of the (second) cohomology of the T-dual $H^2([CS^3/S^1] \times S^1,{{\mathbb Z}})$.
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1. Consider a cyclic subgroup of the group of automorphisms of the trivial gerbe on $[{\protect{\mathrm{pt}}}/{{\mathbb Z}}_k]$. It is enough to prove the result for the generator of this subgroup. An automorphism of the trivial gerbe on $[{\protect{\mathrm{pt}}}/{{\mathbb Z}}_k]$ gives rise to a class in $H^2([{\protect{\mathrm{pt}}}/{{\mathbb Z}}_k],{{\mathbb Z}})$. Consider the proof of Part (2) of Thm. (\[ThmKK\]). The simplicial bundle associated to the principal bundle of stacks $[{\protect{\mathrm{pt}}}/{{\mathbb Z}}_k] \to [{\protect{\mathrm{pt}}}/S^1]$ is $p:B{{\mathbb Z}}_k \to BS^1$. Since we have a class in $H^2([{\protect{\mathrm{pt}}}/{{\mathbb Z}}_k],{{\mathbb Z}})$, we obtain a cohomology class on the simplicial space associated to this stack (See Ref. [@Heinloth], the proof of Prop. (4.7)). In turn this gives a cohomology class on its geometric realization $B{{\mathbb Z}}_k$ in $H^2(B{{\mathbb Z}}_k,{{\mathbb Z}})$.
By the argument in Ref. [@TTDA], Thm. (6.3), this class gives rise to a natural class in the second cohomology group of the T-dual bundle $q:BS^1 \times S^1 \to BS^1$. For all natural numbers $k > 1$, $H^2(B{{\mathbb Z}}_k,{{\mathbb Z}}) \simeq {{\mathbb Z}}/k$. Under T-duality an element of $H^2(B{{\mathbb Z}}_k,{{\mathbb Z}})$ gives a class of the form $kq^{\ast}(a) \simeq k (a \times 1) \in H^2(BS^1 \times S^1,{{\mathbb Z}})$ for some unknown integer $k$ (where $a$ is the generator of $H^2(BS^1,{{\mathbb Z}})$ (See Ref. [@TTDA], Thm. (6.3)). Thus, an automorphism of the gerbe on $[{\protect{\mathrm{pt}}}/{{\mathbb Z}}_k]$ with this characteristic class gives rise to a cohomology class on the T-dual stack $[{\protect{\mathrm{pt}}}/S^1] \times S^1$.
2. This part of the proof is very similar to the previous part. Consider a cyclic subgroup of the group of automorphisms of the trivial gerbe on $[CS^3/{{\mathbb Z}}_k]$. It is enough to prove the result for the generator of this subgroup. The proof is similar to the proof for the previous part, with the principal bundle changed. An automorphism of the trivial gerbe on $[CS^3/{{\mathbb Z}}_k]$ gives rise to a class in $H^2([CS^3/{{\mathbb Z}}_k],{{\mathbb Z}})$. Consider the proof of Part (3) of Thm. (\[ThmKK\]). The simplicial bundle associated to the principal bundle of stacks $[CS^3/{{\mathbb Z}}_k] \to [CS^3/S^1]$ is $(A \underset{*}{\times} B{{\mathbb Z}}_k) \to (B \underset{*}{\times} BS^1)$ (where $A$ and $B$ are defined in Thm. (\[ThmKK\]). Let $P_k= (A \underset{*}{\times} B{{\mathbb Z}}_k)$ and $W= (B \underset{*}{\times} BS^1)$
Since we have a class in $H^2([CS^3/{{\mathbb Z}}_k],{{\mathbb Z}})$, this gives a cohomology class on the simplicial space associated to this stack (See Ref. [@Heinloth], the proof of Prop. (4.7)) and this, in turn, gives a cohomology class on its geometric realization, that is, a class in $H^2(P_k, {{\mathbb Z}})$. The T-dual bundle is $W \times S^1 \to W$ with $H$-flux, and, by Ref. [@TTDA], a class in $H^2(P_k,{{\mathbb Z}})$ under Topological T-duality naturally gives rise to a class in $H^2(W \times S^1, {{\mathbb Z}})$.
Thus a class in $H^2(X,{{\mathbb Z}})$ naturally gives rise to a class in $H^2(X^{\#},{{\mathbb Z}})$. For a $KK$-monopole background, a large gauge transformation of the $B$-field gives rise to a class in $H^2(X,{{\mathbb Z}})$. By the above this induces a class in $H^2(X^{\#},{{\mathbb Z}})$. For the analogy with the dyonic coordinate to be complete, the induced class in $H^2(X^{\#},{{\mathbb Z}})$ should be viewed as an automorphism of the T-dual semi-free space $X_k^{\#}$ which rotates each fiber through $2\pi$. However, it is not clear how to prove this in the stack picture.
A similar construction was made in the $C^{\ast}$-algebraic picture of topological T-duality in Ref. [@Pande] where such a rotation did correspond to a [*nontrivial*]{} [*spectrum-fixing*]{} automorphism of the T-dual $C^{\ast}$-algebra: The T-dual automorphism obtained there was of this type.
The Gysin Sequence {#secGysin}
==================
Consider a principal circle bundle of stacks $p:{{\mathcal E}}\to {{\mathcal F}}$. From Ref. [@Ginot], a Gysin sequence may be constructed for this principal bundle in homology. Can a cohomology Gysin sequence be so constructed? We construct this Gysin sequence below. We also show that for stacks ${{\mathcal E}}$ which are the stacks associated to a semi-free $S^1$-space $E$ a ‘T-dual’ stack may be constructed using this Gysin sequence. We show that that this T-dual stack is the stack associated to the T-dual of $E$ which would be obtained in the formalism of Mathai and Wu (See Ref. [@MaWu]). We show at the end of this section that the stack T-dual defined in this section agrees with the T-dual of Bunke et al. in the examples of Thm. (\[ThmKK\]).
Let $q:{{\mathcal E}}\to {{\mathcal Y}}$ be a principal $S^1$-bundle of stacks. There is a cohomology Gysin sequence for this bundle if $q$ is adequate in the sense of Behrend et al. [@BGNX] Def. (7.4). \[LemGysin\]
Suppose $q$ is adequate, then the cohomology transfer map $T_{S^1}$ (See Definition (8.2) of Ref. [@Ginot]) is well defined since the product $x \cdot u^{\ast}(\theta_S^1)$ in that definition is defined (See Ref. [@BGNX] paragraph after Ex. (7.5)).
The proof of Prop. (8.4) of Ref. [@Ginot] may now be followed with $H_{\ast}$ replaced by $H^{\ast}$. It is clear that the full proof goes through. Let $Z \to [S^1 \backslash {{\mathcal E}}]$ be a classifying space for $[S^1 \backslash {{\mathcal E}}]$ and $Y \to {{\mathcal E}}$ be the classifying space for ${{\mathcal E}}$ obtained by pullback along $q$. Let $c$ be the Euler class of disk bundle associated to the principal bundle $Y \to Z$, then we obtain the following Gysin Sequence: $$\ldots \to H^{i-1}_{S^1}({{\mathcal E}}) \overset{q^{\ast}}{\to} H^{i-1}({{\mathcal E}})
\overset{T_{S^1}}{\to} H^{i-2}_{S^1}({{\mathcal E}}) \overset{\cup c}{\to} H^{i}_{S^1}({{\mathcal E}})
\overset{q^{\ast}} \to H^i({{\mathcal E}}) \to \ldots
\label{CohoGysin}$$ from the cohomology Gysin sequence associated to $Y \to Z$ under the identifications $H^i(Z) \simeq H^i([S^1 \backslash {{\mathcal E}}]) \simeq H^i_{S^1}({{\mathcal E}})$, and $H^i(Y) \simeq H^i({{\mathcal E}})$ exactly as in Ref. [@Ginot], Prop. (8.4).
Recall from Sec. (\[secTDKK\]) above that the semi-free spaces we consider are all total spaces of $KK$-monopoles. In particular, they are all oriented orbifolds. By Prop. (8.35) of Ref. [@BGNX], they are all strongly oriented in the sense of Ref. [@BGNX] Def. (8.21).
Let ${{\mathcal E}}$ be a stack associated to a semi-free space. Let ${{\mathcal Y}}= [S^1 \backslash {{\mathcal E}}]$ and let $p:{{\mathcal E}}\to {{\mathcal Y}}$ be the quotient map. Let ${{\mathcal V}}$ be the associated vector bundle to the principal bundle $p:{{\mathcal E}}\to {{\mathcal Y}}.$ Suppose ${{\mathcal V}}$ is metrizable. Further, suppose ${{\mathcal Y}}$ is strongly oriented in the sense of Ref. [@BGNX], Def. (8.21). Then $p$ is strongly oriented in the sense of Ref. [@BGNX], Def. (8.21). Also, $p$ is adequate. \[LemAdeq\]
Firstly, ${{\mathcal E}}$ is strongly oriented due to Prop. (8.35) of Ref. [@BGNX]. Secondly, ${{\mathcal Y}}$ is strongly oriented by assumption. Thirdly, we will argue that $p$ is strongly proper and normally nonsingular. Hence, $p:{{\mathcal E}}\to {{\mathcal Y}}$ has a strong orientation class by Prop. (8.32) of Ref. [@BGNX].
By Lemma. (\[StLem1\]) above, $p:{{\mathcal E}}\to {{\mathcal Y}}$ is a principal $S^1$-bundle. By Ref. [@Ginot], it is also representable (By Prop. (4.7) of Ref. [@Ginot]). Let $q:{{\mathcal V}}\to {{\mathcal Y}}$ be the associated vector bundle. ${{\mathcal V}}$ is metrizable by assumption. Let $s:{{\mathcal E}}\to {{\mathcal V}}$ be the embedding of ${{\mathcal E}}$ as the unit sphere bundle in ${{\mathcal V}}.$ Then, the following diagram commutes $$\begin{CD}
{{\mathcal V}}@>{id}>> {{\mathcal V}}\\
@AA{s}A @VV{q}V \\
{{\mathcal E}}@>{p}>> {{\mathcal Y}}.
\end{CD}$$ This is the required normally nonsingular diagram for $p.$
It is clear that $p$ is bounded proper from Def. (6.1) of Ref. [@BGNX]. It remains to prove that $p$ is strongly proper. Let $w:{{\mathcal C}}\to {{\mathcal E}}$ be an orientable metrizable vector bundle on ${{\mathcal E}}.$ Choose classifying spaces $Y \to {{\mathcal Y}}$ and $E \to {{\mathcal E}},$ so we obtain a principal bundle $E \to Y.$ Pulling ${{\mathcal V}}$ back to $Y$ we obtain a bundle ${{\mathcal V}}\underset{{{\mathcal Y}}}{\times} Y \to Y.$ The pullback of this bundle along the map $E \to Y$ is trivial since the bundle has the same Euler class as the bundle of stacks ${{\mathcal V}}\to {{\mathcal Y}}$ (this follows from the definition of the Euler class, see Behrend et al. Ref. [@BGNX] Ex. (8.26)). It follows that $p^{\ast}({{\mathcal V}})$ is a trivial bundle. Hence, taking charts, there is an integer $n > 0$ such that ${{\mathcal C}}$ is a direct summand of $(p^{\ast}({{\mathcal V}}))^n \simeq p^{\ast}({{\mathcal V}}^n).$ Now, ${{\mathcal V}}$ is the vector bundle associated to ${{\mathcal E}},$ and, since $p$ is orientable, so is ${{\mathcal V}}.$ Further, ${{\mathcal V}}$ is metrizable by assumption and hence, so is ${{\mathcal V}}^n.$ Hence, $p$ is strongly proper. By the above $p$ is normally nonsingular. Hence, $p$ has a strong orientation class by Prop. (8.32) of Ref. [@BGNX]. By Ex. (7.5) (1) of Ref. [@BGNX], $p$ is adequate.
For all the examples of $KK$-monopoles in Thm. (\[ThmKK\]) above, the total space is the stack $[CS^3/{{\mathbb Z}}_k]$ associated to an oriented orbifold and hence strongly oriented by Prop. (8.35) of Ref. [@BGNX]. From Thm. (\[ThmKK\]) above, the quotient stack by the $S^1$-action is always $[CS^3/S^1]$ and any vector bundle ${{\mathcal V}}$ over $[CS^3/S^1]$ is metrizable by Ex. (3.3) of Ref. [@BGNX]. Also $[CS^3/S^1]$ is strongly oriented by Prop. (8.33) and Def. (8.21) of Ref. [@BGNX]. Hence, in all the examples of $KK$-monopoles calculated above, the bundle maps $p_k:[CS^3/{{\mathbb Z}}_k] \to [CS^3/S^1]$ and $p:\underline{CS^3} \to [CS^3/S^1]$ are adequate. Thus, we may use the Gysin sequence argument above to obtain the T-dual.
Let $E$ be a non-free $S^1$-space. Let ${{\mathcal E}}$ be the underlying stack. Consider the stack $[E/S^1]$. This has a natural presentation as the transformation groupoid ${{\mathbb E}}= [(E \times S^1) \rightrightarrows E]$. This stack has a natural[^6] classifying space $B{{\mathbb E}}$ (the Haefliger-Milnor Classifying Space) associated to this groupoid which is given by the Borel construction $E \times_{S^1} ES^1$. The principal bundle of stacks $p:{{\mathcal E}}\to [E/S^1]$ gives a principal bundle of spaces $E{{\mathbb E}}\simeq (E \times EG) \to B{{\mathbb E}}\simeq (E \times_{S^1} ES^1)$ by pullback and a $2$-cartesian square[^7] $$\begin{CD}
Y=(E \times ES^1) @>f>> E\\
@VqVV @VVV \\
Z=(E \times_{S^1} ES^1) @>{\phi}>> [E/S^1]
\end{CD}$$ where the space in each row is a classifying space[^8]. Also, the map $\phi$ is [*natural*]{} (See Ref. [@Noohi3], before Sec. (4.2)).
In Ref. [@MaWu] Mathai and Wu obtain the T-dual of a non-free $S^1$-space $E$ using the Borel construction: To the space $E$ they associate the same principal circle bundle of spaces $E \times ES^1 \to E \times_{S^1} ES^1$ as described above.
Given a principal bundle of stacks $p:{{\mathcal E}}\to {{\mathcal Y}}$, for which a Gysin sequence exists, one could obtain a T-dual principal bundle of stacks by the following procedure: To $p$ we associate the principal bundle of classifying spaces above $Y \to Z$ (See Ref. [@Ginot], proof of Thm. (8.4)). Note that even if ${{\mathcal E}}$ was the stack associated to a space with a non-free $S^1$-action, the principal bundle of classifying spaces would be a principal [*circle bundle*]{}.
Given a $S^1$-gerbe on ${{\mathcal E}}$, we place an $H$-flux with characteristic class equal to the characteristic class of this gerbe (See Ref. [@Heinloth], Prop. (5.8)) on the total space of this principal bundle of classifying spaces. We can then calculate a Topological T-dual principal circle bundle $Y^{\#} \to Z$ for the associated principal bundle of classifying spaces (possibly with $H$-flux) using the usual Gysin sequence argument (See Ref. [@BEM]). Note that this is similar to the argument of Mathai and Wu in Ref. [@MaWu] for the T-dual of a non-free circle action and $Y^{\#}$ is the T-dual space that they obtain.
If a principal bundle of [*stacks*]{} $p^{\#}:{{\mathcal E}}^{\#} \to {{\mathcal Y}}$ existed which had the spaces $Y^{\#}, Z$ in this bundle as classifying spaces, then we would say that $p^{\#}$ (possibly with a gerbe on it corresponding to the the T-dual $H$-flux above, if any) was the Topological T-dual stack.
Note that the result of the procedure above is the same as calculating the T-dual of $p$ using the argument for principal circle bundles with the ordinary Gysin sequence replaced by the stack Gysin sequence. The only problem is that if the stack Gysin sequence is used directly, unlike the case of a principal circle bundle, there is no guarantee that $p^{\#}$ exists.
It turns out that such a dual bundle of stacks always exists for semi-free spaces because the above T-dual is connected to the T-dual of Ref. [@MaWu]:
Let $E$ be a semi-free space with a $S^1$-action for which the results of Ref. [@MaWu] hold. Suppose $p:\underline{E} \to [E/S^1]$ is adequate in the sense of Ref. [@BGNX]. Then, the T-dual of $\underline{E}$ using the stack Gysin sequence above exists and is $\underline{E^{\#}}$ where $E^{\#}$ is the T-dual of $E$ obtained using the formalism of Ref. [@MaWu]. There will be $H$-flux on the T-dual if the T-dual of Ref. [@MaWu] had $H$-flux. \[ThmStGys\]
Consider a semi-free space $E$. Let $p:{{\mathcal E}}\to [E/S^1]$ be the associated principal bundle of stacks. By Ref. [@MaWu] after Prop. (1), we have a principal circle bundle $\hat{E} \to E$ and the T-dual is $\hat{p}:E^{\#}\simeq \hat{E}/S^1 \to \hat{E}/S^1.$
The map $\hat{E} \to E$ can be replaced by the associated map of stacks $\hat{\underline{E}} \to \underline{E}.$ The $S^1$-action on the stacks in this map gives rise to a quotient map of stacks $\hat{p}:[\underline{\hat{E}}/S^1] \to [E/S^1].$ We claim that $\hat{p}$ is a principal bundle of stacks and that it is the principal bundle of stacks obtained from $p$ using the Gysin sequence argument described above. Also, Eq. (1) of Ref. [@MaWu] is a principal bundle of classifying spaces associated to this principal bundle of stacks.
By the proof of Thm. (6.3) of Ref. [@Noohi3] we have classifying spaces $E \times_{S^1} ES^1 \to [E/S^1]$ and $\hat{E} \times_{S^1} ES^1 \to [\hat{E}/S^1]$ which are atlases for these stacks. Hence, pulling back the map $\hat{p}$ along the atlas $E \times_{S^1} ES^1 \to [E/S^1]$ gives the principal circle bundle $\hat{E} \times_{S^1} ES^1 \to E \times_{S^1} ES^1$ on the atlas. By definition, (see Def. (2.11) of Ref. [@Heinloth]) this implies that $[\hat{E}/S^1] \to [E/S^1]$ is a principal bundle of stacks. It is clear that Eq. (1) of Ref. [@MaWu] is a principal bundle of classifying spaces associated to this principal bundle of stacks.
It remains to prove that this principal bundle of stacks is the one obtained from $p$ using the Gysin sequence argument described above.
From Ref. [@MaWu] second paragraph after Prop. (1), the bundle $\hat{E} \to E$ defines an equivariant characteristic class $c_1^{S^1}(\hat{E}) \in H^2_{S^1}(E,{{\mathbb Z}})$ which is equal to $p_{\ast}([H]).$ By Ref. [@BGNX] before Prop. (2.5), $H^2_{S^1}(E,{{\mathbb Z}}) \simeq H^2([E/S^1]),$ the stack cohomology of the moduli space. Hence, we obtain a class in $H^2([E/S^1])$ from the principal bundle $\hat{E} \to E$ above. We claim that this class is the class of the T-dual bundle $\hat{p}:[\hat{E}/S^1] \to [E/S^1]:$ By definition and the discussion above, the characteristic class of $\hat{p}$ is the characteristic class of the bundle of classifying spaces $\hat{E} \times_{S^1} ES^1 \to E \times_{S^1} ES^1.$ By the discussion above, this can only be the class $c^{S^1}_1 \in H^2_{S^1}(E,{{\mathbb Z}}) \simeq H^2(E \times_{S^1} ES^1,{{\mathbb Z}}),$ the equivariant characteristic class of the bundle $\hat{E} \to E.$
By item (1) before Thm. (1) of Ref. [@MaWu], the T-dual $H$-flux has the property $\hat{p}_{\ast}([\hat{H}]) = e_{S^1} \in H^2_{S^1}(E,{{\mathbb Z}}).$ By the above $H^2_{S^1}(E,{{\mathbb Z}}) \simeq H^2([E/S^1])$ and the class $e_{S^1}$ by definition maps to the characteristic class of the bundle of stacks $p:\underline{E} \to [E/S^1].$ (By Ref. [@BGNX], (see Sec. (2.6), paragraph before Eq. (2.1)) the singular cohomology of a stack is the singular cohomology of its classifying space. Also, the characteristic class of a bundle of stacks ${{\mathcal Y}}\to {{\mathcal X}}$ is the characteristic class of its associated bundle of classifying spaces $Y \to X$ in $H^2(X,{{\mathbb Z}}).$ For the bundle of stacks $p:\underline{E} \to [E/S^1]$ we obtain a bundle of classifying spaces $E \times ES^1 \to E \times_{S^1} ES^1.$ The class $e_{S^1}$ defined above is, by definition (see Ref. [@MaWu] before Prop. (1)) the class of this bundle. This gives the result.) Thus we obtain that $p_{\ast}([H])$ is the class of the T-dual bundle $\hat{p}$ and $\hat{p}_{\ast}([\hat{H}])$ is the class of the bundle $p.$ This implies that the T-dual bundle and $H$-flux are identical to those obtained from the Gysin sequence argument above.
It is not clear what the relation is between the stack T-dual obtained using classifying spaces in this section and that obtained from the formalism of Bunke et al. (Ref. [@Bunke1]). Note first that the simplicial space associated to a groupoid associated to a stack is a classifying space (See Ref. [@Noohi3], Sec. (4.2)). Roughly, to a stack the formalism of Bunke et al. associates a principal bundle of classifying spaces each space of which is the associated simplicial space of a stack in that bundle. The T-dual is obtained by T-dualizing this bundle.
In addition to the simplicial space, it was argued above that one could also associate to a a stack another classifying space the Haefliger-Milnor classifying space. For a principal bundle of stacks, one could pick a groupoid presentation of each stack and pass to the Haefliger-Milnor classifying space of each groupoid. One could define a T-dual stack by T-dualizing this bundle. However, in general this classifying space depends on the groupoid presentation. For the quotient stack of a semi-free space, there is a natural choice of the associated groupoid and hence the associated classifying space.
The two stack T-duals are closely related, and, in most interesting cases, identical. This is because the stack T-dual of Ref. [@Bunke] is connected to the simplicial space associated to a stack while the stack T-dual defined in this section is connected to the Haefliger-Milnor classifying space also associated to that stack (See Ref. [@Noohi3], Sec. (4.2)).
We show that for the T-duals of Thm. (\[ThmKK\]), the two formalisms give the same result:
For every $k > 1$, $k$ a natural number, the T-dual of the principal bundle of stacks $p_k:[CS^3/{{\mathbb Z}}_k] \to [CS^3/S^1]$ using classifying spaces is the principal bundle of stacks $q:([CS^3/S^1]\times S^1) \to [CS^3/S^1]$ with $H$-flux. \[CorKKNew\]
The Borel construction $CS^3 \times_{S^1} ES^1$ is a classifying space for the stack $[CS^3/S^1]$. Also, the Borel construction gives $(CS^3/S^1) \times B{{\mathbb Z}}_k$ as the classifying space for the stack $[CS^3/{{\mathbb Z}}_k]$. By definition, the principal bundle of classifying spaces associated to the principal bundle of stacks $p_k$ above is the principal circle bundle $(CS^3/S^1) \times B{{\mathbb Z}}_k \to (CS^3/S^1) \times BS^1$. Using the Gysin Sequence for this principal bundle of spaces, the T-dual would be the principal circle bundle $q:((CS^3/S^1) \times BS^1) \times S^1 \to ((CS^3/S^1) \times BS^1)$ with $H$-flux. Now, by the above, $((CS^3/S^1) \times BS^1)$ is the classifying space associated to the stack $[CS^3/S^1]$. From the proof of Thm. (\[ThmStGys\]) above, the principal circle bundle $q$ above is the bundle of classifying spaces associated to the principal bundle of stacks $q:([CS^3/S^1] \times S^1)
\to [CS^3/S^1]$. The fact that there is a $H$-flux present on the bundle of classifying spaces implies that there is a gerbe on this bundle of stacks.
By the discussion in the paragraph after Lemma (\[LemAdeq\]) above, each of the maps $p_k$ above is adequate. Hence, one may use the Gysin sequence to calculate the T-dual. It is clear by inspecting the argument that the T-dual will be exactly the same as that obtained in Cor. (\[CorKKNew\]) above.
Final Remarks
=============
In this paper we have studied the Topological T-dual of spaces containing $KK$-monopoles. In Secs. (\[secComp\],\[secTDKK\]) of the paper above, we have explicitly calculated the T-dual of several spaces with $KK$-monopoles. In Sec. (\[secDyon\]) we have attempted to model the ‘Dyonic Coordinate’ associated with $KK$-monopoles within the stack theory formalism. In Ref. [@Pande] we had obtained a model for the same phenomenon using the $C^{\ast}$-algebraic formalism of Topological T-duality. It is interesting that the same phenomenon appears in two completely independent approaches to Topological T-duality.
We now make a few remarks concerning the relation between the three formalisms of Topological T-duality for semi-free spaces with particular reference to the above calculations involving spaces containing $KK$-monopoles: The formalism of Bunke et al. [@Bunke; @Bunke1; @Bunke2] obtains the topological T-dual of the stack associated to a $KK$-monopole by passing to the associated simplicial bundle and taking the Topological T-dual of the simplicial bundle. Obviously, the Topological T-dual of the associated simplicial bundle agrees with the $C^{\ast}$-algebraic T-dual of that bundle. If the stack which one was T-dualizing was the stack associated to the total space of a principal circle bundle, the formalism of Bunke et al. would give the same answer as the $C^{\ast}$-algebraic formalism. Thus, the formalism of Bunke ‘regularizes’ the neighbourhood of the fixed point by passing to the associated simplicial bundle.
Also, as has been argued above, the $C^{\ast}$-algebraic T-dual lifts the $S^1$-action on the space to a ${{\mathbb R}}$-action on the $C^{\ast}$-algebra. The formalism of Bunke et al. does not lift the $S^1$-action to a ${{\mathbb R}}$-action. This causes a difference in T-duals when fixed points are encountered. The formalism of Mathai-Wu [@MaWu] also does not lift the $S^1$-action to a ${{\mathbb R}}$-action.
As has been argued in section Sec. (\[secGysin\]) before Cor. (\[CorKKNew\]), the Topological T-dual obtained from the Gysin sequence formalism and that obtained from the formalism of Bunke et al. should agree for most spaces. Since the T-dual obtained from the formalism of Mathai-Wu agrees with the T-dual obtained from the Gysin sequence formalism by Thm. (\[ThmStGys\]), the T-dual obtained by Bunke et al. should agree with the T-dual obtained by Mathai-Wu in most cases. It should differ from the $C^{\ast}$-algebraic T-dual for the reasons discussed above.
It is interesting to note that the calculation of Topological T-duals for $U(1)$-gerbes on a principal ${{\mathbb T}}^n$-bundle over an arbitrary topological groupoid using the theory of crossed products of groupoid $C^{\ast}$-algebras has also been done by Daenzer in Ref. [@Daenzer]. The formalism can T-dualize non-free group actions. It would be interesting to compare the results of that formalism with the results of this paper.
Acknowledgements {#acknowledgements .unnumbered}
================
[aaaa]{} V. Mathai and J. Rosenberg, ‘T-duality for Torus Bundles with H-fluxes via Noncommutative Topology’, [*Comm. Math. Phy.*]{}, [**253**]{}, pp. 705-721, (2005). U. Bunke and T. Schick, ‘On the Topology of T-duality’, [*Rev. Math. Phys.*]{}, [**17**]{}, 1, pp. 77-112, (2005). Also at [arXiv:math/0405132]{}. Ulrich Bunke and Thomas Schick, ‘T-duality for non-free circle actions’, [*‘Analysis, geometry and topology of elliptic operators’*]{}, pp. 429–466, World Sci. Publ., Hackensack, NJ, 2006. Also at [arXiv:math/0508550]{}. Ulrich Bunke, Markus Spizweck and Thomas Schick, ‘Periodic twisted cohomology and T-duality’, [*Asterisque*]{}, [**337**]{}, (2011). Also at [arXiv:0805.1459]{}. Behrang Noohi, ‘Foundations of Topological Stacks I’, [arXiv:math/0503247]{}. Behrang Noohi, ‘Fibrations of topological stacks’, [arXiv:1010.1748]{}. J. Heinloth, ‘Some Notes on Differentiable Stacks’, [*Mathematisches Institut, Seminars*]{}, Y. Tschinkel, ed., p. 1-32, Universität Göttingen, 2004-05. Ashwin S. Pande, ‘Topological T-duality and $KK$-monopoles’, [*Adv. Theor. Math. Phys.*]{}, [**12**]{}, pp. 185-215, (2007). Reviewed in [*Mathscinet*]{}, Review number [**MR2369414 (2008j:53040)**]{}. Also at [arXiv:math-ph/0612034]{}. Varghese Mathai and Siye Wu, ‘Topology and Flux of T-dual Manifolds with Circle Actions’, [*Comm. Math. Phys.*]{}, [**316**]{}, 1, pp. 279-286, (2012). Also at [arXiv:1108.5045]{}. A. Sen, ‘Kaluza-Klein Dyons in String Theory’, [*Phys. Rev. Lett.*]{}, [**79**]{}, 1619, (1997). J. A. Harvey and S. A. Jensen, ‘Worldsheet Instanton Corrections to the Kaluza-Klein monopole’, [*JHEP*]{}, [**0510**]{}, 028, (2005). Also at [hep-th/0507204]{}. A. Pande, ‘Topological T-duality, Automorphisms and Classifying Spaces’, preprint, submitted for publication. Also at [arXiv:1211.2890]{}. Gregory Ginot and Behrang Noohi, ‘Group actions on stacks and applications to equivariant string topology for stacks’, [arXiv:1206.5603]{}. J. P. May, ‘The Geometry of Iterated Loop Spaces’, [*Lecture Notes in Mathematics*]{}, [**271**]{}, Springer-Verlag, (1972). Glen E. Bredon, ‘Introduction to Compact Transformation Groups’, [*Pure and Applied Mathematics*]{}, [**46**]{}, Academic Press, (1972). I. Raeburn and J. Rosenberg, ‘Crossed product of continuous-trace $C^{\ast}$-algebras by smooth actions’, [*Trans. Am. Math. Soc.*]{}, [**305**]{}, 1, pp. 1–45. Kai Behrend, Gregory Ginot, Behrang Noohi and Ping Xu, ‘String Topology for Stacks’, [*Asterisque*]{}, No. [**343**]{}, (2012); [arXiv:0712.3857]{}. Peter Bouwknegt, Jarah Evslin and Varghese Mathai, ‘T-Duality: Topology Change from $H$-flux’, [*Commun. Math. Phys.*]{}, [**249**]{}:383-415, (2004). Also at [arXiv:hep-th/0306062]{}. Behrang Noohi, ‘Homotopy Types of topological stacks’, [*Adv. Math.*]{} [**230**]{}, no. 4-6, 2014-2047 (2012); [arXiv:0808.3799v2]{}. Calder Daenzer, ‘A groupoid approach to noncommutative T-duality’, [*Commun. Math. Phys.*]{}, [**288**]{}, 1, pp. 55-96, (2009). Also at [arXiv:0704.2592]{} [**\[math.QA\]**]{}.
[^1]: ashwin.s.pande@gmail.com, ashwin@hri.res.in
[^2]: In String Theory, the Taub-NUT metric (see Refs.[@HaJen]) describes a space with one $KK$-monopole. For more details see Ref. [@Pande] and references therein.
[^3]: See Ref. [@Pande] for a detailed discussion.
[^4]: See Ref. [@Heinloth], Def. (2.11)
[^5]: The space $CS^2$ is the coarse moduli space of the stack $[CS^3/S^1]$ (see below) and, as spaces, $CS^3/S^1 \simeq CS^2$. The stack $\underline{CS^2}$ has different stack cohomology groups to the stack $[CS^3/S^1]$, hence they cannot be equivalent.
[^6]: See Ref. [@Noohi3], Sec. (4.3), also the proof of Thm. (6.3) for the definition and properties of classifying spaces.
[^7]: See Ref. [@Noohi3] Sec. (4.3) and cartesian square after Lemma (4.1).
[^8]: See Prop. (6.1) of Ref. [@Noohi3].
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'A model for the evolution of ultrarelativistic heavy-ion collisions at both CERN SPS and RHIC top energies is presented. Based on the assumption of thermalization and a parametrization of the space-time expansion of the produced matter, this model is able to describe a large set of observables including hadronic momentum spectra, correlations and abundancies, the emission of real photons, dilepton radiation and the suppression pattern of charmonia. Each of these obervables provides unique capabilities to study the reaction dynamics and taken together they form a strong and consistent picture of the evolving system. Based on the emission of hard photons measured at SPS, we argue that a strongly interacting, hot and dense system with temperatures above 250 MeV has to be created early in the reaction. Such a system is bound to be different from hadronic matter and likely to be a quark-gluon plasma, and we find that this assumption is in line with the subsequent evolution of the system that is reflected in other observables.'
address: 'Department of Physics, Duke University, PO Box 90305, Durham, NC 27708 , USA'
author:
- Thorsten Renk
title: 'A global description of heavy-ion collisions'
---
Introduction
============
Lattice simulations (e.g. [@lat1]) predict that QCD undergoes a phase transformation at a temperature $T_C \approx 150 - 170$ MeV [@lat2; @lat3] from a confined hadronic phase to a phase, the quark-gluon plasma (QGP), in which quarks and gluons constitute the relevant degrees of freedom and the chiral condensate vanishes. Experimentally, this prediction can only be tested in ultrarelativistic heavy-ion collisions. However, finding evidence, and ultimately proof, for the creation of a quark-gluon plasma faces several difficulties. Arguably the greatest challenge is to link experimental observables to quantities measured on the lattice.
For a large set of observables, the evolution of the expanding medium is a key ingredient for their theoretical description. It is also the place where results from lattice QCD fit in: assuming a thermalized system is created, its evolution is governed by the equation of state (EoS). Hence, in order to test the lattice QCD predictions, one has to start with this assumption and show that it leads to a good description of the experimental data. In the following, we will demonstrate that this is indeed possible by introducing a parametrized evolution model.
The model framework
===================
Assuming that an equilibrated system is created a proper time of order 0.5-1 fm/c after the onset of the collision, we make the ansatz $$s(\tau, \eta_s, r) = N R(r,\tau) \cdot H(\eta_s, \tau)$$ for the entropy density $s$ at given proper time $\tau $ as measured in a frame co-moving with a given volume element, $\eta_s = \frac{1}{2}\ln (\frac{t+z}{t-z})$ the spacetime rapidity and $R(r, \tau), H(\eta_s, \tau)$ two functions describing the shape of the distribution and $N$ a normalization factor. We use Woods-Saxon distributions $$R(r, \tau) = 1/\left(1 + \exp\left[\frac{r - R_c(\tau)}{d_{ws}}\right]\right)$$ and $$H(\eta_s, \tau) = 1/\left(1 + \exp\left[\frac{\eta_s - H_c(\tau)}{\eta_{ws}}\right]\right).$$ for the shapes. Thus, the ingredients of the model are the skin thickness parameters $d_{ws}$ and $\eta_{ws}$ and the parametrizations of the expansion of the spatial extensions $R_c(\tau), H_c(\tau)$ as a function of proper time. From the distribution of entropy density, the thermodynamics can be inferred via the EoS (as obtained in lattice calculations) and the particle emission function is then calculated using the Cooper-Frye formula for a freeze-out hypersurface characterized by a temperature $T_F$. The total entropy is an input parameter whereas the entropy density carried by the different hadronic degrees of freedom is calculated in a statistical hadronization framework. The model is described in greater detail in [@SPS; @RHIC].
An important feature of the model is that it does not require boost invariance but allows for an accelerated longitudinal expansion pattern instead with initial rapidity $\eta_0$ not equal to final rapidity $\eta_f$. This implies that in general the spacetime rapidity $\eta_s$ at some point is not equal to the rapidity $\eta$, rather there’s a non-trivial relation dependent on the trajectory leading to this point. In most relevant cases, however, the relation can be approximated by $\eta = const. \cdot \eta_s$ [@SPS]. Such initial longitudinal compression and re-expansion implies longer lifetime and higher initial temperature and energy density as compared to a Bjorken scenario.
The adjustible parameters of the model are freeze-out temperature $T_F$, the radial expansion velocity $v_\perp$, the initial longitudinal expansion rapidity $\eta_0$ and the skin thickness parameters $d_{ws}, \eta_{ws}$.
Hadronic observables
====================
In the following, we only focus on central collisions. The model parameters are fitted to the recent SPS data for 158 AGeV Pb-Pb collisions obtained by NA49 and CERES [@NA49-1; @CERES-HBT] and 200 AGeV Au-Au collision data obtained by the PHENIX collaboration at RHIC [@Phenix-Spectra; @Phenix-HBT]. The data for both SPS and RHIC favour a comparatively low freeze-out temperature of order 100-110 MeV and a strong transverse flow ($v_\perp = 0.57$ at SPS and $0.65$ at RHIC). In both cases, the initial rapidity inverval is found to be significantly smaller than the finally observed rapidity interval $\eta_f$, we find $\eta_0 = 0.55$, $\eta_f = 1.45$ at SPS and $\eta_0 = 1.8, \eta_f = 3.6$ at RHIC. This is a crucial ingredient for the successful description of the data and the numbers indicate that the actual dynamics of the system could be very different from a Bjorken expansion. The resulting transverse mass ($m_t$) spectra for different hadron species and the pionic Hanbury-Brown Twiss (HBT) correlation radii show good agreement with the data (Figs. \[F-SPEC\], \[F-HBT\] and \[F-HBT\_RHIC\]).
There is a contribution to the $\pi^-$ $m_t$-spectra coming from the vacuum decay of resonances after kineitc decoupling at $\tau_f$. The model is able to calculate the magnitude of this contribution using statistical hadronization but not its $m_t$ distribution, therefore it is left out when we compare with data. This explains the mismatch at low $m_t$ and that the integrated spectrum does not yield the multiplicity indicated by the data. The same is visible in the K$^-$ spectra and the nucleon spectra, albeit less pronounced.
Electromagnetic observables
===========================
For the emission rate of direct photons, we use the complete $O(\alpha_s)$ calculation [@2-2-Kapusta; @2-2-Baier; @Aurenche1; @Aurenche2; @Aurenche3; @Complete1; @Complete2] in the form of the parameterization provided in [@Complete2]. The spectrum of emitted photons can be found by folding the rate [@Complete2] with the fireball evolution. In order to account for flow, the energy of a photon emitted with momentum $k^\mu =(k_t, {\bf k_t}, 0)$ has to be evaluated in the local rest frame of matter, giving rise to a product $k^\mu u_\mu$ with $u_\mu(\eta_s, r, \tau)$ the local flow profile. For comparison with the measured photon spectrum [@PhotonData], we present the differential emission spectrum into the midrapidity slice $y=0$. The resulting photon spectrum is shown in Fig. \[F-EM\] (left). Details of the calculation can be found in [@SPS; @Photons; @Photons_RHIC].
The lepton pair emission rate from a hot domain populated by particles in thermal equilibrium at temperature $T$ is proportional to the imaginary part of the spin-averaged photon self-energy, with these particles as intermediate states. The thermally excited particles annihilate to yield a time-like virtual photon with four-momentum $q$ which decays subsequently into a lepton-antilepton pair. The information about the strong interaction dynamics is encoded in the spectral function which enters the emission rate. For its computation, we have to use different techniques for dealing with partonic and hadronic degrees of freedom. In the partonic phase, we use a leading order calculation using thermal quasiparticles as degrees of freedom. In the hadronic phase we calculate in two different approaches to estimate the theoretical uncertainty - a chiral model [@Dileptons] and a mean-field model [@Amruta].
The resulting invariant mass spectrum agrees with the measured data for both the chiral model and the mean field model of hadronic matter. For SPS, we find photons in the momentum range between 2-3 GeV to be dominated by thermal emission from an initial high-temperature partonic phase whereas the dilepton yield in the invariant mass region below 1 GeV is dominated by emission from hadronic matter. Both quantities are sensitive to the 4-volume of radiating matter and complement each other in supporting the fireball evolution model in its different evolution phases. Most notably, the dilepton result indicates that hadronic matter does not cease to interact after the phase transition, instead substantial modifications of the $\rho$ properties in medium are required by the data. The photon spectrum cannot be reproduced in a Bjorken expansion, thus confirming the findings outlined above based on an analysis of hadronic observables. The sensitivity of the yield to the equilibration time $\tau_0$ allows to obtain $\tau_0 < 3$ fm/c [@SPS; @Photons].
For RHIC, we find that the contribution to the photon spectrum coming from hadronic matter is as large as the yield from partonic matter due to the strong flow. This is unfortunate, as it reduces the capability of a photon measurement to help in the determination of the peak temperature reached in the collision [@Photons_RHIC]. In both calculations, a pre-equilibrium contribution to the photon yield is missing. Such a contribution is expected to be most relevant for the high momentum region and to dominate the yield above 3-4 GeV.
Charmonium
==========
Charmonia yields are calculated by solving rate equations for the interaction of the state with the medium. The initial condition for the rate equations is determined by comparing to the production in p-p and p-A collisions. For SPS conditions, dissociation of the state dominates and we find the equation
$$\frac{d}{d\tau} N^y_\Psi(\tau) = - \lambda_D(\tau)\,N^y_\Psi(\tau)
\,, \quad \lambda_D(\tau) = \sum_n\ \langle\langle\,\sigma^n_D\,v_{rel}
\,\rangle\rangle(\tau)\ \rho_n(\tau)$$
for the rapidity density $N^y_\Psi(\tau)$ as a function of proper time $\tau$ where $\rho_n$ is the medium density [@SPS; @Charm]. Folding this equation with the fireball evolution yields Fig. \[F-CHARM\].
We find good agreement with the data with the exception of the high $E_T$ region where fluctuations in the transverse energy (not included in the model) are relavent and in the low $E_T$ region where the system cannot be expected to be thermalized any more.
Summary
=======
We have presented a thermal description of ultrarelativistic heavy-ion collisions at the CERN SPS for 158 AGeV Pb-Pb which is highly consistent and leads to a good description of a large set of different observable quantities. While this does not provide a conclusive proof for the creation of a thermalized system (and hence a QGP), it constitutes certainly strong evidence for it. It remains to be investigated if the results can be reproduced in a thermal microscopical transport description (like hydrodynamics) and if a non-thermal framework is able to describe the experimental results as well or if thermalization is required by the data.
For RHIC, we have presented a fireball evolution model which is able to simultaneously describe both HBT correlations and transverse mass spectra. This parametrized evolution model will hopefully be able to guide hydrodynamical models (which currently fail to provide such a simultaneous description). We are now in the process of making predictions for other observables based on this evolution model.
I would like to thank B. Müller, S. A. Bass for interesting discussions, helpful comments and support. This work was supported by the DOE grant DE-FG02-96ER40945 and a Feodor Lynen Fellowship of the Alexander von Humboldt Foundation.
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| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'We look for long-living topological solutions of classical nonlinear $(1+1)-$dimensional $\varphi^4$ field theory. To that effect we use the well-known cut-and-match method. In this framework, new long-living states are obtained in both topological sectors. In particular, in one case a highly excited state of a kink is found. We discover several ways of energy reset. In addition to the expected emission of wave packets (with small amplitude), for some selected initial conditions the production of kink-antikink pairs results in a large energy reset. Also, the topological number of a kink in the central region changes in the contrast of conserving full topological number. At lower excitation energies there is a long-living excited vibrational state of the kink; this phenomenon is the final stage of all considered initial states. Over time this excited state of the kink changes to a well-known linearized solution — a discrete kinks excitation mode. This method yields a qualitatively new way to describe the large-amplitude bion, which was detected earlier in the kink-scattering processes in the nontopological sector.'
author:
- 'Alexander E. Kudryavtsev'
- 'Mariya A. Lizunova'
title: 'Search for long-living topological solutions of nonlinear $\varphi^4$ field theory'
---
\[sec:sec1\] Introduction
=========================
We consider the $\lambda\varphi^4$ theory with a real scalar field $\varphi(t,x)$ in $(1+1)$ dimensions [@aek; @bazeia01; @radjaraman]. Its dynamics determined by the following Lagrangian: $$\label{eq:lagrang}
\mathscr{L}=\frac{1}{2} \left( \frac{\partial\varphi}{\partial t} \right)^2-\frac{1}{2} \left( \frac{\partial\varphi}{\partial x} \right) ^2-U(\varphi),$$ where $U(\varphi)$ is a potential, defining the self-interaction of the field in the considered model [@aek], $$\label{eq:potfi4}
U(\varphi)=\frac{\lambda}{4}\left(\frac{m^2}{\lambda}-\varphi^2\right)^2.$$ The plot of Eq. is shown at Fig. \[fig:sec2pic1\] (left panel). We analyze a model with a non-negative potential with two minima, so all static solutions with finite energy split into disjoint classes, so-called topological sectors, according to their asymptotic behavior at very large $x$. Solutions with $\varphi(-\infty)\neq\varphi(+\infty)$ are called topological, while those with $\varphi(-\infty)=\varphi(+\infty)$ are nontopological. Both types of the solutions are of growing interest in physics. In particular, they arise in the questions of three- or two-dimensional domain walls. However, the one-dimensional case also is curious and was considered in different works for diverse models [@gani1; @gani2; @gani3]. In the $\lambda\varphi^4$ model there is a soliton solution called a kink; the phenomenon of “wobbling kink” was studied in [@barashenkov], [@barlit8]. Moreover, a three- or two-dimensional domain wall presents a one-dimensional kink interpolating two different vacua of the model. In some cases these can be solved approximately [@GaKuLi]. The domain walls in the $\lambda\varphi^4$ model can be applied to some cosmological models, for example, during discussions of the dark matter and dark energy [@lit4]. The results of numerical simulations in other models [@GaKuLi] can be applied to solid-state physics [@lit37].
The Lagrangian with yields the equation of motion for $\varphi(t,x)$. After transition to dimensionless variables it reads $$\label{eq:eom1}
\varphi_{tt}-\varphi_{xx}-\varphi+\varphi^3=0.$$ As a next step, we find and study the analytical solutions of Eq. . Note that the vacua of this model $\varphi_{\scriptsize \mbox{vac}}^{(1)}=-1\mbox{ and }\varphi_{\scriptsize \mbox{vac}}^{(2)}=+1$ are stable solutions of . Moreover, there is the unstable permanent solution $\varphi=0$ with infinite energy.
In addition to the previous solutions, there is also a static, nontrivial, topological, solitary wave-like solution [@aek]. It can be easily found by solving the static limit of Eq. , $$\mbox{K}\equiv\displaystyle\varphi_{\scriptsize \mbox{K}}(x-x_0)=\tanh\frac{(x-x_0)}{\sqrt{2}}.
\label{eq:kinkphi4}$$ The antikink $\overline{\mbox{K}}$ is given by minus $\mbox{K}$. The energy functional for the Lagrangian , in static case , is called the mass of the kink $M_{\scriptsize\mbox{K}}=2\sqrt{2}/3$. The plot of Eq. is presented by Fig. \[fig:sec2pic1\] (right panel).
\[sec:sec2lev3\] Excitation spectrum of the kink
------------------------------------------------
In order to analyze the excitation spectrum of the static kink, we add to it a small perturbation $\delta \varphi$ to it. In other words, we make the ansatz $$\varphi(t,x)=\varphi_{\scriptsize \mbox{K}}(x)+\delta\varphi(t,x)=\varphi_{\scriptsize \mbox{K}}(x)+e^{i\omega t}\psi(x).$$ By taking the terms in Eq. linear in $\delta \varphi$, we obtain the following equation: $$\label{eq:hpsiepsi}
\begin{gathered}
\hat{H}\psi=E\psi,\quad \hat{H}=-\frac{d^2}{dx^2}-3\cosh^{-2}\frac{x}{\sqrt{2}}, \\
\quad E=\omega^2-2.
\end{gathered}$$ The eigenvalue $\omega_0=0$ belongs to the discrete part of the excitation spectrum [@aek], but also there is one vibrational excitation given by $$\label{eq:kinkpsi1}
\begin{gathered}
\delta\varphi_1=\psi_1(x)e^{i\omega_1 t},\quad \omega_1=\sqrt{3/2},\\
\psi_1(x)=\left(\frac{3}{2\sqrt{2}}\right)^{1/2}\tanh\frac{x}{\sqrt{2}}\cosh^{-1}\frac{x}{\sqrt{2}}.
\end{gathered}$$
\[sec:sec2lev4\] Analytical solution, depending on $\boldsymbol{x}$
-------------------------------------------------------------------
The above solutions are not a full set of solutions to the $\varphi^4$ model. Let us consider a static wave solution with infinite energy. We consider the static limit of Eq. $$\label{eq:eom_static}
\varphi_{xx}=-\varphi+\varphi^3.$$
![The potential in Eq. . The arrow shows the domain of the solution $\varphi(x)$ for an arbitrarily chosen value $0<\varphi_0<1$.[]{data-label="fig:sec2lev4pic1"}](sec2lev4pic1)
This equation is analogous to Newton’s equation $$\label{eq:v_xvarphi}
\ddot{x}=F(x)=-\nabla V(x),\quad \mbox{where}\quad V(x\to\varphi)=\frac{\varphi^2}{2}-\frac{\varphi^4}{4}+const.$$ Figure \[fig:sec2lev4pic1\] shows a plot of a $V(x\to\varphi)$. In this case $\varphi(x)$ describes a trajectory of an oscillatory movement between points $-\varphi_0$ and $\varphi_0$. In the range $0 < \varphi_0 < 1$ oscillations are periodic. In the limiting case $\varphi_0=1$ the fluctuations disappear, because the time necessary to return to the starting point $\varphi_0=1$ reaches infinity.
Defining the dimensionless variable $\varphi(x)=\varphi_0 \chi(x)$ and the constants $$k^2=\frac{\varphi_0^2/2}{1-\varphi_0^2/2},\quad b^2=1-\frac{\varphi_0^2}{2},$$ where $0\le k^2\le 1$ and $1/2\le b^2\le 1$, leads us to $$\begin{gathered}
\int_0^{\chi(x)}\frac{d\chi}{\sqrt{(1-\chi^2)\left(1-k^2\chi^2\right)}}\\
=\langle\chi =\sin\psi \rangle = \int_0^{\arcsin \chi}\frac{ d\psi}{\sqrt{\left(1-k^2\cos^2\psi\right)}}.
\end{gathered}$$ The last integral is nothing but the elliptic integral of the first kind \[$\mbox{F}(\arcsin\chi,k) = b x$\] [@yanke]. Then, the static periodic solution of Eq. can be written as $$\label{eq:varphiel}
\varphi_{\scriptsize \mbox{el}}(x)=\varphi_0~\mbox{sn}(bx,k),$$ where $\mbox{sn}(bx,k)$ is the elliptic sine [@yanke]. At small $k$ (corresponding to $\varphi_0\ll 1$) there is a concordance $\mbox{sn} (z) \approx \sin (z)$. At $\varphi_0\to 0$ the solution becomes a permanent unstable solution $\varphi\sim 0$, as previously noted. Plots of Eq. are shown in Fig. \[fig:sec2lev4pic2\] for different values of the parameter $\varphi_0$. The elliptic sine period is calculated using the following formula [@yanke]: $$\label{eq:varphielT}
T=\frac{4\mbox{F}(\pi/2,k)}{\sqrt{1-0.5\varphi_0^2}}.$$
\[sec:sec3\] Formulation of the problem
=======================================
The long-living solutions with high amplitude are of growing interest in classical field theory. This type of solution, called bion or breather, was found early in the kink-antikink collisions in the $\varphi^4$ model both in one- and three- dimensional cases [@lit38; @barlit5f; @aeklit65; @Manton1; @aeklit64; @okyn].
Here, we propose to use the popular cut-and-match method to find a long-living field configuration, using previously found solutions $\varphi=0$, Eq. , and Eq. . In this case, a part of the initial state is composed by the kink , which is divided in two equal pieces at $x=0$. These halves of the kink are fixed at $\pm x_0$. Then, one of the solutions ($\varphi=0$ or $\varphi_{\scriptsize \mbox{el}}$ on the finite interval) of is placed in the space between these two halves. An initial state constructed in manner described is shown in Fig. \[fig:sec3pic1\].
Note that if we take $\varphi=0$, the initial state will become unstable. Its energy linearly increases with growing distance $2x_0$.
In another case we make a solution in terms of the elliptic solution for a fixed value $0<\varphi_0<1$. For a smooth gluing of selected solutions one defines the value of $x_0$ as a half of the period $T$ of the elliptic function $\varphi_{\scriptsize \mbox{el}}$. Thus we obtain an initial configuration $$(-1,\varphi_0,0,-\varphi_0,1),$$ which we define to mean the following: in the area $-\infty<x<-T/2$ the initial state consists of the left half of , in $-T/2<x<+T/2$ it is given \[such that $\varphi(x=-T/4)=\varphi_0$, $\varphi(x=0)=0$, and $\varphi(x=T/4)=-\varphi_0$\], and in the area $T/2<x<\infty$ the solution consists of the right part of . There $T$ is the period of the elliptic function . The profile of this type of initial state is shown in Fig. \[fig:sec3pic1\] (for $\varphi_0=0.8$). First, we consider a “static initial state” ($\partial_t\varphi_0=0$), but later we take into account some configurations with dynamics defined by $$\frac{\partial\varphi_0(0,x)}{\partial t}=\frac{\varphi_0(\tau,x)-\varphi_0(0,x)}{\tau}=\delta\varphi_0\neq 0.$$
![The plot of the initial state, which is constructed with the method “cut and match”. A dashed line shows the half-kinks , a solid line shows a solution in terms of elliptic function $\varphi_{\scriptsize \mbox{el}}$ for $\varphi_0=0.8$.[]{data-label="fig:sec3pic1"}](sec3pic1)
\[sec:sec3lev1\] Numerical solution of the equation of motion
-------------------------------------------------------------
We solve the partial differential equation using a convergent difference scheme and with nonfixed boundary conditions, while derivatives are approximated by finite differences. The steps are taken as $h=0.04$ (space step) and $\tau=0.02$ (time step), while the equation is solved from $t=0$ to $t=100$. This choice of steps helps to optimize a ratio accuracy of the obtained results and the duration of computing. During the evolution a check of the conservation of energy is performed by taking into account a flow of energy from fixed boundaries. The initial states are compiled with the use of the computer algebra system [*Mathematica*]{} 8.
\[sec:sec3lev1\] Result for unstable vacuum $\boldsymbol{\varphi=0}$
--------------------------------------------------------------------
The initial condition consists of two halves of the kink, placed in $\pm x_0$, and the unstable zero solution $\varphi=0$ between them. The energy increases linearly with growing value of $x_0$. Two parts in the evolution are observed. First, there is a convergence of both halves of kink with velocity equal to the speed of light. When the halves finally meet each other, two processes alternate: a formation of loops and an emission of waves from the kink (the so-called, “wobbling kink”). The obtained solution $\varphi_{\scriptsize \mbox{sol}}$ is close to the linearized solution of Eq. , where $\varphi_{\scriptsize \mbox{sol}}\approx\varphi_{\scriptsize \mbox{K}}+\delta\varphi$, which is very long lived and is characterized by small emission of waves. These waves carry off some energy from the area of localization. Let us explain this phenomenon. At small values of $k(\varphi_0)$ the solution changes from $\varphi_{\scriptsize \mbox{el}}\approx \mbox{sn} x$ to $\varphi\approx \sin x$. As the $\sin$ is a periodic function, when $2x_0\le 2\pi$ the initial condition (with loops) does not cause an excitation like an excited mode of an elliptic function, but instead an excitation like a high-amplitude vibration of a kink. So, the evolution of the initial state can be described qualitatively by $$\varphi_{\scriptsize \mbox{sol}}\approx \displaystyle\tanh\left(\frac{x}{\sqrt{2}}\right)\left(1+\frac{A(t)}{\cosh\left(x/\sqrt{2}\right)}\right),\quad A(t)=A_0\cos \omega t.
\label{eq:phiwithA}$$ The evolution can be described by this equation as there are two modes in the kink spectrum. One of them, which correlates with Eq. , is responsible for small vibrations across the solution. We have an idea, that even if the observed vibrations stop being small, they still can be described with a periodic function like the $\cos \omega t$ \[we take $\omega=\sqrt{3/2}$ like in Eq. \]. In Eq. the parameter $A_0$ is taken constant, but it is not a constant in the numerical simulations because there is a small emission from area of localization of the solution.
Moreover, there is one precondition to describe qualitatively an obtained solution accurately by Eq. . The function equals zero in $x=0$ one time if $A_0\ge -1$, and three times if $A_0<-1$. Note that a quasiperiodic formation of the loops with period equals $\approx 2\pi$; it is also one reason for using the proposed phenomenological description. This period correlates with $\cos\omega t$, $$T=\frac{2\pi}{\omega}\approx 2\pi,\quad\mbox{as}~\omega=\sqrt{\frac{3}{2}}\approx 1.$$
In [@barashenkov], it is shown that considering the substitution of $\varphi_{\scriptsize \mbox{K}}+\delta\varphi$ in Eq. in the quadratic approximation by $\delta\varphi$ gives us an asymptotically stable solution. Its large amplitude vibrations are characterized by strong suppression. In Figs. \[fig:sec3lev1pic3\] and \[fig:sec3lev1pic5\] two parts of the evolution and a comparison with the analytical solution are shown for two chosen moments of time.
For high values of $x_0$ the observed loops in the evolution are characterized by not-small amplitudes. In this case the final states of evolution can be identify with the elliptic solution $\varphi_{\scriptsize \mbox{el}}$ between two halves of the kink.
\[sec:sec3lev2\] Result for an elliptic function with $\boldsymbol{0<\varphi_0<1}$
----------------------------------------------------------------------------------
### \[sec:sec3lev2A\] Dynamical initial state ($\delta\varphi_0<0$), configuration $(-1,\varphi_0,-1)$
In previous works a long-living configuration has been found, the so-called bion [@aek]. However, an analytical description of the observed process has not been given. In our work we take an initial condition $(-1,\varphi_0,-1)$, composed of one half-kink K and one half-antikink $\bar{\mbox{K}}$ as well as a half of period of $\varphi_{\scriptsize \mbox{el}}$ with fixed $\varphi_0$. This initial state is dynamical ($\varphi_0+\delta\varphi_0,\delta\varphi_0<0$) an is shown in Fig. \[fig:sec3lev2pic1\]. We obtain a long-living state with oscillation of the amplitude of $\varphi$ at $x=0$ (see Fig. \[fig:sec3lev2pic1\]). The observing oscillations in terms of an amplitude $\varphi_0(t)$ are called a regular bion. They also can be a new description of early found bion [@aek].
### \[sec:sec3lev2B\] Statistic initial state ($\delta\varphi_0=0$), configuration $(-1,\varphi_0,0,-\varphi_0,1)$
We take an initial condition $(-1,\varphi_0,0,-\varphi_0,1)$ for $\varphi_0>0.7$ (for smoother stitching), while the observed evolution does not qualitatively depend on $\varphi_0$. We also show the results for the case $\varphi_0=0.8$.
We find two phases in the evolution: the external phase (a loop of high-amplitude formation) and the internal phase (a highly deformed kink). After some time the loops continue forming, but with smaller amplitude. After 4-5 cycles the external phase ends and the solution starts to resemble a long-living excited kink with the wave packet emission from the area of localization. This phenomenon is called a wobbling kink. This state is a final step of the evolution, which is observed for other variants of initial states. The profiles of $\varphi(t,x)$ for $\varphi_0=0.8$ at some particular time are shown in Figs. \[fig:sec3lev2Bpic1\] and \[fig:sec3lev2Bpic3\].
### \[sec:sec3lev2D\] Dynamical initial state ($\delta\varphi_0<0$), configuration $(-1,\varphi_0,0,-\varphi_0,1)$
An addition of $\delta\varphi_0<0$ to the initial state of the configuration $(-1,\varphi_0,0,-\varphi_0,1)$ leads to a faster reduction of the amplitude. At low values of $|\delta\varphi_0|,~ \delta\varphi_0<0$ the loops arise. For the first time during the evolution, the kink-antinkink pairs K$\overline{\mbox{K}}$ turn up. This phenomenon has a threshold. The increase of $|\delta\varphi_0|$ gives us a qualitatively new type of the evolution ($-0.0013<\delta\varphi_0<-0.0044$ for $\varphi_0=0.9$). In the system, we achieve $$\label{eq:1step}
\mbox{{\bf K}}\to\mbox{K}\overline{\mbox{{\bf K}}}\mbox{K},$$ where in the center of the configuration a topological number is changing. The transition is shown in Fig. \[fig:sec3lev2Dpic4\] for $\delta\varphi_0=-0.0040$. The next increasing of $|\delta\varphi_0|$ ($-0.0045\leq\delta\varphi_0<...$ for $\varphi_0=0.9$) gives us the next transition, $$\label{eq:2step}
\mbox{\bf K} \to\mbox{K}\overline{\mbox{K}}\mbox{{\bf K}}\overline{\mbox{K}}\mbox{K}.$$ In this case we observe a conservation of topological number in the center. These transitions are observed for different $\varphi_0$. The transition is shown in Fig. \[fig:sec3lev2Dpic5\] for $\delta\varphi_0=-0.0045$. We expect that with increasing the value of $|\delta\varphi_0|$, similar qualitative changes will be observed in the evolution of the initial state.
Conclusions
===========
In this work we study new long-living solutions in the classical $\lambda\varphi^4$ field theory model in $(1+1)$ dimensions.
We use the cut-and-match method for forming initial states for numerical simulations. Using this method gives us new long-living solutions both for vacuum solutions and solutions with nontrivial topological number.
In previous work [@lit38], a long-living configuration was observed in the kink-antikink scattering and was called a bion. In current work the cut-and-match method gives us an opportunity to describe a bion formation in a new qualitative way.
Furthermore, the highly excited states of the kink are observed in a sector with nontrivial topological number. We find a number of ways to reset this energy from this state. Except for emission of wave packet with small amplitude, firstly, an arising of the kink-antikink pairs has been observed. This phenomenon can perceived as a way to reset energy. At the same time there is a change of the topological number of the kink located in the central zone in the area. At lower excitation energies there is a long-living excited vibrational state of the kink. The phenomenon called the wobbling kink is final state for all considered initial conditions. After some time the excited state of a kink turns to a linearized one, which was formerly known as a discrete mode of exciting kink.
Despite the large number of new results, the cut-and-match method has a number of remaining issues in its application to the $\lambda\varphi^4$ model. In particular, a more detailed study of the dynamic of the initial conditions for the case of $\delta\varphi_0<0$ will be interesting, because in the last case there is the phenomenon of the birth of new kink-antikink pairs.
In the conclusion, we note that this research could be useful in different area of physics and, in particular, could be implemented in the description of the early stages of the evolution of the Universe.
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
===============
We are very grateful to Professor Dr. I.L. Bogolubsky for numerous critical comments during the reading of the manuscript. This work is supported by the Russian Federation Government under the Grant No. 945 from 18.11.2011. M.A. Lizunova also acknowledges the support from the Dynasty Foundation and Edward Lozansky. This work is part of the Delta Institute for Theoretical Physics (DITP) consortium, a program of the Netherlands Organisation for Scientific Research (NWO) that is funded by the Dutch Ministry of Education, Culture and Science (OCW).
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E. Yanke, F. Emde, and F. Lesh, [*Special Functions*]{} (Nauka, Moscow, 1977) (in Russian).
A.E. Kudryavtsev, Pis’ma v Zh. Eksp. Teor. Fiz. [**22**]{}, 178 (1975) \[JETP Letters [**22**]{}, 82 (1975)\].
T.I. Belova, Yad. Fiz. [**58**]{}, 130 (1995) \[Phys. At. Nucl. [**58**]{}, 124 (1995)\].
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I.L. Bogolyubsky and V.G. Makhankov, Pis’ma v Zh. Eksp. Teor. Fiz. [**24**]{}, 15 (1976) \[JETP Letters [**24**]{}, 12 (1976)\].
| {
"pile_set_name": "ArXiv"
} |
---
address: 'School of Physics, University of Melbourne, Parkville 3052, Victoria, Australia'
author:
- 'Neil J. Cornish'
title: The black hole that went away
---
(0,0) (420,160)[[UM-P-96/74]{}]{}
In a recent paper, Kawai [@kawai] describes a remarkable vacuum solution to Einstein’s equations in $(2+1)$-dimensions. The result is remarkable because it goes against all conventional wisdom. The claim is that the solution describes a vacuum black hole with a working Newtonian limit. In contrast, the usual BTZ black hole [@btz] requires a negative cosmological constant, and $(2+1)$ general relativity should not have a Newtonian limit [@many].
The Kawai “black hole” has a metric given by [@kawai; @k2] $$\label{met}
ds^2=-\left[1+a \ln\left({r \over r_0}\right)\right]^2 dt^2
+\left({r_{0} \over r}\right)^2\left(dr^2 +r^2 d\theta^2\right)\, .$$ The coordinates are taken from the conventional range $-\infty < t <\infty$, $0< r <\infty$, $0\leq \theta \leq 2\pi$ and $\theta=\theta+2\pi$. The metric appears to describe a singularity at $r=0$, surrounded by an event horizon at $r=r_{0}\exp(-1/a)$.
A quick check confirms that (\[met\]) is indeed a solution to the vacuum Einstein equations $R_{\mu\nu}=0$. But this is where things start to unravel. Recall that in $(2+1)$-dimensions the full Riemann curvature tensor can always be expressed in terms of the Ricci scalar via the relation [@deser] $$\label{now}
R^{\mu\nu}_{\;\;\; \kappa\lambda}=\epsilon^{\mu\nu\beta}
\epsilon_{\kappa\lambda\alpha}R^{\alpha}_{\beta} \, .$$ Consequently, we know that $R_{\mu\nu\kappa\lambda}=0$ everywhere (there is no conical singularity at $r=0$). As a result, (\[met\]) cannot support a Newtonian limit.
There is another well known spacetime that is everywhere flat, has an event horizon, and a coordinate singularity at the origin – the Rindler wedge. A simple coordinate transform reveals the Kawai solution to be nothing more than a Rindler wedge. Transforming to the cartesian coordinates $x=r_{0}[1+a\ln (r/r_{0})]$, $y=r_{0} \theta$ and rescaling $t$, the metric becomes $$\label{rind}
ds^2=-x^2 dt^2 + dx^2 + dy^2 \, .$$ Both $t$ and $x$ have the usual range $(-\infty,\infty)$, while $y$ is topologically compact with period $2\pi r_{0}$. A further set of standard coordinate transformations reduces the Kawai solution to Minkowski space. The black hole has gone away.
T. Kawai, Prog. Theor. Phys. [**94**]{} 1169, (1995). M. Banados, C. Teitelboim and J. Zanelli, Phys. Rev. Lett. [**69**]{} 1849, (1992). J. R. Gott III and M. Alpert, Gen. Rel. Grav. [**16**]{}, 243 (1984); S. Giddings, J. Abbot and K. Kucha\^ r, Gen. Rel. Grav. [**16**]{}, 751 (1984), N. J. Cornish and N. E. Frankel, Phys. Rev. D[**43**]{}, 2555 (1991). T. Kawai, Phys. Rev. D[**48**]{}, 5668 (1993). S. Deser, R. Jackiw and G ’t Hooft, Ann. Phys. (NY), [**152**]{}, 220 (1984).
| {
"pile_set_name": "ArXiv"
} |
---
abstract: |
Let $\{{\mathcal{M}}_n, n=0,1,\ldots\}$ be the supercritical branching random walk starting with one initial ancestor located at the origin of the real line. For $n=0,1,\ldots$ let $W_n$ be the moment generating function of ${\mathcal{M}}_n$ normalized by its mean. Denote by $AW_n$ any of the following random variables: maximal function, square function, $L_1$ and a.s. limit $W$, ${\underset{n\geq 0}\sup}|W-W_n|$, ${\underset{n\geq 0}\sup}|W_{n+1}-W_n|$. Under mild moment restrictions and the assumption that ${\mathbb{P}}\{W_1>x\}$ regularly varies at $\infty$ it is proved that ${\mathbb{P}}\{AW_n>x\}$ regularly varies at $\infty$ with the same exponent. All the proofs given are non-analytic in the sense that these do not use Laplace-Stieltjes transforms. The result on the tail behaviour of $W$ is established in two distinct ways.
MSC: Primary: 60G42; 60J80; Secondary: 60E99
author:
- |
Aleksander Iksanov[^1] and Sergey Polotskiy[^2]\
,\
title: Regular variation in the branching random walk
---
An introduction, notation and results
=====================================
Let ${\mathcal{M}}$ be a point process on ${\mathbb{R}}$, i.e. random, locally finite counting measure. Explicitly, $${\mathcal{M}}(A)(\omega):=
\sum_{i=1}^{J(\omega)} \delta_{X_i(\omega)}(A),$$ where $J:={\mathcal{M}}({\mathbb{R}})$, $\{X_i: i=\overline{1,J}\}$ are the points of ${\mathcal{M}}$, $A$ is any Borel subset of ${\mathbb{R}}$ and $\delta_x$ is the Dirac measure concentrated at $x$. We assume that ${\mathcal{M}}$ has no atom at $+
\infty$, and the $J$ may be deterministic or random, finite or infinite with positive probability.
Let $\{{\mathcal{M}}_n, n=0,1,\ldots\}$ be the ranching andom alk (BRW), i.e. the sequence of point processes which, for any Borel set $B\subseteq {\mathbb{R}}$, are defined as follows: ${\mathcal{M}}_0(B)=\delta_0 (B)$, $${\mathcal{M}}_{n+1}(B):=\sum_r {\mathcal{M}}_{n,r}(B-A_{n,r}), n=0,1,\ldots,$$ where $\{A_{n,r}\}$ are the points of ${\mathcal{M}}_n$, and $\{{\mathcal{M}}_{n,r}\}$ are independent copies of ${\mathcal{M}}$. More detailed definition of the BRW can be found in, for example, [@BiggKypr; @Iks04; @IksRos].
In the case when ${\mathbb{P}}\{J<\infty\}=1$ we assume that ${\mathbb{E}}J>1$. In the contrary case the condition holds automatically. Thus we only consider the supercritical BRW. As a consequence, ${\mathbb{P}}\{{\mathcal{M}}_n({\mathbb{R}})>0 \ \ \text{for all} \ n\}>0$.
In what follows we use the notation that is generally accepted in the literature on the BRW: $A_u$ denotes the position on ${\mathbb{R}}$ of a generic point $u=i_1\ldots i_n$; the record $|u|=n$ means that the $u$ is a point of ${\mathcal{M}}_n$; the symbol $\sum_{|u|=n}$ denotes the summation over all points of ${\mathcal{M}}_n$; $\mathcal{F}_n=\sigma({\mathcal{M}}_1,\ldots, {\mathcal{M}}_n)$ denotes the $\sigma
$-field generated by $\{{\mathcal{M}}_k, k=1,\ldots, n\}$; $\mathcal{F}_0$ is the trivial $\sigma$-field.
Define the function $$m(y):={\mathbb{E}}\int_{\mathbb{R}}e^{yx}{\mathcal{M}}(dx)={\mathbb{E}}\sum_{|u|=1}e^{y A_u}\in
(0,\infty], y \in {\mathbb{R}},$$ and assume that there exists a $\gamma>0$ such that $m(\gamma)<\infty$. Set $Y_u:=e^{\gamma A_u}/m^{|u|}(\gamma)$ and $$W_n:=m(\gamma)^{-n}\int_{\mathbb{R}}e^{\gamma x}{\mathcal{M}}_n(dx)=\sum_{|u|=n}
Y_u.$$ As is well-known (see, for example, [@Kingman]), the sequence $\{(W_n, \mathcal{F}_n), n=0,1,\ldots\}$ is a non-negative martingale. Notice that $W_0={\mathbb{E}}W_n=1$.
Let $\{d_n, n=1,2,\ldots\}$ be the martingale difference sequence, i.e. $$W_n=1+\sum_{k=1}^n d_k, n=1,2,\ldots$$ The square function $S$ and maximal function $W^\ast$ are defined by $$S:=\left(1+\sum_{k=1}^\infty d_k^2\right)^{1/2} \ \
\text{and} \ \ W^\ast:={\underset{n\geq 0}\sup W_n}.$$ Set also $$S_n:=\left(1+\sum_{k=1}^n d_k^2\right)^{1/2}, n=1,2,\ldots \ \
\text{and} \ \ \Delta:={\underset{n\geq 1}\sup}|d_n|.$$ Recall that since $W_n$ is a non-negative martingale all the defined variables are a.s. finite (for finiteness of $S$ for general $L_1$-bounded martingales we refer to [@Austin] or to Theorem 2 on p.390 [@Chow]).
When the martingale $W_n$ is uniformly integrable, we denote by $W_\infty=W$ its $L_1$ and a.s. limit, and then define $$M:={\underset{n\geq 0}\sup}|W-W_n|={\underset{n\geq 0}\sup}|\sum_{k=n+1}^\infty d_k|.$$
Lemma 1 [@IksNegad] (see also [@Bigg] for a slightly different proof in the case $J<\infty$ a.s.) states that there exist $r\in (0,1)$ and $\theta=\theta(r)>1$ such that whenever $t>1$ $$\label{inn} {\mathbb{P}}\{W>t\} \leq {\mathbb{P}}\{W^\ast>t\} \leq \theta
{\mathbb{P}}\{W>rt\}.$$ This suggests that the tail behaviours of $W$ and $W^\ast$ are quite similar.
Let now $\{f_n:=\sum_{k=0}^n g_k, n=0,1,\ldots\}$ be any martingale. It is well-known that the distributions of maximal $f^\ast:={\underset{n\geq 0}\sup}|f_n|$ and square $S(f):=(\sum_{k=0}^\infty
g^2_k)^{1/2}$ functions are close in many respects. The evidence in favor of such a statement is provided by, for example, the (moment) Burkholder-Gundy-Davis inequality (Theorem 1.1 [@BuDa]) or the distribution function inequalities like (\[bbb\]) of this paper. From [@BuGu] and [@BuDa] and many other subsequent works it follows that there exist a subset $\mathcal{H}$ of the set of all martingales and a class $\mathcal{A}$ of operators on martingales such that the distributions of $A_1h$ and $A_2h$ are close in an appropriate sense whenever $A_i\in \mathcal{A}$ and $h\in \mathcal{H}$. Often can it be possible to express this closeness via moment or distribution function inequalities like those mentioned above. Keeping this in mind, it would not be an unrealistic conjecture that the regular variation of ${\mathbb{P}}\{A_1h>x\}$ is equivalent to that of ${\mathbb{P}}\{A_2h>x\}$, where $A_i$ and $h$ belong to some subsets of operators and martingales respectively that may be different from $\mathcal{A}$ and $\mathcal{H}$. On the other hand, let us notice that as far as we know the conjecture does not follow from previously known results on martingales.
The aim of this paper is to prove a variant of the conjecture for the martingales $W_n$ and operators $A_i, i=\overline{1,5}$ given as follows: $A_1W=W^\ast$, $A_2W=\Delta$, $A_3W=S$, $A_4W=W_\infty$, $A_5W=M$.
In addition to the notation introduced above, other frequently used notation and conventions include: $L(t)$ denotes a function that slowly varies at infinity; $1_{A}$ denotes the indicator function of the set $A$; $f(t) \sim g(t)$ is abbreviation of the limit relation $\underset{t\rightarrow
\infty}{\lim}\dfrac{f(t)}{g(t)}=1$; $x^+:=\max(x,0)$; $x\wedge
y=\min (x,y)$; $x\vee y=\max (x,y)$; we write ${\mathbb{P}}_n\{\cdot\}$ instead of ${\mathbb{P}}\{\cdot|\mathcal{F}_n\}$, and ${\mathbb{E}}_n\{\cdot\}$ instead of ${\mathbb{E}}\{\cdot|\mathcal{F}_n\}$; the record “const” denotes a constant whose value is of no importance and may be different on different appearances.
Now we are ready to state our result.
\[t1\] Assume that there exist $\beta>1$ and $\epsilon>0$ such that $$\label{mom} k_\beta:={\mathbb{E}}\sum_{|u|=1}Y_u^\beta<1,\ \ {\mathbb{E}}\sum_{|u|=1}Y_u^{\beta+\epsilon}<\infty \ \ \text{and}$$ $$\label{var}{\mathbb{P}}\{W_1>x\}\sim x^{-\beta}L(x).$$ Then (I) ${\mathbb{P}}\{W^\ast>x\}\sim
{\mathbb{P}}\{\Delta>x\}\sim {\mathbb{P}}\{S>x\}\sim
(1-k_\beta)^{-1}{\mathbb{P}}\{W_1>x\}$;(II) $W_n$ converges almost surely and in mean to a random variable $W$ and $$\label{ww} {\mathbb{P}}\{W>x\}\sim (1-k_\beta)^{-1}{\mathbb{P}}\{W_1>x\};$$ $${\mathbb{P}}\{M>x\}\sim (1-k_\beta)^{-1}{\mathbb{P}}\{W_1>x\}.$$
We are not aware of any papers on branching processes which investigate the tail behaviour of random variables like $\Delta$, $M$ or $S$. [@IksNegad] is the only paper we know of that deals with the tail behaviour of random variables like $W^\ast$.
When $\gamma=0$ and $J<\infty$ a.s., $W_n$ reduces to the (supercritical) normalized Galton-Watson process. In this case (\[ww\]) was proved in [@BiDo74] for non-integer $\beta$ and in [@Meyer82] for integer $\beta$. When $\gamma>0$, $J<\infty$ a.s. and ${\mathcal{M}}(-\infty, -\gamma^{-1}\log m(\gamma))=0$ a.s., $W$ can be viewed as a limit random variable in the Crump-Mode branching process. In this case (\[ww\]) was established in [@BiDo75] for non-integer $\beta$. The technique used in the last three cited works is purely analytic (based on using the Laplace-Stieltjes transforms and Abel-Tauberian theorems) and completely different from ours. On the other hand, let us notice that the above mentioned analytic approach was successfully employed and further developed by the second-named author. In 2003, in an unpublished diploma paper he proved (\[ww\]) for non-integer $\beta$ for the general case treated here.
Our desire to find a non-analytic proof of (\[ww\]) was a starting point for the development of this paper. In the course of writing two different (non-analytic) proofs were found. One of these proofs given in Section 2 falls within the general scope of the paper. The second given in Section 3 continues a line of research initiated in [@Iks04], [@IksRos], [@Iks06]. Here an underlying idea is that the martingale $W_n$ and so called *perpetuities* have many features in common. In particular, several non-trivial results on perpetuities (however, it seems, only those related to perpetuities with not all moments finite) can be effectively exploited to obtain similar results on the limiting behaviour of $W_n$. Maybe we should recall that, in modern probability, by a perpetuity is meant a random variable $$B_1+\sum_{k=2}^\infty A_1A_2\cdots A_{k-1}B_k,$$ provided the latter series absolutely converges, and where $\{(A_k, B_k): k=1,2,\ldots\}$ are independent identically distributed random vectors.
The paper is structured as follows. In Section 2 we prove Theorem \[t1\]. Here an essential observation is that, given $\mathcal{F}_n$, $W_{n+1}$ looks like a weighted sum of independent identically distributed random variables. This allows us to exploit the well-known result [@MikSam] on the tail behaviour of such sums under the regular variation assumption. The second key ingredient of the proof is using the distribution function inequalities for martingales. In Section 3 we give another proof of (\[ww\]) which rests on a relation between the BRW and perpetuities. Here availability of Grincevičius-Grey [@Grey] result on the tail behaviour of perpetuities is crucial. Finally, in Section 4 we discuss applicability of Theorem \[t1\] to several classes of point processes. The section closes with two remarks which show that (\[mom\]) and (\[var\]) are not necessary conditions for the regular variation of the tails of $W^\ast$, $W$ and a related random variable.
Proof of Theorem \[t1\]
=======================
\(I) We will prove the result for $W^\ast$ and $\Delta$ simultaneously. To this end, let $Q$ and $\tilde{Q}$ be independent identically distributed random variables whose distribution is supported by $(a,\infty), a>-\infty$. Assume that ${\mathbb{P}}\{Q>x\}\sim x^{-\beta}L(x)$ for $\beta>1$. In particular, this assumption ensures that ${\mathbb{E}}|Q|<\infty$ and ${\mathbb{P}}\{|Q|>x\}\sim
{\mathbb{P}}\{Q>x\}$. With a slight abuse of notation, set $Q^s:=|Q|-|\tilde{Q}|$. Then $$\label{sym} 1-F(x):={\mathbb{P}}\{|Q^s|>x\}\sim 2x^{-\beta}L(x).$$ Indeed, $1-F(x)=2\int_0^\infty (1-G(x+y))dG(y)$, where $G(x)={\mathbb{P}}\{|Q|\leq x\}, x\geq 0$. Now (\[sym\]) follows from monotonicity of $1-G$, the relation $1-G(x+y)\sim 1-G(x), y\in
{\mathbb{R}}$ and Fatou’s lemma.
The equality $${\mathbb{E}}t(Z)={\mathbb{E}}\sum_{|u|=1}Y_ut(Y_u),$$ which is assumed to hold for all bounded Borel functions $t$, defines the distribution of a random variable $Z$. More generally, $$\label{ex} {\mathbb{E}}t(Z_1\cdots Z_n)={\mathbb{E}}\sum_{|u|=n}Y_u t(Y_u),$$ where $Z_1, Z_2, \ldots$ are independent copies of the $Z$. Notice that we can permit for (\[ex\]) to hold for any Borel function $t$. In that case we assume that if the right-hand side is infinite or does not exist, the same is true for the left-hand side.
Under the assumptions of the theorem, the function $k_x:={\mathbb{E}}\sum_{|u|=1}Y_u^x$ is log-convex for $x\in (1,\beta)$, $k_1=1$ and $k_\beta<1$. Therefore, $$\label{innn} k_{\beta-\epsilon}<1 \ \ \text{for all} \ \epsilon
\in (0,\beta-1).$$ Also we can pick a $\delta\in (0,\beta-1)$ such that $k_{\beta+\delta}<1$. By using these facts and equality (\[ex\]) we conclude that with this $\delta$ $$\label{useful} {\mathbb{E}}\sum_{|u|=n}Y_u^{\beta-\delta}=k^n_{\beta-\delta}<1 \ \ \text{and}
\ \ {\mathbb{E}}\sum_{|u|=n}Y_u^{\beta+\delta}=k^n_{\beta+\delta}<1.$$ Let us notice, for later needs, that we can choose $\delta$ as small as needed. Among other things, (\[useful\]) implies that for $x\in [1,\beta+\delta]$ $$\label{inter} \sum_{|u|=n}Y_u^x<\infty \ \ \text{a.s.}$$
Until further notice, we fix an arbitrary $n\in {\mathbb{N}}$. Put $$T_n:=|\sum_{|u|=n}Y_u Q_u| \ \ \text{and} \ \ X_n:=\sum_{|u|=n}Y_u |Q_u|.$$ Given $\mathcal{F}_n$, let $\{Q_u:|u|=n\}$ and $\{Q_u^s:|u|=n\}$ be conditionally independent copies of the random variables $Q$ and $Q^s$ respectively. In view of (\[inter\]), an appeal to Lemma A3.7[@MikSam] allows us to conclude that $$\label{basic} {\mathbb{P}}_n\{T_n>x\}\sim \sum_{|u|=n}Y_u^\beta
{\mathbb{P}}\{|Q|>x\} \ \ \text{a.s.}$$ The cited lemma assumes that each term of the series on the left-hand side has zero mean, but this condition is not needed in the proof of the result used here.
Denote by $\mu_n^{\mathcal{F}_n}$ the conditional on $\mathcal{F}_n$ median of $X_n$, i.e. $\mu_n^{\mathcal{F}_n}$ is a random variable that satisfies $${\mathbb{P}}_n\{X_n-\mu_n^{\mathcal{F}_n}\geq 0\}\geq 1/2 \leq {\mathbb{P}}_n\{X_n-\mu_n^{\mathcal{F}_n}\leq 0\} \ \
\text{a.s.}$$ Let also $\mu_n$ denote the usual median of $X_n$. Since $\mu_n^{\mathcal{F}_n}\geq 0$ a.s., we have from (\[basic\]) $$\underset{x\to\infty}{\lim \sup} \dfrac{{\mathbb{P}}_n\{T_n>x+\mu_n^{\mathcal{F}_n}\}}{{\mathbb{P}}\{|Q|>x\}}\leq \sum_{|u|=n}Y_u^\beta \ \
\text{a.s.}$$ If we could prove that for large $x$ $$\label{ineq}
\dfrac{{\mathbb{P}}_n\{T_n>x+\mu_n^{\mathcal{F}_n}\}}{{{\mathbb{P}}\{|Q|>x\}}}\leq
U_n \ \ \text{a.s. \ \ and} \ \ {\mathbb{E}}U_n<\infty,$$ where $U_n$ is a random variable, then using Fatou’s lemma yielded $$\label{inter2} \underset{x\to\infty}{\lim \sup}{\mathbb{E}}\dfrac{{\mathbb{P}}_n\{T_n>x+\mu_n^{\mathcal{F}_n}\}}{{\mathbb{P}}\{|Q|>x\}}=
\underset{x\to\infty}{\lim \sup}
\dfrac{{\mathbb{P}}\{T_n>x+\mu_n\}}{{\mathbb{P}}\{|Q|>x\}}\leq {\mathbb{E}}\sum_{|u|=n}Y_u^\beta\overset{(\ref{ex})}{=}k^n_\beta.$$ Since ${\mathbb{P}}\{|Q|>x+\mu_n\}\sim {\mathbb{P}}\{|Q|>x\}$, (\[inter2\]) implied that $$\underset{x\to\infty}{\lim \sup}
\dfrac{{\mathbb{P}}\{T_n>x\}}{{\mathbb{P}}\{|Q|>x\}}\leq k^n_\beta.$$ On the other hand, by using (\[basic\]) and Fatou’s lemma the reverse inequality for the lower limit follows easily. Therefore, as soon as (\[ineq\]) is established, we get $$\label{main} {\mathbb{P}}\{T_n>x\}\sim k^n_\beta {\mathbb{P}}\{|Q|>x\}.$$
We now intend to show that (\[ineq\]) holds with $$\label{U} U_n=const
\left(\sum_{|u|=n}Y_u^{\beta-\delta}+\sum_{|u|=n}Y_u^{\beta+\delta}
\right),$$ for appropriate small $\delta$ that satisfies (\[useful\]). Notice that $$\label{expec} {\mathbb{E}}U_n=const
(k^n_{\beta-\delta}+k^n_{\beta+\delta})<\infty.$$
By the triangle inequality and conditional symmetrization inequality $$\label{start} (1/2){\mathbb{P}}_n\{T_n>x+\mu_n^{\mathcal{F}_n}\}\leq
(1/2){\mathbb{P}}_n\{X_n>x+\mu_n^{\mathcal{F}_n}\}\leq
{\mathbb{P}}_n\{|\sum_{|u|=n}Y_uQ_u^s|>x\}.$$ Let us show that for $x>0$ $${\mathbb{P}}_n\{|\sum_{|u|=n}Y_uQ_u^s|>x\}\leq$$ $$\label{imp}\leq
{\mathbb{P}}_n\{\underset{|u|=n}{\sup}Y_u|Q_u^s|>x\}+x^{-2}{\mathbb{E}}_n\left(\sum_{|u|=n}Y_u^2(Q_u^s)^21_{\{Y_u|Q_u^s|\leq
x\}}\right):=I_1(n,x)+I_2(n,x).$$ Let $\{Y(k)Q^s(k):k=1,2,\ldots\}$ be any enumeration of the set $\{Y_uQ_u^s:|u|=n\}$. The inequality ${\mathbb{E}}|Q|<\infty$ implies that the series $\sum_{|u|=n}Y_uQ_u$ is absolutely convergent. Therefore $\sum_{|u|=n}Y_uQ_u=\sum_{k=1}^\infty Y(k)Q^s(k)$. Define $$\tau_x:=
\begin{cases}
\inf\{k\geq 1: Y(k)|Q^s(k)|>x\}, \text{\ \ if \ \ }
\underset{k\geq
1}{\sup}Y(k)|Q^s(k)|>x; \\
+\infty, \text{\ \ otherwise }.
\end{cases}$$ For any fixed $m\in {\mathbb{N}}$ and $x>0$ $${\mathbb{P}}_n\{|\sum_{k=1}^m Y(k)Q^s(k)|>x\}\leq$$ $$\leq {\mathbb{P}}_n\{\tau_x\leq m-1\}+{\mathbb{P}}_n\{|\sum_{k=1}^m Y(k)Q^s(k)|>x,
\tau_x\geq m\}\leq$$$$\leq {\mathbb{P}}_n\{\underset{1\leq k\leq
m-1}{\sup}Y(k)|Q^s(k)|>x\}+{\mathbb{P}}_n\{|\sum_{k=1}^{\tau_x\wedge
m}Y(k)Q^s(k)|>x\}\leq$$ (by Markov inequality) $$\leq
{\mathbb{P}}_n\{\underset{1\leq k\leq
m-1}{\sup}Y(k)|Q^s(k)|>x\}+x^{-2}{\mathbb{E}}_n\left(\sum_{k=1}^m
Y(k)Q^s(k)1_{\{\tau_x\geq k\}}\right)^2\leq$$ (${\mathbb{E}}_n Q^s(k)=0$ and, given $\mathcal{F}_n$, $Q^s(k)$ and $1_{\{\tau_x\geq k\}}$ are independent) $$\leq {\mathbb{P}}_n\{\underset{1\leq k\leq
m-1}{\sup}Y(k)|Q^s(k)|>x\}+x^{-2}{\mathbb{E}}_n \sum_{k=1}^m
Y^2(k)(Q^s(k))^21_{\{Y(k)|Q^s(k)|\leq x\}}.$$ If the distribution of $Q^s$ is continuous, sending $m\to\infty$ then completes the proof of (\[imp\]).
Assume now that the distribution of $Q^s$ has atoms. Let $R$ be a random variable with uniform distribution on $[-1,1]$, which is independent of $Q^s$. Given $\mathcal{F}_n$, let $\{R_u:|u|=n\}$ be conditionally independent copies of $R$ which are also independent of $\{Q_u^s:|u|=n\}$. Since for all $t>0$ $${\mathbb{P}}\{|Q^s|>t\}\leq 2{\mathbb{P}}\{|Q^s||R|>t/2\},$$ we have by Theorem 3.2.1[@Woy] $$\label{cont} {\mathbb{P}}_n\{|\sum_{|u|=n}Y_uQ_u^s|>t\}\leq
4{\mathbb{P}}_n\{|\sum_{|u|=n}Y_uQ_u^sR_u|>t/4\},$$ and the distribution of $Q^sR$ is (absolutely) continuous. Now we can apply the already established part of (\[imp\]) to the right-hand side of (\[cont\]). Strictly speaking, when the distribution of $Q^s$ has atoms, (\[imp\]) should be written in a modified form: additional constants should be added, and $Q_u^s$ should be replaced with $Q_u^sR_u$. On the other hand, a perusal of the subsequent proof reveals that only the regular variation of ${\mathbb{P}}\{|Q^s|>x\}$ plays a crucial role. Therefore, for ease of notation we prefer to keep (\[imp\]) in its present form. This does not cause any mistakes as ${\mathbb{P}}\{|Q^sR|>x\}\sim {\mathbb{E}}|R|^\beta
{\mathbb{P}}\{|Q^s|>x\}$.
Assume temporarily that $1-F(x)$ regularly varies with index $-\beta$, $\beta\in (1,2)$. Set $T(x):=\int_0^x y^2dF(y)$. By Theorem 1.6.4 [@BGT] $$T(x)\sim \dfrac{\beta}{2-\beta} x^2(1-F(x))\sim
\dfrac{\beta}{2-\beta}x^{2-\beta}L_1(x).$$ Also by Theorem 1.5.3 [@BGT] there exists a non-decreasing $S(x)$ such that $$\label{s} T(x)\sim S(x).$$ For any $A_i>0$ and $\delta$ defined in (\[useful\]) there exists an $x_i>0$ such that whenever $x\geq x_i, i=1,2,3$ $$\label{A1} x^{\beta+\delta}(1-F(x))\geq 1/A_1;$$ $$\label{A2} x^{\beta-2+\delta}S(x)\geq 1/A_2;$$ $$\label{A3} T(x)\leq
(A_3+\dfrac{\beta}{2-\beta})x^2(1-F(x)):=Bx^2(1-F(x)).$$ Also for any $A_i>1$ and the same $\delta$ as above there exists an $x_i>0$ such that whenever $x\geq x_i$ and $ux\geq x_i,
i=4,5,6$ $$\label{A4} \dfrac{1-F(ux)}{1-F(x)}\leq A_4 (u^{-\beta+\delta}\vee
u^{-\beta-\delta});$$ $$\label{A5} \dfrac{T(ux)}{T(x)}\leq A_5 (u^{2-\beta+\delta}\vee
u^{2-\beta-\delta});$$ $$\label{A6} \dfrac{T(ux)}{T(x)}\leq A_6\dfrac{S(ux)}{S(x)}.$$ (\[A4\]) and (\[A5\]) follows from Potter’s bound (Theorem 1.5.6 (iii)[@BGT]); (\[A6\]) is implied by (\[s\]). Set $x_0:=\underset{1\leq i \leq 6}{\max}x_i$ and assume that $x_0>1$.
To check (\[ineq\]) and (\[U\]), we consider three cases: (a) $\beta\in (1,2)$; (b) $\beta>2, \beta\neq 2^n, n\in{\mathbb{N}}$; (c) $\beta=2^n, n\in {\mathbb{N}}$.
\(a) For any fixed $x\geq x_0$ and $y>0$ $$\dfrac{I_1(n,x/y)}{1-F(x)} \leq \sum_{|u|=n}\dfrac{{\mathbb{P}}_n\{Y_u|Q_u^s|>x/y\}}{1-F(x)}=
\sum_{|u|=n}\dfrac{1-F(x(yY_u)^{-1})}{1-F(x)}=$$ $$=\sum_{|u|=n}\cdots 1_{\{yY_u>x/x_0\}}+\sum_{|u|=n}\cdots 1_{\{yY_u\leq
x/x_0\}}=:I_{11}(n,x,y)+I_{12}(n,x,y).$$ Since $$(yY_u)^{\beta+\delta}\geq
(yY_u)^{\beta+\delta}1_{\{yY_u>x/x_0\}}\geq
(x/x_0)^{\beta+\delta}1_{\{yY_u>x/x_0\}},$$ we get $$I_{11}(n,x,y)\leq
(x_0y)^{\beta+\delta}\dfrac{\sum_{|u|=n}Y_u^{\beta+\delta}}{x^{\beta+\delta}(1-F(x))}\overset{(\ref{A1})}{\leq}
A_1x_0^{\beta+\delta}y^{\beta+\delta}\sum_{|u|=n}Y_u^{\beta+\delta}.$$ Further $$I_{12}(n,x,y)\overset{(\ref{A4})}{\leq}A_4\sum_{|u|=n}(yY_u)^{\beta-\delta}\vee(yY_u)^{\beta+\delta}\leq
A_4\left(y^{\beta-\delta}\sum_{|u|=n}Y_u^{\beta-\delta}+y^{\beta+\delta}\sum_{|u|=n}Y_u^{\beta+\delta}\right);$$ $$\dfrac{I_2(n,x/y)}{1-F(x)}=y^2\sum_{|u|=n}\dfrac{Y^2_u\int_0^{x(yY_u)^{-1}}z^2dF(z)}{x^2(1-F(x))}\overset{(\ref{A3})}{\leq}
By^2\sum_{|u|=n}\dfrac{Y^2_uT(x(yY_u)^{-1})}{T(x)}=$$$$=By^2\left(\sum_{|u|=n}\cdots 1_{\{yY_u>
x/x_0\}}+\sum_{|u|=n}\cdots 1_{\{yY_u\leq
x/x_0\}}\right)=:By^2(I_{21}(n,x,y)+I_{22}(n,x,y)).$$ $$I_{21}(n,x,y)\overset{(\ref{A6})}{\leq}A_6\sum_{|u|=n}\dfrac{Y^2_uS(x(yY_u)^{-1})}{S(x)}1_{\{yY_u>
x/x_0\}}\leq$$ $$\leq
A_6(x_0y)^{\beta-2+\delta}S(x_0)\sum_{|u|=n}\dfrac{Y_u^{\beta+\delta}}{x^{\beta-2+\delta}S(x)}\overset{(\ref{A2})}{\leq}
A_2A_6(x_0y)^{\beta-2+\delta}S(x_0)\sum_{|u|=n}Y_u^{\beta+\delta};$$ $$I_{22}(n,x,y)\overset{(\ref{A5})}{\leq} A_5 \sum_{|u|=n}Y_u^2((yY_u)^{\beta-2-\delta}\vee
(yY_u)^{\beta-2+\delta})\leq$$ $$\leq A_5\left(y^{\beta-2-\delta}\sum_{|u|=n}Y_u^{\beta-\delta}+y^{\beta-2+\delta}\sum_{|u|=n}Y_u^{\beta+\delta}\right).$$ Thus according to (\[imp\]) we have proved that for $x\geq x_0$ and $y>0$ $$\label{impo}
\dfrac{{\mathbb{P}}_n\{|\sum_{|u|=n}Y_uQ_u^s|>x/y\}}{{\mathbb{P}}\{|Q^s|>x\}}\leq
const
\left(y^{\beta-\delta}\sum_{|u|=n}Y_u^{\beta-\delta}+y^{\beta+\delta}\sum_{|u|=n}Y_u^{\beta+\delta}
\right).$$ In particular, since ${\mathbb{P}}\{|Q^s|>x\}\sim 2{\mathbb{P}}\{|Q|>x\}$ then setting in (\[impo\]) $y=1$ and using (\[start\]) leads to (\[ineq\]) with $U_n$ being a multiple of the right-hand side of (\[impo\]).
In the remaining cases we only investigate the situation when $\beta\in (2,4)$ and $\beta=2$. For other values of $\beta$ the inequality (\[ineq\]) with $U_n$ satisfying (\[U\]) follows by induction.
\(b) $\beta\in (2,4)$. Given $\mathcal{F}_n$, let $\{\tilde{N},
N_u:|u|=n\}$ be conditionally independent copies of a random variable $N$ with normal $(0,1)$ distribution. Using the approach similar to that exploited to obtain (\[cont\])(this fruitful argument has come to our attention from [@MikSam]) allows us to conclude that for $x>0$ and appropriate positive constants $c_1, c_2$ $${\mathbb{P}}_n\{|\sum_{|u|=n}Y_uQ_u^s|>x\}\leq$$ $$\label{sam} \leq
c_1{\mathbb{P}}_n\{|\sum_{|u|=n}Y_uN_uQ_u^s|>c_2x\}=c_1{\mathbb{P}}_n\left\{|\tilde{N}|\left(\sum_{|u|=n}Y^2_u(Q_u^s)^2\right)^{1/2}>c_2x\right\}.$$ Notice that ${\mathbb{P}}\{(Q^s)^2>x\}$ regularly varies with index $-\beta/2\in (-2,-1)$. Also it is obvious that, if needed, we can reduce $\delta$ in (\[useful\]) to ensure that $k_{\beta-2\delta}<1$ and $k_{\beta+2\delta}<1$. Therefore we can use (\[impo\]) with $Y_u$ replaced with $Y_u^2$, and $Q_u^s$ replaced with $(Q_u^s)^2$ which gives after a little manipulation that for $x\geq x_0^{1/2}$ $$\dfrac{{\mathbb{P}}_n\{|\sum_{|u|=n}Y_uQ_u^s|>x\}}{{\mathbb{P}}\{|Q^s|>x\}}\leq const\left(\sum_{|u|=n}Y_u^{\beta+2\delta}{\mathbb{E}}|N|^{\beta+2\delta}+
\sum_{|u|=n}Y_u^{\beta-2\delta}{\mathbb{E}}|N|^{\beta-2\delta}\right).$$ By using the same argument as in the case (a) we can check that (\[ineq\]) and (\[U\]) have been proved.
\(c) $\beta=2$. In the same manner as we have established (\[impo\]) it can be proved that for $y>0$ and large $x$ $$\dfrac{{\mathbb{P}}_n\{\sum_{|u|=n}Y^2_u(Q_u^s)^2>x/y\}}{{\mathbb{P}}\{(Q^s)^2>x\}}\leq
const
\left(y^{2-2\delta}\sum_{|u|=n}Y_u^{2-2\delta}+y^{2+2\delta}\sum_{|u|=n}Y_u^{2+2\delta}
\right).$$ Hence an appeal to (\[sam\]) assures that (\[ineq\]) and (\[U\]) hold in this case too.
We have a representation $$\label{repr} W_{n+1}=\sum_{|u|=n}Y_uW_1^{(u)},$$ where, given $\mathcal{F}_n$, $W_1^{(u)}$ are (conditionally) independent copies of $W_1$. Each element of the set $\{W_1^{(u)}:|u|=n\}$ is constructed in the same way as $W_1$, the only exception being that while $W_1$ is defined on the whole family tree, $W_1^{(u)}$ is defined on the subtree with root $u$.
We only give a complete proof for the $\Delta$. The analysis of the $W^\ast$ is similar but simpler, and hence omitted. From (\[repr\]) we conclude that $|d_{n+1}|$ is the same as $T_n$ with $Q_u=W_1^{(u)}-1$. Hence by (\[basic\]) $$\label{co}1_{\{\underset{1\leq k\leq n}{\max}|d_k| \leq
x\}}{\mathbb{P}}_n\{|d_{n+1}|>x\} \sim \sum_{|u|=n}Y_u^\beta
{\mathbb{P}}\{|W_1-1|>x\}\sim \sum_{|u|=n}Y_u^\beta {\mathbb{P}}\{W_1>x\} \ \
\text{a.s.},$$ and by (\[main\]) $$\label{nnn} {\mathbb{P}}\{|d_{n+1}|>x\}\sim k^n_\beta {\mathbb{P}}\{|W_1-1|>x\}\sim
k^n_\beta {\mathbb{P}}\{W_1>x\}.$$ Recall that $${\mathbb{P}}\{\Delta>x\}={\mathbb{P}}\{|d_1|>x\}+\sum_{n=1}^\infty {\mathbb{P}}\{\underset{1\leq k\leq n}{\max}|d_k| \leq
x,|d_{n+1}|>x\}=$$ $$={\mathbb{P}}\{|d_1|>x\}+{\mathbb{E}}\sum_{n=1}^\infty
1_{\{\underset{1\leq k\leq n}{\max}|d_k| \leq
x\}}{\mathbb{P}}_n\{|d_{n+1}|>x\}.$$
Using this, (\[co\]) and applying Fatou’s lemma twice allows us to conclude that $$\underset{x\to\infty}{\lim \inf} \dfrac{{\mathbb{P}}\{\Delta>x\}}{{\mathbb{P}}\{W_1>x\}}\geq 1+\sum_{n=1}^\infty {\mathbb{E}}\underset{x\to\infty}{\lim
\inf} \dfrac{1_{\{\underset{1\leq k\leq n}{\max}|d_k| \leq
x\}}{\mathbb{P}}_n\{|d_{n+1}|>x\}}{{\mathbb{P}}\{W_1>x\}}\geq$$ $$\geq 1+\sum_{n=1}^\infty {\mathbb{E}}\left( \sum_{|u|=n}
Y_u^\beta\right)=(1-k_{\beta})^{-1}.$$
To complete the proof for $\Delta$ we must calculate the corresponding upper limit. For this it suffices to check that for large $x$ and large $n\in {\mathbb{N}}$ $$\label{qqq} \dfrac{{\mathbb{P}}\{|d_{n+1}|>x\}}{{\mathbb{P}}\{W_1>x\}}\leq C_n \ \
\text{and} \ \ C_n \ \ \text{is a summable sequence},$$ and use the dominated convergence theorem. Taking the expectation in (\[ineq\]) allows us to conclude that for $n=1,2,\ldots$ and large $x$ $$\dfrac{{\mathbb{P}}\{|d_{n+1}|>x+\mu_n\}}{{{\mathbb{P}}\{W_1>x\}}}\leq const {\mathbb{E}}U_n,$$ where $\mu_n$ is the median of $V_n:=\sum_{|u|=n}Y_u|W_1^{(u)}-1|$, and ${\mathbb{E}}U_n$ is given by (\[expec\]). The family of distributions of $V_n$ is tight. In view of (\[nnn\]), $${\mathbb{P}}\{|d_{n+1}|>x+y\}\sim {\mathbb{P}}\{|d_{n+1}|>x\}
\ \ \text{locally uniformly in} \ y .$$ Therefore, (\[qqq\]) holds with $C_n=const {\mathbb{E}}U_n$ and the result for $\Delta$ has been proved.
For later needs let us notice here that in the same way as above we can prove that for fixed $n\in {\mathbb{N}}$ $$\label{qq} {\mathbb{P}}\{\underset{m\geq n}{\sup}W_m> x\}\sim
k_\beta^n(1-k_\beta)^{-1}{\mathbb{P}}\{W_1>x\}.$$
Consider now the square function $S$. Since $S\geq \Delta$ a.s., and we have already proved that ${\mathbb{P}}\{\Delta>x\} \sim
(1-k_\beta)^{-1}{\mathbb{P}}\{W_1>x\}$ then $$\underset{x\to\infty}{\lim
\inf}\dfrac{{\mathbb{P}}\{S>x\}}{{\mathbb{P}}\{W_1>x\}}\geq \dfrac{1}{1-k_\beta}.$$ Therefore we must only calculate the upper limit. We begin with showing that for any $n\in {\mathbb{N}}$ $$\label{ss} \underset{x\to\infty}{\lim
\sup}\dfrac{{\mathbb{P}}\{S_n>x\}}{{\mathbb{P}}\{W_1>x\}}\leq
\sum_{m=0}^{n-1}k_\beta^m.$$ We use induction on $n$.(1) If $n=1$ then $S_1\leq W_1$ and (\[ss\]) is obvious.(2) Assume that (\[ss\]) holds for $n=j$ and show that it holds for $n=j+1$. For every $x>0$ and $\epsilon \in (0,1)$ $${\mathbb{P}}\{S_{j+1}>x\}\leq {\mathbb{P}}\{S_j^2>(1-\epsilon)x^2\}+{\mathbb{P}}\{d_{j+1}^2>(1-\epsilon)x^2\}+{\mathbb{P}}\{S_j^2>\epsilon x^2, d_{j+1}^2
>\epsilon x^2\}=$$ write the latter ${\mathbb{P}}$ as ${\mathbb{E}}{\mathbb{P}}_j$ and use $\mathcal{F}_j$-measurability of $S_j$ to get $$={\mathbb{P}}\{S_j>(1-\epsilon)^{1/2}x\}+{\mathbb{P}}\{|d_{j+1}|>(1-\epsilon)^{1/2}x\}+{\mathbb{E}}1_{\{S_j>\epsilon^{1/2}x\}}{\mathbb{P}}_j\{|d_{j+1}|>\epsilon^{1/2}x\}.$$ According to (\[basic\]) with $Q_u$ replaced by $W_1^{(u)}-1$, $${\underset{x\rightarrow\infty}{\lim}}1_{\{S_j>\epsilon^{1/2}x\}}\dfrac{{\mathbb{P}}_j\{|d_{j+1}|>\epsilon^{1/2}x\}}{{\mathbb{P}}\{W_1>x\}}=0 \ \
\text{a.s.},$$ and there exists a $\delta_1>0$ such that for large $x$ $$1_{\{S_j>\epsilon^{1/2}x\}}\dfrac{{\mathbb{P}}_j\{|d_{j+1}|>\epsilon^{1/2}x\}}{{\mathbb{P}}\{W_1>x\}}\leq
\epsilon^{-\beta/2}\sum_{|u|=n}Y_u^\beta+\delta_1 \ \
\text{a.s.}$$ Therefore, by the dominated convergence $${\underset{x\rightarrow\infty}{\lim}}{\mathbb{E}}1_{\{S_j>\epsilon^{1/2}x\}}\dfrac{{\mathbb{P}}_j\{|d_{j+1}|>\epsilon^{1/2}x\}}{{\mathbb{P}}\{W_1>x\}}=0.$$ By the inductive assumption and (\[nnn\]) $$\underset{x\to\infty}{\lim
\sup}\dfrac{{\mathbb{P}}\{S_{j+1}>x\}}{{\mathbb{P}}\{W_1>x\}}\leq
(1-\epsilon)^{-\beta/2}\sum_{m=0}^{j-1}k_\beta^m+(1-\epsilon)^{-\beta/2}k_\beta^j=(1-\epsilon)^{-\beta/2}\sum_{m=0}^jk_\beta^m.$$ Sending $\epsilon\to 0$ proves (\[ss\]).
For $m=0,1,\ldots$ and fixed $n\in {\mathbb{N}}$ set $\tilde{W}_m:=W_{m\vee
n}$ and $\tilde{\mathcal{F}}_m;=\mathcal{F}_{m\vee n}$. Choose $\rho\in (0,\sqrt{3})$ so small that $\nu:=\dfrac{2\rho^2}{3-\rho^2}2^{\beta+1}\in (0,1)$. Applying Theorem 18.2 [@Burk] (in the notation of that paper take $\beta=2$ and $\delta=\rho$) to the non-negative martingale $(\tilde{W}_m, \tilde{\mathcal{F}}_m)$ gives $${\mathbb{P}}\{(\sum_{m=n+1}^\infty d_m^2)^{1/2}>2x\}\leq {\mathbb{P}}\{\underset{m\geq n}{\sup}\tilde{W}_m>\rho x\}+
{\mathbb{P}}\{(\sum_{m=n+1}^\infty d_m^2)^{1/2}>2x, \ \underset{m\geq
n}{\sup}\tilde{W}_m\leq \rho x\}\leq$$ $$\label{bbb} \leq {\mathbb{P}}\{\underset{m\geq n}{\sup}W_m>\rho
x\}+\dfrac{2\rho^2}{3-\rho^2}{\mathbb{P}}\{(\sum_{m=n+1}^\infty
d_m^2)^{1/2}>x\}.$$ By Potter’s bound we can take $y>0$ such that $\dfrac{{\mathbb{P}}\{W_1>x\}}{{\mathbb{P}}\{W_1>2x\}}\leq 2^{\beta+1}$ for $x\geq
y$. Set $A(y):=\underset{x\geq y}{\sup}\dfrac{{\mathbb{P}}\{\underset{m\geq
n}{\sup}W_m>\rho x\}}{{\mathbb{P}}\{W_1>2x\}}$. In view of (\[qq\]), $A(y)<\infty$ and ${\underset{x\rightarrow\infty}{\lim}}A(x)=\dfrac{k_\beta^n}{1-k_\beta}\left(\dfrac{2}{\rho}\right)^\beta$. Now we have for $x\geq y$ $$\dfrac{{\mathbb{P}}\{(\sum_{m=n+1}^\infty d_m^2)^{1/2}>2x\}}{{\mathbb{P}}\{W_1>2x\}}\leq A(y)+\nu
\dfrac{{\mathbb{P}}\{(\sum_{m=n+1}^\infty
d_m^2)^{1/2}>x\}}{{\mathbb{P}}\{W_1>x\}}.$$ Iterating the latter inequality gives that for $k=0,1,\ldots$ $$\underset{x\in [2^k y,
2^{k+1}y]}{\sup}\dfrac{{\mathbb{P}}\{(\sum_{m=n+1}^\infty
d_m^2)^{1/2}>x\}}{{\mathbb{P}}\{W_1>x\}}\leq$$ $$\leq
A(y)(1+\nu+\ldots+\nu^{k-1})+\nu^k \underset{x\in
[y,2y]}{\sup}\dfrac{{\mathbb{P}}\{(\sum_{m=n+1}^\infty
d_m^2)^{1/2}>x\}}{{\mathbb{P}}\{W_1>x\}}.$$ Let $k\to \infty$ to obtain $$\underset{x\to\infty}{\lim \sup}\dfrac{{\mathbb{P}}\{(\sum_{m=n+1}^\infty
d_m^2)^{1/2}>x\}}{{\mathbb{P}}\{W_1>x\}}\leq A(y)(1-\nu)^{-1}.$$ Now sending $y\to\infty$ gives $$\label{ta} \underset{x\to\infty}{\lim
\sup}\dfrac{{\mathbb{P}}\{(\sum_{m=n+1}^\infty
d_m^2)^{1/2}>x\}}{{\mathbb{P}}\{W_1>x\}}\leq
\dfrac{k_\beta^n}{(1-k_\beta)(1-\nu)}\left(\dfrac{2}{\rho}\right)^\beta
=const \ k_\beta^n.$$ For any $\lambda\in (0,1)$ and any $n\in {\mathbb{N}}$ $${\mathbb{P}}\{S>x\}\leq
{\mathbb{P}}\{S_n>(1-\lambda)^{1/2}x\}+{\mathbb{P}}\{(\sum_{k=n+1}^\infty
d^2_k)^{1/2}>\lambda^{1/2}x\}.$$ Therefore $$\underset{x\to\infty}{\lim \sup}\dfrac{{\mathbb{P}}\{S>x\}}{{\mathbb{P}}\{W_1>x\}}\overset{(\ref{ss}),
(\ref{ta})}{\leq}(1-\lambda)^{-\beta/2}\sum_{m=0}^{n-1}k_\beta^m+
const \ \lambda^{-\beta/2}k_\beta^n.$$ Let $n\to\infty$ and then $\lambda\to 0$ to get the desired bound for the upper limit $$\underset{x\to\infty}{\lim
\sup}\dfrac{{\mathbb{P}}\{S>x\}}{{\mathbb{P}}\{W_1>x\}}\leq \dfrac{1}{1-k_\beta}.$$ This completes the proof for $S$.(II) From the already proved relation $$\label{ma} {\mathbb{P}}\{W^\ast>x\}\sim (1-k_\beta)^{-1}{\mathbb{P}}\{W_1>x\}\sim
(1-k_\beta)^{-1}x^{-\beta}L(x),$$ it follows that ${\mathbb{E}}W^\ast<\infty$ which in turn ensures the uniform integrability of $W_n$.
Let us now prove (\[ww\]). Since $W^\ast\geq W$ a.s., ${\mathbb{E}}(W^\ast-x)^+\geq {\mathbb{E}}(W-x)^+$ for any $x\geq 0$. (\[ma\]) together with Proposition 1.5.10 [@BGT] implies that $${\mathbb{E}}(W^\ast-x)^+ \sim
(\beta-1)^{-1}(1-k_\beta)^{-1}x{\mathbb{P}}\{W_1>x\}.$$ Therefore, $$\label{low} \underset{x\to\infty}{\lim
\sup}\dfrac{{\mathbb{E}}(W-x)^+}{x{\mathbb{P}}\{W_1>x\}}\leq
\dfrac{1}{(\beta-1)(1-k_\beta)}.$$ For each $x\geq 1$ define the stopping time $\nu_x$ by $$\nu_x := \left\{\begin{array}{ll}
\inf\{n \geq 1: W_n>x\}, & \hbox{if $W^\ast>x$;} \\
+\infty, & \hbox{otherwise.} \\
\end{array}\right.$$ The random variable $W$ closes the martingale $W_n$. Hence, for each $x\geq 1$ $${\mathbb{E}}(W-x)1_{\{\nu_x<\infty\}}={\mathbb{E}}(W_{\nu_x}-x)1_{\{\nu_x<\infty\}},$$ and hence $${\mathbb{E}}(W-x)^+ \geq
{\mathbb{E}}(W_{\nu_x}-x)^+1_{\{\nu_x<\infty\}}.$$ We now transform the right-hand side into a more tractable form $${\mathbb{E}}(W_{\nu_x}-x)^+1_{\{\nu_x<\infty\}}=\sum_{k=1}^\infty {\mathbb{E}}(W_k-x)^+1_{\{\nu_x=k\}}={\mathbb{E}}\sum_{k=1}^\infty {\mathbb{E}}_{k-1}((W_k-x)^+
1_{\{\nu_x\geq k\}})=$$$$={\mathbb{E}}\sum_{k=1}^\infty 1_{\{\nu_x\geq
k\}}{\mathbb{E}}_{k-1}(W_k-x)^+={\mathbb{E}}\sum_{k=1}^{\nu_x} {\mathbb{E}}_{k-1}(W_k-x)^+.$$ From (\[repr\]) and (\[basic\]) with $Q$ replaced by $W_1$ it follows that for $k=2,3,\ldots$ $${\mathbb{P}}_{k-1}\{W_k>y\}\sim
\sum_{|u|=k-1}Y_u^\beta {\mathbb{P}}\{W_1>y\} \ \ \text{a.s.}$$ An appeal to Proposition 1.5.10 [@BGT] gives that for $k=2,3,\ldots$ $${\mathbb{E}}_{k-1}(W_k-y)^+\sim (\beta-1)^{-1}\sum_{|u|=k-1}Y_u^\beta y{\mathbb{P}}\{W_1>y\} \ \
\text{a.s.}$$ Since ${\underset{x\rightarrow\infty}{\lim}}\nu_x=+\infty$ a.s., using Fatou’s lemma allows us to conclude that $$\underset{x\to\infty}{\lim
\inf}\dfrac{{\mathbb{E}}(W-x)^+}{x{\mathbb{P}}\{W_1>x\}}\geq
\underset{x\to\infty}{\lim \inf}\dfrac{{\mathbb{E}}\sum_{k=1}^{\nu_x}
{\mathbb{E}}_{k-1}(W_k-x)^+}{x{\mathbb{P}}\{W_1>x\}}=$$ $$=
\dfrac{1}{\beta-1}\left(1+\sum_{k=2}^\infty {\mathbb{E}}\left(\sum_{|u|=k-1}Y_u^\beta\right)
\right)=\dfrac{1}{(\beta-1)(1-k_\beta)}.$$ Combining the latter inequality with (\[low\]) yields $${\mathbb{E}}(W-x)^+ \sim
(\beta-1)^{-1}(1-k_\beta)^{-1} x{\mathbb{P}}\{W_1>x\},$$ which by the monotone density theorem (see Theorem 1.7.2 [@BGT]) implies (\[ww\]).
The result for $M$ immediately follows from $${\mathbb{P}}\{W>x\}\sim
{\mathbb{P}}\{W^\ast>x\}\sim (1-k_\beta)^{-1}{\mathbb{P}}\{W_1>x\},$$ as $|W-1|\leq
M\leq W^\ast$ a.s. The proof of the theorem is finished.
The second proof of (\[ww\]) in the case $\beta>2$
==================================================
Assume that the assumptions of Theorem \[t1\] hold with $\beta>2$. By (\[ma\]) and Theorem 1.6.5 [@BGT], $${\mathbb{E}}W^\ast (W^\ast-x)^+ \sim \beta
(\beta-1)^{-1}(\beta-2)^{-1}(1-k_\beta)^{-1}x^{2-\beta}L(x).$$ Since for each $x>0$ ${\mathbb{E}}W^\ast (W^\ast-x)^+\geq {\mathbb{E}}W(W-x)^+$ $$\label{le} \underset{x\to\infty}{\lim \sup} \dfrac{{\mathbb{E}}W(W-x)^+}{x^{2-\beta}L(x)} \leq
\dfrac{\beta}{(\beta-1)(\beta-2)(1-k_\beta)}.$$
Lyons [@Lyons] constructed a probability space with probability measure ${\mathbb{Q}}$ and proved the following equality $${\mathbb{E}}_{{\mathbb{Q}}}(W_n|\mathcal{G})=1+\sum_{k=1}^n \Pi_{k-1}(S_k-1) \ \ \ \ \ {\mathbb{Q}}\ \text{a.s.},$$ where ${\mathbb{E}}_{\mathbb{Q}}$ is expectation with respect to ${\mathbb{Q}}$; $\Pi_0:=1$, $\Pi_k:=M_1M_2\cdots M_k, k=1,2,\ldots$; $\{(M_k, S_k):
k=1,2,\ldots\}$ are $Q$-independent copies of a random vector $(M,S)$ whose distribution is defined by the equality $$\label{eq11}{\mathbb{E}}\sum_{|u|=1}Y_uh(Y_u, \sum_{|v|=1}Y_v)={\mathbb{E}}h(M,S),$$ which is assumed to hold for any nonnegative Borel bounded function $h(x,y)$; $\mathcal{G}$ is the $\sigma$-field that can be explicitly described (we only note that $\sigma((M_k, S_k),
k=1,2,\ldots)\subset \mathcal{G}$). Also for any Borel function $r$ with obvious convention when the right-hand side is infinite or does not exist $$\label{p} {\mathbb{E}}_{\mathbb{Q}}r(W_n):={\mathbb{E}}W_n r(W_n) \ \ \text{and} \ \
{\mathbb{E}}_{\mathbb{Q}}r(W):={\mathbb{E}}W r(W).$$ Lyons explains his clever argument in a quite condensed form. More details clarifying his way of reasoning can be found in [@BigKypr] and [@IksRos].
Since ${\mathbb{P}}\{W_1>x\}$ regularly varies with exponent $-\beta$, $\beta>2$ then ${\mathbb{E}}W_1^2<\infty$. Also by (\[innn\]) ${\mathbb{E}}\sum_{|u|=1}Y_u^2<1$. By Proposition 4 [@Iks04] the last two inequalities together ensure that ${\mathbb{E}}W^2<\infty$. In view of (\[p\]) ${\mathbb{E}}_{{\mathbb{Q}}} W={\mathbb{E}}W^2$ and hence ${\mathbb{E}}_{{\mathbb{Q}}} W<\infty$. In Lemma 4.1 [@IksRos] it was proved that (\[inn\]) holds with ${\mathbb{P}}$ replaced by ${\mathbb{Q}}$. This implies that ${\mathbb{E}}_{{\mathbb{Q}}}
W^\ast<\infty$ iff ${\mathbb{E}}_{{\mathbb{Q}}} W<\infty$. Therefore we have checked that ${\mathbb{E}}_{{\mathbb{Q}}} W^\ast<\infty$ which by the dominated convergence theorem implies that $${\mathbb{E}}_{{\mathbb{Q}}}(W|\mathcal{G})=1+\sum_{k=1}^\infty
\Pi_{k-1}(S_k-1)=:R \ \ \ \ \ {\mathbb{Q}}\ \text{a.s.}$$ By Jensen’s inequality, for any convex function $g$ $${\mathbb{E}}_{{\mathbb{Q}}}(g(W)|\mathcal{G})\geq g({\mathbb{E}}_{{\mathbb{Q}}}(W|\mathcal{G}))=g(R)
\ \ \ \ \ {\mathbb{Q}}\ \text{a.s.}$$ Setting $g(u):=(u-x)^+, x>0$ and taking expectations yields $$\label{tru} {\mathbb{E}}W(W-x)^+\overset{(\ref{p})}{=}{\mathbb{E}}_{{\mathbb{Q}}}(W-x)^+
\geq {\mathbb{E}}_{{\mathbb{Q}}}(R-x)^+.$$ From (\[eq11\]) it follows that ${\mathbb{E}}_{{\mathbb{Q}}}M^{\beta-1}=k_\beta<1$, ${\mathbb{E}}_{{\mathbb{Q}}}M^{\beta-1+\epsilon}=k_{\beta+\epsilon}<\infty$ and $${\mathbb{Q}}\{S-1>t\}=\int_{t+1}^\infty yd{\mathbb{P}}\{W_1\leq y\}.$$ Using the latter equality and Theorem 1.6.5 [@BGT] leads to ${\mathbb{Q}}\{S-1>t\}\sim \beta (\beta-1)^{-1}t^{1-\beta}L(t)$. Therefore Theorem 1 [@Grey] applies to the perpetuity $R$ which gives $${\mathbb{Q}}\{R>t\}\sim
\beta
(\beta-1)^{-1}(1-{\mathbb{E}}_{{\mathbb{Q}}}M^{\beta-1})^{-1}t^{1-\beta}L(t).$$ By Proposition 1.5.10 [@BGT] $${\mathbb{E}}_{{\mathbb{Q}}}(R-x)^+=\int_x^\infty {\mathbb{Q}}\{R>t\}dt \sim
\dfrac{\beta}{(\beta-1)(\beta-2)(1-k_\beta)}x^{2-\beta}L(x).$$ An appeal to (\[tru\]) now results in $$\underset{x\to\infty}{\lim \inf} \dfrac{{\mathbb{E}}W(W-x)^+}{x^{2-\beta}L(x)} \geq
\dfrac{\beta}{(\beta-1)(\beta-2)(1-k_\beta)}.$$ Combining this with (\[le\]) yields $${\mathbb{E}}_{\mathbb{Q}}(W-x)^+\overset{(\ref{p})}{=}{\mathbb{E}}W(W-x)^+\sim
\dfrac{\beta}{(\beta-1)(\beta-2)(1-k_\beta)}x^{2-\beta}L(x).$$ By the monotone density theorem $${\mathbb{Q}}\{W>x\}\sim
\dfrac{\beta}{(\beta-1)(1-k_\beta)}x^{1-\beta}L(x).$$ Since ${\mathbb{Q}}\{W>x\}=\int_x^\infty yd{\mathbb{P}}\{W\leq y\}$, integrating by parts gives $$\dfrac{x{\mathbb{P}}\{W>x\}}{{\mathbb{Q}}\{W>x\}}=1-\dfrac{x}{{\mathbb{Q}}\{W>x\}}\int_x^\infty y^{-2}{\mathbb{Q}}\{W>y\}dy.$$ By Proposition 1.5.10 [@BGT] the right-hand side tends to $(\beta-1)\beta^{-1}$ when $x\to\infty$. Therefore, ${\mathbb{P}}\{W>x\}\sim (\beta-1)(\beta
x)^{-1}{\mathbb{Q}}\{W>x\}\sim (1-k_\beta)^{-1}x^{-\beta}L(x)$ as desired.
This argument seems not to work as just described when $\beta \in
(1,2]$. If $\beta\in (1,2)$ we can get a bound for the upper limit $$\underset{x\to\infty}{\lim \sup} \dfrac{{\mathbb{E}}W(W\wedge x)}{x^{2-\beta}L(x)} \leq
\dfrac{\beta}{(\beta-1)(2-\beta)(1-k_\beta)}.$$ However, we do not know how the corresponding lower limit could be obtained. In fact, we have not been able to find any random variable $\xi$ with appropriate tail behaviour and such that $W\geq \xi$ in some strong or weak sense.
Miscellaneous comments
======================
We begin this section with discussing the following problem: which classes of point processes satisfy both (\[mom\]) and (\[var\]) and which do not.
Let $h:[0,\infty)\to [0,\infty)$ be a nondecreasing and right-continuous function with $h(+0)>0$.
Let $\{\tau_0:=0, \tau_i, i\geq 1\}$ be the renewal times of an ordinary renewal process. In addition to the conditions on $h$ imposed above, assume that $h(0)$ is finite. Consider the point process ${\mathcal{M}}$ with points $\{A_i=\gamma^{-1}\log h(\tau_i),
i=1,2,\ldots\}$, where $\gamma>0$ is chosen so that ${\mathbb{E}}\sum_{i=1}^\infty h(\tau_i)=1$. According to Theorem 1 [@DoBr] $W_1=\sum_{i=1}^\infty h(\tau_i)$ has exponentially decreasing tail. Thus while we can find $h$ and $\{\tau_i\}$ such that (\[mom\]) holds, (\[var\]) always fails.
The situation when ${\mathbb{P}}\{W_1>x\}\sim x^{-\beta}L(x)$ and ${\mathbb{E}}W_1^\beta<\infty$ is not terribly interesting. However, if this is the case Theorem \[t1\] implies the one-way implication of a well-known moment result (see [@Iks04] and [@IksNegad]) $${\mathbb{E}}\sum_{|u|=1}Y_u^\beta<1, \ \ {\mathbb{E}}W_1^\beta<\infty \Leftrightarrow {\mathbb{E}}W^\beta<\infty, \ \ {\mathbb{E}}(W^\ast)^\beta<\infty.$$ In the subsequent examples in addition to (\[mom\]) and (\[var\]) we require that ${\mathbb{E}}W_1^\beta=\infty$. Examples \[e2\] and \[e3\] essentially shows that when the number of points in a point process is infinite, and the points are independent or constitute an (inhomogeneous) Poisson flow, ${\mathbb{E}}\sum_{|u|=1}Y_u^\beta<\infty$ implies ${\mathbb{E}}W_1^\beta<\infty,
\beta>1$.
\[e2\] Assume that ${\mathcal{M}}$ is a point process with independent points $\{C_i\}$, and ${\mathbb{E}}\sum_{i=1}^\infty Y_i=1$ and for some $\beta>1$ ${\mathbb{E}}\sum_{i=1}^\infty Y_i^\beta<\infty$, where $Y_i=e^{\gamma C_i}$ and $\gamma>0$. Then ${\mathbb{E}}W_1^\beta<\infty$.
In this case $W_1=\sum_{i=1}^\infty Y_i$. Hence we must check that ${\mathbb{E}}(\sum_{i=1}^\infty Y_i)^\beta<\infty$. By using the $c_\beta$-inequality let us write the (formal) inequality $$\label{in} {\mathbb{E}}(\sum_{i=1}^\infty Y_i)^\beta={\mathbb{E}}\sum_{i=1}^\infty
Y_i (Y_i+\sum_{k\neq i}Y_k)^{\beta-1}\leq (2^{\beta-2}\vee 1)
({\mathbb{E}}\sum_{i=1}^\infty Y_i^\beta+ ({\mathbb{E}}\sum_{i=1}^\infty Y_i) {\mathbb{E}}(\sum_{i=1}^\infty Y_i)^{\beta-1}).$$ For any $\beta>1$ there exists $n\in {\mathbb{N}}$ such that $\beta\in (n,
n+1]$. We will use induction on $n$. If $\beta \in (1,2]$ then ${\mathbb{E}}\sum_{i=1}^\infty Y_i<\infty$ implies ${\mathbb{E}}(\sum_{i=1}^\infty
Y_i)^{\beta-1}<\infty$. Hence by (\[in\]) ${\mathbb{E}}(\sum_{i=1}^\infty Y_i)^\beta<\infty$. Assume that the conclusion is true for $\beta \in (n, n+1]$ and prove it for $\beta \in (n+1,
n+2]$. Since ${\mathbb{E}}\sum_{i=1}^\infty Y_i<\infty$ and ${\mathbb{E}}\sum_{i=1}^\infty Y_i^\beta<\infty$ we have ${\mathbb{E}}\sum_{i=1}^\infty Y_i^{\beta-1}<\infty$ which, by the assumption of induction, implies ${\mathbb{E}}(\sum_{i=1}^\infty
Y_i)^{\beta-1}<\infty$. It remains to apply (\[in\]).
\[e3\] Let $\{\tau_i, i\geq 1\}$ be the arrival times of a Poisson process with intensity $\lambda>0$. Consider a Poisson point process ${\mathcal{M}}$ with points $\{B_i\}$ and assume that for any $a\in{\mathbb{R}}$ ${\mathcal{M}}(a,\infty)\geq 1$ a.s. Then there exists a function $h$ as described at the beginning of this section that additionally satisfies $h(0)=\infty$, and $\gamma>0$ such that $h(\tau_i)=e^{\gamma B_i}$ and ${\mathbb{E}}W_1={\mathbb{E}}\sum_{i=1}^\infty
h(\tau_i)=\lambda \int_0^\infty h(u)du=1$. The distribution of $W_1$ is infinitely divisible with zero shift and Lévy measure $\nu$ given as follows: $\nu(dx)=\lambda h^\leftarrow (dx)$, where $h^\leftarrow$ is a generalized inverse function. Since $\lambda
{\mathbb{E}}\sum_{i=1}^\infty h^\beta (\tau_i)=\int_0^\infty x^\beta
\nu(dx)$, and as is well-known from the general theory of infinitely divisible distributions, $\int_1^\infty x^\beta
\nu(dx)<\infty$ implies ${\mathbb{E}}W_1^\beta<\infty$, we conclude that ${\mathbb{E}}\sum_{i=1}^\infty e^{\gamma \beta B_i}<\infty$ implies ${\mathbb{E}}W_1^\beta<\infty$.
In the next example we point out a class of point processes which satisfy (\[mom\]) and (\[var\]), and ${\mathbb{E}}W_1^\beta=\infty$.
Let $K$ be a nonnegative integer-valued random variable with ${\mathbb{P}}\{K>x\}\sim x^{-\beta}L(x), \beta>1$, and $\{D_i, i\geq 1\}$ be independent identically distributed random variables which are independent of $K$. If ${\mathcal{M}}$ is a point process with points $\{D_i, i=\overline{1,K}\}$ and there exists a $\gamma>0$ such that ${\mathbb{E}}\sum_{i=1}^K e^{\gamma D_i}=1$ and ${\mathbb{E}}\sum_{i=1}^K
e^{\gamma \beta D_i}<1$ then according to Proposition 4.3 [@Sam] we have ${\mathbb{P}}\{W_1>x\}\sim {\mathbb{E}}e^{\gamma \beta
D_1}x^{-\beta}L(x)$.
We conclude with two remarks that fit the context of the present paper.
The tail behaviour of ${\mathbb{P}}\{W>x\}$ and ${\mathbb{P}}\{W^\ast>x\}$ has been investigated in [@Iks04] and [@IksNegad]. In particular, from those works it follows that when the condition ${\mathbb{E}}\sum_{|u|=1}Y_u^\beta<1$ fails, the condition ${\mathbb{P}}\{W_1>x\}\sim x^{-\beta}L(x)$ is not a necessary one for either ${\mathbb{P}}\{W>x\}\sim x^{-\beta}L(x)$ or ${\mathbb{P}}\{W^\ast>x\}\sim
x^{-\beta}L(x)$ to hold.
Let $\{X_i\}$ be the points of a point process, and let $V$ be a random variable satisfying the following distributional equality $$V\overset{d}{=} \sum_{i=1}^\infty X_iV_i,$$ where $V_1, V_2,
\ldots$ are conditionally on $\{X_i\}$ independent copies of $V$. The distribution of $V$ is called a fixed point of the smoothing transform (see [@Iks04] for more details, and [@IksJur] and [@IksKim] for an interesting particular case). It is known and can be easily checked that the distribution of $W$ is a fixed point of the smoothing transform with $\{X_i:i=1,2,\ldots\}=\{Y_u:|u|=1\}$. Thus (\[ww\]) could be reformulated as a result on the tail behaviour of the fixed points *with finite mean*. The tail behaviour of fixed points with infinite mean deserves a special mention. Typically, their tails regularly vary with index $\alpha\in (0,1)$, or $\int_0^x
{\mathbb{P}}\{V>y\}dy$ slowly varies. This follows from Proposition 1(b) [@Iks04] and Proposition 8.1.7[@BGT].
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[^1]: e-mail address: iksan@unicyb.kiev.ua
[^2]: e-mail address: pilot$\_$ser@mail.ru
| {
"pile_set_name": "ArXiv"
} |
---
abstract: 'Analyzing textual data is a very challenging task because of the huge volume of data generated daily. Fundamental issues in text analysis include the lack of structure in document datasets, the need for various preprocessing steps and performance and scaling issues. Existing text analysis architectures partly solve these issues, providing restrictive data schemas, addressing only one aspect of text preprocessing and focusing on one single task when dealing with performance optimization. Thus, we propose in this paper a new generic text analysis architecture, where document structure is flexible, many preprocessing techniques are integrated and textual datasets are indexed for efficient access. We implement our conceptual architecture using both a relational and a document-oriented database. Our experiments demonstrate the feasibility of our approach and the superiority of the document-oriented logical and physical implementation.'
author:
- 'Ciprian-Octavian Truică$^1$, J[é]{}r[ô]{}me Darmont$^2$, Julien Velcin$^2$'
bibliography:
- 'adma2016.bib'
title: |
A Scalable Document-based Architecture\
for Text Analysis
---
Introduction
============
A vast amount of textual data is generated daily and it is really challenging to develop efficient models and systems to enhance processing performance while doing accurate text analysis. The most fundamental challenges when working with large volumes of heterogeneous text datasets include the lack of structure of textual corpora, the various required preprocessing steps, the need for efficient access and the ability to scale up. Structural issues may be addressed by resorting to textual data warehousing and On-Line Analytical Processing (OLAP). However, such approaches only partially solve the problem because they use a structured schema that falls short when applied to large, heterogeneous volumes of data. Moreover, using a predefined schema makes them extremely dataset-specific. Moreover, when dealing with textual data, we distinguish different preprocessing levels: quite basic operations (e.g., cleaning HTML tags, tokenization, language identification); intermediate operations (e.g., stemming, lemmatization, indexing); and advanced operations (e.g., part of speech tagging, named entity recognition, topic modeling). Each complexity layer in this process requires the previous layer and all operations must remain tractable in terms of memory and CPU time. To the best of our knowledge, no text analysis tool implements all layers, nor any processing workflow.
Finally, when working on performance and scaling issues, state-of-the-art research focuses on one aspect of text analysis, e.g., aggregation, top-k keyword extraction and text indexing. However, text processing techniques used in a single application may be many and, as we mention above, interdependent.
Hence, we present in this paper a scalable text analysis architecture that addresses all these issues. More precisely, we deal with the lack of structure by adopting a novel generic, document-oriented data model that allows storing heterogeneous textural corpora with no predefined structure. We also integrate in our framework all the preprocessing methods that are useful for information retrieval, data mining, text analysis and knowledge discovery. We also propose a new compact data structure to minimize index storage space and the response time of create, read, update and delete (CRUD) operations. Such indexes benefit to text preprocessing, querying and further analysis, and adequately contribute to global scaling.
The remainder of this paper is organized as follows. In Section \[sec:RelatedWorks\], we discuss related works. In Section \[sec:ProposedApproachAndImplementation\], we present the architecture and implementation of our approach. In Section \[sec:ExperimentalValidation\], we experimentally validate our proposal. In Section \[sec:Conclusion\], we finally conclude this paper and hint at future research.
Related Works {#sec:RelatedWorks}
=============
Text Cubes and OLAP
-------------------
Extensive work on information retrieval (IR) and text analysis have been done using OLAP. Most proposals use Text Cubes for OLAPing multidimensional text databases [@DZhang2009]. Lin et al. focus on optimizing query processing and reducing storage costs of Text Cubes [@CXLin2008]. They experimentally show that average query time and storage cost are related to a cube’s number of dimensions. Zhang et al. use Text Cubes for topic modeling [@DZhang2009] and experimentally show that their approach is much faster than computing each topic cube from scratch. Finally, Ding et al. address the problem of keyword search and top-k document ranking using Text Cubes [@BDing2011]. Their algorithms perform well in terms of query response and memory cost when the number of search terms is small.
Ben Kraiem et al. propose a generic multidimensional model for OLAP on tweets [@MBenKraiem2014]. Their experiments show some promising results for knowledge discovery when applying OLAP on a small corpus, but query performance decreases when data volume increases. Bringay et al. propose a data warehouse model to analyze large volumes of tweets [@SBringay2011]. They introduce different operators to identify trends using the top-k most significant words over a period of time for a specific geographical location, as well as the impact of hierarchies on such operators. Unfortunately, no time performance and storage cost analysis is provided.
In conclusion, research done so far on text analysis and OLAP focuses on small, structured datasets and scaling up is not guaranteed.
Text Preprocessing and Analysis
-------------------------------
Managing morphological variation of search terms in IR has been quite extensively studied [@KKettunen2005; @AGJivani2011]. The main successful methods are stemming [@DSharma2012] and lemmatization, which are used to optimize search, minimize the space allocated to inverted indexes (Section \[sec:docindex\]) and, in the case of lemmatization, to add linguistic information. Lemmatization is useful for different types of advanced text analysis, e.g., named entity recognition, automatic domain specific multi-term extraction and part of speech (PoS) tagging. Moreover, lemmatization is easier of use than stemming, saves storage and improves retrieval performance [@KKettunen2005].
Topic modeling is a statistical model for discovering hidden themes that occur in a collections of documents. In recent years, it has been extensively studied, showing the usefulness of analyzing latent topics and discovering topic patterns . Popular approaches for topic modeling are latent semantic indexing (LSI) [@SCDeerwester1990], latent Dirichlet allocation (LDA) [@DMBlei2003], the non-parametric extension hierarchical Dirichlet process (HDP) [@YWTeh2006] and non-negative matrix factorization (NMF) [@SArora2013].
Document Indexing {#sec:docindex}
-----------------
Inverted indexes are data structures used in search engines, whose main purpose is to optimize query response speed. Basic inverted indexes store terms, a list of documents where each term appears and a weight. Weight measures the number of occurrences of the term in a document, e.g., raw term frequency/word co-occurrence (TF), normalized Term Frequency (TF$_n$), etc. In the various methods for managing inverted indexes, great emphasis is put on storage space reduction. For instance, a pruning algorithm based on term frequency-inverse document frequency (TF\*IDF) can be used to minimize index size [@SKVishwakarma2014]. Yet, updating an inverted index is also a problem, because it is dependent on documents. The index must indeed be updated each time documents are added or deleted.
Proposed Approach and Implementation {#sec:ProposedApproachAndImplementation}
====================================
Approach Overview
-----------------
The approach we propose (Figure \[fig:architecture\]) is subdivided into four steps: 1) clean and preprocess documents using natural language processing (NLP) and store the information in a database; 2) construct indexes; 3) analyze data, e.g., with topic modeling, etc.; 4) query and search data, extract top-k most relevant documents, create visualizations and analyses. We construct the inverted index, vocabulary, PoS and named entities (NE) indexes during the index construction step. Indexes may be used afterward by data mining, text analysis, search and visualization. The search engine sorts documents based on a ranking function (e.g., TF\*IDF, Okapi or BM25) to extract the top-k documents.
![System architecture[]{data-label="fig:architecture"}](architecture.png){width="10cm"}
To implement our document-oriented approach, we quite naturally rely on a document-oriented database management system (DODBMS). DODBMSs are a class of NoSQL systems that aim to store, manage and process data using a semi-structured model. DODBMSs encapsulate data in collections of documents [@JHan2011]. A document can contain other nested documents, which turns out to be very flexible [@COTruica2015a].
One feature of DODBMSs is that they are often optimized for create and read operations, while offering reduced functionality for update and delete queries. DODBMSs are designed to work with large amounts of data and the main focus is on the efficiency of data storage, access and analysis [@ERedmond2012]. Another key feature of DODBMSs is the distribution of data across multiple sites. In particular, DODBMSs can horizontally scale CRUD operations throughput [@RCattell2011]. Moreover, decentralized data stores provide good mechanisms for fail-over, removing the single point of failure, due to their scalability and flexibility [@RHecht2011].
We selected MongoDB as our DODBMS, since it beats the best mean time performances for CRUD operations both in single and distributed environments [@COTruica2015a]. Moreover, we also implemented our approach with PostgreSQL, to provide a point of comparison with a well-established, efficient relational database management system (RDBMS) (Section \[sec:ExperimentalValidation\]).
Data Models
-----------
We design a generic model to store heterogeneous text data using a data warehouse snowflake schema (Figure \[fig:postgredb\]). The central component of the model is the *documents* entity, where we store basic information and metadata about a document, e.g., timestamp, title, raw, clean and lemmatized text, etc. The *document\_tags* entity is used to store metadata represented by tags, which can be existing tags, hashtags or at tags. The *vocabulary* entity links documents to information extracted or inferred from the text, which helps enhancing metadata with different weights and tags, e.g., PoS, TF, TF$_n$, lemmas, etc. The *named\_entities* entity stores all the information about entities automatically extracted from the original corpus.
![Conceptual model[]{data-label="fig:postgredb"}](postgresdb.png){width="12cm"}
The DODBMS schemaless design takes all the information presented in the relational schema and stores it for each document in a record of the collection. Using this design, all one-to-many and many-to-many relationships become either vectors (e.g., *hashtags*, *at tags*) or nested documents (e.g., *words*, *named\_entities*). Where the information is not present, these vectors and nested documents may be missing thanks to the flexibility of schemaless database design. A problem that arises is duplication, as multiple records can bear the same metadata, since all the information for a document is stored in one single record. The *vocabulary* entity is constructed as a separate collection. This entity is constructed dynamically, taking user input constraints into account, e.g. date, tags, search words, named-entities.
Interaction with the database is achieved through CRUD operations, aggregation functions and views. We use read operations for information extraction and data visualization. Aggregation functions are used for constructing indexes, searching and preprocessing data for text analysis. We make use of MapReduce for this purpose when using the document-oriented database architecture. Dynamically materialized cubes are constructed using views with aggregation functions, fine-graining query results using different measures, e.g., timestamps, locations, lemmas, tags, named entities.
Text Preprocessing
------------------
The data cleaning module serves three functions: 1) corpus standardization, 2) text preprocessing using NLP to enrich data, and 3) entity creation and information insertion into the database.
The entire corpus is standardized by determining all the fields of a document, including metadata and the labels of *documents*. Then, during the preprocessing step, the following techniques are applied: 1) text cleaning by removing HTML/XML tags and scripts; 2) language identification; 3) expanding contractions; 4) extracting features, e.g., PoS, lemmas and named entities; 5) removing stop words and punctuation; 6) computing term weights. We use a multithreading architecture for data cleaning to cope with large data volumes and scale up vertically. At the end of each thread, the information is stored in a dictionary, together with other metadata. We choose to use asynchronous threads because, after a worker thread finishes, a new job can be assigned to it without waiting for the other worker threads to finish. This is made possible because each task is independent. At the end of this step, a record of the *documents* collection is created and inserted into the database. The record contains all labels from the first step and the information extracted using NLP from the second one.
In the DODBMS implementation, a record stores all the information because its attributes are created dynamically. In contrast, the RDBMS architecture can only store predefined fields due to its rigid schema. Thus, undefined fields are omitted.
The RDBMS approach merges the data cleaning step with the index construction step, because many-to-many relationships between entities, translated as bridge tables, are indexes as well. We could not use a multithreading approach here because information could be lost. Multiple threads could indeed check at the same time whether the information is present and receive a negative response. A constraint violation error could appear and the transaction terminate by a *rollback*. If constraints are missing, then duplicate information could appear and this would impact text analysis.
Index Management
----------------
We propose several indexes for document aggregation, search, extraction of the top-k most signification terms and text analysis, e.g., topic modeling, document clustering. These new indexing structures minimize storage costs and maximize the time performance of CRUD operations.
Index construction in the DODBMS architecture is done using the MapReduce framework. Four indexes are created: 1) an *inverted index* that stores, for each term, a list of corresponding documents; 2) a *vocabulary*, a novel inverted index with additional information for each term in the corpus, e.g., list of documents where the term is found, the TF and TF$_n$ of the term for a document and IDF; 3) a PoS index that stores the part of speech of each term; 4) a named-entity index used for storing named entities. There are no integrity constraints between these collections to improve query response time. Moreover, the structure proposed for the *vocabulary* facilitates query response time, aggregation and search (Figure \[fig:voc\_ex\]). MapReduce is used to construct all indexes. It is also central in aggregation queries needed by the search algorithms. To improve index construction and query response times, we horizontally scale the database, and by doing so add more MapReduce worker. In the RDBMS architecture, indexes are the bridge tables translating many-to-many relationships between entities. The *vocabulary* is the bridge table between the *documents* table and the *words* table. The PoS index is the bridge table between the *vocabulary* table and the *pos* table. In this case, the index also contains the TF and IDF of each term.
![Vocabulary index structure[]{data-label="fig:voc_ex"}](vocabulary2.png){width="11cm"}
The number of entries in the indexes constructed for the DODBMS is equal to the number of terms in the entire corpus. In the RDBMS, the inverted index has more entries, i.e., $\sum\nolimits_{d \in D} \mid t : t \in d \mid$ , where *D* is the corpus and $\mid t : t \in d\mid$ is the number of distinct terms that appear in document $d$.
Updating indexes in the DODBMS is based on document insertion date. The update method we use constructs an intermediary index for new documents, and then it updates the primary index by appending the new documents’ ID and TF to existing labels. Then, the IDF of each term is updated for the whole index. When documents are deleted, we apply a bulk delete operation. In this case, a list of deleted document IDs is stored, which helps update the index structure by removing the deleted documents and then updating the IDF of each term.
Updating indexes in the RDBMS implementation is easier thanks to the database’s structure. When documents are added, indexes are automatically updated based on the insertion date of the last added documents. When documents are removed, the corresponding index entries are also removed. For both operations, the IDF of each term must be recalculated.
Experimental Validation {#sec:ExperimentalValidation}
=======================
In this section, we test each step of our approach and we compare the results achieved by the two instances we developed, i.e., the DODBMS version implemented with MongoDB and the RDBMS version implemented with PostgreSQL. Tests are done using a news corpus consisting of 110,000 articles[^1], a corpus of 5,000,000 tweets[^2] and a scientific corpus of 20,000 abstracts from [@JTang2012]. The size of these corpora, some would argue, is rather small with respect to Big Data. Yet, it is sufficient to illustrate our architecture’s good time performance. Text analysis is indeed not usually done on large corpora. Moreover, other corpora used in the literature are smaller, e.g., 3,000 documents [@DZhang2009], 2,013 records [@CXLin2008], 65,333 tweets [@MBenKraiem2014], 1,801,810 tweets [@SBringay2011]. Our architecture can be deployed in a cloud environment if all the requirements are met, i.e., if Python packages, PostgreSQL, and MongoDB are available. Tests are done on machines that reside in an OpenStack private cloud platform. We purposely selected this hardware architecture and dataset sizes to show that our architecture can achieve good performance even on end-user workstations, as it is sometimes not desirable to send data online due to privacy issues. Moreover, end-users presumably cannot afford very powerful, parallel computers.
News Articles Corpus Experiments
--------------------------------
The first set of experiments are done using two computers with the same hardware configuration: 4 GB RAM and 1 CPU with two 2.2 GHz cores. We choose this hardware architecture to show that our method gives good results on simple computers. Using the initial news articles corpus, seven corpora are created consisting of 100 to 110,000 documents. They are referred to as Corpus $i, i \in \{1, 2, ... , 7\}$. For comparison reasons, experiments are done using a single-thread approach.
Figure \[fig:populatedb\] presents the average time (in seconds) for populating the databases. Duplicate documents are removed in this step. This is done by checking whether an article already exists in the database based on its title. If the document does not exist, then a new record is added. Otherwise, tags are verified so that metadata are not omitted, as the same article could have more tags for different instances found in the corpus. The second set of tests evaluates the efficiency of text cleaning and index construction (Figure \[fig:text\_preprocessing\]).
[.5]{}
+\[color=blue\] table \[x=corpus, y=mongodb\] [populate\_db.txt]{}; +\[color=red\] table \[x=corpus, y=postgresql\] [populate\_db.txt]{};
[.5]{}
+\[color=blue\] table \[x=corpus, y=mongodb\] [text\_cleaning.txt]{}; +\[color=red\] table \[x=corpus, y=postgresql\] [text\_cleaning.txt]{};
[.5]{}
+\[color=blue\] table \[x=corpus, y=mongodb\] [storage.txt]{}; +\[color=red\] table \[x=corpus, y=postgresql\] [storage.txt]{};
[.5]{}
+\[color=blue\] table \[x=terms, y=mongodb\] [search.txt]{}; +\[color=red\] table \[x=terms, y=postgresql\] [search.txt]{};
Figure \[fig:storage\] shows the total storage space (in MB) for all corpora. To respect database normalization in PostgreSQL, bridge tables materializing many-to-many relations have to be added. In MongoDB, such relationships translate into vectors or nested documents inside collections. For example, the *documents* collection contains the authors table as an array of nested documents and the tags table as an array. This brings the issue of duplicates, as we may have the same tags for different document that would be stored in each element of the collection. However, it is a small cost to pay as, using this structure, joins are removed, whereas join is the costliest operation in RDBMSs. Experimental results show that MongoDB efficiently stores the data, minimizing storage space by 30% with respect to PostgreSQL. Moreover, based on the number of records in each collection, from a computational point of view, a select operation performed on a smaller entity shows faster response times than one performed on an entity with a lot of records. For example, it is faster to query the *vocabulary* collection than to interrogate the *vocabulary* table, because the table contains more records than the collection.
[1]{}
---------- --------- -------- -------- --------
Corpus 1 25.36 4.95 5.94 6.57
Corpus 2 103.74 12.63 16.15 19.26
Corpus 3 204.91 20.10 25.36 32.95
Corpus 4 934.56 57.74 81.46 117.15
Corpus 5 1805.27 103.38 141.81 209.14
Corpus 6 2524.33 148.69 212.16 312.61
Corpus 7 3564.07 216.34 311.88 461.88
---------- --------- -------- -------- --------
: After documents are removed[]{data-label="tab:update_idx_delete"}
[1]{}
-- --------------- ---------------- ------------ ---------------- --------------- ---------------
**Update** **Rebuild** **Update** **Rebuild** **Update** **Rebuild**
[**37.65**]{} 123.63 1163.95 [**189.62**]{} [**33.53**]{} 91.52
[**76.54**]{} 126.16 1214.20 [**208.47**]{} [**67.79**]{} 93.08
144.56 [**126.07**]{} 1303.89 [**204.18**]{} 129.44 [**95.24**]{}
201.26 [**130.16**]{} 1395.10 [**201.51**]{} 179.90 [**97.72**]{}
-- --------------- ---------------- ------------ ---------------- --------------- ---------------
: After documents are removed[]{data-label="tab:update_idx_delete"}
[1]{}
-- ------------ ------------- ------------ ------------- ------------ -------------
**Update** **Rebuild** **Update** **Rebuild** **Update** **Rebuild**
-- ------------ ------------- ------------ ------------- ------------ -------------
: After documents are removed[]{data-label="tab:update_idx_delete"}
Figure \[fig:search\_performance\] presents the mean time for extracting the top-k documents. Tests are performed on Corpus 7 with $k=20$. After each search, the database cache and buffers are cleared so that the comparison is accurate. MongoDB is from 86% faster than PostgreSQL for one term-search to over 50% faster for five terms.
Table \[tab:create\_idx\] presents mean text cleaning and index construction times, as index construction is done separately in MongoDB. MapReduce functions were developed to further improve performance. Our results show that text cleaning and index creation is improved by 94% with MongoDB (Figure \[fig:text\_preprocessing\]). Moreover, index update is an important feature in a system where new documents are added or deleted. We use new corpora of 500 to 5,000 articles from Corpus 5 to test this feature in MongoDB. For comparison purposes, for each operation, we tested the performance of updating and rebuilding the entire index. Updating the inverted index and the PoS index (Table \[tab:update\_idx\_add\]) works fast if the number of added documents is small, but time performance shifts for bigger corpora. Then, it is better to rebuild the entire index. If documents are deleted, it is faster to rebuild the inverted index (Table \[tab:update\_idx\_delete\]). Little improvement is seen between updating and rebuilding the PoS index (Table \[tab:update\_idx\_delete\]) when documents are deleted. Concerning vocabulary, it is faster to rebuild the entire index than to update it, because the IDF must be recomputed for each element in the collection (Tables \[tab:update\_idx\_add\] and \[tab:update\_idx\_delete\]).
[.5]{}
+\[color=blue\] table \[x=tweets, y=1\_thread\] [text\_cleaning\_threads.txt]{}; +\[color=red\] table \[x=tweets, y=12\_threads\] [text\_cleaning\_threads.txt]{};
[.5]{}
+\[color=blue\] table \[x=words, y=1\_node\] [search\_tweets.txt]{}; +\[color=red\] table \[x=words, y=5\_nodes\] [search\_tweets.txt]{};
[.5]{}
+\[color=blue\] table \[x=tweets, y=1\_node\_voc\] [voc\_build.txt]{}; +\[color=red\] table \[x=tweets, y=5\_nodes\_voc\] [voc\_build.txt]{};
[.5]{}
+\[color=blue\] table \[x=tweets, y=1\_node\_ne\] [ne\_build.txt]{}; +\[color=red\] table \[x=tweets, y=5\_nodes\_ne\] [ne\_build.txt]{};
Twitter Corpus Experiments {#sec:twitterxp}
--------------------------
This set of experiments is carried out using one machine with the following hardware configuration: 12 GB RAM and 3 CPU with 4 2.6 GHz cores. We choose this hardware configuration to prove that our architecture does not require specialized hardware to have good time performance. We work on 5,000,000 tweets in these experiments.
Figure \[fig:text\_cleaning\_threads\] presents the results obtained when using a multithreading architecture. The improvement obtained from switching from a single thread to a 12-thread implementation is 90%, lowering preprocessing time by a factor of 10. We can observe that the number of nodes used by MongoDB directly impacts performance and enhances response time, especially for large numbers of tweets. The construction time of the vocabulary index improves significantly, by over 59% (Figure \[fig:voc\_build\]). The same happens with the named entities index, with an improvement over 40% (Figure \[fig:ne\_build\]). Keyword search performance remains constant when we scale the database horizontally (Figure \[fig:search\_tweets\]).
Scientific Articles Corpus Experiments
--------------------------------------
This set of experiments uses the scientific corpus and is carried out using the same hardware configuration as in Section \[sec:twitterxp\]. These experiments are designed to test the time performance for constructing the vectorization matrices and extracting topics. Figure \[fig:vectorization\] displays construction time for four different vectorization matrices, namely TF, TF$_n$, TF\*IDF and Okapi BM25. The best performance is obtained by the TF$_n$ vectorization matrix because all the information exists in the *vocabulary* index. TF\*IDF and Okapi BM25 vectorizations are slower because they must be computed for each element during matrix construction. The second set of tests presents the performance time of extracting topics from the entire corpus (\[fig:tm\_comparison\]). LSI is faster then LDA and HDP by a factor of 21 and 13, respectively. NMF achieves the best performance.
[.5]{}
+\[color=blue\] table \[x=test, y=count\_mean \] [vectorization\_times.txt]{}; +\[color=red\] table \[x=test, y=tfidf\_mean \] [vectorization\_times.txt]{}; +\[color=green\] table \[x=test, y=tf\_mean \] [vectorization\_times.txt]{}; +\[color=gray\] table \[x=test, y=okapi\_mean \] [vectorization\_times.txt]{};
[.5]{}
+\[color=blue\] table \[x=algorithm, y=count\_mean \] [tm\_times.txt]{}; +\[color=red\] table \[x=algorithm, y=tfidf\_mean \] [tm\_times.txt]{}; +\[color=green\] table \[x=algorithm, y=tf\_mean \] [tm\_times.txt]{}; +\[color=gray\] table \[x=algorithm, y=okapi\_mean \] [tm\_times.txt]{};
Conclusion {#sec:Conclusion}
==========
In this paper, we present a new, complete architecture for text analysis that improves search performance, minimizes storage cost through efficient document-oriented storage, and scales up horizontally and vertically. Moreover, by exploiting MapReduce to parallelize index construction and by designing new structures for indexing and decreasing the number of records stored in the database, we minimize the number of CRUD operations and further enhance performance. Finally, the algorithm we propose for extracting top-k documents for a given search phrase also considerably improves query response time.
Our experimental results show that a document-oriented architecture is best-suited and improves performances when working with large volumes of text when adding documents into the database, cleaning text and constructing indexes. For all test cases, the mean time for populating the DODBMS is half that of the RDBMS. Cleaning texts and constructing inverted indexes is also faster when using a DODBMS. Although duplicates can be found inside a DODBMS, storage costs are significantly lower than with a RDBMS. A demo application that further shows the capabilities of this architecture is presented in [@COTruica2016]. In future work, we plan to add new features to our framework, such as automatic domain specific multiterm extraction, cross-language IR, word embedding and new topic models, e.g., dynamic topic modeling.From an architectural point of view, we also want to parallelize the algorithms and use a GPU for computations.
[^1]: <http://www.corpora.heliohost.org>
[^2]: Collected with Twitter’s tools at <https://dev.twitter.com>
| {
"pile_set_name": "ArXiv"
} |
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