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abstract: 'We present an in-depth exploration of the module structures of local (co)homology modules (moreover, for complexes) over the completion ${\widehat{R}^{{\mathfrak{a}}}}$ of a commutative noetherian ring $R$ with respect to a proper ideal ${{\mathfrak{a}}}$. In particular, we extend Greenlees-May Duality and MGM Equivalence to track behavior over ${\widehat{R}^{{\mathfrak{a}}}}$, not just over $R$. We apply this to the study of two recent versions of homological finiteness for complexes, and to certain isomorphisms, with a view toward further applications. We also discuss subtleties and simplifications in the computations of these functors.'
address:
- 'Department of Mathematical Sciences, Clemson University, O-110 Martin Hall, Box 340975, Clemson, S.C. 29634 USA'
- 'Richard Wicklein, Mathematics and Physics Department, MacMurray College, 447 East College Ave., Jacksonville, IL 62650, USA'
author:
- 'Sean Sather-Wagstaff'
- Richard Wicklein
title: Extended Local Cohomology and Local Homology
---
[^1]
Introduction {#sec130805a}
============
Throughout this paper let $R$ be a commutative noetherian ring, let ${{\mathfrak{a}}}\subsetneq R$ be a proper ideal of $R$, and let ${\widehat{R}^{{\mathfrak{a}}}}$ be the ${{\mathfrak{a}}}$-adic completion of $R$. Let $K$ denote the Koszul complex over $R$ on a finite generating sequence for ${{\mathfrak{a}}}$. We work in the derived category ${{\mathcal{D}}}(R)$ with objects the $R$-complexes indexed homologically $X=\cdots\to X_i\to X_{i-1}\to\cdots$ and the full subcategory ${{{\mathcal{D}}}_{\text{b}}}(R)$ of complexes with bounded homology. Isomorphisms in ${{\mathcal{D}}}(R)$ are marked by the symbol $\simeq$. The right derived functor of Hom is ${\mathbf{R}\!\operatorname{Hom}_{R}(-,-)}$, and the left derived functor of ${-\otimes_{R}-}$ is ${-\otimes^{\mathbf{L}}_{R}-}$. See, e.g., [@hartshorne:rad; @verdier:cd; @verdier:1] for foundations and Section \[sec140109b\] for background.
This work is part 4 in a series of papers on derived local cohomology and derived local homology. It builds on our previous papers [@sather:afbha; @sather:afcc; @sather:scc], and it is applied in the papers [@sather:afc; @sather:asc].
The starting point for this paper is the following fact. Given an $R$-module $M$, each local cohomology module ${\operatorname{H}}^i_{{\mathfrak{a}}}(M)$ is ${{\mathfrak{a}}}$-torsion, so it has a natural ${\widehat{R}^{{\mathfrak{a}}}}$-module structure. The completion ${\widehat{M}^{{\mathfrak{a}}}}$ also has a natural ${\widehat{R}^{{\mathfrak{a}}}}$-module structure. More generally, given an $R$-complex $X$, the derived local cohomology complex ${\mathbf{R}\Gamma_{{\mathfrak{a}}}(X)}$ and the derived local homology complex ${\mathbf{L}\Lambda^{{\mathfrak{a}}}(X)}$ are naturally complexes over ${\widehat{R}^{{\mathfrak{a}}}}$. These complexes are constructed by applying the torsion and completion functors, respectively, to appropriate resolutions of $X$. For clarity, we write ${\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(X)}$ and ${\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)}$ when we are working over ${\widehat{R}^{{\mathfrak{a}}}}$. See Section \[sec140109b\] for definitions and notation. Note that Section \[sec150922a\] documents some subtleties and a simplification involved in these constructions.
In this paper, we investigate how standard facts for the $R$-complexes ${\mathbf{R}\Gamma_{{\mathfrak{a}}}(X)}$ and ${\mathbf{L}\Lambda^{{\mathfrak{a}}}(X)}$ extend to the ${\widehat{R}^{{\mathfrak{a}}}}$-complexes ${\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(X)}$ and ${\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)}$. Our primary motivation comes from work of Alonso Tarr[í]{}o, Jerem[í]{}as L[ó]{}pez, and Lipman [@lipman:lhcs]; Greenlees and May [@greenlees:dfclh]; Matlis [@matlis:kcd; @matlis:hps]; and Porta, Shaul, and Yekutieli [@yekutieli:hct]. For instance, the main results of Section \[sec150626a\], summarized next, extend Greenlees-May Duality and MGM Equivalence (named for Matlis, Greenlees, and May) to this setting. See Theorems \[thm151002a\], \[thm151003a\], and \[thm151003b\] in the body of the paper.
\[thm151129a\] Let $X,Y\in{{\mathcal{D}}}(R)$ be given.
1. \[thm151129a1\] there are natural isomorphisms in ${{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})$: $$\begin{aligned}
{\mathbf{R}\!\operatorname{Hom}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)},{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(Y)})}
&\simeq
{\mathbf{R}\!\operatorname{Hom}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(X)},{\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(Y)})}\\
&\simeq{\mathbf{R}\!\operatorname{Hom}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(X)},{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(Y)})}.\end{aligned}$$
2. \[thm151129a2\] The functor ${\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}}\colon{{{\mathcal{D}}}(R)_{\text{${{\mathfrak{a}}}$-tor}}}\to{{{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})_{\text{${{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}}$-tor}}}$ is a quasi-equivalence with quasi-inverse given by the forgetful functor $Q\colon {{{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})_{\text{${{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}}$-tor}}}\to{{{\mathcal{D}}}(R)_{\text{${{\mathfrak{a}}}$-tor}}}$.
3. \[thm151129a3\] The functor ${\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}}\colon{{{\mathcal{D}}}(R)_{\text{${{\mathfrak{a}}}$-comp}}}\to{{{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})_{\text{${{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}}$-comp}}}$ is a quasi-equivalence with quasi-inverse given by the forgetful functor $Q\colon {{{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})_{\text{${{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}}$-comp}}}\to{{{\mathcal{D}}}(R)_{\text{${{\mathfrak{a}}}$-comp}}}$.
Here ${{{\mathcal{D}}}(R)_{\text{${{\mathfrak{a}}}$-tor}}}$ and ${{{\mathcal{D}}}(R)_{\text{${{\mathfrak{a}}}$-comp}}}$ are the full subcategories of ${{\mathcal{D}}}(R)$ consisting of the complexes $X$ and $Y$, respectively, such that the natural morphisms ${\mathbf{R}\Gamma_{{\mathfrak{a}}}(X)}\to X$ and $Y\to{\mathbf{L}\Lambda^{{\mathfrak{a}}}(Y)}$ are isomorphisms.
Section \[sec151003a\] investigates the flat and injective dimensions of the complexes ${\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(X)}$ and ${\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)}$ over ${\widehat{R}^{{\mathfrak{a}}}}$. In most cases, we bound these above by flat and injective dimensions of $X$ over $R$.
In Section \[sec151002a\], we use these constructions to explain the connection between the “cohomologically ${{\mathfrak{a}}}$-adically cofinite” complexes of Porta, Shaul, and Yekutieli [@yekutieli:ccc] and our “${{\mathfrak{a}}}$-adically finite” complexes from [@sather:scc]. The first of these notions is only defined when $R$ is ${{\mathfrak{a}}}$-adically complete; in this setting, we show that our notion is equivalent; see Proposition \[prop150626a\]. In general, Theorem \[thm150626a\] shows that the category of ${{\mathfrak{a}}}$-adically finite complexes over $R$ is quasi-equivalent to the category of homologically finite complexes over ${\widehat{R}^{{\mathfrak{a}}}}$, hence to the category of cohomologically ${{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}}$-adically cofinite complexes over ${\widehat{R}^{{\mathfrak{a}}}}$.
The concluding Section \[sec151104b\] exhibits some isomorphisms for use in [@sather:afc; @sather:asc]. For instance, the following result is Theorem \[thm151011a\] from the body of the paper.
\[thm151129c\] Let $R\to S$ be a homomorphism of commutative noetherian rings, and let $X\in{{{\mathcal{D}}}_{\text{b}}}(R)$ be ${{\mathfrak{a}}}$-adically finite over $R$. If ${S\otimes^{\mathbf{L}}_{R}X}\in{{{\mathcal{D}}}_{\text{b}}}(S)$, e.g., if ${\operatorname{fd}}_R(S)<\infty$, then there is an isomorphism in ${{\mathcal{D}}}({{\widehat{S}}^{{{\mathfrak{a}}}S}})$: $${{{\widehat{S}}^{{{\mathfrak{a}}}S}}\otimes^{\mathbf{L}}_{{\widehat{R}^{{\mathfrak{a}}}}}{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)}}\simeq\mathbf{L}\widehat\Lambda^{{{\mathfrak{a}}}S}({S\otimes^{\mathbf{L}}_{R}X}).$$
When $X$ is homologically finite, this is a straightfoward consequence of the isomorphisms ${\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)}\simeq{{\widehat{R}^{{\mathfrak{a}}}}\otimes^{\mathbf{L}}_{R}X}$ and $\mathbf{L}\widehat\Lambda^{{{\mathfrak{a}}}S}({S\otimes^{\mathbf{L}}_{R}X})\simeq{{{\widehat{S}}^{{{\mathfrak{a}}}S}}\otimes^{\mathbf{L}}_{S}({S\otimes^{\mathbf{L}}_{R}X})}$. In general, though, this is more subtle. And while it may seem esoteric, it is key for understanding some base-change properties in [@sather:asc].
Background {#sec140109b}
==========
Derived Categories {#derived-categories .unnumbered}
------------------
In addition to the categories mentioned in Section \[sec130805a\], we also consider the following full triangulated subcategories of ${{\mathcal{D}}}(R)$:
${{\mathcal{D}}}_+(R)$: objects are the complexes $X$ with ${\operatorname{H}}_i(X)=0$ for $i\ll 0$.
${{\mathcal{D}}}_-(R)$: objects are the complexes $X$ with ${\operatorname{H}}_i(X)=0$ for $i\gg 0$.
${{{\mathcal{D}}}^{\text{f}}}(R)$: objects are the complexes $X$ with ${\operatorname{H}}_i(X)$ finitely generated for all $i$.
Doubly ornamented subcategories are defined as intersections, e.g., ${{{\mathcal{D}}}^{\text{f}}}_+(R):={{{\mathcal{D}}}^{\text{f}}}(R)\bigcap{{\mathcal{D}}}_+(R)$.
Resolutions {#resolutions .unnumbered}
-----------
An $R$-complex $F$ is *semi-flat*[^2] if the functor ${F\otimes_{R}-}$ respects quasiisomorphisms and each module $F_i$ is flat over $R$, that is, if ${F\otimes_{R}-}$ respects injective quasiisomorphisms (see [@avramov:hdouc 1.2.F]). A *semi-flat resolution* of an $R$-complex $X$ is a quasiisomorphism $F{\xrightarrow}\simeq X$ such that $F$ is semi-flat. The *flat dimension* of $X$ $${\operatorname{fd}}_R(X):=\inf\{\sup\{i\in{\mathbb{Z}}\mid F_i\neq 0\}\mid\text{$F{\xrightarrow}\simeq X$ is a semi-flat resolution}\}$$ is the length of the shortest bounded semi-flat resolution of $X$, if one exists. The projective and injective versions (semi-projective, etc.) are defined similarly.
For the following items, consult [@avramov:hdouc Section 1] or [@avramov:dgha Chapters 3 and 5]. Bounded below complexes of projective $R$-modules are semi-projective, bounded below complexes of flat $R$-modules are semi-flat, and bounded above complexes of injective $R$-modules are semi-injective. Semi-projective $R$-complexes are semi-flat, and every $R$-complex admits a semi-projective resolution (hence, a semi-flat one) and a semi-injective resolution.
Support and Co-support {#support-and-co-support .unnumbered}
----------------------
The following notions are due to Foxby [@foxby:bcfm] and Benson, Iyengar, and Krause [@benson:csc].
\[defn130503a\] Let $X\in{{\mathcal{D}}}(R)$. The *small support* and *small co-support* of $X$ are $$\begin{aligned}
\operatorname{supp}_R(X)
&=\{\mathfrak{p} \in \operatorname{Spec}(R)\mid {\kappa({{\mathfrak{p}}})\otimes^{\mathbf{L}}_{R}X}\not\simeq 0 \} \\
{\operatorname{co-supp}}_{R}(X)
&=\{\mathfrak{p} \in \operatorname{Spec}(R)\mid {\mathbf{R}\!\operatorname{Hom}_{R}(\kappa({{\mathfrak{p}}}),X)}\not\simeq 0 \} \end{aligned}$$ where $\kappa({{\mathfrak{p}}}):=R_{{\mathfrak{p}}}/{{\mathfrak{p}}}R_{{\mathfrak{p}}}$.
Much of the following is from [@foxby:bcfm] when $X$ and $Y$ are appropriately bounded and from [@benson:lcstc; @benson:csc] in general. We refer to [@sather:scc] as a matter of convenience.
\[cor130528aw\] Let $X,Y\in{{\mathcal{D}}}(R)$. Then we have ${\operatorname{supp}}_R(X)=\emptyset$ if and only if $X\simeq 0$ if and only if ${\operatorname{co-supp}}_R(X)=\emptyset$, because of [@sather:scc Fact 3.4 and Proposition 4.7(a)]. Also, by [@sather:scc Propositions 3.12 and 4.10] we have $$\begin{aligned}
{\operatorname{supp}}_{R}({X\otimes^{\mathbf{L}}_{R}Y})
&= {\operatorname{supp}}_R(X)\bigcap{\operatorname{supp}}_R(Y)\\
{\operatorname{co-supp}}_{R}({\mathbf{R}\!\operatorname{Hom}_{R}(X,Y)})
&= {\operatorname{supp}}_R(X)\bigcap{\operatorname{co-supp}}_R(Y).\end{aligned}$$
Derived Local (Co)homology {#derived-local-cohomology .unnumbered}
--------------------------
The next notions go back to Grothendieck [@hartshorne:lc], and Matlis [@matlis:kcd; @matlis:hps], respectively; see also [@lipman:lhcs; @lipman:llcd]. Let $\Lambda^{{{\mathfrak{a}}}}$ denote the ${{\mathfrak{a}}}$-adic completion functor, and $\Gamma_{{{\mathfrak{a}}}}$ is the ${{\mathfrak{a}}}$-torsion functor, i.e., for an $R$-module $M$ we have $$\Lambda^{{{\mathfrak{a}}}}(M)={\widehat{M}^{{\mathfrak{a}}}}
\qquad
\qquad
\qquad
\Gamma_{{{\mathfrak{a}}}}(M)=\{ x \in M \mid {{\mathfrak{a}}}^{n}x=0 \text{ for } n \gg 0\}.$$ A module $M$ is *${{\mathfrak{a}}}$-torsion* if $\Gamma_{{{\mathfrak{a}}}}(M)=M$.
The associated left and right derived functors (i.e., *derived local homology and cohomology* functors) are ${\mathbf{L}\Lambda^{{\mathfrak{a}}}(-)}$ and ${\mathbf{R}\Gamma_{{\mathfrak{a}}}(-)}$. Specifically, given an $R$-complex $X\in{{\mathcal{D}}}(R)$ and a semi-flat resolution $F{\xrightarrow}\simeq X$ and a semi-injective resolution $X{\xrightarrow}\simeq I$, then we have ${\mathbf{L}\Lambda^{{\mathfrak{a}}}(X)}\simeq\Lambda^{{{\mathfrak{a}}}}(F)$ and ${\mathbf{R}\Gamma_{{\mathfrak{a}}}(X)}\simeq\Gamma_{{{\mathfrak{a}}}}(I)$. Note that these definitions yield natural transformations ${\mathbf{R}\Gamma_{{\mathfrak{a}}}}{\xrightarrow}{{\varepsilon_{{\mathfrak{a}}}}}{\operatorname{id}}{\xrightarrow}{{\vartheta^{{\mathfrak{a}}}}} {\mathbf{L}\Lambda^{{\mathfrak{a}}}}$, induced by the natural morphisms $\Gamma_{{{\mathfrak{a}}}}(I){\xrightarrow}{\iota_{{{\mathfrak{a}}}}^I} I$ and $F{\xrightarrow}{\nu^{{{\mathfrak{a}}}}_F} \Lambda^{{{\mathfrak{a}}}}(F)$. Let ${{{\mathcal{D}}}(R)_{\text{${{\mathfrak{a}}}$-tor}}}$ denote the full subcategory of ${{\mathcal{D}}}(R)$ of all complexes $X$ such that the morphism ${\mathbf{R}\Gamma_{{\mathfrak{a}}}(X)}{\xrightarrow}{{\varepsilon_{{\mathfrak{a}}}^{X}}}X$ is an isomorphism, and let ${{{\mathcal{D}}}(R)_{\text{${{\mathfrak{a}}}$-comp}}}$ denote the full subcategory of ${{\mathcal{D}}}(R)$ of all complexes $Y$ such that the morphism $Y{\xrightarrow}{{\vartheta^{{\mathfrak{a}}}_{Y}}}{\mathbf{L}\Lambda^{{\mathfrak{a}}}(Y)}$ is an isomorphism.
The definitions of ${\mathbf{R}\Gamma_{{\mathfrak{a}}}(X)}$ and ${\mathbf{L}\Lambda^{{\mathfrak{a}}}(X)}$ yield complexes over the completion ${\widehat{R}^{{\mathfrak{a}}}}$, and we denote by ${\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}}$ and ${\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}}$ the associated functors ${{\mathcal{D}}}(R)\to{{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})$. If $Q\colon {{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})\to{{\mathcal{D}}}(R)$ is the forgetful functor, then it follows readily that $Q\circ{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}}\simeq{\mathbf{L}\Lambda^{{\mathfrak{a}}}}$ and $Q\circ{\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}}\simeq{\mathbf{R}\Gamma_{{\mathfrak{a}}}}$.
\[fact130619b\] If $X\in{{{\mathcal{D}}}^{\text{f}}}_+(R)$, then there is a natural isomorphism ${\mathbf{L}\Lambda^{{\mathfrak{a}}}(X)}\simeq {{\widehat{R}^{{\mathfrak{a}}}}\otimes^{\mathbf{L}}_{R}X}$ in ${{\mathcal{D}}}(R)$ by [@frankild:volh Proposition 2.7]. Moreover, the proof of this result shows that there is a natural isomorphism ${\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)}\simeq {{\widehat{R}^{{\mathfrak{a}}}}\otimes^{\mathbf{L}}_{R}X}$ in ${{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})$.[^3] By [@lipman:lhcs Theorem (0.3) and Corollary (3.2.5.i)], there are natural isomorphisms of functors $$\begin{aligned}
{\mathbf{R}\Gamma_{{\mathfrak{a}}}(-)}\simeq{{\mathbf{R}\Gamma_{{\mathfrak{a}}}(R)}\otimes^{\mathbf{L}}_{R}-}&&
{\mathbf{L}\Lambda^{{\mathfrak{a}}}(-)}\simeq{\mathbf{R}\!\operatorname{Hom}_{R}({\mathbf{R}\Gamma_{{\mathfrak{a}}}(R)},-)}.\end{aligned}$$ More generally, from [@shaul:hccac Theorems 3.2 and 3.6] there are natural isomorphisms of functors ${{\mathcal{D}}}(R)\to{{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})$ $$\begin{gathered}
{\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(-)}\simeq{{\widehat{R}^{{\mathfrak{a}}}}\otimes^{\mathbf{L}}_{R}{\mathbf{R}\Gamma_{{\mathfrak{a}}}(-)}}\simeq{{\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(R)}\otimes^{\mathbf{L}}_{R}-}
\\
{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(-)}\simeq{\mathbf{R}\!\operatorname{Hom}_{R}({\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(R)},-)}
\simeq{\mathbf{R}\!\operatorname{Hom}_{R}({\widehat{R}^{{\mathfrak{a}}}},{\mathbf{L}\Lambda^{{\mathfrak{a}}}(-)})}.\end{gathered}$$
Here are Greenlees-May duality and MGM equivalence.
\[fact150626a\] Given $X,Y\in{{\mathcal{D}}}(R)$, we have natural isomorphisms in ${{\mathcal{D}}}(R)$ $$\begin{aligned}
{\mathbf{R}\!\operatorname{Hom}_{R}({\mathbf{R}\Gamma_{{\mathfrak{a}}}(X)},{\mathbf{R}\Gamma_{{\mathfrak{a}}}(Y)})}
&{\xrightarrow}\simeq{\mathbf{R}\!\operatorname{Hom}_{R}({\mathbf{R}\Gamma_{{\mathfrak{a}}}(X)},Y)} \\
&{\xrightarrow}\simeq{\mathbf{R}\!\operatorname{Hom}_{R}({\mathbf{R}\Gamma_{{\mathfrak{a}}}(X)},{\mathbf{L}\Lambda^{{\mathfrak{a}}}(Y)})} \\
&{\xleftarrow}\simeq{\mathbf{R}\!\operatorname{Hom}_{R}(X,{\mathbf{L}\Lambda^{{\mathfrak{a}}}(Y)})} \\
&{\xleftarrow}\simeq{\mathbf{R}\!\operatorname{Hom}_{R}({\mathbf{L}\Lambda^{{\mathfrak{a}}}(X)},{\mathbf{L}\Lambda^{{\mathfrak{a}}}(Y)})}\end{aligned}$$ induced by ${\varepsilon_{{\mathfrak{a}}}}$ and ${\vartheta^{{\mathfrak{a}}}}$; see [@lipman:lhcs Theorem (0.3)$^*$].[^4], From [@lipman:lhcs Corollary to Theorem (0.3)$^*$] and [@yekutieli:hct Theorem 1.2] the next natural morphisms are isomorphisms: $$\begin{aligned}
{\mathbf{R}\Gamma_{{\mathfrak{a}}}}\circ{\operatorname{id}}{\xrightarrow}[\simeq]{{\mathbf{R}\Gamma_{{\mathfrak{a}}}}\circ{\vartheta^{{\mathfrak{a}}}}}{\mathbf{R}\Gamma_{{\mathfrak{a}}}}\circ{\mathbf{L}\Lambda^{{\mathfrak{a}}}}
&&{\mathbf{L}\Lambda^{{\mathfrak{a}}}}\circ{\mathbf{R}\Gamma_{{\mathfrak{a}}}}{\xrightarrow}[\simeq]{{\mathbf{L}\Lambda^{{\mathfrak{a}}}}\circ{\varepsilon_{{\mathfrak{a}}}}}{\mathbf{L}\Lambda^{{\mathfrak{a}}}}\circ{\operatorname{id}}\\
{\mathbf{R}\Gamma_{{\mathfrak{a}}}}\circ{\mathbf{R}\Gamma_{{\mathfrak{a}}}}{\xrightarrow}[\simeq]{{\varepsilon_{{\mathfrak{a}}}}\circ{\mathbf{R}\Gamma_{{\mathfrak{a}}}}}{\operatorname{id}}\circ{\mathbf{R}\Gamma_{{\mathfrak{a}}}}
&&{\operatorname{id}}\circ{\mathbf{L}\Lambda^{{\mathfrak{a}}}}{\xrightarrow}[\simeq]{{\vartheta^{{\mathfrak{a}}}}\circ{\mathbf{L}\Lambda^{{\mathfrak{a}}}}}{\mathbf{L}\Lambda^{{\mathfrak{a}}}}\circ{\mathbf{L}\Lambda^{{\mathfrak{a}}}}.\end{aligned}$$ The second row of isomorphisms here shows that the essential image of ${\mathbf{R}\Gamma_{{\mathfrak{a}}}}$ in ${{\mathcal{D}}}(R)$ is ${{{\mathcal{D}}}(R)_{\text{${{\mathfrak{a}}}$-tor}}}$, and the essential image of ${\mathbf{L}\Lambda^{{\mathfrak{a}}}}$ in ${{\mathcal{D}}}(R)$ is ${{{\mathcal{D}}}(R)_{\text{${{\mathfrak{a}}}$-comp}}}$.
\[cor130528a\] Let $X\in{{\mathcal{D}}}(R)$. Then we know that ${\operatorname{supp}}_R(X)\subseteq{\operatorname{V}}({{\mathfrak{a}}})$ if and only if $X\in{{{\mathcal{D}}}(R)_{\text{${{\mathfrak{a}}}$-tor}}}$ if and only if each homology module ${\operatorname{H}}_i(X)$ is ${{\mathfrak{a}}}$-torsion, by [@sather:scc Proposition 5.4] and [@yekutieli:hct Corollary 4.32].[^5]
\[lem151117a\] Let $Y\in{{\mathcal{D}}}(R)$, and consider the following exact triangles in ${{\mathcal{D}}}(R)$. $$\begin{aligned}
{\mathbf{R}\Gamma_{{\mathfrak{a}}}(Y)}{\xrightarrow}{{\varepsilon_{{\mathfrak{a}}}^{Y}}}Y\to B\to &&
Y{\xrightarrow}{{\vartheta^{{\mathfrak{a}}}_{Y}}}{\mathbf{L}\Lambda^{{\mathfrak{a}}}(Y)}\to C\to \end{aligned}$$ By [@benson:csc Corollary 4.9] one has $$\begin{gathered}
{\operatorname{supp}}_R(B)\bigcap{\operatorname{V}}({{\mathfrak{a}}})=\emptyset={\operatorname{co-supp}}_R(B)\bigcap{\operatorname{V}}({{\mathfrak{a}}})\\
{\operatorname{supp}}_R(C)\bigcap{\operatorname{V}}({{\mathfrak{a}}})=\emptyset={\operatorname{co-supp}}_R(C)\bigcap{\operatorname{V}}({{\mathfrak{a}}}).\end{gathered}$$
\[lem150907a\] The following natural transformations are isomorphisms $$\begin{aligned}
{\mathbf{R}\Gamma_{{\mathfrak{a}}}}\circ{\mathbf{R}\Gamma_{{\mathfrak{a}}}}{\xrightarrow}[\simeq]{{\mathbf{R}\Gamma_{{\mathfrak{a}}}}\circ{\varepsilon_{{\mathfrak{a}}}}}{\mathbf{R}\Gamma_{{\mathfrak{a}}}}\circ{\operatorname{id}}&&{\mathbf{L}\Lambda^{{\mathfrak{a}}}}\circ{\operatorname{id}}{\xrightarrow}[\simeq]{{\mathbf{L}\Lambda^{{\mathfrak{a}}}}\circ{\vartheta^{{\mathfrak{a}}}}}{\mathbf{L}\Lambda^{{\mathfrak{a}}}}\circ{\mathbf{L}\Lambda^{{\mathfrak{a}}}}\end{aligned}$$ by [@benson:lcstc Lemma 3.4(a)], [@benson:csc (4.2)], and [@lipman:llcd Proposition 3.5.3]. Note the slight difference between these and the last two isomorphisms in Fact \[fact150626a\]. Note also that one can obtain these isomorphisms as the special case $X={\mathbf{R}\Gamma_{{\mathfrak{a}}}(R)}$ of the next result.
\[prop151104a\] Let $X\in{{\mathcal{D}}}(R)$ be such that ${\operatorname{supp}}_R(X)\subseteq{\operatorname{V}}({{\mathfrak{a}}})$ or ${\operatorname{co-supp}}_R(X)\subseteq{\operatorname{V}}({{\mathfrak{a}}})$; e.g., $X\simeq K$. Then the following natural transformations are isomorphisms. $$\begin{gathered}
{X\otimes^{\mathbf{L}}_{R}{\mathbf{R}\Gamma_{{\mathfrak{a}}}(-)}}{\xrightarrow}[\simeq]{{X\otimes^{\mathbf{L}}_{R}{\varepsilon_{{\mathfrak{a}}}}}}{X\otimes^{\mathbf{L}}_{R}-}{\xrightarrow}[\simeq]{{X\otimes^{\mathbf{L}}_{R}{\vartheta^{{\mathfrak{a}}}}}}{X\otimes^{\mathbf{L}}_{R}{\mathbf{L}\Lambda^{{\mathfrak{a}}}(-)}}
\\
{\mathbf{R}\!\operatorname{Hom}_{R}(X,{\mathbf{R}\Gamma_{{\mathfrak{a}}}(-)})}\!{\xrightarrow}[\simeq]{\!{\mathbf{R}\!\operatorname{Hom}_{}(X,{\varepsilon_{{\mathfrak{a}}}})}\!}{\mathbf{R}\!\operatorname{Hom}_{R}(X,-)}\!{\xrightarrow}[\simeq]{\!{\mathbf{R}\!\operatorname{Hom}_{}(X,{\vartheta^{{\mathfrak{a}}}})}\!}{\mathbf{R}\!\operatorname{Hom}_{R}(X,{\mathbf{L}\Lambda^{{\mathfrak{a}}}(-)})}
\\
{\mathbf{R}\!\operatorname{Hom}_{R}({\mathbf{L}\Lambda^{{\mathfrak{a}}}(-)},X)}\!{\xrightarrow}[\simeq]{\!{\mathbf{R}\!\operatorname{Hom}_{}({\vartheta^{{\mathfrak{a}}}},X)}\!}
{\mathbf{R}\!\operatorname{Hom}_{R}(-,X)}
\!{\xrightarrow}[\simeq]{\!{\mathbf{R}\!\operatorname{Hom}_{}({\varepsilon_{{\mathfrak{a}}}},X)}\!}{\mathbf{R}\!\operatorname{Hom}_{R}({\mathbf{R}\Gamma_{{\mathfrak{a}}}(-)},X)}\end{gathered}$$
Let $Y\in{{\mathcal{D}}}(R)$, and consider the exact triangle $${\mathbf{R}\Gamma_{{\mathfrak{a}}}(Y)}{\xrightarrow}{{\varepsilon_{{\mathfrak{a}}}^{Y}}}Y\to B\to$$ in ${{\mathcal{D}}}(R)$, and the following induced triangle. $$\label{eq151117b}
{X\otimes^{\mathbf{L}}_{R}{\mathbf{R}\Gamma_{{\mathfrak{a}}}(Y)}}{\xrightarrow}{{X\otimes^{\mathbf{L}}_{R}{\varepsilon_{{\mathfrak{a}}}^{Y}}}}{X\otimes^{\mathbf{L}}_{R}Y}\to {X\otimes^{\mathbf{L}}_{R}B}\to$$ Facts \[cor130528a\] and \[lem151117a\] yield the next sequence $$\begin{aligned}
{\operatorname{supp}}_R({X\otimes^{\mathbf{L}}_{R}B})
&={\operatorname{supp}}_R(X)\bigcap{\operatorname{supp}}_R(B)
\subseteq{\operatorname{V}}({{\mathfrak{a}}})\bigcap{\operatorname{supp}}_R(B)
=\emptyset.\end{aligned}$$ We conclude that ${X\otimes^{\mathbf{L}}_{R}B}\simeq 0$, so the exact triangle implies that ${X\otimes^{\mathbf{L}}_{R}{\varepsilon_{{\mathfrak{a}}}^{Y}}}$ is an isomorphism in ${{\mathcal{D}}}(R)$. The other isomorphisms from the statement of this result follow similarly.
Adic Finiteness {#adic-finiteness .unnumbered}
---------------
The next two items take their cues from work of Hartshorne [@hartshorne:adc], Kawasaki [@kawasaki:ccma; @kawasaki:ccc], and Melkersson [@melkersson:mci].
\[thm130612a\] For $X\in{{\mathcal{D}}}_{\text b}(R)$, the next conditions are equivalent.
1. \[cor130612a1\] One has ${K^R(\underline{y})\otimes^{\mathbf{L}}_{R}X}\in{{{\mathcal{D}}}_{\text{b}}^{\text{f}}}(R)$ for some (equivalently for every) generating sequence $\underline{y}$ of ${{\mathfrak{a}}}$.
2. \[cor130612a2\] One has ${X\otimes^{\mathbf{L}}_{R}R/\mathfrak{a}}\in{{\mathcal{D}}}^{\text{f}}(R)$.
3. \[cor130612a3\] One has ${\mathbf{R}\!\operatorname{Hom}_{R}(R/\mathfrak{a},X)}\in{{\mathcal{D}}}^{\text{f}}(R)$.
4. \[cor130612a4\] One has ${\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)}\in{{{\mathcal{D}}}_{\text{b}}^{\text{f}}}({\widehat{R}^{{\mathfrak{a}}}})$.
\[def120925d\] An $R$-complex $X\in{{{\mathcal{D}}}_{\text{b}}}(R)$ is *$\mathfrak{a}$-adically finite* if it satisfies the equivalent conditions of Fact \[thm130612a\] and $\operatorname{supp}_R(X) \subseteq \operatorname{V}(\mathfrak{a})$.
\[ex160206a\] Let $X\in{{{\mathcal{D}}}_{\text{b}}}(R)$ be given.
(a) \[ex160206a1\] If $X\in{{{\mathcal{D}}}_{\text{b}}^{\text{f}}}(R)$, then we have ${\operatorname{supp}}_R(X)={\operatorname{V}}({{\mathfrak{b}}})$ for some ideal ${{\mathfrak{b}}}$, and it follows that $X$ is ${{\mathfrak{a}}}$-adically finite whenever ${{\mathfrak{a}}}\subseteq{{\mathfrak{b}}}$. (The case ${{\mathfrak{a}}}=0$ is from [@sather:scc Proposition 7.8(a)], and the general case follows readily.)
(b) \[ex160206a2\] $K$ and ${\mathbf{R}\Gamma_{{\mathfrak{a}}}(R)}$ are ${{\mathfrak{a}}}$-adically finite, by [@sather:scc Fact 3.4 and Theorem 7.10].
(c) \[ex160206a3\] The homology modules of $X$ are artinian if and only if there is an ideal ${{\mathfrak{a}}}$ of finite colength (i.e., such that $R/{{\mathfrak{a}}}$ is artinian) such that $X$ is ${{\mathfrak{a}}}$-adically finite, by [@sather:afcc Proposition 5.11].
Computing Derived Functors {#sec150922a}
==========================
Lipman [@lipman:llcd Lemma 3.5.1] shows that, to compute ${\mathbf{R}\Gamma_{{\mathfrak{a}}}(X)}$, one need not use a semi-injective resolution of $X$; it suffices to use an injective resolution of $X$, i.e., a quasiisomorphism $X{\xrightarrow}\simeq I$ where $I$ is a complex of injective $R$-modules. This fact, along with the fact that $\Gamma_{{{\mathfrak{a}}}}$ transforms complexes of injective modules into complexes of injective modules, is the essence of the proof of the first isomorphism in Fact \[lem150907a\]. Our next example shows that [@lipman:llcd Lemma 3.5.1] is crucial here, as it shows that $\Gamma_{{{\mathfrak{a}}}}$ need not respect the class of semi-injective complexes.
\[ex150907a\] We consider the following special case of a construction of Chen and Iyengar [@chen:sirccr Proposition 2.7]. Let $k$ be a field, set $R=k[\![X,Y]\!]/(X^2)$, and let $x$ and $y$ denote the residues in $R$ of $X$ and $Y$, respectively. Set ${{\mathfrak{n}}}=(x,y)R$ and ${{\mathfrak{p}}}=xR$, and consider the injective hull $E=E_R(R/{{\mathfrak{n}}})$. We consider the complexes $$\begin{aligned}
F:=&(\cdots{\xrightarrow}xR{\xrightarrow}xR\to 0) & G&:=\bigoplus_{n\in{\mathbb{Z}}}{\mathsf{\Sigma}}^nF \\
I:=&{\operatorname{Hom}_{R}(F,E)}\cong(0\to E{\xrightarrow}xE{\xrightarrow}x\cdots)&J&:={\operatorname{Hom}_{R}(G,E)}\cong\prod_{n\in{\mathbb{Z}}}{\mathsf{\Sigma}}^nI\end{aligned}$$ The complex $F$ gives a semi-projective (hence, semi-flat) resolution of $R/xR$, and $I$ yields a semi-injective resolution of $M:={\operatorname{Hom}_{R}(R/xR,E)}$. Also, $G$ describes a semi-projective (hence semi-flat) resolution of $\bigoplus_{{{\mathfrak{n}}}\in{\mathbb{Z}}}{\mathsf{\Sigma}}^nR/xR$, and $J$ provides a semi-injective resolution of $\prod_{n\in{\mathbb{Z}}}{\mathsf{\Sigma}}^nM$. Furthermore, ${{\mathfrak{p}}}$ is an associated prime of each module $J_i\cong\prod_{n{\geqslant}i}E$, but $J_{{\mathfrak{p}}}$ is not semi-injective over $R_{{\mathfrak{p}}}$: Chen and Iyengar prove this last claim by noting that $J_{{\mathfrak{p}}}$ is acyclic but not contractible over $R_{{\mathfrak{p}}}$; it follows that $J_{{\mathfrak{p}}}$ is acyclic but not contractible over $R$, so not semi-injective over $R$.
Claim: the complex $\Gamma_{{{\mathfrak{m}}}}(J)$ is not semi-injective over $R$. To see this, note that each module $J_i$ is a direct sum of copies of $E$ and copies of $E_R(R/{{\mathfrak{p}}})$. It follows that we have the following natural short exact sequence of complexes: $$0\to \Gamma_{{{\mathfrak{m}}}}(J)\to J\to J_{{\mathfrak{p}}}\to 0.$$ Since $J$ is semi-injective over $R$, the fact that these are complexes of injective $R$-modules implies that $\Gamma_{{{\mathfrak{m}}}}(J)$ is semi-injective over $R$ if and only if $J_{{\mathfrak{p}}}$ is so; hence, by the previous paragraph, the claim is established.
The following lemma is used in the subsequent example.
\[lem151001a\] Assume that $(R,{{\mathfrak{m}}},k)$ is local and that ${\operatorname{Spec}}(R)=\{{{\mathfrak{m}}},{{\mathfrak{p}}}\}$ with ${{\mathfrak{p}}}\subsetneq{{\mathfrak{m}}}$. Let $L$ be a free $R$-module, and set $L':={\widehat{L}^{{\mathfrak{m}}}}/L$. Then ${\widehat{L}^{{\mathfrak{m}}}}$ is flat over $R$, and $L'$ is a flat $R_{{{\mathfrak{p}}}}$-module (hence, $L'$ is also flat over $R$). If $L$ is not finitely generated over $R$, then $L'\neq 0$.
For the sake of brevity, set ${\widehat{L}}:={\widehat{L}^{{\mathfrak{m}}}}$.
Krull’s Intersection Theorem implies that $L$ is ${{\mathfrak{m}}}$-adically separated, so the natural map $L\to{\widehat{L}}$ is injective. Thus, the definition of $L'$ makes sense. The module ${\widehat{L}}$ is flat over $R$ by [@enochs:rha Theorem 5.3.28]. Also, the proof of [@enochs:rha Proposition 6.7.6] shows that the inclusion $L\to{\widehat{L}}$ is pure. It follows that $L'$ is also flat over $R$.
Claim: $L'$ is naturally an $R_{{{\mathfrak{p}}}}$-module. To see this, first note that [@yekutieli:ccc Corollary 1.9(1)] shows that the natural map $L/{{\mathfrak{m}}}L\to{\widehat{L}}/{{\mathfrak{m}}}{\widehat{L}}$ is an isomorphism. Right-exactness of $-\otimes_Rk$ applied to the natural sequence $$\label{eq151001a}
0\to L{\xrightarrow}{\nu^{{{\mathfrak{m}}}}_L}{\widehat{L}}\to L'\to 0$$ implies that ${L'\otimes_{R}k}=0$. The fact that $L'$ is flat implies that ${\operatorname{Tor}^{R}_{i}(L',k)}=0$ for all $i\neq 0$, and we conclude that ${{\mathfrak{m}}}\notin{\operatorname{supp}}_R(L')$. It follows from [@foxby:bcfm Remark 2.9] that the minimal injective resolution of $L'$ over $R$ contains no summand of the form $E_R(k)$. Thus, the minimal injective resolution of $L'$ over $R$ has the form $$0\to E_R(R/{{\mathfrak{p}}})^{(\mu^0)}{\xrightarrow}{\partial_0}E_R(R/{{\mathfrak{p}}})^{(\mu^1)}\to\cdots.$$ Since each module $E_R(R/{{\mathfrak{p}}})^{(\mu^i)}$ is naturally an $R_{{{\mathfrak{p}}}}$-module, the $R$-module homomorphism $\partial_0$ is $R_{{{\mathfrak{p}}}}$-linear. Since $L'$ is isomorphic to ${\operatorname{Ker}}(\partial_0)$, the claim follows.
We include here a second proof of the claim, as it sheds a different light on the module $L'$, which is somewhat mysterious to us. For this second proof, it suffices to show that $L'\cong{\operatorname{Ext}_{R}^{1}(R_{{{\mathfrak{p}}}},L)}$, since this Ext-module inherits an $R_{{{\mathfrak{p}}}}$-structure from the first slot. (Note that this proof is intimately related to results of [@yekutieli:hct; @yekutieli:sccmc].) Since $L$ is flat over $R$, there is an isomorphism ${\mathbf{L}\Lambda^{{\mathfrak{m}}}(L)}\simeq
{\widehat{L}}$ in ${{\mathcal{D}}}(R)$. Thus, the exact sequence provides an exact triangle $$\label{eq151001b}
L{\xrightarrow}{{\vartheta^{{\mathfrak{m}}}_{L}}}{\mathbf{L}\Lambda^{{\mathfrak{m}}}(L)}\to L'\to$$ in ${{\mathcal{D}}}(R)$. Using the structure of ${\operatorname{Spec}}(R)$ again, as in the proof of the claim in Example \[ex150907a\], we have the next exact triangle $${\mathbf{R}\Gamma_{{\mathfrak{m}}}(R)}{\xrightarrow}{{\varepsilon_{{\mathfrak{m}}}^{R}}}R\to R_{{{\mathfrak{p}}}}\to$$ in ${{\mathcal{D}}}(R)$. In the language of [@yekutieli:hct Section 7], this says that we have $\mathbf{R}\Gamma_{0/{{\mathfrak{m}}}}(R)\simeq R_{{{\mathfrak{p}}}}$. The proof of [@yekutieli:hct Lemma 7.2] exhibits an exact triangle of the following form. $${\mathbf{R}\!\operatorname{Hom}_{R}(\mathbf{R}\Gamma_{0/{{\mathfrak{m}}}}(R),L)}\to L{\xrightarrow}{{\vartheta^{{\mathfrak{m}}}_{L}}}{\mathbf{L}\Lambda^{{\mathfrak{m}}}(L)}\to$$ Rotating this triangle, we obtain the next one. $$L{\xrightarrow}{{\vartheta^{{\mathfrak{m}}}_{L}}}{\mathbf{L}\Lambda^{{\mathfrak{m}}}(L)}\to {\mathsf{\Sigma}}{\mathbf{R}\!\operatorname{Hom}_{R}(\mathbf{R}\Gamma_{0/{{\mathfrak{m}}}}(R),L)}\to$$ Combining this with the triangle in , we conclude that $$L'\simeq {\mathsf{\Sigma}}{\mathbf{R}\!\operatorname{Hom}_{R}(\mathbf{R}\Gamma_{0/{{\mathfrak{m}}}}(R),L)}\simeq{\mathsf{\Sigma}}{\mathbf{R}\!\operatorname{Hom}_{R}(R_{{{\mathfrak{p}}}},L)}.$$ Applying ${\operatorname{H}}_0$, we obtain the next isomorphism $$L'\cong{\operatorname{H}}_0(L')\cong{\operatorname{H}}_0({\mathsf{\Sigma}}{\mathbf{R}\!\operatorname{Hom}_{R}(R_{{{\mathfrak{p}}}},L)})\cong{\operatorname{Ext}_{R}^{1}(R_{{{\mathfrak{p}}}},L)}.$$ This concludes the second proof of the claim.
We conclude the proof of the lemma. Since $L'$ is a flat $R$-module, the localization $L'_{{{\mathfrak{p}}}}\cong L'$ is a flat $R_{{{\mathfrak{p}}}}$-module; the isomorphism follows from the above claim. Finally, assume that $L$ is not finitely generated. To show that $L'\neq 0$, it suffices to show that $L$ is not complete. Since $L$ is free of infinite rank, consider a sequence $e_1,e_2,\ldots$ of distinct elements of a basis of $L$. From our assumption on ${\operatorname{Spec}}(R)$, the nilradical of $R$ is ${{\mathfrak{p}}}$. Let $y\in{{\mathfrak{m}}}{\smallsetminus}{{\mathfrak{p}}}$, which is not nilpotent. It follows that the Cauchy sequence $\{\sum_{i=1}^ny^ie_i\}_{n=1}^\infty$ in $L$ does not converge in $L$, so $L$ is not complete, as desired.
Similar to the previous example, the next one shows that $\Lambda^{{\mathfrak{a}}}$ does not respect the class of semi-flat complexes.
\[ex150907az\] We continue with the set-up of Example \[ex150907a\].
Claim: the complex $\Lambda^{{\mathfrak{m}}}(G)$ is not semi-flat over $R$. Since each module $G_i$ is free over $R$, Lemma \[lem151001a\] provides a short exact sequence $$0\to G{\xrightarrow}{\nu^{{\mathfrak{m}}}_G}\Lambda^{{\mathfrak{m}}}(G)\to G'\to 0$$ where each module $G'_i$ is a non-zero flat $R_{{{\mathfrak{p}}}}$-module. Since $R_{{{\mathfrak{p}}}}$ is Gorenstein and artinian, it follows that $G'_i$ is injective over $R_{{{\mathfrak{p}}}}$, hence $$G_i\cong E_{R_{{{\mathfrak{p}}}}}(\kappa({{\mathfrak{p}}}))^{(\mu_i)}\cong E_R(R/{{\mathfrak{p}}})^{(\mu_i)}\cong R_{{{\mathfrak{p}}}}^{(\mu_i)}$$ for some set $\mu_i$.
Since $G$ is semi-flat, and the modules $\Lambda^{{\mathfrak{m}}}(G)$ and $G'_i$ are flat over $R$ for all $i$, to establish the claim, it suffices to show that $G'$ is not semi-flat over $R$. To accomplish this, we follow the lead of Chen and Iyengar [@chen:sirccr] by showing that $G'$ is exact and minimal. (Recall that a complex $Z$ is *minimal* if every homotopy equivalence $Z\to Z$ is an isomorphism.) If $G'$ were semi-flat, this would imply $G'=0$, which we know to be false.
Note that each homology module ${\operatorname{H}}_i(G)\cong R/xR$ is ${{\mathfrak{m}}}$-adically complete, since $R$ is so. Hence, from [@yekutieli:sccmc Theorem 3] we conclude that the natural morphism ${\vartheta^{{\mathfrak{m}}}_{G}}\colon G\to{\mathbf{L}\Lambda^{{\mathfrak{m}}}(G)}$ is an isomorphism in ${{\mathcal{D}}}(R)$. In other words, the chain map $\nu^{{\mathfrak{m}}}_G\colon G\to\Lambda^{{\mathfrak{m}}}(G)$ is a quasiisomorphism. It follows that $G'$ is exact.
Since $G'$ is a complex of injective $R$-modules, to show that $G'$ is minimal, it suffices to show that the inclusion $\oplus_{i\in{\mathbb{Z}}}{\operatorname{Ker}}(\partial^{G'}_i)\subseteq\oplus_{i\in{\mathbb{Z}}}G'_i$ is an injective envelope over $R$; see, e.g., [@christensen:dcmca Lemma 5.4.16]. As every $R$-module has an injective envelope, we see from [@enochs:rha Corollary 6.4.4] that it suffices to show that each inclusion ${\operatorname{Ker}}(\partial^{G'}_i)\subseteq G'_i$ is an injective envelope over $R$, i.e., over $R_{{\mathfrak{p}}}$ as $G'$ is an $R_{{{\mathfrak{p}}}}$-complex.
Because $(R_{{{\mathfrak{p}}}},{{\mathfrak{p}}}R_{{{\mathfrak{p}}}},\kappa({{\mathfrak{p}}}))$ is a local ring, the inclusion ${\operatorname{Soc}}_{R_{{{\mathfrak{p}}}}}(E_{R_{{{\mathfrak{p}}}}}(\kappa({{\mathfrak{p}}}))^{(\mu_i)})\subseteq E_{R_{{{\mathfrak{p}}}}}(\kappa({{\mathfrak{p}}}))^{(\mu_i)}$ is an injective envelope. That is, the inclusion ${\operatorname{Soc}}_{R_{{{\mathfrak{p}}}}}(G'_i)\subseteq G'_i$ is an injective envelope. On the other hand, the isomorphism $G'_i\cong R_{{{\mathfrak{p}}}}^{(\mu_i)}$ works with the conditions ${{\mathfrak{p}}}R_{{{\mathfrak{p}}}}=xR_{{{\mathfrak{p}}}}\neq 0$ and $x^2=0$ to imply that ${\operatorname{Soc}}_{R_{{{\mathfrak{p}}}}}(G'_i)=xG'_i=(0:_{G_i}x)$. Thus, we are reduced to showing that ${\operatorname{Ker}}(\partial^{G'}_i)=xG'_i$.
By construction, we have $\partial^G_j(G_j)\subseteq xG_{j-1}$ for all $j$, so $\partial^G_j(xG_j)\subseteq x^2G_{j-1}=0$. In other words, the composition $$G_j{\xrightarrow}{x}G_j{\xrightarrow}{\partial^G_j} G_{j-1}$$ is $0$. Applying the functor $\Lambda^{{{\mathfrak{m}}}}$, we see that the composition $$\Lambda^{{{\mathfrak{m}}}}(G_j){\xrightarrow}{x}\Lambda^{{{\mathfrak{m}}}}(G_j){\xrightarrow}{\Lambda^{{{\mathfrak{m}}}}(\partial^G_j)} \Lambda^{{{\mathfrak{m}}}}(G_{j-1})$$ is $0$, that is, the composition $$\Lambda^{{{\mathfrak{m}}}}(G)_j{\xrightarrow}{x}\Lambda^{{{\mathfrak{m}}}}(G)_j{\xrightarrow}{\partial^{\Lambda^{{{\mathfrak{m}}}}(G)}_j} \Lambda^{{{\mathfrak{m}}}}(G)_{j-1}$$ is $0$. Since $\partial^{G'}$ is induced by $\partial^{\Lambda^{{{\mathfrak{m}}}}(G)}_j$, it follows that the composition $$G'_j{\xrightarrow}{x}G'_j{\xrightarrow}{\partial^{G'}_j} G'_{j-1}$$ is $0$. We conclude that $0=\partial^{G'}_j(xG'_j)=x\partial^{G'}_j(G'_j)$. The first equality here implies that ${\operatorname{Ker}}(\partial^{G'}_i)\supseteq xG'_i$. Using the second equality here, we see that ${\operatorname{Ker}}(\partial^{G'}_i)=\partial^{G'}_{i+1}(G'_{i+1})\subseteq(0:_{G_i}x)=xG'_i$. We conclude that ${\operatorname{Ker}}(\partial^{G'}_i)=xG'_i$. This establishes the claim and concludes the example.
As with [@lipman:llcd Lemma 3.5.1], our next result shows that one can use flat resolutions to compute ${\mathbf{L}\Lambda^{{\mathfrak{a}}}}$.
\[prop160202a\] Let $X\in{{\mathcal{D}}}(R)$, and let $F$ be a complex of flat $R$-modules such that $F\simeq X$ in ${{\mathcal{D}}}(R)$. Then we have ${\mathbf{L}\Lambda^{{\mathfrak{a}}}(X)}\simeq\Lambda^{{\mathfrak{a}}}(F)$ in ${{\mathcal{D}}}(R)$.
Claim: given any exact complex $G$ of flat $R$-modules, one has $\Lambda^{{\mathfrak{a}}}(G)\simeq 0$. To establish the claim, let $i\in{\mathbb{Z}}$ be given; we need to show that ${\operatorname{H}}_i(\Lambda^{{\mathfrak{a}}}(G))=0$. Assume that the ideal ${{\mathfrak{a}}}$ is generated by a sequence of length $n$. Then the “telescope complex” $T$ is a projective resolution of ${\mathbf{R}\Gamma_{{\mathfrak{a}}}(R)}$ concentrated in degrees $0,\ldots,-n$; see [@yekutieli:hct]. From this, we conclude that $${\operatorname{H}}_{n+1}({\mathbf{L}\Lambda^{{\mathfrak{a}}}(M)})\cong{\operatorname{H}}_{n+1}({\mathbf{R}\!\operatorname{Hom}_{R}(RG aR,M)})=0$$ for each $R$-module $M$.
Set $M:={\operatorname{Im}}(\partial^G_{i-n-1})$, so we have a flat resolution $$\cdots{\xrightarrow}{\partial^G_{i+1}}G_i{\xrightarrow}{\partial^G_{i}}\cdots{\xrightarrow}{\partial^G_{i-n}}G_{i-n-1}\to M\to 0.$$ From this and the previous paragraph, we conclude that $$0={\operatorname{H}}_{n+1}({\mathbf{L}\Lambda^{{\mathfrak{a}}}(M)})\cong{\operatorname{H}}_i(\Lambda^{{\mathfrak{a}}}(G)).$$ Since $i$ is arbitrary, this establishes the claim.
Now we prove our proposition. Let $P{\xrightarrow}\simeq X$ be a semi-projective resolution. The isomorphism $X\simeq F$ in ${{\mathcal{D}}}(R)$ provides a quasiisomorphism $\phi\colon P{\xrightarrow}\simeq F$. The mapping cone $G:={\operatorname{Cone}}(\phi)$ is an exact complex of flat $R$-modules, so the above claim implies that $$0\simeq\Lambda^{{\mathfrak{a}}}(G)=\Lambda^{{\mathfrak{a}}}({\operatorname{Cone}}(\phi))\simeq{\operatorname{Cone}}(\Lambda^{{\mathfrak{a}}}(\phi)).$$ We conclude that $\Lambda^{{\mathfrak{a}}}(\phi)\colon \Lambda^{{\mathfrak{a}}}(P)\to\Lambda^{{\mathfrak{a}}}(F)$ is a quasiisomorphism, so $\Lambda^{{\mathfrak{a}}}(F)\simeq \Lambda^{{\mathfrak{a}}}(P)\simeq{\mathbf{L}\Lambda^{{\mathfrak{a}}}(X)}$, as desired.
Extended Greenlees-May Duality and MGM Equivalence {#sec150626a}
==================================================
In this section, we extend previous isomorphisms to cover the functors ${\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}}$ and ${\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}}$, beginning with extended versions of parts of Fact \[fact150626a\].
\[lem150805a\] The natural transformations $${\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}}\circ{\mathbf{R}\Gamma_{{\mathfrak{a}}}}{\xrightarrow}[\simeq]{{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}}\circ{\varepsilon_{{\mathfrak{a}}}}}{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}}\circ{\operatorname{id}}{\xrightarrow}[\simeq]{{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}}\circ{\vartheta^{{\mathfrak{a}}}}}{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}}\circ{\mathbf{L}\Lambda^{{\mathfrak{a}}}}$$ are isomorphisms of functors ${{\mathcal{D}}}(R)\to{{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})$.
For the first isomorphism, let $X\in{{\mathcal{D}}}(R)$ be given, and choose semi-projective resolutions $P{\xrightarrow}\simeq X$ and $Q{\xrightarrow}\simeq{\mathbf{R}\Gamma_{{\mathfrak{a}}}(X)}$. Let $\phi\colon Q\to P$ be a chain map representing the natural morphism ${\mathbf{R}\Gamma_{{\mathfrak{a}}}(X)}{\xrightarrow}{{\varepsilon_{{\mathfrak{a}}}^{X}}} X$. Then the induced morphism ${\mathbf{L}\Lambda^{{\mathfrak{a}}}({\mathbf{R}\Gamma_{{\mathfrak{a}}}(X)})}\to{\mathbf{L}\Lambda^{{\mathfrak{a}}}(X)}$ is an isomorphism in ${{\mathcal{D}}}(R)$ by Fact \[fact150626a\], and it is represented by $\Lambda^{{{\mathfrak{a}}}}(\phi)\colon\Lambda^{{{\mathfrak{a}}}}(Q)\to\Lambda^{{{\mathfrak{a}}}}(P)$. It follows that $\Lambda^{{{\mathfrak{a}}}}(\phi)$ is a quasi-isomorphism. Since $\Lambda^{{{\mathfrak{a}}}}(\phi)$ also represents the natural morphism ${\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}({\mathbf{R}\Gamma_{{\mathfrak{a}}}(X)})}\to{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)}$ in ${{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})$, this morphism is also an isomorphism, as desired.
For ${\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}}\circ{\vartheta^{{\mathfrak{a}}}}$, argue similarly with Fact \[lem150907a\] in place of Fact \[fact150626a\].
\[lem150805d\] There is a natural isomorphism ${\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}}\circ{\mathbf{R}\Gamma_{{\mathfrak{a}}}}\simeq{\mathbf{L}\Lambda^{{\mathfrak{a}}{\widehat{R}^{{\mathfrak{a}}}}}}\circ{\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}}$ of functors ${{\mathcal{D}}}(R)\to{{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})$
Let $X\in{{\mathcal{D}}}(R)$ be given, and choose a semi-flat resolution $F{\xrightarrow}\simeq {\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(X)}$ over ${\widehat{R}^{{\mathfrak{a}}}}$. Since ${\widehat{R}^{{\mathfrak{a}}}}$ is flat over $R$, the complex $F$ is also semi-flat over $R$, so it is a semi-flat resolution of $Q({\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(X)})\simeq{\mathbf{R}\Gamma_{{\mathfrak{a}}}(X)}$ over $R$. This explains the isomorphisms in ${{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})$ in the next display $${\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}({\mathbf{R}\Gamma_{{\mathfrak{a}}}(X)})}\simeq
\Lambda^{{{\mathfrak{a}}}}(F)=\Lambda^{{{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}}}(F)\simeq{\mathbf{L}\Lambda^{\mathfrak{a}{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(X)})}$$ The equality comes from the fact that $F$ is an ${\widehat{R}^{{\mathfrak{a}}}}$-complex.
\[lem150805e\] The natural transformation $${\operatorname{id}}\circ{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}}{\xrightarrow}[\simeq]{\vartheta^{{{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}}}\circ{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}}}
{\mathbf{L}\Lambda^{{\mathfrak{a}}{\widehat{R}^{{\mathfrak{a}}}}}}\circ{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}}$$ is an isomorphism of functors ${{\mathcal{D}}}(R)\to{{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})$. In other words, the essential image of ${\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}}$ in ${{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})$ is contained in ${{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})_{\text{${{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}}$}-comp}$.
Let $X\in{{\mathcal{D}}}(R)$ be given. According to Fact \[fact150626a\], it suffices to show that ${\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)}$ is of the form ${\mathbf{L}\Lambda^{\mathfrak{a}{\widehat{R}^{{\mathfrak{a}}}}}(Y)}$ for some $Y\in{{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})$. To this end, the next isomorphisms in ${{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})$, from Lemmas \[lem150805a\] and \[lem150805d\] $${\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)}\simeq{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}({\mathbf{R}\Gamma_{{\mathfrak{a}}}(X)})}\simeq {\mathbf{L}\Lambda^{\mathfrak{a}{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(X)})}$$ show that the complex $Y={\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(X)}$ satisfies this condition.
The next few results are proved like the preceding ones, using semi-injective resolutions for the first two.
\[lem150805b\] The natural transformations $${\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}}\circ{\mathbf{R}\Gamma_{{\mathfrak{a}}}}{\xrightarrow}[\simeq]{{\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}}\circ{\varepsilon_{{\mathfrak{a}}}}}{\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}}\circ{\operatorname{id}}{\xrightarrow}[\simeq]{{\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}}\circ{\vartheta^{{\mathfrak{a}}}}}{\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}}\circ{\mathbf{L}\Lambda^{{\mathfrak{a}}}}$$ are isomorphisms of functors ${{\mathcal{D}}}(R)\to{{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})$.
\[lem150805c\] There is a natural isomorphism ${\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}}\circ{\mathbf{L}\Lambda^{{\mathfrak{a}}}}\simeq{\mathbf{R}\Gamma_{\mathfrak{a}{\widehat{R}^{{\mathfrak{a}}}}}}\circ{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}}$ of functors ${{\mathcal{D}}}(R)\to{{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})$.
\[lem150805f\] The natural transformation $${\mathbf{R}\Gamma_{\mathfrak{a}{\widehat{R}^{{\mathfrak{a}}}}}}\circ{\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}}
{\xrightarrow}[\simeq]{\varepsilon_{{{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}}}\circ{\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}}}
{\operatorname{id}}\circ{\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}}$$ is an isomorphism of functors ${{\mathcal{D}}}(R)\to{{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})$. In other words, the essential image of ${\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}}$ in ${{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})$ is contained in ${{{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})_{\text{${{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}}$-tor}}}$.
\[disc160213a\] After we announced the results of this paper, we learned from Liran Shaul that he has obtained some of the results of this section independently and in more generality. For instance, the isomorphism ${\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}} \simeq {\mathbf{L}\Lambda^{{\mathfrak{a}}{\widehat{R}^{{\mathfrak{a}}}}}}\circ{\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}}$ from Lemmas \[lem150805a\] and \[lem150805d\] is [@shaul:ard Theorem 1.7] in a non-noetherian setting.
Here is a version of Greenlees-May Duality \[fact150626a\] for our extended functors. It is Theorem \[thm151129a\] from the introduction.
\[thm151002a\] Given $X,Y\in{{\mathcal{D}}}(R)$, there are natural isomorphisms in ${{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})$: $$\begin{aligned}
{\mathbf{R}\!\operatorname{Hom}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)},{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(Y)})}
&\simeq
{\mathbf{R}\!\operatorname{Hom}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(X)},{\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(Y)})}\\
&\simeq{\mathbf{R}\!\operatorname{Hom}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(X)},{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(Y)})}.\end{aligned}$$
The first isomorphism follows from the next sequence $$\begin{aligned}
{\mathbf{R}\!\operatorname{Hom}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)},{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(Y)})}
&\simeq{\mathbf{R}\!\operatorname{Hom}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{L}\Lambda^{\mathfrak{a}{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)})},{\mathbf{L}\Lambda^{\mathfrak{a}{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(Y)})})}
\\
&\simeq{\mathbf{R}\!\operatorname{Hom}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{R}\Gamma_{\mathfrak{a}{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)})},{\mathbf{R}\Gamma_{\mathfrak{a}{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(Y)})})}
\\
&\simeq{\mathbf{R}\!\operatorname{Hom}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}({\mathbf{L}\Lambda^{{\mathfrak{a}}}(X)})},{\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}({\mathbf{L}\Lambda^{{\mathfrak{a}}}(Y)})})}
\\
&\simeq{\mathbf{R}\!\operatorname{Hom}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(X)},{\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(Y)})}\end{aligned}$$ wherein the isomorphisms are from Theorem \[lem150805e\], Greenlees-May duality \[fact150626a\], and Lemma \[lem150805c\], and Lemma \[lem150805b\], respectively.
The second isomorphism follows from the next sequence $$\begin{aligned}
{\mathbf{R}\!\operatorname{Hom}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(X)},{\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(Y)})}
&\simeq{\mathbf{R}\!\operatorname{Hom}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}({\mathbf{L}\Lambda^{{\mathfrak{a}}}(X)})},{\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}({\mathbf{L}\Lambda^{{\mathfrak{a}}}(Y)})})}
\\
&\simeq{\mathbf{R}\!\operatorname{Hom}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{R}\Gamma_{\mathfrak{a}{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)})},{\mathbf{R}\Gamma_{\mathfrak{a}{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(Y)})})}
\\
&\simeq{\mathbf{R}\!\operatorname{Hom}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{R}\Gamma_{\mathfrak{a}{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)})},{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(Y)})}
\\
&\simeq{\mathbf{R}\!\operatorname{Hom}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}({\mathbf{L}\Lambda^{{\mathfrak{a}}}(X)})},{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(Y)})}
\\
&\simeq{\mathbf{R}\!\operatorname{Hom}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(X)},{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(Y)})}\end{aligned}$$ which is justified similarly.
\[disc151004a\] It is reasonable to ask whether versions of other isomorphisms from Greenlees-May duality \[fact150626a\] hold in our set-up. For instance, given $X,Y\in{{\mathcal{D}}}(R)$, we have the natural isomorphism $${\mathbf{R}\!\operatorname{Hom}_{R}({\mathbf{R}\Gamma_{{\mathfrak{a}}}(X)},{\mathbf{R}\Gamma_{{\mathfrak{a}}}(Y)})}{\xrightarrow}\simeq{\mathbf{R}\!\operatorname{Hom}_{R}({\mathbf{R}\Gamma_{{\mathfrak{a}}}(X)},Y)}.$$ In our set-up, the naive question would ask whether we have $${\mathbf{R}\!\operatorname{Hom}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(X)},{\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(Y)})}\stackrel{\text{?}}\simeq{\mathbf{R}\!\operatorname{Hom}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(X)},Y)}$$ in ${{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})$. However, this doesn’t make sense as $Y$ does not come equipped with an ${\widehat{R}^{{\mathfrak{a}}}}$-structure. On the other hand, see Propositions \[prop151203a\]–\[cor151004a\] and their proofs for some isomorphisms involving ${\mathbf{R}\!\operatorname{Hom}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(X)},Y)}$ when $Y\in{{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})$.
Another reasonable question to ask would be whether the isomorphism $${\mathbf{R}\!\operatorname{Hom}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)},{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(Y)})}\stackrel{\text{?}}\simeq{\mathbf{R}\!\operatorname{Hom}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)},{\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(Y)})}$$ holds. One sees readily from the example $X=R=Y$ that this fails in general because, one the one hand, we have $${\mathbf{R}\!\operatorname{Hom}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(R)},{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(R)})}
\simeq{\mathbf{R}\!\operatorname{Hom}_{{\widehat{R}^{{\mathfrak{a}}}}}({\widehat{R}^{{\mathfrak{a}}}},{\widehat{R}^{{\mathfrak{a}}}})}\simeq {\widehat{R}^{{\mathfrak{a}}}}$$ which is not in general isomorphic to $$\begin{aligned}
{\mathbf{R}\!\operatorname{Hom}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)},{\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(Y)})}
&\simeq{\mathbf{R}\!\operatorname{Hom}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(R)},{\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(R)})}\\
&\simeq{\mathbf{R}\!\operatorname{Hom}_{{\widehat{R}^{{\mathfrak{a}}}}}({\widehat{R}^{{\mathfrak{a}}}},{\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(Y)})}\\
&\simeq{\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(Y)}.\end{aligned}$$
The next two results are akin to [@shaul:hccac Corollary 3.9].
\[prop151203a\] Let $X\in{{\mathcal{D}}}(R)$ and $Y\in{{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})$ be given. Then there are isomorphisms in ${{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})$ $$\begin{gathered}
{\mathbf{R}\!\operatorname{Hom}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(X)},Y)}\simeq{\mathbf{R}\!\operatorname{Hom}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(X)},{\mathbf{L}\Lambda^{\mathfrak{a}{\widehat{R}^{{\mathfrak{a}}}}}(Y)})} \\
{\mathbf{R}\!\operatorname{Hom}_{{\widehat{R}^{{\mathfrak{a}}}}}(Y,{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)})}\simeq{\mathbf{R}\!\operatorname{Hom}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{R}\Gamma_{\mathfrak{a}{\widehat{R}^{{\mathfrak{a}}}}}(Y)},{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)})} \end{gathered}$$ which are natural in $X$ and $Y$.
We verify the second isomorphism; the verification of the first one is similar. In the following display, the first step is from Theorem \[lem150805e\]: $$\begin{aligned}
{\mathbf{R}\!\operatorname{Hom}_{{\widehat{R}^{{\mathfrak{a}}}}}(Y,{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)})}
&\simeq{\mathbf{R}\!\operatorname{Hom}_{{\widehat{R}^{{\mathfrak{a}}}}}(Y,{\mathbf{L}\Lambda^{\mathfrak{a}{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)})})} \\
&\simeq{\mathbf{R}\!\operatorname{Hom}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{R}\Gamma_{\mathfrak{a}{\widehat{R}^{{\mathfrak{a}}}}}(Y)},{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)})}\end{aligned}$$ The second step is Greenlees-May duality \[fact150626a\].
\[cor151004a\] Let $X\in{{\mathcal{D}}}(R)$ and $Y\in{{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})$ be given, and let $Q\colon {{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})\to{{\mathcal{D}}}(R)$ denote the forgetful functor. Then there are isomorphisms in ${{\mathcal{D}}}(R)$ $$\begin{gathered}
{\mathbf{R}\!\operatorname{Hom}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(X)},Y)}\simeq{\mathbf{R}\!\operatorname{Hom}_{R}(X,{\mathbf{L}\Lambda^{{\mathfrak{a}}}(Q(Y))})} \\
{\mathbf{R}\!\operatorname{Hom}_{{\widehat{R}^{{\mathfrak{a}}}}}(Y,{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)})}\simeq{\mathbf{R}\!\operatorname{Hom}_{R}({\mathbf{R}\Gamma_{{\mathfrak{a}}}(Q(Y))},X)} \end{gathered}$$ which are natural in $X$ and $Y$.
We verify the second isomorphism; the verification of the first one is similar. In the following display, the first step is from Fact \[fact130619b\]: $$\begin{aligned}
{\mathbf{R}\!\operatorname{Hom}_{{\widehat{R}^{{\mathfrak{a}}}}}(Y,{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)})}
&\simeq{\mathbf{R}\!\operatorname{Hom}_{{\widehat{R}^{{\mathfrak{a}}}}}(Y,{\mathbf{R}\!\operatorname{Hom}_{R}({\widehat{R}^{{\mathfrak{a}}}},{\mathbf{L}\Lambda^{{\mathfrak{a}}}(X)})})} \\
&\simeq{\mathbf{R}\!\operatorname{Hom}_{R}({{\widehat{R}^{{\mathfrak{a}}}}\otimes^{\mathbf{L}}_{{\widehat{R}^{{\mathfrak{a}}}}}Y},{\mathbf{L}\Lambda^{{\mathfrak{a}}}(X)})} \\
&\simeq{\mathbf{R}\!\operatorname{Hom}_{R}(Q(Y),{\mathbf{L}\Lambda^{{\mathfrak{a}}}(X)})} \\
&\simeq{\mathbf{R}\!\operatorname{Hom}_{R}({\mathbf{R}\Gamma_{{\mathfrak{a}}}(Q(Y))},X)}. \end{aligned}$$ The second step is Hom-tensor adjointness, and the third one is tensor-cancellation. The last step is an adjointness isomorphism that follows from Fact \[fact130619b\].
The next two results form our extension of MGM equivalence, as described in parts and of Theorem \[thm151129a\] from the introduction; see Remark \[disc151003a\].
\[thm151003a\] The functor ${\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}}\colon{{{\mathcal{D}}}(R)_{\text{${{\mathfrak{a}}}$-tor}}}\to{{{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})_{\text{${{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}}$-tor}}}$ and the forgetful functor $Q\colon {{{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})_{\text{${{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}}$-tor}}}\to{{{\mathcal{D}}}(R)_{\text{${{\mathfrak{a}}}$-tor}}}$ are quasi-inverse equivalences.
Note that ${\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}}$ maps ${{{\mathcal{D}}}(R)_{\text{${{\mathfrak{a}}}$-tor}}}$ into ${{{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})_{\text{${{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}}$-tor}}}$ by Theorem \[lem150805f\]. The forgetful functor $Q$ maps ${{{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})_{\text{${{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}}$-tor}}}$ into ${{{\mathcal{D}}}(R)_{\text{${{\mathfrak{a}}}$-tor}}}$ by Fact \[cor130528a\] and [@sather:afcc Lemma 5.3].
Next, we show that the composition ${\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}}\circ Q$ is equivalent to the identity. For this, consider a complex $X\in{{{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})_{\text{${{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}}$-tor}}}$; it suffices to show that ${\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(Q(X))}\simeq X$ over ${\widehat{R}^{{\mathfrak{a}}}}$. Fix a semi-injective resolution $X{\xrightarrow}\simeq I$ over ${\widehat{R}^{{\mathfrak{a}}}}$. Since ${\widehat{R}^{{\mathfrak{a}}}}$ is flat over $R$, this yields a semi-injective resolution $Q(X){\xrightarrow}\simeq Q(I)$. Also, the torsion functors $\Gamma_{{{\mathfrak{a}}}}$ and $\Gamma_{{{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}}}$ are the same when restricted to ${\widehat{R}^{{\mathfrak{a}}}}$-complexes. By definition of ${{{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})_{\text{${{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}}$-tor}}}$, the inclusion morphism $\Gamma_{{{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}}}(I)\to I$ is a quasiisomorphism. Thus, we have $${\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(Q(X))}\simeq\Gamma_{{{\mathfrak{a}}}}(Q(I))=\Gamma_{{{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}}}(I)\simeq I\simeq X$$ over ${\widehat{R}^{{\mathfrak{a}}}}$, hence the desired conclusion.
Lastly, the composition $Q\circ{\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}}$ is ${\mathbf{R}\Gamma_{{\mathfrak{a}}}}$. When restricted to ${{{\mathcal{D}}}(R)_{\text{${{\mathfrak{a}}}$-tor}}}$, this is isomorphic to the identity by definition of ${{{\mathcal{D}}}(R)_{\text{${{\mathfrak{a}}}$-tor}}}$.
The next result is proved like the previous one, using a semi-flat resolution.
\[thm151003b\] The functor ${\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}}\colon{{{\mathcal{D}}}(R)_{\text{${{\mathfrak{a}}}$-comp}}}\to{{{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})_{\text{${{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}}$-comp}}}$ and the forgetful functor $Q\colon {{{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})_{\text{${{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}}$-comp}}}\to{{{\mathcal{D}}}(R)_{\text{${{\mathfrak{a}}}$-comp}}}$ are quasi-inverse equivalences.
The next result is a consequence of the proofs of Theorems \[thm151003a\] and \[thm151003b\].
\[cor151003a\] There are natural isomorphisms $Q\circ{\mathbf{R}\Gamma_{\mathfrak{a}{\widehat{R}^{{\mathfrak{a}}}}}}\simeq{\mathbf{R}\Gamma_{{\mathfrak{a}}}}\circ Q$ and $Q\circ{\mathbf{L}\Lambda^{{\mathfrak{a}}{\widehat{R}^{{\mathfrak{a}}}}}}\simeq{\mathbf{L}\Lambda^{{\mathfrak{a}}}}\circ Q$ of functors ${{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})\to{{\mathcal{D}}}(R)$.
\[disc151003a\] Theorems \[thm151003a\] and \[thm151003b\] have several consequences. First, they augment Theorems \[lem150805e\] and \[lem150805f\] by showing that the essential images of ${\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}}$ and ${\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}}$ in ${{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})$ are *equal to* ${{{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})_{\text{${{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}}$-tor}}}$ and ${{{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})_{\text{${{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}}$-comp}}}$, respectively.
Second, they show that MGM equivalences over $R$ and ${\widehat{R}^{{\mathfrak{a}}}}$ are essentially the same. These equivalences appear in the rows of the following diagram $$\xymatrix@=15mm{
{{{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})_{\text{${{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}}$-tor}}}\ar@<1ex>[r]^-{{\mathbf{L}\Lambda^{{\mathfrak{a}}{\widehat{R}^{{\mathfrak{a}}}}}}}_-{\simeq}\ar@<1.5ex>[d]^-{Q}_-{\simeq}
&{{{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})_{\text{${{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}}$-comp}}}\ar@<1ex>[l]^-{{\mathbf{R}\Gamma_{\mathfrak{a}{\widehat{R}^{{\mathfrak{a}}}}}}}\ar@<1.5ex>[d]^-{Q}_-{\simeq}
\\
{{{\mathcal{D}}}(R)_{\text{${{\mathfrak{a}}}$-tor}}}\ar@<1ex>[r]^-{{\mathbf{L}\Lambda^{{\mathfrak{a}}}}}_-{\simeq}\ar@<1.5ex>[u]^-{{\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}}}
&{{{\mathcal{D}}}(R)_{\text{${{\mathfrak{a}}}$-comp}}}\ar@<1ex>[l]^-{{\mathbf{R}\Gamma_{{\mathfrak{a}}}}}\ar@<1.5ex>[u]^-{{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}}}
}$$ while our results provide the equivalences in the columns. Various versions of this diagram commute. For instance, the first diagram in the next display commutes by Lemmas \[lem150805a\] and \[lem150805d\]. The second one is from Lemmas \[lem150805b\] and \[lem150805c\]. $$\xymatrix{
{{{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})_{\text{${{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}}$-tor}}}\ar[r]^-{{\mathbf{L}\Lambda^{{\mathfrak{a}}{\widehat{R}^{{\mathfrak{a}}}}}}}_-{\simeq}
&{{{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})_{\text{${{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}}$-comp}}}\\
{{{\mathcal{D}}}(R)_{\text{${{\mathfrak{a}}}$-tor}}}\ar[r]^-{{\mathbf{L}\Lambda^{{\mathfrak{a}}}}}_-{\simeq}\ar[u]^-{{\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}}}_-{\simeq}
&{{{\mathcal{D}}}(R)_{\text{${{\mathfrak{a}}}$-comp}}}\ar[u]_-{{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}}}^-{\simeq}}
\qquad
\xymatrix{
{{{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})_{\text{${{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}}$-tor}}}&{{{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})_{\text{${{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}}$-comp}}}\ar[l]_-{{\mathbf{R}\Gamma_{\mathfrak{a}{\widehat{R}^{{\mathfrak{a}}}}}}}^-{\simeq}
\\
{{{\mathcal{D}}}(R)_{\text{${{\mathfrak{a}}}$-tor}}}\ar[u]^-{{\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}}}_-{\simeq}
&{{{\mathcal{D}}}(R)_{\text{${{\mathfrak{a}}}$-comp}}}\ar[l]_-{{\mathbf{R}\Gamma_{{\mathfrak{a}}}}}^-{\simeq}\ar[u]_-{{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}}}^-{\simeq}
}$$ Corollary \[cor151003a\] explains the commutativity of next two diagrams. $$\xymatrix{
{{{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})_{\text{${{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}}$-tor}}}\ar[r]^-{{\mathbf{L}\Lambda^{{\mathfrak{a}}{\widehat{R}^{{\mathfrak{a}}}}}}}_-{\simeq}\ar[d]_-{Q}^-{\simeq}
&{{{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})_{\text{${{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}}$-comp}}}\ar[d]^-{Q}_-{\simeq}
\\
{{{\mathcal{D}}}(R)_{\text{${{\mathfrak{a}}}$-tor}}}\ar[r]^-{{\mathbf{L}\Lambda^{{\mathfrak{a}}}}}_-{\simeq}
&{{{\mathcal{D}}}(R)_{\text{${{\mathfrak{a}}}$-comp}}}}
\qquad
\xymatrix{
{{{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})_{\text{${{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}}$-tor}}}\ar[d]_-{Q}^-{\simeq}
&{{{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})_{\text{${{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}}$-comp}}}\ar[l]_-{{\mathbf{R}\Gamma_{\mathfrak{a}{\widehat{R}^{{\mathfrak{a}}}}}}}^-{\simeq}\ar[d]^-{Q}_-{\simeq}
\\
{{{\mathcal{D}}}(R)_{\text{${{\mathfrak{a}}}$-tor}}}&{{{\mathcal{D}}}(R)_{\text{${{\mathfrak{a}}}$-comp}}}\ar[l]_-{{\mathbf{R}\Gamma_{{\mathfrak{a}}}}}^-{\simeq}
}$$
Flat and Injective Dimensions {#sec151003a}
=============================
In this section, we provide bounds on the flat and injective dimensions over ${\widehat{R}^{{\mathfrak{a}}}}$ of ${\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)}$ and ${\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(X)}$, for use in [@sather:asc].
\[cor151012a\] Let $X\in{{{\mathcal{D}}}_{\text{b}}}(R)$ be given. Then there are inequalities $$\begin{aligned}
{\operatorname{id}}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)})&{\leqslant}{\operatorname{id}}_R(X)\\
{\operatorname{fd}}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(X)})&{\leqslant}{\operatorname{fd}}_R(X). \end{aligned}$$
The Čech complex over ${\widehat{R}^{{\mathfrak{a}}}}$ on a generating sequence for ${{\mathfrak{a}}}$ shows that we have ${\operatorname{fd}}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(R)}){\leqslant}0$. Thus, by the isomorphism ${\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)}\simeq{\mathbf{R}\!\operatorname{Hom}_{R}({\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(R)},X)}$ in ${{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})$ from Fact \[fact130619b\], we have $${\operatorname{id}}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)})={\operatorname{id}}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{R}\!\operatorname{Hom}_{R}({\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(R)},X)}){\leqslant}{\operatorname{fd}}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(R)})+{\operatorname{id}}_R(X)={\operatorname{id}}_R(X)$$ by [@avramov:hdouc Theorem 4.1(F)]. Similarly, from the isomorphism ${\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(X)}\simeq{{\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(R)}\otimes^{\mathbf{L}}_{R}X}$, we have ${\operatorname{fd}}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(X)}){\leqslant}{\operatorname{fd}}_R(X)$ by [@avramov:hdouc Theorem 4.1(F)].
We end this section with similar bounds for ${\operatorname{id}}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)})$ and ${\operatorname{fd}}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(X)})$, after the following lemma.
\[lem160205a\] Let $N$ be an ${\widehat{R}^{{\mathfrak{a}}}}$-module and is either ${{\mathfrak{a}}}$-adically complete or ${{\mathfrak{a}}}$-torsion. Then one has $$\begin{aligned}
{\operatorname{fd}}_R(N)&={\operatorname{fd}}_{{\widehat{R}^{{\mathfrak{a}}}}}(N)
&
{\operatorname{id}}_R(N)&={\operatorname{id}}_{{\widehat{R}^{{\mathfrak{a}}}}}(N).\end{aligned}$$ In particular, $N$ is flat over $R$ if and only if it is flat over ${\widehat{R}^{{\mathfrak{a}}}}$, and $N$ is injective over $R$ if and only if it is injective over ${\widehat{R}^{{\mathfrak{a}}}}$.
We verify the first displayed equality in the statement of the lemma; the second one is verified similarly, and the subsequent statements follow from these directly. Since ${\widehat{R}^{{\mathfrak{a}}}}$ is flat over $R$, the inequality ${\operatorname{fd}}_R(N){\leqslant}{\operatorname{fd}}_{{\widehat{R}^{{\mathfrak{a}}}}}(N)$ is from [@avramov:hdouc Corollary 4.1(b)(F)]. (Note that this does not use the assumption that $N$ is ${{\mathfrak{a}}}$-adically complete or ${{\mathfrak{a}}}$-torsion.)
Claim. We have $$\label{eq160205a}
{\operatorname{fd}}_{{\widehat{R}^{{\mathfrak{a}}}}}(N)=\sup\{\sup({({\widehat{R}^{{\mathfrak{a}}}}/P)\otimes^{\mathbf{L}}_{{\widehat{R}^{{\mathfrak{a}}}}}N})\mid P\in{\operatorname{V}}({{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}})\}.$$ If $N$ is ${{\mathfrak{a}}}$-adically complete, then this is by [@simon:shpcm Proposition 2.1]. If $N$ is ${{\mathfrak{a}}}$-torsion, it is straightforward to show that we have ${\operatorname{Supp}}_{{\widehat{R}^{{\mathfrak{a}}}}}(N)\subseteq{\operatorname{V}}({{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}})$, so the claim follows from [@avramov:hdouc Proposition 5.3.F]. (The corresponding formula for injective dimension is from [@simon:shpcm Proposition 3.2] and [@avramov:hdouc Proposition 5.3.I].)
Now we complete the proof by verifying the inequality ${\operatorname{fd}}_R(N){\geqslant}{\operatorname{fd}}_{{\widehat{R}^{{\mathfrak{a}}}}}(N)$. The natural isomorphism ${\widehat{R}^{{\mathfrak{a}}}}/{{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}}\cong R/{{\mathfrak{a}}}$ shows that every $P\in{\operatorname{V}}({{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}})$ is of the form $P={{\mathfrak{p}}}{\widehat{R}^{{\mathfrak{a}}}}$ for a unique prime ideal ${{\mathfrak{p}}}\in{\operatorname{V}}({{\mathfrak{a}}})\subseteq{\operatorname{Spec}}(R)$. This also implies that $${\widehat{R}^{{\mathfrak{a}}}}/P={\widehat{R}^{{\mathfrak{a}}}}/{{\mathfrak{p}}}{\widehat{R}^{{\mathfrak{a}}}}\simeq {{\widehat{R}^{{\mathfrak{a}}}}\otimes^{\mathbf{L}}_{R}(R/{{\mathfrak{p}}})}$$ so we find that $${({\widehat{R}^{{\mathfrak{a}}}}/P)\otimes^{\mathbf{L}}_{{\widehat{R}^{{\mathfrak{a}}}}}N}\simeq
{({{\widehat{R}^{{\mathfrak{a}}}}\otimes^{\mathbf{L}}_{R}(R/{{\mathfrak{p}}})})\otimes^{\mathbf{L}}_{{\widehat{R}^{{\mathfrak{a}}}}}N}\simeq{(R/{{\mathfrak{p}}})\otimes^{\mathbf{L}}_{R}N}.$$ From this, we have $$\sup({({\widehat{R}^{{\mathfrak{a}}}}/P)\otimes^{\mathbf{L}}_{{\widehat{R}^{{\mathfrak{a}}}}}N})=\sup({(R/{{\mathfrak{p}}})\otimes^{\mathbf{L}}_{R}N}){\leqslant}{\operatorname{fd}}_R(N).$$ Thus, the inequality ${\operatorname{fd}}_R(N){\geqslant}{\operatorname{fd}}_{{\widehat{R}^{{\mathfrak{a}}}}}(N)$ follows from .
\[prop160205a\] Let $X\in{{{\mathcal{D}}}_{\text{b}}}(R)$ be given.
1. \[prop160205a1\] Then there is an inequality ${\operatorname{id}}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(X)}){\leqslant}{\operatorname{id}}_R(X)$.
2. \[prop160205a2\] If at least one of the following conditions holds
1. ${\operatorname{pd}}_R(X)<\infty$,
2. $\dim(R)<\infty$, or
3. $X$ is ${{\mathfrak{b}}}$-adically finite for some ideal ${{\mathfrak{b}}}$, e.g., $X\in{{{\mathcal{D}}}_{\text{b}}^{\text{f}}}(R)$,
then one has ${\operatorname{fd}}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)}){\leqslant}{\operatorname{fd}}_R(X)$.
Assume without loss of generality that ${\operatorname{id}}_R(X)<\infty$, and let $X{\xrightarrow}\simeq J$ be a bounded semi-injective resolution over $R$ such that $J_{i}=0$ for all $i<-{\operatorname{id}}_R(X)$. It follows that the $R$-complex $\Gamma_{{\mathfrak{a}}}(J)$ is a bounded semi-injective resolution of ${\mathbf{R}\Gamma_{{\mathfrak{a}}}(X)}$ over $R$ such that $\Gamma_{{\mathfrak{a}}}(J)_{i}=\Gamma_{{\mathfrak{a}}}(J_{i})=0$ for all $i<-{\operatorname{id}}_R(X)$. Since each module in this complex is ${{\mathfrak{a}}}$-torsion, the complex $\Gamma_{{\mathfrak{a}}}(J)$ is an ${\widehat{R}^{{\mathfrak{a}}}}$-complex, and Lemma \[lem160205a\] implies that it consists of injective ${\widehat{R}^{{\mathfrak{a}}}}$-modules. Thus, the ${\widehat{R}^{{\mathfrak{a}}}}$-complex $\Gamma_{{\mathfrak{a}}}(J)$ is a bounded semi-injective resolution of ${\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(X)}$ over ${\widehat{R}^{{\mathfrak{a}}}}$ such that $\Gamma_{{\mathfrak{a}}}(J)_{i}=\Gamma_{{\mathfrak{a}}}(J_{i})=0$ for all $i<-{\operatorname{id}}_R(X)$. The inequality ${\operatorname{id}}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(X)}){\leqslant}{\operatorname{id}}_R(X)$ follows.
The argument here is similar to the previous one, but with a twist. As before, assume without loss of generality that $f={\operatorname{fd}}_R(X)<\infty$. Let $P{\xrightarrow}\simeq X$ be a bounded semi-projective resolution over $R$. Truncate $P$ appropriately to obtain a bounded semi-flat resolution $F{\xrightarrow}\simeq X$ over $R$ such that $F_{i}=0$ for all $i>f$ and $F_i=P_i$ for all $i<f$.
Claim: ${\operatorname{pd}}_R(F_f)<\infty$. Indeed, in case ${\operatorname{pd}}_R(X)<\infty$, this is standard. If $X$ is ${{\mathfrak{a}}}$-adically finite, then we have ${\operatorname{pd}}_R(X)<\infty$ by [@sather:afcc Theorem 6.1], so the claim is established in this case. Lastly, a result of Gruson and Raynaud [@raynaud:cpptpm Seconde Partie, Theorem 3.2.6] and Jensen [@jensen:vl Proposition 6] implies that ${\operatorname{pd}}_R(F_f){\leqslant}\dim(R)$, so the claim holds when $\dim(R)<\infty$.
By the claim, a result of Schoutens [@schoutens:lfccm Theorem 5.9] implies that each module $\Lambda^{{\mathfrak{a}}}(F_i)$ is flat over $R$. (If $\dim(R)<\infty$, this is due to Enochs [@enochs:cfm].) From Lemma \[lem160205a\], we conclude that each module $\Lambda^{{\mathfrak{a}}}(F_i)$ is flat over ${\widehat{R}^{{\mathfrak{a}}}}$. Note that this uses the fact that $\Lambda^{{\mathfrak{a}}}(F_i)$ is ${{\mathfrak{a}}}$-adically complete; see, e.g., [@yekutieli:fcigm Corollary 1.7]. So, the ${\widehat{R}^{{\mathfrak{a}}}}$-complex $\Lambda^{{\mathfrak{a}}}(F)$ is a bounded semi-flat resolution of ${\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)}$ over ${\widehat{R}^{{\mathfrak{a}}}}$ such that $\Lambda^{{\mathfrak{a}}}(F)_{i}=0$ for all $i>{\operatorname{fd}}_R(X)$. The inequality ${\operatorname{fd}}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)}){\leqslant}{\operatorname{fd}}_R(X)$ now follows.
\[disc160205a\] To our knowledge, it is not known whether the completion $\Lambda^{{\mathfrak{a}}}(F)$ of a flat $R$-module $F$ is flat over $R$. If this is true, then the extra assumptions (1)–(3) can be removed from Proposition \[prop160205a\].
Cohomological Adic Cofiniteness {#sec151002a}
===============================
Next, we discuss the connection between ${{\mathfrak{a}}}$-adically finite complexes and the following similar notion from [@yekutieli:ccc].
\[defn150626a\] Assume that $R$ is ${{\mathfrak{a}}}$-adically complete. An $R$-complex $X\in{{{\mathcal{D}}}_{\text{b}}}(R)$ is *cohomologically ${{\mathfrak{a}}}$-adically cofinite* if there is a complex $N\in{{{\mathcal{D}}}_{\text{b}}^{\text{f}}}(R)$ such that $X\simeq{\mathbf{R}\Gamma_{{\mathfrak{a}}}(N)}$.
Our first result in this direction, given next, shows that, when it makes sense to compare these two notions, they are the same. It is primarily from [@yekutieli:ccc].
\[prop150626a\] Assume that $R$ is ${{\mathfrak{a}}}$-adically complete. An $R$-complex $X\in{{{\mathcal{D}}}_{\text{b}}}(R)$ is cohomologically ${{\mathfrak{a}}}$-adically cofinite if and only if it is ${{\mathfrak{a}}}$-adically finite.
Assume first that $X$ is cohomologically ${{\mathfrak{a}}}$-adically cofinite, so by definition there is a complex $N\in{{{\mathcal{D}}}_{\text{b}}^{\text{f}}}(R)$ such that $X\simeq{\mathbf{R}\Gamma_{{\mathfrak{a}}}(N)}$. Fact \[fact150626a\] implies that $X$ is in ${{{\mathcal{D}}}(R)_{\text{${{\mathfrak{a}}}$-tor}}}$, so we have ${\operatorname{supp}}_R(X)\subseteq{\operatorname{V}}({{\mathfrak{a}}})$ by Fact \[cor130528a\], and we have ${\mathbf{R}\!\operatorname{Hom}_{R}(R/{{\mathfrak{a}}},X)}\in{{{\mathcal{D}}}^{\text{f}}}(R)$ by [@yekutieli:ccc Theorem 0.4]. Thus, $X$ is ${{\mathfrak{a}}}$-adically finite.
Conversely, assume that $X$ is ${{\mathfrak{a}}}$-adically finite. Then we have ${\operatorname{supp}}_R(X)\subseteq{\operatorname{V}}({{\mathfrak{a}}})$ by definition, so $X$ is in ${{{\mathcal{D}}}(R)_{\text{${{\mathfrak{a}}}$-tor}}}$ by Fact \[cor130528a\]. Also by definition, we have ${\mathbf{R}\!\operatorname{Hom}_{R}(R/{{\mathfrak{a}}},X)}\in{{{\mathcal{D}}}^{\text{f}}}(R)$, so according to [@yekutieli:ccc Theorem 0.4], the complex $X$ is cohomologically ${{\mathfrak{a}}}$-adically cofinite.
The next result gives a similar characterization of ${{\mathfrak{a}}}$-adically finite complexes in the incomplete setting.
\[thm150626a\] Let $Q\colon{{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})\to{{\mathcal{D}}}(R)$ be the forgetful functor, and let ${{\mathcal{D}}}(R)_{\text{${{\mathfrak{a}}}$-fin}}$ be the full subcategory of ${{\mathcal{D}}}(R)$ consisting of all ${{\mathfrak{a}}}$-adically finite $R$-complexes.
1. \[thm150626a1\] An $R$-complex $X\in{{{\mathcal{D}}}_{\text{b}}}(R)$ is ${{\mathfrak{a}}}$-adically finite if and only if there is a complex $N\in{{{\mathcal{D}}}_{\text{b}}^{\text{f}}}({\widehat{R}^{{\mathfrak{a}}}})$ such that $X\simeq Q({\mathbf{R}\Gamma_{\mathfrak{a}{\widehat{R}^{{\mathfrak{a}}}}}(N)})$.
2. \[thm150626a2\] The functor ${\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}}$ induces an equivalence of categories ${{\mathcal{D}}}(R)_{\text{${{\mathfrak{a}}}$-fin}}\to{{{\mathcal{D}}}_{\text{b}}^{\text{f}}}({\widehat{R}^{{\mathfrak{a}}}})$ with quasi-inverse induced by $Q\circ{\mathbf{R}\Gamma_{\mathfrak{a}{\widehat{R}^{{\mathfrak{a}}}}}}$.
3. \[thm150626a3\] If there is a complex $N\in{{{\mathcal{D}}}_{\text{b}}^{\text{f}}}(R)$ such that $X\simeq{\mathbf{R}\Gamma_{{\mathfrak{a}}}(N)}$, then $X\in{{\mathcal{D}}}(R)_{\text{${{\mathfrak{a}}}$-fin}}$.
Claim 1: if $N\in{{{\mathcal{D}}}_{\text{b}}^{\text{f}}}({\widehat{R}^{{\mathfrak{a}}}})$, then we have $N\simeq{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(Q({\mathbf{R}\Gamma_{\mathfrak{a}{\widehat{R}^{{\mathfrak{a}}}}}(N)}))}$ in ${{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})$. Indeed, the first isomorphism in the following sequence is from Corollary \[cor151003a\]. $$\begin{aligned}
{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(Q({\mathbf{R}\Gamma_{\mathfrak{a}{\widehat{R}^{{\mathfrak{a}}}}}(N)}))}
&\simeq{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}({\mathbf{R}\Gamma_{{\mathfrak{a}}}(Q(N))})}
\simeq{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(Q(N))}
\simeq N\end{aligned}$$ The second isomorphism is from Lemma \[lem150805a\]. The third one is from Theorem \[thm151003b\]; to apply this result, we use the conditions $N\in{{{\mathcal{D}}}_{\text{b}}^{\text{f}}}({\widehat{R}^{{\mathfrak{a}}}})\subseteq{{{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})_{\text{${{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}}$-comp}}}$.
Claim 2: if $N\in{{{\mathcal{D}}}_{\text{b}}^{\text{f}}}({\widehat{R}^{{\mathfrak{a}}}})$, then we have $Q({\mathbf{R}\Gamma_{\mathfrak{a}{\widehat{R}^{{\mathfrak{a}}}}}(N)})\in {{\mathcal{D}}}(R)_{\text{${{\mathfrak{a}}}$-fin}}$. Indeed, Corollary \[cor151003a\] implies that $Q({\mathbf{R}\Gamma_{\mathfrak{a}{\widehat{R}^{{\mathfrak{a}}}}}(N)})\simeq {\mathbf{R}\Gamma_{{\mathfrak{a}}}(Q(N))}$, so we have $${\operatorname{supp}}_R(Q({\mathbf{R}\Gamma_{\mathfrak{a}{\widehat{R}^{{\mathfrak{a}}}}}(N)}))={\operatorname{supp}}_R({\mathbf{R}\Gamma_{{\mathfrak{a}}}(Q(N))})\subseteq{\operatorname{V}}({{\mathfrak{a}}})$$ by [@sather:scc Proposition 3.6]. Also, Claim 1 shows that ${\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(Q({\mathbf{R}\Gamma_{\mathfrak{a}{\widehat{R}^{{\mathfrak{a}}}}}(N)}))}\simeq N\in{{{\mathcal{D}}}_{\text{b}}^{\text{f}}}({\widehat{R}^{{\mathfrak{a}}}})$, so $Q({\mathbf{R}\Gamma_{\mathfrak{a}{\widehat{R}^{{\mathfrak{a}}}}}(N)})$ is ${{\mathfrak{a}}}$-adically finite by definition.
Claim 3: if $X\in{{\mathcal{D}}}(R)_{\text{${{\mathfrak{a}}}$-fin}}$, then $X\simeq Q(\mathbf{R}\Gamma_{{{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)}))$ in ${{\mathcal{D}}}(R)$. The first three isomorphisms in the following sequence are from Lemma \[lem150805c\], Lemma \[lem150805b\], and Theorem \[thm151003a\], respectively. $$\begin{aligned}
Q(\mathbf{R}\Gamma_{{{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)}))
&\simeq Q({\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}({\mathbf{L}\Lambda^{{\mathfrak{a}}}(X)})})
\simeq Q({\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}({\mathbf{R}\Gamma_{{\mathfrak{a}}}(X)})})
\simeq{\mathbf{R}\Gamma_{{\mathfrak{a}}}(X)}
\simeq X\end{aligned}$$ The fourth isomorphism is by Fact \[cor130528a\], as we have ${\operatorname{supp}}_R(X)\subseteq{\operatorname{V}}({{\mathfrak{a}}})$ by assumption.
Now we complete the proof of the result. By definition, if $X\in{{\mathcal{D}}}(R)_{\text{${{\mathfrak{a}}}$-fin}}$, then ${\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)}\in{{{\mathcal{D}}}_{\text{b}}^{\text{f}}}({\widehat{R}^{{\mathfrak{a}}}})$, that is, the functor ${\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}}$ maps ${{\mathcal{D}}}(R)_{\text{${{\mathfrak{a}}}$-fin}}$ to ${{{\mathcal{D}}}_{\text{b}}^{\text{f}}}({\widehat{R}^{{\mathfrak{a}}}})$. Claim 2 shows that $Q\circ{\mathbf{R}\Gamma_{\mathfrak{a}{\widehat{R}^{{\mathfrak{a}}}}}}$ maps ${{{\mathcal{D}}}_{\text{b}}^{\text{f}}}({\widehat{R}^{{\mathfrak{a}}}})$ to ${{\mathcal{D}}}(R)_{\text{${{\mathfrak{a}}}$-fin}}$, and Claim 1 shows that the composition ${\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}}\circ Q\circ{\mathbf{R}\Gamma_{\mathfrak{a}{\widehat{R}^{{\mathfrak{a}}}}}}$ is isomorphic to the identity on ${{{\mathcal{D}}}_{\text{b}}^{\text{f}}}({\widehat{R}^{{\mathfrak{a}}}})$. Claim 3 shows that the composition $Q\circ{\mathbf{R}\Gamma_{\mathfrak{a}{\widehat{R}^{{\mathfrak{a}}}}}}\circ{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}}$ is isomorphic to the identity on ${{\mathcal{D}}}(R)_{\text{${{\mathfrak{a}}}$-fin}}$. This establishes part of the theorem, and part follows.
For part , assume that there is a complex $N\in{{{\mathcal{D}}}_{\text{b}}^{\text{f}}}(R)$ such that $X\simeq{\mathbf{R}\Gamma_{{\mathfrak{a}}}(N)}$. Then the complex $N':={\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(N)}\simeq{{\widehat{R}^{{\mathfrak{a}}}}\otimes^{\mathbf{L}}_{R}N}\in{{{\mathcal{D}}}_{\text{b}}^{\text{f}}}({\widehat{R}^{{\mathfrak{a}}}})$ satisfies $$\begin{aligned}
Q({\mathbf{R}\Gamma_{\mathfrak{a}{\widehat{R}^{{\mathfrak{a}}}}}(N')})
&\simeq Q({\mathbf{R}\Gamma_{\mathfrak{a}{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(N)})})\\
&\simeq{\mathbf{R}\Gamma_{{\mathfrak{a}}}(Q({\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(N)}))}\\
&\simeq{\mathbf{R}\Gamma_{{\mathfrak{a}}}({\mathbf{L}\Lambda^{{\mathfrak{a}}}(N)})}\\
&\simeq{\mathbf{R}\Gamma_{{\mathfrak{a}}}(N)}\\
&\simeq X;\end{aligned}$$ see Fact \[fact130619b\]. So, we have $X\simeq Q({\mathbf{R}\Gamma_{\mathfrak{a}{\widehat{R}^{{\mathfrak{a}}}}}(N')})\in{{\mathcal{D}}}(R)_{\text{${{\mathfrak{a}}}$-fin}}$ by part .
The next example shows that the converse of Theorem \[thm150626a\] fails in general. Thus, the characterization in Theorem \[thm150626a\] cannot be simplified (at least not in the naive manner suggested by Theorem \[thm150626a\]).
\[ex150628a\] Let $(R,{{\mathfrak{m}}},k)$ be a local ring that does not admit a dualizing complex. Such a ring exists by work of Ogoma [@ogoma]. Set ${\widehat{R}}:={\widehat{R}^{{\mathfrak{m}}}}$. The injective hull $E:=E_R(k)$ is ${{\mathfrak{m}}}$-adically finite by [@sather:scc Proposition 7.8(b)]. Suppose that there is an $R$-complex $N\in{{{\mathcal{D}}}_{\text{b}}^{\text{f}}}(R)$ such that $E\simeq{\mathbf{R}\Gamma_{{\mathfrak{m}}}(N)}$. In [@sather:asc Example 6.7] we show that this implies that $N$ is dualizing for $R$, contradicting our assumption on $R$.
Induced Isomorphisms {#sec151104b}
====================
This section consists of useful isomorphisms derived from our preceding results. We begin with extended versions of Lemma \[prop151104a\].
\[prop151105a\] Let $X\in{{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})$ be such that ${\operatorname{supp}}_{{\widehat{R}^{{\mathfrak{a}}}}}(X)\subseteq{\operatorname{V}}({{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}})$. Given an $R$-complex $M\in{{\mathcal{D}}}(R)$, there are isomorphisms in ${{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})$ $$\begin{gathered}
{X\otimes^{\mathbf{L}}_{{\widehat{R}^{{\mathfrak{a}}}}}{\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(M)}}\simeq{X\otimes^{\mathbf{L}}_{R}M}\simeq
{X\otimes^{\mathbf{L}}_{{\widehat{R}^{{\mathfrak{a}}}}}({{\widehat{R}^{{\mathfrak{a}}}}\otimes^{\mathbf{L}}_{R}M})}
\\
{\mathbf{R}\!\operatorname{Hom}_{{\widehat{R}^{{\mathfrak{a}}}}}(X,{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(M)})}\simeq{\mathbf{R}\!\operatorname{Hom}_{R}(X,M)}\simeq{\mathbf{R}\!\operatorname{Hom}_{{\widehat{R}^{{\mathfrak{a}}}}}(X,{\mathbf{R}\!\operatorname{Hom}_{R}({\widehat{R}^{{\mathfrak{a}}}},M)})}.\end{gathered}$$
We verify the first two isomorphisms; the others are verified similarly. The first isomorphism in ${{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})$ in the following sequence is from Fact \[fact130619b\]. $$\begin{aligned}
{X\otimes^{\mathbf{L}}_{{\widehat{R}^{{\mathfrak{a}}}}}{\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(M)}}
&\simeq{X\otimes^{\mathbf{L}}_{{\widehat{R}^{{\mathfrak{a}}}}}({{\widehat{R}^{{\mathfrak{a}}}}\otimes^{\mathbf{L}}_{R}{\mathbf{R}\Gamma_{{\mathfrak{a}}}(M)}})}
\\
&\simeq{X\otimes^{\mathbf{L}}_{R}{\mathbf{R}\Gamma_{{\mathfrak{a}}}(M)}}
\\
&\simeq{X\otimes^{\mathbf{L}}_{R}M} \\
&\simeq{X\otimes^{\mathbf{L}}_{{\widehat{R}^{{\mathfrak{a}}}}}({{\widehat{R}^{{\mathfrak{a}}}}\otimes^{\mathbf{L}}_{R}M})}\end{aligned}$$ The second and fourth ones are tensor-cancellation. The third one is the natural one ${X\otimes^{\mathbf{L}}_{R}{\mathbf{R}\Gamma_{{\mathfrak{a}}}(M)}}{\xrightarrow}{{X\otimes^{\mathbf{L}}_{R}{\varepsilon_{{\mathfrak{a}}}^{M}}}}{X\otimes^{\mathbf{L}}_{R}M}$; this is an isomorphism in ${{\mathcal{D}}}(R)$ by Lemma \[prop151104a\], and it respects the ${\widehat{R}^{{\mathfrak{a}}}}$-structure coming from the left.
\[cor151105a\] Let ${\widehat{K}}$ be the Koszul complex over ${\widehat{R}^{{\mathfrak{a}}}}$ on the generating sequence ${\underline{x}}$ for ${{\mathfrak{a}}}$. Given an $R$-complex $M\in{{\mathcal{D}}}(R)$, there are isomorphisms in ${{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})$ $$\begin{gathered}
{{\widehat{K}}\otimes^{\mathbf{L}}_{{\widehat{R}^{{\mathfrak{a}}}}}{\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(M)}}\simeq
{{\widehat{K}}\otimes^{\mathbf{L}}_{{\widehat{R}^{{\mathfrak{a}}}}}({{\widehat{R}^{{\mathfrak{a}}}}\otimes^{\mathbf{L}}_{R}M})}\simeq{{\widehat{K}}\otimes^{\mathbf{L}}_{{\widehat{R}^{{\mathfrak{a}}}}}{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(M)}}
\\
{\mathbf{R}\!\operatorname{Hom}_{{\widehat{R}^{{\mathfrak{a}}}}}({{\widehat{K}}},{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(M)})}\!\simeq{\mathbf{R}\!\operatorname{Hom}_{{\widehat{R}^{{\mathfrak{a}}}}}({{\widehat{K}}},{\mathbf{R}\!\operatorname{Hom}_{R}({\widehat{R}^{{\mathfrak{a}}}},M)})}\simeq{\mathbf{R}\!\operatorname{Hom}_{{\widehat{R}^{{\mathfrak{a}}}}}({{\widehat{K}}},{\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(M)})}.\end{gathered}$$
The next result is Theorem \[thm151129c\] from the introduction. Note that it is straightforward when $X\in{{{\mathcal{D}}}_{\text{b}}^{\text{f}}}(R)$, by Fact \[fact130619b\]. See [@sather:asc Theorems 5.6 and 5.7] for applications.
\[thm151011a\] Let $R\to S$ be a homomorphism of commutative noetherian rings, and let $X\in{{{\mathcal{D}}}_{\text{b}}}(R)$ be ${{\mathfrak{a}}}$-adically finite over $R$. If ${S\otimes^{\mathbf{L}}_{R}X}\in{{{\mathcal{D}}}_{\text{b}}}(S)$, e.g., if ${\operatorname{fd}}_R(S)<\infty$, then there is an isomorphism in ${{\mathcal{D}}}({{\widehat{S}}^{{{\mathfrak{a}}}S}})$: $${{{\widehat{S}}^{{{\mathfrak{a}}}S}}\otimes^{\mathbf{L}}_{{\widehat{R}^{{\mathfrak{a}}}}}{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)}}\simeq\mathbf{L}\widehat\Lambda^{{{\mathfrak{a}}}S}({S\otimes^{\mathbf{L}}_{R}X}).$$
Our finiteness assumption on $X$ implies that $X\in{{{\mathcal{D}}}_{\text{b}}}(R)$ and ${\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)}\in{{{\mathcal{D}}}_{\text{b}}^{\text{f}}}({\widehat{R}^{{\mathfrak{a}}}})$. Thus, we have ${{{\widehat{S}}^{{{\mathfrak{a}}}S}}\otimes^{\mathbf{L}}_{{\widehat{R}^{{\mathfrak{a}}}}}{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)}}\in{{{\mathcal{D}}}^{\text{f}}}_+({{\widehat{S}}^{{{\mathfrak{a}}}S}})$. Also, from [@sather:afcc Theorem 5.10], we have $\mathbf{L}\widehat\Lambda^{{{\mathfrak{a}}}S}({S\otimes^{\mathbf{L}}_{R}X})\in{{{\mathcal{D}}}_{\text{b}}^{\text{f}}}({{\widehat{S}}^{{{\mathfrak{a}}}S}})$. For clarity, we set $K^R:=K=K^R({\underline{x}})$, where ${\underline{x}}$ is a finite generating sequence for ${{\mathfrak{a}}}$, and set $K^{{\widehat{R}^{{\mathfrak{a}}}}}:=K^{{\widehat{R}^{{\mathfrak{a}}}}}({\underline{x}})$, and similarly for $K^S$ and $K^{{{\widehat{S}}^{{{\mathfrak{a}}}S}}}$.
Claim 1: there is an isomorphism ${K^{{\widehat{R}^{{\mathfrak{a}}}}}\otimes^{\mathbf{L}}_{{\widehat{R}^{{\mathfrak{a}}}}}{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)}}\simeq{{\widehat{R}^{{\mathfrak{a}}}}\otimes^{\mathbf{L}}_{R}({K^R\otimes^{\mathbf{L}}_{R}X})}$ in ${{\mathcal{D}}}({\widehat{R}^{{\mathfrak{a}}}})$. This follows from the next sequence of isomorphisms: $$\begin{aligned}
{K^{{\widehat{R}^{{\mathfrak{a}}}}}\otimes^{\mathbf{L}}_{{\widehat{R}^{{\mathfrak{a}}}}}{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)}}
&\simeq {K^{{\widehat{R}^{{\mathfrak{a}}}}}\otimes^{\mathbf{L}}_{{\widehat{R}^{{\mathfrak{a}}}}}({{\widehat{R}^{{\mathfrak{a}}}}\otimes^{\mathbf{L}}_{R}X})}\\
&\simeq {({{\widehat{R}^{{\mathfrak{a}}}}\otimes^{\mathbf{L}}_{R}K^R})\otimes^{\mathbf{L}}_{{\widehat{R}^{{\mathfrak{a}}}}}({{\widehat{R}^{{\mathfrak{a}}}}\otimes^{\mathbf{L}}_{R}X})}\\
&\simeq{{\widehat{R}^{{\mathfrak{a}}}}\otimes^{\mathbf{L}}_{R}({K^R\otimes^{\mathbf{L}}_{R}X})}.\end{aligned}$$ The first isomorphism is from Corollary \[cor151105a\], and the others are routine.
Claim 2: we have ${{{\widehat{S}}^{{{\mathfrak{a}}}S}}\otimes^{\mathbf{L}}_{{\widehat{R}^{{\mathfrak{a}}}}}{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)}}\in{{{\mathcal{D}}}_{\text{b}}^{\text{f}}}({{\widehat{S}}^{{{\mathfrak{a}}}S}})$. To this end, recall that the first paragraph of this proof shows that we have ${{{\widehat{S}}^{{{\mathfrak{a}}}S}}\otimes^{\mathbf{L}}_{{\widehat{R}^{{\mathfrak{a}}}}}{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)}}\in{{{\mathcal{D}}}^{\text{f}}}_+({{\widehat{S}}^{{{\mathfrak{a}}}S}})$. Thus, we need only show that ${{{\widehat{S}}^{{{\mathfrak{a}}}S}}\otimes^{\mathbf{L}}_{{\widehat{R}^{{\mathfrak{a}}}}}{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)}}\in{{{\mathcal{D}}}_{\text{b}}}({{\widehat{S}}^{{{\mathfrak{a}}}S}})$. For this, note first that every maximal ideal of ${{\widehat{S}}^{{{\mathfrak{a}}}S}}$ contains ${{\mathfrak{a}}}{{\widehat{S}}^{{{\mathfrak{a}}}S}}$. In other words, we have ${\operatorname{supp}}_{{{\widehat{S}}^{{{\mathfrak{a}}}S}}}(K^{{{\widehat{S}}^{{{\mathfrak{a}}}S}}})={\operatorname{V}}({{\mathfrak{a}}}{{\widehat{S}}^{{{\mathfrak{a}}}S}})\supseteq{\operatorname{m-Spec}}({{\widehat{S}}^{{{\mathfrak{a}}}S}})$. Thus, according to [@frankild:rrhffd Theorem 4.2(b)], to establish the claim, it suffices to show that we have ${K^{{{\widehat{S}}^{{{\mathfrak{a}}}S}}}\otimes^{\mathbf{L}}_{{{\widehat{S}}^{{{\mathfrak{a}}}S}}}({{{\widehat{S}}^{{{\mathfrak{a}}}S}}\otimes^{\mathbf{L}}_{{\widehat{R}^{{\mathfrak{a}}}}}{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)}})}\in{{{\mathcal{D}}}_{\text{b}}^{\text{f}}}({{\widehat{S}}^{{{\mathfrak{a}}}S}})$. To this end, we consider the following sequence of isomorphisms in ${{\mathcal{D}}}({{\widehat{S}}^{{{\mathfrak{a}}}S}})$. $$\begin{aligned}
{K^{{{\widehat{S}}^{{{\mathfrak{a}}}S}}}\otimes^{\mathbf{L}}_{{{\widehat{S}}^{{{\mathfrak{a}}}S}}}({{{\widehat{S}}^{{{\mathfrak{a}}}S}}\otimes^{\mathbf{L}}_{{\widehat{R}^{{\mathfrak{a}}}}}{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)}})}
&\simeq{({{{\widehat{S}}^{{{\mathfrak{a}}}S}}\otimes^{\mathbf{L}}_{{\widehat{R}^{{\mathfrak{a}}}}}K}^{{\widehat{R}^{{\mathfrak{a}}}}})\otimes^{\mathbf{L}}_{{{\widehat{S}}^{{{\mathfrak{a}}}S}}}({{{\widehat{S}}^{{{\mathfrak{a}}}S}}\otimes^{\mathbf{L}}_{{\widehat{R}^{{\mathfrak{a}}}}}{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)}})}
\\
&\simeq
{{{\widehat{S}}^{{{\mathfrak{a}}}S}}\otimes^{\mathbf{L}}_{{\widehat{R}^{{\mathfrak{a}}}}}({K^{{\widehat{R}^{{\mathfrak{a}}}}}\otimes^{\mathbf{L}}_{{\widehat{R}^{{\mathfrak{a}}}}}{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)}})}
\\
&\simeq
{{{\widehat{S}}^{{{\mathfrak{a}}}S}}\otimes^{\mathbf{L}}_{{\widehat{R}^{{\mathfrak{a}}}}}({{\widehat{R}^{{\mathfrak{a}}}}\otimes^{\mathbf{L}}_{R}({K^R\otimes^{\mathbf{L}}_{R}X})})}
\\
&\simeq
{{{\widehat{S}}^{{{\mathfrak{a}}}S}}\otimes^{\mathbf{L}}_{S}({S\otimes^{\mathbf{L}}_{R}({K^R\otimes^{\mathbf{L}}_{R}X})})}
\\
&\simeq
{{{\widehat{S}}^{{{\mathfrak{a}}}S}}\otimes^{\mathbf{L}}_{S}({({S\otimes^{\mathbf{L}}_{R}K^R})\otimes^{\mathbf{L}}_{S}({S\otimes^{\mathbf{L}}_{R}X})})}
\\
&\simeq
{{{\widehat{S}}^{{{\mathfrak{a}}}S}}\otimes^{\mathbf{L}}_{S}({K^S\otimes^{\mathbf{L}}_{S}({S\otimes^{\mathbf{L}}_{R}X})})}\end{aligned}$$ The third isomorphism is from Claim 1, and the others are standard. Since we have ${S\otimes^{\mathbf{L}}_{R}X}\in{{{\mathcal{D}}}_{\text{b}}}(S)$, by assumption, the condition ${\operatorname{pd}}_S(K^S)<\infty$ implies that ${K^S\otimes^{\mathbf{L}}_{S}({S\otimes^{\mathbf{L}}_{R}X})}\in{{{\mathcal{D}}}_{\text{b}}}(S)$. Thus the flatness of ${{\widehat{S}}^{{{\mathfrak{a}}}S}}$ over $S$ implies that we have $${K^{{{\widehat{S}}^{{{\mathfrak{a}}}S}}}\otimes^{\mathbf{L}}_{{{\widehat{S}}^{{{\mathfrak{a}}}S}}}({{{\widehat{S}}^{{{\mathfrak{a}}}S}}\otimes^{\mathbf{L}}_{{\widehat{R}^{{\mathfrak{a}}}}}{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)}})}
\simeq{{{\widehat{S}}^{{{\mathfrak{a}}}S}}\otimes^{\mathbf{L}}_{S}({K^S\otimes^{\mathbf{L}}_{S}({S\otimes^{\mathbf{L}}_{R}X})})}\in{{{\mathcal{D}}}_{\text{b}}}({{\widehat{S}}^{{{\mathfrak{a}}}S}}).$$ This establishes Claim 2.
Since the complexes $\mathbf{L}\widehat\Lambda^{{{\mathfrak{a}}}S}({S\otimes^{\mathbf{L}}_{R}X})$ and ${{{\widehat{S}}^{{{\mathfrak{a}}}S}}\otimes^{\mathbf{L}}_{{\widehat{R}^{{\mathfrak{a}}}}}{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)}}$ are in ${{{\mathcal{D}}}_{\text{b}}^{\text{f}}}({{\widehat{S}}^{{{\mathfrak{a}}}S}})$, to show that they are isomorphic, Theorem \[thm150626a\] says that it suffices to show that $\mathbf{R}\Gamma_{{{\mathfrak{a}}}{{\widehat{S}}^{{{\mathfrak{a}}}S}}}(\mathbf{L}\widehat\Lambda^{{{\mathfrak{a}}}S}({S\otimes^{\mathbf{L}}_{R}X}))
\simeq\mathbf{R}\Gamma_{{{\mathfrak{a}}}{{\widehat{S}}^{{{\mathfrak{a}}}S}}}({{{\widehat{S}}^{{{\mathfrak{a}}}S}}\otimes^{\mathbf{L}}_{{\widehat{R}^{{\mathfrak{a}}}}}{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)}})$ in ${{\mathcal{D}}}({{\widehat{S}}^{{{\mathfrak{a}}}S}})$. To verify this isomorphism, we compute as follows. $$\begin{aligned}
\mathbf{R}\Gamma_{{{\mathfrak{a}}}{{\widehat{S}}^{{{\mathfrak{a}}}S}}}(\mathbf{L}\widehat\Lambda^{{{\mathfrak{a}}}S}({S\otimes^{\mathbf{L}}_{R}X}))
&\simeq \mathbf{R}\widehat\Gamma_{{{\mathfrak{a}}}S}({S\otimes^{\mathbf{L}}_{R}X})
\\
&\simeq {{{\widehat{S}}^{{{\mathfrak{a}}}S}}\otimes^{\mathbf{L}}_{S}\mathbf{R}\Gamma_{{{\mathfrak{a}}}S}({S\otimes^{\mathbf{L}}_{R}X})}
\\
&\simeq {{{\widehat{S}}^{{{\mathfrak{a}}}S}}\otimes^{\mathbf{L}}_{S}({S\otimes^{\mathbf{L}}_{R}X})}
\\
&\simeq {{{\widehat{S}}^{{{\mathfrak{a}}}S}}\otimes^{\mathbf{L}}_{R}X}\end{aligned}$$ The first isomorphism is by Lemmas \[lem150805b\] and \[lem150805c\] The second isomorphism is from Fact \[fact130619b\]. For the third isomorphism, note that [@sather:afcc Lemma 5.7] shows that ${\operatorname{supp}}_S({S\otimes^{\mathbf{L}}_{R}X})\subseteq{\operatorname{V}}({{\mathfrak{a}}}S)$, so Fact \[cor130528a\] implies that $\mathbf{R}\Gamma_{{{\mathfrak{a}}}S}({S\otimes^{\mathbf{L}}_{R}X})\simeq{S\otimes^{\mathbf{L}}_{R}X}$ in ${{\mathcal{D}}}(S)$. The last isomorphism is tensor-cancellation. The next isomorphisms are justified similarly. $$\begin{aligned}
\mathbf{R}\Gamma_{{{\mathfrak{a}}}{{\widehat{S}}^{{{\mathfrak{a}}}S}}}({{{\widehat{S}}^{{{\mathfrak{a}}}S}}\otimes^{\mathbf{L}}_{{\widehat{R}^{{\mathfrak{a}}}}}{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)}})
&\simeq{\mathbf{R}\Gamma_{{{\mathfrak{a}}}{{\widehat{S}}^{{{\mathfrak{a}}}S}}}({{\widehat{S}}^{{{\mathfrak{a}}}S}})\otimes^{\mathbf{L}}_{{{\widehat{S}}^{{{\mathfrak{a}}}S}}}({{{\widehat{S}}^{{{\mathfrak{a}}}S}}\otimes^{\mathbf{L}}_{{\widehat{R}^{{\mathfrak{a}}}}}{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)}})}
\\
&\simeq{({{{\widehat{S}}^{{{\mathfrak{a}}}S}}\otimes^{\mathbf{L}}_{{\widehat{R}^{{\mathfrak{a}}}}}\mathbf{R}\Gamma_{{{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}}}({\widehat{R}^{{\mathfrak{a}}}})})\otimes^{\mathbf{L}}_{{{\widehat{S}}^{{{\mathfrak{a}}}S}}}({{{\widehat{S}}^{{{\mathfrak{a}}}S}}\otimes^{\mathbf{L}}_{{\widehat{R}^{{\mathfrak{a}}}}}{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)}})}
\\
&\simeq{{{\widehat{S}}^{{{\mathfrak{a}}}S}}\otimes^{\mathbf{L}}_{{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{R}\Gamma_{{{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}}}({\widehat{R}^{{\mathfrak{a}}}})\otimes^{\mathbf{L}}_{{\widehat{R}^{{\mathfrak{a}}}}}{\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)}})}
\\
&\simeq{{{\widehat{S}}^{{{\mathfrak{a}}}S}}\otimes^{\mathbf{L}}_{{\widehat{R}^{{\mathfrak{a}}}}}\mathbf{R}\Gamma_{{{\mathfrak{a}}}{\widehat{R}^{{\mathfrak{a}}}}}({\mathbf{L}\widehat\Lambda^{{\mathfrak{a}}}(X)})}
\\
&\simeq{{{\widehat{S}}^{{{\mathfrak{a}}}S}}\otimes^{\mathbf{L}}_{{\widehat{R}^{{\mathfrak{a}}}}}{\mathbf{R}\widehat\Gamma_{{\mathfrak{a}}}(X)}}
\\
&\simeq{{{\widehat{S}}^{{{\mathfrak{a}}}S}}\otimes^{\mathbf{L}}_{{\widehat{R}^{{\mathfrak{a}}}}}({{\widehat{R}^{{\mathfrak{a}}}}\otimes^{\mathbf{L}}_{R}{\mathbf{R}\Gamma_{{\mathfrak{a}}}(X)}})}
\\
&\simeq{{{\widehat{S}}^{{{\mathfrak{a}}}S}}\otimes^{\mathbf{L}}_{{\widehat{R}^{{\mathfrak{a}}}}}({{\widehat{R}^{{\mathfrak{a}}}}\otimes^{\mathbf{L}}_{R}X})}
\\
&\simeq{{{\widehat{S}}^{{{\mathfrak{a}}}S}}\otimes^{\mathbf{L}}_{R}X}\end{aligned}$$ These two sequences give the desired isomorphism, completing the proof.
Acknowledgments {#acknowledgments .unnumbered}
===============
We are grateful to Srikanth Iyengar, Liran Shaul, and Amnon Yekutieli for helpful comments about this work.
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[to3em]{}, *Des catégories dérivées des catégories abéliennes*, Astérisque (1996), no. 239, xii+253 pp. (1997), With a preface by Luc Illusie, Edited and with a note by Georges Maltsiniotis. [MR ]{}[98c:18007]{}
A. Yekutieli, *On flatness and completion for infinitely generated modules over [N]{}oetherian rings*, Comm. Algebra **39** (2011), no. 11, 4221–4245. [MR ]{}[2855123 (2012k:13058)]{}
[to3em]{}, *A separated cohomologically complete module is complete*, Comm. Algebra **43** (2015), no. 2, 616–622. [MR ]{}[3274025]{}
[^1]: Sean Sather-Wagstaff was supported in part by a grant from the NSA
[^2]: In the literature, semi-flat complexes are sometimes called “K-flat” or “DG-flat”.
[^3]: This is based on the fact that, for a finitely generated free $R$-module $L$, induction on the rank of $L$ shows that the natural isomorphism ${{\widehat{R}^{{\mathfrak{a}}}}\otimes_{R}L}\cong{\widehat{L}^{{\mathfrak{a}}}}$ is ${\widehat{R}^{{\mathfrak{a}}}}$-linear.
[^4]: See \[foot151004a\] also [@yekutieli:hct Thoerem 6.12]. In addition, we have [@yekutieli:hct Remark 6.14] for a discussion of some aspects of this result, and [@yekutieli:ehct] for a correction.
[^5]: The affiliated characterization of ${{{\mathcal{D}}}(R)_{\text{${{\mathfrak{a}}}$-comp}}}$ in terms of co-support is not needed here.
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{
"pile_set_name": "ArXiv"
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---
abstract: 'The physical mechanism responsible for the short outbursts in a recently recognized class of High Mass X-ray Binaries, the Supergiant Fast X-ray Transients (SFXTs), is still unknown. Recent observations performed with $Swift/XRT$, and of the 2007 outburst from IGRJ11215$-$5952, the only SFXT known to exhibit periodic outbursts, suggest a new explanation for the outburst mechanism in this class of transients; the outbursts could be linked to the possible presence of a second wind component in the supergiant companion, in the form of an equatorial wind. The applicability of the model to the short outburst durations of all other Supergiant Fast X-ray Transients, where a clear periodicity in the outbursts has not been found yet, is discussed. The scenario we are proposing also includes the persistently accreting supergiant High Mass X–ray Binaries.'
author:
- Lara Sidoli
- Patrizia Romano
- 'Sandro Mereghetti, Ada Paizis, Stefano Vercellone'
- Vanessa Mangano
- Diego Götz
title: 'A new explanation for the Supergiant Fast X–ray Transients outbursts'
---
Introduction: the SFXTs properties
==================================
Supergiant Fast X–ray Transients (hereafter SFXTs; Smith et al. 2006a) are a new class of hard X–ray sources mostly discovered by the $INTEGRAL$ satellite (Negueruela et al. 2005a, Sguera et al. 2005). They are transient sources which seem to emit X–rays only during “short” outbursts (few hours, as observed with $INTEGRAL$ or $RXTE$) and their optical counterparts are blue supergiant stars. Their X–ray spectra are similar to those of accreting pulsars, thus it is likely that they are High Mass X–ray Binaries (HMXBs) hosting neutron stars. In two SFXTs (among about twenty sources, comprising candidate SFXTs) X–ray pulsations have been indeed observed (IGR J18410-0535/AX J1841.0-0536, P$_{\rm spin}$=4.74 s, Bamba et al. 2001; IGR J11215–5952, P$_{\rm spin}=186.78\pm0.3$s, Smith et al. 2006b, Swank et al. 2007).
SFXTs reach X–ray luminosities up to a few 10$^{36}$ erg s$^{-1}$, while the quiescent level ($\sim$10$^{32}$ erg s$^{-1}$) has been observed only in IGR J17391$-$3021/XTE J1739$-$302 (Smith et al. 2006a; Sakano et al. 2002), IGR J17544$-$2619 (in’t Zand 2005) and probably IGR J18410$-$0535/AX J1841.0$-$0536 (Halpern et al. 2004; Bamba et al. 2001).
It is important to note that none of the SFXT sources has ever been caught to undergo a transition from quiescence to outburst and then back to quiescence in a few hours. The quiescent emission had always been observed well far away from the outbursts, except in one case: only in’t Zand (2005) did observe the transition from quiescence to outburst with $Chandra$ (in IGR J17544$-$2619), but the observation finished before the start of the declining phase to quiescence. Thus the real duration of this outburst could not be measured. The so-called “short” duration (a few hours) of the outbursts from SFXTs is indeed based on observations with instruments not sensitive enough to detect the quiescence level. The instruments onboard $RXTE$ and $INTEGRAL$ could only observe the brightest fast flaring activity (lasting a few hours, less than one day) reaching a few 10$^{36}$ erg s$^{-1}$.
Hence, the definition of SFXTs as transient sources displaying “short” X–ray outbursts lasting only a few hours is strongly biased. This has been observationally demonstrated by our recent deep campaign with $Swift/XRT$ (Romano et al. 2007, hereafter Paper II) of the outburst from the unique SFXT displaying “periodic outbursts”, IGR J11215–5952 (Sidoli et al. 2006, hereafter Paper I). These very sensitive observations showed that the accretion phase in SFXTs lasts longer than what previously thought: a few days instead of only hours.
With these new observations at hand, we report on an alternative model to explain the outbursts from this new class of sources, based on $Swift/XRT$ monitoring observations of IGR J11215–5952 during the last two outbursts (starting on February 9 and July 24, 2007; Sidoli et al. 2007, hereafter Paper III).
Swift/XRT observations of IGR J11215–5952
=========================================
is an X–ray transient discovered by during a fast outburst in April 2005 (Lubinski et al. 2005). The optical counterpart is a B1 supergiant, HD 306414, located at a distance of 6–8 kpc (Negueruela et al. 2005b, Masetti et al. 2006, Steeghs et al. 2006). From the analysis of observations of the source field, we discovered (Paper I) that the outbursts are equally spaced by $\sim$330 days (although a half of this period could not be excluded, due to a lack of observations). This periodicity was later confirmed in March 2006 (outburst after 329 days; Smith et al. 2006c) during a monitoring with $RXTE/PCA$, and was related in a natural way to the orbital period of the system, with the outbursts triggered at (or near to) the periastron passage (Paper I).
Based on this known periodicity, a new outburst was expected for 2007 February 9 and we planned a monitoring campaign with $Swift/XRT$, starting on 2007 February 4 (Romano et al. 2007b). A second monitoring campaign was performed with $Swift/XRT$ in July 2007, in order to monitor the quiescent level and the epoch of the supposed apastron passage (based on the 329 days period; Romano et al. 2007c). These observations led to the detection of a new unexpected outburst starting on 2007, July 24, which reached roughly the same flux as during the February 2007 outburst (Paper III). Details of the $Swift/XRT$ data analysis and spectral/timing results are reported in Paper II and Paper III. Here we concentrate on the shape of the X–ray lightcurve in order to understand the physical mechanism which produces the outbursts.
A new model for the outburst mechanism in SFXTs
===============================================
The IGR J11215–5952 lightcurve observed during the February 2007 outburst represents the most complete set of observations of a SFXT outburst (Fig. \[fig:ecc\], black curve). The first important result of these observations is that the whole outburst phase lasts longer than what previously thought, based on less sensitive instruments: a few days, instead of a few hours. Only the brightest part of the outburst is short (lasts less than 1 day) and would have been seen by the INTEGRAL instruments. Intense flaring activity is also present, both during the bright peak and the declining phase of the outburst, with each single flare lasting minutes or a few hours.
It is natural to associate the clock responsible for the outbursts with the orbital periodicity of the binary system. Since displayed a new outburst after about a half of the 329 days period (Paper III; Romano et al. 2007c), it is possible that 164.5 days is indeed the real orbital period which escaped detection up to now. In both cases (P$_{\rm orb}$=329 days or 164.5 days), the system is a wide binary where the blue supergiant does not fill its Roche lobe, and the system is very likely wind-fed. Applying the Bondi-Hoyle wind accretion scenario, where the neutron star accretes from the wind of the supergiant at different rates depending on the wind density and relative velocity along the orbit, and assuming reasonable parameters for the B-supergiant, we obtain that the observed X–ray lightcurve is always too narrow and steep to be explained with accretion from a spherically symmetric wind, even adopting extreme eccentricities for the binary system (see Fig. \[fig:ecc\]).
This result led us to suggest that the wind from the B supergiant is not spherically symmetric. The alternative viable explanation we propose for the sharpness of the observed X–ray lightcurve is that in the supergiant wind has a second component (besides the polar spherically symmetric one), in the form of an “equatorial disk”, inclined with respect to the orbital plane (see Fig. \[fig:geom\] for an artistic view of the geometry of the system). The short outburst is then produced when the neutron star crosses this equatorial wind component, denser and slower than the polar one. Deviations from spherical symmetry in hot massive star winds are also suggested by optical observations (e.g. Prinja 1990, Prinja et al. 2002) and the presence of equatorial disk components, denser and slower with respect to the polar wind, also results from simulations (ud-Doula et al. 2006).
The thickness [*h*]{} of the densest part of this supergiant equatorial wind can be calculated from the duration of the brightest part of the outburst (which lasts less than 1 day, time needed for the neutron star to cross it) and the neutron star velocity, 100–200 km s$^{-1}$: [*h*]{}$\sim$(0.8–1.7)$\times$sin$(\theta)$$\times$10$^{12}$ cm, where $\theta$ is the inclination angle of the equatorial wind with respect to the orbital plane (with $\theta$=90$\degmark$ if the disk is perpendicular to the orbital plane).
The model we are proposing can explain also the short flares from all the other SFXTs where a clear periodicity in the outbursts recurrence has not been found yet, if a different geometry of the equatorial wind component with respect to the orbital plane is assumed: in , where the outbursts are equally spaced and occur with a fixed periodicity, the inclined equatorial disk wind component should intersect the neutron star at the periastron (or very close to it, see the left panel in Fig. \[fig:geom\]) and can intersect the neutron star orbit once or twice depending both on the extension of the wind disk and on the orbital eccentricity. Instead, it is possible that in the other SFXTs the inclined disk wind intersects twice a wide and highly eccentric orbit, not at the periastron (see the right panel in Fig. \[fig:geom\]), leading to a double periodicity (one shorter than the other) which has not been found yet [*only*]{} because of a lack of a continuous monitoring. This model can also explain the X–ray emission from the persistently accreting HMXBs, if we admit that in this case the neutron star is always moving inside the equatorial wind component which lies on the orbital plane.
In this framework, the sharp X–ray lightcurve observed from can be modelled with different wind parameters (for both polar and equatorial components) depending on the orbital period (164.5 days or 329 days) and the eccentricity of the binary. We assume a blue supergiant with a mass of 39 M$_\odot$ and radius of 42 R$_\odot$, and a polar wind component with a terminal velocity of 1800 km s$^{-1}$. The X–ray lightcurve observed with $Swift/XRT$ is better reproduced assuming a “polar wind” mass loss rate of 5$\times$10$^{-6}$ M$_\odot$ yrs$^{-1}$ (for a P$_{\rm orb}$ of 164.5 days and an eccentricity of 0.4) and 9$\times$10$^{-7}$ M$_\odot$ yrs$^{-1}$ (for a P$_{\rm orb}$ of 329 days and a circular orbit, which is required by the fact that the two consecutive outbursts from reached roughly the same peak flux). The equatorial wind component should have a variable velocity ranging from 750 km s$^{-1}$ to 1400 km s$^{-1}$ (for P$_{\rm orb}$=164.5 days), and from 850 km s$^{-1}$ to 1600 km s$^{-1}$ (for P$_{\rm orb}$=329 days), and a density about 100 times higher than the polar wind component.
Note however that since the X–ray luminosity expected for the wind accretion is proportional to $\mdot$$v_{rel}$$^{-4}$ (where $\mdot$ is the wind mass loss rate, and $v_{rel}$ is the relative velocity of the wind with respect to the neutron star), different combinations of wind density and velocity in the equatorial component can reproduce the X–ray lightcurve as well.
In conclusion, in our model we explain the short recurrent flares if the neutron star intersects an inclined equatorial wind component (once or twice) during its orbit. A different particular geometry and inclination of this equatorial wind with respect to the orbital plane can account for the whole phoenomenology of both SFXTs and persistently accreting HMXBs in general. Both the orbital eccentricity and no-coplanarity can be explained by a substantial supernova kick at birth. This could indicate that SFXTs are likely young systems, probably younger than persistent HMXBs.
Based on observations obtained with XMM-Newton, an ESA science mission with instruments and contributions directly funded by ESA member states and the USA (NASA). Based on observations with INTEGRAL, an ESA project with instruments and the science data centre funded by ESA member states (especially the PI countries: Denmark, France, Germany, Italy, Switzerland, Spain), Czech Republic and Poland, and with the participation of Russia and the USA. We thank the , , and $Swift$ teams for making these observations possible, in particular the duty scientists and science planners. PR thanks INAF-IASFMi for their kind hospitality. DG acknowledges the French Space Agency (CNES) for financial support. This work was supported by contract ASI/INAF I/023/05/0.
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{
"pile_set_name": "ArXiv"
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---
author:
- 'Bo’az Klartag^1^ and Emanuel Milman^2^'
title: |
Centroid Bodies and the Logarithmic Laplace Transform -\
A Unified Approach
---
Introduction
============
This work combines two recent techniques in the study of volumes of high-dimensional convex bodies. The first technique is due to G. Paouris [@Paouris-IsotropicTail], and it relies on properties of the $L_p$-centroid bodies. The second technique was developed by the first named author [@quarter], and it uses the logarithmic Laplace transform.
Suppose that $\mu$ is a Borel probability measure on $\RR^n$ endowed with a Euclidean structure ${\left\vert\cdot\right\vert} = \sqrt{{\left \langle \cdot,\cdot \right \rangle}}$. We say that $\mu$ is a $\psi_\alpha$-measure ($\alpha > 0$) with constant $b_{\alpha}$ if: $$\label{eq:psi-alpha}
{\left(\int_{{\mathbb{R}}^n} {\left\vert{\left \langle x,\theta \right \rangle}\right\vert}^p d\mu(x)\right)}^{\frac{1}{p}} \leq b_\alpha p^{\frac{1}{\alpha}} {\left(\int_{{\mathbb{R}}^n} {\left\vert{\left \langle x,\theta \right \rangle}\right\vert}^2 d\mu(x)\right)}^{\frac{1}{2}} \;\;\; \forall p \geq 2 \;\;\; \forall \theta \in {\mathbb{R}}^n ~.$$ It is well-known that the uniform probability measure $\mu_K$ on any convex body $K \subset \RR^n$ is a $\psi_1$-measure with constant $C$, where $C
> 0$ is a universal constant (this follows from Berwald’s inequality [@ber], see also [@MilmanPajor]). Here, as usual, a convex body in $\RR^n$ means a compact, convex set with a non-empty interior. The isotropic constant $L_K$ of a convex body $K \subset \RR^n$ is the following affine invariant parameter: $$L_K := {\textrm{Vol}_n}(K)^{-\frac{1}{n}} \left( {\textrm{det Cov}}(\mu_K) \right)^{\frac{1}{2n}} ~,$$ where ${\textrm{Vol}_n}$ denotes Lebesgue measure and ${\textrm{Cov}}(\mu_k)$ denotes the covariance matrix of $\mu_K$. The next theorem unifies and slightly improves several known bounds on the isotropic constant.
\[main\_thm\] Let $K \subset \RR^n$ denote a convex body whose barycenter lies at the origin, and suppose that $\mu_K$ is a $\psi_{\alpha}$-measure ($1 \leq \alpha \leq 2$) with constant $b_{\alpha}$. Then: $$L_K \leq C \sqrt{b_\alpha^\alpha n^{1-\alpha/2}} ~,$$ where $C > 0$ is a universal constant.
A central question raised by Bourgain [@bou_amer; @bou_cong] is whether $L_K \leq C$ for some universal constant $C>0$, for any convex body $K \subset \RR^n$ (it is well-known that $L_K \geq c$ for a universal constant $c > 0$). This question is usually referred to as the [*slicing problem*]{} or [*hyperplane conjecture*]{}, see Milman and Pajor [@MilmanPajor] for many of its equivalent formulations and for further background. Plugging $\alpha = 1$ in Theorem \[main\_thm\], we match the best known bound on the isotropic constant, which is $L_K \leq C n^{1/4}$ for any convex body $K \subset \RR^n$ (see Bourgain [@bou_L] and Klartag [@quarter]). In the case $\alpha = 2$, Theorem \[main\_thm\] yields $L_K \leq C b_2$. This slightly improves upon the previously known bound, which is: $$L_K \leq C b_2 \sqrt{\log b_2} ~, \label{eq_2307}$$ due to Dafnis and Paouris [@DP] in the precise form (\[eq\_2307\]) and to Bourgain [@bou_psi2] (with a different power of the logarithmic factor). Here, as elsewhere in this text, we use the letters $c, \tilde{c}, C, \tilde{C}, \bar{C}$ etc. to denote positive universal constants, whose value may not necessarily be the same in different occurrences.
We proceed by recalling the definition of the $L_p$-centroid bodies $Z_p(\mu)$, originally introduced by E. Lutwak and G. Zhang in [@LutwakZhang-IntroduceLqCentroidBodies] (under different normalization), which lie at the heart of Paouris’ remarkable work [@Paouris-IsotropicTail]. Given a Borel probability measure $\mu$ on $\RR^n$ and $p \geq 1$, denote: $$h_{Z_p(\mu)}(\theta) = \left( \int_{\RR^n} {\left\vert{\left \langle x,\theta \right \rangle}\right\vert}^p d\mu(x) \right)^{\frac{1}{p}} \quad , \quad \theta \in \RR^n ~ .$$ The function $h_{Z_p(\mu)}$ is a norm on $\RR^n$, and it is the supporting functional of a convex body $Z_p(\mu) \subseteq \RR^n$ (see e.g. Schneider [@Schneider-Book] for information on supporting functionals). Clearly $Z_p(\mu) \subseteq Z_q(\mu) $ for $p \leq q$.
Now suppose that $K \subset \RR^n$ is a convex body whose barycenter lies at the origin, and denote $Z_p(K) = Z_p(\mu_K)$, where $\mu_K$ is as before the uniform probability measure on $K$. As realized by Paouris, obtaining volumetric and other information on $Z_p(K)$ is very useful for understanding the volumetric properties of $K$ itself. For instance, note that: $$\label{eq:vr-z2}
{\textrm{V.Rad.}}(Z_2(K)) = {\textrm{det Cov}}(\mu_K)^{\frac{1}{2n}} ~,$$ where the volume-radius of a compact set $T \subset \RR^n$ is defined as: $${\textrm{V.Rad.}}(T) = \left( \frac{{\textrm{Vol}_n}(T)}{{\textrm{Vol}_n}(B_n)} \right)^{\frac{1}{n}} ~,$$ measuring the radius of the Euclidean ball whose volume equals the volume of $T$. Here, $B_n = \{ x \in \RR^n ; |x| \leq 1 \}$; note that $c n^{-\frac{1}{2}} \leq {\textrm{Vol}_n}(B_n)^{\frac{1}{n}} \leq C
n^{-\frac{1}{2}}$, as verified by direct calculation. Furthermore, it is known (e.g. [@PaourisSmallBall Lemma 3.6]) that: $$\label{eq:ZnK}
c \cdot Z_\infty(K) \subseteq Z_n(K) \subseteq Z_\infty(K) := conv(K, -K) ~,$$ where $conv(K, -K)$ denotes the convex hull of $K$ and $-K$.
A sharp *lower* bound on the volume of $Z_p(K)$ due to Lutwak, Yang and Zhang [@LYZ] states that ellipsoids minimize ${\textrm{V.Rad.}}(Z_p(K)) / {\textrm{V.Rad.}}(K)$ among all convex bodies $K$, for all $p \geq
1$. An elementary calculation yields: $$\label{eq:LYZ-bound}
{\textrm{V.Rad.}}( Z_p(K) ) \geq c \sqrt{\frac{p}{n}} {\textrm{V.Rad.}}(K) \quad \quad
\quad \text{for} \ 1 \leq p \leq n ~,$$ which is best possible (up to the value of the constant $c>0$) in terms of ${\textrm{Vol}_n}(K)$. However, in view of the slicing problem and (\[eq:vr-z2\]), one may try to strengthen (\[eq:LYZ-bound\]) by replacing its right-hand side by $c \sqrt{p} {\textrm{V.Rad.}}(Z_2(K))$. The next two theorems are a step in this direction. Before formulating the results, we first broaden our scope.
It was realized by K. Ball [@BallPhD; @Ball-kdim-sections] that many questions regarding the volume of convex bodies are better formulated in the broader class of logarithmically-concave measures. A function $\rho: \RR^n \rightarrow [0, \infty)$ is called log-concave if $ -\log \rho: \RR^n \rightarrow (-\infty, \infty] $ is a convex function. A probability measure on $\RR^n$ is log-concave if its density is log-concave. For example, the uniform probability measure on a convex body and its marginals are all log-concave measures (see Borell [@Borell-logconcave] for a characterization).
\[main\_thm2\] Let $\mu$ be a log-concave probability measure on ${\mathbb{R}}^n$ with barycenter at the origin. Let $1 \leq \alpha \leq 2$, and assume that $\mu$ is a $\psi_\alpha$-measure with constant $b_\alpha$. Then: $${\textrm{V.Rad.}}(Z_p(\mu)) \geq c \sqrt{p} {\textrm{V.Rad.}}(Z_2(\mu))~,$$ for all $2 \leq p \leq C n^{\alpha/2} / b_\alpha^\alpha$. Here $c,C >
0$ denote universal constants.
Theorem \[main\_thm\] follows immediately from Theorem \[main\_thm2\]. Indeed, simply observe that for $p$ in the specified range: $$c \sqrt{p} \leq \frac{{\textrm{V.Rad.}}(Z_p(K))}{{\textrm{V.Rad.}}(Z_2(K))} \leq
\frac{{\textrm{V.Rad.}}{(conv(K,-K))}}{{\textrm{V.Rad.}}(Z_2(K)} \leq C \sqrt{n}
\frac{{\textrm{Vol}_n}(K)^{1/n}}{{\textrm{V.Rad.}}(Z_2(K))} = \frac{C \sqrt{n}}{L_K} ~,$$ where the last inequality follows from the Rogers-Shephard inequality [@RS]. This completes the proof of Theorem \[main\_thm\], reducing it to that of Theorem \[main\_thm2\]. We remark here that the proof (of both theorems) only requires that the $\psi_\alpha$ condition (\[eq:psi-alpha\]) hold for $p \geq 2$ so that ${\textrm{diam}}(Z_p(\mu)) \leq c \sqrt{n}$, and only in an average sense (see Subsection \[subsec:generalizations\]).
Our next theorem contains an additional lower bound on the volume of $Z_p(\mu)$ which complements that of Theorem \[main\_thm2\] in some sense. A Borel probability measure $\mu$ on $(\RR^n,{\left\vert\cdot\right\vert})$ is called isotropic when its barycenter lies at the origin, and its covariance matrix equals the identity matrix (i.e. $Z_2(\mu) = B_n$). Any measure with finite second moments and full-dimensional support may be brought into isotropic “position" by means of an affine transformation.
\[main\_thm3\] Let $\mu$ be an *isotropic* log-concave probability measure on ${\mathbb{R}}^n$. Then: $${\textrm{V.Rad.}}(Z_p(\mu)) \geq c \sqrt{p} ~,$$ for all $p \geq 2$ for which: $$\label{eq_0034}
{\textrm{diam}}(Z_p(\mu)) \sqrt{\log p} \leq C \sqrt{n} ~.$$ Here, ${\textrm{diam}}(T) = \sup_{x, y \in T} |x-y|$ stands for the diameter of $T \subset \RR^n$, and $c,C > 0$ are universal constants.
Note that the $\psi_\alpha$-condition (\[eq:psi-alpha\]) is precisely the requirement that $Z_p(\mu) \subseteq b_\alpha
p^{\frac{1}{\alpha}} Z_2(\mu)$ for all $p \geq 2$, and so the conclusion of Theorem \[main\_thm3\] agrees with that of Theorem \[main\_thm2\], up to the logarithmic factor in (\[eq\_0034\]). This discrepancy is explained by the fact that in Theorem \[main\_thm2\], we actually make full use of the growth of ${\textrm{diam}}(Z_p(\mu))$ for all $p \geq 2$, whereas in Theorem \[main\_thm3\] we only assumed this control for the end value of $p$. We emphasize that this constitutes a genuine difference in assumptions, and that the logarithmic factor in (\[eq\_0034\]) is not just a mere technicality: we show in Section \[sec\_counter\] that removing this factor is actually *equivalent* to Bourgain’s original hyperplane conjecture.
We find condition (\[eq\_0034\]) quite interesting from other respects as well. It is very much related to Paouris’ parameter $q^*(\mu)$, to be discussed in Section \[sec4\]. In fact, we show there that the parameter: $$q^{\#}(\mu) := \sup {\left\{q \geq 1 ; {\textrm{diam}}(Z_q(\mu)) \leq c^\sharp \sqrt{n} {\textrm{det Cov}}(\mu)^\frac{1}{2n}\right\}} ~,$$ for a small-enough universal constant $c^\sharp > 0$, is essentially equivalent to and has the same functionality as Paouris’ $q^*(\mu)$ parameter, in addition to being rather convenient to work with.
The lower bounds in Theorem \[main\_thm2\] and Theorem \[main\_thm3\] compare with the matching *upper* bounds on ${\textrm{V.Rad.}}(Z_p(\mu))$, obtained by Paouris [@Paouris-IsotropicTail Theorem 6.2], which are valid for *all* $2 \leq p \leq n$: $$\label{eq:vr-upper}
{\textrm{V.Rad.}}(Z_p(\mu)) \leq C \sqrt{p} {\textrm{V.Rad.}}(Z_2(\mu)) ~.$$ This implies that the lower bounds in both theorems above are sharp, up to constants, and so the only pertinent question is the optimality of the range of $p$’s for which their conclusion is valid. In this direction, Paouris obtained a partial converse to (\[eq:vr-upper\]) in the following range of $p$’s: $$W(Z_p(\mu)) \geq c \sqrt{p} {\textrm{V.Rad.}}(Z_2(\mu)) \quad \quad \forall 2 \leq p \leq q^\#(\mu)
~. \label{eq_152}$$ Here $W(K) = \int_{S^{n-1}} h_K(\theta) d\sigma(\theta)$ denotes half the mean width of $K$, $\sigma$ is the Haar probability measure on the Euclidean unit sphere $S^{n-1}$, and $h_K(\theta) = \sup_{x \in K} {\left \langle x,\theta \right \rangle}$ is the supporting functional of $K$. Note that according to the Urysohn inequality, $W(K) \geq {\textrm{V.Rad.}}(K)$ (see e.g. [@Milman-Schechtman-Book]), and so Theorem \[main\_thm3\] should be thought of as a formal strengthening of (\[eq\_152\]), if it were not for the logarithmic factor in (\[eq\_0034\]).
The rest of this work is organized as follows. We begin with some more or less known preliminaries in Section \[sec2\]. In Section \[sec3\], we deduce a new formula for ${\textrm{V.Rad.}}(Z_p(\mu))$ involving the “tilts" of the measure $\mu$ from [@quarter; @K_psi2], and we relate between the $Z_p$-bodies of the original measure and its tilts. In Section \[sec4\], we deviate from our discussion to review Paouris’ $q^*$-parameter, and compare it with $q^\sharp$; this section may be read independently of this work. In Section \[sec5\], we use projections and the $q^\sharp$-parameter to relate between the determinant of the covariance matrix of $\mu$ and its tilts, and conclude the proofs of Theorems \[main\_thm2\] (in fact, a more general version) and \[main\_thm3\]. In Section \[sec6\], we show that removing the log-factor from Theorem \[main\_thm3\] is equivalent to the slicing problem.
**Acknowledgements**. We thank Grigoris Paouris for interesting discussions.
Preliminaries {#sec2}
=============
Given $1 \leq k \leq n$, the Grassmann manifold of all $k$-dimensional linear subspaces of ${\mathbb{R}}^n$ is denoted by $G_{n,k}$. Given $E \in G_{n,k}$, the orthogonal projection onto $E$ is denoted by $Proj_E$, and given a Borel probability measure $\mu$ on ${\mathbb{R}}^n$, we denote by $\pi_E \mu := (Proj_E)_*(\mu)$ the push-forward of $\mu$ via $Proj_E$. For a convex body $K \subset
\RR^n$ containing the origin in its interior, its polar body is denoted by: $$K^{\circ} = {\left\{ x \in \RR^n \; ; \; {\left \langle x,y \right \rangle} \leq
1 \;\;\; \forall y \in K \right\}} ~.$$ Finally, we denote by $\nabla$ and ${\textrm{Hess}}$ the gradient and Hessian of a sufficiently differentiable function, respectively.
Throughout this text, $x \simeq y$ is an abbreviation for $c x \leq
y \leq C x$ for universal constants $c, C > 0$. Similarly, we write $x \lesssim y$ ($x \gtrsim y$) when $x \leq C y$ ($x \geq c y$). Additionally, for two convex sets $K, T \subset \RR^n$ we write $K \simeq T$ when: $$c K \subseteq T \subseteq C K$$ for universal constants $c, C > 0$.
Extension of the Slicing Problem to log-concave measures {#subsec:slicing}
--------------------------------------------------------
We first recall the well-known extension of the slicing problem from the class of convex bodies to the class of all log-concave measures, due to Ball [@BallPhD; @Ball-kdim-sections]. Given a log-concave probability measure $\mu$ on ${\mathbb{R}}^n$, define its isotropic constant $L_\mu$ by: $$\label{eq:Lmu}
L_\mu := {\left\Vert\mu\right\Vert}_{L_\infty}^\frac{1}{n} {\textrm{det Cov}}(\mu)^{\frac{1}{2n}} ~,$$ where ${\left\Vert\mu\right\Vert}_{L_\infty} := \sup_{x \in {\mathbb{R}}^n} \rho(x)$ and $\rho$ is the log-concave density of $\mu$. It was shown by Ball [@BallPhD; @Ball-kdim-sections] that given $n \geq 1$: $$\sup_\mu {L_\mu} \leq C \sup_{K} L_K ~,$$ where the suprema are taken over all log-concave probability measures $\mu$ and convex bodies $K$ in ${\mathbb{R}}^n$, respectively (see e.g. [@quarter] for the non-even case). Similarly, the following theorem slightly generalizes Theorem \[main\_thm\]:
\[main\_thm1+\] Let $\mu$ denote a log-concave probability measure on ${\mathbb{R}}^n$ with barycenter at the origin. Suppose that $\mu$ is in addition a $\psi_{\alpha}$-measure ($1 \leq \alpha \leq 2$) with constant $b_{\alpha}$. Then: $$L_\mu \leq C \sqrt{b_\alpha^\alpha n^{1-\alpha/2}} ~.$$
As was the case with Theorem \[main\_thm\], deducing Theorem \[main\_thm1+\] from Theorem \[main\_thm2\] is equally elementary. We only require the following additional well-known lemma, which will come in handy in other instances in this work as well. This lemma serves as an extension of (\[eq:ZnK\]) to the class of log-concave measures.
\[lem:Zn\] Let $\mu$ denote a log-concave probability measure on ${\mathbb{R}}^n$ with barycenter at the origin. Then: $${\textrm{V.Rad.}}(Z_n(\mu)) \simeq \frac{\sqrt{n}}{{\left\Vert\mu\right\Vert}_{L_\infty}^\frac{1}{n}} ~.$$
Given Lemma \[lem:Zn\], the reduction of Theorem \[main\_thm1+\] to Theorem \[main\_thm2\] is indeed immediate, since for $p \leq n$ in the range specified in the latter: $$c \sqrt{p} \leq \frac{{\textrm{V.Rad.}}(Z_p(\mu))}{{\textrm{V.Rad.}}(Z_2(\mu))} \leq \frac{{\textrm{V.Rad.}}(Z_n(\mu))}{{\textrm{det Cov}}(\mu)^\frac{1}{2n}} \simeq
\frac{\sqrt{n}}{{\left\Vert\mu\right\Vert}_{L_\infty}^\frac{1}{n} {\textrm{det Cov}}(\mu)^\frac{1}{2n}} = \frac{\sqrt{n}}{L_\mu} ~.$$
Denote by $\rho$ the log-concave density of $\mu$. According to [@PaourisSmallBall Proposition 3.7] (compare with [@K_psi2 Lemma 2.8] and Lemma \[lem:Lambda-Zp\] below): $${\textrm{V.Rad.}}(Z_n(\mu)) \simeq \frac{\sqrt{n}}{\rho(0)^\frac{1}{n}} ~.$$ However, according to Fradelizi [@fradelizi]: $$e^{-n} M \leq \rho(0) \leq M \quad , \quad M := {\left\Vert\mu\right\Vert}_{L_\infty} = \sup_{x \in {\mathbb{R}}^n} \rho(x) ~,$$ and so the assertion immediately follows.
$\Lambda_p$-bodies
------------------
Now suppose that $\mu$ is an arbitrary Borel probability measure on $\RR^n$. Its logarithmic Laplace transform is defined as: $$\Lambda_{\mu}(\xi) := \log \int_{\RR^n} \exp({\left \langle \xi, x \right \rangle})
d\mu(x) \quad \quad , \quad \xi \in \RR^n ~.$$ The function $\Lambda_{\mu}$ is always convex (e.g. by Hölder’s inequality), and clearly $\Lambda_{\mu}(0) = 0$. If in addition the barycenter of $\mu$ lies at the origin, then $\Lambda_\mu$ is non-negative (by Jensen’s inequality). In this case, for any $t \geq
0$ and $\alpha \geq 1$: $$\label{eq:Lambda-inclusion}
\frac{1}{\alpha} {\left\{ \Lambda_\mu \leq \alpha t \right\}} \subseteq {\left\{
\Lambda_\mu \leq t \right\}} \subseteq {\left\{ \Lambda_\mu \leq \alpha t \right\}} ~,$$ where we abbreviate ${\left\{\Lambda_{\mu} \leq t\right\}} = \{ \xi \in \RR^n ; \Lambda_{\mu}(\xi) \leq t \}$. When $\mu$ is log-concave, the convex function $\Lambda_{\mu}$ possesses several additional regularity properties. For instance ${\left\{\Lambda_{\mu} < \infty\right\}}$ is an open set, and $\Lambda_{\mu}$ is $C^{\infty}$-smooth and strictly-convex in this open set (see, e.g., [@K_psi2 Section 2]).
The following lemma describes a certain equivalence, known to specialists, between the $L_p$-centroid bodies and the level-sets of the logarithmic Laplace Transform $\Lambda_\mu$. See Latała and Wojtaszczyk [@LW Section 3] for a proof of a dual version in the symmetric case (i.e., when $\mu(A) = \mu(-A)$ for all Borel subsets $A \subset \RR^n$).
The $\Lambda_p$-body associated to $\mu$, for $p \geq 0$, is defined as: $$\Lambda_p(\mu) := {\left\{\Lambda_{\mu} \leq p\right\}} \cap -{\left\{
\Lambda_{\mu} \leq p \right\}} ~.$$
\[eq\_0000\] \[lem:Lambda-Zp\] Suppose $\mu$ is a log-concave probability measure on $\RR^n$ whose barycenter lies at the origin. Then for any $p \geq 1$: $$\Lambda_p(\mu) \simeq p Z_p(\mu)^{\circ} ~.$$
These two equivalent points of view turn out to complement each other well, and play a synergetic role in this work. Before providing a proof, we illustrate this in the following naive example. Given a log-concave probability measure $\mu$, a well known consequence of Berwald’s inequality (see e.g. [@MilmanPajor]) is that: $$\label{eq:Zp-inclusion}
q \geq p \geq 1 \;\;\; \Rightarrow \;\;\; Z_p(\mu) \subset Z_q(\mu)
\subset C \frac{q}{p} Z_p(\mu) ~.$$ In view of Lemma \[lem:Lambda-Zp\], note that this is nothing else but a reformulation (up to constants) of the trivial set of inclusions in (\[eq:Lambda-inclusion\]).
First, suppose that $\xi \in \Lambda_p(\mu)$. Then: $$\int_{\RR^n} \exp({\left\vert{\left \langle \xi,x \right \rangle}\right\vert}) d \mu(x) \leq \int_{\RR^n}
\exp({\left \langle \xi,x \right \rangle}) d \mu(x) + \int_{\RR^n} \exp(-{\left \langle \xi,x \right \rangle}) d
\mu(x) \leq 2 e^p ~.$$ Using the inequality $t^p / p! \leq e^t$, valid for any $t \geq 0$, we see that: $$h_{Z_p(\mu)}(\xi) = {\left(\int_{\RR^n} {\left\vert{\left \langle \xi,x \right \rangle}\right\vert}^p d
\mu(x)\right)}^{\frac{1}{p}} \leq {\left(2 e^p p!\right)}^\frac{1}{p} \leq C p ~.$$ Since $\xi \in \Lambda_p(\mu)$ was arbitrary, this amounts to $\Lambda_p(\mu) \subseteq C p Z_p(\mu)^\circ$, the first desired inclusion.
For the other inclusion, suppose $\xi \in \RR^n$ is such that $h_{Z_p(\mu)}(\xi) \leq p$, that is: $$\left( \int_{\RR^n} {\left\vert{\left \langle \xi,x \right \rangle}\right\vert}^p d \mu(x) \right)^{1/p}
\leq p ~. \label{eq_540}$$ Write $X$ for the random vector in $\RR^n$ that is distributed according to $\mu$. Then the function: $$\varphi(t) = \PP({\left \langle X,\xi \right \rangle} \geq t) \quad \quad , \quad\quad t \in \RR ~,$$ is log-concave, according to the Prékopa-Leindler inequality (see, e.g., the first pages of [@Pis]). Furthermore, since the barycenter of $\mu$ lies at the origin, we have $1/e \leq
\varphi(0) \leq 1 - 1/e$ by the Grünbaum–Hammer inequality (see e.g. [@bobkov Lemma 3.3]). Using Markov’s inequality, (\[eq\_540\]) entails that: $$\varphi(3 e p) \leq ( 3 e)^{-p} ~.$$ Since $\varphi$ is log-concave, then: $$\PP({\left \langle X,\xi \right \rangle} \geq t) = \varphi(t) \leq \varphi(0) \left( \frac{\varphi(3 ep)}{\varphi(0)}
\right)^{\frac{t}{3 ep}} \leq C \exp(-t / (3e)) \quad , \quad \forall t \geq 3 ep ~.$$ An identical bound holds for $\PP({\left \langle X,\xi \right \rangle} \leq -t)$, and combining the two, we obtain: $$\PP(|{\left \langle X,\xi \right \rangle}| \geq t) \leq C \exp(-t / (3e)) \quad , \quad \forall t \geq 3 ep ~.$$ Therefore: $$\begin{aligned}
\EE \exp \left( \frac{|{\left \langle \xi,X \right \rangle}|}{6e} \right) & = \frac{1}{6e}
\int_0^{\infty}
\exp \left( \frac{t}{6e} \right) \PP(|{\left \langle X,\xi \right \rangle}| \geq t)dt \\
& \leq \frac{1}{6e} \int_0^{3ep} \exp \left( \frac{t}{6e} \right) dt + C \int_{3
ep}^{\infty} \exp(-t / (6 e)) dt \leq \exp\left( \tilde{C} p
\right) ~.\end{aligned}$$ Consequently: $$\max \left \{ \Lambda_{\mu} \left(\frac{1}{6e} \xi \right),
\Lambda_{\mu} \left( -\frac{1}{6e} \xi \right) \right \} \leq \log
\EE \exp \left( \frac{|{\left \langle \xi,X \right \rangle}|}{6e} \right) \leq C p ~,$$ for some $C \geq 1$, and using (\[eq:Lambda-inclusion\]), this implies: $$\max \left \{ \Lambda_{\mu} \left(\frac{1}{6eC} \xi \right),
\Lambda_{\mu} \left( -\frac{1}{6eC} \xi \right) \right \} \leq p ~,$$ for any $\xi \in \RR^n$ with $h_{Z_p(\mu)}(\xi) \leq p$. This is precisely the second desired inclusion $p Z_p(\mu)^\circ \subseteq
C' \Lambda_p(\mu)$, and the assertion follows.
Level Sets of Convex Functions Under Gradient Maps
--------------------------------------------------
The last topic we would like to review pertains to some properties of level sets of convex functions and their gradient images. The possibility to use the gradient image of $\Lambda_{\mu}$ as in [@quarter] is one of the main reasons for additionally employing the logarithmic Laplace transform, rather than working exclusively with the $L_p$-centroid bodies.
\[lem:prod\] Let $F: \RR^n \rightarrow \RR \cup
\{ \infty \}$ be a non-negative convex function, which is $C^1$-smooth in ${\left\{ F < \infty \right\}}$. Let $q,r \geq 0$. Then: $${\left \langle z,\nabla F(x) \right \rangle} \leq q+r \quad \quad \quad \text{for any} \ z
\in {\left\{F \leq r\right\}}, \ x \in \frac{1}{2} \{ F \leq q \}.$$ In other words: $$\nabla F {\left(\frac{1}{2} {\left\{F \leq q\right\}}\right)} \subset (q+r) {\left\{F \leq
r\right\}}^{\circ} ~.$$
Since $F$ is non-negative and its graph lies above any tangent hyperplane, then: $${\left \langle \nabla F(x),\frac{z}{2} \right \rangle} \leq F(x) + {\left \langle \nabla F(x)
,\frac{z}{2} \right \rangle} \leq F(x+z/2) \leq \frac{F(2x) + F(z)}{2} \leq
\frac{q+r}{2} ~.$$
The following lemma was proved in [@K_psi2 Lemma 2.3] for an even function $F$.
\[lem:F-vr\] Let $F: \RR^n \rightarrow \RR \cup \{ \infty \}$ be a non-negative convex function, $C^2$-smooth and strictly-convex in ${\left\{ F < \infty \right\}}$, with $F(0) = 0$. Let $p > 0$, and set: $$F_p := {\left\{F \leq p\right\}} \cap -{\left\{F \leq p\right\}} ~.$$ Assume that: $$\Psi_p := {\left(\frac{1}{{\textrm{Vol}_n}(\frac{1}{2} F_p)} \int_{\frac{1}{2} F_p}
\det {\textrm{Hess}}F(x) dx\right)}^\frac{1}{n} > 0 ~.$$ Then: $${\textrm{V.Rad.}}(F_p) \leq 2 \frac{\sqrt{p}}{\sqrt{\Psi_p}} ~.$$
Applying Lemma \[lem:prod\] with $q=r=p$, and using the change of variables $x = \nabla F(y)$, we obtain: $${\textrm{Vol}_n}(2p (F_p)^\circ) \geq {\textrm{Vol}_n}\left(\nabla F \left(\frac{1}{2} F_p
\right) \right) = \int_{\frac{1}{2} F_p} \det {\textrm{Hess}}F(y) dy = {\textrm{Vol}_n}\left(\frac{1}{2} F_p \right) \Psi_p^n ~.$$ Equivalently, we obtain: $${\textrm{Vol}_n}((F_p)^\circ) \geq {\left(\frac{\Psi_p}{4p}\right)}^{n} {\textrm{Vol}_n}(F_p) ~.$$ Note that $F_p$ is a centrally-symmetric convex body, i.e., $F_p =
-F_p$. The Blaschke–Santaló inequality (see, e.g., [@Schneider-Book]) for a centrally-symmetric convex body $K$ asserts that: $${\textrm{V.Rad.}}(K^\circ) {\textrm{V.Rad.}}(K) \leq 1~.$$ Combining the last two estimates with $K = F_p$, the result immediately follows.
A formula for ${\textrm{V.Rad.}}(Z_p(\mu))$ involving tilted measures {#sec3}
=====================================================================
Let $\mu$ denote a log-concave probability measure on $\RR^n$ with density $\rho$, and let $\xi \in {\left\{\Lambda_{\mu} < \infty\right\}}$. We denote by $\mu_{\xi}$ the “tilt” of $\mu$ by $\xi$, defined via the following procedure. First, define the probability density: $$\rho_\xi(x) := \frac{1}{Z_\xi} \rho(x) \exp({\left \langle \xi,x \right \rangle}) \quad
\quad \quad \text{for} \quad x \in \RR^n ~,$$ where $Z_\xi > 0$ is a normalizing factor. Denoting by $b_\xi \in
{\mathbb{R}}^n$ the barycenter of $\rho_\xi$, we set $\mu_\xi$ to be the probability measure with density $\rho_\xi(\cdot -
b_\xi)$. Note that $\mu_\xi$ is a log-concave probability measure, having the origin as its barycenter. Furthermore, as verified in [@K_psi2 Section 2], we have: $$b_{\xi} = \nabla \Lambda_\mu(\xi) \quad , \quad {\textrm{Cov}}(\mu_{\xi}) =
{\textrm{Hess}}\Lambda_\mu(\xi) ~. \label{eq_2329}$$
The following proposition is one of the main results in this section:
\[prop:vr-formula\] Let $\mu$ denote a log-concave probability measure on ${\mathbb{R}}^n$ whose barycenter lies at the origin. Then, for all $1 \leq p \leq
n$: $$\label{eq:vr-formula}
{\textrm{V.Rad.}}(Z_p(\mu)) \simeq \sqrt{p} \inf_{x \in \frac{1}{2}
\Lambda_p(\mu)} {\textrm{det Cov}}(\mu_x)^{\frac{1}{2n}} .$$
In the proofs of the theorems stated in the Introduction, we will not use the full force of Proposition \[prop:vr-formula\], but rather only the lower bound for ${\textrm{V.Rad.}}(Z_p(\mu))$. This lower bound has a short proof, as the reader will see below. However, the observation that we actually obtain an equivalence seems interesting, hence we provide the arguments for both directions. Before going into the proof, as a testament of its usefulness, we state the following immediate corollary of Proposition \[prop:vr-formula\]:
Let $\mu$ be a log-concave probability measure on $\RR^n$ whose barycenter lies at the origin. Then: $$1 \leq p \leq q \leq n \;\;\; \Rightarrow \;\;\;
\frac{{\textrm{V.Rad.}}(Z_p(\mu))}{\sqrt{p}} \geq c \frac{{\textrm{V.Rad.}}(Z_q(\mu))}{\sqrt{q}}
~.$$
\[rem:LYZ-1\] Using $q = n$ above and the fact that ${\textrm{V.Rad.}}(Z_n(K)) \simeq {\textrm{V.Rad.}}(K)$ for a convex body $K$ whose barycenter lies at the origin, which follows from (\[eq:ZnK\]) as in the Introduction, we immediately verify that: $$\label{eq:our-LYZ-K}
\forall 1 \leq p \leq n \;\;\; {\textrm{V.Rad.}}(Z_p(K)) \geq c \sqrt{\frac{p}{n}}
{\textrm{V.Rad.}}(K)~.$$ This recovers up to a constant the lower bound of Lutwak, Yang and Zhang (\[eq:LYZ-bound\]). Moreover, recalling that ${\textrm{V.Rad.}}(Z_n(\mu))
\simeq \sqrt{n} / {\left\Vert\mu\right\Vert}_{L_\infty}^{\frac{1}{n}}$ by Lemma \[lem:Zn\] and the definition $(\ref{eq:Lmu})$ of $L_\mu$, the same argument yields the following analog of (\[eq:our-LYZ-K\]): $$\forall 1 \leq p \leq n \;\;\; {\textrm{V.Rad.}}(Z_p(\mu)) \geq c \frac{\sqrt{p}}{L_\mu} {\textrm{det Cov}}(\mu)^{\frac{1}{2n}} = c \frac{\sqrt{p}}{L_\mu} {\textrm{V.Rad.}}(Z_2(\mu)) ~.$$ This may also be deduced by only employing the lower-bound in (\[eq:vr-formula\]), as in Remark \[rem:LYZ-2\].
We now turn to the proof of Proposition \[prop:vr-formula\], and begin with the lower bound for ${\textrm{V.Rad.}}(Z_p(\mu))$. In fact, we show a formally stronger statement:
\[lem\_low\] Let $\mu$ denote a log-concave probability measure on ${\mathbb{R}}^n$ whose barycenter lies at the origin. Then, for all $1 \leq p \leq
n$, $${\textrm{V.Rad.}}(Z_p(\mu)) \geq c \sqrt{p} \sqrt{\Psi_p} ~,
$$ where $c > 0$ is a universal constant and: $$\Psi_p := {\left(\frac{1}{{\textrm{Vol}_n}(\frac{1}{2} \Lambda_p(\mu))} \int_{\frac{1}{2} \Lambda_p(\mu)} {\textrm{det Cov}}(\mu_x) dx\right)}^\frac{1}{n} ~.$$
Apply Lemma \[lem:F-vr\] with $F = \Lambda_\mu$. Since $\det {\textrm{Hess}}\Lambda_\mu(x) = {\textrm{det Cov}}(\mu_x)$ according to (\[eq\_2329\]), we deduce that: $$\label{eq_0005}
{\textrm{V.Rad.}}(\Lambda_p(\mu)) \leq 2 \frac{\sqrt{p}}{\sqrt{\Psi_p}} ~.$$ Applying Lemma \[eq\_0000\] in order to pass from $\Lambda_p(\mu)$ to $Z_p(\mu)$, and the Bourgain–Milman inequality (see, e.g., [@Pis]) for a centrally-symmetric convex set $K \subset \RR^n$: $${\textrm{V.Rad.}}(K^\circ) {\textrm{V.Rad.}}(K) \geq c~,$$ we deduce from (\[eq\_0005\]) that: $${\textrm{V.Rad.}}(Z_p(\mu)) \simeq p {\textrm{V.Rad.}}(\Lambda_p(\mu)^\circ) \gtrsim p
{\textrm{V.Rad.}}(\Lambda_p(\mu))^{-1} \gtrsim \sqrt{p} \sqrt{\Psi_p} ~.$$
In order to deduce the upper bound of Proposition \[prop:vr-formula\], and of crucial importance to the main results of this work, is the following elementary observation:
\[prop:Lambda-iso\] Let $\mu$ denote a log-concave probability measure in ${\mathbb{R}}^n$ with barycenter at the origin. Then: $$\forall x \in \frac{1}{2} \Lambda_p(\mu) \quad , \quad \Lambda_p(\mu_x) \simeq \Lambda_p(\mu) ~.$$
Indeed, it is clear that the logarithmic Laplace transform should commute nicely with the tilt operation, and the following identity is verified by direct calculation: $$\label{eq:Lambda-tilt}
\Lambda_{\mu_x}(z) = \Lambda_\mu(z+x) - \Lambda_\mu(x) -
{\left \langle z,b_x \right \rangle} ~, ~ b_x = \nabla \Lambda_\mu(x) ~.$$ Geometrically, this means that the graph of $\Lambda_{\mu_x}$ is obtained from that of $\Lambda_\mu$ by subtracting the tangent plane at $x$ (given by the linear function $z \mapsto \Lambda_\mu(x)
+ {\left \langle z-x,\nabla \Lambda_\mu(x) \right \rangle}$), and translating everything by $-x$ (so that $x$ gets mapped to the origin). In particular, we verify that $\Lambda_{\mu_x}(0) = 0$ and that $\Lambda_{\mu_x} \geq
0$, as required from the logarithmic Laplace transform of a probability measure with barycenter at the origin.
It remains to manipulate level sets of convex functions, once again. We require the following:
\[lem:F-G\] Let $F$ be as in Lemma \[lem:prod\], and let $y \in {\mathbb{R}}^n$ and $D, p > 0$. Define a function $G$ by: $$G(z) := F(z+y) - F(y) - {\left \langle z,\nabla F(y) \right \rangle}~.$$ Then: $$y \in \frac{1}{2} {\left\{F \leq D p \right\}}, \ z \in {\left\{F \leq p\right\}} \cap -{\left\{F \leq p\right\}} \quad \Longrightarrow \quad z \in 2 {\left\{G \leq
(D+1)p\right\}} ~.$$
We apply Lemma \[lem:prod\] with $q = Dp$ and $r=p$. Since $-z
\in {\left\{F \leq p\right\}}$ and $y \in \frac{1}{2} {\left\{F \leq D p\right\}}$, then by the conclusion of that lemma, ${\left \langle -z,\nabla
F(y) \right \rangle} \leq (D+1)p$. Since $F$ is non-negative and convex, we deduce that: $$G(z/2) \leq F(z/2+y) + \frac{D+1}{2} p \leq \frac{F(z) + F(2y)}{2} +
\frac{D+1}{2} p \leq (D+1) p ~.$$
1. If $z \in \Lambda_p(\mu)$, we apply Lemma \[lem:F-G\] with $D=1$ and $y=x$ to $F = \Lambda_\mu$. By (\[eq:Lambda-tilt\]), we deduce that $\Lambda_{\mu_x}(z/2) = G(z/2) \leq 2p$. Using (\[eq:Lambda-inclusion\]), we conclude that $\Lambda_{\mu_x}(z/4) \leq p$. The same argument applies to $-z$ by the symmetry of our assumptions, and so we conclude that $z \in 4 \Lambda_p(\mu_x)$.
2. If $z \in \Lambda_p(\mu_x)$, we would like to apply Lemma \[lem:F-G\] with $y=-x$ to $F = \Lambda_{\mu_x}$, since tilting $\mu_x$ by $-x$ gives back $\mu$. To this end, we must verify that $\Lambda_{\mu_x}(-2x) \leq D p$ for some $D>0$. According to (\[eq:Lambda-tilt\]): $$\Lambda_{\mu_x}(-2x) = \Lambda_\mu(-x) - \Lambda_\mu(x) +
2{\left \langle x,\nabla \Lambda_\mu(x) \right \rangle}~.$$ By Lemma \[lem:prod\], we know that ${\left \langle x,\nabla
\Lambda_\mu(x) \right \rangle} \leq 2p$, and using that $\Lambda_\mu$ is non-negative, convex and vanishes at the origin, we obtain: $$\Lambda_{\mu_x}(-2x) \leq \frac{1}{2} \Lambda_\mu(-2x) + 4p \leq 4.5
p ~.$$ We conclude that we may use $D=4.5$ above, and so Lemma \[lem:F-G\] finally implies that $\Lambda_{\mu}(z/2) = G(z/2) \leq
5.5 p$. As in the first part of the proof, we deduce that $\Lambda_{\mu}(z/11) \leq p$. The same argument applies to $-z$ by the symmetry of our assumptions, and so we conclude that $z \in
11 \Lambda_p(\mu)$.
Using Lemma \[eq\_0000\], we equivalently reformulate Proposition \[prop:Lambda-iso\] as:
\[prop:Zp-iso\] Let $\mu$ denote a log-concave probability measure in ${\mathbb{R}}^n$ with barycenter at the origin. Then: $$\forall x \in \frac{1}{2} \Lambda_p(\mu) \quad , \quad Z_p(\mu_x) \simeq Z_p(\mu) ~.$$
To complete the proof of Proposition \[prop:vr-formula\], we state again Paouris’ upper bound (\[eq:vr-upper\]) on ${\textrm{V.Rad.}}(Z_p(\nu))$:
\[prop:Paouris\] For any log-concave probability measure $\nu$ with barycenter at the origin, and $2 \leq p \leq n$: $${\textrm{V.Rad.}}(Z_p(\nu)) \leq C \sqrt{p} {\textrm{V.Rad.}}(Z_2(\nu))~.$$
The statement is invariant under linear transformations, so we may assume that $\nu$ is isotropic. The claim is then the content of [@Paouris-IsotropicTail Theorem 6.2].
Lemma \[lem\_low\] implies the lower bound: $${\textrm{V.Rad.}}(Z_p(\mu)) \geq c \sqrt{p} \inf_{x \in
\frac{1}{2} \Lambda_p(\mu)} {\textrm{det Cov}}(\mu_x)^{\frac{1}{2n}}~.$$ Since ${\textrm{det Cov}}(\mu_x)^{\frac{1}{2n}} = {\textrm{V.Rad.}}(Z_2(\mu_x))$, then applying Proposition \[prop:Paouris\], we obtain: $$\inf_{x \in
\frac{1}{2} \Lambda_p(\mu)} {\textrm{V.Rad.}}(Z_p(\mu_x)) \leq C \sqrt{p} \inf_{x \in \frac{1}{2}
\Lambda_p(\mu)} {\textrm{det Cov}}(\mu_x)^{\frac{1}{2n}}~. \label{eq_945}$$ But by Proposition \[prop:Zp-iso\], $Z_p(\mu_x) \simeq Z_p(\mu)$ for all $x \in \frac{1}{2}
\Lambda_p(\mu)$, and hence the left-hand side in (\[eq\_945\]) is equivalent to ${\textrm{V.Rad.}}(Z_p(\mu))$, completing the proof.
It follows that all of the inequalities which we used in the proof of Proposition \[prop:vr-formula\] above, are actually equivalences up to numeric constants. This fact has some interesting consequences; we omit a detailed account of these here, and only remark on the following point. Given $1 \leq p \leq n$, denote: $$x_p := \text{argmin}_{x \in \frac{1}{2} \Lambda_p(\mu)} {\textrm{det Cov}}(\mu_x)^{\frac{1}{2n}} ~,$$ so that $\mu_{x_p}$ is the “worst" tilt we need to account for when evaluating ${\textrm{V.Rad.}}(Z_p(\mu))$. It follows that for this tilt: $${\textrm{V.Rad.}}(Z_p(\mu_{x_p})) \simeq \sqrt{p} {\textrm{V.Rad.}}(Z_2(\mu_{x_p})) ~,$$ and in particular, the argument described in Subsection \[subsec:slicing\] implies that $L_{\mu_{x_p}} \leq C \sqrt{n /
p}$. It is interesting to compare this with the approach from [@quarter] for resolving the isomorphic slicing problem. The latter approach is in some sense dual to our current one, since in this work our goal will be to bound ${\textrm{det Cov}}(\mu_{x_p})^{\frac{1}{2n}}$ from below, whereas the goal in [@quarter] was to bound this expression from above. Compare also with Remark \[rem:LYZ-2\].
On Paouris’ definition of $q^*$ {#sec4}
===============================
Given a centrally-symmetric convex body $K \subset {\mathbb{R}}^n$, its “(dual) Dvoretzky-dimension" $k^*(K)$ was defined by V. Milman and G. Schechtman [@MilmanSchechtmanSharpDvorDim] as the largest positive integer $k \leq n$ so that: $$\sigma_{n,k}{\left\{E \in G_{n,k} ; \frac{1}{2} W(K) B_E \subset Proj_E
K \subset 2 W(K) B_E \right\}} \geq \frac{n}{n+k} ~,$$ where $\sigma_{n,k}$ denotes the Haar probability measure on $G_{n,k}$ and $B_E$ denotes the Euclidean unit ball in the subspace $E$. It was shown in [@MilmanSchechtmanSharpDvorDim], following Milman’s seminal work [@Mil71], that: $$\label{eq:k*}
k^*(K) \simeq n {\left(\frac{W(K)}{{\textrm{diam}}(K)}\right)}^2 ~.$$ Define $W_q(K) = \left( \int_{S^{n-1}} h_K(\theta)^q d
\sigma(\theta) \right)^{\frac{1}{q}}$, the $q$-th moment of the supporting functional of $K$. According to Litvak, Milman and Schechtman [@LMS]: $$\label{eq:LMS1}
c_1 W_q(K) \leq \max \left \{ W(K) , \sqrt{q/n} \; \;
{\textrm{diam}}(K) \right \} \leq c_2 W_q(K) ~.$$ The quantity $W_q(Z_q(\mu))$ has a simple equivalent description: a direct calculation as in [@Paouris-Small-Diameter] confirms that for any Borel probability measure $\mu$ on $\RR^n$ and $q \geq 1$: $$\label{WQZQ}
W_q(Z_q(\mu)) \simeq
\frac{\sqrt{q}}{\sqrt{n+q}} I_q(\mu) ~ ~ , ~ ~ I_q(\mu) := {\left( \int_{{\mathbb{R}}^n} |x|^q d\mu(x) \right)}^{\frac{1}{q}} ~.$$ Finally, observe that when the barycenter of $\mu$ is at the origin, then $I_2(\mu)^2 = {\textrm{trace Cov}}(\mu)$.
In [@Paouris-IsotropicTail], Paouris defines $q^*(\mu)$ as follows: $$q^*(\mu) := \sup{\left\{q \in \mathbb{N} ; k^*(Z_q(\mu)) \geq q\right\}} ~.$$ It is straightforward to check that all of Paouris’ results involving $q^*(\mu)$ from [@Paouris-IsotropicTail; @PaourisSmallBall] remain valid when replacing it with $q^*_c(\mu)$ when $c > 0$ is a fixed universal constant, where $q^*_\delta$ is defined as follows: $$q^*_\delta(\mu) := \sup{\left\{q \geq 1 ; k^*(Z_q(\mu)) \geq
\delta^{-2} q\right\}} ~.$$ Although the particular value of $c>0$ seems insignificant for the results of [@Paouris-IsotropicTail; @PaourisSmallBall], the definition we require in this work is essentially that of $q^*_c$ for some *small enough* universal constant $c>0$. Our preference to work with a variant of $q^*_c$ is motivated by Lemma \[lem:P3-equiv\] below and the subsequent remarks.
We proceed as follows. Given a log-concave probability measure $\mu$ on $\RR^n$, $q \geq 1$ and $\delta>0$, consider the following four related properties:
1. $P_1(\delta)$ is the property that $k^*(Z_q(\mu)) \geq \delta^{-2} q$.
2. $P_1'(\delta)$ is the property that ${\textrm{diam}}(Z_q(\mu)) \leq \delta \sqrt{n} \frac{W(Z_q(\mu))}{\sqrt{q}}$.
3. $P_2(\delta)$ is the property that ${\textrm{diam}}(Z_q(\mu)) \leq \delta \sqrt{n} {\textrm{det Cov}}(\mu)^{\frac{1}{2n}}$.
4. $P_W$ is the property that $W(Z_q(\mu)) \geq c \sqrt{q} {\textrm{det Cov}}(\mu)^{\frac{1}{2n}}$, for some specific, appropriately small universal constant $c>0$, as in the proof of Lemma \[lem:P3-equiv\](2) below.
According to (\[eq:k\*\]), we have: $$\label{eq:P_1-P_1'}
P_1(\delta) \Rightarrow P_1'(C_1 \delta) \Rightarrow P_1(C_2 \delta)
~,$$ for all $\delta > 0$, where $C_1, C_2 > 1$ are universal constants. The next lemma relates between the other properties above:
\[lem:P3-equiv\] Suppose $\mu$ is a log-concave probability measure in $\RR^n$ whose barycenter lies at the origin. Let $q \in [1,n]$ and $\delta \in (0,1]$. Then:
1. If $\mu$ is isotropic and $P_1(\delta)$ holds, then $P_2(C_3 \delta)$ holds.
2. 1. If $P_1'(\delta)$ holds, then so does $P_W$.
2. Suppose $\delta < \delta_0$ for a certain appropriately small universal constant $\delta_0 > 0$.\
If $P_2(\delta)$ holds, then so does $P_W$.
3. If $P_2(\delta)$ and $P_W$ hold, then so does $P_1'(C_4 \delta)$.
<!-- -->
1. Clearly $P_1(\delta)$ implies $P_1(1)$. Using (\[WQZQ\]), Paouris’s main result [@Paouris-IsotropicTail Theorem 8.1] and the isotropicity of $\mu$, we know that: $$W_q(Z_q(\mu)) \simeq \frac{\sqrt{q}}{\sqrt{n}} I_q(\mu) \simeq \frac{\sqrt{q}}{\sqrt{n}} I_2(\mu) = \frac{\sqrt{q}}{\sqrt{n}} {\left({\textrm{trace Cov}}(\mu)\right)}^{\frac{1}{2}} = \sqrt{q} ~.$$ In particular, $W(Z_q(\mu)) \leq W_q(Z_q(\mu)) \leq C \sqrt{q}$. Since $P_1(\delta)$ implies $P_1'(C_1 \delta)$, then: $${\textrm{diam}}(Z_q(\mu)) \leq C_1 \delta \sqrt{n} \frac{W(Z_q(\mu))}{\sqrt{q}} \leq C C_1 \delta \sqrt{n} = C_3 \delta \sqrt{n} {\textrm{det Cov}}(\mu)^{\frac{1}{2n}} ~,$$ and $P_2(C_3 \delta)$ holds true.
2. Since all properties are invariant under scaling, we may assume that ${\textrm{det Cov}}(\mu) = 1$. Using (\[WQZQ\]) and the arithmetic-geometric mean inequality: $$\frac{1}{n} I_2(\mu)^2 = \frac{1}{n} {\textrm{trace Cov}}(\mu) \geq {\textrm{det Cov}}(\mu)^{\frac{1}{n}} ~,$$ we see that: $$\label{eq_330}
W_q(Z_q(\mu)) \geq c_0 \frac{\sqrt{q}}{\sqrt{n}} I_q(\mu) \geq c_0 \frac{\sqrt{q}}{\sqrt{n}} I_2(\mu) \geq c_0 \sqrt{q} ~.$$
1. Assuming $P_1'(\delta)$, (\[eq:LMS1\]) implies that $W(Z_q(\mu)) \geq c_1 W_q(Z_q(\mu))$, and together with (\[eq\_330\]), $P_W$ follows.
2. Set $\delta_0 = c_0 c_1$, where $c_0$ is the constant from (\[eq\_330\]) and $c_1$ is the constant from (\[eq:LMS1\]). Using (\[eq\_330\]), the property $P_2(\delta)$ with $0 < \delta < \delta_0$ implies: $$\frac{\sqrt{q}}{\sqrt{n}} {\textrm{diam}}(Z_q(\mu))
\leq \delta \sqrt{q} < c_0 c_1 \sqrt{q} \leq c_1 W_q(Z_q(\mu)) ~.$$ Therefore by (\[eq:LMS1\]), $W(Z_q(\mu)) \geq c_1 W_q(Z_q(\mu))
\geq c_0 c_1 \sqrt{q}$, and $P_W$ follows.
3. This is immediate by plugging the estimates on ${\textrm{diam}}(Z_q(\mu))$ and $W(Z_q(\mu))$ into the definition of $P_1'(\delta)$.
Inspecting the proof, one may check that the assumption that $\delta \leq 1$ is not essential for the proof of parts (1), (2a) and (3), if one allows different dependence on $\delta$ in the conclusion of the assertions. However, the assumption that $\delta < \delta_0$ was *crucially* used in the proof of part (2b).
We conclude from Lemma \[lem:P3-equiv\] and (\[eq:P\_1-P\_1’\]) that $P_1(\delta)$ implies all the other properties if $\mu$ is isotropic, and that $P_2(\delta)$ implies all the other properties if $\delta$ is small enough. Neither of these restrictions are essential for the purposes of this work, but nevertheless we prefer to proceed with the more accessible $P_2(\delta)$ property, since in addition and in contrast to the $P_1(\delta)$ one, it is more stable in the following sense:
1. For any $\mu$, if $P_2(\delta)$ holds for $q$, then it also holds for all $p$ with $1 \leq p < q$.
2. If $\mu$ is isotropic and $P_2(\delta)$ holds for $\mu$ with $q$, then $P_2(\delta \sqrt{n / k})$ holds for $\pi_E \mu$ with $q$, simply because $Z_q(\pi_E \mu) = Proj_E
Z_q(\mu)$ for all $E \in G_{n,k}$.
Consequently, we make the following:
$$\begin{aligned}
q^{\sharp}(\mu) & := & \sup {\left\{q \geq 1 \; ; \; {\textrm{diam}}(Z_q(\mu)) \leq c^\sharp \sqrt{n} {\textrm{det Cov}}(\mu)^{\frac{1}{2n}}\right\}} \\
& = & \Delta_\mu^{-1}(c^\sharp \sqrt{n} {\textrm{det Cov}}(\mu)^{\frac{1}{2n}})
~,
$$
where $[1,\infty) \ni q \mapsto \Delta_\mu(q) := {\textrm{diam}}(Z_q(\mu))$ and $c^\sharp>0$ is a small enough constant, to be prescribed in Lemma \[lem:q-sharp\] below.\
As a convention, if ${\textrm{diam}}(Z_1(\mu)) \geq c^\sharp \sqrt{n} {\textrm{det Cov}}(\mu)^{\frac{1}{2n}}$, we set $q^\sharp(\mu) = 1$.
\[lem:q-sharp\] We may choose the numeric constant $c^\sharp > 0$ small enough so that:
1. $q^\sharp(\mu) \leq n$.
2. $1 \leq q \leq q^\sharp(\mu)$ implies $k^*(Z_q(\mu)) \geq q$ and $W(Z_q(\mu)) \geq c \sqrt{q} {\textrm{det Cov}}(\mu)^{\frac{1}{2n}}$.
Assume first that $q^\sharp(\mu) > 1$. The second point follows immediately from Lemma \[lem:P3-equiv\] and (\[eq:P\_1-P\_1’\]). The first point follows from (\[WQZQ\]), since: $$n \cdot {\textrm{det Cov}}(\mu)^{\frac{1}{n}} \leq {\textrm{trace Cov}}(\mu) = I_2(\mu)^2 \leq I_n(\mu)^{2} \simeq W_n(Z_n(\mu))^2 \leq {\textrm{diam}}(Z_n(\mu))^2 ~.$$ It remains to deal with the degenerate case $q^\sharp(\mu) = 1$. By definition, $k^*(Z_1(\mu)) \geq 1$, and e.g. by (\[eq:k\*\]): $$W(Z_1(\mu)) \geq c \frac{{\textrm{diam}}(Z_1(\mu))}{\sqrt{n}} \geq c c^\sharp \; {\textrm{det Cov}}(\mu)^{\frac{1}{2n}} ~,$$ as required.
Consequently $\lfloor q^\sharp(\mu) \rfloor \leq q^*(\mu)$, and all of Paouris’ results for $q \leq q^*(\mu)$ continue to hold for $q
\leq q^{\sharp}(\mu)$. Similarly, by Lemma \[lem:P3-equiv\], if $\mu$ is isotropic then $q^*_c(\mu) \leq q^\sharp(\mu)$ for some small constant $c>0$. To conclude this section, we reiterate the stability of $q^\sharp(\mu)$ under projections in the following corollary, which is one of the key ingredients in the proof of Theorem \[main\_thm3\]:
\[cor:q-sharp-proj\] Let $\mu$ denote an isotropic log-concave probability measure in ${\mathbb{R}}^n$, let $1 \leq k \leq n, q \geq 1$. Then for all $E \in
G_{n,k}$ with $k \geq (c^{\sharp})^{-2} {\textrm{diam}}^2(Z_q(\mu))$, we have $q^\sharp(\pi_E \mu) \geq q$. In particular $k^*(Proj_E Z_q(\mu))
\geq q$ and $W(Proj_E Z_q(\mu)) \geq c \sqrt{q}$.
Since $\pi_E \mu$ remains isotropic, $Z_q(\pi_E \mu) = Proj_E
Z_q(\mu)$ and ${\textrm{diam}}(Proj_E Z_q(\mu)) \leq {\textrm{diam}}(Z_q(\mu)) \leq
c^\sharp \sqrt{k}$, the assertion follows by definition of $q^\sharp(\pi_E \mu)$ and Lemma \[lem:q-sharp\].
Controlling ${\textrm{det Cov}}(\mu_x)$ via projections {#sec5}
=======================================================
In view of Proposition \[prop:vr-formula\], our goal now is to bound from below ${\textrm{det Cov}}(\mu_x)^{\frac{1}{2n}}$ for the tilted measures $\mu_x$, where $x \in \frac{1}{2}
\Lambda_p(\mu)$. Our only available information is given by Proposition \[prop:Zp-iso\], stating that $Z_p(\mu_x) \simeq Z_p(\mu)$, where $\mu$ itself is assumed isotropic.
Finding a single good direction
-------------------------------
Suppose $\nu$ is a log-concave probability measure on ${\mathbb{R}}^n$ whose barycenter lies at the origin. Recall that its isotropic constant is defined as: $$\label{eq:L_nu}
L_\nu := {\left\Vert\nu\right\Vert}_{L_\infty}^{\frac{1}{n}}
{\textrm{det Cov}}(\nu)^{\frac{1}{2n}} ~.$$ Since the isotropic constant $L_\nu$ satisfies $L_\nu \geq c > 0$ (see e.g. [@MilmanPajor; @K_psi2]), then according to Lemma \[lem:Zn\]: $$\label{eq:basic-est}
{\textrm{det Cov}}(\nu)^{\frac{1}{2n}} \gtrsim \frac{1}{{\left\Vert\nu\right\Vert}_{L_\infty}^{\frac{1}{n}}} \simeq \frac{{\textrm{V.Rad.}}(Z_n(\nu))}{\sqrt{n}} ~.$$
\[rem:LYZ-2\] Since $Z_n(\mu_x) \simeq Z_n(\mu)$ whenever $x \in
\frac{1}{2} \Lambda_n(\mu)$, we immediately see by (\[eq:basic-est\]) and (\[eq:L\_nu\]) that in this case: $${\textrm{det Cov}}(\mu_x)^{\frac{1}{2n}} \gtrsim \frac{{\textrm{V.Rad.}}(Z_n(\mu_x))}{\sqrt{n}}
\simeq \frac{{\textrm{V.Rad.}}(Z_n(\mu))}{\sqrt{n}} \simeq \frac{1}{{\left\Vert\mu\right\Vert}_{L_\infty}^{\frac{1}{n}}} \simeq \frac{{\textrm{det Cov}}(\mu)^{\frac{1}{2n}}}{L_\mu} ~,$$ as already noted in [@K_psi2 Formula (50)]. Using the lower bound on ${\textrm{V.Rad.}}(Z_p(\mu))$ given by Lemma \[lem\_low\], it follows that: $${\textrm{V.Rad.}}(Z_p(\mu)) \gtrsim \frac{\sqrt{p}}{L_\mu} {\textrm{V.Rad.}}(Z_2(\mu)) \quad , \quad \forall 1 \leq p \leq n ~,$$ recovering the extended Lutwak–Yang–Zhang lower-bound from Remark \[rem:LYZ-1\]. This however misses our goal in this section by a factor of $L_\mu$.
We next generalize the basic estimate (\[eq:basic-est\]) to handle other (say integer) values of $k$ between $1$ and $n$, by projecting onto a lower dimensional subspace:
\[lem:basic-est2\] Let $\nu$ denote a log-concave probability measure in ${\mathbb{R}}^n$ with barycenter at the origin, and let $k$ denote an integer between $1$ and $n$. Then: $$\label{eq:basic-est2}
\exists \theta \in S^{n-1} \;\;\; \sqrt{ \int_{\RR^n} {\left \langle x, \theta \right \rangle}^2 d \nu(x) } \geq
\frac{c}{\sqrt{k}} \sup_{E \in G_{n,k}} {\textrm{V.Rad.}}(Proj_E Z_k(\nu)) ~.$$
Given $E \in G_{n,k}$, apply (\[eq:basic-est\]) to $\pi_E \nu$ and note that $Z_k(\pi_E \nu) = Proj_E Z_k(\nu)$.
The idea now is to compare ${\textrm{V.Rad.}}(Proj_E Z_k(\mu_x))$ with ${\textrm{V.Rad.}}(Proj_E
Z_k(\mu))$. Note that if $Z_p(\nu) \simeq Z_p(\mu)$, then by (\[eq:Zp-inclusion\]): $$1 \leq q \leq p \;\;\; \Rightarrow \;\;\; c \frac{q}{p} Z_q(\mu)
\subset Z_q(\nu) \subset C \frac{p}{q} Z_q(\mu) ~,$$ and so ${\textrm{V.Rad.}}(Proj_E Z_k(\nu)) \geq c \frac{k}{p} {\textrm{V.Rad.}}(Proj_E
Z_k(\mu))$ for all $E \in G_{n,k}$, whenever $k \leq p$. To control ${\textrm{V.Rad.}}(Proj_E
Z_k(\mu))$, we have:
\[lem:controling-vr\] Let $\mu$ denote a log-concave probability measure in ${\mathbb{R}}^n$ with barycenter at the origin, and let $1 \leq k \leq q^\sharp(\mu)$. Then: $$\exists E \in G_{n,k} \;\;\; {\textrm{V.Rad.}}(Proj_E Z_k(\mu)) \geq c \sqrt{k}
{\textrm{det Cov}}(\mu)^{\frac{1}{2n}} ~.$$
Lemma \[lem:q-sharp\] asserts that $1 \leq k \leq q^\sharp(\mu)$ implies that $k^*(Z_k(\mu)) \geq k$. Consequently, there exists at least one (in fact, many) $E \in G_{n,k}$ so that: $$\frac{1}{2} W(Z_k(\mu)) B_E \subset Proj_E Z_k(\mu) \subset 2
W(Z_k(\mu)) B_E ~,$$ and hence ${\textrm{V.Rad.}}(Proj_E Z_k(\mu)) \geq \frac{1}{2} W(Z_k(\mu))$. It remains to appeal to Lemma \[lem:q-sharp\] again and deduce from $1 \leq k \leq q^\sharp(\mu)$ that $W(Z_k(\mu)) \geq c \sqrt{k}
{\textrm{det Cov}}(\mu)^{\frac{1}{2n}}$.
Combining all of the preceding discussion, we obtain the following fundamental:
\[prop:det-basic-est\] Let $\nu,\mu$ denote two log-concave probability measures in ${\mathbb{R}}^n$ with barycenters at the origin, and let $1 \leq p \leq n$. Assume that $Z_p(\nu) \simeq Z_p(\mu)$. Then: $$\exists \theta \in S^{n-1} \;\;\; \sqrt{ \int_{\RR^n} {\left \langle x,
\theta \right \rangle}^2 d \nu(x) } \geq c \min \left \{
1,\frac{q^\sharp(\mu)}{p} \right\} {\textrm{det Cov}}(\mu)^{\frac{1}{2n}} ~.$$
To avoid ambiguity of our notation, we explicitly remark that throughout this section, all statements which *assume* that $Z_p(\nu) \simeq Z_p(\mu)$, in fact apply whenever $\frac{1}{B} Z_p(\mu) \subseteq Z_p(\nu) \subseteq B Z_p(\mu)$ for *any* parameter $B \geq 1$, with the resulting constants in the conclusion of those statements depending in addition on $B$.
Controlling the entire ${\textrm{det Cov}}(\nu)$
------------------------------------------------
We can now proceed to control the entire ${\textrm{det Cov}}(\nu)$ by projecting onto the flag of subspaces spanned by the eigenvectors of ${\textrm{Cov}}(\nu)$. To apply Proposition \[prop:det-basic-est\], we require good control over $q^\sharp(\pi_E \mu)$. One way to obtain such control is to make a definition:
The Hereditary-$q^\sharp$ constant of a log-concave probability measure $\mu$ on ${\mathbb{R}}^n$, denoted $q^\sharp_H(\mu)$, is defined as: $$q^\sharp_H(\mu) := n \; \inf_k \inf_{E \in G_{n,k}}
\frac{q^\sharp(\pi_E \mu)}{k}
~.$$
\[rem:qH-iso\] It is useful to note the following alternative formula for $q^\sharp_H(\mu)$, valid only for an [*isotropic*]{}, log-concave probability measure $\mu$ on $\RR^n$. Recalling the definitions of $q^\sharp(\nu)$, $\Delta_\nu(q) = {\textrm{diam}}(Z_q(\nu))$, and using $\sup_{E \in G_{n,k}} {\textrm{diam}}(Proj_E Z_q(\mu)) =
{\textrm{diam}}(Z_q(\mu))$, we obtain: $$\label{eq:q_H-extended}
q^\sharp_H(\mu) = n \; \inf_{1 \leq k \leq n}
\frac{\Delta_\mu^{-1}(c^\sharp \sqrt{k})}{k} \simeq
n \; \inf_{1 \leq q \leq q^\sharp(\mu)} \frac{q}{{\textrm{diam}}(Z_q(\mu))^2} ~,$$ where we use (\[eq:Zp-inclusion\]) and our convention for when $q^\sharp(\nu) = 1$ to justify the last equivalence.
\[prop:Her\] Let $\nu,\mu$ denote two log-concave probability measures in ${\mathbb{R}}^n$ with barycenters at the origin, and assume that $\mu$ is isotropic. Let $1 \leq p \leq A q^\sharp_H(\mu)$ with $A \geq 1$, and assume that $Z_p(\nu) \simeq Z_p(\mu)$. Then: $${\textrm{det Cov}}(\nu)^{\frac{1}{2n}} \geq \frac{c}{A} ~,$$ where $c > 0$ denotes a universal constant.
Let $0 < \lambda_1 \leq \lambda_2 \leq \ldots \leq \lambda_n$ denote the eigenvalues of ${\textrm{Cov}}(\nu)$, and let $E_k \in G_{n,k}$ denote the subspace spanned by the eigenvectors corresponding to $\lambda_1,\ldots,\lambda_k$. Since $Proj_{E_k} Z_p(\nu) \simeq Proj_{E_k} Z_p(\mu)$, Proposition \[prop:det-basic-est\] applied to $\pi_{E_k} \nu$ and $\pi_{E_k}
\mu$ implies that: $$\sqrt{\lambda_k} \geq c \min{\left(1,\frac{q^\sharp(\pi_{E_k}
\mu)}{p}\right)} \geq c \min{\left(1,\frac{q^\sharp_H(\mu)}{p} \frac{k}{n}\right)}
\geq \frac{c}{A} \frac{k}{n} ~.$$ Taking geometric average over the $\lambda_k$’s, the assertion immediately follows.
\[rem:geom-avg\] It is clear from the proof that we may actually replace in the definition of $q_H^\sharp(\mu)$ the infimum over $k$ with a geometric-average over the terms. For future reference, we denote this variant by $q^\sharp_{GH}(\mu)$, and as in Remark \[rem:qH-iso\], obtain the following expression for it when $\mu$ is in addition *isotropic*: $$\label{eq:qGH}
q^\sharp_{GH}(\mu) = n \; {\left(\prod_{k=1}^n
\frac{\Delta_\mu^{-1}(c^\sharp \sqrt{k})}{k}\right)}^{\frac{1}{n}} \simeq
{\left(\prod_{k=1}^n \Delta_\mu^{-1}(c^\sharp
\sqrt{k})\right)}^{\frac{1}{n}} ~.$$
Another way to obtain some (partial) control over $q^\sharp(\pi_E
\mu)$ is to invoke Corollary \[cor:q-sharp-proj\]:
\[prop:log\] Let $\nu,\mu$ denote two log-concave probability measures in ${\mathbb{R}}^n$ with barycenters at the origin, and assume that $\mu$ is isotropic. Let $1 \leq p \leq n$ and $A \geq 1$. Assume that $Z_p(\nu) \simeq Z_p(\mu)$ and that: $$\label{eq:diam-log-assump}
{\textrm{diam}}(Z_p(\mu)) \sqrt{\log(p)} \leq A \sqrt{n} ~.$$ Then: $${\textrm{det Cov}}(\nu)^{\frac{1}{2n}} \geq \exp(-C A^2) ~.$$
We employ the same notation as in the previous proof. Setting: $$k_0 := \lceil (c^{\sharp})^{-2} {\textrm{diam}}^2(Z_p(\mu)) \rceil ~,$$ Corollary \[cor:q-sharp-proj\] states that $q^\sharp(\pi_{E_{k_0}} \mu) \geq p$. Consequently, applying Proposition \[prop:det-basic-est\] to $\pi_{E_{k_0}} \nu$ and $\pi_{E_{k_0}} \mu$, we obtain that $\lambda_{k_0} \geq c > 0$, and hence the largest $n-k_0+1$ eigenvalues of ${\textrm{Cov}}(\nu)$ are bounded below by the same $c > 0$. To bound the contribution of the other eigenvalues, we use (\[eq:Zp-inclusion\]) to obtain the following trivial bound (which may be improved, but ultimately only results in better numeric constants): $$\begin{aligned}
{\textrm{det Cov}}(\pi_{E_{k_0}} \nu)^{\frac{1}{2k_0}} &=& {\textrm{V.Rad.}}(Z_2(\pi_{E_{k_0}} \nu)) \gtrsim \frac{1}{p} {\textrm{V.Rad.}}(Z_p(\pi_{E_{k_0}} \nu)) \\
&\simeq& \frac{1}{p} {\textrm{V.Rad.}}(Z_p(\pi_{E_{k_0}} \mu)) \geq \frac{1}{p}
{\textrm{V.Rad.}}(Z_2(\pi_{E_{k_0}} \mu)) = \frac{1}{p} ~.\end{aligned}$$ Using our estimates separately on $E_{k_0}$ and $E_{k_0}^\perp$, we obtain: $${\textrm{det Cov}}(\nu)^{\frac{1}{2n}} = \left({\textrm{det Cov}}(\pi_{E_{k_0}} \nu)
{\textrm{det Cov}}(\pi_{E_{k_0}^\perp} \nu) \right)^{\frac{1}{2n}} \geq c
{\left(\frac{1}{p}\right)}^{\frac{k_0}{n}} ~.$$ Our assumption (\[eq:diam-log-assump\]) precisely ensures that $k_0 \log(p) \leq C \cdot A^2 n$, and the assertion follows.
Our choice of working in this section with $q^{\sharp}(\mu)$ instead of $q^*_c(\mu)$ is only a matter of convenience and is not of essence, as justified in Section \[sec4\].
Proofs of Main Theorems {#subsec:generalizations}
-----------------------
Theorem \[main\_thm3\] now follows immediately from Proposition \[prop:log\], combined with Propositions \[prop:vr-formula\] and \[prop:Zp-iso\]. Similarly, Proposition \[prop:Her\] and Remark \[rem:geom-avg\], combined with Propositions \[prop:vr-formula\] and \[prop:Zp-iso\], yield:
\[main\_thm2+\] Let $\mu$ denote an isotropic log-concave probability measure in ${\mathbb{R}}^n$. Then: $${\textrm{V.Rad.}}(Z_p(\mu)) \geq c \sqrt{p} \quad \quad , \quad \quad \forall \; 2 \leq p \leq C q^\sharp_H(\mu) ~.$$ Moreover, the same bound remains valid for $2 \leq p \leq C q^\sharp_{GH}(\mu)$.
Now if $\mu$ is a log-concave isotropic measure on ${\mathbb{R}}^n$ which is in addition a $\psi_\alpha$-measure with constant $b_\alpha$ (for $\alpha \in [1,2]$), by definition: $${\textrm{diam}}(Z_p(\mu)) \leq 2 b_\alpha p^{\frac{1}{\alpha}} ~.$$ It therefore follows immediately from (\[eq:q\_H-extended\]) that: $$q^\sharp_H(\mu) \geq \frac{c}{b_\alpha^\alpha} n^{\alpha/2} ~,$$ and thus Theorem \[main\_thm2\] follows from Theorem \[main\_thm2+\].
Lastly, it may be worthwhile to record the following generalization of Theorems \[main\_thm\] and \[main\_thm1+\], which follows immediately, as in Subsection \[subsec:slicing\], from Theorem \[main\_thm2+\] and (\[eq:qGH\]):
Let $\mu$ denote a log-concave probability measure in ${\mathbb{R}}^n$ with barycenter at the origin. Then: $$L_\mu \leq C {\left(\prod_{k=1}^n \frac{k}{\Delta_\mu^{-1}(c^\sharp
\sqrt{k})}\right)}^{\frac{1}{2n}} ~.$$
Observe that in this formulation, we only require an *on-average* control over the growth of $\Delta_\mu(p) = {\textrm{diam}}(Z_p(\mu))$, as opposed to all previously mentioned bounds on $L_\mu$.
Equivalence to the Slicing Problem {#sec_counter}
==================================
\[sec6\]
Denote: $$\label{eq_2246}
L_n := \sup_{K \subseteq \RR^n} L_K ~,$$ where the supremum runs over all convex bodies $K \subset \RR^n$. Recall that $K$ is called isotropic if $\mu_K$, the uniform measure on $K$, is isotropic. Recall also that $Z_p(K) = Z_p(\mu_K)$. In this section, we observe that removing the logarithmic factor in Theorem \[main\_thm3\] is in fact equivalent to Bourgain’s hyperplane conjecture.
\[thm:SlicingEquiv\] Given $n \geq 1$, the following statements are equivalent:
1. There exists $A>0$ so that $L_n \leq A$.
2. There exists $B>0$ so that for any isotropic convex body $K \subset {\mathbb{R}}^n$, we have: $$\label{eq:SlicingEquivAssumption}
{\textrm{V.Rad.}}(Z_p(K)) \geq \sqrt{p} / B \quad \quad \forall 1 \leq p \leq q^{\sharp}(\mu_K) / B ~.$$
The equivalence is in the sense that the parameters $A,B$ above depend solely one on the other, and not on the dimension $n$.
The proof is based on the following construction from Bourgain, Klartag and Milman [@BKM]. Given $m \geq 1$, let $K_m$ denote an isotropic convex body with $L_{K_m} \geq c L_m$. Choosing $c>0$ appropriately, it is well-known (see, e.g., the last remark in [@K_jfa]) that we may assume that $K_m$ is centrally-symmetric and satisfies $K_m \subset 10 \sqrt{m} B_m$. We also set $D_m :=
\sqrt{m+2} B_m$, and note that $D_m$ is isotropic. Given $1/n \leq
\lambda < 1$, consider the cartesian product: $$T_{\lambda} = K_{\lfloor \lambda n \rfloor} \times D_{\lceil (1-\lambda) n \rceil} \subseteq \RR^n ~.$$ Clearly, $T_{\lambda}$ is a centrally-symmetric isotropic convex body, and since $L_{D_m} \simeq 1$, it follows that: $$\label{eq:same-LK}
L_{T_{\lambda}} \simeq L_{\lfloor \lambda n \rfloor}^{\lfloor
\lambda n \rfloor / n} \simeq L_{\lfloor \lambda n \rfloor}^\lambda
~.$$
\[lem\_2335\] For any pair of centrally-symmetric convex bodies $K_1 \subset \RR^{n_1}, K_2 \subset \RR^{n_2}$ and $p \geq 1$, we have: $$\frac{1}{2} (Z_p(K_1) \times Z_p(K_2)) \subset Z_p(K_1 \times K_2) \subset Z_p(K_1) \times Z_p(K_2) ~.$$
Denote $E_1 := {\mathbb{R}}^{n_1} \times {\left\{0\right\}}$ and $E_2 := {\left\{0\right\}}
\times {\mathbb{R}}^{n_2}$. By definition, $Z_p(K_1 \times K_2) \cap E_1 =
Z_p(K_1) \times {\left\{0\right\}}$ and $Z_p(K_1 \times K_2) \cap E_2 =
{\left\{0\right\}} \times Z_p(K_2)$. By the symmetries of $K_1,K_2$ and convexity of $Z_p(K_1 \times K_2)$, it follows that: $$Z_p(K_1 \times K_2) \subseteq Z_p(K_1) \times Z_p(K_2) ~.$$ On the other hand, an elementary argument ensures that: $$Z_p(K_1 \times K_2) \supseteq conv(Z_p(K_1) \times {\left\{0\right\}} , {\left\{0\right\}}
\times Z_p(K_2)) \supseteq \frac{1}{2} {\left( Z_p(K_1) \times
Z_p(K_2) \right)} ~.$$
\[cor\_2344\] For any $1/n \leq \lambda \leq 1/2$: $${\textrm{diam}}(Z_{\lambda n}(T_{\lambda})) \leq C \sqrt{\lambda n} ~.$$
By Lemma \[lem\_2335\] we see that: $${\textrm{diam}}(Z_{\lambda n}(T_{\lambda})) \leq {\textrm{diam}}(Z_{\lambda n}(
K_{\lfloor \lambda n \rfloor})) + {\textrm{diam}}(Z_{\lambda n}(D_{\lceil
(1-\lambda) n \rceil})) ~.$$ Observe that ${\textrm{diam}}(Z_{\lambda n}(K_{\lfloor \lambda n \rfloor})) \leq {\textrm{diam}}(K_{\lfloor \lambda n \rfloor}) \leq 20 \sqrt{\lambda n}$. As for the other summand, a straightforward computation reveals that when $1/n \leq \lambda \leq 1/2$: $$Z_{\lambda n}(D_{\lceil (1-\lambda) n \rceil}) \simeq \sqrt{\lambda} \sqrt{n} B_{\lceil (1-\lambda) n \rceil} ~.$$ The assertion now follows.
Recall that for any isotropic convex body $K \subset \RR^n$: $$q^{\sharp}(K) = q^\sharp(\mu_K) := \sup {\left\{q \geq 1 ; {\textrm{diam}}(Z_q(K)) \leq c^\sharp \sqrt{n}\right\}} ~, \label{eq_2351}$$ where $c^\sharp > 0$ is an appropriate universal constant (as in Section \[sec4\]).
\[cor\_2355\] For any $n \geq 1$, there exists a centrally-symmetric isotropic convex body $K \subset \RR^n$, such that:
1. $\displaystyle q^{\sharp}(K) \geq c n$; and
2. $\displaystyle \log L_K \geq c \log L_n$,
where $c > 0$ is a universal constant.
Take $\lambda_0 := \min \{ (c^\sharp / C)^2,1/2 \}$, where $C$ is the constant from Corollary \[cor\_2344\]. Then $K = T_{\lambda_0}$ satisfies the first assertion in view of the choice of $\lambda_0$, and by (\[eq:same-LK\]): $$L_K \simeq L_{\lfloor \lambda_0 n \rfloor}^{\lambda_0} \gtrsim
L_n^{\lambda_0} ~,$$ where the inequality $L_{\lfloor \lambda n \rfloor} \gtrsim L_n$ for any $0 < c \leq \lambda \leq 1$ follows from the techniques in [@BKM Section 3]. Since $L_K \geq c > 0$, the second assertion follows.
If $L_n \leq A$, then ${\textrm{Vol}_n}(K)^{\frac{1}{n}} \geq 1/A$ for any isotropic convex body $K \subset {\mathbb{R}}^n$. Consequently, by the Lutwak–Yang–Zhang lower-bound (\[eq:LYZ-bound\]), we even have: $${\textrm{V.Rad.}}(Z_p(K)) \geq \frac{c}{A} \sqrt{p} \quad \quad \forall 1 \leq p \leq n ~.$$
For the other direction, apply our assumption (\[eq:SlicingEquivAssumption\]) to the isotropic convex body $K
\subset {\mathbb{R}}^n$ from Corollary \[cor\_2355\], and obtain: $$\frac{\sqrt{p}}{B} \leq {\textrm{V.Rad.}}(Z_p(K)) \leq {\textrm{V.Rad.}}(K) \simeq \frac{\sqrt{n}}{L_K} \quad \quad \forall 1 \leq p \leq q^\sharp(K) / B ~.$$ Corollary \[cor\_2355\] then implies that: $$L_n \leq (L_K)^C \leq {\left(C' B^{\frac{3}{2}} \sqrt \frac{n}{q^\sharp(K)}\right)}^C \leq C_1 B^{C_2} ~,$$ as required.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The Artificial Hormone System (AHS) is a self-organizing middleware to allocate tasks in a distributed system. We extended it by so-called negator hormones to enable conditional task structures. However, this extension increases the computational complexity of seemingly simple decision problems in the system: In [@HutterISORC2020] and [@HutterSENSYBLE2020], we defined the problems / and / and proved their NP-completeness. In this supplementary report to these papers, we show examples of / and /, introduce the novel problem / and explain *why* all of these problems involving negators are hard to solve algorithmically.'
author:
- Eric Hutter
- Mathias Pacher
- Uwe Brinkschulte
bibliography:
- 'Literature.bib'
title: |
On the Hardness of Problems Involving Negator Relationships\
in an Artificial Hormone System
---
Note {#note .unnumbered}
====
This is a supplementary report to [@HutterISORC2020] and [@HutterSENSYBLE2020], extending the results of both papers.
Although this report includes the most relevant information required for understanding, we nevertheless expect the reader to be familiar with at least one of the aforementioned papers.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The star formation rate (SFR) in the Central Molecular Zone (CMZ, i.e. the central 500 pc) of the Milky Way is lower by a factor of $\geq10$ than expected for the substantial amount of dense gas it contains, which challenges current star formation theories. In this paper, we quantify which physical mechanisms could be causing this observation. On scales larger than the disc scale height, the low SFR is found to be consistent with episodic star formation due to secular instabilities or variations of the gas inflow along the Galactic bar. [The CMZ is marginally Toomre-stable when including gas and stars, but highly Toomre-stable when only accounting for the gas, indicating that the condensation of self-gravitating clouds may be limited.]{} On small scales, we find that the SFR in the CMZ is consistent with an elevated critical density for star formation due to the high turbulent pressure – potentially aided by weak magnetic effects and an underproduction of massive stars due to a bottom-heavy initial mass function. The existence of a universal density threshold for star formation is ruled out, as well as the importance of the H[i]{}–H$_2$ phase transition of hydrogen, the tidal field, the magnetic field, radiation pressure, and cosmic ray heating. We propose observational and numerical tests to distinguish between the remaining candidate star formation inhibitors, in which ALMA will play a key role. We conclude the paper by proposing a [self-consistent cycle of star formation in the CMZ, in which the plausible star formation inhibitors are combined]{}. Their ubiquity suggests that the perception of a lowered central SFR should be a common phenomenon in other galaxies. We discuss the implications for [galactic star formation and supermassive black hole growth]{}, [including a prediction that the recently reported bimodality of star formation in high-redshift galaxies may emanate from a difference in the gas inflow rates]{}.'
author:
- |
J. M. Diederik Kruijssen,$^1$[^1] Steven N. Longmore,$^{2,3}$ Bruce G. Elmegreen,$^{4}$ Norman Murray,$^{5}$ John Bally,$^{6}$ Leonardo Testi$^{2,7}$ and Robert C. Kennicutt, Jr.$^{8}$\
$^{1}$Max-Planck Institut für Astrophysik, Karl-Schwarzschild-Straße 1, 85748 Garching, Germany\
$^{2}$European Southern Observatory, Karl-Schwarzschild-Straße 2, 85748, Garching, Germany\
$^{3}$Astrophysics Research Institute, Liverpool John Moores University, Egerton Wharf, Birkenhead CH41 1LD, United Kingdom\
$^{4}$IBM T. J. Watson Research Center, 1101 Kitchawan Road, Yorktown Heights, New York 10598, USA\
$^{5}$Canadian Institute for Theoretical Astrophysics, 60 St. George Street, University of Toronto, Toronto, ON M5S 3H8, Canada\
$^{6}$Center for Astrophysics and Space Astronomy, University of Colorado, Boulder, CO 80309, USA\
$^{7}$INAF-Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, I-50125 Firenze, Italy\
$^{8}$Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK
bibliography:
- 'mybib.bib'
date: 'Accepted X[xxxx]{} XX. Received X[xxxx]{} XX; in original form 2013 March 25.'
title: 'What controls star formation in the central 500 pc of the Galaxy?'
---
\[firstpage\]
Galaxy: centre — galaxies: evolution — galaxies: ISM — galaxies: starburst — galaxies: star formation — stars: formation
Introduction {#sec:intro}
============
Star formation in galactic discs is often described with a power law relation between the star formation rate surface density $\Sigma_{\rm SFR}$ and the gas surface density $\Sigma$ [@kennicutt98b]: $$\label{eq:sflaw}
\Sigma_{\rm SFR}=A_{\rm SK}\Sigma^N ,$$ as was originally proposed for volume densities by @schmidt59. [The typical range of power law indices is $N=1$–$2$, whether $\Sigma$ refers to dense or all gas, and across a range of spatial scales [@kennicutt98; @kennicutt98b; @bigiel08; @liu11; @lada12; @kennicutt12].]{} Star-forming discs in the local Universe follow a tight, ‘Schmidt-Kennicutt’ relation with $A_{\rm SK}=2.5\times10^{-4}$ and $N=1.4\pm0.1$ [@kennicutt98b], with $\Sigma_{\rm SFR}$ in units of ${\mbox{M$_\odot$}}~{\rm yr}^{-1}~{\rm kpc}^{-2}$ and $\Sigma$ in units of ${\mbox{M$_\odot$}}~{\rm pc}^{-2}$. [When using only molecular gas, $A_{\rm mol}=8\times10^{-4}$ and $N=1.0\pm0.2$ [@bigiel08; @bigiel11]. [Since $N>1$ indicates some effect of self-gravity, we refer to these relations as density-dependent star formation relations.]{} Another commonly-used expression is the Silk-Elmegreen [@silk97; @elmegreen87; @elmegreen93; @elmegreen97b] relation]{}: $$\label{eq:sflawomega}
\Sigma_{\rm SFR}=A_{\rm SE}\Sigma\Omega ,$$ with $\Omega$ the angular velocity at the edge of the star-forming disc and $A_{\rm SE}=0.017$ [@kennicutt98b] a proportionality constant, adopting the same units as before and writing $\Omega$ in units of ${\rm Myr}^{-1}$. [As for the case of $N>1$ in equation (\[eq:sflaw\]), the dependence on the angular velocity implies that this star formation relation is density-dependent.]{}
\
These [galactic-scale, global]{} star formation relations have the advantage that they are easily evaluated observationally, but the dependence of surface densities on projection suggests that more fundamental physics drive the observed scaling relations. Recent analyses of star formation in the solar neighbourhood have been used to propose a possibly universal, [local]{} volume density threshold for star formation $n_{\rm Lada}\sim10^4~{\rm cm}^{-3}$ above which most[^2] gas is converted into stars on a $\sim20~{\rm Myr}$ time-scale (@gao04 [@heiderman10; @lada10], although see @gutermuth11 and @burkert13 for an opposing conclusion). It has been argued that this threshold also holds on galactic scales [@lada12], in which case it would connect low-mass star forming regions to high-redshift starburst galaxies.
[It has been shown that $\Sigma_{\rm SFR}$ drops below the relations of equations (\[eq:sflaw\]) and (\[eq:sflawomega\]) beyond a certain galactocentric radius]{} [e.g. @martin01; @bigiel10] – the straightforward detection of cutoff radii is well-suited for testing star formation thresholds. However, these are also the regions of galaxies where the minority of the dense gas resides. The central 500 pc of our Galaxy (the Central Molecular Zone or CMZ) contains a few per cent of the total molecular gas mass in the Galaxy and also a few per cent of the total star formation. At a first glance, this suggests that the star formation rate in the Milky Way is simply directly proportional to the reservoir of molecular gas available to form stars. However, the average density of molecular gas in the CMZ is two orders of magnitude higher than that in the disc. Specifically, the CMZ contains $\sim80\%$ of the NH$_3(1,1)$ integrated intensity in the Galaxy [@longmore13], reflecting an overwhelming abundance of dense gas ($n>{\rm~a~few}\times10^3~{\rm cm}^{-3}$). These two facts are in direct contradiction with [the proposed volumetric and surface density star formation relations that]{} predict that a given mass of gas will form stars more rapidly if the density is higher [e.g. @kennicutt98b; @krumholz05; @padoan11; @krumholz12a]. In @longmore13, we show that these relations do indeed over-predict the measured star formation rate by factors of 10 to 100. [The low SFR in the CMZ is particularly striking because its gas surface density is similar to that observed in starburst galaxies, which seem to follow a sequence with a factor of ten *higher* SFRs than predicted by typical star formation relations [@daddi10b].]{} [If the SFR depends on the density]{}, something is required to slow the rate of star formation in the CMZ compared to the rest of the Milky Way and other galaxies.
[The SFR in the CMZ is also inconsistent with the @lada12 star formation relation, despite the fact that it does not rely on the density. Because most gas in the CMZ is residing at densities larger than the Lada threshold, a gas consumption time-scale of 20 Myr implies a SFR of $1~{\mbox{M$_\odot$}}~{\rm yr}^{-1}$, which is 1–2 orders of magnitude higher than the measured SFR [@longmore13]. In nearby disc galaxies, a simple proportionality of the SFR to the molecular mass is commonplace [@bigiel08; @bigiel11]. Despite the similarities between the CMZ and high-redshift galaxies [@kruijssen13], which do form stars at or above the rate predicted by density-dependent star formation relations, the CMZ is consistent with an extrapolation of the molecular star formation relation with a constant H$_2$ depletion time-scale (i.e. $N=1$ in equation \[eq:sflaw\], also see below). This is surprising – self-gravity implies that dynamical evolution proceeds faster when the density is higher (i.e. $N>1$). If gravity is an important driver of star formation in the CMZ and galaxy discs, then a constant molecular gas depletion time-scale requires that some resistance to the gravitational collapse towards stars must increase at a rate comparable to self-gravity when the density goes up. Due to its extreme characteristics, the CMZ is the prime region to understand the underlying physics.]{}
[In this paper, we take the point of view that starbursts and other regions which suggest $N>1$ (and thereby imply a density-dependence to the SFR) do not seem to apply to the CMZ in the Milky Way. To understand this difference better,]{} we evaluate the global and local processes that affect the rate of star formation in the central region of the Milky Way. Thanks to the strongly contrasting environments of central and outer galactic regions, galaxy centres provide a unique and independent way to study the universality of star formation relations. We exploit this contrast to discuss the implications of the lack of star formation in the CMZ for existing star formation relations with $N>1$.
The paper is concluded by sketching a possible picture of how local and global star formation criteria connect, and we propose observational and numerical tests through which the different components of this picture can be addressed in more detail.
----------------------------------- ------------------- -------------- -------------- ----------- ---------- ---------------- -------------------- ---------------- -------------------- ----------------
Object ID $X_{{\rm CO},20}$ $\Sigma_{2}$ $\kappa$ $\sigma$ $h_2$ $Q_{\rm gas}$ $\Sigma_{\star,2}$ $Q_{\rm tot}$ $\Sigma_{\rm SFR}$ $t_{\rm depl}$
(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)
CMZ 230 pc-integrated - $1.2$ $0.75$–$1.2$ $20$–$50$ $0.5$ $(9.0$–$36)$ $38$ $(1.2$–$2.1)$ $0.20$ $(0.6)$
CMZ 1.3$^\circ$ cloud - $2.0$ $0.88$–$1.5$ $20$–$30$ $0.5$ $(6.4$–$16)$ $20$ $(2.1$–$4.0)$ $0.13$ $(1.5)$
CMZ 100-pc ring - $30$ $1.6$–$3.3$ $25$–$50$ $0.1$ $(0.95$–$4.0)$ $29$ $(0.76$–$2.7)$ $3.0$ $(1.0)$
@kennicutt98b low $\Sigma$ disc $2.8$ $0.05$ $(0.04)$ $(2.6)$ $(0.5)$ $1.5$ - - $0.0024$ $(2.1)$
@kennicutt98b high $\Sigma$ disc $2.8$ $0.20$ $(0.07)$ $(5.9)$ $(0.6)$ $1.5$ - - $0.017$ $(1.2)$
@daddi10b low $\Sigma$ starburst $0.4$ $3.0$ $(0.34)$ $(6.1)$ $(0.05)$ $1.5$ - - $3.9$ $(0.08)$
@daddi10b high $\Sigma$ starburst $0.4$ $10^2$ $(0.80)$ $(86)$ $(0.3)$ $1.5$ - - $560$ $(0.02)$
@daddi10b low $\Sigma$ BzK $1.8$ $2.0$ $(0.05)$ $(55)$ $(6)$ $1.5$ - - $0.28$ $(0.7)$
@daddi10b high $\Sigma$ BzK $1.8$ $10$ $(0.07)$ $(200)$ $(10)$ $1.5$ - - $2.7$ $(0.4)$
----------------------------------- ------------------- -------------- -------------- ----------- ---------- ---------------- -------------------- ---------------- -------------------- ----------------
\
$X_{{\rm CO},20}\equiv X_{\rm CO}/10^{20}~({\rm K}~{\rm km}~{\rm s}^{-1}~{\rm cm}^2)^{-1}\approx0.53\alpha_{\rm CO}/{\mbox{M$_\odot$}}~({\rm K}~{\rm km}~{\rm s}^{-1}~{\rm pc}^2)^{-1}$ is the CO-to-H$_2$ conversion factor, $\Sigma_{2}\equiv\Sigma/10^2~{\mbox{M$_\odot$}}~{\rm pc}^{-2}$ is the gas surface density, $\kappa$ is the epicyclic frequency in units of ${\rm Myr}^{-1}$, $\sigma$ is the velocity dispersion in units of ${\rm km}~{\rm s}^{-1}$, $h_2\equiv h/10^2~{\rm pc}$ is the scale height, $Q_{\rm gas}$ is the @toomre64 disc stability parameter [when only including the self-gravity of the gas, $\Sigma_{\star,2}\equiv\Sigma_\star/10^2~{\mbox{M$_\odot$}}~{\rm pc}^{-2}$ is the stellar surface density, $Q_{\rm tot}$ includes the self-gravity of gas [*and*]{} stars]{}, $\Sigma_{\rm SFR}$ is the star formation rate density in units of ${\mbox{M$_\odot$}}~{\rm yr}^{-1}~{\rm kpc}^{-2}$, and $t_{\rm depl}\equiv\Sigma\Sigma_{\rm SFR}^{-1}/{\rm Gyr}$ is the gas depletion time. [Values in parentheses are implied by the other numbers (see text).]{}
Throughout the paper, we adopt a mean molecular weight of $\mu=2.3$, implying a mean particle mass of $\mu m_{\rm H}=3.9\times10^{-24}~{\rm g}$.
Observational constraints from the Central Molecular Zone {#sec:obs}
=========================================================
@longmore13 present the observational constraints for the suppression of star formation in the CMZ, which are summarised here. [A division is made between [*global*]{} and [*local*]{} physics, where ‘global’ refers to spatial scales larger than the disc scale height ($\Delta R>h$), on which the ISM properties are set by galactic structure, and ‘local’ refers to spatial scales smaller than the disc scale height ($\Delta R<h$), on which the ISM properties are set by microphysics. [For instance, the formation of giant molecular clouds (GMCs) proceeds on global scales, whereas processes internal to the GMCs, such as stellar feedback, the turbulent cascade, or magnetic fields then determine the fraction of the GMC mass that proceeds to star formation.]{}]{}
Global constraints {#sec:globalobs}
------------------
[Figure \[fig:img\] shows a three-colour composite of the central 4$^\circ$ of the Milky Way, corresponding to a spatial scale of $\sim600~{\rm pc}$ at the distance of the Galactic centre [we adopt $8.5$ kpc, which is consistent with @reid09]. We combine the dense gas emission (NH$_3(1,1)$, in red) with infrared imaging (green and blue) to highlight the gas close to (or above) the @lada10 density threshold, young stellar objects, and evolved stars. The legend indicates the objects in the CMZ that are most relevant to this paper. In particular, the 1.3$^\circ$ cloud and the 100-pc, twisted ring of dense gas clouds between Sgr C and Sgr B2 [@molinari11] will be scrutinised in detail. The G0.253+0.016 cloud (the ‘Brick’) was recently identified as a possible progenitor of a young massive cluster [@longmore12], and we will use it as a template for the massive and dense clouds that populate the CMZ.]{}
In Table \[tab:prop\], we list the derived properties of the gas in the CMZ, as well as the typical characteristics of galaxies as defined by existing, [empirical]{} star formation relations. The columns indicate (1) the object ID, (2) the CO–H$_2$ conversion factor [used to derive the gas mass]{}, (3) the gas surface density, (4) the epicyclic frequency, (5) the velocity dispersion, (6) the scale height, (7) the @toomre64 $Q$ stability parameter of the gas disc (see §\[sec:surfacethr\]), [(8) the stellar surface density, (9) the Toomre $Q$ parameter of the entire disc]{}, (10) the observed star formation rate surface density, and (11) the gas depletion time-scale. The first three rows list the properties of the CMZ, where we distinguish two components (the 100-pc ring described by @molinari11 and the $1.3^\circ$ cloud), as well as its properties [smeared out]{} over a radius of 230 pc (corresponding to 1.55$^\circ$, i.e. just beyond the 1.3$^\circ$ cloud). This division into sub-regions is made because the 100-pc ring and the $1.3^\circ$ cloud contain most of the [dense (and thereby total)]{} gas mass within $|l|\la1.5^\circ$.
The total gas mass within 230 pc is taken from @ferriere07, whereas the observed gas surface density in the plane of the sky of the 100-pc ring and the $1.3^\circ$ cloud is derived from the HiGAL data [@molinari10] following the analysis in @longmore13. Note that this estimate [relies on dust emission and is insensitive to]{} the conversion of CO intensity to H$_2$ column density (the ratio of which is represented by the parameter $X_{\rm CO}$), and hence is not affected by any uncertainty in $X_{\rm CO}$ [cf. @sandstrom13 and references therein]. [However, the derived column density does depend on the assumed gas-to-dust ratio, which may be lower in the CMZ than in the Galactic disc due to the higher metallicity (see @longmore13 for a discussion).]{} At the high densities of the CMZ, almost all of the gas is in molecular form. The 3D geometry of the 100-pc ring was taken from @molinari11 [assuming a ring thickness of 10 pc], whereas the face-on projected geometry of the $1.3^\circ$ cloud as an ellipse with semi-major and minor axes of 85 pc and 50 pc is taken from @sawada04. [We adopt the rotation curve of @launhardt02 [Figure 14] to calculate the epicyclic frequency.[^3]]{} The velocity dispersion and scale heights of the dense gas in both the 100-pc ring and the $1.3^\circ$ cloud are determined from the HOPS NH$_3$(1,1) emission [@walsh11; @purcell12]. The value ranges in column 5 of Table \[tab:prop\] represent the maximum and minimum measured velocity dispersions across each region. We calculate $Q_{\rm gas}$ from the preceding columns as in equation (\[eq:Q\]) in §\[sec:surfacethr\] below. [[As can be seen in Table \[tab:prop\], the stellar component dominates the potential. We obtain]{} the stellar surface density from the adopted rotation curve [@launhardt02], by deriving the spherically symmetric volume density profile and only including the stellar mass in a slab of thickness of $2h$. Therefore, the total stellar surface density of the bulge at the same galactocentric radii is higher than listed in Table \[tab:prop\]. Column 9 gives Toomre $Q_{\rm tot}$, which is corrected for the presence of stars as in equation (\[eq:Qtot\]) in §\[sec:surfacethr\] below, using a stellar velocity dispersion of $\sigma_\star\sim100~{\rm km~s}^{-1}$ [@dezeeuw93]. Note that if we assume that the 1.3$^\circ$ cloud is roughly spherically symmetric, its virial parameter [@bertoldi92] $\alpha\sim2$ indicates it is roughly in equilibrium, but only when including the stellar gravity. If the stars would be absent, the cloud would be highly unbound and hence stable against gravitational collapse. A similar effect of the stellar potential is seen when comparing Toomre $Q$ of columns 7 and 9 for the 230 pc-integrated CMZ. By contrast, the $Q$ parameter of the 100-pc ring is hardly affected by the presence of stars.]{} The SFR densities $\Sigma_{\rm SFR}$ are derived using Table 2 of @longmore13, [which lists the SFR for certain parts of the CMZ as derived from free-free emission [@lee12]. For details on the derivation, we refer to the discussion in @longmore13.]{} [The gas depletion time follows from $\Sigma$ and $\Sigma_{\rm SFR}$ as $t_{\rm depl}\equiv\Sigma\Sigma_{\rm SFR}^{-1}$.]{} [While the listed values reflect the depletion times of all gas, i.e. including both H[i]{} and H$_2$, it is important to note that most of the gas is in molecular form. Only for the disc sample from @kennicutt98b this does not hold, where the molecular depletion times are lower than those for all gas listed in Table \[tab:prop\].]{}
The fourth and fifth rows of Table \[tab:prop\] span the star formation relation of equations (\[eq:sflaw\]–\[eq:sflawomega\]) for the disc galaxy sample used in @kennicutt98b. The bottom rows span the high-SFR[^4] and low-SFR sequences of @daddi10b. [At a given surface density, the typical epicyclic frequencies of the @kennicutt98b galaxies are obtained by combining equations (\[eq:sflaw\]) and (\[eq:sflawomega\]), whereas for @daddi10b they follow from their equations (2) and (3). The extragalactic velocity dispersions are obtained by assuming $Q=1.5$ and using the definition of $Q$ (see §\[sec:surfacethr\]). The characteristic disc scale heights are added to distinguish between the aforementioned ‘global’ and ‘local’ regimes, and are calculated by assuming an equilibrium disc including a factor-of-two increase of the disc self-gravity due to the presence of stars [cf. @elmegreen89; @martin01].]{}
We compare the data from Table \[tab:prop\] to the global star formation relations of equations (\[eq:sflaw\]) and (\[eq:sflawomega\]) in Figure \[fig:kslaw\]. The spatially resolved elements of the CMZ (open and closed, black symbols) are forming stars at a rate that is [a factor of 3–20 (i.e. typically an order of magnitude)]{} below either relation. By contrast, the Schmidt-Kennicutt relation does describe the CMZ well when spatially smoothing it over a 230 pc radius (the closed, grey symbol), whereas smoothing hardly affects the agreement with the Silk-Elmegreen relation. It is not obvious whether this simply implies that the global star formation relations break down at spatial scales smaller than $\sim500$ pc. Smoothing is justified on galactic ($\ga500$ pc) scales, because the lifetime of substructure on these scales is typically of the order of (or shorter than) an orbital timescale, and accounting for substructure therefore only introduces spurious stochasticity [e.g. @schruba10]. However, the nuclear rings that appear in numerical simulations of barred galaxies are persistent over many dynamical times [e.g. @piner95; @kim12c], and hence it seems physically incorrect to smear out the 100-pc ring and any other, possibly persistent structure in the CMZ to a much larger scale when comparing to global star formation relations. These structures should be accounted for when describing the star formation (or lack thereof) in the CMZ. Therefore, the physically appropriate representation of the CMZ in Figure \[fig:kslaw\] is given by the open and closed, black symbols.
\
[Also visible in the left-hand panel of Figure \[fig:kslaw\] is that the CMZ (which is nearly completely molecular) agrees well with the @bigiel08 star formation relation between the SFR and the molecular gas mass (shown as the lower dotted line), which was derived for nearby disc galaxies with surface densities $3<\Sigma_{{\rm H}_2}/{\mbox{M$_\odot$}}~{\rm pc}^{-2}<50$. The CMZ extends this range by roughly two orders of magnitude. @bigiel08 [@bigiel11] show that the gas depletion time of the galaxies in their sample is $t_{\rm depl}=1$–2 Gyr, which is indeed consistent with time-scales listed for the CMZ in Table \[tab:prop\]. In the framework of this relation, the CMZ is somehow the norm while both the @daddi10b samples of BzK [@tacconi10; @daddi10] and starburst/submillimeter galaxies [@kennicutt98b; @bouche07; @bothwell10] are the exception. The Bigiel relation is the only global star formation relation that fits the CMZ – the region remains anomalous in the context of equations (\[eq:sflaw\]) and (\[eq:sflawomega\]), or the @lada12 relation (see §\[sec:localobs\]). As will be shown in §\[sec:thresholds\], there are several possible reasons why the CMZ is peculiar.]{}
Local constraints {#sec:localobs}
-----------------
Turning to local physics, in @longmore13 [Figure 4] we have shown that 70–90 per cent of the gas in the CMZ resides at column densities above the Lada column density threshold, and hence should be forming stars. Throughout this paper, we use the corresponding volume density threshold of $n_{\rm Lada}\sim10^4~{\rm cm}^{-3}$ [@lada10]. [While this may apply for the low ($\sim10^2~{\rm cm}^{-3}$) densities of GMCs in the solar neighbourhood, the mean volume density of the gas in the CMZ is $n_0\sim2\times10^4~{\rm cm}^{-3}$ [@longmore13], i.e. more than two orders of magnitudes higher, and it hosts GMCs with typical densities of $\sim10^5~{\rm cm}^{-3}$.]{} [As mentioned in §\[sec:intro\], the threshold for star formation and depletion time-scale from @lada12 would imply a SFR that is 1–2 orders of magnitude higher than the measured value [@longmore13]. If we assume that the SFR is driven by self-gravity, we can use the observed SFR to derive the fraction of the gas mass residing above some volume density threshold for star formation.]{} Given that the free-fall time above $n_{\rm Lada}$ is $t_{\rm ff}<1$ Myr and that the observed depletion time-scale of molecular hydrogen in the CMZ is $\sim1~{\rm Gyr}$ (see the last column of Table \[tab:prop\]), we conclude that if a local density threshold for star formation $n_{\rm th}$ exists, only $\phi_t t_{\rm ff}/\epsilon_{\rm core}t_{\rm depl}\la0.5$ per cent of the gas mass in the CMZ [*should*]{} be above this threshold.[^5]
The density probability distribution function (PDF) ${\rm d}p/{\rm d}n$ of a turbulent interstellar medium (ISM) is often represented by a log-normal, of which the width and dispersion are set by the Mach number [e.g. @vazquez94; @padoan97]. However, the high-density tail can also (at least locally) be approximated by a power law, i.e. ${\rm d}p/{\rm d}n\propto n^{-\beta}$ [e.g. @klessen00b; @kritsuk11; @elmegreen11; @hill12]. This can be used to estimate the required value of $n_{\rm th}$ in the CMZ. As mentioned above, the fraction of gas that is used to form stars is a factor of $f_{{\rm SFR},n}\sim0.005$ lower than the fraction above the Lada threshold. This reduction implies that for the assumption of a power law density PDF $n_{\rm th}=f_{{\rm SFR},n}^{1/(2-\beta)}n_{\rm Lada}$. This relation is shown in Figure \[fig:nth\], which gives $4\times10^8>n_{\rm th}/{\rm cm}^{-3}>10^5$ for exponents $2.5<\beta<4$. The density PDF of the gas in the CMZ is unknown, but the typical high-density slope due to self-gravity in the numerical simulations of @kritsuk11 is $\beta=2.5$–$2.75$, which suggests $n_{\rm th}=10^7$–$4\times10^8~{\rm cm}^{-3}$. [Note that because a power-law tail implies some effect of self-gravity, this functional form should fail to describe the detailed shape of the PDF in the low density-regime where self-gravity is not important. This needs to be kept in mind when choosing a roughly representative value of $\beta$.]{}
For a log-normal PDF with a Mach number of ${\cal M}\sim70$ (cf. Table \[tab:prop\], with a temperature of $T=65$ K as in @ao13 – also see @morris83) and a mean density of $n_0=2\times10^4~{\rm cm}^{-3}$, the required threshold density is even higher at $n_{\rm th}\sim2\times10^9~{\rm cm}^{-3}$. However, this does not account for the influence of the high magnetic field strength [$B\sim100\mu{\rm G}$, @crocker10] near the Galactic Centre. For a thermal-to-magnetic pressure ratio $2c_{\rm s}^2/v_A^2<1$, with $c_{\rm s}$ the sound speed and $v_A$ the Alfvén velocity, the dispersion of the log-normal density PDF is suppressed. Assuming $T=65$ K, we find $2c_{\rm s}^2/v_A^2=0.31$ for the CMZ. If we modify the density PDF accordingly (cf. @padoan11 [@molina12], adopting $B\propto n^{1/2}$ as in @padoan99), the threshold density required by the observed SFR becomes $n_{\rm th}\sim5\times10^8~{\rm cm}^{-3}$. We conclude that the CMZ firmly rules out a density threshold at $n=10^4~{\rm cm}^{-3}$, and adopt a lower limit of $n_{\rm th}=10^7~{\rm cm}^{-3}$. [This is still exceptionally high in comparison to the threshold density of nearby disc galaxies, and is only known to be reached in dense, rapidly star-forming galaxies at high redshift [e.g. @swinbank11].]{}
\
Possible mechanisms for inhibiting star formation {#sec:thresholds}
=================================================
In this section, we summarize and quantify which physical mechanisms may limit the SFR in the central regions of galaxies [with respect to the SFRs predicted by density-dependent star formation relations]{}. The implications of the candidate inhibitors for existing star formation relations are discussed in §\[sec:impl\], where we also propose ways of distinguishing their relative importance observationally. [This section is separated into global and local star formation inhibitors. We discuss in §\[sec:disc\] how these can be connected.]{}
Globally inhibited star formation {#sec:global}
---------------------------------
[We first discuss the possible mechanisms that may limit star formation in the CMZ on spatial scales larger than the disc scale height ($\Delta R>h$). A lack of star formation may be caused by a stability of the gas disc against gravitational collapse, or by global dynamical processes that limit the duration of star formation – either stochastically or due to long-term dynamics.]{}
### Disc stability {#sec:surfacethr}
A recurring question regarding galactic-scale star formation relations has been in which part of the parameter space they apply – is there a threshold density below which star formation is negligible? And if so, by which factor is the SFR reduced? The existence of such a threshold is suggested by the sharply truncated H[ii]{} discs in galaxies, which indicate that beyond a certain galactocentric radius [$\Sigma_{\rm SFR}$ falls off more rapidly than suggested by equations (\[eq:sflaw\]–\[eq:sflawomega\])]{} [@kennicutt89; @martin01; @bigiel10]. [Note that a radial truncation is absent for the molecular ($N=1$) star formation relation [@schruba11].]{} The possible physics behind surface density thresholds are extensively discussed by @leroy08, but we briefly summarize them here.
[The efficiency of galactic star formation may be]{} related to the global gravitational (in)stability of star-forming discs – if the kinematics of a gas disc are such that it can withstand global collapse, star formation is suppressed [@toomre64; @quirk72; @fall80; @kennicutt89; @martin01]. [There are indications]{} that the threshold density for gravitational instability not only [applies globally]{}, but also locally for the spiral arms and the inter-arm regions of M51 [@kennicutt07]. The density threshold for gravitational instability [in galaxy discs]{} is based on the @toomre64 $Q$ parameter: $$\label{eq:Q}
Q_{\rm gas}=\frac{\sigma\kappa}{\pi G\Sigma} ,$$ with $\sigma$ the one-dimensional gas velocity dispersion and $\kappa$ the epicyclic frequency, which is a measure for the Coriolis force [in a condensing gas cloud]{}: $$\label{eq:kappa}
\kappa=\sqrt{2}\frac{V}{R}\left(1+\frac{{{\rm d}}{\ln{V}}}{{{\rm d}}{\ln{R}}}\right)^{1/2},$$ with $V$ the circular velocity, $R$ the galactocentric radius, and $\Omega\equiv V/R$ the angular velocity. Note that for a flat rotation curve we have $\kappa=1.41\Omega$, whereas for solid-body rotation $\kappa=2\Omega$. In gas discs with $Q<1$, [the self-gravity of contracting clouds is sufficient to overcome the Coriolis force and undergo gravitational collapse]{}, whereas $Q>1$ indicates stability by kinetic support. Galaxy discs typically self-regulate to $Q\sim1$ [e.g. @martin01; @hopkins12], with an observed range of 0.5 to 6 for galaxies as a whole [e.g. @kennicutt89; @martin01], and an even larger variation within galaxies (see below and @martin01). [If a substantial mass fraction of the disc is constituted by stars, the disc can be unstable even though $Q_{\rm gas}>1$. In that case we write [cf. @wang94; @martin01]: $$\label{eq:Qtot}
Q_{\rm tot}=Q\left(1+\frac{\Sigma_\star}{\Sigma}\frac{\sigma}{\sigma_\star}\right)^{-1}\equiv Q\psi^{-1} ,$$ with $\sigma_\star$ the stellar velocity dispersion.]{}
[If star formation is driven by global disc instability, then the inversion of]{} equations (\[eq:Q\]) and (\[eq:Qtot\]) leads to a critical surface density for star formation, modulo a proportionality constant $\alpha_Q$: $$\label{eq:surfQ}
\Sigma_{\rm Toomre}=\alpha_Q\frac{\sigma\kappa}{\pi G\psi}\equiv\frac{\sigma\kappa}{\pi GQ_{\rm crit}\psi} ,$$ [where $\psi=1$ if the contribution of stars to the gravitational potential is neglected (in the solar neighbourhood $\psi\sim1.4$, see @martin01).]{} @kennicutt89 and @martin01 empirically determined for nearby disc galaxies that $\alpha_Q=0.5$–$0.85$ with a best value of $\alpha_Q\sim0.7$, indicating that the critical Toomre $Q$ parameter for star formation is $Q_{\rm crit}=1.2$–$2$ or $Q_{\rm crit}\sim1.4$. The threshold density depends on $\sigma$ and $\kappa$, both of which increase towards the Galactic centre [e.g. @morris96; @oka01; @longmore13; @shetty12], and hence it is possible that gravitational stability inhibits star formation in the CMZ.
A different, but closely related form of global disc stability to self-gravity is due to rotational shear, which may compete with self-gravity and prevent the collapse of the disc to form stars [@goldreich65; @elmegreen87; @elmegreen91; @hunter98]. Rather than the epicyclic frequency of equation (\[eq:surfQ\]), this threshold depends on the [shear time, i.e. the time available for gas instabilities to arise during the shear-driven density growth of spirals [@elmegreen87; @elmegreen93; @elmegreen97b]. [This shear condition may be more relevant than the Toomre condition if the angular momentum of a growing gas perturbation is not conserved, as might be the case in the presence of magnetic fields or viscosity.]{} The shear time-scale is the inverse of]{} the @oort27 constant $A$: $$\label{eq:oort}
A_{\rm Oort}=-0.5R\frac{{{\rm d}}\Omega}{{{\rm d}}R} .$$ The critical density for self-gravity to overcome shear is then $$\label{eq:surfshear}
\Sigma_{\rm Oort}=\alpha_A\frac{\sigma A_{\rm Oort}}{\pi G\psi} ,$$ with $\alpha_A\approx2.5$ [@hunter98]. Like $\kappa$, $A_{\rm Oort}$ decreases with galactocentric radius, and hence shear motion may be responsible for the suppression of star formation in the CMZ.
[Before continuing, we should caution that the Toomre or shear stability of a gas disc can only act as a ‘soft’ threshold. There are several ways in which such a threshold may be violated, and as a result it represents a soft separation between star-forming and quiescent gas discs. For instance, small-scale structure (e.g. spirals or bars), turbulent dissipation, magnetic stripping of angular momentum, or a soft equation of state could all lead to local star formation in a disc that is globally stable to star formation.]{}
The overdensities $\Sigma/\Sigma_{\{\rm Toomre,Oort\}}$ with respect to the surface density thresholds for gravitational instability and overcoming rotational shear are shown in Figure \[fig:stability\] as a function of galactocentric radius for a simple model of the Milky Way. For $R\leq0.3$ kpc, we only include the three data points from Table \[tab:prop\], [at galactocentric radii of $R=\{0.08,0.19,0.23\}$ kpc for the 100-pc ring, 1.3$^\circ$ cloud, and the 230-pc integrated CMZ, respectively. The resulting overdensities are shown with and without the stellar contribution to the gravitational potential. For $R\geq3$ kpc, we use the @wolfire03 model for the ISM surface density profile, including a factor of 1.4 to account for the presence of helium, and adopt $\sigma=7~{\rm km}~{\rm s}^{-1}$ [@heiles03]. [At these galactocentric radii, the]{} epicyclic frequency $\kappa$ and the Oort constant $A$ are calculated using the rotation curve from @johnston95. Our conclusions are unaffected when using other Milky Way models. Finally, we correct for the presence of spiral arms by dividing $\Sigma_{\{\rm Toomre,Oort\}}$ by a factor of two [c.f. @balbus88; @krumholz05].]{}
\
Figure \[fig:stability\] shows that the Milky way disc is unstable to star formation in a ring covering $4\la R/{\rm kpc}\la 17$. [Especially the outer edge of the star-forming disc catches the eye]{}. The difference between the Toomre and Oort thresholds across the disc is generally too small to indicate with certainty which mechanism dominates.
Only in the 100-pc ring we tentatively find $\Sigma_{\rm Toomre}>\Sigma_{\rm Oort}$, which indicates a decreasing importance of shear as the
circular velocity decreases for $R<0.1$ kpc, but the difference is less than $1\sigma$. A more substantial change is brought about by including the presence of stars in the calculation of $Q$ (red lines in Figure \[fig:stability\]). When the stellar gravity is ignored, the CMZ outside of the 100-pc ring is highly stable to gravitational collapse with $\Sigma/\Sigma_{\{\rm Toomre,Oort\}}\sim0.1$, suggesting that star formation could potentially be suppressed. However, when the stars are included, the CMZ appears marginally stable – the range of $Q_{\rm tot}$ measured in the CMZ is in fact very similar to that observed in normal disc galaxies [compare Table \[tab:prop\] to @martin01], which suggests a similar degree of self-regulation. The paucity of star formation in the CMZ is therefore not due to the ‘morphological quenching’ [@martig09] of star formation that is considered to enable the long-term presence of quiescent gas reservoirs in galactic spheroids if $Q_{\rm tot}>1$.
The high $Q_{\rm gas}$ may still slow down the condensation of self-gravitating gas clouds and their decoupling from the stellar background potential. [The time-scale for clouds to become gravitationally unstable is $t_{\rm grav}\sim Q_{\rm gas}/\kappa$ [e.g. @jogee05]. If the SFR is limited by slow cloud condensation, this therefore implies a decrease of the SFR by $1/Q_{\rm gas}$. This simple modification of the Silk-Elmegreen relation is consistent with the observed SFR for the 230 pc-integrated CMZ, but it does not explain the other components of the CMZ.]{} Note in particular that the 100-pc ring is always marginally Toomre-stable, both in terms of $Q_{\rm gas}$ and $Q_{\rm tot}$, which is also consistent with its clumpy, beads-on-a-string morphology [@longmore13b]. [We return to this point in §\[sec:total\].]{}
### Episodic star formation {#sec:episodic}
While it is tempting to assume that the CMZ is a steady-state system, the orbital and free-fall time-scales in the CMZ are so short (i.e. a few Myr) that the current reservoir of dense gas may not be related to the observed star formation tracers, which originate from gas that was present at least one dynamical time-scale ago. The CMZ agrees with the Schmidt-Kennicutt relation when averaged over a size scale of $R\sim230$ pc, which corresponds to an orbital period of 14 Myr (see Table \[tab:prop\]). It cannot be ruled out that these are the size and time-scales on which the variability of star formation is sampled well enough to correlate the presently available dense gas with the star formation tracers. [Note that the variation of the SFR cannot be too substantial: the time-integral of the current SFRs in the 100-pc ring and the 230-pc integrated CMZ over a Hubble time gives a total stellar mass that is roughly consistent with the total stellar mass enclosed at these radii [cf. @launhardt02].[^6]]{}
If the star formation in the CMZ is episodic, it needs to be established which physics could be driving the variability. Could the low SFR simply be a stochastic fluctuation? At $0.015~{\mbox{M$_\odot$}}~{\rm yr}^{-1}$ [@longmore13], the 100-pc ring produces about $10^5~{\mbox{M$_\odot$}}$ per dynamical time $t_{\rm dyn}\equiv2\pi/\Omega$, which corresponds to 2–3 young massive clusters (YMCs) and is thus consistent with the presence of the Arches and Quintuplet clusters.[^7] If the SFR were consistently an order of magnitude higher (as predicted by [density-dependent]{} star formation relations), then $\sim25$ such YMCs would be expected. This would imply that the present cluster population in the CMZ is a $\ga4.5\sigma$ deviation. It is thus highly unlikely that the observed SFR is due to simple Poisson noise.
[Episodic star formation should be a common process in the centres of barred galaxies – in the inner Lindblad resonance rings, gravitational instabilities can drive the fragmentation of the nuclear ring and eventually induce a starburst, but this only takes place above a certain threshold density [@elmegreen94]. If the system is steady-state on a global scale, this threshold density is given by $n_{\rm burst}=0.6\kappa^2/G\mu m_{\rm H}$ and hence $n_{\rm burst}=\{0.3,0.5,2.6\}\times10^4~{\rm cm}^{-3}$ for the three regions of the CMZ listed in Table \[tab:prop\]. This is comparable to the current mean density of $n_0=2\times10^4~{\rm cm}^{-3}$, which may indicate that (part of) the CMZ is currently evolving towards a starburst. However, this critical density increases if the gas accretion rate along the bar is substantial. Additionally, orbital curvature causes the density waves in the central regions of galaxies to grow at an increasing rate towards smaller galactocentric radii, even in Toomre-stable discs or cases where the self-gravity of the gas does not set its global geometry [@montenegro99]. It is therefore easy to picture a system in which fresh gas is transported along the bar onto the 100-pc ring, where it builds up until the critical density for instability is reached – possibly at different times throughout the ring. There is a notable population of 24$\mu{\rm m}$ sources at $l\la359.5^\circ$, i.e. beyond the position of Sgr C [see Figure \[fig:img\] and @yusefzadeh09], which may be the remnant of a recent, localised starburst. Variations would typically occur on the dynamical timescale of the system, which is $\sim5$ Myr for the 100-pc ring.]{}
Alternatively, feedback from YMCs might induce a fluctuating SFR. As will be discussed in §\[sec:shocks\], the majority of the [current]{} star formation in the 100-pc ring takes place in the Sgr B2 complex. [In the past, the birth environment of the $24\mu{\rm m}$ sources (possibly also near Sgr B2) may have been the dominant site of star formation]{}. Such asymmetry implies that the feedback from YMCs [originates from discrete locations]{}. This scenario is problematic, because the feedback energy will escape through the path of least resistance and hence YMCs on one side of the ring cannot support gas on the far side (also see §\[sec:rad\]). [While a local starburst such as Sgr B2, or the one responsible for the formation of the $24\mu{\rm m}$ sources, may not impact the entire 100-pc ring, it could blow out the sector containing the starburst. By the time such a burst is $\sim10~{\rm Myr}$ old its H[ii]{} regions will have faded, and much of the gas could be in atomic form, being dispersed to higher latitudes by the combined acceleration of H[ii]{} regions, winds, and supernovae.]{}
The presence of a mere 2–3 clusters would necessarily place the current state of the ring at a minimum SFR, because the scarcity of feedback support suggests it should collapse to form stars over the next few free-fall times. During such a burst of star formation, it could produce $\sim20$ YMCs (which is the number of dense clumps observed by @molinari11) of Arches-like masses, assuming that the progenitor clumps are similar to the cloud G0.253+0.016 [also known as ‘The Brick’, see @longmore12 with $M\sim10^5~{\mbox{M$_\odot$}}$, $R\sim3~{\rm pc}$] and form stars [in gravitationally bound[^8] clusters]{} at a 10–30 per cent efficiency. The timescale for the SFR fluctuations would be set by the half-time of the YMC feedback. Their bolometric luminosity decreases by a factor of two in roughly 8 Myr, implying that if the low SFR in the CMZ is explained by feedback-induced episodicity, a natural timescale would be $\sim10$ Myr.
A 5–10 Myr timescale for episodic star formation in the 100-pc ring places a strong limit on the observability of [any]{} produced YMCs – if their apparent present-day absence is not caused by their possible fading below the detection limit, their disruption timescale would have to be at most 10 Myr. [The dynamical friction timescale of a $10^4~{\mbox{M$_\odot$}}$ cluster orbiting in the 100-pc ring is $t_{\rm df}\sim3$ Gyr,[^9] implying that dynamical friction does not affect YMCs on such a time-scale, and YMCs will therefore keep interacting with the ring regularly after their formation. The disruption of a $10^4~{\mbox{M$_\odot$}}$ cluster in a gas-rich, high-density environment is dominated by tidal shocks due to GMC passages [@elmegreen10b; @kruijssen11; @kruijssen12c]. The time-scale for disruption is then $t_{\rm dis}^{\rm sh}\propto\Sigma_{\rm GMC}^{-1}\rho_{\rm mol}^{-1}\rho_{\rm YMC}$ [@gieles06], with $\Sigma_{\rm GMC}$ the GMC surface density, $\rho_{\rm mol}$ the spatially averaged molecular gas density, and $\rho_{\rm YMC}$ the YMC density. Taking the Brick as a template GMC, we have $\Sigma_{\rm GMC}\sim5\times10^3~{\mbox{M$_\odot$}}~{\rm pc}^{-2}$, whereas $\rho_{\rm mol}=\Sigma_{\rm mol}/2h=150~{\mbox{M$_\odot$}}~{\rm pc}^{-3}$ for the 100-pc ring (cf. Table \[tab:prop\]). Substituting these numbers gives $t_{\rm dis}^{\rm sh}=5.7$ Myr for a $10^4~{\mbox{M$_\odot$}}$ cluster with a half-mass radius of $0.5~{\rm pc}$ [cf. @portegieszwart10], [which is much shorter than the $t_{\rm dis}^{\rm tidal}\sim80$ Myr due to the Galactic tidal field alone [@portegieszwart01].[^10]]{} The disruption of YMCs by tidal shocks is thus capable of ‘hiding’ the evidence left by previous starbursts. At 2 and 4 Myr, the ages of the Arches and Quintuplet clusters are consistent with this disruption process. It therefore remains possible that the star formation in the CMZ is episodic, which would then most likely be driven by local instabilities.]{}
Finally then, there might be a considerable time-variation of the gas inflow from large galactocentric radii onto the CMZ. @kim12c present numerical simulations of the gas flow in the central regions of barred galaxies, and measure the gas flux through a sphere with radius $R=40$ pc, i.e. within their simulated equivalents of the 100-pc ring. They find that the variation of the gas flux is typically less than an order of magnitude in models with pronounced rings, because the inflowing gas is trapped in the nuclear ring before gradually falling to the centre. For those models that do not develop such features, and hence have an uninhibited gas flow to the sphere where the flux is measured, the fluctuations sometimes reach two orders of magnitude. This can be taken as a rough indication of how the gas flow onto the 100-pc ring of the Milky Way may vary. The timescale for the variations in the gas flow corresponds to the dynamical timescale at the end of the bar, i.e. $t_{\rm dyn}\sim100$ Myr, implying a relatively wide window during which the system can be observed at a low SFR. [While this scenario therefore does not require us to put any stringent limits on YMC lifetimes, the preferred period for episodicity is shorter. The presence of two YMCs and the 24$\mu{\rm m}$ sources suggests substantial recent star formation activity, and considering the high densities of the several Brick-like clouds in the CMZ it not clear how long the current dearth of star formation will continue.]{}
### The geometry of the CMZ and tidal shocks {#sec:shocks}
In highly dynamical environments like the CMZ or barred galaxies, it is conceivable that GMCs are disrupted by transient tidal perturbations [‘tidal shocks’, see e.g. @spitzer58; @kundic95] before having been able to form stars. This idea was first introduced by @tubbs82, who suggested that perturbations would limit the star formation time-scale and hence decrease the SFR. [Note that this effect differs from disruption by a steady tidal field (see e.g. @kenney92 for extragalactic examples), to which we turn in §\[sec:tidal\]]{}. The vast majority of the star formation in the 100-pc ring of the CMZ takes place in the complex Sgr B2 [@longmore13]. [It has been proposed by @molinari11 that the 100-pc ring follows the $x_2$ orbit]{} – a family of elliptical orbits with semi-major axis perpendicular to the Galactic bar, which precesses at the same rate as the pattern speed of the bar, resulting in stable, non-intersecting trajectories. The $x_2$ orbits are situated within the $x_1$ orbits, which are elongated along the major axis of the bar [see @athanassoula92]. [In this scenario, Sgr B2 and Sgr C lie at the points where the $x_1$ and $x_2$ orbits touch, and the accumulation of gas occurs due to two density waves]{} with pattern speeds different than the flow velocity of the gas orbiting on the 100-pc ring. [Such a configuration has been observed in other galaxies [e.g. @pan13].]{} The ring upstream of Sgr B2 is fragmented into clouds that have properties suggesting that they could be massive protoclusters [@longmore13b]. [A schematic representation of this configuration is shown in Figure \[fig:cmzschem\].]{} [If Sgr B2 is indeed a standing density wave and the impending encounter of these clouds with Sgr B2 is sufficiently energetic [cf. @sato00], it is possible that star formation is briefly induced due to the tidal compression when they enter Sgr B2, but is subsequently halted when the clouds exit the Sgr B2 region and rapidly expand due to having been tidally heated. [We reiterate that the scenario of Figure \[fig:cmzschem\] is not undisputed (see footnote \[footnote:uncertain\]). However, it does lead to the most extreme cloud-cloud encounters that could take place in the CMZ, and therefore we consider it as an upper limit.]{}]{}
\
It is straightforward to quantify the disruptive effect of a tidal perturbation as the ratio of the energy gain $\Delta E$ to the total energy of the cloud $E$, under the condition that the duration of the perturbation is shorter than the dynamical time of the perturbed cloud [the ‘impulse approximation’, see @spitzer87] – [otherwise the injected energy is simply dissipated]{}. In order to compute the relative energy gain, we consider a head-on collision and approximate the cloud and the perturber with @plummer11 potentials. If we include the correction factors for the extended nature of the perturber [@gieles06] and the second-order energy gain [@kruijssen11], the total relative energy gain becomes: $$\label{eq:deshock}
\left|\frac{\Delta E}{E}\right|=1.1M_7^2 R_{{\rm h},1}^{-4} m_5^{-1} r_{{\rm h},0}^3 V_2^{-2} ,$$ where $M_7\equiv M/10^7~{\mbox{M$_\odot$}}$ is the perturber mass, $R_{{\rm h},1}\equiv R_{\rm h}/10~{\rm pc}$ is its half-mass radius, $m_5\equiv m/10^5~{\mbox{M$_\odot$}}$ is the cloud mass, $r_{{\rm h},0}\equiv r_{\rm h}/1~{\rm pc}$ is its half-mass radius, and $V_2\equiv V/100~{\rm km}~{\rm s}^{-1}$ is the relative velocity between both objects. The cloud is unbound if $|\Delta E/E|\geq1$. To describe Sgr B2 we adopt $M_7\sim0.6$ [@bally88; @goldsmith90] and $R_{{\rm h},1}\sim1$, for the clouds approaching Sgr B2 we use the properties of the Brick, with $m_5\sim1.3$ and $r_{{\rm h},0}\sim2.1$, and the relative velocity is taken to be $V_2\sim1$ ([slightly higher than the line-of-sight streaming velocity in the 100-pc ring]{}). For these numbers, equation (\[eq:deshock\]) gives $|\Delta E/E|=2.8$, suggesting that a Brick-like cloud could be unbound when passing through the gravitational potential chosen to represent Sgr B2. To verify the validity of the impulse approximation, we note that the duration of the perturbation is $\Delta t=2R_{\rm h}/V=0.1$–$0.2$ Myr, whereas the dynamical time is $t_{\rm dyn}=(G\rho_{\rm h})^{-1/2}\sim0.4$ Myr, and hence $\Delta t<t_{\rm dyn}$.
The above approach does not account for the detailed structure of Sgr B2 and the passing clouds, nor does it cover the collisional hydrodynamics of the clouds [e.g. @habe92], [their dissipative nature, or any possible, substantial deviations from spherical symmetry. However, it does make the interesting point that the energy gain during an encounter of a Brick-like cloud with the Sgr B2 region is comparable to the binding energy of the cloud.]{} The clouds in the 100-pc ring are regularly spaced at $\sim25$ pc intervals, and their evolutionary stage seems to be getting progressively older towards Sgr B2, with the clouds closest to Sgr B2 possibly having initiated star formation, [which is possibly caused by the triggering of cloud condensation at the orbital pericentre of the ring near Sgr A$^*$]{} [@longmore13b]. At a circular velocity of $80$–$140~{\rm km}~{\rm s}^{-1}$ (see §\[sec:globalobs\]), the Brick will enter Sgr B2 in 0.2–0.3 Myr and will leave it $\Delta t=0.1$–$0.2$ Myr later. If the cloud is unbound by the interaction, star formation may only proceed for 0.3–0.5 Myr, or $\sim2$ global free-fall times. For a star formation efficiency (SFE) of 1–2% per free-fall time [@krumholz07], the clouds in the 100-pc ring will reach total SFEs of $\sim3\%$, which for the typical cloud mass of $10^5~{\mbox{M$_\odot$}}$ implies that each cloud would typically produce $3000~{\mbox{M$_\odot$}}$ of stars. [This is comparable to the total mass inferred from the $24\mu{\rm m}$ point sources, and hence the above scenario may have applied to their formation too. Alternatively, the hydrodynamic perturbation of the clouds passing through a density wave may also accelerate their collapse, in which case Sgr B2 and Sgr C represent the instigation points of star formation rather than the loci where it is terminated.]{}
We conclude that it is possible that the unique geometry of the CMZ plays a role in [controlling]{} star formation in at least part of the region. While this model may work well for star formation in the 100-pc ring, it remains to be seen to what extent similar effects could inhibit star formation elsewhere in the CMZ. [The above estimate also ignores the possible contraction of the clouds and correspondingly shorter free-fall times during the passage of Sgr B2. We return to this scenario in §\[sec:testshock\], where possible tests are discussed.]{}
Locally inhibited star formation {#sec:local}
--------------------------------
Here we discuss the possible mechanisms on spatial scales smaller than the disc scale height ($\Delta R<h$) that may limit star formation in the CMZ. Volumetric star formation relations generally rely on the free-fall time $t_{\rm ff}\propto n^{-1/2}$ to predict $\Sigma_{\rm SFR}$, in that approximately $1\%$ of the gas mass is converted into stars per free-fall time [@krumholz07]. The relative universality of this number suggests that broadly speaking, there is always a similar fraction of the gas mass that participates in star formation. In star formation theories, this is explained by the idea that the dispersion of the density PDF depends on the Mach number in a similar way to the critical density for star formation – the PDF broadens as the density threshold increases [e.g. @krumholz05; @padoan11]. Considering the broad range of gas densities observed in disc and starburst galaxies, this is a crucial ingredient to allow the SFE per free-fall time to be roughly constant.
The above picture has been challenged by the recent observation by @lada10 that there is a possibly universal, critical volume density $n_{\rm Lada}\sim10^4~{\rm cm}^{-3}$ for converting gas into stars. Such a transition should be expected, as there obviously exists some extreme density above which all gas ends up in stars (modulo the mass lost by protostellar outflows), but it is not clear why such a transition density would be universal. As discussed in §\[sec:obs\], the fraction of the gas mass in the CMZ that is used to form stars is a factor of $f_{{\rm SFR},n}=0.005$ lower than the fraction of the gas with densities above the Lada threshold, [in order for the SFR to be consistent with density-dependent ($N>1$) star formation relations]{}. We showed in §\[sec:obs\] that if it exists, a typical volume density threshold for star formation thus has to be $n_{\rm th}\geq10^7~{\rm cm}^{-3}$ for various parametrizations of the density PDF, i.e. $n_{\rm th}=10^7$–$4\times10^8~{\rm cm}^{-3}$ for a power-law approximation and $n_{\rm th}\sim\{0.5,2\}\times10^9~{\rm cm}^{-3}$ for a log-normal when including and excluding the effect of the magnetic field, respectively.
In the following, we verify which physical mechanisms are consistent with the inhibition of star formation below such densities. Because the turbulent pressure in the CMZ is remarkably high [e.g. @bally88], with $P_{\rm turb}=n\sigma^2\sim10^{-6}~{\rm erg}~{\rm cm}^{-3}$, an important constraint is that potential star formation inhibitors should be able to compete with the turbulence. Therefore, we often use the turbulent pressure as a reference point to calculate the gas volume densities below which the SFR may be suppressed by each mechanism. The mechanism responsible for the low observed SFR needs to be effective up to a critical density of $n_{\rm th}\geq10^7~{\rm cm}^{-3}$ (see §\[sec:obs\]).
### Galactic tides {#sec:tidal}
A first condition for initiating star formation is that the progenitor clouds are not tidally disrupted. The gas needs to be at a density higher than the tidal density, i.e. the density required for a spherical density enhancement to remain bound in a galactic tidal field. While this is not a guarantee that a region will eventually form stars, which requires time and space for it to become strongly self-gravitating, it does represent a key requirement for star formation to proceed. The tidal density is written as $$\label{eq:ntidal}
n_{\rm tidal}=\frac{3A_{\rm pot}\Omega^2}{4\pi \mu m_{\rm H}G} ,$$ which only depends on the angular velocity $\Omega$. [This expression assumes that the cloud orbit is circular. The constant $A_{\rm pot}$ depends on the shape of the galactic gravitational potential, and is $A_{\rm pot}=\{0,2,3\}$ for solid-body rotation, a flat rotation curve, and a point source (i.e. Keplerian) potential, respectively. [Adopting the @launhardt02 potential for the CMZ, we find $A_{\rm pot}=\{2.0,1.9,0.7\}$ for the 230 pc-integrated CMZ, the 1.3$^\circ$ cloud, and the 100-pc ring, respectively.]{} Galactic tides inhibit]{} star formation in regions where $n_{\rm tidal}>n_{\rm th}$, with $n_{\rm th}$ some unknown threshold for star formation. For the three regions of the CMZ that are listed in Table \[tab:prop\], equation (\[eq:ntidal\]) yields $n_{\rm tidal}=\{1.3,1.8,2.1\}\times10^3~{\rm cm}^{-3}$. This is several orders of magnitude lower than the $n_{\rm th}\geq10^7~{\rm cm}^{-3}$ required to explain the SFR in the CMZ, [but it is only an order of magnitude lower than the mean gas density in the CMZ]{}. We did not account for eccentric orbits in this argument, because $n_{\rm tidal}\geq n_{\rm th}$ requires such a high angular velocity ($\Omega_{\rm crit}\geq10^2~{\rm Myr}^{-1}$) that even at fixed circular velocity an eccentricity of $\epsilon\geq0.99$ would be necessary for tides to be the limiting factor. We conclude that star formation in the CMZ is not inhibited by tides.
### Turbulence {#sec:turb}
Turbulence plays a key role in the recent star formation models of @krumholz05 and @padoan11. While the former argue that turbulent pressure support sets the critical density for star formation on the sonic scale, the latter work takes the point that turbulence is only responsible for driving local gravitational instabilities, and that the critical density for star formation is set by the thickness of the post-shock layers in the supersonic ISM. These differences aside, both models do predict a critical overdensity $x\equiv n/n_0$ for star formation[^11] that scales with the GMC virial parameter $\alpha_{\rm vir}$ and Mach number as $$\label{eq:xturb}
x_{\rm turb}=A_x\alpha_{\rm vir}{\cal M}^2 .$$ Both models also have remarkably similar proportionality constants $A_x\sim1$ to within a factor of 1.5.
Adopting a virial parameter of $\alpha_{\rm vir}=1.5$ [cf. @krumholz05; @padoan11], a Mach number of ${\cal M}=70$ as in §\[sec:obs\], and a mean density of $n_0=2\times10^4~{\rm cm}^{-3}$, we see that the critical density for star formation is $n_{\rm turb}\equiv x_{\rm turb}n_0\sim1.5\times10^8~{\rm cm}^{-3}$. This approaches the low end of the required threshold densities $n_{\rm th}\sim\{0.5,2\}\times10^9~{\rm cm}^{-3}$ (see the dotted and dashed lines in Figure \[fig:nth\]), although depending on the shape of the density PDF we indicated a strict lower limit of $n_{\rm th}\geq10^7~{\rm cm}^{-3}$ in §\[sec:obs\]. If we apply equation (\[eq:xturb\]) to the Brick [@longmore12], with $\alpha_{\rm vir}\sim1$, ${\cal M}\sim55$, and $n_0=7.3\times10^4~{\rm cm}^{-3}$, we obtain $n_{\rm turb}\sim2.2\times10^8~{\rm cm}^{-3}$. By contrast, the typical properties of GMCs in the solar neighbourhood are $\alpha_{\rm vir}\sim1.5$, ${\cal M}\sim10$, and $n_0=10^2~{\rm cm}^{-3}$, which gives $n_{\rm turb}\sim1.5\times10^4~{\rm cm}^{-3}$. Interestingly, this equals the @lada10 threshold for star formation to within the uncertainties of this calculation.
The above numbers are very suggestive, in that the ‘normal’ overdensity thresholds for star formation on the one hand predict the Lada threshold for star formation in the solar neighbourhood, and on the other also predict a density threshold in the CMZ of $n_{\rm turb}\sim2\times10^8~{\rm cm}^{-3}$, which gives the best agreement with the observed SFR so far. The difference with the required $n_{\rm th}=5\times10^8~{\rm cm}^{-3}$ for a weakly magnetically confined density PDF is only a factor of three,[^12] and may be offset by uncertainties in the density PDF, in the above numbers, and/or in the observed SFR. [While turbulence is thus capable of increasing the density threshold to the required, extreme densities, we should note that this argument is incomplete, as we have not established what is driving the turbulence. We return to this point in §\[sec:disc\], and will now briefly discuss a particularly interesting uncertainty.]{}
### A bottom-heavy initial mass function {#sec:imf}
[Observational measures of the SFR are strongly biased to the emission from massive stars ($m\ga8~{\mbox{M$_\odot$}}$).]{} The low observed SFR in the CMZ may therefore be spurious due to an overproduction of low-mass stars with respect to the observed massive stars, whether the SFR is determined using massive YSOs [@yusefzadeh09] or the ionising flux from massive stars [@murray10b; @lee12]. Neither technique is capable of reliably detecting stars below $8~{\mbox{M$_\odot$}}$. Recent observational studies of giant elliptical galaxies have found evidence for a bottom-heavy IMF ${{\rm d}}n/{{\rm d}}m\propto m^{-\beta}$ with a power-law slope of $\beta=3$ at the low-mass end [$m\la1~{\mbox{M$_\odot$}}$, see e.g. @vandokkum10; @cappellari12; @goudfrooij13].
It has been suggested that the characteristic mass scale of the core mass function (CMF) is set by the thermal Jeans mass $m_{\rm J}$ [@elmegreen08b] or the sonic mass $m_{\rm sonic}$ [@hopkins12b]. [The Jeans mass is insensitive to the density and the radiation field if the Schmidt-Kennicutt relation is satisfied [@elmegreen08b]. Because in the CMZ it is not, the heating rate should be low compared to the pressure (relative to the Galactic disc) and hence the thermal Jeans mass is low too. This makes the CMZ a prime example of where the CMF could have a lower than normal peak mass.]{} If we assume that the CMF and IMF are related, this suggests that the characteristic turnover of the IMF that is observed in the solar neighbourhood at $m\sim0.5~{\mbox{M$_\odot$}}$ [@kroupa01; @chabrier03] is environmentally dependent. Because $m_{\rm J}$ and $m_{\rm sonic}$ decrease with the pressure, this results in an enhanced population of low-mass stars in the vigorously star-forming galaxies observed at high redshift [e.g. @vandokkum04; @daddi07], which reach Mach numbers of ${\cal M}\sim100$ [e.g. @swinbank11] and may be the progenitors of current giant elliptical galaxies. Such Mach numbers strongly contrast with the ${\cal M}\sim10$ in the Milky Way disc.
For core masses $M\la1~{\mbox{M$_\odot$}}$ (i.e. stellar masses $M\la0.5~{\mbox{M$_\odot$}}$) and a Mach number of ${\cal M}\sim100$, @hopkins12b predicts a steepened mass spectrum with a slope of $\beta\sim3$. Using parameters that are appropriate for the Brick in the CMZ [@longmore12] and adopting a SFE in protostellar cores of $\epsilon=0.5$ [@matzner00], we find $\epsilon m_{\rm sonic}\sim0.01~{\mbox{M$_\odot$}}$, whereas in the solar neighbourhood $\epsilon m_{\rm sonic}\sim0.5~{\mbox{M$_\odot$}}$. [Similarly, at the approximate density threshold for star formation due to turbulence $n_{\rm turb}\sim2\times10^8~{\rm cm}^{-3}$, the thermal Jeans mass in the CMZ ($T=65$ K) is about $\epsilon m_{\rm J}\sim0.06~{\mbox{M$_\odot$}}$, whereas in the solar neighbourhood it is $\epsilon m_{\rm J}\sim0.5~{\mbox{M$_\odot$}}$.]{} These low characteristic masses suggest that the CMZ might be the low-redshift equivalent to the progenitor environment of giant elliptical galaxies, and it is therefore important to verify to what extent any unseen stellar mass at $m<0.5~{\mbox{M$_\odot$}}$ may increase the SFR inferred for a ‘normal’ IMF [@kroupa01; @chabrier03].
We integrate the mass of a @kroupa01 IMF between $m_{\rm min}=0.08~{\mbox{M$_\odot$}}$ and $m_{\rm max}=100~{\mbox{M$_\odot$}}$ and compare it to the mass integral of a similar IMF, but with a power-law slope of $\beta=3$ for masses $m<0.5~{\mbox{M$_\odot$}}$. This increases the total mass by a factor of two, at the same number of massive stars. We conclude that while this is a non-negligible factor, it is (1) comparable to the uncertainty on the SFR in the CMZ and (2) in itself insufficient to explain the factor of $\geq10$ suppression of the SFR in the CMZ. However, if we combine a doubled SFR with a density PDF that is narrowed by the high magnetic field strength in the CMZ (see §\[sec:localobs\]), then the density threshold required by the observed gas depletion time-scale becomes $n_{\rm th}\sim2.5\times10^8~{\rm cm}^{-3}$, which is comparable to the threshold predicted for normal turbulence ($n_{\rm turb}\sim2\times10^8~{\rm cm}^{-3}$, see §\[sec:turb\]).
### The atomic-molecular phase transition of hydrogen {#sec:phase}
[Dense molecular gas as traced by HCN is found to be correlated with star formation tracers [e.g. @gao04; @wu05]. If this relation is causal in nature,]{} molecular gas may be required to form stars (e.g. @schruba11, although see @glover12 and @krumholz12c for an alternative view), implying that a star formation threshold could be related to the phase transition of H[i]{} to H$_2$ [@elmegreen94b; @schaye04; @blitz04; @krumholz09c], which at solar metallicity occurs at $\Sigma\sim10~{\mbox{M$_\odot$}}~{\rm pc}^{-2}$. [This is the only theory that predicts a constant SFR per unit molecular mass (see §\[sec:obs\]).]{} [However, the density scale for the phase transition decreases with increasing metallicity [@krumholz09b], and hence should be even lower than $10~{\mbox{M$_\odot$}}~{\rm pc}^{-2}$ in the central bulge region [e.g. @brown10]. The bulk of the gas mass in the CMZ resides at higher surface densities and is indeed observed to be molecular [e.g. @morris96]. We therefore rule out the atomic-to-molecular transition at $\Sigma\la10~{\mbox{M$_\odot$}}~{\rm pc}^{-2}$ as the cause for a suppressed SFR in the CMZ, because the surface density is much higher in the central $1$ kpc of the Milky Way]{}.
### The Galactic magnetic field {#sec:magnetic}
Another possible explanation would be that the high magnetic field strength in the CMZ [$B\ga100\mu{\rm G}$, @crocker10] inhibits star formation [e.g. @morris89]. Using the condition that the magnetic and turbulent pressure are balanced, this implies a critical density of $$\label{eq:nmag}
n_{\rm mag}=\frac{1}{2\mu_0 \mu m_{\rm H}}\left(\frac{B}{\sigma}\right)^2 ,$$ where $\mu_0=4\pi$ is the vacuum permeability constant. [Using the values for each of the three CMZ regions from Table \[tab:prop\], this gives a critical density of $n_{\rm mag}\sim10~{\rm cm}^{-3}$ above which the magnetic pressure becomes less than the turbulent pressure.]{} Therefore, the magnetic field cannot provide support against the turbulence in the CMZ. [This result is unchanged when adopting the internal velocity dispersions of the clouds in the CMZ rather than the large-scale velocity dispersion.]{} [However, it is important to note that the aforementioned value of the magnetic field strength applies to the low-density intercloud medium, and may be an order of magnitude higher in dense clouds [@morris06]. This could increase the critical density to $n_{\rm mag}\sim10^4~{\rm cm}^{-3}$, which is comparable to the mean gas density, but is still much lower than the density threshold required by density-dependent star formation relations.]{} While it likely does not inhibit star formation, [the presence of a $100\mu{\rm G}$ magnetic field is an important second-order effect. On the one hand, it is capable of narrowing the density PDF of the ISM in the CMZ somewhat, and slows down star formation accordingly (see §\[sec:localobs\] and §\[sec:turb\]). On the other hand, magnetic breaking leads to angular momentum loss during the condensation and contraction of cores, and hence accelerates star formation [@elmegreen87].]{}
### Radiation pressure {#sec:rad}
Considering the high angular velocity of the gas in the CMZ, the 100-pc ring needs to enclose some $10^9~{\mbox{M$_\odot$}}$ of mass. If we make the reasonable assumption that most of this mass is constituted by stars, this implies a high stellar density of the CMZ of $\Sigma_\star\sim3\times10^3~{\mbox{M$_\odot$}}~{\rm pc}^{-2}$ within the gas disc scale height (see Table \[tab:prop\]). [It is therefore worth investigating whether stellar feedback is capable of inhibiting star formation in the CMZ. At first, feedback is dominated by protostellar outflows, followed by radiative feedback, supernovae, and stellar winds. The relative importance of these mechanisms depends on the spatial scale and the environment. It has been shown by @murray10 that in all but the lowest-density environments (e.g. GMCs in the Galactic disc) radiation pressure is the dominant feedback mechanism for disrupting GMCs, whereas on scales $\ga100$ pc the energy deposition by supernovae becomes important. Crucially though, the total energy output from each of these mechanisms is comparable.]{}
We now test the hypothesis that stellar radiation inhibits star formation in the CMZ. [As discussed in §\[sec:episodic\], this scenario has the problem that the CMZ hosts discrete star formation events, implying that the feedback on one side of the CMZ may not be able to affect gas on the opposite side. To test whether feedback is a viable explanation from an energy perspective]{}, we again require pressure equilibrium between radiation pressure and turbulent pressure, which yields a critical stellar surface density of young stars $\Sigma_{\star,{\rm rad}}$ above which radiation pressure is important: $$\label{eq:stelcrit}
\Sigma_{\star,{\rm rad}}=\frac{4c\mu m_{\rm H}n\sigma^2}{\Psi (1+\phi_{\rm tr}\kappa_0T^2\Sigma)} ,$$ where $c$ is the speed of light, $\Psi\sim3\times10^3~{\rm erg}~{\rm s}^{-1}~{\rm g}^{-1}\sim1.5\times10^3~{\rm L}_\odot~{\mbox{M$_\odot$}}^{-1}$ the light-to-mass ratio of a young, well-sampled stellar population [@thompson05], and the term in parentheses indicates the optical depth $1+\phi_{\rm tr}\tau$ with $\tau\sim\kappa_R\Sigma\sim\kappa_0T^2\Sigma$, in which $\phi_{\rm tr}\equiv f_{\rm tr}/\tau\sim0.2$ [@krumholz12b] is a constant that indicates the fraction of infrared radiation that is trapped at an optical depth $\tau=1$, $\kappa_{\rm R}$ is the Rosseland mean dust opacity [cf. @thompson05; @murray10], and $\kappa_0\sim2.4\times10^{-4}~{\rm cm}^2~{\rm g}^{-1}~{\rm K}^{-2}$ is a proportionality constant.
We consider two cases for calculating the critical stellar surface density above which radiation pressure can compete with the turbulent pressure. Firstly, in the 100-pc ring we have $n_0\sim2\times10^4~{\rm cm}^{-3}$, $\sigma\sim35~{\rm km}~{\rm s}^{-1}$ and $\Sigma=3.0\times10^3~{\mbox{M$_\odot$}}~{\rm pc}^{-2}$. Secondly, for the GMC ‘the Brick’ [@longmore12] we adopt $n\sim7.3\times10^4~{\rm cm}^{-3}$, $\sigma\sim16~{\rm km}~{\rm s}^{-1}$ and $\Sigma=5.3\times10^3~{\mbox{M$_\odot$}}~{\rm pc}^{-2}$. In both cases, we assume $T=65~{\rm K}$ as in §\[sec:obs\], which yields an optical depth of $\tau\sim1$. For the 100-pc ring, equation (\[eq:stelcrit\]) gives $\Sigma_{\star,{\rm rad}}\sim1.6\times10^5~{\mbox{M$_\odot$}}~{\rm pc}^{-2}$. Combining this with the surface area taken up by the gas in the ring ($5\times10^3~{\rm pc}^2$) and the lifetime of strongly radiating stars [$\sim4~{\rm Myr}$, @murray10], we see that the 100-pc ring requires a SFR of $\sim200~{\mbox{M$_\odot$}}~{\rm yr}^{-1}$ for radiation pressure to overcome the turbulence. This is four orders of magnitude higher than the measured $\sim0.015~{\mbox{M$_\odot$}}~{\rm yr}^{-1}$. The Brick requires a similar surface density of $\Sigma_{\star,{\rm rad}}\sim1.1\times10^5~{\mbox{M$_\odot$}}~{\rm pc}^{-2}$. Other than implying a critical SFR of $0.7~{\mbox{M$_\odot$}}~{\rm yr}^{-1}$, it also means that a SFE of $\sim20$ (not per cent!) is required to overcome the turbulent pressure, [unless the newly formed stellar population has a five times smaller radius than its parent cloud, in which case a SFE of unity would imply similar turbulent and radiative pressures.]{} The above numbers change by a relatively small amount when also including the combined flux of the more numerous, old stars of the Galactic bulge. [Since radiative feedback is not capable of strongly affecting the gas on the cloud scale, it is also unable to stop the entire CMZ from forming stars (see \[sec:episodic\]).]{}
### Cosmic rays {#sec:rays}
[Cosmic rays pressure could be important in the CMZ, either due to past star formation or black hole activity.]{} For instance, it is possible that star formation in the CMZ is inhibited by the cosmic ray flux from supernovae (SNe). By equating the cosmic ray pressure to the turbulent pressure, and assuming the extreme case in which the cosmic ray energy remains trapped in the CMZ, we obtain the critical volume density above which turbulence overcomes the cosmic ray pressure: $$\label{eq:ncr}
n_{\rm cr}=\frac{\eta_{\rm SN}E_{\rm SN}\Gamma_{\rm SN}\tau_{\rm cr}}{\mu m_{\rm H}V\sigma^2} .$$ In this expression, $\eta_{\rm SN}\sim0.3$ is the fraction of the SN energy that is converted to cosmic rays, $E_{\rm SN}\sim10^{51}~{\rm erg}$ is the energy of a single SN, $V$ is the volume, and $\Gamma_{\rm SN}$ is the SN rate, which for a @kroupa01 or @chabrier03 initial mass function (IMF) is given by $\Gamma_{\rm SN}=0.01~{\rm yr}^{-1}~({\rm SFR}/{\mbox{M$_\odot$}}~{\rm yr}^{-1})$, where the SFR is expressed in ${\mbox{M$_\odot$}}~{\rm yr}^{-1}$. The variable $\tau_{\rm cr}\propto n^{-1}$ indicates the lifetime of cosmic rays to energy loss via collisions with protons in the gas, which is $\tau_{\rm cr}\sim10^8/n_0~{\rm yr}\sim5\times10^3~{\rm yr}$.
Based on Table \[tab:prop\], we have SFRs of $\{3.3,0.06,1.5\}\times10^{-2}~{\mbox{M$_\odot$}}~{\rm yr}^{-1}$ and volumes $V\sim\{3.3,1.4,0.05\}\times10^6~{\rm pc}^3$ for the 230 pc-integrated CMZ, the 1.3$^\circ$ cloud, and the 100-pc ring, respectively. Substitution into equation (\[eq:ncr\]) then gives $n_{\rm cr}=\{0.1,0.005, 3\}~{\rm cm}^{-3}$ (assuming $\sigma\sim35~{\rm km}~{\rm s}^{-1}$), which for all three cases is much lower than the density threshold for star formation required by the observed SFR. Given these numbers, it is very unlikely that cosmic rays from SNe affect the gas dynamics in the CMZ. [For the cosmic ray pressure to compete with turbulence, the SFR would need to be four orders of magnitude higher than is being observed.]{}
Even if all SN energy would be converted to cosmic rays, and their lifetime would be a thousand times longer, cosmic rays would still imply a critical density $n_{\rm cr}<n_0$.
Alternatively, cosmic rays could originate from the activity of the central black hole of the Milky Way, which accretes at a rate of $\dot{M}\la10^{-8}~{\mbox{M$_\odot$}}~{\rm yr}^{-1}$ [@quataert00; @baganoff03]. We assume that 0.5% of the accreted rest mass energy is available to heat the gas [e.g. @dimatteo05], and follow a similar argument as for the case of SN-powered cosmic rays above. This yields a critical volume density of $n_{\rm BH}\sim6\times10^{-3}~{\rm cm}^{-3}$ for the accretion-generated cosmic ray pressure to compete with the turbulence in the volume of the 100-pc ring.
Cosmic rays are also unimportant compared to the thermal pressure anywhere other than the 100-pc ring, where the thermal and cosmic ray pressures are comparable to within a factor of three. [Hence, they may not be important kinematically, but they could be relevant for setting the temperature of the gas [@ao13; @yusefzadeh13]. We conclude that feedback processes in general, and radiative, supernova, and black-hole feedback in particular, are not consistently inhibiting star formation in the CMZ.[^13] [Of course, whether or not this also holds in other, extragalactic cases depends on their recent star formation and black hole activity.]{}]{}
Implications and predictions for future observations of the CMZ {#sec:impl}
===============================================================
We now turn to the implications and possible tests of the remaining plausible star formation inhibitors of §\[sec:thresholds\], which are summarised as follows. On global scales, star formation could be episodic due to gas instabilities or variations in the gas inflow along the Galactic bar. Alternatively, the geometry of the CMZ could cause clouds to be heated by dynamical interactions. On local scales, the reduced SFR is consistent with an elevated critical density for star formation due to the high turbulent pressure in the CMZ. This solution would be aided by a weakly magnetically confined interstellar medium, which would narrow the density PDF, as well as by an underproduction of massive stars due to a possibly bottom-heavy IMF.
Testing episodic star formation {#sec:testepisodic}
-------------------------------
Evidence exists of episodic star formation events in the CMZ [@sofue84; @blandhawthorn03; @yusefzadeh09; @su10] and mechanisms have been proposed to explain how such episodicity can occur. As discussed in §\[sec:episodic\], instabilities can drive the fragmentation of the nuclear ring and eventually induce a starburst.[^14] Gas in barred spiral galaxies like the Milky Way is funnelled from the disc through the bar to the galaxy centre [@sakamoto99; @kormendy04; @sheth05]. Because the conditions in the CMZ effect a higher threshold for star formation than in the disc, the gas needs time to accumulate before initiating star formation. While this by itself can already cause star formation to be episodic, it is also known from numerical simulations [@hopkins10b] that the presence of a bar can cause substantial variations of the gas inflow towards a galaxy centre. The large variation of the central gas concentration of otherwise similar galaxies sketches a similar picture [@sakamoto99].
To constrain the possible episodicity of star formation in the CMZ, it will be necessary to map the structure of the gas flow along the Galactic bar, which is already possible using sub-mm and radio surveys of the Galactic plane [e.g. @molinari10; @walsh11; @purcell12]. [If the CMZ is presently near a low point of an episodic star formation cycle, then the gas needs to be accumulating and hence the inflow rate has to exceed the SFR.]{} An improved 6D map of the CMZ itself would also help to understand the nature of the possible, large-scale instabilities of the gas – the combination of line-of-sight velocities, proper motions, plane-of-the-sky positions, and X-ray light echo timing measurements should lead to a conclusive picture of gas inflow, accumulation, and consumption. Finally, the end result of the star formation process should be considered further. With infrared data and spectral modelling, it is essential to establish the recent ($\sim100~{\rm Myr}$), spatially resolved star formation history of the CMZ. If evidence is found for a statistically significant variation of the SFR, the time-scale of such variations can be used to determine whether stellar feedback, gas instabilities, or a varying gas inflow rate are responsible (although see §\[sec:driver\]–\[sec:total\] for additional constraints).
[Similar measurements could be made for nearby barred galaxies (see §\[sec:extra\]), paying particular attention to spectral features that signify a starburst 50–100 Myr ago with no recent star formation, such as strong Balmer absorption lines and weak nebular emission lines [which is relatively straightforward using fiber spectra from the Sloan Digital Sky Survey, see @wild07].]{}
Testing interaction-limited cloud lifetimes {#sec:testshock}
-------------------------------------------
The suppression of star formation by the unbinding of clouds in dynamical encounters has a simple observational implication: the gas volume density downstream of Sgr B2 should be lower than upstream, where the Brick-like (see Figures \[fig:img\] and \[fig:cmzschem\]) clouds are currently residing. It is shown by @longmore13b that the gas density downstream of Sgr B2 indeed drops, but this could also be the result of star formation, [and is only relevant to the problem at hand if the CMZ is a steady-state system]{}. The option of interaction-limited cloud lifetimes [(or their interaction-induced formation)]{} thus requires a more detailed modelling in numerical simulations.
A key test would be to see what happens to a Brick-like cloud when it enters the Sgr B2 region. A numerical simulation of the fate of the Brick in such a configuration should not be oversimplified – both the gas physics and the global dynamics have to be treated appropriately. If the Sgr B2 complex is indeed situated at the point where the $x_1$ and $x_2$ orbits coincide [[similarly to the central regions in other galaxies, see e.g.]{} @pan13], the latter having a circular velocity of $80$–$140~{\rm km}~{\rm s}^{-1}$, the plausible interpretation is that Sgr B2 is a standing density wave [– much like the spiral arms in the Milky Way disc]{}. It should also be noted that the interaction between the co-moving gas flows on the $x_1$ and $x_2$ orbits may very well influence the properties of the system. This means that the future evolution of the Brick is not accurately modelled by having it cross a fixed, gasless background potential chosen to reflect Sgr B2, not can it be covered by a simulation in which the Brick-like cloud is put on a collision course with a Sgr B2-like complex at the circular velocity of the 100-pc ring. A more feasible approach would be to simulate the CMZ on a larger scale, including the gas flow along the bar. Such a simulation would be follow on existing work done on the gas flow in barred galaxies [e.g. @piner95; @kim12c], or on the type of large-scale, zoom-in simulation of a disc galaxy that has the necessary spatial resolution to resolve the substructure of the gas on the $x_2$ orbit [e.g. @hopkins10b]. The key novelty of a new numerical investigation should be the production (or insertion) of Brick-like clouds on the $x_2$ orbit and tracking their evolution in detail.
We are aiming to address this aspect in a future work (Kruijssen, Dale, Longmore et al. in prep.).
Observationally, it is not straightforward to infer the dynamical processes on the far side of Sgr B2, [and the detailed study of nuclear rings is restricted to nearby (barred) galaxies]{}. However, it is possible to map the properties of clouds like the Brick. ALMA data of the cloud kinematics and density structure will provide a key observational avenue to understand how such clouds may be affected by the high-density environment of the CMZ.
Implications and tests of suppression by turbulence {#sec:testturb}
---------------------------------------------------
In §\[sec:turb\], we show that a very promising explanation for the low SFR in the CMZ is an elevated density threshold for star formation due to the high turbulent pressure (with ${\cal M}\sim70$), based on the star formation models of @krumholz05 and @padoan11. This provides a theoretical justification for the conclusion made in §\[sec:obs\] that the observed SFR is inconsistent with the universal threshold density from @lada12. Even though the CMZ greatly extends the range of environmental conditions that can be probed in the Milky Way, a verification of the turbulence hypothesis requires our analysis to be extended to external galaxies. If the density threshold for star formation is as variable as would be expected on theoretical grounds, then there should also be an environmental variation of which molecular line transitions correlate with star formation tracers. We intend to quantify this in a future work.
[Within our own Galaxy, ALMA observations of clouds in the CMZ are extremely well-suited for addressing the variation of the threshold density for star formation. This threshold needs to be higher in the CMZ than in the disc, and in the high-pressure environment of the CMZ smaller spatial scales need to be probed in order to resolve the final stages of the cloud fragmentation and collapse towards cores and protostars. At the distance of the CMZ, only interferometers like ALMA and the EVLA are capable of reaching the necessary sensitivity and resolution in the wavelength range of interest. By combining star formation tracers with high density tracers, it will be possible to map the conversion of gas into stars in detail. A comparison with similar data from the solar neighbourhood should lead to a conclusive picture of the environmental variation of density thresholds for star formation.]{}
Note that for turbulence to be a conclusive explanation for the suppressed SFR in the CMZ, it is important to constrain the density PDF in the CMZ – preferably well enough to verify the possible influence of weak magnetic fields (see §\[sec:localobs\]). It is possible to map the PDF by comparing the total flux above certain molecular line transitions. The one caveat is that these transitions will need to probe densities as high as the suggested threshold density of a few $10^8~{\rm cm}^{-3}$. It is also relevant to establish whether the IMF in the CMZ may be bottom-heavy due to the high Mach number and correspondingly low sonic mass. Current IMF determinations are only capable of reaching masses of $m\geq5~{\mbox{M$_\odot$}}$ [e.g. @bastian10; @hussman12], but with the new generation of large-scale facilities like the E-ELT it should be possible to probe the IMF at $m<1~{\mbox{M$_\odot$}}$. A presently possible, but less direct method is to perform ALMA observations of the CMF in star-forming clouds in the CMZ. While this assumes a certain mapping of the CMF to an IMF, the advantage is that star formation theories based on the sonic or thermal Jeans mass actually predict a CMF rather than an IMF, implying that a measurement of the CMF enables a more direct verification of our theoretical understanding of star formation in the CMZ.
Discussion {#sec:disc}
==========
Summary {#sec:summ}
-------
We find that the reduction of the star formation rate in the Central Molecular Zone of the Milky Way [with respect to density-dependent ($N>1$) star formation relations]{} can be explained by the inhibition of star formation on both global and local scales.
1. Star formation could be episodic due to gas instabilities or variations in the gas inflow along the Galactic bar, with the present state of the CMZ corresponding to a near-minimum in the episodicity. (§\[sec:episodic\])
2. The geometry of the CMZ may limit cloud lifetimes by dynamical heating, implying lower SFEs, but the phase-space distribution of gas in the CMZ is too uncertain to either confirm or rule out this scenario. (§\[sec:shocks\])
3. Crucially, the high turbulent pressure in the CMZ increases the critical density threshold for star formation orders of magnitude beyond the @lada10 threshold, and hence substantially decreases the SFR with respect to such a threshold. (§\[sec:turb\]) To first order, this explains why the CMZ [@longmore13] and its constituent clouds [@longmore12; @kauffmann13] are presently not efficiently forming stars. We discuss the possible mechanisms for driving the turbulence below.
Interestingly, the density threshold implied by turbulent pressure for solar neighbourhood conditions is $n\sim10^4~{\rm cm}^{-3}$, which coincides with the @lada10 threshold (see §\[sec:localobs\] and §\[sec:turb\]). This agreement theoretically supports the fact that a universal, fixed density threshold for star formation is ruled out by the observations. Second-order effects that could work in conjunction with turbulence are a weak magnetic confinement of the ISM, which would narrow the density PDF (§\[sec:localobs\]), as well as an underproduction of massive stars due to a possibly bottom-heavy IMF (§\[sec:imf\]).
A wide range of mechanisms is found to be unable to explain the dearth of star formation in the CMZ.
1. When including the stellar potential, the gas disc is found to be only marginally Toomre stable, and hence susceptible to gravitational collapse. While this rules out morphological quenching, the gas by itself is not strongly self-gravitating, which may slow the rate at which gas clouds can decouple from the stellar potential and become gravitationally unstable. (§\[sec:surfacethr\])
2. The tidal density is lower than the mean density of the CMZ, indicating that star formation in not inhibited due to clouds being tidally stripped. (§\[sec:tidal\])
3. The atomic-to-molecular phase transition of hydrogen occurs at too low a density to play a role in the CMZ, especially at high bulge metallicities. (§\[sec:phase\])
4. While the magnetic field is capable of affecting the density PDF of the ISM somewhat, the magnetic pressure in the CMZ is much lower than the turbulent pressure. (§\[sec:magnetic\])
5. Radiation pressure should dominate the feedback energy at the surface density of the CMZ, but a SFE larger than 100% would be necessary to have radiation pressure compete with the turbulence. (§\[sec:rad\])
6. Cosmic rays, produced in supernovae or by the central black hole, are unable to overcome the turbulent pressure, even if the SFR was several orders of magnitude higher in the past. By contrast, they do contribute to the thermal pressure in the 100-pc ring. (§\[sec:rays\])
The conclusion that feedback processes are not responsible for the low SFR in the CMZ also agrees with the discrete distribution of the main star formation events – if feedback did play a role, it would only be able to do so locally.
It is interesting that the only global star formation relation that agrees with the low SFR in the CMZ is the @bigiel08 relation (§\[sec:globalobs\]). Is this the ‘fundamental’ star formation relation? The mere existence of high-redshift starburst galaxies with $\Sigma_{\rm SFR}$ two orders of magnitude higher than the CMZ at the same gas surface density [@daddi10b] suggests otherwise. As discussed throughout this paper, there are several reasons why the CMZ could be abnormal, and the extrapolation of the @bigiel08 relation to the CMZ therefore seems premature. The observational and numerical tests of the proposed mechanisms of §\[sec:impl\] will be essential to draw further conclusions on this matter.
In the remainder of this paper, we use the above findings to construct a self-consistent picture of star formation in the CMZ, as well as its implications for star formation in the centres of other galaxies.
What is driving the turbulence? {#sec:driver}
-------------------------------
While we find that (a combination of) more than one mechanism can be tuned to reach a satisfactory estimate of the SFR in the CMZ, it remains to be determined which mechanism is responsible for inhibiting star formation. Out of the possible explanations listed in §\[sec:summ\], the high turbulent pressure of the CMZ is perhaps the most fundamental. Contrary to the possible episodicity of the SFR, or perturbations acting on the cloud scale, it is the only mechanism that works instantaneously, at this moment in time. However, attributing the low SFR to turbulence alone is an incomplete argument, because the turbulent energy dissipates on a vertical disc crossing time. [Using the numbers from Table \[tab:prop\], the dissipation rate in the 230-pc integrated CMZ is roughly $5\times10^{-21}~{\rm erg}~{\rm cm}^{-3}~{\rm s}^{-1}$ [@maclow04]. The dissipation time-scale]{} is a mere $t_{\rm diss}=2h/\sigma\sim3$ Myr, whereas for the 100-pc ring it is about $0.5$ Myr. Something is needed to maintain the turbulence.
If we assume a steady-state CMZ, then the kinetic energy of the gas inflow along the bar is insufficient. The energy flux of the inflow is given by $v_{\rm infl}^2/t_{\rm infl}$, where $v_{\rm infl}$ is the gas inflow speed, and $t_{\rm infl}\equiv M_{\rm gas}/\dot{M}_{\rm infl}$ is the time-scale for accumulating the present gas mass. Under the assumption of a steady-state CMZ, the gas inflow time-scale equals the time-scale for gas depletion $t_{\rm depl}$, which in the CMZ is about a Gyr (see Table \[tab:prop\]). The energy dissipation rate is $\sigma^2/t_{\rm diss}$. Energy balance then gives $$\label{eq:vinfl}
v_{\rm infl}=\sigma\sqrt{\frac{t_{\rm infl}}{t_{\rm diss}}} ,$$ and substituting values representative of the CMZ yields a required $v_{\rm infl}$ for maintaining the turbulence of $600~{\rm km}~{\rm s}^{-1}$, which is much higher than the $\sim200~{\rm km}~{\rm s}^{-1}$ that can be seen in the HOPS NH$_3(1,1)$ data. If we relax the condition of a steady-state CMZ, a gas inflow time-scale of $t_{\rm infl}\sim100~{\rm Myr}$ (rather than the adopted $t_{\rm infl}=1~{\rm Gyr}$) would be needed to maintain the current turbulent pressure at the observed inflow speed. In order for such a scenario to be viable, the peak SFR would have to be more than ten times higher than is presently observed. This would put the CMZ temporarily on the star formation relations of equations (\[eq:sflaw\]) and (\[eq:sflawomega\]).
The gas inflow is only capable of driving the turbulence if star formation in the CMZ is episodic. In that case, the current gas depletion time-scale may not be an accurate estimate for the inflow time-scale. Specific combinations of gas inflow histories and star formation histories can explain the current degree of turbulence as the result of the global gas inflow. For instance, we might be seeing the product of a persistently high inflow rate, which builds up gas until finally a starburst is generated – with the CMZ presently at the stage between two bursts.
Episodic starbursts would periodically drive superbubbles [into the low-density ISM. As mentioned in §\[sec:rays\], massive star feedback can locally blow out parts of the 100-pc ring, producing the observed asymmetries in CO, NH$_3$ and cold dust (see Figure \[fig:img\]). Radio and $\gamma$-ray observations of nested giant bubbles indeed provide evidence that stellar feedback has a profound impact on localized portions of the CMZ [@sofue84; @su10; @carretti13]. On large scales, the bubbles consist of swept up, low-density gas, whereas higher density material can also be driven to a bubble, but only on size scales $\Delta R \la h$.]{}
After cooling in the Galactic halo, these bubbles return to the disc as high-velocity clouds [e.g. @bregman80; @wakker97]. Provided that the resulting energy flow is sufficiently high, this may drive the turbulence [in the [*dense*]{} ISM]{}. However, we estimate in §\[sec:rays\] that even at a star formation maximum, the energy density of feedback is probably smaller than the turbulent pressure. After conservation of energy (or a decrease due to dissipative losses) this implies that in-falling clouds are not likely to provide the necessary energy. While the return of ejected material therefore does not drive the turbulence, feedback itself is unlikely to do so for the same reason. Even if we imagine that feedback would be the source of the turbulence, then the presently low SFR implies that star formation would start after the turbulent energy has dissipated, i.e. in a few Myr from now. The resulting feedback would compress gas, which could trigger even more star formation. Only if the gas were converted to a warm, diffuse phase, star formation could in principle be halted by feedback. However, the resulting blow-out would channel away the energy and a molecular disc would remain, which should still be forming stars.
\
[Quantifying the effects of feedback and other classical turbulence drivers shows that neither are effective. Following the compilation by @maclow04, we find that magnetorotational instabilities (yielding a heating rate of $\sim5\times10^{-24}~{\rm erg}~{\rm cm}^{-3}~{\rm s}^{-1}$), gravitational instabilities ($\sim10^{-24}~{\rm erg}~{\rm cm}^{-3}~{\rm s}^{-1}$), protostellar outflows ($\sim10^{-25}~{\rm erg}~{\rm cm}^{-3}~{\rm s}^{-1}$), ionizing radiation ($\sim10^{-25}~{\rm erg}~{\rm cm}^{-3}~{\rm s}^{-1}$), and supernovae ($\sim2\times10^{-23}~{\rm erg}~{\rm cm}^{-3}~{\rm s}^{-1}$) are all incapable of compensating the dissipation rate. Only in the 100-pc ring, the heating rate due to gravitational instabilities ($\sim10^{-21}~{\rm erg}~{\rm cm}^{-3}~{\rm s}^{-1}$) approaches the local turbulent energy dissipation rate ($\sim10^{-20}~{\rm erg}~{\rm cm}^{-3}~{\rm s}^{-1}$).]{}
A more sustainable solution would be that the turbulence is driven by the (acoustic) gas instabilities themselves. As gas falls into the stellar-dominated potential of the CMZ, it grows unstable due to geometric convergence that compresses the gas to a higher density, even in the absence of self-gravity [@montenegro99], and the turbulent pressure increases accordingly. If the gas density has built up far enough that the compression leads to gravitational collapse, then the rate of energy dissipation increases and the gas eventually forms stars.[^15] In this picture, the inner gas discs of galaxies are driven to a spiral catastrophe by acoustic instabilities, which drive the turbulence and produce irregular gas structures[^16] during an intermittent phase in which a minor role is played by the self-gravity of the gas and star formation. This naturally leads to bursty star formation, and once again it seems that even the explanation of the low SFR in the CMZ by turbulence requires some degree of episodicity. This scenario is consistent with the idea that the central rings of galaxies are prone to gravitational instabilities when a certain density threshold is reached (see §\[sec:episodic\]).
Of course, there exists a global gas inflow rate above which the threshold density criterion for gravitational instability is always satisfied and star formation is no longer episodic. This may be one of the differences between the CMZ and the vigorously star-forming galaxies of @daddi10b. [The precise value of this critical inflow rate varies with galaxy properties, but empirically it is straightforward to distinguish both gas inflow regimes. [If gas mass is building up before undergoing a starburst, then the inflow rate must exceed the SFR (even if the inflow rate is variable), whereas the SFR of a starburst exceeds the inflow rate per definition. Conversely, if the inflow rate is so high that the gas mass does not have to build up before becoming gravitationally unstable and forming stars, the SFR is comparable to the inflow rate. This should also hold instantaneously and would therefore be insensitive to variations of the inflow rate, as long as the inflow rate is high enough to persistently drive star formation. We therefore predict that the transition from episodic star formation to persistent, inflow-driven gravitational instabilities should occur around a mass inflow rate $\dot{M}_{\rm infl}\sim\dot{M}_{\rm SFR}$. If the physics of the CMZ can be extrapolated to the high-redshift regime, then galaxies (or their centres) with $\dot{M}_{\rm infl}\la\dot{M}_{\rm SFR}$ should follow the @daddi10b sequence of galaxies with enhanced SFRs, whereas systems with $\dot{M}_{\rm infl}>\dot{M}_{\rm SFR}$ should statistically be found to have lower SFRs (making excursions to elevated SFRs and hence $\dot{M}_{\rm infl}<\dot{M}_{\rm SFR}$ during their periodic bursts of star formation).[^17]]{}]{}
A self-consistent star formation cycle {#sec:total}
--------------------------------------
We have now discussed all the ingredients for a scenario in which the local and global explanations for the low SFR in the CMZ represent different aspects of the same mechanism. This is crucial for a satisfactory explanation, because the different size scales influence each other. For instance, the overall gas supply is regulated on a global scale ($\Delta R>h$), but a local condition must determine the gas consumption rate if turbulence sets the critical density for star formation.
Based on the discussion in §\[sec:summ\] and §\[sec:driver\], we propose a multi-scale cycle that controls the star formation in the CMZ. It is summarised in Figure \[fig:schem\], which also illustrates the conversion of physical quantities to the observables that part of the analysis in this paper is based on. The different stages of the cycle are as follows.
1. \[pt:inflow\] Gas flows towards the CMZ from larger galactocentric radii, which can be driven by secular evolution (e.g. dust lane or bar transport) or by external torques (e.g. due to minor mergers or galaxy interactions).
2. Geometric convergence causes the inflowing gas to be compressed by acoustic instabilities, without the gas being gravitationally unstable.
3. The acoustic instabilities drive an increased turbulent velocity, which cascades down to smaller scales.
4. \[pt:turb\] The elevated turbulent pressure increases the local density threshold for star formation as $n_{\rm th}\propto n_0{\cal M}^2$ to $n_{\rm th}\sim10^8~{\rm cm}^{-3}$.
5. \[pt:accum\] Due to the high density threshold, the gas is not consumed to form stars and instead accumulates until the global density threshold for gravitational instabilities is reached.
6. The gravitational instabilities drive the gas down to sufficiently high densities to overcome the local density threshold for star formation.
7. \[pt:burst\] At such high densities, the time-scale for turbulent energy dissipation is short, and the gas is rapidly turned into stars. This can occur at different times in different parts of the CMZ, as is exemplified by Sgr B2 undergoing a starburst while the other CMZ clouds are inactive.
8. The cycle repeats itself, starting again from point (i).
Whether or not the system is observed during a starburst or a star formation minimum depends on the relative time-scales of the stages in the above cycle. The inflow and accumulation of gas from stages \[pt:inflow\]–\[pt:turb\] take place on a galactic dynamical time-scale, which is 5 Myr on the spatial scale of the 100-pc ring, and $\sim100$ Myr at the outer edge of the bar. [As stated in §\[sec:surfacethr\], the time-scale for clouds to become gravitationally unstable during stage \[pt:accum\] is $t_{\rm grav}\sim Q_{\rm gas}/\kappa$ [e.g. @jogee05], which in the 100-pc ring is $t_{\rm grav}\sim1~{\rm Myr}$, but in the 230 pc-integrated CMZ it is $t_{\rm grav}\sim20~{\rm Myr}$. This comparison shows that the 100-pc ring may be the unstable phase of the proposed cycle, and that stage \[pt:burst\] may be reached at different times throughout the CMZ.]{} [Because the gas needs to reach high densities before becoming gravitationally unstable and forming stars, it is likely that much of the star formation in the entire CMZ occurs in bound stellar clusters [see §\[sec:episodic\] and @kruijssen12d; @longmore13b].]{}
The gas consumption time-scale during stage \[pt:burst\] should be shorter than the time spanned by stages \[pt:inflow\]–\[pt:accum\] on the scale of individual clouds, because the free-fall time is only a few $10^5$ yr or less at the high density of the clouds in the CMZ. However, the integrated starburst duration for the entire CMZ depends on mixing and hence should be comparable to its dynamical time-scale (i.e. up to 10 Myr), with the gas being resupplied on the dynamical time-scale of the bar (i.e. up to 100 Myr). Such relatively brief episodes of nuclear activity are thought to last up to several Myr, and are supported by HST observations of nearby galaxies [@martini03]. Statistically speaking, the CMZ should therefore be observed at a star formation minimum, although the simultaneous occurrence of mini-bursts of star formation in different parts of the CMZ could temporarily place it on or above galactic star formation relations.
Verifying our proposed scenario using galactic scaling relations alone is not straightforward. One qualitative constraint follows from Figure \[fig:kslaw\], which shows that on a certain length scale ($\Delta R=230~{\rm pc}$) the CMZ fits the empirical Schmidt-Kennicutt relation of equation (\[eq:sflaw\]). It may be that on such a spatial scale, all of the stages \[pt:inflow\]–\[pt:burst\] are being sampled. However, even when averaged over 230 pc, the CMZ does not fit the Silk-Elmegreen relation of equation (\[eq:sflawomega\]). [@krumholz12a recently proposed that a universal star formation relation is obtained by substituting $\Omega\rightarrow1/t_{\rm sf}$ in equation (\[eq:sflawomega\]), where the star formation time-scale $t_{\rm sf}$ is either set by the local (GMC) or global (galactic) dynamical time-scale, whichever is the shortest. If we account for the long condensation time-scale of GMCs and hence increase $t_{\rm sf}$ by a factor of $Q_{\rm gas}$ (as in the Toomre regime of @krumholz12a), we see that the 230 pc-integrated CMZ fits a modified Silk-Elmegreen relation $\Sigma_{\rm SFR}=A_{\rm SE}\Sigma\Omega/Q_{\rm gas}$. The 1.3$^\circ$ cloud and the 100-pc ring do not, and in the framework of the cycle of Figure \[fig:schem\] this is because on such small scales the episodicity of star formation is important.]{}
The conversion of the proposed cycle to observables is well beyond the scope of this paper, but Figure \[fig:schem\] shows a qualitative illustration. The global gas surface density mainly samples stage \[pt:accum\], [whereas the star formation rate density traces stage \[pt:burst\] of the cycle. However, the relation to the actual volume densities of star-forming gas and young stars depends on several factors. Observational limitations such as the spatial resolution, the sensitivity, and possible tracer biases lead to measurement uncertainties in the derivation of the relevant physical quantities. For instance, unresolved stars and gas may not be occupying the same volume [e.g. due to different filling factors or porosities, @silk97], and the sampling of the gas (PDF) and the star formation may be incomplete (the degree of which depends on the adopted tracers). Finally, there may be systematic conversion biases (e.g. $X_{\rm CO}$) when translating measured quantities to physical ones. Before a consensus on these points exists, it is more feasible to verify the proposed scenario as suggested in §\[sec:testepisodic\] and §\[sec:testturb\], i.e. focussing on the parts rather than the sum thereof.]{}
[For example, stages (i)–(iii) may be probed in nearby galaxies, with global gas flow kinematics constraining stages (i) and (ii), and velocity dispersion measurements tracing stage (iii).]{}
The extragalactic regime {#sec:extra}
------------------------
Given that the CMZ can only be observed in its present state, the cycle of Figure \[fig:schem\] is sampled better by including other galaxies where the same process is taking place. The inner regions of galaxies often have dust structures with the number of spiral arms increasing with radius, and power spectra that are consistent with them being a manifestation of acoustic turbulence [e.g. @elmegreen98; @martini99; @elmegreen02b]. These structures host little to no star formation, which if present is often arranged in a circumnuclear ring [e.g. @barth95; @jogee02; @sandstrom10; @vanderlaan13], and their gas by itself is often not strongly self-gravitating [@sani12]. In all respects, the central regions of these galaxies are reminiscent of the CMZ. Indeed, there are known examples of nearby galaxies of which the star formation activity in their central regions falls below galactic scaling relations [@hsieh11; @nesvadba11; @sani12], even in cases which at first sight may be perceived as a starburst system [@kenney93]. By contrast, examples of galaxy centres with a normal or enhanced SFR exist too, which may indicate a different stage in the same process (e.g. the central starbursts in the sample of @jogee05, also see @sakamoto07 [@sakamoto11]). A statistical census of the activity in such galaxies enables a quantitative estimate of the duration of the different stages in Figure \[fig:schem\]. This is a key step, which has been made previously. For instance, @martini03 consider a sample of 123 nearby, active and passive, barred and unbarred galaxies, and estimate that the quiescent stage takes up to ten times longer than the star formation episode, which is consistent with what we propose for the CMZ in §\[sec:total\].
[Following a similar argument, @davies07 find that active galactic nuclei (AGN) typically become active some 50–100 Myr after the onset of prior (but ceased) starburst activity. This is consistent with the scenario of Figure \[fig:schem\] – a starburst and its corresponding feedback may either consume or eject the gas, preventing the black hole from being fed until supernova feedback has ended (after 40 Myr). After that time, the gas flow towards the black hole continues. Because in our scenario the gas is not (yet) self-gravitating, it can proceed to the black hole without forming stars. Due to the gas inflow and preceding starburst activity, this phase should be characterized by centrally concentrated surface brightness and gas mass profiles, which has indeed recently found by @hicks13. We conclude that our results may well have implications for the massive black hole growth and AGN activity. Specifically, the phase of low star formation activity in the cycle of Figure \[fig:schem\] could provide a window for efficient black hole growth.]{}
One of the key questions is how the physics that regulate star formation in the CMZ and the central regions of nearby galaxies translate to other extragalactic environments. An interesting target would be giant elliptical galaxies that contain a sizable, but remarkably quiescent gas reservoir [e.g. @crocker12]. In terms of the gas properties, the CMZ is remarkably similar to the conditions found in ULIRGs and high-redshift, star-forming galaxies [@kruijssen13]. However, these galaxies are undergoing vigorous starbursts, whereas the SFR in the CMZ is much lower than expected. We suggest in §\[sec:driver\] that rapidly star-forming galaxies may have such a high global gas inflow rate that their constituent gas can always become gravitationally unstable, rather than having to build up to a certain density and forming stars episodically. [This hypothesis can be tested with a simple observational prediction. Galaxies or their centres with $\dot{M}_{\rm infl}\la\dot{M}_{\rm SFR}$ should follow the @daddi10b sequence of galaxies with enhanced SFRs, whereas systems with $\dot{M}_{\rm infl}>\dot{M}_{\rm SFR}$ should statistically be found to have lower SFRs (making excursions to elevated SFRs during their periodic bursts of star formation). While observationally challenging, high-resolution ALMA observations should enable this prediction to be tested (see §\[sec:driver\] for a more detailed discussion).]{} Alternatively, the star formation in a subset of such galaxies may take place at sufficiently large galactocentric radii for acoustic instabilities to be unimportant,[^18] i.e. in the regime where the gravitational instability of the gas alone is driving star formation. Another obvious concern is that by specifically considering rapidly star-forming galaxies, we are already biasing ourselves to what may be an episodic peak of star formation activity – in which case star formation in these galaxies could actually be similar to that in the CMZ over longer time-scales.
In summary, [we find that the lower-than-expected SFR in the CMZ can be explained by a self-consistent cycle connecting the galaxy-scale gas flow, acoustic and gravitational instabilities, turbulence, and local star formation thresholds. The cycle is summarized in §\[sec:total\] and Figure \[fig:schem\]. We conclude that a low SFR]{} may well be a common property of the centres of Milky Way-like galaxies, although examples of the opposite should exist too if our proposed explanation is valid. The question remains how the lessons learned about star formation in the central regions of galaxies can be extrapolated to other galactic environments. As discussed in this section and §\[sec:impl\], the combination of future numerical work with data from new observational facilities should allow this topic to be addressed in the necessary detail.
Acknowledgments {#acknowledgments .unnumbered}
===============
JMDK thanks Eli Bressert, Andi Burkert, Cathie Clarke, Richard Davies, Maud Galametz, Phil Hopkins, Mark Morris, Thorsten Naab, Stefanie Walch, Simon White, Farhad Yusef-Zadeh, and Tim de Zeeuw for helpful discussions. We are indebted to Timothy Davis and Maud Galametz for detailed comments on the manuscript. JMDK thanks the Institute of Astronomy in Cambridge, where a large part of this work took place, for their kind hospitality and gratefully acknowledges support in the form of a Visitor Grant.
This research made use of data products from the Midcourse Space Experiment. Processing of the data was funded by the Ballistic Missile Defense Organization with additional support from NASA Office of Space Science. This research has also 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.
\[lastpage\]
[^1]: [email protected]
[^2]: [Except for the mass loss due to protostellar outflows [e.g. @matzner00; @nakamura07].]{}
[^3]: \[footnote:uncertain\][Note that this gives circular velocities of $140$–$200~{\rm km~s}^{-1}$ for the three regions listed in Table \[tab:prop\], whereas the peak line-of-sight velocity of the 100-pc ring as measured from the HOPS NH$_3$(1,1) emission [@walsh11; @purcell12] is $80~{\rm km}~{\rm s}^{-1}$ at the location of Sgr B2. Such a low line-of-sight velocity may be caused by the possible eccentric orbit of the ring [@molinari11], which is thought to lie under such an angle that the edge of the ring lies close to apocentre as seen from Earth. In this scenario, the measured line-of-sight velocity should be lower than the local circular velocity at the position of Sgr B2, a difference that would be amplified further by possible projection effects. An alternative explanation is that the 100-pc ring extends to higher longitudes than stated in @molinari11, in which case the circular velocity would exceed the observed line-of-sight velocities at the presumed tangent points due to projection alone (Bally et al. in prep.). This picture is consistent with the proper motion of Sgr B2, which is $\sim90~{\rm km}~{\rm s}^{-1}$ [@reid09]. Using the measured velocities instead of the @launhardt02 rotation curve gives a factor of $\sim1.6$ lower epicyclic frequencies. Both extremes are used to calculate the possible range of $\kappa$ listed in Table \[tab:prop\], and hence also contribute to the range of $Q_{\rm gas}$ and $Q_{\rm tot}$.]{}
[^4]: This extra sequence of star-forming galaxies contains some galaxies from the nearby starburst sample of @kennicutt98b, but has an elevated SFR with respect to that paper because a different value of $X_{\rm CO}$ is assumed [@daddi10b]. We emphasise that inferring the H$_2$ density using CO emission is an indirect method and hence may introduce substantial uncertainty. However, recent results from the KINGFISH survey of nearby galaxies suggest that $X_{\rm CO}$ weakly decreases with star formation rate density [e.g. @sandstrom13], which is at least qualitatively consistent with @daddi10b.
[^5]: This estimate assumes that the maximum star formation efficiency in protostellar cores is $\epsilon_{\rm core}=0.5$ and the star formation timescale is $\phi_t t_{\rm ff}=1.91t_{\rm ff}$ as in @krumholz05 [Eqs. 19–21]. However, the precise numbers are not important to the order-of-magnitude argument that is made here.
[^6]: [Obviously, the bulge has not been in place for a Hubble time, but this comparison does put the present SFR in an appropriate perspective.]{}
[^7]: We exclude the nuclear cluster of the Milky Way, of which the young stellar component is thought to have a very different origin (see e.g. @antonini13 for a discussion).
[^8]: [Using the cluster formation model of @kruijssen12d we find that at the high gas density of the 100-pc ring about $\sim50\%$ of the stars are expected to form in bound stellar clusters. This idea is supported by the existence of the dispersed population of $24\mu{\rm m}$ sources (see Figure \[fig:img\]), [and the roughly equal numbers of massive stars observed in clusters and in the field of the CMZ [@mauerhan10]]{}.]{}
[^9]: Using equation (30) of @antonini13, with $\gamma=1.8$, $r_0=r_{\rm in}=80~{\rm pc}$, $\rho_0=145~{\mbox{M$_\odot$}}~{\rm pc}^{-3}$, and $m_{\rm cl}=10^4~{\mbox{M$_\odot$}}$.
[^10]: [Note that @portegieszwart01 predict the presence of 50 Arches-like clusters in the CMZ based on a 80-Myr lifetime in the Galactic tidal field. When including tidal shocking due to GMCs, only three or four Arches-like clusters are expected to exist in the CMZ. Within the statistical limits, this is consistent with the presence of the Arches and Quintuplet clusters.]{}
[^11]: For the @krumholz05 model this assumes a typical size-linewidth relation for GMCs of $\sigma\propto R^{0.5}$.
[^12]: Note that the difference is still an order of magnitude when neglecting the (admittedly weak) effect of magnetic fields.
[^13]: [Although locally they may drive arches and bubbles – there will always be some (small) volume $V$ such that the feedback pressure from the enclosed stars competes with the turbulent pressure. For the 100-pc ring (and hence Sgr B2 where most of the star formation takes place), cosmic rays overcome the turbulence for $V\la8~{\rm pc}^{3}$ or $R\la1~{\rm pc}$]{}.
[^14]: [Considering that the high $Q_{\rm gas}$ (see §\[sec:surfacethr\]) may inhibit the decoupling of gas clouds from the stellar background, we note that these are not likely to be gravitational instabilities, but are probably acoustic in nature [@montenegro99].]{}
[^15]: [Note that the regions that reach the threshold density for gravitational collapse need not be randomly distributed throughout the CMZ, but may be corresponding to specific locations in the geometry, such as possibly Sgr B2 (see §\[sec:shocks\] and @elmegreen94 [@boeker08; @sandstrom10]).]{}
[^16]: [There is evidence for the presence of such dust lane structures in the central region of the Milky Way on larger ($\sim~{\rm kpc}$) scales (see e.g. @mccluregriffiths12 and references therein).]{}
[^17]: [Because the detailed variation of galactic inflow rates over long time-scales is unknown, the present-day stellar mass (which gives a time-averaged SFR) can unfortunately not be used to constrain this argument further. For instance, the SFR of a system may presently be persistently high due to a high inflow rate, whereas in the past it could have been episodic at a low inflow rate. In that case, the time-averaged SFR is much lower than the current SFR, and combining the average SFR with the present inflow rate would incorrectly suggest current episodicity]{}.
[^18]: [The circumnuclear starburst of Arp 220 is a clear counterexample.]{}
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Quantum observables can be identified with vector fields on the sphere of normalized states. The resulting [*vector representation*]{} is used in the paper to undertake a simultaneous treatment of macroscopic and microscopic bodies in quantum mechanics. Components of the velocity and acceleration of state under Schr[ö]{}dinger evolution are given for a clear physical interpretation. Solutions to Schr[ö]{}dinger and Newton equations are shown to be related beyond the Ehrenfest results on the motion of averages. A formula relating the normal probability distribution and the Born rule is found.'
author:
- 'A.'
title: On the motion of macroscopic bodies in quantum theory
---
Newtonian dynamics in Hilbert spaces
====================================
Everyday experience shows that macroscopic bodies have well-defined position in space at any time. In the simplest case of a classical particle (material point) position at a given time is provided by vector ${\bf a}$ in the Euclidean space ${\mathbb{R}}^{3}$. Accordingly, the space ${\mathbb{R}}^{3}$ itself can be thought of as the space of all possible positions of a classical particle. In quantum mechanics the state of a spinless particle with a known position ${\bf a} \in {\mathbb{R}}^{3}$ is described by the Dirac delta function $\delta^{3}_{\bf a}({\bf x})=\delta^{3}({\bf x}-{\bf a})$. In particular, the state of a classical particle at any time is such a function. The map $\omega: {\bf a} \longrightarrow \delta^{3}_{\bf a}$ provides a one-to-one correspondence between points ${\bf a} \in {\mathbb{R}}^{3}$ and “state" functions $\delta^{3}_{\bf a}$. The set ${\mathbb{R}}^{3}$ can be then identified with the set $M_{3}$ of all delta functions in the space of state functions of the particle.
The delta functions are, of course, not in the usual $L_{2}({\mathbb{R}}^{3})$ Hilbert space on the measure $d^{3}x$. We study here a way to deal with functions of this type systematically and consistently, and in so doing, establish an interesting connection between the quantum theory and classical mechanics.
The inner product on the usual Hilbert space $L_{2}({{\mathbb{R}}}^{3})$ of state functions of a particle can be formally written for $\varphi, \psi \in L_{2}({\mathbb{R}}^{3})$ in the following way: $$\label{innerdd}
(\varphi, \psi)_{L_{2}}=\int \delta^{3}({\bf x}-{\bf y})\varphi({\bf x}){\overline \psi}({\bf y})d^{3}{\bf x}d^{3}{\bf y}.$$ In particular, the fact that delta functions are not in $L_{2}({\mathbb{R}}^{3})$ is related to the singularity of delta functions. Let us replace the kernel $\delta^{3}({\bf x}-{\bf y})$ by the Gaussian function $\left(\frac{L}{\sqrt {2\pi}}\right)^{3}e^{-\frac{L^{2}}{2}({\bf x}-{\bf y})^{2}}$ for some positive constant $L$. This yields the product $$\label{innerH}
(\varphi, \psi)_{\bf H}=\left(\frac{L}{\sqrt {2\pi}}\right)^{3}\int e^{-\frac{L^{2}}{2}({\bf x}-{\bf y})^{2}} \varphi({\bf x}){\overline \psi}({\bf y})d^{3}{\bf x}d^{3}{\bf y}.$$ One can check Ref.[@KryukovIJMMS] that this is indeed an inner product on $L_{2}({{\mathbb{R}}}^{3})$. Some physical applications of this inner product were studied in Refs.[@KryukovJMP1]-[@KryukovFOP]. The separable Hilbert space ${\bf H}$ obtained by completing the space $L_{2}({{\mathbb{R}}}^{3})$ in this inner product contains delta functions $\delta^{3}({\bf x}-{\bf a})$ and their derivatives. Moreover, by choosing $L$ sufficiently large (or by choosing appropriate units), one can make the norm of any given square-integrable function in this metric as close as desired to its $L_{2}({{\mathbb{R}}}^{3})$-norm.
By dropping the coefficient $(1/{\sqrt {2\pi}})^{3}$ and using $L=\frac{1}{2\sigma}$ for an appropriate $\sigma$ we obtain the product $$\label{hh}
(\varphi, \psi)_{\bf H}=\int e^{-\frac{({\bf x}-{\bf y})^{2}}{8\sigma^{2}}}\varphi({\bf x})\overline{\psi}({\bf y})d^{3}{\bf x}d^{3}{\bf y}.$$
Formally, $$\label{delta1}
\int e^{-\frac{({\bf x}-{\bf y})^{2}}{8\sigma^{2}}}\delta^{3}({\bf x}-{\bf a})\delta^{3}({\bf y}-{\bf a})d^{3}{\bf x}d^{3}{\bf y}=1,$$ so that the norm of the delta function $\delta^{3}({\bf x}-{\bf a})$ in ${\bf H}$ with the metric (\[hh\]) is $1$. The set $M_{3}$ of all delta functions $\delta^{3}_{\bf a}({\bf x})$ with ${\bf a} \in {\mathbb{R}}^{3}$ is therefore a subset of the unit sphere in the Hilbert space ${\bf H}$.
The map $\rho_{\sigma}: {\bf H} \longrightarrow L_{2}({\mathbb{R}}^{3})$ that relates $L_{2}$ and ${\bf H}$-representations is given by the Gaussian kernel $$\label{sigma}
\rho_{\sigma}({\bf x},{\bf y})=\left(\frac{1}{2\pi \sigma^{2}}\right)^{3/4}e^{-\frac{({\bf x}-{\bf y})^{2}}{4\sigma^{2}}}.$$ In terms of $\rho_{\sigma}$, the kernel $G({\bf x}, {\bf y})$ of the metric on ${\bf H}$ is given by $$\label{GG}
G({\bf x}, {\bf y})=(\rho^{\ast}_{\sigma}\rho_{\sigma})({\bf x}, {\bf y})=e^{-\frac{({\bf x}-{\bf y})^{2}}{8\sigma^{2}}},$$ which is consistent with (\[hh\]). The map $\rho_{\sigma}$ transforms delta functions $\delta^{3}_{\bf a}$ to Gaussian functions ${\widetilde \delta^{3}_{\bf a}}=\rho_{\sigma}(\delta^{3}_{\bf a})$, centered at ${\bf a}$, providing an alternative, more common way of dealing with singularity of delta functions. The image $M^{\sigma}_{3}$ of $M_{3}$ under $\rho_{\sigma}$ is a subset of the unit sphere in $L_{2}({\mathbb{R}}^{3})$ made of the functions ${\widetilde \delta^{3}_{\bf a}}$. Both realizations will prove useful in the discussion of motion of macroscopic bodies in quantum mechanics.
To know position ${\bf a}$ of a classical particle in ${\mathbb{R}}^{3}$ is to know the corresponding point $\delta^{3}_{\bf a}$ in $M_{3}$. Consider a path ${\bf r}={\bf a}(t)$ with values in ${\mathbb{R}}^{3}$ and the corresponding path $\varphi=\delta^{3}_{{\bf a}(t)}$ in $M_{3}$. With the use of the chain rule the velocity vector $d \varphi/dt$ can be written as $$\label{chain}
\frac{d \varphi}{dt}=-\frac{\partial}{\partial {\bf x}^{i}}\delta^{3}({\bf x}-{\bf a})\frac {d{\bf a}^{i}}{dt},$$ where the usual summation convention for repeating indices is accepted. It follows that the norm $\left \| \frac{d \varphi}{dt} \right \|^{2}_{H}$ of the velocity in the space ${\bf H}$ is $$\int k({\bf x},{\bf y}) \frac{\partial}{\partial x^{i}}\delta^{3}({\bf x}-{\bf a})\frac {d{\bf a}^{i}}{dt}\frac{\partial}{\partial y^{k}}\delta^{3}({\bf y}-{\bf a})
\frac {d{\bf a}^{k}}{dt}d^{3}{\bf x}d^{3}{\bf y},$$ where $k({\bf x},{\bf y})=e^{-\frac{({\bf x}-{\bf y})^{2}}{8\sigma^{2}}}$. “Integration by parts" in the last expression gives $$\label{parts1}
\left \| \frac{d \varphi}{dt} \right \|^{2}_{H}=
\left.\frac {\partial^{2}k({\bf x},{\bf y})}{\partial x^{i} \partial y^{k}}\right|_{{\bf x}={\bf y}={\bf a}} \frac {d{\bf a}^{i}}{dt}\frac {d{\bf a}^{k}}{dt}.$$ Furthermore, $$\left.\frac {\partial^{2}k({\bf x},{\bf y})}{\partial x^{i} \partial y^{k}}\right|_{{\bf x}={\bf y}={\bf a}}=\frac{1}{4\sigma^{2}}\delta_{ik},$$ where $\delta_{ik}$ is the Kronecker delta symbol. Assuming now that the distance in ${\mathbb{R}}^{3}$ is measured in the units of $2\sigma$ (equivalently, taking $2\sigma=1$) one obtains the equality of the speeds $$\label{Norms}
\left \| \frac{d \varphi}{dt} \right \|_{H}=\left \| \frac{d {\bf a}}{dt} \right \|_{{\mathbb{R}}^{3}}.$$ From this equality of norms it follows that the set $M_{3}$ as a metric subspace of ${\bf H}$ is identical to the Euclidean space ${\mathbb{R}}^{3}$. That is, the one-to-one map $\omega: {\mathbb{R}}^{3} \longrightarrow {\bf H}$ is an isometric embedding Ref.[@KryukovIJMMS]. Notice however that $M_{3}$ is not a vector subspace of ${\bf H}$. Rather, as follows from (\[delta1\]), the metric space $M_{3}$ is a submanifold of the unit sphere $S^{\bf H}$ in ${\bf H}$. Since delta functions $\delta^{3}_{{\bf a}_{k}}$ with different ${{\bf a}_{k}}$, $k=1,...,n$ are linearly independent, the manifold $M_{3}$ “spirals” through dimensions of the sphere, forming a complete subset of ${\bf H}$. This means that no function in ${\bf H}$ is orthogonal to the submanifold $M_{3}$ Ref.[@KryukovIJMMS].
Nevertheless, a vector structure on $M_{3}$ exists. For instance, define the operations of addition $\oplus$ and multiplication by a scalar $\lambda \odot$ via $\omega({\bf a})\oplus\omega({\bf b})=\omega({\bf a}+{\bf b})$ and $\lambda \odot\omega({\bf a})=\omega(\lambda {\bf a})$, where the map $\omega$ is the same as before. The resulting operations are continuous in the topology of $M_{3}\subset {\bf H}$. That is, the metric space $M_{3}$ with this vector structure is isomorphic to the vector space ${\mathbb{R}}^{3}$ with the Euclidean metric. From $$\frac{d}{dt}\delta^{3}_{{\bf a}}({\bf x})=-\frac{\partial}{\partial x^{i}} \delta^{3}_{{\bf a}}({\bf x})\frac{d a^{i}}{dt}$$ and $$\frac{d^{2}}{dt^{2}}\delta^{3}_{{\bf a}}({\bf x}) =
\frac{\partial^{2}}{\partial x^{i}\partial x^{j}} \delta^{3}_{{\bf a}}({\bf x})\frac{d a^{i}}{dt}\frac{d a^{j}}{dt}-\frac{\partial}{\partial x^{i}} \delta^{3}_{{\bf a}}({\bf x})\frac{d^{2} a^{i}}{dt^{2}},$$ together with (\[delta1\]), (\[Norms\]), and the orthogonality of the first and second derivatives of $\delta^{3}_{\bf a}({\bf x})$, it follows that projection of velocity and acceleration of the state $\delta^{3}_{{\bf a}(t)}$ onto $M_{3}$ yields correct Newtonian velocity and acceleration of the classical particle. That is: $$\label{v1}
\left( \frac{d}{dt}\delta^{3}_{{\bf a}}({\bf x}), -\frac{\partial}{\partial x^{i}} \delta^{3}_{{\bf a}}({\bf x})\right)_{\bf H} =\frac{d a^{i}}{dt}$$ and $$\label{a1}
\left( \frac{d^{2}}{dt^{2}}\delta^{3}_{{\bf a}}({\bf x}) , -\frac{\partial}{\partial x^{i}} \delta^{3}_{{\bf a}}({\bf x})\right)_{\bf H} =\frac{d^{2} a^{i}}{dt^{2}}.$$
Furthermore, Newtonian dynamics of the classical particle follows from the principle of least action for the action functional $S$ on paths in ${\bf H}$, defined by $$\int k({\bf x},{\bf y})\left[\frac{m}{2}\frac{d \varphi_{t}({\bf x})}{dt} \frac{d{\overline \varphi_{t}}({\bf y})}{dt}-V({\bf x}) \varphi_{t}({\bf x}) {\overline \varphi_{t}}({\bf y})\right]d^{3}{\bf x}d^{3}{\bf y}dt,$$ where $m$ is the mass of the particle, $V$ is the potential and $k({\bf x}, {\bf y})=e^{-\frac{1}{2}({\bf x}-{\bf y})^{2}}$, as before. Suppose that $\varphi_{t}$ is constrained to take values on the submanifold $M_{3}\subset {\bf H}$, i.e., $\varphi_{t}({\bf x})=\delta^{3}({\bf x}-{\bf a}(t))$. Using (\[chain\]) and integrating by parts as in (\[parts1\]), we immediately obtain $$S=\int\left[\frac{m}{2}\left(\frac{d{\bf a}}{dt}\right)^{2}-V({\bf a})\right]dt,$$ i.e., the usual action functional for a material point in classical mechanics. In these terms, a classical particle is a constrained dynamical system in ${\bf H}$. The same applies to $L_{2}({\mathbb{R}}^{3})$-representation and paths constrained to take values in $M^{\sigma}_{3}=\rho_{\sigma}(M_{3})$ in $L_{2}({\mathbb{R}}^{3})$.
Classical particle mechanics, therefore, has an equivalent realization in terms of the new dynamical variables: the state $\varphi$ of the particle and the velocity $\frac{d\varphi}{dt}$ of the state. A similar realization exists for mechanical systems consisting of any number of classical particles. For example, the map $\omega \otimes \omega: {\mathbb{R}}^{3}\times {\mathbb{R}}^{3} \longrightarrow {\bf H}\otimes {\bf H}$, $\omega \otimes \omega ({\bf a}, {\bf b})=\delta^{3}_{\bf a} \otimes \delta^{3}_{\bf b}$ identifies the configuration space ${\mathbb{R}}^{3}\times {\mathbb{R}}^{3}$ of a two particle system with the embedded submanifold $M_{6}=\omega \otimes \omega({\mathbb{R}}^{3}\times {\mathbb{R}}^{3})$ of the Hilbert space ${\bf H}\otimes {\bf H}$. Consider a path $({\bf a}(t), {\bf b}(t))$ in ${\mathbb{R}}^{3}\times {\mathbb{R}}^{3}$ and the corresponding path $\delta^{3}_{{\bf a}(t)}\otimes \delta^{3}_{{\bf b}(t)}$ with values in $M_{6}$. For any $t$, the vectors $\frac{d}{dt}\delta^{3}_{{\bf a}(t)}\otimes \delta^{3}_{{\bf b}(t)}$ and $\delta^{3}_{{\bf a}(t)}\otimes \frac{d}{dt}\delta^{3}_{{\bf b}(t)}$ are tangent to $M_{6}$ at the point $\delta^{3}_{{\bf a}(t)}\otimes \delta^{3}_{{\bf b}(t)}$ and orthogonal in ${\bf H}\otimes {\bf H}$. The space $M_{6}$ with the induced metric is isometric to the direct product ${\mathbb{R}}^{3}\times {\mathbb{R}}^{3}$ with the natural Euclidean metric. Projection of velocity and acceleration of the state $\varphi(t)=\delta^{3}_{{\bf a}(t)}\otimes \delta^{3}_{{\bf b}(t)}$ onto the basis vectors $\left(- \frac{\partial}{\partial x^i}\delta^{3}_{{\bf a}(t)}\right)\otimes \delta^{3}_{{\bf b}(t)}$ and $\delta^{3}_{{\bf a}(t)}\otimes \left(-\frac{\partial}{\partial x^{k}}\delta^{3}_{{\bf b}(t)}\right)$ yields the velocity and acceleration of the particles by means of the formulas similar to (\[v1\]) and (\[a1\]).
We now turn the attention to quantum theory and explore a useful realization of quantum mechanics in terms of vector fields in the space of states.
Observables as vector fields
============================
Quantum observables can be identified with vector fields on the space of states Ref.[@KryukovUncert]. Namely, given a self-adjoint operator ${\widehat A}$ on a Hilbert space $L_{2}$ of square-integrable functions (it could in particular be the tensor product space of a many body problem) one can introduce the associated linear vector field $A_{\varphi}$ on $L_{2}$ by $$\label{vector}
A_{\varphi}=-i{\widehat A}\varphi.$$ This field is defined on a dense subset $D$ in $L_{2}$ on which the operator ${\widehat A}$ itself is defined. Clearly, to know the vector field $A_{\varphi}$ is the same as to know the operator ${\widehat A}$ itself. Moreover, the commutator of observables and the commutator (Lie bracket) of the corresponding vector fields are related in a simple way: $$\label{comm}
[A_{\varphi},B_{\varphi}]=[{\widehat A},{\widehat B}]\varphi.$$
The field $A_{\varphi}$ associated with an observable, being restricted to the sphere $S^{L_{2}}$ of unit normalized states, is tangent to the sphere. Indeed, the equation for the integral curves of $A_{\varphi}$ has the form $$\label{SchroedA}
\frac{d \varphi_{\tau}}{d\tau}=-i{\widehat A}\varphi_{\tau}.$$ The solution to (\[SchroedA\]) through initial point $\varphi_{0}$ is given by $\varphi_{\tau}=e^{-i{\widehat A}\tau}\varphi_{0}$. Here $e^{-i{\widehat A}\tau}$ denotes the one-parameter group of unitary transformations generated by $-i{\widehat A}$, as described by Stone’s theorem. It follows that the integral curve through $\varphi_{0} \in S^{L_{2}}$ will stay on the sphere. One concludes that, modulo the domain issues, the restriction of the vector field $A_{\varphi}$ to the sphere $S^{L_{2}}$ is a vector field on the sphere.
Under the embedding, the inner product on the Hilbert space $L_{2}$ gives rise to a Riemannian metric (i.e., point-dependent real-valued inner product) on the sphere $S^{L_{2}}$. For this one considers the realization $L_{2R}$ of the Hilbert space $L_{2}$, i.e., the real vector space of pairs $X=({\mathrm Re} \psi, {\mathrm Im} \psi)$ with $\psi$ in $L_{2}$. If $\xi, \eta$ are vector fields on $S^{L_{2}}$, one can define a Riemannian metric $G_{\varphi}: T_{R\varphi}S^{L_{2}}\times T_{R\varphi}S^{L_{2}} \longrightarrow R$ on the sphere by $$\label{Riem}
G_{\varphi}(X,Y)={\mathrm Re} (\xi, \eta).$$ Here the tangent space $T_{R\varphi}S^{L_{2}}$ to $S^{L_{2}}$ at a point $\varphi$ is identified with an affine subspace in $L_{2R}$, $X=({\mathrm Re} \xi, {\mathrm Im} \xi)$, $Y=({\mathrm Re} \eta, {\mathrm Im} \eta)$ and $(\xi, \eta)$ denotes the $L_{2}$-inner product of $\xi, \eta$. Note that the obtained Riemannian metric $G_{\varphi}$ is [*strong*]{} in the sense that it yields an isomorphism ${\widehat G}:T_{R\varphi}S^{L_{2}}\longrightarrow \left (T_{R\varphi}S^{L_{2}}\right)^{\ast}$ of dual spaces.
The Riemannian metric on $S^{L_{2}}$ yields a (strong) Riemannian metric on the projective space $CP^{L_{2}}$. For this, one defines the metric on $CP^{L_{2}}$ so that the bundle projection $\pi: S^{L_{2}} \longrightarrow CP^{L_{2}}$ would be a Riemannian submersion. The resulting metric on $CP^{L_{2}}$ is called the Fubini-Study metric. To put it simply, an arbitrary tangent vector $X \in T_{R\varphi}S^{L_{2}}$ can be decomposed into two components: tangent and orthogonal to the fibre $\{\varphi\}$ through $\varphi$ (i.e., to the plane $C^{1}$ containing the circle $S^{1}=\{\varphi\}$). The differential $d\pi$ maps the tangent component to the zero-vector. The orthogonal component of $X$ can be then identified with $d\pi(X)$. If two vectors $X,Y$ are orthogonal to the fibre $\{\varphi\}$, the inner product of $d\pi(X)$ and $d\pi(Y)$ in the Fubini-Study metric is equal to the inner product of $X$ and $Y$ in the metric $G_{\varphi}$. Note that the obtained Riemannian metrics on $S^{L_{2}}$ and $CP^{L_{2}}$ are invariant under the induced action of the group of unitary transformations on $L_{2}$.
An arbitrary vector in the Hilbert space at a point $\varphi$ can be decomposed onto the radial component (parallel to the radius vector from the origin to the point $\varphi$, i.e., parallel to $\varphi$ itself), and tangential component. The radial component of a vector field $A_{\varphi}$ associated with an observable vanishes. Accordingly, $A_{\varphi}$ can be decomposed into components tangent and orthogonal to the fibre $\{\varphi\}$. These components have a simple physical meaning. In fact, the equality $${\overline A} \equiv (\varphi, {\widehat A}\varphi)=(-i\varphi, -i{\widehat A}\varphi),$$ signifies that the expected value of an observable ${\widehat A}$ in the state $\varphi$ is the projection of the vector $-i{\widehat A}\varphi \in T_{\varphi}S^{L_{2}}$ on the unit vector $-i\varphi=-i I \varphi \in T_{\varphi}S^{L_{2}}$, tangent to the fibre $\{\varphi\}$. Because $$(\varphi, {\widehat A}^{2}\varphi)=({\widehat A}\varphi, {\widehat A}\varphi)=(-i{\widehat A}\varphi, -i{\widehat A}\varphi),$$ the term $(\varphi, {\widehat A}^{2}\varphi)$ is just the norm of the vector $-i{\widehat A}\varphi$ squared. The expected value $(\varphi, {\widehat A}_{\bot}\varphi)$ of the operator ${\widehat A}_{\bot} \equiv {\widehat A}-{\overline A}I$ in the state $\varphi$ is zero. Therefore, the vector $-i{\widehat A}_{\bot}\varphi=-i{\widehat A}\varphi-(-i{\overline A}\varphi)$, which is the component of $-i{\widehat A}\varphi$ orthogonal to $-i\varphi$ is orthogonal to the fibre $\{\varphi\}$. Accordingly, the variance $$\Delta A^{2}=(\varphi, ({\widehat A}-{\overline A}I)^{2}\varphi)=(\varphi, {\widehat A}_{\bot}^{2}\varphi)=(-i{\widehat A}_{\bot}\varphi, -i{\widehat A}_{\bot}\varphi)$$ is the norm squared of the component $-i{\widehat A}_{\bot}\varphi$. As discussed, the image of this vector under $d\pi$ can be identified with the vector itself. It follows that the norm of $-i{\widehat A}_{\bot}\varphi$ in the Fubini-Study metric coincides with its norm in the Riemannian metric on $S^{L_{2}}$ (and in the original $L_{2}$-metric).
Integral curves of the vector field $A_{\varphi}=-i{\widehat A} \varphi$ are solutions to the equation $$\label{evoll}
\frac{d\varphi}{dt}=-i{\widehat A}\varphi$$ for the state $\varphi$ with the initial condition $\left.\varphi \right|_{t=0}=\varphi_{0}$.
Decomposition of $-i{\widehat A}\varphi$ onto the components parallel and orthogonal to the fibre yields the equation $$\label{evolll}
\frac{d\varphi}{dt}=-i{\overline A}\varphi+\left(-i{\widehat A}\varphi+i{\overline A}\varphi\right)=
-i{\overline A}\varphi-i {\widehat A}_{\perp}\varphi.$$ By projecting both sides of this equation by $d\pi$ one obtains $$\label{ddt}
\frac{d\{\varphi\}}{dt}=-i{\widehat A}_{\bot}\varphi.$$
The left hand side of (\[ddt\]) is the velocity of evolution of the projection $\{\varphi\}=\pi(\varphi)$ in $CP^{L_{2}}$. By the above, the norm of the right hand side is the uncertainty of ${\widehat A}$ in the state $\varphi$: $$\label{speed}
\|-i{\widehat A}_{\bot}\varphi\|=\Delta A.$$
In particular, if ${\widehat A}$ is the Hamiltonian ${\widehat h}$, then equation (\[evoll\]) is the Scr[ö]{}dinger equation and the following result is obtained: [*The velocity of evolution of state in the projective space is equal to the uncertainty of energy.*]{} This result was obtained first in Ref.[@AA] by using different methods.
Now let’s decompose the acceleration vector $\frac{d^{2}\varphi}{dt^{2}}=\frac{d}{dt}\left( -i{\widehat h}\varphi\right)=-{\widehat h}^{2}\varphi$. Notice first of all that $${\mathrm Re}(-i\varphi, {\widehat h}^{2}\varphi)=0,$$ so that the parallel tangential component of acceleration of Shr[ö]{}dinger evolution vanishes. This simply means that the phase component of the velocity (i.e., the expected value of energy, see above) does not change. In particular, the tangential component is purely orthogonal. The radial component is given by $-(\varphi, {\widehat h}^{2}\varphi)\varphi=-(-i{\widehat h}\varphi, -i{\widehat h}\varphi)\varphi$. Since $-i{\widehat h}\varphi$ is the velocity of evolution, we recognize in this term the centropidical acceleration ($-\frac{{\bf v}^{2}{\bf r}}{r^{2}}$ with $r=1$).
The tangential component is therefore equal to $$-{\widehat h}^{2}\varphi + (\varphi,{\widehat h}^{2}\varphi)\varphi=-{\widehat h}^{2}_{\perp}\varphi.$$ Therefore, the following result is obtained: [*Acceleration of the Schro[ö]{}dinger evolution of state in the projective space is equal to the uncertainty of the square of energy.*]{}
Components of velocity of state
===============================
Classical and quantum mechanics of a particle are now formulated within the same Hilbert space framework. Recall that the space ${\mathbb{R}}^{3}$ is now identified via the map $\omega$ with the submanifold $M_{3}$ in ${\bf H}$ with the induced Euclidean metric. Alternatively, the map $\omega_{\sigma}=\rho_{\sigma}\omega$ identifies ${\mathbb{R}}^{3}$ with the submanifold $M^{\sigma}_{3}$ in $L_{2}({\mathbb{R}}^{3})$. This later equivalent realization will be used in this section. Note that because all normalized Gaussian functions of a given width $\sigma$ are obtained from a single one by translations in ${\bf x}$, the field ${\bf p}_{\varphi}=-i{\widehat {\bf p}}\varphi$ for $\varphi \in M^{\sigma}_{3}$ is tangent to $M^{\sigma}_{3}$. The goal here is to use the embedding $\omega_{\sigma}$ of ${\mathbb{R}}^{3}$ into the space of states together with the vector representation of observables to study the relation of the Schr[ö]{}dinger evolution with the classical Newtonian motion.
One standard way to describe this relation is via the Ehrenfest theorem (the expected value of the Heisenberg equation of motion): $$\label{Her}
\frac{d}{dt}(\varphi, {\widehat A}\varphi)=-i(\varphi, [{\widehat A}, {\widehat h}]\varphi).$$ Here ${\widehat A}$ does not depend on $t$. For example, for the momentum operator of a free particle we obtain $$\label{para}
\frac{d{\overline{\bf p}}}{dt}=0.$$ Recall that ${\overline{\bf p}}$ is the phase projection of the vector field $p_{\varphi}$. The equation (\[para\]) simply says that this projection is time-independent. Note that the orthogonal projection, i.e. the uncertainty $\Delta{\bf p}$ is also preserved in this case and this is not captured in (\[Her\]).
Compare (\[Her\]) to another equation that follows from the Schr[ö]{}dinger dynamics: $$\label{projj}
2\left(\frac{d \varphi}{dt}, -i {\widehat A} \varphi \right)=
\left( \varphi, \{{\widehat A}, {\widehat h} \}\varphi \right)-\left(\varphi,[{\widehat A}, {\widehat h}]\varphi\right).$$ The Ehrenfest theorem (\[Her\]) for a time-independent observable amounts to using the imaginary part of (\[projj\]), i.e., the part with the commutator $[{\widehat A}, {\widehat h}]$. The left hand side of (\[projj\]) is twice the projection of the velocity of state onto the vector field associated with the observable ${\widehat A}$. The real part of this projection (the term with the anticommutator $\{{\widehat A}, {\widehat h}\}$) is twice the projection in the sense of Riemannian metric (\[Riem\]). This Riemannian projection will be used here.
Suppose that at $t=0$ a microscopic particle is prepared in the state $$\label{initial}
\varphi_{0}({\bf x})=\left(\frac{1}{2\pi\sigma^{2}}\right)^{3/4}e^{-\frac{({\bf x}-{\bf x}_{0})^{2}}{4\sigma^{2}}}e^{i\frac{{\bf p}_{0}({\bf x}-{\bf x}_{0})}{\hbar}},$$ where $\sigma$ is the same as in (\[sigma\]) and ${\bf p}_{0}=m{\bf v}_{0}$ with ${\bf v}_{0}$ being the initial group-velocity of the packet. The set of all initial states $\varphi_{0}$ given by (\[initial\]) form a $6$-dimensional embedded submanifold $M^{\sigma}_{3,3}$ in $L_{2}({\mathbb{R}}^{3})$. The map $\Omega: {\mathbb{R}}^{3}\times {\mathbb{R}}^{3} \longrightarrow M^{\sigma}_{3,3}$, $$\Omega({\bf a},{\bf p})=\left(\frac{1}{2\pi\sigma^{2}}\right)^{3/4}e^{-\frac{({\bf x}-{\bf a})^{2}}{4\sigma^{2}}}e^{i\frac{{\bf p}({\bf x}-{\bf a})}{\hbar}}$$ is a diffeomorphism from the classical phase space of the particle onto the manifold $M^{\sigma}_{3,3}$. For any path $\varphi=\varphi_{\tau}$ in $L_{2}({\mathbb{R}}^{3})$, $\varphi=re^{i\theta}$, the terms of the derivative $$\frac{d\varphi}{d\tau}=\frac{d r}{d\tau}e^{i\theta}+i\frac{d \theta}{d\tau}re^{i\theta}$$ are orthogonal in the Riemannian metric: $$\label{orthog}
\mathrm{Re}\left(\frac{dr}{d\tau}e^{i\theta}, i\frac{d\theta}{d\tau}re^{i\theta}\right)=0.$$ In particular, the vectors $\frac{\partial r}{\partial x^{\alpha}}e^{i\theta}$ and $i\frac{\partial \theta}{\partial p^{\beta}}re^{i\theta}$ tangent to the manifold $M^{\sigma}_{3,3}$ at a point $\varphi_{0}$ are orthogonal and form a basis in the tangent space at that point. For any path $\varphi_{\tau}$ with values in $M^{\sigma}_{3,3}$ the norm of velocity vector $\frac{d \varphi}{d\tau}$ is given by $$\label{phaseMetric}
\left\|\frac{d \varphi}{d \tau}\right\|^{2}_{L_{2}}=\frac{1}{4\sigma^{2}}\left\|\frac{d {\bf a}}{d\tau}\right\|^{2}_{{\mathbb{R}}^{3}}+\frac{\sigma^{2}}{\hbar^{2}}\left\|\frac{d {\bf p}}{d\tau}\right\|^{2}_{{\mathbb{R}}^{3}}.$$ That is, under a proper choice of units, the map $\Omega$ is an isometry, which identifies the Euclidean phase space ${\mathbb{R}}^{3}\times {\mathbb{R}}^{3}$ of the particle with the embedded submanifold $M^{\sigma}_{3,3} \subset L_{2}({\mathbb{R}}^{3})$ furnished with the induced Riemannian metric. The map $\Omega$ is an extension to the phase space of the isometric embedding $\omega_{\sigma}=\rho_{\sigma}\circ\omega$ of the space ${\mathbb{R}}^{3}$ considered in the first section.
Suppose that the state (\[initial\]) evolves according to the Schr[ö]{}dinger equation with the Hamiltonian ${\widehat h}=-\frac{\hbar^{2}}{2m}\Delta+V({\bf x})$. At any point $\varphi_{0} \in M^{\sigma}_{3,3}$, the velocity vector $\frac{d\varphi}{dt}$ is tangent to the unit sphere of states $S^{L_{2}}$ in $L_{2}({\mathbb{R}}^{3})$ and can be decomposed into a sum of components of physical interest. First of all, by (\[evolll\]) $$\frac{d\varphi}{dt}=-\frac{i}{\hbar}{\widehat h}\varphi=-\frac{i}{\hbar}{\overline E}\varphi-\frac{i}{\hbar}{\widehat h}_{\perp}\varphi.$$ So, once again, the component of $\frac{d\varphi}{dt}$ along the vector $i\varphi$ is $\frac{\overline E}{\hbar}$ and the norm of the orthogonal component $\left\|-\frac{i}{\hbar}{\widehat h}_{\perp}\varphi\right\|$ is $\frac{\Delta h}{\hbar}$.
To decompose the orthogonal component $-\frac{i}{\hbar}{\widehat h}_{\perp}\varphi$ of the velocity $\frac{d\varphi}{dt}$, notice that the orthogonal vectors $\frac{\partial r}{\partial x^{\alpha}}e^{i\theta}$ and $i\frac{\partial \theta}{\partial p^{\beta}}re^{i\theta}$ tangent to $M^{\sigma}_{3,3}$ are also orthogonal to vector $i\varphi$: $${\mathrm Re}\left(i\varphi, -\frac{\partial r}{\partial x^{\alpha}}e^{i\theta}\right)=0 \ \textnormal{for all} \ t$$ and $$\left(i\varphi, i\frac{\partial\theta}{\partial p^{\alpha}}\varphi\right)=0 \ \textnormal{for} \ t=0.$$ Calculation of the projection of the velocity $\frac{d \varphi}{dt}$ onto the unit vector $-\widehat{\frac{\partial r}{\partial x^{\alpha}}}e^{i\theta}$ (i.e., the classical space component of $\frac{d\varphi}{dt}$) for any Hamiltonian ${\widehat h}=-\frac{\hbar^{2}}{2m}\Delta+V({\bf x})$ yields $$\label{pproj}
\left.\mathrm{Re}\left(\frac{d \varphi}{dt}, -\widehat{ \frac{\partial r}{\partial x^{\alpha}}}e^{i\theta}\right)\right|_{t=0}=\left.\left(\frac{d r}{dt}, -\widehat{ \frac{\partial r}{\partial x^{\alpha}}}\right)\right|_{t=0}=\frac{v^{\alpha}_{0}}{2\sigma}.$$ Calculation of the projection of velocity $\frac{d \varphi}{dt}$ onto the unit vector $i\widehat{\frac{\partial\theta}{\partial p^{\alpha}}}\varphi$ (momentum space component) gives $$\label{w}
\left.\mathrm{Re} \left(\frac{d\varphi}{dt}, i\widehat{\frac{\partial\theta}{\partial p^{\alpha}}}\varphi\right)\right|_{t=0}=\frac{mw^{\alpha} \sigma}{\hbar},$$ where $$mw^{\alpha}=-\left.\frac{\partial V({\bf x})}{\partial x^{\alpha}}\right|_{{\bf x}={\bf x}_{0}}$$ and $\sigma$ is assumed to be small enough for the linear approximation for $V({\bf x})$ to be valid within intervals of length $\sigma$.
The velocity $\frac{d\varphi}{dt}$ also contains component which is due to the change in $\sigma$ (spreading). The inner product $$\left(i\varphi, i\frac{d\varphi}{d\sigma}\right)=\left(\varphi, \frac{d\varphi}{d\sigma}\right)$$ vanishes at $t=0$, so the vector $i\frac{d\varphi}{d\sigma}$ is also tangent to the sphere $S^{L_{2}}$ and orthogonal to the phase circle. It is also orthogonal to the phase space $M^{\sigma}_{3,3}$. The component of the velocity $\frac{d\varphi}{dt}$ along this vector is given by $$\label{spreadcomp}
\left.\mathrm{Re} \left (\frac{d\varphi}{dt}, i\widehat{\frac{d\varphi}{d\sigma}}\right)\right|_{t=0}=\frac{\sqrt{2}\hbar}{8\sigma^{2}m}.$$ Finally, calculation of the norm of $\frac{d\varphi}{dt}=\frac{i}{\hbar}{\widehat h}\varphi$ at $t=0$ gives $$\label{decomposition}
\left\|\frac{d\varphi}{dt}\right\|^{2}=\frac{{\overline E}^{2}}{\hbar^{2}}+\frac{{\bf v}^{2}_{0}}{4\sigma^{2}}+\frac{m^{2}{\bf w}^{2}{\sigma}^{2}}{\hbar^{2}}+\frac{\hbar^{2}}{32\sigma^{4}m^{2}},$$ which is exactly the sum of squares of the found components. This, therefore, completes a decomposition of the velocity of state at any point $\varphi_{0} \in M^{\sigma}_{3,3}$.
Note that for a closed system the norm of $\frac{d\varphi}{dt}=\frac{i}{\hbar}{\widehat h}\varphi$ is preserved in time. For a system in a stationary state, this amounts to conservation of energy. In fact, in this case $\varphi_{t}({\bf x})=\psi({\bf x})e^{-\frac{iEt}{\hbar}}$, which is a motion along the phase circle, and $$\left\| \frac{d\varphi}{dt}\right\|^{2}=\frac{E^2}{\hbar^2}.$$ As discussed in the section titled Observables as vector fields, for any initial state the norm of the phase component (expected energy) and orthogonal component (energy uncertainty) of the velocity $\frac{d\varphi}{dt}$ are both preserved.
The presence of $i\frac{d\varphi}{d\sigma}$ component of the the velocity in (\[decomposition\]) hints that the classical phase space $M^{\sigma}_{3,3}$ may be usefully extended to include all positive values of $\sigma$. The induced metric on the resulting manifold $M^{\sigma}_{3,3} \times {\mathbb{R}}_{+}$ is then given by the following extension of (\[phaseMetric\]): $$\label{phaseMetric1}
\left\|\frac{d \varphi}{dt}\right\|^{2}_{L_{2}}=\frac{1}{4\sigma^{2}}\left\|\frac{d {\bf a}}{dt}\right\|^{2}_{{\mathbb{R}}^{3}}+\frac{\sigma^{2}}{\hbar^{2}}\left\|\frac{d {\bf p}}{dt}\right\|^{2}_{{\mathbb{R}}^{3}}+\frac{1}{2\sigma^{2}}\left|\frac{d\sigma}{dt}\right|^{2}.$$ With appropriate units, this gives an isometric embedding of the extended phase space ${\mathbb{R}}^{3} \times {\mathbb{R}}^{3} \times {\mathbb{R}}_{+}$ with Euclidean metric into $L_{2}({\mathbb{R}}^{3})$.
The “spreading” component of the velocity admits an interesting interpretation. Suppose that the width of the initial state $\varphi_{0}$ is given by the Compton length $\frac{\hbar}{mc}$, which is a natural limit on the width of state in quantum mechanics. From this and (\[spreadcomp\]) and (\[decomposition\]) it follows that component of velocity of state due to spreading is proportional to the mass $m$ of the particle. So the mass can be thought of as the speed of motion of state in the direction of spreading, orthogonal to the phase space $M^{\sigma}_{3,3}$. The sum of the last three terms in (\[decomposition\]) is equal to the square of the uncertainty $\Delta h$. If ${\bf v}_{0}$ and ${\bf w}$ vanish, then $$\Delta h \ = \ \textnormal{mass term} \ = \ \textnormal{speed of spreading}.$$
In the linear potential approximation, the first term in (\[decomposition\]) is the square of the term $$\label{restmass}
\frac{1}{\hbar}\left(U+K+\frac{\hbar^{2}}{8m\sigma^{2}}\right),$$ where $U=V({\bf x}_{g})$ and $K=\frac{m{\bf v}^{2}_{g}}{2}$ are potential and kinetic energy of the packet considered as a particle with position ${\bf x}_{g}={\bf x}_{0}+{\bf v}_{0}t+\frac{{\bf w}t^{2}}{2}$ and velocity ${\bf v}_{g}={\bf v}_{0}+{\bf w}t$. The last term in parentheses in (\[restmass\]) accounts for the difference in energy of the packets with the same $U$ and $K$, but different values of $\sigma$. Up to a constant factor this term equals the component of velocity due to spreading given by (\[spreadcomp\]). With the unit of length $2\sigma$ given by Compton length and the choice of units that make the metric (\[phaseMetric1\]) for a particle of a given mass Euclidean, this term is equal to the rest energy $mc^{2}$ of the particle.
Calculations show that for $t>0$ the spatial component (\[pproj\]) of velocity of state is given by $\frac{v^{\alpha}_{g}}{2\sigma_{t}}$ while the component (\[spreadcomp\]) due to spreading does not change. Here ${\bf v}_{g}={\bf v}_{0}+{\bf w}t$ is the group velocity and $\sigma_{t}$, given by $$\label{sigmaS}
\sigma^{2}_{t}=\sigma^{2}\left(1+\frac{\hbar^{2}t^{2}}{4m^{2}\sigma^{4}}\right),$$ is the width of the packet at time $t$, and it is assumed that $\sigma_{t}$ is sufficiently small for the linear approximation of $V({\bf x})$ to be valid. The relationship $$\label{Va}
\frac{d}{dt}\left.\left(\frac{d r}{dt}, - \widehat{\frac{\partial r}{\partial{x}^{\alpha}}} \right)\right |_{t=0}
=-\left.\frac{1}{m}\frac{\partial V({\bf x})}{\partial x^{\alpha}}\right|_{x=x_{0}}\frac{1}{2\sigma}$$ together with (\[pproj\]) and (\[w\]) proves that at any point $\varphi_{0} \in M^{\sigma}_{3,3}$, the spatial and momentum space components of $\frac{d\varphi}{dt}$ are related in the same way as their classical counterparts in the phase space. Furthermore, the derived relationships (\[pproj\]), (\[w\]), (\[spreadcomp\]), (\[decomposition\]) and (\[Va\]) remain true at $t=0$ even when the potential $V$ depends on time. In fact, the only expression that contains time derivatives of $V$ is the derivative $\frac{d^{2}r}{dt^{2}}$ in (\[Va\]). However, the corresponding terms $\pm \frac{i}{2r}\frac{dV}{dt}$ cancel out because of the reality of $\frac{d^{2}r}{dt^{2}}$. The immediate consequence of these results and the linear nature of the Schr[ö]{}dinger equation is that under the Schr[ö]{}dinger evolution with the Hamiltonian ${\widehat h}=-\frac{\hbar^{2}}{2m}\Delta+V({\bf x},t)$, the state constrained to $M^{\sigma}_{3,3}$ moves like a point in the phase space representing a particle in Newtonian dynamics. That is, if at each $\varphi_{0} \in M^{\sigma}_{3,3}$, the components of $-\frac{i}{\hbar}{\widehat h}\varphi_{0}$ that are orthogonal to $M^{\sigma}_{3,3}$ are made to vanish while the tangent components are preserved, then the state $\varphi$ will move according to classical physics. So, [*Newtonian dynamics of a particle is the dynamics of one-particle quantum system with state constrained to $M^{\sigma}_{3,3}$*]{}.
On the other hand, there is a unique unitary evolution (one parameter group of unitary operators) on $L_{2}({\mathbb{R}}^{3})$, which, being restricted to $M^{\sigma}_{3,3}$, under projections (\[pproj\]), (\[w\]) yields the Newtonian values of velocity and acceleration. In fact, equations (\[pproj\]), (\[w\]) for the states $\varphi$ given by (\[initial\]) imply the Ehrenfest theorem $$\label{E1}
2\mathrm{Re} \left(\frac{d\varphi}{dt}, {\widehat x} \varphi \right)=\left(\varphi, \frac{\widehat p}{m}\varphi \right)$$ and $$\label{E2}
2\mathrm{Re} \left(\frac{d\varphi}{dt}, {\widehat p} \varphi \right)=\left(\varphi, - \nabla V({\bf x}) \varphi \right).$$ But the set $M^{\sigma}_{3,3}$ of such vectors $\varphi$ is complete in $L_{2}({\mathbb{R}}^{3})$ and on a complete set the Ehrenfest theorem (\[E1\]) and (\[E2\]) together with the condition of unitarity of evolution is known to imply the Schr[ö]{}dinger equation. So formulas (\[pproj\]), (\[w\]) on $M^{\sigma}_{3,3}$ imply the Schr[ö]{}dinger dynamics of the state of the particle on the space of states.
The analogous results can be derived for systems of $n$-classical particles. For instance, consider a system of two distinguishable particles, described by the usual Hamiltonian $$\label{ham2}
{\widehat h}=-\frac{\hbar^2}{2m_1}\Delta_{1}-\frac{\hbar^2}{2m_2}\Delta_{2}+V({\bf x_{1}}, {\bf x_{2}}),$$ where the indices $1$ and $2$ refer to the corresponding particles. The set of states $\varphi_{1}\otimes \varphi_{2}$, where $\varphi_{1}$ and $\varphi_{2}$ for each particle are of the form (\[initial\]) is a $12$-dimensional embedded submanifold $M^{\sigma}_{6,6}$ of the Hilbert space $L_{2}({\mathbb{R}}^{3})\otimes L_{2}({\mathbb{R}}^{3})$ with induced Riemannian metric, isometric to the classical phase space ${\mathbb{R}}_6 \times {\mathbb{R}}_6$ of the two-particle system. Vectors $$\label{set1}
\left(-\frac{\partial r_{1}}{\partial x_{1}^{k}}e^{i\frac{{\bf p}_{1}({\bf x}_{1}-{\bf a}_{1})}{\hbar}}\right)\otimes \varphi_{2}, \quad i\frac{\partial \theta_{1}}{\partial p_{1}^{j}}\varphi_{1}\otimes \varphi_{2}$$ and $$\label{set2}
\varphi_{1}\otimes \left(-\frac{\partial r_{2}}{\partial x_{2}^{k}}\right)e^{i\frac{{\bf p}_{2}({\bf x}_{2}-{\bf a}_{2})}{\hbar}}, \quad \varphi_{1}\otimes i\frac{\partial \theta_{2}}{\partial p_{2}^{j}} \varphi_{2}$$ are tangent to the phase spaces $M^{\sigma}_{3,3}\otimes \varphi_{2}$ and $\varphi_{1}\otimes M^{\sigma}_{3,3}$ of individual particles. These vectors are orthogonal for all values of $k,j=1,2,3$ and form a basis in the space tangent to $M^{\sigma}_{6,6}$.
Suppose now that a two particle quantum system has initial state in $M^{\sigma}_{6,6}$ and evolves by the Hamiltonian (\[ham2\]). Because each operator $\Delta_{k}$ acts on just one function in the tensor product $\varphi_{1}\otimes \varphi_{2}$ and because the inner product in $L_{2}({\mathbb{R}}^{3})\otimes L_{2}({\mathbb{R}}^{3})$ is the product of inner products for individual particles, it follows that the components of the velocity vector $\frac{d}{dt} \left(\varphi_{1}\otimes \varphi_{2}\right)$ in the basis (\[set1\]), (\[set2\]) are given for each particle by their Newtonian values. For instance, $$\left(\frac{d\varphi_{1}}{dt} \otimes \varphi_{2}, -\frac{\partial r_{1}}{\partial x_{1}^{k}}e^{i\frac{{\bf p}_{1}({\bf x}_{1}-{\bf a}_{1})}{\hbar}}\otimes \varphi_{2}\right)=\frac{v_{1}^{k}}{2\sigma_{1}},$$ where ${\bf v_{1}}={\bf p}_{1}/m_{1}$, etc. It follows that: (\[initial\]).
Quantum probability and the classical normal distribution
=========================================================
If a classical experiment for measuring the position of a macroscopic particle is performed, the result is generically a normal probability distribution of the position variable. Now the classical space ${\mathbb{R}}^{3}$ is identified with the submanifold $M^{\sigma}_{3}$ in the Hilbert space $L_{2}$ of states (equivalently, with the submanifold $M_{3}$ in the space ${\bf H}$). A macroscopic particle is identified with a quantum system constrained to the phase space $M^{\sigma}_{3,3}$. Measuring position of a macroscopic particle can be then described in terms of states in $S^{L_{2}}$. Because of this, the normal distribution of position of a macroscopic particle and the probability of transition between quantum states of a microscopic particle become related. It will be shown that, under measurements, macroscopic and microscopic particles obey the same law. Namely:
[*The Born rule for a position measurement of a microscopic particle implies the normal probability distribution of position of a macroscopic particle.*]{}
[*Conversely, suppose that measurements of position of a macroscopic particle are distributed normally. Suppose further that the probability $P(\varphi, \psi)$ for a microscopic particle in an arbitrary state $\varphi\in L_{2}$ to be found under a measurement in a state $\psi$ depends only on the distance $\rho(\pi(\varphi), \pi(\psi))$ between the states, in the Fubini-Study metric on the projective space $CP^{L_{2}}$. Then $P(\varphi, \psi)=\cos^{2}\rho(\pi(\varphi), \pi(\psi))$.*]{}
[To summarize:]{}
[*The normal probability distribution of a position random variable for a particle in the classical space implies the Born rule for transitions between arbitrary quantum states of the particle and vice versa.*]{}
To prove this, note that a macroscopic particle is described in the classical phase space ${\mathbb{R}}^{3}\times {\mathbb{R}}^{3}=M^{\sigma}_{3,3}$, and so its state at a given time is given by the function (see (\[initial\])): $$\label{del}
\varphi_{\bf a}({\bf x})=
\left(\frac{1}{2\pi \sigma^{2}}\right)^{3/4}e^{-\frac{({\bf x}-{\bf a})^{2}}{4\sigma^{2}}}e^{i\frac{{\bf p}({\bf x}-{\bf a})}{\hbar}}$$ Let ${\widetilde \delta^{3}_{\bf a}}({\bf x})$ be the modulus $|\varphi_{\bf a}|$ and let $\delta^{3}_{\bf a}$ denote the usual delta-function. By the Born rule, the probability density $f({\bf b})$ to find the particle at a point ${\bf b}$ is equal to $$\label{Born1}
f({\bf b})=|\varphi_{\bf a}({\bf b})|^{2}=|({\widetilde \delta^{3}_{\bf a}}, \delta^{3}_{\bf b})|^{2}=\left(\frac{1}{2\pi \sigma^{2}}\right)^{3/2}e^{-\frac{({\bf a}-{\bf b})^{2}}{2\sigma^{2}}},$$ which is the normal distribution function. It follows that on the elements of $M^{\sigma}_{3}$, the Born rule [*is*]{} the rule of normal distribution.
Conversely, assume the normal probability distribution of position measurements for macroscopic particles. Here it will be sufficient to deal with particles at rest. A macroscopic particle at rest is represented by the state ${\widetilde \delta^{3}_{\bf a}}({\bf x})$ (zero phase) in the classical space ${\mathbb{R}}^{3}=M^{\sigma}_{3}$, which is a submanifold of $M^{\sigma}_{3,3}$. It was shown that the Born rule and the normal distribution law are the same for the states in $M^{\sigma}_{3,3}$, in particular, for the states ${\widetilde \delta^{3}_{\bf a}}({\bf x})$. Therefore, the normal distribution rule can be also written in the form of the Born rule $$\label{Born2}
P(\tilde{\delta^{3}_{\bf a}}, \tilde{\delta^{3}_{\bf b}})=|(\tilde{\delta^{3}_{\bf a}}, \tilde{\delta^{3}_{\bf b}})|^{2},$$ where $P(\tilde{\delta}^{3}_{\bf a}, \tilde{\delta}^{3}_{\bf b})$ is the probability of transition from the state $\tilde{\delta}^{3}_{\bf a}$ to the state $\tilde{\delta}^{3}_{\bf b}$ under a measurement of an appropriate observable. Note that (\[Born1\]) is the probability density while (\[Born2\]) is the probability of transition. However, assuming ${\widetilde \delta^{3}}_{\bf b}$ is sufficiently sharp, the formulas mean the same thing. In fact, in this case $\delta^{3}_{\bf b}$ in (\[Born1\]) can be replaced with ${\widetilde \delta^{3}}_{\bf b}$. For this recall that ${\widetilde \delta^{3}}_{\bf b}$ is unit-normalized in $L_{2}({\mathbb{R}}^{3})$: $$\int |{\widetilde \delta^{3}}_{\bf b}({\bf x})|^{2}d^{3}{\bf x}=1.$$ Let $h$ be the height ${\widetilde \delta^{3}}_{\bf b}({\bf b})$ of ${\widetilde \delta^{3}}_{\bf b}$ and let $\Delta x$ be defined by $$h^{2}\cdot (\Delta x)^{3}=\int |{\widetilde \delta^{3}}_{\bf b}({\bf x})|^{2}d^{3}{\bf x}=1.$$
Then $h=\frac{1}{(\Delta x)^{3/2}}$ and $$\label{Born3}
|({\widetilde \delta^{3}}_{\bf a}, \delta^{3}_{\bf b})|^{2} \approx \left|{\widetilde \delta^{3}}_{\bf a}({\bf b}) \int \frac{1}{(\Delta x)^{3/2}} d^{3}{\bf x} \right|^{2},$$ where integration is over the cube of side $\Delta x$ centered at ${\bf b}$. As a result, $$\label{Born4}
|({\widetilde \delta^{3}}_{\bf a}, {\widetilde \delta^{3}}_{\bf b})|^{2} \approx \left|{\widetilde \delta^{3}}_{\bf a}({\bf b}) \right|^{2}(\Delta x)^{3}=f({\bf b})(\Delta x)^{3},$$ which relates the probability in (\[Born2\]) to the normal probability density in (\[Born1\]) and identifies $P(\tilde{\delta^{3}_{\bf a}}, \tilde{\delta^{3}_{\bf b}})$ with the probability of finding the macroscopic particle near the point ${\bf b}$.
The Born rule (\[Born2\]) can be also written as $$\label{PP}
P(\tilde{\delta^{3}_{\bf a}}, \tilde{\delta^{3}_{\bf b}})=\cos^{2}\rho(\tilde{\delta^{3}_{\bf a}}, \tilde{\delta^{3}_{\bf b}}),$$ where $\rho(\tilde{\delta^{3}_{\bf a}}, \tilde{\delta^{3}_{\bf b}})$ is the distance between the states $\tilde{\delta^{3}_{\bf a}}, \tilde{\delta^{3}_{\bf b}}$ in the Fubini-Study metric on the projective space $\pi: S^{L_{2}}\longrightarrow CP^{L_{2}}$. Here $\pi(\tilde{\delta^{3}_{\bf a}})$ is identified with $\tilde{\delta^{3}_{\bf a}}$, which is possible because the state is real-valued.
The Fubini-Study distance between the states $\tilde{\delta^{3}_{\bf a}}$, $\tilde{\delta^{3}_{\bf b}}$ takes on all values from $0$ to $\pi/2$, which is the largest possible distance between points in $CP^{L_{2}}$. By assumption, the probability $P(\varphi, \psi)$ of transition between any states $\varphi$ and $\psi$ depends only on the Fubini-Study distance $\rho(\pi(\varphi), \pi(\psi))$ between the states. Given arbitrary states $\varphi, \psi \in S^{L_{2}}$, let then $\tilde{\delta^{3}_{\bf a}}$, $\tilde{\delta^{3}_{\bf b}}$ be two states in $M^{3}_{\sigma}$, such that $$\rho(\pi(\varphi), \pi(\psi))=\rho(\tilde{\delta^{3}_{\bf a}},
\tilde{\delta^{3}_{\bf b}}).$$ From the assumed normal probability distribution for the states $\tilde{\delta^{3}_{\bf a}}$ and the assumption that probability of transition depends only on the Fubini-Study distance between the states, it then follows that $$P(\varphi, \psi)=P(\tilde{\delta^{3}_{\bf a}}, \tilde{\delta^{3}_{\bf b}})=\cos^{2}\rho(\tilde{\delta^{3}_{\bf a}}, \tilde{\delta^{3}_{\bf b}})
=\cos^{2}\rho(\pi(\varphi), \pi(\psi)),$$ which yields the Born rule for arbitrary states. This proves the claim.
This beautiful result is based on a highly non-trivial way in which the classical space is embedded into the Hilbert space of states. Namely, because of the special properties of the embedding, the “classical law” (normal distribution of observation results) becomes a part of the quantum law, which simply extends the classical law to superpositions. The extension is unique if the assumption is made that the probability of transition must only depend on the distance between states in the Fubini-Study metric.
In more detail, denote the distance between two points ${\bf a}, {\bf b}$ in ${\mathbb{R}}^{3}$ by $\left\|{\bf a}-{\bf b}\right\|_{{\mathbb{R}}^{3}}$. Under the embedding of the classical space into the space of states, the variable ${\bf a}$ is represented by the state $\tilde{\delta}^{3}_{\bf a}$. The set of states $\tilde{\delta}^{3}_{\bf a}$ form a submanifold $M^{\sigma}_{3}$ in the Hilbert spaces of states $L_{2}({\mathbb{R}}^{3})$. The manifold $M^{\sigma}_{3}$ is “twisted” in $L_{2}({\mathbb{R}}^{3})$, it belongs to the sphere $S^{L_{2}}$ and spans all dimensions of $L_{2}({\mathbb{R}}^{3})$. Distance between the states $\tilde{\delta}^{3}_{\bf a}$, $\tilde{\delta}^{3}_{\bf b}$ in $L_{2}({\mathbb{R}}^{3})$ or in the projective space $CP^{L_{2}}$ is not equal to $\left\|{\bf a}-{\bf b}\right\|_{{\mathbb{R}}^{3}}$. In fact, the former distance measures length of a geodesic between the states while the latter is obtained using the same metric on the space of states, but applied along a geodesic in the twisted manifold $M^{\sigma}_{3}$. In precise terms the relation between the two distances is given by $$\label{main}
e^{-\frac{({\bf a}-{\bf b})^{2}}{4\sigma^{2}}}=\cos^{2}\rho(\tilde{\delta}^{3}_{\bf a}, \tilde{\delta}^{3}_{\bf b}),$$ where the left hand side is a result of integration in (\[Born2\]). This equation is what accounts for the relation between the normal probability distribution and the Born rule.
Summary
=======
The classical space and classical phase space are now embedded into the space of states of the corresponding quantum system and form a complete set (overcomplete basis) in that space. The dynamics of a classical $n$-particle mechanical system is identified with the Schr[ö]{}dinger dynamics constrained to the classical phase space. Conversely, there is a unique unitary time evolution on the space of states of a quantum system that yields Newtonian dynamics when constrained to the classical phase space. The normal distribution law is derived from the Born rule. Conversely, the Born rule is the only probability law on the the projective space of states that is isotropic and yields the normal distribution on a classical configuration submanifold. These results suggest that other areas of tension between classical and quantum physics can be now fruitfully explored.
[99]{}
A. Kryukov, [*Int. J. Math. & Math. Sci.*]{} [**14**]{}, 2241 (2005)
A. Kryukov, [*J. Math. Phys.*]{} [**49**]{}, 102108 (2008)
A. Kryukov, [*J. Math. Phys.*]{} [**51**]{}, 022110 (2010)
A. Kryukov, [*Found. Phys.*]{} [**41**]{}, 129 (2011)
A. Kryukov, [*Phys. Lett. A*]{} [**370**]{}, 419 (2007)
J. Anandan & Y. Aharonov, [*Phys. Rev. Lett.*]{} [**65**]{}, 1697 (1990)
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'We present a novel mechanism of using solar neutrinos to speed up dark matter, inspired by the fact that neutrinos are the most energetic particles from the Sun with a well-understood spectrum. In a neutrino portal dark sector model, we show that dark matter with sub-GeV mass could be accelerated by the $pp$ neutrinos to velocities well above $10^{-3}c$ and capable of depositing large enough energy at direct detection experiments. A crucial ingredient of this mechanism is the dissociation of stable dark matter bound states that exist in Nature. The resulting dark matter velocity distribution bears a strong resemblance in shape to the solar neutrino spectrum. As an application, we derive a leading limit on light dark matter interaction by reinterpreting a recent PICO experiment result.'
author:
- Yue Zhang
bibliography:
- 'references.bib'
title: Speeding Up Dark Matter With Solar Neutrinos
---
[*Introduction.*]{} Direct detection is an important approach to unveil the nature of dark matter in the Universe. It is widely assumed [@Goodman:1984dc] that most of dark matter particles in our galaxy move non-relativistically, thus the maximal energy transfer per dark matter scattering is limited by its velocity distribution, typically not more than hundreds of keV. Experimentally, in order to hunt signals with such a small energy deposit, tremendous amount of effort has been made to build low-noise detectors deep underground, setting strong limits on dark matter scattering cross sections [@Schumann:2019eaa; @Liu:2017drf]. These limits, however, weaken significantly for dark matter mass below a few GeV, where the available energies due to the scattering fall short of the energy threshold of traditional dark matter detectors. To probe sub-GeV dark matter candidates, one either has to devise new detectors with lower energy thresholds, or consider possibilities where dark matter is made to travel faster [@Bringmann:2018cvk; @Ema:2018bih; @Cappiello:2018hsu; @Wang:2019jtk; @Kouvaris:2015nsa; @An:2017ojc; @Agashe:2014yua; @deNiverville:2011it; @Izaguirre:2013uxa], or novel ways of detection beyond the elastic scattering picture [@Grossman:2017qzw; @Davoudiasl:2011fj; @Dror:2019onn].
In this [*letter*]{} we present a new mechanism of speeding up (a fraction of) dark matter particles and explore its phenomenological consequences. There is room for this possibility thanks to our ignorance of the precise local dark matter velocity distribution [@McCabe:2010zh; @Fox:2010bz; @Green:2011bv; @Frandsen:2011gi]. A simple way to energize dark matter is through its interaction with high-energy cosmic protons [@Bringmann:2018cvk; @Ema:2018bih; @Cappiello:2018hsu; @Wang:2019jtk]. This is not a local effect and relies on assumptions of cosmic ray distributions further away from the solar system. In contrast, given that the Sun is a powerful energy source nearby which we understand well, it is attractive to consider its impact on the dark matter velocities. The dark-matter-electron interaction has been considered to transfer the solar heat to dark matter [@Kouvaris:2015nsa; @An:2017ojc] which, however, is found only effective for very light dark matter with mass close to MeV due to the limited solar temperature. Here, it is worth remarking that the most energetic particles from the Sun are not electrons or photons, but neutrinos, which are directly produced by nuclear reactions and mostly escape without thermalization. The typical solar neutrino energies ($\sim$ MeV scale) are orders of magnitude higher than the solar temperature ($\sim$ keV scale). Thus, solar neutrinos are capable of speeding dark matter up to higher velocities. Moreover, the solar neutrinos also have a much higher flux than that of the diffuse cosmic rays, and their energy spectrum is well understood in standard solar models [@Bahcall:1987jc] and verified experimentally [@Bellerive:2003rj]. Motivated by these observations, we investigate the impact of dark-matter-neutrino interaction on the local dark matter velocity distribution and direct detection experiments.
Consider dark matter mass lying between tens of MeV and GeV scale, the solar neutrino energy satisfies the hierarchy, $m_\chi v \ll E_\nu \ll m_\chi$, where $v\sim 10^{-3}c$ is the virialized halo dark matter velocity. In this case, the solar-neutrino-dark-matter elastic scattering, $$\label{vanillaprocess}
\nu + \chi \to \nu + \chi \ ,$$ is approximately a fixed-target collision, and the velocity of final state dark matter $\chi$ is given by $$\label{normalv}
v_\chi \simeq \frac{2 E_\nu}{m_\chi} \cos\theta \ ,$$ where $\theta$ is the relative angle between the incoming $\nu$ and out going $\chi$ and for simplicity we ignored the halo velocity. When $\chi$ travels to a dark matter detector and strikes on a nucleus target, the energy transfer is at most $$\label{eq:Ereco}
E_r \sim \frac{\left(m_\chi v_\chi \right)^2}{2 m_A} \lesssim \frac{2 E_\nu^2}{m_A} \ ,$$ where $m_A$ is the target nucleus mass, ranging from 10–100GeV. Note the dark matter mass dependence drops out. In order for this recoil energy to surpass the detector’s energy threshold, typically of order keV, we would need $E_\nu \gtrsim 10\,{\rm MeV}$. From the sun, such an energy is only accessible via the $^8B$ and [*hep*]{} neutrinos, which suffer from a much lower flux compared to their low energy counterpart.
This limitation and our ignorance about the nature of dark matter inspire us to go beyond the process in Eq. (\[vanillaprocess\]) but instead consider the following, $$\nu + B \to \chi + \eta \ ,$$ where $B$ is a composite state of dark matter, which in the simplest case is made of two $\chi$ particles. It dissociates after scattering with an energetic neutrino. Because neutrino carries spin, in the final state $\chi$ and $\eta$ must be different particles. Hereafter we assume $\chi$ is a fermion and $\eta$ is a scalar. It is straightforward to work out the kinematics in this case. Assuming $m_\eta \simeq m_\chi$ and $m_B \simeq 2 m_\chi$, the final state dark matter could speed up to $$\label{eq:vafter}
v_\chi \simeq \sqrt\frac{E_\nu}{m_\chi} + \frac{E_\nu}{2m_\chi} \cos\theta \ .$$ Because $E_\nu \ll m_\chi$, the first term dominates and has no $\theta$ dependence. This velocity is parametrically larger than that in Eq. (\[normalv\]). In the kinematic region where binding energy $E_b$ is non-negligible, the above formula is modified to $v_\chi \simeq \sqrt{(E_\nu - E_b)/m_\chi}$. The corresponding recoil energy in the event of dark-matter-nucleus scattering is $$E_{r} \simeq \frac{m_\chi E_\nu}{2m_A} \ .$$ It is also parametrically larger than Eq. (\[eq:Ereco\]). The key reason behind is that the whole solar neutrino energy gets absorbed in this dissociation process.
Through this exercise, we discover a novel way of speeding up sub-GeV dark matter with solar neutrinos which could lead to new prospects for dark matter direct detection experiments. The dark matter particle originally residing in a cosmologically stable bound state gets liberated by a solar neutrino, inheriting an order one fraction of its energy. Next, we show that these ingredients can be accommodated in a simple dark sector model.
We consider a dark sector with fermionic dark matter $\chi$ and a light scalar dark force $\phi$, with the following Yukawa interaction $$\mathcal{L}_{\text{dark-Yukawa}} = y_D \bar\chi \chi \phi \ .$$ Within this simple dark sector, it has been shown [@Wise:2014jva; @Wise:2014ola] possible of accommodating a wide variety of dark matter bound states and offering rich physics cosmologically and experimentally. In particular, it was observed that the exchange $\phi$ among dark matter particles leads to an attractive force regardless of $\chi\chi$ or $\chi\bar\chi$ system. Bound states occur when the mass of $\phi$ is smaller than the Bohr radius, $\sim \alpha_D m_\chi$ where $\alpha_D = y_D^2/(4\pi)$. In the case where $\chi$ is an asymmetric dark matter [@Nussinov:1985xr; @Kaplan:2009ag] where its stability is due to an approximated $U(1)_D$ global symmetry, the $\chi\chi$ and even $n\chi\ (n>2)$ bound states are cosmologically stable. These states could be produced in the early universe and comprise part of the dark matter relic density [@Wise:2014jva; @Gresham:2017zqi; @Gresham:2017cvl; @Gresham:2018anj]. Alternatively, even without an asymmetry, the $\chi\bar\chi$ bound states could also be stable due to a conserved $C$-parity, as pointed out in [@An:2016kie].
In this work, we will focus on the two-body $\chi\chi$ bound state, assuming it comprises an order one fraction of today’s dark matter abundance. We restrict our study to the ground state with zero orbital angular momentum. Fermion spin statistics then implies that the total spin of the ground state much be zero, [*i.e.*]{}, $\chi\chi$ must form a pseudoscalar bound state. We call such a composite state $B$ hereafter, which can be constructed from elementary $\chi$ states, $$\begin{split}
\left| B (\vec{p}) \right\rangle = & \sqrt\frac{1}{2} \sqrt\frac{1}{8m_\chi^3} \sum_{s,s'} \int \frac{d^3 \vec{k}}{(2\pi)^3} \widetilde \Psi (\vec{k}) \\
&\hspace{0.5cm} \times \left| \vec{k}, s \right\rangle \otimes \left| -\vec{k}, s' \right\rangle \bar u_s (\vec{k}) \gamma_5 v_{s'}(-\vec{k}) \ ,
\end{split}$$ where $| \vec{k}, s \rangle$ and $| -\vec{k}, s' \rangle$ are two plane-wave $\chi$ particle states, and $\widetilde \Psi (\vec{k})$ is the bound state wavefunction in the momentum space. $\sqrt{1/2}$ is a symmetry factor accounting for identical particles.
On top of the above dark sector setup, we introduce a neutrino portal interaction for $\chi$, of the form $$\mathcal{L}_{\nu\text{-portal}} = \frac{1}{\Lambda} (LH) (\chi \eta) + {\rm h.c.} \ ,$$ where $\eta$ is a complex scalar and heavier partner of $\chi$. The conservation of global $U(1)_D$ symmetry (for $\chi$ to be asymmetric dark matter) naturally introduces the $\eta$ field which carries opposite charge to $\chi$. Such a neutrino portal interaction was also considered in the context of several interacting dark sector models [@Berryman:2017twh; @Bertoni:2014mva; @Cherry:2014xra; @Falkowski:2009yz; @Berezhiani:1995am].
The key process we consider is depicted in Fig. \[fig:FD\], where a solar neutrino $\nu$ strikes on a $B$ bound state, gets absorbed and dissociate it into two unbounded $\chi$ and $\eta$ particles. The corresponding scattering amplitude takes the form $$\begin{aligned}
&\mathcal{A}_{\nu+ B\to \chi + \eta} = \frac{(v/\Lambda)y_D^2}{(q^2 - m_\phi^2)} \bar u_\chi(p_2) \Pi \frac{1}{\frac{\cancel{p}_1}{2} - \cancel{k} - \cancel{q} - m_\chi} v_\nu(k_1) \ , \nonumber \\
&\Pi = \frac{|\Psi(0)|}{\sqrt{16 m_\chi^3}} \left( \frac{\cancel{p}_1}{2} + \cancel{k} + m_\chi \right) \gamma_5 \left( \frac{\cancel{p}_1}{2} - \cancel{k} - m_\chi \right) \ ,
\end{aligned}$$ where $\Psi(0)$ is the wavefunction at space origin, the relevant momenta in lab frame are $p_1 \simeq [2m_\chi, \vec{0}]$, $p_2 \simeq [m_\chi + |\vec{p}_2|^2/(2m_\chi), \vec{p}_2]$, $k_1 = [ E_\nu, E_\nu \hat z]$. From Eq. (\[eq:vafter\]) we derive $|\vec{p}_2| \simeq \sqrt{m_\chi E_\nu}$. In the limit where the neutrino energy $E_\nu$ is much higher than the binding energy $\approx \alpha_D^2 m_\chi/4$, the momentum transfer $q \simeq [ |\vec{p}_2|^2/(2m_\chi), \vec{p}_2]$ will be much larger than the Bohr momentum $k$. The above amplitude can be further simplified by imposing momentum conservation $p_2 = p_1/2+k+q$. Keeping only the leading terms, the cross section is $$\label{eq:crosssection}
\sigma_{\nu+ B\to \chi + \eta} \simeq \frac{\pi (v/\Lambda)^2 \alpha_D^2|\Psi(0)|^2}{4 \sqrt{m_\chi E_\nu} (m_\chi E_\nu + m_\phi^2)^2} \ ,$$ where it is assumed $m_\eta \simeq m_\chi$ for simplicity. In Coulomb limit ($m_\phi^2 \ll m_\chi E_\nu$), it further simplifies to $$\label{eq:crosssectionCoulomb}
\sigma_{\nu+ B\to \chi + \eta} \simeq \frac{(v/\Lambda)^2 \alpha_D^5 m_\chi^{1/2} }{32 E_\nu^{5/2}} \ .$$ In this derivation we have assumed the momentum transfer $q$ to be much larger than the bound state Bohr momentum $k$. For small enough $E_\nu$, the denominator will be regulated by the non-zero Bohr momentum in the dominator. The phase space shuts off when the neutrino is no longer energetic enough to dissociate the bound state.
![Feynman diagram for solar neutrino to dissociate a dark matter bound state ($B$) and speed up the final state particles, well above the typical halo velocities.[]{data-label="fig:FD"}](FD.pdf){width="38.00000%"}
The purpose of this calculation is to show that the process of dark matter bound state dissociation with solar neutrinos could have a sizable cross section. Indeed, for a set of sample parameters $\alpha_D = 0.05$, $v/\Lambda = 1$, $m_\chi=0.1\,$GeV and $E_\nu = 1\,$MeV, we find $\sigma_{\nu+ B\to \chi + \eta} \simeq 1.2\times 10^{-28}\,{\rm cm}^2$.
So far we have derived the kinematics and dynamics of a new process for solar neutrinos to speed up dark matter by dissociating bound states. The next step is to calculate the resulting flux of dark matter $\chi$. To begin with, we need a distribution of dark matter bound states in the vicinity of the Sun. In this work, we assume that the bound state $B$ comprises an order one fraction of the total dark matter relic abundance, which is shown to be a feasible scenario [@Wise:2014jva], and in turn an order one fraction of the local density of dark matter in the solar system. We do not consider the capture of $B$’s by the Sun because for sub-GeV masses the evaporation effect is strong [@Griest:1986yu; @Gould:1987ju]. The velocity gain of dark matter via evaporation is much smaller than what we consider, Eq. (\[eq:vafter\]).
![Illustration of dark matter velocity distributions due to the acceleration mechanism discussed in this work (red), decomposed into various solar neutrino components, with parameters $\alpha_D = 0.05$, $v/\Lambda = 1$, $m_\chi=0.1\,$GeV, compared with the standard halo velocity distribution (black).[]{data-label="fig:flux"}](Flux.pdf){width="48.00000%"}
The flux of solar neutrinos depends on the distance from the Sun. It could be calculated based on the measured solar neutrino flux by terrestrial experiments on Earth. In the reaction $\nu+ B\to \chi + \eta$ we consider, the angular distribution of the final state dark matter particle is isotropic. Therefore, we integrate over all the space around the Sun for this reaction to occur, and the resulting flux of final state dark matter at Earth is $$\label{eq:Flux}
\frac{d\Phi^\oplus_\chi(E_\nu)}{d E_\nu} = \int d^3 \vec{r} \left[ \frac{\rho_\chi(\vec{r})/(2 m_\chi)}{4\pi d(\vec{r})^2} \right] \left[ \frac{d_\oplus^2}{r^2} \frac{d\Phi^\oplus_\nu(E_\nu)}{d E_\nu} \right] \sigma(E_\nu) \ ,$$ where $d(\vec{r})=\sqrt{r^2+d_\oplus^2-2r d_\oplus \cos\theta}$ is the distance from $\vec{r}$ to the Earth’s position. In this coordinate basis, the Sun is located at the origin, the Earth is on the $\hat z$ axis at $d_\oplus=1\,$AU. $\sigma(E_\nu)$ is the cross section calculated from Eq. (\[eq:crosssectionCoulomb\]) and ${d\Phi^\oplus_\nu(E_\nu)}/{d E_\nu}$ is the differential solar neutrino flux (see [*e.g.*]{}, [@Bellerive:2003rj]. We omit the lines).
The angular part of the above integral can be done analytically. The remaining $r$ integral is dominated by the region $r\lesssim d_\oplus$. Therefore, we can simply set $\rho_\chi(\vec{r}) \simeq \rho_\oplus = 0.3\,{\rm GeV}/{\rm cm}^3$. Under this approximation, the $r$ integral can also be done analytically, yielding $$\label{eq:FluxFormula}
\frac{d\Phi^\oplus_\chi}{d E_\nu} = \frac{\pi^2 \rho_\oplus d_\oplus}{8 m_\chi} \sigma(E_\nu) \frac{d\Phi^\oplus_\nu(E_\nu)}{d E_\nu} \ .$$ In the dissociation process we consider, the incoming solar neutrino energy is related to the final dark matter velocity $v_\chi$ via Eq. (\[eq:vafter\]), $E_\nu \simeq m_\chi v_\chi^2 + E_b$. We derive a differential dark matter flux with respect to its velocity, $$\label{eq:DiffernetialFlux}
\frac{d\Phi^\oplus_\chi}{d v_\chi} = \frac{\pi^2 \rho_\oplus d_\oplus}{8 m_\chi} \sigma(E_\nu) \left[ 2 m_\chi v_\chi \frac{d\Phi^\oplus_\nu(E_\nu)}{d E_\nu} \right] \ ,$$ which serves as new component of dark matter velocity distribution on top of the halo one. In Fig. \[fig:flux\], we plot this differential flux (red curve) for a set of model parameters, $\alpha_D=0.05$ and $m_\chi=0.1\,$GeV, and compare it with the dark matter velocity in the standard halo model (black curve). Clearly, this new component of dark matter particles is characterized by much higher velocities than the halo counterpart. In spite of their overall lower flux, these particles could leave a detectable signal above the threshold of many traditional dark matter detectors as will be shown below.
Moreover, the differential flux with respect to dark matter velocity, shown in Fig. \[fig:flux\], looks very similar in shape to its source, the solar neutrino energy spectrum. Once a positive signal is triggered in direct detection experiments, such a distinct feature may allow us to verify the existence of the above dark matter acceleration mechanism.
As an application of the accelerated dark matter flux derived above, in this section we explore its implication for dark matter direct detection. Here we assume that the dark matter particle $\chi$ has a spin-dependent scattering cross section with the proton (but not with neutron). We take a phenomenological approach and simply parametrize the cross section as $\sigma_{\chi p}^{\rm SD}$. Following the standard formalism [@Jungman:1995df; @An:2010kc], the corresponding detection rate differentiated with respect to the nuclear recoil energy is $$\frac{dR}{d E_r} = N_T \frac{m_A }{2 \mu_{\chi A}^2} \int_{v_{min}}^{v_{esc}} \frac{dv}{v^2} \frac{d\Phi_\chi^\oplus(v)}{d v} \frac{S(\sqrt{2 m_A E_r})}{S(0)} \sigma_{\chi A}^{\rm SD} \ ,$$ where $m_A$ is the mass of the target nucleus, $\mu_{\chi A} = m_\chi m_A/(m_\chi + m_A)$, and $N_T$ is the total number of the nucleus $A$ in the detector. $\sigma_{\chi A}^{\rm SD}$ is the nucleus level cross section, which in the proton-coupling-only case, is equal to $(\mu_{\chi A}^2/\mu_{\chi p}^2) \sigma_{\chi p}^{\rm SD}$. The nuclear form factor is taken from [@Belanger:2008sj]. In the standard case, the differential flux is given by $d\Phi_\chi^\oplus/dv = (\rho_\oplus /m_\chi) v f_1(v)$ where $f_1(v)$ is the standard Maxwellian halo velocity distribution [@Jungman:1995df]. In contrast, when considering the detection of dark matter speeded up by solar neutrinos, the differential flux is given by Eq. (\[eq:FluxFormula\]), with $v_{min}=\sqrt{m_A E_r}/(2\mu_{\chi A}^2)$.
We consider a recent result from the PICO-60 experiment [@Amole:2017dex] at SNOLAB, which is based on a 1167 kg-day exposure using C$_3$F$_8$ and reinterpret the result for the speeding-up scenario. The result is presented in the $\sigma_{\chi p}^{\rm SD}$ versus $m_\chi$ plane in Fig. \[fig:reach\]. The red and blue shaded regions are the new exclusion limits that the PICO-60 experiment could set for if dark matter particles are accelerated by solar neutrinos as considered in this work, for the dark fine-structure constant $\alpha_D=0.05$ and $0.02$, respectively. To set this limit, the dark matter particles need to successfully travel to the underground detector, instead of shielded by the earth above (mainly due to $^{27}$Al in the Earth’s crust for spin-dependent scattering), thus the range of $\sigma_{\chi p}^{\rm SD}$ is shown up to $10^{-26}\,{\rm cm}^2$. For comparison, we also include results from PICO60 [@Amole:2017dex], PICASSO [@Behnke:2016lsk], CDMSlite [@Agnese:2017jvy] and PandaX [@Xia:2018qgs], assuming standard halo model.
The shape of the new exclusion region derived in this work can be understood through the following considerations. Two conditions must be satisfied for the scenario we consider to occur. First, the solar neutrino must be energetic enough to dissociate the dark matter bound state, which requires $$\label{lowerboundvchi}
E_\nu > E_b \simeq \alpha_D^2 m_\chi/4 \ ,$$ in the assumed Coulomb limit. Second, the recoil energy from the resulting dark matter scattering must exceed the experimental threshold, $$m_\chi E_\nu/(2 m_A) > E_{\rm th} \ .$$ For PICO-60, $E_{\rm th}=3.3\,$keV. For a particular solar neutrino component ([*e.g.*]{} $pp$ neutrinos with energies $E_\nu\lesssim 0.4\,$MeV) and the value of $\alpha_D$, there is a window of $m_\chi$ (corresponding to the bumps in the exclusion region in Fig. \[fig:reach\]) to satisfy both conditions, $2 m_A E_{\rm th}/E_\nu < m_\chi < 4 E_\nu/\alpha_D^2$. Therefore, if the dark matter mass is too large or too small, the limit weakens quickly. Moreover, the condition for the above window to exist is $\alpha_D < \sqrt{2 E_\nu^2/ (m_A E_{\rm th})}$. As $\alpha_D$ grows, such a window narrows and finally disappears. In turn, one has to resort to higher energy solar neutrino components at the price of lower fluxes. In Fig. \[fig:reach\], the dashed curves show the potential future reach of PICO experiment assuming a $10^5\,$kg-day exposure [@Levine:2018talk] and the same energy threshold. The reaches can be further improved with a lower detector threshold [@Amole:2019scf].
![A reinterpretation of the existing (and future) PICO search results for sub-GeV dark matter, assuming the latter is accelerated by solar neutrinos via the mechanism presented here. We fix $v/\Lambda=1$ and show results with two benchmark values of $\alpha_D = 0.02$ (blue) and 0.05 (red), respectively.[]{data-label="fig:reach"}](SD-detection.pdf){width="48.00000%"}
In this work, we explore the possible role of solar neutrinos on the dark matter velocity distribution and direct detection experiments. In a simple neutrino portal dark sector model, we show it is possible to speed up dark matter well above $10^{-3}c$. A crucial ingredient for this mechanism to work is the existence of stable bound states of dark matter. The accelerated dark matter particles feature characteristic velocity distribution which could make this mechanism testable. When such a particle strikes on the nucleus target in a dark matter detector, the recoil energy is large enough to render useful limits on sub-GeV dark matter interaction, by reinterpreting the existing direct detection results. Although we only did so based on a spin-dependent search result from PICO-60, it is straightforward to generalize this idea to other dark matter direct detection results.
I would like to thank Eric Dahl for a useful discussion on future spin-dependent dark matter detection experiments. This work is supported by the Arthur B. McDonald Canadian Astroparticle Physics Research Institute.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'One problematic feature of Hall algebras is the fact that the standard multiplication and comultiplication maps do not satisfy the bialgebra compatibility condition in the underlying symmetric monoidal category ${{\rm Vect}}$. In the past this problem has been resolved by working with a weaker structure called a ‘twisted’ bialgebra. In this paper we solve the problem differently by first switching to a different underlying category ${{\rm Vect}}^K$ of vector spaces graded by a group $K$ called the Grothendieck group. We equip this category with a nontrivial braiding which depends on the K-grading. With this braiding, we find that the Hall algebra does satisfy the bialgebra condition exactly for the standard multiplication and comultiplication, and can also be equipped with an antipode, making it a Hopf algebra object in ${{\rm Vect}}^K$.'
author:
- |
Christopher D. Walker\
Department of Mathematics, University of California\
Riverside, CA 92521 USA
title: Hall Algebras as Hopf Objects
---
Introduction
============
Hall algebras have been a popular topic in recent years because of their connection to quantum groups. It is a well known fact, due to Ringel [@Ringel], that the Hall algebra constructed from the representations of a Dynkin quiver over a finite field ${{\mathbb F}}_q$ is isomorphic to ‘half’ of the quantum group associated to the same Dynkin diagram, namely $U_q^+({{\mathfrak g}})$ where ${{\mathfrak g}}$ is the Lie algebra generated by the quiver. This construction provides interesting insight into many structures on the quantum group, but unfortunately does not do everything we hope.
One of the fundamental problems of Hall algebras arises when we try to make the algebra into a Hopf algebra. In the initial definitions of the Hall algebra, we start with a nice associative multiplication. We also find that the Hall algebra is a coalgebra with an equally nice coassociative comultiplication. However, when we try to check that the algebra and coalgebra fit together to form a bialgebra, we see this fails in the standard underlying category ${{\rm Vect}}$ with its usual braiding. Instead, the combination of these maps obeys “Green’s Formula”, a relationship between the multiplication and comultiplication which we describe in detail below (Proposition \[GF\]). This formula basically says that the Hall algebra is ‘almost’ a bialgebra in the standard category ${{\rm Vect}}$. Specifically, we only miss being a bialgebra by a coefficient. To see where this extra coefficient comes from, consider the string diagrams which describe the general bialgebra compatibility axiom. As is standard, we will write multiplication of two elements as: $$\xy
(-5,0);(0,-6)**\crv{(-5,-2)&(0,-4)};
(5,0);(0,-6)**\crv{(5,-2)&(0,-4)};
(0,-6);(0,-9)**\crv{};
\endxy$$ and comultiplication of an element as: $$\xy
(-5,0);(0,6)**\crv{(-5,2)&(0,4)};
(5,0);(0,6)**\crv{(5,2)&(0,4)};
(0,6);(0,9)**\crv{};
\endxy$$ We can then draw the bialgebra axiom as follows. We first to draw multiplication, follow by comultiplication, which looks like: $$\xy 0;/r.10pc/:
(-10,-20)*{}="b1"; (10,-20)*{}="b2";
(-10,20)*{}="T1"; (10,20)*{}="T2";
(0,10)*{}="C";(0,-10)*{}="D";
"T1";"C"**\crv{(-10,17)&(0,13)};
"T2";"C"**\crv{(10,17)&(0,13)};
"b1";"D"**\crv{(-10,-17)&(0,-13)};
"b2";"D"**\crv{(10,-17)&(0,-13)};
"D";"C"**\crv{};
\endxy$$ This should equal the result of comultiplying each element and then multiplying the resulting tensor product of elements. This will look like: $$\xy 0;/r.10pc/:
(-10,-20)*{}="b1"; (10,-20)*{}="b2";
(-10,20)*{}="t1"; (10,20)*{}="t2";
(-10,15)*{}="A";(10,15)*{}="B";
(-10,-15)*{}="E";(10,-15)*{}="F";
"t1";"A"**\crv{};
"t2";"B"**\crv{};
"A";"E"**\crv{(-10,10)&(-18,0)&(-10,-10)};
"B";"F"**\crv{(10,10)&(18,0)&(10,-10)};
"E";"b1"**\crv{};
"F";"b2"**\crv{};
"B";"E"**\crv{(10,10)&(-10,-10)}\POS?(.5)*{\hole}="H";
"A";"H"**\crv{(-10,10)};
"H";"F"**\crv{(10,-10)};
\endxy$$ But there is wrinkle, namely the braiding of the strings halfway down the diagram. This means we must be working in a braided monoidal category. For the Hall algebra, the seemingly natural choice to work in would be ${{\rm Vect}}$. In ${{\rm Vect}}$ this braiding would simply swap elements with no coefficient. However we have already noted that in ${{\rm Vect}}$ the Hall algebra does not satisfy the bialgebra condition as desired.
To ‘fix’ this, a new structure called a ‘twisted’ bialgebra is usually introduced, where swapping the order of elements can still be done, but at the price of an extra coefficient. This coefficient becomes $q^{-\langle A,D\rangle}$ when swapping elements $A$ and $D$, where $\langle A,D\rangle$ is a bilinear form on a group $K$ (called the Grothendieck group) related to the underlying category of the Hall algebra.
To obtain a true (untwisted) bialgebra, one then extends the Hall algebra to some larger algebra and alters the multiplication and comultiplication. This process is interesting in its own right, because the result is isomorphic to a larger piece of a quantum group, namely the universal enveloping algebra of the Borel $\mathfrak{b}$. However, we want to take a different direction to avoid the artificial nature of this fix.
In this paper, we will approach the problem directly. Instead of describing the Hall algebra as a ‘twisted’ bialgebra, we will find a braided monoidal category other than ${{\rm Vect}}$ where the Hall algebra is a true bialgebra object. We accomplish this by giving the category of $K$-graded vector spaces, ${{\rm Vect}}^K$, a braiding that encodes the twisting in the Hall algebra. This works since the extra coefficient $q^{-\langle A,D\rangle}$ from Green’s Formula depends on the crossing strands in the diagram for the bialgebra axiom. We then accomplishes the same task, but in a way that accounts for the correction factor in the underlying structure, rather than including it later. This idea was mentioned by Kapranov [@Kapranov] but details were not provided. Also, Kapranov was working with the same twisted multiplication and comultiplication as Ringel [@Ringel2], where we are using the simpler, non-twisted versions of the maps instead. We will then round out the paper by providing the antipode for this bialgebra to show that the Hall algebra is a Hopf algebra object in our new category.
Hall Algebras {#hall}
=============
In this section we will describe the construction of the Ringel-Hall algebra. We begin with a quiver $Q$ (i.e. a directed graph) whose underlying graph is that of a simply-laced Dynkin diagram. We will then consider the abelian category ${{\rm Rep}}(Q)$ of all finite dimensional representation of the quiver $Q$ over a fixed finite field $\mathbb{F}_q$.
We start by fixing a finite field ${{\mathbb F}}_q$ and a directed graph $D$, which might look like this: $$\xymatrix@=10pt{
&&&&& \bullet \\
\bullet \ar@(ul,dl)[] \ar@/^1pc/[rr]
&& \bullet \ar@/^1pc/[ll] \ar@/^1pc/[rr] \ar@/_1pc/[rr] \ar[rr]
&& \bullet \ar[ur] \ar[dr] \\
&&&&& \bullet
}$$ We shall call the category $Q$ freely generated by $D$ a **quiver**. The objects of $Q$ are the vertices of $D$, while the morphisms are edge paths, with paths of length zero serving as identity morphisms.
By a **representation** of the quiver $Q$ we mean a functor $$R {\colon}Q {\rightarrow}{{\rm FinVect}}_q,$$ where ${{\rm FinVect}}_q$ is the category of finite-dimensional vector spaces over ${{\mathbb F}}_q$. Such a representation simply assigns a vector space $R(d) \in {{\rm FinVect}}_q$ to each vertex of $D$ and a linear operator $R(e) {\colon}R(d) {\rightarrow}R(d')$ to each edge $e$ from $d$ to $d'$. By a **morphism** between representations of $Q$ we mean a natural transformation between such functors. So, a morphism $\alpha {\colon}R
{\rightarrow}S$ assigns a linear operator $\alpha_d {\colon}R(d) {\rightarrow}S(d)$ to each vertex $d$ of $D$, in such a way that $$\xymatrix{
R(d) \ar[d]_{\alpha_d} \ar[r]^{R(e)} & R(d') \ar[d]^{\alpha_{d'}} \\
S(d) \ar[r]_{S(d)} & S(d')
}$$ commutes for any edge $e$ from $d$ to $d'$. There is a category ${{\rm Rep}}(Q)$ where the objects are representations of $Q$ and the morphisms are as above. This is an abelian category, so we can speak of indecomposable objects, short exact sequences, etc. in this category.
In 1972, Gabriel [@Gabriel] discovered a remarkable fact. Namely: a quiver has finitely many isomorphism classes of indecomposable representations if and only if its underlying graph, ignoring the orientation of edges, is a finite disjoint union of Dynkin diagrams of type $A, D$ or $E$. These are called [**simply laced**]{} Dynkin diagrams.
Henceforth, for simplicity, we assume the underlying graph of our quiver $Q$ is a simply laced Dynkin diagram when we ignore the orientations of its edges. Let $X$ be the underlying groupoid of ${{\rm Rep}}(Q)$: that is, the groupoid with representations of $Q$ as objects and *isomorphisms* between these as morphisms. We will use this groupoid to construct the Hall algebra of $Q$.
As a vector space, the Hall algebra is just ${{\mathbb R}}[{\underline{X}}]$. Recall that this is the vector space whose basis consists of isomorphism classes of objects in $X$. In fancier language, it is the zeroth homology of $X$.
We now focus our attention on the Hall algebra product. Given three quiver representations $M,N,$ and $E$, we define the set: $$\mathcal{P}_{MN}^E=
\{(f,g):
0{\rightarrow}N \stackrel{f}{\rightarrow} E \stackrel{g}{\rightarrow} M {\rightarrow}0
\textrm{\; is exact} \}$$ and we call its set cardinality $P_{MN}^E$. In the chosen category this set has a finite cardinality, since each representation is a finite-dimensional vector space over a finite field. The Hall algebra product counts these exact sequences, but with a subtle ‘correction factor’: $$[M] \cdot [N] =\sum_{[E] \in {\underline{X}}}
\frac{P_{MN}^E}{{{\rm aut}}(M) \, {{\rm aut}}(N)}\, [E] \,.$$ Where we call ${{\rm aut}}(M)$ the set cardinality of the group ${{\rm Aut}}(M)$.
Somewhat surprisingly, the above product is associative. In fact, Ringel [@Ringel] showed that the resulting algebra is isomorphic to the positive part $U_q^+ {{\mathfrak g}}$ of the quantum group corresponding to our simply laced Dynkin diagram! So, roughly speaking, the Hall algebra of a simply laced quiver is ‘half of a quantum group’.
This isomorphism also relates to a coalgebra structure on the Hall algebra. Using the same ideas from the multiplication formula, we can define a comultiplication on the Hall algebra to be a carefully weighted sum on ways to ‘factor’ a representation via short exact sequences. Formulaically this becomes: $$\Delta(E)=\sum_{[M],[N] \in {\underline{X}}}
\frac{|\mathcal{P}_{MN}^E|}{ {{\rm aut}}(E)}\, [N]{\otimes }[M] \,.$$ Again, Ringel found that these are the correct factor to make the comultiplication coassociative. However, we immediately run into a problem; these two maps do not satisfy the compatibility condition for a bialgebra.
The Category of K-graded Vector Spaces {#gvs}
======================================
It is interesting to note that the standard multiplication and comultiplication on $U_q^+ {{\mathfrak g}}$ (which the Hall algebra is isomorphic to) also do not satisfy the compatibility axiom of a bialgebra, so we should not expect the Hall algebra to, either. This does not mean there is not an interesting relationship between the multiplication and comultiplication in the Hall algebra. This relationship is often described as being a ‘twisted’ bialgebra, where we do not use the standard extension of the multiplication to the tensor product. We would like to take a different point of view here. It turns out that the bialgebra axiom can be satisfied if we change the category in which we ask for them to be compatible.
In order to describe this new category, we will start with a definition of the Grothendieck group of a general abelian category.
Let $\mathcal{A}$ be an abelian category. We can define an equivalence relation on isomorphism classes of objects in $\mathcal{A}$ by $[A]+[B]=[C]$ if there exists a short exact sequence $0{\rightarrow}A{\rightarrow}C{\rightarrow}B{\rightarrow}0$. The set of equivalence classes under this relation form a group $K_0(\mathcal{A})$ called the [**Grothendieck group**]{}.
$K_0(\mathcal{A})$ has a universal property in the following sense. Given any abelian group $G$, any additive function $f$ from isomorphism classes of $\mathcal{A}$ to the group $G$ will give a unique abelian group homomorphism $\tilde{f}{\colon}K_0(\mathcal{A}){\rightarrow}G$ such that the following diagram commutes:
$$\xymatrix{
\mathcal{A}\ar[rr]\ar[dr]_f & & K_0(\mathcal{A}) \ar[dl]^{\exists !\tilde{f}} \\
& G & \\
}$$
The original purpose of the Grothendieck group was to study Euler characteristics, and this is precisely why we are interested in them here.
In many of the standard references for Hall algebras [@Hubery; @Schiffman] the characteristics of the Grothendieck group of ${{\rm Rep}}(Q)$ are explained explicitly. Many of these properties follow from the fact that ${{\rm Rep}}(Q)$ is hereditary. We can also describe these properties in the general case of an abelian category $\mathcal{A}$ which has finite homological dimension. However, to construct the entire Hall algebra, our abelian category will need to hold to the extra finiteness properties that the groups ${{\rm Ext}}^i(M,N)$ must be finite. This condition is sufficient since it makes the sets $\mathcal{P}^E_{MN}$ finite, and makes the bilinear form in the next proposition well defined.
Let $\mathcal{A}$ be an abelian $k$-linear category for some field $k$. Suppose that $\mathcal{A}$ has finite homological dimension $d$ and $\dim{{\rm Ext}}^i(M,N)$ is finite for all objects $M,N\in \mathcal{A}$. If $K=K_0(\mathcal{A})$ is the Grothendieck group of $\mathcal{A}$, then $K$ admits a bilinear form $\langle\cdot,\cdot\rangle{\colon}K\times K{\rightarrow}{{\mathbb C}}$ given by: $$\langle \underline{m},\underline{n}\rangle = \sum_{i=0}^d (-1)^{i}\dim{{\rm Ext}}^i(M,N)$$
We prove the theorem for $d=1$ (i.e. when the category is hereditary) since this is the main case we will use. The case when $d=0$ is simply bilinearity of ${{\rm Hom}}$, and the cases where $d>1$ follows by a similar argument to $d=1$.\
For $d=1$ we need to show that: $$\dim{{\rm Hom}}(M,N_1\oplus N_2)-\dim{{\rm Ext}}^1(M,N_1\oplus N_2)=$$ $$\dim{{\rm Hom}}(M,N_1)-\dim{{\rm Ext}}^1(M,N_1)+\dim{{\rm Hom}}(M,N_2)-\dim{{\rm Ext}}^1(M,N_2)$$ we begin with the short exact sequence: $$0{\rightarrow}N_1\stackrel{i_1}{\rightarrow} N_1\oplus N_2 \stackrel{\pi_2}{\rightarrow} N_2{\rightarrow}0$$ which, since $d=1$, gives rise to the long exact sequence: $$0{\rightarrow}{{\rm Hom}}(M,N_1) \stackrel{}{\rightarrow} {{\rm Hom}}(M,N_1\oplus N_2) \stackrel{}{\rightarrow} {{\rm Hom}}(M,N_2) \stackrel{h}{\rightarrow}$$ $${{\rm Ext}}^1(M,N_1)\stackrel{}{\rightarrow} {{\rm Ext}}^1(M,N_1\oplus N_2) \stackrel{}{\rightarrow} {{\rm Ext}}^1(M,N_2) {\rightarrow}0.$$ Using a variety of basic equations from the fact that this sequence is exact, as well as some dimension arguments, The left hand sides becomes: $$\dim{{\rm Hom}}(M,N_1\oplus N_2)-\dim{{\rm Ext}}^1(M,N_1\oplus N_2)$$ $$=\dim {{\rm im}}\tilde{\pi_2}+\dim \ker \tilde{\pi_2}-\dim {{\rm im}}\hat{\pi_2}-\dim \ker \hat{\pi_2}$$ and the right hand side turns into: $$\dim{{\rm Hom}}(M,N_1)-\dim{{\rm Ext}}^1(M,N_1)+\dim{{\rm Hom}}(M,N_2)-\dim{{\rm Ext}}^1(M,N_2)$$ $$=\dim {{\rm im}}h+\dim \ker h+\dim{{\rm im}}\tilde{i_1}-\dim {{\rm im}}\hat{i_1}-\dim \ker \hat{i_1}-\dim{{\rm im}}\hat{\pi_2}$$ $$=\dim \ker \hat{i_1}+\dim {{\rm im}}\tilde{\pi_2}+\dim\ker \tilde{\pi_2}-\dim \ker \hat{\pi_2}-\dim \ker \hat{i_1}-\dim{{\rm im}}\hat{\pi_2}$$ $$=\dim {{\rm im}}\tilde{\pi_2}+\dim\ker \tilde{\pi_2}-\dim \ker \hat{\pi_2}-\dim{{\rm im}}\hat{\pi_2}.$$
In general, it is possible to construct a braided monoidal category ${{\rm Vect}}^G$ from any abelian group $G$ equipped with a bilinear form $\langle\cdot,\cdot\rangle$. One common example is the category of super-algebras, which can be thought of in this context in terms of the group ${\ensuremath{\mathbb{Z}}\xspace}_2$ with its unique non-trivial bilinear form. Joyal and Street [@JoyalStreet] described the general idea of constructing a braided monoidal category from a bilinear form. In the next theorem, we will describe how this braiding works in detail for our desired case of the Grothendieck group $K=K_0(\mathcal{A})$ with the previously described bilinear form.
\[BMC\] Let $\mathcal{A}$ be an abelian, $k$-linear category with finite homological dimension. Let $K=K_0(\mathcal{A})$ be its Grothendieck group, and suppose $\dim{{\rm Ext}}^i(M,N)$ is finite for all objects $M,N \in\mathcal{A}$. Then, the category ${{\rm Vect}}^K$ of $K$-graded vector spaces and grade preserving linear operators is a braided monoidal category, with the braiding given by: $$B_{V,W}:V{\otimes }W{\rightarrow}W{\otimes }V$$ $$v{\otimes }w \mapsto q^{-\langle \underline{n},\underline{m}\rangle}w{\otimes }v$$ where $q$ is a non-zero element of $k$.
The monoidal structure on this category is just the tensor product in the category ${{\rm Vect}}$. To define a braiding on this category, we first note that the braiding is defined by isomorphisms in the category which are graded linear operators. Because of linearity, it is enough to define these isomorphisms on a single graded piece. Also note that for any two $K$-graded vector spaces $V$ and $W$, a graded piece of the tensor product $V{\otimes }W$ can be written as the sum of tensor products of graded pieces from $V$ and $W$, or more precisely: $$(V{\otimes }W)_{\underline{d}}=\bigoplus_{\underline{n}\in K} V_{\underline{n}}{\otimes }W_{\underline{d}-\underline{n}}.$$ This lets us define the braiding $B_{V,W}{\colon}V{\otimes }W{\rightarrow}W{\otimes }V$ only on the tensor product of the graded piece $V_{\underline{n}}{\otimes }W_{\underline{m}}$. We thus define the map: $$B_{\underline{n},\underline{m}}{\colon}V_{\underline{n}}{\otimes }W_{\underline{m}}{\rightarrow}W_{\underline{m}}{\otimes }V_{\underline{n}}$$ $$v{\otimes }w \mapsto q^{-\langle \underline{n},\underline{m}\rangle}w{\otimes }v$$ which is easily seen to be an isomorphism. We only need to check the hexagon equations, i.e. ones of the form: $$\xymatrix{
& (W{\otimes }V){\otimes }U\ar[r]^\alpha & W{\otimes }(V{\otimes }U)\ar[dr]^{1{\otimes }B_{V,U}} & \\
(V{\otimes }W){\otimes }U \ar[ur]^{B_{V,W}{\otimes }1}\ar[dr]_\alpha & & & W{\otimes }(U{\otimes }V) \\
& V{\otimes }(W{\otimes }U) \ar[r]_{B_{V,W{\otimes }U}} & (W{\otimes }U){\otimes }V\ar[ur]_\alpha & \\
}$$ We will make the argument for the above hexagon identity, noting the the other versions follow by a similar argument. Now, since we have restricted ourselves to vector spaces with a single grade, it is enough to chase a general element around this diagram. let $v\in V_{\underline{n}}$, $w\in W_{\underline{m}}$, and $u\in U_{\underline{p}}$. The top path of the hexagon diagram yields the composite: $$(v{\otimes }w){\otimes }u \mapsto q^{-\langle\underline{n},\underline{m}\rangle-\langle\underline{n},\underline{p}\rangle}w{\otimes }(u{\otimes }v).$$ For the bottom path we note that $v{\otimes }w\in (V{\otimes }W)_{\underline{m+p}}$, so we get the composite: $$(v {\otimes }w){\otimes }u \mapsto q^{-\langle \underline{n}, \underline{m+p}\rangle} w {\otimes }(u {\otimes }v).$$ Hence, commutativity of the diagram will follow from the equality $$-\langle\underline{m},\underline{n}\rangle-\langle\underline{m},\underline{p}\rangle=-\langle \underline{m}, \underline{n+p}\rangle,$$ which is precisely bilinearity of the form $\langle\cdot, \cdot\rangle$.
The Hopf Algebra Structure
==========================
Now we consider our Hall algebra in the braided monoidal category ${{\rm Vect}}^K$. The concept of a Hopf algebra object in a braided monoidal category was described by Majid [@Majid], where he called it a ‘braided group’, but later [@Majid2] described it in the way we will use here. The basic idea is to ask if the standard defining commutative diagrams for a Hopf algebra hold in some braided monoidal category, instead of the symmetric monoidal category ${{\rm Vect}}$. For the remainder of this section, we will let $Q$ be a simply laced Dynkin quiver. We will focus back on the specific abelian category ${{\rm Rep}}(Q)$ and the category of $K_0({{\rm Rep}}(Q))-$graded vector spaces, which we showed in Section \[gvs\] to be a braided monoidal category. Remember that ${{\rm Rep}}(Q)$ is hereditary and satisfies all the finiteness conditions of Section \[gvs\]. We can now state the main theorem of this paper.
\[HO\] The Hall algebra of ${{\rm Rep}}(Q)$ is a Hopf algebra object in the category $Vect^K$.
To prove this theorem we need to work through the following lemmas. For what follows, we will set $X$ to be the underlying groupoid of ${{\rm Rep}}(Q)$, $\underline{X}$ to be the set of isomorphism classes in $X$, and $K=K_0({{\rm Rep}}(Q))$. Recall that $R[\underline{X}]$ is the vector space of all finite linear combinations of elements of ${\underline{X}}$. This vector space, which is the underlying vector space of the Hall algebra, is easily seen to be $K$ graded.
The vector space ${\mathcal{H}}={{\mathbb R}}[\underline{X}]$ is a $K$-graded vector space, with the grading on each isomorphism class $[M]\in{\underline{X}}$ given by its image in $K$.
For the next two lemmas, we note that the multiplication and comultiplication described were shown to be associative and coassociative in the original category ${{\rm Vect}}$ by Ringel [@Ringel]. This fact passes to our new category since neither axiom requires or depends on the particular braiding on vector spaces, so we will not repeat the argument. After stating both lemmas, we will provide a brief description of why each one is a morphism in the new category ${{\rm Vect}}^K$.
\[mult\] The multiplication map $m:{\mathcal{H}}{\otimes }{\mathcal{H}}{\rightarrow}{\mathcal{H}}$ defined on basis elements by:
$$m([M]{\otimes }[N]) =\sum_{[E]}
\frac{P_{MN}^E}{{{\rm aut}}(M) \, {{\rm aut}}(N)}\, [E]$$ is a morphism in ${{\rm Vect}}^K$.
\[comult\] The comultiplication map $\Delta:{\mathcal{H}}{\rightarrow}{\mathcal{H}}{\otimes }{\mathcal{H}}$ defined on basis elements by:
$$\Delta(E)=\sum_{[M],[N]}
\frac{P_{MN}^E}{ {{\rm aut}}(E)}\, [N]{\otimes }[M]$$ is a morphism in ${{\rm Vect}}^K$.
Note that when Q is a simply-laced Dynkin quiver, the sums in Lemmas \[mult\] and \[comult\] are finite. Both of these lemmas are true for a similar reason. The important fact to note here is that for a fixed $M$, $N$, and $E$ in either sum, there is a short exact sequence $0{\rightarrow}N{\rightarrow}E{\rightarrow}M{\rightarrow}0$. So by the definition of the Grothendieck group $K$, we have that their images obey the identity $[M]+[N] = [E]$. These images determine the grade of the corresponding graded piece they sit in, so the grade is clearly preserved by both maps.
Now we can focus on the compatibility of the new maps, which was the main reason for constructing this new category. We first need an important identity for the multiplication and comultiplication known as Green’s Formula.
\[GF\][(Green’s Formula).]{} For all $M$, $N$, $X$, and $Y$ in ${{\rm Rep}}(Q)$ we have the identity: $$\sum_{[E]}{\frac{P^E_{MN}P^E_{XY}}{{{\rm aut}}(E)}}=\sum_{[A],[B],[C],[D]}{q^{-\langle A,D\rangle}\frac{P^{M}_{AB}P^{N}_{CD}P^{X}_{AC}P^{Y}_{BD}}{{{\rm aut}}(A){{\rm aut}}(B){{\rm aut}}(C){{\rm aut}}(D)}}.$$
The proof of Green’s formula is quite complex, and involves a large amount of homological algebra. It was first presented by Ringel [@RingelGreen], and also appears in [@Hubery] and [@Schiffman] with good explanations. What we are interested in is the consequence of Green’s formula.
We observe in Green’s formula the presence of our braiding coefficient $q^{-\langle A,D \rangle}$. It is important to note that this coefficient depends on what some might view as the “outside” objects $A$ and $D$, and not the “inside” objects $B$ and $C$. We deal with this by using a different comultiplication than the one usually described in the literature [@Hubery; @Schiffman]. In fact, in the category ${{\rm Vect}}$ our chosen comultiplication is the opposite of the standard choice.
In the category ${{\rm Vect}}^K$ the multiplication $m$ and comultiplication $\Delta$ satisfy the bialgebra condition, and thus ${\mathcal{H}}$ is a bialgebra object in ${{\rm Vect}}^K$.
All the hard work for this proof was done in proving Green’s Formula. We now just need to check that Green’s Formula gives us the bialgebra compatibility. First we will multiply two objects, then comultiply the result to get: $$\begin{array}{rl}
\Delta([M]\cdot[N]) & = \displaystyle{\sum_{[E]} \frac{P^E_{MN}}{{{\rm aut}}(M){{\rm aut}}(N)}\Delta([E])} \\
& = \displaystyle{\sum_{[X],[Y]}\sum_{[E]} \frac{P^E_{MN}P^E_{XY}}{{{\rm aut}}(M){{\rm aut}}(N){{\rm aut}}(E)}[Y]{\otimes }[X]}\\
\end{array}$$ On the other hand, if we first comultiply each object, then multiply the resulting tensor products we have: $$\Delta([M])\cdot \Delta([N]) = \sum_{[A],[B],[C],[D]} \frac{P^M_{AB}P^N_{CD}}{{{\rm aut}}(M){{\rm aut}}(N)} ([B]{\otimes }[A])\cdot ([D]{\otimes }[C])$$ To continue, we need to remember the in our category ${{\rm Vect}}^K$ the braiding is non-trivial. This means that if we want to extend the multiplication on $\mathcal{H}$ to $\mathcal{H}{\otimes }\mathcal{H}$ we must include the braiding coefficient. Specifically, we get the formula: $$([B]{\otimes }[A])\cdot ([D]{\otimes }[C]) = q^{-\langle A,D\rangle} [B]\cdot[D]{\otimes }[A]\cdot[C]$$ When substituted above, this yields: $$\begin{array}{c}
\displaystyle{\sum_{[A],[B],[C],[D]} \frac{P^M_{AB}P^N_{CD}}{{{\rm aut}}(M){{\rm aut}}(N)} ([B]{\otimes }[A])\cdot ([D]{\otimes }[C])}\\
= \displaystyle{\sum_{[A],[B],[C],[D]} q^{-\langle A,D\rangle}\frac{P^M_{AB}P^N_{CD}}{{{\rm aut}}(M){{\rm aut}}(N)} [B]\cdot[D]{\otimes }[A]\cdot[C]}\\
= \displaystyle{\sum_{[X],[Y]}\sum_{[A],[B],[C],[D]}\frac{q^{-\langle A,D\rangle} P^{M}_{AB}P^{N}_{CD}P^{X}_{AC}P^{Y}_{BD}}{{{\rm aut}}(M){{\rm aut}}(N){{\rm aut}}(A){{\rm aut}}(B){{\rm aut}}(C){{\rm aut}}(D)}[Y]{\otimes }[X]}\\
\end{array}$$ Thus, Green’s formula give the equality of the two sides.
For completeness, we will also define an antipode for this bialgebra object to make it a Hopf object. This map is also a morphism in ${{\rm Vect}}^K$ since it clearly preserves the grading.
The map $S:{\mathcal{H}}{\rightarrow}{\mathcal{H}}$ defined on generators by: $$S([M])=-[M]$$ is a $K$-grade preserving linear operator, and is an antipode for ${\mathcal{H}}$. Thus ${\mathcal{H}}$ is a Hopf algebra object in ${{\rm Vect}}^K$.
It is possible to generalize these results to other abelian categories, provided they obey the same finiteness properties as ${{\rm Rep}}(Q)$.
Let $\mathcal{A}$ be an abelian, $k$-linear, hereditary category, where $k=\mathbb{F}_q$. Let $K=K_0(\mathcal{A})$ be its Grothendieck group, and suppose $\dim{{\rm Ext}}^i(M,N)$ is finite for all objects $M,N \in\mathcal{A}$. If the sum $$\sum_{[M],[N]}
\frac{P_{MN}^E}{ {{\rm aut}}(E)}\, [N]{\otimes }[M]$$ is finite for all objects $E\in \mathcal{A}$, then the Hall algebra $\mathcal{H}(\mathcal{A})$ is a Hopf object in ${{\rm Vect}}^K$.
Examining the proof of Theorems \[BMC\] and \[HO\], we see these are the conditions that we need to generalize the result from the case $\mathcal{A} = {{\rm Rep}}(Q)$ to other abelian categories. Specifically, we need hereditary to prove Green’s Theorem.
[10]{}
P. Gabriel, Unzerlebare Darstellungen I, *Manuscr. Math. * [**6**]{} (1972), 71–103.
A. W. Hubery, Ringel–Hall algebras, lecture notes available at [$\langle$http://www.maths.leeds.ac.uk/ ahubery/RHAlgs.html$\rangle$](http://www.maths.leeds.ac.uk/~ahubery/RHAlgs.html).
A. Joyal and R. Street, Braided tensor categories, *Adv. Math.* [**102**]{} (1993), 20–78
M. Kapranov, Eisenstein Series and Quantum Affine Algebras, *J. Math. Sci. (New York)* [**84**]{} (1997), 1311–1360.
S. Mac Lane, [*Categories for the Working Mathematician*]{}, Springer, Berlin, 1998.
S. Majid, Braided Groups, [*J. P. App. Alg.*]{} [**86**]{} (1993), 187–221.
S. Majid, Algebras and Hopf algebras in braided categories, in *Advances in Hopf Algebras (Chicago, IL, 1992)*, Lecture Notes in Pure and Appl. Math. **158**, Dekker, New York, 1994, pp. 55–105.
C. Ringel, Hall algebras and quantum groups, *Invent. Math. * [**101**]{} (1990), 583–591.
C. Ringel, Hall algebras revisited, *Israel Math. Conf. Proc. *[**7**]{} (1993), 171–176. Also available at [$\langle$http://www.mathematik.uni-bielefeld.de/$\sim$ringel/opus/hall-rev.pdf$\rangle$](http://www.mathematik.uni-bielefeld.de/~ringel/opus/hall-rev.pdf).
C. Ringel, Green’s theorem on Hall algebras, in *Representation Theory of Algebras and Related Topics (Mexico City, 1994)*, CMS Conf. Proc. [**19**]{}, Amer. Math. Soc., Providence, RI, 1996, pp. 185–225.
O. Schiffman, Lectures on Hall algebras, available as [arXiv:math/0611617](http://arxiv.org/abs/math/0611617).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present a first-principle computation of the mass distribution of jets which have undergone the grooming procedure known as [Soft Drop]{}. This calculation includes the resummation of the large logarithms of the jet mass over its transverse momentum, up to next-to-logarithmic accuracy, matched to exact fixed-order results at next-to-leading order. We also include non-perturbative corrections obtained from Monte-Carlo simulations and discuss analytic expressions for hadronisation and Underlying Event effects.'
author:
- Simone Marzani
- Lais Schunk
- Gregory Soyez
date: December 2017
title: 'The jet mass distribution after [Soft Drop]{}'
---
#### Introduction.
The study of jets at the Large Hadron Collider (LHC) has recently taken a new turn with new substructure observables [@mMDT; @SD] amenable to precise theory calculations [@flsy-letter; @flsy; @our_mmdt], including genuine theory uncertainty bands, and corresponding experimental measurements from both the CMS [@cms_mmdt] and ATLAS [@atlas-sd] collaborations. The substructure techniques we concentrate on are usually referred to as [*grooming*]{} and they aim to reduce sensitivity to non-perturbative corrections and pileup.
A first series of studies has focused on the jet mass after applying the (modified) MassDrop Tagger (mMDT) [@MDT; @mMDT] in dijet events, as measured by the CMS collaboration [@cms_mmdt]. On the theory side, the description of this observable requires to match a resummed calculation, important in the small-mass region, to fixed-order results, which are important for large masses. The former are obtained analytically, including to all orders terms enhanced by the large logarithms of $p_t^2/m^2$ with $p_t$ the jet transverse momentum and $m$ the (groomed) jet mass. The latter is obtained from fixed-order Monte-Carlo simulations. To date, two theory calculations are available: a SCET-based next-to-leading logarithmic (NLL) resummation in the small ${z_{\text{cut}}}$ limit, matched to leading order (LO) results [@flsy], and our previous study matching a leading logarithmic resummation, including finite (but small) ${z_{\text{cut}}}$ effects, to next-to-leading order results [@our_mmdt]. Comparing both predictions, we see a small NLL effect at small mass and non-negligible NLO corrections at large mass.
The goal of the present letter is to extend our mMDT study from Ref. [@our_mmdt] to the case of [Soft Drop]{} [@SD], i.e. allowing for a non-zero value of the angular exponent $\beta$. When $\beta \neq 0$, the logarithmic counting differs from the mMDT case, essentially because [Soft Drop]{}retains soft-collinear radiation, which is always groomed away by mMDT. In this case, the SCET-based calculation from Ref. [@flsy] reaches NNLL accuracy and it is matched, in the dijet case, to LO fixed-order results. Here, we present the results of a NLL resummation matched to NLO fixed-order accuracy.
After a brief review of the [Soft Drop]{}procedure, we will present our results first in the resummation region, then matched to fixed-order. We then provide an analytic estimate of non-perturbative corrections, extending to the [Soft Drop]{}case the analytic results obtained in Ref. [@mMDT] for the mMDT. We conclude by providing and discussing our final predictions, including the theory uncertainty bands. These have already been compared to experimental data in [@atlas-sd], where a good agreement was found, especially in the perturbative region.
#### [Soft Drop]{}.
For a given jet, the [Soft Drop]{}procedure first re-clusters the constituents of the jet with the Cambridge/Aachen algorithm [@cam] into a single jet $j$. Starting from $j$, it then applies the following iterative procedure:
1. undo the last clustering step $j\to j_1,j_2$, with $p_{t1}>p_{t2}$.\[sd-step1\]
2. stop the procedure if the [Soft Drop]{}condition is met: $$\label{eq:sd-cdt}
\frac{\text{min}(p_{t1},p_{t2})}{p_{t1}+p_{t2}}
> {z_{\text{cut}}}\Big(\frac{\theta_{12}}{R}\Big)^\beta,$$ where ${z_{\text{cut}}}$ and $\beta$ are free parameters, $\theta_{12}^2=\Delta y_{12}^2+\Delta \phi_{12}^2$ and $R$ the original jet radius.
3. otherwise, set $j=j_1$ and go back to \[sd-step1\], or stop if $j_1$ has no further substructure.
The limit $\beta\to 0$ corresponds to the mMDT.
#### NLL resummation.
We consider the cumulative cross-section for the ratio $m^2/(p_tR)^2$ to be smaller than some value $\rho$, integrated over the ${\cal{O}}(\alpha_s^2)$ matrix element for the Born-level production of 2 jets, in a given $p_t$ bin: $$\label{eq:nll-resum}
\Sigma_{\text{NLL}}(\rho;p_{t1},p_{t2})=\int_{p_{t1}}^{p_{t2}} \!\!dp_t
\sum_i \frac{d\sigma_{\text{jet,LO}}^{(i)}}{dp_t} \frac{e^{-R_i(\rho)-\gamma_ER_i'(\rho)}}{\Gamma(1+R_i'(\rho))},$$ where we have separated contributions from different flavour channels, $R_i'$ is the derivative of $R_i$ wrt $\log(1/\rho)$ and the radiator $R_i$ is given by $$\begin{aligned}
&R_i(\rho) = \frac{C_i}{2\pi\alpha_s\beta_0^2}
\bigg\{
\Big[
W(1-\lambda_B)-\frac{W(1-\lambda_c)}{1+\beta}-2W(1-\lambda_1)\nonumber\\
&\quad +\frac{2+\beta}{1+\beta}W(1-\lambda_2)
\Big]
-\frac{\alpha_s K}{2\pi}
\Big[\log(1-\lambda_B)-\frac{\log(1-\lambda_c)}{1+\beta}\nonumber\\
&\quad+\frac{2+\beta}{1+\beta}\log(1-\lambda_2) -2\log(1-\lambda_1)
\Big] +\frac{\alpha_s \beta_1}{\beta_0}
\Big[V(1-\lambda_B)\nonumber\\
&\quad-\frac{V(1-\lambda_c)}{1+\beta}-2V(1-\lambda_1)
+\frac{2+\beta}{1+\beta}V(1-\lambda_2)
\Big]
\bigg\}\,,\label{eq:radiator}\end{aligned}$$ where $$\begin{aligned}
\lambda_c = 2\alpha_s\beta_0\log(1/{z_{\text{cut}}}), &&
\lambda_\rho = 2\alpha_s\beta_0\log(1/\rho), \\
\lambda_1 = \frac{\lambda_\rho+\lambda_B}{2}, &&
\lambda_2 = \frac{\lambda_c+(1+\beta)\lambda_\rho}{2+\beta},\end{aligned}$$ and $\lambda_B=2\alpha_S\beta_0B_i$ appears due to hard-collinear splittings, and $W(x)=x\log(x)$, $V(x)=\frac{1}{2}\log^2(x)+\log(x)$.
Note that $\alpha_s$ is calculated using the exact two-loop running coupling, at the scale $p_tR$, and, in order to reach NLL accuracy, it is evaluated in the CMW scheme [@cmw]. Furthermore, compared to the original results [@SD], the hard-collinear contributions have been expressed as corrections to double-logarithm arguments. In practice, this is equivalent to replacing $P_i(z)\to (2C_i/z)\Theta(z<e^{B_i})$. This introduces unwanted NNLL terms but has the advantage to give well-defined and positive resummed distributions which, in turn, makes the matching to fix order easier.
To avoid any potential issue related to the Landau pole, appearing in a region anyway dominated by hadronisation, we have frozen the coupling at a scale $\mu_{\text{fr}}=1$ GeV. Corresponding expressions can be found e.g. in Ref. [@shapes-paper].
#### Matching to NLO.
The [Soft Drop]{}mass distributions for the dijet processes can be calculated at fixed order at ${\cal{O}}(\alpha_s^4)$, i.e. up to NLO accuracy. This is available for example using the NLOJet++ [@nlojet] generator to simulate $2\to 3$ events at LO and NLO. Jets are then clustered with the anti-$k_t$ algorithm [@antikt] as implemented in FastJet-3.2.2 [@fastjet]. In what follows, we have used the CT14 PDF set [@CT14].
NLO mass distributions need to be matched to our NLL resummed results. For this, the LO jet mass distribution needs to be separated in flavour channels, while the flavour separation of the NLO jet mass distribution is instead subleading. At ${\cal{O}}(\alpha_s^3)$ a jet has at most two constituent and the only case where the flavour is ambiguous is when a jet is made of two quarks (or a quark and an anti-quark of different flavours). We (arbitrarily) treat this as a quark jet, an approximation which is valid at our accuracy. To keep the required flavour information in NLOJet++, we have used the patch introduced in Ref. [@bsz-shapes].
To avoid artefacts at large mass, the endpoint of the resummed calculation is matched to the endpoint of the perturbative distribution by replacing $$\log\Big(\frac{1}{\rho}\Big) \to
\log\Big(\frac{1}{\rho}-\frac{1}{\rho_{\text{max},i}}+e^{-{B_q}}\Big)$$ in the resummed results [@match-endpoint]. The endpoints of the LO and NLO distributions are found to be (see Appendix B of Ref. [@our_mmdt]) $\rho_{\text{max,LO}} \approx 0.279303$ and $\rho_{\text{max,NLO}} \approx 0.44974$, for $R=0.8$.
{width="48.00000%"} {width="48.00000%"}
Finally, the matching between NLL and NLO results in each $p_t$ bin can be done using log-R matching given by [@bsz-shapes] $$\begin{aligned}
\label{eq:matched}
&\Sigma_{\text{NLL+NLO}}(\rho) =
\Bigg[ \sum_i
\Sigma_{\text{NLL}}^{(i)}
\exp\bigg(\frac{\Sigma_{\text{LO}}^{(i)}-\Sigma_{\text{NLL,LO}}^{(i)}}
{\sigma_{\text{jet,LO}}^{(i)}}
\bigg)
\Bigg]\\
&\times \exp
\Bigg(
\frac{\bar\Sigma_{\text{NLO}}-\Sigma_{\text{NLL,NLO}}}
{\sigma_{\text{jet,LO}}}
-\sum_i\frac{(\Sigma_{\text{LO}}^{(i)})^2-(\Sigma_{\text{NLL,LO}}^{(i)})^2}{\sigma_{\text{jet,LO}}^{(i)}\sigma_{\text{jet,LO}}}\Bigg).\nonumber\end{aligned}$$ In this expression, $\Sigma_{\text{NLL}}^{(i)}$ is given by Eq. (\[eq:nll-resum\]), trivially split in flavour channels. $\Sigma_{\text{NLL,LO}}^{(i)}$ and $\Sigma_{\text{NLL,NLO}}$ (summed over flavour channels) are the expansion of $\Sigma_{\text{NLL}}^{(i)}$ to LO (${\cal{O}}(\alpha_s^3)$) and NLO (${\cal{O}}(\alpha_s^4)$), respectively. For the fixed-order part $$\begin{aligned}
\Sigma_{\text{LO}}^{(i)}
&=-\int_\rho^1 d\rho'\,\frac{d\sigma_{\text{mass,LO}}^{(i)}}{d\rho'}
+ \sigma_{\text{jet,NLO}}^{(i)},\\
\bar\Sigma_{\text{NLO}}
&=-\int_\rho^1 d\rho'\,\frac{d\sigma_{\text{mass,NLO}}}{d\rho'},\end{aligned}$$ where $d\sigma_{\text{mass,(N)LO}}/d\rho$ denotes the mass distribution at (N)LO as obtained from NLOJet++ and $\sigma_{\text{jet,(N)LO}}$ the (N)LO correction to the inclusive jet cross-section in the $p_t$ bin under consideration. These expressions also require the inclusive jet cross-section, both at LO and NLO, to be split in flavour channels. This is done as for the 3-jet LO distribution above using the flavour-aware NLOJet++ version used in [@bsz-shapes]. Alternatively, we have also used the ($R$-)matching scheme given by Eq. (3.28) of [@bsz-shapes].
From Eq. (\[eq:matched\]) it is trivial to obtain differential distributions in bins of $\rho$. Normalised distributions can then be obtained by dividing the result by the NLO inclusive jet cross-section $\sigma_{\text{jet,LO}}+\sigma_{\text{jet,NLO}}$.[^1]
The uncertainties on the distributions come from four sources: renormalisation and factorisation scales, resummation uncertainty and matching uncertainty. The first two are estimated using the 7-point rule [@7point-rule]. The resummation uncertainties are obtained by varying $\rho$ in Eqs. (\[eq:nll-resum\]) and (\[eq:radiator\]) between $\rho/2$ and $\rho$, introducing the appropriate correction — $\pm \log(2)R'$ in the exponent in (\[eq:nll-resum\]) — to maintain NLL accuracy. The matching uncertainty is estimated by considering both the log-$R$ and $R$ matching schemes. We take the central value from the central scale choice and the uncertainty from the envelope of the 11 scale variations.[^2]
{width="48.00000%"}{width="48.00000%"}
#### Non-perturbative corrections.
Power corrections induced by non-perturbative (NP) effects can be estimated for [Soft Drop]{}using the same approach as the equivalent calculation for mMDT presented in Section 8.3.3 of Ref. [@mMDT]. We have to take into account two effects: (i) the mass of the SD jet will be affected by NP corrections, (ii) NP effects can shift the momentum of the subjets and alter the SD condition.
First, the mass shift can be written as (see [@Dasgupta:2007wa]) $\delta m^2 = C_R{\ensuremath{\Lambda_{\text{hadr}}}}p_t R_{\text{eff}}$, where $R_{\text{eff}}$ is the effective jet radius after grooming, i.e. for a mass $m$ and subjets passing the [Soft Drop]{}condition with a momentum fraction $z$, $R_{\text{eff}}=m/(p_t\sqrt{z(1-z)})$. Following the same steps as in Ref. [@mMDT] we obtain[^3] $$\label{eq:hadr-mshift}
\frac{d\sigma}{dm}\bigg|_{\text{hadr}}^{(m\text{ shift})}
= \frac{d\sigma}{dm}\bigg|_{\text{pert}}
\bigg(1+\frac{C_R{\ensuremath{\Lambda_{\text{hadr}}}}}{m}\:\frac{{z_{\text{SD}}}^{-1/2}-\Delta_i}{{L_{\text{SD}}}+B_i}\bigg),$$ with ${z_{\text{SD}}}={z_{\text{cut}}}^{\frac{2}{2+\beta}}\big(\frac{m}{p_tR}\big)^{\frac{2\beta}{2+\beta}}$, ${L_{\text{SD}}}=\log(1/{z_{\text{SD}}})$ and $$\Delta_q=\frac{3\pi}{8}\quad\text{ and }\quad\Delta_q=\frac{(15C_A-6n_fT_R)\pi}{32\,C_A}.$$
Then, hadronisation will shift the momentum of the softer subjet by an average $\delta p_t=-C_A{\ensuremath{\Lambda_{\text{hadr}}}}/R_{\text{eff}}$, where we have taken into account that the softer subjet typically corresponds to a gluon emission. This means that emissions which were perturbatively passing the [Soft Drop]{}condition, with ${z_{\text{SD}}}<z<{z_{\text{SD}}}-\delta p_t/p_t$, will fail the [Soft Drop]{}condition after hadronisation, leading to a reduction of the cross-section $$\label{eq:hadr-ptshift}
\frac{d\sigma}{dm}\bigg|_{\text{hadr}}^{(p_t\text{ shift})}
= \frac{d\sigma}{dm}\bigg|_{\text{pert}}
\bigg(1-\frac{C_A{\ensuremath{\Lambda_{\text{hadr}}}}}{m}\:\frac{{z_{\text{SD}}}^{-1/2}}{{L_{\text{SD}}}+B_i}\bigg).$$ The final hadronisation correction includes both (\[eq:hadr-mshift\]) and (\[eq:hadr-ptshift\]). Both terms are proportional to $\frac{\Lambda_{\text{hadr}}}{p_t}\big(\frac{p_t}{m}\big)^{\frac{2+2\beta}{2+\beta}}$, which increases with $\beta$ and has the appropriate limits for $\beta\to\infty$ and $\beta\to 0$.
A similar calculation can be carried out for the Underlying Event (UE) contamination. In this case we have $\delta
p_t=\Lambda_{\text{UE}}\pi R_{\text{eff}}^2$ and $\delta
m^2=\frac{1}{2}\Lambda_{\text{UE}}p_tR_{\text{eff}}^4$. Following the same steps as above, we find $$\label{eq:UE-mshift}
\frac{d\sigma}{dm}\bigg|_{\text{UE}}
= \frac{d\sigma}{dm}\bigg|_{\text{pert}}
\bigg(1+\frac{\Lambda_{\text{UE}} m^2}{p_t^3R^3}\:\frac{{z_{\text{SD}}}^{-2}(1-f_{m,i})}{{L_{\text{SD}}}+B_i}\bigg),$$ where the $1$ in the numerator corresponds to the $p_t$ shift and the $f_{m,i}$ term corresponds to mass-shift effects, with $$\begin{aligned}
f_{m,q} &= \frac{1+3{z_{\text{SD}}}+2{z_{\text{SD}}}^2(3{L_{\text{SD}}}-2)}{4},\\
f_{m,g} &= \frac{1+2{z_{\text{SD}}}+3{z_{\text{SD}}}^2(2{L_{\text{SD}}}-1)}{4} + \frac{n_fT_R}{C_A}{z_{\text{SD}}}(1-{z_{\text{SD}}}).\nonumber\end{aligned}$$ This time, both sources of corrections give an effect proportional to $\frac{\Lambda_{\text{UE}}}{p_t}\big(\frac{p_t}{m}\big)^{\frac{2\beta-4}{2+\beta}}$, which increase with $\beta$ and has the expected $\Lambda_{\text{UE}}p_t/m^2$ behaviour in the limit $\beta\to \infty$.
In Fig. \[fig:np-ratio\], we compare our analytic findings (dashed lines) with the Monte-Carlo simulations, obtained with 8.223 [@pythia8] (Monash 13 [@tuneM13] tune, solid lines). We consider both hadronisation corrections (left) and UE effects (right), for a range of $\beta$ values. UE effects are seen to be much smaller than hadronisation corrections. In the region where $\Lambda_{\text{hadr,UE}}\ll m \ll p_t$, our analytic calculation captures the main features observed in the simulations, including the increase with $\beta$ and the global trend in $\rho$. At smaller mass, Pythia simulations exhibit a peak in the hadronisation corrections which is beyond the scope of our power-correction calculation.
Even if the above analytic approach to estimating NP effects is helpful for a qualitative understanding, it is unclear how reliable it would be for phenomenology. For example, hadron masses, which are neglected here, would have an additional effect, even at large mass. Thus, the analytic estimates can, at best, be trusted up to a fudge factor and analytic results can not be trusted at small mass (see also [@Dasgupta:2016bnd]).
As for our mMDT calculation [@our_mmdt], for our final predictions, we have therefore decided to estimate NP corrections using different Monte-Carlo simulations: 6.521 [@herwig] with the tune AUET2 [@tuneAUET2], 6.428 [@pythia6] with the Z2 [@tuneZ2] and Perugia 2011 [@tuneP2011] tunes, and 8.223 [@pythia8] with the 4C [@tune4C] and Monash 13 [@tuneM13] tunes. For each Monte-Carlo, we compute the ratio between the full simulation and the parton level. The average result is taken as the average NP correction, and the envelope as the uncertainty which is added in quadrature to the perturbative uncertainty.
#### Final predictions and conclusions.
Our final predictions, are presented for $\beta=1$ (left) and $\beta=2$ (right) in Fig \[fig:final-prediction\]. To highlight our key observations, we present our final results at NLL matched to NLO and including NP corrections (labelled NLL+NLO+NP), as well as pure perturbative results (NLL+NLO) and results obtained when matching to LO only (NLL+LO). The most striking feature that we observe is that matching to NLO not only affects quite significantly the central value of our prediction, but also significantly reduces the uncertainty across the entire spectrum.
Then, we see that NP corrections remain small over a large part of the spectrum, although they start being sizeable at larger mass when the angular exponent $\beta$ increases. The fact that [Soft Drop]{}observables can be computed precisely in perturbative QCD, with small NP corrections, makes them interesting for further phenomenological studies, including other observables like angularities or attempts to extract the strong coupling constant from fits to the data.
Finally, we note that these predictions have recently been compared to experimental results obtained by the ATLAS collaboration in Ref. [@atlas-sd]. A good overall agreement between data and theory is observed, especially in the region where NP corrections are small.
We thank Ben Nachman and Gavin Salam for many useful discussions. SM and LS would like to thank the IPhT Saclay for hospitality during the course of this project. GS’s work is supported in part by the French Agence Nationale de la Recherche, under grant ANR-15-CE31-0016 and by the ERC Advanced Grant Higgs@LHC (No. 321133).
[99]{}
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[^1]: Note that this normalisation procedure gives consistent results when computing the uncertainties on the matched distributions.
[^2]: Seven factorisation and renormalisation scales, two resummation scales and two matching schemes.
[^3]: Although, instead of averaging $R_{\text{eff}}$ over $z$, we have kept explicit the $z$ dependence of $R_{\text{eff}}$ and averaged the final correction over $z$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We show that the observed scatter in the relations between the mass of supermassive black holes (SMBHs), $M_{\rm bh}$, and the velocity dispersion $\sigma_{\rm sph}$ or mass $M_{\rm sph}$ of their host spheroid, place interesting constraints on the process that regulates SMBH growth in galaxies. When combined with the observed properties of early-type SDSS galaxies, the observed intrinsic scatters imply that SMBH growth is regulated by the spheroid velocity dispersion rather than its mass. The $M_{\rm bh}$–$M_{\rm sph}$ relation is therefore a by-product of a more fundamental $M_{\rm bh}$–$\sigma_{\rm sph}$ relation. We construct a theoretical model for the scatter among baryon modified dark matter halo profiles, out of which we generate a population of spheroid hosts and show that these naturally lead to a relation between effective radius and velocity dispersion of the form $R_{\rm sph}\approx 6{\rm kpc}(\sigma_{\rm
sph}/200\mbox{km}\,\mbox{s}^{-1})^{1.5}$ with a scatter of $\sim$0.2dex, in agreement with the corresponding projection of the fundamental plane for early type galaxies in SDSS. At the redshift of formation, our model predicts the minimum scatter that SMBHs can have at fixed velocity dispersion or spheroid mass under different formation scenarios. We also estimate the additional scatter that is introduced into these relations through collisionless mergers of purely stellar spheroids at $z<1$. We find that the observed scatter in the $M_{\rm bh}$–$\sigma_{\rm sph}$ and $M_{\rm bh}$–$M_{\rm sph}$ relations preclude the properties of dark matter halos from being the governing factor in SMBH growth. The apparent relation between halo and SMBH mass is merely a reflection of the fact that massive halos tend to host massive stellar spheroids (albeit with a large scatter owing to the variance in formation histories). Finally, we show that SMBH growth governed by the properties of the host spheroid can lead to the observed values of scatter in the $M_{\rm bh}$–$\sigma_{\rm sph}$ and $M_{\rm bh}$–$M_{\rm sph}$ relations, only if the SMBH growth is limited by momentum or energy feedback [*over the dynamical time*]{} of the host spheroid.
author:
- 'J. Stuart B. Wyithe, Abraham Loeb'
title: 'Constraints on the Process that Regulates the Growth of Supermassive Black Holes Based on the Intrinsic Scatter in the $M_{\rm bh}$–$\sigma$ Relation'
---
Introduction
============
Supermassive black holes (SMBHs) are a ubiquitous constituent of spheroids in nearby galaxies (e.g. Kormandy & Richstone 1995). A decade ago it became apparent that the masses of SMBHs correlate with the luminosity of the host spheroid (Kormandy & Richstone 1995). Subsequently, other correlations with substantially less intrinsic scatter have been discovered. These include correlations between the SMBH mass ($M_{\rm bh}$), and the mass, $M_{\rm sph}$ (Magorrian et al. 1998; Haering & Rix 2004), stellar velocity dispersion, $\sigma_{\rm sph}$ (Ferrarese & Merritt 2000; Gebhardt et al. 2000), and concentration (Graham et al. 2002) of its host spheroid. The tightest relation, with intrinsic scatter of $\delta\sim0.275\pm0.05$dex, appears to be between SMBH mass and velocity dispersion (Tremaine et al. 2002; Wyithe 2005).
These relations must hold important clues about the astrophysics that regulates the growth of a SMBH and its impact on galaxy formation. While much attention was dedicated towards interpreting the power-law slope of the $M_{\rm bh}$–$\sigma_{\rm sph}$ relation via a slew of analytic, semi-analytic and numerical attempts to reproduce it (e.g. Silk & Rees 1998; Wyithe & Loeb 2003; King 2003; Miralda-Escude & Kollmeier 2005; Adams et al. 2003; Sazonov et al. 2005; Di Matteo et al. 2005), little attention has been directed towards interpreting the constraints that its [*intrinsic scatter*]{} might place on our understanding of SMBH growth (Robertson et al. 2005). Moreover, agreement between data and theory is a necessary but not sufficient condition. A model that reproduces the observations is not necessarily unique. The various successful attempts to model the quasar luminosity function assuming different physical models attest to this fact. Our goals in this paper are to use the observed scatter in the $M_{\rm bh}$–$\sigma_{\rm sph}$ and $M_{\rm bh}$–$M_{\rm
sph}$ relations in addition to their power-law slope as a diagnostic of SMBH formation physics, and to constrain a range of possible models of SMBH formation.
Throughout the paper we adopt the set of cosmological parameters determined by the [*Wilkinson Microwave Anisotropy Probe*]{} (WMAP, Spergel et al. 2003), namely mass density parameters of $\Omega_{m}=0.27$ in matter, $\Omega_{b}=0.044$ in baryons, $\Omega_\Lambda=0.73$ in a cosmological constant, and a Hubble constant of $H_0=71~{\rm km\,s^{-1}\,Mpc^{-1}}$.
Intrinsic scatter in the observed $M_{\rm bh}$–$\sigma_{\rm sph}$ and $M_{\rm bh}$–$M_{\rm sph}$ relations
==========================================================================================================
Tremaine et al. (2002) have compiled a list of spheroids with reliable determinations of both SMBH mass and central velocity dispersion (defined as the luminosity weighted dispersion in a slit aperture of half length $R_{\rm sph}$). These SMBHs show a tight correlation between $M_{\rm bh}$ and $\sigma_{\rm sph}$ (Gebhardt et al. 2000; Ferrarese & Merritt 2000). Recently, Wyithe (2005) found that the $M_{\rm
bh}$–$\sigma_{\rm sph}$ relation within the sample of Tremaine et al. (2002) shows evidence for a deviation from a pure power-law behavior, and obtained an a-posteriori probability distribution for the scatter in the relation of $\delta=0.28\pm0.05$. The best-fit log-quadratic $M_{\rm
bh}$–$\sigma_{\rm sph}$ takes the form $$\begin{aligned}
\label{Msig}
\nonumber
\log_{10}(M_{\rm bh})&=&\alpha+\beta\log_{10}\left(\sigma_{\rm sph}/200\mbox{km/s}\right)\\
&+&\beta_2\left[\log_{10}\left(\sigma_{\rm sph}/200\mbox{km/s}\right)\right]^2,\end{aligned}$$ where $\beta=4.2\pm0.37$ corresponds to the traditional power-law relation and $\beta_{2}=1.6\pm1.3$ quantifies the departure of the local SMBH sample from a pure power-law.
A correlation has also been found between the mass of the spheroid component, $M_{\rm sph}$, and the SMBH (Magorrian et al. 1998). The sample compiled by Haering & Rix (2004) (which overlaps predominantly with the Tremaine et al. 2002 sample) was studied by Wyithe (2005) who found that like the $M_{\rm bh}$–$\sigma_{\rm sph}$ relation, the $M_{\rm bh}$–$M_{\rm sph}$ relation may depart from a uniform power-law, and that the scatter in the $M_{\rm bh}$–$M_{\rm sph}$ relation is larger than in the $M_{\rm bh}$–$\sigma_{\rm sph}$ relation by 50% ($\delta_{\rm sph}=0.41\pm0.07$). The best-fit log-quadratic $M_{\rm bh}$–$M_{\rm sph}$ takes the form $$\begin{aligned}
\label{MM}
\nonumber
\log_{10}(M_{\rm bh})&=&\alpha_{\rm sph}+\beta_{\rm sph}\log_{10}\left(M_{\rm sph}/10^{11}M_\odot\right)\\
&+&\beta_{2,{\rm sph}}\left[\log_{10}\left(M_{\rm sph}/10^{11}M_\odot\right)\right]^2,\end{aligned}$$ where $\beta_{\rm sph}=1.15\pm0.18$ and $\beta_{2,{\rm sph}}=1.12\pm0.14$.
In this paper we use the values of intrinsic scatter ($\delta$ and $\delta_{\rm sph}$) in the local SMBH sample to place constraints on the process that regulates SMBH growth in galaxies.
Which Relation is More Fundamental: $M_{\rm bh}$–$\sigma_{\rm sph}$ or $M_{\rm bh}$–$M_{\rm sph}$?
==================================================================================================
The SMBH mass is observed to correlate tightly with both $\sigma_{\rm sph}$ (Tremaine et al. 2002) and $M_{\rm sph}$ (Haering & Rix 2004), implying that SMBH growth is regulated by properties of the spheroid. It is therefore natural to ask [*which parameter among these two is responsible for setting the SMBH mass?*]{} We address this question empirically without specifying a mechanism for the SMBH growth.
Let us suppose first that the $M_{\rm bh}$–$M_{\rm sph}$ relation is the fundamental one. There would still be a relation between $M_{\rm bh}$ and $\sigma_{\rm sph}$ since $M_{\rm sph}$ and $\sigma_{\rm sph}$ are related. Bernardi et al. (2003) have compiled values of $R_{\rm sph}$, and $\sigma_{\rm sph}$ for the sample of early-type galaxies in the SDSS. We can generate a sub-sample of galaxies within a narrow range of $\sigma_{\rm
sph}$ and find the dynamical mass $M_{\rm sph}=V_{\rm sph}^2R_{\rm
sph}/G=2\sigma_{\rm sph}^2R_{\rm sph}/G$ for each, where $V_{\rm sph}$ is the circular velocity at $R_{\rm sph}$ and we have assumed an isotropic velocity dispersion $\sigma_{\rm sph}=V_{\rm sph}/\sqrt{2}$. We can then use the observed relation $M_{\rm bh}\propto M_{\rm sph}$ with its intrinsic scatter of $\delta_{\rm sph}=0.41\pm0.07$, to find the corresponding scatter in the $M_{\rm bh}$–$\sigma_{\rm sph}$ relation. The resulting distributions of residuals in the SMBH mass are plotted on the left panel of Figure \[fig1\], together with the 1-sigma range in the observed scatter (grey region) of the $M_{\rm bh}$–$\sigma_{\rm sph}$ relation. If the $M_{\rm bh}$–$M_{\rm sph}$ relation were fundamental, then the scatter in the $M_{\rm bh}$–$\sigma_{\rm sph}$ relation would have been $\delta=0.46$dex, well in excess of the observed value $\delta=0.275\pm0.05$. This implies that the $M_{\rm bh}$–$M_{\rm sph}$ relation is not fundamental.
On the other hand if we suppose that it is the $M_{\rm bh}$–$\sigma_{\rm
sph}$ relation that is fundamental, then there would still be a relation between $M_{\rm bh}$ and $M_{\rm sph}$. We generate a sub-sample of galaxies from Bernardi et al. (2003) with $M_{\rm sph}=2\sigma_{\rm
sph}^2R_{\rm sph}/G$ in a narrow range near $M_{\rm
sph}=10^{11}M_\odot$. We can then use the observed $M_{\rm
bh}$–$\sigma_{\rm sph}$ relation (with intrinsic scatter $\delta=0.275\pm0.05$) to find the corresponding scatter in the $M_{\rm
bh}$–$M_{\rm sph}$ relation. The resulting distributions of residuals in SMBH mass are plotted in the right hand panel of Figure \[fig1\], together with the 1-sigma range in the observed scatter (grey region) of the $M_{\rm bh}$–$M_{\rm sph}$ relation. If the $M_{\rm bh}$–$\sigma_{\rm
sph}$ relation is fundamental, then the scatter in the $M_{\rm
bh}$–$M_{\rm sph}$ relation should be $\delta_{\rm sph}=0.40$dex, in good agreement with the observed value $\delta_{\rm sph}=0.41\pm0.07$. This implies that the $M_{\rm bh}$–$\sigma_{\rm sph}$ relation is more fundamental than the $M_{\rm bh}$–$M_{\rm sph}$ relation, with the latter being an incidental consequence of the correlation between $\sigma_{\rm
sph}$ and $M_{\rm sph}$.
Intrinsic Scatter in the $M_{\rm bh}$–$\sigma_{\rm sph}$ Relation and Models for SMBH Evolution
===============================================================================================
In the previous section we showed that the observed scatter in the $M_{\rm
bh}$–$\sigma_{\rm sph}$ and $M_{\rm bh}$–$M_{\rm sph}$ relations, when combined with the scatter among spheroid properties implies that the $M_{\rm bh}$–$\sigma_{\rm sph}$ relation is fundamental for SMBH growth, with the $M_{\rm bh}$–$M_{\rm sph}$ relation being incidental due to correlations between spheroid parameters. This in turn implies that if we have a model for the properties of the host spheroid, then we can deduce the mode of SMBH growth by comparing the calculated scatter in the modeled $M_{\rm bh}$–$\sigma_{\rm sph}$ and $M_{\rm bh}$–$M_{\rm sph}$ relations with observations. In this section we constrain the astrophysics of SMBH growth by computing minimum values for the scatter in the $M_{\rm
bh}$–$\sigma_{\rm sph}$ relation using various models for the regulation of SMBH growth.
Rotation curves, Adiabatic cooling and the Fundamental Plane {#fpsec}
------------------------------------------------------------
Our goal is to model the scatter in the $M_{\rm bh}$–$\sigma_{\rm sph}$ and $M_{\rm bh}$–$M_{\rm sph}$ relations. To accomplish this goal we must have a representative model of the host spheroids as well as the scatter in their parameters. This is achieved by computing the rotation curve that results from cooling of baryons inside a dark-matter halo.
The virial radius $R_{\rm vir}$ and velocity $V_{\rm vir}$ for a halo of mass $M_{\rm halo}$ at redshift $z$ are $$\label{Rvir}
R_{\rm vir} = 109\left(\frac{M_{\rm
halo}}{10^{12}M_\odot}\right)^{1/3}\left(\frac{\Omega_m}{\Omega_m^z}\frac{\Delta_c}{18\pi^2}\right)^{-1/3}\left(\frac{1+z}{2}\right)^{-1}~{\rm
kpc},$$ and $$V_{\rm vir} = 200\left(\frac{M_{\rm
halo}}{10^{12}M_\odot}\right)^{1/3}\left(\frac{\Omega_m}{\Omega_m^z}\frac{\Delta_c}{18\pi^2}\right)^{1/6}\left(\frac{1+z}{2}\right)^{1/2}~{\rm
km~{s}^{-1}},$$ where $\Omega_m^z\equiv[1+(\Omega_\Lambda/\Omega_m)(1+z)^{-3}]^{-1}$, $\Delta_c=18\pi^2+82d-39d^2$ and $d=\Omega_m^z-1$ (see Barkana & Loeb 2001 for more details). The relation of the circular velocity at the virial radius to the velocity dispersion at smaller galactic radii requires specification of the mass density profile. In this work we adopt the Navarro, Frenk & White (1997, hereafter NFW) profile for the dark matter. In addition to $V_{\rm vir}$ and $M_{\rm vir}$ the NFW profile is defined by the concentration parameter $c$, which is the ratio between the virial radius and a characteristic break radius, $c\equiv r_{\rm vir}/r_s$. The median value of $c$ depends on halo mass $M_{\rm halo}$ and redshift $z$. Based on N-body simulations, Bullock et al. (2001) (see also Wechsler et al. 2002) have found a median relation for $c$, $$\label{concparam}
c=7.3\left(\frac{M_{\rm halo}}{10^{12}M_\odot}\right)^{-0.13}\left({1+z\over
2}\right)^{-1},$$ with a scatter of $\Delta \log c=0.14$ dex.
The parameters describing the spheroid are its characteristic radius $R_{\rm sph}$ and velocity dispersion $\sigma_{\rm sph}$. In this paper we assume that the gas available to cool in the halo makes most of the mass within the effective radius of the spheroid $M_{\rm sph}$ with a density distribution described by a Hernquist (1990) profile. The cooling of baryons modifies the density profile of a galaxy and hence its rotation curve. In particular, the velocity dispersion in the luminous portion of the galaxy is substantially larger than would be expected from an NFW profile. Gnedin et al. (2004) have studied the contraction of an NFW halo due to baryon cooling in a cosmological context. They find that traditional adiabatic contraction does not provide a good fit. However they introduce an alternative modified contraction model and provide fitting formulae for contracted profiles as a function of halo mass, concentration parameter, final characteristic radius for the baryons, and the cooled baryon fraction. From this contracted profile we can compute the velocity dispersion $\sigma_{\rm sph}$ at the characteristic radius $R_{\rm sph}$. However the cooled profile, and hence the value of $\sigma_{\rm sph}$ obtained from the cooled profile depends on the value adopted for $R_{\rm
sph}$. Defining $m_{\rm d}$ to be the fraction of the total galaxy mass that makes the spheroid (including the cooled baryons), we can break this degeneracy by identifying the spheroid mass $m_{\rm d}M_{\rm halo}$ with the effective virial mass $M_{\rm sph}$ through $2\sigma_{\rm sph}^2R_{\rm
sph}/G= M_{\rm sph}\equiv m_{\rm d}M_{\rm halo}$. With this second relation we are able to solve uniquely for $R_{\rm sph}$ and $\sigma_{\rm sph}$ within a specified dark matter halo (with parameters $M_{\rm halo}$, $c$, $z$, and $m_{\rm d}$). We choose a probability distribution that is flat in the logarithm of $m_{\rm d}$ over the range $0.015\leq m_{\rm
d}\leq0.15$. The upper value corresponds to the case where all baryons in the halo cool to form the mass of the spheroid \[so that $M_{\rm
sph}=(\Omega_{\rm b}/\Omega_{\rm m})M_{\rm halo}$\], and the range under consideration spans values smaller by up to an order of magnitude relative to this extreme case. Using this formalism we generate a sample of model spheroids. This sample is compared first to the observed fundamental plane of early-type galaxies (below), and then used to discuss the scatter in models for the $M_{\rm bh}$–$\sigma_{\rm sph}$ and $M_{\rm bh}$–$M_{\rm sph}$ relations (§ \[models\]).
Figure \[fig2\] shows a scatter plot of $R_{\rm
sph}$ vs $\sigma_{\rm sph}$ for cooled profiles with randomly selected parameters. The distribution of formation redshift is dictated by the merger trees of halos and corresponding fate of pre-existing galaxies within these halos (Eisenstein & Loeb 1996). It is known empirically that star formation is rather minimal between $0<z<1$ for elliptical galaxies (Bernardi et al. 2003). Since we find that the relation between $R_{\rm sph}$ and $\sigma_{\rm
sph}$ is not very sensitive to the precise probability distribution of collapse redshifts, we adopt in this figure a distribution of collapse redshifts that is flat between $1<z<3$. The resulting points show a correlation with significant scatter. For comparison, the relation between the velocity dispersion $\sigma_{\rm sph}$ of an early type stellar system and its effective radius $R_e$ for galaxies from the SDSS (Bernardi et al. 2003) is shown by the grey dots. A fit to this observed correlation has the form $$\label{Resigma}
\log{\left(\frac{R_e}{6\mbox{kpc}}\right)} \approx
1.5\log{\left(\frac{\sigma_{\rm sph}}{200\mbox{km}\,\mbox{s}^{-1}}\right)},$$ which is plotted as the thick solid line in Figure \[fig2\]. The scatter about the observed relation, $\pm0.2$dex, is bracketed by the pair of dashed lines. Also plotted to guide the eye is a linear relation (thin line). On comparison with the results of Bernardi et al. (2003), we see that the formalism reproduces the observed behavior $R_{\rm sph}\propto\sigma_{\rm sph}^{1.5}$. Moreover the size of the predicted scatter about the mean relation is $\sim0.17$dex, in good agreement with the observed value of $\sim0.2$dex. This result is not very sensitive to the (unknown) distribution chosen for $m_{\rm
d}$. The model does not contain any free parameters. Nevertheless, in addition to the scatter an power-law slope of the $R_{\rm
sph}$–$\sigma_{\rm sph}$ relation, our prescription also predicts its normalization.
For completeness we note that the relation described by equation (\[Resigma\]) depends only on dynamical properties and not on the details of star-formation. However elliptical galaxies follow a 3-parameter [*fundamental plane*]{}, with the scatter around the median $R_{\rm sph}$–$\sigma_{\rm sph}$ relation parameterized by surface brightness rather than a 2 parameter relation (Djorgovski & Davis 1987). The scatter around the [*fundamental plane*]{} is substantially smaller than around the $R_{\rm sph}$–$\sigma_{\rm sph}$ relation shown in Figure \[fig2\]. Bernardi et al (2003) find that the fundamental plane has the form $$\label{fp}
R_{\rm sph}\propto\sigma_{\rm sph}^{1.49\pm0.05}I_{\rm sph}^{-0.75\pm0.01},$$ where the surface brightness is $I_{\rm sph}\propto L/R_{\rm sph}^2$, and show that this requires the mass-to-light $\Gamma$ to have a dependence on $R_{\rm sph}$ of the form $\Gamma\propto R_{\rm sph}^{1/3}$. A complete model of the [*fundamental plane*]{} would need to explain this dependence of the mass-to-light ratio, in addition to the relation $R_{\rm
sph}\propto\sigma_{\rm sph}^{1.5}$. However we do not expect SMBH growth to depend on $\Gamma$, and do not attempt to model this (orthogonal) parameter.
Finally, we note that the non-linearity of the relation between $R_{\rm
sph}\propto\sigma_{\rm sph}^{1.5}$ is surprising since it is derived within the context of CDM dark-matter halos for which the relation between the virial radius and velocity is $R_{\rm vir}\propto V_{\rm vir}$ at any given redshift. In the following section we discuss the growth of SMBHs inside host spheroids in light of the observed scatters in the $M_{\rm
bh}$–$\sigma_{\rm sph}$ and $M_{\rm bh}$–$M_{\rm sph}$ relations.
Models for SMBH Evolution {#models}
-------------------------
Three observations of the relation between SMBHs and their hosts have motivated different classes of models to describe the growth and evolution of SMBHs through accretion. First, the observation of the Magorrian et al. (1998) relation that the mass of the SMBH follows the mass of the spheroid has motivated models where the mass of the SMBH accretes a constant fraction of the available gas following a major merger (e.g. Haiman & Loeb 1998; Kauffmann & Haehnelt 2000). We refer to this scenario as case-I. Second, the observation of the $M_{\rm
bh}$–$\sigma_{\rm sph}$ relation implies that the SMBH growth is regulated by the depth of the gravitational potential well of its host spheroid. We refer to this scenario as case-II. Third, there is evidence from the observation of the $M_{\rm bh}$–$\sigma_{\rm sph}$ relation in quasars, that the $M_{\rm bh}$–$\sigma_{\rm sph}$ relation does not vary with redshift (Shields et al. 2003). This motivates a revision of case-II, to include regulation of the SMBH growth over the dynamical time of the system (e.g. Silk & Rees 1998; Haehnelt, Natarajan & Rees 1998; Wyithe & Loeb 2003). We refer to this as case-III. On the other hand if momentum rather than energy is conserved in the transfer of energy in a quasars outflow to the cold galactic gas, then rather than a SMBH regulated by binding energy, we have a SMBH mass regulated by the binding energy divided by the virial velocity (Silk & Rees 1998; King 2003; Begelman 2004; Murray, Quataert, & Thompson 2005). In analogy with cases-II and III, the SMBH mass may be regulated by the total momentum of the surrounding gas, or by the total momentum divided by the systems dynamical time. We refer to these as cases-IV and V, respectively.
We therefore test five hypotheses for each of the $M_{\rm
bh}$–$\sigma_{\rm sph}$ and $M_{\rm bh}$–$M_{\rm sph}$ relations. In the previous section we described a model that reproduces the observed behavior of $R_{\rm sph}\propto\sigma_{\rm sph}^{1.5}$, with a scatter of $\sim0.2$dex. The agreement with the observed projection of the fundamental plane (Bernardi et al. 2003), gives us confidence that the model provides a sufficiently accurate framework within which we can discuss the scatter in the $M_{\rm bh}$–$\sigma_{\rm sph}$ and $M_{\rm bh}$–$M_{\rm sph}$ relations. Below we list the details of all cases under consideration:
- Case-I: the mass of the SMBH saturates at a constant fraction of the mass of the spheroid, $M_{\rm bh}\propto M_{\rm sph}\propto\sigma_{\rm
sph}^2R_{\rm sph}$.
- Case-II: the mass of the black-hole grows in proportion to the binding energy of baryons in the spheroid. Taking a constant fraction of the cold-gas mass to be material that must be expelled from the spheroid during the feedback we find that the black-hole mass is therefore $M_{\rm bh}\propto M_{\rm sph}\sigma_{\rm
sph}^2\propto\sigma_{\rm sph}^4R_{\rm sph}$.
- Case-III: the mass of the SMBH is determined by the mass for which accretion at the Eddington limit provides a constant fraction of the binding energy of the baryons in the spheroid over a constant fraction of the spheroid’s dynamical time. Thus, the black-hole mass scales as $M_{\rm bh}\propto E_{\rm b}/(R_{\rm sph}/\sigma_{\rm sph})\propto
M_{\rm sph}\sigma_{\rm sph}^3/R_{\rm sph}\propto\sigma_{\rm sph}^5$.
- Case-IV: As in case-II, but with the momentum rather than the energy of the outflow coupling to the gas in the spheroid, yielding $M_{\rm
bh}\propto M_{\rm sph}\sigma_{\rm sph}\propto\sigma_{\rm sph}^3R_{\rm
sph}$.
- Case-V: As in case-III, but with the momentum rather than the energy of the outflow coupling to the gas in the spheroid over a dynamical time, yielding $M_{\rm sph}\sigma_{\rm sph}^2/R_{\rm sph}\propto\sigma_{\rm
sph}^4$.
Studies of the local SMBH inventory suggest that most of mass in SMBHs was accreted during a luminous quasar phase (e.g. Yu & Tremaine 2002; Shankar et al. 2004), with a potentially significant contribution from an additional dust–obscured accretion phase (Martinez-Sansigre et al. 2005). If the fraction of obscured quasars is independent of redshift, then the quasar luminosity function (Fan et al. 2004; Boyle et al. 2000) can be used as a proxy for the distribution of SMBH formation redshifts. Using this redshift distribution and the formalism outlined in the previous sub-section, we can estimate the slope and scatter in the $M_{\rm bh}$–$\sigma_{\rm sph}$ and $M_{\rm bh}$–$M_{\rm sph}$ relations under the five different scenarios for the regulation of SMBH growth.
For each of the five cases we perform Monte-Carlo simulations of SMBH growth. The resulting residuals in the $M_{\rm bh}$–$\sigma_{\rm
sph}$ and $M_{\rm bh}$–$M_{\rm sph}$ relations are plotted in Figure \[fig3\] relative to the best fit log-quadratic relations (equations \[Msig\]-\[MM\]) as a function of $\sigma_{\rm sph}$ (upper row) and $M_{\rm sph}$ (lower row) respectively. For each of the five cases, we also compute distributions of residuals (in dex) relative to the mean SMBH mass at constant $\sigma_{\rm sph}$ and constant $M_{\rm sph}$. The resulting distributions are plotted in Figure \[fig4\]. The values of scatter are listed in the 1st and 2nd columns of Table \[tab1\]. The ratio of the scatter at constant $M_{\rm sph}$ and at constant $\sigma_{\rm sph}$ are listed in column 3. The power-law slopes $\beta$ and $\beta_{\rm sph}$ in each case are listed in columns 4 and 5.
---------- --------------------- --------------------- --------------------------- ------------- ------------------- -------------- -------------------- --------------------------- --------- -------------------
$\delta$ $\delta_{\rm sph}$ $\delta_{\rm sph}/\delta$ $\beta$ $\beta_{\rm sph}$ $\delta$ $\delta_{\rm sph}$ $\delta_{\rm sph}/\delta$ $\beta$ $\beta_{\rm sph}$
Case-I 0.17 (0.275) 0.0 (0.22) 0 (0.8) 3.4 1.0 0.24 (0.275) 0.29 (0.31) 1.2 (1.1) 3.3 1.0
Case-II 0.17 (0.275) 0.10 (0.24) 0.6 (0.87) 5.3 1.6 0.36 0.48 1.3 5.4 1.6
Case-III [**0.0 (0.275)**]{} [**0.25 (0.37)**]{} [**$\infty$ (1.35)**]{} [**5.0**]{} [**1.5**]{} 0.37 0.53 1.4 5.5 1.5
Case-IV 0.17 (0.275) 0.05 (0.22) 0.29 (0.8) 4.2 1.3 0.39 0.38 1.0 4.3 1.3
Case-V [**0.0 (0.275)**]{} [**0.20 (0.34)**]{} [**$\infty$ (1.24)**]{} [**4.0**]{} [**1.2**]{} 0.29 0.42 1.4 4.4 1.3
---------- --------------------- --------------------- --------------------------- ------------- ------------------- -------------- -------------------- --------------------------- --------- -------------------
From the above discussion we see that under the assumption of a unique coupling efficiency between the quasar output and the surrounding spheroid, cases-III and V imply a perfect $M_{\rm bh}$–$\sigma_{\rm sph}$ relation. Cases-I, II and IV do not produce perfect $M_{\rm
bh}$–$\sigma_{\rm sph}$ relations even under this unique coupling assumption due to the scatter in the $R_{\rm sph}$–$\sigma_{\rm sph}$ relation. Similarly, the scatter in the $R_{\rm sph}$–$\sigma_{\rm sph}$ relation leads to scatter in the $M_{\rm bh}$–$M_{\rm sph}$ relation for cases-III and V. As a result, cases-III and V predict a scatter in the $M_{\rm
bh}$–$M_{\rm sph}$ relation that is larger than in the $M_{\rm
bh}$–$\sigma_{\rm sph}$ relation, in agreement with observations.
Obviously in these five cases we have neglected many additional sources of scatter. These include, but are not limited to, the fraction of the Eddington limit at which the SMBH shines during its luminous phase, the efficiency of coupling of feedback energy or momentum to the gas in the host spheroid, and the fraction of the system’s characteristic size for which the dynamical time should be computed. In all five cases the minimum values of the scatter are smaller than the observed $\delta=0.275$. However in cases I, II and IV, the models predict a scatter in the $M_{\rm
bh}$–$M_{\rm sph}$ relation that is smaller than in the $M_{\rm
bh}$–$\sigma_{\rm sph}$ relation while the observations show the opposite. On the other hand if SMBHs are regulated by the binding energy or momentum of gas in the spheroid per dynamical time of the spheroid (cases III & V), then the minimum scatter in the $M_{\rm
sph}$–$\sigma_{\rm sph}$ relation is reduced to zero. Cases-III and V therefore predict that the scatter in the $M_{\rm bh}$–$M_{\rm sph}$ relation should be larger than in the $M_{\rm bh}$–$\sigma_{\rm sph}$ relation as observed.
The scatter in the projection of the fundamental plane onto the $R_{\rm sph}$–$\sigma_{\rm sph}$ plane therefore allows us to differentiate between SMBH growth that is regulated by the mass (Case-I), binding energy (Case-II) or momentum of gas (Case-IV) in the spheroid, and SMBH growth that is regulated by energy or momentum feedback over a dynamical time of the spheroid (Cases-III and V). In the latter cases the scatter in the $M_{\rm bh}$–$M_{\rm sph}$ relation is increased by the large scatter in the spheroid radius, $R_{\rm
sph}$. On the other hand in Cases-III and V, $R_{\rm sph}$ cancels out in the division of mass by dynamical time in the determination of $M_{\rm bh}$ at constant $\sigma_{\rm sph}$.
If the predicted value for the minimum scatter in the $M_{\rm
bh}$–$\sigma_{\rm sph}$ relation is smaller than the observed value of $\delta=0.275$ dex, then there is room in the model for additional random scatter to account for varying Eddington ratio, outflow geometry, dust composition, and other factors. In each case we therefore add random scatter in the formation process at a level which results in the predicted scatter in the $M_{\rm bh}$–$\sigma_{\rm sph}$ relation being equal to the observed value of $\delta=0.275$ dex. This value, the corresponding value predicted for the scatter in the $M_{\rm bh}$–$M_{\rm sph}$ relation ($\delta_{\rm sph}$), and the corresponding ratio ($\delta_{\rm
sph}/\delta$) are listed in parentheses in Table \[tab1\]. While cases-I, II and IV predict ratios of scatter between the $M_{\rm bh}$–$M_{\rm sph}$ and $M_{\rm bh}$–$\sigma_{\rm sph}$ relation that are smaller than unity, case-III predicts a ratio of $\delta_{\rm sph}/\delta\sim1.4$, and case-V a ratio of $\delta_{\rm sph}/\delta\sim1.2$, in good agreement with the observed value ($\delta_{\rm sph}/\delta=1.5\pm0.35$). We therefore conclude that SMBH growth is likely regulated by feedback over the spheroid’s dynamical time.
A further possible discriminant between models is the power-law slope of the $M_{\rm bh}$–$\sigma_{\rm sph}$ relations. Cases III & V, which satisfy constraints from the observed scatter in the $M_{\rm
bh}$–$\sigma_{\rm sph}$ and $M_{\rm bh}$–$M_{\rm sph}$ relations, produce power-law $M_{\rm bh}$–$\sigma_{\rm sph}$ relations, with SMBH mass in proportion to the velocity dispersion raised to the fifth and fourth power respectively. This could be compared with the power-law value from Tremaine et al. (2002) of $\beta=4\pm0.3$ for galaxies with $\sigma_{\rm
sph}\sim200~{\rm km~s^{-1}}$, a comparison which at first sight this appears to support case-V. However Wyithe (2005) has found evidence for a power-law slope that varies with $\sigma_{\rm sph}$ from $\beta\sim4$ near $\sigma_{\rm sph}\sim200~{\rm km~s^{-1}}$, to $\beta\sim5$ near $\sigma_{\rm sph}\sim350~{\rm km~s^{-1}}$ (see Eq. \[Msig\]). The slope of the $M_{\rm bh}$–$M_{\rm sph}$ relation is observed to be close to unity (Haering & Rix 2004). Of the cases (III and V) that produce an acceptably small scatter for the $M_{\rm bh}$–$\sigma_{\rm sph}$ relation, we find that case-III yields $\beta_{\rm sph}\sim1.5$, while case-V leads to a value of $\beta_{\rm sph}\sim1.2$. However Wyithe (2005) also finds evidence for a varying power-law in the $M_{\rm bh}$–$M_{\rm
sph}$ relation, though not at high significance.
Since the models described in this paper each predict power-law relations between $M_{\rm bh}$ and $\sigma_{\rm sph}$, the residuals with respect to the log-quadratic fit therefore show curvature as a function of $\sigma_{\rm sph}$. Cases-III and V both show a power-law slope that agrees with the best-fit relation at some values of $\sigma_{\rm sph}$, which renders discrimination between models based on their predicted power-law slope difficult. It is beyond the scope of this paper to discuss detailed models of the $M_{\rm bh}$–$\sigma_{\rm sph}$ relation; however the observed curvature may point to a turn-over from momentum to energy conservation in the reaction between the outflow and surrounding gas. Alternatively, there could be a velocity dependent efficiency of feedback, which could change the slope of the relation. In summary, based on the observed values of $\delta=0.275\pm0.05$, $\delta_{\rm
sph}=0.41\pm0.07$, only models corresponding to Case-III or Case-V are acceptable. Examples of this class include the models of Silk & Rees (1998); Wyithe & Loeb (2003); King (2003), di Matteo et al. (2005), or Murray et al. (2005).
Additional scatter from dissipationless mergers after the quasar epoch
----------------------------------------------------------------------
Our model computes the minimum scatter in the $M_{\rm bh}$–$\sigma_{\rm
sph}$ relation during a time when the SMBH can grow via gas accretion. At late times a SMBH may find itself in an environment where there is no remaining cold gas. In this regime a merger of two galaxies will proceed via collisionless dynamics. The two SMBHs may coalesce once they enter the gravity-wave dominated regime, but since there is no cold gas the SMBH will not grow during a quasar phase. It would require detailed numerical simulations to discover whether or not the $M_{\rm bh}$–$\sigma_{\rm sph}$ relation holds following a collisionless merger. However, unlike the situation where feedback is still at work, in a collisionless merger the stellar spheroid is not sensitive to the existence of the SMBH (at least at radii comparable to $R_{\rm sph}$). Moreover the total SMBH mass is nearly conserved as long as SMBH binaries coalesce due to interactions with the surrounding stars and the emission of gravitational radiation (Begelman, Blandford, & Rees 1980). The growth of the SMBH and the properties of the spheroid following a collisionless merger should be independent in the sense of the $M_{\rm bh}$–$\sigma_{\rm sph}$ relation. Thus, we would expect collisionless mergers to increase the scatter in the $M_{\rm
bh}$–$\sigma_{\rm sph}$ relation, leaving our estimates of the minimum scatter intact.
N-body simulations of the behavior of stars as collisionless particles have been performed by Gao et al. (2004) who found that the inner mass density profile (interior to some fixed physical radius) is unaffected by mergers, implying that the velocity dispersion interior to any given radius remains the same after a merger. When two spheroids merge, their combined stars cover a larger radius (because their mass increases while the inner mass profile remains unchanged), and this changes the value of $\sigma_{\rm
sph}$. We may therefore use the fact that the inner profile remains invariant in order to estimate the scaling between $R_{\rm sph}$ and $\sigma_{\rm sph}$ in the regime where purely collisionless mergers occur. Based on the simulations of Gao et al. (2004), we assume a universal NFW mass profile for the total mass (dark matter$+$stars) irrespective of the merger history and find what mergers would do to the $M_{\rm bh}$–$\sigma_{\rm sph}$ relation when the total mass in stars and in SMBHs is conserved. If the inner density profile maintains the NFW shape of $\rho\propto1/r$, then $\sigma_{\rm sph}^2\propto r\propto M_{\rm
sph}^{1/2}$, i.e. $M_{\rm sph}\propto \sigma_{\rm sph}^4$, similar to the Faber-Jackson (1976) projection of the fundamental plane of spheroids.
Collisionless mergers will change the average normalization of the $M_{\rm
bh}$–$\sigma_{\rm sph}$ relation. To see why, suppose that we have $N_{\rm
p}$ equal mass progenitors at redshift $z$, each with velocity dispersion $\sigma_{\rm sph,p}$ and spheroid mass $M_{\rm sph,p}$. At redshift $z$, the SMBHs obey $M_{\rm bh,p}=C_{\rm p}\sigma_{\rm sph,p}^{\beta_{\rm p}}$, where $\beta_{\rm p}$ is the slope and $C_{\rm p}$ is a constant. The final SMBH and spheroid masses at $z=0$ are $M_{\rm bh,f}=N_{\rm p}M_{\rm bh,p}$ and $M_{\rm sph,f}=N_{\rm p}M_{\rm sph,p}$ respectively. Using the above scaling the final velocity dispersion is $\sigma_{\rm sph,f}=\sigma_{\rm
sph,p}N_{\rm p}^{1/4}$. We therefore find $M_{\rm bh,f}=N_{\rm p}M_{\rm
bh,p}=N_{\rm p}C_{\rm p}\sigma_{\rm sph,p}^{\beta_{\rm p}}=C_{\rm p}N_{\rm
p}^{1-\beta_{\rm p}/4}\sigma_{\rm sph,f}^{\beta_{\rm p}}$. Thus the normalization of the $M_{\rm bh}$–$\sigma_{\rm sph}$ relation at fixed $\sigma_{\rm sph}$ is changed by a factor $\sim N_{\rm p}^{1-\beta_{\rm
p}/4}$ through collisionless mergers. Note that the slope of the $M_{\rm
bh}$–$\sigma_{\rm sph}$ relation $\beta_{\rm p}$ is preserved if $N_{\rm
p}$ is independent of $\sigma_{\rm sph}$. However the number of progenitors is a function of halo mass and redshift, and so the change in normalization could be a function of $\sigma_{\rm sph}$.
In addition to changing the normalization of the $M_{\rm bh}$–$\sigma_{\rm
sph}$ relation, collisionless mergers will also introduce scatter. In the limit of a perfect correlation between $M_{\rm sph}$ and $\sigma_{\rm
sph}$, and where the number of progenitors ($N_{\rm p}$) is constant, the argument in the previous paragraph shows that collisionless mergers would lead to an $M_{\rm bh}$–$\sigma_{\rm sph}$ relation with no additional scatter beyond that introduced at the formation redshift. However the scatter in the $R_{\rm sph}$–$\sigma_{\rm sph}$ correlation combined with the relation $M_{\rm sph}\propto\sigma_{\rm sph}^2R_{\rm sph}$ implies that there is $\sim0.2$dex of scatter in $M_{\rm sph}$ at fixed $\sigma_{\rm
sph}$. Moreover, different galaxies have different merger histories and therefore a different number of progenitors. Scatter among the properties of the initial building blocks at redshift $z$ therefore leads to scatter in the local $M_{\rm bh}$–$\sigma_{\rm sph}$ relation even if the $M_{\rm
bh}$–$\sigma_{\rm sph}$ relation at $z$ were perfect. This scatter introduced through collisionless mergers will therefore add scatter to the local $M_{\rm bh}$–$\sigma_{\rm sph}$ relation beyond that intrinsic to the formation process itself.
To ascertain the quantitative effect of collisionless mergers on scatter in the $M_{\rm bh}$–$\sigma_{\rm sph}$ relation, we generate merger trees of dark-matter halos using the method described in Vollonteri, Haardt & Madau (2003). Based on the merger trees we find the $N_{\rm p}$ progenitor halos at $z\sim1.5$ that lead to a halo of known mass at $z\sim0$. Using the formalism outlined in § \[fpsec\] we then determine the values of $\sigma_{\rm sph}$, $R_{\rm sph}$ and $M_{\rm sph}$ for the spheroids populating these progenitor halos. We also populate the spheroids with SMBHs of mass $M_{\rm bh}$ according to the perfect $M_{\rm
bh}$–$\sigma_{\rm sph}$ relations that arise from cases-III and V. The final SMBH mass residing in the halo at $z=0$ is $M_{\rm
bh,f}=\sum_{i=0}^{N_{\rm p}}M_{{\rm bh},i}$. It is embedded in a spheroid of mass $M_{\rm sph,f}=\sum_{i=0}^{N_{\rm p}}M_{{\rm sph},i}$. Based on the above scaling for the inner $\rho\propto1/r$ density profile of an NFW halo we may estimate the value of velocity dispersion corresponding to the final spheroid, $\sigma_{\rm sph,f}=\sigma_{\rm sph,0}(M_{\rm sph,f}/M_{\rm
sph,0})^{1/4}$, where $M_{\rm sph,0}$ and $\sigma_{\rm sph,0}$ are the mass and velocity dispersion of the largest progenitor. In Figure \[fig5\] we show the scatter introduced into a perfect $M_{\rm bh}$–$\sigma_{\rm sph}$ relation originating at $z=0.5$ (left), $z=1.5$ (center) and $z=2.5$ (right) by collisionless mergers between those redshifts and $z=0$. As discussed above this scatter arises as a result of scatter in the relation between $\sigma_{\rm sph}$ and $M_{\rm sph}$ in the progenitors. The upper panels show results for Case-III ($\beta=5$), while the lower panels show results for Case-V ($\beta=4$). The scatter introduced is roughly independent of $\sigma_{\rm sph}$ and takes values of $\delta\sim0.1$dex, $\delta\sim0.2$dex and $\delta\sim0.3$dex for mergers originating at $z=0.5$, $z=1.5$ and $z=2.5$, respectively. Thus galaxies that become devoid of gas at higher redshift lead to a larger scatter in the $M_{\rm
bh}$–$\sigma_{\rm sph}$ relation, because these galaxies undergo more collisionless mergers by $z=0$ than a galaxy which becomes devoid of gas only at late times.
The stars that populate massive galaxies appear to be older than those in low mass galaxies (Kauffmann et al. 2003). The cold gas reservoir that made these stars must have been depleted at a higher redshift for the progenitors of high-$\sigma_{\rm sph}$ galaxies. We might therefore expect more scatter in the $M_{\rm bh}$–$\sigma_{\rm sph}$ relation at large $\sigma_{\rm sph}$. For equal mass mergers we have shown that the normalization of the $M_{\rm bh}$–$\sigma_{\rm sph}$ relation at a fixed $\sigma_{\rm sph}$ changes by a factor $\sim N_{\rm p}^{1-\beta_{\rm
p}/4}$. Thus, for $\beta_{\rm p}=5$ (Case-III), we find that the amplitude of the $M_{\rm bh}$–$\sigma_{\rm sph}$ relation should be reduced by collisionless mergers, while for $\beta_{\rm p}=4$ (Case-V) the amplitude should be preserved. This behavior is seen in Figure \[fig5\]. Moreover, more massive galaxies undergo a larger rate of major mergers. Figure \[fig5\] shows that in Case-V the change in normalization of the $M_{\rm
bh}$–$\sigma_{\rm sph}$ relation through collisionless mergers is more significant in high mass than in low mass galaxies, as expected. Collisionless mergers could therefore lead to a reduction in the steepness of the observed $M_{\rm bh}$–$\sigma_{\rm sph}$ relation if $\beta_{\rm p}>4$.
In summary, in order to satisfy the constraint that the $M_{\rm
bh}$–$\sigma_{\rm sph}$ relation have a scatter at $z=0$ that is smaller than $\sim0.3$dex, the intrinsic scatter in the relation at the formation redshift should be smaller than $\sim0.2$ dex, so as to allow for the additional scatter introduced through collisionless mergers. Cases-III and V meet this requirement.
Redshift dependence of the $M_{\rm bh}$–$\sigma_{\rm sph}$ and $M_{\rm bh}$–$M_{\rm sph}$ relations
---------------------------------------------------------------------------------------------------
Recent evidence suggests that the $M_{\rm bh}$–$\sigma_{\rm sph}$ relation is preserved out to high redshift (Shields et al. 2003), while SMBHs make a larger fraction of their host spheroid mass at higher redshift (Rix et al. 2001; Croom et al. 2004; Walter et al. 2004). This behavior is reproduced in models with scenarios of the form case-III or case-V. Figure \[fig6\] shows a scatter plot of the predicted residuals in $M_{\rm bh}$ vs $z$ at constant $\sigma_{\rm
sph}=200~{\rm km~s^{-1}}$ (upper row) and $M_{\rm sph}=10^{11}M_\odot$ (lower row) for each of the five models. The residuals are normalized relative to the mean relation at $z=3$ in each case. While there is no evolution in the $M_{\rm bh}$–$\sigma_{\rm sph}$ relation for cases-III and V, we see that SMBHs are predicted to be an order of magnitude more massive with respect to their host spheroid at $z\sim6$ than they are at $z\sim1$ in agreement with observations. In contrast, models for case-I, II and IV predict that the SMBH mass should decrease by an order of magnitude between $z=0$ and $z=6$ at constant $\sigma_{\rm sph}$, while not evolving significantly at constant $M_{\rm sph}$. The observed evolution of SMBH mass with redshift therefore supports cases III and V as the scenario for SMBH growth. This result supports the findings of § \[models\] based on the scatter in local relations.
Do dark-matter halos play a role in SMBH evolution?
---------------------------------------------------
Attempts to reproduce the observed luminosity function of quasars associate the mass of the SMBH with the properties of the host dark-matter halo. This paradigm allows the abundance of SMBHs to be traced either in semi-analytic or numerical models (e.g. Haiman & Loeb 1998; Haehnelt, Natarajan & Rees 1998; Wyithe & Loeb 2003; Vollonteri, Haardt & Madau 2002). Indeed, Ferrarese (2001) has inferred a relation between the masses of SMBHs and their host dark-matter halo. [*Is it possible that the halo rather than the spheroid regulates SMBH growth?*]{}
Since we have computed spheroid properties within a specified dark matter halo, we are in a position to discuss the role of the dark matter halo in regulating the SMBH growth. In addition to the five cases listed above for SMBH growth within a spheroid, we also try five analogous cases for the formation of SMBHs governed by dark matter halo properties. For each of the additional five cases, we compute the distribution of residuals (in dex) relative to the mean relation as a function of $\sigma_{\rm sph}$ and $M_{\rm sph}$ via a Monte-Carlo algorithm for SMBH formation, and calculate the variance at each of constant $\sigma_{\rm sph}=200~{\rm km~s^{-1}}$ and constant $M_{\rm sph}=10^{11}M_\odot$. Below we list the details of each case. The resulting distributions of residuals are plotted in figure \[fig7\]. The values of scatter are listed in the 6th and 7th columns of Table \[tab1\]. The ratio of the scatter at constant $M_{\rm
sph}$ and at constant $\sigma_{\rm sph}$ are listed in column 8. Values for the slopes $\beta$ and $\beta_{\rm sph}$ are listed in columns 9 and 10.
- Case-I: the mass of the SMBH forms from a constant fraction of the baryonic component of the halo mass, $M_{\rm bh}\propto ({\Omega_{\rm
b}}/{\Omega_{\rm m}})M_{\rm halo}$.
- Case-II: the mass of the black-hole grows in proportion to the binding energy of baryons in the halo. For an NFW profile with a concentration parameter $c$ we get $M_{\rm bh}\propto ({\Omega_{\rm
b}}/{\Omega_{\rm m}})M_{\rm halo} V_{\rm vir}^2 f_{\rm c}$, where $$\label{fc}
f_{\rm c} = \frac{c}{2}\frac{1-1/(1+c)^2-2\ln(1+c)/(1+c)}{[c/(1+c)-\ln(1+c)]^2}.$$
- Case-III: the mass of the SMBH is determined by the mass for which accretion at the Eddington limit provides the binding energy of baryons in the halo over a constant fraction of the halo’s dynamical time (Wyithe & Loeb 2003). We therefore have $M_{\rm bh}\propto \frac{(\Omega_{\rm
b}/\Omega_{\rm m}) M_{\rm halo} V_{\rm vir}^2 f_{\rm c}}{R_{\rm vir}/V_{\rm
vir}}\propto f_{\rm c} V_{\rm vir}^5$.
- Case-IV: same as case-II, but with the momentum rather than the energy of the outflow coupling to the gas in the spheroid. We then find $M_{\rm bh}\propto ({\Omega_{\rm b}}/{\Omega_{\rm m}})M_{\rm halo} V_{\rm vir}
f_{\rm c}$
- Case-V: same as case-IV, but with the momentum rather than the energy of the outflow coupling to the gas in the spheroid over a dynamical time. We then find $M_{\rm bh}\propto \frac{(\Omega_{\rm
b}/\Omega_{\rm m}) M_{\rm halo} V_{\rm vir} f_{\rm c}}{R_{\rm vir}/V_{\rm
vir}}\propto f_{\rm c} V_{\rm vir}^4$.
In these five cases we have again neglected many possible causes of scatter. However in case-I, where the SMBH growth is regulated by halo properties, the minimum value of the scatter in the $M_{\rm
bh}$–$\sigma_{\rm sph}$ relation is slightly smaller than the observed $\delta=0.275$. We have added random scatter to the model for case-I in order to bring the predicted scatter in the $M_{\rm
bh}$–$\sigma_{\rm sph}$ up to the observed value. This value and the corresponding prediction for $\delta_{\rm sph}$ and $\delta/\delta_{\rm sph}$ are listed in parentheses. Case-I predicts a scatter in the $M_{\rm bh}$–$M_{\rm sph}$ relation that is similar to the $M_{\rm bh}$–$\sigma_{\rm sph}$ relation while the observations suggest the latter to be significantly smaller. While case-I cannot be ruled out based only on the predicted minimum scatter in the $M_{\rm bh}$–$\sigma_{\rm sph}$ relation at the formation redshift, the small allowance for additional expected scatter from the aforementioned astrophysical sources renders it unlikely, particularly when the additional scatter of $0.1-0.3$dex from collisionless mergers at low redshift is accounted for. In cases II, III, IV and V where the SMBH growth is regulated by halo properties, the minimum values of the scatter in the $M_{\rm
bh}$–$\sigma_{\rm sph}$ relation are larger than the observed $\delta=0.275$.
While scatter within models of spheroid regulated SMBH growth is not sensitive to the distribution of $m_{\rm d}$, the predicted scatter in models of halo regulated SMBH growth decreases as the assumed range of $m_{\rm d}$ decreases. However, in order to reduce the predicted scatter below the $\sim0.2$dex threshold at the formation redshift (which would allow for additional scatter to be introduced through collisionless mergers), we find that the allowed range around $m_{\rm d}=0.05$ would need to be smaller than $\pm0.025$, which is implausibly narrow. We therefore conclude that it is the spheroid rather than the dark-matter halo which drives the evolution of SMBH mass.
The $M_{\rm bh}$–$M_{\rm halo}$ relation
----------------------------------------
We have demonstrated that the tight relation between $M_{\rm bh}$ and $\sigma_{\rm sph}$ implies that it is the spheroid rather than the halo which governs the growth of SMBHs. However it is clear that since there is an $M_{\rm bh}$–$\sigma_{\rm sph}$ relation, and since larger halos will, on average, host bulges with larger central velocity dispersions, there should also be a correlation between SMBH and halo mass. Ferrarese (2001) has found such a relation. Since it is not possible to measure the dark matter halo mass directly, halo masses for galaxies in the local sample were inferred via a maximum circular velocity estimated from $\sigma_{\rm
sph}$ based on an empirical relation. It is therefore difficult to estimate the scatter in the $M_{\rm bh}$–$M_{\rm halo}$ relation observationally. Here we predict the scatter in the $M_{\rm bh}$–$M_{\rm
halo}$ relation at a fixed value of $M_{\rm halo}$. We compute the distribution of residuals via a Monte-Carlo method as before. We choose $M_{\rm halo}=10^{12}M_\odot$ and find the distribution of values for $\sigma_{\rm sph}$, and hence the distribution of $M_{\rm bh}$ using the observed $M_{\rm bh}$–$\sigma_{\rm sph}$ relation. The resulting distribution is plotted in Figure \[fig8\]. The variance is $\delta_{\rm
halo}=0.4$ dex. Thus the tightness of the $M_{\rm bh}$–$\sigma_{\rm sph}$ relation suggests that the $M_{\rm bh}$–$M_{\rm halo}$ correlation is incidental to the fundamental relation between the SMBH and its host spheroid. Note that this variance is computed at the time of the SMBH formation. The surrounding dark matter halo could continue to grow after the supply of cold gas to the SMBH had ceased. This is consistent with our conclusion that SMBHs grow in proportion to the properties of the spheroid rather than the halo. Indeed one finds massive dark-matter halos in X-ray clusters, which must have increased their velocity dispersion well beyond the corresponding SMBH growth (due to the lack of cooling flows in cluster cores). This late-time growth of dark-matter halos increases considerably the scatter in the $M_{\rm bh}$–$M_{\rm halo}$ relation.
conclusion
==========
We have investigated the implications of intrinsic scatter in the local relations involving SMBHs for models of SMBH formation. Using the sample of spheroid properties from SDSS (Bernardi et al. 2003) we first examined empirically the fundamental parameter describing SMBH growth. The observed scatter in the $M_{\rm bh}$–$\sigma_{\rm sph}$ relation is $\delta=0.275\pm0.05$, while the $M_{\rm bh}$–$M_{\rm sph}$ relation has a larger observed scatter of $\delta_{\rm sph}=0.41\pm0.07$. Assuming that the $M_{\rm bh}$–$M_{\rm sph}$ relation is fundamental, we use the SDSS spheroid sample to compute the resulting scatter in the $M_{\rm
bh}$–$\sigma_{\rm sph}$ relation. We find that this procedure results in a scatter of the $M_{\rm bh}$–$\sigma_{\rm sph}$ relation that is too large to be reconciled with observation. Alternatively, one might assume that SMBH growth is determined by $\sigma_{\rm sph}$ rather than $M_{\rm
sph}$. In this case we used the SMBH sample to compute the resulting scatter in the $M_{\rm bh}$–$M_{\rm sph}$ relation, and found agreement with the observed scatter. We therefore conclude that SMBH growth is governed by $\sigma_{\rm sph}$, and that the observed correlation between $M_{\rm bh}$ and $M_{\rm sph}$ is a by-product of the relation between $M_{\rm sph}$ and $\sigma_{\rm sph}$.
Theoretical models for SMBH formation must reproduce several observational constraints: [*(i)*]{} the scatter in the local $M_{\rm bh}$–$\sigma_{\rm
sph}$ relation is $\delta=0.275\pm0.05$ dex, implying that at the time of formation the scatter should be smaller than $\sim0.2$dex to allow for additional scatter introduced by collisionless mergers of galaxies since $z\sim1$ or earlier; [*(ii)*]{} the scatter in the $M_{\rm bh}$–$M_{\rm
sph}$ relation is larger than in the $M_{\rm bh}$–$\sigma_{\rm sph}$ relation (this result is maintained as additional scatter from collisionless mergers is introduced after SMBH formation); and [*(iii)*]{} The $M_{\rm bh}$–$\sigma_{\rm sph}$ relation is preserved out to high redshift. We find that these constraints are only met by models where SMBH growth is regulated by feedback on the gas feeding the SMBH over the spheroid dynamical time. Other models lead to scatter in the $M_{\rm
bh}$–$\sigma_{\rm sph}$ relation that are too large or scatter in the $M_{\rm bh}$–$M_{\rm sph}$ relation that is smaller than the $M_{\rm
bh}$–$\sigma_{\rm sph}$ relation. In addition, other models lead to a SMBH mass that drops with increasing redshift at a fixed velocity dispersion. The feedback in successful models can be either in the form of energy or momentum transfer between the quasar and the galactic gas, leading to power-law slopes in the $M_{\rm bh}$–$\sigma_{\rm sph}$ relation of $\beta=4$ or $\beta=5$, respectively. Both of these slopes are permitted by the local sample (Wyithe 2005).
The above constraints do [*not*]{} permit SMBH growth to be governed by the properties of the dark-matter halos. Such models lead to scatter in the $M_{\rm bh}$–$\sigma_{\rm sph}$ relation that are too large and/or scatter in the $M_{\rm bh}$–$M_{\rm sph}$ relation that is smaller than the $M_{\rm bh}$–$\sigma_{\rm sph}$ relation. The relation between $M_{\rm
bh}$ and the halo mass (Ferrarese 2001) has a large scatter ($\sim0.4$dex) and is most likely a by-product of the correlation between halo mass and $\sigma_{\rm sph}$.
J.S.B.W. acknowledges the support of the Australian Research Council. This work was supported in part by NASA grants NAG 5-13292, NNG-05-GH54G, and by NSF grants AST-0071019, AST-0204514 (for A.L.).
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'By exploiting a suitable Trudinger-Moser inequality for fractional Sobolev spaces, we obtain existence and multiplicity of solutions for a class of one-dimensional nonlocal equations with fractional diffusion and nonlinearity at exponential growth.'
address: 'Dipartimento di Informatica Università degli Studi di Verona Cá Vignal 2, Strada Le Grazie 15, I-37134 Verona, Italy'
author:
- Antonio Iannizzotto
- Marco Squassina
title: '$\nicefrac{1}{2}$-Laplacian problems with exponential nonlinearity'
---
Introduction and results
========================
Since the seminal results by Trudinger [@T] and Moser [@M] on embeddings of exponential type for the Sobolev spaces $H^1_0(\Omega)$ with $\Omega\subset{{\mathbb R}}^2$, many contributions have appeared related to applications of these results to semi-linear elliptic partial differential equations such as $$\label{prob-loc}
\left\{ \begin{array}{ll}
-\Delta u=f(u) &\mbox{in $\Omega$} \\
u=0 &\mbox{on $\partial\Omega$,}
\end{array}
\right.
\qquad
f(t)\sim e^{4\pi t^{q}}\quad\text{as $t\to+\infty$},
\qquad
0<q\leq 2,$$ where the case $q<2$ is considered a subcritical growth, while the case $q=2$ is known as the critical case with respect to the Trudinger-Moser inequality (see [@M Theorem 1]) $$\label{trudmos}
\sup_{\substack{u\in H^1_0(\Omega) \\ \|\nabla u\|_{L^2(\Omega)}\leq 1}}\int_\Omega e^{4\pi u^2}\leq C|\Omega|.$$ For existence and multiplicity of solutions for problems like via techniques of critical point theory, we refer the reader to De Figueiredo, Miyagaki $\&$ Ruf [@DMR] and to the references therein. Many extensions of inequality have been achieved, for spaces $W^{1,n}_0(\Omega)$ with $\Omega\subset{{\mathbb R}}^n$ and also to higher order Sobolev spaces (see Adams [@adams]). As a consequence, quasi-linear problems involving the $n$-Laplacian on domains $\Omega\subset{{\mathbb R}}^n$ or the linear biharmonic operator $\Delta^2$ for functions of $W^{2,2}_0(\Omega)$ on domains $\Omega\subset{{\mathbb R}}^4$ can be studied. Focusing the attention on nonlinear problems at exponential growth involving linear diffusion, if the dimension four is natural for the biharmonic operator $\Delta^2$ and dimension two is natural for the laplacian $-\Delta$, the natural setting for the fractional diffusion $(-\Delta)^{1/2}$ is dimension one. Fractional Sobolev spaces are well known since the beginning of the last century, especially in the framework of harmonic analysis. More recently, after the paper of Caffarelli $\&$ Silvestre [@caffarelli], a large amount of papers were written on problems involving the fractional diffusion $(-\Delta)^s$, $0<s<1$. Due to its nonlocal nature, working on bounded domains suggests the functions to be defined on the whole ${{\mathbb R}}^n$ and that the problems are formulated as follows (see Servadei $\&$ Valdinoci [@SV]): $$\label{sfrac}
\left\{ \begin{array}{ll}
(-\Delta)^{s} u=f(u) &\mbox{in $\Omega$} \\
u=0 &\mbox{in ${{\mathbb R}}^n\setminus \Omega$.}
\end{array}
\right.$$ For the functional framework of fractional Sobolev spaces and fractional Laplacian, we refer the reader to the survey of Di Nezza, Palatucci $\&$ Valdinoci [@DPV]. Equations like appear in fractional quantum mechanics in the study of particles on stochastic fields modeled by Lévy processes which occur widely in physics and biology and recently the stable Lévy processes have attracted much interest. One dimensional cases have been studied by Weinstein [@weinstein]. In the present paper we will prove the existence and multiplicity of solutions for a Dirichlet problem driven by the $\nicefrac{1}{2}$-Laplacian operator of the following type: $$\label{problema-1d}
\tag{$P$} \ \ \ \ \ \ \ \ \
\left\{ \begin{array}{ll}
(-\Delta)^{\nicefrac{1}{2}} u=f(u) &\mbox{in $(0,1)$} \\
u=0 &\mbox{in ${{\mathbb R}}\setminus (0,1)$,}
\end{array}
\right.$$ equivalently written in $(0,1)$ as the nonlocal equation $$\frac{1}{2\pi}\int_{{{\mathbb R}}}\frac{u(x+y)+u(x-y)-2 u(x)}{|y|^{2}}dy+f(u)=0.$$ It is natural to work on the space $$\label{Xdef}
X=\left\{u\in H^{\nicefrac{1}{2}}({{\mathbb R}}): \ u=0 \ \mbox{in ${{\mathbb R}}\setminus(0,1)$} \right\}, \quad \|u\|_X=[u]_{H^{\nicefrac{1}{2}}({{\mathbb R}})}$$ where $[\cdot]_{H^{\nicefrac{1}{2}}({{\mathbb R}})}$ denotes the Gagliardo semi-norm (see Proposition \[hilb\]). For this space, we state (see Corollary \[mti\]) and exploit the following Trudinger-Moser type inequality: there exists $0<\omega\leq\pi$ such that for all $0<\alpha<2\pi\omega$ we can find $K_\alpha>0$ such that $$\label{ourTM}
\int_0^1 e^{\alpha u^2}dx\leq K_\alpha,\qquad
\text{for all $u\in X$, $\|u\|_X\leq 1$.}$$ We list below our hypotheses on the nonlinearity $f$ in the *subcritical* case:
- Let $f\in C({{\mathbb R}})$ be a function such that $f(0)=0$ and denote $$F(t)=\int_0^t f(\tau)d\tau, \quad \mbox{for all $t\in{{\mathbb R}}$.}$$ Moreover, assume that there exist $t_0,M>0$ such that:
- $0<F(t)\leq M|f(t)|,$ for all $|t|\geq t_0$;
- $0<2F(t)\leq f(t)t$, for all $t\neq 0$;
- $\displaystyle\limsup_{t\to 0}\frac{F(t)}{t^2}<\frac{\lambda_1}{4\pi},$ ($\lambda_1$ provided by Proposition \[poinc\] below);
- $\displaystyle\lim_{|t|\to\infty}\frac{|f(t)|}{e^{\alpha t^2}}=0$, for all $\alpha>0$.
By a (weak) solution of problem we mean a function $u\in X$ satisfying (see Section 3). The following are our main results:
\[subcrit\] If ${\bf H}$ hold, then has a nontrivial solution $u\in H^{\nicefrac{1}{2}}({{\mathbb R}})$. If in addition $f$ is odd, then has infinitely many solutions in $H^{\nicefrac{1}{2}}({{\mathbb R}})$.
On symmetric domains, we also have the following result:
\[subcrit-sym\] If ${\bf H}$ hold, then the problem $$\left\{ \begin{array}{ll}
(-\Delta)^{\nicefrac{1}{2}} u=f(u) &\mbox{in $(-1,1)$} \\
u=0 &\mbox{in ${{\mathbb R}}\setminus (-1,1)$,}
\end{array}
\right.$$ has an even nontrivial solution $u\in H^{\nicefrac{1}{2}}({{\mathbb R}})$ decreasing on ${{\mathbb R}}^+$.
Now we turn to the *critical* case, under the following assumptions:
- Assume ${\bf H}(i)-(iii)$ and:
- there exists $0<\alpha_0<2\pi\omega$ such that $$\lim_{|t|\to\infty}\frac{|f(t)|}{e^{\alpha t^2}}=\left\{
\begin{array}{ll}
\infty & \mbox{if } 0<\alpha<\alpha_0 \\
0 & \mbox{if } \alpha>\alpha_0
\end{array};
\right.$$
- there exists $\psi\in X$ such that $\|\psi\|_X=1$ and $$\sup_{t\in{{\mathbb R}}^+}\Big(\frac{t^2}{4\pi}-\int_0^1 F(t \psi)dx\Big)<\frac{\omega}{2\alpha_0}.$$
For this case we have the following result:
\[C-state\] If ${\bf H'}$ hold, then has a nontrivial solution $u\in H^{\nicefrac{1}{2}}({{\mathbb R}})$.
These results establish a one dimensional fractional counterpart (with the additional information of symmetry and monotonicity of the solution in Theorem \[subcrit-sym\] for symmetric domains) of the results of [@DMR] for the local case in dimension two. As far as the critical case is concerned, typically when $f(t)\sim e^{\alpha_0 t^{2}}$ as $t\to\infty$, it is still unclear how to detect suitable (concentrating) optimizing sequences in $X$ for the fractional Trudinger-Moser inequality . However, we can prove that in this case the functional associated to the problem satisfies the Palais-Smale condition at each level $c<\omega/(2\alpha_0)$ and that the problem has a nontrivial solution under the additional hypothesis ${\bf H'}(v)$. We point out that, with a similar machinery, existence and multiplicity of solutions for fractional non-autonomous problems like $$\left\{ \begin{array}{ll}
(-\Delta)^{\nicefrac{1}{2}} u=f(x,u) &\mbox{in $(a,b)$} \\
u=0 &\mbox{in ${{\mathbb R}}\setminus (a,b)$,}
\end{array}
\right.$$ can be obtained under suitable assumptions on $f:(a,b)\times{{\mathbb R}}\to{{\mathbb R}}$.
Preliminaries
=============
First we recall some basic facts about the $\nicefrac{1}{2}$-Laplacian operator and the related function space $H^{\nicefrac{1}{2}}({{\mathbb R}})$, following mainly [@DPV]. For all $s\in (0,1)$, all measurable $u$ and all $x\in{{\mathbb R}}$ we set $$(-\Delta)^s u(x)=-\frac{C_s}{2}\int_{{{\mathbb R}}}\frac{u(x+y)+u(x-y)-2 u(x)}{|y|^{1+2s}}dy,$$ with the constant $$C_s=\Big[\int_{{\mathbb R}}\frac{1-\cos(\xi)}{|\xi|^{1+2s}}d\xi\Big]^{-1}$$ (see [@DPV Lemma 3.3]). We focus on the case $s=\nicefrac{1}{2}$. Note that $C_{\nicefrac{1}{2}}=\pi^{-1}$. We define $$H^{\nicefrac{1}{2}}({{\mathbb R}})=\left\{u\in L^2({{\mathbb R}}): \ \int_{{{\mathbb R}}^2}\frac{(u(x)-u(y))^2}{|x-y|^2}dxdy<\infty\right\}$$ and for all $u\in H^{\nicefrac{1}{2}}({{\mathbb R}})$ we introduce the Gagliardo seminorm $$[u]_{H^{\nicefrac{1}{2}}({{\mathbb R}})}=\left[\int_{{{\mathbb R}}^2}\frac{(u(x)-u(y))^2}{|x-y|^2}dxdy\right]^\frac{1}{2}$$ and the norm $$\|u\|_{H^{\nicefrac{1}{2}}({{\mathbb R}})}=\left(\|u\|^2_{L^2({{\mathbb R}})}+[u]^2_{H^{\nicefrac{1}{2}}({{\mathbb R}})}\right)^\frac{1}{2}.$$ We know that $(H^{\nicefrac{1}{2}}({{\mathbb R}}),\|\cdot\|_{H^{\nicefrac{1}{2}}({{\mathbb R}})})$ is a Hilbert space. Morever, by [@DPV Proposition 3.6] $$\label{equinorm}
\|(-\Delta)^{\nicefrac{1}{4}}u\|_{L^2({{\mathbb R}})}=(2\pi)^{-\frac{1}{2}}[u]_{H^{\nicefrac{1}{2}}({{\mathbb R}})},\quad
\text{for all $u\in H^{\nicefrac{1}{2}}({{\mathbb R}})$}.$$ Our main tool is a fractional Trudinger-Moser inequality (see Ozawa [@O Theorem 1] and Kozono, Sato $\&$ Wadade [@KSW Theorem 1.1]):
\[oza\] There exists $0<\omega\leq\pi$ with the following property: for all $0<\alpha<\omega$ there exists $H_\alpha>0$ such that $$\int_{{\mathbb R}}\big(e^{\alpha u^2}-1\big)dx\leq H_\alpha\|u\|_{L^2({{\mathbb R}})}^2,$$ for every $u\in H^{\nicefrac{1}{2}}({{\mathbb R}})$ with $\|(-\Delta)^{\nicefrac{1}{4}}u\|_{L^2({{\mathbb R}})}\leq 1$.
We do not possess an explicit formula for the optimal constant $\omega$, and neither we know whether the inequality above holds for $\alpha=\omega$. Now we turn to the space $X$, defined in . Clearly the only constant function in $X$ is $0$, so the seminorm $[\cdot]_{H^{\nicefrac{1}{2}}({{\mathbb R}})}$ turns out to be a norm on $X$, which we denote by $\|\cdot\|_X$. We have the following Poincaré-type inequality:
\[poinc\] There exists $\lambda_1>0$ such that for all $u\in X$ $$\|u\|_{L^2(0,1)}\leq\lambda_1^{-\frac{1}{2}}\|u\|_X.$$ Moreover, equality is realized by some $u\in X$ with $\|u\|_{L^2(0,1)}=1$.
We set $$S=\left\{u\in X: \ \|u\|_{L^2(0,1)}=1\right\}$$ and equivalently prove that $$\label{lam}
\inf_{u\in S}\|u\|_X^2=\lambda_1>0.$$ Clearly $\lambda_1\geq 0$. We first prove that $\lambda_1$ is attained in $S$. Let $(u_n)\subset S$ be a minimizing sequence for . In particular, $\sup_{n\in{{\mathbb N}}}[u]^2_{H^{\nicefrac{1}{2}}({{\mathbb R}})}<\infty$ and $(u_n)$ is bounded in $L^2(0,1)$. In light of [@DPV Theorem 7.1], there exists $u\in L^2(0,1)$ such that, up to a subsequence, $u_n\to u$ in $L^2(0,1)$. We extend $u$ by setting $u(x)=0$ for all $x\in{{\mathbb R}}\setminus(0,1)$, so $u\in L^2({{\mathbb R}})$ and $u_n\to u$ a.e. in ${{\mathbb R}}$. Fatou’s lemma yields $$\int_{{{\mathbb R}}^2}\frac{|u(x)-u(y)|^2}{|x-y|^2}dx\leq\liminf_{n}\int_{{{\mathbb R}}^2}\frac{|u_n(x)-u_n(y)|^2}{|x-y|^2}dx=\lambda_1,$$ hence $u\in X$. Moreover, $\|u\|_{L^2(0,1)}=1$, hence $u\in S$, in particular $u\neq 0$ and $\|u\|_X^2=\lambda_1>0$.
Due to Proposition \[poinc\], we can prove further properties of $X$:
\[hilb\] $(X,\|\cdot\|_X)$ is a Hilbert space.
Clearly the norm $\|\cdot\|_X$ is induced by a inner product, defined for all $u,v\in X$ by $$\langle u,v\rangle_X=\int_{{{\mathbb R}}^2}\frac{(u(x)-u(y))(v(x)-v(y))}{|x-y|^2}dxdy.$$ Moreover, by Proposition \[poinc\] we have for all $u\in X$ $$\label{x-norm}
\|u\|_X\leq\|u\|_{H^{\nicefrac{1}{2}}({{\mathbb R}})}\leq(\lambda_1^{-1}+1)^{\frac{1}{2}}\|u\|_X.$$ So, completeness of $X$ follows at once from that of $H^{\nicefrac{1}{2}}({{\mathbb R}})$.
We specialize Theorem \[oza\] to the space $X$:
\[mti\] For all $0<\alpha<2\pi\omega$ there exists $K_\alpha>0$ such that $$\int_0^1 e^{\alpha u^2}dx\leq K_\alpha$$ for all $u\in X$, $\|u\|_X\leq 1$.
Fix $u\in X$ with $\|u\|_X\leq 1$. Set $v=(2\pi)^{\nicefrac{1}{2}}u$, then $v\in H^{\nicefrac{1}{2}}({{\mathbb R}})$ and by (\[equinorm\]) we have $\|(-\Delta)^{\nicefrac{1}{4}}v\|_{L^2({{\mathbb R}})}\leq 1$. Set $\tilde\alpha=(2\pi)^{-1}\alpha$, so $0<\tilde\alpha<\omega$ and by Theorem \[oza\] and Proposition \[poinc\] we have $$\begin{aligned}
\int_0^1 e^{\alpha u^2}dx & =\int_{{\mathbb R}}\big[e^{\tilde\alpha v^2}-1\big]dx+1 \\
& \leq H_{\tilde\alpha}\|v\|_{L^2(0,1)}^2+1\leq\frac{2\pi H_{\tilde\alpha}}{\lambda_1}+1:=K_\alpha,\end{aligned}$$ which concludes the proof.
We point out a important consequence of the results above:
\[sum\] $e^{u^2}\in L^1(0,1)$ for every $u\in X$.
We follow Trudinger [@T]. Choose $0<\alpha<\omega$ and set for all $t\in{{\mathbb R}}$ $$\phi(t)=\frac{e^{\alpha t^2}-1}{H_\alpha} \quad \mbox{($H_\alpha$ defined as in Theorem \ref{oza}).}$$ We introduce the Orlicz norm induced by $\phi$ putting for all measurable $u:(0,1)\to{{\mathbb R}}$ $$\|u\|_\phi=\inf\Big\{\gamma>0 \ : \ \int_0^1 \phi\Big(\frac{u}{\gamma}\Big)dx\leq 1\Big\},$$ and the corresponding Orlicz space $L_{\phi^*}(0,1)$, see Krasnosel’skiĭ $\&$ Rutickiĭ [@KR p.67] for the definition. We prove (by identifying a function $v\in X$ with its restriction to $(0,1)$) that $$\label{cont-emb}
X\hookrightarrow L_{\phi^*}(0,1) \quad\mbox{continuously.}$$ For all $v\in X\setminus\{0\}$, we set $w=\|v\|_{H^{\nicefrac{1}{2}}({{\mathbb R}})}^{-1}v$, so by (\[equinorm\]) $$\|(-\Delta)^{\nicefrac{1}{4}}w\|_{L^2({{\mathbb R}})}=\frac{[v]_{H^{\nicefrac{1}{2}}({{\mathbb R}})}}{(2\pi)^{\nicefrac{1}{2}}\|v\|_{H^{\nicefrac{1}{2}}({{\mathbb R}})}}\leq (2\pi)^{-\nicefrac{1}{2}}<1.$$ So, in light of Theorem \[oza\], we have $$\int_0^1 \phi\Big(\frac{v}{\|v\|_{H^{\nicefrac{1}{2}}({{\mathbb R}})}}\Big)dx=\int_{{\mathbb R}}\frac{e^{\alpha w^2}-1}{H_\alpha} dx\leq\|w\|_{L^2({{\mathbb R}})}^2\leq 1,$$ hence by $$\|v\|_\phi\leq\|v\|_{H^{\nicefrac{1}{2}}({{\mathbb R}})}\leq(\lambda_1^{-1}+1)^\frac{1}{2}\|v\|_X.$$ Thus, is proved. Now fix $u\in X$ and set $\tilde u=\alpha^{-\nicefrac{1}{2}}u$. By the results of Fiscella, Servadei $\&$ Valdinoci [@FSV], we know that $C^\infty_c(0,1)$ is a dense linear subspace of $X$. So, there exists a sequence $(\psi_n)$ in $C^\infty_c(0,1)$ such that $\psi_n\to\tilde u$ in $X$. By , we have $\psi_n\to\tilde u$ in $L_{\phi^*}(0,1)$ as well. In particular $\tilde u\in E_\phi$, namely the closure of the set of bounded functions of $X$ in $L_{\phi^*}(0,1)$. From a general result on Orlicz spaces (see [@KR formula (10.1), p. 81]) it follows that $$\int_0^1\phi(\tilde u)dx<\infty,$$ which immediately yields the conclusion.
We conclude this section with a technical result which we shall use later:
\[lions\] If $(v_n)$ is a sequence in $X$ with $\|v_n\|_X=1$ for all $n\in{{\mathbb N}}$ and $v_n\rightharpoonup v$ in $X$, $0<\|v\|_X<1$, then for all $0<\alpha<2\pi\omega$ and all $1<p<(1-\|v\|_X^2)^{-1}$ the sequence $(e^{\alpha v_n^2})$ is bounded in $L^p(0,1)$.
By applying the generalized Hölder inequality with exponents $\gamma_1,\gamma_2,\gamma_3>1$ such that $\gamma_1\alpha<2\pi\omega$ and $\gamma_1^{-1}+\gamma_2^{-1}+\gamma_3^{-1}=1$, we have $$\begin{aligned}
\int_0^1 e^{p\alpha v_n^2}dx &=\int_0^1 e^{p\alpha [(v_n-v)^2+2(v_n-v)v+v^2 ]}dx \\
& \leq\left[\int_0^1 e^{\gamma_1 p\alpha(v_n-v)^2}dx\right]^\frac{1}{\gamma_1}\left[\int_0^1 e^{2\gamma_2 p\alpha(v_n-v)v}dx\right]^\frac{1}{\gamma_2}\left[\int_0^1 e^{\gamma_3 p\alpha v^2}dx\right]^\frac{1}{\gamma_3}.\end{aligned}$$ We estimate the three integrals separately. First we note that $$\|v_n-v\|^2_X=1-2\langle v_n,v\rangle_X+\|v\|_X^2\to 1-\|v\|_X^2<\frac{1}{p},$$ so for $n\in{{\mathbb N}}$ big enough we have $\|v_n-v\|_X^2<\nicefrac{1}{p}$. Hence, by Corollary \[mti\] $$\int_0^1 e^{\gamma_1 p\alpha(v_n-v)^2}dx\leq\int_0^1 e^{\gamma_1\alpha\left(\frac{v_n-v}{\|v_n-v\|_X}\right)^2}dx\leq K_{\gamma_1\alpha}.$$ Besides, by Corollary \[mti\] and Proposition \[sum\] we have for some $c_1>0$ $$\begin{aligned}
\int_0^1 e^{2\gamma_2 p\alpha(v_n-v)v}dx &\leq \int_0^1e^{2\left(\frac{\alpha}{2}\right)^{\nicefrac{1}{2}}\frac{v_n-v}{\|v_n-v\|_X}(c_1 v)}dx\leq\int_0^1e^{\frac{\alpha}{2}\left(\frac{v_n-v}{\|v_n-v\|_X}\right)^2+(c_1 v)^2}dx \\
& \leq\left[\int_0^1 e^{\alpha\left(\frac{v_n-v}{\|v_n-v\|_X}\right)^2}dx\right]^{\nicefrac{1}{2}}\left[\int_0^1 e^{2c_1^2 v^2}dx\right]^{\nicefrac{1}{2}}\leq K_\alpha\left[\int_0^1 e^{2c_1^2 v^2}dx\right]^{\nicefrac{1}{2}}.\end{aligned}$$ Finally, clearly $$\int_0^1 e^{\gamma_3 p\alpha v^2}dx<\infty.$$ Thus, $(e^{\alpha v_n^2})$ is bounded in $L^p(0,1)$.
Proofs of Theorems \[subcrit\] and \[subcrit-sym\]
==================================================
In this section we act under ${\bf H}$. We give our problem a variational formulation by setting for all $u\in X$ $$\varphi(u)=\frac{\|u\|_X^2}{4\pi}-\int_0^1 F(u)dx.$$ Proposition \[sum\], ${\bf H}(i)$ and ${\bf H}(iv)$ imply that $\varphi\in C^1(X)$. By , its derivative is given for all $u,v\in X$ by $$\begin{aligned}
\langle\varphi'(u),v\rangle &=\frac{1}{2\pi}\langle u,v\rangle_X-\int_0^1 f(u)v dx \\
&= \int_{{\mathbb R}}(-\Delta)^{\nicefrac{1}{4}}u(-\Delta)^{\nicefrac{1}{4}}v dx-\int_0^1 f(u)v dx.\end{aligned}$$ In particular, if $u\in X$ and $\varphi'(u)=0$, then for all $v\in X$ $$\label{weaks}
\int_{{\mathbb R}}(-\Delta)^{\nicefrac{1}{4}}u(-\Delta)^{\nicefrac{1}{4}}v dx=\int_0^1 f(u)v dx,$$ namely $u$ is a (weak) solution of . First we point out some consequences of ${\bf H}$. By ${\bf H}(iv)$, for all $\alpha>0$ there exists $c_2>0$ such that $$\label{grow}
|f(t)|\leq c_2 e^{\alpha t^2}, \,\,\quad \mbox{for all $t\in{{\mathbb R}}$.}$$ By virtue of ${\bf H}(i)$, there exists $c_3>0$ such that $$\label{expo}
F(t)\geq c_3 e^\frac{|t|}{M}, \quad \mbox{for all $|t|\geq t_0$.}$$ Finally, by ${\bf H}(i)$ and ${\bf H}(ii)$, for all $\varepsilon>0$ there exists $t_\varepsilon>0$ such that $$\label{ar}
F(t)\leq\varepsilon f(t)t, \,\,\quad\text{for all $|t|\geq t_{\varepsilon}$}.$$ The following lemma shows a compactness property of $\varphi$:
\[ps\] $\varphi$ satisfies the Palais-Smale condition at every level $c\in{{\mathbb R}}$.
Let $(u_n)$ be a sequence in $X$ such that $\varphi(u_n)\to c$ ($c\in{{\mathbb R}}$) and $\varphi'(u_n)\to 0$ in $X^*$. We need to show that $(u_n)$ has a convergent subsequence in $X$. By (\[ar\]), for all $0<\varepsilon<\nicefrac{1}{2}$ we can find $c_4>0$ such that for all $t\in{{\mathbb R}}$ $$F(t)\leq\varepsilon f(t)t+c_4.$$ For $n\in{{\mathbb N}}$ big enough we have $\varphi(u_n)\leq c+1$ and $\|\varphi'(u_n)\|_{X^*}\leq 1$, so $$\begin{aligned}
c+1 & \geq\frac{\|u_n\|_X^2}{4\pi}-\int_0^1[\varepsilon f(u_n)u_n+c_4]dx=\left(\frac{1}{2}-\varepsilon\right)\frac{\|u_n\|_X^2}{2\pi}+\varepsilon\langle\varphi'(u_n),u_n\rangle-c_4 \\
& \geq \left(\frac{1}{2}-\varepsilon\right)\frac{\|u_n\|_X^2}{2\pi}-\varepsilon\|u_n\|_X-c_4.\end{aligned}$$ Thus, $(u_n)$ is bounded in $X$. By Proposition \[poinc\], $(u_n)$ is bounded in $H^{\nicefrac{1}{2}}({{\mathbb R}})$ as well. By [@DPV Theorem 7.1 and Theorem 6.10], passing to a subsequence we may assume that $u_n\rightharpoonup u$ in both $X$ and $H^{\nicefrac{1}{2}}({{\mathbb R}})$, and that $u_n\to u$ in $L^q(0,1)$ for all $q\geq 1$ and $u_n(x)\to u(x)$ a.e. in $(0,1)$. In particular, there exists $c_5>0$ such that $\|u_n\|_X^2\leq c_5$, for all $n\in{{\mathbb N}}$. Observe that $(f(u_n))$ is bounded in $L^2(0,1)$. Indeed, by choosing $0<\alpha<\pi\omega/c_5,$ by Corollary \[mti\] and we get $$\label{Lim-2}
\int_0^1 f^2(u_n)dx\leq c_2^2\int_0^1 e^{2\alpha u_n^2}dx\leq
c_2^2\int_0^1 e^{2\alpha c_5 \left(\frac{u_n}{\|u_n\|_X}\right)^2}dx\leq c_2^2K_{2\alpha c_5}.$$ Passing to a subsequence, we have $f(u_n)\rightharpoonup f(u)$ in $L^2(0,1)$. As a consequence, for all $v\in X$ we have $$\langle\varphi'(u),v\rangle=\frac{1}{2\pi}
\langle u,v\rangle_X-\int_0^1 f(u)v dx=\lim_n\langle\varphi'(u_n),v\rangle=0,$$ namely $u$ is a solution of . Observe that $$\lim_n\int_0^1 f(u_n) u_ndx=\int_0^1 f(u) u dx,$$ since by and $f(u_n)\rightharpoonup f(u)$ in $L^2(0,1)$ it holds $$\begin{aligned}
\label{convfss}
&\Big|\int_0^1 f(u_n) u_n dx -\int_0^1 f(u) u dx\Big| \\
&\leq
\|f(u_n)\|_{L^2(0,1)}\|u_n-u\|_{L^2(0,1)}
+\Big|\int_0^1 (f(u_n)-f(u))u dx\Big|. \notag\end{aligned}$$ In turn we have $$\lim_n \frac{\|u_n\|_X^2}{2\pi} =\lim_n\left[\int_0^1 f(u_n)u_n dx+\langle \varphi'(u_n),u_n\rangle\right]
=\int_0^1 f(u)u dx= \frac{\|u\|_X^2}{2\pi},$$ which immediately yields the assertion.
The following lemmas deal with the mountain pass geometry for $\varphi$:
\[mount1\] There exist $\rho,a>0$ such that $\varphi(u)\geq a$ for all $u\in X$ with $\|u\|_X=\rho$.
By ${\bf H}(iii)$ there exist $0<\mu<\lambda_1$ and $\delta>0$ such that for all $|t|<\delta$ we have $F(t)\leq\mu t^2/(4\pi)$. Fix $q>2$, $0<\alpha<2\pi\omega$ and $r>1$ such that $r\alpha<2\pi\omega$ as well. By (\[grow\]) there exists $c_6>0$ such that for all $|t|\geq\delta$ we have $F(t)\leq c_6 e^{\alpha t^2}|t|^q$. Summarizing, for all $t\in{{\mathbb R}}$, we obtain $$F(t)\leq\frac{\mu t^2}{4\pi}+c_6 e^{\alpha t^2}|t|^q.$$ In what follows we use the estimate above, Proposition \[poinc\], Corollary \[mti\] and the continuous embedding $X\hookrightarrow L^{r'q}(0,1)$. For all $u\in X$, $\|u\|_X\leq 1$ we have (for a convenient $c_7>0$) $$\begin{aligned}
\varphi(u) & \geq \frac{\|u\|_X^2}{4\pi}-\int_0^1\Big[\frac{\mu u^2}{4\pi}+c_6 e^{\alpha u^2}|u|^q \Big]dx \\
& \geq \left(1-\frac{\mu}{\lambda_1}\right)\frac{\|u\|_X^2}{4\pi}-c_6\Big(\int_0^1 e^{r\alpha u^2}dx\Big)^{1/r}\Big(\int_0^1 |u|^{r'q}\Big)^{1/r'} \\
& \geq \left(1-\frac{\mu}{\lambda_1}\right)\frac{\|u\|_X^2}{4\pi}-c_7 \|u\|_X^q.\end{aligned}$$ Set for all $t\geq 0$ $$g(t)=\left(1-\frac{\mu}{\lambda_1}\right)\frac{t^2}{4\pi}-c_7 t^q.$$ By a straightforward computation we find $0<\rho<1$ such that $g(\rho)=a>0$. So, for all $u\in X$ with $\|u\|_X=\rho$ we have $\varphi(u)\geq a$.
\[asympt\] If $Y\subset X$ is a linear subspace generated by bounded functions and ${\rm dim}(Y)<\infty$, then $\sup_{u\in Y}\varphi(u)<\infty$ and $$\lim_{\substack{\|u\|_X\to\infty \\ u\in Y}}\varphi(u)=-\infty.$$
Fix $p>2$. By (\[expo\]), we have $|t|^{-p} F(t)\to\infty$ for $|t|\to\infty$, so we can find $c_{8}>0$ such that for all $t\in{{\mathbb R}}$ we have $F(t)\geq |t|^p-c_{8}$. Whence, for some $c_9>0$, we obtain for all $u\in Y$ $$\varphi(u) \leq \frac{\|u\|_X^2}{4\pi}-\|u\|_{L^p(0,1)}^p+c_{8}
\leq \frac{\|u\|_X^2}{4\pi}-c_9\|u\|_X^p+c_{8},$$ which readily yields the assertion.
[*Proof of Theorem \[subcrit\] concluded.*]{} The existence of one solution follows by applying the Mountain Pass Theorem (see Rabinowitz [@rab-book Theorem 2.2]) to $\varphi$ and combining Lemmas \[ps\], \[mount1\] and \[asympt\]. Concerning the multiplicity, we apply [@rab-book Theorem 9.12].
[*Proof of Theorem \[subcrit-sym\] concluded.*]{} Given a nonnegative function $u\in X$ and any $H=(a,\infty)$ with $a<0$, we have the following inequality for the polarization $u^H$ (see Baernstein [@baer Theorem 2, p. 58]) $$\int_{{{\mathbb R}}^2}\frac{(u^H(x)-u^H(y))^2}{|x-y|^2}dxdy
\leq \int_{{{\mathbb R}}^2}\frac{(u(x)-u(y))^2}{|x-y|^2}dxdy,$$ which implies that $\varphi(u^H)\leq \varphi(u)$, for all nonnegative $u$ of $X$. The existence of an even solution on $(-1,1)$, decreasing on $(0,1)$, equal to zero on ${{\mathbb R}}\setminus(-1,1)$ follows by the (symmetric) Mountain Pass Theorem of Van Schaftingen [@vanSch Theorem 3.2] applied to the functional $\varphi$ on $X$ with the $V$ therein chosen as $V=L^2(-1,1)$, and on account of Lemmas \[ps\], \[mount1\] and \[asympt\].
\[ex1\]Fix $1<q<2$ and $0<\mu<\nicefrac{\lambda_1}{2\pi}$. Define $f:{{\mathbb R}}\to{{\mathbb R}}$ by setting, for all $t\geq 0$, $$f(t)=\left\{
\begin{array}{ll}
\mu t & \mbox{if } 0\leq t\leq 1, \\
\mu t^{q-1} e^{t^q-1} & \mbox{if } t>1,
\end{array}
\right.$$ and $f(t)=-f(-t)$ for all $t<0$. It is easily seen that $f$ is continuous, odd and satisfies ${\bf H}$. By Theorem \[subcrit\], then, the corresponding problem admits infinitely many solutions.
Proof of Theorem \[C-state\]
============================
In this section, we consider the critical case, that is, we act under ${\bf H'}$. An important remark here is that (\[grow\]) holds only for $\alpha>\alpha_0$. We prove that the Palais-Smale condition is satisfied only for levels in a certain range:
\[ps-2\] If $f$ satisfies ${\bf H'}$, then $\varphi$ satisfies the Palais-Smale condition at any level $c<\omega/(2\alpha_0)$.
Let $(u_n)$ be a sequence in $X$ such that $\varphi(u_n)\to c$ and $\varphi'(u_n)\to 0$ in $X^*$. Arguing as in the proof of Lemma \[ps\], it is readily seen that there exists a positive constant $c_{10}$ such that, for all $n\in{{\mathbb N}}$, $$\max\Big\{\|u_n\|_X^2,\int_0^1 f(u_n)u_n dx,\int_0^1 F(u_n)dx\Big\}\leq c_{10}.$$ Moreover, up to a subsequence, $u_n\rightharpoonup u$ in $X$ and $u_n\to u$ in $L^q(0,1)$ for all $q\geq 1$. Reasoning as in [@DMR Lemma 2.1] we have $f(u_n)\to f(u)$ in $L^1(0,1)$. Whence, in light of ${\bf H}(i)$ it follows that $\int_0^1 F(u_n)dx\to\int_0^1 F(u)dx$. So we have $$\label{norm-conv}
\frac{\|u_n\|_X^2}{4\pi}\to c+\int_0^1 F(u)dx.$$ Then, since $\varphi'(u_n)\to 0$, we get $$\int_0^1 f(u_n)u_n dx\to 2\Big(c+\int_0^1 F(u)dx\Big).$$ So, by means of ${\bf H}(ii)$, we have $$c=\frac{1}{2}\lim_n\int_0^1\left[f(u_n)u_n-2F(u_n)\right]dx\geq 0.$$ Besides, for all $v\in C^\infty_c(0,1)$ we have $$\langle\varphi'(u),v\rangle=\frac{1}{2\pi}
\langle u,v\rangle_X-\int_0^1 f(u)v dx=\lim_n\langle\varphi'(u_n),v\rangle=0.$$ Recalling again the density result of [@FSV], we have $\langle\varphi'(u),v\rangle=0$ for all $v\in X$, namely $u$ is a solution of . By ${\bf H}(ii)$ and taking $v=u$ we have $$\varphi(u)=\frac{1}{2}\Big(\frac{\|u\|_X^2}{2\pi}-2\int_0^1 F(u)dx\Big)\geq\frac{1}{2}\Big(\frac{\|u\|_X^2}{2\pi}-\int_0^1 f(u)udx\Big)=0.$$ Summarizing, we have $c\geq 0$ and $\varphi(u)\geq 0$. Now we distinguish three cases.
- If $c=0$, then by virtue of and $\varphi(u)\geq 0$, we get $$\frac{\|u\|_X^2}{4\pi}\geq\int_0^1 F(u)dx=\lim_n\frac{\|u_n\|_X^2}{4\pi}.$$ Recalling that $u_n\rightharpoonup u$ in $X$, we conclude that $u_n\to u$ in $X$.
- If $c>0$, $u=0$, then the sequence $(f(u_n))$ is bounded in $L^q(0,1),$ for some $q>1$. Indeed, since $c<\omega/(2\alpha_0)$ we can find $q>1$, ${\varepsilon}>0$ and $\alpha_0<\alpha<2\pi\omega$ such that $2\pi (2c+{\varepsilon})q\alpha:=\beta<2\pi\omega$. Since $\|u_n\|_X^2\to 4\pi c$, for $n\in{{\mathbb N}}$ big enough we have $\|u_n\|_X^2<2\pi(2c+{\varepsilon})$. So, applying and Corollary \[mti\] we have $$\int_0^1|f(u_n)|^qdx\leq c_2^q\int_0^1 e^{q\alpha u_n^2}dx\leq c_2^q\int_0^1 e^{\beta\big(\frac{u_n}{\|u_n\|_X}\big)^2}dx\leq c_2^q K_\beta.$$ Recalling that $u_n\to 0$ in $L^{q'}(0,1)$ and that $$0\leq\int_0^1 f(u_n)u_n dx\leq\|f(u_n)\|_{L^q(0,1)}\|u_n\|_{L^{q'}(0,1)},$$ from $\varphi'(u_n)\to 0$ we have immediately $$\lim_n\frac{\|u_n\|_X^2}{2\pi}=\lim_n\int_0^1 f(u_n)u_ndx=0,$$ whence $u_n\to 0$ in $X$. Thus $\varphi(u_n)\to 0<c$, a contradiction.
- If $c>0$, $u\neq 0$, then we prove that $\varphi(u)=c$. This equality yields the strong convergence by means of . We know that $\varphi(u)\leq c$, so by contradiction assume $\varphi(u)<c$. Then $$\|u_n\|_X^2\to 4\pi\Big(c+\int_0^1 F(u)dx\Big)>\|u\|_X^2.$$ Set $v_n=\|u_n\|_X^{-1}u_n$ and $v=\big(4\pi c+4\pi\int_0^1 F(u)dx\big)^{-\nicefrac{1}{2}}u$. So we have $\|v_n\|_X=1$, $0<\|v\|_X<1$ and $v_n\rightharpoonup v$ in $X$. Since $c<\omega/(2\alpha_0)$, we can find $q>1$, $\alpha_0<\alpha<2\pi\omega$ such that $qc<\omega/(2\alpha)$, hence (recall $\varphi(u)\geq 0$) $$2q\alpha<\frac{\omega}{c-\varphi(u)}.$$ We have $$\lim_n q\alpha\|u_n\|_X^2=4\pi q\alpha\Big(c+\int_0^1 F(u)dx\Big)<2\pi\omega\frac{c+\int_0^1 F(u)dx}{c-\varphi(u)}.$$ We can choose $p>1$, $0<\gamma<2\pi\omega$ such that $$p<\frac{c+\int_0^1 F(u)dx}{c-\varphi(u)}=\frac{1}{1-\|v\|_X^2}$$ and for $n\in{{\mathbb N}}$ big enough $$q\alpha\|u_n\|_X^2<p\gamma.$$ Since $\gamma<2\pi\omega$, by Lemma \[lions\] the sequence $(e^{\gamma v_n^2})$ is bounded in $L^p(0,1)$, so $$\int_0^1|f(u_n)|^q dx\leq c_2^q\int_0^1 e^{q\alpha u_n^2}dx
\leq c_2^q\int_0^1 e^{p(\gamma v_n^2)}dx,$$ which proves that $(f(u_n))$ is bounded in $L^q(0,1)$. Passing if necessary to a subsequence, we have $f(u_n)\rightharpoonup f(u)$ in $L^q(0,1)$ while $u_n\to u$ in $L^{q'}(0,1)$. So, $$\begin{aligned}
\label{convfss}
&\Big|\int_0^1 f(u_n) u_n dx -\int_0^1 f(u) u dx\Big| \\
&\leq
\|f(u_n)\|_{L^q(0,1)}\|u_n-u\|_{L^{q'}(0,1)}
+\Big|\int_0^1 (f(u_n)-f(u))u dx\Big|, \notag\end{aligned}$$ hence $$\lim_n\int_0^1 f(u_n)u_n dx=\int_0^1 f(u)udx.$$ As above, this yields $u_n\to u$ in $X$. This in turn implies $\varphi(u)=c$, a contradiction.
This concludes the proof.
[*Proof of Theorem \[C-state\] concluded.*]{} The conclusions of Lemmas \[mount1\] and \[asympt\] still hold, with small changes in the proofs. Moreover, if we fix $t>0$ such that $\varphi(t\psi)<\varphi(0)$ and denote by $\Gamma$ the set of continuous paths in $X$ joining $0$ and $t\psi$ and set $$c=\inf_{\gamma\in\Gamma}\max_{\tau\in[0,1]}\varphi(\gamma(\tau)),$$ by ${\bf H'}(v)$ we see that $c<\omega/(2\alpha_0)$. Thus, by Lemma \[ps-2\], $\varphi$ satisfies the Palais-Smale condition at level $c$. By the Mountain Pass Theorem, then, has a nontrivial solution.
\[ex2\]Fix $0<\mu<\nicefrac{\lambda_1}{2\pi}$, $0<\alpha_0<2\pi\omega$. Define $f:{{\mathbb R}}\to{{\mathbb R}}$ by setting, for all $t\geq 0$, $$f(t)=\left\{
\begin{array}{ll}
\mu t & \mbox{if } 0\leq t\leq 1, \\
\mu te^{\alpha_0(t^2-1)} & \mbox{if } t>1,
\end{array}
\right.$$ and $f(t)=-f(-t)$ for all $t<0$. It is easily seen that $f$ is continuous and satisfies ${\bf H'}(i)-(iv)$. If there exists $\psi\in X$ satisfying ${\bf H'}(v)$, then by Theorem \[C-state\] the corresponding problem admits a nontrivial solution.
[99]{}
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, Minimax methods in critical point theory with applications to differential equations. CBMS Reg. Conf. Ser. Math. 65, Amer. Math. Soc., Providence, RI, 1986.
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, Symmetrization and minimax principles, Commun. Contemp. Math. [**7**]{} (2005), 463–481.
, Solitary waves of nonlinear dispersive evolution equations with critical power nonlinearities, J. Differential Equations [**69**]{} (1987), 192–203.
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'The Square Kilometre Array (SKA) is a planned large radio interferometer designed to operate over a wide range of frequencies, and with an order of magnitude greater sensitivity and survey speed than any current radio telescope. The SKA will address many important topics in astronomy, ranging from planet formation to distant galaxies. However, in this work, we consider the perspective of the SKA as a facility for studying physics. We review four areas in which the SKA is expected to make major contributions to our understanding of fundamental physics: cosmic dawn and reionisation; gravity and gravitational radiation; cosmology and dark energy; and dark matter and astroparticle physics. These discussions demonstrate that the SKA will be a spectacular physics machine, which will provide many new breakthroughs and novel insights on matter, energy and spacetime.'
bibliography:
- 'refs\_grav.bib'
- 'refs\_astropart.bib'
- 'refs\_cosmology.bib'
- 'refs\_eor.bib'
- 'refs\_other.bib'
title: Fundamental Physics with the Square Kilometre Array
---
astroparticle physics — cosmology — gravitation — pulsars:general — reionization — telescopes
Introduction {#sec:intro}
============
The Square Kilometre Array (SKA) is a large international collaboration, with the goal of building the world’s largest and most powerful radio telescope. The first phase of the SKA (“SKA1”) will begin operations in the early 2020s, and will comprise two separate arrays: SKA1-Low, which will consist of around 130000 low-frequency dipoles in Western Australia, and SKA1-Mid, which will be composed of $\sim$200 dishes in the Karoo region of South Africa [@dewdney2016; @braun2017]. The second phase, SKA2 will be an order of magnitude larger in collecting area than SKA1, and will take shape in the late 2020s.
The science case for the SKA is extensive and diverse: the SKA will deliver spectacular new data sets that are expected to transform our understanding of astronomy, ranging from planet formation to the high-redshift Universe [@aaska2015]. However, the SKA will also be a powerful machine for probing the frontiers of fundamental physics. To fully understand the SKA’s potential in this area, a focused workshop on “Fundamental Physics with the Square Kilometre Array”[^1] was held in Mauritius in May 2017, in which radio astronomers and theoretical physicists came together to jointly consider ways in which the SKA can test and explore fundamental physics.
This paper is not a proceedings from this workshop, but rather is a white paper that fully develops the themes explored. The goal is to set out four broad directions for pursuing new physics with the SKA, and to serve as a bridging document accessible for both the physics and astronomy communities. In §\[sec\_eor\] we consider cosmic dawn and reionisation, in §\[sec\_grav\] discuss strong gravity and pulsars, in §\[sec\_cosmology\] we examine cosmology and dark energy, and in §\[sec\_astropart\] we review dark matter and astroparticle physics. In each of these sections, we introduce the topic, set out the key science questions, and describe the proposed experiments with the SKA.
Conclusions
===========
Physicists seek to understand the nature of matter, energy and spacetime, plus how the three of these have interacted over cosmic time. While great strides have come from terrestrial and solar system experiments, it is increasingly clear that immense progress lies ahead through studying the cosmos.
Modern radio telescopes now have the capability to gather enormous statistical samples of celestial objects, and to make ultra-precise measurements of astrophysical effects. In this paper, we have explained the many ways in which the Square Kilometre Array will push far beyond the current frontiers in these areas, and will allow us to ask and answer new questions about cosmology, gravity, dark matter, dark energy, and more. The SKA will not just be a revolutionary facility for astronomy, but will also be an extraordinary machine for advancing fundamental physics.
[^1]: See [http://skatelescope.ca/fundamental-physics-ska/]{}.
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'L’articolo, dopo una breve introduzione sugli algoritmi genetici e sul loro funzionamento, presenta un tipo di algoritmo genetico chiamato *Viral Search*. Ne vengono presentati i concetti chiave, viene derivato formalmente l’algoritmo e sono eseguiti test numerici volti a illustrarne potenzialità e limiti.'
author:
- 'Matteo Gardini[^1]'
bibliography:
- 'Bibliografia.bib'
title: 'Viral Search: un algoritmo genetico'
---
Introduzione
============
Funzionamento
=============
Viral Search
============
Risultati Numerici
==================
Conclusioni
===========
[^1]: [email protected].
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'CO$_{2}$ ices are known to exist in different astrophysical environments. In spite of this, its physical properties (structure, density, refractive index) have not been as widely studied as those of water ice. It would be of great value to study the adsorption properties of this ice in conditions related to astrophysical environments. In this paper, we explore the possibility that CO$_{2}$ traps relevant molecules in astrophysical environments at temperatures higher than expected from their characteristic sublimation point. To fulfil this aim we have carried out desorption experiments under High Vacuum conditions based on a Quartz Crystal Microbalance and additionally monitored with a Quadrupole Mass Spectrometer. From our results, the presence of CH$_{4}$ in the solid phase above the sublimation temperature in some astrophysical scenarios could be explained by the presence of several retaining mechanisms related to the structure of CO$_{2}$ ice.'
author:
- Ramon Luna
- Carlos Millán
- Manuel Domingo
- Miguel Ángel Satorre
title: |
Thermal desorption of CH$_{4}$\
retained in CO$_{2}$ ice
---
Introduction
============
The physical properties of ices present in astrophysical scenarios are related to their porous structure. One of them is the ice capacity to retain molecules above their characteristic sublimation temperature.
This feature is relevant in the physics of gases present in atmospheres of some astrophysical environments, for instance: Massive stars [@Viti04], giant planets [@Hersant04] and objects of the Solar System such as Triton [@Rubincam03].
It is known that water ice is the most abundant molecule in interstellar icy grain mantles and on some objects of our solar system, but CO$_{2}$ has been revealed as another abundant molecule [@Graauw96; @Gurtler96], and is even the major component in some astrophysical scenarios, such as, for instance, Mars in our Solar System. CO$_{2}$ has been studied from several points of view: Its optical constants (n, k); integrated absorption coefficient (A); spectral properties before and after UV photolysis and ion irradiation, using infrared spectroscopy, and density have been widely derived in previous works [@Wood82; @Hudgins93; @Ehrenfreund97; @Baratta98].
In spite of the results referred to above, it is necessary to continue researching to improve our knowledge of the relationship between the structure and the adsorption properties of CO$_{2}$ at low temperatures. This is important because it can influence the abundance of volatile molecules such as CH$_{4}$, N$_{2}$, CO, etc.
One of the first experiments performed in this area is that of @Schulze80. They studied the density as a function of the deposition temperature, relating this parameter with the adsorption capacity of CO$_{2}$ ice, from 4 to 87 K. They found that the density increases from to 1.7 g cm$^{-3}$ as the deposition temperature increases. As a result, the adsorption capacity decreases with the increasing deposition temperature. They refer to density as the structure including voids (true density). In this paper, we will use the same definition.
{width="120mm"}
To the best of our knowledge, there have been no additional experiments designed to determine the relationship between the structure and the adsorption capacity of CO$_{2}$ ice. With the aim of filling this gap, our laboratory is currently carrying out an exhaustive set of experiments in order to understand the properties of the CO$_{2}$ structure in different conditions.
The experiments presented in this paper are intended to better understand the behaviour of relevant astrophysical molecules co-deposited with CO$_{2}$ and subsequently heated at a fixed rate. A similar technique, Thermal Programmed Desorption (TPD) performed under Ultra High Vacuum (UHV) conditions, has been used to study water ice structure [@Collings03; @Collings04]. Among the structural studies on water it is necessary to highlight those where the porosity of its amorphous form has been studied with other techniques [@Bar-Nun85; @Jenniskens96; @Dohnalek03]. The results and the models used for water by those authors, have been taken as a starting point despite the fact that the differences between both molecules are known.
Concerning to water ice, a first result observed is that when the temperature of deposition increases, the amount of trapped gas decreases. This is a general trend also found by @Schulze80 for CO$_{2}$. It is also found that a first release of molecules is shifted to higher temperatures than their characteristic sublimation temperature; a second release of molecules is also recorded at temperatures related with a structural change (phase change) in water ice ($\sim$ 135 $-$ 150 K); and a last release at the water sublimation temperature ($\sim$ 160 $-$ 175 K).
All these features are related to the structure of water which is known to change with temperature. The experiments of @Stevenson99 and @Jenniskens96 show a continuous variation of the amorphous structure from 10 K up to the crystallization temperature to cubic form at around 140 K.
This variation of structure can be seen in the model presented by @Collings03 based on TPD experiments and RAIR spectra. The desorption process during crystallization was previously studied by @smith97 proposing a mechanism known as “molecular volcano".
Once the main results have been shown, it is important to highlight that these experiments are of great importance in many aspects of astrophysics. An interesting example is that of @Fraser04, who showed the relationship between the mechanisms of chemical and physical adsorption with surface chemistry under interstellar and protostellar conditions. It could be also used to model the sublimation of ices as done by @Viti04 for ices present in solid water near massive protostars.
The models used by @Viti04 combine codeposition and layered deposition experiments performed by @Collings04. In spite of both kinds of experiments have been used it seems from comparing the observational data with laboratory experiments that the water ice shows a compact structure that is best represented by a layered model rather than a mixed ice [@Keane01; @Pontoppidan03; @Fraser04b; @Guillot04; @Palumbo06]. Along the same lines, this leads to the suggestion that species such as CO, CO$_{2}$, OCS, CH$_{3}$OH are partially segregated. However, as @Raut07 have shown, this question could be not completely clear. Then, the possibility exists that at first the ices were mixed. With this in mind, our objective is to improve the means at our disposal with which to improve our knowledge of the chemical and physical interactions of the observed molecules in space.
Since the existence in different astrophysical environments of CO$_{2}$ ices is known, it is relevant to study how their adsorption characteristics can influence the composition of several environments depending on their physical conditions.
In this paper, the desorption behaviour of CH$_{4}$ from CO$_{2}$ during thermal processing is presented. The experimental setup and the experimental procedure are explained in Section 2. In Section 3 the main results are shown and discussed. Finally, the conclusions reached on the influence of temperature on CO$_{2}$ structure are presented in Section 4.
Laboratory experiments
======================
Experimental SETUP {#subsection: experiments}
------------------
The basic components of our experimental configuration (Figure \[setup\]) to carry out these experiments are a high vacuum and low temperature system, a quartz crystal microbalance (QCMB), a laser and a quadrupole mass spectrometer (QMS). The main component is a high vacuum chamber whose pressure conditions are obtained with a rotatory pump ($\sim$ 10$^{-3}$ mbar) backing a Leybold TurboVac 50 pump.
The first stage of a closed-cycle He cryostat (40 K) thermally connected to a shield protector acts as a cryopump providing a pressure in the chamber below 10$^{-7}$ mbar measured with an ITR IoniVac transmiter (5 % in accuracy). The second stage of the cryostat is named [*cold finger*]{} and is able to achieve 10 K. Below this, is located the substrate bearing a QCMB (gold plated surface) in thermal contact with the cold finger.
The temperature in the sample (QCMB) is operated by the Intelligent Temperature Controller ITC 503S (Oxford Instruments), using the feedback of a silicon diode sensor (Scientific Instruments) located just behind it, that lets the temperature vary between 10 to 300 K with an accuracy of 1 K. Another sensor is located at the end of the cryostat second stage, on the edge of the sample holder in order to monitor the behaviour of the system.
Gases or mixtures under study are prepared in a pre-chamber in a proportion estimated from their partial pressures measured with a Ceravac CTR 90 (Leybold Vacuum) whose accuracy is 0.2 %, provided with a ceramic sensor not influenced by the gas type. The gases enter the chamber through a needle valve (Leybold D50968) that regulates the gas flow while the QMS (AccuQuad RGA 100 with a resolution of $\sim$ 0.5 amu) allows us to verify the proportion of gases in the sample (by dividing, in the mass spectra, the area of methane by the area of carbon dioxide).
Experimental procedure
----------------------
Thermal desorption experiments were carried out to analyze sublimation temperature of pure and mixed frozen gases. The following chemicals have been used in this research: CH$_{4}$ $-$ 99.9995, CO$_{2}$ $-$ 99.998 (Praxair), N$_{2}$ $-$ 99.999 (Carburos Metálicos). Pure gases and mixtures of gases are prepared for deposition in the pre-chamber. In all the cases the overall pressure was fixed at 90 mbar.
In order to obtain the desired temperature and to reduce contamination, the procedure to cool down the cold finger is as follows: The cryostat is connected and at the same time the resistor on the cold finger is turned on at a certain voltage to maintain a temperature in the cold finger (200 K) over the deposition temperature of undesired gases (mainly H$_{2}$O and CO$_{2}$). After one hour, when the pressure is around 5$\times$10$^{-8}$ mbar, the current through the resistor is turned off. This procedure allows us to ensure that only a negligible amount of contaminants remain in the chamber, taking into account that a typical experiment lasts 2 hours at maximum. The deposition temperature (15 K) is achieved in a few minutes (to again reduce contamination).
Once the temperature is fixed, the needle valve is opened during 1 minute, to fill the chamber with the selected pure gas or mixture of gases keeping the pumps on. Molecules replenish the chamber randomly and are deposited onto the QCMB (background deposition). The amount of deposit is enough to assume that the continuum is negligible and does not saturate the mass spectrometer (10$^{-4}$ mbar). In all cases the rate of ice deposition is around 1 micrometer per hour, measured using a laser (He-Ne) interferometry and the QCMB frequency variations.
Once deposited, our experiments were performed by heating the substrate at a constant rate of 1 K min$^{-1}$, the vacuum system working continuously, monitoring the molecules present in the chamber during desorption with the QMS and checking the molecules released with the QCMB. The refractive indexes used for calculations were obtained in our laboratory in a series of experiments (Satorre et al. in preparation).
{width="55mm"} {width="55mm"} {width="55mm"}
Results and discussion
======================
In order to study the capacity of CO$_{2}$ to trap CH$_{4}$ in its structure we have performed experiments on thermal desorption for pure CH$_{4}$ and CO$_{2}$ gases and for mixtures of both them. In the first group of experiments, both molecules were deposited as a pure film onto the substrate and in the second set, both gases were co-deposited in a proportion of 95:5 (CO$_{2}$:CH$_{4}$). These three experiments are represented in Figure \[anealing\], where frequency of the QCMB is plotted versus temperature.
For the thermal process of pure CO$_{2}$ (Figure \[anealing\], left panel) we can observe an initial interval (35 to 80 K) where the frequency varies linearly with the temperature as expected for our QCMB (in this specific interval of temperatures) when no release of material takes place. In a second interval, from 80 to 90 K, a sharp increase of frequency is due to the CO$_{2}$ desorption from the QCMB. Finally, at 91 K CO$_{2}$ stops its desorption and the increase in frequency is again caused by a linear temperature effect. Hereafter we will take as the desorption temperature the point in the plot where the slope, after increasing, changes abruptly. In the case of CH$_{4}$ (Figure \[anealing\], center panel), it is shown that the desorption occurs at 38 K, and further on, the variation is again due to temperature effect. In both cases, pure CO$_{2}$ and CH$_{4}$, only one interval of desorption from the substrate is observed during the experiment.
Taking into account the desorption temperatures from the QCMB, our results compare well with those previously published in literature on this area [@Bar-Nun85; @Yoshinobu96; @Collings03].
The results of thermal desorption after co-deposition of CO$_2$:CH$_4$ are presented in Figure \[anealing\] (right panel).
From the plot, we can observe two features from the frequency variations that we can associate to CH$_{4}$ molecules desorbing from the mixture. To isolate the signal due to CH$_{4}$, we remove the contribution of temperature and CO$_{2}$ release subtracting the data showed in Figure 2 (left panel), obtaining Figure \[remnants\]. In this plot, two previously mentioned features (the peaks at 50 and 75 K) are clearly visible. Both signals appear at temperatures higher than the sublimation point of CH$_{4}$ under our experimental conditions. Therefore, in some way, CO$_{2}$ has retained CH$_{4}$ molecules.
Since this technique (QCMB) does not allow us to distinguish different molecules desorbing at the same time, we needed an additional technique to know whether CH$_{4}$ is retained up to the characteristic sublimation temperatures of CO$_{2}$ (91 K in our experimental conditions). Mass spectroscopy allows us to detect that part of CH$_{4}$ desorbs at the same temperature that CO$_{2}$ sublimes (Figure \[em\]). We are able to conclude this because the behavior of CO$_{2}$ and CH$_{4}$ partial pressure during thermal desorption from 90 to 110 K are similar. Additionally, it allowed us to confirm that the peaks at 50 and 75 K are due only to CH$_{4}$ although they appear at slightly shifted temperatures due to the configuration of our system.
![CH$_{4}$ remnants detected by QCMB frequency variations after CO$_{2}$ removal and thermal correction.[]{data-label="remnants"}](lunafig3.eps2){width="80mm"}
To explain CO$_{2}$ matrix ice, in the literature an amorphous CO$_{2}$ structure is generally proposed by authors, some of them arguing that no crystals exist in the whole film when the film is grown at temperatures below 30 K [@Sandford90] but others suggest that this amorphous structure arises from a compilation of small crystallites randomly oriented [@Schulze80].
Taking into account our experiments and the previous models just quoted, below we enumerate and describe the three characteristic temperatures that we have found with the QCMB and the QMS:
TEMPERATURE 1: Around 50 K, the first release of the gas trapped by CO$_2$ takes place. This desorption occurs at higher temperatures than the sublimation point of CH$_4$ and takes place at the temperature reported previously by other authors as the transition between amorphous and crystalline phase of solid CO$_2$ [@Falk87]. This kind of physical process has been named molecular volcano by [@smith97] in the case of CCl$_4$ in water.
In the other model, when the temperature increases the crystallites could undergo a process leading to the structure compacting from a highly porous one to a less porous structure. Molecules of CH$_4$ would be linked to the surface of CO$_2$ (we understand as a surface the upper rough surface and open pores). The increasing density involves a variation in the characteristics of the pores, therefore the temperature increase produces the effect that molecules which were previously retained now sublime.
![Thermal desorption process of CO$_{2}$:CH$_{4}$ mixture recorded as partial pressure versus temperature.[]{data-label="em"}](lunafig4.eps2){width="80mm"}
![CO$_{2}$ porosity from Satorre et al. (in preparation). We have taken our maximum experimental density as the maximum density in the porosity equation. []{data-label="porosity"}](lunafig5.eps2){width="80mm"}
TEMPERATURE 2: A second release occurs at around 75 K. This fact coincides with the compacting of the structure that arises from the continuous variation of the CO$_2$ density, as can be seen when the porosity reaches the minimum value at around 75 K (Figure \[porosity\]). We calculate the porosity as defined by the equation: $p=1-\frac{\rho_{a}}{\rho_{i}}$, (where $\rho_{a}$ is the density obtained at certain temperature of deposition and $\rho_{i}$ is the asymptotic or maximum density), where $\rho_{i}$ = 1.5 g cm$^{-3}$. To perform this plot the maximum density that we have taken in this equation is the maximum experimental density obtained by double angle interferometry and a QCMB when the CO$_2$ is deposited at twelve different temperatures ranging from 15 to 85 K (Satorre et al. in preparation) where the only purpose is to show how the porosity reaches the minimum value at around 75 K, not to give a quantitative value of the porosity.
The work of @Schulze80 supports our findings as they show that the density varies more than 50% between deposits from 10 K to 80 K reaching the maximum density at around 75 K.
TEMPERATURE 3: Finally, CH$_4$ molecules are detected when the CO$_2$ desorbs at 90 K. Those interstitial molecules, which are the most strongly trapped and have remained inside the structure after crystallization, are thus retained until desorption of the CO$_2$ matrix.
Conclusions
===========
Our experiments have been used to study the desorption properties of CH$_4$ in CO$_2$ matrix and to study the structure of CO$_2$ itself and its interaction with different types of molecules. These interactions are very complex and requires further complementary studies (thermal desorption with other molecules and in different proportions, electron diffraction,...) considering that the structure of CO$_2$ accreted at low temperatures is not clear. However some general outlines can be extrapolated from our results. We found different intervals of temperature where CH$_4$ is released at higher temperatures than its sublimation point implying several kinds of interactions responsible for retaining this molecule within the CO$_2$ structure.
CO$_2$ matrices could efficiently retain simple molecules. From the result obtained with CH$_4$ we expect that there are different mechanisms involved producing various temperatures of desorption.
A first mechanism could be associated to the beginning of CO$_2$ crystallization with the adsorbed molecules that bring about an offset to higher temperatures (50 K) than their characteristic sublimation temperature.
Another kind of interaction can be seen from the onset of the peak at 70 K corresponding to the most compacted possible structure of CO$_2$.
Finally, the molecules more strongly retained in the structure are revealed from the last sublimation at 90 K.
Once the shifts in the sublimation temperature are described many astrophysical applications can be found. It is generally assumed that the volatile components (N$_2$, O$_2$, CO, CH$_4$...) on ice layers of interstellar dust grains sublime below 40 K. This assumption may be an oversimplification of the behaviour of such ices. An important proportion of them may be desorbed into the gas phase at higher temperatures as a result of adsorption on the porous surface or entrapment within the closed pores of the hydrogenated layer (H$_2$O) until it desorbs [@Collings04]. The release of material into the gas phase at higher temperatures than previously thought may have a significant impact on the gas-phase chemistry. For example, this has been applied to massive protostars [@Viti04]. But this retarding of the sublimation, as is evident from the results reported in this paper, would not be exclusive to the hydrogenated layer (H$_2$O), but would also be important in the case of the CO$_{2}$ ice. This finding would have to be taken into account in appropriate scenarios.
![Thermal desorption process of CO$_{2}$:N$_{2}$ mixture recorded as partial pressure versus temperature.[]{data-label="co2-n2"}](lunafig6.eps2){width="80mm"}
Furthermore it would be of interest to study the kinetics of chemical processes at temperatures at which it was previously thought to be impossible to retain some volatile elements. As @Fraser04 show, the chemistry is related with the adsorbed molecules and physical processes (diffusion, adsorption,...) in the ices.
It is also possible to study the geographical composition of ices in some solar system satellites such as Triton, where @Quirico99 explored the possibility that CO$_{2}$ is segregated in a separated terrain from the other terrains, one composed of water and the other of a mixture of N$_{2}$, CH$_{4}$ and CO due to difference volatility of molecules. In Triton CO$_{2}$ may be produced from CO by means of, for example, chemical reactions with OH radicals as suggested by @Cruikshank93 or produced by ion irradiation (see for example @Palumbo93). In both cases the possibility that part of the finally segregated CO$_{2}$ retains a small percentage of volatiles within its structure should not be discarded. In the light of our results and taking into account that the model of @Quirico99 fits well but still leaves some discrepancies with the spectra of Triton, new mixtures with low percentages of volatiles could be good laboratory candidates to complete the current models.
Previous applications should be taken as examples, but the experiments presented here are relevant for any astrophysical environment in which the presence of CO$_{2}$ ice is important. It should be taken into account that the conclusions drawn in the present paper could be applied not only to CH$_{4}$ molecules but could be enlarged to other simple molecules with similar characteristics present in astrophysical environments such as N$_{2}$ whose initial preliminary results (See Figure \[co2-n2\]) similar to those of CH$_4$ have corroborated the shift in temperature although further experiments would be necessary to confirm this.
We wish to thank the Ministerio de Educación y Ciencia (Co-financed by FEDER funds) AYA2004-05382. We thank the referee for helpful and constructive comments.
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|
{
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}
|
---
author:
- 'M. Gålfalk'
- 'G. Olofsson'
- 'A.A. Kaas'
- 'S. Olofsson'
- 'S. Bontemps'
- 'L. Nordh'
- 'A. Abergel'
- 'P. André'
- 'F. Boulanger'
- 'M. Burgdorf'
- 'M.M. Casali'
- 'C.J. Cesarsky'
- 'J. Davies'
- 'E. Falgarone'
- 'T. Montmerle'
- |
\
M. Perault
- 'P. Persi'
- 'T. Prusti'
- 'J.L. Puget'
- 'F. Sibille'
date: 'Received 28 November 2003 / Accepted 15 March 2004'
title: 'ISOCAM observations of the star formation region [^1] [^2] '
---
Introduction
============
The dark molecular cloud (Lynds [@lynds]) is one of the nearest and therefore most studied regions of low-mass star formation. It is part of the Taurus-Auriga molecular cloud complex and the distance to its leading edge has been measured to be 140$\pm$10pc (Kenyon et al. [@kenyon]). It shows the usual signs of recent star formation: pre-main-sequence stars (Briceño et al. [@briceno]), Herbig-Haro objects (Devine et al. [@devine99]), reflection nebulosity and an extraordinary bipolar outflow (Snell et al. [@snell]; Rainey et al. [@rainey]; Fridlund & White [@fridlund89a], [@fridlund89b]; Parker et al. [@parker]) that has become a prime example of this outflow type. This outflow emanates from the IR-source (Strom et al. [@strom]; White et al. [@white]; Osorio et al. [@osorio]), a deeply embedded Young Stellar Object (YSO) found to be a binary (Looney et al. [@looney]). Another well known IR-source with a molecular outflow is , a binary (Moriarty-Schieven et al. [@moriarty]) or even triple (Reipurth [@reipurth]) YSO, discovered with the IRAS satellite (Emerson et al. [@emerson]). Among the numerous Herbig-Haro objects of the region, the very compact is a well known YSO with a disk and jets that have been imaged by the HST (Burrows et al. [@burrows]). During the ISO (Infrared Space Observatory) mission several nearby dark clouds were surveyed. In this paper we present mid-IR observations of the region using ISOCAM onboard the ISO-satellite.
The low-mass end of the IMF (Initial Mass Function) is a key objective when investigating star formation, especially for stars lying in the brown dwarf region (below the Hydrogen burning limit of $0.08\ M_{\sun}$) where the IMF is not well known. Since is a nearby star formation cloud, located far away from the crowded Galactic plane ($b=-20^{\circ}$), it should be possible to detect such low-mass stars using ISOCAM. YSO-candidates can generally be found from the ISOCAM data by searching for sources with mid-IR excess, due to heated circumstellar dust. These mid-IR observations have high sensitivity and high spatial resolution, and are therefore suitable for finding and classifying YSOs, as shown for other clouds (Olofsson et al. [@olofsson]; Kaas et al. [@kaas]; Persi et al. [@persi]; Bontemps et al. [@bontemps]).
The young stellar population in has previously been surveyed using several methods: X-ray mapping (Carkner et al. [@carkner]; Favata et al. [@favata]); Optical and near-IR mapping (e.g. Briceño et al. [@briceno]); Optical spectra (Gomez et al. [@gomez]) and H$\alpha$ surveys (e.g. Garnavich et al. [@garnavich]). However, no mid-IR survey as sensitive as this ISOCAM survey (down to $\sim$0.5mJy) had previously been done in .
Lada & Wilking ([@lada]) used observations between 1 and 20$\mu$m to plot the mid-IR Spectral Energy Distribution (SED) of embedded sources in the Ophiuchi dark molecular cloud and found that the population could be divided into three morphological classes. Later (Lada [@lada87]) this YSO classification scheme (Class I–III) was made more quantitative by introducing the spectral index $\alpha$ and approximate limits for the classes. A Class 0 was then introduced (André et al. [@andre93]) and the classification limits were also revised (André & Montmerle [@andre94]). These four YSO classes are defined in a morphological order with Class 0 sources being protostars at the beginning of the main accretion phase, deeply embedded in massive, cold circumstellar envelopes. Class I sources are more evolved and Class II YSOs are T Tauri stars with optically thick IR circumstellar disks. Class III YSOs represent weak line T Tauri stars with at most optically thin disks, they are therefore hard or impossible to detect using mid-IR excess due to their resemblance to field stars. Therefore, using ISOCAM mid-IR observations, mainly classes I & II are detected since Class 0 objects peak in the far-IR spectral region and are probably too weak in the mid-IR to be detected by ISOCAM. The evolution from Class 0 to Class III has classically been assumed to be smooth and gradual, and while this might be the case for isolated young stars it has recently been suggested (Reipurth [@reipurth]) that abrupt class transitions can occur in multiple systems due to violent interactions between its members. Massive and quick disk truncation combined with the possible ejection of light cluster members could then produce highly increased outflow activity as well as transitions from Class 0 or I objects to visible T Tauri stars.
Observations and data reductions
================================
{width="17cm"}
Six regions (total area 0.122sq.deg.) were observed 26–27 September 1997 with ISOCAM, avoiding the mid-IR bright regions around IRS5/NE and HL/XZ Tau due to saturation problems. Two mid-IR filters were used with the LW (Long Wavelength) detector, LW2 (6.7$\mu$m) and LW3 (14.3$\mu$m). For all images, the PFOV is 6$\arcsec$/pixel and the integration time was 2.1s for each transmitted frame ($32\times32$ pixels). During the reduction, around 19 frames were combined at each frame-position to form images with a temporal history. Also, overlaps of half a frame in both equatorial directions gives the much needed spatial redundancy as well as additional temporal history. A total of 286 images were used to put together the 6.7 and 14.3$\mu$m mosaics. Unfortunately the LW detector had a dead column (No24), which however could be covered by using overlapping images for most parts of the final mosaic. Only at some parts close to the southern limit of the mosaic (due to a spacecraft roll angle of $83^{\circ}$) and close to IRS5/NE there is missing information, but not more than about 0.3% of the mapped region. Also, the photometry of sources one pixel from the dead column and at the mosaic edges is unreliable.
For the data reduction, we used the CIA V4.0 package (Ott et al. [@ott]; Delaney et al. [@delaney]) and the SLICE package (Simple & Light ISOCAM Calibration Environment) accessed from inside CIA. The CIA reduction steps consisted of (in order): Extracting useful observations, dark correction (Vilspa dark model, Biviano et al. [@biviano]), glitch removal (multiresolution median transform, Starck et al. [@starck]), short transient correction (Fouks-Schubert model, Coulais & Abergel [@coulais]), flatfielding (constant median flatfield from observations). The data were then processed further using SLICE’s long-term transient and variable flatfield algorithms (Miville-Deschênes et al. [@miville]). Finally the frames were projected into six raster maps for point source detection and photometry. Each detected point source was traced back to its corresponding original frames for temporal and spatial verification (in order to exclude remaining artefacts). For the aperture photometry, the point spread function was used to correct for flux outside the aperture. In total 96 sources were detected with photometry possible for 76 sources at 6.7$\mu$m and 44 sources at 14.3$\mu$m. Regarding photometric comparison, 24 sources had photometry in both filters and could therefore be classified using a colour diagram. A correlation between the two filters is reasonable to assume for a source separation of less than about one pixel ($6\arcsec$), however, to allow for correlations in nebulous regions a limit of has been used. The mean separation of all correlated sources is however $\sim$ .
Even though the integration time was 2.1s for each frame and there are about 19 frames for each image, the total exposure time for a pixel in the mosaic varies between 40s and 160s due to the half frame overlaps in both RA and Dec which makes the total number of frame-pixels available for a mosaic pixel vary by a factor of 1 to 4. Therefore, it is expected that the faintest sources will generally be detected (half a frame, 16 pixels) away from the mosaic edges. From mean uncertainty calculations of all detected ISOCAM sources in , photometric uncertainties are estimated to be ($1\sigma$) 0.4mJy at 6.7$\mu$m and 0.5mJy at 14.3$\mu$m which also approximately represent the detection limits, however as faint sources as these are only detected in low nebulosity regions without artefacts and by using variable flatfielding. On the other hand the surveyed region is not completely mapped down to the $1\sigma$ level, since it is possible that even sources brighter than $3\sigma$ (1.2mJy and 1.5mJy respectively) may have been unnoticed due to varying nebulosity, glitches, memory effects, uncovered dead columns and source confusion close to very bright sources. As for the positional accuracy, by using the 19 LW2 sources also seen in the USNO-A2.0 (Monet) catalogue the uncertainties have been estimated to 0.13s in right ascension and in declination.
In order to convert mJy fluxes into 6.7$\mu$m and 14.3$\mu$m magnitudes, we used the following relations:
$ \begin{array}{lclcl}
m_{6.7} & = & 12.30 & - & 2.5\log_{10} F_{6.7} \\
m_{14.3} & = & 10.69 & - & 2.5\log_{10} F_{14.3}
\end{array} $
Additional observations were made in H$\alpha$ and the $B$, $V$, $I$ and $K$ bands using the 2.56m NOT telescope in La Palma, Canary Islands, Spain. The $K$ band observations were obtained 23, 24 and 27 August 1996 using the Arcetri NICMOS3 camera, ARNICA, a near-IR (1–2.5$\mu$m) $256 \times 256$ pixel HgCdTe array yielding a $2\arcmin \times 2\arcmin$ FOV. The observed region covered about the same region as the ISOCAM mid-IR observations and were divided into four mosaics: NN ($4\times4$ fields), N ($11\times7$ fields), S ($9\times5$ fields) and SS ($5\times5$ fields). The overlap was in both equatorial directions for all fields. Photometry was done on the individual fields down to the limiting magnitude of $K\sim17.5$. There were actually 60 co-added images of 1s each in order not to saturate too many sources, giving a total exposure time of 60s for each field.
The $B$, $V$, and $I$ observations were carried out on 1 and 3 December 2001 using the ALFOSC instrument (Andalucia Faint Object Spectrograph and Camera) mounted on the 2.56m NOT telescope. This instrument has a $2048 \times 2048$ pixel CCD chip and at a PFOV of /pixel it has a FOV of about ${\mbox{6$\stackrel {\prime}{_{\bf \cdot}}$4}} \times {\mbox{6$\stackrel {\prime}{_{\bf \cdot}}$4}}$. We originally set out to do a $5\times5$ field mosaic of the entire ISOCAM region with an overlap of $45\arcsec$, however, due to bad weather on 3 out of the 5 allocated nights we observed the 20 most important frames in the mosaic (regarding ISOCAM source coverage). We used the following NOT-filters and exposure times: $B\#74$ (440nm, 420s), $V\#75$ (530nm, 300s) and $i\#12$ (797nm, 180s and 10s) yielding detection limits of about 22.5, 22.5 and 22.0 mag. respectively. The seeing was typically for both nights. Finally, the H$\alpha$ observations were also observed with ALFOSC on the NOT telescope. Two images were obtained on the night of 22–23 October 2001 using an exposure time of 1200s. These were later put together into a mosaic.
Results
=======
Figure \[spatdist\] illustrates the spatial distribution of all sources detected with ISOCAM, where the visual extinction in magnitude steps is used to indicate the extent of the dark molecular cloud (contours adopted from Minn [@minn]). The distribution appears to be homogeneous, i.e. no apparent correlation between ISOCAM source positions and visual extinction. The $B$, $V$, $I$ and $K$ observations cover roughly the same region as the ISOCAM mosaic, but as the figure shows, one source (ISO-L1551-2) is not included in the optical observations and 15 sources were missed in the near-IR. However, the 2MASS point source catalogue has been used to obtain near-IR ($J$, $H$ and $K_S$) photometry in the whole ISOCAM region.
The final ISOCAM 6.7$\mu$m mosaic is presented in Fig.\[lw2map\] with all 6.7$\mu$m sources circled. As mentioned previously, two regions (close to IRS5/NE and ) were avoided due to saturation issues.
In Table \[ALLtable\], all detected ISOCAM sources are presented along with optical and near-IR photometry. Parenthesis are used to indicate very uncertain sources, where no optical or near-IR counterpart has been found and where typically, there is only one overlap available in the 6.7 or 14.3$\mu$m mid-IR data. Also, bold source numbers indicate sources with an ISOCAM mid-IR excess in the $m_{6.7}-m_{14.3}$ index (i.e. YSO candidates). As stated before, the faintest sources should be detected in regions with the most overlaps. In fact, almost all sources with a LW2 or LW3 flux below 1mJy are located in regions where four image-pixels ($4\times19$ frame-pixels) overlap, giving an exposure time of up to 160s.
In the ISOCAM mid-IR colour/magnitude plot (Fig. \[colour\_iso\]), two populations (‘red’ and ‘blue’) can clearly be seen based on the $m_{6.7}-m_{14.3}$ colour index. These populations are well separated by a colour index gap larger than one magnitude. Sources with negligible uncertainties in the $m_{6.7}-m_{14.3}$ index, with respect to the red-blue dividing line, have no error bars in the plot. Generally, fainter sources at 14.3$\mu$m have larger colour error bars, with ISO-L1551-65 being the most uncertain ‘red’ source in the plot (it is however confirmed as a red source in Fig. \[colour\_k\]). ISO-L1551-3 has larger error bars than might be expected from its 14.3$\mu$m magnitude, it has however only one overlap in the ISOCAM mosaics (making its temporal history quite uncertain) and is located close to a mosaic edge, making its sky background uncertainty larger as well.
Interstellar reddening in the $m_{6.7}-m_{14.3}$ colour index is small or not even present since ‘blue’ ISOCAM (background) sources are also seen in regions with quite large extinction (see Fig. \[spatdist\]), therefore sources without intrinsic 14.3$\mu$m excess should be located close to 0 in the $m_{6.7}-m_{14.3}$ index. For the stars with near-IR photometry, the intrinsic and interstellar reddening effects can easily be separated in a colour/colour diagram using a reddening vector, which indicates the interstellar reddening in the two colours. A set of colour/colour diagrams using near-IR and mid-IR photometry are presented in Figures \[colour\_k\] and \[colour\_hk\]. The reddening vector in Fig. \[colour\_hk\] was calculated by fitting a line to the blue and previously unclassified sources. There is no reddening line in Fig. \[colour\_k\] since this would be an almost vertical line due to the very small interstellar extinction in the $m_{6.7}-m_{14.3}$ colour. As can be seen from these figures, most sources with red $m_{6.7}-m_{14.3}$ colours are also red in the $K-m_{6.7}$ colour and the ‘blue’ sources remain ‘blue’. Thus, these ISOCAM mid-IR excesses are confirmed as intrinsic. The $m_{6.7}-m_{14.3}$ red sources ISO-L1551-3 and 61 have no excess at 6.7$\mu$m (see Fig. \[colour\_hk\]). ISO-L1551-61 is however the known YSO HH30, consisting of an outflow and an optically thick disk seen edge on and ISO-L1551-3 is probably a ClassIII YSO and thus has a small excess (at 14.3$\mu$m). We also note that three sources (ISO-L1551-1, 10 and 72), not detected at 14.3$\mu$m, show intrinsic mid-IR excesses (at 6.7$\mu$m). They are therefore added to our list of sources with mid-IR excesses (i.e. YSO candidates). The ISOCAM ‘red’ source ISO-L1551-2 could not be plotted in the colour/colour diagrams since the mid-IR photometry is very uncertain due to its proximity to an uncovered dead column in the 6.7$\mu$m and 14.3$\mu$m mosaics. However, since roughly the same relative flux is lost on the dead column for both mid-IR filters, ISO-L1551-2 is still included as a YSO candidate.
[crrcclcl]{} ISO-L1551 & $F_{6.7}$\[mJy\] & $F_{14.3}$\[mJy\] & $F_{14.3}/F_{6.7}$ & $\alpha_{6.7-14.3}$ & YSO type$^{\mathrm d}$ & Status & Comments\
\
1 & 1.56 $\pm$ 0.47 & — & — & $+$0.55$^{\mathrm c}$ & Class II$^{\mathrm c}$ & New & Mid-IR excess at 6.7 $\mu$m (S comp.)\
[**2**]{} & 0.79: & 0.97: & 1.24: & $-$0.72: & Class II ? & New & Partially on dead column\
[**3**]{} & 4.95 $\pm$ 0.68 & 3.18 $\pm$ 1.21 & 0.64 & $-$1.58 & Class II / III & New &\
[**5**]{} & 1.44 $\pm$ 0.33 & 1.88 $\pm$ 0.43 & 1.30 & $-$0.65 & Class II & New &\
10 & 2.00 $\pm$ 0.45 & — & — & $-$0.68$^{\mathrm c}$ & Class II$^{\mathrm c}$ & New & Mid-IR excess at 6.7 $\mu$m\
[**13**]{} & 1.76 $\pm$ 0.21 & 2.47 $\pm$ 0.38 & 1.41 & $-$0.55 & Class II & New & Extended + Close to GH 2$^{\mathrm a}$\
[**19**]{} & 0.97 $\pm$ 0.27 & 2.07 $\pm$ 0.43 & 2.12 & $-$0.01 & Class I / II & New &\
[**22**]{} & 0.95 $\pm$ 0.29 & 1.52 $\pm$ 0.40 & 1.59 & $-$0.39 & Class II & New & Extended\
[**27**]{} & 0.87 $\pm$ 0.23 & 2.38 $\pm$ 0.37 & 2.73 & $+$0.32 & Class I & New & Extended\
[**40**]{} & 0.62 $\pm$ 0.23 & 1.03 $\pm$ 0.28 & 1.66 & $-$0.34 & Class II & New & XMM-Newton-15, Favata et al. [@favata]\
[**65**]{} & 0.62 $\pm$ 0.25 & 0.66 $\pm$ 0.47 & 1.05 & $-$0.93 & Class II & New & VLA 21cm - 15$^{\mathrm b}$\
72 & 0.87 $\pm$ 0.33 & — & — & $-$0.68$^{\mathrm c}$ & Class II$^{\mathrm c}$ & New & Mid-IR excess at 6.7 $\mu$m\
[**74**]{} & 1.63 $\pm$ 0.33 & 1.78 $\pm$ 0.45 & 1.09 & $-$0.88 & Class II & New & Extended\
[**85**]{} & 0.61 $\pm$ 0.28 & 1.34 $\pm$ 0.43 & 2.20 & $+$0.04 & Class I / II & New & Double in optical (S comp. extended)\
[**86**]{} & 3.24 $\pm$ 0.38 & 3.86 $\pm$ 0.41 & 1.19 & $-$0.77 & Class II & New & Close to\
\
& 0.89 $\pm$ 0.58 & 1.77 $\pm$ 0.59 & 1.99 & $-$0.09 & Class I / II & & Circumstellar disk seen edge-on\
[**96**]{} & 38.34 $\pm$ 0.71 & 37.55 $\pm$ 0.62 & 0.98 & $-$1.03 & Class II & MHO 5 & Spectral type M6 - M6.5\
\
Graham & Heyer [@graham]
Giovanardi et al. [@giovanardi] - Could be an extragalactic triple radio source (Rodríguez & Cantó [@rodriguez83])
Spectral index $\alpha_{2.2-6.7}$ has been used
As implied by $\alpha_{6.7-14.3}$, assuming that all candidates are YSOs
In Table \[YSOtable\] we list all ISOCAM sources with mid-IR excess and which are therefore YSO candidates. Cross correlations with previous source names and YSO status are given. Using the SED indices $\alpha_{6.7-14.3}$ and $\alpha_{2.2-6.7}$ all YSO candidates have been classified into YSO classes I–III (mostly Class II). For the ISOCAM fluxes, the classical IR spectral index becomes:
$$\alpha_{6.7-14.3} = \frac{d\log(\lambda F_\lambda)} {d\log\lambda} =
-\frac{d\log(\nu F_\nu)} {d\log\nu} = \frac{\log(F_{14.3}/F_{6.7})}{\log(14.3/6.7)}-1$$
and for the $K$ & 6.7$\mu$m fluxes we similarly have:
$$\alpha_{2.2-6.7} = \frac{\log(F_{6.7}/F_{2.2})} {\log(6.7/2.2)} -1$$
Classification limits (Class I–III) for both $\alpha$ indices have been assumed to be the same as in many previous ISOCAM studies (e.g. Kaas et al. [@kaas]; Bontemps et al. [@bontemps]). For the ISOCAM spectral index, we thus have: Class I/II limit at $\alpha_{6.7-14.3}\sim0$, Class II/III limit at $\alpha_{6.7-14.3}\sim-1.6$ and for no excess (simple photospheric blackbody emission) we have $\alpha_{6.7-14.3}\sim-3.0$. For the three YSO candidates without 14.3$\mu$m flux, the classification was done using the $\alpha_{2.2-6.7}$ index. The total flux uncertainties for the YSO candidates in Table \[YSOtable\] were obtained by adding the spatial and temporal errors in quadrature. Temporal errors were estimated from frame photometry (in all overlaps) for each source position, spatial errors were calculated from the background sky variation close to each source (taking into account the aperture size used).
Of the 17 YSO candidates, only two (HH30 and MHO5) were previously known to be YSOs. There is a third previously known YSO (MHO4, Briceño et al. [@briceno]) in the ISOCAM region, however, for which we lack evidence of YSO status. It is detected at both 6.7 and 14.3$\mu$m (ISO-L1551-44) but shows no mid-IR excess. This source has also been observed as an X-ray source (, Carkner et al. [@carkner]). Some of the new YSO candidates in Table \[YSOtable\] are doubtful since they are either extended or lie close to a known Herbig-Haro object. Also, ISO-L1551-1 has an uncertain YSO status since it was not detected at 14.3$\mu$m and its optical/near-IR counterpart is located more than 8$\arcsec$ from the ISOCAM position. ISO-L1551-2 is doubtful since it, as described above, lies partially on an uncovered dead column in both ISOCAM mosaics.
Discussion
==========
{width="17cm"}
Although the ISOCAM mid-IR ($m_{6.7}-m_{14.3}$) colour is efficient in separating field stars from YSO candidates, there are mainly three other types of objects that could ‘contaminate’ the survey and be mistaken for YSOs. These are: very ‘red’ field stars (AGB or M7III spectral types), galaxies and Herbig-Haro objects.
From the Wainscoat et al. ([@wainscoat]) model of the point source mid-IR sky, statistically there should be no AGB or M7III field stars in our field, these could otherwise be mistaken for YSOs due to their mid-IR brightness.
In Fig. \[colour\_iso\], an arrow has been drawn indicating expected $m_{6.7}-m_{14.3}$ colours of galaxies. For a galaxy with no 14.3$\mu$m excess (i.e. $F_{6.7} = F_{14.3}$) we expect a colour of $m_{6.7}-m_{14.3} \sim 1.6$ (marked with an x on the arrow). The arrow was adopted from ISOCAM observations of 46 non-saturated galaxies (Dale et al. [@dale]), including normal star-forming galaxies of many morphological types (Ellipticals, Lenticulars, early to late type Spirals, Irregular & Peculiar types). The bluest of these galaxies are located close to the ‘no 14.3$\mu$m excess mark’ (Quiescent galaxies) while galaxies with higher star formation rates and ‘red’ morphological types are located further along the arrow. There is a probability of about 70% for a randomly observed galaxy to be located within the colours of the arrow, while starburst galaxies may be located at much ‘redder’ colours. From the discussion above, it is clear that differentiation between YSOs and galaxies using only the $m_{6.7}-m_{14.3}$ colour is impossible. According to the ELAIS (European Large Area ISO Survey) ISOCAM is very sensitive to strongly star-forming galaxies at 14.3$\mu$m (Väisänen et al. [@vaisanen]) and based on ELAIS counts about 6 galaxies are expected in a region of this size at the achieved sensitivity.
Also, many of our ISOCAM sources are only detected at 14.3$\mu$m (16 sources; see Table \[ALLtable\]) but since only about 6 galaxies are statistically expected in the region starburst galaxies can only account for some of these sources, most likely the faint ones due to the steep luminosity function and red colours of such galaxies. Our source detection limits at 6.7$\mu$m and $K$ are about 12.5 and 17.5 mag. respectively. Yet many of the sources seen only at 14.3$\mu$m are very bright (e.g. ISO-L1551-51; $m_{14.3}=7.96$) indicating that these sources are most likely asteroids. In fact, no starburst galaxies brighter than about $F_{14.3} \sim 5$mJy are expected in the whole region and the motion of asteroids would explain why these bright sources are only detected in one filter. Given the proximity of to the ecliptic plane and the large number of asteroids with orbits in this region this seems very likely.
There are numerous known Herbig-Haro objects in the observed region. These shock-excited emission line nebulae are associated with outflows from protostellar sources and radiate mainly in H recombination lines and forbidden lines such as \[SII\]($\lambda\lambda6717, 6732$), see e.g. HST observations of (Devine et al. [@devine00]). At infrared wavelengths we could also expect line emission through H$_2$ rovibrational lines and in the ISOCAM filters used for our observations we especially have 5 pure rotational H$_2$ lines (H$_2$0-0S(4) through S(8)) in the 6.7$\mu$m filter and 2 pure rotational lines (S(1) and S(2)) in the 14.3$\mu$m filter. As can be seen in Fig. \[Halfa\] the very bright Herbig-Haro objects and have 6.7$\mu$m peaks that are clearly separated from the corresponding H$\alpha$ peaks. Since the mid-IR and H$\alpha$ traces very different conditions this could be expected, however, there is also the possibility that extended YSO candidates in fact are galaxies seen through the dark molecular cloud. As with galaxies, Herbig-Haro objects cannot be easily separated from YSOs using only the $m_{6.7}-m_{14.3}$ colour. Deep imaging or spectra of the new YSO candidates is needed to confirm their nature. Deep images, preferably in the IR (much deeper than our $K$ images), could show if the YSO candidates look extended or point source like. Also, one could compare an image taken through an \[SII\] interference filter (where HH objects are strong) with a broadband image centred at about 1 micron (where HH objects are weak). Spectra of the YSO candidates could be used to search for strong H$\alpha$ emission and LiI$\lambda6707$ in absorption which would indicate a T Tauri star.
In Figure \[Iband\_imgs\], images are shown for most of the YSO candidates visible in our $I$ or $K$-band observations. Both ISO-L1551-10 and 72 are easily seen in the optical/near-IR but are not included since they are single point sources well within the ISOCAM error circle. Several of the YSO candidates are extended in a spherical or elliptical manner, supporting that these might be galaxies seen through the dark cloud.
At least three of the observed ISOCAM sources have known spectral types from previous studies. ISO-L1551-47 (HD285845) is a binary system with a separation of 73 mas (Schneider et al. [@schneider]), excluded from cloud membership by its radial velocity and proper motion (Walter et al. [@walter]). The primary component is of spectral type G8 and has colour indices consistent with that of a main sequence star (Walter et al. [@walter]). Its distance is implied to be 90pc from its photometric parallax and it is a bright X-ray source (Favata et al. [@favata]). This suggests that ISO-L1551-47 is an active binary system in the foreground of .
ISO-L1551-44 (MHO4, Briceño et al. [@briceno]) and ISO-L1551-96 (MHO5) are known to be very similar YSOs, both very late type TTS of spectral type M6–6.5. They have strong LiI$\lambda6707$ in absorption and H$\alpha$ in emission and similar ages $\sim1$Myr and masses $\sim0.05\,M_{\sun}$, placing them in the brown dwarf region. Both were detected at 6.7 and 14.3$\mu$m, but only ISO-L1551-96 has mid-IR excess at 14.3$\mu$m. ISO-L1551-44, however, shows no mid-IR excess at 6.7$\mu$m nor in the $K$-band, and is therefore a very late type NTTS.
ISO-L1551-21 has previously been detected in X-ray observations (L1551X13, Carknet et al. [@carkner]) where it was suggested to be unrelated to the cloud since it was identified with , a foreground dM star appearing in the Luyten ([@luyten]) proper motion catalogue. It lies close to the Herbig-Haro objects ($17\arcsec$E) and ($20\arcsec$NW) but is definitely a point source.
ISO-L1551-61 (HH30) is a very well known YSO, observed as an optically thick circumstellar disk seen edge on. HST observations (Burrows et al. [@burrows]) have shown clearly that this star is not observed directly, only nebulosity is seen.
ISO-L1551-65 lies at the position of a known triple radio continuum source, probably extragalactic in nature (Rodríguez & Cantó [@rodriguez83]) with an overall extent of $\sim20\arcsec$. It is however unclear if this is the object detected with ISOCAM.
The SEDs for three of the new YSO candidates (ISO-L1551-3, 5, 19) and the known YSO MHO5 (ISO-L1551-96) are shown in Fig. \[SED\]. These sources have at least 7 known broad band magnitudes (see Table \[ALLtable\]), making it worthwile to plot their SEDs. No extinction corrections have been applied for the new candidates (since no spectral classes are known), however, they are all outer members of , located quite far away from the dense central region so the extinction is probably not that large (especially for ISO-L1551-3, see Fig. \[spatdist\]). For MHO5 the extinction is known to be only $A_V = 0.01$ (Briceño et al. [@briceno]) locating it on our side of the cloud with a negligible extinction. For each source a spectral class has been calculated by fitting the observed SEDs with scaled SEDs of M dwarfs as given by Leggett ([@leggett]). The known brown dwarf MHO5 (ISO-L1551-96) is very well fitted using a M6 dwarf, this agrees with the spectral type M6–6.5 found from spectra of this source (Briceño et al. [@briceno]). The three YSO candidates have SEDs that mimics dwarfs of spectral types M3, M6 and M5.5 respectively. All four sources show mid-IR excess when compared to the scaled SEDs. For ISO-L1551-3 the excess is small and only at 14.3$\mu$m which could indicate that the inner part of its accretion disk has been cleared out while the excess originates further out where the dust is cooler.
Conclusions
===========
Based on a deep mid-IR ISOCAM survey (approximately $20\arcmin \times 20\arcmin$) of the dark molecular cloud we have found 14 sources with intrinsic mid-IR excess emission at 14.3$\mu$m which were therefore classified as YSO candidates. Additional observations in $B$, $V$, $I$, $J$, $H$ and $K$ supported the YSO candidate status for most detected candidates and yielded three more candidates. Out of the 17 detected YSO candidates only two were previously known ( and ). This means 15 new candidates, however, several of these are extended and could be background galaxies or Herbig-Haro objects. Assuming a co-eval age of 2-3Myr for our sample (mainly Class II objects) and that all candidates actually are YSOs we only add a few solar masses to the stellar mass component of the Star Formation Efficiency (SFE) in . This is however a small contribution when compared to the previously known YSO population. The new YSO candidates add to the low-mass end of the IMF in , but due to the small number of known YSOs (as expected since has a mass of only $\sim80\,M_{\sun}$, Snell [@snell81]) and the lack of follow-up spectra, the IMF can not be accurately modelled yet. In addition to the 5 known outflow YSOs (, , and ) two brown dwarfs were previously known in the region and one of these, MHO5, belongs to the ISOCAM YSO candidates and was found in our study to have mid-IR excesses compatible with being in a Class II phase of evolution. Most of the YSO candidates, assuming they are real, seem to belong to the YSO class II group (Classical T Tauri Stars) and thus have optically thick circumstellar disks at mid-IR wavelengths. One of the sources (ISO-L1551-61=HH30) has even been seen as an optically thick edge-on disk with the HST. Obviously, more follow-up studies from the ground should be made, especially spectroscopic observations of all the new YSO candidates which is necessary to confirm their YSO status since some of them could well be background galaxies seen through the dark molecular cloud or Herbig Haro objects caused by the known YSOs with outflows, close to the centre of .
The Swedish participation in this research is funded by the Swedish National Space Board. This publication 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, and 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. AAK thanks Carlos Baffa, Mauro Sozzi, Ruggero Stanga, and Lenoardo Testi from the Arnica team for the instrument support at the NOT in 1996.
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{width="18cm"}
{width="17cm"}
{width="18cm"}
[cccrrllllllllll]{}\
\[ALLtable\]\
& & & & & & & & & & & & & &\
\
& & & & & & & & & & & & & &\
\
1 & 04:30:43.1 & 17:58:30 & 1.56 & & & 22.27 & & 19.07 & 16.89 & 16.02 & 15.94 & 11.81 & & Sep S\
& & & & & & & & 20.56 & & & & & & Sep E\
[**2**]{} & 04:30:43.7 & 17:54:25 & 0.79: & 0.97: & 15.9$^{\mathrm d}$ & out & 14.5 & out & 12.87 & 12.43 & 12.23 & 12.56: & 10.72: &\
[**3**]{} & 04:30:48.4 & 18:18:36 & 4.95 & 3.18 & 17.58 & 16.10 & 14.7 & 13.84 & 11.82 & 10.97 & 10.56 & 10.56 & 09.44 &\
4 & 04:30:50.0 & 18:05:20 & 1.87 & & 20.96 & 18.94 & & 15.99 & 13.38 & 12.32 & 11.75 & 11.62 & &\
[**5**]{} & 04:30:50.9 & 18:13:00 & 1.44 & 1.88 & 22.56 & 20.33 & & 17.24 & 14.54 & 13.24 & 12.75 & 11.90 & 10.01 &\
6 & 04:30:51.7 & 18:10:23 & 0.91 & & & 21.06 & & 17.83 & 15.05 & 13.73 & 13.25 & 12.40 & &\
7 & 04:30:52.1 & 17:59:00 & 2.55 & & 21.19 & 19.83 & & 18.21 & & & 15.92$^{\mathrm e}$ & 11.28 & &\
8 & 04:30:52.8 & 18:18:19 & 1.12 & & & 20.01 & & 16.88 & 14.12 & 12.81 & 12.24 & 12.17 & &\
9 & 04:30:53.3 & 17:53:37 & 3.99 & & 16.61 & 15.22 & 14.0 & 13.36 & 11.97 & 11.28 & 11.06 & 10.80 & &\
10 & 04:30:53.8 & 17:59:09 & 2.00 & & 22.62 & 20.74 & & 17.50 & 15.43 & 14.52 & 14.18 & 11.55 & & near HH256\
11 & 04:30:54.2 & 17:58:24 & 17.96 & 2.55 & 16.72 & sat & 13.1 & 12.40 & 10.55 & 09.60 & 09.34 & 09.16 & 09.68 & GSC1269:485\
12 & 04:30:54.6 & 17:55:16 & 19.16 & 2.21 & 12.3$^{\mathrm d}$ & sat & 11.0 & 10.47 & 09.59 & 09.31 & 09.17 & 09.09 & 09.83 & GSC1269:261\
[**13**]{} & 04:30:58.2 & 18:03:29 & 1.76 & 2.47 & & 21.76 & & 18.97 & 16.97 & 15.85 & 15.17 & 11.69 & 09.71 & Galaxy?\
14 & 04:30:59.4 & 18:01:24 & & 2.78 & & & & & & & & & 09.58 &\
15 & 04:30:59.8 & 18:04:25 & 8.01 & 1.84 & 17.96 & 16.16 & 14.7 & 13.71 & 11.61 & 10.66 & 10.28 & 10.04 & 10.03 &\
16 & 04:30:59.8 & 17:59:56 & 3.74 & & 17.25 & 15.66 & 14.2 & 13.59 & 12.04 & 11.28 & 10.97 & 10.87 & &\
17 & 04:31:00.2 & 18:03:45 & & 0.99 & 22.59 & 21.41 & & 18.78 & 16.69 & 15.82 & 15.48 & & 10.70 & Sep. ESE\
& & & & & & 23.17 & & 20.26 & & & & & & Sep. ESE\
18 & 04:31:01.2 & 18:18:23 & 2.00 & & 21.07 & 18.93 & & 16.03 & 13.67 & 12.50 & 11.99 & 11.55 & &\
[**19**]{} & 04:31:01.8 & 18:03:00 & 0.97 & 2.07 & & 21.39 & & 18.69 & 16.42 & 15.80 & 15.25 & 12.33 & 09.90 &\
20 & 04:31:05.7 & 18:14:12 & 1.62 & & & & & & 18.71 & 16.04 & 13.92 & 11.77 & &\
21 & 04:31:05.8 & 18:03:19 & 4.79 & & 17.43 & 15.94 & 15.0 & 13.26 & 11.69 & 11.10 & 10.84 & 10.60 & & L1551X13$^{\mathrm a}$\
[**22**]{} & 04:31:05.9 & 18:01:48 & 0.95 & 1.52 & & & & 19.72 & & & 14.93$^{\mathrm e}$ & 12.35 & 10.24 & Galaxy?\
23 & 04:31:07.2 & 18:14:33 & 0.82 & & & & & & 17.68 & 15.19 & 13.66 & 12.52 & &\
24 & 04:31:08.2 & 18:08:12 & 1.18 & & & & & 19.22 & 15.53 & 14.04 & 13.29 & 12.12 & &\
25 & 04:31:08.7 & 18:18:37 & & 4.94 & & & & & & & & & 08.96 &\
26 & 04:31:08.9 & 18:19:02 & 1.59 & & 19.63 & 17.84 & 16.6 & 15.18 & 13.48 & 12.56 & 12.31 & 11.80 & & Double ()\
[**27**]{} & 04:31:08.9 & 18:01:43 & 0.87 & 2.38 & & & & 19.93 & & & 15.37$^{\mathrm e}$ & 12.45 & 09.75 & Galaxy?\
28 & 04:31:09.0 & 18:03:44 & & 1.51 & & & & & & & 17.7$^{\mathrm e}$ & & 10.24 &\
29 & 04:31:11.2 & 18:11:05 & 0.80 & & & & & & 16.90 & 14.65 & 13.38 & 12.54 & &\
30 & 04:31:11.9 & 18:08:18 & 0.73 & & & & & & & & & 12.64 & &\
31 & 04:31:12.5 & 18:17:49 & & 4.21 & & & & & & & & & 09.13 &\
(32) & 04:31:14.1 & 18:10:22 & & 0.63 & & & & & & & & & 11.20 &\
33 & 04:31:15.9 & 18:09:15 & 5.72 & 0.75 & & 22.61 & & 18.07 & 13.88 & 12.15 & 11.29 & 10.41 & 11.01 &\
34 & 04:31:16.3 & 18:18:34 & 6.10 & & 14.3$^{\mathrm d}$ & sat & 12.5 & 12.37 & 11.24 & 10.65 & 10.45 & 10.34 & & GSC1269:1201\
35 & 04:31:17.6 & 18:06:24 & & 1.82 & & & & & & & & & 10.04 & HH264?\
36 & 04:31:17.6 & 18:01:53 & 1.06 & & 22.36 & 20.06 & & 16.78 & 14.08 & 12.94 & 12.42 & 12.23 & &\
37 & 04:31:19.0 & 18:03:02 & & 0.93 & & & & & & & 16.86$^{\mathrm e}$ & & 10.77 &\
38 & 04:31:19.3 & 18:09:21 & 1.30 & & & & & 20.25 & 15.75 & 13.75 & 12.75 & 12.02 & & Closedouble?\
39 & 04:31:20.2 & 18:01:45 & 2.06 & & 19.79 & 18.05 & 16.9 & 15.41 & 13.19 & 12.24 & 11.83 & 11.52 & &\
[**40**]{} & 04:31:21.2 & 17:59:49 & 0.62 & 1.03 & & & & & & & & 12.82 & 10.66 & X-ray source$^{\mathrm f}$\
41 & 04:31:21.6 & 18:02:50 & 1.25 & & 22.00 & 19.92 & & 16.63 & 14.07 & 12.83 & 12.31 & 12.05 & &\
42 & 04:31:22.5 & 18:14:51 & 0.75 & & 18.60 & 17.10 & 15.9 & 15.13 & 13.56 & 12.67 & 12.44 & 12.61 & &\
43 & 04:31:23.5 & 17:58:52 & 65.63 & 12.87 & 14.1$^{\mathrm d}$ & sat & 11.4 & 10.96 & 09.09 & 08.18 & 07.88 & 07.76 & 07.92 & GSC1269:1017\
44 & 04:31:24.3 & 18:00:25 & 8.53 & 1.27 & 20.38 & 18.67 & & 14.50 & 11.65 & 10.92 & 10.57 & 9.97 & 10.43 & MHO4$^{\mathrm b}$, X15$^{\mathrm a}$\
45 & 04:31:24.6 & 18:13:30 & 0.85 & & 18.44 & 16.98 & 15.9 & 15.05 & 13.49 & 12.78 & 12.50 & 12.47 & & Sep. ENE\
& & & & & & 21.95 & & 20.15 & & & & & & Sep. WSW\
46 & 04:31:25.3 & 18:10:43 & 0.76 & & & & & & & & & 12.60 & &\
47 & 04:31:25.5 & 18:16:19 & 48.56 & 7.62 & 11.0$^{\mathrm d}$ & sat & 09.9 & sat & 08.67 & 08.24 & 08.10 & 08.08 & 08.49 & HD285845\
48 & 04:31:25.6 & 18:17:02 & 0.44 & & 19.92 & 18.21 & 17.2 & 15.55 & 13.99 & 13.36 & 13.11 & 13.20 & &\
49 & 04:31:26.3 & 18:05:52 & 4.25 & & & 21.98 & & 18.28 & 14.75 & 13.30 & 12.58 & 10.73 & & Sep. W\
& & & & & 22.39 & 20.60 & & 17.03 & 13.79 & 12.46 & 11.74 & & & Sep. SE\
50 & 04:31:26.6 & 18:06:59 & 1.40 & & & & & & & & & 11.93 & & HH260?\
51 & 04:31:27.0 & 18:10:41 & & 12.30 & & & & & & & & & 07.96 &\
52 & 04:31:27.7 & 18:06:19 & 3.86 & & & & & & & & & 10.83 & & HH29?\
53 & 04:31:29.0 & 18:17:45 & & 4.23 & & & & & & & & & 09.12 &\
(54) & 04:31:29.1 & 18:18:16 & & 1.29 & & & & & & & & & 10.42 &\
55 & 04:31:29.3 & 18:08:51 & 3.11 & & 17.52 & 16.06 & 15.0 & 13.69 & 12.36 & 11.73 & 11.49 & 11.07 & &\
(56) & 04:31:29.9 & 18:06:10 & 1.01 & & & & & & & & & 12.29 & &\
(57) & 04:31:32.1 & 18:02:49 & 0.74 & & & & & & & & & 12.63 & &\
58 & 04:31:33.4 & 18:04:57 & & 1.24 & & & & & & & 16.71$^{\mathrm e}$ & & 10.46 &\
59 & 04:31:34.0 & 18:05:34 & 1.32 & & & 23.24: & & 19.39 & 15.24 & 13.40 & 12.52 & 12.00 & &\
(60) & 04:31:35.2 & 18:01:54 & & 0.58 & & & & & & & & & 11.27 &\
[**61**]{} & 04:31:37.5 & 18:12:26 & 0.89 & 1.77 & 19.78 & 18.88 & 16.4 & 17.06 & 15.18 & 14.24 & 13.37 & 12.43 & 10.07 & HH30\
62 & 04:31:37.6 & 18:02:58 & 1.67 & & 19.90 & 18.14 & 17.2 & 15.72 & 13.73 & 12.66 & 12.34 & 11.75 & &\
63 & 04:31:39.0 & 18:02:36 & 2.26 & & 17.33 & 15.99 & 15.1 & 14.03 & 12.32 & 11.65 & 11.34 & 11.42 & &\
(64) & 04:31:40.7 & 18:09:03 & 2.36 & & & & & & & & & 11.37 & &\
[**65**]{} & 04:31:44.1 & 18:10:37 & 0.62 & 0.66 & & & & & & & 17.8$^{\mathrm e}$ & 12.82 & 11.15 & VLA 21cm-15$^{\mathrm c}$$^{\mathrm g}$\
(66) & 04:31:44.2 & 18:12:38 & & 1.07 & & & & & & & & & 10.62 &\
67 & 04:31:45.0 & 18:06:12 & & 2.82 & & & & & & & & & 09.57 &\
68 & 04:31:45.3 & 18:14:37 & 11.08 & & 17.55 & 16.28 & 15.1 & 14.13 & 11.86 & 10.67 & 10.06 & 09.69 & &\
69 & 04:31:46.5 & 18:06:40 & & 0.84 & & & & & & & & & 10.88 &\
70 & 04:31:55.7 & 18:11:00 & 1.60 & & 20.18 & 18.33 & 17.0 & 15.27 & 13.31 & 12.45 & 12.12 & 11.79 & &\
71 & 04:31:56.1 & 18:08:55 & 21.28 & 3.39 & 17.08 & 15.13 & 13.9 & 12.70 & 10.63 & 09.54 & 09.18 & 08.98 & 09.37 &\
72 & 04:31:56.4 & 18:13:36 & 0.87 & & 20.82 & 19.48 & 19.1 & 17.60 & 16.21 & 15.26 & 15.09 & 12.45 & &\
73 & 04:31:56.6 & 18:02:52 & 10.02 & 1.68 & 16.42 & 15.11 & 13.6 & 12.96 & 11.23 & 10.34 & 10.07 & 09.80 & 10.13 & GSC1269:559\
[**74**]{} & 04:31:57.2 & 18:17:21 & 1.63 & 1.78 & & 20.50 & & 18.69 & & & 15.73$^{\mathrm e}$ & 11.77 & 10.06 & Galaxy?\
75 & 04:31:57.6 & 18:11:21 & 1.81 & & & & & & & & & 11.66 & &\
76 & 04:31:58.9 & 18:12:01 & 4.29 & & & & & & & & 14.97$^{\mathrm e}$ & 10.72 & &\
77 & 04:31:59.1 & 18:02:52 & 0.85 & & 17.43 & 16.46 & 15.7 & 14.70 & 13.31 & 12.69 & 12.47 & 12.48 & &\
78 & 04:31:59.4 & 18:10:37 & 0.60 & & 17.00 & 15.91 & 15.4 & 14.51 & 13.36 & 12.90 & 12.73 & 12.85 & &\
79 & 04:31:59.5 & 18:13:49 & & 6.67 & 21.33 & 20.10 & 19.2 & 18.47 & & & & & 08.63 & Sep. 11$\arcsec$N\
& & & & & & 22.87 & & 19.90 & & & & & & Sep. 14$\arcsec$NNW\
80 & 04:31:59.9 & 18:08:50 & 2.00 & & 15.51 & sat & 13.9 & 13.25 & 12.13 & 11.67 & 11.48 & 11.55 & & GSC1269:777\
81 & 04:32:00.6 & 18:12:09 & 2.69 & & & & & & & & & 11.23 & &\
82 & 04:32:01.0 & 18:03:55 & & 1.48 & & & & & & & & & 10.26 &\
83 & 04:32:01.1 & 18:12:33 & 2.42 & & & & & & & & & 11.34 & & HH/Galaxy ?\
84 & 04:32:01.1 & 18:06:01 & 1.25 & & 20.25 & 18.42 & 17.4 & 15.86 & 13.63 & 12.50 & 12.11 & 12.06 & &\
[**85**]{} & 04:32:01.1 & 18:05:07 & 0.61 & 1.34 & 22.39 & 20.91 & & 18.88 & 16.63 & 15.49 & 15.05 & 12.84 & 10.37 & Sep. SSE$^{\mathrm h}$\
& & & & & & 22.91 & & 19.86 & & & & & & Sep. NNE\
[**86**]{} & 04:32:01.3 & 18:11:29 & 3.24 & 3.86 & & & & & & & & 11.02 & 09.22 & HH 262 ?\
87 & 04:32:01.5 & 18:11:59 & & 2.22 & & & & & & & & & 09.83 & HHobject ?\
88 & 04:32:02.6 & 18:03:15 & 0.55 & & 17.44 & 16.18 & 15.2 & 14.72 & 13.52 & 12.99 & 12.79 & 12.94 & &\
89 & 04:32:03.6 & 18:06:06 & 25.93 & 6.57 & 17.15 & 15.14 & 13.5 & 12.51 & 10.26 & 09.11 & 08.68 & 08.77 & 08.65 & GSC1270:357\
90 & 04:32:07.2 & 18:14:28 & 0.41 & & 17.70 & 16.39 & 15.4 & 14.74 & 13.45 & 12.75 & 12.57 & 13.28 & &\
91 & 04:32:07.4 & 18:16:28 & 4.54 & & 16.24 & sat & 13.7 & 13.05 & 11.64 & 10.84 & 10.62 & 10.66 & & GSC1270:1234\
92 & 04:32:07.4 & 18:09:05 & 0.82 & & 16.54 & 15.55 & 14.7 & 14.20 & 13.07 & 12.60 & 12.44 & 12.51 & &\
93 & 04:32:09.5 & 18:12:02 & 1.67 & & & & & & & & & 11.74 & & HH/Galaxy ?\
94 & 04:32:10.9 & 18:17:41 & 3.26 & & 14.0$^{\mathrm d}$ & sat & 12.8 & 12.54 & 11.63 & 11.29 & 11.13 & 11.02 & & GSC1270:401\
(95) & 04:32:13.4 & 18:17:51 & & 0.80 & & & & & & & & & 10.93 &\
[**96**]{} & 04:32:15.8 & 18:12:43 & 38.34 & 37.55 & 19.20 & 17.58 & 15.8 & 13.70 & 11.07 & 10.39 & 10.06 & 08.34 & 06.75 & MHO5$^{\mathrm b}$\
Carkner et al. [@carkner]
Briceño et al. [@briceno]
Giovanardi et al. [@giovanardi] – Could be an extragalactic triple radio source (Rodríguez & Cantó [@rodriguez83])
USNO A-2 catalogue has been used
K-band magnitude (ARNICA/NOT)
XMM-Newton-15, Favata et al. [@favata]
XMM-Newton-35
XMM-Newton-49
[^1]: Based on observations with ISO, an ESA project with instruments funded by ESA Member States (especially the PI countries: France, Germany, the Netherlands and the United Kingdom) and with the participation of ISAS and NASA.
[^2]: Based on observations made with the Nordic Optical Telescope, operated on the island of La Palma jointly by Denmark, Finland, Iceland, Norway, and Sweden, in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofisica de Canarias.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Recently, great progress has been made for online handwritten Chinese character recognition due to the emergence of deep learning techniques. However, previous research mostly treated each Chinese character as one class without explicitly considering its inherent structure, namely the radical components with complicated geometry. In this study, we propose a novel trajectory-based radical analysis network (TRAN) to firstly identify radicals and analyze two-dimensional structures among radicals simultaneously, then recognize Chinese characters by generating captions of them based on the analysis of their internal radicals. The proposed TRAN employs recurrent neural networks (RNNs) as both an encoder and a decoder. The RNN encoder makes full use of online information by directly transforming handwriting trajectory into high-level features. The RNN decoder aims at generating the caption by detecting radicals and spatial structures through an attention model. The manner of treating a Chinese character as a two-dimensional composition of radicals can reduce the size of vocabulary and enable TRAN to possess the capability of recognizing unseen Chinese character classes, only if the corresponding radicals have been seen. Evaluated on CASIA-OLHWDB database, the proposed approach significantly outperforms the state-of-the-art whole-character modeling approach with a relative character error rate (CER) reduction of 10%. Meanwhile, for the case of recognition of 500 unseen Chinese characters, TRAN can achieve a character accuracy of about 60% while the traditional whole-character method has no capability to handle them.'
author:
-
bibliography:
- 'refs.bib'
title: 'Trajectory-based Radical Analysis Network for Online Handwritten Chinese Character Recognition'
---
Introduction {#sec:Introduction}
============
Machine recognition of handwritten Chinese characters has been studied for decades [@suen1980automatic]. It is a challenging problem due to a large number of character classes and enormous ambiguities coming from handwriting input. Although some conventional approaches have obtained great achievements [@plamondon2000online; @liu2004online; @zhang2017drawing; @yang2016dropsample; @zhong2015high], they only treated the character sample as a whole without considering the similarity and internal structures among different characters. And they have no capability of dealing with unseen character classes.
However, Chinese characters can be decomposed into a few fundamental structure components, called radicals [@chang1973interactive]. It is an intuitive way to first extract information of radicals that is embedded in Chinese characters and then use this knowledge for recognition. In the past few decades, lots of efforts have been made for radical-based Chinese character recognition. For example, [@wang2001optical] proposed a matching method for radical-based Chinese character recognition. It first detected radicals separately and then employed a hierarchical radical matching method to compose radicals into a character. [@ma2008new] tried to over-segment characters into candidate radicals while the proposed way could only handle the left-right structure and over-segmentation brings many difficulties. Recently, [@wang2017label] proposed a multi-label learning for radical-based Chinese character recognition. It turned a character class into a combination of several radicals and spatial structures. Generally, these approaches have difficulties when dealing with radical segmentation and the analysis of structures among radicals is not flexible. Besides, they did not focus on recognizing unseen Chinese character classes.
In this paper, we propose a novel radical-based approach to online handwritten Chinese character recognition, namely trajectory-based radical analysis network (TRAN). Different from above mentioned radical-based approaches, in TRAN the radical segmentation and structure detection are automatically addressed by an attention model which is jointly optimized with the entire network. The main idea of TRAN is to decompose a Chinese character into radicals and detect the spatial structures among radicals. We then describe the analysis of radicals as a Chinese character caption. A handwritten Chinese character is successfully recognized when its caption matches ground-truth. To be more accessible, we illustrate the TRAN learning way in Fig. \[fig:radical-learning-way\]. The online handwritten Chinese character input is visualized at the bottom-left of Fig. \[fig:radical-learning-way\]. It is composed of four different radicals. The handwriting input is finally recognized as the bottom-right Chinese character caption after the top-down and left-right structures among radicals are detected. Based on analysis of radicals, the proposed TRAN possesses the capability of recognizing unseen Chinese character classes if the radicals have been seen.
![Illustration of TRAN to recognize Chinese characters by analyzing the radicals and the corresponding structures.[]{data-label="fig:radical-learning-way"}](radical-learning-way){width="3.5in"}
The proposed TRAN is an improved version of attention-based encoder-decoder model [@bahdanau2014neural] with RNN [@graves2012supervised]. The attention-based encoder-decoder model has been extensively applied to many applications including machine translation [@cho2014learning; @luong2015effective], image captioning [@xu2015show; @vinyals2015show], speech recognition [@bahdanau2016end] and mathematical expression recognition [@zhang2017watch; @zhang2017multi]. The raw data of online handwritten Chinese character input are variable-length sequence (xy-coordinates). TRAN first employs a stack of bidirectional RNN [@graves2013speech] to encode input sequence into high-level representations. Then a unidirectional RNN decoder converts the high-level representations into output character captions one symbol at a time. For each predicted radical, a coverage based attention model [@zhang2017gru] built in the decoder scans the entire input sequence and chooses the most relevant part to describe a segmented radical or a two-dimensional structure between radicals. Our proposed TRAN is related to our previous work [@zhang2017ran] with two main differences: 1) [@zhang2017ran] focused on the application of RAN on printed Chinese character recognition while this paper focuses on handwritten Chinese character recognition. It is interesting to investigate the performance of RAN on handwritten Chinese character recognition as handwritten characters are much more ambiguous due to the diversity of writing styles. 2) Instead of transforming online handwritten characters into static images and employing convolutional neural network [@krizhevsky2012imagenet] to encode them, we choose to directly encode the raw sequential data by employing an RNN encoder in order to fully exploit the dynamic trajectory information that can not be recovered from static images.
The main contributions of this study are as follows:
- We propose TRAN for online handwritten Chinese character recognition.
- The size of radical vocabulary is largely less than Chinese character vocabulary, leading to decrease of redundancy among output classes and improvement of recognition performance.
- TRAN possess the ability of recognizing unseen or newly created Chinese characters, only if the radicals have been seen.
- We experimentally demonstrate how RAN performs on online handwritten Chinese character recognition compared with state-of-the-arts and show its effectiveness on recognizing unseen character classes.
Description of Chinese character caption {#sec:Description of Chinese character caption}
========================================
In this section, we will introduce how we generate captions of Chinese characters. The character caption is composed of three key components: radicals, spatial structures and a pair of braces (e.g. “{” and “}”). A radical represents a basic part of Chinese character and it is often shared by different Chinese characters. Compared with enormous Chinese character categories, the amount of radicals is quite limited. It is declared in GB13000.1 standard published by National Language Committee of China that nearly 500 radicals consist of over 20,000 Chinese characters. As for the complicated two-dimensional spatial structures among radicals, Fig. \[fig:radical-structure\] illustrates eleven common structures and the descriptions are demonstrated as follows:
- **single-element:** sometimes a single radical represents a Chinese character and therefore we can not find internal structures in such characters
- **a:** left-right structure
- **d:** top-bottom structure
- **stl:** top-left-surround structure
- **str:** top-right-surround structure
- **sbl:** bottom-left-surround structure
- **sl:** left-surround structure
- **sb:** bottom-surround structure
- **st:** top-surround structure
- **s:** surround structure
- **w:** within structure
![Graphical representation of eleven common spatial structures among radicals, different radicals are divided by internal line.[]{data-label="fig:radical-structure"}](radical-structure){width="2.6in"}
After decomposing Chinese characters into radicals and internal spatial structures by following [@cjk-decomp], we use a pair of braces to constrain a single structure. Take “stl” as an example, it is captioned as “stl { radical-1 radical-2 }”. The generation of a Chinese character caption is finished when all radicals are included in the caption.
The proposed approach {#sec:The proposed approach}
=====================
![Overall framework of TRAN for online handwritten Chinese character recognition. It is composed of a bidirectional RNN encoder and a unidirectional RNN decoder.[]{data-label="fig:TRAN-framework"}](TRAN-framework){width="3.2in"}
In this section, we elaborate the proposed TRAN framework, namely generating an underlying Chinese character caption from a sequence of online handwritten trajectory points, as illustrated in Fig. \[fig:TRAN-framework\]. First, we extract trajectory information as the input feature from original trajectory points (xy-coordinates). A stack of bidirectional RNNs are then employed as the encoder to transform the input feature into high-level representations. Since the original trajectory points are a variable-length sequence, the extracted high-level representations are also a variable-length sequence. To associate the variable-length representations with variable-length character caption, we generate a fixed-length context vector via weighted summing the high-level representations and a unidirectional RNN decoder uses the fixed-length context vector to generate the character caption one symbol at a time. We introduce an attention model to produce the weighting coefficients so that the context vector can contain only useful trajectory information at each decoding step.
Feature extraction {#sec:Feature extraction}
------------------
During the data acquisition of online handwritten Chinese character, the pen-tip movements (xy-coordinates) and pen states (pen-down or pen-up) are stored as variable-length sequential data: $$\label{eq:trajectory input}
\left \{\left[ {{x_1},{y_1},{s_1}} \right],\;\left[ {{x_2},{y_2},{s_2}} \right],\; \ldots \;,\;\left[ {{x_N},{y_N},{s_N}} \right] \right\}$$ where $N$ is the length of sequence, ${x_i}$ and ${y_i}$ are the xy-coordinates of the pen movements and ${s_i}$ indicates which stroke the $i^{\textrm{th}}$ point belongs to.
To address the issue of non-uniform sampling by different writing speed and the size variations of the coordinates on different potable devices, the interpolation and normalization to the original trajectory points are first conducted according to [@zhang2017drawing]. Then we extract a 6-dimensional feature vector for each point: $$\label{eq:6dimensional feature}
\left[ {{x_i},{y_i},\Delta {x_i},\Delta {y_i},\delta ({s_i} = {s_{i + 1}}),\delta ({s_i} \ne {s_{i + 1}})} \right]$$ where $\Delta {x_i}={x_{i+1}}-{x_i}$, $\Delta {y_i}={y_{i+1}}-{y_i}$, and $\delta ( \cdot ) = 1$ when the condition is true or zero otherwise. The last two terms are flags which indicate the status of the pen, i.e., $\left[ {1,\;0} \right]$ and $\left[ {0,\;1} \right]$ are pen-down and pen-up respectively. For convenience, in the following sections, we use $\mathbf{X}=\left( {{\mathbf{x}_1},\;{\mathbf{x}_2},\; \ldots ,\;{\mathbf{x}_N}} \right)$ to denote the input sequence of the encoder, where ${\mathbf{x}_i} \in {\mathbb{R}^d}$ ($d=6$).
Encoder {#sec:Encoder}
-------
Given the feature sequence $\left( {{\mathbf{x}_1},\;{\mathbf{x}_2},\; \ldots ,\;{\mathbf{x}_N}} \right)$, we employ RNN as the encoder to encode them into high-level representations as RNN has shown its strength in processing sequential signals. However, a simple RNN has revealed serious problems during training namely vanishing gradient and exploding gradient [@bengio1994learning; @zhang2016rnn]. Therefore, an improved version of RNN named gated recurrent units (GRU) [@chung2014empirical] which can alleviate these two problems is employed in this study as it utilizes an update gate and a reset gate to control the flow of forward information and backward gradient. The GRU hidden state ${\mathbf{h}_t}$ in encoder is computed by: $$\label{eq:GRU function}
{{\mathbf{h}}_t} = \textrm{GRU} \left( {{\mathbf{x}_t}, {\mathbf{h}_{t - 1}}} \right)$$ and the GRU function can be expanded as follows: $$\begin{aligned}
\label{eq:expand GRU}
& {{\mathbf{z}}_t} = \sigma ({{\mathbf{W}}_{xz}}{{\mathbf{x}}_{t}} + {{\mathbf{U}}_{hz}}{{\mathbf{h}}_{t - 1}}) \\
& {{\mathbf{r}}_t} = \sigma ({{\mathbf{W}}_{xr}}{{\mathbf{x}}_{t}} + {{\mathbf{U}}_{hr}}{{\mathbf{h}}_{t - 1}}) \\
& {{\bf{\tilde h}}_t} = \tanh ({{\bf{W}}_{xh}}{{\bf{x}}_{t}} + {{\bf{U}}_{rh}}({{\bf{r}}_t} \otimes {{\bf{h}}_{t - 1}})) \\
& {{\bf{h}}_t} = (1 - {{\bf{z}}_t}) \otimes {{\bf{h}}_{t - 1}} + {{\bf{z}}_t} \otimes {{\bf{\tilde h}}_t}\end{aligned}$$ where $\sigma$ denotes the sigmoid activation function, $\otimes$ denotes an element-wise multiplication operator, ${{\mathbf{z}}_t}$, ${{\mathbf{r}}_t}$ and ${{\bf{\tilde h}}_t}$ are the update gate, reset gate and candidate activation, respectively. ${\mathbf{W}}_{xz}$, ${\mathbf{W}}_{xr}$, ${\bf{W}}_{xh}$, ${\mathbf{U}}_{hz}$, ${\mathbf{U}}_{hr}$ and ${\bf{U}}_{rh}$ are related weight matrices.
Nevertheless, even if the unidirectional GRU can have access to the history of input signals, it does not have the ability of modeling future context. Therefore we exploit the bidirectional GRU by passing the input vectors through two GRU layers running in opposite directions and concatenating their hidden state vectors so that the encoder can use both history and future information. To obtain a high-level representation, the encoder stacks multiple GRU layers on top of each other as illustrated in Fig. \[fig:TRAN-framework\]. In this study, our encoder consists of 4 bidirectional GRU layers. Each layer has 250 forward and 250 backward GRU units. We also add pooling over time axes in high-level GRU layers because: 1) the high-level representations are overly precise and contain much redundant information; 2) the decoder needs to attend less if the number of encoder output reduces, leading to improvement of performance; 3) the pooling operation accelerates the encoding process. The pooling is applied to the top GRU layer by dropping the even output over time.
Assuming the bidirectional GRU encoder produces a high-level representation sequence ${\mathbf{A}}$ with length ${L}$. Because there is one pooling operation in the bidirectional GRU encoder, $L = \frac{N}{2}$. Each of these representations is a $D$-dimensional vector ($D=500$): $$\label{eq:annotation A}
\mathbf{A} = \left\{ {{{\mathbf{a}}_1}, \ldots ,{{\mathbf{a}}_L}} \right\}\;,\;{{\mathbf{a}}_i} \in {\mathbb{R}^D}$$
Decoder with attention model {#sec:Decoder with attention model}
----------------------------
After obtaining high-level representations ${\mathbf{A}}$, the decoder aims to make use of them to generate a Chinese character caption. The output sequence ${\mathbf{Y}}$ is represented as a sequence of one-hot encoded vectors: $$\label{eq:caption Y}
\mathbf{Y} = \left\{ { \mathbf{y}_1, \ldots ,\mathbf{y}_C} \right\}\;,\;{{\mathbf{y}}_i} \in {\mathbb{R}^K}$$ where $K$ is the vocabulary size and $C$ is the length of character caption. Note that, both the length of representation sequence (${L}$) and the length of character caption (${C}$) are variable. To address the mapping from variable-length representation sequence to variable-length character caption, we attempt to compute an intermediate fixed-size vector ${\mathbf{c}_t}$ that incorporates useful information of representation sequence. The decoder then utilizes this fixed-size vector to predict the character caption one symbol at a time. As ${\mathbf{c}_t}$ contains overall information of input sequence, we call it context vector. At each decoding step, the probability of the predicted word is computed by the context vector ${\mathbf{c}_t}$, current decoder state ${\mathbf{s}_t}$ and previous predicted symbol ${\mathbf{y}_{t-1}}$ using a multi-layer perceptron: $$\label{eq:compute Y}
p({{\mathbf{y}}_t}|{{\mathbf{y}}_{t - 1}},{\mathbf{X}}) = g \left ({{\mathbf{W}}_o}h({\mathbf{E}}{{\mathbf{y}}_{t - 1}} + {{\mathbf{W}}_s}{{\mathbf{s}}_t} + {{\mathbf{W}}_c}{{\mathbf{c}}_t})\right )$$ where $g$ denotes a softmax activation function over all the symbols in the vocabulary, $h$ denotes a maxout activation function. Let $m$ and $n$ denote the dimensions of embedding and decoder state, ${{\mathbf{W}}_o} \in {\mathbb{R}^{K \times \frac{m}{2}}}$, ${{\mathbf{W}}_s} \in {\mathbb{R}^{m \times n}}$, ${{\mathbf{W}}_c} \in {\mathbb{R}^{m \times D}}$, and ${\mathbf{E}}$ denotes the embedding matrix.
Since the context vector ${\mathbf{c}_t}$ needs to be fixed-length, it is an intuitive way to produce it by summing all representation vectors ${\mathbf{a}_i}$ at time step $t$. However, average summing is too robust and leads to loss of useful information. Therefore, we adopt weighted summing while the weighting coefficients are called attention probabilities. The attention probability performs as a description that tells which part of representation sequence is useful at each decoding step. We compute the decoder state ${\mathbf{s}_t}$ and context vector ${\mathbf{c}_t}$ as follows: $$\begin{aligned}
\label{eq:decoder with attention}
& {{\mathbf{\hat s}}_t} = \textrm{GRU} \left( {{\bf{y}}_{t-1}}, {{\bf{s}}_{t - 1}} \right) \\
& {\mathbf{F}} = {\mathbf{Q}} * \sum\nolimits_{l=1}^{t - 1} {{{\bm{\alpha}}_l}} \label{eq:coverage} \\
& {e_{ti}} = {\bm{\nu }}_{\text{att}}^{\rm T}\tanh ({{\mathbf{W}}_{\text{att}}}{{\mathbf{\hat s}}_t} + {{\mathbf{U}}_{\text{att}}}{{\mathbf{a}}_i} + {{\mathbf{U}}_f}{{\mathbf{f}}_i}) \\
& {\alpha _{ti}} = \frac{{\exp ({e_{ti}})}}{{\sum\nolimits_{k = 1}^L {\exp ({e_{tk}})} }} \\
& {{\mathbf{c}}_t} = \sum\nolimits_{i=1}^L {{\alpha _{ti}}{{\mathbf{a}}_i}} \\
& {{\mathbf{s}}_t} = \textrm{GRU} \left( {{\mathbf{c}}_t}, {{\mathbf{\hat s}}_t} \right)\end{aligned}$$ Here, we can see that the decoder adopts two unidirectional GRU layers to calculate the decoder state ${\mathbf{s}_t}$. The GRU function is the same one in Eq. . ${{\mathbf{\hat s}}_t}$ denotes the current decoder state prediction, ${{e}_{ti}}$ denotes the energy of ${\mathbf{a}_i}$ at time step $t$ conditioned on ${\mathbf{\hat s}_t}$. The attention probability ${{\alpha}_{ti}}$, which is the $i^{\text{th}}$ element of $\bm{\alpha}_t$, is computed by taking ${{e}_{ti}}$ as input of a softmax function. The context vector ${\mathbf{c}_t}$ is then calculated via weighted summing representation vectors ${\mathbf{a}_i}$ with attention probabilities employed as weighting coefficients. During the computation of attention probability, we also append a coverage vector ${\mathbf{f}_i}$ (the $i^{\text{th}}$ vector of $\mathbf{F}$) in the attention model. The coverage vector is computed based on the summation of all past attention probabilities so that the coverage vector contains the information of alignment history as shown in Eq. . We adopt the coverage vector in order to let the attention model know which part of representation sequence has been attended or not [@tu2016modeling]. Let $n^{'}$ denote the attention dimension. Then ${{\bm{\nu }}_{\text{att}}} \in {\mathbb{R}^{{n^{'}}}}$, ${{\mathbf{W}}_{\text{att}}} \in {\mathbb{R}^{{n^{'}} \times n}}$ and ${{\mathbf{U}}_{\text{att}}} \in {\mathbb{R}^{{n^{'}} \times D}}$.
Training and Testing Details {#sec:Training and Testing Details}
============================
The training objective of the proposed model is to maximize the predicted symbol probability as shown in Eq. and we use cross-entropy (CE) as the objective function: $$\label{eq:objective}
O = - \sum\nolimits_{t=1}^C \log p({w_t}|{\mathbf{y}_{t-1},\mathbf{X}})$$ where $w_t$ represents the ground truth word at time step $t$, $C$ is the length of output string. The implementation details of GRU encoder has been introduced in Section \[sec:Encoder\]. The decoder uses two layers with each using 256 forward GRU units. The embedding dimension $m$, decoder state dimension $n$ and attention dimension $n'$ are all set to 256. The convolution kernel size for computing coverage vector is set to $(5 \times 1)$ as it is a one-dimensional convolution operation, while the number of convolution filters is set to 256. We utilize the adadelta algorithm [@zeiler2012adadelta] with gradient clipping for optimization. The adadelta hyperparameters are set as $\rho = 0.95$, $\varepsilon = {10^{ - 8}}$.
In the decoding stage, we aim to generate a most likely character caption given the input trajectory: $$\label{eq:decoding objective}
{\mathbf{\hat y}} = \mathop {\arg \max }\limits_{\mathbf{y}} \log P\left( {{\mathbf{y}}|{\mathbf{X}}} \right)$$ However, different from the training procedure, we do not have the ground truth of previous predicted word. To prevent previous prediction errors inherited by next decoding step , a simple left-to-right beam search algorithm [@cho2015natural] is employed to implement the decoding procedure. Here, we maintained a set of 10 partial hypotheses beginning with the start-of-sentence $<sos>$. At each time step, each partial hypothesis in the beam is expanded with every possible word and only the 10 most likely beams are kept. This procedure is repeated until the output word becomes the end-of-sentence $<eos>$.
Experiments {#sec:Experiments}
===========
In this section, we present experiments on recognizing seen and unseen online handwritten Chinese character classes by answering the following questions:
Q1
: Is the TRAN effective when recognizing seen Chinese character classes?
Q2
: Is the TRAN effective when recognizing unseen Chinese character classes?
Q3
: How does the TRAN analyze the radicals and spatial structures?
Performance on recognition of seen Chinese character classes (Q1) {#sec:Performance on recognition of seen Chinese character classes (Q1)}
-----------------------------------------------------------------
In this section, we show the effectiveness of TRAN on recognizing seen Chinese character classes. The set of character class is 3,755 commonly used Chinese characters. The dataset used for training is the CASIA [@liu2011casia] dataset including OLHWDB1.0 and OLHWDB1.1. There are totally 2,693,183 samples for training and 224,590 samples for testing. The training and testing data were produced by different writers with enormous handwriting styles across individuals.
**Methods** **Reference** **Accuracy**
----------------------- --------------------- --------------
Human Performance [@yin2013icdar] 95.19%
Traditional Benchmark [@liu2013online] 95.31%
NET4 [@zhang2017drawing] 96.03%
TRAN – 96.43%
: \[tab:1\][Results on CASIA dataset of online handwritten Chinese character recognition.]{}
In Table \[tab:1\], the human performance on CASIA test set and the previous benchmark are both listed. NET4 is the proposed method in [@zhang2017drawing] which represents the state-of-the-art method on CASIA dataset and it belongs to non-radical based methods. NET4 achieved an accuracy of 96.03% while TRAN achieved an accuracy of 96.43%, revealing relative character error rate reduction of 10%. To be fairly comparable, here NET4 and TRAN both did not use the sequential dropout trick as proposed in [@zhang2017drawing] so the performance of NET4 is not as good as the best performance presented in [@zhang2017drawing]. As explained in the contributions of this study in Section \[sec:Introduction\], the main difference between radical based method and non-radical based method for Chinese character recognition is the size of radical vocabulary is largely less than Chinese character vocabulary, yielding decrease of redundancy among output classes and improvement of recognition performance.
Performance on recognition of unseen Chinese character classes (Q2) {#sec:Performance on recognition of unseen Chinese character classes (Q2)}
-------------------------------------------------------------------
The number of Chinese character classes is not fixed as more and more novel characters are being created. Also, the overall Chinese character classes are enormous and it is difficult to train a recognition system that covers them all. Therefore it is necessary for a recognition system to possess the capability of recognizing unseen Chinese characters, called zero-shot learning.
Obviously traditional non-radical based methods are incapable of recognizing these unseen characters since the objective character class has never been seen during training procedure. However TRAN is able to recognize unseen Chinese characters only if the radicals composing unseen characters have been seen. To validate the performance of TRAN on recognizing unseen Chinese character classes, we divide 3755 common Chinese characters into 3255 classes and the other 500 classes. We choose handwritten characters belonging to 3255 classes from original training set as the new training set and we choose handwritten characters belonging to the other 500 classes from original testing set as the new testing set. By doing so, both the testing character classes and handwriting variations have never been seen during training. We explore different size of training set to train TRAN, ranging from 500 to 3255 Chinese character classes and we make sure the radicals of testing characters are covered in training set.
**Train classes** **Train samples** **Test Accuracy**
------------------- ------------------- -------------------
500 359,036 -
1000 717,194 10.74%
1500 1,075,344 26.02%
2000 1,435,295 39.35%
2755 1,975,972 50.45%
3255 2,335,433 60.37%
: \[tab:2\][Results on newly divided testing set based on CASIA dataset of online handwritten unseen Chinese character recognition.]{}
We can see in Table \[tab:2\] the recognition accuracy of unseen Chinese character classes is not available when training set only contains 500 Chinese character classes. We believe it is difficult to train TRAN properly to accommodate large handwriting variations when the number of character classes is quite small. When the training set contains 3255 character classes, TRAN achieves a character accuracy of **60.37**% which is a relatively pleasant performance compared with traditional recognition systems as they can not recognize unseen Chinese character classes which means their accuracies are definitely **0**%. The performance of recognizing unseen Chinese character classes is not as good as the performance presented in [@zhang2017ran] because the handwritten Chinese characters are much more ambiguous compared with printed Chinese characters due to the large handwriting variations.
Attention visualization (Q3) {#sec:Attention visualization (Q3)}
----------------------------
![Examples of attention visualization during the decoding procedure. The red color on trajectory describes the attention probabilities namely the lighter color denotes higher attention probabilities and the darker color denotes lower attention probabilities.[]{data-label="fig:attention_visualization"}](attention_visualization){width="2.8in"}
In this section, we show through attention visualization how TRAN is able to recognize internal radicals and analyze the two-dimensional spatial structure among radicals. Fig. \[fig:attention\_visualization\] illustrates an example of attention visualization. Above the dotted line, there is one Chinese character class and its corresponding character caption. Below dotted line, there are images denoting the visualization of attention probabilities during decoding procedure. We draw the trajectory of input handwritten Chinese character in a two-dimensional greyscale image to visualize attention. Below images there are corresponding symbols generated by decoder at each decoding step.
As we can see in Fig. \[fig:attention\_visualization\], when encountering basic radicals, the attention model generates the alignment well corresponding to the human intuition. Also, it mainly focus on the ending of last radical and the beginning of next radical to detect a spatial structure. Take “d” as an example, by attending to the ending of last radical and the beginning of next radical, the attention model detects a top-bottom direction, therefore a top-bottom structure is analyzed. Immediately after generating a spatial structure, the decoder produces a pair of braces “{}”, which are employed to constrain the two-dimensional structure in Chinese character caption.
Conclusion and future work {#sec:Conclusion and future work}
==========================
In this study we introduce TRAN for online handwritten Chinese character recognition. The proposed TRAN recognizes Chinese character by identifying internal radicals and analyzing spatial structures among radicals. We show from experimental results that TRAN outperforms the state-of-the-art method on recognition of online handwritten Chinese characters and possesses the capability of recognizing unseen Chinese character categories. By visualizing learned attention probabilities, we can observe the alignments of radicals and analysis of structures correspond well to human intuition.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Hand keypoints detection and pose estimation has numerous applications in computer vision, but it is still an unsolved problem in many aspects. An application of hand keypoints detection is in performing cognitive assessments of a subject by observing the performance of that subject in physical tasks involving rapid finger motion. As a part of this work, we introduce a novel hand keypoints benchmark dataset that consists of hand gestures recorded specifically for cognitive behavior monitoring. We explore the state of the art methods in hand keypoint detection and we provide quantitative evaluations for the performance of these methods on our dataset. In future, these results and our dataset can serve as a useful benchmark for hand keypoint recognition for rapid finger movements.'
author:
- Srujana Gattupalli
- Ashwin Ramesh Babu
- James Robert Brady
- Fillia Makedon
- Vassilis Athitsos
title: Towards Deep Learning based Hand Keypoints Detection for Rapid Sequential Movements from RGB Images
---
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In this paper we propose a new state observer design technique for nonlinear systems. It combines the well-known Kazantzis-Kravaris-Luenberger observer and the recently introduced parameter estimation-based observer, which become special cases of it—extending the realm of applicability of both methods. A second contribution of the paper is the proof that these designs can be recast as particular cases of immersion and invariance observers—providing in this way a unified framework for their analysis and design. Simulation results of a physical system that illustrates the superior performance of the proposed observer compared to other existing observers are presented.'
author:
- 'Bowen Yi, Romeo Ortega, and Weidong Zhang[^1] [^2] [^3]'
title: 'On State Observers for Nonlinear Systems: A New Design and a Unifying Framework'
---
[Shell : Bare Demo of IEEEtran.cls for IEEE Journals]{}
State observers, nonlinear systems, parameter estimation.
Introduction and Problem Formulation {#sec1}
====================================
In this paper we are interested in the design of state observers for nonlinear control systems whose dynamics is described by[^4] $$\label{sys}
\begin{aligned}
\dot{x} & = f(x,u) \\
y & = h(x),
\end{aligned}$$ where $x \in \bbr^n$ is the system state, $u \in \bbr^m$ is the control signal, and $y \in \bbr^p$ is the [*measurable*]{} output signal. The problem is to design a dynamical system & = & F(y, , u)\
x & = & H(y,,u) \[dynobs\] with $\chi \in \bbr^{n_\chi}$, such that \_[t]{} |x(t)-x(t)|=0, where $|\cdot|$ is the Euclidean norm. Following standard practice in observer theory we assume that $u$ is such that the state trajectories of are bounded.
Since the publication of the seminal paper [@Luenberger1966TAC], which dealt with linear time-invariant (LTI) systems, this problem has been extensively studied in the control literature. We refer the reader to [@astolfi2008book; @BESbook] for a review of the literature. In this paper we are particularly interested in three recently developed observer design techniques.
The Kazantzis-Kravaris-Luenberger observer (KKLO) first presented in [@Kazantzis1998SCL] as an extension to nonlinear systems of Luenberger’s observer and further developed in [@Andrieu2006SIAM].
Parameter estimation-based observer (PEBO) proposed in [@ortega2015scl], which translates the state observation problem into an on-line *parameter estimation* problem.
Immersion and invariance observer (I$\&$IO), first reported in [@Karagiannis2008TAC] and thoroughly elaborated in [@astolfi2008book], which proposes a more general observer framework based on the generation of attractive and invariant manifolds.
The main contributions of our paper are threefold.
Propose a new observer design technique, called \[KKL+PEB\]O, that combines—in a seamless way—the KKLO and PEBO designs, yielding a new design applicable to a broader class of systems.
Prove that KKLO, PEBO and \[KKL+PEB\]O can be recast as particular cases of generalized I$\&$IO—providing in this way a unified framework for their analysis and design.
Present simulation results of the well-known DC-DC Ćuk converter that illustrate the superior performance of the proposed observer compared to other existing observer designs.
The remainder of the paper is organized as follows. Section \[sec2\] gives some preliminaries on KKLO and PEBO. In Section \[sec3\] we present the new \[KKL+PEB\]O. The unifying framework based on immersion and invariance is given in Section \[sec4\]. In Section \[sec5\] we present two academic examples that illustrate the interest of the new \[KKL+PEB\]O and some simulation results of a physical system that compares the new observer with other observer designs. The paper is wrapped–up with concluding remarks and future research directions in Section \[sec6\].
Preliminaries
=============
In this section we briefly present simple versions of the KKLO and the PEBO that are motivating to generate the new \[KKL+PEB\]O in the next section.
Kazantzis-Kravaris-Luenberger Observer {#subsec21}
--------------------------------------
The KKLO design is based on the following proposition, which is a simplified version of the more general result reported in [@Andrieu2006SIAM; @Kazantzis1998SCL].
\[pro1\] Consider the system satisfying the following assumption.
There exist $n_\xi$ negative real numbers $\lambda_i,\;i=1,2,\dots, n_\xi$, with $n_\xi \geq n$, and mappings $$\begin{aligned}
\phi &: \rea^n \to \rea^{n_\xi}\\
\phil &: \rea^{n_\xi} \times \rea^p \to \rea^{n}\\
B & : \rea^p \times \rea^m \to \rea^{n_\xi}
\end{aligned}$$ with $n_\xi \geq n - p$, satisfying the following.
The KKLO partial differential equation (PDE) $$\label{KKLOPDE}
\nabla \phi^\top(x) f(x,u) = \Lambda \phi(x) + B(h(x),u),$$ where $\nabla:=({\partial \over \partial x})^\top$ and $\Lambda:=\diag\{\lambda_i\}$.
$\phil$ is a left inverse of $\phi$, that is, ((x),h(x))=x. The KKLO & = & + B(y, u)\
& = & (,y), ensures .
The proof of this proposition follows immediately defining the error signal $$e:=\xi - \phi(x)$$ and noting that $\dot e = \Lambda e.$
Parameter Estimation-based Observer {#subsec22}
-----------------------------------
The PEBO design proposed in [@ortega2015scl], although related with the KKLO, aims at formulating the state reconstruction problem as a parameter estimation problem. Towards this end, we are looking for an injection $B(h(x),u)$ and a (left invertible) mapping $\phi(x)$ that transforms the system into[^5] $$\label{dotphi}
\dot \phi(x) = B(h(x),u).$$ In this way, selecting (part of) the observer dynamics as \[dotxi\] = B(y,u), we establish, via simple integration, the key relationship (x(t)) = (t) + , where $\theta$ is a constant vector defined as $\theta:=\phi(x(0))-\xi(0)$. It is clear that, if $\theta$ is known, we have that $$x=\phil(\xi+\theta,y).$$ Hence, the remaining task is to generate an [*estimate*]{} for $\theta$, denoted $\hat \theta$, to obtain the observed state x=(+,y). To achieve this end, we rely on the existence of the regression model
y\
y
=
h((+,y))\
h\^((+,y))f((+,y),u)
, where we underscore that $y,u$ and $\xi$ are, of course, measurable.
The main result in [@ortega2015scl] may be summarized as follows.
\[pro2\] Consider the system satisfying Assumption [**A1**]{} of Proposition \[pro1\] [*with*]{} $\Lambda =0$ and the dynamic extension verifying the following assumption.
There exist mappings $$\begin{aligned}
& M: \rea^{n_\zeta} \times \rea^{n_\xi} \times \rea^p \times \rea^m \to \rea^{n_\zeta}\\
& N: \rea^{n_\zeta} \times \rea^{n_\xi} \times \rea^p \times \rea^m \to \rea^{n_\xi}
\end{aligned}$$ such that the dynamical system $$\label{parest0}
\begin{aligned}
\dot{\zeta} & = M(\zeta, \xi, y, u) \\
\hat{\theta} & = N(\zeta, \xi, y, u),
\end{aligned}$$ defines a stable, [*consistent*]{} parameter estimator for the regression model , that is $\zeta$ is bounded and \_[t]{} | (t) - | = 0. The PEBO , , verifies .
Remarks {#subsec23}
-------
[**R1**]{}Notice that the KKLO , together with the dynamics of the system , admits an [*attractive and invariant*]{} manifold $$\cals:=\{ (x,\xi) \in \rea^n \times \rea^{n_\xi}\;|\; \xi = \phi(x)\},$$ and the state is (asymptotically) reconstructed, via $\phil$, with the knowledge of $\xi$. On the other hand, the PEBO generates an [*invariant*]{} foliation $$\cals_\theta:=\{ (x,\xi) \in \rea^n \times \rea^{n_\xi}\;|\; \xi = \phi(x)+\theta,\;\theta \in \rea^{n_\xi}\}.$$ To reconstruct the state—again via $\phil$—it is necessary to identify the leaf of the foliation where the system evolves. These observations are essential to establish the connection of these observers with the I&IO, which also relies on the generation of an attractive and invariant manifold, defined by an invertible mapping.\
[**R2**]{}Besides the additional difficulty of needing to estimate $\theta$, the main drawback of PEBO is that it relies on the open-loop integration , which might be a problematic operation in practice. In spite of that, PEBO has proven instrumental in the solution of numerous physical systems problems [@bobtsov2015automatica; @bobt2017MagLev; @CHOetal; @PYRetal]—some of them being unsolvable with other observer design techniques—with the open integration problem solved via the addition of “safety nets" similar to the ones used in PID designs or adaptive control.\
[**R3**]{}We underscore the fact that the PDE that needs to be solved for PEBO is [*exactly*]{} the one of KKLO with $\Lambda=0$, that is $$\nabla \phi^\top(x) f(x,u) = B(h(x),u).$$ We refer the reader to [@pebo2017] where the generation of virtual outputs via signal injection technique of [@COMetal] is proposed to simplify the solution of this PDE.
New \[KKL+PEB\] Observers {#sec3}
=========================
In this section we present our first main contribution, namely, a new observer design technique that combines PEBO and KKLO. The key idea of the new observer is to split the states to be estimated in two groups, the first one estimated with a KKLO and the second one with a PEBO.
Main result {#subsec31}
-----------
The following proposition, whose proof follows [*verbatim*]{} from Propositions \[pro1\] and \[pro2\] formalises the discussion above. For ease of presentation, and without loss of generality, we assume that the aforementioned groups are arranged one on top of the other.
\[pro3\] Consider the system satisfying Assumption [**A1**]{} of Proposition \[pro1\] [*with*]{}
& & = &\
& \_L: &= & {\_1,…,\_[q]{}},
where $0 \leq q \leq n_\xi,$ ${\bf 0}_{k \times j}$ is a $k \times j$ matrix of zeros, $\lambda_i < 0,\;i=1,\dots,q$. Partition the mapping $B(y,u)$ as follows $$B(y,u)=\lef[{c} B_L(y,u) \\ B_P(y,u) \rig],\;B_L \in \rea^{q},\;B_P\in \rea^{n_\xi -q}.$$ The \[KKL+PEB\]O
\[pebo-eq\] \_L & = \_L\_L + B\_L(y,u)\
\_P & = B\_P(y,u)\
& = M\_P(, \_P, y, u)\
&=N\_P(, \_P,y,u)\
& = (
\_L\
\_P
+
\_q\
,y ),
where $\mathbf{0}_q$ is a $q$-dimensional vector of zeros, ensures provided the mappings $$\begin{aligned}
& M_P: \rea^{n_\zeta} \times \rea^{n_\xi-q} \times \rea^p \times \rea^m \to \rea^{n_\zeta},\;\\
& N_P: \rea^{n_\zeta} \times \rea^{n_\xi-q} \times \rea^p \times \rea^m \to \rea^{n_\xi-q},
\end{aligned}$$ define a consistent estimator, that is, holds.
Remarks {#subsec32}
-------
[**R4**]{}It is clear that Proposition \[pro3\] contains, as particular cases, Propositions \[pro1\] and \[pro2\]. Indeed, the former is recovered setting $q=n_\xi$ while the latter follows with $q=0$.\
[**R5**]{}The result of Proposition \[pro3\] can be extended in several directions. For instance, it is possible to replace the PDE by $$\nabla \phi^\top(x) f(x,u) = A \phi(x) + B(h(x),u),$$ where $A$ is such that there exists a unitary matrix $P$ ensuring $A=P^\top \Lambda P$ with $\Lambda$ given in .[^6] Clearly, the degree of freedom provided by the inclusion of the matrix $A$ enlarges the set of solutions of the PDE. In this case, the dynamics of $(\xi_L,\xi_P)$ in the observer is replaced by & = & + P B(y, u)\
& = & (P\^+ ,y).\
[**R6**]{}In the case of input-affine systems, [*i.e.*]{}, $f(x,u)=F(x)+g(x)u$, it is possible to decompose the PDE into two, that is, \^(x) F(x) & = & A (x) + B\_F(h(x))\
\^(x) g(x) & = & B\_g(h(x)) and define the observer dynamics via $$B(y,u) := B_F(y) + B_g(y)u.$$ Explicit formulas for the solutions of these equations may be found in [@pebo2017].
I&I Observers: An Unifying Framework {#sec4}
====================================
In this section we show that a mild extension of the I&IO studied in [@Karagiannis2008TAC; @astolfi2008book] allows us to capture, as a particular case the new \[KKL+PEB\]O proposed in this paper—and, consequently, it also contains the KKLO and the PEBO.
Extension of I&I observers {#subsec41}
--------------------------
The main result of the I&IO in [@Karagiannis2008TAC] is extended in the following proposition by relaxing a dimension requirement imposed to some mappings in the original formulation of I&IO—see [**R8**]{} in Subsection \[subsec43\].
\[pro4\] Consider the system . Assume the existence of mappings $$\begin{aligned}
\beta &: \rea^p \times \rea^{n_\chi} \to \rea^{n_z}\\
\phi &: \rea^n \to \rea^{n_z}\\
\phil & : \rea^{n_z} \times \rea^p \to \rea^{n}
\end{aligned}$$ with $\phil(\phi(x),y) = x$ and $n_\chi \ge n_z$, such that the following assumptions hold.
${\mbox{rank }}\nabla_\chi \beta^\top (y,\chi) = n_z$.
The system with state
\_ = & \_y\^( h\^(x) f(x,u) - h\^() f(,u) )\
& - \^(x) f(x,u) + \^() f(,u)
where
& & := &((x)+, h(x))\
& &:= &(y,) - (x),
has an asymptotically stable equilibrium $\dm=0$. The I&IO & = & - \[\_\^\]\^ ( \_y \^h\^() - \^() ) f(,u)\
& & + ( I - \[\_\^\]\^\_\^) Q (y,,u)\
& = & ((y,),y). with $[\cdot]^\dagger$ the pseudoinverse and $Q: \rea^p \times\rea^{n_\chi} \times \rea^m \to \rea^{n_\chi \times n_\chi}$ an arbitrary mapping, verifies .
The dynamics of off-the-manifold coordinate $\dm$ is $$\dot{d}_{\mathcal{M}} = \nabla_y \beta^\top \nabla h^\top(x) f(x,u) + \nabla_\chi \beta^\top \dot{\chi} - \nabla \phi^\top(x) f(x,u).$$ Replacing the dynamics of $\dot{\chi}$ in , we get . According to Assumption [**A5**]{}, we have $$\lim_{t\to \infty} \dm(t) =0.$$ Replacing this limit in and recalling that $\phil(\phi(x),y) = x$ ensures $\lim_{t\to\infty} |\hat{x}(t) - x(t)|=0.$
KKL+PEB observers: An I&I interpretation {#subsec42}
----------------------------------------
In this section we will show that if the system admits a \[KKL+PEB\]O it also admits an I&IO. To unify the notation we define $$\chi := \begin{bmatrix}\xi_L \\ \xi_P \\ \zeta \end{bmatrix}, \;
\xi := \begin{bmatrix}\xi_L \\ \xi_P \end{bmatrix},\;
\upsilon := \begin{bmatrix}\xi_P \\ \zeta \end{bmatrix},$$ introduce the partitions $$\begin{aligned}
\phi(x) & := \begin{bmatrix}\phi_L(x) \\ \phi_P(x) \end{bmatrix} \\
M& := \begin{bmatrix} \mathbf{0}_q \\ M_P(\zeta, \xi_P,y,u) ) \end{bmatrix} \\
N & :=\begin{bmatrix}\mathbf{0}_q,N(\zeta, \xi_P,y,u)\end{bmatrix}
\end{aligned}$$ and define the mapping \[upsilon\]
(x,,u):= & \_y\^( h\^(x) f(x,u) - h\^() f(,u) )\
& - \^(x) f(x,u) + \^() f(,u),
that, according to , defines the dynamics of the off-the-manifold coordinate ${d}_\mathcal{M}$.
\[prop5\] Assume the system admits a \[KKL+PEB\]O with $${\mbox{rank }}\big(\big[I_{n_{{\xi}}}+ \nabla_{{\xi}} N^\top ~\big|~\nabla_{\zeta}N^\top\big]\big) = n_{\xi},$$ and $\{(x,\xi,\zeta)|\theta - \hat{\theta} =0 \}$ is invariant. Then, the system admits an I&IO - with the mappings selected as , and . $$\begin{aligned}
&\beta = \xi + N(\zeta,\xi_P,h(x)) \label{mapping3-1} \\
&\Upsilon = \nonumber\\
& ~~ \begin{bmatrix}
- \Lambda_L \phi_L(x) + \Lambda_L \xi_L \\
\mathbf{0}_{n_\xi - q} \\
\nabla_{\zeta}\beta^\top \big( M_P(\zeta, \phi_P(\hat{x}),h(x),u) - M_P(\zeta, \phi_P(x),h(x),u) \big)
\end{bmatrix}
\label{mapping3-2} \\
& Q = \mathbf{0}. \label{mapping3-3}\end{aligned}$$ If the measurable output signals [*are*]{} partial states, the \[KKL+PEB\]O *exactly* coincides with the I&IO -, and $$\mbox{I\&IO PDE \eqref{upsilon}, \eqref{mapping3-2}}\\
\quad \Longrightarrow \quad
\mbox{[KKL+PEB]O PDE \eqref{KKLOPDE}}.$$
For the sake of clarity, we assume $N_P$ is independent of $u$ to avoid further complicating the notation. Before the proof, we present the following two useful facts.
1. If the output signals are partial states, *i.e.*, $x :={\rm col}(\mathbf{x}_1, \mathbf{x}_2)$ and $y = \mathbf{x}_2$ without loss of generality, we have $$h(\hat{x}) = h(\phil (\xi + N(\zeta,\xi_P,y),y ))
= h( {\rm col}(\hat{\mathbf{x}}_1, \mathbf{x}_2))
= h(x).$$
2. When $\xi_P = \phi_P(x)$, we have $${d\over dt} N_P( \zeta,\phi_P(x), y)
= \mathbf{0}_{n_{\xi} -q},$$ thus yielding the following identity.
\_[\_P]{} N\_P\^\_P\^(x) f(x,u) &+ \_ N\_P\^M\_P(,\_P(x),y,u)\
& + \_y N\_P\^h\^(x) f(x,u) = .
According to , Assumption [**A4**]{} is obviously satisfied. The reminding of the proof is divided into two parts: 1) the selected mappings yield the dynamics -, having the same structure as in \[KKL+PEB\]O; 2) these mappings are solutions of I&IO PDE.
**1)** The dynamics of $\dot{\chi}$ in has the term $[\nabla_\chi \beta^\top]^{\dagger} ( \nabla_y\beta^\top \nabla h^\top - \nabla \phi ^\top ) f(\hat{x},u) $, in which
& \_y \^h\^() f(,u) - \^() f(,u)\
= &
\_q - \_L\^()\
\_y N\_P\^h\^() - \_P\^()
f(, u)
We analyze the above equation in two parts, *i.e.*, $ - \nabla \phi_L^\top f$ and $(\nabla_y N_P^\top \nabla h^\top - \nabla \phi_P^\top) f$. For the first part, the existence of a \[KKL+PEB\] observer yields the PDE $$\Lambda_L\phi_L(x) + B_L(y,u) = \nabla \phi_L^\top (x) f(x,u).$$ Substituting $x$ by $\hat{x} = \phi^{\mathtt{L}}(\xi_L)$, we have \_L\^() f(, u) = \_L \_L + B\_L(y,u). The second partition of verifies the relation below.
& (\_y N\_P\^h\^()- \_P\^()) f(,u)\
& - ( \_[\_P]{} N\_P\^(,\_P(), h()) + I) \_P\^() f(,u)\
& - \_ N\_P\^(,\_P(), h()) M\_P(,\_P(),h(),u)\
= & - \_ \^
B\_P( h(),u)\
M\_P(,\_P ,h(),u)
.
Combining -, we get the mapping $\alpha(\cdot)$ in I&IO is $$\dot{\chi} = \alpha(y,\chi,u) = \begin{bmatrix}
\Lambda_L {\xi }_L + B_L(y,u)\\
B_P( h(\hat{x}),u) \\
M_P(\zeta, \xi_P ,h(\hat{x}),u)
\end{bmatrix},$$ showing that *if* the I&IO PDE has a solution, then the I&IO *asymptotically* coincides the \[KKL+PEB\]O. Furthermore, if the measurable output signals are partial states, due to fact [**F1**]{}, the I&IO *exactly* coincides with the \[KKL+PEB\]O.
**2)** We check the solution existence of I&IO PDE. The first partition of I&IO PDE is verified as follows.
& \_L\_L(x) + B\_L(y,u) = \_L\^(x) f(x,u).\
&- \_L\_L(x) - B\_L (y,u) + \_L\_L + B\_L(y,u)\
& = -\_L\^(x) f(x,u) + \_L\^() f(,u)\
& \_L\_L - \_L\_L(x) = \_L\^() f(,u) -\_L\^(x) f(x,u)
For the second partition of the I&IO PDE, the identity yields . Combining -, it shows that the selecting mappings are solutions of I&IO PDE.
The invariant manifold of \[KKL+PEB\]O is $$\calm = \left\{ (x,\chi) \in \rea^n \times \rea^{n_\chi} \big| \;
\xi + N(\zeta,\xi_P,y) = \phi(x) \right\}.$$ The dynamics of off-the-manifold coordinate is $\dot{d}_{\mathcal{M}} = \Upsilon(x,\chi,u)$, whose convergence is guaranteed by the consistent identification Assumption [**A2**]{} and the fact that the matrix $\Lambda_L$ is Hurwitz. This completes the proof.
& -\_ \^(h(x),)
B\_P(h(x),u)\
M\_P(, \_P(x),h(x),u)
+ \_ \^(h(),)
B\_P(h(),u)\
M\_P(, \_P(),h(),u)
\
= & -\_ \^(h(x),)
\_P\^(x) f(x,u)\
M\_P(, \_P(x),h(x),u)
+ \_ \^(h(),)
\_P\^() f(,u)\
M\_P(, \_P(),h(),u)
\
&\^\_P(x) f(x,u) = B\_P(h(x),u).
We are in position to present the main result of this paper—the unified observer framework captured by I&IO.
For the nonlinear system , a \[KKL+PEB\]O implies the existence of I&I observers. Moreover, the following “set" relationship holds: .
\
} .
Remarks {#subsec43}
-------
[**R7**]{}As discussed above, an I&IO generates the invariant manifold $$\calm = \left\{ (x,\chi) \in \rea^n \times \rea^{n_\chi} \big| \; \beta(h(x),\chi) = \phi(x) \right\},$$ which is made attractive ensuring—via Assumption [**A5**]{}—that the zero equilibrium of the dynamics of the off-the-manifold coordinate , which may be written as [ $$\dot{d}_{\mathcal{M}} = \Upsilon(x,\chi,u) =: \Upsilon_0(x,\dm,u)$$]{} has an asymptotically stable equilibrium at the origin.\
[**R8**]{}In the I&IO proposed in [@Karagiannis2008TAC; @astolfi2008book] we fix $ n_\chi = n_z \le n $. In this case, reduces to $$\dot{\chi} =
- [\nabla_\chi \beta^{-1}]^\top
( \nabla_y\beta^\top \nabla h^\top(\hat{x}) - \nabla \phi^\top (\hat{x}) ) f(\hat{x},u),$$ and [**A4**]{} is equivalent to requiring that $\nabla_\chi\beta$ is a non-singular square matrix. For PEBO and \[KKL+PEB\]O, the dimensions of their corresponding invariant manifolds are less than the dimensions of dynamic extensions. Hence, we generalise I&IO removing the requirement $n_\xi = n_z$ and using the pseudoinverse.\
[**R9**]{}KKLOs and PEBOs are specific cases of \[KKL+PEB\]Os, making them covered by I&IO framework. More specifically the following statements hold.
- the KKLO coincides with the I&IO - with mappings selecting $$\begin{aligned}
n_{\chi} & = n_{z} = q \\
\beta(y,\chi) & = \xi = \chi \\
\Upsilon(x,\chi,u) & = -\Lambda\phi(x) + \Lambda\chi
\end{aligned}$$ with *any* mapping $Q(y,\chi,u)$. The KKLO PDE sacrifices the freedom for $\Upsilon$ by fixing $\Upsilon = - \Lambda\phi(x) + \Lambda\chi$.
- If the measurable output signals are partial states, the PEBO coincides with the I&IO with mappings [^7] $$\begin{aligned}
& \chi ={\mbox{col}}(\xi,\zeta) \\
& \beta = \xi + N(\zeta,\xi,y) \\
&\Upsilon = \nabla_\chi\beta^\top
\begin{bmatrix}
\mathbf{0} \\
M(\zeta, \phi(\hat{x}),h(\hat{x}),u) - M(\zeta, \phi(x),h(x),u)
\end{bmatrix}
\end{aligned}$$ and $Q=0$, where $ \hat{x} = \phil(\beta(y,\chi),y)$.
\
[**R10**]{}Notice that the condition that “the measurable output signals [*are*]{} partial states" in Proposition \[prop5\] is done without loss of generality because it is always possible to do a change of coordinates to verify it.
Examples {#sec5}
========
Proving the interest of the new observer
----------------------------------------
In this section, an academic example for which neither KKLO nor PEBO are applicable, but it is solvable via our new \[KKL+PEB\]O design.
Consider the system \[num\_exmp\]
\_1 & = - x\_1\^3 + e\^[x\_3]{}\
\_2 & = - x\_2 + x\_1\^2 + x\_1\
\_3 & = (x\_1\^2 +1)\^[-1]{} + x\_1 u\
y & = x\_1,
The following facts hold.
The system [*does not admit*]{} a KKLO nor a PEBO.
The system [*admits*]{} a \[KKL+PEB\]O, namely, $$\begin{aligned}
\dot{\xi}_1 & = - \xi_1 + y^2 + \sin y \label{dyn_ext_5.1-1} \\
\dot{\xi}_2 & = u y + ( y^2 +1 )^{-1} \label{dyn_ext_5.1-2}\\
\dot{ \hat{\Theta}} & = \gamma \psi (Y - \psi \hat{\Theta}) \label{exp_estimator5.1-1} \\
\hat{x}_2 & = \xi_1 \\
\hat{x}_3 & = \xi_2 + \ln \hat{\Theta}.\end{aligned}$$ where $\gamma >0$ is an adaptation gain and $Y,\psi$ are obtained via LTI filtering as Y & = & [p p+]{}+ [p +]{}\
& = & [p+]{},(0)>0 with $p:={d \over dt}$ and $\alpha>0$, is a \[KKL+PEB\]O that ensures $$\lim_{t\to\infty} |\hat x_i(t) - x_i(t)|=0,\;i=2,3.$$
\[Proof of [**F3**]{}\] KKLO requires $\phi(x)$ to be injective. To guarantee this property at least one of its three components should depend on $x_3$. Suppose, without loss of generality, that $\phi_2(x)$ depends on $x_3$. Define the three-dimensional vector $\rho$ as (x):=\_2(x). From the PDE we have \^(x)
-x\_1\^3 + e\^[x\_3]{}\
-x\_2 + x\_1\^2 + x\_1\
x\_1 u + ( 1+x\_1\^2)\^[-1]{}
=-\_2 \_2(x) + B\_2(x\_1,u) Since $\phi_2(x)$ dependends of $x_3$, we have $\rho_3(x) \neq 0$. The left hand side term of dependent on $u$ is $\rho_3(x) x_1u$, while the one in the right hand side is $B_2(x_1,u)$, from which we conclude that $\rho_3(x)$ [*only depends*]{} on $x_1$, that is $\rho_3(x) = \rho_3(x_1)$. From Poincare’s lemma we have that holds if and only if the Jacobian $\nabla \rho(x)$ is a [*symmetric*]{} matrix. Applying this condition to the $(1,3)$ element of the Jacobian we conclude that $\rho_1(x)$ should satisfy $$\rho_1(x) :=\rho'_3(x_1) x_3 + L(x_1,x_2),$$ with $L(x_1,x_2)$ to be determined and $(\cdot)'$ the derivative with respect to its argument. From the $(2,3)$ element we also conclude that $\rho_2(x)$ is independent of $x_3$, that is $\rho_2(x) : = \rho_2(x_1,x_2)$.
The terms in the left-hand side of dependent on $x_3$ are $-\rho'_3(x_1) x_3 x_1^3 + \rho'_3(x_1) x_3 e^{x_3} + L(x_1,x_2) e^{x_3}$, while the one on the right-hand side is $-\lambda_2\rho_3(x_1) x_3$. Thus we conclude that $L(x_1,x_2) =0$ and $$-\lambda_2\rho_3(x_1) x_3 = -\rho'_3(x_1) x_3 x_1^3 + \rho'_3(x_1) x_3 e^{x_3}.$$ Hence $$-\lambda_2\rho_3(x_1) = \rho'_3(x_1)(- x_1^3 + e^{x_3}),$$ whose only solution is $\rho_3 =0$, which contradicts with the fact that $\rho_3 \neq 0$, due to $\lambda_2 \neq 0$ in the KKLO. Therefore, it shows that the system does not admit a KKLO.
For the injectivity of $\phi(x)$ in PEBO, assume that $\phi_2(x)$ depends on $x_2$ and $\rho_2(x)\neq 0$. It follows from the argument above that, for the PEBO with $\lambda_2 =0$ we have $\rho_3'(x_1)=0$, yielding $\rho_3(x_1) = c$ and $\rho_1(x)=0$ with a constant $c$. From the (1,2) and (2,1) elements of the Jacobian matrix $\nabla \rho(x)$ we conclude that $\nabla_{x_1}\rho_2=0$ and $\rho_2(x_1,x_2):=\rho_2(x_2)$. Then in terms of , we have $$-\rho_2(x_2)x_2 + \rho_2(x_2) x_1^2 + \rho_2(x_2) \sin x_1 = B_2(x_1,u).$$ Since the first term in the left hand side does not depend on $x_1$, we conclude that $\rho(x_2) =0$ leading to a contradiction. Thus the given system does not admit a PEBO.\
\[Proof of [**F4**]{}\] The \[KKL+PEB\]O PDE with has a solution as $q=1$, $\lambda_1=-1$, $\phi(x) = {\mbox{col}}(x_2,x_3)$ and $$B(h(x),u)=
\begin{bmatrix}
x_1^2 + \sin x_1 \\
x_1 u + {1 \over x_1^2 +1}
\end{bmatrix}.$$ Thus the observer dynamics is given by and . From which we conclude that $$\lim_{t\to \infty} |\xi_1(t) - x_2(t)|=0, \; x_3(t) = \xi_2(t) + \theta,$$ with the constant $\theta$ to be estimated. Noticing the following relationship $\dot{x}_1 = -x_1^3 + e^{x_3}$, we can formulate a linear regression model for the estimation of $\theta$ of the form $$Y = \psi \Theta + \epsilon_t$$ where $\Theta:= \exp(\theta)$, $Y$ and $\psi$ are defined in and $\epsilon_t$ is a (generic) exponentially decaying term that, without loss of generality, we neglect in the sequel.[^8] Finally, the choice of initial condition for $\psi$ ensures that $\psi(t)$ is not square integrable, thus $$\dot{\tilde{\Theta}} = - \gamma \psi^2 \tilde{\Theta}$$ with $\tilde{\Theta}:= \hat{\Theta} - \Theta$ ensures $\lim_{t\to\infty} \hat \Theta(t)=\Theta$ and consequently is guaranteed.
A class of nonlinear systems for \[KKL+PEB\]O {#subsec52}
---------------------------------------------
We identify now a class of systems whose states can be reconstructed with the proposed \[KKL+PEB\]O.
Consider systems of the form $$\begin{aligned}
\label{exa2}
\dot{\mathbf{x}}_1 & = \mathbf{f}_1( \mathbf{x}_1,\mathbf{x}_{2},\mathbf{x}_{3},u) + S({x},u) \nonumber\\
\dot{\mathbf{x}}_2 & = {\mathbf A}_2\mathbf{x}_2 + \mathbf{f}_2 (\mathbf{x}_1, u) \nonumber\\
\dot{\mathbf{x}}_{3} & = {\mathbf A}_3 \mathbf{x}_{3} + \mathbf{f}_{3} (\mathbf{x}_1, \mathbf{x}_{2},u) \\
\dot{\mathbf{x}}_{4} & = \mathbf{f}_4 (\mathbf{x}_1,\mathbf{x}_{2}, \mathbf{x}_{3},u) \nonumber\\
{y} & = \mathbf{x}_1, \nonumber\end{aligned}$$ where ${x}:={\mbox{col}}(\mathbf{x}_{1},\mathbf{x}_{2}, \mathbf{x}_3,\mathbf{x}_{4})$, with $\mathbf{x}_i \in \rea^{n_i},\;i=1,\dots,4$, and $u \in \rea^{m}$, verifying the following assumptions.
There exists $1 \leq k \leq n_1$ such that the corresponding element of the vector $S$ satisfies S\_k([x]{},u)=b\^(\_1,\_[2]{},\_[3]{},u) \_[4]{} for some mapping $b:\rea^{n_1} \times \rea^{n_2} \times \rea^{n_3} \times \rea^m \to \rea^{n_4}$.
The matrices $ {\mathbf A}_2$ and $ {\mathbf A}_3$ are Hurwitz.
The control input guarantees that the trajectories of are bounded and the following persistency of excitation condition is verified \_t\^[t+T]{} b(s) b\^(s)ds I\_[n\_4]{}, for all $t \geq 0$ and some $\delta,T>0$. The system admits a \[KKL+PEB\]O of the form $$\begin{aligned}
\lab{x2}
\dot{\hat {\mathbf{x}}}_2 & = {\mathbf A}_2 \hat {\mathbf{x}}_2 + \mathbf{f}_2 (y, u) \\
\lab{x3}
\dot{\hat {\mathbf{x}}}_{3} & = {\mathbf A}_{3} \hat {\mathbf{x}}_{3} + \mathbf{f}_{3} (y,\hat {\mathbf{x}}_{2},u) \\
\lab{dotxi0}
\dot{\mathbf{\xi}} & = \mathbf{f}_4 (y,\hat{\mathbf{x}}_2,\hat{\mathbf{x}}_{3}, u) \\
\lab{x4}
\hat{\mathbf{x}}_4 & = \mathbf{\xi} + \hat \theta,\end{aligned}$$ with parameter estimator = ( - \^) where $$\begin{aligned}
\hat {Y} & := {\alpha p \over p + \alpha} \big[{y}_k\big] - {\alpha \over p + \alpha}\big[{f}_{1,k}(y,\hat{ \mathbf{x}}_{2},\hat{ \mathbf{x}}_{3},u) \big] \\
& \quad \; - {\alpha \over p + \alpha}\big[b^\top (y,\hat{ \mathbf{x}}_{2},\hat{ \mathbf{x}}_{3},u)\xi \big], \\
\hat \psi & := {\alpha \over p + \alpha}\big[b (y,\hat{ \mathbf{x}}_{2},\hat{ \mathbf{x}}_{3},u) \big]
\end{aligned}$$ with ${f}_{1,k}$ the $k$-th element of the vector $\mathbf{f}_1$.
We first prove boundedness of $ \hat{ \mathbf{x} }_2$ and $ \hat{ \mathbf{x} }_3$. Due to the assumption (iii), we have that $\mathbf{f}_2 (y, u) \in \linf$. Hence, from and the fact that $A_2$ is Hurwitz, we have that $\hat {\mathbf{x}}_2 \in \linf$. Proceeding in the same way with we conclude that $ \hat{ \mathbf{x} }_3 \in \linf$.
Now, we prove that the observation errors $\tilde{x}_i(t) := \hat{x}_i(t) - x_i(t)$ ($i=1,\ldots,3$) converge to zero exponentially fast. It is obvious that $$\dot{\tilde{ \mathbf x}}_2 = {\mathbf A}_2 \tilde{\mathbf x}_2,$$ and $\lim_{t \to \infty} \tilde{ \mathbf x}_2= 0$ (exp.). Similarly, $$\dot{\tilde{ \mathbf x}}_3 = {\mathbf A}_3 \tilde{\mathbf x}_3 + {\mathbf f}_3 (y, \hat{\mathbf x}_2, u) - {\mathbf f}_3 (y, \mathbf{x}_2, u).$$ Consider the function $V(\tilde{ \mathbf x}_3):=\hal \tilde{ \mathbf x}^\top_3P\tilde{ \mathbf x}_3$, with $P>0$ the solution of $P {\mathbf A}_3+ {\mathbf A}_3^\top P = - \ell_1 I <0$. Its time derivative satisfies V & = & -\_1 |\_3|\^2 + 2 \^\_3 P \[ [f]{}\_3 (y, \_2, u) - [f]{}\_3 (y, \_2, u) \]\
& & -\_1 |\_3|\^2 + \_2 |\_3| |\_2|, where we have used the fact that the boundedness of all the arguments of ${\mathbf f}_3(\cdot,\cdot,\cdot)$ ensures $$| {\mathbf f}_3 (y, \hat{\mathbf x}_2, u) - {\mathbf f}_3 (y, \mathbf{x}_2, u)| \leq \ell_3 |\tilde{\mathbf x}_2|$$ for some $\ell_3>0$. From , the comparison lemma and the fact that $ \tilde{ \mathbf{x} }_3 \in \linf$ and $\tilde{ \mathbf x}_2(t) \to 0$ (exp.), we conclude that $\tilde{ \mathbf x}_3(t) \to 0$ (exp.).
It only remains to prove that $\tilde {\mathbf x}_4:= \hat {\mathbf x}_4 - {\mathbf x}_4$ also converges to zero. Consider and define $\tilde {\mathbf \xi}: = \xi - \mathbf{x}_{4}$, which satisfies $$\dot{\tilde{\mathbf \xi}} = \mathbf{f}_{4} (y, \hat {\mathbf{x}}_{2}, \hat {\mathbf{x}}_{3},u) - \mathbf{f}_{4} (y, {\mathbf{x}}_{2}, {\mathbf{x}}_{3},u).$$ Hence, (t) & = & \_0\^t \[ \_[4]{} (y(s), \_[2]{}(s), \_[3]{}(s),u(s))\
&& - \_[4]{} (y(s), \_[2]{}(s), \_[3]{}(s),u(s))\]ds + (0)\
& & \_4 \_0\^t | | ds + (0), for some $\ell_4>0$, where we have used the same argument invoked above to get the second bound. Because of the exponential convergence to zero of its arguments, the integral above converges to a constant as $t \to \infty$, consequently, we can write \_[4]{}(t)= (t) + + , for some constant vector $\theta$—equation corresponds to the key relationship of PEBO with $\phi(x)=\mathbf{x}_4$.
To complete the proof we show now that, under the persistent excitation condition , the proposed estimator is consistent, that is, $\lim_{t\to\infty}\hat \theta(t)=\theta$ that, together with and establishes the claim that $\tilde {\mathbf x}_4(t) \to 0$. Towards this end, notice that replacing in the $k$-th equation of $\dot {\mathbf x}_1$ we get \_[1,k]{} = [f]{}\_[1,k]{}( \_1,\_[2]{},\_[3]{},u) + b\^(\_1,\_[2]{},\_[3]{},u) (+ ), where we have use to get the second equation. On the other hand, $\dot {y}_k= \dot {\mathbf x}_{1,k}$, hence applying the filter ${\alpha \over p + \alpha}$ we get the (ideal) regression form $Y=\psi^\top \theta$ with $Y := {\alpha p \over p + \alpha} \big[{y}_k\big] - {\alpha \over p + \alpha}\big[{f}_{1,k}(y,{ \mathbf{x}}_{2},{ \mathbf{x}}_{3},u) \big] - {\alpha \over p + \alpha}\big[b^\top (y,{ \mathbf{x}}_{2},{ \mathbf{x}}_{3},u)\xi \big],\;
\psi := {\alpha \over p + \alpha}\big[b (y,{ \mathbf{x}}_{2},{ \mathbf{x}}_{3},u) \big],
$ that is, of course, unmeasurable because of the dependence of ${f}_{1,k}$ and $b$ on the unknown states. However, due to the fact that the estimation errors $\tilde{ \mathbf x}_2(t)$ and $\tilde{ \mathbf x}_3(t)$ converge exponentially fast to zero, we have that $\hat Y(t) = Y(t) + \et$ and $\hat \psi(t)=\psi + \et$. Therefore, neglecting the terms $\et$, we get $ \hat{Y} =\hat {\psi}^\top {\theta}$. Replacing the equation above in we get the parameter estimation error equation $$\dot{ \tilde{\theta}} = \Gamma \hat \psi \hat {\psi}^\top\tilde{\theta},$$ where $ \tilde{\theta}:= \hat{\theta}- {\theta}$. The proof of (exponential) convergence of $\tilde \theta(t)$ to zero is completed invoking standard adaptive control arguments.
For the sake of clarity we have presented Proposition \[pro7\] in a very simple form, being possible to extend it in several directions.
Clearly, the number of subsystems of the form $\dot{\mathbf{x}}_{i} = {\mathbf A}_{i} {\mathbf{x}}_{i} + \mathbf{f}_{i} (y, {\mathbf{x}}_{1}, \cdots, {\mathbf{x}}_{n-1},u)$ can be larger than the two taken here.
Invoking the recent results of identification and adaptive control of nonlinearly parameterised systems—see [@LIUetal] and references therein—it is possible to replace Assumption (i) by:\
(i’) There exists $1 \leq k \leq n_1$ such that the corresponding element of the vector $S$ satisfies $$S_k({x},u)=b^\top (\mathbf{x}_1,\mathbf{x}_{2},\mathbf{x}_{3},u)\Phi( \mathbf{x}_{4}),$$ for some [*monotonic*]{} mapping $\Phi:\rea^{n_4} \to \rea^{n_4}$.
Regarding Assumption (i) it is also possible to consider the existence, not just of one element of $S$, but several of them verifying the factorizability condition. This will give rise to a matrix regressor $b$ for which the persistent excitation condition would be easier to satisfy.
For simplicity the unknown parameter $\theta$ is identified in Proposition \[pro7\] with the classical gradient estimator . However, it is possible to replace this estimator with the high-performance dynamic regressor extension and mixing proposed in [@ARAetaltac], see also [@ortega2017sub]. As shown in these papers parameter convergence is ensured without the, often restrictive, persistent excitation condition .
DC-DC Ćuk converter
-------------------
In this section, we consider the widely studied DC-DC Ćuk converter, depicted in Fig. \[fig2\], for which a PEBO and an I&IO were reported in [@ortega2015scl] and [@astolfi2008book], respectively. We also design a KKLO, a \[KKL+PEB\]O and two high gain observers (HGOs) [*à la*]{} [@esfandiari1992ijc]. The purpose of this example is to compare, via simulations, the performance of all these observers from the point of view of gain tuning flexibility and robustness with respect to measurement noise, which is unavoidable in this application.
The averaged model of the system is given as
\_1 & = - (1-u) y\_1 +\
\_2 & = y\_2 - x\_2\
\_1 & = (1-u) x\_1 + u y\_2\
\_2 & = - u y\_1 - x\_2,
where $x:= {\mbox{col}}(i_1,v_4)$, $y: = {\mbox{col}}(v_2,i_3)$, and $L_1,C_2,L_3,C_4,E,G$ are positive constants. $u\in(0,1)$ is a duty cycle. We are interested in estimating $x$ with $y$ measurable. Following the observer designs proposed in this note and the ones reported in the literature, we obtain the observers given in Table \[tab1\], in which $F(p)={\alpha \over p+\alpha}$ and $W(p)={\alpha p \over p+\alpha}$. Notice that for the KKLO, $\Lambda$ is a time-varying stable matrix, since $1 -u \notin \mathcal{L}_2$.
+-----------------------+-----------------------+-----------------------+
| Type | Observer structure | Mappings |
+:======================+:======================+:======================+
| KKLO | $~~~\eqref{kklo}, \;\ | $$\begin{aligned} |
| | hat{x}={\mbox{col}}(L | \Lambda & = \diag (-C |
| | _1^{-1}\xi_2 + L_1^{ | _4^{-1}{G} , - L_1^{- |
| | -1} C_2 y_1, \xi_1)$ | 1}{(1-u)})\\ |
| | | B & = {\ |
| | | mbox{col}}( C_4^{-1} |
| | | y_2 , ( 1+ L_1^{-1} |
| | | C_2) (-1 + u) y_1 + E |
| | | - uy_2 ) |
| | | \end{aligned}$$ |
+-----------------------+-----------------------+-----------------------+
| PEBO [@ortega2015scl] | $$\begin{aligned} | $$\begin{aligned} |
| | \text{ ~\eqref{dotxi} | B & = {\mbox{col}}(L |
| | ,~~} | _1^{-1}(E-(1-u)y_1), |
| | \dot{\hat{\theta}} | C_4^{-1}(y_2+ G uy_1) |
| | & = \Gamma \mathbf | )\\ |
| | {M}^\top(\mathbf{Y} - | \mathbf{M} & = \diag |
| | \mathbf{M} \hat{\the | (C_2^{-1}F [1-u], - L |
| | ta}) \\ | _3^{-1}) |
| | \hat{x} | \\ |
| | & = \hat{\th | \mathbf{Y} & = |
| | eta} + \xi + {\mbox{ | \begin{bmatrix} |
| | col}}(0,C_4^{-1}{GL_3 | W [ y_1] - C_2^{-1 |
| | } y_2) | } F[ \xi_1(1-u) + uy_ |
| | \end{aligned}$$ | 2] \\ |
| | | W [y_2] + F[ L_3 |
| | | ^{-1}(uy_1 + \xi_2 )+ |
| | | C_4^{-1}(GL_3 y_2)] |
| | | \end{bmatrix} |
| | | ,\; |
| | | \alpha ,\Gamma \in \ |
| | | mathbb{R}_+\\ |
| | | \end{aligned}$$ |
+-----------------------+-----------------------+-----------------------+
| \[KKL+PEB\]O | $$\begin{aligned} | $$\begin{aligned} |
| | \eqref{pebo-eq}, | \Lambda & = \diag(0, |
| | ~~ \dot{\hat{\thet | -C_4^{-1}G),\; |
| | a}} & = \gamma M (Y | B = {\m |
| | - M \hat{\theta})\\ | box{col}}(L_1^{-1}({E |
| | \hat{ | -(1-u)y_1}),C_4^{-1} |
| | x} | y_2),\; P=I \\ |
| | & = {\mbox{col}}(\xi_ | Y & = W[y |
| | 1 + \hat{\theta}, \x | _1] - C_2^{-1}F[ {(1- |
| | i_2) | u)\xi_2 + uy_2 }] ,\; |
| | \end{aligned}$$ | M = C_2^ |
| | | {-1}F[1-u ], |
| | | \; |
| | | \alpha , \gamma\in\ma |
| | | thbb{R}_+ |
| | | \end{aligned}$$ |
+-----------------------+-----------------------+-----------------------+
| I&IO | | |
| [@astolfi2008book] | | |
+-----------------------+-----------------------+-----------------------+
| HGO\ | | |
| (time-varying | | |
| dynamics) | | |
+-----------------------+-----------------------+-----------------------+
| HGO\ | | |
| (linear dynamics) | | |
+-----------------------+-----------------------+-----------------------+
Simulations were conduct with measurement noises, which are generated by Matlab/Simulink’s uniform random number block with sampling time of 0.0001s, and the magnitude limitations are \[-0.02,0.02\] for $y_1$ and \[$-2\times 10^{-4},2\times 10^{-4}$\] for $y_2$. The parameters of the converter are $L_1=10$ mH, $C_2$=22.0 $\mu$F, $C_4$=22.9 $\mu$F, G=0.0447 S and $E$=12 v. In order to give a fair comparison study, the system runs with the *ideal* state-feedback with the stabilizing control law given in [@astolfi2008book] $$u = \frac{|V_d|}{|V_d| + E} + \lambda \frac{G|V_d|v_2 + E(x_2 -x_1)}{1 + (G|V_d|v_2 + E(x_2-x_1))^2},$$ where $V_d$ is the set point for the output voltage $v_4$, which was is selected as in [@ortega2015scl]. The observer parameters were taken as $\alpha=0.5,\;\gamma = 0.001, \;\Gamma=\diag(0.001,100), \;\gamma_1=50,\;\gamma_2=1,\; r_1=0.05, \;r_2=0.005, \; \alpha_1=\alpha_3=2$,$\;\alpha_2=\alpha_4=1$, to make the observers have approximate convergence speeds. All the initial values of the dynamic extensions in observers are selected as $0$. The simulation results are given in Fig. \[Doc3\].
The following remarks are in order.
- KKLO and I&IO have two-order dynamics, clearly, the lowest order ones. KLLO has the simplest observer structure. The parameters in PEBO were the easiest to tune with guaranteed convergence speed; KKLO and \[KKL+PEB\]O need to resolve PDEs to tune. Besides, for HGO the achievable convergence speed is severely limited.
- The I&I framework allows to treat in a unified manner the problems of *state and parameter* estimation, see [@astolfi2008book] for the state observation with unknown parameters.
- The KKLO has the best performance in the presence of measurement noise, probably due to the fact that its dynamic extension is a linear system that attenuates the effect of the noise. On the other hand, the dynamics in PEBO, \[KKL+PEB\]O and I&IO are nonlinear, and seem to have a deleterious impact on the noise.
- The first HGO yields a *time-varying* error dynamics, which is stable because of the physical constraint $1-u>0$. It has oscillations in the transient stage. The second HGO has LTI dynamics, where high gain injections are used to estimate the output derivatives, *i.e.*, $\dot{y}_1$ and $\dot{y}_2$. The operating modes of the converter switch at the moments $t=0.2k$ s ($k=1,\ldots,5$), yielding relatively *large* derivatives of the outputs around these moments.
- As expected, the worst performance was systematically observed for the HGOs because of the high-gain injection needed to ensure its stability. It is worth pointing out that this (well-known) deleterious effect of high-gain injection was also observed for mechanical systems in [@ORTetalijc].
{width="7.3cm" height="7cm"} {width="7.3cm" height="7cm"} {width="7.3cm" height="7cm"} {width="7.3cm" height="7cm"}
Concluding Remarks {#sec6}
==================
A new observer design technique, called \[KKL+PEB\]O, which consists of the combination of KKLO and PEBO was introduced—providing more degrees of freedom for the solution of the key PDE. Via the suitable selection of the tuning matrix $\Lambda$ of the form , in the PDE , \[KKL+PEB\]O reduces to PEBO or KKLO. An example that is not solvable with KKLO nor PEBO, but it is via \[KKL+PEB\]O show that the new observer design extends the applicability of PEBO and KKLO. Also we identified a class of nonlinear systems, for which \[KKL+PEB\]O provides a simple constructive solution.
An additional contribution is the proof that, a slight generalisation of the I&IO, allows us to obtain \[KKL+PEB\]O, as well as PEBO and KKLO, as particular cases of I&IO. This provides a unified framework, based on immersion and invariance, to treat the three observer designs and establish the “set" relationship .
Further research is underway in the following directions.
- Exploit the constructive approach to find the free mappings in \[KKL+PEB\]O for some more specific classes of physical systems.
- Generalize the coordinate change from $\phi(x)$ to $\phi(x,u)$ in order to simplify the solution of the PDEs in these observers. Along this line of research, one interesting possibility is extending the theoretical observer existence results in [@Andrieu2006SIAM] to *control* systems with input.
- Study \[KKL+PEB\]O-based output feedback control. In particular, we are currently investigating if, due to the presence of the PEBO part, the resulting controller enjoys a “self-tuning" property similar to model reference adaptive control. That is, if the control objective can be achieved without requiring that the parameter estimation error $\tilde{\theta}$ converges to zero. Such a property would obviate the need of excitation conditions for \[KKL+PEB\]O-based (or PEBO-based) output feedback control.
acknowledgement {#acknowledgement .unnumbered}
===============
The authors would like to thank Stanislav Aranovskiy, Hassan K. Khalil and Laurent Praly for some suggestions and comments in Section \[sec5\], also thank Pauline Bernard and Vincent Andrieu for clarifications on detectability in KKL observer design. They are also grateful to the Editor, Associate Editor and anonymous reviewers for their highly thorough revisions.
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[^1]: This paper is supported by the National Natural Science Foundation of China (61473183, U1509211), China Scholarship Council, the Government of Russian Fedration (074U01, GOSZADANIE 2014/190 (project 2118)), the Ministry of Education and Science of Russian Federation (14.Z50.31.0031). *Corresponding author: W. Zhang*.
[^2]: B. Yi and W. Zhang are with Department of Automation, Shanghai Jiao Tong University, Shanghai 200240, China. [email protected], [email protected]
[^3]: R. Ortega is with LSS, CNRS-CentraleSupélec, Plateau du Moulon, Gif-sur-Yvette 91192, France. [email protected]
[^4]: All mappings in the paper are assumed smooth.
[^5]: To avoid cluttering the notation, whenever clear from context, we use the same symbols to denote mappings playing similar roles in the various observers. The subindex $(\cdot)_P$ or $(\cdot)_L$ is later used to identify the PEBO or KKLO-related mappings in the \[KKL+PEB\]O.
[^6]: With a slight modification it is also possible to consider the case of $A$ with purely imaginary eigenvalues.
[^7]: Here we also assume $N$ is independent of $u$ for the sake of clarity.
[^8]: See Remark 3 in [@ARAetaltac] where the effect of these term is rigorously analyzed.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In this paper, we propose a novel sufficient decrease technique for variance reduced stochastic gradient descent methods such as SAG, SVRG and SAGA. In order to make sufficient decrease for stochastic optimization, we design a new sufficient decrease criterion, which yields sufficient decrease versions of variance reduction algorithms such as SVRG-SD and SAGA-SD as a byproduct. We introduce a coefficient to scale current iterate and satisfy the sufficient decrease property, which takes the decisions to shrink, expand or move in the opposite direction, and then give two specific update rules of the coefficient for Lasso and ridge regression. Moreover, we analyze the convergence properties of our algorithms for strongly convex problems, which show that both of our algorithms attain linear convergence rates. We also provide the convergence guarantees of our algorithms for non-strongly convex problems. Our experimental results further verify that our algorithms achieve significantly better performance than their counterparts.'
author:
- |
Fanhua Shang$^{\dag}$, Yuanyuan Liu$^{\dag}$, James Cheng$^{\dag}$, Kelvin K.W. Ng$^{\dag}$, Yuichi Yoshida$^{\ddag}$\
$^{\dag}\;\!$Department of Computer Science and Engineering, The Chinese University of Hong Kong\
$^{\ddag}\;\!$National Institute of Informatics and Preferred Infrastructure, Inc., Tokyo, Japan
title: Guaranteed Sufficient Decrease for Variance Reduced Stochastic Gradient Descent
---
Introduction
============
Stochastic gradient descent (SGD) has been successfully applied to many large scale machine learning problems [@krizhevsky:deep; @zhang:sgd], by virtue of its low per-iteration cost. However, standard SGD estimates the gradient from only one or a few samples, and thus the variance of the stochastic gradient estimator may be large [@johnson:svrg; @zhao:prox-smd], which leads to slow convergence and poor performance. In particular, even under the strongly convex (SC) condition, the convergence rate of standard SGD is only sub-linear. Recently, the convergence rate of SGD has been improved by various variance reduction methods, such as SAG [@roux:sag], SDCA [@shalev-Shwartz:sdca], SVRG [@johnson:svrg], SAGA [@defazio:saga], Finito [@defazio:Finito], MISO [@mairal:miso], and their proximal variants, such as [@schmidt:sag], [@shalev-Shwartz:prox-sdca] and [@xiao:prox-svrg]. Under the SC condition, these variance reduced SGD (VR-SGD) algorithms achieve linear convergence rates.
Very recently, many techniques were proposed to further speed up the VR-SGD methods mentioned above. These techniques include importance sampling [@zhao:prox-smd], exploiting neighborhood structure in the training data to share and re-use information about past stochastic gradients [@hofmann:vrsg], incorporating Nesterov’s acceleration techniques [@lin:vrsg; @nitanda:svrg] or momentum acceleration tricks [@zhu:Katyusha], reducing the number of gradient computations in the early iterations [@zhu:univr; @babanezhad:vrsg; @zhang:svrg], and the projection-free property of the conditional gradient method [@hazan:svrf]. [@zhu:vrnc] and [@reddi:saga] proved that SVRG and SAGA with minor modifications can asymptotically converge to a stationary point for non-convex problems.
So far the two most popular stochastic gradient estimators are the SVRG estimator independently introduced by [@johnson:svrg; @zhang:svrg] and the SAGA estimator [@defazio:saga]. All these estimators may be very different from their full gradient counterparts, thus moving in the direction may not decrease the objective function anymore, as stated in [@zhu:Katyusha]. To address this problem, inspired by the success of sufficient decrease methods for deterministic optimization such as [@li:apg; @wolfe:sdg], we propose a novel sufficient decrease technique for a class of VR-SGD methods, including the widely-used SVRG and SAGA methods. Notably, our method with partial sufficient decrease achieves average time complexity per-iteration as low as the original SVRG and SAGA methods. We summarize our main contributions below.
- For making sufficient decrease for stochastic optimization, we design a sufficient decrease strategy to further reduce the cost function, in which we also introduce a coefficient to take the decisions to shrink, expand or move in the opposite direction.
- We incorporate our sufficient decrease technique, together with momentum acceleration, into two representative SVRG and SAGA algorithms, which lead to SVRG-SD and SAGA-SD. Moreover, we give two specific update rules of the coefficient for Lasso and ridge regression problems as notable examples.
- Moreover, we analyze the convergence properties of SVRG-SD and SAGA-SD, which show that SVRG-SD and SAGA-SD converge linearly for SC objective functions. Unlike most of the VR-SGD methods, we also provide the convergence guarantees of SVRG-SD and SAGA-SD for non-strongly convex (NSC) problems.
- Finally, we show by experiments that SVRG-SD and SAGA-SD achieve significantly better performance than SVRG [@johnson:svrg] and SAGA [@defazio:saga]. Compared with the best known stochastic method, Katyusha [@zhu:Katyusha], our methods also have much better performance in most cases.
Preliminary and Related Work
============================
In this paper, we consider the following composite convex optimization problem: $$\label{equ1}
\min_{x\in\mathbb{R}^{d}} F(x)\stackrel{\rm{def}}{=}f(x)+r(x)=\frac{1}{n}\!\sum\nolimits_{i=1}^{n}\!f_{i}(x)+r(x),$$ where $f_{i}(x)\!:\!\mathbb{R}^{d}\!\rightarrow\!\mathbb{R},\,i\!=\!1,\ldots,n$ are the smooth convex component functions, and $r(x)$ is a relatively simple convex (but possibly non-differentiable) function. Recently, many VR-SGD methods [@johnson:svrg; @roux:sag; @xiao:prox-svrg; @zhang:svrg] have been proposed for special cases of . Under smoothness and SC assumptions, and $r(x)\!\equiv\!0$, SAG [@roux:sag] achieves a linear convergence rate. A recent line of work, such as [@johnson:svrg; @xiao:prox-svrg], has been proposed with similar convergence rates to SAG but without the memory requirements for all gradients. SVRG [@johnson:svrg] begins with an initial estimate $\widetilde{x}$, sets $x_{0}\!=\!\widetilde{x}$ and then generates a sequence of $x_{k}$ ($k=1,2,\ldots,m$, where $m$ is usually set to $2n$) using $$\begin{aligned}
\label{equ2}
x_{k}=x_{k-\!1}\!-\eta\left[\nabla\! f_{i_{k}}\!(x_{k-\!1})-\nabla\! f_{i_{k}}\!(\widetilde{x})+\widetilde{\mu}\right],\end{aligned}$$ where $\eta\!>\!0$ is the step size, $\widetilde{\mu}\!:=\frac{1}{n}\!\sum^{n}_{i=1}\!\nabla\! f_{i}(\widetilde{x})$ is the full gradient at $\widetilde{x}$, and $i_{k}$ is chosen uniformly at random from $\{1,2,\ldots,n\}$. After every $m$ stochastic iterations, we set $\widetilde{x}\!=\!x_{m}$, and reset $k\!=\!1$ and $x_{0}\!=\!\widetilde{x}$. Unfortunately, most of the VR-SGD methods [@defazio:Finito; @shalev-Shwartz:sdca; @xiao:prox-svrg], including SVRG, only have convergence guarantee for smooth and SC problems. However, $F(\cdot)$ may be NSC in many machine learning applications, such as Lasso. [@defazio:saga] proposed SAGA, a fast incremental gradient method in the spirit of SAG and SVRG, which works for both SC and NSC objective functions, as well as in proximal settings. Its main update rule is formulated as follows: $$\begin{aligned}
\label{equ3}
x_{k}=\textrm{prox}^{r}_{\eta}(x_{k-\!1}-\eta\;\![g^{k}_{i_{k}}-g^{k-1}_{i_{k}}+\frac{1}{n}\!\sum^{n}_{j=1}g^{k-1}_{j}]),\end{aligned}$$ where $g^{k}_{j}$ is updated for all $j\!=\!1,\ldots,n$ as follows: $g^{k}_{j}\!=\!\nabla\!f_{i_{k}}\!(x^{k-\!1})$ if $i_{k}\!=\!j$, and $g^{k}_{j}\!=\!g^{k-\!1}_{j}$ otherwise, and the proximal operator is defined as: $\textrm{prox}^{r}_{\eta}(y)=\arg\min_{x}({1}/{2\eta})\!\cdot\!\|x\!-\!y\|^{2}+r(x)$.
The technique of sufficient decrease (e.g., the well-known line search technique [@more:ls]) has been studied for deterministic optimization [@li:apg; @wolfe:sdg]. For example, [@li:apg] proposed the following sufficient decrease condition for deterministic optimization: $$\label{equ4}
F(x_{k})\leq F(x_{k-1})-\delta\|y_{k}-x_{k-1}\|^{2},$$ where $\delta\!>\!0$ is a small constant, and $y_{k}\!=\!\textrm{prox}^{r}_{\eta_{k}}\!(x_{k-\!1}\!-\!\eta_{k}\nabla\! f(x_{k-\!1}))$. Similar to the strategy for deterministic optimization, in this paper we design a novel sufficient decrease technique for stochastic optimization, which is used to further reduce the cost function and speed up its convergence.
Variance Reduced SGD with Sufficient Decrease
=============================================
In this section, we propose a novel sufficient decrease technique for VR-SGD methods, which include the widely-used SVRG and SAGA methods. To make sufficient decrease for stochastic optimization, we design a sufficient decrease strategy to further reduce the cost function. Then a coefficient $\theta$ is introduced to satisfy the sufficient decrease condition, and takes the decisions to shrink, expand or move in the opposite direction. Moreover, we present two sufficient decrease VR-SGD algorithms with momentum acceleration: SVRG-SD and SAGA-SD. We also give two specific schemes to compute $\theta$ for Lasso and ridge regression.
Our Sufficient Decrease Technique {#sec31}
---------------------------------
Suppose $x^{s}_{k}\!=\!\textrm{prox}^{r}_{\eta}(x^{s}_{k-\!1}\!-\!\eta[\nabla\! f_{i^{s}_{k}}\!(x^{s}_{k-\!1})\!-\!\nabla\! f_{i^{s}_{k}}\!(\widetilde{x}^{s-\!1})\!+\!\widetilde{\mu}^{s-\!1}])$ for the $s$-th outer-iteration and the $k$-th inner-iteration. Unlike the full gradient method, the stochastic gradient estimator is somewhat inaccurate (i.e., it may be very different from $\nabla\! f(x^{s}_{k-\!1})$), then further moving in the updating direction may not decrease the objective value anymore [@zhu:Katyusha]. That is, $F(x^{s}_{k})$ may be larger than $F(x^{s}_{k-\!1})$ even for very small step length $\eta\!>\!0$. Motivated by this observation, we design a factor $\theta$ to scale the current iterate $x^{s}_{k-\!1}$ for the decrease of the objective function. For SVRG-SD, the cost function with respect to $\theta$ is formulated as follows: $$\label{equ5}
\min_{\theta\in\mathbb{R}} F(\theta x^{s}_{k-\!1})\!+\!\frac{\zeta(1\!-\!\theta)^2}{2}\!\|\nabla\!f_{i^{s}_{k}}\!(x^{s}_{k-\!1})-\!\nabla\! f_{i^{s}_{k}}\!(\widetilde{x}^{s-\!1})\|^{2},$$ where $\zeta\!=\!\frac{\delta\eta}{1-L\eta}$ is a trade-off parameter between the two terms, $\delta$ is a small constant and set to 0.1. The second term in involves the norm of the residual of stochastic gradients, and plays the same role as the second term of the right-hand side of . Different from existing sufficient decrease techniques including , a varying factor $\theta$ instead of a constant is introduced to scale $x^{s}_{k-\!1}$ and the coefficient of the second term of , and $\theta$ plays a similar role as the step-size parameter optimized via a line-search for deterministic optimization. However, line search techniques have a high computational cost in general, which limits their applicability to stochastic optimization [@mahsereci:sgd].
For SAGA-SD, the cost function with respect to $\theta$ can be revised by simply replacing $\nabla\! f_{i^{s}_{k}}\!(\widetilde{x}^{s-\!1})$ with $\nabla\! f_{i^{s}_{k}}\!(\phi^{k-\!1}_{i^{s}_{k}})$ defined below. Note that $\theta$ is a scalar and takes the decisions to shrink, expand $x^{s}_{k-\!1}$ or move in the opposite direction of $x^{s}_{k-\!1}$. The detailed schemes to calculate $\theta$ for Lasso and ridge regression are given in Section \[subsec33\]. We first present the following sufficient decrease condition in the statistical sense for stochastic optimization.
\[prop11\] For given $x^{s}_{k-\!1}$ and the solution $\theta_{k}$ of , then the following inequality holds $$\label{equ6}
F(\theta_{k}x^{s}_{k-\!1})\leq F(x^{s}_{k-\!1})-\frac{\zeta(1\!-\!\theta_{k})^2}{2}\|\widetilde{p}_{i^{s}_{k}}\|^{2},$$ where $\widetilde{p}_{i^{s}_{k}}\!=\!\nabla\!f_{i^{s}_{k}}\!(x^{s}_{k-\!1})\!-\!\nabla\! f_{i^{s}_{k}}\!(\widetilde{x}^{s-\!1})$ for SVRG-SD.
It is not hard to verify that $F(\cdot)$ can be further decreased via our sufficient decrease technique, when the current iterate $x^{s}_{k-\!1}$ is scaled by the coefficient $\theta_{k}$. Indeed, for the special case when $\theta_{k}\!=\!1$ for some $k$, the inequality in (\[equ6\]) can be still satisfied. Moreover, Property \[prop11\] can be extended for SAGA-SD by setting $\widetilde{p}_{i^{s}_{k}}\!=\!\nabla\!f_{i^{s}_{k}}\!(x^{s}_{k-\!1})\!-\!\nabla\! f_{i^{s}_{k}}\!(\phi^{k-\!1}_{i^{s}_{k}})$, as well as for other VR-SGD algorithms such as SAG and SDCA. Unlike the sufficient decrease condition for deterministic optimization [@li:apg; @wolfe:sdg], $\theta_{k}$ may be a negative number, which means to move in the opposite direction of $x^{s}_{k-\!1}$.
Momentum Acceleration {#subsec32}
---------------------
In this part, we first design the update rule for the key variable $x^{s}_{k}$ with the coefficient $\theta_{k}$ as follows: $$\label{equ7}
x^{s}_{k}=y^{s}_{k}+(1\!-\!\sigma)(\widehat{x}^{s}_{k}-\widehat{x}^{s}_{k-1}),$$ where $\widehat{x}^{s}_{k}\!=\!\theta_{k}x^{s}_{k-\!1}$, $\sigma\!\in\![0,1]$ is a constant and can be set to $\sigma\!=\!1/2$ which also works well in practice. In fact, the second term of the right-hand side of plays a momentum acceleration role as in batch and stochastic optimization [@zhu:Katyusha; @nesterov:co; @nitanda:svrg]. That is, by introducing this term, we can utilize the previous information of gradients to update $x^{s}_{k}$. In addition, the update rule of $y^{s}_{k}$ is given by $$\label{equ8}
y^{s}_{k}=\textrm{prox}^{r}_{\eta}\!\left(x^{s}_{k-1}-\eta\widetilde{\nabla} f_{i^{s}_{k}}(x^{s}_{k-1})\right),$$ where $\eta\!=\!1/(L\alpha)$, $L\!>\!0$ is a Lipschitz constant (see Assumption \[assum1\] below), $\alpha\!\geq\! 1$ denotes a constant, and $\widetilde{\nabla}\! f_{i^{s}_{k}}\!(x^{s}_{k-\!1})$ can be the two most popular choices for stochastic gradient estimators: the SVRG estimator [@johnson:svrg; @zhang:svrg] for SVRG-SD and the SAGA estimator [@defazio:saga] for SAGA-SD defined as follows: $$\begin{aligned}
\label{equ9}
\widetilde{\nabla}\! f_{i^{s}_{k}}\!(x^{s}_{k-\!1})\!=\!\nabla\! f_{i^{s}_{k}}\!(x^{s}_{k-\!1})\!-\!\nabla\! f_{i^{s}_{k}}\!(\widetilde{x}^{s-\!1})\!+\!\widetilde{\mu}^{s-\!1}\;\textrm{and}\;
\widetilde{\nabla}\! f_{i^{s}_{k}}\!(x^{s}_{k-\!1})\!=\!\nabla\! f_{i^{s}_{k}}\!(x^{s}_{k-\!1})\!-\!\nabla\! f_{i^{s}_{k}}\!(\phi^{k-\!1}_{i^{s}_{k}})\!+\!\overline{\mu}^{s-\!1}\!,\end{aligned}$$ respectively, where $\widetilde{\mu}^{s-\!1}\!:=\!\nabla\! f(\widetilde{x}^{s-\!1})$. For SAGA-SD, we need to set $\phi^{k}_{i^{s}_{k}}\!=\!x_{k-\!1}$, and store $\nabla\! f_{i^{s}_{k}}\!(\phi^{k}_{i^{s}_{k}})$ in the table similar to [@defazio:saga]. All the other entries in the table remain unchanged, and $\overline{\mu}^{s-\!1}\!:=\frac{1}{n}\!\sum^{n}_{j=1}\!\!\nabla\! f_{j}(\phi^{k-\!1}_{j})$ is the table average. From , it is clear that our algorithms can tackle non-smooth problems directly as in [@defazio:saga].
the number of epochs $S$, the number of iterations $m$ per epoch, and step size $\eta$.\
$\widetilde{x}^{0}$ for Case of SC or $\widetilde{x}^{0}\!=\widetilde{y}^{0}$ for Case of NSC. $\overline{x}\!=\!\widetilde{x}^{S}$ (SC) or $\overline{x}\!=\!\widetilde{x}^{S}$ if $F(\widetilde{x}^{S})\leq F(\frac{1}{S}\!\sum^{S}_{s=1}\!\widetilde{x}^{s})$ and $\overline{x}\!=\!\frac{1}{S}\!\sum^{S}_{s=1}\!\widetilde{x}^{s}$ otherwise (NSC)
In summary, we propose a novel variant of SVRG with sufficient decrease (SVRG-SD) to solve both SC and NSC problems, as outlined in **Algorithm** \[alg1\]. For the case of SC, $x^{s}_{0}\!=\!\widehat{x}^{s}_{0}\!=\!\widetilde{x}^{s-\!1}$, while $x^{s}_{0}\!=\!\widehat{x}^{s}_{0}\!=\!\widetilde{y}^{s-\!1}$ and $\widetilde{y}^{s}\!=\![x^{s}_{m}\!-\!(1\!-\!\sigma)\widehat{x}^{s}_{m}]/\sigma$ for the case of NSC. Similarly, we also present a novel variant of SAGA with sufficient decrease (SAGA-SD), as shown in the Supplementary Material. The main differences between them are the stochastic gradient estimators in , and the update rule of the sufficient decrease coefficient in .
Note that when $\theta_{k}\!\equiv\!1$ and $\sigma\!=\!1$, the proposed SVRG-SD and SAGA-SD degenerate to the original SVRG or its proximal variant (Prox-SVRG [@xiao:prox-svrg]) and SAGA [@defazio:saga], respectively. In this sense, SVRG, Prox-SVRG and SAGA can be seen as the special cases of the proposed algorithms. Like SVRG and SVRG-SD, SAGA-SD is also a multi-stage algorithm, whereas SAGA is a single-stage algorithm.
Coefficients for Lasso and Ridge Regression {#subsec33}
-------------------------------------------
In this part, we give the closed-form solutions of the coefficient $\theta$ for Lasso and ridge regression problems. For Lasso problems and given $x^{s}_{k-\!1}$, we have $F(\theta x^{s}_{k-\!1})\!=\!\frac{1}{2n}\!\sum^{n}_{i=1}\!(\theta a^{T}_{i}\!x^{s}_{k-\!1}\!-\!b_{i})^{2}\!+\!\lambda\|\theta x^{s}_{k-\!1}\|_{1}$. The closed-form solution of for SVRG-SD can be obtained as follows: $$\label{equ10}
\theta_{k}=\mathcal{S}_{\tau}\!\left(\frac{\frac{1}{n}b^{T}\!Ax^{s}_{k-\!1}+\zeta\|\widetilde{p}_{i^{s}_{k}}\|^{2}}{\|Ax^{s}_{k-\!1}\|^{2}/n+\zeta\|\widetilde{p}_{i^{s}_{k}}\|^{2}}\right),$$ where $A\!=\![a_{1},\ldots,a_{n}]^{T}\!$ is the data matrix containing $n$ data samples, $b\!=\![b_{1},\ldots,b_{n}]^{T}\!$, and $\mathcal{S}_{\tau}$ is the so-called soft thresholding operator [@donoho:st] with the following threshold, $$\tau\!=\!\frac{\lambda\|x^{s}_{k-\!1}\|_{1}}{\|Ax^{s}_{k-\!1}\|^{2}/n+\zeta\|\widetilde{p}_{i^{s}_{k}}\|^{2}}.$$
For ridge regression problems, and $F(\theta x^{s}_{k-\!1})\!=\!\frac{1}{2n}\!\sum^{n}_{i=1}\!(\theta a^{T}_{i}\!x^{s}_{k-\!1}\!-\!b_{i})^{2}\!+\!\frac{\lambda}{2}\|\theta x^{s}_{k-\!1}\|^{2}$, the closed-form solution of for SVRG-SD is given by $$\label{equ11}
\theta_{k}=\frac{\frac{1}{n}b^{T}\!Ax^{s}_{k-\!1}+\zeta\|\widetilde{p}_{i^{s}_{k}}\|^{2}}{\|Ax^{s}_{k-\!1}\|^{2}/n+\zeta\|\widetilde{p}_{i^{s}_{k}}\|^{2}+\lambda\|x^{s}_{k-\!1}\|^{2}}.$$
In the same ways as in and , we can compute the coefficient $\theta_{k}$ of SAGA-SD for Lasso and ridge regression problems, and revise the update rules in and by simply replacing $\widetilde{p}_{i^{s}_{k}}\!=\!\nabla\!f_{i^{s}_{k}}\!(x^{s}_{k-\!1})\!-\!\nabla\! f_{i^{s}_{k}}\!(\widetilde{x}^{s-\!1})$ with $\widetilde{p}_{i^{s}_{k}}\!=\!\nabla\!f_{i^{s}_{k}}\!(x^{s}_{k-\!1})\!-\!\nabla\! f_{i^{s}_{k}}\!(\phi^{k-\!1}_{i^{s}_{k}})$. We can also derive the update rule of the coefficient for other loss functions using their approximations, e.g., [@bach:sgd] for logistic regression.
Efficient Implementation {#subsec35}
------------------------
Both and require the calculation of $b^{T}\!A$, thus we need to precompute and save $b^{T}\!A$ in the initial stage. To further reduce the computational complexity of $\|Ax^{s}_{k-\!1}\|^{2}$ in and , we use the fast partial singular value decomposition to obtain the best rank-$r$ approximation $U_{r}S_{r}V^{T}_{r}$ to $A$ and save $S_{r}V^{T}_{r}$. Then $\|Ax^{s}_{k-\!1}\|\!\approx\!\|S_{r}V^{T}_{r}x^{s}_{k-\!1}\|$. In practice, e.g., in our experiments, $r$ can be set to a small number to capture 99.5% of the spectral energy of the data matrix $A$, e.g., $r\!=\!10$ for the Covtype data set, similar to inexact line search methods [@more:ls] for deterministic optimization.
The time complexity of each inter-iteration in the proposed SVRG-SD and SAGA-SD with full sufficient decrease is $O(rd)$, which is a little higher than SVRG and SAGA. In fact, we can just randomly select only a small fraction (e.g., $1/10^3$) of stochastic gradient iterations in each epoch to update with sufficient decrease, while the remainder of iterations without sufficient decrease, i.e., $\widehat{x}^{s}_{k}\!=\!x^{s}_{k-\!1}$. Let $m_{1}$ be the number of iterations with our sufficient decrease technique in each epoch. By fixing $m_{1}\!=\!\lfloor m/10^3\rfloor$ and thus without increasing parameters tuning difficulties, SVRG-SD and SAGA-SD[^1] can always converge much faster than their counterparts: SVRG and SAGA, as shown in Figure \[fig1\]. It is easy to see that our algorithms are very robust with respect to the choice of $m_{1}$, and achieve average time complexity per-iteration as low as the original SVRG and SAGA. Thus, we mainly consider SVRG-SD and SAGA-SD with partial sufficient decrease.
![Comparison of SVRG-SD and SAGA-SD with different values of $m_{1}$, and their counterparts for ridge regression on the Covtype data set.[]{data-label="fig1"}](Fig11 "fig:"){width="0.245\columnwidth"} ![Comparison of SVRG-SD and SAGA-SD with different values of $m_{1}$, and their counterparts for ridge regression on the Covtype data set.[]{data-label="fig1"}](Fig12 "fig:"){width="0.245\columnwidth"} ![Comparison of SVRG-SD and SAGA-SD with different values of $m_{1}$, and their counterparts for ridge regression on the Covtype data set.[]{data-label="fig1"}](Fig13 "fig:"){width="0.245\columnwidth"} ![Comparison of SVRG-SD and SAGA-SD with different values of $m_{1}$, and their counterparts for ridge regression on the Covtype data set.[]{data-label="fig1"}](Fig14 "fig:"){width="0.245\columnwidth"}
Convergence Guarantees
======================
In this section, we provide the convergence analysis of SVRG-SD and SAGA-SD for both SC and NSC cases. In this paper, we consider the problem under the following standard assumptions.
\[assum1\] Each convex function $f_{i}(\cdot)$ is $L$-smooth, iff there exists a constant $L\!>\!0$ such that for any $x,y\!\in\! \mathbb{R}^{d}$, $\|\nabla f_{i}(x)-\nabla f_{i}(y)\|\leq L\|x-y\|$.
\[assum2\] $F(\cdot)$ is $\mu$-strongly convex, iff there exists a constant $\mu\!>\!0$ such that for any $x,y\!\in\!\mathbb{R}^{d}$, $$\label{equ15}
F(y)\geq F(x)\!+\!\vartheta^{T}(y\!-\!x)\!+\!\frac{\mu}{2}\|y\!-\!x\|^{2},\;\;\forall\vartheta\in\partial F(x),$$ where $\partial F(x)$ is the subdifferential of $F(\cdot)$ at $x$. If $F(\cdot)$ is smooth, we can revise the inequality by simply replacing the sub-gradient $\vartheta\in\partial F(x)$ with $\nabla\! F(x)$.
Convergence Analysis of SVRG-SD
-------------------------------
In this part, we analyze the convergence property of SVRG-SD for both SC and NSC cases. The first main result is the following theorem, which provides the convergence rate of SVRG-SD.
\[theo1\] Suppose Assumption \[assum1\] holds. Let $x^{*}$ be the optimal solution of Problem , and $\{(x^{s}_{k},y^{s}_{k},\theta^{s}_{k})\}$ be the sequence produced by SVRG-SD, $\eta\!=\!1/(L\alpha)$, and $\frac{2}{\alpha-1}\!<\!\sigma$, then $$\begin{split}
\mathbb{E}\!\left[F(\widetilde{x}^{s})\!-\!F(x^{*})\right]\!\leq&\left(\!\frac{\frac{1-\sigma}{m}\!+\!\frac{2}{\alpha-1}}{\sigma\!-\!\frac{2}{\alpha-1}\!+\!\widehat{\beta}}\right)\!\mathbb{E}\!\left[F(\widetilde{x}^{s-\!1})\!-\!F(x^{*})\right]\\
&\quad+\frac{L\alpha\sigma^{2}}{2m\!\left(\sigma\!-\!\frac{2}{\alpha-1}\!+\!\widehat{\beta}\right)}\mathbb{E}\!\left[\|x^{*}\!-\!z^{s}_{0}\|^{2}\!-\!\|x^{*}\!-\!z^{s}_{m}\|^{2}\right]\!,
\end{split}$$ where $z^{s}_{0}\!=\!\left[x^{s}_{0}\!-\!(1\!-\!\sigma)\widehat{x}^{s}_{0}\right]/\sigma$, $z^{s}_{m}\!=\!\left[x^{s}_{m}\!-\!(1\!-\!\sigma)\widehat{x}^{s}_{m-\!1}\right]/\sigma$, $\widehat{\beta}\!=\!\min_{s=1,\ldots,S}\widehat{\beta}^{s}\!\geq\!0$, and $\widehat{\beta}^{s}\!=\!\mathbb{E}[\sum^{m}_{k=1}\!\!\frac{2c_{k}\beta_{k}}{\alpha-1}(F(\widehat{x}^{s}_{k})\!-\!F(x^{*}))]/\mathbb{E}[\sum^{m}_{k=1}\!(F(\widehat{x}^{s}_{k})\!-\!F(x^{*}))]$.
The proof of Theorem \[theo1\] and the definitions of $c_{k}$ and $\beta_{k}$ are given in the Supplementary Material. The linear convergence of SVRG-SD follows immediately.
\[coro1\] Suppose each $f_{i}(\cdot)$ is $L$-smooth, and $F(\cdot)$ is $\mu$-strongly convex. Setting $\alpha\!=\!19$, $\sigma\!=\!1/2$, and $m$ sufficiently large so that $$\rho=\frac{9}{(7\!+\!18\widehat{\beta})m}+\frac{2}{7\!+\!18\widehat{\beta}}+\frac{171L}{(14\!+\!36\widehat{\beta})m\mu}<1,$$ then SVRG-SD has the geometric convergence in expectation: $$\mathbb{E}\!\left[F(\overline{x})-F(x^{*})\right]\leq\rho^{S}\!\left[F(\widetilde{x}^{0})-F(x^{*})\right]\!.$$
The proof of Corollary \[coro1\] is given in the Supplementary Material. From Corollary \[coro1\], one can see that SVRG-SD has a linear convergence rate for SC problems. As discussed in [@xiao:prox-svrg], $\rho\!\approx\!\frac{L/\mu}{\nu(1-4\nu)m}\!+\!\frac{4\nu}{1-4\nu}$ for the proximal variant of SVRG [@xiao:prox-svrg], where $\nu\!=\!1/\alpha$. For a reasonable comparison, we use the same parameter settings for SVRG and SVRG-SD, e.g., $\alpha\!=\!19$ and $m\!=\!57L/\mu$. Then one can see that $\rho_{\textrm{SVRG}}\!\approx\!31/45$ for SVRG and $\rho_{\textrm{SVRG-SD}}\!\approx\!{7}/{(14\!+\!36\widehat{\beta})}\!<\!{1}/2$ for SVRG-SD, that is, $\rho_{\textrm{SVRG-SD}}$ is smaller than $\rho_{\textrm{SVRG}}$. Thus, SVRG-SD can significantly improve the convergence rate of SVRG in practice, which will be confirmed by the experimental results below.
Unlike most of VR-SGD methods [@johnson:svrg; @xiao:prox-svrg], including SVRG, the convergence result of SVRG-SD for the NSC case is also provided, as shown below.
\[coro2\] Suppose each $f_{i}(\cdot)$ is $L$-smooth. Setting $\alpha\!=\!19$, $\sigma\!=\!1/2$, and $m$ sufficiently large, then $$\mathbb{E}[F(\overline{x})-F(x^{*})]\leq\frac{171L}{(16\!+\!40\widehat{\beta})mS}\|x^{*}\!-\!\widetilde{x}^{0}\|^{2}+\!\left(\frac{9}{(4\!+\!8\widehat{\beta})mS}\!+\!\frac{1}{(2\!+\!4\widehat{\beta})S}\right)\!\left[F(\widetilde{x}^{0})\!-\!F(x^{*})\right]\!.$$
The proof of Corollary \[coro2\] is provided in the Supplementary Material. The constant $\widehat{\beta}\!\geq\!0$ is from the sufficient decrease strategy, which thus implies that the convergence bound in Corollary \[coro2\] can be further improved using our sufficient decrease strategy with an even larger $\widehat{\beta}$.
Convergence Analysis of SAGA-SD
-------------------------------
In this part, we analyze the convergence property of SAGA-SD for both SC and NSC cases. The following lemma provides the upper bound on the expected variance of the gradient estimator in (i.e., the SAGA estimator [@defazio:saga]), and its proof is given in the Supplementary Material.
\[lemm2\] Suppose Assumption \[assum1\] holds. Then the following inequality holds $$\begin{split}
&\mathbb{E}[\|\nabla\!f_{i^{s}_{k}}\!(x^{s}_{k-\!1})\!-\!\!\nabla\!f(x^{s}_{k-\!1})\!-\!\!\nabla\!f_{i^{s}_{k}}\!(\phi^{k-\!1}_{i^{s}_{k}})\!+\!\frac{1}{n}\!\sum^{n}_{j=1}\!\nabla\! f_{j}(\phi^{k-\!1}_{j})\|^{2}]\\
&\leq 4L[F(x^{s}_{k-\!1})-F(x^{*})+\frac{1}{n}\sum^{n}_{j=1}F_{j}(\phi^{k-\!1}_{j})-F(x^{*})].
\end{split}$$
\[theo2\] Suppose $F(\cdot)$ is $\mu$-strongly convex and $f_{i}(\cdot)$ is $L$-smooth. With the same notation as in Theorem \[theo1\], and by setting $\alpha\!=\!19$, $\sigma\!=\!1/2$, and $m$ sufficiently large such that $$\rho=\!\frac{n}{(7\!+\!9\widehat{\beta})m}+\frac{9}{(14\!+\!18\widehat{\beta})m}+\frac{171L}{(28\!+\!36\widehat{\beta})\mu m}<1,$$ then SAGA-SD has the geometric convergence in expectation: $$\mathbb{E}[F(\overline{x})-F(x^{*})]\leq \rho^{S}\!\left[F(\widetilde{x}^{0})-F(x^{*})\right]\!.$$
The proof of Theorem \[theo2\] is provided in the Supplementary Material. Theorem \[theo2\] shows that SAGA-SD also attains linear convergence similar to SVRG-SD. Like Corollary \[coro2\], we also provide the convergence guarantee of SAGA-SD for NSC problems, as shown below.
\[coro3\] Suppose each $f_{i}(\cdot)$ is $L$-smooth. With the same notation as in Theorem \[theo2\] and by setting $\alpha\!=\!19$, $\sigma\!=\!1/2$, and $m\!=\!n$, then $$\mathbb{E}[F(\overline{x})\!-\!F(x^{*})]\leq\!\frac{171L}{(49\!+\!56\widehat{\beta})nS}\|x^{*}\!-\!\widetilde{x}^{0}\|^{2}\!+\!\!\left(\frac{81}{(98\!+\!126\widehat{\beta})nS}\!+\!\frac{9}{(49\!+\!63\widehat{\beta})S}\right)\!\left[F(\widetilde{x}^{0})\!-\!F(x^{*})\right]\!.$$
The proof of Corollary \[coro3\] is provided in the Supplementary Material. Due to $\widehat{\beta}\!\geq\!0$, Theorem \[theo2\] and Corollary \[coro3\] imply that SAGA-SD can significantly improve the convergence rate of SAGA [@defazio:saga] for both SC and NSC cases, which will be confirmed by our experimental results.
As suggested in [@frostig:sgd] and [@lin:vrsg], one can add a proximal term into a non-strongly convex objective function $F(x)$ as follows: $F_{\tau}(x,y)\!=\!f(x)\!+\!\frac{\tau}{2}\|x\!-\!y\|^{2}\!+\!r(x)$, where $\tau\!\geq\!0$ is a constant that can be determined as in [@frostig:sgd; @lin:vrsg], and $y\!\in\! \mathbb{R}^{d}$ is a proximal point. Then the condition number of this proximal function $F_{\tau}(x,y)$ can be much smaller than that of the original function $F(x)$, if $\tau$ is sufficiently large. However, adding the proximal term may degrade the performance of the involved algorithms both in theory and in practice [@zhu:univr]. Therefore, we directly use SVRG-SD and SAGA-SD to solve non-strongly convex objectives.
![Comparison of different VR-SGD methods for solving ridge regression problems ($\lambda\!=\!10^{-4}$) on Ijcnn1 (left), Covtype (center), and SUSY (right). The vertical axis is the objective value minus the minimum, and the horizontal axis denotes the number of effective passes over the data.[]{data-label="fig2"}](Fig21 "fig:"){width="0.326\columnwidth"} ![Comparison of different VR-SGD methods for solving ridge regression problems ($\lambda\!=\!10^{-4}$) on Ijcnn1 (left), Covtype (center), and SUSY (right). The vertical axis is the objective value minus the minimum, and the horizontal axis denotes the number of effective passes over the data.[]{data-label="fig2"}](Fig26 "fig:"){width="0.326\columnwidth"} ![Comparison of different VR-SGD methods for solving ridge regression problems ($\lambda\!=\!10^{-4}$) on Ijcnn1 (left), Covtype (center), and SUSY (right). The vertical axis is the objective value minus the minimum, and the horizontal axis denotes the number of effective passes over the data.[]{data-label="fig2"}](Fig31 "fig:"){width="0.326\columnwidth"}
Experimental Results
====================
In this section, we evaluate the performance of SVRG-SD and SAGA-SD, and compare their performance with their counterparts including SVRG [@johnson:svrg], its proximal variant (Prox-SVRG) [@xiao:prox-svrg], and SAGA [@defazio:saga]. Moreover, we also report the performance of the well-known accelerated VR-SGD methods, Catalyst [@lin:vrsg] and Katyusha [@zhu:Katyusha]. For fair comparison, we implemented all the methods in C++ with a Matlab interface (all codes are made available, see link in the Supplementary Materials), and performed all the experiments on a PC with an Intel i5-2400 CPU and 16GB RAM.
Ridge Regression
----------------
Our experiments were conducted on three popular data sets: Covtype, Ijcnn1 and SUSY, all of which were obtained from the LIBSVM Data website[[^2]]{} (more details and regularization parameters are given in the Supplementary Material). Following [@xiao:prox-svrg], each feature vector of these date sets has been normalized so that $\|a_{i}\|\!=\!1$ for all $i=1,\ldots,n$, which leads to the same upper bound on the Lipschitz constants $L_{i}$. This step is for comparison only and not necessary in practice. We focus on the ridge regression as the SC example. For SVRG-SD and SAGA-SD, we set $\sigma\!=\!1/2$ on the three data sets. In addition, unlike SAGA [@defazio:saga], we fixed $m\!=\!n$ for each epoch of SAGA-SD. For SVRG-SD, Catalyst, Katyusha, SVRG and its proximal variant, we set the epoch size $m\!=\!2n$, as suggested in [@zhu:Katyusha; @johnson:svrg; @xiao:prox-svrg]. Each of these methods had its step size parameter chosen so as to give the fastest convergence.
Figure \[fig2\] shows how the objective gap, i.e., $F(x^{s})\!-\!F(x^{*})$, of all these algorithms decreases for ridge regression problems with the regularization parameter $\lambda\!=\!10^{-4}$ (more results are given in the Supplementary Material). Note that the horizontal axis denotes the number of effective passes over the data. As seen in these figures, SVRG-SD and SAGA-SD achieve consistent speedups for all the data sets, and significantly outperform their counterparts, SVRG and SAGA, in all the settings. This confirms that our sufficient decrease technique is able to accelerate SVRG and SAGA. Impressively, SVRG-SD and SAGA-SD usually converge much faster than the well-known accelerated VR-SGD methods, Catalyst and Katyusha, which further justifies the effectiveness of our sufficient decrease stochastic optimization method.
Lasso and Elastic-Net Regularized Lasso
---------------------------------------
We also conducted experiments of the Lasso and elastic-net regularized (i.e., $\lambda_{1}\|\!\cdot\!\|_{1}\!+\!\lambda_{2}\|\!\cdot\!\|^2$) Lasso problems. We plot some representative results in Figure \[fig3\] (see Figures 3 and 4 in the Supplementary Material for more results), which show that SVRG-SD and SAGA-SD significantly outperform their counterparts (i.e., Prox-SVRG and SAGA) in all the settings, as well as Catalyst, and are considerably better than Katyusha in most cases. This empirically verifies that our sufficient decrease technique can accelerate SVRG and SAGA for solving both SC and NSC objectives.
Conclusion & Future Work
========================
To the best of our knowledge, this is the first work to design an efficient sufficient decrease technique for stochastic optimization. Moreover, we proposed two different schemes for Lasso and ridge regression to efficiently update the coefficient $\theta$, which takes the important decisions to shrink, expand or move in the opposite direction. This is very different from adaptive learning rate methods, e.g., [@kingma:sgd], and line search methods, e.g., [@mahsereci:sgd], all of which cannot address the issue in Section \[sec31\] whatever value the step size is. Unlike most VR-SGD methods [@johnson:svrg; @shalev-Shwartz:sdca; @xiao:prox-svrg], which only have convergence guarantees for SC problems, we provided the convergence guarantees of our algorithms for both SC and NSC cases. Experimental results verified the effectiveness of our sufficient decrease technique for stochastic optimization. Naturally, it can also be used to further speed up accelerated VR-SGD methods such as [@zhu:Katyusha; @zhu:univr; @lin:vrsg].
As each function $f_{i}(\cdot)$ can have different degrees of smoothness, to select the random index $i^{s}_{k}$ from a non-uniform distribution is a much better choice than simple uniform random sampling [@zhao:prox-smd], as well as without-replacement sampling vs. with-replacement sampling [@shamir:sgd]. On the practical side, both our algorithms tackle the NSC and non-smooth problems directly, without using any quadratic regularizer as in [@zhu:Katyusha; @lin:vrsg], as well as proximal settings. Note that some asynchronous parallel and distributed variants [@lee:dsgd; @reddi:sgd] of VR-SGD methods have also been proposed for such stochastic settings. We leave these variations out from our comparison and consider similar extensions to our stochastic sufficient decrease method as future work.
[**Supplementary Materials for “Guaranteed Sufficient Decrease for Variance Reduced Stochastic Gradient Descent”**]{}
In this supplementary material, we give the detailed proofs for some lemmas, theorems and corollaries stated in the main paper. Moreover, we also report more experimental results for both of our algorithms.
Notations {#notations .unnumbered}
=========
Throughout this paper, $\|\!\cdot\!\|$ denotes the standard Euclidean norm, and $\|\!\cdot\!\|_{1}$ is the $\ell_{1}$-norm, i.e., $\|x\|_{1}\!=\!\sum^{d}_{i=1}\!|x_{i}|$. We denote by $\nabla\!f(x)$ the full gradient of $f(x)$ if it is differentiable, or $\partial f(x)$ the subdifferential of $f(\cdot)$ at $x$ if it is only Lipschitz continuous. Note that Assumption 2 is the general form for the two cases when $F(x)$ is smooth or non-smooth[^3]. That is, if $F(x)$ is smooth, the inequality in (12) in Assumption 2 becomes the following form: $$F(y)\geq F(x)+\nabla F(x)(y-x)+\frac{\mu}{2}\|y-x\|^{2}.$$
Appendix A: Proof of Theorem 1 {#appendix-a-proof-of-theorem-1 .unnumbered}
==============================
Although the proposed SVRG-SD is a variant of SVRG, it is non-trivial to analyze its convergence property, as well as that of SAGA-SD. Before proving Theorem 1, we first give the following lemma.
\[lemm1\] Let $x^{*}$ be the optimal solution of Problem (1), then the following inequality holds $$\begin{split}
&\mathbb{E}\!\left[\left\|\nabla\! f_{i^{s}_{k}}\!(x^{s}_{k-\!1})\!-\!\nabla\! f(x^{s}_{k-\!1})\!-\!\nabla\! f_{i^{s}_{k}}\!(\widetilde{x}^{s-\!1})\!+\!\nabla\! f(\widetilde{x}^{s-\!1})\right\|^{2}\right]\\
\leq&\, 4L\!\left[F(x^{s}_{k-\!1})-F(x^{*})+F(\widetilde{x}^{s-\!1})-F(x^{*})\right]\!.
\end{split}$$
Lemma \[lemm1\] provides the upper bound on the expected variance of the variance reduced gradient estimator in (9) (i.e., the SVRG estimator independently introduced in [@johnson:svrg; @zhang:svrg]), which satisfies $\mathbb{E}[\widetilde{\nabla}\! f_{i^{s}_{k}}(x^{s}_{k-\!1})]\!=\!\nabla\!f(x^{s}_{k-\!1})$. This lemma is essentially identical to Corollary 3.5 in [@xiao:prox-svrg]. From Lemma \[lemm1\], we immediately get the following result, which is useful in our convergence analysis.
\[coro4\] For any $\alpha\geq\beta>0$, the following inequality holds $$\begin{split}
&\alpha\mathbb{E}\!\left[\left\|\nabla\! f_{i^{s}_{k}}\!(x^{s}_{k-\!1})-\!\nabla\! f(x^{s}_{k-\!1})\!-\!\nabla\! f_{i^{s}_{k}}\!(\widetilde{x}^{s-\!1})\!+\!\nabla\! f(\widetilde{x}^{s-\!1})\right\|^{2}\right]\!-\!\beta\mathbb{E}\!\left[\left\|\nabla\! f_{i^{s}_{k}}\!(x^{s}_{k-\!1})\!-\!\nabla\! f_{i^{s}_{k}}\!(\widetilde{x}^{s-\!1})\right\|^{2}\right]\\
\leq\, &4L(\alpha\!-\!\beta)\!\left[F(x^{s}_{k-1})-F(x^{*})+F(\widetilde{x}^{s-1})-F(x^{*})\right]\!.
\end{split}$$
$$\begin{split}
&\alpha\mathbb{E}\!\left[\left\|\nabla\! f_{i^{s}_{k}}\!(x^{s}_{k-\!1})\!-\!\nabla\! f(x^{s}_{k-\!1})\!-\!\nabla\! f_{i^{s}_{k}}\!(\widetilde{x}^{s-\!1})\!+\!\nabla\! f(\widetilde{x}^{s-\!1})\right\|^{2}\right]-\beta\mathbb{E}\!\left[\left\|\nabla\! f_{i^{s}_{k}}\!(x^{s}_{k-\!1})\!-\!\nabla\! f_{i^{s}_{k}}\!(\widetilde{x}^{s-\!1})\right\|^{2}\right]\\
=\,&\alpha\mathbb{E}\!\!\left[\left\|[\nabla\! f_{i^{s}_{k}}\!(x^{s}_{k-\!1})\!-\!\nabla\! f_{i^{s}_{k}}\!(\widetilde{x}^{s-\!1})]\!-\![\nabla\! f(x_{k-1})\!-\!\nabla\! f(\widetilde{x}^{s-\!1})]\right\|^{2}\right]-\beta\mathbb{E}\!\!\left[\left\|\nabla\! f_{i^{s}_{k}}\!(x^{s}_{k-\!1})\!-\!\nabla\! f_{i^{s}_{k}}\!(\widetilde{x}^{s-\!1})\right\|^{2}\right]\\
=\,&\alpha\mathbb{E}\!\!\left[\left\|\nabla\! f_{i^{s}_{k}}\!(x^{s}_{k-\!1})\!-\!\nabla\! f_{i^{s}_{k}}\!(\widetilde{x}^{s-\!1})\right\|^{2}\right]\!-\!\alpha\!\left\|\nabla\! f(x^{s}_{k-\!1})\!-\!\nabla\! f(\widetilde{x}^{s-\!1})\right\|^{2}\!-\!\beta\mathbb{E}\!\!\left[\left\|\nabla\! f_{i^{s}_{k}}\!(x^{s}_{k-\!1})\!-\!\nabla\! f_{i^{s}_{k}}\!(\widetilde{x}^{s-\!1})\right\|^{2}\right]\\
\leq\,&\alpha\mathbb{E}\!\left[\left\|\nabla\! f_{i^{s}_{k}}\!(x^{s}_{k-1})-\nabla\! f_{i^{s}_{k}}\!(\widetilde{x}^{s-1})\right\|^{2}\right]\!-\!\beta\mathbb{E}\!\left[\left\|\nabla\! f_{i^{s}_{k}}\!(x^{s}_{k-\!1})\!-\!\nabla\! f_{i^{s}_{k}}\!(\widetilde{x}^{s-\!1})\right\|^{2}\right]\\
=\,&(\alpha\!-\!\beta)\mathbb{E}\!\left[\left\|\left[\nabla\! f_{i^{s}_{k}}\!(x^{s}_{k-1})-\nabla\! f_{i^{s}_{k}}\!(x^{*})\right]-[\nabla\! f_{i^{s}_{k}}\!(\widetilde{x}^{s-1})-\nabla\! f_{i^{s}_{k}}\!(x^{*})]\right\|^{2}\right]\\
\leq\,&2(\alpha\!-\!\beta)\left\{\mathbb{E}\!\left[\left\|\nabla\! f_{i^{s}_{k}}\!(x^{s}_{k-\!1})-\nabla\! f_{i^{s}_{k}}\!(x^{*})\right\|^{2}\right]+\mathbb{E}\!\left[\left\|\nabla\! f_{i^{s}_{k}}\!(\widetilde{x}^{s-\!1})-\nabla\! f_{i^{s}_{k}}\!(x^{*})\right\|^{2}\right]\right\}\\
\leq\, & 4L(\alpha\!-\!\beta)\!\left[F(x^{s}_{k-1})-F(x^{*})+F(\widetilde{x}^{s-1})-F(x^{*})\right]\!,
\end{split}$$
where the second equality holds due to the fact that $\mathbb{E}[\|x\!-\!\mathbb{E}x\|^{2}]\!=\!\mathbb{E}[\|x\|^{2}]\!-\!\|\mathbb{E}x\|^{2}$; the second inequality holds due to the fact that $\|a-b\|^{2}\leq2(\|a\|^{2}+\|b\|^{2})$; and the last inequality follows from Lemma 3.4 in [@xiao:prox-svrg] (i.e., $\mathbb{E}[\left\|\nabla\! f_{i}(x)\!-\!\nabla\! f_{i}(x^{*})\right\|^{2}]\!\leq\! 2L\!\left[F(x)\!-\!F(x^{*})\right]$).
Moreover, we also introduce the following lemmas [@baldassarre:prox; @lan:sgd], which are useful in our convergence analysis.
\[prop1\] Let $\widetilde{F}(x,y)$ be the linear approximation of $F(\cdot)$ at $y$ with respect to $f$, i.e., $$\widetilde{F}(x,y)=f(y)+\left\langle \nabla f(y),\, x-y\right\rangle+ r(x).$$ Then $$F(x)\leq \widetilde{F}(x,y)+\frac{L}{2}\|x-y\|^{2}\leq F(x)+\frac{L}{2}\|x-y\|^{2}.$$
\[prop2\] Assume that $\hat{x}$ is an optimal solution of the following problem, $$\min_{x\in\mathbb{R}^{d}}\frac{\tau}{2}\|x-y\|^{2}+g(x),$$ where $g(x)$ is a convex function (but possibly non-differentiable). Then the following inequality holds for all $x\!\in\!\mathbb{R}^{d}$: $$g(\hat{x})+\frac{\tau}{2}\|\hat{x}-y\|^{2}+\frac{\tau}{2}\|x-\hat{x}\|^{2}\leq g(x)+\frac{\tau}{2}\|x-y\|^{2}.$$
**Proof of Theorem 1:**
Let $\eta=\frac{1}{L\alpha}$ and $p_{i^{s}_{k}}\!=\!\widetilde{\nabla}\! f_{i^{s}_{k}}\!(x^{s}_{k-\!1})\!=\!\nabla\! f_{i^{s}_{k}}\!(x^{s}_{k-1})-\nabla\! f_{i^{s}_{k}}\!(\widetilde{x}^{s-1})+\nabla\! f(\widetilde{x}^{s-1})$. Using Lemma \[prop1\], we have $$\label{equ71}
\begin{split}
F(y^{s}_{k})\leq\,& f(x^{s}_{k-1})+\left\langle\nabla\! f(x^{s}_{k-\!1}),\,y^{s}_{k}\!-\!x^{s}_{k-\!1}\right\rangle+\frac{L\alpha}{2}\!\left\|y^{s}_{k}\!-\!x^{s}_{k-\!1}\right\|^{2}\!-\!\frac{L(\alpha\!-\!1)}{2}\!\left\|y^{s}_{k}\!-\!x^{s}_{k-\!1}\right\|^{2}+r(y^{s}_{k})\\
=\,& f_{i^{s}_{k}}\!(x^{s}_{k-1})+\left\langle p_{i^{s}_{k}},\,y^{s}_{k}-x^{s}_{k-1}\right\rangle+r(y^{s}_{k})+\frac{L\alpha}{2}\!\|y^{s}_{k}-x^{s}_{k-1}\|^2\\
&+\left\langle\nabla\! f(x^{s}_{k-1})-p_{i^{s}_{k}},\,y^{s}_{k}-x^{s}_{k-1}\right\rangle-\frac{L(\alpha\!-\!1)}{2}\|y^{s}_{k}-x^{s}_{k-1}\|^{2}+f(x^{s}_{k-1})-f_{i^{s}_{k}}(x^{s}_{k-1}).
\end{split}$$ Then $$\label{equ72}
\begin{split}
&\left\langle\nabla\! f(x^{s}_{k-1})-p_{i^{s}_{k}},\,y^{s}_{k}-x^{s}_{k-1}\right\rangle-\frac{L(\alpha\!-\!1)}{2}\|y^{s}_{k}-x^{s}_{k-1}\|^{2}\\
\leq\,& \frac{1}{2L(\alpha\!-\!1)}\|\nabla\!f(x^{s}_{k-1})-p_{i^{s}_{k}}\|^{2}+\frac{L(\alpha\!-\!1)}{2}\|y^{s}_{k}\!-\!x^{s}_{k-1}\|^{2}-\frac{L(\alpha\!-\!1)}{2}\|y^{s}_{k}\!-\!x^{s}_{k-1}\|^{2}\\
=\,&\frac{1}{2L(\alpha\!-\!1)}\|\nabla\!f(x^{s}_{k-1})-p_{i^{s}_{k}}\|^{2},
\end{split}$$ where the inequality follows from the Young’s inequality, i.e., $a^{T}b\leq\|a\|^{2}/(2\rho)+\rho\|b\|^{2}/2$ for any $\rho\!>\!0$. Substituting the inequality into the inequality , we have $$\label{equ73}
\begin{split}
F(y^{s}_{k})&\leq f_{i^{s}_{k}}\!(x^{s}_{k-1})+\left\langle p_{i^{s}_{k}},\,y^{s}_{k}-x^{s}_{k-1}\right\rangle+r(y^{s}_{k})+\frac{L\alpha}{2}\|y^{s}_{k}-x^{s}_{k-1}\|^2\\
&\quad+\frac{1}{2L(\alpha\!-\!1)}\|\nabla\!f(x^{s}_{k-1})-p_{i^{s}_{k}}\|^{2}+f(x^{s}_{k-1})-f_{i^{s}_{k}}(x^{s}_{k-1})\\
&\leq f_{i^{s}_{k}}\!(x^{s}_{k-\!1})+r(\widehat{w}^{s}_{k-\!1})+\frac{L\alpha}{2}\!\left(\|\widehat{w}^{s}_{k-\!1}\!-\!x^{s}_{k-1}\|^{2}\!-\!\|\widehat{w}^{s}_{k-\!1}\!-\!y^{s}_{k}\|^{2}\right)+\langle p_{i^{s}_{k}},\,\widehat{w}^{s}_{k-\!1}\!-\!x^{s}_{k-\!1}\rangle\\
&\quad+\frac{1}{2L(\alpha\!-\!1)}\|\nabla\!f(x^{s}_{k-1})-p_{i^{s}_{k}}\|^{2}+f(x^{s}_{k-\!1})-f_{i^{s}_{k}}(x^{s}_{k-\!1})\\
&\leq F_{i^{s}_{k}}\!(\widehat{w}^{s}_{k-1})+\frac{L\alpha}{2}\left(\|\widehat{w}^{s}_{k-1}-x^{s}_{k-1}\|^{2}-\|\widehat{w}^{s}_{k-1}-y^{s}_{k}\|^{2}\right)+f(x^{s}_{k-1})-f_{i^{s}_{k}}(x^{s}_{k-1})\\
&\quad+\frac{1}{2L(\alpha\!-\!1)}\|\nabla\!f(x^{s}_{k-1})-p_{i^{s}_{k}}\|^{2}+\left\langle-\nabla f_{i^{s}_{k}}(\widetilde{x}^{s-1})+\nabla f(\widetilde{x}^{s-1}),\,\widehat{w}^{s}_{k-1}-x^{s}_{k-1}\right\rangle\\
&\leq \sigma F_{i^{s}_{k}}\!(x^{*})+(1-\sigma)F_{i^{s}_{k}}\!(\widehat{x}^{s}_{k-1})+\frac{L\alpha\sigma^{2}}{2}\left(\|x^{*}-z^{s}_{k-1}\|^{2}-\|x^{*}-z^{s}_{k}\|^{2}\right)\\
&\quad+\frac{1}{2L(\alpha\!-\!1)}\|\nabla\!f(x^{s}_{k-1})-p_{i^{s}_{k}}\|^{2}+f(x^{s}_{k-1})-f_{i^{s}_{k}}(x^{s}_{k-1})\\
&\quad+\!\left\langle\nabla f(\widetilde{x}^{s-1})\!-\!\nabla f_{i^{s}_{k}}(\widetilde{x}^{s-1}),\,\widehat{w}^{s}_{k-1}\!-\!x^{s}_{k-1}\right\rangle,
\end{split}$$ where $\widehat{w}^{s}_{k-1}\!=\!\sigma x^{*}+(1\!-\!\sigma)\widehat{x}^{s}_{k-1}$, and $\widehat{x}_{k-\!1}\!=\!\theta_{k-\!1}x_{k-2}$. The second inequality follows from Lemma \[prop2\] with $g(x)\!:=\!\left\langle p_{i^{s}_{k}},\,x\!-\!x^{s}_{k-1}\right\rangle\!+\!r(x)$, $\tau\!=\!L\alpha$, $\hat{x}\!=\!y^{s}_{k}$, $x\!=\!\widehat{w}^{s}_{k-1}$ and $y\!=\!x^{s}_{k-1}$; the third inequality holds due to the convexity of the component function $f_{i^{s}_{k}}(x)$ (i.e., $f_{i^{s}_{k}}\!(x^{s}_{k-\!1})\!+\!\langle\nabla\! f_{i^{s}_{k}}\!(x^{s}_{k-\!1}),\widehat{w}^{s}_{k-\!1}\!-\!x^{s}_{k-\!1}\rangle\!\leq\! f_{i^{s}_{k}}\!(\widehat{w}^{s}_{k-\!1})$); and the last inequality holds due to the convexity of the function $F_{i^{s}_{k}}\!(x)\!:=\!f_{i^{s}_{k}}\!(x)\!+\!r(x)$, and $$z^{s}_{k-1}=[x^{s}_{k-1}-(1\!-\!\sigma)\widehat{x}^{s}_{k-1}]/{\sigma},\;\,z^{s}_{k}=[y^{s}_{k}-(1\!-\!\sigma)\widehat{x}^{s}_{k-1}]/{\sigma},$$ which mean that $\widehat{w}^{s}_{k-\!1}\!-x^{s}_{k-\!1}=\sigma(x^{*}-z^{s}_{k-\!1})$ and $\widehat{w}^{s}_{k-\!1}\!-y^{s}_{k}=\sigma(x^{*}-z^{s}_{k})$.
Using Property 1 with $\zeta=\frac{\delta\eta}{1-L\eta}$ and $\eta=1/L\alpha,$[^4] we obtain $$\label{equ74}
\begin{split}
F(\theta_{k}{x}^{s}_{k-1})=F(\widehat{x}^{s}_{k})&\leq F(x^{s}_{k-1})-\frac{(\theta_{k}\!-\!1)^{2}}{2L(\alpha-1)}\|\nabla\!f_{i^{s}_{k}}\!(x^{s}_{k-\!1})-\!\nabla\! f_{i^{s}_{k}}\!(\widetilde{x}^{s-1})\|^{2}\\
&\leq F(x^{s}_{k-1})-\frac{\beta_{k}}{2L(\alpha\!-\!1)}\|\nabla\!f_{i^{s}_{k}}\!(x^{s}_{k-\!1})-\!\nabla\! f_{i^{s}_{k}}\!(\widetilde{x}^{s-1})\|^{2},
\end{split}$$ where $\beta_{k}=\min\!\left[1/\alpha_{k},\,(\theta_{k}\!-\!1)^{2}\right]$, and $\alpha_{k}$ is defined below. Then there exists $\overline{\beta}_{k}$ such that $$\label{equ83}
\mathbb{E}\!\left[\frac{\beta_{k}}{2L(\alpha\!-\!1)}\|\nabla\!f_{i^{s}_{k}}\!(x^{s}_{k-\!1})-\!\nabla\! f_{i^{s}_{k}}\!(\widetilde{x}^{s-1})\|^{2}\right]=\frac{\overline{\beta}_{k}}{2L(\alpha\!-\!1)}\mathbb{E}\!\left[\|\nabla\!f_{i^{s}_{k}}\!(x^{s}_{k-\!1})-\!\nabla\! f_{i^{s}_{k}}\!(\widetilde{x}^{s-1})\|^{2}\right],$$ where $\overline{\beta}_{k}=\mathbb{E}[\beta_{k}\|\nabla\!f_{i^{s}_{k}}\!(x^{s}_{k-\!1})-\!\nabla\! f_{i^{s}_{k}}\!(\widetilde{x}^{s-1})\|^{2}]/\mathbb{E}[\|\nabla\!f_{i^{s}_{k}}\!(x^{s}_{k-\!1})-\!\nabla\! f_{i^{s}_{k}}\!(\widetilde{x}^{s-1})\|^{2}]$, and $\overline{\beta}_{k}<(\alpha\!-\!1)/{2}$. Using the inequality , then we have $$\label{equ101}
\begin{split}
\mathbb{E}\!\left[F(\widehat{x}^{s}_{k})-F(x^{*})\right]&\leq \mathbb{E}\!\left[F(x^{s}_{k-1})-F(x^{*})-\frac{\beta_{k}}{2L(\alpha\!-\!1)}\|\nabla\!f_{i^{s}_{k}}\!(x^{s}_{k-\!1})-\!\nabla\! f_{i_{k}}\!(\widetilde{x}^{s-1})\|^{2}\right]\\
&= \mathbb{E}\!\left[F(x^{s}_{k-1})-F(x^{*})\right]-\frac{\overline{\beta}_{k}}{2L(\alpha\!-\!1)}\mathbb{E}\!\left[\|\nabla\!f_{i^{s}_{k}}\!(x^{s}_{k-\!1})-\!\nabla\! f_{i^{s}_{k}}\!(\widetilde{x}^{s-1})\|^{2}\right].
\end{split}$$
There must exist a constant $\alpha_{k}\!>\!0$ such that $F(y^{s}_{k})\!-\!F(x^{*})\!=\!\alpha_{k}[F(x^{s}_{k-\!1})\!-\!F(x^{*})]$. Since $\mathbb{E}\!\left[f(x^{s}_{k-\!1})\!-\!f_{i^{s}_{k}}\!(x^{s}_{k-\!1})\right]\!=\!0$, $\mathbb{E}\!\left[\nabla\! f_{i^{s}_{k}}(\widetilde{x}^{s-1})\right]\!=\!\nabla\! f(\widetilde{x}^{s-1})$, $\mathbb{E}\!\left[F_{i^{s}_{k}}\!(x^{*})\right]\!=\!F(x^{*})$, and $\mathbb{E}\!\left[F_{i^{s}_{k}}\!(x^{s}_{k-1})\right]\!=\!F(x^{s}_{k-1})$, and taking the expectation of both sides of (\[equ73\]), we have $$\label{equ75}
\begin{split}
&\alpha_{k}\mathbb{E}\!\left[F(x^{s}_{k-1})-F(x^{*})\right]-\frac{c_{k}\overline{\beta}_{k}}{2L(\alpha\!-\!1)}\mathbb{E}\!\left[\|\nabla\!f_{i^{s}_{k}}\!(x^{s}_{k-\!1})-\!\nabla\! f_{i^{s}_{k}}\!(\widetilde{x}^{s-1})\|^{2}\right]\\
\leq\,&(1-\sigma)\mathbb{E}\!\left[F(\widehat{x}^{s}_{k-1})-F(x^{*})\right]+\frac{L\alpha\sigma^{2}}{2}\mathbb{E}\!\left[\|x^{*}-z^{s}_{k-1}\|^{2}-\|x^{*}-z^{s}_{k}\|^{2}\right]\\
&+\frac{1}{2L(\alpha\!-\!1)}\mathbb{E}\|\nabla\!f(x^{s}_{k-1})-p_{i^{s}_{k}}\|^{2}-\frac{c_{k}\overline{\beta}_{k}}{2L(\alpha\!-\!1)}\mathbb{E}\!\left[\|\nabla\!f_{i^{s}_{k}}\!(x^{s}_{k-\!1})-\!\nabla\! f_{i^{s}_{k}}\!(\widetilde{x}^{s-1})\|^{2}\right]\\
\leq&(1-\sigma)\mathbb{E}\!\left[F(\widehat{x}^{s}_{k-1})-F(x^{*})\right]+\frac{L\alpha\sigma^{2}}{2}\mathbb{E}\!\left[\|x^{*}-z^{s}_{k-1}\|^{2}-\|x^{*}-z^{s}_{k}\|^{2}\right]\\
&+\frac{2(1-c_{k}\overline{\beta}_{k})}{\alpha\!-\!1}\left[F(x^{s}_{k-1})-F(x^{*})+F(\widetilde{x}^{s-1})-F(x^{*})\right],
\end{split}$$ where the second inequality follows from Lemma \[lemm1\] and Corollary \[coro4\]. Here, $c_{k}=\alpha_{k}-[{2(1\!-\!c_{k}\overline{\beta}_{k})}]/({\alpha\!-\!1})$, i.e., $$c_{k}= \frac{\alpha_{k}(\alpha-1)-2}{\alpha-1-2\overline{\beta}_{k}}.$$ Since $\frac{2}{\alpha-1}<\sigma$ with the suitable choices of $\alpha$ and $\sigma$, we have $c_{k}>\alpha_{k}-\frac{2}{\alpha-1}>1-\sigma$. Thus, (\[equ75\]) is rewritten as follows: $$\label{equ76}
\begin{split}
&c_{k}\mathbb{E}\!\left[F(x^{s}_{k-1})-F(x^{*})\right]-\frac{c_{k}\overline{\beta}_{k}}{2L(\alpha-1)}\mathbb{E}\!\left[\|p_{i^{s}_{k}}-\!\nabla\! f_{i^{s}_{k}}\!(\widetilde{x}^{s-1})\|^{2}\right]\\
\leq&\,(1-\sigma)\mathbb{E}[F(\widehat{x}^{s}_{k-1})-F(x^{*})]+\frac{L\alpha\sigma^{2}}{2}\mathbb{E}\!\left[\|x^{*}-z^{s}_{k-1}\|^{2}-\|x^{*}-z^{s}_{k}\|^{2}\right]\\
&\,+\frac{2(1-c_{k}\overline{\beta}_{k})}{\alpha-1}\mathbb{E}\!\left[F(\widetilde{x}^{s-1})-F(x^{*})\right].
\end{split}$$
Combining the above two inequalities (\[equ101\]) and (\[equ76\]), we have $$\begin{split}
&c_{k}\mathbb{E}\!\left[F(\widehat{x}^{s}_{k})-F(x^{*})\right]\\
\leq\,&(1-\sigma)\mathbb{E}\!\left[F(\widehat{x}^{s}_{k-1})-F(x^{*})\right]+\frac{L\alpha\sigma^{2}}{2}\mathbb{E}\!\left[\|x^{*}-z^{s}_{k-1}\|^{2}-\|x^{*}-z^{s}_{k}\|^{2}\right]\\
&+\frac{2(1-c_{k}\overline{\beta}_{k})}{\alpha-1}\mathbb{E}\!\left[F(\widetilde{x}^{s-1})-F(x^{*})\right].
\end{split}$$
Taking the expectation over the random choice of $i^{s}_{1},i^{s}_{2},\ldots,i^{s}_{m}$, summing up the above inequality over $k=1,\ldots,m$, and $\widehat{x}^{s}_{0}=\widetilde{x}^{s-1}$, we have $$\label{equ102}
\begin{split}
&\mathbb{E}\!\left[\sum^{m}_{k=1}\!\left[c_{k}-(1-\sigma)\right][F(\widehat{x}^{s}_{k})-F(x^{*})]\right]\\
\leq\,&(1-\sigma)\mathbb{E}\!\left[F(\widetilde{x}^{s-1})-F(x^{*})\right]+\frac{L\alpha\sigma^{2}}{2}\mathbb{E}\!\left[\|x^{*}-z^{s}_{0}\|^{2}-\|x^{*}-z^{s}_{m}\|^{2}\right]\\
&+\mathbb{E}\!\left[\sum^{m}_{k=1}\frac{2(1-c_{k}\overline{\beta}_{k})}{\alpha-1}[F(\widetilde{x}^{s-1})-F(x^{*})]\right].
\end{split}$$ In addition, there exists $\widehat{\beta}^{s}$ for the $s$-th epoch such that $$\label{equ103}
\begin{split}
&\;\mathbb{E}\!\left[\sum^{m}_{k=1}\left[c_{k}-(1-\sigma)\right][F(\widehat{x}^{s}_{k})-F(x^{*})]\right]\\
=&\;\mathbb{E}\!\left[\sum^{m}_{k=1}\left(\sigma-\frac{2}{\alpha-1}+\frac{2c_{k}\overline{\beta}_{k}}{\alpha-1}\right)[F(\widehat{x}^{s}_{k})-F(x^{*})]\right]\\
=&\;\left(\sigma-\frac{2}{\alpha-1}+\widehat{\beta}^{s}\right)\mathbb{E}\!\left[\sum^{m}_{k=1}[F(\widehat{x}^{s}_{k})-F(x^{*})]\right],
\end{split}$$ where $$\widehat{\beta}^{s}=\frac{\mathbb{E}\!\left[\sum^{m}_{k=1}\frac{2c_{k}\beta_{k}}{\alpha-1}[F(\widehat{x}^{s}_{k})-F(x^{*})]\right]}{\mathbb{E}\!\left[\sum^{m}_{k=1}[F(\widehat{x}^{s}_{k})-F(x^{*})]\right]}.$$ Let $\widehat{\beta}=\min_{s=1,\ldots,S}\widehat{\beta}^{s}$. Using $$\widetilde{x}^{s}=\frac{1}{m}\sum^{m}_{k=1}\widehat{x}^{s}_{k},\;\,F(\widetilde{x}^{s})\leq\frac{1}{m}\sum^{m}_{k=1}F(\widehat{x}^{s}_{k}),$$ (\[equ102\]) and (\[equ103\]), we have $$\begin{split}
&\left(\sigma-\frac{2}{\alpha-1}+\widehat{\beta}\right)m\mathbb{E}\!\left[F(\widetilde{x}^{s})-F(x^{*})\right]\\
\leq\,&\left(1-\sigma+\frac{2m}{\alpha\!-\!1}\right)\mathbb{E}\!\left[F(\widetilde{x}^{s-1})-F(x^{*})\right]\\
&+\frac{L\alpha\sigma^{2}}{2}\mathbb{E}\!\left[\|x^{*}-z^{s}_{0}\|^{2}-\|x^{*}-z^{s}_{m}\|^{2}\right].
\end{split}$$ Therefore, $$\begin{split}
&\mathbb{E}\!\left[F(\widetilde{x}^{s})-F(x^{*})\right]\\
\leq\,&\left(\frac{1-\sigma}{\left(\sigma-\frac{2}{\alpha-1}+\widehat{\beta}\right)m}+\frac{2}{(\alpha\!-\!1)\left(\sigma-\frac{2}{\alpha-1}+\widehat{\beta}\right)}\right)\mathbb{E}\!\left[F(\widetilde{x}^{s-1})-F(x^{*})\right]\\
&+\frac{L\alpha\sigma^{2}}{2m\left(\sigma-\frac{2}{\alpha-1}+\widehat{\beta}\right)}\mathbb{E}\!\left[\|x^{*}-z^{s}_{0}\|^{2}-\|x^{*}-z^{s}_{m}\|^{2}\right].
\end{split}$$ This completes the proof.
Appendix B: Proofs of Corollaries 1 and 2 {#appendix-b-proofs-of-corollaries-1-and-2 .unnumbered}
=========================================
**Proof of Corollary 1:**
For $\mu$-strongly convex problems, and let $x^{s}_{0}=\widehat{x}^{s}_{0}=\widetilde{x}^{s-1}$ and $$z^{s}_{0}=\frac{x^{s}_{0}-(1-\sigma)\widehat{x}^{s}_{0}}{\sigma}=\widetilde{x}^{s-1}.$$ Due to the strong convexity of $F(\cdot)$, we have $$\frac{\mu}{2}\|x^{*}-z^{s}_{0}\|^{2}=\frac{\mu}{2}\|x^{*}-\widetilde{x}^{s-1}\|^{2}\leq F(\widetilde{x}^{s-1})-F(x^{*}).$$ Using Theorem 1, we obtain $$\begin{split}
&\mathbb{E}\!\left[F(\widetilde{x}^{s})-F(x^{*})\right]\\
\leq&\left(\frac{1-\sigma}{m(\sigma\!-\!\frac{2}{\alpha-1}\!+\!\widehat{\beta})}+\frac{2}{(\alpha\!-\!1)\left(\sigma\!-\!\frac{2}{\alpha-1}\!+\!\widehat{\beta}\right)}+\frac{L\alpha\sigma^{2}}{m\mu\left(\sigma\!-\!\frac{2}{\alpha-1}\!+\!\widehat{\beta}\right)}\right)\mathbb{E}\!\left[F(\widetilde{x}^{s-1})-F(x^{*})\right].
\end{split}$$
Replacing $\alpha$ and $\sigma$ in the above inequality with $19$ and $1/2$, respectively, we have $$\begin{split}
&\mathbb{E}\!\left[F(\widetilde{x}^{s})-F(x^{*})\right]\\
\leq&\left(\frac{9}{(7+18\widehat{\beta})m}+\frac{2}{7+18\widehat{\beta}}+\frac{171L}{(14+36\widehat{\beta})m\mu}\right)\mathbb{E}\!\left[F(\widetilde{x}^{s-1})-F(x^{*})\right]\!.
\end{split}$$ This completes the proof.
**Proof of Corollary 2:**
For non-strongly convex problems, and using Theorem 1 with $\alpha=19$ and $\sigma=1/2$, we have $$\label{equ105}
\begin{split}
\mathbb{E}[F(\widetilde{x}^{s})-F(x^{*})]\leq&\;\frac{171L}{(28+72\widehat{\beta})m}\mathbb{E}\!\left[\left\|x^{*}-z^{s}_{0}\right\|^{2}-\left\|x^{*}-z^{s}_{m}\right\|^{2}\right]\\
&\;+\left(\frac{9}{(7+18\widehat{\beta})m}+\frac{2}{7+18\widehat{\beta}}\right)\left[F(\widetilde{x}^{s-1})-F(x^{*})\right]\!.
\end{split}$$
According to the settings of Algorithm 1 for the non-strongly convex case, and let $$x^{s}_{0}=\widehat{x}^{s}_{0}=[x^{s-1}_{m}-(1-\sigma)\widehat{x}^{s-1}_{m}]/\sigma,$$ then we have $$z^{s}_{0}=\frac{x^{s}_{0}-(1-\sigma)\widehat{x}^{s}_{0}}{\sigma}=\frac{x^{s-1}_{m}-(1-\sigma)\widehat{x}^{s-1}_{m}}{\sigma},$$ and $$z^{s-1}_{m}=\frac{x^{s-1}_{m}-(1-\sigma)\widehat{x}^{s-1}_{m}}{\sigma}.$$ Therefore, $z^{s}_{0}=z^{s-1}_{m}$.
Using $z^{0}_{0}=\widetilde{x}^{0}$, and summing up the inequality (\[equ105\]) over all $s=1,\ldots,S$, then $$\begin{split}
\mathbb{E}\!\left[F\!\left(\frac{1}{S}\sum^{S}_{s=1}\widetilde{x}^{s}\right)-F(x^{*})\right]\leq&\;\frac{171L}{(16+40\widehat{\beta})mS}\left\|x^{*}-\widetilde{x}^{0}\right\|^{2}\\
&\;+\left(\frac{9}{(4+8\widehat{\beta})mS}+\frac{1}{(2+4\widehat{\beta})S}\right)\left[F(\widetilde{x}^{0})-F(x^{*})\right]\!.
\end{split}$$
Due to the settings of Algorithm 1 for the non-strongly convex case, we have $$\label{equ106}
\begin{split}
\mathbb{E}\!\left[F(\overline{x})-F(x^{*})\right]\leq&\;\frac{171L}{(16+40\widehat{\beta})mS}\left\|x^{*}-\widetilde{x}^{0}\right\|^{2}\\
&\;+\left(\frac{9}{(4+8\widehat{\beta})mS}+\frac{1}{(2+4\widehat{\beta})S}\right)\left[F(\widetilde{x}^{0})-F(x^{*})\right]\!.
\end{split}$$
This completes the proof.
Appendix C: Proof of Lemma 1 {#appendix-c-proof-of-lemma-1 .unnumbered}
============================
Lemma 1 provides the upper bound on the expected variance of the variance reduced gradient estimator in (9) (i.e., the SAGA estimator introduced in [@defazio:saga]). Before giving the proof of Lemma 1, we first present the following lemmas.
\[lemm12\] Let $x^{*}$ be the optimal solution of Problem (1), then the following inequality holds for all $\phi_{j}$: $$\frac{1}{n}\sum^{n}_{j=1}\left\|\nabla\!f_{j}(\phi_{j})-\nabla\!f_{j}(x^{*})\right\|^{2}\leq 2L\! \left[\frac{1}{n}\sum^{n}_{j=1}f_{j}(\phi_{j})-f(x^{*})-\frac{1}{n}\sum^{n}_{j=1}\left\langle \nabla\!f_{j}(x^{*}),\,\phi_{j}-x^{*}\right\rangle\right]\!.$$
\[lemm13\] $$\begin{split}
\mathbb{E}\!\!\left[\frac{1}{n}\!\sum^{n}_{j=1}\!\left\langle \partial F_{j}(x^{*}),\:\phi^{k}_{j}\!-\!x^{*}\right\rangle\right]\!=\!\frac{1}{n}\!\left\langle \partial F(x^{*}),\,x^{s}_{k-\!1}\!-\!x^{*}\right\rangle+(1\!-\!\frac{1}{n})\frac{1}{n}\!\sum^{n}_{j=1}\!\left\langle \partial F_{j}(x^{*}),\,\phi^{k-\!1}_{j}\!-\!x^{*}\right\rangle\!,
\end{split}$$ where $F_{i}(\cdot)\!=\!f_{i}(\cdot)+r(\cdot)$, and $\partial F_{i}(x^{*})$ denotes a sub-gradient of $F_{i}(\cdot)$ at $x^{*}$.
**Proof of Lemma 1:**
Using Lemma \[lemm13\], we have $$\label{equ107}
\begin{split}
&\,\mathbb{E}\!\left[\frac{1}{n}\sum^n_{j=1}\left\langle \partial F_{j}(x^{*}),\phi^{k-1}_{j}-x^{*}\right\rangle\right]\\
=\,&\mathbb{E}\!\left[\frac{1}{n}\left\langle \partial F(x^{*}),\:x^{s}_{k-2}-x^{*}\right\rangle+(1-\frac{1}{n})\frac{1}{n}\sum^n_{j=1}\left\langle \partial F_{j}(x^{*}),\:\phi^{k-2}_{j}-x^{*}\right\rangle\right]\\
=\,&(1-\frac{1}{n})\mathbb{E}\!\left[\frac{1}{n}\sum^n_{j=1}\left\langle \partial F_{j}(x^{*}),\:\phi^{k-2}_{j}-x^{*}\right\rangle\right]\\
=\,&(1-\frac{1}{n})^{k-1}\mathbb{E}[\frac{1}{n}\sum^n_{j=1}\left\langle \partial F_{j}(x^{*}),\:\phi^{0}_{j}-x^{*}\right\rangle]\\
=\,&(1-\frac{1}{n})^{k-1}\mathbb{E}\!\left[\left\langle \partial F(x^{*}),\:{x}^{s}_{0}-x^{*}\right\rangle\right]\\
=\,&0,
\end{split}$$ where the second and last equalities hold from the optimality of $x^{*}$, the third equality holds due to Lemma \[lemm13\], and the fourth equality is due to $\phi^{0}_{j}\!=\!{x}^{s}_{0}$ for all $j=1,\ldots,n$.
Since $\mathbb{E}[\nabla\! f_{i^{s}_{k}}\!(x^{s}_{k-\!1})]\!=\!\nabla\!f(x^{s}_{k-\!1})$ and $\mathbb{E}[\nabla\! f_{i^{s}_{k}}\!(\phi^{k-\!1}_{i^{s}_{k}})]\!=\!\frac{1}{n}\!\sum^{n}_{i=1}\!\nabla\!f_{i}(\phi^{k-\!1}_{i})$, then for any $i^{s}_{k}\!\in\![n]$, $$\begin{split}
&\mathbb{E}\!\left[\left\|\nabla\! f_{i^{s}_{k}}\!(x^{s}_{k-\!1})-\!\nabla\!f(x^{s}_{k-\!1})-\nabla\! f_{i^{s}_{k}}\!(\phi^{k-1}_{i^{s}_{k}})+\frac{1}{n}\!\sum^{n}_{j=1}\nabla\! f_{j}(\phi^{k-1}_{j})\right\|^{2}\right]\\
=\,&\mathbb{E}\!\left[\left\|\nabla\! f_{i^{s}_{k}}\!(x^{s}_{k-\!1})-\nabla\! f_{i^{s}_{k}}\!(\phi^{k-1}_{i_{k}})\right\|^{2}\right]-\|\nabla\!f(x_{k-\!1})-\frac{1}{n}\!\sum^{n}_{j=1}\nabla\! f_{j}(\phi^{k-1}_{j})\|^{2}\\
\leq\,&\mathbb{E}\!\left[\left\|\nabla\! f_{i^{s}_{k}}\!(x^{s}_{k-\!1})-\nabla\! f_{i^{s}_{k}}\!(\phi^{k-1}_{i^{s}_{k}})\right\|^{2}\right]\\
=\,&\mathbb{E}\!\left[\left\|[\nabla\! f_{i^{s}_{k}}\!(x^{s}_{k-\!1})-\nabla\! f_{i^{s}_{k}}\!(x^{*})]-[\nabla\! f_{i^{s}_{k}}\!(\phi^{k-1}_{i^{s}_{k}})-\nabla\! f_{i^{s}_{k}}\!(x^{*})]\right\|^{2}\right]\\
\leq\,&2\mathbb{E}\!\left[\left\|\nabla\! f_{i^{s}_{k}}\!(\phi^{k-1}_{i^{s}_{k}})-\nabla\! f_{i^{s}_{k}}\!(x^{*})\right\|^{2}\right]+2\mathbb{E}\!\left[\left\|\nabla\! f_{i^{s}_{k}}\!(x^{s}_{k-\!1})-\nabla\! f_{i^{s}_{k}}\!(x^{*})\right\|^{2}\right]\\
\leq\,& 4L\!\left[\frac{1}{n}\!\sum^{n}_{j=1}\!f_{j}(\phi^{k-\!1}_{j})\!-\!f(x^{*})\!+\!\frac{1}{n}\!\sum^n_{j=1}\!\langle\xi^{*}\!,\phi^{k-\!1}_{j}\!-\!x^{*}\rangle\!-\!\frac{1}{n}\!\sum^n_{j=1}\!\left\langle \nabla\! f_{j}(x^{*})\!+\!\xi^{*}\!,\,\phi^{k-\!1}_{j}\!-\!x^{*}\right\rangle\right]\\
&+4L\!\left[F(x^{s}_{k-\!1})\!-\!F(x^{*})\right]\\
\leq\,& 4L\!\left[\frac{1}{n}\!\sum^{n}_{j=1}\!f_{j}(\phi^{k-\!1}_{j})\!-\!f(x^{*})\!+\!\frac{1}{n}\!\sum^n_{j=1}\!r(\phi^{k-\!1}_{j})\!-\!r(x^{*})\right]+4L\!\left[F(x^{s}_{k-\!1})\!-\!F(x^{*})\right]\\
=\,&4L\!\left[\frac{1}{n}\!\sum^{n}_{j=1}F_{j}(\phi^{k-\!1}_{j})-F(x^{*})+F(x^{s}_{k-\!1})-F(x^{*})\right]\!,
\end{split}$$ where $\xi^{*}\!=\!\partial r(x^{*})$, if $r(\cdot)$ is non-smooth, and $\xi^{*}\!=\!\nabla r(x^{*})$ otherwise. The first equality holds due to the fact that $\mathbb{E}[\|x\!-\!\mathbb{E}x\|^{2}]\!=\!\mathbb{E}[\|x\|^{2}]\!-\!\|\mathbb{E}x\|^{2}$; the second inequality holds due to the fact that $\|a-b\|^{2}\leq2(\|a\|^{2}+\|b\|^{2})$; and the third inequality follows from Lemma \[lemm12\] and Lemma 3.4 in [@xiao:prox-svrg]; and the last inequality holds due to the equality in (\[equ107\]) and the convexity of $r(\cdot)$.
Appendix D: Proofs of Theorem 2 and Corollary 3 {#appendix-d-proofs-of-theorem-2-and-corollary-3 .unnumbered}
===============================================
From Lemma 1, we immediately have the following result, which is useful in our convergence analysis below.
\[coro5\] For any $\alpha\geq\beta>0$, we have $$\begin{split}
&\alpha\;\!\mathbb{E}\!\!\left[\left\|\nabla\! f_{i^{s}_{k}}\!(x^{s}_{k-\!1})\!-\!\nabla\! f(x^{s}_{k-\!1})\!-\!\nabla\! f_{i^{s}_{k}}\!(\phi^{k-\!1}_{i^{s}_{k}})\!+\!\frac{1}{n}\!\sum^{n}_{j=1}\!\nabla\! f_{j}(\phi^{k-\!1}_{j})\right\|^{2}\right]\!-\!\beta\;\!\mathbb{E}\!\!\left[\left\|\nabla\! f_{i^{s}_{k}}\!(x^{s}_{k-\!1})\!-\!\nabla\! f_{i^{s}_{k}}\!(\phi^{k-\!1}_{i^{s}_{k}})\right\|^{2}\right]\\
&\leq 4L(\alpha\!-\!\beta)\!\left[F(x^{s}_{k-\!1})-F(x^{*})+\frac{1}{n}\sum^{n}_{j=1}F_{j}(\phi^{k-\!1}_{j})-F(x^{*})\right]\!.
\end{split}$$
**Proof of Theorem 2:**
Let $p_{i^{s}_{k}}=\nabla\! f_{i^{s}_{k}}\!(x^{s}_{k-\!1})-\nabla\! f_{i^{s}_{k}}\!(\phi^{k-\!1}_{i^{s}_{k}})+\frac{1}{n}\!\sum^{n}_{j=1}\!\nabla\! f_{j}(\phi^{k-\!1}_{j})$, and $\widehat{x}^{s}_{k}=\theta_{k}x^{s}_{k-1}$. By the similar derivation for only replacing Lemma 2 and Corollary \[coro4\] with Lemma 1 and Corollary \[coro5\], then the following inequality holds: $$\label{equ78}
\begin{split}
&\alpha_{k}\mathbb{E}\!\left[F(x^{s}_{k-1})-F(x^{*})\right]-\frac{c_{k}\overline{\beta}_{k}}{2L(\alpha\!-\!1)}\mathbb{E}\!\left[\|p_{i^{s}_{k}}-\nabla f(x^{s}_{k-\!1})\|^{2}\right]\\
\leq\,&(1-\sigma)\mathbb{E}\!\left[F(\widehat{x}^{s}_{k-1})-F(x^{*})\right]+\frac{L\alpha\sigma^{2}}{2}\mathbb{E}\!\left[\|x^{*}\!-z^{s}_{k-1}\|^{2}-\|x^{*}\!-z^{s}_{k}\|^{2}\right]\\
&\!\!+\!\frac{2(1\!-\!c_{k}\overline{\beta}_{k})}{\alpha-1}\!\left[F(x^{s}_{k-\!1})\!-\!F(x^{*})\!+\!\frac{1}{n}\!\sum^{n}_{j=1}F_{j}(\phi^{k-\!1}_{j})\!-\!F(x^{*})\right]\!.
\end{split}$$
Given $q>0$, and using the result in the proof of Theorem 1 in [@defazio:saga], we obtain $$\label{equ79}
\begin{split}
\frac{q}{n}\!\left[F(x^{s}_{k-\!1})-F(x^{*})\right]=q\mathbb{E}\!\left[\frac{1}{n}\!\sum^{n}_{j=1}\!F_{j}(\phi^{k}_{j})-F(x^{*})\right]-q(1\!-\!\frac{1}{n})\!\left(\frac{1}{n}\!\sum^{n}_{j=1}\!F_{j}(\phi^{k-\!1}_{j})-F(x^{*})\right)\!.
\end{split}$$
Using (\[equ79\]) and Lemma 1, then (\[equ78\]) is rewritten as follows: $$\label{equ80}
\begin{split}
&\left(\alpha_{k}\!-\!\frac{2(1\!-\!c_{k}\overline{\beta}_{k})}{\alpha-1}\!-\!\frac{q}{n}\right)\mathbb{E}\!\left[F(x^{s}_{k-\!1})\!-\!F(x^{*})\right]\!-\!\frac{c_{k}\overline{\beta}_{k}}{2L(\alpha\!-\!1)}\mathbb{E}\!\left[\|\nabla\! f_{i^{s}_{k}}(x^{s}_{k-\!1})\!-\!\nabla\! f_{i^{s}_{k}}\!(\phi^{k-\!1}_{i^{s}_{k}})\|^{2}\right]\\
\leq\,&(1-\sigma)\mathbb{E}\!\left[F(\widehat{x}^{s}_{k-1})-F(x^{*})\right]+\frac{L\alpha\sigma^{2}}{2}\mathbb{E}\!\left[\|x^{*}-z^{s}_{k-1}\|^{2}-\|x^{*}-z^{s}_{k}\|^{2}\right]\\
&+\left(\frac{2(1\!-\!c_{k}\overline{\beta}_{k})}{\alpha-1}\!+\!q(1\!-\!\frac{1}{n})\right)\!\left[\frac{1}{n}\sum^{n}_{j=1}F_{j}(\phi^{k-\!1}_{j})\!-\!F(x^{*})\right]\!-q\mathbb{E}\!\left[\frac{1}{n}\!\sum^{n}_{j=1}F_{j}(\phi^{k}_{j})\!-\!F(x^{*})\right]\!.
\end{split}$$ Let $$\label{equ81}
\frac{q}{n}=\frac{2}{\alpha-1}\quad \textup{and}\quad c_{k}=\alpha_{k}-\frac{q}{n}-\frac{2(1-c_{k}\overline{\beta}_{k})}{\alpha-1}.$$ Therefore, $$\begin{split}
c_{k}=\frac{\alpha_{k}(\alpha-1)-4}{\alpha-1-2\overline{\beta}_{k}}>0.
\end{split}$$
Using (\[equ80\]) and (\[equ81\]), we have $$\begin{split}
&c_{k}\mathbb{E}\!\left[F(\widehat{x}^{s}_{k})-F(x^{*})\right]\\
\leq\,& c_{k}\mathbb{E}\!\left[F(x^{s}_{k-1})-F(x^{*})\right]-\frac{c_{k}\overline{\beta}_{k}}{2L(\alpha-1)}\mathbb{E}\!\left[\left\|\nabla\! f_{i^{s}_{k}}(x^{s}_{k-1})-\nabla\! f_{i^{s}_{k}}\!(\phi^{k-\!1}_{i^{s}_{k}})\right\|^{2}\right]\\
\leq\,&(1-\sigma)\left[F(\widehat{x}^{s}_{k-1})-F(x^{*})\right]+\frac{L\alpha\sigma^{2}}{2}\mathbb{E}\!\left[\left\|x^{*}-z^{s}_{k-1}\|^{2}-\|x^{*}-z^{s}_{k}\right\|^{2}\right]\\
&+\!\left(\frac{2(1\!-\!c_{k}\overline{\beta}_{k})}{\alpha-1}\!+\!q(1\!-\!\frac{1}{n})\right)\!\left(\frac{1}{n}\!\sum^{n}_{j=1}\!F_{j}(\phi^{k-\!1}_{j})\!-\!F(x^{*})\right)\!-\!q\mathbb{E}\!\!\left[\frac{1}{n}\!\sum^{n}_{j=1}\!F_{j}(\phi^{k}_{j})\!-\!F(x^{*})\right]\\
\leq\,&(1-\sigma)[F(\widehat{x}^{s}_{k-1})-F(x^{*})]+\frac{L\alpha\sigma^{2}}{2}\mathbb{E}\!\left[\left\|x^{*}-z^{s}_{k-1}\right\|^{2}-\left\|x^{*}-z^{s}_{k}\right\|^{2}\right]\\
&+q\left(\frac{1}{n}\sum^{n}_{j=1}F_{j}(\phi^{k-1}_{j})-F(x^{*})\right)-q\mathbb{E}\!\left[\frac{1}{n}\sum^{n}_{j=1}F_{j}(\phi^{k}_{j})-F(x^{*})\right]\!.
\end{split}$$
Taking the expectation over the random choice of the history of $i^{s}_{1},\ldots,i^{s}_{m}$, using Lemma 1, and summing up the above inequality over $k=1,\ldots,m$, then $$\label{equ82}
\begin{split}
&\mathbb{E}\!\left[\sum^{m}_{k=1}(c_{k}-(1-\sigma))\left[F(\widehat{x}^{s}_{k})-F(x^{*})\right]\right]\\
\leq\,&(1-\sigma)[F(\widehat{x}^{s}_{0})-F(x^{*})]+\frac{L\alpha\sigma^{2}}{2}\mathbb{E}\!\left[\left\|x^{*}-z^{s}_{0}\right\|^{2}-\left\|x^{*}-z^{s}_{m}\right\|^{2}\right]\\
&+q\mathbb{E}\!\left[\frac{1}{n}\sum^{n}_{j=1}F_{j}(\phi^{0}_{j})-F(x^{*})\right]-q\mathbb{E}\!\left[\frac{1}{n}\sum^{n}_{j=1}F_{j}(\phi^{m}_{j})-F(x^{*})\right].
\end{split}$$ $c_{k}$ and ${q}/{n}$ are defined in (\[equ81\]) with $\sigma=1/2$ and $\alpha=19$, then there exists $\widehat{\beta}^{s}\geq 0$ for the $s$-th epoch such that $$\label{equ83}
\begin{split}
\mathbb{E}\!\left[\sum^{m}_{k=1}(c_{k}-(1\!-\!\sigma))\left[F(\widehat{x}^{s}_{k})-F(x^{*})\right]\right]&=\mathbb{E}\!\left[\sum^{m}_{k=1}\frac{7+2c_{k}\overline{\beta}_{k}}{9}\left[F(\widehat{x}^{s}_{k})-F(x^{*})\right]\right]\\
&=\left(\frac{7}{9}\!+\!\widehat{\beta}^{s}\right)\mathbb{E}\!\left[\sum^{m}_{k=1}[F(\widehat{x}^{s}_{k})-F(x^{*})]\right],
\end{split}$$ where $\widehat{\beta}^{s}=\mathbb{E}\!\left[\frac{2}{9}\sum^{m}_{k=1}c_{k}\overline{\beta}_{k}\!\left(F(\widehat{x}^{s}_{k})-F(x^{*})\right)\right]/\mathbb{E}[\sum^{m}_{k=1}(F(\widehat{x}^{s}_{k})-F(x^{*}))]$.
Let $\widehat{\beta}\!=\!\min_{s=1,\ldots,S}\widehat{\beta}^{s}$ as in the proof of Theorem 1. Using $\widetilde{x}^{s}\!=\!\frac{1}{m}\sum^{m}_{k=1}\widehat{x}^{s}_{k}$, $F(\widetilde{x}^{s})\!\leq\! \frac{1}{m}\sum^{m}_{k=1}F(\widehat{x}^{s}_{k})$, and (\[equ83\]), then (\[equ82\]) is rewritten as follows: $$\begin{split}
&m\left(\frac{7}{9}+\widehat{\beta}\right)\mathbb{E}\!\left[F(\widetilde{x}^{s})-F(x^{*})\right]\\
\leq\,&\frac{1}{2}\!\left[F(\widetilde{x}^{s-1})-F(x^{*})\right]+\frac{19L}{8}\mathbb{E}\!\left[\left\|x^{*}-\widetilde{x}^{s-1}\right\|^{2}\right]\\
&+q\left(\frac{1}{n}\sum^{n}_{j=1}F_{j}(\phi^{0}_{j})-F(x^{*})\right)-q\:\!\mathbb{E}\!\left[\frac{1}{n}\sum^{n}_{j=1}F_{j}(\phi^{m}_{j})-F(x^{*})\right]\\
\leq\,&\frac{19L}{8}\mathbb{E}\!\left[\|x^{*}-\widetilde{x}^{s-1}\|^{2}\right]+\left(\frac{1}{2}+q\right)\left[F(\widetilde{x}^{s-1})-F(x^{*})\right],
\end{split}$$ where the first and second inequalities hold due to the facts that $\widehat{x}^{s}_{0}=\widetilde{x}^{s-1}$ and $\phi^{0}_{j}=\widetilde{x}^{s-1}$.
Setting $\sigma=1/2$, $\alpha=19$, $\frac{2}{\alpha-1}=\frac{q}{n}$, and using the $\mu$-strongly convex property, we have $$m\left(\frac{7}{9}+\widehat{\beta}\right)\mathbb{E}\!\left[F(\widetilde{x}^{s})-F(x^{*})\right]\leq\left(\frac{1}{2}+\frac{n}{9}+\frac{19L}{4\mu}\right)\left[F(\widetilde{x}^{s-1})-F(x^{*})\right].$$
Therefore, $$\mathbb{E}\!\left[F(\widetilde{x}^{s})-F(x^{*})\right]\leq\left(\frac{n}{(7+9\widehat{\beta})m}+\frac{9}{(14+18\widehat{\beta})m}+\frac{171L}{(28+36\widehat{\beta})\mu m}\right)\mathbb{E}\!\left[F(\widetilde{x}^{s-1})-F(x^{*})\right]\!.$$ This completes the proof.
**Proof of Corollary 3:**
Using the similar derivation in the proof of Theorem 2 for the strongly convex case, and with the same parameter settings (i.e., $\sigma\!=\!1/2$, $\alpha\!=\!19$, and $\frac{2}{\alpha-1}\!=\!\frac{q}{n}$), we have $$\label{equ108}
\begin{split}
&\left(\frac{7}{9}+\widehat{\beta}\right)\mathbb{E}\!\left[F(\widetilde{x}^{s})-F(x^{*})\right]\\
\leq&\;\frac{19L}{8m}\mathbb{E}\!\left[\left\|x^{*}-z^{s}_{0}\right\|^{2}-\left\|x^{*}-z^{s}_{m}\right\|^{2}\right]+\left(\frac{1}{2m}+\frac{n}{9m}\right)\left[F(\widetilde{x}^{s-1})-F(x^{*})\right]\!.
\end{split}$$
According to the settings of Algorithm \[alg12\] for the non-strongly convex case as in Algorithm 1, we have $$z^{s}_{0}=z^{s-1}_{m},\;\,z^{0}_{0}=\widetilde{x}^{0}.$$ Summing up the above inequality (\[equ108\]) over $s=1,\ldots,S$, and setting $m=n$, then $$\begin{split}
&\,\mathbb{E}\!\left[F(\overline{x})-F(x^{*})\right]\leq\mathbb{E}\!\left[F\left(\frac{1}{S}\sum^{S}_{s=1}\widetilde{x}^{s}\right)-F(x^{*})\right]\\
\leq&\,\frac{171L}{(49\!+\!56\widehat{\beta})nS}\|x^{*}\!-\!\widetilde{x}^{0}\|^{2}\!+\!\left(\frac{81}{(98\!+\!126\widehat{\beta})nS}\!+\!\frac{9}{(49\!+\!63\widehat{\beta})S}\right)\!\left[F(\widetilde{x}^{0})\!-\!F(x^{*})\right]\!.
\end{split}$$ This completes the proof.
Appendix E: Codes and Data Sets {#appendix-e-codes-and-data-sets .unnumbered}
===============================
In this section, we first present the detailed descriptions for the three popular data sets: Covtype, SUSY and Ijcnn1, which were obtained from the LIBSVM Data website[[^5]]{}, as shown in Table \[tab\_sim1\]. The C++ code of SVRG [@johnson:svrg] was downloaded from <http://riejohnson.com/svrg_download.html>. For fair comparison, we implemented the proposed SVRG-SD and SAGA-SD (see Algorithm \[alg12\]) algorithms, SAGA [@defazio:saga], Prox-SVRG [@xiao:prox-svrg], Catalyst [@lin:vrsg] (which is based on SVRG and has three important parameters: $\alpha_{k}$, $\kappa$, and the learning rate, $\eta$), and Katyusha [@zhu:Katyusha] in C++ with a Matlab interface[^6], and performed all the experiments on a PC with an Intel i5-2400 CPU and 16GB RAM.
Data sets Sizes $n$ Dimensions $d$ Sparsity
------------ ----------- ---------------- ----------
Ijcnn1 49,990 22 59.09%
Covtype 581,012 54 22.12%
SUSY 5,000,000 18 98.82%
Sido0 12,678 4,932 9.84%
: Data sets and their regularization parameters.[]{data-label="tab_sim1"}
the number of epochs $S$, the number of iterations $m$ per epoch, and step size $\eta$.\
$\widetilde{x}^{0}$. $\overline{x}\!=\!\widetilde{x}^{S}$
![Comparison of different variance reduced SGD methods for solving strongly convex ridge regression problems. The vertical axis is the objective value minus the minimum, and the horizontal axis denotes the running time (seconds) or the number of effective passes over the data.[]{data-label="fig_sim1"}](Fig23 "fig:"){width="0.326\columnwidth"} ![Comparison of different variance reduced SGD methods for solving strongly convex ridge regression problems. The vertical axis is the objective value minus the minimum, and the horizontal axis denotes the running time (seconds) or the number of effective passes over the data.[]{data-label="fig_sim1"}](Fig28 "fig:"){width="0.326\columnwidth"} ![Comparison of different variance reduced SGD methods for solving strongly convex ridge regression problems. The vertical axis is the objective value minus the minimum, and the horizontal axis denotes the running time (seconds) or the number of effective passes over the data.[]{data-label="fig_sim1"}](Fig33 "fig:"){width="0.326\columnwidth"}
![Comparison of different variance reduced SGD methods for solving strongly convex ridge regression problems with different regularization parameters on the Sido0 data set. The vertical axis represents the objective value minus the minimum, and the horizontal axis denotes the running time (top) or the number of effective passes (bottom).[]{data-label="fig_sim2"}](Fig102 "fig:"){width="0.326\columnwidth"} ![Comparison of different variance reduced SGD methods for solving strongly convex ridge regression problems with different regularization parameters on the Sido0 data set. The vertical axis represents the objective value minus the minimum, and the horizontal axis denotes the running time (top) or the number of effective passes (bottom).[]{data-label="fig_sim2"}](Fig104 "fig:"){width="0.326\columnwidth"} ![Comparison of different variance reduced SGD methods for solving strongly convex ridge regression problems with different regularization parameters on the Sido0 data set. The vertical axis represents the objective value minus the minimum, and the horizontal axis denotes the running time (top) or the number of effective passes (bottom).[]{data-label="fig_sim2"}](Fig106 "fig:"){width="0.326\columnwidth"}
Appendix F: More Experimental Results {#appendix-f-more-experimental-results .unnumbered}
=====================================
In this section, we report more experimental results of SVRG [@johnson:svrg], SAGA [@defazio:saga], Catalyst [@lin:vrsg], Katyusha [@zhu:Katyusha], SVRG-SD and SAGA-SD for solving strongly convex ridge regression problems with regularization parameters $\lambda\!=\!10^{-4}$ and $\lambda\!=\!10^{-5}$ in Figure \[fig\_sim1\], where the horizontal axis denotes the number of effective passes over the data set (evaluating $n$ component gradients, or computing a single full gradient is considered as one effective pass) or the running time (seconds). Figure \[fig\_sim2\] shows the performance of all these methods for solving ridge regression problems with different regularization parameters on a sparse data set, Sido0, which can be downloaded from the Causality Workbench website[[^7]]{}. From all the results, we can observe that SVRG-SD and SAGA-SD significantly outperform their counterparts: SVRG and SAGA in terms of both number of effective passes and running time. The accelerated method, Catalyst, usually outperforms the non-accelerated methods, SVRG and SAGA. Moreover, SVRG-SD and SAGA-SD achieve at least comparable performance with the best known stochastic method, Katyusha [@zhu:Katyusha], in terms of number of effective passes. Since SVRG-SD and SAGA-SD have much lower per-iteration complexities than Katyusha, they have more obvious advantage over Katyusha in terms of running time.
![Comparison of different variance reduced SGD methods for solving non-strongly convex Lasso problems. The vertical axis is the objective value minus the minimum, and the horizontal axis denotes the number of effective passes over the data (top) or the running time (seconds, bottom).[]{data-label="fig_sim3"}](Fig51 "fig:"){width="0.326\columnwidth"} ![Comparison of different variance reduced SGD methods for solving non-strongly convex Lasso problems. The vertical axis is the objective value minus the minimum, and the horizontal axis denotes the number of effective passes over the data (top) or the running time (seconds, bottom).[]{data-label="fig_sim3"}](Fig56 "fig:"){width="0.326\columnwidth"} ![Comparison of different variance reduced SGD methods for solving non-strongly convex Lasso problems. The vertical axis is the objective value minus the minimum, and the horizontal axis denotes the number of effective passes over the data (top) or the running time (seconds, bottom).[]{data-label="fig_sim3"}](Fig61 "fig:"){width="0.326\columnwidth"}
![Comparison of different variance reduced SGD methods for solving non-strongly convex Lasso problems. The vertical axis is the objective value minus the minimum, and the horizontal axis denotes the number of effective passes over the data (top) or the running time (seconds, bottom).[]{data-label="fig_sim3"}](Fig53 "fig:"){width="0.326\columnwidth"} ![Comparison of different variance reduced SGD methods for solving non-strongly convex Lasso problems. The vertical axis is the objective value minus the minimum, and the horizontal axis denotes the number of effective passes over the data (top) or the running time (seconds, bottom).[]{data-label="fig_sim3"}](Fig58 "fig:"){width="0.326\columnwidth"} ![Comparison of different variance reduced SGD methods for solving non-strongly convex Lasso problems. The vertical axis is the objective value minus the minimum, and the horizontal axis denotes the number of effective passes over the data (top) or the running time (seconds, bottom).[]{data-label="fig_sim3"}](Fig63 "fig:"){width="0.326\columnwidth"}
![Comparison of different variance reduced SGD methods for solving elastic-net regularized (i.e., $\lambda_{1}\|\!\cdot\!\|_{1}\!+\!\lambda_{2}\|\!\cdot\!\|^2$) Lasso problems on Ijcnn1 (the first column), Covtype (the second column), and SUSY (the last column). The vertical axis is the objective value minus the minimum, and the horizontal axis denotes the number of effective passes over the data (top) or the running time (seconds, bottom).[]{data-label="fig_sim4"}](Fig71 "fig:"){width="0.326\columnwidth"} ![Comparison of different variance reduced SGD methods for solving elastic-net regularized (i.e., $\lambda_{1}\|\!\cdot\!\|_{1}\!+\!\lambda_{2}\|\!\cdot\!\|^2$) Lasso problems on Ijcnn1 (the first column), Covtype (the second column), and SUSY (the last column). The vertical axis is the objective value minus the minimum, and the horizontal axis denotes the number of effective passes over the data (top) or the running time (seconds, bottom).[]{data-label="fig_sim4"}](Fig76 "fig:"){width="0.326\columnwidth"} ![Comparison of different variance reduced SGD methods for solving elastic-net regularized (i.e., $\lambda_{1}\|\!\cdot\!\|_{1}\!+\!\lambda_{2}\|\!\cdot\!\|^2$) Lasso problems on Ijcnn1 (the first column), Covtype (the second column), and SUSY (the last column). The vertical axis is the objective value minus the minimum, and the horizontal axis denotes the number of effective passes over the data (top) or the running time (seconds, bottom).[]{data-label="fig_sim4"}](Fig81 "fig:"){width="0.326\columnwidth"}
![Comparison of different variance reduced SGD methods for solving elastic-net regularized (i.e., $\lambda_{1}\|\!\cdot\!\|_{1}\!+\!\lambda_{2}\|\!\cdot\!\|^2$) Lasso problems on Ijcnn1 (the first column), Covtype (the second column), and SUSY (the last column). The vertical axis is the objective value minus the minimum, and the horizontal axis denotes the number of effective passes over the data (top) or the running time (seconds, bottom).[]{data-label="fig_sim4"}](Fig73 "fig:"){width="0.326\columnwidth"} ![Comparison of different variance reduced SGD methods for solving elastic-net regularized (i.e., $\lambda_{1}\|\!\cdot\!\|_{1}\!+\!\lambda_{2}\|\!\cdot\!\|^2$) Lasso problems on Ijcnn1 (the first column), Covtype (the second column), and SUSY (the last column). The vertical axis is the objective value minus the minimum, and the horizontal axis denotes the number of effective passes over the data (top) or the running time (seconds, bottom).[]{data-label="fig_sim4"}](Fig78 "fig:"){width="0.326\columnwidth"} ![Comparison of different variance reduced SGD methods for solving elastic-net regularized (i.e., $\lambda_{1}\|\!\cdot\!\|_{1}\!+\!\lambda_{2}\|\!\cdot\!\|^2$) Lasso problems on Ijcnn1 (the first column), Covtype (the second column), and SUSY (the last column). The vertical axis is the objective value minus the minimum, and the horizontal axis denotes the number of effective passes over the data (top) or the running time (seconds, bottom).[]{data-label="fig_sim4"}](Fig83 "fig:"){width="0.326\columnwidth"}
Moreover, we report the performance of Prox-SVRG [@xiao:prox-svrg], SAGA [@defazio:saga], Catalyst [@lin:vrsg], Katyusha [@zhu:Katyusha], SVRG-SD and SAGA-SD for solving Lasso and elastic-net regularized Lasso problems with different regularization parameters in Figures \[fig\_sim3\] and \[fig\_sim4\], respectively, from which we can see that SVRG-SD and SAGA-SD also achieve much faster convergence speed than their counterparts: Prox-SVRG and SAGA, respectively. In particular, they also have comparable or better performance than the accelerated VR-SGD methods, Catalyst and Katyusha.
The number of epochs $S$, the number of iterations $m$ per epoch, and step size $\eta$.\
$\widetilde{x}^{0}$.\
The number of epochs $S$, the number of iterations $m$ per epoch, and step size $\eta$.\
$x_{0}=\widetilde{x}^{0}$, $\sigma=0.618-0.382/[1+\exp(-\log(6\lambda)-12)]$ for Case of SC, or $\sigma=1/(S\!+\!3)$ for Case of NSC.\
The number of iterations $K\!=\!S\!\ast\!m$, and step size $\eta$.\
$x_{0}$, $\sigma=0.5-0.5/[1+\exp(-\log\lambda-12)]$ for the cases of SC and NSC. $x_{K}$
Appendix G: Pseudo-Codes of More Algorithms {#appendix-g-pseudo-codes-of-more-algorithms .unnumbered}
===========================================
Recall that the main difference of the original SVRG [@johnson:svrg] (denoted by SVRG-I, see Algorithm \[alg3\] with Option I for completeness) and its variant, SVRG-II (see Algorithm \[alg3\] with Option II) in [@johnson:svrg] is that the former uses the last iterate of the previous epoch as the snapshot point $\widetilde{x}^{s}$, while in the latter, $\widetilde{x}^{s}$ is the average point of the previous epoch, which has been successfully used in [@zhu:Katyusha; @zhu:univr; @xiao:prox-svrg]. In this paper, we observed the following interesting phenomena: When the regularization parameter $\lambda$ is relatively large, e.g., $\lambda\!=\!10^{-3}$, SVRG-II converges significantly faster than SVRG-I, as shown in Figure \[fig\_sim6\] and also suggested in [@zhu:Katyusha; @zhu:univr; @xiao:prox-svrg]; whereas SVRG-I significantly outperforms SVRG-II when $\lambda$ is relatively small, e.g., $\lambda\!=\!10^{-7}$.
As a by-product of SVRG-SD and motivated by the above observations, we propose a momentum acceleration variant of the original SVRG, as outlined in Algorithm \[alg4\], which is called as SVRG-SDI, and can be viewed as a special case of SVRG-SD when $\theta_{k}\!\equiv\!1$ (i.e., without the proposed sufficient decrease technique). Note that SVRG-SDI, as well as SVRG-SD, can use much larger learning rates than SVRG-I and SVRG-II, e.g., $\eta\!=\!1.0$ for SVRG-SD and SVRG-SDI vs. $\eta\!=\!0.4$ for SVRG-I and SVRG-II. From the results in Figure \[fig\_sim6\], where we vary the values of the regularization parameter $\lambda$ from $10^{-3}$ to $10^{-8}$, it is clear that our SVRG-SDI method achieves significantly better performance than both SVRG-I and SVRG-II in most cases, and is comparable to the best known method, Katyusha [@zhu:Katyusha]. Note that Katyusha fails to converge when the regularization parameter is greater than or equal to $10^{-3}$. Moreover, our SVRG-SD method often outperforms the other methods (including Katyusha and SVRG-SDI). Especially in the cases when $\lambda$ is relatively small, e.g., $\lambda\!=\!10^{-6}$ and $\lambda\!=\!10^{-7}$, SVRG-SD converges significantly faster than Katyusha and SVRG-SDI, which further verifies the effectiveness of our sufficient decrease technique for stochastic optimization. Our sufficient decrease technique in SVRG-SD and SVRG-SDI naturally generalizes to other stochastic gradient estimators as in [@defazio:saga; @nguyen:srg; @roux:sag] as well, e.g., Algorithm \[alg5\] (called SAGA-SDI) for the SAGA estimator [@defazio:saga] in (9).
[^1]: Note that SVRG-SD and SAGA-SD with partial sufficient decrease possess the similar convergence properties as SVRG-SD and SAGA-SD with full sufficient decrease because Property \[prop11\] still holds when $\theta_{k}\!=\!1$.
[^2]: <https://www.csie.ntu.edu.tw/~cjlin/libsvm/>
[^3]: Strictly speaking, when the function $F(\cdot)$ is non-smooth, $\vartheta\in \partial F(x)$; while $F(\cdot)$ is smooth, $\vartheta=\nabla F(x)$.
[^4]: Note that our fast versions of SVRG-SD and SAGA-SD (i.e., SVRG-SD and SAGA-SD with randomly partial sufficient decrease) have the similar convergence properties as SVRG-SD and SAGA-SD because Property 1 still holds when $\theta_{k}\!=\!1$. That is, the main difference between their convergence properties is the different values of $\beta_{k}$, as shown below.
[^5]: <https://www.csie.ntu.edu.tw/~cjlin/libsvm/>
[^6]: The codes of all those algorithms can be downloaded by the following link:\
<https://www.dropbox.com/s/5sg7h49qctr9ahi/Code_VD_SGD.zip?dl=0.>
[^7]: <http://www.causality.inf.ethz.ch/home.php>
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NDA-FP-66\
June 1999\
[**Running Gauge Coupling and Quark-Antiquark Potential in Non-SUSY Gauge Theory at Finite Temperature from IIB SG/CFT correspondence**]{}
[Shin’ichi NOJIRI]{}[^1] and [Sergei D. ODINTSOV$^{\spadesuit}$]{}[^2]
[*Department of Mathematics and Physics\
National Defence Academy, Hashirimizu Yokosuka 239, JAPAN*]{}
[*$\spadesuit$ Tomsk Pedagogical University, 634041 Tomsk, RUSSIA\
*]{}
[**abstract**]{}
We discuss the non-constant dilaton deformed ${\rm AdS}_5\times{\rm S}_5$ solutions of IIB supergravity where AdS sector is described by black hole. The investigation of running gauge coupling (exponent of dilaton) of non-SUSY gauge theory at finite temperature is presented for different regimes (high or low $T$, large radius expansion). Running gauge coupling shows power-like behavior on temperature with stable fixed point. The quark-antiquark potential at finite $T$ is found and possibility of confinement is established. It is shown that non-constant dilaton affects the potential, sometimes reversing its behavior if we compare it with the constant dilaton case (${\cal N}=4$ super Yang-Mills theory). Thermodynamics of obtained backgrounds is studied. In particular, next-to-leading term to free energy $F$ is evaluated as $F=-{\tilde V_3 \over 4\pi^2 }\left(
{N^2 \left(\pi T\right)^4 \over 2}
+ {5c^2 \over 768 g_{YM}^6 N{\alpha'}^6
\left(\pi T\right)^4}\right)$. Here $\tilde V_3$ is the volume of the space part in the boundary of AdS, $c$ is the parameter coming from the non-constant dilaton and $N$ is the number of the coincident D3-branes.
Introduction.
=============
One of the unsolved problems in AdS/CFT correspondence [@1] (for an excellent review, see [@AGMOO]) is how to obtain non-SUSY gauge theory with typical running coupling as the boundary side. The related question is about confinement in such theory. It is desirable to answer these questions from supergravity (SG) side as it gives strong coupling regime of boundary quantum field theory (QFT).
There are different proposals to get running gauge coupling in non-SUSY theory: using Type 0 string theory approach [@2], deforming ${\cal N}=4$ theory [@15] (also via AdS orbifolding [@17]) or making non-constant dilaton deformations of ${\rm AdS}_5\times{\rm S}_5$ vacuum in IIB SG [@3; @4; @6; @7; @8; @9; @NO]. In the last case non-constant dilaton breaks conformal invariance and (a part of) supersymmetry of the boundary ${\cal N}=4$ super YM theory. (In the presence of axion (RR-scalar), a part of supersymmetry may be unbroken but dilaton is still non-trivial [@14]). Then, exponent of dilaton actually describes the running gauge coupling with a power-law behavior and UV-stable fixed point. Within such picture the indication to the possibility of confinement is also found. The features of running and confinement depend on the axion [@7], vectors [@8], worldvolume scalar [@9] or curvature of four-dimensional space [@NO].
From another side, it is also realized that planar ${\rm AdS}_5$ BH is dual to a thermal state of ${\cal N}=4$ super YM theory. The corresponding coupling constant dependence has been studied in ref.[@GKT; @AAT] based on earlier study of SG side free energy in ref.[@GKP]. Spherical AdS BH shows the finite temperature phase transition [@HP] which may be used to realize the confinement in large $N$ theory at low temperatures [@witten].
In this paper, we attempt to combine these two approaches, i.e. to find the deformation of IIB SG ${\rm AdS}_5\times{\rm S}_5$ background with non-trivial dilaton where AdS sector is described by BH (hence, temperature appears). Then, running gauge coupling of gauge theory at non-zero temperature is given by exponent of dilaton. We present the class of approximate solutions of IIB SG[^3] with such properties where running coupling shows power-like behavior in the temperature (in the expansion on radius). The quark-antiquark potential for these solutions is also found and possibility of confinement at non-zero temperature is established. Corrections to position of horizon (in near horizon regime) and to the temperature are calculated. Thermodynamics of obtained solutions is also investigated.
The paper is organized as follows. In the next section we present the approximate solution of IIB supergravity. It represents dilatonic perturbation of zero mass hyperbolic AdS BH. The temperature dependence of running gauge coupling (exponent of dilaton) and of corresponding beta-function is derived in different regimes. Quark-antiquark potential which is repulsive unlike to constant dilaton case is analyzed. Section 3 is devoted to the study of the same questions for background representing dilatonic deformation of (non)planar non-zero mass AdS BH. The temperature dependence of running gauge coupling is different from the situation in previous section. Confinement is possible as it follows from the study of quark-antiquark potential. In section 4, we investigate thermodynamic properties of our AdS backgrounds. Free energy, mass and entropy are found with account of non-trivial temperature corrections due to dilaton. This is compared with the leading behaviour of free energy in $N=4$ super Yang-Mills theory. Some outlook is given in the last section.
Perturbative solutions of IIB supergravity, running gauge coupling and potential: zero mass BH case
===================================================================================================
We start from the action of dilatonic gravity in $d+1$ dimensions: \[i\] S=-[1 16G]{}d\^[d+1]{}x (R - - G\^\_\_) . In the following, we assume $\lambda^2\equiv -\Lambda$ and $\alpha$ to be positive. The action (\[i\]) contains the effective action of type IIB string theory. In the type IIB supergravity, which is the low energy effective action of the type IIB string theory, we can consider bosonic background where anti-self-dual five-form is given by the Freund-Rubin-type ansatz and the topology is $M_5\times {\rm S}^5$ with the manifold $M_5$ which is asymptotically ${\rm AdS}_5$. If dilaton only depends on the coordinates in $M_5$, by integrating five coordinates on ${\rm S}^5$, we obtain the effective five dimensional theory, which corresponds to $d=4$ and $\alpha={1 \over 2}$ case in (\[i\]). This will be the case under consideration in this work.
From the variation of the action (\[i\]) with respect to the metric $G^{\mu\nu}$, we obtain[^4] \[iit\] 0=R\_-[1 2]{}G\_R + [2]{}G\_ - (\_\_ -[1 2]{}G\_G\^\_\_) and from that of dilaton $\phi$ \[iiit\] 0=\_(G\^\_) . We now assume the $(d+1)$-dimensional metric is given by \[ii\] ds\^2=-\^[2]{}dt\^2 + \^[2]{}dr\^2 + r\^2 \_[i,j=1]{}\^[d-1]{}g\_[ij]{}dx\^i dx\^j . Here $g_{ij}$ does not depend on $r$ and it is the metric in the Einstein manifold, which is defined by \[vat\] R\_[ij]{}=kg\_[ij]{} . Here $\hat R_{ij}$ is Ricci tensor defined by $g_{ij}$ and $k$ is a constant, especially $k>0$ for sphere , $k=0$ for Minkowski space and $k<0$ for hyperboloid. We also assume $\rho$, $\sigma$ and $\phi$ only depend on $r$. Then the equations (\[iit\]) when $\mu=\nu=t$, $\mu=\nu=r$ and $\mu=i$ and $\nu=j$ give, respectively, \[iii\] 0&=&[(d-1)k\^[2]{} 2r\^2]{} + [(d-1)’ r]{} - [(d-1)(d-2) 2r\^2]{} && + [\^2 2]{}\^[2]{} - [2]{}(’)\^2\
\[iv\] 0&=&-[(d-1)k\^[2]{} 2r\^2]{} + [(d-1)’ r]{} + [(d-1)(d-2) 2r\^2]{} && - [\^2 2]{}\^[2]{} - [2]{}(’)\^2\
\[v\] 0&=&-[(d-3)k\^[2]{} 2r\^2]{} + ” + (’)\^2 - ’’ + [(d-2)(d-3) 2r\^2]{} && - [\^2 2]{}\^[2]{} + [2]{}(’)\^2 . Here $'\equiv {d \over d r}$. Other components give identities. Eq.(\[iiit\]) has the following form \[vi\] 0=(r\^[d-1]{}\^[- ]{}’)’ , which can be integrated to give \[vii\] r\^[d-1]{}\^[- ]{}’=c . Combining (\[iii\]) and (\[iv\]) and substituting (\[vii\]), we obtain \[viii\] 0&=&[(d-1)(’ + ’) r]{} - [c\^2 \^[2- 2]{} r\^[2d-2]{}]{}\
\[ix\] 0&=&[(d-1)k \^[2]{} r\^2]{} + [(d-1)(’ - ’ ) r]{} - [(d-1)(d-2) r\^2]{} + \^2\^[2]{} . If we introduce new variables $U$ and $V$ by \[x\] U\^[+ ]{} , Vr\^[d-2]{}\^[- ]{} , Eqs.(\[viii\]), (\[ix\]) and (\[vii\]) are rewritten as follows \[xi\] 0&=&(d-1)U’-[c\^2 r V\^2]{}U\
\[xii\] 0&=&{[(d-1)k r\^2]{} + \^2 }U - [(d-1) r\^[d-1]{}]{}V’\
\[xiib\] ’&=&[c rV]{} Deleting $U$ from (\[xi\]) and (\[xii\]), we obtain \[xiic\] 0&=&V” + V’ &&- [c\^2V’ (d-1)rV\^2]{} . When $c=0$ the solution is given by \[xviii\] U&=&1 V&=&V\_0 && [kr\^[d-2]{} d-2]{} + [\^2 d(d-1)]{}r\^d - . Here $\mu$ corresponds to the mass of the black hole. $k=0$, positive or negative corresponds to planar, spherical or hyperbolic AdS BH, respectively. Using (\[xviii\]), Eq.(\[xii\]) and (\[xiic\]) can be rewritten as follows: \[xviiib\] U&=&[V’ V\_0’]{} \[xviiic\] 0&=&([V’ V\_0’]{})’-[c\^2V’ (d-1)rV\_0’V\^2]{} .
When $\mu=0$, the solution is isomorphic to AdS. If we choose $k<0$, the metric has the following form: \[xix\] ds\^2=-[(r\^2 - r\_0\^2) l\^2]{}dt\^2 + [l\^2 (r\^2 - r\_0\^2) ]{}dr\^2 + r\^2 \_[i,j=1]{}\^[d-1]{}g\_[ij]{}dx\^i dx\^j . Here \[xx\] l\^2 , r\_0l . The obtained AdS metric has a horizon at $r=r_0$. When $r\sim r_0$, the metric behaves as \[xxi\] ds\^2 \~-[2r\_0 (r - r\_0) l\^2]{}dt\^2 + [l\^2 2r\_0 (r - r\_0)]{}dr\^2 + . Then if we define a new coordinate $\rho$ by \[xxii\] =l the metric has the following form: \[xxiii\] ds\^2 \~-[r\_0 l\^4]{}\^2 dt\^2 + d\^2 + . Therefore when we Wick-rotate $t$ by $t=i\tau$, $\tau$ has a period of ${2\pi l^2 \over r_0}$, whose inverse gives a temperature $T$: \[xxiv\] T=[r\_0 2l\^2]{}=[1 2l]{} .
We now consider the perturbation with respect to $c$. We will concentrate on the case of type IIB SG in $d=4$, by putting $\alpha={1 \over 2}$. Note that in this approximation the radius is away from horizon. Near-horizon regime will be discussed independently.
For $\mu=0$ and $k<0$ case, the leading term for the dilaton $\phi$ is given by substituting $V_0$ in (\[xviii\]) into (\[xiib\]) \[xxv\] &=&\_0 +cl\^2{[1 2r\_0\^4]{}(1 - [r\_0\^2 r\^2]{}) + [1 2r\_0\^2 r\^2]{}} &=& \_0 + c{[1 2l\^6(2T)\^4]{}(1 - [l\^4 (2T)\^2 r\^2]{}) + [1 2l\^2(2T)\^2 r\^2]{}} . which gives the temperature dependent running dilaton. We should note that there is a singularity in the dilaton field at the horizon $r=r_0=2\pi l^2 T$. The fact that dilaton may become singular at IR has been mentioned already in two-boundaries AdS solution of IIB SG in ref.[@3]. It is also interesting that when $r$ is formally less than $r_0$ then dilaton (and also running coupling) becomes imaginary.
Since the string coupling is given by \[ci\] g=g\_s\^ (g\_s : ) , we find the behaviour when $r$ is large and $c$ is small as \[cii\] g\~g\_s{1 + cl\^2(-[1 2r\^4 ]{} - [(2l\^2 T)\^2 3r\^6]{} + [O]{} (r\^[-8]{}) ) + [O]{}(c\^2)} . Here $\phi_0$ has been absorbed into the redefinition of $g_s$. Since $r$ is the length scale corresponding to the radius of the boundary manifold, $r$ can be regarded as the energy scale of the field theory on the boundary [@10]. Therefore the beta-function is given by \[ciii\] (g)=r[dg dr]{}=-4(g-g\_s) + [2\^[5 2]{} 3]{}(2T)\^2l\^3 g\_s ( [g\_s - g c g\_s]{} )\^[3 2]{} . The first term is usual and universal [@4; @7]. The second term defines the temperature dependence.
Let us comment on the case of high $T$. As we consider the behavior near the boundary, first we take $r$ to be large. After that we consider the case of high $T$. In this case $r\gg Tl^2$ and we can consider the large $T$ case in the expression (\[ciii\]). The problem might happen when $r\sim Tl^2$. In this case, we need to solve Eq.(\[xxv\]) with respect to $r$ as a function of $T$ and $\phi$ or coupling: $r=r(g,T)$. Then from (\[xxv\]) and (\[ci\]), we find the following expression of the beta-function: \[gTii\] (g) \~.r[dg dr]{}|\_[r=r(g,T)]{} =[g\_s c l\^2 r(g,T)\^4 (1 - [l\^4 (2T)\^2 r(g,T)\^2]{}) ]{} . In case $r$ is large, the above equation reproduces (\[ciii\]). We can also consider the case that the last term in (\[xxv\]) is larger than the second term which contains $\ln (\cdots)$. In this case, the coupling is given by \[gTib\] g\~g\_s( 1 + [c 2l\^2 (2T)\^2 r\^2]{} ) , which changes the leading behavior of the beta-function: \[gTiib\] (g)\~- 2 (g-g\_s) + . This beta-function presumably defines strong coupling regime of non-SUSY gauge theory at high temperature. It is interesting to note that in perturbative gauge theory at non-zero temperature the running gauge coupling contains not only standard logarithms of $T$ but also terms linear on $T$ (see ref.[@volodya] and references therein). Of course, in our case we have not AF theory but the one with stable fixed point.
Now we consider the correction for $V$ and $U$ , writing them in the following form: \[xxvi\] V=V\_0+c\^2 v ,U=1+c\^2 u . Substituting (\[xxvi\]) and neglecting the higher orders in $c^2$, we obtain \[xxvii\] u&=&[v’ V\_0’]{} \[xxviii\] 0&=&([v’ V\_0’]{})’-[1 6rV\_0\^2]{} . With $\mu=0$ and $k<0$ in the above equations one gets, \[xxix\] u&=&[4 3 k\^4 l\^4]{}{-[1 2s\^2]{} - [2 s]{} . && . -3(1 - [1 s]{}) - [1 (s-1)]{} + c\_1 }\
\[xxx\] v&=&[2 3k\^2 l\^2]{}{ -[1 2]{}(3s\^2 - 3s +1 )(1 - [1 s]{}) . && . -[3s 2]{} + [3 4]{} -[1 4s]{} +[c\_1 2]{}(s\^2 - s) + c\_2} Here \[xxxi\] s=-[2 r\^2 kl\^2]{} and $c_1$ and $c_2$ are constants of the integration, which should vanish if we require $u$, $v\rightarrow 0$ when $r\rightarrow \infty$.
From (\[xxix\]) and (\[xxx\]), we find that $U$ and $V$ or $\e^{2\rho}$ and $\e^{2\sigma}$ have the singularity at the unperturbative horizon corresponding to $s=1$. Eq.(\[xxv\]) tells also that the dilaton field is also singular there. In other words, the expansion with respect to $c^2$ breaks down when $s\sim 0$. Therefore the singularity in $U$, $V$ would not be real one.
In order to investigate the behavior in near-horizon regime we assume that the radius of the horizon is large and use ${1 \over r}$ expansion: \[r1\] V=[r\^4 l\^2]{} + [kr\^2 2]{} + [a r\^4]{} + [O]{}(r\^[-6]{}) . We put the constant term to be zero assuming that the black hole mass vanishes. The absence of ${1 \over r^2}$ term can be found from (\[xviiib\]). Eq.(\[xviiib\]) also tells that \[r2\] a=[c\^2l\^2 48]{} and $\e^\phi$, $V$ and $U$ have the following forms: \[r3\] \^&=&\^[\_0]{}(1 - [cl\^2 4 r\^4]{} + [O]{}(r\^[-6]{})) V&=&[r\^4 l\^2]{} + [kr\^2 2]{} + [c\^2l\^2 48 r\^4]{} + [O]{}(r\^[-6]{}) U&=&1-[c\^2l\^4 192 r\^8]{} + [O]{}(r\^[-10]{}) . From the equation $V=0$ we find the position of the horizon \[r4\] r=r\_hl(1 - [c\^2 6 k\^4 l\^4]{}) , which gives the correction to the temperature: \[r5\] T=[1 2l]{} - [c\^2(-[k 2]{})\^[-[7 2]{}]{} 192 l\^5]{} .
Let us turn now to the analysis of the potential between quark and anti-quark[@5]. We evaluate the following Nambu-Goto action \[rg5\] S=[1 2]{}dd . with the “string” metric $g^s_{\mu\nu}$, which could be given by multiplying a dilaton function $\e^\phi$ to the metric tensor in (\[ii\]). We consider the static configuration $x^0=\tau$, $x^1\equiv x=\sigma$, $x^2=x^3=\cdots=x^{d-1}=0$ and $r=r(x)$. Choose the coordinates on the boundary manifold so that the line given by $x^0=$constant, $x^1\equiv x$ and $x^2=x^3=\cdots=x^{d-1}=0$ is geodesic and $g_{11}=1$ on the line. Substituting the configuration into (\[rg5\]), we find \[rg7\] S=[[T]{} 2]{}dx \^(r) . Here ${\cal T}$ is the length of the region of the definition of $\tau$ and we choose $\phi_0=0$ for simplicity. The orbit of $r$ can be obtained by minimizing the action $S$ or solving the Euler-Lagrange equation ${\delta S \over \delta r}- \partial_x\left({\delta S
\over \delta\left(\partial_x r\right)}\right)=0$. The Euler-Lagrange equation tells that \[rg8\] E\_0=\^(r) is a constant. If we assume $r$ has a finite minimum $r_{\rm min}$, where $\partial_x r|_{r=r_{\rm min}}=0$, $E_0$ is given by \[rg9b\] E\_0=\^[(r\_[min]{})]{} . Introducing a parameter $t$, we parametrize $r$ by \[rg9\] r=r\_[min]{}t . Then we find \[rg10\] [dx dt]{}&=& [l r\_[min]{}\^2t(\^2t + 1 )\^[1 2]{}]{} && {1 + [kl\^2 4r\_[min]{}\^2]{} [\^4 t - \^2 t -1 (\^2 t + 1 )\^2 t ]{} + [O]{}(r\_[min]{}\^[-4]{}) } . Taking $t\rightarrow +\infty$, we find the distance $L$ between “quark” and “anti-quark” \[rg11\] L&=& [lA r\_[min]{}]{} + [kl\^3 B 4r\_[min]{}\^3]{} + [O]{}( r\_[min]{}\^[-5]{})\
A&& \_[-]{}\^ =1.19814... B&& \_[-]{}\^dt [\^4t - \^2t -1 \^4t (\^2t + 1)\^[3 2]{}]{} =-0.162061... .As one sees the next-to-leading correction to distance depends on the curvature of space-time[@NO] or temperature.
Eq.(\[rg11\]) can be solved with respect to $r_{\rm min}$ and we find \[rg12\] r\_[min]{}=[lA L]{} + [klBL 4A\^2]{} + [O]{}(L\^3) . Using (\[rg8\]), (\[rg9\]) and (\[rg11\]), we find the following expression for the action $S$ \[rg13\] S&=&[[T]{} 2]{}E(L)\
E(L)&=&\_[-]{}\^dt [\^2 t (\^2 t + 1)\^[1 2]{}]{}{1 + [kl\^2 4 r\_[min]{}\^2]{} [1 \^2 t (\^2 t + 1)]{} + [O]{}(r\_[min]{}\^[-4]{})} . Here $E(L)$ expresses the total energy of the “quark”-“anti-quark” system. The energy $E(L)$ in (\[rg13\]), however, contains the divergence due to the self energies of the infinitely heavy “quark” and “anti-quark”. The sum of their self energies can be estimated by considering the configuration $x^0=\tau$, $x^1=x^2=x^3=\cdots
=x^{d-1}=0$ and $r=r(\sigma)$ (note that $x_1$ vanishes here) and the minimum of $r$ is $r_D$, where branes would lie : $r_D\gg r_{\rm min}$. We devide the region for $r$ to two ones, $\infty>r>r_{\rm min}$ and $r_{\rm min}<r<r_D$. Using the parametrization of (\[rg9\]) for the region $\infty>r>r_{\rm min}$, we find the following expression of the sum of self energies: \[rg14\] E\_[self]{}=2r\_[min]{}\_0\^dt t + 2(r\_[min]{} - r\_D ) + [O]{}(r\_[min]{}\^[-3]{}) . Then the finite potential between “quark” and “anti-quark” is given by \[rg15\] E\_[q|q]{}(L)&&E(L) - E\_[self]{} &=&r\_[min]{}(C + [kl\^2D 4r\_[min]{}\^2]{} + [O]{}(r\_[min]{}\^[-4]{})) &=&[lAC L]{} + [kl 4]{}([BC A\^2]{} + [D A]{} )L + [O]{}(L\^3)\
&=&[lAC L]{} - [l\^3 (2T)\^2 2]{} ([BC A\^2]{} + [D A]{} )L + [O]{}(L\^3) C&=&2\_0\^dt{ [\^2 t (\^2 t + 1)\^[1 2]{}]{} -t} -2 =-1.19814... D&=& 2\_0\^ =0.711959 .Here we neglected the $r_{\rm min}$ or $L$ independent term. We should note that next-to-leading term is linear in $L$, which might be relevant to the confinement. For the confinement, it is necessary that the quark-antiquark potential behaves as \[cnfpt\] E\_[q|q]{}\~a L with some positive constant $a$ for large $L$. For high temperature, it is usually expected that there occurs the phase transition to the deconfinement phase, where the potential behaves as Coulomb force, \[dcnfpt\] E\_[q|q]{} \~[a’ L]{} . Since ${BC \over A^2} + {D \over A}>0$ and $k<0$, the contribution from next-to-leading term in the potential is repulsive. The leading term expresses the repulsive but shows the Coulomb like behavior. The next-leading-term tells that the repulsive force is long-range than Coulomb force.
The expression (\[rg15\]) is correct even at high temperature if $L$ is small or $r_{\rm min}$ is large. If $r_{\rm min}$ is small and the orbit of string approaches to the horizon and/or enters inside the horizon, the expression would not be valid. Since the horizon is given by (\[xx\]), the expression (\[rg15\]) would be valid if \[vali\] r\_[min]{}r\_0=l or using (\[xxiv\]) and (\[rg12\]), \[valii\] LA=2A l T . The above condition (\[valii\]) makes difficult to evaluate the potential quantitively by the analytic calculation when $L$ is large and numerical calculation would be necesssary. In order to investigate the qualitive behavior of the potential when $L$ is large, we consider the background where the dilaton is constant $\phi=\phi_0$, which would tell the effect of the horizon or finite temperature. As $c=0$ when the dilaton is constant, we can use the solution in (\[xviii\]). Then by the calculation similar to (\[rg15\]) but without assuming $L$ is small or $r_{\rm min}$ is large, we obtain the following expression of the quark-antiquark potential: \[PotlL\] E\_[q|q]{}&=&r\_[min]{} \_[-]{}\^dt t { ( 1 - [1 \^2 t]{} )\^[-[1 2]{}]{} -1 } &&+ 2(r\_D - r\_[min]{}) . Constant $-1$ in $\{\ \}$ and the last term correspond to the subtraction of the self-energy. The integration in (\[PotlL\]) converges and the integrand is monotonically decreasing function of ${1 \over r_{\rm min}}$ if $r_{\rm min}$ is larger than the radius of the horizon $r_0$ : $r_{\rm min}>r_0$ and vanishes in the limit of $r_{\rm min}\rightarrow r_0$. Therefore if $r_{\rm min}$ decreases and approaches to $r_0$ when $L$ is large, which seems to be very natural, the potential $E_{q\bar q}$ approaches to a constant $E_{q\bar q}\rightarrow 2\left(r_D - r_0\right)$ and do not behaves as a linear function of $L$. This tells that the quark is not confined. This effect would corresponds to deconfining phase of QCD in the finite temperature.
We can also evaluate the potential between monopole and anti-monopole using the Nambu-Goto action for $D$-string instead of (\[rg5\]) (cf.ref.[@13]): \[rg5m\] S=[1 2]{}dd\^[-2]{} . For the static configuration $x^0=\tau$, $x^1\equiv x=\sigma$, $x^2=x^3=\cdots=x^{d-1}=0$ and $y=y(x)$, we find, instead of (\[rg7\]) \[rg7m\] S=[[T]{} 2]{}dx \^[-(r)]{} . Since $\phi$ is proportional to $c$ and $V$ and $U$ contain $c$ in the form of its square $c^2$, the potential between monopole and anti-monopole is given by changing $c$ by $-c$ in the potential between quark and anti-quark. Since the expression (\[rg15\]) does not contain $c$ in the given order, the potential $E_{m\bar m}(L)$ for monopole and anti-monopole is identical with that of quark and anti-quark in this order: \[rg8m\] E\_[m|m]{}(L)=E\_[q|q]{}(L) . Hence, we showed that non-constant dilaton deformation of IIB SG vacuum changes the structure of potential and confinement is becoming non-realistic.
Running coupling and quark-antiquark potential at finite temperature: non-zero mass BH case
===========================================================================================
In this section we consider another interesting case that $k=0$ and $\mu\neq 0$, which corresponds to the throat limit of D3-brane [@GKT; @GKP][^5] and $V_0$ has the following form: \[ki\] V\_0=[r\^4 l\^2]{} - . $\e^{2\rho}$ and $\e^{2\sigma}$ have the following form: \[kibb\] \^[2]{}=\^[-2]{}=[1 r\^2]{}( [r\^4 l\^2]{} - ) Therefore when $c=0$, the horizon is given by [@GKT] \[kib\] r=\^[1 4]{}l\^[1 2]{} and the black hole temperature is \[kii\] T=[\^[1 4]{} l\^[3 2]{}]{} . In a way similar to $k<0$ and $\mu=0$ case, we obtain \[xxxii\] &=&\_0 + [c 4]{}(1 - [1 q\^2]{}) u&=&-[12 \^2]{}{ [1 q\^2 - 1]{} + (1 - [1 q\^2]{}) + c\_1’} v&=&[1 12]{}{ -q\^2 (1 - [1 q\^2]{}) - 1 + [c\_1’ q\^2 2]{} + c\_2’} . Here \[xxxiii\] q and $c_1'$ and $c_2'$ are constants of the integration, which should vanish if we require $u$, $v\rightarrow 0$ when $r\rightarrow \infty$. The approximation when $r$ is far from horizon is again employed.
Using (\[kii\]) and (\[xxxii\]), we find the behaviour of the string coupling (\[ci\]) when $r$ is large and $c$ is small ($\phi_0$ is absorbed into the redefinition of $g_s$) : \[civ\] g=g\_s{1 + [cl\^2 4]{}(-[1 r\^4]{} - [(T)\^4 l\^8 r\^8]{} + [O]{} (r\^[-12]{})) + [O]{}(c\^2) } . The behavior of the second term is characteristic for $k=0$ case since the second term behaves as ${\cal O}\left(r^{-6}\right)$ for $k\neq 0$.[^6] Eq.(\[civ\]) gives the following beta-function \[cv\] (g)=r[dg r]{}=-4 (g - g\_s) + [8 (T )\^4 l\^6 g\_s c]{} (g - g\_s)\^2 + . The first term is universal one [@4; @7] but the behavior of the second temperature dependent term is characteristic for $k=0$.
We now consider the high temperature and $r\sim Tl^2$ case. For this purpose, we write the coupling as follows: \[hTi\] g=g\_s(1 - [l\^2r\^4]{})\^[c 4]{} . Here we used (\[xxxii\]). Eq.(\[hTi\]) can be solved with respect to ${1 \over r^4}$: \[hTii\] [l\^2r\^4]{}=1-( [g g\_s]{} )\^[4c]{} . On the other hand, Eq.(\[hTi\]) gives \[hTiii\] r[dg dr]{}=[g\_s c l\^2 r\^4]{}(1 - [1 r\^4]{} )\^[[c 4]{} -1]{} . Substituting (\[hTii\]) into (\[hTiii\]), we obtain the following expression: \[hTiv\] (g)=r[dg dr]{} =[gc ]{}( [g g\_s]{} )\^[-[4c]{}]{} ( 1 - ( [g g\_s]{} )\^[4c]{}) . Using (\[kii\]), we find the following expression of the temperature dependent beta-function: \[hTv\] (g)=[gc \^2 l\^6T\^4]{} ( [g g\_s]{} )\^[-[4\^2 l\^6T\^4 c]{}]{} ( 1 - ( [g g\_s]{} )\^[4\^2 l\^6T\^4 c]{}) . Since $T$ always appears in the combination of ${c \over T^4}$, the high temperature is consistent with the small $c$. As one can see $T$-dependence is quite complicated. It is qualitatively different from the case of low temperature.
In order to investigate the corrections to the position of the horizon and the temperature, we use ${1 \over r}$ expansion as in the previous section assuming the black hole is large. Then we find \[rii1\] \^&=&\^[\_0]{}(1 - [cl\^2 4 r\^4]{} + [O]{}(r\^[-8]{})) U&=&1 - [c\^2 l\^4 48 r\^8]{} + [O]{}(r\^[-12]{}) V&=&[r\^4 l\^2]{} - + [c\^2 l\^2 48 r\^4]{} + [O]{}(r\^[-8]{}) . Then corrections to the position of the horizon and the temperature are given as \[rii2\] r&=&r\_hl\^[1 2]{}\^[1 4]{}(1 - [c\^2 192 \^2]{}) T&=&[\^[1 4]{} l\^[3 2]{}]{}(1 - [5c\^2 192\^2]{}) .
The discussion of potential between quark and anti-quark for above case may be done similarly to the situation when $k<0$ and $\mu=0$. Instead of Eqs.(\[rg10\]), (\[rg11\]), (\[rg12\]), we obtain \[Org10\] [dx dt]{}&=& [l r\_[min]{}\^2t(\^2t + 1 )\^[1 2]{}]{}&& {1 - [l\^2 2r\_[min]{}\^4]{} [\^4 t -1 \^4 t]{} - [c l\^2 4r\_[min]{}\^4]{} [\^4 t + 1 \^4 t]{} + [O]{}(r\_[min]{}\^[-8]{}) }\
\[Org11\] L&=& [lA r\_[min]{}]{} - [l\^3 B\_1 2r\_[min]{}\^5]{} - [c l\^3 B\_2 4r\_[min]{}\^5]{} + [O]{}(r\_[min]{}\^[-9]{})\
B\_1&& \_[-]{}\^dt [\^2 t (\^2t + 1)\^[1 2]{} \^6t]{} = 0.479256... B\_2&& \_[-]{}\^dt [\^4t + 1 \^6t(\^2t + 1)\^[1 2]{}]{} = 1.91702...\
\[Org12\] r\_[min]{}&=&[lA L]{} - [B\_1 L\^3 2l A\^4]{} - [c B\_2 L\^3 4l A\^4]{} + [O]{}(L\^7) . Then we find the finite potential between “quark” and “anti-quark” is given by \[Org15\] E\_[q|q]{}(L) &=&r\_[min]{}{C + [l\^2 A r\_[min]{}\^4]{}( [2]{} - [5c 12]{}) + [O]{}(r\_[min]{}\^8)}\
&=&[lAC L]{} + [L\^3 lA\^3]{} {([A 2]{} - [C B\_1 2]{} ) + c (-[5A 12]{} - [C B\_2 8]{} )} + [O]{}(L\^7) &=&[lAC L]{} + [L\^3 lA\^3]{} {l\^6(T)\^4 ([A 2]{} - [C B\_1 2]{} ) + c (-[5A 12]{} - [C B\_2 8]{} )} && + [O]{}(L\^7) .Here we choose $\phi_0=0$ and neglect $r_{\rm min}$ or $L$ independent terms, again. The behavior of the potential is qualitatively identical with that in [@BISY] except $L^3$-term in potential (next-to-leading term) contains the contribution from dilaton. Since \[TTi\] [A 2]{} - [CB\_1 2]{} = 0.886178... , -[A 4]{} - [CB\_2 2]{} = 0.649204... , the $L^3$ potential becomes attractive if $l^6\left(\pi T\right)^4 > \gamma c$ (high temperature or small dilaton) and repulsive if $l^6\left(\pi T\right)^4 < \gamma c$ (low temperature or large dilaton). Here \[TTii\] = -0.732589... . Hence, we proved the possibility of confinement at finite temperature. The potential (\[Org15\]) is valid if $r_{\rm min}$ is much larger than the radius of the horizon: \[TTiii\] r\_[min]{}\^[1 4]{}l\^[1 2]{} or \[TTiv\] L=l T . The potential between monopole and anti-monopole is given by changing $c$ into $-c$ in (\[Org15\]): \[Org15m\] E\_[m|m]{}(L) &=&[lAC L]{} + [L\^3 lA\^3]{} {l\^6(T)\^4 ([A 2]{} - [C B\_1 2]{} ) - c (-[5 12]{} - [C B\_2 8]{} )} && + [O]{}(L\^7) .Therefore the $L^3$ potential becomes attractive if $l^6\left(\pi T\right)^4 > -\gamma c$ and repulsive if $l^6\left(\pi T\right)^4 < -\gamma c$. In other words, the behavior of monopole-antimonopole potential is reversed.
We now consider more general case where either $k$ or $\mu$ do not vanish. If we define $r_\pm^2$ by \[gi\] r\_\^2( -1 ) $V_0$ has the following form: \[gii\] V\_0=[1 l\^2]{}( r\^2 - r\_+\^2 ) ( r\^2 - r\_-\^2 ) . Since $\mu$ corresponds the black hole mass, $\mu$ should not be negative. If $\mu>0$, $r_+^2$ is positive and $r_-^2$ is negative when $k>0$ and $r_-^2$ is positive and $r_+^2$ is negative when $k<0$. Then $r=r_+$ corresponds to the horizon for $k>0$ in $c=0$ case and $r=r_-$ corresponds to the horizon for $k<0$. Therefore there is only one horizon when $\mu>0$. On the other hand, when $\mu<0$ although it might look unphysical, there are two horizons corresponding to $r=r_\pm$ when $k<0$.
When $c=0$, the temperature corresponding to the horizon at $r=r_\pm$ is given by \[giib\] T= .
When $c$ is small but does not vanish, the leading behavior of $\phi$ is given by, \[giii\] &=& \_0 + [cl\^2 2]{}{ - [1 r\_+\^2 r\_-\^2]{}r\^2 . && . + [1 r\_+\^2 (r\_+\^2 - r\_-\^2 )]{} (r\^2 - r\_+\^2 ) - [1 r\_-\^2 (r\_+\^2 - r\_-\^2 )]{} (r\^2 - r\_-\^2 )} . Then the behavior of the string coupling (\[ci\]) when $r$ is large and $c$ is small ($\phi_0$ is absorbed into the redefinition of $g_s$) : \[cvi\] g=g\_s{1 + [cl\^2 2]{}(-[1 2r\^4]{} - [r\_+\^2 + r\_-\^2 3r\^6]{} + [O]{}(r\^[-8]{})) + [O]{}(c\^2)} , and the beta-function is given by \[cvii\] (g)=r[dg dr]{}=-4(g-g\_s) + [8 3]{}(r\_+\^2 + r\_-\^2)[cg\_s l]{} ([g-g\_s cg\_s]{})\^[3 2]{} . Note that the second term vanishes when $k=0$, which is the reason why the behavior of the next-to-leading term in $k=0$ is different from that in $k\neq 0$. Eq.(\[giib\]) gives the temperature dependence in the coupling (\[cvi\]) and the beta-function (\[cvii\]). The next-to-leading term shows again power-like behavior on $g$ as it happened already in IIB SG solutions of refs.[@3; @4; @6; @7; @8; @9; @NO] (no temperature) and in GUTs with large internal dimensions [@12]. Note that two of the parameters $k$, $\mu$ and $T$ are independent. If we consider the high temperature regime by fixing $k$, $\mu$ becomes large and the behavior approaches to $k=0$ case in (\[hTiv\]). On the other hand, if we consider the high temperature regime by fixing $\mu$, $k$ is positive and becomes large and the behavior approaches to $k<0$ and $\mu=0$ case in (\[gTii\]). One can also find quark-antiquark potential which looks very complicated so we do not write it explicitly.
The corrections to $U$ and $V$ coming from the non-trivial dilaton are given by \[giv\] u&=&c\_1” + [l\^4 (r\_+\^2 - r\_-\^2 )]{} { ([1 r\_+\^2]{} - [1 r\_-\^2]{} )\^2 r\^2 . && - [1 r\_+\^2]{}[1 r\^2 - r\_+\^2]{} - [3r\_+\^2 - r\_-\^2 r\_+\^4 (r\_+\^2 - r\_-\^2 )]{} ( r\^2 - r\_+\^2 ) && . - [1 r\_-\^2]{}[1 r\^2 - r\_-\^2]{} + [3r\_-\^2 - r\_+\^2 r\_-\^4 (r\_+\^2 - r\_-\^2 )]{} ( r\^2 - r\_-\^2 ) }\
\[gv\] v&=&[1 l\^2]{} . Here $c_1''$ and $c_2''$ are constants of the integration, which should vanish if we require $u$, $v\rightarrow 0$ when $r\rightarrow \infty$.
Thermodynamics of approximate AdS backgrounds of IIB supergravity
=================================================================
In the present section we will be interesting in the thermodynamical quantities like free energy. After Wick-rotating the time variables by $t\rightarrow i\tau$, the free energy $F$ can be obtained from the action $S$ in (\[i\]) where the classical solution is substituted: \[F1\] F=[1 T]{}S . Multiplying $G^{\mu\nu}$ with the equation of motion in (\[iit\]), we find \[F2\] R - [1 2]{} G\^\_\_ =[5 3]{} . Here we only consider the case of $d=4$ and $\alpha={1 \over 2}$. Substituting (\[F2\]) into (\[i\]), we find after the Wick-rotation \[F3\] S=[1 2G l\^2]{}d\^5 x . Here we used (\[xx\]). From the expressions of the metric $G_{\mu\nu}$ in (\[ii\]) and (\[x\]), Eq.(\[F3\]) is rewritten as follows: \[F4\] S=[1 2G l\^2]{}[V\_3 T]{}\_[r\_h]{}\^ dr r\^3 U . Here $V_3$ is the volume of 3d Einstein manifold and $r_h$ is the radius of the horizon and we assume $\tau$ has a period of ${1 \over T}$. Since $U$ has a singularity at $r=r_h$ in the expansion with respect to $c$, we use ${1 \over r}$ expansion. Furthermore the expression of $S$ contains the divergence coming from large $r$. In order to subtract the divergence, we regularize $S$ in (\[F4\]) by cutting off the integral at a large radius $r_{\rm max}$. After that we subtract the divergent part. In case of $k<0$ and $\mu=0$, we subtract it by using the extremal solution with $c=0$ ($U=1$): \[F4b\] S\_[reg]{}=[1 2G l\^2]{}[V\_3 T]{}( \_[r\_h]{}\^[r\_[max]{}]{} dr r\^3 U - \_[r\^[ex]{}\_h]{}\^[r\_[max]{}]{} dr r\^3 ) . Here \[F4c\] V\^[ex]{}(r\^2 - [r\^[ex]{}\_h]{}\^2 )\^2 ,r\^[ex]{}\_h=[l 2]{} , which corresponds to the extremal solution (the solution has negative mass parameter $\mu=-{k^2 l^2 \over 4}$). The factor $\sqrt{V(r=r_{\rm max})
\over V^{\rm ex}(r=r_{\rm max}) }$ is chosen so that the proper length of the circle which corresponds to the period ${1 \over T}$ in the Euclidean time at $r=r_{max}$ coincides with each other in the two solutions with $\mu=0$ and $k<0$ one and extremal one in (\[F4c\]). Then we obtain \[F5\] F=[V\_3 2G l\^2]{}(-[5k\^2 l\^4 128]{} + [c\^2 48 k\^2]{}) . With the help of (\[r5\]), we find the following expression \[F6\] F=-[V\_3 2G l\^2]{}( [ 5l\^8(2T)\^4 32]{} + [c\^2 768 l\^4 (2T)\^4]{}) . In order to get the entropy ${\cal S}$, we need to know $T$ dependence of $V_3$ although $V_3$ is infinite. Since $k$ is proportional the curvature, $V_3$ would be proportional to $k^{-{3 \over 2}}$. Then we find \[F7\] [dV\_3 dT]{}&=&[1 k]{}[dk dT]{}k[dV\_3 dk]{} &=&-[3V\_3 2T]{}(1 - [c\^2 6l\^[12]{} (2T)\^8]{} + ) . Therefore the entropy ${\cal S}$ and mass (energy) $E$ are given by \[F8\] [S]{}&=&-[dF dT]{}=[V\_3 2G l\^2 T ]{}( [ 25l\^8(2T)\^4 64]{} + [49c\^2 1536 l\^4 (2T)\^4]{}) E&=&F+T[S]{}=[V\_3 2G l\^2]{}( [ 15 l\^8(2T)\^4 64]{} + [47 c\^2 1536 l\^4 (2T)\^4]{}) . In terms of string theory correspondence[@GKT], the parameters $G$ and $l$ are given by \[spara\] l\^4&=&2g\_[YM]{}\^2 N[’]{}\^2 Gl&=&[g\_[YM]{}\^2[’]{}\^2 N]{} . Here the Yang-Mills coupling $g_{YM}$ is given by the string coupling $g_s$: $g_{YM}^2=2\pi g_s$ and $N$ is the number of the coincident D3-branes. As $V_3$ is now dimensionless, we multiply $l^3$ with $V_3$: \[F10\] V\_3l\^3 V\_3 . Then Eqs.(\[F6\]) and (\[F8\]) can be rewritten as follows: \[F10b\] F&=&-[V\_3 2\^2 ]{}( [ 5N\^2 (2T)\^4 16]{} + [c\^2 3072 l\^4 g\_[YM]{}\^6 N[’]{}\^6 (2T)\^4]{})&=&[V\_3 2\^2 T ]{}( [ 25N\^2(2T)\^4 32]{} + [49 c\^2 6144 g\_[YM]{}\^6 N[’]{}\^6 (2T)\^4]{}) E&=&[V\_3 2G l\^2]{}( [ 15 N\^2(2T)\^4 32]{} + [47 c\^2 6144 g\_[YM]{}\^6 N[’]{}\^6 (2T)\^4]{}) .
For $k=0$ and $\mu>0$ case, we can obtain the thermodynamical quantities in a similar way using Eqs. (\[rii1\]) and (\[rii2\]) in ${1 \over r}$ expansion. We regularize $S$ in (\[F4\]) by subtracting the solution with $\mu=0$ and $c=0$ ($U=1$): \[F4bb\] S\_[reg]{}=[1 2G l\^2]{}[V\_3 T]{}( \_[r\_h]{}\^[r\_[max]{}]{} dr r\^3 U - \_0\^[r\_[max]{}]{} dr r\^3 ) . We can assume here that $V_3$ does not depend on $T$ since $k$ is fixed to vanish. Then we obtain for the case \[F9\] F&=&-[V\_3 4G l\^2]{}( [ l\^8(T)\^4 4]{} + [5c\^2 192 l\^4 (T)\^4]{}) &=&[V\_3 4G l\^2 T]{}( l\^8(T)\^4 - [5c\^2 48 l\^4 (T)\^4]{}) E&=&[V\_3 4G l\^2]{}( [ 3l\^8(T)\^4 4]{} - [25c\^2 192 l\^4 (T)\^4]{}) . Note that the leading term in ${\cal S}$ is the volume of 3d manifold at horizon (${V_3 r_h^3 \over l^3}$) divided by $4G$. Then by using (\[spara\]) and (\[F10\]), we find \[F11\] F&=&-[V\_3 4\^2]{}( [ N\^2 (T)\^4 2]{} + [5c\^2 768 g\_[YM]{}\^6 N[’]{}\^6 (T)\^4]{}) &=&[V\_3 4\^2 T]{}( 2 N\^2 (T)\^4 - [5c\^2 192 g\_[YM]{}\^6 N[’]{}\^6 (T)\^4]{}) E&=&[V\_3 4\^2 ]{}( [ 3 N\^2 (T)\^4 2]{} - [25c\^2 768 g\_[YM]{}\^6 N[’]{}\^6 (T)\^4]{}) . The leading behaviours of $F$ and $S$ are consistent with [@GKT]. As we used ${1 \over r}$ expansion, the second terms in (\[F11\]) become dominant when the radius of horizon $r_h$ is large and the parameter $c$ specifying non-trivial dilaton is not very small. Notice that in other temperature regimes (or using another schemes for approximated solutions of gravitational equations) one can get also qualitatively different thermodynamical next-to-leading terms (on temperature). One has to remark that leading term in above free energy describes the strong coupling regime free energy for ${\cal N}=4$ super YM theory with the usual mismatch factor 3/4 if compare with perturbative free energy (for a detailed discussion of this case, see[@GKT]).
Discussion
==========
We studied the approximate (dilaton perturbed) solutions of IIB SG near BH-like ${\rm AdS}_5\times{\rm S}_5$ background. Thanks to presence of dilaton, the running gauge coupling of non-SUSY gauge theory at finite temperature may be extracted from these solutions. It is interesting that corresponding strong coupling regime beta-function may depend on the temperature in the complicated way (mainly, power-like behavior). We also estimated quark-antiquark (and monopole-antimonopole) potential at finite temperature from SG side. Its comparison with the potential of ${\cal N}=4$ super YM theory at finite temperature is also done. It is remarkable that confinement depending on features of geometry and dilaton may occur.
As our IIB SG solutions are approximate (actually large radius expansion) it is clear that one is able to develop other schemes to search for similar solutions. Unfortunately, it is not yet clear how to identify explicitly boundary non-SUSY thermal gauge theory corresponding to these solutions. One possibility is to calculate free energy from SG side (with non-trivial dilaton) and compare it with free energy of perturbative thermal gauge theories with running gauge coupling. The last quantity is available in QFT.
Note also that one can generalize our solutions via adding RR-scalar(axion) to bosonic sector of IIB SG as it was discussed in refs.[@14]. As it follows from results of refs.[@7; @NO] the structure of running gauge coupling changes drastically in this case. In particular, part of supersymmetries may be unbroken [@14] but asymptotic freedom may be realized in strong coupling phase [@7; @NO]. We expect that at finite temperature this property may survive.
[**Acknoweledgements**]{} We thank J. Ambjørn for the interest to this work.
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[^1]: e-mail: [email protected]
[^2]: e-mail: [email protected]
[^3]: These solutions presumably describe thermal states of non-SUSY gauge theory which descends from ${\cal N}=4$ super YM after breaking of SUSY and conformal invariance.
[^4]: The conventions of curvatures are given by $$\begin{aligned}
R&=&G^{\mu\nu}R_{\mu\nu} \\
R_{\mu\nu}&=& -\Gamma^\lambda_{\mu\lambda,\kappa}
+ \Gamma^\lambda_{\mu\kappa,\lambda}
- \Gamma^\eta_{\mu\lambda}\Gamma^\lambda_{\kappa\eta}
+ \Gamma^\eta_{\mu\kappa}\Gamma^\lambda_{\lambda\eta} \\
\Gamma^\eta_{\mu\lambda}&=&{1 \over 2}G^{\eta\nu}\left(
G_{\mu\nu,\lambda} + G_{\lambda\nu,\mu} - G_{\mu\lambda,\nu}
\right)\ .\end{aligned}$$
[^5]: In ref.[@GKT] $\alpha'$-corrections to leading term ($T^4$) of free energy for above AdS BH have been derived. The temperature was actually fixed. In the case under discussion we consider dilatonic deformation of such AdS BH using tree level bosonic sector of IIB SG. Thus, we define the corrections (next-to-the leading term on the temperature) to solution (and free energy).
[^6]: The case that the boundary is the Einstein manifold with $k\neq 0$ has been discussed in [@NO].
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'Frédéric Menous[^1] and Frédéric Patras[^2]'
title: 'Renormalization: a quasi-shuffle approach.'
---
Introduction. {#introduction. .unnumbered}
=============
In the early 2000s, the usual BPHZ algorithm for renormalization in perturbative quantum field theory has been interpreted, after dimensional regularization, as a Birkhoff decomposition of characters on the Hopf algebra of Feynman graphs, with values in a Rota-Baxter algebra of amplitudes [@ck1; @ck2; @guo]. This idea was later shown to be meaningful in a broad variety of contexts: in the theory of dynamical systems, in analysis and numerical analysis (Rayleigh-Schrödinger series) or, more recently, in the theory of regularity structures and the study of very irregular stochastic differential equations or stochastic partial differential equations, see e.g. [@men; @men06; @men13; @hai; @hz]
In this context, P. Cartier suggested the existence of a hidden universal symmetry group (the “cosmical Galois group”) that would underlie renormalization. Using geometrical tools such as universal singular frames, Connes and Marcolli constructed a candidate group in 2004 [@CM]. Their construction was translated in the langage of Hopf algebras in [@KGP] and the group shown to coincide with the prounipotent group of group-like elements in the completion with respect to the grading of the descent algebra -a Hopf algebra that, as an algebra, is the free associative algebra generated by the Dynkin operators [@Pat].
However, the action of this group or of the descent algebra on the Hopf algebras of Feynman diagrams showing up in pQFT does not actually perform renormalization. It captures nicely certain phenomena related to Lie theory and the behaviour of the Dynkin operators: for example, the structure of certain renormalization group equations and the algebraic properties of beta functions (see the original article by Connes and Marcolli [@CM] and the detailed algebraic and combinatorial analysis of these phenomena in [@patrasC]. Further insights on the role of (generalized) Dynkin operators in the theory of differential equations can be found in [@MP]). However, the group and the descent algebra act on Feynman diagrams and do not encode operations that occur at the level of the target algebra of amplitudes. They fail therefore to capture typical renormalization operations such as projections on divergent or regular components of amplitudes. Substraction maps, for example, cannot be encoded in it, and neither are more advanced operations such as the construction of the counterterm.
In the present article, we follow a different approach that complements Connes-Marcolli’s and its Hopf algebraic and combinatorial interpretation by showing show how a semigroup of operators can be associated to the algebra of coefficients of a given regularization and renormalization scheme in pQFT. Its construction relies heavily on the universal properties of commutative and noncommutative quasi-shuffle algebras. This semigroup acts in a natural way on regularized amplitudes and perform the expected operations: preparation map, extraction of counterterms, renormalization. Notice that many of our results and constructions do not require the algebra of coefficients to be commutative.
Let us sketch up the ideas and results. Concretely we deal with conilpotent bialgebras $H = k \oplus H^+$. These bialgebras are Hopf algebras and the coalgebra structure on $H$ induces a convolution product on the space $\mathcal{L}( H, A )$ of linear morphisms from $H$ to an associative algebra $A$. If $A$ is unital, then the subset $\mathcal{U}( H, A )$ of linear morphisms that send the unit $1_H$ of $H$ on the unit $1_A$ of $A$ is a group for the convolution and, if A is commutative, the subset $\mathcal{C}( H, A )$ of characters (i.e. algebra morphisms) is a subgroup of $\mathcal{U}( H, A )$.
In pQFT, the algebra $A$ is often called the algebra of (regularized) amplitudes, and we will often use this terminology. In this context, the renormalization process equips the target unital algebra $A$ with a projection operator $p_+$ such that $$A = {\ensuremath{\operatorname{Im}}} p_+ \oplus {\ensuremath{\operatorname{Im}}} p_- = A_+ \oplus A_-,$$ where $p_- =
{\ensuremath{\operatorname{Id}}} - p_+$ and $A_+$ and $A_-$ are subalgebras. Here, $p_-$ should be thought of as a projection on the “divergent part”, so that $p_+$ substract divergences. For example, in dimensional regularization, $A$ identifies with the algebra of Laurent series, $\CC[[\varepsilon,\varepsilon^{-1}]$, and $p_-$ (resp. $p_+$) is the projection on $\varepsilon^{-1} \CC[\varepsilon^{-1}]$ (resp. $\CC[[\varepsilon ]]$). As was first observed by Ebrahimi-Fard, building on previous results by Brouder and Kreimer, these data define a Rota-Baxter algebra structure on $A$ and $\mathcal{L}( H, A )$.
The choice of the subtraction operator is not always unique –for example when using momentum subtraction schemes. How this phenomenon impacts the combinatorics and Rota-Baxter structures was investigated in [@EFP2]. Although we do not investigate it further here, the tools we develop in the present article should be useful in that context since they put forward the idea that one should study for its own the combinatorial structure of the target algebra of amplitudes $A$, independently of the choice of a particular subtraction map $p_+$.
It is then well-know that, given $p_+$, there exists a unique Birkhoff decomposition of any morphism $\varphi \in
\mathcal{U}( H, A )$ $$\varphi_- \ast \varphi = \varphi_+ \hspace{2em} \varphi_+, \varphi_- \in
\mathcal{U}( H, A )$$ where $\varphi_+ ( H^+ ) \subset A_+$ and $\varphi_- ( H^+ ) \subset A_-$. Moreover, if $A$ is commutative, this decomposition is defined in the subgroup $\mathcal{C}( H, A )$. The classical proofs of this result are recursive, using the filtration on $H$ (they rely ultimately on the Bogoliubov recursion [@emp]).
We propose to develop here a “universal” framework to handle the combinatorics of renormalization and to give in this framework explicit, and in some sense universal, formulas for $\varphi_+$ and $\varphi_-$. To do so, we consider the quasi-shuffle Hopf algebra $QSh(A)$ over an algebra $A$, that is, the standard tensor coalgebra over $A$ equipped with the quasi-shuffle (or stuffle) product. Using the properties of the functor $QSh$ (including the surprising property, for any Hopf algebra $H$ to be canonically embedded into $QSh(H^+)$), we compute then the inverse and the Birkhoff decomposition of a fundamental element $j \in \mathcal{U}( QSh(A), A )$ defined by $$j ( 1 ) = 1_A, \hspace{1em} j ( a_1 ) = a_1, \hspace{1em} j ( a_1
\otimes \ldots \otimes a_s ) = 0 \hspace{1em} {\ensuremath{\operatorname{if}}} \hspace{1em} s \geq
2 .$$ We show then the existence of an action of $\mathcal{U}(
QSh(A), A )$ on $\mathcal{U}( H, A )$. More precisely we define a map $$\mathcal{U}( QSh(A), A ) \times \mathcal{U}( H, A ) \rightarrow
\mathcal{U}( H, A )$$ $$(f,\varphi)\mapsto f\odot\varphi,$$ such that $$j\odot \varphi = \varphi \hspace{1em} {\ensuremath{\operatorname{and}}} \hspace{1em} ( f \ast
g)\odot \varphi = ( f\odot \varphi ) \ast ( g\odot \varphi ),$$ and obtain explicit formulas such as:
1. If $j^{\ast - 1}$ is the inverse of $j$, then $\varphi^{\ast - 1} =
j^{\ast - 1}\odot \varphi $.
2. If $j_- \ast j = j_+$ (Birkhoff decomposition), then $\varphi_- \ast \varphi
= \varphi_+$ where $\varphi_{\pm} = j_{\pm}\odot \varphi$.
The article is organized as follows. After a preliminary section fixing notations and recalling general properties of Hopf algebras, section \[sect:1\] analyses the algebraic properties of algebras of regularized amplitudes and explains how they give rise to quasi-shuffle algebra structures. Section \[sect:2\] introduces Hoffman’s quasi-shuffle functor (i.e. the notion of quasi-shuffle algebra over an algebra -in the commutative case, it is the left adjoint to the forgetful functor from quasi-shuffle algebras to commutative algebras). Section \[sect:3\] investigates its categorical properties, including a surprising right adjoint property (Thm. \[mainthm\]). Section \[sect:4\] studies, using these techniques, the map $j$ (mapping a cofree coalgebra to its cogenerating vector space). This is the key to latter applications to renormalization which are the purpose of Section \[sect:5\], as well as the construction, for each algebra of amplitudes, of a “universal semigroup” in which the operations characteristic of renormalization are encoded. The last two sections survey various applications, in particular to Dynamics and Analysis.
We acknowledge support from the CARMA grant ANR-12-BS01-0017, “Combinatoire Algébrique, Résurgence, Moules et Applications”and the CNRS GDR “Renormalisation”. We thank warmly K. Ebrahimi-Fard, from whom we learned some years ago already the meaningfulness of Rota–Baxter algebras and their links with quasi–shuffle algebras.
Notation and Hopf algebra fundamentals {#section:0}
======================================
Everywhere in the article, algebraic structures are defined over a fixed ground field $k$ of characteristic $0$. We fix here the notations relative to bialgebras and Hopf algebras, following [[@fig]]{} (see also [@Cartier2], [[@maj]]{} and [[@sweed]]{}) and refer to these articles and surveys for details and generalities on the subject. Recall that a bialgebra $B$ is an associative algebra with unit and a coassociative coalgebra with counit such that the product is a morphism of coalgebras (or, equivalently, the coproduct is a morphism of algebras). We will usually write $m$ the product, $\Delta$ the coproduct, $u:k\to B$ the unit and $\eta:B\to k$ the counit. When ambiguities might arise we put an index (and denote e.g. $m_B$ the product instead of $m$).
We use freely the Sweedler notation and write $$\Delta h = \sum h_{( 1 )} \otimes h_{( 2 )}.$$ Thanks to coassociativity, we can define recursively and without any ambiguity the linear morphisms $\Delta^{[ n ]} : B \rightarrow B^{\otimes n}$ ($n \geq 1$) by $\Delta^{[ 1 ]} = {\ensuremath{\operatorname{Id}}}$ and, for $n \geq 1$, $$\Delta^{[ n + 1 ]} = ( {\ensuremath{\operatorname{Id}}} \otimes \Delta^{[ n ]} ) \circ \Delta = (
\Delta^{[ n ]} \otimes {\ensuremath{\operatorname{Id}}} ) \circ \Delta = ( \Delta^{[ k ]} \otimes
\Delta^{[ n + 1 - k ]} ) \circ \Delta \hspace{1em} ( 1 \leq k \leq n )$$ and write $$\Delta^{[ n ]} h = \sum h_{( 1 )} \otimes \ldots \otimes h_{( n )}$$ In the same way, for $n \geq 1$, we define $m^{[ n ]} : B^{\otimes^n}
\rightarrow B$ by $m^{[ 1 ]} = {\ensuremath{\operatorname{Id}}}$ and $$m^{[ n + 1 ]} = m \circ ( {\ensuremath{\operatorname{Id}}} \otimes m^{[ n ]} ) = m \circ ( m^{[ n
]} \otimes {\ensuremath{\operatorname{Id}}} )$$
The reduced coproduct $\Delta'$ on $H^+:=Ker \ \eta$ is defined by $$\Delta' h = \Delta h - 1 \otimes h - h \otimes 1$$ Its iterates (defined as for $\Delta$) are written $\Delta'^{[ n ]}$. A bialgebra is conilpotent (or, more precisely, locally conilpotent) is for any $h\in H^+$ there exists a $n\geq 1$ (depending on $h$) such that $\Delta'^{[ n ]}(h)=0.$
A bialgebra $H$ is a Hopf algebra if there exists an antipode $S$, that is to say a linear map $S : H \rightarrow H$ such that : $$m \circ ( {\ensuremath{\operatorname{Id}}} \otimes S ) \circ \Delta = m \circ ( S \otimes {\ensuremath{\operatorname{Id}}}
) \circ \Delta = u \circ \eta : H \rightarrow H$$ In this article, we will consider only conilpotent bialgebras, which are automatically Hopf algebras.
Given a connected bialgebra $H$ and an algebra $ A$ with product $ m_A$ and unit $ u_A$, the coalgebra structure of $H$ induces an associative convolution product on the vector space $\mathcal{L}( H, A )$ of $k$–linear maps : $$\forall ( f, g ) \in \mathcal{L}( H, A ) \times \mathcal{L}( H, A ),
\hspace{1em} f \ast g = m_A \circ ( f \otimes g ) \circ \Delta$$ with a unit given by $u_A \circ \eta$, such that $(\mathcal{L}( H, A ), \ast,
u_A \circ \eta )$ is an associative unital algebra.
Let $H$ be a conilpotent bialgebra (and therefore a Hopf algebra) and set $$\mathcal{U}( H, A ) = \{ f \in \mathcal{L}( H, A ) \hspace{1em} ;
\hspace{1em} f ( 1_H ) = 1_A \}$$ then $\mathcal{U}( H, A )$ is a group for the convolution product.
$\mathcal{U}( H, A )$ is obviously stable for the convolution product. Following [[@fig]]{}, we will remind why any element $f \in \mathcal{U}(
H, A )$ as a unique inverse $f^{\ast - 1}$ in $\mathcal{U}( H, A )$. One can write formally $$f^{\ast - 1} = ( u_A \circ \eta - ( u_A \circ \eta - f ) )^{\ast - 1} =
u_A \circ \eta + \sum_{k \geq 1} ( u_A \circ \eta - f )^{\ast k}$$ This series seems to be infinite but, because of the conilpotency assumption, for any $h \in H'$ $$( u_A \circ \eta - f )^{\ast k} ( h )=( - 1 )^k m_A^{[ k ]} \circ f^{\otimes k} \circ
\Delta'^{[ k ]} ( h ) \label{eq15}$$ vanishes for $k$ large enough.
When this result is applied to ${\ensuremath{\operatorname{Id}}} : H \rightarrow H \in
\text{$\mathcal{U}( H, H )$}$, then its convolution inverse is the antipode $S$ (this is the usual way of proving that any conilpotent bialgebra is a Hopf algebra).
If $B \subset A$ is a subalgebra of $A$ which is not unital, then we write $$\mathcal{U}( H, B ) = \{ f \in \mathcal{L}( H, A ) \hspace{1em} ;
\hspace{1em} f ( 1_H ) = 1_A \hspace{1em} {\ensuremath{\operatorname{and}}} \hspace{1em} f ( H^+
) \subset B \}$$ This is a subgroup of $\mathcal{U}( H, A )$.
Let now $\mathcal{C}( H, A )$ be the subset of $\mathcal{L}( H, A )$ whose elements are algebra morphisms (also called characters over $A$). Of course, $$\mathcal{C}( H, A ) \subset \text{$\mathcal{U}( H, A )$}$$ but this shall not be a subgroup: if $A$ is not commutative, there is no reason why it should be stable for the convolution product. Nonetheless if $A$ is commutative, the product from $A\otimes A$ to $A$ is an algebra map: it follows that the convolution of algebra morphisms is an algebra morphism and $\mathcal{C}( H, A )$ is a subgroup of $\text{$\mathcal{U}( H, A )$}$.
Moreover if $f
\in \text{$\mathcal{U}( H, A )$}$ is an algebra map, then its inverse $f^{\ast
- 1}$ in $\mathcal{U}( H, A )$ is an antialgebra map given by $f^{\ast - 1} = f \circ S$ : $$\begin{array}{ccc}
f \ast f \circ S & = & m_A \circ ( f \otimes f \circ S ) \circ \Delta\\
& = & m_A \circ ( f \otimes f ) \circ ( {\ensuremath{\operatorname{Id}}} \otimes S ) \circ
\Delta\\
& = & f \circ m \circ ( {\ensuremath{\operatorname{Id}}} \otimes S ) \circ \Delta\\
& = & f \circ u \circ \eta\\
& = & u_A \circ \eta ,
\end{array}$$ where we recall that the antipode is an antialgebra morphism: $$S(gh)=S(h)S(g).$$
From renormalization to quasi-shuffle algebras {#sect:1}
==============================================
The fundamental ideas of renormalization in pQFT were already alluded at in the introduction, we recall them very briefly and refer to textbooks for details (this first paragraph is mainly motivational, we will move immediately after to an algebraic framework that can be understood without mastering the quantum field theoretical background). Starting from a given quantum field theory, one expands perturbatively the quantities of interest (such as Green’s functions). This expansion is indexed by Feynman diagrams, and to each of these diagrams is associated a quantity computed by means of certain integrals. Very often, these integrals are divergent and need to be regularized and renormalized. Typically, a quantity such as $$\phi(c):=\int_{0}^{\infty} \frac{dy}{y+c}$$ is divergent, but becomes convergent up to the introduction of an arbitrary small regularizing parameter $\varepsilon$ (for dimensional reasons, one also introduces a mass term $\mu$) $$\phi(c;\varepsilon):=\int_{0}^{\infty} \frac{\mu^{\varepsilon}d y}{(y+c)^{1+\varepsilon}}
=\frac{1}{\varepsilon} + \log(\mu/c) +O(\varepsilon).$$ In that toy model case, close to the dimensional regularization method, the “regularized amplitude” $\phi(c;\varepsilon)$ lives in $A=\CC[[\varepsilon,\varepsilon^{-1} ]$ and is renormalized by removing the divergency $\frac{1}{\varepsilon}$ (the component of the expansion in $\varepsilon^{-1}\CC[\varepsilon^{-1}]$).
These ideas are axiomatized using the notion of Rota–Baxter algebras as follows. Following [[@kur]]{}, let $p_+$ an idempotent of $\mathcal{L}( A, A )$ where $A$ is a unital algebra (in our toy model example, $p_+$ would stand for the projection on $\CC[[\varepsilon ]]$). If we have for $x, y$ in $A$ : $$p_+ ( x ) p_+ ( y ) + p_+ ( x y ) = p_+ ( x p_+ ( y ) ) + p_+ ( p_+ ( x ) y
) ),$$ then $p_+$ is a Rota-Baxter operator, $( A, p_+ )$ is a Rota-Baxter algebra and if $p_- = {\ensuremath{\operatorname{Id}}} - p_+$, $A_+ = {\ensuremath{\operatorname{Im}}} p_+$ and $A_- = {\ensuremath{\operatorname{Im}}}
p_-$ then
- $A = A_+ \oplus A_-$.
- $p_-$ satisfies the same relation.
- $A_+$ and $A_-$ are subalgebras.
Conversely if $A = A_+ \oplus A_-$ and $A_+$ and $A_-$ are subalgebras, then the projection $p_+$ on $A_+$ parallel to $A_-$ defines a Rota-Baxter algebra $( A, p_+ )$.
The idempotency condition is not required to define a Rota–Baxter algebra. In general:
A Rota–Baxter (RB) algebra is an associative algebra $A$ equipped with a linear endomorphism $R$ such that $$\forall x,y\in A, R(x)R(y)=R(R(x)y+xR(y)- xy).$$ It is an idempotent RB algebra if $R$ is idempotent (in that case we will set $p_+:=R$ to emphasize that we are in the framework typical for renormalization). It is a commutative Rota–Baxter algebra if it is commutative as an algebra.
The notion of Rota–Baxter algebra is actually slightly more general: a Rota–Baxter algebra of weight $\theta$ is defined by the identity $$\forall x,y\in A, R(x)R(y)=R(R(x)y+xR(y)+\theta xy).$$ We restrict here the definition to the weight $-1$ case, which is the one meaningful for renormalization.
Using Rota–Baxter algebras of amplitudes, the principle of renormalization in physics can be formulated algebraically in the following way.
Let $H$ be a conilpotent bialgebra and $( A, p_+ )$ an idempotent Rota-Baxter algebra (so that $A=A_-\oplus A_+$). Then for any $\varphi \in \mathcal{U}( H, A )$ there exists a unique pair $(
\varphi_+, \varphi_- ) \in \mathcal{U}( H, A_+ ) \times \mathcal{U}( H, A_-
)$ such that $$\varphi_- \ast \varphi = \varphi_+$$ Moreover, if $A$ is commutative and $\varphi$ is a character, then $\varphi_+$ and $\varphi_-$ are also characters. This factorization is called the Birkhoff decomposition of $\varphi$.
Let us postpone the assertion on characters and prove the existence and unicity -notions such as the one of Bogoliubov’s preparation map will be useful later. As $A_+$ and $A_-$ are subalgebras of $A$, $\mathcal{U}( H, A_+ )$ and $\mathcal{U}( H, A_- )$ are subgroups of $\mathcal{U}( H, A )$. If such a factorization exists, then it is unique : If $\varphi =
\varphi_-^{\ast - 1} \ast \varphi_+ = \psi_-^{\ast - 1} \ast \psi_+$, then $$\phi = \psi_+^{} \ast \varphi^{\ast - 1}_+ = \psi_- \ast \varphi_-^{\ast
- 1} \in \mathcal{U}( H, A_+ ) \cap \mathcal{U}( H, A_- )$$ thus for $h \in H^+$, $\phi ( h ) \in A_+ \cap A_- = 0$. We finally get that $$\psi_+^{} \ast \varphi^{\ast - 1}_+ = \psi_- \ast \varphi_-^{\ast - 1} =
u_A \circ \eta$$ and $\varphi_+ = \psi_+$, $\varphi_- = \psi_-$.
Let us prove now that the factorization exists. Let $\varphi \in
\mathcal{U}( H, A )$, we must have $\varphi_+ ( 1_H ) = \varphi_- ( 1_H ) =
1_A$. Let $\bar{\varphi} \in \mathcal{U}( H, A )$ the Bogoliubov preparation map defined recursively on the increasing sequence of vector spaces $H^+_n:=Ker {\Delta'}^{[n]}$ ($n \geq 1$) by $$\label{eq:Bog}
\bar{\varphi} ( h ) = \varphi ( h ) - m_A \circ ( p_- \otimes {\ensuremath{\operatorname{Id}}}
) \circ ( \bar{\varphi} \otimes \varphi ) \circ \Delta' ( h )$$ (since $H$ is conilpotent, $H^+=\cup_nH^+_n$). Now if $\varphi_+$ and $\varphi_-$ are the elements of $\mathcal{U}( H, A )$ defined on $H^+$ by $$\varphi_+ ( h ) = p_+ \circ \bar{\varphi} ( h ) \hspace{1em},
\hspace{1em} \varphi_- ( h ) = - p_- \circ \bar{\varphi} ( h )
\hspace{1em} ( \bar{\varphi} ( h ) = \varphi_+ ( h ) - \varphi_- ( h ) ),$$ then $$\text{} \varphi_+ \in \mathcal{U}( H, A_+ ) \hspace{1em}, \hspace{1em}
\varphi_- \in \text{$\mathcal{U}( H, A_- )$} \hspace{1em}, \hspace{1em}
\varphi_- \ast \varphi_{} = \varphi_+$$
We turn now to another algebraic structure, induced by the one of RB algebras, but weaker –the one we will be concerned later on: quasi-shuffle algebras. Concretely, the target algebras of amplitudes (such as the algebra of Laurent series) happen to be quasi-shuffle algebras, whereas the algebras of linear forms on Feynman diagrams with values in a commutative RB algebra of amplitudes happen to be noncommutative quasi-shuffle algebras.
Indeed, a RB algebra is always equipped with an associative product, the RB double product $\star$, defined by: $$x\star y:=R(x)y+xR(y)-xy$$ so that: $R(x)R(y)=R(x\star y)$. Setting $x\prec y:=xR(y),\ x\succ y:=R(x)y$, one gets $$(x y)\prec z=xyR(z)=x (y\prec z),$$ $$(x\prec y)\prec z=xR(y)R(z)=x\prec (y\star z),$$ $$(x\succ y)\prec z=R(x)yR(z)=x\succ (y\prec z),$$ and so on. These observations give rise to the axioms of noncommutative quasi-shuffle algebras (NQSh, also called tridendriform, algebras). On an historical note, we learned recently from K. Ebrahimi-Fard that the following axioms and relations seem to have first appeared in the context of stochastic calculus, namely in the work of Karandikar in the early 80’s on matrix semimartingales, see e.g. [@Kandihar]. See also [@fpQSh] for details and other references.
A noncommutative quasi-shuffle algebra (NQSh algebra) is a nonunital associative algebra (with product written $\bullet$) equipped with two other products $\prec, \succ$ such that, for all $x,y,z\in A$: $$\begin{aligned}
\label{E1}(x\prec y)\prec z=x\prec(y\star z),&\ \ (x\succ y)\prec z=x\succ(y\prec z)\\
\label{E3}(x\star y) \succ z=x\succ (y\succ z),& \ \ (x\prec y)\bullet z=x\bullet (y\succ z)\\
\label{E6}(x \succ y)\bullet z=x\succ (y\bullet z),& \ \ (x\bullet y)\prec z=x\bullet (y\prec z).\end{aligned}$$ where $x\star y:=x\prec y+x\succ y+x\bullet y$.
Notice that $
(x\bullet y)\bullet z= x\bullet (y\bullet z)$ and $(\ref{E1})+(\ref{E3})+(\ref{E6})$ imply the associativity of $\star$: $$(x\star y)\star z=x\star (y\star z).$$
When the RB algebra is commutative, the relations between the three products $\prec , \succ , \bullet$ simplify (since $x\prec y=xR(y)=y\succ x$) and one arrives at the definition:
A quasi-shuffle (QSh) algebra $A$ is a nonunital commutative algebra (with product written $\bullet$) equipped with another product $\prec$ such that $$\begin{aligned}
\label{QS1}(x\prec y)\prec z&=x\prec(y\star z)\\
\label{QS2}(x\bullet y)\prec z&=x\bullet(y\prec z).\end{aligned}$$ where $x\star y:=x\prec y+y\prec x+x\bullet y$.
We also set for further use $x\succ y:=y\prec x$ (this makes a QSh algebra a NQSh algebra). The product $\star$ is automatically associative and commutative and defines another commutative algebra structure on $A$.
It is sometimes convenient to equip NQSh and QSh algebras with a unit. The phenomenon is exactly similar to the case of shuffle algebras [@schutz]. Given a NQSh algebra, one sets $B:=k\oplus A$, and the products $\prec$, $\succ$, $\bullet$ have a partial extension to $B$ defined by, for $x\in A$: $$1\bullet x=x\bullet 1:=0, \ 1\prec x:=0,\ x\prec 1:=x, \ 1\succ x:=x, \ x\succ 1:=0.$$ The products $1\prec 1$, $1\succ 1$ and $1\bullet 1$ cannot be defined consistenly, but one sets $1\star 1:=1$, making $B$ a unital commutative algebra for $\star$. The categories of NQSh/QSh and unital NQSh/QSh algebras are equivalent (under the operation of adding or removing a copy of the ground field).
Formally, the relations between RB algebras and NQSh algebras are encoded by the Lemma:
The identities $x\prec y:=xR(y),\ x\succ y:=R(x)y, x\bullet y:=xy$ induce a forgetful functor from RB algebras to NQSh algebras, resp. from commutative RB algebras to QSh algebras.
We already alluded to the fact that, in a given quantum field theory, the set of linear forms from the linear span of Feynman diagrams (or equivalently algebra maps from the polynomial algebra they generate) to a commutative RB algebra of amplitudes carries naturally the structure of a noncommutative RB algebra. In the context of QSh algebras, this result generalizes as follows:
\[convol\] Let $C$ be a (coassociative) coalgebra with coproduct $\Delta$ and $A$ be a NQSh algebra. Then the set of linear maps $Hom(C,A)$ is naturally equipped with the structure of a NQSh algebra by the products: $$f\prec g (c):=f(c^{(1)})\prec g(c^{(2)}),$$ $$f\succ g (c):=f(c^{(1)})\succ g(c^{(2)}),$$ $$f\bullet g (c):=f(c^{(1)})\bullet g(c^{(2)}),$$ where we used Sweedler’s notation $\Delta(c)=c^{(1)}\otimes c^{(2)}$.
The proposition follows from the fact that the relations defining NQSh algebras are non-symmetric (in the sense that they do not involve permutations: for example, in the equation $(x\prec y)\prec z=x\prec(y\star z)$, the letters $x,y,z$ appear in the same order in the left and right hand side, and similarly for the other defining relations).
The quasi-shuffle Hopf algebra $QSh(A)$. {#sect:2}
========================================
For details on the constructions in this section, we refer the reader to [[@hof; @fpQSh; @hi]]{}. Let $A$ be an associative algebra. We write $QSh(A)$ for the graded vector space $QSh(A) = \bigoplus_{n \geq 0} QSh(A)_{ n }=k\oplus\bigoplus_{n \geq 1} QSh(A)_{ n }=:k\oplus QSh^+(A)$ where, for $n
\geq 1$, $QSh(A)_{n} = A^{\otimes n}$ and $QSh(A)_{0} =
k$ (notice that when $A$ is unital, one has to distinguish between $1\in k=QSh(A)_{0}$ and $1_A\in A\subset QSh(A)_1$). We denote $l ({\ensuremath{\boldsymbol{a}}}) = n$ the length of an element ${\ensuremath{\boldsymbol{a}}}$ of $QSh(A)_{n}$.
For convenience, an element ${\ensuremath{\boldsymbol{a}}}= a_1 \otimes \ldots \otimes a_n$ of $QSh(A)$ will be called a word and will be written $a_1\dots a_n$ (it should not be confused with the product of the $a_i$ in $A$). We will reserve the tensor product notation for the tensor product of elements of $QSh(A)$ (so that for example, $a_1a_2\otimes a_3\in QSh(A)_2\otimes QSh(A)_1$). Also, we distinghish between the concatenation product of words (written $\cdot$: $a_1a_2a_3\cdot b_1b_2=a_1a_2a_3b_1b_2$) and the product in $A$ by writing $a\cdot_A b$ the product of $a$ and $b$ in $A$ (whereas $a\cdot b$ would stand for the word $ab$ of length 2).
The graded vector space $QSh^+(A)$ (resp. $QSh(A)$) is given a graded (resp. unital) NQSh algebra structure by induction on the length of tensors such that for all $a,b \in A$, for all $v,w \in QSh(A)$: $$av\prec bw=a(v\star bw),$$ $$av \succ bw=b(av \star w),$$ $$av \bullet bw=(a._Ab)(v \star w),$$ where $\qshuffle:=\star=\prec+\succ+\bullet$ is usually called the quasi-shuffle (or stuffle) product (by definition: $\forall v\in QSh(A), 1\qshuffle v=v=v\qshuffle 1$). Notice that this product $\qshuffle$ can be defined directly by the two equivalent inductions $$av\qshuffle bw:=a(v\qshuffle bw)+b(av\qshuffle w)+a\cdot_Ab (v\qshuffle w)$$ or $$va\qshuffle wb:=(v\qshuffle bw)a+(av\qshuffle w)b+ (v\qshuffle w)a\cdot_Ab.$$
When $A$ is commutative, $QSh(A)$ is a unital quasi-shuffle algebra
For example : $$a_1 a_2 \qshuffle b = a_1
a_2 b + a_1 b a_2 + b a_1 a_2
+ a_1 ( a_2\cdot_A b)+ (a_1\cdot_A b)a_2$$ Notice at last that, under the action of the four products $\prec, \succ,\star,\bullet$, the image of $ QSh(A)_{r } \otimes QSh(A)_{
s } $ is contained in $ \bigoplus_{t = \max ( r, s )}^{r + s} QSh(A)_{ t }$
One can also define :
- a counit $\eta : QSh(A) \rightarrow k$ by $\eta ( 1 ) := 1$ and for $s \geq 1$, $\eta
( a_1 \ldots a_s ) = 0$,
- a coproduct (called deconcatenation coproduct) $\Delta : QSh(A) \rightarrow
QSh(A) \otimes QSh(A)$ such that $\Delta( 1) = 1\otimes 1$ and for $s \geq 1$ and ${\ensuremath{\boldsymbol{a}}}= a_1 \ldots a_s \in
QSh(A)_{s}$, $$\Delta ( {\ensuremath{\boldsymbol{a}}} ) = {\ensuremath{\boldsymbol{a}}} \otimes
1 + 1 \otimes{\ensuremath{\boldsymbol{a}}} + \sum_{r = 1}^{s -
1} ( a_1 \ldots a_r ) \otimes( a_{r + 1}
\ldots a_s )$$
making $QSh(A)$ a graded coalgebra. It is a matter of fact to check that $QSh(A)$ is a unital conilpotent bialgebra (and thus a Hopf algebra, see e.g. [@Cartier2]), which is called the quasi-shuffle or stuffle Hopf algebra on $A$ (this terminology, that we adopt, is convenient, usual, but slightly misleading because when $A$ is only associative, $QSh(A)$ is a unital noncommutative quasi-shuffle algebra).
Operations and universal properties {#sect:3}
===================================
Let us focus now in the first part of this section on the case relevant to renormalization, that is when $A$ is commutative but not necessarily unital. It follows then from standard arguments in universal algebra that, given a quasi-shuffle algebra $B$, morphisms of quasi-shuffle algebras from $QSh^+(A)$ to $B$ are naturally in bijection with morphisms of (non unital) algebras from $A$ to $B$: $$Hom_{QSh}(QSh^+(A),B)\cong Hom_{Alg}(A,B).$$ In categorical terms (see [@fpQSh] for a direct and elementary proof):
The left adjoint $U$ of the forgetful functor from the category of quasi-shuffle algebras $QSh$ to the category of non unital commutative algebras $Com$, or “quasi-shuffle enveloping algebra” functor from $Com$ to $QSh$, is Hoffman’s quasi-shuffle algebra functor $A\longmapsto QSh^+(A)$.
It is interesting to analyse the concrete meaning of this Proposition. Let us consider first the counit of the adjunction, that is the quasi-shuffle algebra map from $QSh^+(A)$ to $A$, when $A$ is a quasi-shuffle algebra. By definition of $\prec$, the element $a_1\dots a_n\in QSh(A)_n$ can be rewritten (in $QSh(A)$) $a_1\prec (a_2\prec \dots (a_{n-1}\prec a_n))$. The trick goes back to Schützenberger who used it in his seminal but not enough acknowledged study of shuffle algebras [@schutz]. It follows that the counit of the adjunction maps $a_1\dots a_n\in QSh(A)_n$ to $a_1\prec (a_2\prec \dots (a_{n-1}\prec a_n))$ (computed now in $A$).
Let us move now to the case when $A$ is a commutative RB algebra. Then, $A$ is in particular a quasi-shuffle algebra with $a\prec b:=aR(b)$. The counit of the same adjunction is then the map that sends $a_1\dots a_n\in QSh(A)_n$ to $a_1R(a_2 R(a_3 \dots R(a_{n-1}R( a_n)))$. In particular, $a^n$ is mapped to $aR(a R(a \dots R(aR( a)))$ -a term that is known to play a key role in renormalization, see in particular [@emp].
This relatively standard adjunction analysis can be completed in the case we are interested in (maps from $QSh^+(A)$ to $B$, when $B$ is a quasi-shuffle algebra), due to the existence of a Hopf algebra structure on $QSh(A)$. According to Proposition \[convol\], we have first that
Let $A$ be an associative algebra and $B$ a NQSh algebra, the vector space of linear morphisms ${\mathcal L}(
QSh(A), B )$ is a NQSh algebra.
Furthermore, by properties that hold for arbitrary maps from a conilpotent Hopf algebra to an algebra, if $B$ is unital, the set of linear maps that map the unit of $QSh(A)$ to the unit of $B$, $\mathcal{U}( QSh(A), B )$ is a group for the product $\star$. Moreover, when $B$ is commutative, the subset of algebra maps from $QSh(A)$ to $B$, $\mathcal{C} ( QSh(A), B )$, is a subgroup.
Next, notice that the functor $QSh$ is compatible with Hopf algebra structures: an algebra map $l$ from $A$ to $B$ induces a map $QSh(l)$ of quasi-shuffle algebras from $QSh(A)$ to $QSh(B)$ defined by $$QSh(l)( 1 ) = 1 \hspace{1em} {\ensuremath{\operatorname{and}}}
\hspace{1em} QSh(l) ( a_1 \ldots a_r ) = l ( a_1 )
\ldots l ( a_r ) \hspace{1em} ( r \geq 1 )$$ and therefore $\Delta \circ QSh(l) = ( QSh(l)
\otimes QSh(l )) \circ \Delta$. In particular, $QSh(l)$ is a Hopf algebra morphism.
The last universal property of the $QSh$ functor that we would like to emphasize is more intriguing and does not seem to have been noticed before. Whereas $QSh$ is naturally a left adjoint, it also happens indeed to be a right adjoint, a property that will prove essential in our later developments.
\[mainthm\] Let $H$ be a conilpotent Hopf algebra and $A$ be a unital associative algebra, then we have a natural isomorphism between (unital) algebra maps from $H$ to $A$ and Hopf algebra maps from $H$ to $QSh(A)$: $$Hom_{Alg}(H,A)\cong Hom_{Hopf}(H,QSh(A)).$$
Indeed, $QSh(A)$ is, as a coalgebra, the cofree coalgebra over $A$ (viewed as a vector space) in the category of conilpotent coalgebras. These properties are dual to the ones of tensor algebras (more familiar, but equivalent up to the fact that the dual of a coalgebra is an algebra but the converse is not always true -this is the reason for the conilpotency hypothesis): the tensor algebra over a vector space $V$ is, when equipped with the concatenation product, the free associative algebra over $V$. There is therefore a natural isomorphism between linear maps from the kernel $C^+$ of the counit of a coaugmented conilpotent coalgebra $C$ to $A$ and coalgebra maps from $C$ to $QSh(A)$ $${\mathcal L}(C^+,A)\cong Hom_{Coalg}(C,QSh(A)).$$ Coaugmented means that there is a coalgebra map from the ground field to $C$, insuring that $C$ decomposes as the direct sum of $k$ and of the kernel of the counit (as happens for a Hopf algebra, for which the composition of the unit and the counit is a projection on the ground field orthogonally to the kernel of the counit).
The isomorphism is given explicitly as follows: it maps $\phi\in {\mathcal L}(C^+,A)$ to $\tilde\phi :=\sum\limits_{i=0}^\infty \phi^{\otimes n}\circ {\Delta'}^{[n]}$ (where $\phi^{\otimes 0}\circ{\Delta'}^{[0]}$ stands for the composition of the counit of $C$ with the unit of $QSh(A)$). In particular, the map $\phi$ factorizes as (the restriction to $C^+$ of) $j\circ \tilde \phi$, where $j \in {\mathcal L}( QSh(A), A )$ is defined by $j (
1 ) = 1_A$, $j ( a_1 ) = a_1$ and $j ( a_1 \ldots a_r
) = 0$ if $r \geq 2$.
To prove the Theorem, it is therefore enough to show that, when a linear map $\phi$ from $H^+$ to $A$ is the restriction to $H^+$ of an algebra map from $H$ to $A$, then the induced map $\tilde\phi$ is also an algebra map (since we already know it is a coalgebra map). Concretely, we have to prove that, for $h,h'\in H^+$, $\tilde\phi (hh')=\tilde\phi(h)\qshuffle \tilde\phi (h')$. The Theorem will then follow if we prove that $$\sum\limits_{n=1}^\infty \phi^{\otimes n}\circ {\Delta'}^{[n]}(hh')=\sum\limits_{p=1}^\infty\phi^{\otimes p}\circ {\Delta'}^{[p]}(h)\qshuffle \sum\limits_{q=1}^\infty \phi^{\otimes p}\circ {\Delta'}^{[q]}(h').$$ Using that $\phi$ and that $\Delta$ are algebra maps, this follows from the following Lemma (where, to avoid ambiguities, we use the notation ${\Delta'}^{[p]}(h)=h_{(1,p)}'\otimes \dots \otimes h_{(p,p)}'$) by identification of the terms in the left and right hand side.
We have, for the iterated coproduct and $h\in H^+$, $$\Delta^{[n]}(h)=\sum_{i=1}^n \sum_{f\in Inj(i,n)}f_\ast (h_{(1,i)}'\otimes \dots \otimes h_{(i,i)}'),$$ where $Inj(i,n)$ stands for the set of increasing injections from $[i]:=\{1,\dots ,i\}$ to $[n]$ and $$f_\ast (h_{(1,i)}'\otimes \dots \otimes h_{(i,i)}')=l_{(1)}\otimes \dots \otimes l_{(n)}$$ with $l_{(q)}:=h'_{(p,i)}$ if $q=f(p)$ and $l_{(q)}:=1$ if $q$ is not in the image of $f$.
For example, $\Delta^{[1]}(h)={\Delta'}^{[1]}(h)=h=h_{(1,1)}'$, $$\Delta^{[2]}(h)=\Delta(h)=h_{(1,1)}'\otimes 1+1\otimes h_{(1,1)}'+h_{(1,2)}'\otimes h_{(2,2)}'$$ and $$\begin{array}{rcl}
\Delta^{[2]}(hk)&=&\Delta^{[2]}(h)\Delta^{[2]}(k) \\
&=& (h_{(1,1)}'\otimes 1+1\otimes h_{(1,1)}'+h_{(1,2)}'\otimes h_{(2,2)}')\\
& &\ \ \times \ (k_{(1,1)}'\otimes 1+1\otimes k_{(1,1)}'+k_{(1,2)}'\otimes k_{(2,2)}'),\end{array}$$ so that $${\Delta'}^{[2]}(hk)=h_{(1,1)}'\otimes k_{(1,1)}'+k_{(1,1)}'\otimes h_{(1,1)}'+h_{(1,1)}'k_{(1,2)}'\otimes k_{(2,2)}'+
k_{(1,2)}'\otimes h_{(1,1)}'k_{(2,2)}'$$ $$+h_{(1,2)}'k_{(1,1)}'\otimes h_{(2,2)}'
+h_{(1,2)}'\otimes h_{(2,2)}'k_{(1,1)}'+h_{(1,2)}'k_{(1,2)}'\otimes h_{(2,2)}'k_{(2,2)}',$$ where one recognizes the tensor degree $2$ component of $$({\Delta'}^{[1]}(h)+{\Delta'}^{[2]}(h))\qshuffle ({\Delta'}^{[1]}(k)+{\Delta'}^{[2]}(k)).$$
The Theorem has an important corollary, that we state also as a Theorem in view of its importance for our approach to renormalization.
Let $H$ be a conilpotent bialgebra, then, the unit, written $\iota$, of the adjunction in the previous Theorem, ($\iota(1):=1$ and $\forall h\in H^+, \iota(h)=\sum\limits_{k\geq 1}{\Delta'}^{[k]}(h))$ defines an injective Hopf algebra morphism from $H$ to $QSh(H^+)$. In particular, any conilpotent (resp. conilpotent commutative) Hopf algebra embeds into a noncommutative quasi-shuffle (resp. a quasi-shuffle) Hopf algebra.
We let the reader check the following Lemma, that will be important later in the article and makes Theorem \[mainthm\] more precise:
The map $j \in {\mathcal L}( QSh(A), A )$ is a morphism of algebras.
The map $j \in \mathcal{U}( QSh(A), A )$. {#sect:4}
=========================================
We shall now illustrate the ideas of the previous section on the map $j \in \mathcal{U}( QSh(A), A )$ (recall it is defined by $j (
1) = 1_A$, $j ( a_1 ) = a_1$ and $j ( a_1 \ldots a_r
) = 0$ if $r \geq 2$). In a sense, this will be the only computation of inverse and of Birkhoff decomposition we will need. This map $j$ plays a fundamental role. We already saw that it appears in the adjunction ${\mathcal L}(C^+,A)\cong Hom_{Coalg}(C,QSh(A)).$ It will also appear later to be the unit of a semigroup structure on $\mathcal{U}( QSh(A), A )$ to be introduced in the next section.
For the inverse, we get $j^{\ast - 1}$ : $$j^{\ast - 1} = u_A \circ \eta + \sum_{k \geq 1} ( u_A
\circ \eta - j )^{\ast k}$$ Which means that $j^{\ast - 1} ( 1 ) = 1_A$ and for ${\ensuremath{\boldsymbol{a}}}= a_1 \ldots a_s \in {QSh(A)}^+$, $$\begin{array}{ccc}
j^{\ast - 1} ({\ensuremath{\boldsymbol{a}}}) & = &\displaystyle \sum_{k \geq 1} ( - 1 )^k m_A^{[ k ]}
\circ j^{\otimes_{} k} \circ {\Delta'}^{[ k ]} ({\ensuremath{\boldsymbol{a}}})\\
& = &\displaystyle \sum_{k \geq 1} ( - 1 )^k \sum_{{\ensuremath{\boldsymbol{a}}}^1 \cdot \ldots
\cdot {\ensuremath{\boldsymbol{a}}}^k ={\ensuremath{\boldsymbol{a}}} \atop {\ensuremath{\boldsymbol{a}}}^i\in {QSh(A)}^+} m_A^{[ k ]} \circ j^{\otimes k}
({\ensuremath{\boldsymbol{a}}}^1 \otimes \ldots \otimes
{\ensuremath{\boldsymbol{a}}}^k )\\
& = &\displaystyle ( - 1 )^s m_A^{[ s ]}
({a}_1 \otimes \ldots \otimes
{a}_s )\\
& = & ( - 1 )^s a_1\cdot_A \ldots\cdot_A a_s=j\circ S({\ensuremath{\boldsymbol{a}}})
\end{array}$$ where $$\begin{array}{lll}
S ({\ensuremath{\boldsymbol{a}}}) & = & \displaystyle \sum_{k \geq 1} ( - 1 )^k m^{[ k ]}
\circ \Delta'^{[ k ]} ({\ensuremath{\boldsymbol{a}}})\\
& = & \displaystyle\sum_{k \geq 1} ( - 1 )^k \sum_{{\ensuremath{\boldsymbol{a}}}^1 \cdot \ldots
\cdot {\ensuremath{\boldsymbol{a}}}^k ={\ensuremath{\boldsymbol{a}}} \atop {\ensuremath{\boldsymbol{a}}}^i\in {QSh(A)}^+}
{\ensuremath{\boldsymbol{a}}}^1 \qshuffle \ldots \qshuffle
{\ensuremath{\boldsymbol{a}}}^k .
\end{array}$$ Note that the previous sums run over all the possible factorizations in nonempty words of ${\ensuremath{\boldsymbol{a}}}$ for the concatenation product.
If $( A, p_+ )$ is a Rota-Baxter algebra then the Bogoliubov preparation map $\bar{j}$ associated to $j$, see equation (\[eq:Bog\]), is such that $\bar{j} (1 ) = 1_A$ and can be defined recursively on vector spaces $QSh(A)_n$ ($n \geq 1$) by $$\bar{j} ( h ) = j ( h ) - m_A \circ ( p_- \otimes {\ensuremath{\operatorname{Id}}} ) \circ (
\bar{j} \otimes j ) \circ \Delta' ( h )$$ Let us begin the recursion on the length of the sequence. If ${\ensuremath{\boldsymbol{a}}}=
a_1$ then $\bar{j} ( a_1 ) = j ( a_1 ) = a_1$. Now, if ${\ensuremath{\boldsymbol{a}}}=
a_1 \cdot a_2=a_1 a_2$, $$\bar{j} ( a_1 a_2 ) = j ( a_1 a_2 ) - m_A \circ ( p_-
\otimes {\ensuremath{\operatorname{Id}}} ) \circ ( \bar{j} \otimes j ) ( ( a_1 )
\otimes ( a_2 ) ) = - p_- ( a_1 )\cdot_A a_2$$ and $$\begin{array}{ccc}
\bar{j} ( a_1 a_2 a_3 ) & = & - m_A \circ ( p_- \otimes {\ensuremath{\operatorname{Id}}} )
\circ ( \bar{j} \otimes j ) ( ( a_1 a_2 ) \otimes (
a_3 ) )\\
& = & p_- ( p_- ( a_1 )\cdot_A a_2 )\cdot_A a_3
\end{array}$$ Thus, for $r \geq 2$, $$\bar{j} ( a_1 \ldots a_r ) = - p_- ( \bar{j} ( a_1 \ldots
a_{r - 1} ) )\cdot_A a_r$$ It is then easy to prove that in general (see e.g. [@emp] for a systematic study of combinatorial approaches and closed solutions to the Bogoliubov recursion)
The Birkhoff decomposition $$( j_+, j_- ) \in \mathcal{U}(
QSh(A), A_+ ) \times \mathcal{U}( QSh(A), A_- )$$ such that $$j_- \ast j = j_+$$ is given by the formula : for $r \geq 1$ and ${\ensuremath{\boldsymbol{a}}}= a_1 \otimes
\ldots \otimes a_r \in {QSh(A)}^+$, $$\label{eq:Birkj}
\hspace{1em} \left\{\begin{array}{lllll}
j_+ ({\ensuremath{\boldsymbol{a}}}) & = & p_+ ( \bar{j} ({\ensuremath{\boldsymbol{a}}}) ) & = & ( - 1 )^{r
- 1} p_+ ( p_- ( \ldots ( p_- ( a_1 )\cdot_A a_2 ) \ldots \cdot_A a_{r - 1} )\cdot_A a_r )\\
j_- ({\ensuremath{\boldsymbol{a}}}) & = & - p_- ( \bar{j} ({\ensuremath{\boldsymbol{a}}}) ) & = & ( - 1
)^r p_- ( p_- ( \ldots ( p_- ( a_1 )\cdot_A a_2 ) \ldots \cdot_Aa_{r - 1} ) \cdot_A a_r )
\end{array}\right.$$ Moreover, if $A$ is commutative then $\mathcal{C}( QSh(A), A )$ is a group and $j_{_+}$ and $j_-$ are characters.
Let us prove the last assumption, when $A$ is commutative. Since $j$ is a character it is sufficient to prove that $j_-$ is a character. By induction on $t \geq 0$ we will show that for two tensors ${\ensuremath{\boldsymbol{a}}}$ and ${\ensuremath{\boldsymbol{b}}}$ in $QSh(A)$, if $l ({\ensuremath{\boldsymbol{a}}}) +
l ({\ensuremath{\boldsymbol{b}}}) = t$, then $$j_- ({\ensuremath{\boldsymbol{a}}} \qshuffle
\text{${\ensuremath{\boldsymbol{b}}}$} ) = j_- ({\ensuremath{\boldsymbol{a}}}) j_- ({\ensuremath{\boldsymbol{b}}})$$ This identity is trivial for $t = 0$ and $t = 1$ since at least one of the sequences is the empty sequence. This also trivial for any $t$ if one of the sequences is empty. Now suppose that $t \geq 2$ and that ${\ensuremath{\boldsymbol{a}}}= a_1
\ldots a_r \in QSh(A)_{ r }$ and ${\ensuremath{\boldsymbol{b}}} =
b_1 \ldots b_s \in QSh(A)_{s }$ with $r \geq 1$, $s \geq 1$ and $r + s = t$. Let $\tilde{{\ensuremath{\boldsymbol{a}}}} = a_1 \ldots
a_{r - 1} \in QSh(A)_{r - 1 }$ ($\tilde{{\ensuremath{\boldsymbol{a}}}} =
1$ if $r = 1$) and ${\ensuremath{\boldsymbol{\tilde{b}}}} = b_1 \ldots
b_{s - 1} \in QSh(A)_{s - 1 }$ (${\ensuremath{\boldsymbol{\tilde{b}}}} =
1$ if $s = 1$), then : $${\ensuremath{\boldsymbol{a}}} \qshuffle {\ensuremath{\boldsymbol{b}}} =
( \tilde{{\ensuremath{\boldsymbol{a}}}} \qshuffle {\ensuremath{\boldsymbol{b}}} )
\cdot a_r + ({\ensuremath{\boldsymbol{a}}} \qshuffle
{\ensuremath{\boldsymbol{\tilde{b}}}} ) \cdot b_s + ( \tilde{{\ensuremath{\boldsymbol{a}}}}
\qshuffle{\ensuremath{\boldsymbol{\tilde{b}}}} ) \cdot (a_r \cdot_A b_s)$$ Now we have $$j_- ({\ensuremath{\boldsymbol{a}}}) = - p_- ( j_- ( \tilde{{\ensuremath{\boldsymbol{a}}}} ) \cdot_A a_r ) = - p_- (
x ) \hspace{1em} {\ensuremath{\operatorname{and}}} \hspace{1em} j_- ({\ensuremath{\boldsymbol{b}}}) = - p_- ( j_-
( {\ensuremath{\boldsymbol{\tilde{b}}}} ) \cdot_A b_s ) = - p_- ( y ),$$ where $x:=j_- ( \tilde{{\ensuremath{\boldsymbol{a}}}} ) \cdot_A a_r$ and $y:=j_-
( {\ensuremath{\boldsymbol{\tilde{b}}}} ) \cdot_A b_s$. Thanks to the Rota-Baxter identity, and omitting $\cdot_A$ in the following computations in $A$, $$\begin{array}{ccc}
j_- ({\ensuremath{\boldsymbol{a}}}) j_- ({\ensuremath{\boldsymbol{b}}}) & = & p_- ( x ) p_- ( y )\\
& = & p_- ( x p_- ( y ) ) + p_- ( p_- ( x ) y ) - p_- ( x y )\\
& = & p_- ( j_- ( \tilde{{\ensuremath{\boldsymbol{a}}}} ) a_r p_- ( j_- (
{\ensuremath{\boldsymbol{\tilde{b}}}} ) b_s ) ) + p_- ( p_- ( j_- ( \tilde{{\ensuremath{\boldsymbol{a}}}}
) a_r ) j_- ( {\ensuremath{\boldsymbol{\tilde{b}}}} ) b_s ) \\
& & \quad - p_- ( j_- (
\tilde{{\ensuremath{\boldsymbol{a}}}} ) a_r j_- ( {\ensuremath{\boldsymbol{\tilde{b}}}} ) b_s )
\end{array}$$ but as $A$ is commutative, by induction we get $$\begin{array}{ccc}
j_- ({\ensuremath{\boldsymbol{a}}}) j_- ({\ensuremath{\boldsymbol{b}}}) & = & - p_- ( j_- (
\tilde{{\ensuremath{\boldsymbol{a}}}} ) j_- ({\ensuremath{\boldsymbol{b}}}) a_r ) - p_- ( j_-
({\ensuremath{\boldsymbol{a}}}) j_- ( {\ensuremath{\boldsymbol{\tilde{b}}}} ) b_s ) - p_- ( j_- (
\tilde{{\ensuremath{\boldsymbol{a}}}} ) j_- ( {\ensuremath{\boldsymbol{\tilde{b}}}} ) a_r b_s )\\
& = & - p_- ( j_- ( \tilde{{\ensuremath{\boldsymbol{a}}}} \qshuffle
{\ensuremath{\boldsymbol{b}}} ) a_r ) - p_- ( j_- ({\ensuremath{\boldsymbol{a}}}
\qshuffle {\ensuremath{\boldsymbol{\tilde{b}}}} ) b_s ) - p_- ( j_- (
\tilde{{\ensuremath{\boldsymbol{a}}}} \qshuffle {\ensuremath{\boldsymbol{\tilde{b}}}} ) a_r b_s )\\
& = & j_- ( ( \tilde{{\ensuremath{\boldsymbol{a}}}} \qshuffle {\ensuremath{\boldsymbol{b}}}
) \cdot a_r ) + j_- ( ({\ensuremath{\boldsymbol{a}}} \qshuffle
{\ensuremath{\boldsymbol{\tilde{b}}}} ) \cdot b_s ) + j_- ( (
\tilde{{\ensuremath{\boldsymbol{a}}}} \qshuffle {\ensuremath{\boldsymbol{\tilde{b}}}} ) \cdot (a_r b_s ))\\
& = & j_- ( ( \tilde{{\ensuremath{\boldsymbol{a}}}} \qshuffle {\ensuremath{\boldsymbol{b}}}
) \cdot a_r + ({\ensuremath{\boldsymbol{a}}} \qshuffle
{\ensuremath{\boldsymbol{\tilde{b}}}} ) \cdot b_s + (
\tilde{{\ensuremath{\boldsymbol{a}}}} \qshuffle {\ensuremath{\boldsymbol{\tilde{b}}}} ) \cdot
(a_r b_s) )\\
& = & j_- ( {\ensuremath{\boldsymbol{a}}} \qshuffle {\ensuremath{\boldsymbol{b}}}
)
\end{array}$$
In the sequel, when there is no ambiguity, we shall omit the notation $\cdot_A$ when applying formula (\[eq:Birkj\]).
As we will see these formulas are almost sufficient to compute the Birkhoff decomposition for any conilpotent bialgebra.
The universal semigroup and renormalization. {#sect:5}
============================================
Let $A$ be a unital algebra. Then, by adjunction we know that $$\mathcal{U}(
QSh(A), A )\cong Hom_{Coalg}(
QSh(A), QSh(A) ).$$ In particular, the composition of coalgebra endomorphisms of $QSh(A)$ equips $\mathcal{U}(
QSh(A), A )$ with a semigroup structure.
The universal semigroup associated to a unital algebra $A$ is the set $\mathcal{U}(
QSh(A), A )$ equipped with the associative unital product induced by composition of coalgebra endomorphisms of $QSh(A)$: for $f$ and $g$ in $\mathcal{U}(
QSh(A), A )$ $$f \odot g := f\circ QSh(g)\circ \iota .$$ Its unit is the map $j$: $$f \odot j = f \circ QSh(j) \circ \iota = f\circ Id=f .$$
This semigroup structure generalizes to an action on linear maps from a Hopf algebra to $A$ as follows.
Let $H$ be a conilpotent bialgebra. For $\varphi \in \mathcal{U}( H, A )$ and $f \in \mathcal{U}( QSh(A), A )$ we set $$f\odot \varphi : = f \circ QSh(\varphi) \circ \iota .$$ This morphism $f\odot \varphi $ is linear from $H$ to $A$ and unital: $$f\odot \varphi ( 1_H ) = f \circ QSh(\varphi) \circ \iota ( 1_H )
= f \circ QSh(\varphi) ( 1 ) = f ( 1 ) = 1_A .$$ We get a left action of $\mathcal{U}( QSh(A), A )$ on $\mathcal{U}( H, A )$: $$\odot : \mathcal{U}( QSh(A), A )\times \mathcal{U}( H, A )\to \mathcal{U}( H, A ).$$ Moreover, when $A$ is commutative, if $\varphi \in
\mathcal{C}( H, A )$ and $f \in \mathcal{C}( QSh(A), A )$ it is clear, by composition of algebra morphisms, that $ f\odot \varphi \in \mathcal{C}( H,
A )$.
That $j$ acts as the identity map on $\mathcal{U}( H, A )$ follows from: for $h \in H^+$, $$\begin{array}{ccc}
j\odot \varphi ( h ) & = &\displaystyle j \circ QSh(\varphi)\left( h +
\sum_{k \geq 2} \sum h'_{( 1 )} \otimes \ldots \otimes h'_{( k )}
\right)\\
& = &\displaystyle j \left( \varphi ( h ) + \sum_{k \geq 2} \varphi ( h'_{( 1 )} )
\cdot \ldots \cdot \varphi ( h'_{( k )} ) \right)\\
& = & \varphi ( h )
\end{array}$$
The action $\odot$ and the convolution product $\ast$ (recall that $QSh(A)$ is a Hopf algebra) satisfy the distributivity relation: For $f$ and $g$ in $\mathcal{U}( QSh(A), A )$ and $\varphi$ in $\mathcal{U}( H, A )$, $$(f\ast g)\odot \varphi=(f\odot \varphi)\ast (g\odot \varphi).$$
Indeed, $$\begin{array}{ccc}
(f\ast g)\odot \varphi & = & m_A \circ ( f \otimes g ) \circ
\Delta \circ QSh(\varphi) \circ \iota\\
& = & m_A \circ ( f \otimes g ) \circ ( QSh(\varphi) \otimes
QSh(\varphi) ) \circ \Delta \circ \iota\\
& = & m_A \circ ( f \otimes g ) \circ ( QSh(\varphi) \otimes
QSh(\varphi) ) \circ ( \iota \otimes \iota ) \circ \Delta\\
& = & m_A ( f\odot \varphi \otimes g\odot \varphi ) \circ \Delta\\
& = & (f\odot \varphi ) \ast ( g\odot \varphi )
\end{array}$$
Note that, in the case $H=QSh(A)$, $\mathcal{U}( QSh(A), A )$ is equipped with two products $\ast$ and $\odot$ that look similar, in their interactions, to the product and composition of power series.
These constructions generalize as follows. Let $B$ be another unital algebra. For $\varphi \in \mathcal{U}( H, A )$ and $f \in \mathcal{U}( QSh(A), B )$ we define $$f\odot \varphi = f \circ QSh(\varphi) \circ \iota .$$ The morphism $f\odot\varphi $ is linear from $H$ to $B$ and $$f\odot \varphi ( 1_H ) = f \circ QSh(\varphi)\circ \iota ( 1_H )
= f \circ QSh(\varphi) ( 1 ) = f ( 1 ) = 1_B .$$ thus $f\odot \varphi \in \mathcal{U}( H, B )$. Moreover, when $A$ and $B$ are commutative, if $\varphi \in
\mathcal{C}( H, A )$ and $f \in \mathcal{C}( QSh(A), B )$ it is clear, by composition of algebra morphisms that $ f\odot \varphi \in \mathcal{C}( H,
B )$.
Let $\varphi \in \mathcal{U}( H, A )$, then its convolution inverse if given by $$\varphi^{\ast -1}=j^{\ast -1}\odot \varphi.$$
Indeed, since $j\odot \varphi = \varphi$, if $\psi := j^{\ast - 1} \odot\varphi $, then $$\psi \ast \varphi = ( j^{\ast - 1}\odot \varphi ) \ast ( j\odot \varphi ) = (
j^{\ast - 1} \ast j)\odot \varphi = ( u_A \circ \eta)\odot \varphi
= u_A \circ \eta .$$
For example, if $h \in H^+$ with $\Delta'^{[4]}(h)=0$, then $$\iota ( h ) = h + \sum h'_{( 1 )} \otimes h'_{( 2 )} + \sum h'_{( 1 )} \otimes h'_{(
2 )} \otimes h'_{( 3 )}$$ so, $$QSh(\varphi) \circ \iota ( h ) = \varphi ( h ) + \sum\varphi ( h'_{( 1
)} ) \cdot \varphi ( h'_{( 2 )} ) +\sum \varphi ( h'_{( 1 )} ) \cdot
\varphi ( h'_{( 2 )} ) \cdot \varphi ( h'_{( 3 )} )$$ and finally $$\varphi^{\ast - 1} ( h ) = j^{\ast - 1} \circ QSh(\varphi) \circ
\iota(h) = - \varphi ( h ) +\sum \varphi ( h'_{( 1 )} ) \varphi ( h'_{( 2 )} ) -
\sum \varphi ( h'_{( 1 )} ) \varphi ( h'_{( 2 )} ) \varphi ( h'_{( 3 )} )$$ We recover the usual formula for the inverse.
Assume now that $A$ is an idempotent Rota–Baxter algebra. Let $\varphi \in \mathcal{U}( H, A )$. Then the Birkhoff-Rota-Baxter decomposition of $\varphi$ is given by $$\varphi_-=j_-\odot \varphi,\ \ \varphi_+=j_+\odot\varphi .$$
Indeed, since $ j\odot \varphi =
\varphi$, we have $$\varphi_- \ast \varphi = ( j_-\odot \varphi ) \ast ( j\odot \varphi ) = ( j_-
\ast j)\odot \varphi = j_+\odot \varphi = \varphi_+$$ and, of course, $\varphi_{\pm} \in \mathcal{U}( H, A_{\pm} )$.
For example, if $h \in H'$ with $\Delta'^{[4]}(h)=0$, then $$\begin{array}{ccc}
\varphi_+ ( h ) & = & p_+ ( \varphi ( h ) ) -\sum p_+ ( p_- ( \varphi ( h'_{(
1 )} ) ) \varphi ( h'_{( 2 )} ) ) + \sum p_+ ( p_- ( p_- ( \varphi ( h'_{( 1
)} ) ) \varphi ( h'_{( 2 )} ) ) \varphi ( h'_{( 3 )} ) )\\
& & \\
\varphi_- ( h ) & = & - p_- ( \varphi ( h ) ) +\sum p_- ( p_- ( \varphi (
h'_{( 1 )} ) ) \varphi ( h'_{( 2 )} ) ) -\sum p_- ( p_- ( p_- ( \varphi (
h'_{( 1 )} ) ) \varphi ( h'_{( 2 )} ) ) \varphi ( h'_{( 3 )} ) )
\end{array}$$
Needless to say that if $A$ is commutative, these computations works in the subgroup $\mathcal{C}( H, A )$.
Once these formulas are given, we get formulas in the different contexts where renormalization, or rather Birkhoff decomposition, is needed. We end this paper with two sections that illustrate how these formulas could be used :
- to perform inversion and Birkhoff decomposition of diffeomorphisms that correspond to characters on the Faà di Bruno Hopf algebra,
- to perform the Birkhoff decomposition with the same formula in various cofree Hopf algebras that differ by their algebra structures, but for which the map $\iota$ is the same as these Hopf algebras are tensor coalgebras.
Renormalizing diffeomorphisms in pQFT and Dynamics {#sect:6}
==================================================
Let us focus in this section on the example of the Faà di Bruno Hopf algebra $\mathcal{H}_{{\ensuremath{\operatorname{FdB}}}}$ (see [@BF; @fig; @MF; @men]) whose group of characters corresponds to the group of formal identity-tangent diffeomorphisms. We will first express the reduced coproduct and then the map $\iota$ from this Hopf algebra to its associated quasi-shuffle Hopf algebra and then focus on the Birkhoff decomposition of characters with values in the Laurent series that appear in several areas, as a factorisation of diffeomorphisms for the composition.
Recall that the decomposition is unique: the same results could be obtained by induction using the classical renormalization process (the Bogoliubov recursion). One advantage of the present approach is to encode the combinatorics of renormalization into a universal framework, probably similar to the one P. Cartier suggested when advocating the existence of a “Galois group” underlying renormalization. Compare in particular our approach with [@ck1; @guo; @emp].
Consider the group of formal identity tangent diffeomorphisms with coefficients in a commutative $\mathbb{C}$–algebra $A$: $$G(A) = \{ f ( x ) = x + \sum_{n \geq 1} f_n x^{n + 1} \in A[ [ x ]
] \}$$ with its product $\mu : G(A) \times G(A) \rightarrow G(A)$ : $$\mu ( f, g ) = f \circ g.$$ For $n \geq 0$, the functionals on $G(A)$ defined by $$a_n ( f ) = \frac{1}{( n + 1 ) !_{}} ( \partial_x^{n + 1} f ) ( 0 ) = f_n
\hspace{1em} a_n : G(A) \rightarrow A$$ are called de Faà di Bruno coordinates on the group $G(A)$ and $a_0 = 1$ being the unit, they generates a graded unital commutative algebra $$\mathcal{H}_{{\ensuremath{\operatorname{FdB}}}} =\mathbbm{C}[ a_1, \ldots, a_n, \ldots ]
\hspace{1em} ( {\ensuremath{\operatorname{gr}}} ( a_n ) = n )$$ The action of these functionals on a product in $G(A)$ defines a coproduct on $\mathcal{H}_{{\ensuremath{\operatorname{FdB}}}}$ that turns to be a graded connected Hopf algebra (see [[@fig]]{} for details). For $n \geq 0$, the coproduct is defined by $$a_n \circ \mu = m \circ \Delta ( a_n ) \label{copfdb}$$ where $m$ is the usual product in $A$, and the antipode reads $$S \circ a_n = a_n \circ {\ensuremath{\operatorname{inv}}}$$ where ${\ensuremath{\operatorname{inv}}} ( \varphi ) = \varphi^{\circ {- 1}}$ is the composition inverse of $\varphi$.
For example if $f ( x ) = x + \sum_{n \geq 1} f_n x^{n + 1}$ and $g ( x ) = x
+ \sum_{n \geq 1} g_n x^{n + 1}$ then if $h(x)=f \circ g(x)= x + \sum_{n \geq 1} h_n x^{n + 1}$, $$\begin{array}{ccccccc}
a_0 ( h ) & = & 1 = a_0 ( f ) a_0 ( g ) & \rightarrow & \Delta a_0 & = &
a_0 \otimes a_0\\
a_1 ( h ) & = & f_1 + h_1 & \rightarrow & \Delta a_1 & = & a_1 \otimes
a_0 + a_0 \otimes a_1\\
a_2 ( h ) & = & f_2 + 2f_1 g_1 + g_2 & \rightarrow & \Delta a_2 & = & a_2
\otimes a_0 + 2a_1 \otimes a_1 + a_0 \otimes a_2.
\end{array}$$
More generally, using classical formulas on the composition of diffeomorphisms (see [@BF; @EFP1; @MF; @men11]), we have $$\Delta (a_n)=\sum_{k=0}^{n} \sum_{l_0+\dots l_k=n-k\atop l_i\geq 0} a_k \otimes a_{l_0}\dots a_{l_k}$$
Let us consider sequences of positive integers $$\mathcal{N}= \{ {\ensuremath{\boldsymbol{n}}}= ( n_1, \ldots, n_s ) \in (\mathbbm{N}^{\ast}
)^s, \hspace{1em} s \geq 1 \}$$ For ${\ensuremath{\boldsymbol{n}}}= ( n_1, \ldots, n_s ) \in \mathcal{N}$, we denote $$\| {\ensuremath{\boldsymbol{n}}} \| = n_1 + \ldots + n_s, \hspace{1em} l ( {\ensuremath{\boldsymbol{n}}} ) =
s, \hspace{1em} a_{{\ensuremath{\boldsymbol{n}}}}=a_{n_1}\dots a_{n_s}$$ and, if $n \geq 1$, $$\mathcal{N}_n = \{ {\ensuremath{\boldsymbol{n}}} \in \mathcal{N} \hspace{1em} ; \hspace{1em}
\| {\ensuremath{\boldsymbol{n}}} \| = n \}$$
With these notations, the reduced coproduct (with $a_0=1$) reads $$\Delta'(a_n)=\sum_{k=1}^{n-1} \sum_{{\ensuremath{\boldsymbol{n}}} \in \mathcal{N}_{n-k}} \left( \begin{array}{c}
k+1 \\
l({\ensuremath{\boldsymbol{n}}})
\end{array} \right)a_k \otimes a_{{\ensuremath{\boldsymbol{n}}}}$$ and when iterating the coproduct, we get,
\[prop:iotafdb\] For $n\geq 1$, $$\label{eq:iotafdb}
\iota(a_n)=\sum_{{\ensuremath{\boldsymbol{n}}}\in \mathcal{N}_n}\sum_{{\ensuremath{\boldsymbol{n}}}^1\dots {\ensuremath{\boldsymbol{n}}}^t={\ensuremath{\boldsymbol{n}}}\atop
t\geq 1, l({\ensuremath{\boldsymbol{n}}}^1)=1} \lambda({\ensuremath{\boldsymbol{n}}}^1,\dots, {\ensuremath{\boldsymbol{n}}}^t) a_{{\ensuremath{\boldsymbol{n}}}^1}\otimes \cdots \otimes a_{{\ensuremath{\boldsymbol{n}}}^t}$$ where the sums run over all the decompositions in non empty sequences ${\ensuremath{\boldsymbol{n}}}^1\dots {\ensuremath{\boldsymbol{n}}}^t={\ensuremath{\boldsymbol{n}}}$ and $$\lambda({\ensuremath{\boldsymbol{n}}}^1,\dots, {\ensuremath{\boldsymbol{n}}}^t)=\prod_{i=2}^t \left( \begin{array}{c}
\|{\ensuremath{\boldsymbol{n}}}^1\dots {\ensuremath{\boldsymbol{n}}}^{i-1}\| +1 \\
l({\ensuremath{\boldsymbol{n}}}^i) \end{array} \right)$$
Note that we kept in formula (\[eq:iotafdb\]) the tensor product notation to avoid confusion since we deal with words whose letters are monomials. The proof is simply based on the recursive definition of reduced iterated coproduct and already provides a formula for the composition inverse of a diffeomorphism in $G(A)$.
Let $f ( x ) = x + \sum_{n \geq 1} f_n x^{n + 1}\in G(A)$, we can consider its associated character defined by $\varphi(a_n)=f_n$ and then, using our previous formulas, the coefficients of the composition inverse $g$ of $f$ are given by $$g_n=\varphi^{*{-1}}(a_n)=\sum_{{\ensuremath{\boldsymbol{n}}}=( n_1, \ldots, n_s )\in \mathcal{N}_n}\left( \sum_{{\ensuremath{\boldsymbol{n}}}^1\dots {\ensuremath{\boldsymbol{n}}}^t={\ensuremath{\boldsymbol{n}}}\atop
t\geq 1, l({\ensuremath{\boldsymbol{n}}}^1)=1} (-1)^t \lambda({\ensuremath{\boldsymbol{n}}}^1,\dots, {\ensuremath{\boldsymbol{n}}}^t)\right) f_{n_1}\dots f_{n_s}$$
This result, as the following one, uses the obvious isomorphism between $G(A)$ and $\mathcal{C}(\mathcal{H}_{{\ensuremath{\operatorname{FdB}}}},A)$. One can also compute the Birkhoff decomposition in the group of formal identity-tangent diffeomorphism with coefficients in the a Rota-Baxter algebra of Laurent series $A =\mathbbm{C}[ [ \varepsilon, \varepsilon^{- 1} ]$ with its usual projections $p_+$ and $p_-$ on the regular and polar parts of such series. Any element $$f ( x ) = x + \sum_{n \geq 1} f_n ( \varepsilon ) x^{n+1} \hspace{1em},
\hspace{1em} f_n ( \varepsilon ) \in \mathbbm{C}[ [ \varepsilon, \varepsilon^{- 1} ]$$ can be decomposed as $f_- \circ f = f_+$ with $$\begin{array}{ccccc}
f_- ( x ) & = & \displaystyle x + \sum_{n \geq 1} f_{-, n} ( \varepsilon ) x^{n+1} & &
f_{-, n} ( \varepsilon ) \in \varepsilon^{- 1} \mathbbm{C}[
\varepsilon^{- 1} ]\\
f_+ ( x ) & = & \displaystyle x + \sum_{n \geq 1} f_{+, n} ( \varepsilon ) x^{n+1} & &
f_{+, n} ( \varepsilon ) \in \mathbbm{C}[ [ \varepsilon ] ].
\end{array}$$ Using proposition \[prop:iotafdb\], we get for $n\geq 1$,
The coefficients of the Birkhoff decomposition of a formal identity-tangent diffeomorphism are given by $$\label{bddiff}
\begin{array}{rcl}
\varphi_+(a_n) &=& {\displaystyle}\sum_{{\ensuremath{\boldsymbol{n}}}\in \mathcal{N}_n}\sum_{{\ensuremath{\boldsymbol{n}}}^1\dots {\ensuremath{\boldsymbol{n}}}^t={\ensuremath{\boldsymbol{n}}}\atop
t\geq 1, l({\ensuremath{\boldsymbol{n}}}^1)=1} \lambda({\ensuremath{\boldsymbol{n}}}^1,\dots, {\ensuremath{\boldsymbol{n}}}^t)(-1)^{t-1} p_+ ( p_- ( \ldots ( p_- ( \varphi(a_{{\ensuremath{\boldsymbol{n}}}^1}) ) \varphi(a_{{\ensuremath{\boldsymbol{n}}}^2}) ) \ldots )\varphi(a_{{\ensuremath{\boldsymbol{n}}}^t}) )\\
\varphi_-(a_n) &=& {\displaystyle}\sum_{{\ensuremath{\boldsymbol{n}}}\in \mathcal{N}_n}\sum_{{\ensuremath{\boldsymbol{n}}}^1\dots {\ensuremath{\boldsymbol{n}}}^t={\ensuremath{\boldsymbol{n}}}\atop
t\geq 1, l({\ensuremath{\boldsymbol{n}}}^1)=1} \lambda({\ensuremath{\boldsymbol{n}}}^1,\dots, {\ensuremath{\boldsymbol{n}}}^t)(-1)^{t} p_- ( p_- ( \ldots ( p_- ( \varphi(a_{{\ensuremath{\boldsymbol{n}}}^1}) ) \varphi(a_{{\ensuremath{\boldsymbol{n}}}^2}) ) \ldots )\varphi(a_{{\ensuremath{\boldsymbol{n}}}^t}) )
\end{array}$$ where $\varphi$, $\varphi_+$ and $\varphi_-$ are the characters associated to $f$, $f_+$ and $f_-$ ($\varphi(a_n)=f_n$).
Let us explain how such diffeomorphisms appear in various area, where there Birkhoff decomposition makes sense.
Such a factorization appears first classicaly in quantum field theory: after dimensional regularization, the unrenormalized effective coupling constants are the image by a formal identity-tangent diffeomorphism of the coupling constants of the theory (see [@ck2; @EFP1] for a Hopf algebraic approach). Moreover, the coefficients of this diffeomorphism are Laurent series in the parameter $\varepsilon$ associated to the dimensional regularization process and the Birkhoff decomposition of this diffeomorphism gives directly the bare coupling constants and the renormalized coupling constants.
As proved in [@ck2], in the case of the massless $\phi^3_6$ theory, the unrenormalized effective coupling constant can be written as a diffeomorphism $f ( x ) = x + \sum_{n \geq 1} f_n ( \varepsilon ) x^{n+1}$ where $x$ is the initial coupling constant. From the physical point of view, the decomposition $f_- \circ f = f_+$ is such that, $x + \sum_{n \geq 1} f_{+, n} ( 0) x^{n+1}$ is the renormalized effective constant of the theory.
Such diffeomorphisms (and the need for renormalization) also appear in the classification of dynamical systems, especially when dealing with dynamical systems that cannot be analytically of formally linearized. Let us illustrate this on a very simple example (see [@men13] for a general approach). The following autonomous analytic dynamical system $$\left\lbrace
\begin{array}{rcl}
{\displaystyle}\dot{x} &=&\alpha x\\
{\displaystyle}\dot{z} &=&\beta z +b(x)z^2
\end{array}\right.$$ can be considered as a perturbation of the linear system $$\left\lbrace
\begin{array}{rcl}
{\displaystyle}\dot{x} &=&\alpha x\\
{\displaystyle}\dot{y} &=&\beta y
\end{array} \right.$$ so that one could expect that a change of coordinate $(x,y)=\psi(x,z)=\left(x,f(x,z) \right)$ allows to go from one system to the other one, that is to linearize the first system. In this simple case (see [@men13] for details) the solution should be $f(x,z)=\frac{z}{1-a(x)z}$ where $$\alpha xa'(x)+\beta a(x)+b(x)=0$$ that yields formally, if $b(x)=\sum_{n\geq 0}b_n x^n$, $$a(x)=-\sum_{n\geq 0}\frac{b_n}{\alpha n + \beta} x^n.$$ This series could be ill-defined whenever there exists $n_0$ such that $ \alpha n_0 + \beta=0$. This happens for example with $n=0$ for $(\alpha,\beta)=(1,0)$ and, in this case, we could regularize by considering the system with linear part $(\alpha,\beta)=(1+\varepsilon,\varepsilon)$. As a function of $z$, $f(x,z)$ is then an identity-tangent diffeomorphism whose coefficients are in $\mathbbm{C}[ [x ]][[ \varepsilon ,\varepsilon^{- 1} ]$: $$f(x,z)=\frac{z}{1-a(x)z}=z+\sum_{n\geq 1}a(x)^n z^{n+1} ,\quad a(x)=-\frac{b(0)}{\varepsilon}-\sum_{n\geq1}\frac{b_n}{n(1+\varepsilon)+\varepsilon}x^n.$$ This very simple case can be handled directly and, after Birkhoff decomposition, the regular part in $\varepsilon$ is $$f_+(x,z)=\frac{z}{1-a_+(x)z}=z+\sum_{n\geq 1}a_+(x)^n z^{n+1} ,\quad a_+(x)=-\sum_{n\geq1}\frac{b_n}{n(1+\varepsilon)+\varepsilon}x^n$$ and, for $\varepsilon=0$, the corresponding change of coordinate conjugates the system $$\left\lbrace
\begin{array}{rcl}
{\displaystyle}\dot{x} &=& x\\
{\displaystyle}\dot{z} &=& b(x)z^2
\end{array}\right.$$ to $$\left\lbrace
\begin{array}{rcl}
{\displaystyle}\dot{x} &=& x\\
{\displaystyle}\dot{y} &=&b(0)y^2
\end{array} \right. .$$ This approach can be generalized to more general systems for which the Birkhoff decomposition is not so obvious, so that formula (\[bddiff\]) could be useful. For instance, the same process of regularization/factorization allows to conjugate the system $$\left\lbrace
\begin{array}{rcl}
{\displaystyle}\dot{x} &=& x\\
{\displaystyle}\dot{z} &=& \sum_{k\geq 1} b_k(x)z^{k+1}
\end{array}\right.$$ to a system $$\left\lbrace
\begin{array}{rcl}
{\displaystyle}\dot{x} &=& x\\
{\displaystyle}\dot{y} &=& \sum_{k\geq 1} c_k y^{k+1}
\end{array}\right.$$ which is called a “normal form”, with coefficients $c_k$ that do not depend any more on $x$.
Diffeomorphisms in higher dimension (and thus the corresponding Hopf algebra) appear as well in physics (with more than one coupling constant) and in dynamics: let us consider vector fields given by $\nu$ series ${\ensuremath{\boldsymbol{u}}}({\ensuremath{\boldsymbol{x}}})=(u_1({\ensuremath{\boldsymbol{x}}}),\dots,u_{\nu}({\ensuremath{\boldsymbol{x}}}))\in \CC_{\geq 2}\lbrace {\ensuremath{\boldsymbol{x}}}\rbrace$ of $\nu$ variables ${\ensuremath{\boldsymbol{x}}}=(x_1,\dots,x_{\nu})$ that can be seen as “perturbations” of linear vector fields $(\lambda_1 x_1,\dots,\lambda_{\nu}x_{\nu})$: $$\label{eq:vf}
\frac{dx_i}{dt}=\lambda_i x_i + u_i({\ensuremath{\boldsymbol{x}}})=X_i({\ensuremath{\boldsymbol{x}}}),\quad i=1,\dots, \nu.$$ The linearization problem consists in finding an identity-tangent diffeomorphism $\varphi$ in dimension $\nu$ such that the change of coordinates ${\ensuremath{\boldsymbol{x}}}=\varphi({\ensuremath{\boldsymbol{y}}})$ transforms the previous object into its linear part. For differential equations, this reads, for $i=1,\dots, \nu$: $$\frac{dx_i}{dt}=\sum_{j=1}^{\nu} \frac{dy_j}{dt}\frac{\partial \varphi_i}{\partial y_j}({\ensuremath{\boldsymbol{y}}})=\sum_{j=1}^{\nu} \lambda_j y_j \frac{\partial \varphi_i}{\partial y_j}({\ensuremath{\boldsymbol{y}}})=\lambda_i \varphi_i({\ensuremath{\boldsymbol{y}}})+u_i(\varphi({\ensuremath{\boldsymbol{y}}}))=\lambda_i x_i + u_i({\ensuremath{\boldsymbol{x}}}).\label{eq:homVF}$$ When trying to solve these so-called “homological equations”, some obstructions can occur, independently on any assumption on the analycity of $\varphi$. These equations cannot be formally systematically solved when some combinations $ m_1 \lambda_1 + \ldots m_{\nu} \lambda_{\nu} - \lambda_i$ vanish (here $ i \in \{ 1, \ldots, \nu \},\ m_j\geq0,\ \sum m_j \geqslant 2$):
Such cancellations, which are called *resonances*, prevent from linearizing the differential and one can once again use regularization of the linear part and Birkhoff decomposition to get a change of coordinate that conjugate the vector field to a so-called normal form, see [@men13].
Tensor coalgebras, MZVs, Analysis
=================================
If $X$ be an alphabet (that is a set), its associated tensor vector space $T(X)$ inherits a coalgebra structure related to the concatenation. If we note tensors products as words ${\ensuremath{\boldsymbol{x}}}=x_1\otimes \dots\otimes x_s=x_1\dots x_s$, $$\Delta({\ensuremath{\boldsymbol{x}}})=1\otimes {\ensuremath{\boldsymbol{x}}} +\sum_{{\ensuremath{\boldsymbol{x}}}^1{\ensuremath{\boldsymbol{x}}}^2={\ensuremath{\boldsymbol{x}}}} {\ensuremath{\boldsymbol{x}}}^1\otimes {\ensuremath{\boldsymbol{x}}}^2 + {\ensuremath{\boldsymbol{x}}} \otimes 1$$ where the central sum, that corresponds to the reduced coproduct, is over nonempty words ${\ensuremath{\boldsymbol{x}}}^1,{\ensuremath{\boldsymbol{x}}}^2$ whose concatenation is ${\ensuremath{\boldsymbol{x}}}$.
The quasi-shuffle Hopf algebras $QSh(A)$ are examples of such coalgebras (choose simply a linear basis $X$ of $A$!). There are however many Hopf algebras with such a coalgebra structure that differ as algebras –but the associated map $\iota$ and the associated formula for the Birkhoff decomposition of characters, does not depend on the algebra structure. For the map $\iota$, we obviously get: $$\iota({\ensuremath{\boldsymbol{x}}})= \sum_{{\ensuremath{\boldsymbol{x}}}^1{\ensuremath{\boldsymbol{x}}}^2\dots {\ensuremath{\boldsymbol{x}}}^t={\ensuremath{\boldsymbol{x}}}\atop t\geq 1\ ;\ {\ensuremath{\boldsymbol{x}}}^i\not= \emptyset} {\ensuremath{\boldsymbol{x}}}^1\otimes {\ensuremath{\boldsymbol{x}}}^2 \otimes \dots \otimes {\ensuremath{\boldsymbol{x}}}^t$$ and if $\varphi$ is a character from a Hopf algebra with such a coalgebra structure, with values in a commutative Rota-Baxter algebra $(A,p_+)$, the factorization $\varphi_- \ast \varphi=\varphi_+$ is given for any ${\ensuremath{\boldsymbol{x}}}\in T(X)$ by
$$\label{eq:wordbirk}
\begin{array}{ccc}
\varphi_+ ( {\ensuremath{\boldsymbol{x}}} ) & = & {\displaystyle}\sum_{{\ensuremath{\boldsymbol{x}}}^1{\ensuremath{\boldsymbol{x}}}^2\dots {\ensuremath{\boldsymbol{x}}}^t={\ensuremath{\boldsymbol{x}}}\atop t\geq 1\ ;\ {\ensuremath{\boldsymbol{x}}}^i\not= \emptyset} (-1)^{t-1} p_+ ( p_- ( \ldots ( p_- ( \varphi({\ensuremath{\boldsymbol{x}}}^1)) \varphi( {\ensuremath{\boldsymbol{x}}}^2) ) \ldots ) \varphi({\ensuremath{\boldsymbol{x}}}^t)) \\
\varphi_- ( {\ensuremath{\boldsymbol{x}}} ) & = & {\displaystyle}\sum_{{\ensuremath{\boldsymbol{x}}}^1{\ensuremath{\boldsymbol{x}}}^2\dots {\ensuremath{\boldsymbol{x}}}^t={\ensuremath{\boldsymbol{x}}}\atop t\geq 1\ ;\ {\ensuremath{\boldsymbol{x}}}^i\not= \emptyset} (-1)^{t} p_- ( p_- ( \ldots ( p_- ( \varphi({\ensuremath{\boldsymbol{x}}}^1)) \varphi( {\ensuremath{\boldsymbol{x}}}^2) ) \ldots ) \varphi({\ensuremath{\boldsymbol{x}}}^t))
\end{array}$$
Let us list some example where this formula appear or can be used.
In [@GZ Section 3] Guo and Zhang consider regularized MZV as characters on a quasi-shuffle algebra $\mathcal{H}_{\mathfrak{M}}=T(\mathfrak{M})$ whose quasi-shuffle product stems from the additive semigroup structure of the alphabet $$\mathfrak{M}=\left\{[ \begin{array}{c}
s \\
r
\end{array} ] \ ;\ (s, r) \in \mathbb{Z}\times \mathbb{R}^{+*} \right\rbrace.$$ They propose then a directional regularization of MZV defined on words $$Z([ \begin{array}{c}
s_1 \\
r_1
\end{array} ]\dots [ \begin{array}{c}
s_k \\
r_k
\end{array} ] ;\varepsilon) =\sum_{n_1>\dots>n_k>0}\frac{e^{n_1 r_1 \varepsilon}\dots e^{n_k r_k \varepsilon}}{n_1^{s_1}\dots n_k^{s_k}}$$ that defines a character on $\mathcal{H}_{\mathfrak{M}}$ with values in an algebra of Laurent series. The formula they give for the Birkhoff decomposition (Theorem 3.8) coincide equation (\[eq:wordbirk\]).
As a toy model for applications in physics [@EFP1 section 4.2] considers a character on the polynomial commutative Hopf algebra $\mathcal{H}^{\text{lad}}$ of ladder trees. If the ladder tree with $n$ nodes is $t_n$, then $$\Delta(t_n)=t_n \otimes 1+\sum_{k=1}^{n-1} t_k \otimes t_{n-k} +1 \otimes t_n.$$ It is a matter of fact to identify the coalgebra structure of $\mathcal{H}^{\text{lad}}$ with the tensor deconcatenation coalgebra $T(\{x\})$ over an alphabet with one letter, where $t_n$ corresponds to the word $\underbrace{x \dots x}_n$. Formula (\[eq:wordbirk\]) can be applied to the character mapping the tree $t_n$ to an $n$-fold Chen’s iterated integral defined recursively by $$\psi(p ; \varepsilon, \mu)(t_n)= \mu^{\varepsilon}\int_p^{\infty} \psi(x ; \varepsilon, \mu)(t_{n-1})\frac{dx}{x^{1+\varepsilon}}=\frac{e^{-n\varepsilon \log(p/\mu)}}{n!\varepsilon^n}=f_n(\varepsilon)$$ with values in the Laurent series in $\varepsilon$. We get for the couterterms: $$\psi_-(p ; \varepsilon, \mu)(t_n))=\sum_{n_1+\dots+n_t=n \atop t\geq 1 \ ,\ n_i >0}(-1)^{t} (-1)^{t} p_- ( p_- ( \ldots ( p_- ( f_{n_1}(\varepsilon)) f_{n_2}(\varepsilon) ) \ldots ) f_{n_t}(\varepsilon))$$
When dealing with differential equations and associated diffeomorphisms (flow, conjugacy map), characters on shuffle Hopf algebras appear almost naturally. For instance, such characters correspond to:
- the coefficients of word series in [@MSS],
- “symmetral moulds” in mould calculus (see [@FM; @snag])
- or Chen’s iterated integrals (see for instance [@kre; @manchon]).
Let us just give the example of a simple differential equation related to mould calculus (see [@men06]). Let $b ( x, y )=\sum_{n\geq 0}x^n b_n(y) \in y^2 \mathbbm{C}[ [ x, y ] ]$ and $d \in \mathbbm{N}$. If one looks for a formal identity tangent diffeomorphism $\varphi ( x, y )$ in $y$, with coefficients in $\mathbbm{C}[ [ x ] ]$ such that, if $y$ is a solution of $$( E_{b, d} ) \hspace{2em} x^{1 - d} \partial_x y = b ( x, y )$$ then $z = \varphi ( x, y )$ is a solution of $$( E_{0, d} ) \hspace{2em} x^{1 - d} \partial_x z = 0.$$ One can try to compute this diffeomorphism as a “mould series”: $$\label{eq:moulddiff}
\varphi_d ( x, y )=y + \sum_{s \geq 1} \sum_{n_1, \ldots n_s \in
\mathbbm{N}} V_d(n_1, \ldots, n_s) \mathbbm{B}_{n_s} \ldots
\mathbbm{B}_{n_1}.y \quad (\mathbbm{B}_n = b_n ( y ) \partial_y )$$ where $V_d$ is a character on the shuffle algebra $T(\mathbbm{N})$, with values in $\mathbbm{C}[ [ x ] ]$. Whenever $d$ is a positive integer, this character can be computed and for any word $( n_1, \ldots, n_s )$
$$V_d(n_1, \ldots, n_s) = \frac{( - 1 )^s x^{n_1 + \ldots + n_s + s
d}}{( \check{n}_1 + d ) ( \check{n}_2 + 2 d ) \ldots ( \check{n}_s + s
d )} \quad ( \check{n}_i = n_1 + \ldots + n_i ).$$
The map $\varphi_d ( x, y )\in \mathbb{C}[[x,y]]$ is then well defined and conjugates $( E_{b, d} )$ to $(E_{0, d} )$. For $d=0$, there may be divisions by $0$ and, in this case, one can consider $d=\varepsilon$ as a real parameter and use the expansion $x^{\varepsilon}=\sum \frac{(\varepsilon \log x))^n}{n!}$ so that the character $V_{\varepsilon}$ has its values in $\mathfrak{B}[ [
\varepsilon ] ] [ \varepsilon^{- 1} ]$ where $\mathfrak{B}=\mathbbm{C}[[\log x,
x]$. If one uses the same formula (\[eq:wordbirk\]) to perform the Birkhoff decomposition, the regular character $V_{\varepsilon,+}$, evaluated at $\varepsilon=0$ allows to find a diffeomorphism (as in equation (\[eq:moulddiff\])) that conjugates $x \partial_x
y = b ( x, y )$ to $x \partial_x z = 0$ with a price to pay : it contains monomials in $x$ and $\log x$. See [@men06] for details.
Not also that the same ideas can be used for the the even-odd factorization of characters in combinatorial Hopf algebras (see [[@ag1]]{}, [[@ag2]]{} and [[@guo]]{}).
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[^1]: Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France. [[[*Email*]{}: [[email protected]]{}]{}]{}
[^2]: Laboratoire J.A. Dieudonné, Université de la Côte d’Azur, CNRS, UMR 7531, Parc Valrose, 06108 Nice Cedex 2, France. [[[*Email*]{}: [[email protected]]{}]{}]{}
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'This paper deals with the massive three-dimensional Dirac operator coupled with a Lorentz scalar shell interaction supported on a compact smooth surface. The rigorous definition of the operator involves suitable transmission conditions along the surface. After showing the self-adjointness of the resulting operator we switch to the investigation of its spectral properties, in particular, to the existence and non-existence of eigenvalues. In the case of an attractive coupling, we study the eigenvalue asymptotics as the mass becomes large and show that the behavior of the individual eigenvalues and their total number are governed by an effective Schrödinger operator on the boundary with an external Yang-Mills potential and a curvature-induced potential.'
address:
- 'Institut für Numerische Mathematik, Technische Universität Graz, Steyrergasse 30, 8010 Graz, Austria'
- 'Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France'
- 'Laboratoire de Mathématiques d’Orsay, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405 Orsay, France'
author:
- Markus Holzmann
- 'Thomas Ourmières-Bonafos'
- Konstantin Pankrashkin
title: |
Dirac operators\
with Lorentz scalar shell interactions
---
Introduction
============
Motivations and main results
----------------------------
The Dirac operator was introduced to give a quantum mechanical framework that takes relativistic properties of particles of spin $\frac{1}{2}$ into account. This operator can be seen as a relativistic counterpart of the Schrödinger operator and, as for this latter, the behavior of physical systems can be deduced from a thorough spectral analysis [@T92].
In the present paper we focus on a class of Dirac operators with potentials supported on zero measure sets (the so-called $\delta$-potentials). Such interactions are often used in mathematical physics as idealizations for regular potentials located in a neighborhood of this zero set. While such operators are well understood in the one-dimensional case, see e.g. [@AGHH; @CMR; @GS; @PR] as well as for the closely related radial mutidimensional case [@DES89], the systematic study in higher dimension appeared to be much more involved and attracted a lot of attention recently. It seems that the first results on Dirac operators with interactions supported on general smooth surfaces (shells) were obtained in [@AMV14; @AMV15; @AMV15bis], where the self-adjointness and the discrete spectrum were discussed. The analysis was based mostly on the usage of potential operators involving the fundamental solution of the unperturbed Dirac equation. In [@BEHL17; @BH17; @OV16], the study was pushed further in order to understand the Sobolev regularity of functions in the domain, the $\delta$-shell potential being then encoded by a transmission condition at the shell. Furthermore, as for Schrödinger operators with $\delta$-potentials [@BEHL17_1], the shell interactions in the Dirac setting can be understood as suitable limits of regular potentials localized near the surface, as it was shown recently in [@MP16; @MP17]. One of the main motivations for the present paper is the recent work [@ALTR16], where the closely related MIT bag model dealing with Dirac operators in bounded domains and special boundary conditions were studied. In fact, it is shown in [@ALTR16] that for large negative masses the asymptotics of the MIT bag eigenvalues is determined by an effective operator acting on the boundary, and it is one of our objectives to study the related problem for scalar shell interactions.
We are going to study the specific case of the three-dimensional Dirac operator coupled with a Lorentz scalar shell interaction of strength $\tau\in{\mathbb{R}}$ supported on a smooth compact surface $\Sigma$. The operator acts in $L^2({\mathbb{R}}^3,\mathbb{C}^4)$ and writes formally as $$\label{eqn:op_form}
A_{m,\tau}:= -{\mathrm{i}}\big(\alpha_1\partial_1+\alpha_2\partial_2+\alpha_3\partial_3\big) +m\beta + \tau \beta\delta_{\Sigma},$$ where $\alpha_1,\alpha_2,\alpha_3,\beta$ are the standard $\mathbb{C}^{4\times4}$ Dirac matrices written down explicitly in , $m\in\mathbb{R}$ is the mass of the particle and $\delta_\Sigma$ is the Dirac distribution on $\Sigma$. The expression is formal due to the presence of the singular term $\delta_\Sigma$, the rigorous definition of $A_{m,\tau}$ is given below in using suitable transmission conditions at $\Sigma$. We remark that the special value $\tau=0$ corresponds to the free Dirac operator, whose properties are well known (see Section \[section\_free\_Operator\]). Furthermore, the values $\tau=\pm 2$ play a special role as they correspond to “hard walls” at $\Sigma$, i.e. $A_{m,\pm2}$ is decoupled and represent the direct sum of two operators acting inside and outside of $\Sigma$; this corresponds to the so-called MIT bag model already considered in [@ALTR16], see Remark \[rem-mit\] below.
In what follows we exclude the above special values of $\tau$. Our main results can be roughly summed up as follows.
- The operator $A_{m,\tau}$ defined as in below is self-adjoint, its spectrum is symmetric with respect to $0$, and its essential spectrum is $(-\infty,-|m|]\cup[|m|,+\infty)$.
- The operator $A_{m,\tau}$ is unitarily equivalent to $A_{m,\frac{4}{\tau}}$ and to $A_{-m,-\tau}$.
In view of the preceding symmetry, without loss of generality for the subsequent points we assume that $m\ge 0$.
- If $\tau \geq 0$, the discrete spectrum of $A_{m,\tau}$ is empty.
- For any $m>0$ there exists $\tau_m>0$ such that the discrete spectrum of $A_{m,\tau}$ is empty for $|\tau|<\tau_m$ and for $|\tau|>\dfrac{4}{\tau_m}$.
Finally, being motivated by the analysis of [@ALTR16] we provide an asymptotic study of the discrete spectrum for the case when $$\label{eq-as000}
\tau<0 \text{ with }\tau\ne -2\text{ is fixed }, \quad m\to +\infty$$ and obtain the following results:
- The total number of discrete eigenvalues of $A_{m,\tau}$ counted with multiplicities obeys a Weyl-type law and behaves as $$\frac{16}{\pi} \frac{\tau^2}{(\tau^2 + 4)^2}|\Sigma|m^2+{\mathcal{O}}(m\log m),$$ with $|\Sigma|$ being the surface area of $\Sigma$.
- Denote the eigenvalues of $A_{m,\tau}$ by $\pm \mu_j(m)$ with $\mu_j(m)\ge 0$ enumerated in the non-decreasing order, then for each fixed $j\in{\mathbb{N}}$ there holds $$\label{eq-eig00}
\mu_j(m) = \frac{|\tau^2 -4|}{\tau^2+4}m + \frac{\tau^2+4}{|\tau^2 - 4|}\frac{E_j(\Upsilon_\tau)}{2m} + \mathcal{O}\Big(\frac{\log m}{m^2}\Big),$$ where $E_j(\Upsilon_\tau)$ is the $j$-th eigenvalue of the $m$-independent Schrödinger operator $\Upsilon_\tau$ with an external Yang-Mills potential in $L^2(\Sigma,{\mathbb{C}}^2)$, $$\Upsilon_\tau = \Big({\mathrm{d}}+ {\mathrm{i}}\dfrac{4}{\tau^2+4}\,\omega\Big)^*\Big({\mathrm{d}}+ {\mathrm{i}}\dfrac{4}{\tau^2+4}\,\omega\Big) - \Big(\dfrac{\tau^2-4}{\tau^2+4}\Big)^2 M^2 + \dfrac{\tau^4+16}{(\tau^2+4)^2} K,$$ where $K$ and $M$ are respectively the Gauss and mean curvature and the $1$-form $\omega$ is given by the local expression $\omega:=\sigma\cdot (\nu\times \partial_1 \nu){\mathrm{d}}s_1+\sigma\cdot (\nu\times \partial_2 \nu){\mathrm{d}}s_2$ with $\nu$ being the outer unit normal on $\Sigma$. (The precise definition of $\Upsilon_\tau$ is given in Subsection \[sec-eff\].)
We remark that by setting formally $\tau=\pm 2$ in one recovers the eigenvalue asymptotics for the MIT bag model as obtained in [@ALTR16 Thm. 1.13] with the effective operator written in an alternative way.
Let us describe the structure of the paper. In the following Section \[ssec-not\] we introduce first a couple of conventions used throughout the text. Section \[sec-qual\] is devoted to the definition of the operator and to the proof of the assertions (A) and (B), see Theorem \[thm-basic\]. The proofs are mostly based on the use of singular integral operators previously studied in [@OV16] and some resolvent machineries already used in a similar (but different) context in [@BEHL17; @BH17]. In Section \[sec-var\] we deal with a more detailed study of the discrete spectrum. The key idea of the analysis is to obtain the sesquilinear form for the square of $A_{m,\tau}$. The squared operator clearly acts as the (shifted) Laplacian away from $\Sigma$, and the main difficulty is to understand how the transmission condition translates to $A_{m,\tau}^2$, which is settled in Proposition \[theorem\_form\_B\_tau\_square\]. The approach is reminiscent of [@HMR02 p. 379] and [@ALTR16] for other types of Dirac operators. It turns out that the quadratic form for $A_{m,\tau}^2$ is given by the same expression as the one for the so-called $\delta'$-potential, see e.g. [@BLL12 Prop. 3.15], but defined on a smaller domain. Hence, our construction delivers a new type of generalized surface interactions [@ER]. Nevertheless, an additional geometry-induced constraint along $\Sigma$ leads to a much more involved analysis and a completely different behavior when compared to the $\delta'$-interaction studied, e.g., in [@EJ14]. In particular, Propositions \[prop3.4\] and \[thm-disc1\] cover the above points (C) and (D). Section \[sec:dis-spec\] is then devoted to the study of the asymptotic regime , and the points (E) and (F) follow from Corollaries \[cor32\] and \[cor33\], which are both consequences of a central estimate given in Theorem \[thm-ev1\]. In fact, the asymptotic analysis does not use the above operator $\Upsilon_\tau$ but another unitary equivalent operator introduced in Section \[ssec42\] which is easier to deal with and which implies an equivalent reformulation given in Proposition \[prop-main1\]. The upper and lower bounds for the eigenvalues are then obtained separately in Subsections \[sec-upp1\] and \[sec-low1\] respectively, by comparing the operator $A_{m,\tau}$ first with operators in thin neighborhoods of $\Sigma$ and then, using a change of variable, with operators with separated variables in $\Sigma\times I$ with $I$ being a one-dimensional interval, whose one-dimensional part is analyzed directly similar to, e.g., [@EY; @PP]. Contrary to the approach of [@ALTR16] our study does not use semi-classical type estimates, which allows a self-contained proof.
Notations {#ssec-not}
---------
For a Hilbert space ${\mathcal{H}}$, one denotes by $\langle\cdot,\cdot\rangle_{{\mathcal{H}}}$ the scalar product on ${\mathcal{H}}$ and by $\|\cdot\|_{{\mathcal{H}}}$ the associated norm. As there is no risk of confusion and for the sake of readability, we simply set $\|\cdot\|_{\mathbb{C}^4} = |\cdot|$ and $\|\cdot\|_{\mathbb{R}^3\otimes\mathbb{C}^4} = |\cdot|$.
By ${\mathbf{B}}({\mathcal{H}})$ we denote the Banach space of bounded linear operators in ${\mathcal{H}}$. If $T$ is a self-adjoint operator in ${\mathcal{H}}$, then we denote by ${\mathop{\mathcal{D}}}(T)$ its domain, by $\ker(T)$ and ${\mathop{\mathrm{ran}}}(T)$ its kernel and range respectively, and $E_n(T)$ will stand for the $n$-th eigenvalue of $T$ when enumerated in the non-decreasing order and counted according to multiplicities. The spectrum of $T$ is denoted by ${\mathop{\mathrm{spec}}\nolimits}(T)$, the essential spectrum by ${\mathop{\mathrm{spec}}\nolimits}_{\mathrm{ess}}(T)$ and the resolvent set by ${\mathop{\mathrm{res}}\nolimits}(T)$. If the operator $T$ in ${\mathcal{H}}$ is generated by a closed lower semibounded sesquilinear form $t$ defined on the domain ${\mathop{\mathcal{D}}}(t)$, then the following variational characterization of the eigenvalues holds (min-max principle): for $n\in{\mathbb{N}}$ set $$\varepsilon_n(T):=\inf_{\substack{V\subset {\mathop{\mathcal{D}}}(t)\\ \dim V=n}}\sup_{\substack{u\in V\\u\ne 0}} \dfrac{t(u,u)}{\|u\|^2_{{\mathcal{H}}}},$$ then $E_n(T)=\varepsilon_n(T)$ if $\varepsilon_n(T)<\inf{\mathop{\mathrm{spec}}\nolimits}_{\mathrm{ess}}(T)$, otherwise one has $\varepsilon_m(T)=\inf{\mathop{\mathrm{spec}}\nolimits}_{\mathrm{ess}}(T)$ for all $m\ge n$. We sometimes write $E_n(t):=E_n(T)$ and $\varepsilon_n(t):=\varepsilon_n(T)$. Furthermore, for $E\in{\mathbb{R}}$ we denote by ${\mathcal{N}}(T,E)$ the number of eigenvalues of $T$ in $(-\infty,E)$ and set ${\mathcal{N}}(t,E):={\mathcal{N}}(T,E)$.
For two closed and semibounded from below sesquilinear forms $t_1$ and $t_2$ their direct sum $t_1\oplus t_2$ is the sesquilinear form defined on ${\mathop{\mathcal{D}}}(t_1\oplus t_2):={\mathop{\mathcal{D}}}(t_1) \times {\mathop{\mathcal{D}}}(t_2)$ by $$(t_1\oplus t_2)\big((u_1,u_2),(u_1,u_2)\big):=t_1(u_1,u_1)+t_2(u_2,u_2),
\quad (u_1, u_2) \in {\mathop{\mathcal{D}}}(t_1) \times {\mathop{\mathcal{D}}}(t_2).$$ If $T_1$ and $T_2$ are the operators associated with $t_1$ and $t_2$, then the operator associated with $t_1\oplus t_2$ is $T_1\oplus T_2$, and ${\mathcal{N}}(t,E)={\mathcal{N}}(t_1,E)+{\mathcal{N}}(t_2,E)$. The form inequality $t_1\ge t_2$ means that ${\mathop{\mathcal{D}}}(t_1)\subseteq {\mathop{\mathcal{D}}}(t_2)$ and $t_1(u)\ge t_2(u)$ for all $u\in {\mathop{\mathcal{D}}}(t_1)$. By the min-max principle the form inequality implies the respective inequality for the Rayleigh quotients, $\varepsilon_n(t_1)\ge \varepsilon_n(t_2)$ for any $n\in{\mathbb{N}}$, and the reverse inequality for the eigenvalue counting functions, ${\mathcal{N}}(t_1,E)\le {\mathcal{N}}(t_2,E)$ for all $E\in{\mathbb{R}}$.
Let $\alpha_1$, $\alpha_2$, $\alpha_3$, $\beta$ and $\gamma_5$ be the $4\times 4$ Dirac matrices $$\label{def_Dirac_matrices}
\alpha_j := \begin{pmatrix} 0 & \sigma_j \\ \sigma_j & 0 \end{pmatrix},
\quad \beta := \begin{pmatrix} I_2 & 0 \\ 0 & -I_2 \end{pmatrix}, \quad
\gamma_5: = \begin{pmatrix} 0 & I_2 \\ I_2 & 0 \end{pmatrix},$$ where $I_k$ denotes the $k \times k$ identity matrix and $\sigma_j$ are the $2\times2$ Pauli spin matrices, $$\sigma_1 := \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, \qquad
\sigma_2 := \begin{pmatrix} 0 & -{\mathrm{i}}\\ {\mathrm{i}}& 0 \end{pmatrix}, \qquad
\sigma_3 := \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}.$$ The Dirac matrices fulfill the anti-commutation relations $$\begin{gathered}
\label{eq_commutation}
\alpha_j\alpha_k + \alpha_k\alpha_j = 2\delta_{jk}I_4,\qquad
j,k\in\{0,1,2,3\}, \quad \alpha_0 := \beta,\\
\label{commutation_gamma_5}
\gamma_5 \alpha_j = \alpha_j \gamma_5, ~j \in \{ 1, 2, 3\}, \quad \gamma_5 \beta = -\beta \gamma_5.\end{gathered}$$ For vectors $x = (x_1, x_2, x_3)\in{\mathbb{R}}^3$ we employ the notation $$\alpha \cdot x := \alpha_1 x_1 + \alpha_2 x_2 + \alpha_3 x_3,
\quad
\sigma \cdot x := \sigma_1 x_1 + \sigma_2 x_2 + \sigma_3 x_3.$$
Qualitative spectral properties {#sec-qual}
===============================
Definition of the operator
--------------------------
Let $\Omega_+\subset {\mathbb{R}}^3$ be a bounded $C^4$ smooth domain. We set $$\Sigma:=\partial\Omega_+, \quad \Omega_-:={\mathbb{R}}^3\setminus \overline{\Omega_-},$$ and denote by $\nu$ the unit normal vector field on $\Sigma$ pointing outwards of $\Omega_+$. For $s \in \Sigma$ and $\tau\in{\mathbb{R}}$ we set $$\label{def_M}
{\mathcal{B}}(s) := - {\mathrm{i}}\beta \alpha \cdot \nu(s), \quad {\mathcal{P}}_\tau^\pm(s) := \frac{\tau}{2} \pm {\mathcal{B}}(s).$$ Note that for any $s \in \Sigma$ the matrix ${\mathcal{B}}(s)$ is self-adjoint and unitary by .
For $m\in{\mathbb{R}}$ and $\tau\in{\mathbb{R}}$, we denote by $A_{m,\tau}$ the operator in $L^2({\mathbb{R}}^3,{\mathbb{C}}^4)\simeq L^2(\Omega_+,{\mathbb{C}}^4)\oplus L^2(\Omega_-,{\mathbb{C}}^4)$ acting as $$\label{equation_dom_B_tau}
\begin{aligned}
A_{m,\tau} u &= \Big((-{\mathrm{i}}\alpha \cdot \nabla + m \beta) u_+ , (-{\mathrm{i}}\alpha \cdot \nabla + m \beta) u_-\Big),\\
{\mathop{\mathcal{D}}}(A_{m,\tau}) &= \big\{
u = (u_+,u_-): u_\pm\in H^1(\Omega_\pm, \mathbb{C}^4), \
{\mathcal{P}}_\tau^- u_+ +{\mathcal{P}}_\tau^+ u_-= 0 \text{ on } \Sigma\big\}.
\end{aligned}$$ For $\tau \in \mathbb{R} \setminus \{ -2,2\}$ we set $$\label{def_R_tau}
{\mathcal{R}}_\tau^\pm := -({\mathcal{P}}_\tau^\mp )^{-1} {\mathcal{P}}_\tau^\pm= \dfrac{4+\tau^2}{4-\tau^2} I_4 \pm \dfrac{4\tau}{4-\tau^2}\,{\mathcal{B}}.$$ Then one has the commutation relations $$\label{eqcommr1}
{\mathcal{R}}_\tau^\pm {\mathcal{B}}={\mathcal{B}}{\mathcal{R}}_\tau^\pm, \quad
{\mathcal{R}}_\tau^\pm\gamma_5= \gamma_5 {\mathcal{R}}_\tau^\mp.$$ For $\tau \notin\{ -2, 0, 2 \}$, the transmission condition for $u\in {\mathop{\mathcal{D}}}(A_{m,\tau})$ can equivalently be rewritten as $$\label{eqtrans2}
u_+={\mathcal{R}}_\tau^+ u_- \quad \text{ or } \quad
u_-={\mathcal{R}}_\tau^- u_+
\quad \text{ or } \quad
u_+ + u_- = \dfrac{2}{\tau} {\mathcal{B}}(u_+ - u_-).$$
\[rem-mit\] For $|\tau|=2$ the transmission condition in decomposes as $$\begin{aligned}
u_+ &= {\mathcal{B}}u_+, &\qquad u_- &= - {\mathcal{B}}u_- && \text{ for }\tau = 2,\\
u_+ &= -{\mathcal{B}}u_+, &\qquad u_- &= {\mathcal{B}}u_- && \text{ for } \tau = -2,\end{aligned}$$ i.e. $A_{m, \pm 2}$ is the orthogonal sum of Dirac operators in $\Omega_\pm$ with MIT bag boundary conditions as studied, e.g., in [@ALTR16; @OV16]. Using the language of [@ALTR16], for $\tau=2$ and $m>0$ one recovers the MIT bag operator with the positive mass $m$ in $\Omega_+$ and the one with a negative mass $(-m)$ in $\Omega_-$ (and vice versa for $\tau=-2$).
As mentioned in the introduction it was shown in [@MP16; @MP17] that, under some technical assumptions, the operators $A_{m,\tau}$ can be approximated by Dirac operators with regular potentials. As $A_{m,\tau}$ approximates $A_{m,\pm 2}$ for $\tau$ tending to $\pm2$, this could provide a new interpretation and regularization of MIT bag operators with negative masses, namely as the restriction of the limit of Dirac operators with suitable squeezed potentials and positive mass. The missing point in this program is the fact that the technical restrictions of [@MP16] do not allow to study the values of $\tau$ close to $\pm 2$.
The transmission condition in corresponds to the operator acting as formally written in , cf. [@AMV15 Section 5]. Indeed, for $u =(u_+,u_-)\in H^1(\Omega_+,\mathbb{C}^4)\times H^1(\Omega_-,\mathbb{C}^4)$ let us define the distribution $\delta_\Sigma u$ by its action $$\langle \delta_\Sigma u,\varphi\rangle = \frac12\iint_{\Sigma}(u_+ + u_-)\varphi \,{\mathrm{d}}\Sigma,\quad\varphi\in C_0^\infty(\mathbb{R}^3,\mathbb{C}^4).$$ with ${\mathrm{d}}\Sigma$ being the surface measure. When computing $A_{m,\tau} u$ in the distributional sense using the above definition of $\delta_\Sigma u$ and the expression given in , one sees that the transmission condition in ensures that $A_{m,\tau} u$ belongs to $L^2(\mathbb{R}^3,\mathbb{C}^4)$.
Let us list some basic properties of the operator $A_{m, \tau}$:
\[thm-basic\] The operator $A_{m,\tau}$ defined in is self-adjoint, and the following assertions hold true:
1. the essential spectrum of $A_{m,\tau}$ is $\big(-\infty,-|m|\big]\mathop{\cup}\big[|m|,+\infty)$,
2. the spectrum of $A_{m,\tau}$ is symmetric with respect to $0$,
3. each eigenvalue of $A_{m,\tau}$ has an even multiplicity,
4. for $\tau\neq0$, the operator $A_{m,\tau}$ is unitarily equivalent to $A_{m,\frac{4}{\tau}}$,
5. the operator $A_{-m,-\tau}$ is unitarily equivalent to $A_{m,\tau}$.
The results will be deduced from [@BEHL17; @BH17] by applying the abstract machinery developed there for suitable boundary conditions. To keep the paper self-contained we give a complete proof in the rest of this section. We first introduce some related integral operators in Section \[section\_free\_Operator\], and with their help we prove the self-adjointness of $A_{m,\tau}$ in Proposition \[theorem\_basic\_properties\]. The points (a)–(e) are justified in Section \[sec-ae\].
Auxiliary integral operators {#section_free_Operator}
----------------------------
First, we define the free Dirac operator and discuss some of its properties which will be needed for our further considerations. Recall the definition of the Dirac matrices $\alpha_j$ and $\beta$ from . Then, the free Dirac operator $A_{m, 0}$ is given by $$\label{def_free_Dirac}
A_{m, 0} u := -{\mathrm{i}}\sum_{j=1}^3 \alpha_j \partial_j u + m \beta u = -{\mathrm{i}}(\alpha \cdot \nabla) u + m \beta u,
\quad {\mathop{\mathcal{D}}}(A_{m, 0}) = H^1(\mathbb{R}^3, \mathbb{C}^4).$$ With the help of the Fourier transform one easily sees that $A_{m, 0}$ is self-adjoint and that $$\label{spectrum_A_{m, 0}}
{\mathop{\mathrm{spec}}\nolimits}(A_{m, 0}) = {\mathop{\mathrm{spec}}\nolimits}_{\mathrm{ess}}(A_{m, 0}) = (-\infty, -|m|] \cup [|m|, \infty).$$ For $\lambda \in {\mathop{\mathrm{res}}\nolimits}(A_{m, 0}) = \mathbb{C} \setminus \big( (-\infty, -|m|] \cup [|m|, \infty) \big)$ the resolvent of $A_{m, 0}$ is given by $$\begin{gathered}
(A_{m, 0} - \lambda)^{-1} u(x) = \iiint_{\mathbb{R}^3} G_\lambda(x-y) u(y) {\mathrm{d}}y,\\
G_\lambda(x) = \left( \lambda I_4 + m \beta
+ \left( 1 - {\mathrm{i}}\sqrt{\lambda^2 - m^2} |x| \right) \frac{{\mathrm{i}}(\alpha \cdot x )}{|x|^2} \right)
\frac{e^{{\mathrm{i}}\sqrt{\lambda^2 - m^2} |x|}}{4 \pi |x|};\end{gathered}$$ cf. [@T92 Section 1.E] or [@AMV14 Lemma 2.1]. In this formula we use the convention $\Im \sqrt{\lambda^2 - m^2} > 0$. The resolvent of $A_{m, 0}$ and the particular form of its integral kernel will be important later for the basic spectral analysis of the Dirac operator with a Lorentz scalar $\delta$-shell interaction.
Now we are going to discuss some integral operators which are related to the Green’s function $G_\lambda$. For $\lambda \in {\mathop{\mathrm{res}}\nolimits}(A_{m, 0})$ we define $\Phi_\lambda: L^2(\Sigma, \mathbb{C}^4) \rightarrow L^2(\mathbb{R}^3, \mathbb{C}^4)$ acting as $$\label{def_gamma_lambda}
\Phi_\lambda \varphi(x) := \iint_\Sigma G_\lambda(x-y) \varphi(y) {\mathrm{d}}\Sigma(y),
\quad \varphi \in L^2(\Sigma, \mathbb{C}^4),~x \in \mathbb{R}^3,$$ and $\mathcal{C}_\lambda: H^{\frac{1}{2}}(\Sigma, \mathbb{C}^4) \rightarrow H^{\frac{1}{2}}(\Sigma, \mathbb{C}^4)$, $$\label{def_M_lambda}
\mathcal{C}_\lambda \varphi(x) := \lim_{\varepsilon \searrow 0} \iint_{\Sigma \setminus B(x, \varepsilon)}
G_\lambda(x-y) \varphi(y) {\mathrm{d}}\Sigma(y), \quad
\varphi \in H^{\frac{1}{2}}(\Sigma, \mathbb{C}^4),~x \in \Sigma,$$ where ${\mathrm{d}}\Sigma$ is the surface measure on $\Sigma$ and $B(x,\varepsilon)$ is the ball of radius $\varepsilon$ centered at $x$. Both operators $\Phi_\lambda$ and $\mathcal{C}_\lambda$ are well-defined and bounded, see [@BEHL17 Proposition 3.4] and [@BH17 Proposition 4.2 (ii)] or [@OV16 Sections 2.1 and 2.2], and $\Phi_\lambda$ is injective by [@BEHL17 Proposition 3.4 and Definition 2.3]. We also note the useful property $$\label{equation_gamma_smooth}
\Phi_\lambda \varphi \in H^1(\Omega_+, \mathbb{C}^4) \oplus H^1(\Omega_-, \mathbb{C}^4)
\quad \text{for} \quad \varphi \in H^{\frac{1}{2}}(\Sigma, \mathbb{C}^4);$$ cf. [@BH17 Proposition 4.2 (i)]. Moreover, if $\lambda \in {\mathop{\mathrm{res}}\nolimits}(A_{m, 0})$, then a function $u_\lambda \in H^1(\Omega_+, \mathbb {C}^4) \oplus H^1(\Omega_-, \mathbb{C}^4)$ satisfies $$(-{\mathrm{i}}\alpha \cdot \nabla + m \beta - \lambda) u_\lambda = 0 \text{ in } \Omega_\pm$$ iff there exists $\varphi \in H^{\frac{1}{2}}(\Sigma, \mathbb{C}^4)$ such that $$\label{equation_kernel}
u_\lambda = \Phi_\lambda \varphi;$$ cf. [@BH17 Proposition 4.2]. The adjoint $\Phi_\lambda^*: L^2(\mathbb{R}^3, \mathbb{C}^4) \rightarrow L^2(\Sigma, \mathbb{C}^4)$ of $\Phi_\lambda$ acts as $$\label{equation_gamma_lambda_star}
\Phi_\lambda^* u = \big( (A_{m, 0} - \overline{\lambda})^{-1} u \big)\big|_\Sigma$$ and it has the more explicit representation $$\Phi_\lambda^* u(x) = \iiint_{\mathbb{R}^3} G_{\overline{\lambda}}(x-y) u(y) {\mathrm{d}}y, \quad
u \in L^2(\mathbb{R}^3, \mathbb{C}^4),~x \in \Sigma.$$
Let $\varphi \in H^{\frac{1}{2}}(\Sigma, \mathbb{C}^4)$ and $\lambda \in {\mathop{\mathrm{res}}\nolimits}(A_{m, 0})$. Then, the trace on $\Sigma$ of $$\Phi_\lambda \varphi = \big( (\Phi_\lambda \varphi)_+, (\Phi_\lambda \varphi)_- \big)
\in H^1(\Omega_+, \mathbb{C}^4) \oplus H^1(\Omega_-, \mathbb{C}^4)$$ is $\big((\Phi_\lambda \varphi)_\pm\big) \big|_\Sigma = \mathcal{C}_\lambda \varphi \mp \dfrac{{\mathrm{i}}}{2} (\alpha \cdot \nu) \varphi$, see [@AMV15 Lemma 2.2] for $\lambda \in (-|m|, |m|)$; the case $\lambda \in \mathbb{C} \setminus \mathbb{R}$ can be shown in the same way. In particular, we have $$\begin{gathered}
\label{jump1}
\frac{1}{2} \left( (\Phi_\lambda \varphi)_+ + (\Phi_\lambda \varphi)_- \right)
= \mathcal{C}_\lambda \varphi \text{ on } \Sigma,\\
\label{jump2}
{\mathrm{i}}\alpha \cdot \nu \left( (\Phi_\lambda \varphi)_+ - (\Phi_\lambda \varphi)_- \right) = \varphi
\text{ on } \Sigma.\end{gathered}$$ The operator $\mathcal{C}_\lambda^2 - \frac{1}{4}I_4$ can be extended to a bounded operator $$\label{C_lambda_compact}
\widetilde{\mathcal{C}}_\lambda^2 - \frac{1}{4} I_4: H^{-\frac{1}{2}}(\Sigma, \mathbb{C}^4) \rightarrow H^{\frac{1}{2}}(\Sigma, \mathbb{C}^4),$$ see [@BH17 Proposition 4.4 (iii)] and also [@OV16 Proposition 2.8]. In particular, the operator $\big(\mathcal{C}_\lambda^2 - \frac{1}{4}I_4 \big)$ is compact in $H^{\frac{1}{2}}(\Sigma, \mathbb{C}^4)$.
We end this section with a variant of the Birman-Schwinger principle for the operator $A_{m, \tau}$. It is a special variant of the general result stated in [@BEHL17 Theorem 2.4] or [@AMV15 Proposition 3.1]; to keep the presentation self-contained, we add a short and simple proof here.
\[lemma\_Birman\_Schwinger\] Let $A_{m, \tau}$ be defined as in and let $\tau \in \mathbb{R}$. Then $\lambda \in {\mathop{\mathrm{res}}\nolimits}(A_{m, 0})$ is an eigenvalue of $A_{m, \tau}$ if and only if $-1$ is an eigenvalue of $\tau \beta \mathcal{C}_\lambda$.
Assume that $\lambda \in {\mathop{\mathrm{res}}\nolimits}(A_{m, 0})$ is an eigenvalue of $A_{m, \tau}$ with eigenfunction $u_\lambda$. Then, by there exists a $0 \neq \varphi \in H^{\frac{1}{2}}(\Sigma, \mathbb{C}^4)$ such that $u_\lambda = \Phi_\lambda \varphi$. Since $u_\lambda \in {\mathop{\mathcal{D}}}(A_{m, \tau})$ it holds $\mathcal{P}_\tau^- u_{\lambda, +} + \mathcal{P}_\tau^+ u_{\lambda, -} = 0$. Using the definitions of the matrices $\mathcal{P}_\tau^\pm$ from and and this yields $$\begin{split}
0 &= \mathcal{P}_\tau^- (\Phi_\lambda \varphi)_+ + \mathcal{P}_\tau^+ (\Phi_\lambda \varphi)_-\\
&= {\mathrm{i}}\beta \alpha \cdot \nu \big( (\Phi_\lambda \varphi)_+ - (\Phi_\lambda \varphi)_-\big)
+ \frac{\tau}{2} \big( (\Phi_\lambda \varphi)_+ + (\Phi_\lambda \varphi)_-\big) \\
&= \beta (I_4 + \tau \beta \mathcal{C}_\lambda) \varphi,
\end{split}$$ i.e. $-1$ is an eigenvalue of $\tau \beta \mathcal{C}_\lambda$.
Conversely, if $-1$ is an eigenvalue of $\tau \beta \mathcal{C}_\lambda$ with non-trivial eigenfunction $\varphi$, then $u_\lambda := \Phi_\lambda \varphi \neq 0$ satisfies $u_\lambda \in H^1(\Omega_+, \mathbb{C}^4) \oplus H^1(\Omega_-, \mathbb{C}^4)$ by . Moreover, employing again and we obtain $$\begin{split}
\mathcal{P}_\tau^- u_{\lambda, +} + \mathcal{P}_\tau^+ u_{\lambda, -}
&= \mathcal{P}_\tau^- (\Phi_\lambda \varphi)_+ + \mathcal{P}_\tau^+ (\Phi_\lambda \varphi)_-\\
&= {\mathrm{i}}\beta \alpha \cdot \nu \big( (\Phi_\lambda \varphi)_+ - (\Phi_\lambda \varphi)_-\big)
+ \frac{\tau}{2} \big( (\Phi_\lambda \varphi)_+ + (\Phi_\lambda \varphi)_-\big) \\
&= \beta (I_4 + \tau \beta \mathcal{C}_\lambda) \varphi = 0,
\end{split}$$ as $\varphi \in \ker(I_4 + \tau \beta \mathcal{C}_\lambda)$. This shows $u_\lambda \in {\mathop{\mathcal{D}}}(A_{m, \tau})$. Eventually, equation implies $$(A_{m, \tau} - \lambda) u_\lambda = (A_{m, \tau} - \lambda) \Phi_\lambda \varphi = 0$$ and hence $\lambda$ is an eigenvalue of $A_{m, \tau}$.
Using Lemma \[lemma\_Birman\_Schwinger\] and a result from [@AMV15] we deduce finally, that $A_{m, \tau}$ has no eigenvalues in $(-|m|, |m|)$, if the interaction strength $\tau$ is small.
\[corollary\_no\_discrete\_spectrum\_in\_gap\] There exists $\tau_m > 0$ such that $A_{m, \tau}$ has no eigenvalues in $(-|m|, |m|)$ for all $|\tau| < \tau_m$.
First, by [@AMV15 Lemma 3.2] there exists a constant $C(m):=C > 0$ independent of $\lambda$ such that $$\| \mathcal{C}_\lambda \varphi \|_{L^2(\Sigma, \mathbb{C}^4)}
\leq C \| \varphi \|_{L^2(\Sigma, \mathbb{C}^4)}
\qquad \forall \varphi \in H^{\frac{1}{2}}(\Sigma, \mathbb{C}^4), \lambda \in (-|m|, |m|).$$ Hence, if $\tau < \tau_m := \frac{1}{C}$, then $-1$ can not be an eigenvalue of $\tau \beta \mathcal{C}_\lambda$. From Lemma \[lemma\_Birman\_Schwinger\] we conclude that $A_{m, \tau}$ can not have eigenvalues in $(-|m|, |m|)$ for $\tau < \tau_m$.
Proof of self-adjointness
-------------------------
First, we prove that $A_{m, \tau}$ is symmetric:
\[lemma\_symmetric\] Let $m, \tau \in \mathbb{R}$, then the operator $A_{m, \tau}$ given by is symmetric.
Let $u \in {\mathop{\mathcal{D}}}(A_{m, \tau})$. Employing an integration by parts we have $$\begin{gathered}
\langle A_{m, \tau} u, u \rangle_{L^2(\mathbb{R}^3, \mathbb{C}^4)}
- \langle u, A_{m, \tau} u \rangle_{L^2(\mathbb{R}^3, \mathbb{C}^4)} = \langle -{\mathrm{i}}\alpha \cdot \nu u_+, u_+ \rangle_{L^2(\Sigma, \mathbb{C}^4)} -\langle -{\mathrm{i}}\alpha \cdot \nu u_-, u_- \rangle_{L^2(\Sigma, \mathbb{C}^4)} \\
= \frac{1}{2} \langle -i \alpha \cdot \nu (u_+ - u_-), u_+ + u_- \rangle_{L^2(\Sigma, \mathbb{C}^4)}
- \frac{1}{2} \langle u_+ + u_-, -i \alpha \cdot \nu (u_+ - u_-) \rangle_{L^2(\Sigma, \mathbb{C}^4)}.
\end{gathered}$$ Using the transmission condition , the anti-commutation relation and $\beta^2 = I_4$ the last term can be rewritten $$\begin{gathered}
\frac{1}{2} \langle -i \alpha \cdot \nu (u_+ - u_-), u_+ + u_- \rangle_{L^2(\Sigma, \mathbb{C}^4)}
- \frac{1}{2} \langle u_+ + u_-, -i \alpha \cdot \nu (u_+ - u_-) \rangle_{L^2(\Sigma, \mathbb{C}^4)} \\
= \frac{\tau}{4} \langle \beta (u_+ + u_-), u_+ + u_- \rangle_{L^2(\Sigma, \mathbb{C}^4)}
- \frac{\tau}{4} \langle u_+ + u_-, \beta (u_+ + u_-) \rangle_{L^2(\Sigma, \mathbb{C}^4)} = 0,
\end{gathered}$$ which shows that $\langle A_{m, \tau} u, u \rangle_{L^2(\mathbb{R}^3, \mathbb{C}^4)} \in \mathbb{R}$. Since $u \in {\mathop{\mathcal{D}}}(A_{m, \tau})$ was arbitrary, the claim of this lemma follows.
The following technical result will play a crucial role in the proof of the self-adjointness of $A_{m, \tau}$:
\[lemma\_inverse\_M\] Let $\tau \in \mathbb{R}$ and let for $\lambda \in \mathbb{C} \setminus \mathbb{R}$ the operator $\mathcal{C}_\lambda$ be defined by . Then the operator $I_4 + \tau \beta \mathcal{C}_\lambda$ admits a bounded and everywhere defined inverse in $H^{\frac{1}{2}}(\Sigma, \mathbb{C}^4)$.
First, we note that $I_4 + \tau \beta \mathcal{C}_\lambda$ is injective, as otherwise the symmetric operator $A_{m, \tau}$ would have the non-real eigenvalue $\lambda$ by Lemma \[lemma\_Birman\_Schwinger\]. To show that $I_4 + \tau \beta \mathcal{C}_\lambda$ is also surjective note $$\begin{split}
{\mathop{\mathrm{ran}}}\big( I_4 + \tau \beta \mathcal{C}_\lambda \big)
&\supset {\mathop{\mathrm{ran}}}\big[ \big( I_4 + \tau \beta \mathcal{C}_\lambda \big)
\big( I_4 - \tau \beta \mathcal{C}_\lambda \big) \big] \\
&= {\mathop{\mathrm{ran}}}\big[ \big( I_4 + \tau \beta \mathcal{C}_\lambda \big)
\big( I_4 + \mathcal{C}_\lambda \tau \beta - \tau ( \mathcal{C}_\lambda \beta + \beta \mathcal{C}_\lambda) \big) \big]\\
&= {\mathop{\mathrm{ran}}}\big[ I_4 +\tau^2 \beta \mathcal{C}_\lambda^2 \beta
- \tau^2 \beta \mathcal{C}_\lambda ( \beta \mathcal{C}_\lambda + \mathcal{C}_\lambda \beta )\big].
\end{split}$$ Using the anti-commutation relations we obtain that $\beta \mathcal{C}_\lambda + \mathcal{C}_\lambda \beta$ is an integral operator with kernel $$K(x, y) = \big( \lambda \beta + m I_4 \big) \frac{e^{{\mathrm{i}}\sqrt{\lambda^2 - m^2} |x-y|}}{2 \pi |x-y|},$$ i.e. $\beta \mathcal{C}_\lambda + \mathcal{C}_\lambda \beta$ is a constant matrix times the single layer boundary integral operator for $-\Delta + m^2 - \lambda^2$ which is compact in $H^{\frac{1}{2}}(\Sigma, \mathbb{C}^4)$; cf., e.g., [@M00 Theorem 6.11]. Moreover, by also $\mathcal{C}_\lambda^2 - \frac{1}{4} I_4$ is compact in $H^{\frac{1}{2}}(\Sigma, \mathbb{C}^4)$. Since $\mathcal{C}_\lambda$ is bounded in $H^{\frac{1}{2}}(\Sigma, \mathbb{C}^4)$ we deduce that $$\mathcal{K}_\lambda := \tau^2 \beta \left( \mathcal{C}_\lambda^2 - \frac{1}{4} I_4 \right) \beta
- \tau^2 \beta \mathcal{C}_\lambda ( \beta \mathcal{C}_\lambda + \mathcal{C}_\lambda \beta)$$ is compact in $H^{\frac{1}{2}}(\Sigma, \mathbb{C}^4)$. Note that both operators $I_4 + \tau \beta \mathcal{C}_\lambda$ and $I_4 - \tau \beta \mathcal{C}_\lambda$ are injective, as otherwise one of the symmetric operators $A_{m, \pm \tau}$ would have the non-real eigenvalue $\lambda$ by Lemma \[lemma\_Birman\_Schwinger\]. Hence, we get finally by Fredholm’s alternative that $${\mathop{\mathrm{ran}}}\big[ I_4 + \tau^2 \beta \mathcal{C}_\lambda^2 \beta
- \tau^2 \beta \mathcal{C}_\lambda ( \beta \mathcal{C}_\lambda + \mathcal{C}_\lambda \beta) \big]
= {\mathop{\mathrm{ran}}}\left[ \left(1 + \frac{\tau^2}{4} \right) I_4 + \mathcal{K}_\lambda \right]
= H^{\frac{1}{2}}(\Sigma, \mathbb{C}^4).$$ Therefore, we deduce eventually $$\begin{split}
{\mathop{\mathrm{ran}}}\big( I_4 + \tau \beta \mathcal{C}_\lambda \big)
&\supset {\mathop{\mathrm{ran}}}\big[ I_4 + \tau^2 \beta \mathcal{C}_\lambda^2 \beta
- \tau^2 \beta \mathcal{C}_\lambda ( \beta \mathcal{C}_\lambda + \mathcal{C}_\lambda \beta) \big]
= H^{\frac{1}{2}}(\Sigma, \mathbb{C}^4)
\end{split}$$ and thus, $I_4 + \tau \beta \mathcal{C}_\lambda$ is surjective. This shows that the closed operator $I_4 + \tau \beta \mathcal{C}_\lambda$ is bijective and hence, it admits a bounded and everywhere defined inverse by the closed graph theorem.
Now, we are prepared to prove the self-adjointness of $A_{m, \tau}$ which is the central point of Theorem \[thm-basic\]:
\[theorem\_basic\_properties\] Let $m, \tau \in \mathbb{R}$ and let $A_{m, \tau}$ be defined by . Then, $A_{m, \tau}$ is self-adjoint and for any $\lambda \in \mathbb{C} \setminus \mathbb{R}$ one has the resolvent formula $$(A_{m, \tau} - \lambda)^{-1} = (A_{m, 0} - \lambda)^{-1} - \Phi_\lambda \big( I_4 + \tau \beta \mathcal{C}_\lambda \big)^{-1}
\tau \beta \Phi_{\overline{\lambda}}^*.$$
Since $A_{m, \tau}$ is symmetric by Lemma \[lemma\_symmetric\] it suffices to show that ${\mathop{\mathrm{ran}}}(A_{m, \tau} - \lambda) = L^2(\mathbb{R}^3, \mathbb{C}^4)$ for $\lambda \in \mathbb{C} \setminus \mathbb{R}$. Let $\lambda \in \mathbb{C} \setminus \mathbb{R}$ and $v \in L^2(\mathbb{R}^3, \mathbb{C}^4)$ be fixed. We define $$\label{Krein1}
u := (A_{m, 0} - \lambda)^{-1} v - \Phi_\lambda \big( I_4 + \tau \beta \mathcal{C}_\lambda \big)^{-1}
\tau \beta \Phi_{\overline{\lambda}}^* \, v.$$ Note that $u$ is well-defined, as $\Phi_{\overline{\lambda}}^* \, v = \big( (A_{m, 0} - \lambda)^{-1} v \big) \big|_\Sigma \in H^{\frac{1}{2}}(\Sigma, \mathbb{C}^4)$, see , and $I_4 + \tau \beta \mathcal{C}_\lambda$ is bijective in $H^{\frac{1}{2}}(\Sigma, \mathbb{C}^4)$ by Lemma \[lemma\_inverse\_M\]. We are going to prove that $u \in {\mathop{\mathcal{D}}}(A_{m, \tau})$ and $(A_{m, \tau} - \lambda) u = v$. Then, this implies the claim on the range of $A_{m, \tau} - \lambda$ and the resolvent formula.
Due to the mapping properties of $\Phi_{\overline{\lambda}}^*$ and $\mathcal{C}_\lambda$ we have $$\big( I_4 + \tau \beta \mathcal{C}_\lambda \big)^{-1}
\tau \beta \Phi_{\overline{\lambda}}^* \, v \in H^{\frac{1}{2}}(\Sigma, \mathbb{C}^4).$$ Therefore, we have by that $u \in H^1(\Omega_+, \mathbb{C}^4) \oplus H^1(\Omega_-, \mathbb{C}^4)$. Moreover, using , , and ${\mathop{\mathcal{D}}}(A_{m, 0}) = {\mathop{\mathrm{ran}}}(A_{m, 0} - \lambda) = H^1(\mathbb{R}^3, \mathbb{C}^4)$ we deduce $$\begin{split}
\mathcal{P}_\tau^-& u_+ + \mathcal{P}_\tau^+ u_-
= \frac{\tau}{2} (u_+ + u_-) + {\mathrm{i}}\beta \alpha \cdot \nu (u_+ - u_-) \\
&= \tau \big( (A_{m, 0} - \lambda)^{-1} v \big) \big|_\Sigma
- \tau \mathcal{C}_\lambda \big( I_4 + \tau \beta \mathcal{C}_\lambda \big)^{-1} \tau \beta \Phi_{\overline{\lambda}}^*\, v
- \beta \big( I_4 + \tau \beta \mathcal{C}_\lambda \big)^{-1} \tau \beta \Phi_{\overline{\lambda}}^*\, v \\
&= \tau \Phi_{\overline{\lambda}}^* \,v - \beta (I_4 + \tau \beta \mathcal{C}_\lambda)
\big( I_4 + \tau \beta \mathcal{C}_\lambda \big)^{-1} \tau \beta \Phi_{\overline{\lambda}}^*\, v = 0,
\end{split}$$ i.e. $u \in {\mathop{\mathcal{D}}}(A_{m, \tau})$. Using we have $(A_{m, \tau} - \lambda) u = v$. Hence, the theorem is shown.
Basic properties {#sec-ae}
----------------
In this section we are going to prove the points (a)–(e) of Theorem \[thm-basic\]. To prove (a) take any $\lambda \in \mathbb{C} \setminus \mathbb{R}$. First, we note that by Lemma \[lemma\_inverse\_M\] the inverse $(I_4 + \tau \beta \mathcal{C}_\lambda)^{-1}$ is a bounded operator in $H^{\frac{1}{2}}(\Sigma, \mathbb{C}^4)$. Moreover, since ${\mathop{\mathrm{ran}}}\Phi_{\overline{\lambda}}^* = H^{\frac{1}{2}}(\Sigma, \mathbb{C}^4)$, see , and since $\Phi_{\overline{\lambda}}^*: L^2(\mathbb{R}^3, \mathbb{C}^4) \rightarrow L^2(\Sigma, \mathbb{C}^4)$ is bounded, it follows from the closed graph theorem that the product $$\big( I_4 + \tau \beta \mathcal{C}_\lambda \big)^{-1} \tau \beta \Phi_{\overline{\lambda}}^*:
L^2(\mathbb{R}^3, \mathbb{C}^4) \rightarrow H^{\frac{1}{2}}(\Sigma, \mathbb{C}^4)$$ is bounded. As the embedding $\iota: H^{\frac{1}{2}}(\Sigma, \mathbb{C}^4) \rightarrow L^2(\Sigma, \mathbb{C}^4)$ is compact and $\Phi_\lambda$ is bounded, we deduce with the help of Theorem \[theorem\_basic\_properties\] that $$(A_{m, \tau} - \lambda)^{-1} - (A_{m, 0} - \lambda)^{-1}
= -\Phi_\lambda \big( I_4 + \tau \beta \mathcal{C}_\lambda \big)^{-1} \tau \beta \Phi_{\overline{\lambda}}^*$$ is compact in $L^2(\mathbb{R}^3, \mathbb{C}^4)$. Hence, we find $${\mathop{\mathrm{spec}}\nolimits}_{\mathrm{ess}}(A_{m, \tau}) = {\mathop{\mathrm{spec}}\nolimits}_{\mathrm{ess}}(A_{m, 0}) = (-\infty, -|m|] \cup [|m|, \infty).$$ This is statement (a) of Theorem \[thm-basic\].
Next, we define the charge conjugation operator ${\mathcal{C}}$ and the time reversal operator ${\mathcal{T}}$ by ${\mathcal{C}}u := {\mathrm{i}}\beta \alpha_2 \overline{u}$ and ${\mathcal{T}}u := -{\mathrm{i}}\gamma_5 \alpha_2 \overline{u}$. Then a simple computation shows that ${\mathcal{C}}^2 = -{\mathcal{T}}^2 = \mathrm{Id}$. Furthermore, $\mathcal{C}$ and $\mathcal{T}$ leave ${\mathop{\mathcal{D}}}(A_{m, \tau})$ invariant and $$A_{m,\tau} {\mathcal{C}}= -{\mathcal{C}}A_{m,\tau} \qquad \text{ and } \qquad A_{m,\tau} {\mathcal{T}}= {\mathcal{T}}A_{m,\tau}.$$
Assume that $\lambda\in {\mathop{\mathrm{spec}}\nolimits}(A_{m,\tau})$. Then there exists a sequence $(u_j)\in{\mathop{\mathcal{D}}}(A_{m,\tau})$ with $\|u_j\|=1$ and $(A_{m,\tau}-\lambda)u_j\to 0$ when $j\rightarrow+\infty$. Then for $v_j:={\mathcal{C}}u_j$ one has $\|v_j\|=1$ and $(A_{m,\tau}+\lambda)v_j = -{\mathcal{C}}(A_{m,\tau}-\lambda)u_j\to 0$, i.e. $-\lambda\in {\mathop{\mathrm{spec}}\nolimits}(A_{m,\tau})$. This proves the point (b).
Furthermore, if $u\in \ker(A_{m,\tau}-\lambda)$, then also ${\mathcal{T}}u\in \ker(A_{m,\tau}-\lambda)$. Moreover ${\mathcal{T}}^2 u=-u$ and a simple calculation using the definition of ${\mathcal{T}}$ shows $\langle u,{\mathcal{T}}u\rangle_{L^2({\mathbb{R}}^3,{\mathbb{C}}^4)}= - \langle u,{\mathcal{T}}u\rangle_{L^2({\mathbb{R}}^3,{\mathbb{C}}^4)}$ and hence $\langle u,{\mathcal{T}}u\rangle_{L^2({\mathbb{R}}^3,{\mathbb{C}}^4)}=0$. This proves (c).
In order to prove (d), we note first that the claim is trivial for $\tau = \pm 2$. For $\tau \neq \pm 2$ consider the unitary transform $$V :\left\{\begin{array}{ccc} L^2(\Omega_+,\mathbb{C}^4) \oplus L^2(\Omega_-,\mathbb{C}^4) &\longrightarrow& L^2(\Omega_+,\mathbb{C}^4) \oplus L^2(\Omega_-,\mathbb{C}^4)\\
(u_+,u_-) & \mapsto & (u_+,-u_-).
\end{array}\right.$$ Let $\tau \neq 0$. For $u\in{\mathop{\mathcal{D}}}(A_{m,\tau})$ we have $(Vu) \in {\mathop{\mathcal{D}}}(A_{m,\frac4\tau})$ because ${\mathcal{R}}_\tau^+ = - {\mathcal{R}}_{\frac4\tau}^+$. Hence, we have $ A_{m,\tau} = V^{-1} A_{m,\frac4\tau} V$ which yields that $A_{m,\tau}$ and $A_{m,\frac4\tau}$ are unitarily equivalent.
Finally, the point (e) follows from the pointwise equality $\gamma_5 A_{m,\tau}= A_{-m,-\tau}\gamma_5$.
Variational approach {#sec-var}
====================
Quadratic form for the square of the operator
---------------------------------------------
In order to proceed with a more detailed study, let us introduce some geometric quantities. Throughout this section assume that $\Sigma$ be the boundary of a bounded $C^4$ smooth domain. Recall that at each point $s\in\Sigma$ the Weingarten map $S:T_s\Sigma\to T_s\Sigma$ is defined by $S:={\mathrm{d}}\nu(s)$. Its eigenvalues $\kappa_1$ and $\kappa_2$ are called the principal curvatures, and we denote by $$\label{def_curvature}
M:=\dfrac{\kappa_1+\kappa_2}{2}, \quad K:=\kappa_1 \kappa_2,$$ the mean curvature and the Gauss curvature of $\Sigma$, respectively.
The following result will be of crucial importance for the subsequent analysis:
\[theorem\_form\_B\_tau\_square\] Let $m\in{\mathbb{R}}$ and $\tau \in {\mathbb{R}}\setminus\{0,-2,2\}$, then for any $u\in {\mathop{\mathcal{D}}}(A_{m,\tau})$ there holds $$\begin{gathered}
\label{equation_form}
\| A_{m,\tau} u \|^2_{L^2({\mathbb{R}}^3,{\mathbb{C}}^4)} = \iiint_{{\mathbb{R}}^3\setminus \Sigma} \big|\nabla u\big|^2\, {\mathrm{d}}x + m^2 \iiint_{{\mathbb{R}}^3} | u|^2\, {\mathrm{d}}x\\
+ \frac{2 m}{\tau} \iint_\Sigma | u_+ - u_-|^2 {\mathrm{d}}\Sigma + \iint_\Sigma M |u_+|^2 {\mathrm{d}}\Sigma - \iint_\Sigma M |u_-|^2 {\mathrm{d}}\Sigma
\end{gathered}$$ with ${\mathrm{d}}\Sigma$ being the surface measure on $\Sigma$.
The proof of Proposition \[theorem\_form\_B\_tau\_square\] will use a couple of preliminary computations. First recall the elementary equality $$\label{lemma_alpha_nu}
(\alpha \cdot \nu) \cdot (\alpha \cdot \nabla) - \nu \cdot \nabla I_4
= {\mathrm{i}}\gamma_5 \alpha \cdot (\nu \times \nabla).$$ Other important identities are summarized in the following lemma. Recall that for two operators $A$ and $B$ one denotes by $[A,B]:=AB-BA$ their commutator.
\[lemma\_intergation\_by\_parts\] Let $\Omega \subset \mathbb{R}^3$ be an open set with compact $C^4$ smooth boundary, let $\nu$ be the outward pointing normal vector field on the boundary, let $M$ be the mean curvature on $\partial \Omega$, and let ${\mathcal{B}}$ be defined by . Then for $u \in H^2(\Omega,{\mathbb{C}}^4)$ the following identities hold: $$\begin{gathered}
\label{lemma_commutation_diff_B}
\big[\alpha \cdot (\nu\times \nabla), {\mathcal{B}}\big] u = -2 {\mathrm{i}}M \gamma_5 {\mathcal{B}}u \text{ on } \partial\Omega,\\
\label{equation_alpha_term}
\| \alpha \cdot \nabla u \|_{L^2(\Omega,{\mathbb{C}}^4)}^2 = \| \nabla u \|^2_{L^2(\Omega,{\mathbb{R}}^3\otimes{\mathbb{C}}^4)}
+ \big\langle u, {\mathrm{i}}\gamma_5 \alpha \cdot (\nu \times \nabla) u \big\rangle_{L^2(\partial \Omega,{\mathbb{C}}^4)},\\
\label{equation_beta_term1} 2 \Re \langle -{\mathrm{i}}\alpha \cdot \nabla u, \beta u\rangle_{L^2(\Omega,{\mathbb{C}}^4)} = \langle -{\mathrm{i}}\alpha \cdot \nu u, \beta u\rangle_{L^2(\partial\Omega,{\mathbb{C}}^4)}.
\end{gathered}$$ In particular, $$\begin{gathered}
\label{eqpart1}
\| (-{\mathrm{i}}\alpha \cdot \nabla + m \beta) u \|_{L^2(\Omega,{\mathbb{C}}^4)}^2 = \| \nabla u \|_{L^2(\Omega,{\mathbb{R}}^3\otimes{\mathbb{C}}^4)}^2 + m^2 \| u \|_{L^2(\Omega,{\mathbb{C}}^4)}^2\\
+ \langle -{\mathrm{i}}\alpha \cdot \nu u, m \beta u \rangle_{L^2(\partial\Omega,{\mathbb{C}}^4)}
+ \big\langle u, {\mathrm{i}}\gamma_5 \alpha \cdot (\nu \times \nabla) u \big\rangle_{L^2(\partial\Omega,{\mathbb{C}}^4)}. \end{gathered}$$
The identity was obtained in [@ALTR16 Lemma A.3]. By applying Green’s formula and the equality $(\alpha \cdot \nabla)^2 = \Delta$ we obtain $$\begin{split}
\| \alpha \cdot \nabla u \|_{L^2(\Omega,{\mathbb{C}}^4)}^2
&= \langle \alpha \cdot \nu u, \alpha \cdot \nabla u\rangle_{L^2(\partial \Omega,{\mathbb{C}}^4)}
-\langle u, (\alpha \cdot \nabla)^2 u \big\rangle_{L^2(\Omega,{\mathbb{C}}^4)}\\
&=\big\langle u, (\alpha \cdot \nu) \cdot (\alpha \cdot \nabla) u\big\rangle_{L^2(\partial \Omega,{\mathbb{C}}^4)}
-\langle u, \Delta u \rangle_{L^2(\Omega,{\mathbb{C}}^4)} \\
&= \big\langle u, (\alpha \cdot \nu) \cdot (\alpha \cdot \nabla) u\big\rangle_{L^2(\partial \Omega,{\mathbb{C}}^4)}
- \langle u, \nu \cdot \nabla u\rangle_{L^2(\partial \Omega,{\mathbb{C}}^4)} + \| \nabla u \|_{L^2(\Omega,{\mathbb{R}}^3\otimes{\mathbb{C}}^4)}^2,
\end{split}$$ and one arrives at with the help of . Furthermore, an integration by parts and the anti-commutation rule show that $$\begin{split}
\langle-{\mathrm{i}}\alpha \cdot \nabla u, \beta u\rangle_{L^2(\Omega,{\mathbb{C}}^4)} &= \langle-{\mathrm{i}}\alpha \cdot \nu u, \beta u\rangle_{L^2(\partial\Omega,{\mathbb{C}}^4)}
- \langle-{\mathrm{i}}u, \alpha \cdot \nabla \beta u\rangle_{L^2(\Omega,{\mathbb{C}}^4)} \\
&= \langle-{\mathrm{i}}\alpha \cdot \nu u, \beta u\rangle_{L^2(\partial\Omega,{\mathbb{C}}^4)} - \langle\beta u, -{\mathrm{i}}\alpha \cdot \nabla u\rangle_{L^2(\Omega,{\mathbb{C}}^4)},
\end{split}$$ which implies $$2 \Re \langle -{\mathrm{i}}\alpha \cdot \nabla u, \beta u\rangle_{L^2(\Omega,{\mathbb{C}}^4)}
= \langle-{\mathrm{i}}\alpha \cdot \nu u, \beta u\rangle_{L^2(\partial\Omega,{\mathbb{C}}^4)}$$ and proves . Finally, $$\begin{gathered}
\big\| (-{\mathrm{i}}\alpha \cdot \nabla + m \beta) u \big\|_{L^2(\Omega,{\mathbb{C}}^4)}^2 \\
= \| \alpha \cdot \nabla u \|_{L^2(\Omega,{\mathbb{C}}^4)}^2 + m^2 \| \beta u \|_{L^2(\Omega,{\mathbb{C}}^4)}^2
+ 2 \Re \langle-{\mathrm{i}}\alpha \cdot \nabla u, m \beta u\rangle_{L^2(\Omega,{\mathbb{C}}^4)},
\end{gathered}$$ and using that $\beta$ is unitary, and we arrive at .
We will also use the following assertion, which is a rather standard application of the elliptic regularity argument, but we prefer to give a proof for the sake of completeness.
\[lem:dens\_domtilde\] For $\tau\notin\{-2,2\}$ the subspace $\widetilde{{\mathop{\mathcal{D}}}}(A_{m,\tau}) := {\mathop{\mathcal{D}}}(A_{m,\tau}) \cap H^2(\mathbb{R}^3\setminus\Sigma,{\mathbb{C}}^4)$ is dense in ${\mathop{\mathcal{D}}}(A_{m,\tau})$ in the $H^1(\mathbb{R}^3\setminus\Sigma,{\mathbb{C}}^4)$-norm.
It is well-known, see, e.g., , that there exists a bounded linear operator $E : H^{\frac{1}{2}}(\Sigma,{\mathbb{C}}^4) \longrightarrow H^1(\Omega_+,{\mathbb{C}}^4)$ such that for any $\xi \in H^{\frac{1}{2}}(\Sigma,{\mathbb{C}}^4)$ one has $(E\xi)|_\Sigma = \xi$ and $E\big(H^{\frac{3}{2}}(\Sigma)\big) \subset H^2(\Omega_+)$.
Let $(u_+,u_-)\in{\mathop{\mathcal{D}}}(A_{m,\tau})$. As $H^2(\Omega_\pm,\mathbb{C}^4)$ is dense in $H^1(\Omega_\pm,\mathbb{C}^4)$ with respect to the $H^1$-norm, there exist $v_\pm^\varepsilon \in H^2(\Omega_\pm,{\mathbb{C}}^4)$ such that $\lim_{\varepsilon\rightarrow0}\|v_\pm^\varepsilon - u_\pm\|_{H^1(\Omega_\pm,{\mathbb{C}}^4)} = 0$. Set $w_-^\varepsilon = v_-^\varepsilon$ and $w_+^\varepsilon = v_+^\varepsilon + E\varphi^\varepsilon$, where $\varphi^\varepsilon$ is given by $\varphi^\varepsilon = -(\mathcal{P}_\tau^-)^{-1}(\mathcal{P}_\tau^-v_+^\varepsilon + \mathcal{P}_\tau^+v_-^\varepsilon)$. Note that $\varphi^\varepsilon \in H^{3/2}(\Sigma, \mathbb{C}^4)$ as $v_\pm^\varepsilon\in H^2(\Omega_\pm,{\mathbb{C}}^4)$ and $\mathcal{P}_\tau^\pm, (\mathcal{P}_\tau^-)^{-1} \in C^2(\Sigma,{\mathbb{C}}^{4\times 4})$. Thus, we have $w_\pm^\varepsilon \in H^2(\Omega_\pm,{\mathbb{C}}^4)$ due to the above properties of $E$.
We claim that $\lim_{\varepsilon \rightarrow 0} \|w_\pm^\varepsilon - u_\pm\|_{H^1(\Omega_\pm,{\mathbb{C}}^4)} = 0$. By definition, it is clear that this is true for $w_-^\varepsilon$ so, we focus on $w_+^\varepsilon$. We have $$\begin{aligned}
\|w_+^\varepsilon - u_+\|_{H^1(\Omega_+,{\mathbb{C}}^4)} &\leq \|v_+^\varepsilon - u_+\|_{H^1(\Omega_+,{\mathbb{C}}^4)} + \|E\varphi^\varepsilon\|_{H^1(\Omega_+,{\mathbb{C}}^4)}\\
& \leq \|v_+^\varepsilon - u_+\|_{H^1(\Omega_+,{\mathbb{C}}^4)} + C\|\varphi^\varepsilon\|_{H^{\frac{1}{2}}(\Sigma,{\mathbb{C}}^4)},\end{aligned}$$ with a constant $C>0$ thanks to the boundedness of $E$. The first term in the right-hand side converges to zero by definition. This is also true for the second term because using the transmission condition $\mathcal{P}^-_\tau u_+ + \mathcal{P}^+_\tau u_- = 0$ we get $$\begin{aligned}
\|\varphi^\varepsilon\|_{H^{\frac{1}{2}}(\Sigma,{\mathbb{C}}^4)} &= \|(\mathcal{P}_\tau^+)^{-1}\big(\mathcal{P}_\tau^-(v_+^\varepsilon - u_+) + \mathcal{P}_\tau^+(v_-^\varepsilon - u_-)\big)\|_{H^{\frac{1}{2}}(\Sigma,{\mathbb{C}}^4)}\\
&\leq K \big(\|v_+^\varepsilon - u_+\|_{H^1(\Omega_+,{\mathbb{C}}^4)} + \|v_-^\varepsilon - u_-\|_{H^1(\Omega_-,{\mathbb{C}}^4)}\big),\end{aligned}$$ with a constant $K>0$. Thus, the right-hand side converges to zero by hypothesis and we get $\lim_{\varepsilon \rightarrow 0} \|w_\pm^\varepsilon - u_\pm\|_{H^1(\Omega_\pm,{\mathbb{C}}^4)} = 0$.
The only thing left to prove is that $(w_+^\varepsilon,w_-^\varepsilon)\in\widetilde{{\mathop{\mathcal{D}}}}(A_{m,\tau})$ which is true if the transmission condition is verified. Indeed, we have $$\begin{split}
\mathcal{P}_\tau^- w_+^\varepsilon + \mathcal{P}_\tau^+ w_-^\varepsilon &= \mathcal{P}_\tau^-\big(v_+^\varepsilon + \varphi^\varepsilon\big) + \mathcal{P}_\tau^+v_-^\varepsilon
=\mathcal{P}_\tau^-v_+^\varepsilon + \mathcal{P}_\tau^+v_-^\varepsilon + \mathcal{P}_\tau^-\varphi^\varepsilon\\
&=\mathcal{P}_\tau^-v_+^\varepsilon + \mathcal{P}_\tau^+v_-^\varepsilon - \mathcal{P}_\tau^-v_+^\varepsilon - \mathcal{P}_\tau^+v_-^\varepsilon=0.
\end{split}$$ Hence, the lemma is proved.
Due to Lemma \[lem:dens\_domtilde\] it is sufficient to prove the result for the functions $u\in \widetilde{{\mathop{\mathcal{D}}}}(A_{m,\tau})$. Using for $\Omega=\Omega_\pm$ we obtain $$ \begin{split}
\| A_{m,\tau} u \|_{L^2({\mathbb{R}}^3,{\mathbb{C}}^4)}^2
&= \big\|\big (-{\mathrm{i}}\alpha \cdot \nabla + m \beta) u_+ \big\|_{L^2(\Omega_+,{\mathbb{C}}^4)}^2
+ \big\| (-{\mathrm{i}}\alpha \cdot \nabla + m \beta) u_- \big\|_{L^2(\Omega_-,{\mathbb{C}}^4)}^2 \\
&= \| \nabla u \|_{L^2({\mathbb{R}}^3\setminus\Sigma,{\mathbb{R}}^3\otimes{\mathbb{C}}^4)}^2 + m^2 \| u \|_{L^2(\mathbb{R}^3,{\mathbb{C}}^4)}^2+J_1+J_2
\end{split}$$ with $$\begin{aligned}
J_1&= \langle -{\mathrm{i}}\alpha \cdot \nu u_+, m \beta u_+ \rangle_{L^2(\Sigma,{\mathbb{C}}^4)} - \langle -{\mathrm{i}}\alpha \cdot \nu u_-, m \beta u_- \rangle_{L^2(\Sigma,{\mathbb{C}}^4)},\\
J_2&= \big\langle u_+, {\mathrm{i}}\gamma_5 \alpha \cdot (\nu \times \nabla) u_+ \big\rangle_{L^2(\Sigma,{\mathbb{C}}^4)}
- \big\langle u_-, {\mathrm{i}}\gamma_5 \alpha \cdot (\nu \times \nabla) u_- \big\rangle_{L^2(\Sigma,{\mathbb{C}}^4)}.
\end{aligned}$$ To simplify $J_1$ we remark first that $$\langle -{\mathrm{i}}\alpha \cdot \nu u_\pm, m\beta u_\pm\rangle_{L^2(\Sigma,{\mathbb{C}}^4)}\\
= m \big\langle u_\pm, {\mathrm{i}}(\alpha \cdot \nu)\beta u_\pm\big\rangle_{L^2(\Sigma,{\mathbb{C}}^4)}=m \langle u_\pm, {\mathcal{B}}u_\pm\rangle_{L^2(\Sigma,{\mathbb{C}}^4)},$$ which yields $J_1=m \big( \langle u_+, {\mathcal{B}}u_+\rangle_{L^2(\Sigma,{\mathbb{C}}^4)}- \langle u_-, {\mathcal{B}}u_-\rangle_{L^2(\Sigma,{\mathbb{C}}^4)}\big)$. By and we get $$\begin{aligned}
J_1&=m \big( \langle u_+, {\mathcal{B}}u_+\rangle_{L^2(\Sigma,{\mathbb{C}}^4)}- \langle u_-, {\mathcal{B}}u_-\rangle_{L^2(\Sigma,{\mathbb{C}}^4)}\big)\\
&= m\big[ \langle u_+, {\mathcal{B}}u_+\rangle_{L^2(\Sigma,{\mathbb{C}}^4)}- \langle u_-, {\mathcal{B}}u_-\rangle_{L^2(\Sigma,{\mathbb{C}}^4)}\\
&\quad \qquad- \langle {\mathcal{R}}_\tau^+ u_-, {\mathcal{B}}u_-\rangle_{L^2(\Sigma,{\mathbb{C}}^4)}
+ \langle u_-, {\mathcal{B}}{\mathcal{R}}_\tau^+ u_-\rangle_{L^2(\Sigma,{\mathbb{C}}^4)}\big] \\
&= m\big[ \langle u_+, {\mathcal{B}}u_+\rangle_{L^2(\Sigma,{\mathbb{C}}^4)}- \langle u_-, {\mathcal{B}}u_-\rangle_{L^2(\Sigma,{\mathbb{C}}^4)}\\
&\qquad \quad- \langle u_+, {\mathcal{B}}u_-\rangle_{L^2(\Sigma,{\mathbb{C}}^4)}
+ \langle u_-, {\mathcal{B}}u_+\rangle_{L^2(\Sigma,{\mathbb{C}}^4)}
\big] \\
&= m \big\langle u_+ + u_-, {\mathcal{B}}(u_+ - u_-) \big\rangle_{L^2(\Sigma,{\mathbb{C}}^4)}\\
&= \dfrac{2m}{\tau } \langle {\mathcal{B}}(u_+ - u_-), {\mathcal{B}}(u_+ - u_-) \big\rangle_{L^2(\Sigma,{\mathbb{C}}^4)}\\
&= \dfrac{2m}{\tau } \|u_+ - u_-\|_{L^2(\Sigma,{\mathbb{C}}^4)}^2.
\end{aligned}$$ It remains to analyze the term $J_2$. Making again use of the transmission condition and the commutation relation we obtain $$\begin{split}
J_2&=\big\langle {\mathcal{R}}_\tau^+ u_-, {\mathrm{i}}\gamma_5 \alpha \cdot (\nu \times \nabla) u_+ \big\rangle_{L^2(\Sigma,{\mathbb{C}}^4)}
- \big\langle u_-, {\mathrm{i}}\gamma_5 \alpha \cdot (\nu \times \nabla) {\mathcal{R}}_\tau^- u_+ \big\rangle_{L^2(\Sigma,{\mathbb{C}}^4)} \\
&= \big\langle\gamma_5 u_-, {\mathrm{i}}\big({\mathcal{R}}_\tau^- \alpha \cdot (\nu \times \nabla) - \alpha \cdot (\nu \times \nabla) {\mathcal{R}}_\tau^-\big) u_+ \big\rangle_{L^2(\Sigma,{\mathbb{C}}^4)} \\
&= \frac{4\tau}{4-\tau^2} \Big\langle \gamma_5 u_-, {\mathrm{i}}\big[\alpha \cdot (\nu \times \nabla), {\mathcal{B}}\big] u_+ \Big\rangle_{L^2(\Sigma,{\mathbb{C}}^4)}.
\end{split}$$ With the help of we arrive at $$J_2=\frac{4\tau}{4-\tau^2} \langle \gamma_5 u_-, 2M \gamma_5{\mathcal{B}}u_+ \rangle_{L^2(\Sigma,{\mathbb{C}}^4)}
=\frac{4\tau}{4-\tau^2} \langle u_-, 2M {\mathcal{B}}u_+ \rangle_{L^2(\Sigma,{\mathbb{C}}^4)}.$$ Finally, using the expressions of ${\mathcal{R}}_\tau^\pm$ and the transmission conditions we conclude $$\begin{aligned}
J_2&=\frac{4\tau}{4-\tau^2} \langle {\mathcal{B}}u_-, M u_+ \rangle_{L^2(\Sigma,{\mathbb{C}}^4)}+\frac{4\tau}{4-\tau^2} \langle u_-, M {\mathcal{B}}u_+ \rangle_{L^2(\Sigma,{\mathbb{C}}^4)}\\
&=\langle {\mathcal{R}}_\tau^+ u_-, M u_+ \rangle_{L^2(\Sigma,{\mathbb{C}}^4)}
- \langle u_-, M {\mathcal{R}}_\tau^- u_+ \rangle_{L^2(\Sigma,{\mathbb{C}}^4)}\\
&= \langle u_+, M u_+ \rangle_{L^2(\Sigma,{\mathbb{C}}^4)}- \langle u_-, M u_- \rangle_{L^2(\Sigma,{\mathbb{C}}^4)},
\end{aligned}$$ which completes the proof of for $u\in \widetilde{{\mathop{\mathcal{D}}}}(A_{m,\tau})$.
First estimates for the discrete spectrum
-----------------------------------------
First remark that as a direct consequence of Corollary \[corollary\_no\_discrete\_spectrum\_in\_gap\] and Theorem \[thm-basic\] (d) we have the following assertion:
\[prop3.4\] Let $m \in \mathbb{R}$ be fixed. Then there exists $\tau_m>0$ such that $A_{m,\tau}$ has no discrete spectrum for $|\tau|<\tau_m$ and $|\tau|>\frac{4}{\tau_m}$.
The following assertion holds true due to the unique continuation principle, see Theorem 3.7 in [@AMV15] and the discussion thereafter to obtain the result in our setting:
\[lem-amv\] Assume that $\tau\notin\{-2,2\}$ and that $\Omega_-$ is connected. Then $A_{m,\tau}$ has no eigenvalues in ${\mathbb{R}}\setminus[-m,m]$.
Now we use Proposition \[theorem\_form\_B\_tau\_square\] to obtain first estimates on the discrete spectrum of $A_{m,\tau}$.
\[thm-disc1\] Assume that $\tau\notin\{-2,2\}$, then:
- the discrete spectrum of $A_{m,\tau}$ is finite,
- if $m\tau\ge 0$, then the discrete spectrum of $A_{m,\tau}$ is empty,
- if $m\tau>0$, then $\pm m$ are not eigenvalues of $A_{m,\tau}$,
- if $m\tau>0$ and $\Omega_-$ is connected, then $A_{m,\tau}$ has no eigenvalues.
\(a) It is sufficient to show that the discrete spectrum of $A:=A_{m,\tau}^2$ is finite, i.e. that $A$ has at most finitely many eigenvalues in $[0,m^2)$. Recall that $A$ is the self-adjoint operator associated with the sesquilinear form $$a(u,u)=\|A_{m,\tau} u\|^2_{L^2({\mathbb{R}}^3,{\mathbb{C}}^4)}, \quad u\in {\mathop{\mathcal{D}}}(A_{m,\tau}).$$ Let $\Omega\subset{\mathbb{R}}^3$ be a large ball containing $\overline{\Omega_+}$ and set $\Omega^c = {\mathbb{R}}^3\setminus\overline{\Omega}$. Using the natural identification $$L^2({\mathbb{R}}^3,{\mathbb{C}}^4)\simeq L^2(\Omega_+,{\mathbb{C}}^4)\oplus L^2(\Omega_-\cap \Omega,{\mathbb{C}}^4)\oplus L^2(\Omega^c,{\mathbb{C}}^4),
\quad u\simeq (u_+,u_-,u_c),$$ consider the sesquilinear form $$\begin{gathered}
b(u,u)=\iiint_{{\mathbb{R}}^3\setminus (\Sigma \cup\partial \Omega)} \big| \nabla u \big|^2\, {\mathrm{d}}x + m^2 \iiint_{{\mathbb{R}}^3} | u|^2\, {\mathrm{d}}x\\
+ \frac{2 m}{\tau} \iint_\Sigma | u_+ - u_-|^2 {\mathrm{d}}\Sigma + \iint_\Sigma M |u_+|^2 {\mathrm{d}}\Sigma
- \iint_\Sigma M |u_-|^2 {\mathrm{d}}\Sigma\end{gathered}$$ defined for $u$ satisfying $$\begin{gathered}
u_+\in H^1(\Omega_+,{\mathbb{C}}^4), \quad u_-\in H^1(\Omega_-\cap \Omega,{\mathbb{C}}^4), \quad
u_c\in H^1(\Omega^c,{\mathbb{C}}^4),\\
{\mathcal{P}}_\tau^- u_+ + {\mathcal{P}}_\tau^+ u_-=0 \text{ on } \Sigma.\end{gathered}$$ Then $b$ is closed, lower semibounded, and, moreover, it is an extension of the form $a$. Let $B$ be the self-adjoint operator in $L^2({\mathbb{R}}^3,{\mathbb{C}}^4)$ associated with $b$, then due to the min-max principle one has $\varepsilon_n(A)\ge \varepsilon_n(B)$ for all $n$. Therefore, it is sufficient to show that $B$ has at most finitely many eigenvalues in $(-\infty,m^2)$.
One easily remarks that $B=B_0\oplus B_c$, where $B_0$ is the self-adjoint operator in $L^2(\Omega,{\mathbb{C}}^4)$ generated by the sesquilinear form $$\begin{gathered}
b_0(u,u)=\iiint_{\Omega\setminus \Sigma } \big| \nabla u \big|^2\, {\mathrm{d}}x + m^2 \iiint_{\Omega} | u|^2\, {\mathrm{d}}x\\
+ \frac{2 m}{\tau} \iint_\Sigma | u_+ - u_-|^2 {\mathrm{d}}\Sigma + \iint_\Sigma M |u_+|^2 {\mathrm{d}}\Sigma
- \iint_\Sigma M |u_-|^2 {\mathrm{d}}\Sigma\end{gathered}$$ with $$\begin{gathered}
{\mathop{\mathcal{D}}}(b_0)=\big\{ u=(u_+,u_-):
u_+\in H^1(\Omega_+,{\mathbb{C}}^4), \quad u_-\in H^1(\Omega_-\cap \Omega,{\mathbb{C}}^4),\\
\quad {\mathcal{P}}_\tau^- u_+ + {\mathcal{P}}_\tau^+ u_-=0 \text{ on } \Sigma
\big\},\end{gathered}$$ and $B_c$ is the self-adjoint operator in $L^2(\Omega^c,{\mathbb{C}}^4)$ given by the sesquilinear form $$b_c(u_c,u_c)=\iiint_{\Omega^c } \big| \nabla u_c \big|^2\, {\mathrm{d}}x + m^2 \iiint_{\Omega^c} | u_c|^2\, {\mathrm{d}}x,\quad
{\mathop{\mathcal{D}}}(b_c)=H^1(\Omega^c,{\mathbb{C}}^4).$$ One has $B_c\ge m^2$ and, therefore, the number of eigenvalues of $B$ in $(-\infty,m^2)$ is the same as that of $B_0$. On the other hand, the domain of $b_0$ is compactly embedded in $L^2(\Omega,{\mathbb{C}}^4)$ and hence, $B_0$ has compact resolvent. As $B_0$ is lower semibounded, its eigenvalues form a sequence converging to $+\infty$ and there are at most finitely many eigenvalues in $(-\infty,m^2)$.
\(b) It is sufficient to show that $A_{m,\tau}^2$ has no discrete spectrum. As the essential spectrum of $A_{m,\tau}^2$ is $[m^2,+\infty)$, it is sufficient to show that $A_{m,\tau}^2\ge m^2$. The case $\tau=0$ is obvious and corresponds to the free Dirac operator, cf. Section \[section\_free\_Operator\], so we may assume that $\tau\ne 0$ and $m\tau\ge 0$. By Proposition \[theorem\_form\_B\_tau\_square\] we have for any $u\in {\mathop{\mathcal{D}}}(A^2_{m,\tau})\subset {\mathop{\mathcal{D}}}(A_{m,\tau})$ $$\begin{gathered}
\label{eq-amt}
\langle u, A^2_{m,\tau} u\rangle_{L^2({\mathbb{R}}^3,{\mathbb{C}}^4)}=\|A_{m,\tau}u\|^2_{L^2({\mathbb{R}}^3,{\mathbb{C}}^4)}\\
=\|A_{0,\tau}u\|^2_{L^2({\mathbb{R}}^3,{\mathbb{C}}^4)}
+m^2\|u\|^2_{L^2({\mathbb{R}}^3,{\mathbb{C}}^4)} +\frac{2 m}{\tau} \| u_+ - u_-\|^2_{L^2(\Sigma,{\mathbb{C}}^4)}
\ge m^2\|u\|^2_{L^2({\mathbb{R}}^3,{\mathbb{C}}^4)}\end{gathered}$$ and thus, the claim is also true for $\tau \neq 0$.
\(c) It is sufficient to show that $\ker(A^2_{m,\tau}-m^2)=\{0\}$. Let $u\in \ker(A^2_{m,\tau}-m^2)$, then similar to one has $$\begin{aligned}
0&=\langle u, (A^2_{m,\tau}-m^2) u\rangle_{L^2({\mathbb{R}}^3,{\mathbb{C}}^4)}\\
&=\|A_{0,\tau}u\|^2_{L^2({\mathbb{R}}^3,{\mathbb{C}}^4)} +\frac{2 m}{\tau} \| u_+ - u_-\|^2_{L^2(\Sigma,{\mathbb{C}}^4)}
\ge \frac{2 m}{\tau} \| u_+ - u_-\|^2_{L^2(\Sigma,{\mathbb{C}}^4)}\ge 0\end{aligned}$$ implying $u_+=u_-$ on $\Sigma$. Together with the condition ${\mathcal{P}}_\tau^- u_+ +{\mathcal{P}}_\tau^+ u_-= 0$ this implies that $u_+=u_-=0$ on $\Sigma$. Using again Proposition \[theorem\_form\_B\_tau\_square\] we arrive at $$0=\langle u, (A^2_{m,\tau}-m^2) u\rangle_{L^2({\mathbb{R}}^3,{\mathbb{C}}^4)}= \iiint_{\Omega_+} |\nabla u_+|^2{\mathrm{d}}x+\iiint_{\Omega_-} |\nabla u_-|^2{\mathrm{d}}x$$ and deduce that $u_\pm$ are constant on each connected component of $\Omega_\pm$. As $u_\pm=0$ on $\Sigma=\partial\Omega_\pm$, the functions $u_\pm$ vanish identically.
\(d) Follows from the points (b) and (c) and Proposition \[lem-amv\].
Discrete spectrum in the large mass limit {#sec:dis-spec}
=========================================
Effective operator on the shell {#sec-eff}
-------------------------------
By Theorem \[thm-disc1\] $A_{m, \tau}$ can only have discrete spectrum when $\tau$ and $m$ have opposite signs. As seen in Theorem \[thm-basic\] (e), the operators $A_{m,-\tau}$ and $A_{-m,\tau}$ are unitarily equivalent, hence in this section we assume without loss of generality that $$\tau<0 \text{ with } \tau\ne -2 \text{ is fixed}$$ and we are going to study the behavior of the discrete spectrum as $m\to +\infty$.
In order to state the main result, we need to introduce an effective operator on $\Sigma$, which appears to be a Schrödinger operator with an external Yang-Mills potential, cf. [@sred Section 69]. Namely, consider the (matrix-valued) $1$-form $\omega$ on $\Sigma$ given by $\omega=\sigma\cdot (\nu\times{\mathrm{d}}\nu)$, i.e. by the local expression $$\label{eq-omega0}
\omega=\omega_1{\mathrm{d}}s_1+\omega_2{\mathrm{d}}s_2 \in T^*\Sigma\otimes {\mathbf{B}}({\mathbb{C}}^2), \quad
\omega_j=\sigma\cdot(\nu\times \partial_j \nu).$$ For a parameter (coupling constant) $\theta\in {\mathbb{R}}$, denote $$\Lambda(\theta)=({\mathrm{d}}+ {\mathrm{i}}\theta \omega)^*({\mathrm{d}}+ {\mathrm{i}}\theta \omega)$$ the associated Bochner Laplacian in $L^2(\Sigma,{\mathbb{C}}^2)$. Recall that by definition this operator is given by the local expression $$\Lambda(\theta)=
-\dfrac{1}{\sqrt{\det g}}\sum_{j,k} (\partial_j + {\mathrm{i}}\theta\omega_j)g^{jk}
\sqrt{\det g} (\partial_k + {\mathrm{i}}\theta\omega_k), \quad {\mathop{\mathcal{D}}}\big(\Lambda(\theta)\big)=H^2(\Sigma,{\mathbb{C}}^2),$$ where $(g_{jk})$ is the metric tensor on $\Sigma$, $(g^{jk}):=(g_{jk})^{-1}$, and it is the unique self-adjoint operator associated with the closed sesqulinear form $\lambda_\theta$ given by $$\lambda_\theta(u,u)= \iint_\Sigma \sum_{j,k}g^{jk} \big\langle \partial_j u+{\mathrm{i}}\theta \omega_j u,
\partial_k u+{\mathrm{i}}\theta \omega_k u\big\rangle_{{\mathbb{C}}^2} {\mathrm{d}}\Sigma,
\quad {\mathop{\mathcal{D}}}(\lambda_\theta)=H^1(\Sigma,{\mathbb{C}}^2).$$ Finally, consider the Schrödinger operator with an additional (bounded) scalar potential induced by curvatures given by $$\Upsilon_\tau=\Lambda\Big(\frac{4}{\tau^2+4}\Big) - \Big(\dfrac{\tau^2-4}{\tau^2+4}\Big)^2 M^2 + \dfrac{\tau^4+16}{(\tau^2+4)^2} K,$$ which acts on $L^2(\Sigma,{\mathbb{C}}^2)$ as well.
We will often use the shorthand $$\label{eq-aaa}
\mu=\dfrac{4|\tau|}{\tau^2+4}\in (0,1).$$
The aim of the present section is prove the following main result:
\[thm-ev1\] Assume that $\delta\equiv \delta(m)>0$ is chosen in such a way that $$\label{eq-mdelta}
\delta\to 0 \text { and }
m\delta\to +\infty
\text{ for } m\to +\infty.$$ Then there exist constants $b>0$, $c>0$ and $m_0>0$ such that for all $m>m_0$ and $j\in\big\{1,\dots,{\mathcal{N}}(A^2_{m,\tau},m^2)\big\}$ one has $$\begin{gathered}
\Big( \dfrac{\tau^2-4}{\tau^2+4}\Big)^2 m^2 + (1-b\delta)E_j(\Upsilon_\tau\oplus\Upsilon_\tau)- c(\delta+m^2e^{-2\mu m\delta})
\le E_j(A_{m,\tau}^2)\\
\le\Big( \dfrac{\tau^2-4}{\tau^2+4}\Big)^2 m^2 + (1+b\delta)E_j(\Upsilon_\tau\oplus\Upsilon_\tau)+ c(\delta+m^2e^{-2\mu m\delta}).\end{gathered}$$
Let us present first some important consequences.
\[cor11\] For any fixed $j\in{\mathbb{N}}$ there holds $$E_j(A_{m,\tau}^2)= \Big(\dfrac{\tau^2-4}{\tau^2+4}\Big)^2 m^2 + E_j(\Upsilon_\tau\oplus\Upsilon_\tau)+{\mathcal{O}}\Big(\dfrac{\log m}{m}\Big)
\text{ as } m\to+\infty.$$
As the $j$-th eigenvalue of $\Upsilon_\tau\oplus\Upsilon_\tau$ does not depend on $m$, it is sufficient to use Theorem \[thm-ev1\] with $\delta= k m^{-1} \log m$ and a sufficiently large $k>0$.
\[cor32\] Denote the eigenvalues of $A_{m,\tau}$ by $\pm \mu_j(m)$ with $\mu_j(m)\ge 0$ enumerated in the non-decreasing order according to the multiplicities, then for any fixed $j\in{\mathbb{N}}$ there holds $$\mu_j(m)=\dfrac{|\tau^2-4|}{\tau^2+4}\, m+ \dfrac{\tau^2+4}{|\tau^2-4|}\, \dfrac{E_{j}(\Upsilon_\tau)}{2 m} + {\mathcal{O}}\Big(\dfrac{\log m}{m^2}\Big)
\text{ as } m\to+\infty.$$
One has $\mu_j(m)^2=E_{2j}(A_{m,\tau}^2)$ due to the degeneracy, see Theorem \[thm-basic\](c). Now it is sufficient to apply Taylor expansion to $\sqrt{E_{2j}(A_{m,\tau}^2)}$ using the asymptotics of Corollary \[cor11\] and to remark that $E_{2j}(\Upsilon_\tau\oplus\Upsilon_\tau)=E_j(\Upsilon_\tau)$.
Finally, the following Weyl-type asymptotics holds:
\[cor33\] The total number ${\mathcal{N}}(A^2_{m,\tau},m^2)$ of discrete eigenvalues of $A_{m,\tau}$ satisfies $${\mathcal{N}}(A^2_{m,\tau},m^2)= \dfrac{16}{\pi} \dfrac{\tau^2}{(\tau^2+4)^2} \,|\Sigma| \,m^2+{\mathcal{O}}(m\log m) \text{ as } m\to +\infty.$$
Using Theorem \[thm-ev1\] with $\delta= k m^{-1} \log m$ and a sufficiently large $k>0$ one concludes that there exist constants $C>0$ and $m_0>0$ such that for $m>m_0$ there holds $$\begin{gathered}
\label{eq-weyl1}
{\mathcal{N}}\Big(\Upsilon_\tau\oplus\Upsilon_\tau,\dfrac{16 \tau^2}{(\tau^2+4)^2} m^2-Cm \log m\Big)\\
\le{\mathcal{N}}(A^2_{m,\tau},m^2) \le
{\mathcal{N}}\Big(\Upsilon_\tau\oplus\Upsilon_\tau, \dfrac{16 \tau^2}{(\tau^2+4)^2} m^2+Cm \log m\Big).\end{gathered}$$ Due to the obvious identity ${\mathcal{N}}(\Upsilon_\tau\oplus\Upsilon_\tau,E)\equiv 2{\mathcal{N}}(\Upsilon_\tau,E)$ it is sufficient to study the behavior of ${\mathcal{N}}(\Upsilon_\tau,E)$ for large $E$. As $\Upsilon_\tau$ is an elliptic differential operator on a compact manifold having the same principal symbol as the Laplacian, the classical Weyl asymptotics, see e.g. Section 16.1 in [@S01], gives $${\mathcal{N}}(\Upsilon_\tau,E)=2\cdot \dfrac{|\Sigma|}{4\pi }\,E + {\mathcal{O}}(\sqrt{E}), \quad E\to+\infty,$$ and the substitution into gives the result. We remark that the latter result on ${\mathcal{N}}(\Upsilon_\tau,E)$ is indeed very standard for the operators with $C^\infty$ coefficients, but in our case the coefficients are only supposed to be $C^2$. For the extension of the Weyl asymptotics to $C^k$ coefficients see e.g. Theorem 1.1 in [@Z99].
One easily sees that $\Upsilon_\tau$ commutes with the charge conjugation operator $u \mapsto \sigma_2 \overline{u}$ satisfying $\langle u, \sigma_2 \overline{u}\rangle_{L^2(\Sigma,{\mathbb{C}}^2)}=0$ for any $u\in {L^2(\Sigma,{\mathbb{C}}^2)}$. This implies that any eigenvalue of $\Upsilon_\tau$ has an even multiplicity, which is in agreement with Theorem \[thm-basic\] (c). Furthermore, a short direct computation shows that the operators $\Lambda(\theta)$ and $\Lambda(1-\theta)$ are unitarily equivalent, the associated unitary operator being $u\mapsto (\sigma\cdot \nu) u$. As a result, the operator $\Upsilon_\tau$ is unitarily equivalent to $\Upsilon_{\frac{4}{\tau}}$, which is in agreement with Theorem \[thm-basic\] (d).
Intermediate operator {#ssec42}
---------------------
In what follows, it will be more comfortable to work with another operator which is unitarily equivalent to $\Upsilon_\tau\oplus\Upsilon_\tau$ but acts in a different space. Namely, consider the following Hilbert space: $$\label{def_Hilbert_space_H}
\begin{split}
{\mathcal{H}}\equiv{\mathcal{H}}_\tau=&\big\{v=(v_+,v_-): v_\pm \in L^2(\Sigma,{\mathbb{C}}^4): \quad
{\mathcal{P}}^-_\tau v_+ + {\mathcal{P}}^+_\tau v_-=0\big\},\\
&\langle u,v\rangle_{{\mathcal{H}}}=\langle u_+,v_+\rangle_{L^2(\Sigma,{\mathbb{C}}^4)}+\langle u_-,v_-\rangle_{L^2(\Sigma,{\mathbb{C}}^4)},
\end{split}$$ and denote by ${\mathcal{L}}^\tau_0$ the self-adjoint operator associated with the sesquilinear form $$\label{eq-ll1}
\begin{aligned}
\ell^\tau_0(v,v)&=\iint_\Sigma \Big(\|\nabla_s v_+\|^2_{T_s\Sigma\otimes{\mathbb{C}}^4}+\|\nabla_s v_-\|^2_{T_s\Sigma\otimes{\mathbb{C}}^4} \Big){\mathrm{d}}\Sigma,\\
{\mathop{\mathcal{D}}}(\ell^\tau_0)&=\big\{v=(v_+,v_-)\in {\mathcal{H}}_\tau: v_\pm \in H^1(\Sigma,{\mathbb{C}}^4)\big\},
\end{aligned}$$ where $\nabla_s v$ stands for the gradient of $v$ on $\Sigma$.
\[prop45\] The operators $\Upsilon_\tau\oplus\Upsilon_\tau$ and ${\mathcal{L}}^\tau:={\mathcal{L}}^\tau_0+K-M^2$ are unitarily equivalent.
As the matrices ${\mathcal{P}}^\pm_\tau(s)$ are invertible for any $s \in \Sigma$, the map $V:L^2(\Sigma,{\mathbb{C}}^4)\to {\mathcal{H}}$ given by $$\begin{split}
(V f)_\pm(s)&=\mp {\mathcal{P}}^\pm_\tau(s) f(s) = \mp \Big(\dfrac{\tau}{2}\pm {\mathcal{B}}(s)\Big)f(s)\\
&= \Big(-{\mathcal{B}}(s)\mp \dfrac{\tau}{2}\Big) f(s) = \Big(i\beta \alpha \nu (s)\mp \dfrac{\tau}{2}\Big) f(s)
\end{split}$$ is bijective. Furthermore, everywhere on $\Sigma$ one has $$\begin{aligned}
|(V f)_\pm|^2=\Big|\big({\mathcal{B}}\pm \dfrac{\tau}{2}\big) f \Big|^2& =|{\mathcal{B}}f|^2+ \dfrac{\tau^2}{4} |f|^2 \pm \tau \Re \langle {\mathcal{B}}f,f\rangle\\
&=\Big(1+\dfrac{\tau^2}{4}\Big) |f|^2 \pm \tau \Re \langle {\mathcal{B}}f,f\rangle,\end{aligned}$$ and then $$\begin{aligned}
\|(V f)_\pm\|^2_{L^2(\Sigma,{\mathbb{C}}^4)}&=\dfrac{\tau^2+4}{4} \|f\|^2_{L^2(\Sigma,{\mathbb{C}}^4)}\pm \tau \Re \langle {\mathcal{B}}f,f\rangle_{L^2(\Sigma,{\mathbb{C}}^4)},\\
\|V f\|^2_{\mathcal{H}}&=\|(V f)_+\|^2_{L^2(\Sigma,{\mathbb{C}}^4)}+\|(V f)_-\|^2_{L^2(\Sigma,{\mathbb{C}}^4)}= \dfrac{\tau^2+4}{2} \|f\|^2_{L^2(\Sigma,{\mathbb{C}}^4)}.\end{aligned}$$ Therefore, the operator $$U:=\sqrt{\dfrac{2}{\tau^2+4}}\, V:L^2(\Sigma,{\mathbb{C}}^4)\to {\mathcal{H}}$$ is unitary. We are going to show that $U^*{\mathcal{L}}^\tau U=\Upsilon_\tau \oplus \Upsilon_\tau$. As the operators $K$, $M$, $U$ act pointwise, they commute and thus $$\label{eq-ula}
U^*{\mathcal{L}}^\tau U=U^*{\mathcal{L}}^\tau_0 U+K-M^2.$$
In order to obtain an expression for $U^*{\mathcal{L}}^\tau_0 U$ let us transform the expression $\ell^\tau_0(U f, Uf)$ for $f\in H^1(\Sigma,{\mathbb{C}}^4)$. In the local coordinates of $\Sigma$ one has $$\begin{gathered}
\big\|\nabla_s (Uf)_\pm\big\|^2_{T_s\Sigma\otimes{\mathbb{C}}^4}
=\sum_{j,k}g^{jk} \big\langle \partial_j (Uf)_\pm, \partial_k (Uf)_\pm\big\rangle_{{\mathbb{C}}^4}\\
=\dfrac{2}{\tau^2+4}\sum_{j,k}g^{jk}
\big\langle {\mathrm{i}}\beta \alpha \cdot \nu \partial_j f + {\mathrm{i}}\beta \alpha \cdot \partial_j\nu f \mp \dfrac{\tau}{2}\partial_j f,
{\mathrm{i}}\beta \alpha \cdot \nu \partial_k f + {\mathrm{i}}\beta \alpha \cdot \partial_k\nu f \mp \dfrac{\tau}{2}\partial_k f\big\rangle_{{\mathbb{C}}^4}\end{gathered}$$ and $$\begin{gathered}
\big\langle {\mathrm{i}}\beta \alpha \cdot \nu \partial_j f + {\mathrm{i}}\beta \alpha \cdot \partial_j\nu f \mp \dfrac{\tau}{2}\partial_j f,
{\mathrm{i}}\beta \alpha \cdot \nu \partial_k f + {\mathrm{i}}\beta \alpha \cdot \partial_k\nu f \mp \dfrac{\tau}{2}\partial_k f\big\rangle_{{\mathbb{C}}^4}\\
=\big\langle {\mathrm{i}}\beta \alpha \cdot \nu \partial_j f + {\mathrm{i}}\beta \alpha \cdot \partial_j\nu f,
{\mathrm{i}}\beta \alpha \cdot \nu \partial_k f + {\mathrm{i}}\beta \alpha \cdot \partial_k\nu f\big\rangle_{{\mathbb{C}}^4}
+\dfrac{\tau^2}{4} \langle \partial_j f, \partial_k f\rangle_{{\mathbb{C}}^4}\\
\mp \dfrac{\tau}{2}\Big(\big\langle \partial_j f, {\mathrm{i}}\beta \alpha \cdot \nu \partial_k f + {\mathrm{i}}\beta \alpha \cdot \partial_k\nu f\big\rangle_{{\mathbb{C}}^4}
+\big\langle {\mathrm{i}}\beta \alpha \cdot \nu \partial_j f + {\mathrm{i}}\beta \alpha \cdot \partial_j\nu f, \partial_k f\big\rangle_{{\mathbb{C}}^4}\Big).\end{gathered}$$ It follows that $$\begin{split}
&\big\|\nabla_s (Uf)_+\big\|^2_{T_s\Sigma\otimes{\mathbb{C}}^4}+\big\|\nabla_s (Uf)_-\big\|^2_{T_s\Sigma\otimes{\mathbb{C}}^4}\\
&=\dfrac{4}{\tau^2+4}\sum_{j,k}g^{jk} \Big(\big\langle {\mathrm{i}}\beta \alpha \cdot \nu \partial_j f + {\mathrm{i}}\beta \alpha \cdot \partial_j\nu f,
{\mathrm{i}}\beta \alpha \cdot \nu \partial_k f + {\mathrm{i}}\beta \alpha \cdot \partial_k\nu f\big\rangle_{{\mathbb{C}}^4}
+\dfrac{\tau^2}{4} \langle \partial_j f, \partial_k f\rangle_{{\mathbb{C}}^4}\Big).
\end{split}$$ We further use the unitarity of $\beta$ and of $\alpha\cdot\nu$ to transform $$\begin{split}
\big\langle {\mathrm{i}}\beta \alpha \cdot \nu \partial_j f + &{\mathrm{i}}\beta \alpha \cdot \partial_j\nu f,
{\mathrm{i}}\beta \alpha \cdot \nu \partial_k f + {\mathrm{i}}\beta \alpha\cdot \partial_k\nu f\big\rangle_{{\mathbb{C}}^4} \\
&=\big\langle \alpha \cdot \nu \partial_j f + \alpha \cdot \partial_j\nu f,
\alpha \cdot \nu \partial_k f + \alpha \cdot \partial_k\nu f\big\rangle_{{\mathbb{C}}^4}\\
&=\big\langle \partial_j f + (\alpha \cdot \nu ) (\alpha \cdot \partial_j\nu) f,
\partial_k f + (\alpha \cdot \nu ) (\alpha \cdot \partial_k\nu) f\big\rangle_{{\mathbb{C}}^4}=:J.
\end{split}$$ Now we use the identity $$\label{eq-axy}
(\alpha\cdot x) (\alpha\cdot y)=(x\cdot y) I_4 + {\mathrm{i}}\gamma_5 \alpha \cdot (x\times y)$$ and the equality $\nu \cdot \partial_j \nu=0$, which holds due to $|\nu|=1$, to find $$J=\big\langle \partial_j f + {\mathrm{i}}\gamma_5 \alpha \cdot (\nu\times \partial_j \nu) f,
\partial_k f + {\mathrm{i}}\gamma_5 \alpha \cdot (\nu\times \partial_k \nu) f\big\rangle_{{\mathbb{C}}^4}.$$ Denote $$\label{eq-aaj}
A_j:=\gamma_5 \alpha \cdot (\nu\times \partial_j \nu),$$ then we have $$\begin{gathered}
\big\|\nabla_s (Uf)_+\big\|^2_{T_s\Sigma\otimes{\mathbb{C}}^4}+\big\|\nabla_s (Uf)_-\big\|^2_{T_s\Sigma\otimes{\mathbb{C}}^4}\\
=\dfrac{4}{\tau^2+4}\sum_{j,k}g^{jk} \Big(
\big\langle \partial_j f + {\mathrm{i}}A_j f, \partial_k f + {\mathrm{i}}A_k f\big\rangle_{{\mathbb{C}}^4}
+\dfrac{\tau^2}{4} \langle \partial_j f, \partial_k f\rangle_{{\mathbb{C}}^4}\Big).\end{gathered}$$ Because of $$\begin{split}
&\big\langle \partial_j f + {\mathrm{i}}A_j f, \partial_k f + {\mathrm{i}}A_k f\big\rangle_{{\mathbb{C}}^4}
+\dfrac{\tau^2}{4} \langle \partial_j f, \partial_k f\rangle_{{\mathbb{C}}^4}\\
&\qquad \qquad \qquad =\dfrac{\tau^2+4}{4} \langle \partial_j f, \partial_k f\rangle_{{\mathbb{C}}^4} +
\big\langle \partial_j f , {\mathrm{i}}A_k f\big\rangle_{{\mathbb{C}}^4}+\big\langle {\mathrm{i}}A_j f, \partial_k f \big\rangle_{{\mathbb{C}}^4}
+\big\langle {\mathrm{i}}A_j f, {\mathrm{i}}A_k f\big\rangle_{{\mathbb{C}}^4}\\
&\qquad \qquad \qquad =\dfrac{\tau^2+4}{4}\Big( \langle \partial_j f, \partial_k f\rangle_{{\mathbb{C}}^4}
+\big\langle \partial_j f , {\mathrm{i}}\dfrac{4}{\tau^2+4}A_k f\big\rangle_{{\mathbb{C}}^4}+\big\langle {\mathrm{i}}\dfrac{4}{\tau^2+4} A_j f, \partial_k f \big\rangle_{{\mathbb{C}}^4}\\
&\qquad \qquad \qquad \qquad \qquad \qquad +\dfrac{\tau^2+4}{4}\big\langle {\mathrm{i}}\dfrac{4}{\tau^2+4} A_j f, {\mathrm{i}}\dfrac{4}{\tau^2+4}A_k f\big\rangle_{{\mathbb{C}}^4}\Big)\\
&=\dfrac{\tau^2+4}{4} \bigg( \Big\langle \partial_j f+{\mathrm{i}}\dfrac{4}{\tau^2+4} A_j f, \partial_k f+{\mathrm{i}}\dfrac{4}{\tau^2+4} A_k f\Big\rangle_{{\mathbb{C}}^4}
+\dfrac{\tau^2}{4}\cdot \Big( \dfrac{4}{\tau^2+4}\Big)^2\big\langle {\mathrm{i}}A_j f, {\mathrm{i}}A_k f\big\rangle_{{\mathbb{C}}^4}\bigg),
\end{split}$$ we obtain $$\label{eq-ll3}
\begin{split}
\big\|&\nabla_s (Uf)_+\big\|^2_{T_s\Sigma\otimes{\mathbb{C}}^4}+\big\|\nabla_s (Uf)_-\big\|^2_{T_s\Sigma\otimes{\mathbb{C}}^4}\\
&=\sum_{j,k}g^{jk} \bigg( \big\langle \partial_j f+{\mathrm{i}}\dfrac{4}{\tau^2+4} A_j f, \partial_k f+{\mathrm{i}}\dfrac{4}{\tau^2+4} A_k f\big\rangle_{{\mathbb{C}}^4}
+\dfrac{4\tau^2}{(\tau^2+4)^2}\big\langle A_j f, A_k f\big\rangle_{{\mathbb{C}}^4}\bigg)\\
&=\sum_{j,k}g^{jk} \bigg( \big\langle \partial_j f+{\mathrm{i}}\dfrac{4}{\tau^2+4} A_j f, \partial_k f+{\mathrm{i}}\dfrac{4}{\tau^2+4} A_k f\big\rangle_{{\mathbb{C}}^4}\bigg)
+ \dfrac{4\tau^2}{(\tau^2+4)^2} \big\langle f, W f\big\rangle_{{\mathbb{C}}^4}
\end{split}$$ with $$W:=\sum_{j,k}g^{jk} A_j A_k.$$ where we used $A_j^*=A_j$, which holds by and . Using the expression for $A_j$ and we obtain $$\label{eq-ajak}
\begin{split}
A_j A_k&=\gamma_5 \big(\alpha \cdot(\nu\times\partial_j \nu)\big)\gamma_5 \big(\alpha \cdot(\nu\times\partial_k \nu)\big)\\
&=\big(\alpha \cdot(\nu\times\partial_j \nu)\big)\big(\alpha \cdot(\nu\times\partial_k \nu)\big)
= a_{jk} I_4+ {\mathrm{i}}\gamma_5 \alpha\cdot b_{jk}
\end{split}$$ with $$a_{jk}:=(\nu\times\partial_j \nu)\cdot (\nu\times\partial_k \nu),
\quad
b_{jk}= (\nu\times\partial_j \nu)\times(\nu\times\partial_k \nu).$$ Due to $g^{jk}=g^{kj}$ and $b_{kj}=-b_{jk}$ we have $\sum_{jk}g^{jk}b_{jk}=0$, which shows that $W$ is a scalar potential, $$W=\sum_{j,k}g^{jk} a_{jk} I_4.$$ Recall the elementary identities $$\label{cross1}
\begin{aligned}
(a\times b)\cdot (c\times d)&= (a\cdot c) (b\cdot d) -(a\cdot d)(b\cdot c),\\
(a\times b)\times (a\times c)&= \big(a\cdot(b\times c)\big) a, \quad a,b,c,d\in{\mathbb{R}}^3,
\end{aligned}$$ then $a_{jk}= |\nu|^2 (\partial_j\nu\cdot\partial_k\nu) - (\nu \cdot\partial_j\nu) (\nu \cdot\partial_k\nu) = \partial_j\nu\cdot\partial_k\nu$, as $|\nu|=1$. In order to give a more explicit expression for $W$ we assume that the local coordinates are chosen in such a way that the associated tangent vectors $t_j$ correspond to the principal directions, i.e. that $\partial_j \nu=\kappa_j t_j$ with $\kappa_j$ being the principal curvatures, then $g_{jk}$ and $g^{jk}$ are diagonal, $\partial_j\nu\cdot\partial_k\nu=\kappa_j \kappa_k g_{jk} \delta_{jk}$, and $$W=\sum_{j,k} \kappa_j \kappa_k\delta_{jk} g^{jk} g_{jk}=\kappa_1^2+\kappa_2^2 = 4M^2-2K.$$ Therefore, it follows from that $$\begin{gathered}
\label{eq-elluu}
\ell^\tau_0(Uf,Uf)=\iint_\Sigma \sum_{j,k}g^{jk} \big\langle \partial_j f+{\mathrm{i}}\dfrac{4}{\tau^2+4} A_j f, \partial_k f+{\mathrm{i}}\dfrac{4}{\tau^2+4} A_k f\big\rangle_{{\mathbb{C}}^4}{\mathrm{d}}\Sigma\\
+\dfrac{8\tau^2}{(\tau^2+4)^2} \big\langle f,(2 M^2 - K)f\big\rangle_{L^2(\Sigma,{\mathbb{C}}^4)}.\end{gathered}$$ Furthermore, using in the expressions for the Dirac matrices one has $$\label{eq-aomega}
\gamma_5 \alpha_j=\begin{pmatrix}
\sigma_j & 0\\
0 & \sigma_j
\end{pmatrix}, \quad
A_j= \begin{pmatrix}
\omega_j & 0\\
0 & \omega_j
\end{pmatrix}$$ with $\omega_j$ given in . Therefore, using the natural unitary identification operator $J:L^2(\Sigma,{\mathbb{C}}^2)\otimes L^2(\Sigma,{\mathbb{C}}^2)\to L^2(\Sigma,{\mathbb{C}}^4)$ one may rewrite as $$\begin{gathered}
\ell^\tau_0(Uf,Uf)=(\lambda_\frac{4}{\tau^2+4} \oplus \lambda_\frac{4}{\tau^2+4})(J^*f,J^*f)\\
+\dfrac{8\tau^2}{(\tau^2+4)^2} \big\langle J^*f,(2 M^2 - K)J^*f\big\rangle_{L^2(\Sigma,{\mathbb{C}}^2)\otimes L^2(\Sigma,{\mathbb{C}}^2)},\end{gathered}$$ which yields $$(U J)^*{\mathcal{L}}_0^\tau (UJ) = \Lambda\Big(\frac{4}{\tau^2+4}\Big)\oplus \Lambda\Big(\frac{4}{\tau^2+4}\Big)+\dfrac{8\tau^2}{(\tau^2+4)^2} (2M^2-K).$$ As $K$, $M$ and $J$ commute, the substitution into completes the proof.
As both $K$ and $M$ are bounded, one can set $c_0:=\|K-M^2\|_\infty$ and remark that for all $c>0$, $\delta>0$ and $j\in {\mathbb{N}}$ there holds $$\begin{aligned}
E_j\big((1+c\delta) {\mathcal{L}}^\tau_0+K-M^2\big)&\equiv E_j\Big((1+c\delta) ({\mathcal{L}}^\tau_0+K-M^2) -c\delta(K-M^2)\Big)\\
&\le (1+c\delta) E_j(\Upsilon_\tau \oplus \Upsilon_\tau) c_0 c\delta,\\
E_j\big((1-c\delta) {\mathcal{L}}^\tau_0+K-M^2\big)&\equiv E_j\Big((1-c\delta) ({\mathcal{L}}^\tau_0+K-M^2) +c\delta(K-M^2)\Big)\\
&\ge (1-c\delta) E_j(\Upsilon_\tau \oplus \Upsilon_\tau)- c_0 c\delta.\end{aligned}$$ Therefore, Theorem \[thm-ev1\] becomes a consequence of the following two-side asymptotic estimate:
\[prop-main1\] Assume that $\delta\equiv \delta(m)>0$ is chosen in order to satisfy , then there exist constants $c>0$ and $m_0>0$ such that for any $m>m_0$ and $j\in\big\{1,\dots,{\mathcal{N}}(A^2_{m,\tau},m^2)\big\}$ it holds $$\begin{gathered}
\label{main_estimates}
\Big( \dfrac{\tau^2-4}{\tau^2+4}\Big)^2 m^2 + E_j\Big( (1-c\delta){\mathcal{L}}^\tau_0 + K-M^2\Big)-c(\delta+ m^2e^{-2\mu m\delta}) \le E_j(A_{m,\tau}^2)\\
\le \Big( \dfrac{\tau^2-4}{\tau^2+4}\Big)^2 m^2 + E_j\Big( (1+c\delta){\mathcal{L}}^\tau_0 + K-M^2\Big)+c(\delta+m^2e^{-2\mu m\delta}).\end{gathered}$$
The proof of Proposition \[prop-main1\] occupies the rest of the paper and is split into several parts. In Subsection \[sec-tub\] we give first a two-side estimate for the eigenvalues $E_j(A_{m,\tau}^2)$ in terms of operators in $\Sigma\times(-\delta,\delta)$ by using tubular coordinates. In Subsection \[sec-upp1\] we obtain the right-hand side inequality of , and Subsection \[sec-low1\] is devoted to the lower bound.
One may use a part of the computation of Proposition \[prop45\] to give some additional information on the external Yang-Mills potential given by the form $\omega$ and appearing in the definition of the effective operator $\Upsilon_\tau$.
\[prop-field1\] Let $\omega$ be given by and let $\theta \in {\mathbb{R}}$. Then the field strength $F_\theta$ defined by $\theta\,\omega$ is $F^\theta= 2\theta (1-\theta) K (\sigma \cdot \nu) \operatorname{vol}_\Sigma$ with $\operatorname{vol}_\Sigma$ being the volume form on $\Sigma$.
By definition, see [@sred Section 69], the field strength $F\equiv F_\theta$ defined by the form $\theta \omega$ is given by $F=F_{12}\,{\mathrm{d}}s_1\wedge {\mathrm{d}}s_2$, where $F_{jk}=\theta(\partial_j \omega_k-\partial_k \omega_j) + {\mathrm{i}}\theta^2 (\omega_j \omega_k - \omega_k \omega_j)$. One easily computes $\partial_j \omega_k= \sigma \cdot (\partial_j \nu \times \partial_k \nu)+ \sigma \cdot (\nu \times \partial_j \partial_k \nu)$, which gives $\partial_1 \omega_2-\partial_2 \omega_1= 2\sigma\cdot x$ with $x:=\partial_1 \nu \times \partial_2 \nu$, and ${\mathrm{i}}(\omega_j \omega_k - \omega_k \omega_j)=-2 \sigma \cdot b_{jk}$ in view of and of the block representation . To obtain a readable expression for $b_{jk}$ we use , then $b_{jk}=\big(\nu \cdot (\partial_j \nu \times \partial_k \nu)\big)\nu$, and $b_{12}=(\nu\cdot x) \nu$ is the orthogonal projection of $x$ onto the line directed by $\nu$. As the vectors $\partial_j \nu$ are orthogonal to $\nu$, the vector $x$ is a multiple of $\nu$, therefore, we have $b_{12}=x$ and $F_{12}= 2\theta(1-\theta) \sigma \cdot x$. In order to compute the vector $x$ we assume that the local coordinates are chosen in such a way that the triple $(t_1,t_2,\nu)$ is direct and recall that $\partial_j \nu=S t_j$ with $S$ being the Weingarten map, then $x=\partial_1 \nu \times \partial_2 \nu= (\det S) (t_1\times t_2)\equiv K |t_1\times t_2|\nu$. Having in mind that the volume form is $\operatorname{vol}_\Sigma=|t_1\times t_2|{\mathrm{d}}s_1\wedge {\mathrm{d}}s_2$, we arrive at the sought representation.
Reduction to tubular neighborhoods {#sec-tub}
----------------------------------
The proof of Proposition \[prop-main1\] is based on a variant of rather standard estimates in thin neighborhoods of $\Sigma$. We are going to start with the following result:
\[lem-qnd\] There exist $\delta_0>0$ and $c>0$ such that for any $\delta\in(0,\delta_0)$, any $m\in{\mathbb{R}}$ and any $j\in\big\{1,\dots,{\mathcal{N}}(A^2_{m,\tau},m^2)\big\}$ there holds $$E_j(q^N_{m,\tau,\delta})+m^2\le E_j(A^2_{m,\tau})\le E_j(q^D_{m,\tau,\delta})+m^2,$$ where the sesquilinear forms $q^{N/D}_{m,\delta}$ in $L^2\big(\Sigma\times(-\delta,\delta),{\mathbb{C}}^4\big)$ are given by $$\begin{gathered}
q^N_{m,\tau,\delta}(u,u)=\iiint_{\Sigma\times(-\delta,\delta)} \Big((1-c\delta)\|\nabla_s u\|^2_{T_s\Sigma\otimes{\mathbb{C}}^4}
+ (K-M^2-c\delta)\, |u|^2\Big){\mathrm{d}}\Sigma{\mathrm{d}}t\\
+ \iint_\Sigma \bigg(\int_{-\delta}^\delta |\partial_t u|^2 {\mathrm{d}}t + \frac{2m}{\tau} \big|u(\cdot,0^+)-u(\cdot,0^-)\big|^2 -c \big|u(\cdot,\delta)\big|^2-c\big|u(\cdot,-\delta)\big|^2\bigg) {\mathrm{d}}\Sigma\end{gathered}$$ with domain $$\begin{gathered}
{\mathop{\mathcal{D}}}(q^N_{m,\tau,\delta})=\Big\{ u\in H^1\Big((\Sigma\times(-\delta,\delta)) \setminus (\Sigma\times\{0\}), \mathbb{C}^4\Big):
{\mathcal{P}}_\tau^- u(\cdot,0^+)+{\mathcal{P}}_\tau^+ u(\cdot,0^-)=0\Big\}\end{gathered}$$ and $$\begin{gathered}
q^D_{m,\tau,\delta}(u,u)=\iiint_{\Sigma\times (-\delta,\delta)} (1 + c \delta) \|\nabla_s u\|^2_{T_s\Sigma\otimes{\mathbb{C}}^4}
+ (K-M^2+c\delta)\,|u|^2\Big){\mathrm{d}}\Sigma{\mathrm{d}}t\\
+ \iint_\Sigma \bigg(\int_{-\delta}^\delta |\partial_t u|^2 {\mathrm{d}}t + \frac{2m}{\tau} \big|u(\cdot,0^+)-u(\cdot,0^-)\big|^2 \bigg) {\mathrm{d}}\Sigma\end{gathered}$$ with domain $$\begin{gathered}
{\mathop{\mathcal{D}}}(q^D_{m,\tau,\delta})=\Big\{ u\in H^1\Big(\big(\Sigma\times(-\delta,\delta)\big) \setminus (\Sigma\times\{0\}), \mathbb{C}^4\Big):\\
{\mathcal{P}}_\tau^- u(\cdot,0^+)+{\mathcal{P}}_\tau^+ u(\cdot,0^-)=0, \ u(\cdot,\delta)=u(\cdot,-\delta)=0\Big\}.\end{gathered}$$
The computations are quite standard, but we prefer to give full details for the sake of completeness. Consider the map $$\label{eqn:chang_var}
\Phi: \Sigma\times{\mathbb{R}}\ni (s,t)\mapsto s - t\nu (s) \in {\mathbb{R}}^3.$$ According to a classical result of differential geometry there is some $\delta_0>0$ such that for all $0<\delta<\delta_0$ the mapping $\Phi:\Sigma\times (-\delta,\delta)\mapsto \Omega^\delta:=\big\{x\in {\mathbb{R}}^3: \operatorname{dist}(x,\Sigma)<\delta)\big\}$ is a diffeomorphism with $\operatorname{dist}\big(\Phi(s,t), \Sigma\big)=|t|$. From now we assume that $\delta\in(0,\delta_0)$ and define $$\Phi\big(\Sigma\times(0,\delta)\big):=\Omega^\delta_+,
\quad
\Phi\big(\Sigma\times(-\delta,0)\big):=\Omega^\delta_-,
\quad
\Omega^\delta_\pm:=\Omega^\delta\cap \Omega_\pm.$$
Denote by $a$ the sesquilinear form defined on ${\mathop{\mathcal{D}}}(a)={\mathop{\mathcal{D}}}(A_{m,\tau})$ by $$\begin{gathered}
a(u,u)=\|A_{m,\tau} u\|^2_{L^2({\mathbb{R}}^3,{\mathbb{C}}^4)}-m^2\|u\|^2_{L^2({\mathbb{R}}^3,{\mathbb{C}}^4)}\\
=\iiint_{{\mathbb{R}}^3\setminus \Sigma} \big|\nabla u\big|^2\, {\mathrm{d}}x + \frac{2 m}{\tau} \iint_\Sigma | u_+ - u_-|^2 {\mathrm{d}}\Sigma + \iint_\Sigma M |u_+|^2 {\mathrm{d}}\Sigma - \iint_\Sigma M |u_-|^2 {\mathrm{d}}\Sigma.\end{gathered}$$ Furthermore, using the natural identification $$L^2({\mathbb{R}}^3,{\mathbb{C}}^4)\simeq L^2(\Omega^\delta_+,{\mathbb{C}}^4)\oplus L^2(\Omega^\delta_-,{\mathbb{C}}^4)\oplus L^2({\mathbb{R}}^3\setminus \overline{\Omega^\delta},{\mathbb{C}}^4),
\quad u\simeq (u_+,u_-,u_c),$$ consider the sesquilinear form $$\begin{gathered}
b^N(u,u)=\iiint_{\Omega^\delta_+\cup \Omega^\delta_-\cup({\mathbb{R}}^3\setminus \partial\Omega^\delta)} \big| \nabla u \big|^2\, {\mathrm{d}}x \\
+ \frac{2 m}{\tau} \iint_\Sigma | u_+ - u_-|^2 {\mathrm{d}}\Sigma + \iint_\Sigma M |u_+|^2 {\mathrm{d}}\Sigma
- \iint_\Sigma M |u_-|^2 {\mathrm{d}}\Sigma\end{gathered}$$ defined on the functions $u$ with $$\begin{gathered}
u_+\in H^1(\Omega^\delta_+,{\mathbb{C}}^4), \quad u_-\in H^1(\Omega^\delta_-\cap \Omega,{\mathbb{C}}^4), \quad
u_c\in H^1({\mathbb{R}}^3\setminus \overline{\Omega^\delta},{\mathbb{C}}^4),\\
{\mathcal{P}}_\tau^- u_+ + {\mathcal{P}}_\tau^+ u_-=0 \text{ on } \Sigma,\end{gathered}$$ and denote by $b^D$ its restriction to the functions vanishing at $\partial \Omega^\delta$, then in the sense of forms one has $b^N\le a \le b^D$. Furthermore, one has the representations $$b^{N/D}=b_0^{N/D} \oplus b_c^{N/D},$$ where $b^N_0$ is the sesquilinear form in $L^2(\Omega^\delta,{\mathbb{C}}^4)$ given by $$\begin{gathered}
b^N_0(u,u)=\iiint_{\Omega^\delta \setminus \Sigma} \big| \nabla u \big|^2\, {\mathrm{d}}x
+ \frac{2 m}{\tau} \iint_\Sigma | u_+ - u_-|^2 {\mathrm{d}}\Sigma\\
+ \iint_\Sigma M |u_+|^2 {\mathrm{d}}\Sigma - \iint_\Sigma M |u_-|^2 {\mathrm{d}}\Sigma,
\quad u_\pm:=u|_{\Omega^\delta_\pm},\end{gathered}$$ on the functions $u$ such that $u_\pm\in H^1(\Omega^\delta_\pm,{\mathbb{C}}^4)$ and ${\mathcal{P}}_\tau^- u_+ + {\mathcal{P}}_\tau^+ u_-=0$ on $\Sigma$, the sesquilinear form $b^N_c$ is given by $$b^N_c(u_c,u_c)=\iiint_{{\mathbb{R}}^3\setminus \overline{\Omega^\delta}}
\big| \nabla u_c \big|^2\, {\mathrm{d}}x,\quad
u_c\in H^1(\Omega^c,{\mathbb{C}}^4),$$ and the forms $b^D_0$ and $b^D_c$ are the restrictions of $b^N_0$ and $b^N_c$, respectively, on functions vanishing on $\partial\Omega^\delta\equiv \partial({\mathbb{R}}^3\setminus \overline{\Omega^\delta})$. Due to the obvious inequalities $b^{N/D}_c\ge 0$ and to the fact that $b^{N/D}_0$ define operators with compact resolvents, one has then $$\label{eq-bab}
E_n(b^N_0)\le E_n(a) \le E_n(b^D_0) \quad \text{ for $n$ with $E_n(a)<0$}$$ We are now going to give a lower bound of the form $b^N_0$ and an upper bound of the form $b^D_0$ using the above diffeomorphism $\Phi$. The metric $G$ on $\Sigma\times(-\delta,\delta)$ induced by $\Phi$ takes the form $$G(s,t)=\Tilde g (s,t)+ {\mathrm{d}}t^2, \quad \Tilde g(s,t):= g(s)\circ(I_s-t S)^2$$ where $I_s:T_s \Sigma\to T_s\Sigma$ is the identity map, $S:T_s \Sigma\to T_s \Sigma$ is the Weingarten map, $S=\mathrm{d}\, \nu(s)$, and $g$ is the metric of $\Sigma$ induced by the embedding in ${\mathbb{R}}^3$. The associated volume form on $\Sigma\times(-\delta,\delta)$ is given by $$\begin{gathered}
{\mathrm{d}}\operatorname{vol}(s,t)=\sqrt{\det G(s,t)}\,{\mathrm{d}}s{\mathrm{d}}t\equiv
\varphi(s,t) \sqrt{\det g} \,{\mathrm{d}}s {\mathrm{d}}t\equiv \varphi(s,t) {\mathrm{d}}\Sigma(s){\mathrm{d}}t,\\
\varphi(s,t):=\det(I_s-t S)=1-2t M(s)+t^2K(s),\end{gathered}$$ and we may assume that $\delta$ is sufficiently small to have $\varphi\ge \frac{1}{2}$.
Let us start by considering the unitary transform $$U : L^2(\Omega^\delta) \ni u\mapsto (U u) := u \circ \Phi \in L^2\big(\Sigma \times (-\delta,\delta),{\mathrm{d}}\operatorname{vol}\big).$$ Then the standard change of variables with the help of the above expressions for the metric tensor show that for $\Tilde u:=U u$ one has in the local coordinates $$\iiint_{\Omega^\delta_\pm} |\nabla u|^2{\mathrm{d}}x = \pm\iint_{\Sigma}\int_0^{\pm\delta} \sum_{j,k=1}^3 G^{jk} \langle \partial_j \Tilde u, \partial_k \Tilde u\rangle \,{\mathrm{d}}\operatorname{vol}(s,t),
\quad (G^{jk}):=(G_{jk})^{-1}.$$ Therefore, if we define the sesquilinear forms $b_1^{N/D}$ in $L^2\big(\Sigma \times (-\delta,\delta),{\mathrm{d}}\operatorname{vol}\big)$ by $b^{N/D}_0(u,u)=b^{N/D}_1(U u, U u)$, then $b_1^N$ is given explicitly by $$\begin{split}
b^N_1(u,u)&=\iiint_{\Sigma\times(-\delta,\delta)}
\sum_{j,k=1}^3 G^{jk} \langle \partial_j u, \partial_k u\rangle \varphi{\mathrm{d}}\Sigma{\mathrm{d}}t\\
&~+\dfrac{2m}{\tau}\iint_\Sigma \big| u(\cdot,0^+)-u(\cdot,0^-)\big|^2{\mathrm{d}}\Sigma +\iint_\Sigma M\big| u(\cdot,0^+)\big|^2{\mathrm{d}}\Sigma - \iint_\Sigma M\big| u(\cdot,0^-)\big|^2{\mathrm{d}}\Sigma
\end{split}$$ on the domain $$\begin{gathered}
{\mathop{\mathcal{D}}}(b^N_1)=U{\mathop{\mathcal{D}}}(b^N_0)=\Big\{ u\in H^1\big( \big(\Sigma\times(-\delta,0), \mathbb{C}^4\big)
\cup \big(\Sigma\times(0,\delta), \mathbb{C}^4\big),
\varphi{\mathrm{d}}\Sigma {\mathrm{d}}t\big): \\
{\mathcal{P}}_\tau^- u(\cdot,0^+)+{\mathcal{P}}_\tau^+ u(\cdot,0^-)=0\Big\}
\end{gathered}$$ and $b^D_1$ is its restriction to the functions vanishing at $\Sigma\times\{\pm\delta\}$. By construction one has $E_n(b^{N/D}_1)=E_n(b^{N/D}_0)$ for all $n$, and due to there holds $$\label{eq-bab2}
E_n(b^N_1)\le E_n(a) \le E_n(b^D_1) \text{ for any $n$ with $E_n(a)<0$.}$$
Due to the above expression for $\Tilde g$ one can estimate, with some $C>0$ that for all for $u\in{\mathop{\mathcal{D}}}(b_1^{N/D})$ we have $$\begin{split}
(1-C\delta) \sum_{j,k=1}^2 g^{jk} \langle \partial_j u, \partial_k u\rangle
+|\partial_t u|^2 &\le \sum_{j,k=1}^3 G^{jk} \langle \partial_j u, \partial_k u\rangle\\
&\le
(1+C\delta) \sum_{j,k=1}^2 g^{jk} \langle \partial_j u, \partial_k u\rangle
+|\partial_t u|^2,
\end{split}$$ and then $b^N_2\le b^N_1$ and $b^D_1\le b^D_2$, where the form $b^N_2$ is given by $$\begin{split}
b^N_2&(u,u)=\iiint_{\Sigma\times(-\delta,\delta)}
\Big((1-C\delta)\| \nabla_s u\|^2_{T_s\Sigma\otimes{\mathbb{C}}^4} + |\partial_t u|^2\Big) \varphi\, {\mathrm{d}}\Sigma{\mathrm{d}}t\\
&+\dfrac{2m}{\tau}\iint_\Sigma \big| u(\cdot,0^+)-u(\cdot,0^-)\big|^2{\mathrm{d}}\Sigma
+\iint_\Sigma M\big| u(\cdot,0^+)\big|^2{\mathrm{d}}\Sigma - \iint_\Sigma M\big| u(\cdot,0^-)\big|^2{\mathrm{d}}\Sigma
\end{split}$$ on the domain ${\mathop{\mathcal{D}}}(b^N_2)={\mathop{\mathcal{D}}}(b^N_1)$, and the form $b^D_2$ is given by $$\begin{split}
b^D_2&(u,u)=\iiint_{\Sigma\times(-\delta,\delta)}
\Big((1+C\delta)\| \nabla_s u\|^2_{T_s\Sigma\otimes{\mathbb{C}}^4} + |\partial_t u|^2\Big) \varphi\, {\mathrm{d}}\Sigma{\mathrm{d}}t\\
&+\dfrac{2m}{\tau}\iint_\Sigma \big| u(\cdot,0^+)-u(\cdot,0^-)\big|^2{\mathrm{d}}\Sigma
+\iint_\Sigma M\big| u(\cdot,0^+)\big|^2{\mathrm{d}}\Sigma - \iint_\Sigma M\big| u(\cdot,0^-)\big|^2{\mathrm{d}}\Sigma
\end{split}$$ on the domain $ {\mathop{\mathcal{D}}}(b^D_2)=\big\{u\in {\mathop{\mathcal{D}}}(b^N_2): u(\cdot,\pm\delta)=0\big\}$. Then for any $n$ one has $E_n(b^N_2)\le E_n(b^N_1)$ and $E_n(b^D_1)\le E_n(b^D_2)$ and, due to , $$\label{eq-bab3}
E_n(b^N_2)\le E_n(a) \le E_n(b^D_2) \text{ for any $n$ with $E_n(a)<0$}.$$
In order to remove the weight $\varphi$ in the above expressions, let us introduce the unitary transform $$V : L^2\big(\Sigma\times(-\delta,\delta),\varphi {\mathrm{d}}\Sigma {\mathrm{d}}t)\to L^2\big(\Sigma\times(-\delta,\delta)\big),
\quad (Vu)(s,t) := \varphi(s,t)^{\frac{1}{2}} u(s,t)$$ and the sesquilinear forms $b^{N/D}_3(u,u)=b^{N/D}_2(V^{-1}u,V^{-1}u)$ defined on ${\mathop{\mathcal{D}}}(b^{N/D}_3)=V\big( {\mathop{\mathcal{D}}}(b^{D/N}_2)\big)$. One sees easily that ${\mathop{\mathcal{D}}}(b^{N/D}_3)={\mathop{\mathcal{D}}}(q^{N/D}_{m,\tau,\delta})$. Furthermore, to have a more explicit expression for $b_3^{N/D}$ we remark that for $$v(s,t):=V^{-1}u(s,t) = \varphi(s,t)^{-\frac12}u(s,t)$$ one has $$\partial_t v = \varphi^{-\frac{1}{2}}\partial_t u - \frac{1}{2}\partial_t \varphi \cdot \varphi^{-\frac{3}{2}} u
=\varphi^{-\frac{1}{2}}\partial_t u +(M-tK) \varphi^{-\frac{3}{2}} u.$$ Hence, we get $$\begin{aligned}
|\partial_t v|^2 & = \varphi^{-1}|\partial_t u|^2 + \varphi^{-3}(M - t K)^2 |u|^2 + \varphi^{-2}(M- tK) \cdot 2\Re \langle\partial_t u,u\rangle\\
&=\varphi^{-1}|\partial_t u|^2 + \varphi^{-3}(M - t K)^2 |u|^2 + \varphi^{-2}(M- tK) \partial_t \big(|u|^2\big),
\end{aligned}$$ which implies $$\begin{gathered}
\int_{-\delta}^\delta |\partial_t v|^2\varphi {\mathrm{d}}t =
\int_{-\delta}^\delta |\partial_t u|^2 {\mathrm{d}}t
+\int_{-\delta}^\delta \varphi^{-2} (M-tK)^2 |u|^2{\mathrm{d}}t\\
+ \int_{-\delta}^\delta \varphi^{-1}(M- tK) \partial_t \big(|u|^2\big){\mathrm{d}}t=:J_1+J_2+J_3.
\end{gathered}$$ Using the integration by parts on $(-\delta,0)$ and $(0,\delta)$ we get $$\begin{gathered}
J_3=-\int_{-\delta}^{\delta} \partial_t \big(\varphi^{-1}(M- tK)\big)|u|^2{\mathrm{d}}t +\frac{M-\delta K}{1-2\delta M+\delta^2K} \big|u(\cdot,\delta)\big|^2- M \big|u(\cdot,0^+)\big|^2\\
+M \big|u(\cdot,0^-)\big|^2- \frac{M+\delta K}{1+2\delta M + \delta^2K}\big|u(\cdot,-\delta)\big|^2 .\end{gathered}$$ In view of the expression for $\varphi$ one sees that uniformly on $\Sigma$ when $\delta$ tends to $0$, one has $$\begin{gathered}
\frac{M\pm\delta K}{1\pm2\delta M+\delta^2K}=M + {\mathcal{O}}(\delta), \quad ,\varphi^{-2} (M-tK)^2=M^2+{\mathcal{O}}(\delta)\\
-\partial_t \big(\varphi^{-1}(M- tK)\big)=-2M^2+K +{\mathcal{O}}(\delta).\end{gathered}$$ Therefore, for $u\in {\mathop{\mathcal{D}}}(b^{N}_3)$ we can estimate, with a suitable $C>0$, $$\begin{gathered}
\label{eq-au1}
\int_{-\delta}^\delta |\partial_t V^{-1} u|^2 \varphi{\mathrm{d}}t
\ge \int_{-\delta}^\delta \big( |\partial_t u|^2
+(K-M^2 -C\delta) |u|^2 \big) {\mathrm{d}}t \\
+ M \big|u(\cdot,0^-)\big|^2 - M \big|u(\cdot,0^+)\big|^2
-C \big|u(\cdot,-\delta)\big|^2 -C \big|u(\cdot,\delta)\big|^2,\end{gathered}$$ while for $u\in {\mathop{\mathcal{D}}}(b^{D}_3)$ the terms with $u(\cdot,\pm\delta)$ vanish, thus, $$\begin{gathered}
\label{eq-au2}
\int_{-\delta}^\delta |\partial_t V^{-1} u|^2\varphi
\le \int_{-\delta}^\delta \big(|\partial_t u|^2{\mathrm{d}}t
+(K-M^2 +C\delta) |u|^2 \big) {\mathrm{d}}t \\
+ M \big|u(\cdot,0^-)\big|^2 - M \big|u(\cdot,0^+)\big|^2.\end{gathered}$$
In order to control the integral of $ \| \nabla_s v\|^2_{T_s\Sigma\otimes{\mathbb{C}}^4} \varphi$ we remark that due to the Cauchy-Schwarz inequality one has $$\begin{split}
\| \nabla_s v\|^2_{T_s\Sigma\otimes{\mathbb{C}}^4}
&=\Big\| \varphi^{-\frac12}\nabla_s u-\dfrac{1}{2} \varphi^{-\frac32}\nabla_s \varphi \cdot u\Big\|^2_{T_s\Sigma\otimes{\mathbb{C}}^4}\\
&\ge (1-\delta)\big\| \varphi^{-\frac12}\nabla_s u\big\|^2_{T_s\Sigma\otimes{\mathbb{C}}^4} +\Big(1-\dfrac{1}{\delta}\Big) \Big\|\dfrac{1}{2} \varphi^{-\frac32}\nabla_s \varphi \cdot u\Big\|^2_{T_s\Sigma\otimes{\mathbb{C}}^4}\\
&\ge (1-\delta)\varphi^{-1} \|\nabla_s u\|^2_{T_s\Sigma\otimes{\mathbb{C}}^4}- \dfrac{t^2}{\delta \varphi^3} \left\|\nabla_s M-\frac{t}{2} \nabla_s K\right\|^2_{T_s\Sigma} |u|^2,
\end{split}$$ which results in $$\begin{gathered}
\label{eq-au3}
\iiint_{\Sigma\times(-\delta,\delta)} \| \nabla_s v\|^2_{T_s\Sigma\otimes{\mathbb{C}}^4} \varphi {\mathrm{d}}\Sigma {\mathrm{d}}t\\
\ge (1-\delta)\iiint_{\Sigma\times(-\delta,\delta)} \| \nabla_s u\|^2_{T_s\Sigma\otimes{\mathbb{C}}^4} {\mathrm{d}}\Sigma {\mathrm{d}}t
-C'\delta \iiint_{\Sigma\times(-\delta,\delta)} | u|^2 {\mathrm{d}}\Sigma {\mathrm{d}}t\end{gathered}$$ with a suitable $C'>0$. Analogous estimates give $$\begin{gathered}
\label{eq-au4}
\iiint_{\Sigma\times(-\delta,\delta)} \| \nabla_s v\|^2_{T_s\Sigma\otimes{\mathbb{C}}^4} \varphi {\mathrm{d}}\Sigma {\mathrm{d}}t\\
\le (1+\delta)\iiint_{\Sigma\times(-\delta,\delta)} \| \nabla_s u\|^2_{T_s\Sigma\otimes{\mathbb{C}}^4} {\mathrm{d}}\Sigma {\mathrm{d}}t
+C''\delta \iiint_{\Sigma\times(-\delta,\delta)} |u|^2 {\mathrm{d}}\Sigma {\mathrm{d}}t\end{gathered}$$ with some $C''>0$. The substitution of and into the expression for $b^N_3$ and of and into the expression for $b^D_3$ give the result.
For the rest of the section we always assume that $\delta$ is any function of $m$ satisfying , then the assumptions of Lemma \[lem-qnd\] are satisfied for large $m$. Recall that the parameter $\mu\in(0,1)$ was introduced in .
Upper bound {#sec-upp1}
-----------
In this section we derive an upper bound for the eigenvalues of $q_{m, \tau, \delta}^D$ from Lemma \[lem-qnd\]. Let us start with the analysis of an auxiliary one-dimensional operator.
\[lem1dd\] Let $s\in\Sigma$, $m>0$, consider the following sesquilinear form $t^D_{s,m}$ in $L^2\big((-\delta,\delta),{\mathbb{C}}^4\big)$ given by $$\begin{aligned}
t^D_{s,m}(u,u)&=\int_{-\delta}^\delta |u'|^2 {\mathrm{d}}t + \frac{2m}{\tau} \big|u(0^+)-u(0^-)\big|^2,\\
{\mathop{\mathcal{D}}}(t^D_{s,m})&=\Big\{ u\in H^1\big((-\delta,\delta)\setminus\{0\},{\mathbb{C}}^4\big): {\mathcal{P}}^-_\tau(s) u(0^+)+{\mathcal{P}}^+_\tau(s) u(0^-)=u(\pm\delta)=0\Big\},
\end{aligned}$$ and let $T^D_{s,m}$ be the associated self-adjoint operator in $L^2\big((-\delta,\delta),{\mathbb{C}}^4\big)$. Then for $m\to +\infty$ the first eigenvalue of $T^D_{s,m}$ is independent of $s$, has the multiplicity $4$ and is given by $$E_1(T^D_{s,m})= -\mu^2m^2 \big(1+{\mathcal{O}}(e^{-2\mu m\delta})\big). \label{eq-e1d}$$ Furthermore, one can represent, with a suitable smooth function $\psi_m:(0,\delta)\to{\mathbb{R}}$ independent of $s$, $$\begin{gathered}
\label{eq-vv1}
\ker\big( T^D_{s,m} -E_1(T^D_{s,m})\big)
=\Big\{v:
v(t)=v_\pm \psi_m(|t|) \text{ for } \pm t>0\\ \text{ with }
v_\pm\in{\mathbb{C}}^4 \text{ such that } {\mathcal{P}}^-_\tau(s) v_+ +{\mathcal{P}}^+_\tau(s) v_-=0
\Big\}.
\end{gathered}$$
Let us start by giving a more precise description of $T^D_{s,m}$. It is standard to see that ${\mathop{\mathcal{D}}}(T^D_{s,m})\subset H^2\big((-\delta,\delta)\setminus\{0\}\big)$ and that $T^D_{s,m}$ acts as $T^D_{s,m} u=-u''$. Therefore, it is sufficient to specify the boundary conditions at $0$ and $\pm\delta$. Let $v\in {\mathop{\mathcal{D}}}(T^D_{s,m})$, then $v$ belongs to ${\mathop{\mathcal{D}}}(t^D_{s,m})$, i.e. $$\begin{aligned}
\label{eq-p1d}
v(0^+)&={\mathcal{R}}^+_\tau v(0^-),\\
\label{eq-p2d}
v(\pm\delta)&=0,\end{aligned}$$ and $t^D_{s,m}(u,v)=\langle u,T^D_{s,m}v\rangle_{L^2((-\delta,\delta),{\mathbb{C}}^4)}$ for all $u\in {\mathop{\mathcal{D}}}(t^D_{s,m})$. Using integration by parts on $(-\delta,0)$ and $(0,\delta)$ we conclude that $$\begin{aligned}
t^D_{s,m}(u,v)&=\int_{-\delta}^\delta \langle u, -v''\rangle{\mathrm{d}}t +s^D_{s,m}(u,v),\\
s^D_{s,m}(u,v)&=\big\langle u(0^-),v'(0^-)\big\rangle
-\big\langle u(0^+),v'(0^+)\big\rangle + \frac{2m}{\tau} \big \langle u(0^+)-u(0^-), v(0^+)-v(0^-)\big\rangle.\end{aligned}$$ Therefore, it is sufficient to check for which $v$ one has $s^D_{s,m}(u,v)=0$ for all $u\in {\mathop{\mathcal{D}}}(t^D_{s,m})$. Due to the fact that $u(0^+) = {\mathcal{R}}_\tau^+ u(0^-)$, $u(0^-)\in{\mathbb{C}}^4$ is arbitrary and to $$s^D_{s,m}(u,v)
=\big\langle u(0^-),v'(0^-)-{\mathcal{R}}^+_\tau v'(0^+)\big\rangle
+ \frac{2m}{\tau} \Big \langle u(0^-), ({\mathcal{R}}^+_\tau-I_4)\big(v(0^+)-v(0^-)\big)\Big\rangle,$$ we have then $$\label{eq-p3d}
{\mathcal{R}}^+_\tau v'(0^+)-v'(0^-)=\frac{2m}{\tau} ({\mathcal{R}}^+_\tau-I_4)\big(v(0^+)-v(0^-)\big).$$ Therefore, the domain of $T^D_{s,m}$ consists of the functions $v\in H^2\big((-\delta,\delta)\setminus\{0\}\big)$ satisfying the boundary conditions , and .
One then concludes that a negative number $E=-k^2$ is an eigenvalue of $T^D_{s,m}$ iff one can find $a_\pm, b_\pm\in {\mathbb{C}}^4$, not all zero, such that that the function $v$ given by $$v(t)=\begin{cases}
a_+ e^{-kt} + b_+ e^{kt}, & t>0,\\
a_- e^{kt} + b_- e^{-kt}, & t<0,
\end{cases}$$ satisfies the above boundary conditions. From we deduce $$a_\pm = \theta b_\pm, \quad \theta:=-e^{2k\delta}$$ and hence $$v(t)=\begin{cases}
(\theta e^{-kt} + e^{kt})b_+ , & t>0,\\
(\theta e^{kt} + e^{-kt})b_- , & t<0.
\end{cases}$$ It follows then from that $b_+={\mathcal{R}}^+_\tau b_-$ and $$\label{equation_eigenfunction_upper_bound}
v(t)=\begin{cases}
(\theta e^{-kt} + e^{kt}){\mathcal{R}}^+_\tau b_- , & t>0,\\
(\theta e^{kt} + e^{-kt})b_- , & t<0.
\end{cases}$$ Then $$\begin{aligned}
v(0^+)&=(\theta+1){\mathcal{R}}^+_\tau b_-,& v(0^-)&=(\theta+1)b_-,\\
v'(0^+)&=-k(\theta-1){\mathcal{R}}^+_\tau b_-, & v'(0^-)&=k(\theta-1)b_-,\end{aligned}$$ and the substitution into shows that $E=-k^2$ is an eigenvalue iff the equation $$-k(\theta-1)\big(({\mathcal{R}}^+_\tau)^2+I_4\big)b_-=\dfrac{2m}{\tau}(\theta+1)({\mathcal{R}}^+_\tau -I_4)^2 \ b_-$$ admits a solution $b_-\ne 0$. A straightforward calculation shows that $$({\mathcal{R}}_\tau^+)^2 + I_4 = \frac{2 (\tau^2 + 4)}{4 - \tau^2} {\mathcal{R}}_\tau^+ \quad \text{and} \quad
({\mathcal{R}}_\tau^+ - I_4)^2 = \tau \cdot \frac{4 \tau}{4 - \tau^2} {\mathcal{R}}_\tau^+$$ and therefore, one may rewrite the last condition as $$k \dfrac{\theta-1}{\theta+1}\,b_-=-\dfrac{2m}{\tau} \cdot \big(({\mathcal{R}}^+_\tau)^2+I_4\big)^{-1}({\mathcal{R}}^+_\tau-I_4)^2 \ b_- = \mu m b_-$$ with $\mu$ given by . Therefore, a solution $b_-\ne 0$ to the preceding equation exists iff $k$ satisfies $$\label{eq-kk1d}
k\dfrac{\theta-1}{\theta+1}=\mu m,$$ and in that case the first eigenvalue is four times degenerate due to the arbitrary choice of $b_-\in{\mathbb{C}}^4$, and the representation follows from the preceding representation for $v$ in . In order to show the uniqueness of the lowest eigenvalue and to study the behavior with respect to $m$ and $\delta$, let us rewrite in the form $$F(k\delta)=\mu m \delta, \quad F(x)= x \coth x.$$ Since $$F'(x) = \frac{\sinh x \cosh x - x}{\sinh^2 x} > 0, \quad x > 0,$$ one remarks that $F:(0,+\infty)\to(1,+\infty)$ is a diffeomorphism, with $F(0^+)=1$ and $F(+\infty)=+\infty$, which shows that the solution $k$ is unique for $\mu m\delta>1$. Furthermore, for $m\delta\to+\infty$ one has obviously $k\delta\to+\infty$, which implies that $\theta\to -\infty$. The substitution into shows that $k\sim \mu m$, and another use of gives .
Now we are going to use the preceding lemma in order to establish an upper estimate for the eigenvalues defined by the form $q^D_{m,\tau,\delta}$ from Lemma \[lem-qnd\]:
\[lem-upb\] There exists $C>0$ and $m_0>0$ such that for $m>m_0$ and any $j\in{\mathbb{N}}$ there holds $$E_j(q^D_{m,\tau,\delta})\le -\Big( \dfrac{4 \tau}{\tau^2+4}\Big)^2 m^2 + E_j\Big( (1+C\delta){\mathcal{L}}^\tau_0 + K-M^2\Big)
+C\delta+ C m^2e^{-2\mu m\delta}.$$
Recall that ${\mathop{\mathcal{D}}}(\ell^\tau_0)$ is defined in . Define for $v = (v_+, v_-) \in{\mathop{\mathcal{D}}}(\ell^\tau_0)$ $$u_v(s,t)=
c_m\psi_m\big(|t|\big)\begin{cases}
v_+(s), & t>0,\\
v_-(s), & t<0,
\end{cases}$$ with the function $\psi_m$ as in , where the constant $c_m>0$ is independent of $s$ and chosen by $c_m^2\|\psi_m\|^2_{L^2(0,\delta)}=1$. Then $u_v\in {\mathop{\mathcal{D}}}(q^D_{m,\tau,\delta})$ with $\|u_v\|_{L^2(\Sigma\times(-\delta,\delta),{\mathbb{C}}^4)}=\|v\|_{\mathcal{H}}$. Due to the choice of $\psi_m$ and $v$ one has then $$\begin{gathered}
\iint_\Sigma \bigg(\int_{-\delta}^\delta |\partial_t u_v|^2 {\mathrm{d}}t + \frac{2m}{\tau} \big|u_v(\cdot,0^+)-u_v(\cdot,0^-)\big|^2 \bigg) {\mathrm{d}}\Sigma\\
=E_1(T^D_{s,m}) \|u_v\|^2_{L^2(\Sigma\times(-\delta,\delta),{\mathbb{C}}^4)}=E_1(T^D_{s,m})\|v\|^2_{\mathcal{H}}\end{gathered}$$ with the operator $T^D_{s,m}$ from Lemma \[lem1dd\]. One also has $$\begin{gathered}
\iiint_{\Sigma\times (-\delta,\delta)} \Big((1+c\delta)\|\nabla_s u_v\|^2_{T_s\Sigma\otimes{\mathbb{C}}^4}
+ (K-M^2+c\delta)\,|u_v|^2\Big){\mathrm{d}}\Sigma{\mathrm{d}}t\\
\begin{aligned}
&=\int_0^\delta c_m^2 \big|\psi_m(t)\big|^2 {\mathrm{d}}t \cdot
\iint_\Sigma \Big((1+c\delta) \|\nabla_s v_+\|^2_{T_s\Sigma\otimes{\mathbb{C}}^4}
+ (K-M^2+c\delta)\,|v_+|^2\Big){\mathrm{d}}\Sigma\\
&\quad +\int_{-\delta}^0 c_m^2 \big|\psi_m(-t)\big|^2 {\mathrm{d}}t \cdot
\iint_\Sigma \Big((1+c\delta) \|\nabla_s v_-\|^2_{T_s\Sigma\otimes{\mathbb{C}}^4}
+ (K-M^2+c\delta)\,|v_-|^2\Big){\mathrm{d}}\Sigma\\
&=\iint_\Sigma \Big((1+c\delta) \|\nabla_s v_+\|^2_{T_s\Sigma\otimes{\mathbb{C}}^4}
+ (K-M^2+c\delta)\,|v_+|^2\Big){\mathrm{d}}\Sigma\\
&\quad+\iint_\Sigma \Big((1+c\delta) \|\nabla_s v_-\|^2_{T_s\Sigma\otimes{\mathbb{C}}^4}
+ (K-M^2+c\delta)\,|v_-|^2\Big){\mathrm{d}}\Sigma\\
&=(1+c\delta)\ell^\tau_0(v,v)+ \big\langle v,(K-M^2+c\delta) v\big\rangle_{\mathcal{H}},
\end{aligned}\end{gathered}$$ i.e. $$q^D_{m,\tau,\delta}(u_v,u_v)=(1+c\delta)\ell^\tau_0(v,v)+ \big\langle v,(K-M^2+c\delta) v\big\rangle_{\mathcal{H}}+ E_1(T^D_{s,m})\|v\|^2_{\mathcal{H}}.$$ By Lemma \[lem1dd\] one can find $m_0>0$ and $C>0$ independent of $s$ such that $$E_1(T^D_{s,m,\delta})\le -\mu^2m^2+Cm^2 e^{-2\mu m\delta} \text{ for } m>m_0,$$ and then $$\dfrac{q^D_{m,\tau,\delta}(u_v,u_v)}{\|u_v\|_{L^2(\Sigma\times(-\delta,\delta),{\mathbb{C}}^4)}}
\le -\mu^2m^2 +\dfrac{(1+c\delta)\ell^\tau_0(v,v)+ \big\langle v,(K-M^2) v\big\rangle_{\mathcal{H}}}{\|v\|^2_{\mathcal{H}}} + Cm^2 e^{-2\mu m\delta} + c\delta.$$ If $F_j$ is a $j$-dimensional subspace of ${\mathop{\mathcal{D}}}(\ell^\tau_0)$, then ${\mathcal{F}}_j:=\{u_v: v\in F_j\}$ is a $j$-dimensional subspace of ${\mathop{\mathcal{D}}}(q^D_{m,\tau,\delta})$, and by the min-max-principle one has, by estimating all constants by a generic constant $C$, $$\begin{split}
E_j&(q^D_{m,\tau,\delta})\le \min_{{\mathcal{F}}_j}\max_{u\in {\mathcal{F}}_j} \dfrac{q^D_{m,\tau,\delta}(u,u)}{\|u\|_{L^2(\Sigma\times(-\delta,\delta),{\mathbb{C}}^4)}}
\leq\min_{F_j}\max_{v\in F_j} \dfrac{q^D_{m,\tau,\delta}(u_v,u_v)}{\|u_v\|_{L^2(\Sigma\times(-\delta,\delta),{\mathbb{C}}^4)}}\\
&\le -\mu^2m^2 +\min_{F_j}\max_{v\in F_j} \dfrac{(1+C\delta)\ell^\tau_0(v,v)+ \big\langle v,(K-M^2) v\big\rangle_{\mathcal{H}}}{\|v\|^2_{\mathcal{H}}}
+ Cm^2 e^{-2\mu m\delta} + C\delta\\
&=-\mu^2m^2 +E_j\Big((1+C\delta){\mathcal{L}}^\tau_0+K-M^2\Big) + Cm^2 e^{-2\mu m\delta} + C\delta. \qedhere
\end{split}$$
It is sufficient to substitute the estimate of Lemma \[lem-upb\] into the upper bound of Lemma \[lem-qnd\] and to use $$m^2-\mu^2m^2=(1-\mu^2)m^2=\Big(\dfrac{\tau^2-4}{\tau^2+4}\Big)^2 m^2. \qedhere$$
Lower bound {#sec-low1}
-----------
We start with an estimate for another auxiliary one-dimensional operator.
\[lem1dn\] For $m, c>0$ let $h_{m,c}$ be the sesquilinear form in $L^2\big((-\delta,\delta),{\mathbb{C}}^4\big)$ given by $$\begin{aligned}
h_{m,c}(u,u)&=\int_{-\delta}^\delta |u'|^2 {\mathrm{d}}t + \frac{2m}{\tau} \big|u(0^+)-u(0^-)\big|^2 -c \big|u(\delta)\big|^2-c\big|u(-\delta)\big|^2,\\
{\mathop{\mathcal{D}}}(h_{m,c})&=\Big\{ u\in H^1\big((-\delta,\delta)\setminus\{0\},{\mathbb{C}}^4\big): \widetilde{P}^-_\tau u(0^+)+\widetilde{P}^+_\tau u(0^-)=0\Big\},\\
\widetilde{P}^\pm_\tau&:=\dfrac{\tau}{2}\pm \beta,
\end{aligned}$$ and let $H_{m,c}$ be the associated self-adjoint operator in $L^2\big((-\delta,\delta),{\mathbb{C}}^4\big)$. Then for $m\to +\infty$ the first eigenvalue has the multiplicity $4$ and $$\begin{gathered}
E_1(H_{m,c})= -\mu^2m^2 \big(1+{\mathcal{O}}(e^{-2\mu m\delta})\big), \label{eq-e1}\\
E_5(H_{m,c})\ge \dfrac{b^2}{\delta^2} \quad \text{ for some } b>0. \label{eq-e5}
\end{gathered}$$ Furthermore, one can represent, with a suitable smooth function $\psi_{m,c}:(0,\delta)\to{\mathbb{R}}$, $$\begin{gathered}
\label{eq-vv2}
\ker\big( H_{m,c} -E_1(H_{m,c})\big)
=\Big\{u:
u(t)=v_\pm \psi_{m,c}\big(|t|\big) \text{ as } \pm t>0\\ \text{ with }
v_\pm\in{\mathbb{C}}^4 \text{ such that } \widetilde{P}^-_\tau v_+ +\widetilde{P}^+_\tau v_-=0
\Big\}.
\end{gathered}$$
In the proof we rewrite the condition $\widetilde{P}^-_\tau u(0^+)+\widetilde{P}^+_\tau u(0^-)=0$ as $u(0^+)=\widetilde{R}^+_\tau u(0^-)$ with $\widetilde{R}^+_\tau=-(\widetilde{P}^-_\tau)^{-1}\widetilde{P}^+_\tau$.
Let us give first a more precise description of $H_{m,c}$. It is standard to see that ${\mathop{\mathcal{D}}}(H_{m,c})\subset H^2\big((-\delta,\delta)\setminus\{0\},{\mathbb{C}}^4\big)$ and that $H_{m,c}$ acts as $H_{m,c} u=-u''$. Therefore, it is sufficient to specify the boundary conditions at $0$ and $\pm\delta$. Let $v\in {\mathop{\mathcal{D}}}(H_{m,c})$, then $v$ belongs to ${\mathop{\mathcal{D}}}(h_{m,c})$, i.e. $$\label{eq-p1}
v(0^+)=\widetilde{R}^+_\tau v(0^-),$$ and $h_{m,c}(u,v)=\langle u,H_{m,c} v\rangle_{L^2}$ for all $u\in {\mathop{\mathcal{D}}}(h_{m,c})$. Using integration by parts on $(-\delta,0)$ and $(0,\delta)$ we conclude that $$\begin{aligned}
h_{m,c}(u,v)&=\int_{-\delta}^\delta \langle u, -v''\rangle{\mathrm{d}}t +s_{m, c}(u,v),\\
s_{m,c}(u,v)&=-\big\langle u(-\delta), v'(-\delta)\big\rangle+ \big\langle u(0^-),v'(0^-)\big\rangle
-\big\langle u(0^+),v'(0^+)\big\rangle\\
&\qquad+\big\langle u(\delta),v'(\delta)\big\rangle
+ \frac{2m}{\tau} \big \langle u(0^+)-u(0^-), v(0^+)-v(0^-)\big\rangle\\
&\qquad-c \big\langle u(\delta),v(\delta)\big\rangle-c \big\langle u(-\delta),v(-\delta)\big\rangle.\end{aligned}$$ Therefore, it is sufficient to check for which $v$ one has $s_{m,c}(u,v)=0$ for all $u\in {\mathop{\mathcal{D}}}(h_{m,c})$. Testing on $u$ localized near $\pm\delta$ one concludes that $v$ must satisfy $$\label{eq-p2}
v'(\pm\delta)=\pm c v(\pm\delta).$$ Now assume that $u$ vanishes at $\pm \delta$, then using one rewrites $$\begin{aligned}
s_{m,c}(u,v)&
=\big\langle u(0^-),v'(0^-)\big\rangle -\big\langle u(0^+),v'(0^+)\big\rangle \\
&\qquad + \frac{2m}{\tau} \big \langle u(0^+)-u(0^-), v(0^+)-v(0^-)\big\rangle\\
&=\big\langle u(0^-),v'(0^-)-\widetilde{R}^+_\tau v'(0^+)\big\rangle
+ \frac{2m}{\tau} \Big \langle u(0^-), (\widetilde{R}^+_\tau-I_4)\big(v(0^+)-v(0^-)\big)\Big\rangle\end{aligned}$$ implying $$\label{eq-p3}
\widetilde{R}^+_\tau v'(0^+)-v'(0^-)=\frac{2m}{\tau} (\widetilde{R}^+_\tau-I_4)\big(v(0^+)-v(0^-)\big).$$ By summarizing the above, the domain of $H_{m,c}$ consists of the functions $v\in H^2\big((-\delta,\delta)\setminus\{0\},{\mathbb{C}}^4\big)$ satisfying the boundary conditions , and .
One then concludes that a negative number $E=-k^2$ with $k>0$ is an eigenvalue of $H_{m,c}$ iff one can find $a_\pm, b_\pm\in {\mathbb{C}}^4$, not all zero, such that the associated eigenfunction $$v(t)=\begin{cases}
a_+ e^{-kt} + b_+ e^{kt}, & t>0,\\
a_- e^{kt} + b_- e^{-kt}, & t<0,
\end{cases}$$ satisfies the above boundary conditions. From we deduce $$a_\pm = \theta b_\pm, \quad \theta:=\dfrac{k-c}{k+c}e^{2k\delta},
\quad \text{i.e.} \quad
v(t)=\begin{cases}
(\theta e^{-kt} + e^{kt})b_+ , & t>0,\\
(\theta e^{kt} + e^{-kt})b_- , & t<0.
\end{cases}$$ It follows then from that $b^+=\widetilde{R}^+_\tau b_-$ and $$v(t)=\begin{cases}
(\theta e^{-kt} + e^{kt})\widetilde{R}^+_\tau b_- , & t>0,\\
(\theta e^{kt} + e^{-kt})b_- , & t<0.
\end{cases}$$ Then $$\begin{aligned}
v(0^+)&=(\theta+1)\widetilde{R}^+_\tau b_-,& v(0^-)&=(\theta+1)b_-,\\
v'(0^+)&=-k(\theta-1)\widetilde{R}^+_\tau b_-, & v'(0^-)&=k(\theta-1)b_-,\end{aligned}$$ and the substitution into shows that $E=-k^2$ is an eigenvalue iff the equation $$-k(\theta-1)\big((\widetilde{R}^+_\tau)^2+I_4\big)b_-=\dfrac{2m}{\tau}(\theta+1)(\widetilde{R}^+_\tau-I_4)^2 \ b_-$$ admits a solution $b_-\ne 0$. One may rewrite the last condition as $$k \dfrac{\theta-1}{\theta+1}\,b_-=-\dfrac{2m}{\tau} \big((\widetilde{R}^+_\tau)^2+I_4\big)^{-1}\cdot (\widetilde{R}^+_\tau-I_4)^2 b_-,$$ and using the equality $\beta^2=I_4$ we compute $$\label{eq-err1}
\big((\widetilde{R}^+_\tau)^2+I_4\big)^{-1}\cdot (\widetilde{R}^+_\tau-I_4)^2 =\dfrac{2\tau^2}{4+\tau^2} I_4.$$ Therefore, a solution $b_-\ne 0$ to the above equation exists iff $k$ satisfies $$\label{eq-kk1}
k\dfrac{\theta-1}{\theta+1}=m \mu$$ with $\mu$ given by , and in that case the first eigenvalue is four times degenerate due to the arbitrary choice of $b_-$, and the representation follows from the preceding constructions of the function $v$. In order to show the uniqueness of $k$ and to study its behavior with respect to $m$, let us rewrite as $$F_{c\delta}(k\delta)=\mu m \delta, \quad F_\varepsilon(x):= x \dfrac{x \tanh x - \varepsilon}{x-\varepsilon \tanh x}.$$ One remarks that for $\varepsilon\in(0,1)$ the function $F_\varepsilon:(0,+\infty)\to {\mathbb{R}}$ is well-defined and $$F'_\varepsilon (x)=x\dfrac{\varepsilon(1-\tanh^2 x) + \dfrac{x^2-\varepsilon^2}{\cosh^2 x}}{(x-\varepsilon \tanh x)^2}
+ \dfrac{1}{x} F_\varepsilon(x),$$ and $F'_\varepsilon(x)>0$ provided $x>\varepsilon$ and $F_\varepsilon(x)>0$. Furthermore, $F_\varepsilon(x)>0$ if and only if $x\tanh x>\varepsilon$, therefore, $F^{-1}_\varepsilon\big((0,+\infty)\big)$ is a subinterval of $(\varepsilon,+\infty)$. It follows that $F_\varepsilon:F^{-1}_\varepsilon\big((0,\infty)\big)\to (0,\infty)$ is a diffeomorphism, and there exists a unique solution $k$ provided $c\delta<1$, which is satisfied for large $m$ due to , as $\mu m \delta > 0$. On the other hand, $F_\varepsilon(x)$ is decreasing in $\varepsilon$ due to $$\dfrac{\partial F_\varepsilon(x)}{\partial\varepsilon}=-x^2\,\dfrac{1-\tanh^2 x}{(x-\varepsilon \tanh x)^2}<0,$$ which implies $k\delta\ge k_0\delta$ with $k_0>0$ being the solution to $F_0(k_0\delta)=\mu m\delta$. As $F_0(x)=x\tanh x$, one easily checks that $k_0\delta\to +\infty$ for $m\delta\to +\infty$, and then $k\delta\to+\infty$ and $\theta\to +\infty$. Therefore, $k\sim m \mu$ for large $m$ due to , and another iteration of gives .
In order to estimate the next eigenvalue of $H_{m,c}$ we proceed first in the same way and show that $E=k^2$ with $k>0$ is an eigenvalue iff $$\label{eq-gkd}
G_{c\delta}(k\delta)=\mu m\delta, \quad G_\varepsilon(x):=F_\varepsilon(ix)=-x\dfrac{x\tan x+\varepsilon}{x-\varepsilon\tan x}.$$ Using the convexity of $x\mapsto \tan x$ one sees that $0<\tan x< \frac{4}{\pi} x$ for $x\in\big(0,\frac{\pi}{4}\big)$, hence, $G_\varepsilon(x)<0$ for all $x\in\big(0,\frac{\pi}{4}\big)$ and $\varepsilon\in\big(0,\frac{4}{\pi}\big)$. As $\mu m\delta>0$, it follows that admits no solution $k$ with $k\delta\in \big(0,\frac{\pi}{4}\big)$ as $m$ is large, in other words, the operator $H_{m,c}$ has no eigenvalues in $\big(0, \frac{\pi^2}{16 \delta^2}\big)$ for $m\to +\infty$. In order to complete the proof of it remains to check that $0$ is not an eigenvalue of $H_{m,c}$ for $m\to +\infty$. If $0$ were an eigenvalue, then there would exist $a_\pm,b_\pm\in {\mathbb{C}}^4$, not all zero, for which the function $$v(t)=\begin{cases}
a_+ + b_+t, & t>0,\\
a_- - b_-t, & t<0,
\end{cases}$$ would satisfy the boundary conditions , , . The condition gives $$a_+=\widetilde{R}^+_\tau a_- \quad \text{and} \quad v(0^+)=\widetilde{R}^+_\tau a_-, \quad v(0^-)=a_-,$$ and implies $$b_\pm = \dfrac{c}{1-c\delta} a_\pm,$$ hence we deduce $$v(t)=\dfrac{1}{1-c\delta}\begin{cases}
(1-c\delta+ct)\widetilde{R}^+_\tau a_-, & t>0,\\
(1-c\delta-ct)a_-, & t<0.
\end{cases}$$ This yields $$v'(0^+)=\dfrac{c}{1-c\delta}\widetilde{R}^+_\tau a_- \quad \text{and} \quad v'(0^-)=-\dfrac{c}{1-c\delta}a_-$$ and the substitution into together with the identity imply $$\dfrac{c}{1-c\delta}\big((\widetilde{R}^+_\tau)^2+1\big)a_-=\dfrac{2m}{\tau}(\widetilde{R}^+_\tau-1)^2a_-,
\text{ i.e. } \Big(\dfrac{c}{1-c\delta}+\dfrac{4m|\tau|}{\tau^2+4}\Big)a_-=0.$$ As the number in the parentheses is non-zero for large $m$, the only solution is the trivial one $a_-=0$, which then implies that $0$ is not an eigenvalue of $H_{m,c}$.
For what follows we need a special representation for the matrices ${\mathcal{B}}$ from :
\[lem-btheta\] For each $s\in\Sigma$ there holds ${\mathcal{B}}(s)=\Theta_0(s) \beta \Theta_0(s)^*$ with the unitary matrices $\Theta_0(s) \in {\mathbb{C}}^{4 \times 4}$ given by $$\label{eq-theta0}
\Theta_0(s)=\dfrac{1}{\sqrt{2}}\Big( I_4+{\mathrm{i}}\alpha\cdot\nu(s)\Big).$$
Using $(\alpha\cdot \nu)^2=I_4$ one easily checks that $\Theta_0(s)^*=\frac{1}{\sqrt{2}}\big( I_4-{\mathrm{i}}\alpha\cdot\nu(s)\big)$ and that $\Theta^*_0(s) \Theta_0(s)=I_4$, i.e. that $\Theta_0$ is unitary. Moreover, using the commutation relations we have $$\begin{gathered}
\Theta_0 \beta \Theta^*_0=\dfrac{1}{2}( 1+{\mathrm{i}}\alpha\cdot\nu)\beta ( 1-{\mathrm{i}}\alpha\cdot\nu)=\dfrac{1}{2}( 1+{\mathrm{i}}\alpha\cdot\nu)( 1+{\mathrm{i}}\alpha\cdot\nu) \beta\\
=\dfrac{1}{2}\big( I_4+ 2{\mathrm{i}}\alpha\cdot \nu - (\alpha\cdot \nu)^2\big)\beta ={\mathrm{i}}(\alpha\cdot \nu) \beta=-{\mathrm{i}}\beta \alpha\cdot \nu={\mathcal{B}}. \qedhere\end{gathered}$$
An explicit computation with the help of Lemma \[lem-btheta\] gives then the following result.
\[lem1dn1\] For $s\in\Sigma$, $m>0$, and $c>0$ consider the following sesquilinear form $t^N_{s,m,c}$ in $L^2\big((-\delta,\delta),{\mathbb{C}}^4\big)$: $$\begin{aligned}
t^N_{s,m,c}(u,u)&=\int_{-\delta}^\delta |u'|^2 {\mathrm{d}}t + \frac{2m}{\tau} \big|u(0^+)-u(0^-)\big|^2 -c \big|u(\delta)\big|^2-c\big|u(-\delta)\big|^2,\\
{\mathop{\mathcal{D}}}(t^N_{s,m,c})&=\Big\{ u\in H^1\big((-\delta,0)\cup(0,\delta),{\mathbb{C}}^4\big): {\mathcal{P}}^-_\tau(s) u(0^+)+{\mathcal{P}}^+_\tau(s) u(0^-)=0\Big\},
\end{aligned}$$ then the associated self-adjoint operator $T^N_{s,m,c}$ in $L^2\big((-\delta,\delta),{\mathbb{C}}^4\big)$ is unitarily equivalent to the operator $H_{m,c}$ from Lemma \[lem1dn\], $$T^N_{s,m,c}=\Theta(s) H_{m,c} \Theta(s)^*,$$ where $\Theta(s)$ is the unitary map in $L^2\big((-\delta,\delta),{\mathbb{C}}^4\big)$ defined by $\big(\Theta(s) u\big) (t):=\Theta_0(s) u(t)$ with $\Theta_0$ given by , and $s\mapsto \Theta(s)$ is a $C^2$ map in the operator norm topology. Furthermore, one can represent, with a suitable smooth function $\psi_{m,c}:(0,\delta)\to{\mathbb{R}}$ independent of $s$, $$\begin{gathered}
\label{eq-vv3}
\ker\big( T^N_{s,m,c} -E_1(T^N_{s,m,c})\big)
=\Big\{v:
v(t)=v_\pm \psi_{m,c}(|t|) \text{ as } \pm t>0\\ \text{ with }
v_\pm\in{\mathbb{C}}^4 \text{ such that } {\mathcal{P}}^-_\tau(s) v_+ +{\mathcal{P}}^+_\tau(s) v_-=0
\Big\}.
\end{gathered}$$
Now let us recall some standard constructions, for which it is useful to use the identification $$L^2\big(\Sigma\times(-\delta,\delta),{\mathbb{C}}^4)\simeq L^2(\Sigma, {\mathcal{G}}), \quad
{\mathcal{G}}:=L^2\big((-\delta,\delta),{\mathbb{C}}^4).$$ Recall that for any Banach space $B$ the gradient $\nabla_s:C^1(\Sigma, B)\to C^0(T\Sigma,B)$ acts in local coordinates of $\Sigma$ as $$(\nabla_s f)_j=\sum_{k} g^{jk} \partial_k f.$$ In particular, for the $C^2$ maps $\Theta:\Sigma\to {\mathbf{B}}({\mathcal{G}})$ and $\Theta^*:\Sigma\to {\mathbf{B}}({\mathcal{G}})$ from Lemma \[lem1dn1\] one can find a constant $C>0$ such that for every $u\in C^0(\Sigma,{\mathcal{G}})$ at every point $s\in\Sigma$ there holds $$\label{eq-noth}
\big\|(\nabla_s \Theta) u\big\|_{T_s\Sigma\otimes{\mathcal{G}}}\le C \|u\|_{{\mathcal{G}}}, \quad
\big\|(\nabla_s \Theta^*) u\big\|_{T_s\Sigma\otimes{\mathcal{G}}}\le C \|u\|_{{\mathcal{G}}},$$ and $C$ is independent of $m$ and $\delta$. Furthermore, let $\pi(s)$ be the orthogonal projector in $L^2\big((-\delta,\delta),{\mathbb{C}}^4\big)$ on the subspace $\ker\big(T^N_{s,m,c}-E_1(T^N_{s,m,c})\big)$. Denote by $\Pi$ the orthogonal projector in $L^2\big(\Sigma\times(-\delta,\delta),{\mathbb{C}}^4)$ given by $$(\Pi u)(s,t)=\pi(s) u(s,\cdot) (t)$$ and set $\Pi^\perp:=1-\Pi$. Due to the fibered structure, both $\Pi$ and $\Pi^\perp$ also define in the canonical way bounded operators in $L^2(T\Sigma)\otimes L^2\big((-\delta,\delta),{\mathbb{C}}^4)$, to be denoted by the same symbols.
\[lem29\] The map $[\nabla_s,\Pi] u:=\nabla_s (\Pi u) -\Pi(\nabla_s u)$ defined for $u\in C^1(\Sigma) \otimes L^2\big((-\delta,\delta),{\mathbb{C}}^4)$ extends by density to a bounded operator $$[\nabla_s,\Pi]: L^2(\Sigma) \otimes L^2\big((-\delta,\delta),{\mathbb{C}}^4)\to L^2(T\Sigma) \otimes L^2\big((-\delta,\delta),{\mathbb{C}}^4),$$ whose norm remains uniformly bounded for $\delta\to 0^+$ and $m\delta\to+\infty$. Moreover we have $[\nabla_s,\Pi]\Big(H^1(\Sigma) \otimes L^2\big((-\delta,\delta)\Big) \subset H^1(T\Sigma) \otimes L^2\big((-\delta,\delta),{\mathbb{C}}^4)$. The same conclusion holds for $[\nabla_s,\Pi^\perp]\equiv -[\nabla_s,\Pi]$.
By Lemma \[lem-btheta\] one can represent $\pi(s)=\Theta(s)\pi_0 \Theta(s)^*$, where $\pi_0$ is the orthogonal projector in $L^2\big((-\delta,\delta),{\mathbb{C}}^4)$ on $\ker\big(H_{m,c}- E_1(H_{m,c})\big)$ with the operator $H_{m,c}$ from Lemma \[lem1dn\]. As $\pi_0$ does not depend on $s$, a direct computation in the local coordinates shows that at each point $s\in\Sigma$ one has $$\label{eqn:expcommut}
[\nabla_s,\Pi] u= (\nabla_s \Theta) \pi_0 \Theta^*u + \Theta \pi_0 (\nabla_s \Theta^*)u.$$ Using we estimate $$\begin{aligned}
\big\| (\nabla_s \Theta) \pi_0 \Theta^*u\big\|_{T_s \Sigma \otimes{\mathcal{G}}}&\le
C \|\pi_0 \Theta^*u\|_{{\mathcal{G}}} \le C\|\pi_0\|_{{\mathbf{B}}(G)}\| \Theta^*\|_{{\mathbf{B}}({\mathcal{G}})} \|u\|_{{\mathcal{G}}}\le C \|u\|_{{\mathcal{G}}},\\
\big\| \Theta \pi_0 (\nabla_s \Theta^*)u\big\|_{T_s \Sigma \otimes{\mathcal{G}}}&
\le
\big\| \Theta\|_{{\mathbf{B}}(T_s \Sigma \otimes{\mathcal{G}})}
\big\|\pi_0\|_{{\mathbf{B}}(T_s \Sigma \otimes{\mathcal{G}})} \|(\nabla_s \Theta^*) u\|_{T_s \Sigma \otimes{\mathcal{G}}} \le C \|u\|_{{\mathcal{G}}},\end{aligned}$$ then $$\begin{gathered}
\big\|[\nabla_s,\Pi] u\big\|^2_{L^2(T\Sigma)\otimes{\mathcal{G}}}
=\iint_\Sigma \big\| (\nabla_s \Theta) \pi_0 \Theta^*u + \Theta \pi_0 (\nabla_s \Theta^*) u\big\|^2_{T_s \Sigma \otimes{\mathcal{G}}}{\mathrm{d}}\Sigma(s)\\
\le
2\iint_\Sigma \big\| (\nabla_s \Theta) \pi_0 \Theta^*u\big\|_{T_s \Sigma \otimes{\mathcal{G}}}^2{\mathrm{d}}\Sigma(s)
+
2\iint_\Sigma \big\| \Theta \pi_0 (\nabla_s \Theta^*)u\big\|_{T_s \Sigma \otimes{\mathcal{G}}}^2{\mathrm{d}}\Sigma(s)\\
\le 4C^2 \iint_\Sigma \| u\|^2_{\mathcal{G}}{\mathrm{d}}\Sigma(s)=4C^2\|u\|^2_{L^2(\Sigma)\otimes {\mathcal{G}}}.\end{gathered}$$ As the constant $C$ is independent of $m$ and $\delta$, the continuity result follows. To prove the mapping properties of $[\nabla_s,\Pi]$ between the Sobolev spaces of order $1$, it is enough to remark that is differentiable with respect to $s$ because $\Theta$ is $C^2$.
\[lem-lowb\] Let the form $q^N_{m,\tau,\delta}$ be as in Lemma \[lem-qnd\] and let $\mu$ be given by . Then there are constants $b>0$ and $m_0>0$ such that for all $m>m_0$ and $j\in\big\{1,\dots,{\mathcal{N}}(q^N_{m,\tau,\delta},0)\big\}$ it holds $$E_j(q^N_{m,\tau,\delta})\ge -\mu^2m^2+E_j\big((1-b\delta){\mathcal{L}}^\tau_0 +K-M^2\big)- b m^2e^{-2\mu m\delta}-b\delta.$$
Let $c>0$ be as in the expression for $q^N_{m,\tau,\delta}$. Then by Lemmas \[lem1dn\] and \[lem1dn1\] one may estimate, with some $b_0, b_1, m_0>0$ independent of $s$, $$\label{eq-lambdas}
E_1(T^N_{s,m,c})\ge -\mu^2m^2 - b_0 m^2e^{-2\mu m\delta}, \quad
E_5(T^N_{s,m,c})\ge \dfrac{b_1^2}{\delta^2}
\quad \text{ for } m>m_0.$$
Let $u \in {\mathop{\mathcal{D}}}(q^N_{m,\tau,\delta})$ be fixed. Due to the definition of $\Pi$ and with the help of the min-max principle one obtains $$\begin{gathered}
\iint_\Sigma \bigg(\int_{-\delta}^\delta |\partial_t u|^2 {\mathrm{d}}t + \frac{2m}{\tau} \big|u(\cdot,0^+)-u(\cdot,0^-)\big|^2 -c \big|u(\cdot,\delta)\big|^2-c\big|u(\cdot,-\delta)\big|^2\bigg) {\mathrm{d}}\Sigma\\
\ge E_1(T^N_{s,m,c}) \|\Pi u\|^2_{L^2(\Sigma\times(-\delta,\delta),{\mathbb{C}}^4)}
+ E_5(T^N_{s,m,c}) \|\Pi^\perp u\|^2_{L^2(\Sigma\times(-\delta,\delta),{\mathbb{C}}^4)},\end{gathered}$$ and using the pointwise orthogonality $\big\langle \Pi u(s,\cdot), \Pi^\perp u(s,\cdot)\big\rangle_{L^2((-\delta,\delta),{\mathbb{C}}^4)}=0$, $s\in \Sigma$, one gets $$\begin{aligned}
\iint_{\Sigma\times(-\delta,\delta)} (K-M^2-c\delta)\, |u|^2{\mathrm{d}}\Sigma{\mathrm{d}}t
&= \iint_{\Sigma\times(-\delta,\delta)} (K-M^2-c\delta)\, |\Pi u|^2{\mathrm{d}}\Sigma{\mathrm{d}}t\\
&\quad +\iint_{\Sigma\times(-\delta,\delta)} (K-M^2-c\delta)\, |\Pi^\perp u|^2{\mathrm{d}}\Sigma{\mathrm{d}}t,\end{aligned}$$ implying $$\begin{gathered}
\label{eq231}
q^N_{m,\tau,\delta}(u,u)
\ge (1-c\delta) \iint_{\Sigma\times(-\delta,\delta)} \|\nabla_s u\|^2_{T_s\Sigma \otimes {\mathbb{C}}^4}{\mathrm{d}}\Sigma{\mathrm{d}}t\\
+ \big\langle \Pi u, (K-M^2) \Pi u\big\rangle_{L^2(\Sigma\times(-\delta,\delta),{\mathbb{C}}^4)} +(-\mu^2m^2 - b_0 m^2e^{-2\mu m\delta}-c\delta) \|\Pi u\|^2_{L^2(\Sigma\times(-\delta,\delta),{\mathbb{C}}^4)}\\
+ \big\langle \Pi^\perp u, (K-M^2) \Pi^\perp u\big\rangle_{L^2(\Sigma\times(-\delta,\delta),{\mathbb{C}}^4)} + \Big(\dfrac{b_1^2}{\delta^2}-c\delta\Big) \|\Pi^\perp u\|^2_{L^2(\Sigma\times(-\delta,\delta),{\mathbb{C}}^4)}.\end{gathered}$$ Now we would like to separate the terms with $\Pi u$ and $\Pi^\perp u$ in the first term on the right-hand side. One has, with the norms and scalar products taken in $L^2\Big(T\Sigma, L^2\big((-\delta,\delta),{\mathbb{C}}^4\big)\Big)$, $$\label{eq-nu1}
\| \nabla_s u\|^2=\|\nabla_s(\Pi u)\|^2+\|\nabla_s(\Pi^\perp u)\|^2+
2\Re \langle \nabla_s (\Pi u), \nabla_s(\Pi^\perp u)\rangle,$$ and $$\begin{aligned}
\Big\langle \nabla_s (\Pi u), \nabla_s(\Pi^\perp u)\Big\rangle &= \Big\langle \nabla_s \Pi \Pi u, \nabla_s \Pi^\perp \Pi^\perp u \Big\rangle\\
&=\Big\langle \big([\nabla_s,\Pi] + \Pi\nabla_s\big) \Pi u, \big([\nabla_s,\Pi^\perp] + \Pi^\perp\nabla_s\big) \Pi^\perp u\Big\rangle\\
&=\Big\langle [\nabla_s,\Pi] \Pi u, [\nabla_s,\Pi^\perp] \Pi^\perp u\Big\rangle
+\Big\langle \Pi\nabla_s \Pi u, [\nabla_s,\Pi^\perp] \Pi^\perp u\Big\rangle\\
&\quad +\Big\langle [\nabla_s,\Pi] \Pi u, \Pi^\perp\nabla_s\Pi^\perp u\Big\rangle
+\Big\langle \Pi\nabla_s \Pi u, \Pi^\perp\nabla_s \Pi^\perp u\Big\rangle\\
&=:J_1+J_2+J_3+J_4.\end{aligned}$$ Due to the definition of $\Pi$ and $\Pi^\perp$ one has $J_4=0$. By Lemma \[lem29\] we estimate, with some $c_0,c_1>0$ independent of $m$ and $\delta$: $$\begin{aligned}
|J_1|&\le c_0 \|\Pi u\|\cdot \|\Pi^\perp u\|\le c_0 \delta \|\Pi u\|^2+ \dfrac{c_0}{\delta}\|\Pi^\perp u\|^2,\\
|J_2|&\le c_1\|\nabla_s \Pi u\|\cdot\|\Pi^\perp u\|
\le c_1\delta \|\nabla_s \Pi u\|^2+ \dfrac{c_1}{\delta}\|\Pi^\perp u\|^2.\end{aligned}$$ Finally, using the self-adjointness of $\Pi^\perp$ and that by Lemma \[lem29\] we have $\Pi^\perp [\nabla_s,\Pi] \Pi u \in H^1\Big(T\Sigma, L^2\big((-\delta,\delta),{\mathbb{C}}^4\big)\Big)$, we can perform an integration by parts to obtain $$\begin{aligned}
J_3&=\Big\langle \Pi^\perp [\nabla_s,\Pi] \Pi u, \nabla_s\Pi^\perp u\Big\rangle\\
&=\iint_\Sigma \big\langle \Pi^\perp [\nabla_s,\Pi] \Pi u, \nabla_s \Pi^\perp u\big\rangle_{T_s\Sigma \otimes L^2((-\delta,\delta),{\mathbb{C}}^4)} {\mathrm{d}}\Sigma(s)\\
&=-\iint_\Sigma \big\langle \operatorname{div}_s \big(\Pi^\perp [\nabla_s,\Pi] \Pi u\big), \Pi^\perp u\big\rangle_{L^2((-\delta,\delta),{\mathbb{C}}^4)} {\mathrm{d}}\Sigma(s)\\
&=-\Big\langle \operatorname{div}_s \big(\Pi^\perp [\nabla_s,\Pi] \Pi u\big), \Pi^\perp u\Big\rangle,\end{aligned}$$ which yields $$|J_3| \le \Big\|\operatorname{div}_s \big(\Pi^\perp [\nabla_s,\Pi] \Pi u\big)\Big\|_{L^2(\Sigma\times(-\delta,\delta),{\mathbb{C}}^4)}
\cdot\big\|\Pi^\perp u\big\|_{L^2(\Sigma\times(-\delta,\delta),{\mathbb{C}}^4)}.$$ Recall that in the local coordinates on $\Sigma$ for a vector field $A=(A_j)$ one has $$\operatorname{div}_s A=\sum_{j} \big( \partial_j A_j +\sum_k \Gamma^j_{kj} A_k\big),$$ with $\Gamma^j_{kj}$ being the Cristoffel symbols depending on the choice of coordinates only. In our case the $j$-th component of the vector $\Pi^\perp [\nabla_s,\Pi] \Pi u$ can be computed using and is $$\big(\Pi^\perp [\nabla_s,\Pi] \Pi u\big)_j=\Theta \pi_0^\perp \Theta^* \sum_{k} g^{jk} \big(\partial_k\Theta \cdot \pi_0 \Theta^*+
\Theta \pi_0 \partial_k\Theta^*\big) (\Pi u).$$ Furthermore, the projector $\pi_0$ does not depend on $s$ while $\Theta$ is $C^2$ in $s$ (see Lemma \[lem-btheta\]) and does not depend on $m$ and $\delta$. Therefore, with suitable $c_2>0$ one may estimate $$\begin{gathered}
\big\|\operatorname{div}_s \big(\Pi^\perp [\nabla_s,\Pi] \Pi u\big)\big\|_{L^2(\Sigma\times(-\delta,\delta),{\mathbb{C}}^4)}\\
\le c_2 \Big(\|\Pi u\|_{L^2(\Sigma\times(-\delta,\delta),{\mathbb{C}}^4)} +\|\nabla_s(\Pi u)\|_{L^2(T\Sigma, L^2((-\delta,\delta),{\mathbb{C}}^4))}\Big),\end{gathered}$$ which gives $$|J_3|\le 2 c_2 \delta \Big(\|\Pi u\|^2_{L^2(\Sigma\times(-\delta,\delta),{\mathbb{C}}^4)} +\|\nabla_s(\Pi u)\|^2_{L^2(T\Sigma, L^2((-\delta,\delta),{\mathbb{C}}^4))}\Big)+ \dfrac{c_2}{\delta}\big\|\Pi^\perp u\big\|^2_{L^2(\Sigma\times(-\delta,\delta),{\mathbb{C}}^4)}.$$ Therefore, from we obtain, with a suitable $c_3>0$, $$\begin{aligned}
\|\nabla_s u\|_{L^2(T\Sigma, L^2((-\delta,\delta),{\mathbb{C}}^4))}^2 &\ge \|\nabla_s (\Pi u)\|^2_{L^2(T\Sigma, L^2((-\delta,\delta),{\mathbb{C}}^4))}\\
& \quad-2 \Big|\big\langle \nabla_s (\Pi u),\nabla_s(\Pi^\perp u)\big\rangle_{L^2(T\Sigma, L^2((-\delta,\delta),{\mathbb{C}}^4))}\Big|\\
&\ge (1-c_3 \delta)\|\nabla_s (\Pi u)\|^2_{L^2(T\Sigma, L^2((-\delta,\delta),{\mathbb{C}}^4))}\\
&\quad -c_3 \delta\|\Pi u\|^2_{L^2(\Sigma, L^2((-\delta,\delta),{\mathbb{C}}^4))}
-\dfrac{c_3}{\delta}\|\Pi^\perp u\|^2_{L^2(\Sigma, L^2((-\delta,\delta),{\mathbb{C}}^4))}.\end{aligned}$$ Let us substitute all the estimates obtained into . One remarks that all terms $\Pi^\perp u$ can be minorated by $$\Big(\dfrac{b_1^2}{\delta^2}-c\delta-\dfrac{(1-c\delta)c_3}{\delta}\Big)
\big\| \Pi^\perp u\big\|^2_{L^2(\Sigma\times(-\delta,\delta),{\mathbb{C}}^4)}
+\big\langle \Pi^\perp u, (K-M^2) \Pi^\perp u\big\rangle_{L^2(\Sigma\times(-\delta,\delta),{\mathbb{C}}^4)}.$$ Therefore, one can increase the value of $m_0$ such that for $m>m_0$ the term becomes non-negative (as $\delta$ becomes small). Therefore, for large $m>m_0$ we may simply estimate $$\label{eq-u0}
q^N_{m,\tau,\delta}(u,u)\ge q_0(\Pi u,\Pi u),$$ where $q_0$ is the sesquilinear form in the Hilbert space ${\mathop{\mathrm{ran}}}\Pi$ defined on $\Pi\big({\mathop{\mathcal{D}}}(q^N_{m,c})\big)$ by $$\begin{gathered}
q_0(u,u)=
(1-b\delta) \iint_{\Sigma\times(-\delta,\delta)} \|\nabla_s u\|^2_{T_s\Sigma \otimes {\mathbb{C}}^4}{\mathrm{d}}\Sigma{\mathrm{d}}t\\
+ \big\langle u, (K-M^2) u\big\rangle_{L^2(\Sigma\times(-\delta,\delta),{\mathbb{C}}^4)}-(\mu^2m^2 + b m^2e^{-2\mu m\delta}+b\delta) \|u\|^2_{L^2(\Sigma\times(-\delta,\delta),{\mathbb{C}}^4)}\end{gathered}$$ and $b>0$ is a suitable constant.
Now define a sesquilinear form $q$ on ${\mathop{\mathrm{ran}}}(\Pi) \times {\mathop{\mathrm{ran}}}(\Pi^\perp)$ by $q\big( (u,u^\perp),(u,u^\perp)\big)=q_0(u,u)$. Then the inequality takes the form $q_{m, \tau, \delta}^N(u,u)\ge q_0(\Pi u,\Pi u) = q(Uu,Uu)$, where $U u=(\Pi u,\Pi^\perp u)$. As $U$ is unitary, one has by the min-max principle $E_j(q^N_{m,\tau,c})\ge E_j(q)$ for all $j$. On the other hand, due to the representation $q=q_0\oplus 0$ we have $E_j(q)=E_j(q_0)$ for all $j\in{\mathbb{N}}$ with $E_j(q_0)<0$. Therefore, $E_j(q^N_{m,\tau,\delta})\ge E_j(q_0)$ for all $j\in \big\{1,\dots, {\mathcal{N}}(q_0,0)\big\}$. But again due to the form inequality one has ${\mathcal{N}}(q^N_{m,\tau,\delta},0)\le {\mathcal{N}}(q_0,0)$, therefore, $$E_j(q^N_{m,\tau,\delta})\ge E_j(q_0) \text{ for all } j\in\big\{1,\dots,{\mathcal{N}}(q^N_{m,\tau,\delta},0)\big\}.$$
It remains to find a suitable expression for $E_j(q_0)$. Let ${\mathcal{H}}$ be defined by . Using the representation and choosing a constant $c_m>0$ such that $c_m^2\|\psi_{m,c}\|^2_{L^2(0,\delta)}=1$ one concludes that the map $$V: {\mathcal{H}}\to {\mathop{\mathrm{ran}}}(\Pi), \quad (V v)(s,t)=c_m v_\pm(s) \psi\big(|t|\big) \text{ for } \pm t>0,$$ is unitary, and with the form $\ell^\tau_0$ from we have $$q_0(Vv,Vv)=(1-b\delta) \ell^\tau_0(v,v) + \big\langle v, (K-M^2) v\big\rangle_{{\mathcal{H}}} +(-\mu^2m^2 - b m^2e^{-2\mu m\delta}-b\delta) \|v\|^2_{\mathcal{H}},$$ which shows $$E_j(q_0)=-\mu^2m^2+E_j\big((1-b\delta){\mathcal{L}}^\tau_0 +K-M^2\big)- b m^2e^{-2\mu m\delta}-b\delta$$ for all $j\in{\mathbb{N}}$ and concludes the proof of this lemma.
It is sufficient to use the estimate of Lemma \[lem-lowb\] in the left-hand inequality of Lemma \[lem-qnd\].
Acknowledgments {#acknowledgments .unnumbered}
===============
Markus Holzmann was supported by the Austrian Agency for International Cooperation in Education and Research (OeAD). Thomas Ourmières-Bonafos was supported by a public grant as part of the “Investissement d’avenir” project, reference ANR-11-LABX-0056-LMH, LabEx LMH. Thomas Ourmières-Bonafos and Konstantin Pankrashkin are supported by the PHC Amadeus 2017–2018 37853TB funded by the French Ministry of Foreign Affairs and the French Ministry of Higher Education, Research and Innovation. The authors thank Yuri Kordyukov for comments on a preliminary version of the text.
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{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Let $G$ be a graph which satisfies $c^{-1}\,a^r\le |B(v,r)|\le c\,a^r$, for some constants $c,a>1$, every vertex $v$ and every radius $r$. We prove that this implies the isoperimetric inequality $|\partial A| \ge C |A| / \log(2+ |A|)$ for some constant $C=C(a,c)$ and every finite set of vertices $A$.'
author:
- Itai Benjamini
- Oded Schramm
title: |
Pinched exponential volume growth implies\
an infinite dimensional isoperimetric inequality
---
A graph $G=\bigl(V(G),E(G)\bigr)$ has pinched growth $f(r)$ if there are two constants $0 < c < C < \infty$ so that every ball $B(v,r)$ of radius $r$ centered around a vertex $v\in V(G)$ satisfies $$c \,f(r) < \bl|B(v,r)\br| < C\, f(r)\,.$$ For example, Cayley graphs and vertex-transitive graphs have pinched growth.
It is easy to come up with an example of a tree for which every ball satisfies $\bl|B(v,r)\br| \ge 2^{r/2}$, yet there are arbitrarily large finite subsets of $G$ with one boundary vertex. For example, start with ${\ensuremath{\mathbb{N}}}$ (an infinite one-sided path) and connect every vertex $n$ to the root of a binary tree of depth $n$. This tree does not have pinched growth.
We will see that, perhaps surprisingly, the additional assumption of pinched exponential growth (that is, pinched growth $a^r$, for some $a>1$) implies an infinite dimensional isoperimetric inequality. For a set $A\subset V(G)$ of vertices denote by $\partial A $ the (vertex) boundary of $A$, consisting of vertices outside of $A$ which have a neighbor in $A$.
\[main\] Let $G$ be an infinite graph with pinched growth $a^r$, where $a>1$. Then there is a constant $c>0$ such that for every finite set of vertices $A\subset V(G)$, $$\label{e.ii}
\bl|\partial A\br| \ge c\, |A| / \log(2+ |A|)\,.$$
We say that $G$ satisfies an $s$-dimensional isoperimetric inequality if there is a $c>0$ such that $\bl|{{\partial}}A\br|\ge c\,|A|^{(s-1)/s}$ holds for every finite $A\subset V(G)$. Thus, (\[e.ii\]) may be considered an infinite dimensional isoperimetric inequality.
Coulhon and Saloff-Coste [@CS] proved that when $G$ is a Cayley graph of an infinite, finitely-generated, group, the isoperimetric inequality $$\label{e.cs}
\bl|{{\partial}}A\br| \ge \frac{|A|}{4\,m\,\phi\bl(2|A|\br)}$$ holds for every finite $A\subset V(G)$, where $m$ is the number of neighbors every vertex has and $\phi(n)=\inf\bl\{r\ge 1:|B(v,r)|\ge n\br\}$ (here, $v\in V(G)$ is arbitrary). This result implies Theorem \[main\] for the case where $G$ is a Cayley graph, even when the upper bound in the pinched growth condition is dropped. The tree example discussed above shows that Theorem \[main\] is not valid without the upper bound. Thus, the (short and elegant) proof of (\[e.cs\]) from [@CS] does not generalize to give Theorem \[main\], and, in fact, the proof below does not seem related to the arguments from [@CS].
It is worthwhile to note that (\[e.cs\]) is also interesting for Cayley graphs with sub-exponential growth. For example, it shows that ${\ensuremath{\mathbb{Z}}}^d$ satisfies a $d$-dimensional isoperimetric inequality.
Another related result, with some remote similarity in the proof, is due to Babai and Szegedy [@BaS]. They prove that for a finite vertex transitive graph $G$, and $A \subset V(G)$, $0 < |A| <
|G|/2$, $$|{{\partial}}A\br| \ge {|A|} /(1+\diam G)\,.$$
The isoperimetric inequality (\[e.ii\]) is sharp up to the constant, since there are groups with pinched growth $a^r$ where (\[e.ii\]) cannot be improved. Examples include the lamplighter on ${\ensuremath{\mathbb{Z}}}$ [@LPP:lamplighter]. See [@Ha] for a discussion of growth rates of groups and many related open problems.
Regarding pinched polynomial growth, it is known that for every $d>1$ there is a tree with pinched growth $r^d$ containing arbitrarily large sets $A$ with $|{{\partial}}A|=1$, see, e.g., [@BS].
\[pr.tree\] Does every graph of a pinched exponential growth contain a tree with pinched exponential growth?
In [@BS:Cheeger] it was shown that every graph satisfying the linear isoperimetric inequality $|{{\partial}}A|\ge c\,|A|$ ($c>0$) contains a tree satisfying such an inequality, possibly with a different constant. The question whether one can find a [**spanning**]{} tree with a linear isoperimetric inequality was asked earlier [@DSS]. It follows from Theorem \[main\] that a tree with pinched exponential growth satisfies the linear isoperimetric inequality. (If a tree satisfies $|{{\partial}}A| \ge 3$ for every vertex set $A$ of size at least $k$, then every path of $k$ vertices in the tree must contain a branch point, a point whose removal will give at least $3$ infinite components. Consequently the tree contains a modified infinite binary tree, where every edge is subdivided into at most $k$ edges.) Consequently, Problem \[pr.tree\] is equivalent to the question whether every graph with pinched exponential growth contains a tree satisfying a linear isoperimeteric inequality.
As a warm up for the proof of Theorem \[main\], here is an easy argument showing that when $G$ has pinched growth $a^r$ it satisfies a two-dimensional isoperimetric inequality. Let $A\subset V(G)$ be finite. Let $v$ be a vertex of A that is farthest from $\partial A$, and let $r$ be the distance from $v$ to $\partial A$ . Note that $B(v,2r)\subset \bigcup_{u \in \partial
A} B(u,r)$. This gives, $a^{2r} \le O(1)\, |\partial A |\, a^r$, and therefore $O(1)\,|\partial A | \ge a^r$. On the other hand, $\bigcup_{u \in \partial A } B(u,r)\supset A$, which gives $O(1)\, |\partial A|\, a^r \ge |A|$. Hence, $O(1)\, |\partial A|^2 \ge |A|$.
For vertices $v,u$ set $z(v,u) := a^{-d(v,u)}$, where $d(v,u)$ is the graph distance between $v$ and $u$ in $G$.
We estimate in two ways the quantity $$Z=Z_A := \sum_{v \in A} \sum_{u \in \partial
A} z(v,u)\,.$$
Fix $v \in A$. For every $w \notin A$, fix some geodesic path from $v$ to $w$, and let $w'$ be the first vertex in $\partial A$ on this path. Let $R$ be sufficiently large so that $\bl|B(v,R)\br|\ge 2\,|A|$, and set $W:=B(v,R)\setminus A$. Then $$\bl| \{ (w,w') : w \in W \} \br| = | W| \ge a^R/O(1)\,.$$ On the other hand, we may estimate the left hand side by considering all possible $u \in \partial A$ as candidates for $w'$. If $w\in W $, then $d(v,w')+d(w',w)\le R$. Thus, each $u$ is equal to $w'$ for at most $O(1)\, a^{R-d(v,u)}$ vertices $w\in W$. This gives $$\bl| \{ (w,w') : w \in W \} \br| \le O(1) \sum_{u \in \partial A}
a^{R-d(u,v)} = O(1)\,a^R\sum_{u\in{{\partial}}A} z(v,u)\,.$$ Combining these two estimates yields $O(1)\sum_{u\in {{\partial}}A} z(v,u) \ge 1$. By summing over $v$, this implies $$\label{e.Z1}
O(1)\,Z \ge |A|\,.$$
Now fix $u \in \partial A$, set $m_r:=\bl|\{v\in A:d(v,u)=r\}\br|$, and consider $$\label{e.Zu}
Z(u):=\sum_{v \in A} z(v,u)=\sum_r m_r\,a^{-r}\,.$$ For $r\le \log|A|/\log a$, we use the inequality $m_r\le \bl|B(u,r)\br|=O(1)\,a^r$, while for $r>\log|A|/\log a$, we use $m_r\le |A|$. We apply these estimates to (\[e.Zu\]), and get $Z(u)\le O(1)\,\log(2+|A|)$, which gives $Z=\sum_{u\in{{\partial}}A} Z(u) \le O(1)\, |\partial A|\, \log(2+ |A|)$. Together with (\[e.Z1\]), this gives (\[e.ii\]).
Next, we present a slightly different version of Theorem \[main\], which also applies to finite graphs.
\[finite\] Let $G$ be a finite or infinite graph, $c>0$, $a>1$, $R\in{\ensuremath{\mathbb{N}}}$, and suppose that $c^{-1}\,a^r\le |B(v,r)|\le c\,a^r$ holds for all $r=1,2,\dots,R$ and for all $v\in V(G)$. Then there is a constant $C=C(a,c)$, depending only on $a$ and $c$, such that $$C\,\bl|\partial A\br| \ge |A| / \log(2+ |A|)$$ holds for every finite $A\subset V(G)$ with $|A|\le C^{-1}\, a^R$.
The proof is the same. A careful inspection of the proof shows that one only needs the inequality $c^{-1}\,a^r\le |B(v,r)|$ to be valid for $v\in A$ and the inequality $|B(v,r)|\le c\,a^r$ only for $v\in{{\partial}}A$.
[Acknowledgements:]{} We thank Thierry Coulhon and Iftach Haitner for useful disscusions.
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I. Benjamini and O. Schramm, Recurrence of distributional limits of finite planar graphs. Electron. J. Probab. 6 (2001), no. 23, 13 pp. (electronic).
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P. de la Harpe, Topics in geometric group theory. Chicago Lectures in Mathematics. University of Chicago Press, Chicago, IL, 2000. vi+310 pp.
R. Lyons, R. Pemantle and Y. Peres, Random walks on the lamplighter group. Ann. Probab. 24 (1996), no. 4, 1993–2006.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present results from observations of the most famous starburst galaxy M82 with the High-Resolution Camera onboard the [*Chandra X-Ray Observatory*]{}. We found nine sources in the central $1\arcmin\times1\arcmin$ region, but no source was detected at the galactic center. Comparing the observations on 1999 October 28 and on 2000 January 20, we found four of the nine sources showed significant time variability. In particular, , which is $9\arcsec$ away from the galactic center, showed extremely large time variability. We conclude that this source is the origin of the hard X-ray time variability of M82 detected with [*ASCA*]{}. Assuming a spectral shape obtained by the [*ASCA*]{} observation, its luminosity in the 0.5 – 10 keV band changed from 1.2 on 1999 October 28 to 8.7 on 2000 January 20.'
author:
- |
H. Matsumoto, T. G. Tsuru, K. Koyama, H. Awaki, C. R. Canizares, N. Kawai,\
S. Matsushita, A. Prestwich, M. Ward, A. L. Zezas, and R. Kawabe
title: 'Discovery of a Luminous, Variable, Off-Center Source in the Nucleus of M82 with the [*Chandra*]{} HRC'
---
Introduction
============
The X-ray spectrum of the starburst galaxy M82 measured with [*ASCA*]{} [[@Tanaka1994]]{} consists of soft, medium, and hard components [[@Tsuru1997]]{}. The hard component, which dominates the X-ray spectrum above 2 keV, was found to show time variability by monitoring M82 with [*ASCA*]{} in 1996 [[@Matsumoto1999; @Ptak1999]]{}. Assuming a distance of 3.9 Mpc (i.e. $1\arcsec \sim$ 19 pc) [[@Sakai1999]]{}, its luminosity in the 0.5 – 10 keV band changed between 4.5 and 1.6 at various time scales from 1 s to a month. The most plausible explanation for the variability is that a low-luminosity AGN exists in M82. If this explanation is correct, M82 would be one of the most important objects for the study of a relation between an AGN and starburst activity [[e.g. @Umemura1997]]{} because of its proximity. However, hard X-ray spectra of M82 obtained with [*Ginga*]{} and [*BeppoSAX*]{}, which can detect the X-ray photons of higher energy than [*ASCA*]{}, differ from typical AGN spectra, since they cannot be fitted by a power-law model [[@Tsuru1992; @Cappi1999]]{}. Therefore, it is also possible that the origin of the hard component is a new type of compact X-ray source.
Furthermore, [@Matsumoto1999]{} compared the [*ASCA*]{} image with the [*ROSAT*]{} HRI image [[@Strickland1997]]{}, and showed that the hard component probably comes from the X-ray brightest source detected with the [*ROSAT*]{} HRI. The HRI source was found to show time variability [[@Collura1994]]{}. Since [@Stevens1999]{} showed that the source is away from the dynamical center of M82, this galaxy may harbor an “off-center” AGN.
Thus, the source of the hard component of M82 would be a quite interesting and important object. For further investigation, it is necessary to determine the position of the hard component precisely and to find a counterpart in other wavelengths if possible. Therefore, we analyzed the data of M82 obtained with the High-Resolution Camera (HRC) [[e.g. @Murray1987; @Murray1997]]{} onboard the [*Chandra X-Ray Observatory (CXO)*]{}[[@Weisskopf1995]]{}. The HRC has sensitivity in the 0.1 – 10 keV band with a peak at about 1 keV, and it has the highest angular resolution (FWHM $\sim0\farcs5$) of all instruments onboard X-ray observatories to date. Uncertainties in this paper refer to 1$\sigma$ confidence limits.
Data Analysis and Results
=========================
M82 was observed twice with [*CXO*]{} as a calibration target using the HRC-I on 1999 October 28 and 2000 January 20. The HRC data were processed with the standard procedure by the [*Chandra X-ray Center (CXC)*]{} [[^1]]{}, and we used only the fully processed science products (level = 2) as event files. We also applied an event screening procedure developed by the HRC team to eliminate “ghost” events. The exposure times are 2788 s for the first observation and 17684 s for the second.
We found that all X-ray bright sources which can be the origin of the [*ASCA*]{} hard component exist within the central $1\arcmin\times1\arcmin$ (= 1.1 kpc $\times$ 1.1 kpc) region, which is shown in Figure [\[fig:HRC\]]{}. The position of the bright off-center source found with the [*ROSAT*]{} HRI [[@Stevens1999]]{} is also within the field. Therefore, we concentrate on the central $1\arcmin\times1\arcmin$ region. In Figure \[fig:HRC\], we see nine prominent sources, which are designated with circles and numbers. More detailed analysis including diffuse emission and other fainter sources will be presented in a forthcoming paper (M. Ward et al., in preparation). We determined the positions of the peaks of the nine HRC sources with the wavelet algorithm using [*Chandra Interactive Analysis of Observations (CIAO)*]{}, and named the HRC sources by using these positions (Table [\[tbl:src\]]{}). The position uncertainty is less than $\sim 0\farcs1$.
According to “the [*CXC*]{} memo on astrometry problems” [^2], the HRC event files we used could have offsets up to $10\arcsec$ in the celestial coordinates. To check the reliability of the coordinates in the event files, we compared the coordinates of the HRC sources with the Two Micron All Sky Survey (2MASS) Point Source Catalog. There are 17 2MASS sources in the field of Figure [\[fig:HRC\]]{}. According to the [*CXC*]{}, the offset of the celestial coordinate in ordinary CXO event files is $1\arcsec$ (RMS). The position uncertainty in the 2MASS Catalog is $\sim 0\farcs1$. Considering these uncertainties, we found that three 2MASS sources out of 17 agree with the HRC sources (Nos. 1, 6, and 8). Since the probability of a chance coincidence is less than 0.1 %, we can assume that the HRC coordinates are reliable. We then compared the coordinates of the HRC sources with the 5 GHz sources [[@Muxlow1994]]{} whose position accuracy is $\sim0\farcs5$. The fact that four HRC sources (Nos. 4, 5, 6, and 7) have counterparts on the 5 GHz map also supports the reliability of the HRC coordinates. The identifications of the HRC sources are shown in Table [\[tbl:src\]]{}.
We estimated the counting rates of the HRC sources using the X-ray events within the circular regions shown in Figure \[fig:HRC\]. The backgrounds including the diffuse emission were estimated using source-free regions around the sources. The counting rates are listed in Table \[tbl:src\]. Four HRC sources (Nos. 5, 7, 8, and 9) show significant time variability. In particular, No. 5 disappeared in the second observation, and No. 7 became brighter by a factor of 7. The counting rates of No. 1 and No. 4 did not change significantly between the two observations.
The dynamical center at $(\alpha, \delta)_{\rm J2000}$ = $(9^{\rm h}55^{\rm m}51.9^{\rm s},
69{\degr}40{\arcmin}47.1{\arcsec})$, which is determined by the radio observation of the motion of gas, is shown as the green cross in Figure \[fig:HRC\] [[@Weliachew1984]]{}. The position error circle with a radius of $2\arcsec$ is also shown as the green circle. We should note that all nine HRC sources are clearly away from the dynamical center. The counting rate of the dynamical center estimated with the extraction radius of $2\arcsec$ is ($1.79\pm1.48$) c s$^{-1}$ for 1999 October 28, and ($1.12\pm6.84$) c s$^{-1}$ for 2000 January 20. Thus, we found no significant source at the dynamical center.
We estimated the HRC counting rate of the [*ASCA*]{} hard component [[@Matsumoto1999]]{} using [*W3PIMMS*]{} v3.0 [^3]: the expected counting rate is 0.72 c s$^{-1}$ in the highest state and 9.7 c s$^{-1}$ in the lowest. Only the counting rate of No. 7 is consistent with the expected counting rate. Furthermore, the position of the [*ASCA*]{} hard component is consistent with that of No. 7 [[@Matsumoto1999]]{}. Therefore, we can conclude that the variability of the [*ASCA*]{} hard component is due to No. 7 (). The separation between No. 7 and the dynamical center is $9\arcsec$ ($\sim$ 170 pc).
Discussion
==========
We found that 41.5+59.7 [[@Kronberg1985]]{} is a candidate of the radio counterpart of No. 7 as well as 41.30+59.6 [[@Muxlow1994]]{}. 41.5+59.7 is $0\farcs76$ away from No. 7, while the separation between 41.30+59.6 and No. 7 is $0\farcs96$. According to the morphology and spectral shape in the 5 GHz radio band, [@Muxlow1994]{} suggested that 41.30+59.6 is a young supernova remnant (SNR). 41.5+59.7 show a 100 % drop in the radio flux within a year, and its radio decay time scale and spectrum are very similar to SN 1983n [[@Kronberg1985b; @Kronberg2000]]{}. Therefore, this source may also be a SNR. The hard X-ray emission of M82 was found to show the short-term variability [[@Matsumoto1999; @Ptak1999]]{}. Since it is rather difficult to explain the short-term variability in terms of a SNR origin, both radio sources may not be a real counterpart of No. 7.
If we assume the spectral shape of No. 7 is an absorbed thermal bremsstrahlung model with a temperature of 10 keV and a column density of $10^{22}$ cm$^{-2}$, which is the typical spectral shape of the [*ASCA*]{} hard component [[@Matsumoto1999]]{}, the peak X-ray flux of No. 7 is 6.6 in the 2 – 10 keV band. According to the the $\log N - \log S$ relation [[@Ueda1999]]{}, the probability that a source as bright as No. 7 exists in the $1\arcmin\times1\arcmin$ field is $\sim$ 0.3 %. Therefore, No. 7 is probably not a background AGN. The [*ASCA*]{} spectrum of the variable source obtained by subtracting the spectrum of the lowest state from the highest state shows heavy absorption (the column density is $\sim$ $10^{22}$ cm$^{-2}$) [[@Matsumoto1999]]{}. Since the Galactic absorption toward M82 is 4 cm$^{-2}$ [[@Dickey1990]]{}, the variable source is embedded deeply in M82, and hence No. 7 is probably not a foreground source, unless the source has extremely large intrinsic absorption.
Assuming the absorbed bremsstrahlung model of 10 keV and $10^{22}$ cm$^{-2}$, unabsorbed X-ray luminosity in the 0.5 – 10 keV band ($L_{\rm X}^{\rm 0.5-10 keV}$) is expressed using an HRC counting rate ($C_{\rm HRC}$) as $$L_{\rm X}^{\rm 0.5-10 keV}
= 1.67{\mbox{$\times10^{38}$}}\ \mbox{erg s$^{-1}$} \times
\left[\frac{C_{\rm HRC}}{10^{-3}\ \mbox{c s$^{-1}$}}\right].$$ The $L_{\rm X}^{\rm 0.5-10 keV}$ of No. 7 was estimated to be 1.2 on 1999 October 28 and 8.7 on 2000 January 20.
If we assume that No. 7 is a black hole (BH) and that the maximum luminosity does not exceed the Eddington luminosity, the mass of the BH must be larger than 700 $M_\sun$, and No. 7 is not a stellar-mass BH ($\sim$ 10 $M_\sun$). Since No. 7 is $9\arcsec$ away from the dynamical center, the mass of No. 7 must be much smaller than the gravitational mass within $9\arcsec$ from the center which is 4 $M_\sun$ [[@McLeod1993]]{}, otherwise the dynamical center would be shifted from the current position. Therefore, No. 7 is at the low end of the mass distribution of super-massive BHs ($10^6$ – $10^9$ $M_\sun$) or a medium-massive BH ($10^3$ – $10^6$ $M_\sun$). The possibility of the medium-massive BH is discussed in detail in [@Matsushita2000b]{} along with the discovery of an expanding molecular superbubble surrounding No. 7 [[@Matsushita2000]]{}. Other possibilities such as an X-ray binary source whose jet is strongly beamed at us cannot be excluded. Further investigation including other wavelengths is strongly encouraged to reveal the true character of No. 7 ().
If we use equation (1) to the sources other than No. 7, their $L_{\rm X}^{\rm 0.5-10 keV}$s are much greater than the Eddington luminosity for a 1.4 $M_\sun$ object. Therefore, these sources may also be BHs.
Though we found no significant source at the dynamical center, it is still possible that a faint X-ray source such as Sgr A$^*$ exists [[@Koyama1996]]{}. Assuming an absorbed power-low model with a photon index of 1.7 and a column density of $10^{22}$ cm$^{-2}$, the upper limit of $L_{\rm X}^{\rm 0.5-10 keV}$ of the dynamical center is 5.4 for 1999 October 28 and 1.3 for 2000 January 20.
This paper 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. The authors are grateful to Miss Deborah Gage for careful review of the manuscript. We also thank Dr. P. Kaaret for valuable comments. H. M. and S. M. are supported by the JSPS postdoctoral Fellowships for Research Abroad.
Cappi, M. et al., 1999, , 350, 777
Collura, A., Reale, F., Schulman, E., Bregman, J. N. 1994, , 420, L63
Dickey, J. M., & Lockman, F. J. 1990, , 28, 215
Koyama, K., Maeda, Y., Sonobe, T., Takeshima, T., Tanaka, Y., & Yamauchi, S. 1996, , 48, 249
Kronberg, P. P., Biermann, P., & Schwab, F. R. 1985, , 291, 693
Kronberg, P. P., & Sramek, R. A. 1985, Science, 227, 28
Kronberg, P. P., Sramek, R. A., Birk, G. T., Dufton, W., Clarke, T. E., Allen, M. L. 2000, , 535, 706
McLeod, K. K., Rieke, G. H., Rieke, M. J., & Kelly, D. M. 1993, , 412, 111
Matsumoto, H., & Tsuru, T. G. 1999, , 51, 321
Matsushita, S. 2000, Ph.D. thesis, The Graduate University for Advanced Studies, Japan
Matsushita, S. et al. 2000, , submitted
Murray, S. S., Chappell, J. H., Elvis, M. S., Forman, W. R., & Grindlay, J. E. 1987, Astrophys. Lett. Commun., 26, 113
Murray, S. S., et al. 1997, Proc. SPIE, 3114, 11
Muxlow, T. W. B., Pedlar, A., Wilkinson, P. N., Axon, D. J., Sanders, E. M. & de Btuyn, A. G. 1994, , 266, 455
Ptak, A., & Griffiths, R. 1999, , 517, L85
Sakai, S., & Madore, B. F., 1999, , 526, 599
Stevens, I. R., Strickland, D. K., & Wills, K. A., 1999, , 308, L23
Strickland, D. K., Ponman, T. J., & Stevens, I. R., 1997, , 320, 378
Tanaka, Y. Inoue, H., & Holt, S. S. 1994, , 46, L37
Tsuru T. G. 1992, Ph.D. thesis, The University of Tokyo
Tsuru T. G., Awaki H., Koyama K., Ptak A. 1997, , 49, 619
Ueda, Y. et al. 1999, , 518, 656
Umemura, M., Fukue, J., & Mineshige, S., 1997, , 479, L97
Weisskopf, M. C., O’Dell, S. L., Elsner, R. F., & Van Speybroeck, L. P. 1995, Proc. SPIE, 2515, 312
Weliachew, L., Formalont, E. B., & Greisen, E. W. 1984, , 137, 335
[ccccccc]{} 1&&0955547+694100 &&$2\farcs0$ &2.51$\pm$1.08 &2.42$\pm$0.40\
2&&&&$2\farcs0$ &6.10$\pm$2.00 &3.15$\pm$0.67\
3&&&&$2\farcs0$ &4.30$\pm$2.43 &8.43$\pm$0.95\
4&&&42.65+57.8 &$1\farcs3$ &8.04$\pm$2.14 &6.27$\pm$0.72\
5&&&42.21+59.0 &$1\farcs4$ &20.9$\pm$3.1 &0.292$\pm$0.505\
6&&0955507+694043 &41.95+57.5 &$1\farcs1$ &2.99$\pm$1.54 &5.20$\pm$0.65\
7&& &41.30+59.6 or 41.5+59.7 &$2\farcs0$ &71.1$\pm$5.9 &520$\pm$5\
8&&0955475+694036 & &$2\farcs0$ &24.8$\pm$3.1 &4.89$\pm$0.66\
9&&& &$2\farcs0$ &4.63$\pm$1.48 &22.8$\pm$1.2\
&dynamical center &&&$2\farcs0$ &1.79$\pm$1.48 &0.112$\pm$0.684\
(a)[7cm]{}(b)\
[^1]: <http://asc.harvard.edu/>
[^2]: <http://asc.harvard.edu/ciao/caveats/aspect4.html>
[^3]: <http://heasarc.gsfc.nasa.gov/Tools/w3pimms.html>
|
{
"pile_set_name": "ArXiv"
}
|
---
address: 'c/o IBM T.J. Watson Research Center, Yorktown Heights, NY 10598'
author:
- Daniel Braun
title: Quantum Chaos and Quantum Algorithms
---
[2]{}
The problem of quantum chaos in quantum computers (QC) [@Deutsch85] has recently attracted considerable attention after a pioneering work by Georgeot and Shepelyansky [@Georgeot99]. These authors pointed out that residual, uncontrolled interactions between qubits might induce quantum chaos in the QC if the strength of the interaction exceeds a certain critical level, and they argued that this might destroy the operability of the QC. While the model considered by Georgeot and Shepelyansky [@Georgeot99] does not describe a particular physical realization of a QC, and in particular does not allow for time dependent operating of quantum gates, it is sufficiently generic to mimic a quantum register, i.e. a (static) memory of the QC, in which a state can be stored and from which it should be retrievable again at a later time.
It is clear that residual interactions between the qubits will in general lead to eigenstates of the quantum register that are not the original product states $|0\rangle\ldots|0\rangle|0\rangle$, $|0\rangle\ldots|0\rangle|1\rangle$, $\ldots$ $|1\rangle\ldots|1\rangle|1\rangle$ (called multi–qubit states in the following). Therefore, information stored in the register will evolve with time. If the exact eigenstates of the register are superpositions of only a few multi–qubit states, the register will oscillate quasiperiodically between these. However, in the case of quantum chaos, the eigenstates of the register are superpositions of practically [*all*]{} $2^n$ multi–qubit states. The quasiperiodic oscillations due to the interaction between qubits degradate then for all practical times to a decay of the original register state into all multi–qubit states. The time scale $\tau_\chi$ for this decay is set by the inverse width of the distribution of eigenenergies. Georgeot and Shepelyansky concluded that all calculations of the quantum computer should have finished long before the time $\tau_\chi$ and that this would limit the operability of the computer in a very similar fashion as decoherence. The critical interaction strength $J_c$ between qubits at which quantum chaos sets in decreases like $J_c\propto 1/n$ with the number $n$ of qubits, and the transition to quantum chaos becomes more and more abrupt with increasing $n$ [@Georgeot99].
Later, Silvestrov et al. questioned Georgeot’s and Shepelyansky’s conclusions [@Silvestrov00]. They showed that even in the case of quantum chaos error correction schemes [@Shor95; @Steane96] are capable of dealing with errors generated by quantum chaos. However, much more error correction is needed than in the absence of quantum chaos. Their model again included only static interactions between qubits.
In this work I examine a question that is somewhat complementary to the one investigated in [@Georgeot99; @Silvestrov00]: Is it possible that already the interactions introduced on purpose between qubits in order to operate the quantum computer lead to quantum chaos, even if there are no residual parasitic interactions between qubits? This is an intriguing question, since it has implications for the amount of resources necessary for implementing a given algorithm. Already classically chaotic algorithms (e.g. for calculating the time evolution of a classical chaotic system) require much more computing resources than integrable ones: Since errors in the initial conditions amplify exponentially in time, one needs to spend many more bits than for calculating an integrable time evolution over the same time.
In quantum mechanics chaos manifests itself by a sensitivity not to the initial state but to the [*control parameters*]{} [@Peres91] (amongst other signatures, see below). The fidelity $|\langle
\psi(k)|\tilde{\psi}(k)\rangle|^2$ of a wave–function $|\psi(k)\rangle=U^k(\lambda)|\psi\rangle$ with respect to a wave–function $|\tilde{\psi}(k)\rangle=U^k(\tilde{\lambda})|\psi\rangle$ decreases exponentially with the discrete time $k$, if $\lambda$ and $\tilde{\lambda}$ are two slightly different system control parameters and the time evolution $U(\lambda)$ is chaotic. In the case of integrable quantum dynamics the fidelity typically shows quasiperiodic oscillations in $k$. Therefore a chaotic quantum algorithm will need more resources in the form of more precise quantum gates, or as pointed out in [@Silvestrov00], more error correction.
A quantum algorithm (QA) is uniquely defined by the unitary transformation $U$ it induces in the entire multi-qubit Hilbert space, and the question of quantum chaos can be studied directly on the level of that unitary transformation, without the need to deal with the time dependent Hamiltonian by which it is generated. This is what I am going to do in this work, therefore being able to go beyond the models with time-independent Hamiltonians studied in [@Georgeot99; @Silvestrov00].
Obviously, the answer to the question posed depends on the quantum algorithm. A quantum algorithm simulating a quantum chaotic system [@Song01; @Georgeot01] is by definition a unitary transformation showing quantum chaos, and will thus need very precise tuning of the control parameters. In this paper I focus, however, on two of the most well–known quantum algorithms, where the answer is less clear from the beginning, namely Grover’s search algorithm [@Grover97] and the quantum Fourier transform (QFT). The latter is the center piece of several important QAs, like phase estimation, order-finding, the hidden subgroup problem (see [@Nielsen00]), and, most prominently, Shor’s factoring algorithm [@Shor94]. Also from the pure quantum chaos point of view the question of quantum chaos in these algorithms is very interesting. In fact it turns out that both algorithms have symmetry properties that lead to a remarkable and very non generic mixture of signatures of quantum chaos and quantum integrability.
The first thing that comes into mind for checking for quantum chaos, is the eigenvalue and eigenvector statistics of $U$: It is believed that an eigenvalue– and eigenvector statistics of $U$ corresponding to Dyson’s circular ensembles indicates quantum chaos [@Bohigas84; @Berry84]. That is, the eigenvalues $\lambda_i=\exp(-{{\rm i}}\varphi_i)$ should show universal level repulsion [@proof]. In the limit of large $N$ one expects a distribution $P(s)$ of nearest neighbor spacings $s_i=N(\varphi_{i+1}-\varphi_i)/2\pi$ that is well described by the universal Wigner–Dyson statistics [@Mehta91]. If $U$ is covariant under any anti–unitary operation $T$ that squares to unity there is always a basis in which the eigenvectors of $U$ can be chosen real. The relevant random matrix ensemble is then the circular orthogonal ensemble (COE) with a $P(s)$ very well approximated by $$\label{WD}
P(s)=\frac{s\pi}{2}{{\rm e}}^{-s^2\pi/4}\,.$$ The best known example is conventional time reversal symmetry, in which case $T$ is the complex conjugation operator [@Haake91]. The eigenvector statistics is usually described in terms of a distribution of eigenvector components. Picking any component $c_{i}$ of any eigenvector at random, random matrix theory (RMT) predicts for $y=N|c_{i}|^2$ in the COE case the so–called Porter–Thomas distribution, $$\label{PT}
R_{\rm COE}(y)=\frac{1}{\sqrt{2\pi y}}{{\rm e}}^{-y/2}\,.$$
Let us now have a look at Grover’s algorithm [@Grover97]. It allows to find an entry with index $\xi$ in a unsorted quantum database that is distinguished from the others by a given property. The distinction may be formalized by an oracle query $O$ which in the multi–qubit basis is a unitary diagonal matrix with entries $O_{ii}=1-2\delta_{i\xi}$ where $\delta_{i\xi}$ is the Kronecker delta. Thus, presented a register state the oracle always gives back the same state unless it is the searched one in which case the oracle changes the state’s phase by $\pi$. Grover’s algorithm commences with the Hadamard transformation $$\label{hada}
H=H_0\otimes H_2\otimes\ldots\otimes H_{n-1}\,,$$ where $$\label{hi}
H_i=\frac{1}{\sqrt{2}}\left(
\begin{array}{cc}
1&1\\
1&-1
\end{array}
\right)$$ is a Hadamard transformation for the $i$th qubit. Starting from the register state $|0\rangle \ldots |0\rangle$, the Hadamard transformation brings the system into a superposition of all multi–qubit states with equal weight $1/\sqrt{N}$. Next comes an iteration of the oracle query followed by a “diffusion matrix” $D$. The latter has matrix elements $D_{ij}=2/N-\delta_{ij}$. The total algorithm thus reads $$\label{Grover}
U_G=(DO)^pH\,.$$ The optimal value for the integer $p$ is given by $p=[\pi/(4\theta)]$ with $\sin^2\theta=1/N$ ($[.]$ denotes the integer value) [@Boyer96], and I have chosen this value for all calculations. Remarkably, the quantum computer finds the searched element with $\sim \sqrt{N}$ queries, whereas a classical computer when presented with the same problem would have to ask the oracle on the average $N/2$ times.
From the definition of $U_G$ it is clear that $U_G$ is real. Covariance under conventional time reversal symmetry, $$\label{cv}
KU_GK^{-1}=U_G^\dagger$$ thus is fulfilled if $U_G$ is symmetric, $U_G=U_G^T$, where $U_G^T$ denotes the transposed matrix. Using the commutation relations between $H,O$ and $D$ (or by evaluating the matrix numerically, see fig.\[fig.syG\]) one convinces oneself that $U_G$ is [*almost*]{} symmetric.
\
\
In fact the average absolute value of the matrix elements of the symmetric part of $U_G$ decays like $1/\sqrt{N}$, whereas the average absolute value of the antisymmetric part decays like $1/N$. Just one element escapes from the general rule set by the average: $U_{G\xi 0}$, the element in the zeroth column pertaining to the searched element is of order unity minus a $1/\sqrt{N}$ correction (that is how the whole algorithm works), whereas $U_{G0\xi}$ plays no such special role. Nevertheless, for $N\to\infty$ the weight of this particular element is zero and therefore $U_G$ has the unusual property that it is at the same time unitary as well as real and (almost) symmetric. Thus, all eigenvalues have to be at the same time on the unit circle, and have to be real. Therefore all eigenvalues are expected to be either unity or minus unity! This is well confirmed numerically (see FIG.\[fig.ewG\]). In fact it turns out that all eigenvalues, even those which are not plus or minus one, are to very good approximation 6th roots of one, and Grover’s algorithm is therefore approximately a 6th root of the unit operation!
The resulting high degeneracy of the eigenphases is in strong contrast to the level repulsion that goes along with quantum chaos. However, absence of level repulsion in $U$ does not exclude quantum chaos. Indeed, suppose some $U$ did show level repulsion, then $U^M$ with $M\gg 1$ will in general not, since the spectrum gets completely mixed by winding it $M$ times around the unit circle. A Poissonian statistics will be the consequence. Nevertheless, if $U$ has a chaotic classical counterpart, so will $U^M$, and to conclude that $U^M$ is [not]{} chaotic just from the level statistics is therefore not possible. If the phases of $U$ are commensurate one can even find an $M$ so that all eigenvalues are degenerate.\
\
Let us look at the eigenvector statistics of $U_G$. FIG.\[fig.evG\] shows that is obeys almost perfectly RMT! This is, however, a pure consequence of the high degeneracy. Indeed, any linear combination of eigenstates in the subspace pertaining to the degenerate eigenvalue $\lambda=1$ is again an eigenstate to the same eigenvalue (and correspondingly for $\lambda=-1$). Thus, a set of eigenvectors can be oriented in a completely arbitrary way in this subspace with the only restriction that they be all normalized and mutually orthonormal. This is just as in RMT, where the eigenvectors are all statistically independent from the eigenvalues, and their orientation is statistically uniform on the whole possible hyper-sphere. In the present situation the diagonalization routine picks initial orientations at random and thus mimics perfect RMT behavior in both of the two subspaces.
Given the unclear picture presented by the eigenvalue and eigenvector statistics, it seems reasonable to examine directly the sensitivity with respect to slight variations in $U_G$. One might do so by applying Peres’ original scheme, i.e. creating one perturbed algorithm $U_G'$ and studying the decay of overlap between $U_G^n|\psi\rangle$ and $(U_G')^n|\psi\rangle$ as function of $n$ for some random initial $|\psi\rangle$. A natural parameter in $U_G$ that might be varied is the number of iterations $p$ of the transformation $DO$. For $p\sim \sqrt{N}$ going from $p$ to $p+1$ appears indeed as a small perturbation. However, all that has been said about the spectrum of $U_G$ applies to the such perturbed $U_G'$ as well, i.e. all eigenvalues will generically be $\pm 1$. A spectral decomposition of $U_G=\sum_i|u_i^{(+)}\rangle\langle
u_i^{(+)}|-\sum_l|u_l^{(-)}\rangle\langle
u_l^{(-)}|+\sum_k{{\rm e}}^{{{\rm i}}\varphi_k}|u_k\rangle\langle u_k|$ (where $\varphi_k$ are the phases different from $0$ and $\pm\pi$) and correspondingly for $U_G'$ shows that for even $n$ the overlap $\langle
\psi|(U_G^\dagger)^n (U_G')^n|\psi\rangle$ depends on $n$ only due to the exceptional eigenvalues that differ from $\pm 1$. All the other eigenvalues contribute only two different values to the overlaps, depending on whether $n$ is even or not. Therefore, varying $p$ (or introducing any other perturbation that does not lift the degeneracy of the spectrum) does not lead to exponentially decaying overlap.
I have therefore applied another perturbation: All $DO$ factors in $U_G$ are multiplied with a random matrix $V_i$ (drawn independently for each factor) close to unity, $$\label{}
U_G'=(DO)V_1\ldots(DO)V_pH\,.$$ I constructed all of these random matrices $V_i$ as tensor products of $2\times 2$ orthogonal matrices $O(2,\varphi)$ close to ${\rm diag}(1,1)$ in the Hilbert space of each qubit, $V_i=O(2,\varphi_{1i})\otimes\ldots\
\otimes O(2,\varphi_{pi})$, where $\varphi_{1i},\ldots,\varphi_{pi}$ are chosen randomly and independently from a uniform distribution in the interval $-\epsilon/2\ldots \epsilon/2$, and $O(2,\varphi)$ is a $2\times 2$ orthogonal transformation acting on a single qubit as $$\label{ri}
O(2,\varphi)=\left(\begin{array}{cc}
\cos\varphi & \sin\varphi\\
-\sin\varphi & \cos\varphi
\end{array}
\right)\,.$$ Fig.\[fig.Groverolap\] shows quasiperiodic overlap with almost perfect revival within 500 iterations for a 5 qubit Grover algorithm with $\xi=2$ as selected index and $\epsilon=0.1$. For iteration numbers exceeding 5000 one notices a slight decay, but nevertheless the Fourier spectrum of the decay shows a few very sharp and strong peaks, indicating the quasi-periodic nature of the function. We conclude that according to Peres’ criterion, Grover’s search algorithm is free of quantum chaos.
One might object that as a quantum algorithm, $U_G$ will typically not be iterated, though. I therefore also applied another criterion proposed by Schack et al. [@Schack94], which seems to be more appropriate in the current situation. These authors examined the distribution of angles between vectors propagated by slightly perturbed unitary matrices, and embedded Peres’ original sensitivity criterion into an information theoretical framework. They showed for the example of a kicked top that in the case of a chaotic quantum map the distribution $P(\alpha)$ of angles $\alpha$ between Hilbert space vectors propagated by many slightly and randomly perturbed unitary transformations corresponds to that of randomly chosen vectors, which resembles a Gaussian peak. An overall average angle can be steered with a deterministic part of the random vectors and adapted to $P(\alpha)$ of the Hilbert space vectors. On the other hand, for integrable quantum maps the random perturbations lead to many more or less degenerate angles, as the propagated vectors do not explore all Hilbert space dimensions. The angle distribution therefore typically contains several more or less pronounced peaks.
\
Instead of adapting the deterministic part of the random vectors, I chose to “unfold” the angles, i.e. to rescale them by the average angle. It turned out that “universal” distributions are obtained by this procedure, both for the randomly drawn vectors as well as those propagated by $U_G'$. In the latter case one obtains a distribution which over several orders of magnitude of the perturbation strength $\epsilon$ is independent of $\epsilon$, [^1] in the former case the unfolded angle distribution is independent of the deterministic part of the random vectors.
I have constructed two particular classes of perturbations. The first one closely corresponds to the original recipe by Schack et al. [@Schack94]. The perturbed algorithm is obtained by multiplying all $DO$ factors in $U_G$ with only one out of two random orthogonal matrices close to unity, namely $V_+=O(2,\varphi_1)\otimes\ldots\
,O(2,\varphi_p)$ or $V_-=V_+^{-1}$, where again $\varphi_1,\ldots,\varphi_p$ are chosen randomly and independently from a uniform distribution in the interval $-\epsilon/2\ldots \epsilon/2$, but are kept fixed for all factors; i.e. we obtain $2^p$ perturbed Grover algorithms $DOV_+DOV_+\ldots DOV_+$, $DOV_+DOV_+\ldots DOV_-$, $\ldots$, $DOV_-DOV_-\ldots DOV_-$. A random initial vector is then propagated by these $2^p$ perturbed matrices and the angle distribution between the resulting vectors is analyzed.
Fig. \[fig.pofphiG\] shows that this sort of ’digital’ perturbation does not entirely randomize the propagated vector. Rather the distribution of angles shows many pronounced peaks which means that different perturbations lead to similar Hilbert space vectors. Thus, from this plot one would conclude that the behavior is integrable.
\
\
However, the situation is very different for the second form of perturbation which corresponds to the one introduced for studying the decaying overlap, i.e. instead of choosing between just two matrices $V_-$ and $V_+$ each factor $DO$ is multiplied with an independent random matrix of the form $V(\varphi_{1i},\ldots,\varphi_{pi})=O(2,\varphi_{1i})\otimes\ldots\,\otimes
O(2,\varphi_{pi})$. Fig. \[fig.pofphiG2\] shows that in this case a broad, Gaussian like distribution without much further structure arises, much broader in fact (after unfolding) than the angle distribution obtained from the random vectors.
We may therefore conclude that for certain perturbations and according to the criterion by Schack et al. Grover’s search algorithm does show hypersensitivity with respect to perturbations. This is quite surprising in the light of Grover’s earlier finding that almost any unitary transformation substituted for the Hadamard matrix still leads to a functioning search algorithm [@Grover98], and it seems, indeed, that the criterion by Schack et al. is singled out compared to the other criteria examined. Note, however, that there is not necessarily a contradiction to Grover’s finding, since for Grover’s algorithm to work it is enough that only the first column of $U_G$ be largely unaffected by the perturbations (and the same argument applies to all quantum algorithms that start from just one initial state, typically the state $|00\ldots 0\rangle$)! The present result is more general in as much as it makes a statement about the sensitivity of the entire matrix with respect to perturbations.
How rapidly does the average fidelity decrease with the perturbation strength? If we measure lack of fidelity as average absolute error of all matrix elements, the answer is: linearly, over several orders of magnitude of $\epsilon$ ($0.0001\le\epsilon\le 0.1$), as can be seen from Fig.\[fig.errofeps\], and as would have been expected from perturbation theory.
Let me now come to the quantum Fourier transform (QFT). Since it is a universal part of several proposed quantum algorithms including Shor’s algorithm [@Shor94], it makes sense to give the QFT special attention. The quantum Fourier transform $U_{FT}$ on a $n$ bit register (with qubits indexed as $0\ldots n-1$) can be constructed from one– and two–qubit operations as [@Shor94] $$\begin{aligned}
\label{shorc}
U_{\rm FT}&=&FH_0S_{0,1}S_{0,2}\ldots
S_{0,n-1}H_1\ldots H_{n-3}S_{n-3,n-2}\nonumber\\
&&S_{n-3,n-1}H_{n-2}S_{n-2,n-1}H_{n-1}\,, \end{aligned}$$ where $S_{j,k}$ is a conditional phase shift matrix between qubits $j,k$ defined by $S_{j,k}={\rm diag}(1,1,1,\exp({{\rm i}}\pi/2^{k-j}))$ in the basis $00$, $01$, $10$, and $11$ formed by the two qubits $j$ and $k$. The matrix $F$ flips all qubits. With all the two–qubit interactions introduced, one would naively expect quantum chaos. However, the QFT is constructed such that in the whole Hilbert space $U_{\rm FT}$ is very simple and symmetric, $$\label{QFT}
U_{{\rm FT}lk}=\frac{1}{\sqrt{N}}{{\rm e}}^{{{\rm i}}2\pi lk/N}\,.$$ One easily convinces oneself that $U^4={\bf 1}$! Thus, all possible eigenphases are $0$, $\pi$, and $\pm\pi/2$. The situation is therefore even simpler than in Grover’s algorithm: There is an exact relation that dictates a high degeneracy of only four possible eigenphases. So again, there is no level repulsion.
\
The matrix $U_{\rm FT}$ is covariant under conventional time–reversal: $KU_{\rm FT}K^{-1}=U^*=U^\dagger$ since $U=U^T$. And a numerical evaluation of the eigenvector statistics leads again to good agreement with the Porter–Thomas distribution, which, however, results once more from the high degeneracy of the eigenvalues and not from quantum chaos.
\
For studying the sensitivity of $U_{\rm FT}$ with respect to perturbations, I slightly perturbed all phases in the conditional phase gates by adding an additional random phase with the same (on the average) relative amount for all gates [@relphase]. Fig. \[fig.qftolap\] shows that the overlap of a random state propagated with a slightly perturbed matrix $U_{\rm FT}'$ and the same state propagated with the original $U_{\rm FT}$ shows again quasiperiodic oscillations. The basic period is two, as could be expected from $(U_{\rm
FT}^2)_{ij}=\delta_{i,N-j}$, i.e. the exact QFT leads back to the same starting vector up to a relabeling of the indices when applied twice. The perturbed QFT only leads to a small deviation from this relabeled initial vector, and therefore almost perfect overlap is restored every second iteration. Thus, according to Peres’ criterion the QFT is free of quantum chaos.
Fig. \[fig.pofphiG2\] shows the angle distribution resulting from propagating a random initial vector by 100 differently perturbed matrices for an algorithm running on five qubits. The Hilbert space vectors are complex now, so I defined the angles $\alpha_{i,k}$ as $\alpha_{i,k}=\arccos(|\langle \psi_i|\psi_k\rangle|/\sqrt{\langle
\psi_i|\psi_i\rangle\langle \psi_k|\psi_k\rangle})$, and all angles were again rescaled according to $\alpha_{i,k}\longrightarrow \alpha_{i,k}/\langle \alpha \rangle$, where $\langle \alpha \rangle$ is the average angle for all pairs $i\ne k$. The resulting distribution depends for a small number $q$ of qubits still substantially on $q$. For $q=3$ a distribution is obtained that is remarkably close to the famous Wigner Dyson surmise for the orthogonal ensemble (\[WD\]), but for larger numbers of qubits the distribution approaches more and more a Gaussian, not too different from the distribution obtained from the Grover algorithm (see Fig. \[fig.pofphiG2\]). Again the distribution is stable in the range I examined it, $0.0001\le \epsilon\le 0.1$ . Thus, also the QFT shows hypersensitivity with respect to random perturbations as judged by the criterion of Schack et al., and in contrast to Peres’ original criterion!
On the other hand it is known that if the controlled phase gates are performed to a precision $\Delta=1/p(n)$ where $p$ is a polynomial in the number of qubits, then the maximum error of the final state $U_{QFT}|\psi\rangle$ for all input states $|\psi\rangle$ is of order $n^2/p(n)$ [@Nielsen00]. Thus, only polynomial precision is needed for the controlled phase gates. Coppersmith has shown, indeed, that an approximate QFT can be obtained by dropping the gates with the exponentially small phases altogether [@Coppersmith94].
[*In summary*]{} I have shown that both Grover’s search algorithm and the QFT give rise to the same unusual combination of quantum signatures of chaos and of integrability. Strong symmetries lead to a large degeneracy of the spectra of eigenvalues of the unitary matrices representing these algorithms. In fact, the QFT is a fourth root of the unity matrix, and Grover’s algorithm is to a good approximation a 6th root of unity! The corresponding lack of level repulsion would be commonly interpreted as absence of quantum chaos. The eigenvector statistics closely follows RMT predictions, and one would commonly interprete this as a signature of quantum chaos. However, as shown above, it is here but an artefact arising from the highly degenerate spectrum. The overlap between a random state propagated by a perturbed algorithm and the same state propagated by the corresponding unperturbed algorithm shows quasiperiodic oscillations both for Grover’s algorithm and the QFT, thus signaling absence of quantum chaos, in agreement with earlier studies addressing the stability of these codes. The only criterion indicating quantum chaos and not evidently explainable by an artefact is the distribution of angles between vectors propagated once by many slightly disturbed algorithms. After unfolding the angles universal distributions are obtained for large enough Hilbert spaces, both for Grover’s algorithm and for the QFT, that resemble the one for random vectors.
[*Acknowledgment:*]{} I would like to thank Henning Schomerus for a useful discussion. This work was supported by the Sonderforschungsbereich 237 “Unordnung und gro[ß]{}e Fluktuationen".
D. Deutsch, Proc. R. Soc. A [**400**]{}, 97 (1985); [*ibid.*]{}, [**425**]{}, 73 (1989). B. Georgeot and D. L. Shepelyansky, quant-ph/9909074 nad quant-ph/0005015. D. L. Shepelyansky, quant-ph/0006073. P. G. Silvestrov, H. Schomerus, and C. W. J. Beenakker, quant-ph/0012119. P. W.Shor, Phys. Rev. A [**52**]{}, 2493 (1995). A. M. Steane, Phys. Rev.Lett. [**77**]{}, 793 (1996). A. Peres in [*Quantum Chaos*]{}, ed. by H. A. Cerdeira, R. Ramaswamy, M. C. Gutzwiller, and G. Casati (World Scientific, Singapore, 1991). P. H. Song and D. L. Shepelyansky, Phys. Rev. Lett. [**86**]{}, 2162 (2001). B. Georgeot and D. L. Shepelyansky, Phys. Rev. Lett. [**86**]{}, 2890 (2001). L. K. Grover, Phys. Rev. Lett. [**79**]{}, 325 (1997). M. A. Nielsen and I. L. Chuang, [*Quantum Computation and Quantum Information*]{}, Cambridge University Press, Cambridge, UK (2000). P. W. Shor in [*Proc. 35th Annual Symp. on Foundations of Computer Science*]{} (Santa Fe, NM: IEEE Computer Society Press); see also quant-ph/9508027. O. Bohigas, M. J. Giannoni and C. Schmitt, Phys.Rev.Lett [**52**]{}, 1 (1984). M.V.Berry and M.Robnik, J.Phys.A: Math.Gen.[**17**]{}, 2413 (1984). No definitive proof of this so–called RMT–hypothesis is known, but overwhelming experimental and numerical evidence exists (see T. Guhr A. Müller–Gröling, H.A. Weidenmüller, Phys.Rep.[**299**]{}, 190 (1998)). Recently a strong analytical argument has been found (P.A.Braun, S. Gnutzmann, F. Haake, M. Kus, K. [Ż]{}yczkowski; nlin.CD/0006022). M.L. Mehta, [*Ramdom Matrices*]{}, 2nd edn. (Academic Press, New York, 1991). F. Haake, [*Quantum Signatures of Chaos*]{}, Springer, Berlin (1991). M. Boyer, G. Brassard, P. Hoyer, and A. Tapp, in [*Proceedings of the 4th Workshop on Physics and Computation — PhysComp’96*]{} (1996); quant-ph/9605034. R. Schack, G. M. D’Ariano, and C. M. Caves, Phys. Rev. E [**50**]{}, 972 (1994). L.K. Grover, Phys.Rev.Lett. [**80**]{} 4329 (1998). D. A. Lidar and O. Biham, Phys. Rev. E [**56**]{} 3661 (1997). Note that the original phases vary between $\pi/2$ and $\pi/2^{n-1}$, so perturbing with the same [*relative*]{} error is essential. D. Coppersmith, IBM Research Report RC 19642.
[^1]: This is not true, though, for the perturbed Grover algorithm using the ’digital’ form of perturbation; see below.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Let $\Gamma$ be a torsion-free hyperbolic group. We show that the set of solutions of any system of equations with one variable in $\Gamma$ is a finite union of points and cosets of centralizers if and only if any two-generator subgroup of $\Gamma$ is free.'
address: 'Université de Lyon; Université Lyon 1; INSA de Lyon, F-69621; Ecole Centrale de Lyon; CNRS, UMR5208, Institut Camille Jordan, 43 blvd du 11 novembre 1918, F-69622 Villeurbanne-Cedex, France'
author:
- Abderezak OULD HOUCINE
bibliography:
- 'biblio.bib'
title: 'One variable equations in torsion-free hyperbolic groups'
---
[^1]
[^2]
Introduction
============
Equations with one variable in free groups have been studied by Lyndon [@Lyndon-equa], Lorents [@Lorents1; @Lorents2], and Appel [@Appel-equation], among others, and the conclusion is that the set of solutions of a finite system of equations with one variable is a finite union of points and cosets of centralizers. However, Lorents announced his result without proof and the proof of Appel contains a gap [@baumslag-review]. In [@chis-rem], Chiswell and Remeslennikov gave a proof of this result, by using coordinate groups and Lyndon length functions in ultrapowers of free groups. In this paper we shall be concerned with a description of equations with one variable in a more larger class of groups.
\[thmmain\] Let $\Gamma$ be a nonabelian torsion-free hyperbolic group such that any two-generator subgroup of $\Gamma$ is free. Then the set of solutions of a system of equations with one variable in $\Gamma$ is a finite union of points and cosets of centralizers.
We notice that nonfree torsion-free hyperbolic groups whose two-generator subgroups are free exist. For instance by taking a nonfree hyperbolic group which is a limit group of free groups, we obtain such examples. The precedent theorem is no longer true if we drop the assumption that two-generator subgroups of $\Gamma$ are free, and in fact we have the following equivalence.
\[thmmain2\] Let $\Gamma$ be a nonabelian torsion-free hyperbolic group. Then the following properties are equivalent:
$(1)$ the set of solutions of a system of equations with one variable in $\Gamma$ is a finite union of points and cosets of centralizers;
$(2)$ any two-generator subgroup of $\Gamma$ is free.
As a consequence of Theorem \[thmmain\] is that any quantifier-free formula in a torsion-free hyperbolic group satisfying the hypothesis of the theorem, is a boolean combination of cosets of centralizers. It follows in particular that any proper subgroup of a torsion-free hyperbolic group, under the hypothesis of the theorem, which is definable by a quantifier-free formula, is cyclic. However this property is steal true in any torsion-free hyperbolic group and has a simple proof (see the end of the appendix).
Our approach to prove Theorem \[thmmain\] is to use coordinate groups of varieties as in [@chis-rem], and the structure of restricted $\Gamma$-limit groups obtained from Sela’s work on limit groups of torison-free hyperbolic groups [@Sela-hyp]. In the next section we prove the main result, while the appendix is devoted to the proof of an intermediate result on the structure of restricted $\Gamma$-limit groups.
Equations with one variable
===========================
Let $G$ be a group. A *$G$-group* $H$ is a group having an isomorphic fixed copy of $G$ which we will identify with $G$. A homomorphism $h : H_1
\rightarrow H_2$ between two $G$-groups is called a *$G$-homomorphism* if for any $g \in G$, $h(g)=g$. A *$G$-isomorphism* is defined analogously and we use the notation $H_1 \cong_G H_2$.
If $X$ is set, we denote by $G[X]$ the free product $G*F(X)$ where $F(X)$ is the free group on $X$. If $X=\bar x=\{x_1, \dots, x_n\}$ we use the notation $G[\bar
x]$.
For an element $w(\bar x) \in G[\bar x]$ and a tuple $\bar g
=(g_1, \dots, g_n)\in G^n$ we denote by $w(\bar h)$ the element of $G$ obtained by replacing each $x_i$ by $g_i$ ($1 \leq i \leq n$). A *variety* in $G^n$ is a set of the form $$V(S)=\{\bar g \in G^n ~|~w(\bar g)=1 \hbox{ in $G$ for all }w \in S\},$$ for some $S \subseteq G[ \bar x]$. For any $S \subseteq G[ \bar x]$ we use the notation $S(\bar x)=1$ as an abreviation for the system of equations $\{w(\bar x)=1| w\in S\}$. The group $G$ is called *equationally noetherian* if for any $n \geq 1$ and any subset $S$ of $G[\bar x]$ there exists a finite subset $S_0
\subseteq S$ such that ${V(S)=V(S_0)}$. A subset of $G^n$ is *closed* if it is the intersection of finite union of varieties. This defines a topology on $G^n$, called the *Zariski topology*. Then $G$ is equationally noetherian if and only if for any $n \geq 1$, the Zariski topology on $G^n$ is noetherian [@Baum-Rem-algebraic].
If $G$ is equationally noetherian, then for any variety $V(S)$ in $G^n$ one associates to it its irreducibles components, which are also varieties. For more details on these notions we refer the reader to [@Baum-Rem-algebraic].
[@Sela-hyp Theorem 1.22] states that any system of equations (without parameters) in finitely many variables is equivalent in a trosion-free hyperbolic group to a finite subsystem. This property is equivalent, when the group under consideration $G$ is finitely generated, to the fact that $G$ is equationally noetherian. Hence, we have the following.
[@Sela-hyp Theorem 1.22] A torsion-free hyperbolic group is equationally noetherian.
It follows that the set of solutions of a system of equations is a finite union of irreducible varieties, and thus the study of such sets is reduced to the study of irreducible varieties. This section is devoted to prove the following theorem which is a generalization of [@chis-rem Theorem 5.3].
\[thm1\] Let $\Gamma$ be a nonabelian torsion-free hyperbolic group such that any two-generator subgroup of $\Gamma$ is free. If $V$ is a proper nonempty irreducible variety in $\Gamma$ then either $V$ is a singleton or $V$ is a coset of a centralizer.
Theorem \[thmmain\] is a mere consequence of Theorem \[thm1\]. Concerning quantifier-free formulas, since $\Gamma$ is a nonabelian CSA-group, there exist two elements $c, d \in \Gamma$ such that $C_\Gamma(c,d)=C_\Gamma(c) \cap C_\Gamma(d)=1$, and thus we can write $V=aC_\Gamma(c) \cap aC_\Gamma(d)$ when $V=\{a\}$.
Let $G$ be a group and $S$ a subset of $G[x_1, \dots, x_n]$. We let $$G_S(\bar x)={\langle}G[x_1,\dots, x_n]| w(x_1, \dots, x_n)=1, \;\; w\in S{\rangle},$$ and we let $$\bar S=\{w(\bar x) \in G[\bar x] | G \models \forall \bar x (S(\bar x)=1 \Rightarrow w(\bar w)=1)\}.$$
The group $G_{\bar S}(\bar x)$ is called the *coordiante group* associated to $S$ or to $V(S)$. We notice that for any $w \in G[\bar x ]$, $G_{\bar S}(\bar x) \models w(\bar x)=1$ if and only if $w \in \bar S$.
In order to prove the above theorem we shall need Lemma \[lem-reduction\] below, which connects the structure of a variety to the structure of its coordiante group. First, we prove the following technical proposition of independent interest.
\[lem-hnn\] Let $G$ be a group, $H$ a subgroup of $G$ and suppose that $G$ is generated by $H \cup\{s\}$ for some $s \in G$.
$(1)$ Let $G={\langle}H, t| [A,t]=1{\rangle}$ and suppose that:
$(i)$ $A$ is malnormal in $H$;
$(ii)$ for any $u \in H$ either $u \in A$ or ${\langle}u,A{\rangle}$ is the natural free product ${\langle}u{\rangle}*A$.
Then there exist $h_1, h_2 \in H$ such that $s^{\pm 1}=h_1th_2$.
$(2)$ Let $G={\langle}H, t| A^t=B{\rangle}$ and suppose that:
$(i)$ $A$ and $B$ are malnormal in $H$ and $G$ is seperated;
$(ii)$ for any $u \in H$ either $u \in A$ or ${\langle}u,A{\rangle}$ is the natural free product ${\langle}u{\rangle}*A$ and similarly for $B$.
Then there exist $h_1, h_2 \in H$ such that $s^{\pm 1}=h_1th_2$.
$(3)$ Let $G=H*_{A=B}K$ and suppose that:
$(i)$ $A$ is malnormal in $H$ and $B$ is malnormal in $K$;
$(ii)$ for any $u \in H$ either $u \in A$ or ${\langle}u,A{\rangle}$ is the natural free product ${\langle}u{\rangle}*A$, and similarly for $B$ in $K$.
Then there exist $h \in H, k \in K$ such that $s^{\pm 1}=hk$.
$\;$
$(1)$ If $v(\bar x,y)$ is a word in the free group with basis $\bar x \cup\{y\}$, we denote by $exp_y(v)$ the exponent sum of $y$ in $v$.
Let $s'$ be a cyclically reduced conjugate of $s$. Since $H \cup\{s\}$ generates $G$, there is a word $w(\bar x, y)$ such that $t=w(\bar h, s)$ for some tuple $\bar h$ of $H$. By the abelianization of $G$, we have $exp_t(w(\bar h, s))=exp_t(s')exp_s(w)$. Therefore $exp_t(s')=\pm 1$. Hence the number of occurrences of $t$ in $s'$ is odd. We have $s=s'^g$ for some $g \in G$. We claim now that $g \in H$ and $|s'|=1$, where $|.|$ denotes the length of normal forms.
Suppose towards a contradiction that $|s'| \geq 1$ and thus $|s'|\geq 3$. Using the fact that $exp_t(s')=\pm 1$ and the malnormality of $A$, a simple count shows that $$|s^2|>|s|, \leqno (1)$$ and using also calculations with normal forms, we get for any $h,h' \in H$, with $h \neq 1, h'\neq 1$, that $$|hs^{\pm 1}h's^{\pm 1}|> |hs^{\pm 1}|, |h's^{\pm 1}|. \leqno (2)$$
Using (1) and (2), by [@AHoucine Lemma 4.2], we get that or any sequence $h_1, \dots, h_n$ of nontrivial elements of $H$, for any sequence $\varepsilon_0, \dots, \varepsilon_{n}$ of $\mathbb Z$, $\varepsilon_i \neq 0$, $$|s^{\varepsilon_0}h_1 s^{\varepsilon_1}h_2 \cdots h_ns^{\varepsilon_{n}}|\geq |s|>1,$$ and thus $t \not \in {\langle}H,s{\rangle}$; which is a contradiction.
Therefore $|s'|=1$ and we write $s'=ut^{\varepsilon}v$, where $\varepsilon=\pm 1$. To simplify, we may assume that $\varepsilon=1$. Write $g = h_0t^{\varepsilon_1}h_1\cdots t^{\varepsilon_n}h_n$ in normal form. Replacing $s$ by $h_n^{-1}sh_n$ and $s'$ by $h_1^{-1}utvh_1$ we may assume without loss of generality that $h_0=h_n=1$.
We claim now that $h_1, \dots, h_{n-1} \in A$. Suppose that for some $i$, $h_i \in A$. Then proceeding as above, a simple count with normal forms, shows that for any $h,h' \in H$, with $h \neq 1, h'\neq 1$, that $|hs^{\pm 1}h's^{\pm 1}|> |hs^{\pm 1}|, |h's^{\pm 1}|$, and we get a contradiction as above by [@AHoucine Lemma 4.2].
Hence we get $g=a t^p$ for some $p \in \mathbb Z$ and $a \in A$. Replacing again $u$ be $a^{-1}u$ and $v$ by $va$, we may assume that $a=1$. Hence $s=t^{-p}utvt^p$.
We claim that either $v \in A$ or $u \in A$. Suppose that $v \not \in A$ and $u \not \in A$. Then proceeding as above, we see also that a simple calculation with normal forms, shows that for any $h,h' \in H$, with $h \neq 1, h'\neq 1$, that $|hs^{\pm 1}h's^{\pm 1}|> |hs^{\pm 1}|, |h's^{\pm 1}|$, which is a contradiction by [@AHoucine Lemma 4.2].
Hence $v \in A$ or $u \in A$. We treat only the case $v \in A$, the other case being similar. Replacing again $u$ by $ua$, we may assume that $v=1$. Therefore $s= t^{-p}ut^{p+1}$. Clearly by $(ii)$ we have ${\langle}u,A{\rangle}={\langle}u{\rangle}*A$.
We claim that for any sequence $h_1, \dots, h_n$ of nontrivial elements of $H$, for any sequence $\varepsilon_0, \dots, \varepsilon_{n}$ of $\mathbb Z$, $\varepsilon_i \neq 0$, the normal form of the product $$s^{\varepsilon_0}h_1 s^{\varepsilon_1}h_2 \cdots h_ns^{\varepsilon_{n}},$$ is of the form $$t^{\delta_1}d_1 \cdots t^{\delta_p}d_pt^{q},$$ where $\delta_i=\pm 1$, $q \in \{p, p+1\}$, $d_i \in H$, and $d_p \in {\langle}u,A{\rangle}$ with the property that the last element of the normal form of $d_p$, with respect to the structure ${\langle}u,A{\rangle}={\langle}u{\rangle}*A$, is $u^{\pm 1}$.
The proof is by induction on $n$ and the detailled verification is left to the reader.
Hence we conclude that for any sequence $h_1, \dots, h_n$ of nontrivial elements of $H$, for any sequence $\varepsilon_0, \dots, \varepsilon_{n}$ of $\mathbb Z$, $\varepsilon_i \neq 0$, $$|s^{\varepsilon_0}h_1 s^{\varepsilon_2}h_2 \cdots h_ns^{\varepsilon_{n}}|\geq 2,$$ and thus $t \not \in {\langle}H,s{\rangle}$; a final contradiction.
\(2) This case is similar to (1). Proceeding as above, we conclude that $s=g^{-1}utvg$, and we suppose that $g \not \in H$. Then, as before we may assume that $g=t^{\varepsilon_1}h_1 \cdots h_{n_1}t^{\varepsilon_n}$. At this stage, by using the fact that $G$ is seperated, we get $g=t^{\pm 1}$ and we assume without loss of generality that $g=t$. Hence $s=t^{-1}utvt$. Then, as above, we may assume that $v \in B$ and, without loss of generality $v=1$ and thus $s=t^{-1}ut^2$. Then as before, for any sequence $h_1, \dots, h_n$ of nontrivial elements of $H$, for any sequence $\varepsilon_0, \dots, \varepsilon_{n}$ of $\mathbb Z$, $\varepsilon_i \neq 0$, the normal form of the product $$s^{\varepsilon_0}h_1 s^{\varepsilon_2}h_2 \cdots h_ns^{\varepsilon_{n}},$$ is of the form $$t^{\delta_1}d_1 \cdots t^{\delta_p}d_pt^{q},$$ where $\delta_i=\pm 1$, $q \geq 1$, $d_i \in H$, and $d_p \in {\langle}u,A{\rangle}$ with the property that the last element of the normal form of $d_p$, with respect to the structure ${\langle}u,A{\rangle}={\langle}u{\rangle}*A$, is $u^{\pm 1}$. The proof is by induction on $n$ and the detailled verification is left to the raider. We conclude that $t \not \in {\langle}H,s{\rangle}$; a final contradiction.
\(3) This case is also similar to (1) and (2). Write $s =y_1y_2 \dots y_n$ in normal form. We claim that $n \leq 2$.
Suppose first by contradiction that $n \geq 4$. Using calculations with normal forms, we find that $| h y^{\pm 1} h ' y^{\pm 1}|>|h y^{\pm 1}|, |h' y^{\pm 1}|$ for any $h,h'$ with $h \neq 1$ and $h' \neq 1$. Hence by [@AHoucine Lemma 4.2], for any nontrivial elements $h_1, \dots, h_n \in H$, for any sequence $p_1, \dots, p_n$ of $\mathbb Z$, $p_i \neq 0$, we have $$|h_1 y ^{p_1} \cdots h_n y^{p_n}|>|y|>1,$$ which is clearly a contradiction. Therefore $n \leq 3$. Suppose that $n=3$. We treat only the case $y_1 \in H$, the other case being similar.
We claim now that for any sequence $h_1, \dots, h_n$ of nontrivial elements of $H$, for any sequence $\varepsilon_0, \dots, \varepsilon_{n}$ of $\mathbb Z$, $\varepsilon_i \neq 0$, the normal form of the product $$s^{\varepsilon_0}h_1 s^{\varepsilon_2}h_2 \cdots h_ns^{\varepsilon_{n}},$$ is of the form $$d_1 \cdots d_p d_{p+1},$$ where $ p \geq 2$, $d_{p+1} \in \{y_1^{\pm1}, y_3^{\pm 1}\}$, and $d_p \in {\langle}y_2,B{\rangle}$ with the property that the last element of the normal form of $d_p$, with respect to the structure ${\langle}y_2,B{\rangle}={\langle}y_2{\rangle}*B$, is $y_2^{\pm 1}$. Which is a contradiction.
\[lem-reduction\] Let $G$ be a group and $S \subseteq G[x]$.
$(1)$ If $G_{\bar S}(x) \cong_G G$ then $V(S)$ is a singleton.
$(2)$ If $G_{\bar S}(x) \cong_G G*\mathbb Z$ then $V(S)=G$.
$(3)$ If $G_{\bar S}(x) \cong_G {\langle}G,t | [A,t]=1{\rangle}$, where $A$ is a nontrivial malnormal cyclic subgroup of $G$, and $G$ satisfies the property that any two-generator subgroup of $G$ is free, then $V(S)=uC_G(A)^v$ for some $u,v \in G$.
$(1)$ Let $h : G_{\bar S}(x) \rightarrow G$ be a $G$-isomorphism. Then $h(x) \in G$ and thus $x \in G$. Hence $xg^{-1} \in \bar S$ for some $g \in G$. Therefore $V(S)=\{g\}$.
$(2)$ Let $h : G_{\bar S}(x) \rightarrow G*\mathbb Z$ be a $G$-isomorphism. Clearly $h(x) \not \in G$. Hence the subgroup ${\langle}G, h(x){\rangle}$ is the natural free product $ G*{\langle}h(x){\rangle}$. Therefore if $G_{\bar S}(x) \models w(x)=1$, where $w \in S$, then $G*{\langle}t|{\rangle}\models w(t)=1$. Hence $G \models \forall t w(t)=1$ and thus $V(S)=G$.
$(3)$ Let $h : G_{\bar S}(x) \rightarrow {\langle}G, t |[A,t]=1{\rangle}$ be a $G$-isomorphism. Since $G \cup\{h(x)\}$ generates the HNN-extension under consideration, $h(x)=u_0t^\varepsilon v$ where $ \varepsilon =\pm 1$ and $u_0,v \in G$ by Proposition \[lem-hnn\]. Without loss of generality we may assume that $\varepsilon =1$.
We claim that $V(S)=uC_G(A)^v$, where $u=u_0v$. Let $g \in G$ be a solution of the system $S(x)=1$. Then there exists a $G$-homomorphism $f : G_{\bar S}(x) \rightarrow G$ such that $f(x)=g$. Hence $g=u_0f(t)v$. Since $[t,a]=1$ for all $a \in A$ we get $[f(t),a]=1$ for all $a \in A$ and thus $g \in uC_G(A)^v$.
We have $G_{\bar S}(x)={\langle}G, x | x=u_0tv, [A,t]=1{\rangle}$. Hence for any $w \in S$, $w(x)=1$ is a consequence of the precedent presentation. Let $g \in uC_G(A)^v$. Then, using the precedent presentation, there exists a $G$-homomorphism $f : G_{\bar S}(x) \rightarrow G$ such that $f(x)=g$. Hence for any $w \in S$, $w(g)=1$, by the precedent observation. We conclude that $V(S)=uC_G(A)^v$ and this terminates the proof.
Using Lemma \[lem-reduction\], the proof of Theorem \[thm1\] is reduced to the proof of the following theorem which is a generalization of [@chis-rem Theorem 5.1].
\[thm2\]Let $\Gamma$ be a nonabelian torsion-free hyperbolic group such that any two-generator subgroup of $\Gamma$ is free. The coordinate group $\Gamma_{\bar S}(x)$ of the nonempty irreducible variety $V(S)\subseteq \Gamma$ satisfies one of the following:
$(1)$ $\Gamma_{\bar S}(x) \cong_\Gamma \Gamma$;
$(2)$ $\Gamma_{\bar S}(x) \cong_\Gamma \Gamma*\mathbb Z$;
$(3)$ $\Gamma_{\bar S}(x) \cong_\Gamma {\langle}\Gamma, t | [u, t] = 1{\rangle}$, for some nontrivial element $u$ in $\Gamma$.
The remainder of this section is devoted to prove Theorem \[thm2\]. In the sequel we let $\Gamma$ to be a fixed nonabelian torsion-free hyperbolic group.
$\;$
$\bullet$ A sequence of homomorphisms $(f_n)_{n \in \mathbb N}$ from $H$ to $\Gamma$ is called *stable* if for any $h \in H$ either $f_n(h)=1$ for all but finitely many $n$, or $f_n(h) \neq 1$ for all but finitely many $n$. The *stable kernel* of $(f_n)_{n \in \mathbb N}$, denoted $Ker_{\infty}((f_n)_{n \in \mathbb N})$, is the set of elements $h \in H$ such that $f_n(h)=1$ for all but finitely many $n$.
$\bullet$ A *restricted $\Gamma$-limit group* is a $\Gamma$-group $G$ such that there exists a $\Gamma$-group $H$ and a stable sequence of $\Gamma$-homomorphisms $(f_n)_{n \in \mathbb N}$ from $H$ to $\Gamma$ such that $G=H/ Ker_{\infty}((f_n)_{n \in \mathbb N})$.
The proof of the following lemma is straightforward and it is left to the reader.
\[lem1\] Let $S \subseteq \Gamma[x]$ such that $V(S)$ is irreducible and nonempty.
$(1)$ The group $\Gamma_{\bar S}(x)$ is a $\Gamma$-group and for any finite subset $A$ of $\Gamma_{\bar S}(x)$ such that $1 \not \in A$ there exists a $\Gamma$-homomorphism $f : \Gamma_{\bar S}(x) \rightarrow \Gamma$ such that $1 \not \in f(A)$.
$(2)$ Let $\varphi : \Gamma_{\bar S}(x) \rightarrow L$ be a $\Gamma$-epimorphism where $L$ is a restricted $\Gamma$-limit group. Then there exists $U \subseteq \Gamma[x]$ such that $V(U)$ is irreducible and nonmepty and $L=\Gamma_{\bar U}(x)$.
It follows in particular, by Lemma \[lem1\](1), that if $V(S)$ is irreducible and nonempty then $\Gamma_{\bar S}(x)$ is a restricted $\Gamma$-limit group. Lemma \[lem1\](1) implies also that $\Gamma$ is existentially closed in $\Gamma_{\bar S}(x)$.
[@Champ-Guirardel Definition 4.16] Let $G$ be a group which is the fundamental group of a graph of groups $\Lambda$. Let $H$ be a nontrivial elliptic subgroup of $G$ with respect to $\Lambda$. The *elliptic abelian neighbourhood* of $H$ is the subgroup $\hat H$ generated by all the elliptic elements of $G$ which commute with a nontrivial element of $H$.
\[weakly-constructible\] A restricted $\Gamma$-limit group $G$ is said *weakly constructible* if one of the following cases holds:
$(1)$ $G=H*_CK$, where $\Gamma \leq H$ and $C$ is a notrivial cyclic group and $K$ is noncyclic;
$(2)$ $G={\langle}H, t| C_1^t=C_2{\rangle}$, $\Gamma \leq H$ and $C_1$ is a notrivial cyclic group, and there exists a proper quotient restricted $\Gamma$-limit group $L$ of $G$ where the corresponding $\Gamma$-epimorphism $\varphi : G \rightarrow L$ is one-to-one in restriction to the elliptic abelian neighbourhood of $H$.
We will use the following theorem, which is sufficient for our purpose, and whose proof proceeds in a similar way to that of [@Sela-Diophan1; @Champ-Guirardel]. For completeness, the proof is given in the appendix.
\[thm-princip2\] Let $G$ be a restricted $\Gamma$-limit group. If $G$ is not $\Gamma$-isomorphic to $\Gamma$ and if it is freely indecomposable relative to $\Gamma$ then $G$ is weakly constructible.
**Proof of Theorem \[thm2\].**
Let $S \subseteq \Gamma[x]$ such that $V(S)$ is irreducible and nonempty. We may assume that $\Gamma_{\bar S}(x)$ is not $\Gamma$-isomorphic to $\Gamma$ and it is freely indecomposable relative to $\Gamma$. By Lemma \[lem1\], every proper quotient of $\Gamma_{\bar S}(x)$, which is a restricted $\Gamma$-limit group, is of the form $\Gamma_{\bar U}(x)$. Hence, by the descending chain condition on $\Gamma$-limit groups, we may assume that the theorem holds for all proper quotients of $\Gamma_{\bar S}(x)$ which are restricted $\Gamma$-limit groups. By Theorem \[thm-princip2\], we treat the two cases (1) and (2) of Definition \[weakly-constructible\].
Since any two-generator subgroup of $\Gamma$ is free, we have the following claim whose proof proceeds in a similar way to that of [@Champ-Guirardel Claim 4.25] and it is left to the reader.
Claim 1. *Let $G$ be a $\Gamma$-limit group and $T_1, T_2$ two abelian subgroups of $G$. Then either ${\langle}T_1,T_2{\rangle}$ is abelian or ${\langle}T_1, T_2{\rangle}=T_1*T_2$.*
We now prove the following claim.
Claim 2. *Suppose that $\Gamma_{\bar V}(x)={\langle}H, t|C_1^t=C_2{\rangle}$, where $\Gamma \leq H$ and $C_1$ is cyclic, is a restricted $\Gamma$-limit group which is freely indecomposable relative to $\Gamma$ and such that:*
$(i)$ *there exists a proper quotient restricted $\Gamma$-limit group $L$ of $\Gamma_{\bar V}(x)$ where the corresponding $\Gamma$-epimorphism $\varphi : \Gamma_{\bar V}(x) \rightarrow L$ is one-to-one in restriction to the elliptic abelian neighbourhood of $H$;*
$(ii)$ *every proper restricted $\Gamma$-limit group quotient of $\Gamma_{\bar V}(x)$ satisfies the conclusion of the theorem.*
*Then $H$ is $\Gamma$-isomorphic to $\Gamma$.*
Since $\Gamma_{\bar V}(x)$ is a CSA-group either $C_1$ or $C_2$ is malnormal in $H$. We treat the case $C_1$ is malnormal in $H$, the other case being similar.
Let $D=C_H(C_2)$. We make the following two assumptions:
$(a)$ $C_1$ and $C_2$ are not conjugate in $H$;
$(b)$ $D$ is noncyclic;
and we show that we obtain a contradiction.
By $(a)$ the HNN-extension is seperated, and thus, since $C_1$ is malnormal, we have $tDt^{-1}=C_{\Gamma_{\bar V}(x)}(C_1)$ and $D= C_{\Gamma_{\bar V}(x)}(C_2)$. By putting $D'=tDt^{-1}$, and since $\hat H={\langle}H, \hat C_1, \hat C_2{\rangle}={\langle}H, D'{\rangle}$, we get $$\hat H=H*_{C_1}D', \; \Gamma_{\bar V}(x)={\langle}\hat H, t | D'^t=D{\rangle}.$$
We notice that $D$ and $D'$ are steal not conjugate in $\hat H$. By construction $D$ and $D'$ are malnormal in $\hat H$. Hence, by Proposition \[lem-hnn\], $\Gamma_{\bar V}(x)$ is generated by $\Gamma \cup \{k_1tk_2\}$ for some $k_1, k_2 \in \hat H$. We replace $D'$ by $U=D'^{k_1^{-1}}$, $D$ by $V=D^{k_2}$ and $t$ by $r=k_1tk_2$, and thus we get $$\Gamma_{\bar V}(x)={\langle}\hat H, r| U^r=V{\rangle},$$ and $\Gamma_{\bar V}(x)$ is generated by $\Gamma \cup \{r\}$. Using normal forms, we conclude that $\hat H={\langle}\Gamma, U, V{\rangle}$. We notice also that $U$ and $V$ are steal not conjugate in $\hat H$. Using normal forms, we conclude that either $U \cap \Gamma \neq 1$ or $V \cap \Gamma \neq 1$. Without loss of generality we assume that $U \cap \Gamma \neq 1$.
Clearly $L$ is freely indecomposable with respect to $\Gamma$. If $L$ is $\Gamma$-isomorphic to $\Gamma$ then, since $\varphi$ is one-to-one in restriction to $H$, we get the required conclusion. Hence we assume that $L={\langle}\Gamma, s|u^s=u{\rangle}$.
Since $L$ is generated by $\Gamma \cup\{\varphi(r)\}$, by Proposition \[lem-hnn\], $\varphi(r)=\gamma_1s^{\varepsilon}\gamma_2$ for some $\gamma_1, \gamma_2 \in \Gamma$, $\varepsilon =\pm 1$, and without loss of generality we assume that $\varepsilon =1$.
We claim that $U^{\gamma} \leq {\langle}u,s{\rangle}$ for some $\gamma \in \Gamma$. Since $U \cap \Gamma \neq 1$, we let $\gamma_0 \in U \cap \Gamma$. Then $U \leq C_L(\gamma_0)$ and since $U$ is noncyclic, we conclude that $C_L(\gamma_0)={\langle}u,s{\rangle}^{\gamma}$ for some $\gamma \in \Gamma$; and we obtain the required conclusion.
Replacing $U$ by $U^\gamma$,we assume that $\gamma=1$. Replacing also $V$ by some of its conjugates, we assume that $\gamma_2=1$. Hence we conclude $$U={\langle}u, s^p{\rangle}, \; p \in \mathbb Z, \; \; V=U^{\gamma s}.$$
Suppose towards a contradiction that $p \neq \pm 1$. We claim that $\hat H={\langle}\Gamma, U{\rangle}*V$, which gives $$\Gamma_{\bar V}(x)={\langle}\Gamma, U{\rangle}*_{U=V^t}{\langle}V^t,t{\rangle}=B* \mathbb Z, \; \Gamma \leq B,$$ which is a contradiction.
Celarly ${\langle}u, \gamma{\rangle}$ is free of rank 2, as otherwise we obtain that $U$ and $V$ are conjugate in $\hat H$; which is a contradiction. It follows that ${\langle}\Gamma, V{\rangle}=\Gamma*V$.
Clearly we also have ${\langle}\Gamma, U{\rangle}={\langle}\Gamma, s^p| u^{s^p}=u{\rangle}$. Since $p \neq \pm 1$, we get that the length $$|\gamma_1 s^{\varepsilon _1 p}\gamma_2 s^{\varepsilon_2p} \cdots \gamma_n s^{\varepsilon_np}\gamma_{n+1}. s^{-1}\gamma^{-1}d \gamma s|$$ is greater than $$|\gamma_1 s^{\varepsilon _1 p}\gamma_2 s^{\varepsilon_2p} \cdots \gamma_n s^{\varepsilon_np}\gamma_{n+1}|, \; | s^{-1}\gamma^{-1}d \gamma s|,$$ for any reduced sequence $\gamma_1 s^{\varepsilon _1 p}\gamma_2 s^{\varepsilon_2p} \cdots \gamma_n s^{\varepsilon_np}\gamma_{n+1}$ of ${\langle}\Gamma, U{\rangle}$ and for any nontrivial element $d$ of $U$. Thus by [@AHoucine Lemma 4.2], we get the required contradiction.
Therefore $p=\pm 1$ and thus $U={\langle}u,s{\rangle}$. But this implies $\varphi(H)=L$ and thus $U$ and $V$ are conjugate in $\hat H$; which is also a contradiction.
So finally we conclude that one of the following two cases holds:
$(1)$ $C_1$ and $C_2$ are conjugate in $H$;
$(2)$ $C_H(C_1)$ and $C_H(C_2)$ are cyclic.
We claim that in each case we have $$\Gamma_{\bar V}(x)={\langle}{\langle}\Gamma, c{\rangle}, t| c'^t=c{\rangle},$$ where $c' \in \Gamma$ and $H={\langle}\Gamma, c{\rangle}$.
Suppose that $(1)$ holds. By rewritting the HNN-extension, we may assume that $C_1=C_2$. By Proposition \[lem-hnn\], $\Gamma_{\bar V}(x)$ is generated by $h_1th_2$ for some $h_1, h_2 \in H$. Hence $$\Gamma_{\bar V}(x)={\langle}H, s| D_1^s=D_2{\rangle},$$ where $s=h_1th_2$, $D_1=C_1^{h_1^{-1}}$, $D_2=C_2^{h_2}$. Using normal forms we conclude that $H={\langle}\Gamma, D_1, D_2{\rangle}$.
Let $d_1$ (resp. $d_2$) generates $D_1$ (resp. $D_2$). We claim that either $d_1 \in \Gamma$ or $d_2 \in \Gamma$. Since $d_1$ can be written as a word on $\Gamma \cup \{s\}$, using normal forms we get $d_1^n \in \Gamma$ or $d_2^n \in \Gamma$ for some $n \in \mathbb Z$, $n\neq 0$. Suppose that $d_1^n \in \Gamma$ for some $n \in \mathbb Z$, $n\neq 0$. Since $\Gamma$ is existentially closed in $\Gamma_{\bar V}(x)$ (Lemma \[lem1\](1)), there exists $\gamma \in \Gamma$ such that $\gamma^n=d_1^n$. Since $G$ is torsion-free and commutative transitive, we get $d_1=\gamma$ as claimed. Therefore $$\Gamma_{\bar V}(x)={\langle}{\langle}\Gamma, c{\rangle}, t| c'^t=c{\rangle},$$ where $c' \in \Gamma$ and $H={\langle}\Gamma, c{\rangle}$ as required.
Now suppose that $(2)$ holds with $(1)$ does not hold. Proceeding as before, we have $$\Gamma_{\bar V}(x)={\langle}\hat H, r| U^r=V{\rangle},$$ where in this case $U$ and $V$ are cyclic and malnormal, and the HNN-extension is seperated. Proceeding as above we conclude that $\hat H={\langle}\Gamma, U, V{\rangle}$ and also that $U \leq \Gamma$ or $V \leq \Gamma$, and without loss of generality we assume that $U \leq \Gamma$.
We claim that $H=\hat H$. Since $\Gamma_{\bar V}(x)$ is a CSA-group, either $C_1={\langle}c_1{\rangle}$ is malnormal or $C_2={\langle}c_2{\rangle}$ is malnormal in $H$. We suppose that $C_1$ is malnormal the other case can be treated similalrly. Without loss of generality, after conjugation, we may also suppose that $C_1\leq U$. Since $V$ is cyclic we get $c_2=d_1^p$ for some $p \in \mathbb Z$. Hence, proceeding as above, since $c_1$ and $c_2$ are conjugate and $c_1 \in \Gamma$ we conclude that $p=\pm 1$. Therefore $C_2$ is also malnormal. Finally we conclude that $$\Gamma_{\bar V}(x)={\langle}{\langle}\Gamma, c{\rangle}, t| c'^t=c{\rangle},$$ where $c' \in \Gamma$ and $H={\langle}\Gamma, c{\rangle}$ as required.
Hence in each case, we have $$\Gamma_{\bar V}(x)={\langle}{\langle}\Gamma, c{\rangle}, t| c'^t=c{\rangle},$$ where $c' \in \Gamma$ and $H={\langle}\Gamma, c{\rangle}$. By Lemma \[lem1\](2), $L=\Gamma_{\bar U}(x)$ for some $U \subseteq \Gamma[x]$. Clearly $\Gamma_{\bar U}(x)$ is freely indecomposable relative to $\Gamma$. If $\Gamma_{\bar U}(x)$ is $\Gamma$-isomorphic to $\Gamma$ then we get the required conclusion as $\varphi$ is one-to-one in restriction to $H$.
Therefore $\Gamma_{\bar U}(x)={\langle}\Gamma, s|u^s=u{\rangle}$. We claim that $\varphi(c) \in \Gamma$ and this will ends the proof. We steal denote by $c$ the image of $c$ in $\Gamma_{\bar U}(x)$. Recall that $
\Gamma_{\bar S}(t)={\langle}{\langle}\Gamma, c{\rangle},t| c'^t=c{\rangle},
$ and thus $c'^{\varphi(t)}=c$. Set $t'=\varphi(t)$. Since $\Gamma \cup\{t'\}$ generates $\Gamma_{\bar U}(x)$, without loss of generality, $t'=\gamma_1s\gamma_2$ for some $\gamma_1,\gamma_2 \in \Gamma$ by Proposition \[lem-hnn\].
Replacing $s$ by $s^{\gamma_2}$ and $u$ by $u^{\gamma_2}$ we may assume that $\gamma_2=1$ and we write $t'=\gamma s$. Therefore $c= s^{-1}\gamma^{-1}c' \gamma s$. It follows that ${\langle}\Gamma,c{\rangle}=\Gamma*_{u=u^s }{\langle}u^s, c{\rangle}$.
By Claim 1, ${\langle}u^s, c{\rangle}$ is either free of rank 2 or abelian. If ${\langle}u^s, c{\rangle}$ is free of rank $2$, then $\Gamma_{\bar S}(x)$ will be freely decomposable with respect to $\Gamma$; a contradiction to our assumption. If ${\langle}u^s, c{\rangle}$ is abelian then $[s,c]=1$ and thus $c= \gamma^{-1}c' \gamma \in \Gamma$ as claimed. This ends the proof of the claim.
Now we treat the two cases of Definition \[weakly-constructible\].
Case (1). Let $G=H*_CK$ be the given splitting with $K$ is noncyclic and $\Gamma \leq H$. Since $G$ is a CSA-group, $C$ is malnormal either in $H$ or $K$. We claim that $K$ is abelian. Let $$H'=H*_CC', K'= K, C'=C_K(C),$$ whenever $C$ is malnormal in $H$ and $$H'=H, K'=C'*_CK, C'=C_H(C),$$ whenever $C$ is malnormal in $K$. We get $G=H'*_{C'}K'$ with $\Gamma \leq H'$ and $C'<K'$ is malnormal in both $H'$ and $K'$. By Proposition \[lem-hnn\], and without loss of generality, $x=hk$ where $h \in H'$ and $k \in K'$.
Let $v \in K$. Then $v$ can be written as a reduced word on $\Gamma \cup\{hk\}$. By reducing this word with respect to the structure of the free product with amalgamation, we get $v \in {\langle}k,C'{\rangle}$ and thus $K'={\langle}k, C'{\rangle}$.
By Claim 1, either $K'$ is abelian or $K'=C'*{\langle}k{\rangle}$. Clearly the later case is impossible as otherwise $\Gamma_{\bar S}(x)$ will be freely decomposable relative to $\Gamma$; a contradiction with our assumption.
Therefore $K'$ is abelian and in particular $K$ is abelian as claimed. Since $C$ has an infinite index in $K$ we can write $K=C_0 \times C_1 \times \cdots \times C_n$, where $C_0={\langle}t_0{\rangle}$, $C={\langle}c{\rangle}$ with $c=t_0^p$ for some $p \in \mathbb Z$, $n \geq 1$ and each $C_i$ is cyclic. We let, for $0 \leq i \leq n$, $$L_0=H*_CC_0, \;L_1={\langle}L_0, t_1| C_0^{t_1}=C_0{\rangle}, \;$$$$L_i={\langle}L_{i-1}, t_{i}| (C_0\times \dots \times C_{i-1})^{t_i}=(C_0\times \dots \times C_{i-1}){\rangle}.$$
We see that each $L_{i}$ is a proper quotient of $L_{i+1}$ and $\Gamma_{\bar S}(x)=L_n$. Hence by induction each $L_i$ satisfies conclusions of the theorem for $0 \leq i \leq n-1$.
We claim that $L_0$ is $\Gamma$-isomorphic to $\Gamma$. We see that $L_1$ satisfies all the assumptions of Claim 2 and thus $L_0$ is $\Gamma$-isomorphic to $\Gamma$ as desired.
We claim that $n=1$. Suppose towards a contradiction that $n \geq 2$.
We have $$L_2={\langle}{\langle}L_0, t_1| t_0^{t_1}=t_0{\rangle}, t_2| t_0^{t_2}=t_0, t_1^{t_2}=t_1{\rangle}.$$
By Proposition \[lem-hnn\], $L_2$ is generated by $\Gamma \cup \{h_1t_2h_2\}$ where $h_1,h_2 \in L_1$. Again, since $L_1$ is generated by $\Gamma \cup\{h_1h_2\}$ we find, by Proposition \[lem-hnn\], $h_1h_2= \gamma_1t_1^{\pm 1}\gamma_2$ for some $\gamma_1, \gamma_2 \in L_0$. Now there exists a word $w(\bar x;y)$ such that $t_1=w(\bar \gamma; (h_1t_2h_2))$, and thus in $L_0 \times {\langle}t_1{\rangle}\times {\langle}t_2{\rangle}$ we have $$w(\bar \gamma; h_1t_2 h_2)=w(\bar \gamma; h_1h_2t_2)=v(\bar \gamma; \gamma_1 \gamma_2)(t_1^{\pm 1}t_2)^p=t_1,$$ for some $p\neq 0$, $p\in \mathbb Z$, which is clearly a contradiction.
Hence $n=1$ as claimed and finally $$\Gamma_{\bar s}(x)={\langle}\Gamma, t_1| t_0^{t_1}=t_0{\rangle}.$$
Case $(2)$. Let $\Gamma_{\bar S}(x)={\langle}H,t |c_1^t=c_2{\rangle}$. $\Gamma_{\bar S}(x)$ satisfies all the assumptions of Claim 2, and thus $H$ is $\Gamma$-isomorphic to $\Gamma$. Therefore $\Gamma_{\bar S}(x)={\langle}\Gamma, t| c'^t=c{\rangle}$. Since $\Gamma$ is existentially closed in $\Gamma_{\bar S}(x)$, $c$ and $c'$ are conjugate in $\Gamma$. Thus $\Gamma_{\bar S}(x)$ can be rewritten as ${\langle}\Gamma, s| u^s=u{\rangle}$ and we obtain the required conclusion. This ends the proof in this case and the proof of the theorem.
**Proof of Theorem \[thmmain2\].**
Theorem \[thmmain\] shows $(2) \Rightarrow (1)$, so we show $(1) \Rightarrow (2)$. Let $H={\langle}a,b{\rangle}$ be a nontrivial two-generator subgroup of $\Gamma$. We may suppose without loss of generality that $a$ is root-free. We claim that the group $\Gamma*_a{\langle}a, b'{\rangle}$ is a restricted $\Gamma$-limit group, where ${\langle}a, b{\rangle}\cong {\langle}a,b'{\rangle}$. Since $a$ is root-free, by applying [@groves-2007 Lemma 5.4], we see that the group ${\langle}\Gamma, t | a^t=a{\rangle}$ is a restricted $\Gamma$-limit group. We have ${\langle}\Gamma, b^t{\rangle}=\Gamma*_a{\langle}a,b^t{\rangle}$ with ${\langle}a,b^t{\rangle}\cong {\langle}a,b{\rangle}$. Hence $\Gamma*_a{\langle}a,b'{\rangle}$ is a restricted $\Gamma$-limit group, where we can take $b'=b^t$.
Let $$S(x)=\{w(x) \in \Gamma[x]| \Gamma*_a{\langle}a,b'{\rangle}\models w(b')=1 \}.$$
It is not hard to see that $$\Gamma_{\bar S}(x) \cong_{\Gamma} \Gamma*_a{\langle}a,x{\rangle},$$ with ${\langle}a,x{\rangle}\cong {\langle}a,b{\rangle}$.
Suppose that $[a,b] \neq 1$. Then $V(S)$ is infinite and irreducible. Hence, by (1), $V(S)$ is a coset of a centralizer. So let $u,v \in \Gamma$ such that $V(S)=v C_\Gamma(u)$. By applying [@groves-2007 Lemma 5.4], we conclude that $$\Gamma_{\bar S}(x) \cong_{\Gamma} {\langle}\Gamma, s | u^s=u{\rangle},$$ where $x=vs$.
Suppose that ${\langle}a,x{\rangle}$ is not free of rank $2$. Then there exists a nontrivial relation and using normal forms, we conclude that either $a^p=u^q$ or $v^{-1}a^pv=u^q$ for some $p, q \in \mathbb Z$. If the latter case holds then we may replace $u$ by $vuv^{-1}$ and $s$ by $vsv^{-1}$ and thus we get $x=sv$. Thus we conclude that we may assume $a^p=u^q$ for some $p, q \in \mathbb Z$ and $x=vs$ or $x=sv$. Since $a$ and $u$ are root-free, we get $a=u^{\pm 1}$ and without loss of generality, we assume that $a=u$.
Returning to our first construction, we get $${\langle}\Gamma, b^t{\rangle}=\Gamma*_a{\langle}a,b^t{\rangle}={\langle}\Gamma, s|a^s=a{\rangle}\leq {\langle}\Gamma, t|a^t=a{\rangle},$$ and thus $s=a^pt^q$ for some $p, q \in \mathbb Z$. Hence $b^t= v a^pt^q$ or $b^t=t^q a^p v$. In the group $\Gamma \times {\langle}t|{\rangle}$ we get $q=0$. Hence we find $b^t=va^p$ and thus $b \in {\langle}a{\rangle}$, which is a contradiction. Therefore ${\langle}a,x{\rangle}$ is free of rank $2$.
Appendix
========
In this appendix, we give a proof of the following theorem, where $\Gamma$ is steal a torsion-free hyperbolic group. For the notions used here, and which are not defined, we refer the reader to [@Champ-Guirardel].
\[athm1\]Let $G$ be a restricted $\Gamma$-limit group. If $G$ is not $\Gamma$-isomorphic to $\Gamma$ and if it is freely indecomposable relative to $\Gamma$, $G$ is weakly constructible.
A cyclic splitting (relative to $\Gamma$) is *essential* if any edge group is of infinite index in any vertex group.
The proof of the following proposition is similar to that of [@groves-2007 Theorem 3.7] and it is left to the raider.
\[aprop1\]A restricted $\Gamma$-limit group which is not $\Gamma$-isomorphic to $\Gamma$ and which is freely indecomposable relative to $\Gamma$ admits an essential cyclic splitting (relative to $\Gamma$).
[@Sela-Diophan1 Definition 8.3] Let $G$ be a restricted $\Gamma$-limit group which is not $\Gamma$-isomorphic to $\Gamma$ and which is freely indecomposable relative to $\Gamma$. The *restricted modular group $RMod(G)$* is the subgroup of $Aut(G)$ generated by the following families of automorphisms of $G$, which fixe pointwise the vertex group stabilized by $\Gamma$ in the restricted cyclic JSJ-decomposition of $G$ with respect to $\Gamma$:
$(1)$ Dehn twists along edges of the restricted cyclic JSJ-decomposition of $G$.
$(2)$ Dehn twists along essential s.c.c. in CMQ vertex groups in the restricted cyclic JSJ-decomposition of $G$.
$(3)$ Let $A$ be an abelian vertex group in the restricted cyclic JSJ-decomposition of $G$. Let $A_1<A$ be the subgroup generated by all edge groups connected to the vertex stabilized by $A$ in the cyclic JSJ-decomposition of $G$. Every automorphism of $A$ which fixes pointwise $A_1$ can be extended to an automorphism of $G$ which fixes the vertex stabilized by $\Gamma$. We call these generalized Dehn twists and they form the third family of automorphisms that generate $RMod(G)$.
(Shortening quotients) Let $G$ be a restricted $\Gamma$-limit group endowed with a finite generating set $B$.
$(1)$ A $\Gamma$-homomorphism $h : G \rightarrow \Gamma$ is said *short* if $$\max_{b \in B}|h(b)|\leq \max_{b \in B}|h(\tau(b))|,$$ for any restricted modular automorphism $\tau \in RMod(G)$. Here $|.|$ denotes the word length with respect to some fixed, for all the rest of this section, finite generating set of $\Gamma$.
$(2)$ Let $(h_n : G \rightarrow \Gamma)_{n \in \mathbb N}$ be a sequence of short $\Gamma$-homomorphisms. The group $G/ Ker_{\infty}((h_n)_{n \in \mathbb N})$ is called a *shortening quotient* of $G$.
\[athm2\] [@Sela-Diophan1 Claim 5.3][@Sela-hyp Proposition 1.15] Let $G$ be a restricted $\Gamma$-limit group which is not $\Gamma$-isomorphic to $\Gamma$ and which is freely indecomposable relative to $\Gamma$. Then every shortening quotient of $G$ is a strict quotient.
\[aprop2\] Let $G$ be a restricted $\Gamma$-limit group which is not $\Gamma$-isomorphic to $\Gamma$ and which is freely indecomposable relative to $\Gamma$. Then either $G$ is a free extension of a centralizer or $G$ is weakly constructible.
We begin first with the following lemma which is analogous to [@Champ-Guirardel Proposition 4.12].
\[alem1\]Let $H$ be a restricted $\Gamma$-limit group with a one edge cyclic splitting $H=A*_CB$ or $A*_C$ satisfying the following property: there exists a $\Gamma$-epimorphism $\varphi : H \rightarrow L$, where $L$ is a restricted $\Gamma$-limit group, such that $\varphi$ is one-to-one in restriction to the elliptic abelian neighbourhood of each vertex group.
Then there exists a sequence of Dehn twists $(\tau_i)_{i \in \mathbb N}$ on $H$, fixing pointwise $A$, such that $(\varphi \circ \tau_i)_{i \in \mathbb N}$ converges to the identity of $H$.
Proceeding as in the proof of Theorem \[thm2\], one first transform the given splitting to another one which is either 1-acylindrical or a free extension of a centralizer. Then the rest of the proof proceeds in a similar way to that of [@Champ-Guirardel Proposition 4.12] and [@Sela-Diophan1 Theorem 5.12], by using [@groves-2007 Lemma 5.4] instead of Baumslag’s lemma [@Champ-Guirardel Lemma 3.5] and by choosing the Dehn twist along some $c \in C$.
**Proof of Proposition \[aprop2\].**
The proof proceeds in a similar way to that of [@Champ-Guirardel Proposition 4.18, Proposition 4.18] and [@Sela-Diophan1 Proposition 5.10]. We suppose that $G$ is not a free extension of a centralizer and that it does not satisfy (1) of Definition \[weakly-constructible\].
Let $\Lambda$ to be the cyclic JSJ-decomposition of $G$ which is nontrivial by Proposition \[aprop1\]. Suppose first that $\Lambda$ has an abelian vertex group $G_v$ such that $A_1$ has an infinite index in $G_v$, where $A_1$ is the group generated by incident edge groups. Then in that case $G$ can be written as a nontrivial free extension of a centralizer.
Thus we may assume that for each abelian vertex group $G_v$, the subgroup generated by incident edge groups has finite index in $G_v$. Hence by definition each restricted modular automorphism $\tau$ is a conjugation in restriction to each nonsurface type vertex group, to each edge group and the identity on the vertex group containing $\Gamma$.
Let $(f_i : G \rightarrow \Gamma)_{i \in \mathbb N}$ be a sequence of $\Gamma$-homomorphisms converging to the identity of $G$. For each $i \in \mathbb N$ choose $\tau_i$ to be a restricted modular automorphism such that $f_i \circ \tau_i$ is short. Up to extracting a subesequence, we may assume that $(f_i \circ \tau_i)_{i \in \mathbb N}$ converges to a restricted $\Gamma$-limit group $L$ and we let $\varphi :G \rightarrow L$ to be the natural map. By Theorem \[athm2\], $L$ is a proper quotient.
Proceeding as in [@Champ-Guirardel Proposition 4.18], we conclude that $\varphi$ is one-to-one in restriction to the elliptic abelian neighboorhood of each nonsurface vertex group and of the vertex group containing $\Gamma$
Let $A$ be the vertex group containing $\Gamma$ and let $e$ be an edge incident to $A$. Write $H=A*_CB$ or $H=A*_C$ the subgroup of $G$ corresponding to the amalgam or HNN-extension carried by $e$.
Suppose that $H=A*_CB$ and $B$ is abelian. Since $H \leq \hat A$, it follows that $\varphi$ is one-to-one in restriction to $H$. Let $\bar \Lambda$ be the graph of groups obtained by collapsing $e$. Then $H$ is a vertex group and $\varphi$ is one-to-one in restriction to elliptic abelian neighboorhood of each vertex group of $\bar \Lambda$. If there is a another vertex abelian group $H'$ connected to $H$, we do the same construction. At the end of the procedure we get a cyclic splitting $\Lambda'$ such that if an edge is connected to the vertex group containing $\Gamma$ in $\Lambda'$ with different end points then in the corresponding amalgam $H=A*_CK$, $K$ is nonabelian. But this contradicts our hypothesis; because in that case $G=D_1*_CD_2$ whith $D_2$ noncyclic and $\Gamma \leq D_1$.
Hence $G$ can be written as $G={\langle}K, t_1, \dots, t_n | C_i^{t_i}=C'_i{\rangle}$ for some cyclic subgroups $C_1, \dots, C_n, C'_1, \dots, C'_n$ of $K$ and $\Gamma \leq A \leq K \leq \hat A$ and $\varphi$ is one-to-one in restriction to $\hat A$. If for some $i$ and $a \in K$, $C_i \cap {C'_i}^a\neq 1$ then $G$ can be written as a free extension of a centralizer. Hence for any $i$ and $a \in K$, $C_i \cap {C'_i}^a= 1$.
If $n=1$ we get the required conclusion. So we suppose that $n \geq 2$.
Let $H={\langle}K, t_1|C_1^t=C'_1{\rangle}$ and let $e$ be the edge corresponding to $C_1$. Lemma \[alem1\] applies in this case and we get a sequence of Dehn twists $(\tau_i)_{i \in \mathbb N}$ on $H$ such that $(\varphi \circ \tau_i)_{|H}$ converges to the identity of $H$. Up to exctracting a subsequence, we may assume that $(\varphi \circ \tau_i)_{i \in \mathbb N}$ converges to a $\Gamma$-epimorphism $\phi : G \rightarrow L'$, where $L'$ is a restricted $\Gamma$-limit group and where we identify $\tau_i$ with its natural extension to the entire group $G$. Let $\bar \Lambda$ be the graph of groups obtained by collapsing $e$. By construction $\phi$ is one-to-one in restriction to the elliptic abelian neighboorhood of the vertex group. If the obtained $\phi$ is not one-to-one, we conclude by induction on $n$.
So suppose that $\phi$ is one-to-one. We consider in this case the connected component $\Lambda_1$ of $\Lambda\setminus e$. Then $\varphi$ is one-to-one in restriction to the elliptic abelian neighboorhood of the fundamental group of $\Lambda_1$. Hence we obtain a one edge cyclic splitting of $G$ such that $\varphi$ is one-to-one in restriction to the elliptic abelian neighboorhood of the vertex group.
**Proof of Theorem \[athm1\].**
Let $G$ be a restricted $\Gamma$-limit group which is not $\Gamma$-isomorphic to $\Gamma$ and which is freely indecomposable relative to $\Gamma$. By the descending chain condition on restricted $\Gamma$-limit groups, we may asssume that every restricted $\Gamma$-limit proper quotient of $G$ satisfies the conclusion of the theorem if it satisfies its hypothesis.
By Proposition \[aprop2\], we may assume that $G$ is a nontrivial free extension of a centralizer. Set $G={\langle}H,t|C^t=C{\rangle}$ where $C$ is a nontrivial abelian subgroup of $H$ and $\Gamma \leq H$.
Define $\phi : G \rightarrow H$ by $\phi(t)=1$ and the identity on $H$.
If $C$ is cyclic then we get the required conclusion. So we suppose that $C$ is noncyclic. Clearly $H$ is not $\Gamma$-isomorphic to $\Gamma$. Similarly if $H$ is freely decomposable with respect to $\Gamma$ then $C$ is contained in some conjugate of a factor and thus $G$ is itself freely decomposable with respect to $\Gamma$.
Hence $H$ satisfies the hypothesis of the theorem and by induction we conclude that $H$ is weakly constructible.
Suppose that $H=A*_TB$ where $\Gamma \leq A$, $T$ is nontrivial and cyclic and $B$ is noncyclic. Since $C$ is noncyclic we conclude, up conjugation, that $C \leq A$ or $C \leq B$. Therefore $G$ can be written as $A'*_TB'$ with $B'$ is noncyclic.
Now suppose that $H=A*_T$. Let $L$ be the proper restricted $\Gamma$-limit quotient of $H$ given by the definition and let $\varphi : H \rightarrow L$ be the corresponding $\Gamma$-epimorphism.
Suppose first that $C$ is not elliptic in the splitting $H=A*_T$. Since $C$ is noncyclic, we conclude that $H$ can be written $H=A*_TC'$ where $C'$ is a conjugate of $C$. Hence $G$ can be written as $G=A*_TC''$, with $C''$ is noncyclic and we get the required conclusion.
Suppose now that $C$ is elliptic in the splitting $H=A*_T$ and without loss of generality that $C \leq A$.
Let $C'=C_L(\varphi(C))$ and let $L'= {\langle}L, s|C'^s=C'{\rangle}$. Then $L'$ is a restricted $\Gamma$-limit group. Define $\varphi' : G \rightarrow L'$ by $\varphi'_{|H}$ to be $\varphi$ and $\varphi'(t)=s$. Now $L'$ is a strict quotient of $G$ as $L$ is a proper quotient of $H$.
Then $G={\langle}A,t{\rangle}*_T={\langle}A,t|C^t=C{\rangle}*_T$ with $T \leq A$. Hence $G$ has a cyclic splitting and with $\varphi'$ is one-to-one in restriction to the elliptic abelian neighboorhood of ${\langle}A,t|C^t=C{\rangle}$.
We close this appendix with the following proposition.
Let $\Gamma$ be a torsion-free hyperbolic group and let $H \leq \Gamma$ be a proper subgroup definable by a quantifier-free formula. Then $H$ is abelian.
Since $\Gamma$ is equationally noetherian, $H$ is closed in the Zariski topology. Hence $H$ is definable by a finite union of varieties. Without loss of generality we assume that $H$ is definable by an equation, the general case can be treated similarly. So suppose that $H$ is definable by $w(c_1, \dots, c_p;x)=1$. Let $a \in H$. By [@groves-2007 Lemma 5.4], since for any $n \in \mathbb N$, $w(c_1, \dots, c_p; a^n)=1$, we obtain $a \in C_\Gamma(c_1) \cup \dots \cup C_{\Gamma}(c_p)$. But if $H$ is nonabelian, $H$ contains a nonabelian free subgroup and we get a contradiction.
[^1]: This research was supported by the “ANR” Grant in the framework of the project “GGL”(ANR-05-JC05-47038).
[^2]:
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Data from the Galileo Probe, collected during its descent into Jupiter’s atmosphere, is used to obtain a vertical profile of the zonal wind from $\mathbf{\sim\! 0.5}$ bar (upper troposphere) to $\mathbf{\sim\! 0.1\, \mu\mbox{bar}}$ (lower thermosphere) at the probe entry site. This is accomplished by constructing a map of gravity wave Lomb-Scargle periodograms as a function of altitude. The profile obtained from the map indicates that the wind speed above the visible cloud deck increases with height to $\mathbf{\sim\!150}$ ms$\mathbf{^{-1}}$ and then levels off at this value over a broad altitude range. The location of the turbopause, as a region of wide wave spectrum, is also identified from the map. In addition, a cross-equatorial oscillation of a jet, which has previously been linked to the quasi-quadrennial oscillation in the stratosphere, is suggested by the profile.'
title: 'The vertical structure of Jupiter’s equatorial zonal wind above the cloud deck, derived using mesoscale gravity waves'
---
Introduction
============
The horizontal structure of Jupiter’s zonal (east-west) winds at the level of the visible cloud deck on Jupiter have been observed for many decades [e.g., @Limaye1986; @Ingersoll2004; @Garcia2001; @Vasavada2005; @Li2006]. In contrast, the vertical structure of zonal winds is not well known – especially away from the cloud deck and in the equatorial region. This is because of the lack of usable tracers and the breakdown of the thermal wind relation near the equator, preventing latitudinal temperature measurements to be related to the vertical wind shear.
In December 1995 the Galileo probe entered Jupiter’s upper atmosphere at 6$^{\circ}\!$ N, above a $5 \mu$m hot spot in the north equatorial belt. During the descent the Atmospheric Structure Instrument on board collected information about the density, pressure, and temperature of the atmosphere [@Seiff1998]. Analysis of the temperature profile in the thermospheric and stratospheric regions identified discernible perturbations which have been interpreted as manifestations of internal, or vertically propagating, gravity waves [@Young1997; @Young2005; @Matcheva1999]. Vertical temperature profiles for Jupiter’s atmosphere have also been obtained from occultation studies using stars and other spacecraft. A number of these profiles contain oscillations that have been characterised as manifestations of internal gravity waves as well [@French1974; @Lindal1992; @Hubbard1995; @Raynaud2003; @Raynaud2004].
Gravity waves are oscillations of fluid parcels about their altitudes of neutral buoyancy [@Gossard1975]. The waves are a common feature of stably stratified atmospheres. They have been captured in images of Jupiter’s clouds, typically near the planet’s equator [@Flasar1986; @Arregi2009; @Reuter2007]. Fig. 1 shows an example. Internal gravity waves (hereafter simply gravity waves) grow in amplitude, due to the fall in background atmospheric density. The dynamics of such waves is described by the Taylor–Goldstein equation [@Taylor1931; @Goldstein1931],
$$%\begin{eqnarray*}
\frac{{\rm d}^2 w}{{\rm d}z^2} + m^2(z)\, w\ =\ F(z)\, .
%\end{eqnarray*}$$
Here $w$ is the perturbation in the vertical wind, which is adjusted for the amplitude growth and also assumed to be oscillatory (wave-like) in the horizontal direction and time; $F$ is a function that represents the source and dissipation of waves; and, $m$ is the vertical wavenumber, which depends on the physical properties of the medium in which the waves propagate. Specifically,
$$\begin{aligned}
m(z)\ = \hspace*{6.35cm} \nonumber \\
\left[\,\frac{N^2}{I^2} - \frac{1}{I}\,\frac{{\rm d}^2 I}{{\rm
d}z^2} - \frac{1}{HI}\,\frac{{\rm d} I}{{\rm d}z} -
\frac{1}{4H^{2}}\left(1 - 2\frac{{\rm d}H}{{\rm d}z}\right) -
k^2\,\right]^{1/2}\hspace*{-.2cm} ,
\end{aligned}$$
where $N$ is the Brunt–Väisälä (buoyancy) frequency; $I$ is the intrinsic phase speed, $c - u_0$, where $c$ is the horizontal phase speed and $u_0$ is the zonal wind; $H$ is the density scale height; and, $k$ is the horizontal wavenumber. In general, all of the variables depend on altitude $z$. Crucially, the zonal wind profile $u_0 (z)$ can be obtained by solving for $I = I(H,k,N,m)$ in equation (2). In this letter we present a profile from this inversion that span much larger range of altitude above the 1 bar level (range of $\sim\! 500$ km) than in past analyses.
Method
======
For the inversion we compute $H$ from the collected density profile. We also use $k = 2\pi/300$ km$^{-1}$ from the average value obtained in analysis of Voyager images [@Flasar1986]. Other analyses have reported slightly different values (e.g., average of $2\pi/165$ km$^{-1}$ by @Arregi2009); however, as long as $k^2
\ll m^2$ (as is the case here), the value of $k$ does not significantly alter the result of the inversion. $N$ and $m$ are obtained from the potential temperature, $\theta = T(p_{\rm
ref}/p)^\kappa$, related to specific entropy of the atmosphere; here $T$ is the temperature, $p$ is the pressure, $p_{\rm ref} = 1$ bar is a constant reference pressure and $\kappa = R/c_p$ with $R$ the specific gas constant and $c_p$ the specific heat at constant pressure. To obtain $N$ and $m$, we decompose $\theta$ into a mean background $\bar{\theta}$ (extracted using a spatially-moving fitting window) and a small perturbation $\Delta \theta$ about the mean (presumed caused by mesoscale gravity waves). Such decomposition of temperature is standard in gravity wave studies [@Lindzen1990; @Nappo2002; @Young1997], and is reasonable here given that the spatial scales of the background and waves are well separated. Further, the horizontal distance travelled by the probe in the region studied, 3300 km, is small compared to Jupiter’s circumference and is completely within the hot-spot entered by probe [@Orton1998]; hence, zonal variation in the background over this distance is not expected to be significant.
Fig. 2a shows the relative perturbation, $\Delta\theta/\bar{\theta}$, resulting from a 75 km- wide moving window. We have checked that the result is not affected by the choice of window size, by varying the size from 55 km to 85 km. Fig. 2b shows $N = [g\,{\rm
d}(\ln\bar{\theta})/{\rm d}z]^{1/2}$ resulting from the decomposition; here $g = g(z)$ is the gravity. In the figure note that $N$ is much smaller in the region below $\sim$20 km than that above (the stratosphere)—in agreement with @Magalhaes2002, whose observations give $N \approx 6 \times 10^{-3}$ s$^{-1}$ in the lower region. This represents a possible ducting region, a source for the waves in the probe data.
Although $w$ information was not collected by the probe, $m(z)$ can still be obtained from $\Delta\theta$ through the polarization relation [e.g., @Watkins2010]. For this we generate a series of Lomb-Scargle periodograms [@Scargle1982] of the $\Delta\theta$ data, which is non-uniformly spaced in $z$. The non-uniform spacing renders analysis by standard Fourier or wavelet transforms unsuitable. One periodogram at each $z$ is generated using a smoothing window. We have also verified in this procedure that the obtained result varies little between different sized smoothing windows. All the periodograms are subsequently combined to produce a two-dimensional map of the wave spectral energy density as a function of wavenumber and altitude, ${\cal E} = {\cal E}(m^\ast,z)$, shown in Fig. 3; here $m^\ast$ is the vertical wavenumber $m$ prior to an adjustment for wave propagation geometry. Before discussing this adjustment, we discuss three features which are already apparent in the ${\cal E}$ map.
Results
=======
First, gravity waves identified previously [@Young2005] are recovered (W1 and W2 in the map). This agreement gives confidence in our procedure. In addition to those waves, we identify in our analysis new gravity waves throughout the analyzed domain. In particular, note the high energy density (dark red) regions near the 50 km, 300 km and 400 km altitudes all with $m^\ast \approx 0.25$. These constitute new waves.
Second, we identify a region consistent with a turbopause $\sim$50 km thick, centered at approximately 400 km altitude. This is the region where the width of the sub-spectrum containing significant energy increases markedly. In this region, molecular diffusion becomes comparable to eddy diffusion and gravity waves—growing in amplitude as they propagate upward—break, transferring energy into the higher wavenumbers. Above this region, energy is lower across the entire spectrum and the spectrum itself is much steeper (i.e., narrower) than that for the turbopause region. Above the turbopause region, the atmosphere becomes inhomogeneous, separating out into layers of different molecular species. Past studies have placed Jupiter’s turbopause at the $\sim\! 5~\mu$bar [@Festou1981] and $\sim\!
0.5~\mu$bar [@Yelle1996] levels, based on observations and modeling. Our result supports the latter location.
Third, in the lower part of the analyzed domain there appears to be a ducting region, a region with a sharp jump in $N$. As already noted, such a region can serve as a source of gravity waves. Horizontally propagating gravity waves in this region have previously been observed (see, e.g., Fig. 1), which have been suggested as waves trapped in a “leaky” duct. The ducted wave travels horizontally by undergoing internal reflections at the boundaries; however, part of the wave escapes the duct to propagate vertically. Because the wavenumber with maximum energy in the ${\cal E}$ map can be traced down to the ducting region, it is likely that the waves have come from there.
Note that above the ducting region the number of local peaks in the spectrum generally reduces with altitude (the centroid of the spectrum at each height is shifted to lower wavenumber). Also, the magnitude of the peak energy in the low wavenumber (white line in Fig. 3) is high at first, then decreases, and then increases again along the white line, until the topside of the turbopause region at $\sim\!
425$ km above the 1 bar level. This is indicative of wave saturation or encounters with a critical layer, where $I = 0$ locally, for high number wavenumbers as they propagate upward. The second or third multi-peaked altitude regions, between $\sim\! 200$ km and $\sim\!
300$ km above the 1 bar level, may be due to wave breaking and secondary wave generation from breaking layers.
As alluded to earlier, $m^\ast$ differs from the required wavenumber $m$. The latter is the wavenumber that would be observed by a probe travelling in a vertical direction. However, throughout most of its entry phase, the probe had a shallow angle of attack ($\sim\! 7^{\circ}$ below the horizontal), which changes the wavenumber observed by the probe during its passage in the upper part of the analyzed domain. We use the geometry of the probe’s path to obtain the true vertical wavenumber,
$$m\ =\
\left(\frac{\tan\gamma\,\tan\beta}{1-\tan\gamma\,\tan\beta}\right)\,
m^\ast,$$
where $\gamma$ is the angle of attack and $\beta$ is the angle the wavevector makes with the horizontal. We make no correction for the relative motion of the probe with respect to the wave since the probe’s velocity is supersonic (indeed, hypersonic, with up to Mach 51) for much of the entry phase. Gravity wave phase speeds are subsonic.
Although $\gamma = \gamma(z)$ is known from the probe’s trajectory [@Seiff1998], $\beta$ is not. To estimate this quantity, we assume that the probe’s trajectory is vertical in the period just before parachute deployment. This is not far from the actual situation since the probe’s angle of attack was $83^{\circ}$ just before the parachute was deployed, near the lower boundary of our domain. In this region, $m^\ast \approx m$; and, since
$$\beta = \arccos\left( \frac{k}{\sqrt{k^2 + m^2}}\right)\, ,$$
we have $\beta\approx 85^\circ$. Also, in this region $m \approx
0.25$ km$^{-1}$ (giving vertical wavelength of $\approx\! 25$ km), in good agreement with previously recovered values [@Magalhaes2002; @Arregi2009]. Now, all the parameters required to recover $I$ have been obtained, and a vertical profile for $I$ can be estimated.
Finally, to obtain $u_0$ from $I$ a value for the horizontal phase speed $c$ is required. This is not well known. However, equatorial gravity waves with a phase speed of about 100 m s$^{-1}$ greater than the winds have been observed in images of Jupiter’s clouds returned by New Horizons [@Reuter2007]. This gives $c \approx
180$ m s$^{-1}$. The Doppler Wind Experiment (DWE) on the probe has measured the zonal winds to be approximately this speed in the deeper atmosphere [@Atkinson1998], suggesting that the waves could have been driven by convective overshoot below the clouds. This value also agrees well with the 140–195 m s$^{-1}$ range reported for the similar region, at the probe entry site [@Magalhaes2002]. Somewhat smaller values (approximately 70–145 m s$^{-1}$) have been obtained for waves at different longitudes [@Arregi2009]. Such variation in $c$ does not change the shape of the $u_0$ profile we recover, but the magnitude would be reduced. Variations of $c$ with altitude would increase the uncertainty as well, but they are expected to be small and not fundamentally change the profile. This gives the profile for $u_0(z)$ shown in Fig. 4a. Here we have used $c
= 180$ m s$^{-1}$. The standard deviation shown indicates the variation in the profile given by the various smoothing window sizes considered in constructing ${\cal E}$.
Discussion
==========
The DWE reported flow speeds that increase with depth reaching $170$ m s$^{-1}$ at the $5$ bar level and then remaining high at lower levels [@Atkinson1998]. The upper part of the DWE profile is shown in Fig. 4a. The wind speed at the bottom of our profile agrees well with that at the top of the DWE profile. Our profile shows increasing zonal wind speed with altitude (up to $\sim\! 100$ km above the 1 bar level). This implies that the wind speed is a minimum near the cloud-top level. This is similar to what has been observed from studies using the thermal wind equation [@Flasar2004; @SimonMiller2006]. However, in our profile this high speed wind is not a jet, as the zonal wind speed does not diminish appreciably with altitude near this level. In fact, the speed is roughly constant throughout the stratosphere, starting from this level. There are some fluctuations of the order of 20 m s$^{-1}$ in the wind speed in the thermosphere. The Richardson number, $Ri =
N^2\,({\rm d}u_0/{\rm d}z)^{-2}$, is greater than $1/4$ for the entire profile, which indicates the flow is stable with respect to Kelvin-Helmholtz instability.
Temporal variations in the temperature profile of Jupiter’s equatorial stratosphere have been observed to have a period of $4$ to $5$ years and are therefore known as the quasi-quadrennial oscillation (QQO) [@Leovy1991; @Friedson1999]. These oscillations have been linked to variations in the zonal wind observed via cloud-tracking on Jupiter with a period of $\sim$4.4 years [@SimonMiller2007]. Further, a jet located just north of the equator, derived using the thermal wind equation applied to data gathered in December 2000 and January 2001, has been linked to the QQO [@Flasar2004]. A jet of similar magnitude, just south of the equator, has been observed in observations gathered in 1979 [@SimonMiller2006] (which fits well with a period of 4.4 years), suggesting this jet may be oscillating about the equator. No such jet is visible in our profile. However, the probe entered Jupiter’s atmosphere 5 years before the northern jet was observed. Thus, our profile is consistent with such an oscillation as the jet would be located south of the equator, away from the probe entry site, at the time the probe entered.
Gravity wave creation and dissipation is associated with heating and cooling of the atmosphere. Ignoring molecular viscosity, which is only important above the turbopause, the heating rate is proportional to the vertical gradient of energy flux—specifically,
$$\frac{\partial T}{\partial t} = - \frac{1}{\rho c_p} \frac{\partial F_z}
{\partial z}$$
We estimate the energy flux, $F_z=\rho \overline{\phi w}$, from $\Delta \theta$ using the polarization relations to derive the perturbations in the geopotential $\phi$ and the vertical velocity $w$. The flux comes from the zonal average of the product of these quantities. Since the product is not available to us we average over a single vertical wavelength instead. The heating rate profile thus obtained is shown in Fig 4b. It can be seen that heating is small through the stratosphere. Indeed, on average it is very close to zero. It is only in the thermosphere that the average heating rate deviates from zero with a magnitude of $\sim\!$ 0.5 K per Jupiter rotation. The flux responsible for heating in this region is small ($F_z < 10^{-3}$ W m$^{-2}$), compared to radiation fluxes measured in the troposphere ($3 < F_z < 16$ W m$^{-2}$) [@Sromovsky1998]. Nevertheless, the peak heating rate magnitude is about 50 times larger, partly because the lower thermsopshere is much less dense than the troposphere.
The circulation of Jupiter’s stratosphere is important for understanding the planet’s circulation as a whole. The location of the turbopause is essential for understanding the coupling between Jupiter’s upper atmosphere and the circulation in the lower atmosphere. The Juno mission will explore the troposphere of Jupiter to depths of $100$ bar or more. This will provide better insight to the source and behaviour of gravity waves in the troposphere, possibly allowing better limits to be derived for the wave-vector angle $\beta$ in our analysis and better understanding of the mechanisms that generate the gravity waves. The planned Jupiter Icy Moon Explorer (JUICE) mission will directly study gravity wave activity and zonal winds in the stratosphere of Jupiter, extending the result present here.
C.W. is supported by the Science and Technology Facilities Council (STFC), and J.Y-K.C. is supported by the STFC PP/E001858/1 grant. The authors acknowledge H. Thrastarson, I. Polichtchouk, and C. Agnor for useful discussions. We thank the reviewers for helpful suggestions.
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![Gravity wave in Jupiter’s atmosphere (Voyager image 16316.34). This wave is propagating horizontally in the troposphere. Many such waves are captured in the Voyager images [@Flasar1986]. Image contrast has been enhanced to improve the visibility of the wave. The wave’s location is indicated by the arrow. Image courtesy NASA/JPL-Caltech.](JupGWFig1.eps){width="20pc"}
![Relative potential temperture perturbation and Brunt–Väisälä (buoyancy) frequency profiles for Jupiter’s atmosphere. All altitudes are relative to the 1 bar pressure level. The profiles are based on data gathered from $t =
-173.055$ s to $t = 218.170$ s; $t = 0$ s is the point of parachute deployment. Our analysis used the complete set of acceleration data collected from both accelerometers with the exception of one outlier at $t = -157.742$ s. The acceleration data collected by the probe includes a small spurious oscillation[@Seiff1998] which we did not smooth out as its effects on the calculation was negligible in the atmospheric region we analyzed. The bottom 25 km of the profile is from direct measurement of the temperature with the first 15 s of direct temperature measurements removed as these were anomalously high. **a,** The vertical profile of potential temperature perturbations, $\Delta \theta$, scaled by the background value, $\bar{\theta}$. The perturbations show wavelike oscillations throughout the stratosphere and lower thermosphere. Note that the short wavelength oscillations in the layer between 25 km and 50 km are due to accelerations caused by buffeting of the probe as its velocity became subsonic. **b,** The Brunt–Väisälä frequency is the maximum frequency that a gravity wave can have and shows the stability of the atmosphere against convective stability.](JupGWFig2.eps){width="20pc"}
![Moving Lomb-Scargle periodogram of the potential temperature perturbation from Fig. 2a. Regions of high spectral energy are in red while low energy is in blue. The variation of the observed vertical wavenumber $m^\ast$ is shown (white line). The line is produced by identifying local energy maxima and constructing a line joining them, avoiding local minima, starting at the level of the duct in the troposphere. Note that there are other waves within the periodogram we do not consider in our analysis. One such wave, the region of high energy at $m^\ast\approx 0.61$ km$^{-1}$ between 180 km and 210 km altitude (labelled W1) has been previously identified as a saturating gravity wave[@Young2005]. Another region (labelled W2) has also been previously identified[@Young2005]. The region at an altitude of around 400 km shows a broadening of the range of wavenumbers with increased spectral energy. This is indicative of the turbopause, the region where waves break and the atmosphere begins to become heterogeneous.](JupGWFig3.eps){width="20pc"}
{width="20pc"}
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'Several studies indicate that attracting students to research careers requires to engage them from early undergraduate years. Following this paradigm, our Engineering School has developed an undergraduate research program that allows students to enroll in research in exchange for course credits. Moreover, we developed a web portal to inform students about the program and the opportunities, but participation remains lower than expected. In order to promote student engagement, we attempt to build a personalized recommender system of research opportunities to undergraduates. With this goal in mind we investigate two tasks. First, one that identifies students that are more willing to participate on this kind of program. A second task is generating a personalized list of recommendations of research opportunities for each student. To evaluate our approach, we perform a simulated prediction experiment with data from our School, which has more than 4,000 active undergraduate students nowadays. Our results indicate that there is a big potential to create a personalized recommender system for this purpose. Our results can be used as a baseline for colleges seeking strategies to encourage research activities within undergraduate students.'
author:
- Felipe del Rio
- Denis Parra
- Jovan Kuzmicic
- Erick Svec
bibliography:
- 'sigproc.bib'
title: Towards a Recommender System for Undergraduate Research
---
Introduction
============
In a globalized world, academic institutions are compelled to offer rich learning experiences to their students, with a complex curriculum that include extra academic activities [@bauer2003alumni]. In order to address this issue, our School of Engineering[^1] established an undergraduate research program in 2011, known as IPre (in Spanish *Investigación en Pregrado*), which allows students to receive course credits when joining a research project with faculty advice. The mission of the IPre program is to contribute to the academic and professional development of engineering undergraduates by enhancing their research skills [@harsh2011undergraduate].
**Context and Problem**. Nowadays, the IPre program has an offer-demand system focused on student-faculty interaction on a web platform. Herein, professors offer *Research Opportunities* to a general board where students can browse and apply to available projects. In this way, students have access to research topics that are new to them and work in different attractive areas. Although this platform promotes exchange of ideas, student engagement in undergraduate research programs faces major challenges [@merkel2001undergraduate], and IPre is not an exception. In order to promote these programs, recent literature has aimed to identify undergraduates’ motivation with research activities [@douglass2013undergraduate; @zimbardi2014embedding]. In this line, we have detected lack of knowledge about the IPre program and the available research opportunities as a major factor, thus we herein propose a personalized approach to enroll students in undergraduate research.
**Objective and Tasks**. In order to address the challenge of promoting student engagement in our undergraduate research, and considering the success of personalization for increasing user engagement in several areas and communities, we decided to explore the potential of a recommender systems. In this work we study the feasibility of such system studying two tasks, using data collected from the current online IPre system over the last five years: (i) Identifying Students who would be likely to participate in the undergraduate research program, and (ii) recommending relevant research opportunities to undergraduate Engineering students.
**Results and Contributions**. Our results indicate that it is possible to identify which students will be more likely to participate, with a precision up to 72.7%. Moreover, the task of recommending is indeed more challenging. We compared several methods and parameters and we were able to obtain a model which close to MAP=0.2, but it requires further research to get to a more accurate recommendation approach. Nonetheless, these results set an appropriate baseline to improve further our current IPre system.
Dataset & Features
==================
We used a dataset from the IPre program over 2012-2016 period, representing applications of students to undergraduate research opportunities. The dataset comprises user profiles of $10,546$ undergraduate students of the Engineering School, among them $1,134$ students applied to $1,017$ available research opportunities. Students could apply to more than one opportunity, so we recorded $1,624$ applications in total, having 81.4% of the applications accepted.
[*Task 1*]{} was about predicting whether student $u_i$ applied to research opportunities or not (1:applied, 0:did not apply). In this task we compared three feature sets: (a) [*Base*]{}: semesters enrolled, number of credits approved, (b) [*Base + ipre:*]{} features in (a) plus a boolean indicating previous applications to IPRE, and (c) [*Base + ipre + gpa:*]{} features in (b) plus GPA.
For [*Task 2*]{}–predicting which research opportunities the students applied– we made recommendation as a classification task, i.e., predict whether student $u_i$ would apply to a research opportunity $o_j$ (1:positive, 0:negative). We used three feature sets: (a) [*Base*]{}: cosine similarity between research opportunity abstract and descriptions of courses approved, (b) [*Base + ht*]{}: features in (a) plus a boolean indicating that the student was taught by the faculty offering the opportunity, and (c) [*Base + ht + dept*]{} features in (b) plus the percentage of courses approved taught by the same department as the faculty offering the opportunity (e.g. computer science).
Evaluation Methodology & Results
================================
All data before 2014 is used for training and everything afterwards for testing. In both tasks we test a baseline classifier, logistic regression (LogReg), gradient boosted trees (GBT) and support vector machines (SVM). For **task 1**, predicting whether the user applies to opportunities or not, the dataset is highly unbalanced since 89.7% of the students do not apply to opportunities. We measure classifier performance with accuracy, precision and F-1 score. As a baseline we use a model that predicts the most common class.
For **task 2**, predicting which opportunities a student actually applied to, we classify several opportunities for each student and we rank them based on their prediction score. Then, we used the ranking metric Mean Average Precision (MAP) [@parra2013recommender] to evaluate the performance. The baseline method consisted on generating a random list of recommendations. In this task, we analyzed: recommendation list size ($k$), feature sets and algorithm used.
Results
=======
**Task 1**: [*Predict is student applies to opportunities*]{}. Table \[table:is-ipre-student\] shows the results in two groups: (a) comparing methods (using all features), and (b) comparing features (using the best method). Here we see that all methods (LogReg, GBT and SVM) outperform the baseline in all metrics. The best methods though are GBT (accuracy=92%, precision=72.7%, F-1=0.54) and LogReg (accuracy=91.2%, precision=62.4%, F-1=0.55). This result is very high considering the class imbalance. In terms of feature sets, the baseline (semesters enrolled and number of credits approved) is boosted specially by considering if the student previously applied to an IPre opportunity in the past; i.e., most likely will apply again.
**Task 2**: [*Recommending research opportunities*]{}. We analyze this task in two stages. First, using all the features we compare methods, as seen in Figure \[fig:map-by-type\]. We found that all methods outperform a random baseline, but LogReg and GBT perform the best, getting to a MAP up to $0.20$. Our top method scored 14.6 times higher that the baseline for $k=20$ and closer to 10 times on a longer recommendation list. Then, using LogReg method, we study different features set as seen in Figure \[fig:map-by-feat\]. We observe that knowing if the student had a class with the professor offering the research opportunity increases significantly the prediction compared to only matching content description of courses and research opportunity. A smaller yet important boost on the recommendation is also given by matching department information in the model
![Task2 MAP by classifier using all features.[]{data-label="fig:map-by-type"}](map-by-type.png)
![Task 2 MAP by feature sets using LR.[]{data-label="fig:map-by-feat"}](map-by-feat.png)
Conclusion
==========
In this work we showed feasibility of: (a) identifying students prone to apply to research opportunities, and (b) recommending research opportunities for undergraduate students. There is still room for improvement by adding new features and other recommendation approaches (such as factorization machines or neural networks). We are currently conducting a user study to verify the generalizability of our results. We expect to serve as a baseline for institutions implementing these features in their academic systems.
[^1]: institution not disclosed to maintain blind revision requirement
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
In this paper, we will study some properties of b-weakly compact operators and we will investigate their relationships to some variety of operators on the normed vector lattices. With some new conditions, we show that the modulus of an operator $T$ from Banach lattice $E$ into Dedekind complete Banach lattice $F$ exists and is $b$-weakly operator whenever $T$ is a $b$-weakly compact operator. We show that every Dunford-Pettis operator from a Banach lattice $E$ into a Banach space $X$ is b-weakly compact, and the converse holds whenever $E$ is an $AM$-space or the norm of $E^\prime$ is order continuous and $E$ has the Dunford-Pettis property. We also show that each order bounded operator from a Banach lattice into a $KB$-space admits a $b$-weakly compact modulus.\
**Keywords:** Banach lattice, order continuous norm, b-weakly compact operator, Dunford-Pettis operator.\
**MSC(2010):** Primary 46B42; Secondary 47B60.
address:
- 'Department of Mathematics, University of Mohaghegh Ardabili, Ardabil, Iran.'
- 'Department of Mathematics, University of Mohaghegh Ardabili, Ardabil, Iran.'
author:
- Masoumeh Mousavi Amiri
- 'Kazem Haghnejad Azar $^*$'
date: 'Received: , Accepted: .'
title: 'Some notes on $b$-weakly compact operators'
---
[^1]
Introduction
============
An operator $T$ from a Banach lattice $E$ into a Banach space $X$ is said to be b-weakly compact, if it maps each subset of $E$ which is b-order bounded (i.e. order bounded in the topological bidual $E^{\prime\prime}$) into a relatively weakly compact subset of $X$. Note that in [@2], the class of b-weakly compact operators is introduced by Alpay-Altin-Tonyali and several interesting characterizations were given in [@2; @4; @9; @10]. After that, a series of papers which gave different characterizations of this class of operators were published [@2; @3; @4; @5; @6; @7; @8; @9; @10]. The most beautiful property of the class of b-weakly compact operators is that it satisfies the domination property as proved in [@2]. But one of shortcomings of this class is that the modulus of a $b$-weakly compact operator need not be $b$-weakly compact. Note that each weakly compact operator is b-weakly compact, but the converse is not true in general. In [@8], authors characterized Banach lattices on which each b-weakly compact operator is weakly compact. In [@10], authors proved that an operator $T$ from a Banach lattice $E$ into a Banach space $X$ is $b$-weakly compact if and only if $(Tx_n)$ is norm convergent for every positive increasing sequence $(x_n)$ of the closed unit ball $B_E$ of $E$.
Some basic definitions
----------------------
An element $e>0$ in a Riesz space $E$ is said to be an order unit whenever for each $x\in E$ there exists some $\lambda >0$ with $|x| \leq \lambda e$. For example $\ell^\infty$ has order unit, but $c_0$ has not. A sequence $(x_n)$ in a vector lattice is said to be disjoint whenever $|x_n| \wedge |x_m| =0$ holds for $n\neq m$. Let $E$ be a Riesz space. The subset $ E^+=\{x\in E:x\geqslant0\}$ is called the positive cone of $E$ and the elements of $E^+$ are called the positive elements of $E$. For an operator $T:E\rightarrow F$ between two Riesz spaces we shall say that its modulus $|T|$ exists ( or that $T$ possesses a modulus) whenever $|T|:=T\vee (-T)$ exists in the sense that $|T|$ is the supremum of the set $\{-T,T\}$ in $\mathcal{L}(E,F)$. An operator $T:E\rightarrow F$ between two Riesz spaces is called order bounded if it maps order bounded subsets of $E$ into order bounded subsets of $F$. An operator $T: E\rightarrow F$ between two Riesz spaces is positive if $T(x)\geq 0$ in $F$ whenever $x\geq 0$ in $E$. Note that each positive linear mapping on a Banach lattice is continuous. It is clear that every positive operator is order bounded, but the converse is not true in general. A Banach lattice $E$ has order continuous norm if $\| x_\alpha\|\rightarrow 0$ for every decreasing net $(x_\alpha)_\alpha$ with $\inf_\alpha x_\alpha=0$. If $E$ is a Banach lattice, its topological dual $E^\prime$, endowed with the dual norm and dual order is also a Banach lattice. A Banach lattice $E$ is said to be an $AM$-space if for each $x,y\in E$ such that $|x|\wedge |y|=0$, we have $\|x+y\|= max \{\|x\|, \|y\|\}$. A Banach lattice $E$ is said to be $KB$-space whenever each increasing norm bounded sequence of $E^+$ is norm convergent. A subset $A$ of a Riesz space $E$ is called b-order bounded in $E$ if it is order bounded in the bidual $E^{\thicksim\thicksim}$. An operator $T:E\rightarrow F$ between two Banach spaces is called a Dunford-Pettis operator whenever $x_n\xrightarrow{w} 0$ implies $Tx_n\xrightarrow{\|\cdot \|} 0$. Recall that an operator $T$ from a Banach lattice $E$ into a Banach space $X$ is said to be order weakly compact, if it maps each order bounded subset of $E$ into a relatively weakly compact subset of $X$, i.e., $T[-x,x]$ is relatively weakly compact in $X$ for each $x\in E^+$. Assume that $E$ and $F$ are normed lattice. A positive linear operator $T:E\rightarrow F$ is called almost interval preserving if $T[0,x]$ is dense in $[0,Tx]$ for every $x\in E^+$. Let $E$ be a vector lattice. A sequence $\{x_n\}_1^ \infty\subset E$ is called order convergent to $x$ as $n \to \infty$ if there exists a sequence $\{y_n\}^\infty_1$ such that $y_n \downarrow 0$ as $n \rightarrow \infty$ and $|x_n -x| \leq y_n$ for all $n$. We will write $x_n\xrightarrow{o_1}x$ when $\{x_n\} $ is order convergent to $x$. A sequence $\{x_n\}$ in a vector lattice $E$ is strongly order convergent to $x\in E$, denoted by $x _n \xrightarrow{o_2}x$ whenever there exists a net $\{y_\beta\}_{\beta\in \mathcal{B}}$ in $E$ such that $y_\beta \downarrow 0$ and that for every $\beta\in \mathcal{B}$, there exists $n_0$ such that $|x_n -x| \leq y_\beta$ for all $n\geq n_0$. It is clear that every order convergent sequence is strongly order convergent, but two convergence are different, for information see, [@1b]. A net $ (x_{\alpha})_{\alpha}$ in Banach lattice $E$ is unbounded norm convergent (or, $un$-convergent for short) to $ x \in E $ if $ | x_{\alpha} - x | \wedge u\xrightarrow{\|\cdot \|} 0$ for all $ u \in E^{+} $. We denote this convergence by $ x_{\alpha} \xrightarrow{un}x $. This convergence has been introduced and studied in [@9d; @10a]. For terminology concerning Banach lattice theory and positive operators, we refer the reader to the excellent books of [@1; @11; @12].
**Main Results**
================
The collections of b-weakly compact operators, order weakly compact operators, weakly compact operators and compact operators will be denoted by $W_b(E,X)$, $W_o(E,X)$, $W(E,X)$ and $K(E,X)$, respectively, whenever there is not confused. We have the following relationships between these spaces: $$K(E,X) \subseteq W(E,X)\subseteq W_b(E,X)\subseteq W_{o}(E,X).$$ In [@2], authors show that the above inclusion may be proper. For example, note that each weakly compact operator is a $b$-weakly operator, but the converse may be false in general. Of course the identity operator $I:L^1 [0,1]\rightarrow L^1 [0,1]$ is a $b$-weakly operator, but is not weakly compact. Now let $E$ be a Banach lattice such that the norm of $E^ \prime $ is order continuous and let $X$ be a Banach space. Then, by [@8 Theorem 2.2], it is obvious that each $b$-weakly operator $T:E\rightarrow X$ is weakly compact.\
The next example due to Z. L. Chen and A. W. Wickstead in [@13] shows that the order bounded $b$-weakly compact operators from a Banach lattice into a Dedekind complete Banach lattice do not form a lattice, i.e., a modulus of a $b$-weakly compact operator need not be $b$-weakly compact.
\[2.13\] Let $E=C[0,1]$, $F=l_\infty (F_n)$ where $F_n=(l_\infty,\|\cdot \|_n)$ and $\|(\lambda_k)\|_n=\max\{\|(\lambda_k)\|_\infty ,n\limsup(|\lambda_k|)\}$ for all $(\lambda_k)\in l_\infty$. Then for each $n\in \mathbb{N}$, $F_n$ is a Dedekind complete $AM$-space, hence so is $F$. Define $T_n:E\rightarrow F_n$ by $T_n(f)=(2^n.\int _{I_n} f.r_kdt)_{k=1}^{\infty}\in F_n$ for all $f\in E$, where $r_n$ is the $n^,$th Rademacher function on $[0,1]$ and $I_n=(2^{-n},2^{-n+1})$.\
Now define $T:E\rightarrow F$ by $T(f)=(\frac{1}{n}T_n(f))_{n=1}^{\infty}$. Then $T$ is a weakly compact operator. So, $T$ is a $b$-weakly compact operator and its modulus $|T|$ exists and $|T|$ is not order weakly compact hence not $b$-weakly compact. So, $W_{b}(E,F)$ is not a Riesz space.
Alpay and Altin in [@4] show that for $b$-weakly compact operator $T$ from a Banach lattice $E$ into a Dedekind complete Banach lattice $F$, the modulus of $T$ exist and is $b$-weakly compact operator whenever $F$ is $AM$-space with order unit. Now in the following theorems, we show that $T\in W_b(E,F)$ whenever $\vert T\vert\in W_b(E,F)$ and with some new conditions, we show that the modulus of $T$ exists and is $b$-weakly operator whenever $T$ is a $b$-weakly compact operator.
\[t:2.6\] Let $E$ and $F$ be normed vector lattices. We have the following assertions.
1. If $T:E\rightarrow F$ is an order bounded operator and $F$ is $KB$-space, then $T$ and $\vert T\vert$ are $b$-weakly operator.
2. If $\vert T \vert$ is a $b$-weakly compact operator, then $$T \in W_{b}(E,F)$$
<!-- -->
1. Since every $KB$-space has order continuous norm, so $F$ is a Dedekind complete Banach lattice. Then, by Theorem 1.18 from [@1], $|T|$ exists. Now, let $(x_n)$ be a positive increasing sequence in $B_E$. By Theorem 4.3 of [@1], $T^+(x_n)$ is a Positive increasing norm bounded sequence in $F$. Since $F$ is a $KB$-space, $T^+(x_n)$ is norm convergent. Thus, $T^+$ is $b$-weakly compact. Similarly, $T^-$ is $b$-weakly compact. It follows from the identities $T=T^++T^-$ and $|T|=T^++T^-$ that $T$ and $|T|$ are $b$-weakly compact operators, so we are done.
2. Since $0 \leq T^- , T^{+}\leq \vert T \vert$, then by using Corollary 2.9 from [@2], $T^-$ and $T^+$ are two $b$-weakly compact operators, so $T$ is a $b$-weakly compact operator.
Let $T$ be an order bounded operator from Banach lattice $E$ into Dedekind complete Banach lattice $F$. If $c_0$ dose not embed in $F$, then $T$ and $\vert T \vert$ are $b$-weakly compact operators.
At first, assume that $T$ is a positive operator. By Theorem 4.3 [@1], $T$ is continuous. Thus by Proposition 1 [@4], it suffices to show that $(Tx_n)$ is norm convergent to zero for each $b$-order bounded disjoint sequence $(x_n)$ in $E^+$. Now let $(x_n)$ be a $b$-order bounded and disjoint sequence in $E^+$. It follows that there is a $0\leq x^{\prime\prime}\in E^{\prime\prime}$ such that $0\leq x_n\leq x^{\prime\prime}$ for all $n$. Then for each $0\leq x^\prime \in E^\prime$, we have $$\begin{aligned}
x^\prime (\sum_{i=1}^k x_n)=x^\prime (\bigvee_{i=1}^k x_n)\leq x^{\prime\prime}(x^\prime), \quad\text{for~ all}~ k~ \text{holds.}\end{aligned}$$ Hence $x^\prime (\sum_{i=1}^\infty x_n)<\infty$ for all $0\leq x^\prime \in E^\prime$. Then for each $0\leq y^\prime\in F^\prime$ we have $\sum_{i=1}^k y^\prime (T x_n)=\sum_{i=1}^k T^\prime y^\prime (x_n)<+\infty$. It follows that the sequence $(s_m=\sum_{n=1}^m Tx_n)$ is weakly bounded, and by Theorem 2.5.5 [@Megginson], it is norm bounded. Since $c_0$ dose not embed in $F$, by Theorem 4.60 [@1], $F$ is a $KB$-space. Since the sequence $(s_m)$ is positive, increasing and norm bounded, so it is norm convergent, and so the sequence $(\sum_{n=m}^k Tx_n)$ is norm convergent to zero. It follows that $\lim \Vert Tx_n\Vert=0$. Therefore, $T$ is a $b$-weakly compact operator. Now, let $T$ be an order bounded operator. By the identities $T=T^+-T^-$ and $|T|=T^++T^-$, it follows that $T$ and $|T|$ are $b$-weakly compact operators.
\[2.1\] Let $E$ and $F$ be two Banach lattices. Then we have the following assertions.
1. If $T$ is a positive and order weakly compact operator from $E$ onto $F$, then $F$ has $\sigma$-order continuous norm.\[2.1.a\]
2. If a positive $b$-weakly compact operator $T:E \rightarrow F$ between Banach lattices is surjective, then the norm of $F$ is $\sigma$-order continuous.
<!-- -->
1. Assume that $\{y_n\}_n\subseteq F$ with $y_n\downarrow 0$. Since $T$ is surjective, there is an element $x_1\in E$ such that $Tx_1=y_1$. It is clear that $\{y_n\}_n\subseteq [0,y_1]\subseteq T([0,x_1])$. Since $T([0,x_1])$ is relatively weakly compact, there is a subsequence $\{y_{n_j}\}_j$ of $\{y_n\}_n$ such that $y_{n_j} \xrightarrow{w} y_0 \in F$. Since $\{y_{n_j}\}_j$ is a decreasing sequence, by Theorem 3.52 from [@1], $y_{n_j} \xrightarrow{\|\cdot \|} y_0 \in F$. It follows from $y_n\downarrow 0$ that $y_0=0$. Hence $\|y_n\| \rightarrow 0$. Thus $F$ has order continuous norm.
2. Follows from (\[2.1.a\]).
Now by [@1 Theorem 4.11] we have the following result:
\[2.3\] Let $E$ be a normed lattice with order continuous norm. Then the norm completion of $E$ has order continuous norm.
We prove that Theorem 4.11 (2) of [@1] holds. Let $0\leq x_n\uparrow\leq x$ holds in $E$. It follows from [@1 Corollary 4.10] that $E$ is Dedekind complete. Put $y=\sup{x_n}$. So, $y-x_n\downarrow 0$ in $E$. Therefore, $\| y-x_n\|\to 0$. We have $$\|x_n-x_m\|\leq \|x_n-y\|+\| y-x_m\| \to 0,$$ hence $\{x_n\}$ is a norm Cauchy sequence.
\[2.5\] Let $E$ and $F$ be two Banach lattices such that $E$ has order unit and $F$ has order continuous norm. Then every order bounded operator $T : E \rightarrow F$ is b-weakly compact.
Let $A$ be a b-order bounded subset of $E$. Since $E$ has an order unit, $A$ is order bounded in $E$. Therefore, $T(A)$ is order bounded in $F$. Since $F$ has an order continuous norm, $T(A)$ is a relatively weakly compact. Thus $T$ is b-weakly compact operator.
\[1.1, 2.7\] Every order bounded operator from $\ell^ \infty$ into $c_0$ is b-weakly compact.
Let $E$ and $F$ be two Banach lattices where $E$ has order continuous norm. Let $G$ be an order dense sublattice of $E$ and let $T$ be a positive operator from $E$ into $F$. If $T\in W_b(G,F)$, then $T\in W_b(E,F)$.
Let $\{x_n\}$ be a positive increasing sequence in $E$ with $\sup_n\Vert x_n \Vert<\infty$. Since $G$ is order dense in $E$, by Theorem 1.34 from [@1], we have $$\{y\in G:~0\leq y\leq x_n\}\uparrow x_n,$$ for each $n$. Let $\{y_{mn}\}_{m=1}^\infty\subset G$ with $0\leq y_{mn}\uparrow x_n$ for each $n$. Put $z_{mn}=\vee_{i=1}^n y_{mi}$ and $0\leq T\in L(E,F)$. It follows that $z_{mn}\uparrow_m x_n$ and $\sup_{m,n}\Vert z_{mn}\Vert\leq \sup_n\| x_n\|<\infty$. Now, if $T\in W_b(G,F)$, then $\{Tz_{mn}\}$ is norm convergent to some point $y\in F$. Now, we have the following inequalities $$\|Tx_n-Tz_{mn}\|\leq\|T\|\|x_n-z_{mn}\|\leq \|T\|\|x_n-y_{mn}\|\rightarrow 0.$$ Thus by the following inequality proof holds $$\|Tx_n-y\|\leq\|Tx_n-Tz_{mn}\|+\|Tz_{mn}-y\|.$$
Let $E$ and $F$ be two vector lattices. $L^{(1)}_c(E,F)$ (resp. $L^{(2)}_c(E,F)$) is the collection of operators $T\in L_b(E,F)$, which $x_n\xrightarrow{o_1}0$ ($x_n\xrightarrow{o_2}0$) $Tx_{n_k} \xrightarrow{o_1}0$ (resp. $Tx_{n_k} \xrightarrow{o_2}0$) $\{x_{n_k}\}$ is a subsequence of $\{x_{n}\}$.
In [@1b], there are some examples which shows that two classifications of operators $L^{(1)}_c(E,F)$ and $L^{(2)}_c(E,F)$ are different.
\[2.8a\] Let $E$, $F$ be two Banach lattices such that $E$ has order continuous norm. Then
1. $W_b(E,F)^+ \subseteq L^{(2)}_c(E,F)$.
2. If $W_b(E,F)$ is vector lattice and $F$ Dedekind complete, then $W_b(E,F)$ is an order ideal of $ L^{(1)}_c(E,F)= L^{(2)}_c(E,F)$.
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1. Let $T$ be a positive $b$-weakly compact operator and let $\{x_n\}\subset E$ be a strongly order convergence sequence in $E$. Without lose of generality, we set $0\leq x_n\xrightarrow{o_2} 0$, which follows $\{x_n\}$ is norm convergent to zero. Set $\{x_{n_j}\}$ as subsequence with $\sum_{k=1}^{+\infty}\Vert x_{n_j} \Vert< +\infty$. Define $y_m=\sum_{j=1}^m x_{n_j}$. Then $0\leq y_m\uparrow$ and $\sup_m\Vert y_m\Vert<\infty$. Since $T$ is $b$-weakly compact operator, $\{Ty_m\}$ is norm convergent to some point $z\in F$. Now by [@9g], page 7, it has a subsequence as $\{Ty_{m_k}\}$ which is strongly order convergent to $z\in F$. Thus there is $\{z_\beta\}\subset F^+$ and that for each $\beta$ there exists $n_0$ $\vert Ty_{m_k}-z\vert\leq z_\beta\downarrow 0$ whenever $k\geq n_0$. If we set $k\geq k' \geq n_0$, then we have the following inequalities $$\begin{aligned}
0&\leq Tx_{n_{m_k}}\leq \vert Ty_{m_k}-Ty_{m_{k'}}\vert\\
&\leq\vert Ty_{m_k}-z\vert +\vert Ty_{m_{k'}}-z\vert\leq z_\beta+z_\beta\downarrow 0, \end{aligned}$$ which shows that $T\in L^{(2)}_c(E,F)$ and proof immediately follows.
2. By equality $T=T^+-T^-$ and Theorem 1.7 from [@1b], we have $ L^{(2)}_c(E,F)=L^{(1)}_c(E,F)$. Since $W_b(E,F)$ is a vector lattice, it follows from part (1) that $W_b(E,F)$ is a subspace of $ L^{(1)}_c(E,F)$. Now proof follows from the fact that $W_b(E,F)$ satisfies the domination property.
By using conditions of Theorem \[2.8a\], we can design the following question.\
[**Question.**]{} Is $W_b(E,F)$ a band in $ L^{(1)}_c(E,F)=L^{(2)}_c(E,F)$?
\[2.6\] Let $E$ and $F$ be two Banach lattices such that $F$ is a $KB$-space. Then every bounded operator $T : E \rightarrow F$ is b-weakly compact.
By using [@4 Proposition 1], it is enough to show that $\{Tx_n\}_n$ is norm convergent for each b-order bounded increasing sequence $\{x_n\}_n$ in $E^+$. Let $\{x_n\}_n$ be a b-order bounded increasing sequence in $E^+$. Since $F$ is a $KB$-space, by [@1 Theorem 4.63], there exists a $KB$-space $G$, a lattice homomorphism $R : E \rightarrow G$ and an operator $S : G \rightarrow F$ such that $T=S\circ R$. Since $R$ is a lattice homomorphism, $R$ is a positive operator and therefore is b-order bounded. Then $R(x_n)$ is a b-order bounded increasing sequence in $G$. Since $G$ is a $KB$-space, $R(x_n)$ is norm convergent in $G$. It follows that $S\circ R(x_n)$ is also norm convergent in $F$. Hence $\{T(x_n)\}$ is norm convergent in $F$. This completes the proof.
\[2.7\] Let $E$ be a Banach lattice. Then every bounded operator from $E$ into $\ell^1$ is b-weakly compact.
In the following proposition, we show that each Dunford-Pettis operator is b-weakly compact, the converse is not always true. In fact, the identity operator of the Banach lattice $\ell^2$ is b-weakly compact, but it is not Dunford-Pettis. Recall that if $E$ is a Banach lattice and if $0\leqslant{ x^{\prime\prime}} \in{ E^{\prime\prime}}$, then the principal ideal $I_{x^{\prime\prime}}$ generated by $x^{\prime\prime} \in E^{\prime\prime}$ under the norm ${\|\cdot\|}_\infty$ defined by $$\| y^{\prime\prime}\|_\infty =\inf \{\lambda >0: ~| y^{\prime\prime}| \leq \lambda x^{\prime\prime}\},\ y^{\prime\prime} \in {I_{x^{\prime\prime}}},$$ is an $AM$-space with unit $x^{\prime\prime}$, which its closed unit ball coincides with the order interval $[-x^{\prime\prime},x^{\prime\prime}]$.
\[2.8\] Let $E$ be a Banach lattice. Then every b-order bounded disjoint sequence in $E$ is weakly convergent to zero.
Let $\{x_n\}_n$ be a disjoint sequence in $E$ such that $\{x_n\}_n \subseteq{[-x^{\prime\prime},x^{\prime\prime}]}$ for some $x^{\prime\prime} \in E^{\prime\prime}$. Let $Y=I_{x^{\prime\prime}} \cap E$ and equip $Y$ with the order unit norm ${\|\cdot\|}_\infty$ generated by $x^{\prime\prime}$. The space $(Y,{\|\cdot\|}_\infty)$ is an $AM$-space, so, $Y^\prime$ is an $AL$-space and hence its norm is order continuous. Now, by [@11 Theorem 2.4.14] we see that $x_n\xrightarrow{w} 0$.
\[2.9\] Every Dunford-Pettis operator from a Banach lattice $E$ into a Banach space $X$ is b-weakly compact.
Let $T$ be a Dunford-Pettis operator from a Banach lattice $E$ into a Banach space $X$. By [@4 Proposition 1], it is enough to show that $\{Tx_n\}_n$ is norm convergent to zero for each b-order bounded disjoint sequence $\{x_n\}_n$ in $E^+$. Let $\{x_n\}_n$ be a b-order bounded disjoint sequence in $E^+$. As the canonical embedding of $E$ into $E^{\prime\prime}$ is a lattice homomorphism, $\{x_n\}_n$ is an order bounded disjoint sequence in $E^{\prime\prime}$. Thus by preceding lemma, $\{x_n\}_n$ is $\sigma(E, E^{\prime})$ convergent to zero in $E$. Now, since $T$ is Dunford-Pettis, $\{Tx_n\}_n$ is norm convergent to zero. This completes the proof.
As a consequence of [@1 Theorem 5.82], [@8 Theorem 2.2] and [@10 Theorem 2.3], we have the following results.
\[2.10\] Let $E$ be a Banach lattice and let $X$ be a Banach space. Then each b-weakly compact operator from $E$ into $X$ is Dunford-Pettis, if one of the following assertions is valid:
1. $E$ is an $AM$-space.
2. The norm of $E^\prime$ is order continuous and $E$ has the Dunford-Pettis property (i.e. each weakly compact operator from a Banach space $E$ into another $F$ is Dunford-Pettis).
For the next results we need the following lemma:
\[2.15\]
1. If an operator $T$ from a Banach space $X$ into a Banach space $Y$ is compact and $T(X)$ is closed, then $T(X)$ is finite-dimensional. As a consequence, if $T:X\rightarrow Y$ is a surjective compact operator between Banach spaces, then $Y$ is finite-dimensional.
2. If $T:X\rightarrow Y$ is a weakly compact operator between Banach spaces and $T(X)$ is closed, then $T(X)$ is reflexive. As a consequence, if $T:X\rightarrow Y$ is a surjective weakly compact operator between Banach spaces, then $Y$ is reflexive.
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1. Let $T:X\rightarrow Y$ be a compact operator between Banach spaces. Since $\overline{T(X)}=T(X)$, $T(X)$ is a Banach space. If $U$ denotes the open ball of $X$, then $T(U)$ is an open set in $T(X)$. On the other hand, $\overline{T(U)}$ is compact so, $T(X)$ is locally compact and hence $T(X)$ is finite-dimensional.
2. Let $T:X\rightarrow Y$ be a weakly compact operator between Banach spaces. Since $T(X)$ is closed, $T(X)$ is a Banach space and from equality $T(X)=\bigcup_{n\in\mathbb{N}}nT(B_X)$, we see that $T(B_X)$ contains a closed ball of $T(X)$. On the other hand, $\overline{T(B_X)}$ is weakly compact, so, that closed ball is weakly compact, therefore $T(X)$ is reflexive.
\[2.16\] Let $E$ be a Banach lattice and let $X$ be a non-reflexive Banach space. If $T:E\rightarrow X$ is a surjective b-weakly compact operator, then the norm of $E^\prime$ is not order continuous.
If the norm of $E^\prime$ is order continuous then by using [@8 Theorem 2.2] and Lemma \[2.15\], $X$ is reflexive which is a contradiction.
\[2.17\] Let $E$, $F$ be two Banach lattices and $F^\prime$ has order continuous norm. Suppose that $T:E \rightarrow F$ is an almost interval preserving, injective and b-weakly compact operator which has a closed range. Then $E$ is reflexive.
Since $T(E)$ is closed, $T(E)$ is a Banach space, so, $T_1 :E\rightarrow T(E)$ is a bijective operator between two Banach spaces. Then $T_1^\prime :T(E)^\prime \rightarrow E^\prime$ is bijective. Without lose generality we replace $T^\prime$ with $T_1^\prime$. On the other hand, since $T$ is an almost interval preserving, by Theorem 1.4.19 [@11], $T^\prime$ is a lattice homomorphism, and so by Theorem 2.15 [@1], $T^\prime$ and ${(T')^{-1}}$ are both positive operators. Since $F^\prime$ has order continuous norm, by [@1 Theorem 4.59], $F^\prime$ is a $KB$-space, and so $T^\prime$ is b-weakly compact operator and the norm of $E^\prime$ is order continuous. Since $T$ is a b-weakly compact, by [@8 Theorem 2.2], $T$ is weakly compact. So, $T^\prime$ is weakly compact. Now, by Lemma \[2.15\], $E^\prime$ is reflexive and then $E$ is reflexive. This completes the proof.
Recall that a nonzero element $x$ of a Banach lattice $E$ is discrete if the order ideal generated by $x$ is equal to the subspace generated by $x$. The vector lattice $E$ is discrete if it admits a complete disjoint system of discrete elements. For example the Banach lattice $\ell^2$ is discrete but $L^1[0,1]$ is not.
\[2.18\] Let $E$ be a Banach lattice and let $X$ be a Banach space and let $T:E\rightarrow X$ be an injective b-weakly compact operator which its range is closed. If one the following conditions holds, then $E$ is finite dimensional.
1. $E$ is an $AM$-space with order continuous norm.
2. $E$ is an $AM$-space and $E^\prime$ is discrete.
According to the proof of Proposition \[2.17\], $T^\prime:X^\prime \rightarrow E^\prime$ is a surjective operator. By [@10 Proposition 2.3], if one of the above conditions holds, then $T$ is a compact operator. Thus, $T^\prime$ is compact. Now, by Lemma \[2.15\], $E^\prime$ is finite dimensional. Hence, $E$ is finite dimensional.
An operator $T:E\rightarrow F$ between two normed vector lattices is unbounded $b$-weakly compact if $\{Tx_n\}$ is $un$-convergent for every positive increasing sequence $\{x_n\}_n$ in the closed unit ball $B_E$ of $E$.
For normed vector lattices $E$ and $F$, the collection of unbounded $b$-weakly compact operators will be denoted by $UW_b(E,F)$. If a Banach lattice $F$ has strong unit, by using Theorem 2.3 [@10a], we have $W_b(E,F)=UW_b(E,F)$. It is obvious that every $b$-weakly compact operator is unbounded $b$-weakly compact, but the following example shows that the converse does not hold, in general.
\[Ex:12\] Let $I_{c_0}$ be the identity mapping from $c_0$ into itself. Then $I_{c_0}$ is an unbounded $b$-weakly compact operator. But $I_{c_0}$ is not a $b$-weakly compact operator.
The following characterization is obvious.
Let $E$ and $F$ be two normed vector lattices and let $T$ be an operator from $E$ into $F$. Then the following are equivalent:
1. $T$ is unbounded $b$-weakly compact.
2. $\{Tx_n\}$ is norm convergent for each $b$-order bounded increasing sequence $\{x_n\}$ in $E^+$.
It is easy to see that the class of unbounded $b$-weakly compact operators satisfies the domination property.
Let $E$ and $F$ be two normed vector lattices and let $S$ and $T$ be two operators from $E$ into $F$ with $0\leq S\leq T$. If $T$ is unbounded $b$-weakly compact then $S$ is likewise unbounded $b$-weakly compact.
Let $E$, $F$ be two Banach lattices and let $T\in L(E,F)$. If for an ideal $I$ of $E$ the restriction $T\vert_I:I\rightarrow F$ is a surjective homomorphism which is also a $b$-weakly compact operator, then $T\in UW_b(E,F)$.
Let $\{x_n\}$ be a positive increasing sequence in $E$ with $\sup_n\Vert x_n \Vert<\infty$ and let $x\in I$. Then $ x_n\wedge x\in I$, $ 0\leq x_n\wedge x\uparrow$ and $\sup_n\Vert x_n\wedge x \Vert\leq\Vert x_n\Vert<\infty$. Since $T\in W_b(I,F)$, $\{T(x_{n}\wedge x)\}$ is convergent for each $x\in I$. As $T$ is homomorphism and surjective, $\{T(x_{n})\wedge y\}$ is convergent for all $y\in F$ and proof follows.
Let $E$ and $F$ be two Banach lattices and let $T: E\rightarrow F$ be a surjective homomorphism. Then by one of the following conditions we have $T\in UW_b(E,F)$.
1. $E$ is a $KB$-space.
2. $F$ has order continuous norm.
If $E$ is a $KB$-space, then $T\in W_b(E,F)$ and proof follows.\
Assume that $F$ has order continuous norm. Let $\{x_n\}\subset E^+$ be an increasing sequence such that $\sup\Vert x_n\Vert<\infty$ and let $x\in E^+$. Set $y_n=x_n\wedge x$, which follows that $y_n\uparrow\leq x$ and $\sup_n\Vert y_n\Vert\leq \Vert x \Vert$. Since $T$ is lattice homomorphism, $T$ is positive, which follows $Ty_n\uparrow\leq Tx$. By using Theorem 4.11 [@1], $\{Ty_n\}$ is norm Cauchy, and so is norm convergence in $F$. On the other hand, $T(x_n\wedge x)=Tx_n\wedge Tx$. Therefore, $Tx_n\wedge y$ is norm convergent for each $y\in F$.
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[^1]: $^*$Corresponding author
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In this paper, we share our approach to real-time segmentation of fire perimeter from aerial full-motion infrared video. We start by describing the problem from a humanitarian aid and disaster response perspective. Specifically, we explain the importance of the problem, how it is currently resolved, and how our machine learning approach improves it. To test our models we annotate a large-scale dataset of $400,000$ frames with guidance from domain experts. Finally, we share our approach currently deployed in production with inference speed of $20$ frames per second and an accuracy of $92$ (F1 Score).'
author:
- |
Jigar Doshi\
CrowdAI\
`[email protected]`\
Dominic Garcia\
Joint Artificial Intelligence Center\
`[email protected]`\
Cliff Massey\
CrowdAI\
`[email protected]`\
Pablo Llueca\
CrowdAI\
`[email protected]`\
Nicolas Borensztein\
CrowdAI\
`[email protected]`\
Michael Baird\
California Air National Guard\
`[email protected]` Matthew Cook\
Joint Artificial Intelligence Center\
`[email protected]`\
Devaki Raj\
CrowdAI\
`[email protected]`
bibliography:
- 'neurips\_2019.bib'
title: 'FireNet: Real-time Segmentation of Fire Perimeter from Aerial Video'
---
Introduction
============
The effects of climate change continue to be felt around the globe. As the global climate is a complex and interconnected system, more and more of these effects are becoming evident that may not have been noticed before, even in the last few years. One particular impact of climate change that is clear, however, is the increase in wildfires—both in terms of frequency and intensity. Previously, wildfires were a perennial issue only in certain fire-prone locations, such as parts of Australia and the western U.S. Now, however, new geographies and biomes are dealing with wildfire due primarily to the drying of forests, which makes them more susceptible to fire [@ClimateAssessment]. Climate change is not the only variable, however, as human activity is also responsible for the increased spread of wildfire. As more and more homes and infrastructure are built along forest boundaries—or even within them—additional fuel is available for wildfires once they occur.
With increased intensity of wildfire comes increased intensity of the level of damage they do to the natural and built environment. In California alone, the 310 separate incidents in the 2018 wildfire season resulted in over 1.6 million acres burned, 93 fatalities, and more than 23,000 burned structures [@CalFire]. Damage to insured structures just in November 2018 was estimated to top $\$12$ billion, according to the California Insurance Commissioner’s office [@CalDamage1]. The monetary costs of wildfire have an enormous impact on a region, as property insurance companies, reinsurance companies, and individual policyholders must all attempt to cover losses. This has made it more difficult for those returning and rebuilding to find property insurance again, with a six percent increase in cases where the insurer did not renew property insurance from 2017 to 2018 [@CalDamage2; @gupta2019creating; @ioffe2015batch].
Environmental impacts, too, are of concern after a wildfire. By burning hundreds or sometimes thousands of acres of vegetation, wildfires release the carbon trapped within that vegetation into the air, mainly as carbon dioxide. Those same 2018 wildfires in California released an estimated 68 million metric tons of carbon dioxide, roughly the same as the state’s annual emissions from electricity generation [@CalDamage3]. This by-product of the fire reinforces the warming effects of climate change, exacerbating the underlying problem.
Even with potential carbon mitigation and active forestry management efforts in the near- and mid-term, we can expect the upward trend in wildfire frequency, intensity, and damage to continue. With this in mind, it is clear that firefighters and other first responders will need improved or new tools to better understand the extent and behavior of wildfire so that they may plan their response accordingly. In California, the Department of Forestry and Fire Protection (CAL FIRE) is the agency responsible for the stewardship and protection of the roughly 31 million acres of the state’s wildland.
The current procedure for CAL FIRE is to request aerial data to identify and monitor the extent of an active fire is a multi-step process that enables the agency to make a request through government channels for assistance from the California Air National Guard (CA ANG). Once a CA ANG aerial asset and sensor is within range of the fire, the CA ANG team uses near-real-time infrared (IR) full-motion video (FMV) feed to locate the fire perimeter, placing the sensor’s reticle on and around the perimeter of the fire. A team of intelligence analysts, including all source analysts, imagery analysts, and incident awareness and assessment coordinators assess the FMV frames looking for the fire perimeter. The sensor point of interest (where the sensor is looking on the ground) is synced with a mapping tool on a 2D geospatial mapping platform. From here, an analyst must manually point-and-click for hours on-end to create a fire perimeter polygon, often while working around-the-clock shifts. The fire perimeter analysts will analyze the sensor crosshairs, intensity of the IR returns, digital terrain elevation data, and geographical and topographical features, all while looking for possible false positives (such as roads, bodies of water, and flares) in order to make an assessment of where they think the fire perimeter is currently located. Depending on the size and extent of the fire, this process can take hours to complete—even for just a single section of fire perimeter.
This is where automation—specifically in the form of machine learning—can provide a force multiplier, as human/machine teaming reduces the cognitive burden on analysts, freeing them up to focus on higher-order tasks—like contextual assessments, sensemaking, and decision-making—that machines currently cannot accomplish. Because speed and accuracy are both critical to response efforts, machine learning algorithms and a more automated process can be used to provide first responders with more accurate information about the fire perimeter more often and with higher accuracy. Wildfires can spread at rates of 7-10 miles per hour in forested regions, and even more quickly in grasslands. Near-real time updates of the fire perimeter are therefore needed to inform the public and to coordinate response efforts. Accuracy, too, is important, especially as it relates to the precise geolocation of the fire perimeter. Not only will automating the fire perimeter speed up the analysis process, it will also mean that high demand, low density FMV assets can cover more ground. By taking the human out of the loop and putting them on the loop we anticipate a return on investment in both human capital as well as sensing resources in flight resulting in overall better situational awareness
Approach
========
We cast this fire perimeter detection as a segmentation task. Our task is to compute a binary mask of the fire for every input frame of the video. For video segmentation applications, in order to achieve temporal consistency, LSTM- or RNN-like layers are used. However, they are too computationally expensive for an on-device, real-time segmentation application. We build our model in two steps. First, we train a standard over-parameterized model to achieve the best performance. Following that, we prune our models to trade off accuracy and inference speed.
One of the key issues for video segmentation over naive frame-by-frame segmentation is temporal consistency, which causes poor overall performance [@hairSegmentation]. To address these issues, we explore the following two strategies: 1) Adding more frames for context using 3D Convs and 2) Adding the previous prediction as additional channels.
The second goal is to make these models lightweight enough to run above 15 frames per second on a standard K80 Nvidia GPU. This is a hard requirement given that these models are built for real-time segmentation applications. In order to achieve this inference speed, we prune our models along these three dimensions: 1) Input Resolution, 2) Model Thickness, and 3) Model Depth
A typical pruning algorithm in the literature [@net-compression-suau; @net-compression-hu] is a three-stage pipeline, i.e. training (a large model), pruning, and fine-tuning. During pruning, according to a certain criterion, redundant weights are pruned and important weights are kept to best preserve the accuracy. However we found in our experiments that fine-tuning a pruned model only gives comparable or worse performance than training that model with randomly initialized weights. This observation was also validated on wide range of datasets by [@liu2018rethinking]. Hence all our pruning experiments do not follow this three stage approach; we always train the pruned model from scratch.
Dataset
-------
Our dataset consist of short videos (150 frames) captured over the past few years for various wildfires around the U.S. This massive dataset has roughly 400,000 frames annotated. Out of these 400K frames around 100K frames have an active fire in the video. All the videos are captured via on-board infrared (IR) sensor, since the fire perimeter would rarely be visible in the standard RGB spectrum due to smoke and other factors.
The specific computer vision task is to segment out burning or burnt regions in the video. The technical term is fire perimeter, which means entire length of the outer edge of the fire [@FireTerms]. To annotate these videos we consulted subject matter experts from CAL FIRE and the California Air National Guard. Subsequently, once the definitions were agreed upon, we used our annotation vendors to annotate this dataset with constant QA from the experts.
   
Network Architecture
====================
Our main model was inspired by the family of U-Net architectures [@Ronneberger2015-ia; @doshi2018residual], where low-level feature maps are combined with higher-level ones, which enables precise localization. This type of network architecture was designed to effectively solve image segmentation problems, particularly in the medical imaging field. This hour-glass model is generally a default choice for segmentation challenges in Kaggle.
The encoder of the model consists of resnet blocks [@He2015-cq] with the addition of batch norm [@Szegedy2015-bn] and a total of 8 downsampling layers. We decided to keep a constant number of 128 feature maps throughout the network; we will vary this parameter while pruning the model. The decoder is similar to the encoder, where instead of max-pooling, we use deconvolution layers to upsample with a skip connection from the corresponding encoder, combining deep representations of the prior decoder layer with more precise spatial representations from the corresponding encoder layer. All weights are initialized with the He norm [@He2015-td] and all convs are followed by batch norm and the activation function. We used a leaky RELU with a slope of $-0.1x$ as our activation function.
The final head consists of a $1\times1$ convolution followed by a hard sigmoid activation function. We use this activation function instead of a standard sigmoid because of our choice of a sensitive dice loss function. In the case of the soft sigmoid function, the model activation only approaches the extreme values (0 and 1) values which leave a lot of small residual error that adds up and diffuses the model’s ability to focus on ‘harder’ error signals.
3D UNet Model
-------------
We take our existing UNet architecture and add an additional dimension of input. We add the previous 8 frames as input to the model. All the 2D Convs in the encoder and decoders were converted into 3D and the final head was appropriately reshaped to get an output for the latest frame. Surprisingly, this approach did not yield any significant performance improvement however the model was very slow. There is another logical option of both input and output as 8 frames at a time. However, due to the real-time requirements of the system, we cannot have any latency of this nature. Due to the above-mentioned reasons, we abort this line of model exploration.
Adding Previous Frame Predictions (PrevPred)
--------------------------------------------
The idea is to add the previous predictions of the model back as input for the current frame’s prediction. This technique has shown to improve model consistency in video segmentation tasks\[hair seg paper\]. After some hyperparameter search, we found the best result when we added $t-1, t-3, t-5$ frames’ prediction into the model. Adding these frames slightly slows down the model. However, we see a consistent improvement in the performance as shown in table \[Results\].
Pruning
-------
We prune the model’s depth from 8 to 4 without too much loss of performance. In addition, we pruned the model’s thickness down. In the original architecture, we had 128 channels and we prune it down to 64 for the first layer and 32 for the rest. In both the pruning mentioned above pruning any further caused a sharp dip in model performance. We also experimented in reducing the input resolution to the model however that always led to a significant drop in performance, so we kept it at the original resolution.
Training Setup
==============
Data Augmentation
-----------------
We apply a host of data augmentations to the training data during training. We augment the images fed into all the models by randomly scaling between 1/1.05 and 1, rotate, flip, salt and pepper noise, shearing between −5 and 5 degrees. Each image had a 10% chance of having the above augmentations applied. Not surprisingly, adding various data augmentation improved the model performance. It especially improved the model’s temporal consistency. However, even stronger data augmentation reduced model generalization. We think this might be due to the over-perturbation of the training distribution away from the real distribution.
When we add the previous predictions mask back into the model, we apply a few data augmentation techniques.
- **Empty Mask:** We train the model to work well when starting a new clip by adding blank masks for previous frames
- **Affine Transformation:** We perturb the masks with random transformations during training. Small perturbations train the network to be robust to noise, while large perturbation trains the model to ignore these masks and just use the pixels.
- **Ground Truth Masks**: Early on during the training instead of using the model’s predictions from previous frames we use the ground truth annotations. This approach helps the model to learn quickly and also leads to slightly better performance.
Loss Function
-------------
The dice similarity coefficient (DSC) measures the amount of agreement between the model prediction and the ground truth. It is a widely used metric in high class imbalance segmentation tasks [@Milletari2016-jm; @Shen2018-id]
We use a continuous version of the Dice score that allows differentiation and can be used as a loss function in optimization of our network using stochastic gradient based methods.
$$\mathcal { L } _ { D S C } = - \frac { 2\sum _ { i } ^ { N } s _ { i } r _ { i } } { \sum _ { i } ^ { N } s _ { i } + \sum _ { i } ^ { N } r _ { i } + \epsilon}$$
where $s_i$ represents a continuous value from the model for each pixel which is typically an output from a $sigmoid$ or $softmax$ activation function. $r_i$ represents the ground truth annotation for each pixel. $\epsilon$ is a smoothing factor typically set to $1.0$
We experimented with a baseline binary cross-entropy loss and found it to be consistently worse than dice loss across all the configurations.
Optimization
------------
We trained all the models with ADAM [@Kingma2014-vd] using $\beta1$ as 0.9 and $\beta2$ as 0.999 without any weight decay. Unless stated otherwise, we started the training with a learning rate of $2e^{-4}$. We reduce the learning rate by a factor of $2$ after observing no improvement in the validation loss for 10 epochs. All the models were trained for $150$ epochs.
Results
=======
In Table \[Results\], we show our quantitative results. We measure the inference speed in terms of number of frames per sec ($fps$) forward propagated on a single gpu (Nvidia K80). Each column represents a model setup. First column (Basic UNet) is the baseline model which we would like to ideally match in accuracy while achieving a much higher inference speed. The second column is the model where we add previous predictions back into the model for subsequent predictions. Unsurprisingly, the inference speed slows down since the input dimension was increased. The third column is our best pruned model without the addition of previous predictions. This setup achieves the best inference speed however it suffers in terms of accuracy. Finally, the model we use in production is the fourth column model where we add those previous predictions back into the pruned model. This model gives us the right balance between inference speed and accuracy.
Model Basic UNet UNet+ PrevPred Pruned w/o PrevPred Pruned + PrevPred
----------------- ------------ ---------------- --------------------- ------------------- --
Inference Speed $5 fps$ $3 fps$ $22 fps$ $20 fps$
F1 Accuracy $94$ $95$ $86$ $92$
: Each column represents different model and their corresponding inference speed measured in frames per second (fps) and accuracy as measured in F1 score[]{data-label="Results"}
Conclusion and Future Work
==========================
In this work, we show one of the first large-scale approaches to real-time mapping of fire perimeter in disaster situations. We motivate it by sharing the current state of affairs and why automating it would be extremely beneficial during disaster situations. We share our current approach to the problem, which shows good performance both in terms of accuracy and inference speed. In the future, after necessary considerations, we hope to share this dataset and let the community benefit from it.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'This paper considers volume averaging in the quasispherical Szekeres model. The volume averaging became of considerable interest after it was shown that the volume acceleration calculated within the averaging framework can be positive even when the local expansion rate decelerates. This issue was intensively studied within spherically symmetric models. However, since our Universe is not spherically symmetric similar analysis is needed in non-symmetrical models. This papers presents the averaging analysis within the quasispherical Szekeres model which is a non-symmetrical generalisation of the spherically symmetric Lemaître–Tolman family of models. In the quasispherical Szekeres model the distribution of mass over a surface of constant $t$ and $r$ has the form of a mass-dipole superposed on a monopole. This paper shows that when calculating the volume acceleration, $\ddot{a}$, within the Szekeres model, the dipole does not contribute to the final result, hence $\ddot{a}$ only depends on a monopole configuration. Thus, the volume averaging within the Szekeres model leads to literally the same solutions as those obtained within the Lemaître–Tolman model.'
author:
- 'Krzysztof Bolejko$^{1,2}$'
date: 'Received: date / Accepted: date'
title: Volume averaging in the quasispherical Szekeres model
---
Introduction
============
In the standard approach to cosmology it is assumed that the Universe can be described by the homogeneous Friedmann model. Within such a framework in order to correctly describe cosmological observations one needs to postulate the existence of dark energy, which in its simplest form can be represented by a cosmological constant. However, our Universe on scales up to at least 100 Mpc is very inhomogeneous. Thus, it can evolve differently than the homogeneous model. The difference between evolution of the homogeneous model and the inhomogeneous Universe is known as a backreaction effect. The direct study of the dynamical effects of inhomogeneities is difficult since both general matter distribution and the numerical evolution of cosmological models employing the full Einstein equations are unavailable at the level of detail which would make them useful in studying this problem. Currently, therefore, one of the most popular approaches to backreaction is via averaging methods.
In the averaging approach to backreaction, one considers a solution to the Einstein equations for a general matter distribution and then an average of various observable quantities is taken. If several assumptions are introduced (see Sec. \[SBE\]), the averaging leads to the Buchert equations. For a review on backreaction and the Buchert averaging scheme the reader is referred to [@R06; @B08]. Within the Buchert averaging scheme there are a number of examples where it was shown by using spherically symmetric inhomogeneous models that one can obtain negative values of the volume deceleration parameter even if $\Lambda = 0$ [@NT05; @PS06; @KKNNY07; @CGH08; @S08; @BA08].
However, we should be aware that the results of averaging can be gauge-depended [@KKM]. Another problem regarding an application of spherical symmetric models is the problem with the age of the Universe. Within the models studied in [@BA08] those of realistic density distribution and with $q<0$ had very large values of the $t_B$ function, of amplitude $10^{10}$ y (this means that the age of the Universe within such models is unrealistically small[^1]). This feature, however, can be an artefact of assumed spherical symmetry. Therefore, it is of great importance to study averaging in non-symmetrical inhomogeneous models. One of the immediate candidates is the Szekeres model [@S75a]. The Szekeres model is a generalisation of the Lemaitre–Tolman model that has no symmetry [@BST77]. Within the quasispherical Szekeres model one can describe two [@B06] or even three structures [@B07]. Thus, the Szekeres model not only allows us to study how cosmic structures affect their evolution but also enables the examination of the volume acceleration of such systems which consist of several structures. This paper, therefore, addresses the subject of volume averaging in the Szekeres model.
Buchert equations {#SBE}
=================
Since the Buchert averaging scheme involves the volume average it applies to averaging of scalars only – volume averaging of tensors leads to noncovariant quantities (for a review and a detailed discussion about tensor averaging the reader is referred to [@AK97]). The average of a scalar $\Psi$ is then equal to
$${\langle{\Psi}\rangle_{\mathcal{D}}} = \frac{1}{V_{\mathcal{D}}} \int_{\mathcal{D}} {\rm d}^3x \sqrt{-h}
~ \Psi.$$
where $h$ is a determinant of a 3D spatial metric, $h_{\alpha \beta} = g_{\alpha \beta} - u_{\alpha} u_{\beta}$; ${\mathcal{D}}$ is a domain of averaging, and $V_D$ its volume,
$$V_{\mathcal{D}} = \int_{\mathcal{D}} {\rm d}^3x \sqrt{-h}.$$
The Buchert equations are obtained if the following are assumed
- the Universe is filled with an irrotational dust only,
- the metric is of the form $ds^2 = dt^2 - g_{ij} dx^i dx^j$ (3+1 ADM space-time foliation with a constant lapse and a vanishing shift vector).
Then by averaging the Raychaudhuri equation we obtain [@B00]
$$3 \frac{\ddot{a}_{\mathcal{D}}}{a_{\mathcal{D}}} = - 4 \pi {\langle{\rho}\rangle_{\mathcal{D}}} + \mathcal{Q}_{\mathcal{D}},
\label{bucherteq1}$$
where a dot ($\dot{}$) denotes $\partial_t$; the scale factor $a_{\mathcal{D}}$ is defined as a cube root of the volume
$$a_{\mathcal{D}} = (V_{\mathcal{D}}/V_{\mathcal{D}_i})^{1/3},
\label{aave}$$
(where $V_{\mathcal{D}_i}$ is an initial volume); and the backreaction term $\mathcal{Q}_{\mathcal{D}}$ is given by $$\mathcal{Q}_{\mathcal{D}} \equiv \frac{2}{3}\left( {\langle{{\Theta^2}}\rangle_{\mathcal{D}}} - {\langle{ \Theta }\rangle_{\mathcal{D}}}^2 \right)
- 2 {\langle{ \sigma^2}\rangle_{\mathcal{D}}},
\label{qdef}$$ where $\Theta$ is the scalar of expansion and $\sigma$ is the shear scalar. Averaging of the Hamiltonian constraint leads to [@B00]
$$3 \frac{\dot{a}_{\mathcal{D}}^2}{a_{\mathcal{D}}^2} = 8 \pi {\langle{ \rho}\rangle_{\mathcal{D}}} -
\frac{1}{2} {\langle{ \mathcal{R} }\rangle_{\mathcal{D}}} - \frac{1}{2} \mathcal{Q}_{\mathcal{D}}, \label{bucherteq2}$$
where ${\langle{ \mathcal{R} }\rangle_{\mathcal{D}}}$ is an average of the spatial Ricci scalar $^{(3)}
\mathcal{R}$. The above is compatible with (\[bucherteq1\]) if the integrability condition holds
$$\frac{1}{a_{\mathcal{D}}^6} \partial_t \left( \mathcal{Q}_{\mathcal{D}} a_{\mathcal{D}}^6 \right) +
\frac{1}{a_{\mathcal{D}}^2} \partial_t \left( {\langle{R}\rangle_{\mathcal{D}}} a_{\mathcal{D}}^2 \right) = 0.
\label{intcond}$$
Equations (\[bucherteq1\]) and (\[bucherteq2\]) are very similar to the Friedmann equations, where $\mathcal{Q}_{\mathcal{D}}=0$, and $\rho$ and $\mathcal{R}$ depend on time only. In fact, it can be shown that they are kinematically equivalent to Friedmann equations with a scalar field [@BLA06].
Using (\[bucherteq1\]) and (\[bucherteq2\]) we can calculate the deceleration parameter
$$q \equiv - \frac{\ddot{a}_{\mathcal{D}}a_{\mathcal{D}}}{\dot{a}_{\mathcal{D}}^2} = - \frac{- 4 \pi G {\langle{ \rho }\rangle_{\mathcal{D}}} + \mathcal{Q}_{\mathcal{D}}}{8 \pi G {\langle{ \rho }\rangle_{\mathcal{D}}} -
\frac{1}{2} {\langle{ \mathcal{R} }\rangle_{\mathcal{D}}} - \frac{1}{2} \mathcal{Q}_{\mathcal{D}}}.
\label{decparave}$$
In the standard approach to cosmology where Friedmann models are employed the case of $q<0$ implies that $\Lambda > 0$. However, as it was shown within inhomogeneous but isotropic models [@NT05; @PS06; @KKNNY07; @CGH08; @S08; @BA08] we can have $q<0$ even if $\Lambda = 0$. This suggest that the apparent acceleration of the Universe might be explained not by invoking dark energy but by taking into account matter inhomogeneities. In Sec. \[sec4\] we will examine this issue by employing the non-symmetrical Szekeres model.
The Szekeres model {#SZsec}
==================
The metric of the Szekeres model is of the following form [@S75a]
$${\rm d} s^2 = {\rm d} t^2 - X^2 {\rm d} r^2 - Y^2 ( {\rm d}x^2 + {\rm d}y^2). \label{ds2}$$
For our purpose it is more convenient to adopt a pair of complex conjugate coordinates
$$\zeta = x + i y, \quad \bar{\zeta} = x - i y,$$
so that the metric becomes
$${\rm d} s^2 = {\rm d} t^2 - X^2 {\rm d} r^2 - Y^2 {\rm d} \zeta {\rm d} \bar{\zeta}. \label{ds21}$$
where $$X = \frac{{\cal E}(r,\zeta,\bar{\zeta}) Y'(t,r,\zeta,\bar{\zeta})}{\sqrt{\varepsilon-k(r)}}, \quad Y = \frac{\Phi(t,r)}{{\cal E}(r,\zeta,\bar{\zeta})},$$ and $${\cal E}= a(r) \zeta \bar{\zeta} + b(r) \zeta + c(r) \bar{\zeta} + d(r), \quad \varepsilon = 0, \pm 1.$$ Here a prime (’) denotes $\partial_r$.
The case where $\varepsilon = -1$ is often called the quasihyperbolic Szekeres model (for a detailed discussion on the quasihyperbolic Szekeres models see [@HK08]), $\varepsilon = 0$ quasiplane (for details see [@HK08; @K08]), and $\varepsilon = 1$ quasispherical (for details see [@HK02]). Although it is possible to have within one model quasispherical and quasihyperbolic regions separated by the quasiplane region [@HK08], only the quasispherical case will be considered here. This is because the averaging within the quasihyperbolic and quasiplane requires a special treatment. Firstly, an area of a surface of constant $t$ and $r$ in the quasihyperbolic and quasiplane models is infinite. Secondly, there is no origin – in the quasihyperbolic model $r$ cannot be equal to $0$, and in the quasiplane $r$ can only asymptotically approach the origin, $r \rightarrow 0$ [@HK08].
In the quasispherical Szekeres model a surface of constant $t$ and $r$ is a sphere of radius $\Phi(r,t)$ [@S75a]. Thus, the quasispherical Szekeres model is a generalisation of the Lemaître–Tolman model [@L33; @T34]. Within the Szekeres model shells of matter, however, are not concentric. The quasispherical Szekeres model becomes the Lemaître–Tolman (LT) model when ${\cal E}'=0$. In this case
$$X = \frac{\Phi'}{\sqrt{1-k(r)}}, \quad Y = \frac{\Phi}{{\cal E}}, \quad \frac{{\rm d} \zeta {\rm d} \bar{\zeta}}{{\cal E}^2} = {\rm d}{\theta}^2 + \sin^2\theta {\rm d} {\phi}^2.$$
Originally, Szekeres considered only a case of $p=0=\Lambda$. This result was generalised by Szafron [@Szf77] to the case of uniform pressure, $p=p(t)$. A special case of this solution, the cosmological constant, was discussed in detail by Barrow and Stein-Schabes [@BSS84]. In the case of $p=0=\Lambda$, the Einstein equations reduce to
$$\dot{\Phi}^2 = \frac{2M}{\Phi} - k,
\label{vel}$$
$$4 \pi \rho = \frac{M' - 3 M {\cal E}'/{\cal E}}{\Phi^2 ( \Phi' - \Phi {\cal E}'/{\cal E})},
\label{rho}$$
where $M(r)$ is another arbitrary function. In a Newtonian limit $M$ is equal to the mass inside the shell of radial coordinate $r$. Although the $\rho$ function, as seen from (\[rho\]), is a function of all coordinates, it can be shown that the distribution of mass over each single sphere of constant $t$ and $r$ has the structure of a mass dipole superposed on a monopole, $\rho(t,r,\zeta, \bar{\zeta}) = \rho_{mon}(t,r) + \rho_{dip}(t,r,\zeta, \bar{\zeta})$ [@S75b; @dS85; @PK06]. In general case, the orientation of the dipole axis is different on every constant-($t,r$) sphere.
The dipole contribution vanishes when ${\cal E}'=0$ and then the Szekeres model reduces to the LT model.
The scalar of expansion is equal to $$\Theta = u^{\alpha}{}_{;\alpha} =
\frac{\dot{\Phi}' + 2 \dot{\Phi} \Phi'/ \Phi
- 3 \dot{\Phi} {\cal E}'/{\cal E}}{
\Phi' - \Phi {\cal E}'/{\cal E}}.$$ The scalar of shear is
$$\sigma^2 = \frac{1}{3} \left( \frac{\dot{\Phi}' - \dot{\Phi} \Phi'/ \Phi}{\Phi' - \Phi {\cal E}'/{\cal E}} \right)^2.$$
The spatial Ricci scalar $^{(3)}\mathcal{R}$ is equal to
$$^{(3)}\mathcal{R} =
2 \frac{k}{\Phi^2} \left( \frac{ \Phi k'/k - 2 \Phi {\cal E}'/{\cal E}}{
\Phi' - \Phi {\cal E}'/{\cal E}} + 1 \right).
\label{qsz3R}$$
In the LT limit these scalars reduce to
$$\Theta = \frac{\dot{\Phi}'}{\Phi'} + 2 \frac{\dot{\Phi}}{\Phi},
\quad
\sigma^2 = \frac{1}{3} \left( \frac{\dot{\Phi}'}{\Phi'} - \frac{\dot{\Phi}}{\Phi} \right)^2,
\quad
^{(3)}\mathcal{R} = 2 \frac{( \Phi k )'}{\Phi^2 \Phi'}.
\label{thtlt}$$
Averaging in the quasispherical Szekeres model {#sec4}
==============================================
This section considers the volume averaging within the quasispherical Szekeres model. The volume is calculated around the observer located at the origin. It will be shown that the dipole configuration does not affect such quantities as ${\langle{\rho}\rangle_{\mathcal{D}}}$, $\mathcal{Q}_{\mathcal{D}}$, or ${\langle{\mathcal{R}}\rangle_{\mathcal{D}}}$. These functions only depend on the monopole configuration. First let us notice that the volume in the Szekeres model is exactly the same as in the LT model (i.e. if ${\cal E}' = 0$). Following the method presented in [@S75b] we obtain
$$\begin{aligned}
&& V_{\mathcal{D}} = \int\limits\limits_0^{r_{\mathcal{D}}} {\rm d} {r} \int \int {\rm d} \zeta {\rm d} \bar{\zeta} ~
X Y^2 = \int\limits_0^{r_{\mathcal{D}}} {\rm d} {r} \int \int {\rm d} \zeta {\rm d} \bar{\zeta}
\frac{\Phi^2}{\sqrt{1-k}} \left( \Phi' - \Phi \frac{{\cal E}'}{{\cal E}} \right) \frac{1}{{\cal E}^2} \nonumber \\
&& =
\int\limits_0^{r_{\mathcal{D}}} {\rm d} {r}
\left[ \frac{\Phi^2 \Phi'}{\sqrt{1-k}}
\int \int \frac{{\rm d} \zeta {\rm d} \bar{\zeta}}{{\cal E}^2} +
\frac{1}{2} \frac{\Phi^3 }{\sqrt{1-k}} \frac{\partial}{\partial {r}} \left(
\int \int \frac{{\rm d} \zeta {\rm d} \bar{\zeta}}{{\cal E}^2} \right) \right].\end{aligned}$$
Since ${\rm d} \zeta {\rm d} \bar{\zeta}/{\cal E}^2$ is the metric of a unit sphere, hence
$$\int \int \frac{{\rm d} \zeta {\rm d} \bar{\zeta}}{{\cal E}^2} = 4 \pi.$$
Thus
$$V_{\mathcal{D}} = 4 \pi \int\limits_0^{r_{\mathcal{D}}} {\rm d} {r} \frac{\Phi^2 \Phi'}{\sqrt{1-k}}.
\label{vol}$$
The same result is obtained if initially ${\cal E}'$ is set to zero. Thus, the dipole component does not contribute to total volume. As it will be shown below it also does not contribute to ${\langle{\rho}\rangle_{\mathcal{D}}}$.
$$\begin{aligned}
&& {\langle{\rho}\rangle_{\mathcal{D}}} = \frac{1}{V_{\mathcal{D}}} \int\limits_0^{r_{\mathcal{D}}} {\rm d} {r} \int \int {\rm d} \zeta {\rm d} \bar{\zeta} ~
X Y^2 \rho \nonumber \\
&& = \frac{1}{4 \pi V_{\mathcal{D}}} \int\limits_0^{r_{\mathcal{D}}} {\rm d} {r} \int \int \frac{{\rm d} \zeta {\rm d} \bar{\zeta}}{{\cal E}^2} \frac{\Phi^2}{\sqrt{1-k}}
\left( \Phi' - \Phi \frac{{\cal E}'}{{\cal E}} \right)
\frac{M' - 3 M {\cal E}'/{\cal E}}{\Phi^2 \left( \Phi' - \Phi {\cal E}'/{\cal E} \right)} \nonumber \\
&& = \frac{1}{V_{\mathcal{D}}} \int\limits_0^{r_{\mathcal{D}}} {\rm d} {r} \frac{M'}{\sqrt{1-k}} +
\frac{3}{8 \pi V_{\mathcal{D}}} \int\limits_0^{r_{\mathcal{D}}} {\rm d} {r} \frac{M}{\sqrt{1-k}}
\frac{\partial}{\partial {r}} \left(
\int \int \frac{{\rm d} \zeta {\rm d} \bar{\zeta}}{{\cal E}^2} \right) \nonumber \\
&& = \frac{1}{V_{\mathcal{D}}} \int\limits_0^{r_{\mathcal{D}}}{\rm d} {r} \frac{M'}{\sqrt{1-k}}.
\label{arho}\end{aligned}$$
The average of the scalar of the expansion is
$$\begin{aligned}
&& {\langle{\Theta}\rangle_{\mathcal{D}}} = \frac{1}{V_{\mathcal{D}}} \int\limits_0^{r_{\mathcal{D}}}{\rm d} {r} \int \int \frac{ {\rm d} \zeta {\rm d} \bar{\zeta}}{{\cal E}^2}
\frac{\Phi^2}{\sqrt{1-k}} \left( \Phi' - \Phi \frac{{\cal E}'}{{\cal E}} \right) \frac{\dot{\Phi'} + 2 \dot{\Phi}{\Phi'}/ \Phi - 3 \dot{\Phi} {\cal E}'/{\cal E}}{\Phi' - \Phi {\cal E}'/{\cal E}} \nonumber \\
&& = \frac{4 \pi}{V_{\mathcal{D}}} \int\limits_0^{r_{\mathcal{D}}}{\rm d} {r}
\frac{\Phi^2 \Phi'}{\sqrt{1-k}} \left(
\frac{\dot{\Phi}'}{\Phi'} + 2 \frac{\dot{\Phi}}{\Phi} \right).
\label{atht}\end{aligned}$$
As above, ${\cal E}'/{\cal E}$ does not contribute to the final result and, as seen from (\[thtlt\]), the result is the same as in the LT model. To calculate the backreaction term $\mathcal{Q}_{\mathcal{D}}$ we still need to find the average of the following quantity
$$\begin{aligned}
&& \frac{2}{3} {\langle{\Theta^2}\rangle_{\mathcal{D}}} - 2 {\langle{\sigma^2}\rangle_{\mathcal{D}}} = \frac{2}{3} \frac{1}{V_{\mathcal{D}}} \int\limits_0^{r_{\mathcal{D}}}{\rm d} {r} \int \int \frac{ {\rm d} \zeta {\rm d} \bar{\zeta}}{{\cal E}^2}
\frac{\Phi^2}{\sqrt{1-k}} \left( \Phi' - \Phi \frac{{\cal E}'}{{\cal E}} \right) \nonumber \\
&& \times \frac{ \left( \dot{\Phi}' + 2 \dot{\Phi} \Phi' / \Phi - 3 \dot{\Phi} {\cal E}'/{\cal E}
\right)^2 - \left( \dot{\Phi}' - \dot{\Phi} \Phi' / \Phi \right)^2}{
\left( \Phi' - \Phi {\cal E}'/{\cal E} \right)^2}
= \nonumber \\
&& = \frac{2}{3} \frac{1}{V_{\mathcal{D}}} \int\limits_0^{r_{\mathcal{D}}}{\rm d} {r}
\int \int \frac{ {\rm d} \zeta {\rm d} \bar{\zeta}}{{\cal E}^2}
\frac{\Phi^2}{\sqrt{1-k}}
\left( 3 \frac{\dot{\Phi}^2}{\Phi^2} \Phi' + 6 \frac{\dot{\Phi} \dot{\Phi}'}{\Phi}
+ 3 \frac{\dot{\Phi}^2}{\Phi^2} \Phi \frac{{\cal E}'}{{\cal E}} - 12 \frac{\dot{\Phi}^2}{\Phi} \frac{{\cal E}'}{{\cal E}} \right) \nonumber \\
&& = \frac{8 \pi}{3} \frac{1}{V_{\mathcal{D}}} \int\limits_0^{r_{\mathcal{D}}}{\rm d} {r}
\frac{\Phi^2 \Phi'}{\sqrt{1-k}}
\left( 3 \frac{\dot{\Phi}^2}{\Phi^2} + 6 \frac{\dot{\Phi} \dot{\Phi}'}{\Phi \Phi'}
\right).
\label{tms2} \end{aligned}$$
This result is also the same if initially ${\cal E}'$ is set to zero. Finally let us notice that ${\langle{\mathcal{R}}\rangle_{\mathcal{D}}}$ is also the same as in the case of ${\cal E}'=0$. Averaging relation (\[qsz3R\]) yields
$$\begin{aligned}
&& {\langle{\mathcal{R}}\rangle_{\mathcal{D}}} = \frac{1}{V_{\mathcal{D}}} \int\limits_0^{r_{\mathcal{D}}}{\rm d} {r} \int \int \frac{ {\rm d} \zeta {\rm d} \bar{\zeta}}{{\cal E}^2}
\left( \Phi' - \Phi \frac{{\cal E}'}{{\cal E}} \right) \frac{2 k}{\sqrt{1-k}}
\left( \frac{ \Phi k'/k - 2 \Phi {\cal E}'/{\cal E}}{
\Phi' - \Phi {\cal E}'/{\cal E}} + 1 \right) \nonumber \\
&& = \frac{8 \pi}{V_{\mathcal{D}}} \int\limits_0^{r_{\mathcal{D}}}{\rm d} {r} \frac{( \Phi k )'}{\sqrt{1-k}} .
\label{a3r}\end{aligned}$$
Inserting (\[arho\])–(\[a3r\]) into (\[bucherteq1\]) and (\[bucherteq2\]) we see that the dipole configuration contributes neither to $\dot{a}_{\mathcal{D}}$ nor to $\ddot{a}_{\mathcal{D}}$. The volume acceleration, as well as the volume deceleration parameter, depends only on the monopole distribution — it depends only on functions of $t$ and $r$ \[i.e. $M(r)$, $k(r)$ and $\Phi(t,r)$\], and not on $\zeta$ and $\bar{\zeta}$ \[the dependence on ${\cal E}(t,r,\bar{\zeta},{\zeta})$) vanishes\]. Moreover, ${\langle{\rho}\rangle_{\mathcal{D}}}$, $\mathcal{Q}_{\mathcal{D}}$, and ${\langle{\mathcal{R}}\rangle_{\mathcal{D}}}$ are literally of the same form as if initially ${\cal E}'$ is set to zero (the LT case).
Conclusions
===========
In this paper the volume averaging within the quasispherical Szekeres model has been investigated. The Szekeres model is a generalisation of the LT model. The density distribution in the quasispherical Szekeres model has the structure of a time-dependent mass dipole superposed on a monopole. When calculating the volume acceleration ($\ddot{a}$ to be more exact) or volume deceleration parameter ($q$) within the quasispherical Szekeres, the dipole does not contribute to the final result and $\ddot{a}$ only depends on a monopole configuration. The solutions are the same if initially ${\cal E}'$ was set to zero, thus the results and conclusions found when studying averaging within the LT models also apply to the Szekeres models. For example, we can, without any further calculations, conclude that $\mathcal{Q}_{\mathcal{D}} =0$ when $k=0$. This is an implication of the result obtained by Paranjape and Singh [@PS06] who showed that in the parabolic ($k=0$) LT model the backreaction term, $\mathcal{Q}_{\mathcal{D}}$, vanishes. Another result obtained within the LT models which, as has been shown, also appears to apply to the Szekeres models is that $\mathcal{Q}_{\mathcal{D}} >0$ is only possible for unbounded systems, $k<0$ [@S08]. However, as shown in [@BA08], in most cases this also requires that the bang time function, $t_B$, is of amplitude of $10^{10}$ y.
This research has been supported by The Peter and Patricia Gruber Foundation and the International Astronomical Union. I thank Paulina Wojciechowska, Catherine Buchanan and Henk van Elst and the referees for their useful comments and suggestions.
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[^1]: This, however, does not apply to the two-scale averaging approach. For details on the two-scale models see works by Räsänen [@SR06a] or Wiltshire [@W07a; @W07b; @LNW08].
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'We study how the number $c(X)$ of components of a graph $X$ can be expressed through the number and properties of the components of a quotient graph $X/\hspace{-1mm}\sim.$ We partially rely on classic qualifications of graph homomorphisms such as locally constrained homomorphisms and on the concept of equitable partition and orbit partition. We introduce the new definitions of pseudo-covering homomorphism and of component equitable partition, exhibiting interesting inclusions among the various classes of considered homomorphisms. As a consequence, we find a procedure for computing $c(X)$ when the projection on the quotient $X/\hspace{-1mm}\sim$ is pseudo-covering. That procedure becomes particularly easy to handle when the partition corresponding to $X/\hspace{-1mm}\sim$ is an orbit partition.'
author:
- Daniela Bubboloni
title: Graph homomorphisms and components of quotient graphs
---
[^1]
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**Introduction and main results**
=================================
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In algebra it is very common to study the properties of a set, endowed with some structure, by its quotients. Passing to a quotient reduces the complexity and allows one to focus only on certain properties, disregarding inessential details. That idea has revealed to be immensely fruitful especially in group theory, where manageable theorems describe the link between group homomorphisms and quotient groups. In graph theory the notion of quotient graph appears less natural to deal with ([@hahn], [@kna]). That depends in large part on the fact that no notion of kernel is possible for a graph homomorphism. As a consequence it is often difficult to understand which properties are preserved in passing from a graph to a quotient graph. In this paper, developing a theory of graph homomorphisms, we show how to use information on the quotient graph components to get information on the graph components. All the considered graphs are finite, undirected, simple and reflexive, that is, they have a loop on each vertex. Reflexivity simplifies the study of graph homomorphisms without affecting connectivity. Let $X$ and $Y$ be two graphs and let $\varphi$ be a homomorphism from $X$ to $Y.$ Recall that $\varphi$ is called complete if it maps both the vertices and edges of $X$ onto those of $Y$. Our starting point is that dealing with the quotients of a given graph is equivalent to dealing with the complete homomorphisms from it to any possible target graph (Lemmata \[nat-comp\] and \[h\]). Unfortunately, $\varphi$ being complete does not guarantee the image of a component of $X$ being a component of $Y.$ In our opinion, the property which we call “the natural migration of the components”, is mandatory in order to control the number of components of $X$ by means of those in $Y.$ A first type of homomorphisms for which the components naturally migrate is given by those $\varphi$ which we call [*tame*]{}, for which vertices with the same image are connected. In that case the number of components of $X$ and $Y$ is the same (Sections \[general\] and \[hom-pa\]). Moreover, there exist classic qualifications of graph homomorphisms which fit well. Recall that $\varphi$ is called locally surjective if it maps the neighborhood of each vertex of $X$ onto the neighborhood in $Y$ of its image. Locally surjective homomorphisms have a long history in the scientific literature. Everett and Borgatti [@Ev] introduced them, with the name of role colorings, for the analysis of social behavior. Recently this class of homomorphisms has received a lot of attention in theoretical computer science ([@FI; @Ler]).
We state our main results after establishing some notation. Denote by $V_X$ the vertex set of $X$ and by $E_X$ its edge set; by $C_{X}(x)$ the component containing $x\in V_X$; by $\mathcal{C}(X)$ the set of components and by $c(X)$ their number. For every $C'\in \mathcal{C}(Y)$, set $\mathcal{C}(X) _{C'}=\{C\in \mathcal{C}(X)\ :\ \varphi(C)\subseteq C' \}$; for every $y\in V_Y$ and $\hat{X}=(V_{\hat{X}},E_{\hat{X}})$ subgraph of $X$, put $k_{\hat{X}}(y)=|V_{\hat{X}}\cap \varphi^{-1}(y) |$, $\mathcal{C}(X)_{y}=\{C\in\mathcal{C}(X)\ :\ k_C(y)>0\}|$ and $c(X)_{y}=|\mathcal{C}(X)_{y}|$. Denote by $\sim_{\varphi}$ the equivalence relation induced by $\varphi$ and, for every $x\in V_X$, by $C_{X}(x)/\hspace{-1mm}\sim_{\varphi}$ the quotient graph of $C_{X}(x)$ with respect to $\sim_{\varphi}$.
\[loc-sur-com\] Let $X$, $Y$ be graphs and $\varphi:X \rightarrow Y$ be a locally surjective homomorphism.
- If $C\in \mathcal{C}(X)$, then $\varphi(C)\in \mathcal{C}(Y)$. In particular, the image of $X$ is a union of components of $Y$.
- For every $x\in V_X$, $\varphi(C_{X}(x))=C_{Y}(\varphi(x))\cong C_{X}(x)/\hspace{-1mm}\sim_{\varphi}.$
- For every $C\in\mathcal{C}(X)$, $\varphi^{-1}(\varphi(V_C))=\displaystyle{\bigcup_{\hat{C}\in \mathcal{C}(X)_{\varphi(C)}}V_{\hat{C}}}$.
- For $1\leq i\leq c(Y)$, let $y_i\in V_Y$ be such that $\mathcal{C}(Y)=\{C_Y(y_i): 1\leq i\leq c(Y)\}$. Then $$\label{formula}
c(X)=\sum_{i=1}^{c(Y)}c(X)_{y_i}$$
While the numbers $c(X)_{y_i}$ in Formula are generally difficult to compute explicitly, in a number of applications the following property gives a more manageable formula. We call $\varphi$ [*component equitable*]{} whenever, for every $y\in V_Y$, every component in $\mathcal{C}(X)_{y}$ intersects the fibre $\varphi^{-1}(y)$ in a set of the same size (Section \[equi\]). Component equitability transfers the well known notion of an equitable partition ([@god2 Section 5.1]) to components rather than neighborhoods. If $\varphi$ is both locally surjective and component equitable, then $c(X)_{y_i}=k_{X}(y_i)/k_{C_i}(y_i),$ where $k_{X}(y_i)=|\varphi^{-1}(y_i)|$ and $C_i\in \mathcal{C}(X)_{y_i}$ (Proposition \[union2\]).
The most important subset of component equitable homomorphisms is given by the so called [*orbit homomorphisms*]{}, that is, those homomorphisms $\varphi$ for which the equivalence classes of $\sim_{\varphi}$ in $V_X$ coincide with the orbits of a suitable group of graph automorphisms of $X$. Since the complete orbit homomorphisms are necessarily locally surjective (Proposition \[union2cons\]), as a consequence of Theorem A, we get the following important result.
\[summary\] Let $X$, $Y$ be graphs and $\varphi:X \rightarrow Y$ be a complete orbit homomorphism. For $1\leq i\leq c(Y)$, let $y_i\in V_Y$ be such that $\mathcal{C}(Y)=\{C_Y(y_i): 1\leq i\leq c(Y)\}$ and $C_i\in \mathcal{C}(X)_{y_i}.$ Then $$\label{best-for}
c(X)=\sum_{i=1}^{c(Y)}\frac{k_{X}(y_i)}{k_{C_i}(y_i)}$$
We exhibit a precise algorithmic procedure (Procedure \[procedure\]) for the computation of Formula . Moreover, we give some results to control the isomorphism class and the properties of the components (Corollaries \[isolated\] and \[connection\], Proposition \[union2cons\](i) and Section \[iso-class\]).
One of the motivations of our research is to produce a rigorous method to count the components of the proper power graph of a finite group $G$ through the knowledge of the components of some of its quotients. Recall that the power graph of $G$ is the graph $P (G)$ with $V_{P (G)}=G$ and $\{x,y\}\in E_{P (G)}$, for $x,y\in G$, if there exists $m\in\mathbb{N}$ such that $x=y^m$ or $y=x^m$. The proper power graph $P_0 (G)$ is defined as the $1$-deleted subgraph of $P(G).$ While $P (G)$ is obviously connected, $P_0 (G)$ may not be, and the counting of its components is an interesting topic. The reader is referred to [@sur] for survey about power graphs. In two forthcoming papers, [@BIS1] and [@BIS2], we will apply the general method developed here to that issue, with particular attention to permutation groups. Actually, if $G$ is the symmetric or the alternating group there exists a complete orbit homomorphism which is very natural to be considered for an application of Theorem B. Those results seem to also have promising applications to simple and almost simple groups.
In addition to developing the tools for counting components of a graph using homomorphisms, we also compare various classes of homomorphisms (Lemma \[intersection\], Propositions \[propc\] and \[union2cons\]).
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**Graphs**
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0.4 true cm For a finite set $A$ and $k\in \mathbb{N}$, let $\binom{A}{k}$ be the set of the subsets of $A$ of size $k.$ A [*graph*]{} $X=(V_X,E_X)$ is a pair of finite sets such that $V_X\neq\varnothing$ is the set of vertices, and $E_X$ is the set of edges which is the union of the set of [*loops*]{} $L_X=\binom{V_X}{1}$ and a set of [*proper edges*]{} $E^*_X\subseteq \binom{V_X}{2}$. Note that $E^*_X$ may be empty. We usually specify the edges of a graph $X$ giving only $E^*_X$.
Let $X$ be a graph. A [*subgraph*]{} $\hat{X}=(V_{\hat{X}},E_{\hat{X}})$ of $X$ is a graph such that $V_{\hat{X}}\subseteq V_{X}$ and $E_{\hat{X}}\subseteq E_{X}$. If $\hat{X}$ is a subgraph of $X$, we write $\hat{X}\subseteq X.$ For $s\in\mathbb{N}\cup\{0\}$, a subgraph $\gamma$ of $X$ such that $V_{\gamma}=\{x_i: 0\leq i\leq s\}$ with distinct $x_i\in V_X$ and $E^*_{\gamma}=\{\{x_i,x_{i+1}\} : 0\leq i\leq s-1\}$, is called a [*path*]{} of length $s$ between $x_0$ and $x_s$. Given $U\subseteq V_X$, the [*subgraph induced*]{} by $U$ is the subgraph $\hat{U}$ of $X$ having $V_{\hat{U}}=U$ and $E_{\hat{U}}=\{\{x_1,x_2\}\in E_X:x_1,x_2\in U\}$. A subgraph is called [*induced*]{} if it is the subgraph induced by some subset of vertices. Two vertices $x_1,x_2\in V_X$ are said to be [*connected*]{} in $X$ if there exists a path between $x_1$ and $x_2.$ $X$ is called [*connected*]{} if every pair of its vertices is connected. It is well known that connectedness is an equivalence relation on $V_X.$ Any subgraph of $X$ induced by a connectedness equivalence class, is called a [*component*]{} of $X$. Equivalently, a component of $X$ is a maximal connected subgraph of $X$. It is easily checked that the vertices (the edges) of the components of $X$ give a partition of $V_X$ ($E_X$); a connected subgraph $\hat{X}$ of $X$ is a component if and only if $x_1\in V_{\hat{X}}$ and $\{x_1,x_2\}\in E_X$ imply $\{x_1,x_2\}\in E_{\hat{X}}.$ The component of $X$ containing $x\in V_X$ is denoted by $C_{X}(x)$. If the only vertex of $C_{X}(x)$ is $x$, we say that $x$ (the component $C_{X}(x)$) is an [*isolated vertex*]{}. The set of components of $X$ is denoted by $\mathcal{C}(X)$ and its size by $c(X).$ Given $x\in V_X$, the [*neighborhood*]{} of $x$ is the subset of $V_X$ defined by $N_X(x)=\{u\in V_X: \{x,u\}\in E_X\}.$ Note that $x\in N_X(x)$ by reflexivity. When dealing with a unique fixed graph $X$, we usually omit the subscript $X$ in all the above notation. The terminology not explicitly introduced is standard and can be find in [@dl].
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**Quotient graphs and number of components** {#general}
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Let $X=(V,E)$ be a graph and $\sim$ be an equivalence relation on $V$. For every $x\in V,$ denote by $[x]$ the equivalence class of $x$ and call it a [*cell*]{}. Thus, for $x,y\in V$, we have $[x]=[y]$ if and only if $x\sim y$ and the elements of the partition $V/\hspace{-1mm}\sim$ of $V$ associated to $\sim$ are represented by $[x]$, for $x\in V.$ The [*quotient graph*]{} of $X$ with respect to $\sim$, denoted by $X/\hspace{-1mm}\sim,$ is the graph with vertex set $[V]=V/\hspace{-1mm}\sim$ and edge set $[E]$ defined as follows: for every $[x_1]\in [V]$ and $[x_2]\in [V]$, $\{[x_1],[x_2]\}\in [E]$ if there exist $\tilde{x}_1,\tilde{x}_2\in V$ such that $\tilde{x}_1\sim x_1,\ \tilde{x}_2\sim x_2$ and $ \{\tilde{x}_1,\tilde{x}_2\}\in E.$
Passing from a graph $X$ to a quotient graph $X/\hspace{-1mm}\sim$ reduces the complexity and obviously different equivalence relations imply different levels of complexity reduction. For instance, in the extreme case of the total equivalence relation, which reduces $X$ to a single vertex, all information about the graph $X$ is lost. By an appropriate choice of the equivalence relation, we may produce a less complex quotient graph while maintaining a relationship between components of the graph and its quotient. The easiest case is when the equivalence classes are each contained in a single component, in which case $c(X)=c(X/\hspace{-1mm}\sim).$
\[sim\] [Let $X=(V,E)$ be a graph and $\sim$ be an equivalence relation on $V$. We say that $\sim$ is [*tame*]{} if for every $x,\tilde{x}\in V$, $[x]=[\tilde{x}]$ implies $C_X(x)=C_X(\tilde{x}).$ We say that $X/\hspace{-1mm}\sim$ is a [*tame*]{} quotient of $X$ if $\sim$ is tame.]{}
Obviously every graph $X$ admits tame equivalence relations on its vertex set. One example is given by the relation identifying all the vertices in the same component. Note also that, if $X$ is connected, each equivalence relation $\sim$ on $V$ is tame.
\[quotient-graph\] Let $X=(V,E)$ be a graph and $\sim$ be an equivalence relation on $V$. Then:
- $c(X/\hspace{-1mm}\sim)\leq c(X)$;
- $c(X/\hspace{-1mm}\sim)=c(X)$ if and only if $\sim$ is tame;
- $X$ is connected if and only if $X/\hspace{-1mm}\sim$ is connected and tame.
Note first that the map $f: \mathcal{C}(X)\rightarrow \mathcal{C}(X/\hspace{-2mm}\sim)$ defined by $f(C_{X}(x))=C_{X/\sim}([x])$ for all $x\in V$ is well defined as the quotient construction respects adjacency, and hence connectedness of any pair of vertices.
\(i) The map $f$ is obviously surjective, so $$c(X/\hspace{-1mm}\sim)=|\mathcal{C}(X/\hspace{-1mm}\sim)|\leq |\mathcal{C}(X)|=c(X).$$
\(ii) By (i) and by the definition of $f$, $c(X/\hspace{-1mm}\sim)=c(X)$ holds if and only if $C_{X/\sim}([x])=C_{X/\sim}([y])$ implies $C_{X}(x)=C_{X}(y)$ for all $x,y\in V.$ Suppose $c(X/\hspace{-1mm}\sim)=c(X)$ and let $[x]=[y]$, for some $x,y\in V.$ Then $C_{X/\sim}([x])=C_{X/\sim}([y])$ and therefore $C_{X}(x)=C_{X}(y)$, so $\sim$ is tame. Conversely, suppose $\sim$ is tame and let $C_{X/\sim}([x])=C_{X/\sim}([y])$ for some $x,y\in V.$ Then in $X/\hspace{-1mm}\sim$ there is a path $\gamma$ between $[x]$ and $[y]$. Observe first that if $u,v\in V$ are such that $\{[u],[v]\}\in [E]$, then $u$ and $v$ are connected in $X$. Indeed, by definition of edge in a quotient graph, there exist $\tilde{u},\tilde{v}\in V$ such that $[\tilde{u}]=[u],\ [\tilde{v}]=[v]$ and $ \{\tilde{u},\tilde{v}\}\in E.$ Thus $\tilde{u}$ and $\tilde{v}$ are connected in $X$ and, $\sim$ being tame, $u$ and $\tilde{u}$ as well as $v$ and $\tilde{v}$ are connected in $X$. By transitivity of connectedness, we then also have $u$ and $v$ connected in $X$. Now, by an obvious inductive argument on the length of $\gamma$, we deduce that $x$ and $y$ are connected in $X$. Thus $C_{X}(x)=C_{X}(y).$
\(iii) Let $X$ be connected. Then $\sim$ is trivially tame. Moreover $c(X)=1$ so that, by (i), $c(X/\hspace{-1mm}\sim)=1$ which says that $X/\hspace{-1mm}\sim$ is connected. Conversely, let $X/\hspace{-1mm}\sim$ be connected and tame. Then, by (ii), $c(X)=c(X/\hspace{-1mm}\sim)=1$ and so $X$ is connected.
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**Homomorphisms of graphs and partitions** {#hom-pa}
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Let $X$ be a graph and suppose that you want to compute $c(X)$ by looking at the components of a quotient $X/\hspace{-1mm}\sim$ whose components are easier to interpret. To that end, dealing only with tame quotients is surely too restrictive. It turns out to be useful to introduce quotients which substantially reduce the complexity of $X$ at the cost of changing, in some controlled way, the number of components. To develop this idea we must isolate a set of crucial definitions qualifying the graph homomorphisms. Recall that the word graph always means a finite, undirected, simple and reflexive graph. Throughout the next sections, let $X$, $Y$ be fixed graphs. We do not explicitly repeat that assumption any more.
Maps and admissibility
----------------------
Let $A$ be a set and $\varphi:V_X \rightarrow A$ be a map. For every $y\in A$ the subset of $V_X$ given by $\varphi^{-1}(y)$ is called the [*fibre*]{} of $\varphi$ on $y$. The relation $\sim_{\varphi}$ on $V_X$ defined, for every $x,y\in V_X$, by $x\sim_{\varphi}y$ if $\varphi(x)=\varphi(y),$ is an equivalence relation. The equivalence classes of $\sim_{\varphi}$ are called $\varphi$-[*cells*]{} and coincide with the nonempty fibres of $\varphi$. We call $\sim_{\varphi}$ the equivalence relation induced by $\varphi$ and denote the corresponding quotient graph by $X/\hspace{-1mm}\sim_{\varphi}$. The above considerations allow us to transfer terminology from partitions to maps.
Given $U\subseteq V_X$ and $y\in A$, define the [*multiplicity*]{} of $y$ in $U$ by the non-negative integer $$k_{U}(y)=|U\cap \varphi^{-1}(y) |.$$ In other words $k_{U}(y)$ is the size of the intersection between $U$ and the fibre of $\varphi$ on $y$. We say that $y$ is [*admissible*]{} for $U$ (or $U$ is admissible for $y$), if $k_{U}(y)>0$. Note that $y$ is admissible for $U$ if and only if $y\in \varphi(U).$ Thus $\varphi(U)$ is the subset of elements of $A$ admissible for $U.$ If $\hat{X}$ is a subgraph of $X$ we adopt the same language referring to $V_{\hat{X}}$ and we define $k_{\hat{X}}(y)$ by $k_{V_{\hat{X}}}(y).$ In the sequel, the concepts of admissibility and of multiplicity reveal themselves very useful when the subgraph under consideration is a component of $X$. Note that $k_{X}(y)$ is simply the size of the fibre $\varphi^{-1}(y)$. We will usually apply the above ideas when $A$ is the vertex set of some graph.
Homomorphisms
-------------
Let $\varphi:V_X \rightarrow V_Y$ be a map. Then $\varphi$ is called a [*homomorphism*]{} from $X$ to $Y$ if, for each $x_1,x_2\in V_X,$ $\{x_1,x_2\}\in E_X$ implies $\{\varphi(x_1),\varphi(x_2)\}\in E_Y$. The set of the homomorphisms from $X$ to $Y$ is denoted by $\mathrm{Hom}(X,Y).$ $\varphi\in\mathrm{Hom}(X,Y)$ is called [*surjective*]{} ([*injective, bijective*]{}) if $\varphi:V_X\rightarrow V_Y$ is surjective (injective, bijective). We denote the set of surjective homomorphisms from $X$ to $Y$ by $\mathrm{Sur}(X,Y).$
Note that a map $\varphi:V_X \rightarrow V_Y$ is a homomorphism from $X$ to $Y$ if and only if $$\label{hom-neigh}
\forall x\in V_X,\ \varphi(N_X(x))\subseteq N_Y(\varphi(x)).$$
Let $\varphi\in\mathrm{Hom}(X,Y).$ Observe that $\varphi$ may map a proper edge of $X$ to a loop of $Y.$ Moreover, $\varphi$ induces a map between $E_X$ and $E_Y,$ associating to every edge $e=\{x_1,x_2\}\in E_X$, the edge $\varphi(e)=\{\varphi(x_1),\varphi(x_2)\}\in E_Y$. We denote that map between $E_X$ and $E_Y$ again with $\varphi.$ We also use the notation $\varphi:X \rightarrow Y$ to indicate the homomorphism $\varphi.$
An important example of surjective homomorphism is given by the projection on the quotient. Consider a quotient graph $X/\hspace{-1mm}\sim$ and let $\pi:V_X\rightarrow [V_X]$ be the map defined by $\pi(x)=[x],$ for all $x\in V_X$. If $\{x_1,x_2\}\in E_X$, then we surely have $\{[x_1],[x_2]\}\in [E_X]$. Thus $\pi\in \mathrm{Sur}(X,X/\hspace{-1mm}\sim)$ and $\pi$ is called the [*projection*]{} on the quotient graph.
If $\hat{X}$ is a subgraph of $ X$, then the [*image*]{} of $\hat{X}$ by $\varphi\in\mathrm{Hom}(X,Y)$ is defined as the subgraph of $Y$ given by $\varphi(\hat{X})=(\varphi(V_{\hat{X}}), \varphi(E_{\hat{X}}))$. Observe that, generally, if $\hat{X}\subseteq X$ then $\varphi(\hat{X})$ is not an induced subgraph of $Y$. In particular, the condition $\varphi\in \mathrm{Sur}(X,Y)$ is weaker than $\varphi(X)=Y,$ because the surjectivity requires only $\varphi(V_X)=V_Y$ while $\varphi(X)=Y$ requires both $\varphi(V_X)=V_Y$ and $\varphi(E_X)=E_Y.$
\[comp\]
Let $\varphi\in \mathrm{Hom}(X,Y)$. Then $\varphi$ is called:
- [*complete*]{} if $\varphi(X)=Y$. We denote the set of complete homomorphisms from $X$ to $Y$ by $\mathrm{Com}(X,Y)$;
- an [*isomomorphism*]{} if $\varphi$ is bijective and complete. We denote the set of isomorphisms from $X$ to $Y$ by $\mathrm{Iso}(X,Y)$. If $ \mathrm{Iso}(X,Y)\neq \varnothing,$ we say that $X$ and $Y$ are isomorphic and we write $X\cong Y$;
- [*tame*]{} if $\sim_{\varphi}$ is tame. We denote the set of tame homomorphisms from $X$ to $Y$ by $\mathrm{T}(X,Y).$
We make a few comments on these definitions. First of all note that $\varphi$ is tame if and only if every fibre of $\varphi$ is connected. Note also that the composition of complete homomorphisms is a complete homomorphism and that $$\label{sub}
\mathrm{Iso}(X,Y)\subseteq \mathrm{Com}(X,Y)\subseteq \mathrm{Sur}(X,Y)\subseteq \mathrm{Hom}(X,Y).$$ Finally note that, each homomorphism $\varphi\in \mathrm{Hom}(X,Y)$ induces a complete homomorphism from $X$ to $\varphi(X)$. Our strong interest in completeness is motivated by the fact that the projection on the quotient graph is a complete surjective homomorphism.
\[nat-comp\] Let $X$ be a graph and $\sim$ an equivalence relation on $V_X$. Then $\pi\in\mathrm{Com}(X,X/\hspace{-1mm}\sim).$
Since $\pi\in \mathrm{Sur}(X,X/\hspace{-1mm}\sim),$ we need only check that $[E_X]\subseteq \pi(E_X).$ Pick $e=\{[x_1],[x_2]\}\in [E_X]$, with $x_1,x_2\in V_X$. Then, by definition of quotient graph, there exist $\tilde{x}_1,\tilde{x}_2\in V_X$ such that $\tilde{x}_1\sim x_1,\ \tilde{x}_2\sim x_2$ and $ \{\tilde{x}_1,\tilde{x}_2\}\in E_X.$ Thus, we have $\pi(\{\tilde{x}_1,\tilde{x}_2\})=\{\pi(\tilde{x}_1),\pi(\tilde{x}_2)\}=e.$
The following lemma shows that $X/\hspace{-1mm}\sim_{\varphi}$ is isomorphic to $Y$ when $\varphi\in \mathrm{Com}(X,Y)$ and enables us to interpret every quotient graph of $X$ as the image of $X$ under a complete homomorphism.
\[h\] Let $\varphi\in \mathrm{Hom}(X,Y)$ and let $$\tilde{\varphi}:X/\hspace{-1mm}\sim_{\varphi}\rightarrow Y$$ be the map defined by $\tilde{\varphi}([x])=\varphi(x)$ for all $[x]\in [V_X].$ Then:
- $\tilde{\varphi}$ is an injective homomorphism, and $\tilde{\varphi}$ is surjective if and only if $\varphi$ is surjective;
- $\tilde{\varphi}$ is an isomorphism if and only if $\varphi$ is complete.
\(i) This is just .
\(ii) Suppose $\varphi$ is complete. Thus $\varphi$ is also surjective and, by (i), $\tilde{\varphi}$ is a bijective homomorphism. On the one hand, due to $\tilde{\varphi}([E_X])=\varphi(E_X)=E_Y,$ $\tilde{\varphi}$ is also complete and hence an isomorphism. Assume now that $\tilde{\varphi}$ is an isomorphism. By definition of $\tilde{\varphi}$, we have $\varphi=\tilde{\varphi}\circ \pi$. On the other hand, by Lemma \[nat-comp\], $\pi$ is complete and by , also $\tilde{\varphi}$ is complete. Thus $\varphi$ is complete because it is a composition of complete homomorphisms.
Equitable and orbit partitions {#equi-orb}
------------------------------
We recall some classic types of partitions and extend the definitions to the context of homomorphisms.
\[def-eq\]
Let $\mathcal{P}=\{P_1,\dots, P_k\}$ be a partition of $V_X$. Then $\mathcal{P}$ is called:
- an [*equitable partition*]{} if, for every $i,j\in \{1,\dots, k\}$, the size of $N_X(x)\cap P_j$ is the same for all $x\in P_i$. We call $\varphi\in \mathrm{Hom}(X,Y)$ an [*equitable homomorphism*]{} if the partition into $\varphi$-cells is equitable. The set of equitable homomorphisms is denoted by $ \mathrm{E}(X,Y)$;
- an [*orbit partition*]{} if $\mathcal{P}$ is the set of orbits of some $\mathfrak{G}\leq \mathrm{Aut}(X)$. We call $\varphi\in \mathrm{Hom}(X,Y)$ an [*orbit homomorphism*]{} (with respect to $\mathfrak{G}$) if the partition into $\varphi$-cells is an orbit partition (with respect to $\mathfrak{G}$). The set of orbit homomorphisms is denoted by $ \mathrm{O}(X,Y).$ If $\varphi\in \mathrm{O}(X,Y)$ is an orbit homomorphism with respect to $\mathfrak{G}$ we say briefly that $\varphi$ is $\mathfrak{G}$-[*consistent*]{} or that $\mathfrak{G}$ is $\varphi$-[*consistent*]{}.
It is well known that any orbit partition is an equitable partition but the converse does not hold ([@Ler Proposition 9.3.5]). Thus we have $ \mathrm{O}(X,Y)\subseteq \mathrm{E}(X,Y)$ with a proper inclusion in general. Since the partition with each cell containing just a vertex is the orbit partition relative to the identity subgroup of $\mathrm{Aut}(X)$, we also have $\mathrm{Iso}(X,Y)\subseteq \mathrm{O}(X,Y)\cap \mathrm{Com}(X,Y).$ Once the graph $X$ is fixed, the homomorphisms $\varphi\in \mathrm{O}(X,Y)\cap\mathrm{Com}(X,Y),$ for some graph $Y,$ can be easily described in terms of graph automorphisms of $X$. Indeed, pick $\mathfrak{G}\leq \mathrm{Aut}(X)$ and let $\sim_\mathfrak{G}$ be the corresponding orbit partition of $V_X.$ Then the projection onto the quotient graph $Y=X/\hspace{-1mm}\sim_\mathfrak{G}$ belongs to $\mathrm{O}(X,Y)\cap\mathrm{Com}(X,Y)$. Conversely let $\varphi\in \mathrm{O}(X,Y)\cap\mathrm{Com}(X,Y)$ be an orbit homomorphism with respect to $\mathfrak{G}\leq \mathrm{Aut}(X)$. Then, by Lemma \[h\], $\varphi$ coincides up to an isomorphism with the projection on $X/\hspace{-1mm}\sim_{\varphi}=X/\hspace{-1mm}\sim_\mathfrak{G}.$
The following equivalent formulation for the $\varphi$-consistency is immediate.
\[phi-con\] Let $\varphi\in\mathrm{Hom}(X,Y).$ A group $\mathfrak{G}\leq {\mathrm {Aut}}(\Gamma)$ is $\varphi$-consistent if and only if the following two conditions are satisfied:
- $\varphi\circ f=\varphi,\ \forall f\in \mathfrak{G};$
- for each $x_1,x_2\in V_X$ with $\varphi(x_1)=\varphi(x_2)$, there exists $f\in \mathfrak{G}$ such that $x_2=f(x_1).$
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**Homomorphisms and components**
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Given a generic $\varphi\in \mathrm{Hom}(X,Y)$, the relation between the components in the graphs $X$ and $Y$ is quite poor. Obviously, the following fact holds.
\[quasi-path\] Let $\varphi\in \mathrm{Hom}(X,Y)$. If $\hat{X}$ is a connected subgraph of $X$ then $\varphi(\hat{X})$ is connected.
Thus, if $C\in\mathcal{C}(X)$, then $\varphi(C)$ is a connected subgraph of $Y$ but it is not necessarily a component. The best we can say is that there exists a unique component $C'\in\mathcal{C}(Y)$ such that $\varphi(C)\subseteq C'.$ Unfortunately things do not improve if $\varphi\in \mathrm{Com}(X,Y).$ Consider as a very basic example, the graph $X$ with $$V_X=\{1a,1b, 2,3\}, \quad E^*_X=\{\{1a,3\}, \{1b,2\}\}$$ and the equivalence relation $\sim$ on $V_X$ defined only by $1a\sim 1b$. Then $Y=X/\hspace{-1mm}\sim$ is connected and is a path of length $2.$ Now look at the complete homomorphism $\pi:X\rightarrow Y$ given by the natural projection. $\pi$ takes the component $C$ of $X$ having $V_C=\{1a,3\}$ into the connected subgraph $\pi(C)$ such that $V_{\pi(C)}=\{[1a],[3]\}$ and $E^*_{\pi(C)}= \{\{[1a],[3]\}\}$. Thus $\pi(C)$, being a path of length $1,$ is different from the only component of $Y$. Nevertheless there is a specific situation which is worth discussing.
\[main-component\] Let $\varphi\in \mathrm{Com}(X,Y)$ and assume that every component of $X$ apart from a unique $C\in \mathcal{C}(X)$ is an isolated vertex. Let $C'\in \mathcal{C}(Y)$ be the only component of $Y$ such that $\varphi(C)\subseteq C'$. If $V_{C'}=V_{\varphi(C)},$ then $\varphi(C)=C'.$
We know that $C'$ and $\varphi(C)$ have the same vertices so that we just need to show that they also have the same edges. Since a component is always an induced subgraph, we trivially have $E_{\varphi(C)}\subseteq E_{C'}.$ To show the other inclusion it is enough to show that $E^*_{C'}\subseteq E_{\varphi(C)}.$ Let $e'=\{y_1,y_2\}\in E^*_{C'}$, for some distinct $y_1,y_2\in V_{C'}$. Then, by the completeness of $\varphi$, there exist $x_1,x_2\in V_X$ such that $\varphi(x_1)=y_1,\varphi(x_2)=y_2 $ and $e=\{x_1,x_2\}\in E_X.$ As $y_1\neq y_2$ we also have $x_1\neq x_2$. Thus $e\in E^*_X$, which implies that $x_1$ and $x_2$ are not isolated in $X$. But if a component of $X$ is not an isolated vertex, it coincides with $C$. It follows that $x_1,x_2\in V_{C}$ and so $e\in E_{C}.$ Hence $e'=\varphi(e)\in E_{\varphi(C)}.$
We now consider some well known types of homomorphisms. By , every graph homomorphism $\varphi\in \mathrm{Hom}(X,Y)$ maps $N_X(x)$ into $N_Y(\varphi(x))$ for all $x\in V_X.$ Denoting by $\varphi_{\mid N_X(x)}:N_X(x)\rightarrow N_Y(\varphi(x))$ the corresponding restriction homomorphism, the [*locally constrained graph homomorphisms*]{} are those requiring an additional condition on the map $\varphi_{\mid N_X(x)}$ for all $x\in V_X.$
\[classic\] [Let $\varphi\in \mathrm{Hom}(X,Y)$. Then $\varphi$ is called [*locally surjective (injective, bijective)*]{} if, for every $x\in V_X$, $\varphi_{\mid N_X(x)}$ is surjective (injective, bijective). We denote the set of the locally surjective (injective, bijective) homomorphisms by $\mathrm{LSur}(X,Y)$ (by $\mathrm{LIn}(X,Y)$, $\mathrm{LIso}(X,Y)$).]{}
An exhaustive survey of the three types of locally constrained graph homomorphisms defined above is given in [@FIKRA] to which we refer the reader for a wide overview on the many applications in different areas, from graph theory and combinatorial topology to computer science and social behaviour. We will be particularly interested in the locally surjective homomorphisms because they represent a manageable and wide class of homomorphisms which guarantee the natural migration of the components (see Proposition \[hom-con\]). Note that, by , $\varphi\in \mathrm{Hom}(X,Y)$ is locally surjective if and only if $$\label{hom-neigh-ls}
\forall x\in V_X,\ N_Y(\varphi(x))\subseteq \varphi(N_X(x)).$$ Note also that being locally surjective does not imply being surjective.
We next recall the class of locally strong homomorphisms which, appearint for the first time in [@Pul], were later used in the study of the endomorphism spectrum of a graph [@kna-paper].
\[locstr\] [Let $\varphi\in \mathrm{Hom}(X,Y)$. Then $\varphi$ is called [*locally strong*]{} if, for every $x_1,x_2\in V_X$, $\{\varphi(x_1),\varphi(x_2)\}\in E_Y$ implies that, for every $\tilde{x}_1\in \varphi^{-1}(\varphi(x_1))$, there exists $\tilde{x}_2\in \varphi^{-1}(\varphi(x_2))$ such that $\{\tilde{x}_1,\tilde{x}_2\}\in E_X.$ We denote the set of the locally strong homomorphisms by $\mathrm{LS}(X,Y).$]{}
We show that being locally surjective implies being locally strong and that these two classes coincide in the context of surjective homomorphisms. To this end, we first present a useful characterisation of the locally strong homomorphisms.
\[formulation\] $\varphi\in \mathrm{LS}(X,Y)$ if and only if, for every $x_1,x_2\in V_X$, $\{\varphi(x_1),\varphi(x_2)\}\in E_Y$ implies that there exists $\tilde{x}_2\in \varphi^{-1}(\varphi(x_2))$ such that $\{x_1,\tilde{x}_2\}\in E_X.$
Let $\varphi\in \mathrm{LS}(X,Y)$ and let $x_1,x_2\in V_X$ be such that $\{\varphi(x_1),\varphi(x_2)\}\in E_Y$. Since $x_1\in \varphi^{-1}(\varphi(x_1))$, $\varphi$ locally strong implies that there exists $\tilde{x}_2\in \varphi^{-1}(\varphi(x_2))$ with $\{x_1,\tilde{x}_2\}\in E_X.$
Assume next that, for every $x_1,x_2\in V_X$, $\{\varphi(x_1),\varphi(x_2)\}\in E_Y$ implies that there exists $\tilde{x}_2\in \varphi^{-1}(\varphi(x_2))$ such that $\{x_1,\tilde{x}_2\}\in E_X.$ We show that $\varphi\in \mathrm{LS}(X,Y)$. Let $x_1,x_2\in V_X$ be such that $e=\{\varphi(x_1),\varphi(x_2)\}\in E_Y$ and pick any $\tilde{x}_1\in \varphi^{-1}(\varphi(x_1))$. Then $e=\{\varphi(\tilde{x}_1),\varphi(x_2)\}$ and so, applying the assumption to $\tilde{x}_1, x_2$, we obtain the existence of $\tilde{x}_2\in \varphi^{-1}(\varphi(x_2))$ such that $\{\tilde{x}_1,\tilde{x}_2\}\in E_X.$
\[intersection\] Let $X$ and $Y$ be graphs. Then the following hold:
- $\mathrm{LSur}(X,Y)\subseteq \mathrm{LS}(X,Y)$;
- $\mathrm{LS}(X,Y)\cap \mathrm{Sur}(X,Y)=\mathrm{LSur}(X,Y)\cap \mathrm{Sur}(X,Y).$
\(i) Let $\varphi\in\mathrm{LSur}(X,Y)$. By Lemma \[formulation\], we need to show that for every $x_1,x_2\in V_X$, $\{\varphi(x_1),\varphi(x_2)\}\in E_Y$ implies that there exists $\tilde{x}_2\in \varphi^{-1}(\varphi(x_2))$ such that $\{x_1,\tilde{x}_2\}\in E_X.$ Indeed, if $\{\varphi(x_1),\varphi(x_2)\}\in E_Y$, we have that $\varphi(x_2)\in N_Y(\varphi(x_1))$ and, since $\varphi\in\mathrm{LSur}(X,Y)$, we have that $N_Y(\varphi(x_1))=\varphi(N_X(x_1)).$ Hence there exists $\tilde{x}_2\in N_X(x_1)$ such that $\varphi(\tilde{x}_2)=\varphi(x_2)$, which means $\{x_1,\tilde{x}_2\}\in E_X$ and $\tilde{x}_2\in \varphi^{-1}(\varphi(x_2))$.
\(ii) By (i), it is enough to show that $\mathrm{LS}(X,Y)\cap \mathrm{Sur}(X,Y)\subseteq\mathrm{LSur}(X,Y)\cap \mathrm{Sur}(X,Y).$ Let then $\varphi\in \mathrm{LS}(X,Y)\cap \mathrm{Sur}(X,Y)$ and show that $\varphi\in\mathrm{LSur}(X,Y).$ We need to see that, for every $x\in V_X$, $\varphi(N_X(x))=N_Y(\varphi(x)).$ One inclusion is obvious by and therefore we need only to show that $N_Y(\varphi(x))\subseteq \varphi(N_X(x)).$ Let $y\in N_Y(\varphi(x))$. Then $\{y,\varphi(x)\}\in E_Y$ and, $\varphi$ being surjective, there exists $x'\in V_X$ such that $y=\varphi(x').$ Thus, as $\{\varphi(x), \varphi(x')\}\in E_Y$ and $\varphi$ is locally strong, there exists $\tilde{x}'\in V_X$ such that $\{x,\tilde{x}'\}\in E_X$ and $\varphi(\tilde{x}')=\varphi(x')=y.$ Hence $y\in \varphi(N_X(x)).$
Generally, $\mathrm{LSur}(X,Y)\subsetneq\mathrm{LS}(X,Y).$ Consider, for instance, the graph $X$ with $V_X=\{1\}$ and $E^*_X=\varnothing$; the graph $Y$ with $V_Y=\{a,b\}$ and $E^*_Y=\{\{a,b\}\}$; $\varphi\in\mathrm{Hom}(X,Y)$ defined by $\varphi(1)=a$. Then, trivially, $\varphi\in \mathrm{LS}(X,Y)$ but $\varphi\notin \mathrm{LSur}(X,Y).$
\[pseudo\] [$\varphi\in \mathrm{Hom}(X,Y)$ is called [*pseudo-covering*]{} if $\varphi\in \mathrm{LS}(X,Y)\cap \mathrm{Sur}(X,Y)$. We denote the set of the pseudo-covering homomorphisms from $X$ to $Y$ by $\mathrm{PC}(X,Y).$]{}
Observe that for a projection on a quotient graph, being pseudo-covering is equivalent to being locally strong as well as to being locally surjective. Lemma \[intersection\] makes clear two good reasons to adopt the term pseudo-covering. First of all in [@pr Definition 1.7] a graph is called a pseudo-cover of its quotient graph when the natural projection is locally strong. Secondly the word covering is typically used in the context of locally constrained graph homomorphisms. More precisely, if $\varphi\in \mathrm{LIso}(X,Y)\cap \mathrm{Sur}(X,Y)$, then $\varphi$ is called a covering ([@god Section 6.8]); if $\varphi\in \mathrm{LIn}(X,Y)$, then $\varphi$ is called a partial covering ([@FI2]). So, in some sense, we are filling a vacancy of terminology, with respect to the concept of covering, in the locally surjective case. Note also that pseudo-covering homomorphisms are considered in [@FI] with the name of global role assignments. There it is proved that the problem of deciding if, given a graph $Y$, we have $\mathrm{PC}(X,Y)\neq\varnothing$, for some input graph $X$, is $\mathrm{NP}$-complete, with the exception of the case in which all the components of $Y$ have at most two vertices.
\[h1\] Let $\varphi\in \mathrm{Com}(X,Y)$ and let $\pi$ the projection of $X$ onto $X/\hspace{-1mm}\sim_{\varphi}$. Then $\pi$ is pseudo-covering (locally surjective, locally injective, locally bijective, locally strong) if and only if $\varphi$ is.
Using the notation of Lemma \[h\] we have $\tilde{\varphi}\circ\pi=\varphi$ and, since $\varphi\in \mathrm{Com}(X,Y)$, $\tilde{\varphi}$ is an isomorphism. Since the composition of a pseudo-covering (locally surjective, locally injective, locally bijective, locally strong) homomorphism with an isomorphism is pseudo-covering (locally surjective, locally injective, locally bijective, locally strong), the assertion follows.
\[propc\] Let $X$, $Y$ and $Z$ be graphs.
- If $\varphi\in \mathrm{PC}(X,Y)$ and $\psi\in \mathrm{PC}(Y,Z),$ then $\psi\circ \varphi\in \mathrm{PC}(X,Z);$
- $$\label{sub2}
\begin{array}{l}
\mathrm{Iso}(X,Y)\subseteq \mathrm{O}(X,Y)\cap \mathrm{Com}(X,Y)\subseteq \mathrm{E}(X,Y)\cap \mathrm{Com}(X,Y)\subseteq \\
\mathrm{LS}(X,Y)\cap \mathrm{Com}(X,Y)= \mathrm{PC}(X,Y)=\mathrm{LSur}(X,Y)\cap \mathrm{Com}(X,Y)\subseteq \mathrm{Com}(X,Y).
\end{array}$$
\(i) Straightforward.
\(ii) The first two inclusions follow from the discussion in Section \[equi-orb\]. We show that $\mathrm{E}(X,Y)\cap \mathrm{Com}(X,Y)\subseteq \mathrm{LS}(X,Y)\cap \mathrm{Com}(X,Y).$ Let $\varphi\in \mathrm{E}(X,Y)\cap \mathrm{Com}(X,Y)$ and show that $\varphi\in \mathrm{LS}(X,Y).$ By Lemma \[h1\], it is enough to show that the natural projection $\pi:X\rightarrow X/\hspace{-1mm}\sim_{\varphi}$ is locally strong. By Lemma \[formulation\], we need to see that for every $x_1,x_2\in V_X$, $\{[x_1],[x_2]\}\in E_{ X/\sim_{\varphi}}$ implies that there exists $\tilde{x}_2\in \varphi^{-1}(\varphi(x_2))$ such that $\{x_1,\tilde{x}_2\}\in E_X.$ Now, $\{[x_1],[x_2]\}\in E_{ X/\sim_{\varphi}}$ means that there exist $x'_1, x'_2\in V_X$ such that $\{x'_1,x'_2\}\in E_X$, $\varphi(x'_1)=\varphi(x_1)$ and $\varphi(x'_2)=\varphi(x_2)$. Thus $x'_2\in N_X(x'_1)\cap \varphi^{-1}(\varphi(x_2))$ and $x_1,x'_1$ belong to the same $\varphi$-cell. Since the partition into $\varphi$-cells is equitable, we then have $N_X(x_1)\cap \varphi^{-1}(\varphi(x_2))\neq\varnothing$ and, to conclude, it suffices to pick any $\tilde{x}_2\in N_X(x_1)\cap \varphi^{-1}(\varphi(x_2))$.
Next we see that $\mathrm{PC}(X,Y)= \mathrm{LS}(X,Y)\cap \mathrm{Com}(X,Y)$. By definition of $\mathrm{PC}(X,Y)$ we have $\mathrm{LS}(X,Y)\cap\supseteq\mathrm{PC}(X,Y)\supseteq \mathrm{LS}(X,Y)\cap \mathrm{Com}(X,Y)$. Moreover as an obvious consequence of Lemma \[formulation\], we have $\mathrm{PC}(X,Y)\subseteq \mathrm{Com}(X,Y)$ and so $\mathrm{PC}(X,Y)\subseteq \mathrm{LS}(X,Y)\cap\mathrm{Com}(X,Y).$
The fact that $\mathrm{PC}(X,Y)=\mathrm{LSur}(X,Y)\cap \mathrm{Com}(X,Y)$ is now a consequence of Lemma \[intersection\].
\[phicon2\] [Let $\varphi\in \mathrm{Hom}(X,Y)$. For $C'\in \mathcal{C}(Y)$, put $$\mathcal{C}(X) _{C'}=\{C\in \mathcal{C}(X)\ :\ \varphi(C)\subseteq C' \},\qquad c (X) _{C'}=|\mathcal{C}(X) _{C'}|.$$]{}
\[hom-con\] Let $\varphi\in \mathrm{LSur}(X,Y)$.
- If $C\in \mathcal{C}(X)$, then $\varphi(C)\in \mathcal{C}(Y)$. In particular, the image of $X$ is a union of components of $Y$.
- For every $x\in V_X$, $\varphi(C_{X}(x))=C_{Y}(\varphi(x))\cong C_{X}(x)/\hspace{-1mm}\sim_{\varphi}.$
- For every $C\in\mathcal{C}(X)$, $\varphi^{-1}(\varphi(V_C))=\displaystyle{\bigcup_{\hat{C}\in \mathcal{C}(X)_{\varphi(C)}}V_{\hat{C}}}$.
\(i) We first consider the case $\varphi\in \mathrm{PC}(X,Y)$. Let $C\in \mathcal{C}(X)$: we shall show that $\varphi(C)\in \mathcal{C}(Y).$ By Lemma \[quasi-path\], $\varphi(C)$ is a connected subgraph and we need to see that it is maximal connected. Assume the contrary. Then there exists an edge $\{y,y'\}\in E_Y\setminus \varphi(E_C),$ with $y\in V_{\varphi(C)}=\varphi(V_C)$ and $y'\in V_Y$. Let $x\in V_C$ with $y=\varphi(x).$ $\varphi$ being surjective and locally strong, there also exists $x'\in V_X$ such that $\varphi(x')=y'$ and $\{x,x'\}\in E_X.$ Since $x\in V_C$, with $C$ a component, we then get $x'\in V_C$ and so $e=\{x,x'\}\in E_C.$ Thus $\varphi(e)= \{\varphi(x),\varphi(x')\}\in \varphi(E_C),$ that is, $\{y,y'\}\in \varphi(E_C),$ a contradiction.
We now consider the case $\varphi\in \mathrm{LSur}(X,Y)$. Let $C\in \mathcal{C}(X)$ and $C'\in \mathcal{C}(Y)$ be the unique component of $Y$ containing $\varphi(C).$ Then it is easily checked that $\varphi_{\mid C}\in \mathrm{LSur}(C,C')$. By [@FI Observation 2.4], $C'$ being connected, we also have that $\varphi_{\mid C}\in \mathrm{Sur}(C,C')$ and thus $\varphi_{\mid C}\in \mathrm{PC}(C,C')$. Since the result has been proved for pseudo-covering homomorphisms and $C$ is connected, we deduce that $\varphi(C)=C'$.
\(ii) Let $x\in V_X$. By (i), $\varphi(C_{X}(x))$ is a component of $Y$ that contains the vertex $\varphi(x)$ and thus $\varphi(C_{X}(x))=C_{Y}(\varphi(x)).$ Next observe that $\varphi$ restricted to the subgraph $C_{X}(x)$ defines a complete homomorphism onto $C_{Y}(\varphi(x)) $ and apply Lemma \[h\].
\(iii) The fact that if $\hat{C}\in \mathcal{C}(X)_{\varphi(C)}$ then $V_{\hat{C}}\subseteq \varphi^{-1}(\varphi(V_C))$ is obvious. Let $x\in \varphi^{-1}(\varphi(V_C))$, for some $C\in\mathcal{C}(X)$. To conclude it is enough to show that $\varphi(C_{X}(x))=\varphi(C).$ From $\varphi(x)\in
\varphi(V_C)$, it follows that there exists $\overline{x}\in V_C$ with $\varphi(x)=\varphi(\overline{x})$. Thus, by (ii), we get $$\varphi(C_{X}(x))=C_{Y}(\varphi(x))=C_{Y}(\varphi(\overline{x}))=\varphi(C_{X}(\overline{x}))=\varphi(C).$$
As an interesting consequence, we have a comparison between the isolated vertices of $X$ and those in $Y$ and a general link between the components of $X$ and $Y$ in the tame case.
\[isolated\] Let $\varphi\in \mathrm{LSur}(X,Y)$ and $x\in V.$ If $x$ is isolated in $X$, then $\varphi(x)$ is isolated in $Y$.
If $V_{C_{X}(x)}=\{x\}$ then, by Proposition \[hom-con\] (ii), $V_{C_{Y}(\varphi(x))}=\{\varphi(x)\}.$
\[tame-component\] Let $\varphi\in \mathrm{PC}(X,Y)\cap \mathrm{T}(X,Y).$ Then $\varphi$ induces a bijection between $\mathcal{C}(X)$ and $\mathcal{C}(Y)$. Given $C'\in \mathcal{C}(Y)$, if $C$ is the unique component of $X$ such that $\varphi(C)=C'$, then $V_C=\varphi^{-1}(V_{C'}).$
By Proposition \[hom-con\], we can define the map $\varphi_{\mathcal{C}}:\mathcal{C}(X)\rightarrow \mathcal{C}(Y)$ by $\varphi_{\mathcal{C}}(C)=\varphi(C)$ for all $C\in \mathcal{C}(X)$. $\varphi$ being surjective, $\varphi_{\mathcal{C}}$ is surjective too. Since $\sim_{\varphi}$ is tame, Proposition \[quotient-graph\], gives $c(X)=c(X/\hspace{-1mm}\sim)$. On the other hand, $\varphi$ being complete, Lemma \[h\], guarantees that $Y\cong X/\hspace{-1mm}\sim$ and thus $c(Y)=c(X/\hspace{-1mm}\sim)$, so that $c(X)=c(Y).$ It follows that $\varphi_{\mathcal{C}}$ is injective.
Let next $C'\in \mathcal{C}(Y)$ and $C\in \mathcal{C}(X)$ be the unique component such that $\varphi(C)=C'$. Surely we have $V_C\subseteq \varphi^{-1}(V_{C'}).$ To get the other inclusion let $x_1\in \varphi^{-1}(V_{C'})$ and choose $x_2\in V_C$. Since both $\varphi(x_1)$ and $\varphi(x_2)$ belong to $V_{C'},$ we have $C'=C_Y(\varphi(x_1))=C_Y(\varphi(x_2))$. Hence, by Proposition \[hom-con\], we have $\varphi(C_X(x_1))=\varphi(C_X(x_2))$, that is, $\varphi_{\mathcal{C}}(C_X(x_1))=\varphi_{\mathcal{C}}(C_X(x_2)).$ Since $\varphi_{\mathcal{C}}$ is a bijection, we then get $C_X(x_1)=C_X(x_2)=C$ so that $x_1\in V_C.$
\[quot-pse\][Let $X$ be a graph and $\sim$ an equivalence relation on $V_X$. We say that the quotient graph $X/\hspace{-1mm}\sim$ is [*pseudo-covered*]{} by $X$ (is an [*orbit quotient*]{} of $X$), with respect to $\sim,$ if the projection $\pi:X \rightarrow X/\hspace{-1mm}\sim$ is pseudo-covering (is an orbit homomorphism). ]{}
Note that $X/\hspace{-1mm}\sim$ is pseudo-covered if and only if, for each $x_1,x_2\in V_X$ such that $\{[x_1],[x_2]\}\in [E_X]$ there exists $\tilde{x}_2\in V_X$ with $\{x_1,\tilde{x}_2\}\in E_X$ and $[\tilde{x}_2] =[x_2]$. We establish next a useful criterium of connectedness for $X$ relying on that of $X/\hspace{-1mm}\sim$.
\[connection\] Assume that $X/\hspace{-1mm}\sim$ is connected and pseudo-covered. If there exists $[x]\in [V_X]$ such that $[x]\subseteq V_{C_{X}(x)}$, then $X$ is connected.
By Lemma \[nat-comp\], we can apply Proposition \[hom-con\] to $\pi:X \rightarrow X/\hspace{-1mm}\sim$ obtaining that, for each $C\in \mathcal{C}(X),$ $\pi(C)=X/\hspace{-1mm}\sim.$ In particular $\pi(V_C)=[V_X]$ and thus each component contains at least one vertex in each equivalence class with respect to $\sim.$ Since $[x]\subseteq V_{C_{X}(x)}$, we therefore have a common vertex for $C$ and $C_{X}(x)$. Thus $C_{X}(x)=C$ is the only component in $X$.
0.6 true cm
**Counting the components** {#new}
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0.4 true cm
Our goal is to count components of a graph $X$ by counting those of a less complex homomorphic image $Y$. We begin with a rough link between the two.
\[phicon\] [Let $\varphi\in \mathrm{Hom}(X,Y)$. We denote the set of components of $X$, admissible for a fixed $y\in V_Y,$ by $$\mathcal{C}(X)_{y}=\{C\in\mathcal{C}(X)\ :\ k_C(y)>0\}$$ and its size by $c(X)_{y}.$ ]{}
Observe that no ambiguity arises between the definition above and Definition \[phicon2\], because the indices are taken in different sets.
\[rough\] Let $\varphi\in \mathrm{Hom}(X,Y)$, and let $\mathcal{C}(Y)=\{C_i': i\in\{1,\dots,c(Y)\}\}.$ Then $$\label{rough-for}
c(X)=\sum_{i=1}^{c(Y)}c(X)_{C'_i}.$$
Define the map $\varphi_{ \mathcal{C}(X)}: \mathcal{C}(X)\rightarrow \mathcal{C}(Y)$ by $ \varphi_{ \mathcal{C}(X)}(C)=C'$ for all $C\in \mathcal{C}(X),$ where $C'$ is the unique component of $Y$ with $\varphi(C)\subseteq C'$. Then $\mathcal{C}(X) _{C_i'}=\varphi_{ \mathcal{C}(X)}^{-1}(C_i')$, for $i\in \{1,\dots,c(Y)\}$. Thus $ \mathcal{C}(X)=\bigcup_{i=1}^{c(Y)}\mathcal{C}(X) _{C_i'}$ and, since the union is disjoint, we get the desired equality.
**Counting the components for locally surjective homomorphisms** {#comp-Lsur}
-----------------------------------------------------------------
Formula is generally of little help in computing $c(X)$ from $c(Y)$ since the numbers $c(X)_{C'_i}$ are hard to determine. If $\varphi$ is locally surjective, by Proposition \[hom-con\], we have $\mathcal{C}(X) _{C'}=\{C\in \mathcal{C}(X)\ :\ \varphi(C)=C' \}$ and we can write a more expressive formula.
\[second\] Let $\varphi\in \mathrm{LSur}(X,Y)$.
- For each $y\in V_Y,\
\mathcal{C}(X) _{C_{Y}(y)}=\mathcal{C}(X)_{y}.$ In particular $c(X) _{C_{Y}(y)}=c(X)_{y}.$
- If $y, \overline{y}\in V_{C'}$, for some $C'\in \mathcal{C}(Y)$, then $c(X)_{y}=c(X)_{\overline{y}}.$
- For $1\leq i\leq c(Y)$, let $y_i\in V_Y$ be such that $\mathcal{C}(Y)=\{C_Y(y_i): 1\leq i\leq c(Y)\}$. Then $$\label{genfor}
c(X)=\sum_{i=1}^{c(Y)}c(X)_{y_i}.$$
\(i) Let $C\in \mathcal{C}(X) _{C_{Y}(y)}$. Thus, as $\varphi\in \mathrm{LSur}(X,Y)$, $\varphi(C)=C_{Y}(y)$ so that, in particular, there exists $x\in V_C$ with $\varphi(x)=y$ and so $k_C(y)>0$. Conversely if $C\in\mathcal{C}(X)$ and $ k_C(y)>0,$ then there exists $x\in V_C$ with $\varphi(x)=y$. Thus $C=C_{X}(x)$ and, by Proposition \[hom-con\], $\varphi(C)=\varphi(C_{X}(x))=C_{Y}(\varphi(x))=C_{Y}(y)$.
(ii)-(iii) They follow immediately as an application of (i) and of Lemma \[rough\].
Note that the integers $c(X)_{y_i}$ in are non-negative, and that $c(X)_{y_i}=0$ if and only if the component $C_i'$ is not included in the image of $X$ by $\varphi$.
Combine Proposition \[hom-con\] and Lemma \[second\] (iii).
**The component equitable homomorphisms** {#equi}
------------------------------------------
While Formula \[genfor\] improves Formula \[rough-for\] allowing to pass from $C_i'$ to one of its vertices $y_i, $ the computation of $c(X)_{C'_i}$ often remains challenging. Fortunately, in many applications, we have the following property: every component of $X$ admissible for $y\in V_Y$ intersects the fibre $\varphi^{-1}(y)$ in sets of the same size.
\[comp-eq\] [$\varphi\in \mathrm{Hom}(X,Y)$ is called [*component equitable*]{} if for every $y\in V_Y$ and every $C,\hat{C}\in \mathcal{C}(X)_{y}$, we have $k_{C}(y)=k_{\hat{C}}(y)$. We denote the set of the component equitable homomorphisms from $X$ to $Y$ by $\mathrm{CE}(X,Y)$. ]{}
We exhibit examples showing that generally, among the classes $\mathrm{CE}(X,Y), \mathrm{E}(X,Y), \mathrm{PC}(X,Y)$, no further inclusion apart from $\mathrm{E}(X,Y)\cap \mathrm{Com}(X,Y)\subseteq \mathrm{PC}(X,Y)$, proved in , holds. In all of the following examples let $Y$ be defined by $V_Y=\{y,z\}$, $E^*_Y=\{y,z\}$.
\[PC+CE-E\] [Let $X$ be defined by $$V_X=\{1,2,3,4,5,6,7,8\},\quad E^*_X=\{\{1,2\},\{1,5\},\{1,6\},\{2,6\},\{3,4\},\{3,7\},\{4,8\}\}$$ and consider $\varphi:V_X\rightarrow V_Y$ given by $\varphi(x)=y$ for all $1\leq x\leq 4$, $\varphi(x)=z$ for all $5\leq x\leq 8.$ Then $\varphi\in (\mathrm{PC}(X,Y)\cap \mathrm{CE}(X,Y))\setminus \mathrm{E}(X,Y)$.]{}
\[PC-E-CE\] [Let $X$ be defined by $$V_X=\{1,2,3,4,5,6\},\quad E^*_X=\{\{1,2\}, \{1,4\},\{2,5\}, \{3,6\}\}$$ and consider $\varphi:V_X\rightarrow V_Y$ given by $\varphi(x)=y$ for all $1\leq x\leq 3$, $\varphi(x)=z$ for all $4\leq x\leq 6.$ Then $\varphi\in \mathrm{PC}(X,Y)\setminus\left( \mathrm{CE}(X,Y)\cup\mathrm{E}(X,Y)\right)$.]{}
\[E+Com-CE\][Let $X$ be defined by $V_X=\{x\in\mathbb{N}: 1\leq x\leq 14\}$ and $$\begin{array}{l} E^*_X=\{\{1,2\}, \{1,3\},\{1,8\}, \{2,3\},\{2,9\},\{3,10\},\{4,5\},\{4,7\},\{4,11\},\{5,6\},\{5,12\},\\
\{6,7\},\{6,13\},\{7,14\},\{8,9\},\{8,10\},\{9,10\},\{11,12\},\{11,14\},\{12,13\}, \{13,14\}\}.
\end{array}$$ Consider $\varphi:V_X\rightarrow V_Y$ given by $\varphi(x)=y$ for all $1\leq x\leq 7$, $\varphi(x)=z$ for all $8\leq x\leq 14.$ Then $\varphi\in (\mathrm{E}(X,Y)\cap\mathrm{Com}(X,Y) )\setminus \mathrm{CE}(X,Y)$.]{}
\[union2\] Let $\varphi\in \mathrm{LSur}(X,Y)\cap \mathrm{CE}(X,Y)$ and $y\in V_Y$. Then $c(X)_{y}=\frac{k_{X}(y)}{k_{C}(y)}$ for all $C\in \mathcal{C}(X)_{y}.$ In particular $k_{C}(y)$ divides $k_{X}(y).$
$c(X)_{y}$ is the number of components of $X$ admissible for $y$ and, since $\varphi\in \mathrm{CE}(X,Y)$, each of those components admits the same number of vertices mapped by $\varphi$ into $y$. Thus, for each $C \in \mathcal{C}(X)_{y}$, we have $c(X)_{y} k_{C}(y)=k_{X}(y)$, where the factors are positive integers.
**Counting the components for orbit homomorphisms** {#automorphic}
----------------------------------------------------
\[union2cons\] Let $\varphi\in \mathrm{O}(X,Y)$ be $\mathfrak{G}$-consistent and let $y\in V_Y$.
- For each $C\in \mathcal{C}(X)_{y},\ \mathcal{C}(X)_{y}=\{f(C)\ : f\in \mathfrak{G}\}.$ In particular, the components of $X$ admissible for $y$ are isomorphic through a graph automorphism of $X$.
- $\mathrm{O}(X,Y)\subseteq \mathrm{CE}(X,Y).$
- If $\varphi\in \mathrm{O}(X,Y)\cap \mathrm{Com}(X,Y)$, then $c(X)_{y}=\frac{|\varphi^{-1}(\varphi(V_C))|}{|V_C|}$ for all $C\in \mathcal{C}(X)_{y}.$
\(i) Let $C\in \mathcal{C}(X)_{y}$ and $f\in \mathfrak{G}$. Then there exists $x\in V_X$ such that $\varphi(x)=y$ and, as $f\in \mathrm{Aut}(X),$ we have $f(C)=f(C_X(x))=C_Y(f(x))$. By condition (a) in Lemma \[phi-con\], we have that $\varphi\circ f=\varphi.$ Thus $\varphi(f(x))=\varphi(x)=y$ which gives $f(C)\in \mathcal{C}(X)_{y}$. So $\{f(C)\ : f\in \mathfrak{G}\}\subseteq \mathcal{C}(X)_{y}.$ Note also that $f(\varphi^{-1}(y)\cap V_C)= \varphi^{-1}(y)\cap V_{f(C)}$. Since $f$ is a bijection, that implies $$\label{A} k_{f(C)}(y)=k_{C}(y).$$ We next show $\mathcal{C}(\Gamma)_{y}\subseteq \{f(C): f\in \mathfrak{G}\}.$ Let $\hat{C}\in \mathcal{C}(X)_{y}$ and let $\hat{x}\in V_{\hat{C}}$ such that $\varphi(\hat{x})=y$. Thus we have $\varphi(x)=\varphi(\hat{x})$ and, by condition (b) in Lemma \[phi-con\], there exists $f\in \mathfrak{G}$ with $\hat{x}=f(x).$ It follows that $\hat{x}\in f(V_C)=V_{f(C)}.$ Hence $\hat{C}$ and $f(C)$ are components with a vertex in common, which implies that $\hat{C}=f(C)$.
\(ii) Use (i) and .
\(iii) Let $C\in \mathcal{C}(X)_{y}$ and $x\in V_C$ with $\varphi(x)=y$. By (i), all the components in $\mathcal{C}(X)_{y}$ have the same number of vertices, so that, $c(X)_{y} |V_C|$ counts the vertices of all the components in $\mathcal{C}(X)_{y}$, that is, the size of the set $\bigcup_{\hat{C}\in \mathcal{C}(X)_{y}} V_{\hat{C}}$. By , we have $\varphi\in \mathrm{O}(X,Y)\cap \mathrm{Com}(X,Y)\subseteq\mathrm{LSur}(X,Y)$. Thus we can apply Proposition \[hom-con\] (ii) to $\varphi,$ obtaining $C_{Y}(y)=\varphi(C).$ So, by Lemma \[second\] (i), we get $\mathcal{C}(X)_{y}=\mathcal{C}(X)_{C_{Y}(y)}=\mathcal{C}(X)_{\varphi(C)}$. Hence, by Proposition \[hom-con\] (iii), $$\displaystyle{\bigcup_{\hat{C}\in \mathcal{C}(X)_{y}} V_{\hat{C}}}=\displaystyle{\bigcup_{\hat{C}\in \mathcal{C}(X)_{\varphi(C)}}V_{\hat{C}}}=\varphi^{-1}(\varphi(V_C)),$$ which gives $c(X)_{y} |V_C|=|\varphi^{-1}(\varphi(V_C))|.$
By Propositions \[propc\] and \[union2cons\] we have $$ \mathrm{O}(X,Y)\cap\mathrm{Com}(X,Y)\subseteq \mathrm{CE}(X,Y)\cap \mathrm{LSur}(X,Y)\cap \mathrm{Com}(X,Y).$$ Thus the assertion follows, combining Lemma \[second\] and Proposition \[union2\].
Note that Formula is more manageable than Formula due to its high level of symmetry. Moreover the terms in the summand are easily computable in many contexts. A remarkable case is given when $X$ is the quotient proper power graph and $Y$ is the proper power type graph of a fusion controlled permutation group. That case will be examined in [@BIS1] and [@BIS2]. We now write an explicit procedure for computing $c(X)$ based upon our results.
[**Procedure to compute $c(X)$ for $\varphi\in \mathrm{O}(X,Y)\cap\mathrm{Com}(X,Y)$** ]{} \[procedure\]
- [*Selection of $y_i$ and $C_i.$*]{}
<!-- -->
- : Pick arbitrary $y_1\in V_Y$ and choose any $C_1\in\mathcal{C}(X)_{y_1}$.
- : Given $y_1,\dots, y_i\in V_Y$ and $C_1,\dots,C_i\in\mathcal{C}(X)$ such that $C_j\in \mathcal{C}(X)_{y_j}\ (1\leq j\leq i)$, choose any $y_{i+1}\in V_Y\setminus \bigcup_{j=1}^iV_{C_{Y}(y_j)}=V_Y\setminus \bigcup_{j=1}^iV_{\varphi(C_j)}$ and any $C_{i+1}\in \mathcal{C}(X)_{y_{i+1}}.$
- : The procedure stops in $l=c(Y)$ steps.
<!-- -->
- [*The value of $c(X)$.* ]{}
Compute the integers $\frac{k_{X}(y_j)}{k_{C_j}(y_j)}\ (1\leq j\leq c(Y))$ and sum them up to get $c(X)$.
Given a graph $X$, Procedure 6.10 may be applied to any graph $Y$ such that $ \mathrm{O}(X,Y)\cap\mathrm{Com}(X,Y)\neq \varnothing$ once $\varphi\in \mathrm{O}(X,Y)\cap\mathrm{Com}(X,Y)$ is chosen. Such $Y$, as explained in Section \[equi-orb\], are the quotients of $X$ with respect to the orbit partitions of the possible $\mathfrak{G}\leq \mathrm{Aut}(X)$, and $\varphi$ are the corresponding projection on $Y.$ Choices of $\mathfrak{G}$ with different sets of orbits lead to different computations of the coefficients $\frac{k_{X}(y_j)}{k_{C_j}(y_j)}$, with the computation easier when $\mathfrak{G}$ is “large”.
0.6 true cm
**The isomorphism class of the components** {#iso-class}
============================================
0.4 true cm Under the assumption $\varphi\in \mathrm{PC}(X,Y)$, Proposition \[hom-con\] guarantees that each component $C$ of $X$ admits as quotient the component $\varphi(C)$ of $Y$. In this short section, we study when $C$ is actually isomorphic to $\varphi(C)$.
\[iso\] Let $\varphi\in \mathrm{PC}(X,Y)$. Given $C\in \mathcal{C}(X),$ we have $C\cong \varphi(C)$ if and only if $k_C(y)=1$ for all $y\in \varphi(C).$
Since $\varphi_{\mid C}:C\rightarrow \varphi(C)$ is always a complete homomorphism, $\varphi_{\mid C}$ is an isomorphism if and only if it is injective, that is, $k_C(y)=|V_C\cap \varphi^{-1}(y)|=1$ for all $y\in \varphi(C).$
\[independence\] Let $\varphi\in \mathrm{O}(X,Y)\cap \mathrm{Com}(X,Y)$ and let $C\in \mathcal{C}(X)$.
- If $y,\overline{y}\in V_{\varphi(C)}$, then $\frac{k_{X}(y)}{k_{C}(y)}=\frac{k_{X}(\overline{y})}{k_{C}(\overline{y})}.$
- $C\cong \varphi(C)$ if and only if there exists $y\in V_{\varphi(C)}$ such that $k_{C}(y)=1$ and, for every $\overline{y}\in V_{\varphi(C)}$, $k_{X}(y)=k_{X}(\overline{y}).$
- If there exists $y\in V_{\varphi(C)}$ such that $k_{C}(y)=k_{X}(y),$ then for every $\overline{y}\in V_{\varphi(C)}$, $k_{C}(\overline{y})=k_{X}(\overline{y}).$
- If there exists $y\in V_{\varphi(C)}$ such that $k_{C}(y)=k_{X}(y)>1,$ then $C\not\cong \varphi(C).$
\(i) Since $C\in \mathcal{C}(X)_{y}\cup \mathcal{C}(X)_{\overline{y}},$ by Proposition \[union2\], we get $c(X)_{y}=\frac{k_{X}(y)}{k_{C}(y)}$ as well as $c(X)_{\overline{y}}=\frac{k_{X}(\overline{y})}{k_{C}(\overline{y})}.$ Now, by , $\mathrm{O}(X,Y)\cap \mathrm{Com}(X,Y)\subseteq \mathrm{LSur}(X,Y)\cap \mathrm{Com}(X,Y)$. Thus Lemma \[second\] (ii) applies giving $c(X)_{y}=c(X)_{\overline{y}}$ and the equality follows.
(ii)-(iv) They are immediate from (i) and Lemma \[iso\].
[**Acknowledgements**]{} The author wishes to thank Gena Hahn, Cheryl Praeger, Jiří Fiala, Jürgen Lerner and Ulrich Knauer for helpful comments on a preliminary version of the paper. A particular thank to the anonymous referee whose advices led to a significative improvement of the paper. The author is partially supported by GNSAGA of INdAM.
[20]{} J. Abawajy, A. Kelarev and M. Chowdhury, Power Graphs: A Survey, [*Electronic Journal of Graph Theory and Applications* ]{}[**1**]{} (2) (2013), 125-147. D. Bubboloni, Mohammad A. Iranmanesh, S. M. Shaker, Quotient graphs for power graphs, arXiv: 1502.02966v3 (2016). D. Bubboloni, Mohammad A. Iranmanesh, S. M. Shaker, On some graphs associated with the finite alternating groups, arXiv: 1412.7324v2 (2016). R. Diestel, [*Graph Theory*]{}, Graduate Texts in Mathematics Vol. 173, Springer-Verlag Heidelberg, 2010. M. G. Everett, S. Borgatti, Role colouring a graph, [ *Math. Social Sci.*]{} [**21**]{}(2) (1991), 183-188. J. Fiala, D. Paulusma, A complete complexity classification of the role assignment problem, [*Theoret. Comput. Sci.*]{} [**349**]{} (2005), 67-81. J. Fiala, J. Kratochvíl, Partial covers of graphs, [ *Discussione Mathematicae*]{} [**22**]{} (2002), 89-99. J. Fiala, J. Kratochvíl, Locally Constrained Graph Homomorphisms - Structure, Complexity, and Applications, [*Journal Computer Science Review*]{} [**2**]{} (2008), 97-111. C. D. Godsil, G. Royle, [*Algebraic Graph Theory*]{}, Graduate Texts in Mathematics 207, Springer-Verlag New York Berlin Heidelberg, 2001. C.D. Godsil, [*Algebraic Combinatorics*]{}, Chapman & Hall, New York, 1993. G. Hahn, C. Tardif, Graph homomorphisms: structure and symmetry, in [*Graph symmetry*]{}, G. Hahn, G. Sabidussi, NATO ASI Series C, Vol. 497, Kluwer 1997, 107-166. U. Knauer, [*Algebraic Graph Theory: Morphisms, Monoids and Matrices*]{}, Walter de Gruyter, 2011. U. Knauer, Endomorphism spectra of graphs, [*Discrete Mathematics*]{} [**109**]{} (1992), 45-57. J. Lerner, Role assignments, pp. 216-252 in [*Network Analysis: Methodological Foundations*]{}, Ed. U. Brandes and T. Erlebach, Springer-Verlag Berlin, 2005. C.E. Praeger, Imprimitive symmetric graphs, [*Ars Combin.*]{} [**19**]{} A (1985), 149-163. A. Pultr, V. Trnková, [*Combinatorial, Algebraic and Topological Representations of Groups, Semigroups and Categories*]{}, Vol. 22 of Mathematical Studies, North-Holland Publishing Company, 1980.
[\
[Dipartimento di Matematica e Informatica U.Dini]{},\
[ Viale Morgagni 67/a]{}, 50134 Firenze, Italy]{}\
[Email: [email protected]]{}\
[^1]: [-0.4 true cm MSC(2010): 05C60, 05C70, 05C40. Keywords: Graph homomorphism, Quotient graph, Component, Power graphs.\
]{}
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We prove a mixed version of a conjecture of Griffiths: that the closure of the image of any admissible mixed period map is quasiprojective, with a natural ample bundle. Specifically, we consider the map from the image of the mixed period map to the image of the period map of the associated graded. On the one hand, we show in a precise manner that the parts of this map parametrizing extension data of non-adjacent-weight pure Hodge structures are quasi-affine. On the other hand, extensions of adjacent-weight pure polarized Hodge structures are parametrized by a compact complex torus (the intermediate Jacobian) equipped with a natural theta bundle which is ample in Griffiths transverse directions.
Our proof makes heavy use of o-minimality, and recent work with B. Klingler associating a ${\mathbb{R}}_{{\mathrm{an}},\exp}$-definable structure to mixed period domains and admissible mixed period maps.
address:
- 'B. Bakker: Dept. of Mathematics, University of Georgia, Athens, USA.'
- 'Y. Brunebarbe: Dept. of Mathematics, Univ. Bordeaux, Talence, France.'
- 'J. Tsimerman: Dept. of Mathematics, University of Toronto, Toronto, Canada.'
author:
- Benjamin Bakker
- Yohan Brunebarbe
- Jacob Tsimerman
bibliography:
- 'biblio.mixed.bib'
title: Quasiprojectivity of images of mixed period maps
---
Introduction
============
Let $X$ be an algebraic space and ${\mathcal{M}}$ a moduli space of graded-polarized integral mixed Hodge structures, henceforth referred to as a period space. There is a period space ${\mathcal{D}}$ parametrizing the associated graded objects of the points in ${\mathcal{M}}$ with a map ${\mathcal{M}}\to {\mathcal{D}}$, and to any period map ${\varphi}:X\to {\mathcal{M}}$ corresponding to a variation of graded-polarized integral mixed Hodge structures there is a period map $\gr{\varphi}:X\to {\mathcal{D}}$ for the associated graded variation. By the results of [@bbt] we have a factorization $$\begin{tikzcd}
X\arrow{rr}{\gr{\varphi}}\arrow{dr}[swap]{g}&&{\mathcal{D}}\\
&Z\arrow[ru,hook,swap,"\epsilon"]&
\end{tikzcd}$$ where $g$ is algebraic and dominant (meaning ${\mathcal{O}}_Z\to g_*{\mathcal{O}}_X$ is injective) and $\epsilon$ is a closed immersion. Moreover, the Griffiths bundle on $Z$ is algebraic and ample, and in particular $Z$ is quasiprojective.
Our main result is to extend this picture to the mixed case. Precisely, we show:
\[thm main\] Let $X$ be a separated algebraic space of finite type over ${\mathbb{C}}$ and ${\varphi}:X\to {\mathcal{M}}$ the period map associated to an admissible[^1] variation of (graded-polarized integral) mixed Hodge structures. Then there is a factorization$$\begin{tikzcd}
X\arrow{rr}{{\varphi}}\arrow{dr}[swap]{f}&&{\mathcal{M}}.\\
&Y\arrow[ru,hook,swap,"\iota"]&
\end{tikzcd}$$ where $f$ is dominant algebraic and $\iota$ is a closed immersion. Moreover, the natural theta bundle on $Y$ is algebraic and relatively ample over the image $Z$ of the period map of the associated graded. In particular, $Y$ is quasiprojective.
The period space ${\mathcal{M}}$ is naturally a quotient $\Gamma\backslash M$ of a graded-polarized integral mixed period domain $M$ by an arithmetic group $\Gamma$, but the same result for the quotient[^2] $\Gamma_\mathrm{mon}\backslash M$ by the image of the monodromy representation easily follows from Theorem \[thm main\] (see Corollary \[cor mon image\]).
As in [@bbt], we for instance obtain as a corollary the following:
\[introcoarse\] Let $\mathcal{X}$ be a separated Deligne–Mumford stack of finite type over ${\mathbb{C}}$ admitting a quasi-finite admissible^*\[footnote\]*^ mixed period map. Then the coarse moduli space of $\mathcal{X}$ is quasi-projective.
The factorization statement in Theorem \[thm main\] follows easily from [@bbkt] and [@bbt], and the main content is the relative ampleness of the theta bundle. This is especially interesting compared to the corresponding result in the pure case as the positivity does not stem from the negative curvature of ${\mathcal{M}}$; indeed, the fibers of ${\mathcal{M}}\to{\mathcal{D}}$ are flat.
The theta bundle is loosely constructed as follows (see section \[sect theta\] for details). For any polarized integral pure Hodge structure $V=(V_{\mathbb{Z}},F^\bullet V,q_{\mathbb{Z}})$ of weight $-1$, extensions of the form $$0\to V\to E\to{\mathbb{Z}}(0)\to0\label{eq one step}$$ are parametrized by the intermediate Jacobian $$J(V)=V_{\mathbb{C}}/F^0V+V_{\mathbb{Z}}$$ which is a (compact) complex torus. The polarization of $V$ endows $J(V)$ with a natural theta bundle which is positive in Griffiths transverse directions. Now for a general variation of graded-polarized integral mixed Hodge structures $E$, we obtain from each variation $\gr^W_{[w-1,w]}E:=W_{w}E/W_{w-2}E$ an extension of the form via the natural pullback: $$\begin{tikzcd}[column sep=small]
0\arrow[r]&\gr_{w-1}^WE\otimes (\gr_{w}^WE)^\vee\arrow[d,equal]\arrow{r}&E' \ar[r]\arrow{d}&{\mathbb{Z}}(0)\arrow{r}\arrow{d}&0\\
0\arrow{r}&\gr_{w-1}^WE\otimes (\gr_{w}^WE)^\vee\arrow{r}&\gr^W_{[w-1,w]}E \otimes (\gr_{w}^WE)^\vee\arrow{r}&\gr^W_{w}E\otimes (\gr_{w}^WE)^\vee\arrow{r}&0
\end{tikzcd}$$ The theta bundle of Theorem \[thm main\] is then the product $\Theta:=\bigotimes_w \Theta_{[w-1,w]}$ of the theta bundles $\Theta_{[w-1,w]}$ associated to each of the $\gr^W_{[w-1,w]}E$. In fact, it is easy to see that $\bigotimes_i \Theta_{[w-1,w]}^{a_w}$ is $f$-ample for any $a_w>0$.
There are two main difficulties in establishing the relative ampleness of $\Theta$. First, we must show $\Theta$ is algebraic. This follows for $X$ smooth by work of Brosnan–Pearlstein [@bparch] and in general by definable GAGA [@bbt]. We also give a new proof of the result of Brosnan–Pearlstain, see Remark \[rmk new proof\].
Second, the theta bundle only accounts for the compact parts of the extension data, and the rest of the argument is devoted to showing that the remaining extension data is affine. More precisely, there are period maps for which $Y\to Z$ has positive-dimensional fibers but for which all of the $\gr^W_{[w-1,w]}E$ are locally constant on the fibers, and in this case the theorem requires ${\mathcal{O}}_Y$ to be relatively ample—i.e., that $Y$ is quasiaffine over $Z$. This ultimately relies on the geometry of mixed period spaces parametrizing extensions as in with $V$ of weight $\leq -2$ and our argument critically uses the work of Saito.
Outline
-------
In §\[sect alg\] we prove the factorization part of Theorem \[thm main\] and the algebraicity of the theta bundle. In §\[sect setup\] we prove an ampleness criterion in terms of point separation by definable sections. We also apply the work of Saito to prove some results on the local monodromy of the unipotent part of a variation of mixed Hodge structures. In §\[sect proof\] we prove the quasiprojectivity part of Theorem \[thm main\].
Acknowledgements
----------------
Y.B. would like to thank P. Brosnan for an interesting discussion related to the biextension bundle. B.B. was partially supported by NSF grants DMS-1702149 and DMS-1848049.
Notation
--------
Unless otherwise stated, by definable we mean definable in the o-minimal structure ${\mathbb{R}}_{{\mathrm{an}},\exp}$. All algebraic spaces are assumed to be separated and of finite type over ${\mathbb{C}}$; all definable analytic spaces (resp. analytic spaces) are complex definable analytic spaces (resp. complex analytic spaces). We generally do not distinguish notationally between algebraic spaces, their associated definable analytic spaces, or their associated analytic spaces.
Algebraicity of period maps and theta bundles {#sect alg}
=============================================
Throughout, we use the following terminology. Let $(V_{\mathbb{Z}},W_\bullet V_{\mathbb{Q}})$ be a free finite rank ${\mathbb{Z}}$-module with an increasing rational filtration. We denote by $\gr^W_wV_{\mathbb{Z}}$ the $w$th graded object $\gr^W_wV_{\mathbb{Q}}$ with the integral structure induced by $V_{\mathbb{Z}}$. For each $w$ let a $(-1)^w$-symmetric form $q_w$ on $\gr^W_wV_{\mathbb{Z}}$ be given. There is an associated graded-polarized mixed period domain $M$ parametrizing graded-polarized mixed Hodge structures on $(V_{\mathbb{Z}},W_\bullet V_{\mathbb{Q}},q_\bullet)$. By a graded-polarized mixed period space we mean the quotient ${\mathcal{M}}=\Gamma\backslash M$ by an arithmetic subgroup $\Gamma\subset {\mathbf{G}}({\mathbb{Z}}):=\Aut(V_{\mathbb{Z}},W_\bullet V_{\mathbb{Q}},q_\bullet)$. We have that ${\mathcal{M}}$ is naturally an ${\mathbb{R}}_{\alg}$-definable analytic space by [@bbkt]. When the weight filtration has one nonzero graded piece we refer to ${\mathcal{M}}$ (resp. $M$) as a polarized pure period space (resp. domain), and usually denote it by ${\mathcal{D}}$ (resp. $D$). We also denote by $\check{M}$ the “compact" dual of $M$—the space of filtrations $F^\bullet$ on $V_{\mathbb{C}}$ with fixed $\dim \gr^p_F\gr^W_wV_{\mathbb{C}}$ such that the induced filtration $F^\bullet \gr^W_wV_{\mathbb{C}}$ is $q_w$-isotropic—which is naturally a complex algebraic variety. See for instance [@usui; @pearlstein; @bbkt] for background on mixed period spaces.
Admissible period maps
----------------------
For a definable analytic space $X$, by a definable period map we mean a definable locally liftable map ${\varphi}:X\to {\mathcal{M}}$ which is tangent to the Griffiths transverse foliation of ${\mathcal{M}}$ on the reduced[^3] regular locus. A definable period map is equivalent to a variation of graded-polarized integral mixed Hodge structures, which consists of: a filtered local system $({\mathcal{V}}_{\mathbb{Z}},W_\bullet {\mathcal{V}}_{\mathbb{Q}},q_\bullet)$ locally modeled on $(V_{\mathbb{Z}},W_\bullet V_{\mathbb{Q}},q_\bullet )$ and a locally split filtration $F^\bullet$ of ${\mathcal{V}}_{\mathbb{Z}}\otimes_{{\mathbb{Z}}}{\mathcal{O}}_X$ by definable coherent subsheaves which satisfies Griffiths transversality on the reduced regular locus and which is fiberwise a graded-polarized integral mixed Hodge structure.
We briefly recall the notion of admissible variations; see for instance [@kashiwara] for details. Let $({\mathcal{V}}_{\mathbb{Z}}, W_\bullet {\mathcal{V}}_{\mathbb{Q}}, F^\bullet)$ be a variation of graded-polarizable integral mixed Hodge structures on the punctured disk $\Delta^*$ with unipotent monodromy. Let ${\overline{V}} $ and ${\overline{W}}_\bullet {\overline{V}}$ denote the canonical extensions of ${\mathcal{V}}_{\mathbb{Z}}\otimes_{{\mathbb{Z}}} {\mathcal{O}}_{\Delta^*}$ and $W_\bullet{\mathcal{V}}_{\mathbb{Q}}\otimes_{{\mathbb{Q}}} {\mathcal{O}}_{\Delta^*}$ to $\Delta$ respectively, equipped with their logarithmic connections. Recall that the variation $({\mathcal{V}}_{\mathbb{Z}}, W_\bullet {\mathcal{V}}_{\mathbb{Z}}, F^\bullet)$ is called pre-admissible if the following conditions hold:
1. The residue at the origin of the logarithmic connection on ${\overline{V}}$, which is an endomorphism of the fiber ${\overline{V}}_{|0}$ of ${\overline{V}}$ at the origin, admits a weight filtration relative to ${\overline{W}}_{\bullet}{\overline{V}}_{|0}$.
2. The Hodge filtration $F^\bullet$ extends to a locally split filtration ${\overline{F}}^\bullet$ of ${\overline{V}} $ such that $\gr_ {{\overline{F}}}^p \gr_k^{{\overline{W}}} {\overline{V}}$ is locally-free for all $p$ and $k$.
Given a Zariski-open subset $S$ in a reduced complex analytic space ${\overline{S}}$, we say that a graded-polarizable variation $({\mathcal{V}}_{\mathbb{Z}}, W_\bullet{\mathcal{V}}_{\mathbb{Q}},F^\bullet)$ on $S$ is admissible with respect to the inclusion $S \subset {\overline{S}}$ if for any holomorphic map $f : \Delta \to {\overline{S}}$ such that $f(\Delta^*) \subset S$ and $f^* {\mathcal{V}}_{\mathbb{Z}}$ has unipotent monodromy, the pull-back variation on $\Delta^*$ is pre-admissible. One easily verifies that a variation on $\Delta^*$ with unipotent monodromy which is pre-admissible is admissible with respect to the inclusion $\Delta^* \subset \Delta$. Moreover, if a variation over a complex algebraic variety $S$ is admissible with respect to an algebraic compactification ${\overline{S}}$ of $S$, then it is admissible with respect to any other algebraic compactification of $S$.
\[defn admissible\] Let $X$ be an algebraic space. We say a period map ${\varphi}:X\to {\mathcal{M}}$ is *admissible* if it is definable and the reduced map ${\varphi}^{\reduced}:X^{\reduced}\to{\mathcal{M}}$ is admissible.
See Corollary \[cor tfae\] for some further discussion on the admissibility condition in the nilpotent directions.
Properness of admissible period maps
------------------------------------
We will need an extension property for mixed period maps in Lemma \[lemma make proper\] that is analogous to Griffiths’ result in the pure case [@Giii Theorem 9.5]. This is most likely known to experts, but we include a full argument for the reader’s convenience.\
We first prove a criterion of properness for definable analytic maps analogous to the valuative criterion of properness for algebraic maps.
\[valuative criterion of properness\] Let $X$ be an algebraic space, ${\mathcal{M}}$ a definable analytic space and ${\varphi}: X \to {\mathcal{M}}$ a definable analytic map. Then the map ${\varphi}$ is proper if, and only if, the following property holds: a definable holomorphic map $v: \Delta^* \to X $ extends to $\Delta$ as soon as ${\varphi}\circ v : \Delta^* \to {\mathcal{M}}$ does.
Clearly we can assume that both $X$ and ${\mathcal{M}}$ are reduced. Let ${\overline{X}}$ be an algebraic space compactifying $X$ and let $\tilde X$ denote the topological closure of $X$ in ${\overline{X}} \times {\mathcal{M}}$. Then $\tilde X$ is definable and analytic by Bishop’s theorem [@bishop Theorem 3], as definable sets have locally bounded volume. Since ${\overline{X}}$ is proper the induced holomorphic map $\tilde {\varphi}: \tilde X \to {\mathcal{M}}$ is proper, and the map ${\varphi}: X \to {\mathcal{M}}$ is proper exactly when $X= \tilde X$. Assume first that ${\varphi}$ is not proper, so that there exists $\tilde x \in \tilde X - X$. Since $\tilde X - X$ is a closed analytic subset of $\tilde X$ (as it is the intersection of $({\overline{X}} - X) \times {\mathcal{M}}$ with $\tilde X$), there exists $v : \Delta \to \tilde X$ a definable holomorphic map such that $v(\Delta^\ast) \subset X$ and $v(0) = \tilde x$. Then the map ${\varphi}\circ v : \Delta^* \to {\mathcal{M}}$ does extend to $\Delta$ but $v$ does not. Conversely, let $v : \Delta^* \to X$ be a definable holomorphic map such that ${\varphi}\circ v $ extends to $\Delta$. The induced definable holomorphic map $\Delta^* \to X \times {\mathcal{M}}$ extends to $\Delta \to {\overline{X}} \times {\mathcal{M}}$ and takes values in $\tilde X = X$, hence we are done.
We now apply this criterion to our situation. Let $X$ be a smooth algebraic space, ${\varphi}:X\to {\mathcal{M}}$ an admissible period map, and let $X\subset {\overline{X}}$ be a smooth compactification such that ${\overline{X}} \backslash X=\bigcup_i D_i$ is a normal crossing divisor. Note that for any $i$ the local monodromy around $D_i$ is quasi-unipotent. We may cover ${\overline{X}}$ by polydisks $P\cong \Delta^{n_P}$ such that $P\cap X\cong (\Delta^*)^{r_P}\times\Delta^{s_P}$. For each polydisk $P$ we choose a basepoint $x_P\in P$, and let $N_i^P$ be the logarithm of the unipotent part of the local monodromy operator associated to the $D_i$ meeting $P$. For each $P$ let $C_P$ be the cone generated by $\{N^P_i\}$.
\[lemma proper cond\] The period map ${\varphi}$ is proper if, and only if none of the cones $C_P$ contains $0$.
When ${\mathcal{M}}$ is a pure period space, it follows from the strong version of the nilpotent orbit theorem [@hodgeasymp Theorem 1.15] that the cone $C_P$ contains $0$ only if one of the $N^P_i$’s is zero, in accordance with Griffiths’ result.
Let $v: \Delta^* \to X $ be a definable holomorphic map. Thanks to the nilpotent orbit theorem [@bbkt Proposition 4.3], the composition ${\varphi}\circ v : \Delta^* \to {\mathcal{M}}$ extends to $\Delta$ exactly when the monodromy around $0$ is zero. On the other hand, after shrinking one can assume that $v: \Delta^* \to X $ takes values in one of the polydisk $P$. Then the logarithm of the unipotent part of the monodromy around $0$ is of the form $\sum_i a_i \cdot N_i^P$ for some non-negative integers $a_i$, and conversely every integral element of $C_p$ arises from a definable holomorphic map $\Delta^* \to P $. Since the map $v: \Delta^* \to X$ extends to $\Delta \to X$ exactly when all the $a_i$’s are zero, we conclude using Lemma \[valuative criterion of properness\].
\[lemma make proper\]Let $X$ be a smooth algebraic space and ${\varphi}:X\to {\mathcal{M}}$ an admissible period map. Then there exists a log smooth partial compactification $X\subset \tilde X$ for which the period map extends to a proper map ${\overline}{\varphi}: \tilde X\to{\mathcal{M}}$.
Let ${\overline{X}}$ be a log smooth compactification of $X$. For any polydisk $P$, consider the positive octant ${\mathbb{R}}_{\geq 0}^{r_P}$. The assignment of the monodromy operator $N_i$ to the standard basis vector $e_i$ yields a linear map ${\mathbb{R}}^{r_P}\to\frak{g}$ to the Lie algebra. Its kernel is an integral linear subspace of ${\mathbb{R}}^{r_P}$, and we denote by $K$ its intersection with ${\mathbb{R}}^{r_P}_{\geq 0}$. We may find an integral simplicial subdivision of the standard fan on ${\mathbb{R}}_{\geq 0}^{r_P}$ for which $K$ is a union of facets. This subdivision corresponds to a (global) monomial modification ${{\overline{X}}}_P\to {\overline{X}}$ for which the condition of Lemma \[lemma proper cond\] is satisfied on the preimage of $P$, once we extend the period map over the boundary components with no monodromy using the nilpotent orbit theorem [@bbkt Proposition 4.3]. Notice that any further monomial modification ${\overline{Z}} \to {{\overline{X}}}_P$ will also satisfy the condition above $P$. Thus, taking ${\overline{Z}}$ to be a monomial modification of ${\overline{X}}$ that dominates each of the ${{\overline{X}}}_P$ and extending the period map over the boundary components with no monodromy, the condition of Lemma \[lemma proper cond\] is satisfied.
Unlike in the pure case, some blow-ups may be necessary. Consider the mixed period space ${\mathbf{G}}_m=\Ext^1_{{\mathbb{Z}}MHS}({\mathbb{Z}}(0),{\mathbb{Z}}(1))$ and the period map ${\mathbb{A}}^1\times{\mathbf{G}}_m\to{\mathbf{G}}_m$ which is just the second projection. Take ${\mathbb{A}}^1\times{\mathbf{G}}_m\subset{\mathbb{A}}^2\subset {\mathbb{P}}^2$ as a log smooth compactification. On each vertical line ${\mathbb{A}}^1\times\{z\}$ the period map extends as it is trivial, but the period map does not globally extend over the line at infinity. In this case if the monodromy logarithm around ${\mathbb{A}}^1\times\{0\}$ is $ N$, then the monodromy around infinity is $-N$.
Algebraicity of the Hodge filtration
------------------------------------
For any definable analytic space $X$ and a ${\mathbb{C}}$-local system ${\mathcal{E}}$, ${\mathcal{E}}\otimes_{{\mathbb{C}}_X}{\mathcal{O}}_X$ is naturally a definable analytic coherent sheaf by taking flat trivialization on a definable cover by simply-connected open sets. The following is a nonreduced version of the Deligne extension which is essentially contained in [@bbt §5].
Let $X$ be an algebraic space and ${\mathcal{E}}$ a ${\mathbb{C}}$-local system whose local monodromy has unit norm eigenvalues. Then the definable analytic coherent sheaf $E_X:={\mathcal{E}}\otimes_{{\mathbb{C}}_X}{\mathcal{O}}_{X}$ is algebraic.
First assume $X^{\reduced}$ is smooth. Let $Z$ be a compactification of $X$ for which $(Z^\reduced,Z^\reduced \backslash X^{\reduced})$ is log smooth. We may take a ${\mathbb{R}}_{\mathrm{an}}$-definable cover of $Z^{\mathrm{an}}$ by open sets $P$ for which $P^\reduced\cong \Delta^n$ is a polydisk and $(P^*)^\reduced\cong(\Delta^*)^r\times\Delta^s$ where $P^*=P\cap X$. As $P$ is Stein (since $\Delta^n$ is), we may lift the coordinates to functions $q_i$ on $P$, which are ${\mathbb{R}}_{\mathrm{an}}$-definable after shrinking $P$. Now the ${\mathbb{R}}_{{\mathrm{an}},\exp}$-definable analytic space structure on $P^*$ induces one on any chosen ${\mathbb{R}}_{{\mathrm{an}},\exp}$-definable simply-connected fundamental set of the covering map ${\mathbb{H}}^r\times \Delta^s\to(\Delta^*)^r\times \Delta^s$; call this space $\Phi$. The $q_i$ are ${\mathbb{R}}_{{\mathrm{an}}}$-definable morphisms $P\to{\mathbb{C}}$ and as the multivalued function $\log:{\mathbb{C}}^*\to{\mathbb{C}}$ is definable on angular sectors, the logarithms $z_i=\log q_i$ are ${\mathbb{R}}_{{\mathrm{an}},\exp}$-definable on $\Phi$. We then define a Deligne extension ${\overline{E}}$ of $E_X$ locally using the lattice $\tilde v:=\exp(\sum_iz_iN_i)v$ for flat sections $v$ of ${\mathcal{E}}$ where the $N_i$ are logarithms of the local monodromy, and the same proof as in the reduced case shows these extensions patch (see for instance [@Deligne_book Proposition 5.4]). Now by ordinary GAGA [@gaga], the extension ${\overline{E}}$ is algebraic, and an algebraic frame can be written analytically (hence ${\mathbb{R}}_{\mathrm{an}}$-definably) in terms of the $\tilde v$, while the change-of-basis to the flat frame $\exp(\sum_i z_iN_i)$ is ${\mathbb{R}}_{{\mathrm{an}},\exp}$-definable as the $N_i$ are imaginary.
In the general case, by performing blow-ups along reduced centers we may produce a proper map $\pi:Y\to X$ which is dominant on an open set $U$ of $X$ and for which $Y^\reduced$ is smooth. Let $X'$ be the image of $Y$ in $X$. For a sufficiently big thickening $S$ of the reduced complement $X^\reduced\backslash U^\reduced$, the following square is a pushout $$\xymatrix{
S\times_X X'\ar[d]\ar[r]&X'\ar[d]\\
S\ar[r]&X.
}$$ As $\pi':Y\to X'$ is dominant and $E_{X'}\subset \pi'_*E_Y$, by definable GAGA [@bbt Theorem 3.1] $E_{X'}$ is algebraic, while by Noetherian induction $E_{S}$ is algebraic. $E_X$ is the pushout of $E_S$ and $E_{X'}$, hence algebraic.
\[cor tfae\]Let $X$ be an algebraic space with an analytic period map ${\varphi}:X\to{\mathcal{M}}$ whose reduction ${\varphi}^{\reduced}:X^{\reduced}\to{\mathcal{M}}$ is admissible. Then the following are equivalent:
1. ${\varphi}$ is definable (or equivalently admissible);
2. The Hodge filtration pieces $F^\bullet_X$ are definable analytic subbundles of the ambient flat vector bundle;
3. The Hodge filtration pieces $F^\bullet_X$ are algebraic subbundles of the ambient flat vector bundle.
$(2)\Leftrightarrow(3)$ is immediate given the proposition and definable GAGA. For $(1)\Leftrightarrow(2)$, let $U_i$ be a definable cover of $X$ by simply-connected open sets. The definability of ${\varphi}$ (given the definability of ${\varphi}^{\reduced}$) is equivalent to the definability of the lifts $U_i\to M$ to the universal cover $M$ of ${\mathcal{M}}$, which is in turn clearly equivalent to the definability of $F^\bullet_X$ as a subbundle of the ambient flat bundle with its flat definable structure.
Recall by [@sz] that all variations of graded-polarized integral mixed Hodge structures coming from geometry are admissible. From the corollary it is clear that this is true over possibly non-reduced bases as well.
To algebraize theta bundles in Section \[sect theta\], we will need the following result, which formalizes the idea that the deformation theory of variations of Hodge structures is algebraic, even in the singular setting.
\[prop nbhd\] Let ${\mathcal{M}}$ be a graded-polarized mixed period space and $X\subset{\mathcal{M}}$ an algebraic Griffiths-transverse closed definable analytic subspace. Then for any $n$ the $n$th order thickening of $X$ in ${\mathcal{M}}$ is algebraic.
By definable GAGA we may assume $X$ is reduced. Let $(\mathcal{V}_{\mathbb{C}},W_\bullet \mathcal{V}_{\mathbb{C}})$ be the filtered ${\mathbb{C}}$-local system underlying the mixed variation on $X$, and let $(V,W_\bullet V,F^\bullet V)$ be the associated bifiltered vector bundle with its canonical algebraic structure. Consider $\mathrm{Fl}=\mathrm{Fl}(W_\bullet V)$ the relative flag variety of filtrations $F'^\bullet V$ of $V$ which intersect $W_\bullet V$ with the same dimensions as $F^\bullet V$ and for which the induced filtration $F'^\bullet\gr_k^WV$ is $q_k$-isotropic for each $k$. We have a section $s:X\to \mathrm{Fl}$ of the natural map $\pi:\mathrm{Fl}\to X$ given by $F^\bullet V$; let $S_n\subset \mathrm{Fl}$ be the $n$th order thickening of $s(X)$ in $\mathrm{Fl}$, which is clearly algebraic.
There is a natural admissible period map $\iota:S_n\to {\mathcal{M}}$ extending the inclusion $X\subset{\mathcal{M}}$ which we claim is the closed embedding of the $n$th order thickening. Indeed, $S_n$ can also be analytically constructed as follows. Let $\tilde X$ be the universal cover of $X$, so we have a closed embedding $\tilde X\subset M$ where $M$ is the universal cover of ${\mathcal{M}}$. Let $\tilde S_n$ be the $n$th order thickening of $X$ in $M$, considered as a subspace of $M\times\check{M}$ via the diagonal embedding $M\to M\times \check{M}$ where $\check M$ is the “compact" dual. Now if $\Gamma$ is the image of the monodromy representation, then we have an embedding $S^{\mathrm{an}}_n=\Gamma\backslash \tilde S_n\subset \mathrm{Fl}=\Gamma\backslash(M\times\check{M})$ where $\Gamma$ acts diagonally on $M\times \check{M}$.
That $S_n$ is analytically the closed embedding of the $n$th order thickening is now obvious, and the definability of $\iota:S_n\to {\mathcal{M}}$ follows from Corollary \[cor tfae\].
Algebracity of images
---------------------
\[prop images\]Let $X$ be an algebraic space and ${\varphi}:X\to {\mathcal{M}}$ a definable mixed period map. Then there is a factorization $$\begin{tikzcd}
X\arrow{rr}{{\varphi}}\arrow{dr}[swap]{f}&&{\mathcal{M}}.\\
&Y\arrow[ru,hook,swap,"\iota"]&
\end{tikzcd}$$ where $f$ is dominant algebraic and $\iota$ is a closed immersion.
First assume $X$ reduced and let $\pi: X'\to X$ be a resolution. By Lemma \[lemma make proper\] there is a partial compactification $X'\subset Z'$ for which the period map of $X'$ extends to a proper map ${\overline{{\varphi}}}: Z'\to{\mathcal{M}}$. Now apply [@bbt Theorem 4.2].
In general, let $Y$ be the closure of the image of $X^\reduced$, which is algebraic by the above. By definable GAGA, the pullback $X_n\subset X$ of the $n$th order thickening of $Y$ to $X$ is an increasing sequence of subspaces set-theoretically supported on all of $X^\reduced$, which by Noetherian induction [@bbt Cor. 2.32] on the supports of the ideal sheaves $I_{X_n}$ is eventually all of $X$. By Proposition \[prop nbhd\] and definable GAGA, the claim for $X$ follows.
\[cor mon image\] Let $\Gamma_\mathrm{mon}\subset{\mathbf{G}}({\mathbb{Z}})$ be the image of the monodromy representation of the variation of mixed Hodge structures associated to ${\varphi}$, and ${\varphi}_\mathrm{mon}:X\to \Gamma_\mathrm{mon}\backslash M$ the corresponding lift of ${\varphi}$. Then there is a factorization $$\begin{tikzcd}
X\arrow{rr}{{\varphi}_\mathrm{mon}}\arrow{dr}[swap]{f}&&\Gamma_\mathrm{mon}\backslash M.\\
&Y\arrow[ru,hook,swap,"\iota"]&
\end{tikzcd}$$ where $f$ is dominant algebraic and $\iota$ is a closed immersion.
As in the above proof we may assume ${\varphi}$ and therefore ${\varphi}_\mathrm{mon}$ is proper. Taking $\Gamma'_\mathrm{mon}\subset\Gamma_\mathrm{mon}$ to be a finite-index torsion-free normal subgroup, $Y$ will be the quotient of the image of $X$ in $\Gamma_\mathrm{mon}'\backslash M$ by $\Gamma_\mathrm{mon}/\Gamma_\mathrm{mon}'$, so we may assume $\Gamma_\mathrm{mon}$ to be contained in a torsion-free normal arithmetic subgroup $\Gamma\subset{\mathbf{G}}({\mathbb{Z}})$. Now $Y$ is a finite étale cover of the image in $\Gamma\backslash M$ and therefore algebraic.
Theta bundles {#sect theta}
-------------
Let ${\mathcal{D}}$ be a polarized pure period space parametrizing polarized weight $-1$ Hodge structures $V$ on $(V_{\mathbb{Z}},q_{\mathbb{Z}})$. We can consider the graded-polarized mixed period spaces ${\mathcal{M}}$ resp. ${\mathcal{M}}'$ of extensions $$0\to V\to E\to {\mathbb{Z}}(0)\to 0$$ resp. $$0\to {\mathbb{Z}}(-1)\to E\to V\to 0$$ both of which map to ${\mathcal{D}}$. We may also consider the graded-polarized mixed period space ${\mathcal{B}}$ parametrizing mixed Hodge structures $E$ with weights $[-2,0]$ with $$\begin{aligned}
\gr^W_{-2}E&={\mathbb{Z}}(-1)\\
\gr^W_{-1}E&=V\\
\gr^W_{0}E&={\mathbb{Z}}(0).\end{aligned}$$ The natural map ${\mathcal{B}}\to{\mathcal{M}}\times_{\mathcal{D}}{\mathcal{M}}'$ is canonically an analytic $\Ext^1_{{\mathbb{Z}}MHS}({\mathbb{Z}}(0),{\mathbb{Z}}(-1))\cong{\mathbf{G}}_m$ torsor which we call the biextension torsor; the associated analytic line bundle ${\mathcal{P}}$ on ${\mathcal{M}}\times_{\mathcal{D}}{\mathcal{M}}'$ we call the biextension bundle. Viewing ${\mathcal{M}}\to{\mathcal{D}}$ as the universal intermediate Jacobian $$J({\mathcal{V}}):={\mathcal{V}}_{\mathbb{C}}/F^0{\mathcal{V}}+{\mathcal{V}}_{\mathbb{Z}}$$ and ${\mathcal{M}}'\to{\mathcal{D}}$ as the dual $J({\mathcal{H}}^\vee)$, the biextension bundle ${\mathcal{P}}$ is naturally thought of as the universal Poincaré bundle.
While the total space ${\mathcal{B}}$, the map to ${\mathcal{M}}\times_{\mathcal{D}}{\mathcal{M}}'$, and the ${\mathbf{G}}_m$-action are all definable analytic, it is not clear that ${\mathcal{B}}$ is a definable analytic ${\mathbf{G}}_m$-torsor as it is not clearly that it is definably locally trivial.
\[prop theta alg gen\] Let $X$ be an algebraic space and ${\varphi}: X\to {\mathcal{M}}\times_{\mathcal{D}}{\mathcal{M}}'$ an admissible period map. Then the pullback $B_X$ of the biextension torsor has a natural algebraic étale ${\mathbf{G}}_m$-torsor structure for which the map $B_X\to{\mathcal{B}}$ is definable.
We start by making some preliminary observations. First, note that $B_X$ has a natural definable structure, as it is the base-change of ${\mathcal{B}}$. Thus, it suffices to show that the space underlying $B_X$ has an algebraic structure compatible with the definable structure. Indeed, this algebraic structure is unique by definable GAGA, hence the naturality. Since both the map to $X$ and the ${\mathbf{G}}_m$-action are pulled back from ${\mathcal{B}}$, they are likewise algebraic, and as $B_X\to X$ clearly admits an fppf-local section (over $B_X$ for instance) it follows that it is an étale ${\mathbf{G}}_m$-torsor.
Now to show that the underlying space of $B_X$ is algebraic we proceed by considering successively more general cases.
### Step 1 {#step-1 .unnumbered}
For $X$ smooth, the proposition is a result of Brosnan–Pearlstein:
\[thm theta alg smooth case\] In the above situation and assuming $X$ smooth, $B_X$ (as a sheaf) admits a natural meromorphic extension to any log smooth compactification ${\overline{X}}$ whose sections correspond to admissible liftings $\tilde{\varphi}:X\to{\mathcal{B}}$ of ${\varphi}$. In particular, $B_X$ is an étale ${\mathbf{G}}_m$-torsor.
Note that this algebraic structure is indeed compatible with the definable structure: étale locally the map $B_X\to {\mathcal{B}}$ is identified with $$B_{X}\cong {\mathbf{G}}_m\times X\to{\mathbf{G}}_m\times {\mathcal{B}}\to{\mathcal{B}}$$ where the left isomorphism is induced by a local section of $B_{X}\to X$, the middle map comes from the corresponding admissible lift $X\to {\mathcal{B}}$, and the right map is the action.
### Step 2 {#step-2 .unnumbered}
For $X$ reduced, by taking the closure of the image we may assume ${\varphi}:X\to {\mathcal{M}}\times_{\mathcal{D}}{\mathcal{M}}'$ is a closed immersion by Proposition \[prop images\]. Let $\pi:Y\to X$ be a resolution. By the previous step, $B_{Y}$ is algebraic and the natural map $B_{Y}\to{\mathcal{B}}$ is an admissible period map. It follows by Proposition \[prop images\] again that the image of $B_{Y}\to{\mathcal{B}}$ is algebraic, and this is just the underlying space of $B_X$.
### Step 3 {#step-3 .unnumbered}
For general $X$ we may still assume ${\varphi}:X\to {\mathcal{M}}\times_{\mathcal{D}}{\mathcal{M}}'$ a closed immersion. By the previous step $B_{X^\reduced}$ is algebraic, and by Proposition \[prop nbhd\] and definable GAGA we conclude that the total space $B_X$ is algebraic, hence $B_X$ is algebraic.
In the setup of the proposition, the pullback $P_X$ of the biextension bundle is naturally an algebraic line bundle.
Note that we have a natural definable map $$\sigma:J({\mathcal{V}})\to J({\mathcal{V}}^\vee)$$ commuting with the projection to ${\mathcal{D}}$ which on fibers is the map $\Ext^1_{{\mathbb{Z}}MHS}({\mathbb{Z}}(0),V)\to\Ext^1_{{\mathbb{Z}}MHS}({\mathbb{Z}}(0),V^\vee)$ coming from the polarizing form $q:V\to V^\vee$.
Let $X$ be an algebraic space and ${\varphi}:X\to{\mathcal{M}}$ an admissible period map. The line bundle $\Theta_X$ on $X$ which is the pullback of ${\mathcal{P}}$ along ${\varphi}\times(\sigma\circ{\varphi}):X\to{\mathcal{M}}\times_{\mathcal{D}}{\mathcal{M}}'$ endowed with the natural algebraic structure of Proposition \[prop theta alg gen\] is the *theta bundle* of ${\varphi}$.
Let $X$ be an algebraic space, $V$ a polarized pure Hodge structure of weight $-1$, and ${\varphi}:X\to J(V)$ an admissible quasifinite period map. Then the image ${\varphi}(X)$ is contained in a translate of a subtorus which is an abelian variety.
By Proposition \[prop images\] we may assume $X$ is proper, reduced, irreducible, and a closed subspace $X\subset J(V)$ containing the split point $0\in J(V)$. By replacing $X$ with the image of the difference map $X\times X\to J(V)$ we may eventually assume $X$ is a sub-group. If $H_{\mathbb{Z}}$ is the image of the monodromy $H_1(X,{\mathbb{Z}})\to V_{\mathbb{Z}}$ and $H_{\mathbb{C}}\subset V_{\mathbb{C}}$ the complex span, then $X=H_{\mathbb{C}}/F^0V\cap H_{\mathbb{C}}+H_{\mathbb{Z}}$. The tangent bundle of $J(V)$ is canonically $V_{\mathbb{C}}/F^0V$ and the Griffiths transverse subbundle is $F^{-1}V/F^0V$, so we have $H_{\mathbb{Z}}\subset F^{-1}V$. As $X$ is definable, it must be a compact real torus, so we must have $H_{\mathbb{R}}\cong H_{\mathbb{C}}/F^0V\cap H_{\mathbb{C}}$ via the quotient map. It follows that $H_{\mathbb{Z}}$ underlies a polarized sub Hodge structure of level one.
\[cor theta ample one\] In the setup of the proposition, $\Theta_X$ is ample.
The theta bundle on $J(V)$ is clearly the line bundle associated to the hermitian form $q(u,{\overline}v)$, and restricts to the usual theta bundle of $H_{\mathbb{C}}/F^0H+H_{\mathbb{Z}}$.
\[rmk new proof\]Considerations as in the previous proposition can be used to give a new proof of Theorem \[thm theta alg smooth case\] as follows. Consider a diagram $$\xymatrix{
X\ar[r]\ar[d]&{\mathcal{M}}\times_{\mathcal{D}}{\mathcal{M}}'\ar[d]\\
Y\ar[r]&{\mathcal{D}}}$$ where the horizontal maps are Griffiths transverse closed immersions and $X,Y$ are reduced. After base-changing along an étale map $Y'\to Y$ with dense image, $X':=X\times_YY'\to Y'$ admits a section. As in the proof of the proposition, using Proposition \[prop images\] we may replace $X'$ with the image of the difference map $X'\times_{Y'}X'\to {\mathcal{M}}\times_{\mathcal{D}}{\mathcal{M}}'$, and after finitely many iterations we may assume (after shrinking $Y'$) there is a factorization $$\xymatrix{
X'\ar[rd]\ar[r]&A\times_{Y'}A^\vee\ar[r]\ar[d]&{\mathcal{M}}\times_{\mathcal{D}}{\mathcal{M}}'\ar[d]\\
&Y'\ar[r]&{\mathcal{D}}}$$ where $A\to Y'$ is a smooth definable analytic family of polarized abelian varieties whose fibers are the subgroups generated by the corresponding fibers of $X'\to Y'$. Then $A$ is pulled back along a definable hence algebraic map of $Y'$ to a Shimura variety and the universal Poincaré bundle is algebraic, so $A$, $B_{A\times_{Y'}A^\vee}$, and therefore $B_{X'}$ are all algebraic. The closure of the image of $B_{X'}$ in ${\mathcal{B}}$ is $B_X$, and therefore algebraic by Proposition \[prop images\].
While the argument in the remark directly shows that $B_X$ is algebraic on a stratification, the global definable structure is needed to glue these algebraic structures together since the fiber dimension may jump, as shown in the example(s) below.
It is not hard to provide examples where fiber dimensions jump, even in the mixed Shimura setting. In fact, consider the universal abelian variety $\mathcal{X}_g$ over the moduli stack ${\mathcal{A}}_g$ (one may add level structure to rigidify everything into schemes). Now one may simply take a curve $C$ inside a fiber over ${\mathcal{A}}_g$, and a generic surface $S$ containing it. This is an example of a mixed variation of weights 0,1 with jumping fiber dimensions over the associated graded.
We give below what we consider a more interesting example, where the fibers over the graded are generically finite simply for lack of hodge classes in the associated pure variation, but then over points in the graded which acquire hodge classes, the fiber dimension jumps. This kind of example is harder to construct “artificially” in the manner above, and appears to be a more intrinsic geometric phenomenon.
Let $K$ be a sufficiently high level cover of the moduli space of K3 surfaces polarized by the lattice ${\left(\begin{smallmatrix} 0 & 2\\#3 & 0 \end{smallmatrix}\right)}$, so that it is an irreducible quasiprojective variety. Let $f_1,f_2$ be the divisor classes of the two elliptic pencils and let $S$ be the moduli space of pairs $(X,E)$ with $X\in K$ and $E$ a smooth section of $f_1$. $S$ is also irreducible and admits a forgetful map $S\to K$. Consider the cohomology of the complement $H^2(X\backslash E,{\mathbb{Z}})$ which sits in an extension $$0\to H^2(X,{\mathbb{Z}})/{\mathbb{Z}}f_1\to H^2(X\backslash E,{\mathbb{Z}})\to H^1(E,{\mathbb{Z}})(-1)\to 0.$$ Let $H^2_v(X,{\mathbb{Z}}):=H^2(X,{\mathbb{Z}})/({\mathbb{Z}}f_1+ {\mathbb{Z}}f_2)$, which yields an extension $$0\to H^2_v(X,{\mathbb{Z}})\to H^2_v(X\backslash E,{\mathbb{Z}})\to H^1(E,{\mathbb{Z}})(-1)\to 0.$$
Let ${\varphi}:S\to {\mathcal{M}}$ be the resulting mixed period map for the variation $H^2_v(X\backslash E,{\mathbb{Z}})$, and $\gr{\varphi}: S\to {\mathcal{D}}$ that of the associated graded. It is easy to see that $H^2_v(X,{\mathbb{Z}})\otimes H^1(E,{\mathbb{Z}})^\vee(1)$ generically has no nontrivial sub Hodge structures, so the generic fiber of $\gr{\varphi}$ is 0-dimensional. On the other hand, if $X$ is the Kummer surface of $E\times E'$ with the elliptic pencil given by the first factor $E\times\{0\}$, then for $p\in E\backslash E[2]$ we have $$\begin{aligned}
H^2(X\backslash E\times\{p\},{\mathbb{Q}})&=H^2(\mathrm{Bl}_{E[2]\times E'[2]}(E\times E'\backslash\{\pm p\}),{\mathbb{Q}})^{\pm 1}\\
&={\mathbb{Q}}(-1)^{17}\oplus H^1(E,{\mathbb{Q}})\otimes H^1(E'\backslash\{\pm p\},{\mathbb{Q}}) \end{aligned}$$ so the associated graded is constant on the fiber of $S\to K$ above $X$.
Setup for the proof of quasiprojectivity {#sect setup}
========================================
In this section we collect several results that will be needed for the proof of the quasiprojectivity part of Theorem \[thm main\]. The first is an ampleness criterion in terms of definable sections; the second allows us to endow the cohomology groups of variations of mixed Hodge structures with mixed Hodge structures over arbitrary bases; the third gives some control on the monodromy of extensions of variations of mixed Hodge structures, again over arbitrary bases.
Definable-analytically quasiaffine maps
---------------------------------------
We first show that to prove $f$-ampleness of an algebraic line bundle $L$ and an algebraic map $f:X\to Y$, it suffices to show the definable stalks of $f_*L^n$ separate points. Note that this is weaker than the assumption that $f_*L^n$ separates points definably locally on $Y$.
\[lemma definably ample\]Let $f:X\to Y$ be a map of reduced algebraic spaces, $L$ a line bundle on $X$. Assume for any point $y\in Y$ and any 0-dimensional subspace $P\subset X$ supported on the fiber $X_y$ above $y$ that the restriction on stalks $$(f^{\mathrm{def}}_*(L^n)^{\mathrm{def}})_y\to (f^{\mathrm{def}}_*(L^n|_P)^{\mathrm{def}})_y$$ is surjective for $n\gg 1$. Then $L$ is $f$-ample.
By Zariski’s main theorem, it is enough to show for all $y$ and $P$ as in the statement of the theorem that the restriction map $$(f_*(L^n))_y\to (f_*(L^n|_P))_y$$ is surjective for $n\gg1$. Let $g:Z\to Y$ be a relative compactification of $X$, so $g$ is proper and there is an open immersion $X\to Z$ over $Y$. Let $S$ be the complement of $X$ in $Z$. By assumption there is an $n$, an analytic open neighborhood $U\subset Y^{\mathrm{an}}$ of $y$ and finitely many sections of $(L^n)^{\mathrm{def}}(f^{-1}(U))$ separating $P$, since $(f^{\mathrm{def}}_*(L^n|_P)^{\mathrm{def}})_y$ is a finite-dimensional vector space. We may assume there is a line bundle $M$ extending $L^n$, and by the following lemma definable sections extend meromorphically.
Let $Z$ be a reduced definable analytic space and $S\subset Z$ a closed definable analytic subspace. Any definable analytic $f:Z\backslash S\to {\mathbb{C}}$ extends meromorphically to $Z$.
The closure of the graph $\Gamma(f)\subset (Z\backslash S)\times{\mathbb{C}}$ in $Z\times{\mathbb{P}}^1$ is definable and analytic by for example Bishop’s theorem [@bishop Theorem 3], as definable sets have locally bounded volume.
It thus follows that $$(g^{\mathrm{an}}_*(\sHom(I_S^m,M)^{\mathrm{an}})_y\to (g^{\mathrm{an}}_*(L_P^n)^{\mathrm{an}})_y$$ is surjective for $m\gg0$, and by ordinary GAGA this means the horizontal map below is surjective, finishing the proof. $$\xymatrix{
(g_*(\sHom(I_S^m,M))_y\ar[rd]\ar[rr]&& (g_*L_P^n)_y\\
&(f_*L^n)_y\ar[ur]&
}$$
Lemma \[lemma definably ample\] provides a particularly easy criterion for $X\to Y$ to be quasiaffine.
We say that a map $X\to Y$ of definable analytic spaces is definable-analytically quasiaffine if *analytically* locally on $Y$ it factors as $$\xymatrix{
X\ar[rd]\ar[r]^\iota&{\mathbb{C}}^N\times Y\ar[d]^{\pi_2}\\
&Y
}$$ where $\iota$ is a definable analytic locally closed immersion and $\pi_2$ the second projection.
Recalling that an algebraic map $X\to Y$ is quasiaffine if and only if ${\mathcal{O}}_X$ is relatively ample [@egaii II.5.1.6], we have:
\[cor qa crit\]Let $f:X\to Y$ be a map of algebraic spaces which is definable-analytically quasiaffine. Then $f$ is quasiaffine.
Hodge modules and period maps
-----------------------------
To equip the cohomology of variations of mixed Hodge structures over arbitrary bases with functorial mixed Hodge structures, we will rely crucially on Saito’s formalism of mixed Hodge modules [@saito88; @saito90]. Briefly, for any reduced algebraic space $X$ there is an abelian category $\MHM(X)$ of graded polarizable mixed Hodge modules and a faithful functor $$\rat:D^b\MHM(X)\to D^b_c({\mathbb{Q}}_X)$$ which is exact with respect to the perverse $t$-structure and such that the usual functors $Rf_*, f^*, f_!, f^!,\otimes^L,R\sHom$ on derived categories of constructible sheaves lift to functors $f_*, f^*, f_!, f^!,\otimes,\sHom$. For $X$ smooth, a mixed Hodge module consists of a filtered $D$-module $M$ and a ${\mathbb{Q}}$-perverse sheaf $P$ with a quasi-isomorphism $\mathrm{DR}(M)\xrightarrow{\cong}P_{\mathbb{C}}$ where $\mathrm{DR}(M)$ is the de Rham complex of $M$, while in the general case they are patched together from such objects via local embeddings into smooth ambient spaces.
For $X$ a reduced algebraic space we say $E\in D^b\MHM(X)$ is *smooth* if its underlying rational structure is a local system in degree 0 (with respect to the standard $t$-structure on $D^b_c({\mathbb{Q}}_X)$).
For smooth $X$, there is a natural equivalence of categories [@saito90 Theorem 3.27] $$\left\{\parbox{5.5cm}{\centering admissible variations of rational mixed Hodge structures on $X$}\right\}\label{eq equiv}\to\left\{\mbox{smooth objects of $D^b\MHM(X)$}\right\}$$ which is compatible with pull-backs along algebraic maps $f:X\to Y$.
\[prop mhm\] The functor uniquely extends to a fully faithful functor for any reduced algebraic space $X$ which is compatible with pull-backs along algebraic maps $f:X\to Y$. If $X$ is moreover seminormal then the extension is an equivalence of categories.
In particular, to every admissible period map $X\to{\mathcal{M}}$ we obtain a “pullback object"[^4] $E^H_X\in D^b\MHM(X)$ whose underlying rational structure is the local system ${\mathcal{E}}_X$.
The uniqueness and functoriality are consequences of the uniqueness and functoriality for smooth $X$ and the following fact:
\[lemma mhm morph\] Let $X$ be a reduced algebraic space. For smooth $E,F\in D^b\MHM(X)$ and any dense open set $j:U\to X$ we have $${\textrm{Hom}}(E,F)\cong {\textrm{Hom}}(j^*E,j^*F)$$ via the natural map.
Let ${\mathcal{E}}=\rat(E)$ and ${\mathcal{F}}=\rat(F)$. On the level of sheaves $${\textrm{Hom}}({\mathcal{E}},{\mathcal{F}})\cong {\textrm{Hom}}(j^*{\mathcal{E}},j^*{\mathcal{F}})$$ via the natural map, while $$\begin{aligned}
{\textrm{Hom}}(E,F)&={\textrm{Hom}}({\mathbb{Q}}_X^H,\sHom(E,F))\notag\\
&={\textrm{Hom}}({\mathbb{Q}}(0),\pt_*\sHom(E,F))\label{eq saito}\\
&=\Hdg_0({\textrm{Hom}}({\mathcal{E}},{\mathcal{F}}))_{\mathbb{Q}}\notag\end{aligned}$$ where we equip ${\textrm{Hom}}({\mathcal{E}},{\mathcal{F}})$ with its mixed Hodge structure as $\rat(H^0\pt_*\sHom(E,F))$ and define $\Hdg_k(H)_{\mathbb{Q}}:={\textrm{Hom}}({\mathbb{Q}}(-k),H)=F^kH\cap W_{2k}H_{\mathbb{Q}}$ in general for a rational mixed Hodge structure $H$. Likewise for ${\textrm{Hom}}(j^*E,j^*F)$.
It therefore suffices to show the existence of an extension of $E^H_{X^{\mathrm{reg}}}$ on the regular locus $X^{\mathrm{reg}}\subset X$ to an object $E\in D^b\MHM(X)$ with rational structure ${\mathcal{E}}_X$. We proceed by induction on $\dim X$, the 0-dimensional case being obvious.
Let $\pi:X'\to X$ be a resolution, let $S\subset X$ be the singular locus, and let $S'\subset X'$ be the reduced preimage of $S$. By induction we may assume $E^H_S$ and $E^H_{S'}\cong \pi^*E_S$ exist, and by stipulation $E^H_{X'}$ exists. We have a triangle in $D^b\MHM(X)$ $$\label{eq q cone}{\mathbb{Q}}_X^H\to \pi_*{\mathbb{Q}}_{X'}^H\to A\to{\mathbb{Q}}_X^H[1].$$ As $\pi$ is proper the middle map has an adjoint $$\notag\label{eq q map}{\mathbb{Q}}_{X'}^H\to \pi^!A.$$ Consider the map $$E^H_{X'}\to E^H_{X'}\otimes \pi^!A$$ obtained from tensoring by $E^H_{X'}$. We have natural identifications $$E^H_{X'}\otimes \pi^!A\cong E^H_{S'}\otimes \pi^!A\cong \pi^!(E^H_S\otimes A)$$ the first because $\pi^!A$ is supported on $S'$ and the second because $E^H_{S}$ is smooth. We therefore obtain a map $E^H_{X'}\to \pi^!(E^H_S\otimes A)$, and we define $E$ to be the cone of the adjoint: $$\label{eq E cone}E\to \pi_*E^H_{X'}\to E^H_S\otimes A\to E[1].$$ The image of under $\rat$ is easily seen to be isomorphic to the natural sequence $${\mathcal{E}}_X\to R\pi_*{\mathcal{E}}_{X'}\to {\mathcal{E}}_S\otimes \rat(A)\to{\mathcal{E}}_X[1].$$ obtained by tensoring the $\rat$ of by ${\mathcal{E}}_X$. Moreover, restricting to the regular locus we see (by proper base-change) that $E^H_{X^\mathrm{reg}}\cong E^H_{X'}|_{X^\mathrm{reg}}\cong E|_{X^\mathrm{reg}}$.
For the second claim, assume that $X$ is seminormal and let $\pi:X'\to X$ be a resolution. Let $E\in D^b\MHM(X)$ be a smooth object and let us prove that it comes from an admissible variation of rational mixed Hodge structures on $X$. Since we can argue Zariski-locally, let’s assume that $\pi^*E$ is associated to a period map ${\varphi}:X'\to{\mathcal{M}}$. Clearly ${\varphi}$ is pointwise constant on any fiber of $\pi$. Since ${\mathcal{M}}$ is smooth and $X$ seminormal, it follows that ${\varphi}$ factors through $X$. Here we’ve used that the analytification of a seminormal algebraic space is weakly normal [@gtsn Cor. 6.14] so that the regular functions are continuous meromorphic functions.
\[remark should\] The seminormality hypothesis is necessary in the second statement of Proposition \[prop mhm\] as in general the seminormalization $X'\to X$ is a universal homeomorphism and the functor $\rat :D^b\MHM(X)\to D_c^b({\mathbb{Q}}_X)$ is faithful.
Monodromy of extensions
-----------------------
Let $Y$ be a reduced algebraic space with an admissible variation of rational mixed Hodge structures $V_Y$. Let $X$ be a reduced algebraic space with a map $f:X\to Y$ and an admissible variation of rational mixed Hodge structures $E_X$ which sits in an extension $$0\to V_X\to E_X\to {\mathbb{Q}}_X(0)\to0$$ where $V_X=f^*V_Y$. We have a corresponding exact sequence of rational structures $$0\to {\mathcal{V}}_X\to{\mathcal{E}}_X\to{\mathbb{Q}}_X\to0.$$ Given a point $y\in Y$, let $U\subset Y$ be a small neighborhood. The local system ${\mathcal{E}}_X$ restricted to $X_U:=f^{-1}(U)$ has monodromy landing in ${\mathcal{V}}_{Y,y}$, and we will need two lemmas controlling the image.
\[lemma saito\]In the above situation, the image of the extension class of ${\mathcal{E}}_X$ in $\Ext^1({\mathbb{Q}}_X,{\mathcal{V}}_X)\cong H^1(X,{\mathcal{V}}_X)$ under the composition $$\label{eq sequence}H^1(X,{\mathcal{V}}_X)\to H^1(Y,Rf_*{\mathcal{V}}_X)\to (R^1f_*{\mathcal{V}}_X)_y$$ is Hodge of weight 0 for any $y\in Y$.
Note that in the statement of the lemma we are using that ${\mathcal{V}}_X$ underlies an object $V^H_X\in D^b\MHM(X)$ by Proposition \[prop mhm\], and that the sequence underlies $$H^1\pt_*(V^H_X)\to H^1\pt_*(f_*V^H_X)\to H^1(i^*_yf_*V^H_X)$$ where $i_y:y\to Y$ is the inclusion. All three groups in are therefore equipped with mixed Hodge structures and the maps are morphisms of mixed Hodge structures.
The triangle $${\mathcal{V}}_X\to{\mathcal{E}}_X\to{\mathbb{Q}}_X\to{\mathcal{V}}_X[1]$$ lifts to a triangle $$V^H_X\to E^H_X\to {\mathbb{Q}}^H_X\to V_X^H[1]$$ in $D^b\MHM(X)$ as the morphisms exist by Lemma \[lemma mhm morph\] and exactness can be checked on the underlying rational structures. Moreover, $${\textrm{Hom}}({\mathbb{Q}}(0),H^1\pt_*V_X)={\textrm{Hom}}({\mathbb{Q}}(0),\pt_*V^H_X[1])=H^1\pt_*(V^H_X)$$ and the group on the left is the weight 0 Hodge classes of $H^1(X,{\mathcal{V}}_X)$.
Note that $f_*V^H_X=V^H_Y\otimes f_*({\mathbb{Q}}^H_X)$ and that $$H^1(i_y^*f_*V^H_X)\cong i_y^*V^H_Y\otimes H^1(i_y^*f_*{\mathbb{Q}}^H_X)\cong {\textrm{Hom}}(H_1(X_U,{\mathbb{Q}}),{\mathcal{V}}_{Y,y})$$ as mixed Hodge structures. Furthermore, under this identification the image of the extension class of ${\mathcal{E}}_X$ under is precisely the monodromy representation of ${\mathcal{E}}_X$ restricted to $X_U$.
The previous lemma therefore implies that the monodromy representation $H_1(X_U,{\mathbb{Q}})\to {\mathcal{V}}_{Y,y}$ is a morphism of mixed Hodge structures; the following lemma controls the Hodge numbers of $H_1(X_U,{\mathbb{Q}})$.
\[lemma numbers\] For any map $f:X\to Y$ of reduced algebraic spaces and any $y\in Y$, the nonzero Hodge numbers $h^{p,q}$ of $(R^nf_*{\mathbb{Q}}_X)_y$ satisfy $0\leq p,q\leq n$.
\[rmk Y pt\]The claim is true for $Y$ a point (for instance see [@PSmix Thm. 5.39]), and therefore for proper $f$ by proper base-change.
In the following proof the sheaves/morphisms between them naturally underlie objects in the derived category of mixed Hodge modules (and thus possess/preserve natural mixed Hodge structures), but we phrase the argument entirely in terms of the rational structures for simplicity.
We proceed by induction on $\dim X$, the claim being obvious if $X$ is 0-dimensional. Choose a relative compactification $Z$ and let $F$ be the fiber over $y$. $$\xymatrix{
X\ar[dr]_{f}\ar[r]^j&Z\ar[d]^{g}&\ar[l]_i F\ar[d]\\
&Y&\ar[l]^{i_y}y
}$$ Let $Z'$ be a log resolution of $(Z,Z\setminus X)$ and let $X',F'$ be the reduced preimages of $X,F$. $$\xymatrix{
S'\ar[r]^{\iota'}\ar[d]^\gamma&X'\ar[d]^{\varphi}\ar[r]^{j'}&Z'\ar[d]^{\pi}&\ar[l]_{i'} F'\ar[d]^p\\
S\ar[r]_\iota&X\ar[r]_j&Z&\ar[l]^i F\\
}$$ The map ${\varphi}$ is an isomorphism on a dense Zariski open set $V\subset X$; let $\iota:S\to X$ be the inclusion of the complement of $V$ and $\iota':S'\to X'$ the preimage.
There is a natural morphism ${\mathbb{Q}}_X\to R{\varphi}_*{\mathbb{Q}}_{X'}$. Let $A$ be the cone, so we have a triangle $${\mathbb{Q}}_X\to R{\varphi}_*{\mathbb{Q}}_{X'}\to A\to{\mathbb{Q}}_X[1]\label{eq cone}$$ and note that ${\mathscr{H}}^i(A)=0$ for $i<0$. Applying $R\Gamma i^*Rj_*$ we obtain an exact sequence $$\label{eq les}\xymatrix{
H^{n-1}(i^*Rj_*A)\ar[r]& H^n(i^*Rj_*{\mathbb{Q}}_X)\ar[r]& H^n(i^*Rj_*R{\varphi}_*{\mathbb{Q}}_{X'})&\\
&(R^nf_*{\mathbb{Q}}_X)_y\ar@{=}[u]&H^n(i'^*Rj'_*{\mathbb{Q}}_{X'})\ar@{=}[u]&
}$$ where the vertical identifications are by proper base-change. The object $A$ is supported on $S$, so pulling back to $S$ we obtain a triangle $${\mathbb{Q}}_S\to R\gamma_*{\mathbb{Q}}_{S'}\to \iota^*A\to{\mathbb{Q}}_S[1]$$ and after applying $R\Gamma i^*Rj_*\iota_*$ an exact sequence $$\xymatrix{
H^{n-1}(i^*Rj_*\iota_*R\gamma_*{\mathbb{Q}}_{S''})\ar@{=}[d]\ar[r]& H^{n-1}(i^*Rj_*A)\ar[r]& H^n(i^*Rj_*\iota_*{\mathbb{Q}}_S)\ar@{=}[d]\\
(R^{n-1}(f\circ\iota\circ\gamma)_*{\mathbb{Q}}_{S''})_y&& (R^n(f\circ \iota)_*{\mathbb{Q}}_S)_y.}$$ By the induction hypothesis, it follows that the nonzero Hodge numbers $h^{p,q}$ of $H^{n-1}(i^*Rj_*A)$ have $0\leq p,q\leq n$, and by it is therefore enough to prove:
The nonzero Hodge numbers $h^{p,q}$ of $H^n(i'^*Rj'_*{\mathbb{Q}}_{X'})$ satisfy $0\leq p,q\leq n$.
Recall that $Rj'_*$ is exact in the perverse $t$-structure; let $M=j'_*{\mathbb{Q}}_{X'}^H[d]$ where $d=\dim X$ (we may assume $X'$ irreducible). Recall that $$\gr^W_{d+k}M = \bigoplus_{|I|=k}{\mathbb{Q}}^H_{D_I}(-k)[d-k]$$ where $D_I=\bigcap_{i\in I}D_i$ is the $I$th boundary stratum of $Z'\backslash X'$. Defining $F'_I:=F'\cap D_I$ to be the reduced intersection, we have $i'^*{\mathbb{Q}}^H_{D_I}={\mathbb{Q}}^H_{F'_I}$ (as this is clearly true on the level of underlying rational structures), so $$i'^*\gr^W_{d+k}M=\bigoplus_{|I|=k}{\mathbb{Q}}^H_{F'_I}(-k)[d-k].$$ To prove the claim, it suffices to show the claimed vanishing of Hodge numbers for $H^{n-d}(i'^*\gr^W_{d+k}M)$ for all $k$. This in turn follows because $H^{n-k}(F'_I,{\mathbb{Q}})(-k)$ satisfies the vanishing for all $k$ (see Remark \[rmk Y pt\]).
Proof of quasiprojectivity {#sect proof}
==========================
Induction step
--------------
Let ${\mathcal{M}}_{w}$ be a mixed period space parametrizing graded-polarized integral mixed Hodge structures $E$ with weights $\leq w$ and let $\mathcal{M}_{ w-1}$ (resp. ${\mathcal{M}}_{[w-1,w]}$, ${\mathcal{D}}_{w-1}$) be the graded-polarized mixed period space parametrizing mixed Hodge structures $H$ of the form $W_{w-1}E$ (resp. $\gr_{[w-1,w]}^WE$, $\gr_{w-1}^WE$). We naturally have a definable analytic morphism $${\mathcal{M}}_{ w}\to{\mathcal{M}}_{w-1}\times_{{\mathcal{D}}_{w-1}} {\mathcal{M}}_{[w-1,w]}$$ From section \[sect theta\] we have an analytic theta bundle $\Theta_{[w-1,w]}$ on ${\mathcal{M}}_{[w-1,w]}$; we also denote by $\Theta_{[w-1,w]}$ the pullback to ${\mathcal{M}}_{w}$. As no confusion is likely to occur, for any period map ${\varphi}:X\to {\mathcal{M}}_{w}$ we also denote by $\Theta_{[w-1,w]}$ the pullback to $X$ together with its natural algebraic structure as in Proposition \[prop theta alg gen\].
The main result of this section is:
\[prop induct step\]Let $X,Y,Z$ be algebraic spaces together with a diagram $$\xymatrix{
X\ar[dd]_{f}\ar[rd]^g\ar[rr]&&{\mathcal{M}}_w\ar[dd]\ar[rd]&\\
&Y\ar[rr]\ar[ld]^h&&{\mathcal{M}}_{w-1}\times_{{\mathcal{D}}_{w-1}}{\mathcal{M}}_{[w-1,w]}\ar[ld]\\
Z\ar[rr]&&{\mathcal{M}}_{w-1}&
}$$ whose horizontal maps are definable Griffiths transverse closed immersions. Then ${\mathcal{O}}_X$ is $g$-ample and the theta bundle $\Theta_{[w-1,w]}$ is $h$-ample.
\[cor theta ample\]In the above situation, $\Theta_{[w-1,w]}$ is $f$-ample.
Before the proof we make some observations. As in the introduction, ${\mathcal{M}}_{w}$ embeds into the mixed period space ${\mathcal{M}}'_{ 0}$ parametrizing extensions of the form $$0\to W_{w-1}E\otimes \gr_w^WE^\vee \to E'\to{\mathbb{Z}}(0)\to0$$ and likewise we have embeddings of ${\mathcal{M}}_{w-1}, {\mathcal{M}}_{[w-1,w]}, {\mathcal{D}}_{w-1},{\mathcal{M}}_{w-1}\times_{{\mathcal{D}}_{w-1}}{\mathcal{M}}_{[w-1,w]}$ into the corresponding spaces ${\mathcal{M}}_{-1}', {\mathcal{M}}_{[-1,0]}', {\mathcal{D}}_{-1}',{\mathcal{M}}'_{-1}\times_{{\mathcal{D}}'_{-1}}{\mathcal{M}}'_{[-1,0]}$ associated to ${\mathcal{M}}'_{0}$ that are compatible with the obvious maps. Finally, $\Theta_{[w-1,w]}$ is pulled back from ${\mathcal{M}}'_{0}$.
Thus we may assume $w=0$ and $\gr_0^WE\cong{\mathbb{Z}}(0)$, and ${\mathcal{M}}_0$ parametrizes extensions of the form $$0\to V\to E\to {\mathbb{Z}}(0)\to0$$ for $V$ in ${\mathcal{M}}_{-1}$. In this case, $\mathcal{M}_{ 0}$ is isomorphic to the intermediate Jacobian $$J({\mathcal{V}}):={\mathcal{V}}_{\mathbb{C}}/F^0{\mathcal{V}}+{\mathcal{V}}_{\mathbb{Z}}$$ over ${\mathcal{M}}_{-1}$ where ${\mathcal{V}}_{\mathbb{Z}}$ is the universal ${\mathbb{Z}}$-local system and $F^\bullet{\mathcal{V}}$ is the universal Hodge filtration. Moreover, we have maps $$J(W_{-2}{\mathcal{V}})\to J({\mathcal{V}})\xrightarrow{\pi} J(\gr^W_{-1}{\mathcal{V}})$$ of definable analytic spaces by interpreting each as mixed period spaces. In particular, ${\mathcal{M}}_{-1}\times_{{\mathcal{D}}_{-1}}{\mathcal{M}}_{[-1,0]}$ is isomorphic to $J(\gr^W_{-1}{\mathcal{V}})$ over ${\mathcal{M}}_{-1}$ and $\pi$ has a definable action by $J(W_{-2}{\mathcal{V}})$ which around every point of $J(\gr^W_{-1}{\mathcal{V}})$ admits an analytic (hence definable, after shrinking) trivializing section.
For any $y\in J(\gr_{-1}^W{\mathcal{V}})$, let $U'\subset J(\gr_{-1}^W{\mathcal{V}})$ be a small ball neighborhood, and for $z\in {\mathcal{M}}_{-1}$ the image of $y$ let $U\subset{\mathcal{M}}_{-1}$ be the (open) image of $U'$. Denote $$J^{\Hdg}_z(W_{-2}{\mathcal{V}}_U):=(W_{-2}{\mathcal{V}}_U)_{{\mathbb{C}}}/F^0W_{-2}{\mathcal{V}}_U+\Hdg_{-1}(W_{-2}V_z)_{\mathbb{Z}},$$ where $\Hdg_{k}(H)_{\mathbb{Z}}:=F^{k}H\cap W_{2k}H_{\mathbb{Z}}$. We have a diagram $$J^{\Hdg}_z(W_{-2}{\mathcal{V}}_U)\to J(W_{-2}{\mathcal{V}}_U)\to U.$$ Note that independently of the choice of a basepoint there is an identification of the fundamental group of $J(W_{-2}{\mathcal{V}}_U)$ with $W_{-2}V_{\mathbb{Z}}$, that of $J^{\Hdg}_z(W_{-2}{\mathcal{V}}_U)$ with $\Hdg_{-1}(W_{-2}V_z)_{\mathbb{Z}}$, and that the first map is a covering map with covering group $V_{\mathbb{Z}}/\Hdg_{-1}(W_{-2}V_z)_{\mathbb{Z}}$.
We endow $J^{\Hdg}_z(W_{-2}{\mathcal{V}}_U)$ with a definable structure as follows. We may choose a definable analytic splitting$$W_{-2}{\mathcal{V}}_U\otimes {\mathcal{O}}_U=F^0W_{-2}{\mathcal{V}}_U\oplus (\Hdg_{-1}(W_{-2}V_z)_{\mathbb{Z}}\otimes{\mathcal{O}}_U)\oplus Q$$ where $\Hdg_{-1}(W_{-2}V_z)_{\mathbb{Z}}\otimes{\mathcal{O}}_U\subset{\mathcal{V}}_U\otimes{\mathcal{O}}_U$ is the constant subbundle and $Q$ is a definable analytic subbundle. We may therefore definably identify $$J_z^{\Hdg}(W_{-2}{\mathcal{V}}_U)\cong \Hdg_{-1}(W_{-2}V_z)_{{\mathbf{G}}_m}\times \mathbb{C}^N\times U$$ over $U$, where for a mixed Hodge structure $H$ $$\Hdg_{-1}(H)_{{\mathbf{G}}_m}:=(\Hdg_{-1}(H)_{{\mathbb{Z}}}\otimes{\mathbb{C}})/\Hdg_{-1}(H)_{{\mathbb{Z}}}\cong{\mathbf{G}}_m^n$$ with its canonical definable structure.
Choosing a definable section of $\pi$, we identify $$\pi^{-1}(U')\cong J(W_{-2}{\mathcal{V}}_U)\times_U U'.$$ The fundamental group of $\pi^{-1}(U')$ (after choosing a basepoint) is canonically identified with $(W_{-2}V_z)_{\mathbb{Z}}$ via the action of $J(W_{-2}{\mathcal{V}}_U)$. Denote by $$J^{\Hdg}_{U',y}:=J^{\Hdg}_z(W_{-2}{\mathcal{V}}_U)\times_U U'$$ and note that $J^{\Hdg}_{U',y}$ and its definable structure don’t depend on the choices. Furthermore, since $J_z^{\Hdg}(W_{-2}{\mathcal{V}}_U)\to U$ is clearly definable-analytically quasiaffine (even affine), we have:
\[lemma def an affine\]$J^{\Hdg}_{U',y}\to U'$ is definable-analytically quasiaffine.
We are now ready to prove Proposition \[prop induct step\].
By [@egaii II.4.6.16] we may assume $X,Y,Z$ are all reduced. We first show that $\Theta_{[-1,0]}$ is $h$-ample. $\Theta_{[-1,0]}$ is algebraic by Proposition \[prop theta alg gen\], and since $h$ is proper it suffices to show it is ample on fibers, which is Corollary \[cor theta ample one\].
It remains to prove that ${\mathcal{O}}_X$ is $g$-ample. Using Lemmas \[cor qa crit\] and \[lemma def an affine\], it is sufficient to show the following:
For any $y\in Y$ and any sufficiently small open ball neighborhood $y\in U'\subset J(\gr^W_{-1}{\mathcal{V}})$ as above we have a definable analytic lifting $$\xymatrix{
&J^{\Hdg}_{U',y}\ar[d]\\
X_{U'}\ar@{-->}[ru]\ar[r]&\pi^{-1}(U')
}\label{eq lift}$$ where $X_{U'}:=X\cap \pi^{-1}(U')$.
While $J^{\Hdg}_{U',y}\to \pi^{-1}(U')$ is of course not definable, we first claim:
Any analytic lift as in is definable.
Let $\Xi_0\subset M_0$ be a definable fundamental set for ${\mathcal{M}}_0$ and let $\Xi'$ be the preimage of $X_{U'}$ in $\Xi_0$. Its enough to show that $\Xi'\to J^{\Hdg}_{U',y}$ is definable. Recall that $M_0$ is identified with $V_{\mathbb{C}}/F^0V$ over $M_{-1}$. Taking a resolution of $X$, there are finitely many nilpotent orbits approximating the preimage of $X$ in $\Xi_0$; by [@hpmixnil Theorem 6.2], outside of a bounded set $\Xi'$ is within a finite distance (with respect to the standard metric on $V_{\mathbb{C}}/F^0V$) of finitely many nilpotent orbits whose monodromy is trivial in $M_{-1}$. It thus suffices to verify that each such nilpotent orbit (restricted to a product of bounded vertical strips) has definable image in $J^{\Hdg}_{U',y}$. But possibly after shrinking $U'$, each such nilpotent is $v+\sum_i t_in_i$ for $v\in V_{\mathbb{C}}/F^0V$ and $n_i\in \Hdg_{-1}(V_z)_{\mathbb{Z}}=\Hdg_{-1}(W_{-2}V_z)_{\mathbb{Z}}$, for which the claim is obvious.
By Lemma \[lemma saito\] the monodromy of the extension $$0\to {\mathcal{V}}_X\to{\mathcal{E}}_X\to{\mathbb{Z}}_X$$ restricted to $X_{U'}$ is an element $\xi$ of $(R^1g_*{\mathcal{V}}_X)_y$ which is Hodge of weight 0. We have an exact sequence $$\label{eq 1}(R^1g_*W_{-2}{\mathcal{V}})_y\to (R^1g_*{\mathcal{V}})_y\to (R^1g_*\gr^W_{-1}{\mathcal{V}})_y$$ and as the extension $$0\to \gr^W_{-1}{\mathcal{V}}_X\to \gr_{[-1,0]}^W E\to{\mathbb{Z}}_X\to 0$$ is pulled back from $Y$, $\xi$ maps to $0$ under the right map of . Thus, $\xi$ comes from a class of $(R^1g_*W_{-2}{\mathcal{V}})_y$ which is Hodge of weight 0, and by Lemma \[lemma numbers\] this is an element of $${\textrm{Hom}}(H_1(X_{U'},{\mathbb{Q}}),\Hdg_{-1}(W_{-2}V_y)_{\mathbb{Q}}).$$
General case
------------
Let ${\mathcal{M}}$ be a mixed period space parametrizing graded-polarized integral Hodge structures $E$. Let ${\mathcal{D}}_w$ be the polarized pure period space of the associated graded object $ \gr^W_w E$ and ${\mathcal{M}}_{[w-1,w]}$ the graded-polarized mixed period space of $\gr^W_{[w-1,w]}E$. We have a diagram $$\xymatrix{
{\mathcal{M}}\ar[dd]\ar[rd]&\\
&\prod_w {\mathcal{M}}_{[w-1,w]}\ar[ld]\\
{\mathcal{D}}:=\prod_w{\mathcal{D}}_w&
}$$ where the bottom diagonal map arises from the fact that the natural map $\prod_w {\mathcal{M}}_{[w-1,w]}\to{\mathcal{D}}\times{\mathcal{D}}$ factors through the diagonal.
Consider a diagram $$\xymatrix{
X\ar[dd]_{f}\ar[rd]^g\ar[rr]&&{\mathcal{M}}\ar[dd]\ar[rd]&\\
&Y\ar[rr]\ar[ld]^h&&\prod_w {\mathcal{M}}_{[w-1,w]}\ar[ld]\\
Z\ar[rr]&&{\mathcal{D}}&
}$$ where $X,Y,Z$ are algebraic spaces and the horizontal maps are Griffiths transverse closed immersions. The proof of the quasiprojectivity claim of Theorem \[thm main\] is completed by the following:
1. ${\mathcal{O}}_X$ is $g$-ample;
2. $\Theta_X:=\bigotimes_w\Theta_{[w-1,w]}$ is $h$-ample.
<!-- -->
1. If $w_{\min}$ (resp. $w_{\max}$) is the minimum (resp. maximum) weight $w$ for which $\gr_w^WE\neq 0$, then by taking images we have diagrams $$\xymatrix{
X_w\ar[r]\ar[d]_{f_w}&{\mathcal{M}}_{w}\ar[d]\ar[r]&{\mathcal{M}}_{[w-1,w]}\\
X_{w-1}\ar[r]&{\mathcal{M}}_{w-1}&
}$$ with $X=X_{w_{\max}}$, ${\mathcal{M}}= {\mathcal{M}}_{w_{\max}}$, $Z=X_{w_{\min}}$, and ${\mathcal{D}}= {\mathcal{M}}_{w_{\min}}$. By Corollary \[cor theta ample\] the theta bundle $\Theta_{[w-1,w]}$ is $f_w$-ample, and it follows that $L:=\bigoplus_w \Theta_{[w-1,w]}^{a_w}$ is $f$-ample for some $a_w>0$ [@egaii II.4.6.13]. As $L$ is pulled back from $Y$, it follows that ${\mathcal{O}}_X$ is $g$-ample.
2. $\Theta_X$ is ample on the fibers of $h$ by Corollary \[cor theta ample one\], and since $h$ is proper, it follows that it is $h$-ample.
[^1]: \[footnote\]See Definition \[defn admissible\] for the definition of admissibility over nonreduced bases.
[^2]: Such quotients are not good moduli spaces however as they do not in general have a tame geometry.
[^3]: Note in particular that we do not require the nilpotent tangent directions to be Griffiths transverse, though it is not clear that this level of generality is useful: variations coming from geometry will satisfy Griffiths transversality in the nilpotent directions as well.
[^4]: $E^H_X$ of course depends on ${\varphi}$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present evidence for a skewed distribution of *UV* Fe <span style="font-variant:small-caps;">ii</span> emission in quasars within candidate overdense regions spanning spatial scales of $\sim50$ Mpc at $1.11 < z < 1.67$, compared to quasars in field environments at comparable redshifts. The overdense regions have an excess of high equivalent width sources (W2400 $>$ 42 Å), and a dearth of low equivalent width sources. There are various possible explanations for this effect, including dust, Ly$\alpha$ fluorescence, microturbulence, and iron abundance. We find that the most plausible of these is enhanced iron abundance in the overdense regions, consistent with an enhanced star formation rate in the overdense regions compared to the field.'
author:
- |
Kathryn A. Harris$^{1}$[^1], G.M. Williger$^{2,3,4}$, L. Haberzettl$^2$, S. Mitchell$^{2,5}$, D. Farrah$^1$, M.J. Graham$^6$, R. Dav[é]{}$^{7}$, M.P. Younger$^8$, I.K. S[ö]{}chting$^9$\
$^1$ Department of Physics, Virginia Tech, Blacksburg, VA 24061, USA\
$^2$Department of Physics and Astronomy, University of Louisville, Louisville, KY 40292, USA\
$^3$Lab. Lagrange, U. de Nice, UMR 7293, 06108 Nice Cedex 2, France\
$^4$Institute for Astrophysics and Computational Sciences, Catholic University of America, Washington DC 20064, USA\
$^5$Department of Aerospace Engineering ACCEND, University of Cincinnati, Cincinnati OH, USA\
$^6$Center for Advanced Computing Research, California Institute of Technology, 1200 E California Blvd, Pasadena CA 91125, USA\
$^{7}$Department of Astronomy, University of Arizona, 933 North Cherry Ave, Tucson, AZ 85721, USA\
$^8$Jeremiah Horrocks Institute, University of Central Lancashire, Preston, PR1 2HE\
$^9$Astrophysics, Denys Wilkinson Building, Keble Road, University of Oxford, Oxford OX1 3RH\
date: 'Accepted xxxx. Received xxxx; in original form xxxx'
title: 'Evidence of Increased *UV* Fe Emission in Quasars in Candidate Overdense Regions'
---
galaxies:active - quasars:emission lines - large-scale structure of Universe - galaxies:abundances
INTRODUCTION
============
There is significant controversy over the stellar mass-metallicity (M-Z) relation as a function of environment and redshift. The general expectation might be that metallicity is higher in overdense regions at a given redshift, since high redshift starburst galaxies seem to prefer such regions [@Gomez2003; @Blain2004; @Farrah2006; @Cooper2008]. Earlier star formation would give rise to earlier metal enrichment of the ISM/IGM. For example, supernovae [e.g. @Adelberger2005; @Domainko2004] may efficiently enrich the IGM over Mpc scales.
Conversely, direct observational studies are ambiguous. At low redshift, some authors [e.g. @Hughes2013] find no relation between metallicity and environment, while others [@Skillman1996; @Cooper2008] claim a weak but significant trend for galaxies in groups or clusters to have higher metallicities than field galaxies. At higher redshifts, there is even more uncertainty [e.g. @Hamann1993] with few studies considering environment.
A potentially powerful way to constrain star formation histories in different environments at high redshifts is to use the ratio of Fe <span style="font-variant:small-caps;">ii</span>\[*UV*\] to Mg <span style="font-variant:small-caps;">ii</span>\[$\lambda$2798\]. To first order, Fe <span style="font-variant:small-caps;">ii</span> is produced from SNeIa roughly one Gyr after the initial burst of star formation, while Mg <span style="font-variant:small-caps;">ii</span> is created in SNeII. Hence their ratio can be used as a cosmological clock [@Hamann1993] to age-date the initial star formation. Moreover, both emission lines are seen in quasars, where the quasar illuminates the metal rich gas. This allows the lines and therefore the metallicities to be observed to potentially very high redshifts. However there is a large amount of scatter seen in this ratio, the reasons for which are not fully understood.
*UV* Fe <span style="font-variant:small-caps;">ii</span> has been observed in different objects such as symbiotic stars [e.g @Hartman2000], young stellar objects [e.g. @Cooper2013], novae [e.g @Johansson1984] and the Broad Line Region (BLR) of active galactic nuclei (AGN) [@Sigut1998]. In AGN, *UV* Fe <span style="font-variant:small-caps;">ii</span> is seen at varying strengths, though the reasons for this variation are still debated. A number of Fe <span style="font-variant:small-caps;">ii</span>-bright quasars have been found and studied in detail over a wide redshift range [e.g. @Osterbrock1976; @Weymann1991; @Graham1996; @Vestergaard2001; @Bruhweiler2008].
While several mechanisms likely affect the observed iron emission (e.g. abundance, collisional excitation, microturbulence and Ly$\alpha$ fluorescence, see e.g. @Netzer1983 [@Sigut2003; @Baldwin2004; @Matsuoka2007]), it is plausible (given that all but abundance are small $<$pc scale mechanisms and unlikely to be effected by the $>$Mpc scale environment) that this emission is a reasonably proxy for the metallicity build up in galaxies.
In this paper, we explore the use of the *UV* Fe <span style="font-variant:small-caps;">ii</span> in high redshift quasar spectra to consider differences in SFHs in different environments at high redshift. To do so, we consider the overdense regions of quasars in Large Quasar Groups (LQGs).
LQGs are some of the largest candidate overdensities seen in the Universe, spanning 50-200 $h^{-1}$ Mpc, have been found at $z > 1$, and are potentially the precursors of the large overdensities seen at the present epoch, such as super-clusters [@Komberg1996; @Wray2006]. These LQGs exist at high redshifts and presumably trace the mass distribution. There are $\sim$ 40 published examples of LQGs [@Clowes2012 (CCGS12) and references therein].
By using LQGs we can quickly assemble statistically significant numbers of quasars in overdense regions, to compare to field samples. The observations for this paper were taken in the direction of the Clowes-Campusano LQG (CCLQG; @Clowes1991 [@Clowes1994]) which lies at a redshift of $z \sim 1.3$, and spans $\sim$100-200 $h^{-1}$ Mpc.
We compare the *UV* Fe <span style="font-variant:small-caps;">ii</span> in quasars in LQGs at $z > 1$ to the same emission seen in quasars in the field over similar redshifts to search for differences in star formation history as a function of environment. We will present 12 AGN at $z$ = \[1.159,1.689\] with increased *UV* Fe <span style="font-variant:small-caps;">ii</span> emission (W2400 $>$ 32Å) evident in their spectra. All of the quasars are within an area of 1.6 deg$^{2}$, and lie within the redshift range of the overdensity previously described. The cosmology used is $H_0=70$ kms$^{-1}$Mpc$^{-1}$, $\Omega_m=0.27$ and $\Omega_{\Lambda}=0.73$.
ANALYSIS
========
We treat the LQG region as a potential high density environment.
By comparing the measurements of the Fe <span style="font-variant:small-caps;">ii</span> emission in these quasars to the emission from a control sample of randomly selected quasars, we examine any differences between the samples. Due to the limits of the observations, we do not study the whole LQGs field, using only two 0.8 deg$^2$ of the area (which is covered by our additional observations described later in this section). These fields are centred on RA = 162.146, Dec = 5.406, and RA = 162.514, Dec = 4.528.
[FE ]{} MEASUREMENT TECHNIQUES {#sect:spectra}
------------------------------
To measure the Fe <span style="font-variant:small-caps;">ii</span> emission, we used the method described in @Weymann1991. We use this method to provide an estimate of the overall emission as opposed to, for example, the @Hartig1986 method which gives an estimate at a single wavelength. The continuum level is found at the central wavelength within two wavelength ranges, 2240–2255 [Å]{} and 2665–2695 [Å]{}. A straight line is then drawn between the centres of these two wavelength ranges to create the effective continuum. @Weymann1991 calculate the equivalent width (EW) between 2255 and 2650 [Å]{} (W2400) with respect to this effective continuum level. The errors on the measurements are estimated based on the noise across the continuum which has the greatest effect and therefore the dominant error in the Fe <span style="font-variant:small-caps;">ii</span> measurement. (The values are estimated in Section \[sect:control\].)
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{width="45.00000%"} {width="45.00000%"}
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LQG FIELD SAMPLE
----------------
Two LQGs and an additional quasar set have been found in the area studied in this paper. The overdensity was estimated using $(\rho - \langle\rho\rangle)/\langle\rho\rangle$ (CCGS12).
1. L1.28: The CCLQG lies at $z$ = \[1.187,1.423\], contains 34 members, and has an estimated overdensity of 0.83 and a statistical significance of 3.57$\sigma$ (CCGS12).
2. L1.11: There is another LQG at $z$ = \[1.004,1.201\], containing 38 members (CCGS12). This group has an estimated overdensity of 0.55 and a statistical significance of 2.95$\sigma$.
3. L1.54: There is an “enhanced set” of quasars with 21 members at $z$ = \[1.477,1.614\]. This group has an overdensity of 0.49, and a statistical significance of 1.75$\sigma$, which though suggestive, is not high enough to be statistically significant for a large structure (@Newman1999, CCGS12).
The original LQG members were selected from the SDSS DR7QSO catalogue [@Schneider2010]. A magnitude cut of $i$-mag = 19.1 [@Schneider2010] was applied to create a uniform sample and quasars are within a 3D linking length of 100 Mpc. A convex hull is created around the members, giving the total volume covered by the LQG. See CCGS12 for more details on the method used to select LQG members.
The latest discussion of these LQGs can be found in CCGS12. Due to uncertainties over LQG membership caused by the member selection criteria and sample completeness, for the rest of the paper we will not be discussing the LQG or which quasars are classed as members. We will assume that quasars trace the mass distribution and therefore this area space and redshift range is therefore a candidate overdense region. @Martini2013 [and references therein] found for $1 < z < 1.5$ the fraction of AGN lying in clusters is increased compared to lower redshifts, making this a reasonable assumption.
There are 10 quasars at $1.1 < z < 1.7$ from the SDSS DR7 QSO catalogue [@Schneider2010] which have SDSS spectra in the area of the LQGs we are studying. The spectra cover the wavelength range 3800–9200 [Å]{} and have a resolution of 2.5 [Å]{} [@SDSSprojectbook].
To improve statistics and better sample the overdensity, we increased the sample size. We start with a sample of quasars with photometric redshifts from the DR7 catalogue by @Richards2009 which place them within the redshift range of the LQGs. We then randomly selected a subset of 32 for followup spectroscopy (observed as part of a larger observing project), dependent on available fiber positioning. We used the Hectospec instrument [@Fabricant2005] a multi-object optical spectrograph, mounted at the 6.5-m MMT on Mount Hopkins, Arizona. The spectra were taken over nine nights and, due to inaccuracies in photometric redshifts, produced 18 quasar spectra within the required redshift range. The remaining objects were a mixture of quasars (generally at lower redshifts) and star forming galaxies.
The Hectospec data cover 3900 to 9100 [Å]{} and have a resolution of 1.2 [Å]{}. These spectra were reduced using the IDL based pipeline, <span style="font-variant:small-caps;">HSRED</span>[^2]. Table \[tab:feIIobslog\] shows the dates, fields, and exposures times for the Hectospec observations.
Date RA (J2000) Dec (J2000) Exposure (s)
------------ ------------ ------------- --------------
17.02.2010 10:50:16.9 +04:37:12 5400
18.02.2010 10.50:16.9 +04:37:12 5400
19.02.2010 10:50:06.9 +04:29:16 5094
06.04.2010 10:50:06.9 +04:29:16 5400
07.04.2010 10:48:31.8 +05:23:29 7200
09.04.2010 10:48:31.8 +05:23:29 5400
10.04.2010 10:48:38.9 +05:25:57 5400
11.04.2010 10:48:38.9 +05:25:57 5400
11.04.2010 10:49:57.0 +04:30:01 5400
12.04.2010 10:49:57.0 +04:30:01 1800
\[tab:feIIobslog\]
The final catalogue of quasars (see Table \[tab:Fedata\]) contains 18 quasars from Hectospec and 6 quasars from SDSS spectra within the redshift range $1.1 < z < 1.7$. Four of the SDSS quasars were removed due to low signal-to-noise spectra but are included in Table \[tab:Fedata\] for completeness. The area occupied by these quasars covers 1.6 deg$^2$ of the LQGs. An example of the spectra is shown in Fig. \[fig:spec\].
### COMPLETENESS AND LQG MEMBERS {#sect:complete}
The Hectospec quasars, though not a complete sample, were randomly selected across area and redshift range, with no bias towards strong or weak Fe <span style="font-variant:small-caps;">ii</span> emission, magnitude, or location (beyond being within the field of the LQG overdensities). The quasars were observed as part a larger project which observed lower redshift luminous red galaxies. Therefore there was no biasing on the placement of the available fibers for observing these quasars.
Because of the data and the above described member selection method, we can say which quasars are part of the LQG as it is defined in CCGS12 but can not say whether these are the only members. If the sample used to determine members were complete down to the magnitude of $g$-mag = 21.1 (limit of the Hectospec data), additional members may be found and the shape of the convex hull would change.
For the purposes of this paper, we will assume that the LQGs indicate a general overdensity within this region. When mentioning the LQGs region, we refer to a region of space with a potential mass overdensity.
CONTROL SAMPLE {#sect:control}
--------------
The control sample was taken from Stripe 82 from SDSS [@York2000], which has a similar limiting magnitude (complete down to *g*-mag=21 to match the general completeness in the area of the LQGs) and taken from areas which do not contain any previously known LQGs. The samples were run through the program used to find the LQG and were determined not to be within a LQG within a 2$\sigma$ significance. We took multiple two deg$^{2}$ samples across the length of the stripe to reduce the impact of any marginal overdensities in a single area. The initial sample contains in total 394 field quasars within the redshift range $1.1 < z < 1.7$.
The errors were estimated across a range of objects and compared to the measured SNR. Spectra with SNR$ < 5$ per pixel rest EW had errors of $\pm$ 8.4 Å. This decreases to $\pm$ 4.8 Å for $5 <$ SNR $< 10$ per pixel and $\pm$ 2.85 Å for SNR $> 10$ per pixel. Therefore, to reduce the effects of errors in measurements due to low SNR, spectra with an average SNR $<$ 5 per pixel were rejected. This removes four quasars from the LQGs field leaving 24, and reduces the control sample to 178 quasars, removing more control quasars due to generally lower SNR in SDSS spectra. The rejected quasars cover a range of W2400 EW values and do not favour any strength.
Fig. \[fig:W2400\_mag\] shows the distribution of W2400 EW as a function of the $g$ band magnitude for both the control sample (circles, red) and the LQG field quasars (triangles, blue). Though some of the Hectospec quasars are fainter than the control sample quasars, there is no obvious relation between the magnitude and the W2400 EW emission. This is discussed further in section \[sect:discuss\].
Group Quasar Redshift RA (J2000) DEC (J2000) membership$^a$ W2400$^b$ (Å) *g*-mag
-------------- ---------------------------------- ---------- ------------- -------------- ---------------- --------------- ---------
Ultra-strong SDSS J104947.34+041746.2/qso412 1.159 10:49:47.35 +04:17:46.35 L1.11 56.08 20.51
SDSS J104800.40+052209.7/qso425 1.230 10:48:00.41 +05:22:09.90 L1.28 56.01 19.70
SDSS J104914.32+041428.6\* 1.607 10:49:14.33 +04:14:28.65 L1.54 54.34 19.14
SDSS J104930.44+054046.1/qso27 1.315 10:49:30.46 +05:40:46.20 L1.28 53.75 21.08
SDSS J104815.93+055007.8/qso421 1.665 10:48:15.94 +05:50:07.80 53.32 20.60
SDSS J104926.83+042334.6/qso417 1.653 10:49:26.83 +04:23:34.80 49.03 20.21
SDSS J104921.05+050948.3/qso29 1.417 10:49:21.07 +05:09:48.30 47.78 19.58
SDSS J105131.95+045124.7/qso41 1.434 10:51:31.94 +04:51:24.90 47.53 19.88
Strong SDSS J104958.91+042723.3/qso217 1.622 10:49:58.92 +04:27:23.40 L1.54 37.64 20.79
SDSS J105010.05+043249.1/qso48\* 1.217 10:50:10.06 +04:32:49.20 L1.28 35.33 18.56
SDSS J104933.41+054840.3/qso219 1.349 10:49:34.71 +05:48:36.00 L1.28 35.15 20.93
SDSS J105255.65+055112.9$^{c}$ 1.678 10:52:55.65 +05:51:12.93 32.8 20.03
SDSS J104937.47+045757.0/qso416 1.154 10:49:37.48 +04:57:57.10 32.72 20.98
Average SDSS J105000.36+045157.8/qso410 1.418 10:50:00.36 +04:51:57.90 31.39 20.88
SDSS J105154.14+041059.5$^{c}$ 1.552 10:51:54.14 +04:10:59.55 L1.54 29.94 21.29
SDSS J105141.89+045831.8\* 1.608 10:51:41.91 +04:58:27.90 L1.54 29.42 19.52
SDSS J105007.90+043659.7/qso49 1.131 10:50:07.90 +04:36:59.70 L1.11 28.46 19.42
SDSS J105036.09+045608.3/qso45 1.317 10:50:36.10 +04:56:11.40 L1.28 27.81 20.95
SDSS J105352.75+043055.0/qso22 1.216 10:50:30.77 +04:30:55.05 L1.28 26.5 19.85
SDSS J104656.71+054150.3\* 1.228 10:46:56.71 +05:41:50.25 L1.28 24.57 17.99
SDSS J104751.88+043709.9 1.696 10:47:51.89 +04:37:09.90 24.49 19.51
SDSS J104840.85+040938.3/qso420 1.238 10:48:40.85 +04:09:38.55 24.42 20.46
SDSS J105249.68+040046.3$^{c}$ 1.193 10:52:49.68 +04:00:46.50 L1.11 24.12 19.27
SDSS J104932.22+050531.7/qso26\* 1.111 10:49:32.23 +05:05:31.50 L1.11 23.16 18.84
SDSS J104733.16+052454.9\* 1.334 10:47:33.17 +05:24:55.05 L1.28 20.42 17.98
SDSS J104943.28+044948.8/qso413 1.295 10:49:43.30 +04:49:48.75 L1.28 19.37 19.53
SDSS J104938.35+052932.0\*$^{c}$ 1.517 10:49:38.35 +05:29:31.95 L1.54 18.83 19.48
SDSS J105018.10+052826.4\* 1.307 10:50:18.12 +05:28:26.40 L1.28 18.46 19.39
\[tab:Fedata\]\
\
\
\
RESULTS
=======
Table \[tab:Fedata\] summarises the data for the sample. The quasars with an SDSS name as well as qsoXXX are those quasars selected from the photometric catalogue and re-observed using Hectospec. The spectra for these objects is available in the online material for this paper. The four quasars removed from the LQG sample due to low SNR have been included for completeness (denoted by “c”) but are not included in the analysis.
Table \[tab:sample\_props\] shows the median, mean and standard deviation of the control and LQGs samples. These data were used to define boundaries; the representative average range for the Fe <span style="font-variant:small-caps;">ii</span> equivalent width was taken as 10 - 32 Å, anything between 32 and 43 Å EW was classed as strong and greater than 43 Å EW was classed as ultra-strong Fe <span style="font-variant:small-caps;">ii</span>.
Using this system, eight quasars were classed as ultra-strong and four were classed as strong Fe <span style="font-variant:small-caps;">ii</span> emitters from a sample size of 24 quasars within 1.6 deg$^{2}$, in the redshift interval of $1.1 < z < 1.7$.
A SIGNIFICANT DIFFERENCE IN THE DISTRIBUTION OF ULTRA-STRONG EMITTERS
---------------------------------------------------------------------
Table \[tab:cont\_samples\] shows the number of quasars (and percentage) with different *UV* Fe <span style="font-variant:small-caps;">ii</span> strengths in the LQGs field and the control fields. We show both the complete sample and a magnitude limited sample where all the quasars are within the same magnitude range ($17.98 < g < 20.56$). The LQGs field has a large percentage of quasars with strong and ultra-strong Fe <span style="font-variant:small-caps;">ii</span> emission. 33.3 $\pm~11.8$ per cent of the LQG field sample show ultra-strong Fe <span style="font-variant:small-caps;">ii</span> emission and 16.7 $\pm~8.3$ per cent show strong emission. This compares to the control sample which has 3.4 $\pm~1.4$ per cent of quasars showing ultra-strong emission and 15.7 $\pm~3.1$ per cent showing strong emission. Thus there is a statistical difference for the ultra-strong emitting quasars, which is also seen to the magnitude limited samples. For the magnitude limited samples, the percentage of strong quasars in the LQG field drops to 5.9 $\pm~5.9$ per cent, compared to the control sample value of 16.0 $\pm~3.0$ per cent, which are no longer within the errors.
However as the definitions of strong and ultra-strong are arbitrary and dependent on the control sample, for the rest of the paper, we will concentrate on the differences in the full distribution from the data and control samples.
----------------------- -------- ------- -----------
Sample median mean standard
deviation
control sample 21.20 22.59 10.86
LQG field 32.05 35.71 12.71
\[tab:sample\_props\]
----------------------- -------- ------- -----------
----------------------- ------------- --------------- -------------- ---------------
Strength LQGs field control field LQGs field control field
Ultra-strong 5.0 (33.3%) 0.23 (3.37%) 3.75 (35.3%) 0.23 (3.45%)
Strong 2.5 (16.7%) 1.08 (15.7%) 0.63 (5.9%) 1.08 (16.2%)
Average 7.5 (50.0%) 4.78 (69.7%) 6.25 (58.8%) 4.62 (69.4%)
Weak 0.0 (0.0%) 0.77 (11.2%) 0.0 (0.0%) 0.73 (11.0%)
\[tab:cont\_samples\]
----------------------- ------------- --------------- -------------- ---------------
W2400 distribution {#sect:w2400dist}
------------------
Fig. \[fig:hist\_dist\_both\] (bin size= 3 [Å]{}) shows the distribution of the W2400 EW for the LQGs field (solid, blue online) and the control sample (hatched, red online). The relative excess of *UV* Fe <span style="font-variant:small-caps;">ii</span> emission in the LQGs field can be clearly seen for W2400 EW $> 45$ Å. For W2400 EW $< 20$ Å, the histogram shows the lack of low emission quasars within the LQGs field compared to the control sample. Fig. \[fig:hist\_err\] shows a selection of histograms from a Monte-Carlo method. For each histogram, a point is randomly selected for each object across the whole distribution with appropriate weighting. This figure shows at the upper end of the emission, there is again an excess of quasars with W2400 $>$ 45, indicating this result is not affected by the errors. There is also a lack at W2400 $<$ 20 Å.
To quantify the difference in distributions, we employ the Mann-Whitney test, a powerful non-parametric test for comparing two populations. The Mann-Whitney test does not require any assumptions about the forms of the distributions, and is less likely to apply significance to outliers due to the ranking method used. This test is however sensitive to rounding, which can create ties in ranks in the data, therefore we have measured to two decimal places, though the data is not accurate to this level, and do not rounded our data at any point [@DeGroot1986]. Median latencies in the LQG field and control sample are 32.05 and 21.20 respectively. Using a one-tailed Mann-Whitney test, with 24 LQG quasars and 178 control sample quasars, the distributions in the two groups differ significantly with a p-value of 99.996%.
To estimate the effects of the errors on the W2400 measurements, a Monte-Carlo method was used to resample points from within the error limits for each measurement across the whole distribution with appropriate weighting and Mann-Whitney test repeated, using the same parameters as above. In each case, P $<$ 0.05. Therefore taking into account errors, the two distributions are still differ significantly.
To investigate the lack of weak Fe <span style="font-variant:small-caps;">ii</span> emitting (W2400 $<$ 20 Å) which could be due to the limit sample size, the Mann-Whitney statistical test was repeated using the samples with only W2400 $>$ 20 Å. This test gives P = $0.013 \pm ~0.05$, indicating that removing the weak emitters does have a significant effect on the result. However, this artificially truncates the values, creating an artificial distribution. To properly test this lack of weak emitters, a larger sample of quasars within overdense regions would be needed.
DISCUSSION {#sect:discuss}
==========
We have shown there is an increase in the Fe <span style="font-variant:small-caps;">ii</span> emission within quasars within the LQGs compared to our control sample. There are various possible explanations:
1. a selection effect - created by the selection of LQG quasars and magnitude limits,
2. dust - different amounts of dust within the LQG sample and the control sample causing the difference in the observed EW distributions,
3. Ly$\alpha$ fluorescence - Ly$\alpha$ pumping can cause an increase the Fe <span style="font-variant:small-caps;">ii</span>,
4. microturbulence - motions within the cloud line emitting region,
5. iron abundance - an enhanced Milky Way-like star formation creating an increased iron abundance.
We do not believe the observed distribution differences are due to selection effects. The quasars observed with Hectospec were randomly selected from the photometric catalogues. The control sample was selected to match the redshift and magnitude distributions of LQG quasars. However, there is a slight difference in the magnitude ranges, due to the magnitude limit of SDSS, shown in Fig. \[fig:W2400\_mag\]. Seven quasars within the LQG are fainter than the control sample by $<$ 0.5 magnitudes. However, the correlation between *UV* Fe <span style="font-variant:small-caps;">ii</span> and the quasar luminosity is still debated. Some studies have found an inverse Baldwin effect in the optical Fe <span style="font-variant:small-caps;">ii</span> emission, with the EW Fe <span style="font-variant:small-caps;">ii</span> emission increasing with the continuum emission [@Kovacevic2010; @Dong2011; @Han2011]. For the *UV* Fe <span style="font-variant:small-caps;">ii</span>, no significant correlation has been observed between the *UV* Fe <span style="font-variant:small-caps;">ii</span> and the quasar luminosity or $L/L_{Edd}$ [@Dong2011; @Sameshima2009].[^3]
To investigate any effect of the magnitude on our data, a magnitude limit of $17.98 < g < 20.56$ was applied to both samples. The Mann-Whitney test gives a P = 0.0007 showing that even with a magnitude-limited sample which further limits the sample size, the distributions of the LQGs field and control samples are still different.
The second possible explanation is an difference in dust properties between the LQG and the control sample causes the differences observed. As an excess of dust in the LQG region would reduce the UV emission, we do not believe this difference is due to dust emission. For dust emission to have an effect on our results, the control sample would have to see evidence of an steeper extinction law. However as the control sample consists of quasars from 13 different areas, it would require large scale special dust properties with the LQG field, which is unlikely. Since there is now a consensus that higher rates of star formation are seen in overdense environments at z $>$ 1 [e.g. @Farrah2006; @Amblard2011], we think it very unlikely that ISM dust is the cause of this difference, since if dust were causing the effect we’d expect the very high EW systems to be found in the field.
The third and fourth options are Ly$\alpha$ fluorescence and microturbulence, which are additional mechanisms within the BLR believed to increase the Fe <span style="font-variant:small-caps;">ii</span> emission. Again we do not believe this is the case as the control sample was selected to have similar quasar properties. As mentioned above, the small differences in magnitude are unlikely to be the cause of the distribution differences.
An increase in Ly$\alpha$ emission can cause an increase in the *UV* Fe <span style="font-variant:small-caps;">ii</span> emission [@Sigut2003; @Sigut2004; @Verner2004]. As the width of the Ly$\alpha$ increases, it overlaps with numerous Fe <span style="font-variant:small-caps;">ii</span> lines within the wavelength range 1212-1218 Å. These lines are excited, and when they decay produce emission in the *UV* Fe <span style="font-variant:small-caps;">ii</span> region, 2200-2700 Å. Increasing the Ly$\alpha$ emission will therefore increase the *UV* Fe <span style="font-variant:small-caps;">ii</span> emission. In fact, @Sigut1998 found that Ly$\alpha$ fluorescent excitation can more than double the *UV* Fe <span style="font-variant:small-caps;">ii</span> flux.
Low resolution $R\sim 90$ *GALEX UV* spectra which cover the Ly$\alpha$ emission exist for six of the quasars (program GI5-059, Williger et al.). Fig. \[fig:LyA\] shows the correlation between the Fe <span style="font-variant:small-caps;">ii</span> EW measurements and the equivalent widths of the Ly$\alpha$ emission line. The line drawn is the weighted (using both sets of errors) least squares best-fit. The Pearson correlation coefficient between the Ly$\alpha$ and the Fe <span style="font-variant:small-caps;">ii</span> is 0.830 $\pm ~0.14$. There is a suggestive trend for quasars with higher Ly$\alpha$ emission to have stronger Fe <span style="font-variant:small-caps;">ii</span> emission, as predicted [e.g @Sigut2003; @Sigut2004; @Verner2004]. However, there are only six spectra here with *GALEX* Ly$\alpha$ emission. This fit is highly dependent on the presence of qso425 (which has the largest W2400 EW) and not robust.
Though the Ly$\alpha$ emission may influence the observed Fe <span style="font-variant:small-caps;">ii</span> emission, there is no reason to believe the quasars within the LQG field have increased Ly$\alpha$ emission compared to randomly selected quasars. However, more data of quasar Ly$\alpha$ emission in various environments would be needed to fully investigate this. Equally with an overdense environments, the effect of other quasars and nearby galaxies is negligible compared to the emission from the accretion disc of the quasar.
The Fe <span style="font-variant:small-caps;">ii</span> flux strength can also be increased by microturbulence around the AGN [@Vestergaard2001; @Sigut2003; @Sigut2004; @Verner2003; @Verner2004; @Bruhweiler2008]. Microturbulence (non-thermal random motions within a cloud’s line emitting region; @Bottorff2000a [@Bottorff2000b]) spreads the line absorption coefficient over a larger wavelength range [@Bruhweiler2008], broadens the Ly$\alpha$ emission lines, and therefore increases the *UV* Fe <span style="font-variant:small-caps;">ii</span> emission observed. Microturbulence is occurs within the BLR. Large scale dynamic effects due to the large scale environment are unlikely to have an affect on the BLR without causing observable differences in the host galaxy, such star formation rates and luminosity, which is not seen here as our control was designed to match the field sample.
Although these factors have been shown to influence the *UV* Fe <span style="font-variant:small-caps;">ii</span> emission, modelling needs to be completed for quasars in environment over a range of densities to study how Ly$\alpha$ fluorescence and microturbulence can change with environment.
The final option is that the observed difference is due to the host galaxy and the quasars simply illuminate this difference. As previously noted, the dependence of metallicity with environment is still highly debated, with some studies showing a weak but significant trend for galaxies in higher density regions (such as groups or clusters) to have higher metallicities. Therefore if, as we assume the quasars in LQGs trace the overdense regions, we would expect the host galaxies to have greater metallicities.
Galaxies with old stellar populations have been found to favour higher density environments at z $\sim$ 0 [e.g. @Balogh2004; @Blanton2005] and z $\sim$ 1 [e.g. @Cooper2006]. @Martini2013 found AGN have evolved more rapidly in higher density environments than the field population. This suggests, at high redshifts, star formation may occur in high density environments [e.g. @Cooper2008]. If so, this will increase the metals available in the vicinity of these quasars. To produce the observed Fe <span style="font-variant:small-caps;">ii</span> (assuming abundance is the main factor), the hosts would have gone through a period of enhanced star formation between $2 < z < 3$, assuming it takes between 0.3 Gyr and 1 Gyr [@Hamann1993] for the required number of SNeIa to occur to create significant amounts of iron. This is during the peak epoch of star formation [e.g @Lilly1996; @Madau1996; @Sobral2013].
There is no significant enhancement in Mg <span style="font-variant:small-caps;">ii</span> in the LQG quasars compared to the control sample. This is consistant with a Milky Way-like star formation as opposed to a starburst. A increase in quiescent star formation in some of the galaxies within the LQGs would produce an increase in the iron abundance with respect to the Mg <span style="font-variant:small-caps;">ii</span>.
Within an overdense region, there could also be additional metal enrichment of the quasars from supernovae occurring the inter-cluster medium and within nearby galaxies. Supernovae have been shown to efficiently enrich the IGM over Mpc scales [e.g. @Domainko2004; @Adelberger2005]. These metals may then accrete the quasar, further enriching the quasar host.
SUMMARY
=======
There is a increase in Fe <span style="font-variant:small-caps;">ii</span> emission in a candidate overdense region, indicating there may be a build up of iron. It is consistant with an increase in star formation in overdense region at high redshift. This star formation must have occurred at 2 $<$ z $<$ 3 for iron to be observed in these quasars. Additionally surrounding galaxies in this dense region will release metals into the IGM, which can fall onto the quasar, producing an observed metal increase.
This will make published LQGs interesting regions in which to study the evolution of metals in high density regions and at high redshifts.
Acknowledgments
===============
KAH would like to acknowledge and thank the STFC, the University of Central Lancashire and the Obs. de la Côte d’Azur for their funding, and hospitality. Also KAH would like thank Roger Clowes and Luis Campusano for their communications
This research uses data from the Hectospec instrument on the MMT.
The authors acknowledge support from the NASA GALEX program GI5-059, grant NNX09AQ13G.
This research has used the SDSS DR7QSO catalogue [@Schneider2010]. Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org/.
The SDSS is managed by the Astrophysical Research Consortium for the Participating Institutions. The Participating Institutions are the American Museum of Natural History, Astrophysical Institute Potsdam, University of Basel, University of Cambridge, Case Western Reserve University, University of Chicago, Drexel University, Fermilab, the Institute for Advanced Study, the Japan Participation Group, Johns Hopkins University, the Joint Institute for Nuclear Astrophysics, the Kavli Institute for Particle Astrophysics and Cosmology, the Korean Scientist Group, the Chinese Academy of Sciences (LAMOST), Los Alamos National Laboratory, the Max-Planck-Institute for Astronomy (MPIA), the Max-Planck-Institute for Astrophysics (MPA), New Mexico State University, Ohio State University, University of Pittsburgh, University of Portsmouth, Princeton University, the United States Naval Observatory, and the University of Washington.
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[^1]: E-mail: [email protected]
[^2]: HSRED is an IDL based reduction package for Hectospec spectra created by Richard Cool and hosted at http://www.astro.princeton.edu/$\sim$rcool/hsred/
[^3]: The significance does increase, though still weak, if the *UV* continuum is used to calculate the luminosity which is expected as the *UV* Fe <span style="font-variant:small-caps;">ii</span> is powered by the continuum at shorter wavelengths to the optical continuum.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We review two standard methods for structural classification in simulations of crystalline phases, the Common Neighbour Analysis and the Centrosymmetry Parameter. We explore the definitions and implementations of each of their common variants, and investigate their respective failure modes and classification biases. Simple modifications to both methods are proposed, which improve their robustness, interpretability, and applicability. We denote these variants the *Interval Common Neighbour Analysis*, and the *Minimum-Weight Matching Centrosymmetry Parameter*.'
address: 'Department of Physics, Technical University of Denmark, 2800 Kgs. Lyngby, Denmark'
author:
- Peter M Larsen
title: Revisiting the Common Neighbour Analysis and the Centrosymmetry Parameter
---
Introduction
============
Extraction of useful material properties in a molecular dynamics (MD) simulation of condensed phases requires that we can classify crystal structures. The lowest level of this analysis is to resolve the structure at the atomic level, which is a prerequisite for determining meso- and macro-scale properties such yield strength [@schiøtz1998softening; @li2010nanotwins] and hardness [@kallman1993silicon], evolution in dislocation structures [@stukowski2012automated; @zepedaruiz2017probing] and grain structures [@panzarino2015quantitative; @chesser2020distinct], and irradiation damage [@marian2003modelling].
Two structural analysis methods are particularly notable for their ubiquity: *Common Neighbour Analysis* (CNA) [@faken1994systematic] and the *Centrosymmetry Parameter* (CSP) [@kelchner1998dislocation]. These are amongst the first structural analysis methods to exploit crystal geometry, and remain the most widely used despite the development of more advanced methods based on topology [@malins2013tcc; @lazar2017vorotop], geometry [@ackland2006bond; @larsen2016robust; @martelli2018localorder], statistical geometric (or machine-learning) descriptors [@steinhardt1983boo; @bartok2013soap; @spellings2018machine; @ceriotti2019unsupervised; @adorf2020analysis], and persistent homology [@buchet2018persistent; @maroulas2019persistent]. To a large extent, these methods have endured due to their speed, simplicity, and being either parameter-free (CNA) or single-parameter (CSP) methods. Even when not used directly, they commonly serve as benchmarks for the development of new structural analysis methods.
In this paper we revisit the CNA and CSP methods. We describe their functionality and perform a comprehensive analysis of their respective failure modes, and propose some simple modifications which fix most of them. In doing so, we extend the applicability and usefulness of both methods, and raise the baseline against which new methods can be compared.
![Ball-and-stick models and CNA signatures of the local environments of the condensed phases recognized by CNA. The models consist of the central atom (white), the atoms of the first neighbour shell (blue), and (for bcc) the atoms of the second neighbour shell (green).[]{data-label="fig:cna_schematics"}](figure1a.png "fig:"){width="100.00000%"}\
[FCC]{}\
[$12 \times (421)$]{}\
[$ $]{}\
![Ball-and-stick models and CNA signatures of the local environments of the condensed phases recognized by CNA. The models consist of the central atom (white), the atoms of the first neighbour shell (blue), and (for bcc) the atoms of the second neighbour shell (green).[]{data-label="fig:cna_schematics"}](figure1b.png "fig:"){width="100.00000%"}\
[HCP]{}\
[$6 \times (421)$]{}\
[$6 \times (422)$]{}
![Ball-and-stick models and CNA signatures of the local environments of the condensed phases recognized by CNA. The models consist of the central atom (white), the atoms of the first neighbour shell (blue), and (for bcc) the atoms of the second neighbour shell (green).[]{data-label="fig:cna_schematics"}](figure1c.png "fig:"){width="100.00000%"}\
[Icosahedral]{}\
[$12 \times (555)$]{}\
[$ $]{}\
![Ball-and-stick models and CNA signatures of the local environments of the condensed phases recognized by CNA. The models consist of the central atom (white), the atoms of the first neighbour shell (blue), and (for bcc) the atoms of the second neighbour shell (green).[]{data-label="fig:cna_schematics"}](figure1d.png "fig:"){width="100.00000%"}\
[BCC]{}\
[$8 \times (666)$]{}\
[$6 \times (444)$]{}\
Common Neighbour Analysis {#sec:cna}
=========================
The CNA method has its origins in the work of Blaisten-Farojas and Andersen [@blaistenbarojas1985effects], who encoded $n$-body clusters using their common neighbour relationships. Honeycutt and Andersen [@honeycutt1987molecular] generalized this approach to a larger set of clusters and used their relative abundances to characterize structural order in nanoparticles. The currently used form of CNA is due to Faken and J[ó]{}nsson [@faken1994systematic], who further extended the method to characterize the local environment of a central atom.
The CNA method employs a ball-and-stick model: each neighbour atom (ball) is joined to its nearest neighbours by a bond (stick). To identify the structure of an atom in a simulation, its ball-and-stick model is constructed and compared against the reference structures. This comparison is best described as a graph isomorphism test, but for historical reasons the comparison is made by computing a signature of common neighbour relationships. The signature consists of three indices per neighbour atom: the number of bonded neighbours shared with the central atom, the number of bonds between shared neighbours, and the number of bonds in the longest chain formed by shared neighbours. The ball-and-stick models and their associated CNA signatures are shown in Figure \[fig:cna\_schematics\].
In a perfect crystal, the radial distribution function (RDF) contains well-separated peaks, which makes it easy to identify the appropriate bonded pairs. In a MD simulation, however, the atoms must necessarily move from their ideal positions. To account for the resulting variability in the distance between bonded pairs, a threshold $r_\text{c}$ is introduced which defines nearest neighbours as those whose distance satisfies $r < r_\text{c}$. Honeycutt and Andersen set $r_\text{c}$ to the first minimum in the (RDF), which lies between the first and second neighbour shells. The use of a single, global threshold is commonly referred to as the *Conventional Common Neighbour Analysis* (c-CNA) method.
For some simulations it is not possible to define a single consistent threshold. For example, in multi-phase systems with differing lattice constants, the peaks in the RDF are not well-separated. To account for this variation in scale, Stukowski introduced the *Adaptive Common Neighbour Analysis* (a-CNA) [@stukowski2012structure], which computes a local threshold that is specific to each atom. For reference structures containing 12 atoms, such as the fcc structure, a local length scale is calculated by averaging the distances of the atoms in the first neighbour shell: $$l^\text{fcc} = \frac{1}{12}\sum\limits_{i=1}^{12} {\ensuremath{\left|\left|{\vec{p}_i}\right|\right|}}$$ This length scale is also employed for the hcp and icosahedral structures. For bcc crystals, the local length scale is calculated as a weighted average of the eight atoms of the second shell and the six atoms of the third neighbour shell, $$l^\text{bcc} =
\frac{1}{14} \left( \frac{2}{\sqrt{3}} \sum\limits_{i=1}^{8} {\ensuremath{\left|\left|{\vec{p}_i}\right|\right|}}
+ \sum\limits_{i=9}^{14} {\ensuremath{\left|\left|{\vec{p}_i}\right|\right|}} \right)$$ which provides an estimate of the second neighbour shell distance[^1]. The thresholds are chosen to lie, respectively, halfway between the first and second shells, and halfway between the second and third shells: $$r_\text{c}^\text{fcc} = \frac{1 + \sqrt{2}}{2} l^\text{fcc},
\hspace{8mm}
r_\text{c}^\text{bcc} = \frac{1 + \sqrt{2}}{2} l^\text{bcc}$$ Calculation of a per-atom threshold renders the a-CNA both scale-invariant and parameter-free, and is a significant improvement over the c-CNA with a global threshold. Nonetheless, the threshold is not always optimal. In fact, all existing CNA methods can fail in the following ways:
1. \[failure:1\] The threshold is too low, leaving some neighbour atoms under-coordinated.
2. \[failure:2\] The threshold is too high, leaving some neighbour atoms over-coordinated.
3. \[failure:3\] At a chosen threshold there are multiple signature matches, and the classification is ambiguous.
4. \[failure:4\] No consistent threshold exists: every threshold choice is simultaneously too low for some atoms and too high for other atoms.
Here we will address points (\[failure:1\])-(\[failure:3\]) by considering the intervals of possible threshold choices. Point (\[failure:4\]) is not possible to address with a CNA-type algorithm. Similarly, CNA requires that the ‘correct’ neighbours are those which lie closest to the central atom, which is often not the case at high temperatures [@larsen2016robust]. Both of these issues, though, require a more advanced structural classification approach and are not treated here.
![Histogram of the intervals in which fcc matches can be found in a high temperature Pd simulation. The $x$-axis shows the threshold normalized by the local length scale. In a perfect fcc crystal the interval start and end points lie at $1$ and $\sqrt{2}$ respectively. The a-CNA threshold lies exactly halfway between these values. In perturbed environments the start and end points can lie on the wrong side of this threshold, resulting in miscoordinated neighbour atoms.[]{data-label="fig:interval_histogram"}](figure2.png "fig:"){width="100.00000%"}\
$r / l^\text{fcc}$\
Interval Common Neighbour Analysis
----------------------------------
The threshold, $r_\text{c}$, is continuous parameter. However, for any local environment, the number of threshold choices which affect the coordination environment is limited by the number of pairwise distances. We can investigate every meaningful threshold by inserting bonds one at a time, in sorted order (short to long). After each bond is inserted, we test for a match against a reference structure. By enumerating all thresholds choices, we map out the *intervals* in which the structure is unchanged. For this reason we call this method *Interval Common Neighbour Analysis* (i-CNA).
![Recognition rates of the Interval CNA, Adaptive CNA, and Conventional CNA methods as a function of atomic perturbation. The atoms of a large fcc crystal with a lattice constant of $2\text{\AA}$ are perturbed by adding noise from a normal distribution $N\left( 0, \sigma \right)$. The global threshold used in the conventional CNA analysis is calculated by finding the first minimum of the RDF.[]{data-label="fig:fcc_fractions"}](figure3.png "fig:"){width="100.00000%"}\
$\sigma$\
Figure \[fig:interval\_histogram\] shows the intervals in which fcc matches are found in a simulation of a Pd polycrystal containing 1 million atoms at $80\%$ of the melting temperature. The histograms in blue and orange show the starts and ends of the intervals. It can be seen that a significant fraction of the interval start points (end points) lie at a larger (smaller) distance than the a-CNA threshold, in which case the a-CNA threshold produces an under-coordinated (over-coordinated) structure. The i-CNA method avoids these issues by exhaustively testing the threshold intervals, which results in an optimal threshold selection for each atom.
Since the i-CNA method tests a greater number of thresholds (including an equivalent threshold to that of a-CNA), its structure recognition rate is guaranteed to be at least as good a-CNA. In order to ensure that the method does not produce false positives, we impose restrictions on the maximum threshold values: $$r_\text{c}^\text{fcc} < \frac{1 + 2 \sqrt{2}}{3} l^\text{fcc},
\hspace{8mm}
r_\text{c}^\text{bcc} < \frac{1 + 2 \sqrt{2}}{3} l^\text{bcc}$$ In the fcc (bcc) crystal, these upper bounds lie two thirds of the distance between the first and second (second and third) neighbour shells. This choice is somewhat arbitrary, but it lies at the tail end of the interval start histogram, and experimentation with a variety of simulation data reveals that a threshold which exceeds this choice tends to produce spurious classifications only. Figure \[fig:fcc\_fractions\] compares the recognitions rates of all three CNA methods as a function of atomic perturbation, and demonstrates that i-CNA method has a better recognition rate at all perturbation levels. This is illustrated in MD simulation data in Figure \[fig:fcc\_renders\]. Due to the need to investigate multiple threshold intervals, the improved recognition comes at an additional computational cost of approximately $35\%$. The overhead is kept low by keeping track of each atom’s coordination number; a full comparison is only necessary if the coordination numbers match those of a reference structure.
![Fraction of atoms identified as fcc along the Bain transformation path in a large single crystal with small atomic perturbations. The transformation is a uniaxial strain $(1-t) + t / \sqrt{2}$; the crystal is fcc at $t=0$ and bcc at $t=1$. All atoms are either fcc or bcc (a fcc fraction of 0 means all atoms are bcc). The classification of the c-CNA method is biased towards bcc; the a-CNA method is biased towards fcc; the i-CNA method is unbiased.[]{data-label="fig:bain_fractions"}](figure5.png "fig:"){width="100.00000%"}\
Resolving Ambiguous Classifications
-----------------------------------
In addition to a having a good recognition rate, a structural classification method should be unbiased. This has particular importance in simulations of phase transitions, in which a biased classification can affect the conclusions drawn.
The a-CNA method tests for structure matches using two thresholds, $r_\text{c}^\text{fcc}$ and $r_\text{c}^\text{bcc}$, each of which may produce a match. The method does not have any extra information to use to resolve this ambiguity. Instead, the method (as implemented in OVITO [@stukowski2010visualization]) resolves ambiguities in favour of the smaller structure. In the i-CNA method, ambiguities can be resolved by selecting the match with the widest interval.
The effect of ambiguity resolution is studied in Figure \[fig:bain\_fractions\]. The figure shows classification rates along the Bain transformation [@bain1924nature], which is a martensitic transformation from the fcc to the bcc crystal structure that can be achieved by application of a uniaxial strain along any of the principal axes in the conventional fcc cell. The i-CNA method places the phase transition halfway along the transformation path. By favouring smaller structures, the a-CNA method places the phase transition much further along the path, and is effectively biased towards a fcc classification. In the c-CNA method, a fcc classification is not possible as soon as 14 neighbours lie within the threshold distance. Here, there is no ambiguity to resolve but the method is strongly biased towards a bcc classification.
The Centrosymmetry Parameter {#sec:csp}
============================
The CSP is a structural analysis method which takes an opposite approach to CNA. Rather than using a set of reference structures to classify the topology of the local atomic environment, it computes an order parameter which quantifies the degree of inversion (or *centro*) symmetry of the local environment. In the original formulation of Kelchner *et al.*, the CSP is defined using the $N=12$ nearest neighbours of a central atom: $$\text{CSP}\left({\ensuremath{\mathbf{R}}}\right) = \sum_{i=1}^{6} {\ensuremath{\left|\left|{ \vec{r}_{i} + \vec{r}_{i+6} }\right|\right|}}^2
\label{eq:csp}$$ where $\vec{r}_{i}$ and $\vec{r}_{i+6}$ are the vectors ‘corresponding to the six pairs of opposite nearest neighbours in the fcc lattice’ [@kelchner1998dislocation]. For the bcc lattice the summation is replaced by the eight nearest neighbours. For other centrosymmetric structures the appropriate summation is similarly intuitive, as each nearest neighbour atom has a clearly defined opposite neighbour.
There are two commonly used algorithms for calculating the CSP. The first algorithm (described in ref. [@stukowski2012structure]) calculates a weight $w_{ij} = {\ensuremath{\left|\left|{ \vec{r}_{i} + \vec{r}_{j} }\right|\right|}}$ for each of the $N(N-1)/2$ pairs of neighbour atoms, and calculates the CSP as the summation over the $N/2$ smallest weights. Reproduction of the Au defects described in ref. [@kelchner1998dislocation] reveals that this approach was used by Kelchner *et al.* The second algorithm (described in ref. [@bulatov2006computer]) proceeds by ordering the atoms by their distance from the central atom; opposite pairs are found by (i) pairing the inner-most atom with its minimal-weight partner, (ii) removing this pair, and (iii) repeating the process until no atoms are left.
The first method is implemented in LAMMPS [@plimpton1995fast] and OVITO [@stukowski2010visualization]. We will describe this method as the *Greedy Edge Selection* (GES) CSP. The second method is implemented in AtomEye [@li2003atomeye] and Atomsk [@hirel2015atomsk]. We will describe this method as the *Greedy Vertex Matching* (GVM) CSP. The GVM implementation typically employs a normalization constant $$\frac{1}{2 \sum\limits_i {\ensuremath{\left|\left|{\vec{r}_i}\right|\right|}}^2}$$ which renders the CSP invariant to scale, but, in highly centrosymmetric structures at least, these methods are otherwise equivalent.
![Comparison of the MWM, GES, and GVM methods, for the nearest-neighbours of an atom in a perturbed hexagonal lattice. The vertex labels are ordered by distance from the central atom. []{data-label="fig:csp_example"}](figure6a.png "fig:"){width="85.00000%"}\
[Minimum-weight matching\
CSP=0.39]{}
![Comparison of the MWM, GES, and GVM methods, for the nearest-neighbours of an atom in a perturbed hexagonal lattice. The vertex labels are ordered by distance from the central atom. []{data-label="fig:csp_example"}](figure6b.png "fig:"){width="85.00000%"}\
[Greedy edge selection\
CSP=0.32]{}
![Comparison of the MWM, GES, and GVM methods, for the nearest-neighbours of an atom in a perturbed hexagonal lattice. The vertex labels are ordered by distance from the central atom. []{data-label="fig:csp_example"}](figure6c.png "fig:"){width="85.00000%"}\
[Greedy vertex matching\
CSP=4.36]{}
![Illustration of the failure modes of greedy CSP calculation methods. **Left** The centrosymmetry of a slightly perturbed hexagonal structure is changed by rotating a single vertex about the central atom through an angle of $2\pi$. Vertex labels are ordered by distance from the central atom. **Right** The CSP values at every angle, calculated using all three methods. GVM is not a continuous function of rotation. GES consistently underestimates the actual CSP.[]{data-label="fig:csp_rotation"}](figure7a.png){width="60.00000%"}
![Illustration of the failure modes of greedy CSP calculation methods. **Left** The centrosymmetry of a slightly perturbed hexagonal structure is changed by rotating a single vertex about the central atom through an angle of $2\pi$. Vertex labels are ordered by distance from the central atom. **Right** The CSP values at every angle, calculated using all three methods. GVM is not a continuous function of rotation. GES consistently underestimates the actual CSP.[]{data-label="fig:csp_rotation"}](figure7b.png){width="100.00000%"}
The Graph Matching Centrosymmetry Parameter
-------------------------------------------
The definition in Equation (\[eq:csp\]) describes the CSP for centrosymmetric structures, but does not specify how the CSP should be calculated in the general case. The CSP is a function $f : {\ensuremath{\mathbb{R}^{N \times d}}\xspace} \rightarrow {\ensuremath{\mathbb{R}^{}}\xspace}$ of $N$ points in $d$ dimensions, where $N$ is even. In order to generalize the CSP to arbitrary structures, we impose two conditions which must be fulfilled:
1. Each atom must have exactly one opposite neighbour.
2. The sum of weights must be minimal.
If both conditions are satisfied $f$ is continuous (albeit non-differentiable) function of the input coordinates. The conditions define the CSP as a *minimum-weight matching* [@edmonds1965maximum] on the perfect graph with atoms as nodes and squared pairwise distances as edge weights. We will describe the calculation of the CSP satisfying the above conditions as the *Minimum-Weight Matching* (MWM) CSP. The differences in the CSP algorithms are shown for a two-dimensional example in Figure \[fig:csp\_example\]. The MWM method produces the intuitive result. The GES method violates condition (i): some atoms have no opposite neighbours and others have multiple opposite neighbours. The GVM method violates condition (ii): each atom has exactly one neighbour, but the weight sum is not minimal. In this example, a relatively small deviation from perfect centrosymmetry is sufficient to induce failure of the greedy methods. Furthermore, they fail in different ways and the calculated CSP values are inconsistent.
The failure modes of the greedy methods are explored in further detail in Figure \[fig:csp\_rotation\]. The first atom is rotated through a $2\pi$ angle range, and the CSP values for each method is calculated at every rotation. The GVM method is not a continuous function of the input coordinates; small changes in geometry cause large changes in the calculated CSP value. The GES method, on the other hand, assigns similar CSP values for very different structures. Calculation of the CSP as a MWM remedies both of these problems.
![Histograms of the CSP using the minimum-weight matching, greedy edge selection, and greedy vertex matching methods in a polycrystalline HCP Ru sample. The CSP values have been normalized by the square of the lattice constant ($a$). The GVM has a further peak at $\text{CSP} / a^2 = 2.5$, omitted here for clarity.[]{data-label="fig:csp_simulation"}](figure8.png){width="100.00000%"}
The above example is physically implausible from an energetic perspective, but the effects in a MD simulation are the same. Figure \[fig:csp\_simulation\] shows the CSP distribution in a simulation of polycrystalline Ru in the hcp phase. The GVM method produces a distribution with multiple peaks, despite the local environments being very similar. The GES method performs better: it has a single peak, but the peak is narrower than that of the MWM method, which indicates that different local environments are mapped onto a smaller range of CSP values, and has a lower mode, which results in a poorer separation from centrosymmetric structures.
Using a standard graph matching library we achieve approximately $30,000$ CSP calculations per second on a single thread of a standard laptop computer. This is significantly slower than the greedy methods, but we can improve the running time with a hybrid strategy: first we calculate the CSP using the GES method; if the assigned graph edges constitute a valid matching then the calculated CSP is equal to the MWM method; otherwise the full MWM method is employed. With this approach the graph matching code is only invoked in case of failure.
Conclusions
===========
We have analyzed the behaviour of the CNA and CSP methods, and presented some simple extensions which improve their usefulness without introducing any extra parameters. By performing an exhaustive threshold search, the i-CNA method achieves a better structural recognition rate in local environments with larger atomic perturbations. Furthermore, we introduced the Bain transformation as a test for classification bias, and it was shown that i-CNA is unbiased.
For the CSP, we demonstrated that the existing calculation methods do not produce consistent results. Using methods from graph theory, we proposed a solution which is continuous with respect to atomic perturbations and produces an intuitive result at any level of centrosymmetry.
Acknowledgments {#acknowledgments .unnumbered}
===============
The author thanks for Alexander Stukowski for helpful discussions and Daniel Utt for providing Pd simulation data. This work was supported by Grant No. 7026-00126B from the Danish Council for Independent Research.
References {#references .unnumbered}
==========
[^1]: This formula corrects a minor error in the original description by Stukowski.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We investigate the advantages of using two independent, linear detectors for continuous quantum measurement. For single-shot quantum measurement, the measurement is maximally efficient if the detectors are twins. For weak continuous measurement, cross-correlations allow a violation of the Korotkov-Averin bound for the detector’s signal-to-noise ratio. A vanishing noise background provides a nontrivial test of ideal independent quantum detectors. We further investigate the correlations of non-commuting operators, and consider possible deviations from the independent detector model for mesoscopic conductors coupled by the screened Coulomb interaction.'
author:
- 'Andrew N. Jordan and Markus Büttiker'
date: 'May 2, 2005'
title: 'Continuous quantum measurement with independent detector cross-correlations'
---
There has recently been intensive research, both experimental and theoretical, into the development of quantum detectors in the solid state. Mesoscopic structures, such as the quantum point contact, single electron transistor, and SQUID have been used for fast qubit read-out. Contrary to the historical assumption that the quantum measurement occurs instantaneously, in the modern theory of quantum detectors the continuous nature of the measurement process is essential to the understanding and optimization of how quantum information is collected.
The ultimate goal for quantum computation is the development of “single-shot” detectors, where in one run the qubit’s state is unambiguously determined. An important figure of merit is the detector’s efficiency, defined as the product of the time taken to measure the state of the qubit, with the measurement induced dephasing rate. This efficiency is minimized for detectors that do not lose information about the quantum state as the measurement is being performed. Single-shot detectors are difficult to realize because of the fast time resolution required. Another approach is that of weak measurement, where detector backaction renders the state of the qubit invisible in the average output of the detector, but quantum coherent oscillations are uncovered in the spectral density of the detector. These measurements are easier to preform because both detector and qubit averaging are permitted, and the experiments only require a bandwidth resolution of the qubit energy splitting. One important result in weak measurement theory is the Korotkov-Averin bound. It states that as the weak measurement is taking place, the detector backaction quickly destroys the quantum oscillations, so the maximum detector signal-to-noise ratio is fundamentally limited at 4. This result was derived in Refs. [@stn], confirmed in Refs. [@verifystn], generalized in Refs. [@generalized], and measured in Ref. [@expt].
In this Letter, the theoretical advantages of considering the cross-correlated output of independent quantum detectors are investigated. It is clear that cross-correlations bestow an experimental advantage [@beuhler] in quantum measurement, because the procedure filters out any noise not shared by the two detectors. Thus, extraneous noise produced by sources such as charge traps in one detector will be removed. This technique is also used in quantum noise measurements for this same advantage [@noise]. We demonstrate that although the two detectors cannot improve the efficency of the detection process, the cross-correlated output can violate the Korotkov-Averin bound. As a solid-state implementation of the results derived, Fig. 1 depicts two quantum point contacts capacitively coupled to the same double quantum-dot representing a charge qubit. It should be stressed that such structures have already been fabricated [@DD].
[*Detector assumptions and linear response.*]{}– We employ the linear response approach to quantum measurement because of its elegant simplicity, and general applicability to a wide range of detectors [@stn; @pilgram; @stone]. The quantum operator to be measured is $\sigma_z$. The Hamiltonian is $$H = -(\epsilon\, \sigma_z +\Delta \sigma_x)/2 +
H_1 + H_2 + Q_1 \sigma_z/2 + Q_2 \sigma_z/2,
\label{ham}$$ where $Q_{1,2}$ are the bare input variables of detector $1$ and $2$, and $H_{1,2}$ are their Hamiltonians, and $I_{1,2}$ are the bare output variables of the detectors. The small coupling constants are incorporated into the definition of $Q_{1,2}$. We assume that the detector is much faster than all qubit time scales, so the relevant detector correlation functions are the stationary zero frequency correlators:
$$\begin{aligned}
&&\langle I_{i}(t + \tau ) I_{j}(t) \rangle \;= \;
S_{I}^{(i)}\delta_{ij} \delta (\tau),
\label{SI}\\
&&\langle Q_{i}(t + \tau ) Q_{j}(t) \rangle =
S_{Q}^{(i)}\delta_{ij} \delta (\tau),
\label{SQ}\\
&&\langle Q_{i}(t + \tau ) I_{j}(t) \rangle =
({\rm Re}S^{(i)}_{Q I} + i\, {\rm Im}S^{(i)}_{Q I})
\delta_{ij} \delta (\tau),
\label{SQI}\\
&&\langle I_{i}(t + \tau ) Q_{j}(t) \rangle =
({\rm Re}S^{(i)}_{Q I}
- i\, {\rm Im}S^{(i)}_{Q I})\delta_{ij} \delta (\tau).
\label{SIQ}\end{aligned}$$
where the time delta functions have a small shift $\delta (\tau-0)$, reflecting the finite response time of the detector. Linear response theory tells us that the response coefficients $\lambda_{1,2}$ are given by $\lambda_{i} = -2\, {\rm Im}S^{(i)}_{Q I} /\hbar$, so the output of the detectors (with the average subtracted) is ${\cal O}_{i} = I_{i} + \lambda_{i} \sigma_z/2$. As the detector is turned on, it ideally collects information about the operator $\sigma_z$ while destroying information about $\sigma_{x,y}$. The measurement is complete when the integrated difference in qubit signal exceeds the detector noise, so the state of the qubit may be determined. In the simplest case of $\Delta=0$, the standard expressions for the dephasing rate $\Gamma$ and measurement time $T_M$ are [@rmp] = S\_Q/(2\^2), T\_M =4 S\_I/ \^2. \[relations\] Let us next observe $$\hbar^2 \lambda^{2}= 4({\rm Im} S_{QI})^2 \le 4 \vert S_{QI} \vert^2 \le
4 S_Q S_I,
\label{lr}$$ where we have used the Cauchy-Schwartz inequality. For a lone detector the above relations imply $\Gamma T_M \ge 1/2$, where equality is reached for quantum limited detectors. The two conditions needed to reach the “Heisenberg efficiency” are, (a) S\_[QI]{} = 0, (b) S\_[QI]{} \^2 = S\_Q S\_I. \[he\] In the context of mesoscopic scattering detectors, condition (b) is related to the energy dependence of the transmission of the scatterer, while (a) is related to the symmetry of the scatterer [@stn]. Pilgram and one of the authors derived Eqs. (\[he\]) for arbitrary detectors described by scattering matrices [@pilgram]. Clerk, Girvin, and Stone interpreted these conditions as no lost information either through (a) phase or (b) energy averaging [@stone].
[*Can we do better with two detectors?*]{}– By adding an additional detector to the qubit, the signals may be averaged, ${\cal O}=({\cal
O}_1+{\cal O}_2)/2$, so the measurement time may be reduced. On the other hand, the new detector dephases the qubit more quickly. For statistically independent detectors, the measurement-induced dephasing rate is simply the sum of the individual dephasing rates, so the two-detector efficiency is T\_M = 2 (S\_I\^[(1)]{}+S\_I\^[(2)]{})(S\_Q\^[(1)]{}+S\_Q\^[(2)]{})/\^2 (\_1+\_2)\^2 1/2, \[averaged\] where equality is reached for twin detectors that are themselves quantum limited. This condition may also be interpreted as no lost information between the two detectors. Rather than average the signals, we could instead cross-correlate them. However, this also brings no advantage because the new signal obtained by multiplying the output from the two detectors, ${\cal O}_{1} (t_1) {\cal O}_{2}(t_2 )$, has its own noise. If we could average over many trials the noise could be eliminated, but for single-shot measurement the efficiency is still intrinsically limited.
[*Violation of the Korotkov-Averin bound.*]{}– But consider next Korotkov and Averin’s bound on the signal-to-noise ratio for a weakly measured qubit [@stn]. It states that the ratio of the measured qubit signal to detector noise, $\cal R$, is fundamentally limited by 4. This bound can be overcome with quantum nondemolition measurements by increasing the signal [@qnd]. In this Letter, we are concerned with reducing the noise. To see how this bound emerges, we briefly derive this inequality for one detector. The Hamiltonian is given by Eq. (\[ham\]) with $Q_2=0$. The time averaged output of the detector is $\la{\cal O}\ra= (\lambda/2) (1/T) \int_{0}^{T} dt \langle
\sigma_z (t) \rangle$. For a weakly measured qubit, the statistical average over $\sigma_z$ is taken with respect to the stationary, mixed, density matrix of the qubit, $\rho=(1/2)\openone$, and therefore the qubit makes [*no*]{} contribution to the average output current. The detector’s spectral density is $S(\omega) = S_{I}
+ (\lambda^{2}/4) S_{zz}(\omega)$, where S\_[ij]{}() = 2 \_[0]{}\^ dt (t) \_i (0) \_j (t) . \[Sij\] The qubit dynamics may be found by expanding the evolution operator to second order in the coupling constant, and averaging over the $\delta$-correlated $Q$ fluctuations to obtain equations of motion, with dephasing rate $\Gamma$. For the special case of $\epsilon=0$, the noise spectrum in the vicinity of $\omega=\Delta/\hbar \equiv \Omega$ is [@stn] $$S(\omega) = S_{I} + \frac{\lambda^2 \Gamma}{2} \frac{
\Omega^2}{(\omega^2-\Omega^2)^2 + \omega^2 \Gamma^2}.$$ At the qubit frequency, $\omega = \Omega$, the signal has a maximum of $S_{\rm max} = \lambda^2/(2 \Gamma)= \hbar^2\lambda^2/S_{Q}$. We use again the linear response relation (\[lr\]) to bound the signal-to-noise ratio of the detector as = S\_[max]{}/S\_I 4. \[STN\] This is the Korotkov-Averin bound.
Consider now the cross-correlation of the outputs from two independent detectors, both measuring the same qubit operator $\sigma_z$. The qubit dynamics is the same, except that $\Gamma=\Gamma_1+\Gamma_2$. The spectral density of the cross-correlation $S_{1,2}(\omega)$ contains four terms, $$\begin{aligned}
S_{1,2}(\omega)&=&\int_{0}^{\infty}dt\cos(\omega t)
[2 \langle I_{1} (0) I_{2} (t) \rangle
+ \lambda_1 \langle\sigma_z (0) I_2(t) \ra \nonumber \\
&+& \lambda_2\la I_1(0) \sigma_z (t) \rangle
+(\lambda_1\lambda_2/2) \langle \sigma_z (0) \sigma_z (t) \rangle ].
\label{s12}\end{aligned}$$ According to Eq. (\[SI\]) the bare detector noise of the two detectors are uncorrelated, the qubit dynamics is uncorrelated with the bare detector noise, so only the qubit signal (\[Sij\]) contributes to the correlation function (\[s12\]). The remaining question is the detector configuration that maximizes the signal. The maximum signal at $\omega =\Omega$ is $S_{\rm max}=\lambda_1\lambda_2
/[2 (\Gamma_1+\Gamma_2)]$, and we may use the relations (\[lr\]) to bound the cross-correlated signal in relation to the noise of the individual detectors as S\_[max]{} 2 , \[maxS\] where equality is reached for $S_Q^{(1)}=S_Q^{(2)}$. For twin detectors, (\[maxS\]) is half of the single detector signal, because of the doubled dephasing rate [@note1].
We have successfully removed the background noise, and can now see the naked destruction of the qubit. The signal-to-noise ratio ${\cal R}$ is divergent, violating the Korotkov-Averin bound. Why did cross-correlations help here, but not in the quantum efficency? The reason is that the spectral density, in contrast to the measurement efficiency, is not protected by the uncertainty principle, so there is no fundamental limitation on its measurement.
[*Detecting the detector.*]{}— Although the above result is very appealing, one might worry that it can be spoiled by some weak direct coupling between the detectors. We now take this into account by introducing another term in the Hamiltonian, $H_{1,2} = \alpha Q_1
Q_2$, where $\alpha$ is a relative coupling constant between the two detectors. The additional contribution to the zero-frequency cross-correlator, $\delta S_{1,2} = \int dt \langle \delta {\cal
O}_{1}(0) \delta {\cal O}_{2}(t) \rangle$, consists of four terms, $$\begin{aligned}
&&\delta S_{1,2}=\int dt [\langle I_1(0) I_2(t) \rangle
+ \alpha \lambda_1 \langle
Q_2(0) I_2(t) \rangle \nonumber \\
&& + \alpha \lambda_2 \langle I_1(0) Q_1(t) \rangle
+ \alpha^2 \lambda_1 \lambda _2 \langle Q_1(0) Q_2(t) \rangle].\end{aligned}$$ The first and last term vanish for independent detectors, leaving the middle two terms. These middle terms may be interpreted as a fluctuation in one detector causing a response in the other detector, which is then correlated back with the original bare detector variable in the signal multiplication. Using the correlators Eq. (\[SQI\],\[SIQ\]), we find S\_[1,2]{} = \_1 [Re]{} S\^[(2)]{}\_[QI]{} + \_2 [Re]{} S\^[(1)]{}\_[QI]{}, \[addcross\] where we have substituted the response coefficient for the imaginary part of the $Q$-$I$ correlator, which causes the imaginary part of the additional cross-correlated signal to vanish. An interesting aspect of the above equation is that if the detectors are both quantum limited, we saw in Eq. (\[he\]a) that one condition was that the real part of the $Q$-$I$ correlator should vanish. Eq. (\[addcross\]) provides a simple experimental test to check this condition: background cross-correlations should vanish. If this is true, then the direct coupling between detectors does not give any noise pedestal to overcome for weak measurement.
[*Weak measurement of non-commuting observables*]{}.— Once we have two detectors involved, there is no reason why they both have to measure the same observable (or one that commutes with it). We now consider an experiment where one detector measures $\sigma_z$ and the other measures $\sigma_x$, and the outputs are cross-correlated. The measured spectrum is $S_{1,2}(\omega) = (\lambda_1 \lambda_2/4)
S_{zx}(\omega)$. This experiment could be implemented with a split Cooper-pair box [@CPB], where a SQUID is weakly measuring the persistent current, and a quantum point contact is weakly measuring the electrical charge. In standard measurement theory, the question of a simultaneous measurement of non-commuting observables cannot even be posed.
The coupling part of the Hamiltonian is altered to be $H_c= (1/2) Q_1 \sigma_z + (1/2) Q_2 \sigma_x$. We parameterize any traceless qubit operator as $\sigma(t)
=\sum_i x_i(t) \sigma_{x_i}$, and the density matrix $\rho =
(1/2)\openone + \sigma(t)$, so $(x, y, z)$ also represent coordinates on the Block sphere. Defining $\Gamma_z = S_Q^{(2)}/(2 \hbar^2)$, $\Gamma_x = S_Q^{(1)}/(2 \hbar^2)$, the equations of motion for $x_i$, averaged over the white noise of $Q_1$ and $Q_2$, are
x\
y\
z
=
-\_z & -/& 0\
/& -\_x -\_z & -/\
0 & /& -\_x
x\
y\
z
. \[block-red\] Diagonalization of the transition matrix in the case $\Gamma_x
=0$ gives the usual expressions for the dephasing and relaxation rates. This set-up is always far away from the Heisenberg efficiency because one detector is destroying the signal the other is trying to measure. However, this situation has interesting behavior in the weak measurement case. The cross-correlation $S_{1,2}(\Omega)$ attains its maximum signal at the symmetric point $\epsilon = \Delta$, $\Gamma_x=\Gamma_z=\Gamma$, so the qubit frequency is $\Omega=\sqrt{2}\Delta/\hbar$. The master equation may be solved in the weak dephasing limit, giving the correlation (for positive frequencies) S\_[xz]{}() = S\_[zx]{}() = - . \[Szx\] The first term has a peak at zero frequency, while the second term has a peak at $\omega =\Omega$, with width $3\Gamma/2$, and signal $-1/3 \Gamma$. Bounding this signal in relation to the noise in the individual twin detectors gives $\vert S_{1,2}(\Omega) \vert \le (2/3) S_I$. The interesting feature of this correlator is that it changes sign as a function of frequency. The low frequency part describes the incoherent relaxation to the stationary state, while the high frequency part describes the out of phase, coherent oscillations of the $z$ and $x$ degrees of freedom. The measured correlator $S_{zx}$, as well as $S_{xx}, S_{zz}$ are plotted as a function of frequency in Fig.\[combo\](b,c,d) for different values of $\epsilon$. These correlators all describe different aspects of the time domain destruction of the quantum state by the weak measurement, visualized in Fig.\[combo\]a. We note that the cross-correlator changes sign for $\epsilon = -\Delta$.
[*Implementation.*]{}– We now consider two quantum point contacts, measuring a double quantum dot qubit. The point contact perfectly obeys conditions (\[he\]) and is thus an ideal detector [@stn; @pilgram; @stone]. The bare input detector variable $Q$ is identified with the electrical charge in the point contact, while the bare output variable $I$ is identified with the shot noise. The conductance of the QPC is sensitive to the electron’s position on the double dot. A measurement of the quantum state occurs when the integrated current difference exceeds the shot noise power. In the geometry shown in Fig. 1, one detector measures $\sigma_z$, while the other detector measures $-\sigma_z$, so the qubit signal will be anti-correlated. The charges on the two detectors are not independent, but rather must be the opposite of each other to have charge neutrality in the system. This electrical screening generates correlations between the potentials of the two quantum point contacts, increasing the dephasing rate, hurting the efficiency, and producing some background noise for the cross-correlator. This situation is markedly in contract with the single detector case [@pilgram], where screening simply renormalized the coupling constant. However, in realistic detectors there will always be other gates to control the quantum double dot, creating a larger capacitance matrix than the minimal one shown in Fig. 1. In this extended geometry, a charge fluctuation in one detector will be screened by the surrounding metallic gates, not by the other detector, justifying the independent detector model. We mention that in the experiment already done by Buehler [*et al.*]{} [@beuhler], the detectors seem to be completely independent.
[*Conclusions.*]{}– We considered the advantages that two independent quantum detectors measuring the same qubit can bring to the quantum measurement problem. The Heisenberg efficiency could be reached with quantum limited twin detectors. The asymmetry of the detector, related to phase information in the case of mesoscopic scattering detectors, could be measured with low-frequency cross-correlations, and thus provides a non-trivial experimental test for quantum efficiency. For weak continuous measurement, the cross-correlated signal removes the noise pedestal, and allows a violation of the Korotkov-Averin bound on the signal-to-noise ratio. The cross-correlation of non-commuting operators were also investigated, which showed a cross-over from positive to negative correlation as a function of frequency. Although we have focused on mesoscopic qubits, this technique easily extends to other systems where similar bounds have been derived, such as single spins and nano-mechanical oscillators [@more].
We thank A. N. Korotkov and D. V. Averin for correspondence and suggestions. This work was supported by the SNF and MaNEP.
[99]{}
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The response functions may be quite different, so the cross-correlated signal $S_{\rm max} \le
2 S_I^{(2)} \lambda_1/\lambda_2 = 2 S_I^{(1)} \lambda_2/\lambda_1$ may be much larger than the noise in one detector, provided it is much smaller than the noise in the other detector.
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|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'Philip Bechtle[^1]'
date: |
Received: 1 Oct 2003 / Accepted: 20 Nov 2003 /\
Published Online: 26 Nov 2003 – © Springer-Verlag / Società Italiana di Fisica 2003
title: Searches for Neutral Higgs Boson and Interpretations in the MSSM at LEP
---
Introduction {#sect:intro}
============
In the Standard Model (SM) it is generally assumed that the Higgs mechanism is resposible for the breaking of electroweak symmetry and for the generation of elementary particle masses. The Minimal Supersymmetric Standard Model (MSSM) is the SUSY extension of the SM with minimal new particle content. It introduces two complex Higgs field doublets. The MSSM predicts five Higgs bosons: three neutral and two charged ones. At least one of the neutral Higgs bosons is predicted to have its mass close to the electroweak energy scale, providing a high motivation to the searches at current and future colliders.
In the MSSM the Higgs potential is assumed to be invariant under CP transformation at tree level. However, it is possible to break CP symmetry in the Higgs sector by radiative corrections, especially by contributions from complex trilinear couplings $A_{\mathrm{t,b}}$ of third generation scalar-quarks [@Pilaftsis:1999qt]. Since the input parameter space is generally too large to be scanned completely, so called benchmark scenarios (cf. Tab. \[tab:scenarios\]) have been proposed [@newbenchmarks; @lhc_benchmarks; @Carena:2000ks], each emphasising a certain phenomenological situation. The parameters ${\mbox{$\tan\beta$}}=v_2/v_1$ and $m_{{\ensuremath{\mathrm{A}}}}$ governing the Higgs sector on tree-level are scanned, while all parameters on loop level are kept fixed for one scenario. CP conserving (CPC) and CP violating (CPV) scenarios exist.
[ll]{}\
No Mixing & No mixing in the stop-sbottom sector\
$\mathrm{m}_h$max & Maximum ${\mbox{$m_{\mathrm{h}}$}}$ for given ${\mbox{$\tan\beta$}},{\mbox{$m_{\mathrm{A}}$}}$\
Large $\mu$ & Always kinematically accessible,\
& but ${\ensuremath{\mathrm{h}}}{\mbox{$\rightarrow$}}{\ensuremath{\mathrm{b}\bar{\mathrm{b}}}}$ suppressed\
$\mathrm{m}_h$max$^{+}$ & like $\mathrm{m}_h$max, but favoured by $(g-2)_{\mu}$.\
constr. $\mathrm{m}_h$max & like $\mathrm{m}_h$max, but favoured by $(\mathrm{b}{\mbox{$\rightarrow$}}\mathrm{s}\gamma)$\
gluophobic & ${\ensuremath{\mathrm{h}}}\mathrm{gg}$ coupling suppressed,\
& reduced LHC production cross section\
small $\alpha_{\mathrm{eff}}$ & ${\ensuremath{\mathrm{h}}}{\mbox{$\rightarrow$}}{\ensuremath{\mathrm{b}\bar{\mathrm{b}}}}$ suppressed by cancellation of\
& $\tilde{\mathrm{b}}-\tilde{\mathrm{g}}$ loops\
\
CPX & Mixing of CP- and mass-eigensates\
& several derivates under study\
Depending on the parameters of the MSSM, Higgs Bosons can be produced in Higgstrahlung ${\mbox{${\mathrm{e}}^+ {\mathrm{e}}^-$}}{\mbox{$\rightarrow$}}{\ensuremath{\mathrm{h}}}{\ensuremath{\mathrm{Z}}}$, as in the SM, or in pair production ${\mbox{${\mathrm{e}}^+ {\mathrm{e}}^-$}}{\mbox{$\rightarrow$}}{\ensuremath{\mathrm{h}}}{\ensuremath{\mathrm{A}}}$. Flavour independent Higgs decays, Higgs decays into invisible particles or decays of the type ${\ensuremath{\mathrm{h}}}{\mbox{$\rightarrow$}}{\ensuremath{\mathrm{A}}}{\ensuremath{\mathrm{A}}}$ additionally present new topologies.
Since the end of LEP running, the LEP collaborations have developed new searches closing some of these unexcluded areas in the parameter space, thereby giving important new information about the tasks left over for future experiments. This publication will focus on new searches dedicated to formerly uncovered final states described in Section \[sect:searches\], on the consequences of new theoretical developments in Section \[sect:newbenchmarks\] and on the interpetation of the MSSM Higgs searches in the benchmark scenarios in Section \[sect:interpretations\].
Searches for Higgs bosons in the MSSM {#sect:searches}
=====================================
The search for Higgs bosons in the MSSM uses a large variety of channels.The SM production channels Higgsstrahlung ${\mbox{${\mathrm{e}}^+ {\mathrm{e}}^-$}}{\mbox{$\rightarrow$}}{\ensuremath{\mathrm{h}}}{\ensuremath{\mathrm{Z}}}$ and Boson fusion are reintrerpreted in the MSSM. Additionally, dedicated searches for Higgsstrahlungschannels with Higgs decays in the MSSM exist. Also, pair production ${\mbox{${\mathrm{e}}^+ {\mathrm{e}}^-$}}{\mbox{$\rightarrow$}}{\ensuremath{\mathrm{h}}}{\ensuremath{\mathrm{A}}}$ and Yukawa production ${\mbox{${\mathrm{e}}^+ {\mathrm{e}}^-$}}{\mbox{$\rightarrow$}}{\ensuremath{\mathrm{b}\bar{\mathrm{b}}}}{\ensuremath{\mathrm{h}}}/{\ensuremath{\mathrm{A}}}$ channels are used. New searches comprise the search for ${\mbox{${\mathrm{e}}^+ {\mathrm{e}}^-$}}{\mbox{$\rightarrow$}}{\ensuremath{\mathrm{h}}}{\ensuremath{\mathrm{Z}}}$ with ${\ensuremath{\mathrm{h}}}{\mbox{$\rightarrow$}}{\ensuremath{\mathrm{A}}}{\ensuremath{\mathrm{A}}}$, with ${\mbox{$m_{\mathrm{A}}$}}<10$ GeV below the ${\ensuremath{\mathrm{b}\bar{\mathrm{b}}}}$ production threshold [@opal:lowma] by OPAL. The same final state with heavier ${\mbox{$m_{\mathrm{A}}$}}$ has been sought-after for the interpretation in CPV models [@opal:mssmnote].
The search from DELPHI [@delphi:invis] for invisibly decaying Higgs bosons has been interpreted in a modified ${\mbox{$m_{\mathrm{h}}{\mathrm{-max}}$}}$ scenario with $M_2=\mu=150$ GeV. It shows that the benchmark scenarios from Tab. \[tab:scenarios\] do not cover the full range of possible MSSM topologies in the Higgs sector, since the search for invisibly decaying Higgs bosons is needed to cover areas unexcluded by the standard searches.
New benchmark scenarios and Higgs mass calculations in the MSSM {#sect:newbenchmarks}
===============================================================
The calculations of the observables of the Higgs sector, the masses, branching ratios and cross-sections, depending on the choice of SUSY parameters have been performed using two different calculation tools. FEYNHIGGS [@feynhiggs; @Heinemeyer:CFeyn] is based on the two-loop diagrammatic approach of [@MSSMMHBOUND7], and SUBHPOLE/CPH [@Carena:2000ks] is based on the one-loop renormalization-group improved calculation of [@MSSMMHBOUND5; @carenamrennawagner].
The first three CPC benchmark scenarios of Tab. \[tab:scenarios\] have traditionally been considered in the past and have now been extended to scenarios 4 and 5, motivated by limits on the branching ratio of the inclusive decay of a B meson into strange particle states and a photon $\mathrm{B}{\mbox{$\rightarrow$}}\mathrm{X}_{\mathrm{s}}\gamma$ and muon anomalous magnetic moment $(g-2)_{\mu}$ measurements. The last two CPC benchmark scans are aiming to set the stage for future Higgs searches at the LHC. There, some of the dominant search channels would be suppressed, resulting in a reduced search sensitivity. The CPV scenario CPX maximises the mixing of CP- and masseigenstates and has been tested wit hseveral different parameter settings [@opal:mssmnote].
With respect to the calculations used for the MSSM Higgs LEP combination in 2001 [@lephiggs:2001], new 2-loop calculations of top loop corrections to the Higgs boson mass have become available [@newcalc]. They shift the maximal ${\mbox{$m_{\mathrm{h}}$}}$ achievable in the ${\mbox{$m_{\mathrm{h}}{\mathrm{-max}}$}}$ scenario upwards by up to $5$ GeV. The maximal ${\mbox{$m_{\mathrm{h}}$}}$ lies at about 135 GeV. While the expected experimental lower limit on ${\mbox{$m_{\mathrm{h}}$}}$ for low ${\mbox{$\tan\beta$}}$ is not expected to change much with respect to the latest LEP exclusion (cf. Fig. \[fig:mhmax\]), the theoretical upper limit on ${\mbox{$m_{\mathrm{h}}$}}$ shifted from the border of the black area to the border of the light grey (yellow) area. If the top mass would additionally shift upwards from its current central value of $m_{\mathrm{t}}=174.3\pm5.1$ GeV [@RPP2000] by only one sigma (which could well be the case, given latest measurements from D0 in the leptonic decay channel [@d0:newmtop]), then the upper limit on ${\mbox{$m_{\mathrm{h}}$}}$ would increase again by almost 5 GeV, as indicated by the border of the dark grey (red) area in Fig. \[fig:mhmax\].
This example shows that new theoretical developments and a higher precision on $m_{\mathrm{t}}$ could well influence the exclusion of low ${\mbox{$\tan\beta$}}$ by LEP. A higher precision on $m_{\mathrm{t}}$ is therefore highly desireable. This could also have implication on the search channels that have to be investigated at future accelerator experiments at LHC searching for a MSSM Higgs boson, where the region of small ${\mbox{$\tan\beta$}}$ can not be regarded as excluded by LEP.
Interpretation of the Higgs searches in the MSSM {#sect:interpretations}
================================================
The combination of all Higgs searches of one experiment is used to derive [@delphi:mssm; @l3:mssm; @opal:mssmnote] exclusions in the MSSM parameter space for the scenarios in Tab. \[tab:scenarios\]. No major changes with respect to previous interpretations are recieved for the no-mixing, ${\mbox{$m_{\mathrm{h}}{\mathrm{-max}}$}}$ and large-$\mu$ scenarios. The latter now can be almost completely exluded by one experiment alone [@delphi:mssm], thanks to flavour independent searches [@eps:flavindep_proc]. OPAL has also studied [@opal:mssmnote] the new CPC scenarios 4 to 7 from Tab. \[tab:scenarios\].
In summary, the parameter choices of the new CPC benchmark scenarios introduce no need for new searches at LEP. Latest in a LEP combination, all topologies are covered up to the kinematic limits of the production channels. The limits on ${\mbox{$m_{\mathrm{h}}$}}$ and ${\mbox{$m_{\mathrm{A}}$}}$ are around 85 to 90 GeV for all CPC scenarios [@aleph:mssm; @delphi:mssm; @l3:mssm; @opal:mssmnote].
The CPX scenario with maximal CP violation in the Higgs sector shows a decoupling of the lightest Higgs bosons ${\ensuremath{\mathrm{H}_1}}$ from the ${\ensuremath{\mathrm{Z}}}$ in the intermediate ${\mbox{$\tan\beta$}}$ range from 4 to 10 (${\ensuremath{\mathrm{H}_1}}$ and ${\ensuremath{\mathrm{H}_2}}$ being the lightest and next-to-lightest Higgs boson mass eigenstates). There anyhow ${\ensuremath{\mathrm{H}_2}}$ couples to the ${\ensuremath{\mathrm{Z}}}$ and is heavier than around $100$ GeV. Where kinematically accessible, the decay ${\ensuremath{\mathrm{H}_2}}{\mbox{$\rightarrow$}}{\ensuremath{\mathrm{H}_1}}{\ensuremath{\mathrm{H}_1}}$ is dominant. Another difference to the common CPC scenarios is the large mass difference ${\mbox{$m_{\mathrm{H}_{2}}$}}-{\mbox{$m_{\mathrm{H}_{1}}$}}$ in the range with dominant pair production.
Fig. \[fig:cpx\] [@opal:mssmnote] shows the exclusion areas of the CPX scenario. In the region with dominant ${\mbox{${\mathrm{e}}^+ {\mathrm{e}}^-$}}{\mbox{$\rightarrow$}}{\ensuremath{\mathrm{H}_2}}{\ensuremath{\mathrm{Z}}}{\mbox{$\rightarrow$}}{\ensuremath{\mathrm{H}_1}}{\ensuremath{\mathrm{H}_1}}{\ensuremath{\mathrm{Z}}}$ production at intermediate ${\mbox{$\tan\beta$}}$ and ${\mbox{$m_{\mathrm{h}}$}}<50$ GeV open areas emerge. It is expected that a LEP combination will be able to close these holes. Also the lower limit on ${\mbox{$m_{\mathrm{H}_{1}}$}}$ in the large ${\mbox{$\tan\beta$}}$ region, where pair production dominates, is reduced due to the large ${\mbox{$m_{\mathrm{H}_{2}}$}}-{\mbox{$m_{\mathrm{H}_{1}}$}}$. At ${\mbox{$\tan\beta$}}>5$ and ${\mbox{$m_{\mathrm{H}_{1}}$}}<10$ GeV, below the ${\ensuremath{\mathrm{b}\bar{\mathrm{b}}}}$ production threshold and in the pair production region, hardly any experimental constraints exist, since no pair production searches for ${\mbox{$m_{\mathrm{H}_{2}}$}}\approx100$ GeV and ${\mbox{$m_{\mathrm{H}_{1}}$}}<10$ GeV exist. Only at large ${\mbox{$\tan\beta$}}>20$ Yukawa production searches can be used.
Conclusions {#sect:conclusions}
===========
The developments in the MSSM Higgs searches at LEP after the end of LEP data taking in November 2000 exhibit four important lessons. First, also the increased set of CPC and CPV benchmark scenarios do not cover the full range of possible experimental phenomena in the MSSM Higgs sector. Therefore, secondly, a large variety of individual searches is necessary to cover the rich physics spectrum of the MSSM Higgs sector, which only now become fully available.
Third, new theoretical developments can influence limits on MSSM parameters. Especially the ${\mbox{$\tan\beta$}}$ exclusion of the final LEP combination could be affected. This is also important for the possible MSSM topologies in Higgs searches at the LHC. The importance of external measurements like $m_{\mathrm{t}}$ from the Tevatron becomes evident. A greater precision on $m_{\mathrm{t}}$ would be benefitial. Fourth, CPV scenarios show that there is still no strict lower limit on the Higgs mass from LEP. Especially in regions with low ${\mbox{$m_{\mathrm{H}_{1}}$}}$, but either dominant ${\mbox{${\mathrm{e}}^+ {\mathrm{e}}^-$}}{\mbox{$\rightarrow$}}{\ensuremath{\mathrm{H}_2}}{\ensuremath{\mathrm{Z}}}$ or dominant ${\mbox{${\mathrm{e}}^+ {\mathrm{e}}^-$}}{\mbox{$\rightarrow$}}{\ensuremath{\mathrm{H}_1}}{\ensuremath{\mathrm{H}_2}}$ production no ${\mbox{$\tan\beta$}}$ independent limit on the Higgs mass exists. Also these regions must probably be sought by future colliders.
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OPAL Collaboration, *Search for Neutral Higgs Bosons Predicted by CP Conserving and CP Violating MSSM Scenarios with the OPAL detector at LEP*, 2003, OPAL PN524
\[DELPHI Collaboration\], *Searches for invisibly decaying Higgs bosons with the DELPHI detector at LEP*, DELPHI 2003-036 CONF 656.
S. Heinemeyer, W. Hollik and G. Weiglein, Comp. Phys. Comm. [**124**]{} (2000) 76; Also see [http://www.feynhiggs.de]{}.
M. Frank, S. Heinemeyer, W. Hollik and G. Weiglein, hep-ph/0212037.\
Also see [www.feynhiggs.de]{}.
S. Heinemeyer, W. Hollik and G. Weiglein, Eur. Phys. Jour. [**C9**]{} (1999) 343.
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ALEPH, DELPHI, L3 and OPAL Collaborations, OPAL TN699
A. Brignole, G. Degrassi, P. Slavich and F. Zwirner, Nucl. Phys. B [**631**]{} (2002) 195;\
G. Degrassi, S. Heinemeyer, W. Hollik, P. Slavich and G. Weiglein, Eur. Phys. J. C [**28**]{} (2003) 133.
D.E. Groom [*et al*]{}, Eur. Phys. J. [**C15**]{} (2000) 1, available on the PDG WWW pages (URL: [http://pdg.lbl.gov/]{}).
M. Coca \[CDF & D0 Collaborations\], FERMILAB-CONF-03-238-E [*Presented at Flavor Physics and CP Violation (FPCP 2003), Paris, France, 3-6 Jun 2003*]{}
J. Fernandez \[DELPHI Collaboration\], DELPHI 2003-045-CONF-665, contributed paper no. 320
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[^1]: Speaker; on behalf of the LEP collaborations
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'A partial wave analysis of pion photoproduction has been obtained in the framework of fixed-$t$ dispersion relations valid from threshold up to 500 $MeV$. In the resonance region we have precisely determined the electromagnetic properties of the $\Delta(1232)$ resonance, in particular the E2/M1 ratio $R_{EM}=(-2.5 \pm 0.1) \%$. For pion electroproduction recent experimental data from Mainz, Bates and JLab for $Q^2$ up to 4.0 (GeV/c)$^2$ have been analyzed with two different models, an isobar model (MAID) and a dynamical model. The E2/M1 ratios extracted with these two models show, starting from a small and negative value at the real photon point, a clear tendency to cross zero, and become positive with increasing $Q^2$. This is a possible indication of a very slow approach toward the pQCD region. The C2/M1 ratio near the photon point is found as $R_{SM}(0)=(-6.5 \pm 0.5) \%$. At high $Q^2$ the absolute value of the ratio is strongly increasing, a further indication that pQCD is not yet reached.'
address:
- 'Institut für Kernphysik, Universtät Mainz, D-55099 Mainz'
- 'JINR Dubna, 141980 Moscow Region, Russia'
- 'Department of Physics, National Taiwan University, Taipei 10617, Taiwan'
author:
- 'L. Tiator, D. Drechsel, O. Hanstein, S.S. Kamalov and S.N. Yang'
title: |
\
The [*E2/M1*]{} and [*C2/M1*]{} ratios and form factors in $N\rightarrow \Delta$ transitions
---
INTRODUCTION {#sec:intro}
============
The determination of the quadrupole excitation strength $E_{1+}^{(3/2)}$ in the region of the $\Delta (1232)$ resonance has been the aim of considerable experimental and theoretical activities. Within the harmonic oscillator quark model, the $\Delta$ and the nucleon are both members of the symmetrical 56-plet of $SU(6)$ with orbital momentum $L = 0$, positive parity and a Gaussian wave function in space. In this approximation the $\Delta$ may only be excited by a magnetic dipole transition $M_{1+}^{(3/2)}$ [@Bec65]. However, in analogy with the atomic hyperfine interaction or the forces between nucleons, also the interactions between the quarks contain a tensor component due to the exchange of gluons. This hyperfine interaction admixes higher states to the nucleon and $\Delta$ wave functions, in particular $d$-state components with $L = 2$, resulting in a small electric quadrupole transition $E_{1+}^{(3/2)}$ between nucleon and $\Delta$ [@Kon80; @Ger81; @Isg82]. In addition quadrupole transitions are possible by mesonic and gluonic exchange currents [@Buch97]. Therefore an accurate measurement of $E_{1+}^{(3/2)}$ is of great importance in testing the forces between the quarks and, quite generally, models of nucleons and isobars.
The E2/M1 ratio, $R_{EM} =E_{1+}^{(3/2)}/M_{1+}^{(3/2)}$ has been predicted to be in the range $- 3\%\le R_{EM} < 0\%$ in the framework of constituent quark [@Kon80; @Isg82; @Buch97; @Dre84], relativized quark [@Cap90; @Cap92] and chiral bag models [@Ber88]. Considerably larger values have been obtained in Skyrme models [@Wir87]. A first lattice QCD calculation resulted in a small value with large error bars $(- 6 \% \le R_{EM} \le 12\%)$ [@Lei92]. However, the connection of the model calculations with the experimental data is not evident. Clearly, the $\Delta$ resonance is coupled to the pion-nucleon continuum and final-state interactions will lead to strong background terms seen in the experimental data, particularly in case of the small $E_{1+}$ amplitude. The question of how to ”correct” the experimental data to extract the properties of the resonance has been the topic of many theoretical investigations. Unfortunately it turns out that the analysis of the small $E_{1+}$ amplitude is quite sensitive to details of the models, e.g. nonrelativistic vs. relativistic resonance denominators, constant or energy-dependent widths and masses of the resonance, sizes of the form factor included in the width etc. In other words, by changing these definitions the meaning of resonance vs. background changes, too.
In order to study the $\Delta$ deformation, pion photoproduction on the proton has been measured by the LEGS collaboration [@Bla97] at Brookhaven and by the A2 collaboration [@Beck97] at Mainz using transversely polarized photons, i.e. by measuring the polarized photon asymmetry $\Sigma$. In particular, the cross section $d\sigma_{\parallel}$ for photon polarization in the reaction plane turns out to be very sensitive to the small $E_{1+}$ amplitude. Assuming for simplicity that only the $P$-wave multipoles contribute, the differential cross section is $$\frac{d\sigma_{\parallel}}{d\Omega} = \frac{q}{k} (A_{\parallel} + B_{\parallel}
\cos \Theta_{\pi} + C_{\parallel} \cos^2 \Theta_{\pi}),$$ where $q$ and $k$ are the pion and photon momenta and $\Theta_{\pi}$ is the pion emission angle in the $c.m.$ frame. Neglecting the (small) contributions of the Roper multipole $M_{1-}$, one obtains [@Beck97] $$C_{\parallel}/A_{\parallel} \approx 12 R_{EM} ,$$ because the isospin $\frac{3}{2}$ amplitudes strongly dominate the cross section for $\pi^{0}$ production.
In order to obtain the C2/M1 ratio and the form factors as functions of $Q^2$ pion electroproduction has been studied. At Mainz, Bonn, Bates and JLab different experiments have been performed, without polarization as well as single and double polarization.
While the experiments at Mainz and Bates measured at $Q^2\sim 0.1$ GeV$^2$ in order to get the C2/M1 ratio close to the photon point, the JLab experiment was motivated by the possibility of determining the range of momentum transfers where perturbative QCD (pQCD) would become applicable. In the limit of $Q^2 \rightarrow
\infty$, pQCD predicts [@Brodsky81] that only helicity-conserving amplitudes contribute, leading to $R_{EM} = E_{1+}^{(3/2)}/M_{1+}^{(3/2)} \rightarrow 1$ and $R_{SM} = S_{1+}^{(3/2)}/M_{1+}^{(3/2)} \rightarrow const$.
PION PHOTOPRODUCTION {#sec:real}
====================
Starting from fixed-$t$ dispersion relations for the invariant amplitudes of pion photoproduction, the projection of the multipole amplitudes leads to a well known system of integral equations, $$\label{inteq}
\mbox{Re}{\cal M}_{l}(W) = {\cal M}_{l}^{\mbox{\scriptsize P}}(W)
+ \frac{1}{\pi}\sum_{l'}{\cal P}\int_{W_{\mbox{\scriptsize thr}}}^{\infty}
K_{ll'}(W,W')\mbox{Im}{\cal M}_{l'}(W')dW',$$ where ${\cal M}_l$ stands for any of the multipoles $E_{l\pm}, M_{l\pm},$ and ${\cal M}_{l}^{\mbox{\scriptsize P}}$ for the corresponding (nucleon) pole term. The kernels $K_{ll'}$ are known, and the real and imaginary parts of the amplitudes are related by unitarity. In the energy region below two-pion threshold, unitarity is expressed by the final state theorem of Watson, $$\label{watson}
{\cal M}_l^I (W) = \mid {\cal M}_{l}^{I} (W)\mid
e^{i(\delta_{l}^{I} (W) + n\pi)},$$ where $\delta_{l}^{I}$ is the corresponding $\pi N$ phase shift and $n$ an integer. We have essentially followed the method of Schwela et al [@Sch69; @Pfe72] to solve Eq. (\[inteq\]) with the constraint (\[watson\]). In addition we have taken into account the coupling to some higher states neglected in that earlier reference. At the energies above two-pion threshold up to $W = 2$ GeV, Eq. (\[watson\]) has been replaced by an Ansatz based on unitarity [@Sch69]. Finally, the contribution of the dispersive integrals from $2$ GeV to infinity has been replaced by $t$-channel exchange, parametrized by certain fractions of $\rho$- and $\omega$-exchange. Furthermore, we have to allow for the addition of solutions of the homogeneous equations to the coupled system of Eq. (\[inteq\]). The whole procedure introduces 9 free parameters, which have been determined by a fit to the data. [@Han98]
In Figure 1 we show the $P_{33}$ multipoles $M^{3/2}_{1+}$ and $E^{3/2}_{1+}$. Our dispersion theoretical analysis (solid line) agrees very well with our single-energy fit and with the single-energy fit of Beck et al. [@Beck00]. The only systematic deviation becomes visible in the electric multipole above the resonance position. This can be due either to our truncation of partial waves or to systematics in the experiment at the highest energies. In a new experiment a full angular coverage of the differential cross section for $p(\gamma,\pi^0)p$ over a wide range of energy from threshold up to $E^{lab}_\gamma =440
MeV$ has been taken and in the near future a new and very precise multipole analysis should also clarify this small deviation.
According to the Watson theorem, at least up to the two-pion threshold, the ratio $E_{1+}^{(3/2)}/M_{1+}^{(3/2)}$ is a real quantity. However, it is not a constant but even a rather strongly energy dependent function. If we determine the resonance position as the point, where the phase $\delta_{1+}^{(3/2)}(W=M_\Delta)=90^\circ$, we can define the so-called “full” ratio $$R_{EM} = \left. \frac{E_{1+}^{(3/2)}}{M_{1+}^{(3/2)}}\right|_{W=M_\Delta}
= \left. \frac{\mbox{Im}E_{1+}^{(3/2)}}{\mbox{Im}M_{1+}^{(3/2)}}
\right|_{W=M_\Delta}\,.$$ We note that this ratio is identical to the ratio obtained with the $K$-matrix at the $K$-matrix pole $W=M_\Delta$. This can be seen by using the relation between the $T$- and the $K$-matrix, $$T=K\cos\delta e^{i\delta} \quad \mbox{and consequently} \quad
K=\mbox{Re}T+\mbox{Im}T\, tan\delta\,.$$ Therefore, at $W=M_\Delta$ we find $$K(E_{1+}^{(3/2)})/K(M_{1+}^{(3/2)})=
\mbox{Im}E_{1+}^{(3/2)}/\mbox{Im}M_{1+}^{(3/2)}=R_{EM}\,.$$ The recent, nearly model-independent value of the Mainz group at $W =M_{\Delta}=1232$ MeV is $(-2.5 \pm 0.1 \pm 0.2)\%$ [@Beck00] is in excellent agreement with our dispersion theoretical calculation that gives $(-2.54\pm 0.10)\%$, see Table 1.
$R_{EM} [\%]$ Reference
----------------------------------------------- ----------------------------
$ -2.54 \pm 0.10 $ Hanstein et al. [@Han98]
$ -2.5 \pm 0.1_{stat.} \pm 0.2_{syst.} $ Beck et al. [@Beck00]
$ -3.0 \pm 0.3_{stat.+syst.} \pm 0.2_{mod.} $ Blanpied et al. [@Bla97]
$ -1.5 \pm 0.5 $ Arndt et al. [@Arn97]
$ -3.19 \pm 0.24 $ Davidson et al. [@Dav97]
$ -2.5 \pm 0.5 $ PDG 2000 estimate [@PDG00]
: E2/M1 ratios for $Q^2$=0 from different analyses.
As it was demonstrated in different approaches [@Han98; @Dav99], the precise E2/M1 ratio is very sensitive to the specific database used in the fit. Therefore, the SAID value, obtained with the full database is rather low ($-1.5 \%$) and the values obtained with the LEGS differential cross sections are twice as large, around $-3\%$.
The ratio so far discussed above, is the ratio at the K-matrix pole on the real energy axis. In scattering theory, the T-matrix pole in the complex plane, however, is more fundamental. The analytic continuation of a resonant partial wave as function of energy into the second Riemann sheet should generally lead to a pole in the lower half-plane. A pronounced narrow peak reflects a time-delay in the scattering process due to the existence of an unstable excited state. This time-delay is related to the speed $SP$ of the scattering amplitude $T$, defined by [@Hoe92] $$SP(W) = \left\vert \frac{dT(W)}{dW}\right\vert ,$$ where $W$ is the total $c.m.$ energy. In the vicinity of the resonance pole, the energy dependence of the full amplitude $T = T_{B} + T_{R}$ is determined by the resonance contribution, $$\label{T_res}
T_{R} (W) = \frac{r\Gamma_{R} e^{i\phi}}{M_{R}- W-
i\Gamma_{R}/2}\,\,,$$ while the background contribution $T_B$ should be a smooth function of energy, ideally a constant. We note in particular that $W_R = M_{R}- i\Gamma_{R}/2$ indicates the position of the resonance pole in the complex plane, i.e. $M_{R}$ and $\Gamma_{R}$ are constants and differ from the energy-dependent widths, and possibly masses, derived from fitting certain resonance shapes to the data. In the limit where the derivative of the smooth background can be neglected, the speed takes the simple form $$\label{Speed}
SP(W) = \frac{r\Gamma_{R}}{(M_{R}- W)^2+
\Gamma_{R}^2/4}\,.$$ From this form, the position of the pole as well as the absolute magnitude of the residue can be easily obtained. Furthermore, in our dispersion approach we have also checked the validity of the assumption to neglect the background and found that this procedure works very well for the Delta resonance.
Applying this technique to our $P_{33}$ amplitudes we find the pole at $W_{R} = M_{R} - i \Gamma_{R}/2 = (1211 - 50 i)$ MeV in excellent agreement with the results obtained from $\pi N$ scattering, $M_{R}=
(1210\pm 1)$ MeV and $\Gamma_{R} = 100$ MeV [@Hoe92]. The complex residues and the phases are obtained as $r_E=1.23\cdot
10^{-3}/m_{\pi}, \phi_E=-154.7^{\circ}, r_M=21.16\cdot
10^{-3}/m_{\pi}$ and $\phi_M=-27.5^{\circ}$, yielding a complex ratio of the residues $$R_{\Delta} = \frac{r_{E} e^{i\phi_{E}}}{r_{M} e^{i\phi_{M}}}
= - 0.035 - 0.046 i.$$ While the experimentally observed ratio $R_{EM}$ is real and very sensitive to small changes in energy, the ratio $R_{\Delta}$ is a complex number defined by the residues at the pole, therefore, it does not depend on energy.
It should be noted, however, that a resonance without the accompanying background terms is unphysical, in the sense that only the sum of the two obeys unitarity. Furthermore we want to point out that the speed-plot technique does not give information about the strength parameters of a “bare” resonance, i.e. in the case where the coupling to the continuum is turned off. Both the pole position and the residues at the pole will change for such a hypothetical case, but the exact values for the “bare” resonance can only be determined by a model calculation and as such will depend on the ingredients of the model.
PION ELECTROPRODUCTION {#sec:virtual}
======================
In the dynamical approach to pion photo- and electroproduction [@Yang85], the t-matrix is expressed as $$\begin{aligned}
t_{\gamma\pi}(E)=v_{\gamma\pi}+v_{\gamma\pi}\,g_0(E)\,t_{\pi
N}(E)\,, \label{eq:tgamapi}\end{aligned}$$ where $v_{\gamma\pi}$ is the transition potential operator for $\gamma^*N \rightarrow \pi N$, and $t_{\pi N}$ and $g_0$ denote the $\pi N$ t-matrix and free propagator, respectively, with $E
\equiv W$ the total energy in the CM frame. A multipole decomposition of Eq. (\[eq:tgamapi\]) gives the physical amplitude in channel $\alpha$ [@Yang85], $$\begin{aligned}
t_{\gamma\pi}^{(\alpha)}(q_E,k;E+i\epsilon)
&=&\exp{(i\delta^{(\alpha)})}\,\cos{\delta^{(\alpha)}}
\nonumber\\&\times&
\left[v_{\gamma\pi}^{(\alpha)}(q_E,k) + P\int_0^{\infty}
dq' \frac{q'^2R_{\pi
N}^{(\alpha)}(q_E,q';E)\,v_{\gamma\pi}^{(\alpha)}(q',k)}{E-E_{\pi
N}(q')}\right], \label{eq:backgr}\end{aligned}$$ where $\delta^{(\alpha)}$ and $R_{\pi N}^{(\alpha)}$ are the $\pi
N$ scattering phase shift and reaction matrix in channel $\alpha$, respectively; $q_E$ is the pion on-shell momentum and $k=|{\bf
k}|$ is the photon momentum. The multipole amplitude in Eq. (\[eq:backgr\]) manifestly satisfies the Watson theorem and shows that the $\gamma\pi$ multipoles depend on the half-off-shell behavior of the $\pi N$ interaction.
In a resonant channel like (3,3) in which the $\Delta(1232)$ plays a dominant role, the transition potential $v_{\gamma\pi}$ consists of two terms, $$\begin{aligned}
v_{\gamma\pi}(E)=v_{\gamma\pi}^B + v_{\gamma\pi}^{\Delta}(E)\,,
\label{eq:tranpot}\end{aligned}$$ where $v_{\gamma\pi}^B$ is the background transition potential and $v_{\gamma\pi}^{\Delta}(E)$ corresponds to the contribution of the bare $\Delta$.
It is well known that for a correct description of the resonance contributions we need, first of all, a reliable description of the nonresonant part of the amplitude.
In the new version of MAID (MAID2000), the $S$, $P$, $D$ and $F$ waves of the background contributions are complex numbers defined in accordance with the K-matrix approximation, $$t_{\gamma\pi}^{B,\alpha}({\rm MAID})=
\exp{(i\delta^{(\alpha)})}\,\cos{\delta^{(\alpha)}}
v_{\gamma\pi}^{B,\alpha}(W,Q^2).
\label{eq:bg00}$$ From Eqs. (\[eq:backgr\]) and (\[eq:bg00\]), one finds that the difference between the background terms of MAID and of the dynamical model is that off-shell rescattering contributions (principal value integral) are not included in MAID. To take account of the inelastic effects at the higher energies, we replace $\exp{(i\delta^{(\alpha)})} \cos{\delta^{(\alpha)}} =
\frac 12 (\exp{(2i\delta^{(\alpha)})} +1)$ in Eqs. (\[eq:backgr\]) and (\[eq:bg00\]) by $\frac 12
(\eta_{\alpha}\exp{(2i\delta^{(\alpha)})} +1)$, where $\eta_{\alpha}$ is the inelasticity. In our actual calculations, both the $\pi N$ phase shifts $\delta^{(\alpha)}$ and inelasticity parameters $\eta_{\alpha}$ are taken from the analysis of the GWU group [@Arn97].
Following Ref. [@Maid], we assume a Breit-Wigner form for the resonance contribution ${\cal A}^{R}_{\alpha}(W,Q^2)$ to the total multipole amplitude, $${\cal A}_{\alpha}^R(W,Q^2)\,=\,{\bar{\cal A}}_{\alpha}^R(Q^2)\,
\frac{f_{\gamma R}(W)\Gamma_R\,M_R\,f_{\pi
R}(W)}{M_R^2-W^2-iM_R\Gamma_R} \,e^{i\phi}, \label{eq:BW}$$ where $f_{\pi R}$ is the usual Breit-Wigner factor describing the decay of a resonance $R$ with total width $\Gamma_{R}(W)$ and physical mass $M_R$. The expressions for $f_{\gamma R}, \, f_{\pi R}$ and $\Gamma_R$ are given in Ref. [@Maid]. The phase $\phi(W)$ in Eq. (\[eq:BW\]) is introduced to adjust the phase of the total multipole to equal the corresponding $\pi N$ phase shift $\delta^{(\alpha)}$. Because $\phi=0$ at resonance, $W=M_R$, this phase does not affect the $Q^2$ dependence of the $\gamma N R$ vertex.
In the dynamical model of Ref. [@KY99], a scaling assumption was made concerning the (bare) form factors ${\bar{\cal
A}}_{\alpha}^\Delta(Q^2)$, namely, that all of them have the same $Q^2$ dependence, $$\begin{aligned}
{\bar{\cal A}}_{\alpha}^{\Delta}(Q^2)={\bar{\cal
A}}_{\alpha}^{\Delta}(0) \frac{ k}{k_W}\,F(Q^2),\,\end{aligned}$$ where $\alpha = M, E,$ and $S$, $k_W = (W^2 - m_N^2)/2W$, ${
k}^2=Q^2+((W^2-m_N^2-Q^2)/2W)^2$, and $F$ is normalized to $F(0) =
1$. The values of ${\bar{\cal A}}_M^{\Delta}(0)$ and ${\bar{\cal
A}}_E^{\Delta}(0)$ were determined by fitting to the multipoles obtained in the recent analyses of the Mainz [@Han98] and GWU [@VPI97] groups. The $Q^2$ evolution of the form factor $F$ was assumed to take the form $ F(Q^2)=(1+\beta\,Q^2)\,e^{-\gamma
Q^2}\,G_D(Q^2),$ where $G_D(Q^2)=1/(1+Q^2/0.71)^2$ is the usual dipole form factor. The parameters $\beta$ and $\gamma$ were determined by fitting ${\bar{\cal A}}_{M}^{\Delta}(Q^2)$ to the data for $G_M^*$ defined by [@Maid; @KY99], $$\begin{aligned}
M_{1+}^{(3/2)}(M_{\Delta},Q^2)=\frac {k}{m_N}
\sqrt{\frac{3\alpha_{em}}{8\Gamma_{exp} q_{\Delta}}}\, G_M^*(Q^2),
\label{eq:gmform2}\end{aligned}$$ where $\alpha_{em}=1/137$, $\Gamma_{exp}=115$ MeV, and $q_{\Delta}$ is the pion momentum at $W=M_\Delta$. With the relation ${\bar{\cal A}}_E^{\Delta}(0)= {\bar{\cal
A}}_S^{\Delta}(0)$, the ratios $R_{EM}$ and $R_{SM}$ between the full multipoles were then evaluated [@KY99] and found to agree with the values extracted in Ref. [@Frolov99].
In the present analysis, we do not impose the scaling assumption and write, for electric ($\alpha=E$) and Coulomb ($\alpha=S$) multipoles, $$\begin{aligned}
{\bar{\cal A}}_{\alpha}^{\Delta}(Q^2)=X_{\alpha}^{\Delta}(Q^2)\,{\bar{\cal
A}}_{\alpha}^{\Delta}(0) \frac{ k}{k_W}\,F(Q^2),\end{aligned}$$ with $X_{\alpha}^{\Delta}(0) = 1$, and we allow both $X_E$ and $X_S$ to be determined by the experiment.
\[fig:gm\] The dynamical model and MAID are used to analyze the recent JLab differential cross section data on $p(e,e'p)\pi^0$ at high $Q^2$. All measured data, 751 points at $Q^2$=2.8 and 867 points at $Q^2$=4.0 (GeV/c)$^2$ covering the entire energy range $1.1 < W < 1.4$ GeV, are included in our global fitting procedure using the MINUIT code and we obtain a very good fit to the measured differential cross sections. Our results for the $G_M^*$ form factor are shown in Figure 2. Here the best fit is obtained with $\gamma=0.21$ (GeV/c)$^{-2}$ and $\beta=0$ in the case of MAID, and $\gamma=0.40$ (GeV/c)$^{-2}$ and $\beta=0.52$(GeV/c)$^{-2}$ in the case of the dynamical model. It is worth noting that in the definition of Eq. (\[eq:gmform2\]), $G_M^*(0)/3$ takes a value of 1 to an accuracy of $1\%$. This very precise value is extracted from the recent Mainz experiment [@Beck00]. With this number we can also determine a very precise $N \rightarrow \Delta$ magnetic transition moment, $\mu_{N\Delta}=3.46 \pm 0.03$ in units of nuclear magnetons.
ratios $Q^2(GeV^2)$ MAID DM Ref. [@Frolov99]
-------------------- -------------- ------------------- ------------------ ------------------
$ G_M^*\times 100$ 2.8 $ 6.78\pm 0.05 $ $ 7.00\pm 0.04$ $ 6.9\pm 0.4$
4.0 $ 2.86\pm 0.02 $ $ 3.04\pm 0.02$ $ 2.9\pm 0.2$
$R_{EM}$ 2.8 $ -0.56\pm 0.33 $ $-1.28\pm 0.32$ $-2.00\pm 1.7$
4.0 $ 0.09\pm 0.50 $ $-0.84\pm 0.46$ $-3.1\pm 1.7$
$R_{SM}$ 2.8 $ -9.14\pm 0.54 $ $-11.65\pm 0.52$ $-11.2\pm 2.3$
4.0 $-13.37\pm 0.95 $ $-17.70\pm 1.0 $ $-14.8\pm 2.3 $
: Our results for the magnetic transition form factor $G_M^*$ and for the ratios $R_{EM}$ and $R_{SM}$, at $Q^2$=2.8 (upper row) and 4.0 (lower row) (GeV/c)$^2$, extracted from a global fit to the data with MAID and the dynamical model as discussed in the text. Results from Ref. [@Frolov99] are listed for comparison. Ratios are given in (%).
We have also re-analyzed older DESY [@Haidan79] data measured at $Q^2=3.2 (GeV/c)^2$ and found significantly different results for the ratios compared to a previous analysis [@Burkert95]. However, since these data also give a $G_M^*$ value $20 \%$ below our fit curves in Figure 2, we did not include them in our fits of the ratios.
In a similar way we also analyzed recent Bates measurements [@Mertz99] for unpolarized differential cross sections, $R_{LT}$ response function and $A_{LT}$ asymmetry for $p(e,e'\pi^0)p$ at $Q^2=0.126 (GeV/c)^2$ and obtained the following preliminary values for the E2/M1 and C2/M1 ratios $$R_{EM}=(-2.1\pm 0.2)\% \quad \mbox{and} \quad R_{SM}=(-6.3\pm 0.2)\% \,.$$ Finally, in a double polarization experiment at Mainz [@Schmieden99], measuring the recoil polarization of the proton, $p(\vec{e},e'\vec{p})\pi^0$ at $Q^2=0.121 (GeV/c)^2$, a preliminary value of $$R_{SM}=(-5.8\pm 1.0)\% \,.$$ could be extracted in a rather model-independent way from the x-component of the recoil polarization $P_x$, which is very sensitive to the resonant $S_{1+}$ multipole.
Our extracted values for $R_{EM}$ and $R_{SM}$ and a comparison with the results of Ref. [@Frolov99] are presented in Table 2 and shown in Figure 3. The main difference between our results and those of Ref. [@Frolov99] is that our values of $R_{EM}$ show a clear tendency to cross zero and change sign as $Q^2$ increases. This is in contrast with the results obtained in the original analysis [@Frolov99] of the data which concluded that $R_{EM}$ would stay negative and tend toward more negative values with increasing $Q^2$.
CONCLUSIONS {#sec:concl}
===========
In the framework of fixed-t dispersion relations with the new and very precise data obtained at MAMI in Mainz we have obtained a new partial wave analysis for pion photoproduction. The uncertainties in most multipoles could be considerably improved compared to previous analyses. At the resonance position, where the phase passes $90^{\circ}$, we obtain an E2/M1 ratio of $R_{EM}=(-2.5 \pm
0.1) \%$. At the pole in the complex plane we obtain the ratio of the resonant electric and magnetic multipoles as $R_{\Delta} = -
0.035 - 0.046 i$. This is a further model-independent ratio that can be determined in any analysis or calculation of pion photoproduction.
For pion electroproduction, we have re-analyzed the recent JLab data for electroproduction of the $\Delta(1232)$ resonance via $p(e,e'p)\pi^0$ with two models for pion electroproduction, both of which give excellent descriptions of the existing data. In contrast to previous findings, our models indicate that $R_{EM}$, starting from a small and negative value at the real photon point, actually exhibits a clear tendency to cross zero and change sign as $Q^2$ increases. It will be most interesting to have data at yet higher momentum transfer in order to see whether such a trend continues, which would be a sign for a rather slow approach towards the pQCD region. Furthermore, the absolute value of $R_{SM}$ is strongly increasing, which indicates that the pQCD prediction of $R_{SM}\rightarrow constant$ is not yet reached.
ACKNOWLEDGMENTS
===============
We wish to thank R. Beck, R. Leukel, H. Schmieden, R. Gothe and C. Papanicolas for their contribution on the experimental data. This work was supported in part by NSC under Grant No. NSC89-2112-M002-038, by Deutsche Forschungsgemeinschaft (SFB443) and by a joint project NSC/DFG TAI-113/10/0.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Proposals that quantum gravity gives rise to non-commutative spacetime geometry and deformations of Poincaré symmetry are examined in the context of (2+1)-dimensional quantum gravity. The results are expressed in five lessons, which summarise how the gravitational constant, Planck’s constant and the cosmological constant enter the non-commutative and non-cocommutative structures arising in (2+1)-dimensional quantum gravity. It is emphasised that the much studied bicrossproduct $\kappa$-Poincaré algebra does not arise directly in (2+1)-dimensional quantum gravity'
---
EMPG-07-21
24 pt [ **Lessons from (2+1)-dimensional quantum gravity**]{}
16 pt
B. J. Schroers[^1]\
Department of Mathematics and Maxwell Institute for Mathematical Sciences\
Heriot-Watt University\
Edinburgh EH14 4AS, United Kingdom
[October 2007]{}
[Based on talk given at the conference\
“From Quantum to Emergent Gravity: Theory and Phenomenology”\
June 11-15 2007, Trieste, Italy]{}
Introduction and motivation
===========================
A key motivation for the study of (2+1)-dimensional quantum gravity is to shed light on general and conceptual issues associated with quantising gravity [@Carlipbook]. The goal of this talk is to focus on two closely related issues, namely the role of non-commutative geometry and the emergence of deformed versions of special relativity in quantum gravity, and to extract lessons regarding these issues from (2+1)-dimensional quantum gravity.
In the course of the talk I need to refer to some of the technical tools which make classical and quantum gravity in 2+1 dimensions tractable, such as the formulation as a Chern-Simons theory and techniques from the theory of quantum groups. However, I shall try to express each of the lessons in simple, physical terms. There will be a total of five lessons, and all of them involve in an essential way the physical constants which enter quantum gravity, namely the speed of light $c$, the gravitational constant $G$, the cosmological constant $\Lambda_c$ and Planck’s constant $\hbar$. We will mostly set the speed of light to one, but exhibit the other constants explicitly. A special feature of 2+1 dimensions is that the Planck mass can be expressed in terms of $G$ only - without involving $\hbar$. The reason for this is that the dimension of $G$ in 2+1 dimensions is that of an inverse mass. It follows that we can form two length parameters \_P=G \_C= and one dimensionless ratio $\ell_P/\ell_c$.
The talk is based on previous work with Catherine Meusburger [@we1; @we2; @we3] and on current work with Shahn Majid [@MS]. I begin be reviewing basic properties of the Poincaré group and special relativity in 2+1 dimensions. Promoting global Poincaré symmetry to a local symmetry leads to the formulation of gravity in 2+1 dimensions as a Chern-Simons gauge theory. The cosmological constant can be introduced in this picture as a deformation parameter which changes the gauge group of the Chern-Simons theory. Quantisation deforms the gauge group to a Hopf algebra which is neither commutative nor cocommutative. However, one important lesson one learns in 2+1 dimensions is that the so-called bicrossproduct $\kappa$-Poincaré algebra, much discussed in the recent literature on deformed or doubly-special relativity (see e.g. [@KGintro1] for a review), is not isomorphic to any of the Hopf algebras arising directly in the quantisation of 2+1 gravity, contrary to what is sometimes claimed. I will conclude the talk with a careful explanation and discussion of this statement.
Special relativity in 2+1 dimensions
====================================
Minkowski space and it symmetries
---------------------------------
I denote vectors in three-dimensional Minkowski space by $\bx$ with coordinates $x^a$, $a=0,1,2$. The metric is $\eta_{ab} =$diag$ (+,-,-)$, so that the totally antisymmetric tensor $\epsilon_{abc}$ satisfies $\epsilon^{abc} = \epsilon_{abc}$. The identity component of the Lorentz group is $SO^+(2,1)$, which is isomorphic to $SL(2,{\mathbb{R}})/{\mathbb{Z}}_2\simeq SU(1,1)/{\mathbb{Z}}_2$. I denote the Lie algebra of the Lorentz group by ${\mathfrak su}(1,1)$ in this talk; its generators are the rotation generator $J_0$ and the boost generators $J_1$ and $J_2$ with commutators =\_[abc]{} J\^c.
The isometry group of Minkowski space is the Poincaré group, which plays a key role this talk. I will work with the double cover of the identity component of the Poincaré group P\_3= SU(1,1)\^3 with multiplication law (v\_1,\_1)(v\_2,\_2)=(v\_1v\_2,\_1+(v\_1)\_2), where the notation exploits the identification of ${\mathbb{R}}^3$ with the Lie algebra ${\mathfrak su}(1,1)$. The Lie algebra ${\mathfrak p}_3$ of the Poincaré group $P_3$ is generated by the Lorentz generators $J_a$ and translation generators $P_a$, with brackets =\_[abc]{}J\^c, =\_[abc]{}P\^c, =0. This algebra has the invariant, non-degenerate inner product \[inprod\] J\_a,P\_b= \_[ab]{} which will be crucial in what follows.
Phase space of a free point particle=(Co)adjoint orbit
------------------------------------------------------
An excellent way to think about the phase space of any dynamical system is as the space of all solutions of the equations of motion. For a free relativistic particle the phase space is then the space of all timelike straight lines in Minkowski space. A given line can be parametrised by giving its direction $\hat \bp$ and one point $\bx$ on it. Since the points $\bx $ and $\bx + \tau \hat \bp$ lie on the same line for any $\tau \in {\mathbb{R}}$, it is convenient to use, instead of $\bx$, the vector \[kpx\] = + s, for arbitrary but fixed $s \in {\mathbb{R}}$. Clearly, $\bk$ is invariant under $\bx \mapsto \bx + \tau \hat \bp$. A given line is then uniquely characterised by two vectors $\bp$ and $\bk$ with fixed values for $\bp^2=m^2$ and $\bp{\!\cdot\!}\bk=ms$. The space of all such lines (for given $m,s$) is four dimensional.
{width="10truecm"}
In order to derive the symplectic structure on the phase space we require an action. In the case of the free relativistic particle this action has a geometrical interpretation in terms of the coadjoint orbit method [@Kirillov]. For convenience I use the inner product to identify the dual vector space ${\mathfrak p}^*_3$ with ${\mathfrak p}_3$ , and consider adjoint orbits instead of coadjoint orbits. In particular, I identify P\^\*\_0J\_0, J\^\*\_0P\_0 and consider the adjoint orbit of the Lie algebra element $mJ_0 + sP_0$. For $g=(v,\bx)\in P_3$ we define three-vectors $\bp$ and $\bk$ via g(mJ\_0 + sP\_0)g\^[-1]{} =p\_aJ\^a + k\_aP\^a, which is equivalent to $p_aJ^a=\Ad(v)J_0$ and the relation between $\bx,\bp$ and $\bk$. Then the action of a and spinning point particle [@SousaGerbert] can be written as I\_= d p\_a \^a +sP\_0,v\^[-1]{} = d mJ\_0 +s P\_0, g\^[-1]{}. The resulting Poisson brackets are the canonical Kirillov-Kostant-Souriau brackets [@Kirillov] of the coordinate functions $p_a$ and $k_a$: \[coadbrackets\] { k\_a,k\_b }=-\_[abc]{}k\^c, {k\_a,p\_b}=-\_[abc]{}p\^c, { p\_a,p\_b } =0.
2+1 gravity as a Poincaré gauge theory
======================================
The Chern-Simons formulation
----------------------------
The starting point for the Chern-Simons formulation of 2+1 gravity is Cartan’s trick of combining the dreibein $e_a$ and spin connection $\omega = \omega_a J^a$ into the one-form A=e\_aP\^a + \_aJ\^a, with values in ${\mathfrak p}_3$. As observed in [@AT; @Witten] the Einstein-Hilbert action of 2+1 gravity can then be written as a Chern-Simons action: \[action\] I\_ = \_[M\_3]{} A dA + , A . Note that the definition of the action (but not of the connection) requires the inner product . The equation of motion following from is the flatness condition for the curvature of the connection $A$: F\_A=0. This is equivalent to requiring the spin connection to be flat and torsion free, and hence to the Einstein equations.
Introducing point particles
---------------------------
We consider a spacetime of topology M\_3= S\_[gn]{}, where $S_{gn} $ is a surface of genus $g$ with $n$ marked points, and introduce local coordinates $x=(x_1,x_2)$ on the surface $S_{gn}$ as well as a coordinate $\tau$ for ${\mathbb{R}}$. Each of the marked points $S_{gn}$ is then decorated with a (co)adjoint orbit of the Poincaré group which is coupled to the gauge field via minimal coupling. Concentrating on one marked point, with coordinate $x^*$, the coupling is I\_= d mJ\_0 +s P\_0, g\^[-1]{}( +A\_(,x\^\*))g . The equation of motion is now F\_A=-g(J\_0 +P\_0)g\^[-1]{}dx\_1dx\_2\^2(x-x\^\*), with $\mu = 8\pi m G$ and $\sigma = 8\pi s G $. This forces the holonomy around a given puncture to lie in a fixed conjugacy class \_:={ g e\^[-J\_0 -P\_0]{} g\^[-1]{}| g P\_3}.
Holonomies and phase space
--------------------------
In the Chern-Simons formulation, the phase space of (2+1)-dimensional gravity can be parameterised by holonomies around non-contractible loops on $S_{gn}$, see Fig. 2 and [@we1] for further details. Defining the extended phase space via \[extspace\] = P\_3\^[2g]{}\_[\_n \_n]{}…[C]{}\_[\_1 \_1]{}, the physical phase space is obtained as a finite quotient: $$\begin{aligned}
\label{pspace}
{\cal P} & = \{(A_g,B_g,\ldots, A_1,B_1, M_n,\ldots M_1)\in \tilde {\cal P}| \nonumber \\
& \quad [A_g,B_g^{-1}]\ldots[ A_1,B_1^{-1}]M_n\ldots M_1=1\}/\mbox{conjugation}.\end{aligned}$$ The space $\cal P$ inherits a symplectic structure from the infinite-dimensional affine space of connections $A$, of which it is an infinite-dimensional symplectic quotient. The resulting symplectic structure on the phase space $\cal P$ (called Atiyah-Bott structure) can be described explicitly in a framework introduced by Fock and Rosly [@FR], and developed in [@AMII] and [@AS], see also [@we1; @we2] for its application in (2+1)-dimensional gravity. The basic idea is to work on the extended phase space $\tilde {\cal P} $, and to define a symplectic structure on it in such a way that the induced symplectic structure on the quotient agrees with the Atiyah-Bott structure. As emphasised particularly in [@AMII], the Fock-Rosly symplectic structure on $\tilde {\cal P}$ is isomorphic, via a “decoupling transformation”, to a direct product symplectic structure consisting of building blocks associated to the Poisson-Lie structure of the gauge group (for us $P_3$), namely a copy of the so-called Heisenberg double for every handle on $S_{gn}$, and a symplectic leaf of the dual or Semenov-Tian-Shansky structure for every particle. For details regarding this structures see [@AMI] and also [@K-S; @CP] for further background.
{width="11truecm"}
$P_3$ as Poisson-Lie group
--------------------------
A fundamental ingredient of the Fock-Rosly construction is an $r$-matrix whose defining feature is that it satisfies the classical Yang-Baxter equation, and that its symmetric part agrees (after dualising) with the inner product used in the definition of the Chern-Simons action. It is easy to check that the r-matrix \[rmat\] r= P\_aJ\^a\_3\_3. satisfies these requirements. Given , the group $P_3$ can be equipped with the Sklyanin bracket, thus turning it into a Poisson-Lie group. The bracket takes the following form in terms of the parametrisation $(v,\bx)\in P_3$: \[Sklyanin\] {x\_a,x\_b}=G\_[abc]{}x\^c, {x\_a,f(v)}={f(v),g(v)}=0.
As mentioned above, it is not the Sklyanin bracket itself which enters the symplectic structure of the phase space but the associated Heisenberg double and dual Poisson structures. We focus on the latter here, and note that, as a group, the dual Poisson-Lie group of $P_3$ is P\_3\^\*=SU(1,1)\^3. To write down the Poisson structure of $P^*_3$ explicitly, we write elements as $(u,-\bj)$, with u=(-8G p\_aJ\_a). Then one finds the following brackets of coordinate functions: \[dualbrackets\] {j\_a,j\_b}=-\_[abc]{}j\^c, {j\_a,p\_b}=-\_[abc]{}p\^c, { p\_a,p\_b } =0. The Poisson manifold $P^*_3$ is a non-linear or deformed version of the linear Poisson manifold ${\mathfrak p}_3^*$. The brackets are precisely the same brackets as those of the coordinate functions on ${\mathfrak p}_3^*$ . However, it is important to keep in mind that for $P^*_3$, the coordinates $p_a$ are functions on the non-linear space $SU(1,1)$, whereas for ${\mathfrak p}_3^*$ they are functions on a linear space.
Conjugacy class as particle phase space
---------------------------------------
One of the results from the theory of Poisson-Lie groups which fits very beautifully into the current story is that the symplectic leaves of $P^*_3$ are conjugacy classes in $P_3$. We saw earlier that holonomies around a given puncture are forced to lie in a fixed conjugacy class ${\cal C}_{\mu \sigma}$ of $P_3$, labelled by the mass and spin of the particle associated with the puncture. As shown in [@AMII], the induced symplectic structure on those conjugacy classes is precisely that of the dual Poisson-Lie group of the gauge group, in our case $P^*_3$. The map between the conjugacy class in $P_3$ and the dual group $P^*_3$ is explicitly given by (v,)e\^[-J\_0 -P\_0]{}(v,)\^[-1]{}=(u,-(u))(u,-), with the brackets between the coordinates $j^a$ and $p^a$ as in . On the basis of those brackets we interpret $\bj$ as the “angular momentum” associated to the particle, and the element $u$ as a “group-valued momentum”. We thus arrive at the following formulae for angular momentum and momentum in terms of the Poincaré element $(v,\bx)$: \[pv\] u&=&ve\^[-J\_0]{} v\^[-1]{} =e\^[-8G p\_a J\^a]{}\
\[jpx\] &=& (1-(u\^[-1]{}))+ sp\_a P\_a = \[,p\_aJ\^a\] + sp\_a P\_a +(\^2). The last line shows that the formula for $\bj$ can be viewed as a deformed version of the relation for a free relativistic particle. Following this analogy we think of $x^a$ as position coordinates. There is an important connection between the brackets of the position coordinates and the brackets of momentum and angular momentum: the conjugation action of $(v,\bx)$ on $(u,-\Ad(u)\bj)$ is a Poisson action only if we take into account the non-trivial Poisson brackets of the position coordinates $x^a$ given in . We thus arrive at
[|l|]{}\
**LESSON 1: particle phase space in 2+1 gravity**
\
\
$\bullet$ Momentum space has curvature radius $\propto \frac 1 G $\
$\bullet$ Position coordinates do not Poisson commute $\propto G$\
$\bullet$ The angular momentum Poisson algebra is unchanged - but the relation between position,\
momentum and angular momentum is changed\
\
Introducing the cosmological constant
=====================================
Lie groups and Lie algebras
---------------------------
In 2+1 gravity, solutions of the Einstein equations are locally isometric to a model spacetime which is determined by the signature of spacetime (Euclidean or Lorentzian) and the cosmological constant [@Carlipbook]. The isometry groups of these model spacetimes are therefore local isometry groups in 2+1 gravity. In the formulation as a Chern-Simons gauge theory [@AT; @Witten], the local isometry groups play the role of gauge groups. We list the groups arising for different signatures and signs of the cosmological constant in Table 1.
----------------- ----------------------------------------------------------- -----------------------------------------------------------------
Cosmological Euclidean signature Lorentzian signature
constant
$\Lambda_c = 0$ $E_3$ $P_3$
$\Lambda_c > 0$ $ SO(4) \simeq \frac{SU(2)\times SU(2)}{{\mathbb{Z}}_2}$ $SO(3,1) \simeq SL(2,{\mathbb{C}})/{\mathbb{Z}}_2$
$\Lambda_c < 0$ $ SO(3,1) \simeq SL(2,{\mathbb{C}})/{\mathbb{Z}}_2 $ $SO(2,2) \simeq \frac{SU(1,1)\times SU(1,1)}{ {\mathbb{Z}}_2} $
----------------- ----------------------------------------------------------- -----------------------------------------------------------------
Table 1: Local isometry groups in 2+1 gravity
The Lie brackets of the associated Lie algebras can be written in unified fashion by introducing = {
[l l]{} \_c &\
-\_c &.
. They take the following form in terms of generators $J_a$ and $P_a$ adapted to the Cartan decomposition: =\_[abc]{}J\^c, =\_[abc]{}P\^c \[P\_a,P\_b\]=\_[abc]{}J\^c. The invariant pairing remains regardless of the value of $\Lambda$. Later we will also need the Iwasawa decomposition of the Lie algebras. As explained in [@we3], the generators P\_a=P\_a+\_[abc]{}n\^b J\^c, \^2=-, together with $J_a$ provide this decomposition. In particular one has =n\_aP\_b-n\_b P\_a.
It is explained in [@we3] how to write down Sklyanin, dual and Heisenberg double brackets for the gauge groups listed in Table 1; as explained earlier, this amounts to a complete description of the symplectic structure on the phase space in the Fock-Rosly framework. The cosmological constant introduces curvature into the model spacetimes of 2+1 gravity; it is therefore not surprising that momenta, which generate translations in the model spacetime, no longer Poisson commute when the cosmological constant is non-vanishing.
[|l|]{}\
**LESSON 2: the effect of the cosmological constant**
\
\
$\bullet$ If $\Lambda \neq 0$ position space has curvature radius $\propto$ $\ell_c$\
$\bullet$ $\Lambda \neq 0$ momenta do not Poisson commute $\propto \frac 1 \ell_c $\
$\bullet$ LESSON 1 still applies.\
\
Quantisation
============
Quantisation of free point-particle (coadjoint orbit) brackets
--------------------------------------------------------------
The quantisation of the Poisson algebra of momenta and angular momenta of a free particle leads to the associative algebra generated by $J_0, J_1, J_2$ and $P_0,P_1,P_2$ with relations =\_[abc]{} J\^c, =\_[abc]{}P\^c, =0 The resulting algebra is the universal enveloping algebra $U({\mathfrak p}_3)$ [@Dixmier]. Alternatively, one can think of the momenta as coordinate functions $p_a$ on momentum space $({\mathbb{R}}^*)^3$. The ${\mathfrak su}(1,1)$ generators act on $({\mathbb{R}}^*)^3$ by infinitesimal rotations or boosts, and hence on the polynomial algebra Pol$(({\mathbb{R}}^*)^3)$. One can view $U({\mathfrak p}_3)$ therefore also as the semi-direct product of algebras \[freepaquant\] U((1,1))((\^\*)\^3). The description of the momentum algebra as a function algebra offers certain advantages which become manifest when one writes down the coalgebra structure which turns into a Hopf algebra. The coalgebra structure encodes how momenta and angular momenta of several particles are combined, see [@bamus] for details on this point of view. For a free particle this is through simple addition i.e. J\_a= J\_a 1 + 1 J\_a \[deltap\] (p\_a)= p\_a1 + 1 p\_a. The last formula is a special case of the following general construction. Suppose $G$ is any Lie group, and ${\mathbb{C}}(G)$ is the abelian algebra of complex valued functions on $G$, with pointwise multiplication[^2]. Then we can define a coproduct via \[comult\] :(G)(GG) ,f (g,h)=f(gh). For $G=({\mathbb{R}}^*)^3$ this leads to the rule for the coordinate functions $p_a$.
The Lorentz double
------------------
It is explained in detail in [@we2] how the quantisation of the Poisson brackets of the momentum and angular momentum of a gravitating particle in 2+1 dimensions leads to the Hopf algebra \[lordouble\] D(U(su(1,1)):=U((1,1))(SU(1,1)). This Hopf algebra is a particular example of a quantum double, and was called Lorentz double in [@bamus]. Following our discussion of the phase of a particle in 2+1 gravity, it is not difficult to appreciate how this algebra arises. The angular momentum algebra is unchanged compared to the free relativistic particle, but the momentum coordinates are now functions on the group manifold $SU(1,1)$ rather than the linear space $({\mathbb{R}}^*)^3$. To go from to we simply replace $\mbox{Pol}(({\mathbb{R}}^*)^3)$ by the function algebra ${\mathbb{C}}(SU(1,1))$. Since the group $G=SU(1,1)$ is non-abelian it follows immediately that the momentum addition according to the general rule is not cocommutative (i.e. depends on the order in the tensor product). Applying the rule to group elements parametrised as in , and expanding in powers of $G$ one computes the leading order in non-cocommutativity. In Lesson 3 we combine this result with the usual quantisation of the Poisson brackets for the position coordinates. Note that the lack of cocommutativity is independent of $\hbar$ and therefore really a classical effect; it is merely the manifestation of the momentum space curvature in the language of the Hopf algebra .
[|l|]{}\
**LESSON 3: quantisation with vanishing cosmological constant**
\
\
$\bullet $ $ [J_a,J_b] = \hbar \;\; \epsilon_{abc}J^c$: Angular momentum coordinates do not commute $\propto \hbar$\
$\bullet $ $ \Delta(p_a)=1\otimes p_a + p_a\otimes 1 + G \;\;
\epsilon_{abc}p^b\otimes p^c +\ldots $: Momenta do not cocommute $\propto G$\
$\bullet $ $[X_a,X_b] = l_P\;\; \epsilon_{abc}X^c$: Position coordinates do not commute $\propto l_P$\
\
Quantisation when $\Lambda \neq 0$
----------------------------------
The quantisation of 2+1 gravity has been been studied in the so-called combinatorial or Hamiltonian framework for the cases $\Lambda=0$ (for both Euclidean and Lorentzian signature, see [@Schroers] and [@we2]) and $\Lambda <0$ [@BNR]. Table 2 lists the quantum groups which (are believed to) play a role analogous to that of the Lorentz double in the case $\Lambda =0$ (and Lorentzian signature). For $\Lambda >0$ the quantisation along the lines described in this talk has not been carried out in detail, so the corresponding entries are conjectural. The parameter $q$ in the table is q=e\^[-G]{}, and combines all three physical constants which enter 2+1 dimensional quantum gravity. As usual, $D(H)$ stands for the quantum double of a Hopf algebra $H$.
All of the quantum groups in Table 2 are non-commutative and non-cocommutative. By studying the algebra and coalgebra structure of the quantum groups in Table 2, one can extract the parameters which control the failure of commutativity and cocommutativity to leading order. The results are summarised in the table below; it also includes a row for the position algebra, which I obtained by considering the dual Hopf algebra of the momentum algebra. See [@BM] for a detailed discussion of the postion algebra for the quantum double of $SU(2)$.
------------------ --------------------------- ------------------------
[**LESSON 4**]{} Commutator Co-commutator
Angular momentum $\hbar$ $\frac {G} {\ell_c} $
Momentum $ \frac{\hbar}{\ell_c} $ $G$
Position $\hbar G $ $\frac{1}{\ell_c}$
------------------ --------------------------- ------------------------
The $\kappa$-Poincaré algebra
=============================
The so-called $\kappa$-Poincaré Hopf algebra was one of the first deformations of Poincaré symmetry proposed in the literature [@LNRT; @MR]. At first sight, the $\kappa$-Poincaré algebra shares certain structural features with the Lorentz double: it has a deformation parameter with the dimension of mass, it involves a curved momentum space, and it is isomorphic to the universal enveloping algebra $U({\mathfrak p}_3)$ as an algebra (though not as a Hopf algebra). As we have seen, momentum space in the Lorentz double is the group manifold $SU(1,1)$, which, as a Lorentzian manifold, is isomorphic to anti-de Sitter space. In the standard version of the (2+1)-dimensional $\kappa$-Poincaré Hopf algebra, by contrast, momentum space is de Sitter space ={(,\_3)\^4|-\_0\^2+\_1\^2+\_2\^2 +\_3\^2=\^2}. The group $SU(1,1)$, and hence the Lie algebra ${\mathfrak su}(1,1)$, act on de Sitter space. One can therefore define the semidirect product of $ U({\mathfrak su}(1,1))$ with the algebra of complex-valued functions on $dS$: U([su]{}(1,1))(). However, since de Sitter space (unlike anti de Sitter space) is not a group manifold, we cannot use our standard construction to define a coproduct.
In order to understand the construction of the coproduct we need to take a (short) detour and review the bicrossproduct construction [@Majidbicross; @Majid] of which the Hopf algebra structure of the $\kappa$-Poincaré algebra is a special case [@MR]. The starting point of the construction is the following factorisation of elements of the group $SL(2,{\mathbb{C}})$ (strictly speaking this only holds for elements which obey a certain condition, see [@we3] for details): \[factorise\] gSL(2,)g=us= rv, u,v SU(1,1) , r,s AN. where $AN\simeq {\mathbb{R}}\ltimes {\mathbb{R}}^2$ is group of matrices of form r=e\^[-]{} & +i\
0 & e\^. Here $p_0,p_1,p_2$ are real parameter which we will eventually interpret as momentum coordinates; the constant $\kappa$ has the dimension of mass, and is introduced at this point for purely dimensional reasons. Next recall that (3+1)-dimensional Minkowski space can naturally be identified with the vector space of Hermitian 2$\times$2 matrices, and that the action of $g\in SL(2,{\mathbb{C}})$ on Hermitian matrices $h\mapsto ghg^\dagger$ implements (3+1)-dimensional Lorentz transformations. The de Sitter manifold can be realised as a submanifold of the space of Hermitian $2\times 2$ matrices via ={\_0+\_1\_1+\_2\_2 +\_3\_3| -\_0\^2+\_1\^2+\_2\^2 +\_3\^2=\^2}, where $\sigma_1,\sigma_2$ and $\sigma_3$ are the Pauli matrices. Now note that dS is the orbit of $\kappa\sigma_3$ under the $SL(2,{\mathbb{C}})$ action, and that the subgroup $SU(1,1)$ of $SL(2,{\mathbb{C}})$ is precisely the stabiliser group of $\kappa\sigma_3$. Thus, provided the second factorisation in holds, we obtain a map \[ands\] AN,rr \_3 r\^. In fact, as explained in [@we3], the image of this map is only “half” of de Sitter space. However, if we use the image of the map instead of all of de Sitter space as momentum space we obtain a curved momentum manifold which has a group structure (that of AN). Moreover, since de Sitter space is acted on by (2+1)-dimensional Lorentz transformations, we have an action of infinitesimal Lorentz transformations on the functions on “half” of de Sitter space. Thus we can define P\_= U([su]{}(1,1))(AN) which is a semi-direct product of algebras, and has a non-cocommutative momentum coproduct (p\_i)=p\_i1 + e\^[-]{}p\_i which uses the group structure of $AN$. The symbol $\rlbicross $ indicates that there is a twist in the angular momentum comultiplication, but this will not concern us here.
Relation with 2+1 gravity?
==========================
We saw that the $\kappa$-Poincaré algebra is a bicrossproduct Hopf algebra; it has some structural similarities with the Lorentz double, but is certainly not isomorphic to it. I will end this talk by sketching some observations about how these two Hopf algebras can be related by a process called semidualisation [@Majid]. Consider two Hopf algebras which are each other’s dual as Hopf algebra, e.g. \[dual\] U\_q([an]{})\_q(AN). Semidualisation can be applied to Hopf algebras that factorise, and replaces one of the factors by its dual. For example, starting with U([sl]{}(2,))U([su]{}(1,1)) U([an]{}), and using the classical ($q=1$) version of the semidualisation map is U([sl]{}(2,)) U([su]{}(1,1))(AN)= P\_ Combining the semidualisation with the quantum duality principle [@Drinfeld; @STS] \_q(SU(1,1))U\_q([an]{}), which holds only when $q\neq 1$, we obtain the following diagram [@MS]
$
\begin{array}{ccccc}
U_q({\mathfrak su}(1,1))\dcross {\mathbb{C}}_q(SU(1,1)) &\stackrel{q\neq 1}{\simeq}&
U_q({\mathfrak sl}(2,{\mathbb{C}}))&\stackrel{S}{\mapsto} & U_q({\mathfrak su}(1,1))\rlbicross {\mathbb{C}}_q(AN)\\
&&&&\\
\downarrow q\rightarrow 1 &&&&\downarrow q\rightarrow 1\\
&&&&\\
D(U(su(1,1)))&&&& P_\kappa
\end{array}
$
Summarising the comparison between the bicrossproduct $\kappa$-Poincaré algebra and the quantum doubles arising in 2+1 quantum gravity is the subject of the fifths and last lesson. Before coming to that summary, I should comment on the suspiciously vague word “arising” in the previous sentence.
In this talk I have explained the technical origin of the quantum groups in Table 2 in the Fock-Rosly description of the phase space. However, the $r$-matrices used in the Fock-Rosly construction are really auxiliary objects, used to define a Poisson structure on the extended phase space ; the induced Poisson structure on the physical phase space only depends on the symmetric part of these $r$-matrices. Correspondingly, the quantum groups in Table 2 are auxiliary objects in the quantisation, and not uniquely associated to the quantum theory. Fortunately, there is independent evidence that the quantum doubles of Table 2 play an essential role in 2+1 quantum gravity, which does not make use of the Fock-Rosly construction [@Noui]. Thus I think it is fair to say that 2+1 quantum gravity does provide evidence for the general idea that quantum gravity leads to a deformation of Poincaré symmetry, with a deformation parameter of dimension mass; the Lorentz double provides a specific realisation of this. I should stress that this is a deformation of Hopf algebras. A meaningful discussion must take into account both the algebra and the coalgebra structure. In the standard basis for the Lorentz double, for example, the algebra remains unchanged, and all the deformation takes place in the coalgebra.
By contrast, the role of the bicrossproduct $\kappa$-Poincaré Hopf algebra $P_\kappa$ in 2+1 quantum gravity remains, to my mind, unclear. It is possible to obtain the $\kappa$-Poincaré algebra in 3+1 dimensions by a contraction procedure from $U_q(so(3,2))$ in the limit $\Lambda \rightarrow 0$ [@LNRT]. This contraction procedure is sometimes interpreted as evidence for the emergence of $P_\kappa$ in a low energy limit of gravity in both 2+1 and 3+1 dimensions, see e.g. [@ACSS]. However, I am not aware of a careful version of this argument which takes into account both the algebra and the co-algebra structure, and also keeps track of the $*$-structure (the analogue of a real structure for Hopf algebras, see e.g. [@Majid]). It is not sufficient to consider the algebra alone, since the bicrossproduct $\kappa$-Poincaré algebra, like the Lorentz double, is isomorphic to the Poincaré algebra as an algebra (see e.g. [@KGN2]). The $*$-structure matters because it distinguishes, for example, $\mathfrak{su}(2)$ from $\mathfrak{su}(1,1)$, and therefore Euclidean from Lorentzian physics. I indicated above another way of obtaining $P_\kappa$ by a sequence of mathematical steps from one of the quantum doubles in Table 2; interpreting these steps physically and relating them to the contraction procedure in [@LNRT] is the subject of [@MS]. However, at this stage the arguments for a role of $P_\kappa$ in 2+1 gravity seem far less convincing to me than those for the Lorentz double.
In relation to 3+1 dimensions, the situation in 2+1, as I see it, presents a dilemma. The quantum groups which arise are all quantum doubles whose construction goes back to the essentially (2+1)-dimensional pairing . Other constructions which do generalise to higher dimensions, like the bicrossproduct construction, by contrast, do not arise naturally in 2+1 quantum gravity.
[|l|]{}\
**LESSON 5: $\kappa$-Poincaré versus quantum doubles**
\
\
$\bullet$ In 2+1 gravity momentum space is either Euclidean and positively curved\
(three-sphere) or Lorentzian and negatively curved (anti-de Sitter).\
The position algebra is $[X_a,X_b]=\ell_P \;\epsilon_{abc}X^c.$\
$\bullet$ In the standard bicrossproduct construction of $\kappa$-Poincaré, momentum space\
is Lorentzian and positively curved (de Sitter).\
The position algebra is $[X_0,X_i]=\ell_PX_i.$\
$\bullet$ Lorentz double and $\kappa$-Poincaré are [*different*]{} Hopf algebras arising as $q\rightarrow 0$ limits\
of semidual Hopf-algebras .\
\
[99]{} S. Carlip, [ Quantum gravity in 2+1 dimensions]{}, Cambridge University Press, Cambridge, 1998.
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[^1]: [email protected]
[^2]: I do not discuss analytical aspects of this algebra in the current talk, and therefore will not specify the class of functions further; however, we do require the functions to be differentiable
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We introduce new method of optimization for finding free parameters of affine iterated function systems (IFS), which are used for fractal approximation. We provide the comparison of effectiveness of fractal and quadratic types of approximation, which are based on a similar optimization scheme, on the various types of data: polynomial function, DNA primary sequence, price graph and graph of random walking.'
author:
- 'K. Igudesman and G. Shabernev'
title: Novel method of fractal approximation
---
Introduction
============
It is well known that approximation is a crucial method for making complicated data easier to describe and operate. In many cases we have to deal with irregular forms, which can’t be approximate with desired precision. Fractal approximation become a suitable tool for that purpose. Ideas for interpolation and approximation with the help of fractals appeared in works of M. Barnsley [@barnsley_fractals_everywhere] and was developed by P. Massopust [@Massop] and C. Bandt and A. Kravchenko [@bandt_kravchenko].
Today we can apply fractals to approximate such interesting and interdisciplinary data as graphs of DNA primary sequences of different species and interbeat heart intervals [@landscapes], price waves and many others.
Section \[S:fract\_interp\] of this work is devoted to the construction of fractal interpolation functions. Necessary condition on free parameters $d_i$ of affine iterated function systems is shown. One graphical example is given.
In section \[S:approximation\] we give the common scheme of approximation of general function $g\in L^2[a,b]$ and obtain the equation for direct calculation of free parameters $d_i$.
In section \[S:examples\] we illustrate the results on concrete examples.
Fractal Interpolation Functions {#S:fract_interp}
===============================
There are two methods for constructing fractal interpolation functions. In 1986 M. Barnsley [@barnsley_fractals_everywhere] defined such functions, as attractors of some specific iterated function systems. In this work we use common approach, which was developed by P. Massopust [@Massop].
Let $[a,b]\subset \mathbb{R}$ be a nonempty interval, $1<N\in\mathbb{N}$ and $\{(x_i,y_i)\in[a,b]\times\mathbb{R}\mid a=x_0<x_1<\cdots<x_{N-1}<x_N=b\}$ — are points of interpolation. For all $i=\overline{1,N}$ consider affine transformations of the plane $$A_i:\mathbb{R}^2\rightarrow\mathbb{R}^2,\quad
A_i
\left(
\begin{array}{c}
x \\
y \\
\end{array}
\right)
:=
\left(
\begin{array}{cc}
a_i & 0 \\
c_i & d_i \\
\end{array}
\right)
\left(
\begin{array}{c}
x \\
y \\
\end{array}
\right)
+
\left(
\begin{array}{c}
e_i \\
f_i \\
\end{array}
\right).$$ We require following two conditions hold true for all $i$: $$A_i(x_0,y_0)=(x_{i-1},y_{i-1}),\quad A_i(x_N,y_N)=(x_{i},y_{i}).$$ In this case $$\label{E:coeff}
\begin{array}{ll}
\displaystyle a_i=\frac{x_i-x_{i-1}}{b-a},&
\displaystyle c_i=\frac{y_i-y_{i-1}-d_i(y_N-y_0)}{b-a},\\
\displaystyle e_i=\frac{bx_{i-1}-ax_i}{b-a},& \displaystyle
f_i=\frac{by_{i-1}-ay_i-d_i(by_0-ay_N)}{b-a},
\end{array}$$ there $\{d_i\}_{i=1}^N$ act like family of parameters. Notice, that for all $i$ operator $A_i$ takes the line segment between $(x_0,y_0)$ and $(x_N,y_N)$ to the line segment passes through points of interpolation $(x_{i-1},y_{i-1})$ and $(x_i,y_i)$.
Let $\mathcal{K}$ be a space of nonempty compact subsets $\mathbb{R}^2$ with Hausdorff metric. Define the Hutchinson operator [@hutchinson_fractals] $$\Phi:\mathcal{K}\rightarrow\mathcal{K},\qquad \Phi(E)=\bigcup_{i=1}^NA_i(E).$$ It is easily seen [@barnsley_fractals_everywhere], that the Hutchinson operator $\Phi$ take a graph of any continuous function on a segment $[a,b]$ to a graph of a continuous function on the same segment. Thus, $\Phi$ can be treated as operator on the space of continuous functions $C[a,b]$.
For all $i=\overline{1,N}$ denote $$\label{E:fun}
\begin{array}{l}
u_i:[a,b]\rightarrow[x_{i-1},x_i],\quad u_i(x):=a_ix+e_i,\\
p_i:[a,b]\rightarrow\mathbb{R},\quad p_i(x):=c_ix+f_i.
\end{array}$$ Massopust [@Massop] has shown, that $\Phi$ acts on $C[a,b]$ according to the rule $$\label{E:operator}
(\Phi g)(x)=\sum_{i=1}^N\left((p_i\circ u_i^{-1})(x)+d_i(g\circ u_i^{-1})(x)\right)\chi_{[x_{i-1},x_i]}(x).$$ Moreover, if $|d_i|<1$ for all $i=\overline{1,N}$, then operator $\Phi$ is contractive on the Banach space $(C[a,b],\|\ \|_{\infty})$ with contractive constant $d\leq\max\{|d_i|\mid i=\overline{1,N}\}$. By the fixed-point theorem there exists unique function $g^\star\in C[a,b]$, such that $\Phi g^\star=g^\star$ and for all $g\in C[a,b]$ we have $$\lim_{n\to\infty}\|\Phi^n(g)-g^\star\|_\infty=0.$$ We will call $g^\star$ fractal interpolation function. It is clear, that if $g\in C[a,b]$, $g(x_0)=y_0$ and $g(x_N)=y_N$, then $\Phi(g)$ passes through points of interpolation. In this case we will call $\Phi^n(g)$ pre-fractal interpolation functions of order $n$.
Picture shows fractal interpolation function, which was constructed on points of interpolation $(0,0)$, $(0.5,0.5)$ è $(1,0)$ with parameters $d_1=d_2=0.5$.
![Fractal interpolation function.[]{data-label="F:ex1"}](ex1.eps)
Approximation {#S:approximation}
=============
From now on we assume, that $|d_i|<1$ for all $i=\overline{1,N}$. We try to approximate function $g\in C[a,b]$ by the fractal interpolation function $g^\star$, which is constructed on points of interpolation $\{(x_i,y_i)\}_{i=0}^N$. Thus, it is sufficient to fit parameters $d_i\in(-1,1)$ to minimize the distance between $g$ and $g^\star$.
We use methods that have been developed for fractal image compression [@barnsley_and_Hurd]. Notice, that from (\[E:operator\]), (\[E:fun\]) and (\[E:coeff\]) follows, that for all $g,h\in L^2[a,b]$ $$\begin{split}
\|\Phi g-\Phi h\|_2=\sqrt{\int_a^b(\Phi g-\Phi h)^2\,\mathrm{d}x}
=\sqrt{\sum_{i=1}^Nd_i^2\int_{x_{i-1}}^{x_i}(g\circ u_i^{-1}(x)-h\circ u_i^{-1}(x))^2\,\mathrm{d}x}\\
\leq \max_{i=\overline{1,N}}\{|d_i|\}\cdot\sqrt{\sum_{i=1}^N a_i\int_a^b(g-h)^2\,\mathrm{d}x}
=\max_{i=\overline{1,N}}\{|d_i|\}\cdot\|g-h\|_2.
\end{split}$$ Thus, $\Phi:L^2[a,b]\rightarrow L^2[a,b]$ is contractive operator with a fixed point $g^\star$.
Furthermore, instead of minimization of $\|g-g^\star\|_2$ we will minimize $\|g-\Phi g\|_2$, that makes the problem of optimization much easier. The collage theorem provides validity of such approach.
Let $(X, d)$ be a non-empty complete metric space. Let $T : X\to X$ be a contraction mapping on $X$ with contractivity factor $c<1$. Then for all $x\in X$ $$d(x,x^\star)\leq\frac{d(x,T(x))}{1-c}$$ where $x^\star$ is the fixed point of $T$.
$\blacktriangleright$ For all integer $n$ we have $$\begin{split}
d(x,x^\star)\leq d(x,T(x))+d(T(x),T^2(x))+\cdots +d(T^{n-1}(x),T^n(x))+d(T^n(x),x^\star)\\
\leq d(x,T(x))(1+c+c^2+\cdots+c^{n-1})+d(T^n(x),x^\star).
\end{split}$$ Letting $n\rightarrow\infty$ we establish the formula. $\blacktriangleleft$
Considering (\[E:coeff\]) and (\[E:fun\]), we rewrite (\[E:operator\]): $$\label{E:operator1}
(\Phi g)(x)=\sum_{i=1}^N\Big(\alpha_i(x)-d_i\big(\beta_i(x)-g\circ\gamma_i(x)\big)\Big)\chi_{[x_{i-1},x_i]}(x),$$ where $$\alpha_i(x)=\frac{(y_i-y_{i-1})x+(x_iy_{i-1}-x_{i-1}y_i)}{x_i-x_{i-1}},$$ $$\label{E:abg}
\beta_i(x)=\frac{(y_N-y_{0})x+(x_iy_{0}-x_{i-1}y_N)}{x_i-x_{i-1}},$$ $$\gamma_i(x)=\frac{(b-a)x+(x_ia-x_{i-1}b)}{x_i-x_{i-1}}.$$ Thus, we have to minimize functional $$(\|g-\Phi g\|_2)^2=\sum_{i=1}^N\int_{x_{i-1}}^{x_i}\Big(g(x)-\alpha_i(x)+d_i\big(\beta_i(x)-g\circ\gamma_i(x)\big)\Big)^2\,\mathrm{d}x.$$ Setting partial derivatives with respect to $d_i$ to zero we obtain $$\label{E:d}
d_i=\frac{\int_{x_{i-1}}^{x_i}\big(\alpha_i(x)-g(x)\big)\big(\beta_i(x)-g\circ\gamma_i(x)\big)\,\mathrm{d}x}
{\int_{x_{i-1}}^{x_i}\big(\beta_i(x)-g\circ\gamma_i(x)\big)^2\,\mathrm{d}x}, \qquad i=1,\ldots,N.$$
Discretization and results {#S:examples}
==========================
In this section we will approximate discrete data $Z=\{(z_m,w_m)\}_{m=1}^M$, $a=z_1<z_2<\cdots <z_M=b$ by the fractal interpolation function $g^\star$, which is constructed on points of interpolation $X=\{(x_i,y_i)\}_{i=0}^N$, $N\ll M$. Taking $X\subset
Z$, $(x_0,y_0)=(z_1,w_1)$ and $(x_N,y_N)=(z_M,w_M)$ we fit parameters $d_i\in(-1,1)$ to minimize $$\sum_{m=1}^M(w_m-g^\star(z_m))^2.$$
Let us approximate $Z$ by the piecewise constant function $g:[a,b]\to\mathbb{R}$. More precisely $g(z)=w_m$, where $(z_m,w_m)\in Z$ and $z_m$ is a nearest neighbor of $z$. From (\[E:d\]) we obtain the discrete formulas for $d_i$: $$\label{E:d_discrete}
d_i=\frac{\sum\limits_{z_m\in[x_i,x_{i+1}]}\big(\alpha_i(z_m)-w_m\big)\big(\beta_i(z_m)-g\circ\gamma_i(z_m)\big)}
{\sum\limits_{z_m\in[x_i,x_{i+1}]}\big(\beta_i(z_m)-g\circ\gamma_i(z_m)\big)^2\,}, \qquad i=1,\ldots,N-1.$$
After finding $d_i$ we obtain formulas for affine transformations $A_i$ and we are able to construct fractal interpolation function $g^\star$ for $g$.
Our aim is to compare fractal approximation with a piecewise quadratic approximation function which is based on the same discretization. On each segment $[x_{i-1},x_i]$ approximating function has the quadratic form $q_i(x)=k_i x^2+r_i x+l_i$. To get a continuous function we claim that $q_i(x_{i-1})=g(x_{i-1})$ and $q_{i}(x_{i})=g(x_{i})$. From this we find coefficients $k_i$ and $l_i$. To find free parameter $r_i$ we minimize functional $$\sum\limits_{z_m \in [x_{i-1},x_{i}]}(w_m-q_i(z_m))^2$$ with respect to $r_i$ on each segment $[x_{i-1},x_{i}],i=\overline{1,N}$. The approximating function $q(x)$ will have following form: $$q(x)=\left\{
\begin{array}{ll}
q_1(x)=k_1 x^2+r_1 x+l_1, & x\in [a=x_0,x_1]; \\
q_2(x)=k_2 x^2+r_2 x+l_2, & x\in [x_1,x_2]; \\
\qquad \vdots\\
q_N(x)=k_N x^2+r_N x+l_N, & x\in [x_{N-1},x_N=b].
\end{array}
\right.$$ Since there is one free parameter $r_i$ in each function $q_i(x)$ and one parameter $d_i$ for each affine transformation $A_i$ it makes the comparison correct.
To compare fractal and quadratic approximations we consider four types of data.
1. Polynomial function.
2. DNA sequence.
3. Price graph.
4. Random walking graph.
For all types of data $M=10000$, $z_m=m$, $[a,b]=[1,M]$, $\{w_m\}_{m=1}^M$ are normalized sequences, that is $E(\{w_m\})=0$ and $E(\{w_m^2\})=1$. For all cases we choose $(x_0,y_0)=(1,w_1)$, $(x_N,y_N)=(M,w_M)$ and other interpolation points $(x_i,y_i)$, $i=\overline{1,N-1}$ are local extremums of the given data.
Let $f(x)=-6x + 5x^2 + 5x^3 - 5x^4 + x^5,\, x\in [-1,2.5]$. As we work with the segment $[1,M]$ we map $[-1,2.5]$ to it. Consider sequence $v_m=f\left(\frac{7(m-1)}{2(M-1)}-1\right)$, $m=\overline{1,M}$. Set $w_m=(v_m-s_1)/s_2$, where $s_1$ and $s_2$ are mean and deviation of $\{v_m\}_{m=1}^M$. Figure \[F:FuncGraph\] shows the normalized sequence $\{w_m\}$.
![The graph of original function $g$.[]{data-label="F:FuncGraph"}](FuncGraph.eps)
Choose five interpolation points $x_0=1$, $x_1=500$, $x_2=4000$, $x_3=7500$, $x_4=10000$. Applying (\[E:d\_discrete\]) we obtain $d_1=0.066$, $d_2=0.155$, $d_3=0.033$, $d_4=0.096$. The small values of $|d_i|$ mean that on segments $[x_{i-1},x_i]$ fractal approximation function looks as a straight line. Figure \[F:FuncInterp\] shows the graphs of fractal and quadratic approximating functions.
A DNA sequence can be identified with a word over an alphabet $\mathcal{N}=\{A,C,G,T\}$. Here we have the sequence of 10000 nucleotides of Edwardsiella tarda. The graph represented by the formula $$v_1=0, \, v_m=v_{m-1}+\left\{
\begin{array}{ll}
+1, & \mbox{if} \,\, m^{th} \mbox{nucleotide belongs to (A,G)};\\
-1, & \mbox{if} \,\, m^{th} \mbox{nucleotide belongs to (C,T)}.
\end{array}
\right.$$ For full description of representation of DNA primary sequences see [@prime_sequence]. Figure \[F:DNAGraph\] shows the sequence $\{w_m\}$ after normalization of $\{v_m\}$ according to the formula in the previous example.
![Picture shows DNA Graph of 10000 nucleotides of Edwardsiella tarda.[]{data-label="F:DNAGraph"}](DNAGraph.eps)
Interpolation points are $x_0=1$, $x_1=1000$, $x_2=2500$, $x_3=3000$, $x_4=3500$,$x_5=5000$, $x_6=6500$, $x_7=7000$, $x_8=8000$, $x_9=9000$, $x_{10}=10000$. Applying (\[E:d\_discrete\]) we obtain $d_1=-0.001$, $d_2=0.274$, $d_3=0.31$, $d_4=0.24$, $d_5=-0.057$, $d_6=0.211$, $d_7=-0.42$, $d_8=-0.121$, $d_9=0.215$, $d_{10}=0.158$. Figure \[F:DNAInterp\] shows the graphs of fractal and quadratic approximating functions.
We take price wave of 10000 prices $v_m,\,m=\overline{1,M}$ of one day period for EUR/USD, then normalize it (Figure \[F:PriceGraph\]).
![Picture shows Price Graph for EUR/USD.[]{data-label="F:PriceGraph"}](PriceGraph.eps)
Interpolation points are $x_0=0$, $x_1=500$, $x_2=1500$, $x_3=2000$, $x_4=2500$,$x_5=3000$,$x_6=4000$, $x_7=5000$, $x_8=6000$, $x_9=8000$, $x_{10}=10000$. Applying (\[E:d\_discrete\]) we obtain $d_1=-0.334$, $d_2=-0.004$, $d_3=0.315$, $d_4=0.307$, $d_5=0.333$, $d_6=-0.28$, $d_7=-0.067$, $d_8=0.027$, $d_9=0.047$, $d_{10}=-0.33$. Figure \[F:PriceInterp\] shows the graphs of fractal and quadratic approximating functions.
Picture shows Random Walking Graph. It represented by the formula $v_0=0, \, v_i=v_{i-1}+\xi_i$, where $\xi_i$ is a random value with normal distribution.
![Normalized Random Walking graph.[]{data-label="F:RandomGraph"}](RandomWalkingGraph.eps)
Interpolation points are $x_0=0$, $x_1=1500$, $x_2=2000$, $x_3=3000$, $x_4=4000$, $x_5=5500$, $x_6=6300$, $x_7=7600$, $x_8=8000$, $x_9=9000$, $x_{10}=10000$. Applying (\[E:d\_discrete\]) we obtain $d_1=-0.237$, $d_2=0.14$, $d_3=-0.020$, $d_4=-0.105$, $d_5=0.105$, $d_6=0.0545$, $d_7=-0.184$, $d_8=-0.368$, $d_9=0.081$, $d_{10}=-0.111$. Figure \[F:RandomWalkingInterp\] shows the graphs of fractal and quadratic approximating functions.
To compare the results we calculate approximation errors for each type of data. Let $h(x)$ be the approximating function for data $\{w_m\}_{m=1}^M$. Then approximation error is $$\sqrt{\sum\limits_{m=1}^M \frac{(h(x_m)-w_m))^2}{M}}.$$
Here we represent the table of approximation errors for each type
$$\begin{array}{ccc}
\quad & Fractal\,& Quadratic\,\\
Polynomial Function & 0.0359037 & 0.0245094 \\
DNA\, Primary\, Sequence & 0.0692072 & 0.0624714 \\
Price\, Graph & 0.0501345 & 0.0533686 \\
Random\, Walking & 0.1015339 & 0.101438
\end{array}$$
From it we see, that fractal approximation is better for price graph and nearly equal for random walking, but much worse for smooth function and slightly for DNA sequence. Different results were appearing during calculations of errors. We assume that some conditions could give us more exact approximation results from fractal interpolation function and for that extra observations should be established.
[6]{}
C. Bandt, A. Kravchenko. . , 24(10):2717–2728, 2011.
M. F. Barnsley. . Academic Press Inc., MA, 1988.
M. F. Barnsley and L. P. Hurd. . Wellesley, MA:AK Peters, 1993.
Feng-lan Bai, Ying-zhao Liu, Tian-ming Wang. .
J. Hutchinson. , 30:713–747, 1981.
P. Massopust. Oxford University Press, Oxford, 2010.
H. E. Stanley, S. V. Buldyrev, A. L. Goldbergerb, J. M. Hausdorff, S. Havlin, J. Mietusb, C. K. Peng, F. Sciortino and M. Simons. .
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present microscopic derivations of the one-dimensional low-energy boson effective Hamiltonians of quantum wire and quantum Hall bar systems. The quantum Hall system is distinguished by its spatial separation of oppositely directed electrons. We discuss qualitative differences in the plasmon collective mode dispersions and the ground state correlation functions of the two systems which are consequences of this difference. The slowly-decaying quasi-solid correlations expected in a quantum wire are strongly suppressed in quantum Hall bar systems.'
address: ' Department of Physics, Indiana University, Bloomington, Indiana 47405, U.S.A.'
author:
- 'U. Zülicke and A. H. MacDonald'
title: Plasmon Modes and Correlation Functions in Quantum Wires and Hall Bars
---
Introduction
============
The quantum Hall effect occurs in two-dimensional (2D) electron systems (ES) when the chemical potential lies in a charge gap which occurs at a density ($n^{*}$) which is dependent on magnetic field ($B$). The $B$ dependence of $n^{*}$ requires[@ahm:braz:96] gapless excitations localized at the edge of the 2D ES. The low-energy effective Hamiltonian which describes this edge system is simplest when the edge is sharp[@smoothedge] on a microscopic length scale and the bulk Landau level filling factor $\nu^{*} = 2 \pi \ell^2
n^{*} =1/m$ with $m$ being an odd integer. In this case, the low-energy edge excitations can be mapped to those of a one-dimensional (1D) fermion system[@bih:prb:82; @ahm:prl:90] and described[@wen:prb:90; @wen:int:92] by a version of the Tomonaga-Luttinger (TL) model for 1D fermion systems[@emery; @sol:adv:79; @fdmh:jpc:81] modified to account for the magnetic field and the long-range of the electron-electron interaction. Edge excitations in quantum Hall (QH) bars are analogous to the excitations of electron systems in quantum wires[@somerecentrefs] which are also described at low energies by a TL model modified to account for long-range interactions. Close relationships exist between studies of the effect of Coulomb interactions on the transport properties of QH systems[@oreg:prl:95; @newmoon] and analogous studies of quantum wires.[@fab:prl:94; @gia:prb:95] There are, however, important distinctions between these two systems which result from the spatial separation of oppositely directed electrons in the QH case and are the subject of this paper. As we explain below, the energy-wavevector relationship for the plasmon boson states of quantum wires and QH bars have quite different microscopic underpinnings. In addition, the strong quasi-solid correlations expected[@hjs:prl:93] in a quantum wire are suppressed in typical Hall bar systems.[@brey:95]
Non-Interacting Electrons
=========================
The analogy between quantum wires and quantum Hall bars is most direct for $\nu=1$ and we begin by discussing this case, first for non-interacting electrons. To form a quantum wire, electrons in a 2D ES are confined in an additional direction, say the $\hat y$-direction, while the motion in the remaining ($\hat x$-)direction stays free. With periodic boundary conditions applied over a length $L$ in the $\hat x$-direction, the electronic single-particle wave functions then have the form $$\label{wavefunct}
\psi^{\mbox{\tiny QW}}_{n, k} (x, y) = \frac{1}{\sqrt{L}} \,
e^{i k x} \,\, \chi^{\mbox{\tiny QW}}_{n}(y)$$ where $\chi^{\mbox{\tiny QW}}_{n}(y)$ is the $n$-th discrete subband state for the quantum wire. The single-particle eigenenergy $\varepsilon_{n,k} = E_n + \hbar^2 k^2 / 2 m^{*}$. For wires widths comparable to or smaller than the typical distance between electrons, the energy spacing between different subbands will be larger than the Fermi energy and only the lowest ($n = 0$) subband will be occupied in the ground and low-energy excited states. The ground state of the non-interacting many-electron system is a 1D Fermi gas state in which all single-particle states with $n = 0$ and $|k| \le k_F$ are occupied. The subband wavefunction leads only to form factors which will modify the electron-electron interaction in the system at short distances. The situation for a 2D ES in a strong perpendicular magnetic field is similar. For a Hall bar geometry where the system is confined in the $\hat y$-direction and periodic boundary conditions are applied in the $\hat x$-direction, the Landau gauge \[$\vec A = (-By, 0, 0)$\] wavefunctions have the form[@caveatsf] $$\psi^{\mbox{\tiny HB}}_{n, k} (x, y) = \frac{1}{\sqrt{L}} \, e^{i k x}
\,\, \chi^{\mbox{\tiny HB}}_{n}(y - \ell^2 k)\mbox{ .}$$ Here $\ell:=[\hbar c/(e B)]^{1/2}$ is the magnetic length, $n$ is the Landau level index, and $\chi^{\mbox{\tiny HB}}_{n}(y)$ is the wave function of a harmonic oscillator with frequency $\omega_c = e B/(
m^* c)$ which is localized to a length $\sim \ell$ around $y=0$. The oscillator wavefunctions play the role of the subband wavefunction in a quantum wire, but in the Hall bar case they are displaced from the origin by a distance proportional to wave vector $k$. In addition, the dependence of the single-particle eigenenergy, $\varepsilon_{n,k}
= \hbar \omega_c (n+1/2) + V^{\mbox{\tiny ext}}(\ell^2 k)$, on $k$ is due to the confinement potential $V^{\mbox{\tiny ext}}(y)$ rather than to the kinetic energy. In the strong magnetic field limit, only $n=0$ states will be occupied at $\nu =1$, even when the width of the Hall bar is macroscopic, and states at the Fermi energy with $k = \pm
k_F$ will be localized at opposite edges of the sample, as illustrated in Fig. \[wirebar\]. This property of Hall bar systems plays the central role role in the edge state picture of the integer quantum Hall effect for non-interacting electrons.[@bih:prb:82; @butt:prb:88] The sample width $W = 2 k_F
\ell^2$ is assumed to be much larger than the magnetic length $\ell$ throughout this paper. When this condition is not satisfied the distinction between a quantum wire and a quantum Hall bar blurs.
Coulomb Matrix Elements and Low-Energy Hamiltonian
==================================================
Quantum wire and QH bar systems are described by microscopic Hamiltonians of the same form
\[microscop\] $$\begin{aligned}
H &=& H^{0} + H^{\mbox{\tiny int}} \\
H^{0} &=& \sum_{k} \, \hbar (|k|-k_F)\, v_{\mbox{\tiny F}} \,
c_{k}^\dagger c_k \\ \label{interact}
H^{\mbox{\tiny int}} &=& \frac{1}{2L} \sum_{k, p, q} V(k, p, q) \,\,
c_{k+q}^\dagger c_{p}^\dagger c_{p+q} c_k\end{aligned}$$
where $H^{0}$ is the one-body term in the Hamiltonian and the single-particle energy has been linearized around $k = \pm k_F$ so that $v_{\mbox{\tiny F}} = \hbar k_F / m^{*}$ in the quantum wire case while
3.5in
$$\label{fermvel}
v_{\mbox{\tiny F}} = \frac{\ell^2}{\hbar} \left. \frac{d
V^{\mbox{\tiny ext}}}{d y} \right|_{W/2}$$
in the Hall bar case. The $y$-dependent part of the wavefunctions enters crucially into the form of the matrix element $V(k, p, q)$. For quantum wires,[@dassarma; @gold:prb:90] the function $\chi^{\mbox{\tiny QW}}_{n}(y)$ does not depend on $k$, and the interaction matrix element is a function of momentum transfer $q$ only. Up to an irrelevant constant, it is then possible to rewrite Eqs. (\[microscop\]) as $$\label{1Dhamilt}
H_{\mbox{\tiny eff}}^{\mbox{\tiny QW}} = H^{0} + \frac{1}{2 L}
\sum_{q \ne 0} V^{\mbox{\tiny QW}}_q \varrho_q \varrho_{-q} \mbox{ .}$$ Here $\varrho_q = \sum_k c_{k+q}^{\dagger} c_k$ is the 1D Fourier transform of the density operator. For Coulomb interaction, the effective 1D potential at small $q$ is[@dassarma; @gold:prb:90] $$V^{\mbox{\tiny QW}}_q = \frac{e^2}{\epsilon } (-2) \ln{\left[\alpha
\, q\, W \right]}$$ where $W$ is the width of the quantum wire and $\alpha$ is a constant of order unity which depends on the details of the confining potential. In the QH bar, however, single-particle states with different relative momentum are separated in the $\hat y$-direction. (See Fig. \[wirebar\].) Consequently, the matrix element $V(k, p, q)$ depends on both $q$ and $k-p$. If the originally 2D interaction is $U(\vec r)$ and has the 2D Fourier transform $U(\vec q) = U(q_x, q_y)$, we find that $$\label{intmatel}
V(k, p, q) = \frac{e^{-\frac{1}{2}(q\ell)^2}}{\ell }\,\,
\int_{-\infty}^{\infty} d\kappa \,\, e^{-\frac{1}{2} \kappa^2}\,
U(q, \kappa \ell^{-1})\, e^{i\kappa (k-p) \ell}.$$ For the physically relevant Coulomb interaction in the limit of small momentum transfer $q\le\ell^{-1}$ we obtain (see also Fig. \[coulmat\]) $$\label{coulomb}
V(k, p, q) = \frac{e^2}{\epsilon } \left\{
\begin{tabular}{ll}
$2\,\mbox{K}_0(|q (k-p) \ell^2|)$ & $\mbox{for } |k-p| > \ell^{-1}$\\
$-2\ln{\left[\sqrt{\frac{\gamma}{8}} \, q\ell\right]}$ & $\mbox{for }
|k-p| \le \ell^{-1}$
\end{tabular}
\right.$$ where $\epsilon$ is the host semiconductor dielectric constant, and $\gamma \sim 1.78$ is the exponential of Euler’s constant. Corrections to Eq. (\[coulomb\]) are analytic in $q$, in $|k-p|^{-1}$ for large $|k-p|$, and in $|k-p|$ for small $|k-p|$ and are negligible for our purposes. Similar expressions for two-particle Coulomb matrix elements in a Hall bar have been reported previously.[@lee:prb:90; @wen:prb:91a; @bih:prl:93; @wen:prb:94] Here we want to use this expression to derive a 1D effective Hamiltonian similar to Eq. (\[1Dhamilt\]), to describe the low-energy excitations of a Hall bar. It is useful to separate the $q=0$ term in the Hamiltonian which, in contrast to the quantum wire case, is not a constant. Defining $\tilde{V}(k-p) := V(k, p, 0)$ we find that $$H^{\mbox{\tiny HB}} = H^{0} + H^{q=0} + H^{\mbox{\tiny TL}}$$ where
$$\begin{aligned}
H^{q=0} &=& \frac{1}{2 L}\sum_{k, p} \tilde{V}(k-p)c_{k}^\dagger
c_{p}^\dagger c_p c_k \\ &\simeq& \frac{1}{L} \sum_{k}\left[
\sum_{p=-k_F}^{k_F} \tilde{V}(k - p) \right]\, c_{k}^\dagger c_k\end{aligned}$$
The last line holds in the low-energy sector of the Hamiltonian where number operator fluctuations are negligible except for single-particle states close to the edge of the system which do not contribute importantly to the sum. $H^{q=0}$ simply adds the electrostatic (Hartree) contribution to the single-particle energies which would be present in a Hartree self-consistent field theory. This contribution is an irrelevant constant for a quantum wire but is state dependent in a Hall bar. The Hartree energy is positive and smaller in magnitude for states with $|k| > k_F$ since they are localized farther from the other electrons. Linearizing the Hartree single-particle energy we find that for low energies and $W \gg \ell$ $$\label{elstatic}
H^{q=0}_{\mbox{\tiny eff}} = \sum_{k} \left(-\frac{e^2}{\epsilon \pi}
\right) \, (|k| - k_F) \, \ln{\left[\sqrt{2\gamma} \frac{W}{\ell}
\right]} c_{k}^{\dagger} c_k \mbox{ .}$$
The low-energy physics of a 1D ES can be captured in an approximation where the Hamiltonian is projected onto sectors where the number of right going fermions $\big( N_R := \sum_{k>0} \big< c_k^\dagger c_k
\big> \big)$ and the number of left going fermions $\big( N_L :=
\sum_{k<0} \big< c^\dagger_kc_k \big> \big)$ are fixed. As is well known from studies of TL models[@emery; @sol:adv:79; @fdmh:jpc:81] for 1D ES, this projection permits one-body terms linearized around the Fermi points to be expressed in terms of density operators $\varrho^{\mbox{\tiny R,L}}_q := \sum_{k > 0 \atop k < 0}
c^\dagger_{k + q} c_k$. $\big(\varrho_q = \varrho^{\mbox{\tiny L}}_q
+ \varrho^{\mbox{\tiny R}}_q \big)$ Using this procedure, $H^{q=0}_{\mbox{\tiny eff}}$ can be lumped with the $q \ne 0$ interaction terms in the Hamiltonian to obtain $$\begin{aligned}
\label{1Deffhamilt}
H^{\mbox{\tiny HB}}_{\mbox{\tiny eff}} &=& H^{0} + \frac{1}{L}
\sum_{q>0} \left\{ V^{\mbox{\tiny intra}}_q \left[
\varrho^{\mbox{\tiny L}}_q \varrho^{\mbox{\tiny L}}_{-q} +
\varrho^{\mbox{\tiny R}}_q \varrho^{\mbox{\tiny R}}_{-q} \right]
\right. \nonumber \\ && \hspace{2cm} + \left.
V^{\mbox{\tiny inter}}_q \left[ \varrho^{\mbox{\tiny L}}_q
\varrho^{\mbox{\tiny R}}_{-q} + \varrho^{\mbox{\tiny R}}_q
\varrho^{\mbox{\tiny L}}_{-q} \right] \right\}\end{aligned}$$ where the effective 1D intra-edge and inter-edge interactions are $$V^{\mbox{\tiny intra}}_q := \frac{e^2}{\epsilon } (-2) \ln{\left[
\frac{\gamma}{2} \, q\, W \right]} \mbox{ and }
V^{\mbox{\tiny inter}}_q := \frac{e^2}{\epsilon } 2 \mbox{K}_0(q W)
\mbox{ .}$$ Microscopically, the $q \ne 0$ terms in the interaction Hamiltonian represent the loss of exchange energy when a density wave is created in the system. To obtain this result we have used the fact that the dependence of $V(k,p,q)$ on $k$ and $p$ is negligible at small $q$ when $k$ and $p$ are near the same Fermi point and appealed to linearization in setting $|k-p| = W/\ell^2$ when $k$ and $p$ are near opposite Fermi points.
Previous studies addressing the effect of Coulomb interaction in QH systems have used a Hamiltonian of the form shown in Eq. (\[1Deffhamilt\]) as starting point. In this work the Hamiltonian implicitly or explicitly contains two undetermined parameters: the bare Fermi velocity $v_{\mbox{\tiny F}}$ appearing in $H_0$ and a cut-off length, generally assumed to be microscopic, appearing in the expression for $V^{\mbox{\tiny intra}}_q$. In our microscopic analysis, the length appearing in $V^{\mbox{\tiny intra}}_q$ is the macroscopic sample width $W$ and $v_{\mbox{\tiny F}}$, defined by Eq. (\[fermvel\]), is dependent on the external potential. However, it is important to realize that the projection onto fixed $N_R$ and $N_L$ sectors obviates the distinction we have made between one-body and two-body terms in the underlying microscopic Hamiltonian. The identification of a bare Fermi velocity associated with the one-body term plays [*no role*]{} in the physics. In our analysis, the intra-edge interaction represents the sum of terms originating from the one-body Hartree energy which leads to an attractive effective interaction and the $q \ne 0 $ terms in the intra-edge interaction which are repulsive. We could as well have grouped the Hartree term with the one-body term in the Hamiltonian. With this choice, the one-body term would vanish if the external potential originated from a positively-charged background which precisely cancelled the ground state electron charge density. In typical experimental situations the external potential which attracts electrons to the Hall bar is weaker near the edge than in the neutralizing background model frequently used in theoretical model calculations so that the Fermi velocity is negative when the Hartree term is grouped with the one-body terms. The negative Fermi velocity needn’t have any physical consequences, however, since in the case of interest the $q \ne 0$ terms in the Hamiltonian stabilize all excitations.
3.7in
For smoother edges this Fermi velocity becomes more and more negative and eventually the system will become unstable to edge reconstructions,[@wen:prb:94; @ahm:aust:93; @edgerecon] signaling a phase transition to a state with a different and more complicated low-energy effective Hamiltonian. However, this instability involves ‘ultraviolet’ physics which is beyond the scope of the present study.
We see from the expression for $V^{\mbox{\tiny inter}}_q$ that the inter-edge interaction is important only if $q W < 1$, because the modified Bessel function K$_0(x)$ decays rapidly for $x > 1$. Physically, the Coulomb potential due to a 1D density wave is weak when viewed from a point removed from the 1D system by a distance longer than the period of the wave. For small $x$, K$_0(x) \to -
\ln{\left[\frac{\gamma}{2} x \right]}$ so that in the limit $q \ll
W^{-1}$, we have $V^{\mbox{\tiny intra}}_q = V^{\mbox{\tiny inter}}_q
= V^{\mbox{\tiny QW}}_q$. The contribution to the Hamiltonian of a quantum Hall bar from modes with a wavelength exceeding $W$ is identical to the corresponding contribution to the effective 1D Hamiltonian for the quantum wire.
Bosonization
============
TL models described by a Hamiltonian displayed in Eq. (\[1Deffhamilt\]) can be solved by means of bosonization.[@emery; @sol:adv:79; @fdmh:jpc:81] This is possible because within the restricted Hilbert space of low-energy, small-$q$ excitations around a uniform ground state, the density operators $\varrho^{\mbox{\tiny L,R}}_q$ obey simple bosonic commutation relations. If $\nu$ denotes the occupation number of single-particle states in the uniform many-particle ground state around which the excitations occur, the commutation relations are
\[anomaly\] $$\begin{aligned}
\left[ \varrho^{\mbox{\tiny L}}_q, \varrho^{\mbox{\tiny L}}_{q'}
\right] &=& \nu \, \frac{q L}{2 \pi} \, \delta_{q + q' , 0} \\
\left[ \varrho^{\mbox{\tiny R}}_q, \varrho^{\mbox{\tiny R}}_{q'}
\right] &=& - \nu \, \frac{q L}{2 \pi} \, \delta_{q + q' , 0} \\
\left[ \varrho^{\mbox{\tiny L}}_q, \varrho^{\mbox{\tiny R}}_{q'}
\right] &=& 0\end{aligned}$$
So far, we have only considered the case of $\nu = 1$, which is the generic case for a quantum wire. Below we identify $\nu$ as the Landau level filling factor of the QH bar and comment on the validity of the Luttinger liquid description of QH edges[@wen:prb:90] at $\nu < 1$.
In order to understand the intriguing features which are special to the physics of a 1D ES, it has proven to be useful[@fdmh:jpc:81] to introduce two new bosonic fields $\theta(x)$ and $\phi(x)$, called [*phase fields*]{}. In what follows, we assume that $N_R$ and $N_L$ are fixed. It turns out that the Hamiltonian (\[1Deffhamilt\]) can then be rewritten (uniquely up to a constant) as a quadratic Hamiltonian in these phase fields. Here, we have an additional motivation for following this procedure; the generalization of the TL picture to fractional filling factors $\nu$ is straightforward once the phase fields are introduced. The theory can be formulated in the phase field formalism for an arbitrary filling factor, and the value of $\nu$ enters the theory only when calculating physical observables like densities and currents. Before going into algebraic details, we comment on the justification of the TL model for fractional QH edges. (For a related discussion see Ref. .) The explicit derivation for a quantum Hall bar outlined in the previous section does not generalize to the $\nu < 1$ case. However, using arguments based on the analytical structure of many-body wave functions that describe the 2D ES in the fractional QH regime, it can be shown that for abrupt edges the one-to-one correspondence between the low-lying excitations in this system and the excitations of a 1D boson system holds[@ahm:prl:90; @ahm:braz:96] for $\nu =1/m$ where $m$ is an odd integer. The form of the theory is essentially fixed[@ahm:braz:96] by this bosonization property and the requirement that the theory recover the fractional quantum Hall effect under appropriate circumstances. Therefore we believe that, for abrupt edges, the low-energy Hamiltonian for a QH bar can be written in the form of Eq. (\[1Deffhamilt\]) and that the commutation relations Eqs. (\[anomaly\]) are valid for the case of $\nu<1$ also. An exception occurs for large $m$ when the ground state of the 2D ES is expected[@wigcrys] to be a Wigner crystal.
The relation of the phase fields to the densities of left going and right going fermions is given in reciprocal space
$$\begin{aligned}
\theta_q &=& \sqrt{\frac{\pi}{\nu}} \frac{i}{q} \, \left[
\varrho^{\mbox{\tiny R}}_q + \varrho^{\mbox{\tiny L}}_q \right] \\
\phi_q &=& \sqrt{\frac{\pi}{\nu}} \frac{i}{q} \, \left[
\varrho^{\mbox{\tiny R}}_q - \varrho^{\mbox{\tiny L}}_q \right]\end{aligned}$$
We use reciprocal space language for the following discussion because it is convenient for dealing with anomalies caused by the long-range of the electron-electron interaction. The Hamiltonian (\[1Deffhamilt\]) is quadratic in the phase fields: $$\label{phaseham}
H = \frac{1}{2 L} \sum_{q} \, |q| \, E_q \left\{ \frac{1}{g_q}
| \theta_q |^2 + g_q \, | \phi_q |^2 \right\}.$$ Here $E_q$ is the energy of the low-lying bosonic excitations of the system, generally referred to as plasmons in the quantum wire case and as edge magnetoplasmons in a QH bar. The dispersion relation $E_q$ for the plasmon excitations reads $$\label{disper}
E_q = |q| \, \left[ \hbar v_{\mbox{\tiny F}} + \frac{\nu }{2 \pi}
V^{\mbox{\tiny intra}}_q \right] \sqrt{ (1 + \xi_q) (1 - \xi_q) }$$ and the interaction parameter $g_q$ is defined as $$\label{gfactor}
g_q = \sqrt{ \frac{1 - \xi_q}{1 + \xi_q}}\mbox{ .}$$ The parameter $\xi_q$ measures the relative strengths of inter- and intra-edge contributions to the plasmon energy, and its formal expression is $$\label{interintra}
\xi_q := \frac{V^{\mbox{\tiny inter}}_q}{V^{\mbox{\tiny intra}}_q +
2 \pi \hbar v_{\mbox{\tiny F}} / \nu} \mbox{ .}$$ Obviously, $\xi_q = 0$ in the absence of inter-edge interaction. For larger $v_{\mbox{\tiny F}}$, [*i.e. *]{} sharper confining potential, $\xi_q$ is smaller, and we expect corrections due to $V^{\mbox{\tiny inter}}_q$ to be less important. Expressions (\[disper\]) and (\[interintra\]) reflect the interchangeability of one-body and two-body contributions to the plasmon dispersion and $V^{\mbox{\tiny intra}}_q$. Note that in samples with aspect ratios close to unity the value of $g_q$ is close to $1$ even at the smallest physically-relevant values of $q \sim L^{-1}$. Using these results we can now compare the physical properties of excitations in quantum wire and QH bar systems.
Plasmon Dispersion
------------------
In order to do detailed calculations, we must specify the confining potential of the QH bar. In what follows, we adopt the neutralizing background model which produces sharp confinement and diminishes corrections due to the inter-edge interaction as explained above. To calculate the Fermi velocity resulting from a uniform neutralizing background, we can use Eq. (\[elstatic\]) apart from a sign change since the Hartree energy of the Fermi sea is simply the electrostatic potential due to the electron charge density in the ground state. With the fractional filling factor incorporated properly, we find that $$v_{\mbox{\tiny F}} = \nu \, \frac{e^2}{\hbar \epsilon \pi}\,
\ln{\left[ \sqrt{2\gamma} \frac{W}{\ell} \right]} \mbox{ .}$$ For a macroscopic QH sample, $W / \ell$ is rather large (hence $v_{\mbox{\tiny F}}$ is big), and $W \sim L$. Therefore, the bare Fermi velocity in this system is larger than the renormalization terms arising from interaction effects ($V^{\mbox{\tiny intra}}_q$ and $V^{\mbox{\tiny inter}}_q$). It is interesting to compare the $W \sim L$ limit with the case of a perfectly circular quantum dot.[@ahm:aust:93; @pit:prl:94; @bla:int:95] In that case it follows purely from symmetry arguments that in the long-wavelength limit the plasmon energy is determined completely by the external potential term in the Hamiltonian. We expect this to be approximately true for all samples with aspect ratios close to one. The plasmon dispersion in this limit is
quantum wire QH bar
------------------------------ -------------- --------
kinetic energy $+$ $0$
external confining potential $0$ $+$
Hartree energy $0$ $-$
exchange/correlation energy $+$ $+$
: Contributions to the plasmon dispersion in quantum wire and quantum-Hall (QH) bar systems. This Table is based on the common separation of energies in electronic systems into kinetic, Hartree, external potential, and exchange-correlation contributions. Table entries indicate whether the energy mentioned in the left column gives a positive ($+$), negative ($-$), or zero ($0$) contribution to the energy of long wavelength plasmons.
\[compare\]
$$\label{macrodisperse}
E_q = - \nu \, \frac{e^2}{\epsilon \pi} \, |q| \, \ln{\left[
\sqrt{\gamma /8} \,\, |q| \ell \right]}\mbox{ .}$$
Expressions of the form of Eq. (\[macrodisperse\]) for the edge magnetoplasmon dispersion have been successfully applied to interpret experimental data[@hls:prb:83; @tal:jetp:86; @heit:prl:90; @wass:prb:90; @heit:prl:91; @ray:prb:92; @zhi:prl:93; @hel1:85; @hel2:85; @hel:91] and were originally derived classically[@vav-sam:jetp:88; @bla:phyb:92]. That Eq. (\[macrodisperse\]) is the correct result for experimentally-realistic QH samples is one of the main points of our discussion here. This result should be contrasted with the plasmon dispersion relation at long wavelengths in a quantum wire where (see also Ref. ) it is found[@dassarma] that $E_q
\sim |q| \sqrt{|\ln [|q|]|}$. More generally the underlying microscopic terms in the Hamiltonian differ qualitatively in their influence on the magnetoplasmon dispersion in the two cases as summarized in Table \[compare\] which is based on the common separation of energies into kinetic, Hartree, external potential, and exchange-correlation contributions.
Correlation Functions
---------------------
The phase field formalism facilitates the straightforward calculation of electronic correlation functions. A hallmark of the bosonization approach in the study of 1D systems[@emery; @sol:adv:79; @fdmh:jpc:81] is the possibility to express the electron field operator $\psi$ in terms of the phase fields. Electron Green’s functions can then be written in terms of Green’s functions of the phase fields, which are readily calculated because the Hamiltonian (\[phaseham\]) is quadratic.
In truly 1D systems, the electron field operator depends on [*one*]{} spatial coordinate ($x$) only. Incorporating the 2D aspect of a quasi-1D ES and, in particular, a QH bar, we have to consider its dependence on the lateral coordinate ($y$) as well: $$\label{electron}
\psi_{\mbox{\tiny L,R}}(x, y) = \Phi_{\mbox{\tiny L,R}}(x, y) \exp{
\left[ \pm i \sqrt{\frac{\pi}{\nu}} \theta(x) + i
\sqrt{\frac{\pi}{\nu}} \phi(x) \right] }$$ where $\Phi^{\mbox{\tiny QW}}_{\mbox{\tiny L,R}}(x, y) :=
\psi^{\mbox{\tiny QW}}_{0, \mp k_F} (x, y)$ for the quantum wire and $\Phi^{\mbox{\tiny HB}}_{\mbox{\tiny L,R}}(x, y) :=
\psi^{\mbox{\tiny HB}}_{0, \mp k_F} (x, y)$ for the QH bar. The operator for the total density of electrons at some position along a quantum Hall bar can be obtained by integrating over the transverse ($y$) coordinate: $$\label{totaldensity}
\varrho(x) = \int dy \left[ \psi_{\mbox{\tiny L}}(x,y)
+\psi_{\mbox{\tiny R}}(x,y)\right]^{\dagger} \left[
\psi_{\mbox{\tiny L}}(x,y) +\psi_{\mbox{\tiny R}}(x,y)\right]\mbox{ .}$$ In terms of the phase fields we find that
$$\label{osccharge}
\varrho(x) = \sqrt{\frac{\nu}{\pi}} \partial_x \theta(x) +
\hat O_{\mbox{\tiny CDW}} (x)$$
where $\hat O_{\mbox{\tiny CDW}}$ represents the portion of the charge density which oscillates with period $\pi / k_F $. The slow decay of correlations associated with this part of the charge density[@hjs:prl:93] is the basis of the quasi-crystalline character of the electrons in a quantum wire. The spatial separation of oppositely directed electrons in quantum Hall bars is not important in the first term of Eq. (\[osccharge\]) because the $y$-dependent part of $\Phi_{\mbox{\tiny L,R}}(x, y)$ is normalized. However in the $2 k_F$ term, the overlap of $\chi_0(y)$’s at $\pm k_F$ enters and the spatial separation changes the result. We find that $$\hat O_{\mbox{\tiny CDW}}(x) = \frac{1}{\pi \ell} \exp{\left[ \left(
-\frac{W}{2\ell} \right)^2 \right]} \cos{\left[2 k_F x + 2
\sqrt{\frac{\pi}{\nu}} \theta(x) \right]} \mbox{ .}$$
The quantum wire case can be recovered by setting $W =0$. In the Hall bar case $\hat O_{\mbox{\tiny CDW}}$ acquires the prefactor $\exp{[
- (W/\ell)^2 ]}$ which, for realistic sample dimensions ($W/\ell \sim
10\dots 100$), is extremely small.
Using the bosonization technique, we find for the CDW correlations $$\begin{aligned}
&&\left< \hat O_{\mbox{\tiny CDW}}(x) \, \hat O_{\mbox{\tiny CDW}}(0)
\right> = \frac{1}{2\pi^2 \ell^2} \exp{\left[-\left(W/\ell\right)^2
\right]} \cos{(2 k_F x)} \nonumber \\ && \hspace{2.5cm} \times \exp{
\left\{ -\frac{2\pi}{\nu} \left< \left[ \theta(x) - \theta(0)
\right]^2 \right> \right\} } \mbox{ .}\end{aligned}$$ The phase field correlation function is determined by the interaction parameter $g_q$ which reflects the non-Fermi-liquid properties of the system[@emery; @sol:adv:79; @fdmh:jpc:81] resulting from inter-edge interaction. A standard calculation[@hjs:prl:93] gives the following result $$\left< \left[ \theta(x) - \theta(0) \right]^2 \right> =
\frac{2}{L}\sum_{q > 0}\,\, g_q\,\frac{1 - \cos{(q x)}}{q} \mbox{ .}$$ For the case of the Coulomb interaction we find[@hjs:prl:93] (for $x\geq W$) that $$\begin{aligned}
\label{cdwcorr}
&& \left< \hat O_{\mbox{\tiny CDW}}(x) \, \hat O_{\mbox{\tiny CDW}}
(0) \right> = \frac{1}{2\pi^2 \ell^2} \exp{\left[-\left(W/\ell
\right)^2\right]} \cos{(2 k_F x)} \nonumber \\ && \hspace{2cm} \times
\exp{\left[ -\frac{2}{\nu} \sqrt{ \left| \ln[W/\ell] \, \ln [x^2/(W
\ell)] \right|} \right]} \mbox{ .}\end{aligned}$$ The very weak dependence on $x$ at $x \gg W$ is associated with the vanishing of $g_q$ for $qW \ll 1$. The factor $\exp{\left[ -
\frac{2}{\nu} \sqrt{ \left| \ln[W/\ell] \, \ln [x^2/(W \ell)]
\right|} \right]}$ on the r.h.s. of Eq. (\[cdwcorr\]) is plotted as a function of $x$ for two different values of $W$ in Fig. \[cdwfig\]. For comparison, the power-law expected when inter-edge interactions are absent $$\label{powerlaw}
\left< \hat O_{\mbox{\tiny CDW}}(x) \, \hat O_{\mbox{\tiny CDW}}(0)
\right> \sim \left(\frac{x}{\ell}\right)^{-2/ \nu} \mbox{ if }
V^{\mbox{\tiny inter}}_q \equiv 0$$ is also plotted. The correlations shown in Eq. (\[cdwcorr\]) fall off more slowly than they would if inter-edge interactions were neglected. However, for large $W/\ell$, which is the case applicable to QH samples, the relative difference between Eq. (\[cdwcorr\]) and the power-law limit \[Eq. (\[powerlaw\])\] remains small even when $x$ is several times larger than $W$, see Fig. \[cdwfig\].
3.6in
For Hall bar samples with aspect ratios $L/W$ of order less than $\sim 10$, even the minimum value of $g_q$ (for $q=2\pi/L$) is close to unity. Therefore, there is no regime where quasi-solid correlations occur.
Summary
=======
In conclusion, we have calculated microscopically the dispersion relation for edge magnetoplasmons in a QH bar, emphasizing distinctions between these excitations and the plasmon excitations of a quantum wire. We have carefully examined the influence of different terms in the total Hamiltonian on plasmon excitations and obtain quite different results in Hall bar and quantum wire cases. Whereas the plasmon energy in the quantum wire case has contributions only from kinetic energy gain and exchange energy loss in the underlying electron system, the energy of edge magnetoplasmons in a QH bar has additional contributions arising from electrostatic (Hartree) and external potential terms but no contribution due to kinetic energy gain. Despite these differences, the low-energy effective Hamiltonian for both systems is identical for $q W \ll 1$.
Typical sample geometries of QH bars have an aspect ratio close to unity. Therefore, in this case, $q W \geq 1$ even for the smallest possible $q$. In this limit, inter-edge interaction is negligibly small. As a result we find that for typical sample geometries, the classical magnetoplasmon dispersion relation, Eq. (\[macrodisperse\]), which differs from the plasmon dispersion relation for a quantum wire, is accurate. Important corrections to the plasmon dispersion due to the interaction between left and right moving parts of the electron density occur only for long narrow Hall bar samples. We also find that typical QH samples are not in the regime where the quasi-solid behavior expected for charge density correlations in a quantum wire is important. Furthermore, the spatial separation of left movers and right movers in a QH bar leads to a suppression of the CDW-fluctuations which is a Gaussian function of the width $W$ of the Hall bar.
The authors acknowledge many helpful discussions with S. M. Girvin, J. J. Palacios, and R. Haussmann. This work was funded in part by NSF grant DMR-9416906. U.Z. thanks Studienstiftung des deutschen Volkes (Bonn, Germany) for financial support.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The role of spin-orbit interaction, arises from the Dzyaloshinski-Moriya anisotropic antisymmetric interaction, on the entanglement transfer via an antiferromagnetic XXZ Heisenberg chain is investigated. From symmetrical point of view, the XXZ Hamiltonian with Dzyaloshinski-Moriya interaction can be replaced by a modified XXZ Hamiltonian which is defined by a new exchange coupling constant and rotated Pauli operators. The modified coupling constant and the angle of rotations are depend on the strength of Dzyaloshinski-Moriya interaction. In this paper we study the dynamical behavior of the entanglement propagation through a system which is consist of a pair of maximally entangled spins coupled to one end of the chain. The calculations are performed for the ground state and the thermal state of the chain, separately. In both cases the presence of this anisotropic interaction make our channel more efficient, such that the speed of transmission and the amount of the entanglement are improved as this interaction is switched on. We show that for large values of the strength of this interaction a large family of XXZ chains becomes efficient quantum channels, for whole values of an isotropy parameter in the region $-2\leq\Delta\leq2$.'
author:
- 'Morteza Rafiee$^1$ [^1], Morteza Soltani$^2$[^2], Hamidreza Mohammadi$^3$ [^3] and Hossein Mokhtari$^4$ [^4]'
title: Entanglement Transfer via XXZ Heisenberg chain with DM Interaction
---
Introduction
============
Recently transmitting a quantum state is a most important task in quantum information and computation processing [@bennet]. Today this purpose could be achieved in two manner: i) using standard teleportation protocols and, ii) information transfer via quantum networks. Bayat et.al [@bayat] have shown that the fidelity of transmission is the same for both cases. Additionally, the former case employs the flying qbits as quantum channel and hence the fidelity of transmission reduces due to incompleteness of stationary-to-flying qbits conversion process [@vincenzo]. Thus the later case is superior to the former, particularly in solid state devices. Among the numerous quantum systems suitable for quantum networks implementation, the spin chains offer a great advantages. One of the most interesting art of the spin chains is their ability to use as quantum wires in the information transfer protocols over the short distances [@bose; @eisert; @giovannetti; @osborne; @bayat3; @Burgarth; @Burgarth2; @bayat2]. Tunable spin interaction in these systems plays the key role to motivate one for using this permanently potential in the quantum information transfer processing. The major works on spin chain are in the Ferromagnetic(FM) phase [@osborne] and the effects of temperature [@bayat3] and decoherence [@Burgarth] have been investigated for FM channels. There are lesser works on Antiferromagnetic(AFM) phase [@asoudeh; @eckert], but as mentioned in Refs. [@bayat2; @bayat], AFM spin chain have higher ability in transfer of information and thus is a better and faster alternative. So we prefer to study AFM spin chain. Fortunately, Antiferromagnetic spin chains with short length(up to 10 spins) have been built experimentally [@Hirjibehedin]. However, much attention has been paid to the entanglement in spin systems with only spin-spin interaction and spin-orbit interaction has been leaved. The spin-orbit interaction produces and anisotropic part of exchange interaction between localized conduction band electrons in crystals with lack of inversion symmetry, including all low dimensional structures and also bulk semiconductors with zinc-blende and wurtzite type of crystal lattice [@kavokin]. The main part of the anisotropic interaction has the form of Dzyaloshinski-Moriya interaction (DMI) [@Dzyaloshinski; @Moriya1; @Moriya2] which have explained the weak ferromagnetism of antiferromagnetic crystals ($\alpha-Fe_2O_3,MnCO_3$ and $CrF_3$). As was shown [@hamid; @hamid1; @wang] in the two-qubit Heisenberg systems the (DMI) plays an important role. The (DMI) express by $$\vec{D}.[\vec{S}_1\otimes\vec{S}_2].\nonumber$$ This interaction arises from extending the Anderson’s theory of superexchange interaction by including the spin-orbit coupling effects[@Moriya1]. In this paper we interested to consider the model with DMI and also investigated the effects of this interaction on the information transfer processing.\
Following the approach proposed by S. Bose[@bose], we place a spin encoding the state at one end of the chain (which is now equipped with DMI) and wait for specific amount of time to let this state propagate to the other end. Therefore, entanglement could be transferred from one end of the chain to the other. We will show that the XXZ chain with DMI can be reduced to the modified XXZ chain with new coupling constant and rotated Pauli operators. The modified coupling constant and the angle of rotations are depend on the strength of DMI. Consequently, the entanglement transfer protocol through the XXZ+DMI chains becomes the same as a protocol via a general form of XXZ chain with new definition of coupling constant and Pauli matrices, but it should be noted that at this case the initial state may be changed. Indeed, both modified coupling constant and the angle of rotations, which is referred by “phase factor” through the text, are the efficient parameters for entanglement transfer. For the sake of clarity, the effects of phase factor and coupling constant have been separated. Our calculation shows that this phase factor does not affect on the entanglement transfer when $\Delta=0$ for the XXZ chain. Whereas, this phase factor has desirable effects on the parameters of entanglement transfer for the case of $\Delta<0$ and undesirable effects when $\Delta>0$. However, there is a contest between the phase factor and coupling constant. For large amounts of the strength of DMI($D$), the effects of coupling constant are more dominant than the effects of phase factor and has desirable effects. Also, in the whole range of $\Delta$, the effects of the modified coupling constants and the phase factor simultaneously investigated on our main goal, i.e, information or entanglement transmission. Furthermore, the other advantage of DMI is that the speed of information transmission increases as increasing the strength of this interaction. Also, at nonzero temperature this interaction (DMI) improves both $E_{max}$ and $t_{opt}$ as increasing the strength of DMI ($D$) for both positive and negative amounts of $\Delta$.\
The paper is organized as follows. In Sec. II we introduce the XXZ Hamiltonian with DMI and discuss the way to change the form of Hamiltonian to the general XXZ Hamiltonian. The new set of eigenstates have been introduced in this Sec. In Sec. III we introduce our state transmission. In Sec. IV we show analytically the dependence of concurrence on $D$ and pase factor. In Sec. V we show our numerical calculation of entanglement transfer and corresponding times in both zero and nonzero temperature and calculated the speed of information transfer while, we summarize our results in Sec. VI.
HAMILTONIAN AND MODEL
=====================
The Hamiltonian of the open XXZ chain in presence of spin-orbit interaction is defined as $$\label{1}
H_{ch}=\sum\limits_{i=1}^{N_{ch}-1}\{J[\sigma_i^x\sigma_{i+1}^x+\sigma_i^y\sigma_{i+1}^y]+\Delta\sigma_i^z\sigma_{i+1}^z\}
+\sum\limits_{i=1}^{N_{ch}-1}\{\vec{D}.(\vec\sigma_i\times\vec\sigma_{i+1})\},$$ Where $N_{ch}$ is the number of spin in $1D$ chain, $\vec\sigma_i$=($\sigma_i^x$,$\sigma_i^y$,$\sigma_i^z$) is the vector of pauli matrices, J is the exchange coupling, the parameter $\Delta$ is the anisotropy exchange coupling in the z direction and D is the Dzyaloshinski-Moriya vector. Different phases of chain is depending on different range of J and $\Delta$. For the case of $J<0$ the chain is called Ferromagnetic(FM) Heisenberg chain and the case $J>0$ is Antiferromagnetic(AFM) Heisenberg chain. The AFM Heisenberg chain includes FM phase($\frac{\Delta}{J}<0$), AFM phase($0<\frac{\Delta}{J}<1$) and Néel phase($\frac{\Delta}{J}>1$)[@14]. If we take $\vec{D}=D\hat{z}$, then the above Hamiltonian can be written as $$\label{2}
H_{ch}=\sum\limits_{i=1}^{N_{ch}-1}\{J[\sigma_i^x\sigma_{i+1}^x+\sigma_i^y\sigma_{i+1}^y]+\Delta\sigma_i^z\sigma_{i+1}^z\}
+D\sum\limits_{i=1}^{N_{ch}-1}\{\sigma_i^x\sigma_{i+1}^y-\sigma_i^y\sigma_{i+1}^x\}.$$ This Hamiltonian is invariant under z-axis rotation, i.e, $[H,S_z]=0$, where $S_z={\mbox{$\textstyle \frac{1}{2}$}}{\sum\limits_{i=1}^{N_{ch}}\sigma_i^z}$. By this property the Hamiltonian can be express in the form of usual XXZ model without explicit DMI. For this purpose, the pauli operators in x-y directions are manipulated by the unitary transformation which is depend on the spin sites[@15] $$\begin{aligned}
\label{3}
U_{N_{ch}}&=&\exp^{-i\sum\limits_{m=2}^{N_{ch}}(m-1)\phi\sigma_m^z},\end{aligned}$$ where $\phi=tang^{-1}(D{/}J)$. Hence these spin coordinate transformations reads $$\begin{aligned}
\label{4}
\tilde{\sigma}_i^x&=&\sigma_{i}^x\cos{\phi_i}+\sigma_i^y\sin{\phi_i},\nonumber\\
\tilde{\sigma}_{i}^y&=&-\sigma_{i}^x\sin{\phi_i}+\sigma_i^y\cos{\phi_i},\nonumber\\
\tilde{\sigma}_i^z&=&\sigma_{i}^z,\end{aligned}$$ where $\phi_i=(i-1)\phi$. So, the modified Hamiltonian is express as $$\begin{aligned}
\label{5}
\tilde{H}_{ch}&=&U_{N_{ch}}H_{ch}U^\dag_{N_{ch}}=\tilde{J}\sum\limits_{i=1}^{N_{ch}-1}\{\tilde{\sigma}_i^x\tilde{\sigma}_{i+1}^x+
\tilde{\sigma}_i^y\tilde{\sigma}_{i+1}^y\}
+\Delta\sum\limits_{i=1}^{N_{ch}-1}\sigma_i^z\sigma_{i+1}^z,\\
\tilde{J}&=&sgn(J)\sqrt{J^2+D^2}.\nonumber\end{aligned}$$ The eigenstates of the new Hamiltonian (${|\tilde{\psi_n}\rangle}$) are related to the earlier one (${|\psi_n\rangle}$) via ${|\tilde{\psi_n}\rangle}=U_{N{ch}}{|\psi_n\rangle}$. As the model of the system is specified, we can investigate the information transmission processing in this system.
Entanglement Transmission
=========================
The quantum information transmission of one part of a two-spin maximally entangled state ($0^\prime0$) via XXZ+DMI spin chain is investigated while the spin chain is in its ground state (${|\psi_{gs}\rangle}_{ch}$). At t=0 we interact the spin $0$ with the first spin of the chain. We suppose that the chain is prepare in a unique grand state, initially. The preparation could be performed by applying a small magnetic field, if it is required. Furthermore, the interaction between spin $0$ and first spin of the chain has the same form of the rest of interaction, $$\label{6}
H_{I}=J(\sigma_0^x\sigma_{1}^x+\sigma_0^y\sigma_{1}^y+\Delta\sigma_0^z\sigma_{1}^z)
+D(\sigma_0^x\sigma_{1}^y-\sigma_0^y\sigma_{1}^x).$$ Indeed, the system is consist of $0^\prime0$ and $N_{ch}$ spins and hence the total length of the system is $N=N_{ch}+2$. The initial state of the system is $$\label{7}
{|\psi(0)\rangle}={|\psi^-\rangle}_{0^\prime0}\otimes{|\psi_{gs}\rangle}_{ch},$$ where $$\label{8}
{|\psi^-\rangle}_{0^\prime0}=\frac{{|01\rangle}-{|10\rangle}}{\sqrt{2}}.$$ This ${|\psi(0)\rangle}$ is used as a channel which transfer the entanglement. Therefore, the total Hamiltonian being $$\label{9}
H=I_{0^\prime}\otimes(H_{ch}+H_I),$$ By this Hamiltonian, the initial state evolves to the state ${|\psi(t)\rangle}=e^{-iHt}{|\psi(0)\rangle}$ and the two sites reduced density matrix of can be computed as $\rho_{mn}(t)=tr_{\widehat{mn}}\{{|\psi(t)\rangle}{\langle \psi(t)|}\}$, where $tr_{\widehat{mn}}$ is the partial trace over the system except sites m and n. The two sites reduced density matrix in computational basis (${|00\rangle},{|01\rangle},{|10\rangle},{|11\rangle}$) has the general form as[@hamid] $$\label{10}
\rho_{mn}(t) =\left(\begin{array}{*{20}c}
{{a(t)}} & {{0}} & {{0}} & {{0}} \\
{{0}} & {{x(t)}} & {{z(t)}} & {{0}} \\
{{0}} & {{z^*(t)}} & {{y(t)}} & {{0}} \\
{{0}} & {{0}} & {{0}} & {{b(t)}} \\
\end{array}\right).$$ Although, the density matrix has been written in the Schr$\ddot{\text o}$dinger picture, but $\rho_{mn}(t)$ in terms of spin-spin correlation function could be expressed in the Heisenberg picture as follows[@16] $$\begin{aligned}
\label{11}
a(t)&=&1+{\mbox{$\textstyle \frac{1}{2}$}}\langle\sigma_m^z(t)+\sigma_n^z(t)\rangle+\langle\sigma_m^z(t)\sigma_n^z(t)\rangle,\nonumber\\
x(t)&=&1+{\mbox{$\textstyle \frac{1}{2}$}}\langle\sigma_m^z(t)-\sigma_n^z(t)\rangle-\langle\sigma_m^z(t)\sigma_n^z(t)\rangle,\nonumber\\
y(t)&=&1-{\mbox{$\textstyle \frac{1}{2}$}}\langle\sigma_m^z(t)-\sigma_n^z(t)\rangle-\langle\sigma_m^z(t)\sigma_n^z(t)\rangle,\\
b(t)&=&1-{\mbox{$\textstyle \frac{1}{2}$}}\langle\sigma_m^z(t)+\sigma_n^z(t)\rangle+\langle\sigma_m^z(t)\sigma_n^z(t)\rangle,\nonumber\\
z(t)&=&\langle\sigma_m^x(t)\sigma_n^x(t)\rangle+\langle\sigma_m^y(t)\sigma_n^y(t)\rangle+
i(\langle\sigma_m^x(t)\sigma_n^y(t)\rangle-\langle\sigma_m^y(t)\sigma_n^x(t)\rangle),\nonumber\end{aligned}$$ these correlations are computed in terms of the initial state (Eq.\[7\]) and $\sigma_m^{\alpha}(t)=e^{-i H
t}\sigma_m^{\alpha}e^{-i H t}$ where $\alpha=\{x,y,z\}$. Since the concurrence is directly defined in terms of the density matrix and so any minimization procedure is not necessary, the concurrence is used as a measure of entanglement for arbitrary mixed state of two qubits[@17], $$\begin{aligned}
\label{12}
C&=&max\{0,2\lambda_{max}-tr\sqrt{R}\},\\
R&=&\rho\sigma^y\otimes\sigma^y\rho^*\sigma^y\otimes\sigma^y,\end{aligned}$$ where $\lambda_{max}$ is the largest eigenvalues of the matrix $\sqrt{R}$. For our density matrix the concurrence results to be $$\label{13}
C=2max\{0,C^{(1)},C^{(2)}\},$$ where $C^{(1)}=-\sqrt{xy}$ and $C^{(2)}=|z|-\sqrt{ab}$. Because, $C^{(1)}$ is always negative here the concurrence is[@14] $$\label{14}
C=2max\{0,|z|-\sqrt{ab}\}.$$ For our main goal the first site refers to $0^\prime$ and other site refers to the spin located at the end of the chain, say (j). So in terms of density matrix the subscript m is change to $0^\prime$ and n is change to j. The singlet fraction of the state, $\rho_{0'j}$, as an indicator of the average fidelity of state transferring could be obtain as $$\begin{aligned}
\label{frac}
F={\langle\psi^-|\rho_{0'j}|\psi^-\rangle}={\mbox{$\textstyle \frac{1}{2}$}}(x+y-2z).\end{aligned}$$ With the aid of $\tilde{H}_{ch}=U_{N_{ch}}H_{ch}U^\dag_{N_{ch}}$ we have $$\begin{aligned}
\label{20}
\langle\sigma_m^\alpha(t)\rangle&=&\langle e^{-i H
t}\sigma_m^\alpha e^{i H t}\rangle\nonumber\\
&=&{\langle 0^\prime0|}{\langle \psi_{gs}|}_{ch}U_{N_{ch}+1}e^{-i\tilde{H}t}\tilde{\sigma}_m^\alpha
e^{i\tilde{H}t}U_{N_{ch}+1}^\dag{|\psi_{gs}\rangle}_{ch}{|0^\prime0\rangle},\end{aligned}$$ with use of the ${|\tilde{\psi_n}\rangle}=U_{N{ch}}{|\psi_n\rangle}$, the expectation value can be express as $$\begin{aligned}
\label{22}
\langle\sigma_m^\alpha(t)\rangle&=&{\langle 0^\prime0|}{\langle \tilde{\psi}_{gs}|}_{ch}U_{N_{ch}}U_{N_{ch}+1}\tilde{\sigma}_m^\alpha(t)
U_{N_{ch}+1}^\dag
U_{N_{ch}}^\dag{|\tilde{\psi}_{gs}\rangle}_{ch}{|0^\prime0\rangle},\end{aligned}$$ where $U_{N_{ch}}U_{N_{ch}+1}=\exp^{-i\sum\limits_{m=1}^{N_{ch}}(2m-1)\phi\sigma_m^z}$ is phase factor which modify the states on the right hand of above equation. So, we can conclude that this model is similar to usual XXZ model with the new strength coupling in XY direction ($\tilde
J$) and the new set of states which are multiplied by the phase factor. In the following, the effects of these parameters (phase factor and $\tilde J$) will investigated on entanglement transfer processing.
Analytical Calculation
======================
To more clarifying, the concurrence between the $0^\prime$ site and the end of the chain with the length of($N_{ch}=2$) have been calculated analytically in the appendix. In these calculations the explicit form of $\rho_{mn}(t)=tr_{mn}\{{|\psi(t)\rangle}{\langle \psi(t)|}\}$ was used and these results are quality compatible with the numerical results for higher N. From the relation(\[23\]) the concurrence for the case of ($\Delta=0$) is $$\begin{aligned}
\label{24}
C(t)&=&\frac{1}{8}\big{(}4\big{|}\cos{(\xi t)}-1\big{|}-\big{|}e^{2i\phi}(1+\cos{(\xi t)})\nonumber\\
&+&i\sqrt{2}\sin{(\xi t)}\big{|}\times\big{|}1+\cos{(\xi
t)}+i\sqrt{2}e^{2i\phi}\sin{(\xi t)}\big{|}\big{)},\end{aligned}$$ where $\xi=2\sqrt{2}\tilde{J}$ and we have the maximum entanglement(C=1) for $\xi t=\pi$. As we can see, amounts of entanglement at the first peak ($E_{max}$) and corresponding time ($t=t_{opt}$) in this case are independent on the $\phi$.
Furthermore, for the case of $\Delta\neq0$ the concurrence is obtained in Eq.(\[25\]). In this case $E_{max}$ and $t_{opt}$ depend on $\phi$ as indicated in Fig. 1. In these figure, for clarifying the role of $\phi$, individually, we fixed the size of $\tilde J$. The results show that, in the case of $\Delta<0$, increasing $\phi$ improves both $E_{max}$ and $t_{opt}$ and for the case $\Delta>0$, the increase of $\phi$ has undesirable effects on both $E_{max}$ and $t_{opt}$. The calculation becomes more involved when $N$ exceeds 4, this prevents one from writing an analytical expression for the concurrence. So, we solve this problem numerically.
Numerical Calculation
=====================
Entanglement at T=0
-------------------
Whereas, in the XXZ spin chain the AFM Heisenberg chains ($J>0$) is a better candidate for information transfer than FM chains ($J<0$) [@bayat2], we confined our calculations to the case $J>0$ and take $J=1$ to simplify the calculation. The entanglement is calculated between $0^\prime$ and spin located at the end of the chain which has the length of N=8. In Fig. 2(a), $E_{max}$, as measured by concurrence, have been plotted in the domain of ($-2\leq\Delta\leq2$) for different values of $D$ and Fig. 2(b) illustrates the behavior of $E_{max}$ as a function of $\Delta$ for different values of $\phi$, where $\tilde J$ is fixed. For the special case D=0 (i.e, $\tilde J=J$ and $\phi=0$) the results are the same as in Ref. [@bayat], qualitatively. In this case $E_{max}$ vanishes at the quantum phase transition (QPT) point ($\frac{\Delta}{J}=-1$) [@14]. In the presence of $D$, $\tilde
J\neq J$ and hence the QPT point shifts to the $\frac{\Delta}{\tilde J}=-1$, e.g, for the case of $D=1$ the QPT occurs at $\Delta=-\sqrt 2$ [^5]. Despite to the results of [@bayat], at this modified transition point $E_{max}$ does not vanish, this is due to the presence of the phase factor $\phi$ as indicated in Fig. 2(b). Furthermore, the amount of $\tilde J$ is so large for large values of $D$ and so the QPT point disappear in the frame of $-2\leq\Delta\leq2$. In this range of $\Delta$ and $D$, the amount of $E_{max}$ is unsensible to the values of $\phi$ and hence $\tilde J$ plays the main role to quantifying $E_{max}$. Since, $\frac{\Delta}{\tilde J}$ approaches zero as $D$ becomes large, the system treat as the case $\Delta=0$, i.e, $E_{max}\longrightarrow E_{max}(\Delta=0)$. Fig. 3(a) reveals that the behavior of $t_{opt}$ is compatible with the result of Fig. 2(a). This figure shows that $t_{opt}$ decreases as increasing $D$, hence the presence of DMI enhances the speed of information transmission. Also, Figs. 2(b) and 3(b) show that the effect of $\phi$ on $E_{max}$ and $t_{opt}$ is undesirable for the case of $\Delta>0$. In contrast, for the case $\Delta<0$, the maximum entanglement at the first peak and speed of transmission enhances with $\phi$. In summary, all of the chains with $-2\leq\Delta\leq2$ can be used as protocol for information transfer processing with the same cost.
Fig. 4 depicts the singlet fraction in terms of $\Delta$ for different values of $D$. The result of this picture confirms the above mentioned effects of DMI on entanglement transfer properties.
In order to better illustrate the effects of $D$ on the entanglement transfer properties, $E_{max}$ is plotted in terms of $D$ and $\phi$ in Figs. 5(a)-5(d) for different values of $\Delta$ respectively. The dips appearing in Figs. 5(a) and 5(c) are due to the effects of the phase factor ($\phi$) which is obvious from Figs. 5(b) and 5(d). The effects of DMI on speed of state transmission are plotted in Figs. 6(a) and 6(b), more obviously.
Thermal Entanglement
--------------------
Since, preparing the system at $T=0$ is far from access, the presence of thermal excitations is unavoidable. So, we consider the state of channel as a thermal state instead of ground state. The thermal state of channel at temperature $T$ is given by density matrix $\rho_{ch}=\frac{e^{-\beta H_{ch}}}{Z}$, where $\beta=\frac{1}{k_BT}$, $k_B$ is the Boltzman constant and $Z=tr(e^{-\beta H_{ch}})$ is the partition function. So the initial state of system is $$\label{15}
\rho(0)={|\psi^-\rangle}{\langle \psi^-|}\otimes\frac{e^{-\beta
H_{ch}}}{Z}.$$ Employing the system parameters as before, we repeat the calculations for this new initial state and the results have been shown in Figs 7 and 8 for $\Delta=1$ and $\Delta=-1$, respectively. Two considerations are in order at this stage. Firstly, at hight temperatures, thermal fluctuations suppress the quantum correlations and hence the entanglement vanishes and also, $t_{opt}$ becomes so large. Secondly, the presence of DMI amplify the quantum correlations, so by increasing $D$ we can obtain nonzero entanglement at larger temperatures. For instance Fig 8(a) shows that for the case of $\Delta=-1$ the entanglement can be exist at higher temperatures in the presence of DMI while it is zero for all temperature in the absence of DMI. Also, the speed of transfer improves as $D$ increases.
Information speed
-----------------
As mentioned before, the DMI imposes desirable effects on the speed of transmission. In order to better clarify, we compare $t_{opt}$ with $\emph{v}^{-1}$ (which $\emph{v}$ is the spin-wave velocity) which is obtained with the aid of field theoretic techniques [@field]. Note that, the later get the qualitative behavior of correlations in the thermodynamic limit while the former is calculated for very short chain ($N=8$). Also, in the field theoretic techniques the dynamical correlations are computed for the ground state of the system while in our problem it is not the case. Despite difference, we still can use some well-known results of the field theoretic techniques. For instance, in our modified XXZ model(XXZ model equipped with DMI) and for the range $-1<\Delta<1$, the spin-spin correlation function in the asymptotic thermodynamical limit have the following form [@field] $$\begin{aligned}
\label{17}
\langle{\sigma^\mu_l(t)}{\sigma^\mu_{l+n}}&\rangle\sim&(-1)^n[n^2-\emph{v}^2t^2]^{-(1/2)\eta_\mu},\\
\eta_x=\eta_y&=&\eta_z^{-1}=1-\frac{\gamma}{\pi}\nonumber,\end{aligned}$$ where $$\label{18}
\Delta=\cos{(\gamma)},$$ and the propagation velocity of excitation in the chain can be written as $$\label{19}
\emph{v}\sim\frac{\pi\tilde{J}\sin{(\gamma)}}{\gamma}.$$ This velocity is proportional to the strength of DMI ($D$), which is included in $\tilde J$, and also it depends on the size of $\Delta$. According to the Ref. [@field], $\emph v\,$ refers to the velocity of propagation of the correlations which is related to the entanglement transmission speed. Fig. 9 shows $\emph{v}^{-1}$, which is obtained from Eq. (\[19\]), and $t_{opt}$, which is calculated in previous section, in terms of $D$ for two different values of $\Delta$. As we can see both quantities have similar behavior qualitatively, such that they descent with increasing of $D$. As this figure illustrates, despite of the field theory calculation for spin-wave propagation time ($\emph v ^{-1}$), the curves of $t_{opt}$ cross each other, indeed at the cross point the effect of the phase factor ($\phi$) becomes considerable. As a consequence, for large values of $D$ the speed of information transmission raises up and ultimately reaches the asymptotic value. Faster dynamics in large values of $D$ stems to the entanglement enhancement of the channel in this region.
Conclusions
===========
In this paper we examined a XXZ Heisenberg chain equipped with the Dzyaloshinski-Moriya interaction (DMI) as a quantum channel for investigation of the entanglement transfer properties. We had shown that the presence of DMI enhances the amount of coupling constant, J, and also imposes a phase factor on the state of the channel. In order to clarifying the role of DMI, we trace the effects of the new coupling constant ($\tilde J$) and phase factor ($\phi$) for a wide range of the anisotropy parameter, $\Delta$, separately. Indeed, increasing $\tilde J$ with $D$ leads to increase the strength of spin-spin correlations and ultimately improves the amount of the entanglement. For the case $\Delta<0$ the effects of $\phi$ on the entanglement properties of transmission is more desirable. In contrast, for the case $\Delta>0$ increasing $\phi$, individually, destroys the entanglement of the channel. The effects of $\tilde J$ dominate for large values of $D$ and hence the channel becomes more efficient for all values of anisotropy parameter in the region $-2\leq\Delta\leq2$. We calculated the entanglement properties for the ground state and the thermal state of the chain, separately. Our results show that the amount of entanglement and the speed of transmission increase as $D$ increases. Also, we show that the entanglement can be exist at higher temperatures as $D$ increases and hence the transmission channel could be work more efficiently at higher temperatures.
Analytical Calculation for N=4
==============================
In this appendix, we give analytical calculation for the XXX and XXZ chains with $N=4$, separately. For XXX chain ($\Delta=0$), the initial state ${|\psi(0)\rangle}$ in the basis of the Hamiltonian (\[5\]) for $N=3$ can be written as [^6] $$\begin{aligned}
{|\psi(0)\rangle}&=&\frac{e^{-i\phi}}{2}\big{[}{|0\rangle}_{0^\prime}\otimes(\frac{e^{2i\phi}}{\sqrt{2}}{|\psi_2\rangle}+
\frac{1}{2}(\sqrt{2}+e^{2i\phi}){|\psi_5\rangle}+\frac{1}{2}(-\sqrt{2}+e^{2i\phi}){|\psi_7\rangle})\nonumber\\
&+&{|1\rangle}_{0^\prime}\otimes(-\frac{1}{\sqrt{2}}{|\psi_3\rangle}+\frac{1}{2}(1+\sqrt{2}e^{2i\phi}){|\psi_6\rangle}+
\frac{1}{2}(1-\sqrt{2}e^{2i\phi}){|\psi_8\rangle})\big{]},\end{aligned}$$ where $$\begin{aligned}
{|\psi_2\rangle}&=&\frac{1}{\sqrt{2}}(-{|011\rangle}+{|110\rangle}),\nonumber\\
{|\psi_3\rangle}&=&\frac{1}{\sqrt{2}}(-{|001\rangle}+{|100\rangle}),\nonumber\\
{|\psi_5\rangle}&=&\frac{1}{2}({|011\rangle}-\sqrt{2}{|101\rangle}+{|110\rangle}),\nonumber\\
{|\psi_6\rangle}&=&\frac{1}{2}({|001\rangle}-\sqrt{2}{|010\rangle}+{|100\rangle}),\\
{|\psi_7\rangle}&=&\frac{1}{2}({|011\rangle}+\sqrt{2}{|101\rangle}+{|110\rangle}),\nonumber\\
{|\psi_8\rangle}&=&\frac{1}{2}({|001\rangle}+\sqrt{2}{|010\rangle}+{|100\rangle}),\nonumber\end{aligned}$$ are the relevant eigenstates of $H_3(\Delta=0)$ and the corresponding eigenvalues are $E_2=E_3=0$, $E_5=E_6=-\xi$ and $E_7=E_8=\xi$, here we define $\xi=2\sqrt{2}\tilde{J}$. The state of system at later times can be obtained as $$\begin{aligned}
{|\psi(t)\rangle}&=&\frac{e^{-i\phi}}{2}[{|0\rangle}_{0^\prime}\otimes(\frac{e^{2i\phi}}{\sqrt{2}}{|\psi_2\rangle}+
\frac{1}{2}(\sqrt{2}+e^{2i\phi})e^{i\xi t}{|\psi_5\rangle}\nonumber\\
&+& \frac{1}{2}(-\sqrt{2}+e^{2i\phi})e^{-i\xi t}{|\psi_7\rangle})+
{|1\rangle}_{0^\prime}\otimes(-\frac{1}{\sqrt{2}}{|\psi_3\rangle}\nonumber\\
&+&\frac{1}{2}(1+\sqrt{2}e^{2i\phi})e^{i\xi
t}{|\psi_6\rangle}+\frac{1}{2}(1-\sqrt{2}e^{2i\phi})e^{-i\xi
t}{|\psi_8\rangle})].\end{aligned}$$ The corresponding reduced density matrix, $\rho_{0'2}(t)$ is in the form of Eq. (\[10\]) with the following components $$\begin{aligned}
a&=&\frac{1}{16}\big{|}e^{2i\phi}(1+\cos{(\xi t)})+i\sqrt{2}\sin{(\xi t)}\big{|}^2,\nonumber\\
x&=&\frac{1}{16}\big{|}e^{2i\phi}(-1+\cos{(\xi t)})+i\sqrt{2}\sin{(\xi t)}\big{|}^2,\nonumber\\
&+&\frac{1}{8}\big{|}\sqrt{2}\cos{(\xi t)}+ie^{2i\phi}\sin{(\xi t)}\big{|}^2,\nonumber\\
y&=&\frac{1}{8}\big{|}\sqrt{2}e^{2i\phi}\cos{(\xi t)}+i\sin{(\xi t)}\big{|}^2,\\
&+&\frac{1}{16}\big{|}-1+\cos{(\xi t)}+i\sqrt{2}e^{2i\phi}\sin{(\xi t)}\big{|}^2,\nonumber\\
b&=&\frac{1}{16}\big{|}1+\cos{(\xi t)}+i\sqrt{2}e^{2i\phi}\sin{(\xi t)}\big{|}^2,\nonumber\\
z&=&\frac{1}{4}(-1+\cos{(\xi t)}).\nonumber\end{aligned}$$ Therefore the concurrence could be computed as $$\begin{aligned}
\label{23}
C(t)&=&\frac{1}{8}\big{(}4\big{|}-1+\cos{(\xi t)}\big{|}-\big{|}e^{2i\phi}(1+\cos{(\xi t)})\nonumber\\
&+&i\sqrt{2}\sin{(\xi t)}\big{|}\times\big{|}1+\cos{(\xi t)}\nonumber\\
&+&i\sqrt{2}e^{2i\phi}\sin{(\xi t)}\big{|}\big{)}.\end{aligned}$$ Following the same procedure for XXZ chain ($\Delta\neq0$), the initial state ${|\psi(0)\rangle}$ in the basis of corresponding Hamiltonian is $$\begin{aligned}
{|\psi(0)\rangle}&=&\frac{e^{-i\phi}}{2}\big{[}{|0\rangle}_{0^\prime}\otimes(\frac{e^{2i\phi}}{\sqrt{2}}{|\psi_2\rangle}+
\frac{\alpha+e^{2i\phi}}{\sqrt{2+\alpha^2}}{|\psi_5\rangle}+\frac{-\beta+e^{2i\phi}}{\sqrt{2+\beta^2}}{|\psi_7\rangle})\nonumber\\
&+&{|1\rangle}_{0^\prime}\otimes(-\frac{1}{\sqrt{2}}{|\psi_3\rangle}+\frac{1+\alpha
e^{2i\phi}}{\sqrt{2+\alpha^2}}{|\psi_6\rangle}+\frac{1-\beta
e^{2i\phi}}{\sqrt{2+\beta^2}}{|\psi_8\rangle})\big{]},\end{aligned}$$ where $\alpha=\frac{\Delta+\sqrt{8\tilde{J}^2+\Delta^2}}{2\tilde{J}}$, $\beta=-\frac{\Delta-\sqrt{8\tilde{J}^2+\Delta^2}}{2\tilde{J}}$ and $$\begin{aligned}
{|\psi_2\rangle}&=&\frac{1}{\sqrt{2}}(-{|011\rangle}+{|110\rangle}),\nonumber\\
{|\psi_3\rangle}&=&\frac{1}{\sqrt{2}}(-{|001\rangle}+{|100\rangle}),\nonumber\\
{|\psi_5\rangle}&=&\frac{1}{\sqrt{2+\alpha^2}}({|011\rangle}-\alpha{|101\rangle}+{|110\rangle}),\nonumber\\
{|\psi_6\rangle}&=&\frac{1}{\sqrt{2+\alpha^2}}({|001\rangle}-\alpha{|010\rangle}+{|100\rangle}),\\
{|\psi_7\rangle}&=&\frac{1}{\sqrt{2+\beta^2}}({|011\rangle}+\beta{|101\rangle}+{|110\rangle}),\nonumber\\
{|\psi_8\rangle}&=&\frac{1}{\sqrt{2+\beta^2}}({|001\rangle}+\beta{|010\rangle}+{|100\rangle}),\nonumber\end{aligned}$$ are the relevant eigenstates of $H_3(\Delta\neq0)$ and the corresponding eigenvalues are $E_2=E_3=0$, $E_5=E_6=-2\tilde{J}\alpha$ and $E_7=E_8=2\tilde{J}\beta$. Therefore, the initial state, ${|\psi(0)\rangle}$ evolves to the state $$\begin{aligned}
{|\psi(t)\rangle}&=&\frac{e^{-i\phi}}{2}\big{[}{|0\rangle}_{0^\prime}\otimes(\frac{e^{2i\phi}}{\sqrt{2}}{|\psi_2\rangle}+
\frac{(\alpha+e^{2i\phi})e^{-iE_5t}}{\sqrt{2+\alpha^2}}{|\psi_5\rangle}\nonumber\\
&+&\frac{(-\beta+e^{2i\phi})e^{-iE_7t}}{\sqrt{2+\beta^2}}{|\psi_7\rangle})
+{|1\rangle}_{0^\prime}\otimes(-\frac{1}{\sqrt{2}}{|\psi_3\rangle}\nonumber\\
&+&\frac{(1+\alpha
e^{2i\phi})e^{-iE_6t}}{\sqrt{2+\alpha^2}}{|\psi_6\rangle}+\frac{(1-\beta
e^{2i\phi})e^{-iE_8t}}{\sqrt{2+\beta^2}}{|\psi_8\rangle})\big{]}.\end{aligned}$$ The corresponding reduced density matrix, $\rho_{0'2}(t)$ is in the form of Eq. (\[10\]) with the following components $$\begin{aligned}
a&=&\frac{1}{4}|\frac{e^{2 i \tilde{J} t \alpha } (\alpha +e^{2 i
\phi})}{\alpha ^2+2}+\frac{1}{2} e^{2 i \phi }+\frac{e^{-2 i
\tilde{J} t\beta } (e^{2 i \phi }-\beta )}{\beta^2+2}|^2,\nonumber\\
x&=&\frac{1}{4}\big{|}-\frac{1}{2}e^{2i\phi}+\frac{(\alpha+e^{2i\phi})e^{2i\alpha \tilde{J} t}}{2+\alpha^2}+
\frac{(-\beta+e^{2i\phi})e^{-2i\beta \tilde{J} t}}{2+\beta^2}\big{|}^2\nonumber\\
&+&\frac{1}{4}\big{|}-\frac{\alpha(\alpha+e^{2i\phi})e^{2i\alpha \tilde{J} t}}{2+\alpha^2}+
\frac{\beta(-\beta+e^{2i\phi})e^{-2i\beta \tilde{J} t}}{2+\beta^2}\big{|}^2,\nonumber\\
y&=&\frac{1}{4}\big{(}\big{|}-\frac{1}{2}+\frac{(1+\alpha
e^{2i\phi})e^{2i\alpha \tilde{J} t}}{2+\alpha^2}+\frac{(1-\beta
e^{2i\phi})e^{-2i\beta \tilde{J} t}}{2+\beta^2}\big{|}^2\nonumber\\
&+&\big{|}-\frac{\alpha(1+\alpha e^{2i\phi})e^{2i\alpha \tilde{J}
t}}{2+\alpha^2}+\frac{\beta(1-\beta e^{2i\phi})e^{-2i\beta \tilde{J} t}}{2+\beta^2}\big{|}^2\big{)},\\
b&=&\frac{1}{4}\big{|}\frac{1}{2}+\frac{(1+\alpha e^{2i\phi})e^{2i\alpha \tilde{J} t}}{2+\alpha^2}+
\frac{(1-\beta e^{2i\phi})e^{-2i\beta \tilde{J} t}}{2+\beta^2}\big{|}^2,\nonumber\\
z&=&\frac{1}{4} ((\frac{\beta (-\beta +e^{-2 i \phi })
e^{2i\tilde{J} t \beta }}{\beta ^2+2}-\frac{\alpha (\alpha +e^{-2
i \phi }) e^{-2 i \tilde{J} t \alpha}}{\alpha ^2+2})\nonumber\\
&\times&(\frac{(1+\alpha e^{2 i \phi }) e^{2 i \tilde{J} t \alpha }}{\alpha ^2+2}+
\frac{(1-\beta e^{2 i \phi }) e^{-2 i \tilde{J} t \beta }}{\beta ^2+2}-\frac{1}{2})\nonumber\\
&+&(\frac{(\alpha +e^{-2 i \phi }) e^{-2 i \tilde{J} t \alpha }}{\alpha ^2+2}+
\frac{(-\beta +e^{-2 i \phi }) e^{2 i \tilde{J} t \beta }}{\beta ^2+2}-\frac{1}{2} e^{-2 i \phi })\nonumber\\
&\times& (\frac{\beta (1-\beta e^{2 i \phi }) e^{-2 i J t \beta }}{\beta ^2+2}-\frac{\alpha (1+\alpha
e^{2 i \phi }) e^{2 i \tilde{J} t \alpha }}{\alpha ^2+2})),\nonumber\end{aligned}$$ and hence $$\begin{aligned}
\label{25}
C(t)&=&{\mbox{$\textstyle \frac{1}{2}$}}(|(\frac{e^{2 i \tilde{J} t \beta } (e^{-2 i \phi
}-\beta ) \beta }{\beta ^2+2}-\frac{e^{-2 i \tilde{J} t \alpha }
\alpha (\alpha +e^{-2 i \phi
})}{\alpha ^2+2})\nonumber\\
&\times&(\frac{e^{2 i \tilde{J} t \alpha } (e^{2 i \phi } \alpha +1)}{\alpha ^2+2}+
\frac{e^{-2 i \tilde{J} t \beta } (1-e^{2 i \phi } \beta
)}{\beta ^2+2}-\frac{1}{2})\nonumber\\
&+&(\frac{e^{-2 i \tilde{J} t \alpha } (\alpha +e^{-2 i \phi })}{\alpha ^2+2}-\frac{1}{2} e^{-2 i \phi }+
\frac{e^{2 i \tilde{J} t \beta
} (e^{-2 i \phi }-\beta )}{\beta ^2+2})\\
&\times&(\frac{e^{-2 i \tilde{J} t \beta } \beta (1-e^{2 i \phi } \beta )}{\beta ^2+2}-
\frac{e^{2 i \tilde{J} t \alpha }
\alpha (e^{2 i \phi } \alpha +1)}{\alpha ^2+2})|\nonumber\\
&-&|\frac{e^{2 i \tilde{J} t \alpha } (\alpha +e^{2 i \phi })}{\alpha
^2+2}+\frac{1}{2} e^{2 i \phi }+\frac{e^{-2 i \tilde{J} t \beta } (e^{2 i \phi }-\beta )}{\beta ^2+2}|\nonumber\\
&\times&|\frac{e^{2 i \tilde{J} t \alpha } (e^{2 i \phi } \alpha
+1)}{\alpha ^2+2}+\frac{e^{-2 i \tilde{J} t \beta } (1-e^{2 i \phi } \beta )}{\beta
^2+2}+\frac{1}{2}|).\nonumber\end{aligned}$$
[99]{} C. H. Bennett and D. P. DiVincenzo, Nature (London) **404**, 247 (2000). A. Bayat and S. Bose, Phys. Rev. A **81**, 012304 (2010). D. P. DiVincenzo, Fortschr. Phys. **48**, 771 (2000). S. Bose, Phys. Rev. Lett. [**91**]{}, 207901 (2003). J. Eisert [*et. al.*]{}, Phys. Rev. Lett. [**93**]{}, 190402 (2004); M. Christandl [*et. al.*]{}, Phys. Rev. Lett. **92**, 187902 (2004). V. Giovannetti and D. Burgarth, Phys. Rev. Lett. [**96**]{}, 030501 (2006); J. Fitzsimons and J. Twamley, Phys. Rev. Lett. [**97**]{}, 090502 (2006); A. Kay, Phys. Rev. Lett. [**98**]{}, 010501 (2007). T. J. Osborne and N. Linden, Phys. Rev. A **69**, 052315 (2004); A. Lyakhov and C. Bruder, Phys. Rev. B [**74**]{}, 235303 (2006); A. Bayat and V. Karimipour, Phys. Rev. A [**71**]{}, 042330 (2005); D. Burgarth, S. Bose, Phys. Rev. A 73, 062321 (2006); L. Zhou, J. Lu, T. Shi and C. P. Sun, quant-ph/0608135. D. Burgarth and S. Bose, Phys. Rev. A [**71**]{}, 052315 (2005); M. Avellino, A. J. Fisher, S. Bose, Phys. Rev. A [**74**]{}, 012321 (2006); A. Bayat and S. Bose, Advances in Mathematical Physics, **2010**, 127182 (2010); A. Bayat, D. Burgarth, S. Mancini and S. Bose, Phys. Rev. A **77**, 050306(R) (2008). M. Asoudeh and V. Karimipour, Phys. Rev. A **73**, 062109 (2006) K. Eckert, O. R. Isart, and A. Sanpera, New J. Phys. **9**, 155 (2007); J. Ren and S. Zhu, Phys. Rev. A **81**, 014302 (2010). C. F. Hirjibehedin, C. P. Lutz and A. J. Heinrich , Science **312**, 1021 (2006). K. V. Kavokin, Phys. Rev. B **64**, 075305 (2001). I. Dzyaloshinski, J. Phys. Chem. Solids **4**, 241 (1958). T. Moriya, Phys. Rev. **117**, 635 (1960). T. Moriya, Phys. Rev. Lett. **4**, 228 (1960). F. Kheirandish, S. J. Akhtarshenas, and H. Mohammadi, Phys. Rev. A **77**, 042309 (2008). F. Kheirandish, S. J. Akhtarshenas, and H. Mohammadi, Eur. Phys. J. D **57**, 1 (2010). X. Wang, Phys. Lett. A **281**, 101 (2001) H. Mikeska and A. Kolezhuk, Lect. Notes Phys. **645**, 1 (2004). O. Derzhko and T. Verkholyak, e-print arXiv:0712.2507. L. Amico, A. Osterloh, F. Plastina, G. Palma, and R. Fazio, Phys. Rev. A **69**, 022304 (2004). W. K. Wootters, Phys. Rev. Lett. **80**, 2245 (1998). R. Jafari and A. Langari arXiv:0812.1862. K. Fabricius, U. Low, and J. Stolze, Phys. Rev. B **55**, 5833 (1997).
[^1]: [email protected]
[^2]: [email protected]
[^3]: [email protected]
[^4]: [email protected]
[^5]: It is important to note that, in the presence of DM interaction the phase diagram of the chain may be changed [@langari] and hence we must be care on employing the words such that phase transition and so on. Whereas, in this paper, we modeled the XXZ chain with DM interaction by modified with XXZ chain and hence we can still borrow this words from the phase transition terminology.
[^6]: since the spin $0^\prime$ does not coupled to the chain it remains untouched during the evolution of the system. Thus, unitary transformation affects only on the sites (0, 1, 2).
|
{
"pile_set_name": "ArXiv"
}
|
---
address:
- 'Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA'
- 'Institute for Mathematical Problems in Biology, Russian Academy of Sciences, Pushchino, Moscow Region, Russia'
author:
- 'Yu. G. Zarhin'
title: Del Pezzo surfaces of degree $2$ and jacobians without complex multiplication
---
Notations and Statements
========================
In a series of his articles [@ZarhinMRL; @ZarhinMRL2; @ZarhinTexel; @ZarhinMMJ; @ZarhinBSMF] the author constructed explicitly $m$-dimensional abelian varieties without non-trivial endomorphisms for every $m>1$. This construction may be described as follows. Let $K_a$ be an algebraic closure of a perfect field $K$ with $\fchar(K)\ne 2$. Let $n=2m+1$ or $2m+2$. Let us choose an $n$-element set $\RR \in
K_a$ that constitutes a Galois orbit over $K$ and assume, in addition, that the Galois group of $K(\RR)$ over $K$ is “big" say, coincides with full symmetric group $\Sn$ or the alternating group $\An$. Let $f(x) \in K[x]$ be the irreducible polynomial of degree $n$, whose set of roots coincides with $\RR$. Let us consider the hyperelliptic curve $C_f:y^2=f(x)$ over $K_a$ and let $J(C_f)$ be its jacobian which is the $m$-dimensional abelian variety. Then the ring $\End(J(C_f))$ of all $K_a$-endomorphisms of $J(C_f)$ coincides with $\Z$ if either $n>6$ or $\fchar(K)\ne
3$.
The aim of this paper is to construct abelian threefolds without complex multiplication, using jacobians of non-hyperelliptic curves of genus $3$. It is well-known that these curves are smooth plane quartics and closely related to Del Pezzo surfaces of degree $2$. (We refer to [@Manin; @Iskovskikh; @IskovskikhSh; @Demazure; @Dolgachev; @Dolgachev2; @T] for geometric and arithmetic properties of Del Dezzo surfaces. In particular, relations between the degree $2$ case and plane quartics are discussed in detail in [@Demazure; @Dolgachev; @Dolgachev2]). On the other hand, Del Pezzo surfaces of degree $2$ could be obtained by blowing up seven points on the projective plane $\P^2$ when these points are in [*general position*]{}, i.e., no three points lie on a one line, no six on a one conic ([@Iskovskikh §3], [@Demazure Th. 1 on p. 27]).
In order to describe our construction, let us start with the projective plane $\P^2$ with homogeneous coordinates $(x:y:z)$. Let us consider a $7$-element set $B\subset \P^2(K_a)$ of points in general position and assume that the absolute Galois group $\Gal(K)$ of $K$ permutes elements of $B$ in such a way that $B$ constitutes a Galois orbit. We write $Q_B$ for the $6$-dimensional $\F_2$-vector space of maps $\varphi:B \to \F_2$ with $\sum_{b\in
B}\varphi(b)=0$. The action of $\Gal(K)$ on $B$ provides $Q_B$ with the natural structure of $\Gal(K)$-module. Let $G_B$ be the image of $\Gal(K)$ in the group $\Perm(B)\cong \SS_7$ of all permutations of $B$. Clearly, $Q_B$ carries a natural structure of faithful $\Perm(B)$-module and the structure homomorphism $\Gal(K)
\to \Aut(Q_B)$ coincides with the composition of $\Gal(K)\twoheadrightarrow G_B$ and $G_B\subset\Perm(B)\hookrightarrow \Aut(Q_B)$.
Let $H_B$ be the $K_a$-vector space of homogeneous cubic forms in $x,y,z$ that vanish on $B$. It follows from proposition 4.3 and corollary 4.4(i) in Ch. 5, §4 of [@Har] that $H_B$ is $3$-dimensional and $B$ coincides with the set of common zeros of elements of $H_B$. Since $B$ is $\Gal(K)$-invariant, $H_B$ is defined over $K$, i.e., it has a $K_a$-basis $u,v,w$ such that the forms $u,v,w$ have coefficients in $K$.
We write $V(B)$ for the Del Pezzo surface of degree $2$ obtained by blowing up $B$. Then $V(B)$ is a smooth projective surface that is defined over $K$ (see Remark 19.5 on pp. 89–90 of [@Manin]). We write $$g_B:V(B)\to \P^2$$ for the corresponding birational map defined over $K$. Recall that for each $b\in B$ its preimage $E_b$ is a a smooth projective rational curve with self-intersection number $-1$. By definition, $g_B$ establishes a $K$-biregular isomorphism between $V(B)\setminus \bigcup_{b\in B}E_b$ and $\P^2\setminus B$. Clearly, $$\sigma(E_b)=E_{\sigma(b)}\quad \forall \ b\in B,
\sigma\in\Gal(K).$$ Let $\Omega_{V(B)}$ be the canonical (invertible) sheaf on $V(B)$. Let us consider the line $L:z=0$ as a divisor in $\P^2$. Clearly, $B$ does not meet the $K$-line $L$; otherwise, the whole $\Gal(K)$-orbit $B$ lies in $L$ which is not true, since no $3$ points of $B$ lie on a one line. It is known that $$K_{V(B)}:=-3g_B^*(L)+\sum_{b\in B}E_b=-g_B^*(3L)+\sum_{b\in B}E_b$$ is a [*canonical divisor*]{} on $V(B)$. Clearly, for each form $q\in
H_B$ the rational function $\frac{q}{z^3}$ on $\P^2$ satisfies $\div(\frac{q}{z^3})+3L\ge 0$, i.e., $\frac{q}{z^3}\in
\Gamma(\P^2,3L)$. Also $\frac{q}{z^3}$ is defined and vanishes at every point of $B$. It follows easily that $\frac{q}{z^3}$ (viewed as rational function on $V(B)$) lies in $\Gamma(V(B),3g_B^*(L)-\sum_{b\in B}E_b)=\Gamma(V(B), -K_{V(B)})$. Since $\Gamma(V(B), -K_{V(B)})$ is $3$-dimensional [@Manin theorem 24.5 on p. 121], $$\Gamma(V(B), -K_{V(B)})=K_a \cdot \frac{u}{z^3}\oplus K_a \cdot \frac{v}{z^3}\oplus
K_a \cdot \frac{w}{z^3}\ .$$ Using proposition 4.3 in [@Har Ch. 5, §4], one may easily get a well-known fact that the sections of $\Gamma(V(B), -K_{V(B)})$ have no common zeros on $V(B)$. This gives us a regular anticanonical map $$\pi:V(B)\stackrel{g_B}{\longrightarrow}\P^2
\stackrel{(u:v:w)}{\longrightarrow}\P^2$$ which is obviously defined over $K$. It is known that $\pi$ is a regular double cover map, whose ramification curve is a smooth quartic $$C_B \subset \P^2$$ (see [@Demazure pp. 67–68], [@Dolgachev Ch. 9]). Clearly, $C_B$ is a genus $3$ curve defined over $K$. Let $J(B)$ be the jacobian of $C_B$; clearly, it is a three-dimensional abelian variety defined over $K$. We write $\End(J(B))$ for the ring of $K_a$-endomorphisms of $J(B)$.
The following assertion is based on Lemmas 1-2 on pp. 161–162 of [@Dolgachev].
\[points\] Let $J(B)_2$ be the kernel of multiplication by $2$ in $J(B)(K_a)$. Then the Galois modules $J(B)_2$ and $Q_B$ are canonically isomorphic.
Using Lemma \[points\] and results of [@ZarhinMRL; @ZarhinBSMF], one may obtain the following statement.
\[endo\] Let $B\subset \P^2(K_a)$ be a $7$-element set of points in general position. Assume that $\Gal(K)$ permutes elements of $B$ and the image of $\Gal(K)$ in $\Perm(B)\cong \SS_7$ coincides either with the full symmetric group $\SS_7$ or with the alternating group $\A_7$. Then $\End(J(B))=\Z$.
This leads to a question: how to construct such $B$ in general position? The next lemma provides us with desired construction.
\[general\] Let $f(t) \in K[t]$ be a separable irreducible degree $7$ polynomial, whose Galois group $\Gal(f)$ is either $\SS_7$ or $\A_7$. Let $\RR_f\subset K_a$ be the $7$-element set of roots of $f$. Then the $7$-element set $$B_f =\{(\alpha^3:\alpha:1)\mid \alpha\in\RR_f\}\subset
\P^2(K_a)$$ is in general position.
Clearly, $\Gal(K)$ permutes transitively elements of $B_f$ and the image of $\Gal(K)$ in $\Perm(B)$ coincides either with $\SS_7$ or with $\A_7$; in particular, $B_f$ constitutes a Galois orbit. This implies the following statement.
Let $f(t) \in K[t]$ be a separable irreducible degree $7$ polynomial, whose Galois group $\Gal(f)$ is either $\SS_7$ or $\A_7$. Then $\End(J(B_f))=\Z$.
Proofs
======
Let $\Pic(V(B))$ be the Picard group of $V(B)$ over $K_a$. It is known that $\Pic(V(B))$ is a free commutative group of rank $8$ provided with the natural structure of Galois module. More precisely, it has canonical generators $l_0=$ the class of $g_B^*(L)$ and $\{l_b\}_{b\in B}$ where $l_b$ is the class of the exceptional curve $E_b$. Clearly, $l_0$ is Galois invariant and $$\sigma(l_b)=l_{\sigma(b)} \quad \forall \ b\in B, \sigma\in
\Gal(K).$$ Clearly, the class of $K_{V(B)}$ equals $-3l_0+\sum_{b\in B}l_b$ and obviously is Galois-invariant. There is a non-degenerate Galois invariant symmetric intersection form $$(, ): \Pic(V(B)) \times \Pic(V(B)) \to \Z.$$ In addition (ibid), $$(l_0, l_0)=1, (l_b,l_0)=0, (l_b,l_b)=-1, (l_b,l_{b'})=0 \quad
\forall\ b \ne b'.$$ Clearly, the orthogonal complement $\Pic(V(B))_0$ of $K_{V(B)}$ in $\Pic(V(B))$ coincides with $$\{a_0 l_0+\sum_{b\in B}a_b l_b\mid a_0,a_b \in\Z, -3 a_0+\sum_{b\in B}a_b
=0\};$$ it is a Galois-invariant [*pure*]{} free commutative subgroup of rank $7$.
Notice that one may view $C_B$ as a $K$-curve on $V(B)$ [@Dolgachev p. 160]. Then the inclusion map $C_B\subset V(B)$ induced the homomorphism of Galois modules $$r:\Pic(V(B)) \to \Pic(C_B)$$ where $\Pic(C_B)$ is the Picard group of $C_B$ over $K_a$. Recall that $J(B)(K_a)$ is a Galois submodule of $\Pic(C_B)$ that consists of divisor classes of degree zero. In particular, $J(B)_2$ coincides with the kernel $\Pic(C_B)_2$ of multiplication by $2$ in $\Pic(C_B)$. It is known (Lemma 1 on p. 161 of [@Dolgachev]) that $$r(\Pic(V(B))_0)\subset \Pic(C_B)_2=J(B)_2.$$ This gives rise to the homomorphism $$\bar{r}:\Pic(C_B)_0/2\Pic(C_B)_0 \to J(B)_2, \quad
D+2\Pic(C_B)_0\mapsto r(D)$$ of Galois modules. By Lemma 2 on pp. 161-162 of [@Dolgachev], the kernel of $\bar{r}$ is as follows. The intersection form on $\Pic(V(B))$ defines by reduction modulo $2$ a symmetric bilinear form $$\begin{gathered}
\psi:\Pic(V(B))/2\Pic(V(B)) \times \Pic(V(B))/2\Pic(V(B))
\to\Z/2\Z=\F_2,\\ \quad D+2\Pic(V(B)) ,D'+2\Pic(V(B))\mapsto
(D,D')+2\Z\end{gathered}$$ and we write $$\psi_0:\Pic(V(B))_0/2\Pic(V(B))_0 \times \Pic(V(B))_0/2\Pic(V(B))_0
\to\F_2$$ for the restriction of $\psi$ to $\Pic(V(B))_0$. Then the kernel (radical) of $\psi_0$ coincides with $\ker(\bar{r})$. (The same Lemma also asserts that $\bar{r}$ is surjective.)
Let us describe explicitly the kernel of $\psi_0$. Since $\Pic(V(B))_0$ is a pure subgroup of $\Pic(V(B))$, we may view $\Pic(V(B))_0/2\Pic(V(B))_0$ as a $7$-dimensional $\F_2$-vector subspace (even Galois submodule) in $\Pic(V(B))/2\Pic(V(B))$. Let $\bar{l}_0$ (resp. $\bar{l}_b$) be the image of $l_0$ (resp. $l_b$) in $\Pic(V(B))/2\Pic(V(B))$. Then $\{\bar{l}_0,\{\bar{l}_b\}_{ b\in B}\}$ constitute an [*orthonormal*]{} (with respect to $\psi$) basis of the $\F_2$-vector space $\Pic(V(B))/2\Pic(V(B))$. Clearly, $\psi$ is non-degenerate. It is also clear that $$\Pic(V(B))_0/2\Pic(V(B))_0=\{a_0 \bar{l}_0+\sum_{b\in B}a_b \bar{l}_b\mid a_0,a_b \in\F_2, a_0+\sum_{b\in B}a_b
=0\}$$ is the orthogonal complement of [*isotropic*]{} $$\bar{v}_0=\bar{l}_0+\sum_{b\in B}\bar{l}_b$$ in $\Pic(V(B))/2\Pic(V(B))$ with respect to $\psi$. Notice that $\bar{v}_0$ is Galois-invariant. The non-degeneracy of $\psi$ implies that the kernel of $\psi_0$ is the Galois-invariant one-dimensional $\F_2$-subspace generated by $\bar{v}_0$.
This gives us the injective homomorphism $$(\Pic(V(B))_0/2\Pic(V(B))_0)/\F_2 \bar{v}_0 \hookrightarrow
J(B)_2$$ of Galois modules; dimension arguments imply that it is an isomorphism. So, in order to finish the proof, it suffices to construct a surjective homomorphism $\Pic(V(B))_0/2\Pic(V(B))_0
\twoheadrightarrow Q_B$ of Galois modules, whose kernel coincides with $\F_2 \bar{v}_0$. In order to do that, let us consider the homomorphism $$\kappa:\Pic(V(B))_0/2\Pic(V(B))_0 \to Q_B$$ that sends $z=a_0 \bar{l}_0+\sum_{b\in B}a_b \bar{l}_b$ to the function $\kappa(z):b \mapsto a_b+a_0$. Since $$a_0+\sum_{b\in
B}a_b=0 \ \text{and} \ \#(B)a_0 =7a_0=a_0\in\F_2,$$ indeed we have $\kappa(z)\in Q_B$. It is also clear that $\kappa(z)$ is identically zero if and only if $a_0=a_b \ \forall \ b$, i.e. $z=0$ or $\bar{v}_0$. Clearly, $\kappa$ is a surjective homomorphism of Galois modules and $\ker(\kappa)=\F_2 \bar{v}_0$.
We will use a notation $(x:y:z)$ for homogeneous coordinates on $\P^2$. Suppose that here are three points of $B_f$ that lie on a line $ax +by+cz=0$. This means that there are distinct roots $\alpha_1,\alpha_2,\alpha_3$ of $f$ and elements $a,b,c \in K_a$ such that all $a \alpha_i^3+b \alpha_i+c=0$ and, at least, one of $a,b,c$ does not vanish. It follows that the polynomial $at^3+bt+c\in K_a[t]$ is not identically zero and has three distinct roots $\alpha_1,\alpha_2,\alpha_3$. This implies that $a\ne 0$ and $$at^3+bt+c=a(t-\alpha_1)(t-\alpha_2)(t-\alpha_3).$$ It follows that $\alpha_1+\alpha_2+\alpha_3=0$. Let us denote the remaining roots of $f$ by $\alpha_4,\alpha_5,\alpha_6,\alpha_7$. Clearly, $\Gal(K)$ acts $3$-transitively on $\RR_f$. This implies that there exists $\sigma\in\Gal(K)$ such that $$\sigma(\alpha_1)=\alpha_4,\sigma(\alpha_2)=\alpha_2,\sigma(\alpha_3)=\alpha_3$$ and therefore $\alpha_2+\alpha_3+\alpha_4=\sigma(\alpha_2+\alpha_3+\alpha_1)=0$ and therefore $\alpha_1=\alpha_4$ which is not the case. The obtained contradiction proves that no three points of $B_f$ lie on a one line.
Suppose that six points of $B_f$ lie on a one conic. Let $$a_0 z^2+a_1 yz+a_2 y^2+a_3 xz+a_4 xy + a_6x^2 =0$$ be an equation of the conic. Then not all $a_i$ do vanish and there are six distinct roots $\alpha_1, \cdots ,\alpha_6$ of $f$ such that all $a_6 \alpha_k^6+\sum_{i=1}^4 a_i\alpha_k^i=0$. This implies that the polynomial $a_6t^6+\sum_{i=1}^4 a_i t^i$ is not identically zero and has $6$ distinct roots $\alpha_1, \cdots \alpha_6$. It follows that $a_6 \ne 0$ and $$f(t)=a_6 \prod_{i=1}^6 (t-\alpha_i).$$ This implies that $\sum_{i=1}^6 \alpha_i=0$. Since the sum of all roots of $f$ lies in $K$, the remaining seventh root of $f$ lies in $K$. This contradicts to the irreducibility of $f$. The obtained contradiction proves that no six points of $B_f$ lie on a one conic.
\[endo1\] Let $B\subset \P^2(K_a)$ be a $7$-element set of points in general position. Assume that $\Gal(K)$ permutes elements of $B$ and the image of $\Gal(K)$ in $\Perm(B)\cong \SS_7$ coincides either with the full symmetric group $\SS_7$ or with with the alternating group $\A_7$; in particular, $B$ consitutes a Galois orbit. Then either $\End(J(B))=\Z$ or $\fchar(K)>0$ and $J(B)$ is a supersingular abelian variety.
Recall that $G_B$ is the image of $\Gal(K)$ in $\Perm(B)$. By assumption, $G_B=\SS_7$ or $\A_7$. It is known [@ZarhinTexel Ex. 7.2] that the $G_B$-module $Q_B$ is [*very simple*]{} in the sense of [@ZarhinTexel; @ZarhinVery; @ZarhinMMJ]. In particular, $$\End_{G_B}(Q_B)=\F_2.$$ The surjectivity of $\Gal(K)\twoheadrightarrow G_B$ implies that the $\Gal((K)$-module $Q_B$ is also very simple. Applying Lemma \[points\], we conclude that the $\Gal((K)$-module $J(B)_2$ is also very simple. Now the assertion follows from lemma 2.3 of [@ZarhinTexel].
In light of Lemma \[endo1\], we may and will assume that $\fchar(K)>0$ and $J(B)$ is a supersingular abelian variety. We need to arrive to a contradiction. Replacing if necessary $K$ by its suitable quadraric extension we may and will assume that $G_B=\A_7$. Adjoining to $K$ all $2$-power roots of unity, we may and will assume that $K$ contains all $2$-power roots of unity and still $G_B=\A_7$. It follows from Lemma \[points\] that $\A_7$ is isomorphic to the image of $\Gal(K) \to \Aut_{\F_2}(J(B)_2)$ and the $\A_7$-module $J(B)_2$ is very simple; in particular, $\End_{\A_7}(J(B)_2)=\F_2$. Applying Theorem 3.3 of [@ZarhinBSMF], we conclude that there exists a central extension $G_1 \twoheadrightarrow \A_7$ such that $G_1$ is perfect, $\ker(G_1\twoheadrightarrow \A_7)$ is a central cyclic subgroup of order $1$ or $2$ and there exists a symplectic absolutely irreducible $6$-dimensional representation of $G_1$ in characteristic zero. This implies (in notations of [@Atlas]) that either $G_1\cong \A_7$ or $G_1\cong 2.\A_7$. However, the table of characters on p. 10 of [@Atlas] tells us that neither $\A_7$ nor $2.\A_7$ admits a [*symplectic*]{} absolutely irreducible $6$-dimensional representation in characteristic zero. The obtained contradiction proves the Theorem.
Explicit formulas
=================
In this section we describe explicitly $H_B$ when $B=B_f$. We have $$f(t)=\sum_{i=0}^7 c_i t^i \in K[t], \ c_7 \ne 0.$$ We are going to describe explicitly cubic forms that vanish on $B_f$. Clearly, $u:=xz^2-y^3$ and $v:=c_7 x^2 y+ c_6 x^2 z+ c_5 x y^2+c_4 xyz+c_3
x z^2+ c_2 y^2 z +c_1 y z^2 + c_0 z^3$ vanish on $B_f$. In order to find a third vanishing cubic form, let us define a polynomial $h(t)\in K[t]$ as a (non-zero) remainder with respect to division by $f(t)$: $$t^9 -h(t) \in f(t) K[t],\ \deg(h)<\deg(f)=7.$$ We have $$h(t)=\sum_{i=0}^6 d_i t^i\in K[t].$$ For all roots $\alpha$ of $f$ we have $$0=\alpha^9-h(\alpha)=\alpha^9-\sum_{i=6}^6 d_i \alpha^i.$$ This implies that the cubic form $w=x^3-d_6 x^2 z-d_5 x
y^2-d_4xyz-d_3 x z^2- d_2 y^2 z -d_1 y z^2 - d_0 z^3$ vanishes on $B_f$. Since $u,v,w$ have $x$-degree 1,2,3 respectively, they are linearly independent over $K_a$ and therefore constitute a basis of $3$-dimensional $H_{B_f}$.
Now assume (till the end of this Section) that $\fchar(K)\ne
3$.[^1] Since $C_{B_f}$ is the ramification curve for $\pi$, it follows that $$g_B(C_{B_f})=\left\{(x:y:z), \begin{vmatrix}
u_x & u_y & u_z\\
v_x & v_y & v_z\\
w_x & w_y & w_z
\end{vmatrix}=0\right\}\subset \P^2$$ is a singular sextic which is $K$-birationally isomorphic to $C_{B_f}$. (See also [@Dolgachev proposition 2 on p. 167].)
Another proof
=============
The aim of this Section is to give a more elementary proof of Theorem \[endo\] that formally does not refer to Lemma 2 of [@Dolgachev Lemma 2 on pp. 161–162] (and therefore does not make use of the Smith theory. However, our arguments are based on ideas of [@Dolgachev Ch. IX].) In order to do that, we just need to prove Lemma \[points\] under an additional assumption that the image of $\Gal(K)$ in $\Perm(B)$ is “very big".
\[pointsB\] Let $J(B)_2$ be the kernel of multiplication by $2$ in $J(B)(K_a)$. Suppose that $G_B$ coincides either with $\Perm(B)$ or with $\A_7$. Then the Galois modules $J(B)_2$ and $Q_B$ are isomorphic.
Let $g_0: V(B) \to V(B)$ be the Geiser involution [@Demazure p. 66– 67], i.e., the biregular covering transformation of $\pi$. Clearly, $g_0$ is defined over $K$. This implies that if $E$ is an irreducible $K_a$-curve on $V(B)$ then $E$ and $g_0(E)$ have the same stabilizers in $\Gal(K)$. Clearly, different points $b_1$ and $b_2$ of $B$ have different stabilizers in $G_B$ and therefore in $\Gal(K)$. This implies that $g_0(E_{b_1}) \ne E_{b_2}$, since the stabilizers of $g_0(E_{b_1})$ and $E_{b_2}$ coincide with the stabilizers of $b_1$ and $b_2$ respectively. This implies that the lines $$\pi(E_{b_1}), \pi(E_{b_2}) \subset \P^2,$$ which are bitangents to $C_B$ [@Demazure p. 68], do not coincide.
For each $b\in B$ we write $D_b$ for the effective degree $2$ divisor on the plane quartic $C_B$ such that $2 D_b$ coincides with the intersection of $C_B$ and $\pi(E_{b})$; it is well known that (the linear equivalence class of) $D_b$ is a theta characteristic on $C_B$. It is also clear that $$\sigma(D_b)=D_{\sigma(b)} \quad \forall\ \sigma\in\Gal(K),\ b\in B.$$ Clearly, if $b_1\ne b_2$ then $D_{b_1}\ne D_{b_2}$ and the divisors $2D_{b_1}$ and $2D_{b_2}$ are linearly equivalent. On the other hand, $D_{b_1}$ and $D_{b_2}$ are [*not*]{} linearly equivalent. Indeed, if $D_{b_1}-D_{b_2}$ is the divisor of a rational function $s$ then $s$ is a non-constant rational function on $C_B$ with, at most, two poles. This implies that either $C_B$ is either a rational (if $s$ has exactly one pole) or hyperelliptic (if $s$ has exactly two poles). Since a smooth plane quartic is neither rational nor hyperelliptic, $D_{b_1}-D_{b_2}$ is not a principal divisor.
Let $(\Z^B)^0$ be the free commutative group of all functions $\phi:B \to\Z$ with $\sum_{b\in B}\phi(b)=0$. Clearly, $(\Z^B)^0$ is provided with the natural structure of $\Gal(K)$-module and there is a natural isomorphism of $\Gal(K)$-modules $$(\Z^B)^0/2 (\Z^B)^0 \cong Q_B.$$ Let us consider the homomorphism of commutative groups ${\mathfrak r}: (\Z^B)^0 \to \Pic(C_B)$ that sends a function $\phi$ to the linear equivalence class of $\sum_{b\in B} \phi(b)D_b$. Clearly, $${\mathfrak
r}((\Z^B)^0)\subset J(B)_2\subset \Pic(B)$$ and therefore ${\mathfrak r}$ kills $2\cdot (\Z^B)^0$. On the other hand, the image of ${\mathfrak r}$ contains the (non-zero) linear equivalence class of $D_{b_1}-D_{b_2}$. This implies that ${\mathfrak r}$ is not identically zero and we get a non-zero homomorphism of $\Gal(K)$-modules $$\bar{\mathfrak r}: Q_B \cong (\Z^B)^0/2 (\Z^B)^0 \to J(B)_2.$$ It is well-known that our assumptions on $G_B$ imply that the $G_B$-module $Q_B$ is (absolutely) simple and therefore $Q_B$, viewed as Galois module, is also simple. This implies that $\bar{\mathfrak r}$ is injective. Since the $\F_2$-dimensions of both $Q_B$ and $J(B)_2$ equal to $6$ and therefore coincide, we conclude that $\bar{\mathfrak r}$ is an isomorphism.
Added in translation
====================
The following assertion is a natural generalization of Lemma \[general\].
\[General\] Suppose that $E\subset \P^2$ is an absolutely irreducible cubic curve that is defined over $K$. Suppose that $B\subset E(K_a)$ is a a $7$-element set that is a $\Gal(K)$-orbit. Let us assume that the image $G_B$ of $\Gal(K)$ in the group $\Perm(B)$ of all permutations of $B$ coincides either with $\Perm(B)\cong\SS_7$ or with the alternating group $\A_7$. Then $B$ is in general position.
Clearly, $\Gal(K)$ acts $3$-transitively on $B$.
Step 1. Suppose that $D$ is a line in $\P^2$ that contains three points of $B$ say, $$\{P_1,P_2,P_3\}\subset \{P_1,P_2,P_3, P_4,P_5,P_6,P_7\}=B.$$ Clearly, $D\bigcap E=\{P_1,P_2,P_3\}$. There exists $\sigma
\in\Gal(K)$ such that $\sigma(\{P_1,P_2,P_3\})=\{P_1,P_2,P_4\}$. It follows that the line $\sigma(D)$ contains $\{P_1,P_2,P_4\}$ and therefore $\sigma(D)\bigcap E=\{P_1,P_2,P_4\}$. In particular, $\sigma(D)\ne D$. However, the distinct lines $D$ and $\sigma(D)$ meet each other at [*two*]{} distinct points $P_1$ and $P_2$. Contradiction.
Step 2. Suppose that $Y$ is a conic in $\P^2$ such that $Y$ contains six points of $B$ say, $\{P_1,P_2,P_3,
P_4,P_5,P_6\}=B\setminus\{P_7\}$. Clearly, $Y\bigcap
E=B\setminus\{P_7\}$. If $Y$ is reducible, i.e., is a union of two lines $D_1$ and $D_2$ then either $D_1$ or $D_2$ contains (at least) three points of $B$, which is not the case, thanks to Step 1. Therefore $Y$ is [*irreducible*]{}.
There exists $\sigma \in\Gal(K)$ such that $\sigma(P_1)=P_7$. Then $\sigma(P_7)=P_i $ for some positive integer $i\le 6$. This implies that $\sigma(B\setminus\{P_7\})=B\setminus\{P_i\}$ and the irreducible conic $\sigma(Y)$ contains $B\setminus\{P_i\}$. Clearly, $\sigma(Y)\bigcap
E=B\setminus\{P_i\}$ contains $P_7$. In particular, $\sigma(Y)\ne
Y$. However, both conics contain the $5$-element set $B\setminus\{P_i,P_7\}$. Contradiction.
[99]{}
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, R. A. Wilson, Atlas of finite groups. Clarendon Press, Oxford, 1985.
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I. Dolgachev, D. Ortland, Point sets in projective spaces and theta functions, Astérisque [**165**]{} (1986).
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R. Hartshorne, Algebraic Geometry, GTM [**52**]{}, Springer-Verlag, 1977.
V. A. Iskovskikh, [*Minimal models of rational surfaces over arbitrary fields*]{}. Izv. Akad. Nauk Ser. Mat. [**43**]{} (1979), 19–43; Math. USSR-Izv. [**14**]{} (1980), 17–39.
V. A. Iskovskikh, I.R. Shafarevich, [*Algebraic surfaces*]{}. Algebraic geometry, II, 127–262, Encyclopaedia Math. Sci., [**35**]{}, Springer, Berlin, 1996.
Yu. I. Manin, Cubic forms, second edition, North Holland, 1986.
Yu. I. Manin, M. A. Tsfasman, [*Rational varieties: algebra, geometry, arithmetic*]{}. Uspekhi Mat. Nauk [**41**]{} (1986), no. 2(248), 43–94; Russian Math. Surveys [**41**]{} (1986), no. 2, 51–116.
Yu. G. Zarhin, [*Hyperelliptic jacobians without complex multiplication*]{}. Math. Res. Letters [**7**]{} (2000), 123–132.
Yu. G. Zarhin, [*Hyperelliptic jacobians and modular representations*]{}. In: Moduli of abelian varieties (eds. C. Faber, G. van der Geer and F. Oort). Progress in Math., vol. [**195**]{} (Birkhäuser, 2001), pp. 473–490.
Yu. G. Zarhin, [*Hyperelliptic jacobians without complex multiplication in positive characteristic*]{}. Math. Res. Letters [**8**]{} (2001), 429–435.
Yu. G. Zarhin, [*Very simple $2$-adic representations and hyperelliptic jacobians*]{}. Moscow Math. J. [**2**]{} (2002), issue 2, 403-431.
Yu. G. Zarhin, [*Very simple representations: variations on a theme of Clifford*]{}. In: Progress in Galois Theory (H. Völklein, T. Shaska eds.), Springer Verlag, 2005, pp. 151–168.
Yu. G. Zarhin, [*Non-supersingular hyperelliptic jacobians*]{}. Bull. Soc. Math. France [**132**]{} (2004), 617–634.
Yu. G. Zarhin, [*Del Pezzo surfaces of degree $2$ and jacobians without complex multiplication*]{} (Russian). Trudy St. Petersburg Mat. Obsch. [**11**]{} (2005), 81–91.
[^1]: This condition was inadvertently omitted in the Russian version [@ZarhinVR].
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- 'Weslem L. Silva and Rafael M. Souza'
title: 'Periodic points on T - fiber bundles over the circle'
---
Introduction
============
Let $f: M \to M$ be a map and $x \in M$, where $M$ a compact manifold. The point $x$ is called a periodic point of $f$ if there exists $n \in \mathbb{N}$ such that $f^{n}(x) = x,$ in this case $x$ a periodic point of $f$ of period $n.$ The set of all $\{x \in M| \hbox{ x is periodic} \}$ is called the set of periodic points of $f$ and is denoted by $P(f).$
If $M$ is a compact manifold then the Nielsen theory can be generalized to periodic points. Boju Jiang introduced (Chapter 3 in [@Jiang] ) a Nielsen-type homotopy invariant $NF_{n}(f)$ being a lower bound of the number of n-periodic points, for each $g$ homotopic to $f;$ $ Fix(g^{n}) \geq NF_{n}(f).$ In case $dim(M) \geq 3$, $M$ compact PL- manifold, then any map $f: M \to M$ is homotopic to a map $g$ satisfying $ Fix(g^{n}) = NF_{n}(f) $, this was proved in [@Jezierski].
Consider a fiber bundle $F \to M \stackrel{p}{\to} B$ where $F,M, B$ are closed manifolds and $f: M \to M$ a fiber-preserving map over $B.$ In natural way is to study periodic points of $f$ on $M,$ that is, given $n \in \mathbb{N}$ we want to study the set $\{x \in M| f^{n}(x) = x \}.$ The our main question is; when $f$ can be deformed by a fiberwise homotopy to a map $g: M \to M$ such that $Fix(g^{n}) = \emptyset$ ?
In this paper we study periodic points of $f$ when the fiber is the torus $T$ and the base is the circle $S^{1}.$
General problem
===============
Let $F \to M \stackrel{p}{\to} B$ be a fibration and $f: M \to M$ a fiber-preserving map over $B$, where $F,M,B$ are closed manifolds. Given $n \in \mathbb{N}$, from relation $p \circ f = p,$ we obtain $p \circ f^{n} = p$, thus $f^{n}: M \to M$ is also a fiber-preserving map for each $n \in \mathbb{N}$. We want to know when $f$ can be deformed by a fiberwise homotopy to a map $g: M \to M$ such that $Fix(g^{n}) = \emptyset.$ The the following lemma give us a necessary condition to a positive answer the question above.
\[lemma1\] Let $f: M \to M$ be a fiber-preserving map. If some $k$, where $k$ divides $n$, the map $f^{k}: M \to M$ can not be deformed to a fixed point free map, by a fiberwise homotopy, then can not exists, $g \sim_{B} f ,$ such that $g^{n}: M \to M$ is a fixed point free map.
In fact, suppose that exists $g \sim_{B} f$ such that $Fix(g^{n}) = \emptyset.$ Since $Fix(g^{k}) \subset Fix(g^{n})$ and $Fix(g^{k}) \neq \emptyset$ then we have a contradiction.
Therefore, a necessary condition is that for all $k$, where $k$ divides $n$, the map $f^{k}: M \to M$ must be deformed by a fiberwise homotopy to a fixed point free map over $B.$
Note that for each $n$ the square of the following diagram is commutative; $$\xymatrix{
\ldots \ar[r] & \pi_{1}(F,x_{0}) \ar[r]^{i_{\#}} \ar[d]^{(f^{n}|_{F})_{\#}} & \pi_{1}(M,x_{0}) \ar[r]^{p_{\#}} \ar[d]^{f^{n}_{\#}}
& \pi_{1}(B,p(x_{0})) \ar[r] \ar[d]^{Id} & 0 \\
\ldots \ar[r] & \pi_{1}(F,f^{n}(x_{0})) \ar[r]^{i_{\#}} & \pi_{1}(M,f^{n}(x_{0})) \ar[r]^{p_{\#}}
& \pi_{1}(B,p(x_{0})) \ar[r] & 0 \\
}$$
Since in our case all spaces are path-connected then we will represent the generators of the groups $\pi_{1}(M,f^{n}(x_{0}))$ for each $n,$ with the same letters. The same thing we will do with $\pi_{1}(T,f^{n}(0)).$
Let ${M \times }_{B} M$ be the pullback of $p:M \to B$ by $p:M \to B$ and $p_{i}:{M \times }_{B} M \to M, i=1,2,\,$ the projections to the first and the second coordinates, respectively.
The inclusion $M \times_{B} M - \Delta
\hookrightarrow M \times_{B} M,$ where $\Delta $ is the diagonal in $M \times_{B} M,$ is replaced by the fiber bundle $q:E_{B}(M) {\rightarrow}M \times_{B}
M,$ whose fiber is denoted by $\mathcal{ F}.$ We have $\pi_{m}(E_{B}(M)) \approx \pi_{m}(M \times_{B}
M -\Delta )$ where $E_{B}(M) =\{(x,\omega) \in B
\times A^{I} | i(x) = \omega(0) \}, $ with $A = M
\times_{B} M,$ $B = M \times_{B} M - \Delta$ and $q$ is given by $q(x,\omega)= \omega(1).$
E. Fadell and S. Husseini in [@fadell] studied the problem to deform the map $f^{n},$ for each $n \in \mathbb{N},$ to a fixed point free map. They supposed that $dim(F)\geq 3$ and that $F,M,B$ are closed manifolds. The necessary and sufficient condition to deform $f^{n}$ is given by the following theorem that the proof can be find in [@fadell].
\[theorem-fadell\] Given $n \in \mathbb{N}$, the map $f^{n}: M \to M$ is deformable to a fixed point free map if and only if there exists a lift $\sigma(n)$ in the following diagram;
$$\label{diagram1.1}
\xymatrix { & \mathcal{F} \ar[d] & \mathcal{F} \ar[d] \\
& E_{B}(f^{n}) \ar[d]^{q_{f^{n}}} \ar[r]^{\bar{q}_{f^{n}}} & E_{B}(M) \ar[d]^{q} \\
M \ar[r]^{1} \ar@{-->}[ru]^{\sigma(n)} & M \ar[r]^-{(1,f^{n})} & M \times_{B} M \\ }$$
where $ E_{B}(f^{n}) \to M$ is the fiber bundle induced from $q$ by $(1,f^{n}).$
In the Theorem \[theorem-fadell\] we have $\pi_{j-1}(\mathcal{F}) \cong \pi_{j}( M \times_{B} M, M \times_{B} M - \Delta) $ $ \cong \pi_{j}(F,F-x)$ where $x$ is a point in $F.$ In this situation, that is, $dim(F) \geq 3$ the classical obstruction was used to find a cross section.
When $F$ is a surface with Euler characteristic $ \leq 0$ then by Proposition 1.6 from [@daci1] we have necessary e sufficient conditions to deform $f^{n}$ to a fixed point free map over $B.$ The next proposition gives a relation between a geometric diagram and our problem.
\[prop-1\] Let $f: M \to M$ be a fiber-preserving map over $B.$ Then there is a map $g$, $g \sim_{B} f,$ such that $Fix(g^{n}) = \emptyset$ if and only if there is a map $h_{n}:M \to M \times_{B} M - \Delta$ of the form $h_{n} = (Id,s^{n})$, where $s: M \to M,$ is fiberwise homotopic to $f$ and makes the diagram below commutative up to homotopy.
$$\label{diagrama1.1}
\xymatrix{ & M \times_{B} M - \Delta \ar @{_{(}->}[d] ^{i} \\
M \ar[r]_-{(1, f^{n})} \ar @{-->} [ru]^{h_{n}} & M \times_{B} M \\ }$$
[*Proof.*]{} $(\Rightarrow)$ Suppose that exists $g: M \to M$, $g \sim_{B} f,$ with $Fix(g^{n}) = \emptyset.$ Is enough to define $h_{n} = (Id, g^{n}),$ that is, $s = g.$ $\phantom{)}$
$(\Leftarrow)$ If there is $h_{n}$ then we have $(Id, f^{n}) \sim (Id, s^{n})$, where $s \sim_{B} f$. Therefore $Fix(s^{n}) = \emptyset.$ Thus, takes $s=g.$
Torus fiber-preserving maps {#section-2}
===========================
Let $T$ be, the torus, defined as the quotient space $ {\mathbb{R} \times \mathbb{R}}/{\mathbb{Z} \times \mathbb{Z}} $. We denote by $(x, y)$ the elements of $\mathbb{R} \times \mathbb{R}$ and by $[(x,y)]$ the elements in T.
Let $MA = \frac{T \times [0,1]}{([(x,y)],0) \sim (\left[A \left(^{x}_{y} \right) \right],1)} $ be the quotient space, where $A$ is a homeomorphism of $T$ induced by an operator in $\mathbb{R}^{2}$ that preserves $ \mathbb{Z} \times \mathbb{Z}$. The space $MA$ is a fiber bundle over the circle $S^{1}$ where the fiber is the torus. For more details on these bundles see [@daci1].
Given a fiber-preserving map $f: MA \to MA $, i.e. $p \circ f = p$ we want to compute the number $Fix(g^{n})$ for each map $g$ fiberwise homotopic to $f$.
Consider the loops in $MA$ given by; $a(t) = <[(t,0)],0> $, $b(t) = $ $ <[(0,t)],0> $ and $c(t) = <[(0,0)],t> $ for $t \in [0,1]$. We denote by $B$ the matrix of the homomorphism induced on the fundamental group by the restriction of $f$ to the fiber $T.$ From [@daci1] we have the following theorem that provides a relationship between the matrices $A$ and $B$, where $$A = { \left( \begin{array}{cc} a_{1} & a_{3} \\ a_{2} & a_{4} \\ \end{array} \right)}$$ From [@daci1] the induced homomorphism $f_{\#}: \pi_{1}(MA) \to \pi_{1}(MA) $ is given by; $f_{\#}(a) = a^{b_{1}} b^{b_{2}} $, $f_{\#}(b) = a^{b_{3}} b^{b_{4}} $, $f_{\#}(c) = a^{c_{1}} b^{c_{2}} c $. Thus $$B = { \left( \begin{array}{cc} b_{1} & b_{3} \\ b_{2} & b_{4} \\ \end{array} \right)}$$
\[daci1theorem\] $(1) \,\, \pi_{1}(MA,0) = \langle a,b,c | [a,b] = 1, cac^{-1} = a^{a_{1}} b^{a_{2}} ,
cbc^{-1} $ $ = a^{a_{3}} b^{a_{4}} \rangle $
$(2)\, B $ commutes with $A$.
$(3) \, $ If $f$ restricted to the fiber is deformable to a fixed point free map then the determinant of $B - I$ is zero, where $I$ is the identity matrix.
$(4) \, $ If $v$ is an eigenvector of $B$ associated to 1 $(for \, B \neq Id)$ then $A(v)$ is also an eigenvector of $B$ associated to 1.
$(5) \, $ Consider $w = A(v)$ if the pair $v,w$ generators $ \mathbb{Z} \times \mathbb{Z}$, otherwise let $w$ be another vector so that $v,w$ span $\mathbb{Z} \times \mathbb{Z}$. Define the linear operator $P:\mathbb{R} \times \mathbb{R} \to \mathbb{R} \times \mathbb{R} $ by $P(v) = \left( ^{1}_{0} \right)$ and $P(w) = \left( ^{0}_{1} \right)$. Consider an isomorphism of fiber bundles, also denoted by $P$, $P: MA \to M(A^{1})$ where $A^{1} = P \cdot A \cdot P^{-1}$. Then $MA$ is homeomorphic to $M(A^{1})$ over $S^{1}$. Moreover we have one of the cases of the table below with $B^{1} = P \cdot A \cdot P^{-1}$ and $B^{1} \neq Id$, except in case $I$:
------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------
$ Case \,\, I $ $ A^{1} = \left( \begin{array}{cc} a_{1} & a_{3} \\ a_{2} & a_{4} \\ \end{array} \right) $ , $ B^{1} = \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \\ \end{array} \right) $
$a_{3} \neq 0 $
$ Case \,\, II $ $ A^{1} = \left( \begin{array}{cc} 1 & a_{3}\\ 0 & 1 \\ \end{array} \right) $ , $ B^{1} = \left( \begin{array}{cc} 1 & b_{3} \\ 0 & b_{4} \\ \end{array} \right) $
$a_{3}(b_{4}-1) = 0 $
$ Case \,\, III $ $ A^{1} = \left( \begin{array}{cc} 1 & a_{3} \\ 0 & -1 \\ \end{array} \right) $ , $ B^{1} = \left( \begin{array}{cc} 1 & b_{3} \\ 0 & b_{4} \\ \end{array} \right) $
$a_{3}(b_{4}-1) = -2b_{3} $
$ Case \,\, IV $ $ A^{1} = \left( \begin{array}{cc} -1 & a_{3} \\ 0 & -1 \\ \end{array} \right) $ , $ B^{1} = \left( \begin{array}{cc} 1 & b_{3} \\ 0 & b_{4} \\ \end{array} \right) $
$a_{3}(b_{4}-1) = 0 $
$ Case \,\, V $ $ A^{1} = \left( \begin{array}{cc} -1 & a_{3} \\ 0 & 1 \\ \end{array} \right) $ , $ B^{1} = \left( \begin{array}{cc} 1 & b_{3} \\ 0 & b_{4} \\ \end{array} \right) $
$a_{3}(b_{4}-1) = 2b_{3} $
------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------
From [@daci1] we have the following theorem:
\[main-theorem-daci1\] A fiber-preserving map $f : MA \to MA$ can be deformed to a fixed point free map by a homotopy over $S^{1}$ if and only if one of the cases below holds:
\(1) $MA$ is as in case I and $f$ is arbitrary
\(2) $MA$ is as in one of the cases $II$ or $III$ and $c_{1}(b_{4}-1)-c_{2}b_{3}=0$
\(3) $MA$ is as in case $IV$ and $n(b_{4}(b_{3}+1) -1 -c_{1}(b_{4}-1) + b_{3}c_{2}) - (n-1)(b_{4}-1) \equiv_{2} 0$ except when:
$a_{3}$ is odd and $[(c_{1}+b_{3}c_{2}\dfrac{(n-1)}{2},nc_{2})]=[(0,0)]\in\dfrac{\mathbb{Z}\oplus\mathbb{Z}}{\langle(1,2),(0,4)\rangle}$ or
$a_{3}$ is even and $[(nc_{1} +\dfrac{n(n-1)}{2}b_{3}b_{4}c_{2}, c_{2}+(n-1)b_{4}c_{2})]=[(0,0)]$, with $[(0,0)]\in\dfrac{\mathbb{Z}\oplus\mathbb{Z}}{\langle(2,0),(0,2)\rangle}.$
\(4) $MA$ is as in case $V$ and either
$a_{3}$ is even and $(b_{4}-1)(c_{1}-\frac{a_{3}}{2}c_{2}-1) \equiv 0 \,\, mod \,\, 2$, except when $c_{1}-\frac{a_{3}}{2}c_{2}-1$ and $\frac{b_{4}-1}{L}$ are odd, or
$a_{3}$ is odd and $\frac{b_{4}-1}{2}(1+c_{2}) \equiv 0 \,\, mod \,\, 2$ except when $1+c_{2}$ and $\frac{b_{4}-1}{L}$ are odd, where $L:= mdc(b_{4}-1,c_{2})$.
Given $n \in \mathbb{N}$ we denote the induced homomorphism $f^{n}_{\#}: \pi_{1}(MA) \to \pi_{1}(MA) $ by; $f_{\#}(a) = a^{b_{1n}} b^{b_{2n}} $, $f_{\#}(b) = a^{b_{3n}} b^{b_{4n}} $, $f_{\#}(c) = a^{c_{1n}} b^{c_{2n}} c $, where $b_{j1} = b_{j}, j=1,...,4$ and $c_{j1}=c_{j}, j=1,2.$ Thus the matrix of the homomorphism induced on the fundamental group by the restriction of $f^{n}$ to the fiber $T$ is given by; $$B_{n} = { \left( \begin{array}{cc} b_{1n} & b_{3n} \\ b_{2n} & b_{4n} \\ \end{array} \right)}$$ where $B_{1} = B$ is the matrix of $(f_{|T})_{\#}$ and $B_{n} = B^{n}.$ From [@K-94] we have $$N(h^{n}) = |L(h^{n})| = |det([h_{\#}]^{n} -I)|$$ for each map $h: T \to T$ on torus, where $[h_{\#}]$ is the matrix of induced homomorphism and $I$ is the identity.
Since $(B^{n}-I) = (B-I)(B^{n-1}+...+B+I)$ then $det(B^{n}-I) = det(B-I)det(B^{n-1}+...+B+I)$. Therefore if $f_{|T}$ is deformable to a fixed point free map then $f^{n}_{|T}$ is deformable to a fixed point free map.
In the Theorems \[daci1theorem\] and \[main-theorem-daci1\] putting $f^{n}$ in the place of $f$ we will get conditions to $f^{n}.$ The conditions in Theorem \[daci1theorem\] to $f^{n}$ is the same of $f$ but the conditions to $f^{n}$ in the Theorem \[main-theorem-daci1\] are different of $f$ and are in the Theorem \[theorem-1\].
Fixed points of $f^{n}$
=======================
Given a fiber-preserving map $f: MA \to MA$, if $f \sim_{S^{1}} g$ then $f^{n} \sim_{S^{1}} g^{n}$. Therefore, if $Fix(g^{n}) = \emptyset$ then the homomorphism $f^{n}_{\#}: \pi_{1}(M) \to \pi(M)$ satisfies the condition of deformability gives in [@daci1].
\[proposition-1\] Let $f: MA \to MA$ be a fiber-preserving map, where $M$ is a T-bundle over $S^{1}.$ Suppose which $f$ restrict to the fiber can be deformed to a fixed point free map. This implies $L(f|_{T})=0.$ From Theorem \[daci1theorem\] we can suppose that the induced homomorphism $f_{\#}: \pi_{1}(MA) \to \pi_{1}(MA) $ is given by; $f_{\#}(a) = a $, $f_{\#}(b) = a^{b_{3}} b^{b_{4}} $, $f_{\#}(c) = a^{c_{1}} b^{c_{2}} c .$ Given $n \in \mathbb{N}$ then from relation $(f_{\#})^{n} = f^{n}_{\#}$ we obtain; $$f^{n}_{\#}(a)=a,$$ $$f^{n}_{\#}(b)=a^{b_{3}\sum_{i=0}^{n-1}b_{4}^{i}}b^{b_{4}^{n}},$$ $$f^{n}_{\#}(c)=a^{nc_{1}+b_{3}c_{2}\sum_{i=0}^{n-2}(n-1-i)b_{4}^{i}}b^{c_{2}\sum_{i=0}^{n-1}b_{4}^{i}}c.$$
[*Proof.*]{} In fact, $f^{2}_{\#}(b) = f_{\#}(a^{b_{3}} b^{b_{4}}) = a^{b_{3}}(a^{b_{3}} b^{b_{4}})^{b_{4}} = a^{b_{3}+ b_{3}b_{4}}b^{b_{4}^{2}}$ and $f^{2}_{\#}(c) = f_{\#}(a^{c_{1}} b^{c_{2}} c) = a^{c_{1}}(a^{b_{3}} b^{b_{4}})^{c_{2}}(a^{c_{1}} b^{c_{2}} c) = a^{2c_{1}+b_{3}c_{2}}b^{c_{2} + c_{2}b_{4}}c$. Suppose that $f^{n}_{\#}(b) = a^{b_{3}\sum_{i=0}^{n-1}b_{4}^{i}}b^{b_{4}^{n}}$ and $f^{n}_{\#}(c) = a^{nc_{1}+b_{3}c_{2}\sum_{i=0}^{n-2}(n-1-i)b_{4}^{i}}b^{c_{2}\sum_{i=0}^{n-1}b_{4}^{i}}c$. Then, $$\begin{array}{lclcl}
f^{n+1}_{\#}(b) & = & f_{\#}(a^{b_{3}\sum_{i=0}^{n-1}b_{4}^{i}}b^{b_{4}^{n}}) & = & a^{b_{3}\sum_{i=0}^{n-1}b_{4}^{i}}(a^{b_{3}} b^{b_{4}})^{b_{4}^{n}} \\
& = & a^{b_{3}\sum_{i=0}^{n-1}b_{4}^{i}}(a^{b_{3}b_{4}^{n}} b^{b_{4}^{n+1}}) & = & a^{b_{3}\sum_{i=0}^{n}b_{4}^{i}}b^{b_{4}^{n+1}};
\end{array}$$ $$\begin{array}{lcl}
f^{n+1}_{\#}(c) & = & f_{\#}(a^{nc_{1}+b_{3}c_{2}\sum_{i=0}^{n-2}(n-1-i)b_{4}^{i}}b^{c_{2}\sum_{i=0}^{n-1}b_{4}^{i}}c) \\
& = & a^{nc_{1}+b_{3}c_{2}\sum_{i=0}^{n-2}(n-1-i)b_{4}^{i}}(a^{b_{3}} b^{b_{4}})^{c_{2}\sum_{i=0}^{n-1}b_{4}^{i}}(a^{c_{1}} b^{c_{2}} c)\\
& = & a^{(nc_{1}+b_{3}c_{2}\sum_{i=0}^{n-2}(n-1-i)b_{4}^{i})+ (b_{3}c_{2}\sum_{i=0}^{n-1}b_{4}^{i})+ (c_{1})} b^{(c_{2}\sum_{i=1}^{n}b_{4}^{i})+(c_{2})} c\\
& = & a^{(n+1)c_{1}+b_{3}c_{2}\sum_{i=0}^{n-1}(n-1-i)b_{4}^{i}} b^{c_{2}\sum_{i=0}^{n}b_{4}^{i}} c.
\end{array}$$
\[theorem-1\] Let $f: MA \to MA$ be a fiber-preserving map, where $MA$ is a T-bundle over $S^{1}.$ Suppose which $f$ restrict to the fiber can be deformed to a fixed point free map and that the induced homomorphism $f_{\#}: \pi_{1}(MA) \to \pi_{1}(MA) $ is given by; $f_{\#}(a) = a $, $f_{\#}(b) = a^{b_{3}} b^{b_{4}} $, $f_{\#}(c) = a^{c_{1}} b^{c_{2}} c $ as in cases of the Theorem \[main-theorem-daci1\]. Then $f^{n}: MA \to MA$ can be deformed to a fixed point free map over $S^{1}$ if and only if the following conditions are satisfies;
1\) $MA$ is as in case $I$ and $f$ is arbitrary.
2\) $MA$ is as in cases $II$, $III$ and $c_{1}(b_{4}-1)-c_{2}b_{3} = 0$ or $n$ even and $b_{4} = -1.$
3\) $MA$ is as in case $IV$ and $n(b_{4}(b_{3}+1) -1 -c_{1}(b_{4}-1) + b_{3}c_{2}) - (n-1)(b_{4}-1) \equiv_{2} 0$ except when:
$a_{3}$ is odd and $[(nc_{1}+\frac{n(n-1)}{2}b_{3}c_{2},nc_{2})]=[(0,0)]\in\frac{\mathbb{Z}\oplus\mathbb{Z}}{\langle(1,2),(0,4)\rangle}$ or
$a_{3}$ is even and $[(nc_{1} +\frac{n(n-1)}{2}b_{3}b_{4}c_{2}, c_{2}+(n-1)b_{4}c_{2})]=[(0,0)]\in\frac{\mathbb{Z}\oplus\mathbb{Z}}{\langle(2,0),(0,2)\rangle}.$
4\) $MA$ is as in case $V$ and either
$a_{3}$ is even and $n(b_{4}-1)(c_{1}-\frac{a_{3}}{2}c_{2}-1) + (n-1)(b_{4}-1)\equiv 0 \,\, mod \,\, 2$, except when $n(c_{1}-\frac{a_{3}}{2}c_{2}-1) + (n-1)$ and $\frac{b_{4}-1}{L}$ are odd, or
$a_{3}$ is odd and $\frac{b_{4}-1}{2}((1+c_{2})(1+(n-1)b_{4})) \equiv 0 \,\, mod \,\, 2$ except when $(1+c_{2})(1+(n-1)b_{4})$ and $\frac{b_{4}-1}{L}$ are odd, where $L:= mdc(b_{4}-1,c_{2})$.
In proof of this theorem we will denote $f^{n}_{\#}(b)=a^{b'_{3}}b^{b'_{4}}$ and $f^{n}_{\#}(c)=a^{c'_{1}}b^{c'_{2}}c.$
\(1) From Theorem \[main-theorem-daci1\] each map $f: MA \to MA$ is fiberwise homotopic to a fixed point free map over $S^{1}$ in particular that happens with $f^{n}: MA \to MA$ for each $n \in \mathbb{N}.$
\(2) If $b_{4}= 1$ then $b'_{3}=nb_{3}$, $b'_{4}=1$, $c'_{1}=nc_{1}+b_{3}c_{2}\dfrac{n(n-1)}{2}$ and $c'_{2}=nc_{2}$. In this sense, following Theorem \[main-theorem-daci1\] of [@daci1], in cases $II$ and $III$, $f^{n}$ can be deformed, by a fiberwise homotopy, to a fixed point free map if and only if $c'_{1}(b'_{4}-1)-c'_{2}b'_{3}=0$. However, $c'_{1}(b'_{4}-1)-c'_{2}b'_{3} \ = \ n^{2}c_{2}b_{3}$, and $n^{2}c_{2}b_{3}=0$ if and only if $c_{2}=0$ or $b_{3}=0$.
For $b_{4}\neq 1$ we have $b'_{3}=b_{3}\displaystyle{\sum_{i=0}^{n-1}}b_{4}^{i} = b_{3}\dfrac{b_{4}^{n}-1}{b_{4}-1}$, $b'_{4}=b_{4}^{n}$, $c'_{1}=nc_{1}+b_{3}c_{2}\displaystyle{\sum_{i=0}^{n-2}}(n-1-i)b_{4}^{i} = nc_{1}+b_{3}c_{2}\dfrac{b_{4}^{n}-nb_{4}+n-1}{(b_{4}-1)^{2}}$ and $c'_{2}=c_{2}\displaystyle{\sum_{i=0}^{n-1}}b_{4}^{i} = c_{2}\dfrac{b_{4}^{n}-1}{b_{4}-1}$. Then, $ c'_{1}(b'_{4}-1)-c'_{2}b'_{3} \ = \ \dfrac{n(b_{4}^{n}-1).(c_{1}(b_{4}-1)-c_{2}b_{3})}{b_{4}-1}.$ In fact, $$\begin{array}{rcl}
c'_{1}(b'_{4}-1) & = & \left( nc_{1}+c_{2}b_{3}\dfrac{b_{4}^{n}-nb_{4}+n-1}{(b_{4}-1)^{2}}\right) (b_{4}^{n}-1)\\
& = & nc_{1}(b_{4}^{n}-1) + c_{2}b_{3}\left(\dfrac{(b_{4}^{n}-1)-n(b_{4}-1)}{(b_{4}-1)^{2}}\right) (b_{4}^{n}-1)\\
& = & nc_{1}(b_{4}^{n}-1) + c_{2}b_{3} \left(\dfrac{b_{4}^{n}-1}{b_{4}-1} \right)^{2} -n c_{2}b_{3}\left(\dfrac{b_{4}^{n}-1}{b_{4}-1} \right);\\
c'_{2}b'_{3} & = & \left(c_{2}\dfrac{b_{4}^{n}-1}{b_{4}-1} \right) \left( b_{3}\dfrac{b_{4}^{n}-1}{b_{4}-1}\right)
= c_{2}b_{3} \left(\dfrac{b_{4}^{n}-1}{b_{4}-1} \right)^{2}.\\
\end{array}$$ Therefore, $$\begin{array}{ccl}
c'_{1}(b'_{4}-1)-c'_{2}b'_{3} & = & nc_{1}(b_{4}^{n}-1) -n c_{2}b_{3}\left(\dfrac{b_{4}^{n}-1}{b_{4}-1} \right)\\
& = & n (b_{4}^{n}-1)\left(c_{1} - \dfrac{c_{2}b_{3}}{b_{4}-1} \right)\\
& = & n \left(\dfrac{b_{4}^{n}-1}{b_{4}-1}\right)(c_{1}(b_{4}-1) - c_{2}b_{3})\\
& = & n (c_{1}(b_{4}-1) - c_{2}b_{3}) \left(\displaystyle{\sum_{i=0}^{n-1}}b_{4}^{i}\right) .
\end{array}$$
Note that $cos\left(\dfrac{2k\pi}{n}\right) +i.sin\left(\dfrac{2k\pi}{n}\right)$, for $k=0,1,\dots,n-1$, are the roots of $b_{4}^{n}-1=0$. So, $+1$ and $-1$ are the only two possible integer solutions $b_{4}^{n}-1=0$. Since $b_{4}^{n}-1\neq 0$ for $n$ odd and $b_{4}\neq1$, we may assume that $\dfrac{n(b_{4}^{n}-1).(c_{1}(b_{4}-1)-c_{2}b_{3})}{b_{4}-1}=0$ if and only if $c_{1}(b_{4}-1)-c_{2}b_{3}=0$. Then, by Theorem \[main-theorem-daci1\] again, in cases $II$ and $III$ and $n$ odd, $f^{n}$ can be deformed, by a fiberwise homotopy, to a fixed point free map if and only if $f$ can be deformed, by a fiberwise homotopy, to a fixed point free map. For $n$ even and $b_{4}\neq 1$, $b_{4}^{n}-1 = 0$ if and only if $b_{4}=-1$.
\(3) Following Theorem \[main-theorem-daci1\] of [@daci1], in cases $IV$, $f^{n}$ can be deformed, by a fiberwise homotopy, to a fixed point free map iff $b'_{4} (b'_{3} + 1) - 1 - c'_{1} (b'_{4} - 1) + c'_{2} b'_{3} \equiv_{2} 0$ except when $a_{3}$ even and $[(c'_{1},c'_{2})]=[(0,0)] \in\dfrac{\mathbb{Z}\oplus\mathbb{Z}}{\langle(2,0),(0,2)\rangle}$, or $a_{3}$ odd and $[(c'_{1},c'_{2})]=[(0,0)]\in\dfrac{\mathbb{Z}\oplus\mathbb{Z}}{\langle(1,2),(0,4)\rangle}$.
As in case $(2)$, we have $-c'_{1}(b'_{4}-1)+c'_{2}b'_{3} = -n(c_{1}(b_{4}-1)-c_{2}b_{3})\left(\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}}\right)$ and $ b'_{4}(b'_{3}+1)-1 = b_{4}^{n}\left(1 + b_{3}\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}}\right)-1 = (b_{4}^{n}-1) + b_{4}^{n}b_{3}\left(\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}}\right) = (b_{4}-1)\left(\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}}\right) + b_{4}^{n}b_{3}\left(\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}}\right)$. Then $$\begin{array}{c}
b'_{4}(b'_{3}+1)-1-c'_{1}(b'_{4}-1)+c'_{2}b'_{3} \ \ = \\
(b_{4}-1)\left(\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}}\right) + b_{3}b_{4}^{n}\left(\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}}\right) -n(c_{1}(b_{4}-1)-c_{2}b_{3})\left(\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}}\right) \ \ = \\
(b_{4}^{n}-1)(1-nc_{1}) +b_{3}(b_{4}^{n}+nc_{2})\left(\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}}\right) \ \ \equiv_{2}\\
(b_{4}-1)(1-nc_{1}) +b_{3}(b_{4}+nc_{2})(1+(n-1)b_{4}) \ = \\
(b_{4}-1)(1-nc_{1}) +b_{3}(b_{4}+(n-1)b_{4}^{2} +nc_{2} +n^{2}b_{4}c_{2}-nb_{4}c_{2}) \ \equiv_{2} \\
(b_{4}-1)(1-nc_{1}) +b_{3}(b_{4}+(n-1)b_{4} +nc_{2} +nb_{4}c_{2}-nb_{4}c_{2}) \ = \\
(b_{4}-1)(1-nc_{1}) +b_{3}(nb_{4} +nc_{2}) \ = \\
n(b_{4}(b_{3}+1) -1 -c_{1}(b_{4}-1) + b_{3}c_{2})-(n-1)(b_{4}-1).
\end{array}$$
The exceptions holds for $a_{3}$ even and $[(c'_{1},c'_{2})]=[(0,0)] \in\dfrac{\mathbb{Z}\oplus\mathbb{Z}}{\langle(2,0),(0,2)\rangle}$, or $a_{3}$ odd and $[(c'_{1},c'_{2})]=[(0,0)]\in\dfrac{\mathbb{Z}\oplus\mathbb{Z}}{\langle(1,2),(0,4)\rangle}$. In this sense, we have $$(c'_{1},c'_{2})= \displaystyle{\left(nc_{1}+b_{3}c_{2}\sum_{i=0}^{n-1}(n-1-i)b_{4}^{i}, c_{2}\sum_{i=0}^{n-1}b_{4}^{i}\right)}.$$ If $a_{3}$ is odd then $b_{4}=1$, $c_{2}\displaystyle{\sum_{i=0}^{n-1}1^{i}}= nc_{2}$ and $nc_{1}+b_{3}c_{2}\displaystyle{\sum_{i=0}^{n-1}(n-1-i)1^{i}} = nc_{1}+b_{3}c_{2}\dfrac{n(n-1)}{2}$. If $a_{3}$ is even then $c_{2}\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}}\equiv_{2} c_{2}(1+(n-1)b_{4})$ and $nc_{1}+b_{3}c_{2}\displaystyle{\sum_{i=0}^{n-1}(n-1-i)b_{4}^{i} }\equiv_{2}nc_{1}+ \dfrac{n(n-1)}{2}b_{3}b_{4}c_{2}$.
\(4) From Theorem \[main-theorem-daci1\] the map $f^{n}$ can be deformed, over $S^{1}$, to a fixed point free map if and only if the following condition is satisfy;
$a_{3}$ is even and $(b^{'}_{4}-1)(c^{'}_{1}-\frac{a_{3}}{2}c^{'}_{2}-1) \equiv 0 \,\, mod \,\, 2$, except when $c^{'}_{1}-\frac{a_{3}}{2}c^{'}_{2}-1$ and $\frac{b^{'}_{4}-1}{L}$ are odd, or $a_{3}$ is odd and $\frac{b^{'}_{4}-1}{2}(1+c^{'}_{2}) \equiv 0 \,\, mod \,\, 2$ except when $1+c^{'}_{2}$ and $\frac{b^{'}_{4}-1}{L}$ are odd, where $L:= mdc(b^{'}_{4}-1,c^{'}_{2})$.
Note that if $b_{4}=1$ then from Theorem \[daci1theorem\] we must have $b_{3}=0$ and this situation return in the case $I$. Therefore let us suppose $b_{4} \neq 1.$
From previous calculation we have; $b^{'}_{4} = b^{n}_{4} $, $b^{'}_{3} = b_{3}\frac{b^{n}_{4}-1}{b_{4}-1}$, $c^{'}_{2} = c_{2}\frac{b^{n}_{4}-1}{b_{4}-1}$ and $c^{'}_{1} = nc_{1} + b_{3}c_{2}\frac{b^{n}_{4}-nb_{4}+n-1}{(b_{4}-1)^{2}}.$ From Theorem \[daci1theorem\] we have $a_{3}(b_{4}-1) = 2b_{3}.$
Suppose $a_{3}$ even. Since $c^{'}_{1}(b^{'}_{4}-1)-c^{'}_{2}b^{'}_{3} = \frac{n(b^{n}_{4}-1)(c_{1}(b_{4}-1)-c_{2}b_{3})}{b_{4}-1} $ Then $(b^{'}_{4}-1)(c^{'}_{1}-\frac{a_{3}}{2}c^{'}_{2}-1) = n(b^{n}_{4}-1)(c_{1}-\frac{a_{3}}{2}c_{2}-1)+(n-1)(b^{n}_{4}-1).$ In fact, $$\begin{array}{lll}
c^{'}_{1}-\frac{a_{3}}{2}c^{'}_{2} & = & nc_{1} + b_{3}c_{2}\frac{b^{n}_{4}-nb_{4}+n-1}{(b_{4}-1)^{2}}
- \frac{a_{3}}{2}c_{2}\frac{b^{n}_{4}-1}{b_{4}-1} \\
& = & nc_{1} + b_{3}c_{2}\frac{(b^{n}_{4}-1)-n(b_{4}-1)}{(b_{4}-1)^{2}} - b_{3}c_{2}\frac{b^{n}_{4}-1}{(b_{4}-1)^{2}} \\
& = & nc_{1} - \frac{b_{3}c_{2}n}{b_{4}-1} \\
& = & n(c_{1}-\frac{a_{3}}{2}c_{2}).\\
\end{array}$$
We know that if $L:=mdc(b_{4}-1,c_{2})$ then $kL:= mdc(k(b_{4}-1),kc_{2}).$ Thus, $mdc(b^{'}_{4}-1,c^{'}_{2}) = L^{'} =
\frac{b^{n}_{4}-1}{(b_{4}-1)}L = $ where $L = mdc(b_{4}-1,c_{2})$ because $b^{'}_{4}-1 = \frac{b^{n}_{4}-1}{(b_{4}-1)} (b_{4}-1)$ and $c^{'}_{2} =c_{2} \frac{b^{n}_{4}-1}{(b_{4}-1)}.$ Furthermore, $\frac{b^{'}_{4}-1}{L^{'}} = \frac{b^{'}_{4}-1}{L} \frac{b_{4}-1}{(b^{n}_{4}-1)} = \frac{b_{4}-1}{L}. $ With these calculations we obtain the conditions statements on the theorem.
In the case $a_{3}$ odd we must have; $\frac{b^{n}_{4}-1}{2}(1+c_{2}\frac{b^{n}_{4}-1}{b_{4}-1}) \equiv 0 \,\, mod \,\, 2$ except when $1+c_{2}\frac{b^{n}_{4}-1}{b_{4}-1}$ and $\frac{b_{4}-1}{L}$ are odd, where $L:= mdc(b_{4}-1,c_{2})$.
Note that $\frac{b^{n}_{4}-1}{b_{4}-1}$ is even if and only if $1+(n-1)b_{4}$ is even, and $b^{n}_{4}-1$ is even if and only if $b_{4}-1$ is even, for all $n \in \mathbb{N}.$ With this we obtain the enunciate of the theorem.
\[corollary-1\] From Theorem \[theorem-1\], if $f: MA \to MA$ is deformed to a fixed point free map over $S^{1}$ and $n \in \mathbb{N}$ is odd then the map $f^{n}: MA \to MA$ can be deformed to a fixed point free map over $S^{1}.$
[*Proof.*]{} If $f: MA \to MA$ is deformed to a fixed point free map over $S^{1}$ then the conditions of the Theorem \[main-theorem-daci1\] are satisfied. Suppose $n$ odd then the conditions of the Theorem \[theorem-1\] also are satisfied. Thus $f^{n}: MA \to MA$ can be deformed to a fixed point free map over $S^{1}.$
In the corollary above if $n$ is even the above statement may not holds, for example in the case V of the Theorem \[theorem-1\] if $n$, $b_{4}$, $a_{3}$ and $c_{1}- \frac{a_{3}}{2}c_{2} -1$ are even then $f: MA \to MA$ is deformed to a fixed point free map over $S^{1}$ but $f^{n}$ is not.
\[proposition-2\] Let $f: MA \to MA$ be a fiber-preserving such that the induced homomorphism $f_{\#}: \pi_{1}(MA) \to \pi_{1}(MA) $ is given by; $f_{\#}(a) = a $, $f_{\#}(b) = a^{b_{3}} b^{b_{4}} $, $f_{\#}(c) = a^{c_{1}} b^{c_{2}} c .$ Suppose that for some $n$ odd, $ n \in \mathbb{N}$ the fiber-preserving map $f^{n}: MA \to MA$ is deformed to a fixed point free map over $S^{1}.$ If $k$ is a divisor of $n$ then the map $f^{k}: MA \to MA$ can be deformed by a fiberwise homotopy to a fixed point free map over $S^{1}.$
[*Proof.*]{} Is enough to verify that if the conditions of the Theorem \[theorem-1\] are satisfied for some $n$ odd then those conditions are also satisfied for any $k$ divisor of $n.$ We will analyze each case of the Theorem \[theorem-1\].
Case I. In this case for each $n \in \mathbb{N}$ the fiber-preserving map can be deformed over $S^{1}$ to a fixed point free map.
Cases II and III. In these cases if for some $n$ odd the fiber-preserving map $f^{n}: MA \to MA$ is deformed to a fixed point free map over $S^{1}$ then we must have; $c_{1}(b_{4}-1)-c_{2}b_{3} = 0.$ Thus for all $k \leq n$ the $f^{k}$ can be deformed to a fixed point free map over $S^{1},$ in particular when $k$ divides $n.$
Case IV. Suppose that for some odd positive integer $n$ the fiber-preserving map $f^{n}: MA \to MA$ is deformed to a fixed point free map over $S^{1}$, then $n(b_{4}(b_{3}+1) -1 -c_{1}(b_{4}-1) + b_{3}c_{2}) - (n-1)(b_{4}-1) \equiv_{2} 0$ and
if $a_{3}$ is odd then $[(nc_{1}+\frac{n(n-1)}{2}b_{3}c_{2},nc_{2})]\neq[(0,0)]\in\frac{\mathbb{Z}\oplus\mathbb{Z}}{\langle(1,2),(0,4)\rangle}$ or
if $a_{3}$ is even then $[(nc_{1} +\frac{n(n-1)}{2}b_{3}b_{4}c_{2}, c_{2}+(n-1)b_{4}c_{2})]\neq[(0,0)]\in\frac{\mathbb{Z}\oplus\mathbb{Z}}{\langle(2,0),(0,2)\rangle}.$
Note that if $a_{3}$ is odd then $b_{4}=1$ and $$n(b_{4}(b_{3}+1) -1 -c_{1}(b_{4}-1) + b_{3}c_{2}) - (n-1)(b_{4}-1) \equiv_{2} b_{3} + b_{3}c_{2}=b_{3}(1 + c_{2});$$ $$\begin{array}{ccl}
[(nc_{1}+\frac{n(n-1)}{2}b_{3}c_{2},nc_{2})] & = & [(0,nc_{2}- 2(nc_{1}+\frac{n(n-1)}{2}b_{3}c_{2}))]\\
& = & [(0,n(c_{2} - 2c_{1}-(n-1)b_{3}c_{2} )))] \in\frac{\mathbb{Z}\oplus\mathbb{Z}}{\langle(1,2),(0,4)\rangle};\\
&\Rightarrow& n(c_{2} - 2c_{1}-(n-1)b_{3}c_{2})\not\equiv_{4}0\\
&\Rightarrow& c_{2} - 2c_{1}-(n-1)b_{3}c_{2}\not\equiv_{4}0.
\end{array}$$ If $b_{3}$ is even then $c_{2} - 2c_{1}-(n-1)b_{3}c_{2}\equiv_{4}c_{2} - 2c_{1}$. If $c_{2}$ is odd then $c_{2} - 2c_{1}-(n-1)b_{3}c_{2}$ is odd and $c_{2} - 2c_{1}-(k-1)b_{3}c_{2}$ is odd for each odd $k$. If $a_{3}$ is even we have $$\left[\left(nc_{1} +\frac{n(n-1)}{2}b_{3}b_{4}c_{2}, c_{2}+(n-1)b_{4}c_{2}\right)\right]=\left[\left(c_{1} +\frac{(n-1)}{2}b_{3}b_{4}c_{2}, c_{2}\right)\right].$$ Then, $c_{2}\equiv_{2}1$ or $c_{1} +\frac{(n-1)}{2}b_{3}b_{4}c_{2}\equiv_{2}1$. So, if $c_{2}\equiv_{2}0$ then $c_{1} +\frac{(n-1)}{2}b_{3}b_{4}c_{2}\equiv_{2}c_{1}$. Therefore, $c_{1}\equiv_{2}1$ or $c_{2}\equiv_{2}1$.
Let $k$ be an integer such that $k$ divides $n$ then $k$ must be odd and $$\begin{array}{ccc}
k(b_{4}(b_{3}+1) -1 -c_{1}(b_{4}-1) + b_{3}c_{2}) - (k-1)(b_{4}-1) & \equiv_{2} & \\
n(b_{4}(b_{3}+1) -1 -c_{1}(b_{4}-1) + b_{3}c_{2}) - (n-1)(b_{4}-1) & \equiv_{2} & 0.
\end{array}$$ If $a_{3}$ is odd then $b_{3}$ is even or $c_{2}$ is odd, $b_{4}=1$ and $$\begin{array}{ccl}
[(kc_{1}+\frac{k(k-1)}{2}b_{3}c_{2},kc_{2})] & = & [(0,k(c_{2} - 2c_{1}-(k-1)b_{3}c_{2} )))] \\
& = & [(0,c_{2} - 2c_{1}-(k-1)b_{3}c_{2} )))] \\
& = & [(0,c_{2} - 2c_{1}-(n-1)b_{3}c_{2} )))] \\
&\neq & [(0,0)] \in\frac{\mathbb{Z}\oplus\mathbb{Z}}{\langle(1,2),(0,4)\rangle}.\\
\end{array}$$ Then, $f^{k}: MA \to MA$ can be deformed to a fixed point free map over $S^{1}$.
If $a_{3}$ is even then $$\begin{array}{ccl}
[(kc_{1} +\frac{k(k-1)}{2}b_{3}b_{4}c_{2}, c_{2}+(k-1)b_{4}c_{2})] & = & [(c_{1} +\frac{(k-1)}{2}b_{3}b_{4}c_{2}, c_{2})] \\
&\neq & [(0,0)]\in\frac{\mathbb{Z}\oplus\mathbb{Z}}{\langle(2,0),(0,2)\rangle}.\\
\end{array}$$ Then, $f^{k}: MA \to MA$ can be deformed to a fixed point free map over $S^{1}$.
Case V. Suppose that for some $n$ odd, $ n \in \mathbb{N}$ the fiber-preserving map $f^{n}: MA \to MA$ is deformed to a fixed point free map over $S^{1}.$ If $k$ divides $n$ then there is $l \in \mathbb{N}$ such that $kl=n,$ in particular $l$ must be odd. The conditions to deform $f^{n}$ in this case with $a_{3}$ even are;
$a_{3}$ is even and $n(b_{4}-1)(c_{1}-\frac{a_{3}}{2}c_{2}-1) + (n-1)(b_{4}-1)\equiv 0 \,\, mod \,\, 2$, except when $n(c_{1}-\frac{a_{3}}{2}c_{2}-1) + (n-1)$ and $\frac{b_{4}-1}{L}$ are odd, where $L:=mdc(b_{4}-1,c_{2}).$
We have; $l[k(b_{4}-1)(c_{1}-\frac{a_{3}}{2}c_{2}-1) + (k-1)(b_{4}-1)] = $ $ lk(b_{4}-1)(c_{1}-\frac{a_{3}}{2}c_{2}-1) + (lk-l)(b_{4}-1) =$ $ n(b_{4}-1)(c_{1}-\frac{a_{3}}{2}c_{2}-1) + (n-1)(b_{4}-1) + (1-l)(b_{4}-1).$
From hypothesis $n(b_{4}-1)(c_{1}-\frac{a_{3}}{2}c_{2}-1) + (n-1)(b_{4}-1)$ is even. Since $l$ is odd then $(1-l)(b_{4}-1)$ is even. Therefore $l[k(b_{4}-1)(c_{1}-\frac{a_{3}}{2}c_{2}-1) + (k-1)(b_{4}-1)]$ is even. Since $l$ is odd then we must have $k(b_{4}-1)(c_{1}-\frac{a_{3}}{2}c_{2}-1) + (k-1)(b_{4}-1) \equiv 0 \,\, mod \,\, 2.$
Note that $l[k(c_{1}-\frac{a_{3}}{2}c_{2}-1) + (k-1)] = $ $[n(c_{1}-\frac{a_{3}}{2}c_{2}-1) + (n-1)]+ (1-l).$ Thus $l[k(c_{1}-\frac{a_{3}}{2}c_{2}-1) + (k-1)]$ is odd because $[n(c_{1}-\frac{a_{3}}{2}c_{2}-1) + (n-1)]$ is odd by hypothesis and $(1-l)$ is even. Since $l$ is odd then we must have $k(c_{1}-\frac{a_{3}}{2}c_{2}-1) + (k-1)$ odd. Therefore the conditions to deform $f^{k}$ to a fixed point free map over $S^{1}$, in the Theorem \[theorem-1\], are satisfied. The case $a_{3}$ odd is analogous.
\[proposition-3\] Let $f: MA \to MA$ be a fiber-preserving. If $m,n$ are odd, $m,n \in \mathbb{N},$ then $f^{m}$ is deformable to a fixed point free map over $S^{1}$ if and only if $f^{n}$ is deformable to a fixed point free map over $S^{1}.$
[*Proof.*]{} If $m,n$ are odd and $f^{m}$ is deformable to a fixed point free map over $S^{1}$ then by Proposition \[proposition-2\] $f$ is deformable to a fixed point free map over $S^{1}$. From Corollary \[corollary-1\] $f^{n}$ is deformable to a fixed point free map over $S^{1}.$
We have a analogous result to $n$ even;
\[proposition-4\]
Let $f: MA \to MA$ be a fiber-preserving map, where $MA$ is a T-bundle over $S^{1}.$ Suppose which $f$ restrict to the fiber can be deformed to a fixed point free map and that the induced homomorphism $f_{\#}: \pi_{1}(MA) \to \pi_{1}(MA) $ is given by; $f_{\#}(a) = a $, $f_{\#}(b) = a^{b_{3}} b^{b_{4}} $, $f_{\#}(c) = a^{c_{1}} b^{c_{2}} c $ as in cases of the Theorem \[main-theorem-daci1\]. Given an even positive integer $n$ such that $f^{n}$ is deformable to a fixed point free map over $S^{1}$ then $f^{k}$ is deformable to a fixed point free map over $S^{1}$, for all even positive integer $k$, except when $MA$ is as in case $IV$ and
$a_{3}$ is odd and $k\equiv_{4}0$ or
$a_{3}$ is even, $k\equiv_{4}0$ and $b_{3}b_{4}c_{2}\equiv_{2}1$.
[*Proof.*]{} Is enough to verify that if the conditions of the Theorem \[theorem-1\] are satisfied for some $n$ even then those conditions are also satisfied by every even $k$. We will analyze each case of the Theorem \[theorem-1\].
Case I. In this case for each $n \in \mathbb{N}$ the fiber-preserving map can be deformed over $S^{1}$ to a fixed point free map.
Cases II and III. In these cases if for some $n$ odd the fiber-preserving map $f^{n}: MA \to MA$ is deformed to a fixed point free map over $S^{1}$ then we must have; $c_{1}(b_{4}-1)-c_{2}b_{3} = 0$ or $b_{4}=-1$. Thus, for all even $k$, $f^{k}$ can be deformed to a fixed point free map over $S^{1}$.
Case IV. If $n$ is an even positive integer and $f^{n}: MA \to MA$ is deformed to a fixed point free map over $S^{1}$, then $n(b_{4}(b_{3}+1) -1 -c_{1}(b_{4}-1) + b_{3}c_{2}) - (n-1)(b_{4}-1) \equiv_{2} 0$ and
if $a_{3}$ is odd then $[(nc_{1}+\frac{n(n-1)}{2}b_{3}c_{2},nc_{2})]\neq[(0,0)]\in\frac{\mathbb{Z}\oplus\mathbb{Z}}{\langle(1,2),(0,4)\rangle}$ or
if $a_{3}$ is even then $[(nc_{1} +\frac{n(n-1)}{2}b_{3}b_{4}c_{2}, c_{2}+(n-1)b_{4}c_{2})]\neq[(0,0)]\in\frac{\mathbb{Z}\oplus\mathbb{Z}}{\langle(2,0),(0,2)\rangle}.$
Note that if $a_{3}$ is odd then $b_{4}=1$ and $$\begin{array}{ccl}
[(nc_{1}+\frac{n(n-1)}{2}b_{3}c_{2},nc_{2})] & = & [(0,nc_{2}- 2(nc_{1}+\frac{n(n-1)}{2}b_{3}c_{2}))]\\
& = & [(0,n(c_{2} - 2c_{1}-(n-1)b_{3}c_{2} )))] \in\frac{\mathbb{Z}\oplus\mathbb{Z}}{\langle(1,2),(0,4)\rangle};\\
&\Rightarrow& n(c_{2} - 2c_{1}-(n-1)b_{3}c_{2})\not\equiv_{4}0\\
&\Rightarrow& c_{2} - 2c_{1}-(n-1)b_{3}c_{2}\equiv_{2}1 \ and \ n\equiv_{4}2.
\end{array}$$ If $a_{3}$ is even we have $$\left[\left(nc_{1} +\frac{n(n-1)}{2}b_{3}b_{4}c_{2}, c_{2}+(n-1)b_{4}c_{2}\right)\right]=\left[\left(\frac{n}{2}b_{3}b_{4}c_{2}, c_{2}(1+b_{4})\right)\right].$$
Let $k$ be an even positive integer then $k$ must be odd and $$\begin{array}{ccc}
k(b_{4}(b_{3}+1) -1 -c_{1}(b_{4}-1) + b_{3}c_{2}) - (k-1)(b_{4}-1) & \equiv_{2} & 0\\
\end{array}$$ Then, $f^{k}: MA \to MA$ can be deformed to a fixed point free map over $S^{1}$ except when:
$a_{3}$ is odd and $$\begin{array}{ccl}
[(kc_{1}+\frac{k(k-1)}{2}b_{3}c_{2},kc_{2})] & = & [(0,k(c_{2} - 2c_{1}-(k-1)b_{3}c_{2} )))] \\
& = & [(0,k )))] \\
& = & [(0,0)] \in\frac{\mathbb{Z}\oplus\mathbb{Z}}{\langle(1,2),(0,4)\rangle}.\\
\end{array}$$
or $a_{3}$ is even and $$\begin{array}{ccl}
[(kc_{1} +\frac{k(k-1)}{2}b_{3}b_{4}c_{2}, c_{2}+(k-1)b_{4}c_{2})] & = & \left[\left(\frac{k}{2}b_{3}b_{4}c_{2}, c_{2}(1+b_{4})\right)\right] \\
&= & [(0,0)]\in\frac{\mathbb{Z}\oplus\mathbb{Z}}{\langle(2,0),(0,2)\rangle}.\\
\end{array}$$ Note that, if $f^{n}$ can be deformed to a fixed point free map over $S^{1}$ but $f^{k}$ does not then we have $b_{3}b_{4}c_{2}\equiv_{2}1$ and $k\equiv_{4}0$.
Case V. If $n$ is an even positive integer and $f^{n}: MA \to MA$ is deformed to a fixed point free map over $S^{1}$, then
if $a_{3}$ is odd then $\frac{b_{4}-1}{2}((1+c_{2})(1+(n-1)b_{4})) \equiv_{2} 0 $ and at least one of $(1+c_{2})(1+(n-1)b_{4})$ and $\frac{b_{4}-1}{L}$ is even, where $L:= mdc(b_{4}-1,c_{2})$, or
if $a_{3}$ is even then $n(b_{4}-1)(c_{1}-\frac{a_{3}}{2}c_{2}-1) + (n-1)(b_{4}-1)\equiv_{2} 0$ and at least one of $n(c_{1}-\frac{a_{3}}{2}c_{2}-1) + (n-1)$ and $\frac{b_{4}-1}{L}$ is even, where $L:=mdc(b_{4}-1,c_{2}).$
Let $a_{3}$ odd and $k$ an even positive integer then $$\begin{array}{crcl}
& (1+(k-1)b_{4})) & \equiv_{2} & (1+(n-1)b_{4})) \\
\Rightarrow & \frac{b_{4}-1}{2}((1+c_{2})(1+(k-1)b_{4})) & \equiv_{2} & \frac{b_{4}-1}{2}((1+c_{2})(1+(n-1)b_{4})) \\
& & \equiv _{2}& 0 ; \\
& (1+c_{2})(1+(k-1)b_{4}) & \equiv_{2} & (1+c_{2})(1+(n-1)b_{4}).
\end{array}$$ Then, $f^{k}: MA \to MA$ can be deformed to a fixed point free map over $S^{1}$ for $a_{3}$ odd. Let $a_{3}$ even and $k$ an even positive integer then $$\begin{array}{crcl}
& n(b_{4}-1)(c_{1}-\frac{a_{3}}{2}c_{2}-1) + (n-1)(b_{4}-1) & \equiv_{2} &b_{4}-1;\\
& n(c_{1}-\frac{a_{3}}{2}c_{2}-1) + (n-1) & \equiv_{2} & 1;\\
\Rightarrow & k(b_{4}-1)(c_{1}-\frac{a_{3}}{2}c_{2}-1) + (k-1)(b_{4}-1) & \equiv_{2} & 0.
\end{array}$$ Then, $f^{k}: MA \to MA$ can be deformed to a fixed point free map over $S^{1}$ for $a_{3}$ even.
Given $n \in \mathbb{N}$ and $f: MA \to MA$ a fiber-preserving then from Propositions \[proposition-3\] and \[proposition-4\] the conditions to deform $f$ and $f^{2}$ to a fixed point free map over $S^{1}$ are enough to deform $f^{k}$ to a fixed point free map over $S^{1}$ for all $k$ divisor of $n.$
\[theorem-2\] Let $f: T \times I \to T \times I$ be the map defined by; $$f(x,y,t)=( x+b_{3}y+c_{1}t+\varepsilon , b_{4}y+c_{2}t+\delta , t).$$ Denoting $f^{n}(x,y,t) = (x_{n},y_{n},t)$ then $x_{n}$ and $y_{n}$ are given by $$\begin{array}{ccl}
x_{n} & = & x +b_{3}y\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}} + (nc_{1}+b_{3}c_{2}\sum_{i=0}^{n-1}ib_{4}^{n-1-i})t+b_{3}\delta\sum_{i=0}^{n-1}ib_{4}^{n-1-i}+n\varepsilon\\
y_{n} & = & b_{4}^{n}y+c_{2}t\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}}+\delta\sum_{i=0}^{n-1}b_{4}^{i},
\end{array}$$ and for each $n \in \mathbb{N}$ and $\epsilon, \delta$ appropriates the map $f^{n}$ induces a fiber-preserving map in the fiber bundle $MA$, as in Theorem \[daci1theorem\], which we will represent by $f^{n}(<x,y,t>) = <x_{n},y_{n},t>,$ such that the induces homomorphism $(f^{n})_{\#}$ is as in the Proposition \[proposition-1\]. Note that the induce homomorphism $f_{\#}$ is given by; $f_{\#}(a) = a $, $f_{\#}(b) = a^{b_{3}} b^{b_{4}} $, $f_{\#}(c) = a^{c_{1}} b^{c_{2}} c $.
[*Proof.*]{} Denote $f^{n}(x,y,t)= ( x_{n},y_{n},t)$ for each $n \in \mathbb{N}.$ We have $$\begin{array}{ccl}
x_{2} & = & x_{1}+b_{3}y_{1}+c_{1}t+\varepsilon\\
& = & (x+b_{3}y+c_{1}t+\varepsilon )+b_{3}(b_{4}y+c_{2}t+\delta)+c_{1}t+\varepsilon\\
& = & x +b_{3}y(b_{4}+1) + (2c_{1}+b_{3}c_{2})t+b_{3}\delta+2\varepsilon;\\
y_{2} & = & b_{4}y_{1}+c_{2}t+\delta\\
& = & b_{4}(b_{4}y+c_{2}t+\delta)+c_{2}t+\delta\\
& = & b_{4}^{2}y+c_{2}(b_{4}+1)t+(b_{4}+1)\delta.
\end{array}$$
Suppose that $f^{n}(x,y,t) = (x_{n},y_{n},t)$ as in hypothesis, then $$f^{n+1}(x,y,t)=( x_{n}+b_{3}y_{n}+c_{1}t+\varepsilon , b_{4}y_{n}+c_{2}t+\delta , t)=(x_{n+1},y_{n+1},t),$$ where $$\begin{array}{ccl}
x_{n+1} & = & x_{n}+b_{3}y_{n}+c_{1}t+\varepsilon\\
& = & (x +b_{3}y\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}} + (nc_{1}+b_{3}c_{2}\sum_{i=0}^{n-1}ib_{4}^{n-1-i})t+
b_{3}\delta\sum_{i=0}^{n-1}ib_{4}^{n-1-i}+n\varepsilon) + \\
& & + b_{3}(b_{4}^{n}y+c_{2}t\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}}+\delta\sum_{i=0}^{n-1}b_{4}^{i})+c_{1}t+\varepsilon\\
& = & \displaystyle{x + ( b_{3}y\sum_{i=0}^{n-1}b_{4}^{i} + b_{3}yb_{4}^{n} ) + ((nc_{1}+b_{3}c_{2}\sum_{i=0}^{n-1}ib_{4}^{n-1-i})t +c_{1}t+} \\
& & \displaystyle{ b_{3}c_{2}t\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}}) } \displaystyle{+ (b_{3}\delta\sum_{i=0}^{n-1}ib_{4}^{n-1-i}+ b_{3}\delta\sum_{i=0}^{n-1}b_{4}^{i}) +(n\varepsilon+\varepsilon)} \\
& = & x +b_{3}y\displaystyle{\sum_{i=0}^{n}b_{4}^{i}} + ((n+1)c_{1}+b_{3}c_{2}\sum_{i=0}^{n}ib_{4}^{n-i})t+b_{3}\delta\sum_{i=0}^{n}ib_{4}^{n-i}+(n+1)\varepsilon;\\
\end{array}$$
$$\begin{array}{ccl}
y_{n+1} & = & b_{4}y_{n}+c_{2}t+\delta\\
& = & b_{4}(b_{4}^{n}y+c_{2}t\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}}+\delta\sum_{i=0}^{n-1}b_{4}^{i})+c_{2}t+\delta\\
& = & \displaystyle{b_{4}^{n+1}y+ (c_{2}t\sum_{i=1}^{n}b_{4}^{i} +c_{2}t)+(\delta\sum_{i=1}^{n}b_{4}^{i}+\delta)}\\
& = & b_{4}^{n+1}y+c_{2}t\displaystyle{\sum_{i=0}^{n}b_{4}^{i}}+\delta\sum_{i=0}^{n}b_{4}^{i},
\end{array}$$
as we wish. Now, to verify that $f^{n}(<x,y,0>) = f^{n}(<A\left(^{x}_{y}\right),1>)$ for each $n \in \mathbb{N}$, in $T \times I$, where $T$ is the torus, firstly we will verify this condition for $n = 1$. We have $$f<x,y,0>=< x+b_{3}y+\varepsilon , b_{4}y+\delta , 0> \,\,\,\,\, and$$ $$f<A\left(^{x}_{y}\right),1> = <(a_{1}+a_{2}b_{3})x +(a_{3}+b_{3}a_{4})y +c_{1}+\varepsilon,
b_{4}a_{2}x+b_{4}a_{4}y+c_{2} +\delta, 1 >$$ But in $MA$ we have $<x,y,0> = <A\left(^{x}_{y}\right),1>$, that is, $<x,y,0> = <a_{1}x+a_{3}y, a_{2}x+a_{4}y,1>.$ Now we will analyze each case of the Theorem \[daci1theorem\].
Case I. In this case we need consider $b_{3} = 0$ and $b_{4}=1.$ Thus, in $MA$ $ f<x,y,0> = <x+\epsilon, y + \delta,0> = $ $<a_{1}x + a_{3}y + a_{1}\epsilon+a_{3}\delta, a_{2}x + a_{4}y + a_{2}\epsilon+a_{4}\delta,1>. $ Note that, $f<A\left(^{x}_{y}\right),1> =
<a_{1}x +a_{3}y +c_{1}+\epsilon, a_{2}x+a_{4}y+c_{2} +\delta, 1 > . $ Therefore $f<x,y,0> = f<A\left(^{x}_{y}\right),1>$ if $a_{1}\epsilon+a_{3}\delta = \epsilon + k$ and $a_{2}\epsilon+a_{4}\delta = \delta + l$ where $k,l \in \mathbb{Z}.$
Case II. In this case we have $a_{1} = a_{4}=1$, $a_{2}=0$ and $a_{3}(b_{4}-1)=0.$ Therefore, $f<A\left(^{x}_{y}\right),1> = <x +(a_{3}+b_{3})y +c_{1}+\epsilon, b_{4}y+c_{2} +\delta, 1 >.$ Thus, $f<x,y,0> = <x+b_{3}y+\epsilon,b_{4}y+\delta,0> = <x +(a_{3}+b_{3})y +\epsilon+a_{3}\delta, b_{4}y+\delta,1>$ $= <x +(a_{3}+b_{3})y +\epsilon+ a_{3}\delta, b_{4}y+\delta,1>.$ Therefore $f<x,y,0> = f<A\left(^{x}_{y}\right),1>$ if $a_{3}\delta \in \mathbb{Z}.$
Case III. In this case we have $a_{1} = 1,$ $ a_{4}=-1$, $a_{2}=0$ and $a_{3}(b_{4}-1)=-2b_{3}.$ Therefore $f<A\left(^{x}_{y}\right),1> = <x +(a_{3}-b_{3})y +c_{1}+\epsilon, -b_{4}y+c_{2} +\delta, 1 >.$ We have $f<x,y,0> = <x+b_{3}y+\epsilon,b_{4}y+\delta,0> = <x+(a_{3}b_{4}+b_{3})y+\epsilon +a_{3}\delta,-b_{4}y-\delta,1>.$ Thus $f<x,y,0> = <x+(a_{3}-b_{3})y+\epsilon +a_{3}\delta,-b_{4}y-\delta,1>.$ Then $f<x,y,0> = f<A\left(^{x}_{y}\right),1>$ if $a_{3}\delta \in \mathbb{Z}$ and $\delta = \frac{k}{2}, k \in \mathbb{Z}.$
Case IV. In this case we have $a_{1} = -1,$ $ a_{4}=-1$, $a_{2}=0$ and $a_{3}(b_{4}-1)=0.$ Thus $f<A\left(^{x}_{y}\right),1> = <-x +(a_{3}-b_{3})y +c_{1}+\epsilon, -b_{4}y+c_{2} +\delta, 1 >.$ We have $f(x,y,0) = (x+b_{3}y+\epsilon,b_{4}y+\delta,0) = (-x+(a_{3}b_{4}-b_{3})y-\epsilon +a_{3}\delta,-b_{4}y-\delta,1)$ Thus $f<x,y,0> = <-x+(a_{3}-b_{3})y-\epsilon +a_{3}\delta,-b_{4}y-\delta,1>.$ Then $f<x,y,0> = f<A\left(^{x}_{y}\right),1>$ if $\epsilon = \frac{a_{3}m +2k}{4}$ and $ \delta = \frac{m}{2}$ where $m,k \in \mathbb{Z}.$
Case V. In this case we have $a_{1} = -1,$ $ a_{4}=1$, $a_{2}=0$ and $a_{3}(b_{4}-1)= 2b_{3}.$ Therefore $f<A\left(^{x}_{y}\right),1> = <-x +(a_{3}+b_{3})y +c_{1}+\epsilon, b_{4}y+c_{2} +\delta, 1 >.$ We have $f(x,y,0) = (x+b_{3}y+\epsilon,b_{4}y+\delta,0) = (-x+(a_{3}b_{4}-b_{3})y-\epsilon +a_{3}\delta,b_{4}y+\delta,1)$ Thus $f<x,y,0> = <x+(a_{3}+b_{3})y-\epsilon +a_{3}\delta,b_{4}y+\delta,1>.$ Then $f<x,y,0> = f<A\left(^{x}_{y}\right),1>$ if $ \epsilon = \frac{a_{3}\delta+k}{2}$ where $k \in \mathbb{Z}.$
Now we will verify the condition for all $n \in \mathbb{N}.$
Case I. In $MA$ we have $$\begin{array}{ccl}
f^{n}<x,y,0> & = & <x +n\varepsilon, y +n\delta,0>\\
& = & <a_{1}(x+n\varepsilon) + a_{3}(y + n\delta), a_{2}(x+n\varepsilon) + a_{4}(y+n\delta) ,1>\\
& = & <a_{1}x + a_{3}y + na_{1}\varepsilon+na_{3}\delta, a_{2}x + a_{4}y + na_{2}\varepsilon+na_{4}\delta,1>.
\end{array}$$ But, $f^{n}\left<A \left(^{x}_{y}\right),1\right> = <a_{1} x + a_{3} y + n c_{1} +n\varepsilon, a_{2}x + a_{4}y + nc_{2}+ n\delta,1>.$ Then, $f^{n}<x,y,0> = f^{n}\left<A \left(^{x}_{y}\right),1\right>$ if $na_{1}\varepsilon +n a_{3}\delta= n\varepsilon+k$, $k\in\mathbb{Z}$, and $na_{2}\varepsilon+na_{4}\delta = n\delta +l$, $l \in\mathbb{Z}$.
Case II. $$\begin{array}{l}
f^{n}<x,y,0> = \\
= <x +b_{3}y\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}} +b_{3}\delta\sum_{i=0}^{n-1}ib_{4}^{n-1-i}+n\varepsilon, b_{4}^{n}y+\delta\sum_{i=0}^{n-1}b_{4}^{i},0 >\\
= <\left(x +b_{3}y\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}} +b_{3}\delta\sum_{i=0}^{n-1}ib_{4}^{n-1-i}+n\varepsilon\right) + a_{3}\left( b_{4}^{n}y+\delta\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}}\right) , \\
b_{4}^{n}y+ \delta\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}},1>\\
= <x + \left(a_{3}b_{4}^{n}+b_{3}\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}} \right)y +
\left(a_{3}\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}}+b_{3}\displaystyle{\sum_{i=0}^{n-1}ib_{4}^{n-1-i}} \right)\delta +
n\varepsilon, \\
b_{4}^{n}y+\delta\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}},1 >
\end{array}$$ But, $f^{n}\left(<A \left(^{x}_{y}\right),1>\right)=$ $$\begin{array}{l}
=<x + \left(a_{3}+b_{3}\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}}\right)y +
b_{3}\delta\displaystyle{\sum_{i=0}^{n-1}ib_{4}^{n-1-i}}+ n\varepsilon+
b_{3}c_{2}\displaystyle{\sum_{i=0}^{n-1}ib_{4}^{n-1-i}}
+ nc_{1}, \\ b_{4}^{n}y+\delta\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}} + c_{2}\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}},1>.
\end{array}$$ Thus $f^{n}\left(<A \left(^{x}_{y}\right),1>\right) = f^{n}<x,y,0> $ if $\delta a_{3} \displaystyle{\sum_{i=0}^{n-1} b_{4}^{i}}\in\mathbb{Z}$.
Case III. $f^{n}<x,y,0> = $ $$\begin{array}{l}
=< x +b_{3}y\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}} + b_{3}\delta\sum_{i=0}^{n-1}ib_{4}^{n-1-i}+n\varepsilon , b_{4}^{n}y+\delta\sum_{i=0}^{n-1}b_{4}^{i},0 >\\
= <\left(x +b_{3}y\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}} +b_{3}\delta\sum_{i=0}^{n-1}ib_{4}^{n-1-i}+n\varepsilon\right) + a_{3}\left( b_{4}^{n}y+\delta\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}}\right) ,\\
-b_{4}^{n}y-\delta\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}},1 >\\
= <x + \left(a_{3}b_{4}^{n}+b_{3}\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}} \right)y +
\left(a_{3}\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}}+b_{3}\displaystyle{\sum_{i=0}^{n-1}ib_{4}^{n-1-i}}
\right)\delta + n\varepsilon, \\
-b_{4}^{n}y-\delta\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}},1>
\end{array}$$ Note that, $f^{n}\left(<A \left(^{x}_{y}\right),1>\right) = $ $$\begin{array}{l}
=<x + \left(a_{3}-b_{3}\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}}\right)y - b_{3}\delta\displaystyle{\sum_{i=0}^{n-1}ib_{4}^{n-1-i}}+ n\varepsilon-b_{3}c_{2}\displaystyle{\sum_{i=0}^{n-1}ib_{4}^{n-1-i}} + nc_{1}, \\
-b_{4}^{n}y+\delta\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}} + c_{2}\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}},1>.
\end{array}$$ Therefore, $f^{n}<x,y,0> = f^{n}\left(<A \left(^{x}_{y}\right),1>\right) $ if $2\delta\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}}
\in\mathbb{Z}$ and $\left(a_{3}\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}}+2b_{3}\displaystyle{\sum_{i=0}^{n-1}ib_{4}^{n-1-i}}
\right)\delta \in \mathbb{Z}$.
Case IV. $f^{n}<x,y,0> = $ $$\begin{array}{l}
= < x +b_{3}y\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}} + b_{3}\delta\sum_{i=0}^{n-1}ib_{4}^{n-1-i}+n\varepsilon , b_{4}^{n}y+\delta\sum_{i=0}^{n-1}b_{4}^{i},0>\\
= < -\left(x +b_{3}y\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}} +b_{3}\delta\sum_{i=0}^{n-1}ib_{4}^{n-1-i}+n\varepsilon\right) + a_{3}\left( b_{4}^{n}y+\delta\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}}\right) , \\
-b_{4}^{n}y-\delta\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}},1>\\
\end{array}$$ $$\begin{array}{l}
= <-x + \left(a_{3}b_{4}^{n}-b_{3}\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}} \right)y +
\left(a_{3}\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}}-b_{3}\displaystyle{\sum_{i=0}^{n-1}ib_{4}^{n-1-i}} \right)\delta
- n\varepsilon, \\
-b_{4}^{n}y-\delta\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}},1>
\end{array}$$ We have, $f^{n}\left(<A \left(^{x}_{y}\right),1>\right) = $ $$\begin{array}{l}
= <-x + \left(a_{3}-b_{3}\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}}\right)y - b_{3}\delta\displaystyle{\sum_{i=0}^{n-1}ib_{4}^{n-1-i}}+
n\varepsilon-b_{3}c_{2}\displaystyle{\sum_{i=0}^{n-1}ib_{4}^{n-1-i}} + nc_{1},\\
-b_{4}^{n}y+\delta\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}} + c_{2}\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}},1 >
\end{array}$$ Thus, $f^{n}<x,y,0> = f^{n}\left(<A \left(^{x}_{y}\right),1>\right) $ if $2\delta\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}}\in\mathbb{Z}$ and $2n\varepsilon=a_{3}\delta\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}}+ k$, $k\in\mathbb{Z}$.
Case V. $f^{n}<x,y,0> = $ $$\begin{array}{l}
= < x +b_{3}y\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}} + b_{3}\delta\sum_{i=0}^{n-1}ib_{4}^{n-1-i}+n\varepsilon ,
b_{4}^{n}y+\delta\sum_{i=0}^{n-1}b_{4}^{i},0>\\
= <-\left(x +b_{3}y\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}} +b_{3}\delta\sum_{i=0}^{n-1}ib_{4}^{n-1-i}+n\varepsilon\right)
+ a_{3}\left( b_{4}^{n}y+\delta\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}}\right) , \\
b_{4}^{n}y+ \delta\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}},1>\\
= < -x + \left(a_{3}b_{4}^{n}-b_{3}\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}} \right)y +
\left(a_{3}\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}}-b_{3}\displaystyle{\sum_{i=0}^{n-1}ib_{4}^{n-1-i}} \right)\delta
- n\varepsilon, \\
b_{4}^{n}y+\delta\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}},1 >
\end{array}$$ We have, $f^{n}\left(<A \left(^{x}_{y}\right),1>\right) = $ $$\begin{array}{l}
=<-x + \left(a_{3}+b_{3}\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}}\right)y - b_{3}\delta\displaystyle{\sum_{i=0}^{n-1}ib_{4}^{n-1-i}}+ n\varepsilon-b_{3}c_{2}\displaystyle{\sum_{i=0}^{n-1}ib_{4}^{n-1-i}} +
nc_{1}, \\
+b_{4}^{n}y+\delta\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}} + c_{2}\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}},1>.
\end{array}$$ Therefore, $f^{n}<x,y,0> = f^{n}\left(<A \left(^{x}_{y}\right),1>\right) $ if $2n\varepsilon=a_{3}\delta\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}}+ k$, $k\in\mathbb{Z}$.
Thus for each $n \in \mathbb{N}$ and $\epsilon, \delta $ satisfying the conditions above the map $f^{n}: T \times I \to T \times I$ induces a fiber-preserving map on $MA$ which will be represent by the same symbol.
The next result we will help us to study the fixed points of $f^{n}$ given in Theorem \[theorem-2\].
\[prop-2\] Let $n, \ b_{3}, \ b_{4}, \ c_{1}, \ c_{2} \in \mathbb{Z}$, $n\geq1$. If $c_{1}(b_{4}-1)-c_{2}b_{3}\neq 0$ then for all $\varepsilon , \ \delta \in \mathbb{R}$ there are $a,\ b \in \mathbb{Z}$ such that the system bellow has solution $(x,y,t) \in \mathbb{R}^{2}\times I$: $$\left\{
\begin{array}{lll}
x + a & = & x +b_{3}y\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}} + (nc_{1}+b_{3}c_{2}\sum_{i=0}^{n-1}ib_{4}^{n-1-i})t+b_{3}\delta\sum_{i=0}^{n-1}ib_{4}^{n-1-i}+n\varepsilon;\\
y + b & = & b_{4}^{n}y+c_{2}t\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}}+\delta\sum_{i=0}^{n-1}b_{4}^{i}.
\end{array} \right.$$
[*Proof:*]{} Suppose $b_{4}\neq 1$ and $b_{4}\neq-1$ with $n$ even ($b_{4}=-1$ with $n$ odd is allowed) and $c_{1}(b_{4}-1)-b_{3}c_{2}\neq 0$ then given $\varepsilon, \ \delta\in\mathbb{R}$ we have the solutions $x\in\mathbb{R}$ and: $$\begin{array}{ccl}
t & = & \dfrac{nb_{3}\delta-n(b_{4}-1)\varepsilon - (b_{4}-1)a - b_{3}b}{n(c_{1}(b_{4}-1)-b_{3}c_{2})}\in I;\\
y & = & \dfrac{nc_{2}\varepsilon-nc_{1}\delta - ac_{2}}{n(c_{1}(b_{4}-1)-b_{3}c_{2})} + b\left(\dfrac{1}{b^{n}_{4}-1}+ \dfrac{b_{3}c_{2}}{n(b_{4}-1)(c_{1}(b_{4}-1)-b_{3}c_{2})}\right)\in\mathbb{R}.
\end{array}$$ Thus, we need to find $a, \ b\in \mathbb{Z}$ such that $0\leq t\leq 1$. Let $k_{0}=n(c_{1}(b_{4}-1)-b_{3}c_{2})\in \mathbb{Z}$, $k_{0}\neq0$, and $k_{1}= nb_{3}\delta-n(b_{4}-1)\varepsilon\in \mathbb{R}$, $t=\dfrac{k_{1} - (b_{4}-1)a - b_{3}b}{k_{0}}$. If $0\leq k_{1}\leq k_{0}$ or $k_{0}\leq k_{1}\leq 0$ let $a=b=0$, then $t=\dfrac{k_{1}}{k_{0}}$. If $0< k_{0}\leq k_{1}$ or $k_{1}\leq0< k_{0}$ then there are $d, \ q \in \mathbb{Z}$ such that $k_{1}=dk_{0}+q$ with $0\leq q <k_{0}$. Let $a=nc_{1}d$ and $b=nc_{2}d$, then $$t=\dfrac{dk_{0}+q - (b_{4}-1)nc_{1}d - b_{3}nc_{2}d}{k_{0}}=d +\dfrac{q}{k_{0}}- \dfrac{dk_{0}}{k_{0}}=\dfrac{q}{k_{0}}.$$ If $ k_{1}\leq k_{0}<0$ or $k_{0}<0\leq k_{1}$ then there are $d, \ q \in \mathbb{Z}$ such that $k_{1}=dk_{0}+q$ with $0\leq q <|k_{0}|$. Let $k\in\mathbb{Z}$ the least integer greater than $\dfrac{-q}{k_{0}}$, $a=nc_{1}(d-k)$ and $b=nc_{2}(d-k)$, then $$t=\dfrac{dk_{0}+q - (b_{4}-1)nc_{1}(d-k) - b_{3}nc_{2}(d-k)}{k_{0}}=d +\dfrac{q}{k_{0}}- (d-k)=\dfrac{q}{k_{0}}+k.$$ Then, $0\leq t\leq 1$.
If $b_{4}= 1$ and $c_{1}(b_{4}-1)-b_{3}c_{2}\neq 0$ then $b_{3}c_{2}\neq 0$. Thus, given $\varepsilon, \ \delta\in\mathbb{R}$ we have the solutions $x\in\mathbb{R}$ and: $$\begin{array}{ccl}
t & = & \dfrac{b}{nc_{2}}-\dfrac{\delta}{c_{2}}\in I;\\
y & = & \dfrac{-nc_{2}\varepsilon+nc_{1}\delta + ac_{2}}{nb_{3}c_{2}} - b\left(\dfrac{c_{1}}{nb_{3}c_{2}}+ \dfrac{n-1}{2n}\right)\in\mathbb{R}.
\end{array}$$ We need to find $b\in \mathbb{Z}$ such that $0\leq t\leq 1$. If $c_{2}>0$ take $n\delta\leq b\leq n(c_{2}+\delta)$ and if $c_{2}<0$ take $n\delta\geq b\geq n(c_{2}+\delta)$.
If $b_{4}= -1$, $n$ even and $c_{1}(b_{4}-1)-b_{3}c_{2}\neq 0$ then $2c_{1}+ b_{3}c_{2}\neq 0$. Thus, given $\varepsilon, \ \delta\in\mathbb{R}$ we have the solutions $x,\ y\in\mathbb{R}$ and: $$\begin{array}{ccl}
t & = & \dfrac{2a-nb_{3}\delta-2n\varepsilon}{n(2c_{1}+ b_{3}c_{2})}\in I;\\
\end{array}$$ We need to find $a\in \mathbb{Z}$ such that $0\leq t\leq 1$. Let $n=2k$, $k_{0}=k(2c_{1}+b_{3}c_{2})\in \mathbb{Z}$, $k_{0}\neq0$, and $k_{1}= -kb_{3}\delta-2k\varepsilon\in \mathbb{R}$, $t=\dfrac{a+k_{1}}{k_{0}}$. If $0\leq k_{1}\leq k_{0}$ or $k_{0}\leq k_{1}\leq 0$ let $a=0$, then $t=\dfrac{k_{1}}{k_{0}}$. If $0< k_{0}\leq k_{1}$ or $k_{1}\leq0< k_{0}$ then there are $d, \ q \in \mathbb{Z}$ such that $k_{1}=dk_{0}+q$ with $0\leq q <k_{0}$. Let $a=-k_{0}d$, then $t=\dfrac{-k_{0}d + dk_{0}+q}{k_{0}}=\dfrac{q}{k_{0}}$. If $ k_{1}\leq k_{0}<0$ or $k_{0}<0\leq k_{1}$ then there are $d, \ q \in \mathbb{Z}$ such that $k_{1}=dk_{0}+q$ with $0\leq q <|k_{0}|$. Let $l\in\mathbb{Z}$ the greatest integer lower than $q$ and $a=-l-k_{0}d$, then $t=\dfrac{-l-k_{0}d+dk_{0}+q}{k_{0}}=\dfrac{q-l}{k_{0}}$. Then, $0\leq t\leq 1$.
\[maintheorem\] Let $f: MA \to MA$ be a fiber-preserving map, where $MA$ is a T-bundle over $S^{1},$ and $f_{\#}(a) = a $, $f_{\#}(b) = a^{b_{3}} b^{b_{4}} $, $f_{\#}(c) = a^{c_{1}} b^{c_{2}} c $. Suppose that $f^{n}: MA \to MA$ can be deformed to a fixed point free map over $S^{1}.$ Then there exits a map $g$ fiberwise homotopic to $f$ such that $g^{n}$ is a fixed point free map, in the cases $I$ and $II,$ if and only if the following conditions are satisfies;
1\) $MA$ is as in case $I$ and $f$ is arbitrary.
2\) $MA$ is as in cases $II$, $III$ and $c_{1}(b_{4}-1)-c_{2}b_{3} = 0$.
[*Proof.*]{} (1) For each map $f$ such that $(f_{|T})_{\#} = Id$ consider the map $g$ fiberwise homotopic to $f$ given by; $g^{'}(<x,y,t>) = <x+c_{1}t+\epsilon, y+c_{2}t+\delta, t>$, with $\epsilon , \delta \in \mathbb{Q}-\mathbb{Z} $ satisfying the condition; $a_{1}\epsilon + a_{3}\delta = \epsilon + k$ and $a_{2}\epsilon + a_{4}\delta = \epsilon + k$ for some $k,l \in \mathbb{Z}.$ Note that $g^{'}$ is fiberwise homotopic to the map $g$ defined by; $$g(<x,y,t>) = \left \{
\begin{array}{lll}
<x+2c_{1}t+\epsilon, y + \delta, t> & if & 0 \leq t \leq \frac{1}{2} \\
<x+ c_{1}+\epsilon, y + c_{2}(2t-1)+\delta, t> & if & \frac{1}{2} \leq t \leq 1 \\
\end{array} \right.$$ In fact, $H: MA \times I \to MA$ defined by; $$H(<x,y,t>,s) = \left \{
\begin{array}{lll}
<x+c_{1}t+\epsilon, y + c_{2}t + \delta, t> & if & 0 \leq t \leq s \\
<x+ c_{1}(2t-s)+\epsilon, y + c_{2}s+\delta, t> & if & s \leq t \leq \frac{s+1}{2} \\
<x+ c_{1}+\epsilon, y + c_{2}(2t-1)+\delta, t> & if & \frac{s+1}{2} \leq t \leq 1 \\
\end{array} \right.$$ is a homotopy between $g^{'}$ and $g.$ Note that, $$g^{n}(<x,y,t>) = \left \{
\begin{array}{lll}
<x+n2c_{1}t+n\epsilon, y + n\delta, t> & if & 0 \leq t \leq \frac{1}{2} \\
<x+ nc_{1}+n\epsilon, y + nc_{2}(2t-1)+n\delta, t> & if & \frac{1}{2} \leq t \leq 1 \\
\end{array} \right.$$ In this case we have $Fix(g^{n}) = \emptyset$.
\(2) From Theorem \[theorem-1\] we must consider two situations; $c_{1}(b_{4}-1)-c_{2}b_{3} = 0$ or $n$ even and $b_{4} = -1.$ First we will suppose $c_{1}(b_{4}-1)-c_{2}b_{3} = 0$. Note that if $c_{1}(b_{4}-1)-c_{2}b_{3} \neq 0$ then $g$ can not be deformed to a fixed point free map. Therefore can not exits $f$ fiberwise homotopic to $g$ such that $Fix(f^{n}) = \emptyset$ for all $n \geq 1$ by Lemma \[lemma1\].
Suppose $b_{4}=1$ and $f^{n}(x,y,t) = (x_{n},y_{n},t)$ for each $n \in \mathbb{N}$ gives in the Theorem \[theorem-2\]. Then: $$\left\{\begin{array}{ccl}
x_{n} & = & \displaystyle{x +nb_{3}y + \left(c_{1}+b_{3}c_{2}\frac{n-1}{2}\right)nt+\frac{n(n-1)}{2}b_{3}\delta+n\varepsilon},\\
y_{n} & = & \displaystyle{y+nc_{2}t+n\delta}.
\end{array}\right.$$ For $c_{2}b_{3} = 0$, $f^{n}(x,y,t)$ is a fixed point free map for each $n$ choosing $\epsilon \in \mathbb{R}-\mathbb{Q}$ and $\delta = 0$ if $b_{3} = 0$ or $\delta \in \mathbb{R}-\mathbb{Q}$ if $c_{2} = 0.$ In fact, if $b_{3} = 0$ and $c_{2} \neq 0$ choose $\epsilon \in \mathbb{R}-\mathbb{Q}$ and $\delta = 0$, then $$\left\{
\begin{array}{lll}
x + k_{n} & = & x + nc_{1}t+ n\varepsilon \\
y + l_{n} & = & y + nc_{2}t
\end{array} \right.$$ for some $k_{n},l_{n} \in \mathbb{Z}.$ From equations we obtain $t = \dfrac{l_{n}}{nc_{2}} \in \mathbb{Q}$ and $ \epsilon = \dfrac{k_{n}}{n} - \dfrac{c_{1}l_{n}}{nc_{2}} \in \mathbb{Q}$, but this is a contradiction because $\epsilon $ in $\mathbb{R}-\mathbb{Q}.$ Therefore $f^{n}$ has no fixed point for all $n$. If $c_{2} = 0$ we choose $\delta \in \mathbb{R}-\mathbb{Q} $ then $y +n\delta = y+l_{n}$ which implies $\delta = \frac{l_{n}}{n} \in \mathbb{Q}$ that is a contradiction because $\delta $ in $ \mathbb{R}-\mathbb{Q} $. Thus $f^{n}$ has no fixed point for all $n$.
Now we suppose $b_{4}\neq1$, $\delta = 0$ and $c_{1}(b_{4} - 1)=c_{2}b_{3}$. Then $$\begin{array}{ccl}
x_{n} & = & x +b_{3}y\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}} + (nc_{1}+b_{3}c_{2}\sum_{i=0}^{n-1}ib_{4}^{n-1-i})t+n\varepsilon\\
& = & x + \left( \dfrac{b_{3}(b_{4}^{n}-1)}{b_{4}-1}\right) y + \left( n c_{1}+\dfrac{b_{3}c_{2}(b_{4}^{n}-1+n(1-b_{4}))}{(b_{4}-1)^{2}}\right) t + n\varepsilon\\
& = & x + \left( \dfrac{b_{4}^{n}-1}{b_{4}-1}\right)b_{3} y + \left( \dfrac{b_{4}^{n}-1}{b_{4}-1}\right) c_{1} t + n\varepsilon;\\
y_{n} & = & b_{4}^{n}y+c_{2}t\displaystyle{\sum_{i=0}^{n-1}b_{4}^{i}}\\
& = & b_{4}^{n}y+ \left( \dfrac{c_{2}(b_{4}^{n}-1)}{b_{4}-1}\right)t.
\end{array}$$
We are interested at the solutions of the system above: $$\begin{aligned}
x_{n} & = & x + k_{n}; \label{eq1}\\
y_{n} & = & y + l_{n}, \label{eq2}\end{aligned}$$ where $k_{n}, l_{n}\in \mathbb{Z}$ and $n\in\mathbb{N}$. With the equation \[eq2\] and $c_{2}\neq 0$, we have: $$t=\dfrac{l_{n}(b_{4}-1)}{c_{2}(b_{4}^{n}-1)} + \dfrac{1-b_{4}}{c_{2}}y.$$ For this $t$ we obtain: $$\begin{array}{ccl}
x_{n} & = & x + \left( \dfrac{b_{4}^{n}-1}{b_{4}-1}\right)b_{3} y + \left( \dfrac{b_{4}^{n}-1}{b_{4}-1}\right) c_{1} t + n\varepsilon\\
& = & x -\left( \dfrac{(b_{4}^{n}-1) ((b_{4}-1)c_{1}-c_{2}b_{3})}{(b_{4}-1) c_{2}}\right) y + n\varepsilon +\dfrac{c_{1}l_{n}}{c_{2}}\\
& = & x + n\varepsilon +\dfrac{c_{1}l_{n}}{c_{2}}
\end{array}$$
But, for $\varepsilon\in\mathbb{R}- \mathbb{Q}$ the equation \[eq2\] has no solution for all $n\in\mathbb{N}$. In fact, for any $k_{n}\in\mathbb{Z}$ we have: $$x_{n} = x + k_{n}
\Rightarrow x + n\varepsilon +\dfrac{c_{1}l_{n}}{c_{2}} = x + k_{n}
\Rightarrow \underbrace{\varepsilon}_{\in\mathbb{R}-\mathbb{Q}}= \underbrace{\dfrac{k_{n}}{n} - \dfrac{c_{1}l_{n}}{nc_{2}}}_{\in\mathbb{Q}}.$$
In the other side, if $c_{2}= 0$ then $c_{1}=0$ because $b_{4}\neq 1$ and: $$\begin{array}{ccl}
x_{n} & = & x + \left( \dfrac{b_{4}^{n}-1}{b_{4}-1}\right)b_{3} y + \left( \dfrac{b_{4}^{n}-1}{b_{4}-1}\right) c_{1} t + n\varepsilon\\
& = & x + \left( \dfrac{b_{4}^{n}-1}{b_{4}-1}\right)b_{3} y + n\varepsilon;\\
y_{n} & = & b_{4}^{n}y+ \left( \dfrac{c_{2}(b_{4}^{n}-1)}{b_{4}-1}\right)t\\
& = & b_{4}^{n}y.
\end{array}$$
Thus, $y=\dfrac{l_{n}}{b_{4}^{n}-1}$ by equation \[eq2\]. Then, $x_{n} = x + n\varepsilon + \dfrac{b_{3}l_{n}}{b_{4}-1}$ and the equation \[eq1\] has no solution for $\varepsilon\in\mathbb{R}- \mathbb{Q}$ and for $n\in\mathbb{N}$. In fact, for any $k_{n}, l_{n}\in\mathbb{Z}$ we have: $$x_{n} = x + k_{n}
\Rightarrow x + n\varepsilon +\dfrac{b_{3}l_{n}}{b_{4}-1} = x + k_{n}
\Rightarrow \underbrace{\varepsilon}_{\in\mathbb{R}-\mathbb{Q}}= \underbrace{\dfrac{k_{n}}{n} - \dfrac{b_{3}l_{n}}{n(b_{4}-1)}}_{\in\mathbb{Q}}.$$
If $b_{4}=- 1$ and $n$ even then $f^{n}(x,y,t) = (x_{n},y_{n},t)$ such that: $$\begin{array}{ccl}
x_{n} & = & x + b_{3}\delta\left(\frac{n}{2}\right)+n\varepsilon;\\
y_{n} & = & y.
\end{array}$$ Thus, choosing $\delta = 0$ and $\varepsilon \in \mathbb{R}-\mathbb{Q},$ the first equation has no solution, that is, $Fix(f) = \emptyset.$
With a similar way of the Theorem \[maintheorem\] we obtain the following result.
Let $f: MA \to MA$ be a fiber-preserving map, where $MA$ is a T-bundle over $S^{1},$ and $f_{\#}(a) = a $, $f_{\#}(b) = a^{b_{3}} b^{b_{4}} $, $f_{\#}(c) = a^{c_{1}} b^{c_{2}} c $. Suppose that $f^{n}: MA \to MA$ can be deformed to a fixed point free map over $S^{1}$ as in the Theorem \[theorem-1\]. If the following conditions are satisfied, in the cases $IV$ and $V,$ then there exits a map $g$ fiberwise homotopic to $f$ such that $g^{n}$ is a fixed point free map.
$MA$ is as in case $IV,$ $V$ and $$c_{1}(b_{4}-1)-c_{2}b_{3} = 0 \,\, and \,\, n \,\, odd \,\, or$$ $$c_{1}(b_{4}-1)-c_{2}b_{3} = 0, \,\, b_{4} \,\, odd \,\, and \,\, n = 4k + 2, k \geq 0.$$
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---
abstract: 'This paper studies the static economic optimization problem of a system with a single aggregator and multiple prosumers associated with the aggregator in a real-time balancing market. The aggregator, as the agent responsible for portfolio balancing, needs to minimize the cost for imbalance satisfaction in real-time by proposing a set of optimal incentivizing prices for the prosumers. On the other hand, the prosumers, as price taker and self-interest agents, want to maximize their profit by changing their supplies or demands and providing flexibility based on the proposed incentivizing prices. We model this problem as a bilevel optimization problem. The state-of-the-art approach to solve a bilevel optimization problem is to reformulate it as a problem. Despite recent developments in the solvers for problems, the computation time for a problem with a large number of decision variables may not be appropriate for real-time applications. We propose a convex equivalent optimization problem for the original bilevel one and prove that the global optimum of the prosumers/aggregator bilevel problem can be found by solving a convex problem. Also, we demonstrate the efficiency of our convex equivalent with respect to an formulation in terms of computation time and optimality.'
author:
- 'Koorosh Shomalzadeh, Jacquelien M. A. Scherpen, and M. Kanat Camlibel[^1] [^2] [^3]'
bibliography:
- 'IEEEabrv.bib'
- 'mybib.bib'
title: A Convex Formulation of a Bilevel Optimization Problem for Energy Markets
---
Real-time balancing market , convex optimization, bilevel optimization, flexibility management.
Introduction
============
In recent years, the increase in the penetration of s at the demand side has drastically changed the structure of our power system. As a result, the old passive households, which only consumed energy, found a more active role with the help of the demand side generation. The new term *prosumer* was introduced in the energy community to represent this transition for households [@parag2016electricity].
The emergence of prosumers calls for a new real-time market structure in contrast to the existing day ahead and intraday markets. Since output power of many s is volatile due to their intrinsic environmental dependency, planning for supply and demand matching needs to be done as close as possible to real-time to keep the system stable and economically efficient. Therefore, an [@pineda2013using] that incorporates available unused capacity of prosumers’ controllable s and flexible loads, which together we denote here as controllable units, should be developed to address the supply volatility by incentivizing prosumers.
Currently, there is only an ex-post financial settlement procedure in the Netherlands and most of Europe, and no actual or physical real-time balancing occurs [@wang2015review]. Communication infrastructure in the new paradigm of smart grid [@conejo2010real] facilitates the participation of the prosumers with controllable units in an . Moreover, to prevent direct interaction of the prosumers with higher level agents in the market and aggregate them, a market participant, the *aggregator*, has been introduced [@gkatzikis2013role]. The aggregators have different roles in different market structures.
The goal of an aggregator in an is to optimize its operational costs for balancing by incentivizing the prosumers to utilize their unused assets. There are many approaches which an aggregator can employ to steer its associated prosumers to an optimal operation point [@vardakas2014survey]. One of the most popular approaches is to consider the aggregator as a leader, who can anticipate the reaction of the prosumers, proposes some prices to the following prosumers such that their reactions would be optimal for the aggregator. This price incentive oriented setup falls into the category of *bilevel optimization problems* [@colson2007overview] and *Stackelberg games* [@von2010market], where the lower level problems and the upper level problem are the problems related to the prosumers and the aggregator, respectively.
Bilevel optimization problems have already been employed in the literature for electricity markets. The well-known paper [@hobbs2000strategic] models strategic offering of a dominant generating firm as a two level optimization problem, where at the upper level a generator firm maximizes its profit and at the lower level a system operator maximizes social welfare or minimizes total system cost. It is assumed that the dominant firm knows about the other non-dominant firms bids and offers. More recent papers [@du2018hierarchical] and [@henriquez2017participation] propose a similar approach for participation of microgrids and aggregators in an . In these types of models, the price for the electricity is equal to the dual variable corresponding to the clearing constraint in the lower level problem.
In contrast to dealing with the interaction between an aggregator and a system operator as in [@hobbs2000strategic; @du2018hierarchical; @henriquez2017participation], [@zugno2013bilevel] investigates the interaction between prosumers and an aggregator. In this setup, the aggregator minimizes its cost at the upper level and the prosumers minimize the their electricity consumption cost based on the price proposed by the aggregator. Also, the papers [@yang2018model] and [@yang2017framework] investigate the prosumers/aggregator interaction albeit with a different approach. Indeed, these papers deal with *personalized price* for individual prosumers based on their past consumption behavior.
The state-of-the-art approach to solve these types of bilevel optimization problems is to solve them as s by using commercial off-the-shelf software packages. However, implementing the mentioned setup in real-time requires very fast computations. The time intervals for a real-time balancing market can often be as low as $5$ minutes [@vlachos2013demand]. Therefore, the solution for each interval has to be computed and executed within seconds or even less. While papers like [@ghamkhari2016strategic] have studied the computational efficiency for generating firms strategic offering by introducing a convex relaxation for optimization problem considered in [@hobbs2000strategic], to the best of our knowledge, no study addressed the computation time for the prosumers/aggregator setup with personalized prices.
In this paper, we define the problem of static economic optimization of an aggregator and the corresponding prosumers for participation in the as a bilevel optimization problem with personalized prices. This problem, in general, is non-convex [@luo1996mathematical]. By assuming that the units are modular, we show that the global optimum for the non-convex problem can be obtained as a solution of a certain convex optimization problem. This convex equivalent formulation has two main advantages. On the one hand, it guarantees global optimality. On the other hand, a convex formulation is attractive in real-time applications since the computation time is linear in the number of variables whereas it is exponential for s [@ruiz2009pool]. In addition, this significant reduction in computation time has the potential to help the aggregator to participate in the ancillary service markets such as primary and secondary reserve markets which have very short time intervals [@wang2015review].
The paper is organized as follows. Section \[sec:pf\] explains the prosumers/aggregator interaction model in a real-time balancing market and introduces the bilevel problem. In Section \[sec:ce\], we show that the bilevel optimization problem is equivalent to a certain convex problem. The efficiency of the proposed method is illustrated by means of simulations in Section \[sec:sim\]. Finally, the papers closes with the conclusions in Section \[sec:con\].
Problem formulation {#sec:pf}
===================
In this section, we formulate the static bilevel economic optimization problem of an aggregator and its portfolio for participation in an . Each aggregator has a set of prosumers under contract and each prosumer is on a contract with only one aggregator. There are many types of aggregators in an electricity market. In this paper, we consider a commercial aggregator which also acts as a [@ding2013real]. Therefore, the aggregator here is also responsible for balancing its portfolio. To do so, the aggregator receives a real-time price from the , who usually has the highest role in the market hierarchy, and incentivizes the prosumers to supply or consume more or less based on that. The change in each prosumer electrical energy supply or demand in a time interval is referred as *flexibility*. Next, we explain the problem setting and market structure in detail.
Prosumers are equipped with various kinds of units. They consist of two prominent categories, namely controllable and uncontrollable units. units and units are examples of controllable active supply and demand units of electricity, respectively. Output generation of units such as solar cells and wind turbines is dependent on environmental conditions. Thus these are uncontrollable supply units. Throughout this paper, we assume that each prosumer has either a modular or as a controllable unit and it might have a solar panel or a wind turbine as an uncontrollable one. Each prosumer heat demand is also assumed to be flexible by considering a loss of comfort factor, that is, it is willing to consume more or less heat if its loss of comfort is compensated by the aggregator. Since heat is an output for both and , prosumers are able to alter their controllable units output level to participate in the balancing market. In this paper, we focus on a static and one time-step optimization problem without considering storage devices.
Due to the uncertain nature and volatility of both the uncontrollable DS units and the prosumers demand, there could be a mismatch between the pre-planned supply and demand schedules in the real-time. To balance this mismatch and to participate in the , the aggregator incentivizes the prosumers with personalized prices [@yang2017decision] in a centralized way to consume or supply more energy using their controllable units. Before providing a precise mathematical formulation, we elaborate on some technical notions.
The aggregator is in *up-regulation* if its prosumers’ demand is lower than its supply. Similarly, the aggregator is in *down-regulation* if the demand is higher than the supply for its prosumers. Likewise, the is in *up-regulation* if the total system demand is lower than the total system generation. Otherwise, it is in *down-regulation*. Based on these definitions, we distinguish the following four cases:
\[cs:1\] The aggregator and the both are in up-regulation:
\[cs:2\] The aggregator is in up-regulation and the is in down-regulation:
\[cs:3\] The aggregator and the both are in down-regulation:
\[cs:4\] The aggregator is in down-regulation and the is in up-regulation:
In both Case \[cs:2\] and Case \[cs:4\] the solution for the optimal strategy of the aggregator is trivial: sell the requested flexibility to the . However, in Case \[cs:1\] and Case \[cs:3\] the aggregator needs to find a trade-off between the possible options for the optimal strategy. In the following subsection, we focus on modeling Case \[cs:1\] and Case \[cs:3\] as a bilevel optimization problem.
The prosumers/aggregator model
------------------------------
We consider both the aggregator and the prosumer as self-interest agents. The aggregator tries to minimize its cost to settle the imbalance and the prosumer’s goal is to maximize its revenue and minimize its cost and discomfort by altering its demand or supply given the personalized price proposed by the aggregator.
We consider one aggregator and $n$ prosumers each has an or . We denote the proposed personalize price by the aggregator to the $i$th prosumer by $x_i$ and the prosumer $i$’s optimal flexibility response by $y_i$ for $i\in N=\{1,\dots,n\}$. To model both Case \[cs:1\] and Case \[cs:3\], we employ the following optimization problem for each prosumer:
\[prob00\] $$\begin{aligned}
{2} \label{prob001}
& \underset{y_i}{\mathrm{max}} \quad && x_iy_i-(\frac{1}{2}a_iy_i^2+b_iy_i)\\ \label{prob002}
&\mathrm{subject \ to} \quad && 0 \le y_i \le m_i,
\end{aligned}$$
where $m_i$ is the maximum available flexibility, $b_i$ is the price of providing flexibility and $a_i\ge 0$ models the discomfort for the prosumer $i$. Next, we elaborate further on the model and parameters.
In , the first term corresponds to the received payment by the prosumer $i$ from the aggregator. The second term models the discomfort of the prosumer $i$ for providing flexibility $y_i$. Finally, the last term captures the payment the prosumer $i$ should receive or the cost it should pay with respect to the intraday market plannings for providing flexibility $y_i$.
The parameter $b_i$ for a prosumer with in both the aggregator up-regulation (Case \[cs:1\]) and down-regulation (Case \[cs:3\]) is as follows: $$b_i=\begin{cases*}
\pi_e & \textrm{if aggregator in up-regulation,} \\
-\pi_g & \textrm{if aggregator in down-regulation,}
\end{cases*}$$ where $\pi_e \ge 0$ and $\pi_g \ge 0$ are fixed electricity and gas prices charged by electricity and gas suppliers, respectively. Likewise, for a prosumer with this parameter is defined as follows: $$b_i=\begin{cases*}
-c_i \pi_e & \textrm{if aggregator in up-regulation,} \\
c_i \pi_g & \textrm{if aggregator in down-regulation,}
\end{cases*}$$ where $c_i$ is dependent on the technology of the prosumer $i$ and is given by $$c_i=\frac{\textrm{nominal input power }}{\textrm{nominal electricity output power}}\cdot$$
Further, we define the maximum available flexibility $m_i$ as follows. For prosumer $i$, let $P_i\ge 0$ denote the electrical power of its device. As such, $P_i$ is the input electrical power to an device or the output electrical power of an device. Also, let $P_i^{\mathrm{max}}$ denote the maximum electrical power for prosumer $i$. Then, the maximum available flexibility of the prosumer $i$ with an is given by $$m_i=\begin{cases*}
(P_i^{\mathrm{max}}-P_i)\Delta t & \text{if aggregator in up-regulation,} \\
P_i\Delta t & \text{if aggregator in down-regulation,} \\
\end{cases*}$$ where $\Delta t$ is the duration of each time step for the and assumed to be equal to $300$ seconds in this paper. Similarly, we define $m_i$ for a prosumers which owns an as follows: $$m_i=\begin{cases*}
P_i\Delta t & \text{if aggregator in up-regulation,} \\
(P_i^{\mathrm{max}}-P_i)\Delta t & \text{if aggregator in down-regulation.} \\
\end{cases*}$$
(TSO) at (4,8) [TSO]{}; (Agg) at (4,4) [Aggregator]{}; (pro1) at (0,0) [ Pro. $1$]{}; (pro2) at (8,0) [ Pro. $n$]{}; (pron) at (4,0) […]{}; (Agg) to\[bend right\] node\[label=above:$x_1$\] (pro1); (Agg) to\[bend left\] node\[label=above:$x_{n}$\] (pro2); (pro1) to\[bend right\] node\[label=below:$y_1$\] (Agg); (pro2) to\[bend left\] node\[label=below:$y_{n}$\] (Agg); (TSO) to\[bend right\] node\[label=left:$p$\] (Agg); (TSO) to\[bend left\] node\[label=right:$f-\sum_{i}y_i$\] (Agg); (7,7.5) rectangle (11,9.5); (7.5,8) – (8,8) node\[anchor= west\] [Flexibility flow ]{}; (7.5,9) – (8,9) node\[anchor= west\] [Price signal]{};
As the agent responsible for supply and demand balancing in the , the aggregator has two options to accomplish its goal, namely, to incentivize the prosumers for flexibility provision with the associated cost of $x_iy_i$ or to buy flexibility from the with the price $p \ge 0$. The aggregator’s problem is to find the best strategy given these two options.
Considering the above model, $x_i$ being nonnegative and the prosumers’ optimality conditions, we obtain the bilevel optimization problem which has the problem as a constraint for each prosumer:
\[prob0\] $$\begin{aligned}
{2}
& \underset{x,y}{\mathrm{min}} \quad &&\ \sum_{i} x_iy_i+p(f-\sum_{i}y_i) \\
& \mathrm{subject \ to}\quad && x_i \ge 0, \qquad \forall i\in N,\\ \label{0in}
&&& \sum_{i}y_i \le f,\\
&&& \mkern-36mu \begin{cases} \label{prob01}
\underset{y_i}{\mathrm{max}} & x_iy_i-\frac{1}{2}a_iy_i^2-b_iy_i \\
\mathrm{subject \ to} \quad & 0 \le y_i \le m_i,
\end{cases}\quad \forall i\in N,
\end{aligned}$$
where $x$ and $y$ are vectors with components $x_i$ and $y_i$, respectively. Also, $f \ge 0$ denotes the mismatch between supply and demand in both up- and down-regulation. If the flexibility provided by the prosumers is $\sum_{i}y_i$. Then, the aggregator needs to trade $(f-\sum_{i}y_i)$ with the . Figure \[fig:interactions\] shows these interactions. We consider an ex-ante pricing scheme, that is, the informs the aggregator about the price $p$ prior to the start of each 5-minute interval.
These types of bilevel problems are very similar to Stackelberg games [@von2010market], where a leader announces a policy to its followers and then the followers, who are unaware of the outside world, react by their best response strategy. In other words, the leader has the advantage of anticipating the followers reactions.
In the setup we consider in this paper, the aggregator’s goal is to satisfy its internal imbalance in real-time. However, in other possible settings beyond the scope of this paper, helping the to satisfy the total system imbalance can also be a goal for the aggregator. Therefore, in that setting the problem formulation for Case \[cs:1\] and Case \[cs:3\] is given by without considering . In this situation, if $\sum_{i}y_i -f\le 0$, the aggregator pays $p(f-\sum_{i}y_i)$ to the and if $\sum_{i}y_i - f >0$, then the aggregator receives $p(f-\sum_{i}y_i)$ from the for providing flexibility.
The bilevel market optimization problem and its solution
--------------------------------------------------------
The model above for the aggregator and the prosumers interactions is very close to the bilevel electricity market models in [@zugno2013bilevel; @yang2018model; @yang2017framework], where different market technicalities have been considered. Furthermore, we restrict our model to a static case. Despite these differences, our model captures the basic properties of a bilevel market.
In general, bilevel optimization problems are very difficult to solve. They have been extensively studied in the framework of . We refer to [@luo1996mathematical] for a full investigation of s. The simplest case of a bilevel optimization problem is when both the upper and lower level problems are linear. Even in this simplest case, [@hansen1992new] has shown that the problem is strongly NP-hard. Some classes of bilevel optimization problems can be reformulated as problems and solved by commercial software packages [@fortuny1981representation]. This approach has been extensively used to solve electricity market optimization problems as a state-of-the-art approach [@li2018participation], [@wang2017strategic].
An aggregator can have up to several thousands of prosumers under its contract. To implement an with 5-minute time intervals, the optimal solution of the problem should be found as fast as possible. The increase in the number of the optimization variables, as a result of the growth in the number of the prosumers, leads to an unacceptable computation time in real-time applications for combinatorial optimization problems such as problems. In the following section, we introduce a convex equivalent for the problem .
On the convexity of the bilevel electricity market problem {#sec:ce}
==========================================================
One can parameterize the solution of by using a piece-wise linear map. Indeed, given $x_i$, is a concave quadratic problem in $y_i$. Solving this problem analytically leads to the following piece-wise linear map from $x_i$ to $y_i$: $$\label{sol01}
y_i=\begin{dcases}
0 \quad & x_i < b_i, \\
\frac{x_i-b_i}{a_i} \quad & b_i \le x_i \le a_im_i+b_i, \\
m_i \quad & x_i > a_im_i+b_i,
\end{dcases}$$ which is depicted in Figure \[pwl\]. The interpretation for the sign of $b_i$ was given in the previous section. As can be seen in Figure \[pwl2\], a prosumer can provide flexibility $y_i=-\displaystyle\frac{b_i}{a_i}$ without any incentive ($x_i=0$) when $b_i <0$. The following assumption on the parameters $a_i$, $b_i$ and $f$ guarantees feasibility of the problem .
\[ass\] The total flexibility provided by the prosumers with negative $b_i$ without any incentive is less than or equal to the total requested flexibility, i.e., $$\underset{i\in\{j\in N|b_j < 0\}}{\sum} -\frac{b_i}{a_i} \le f\mathrm{.}$$
A prosumer with an () in up-regulation (down-regulation) is not able to decrease its supply (demand) drastically in real-time and provide flexibility without any incentive. This is mainly due to decisions the prosumers made in the intraday market. No incentive strategy is designed for the prosumers in the intraday market. Because of this, $\underset{i\in\{j\in N|b_j < 0\}}{\sum} -\frac{b_i}{a_i}$ is much less than $f$ in practice. Thus, Assumption \[ass\] is satisfied.
Having as the solution to the problem , let us rewrite the bilevel optimization problem as the piece-wise quadratic optimization problem:
\[prob1\] $$\begin{aligned}
{2}
& \underset{x,y}{\mathrm{min}} \quad && \phi(x,y) = \sum_{i} x_iy_i+p(f-\sum_{i}y_i) \\ \label{prob1b}
& \mathrm{subject \ to}\quad && x_i \ge 0, \qquad \forall i\in N\\ \label{prob1c}
&&& \sum_{i}y_i \le f,\\ \label{prob1d}
&&& \mkern-44mu y_i=\begin{cases}
0 \quad & x_i < b_i, \\
\frac{x_i-b_i}{a_i} \quad & b_i \le x_i \le a_im_i+b_i, \\
m_i \quad & x_i > a_im_i+b_i,
\end{cases} \quad \forall i \in N.
\end{aligned}$$
In general, this problem is non-convex as illustrated by the following example.
Suppose a two-dimensional case of the problem where $a_1=a_2=1$, $b_1=b_2=2$, $m_1=m_2=6$, $p=10$, $f=30$. Figure \[nonconvex\] depicts objective function of the problem with these parameters. As can be seen, the objective function is non-convex. Note that its minimum coincides with the minimum of the convex quadratic problem obtained from by taking $y_i=\frac{x_i-b_i}{a_i}$ with $b_i\le x_i \le a_im_i+b_i$ for $i\in\{1,2\}$.
Motivated by this example, we consider the following convex quadratic problem:
\[prob2\] $$\begin{aligned}
{2}
& \underset{x,y}{\mathrm{min}} \quad && \psi(x,y) = \sum_{i} {x}_i{y}_i+p(f-\sum_{i}{y}_i) \\ \label{prob2b}
& \mathrm{subject \ to}\quad && {x}_i\ge 0, \qquad \forall i \in N\\ \label{prob2c}
&&&\sum_{i}{y}_i \le f,\\ \label{prob2d}
&&& \mkern-50mu {y}_i=\frac{{x}_i-b_i}{a_i}, \quad b_i \le {x}_i \le a_im_i+b_i, \quad \forall i \in N.
\end{aligned}$$
It turns out that the global minimum of the nonconvex problem can be found by solving the convex problem .
\[the1\] Let $x^*$ and $y^*$ be the minimizers of the convex quadratic minimization problem . Also, let $\psi^*$ be the minimum value, that is $\psi^*=\psi(x^*,y^*)$. Then, $x^*$ and $y^*$ are also the minimizers for the problem , i.e., $$\phi(x,y) \ge \psi^*, \quad \text{for all feasible} \ x,y \ \text{of the problem \eqref{prob1}},$$ or equivalently $$\begin{gathered}
\sum_{i}x_iy_i+p(f-\sum_{i}y_i) \ge \sum_{i}{x}_i^*{y}_i^*+p(f-\sum_{i}{y}_i^*),\quad \\ \text{for all feasible} \ x,y \ \text{of the problem \eqref{prob1}}.
\end{gathered}$$
Clearly, $x^*$ and $y^*$ are feasible for . Suppose, for the sake of contradiction, that $\phi(x,y) < \psi^*$ for some feasible $x,y$ of the problem . This means that $$\label{eqcont}
\sum_{i}x_iy_i+p(f-\sum_{i}y_i) < \sum_{i}{x}_i^*{y}_i^*+p(f-\sum_{i}{y}_i^*),$$ Define $\bar{x}_{i}$ as follows: For all ${i\in\{j\in N|b_j \ge 0\}}$ $$\label{eq:xbar1}
\bar{x}_{i}=\begin{cases}
b_i \quad &0 \le x_i < b_i, \\
x_i \quad &b_i \le x_i \le a_im_i+b_i,\\
a_im_i+b_i \quad& x_i>a_im_i+b_i,
\end{cases}$$ and for all $i\in\{j\in N|b_j < 0\}$ $$\label{eq:xbar2}
\bar{x}_{i}=\begin{cases}
x_i \quad &0 \le x_i \le a_im_i+b_i,\\
a_im_i+b_i \quad& x_i>a_im_i+b_i.
\end{cases}$$ It follows from and that $$\begin{aligned}
\label{eq:xbar11}
b_i \le \bar{x}_{i} \le a_im_i+b_i, \qquad \forall i\in\{j\in N|b_j \ge 0\}, \\ \label{eq:xbar22}
0 \le \bar{x}_{i} \le a_im_i+b_i, \qquad \forall i\in\{j\in N|b_j < 0\}.
\end{aligned}$$ Furthermore, we define for all $i$ $$\label{eq:ybar}
\bar{y}_i=\frac{\bar{x}_i-b_i}{a_i}.$$ Then, , and imply that $\bar{x}$ and $\bar{y}$ satisfy and . Since $y=\bar{y}$ due to , implies . Therefore, $\bar{x}$ and $\bar{y}$ are feasible for the problem .\
Now, suppose that $b_i\ge 0$ for some $i$. We consider the following three cases.\
[Case 1]{}: $x_i <b_i$. Then, we have $$\label{case1}
\begin{aligned}
x_i < \bar{x}_{i}=b_i \\
y_i=\bar{y}_{i}=0\\
\end{aligned} \quad \Longrightarrow \quad x_iy_i=\bar{x}_{i}\bar{y}_{i}=0.$$ Case 2: $b_i \le x_i \le a_im_i+b_i$. In this case, we have $$\label{case2}
\begin{aligned}
x_i = \bar{x}_{i} \\
y_i=\bar{y}_{i}\\
\end{aligned}\quad \Longrightarrow \quad x_iy_i=\bar{x}_{i}\bar{y}_{i}.$$ Case 3: $x_i \ge a_im_i+b_i$. This leads to $$\label{case3}
\begin{aligned}
x_i >\bar{x}_{i}=a_im_i+b_i \\
y_i=\bar{y}_{i}=m_i\\
\end{aligned}\quad \Longrightarrow \quad x_iy_i>\bar{x}_{i}\bar{y}_{i}.$$\
If $b_i<0$, we can follow a similar line of reasoning. Suppose $0 \le x_i \le a_im_i+b_i$ which results in $$\label{case4}
\begin{aligned}
x_i = \bar{x}_{i} \\
y_i=\bar{y}_{i}\\
\end{aligned}\quad \Longrightarrow \quad x_iy_i=\bar{x}_{i}\bar{y}_{i},$$ and now suppose $x_i >a_im_i+b_i$. Then we can write $$\label{case5}
\begin{aligned}
x_i >\bar{x}_{i}=a_im_i+b_i \\
y_i=\bar{y}_{i}=m_i\\
\end{aligned}\quad \Longrightarrow \quad x_iy_i>\bar{x}_{i}\bar{y}_{i}.$$\
Based on -, we can conclude that $$\begin{aligned}
\sum_{i}\bar{x}_{i}\bar{y}_{i}+p(f-\sum_{i}\bar{y}_{i}) <\sum_{i}x_iy_i+p(f-\sum_{i}y_i).
\end{aligned}$$ Using , we have $$\sum_{i}\bar{x}_{i}\bar{y}_{i}+p(f-\sum_{i}\bar{y}_{i}) < \sum_{i}x_{i}^*y_{i}^*+p(f-\sum_{i}y_{i}^*),$$ which is a contradiction since $x^*$ and $y^*$ are the minimizers of the problem .
The piece-wise linear constraint makes the problem a piece-wise quadratic optimization problem with $3^n$ quadratic problems where $n$ is the number of prosumers. Theorem \[the1\] proves that one of these $3^n$ problems always attains the global optimum.
Let $x_i^*=b_i \ge 0$ and $y_i^*=\displaystyle\frac{x_i^*-b_i}{a_i}=0$ for some $i$. Then we have $x_i^*y_i^*=0$. Based on , we can conclude $x_i^* \in [0,b_i]$ and $y_i^*=0$ are the corresponding optimal solutions for the problem .
In our bilevel optimization problem, the aggregator needs to have all the information about the prosumers to solve in a centralized way. This is rather an unrealistic assumption, since the prosumers are not willing to share their information with third parties. Our reformulation, i.e., being convex now allows a distributed solution to find the optimum, see [@Bertsekas/99].
Simulations {#sec:sim}
===========
In this section, we evaluate the performance of our convex equivalent problem for the in terms of computation time and optimality. We use the state-of-the-art -based approach in [@fortuny1981representation] as a benchmark for this evaluation. The following subsection briefly explains how to convert the problem to an problem.
-based approach
---------------
type Nominal electricity input power parameter $\abs{b_i}$
------ --------------------------------- -----------------------------
$1 $ $1.1 \ kW$ $0.1707 \ \text{\euro}/kWh$
type Nominal input power Nominal electricity output power $\abs{b_i}$
------ --------------------- ---------------------------------- --------------------------------------
$1$ $8 \ kW$ $1 \ kW$ $0.6888 \ \text{\euro}/\mathrm{kWh}$
$2$ $4.7 \ kW$ $0.8 \ kW$ $0.5088 \ \text{\euro}/\mathrm{kWh}$
The lower level optimization problems in are concave maximization problems with strong duality [@boyd2004convex]. Therefore, they can be replaced by the necessary and sufficient [Karush-Kuhn-Tucker]{} optimality conditions as
\[MPEC\] $$\begin{aligned}
{3}
& \underset{x,y,\lambda_1,\lambda_2}{\mathrm{min}} \quad && \sum_{i} x_iy_i+p(f-\sum_{i}y_i) \\
& \mathrm{subject \ to}\quad && x_i\ge 0, \quad \forall i\in N,\\
&&& \sum_{i}y_i \le f,\\
&&& \mkern-55mu \begin{cases}
x_i-a_iy_i-b_i+\lambda_{1i}-\lambda_{2i}=0,\\
0 \le y_i \perp \lambda_{1i} \ge 0,\\
0 \le m_i-y_i \perp \lambda_{2i} \ge 0,
\end{cases}\quad \forall i\in N,
\end{aligned}$$
where $\lambda_{1i}$ and $\lambda_{2i}$ are dual coefficient for the constraints on $y_i$. By introducing auxiliary binary variables $z_i$ and $w_i$ and a sufficiently large constant $M$, the problem can be turned to an problem as
\[MIP\] $$\begin{aligned}
{3}
& \underset{w,y,z,\lambda_{1},\lambda_{2}}{\mathrm{min}} \quad && \sum_{i} (a_iy_i^2+(b_i-p)y_i+m_i\lambda_{2i})+pf \\
& \mathrm{subject \ to}\quad && \sum_{i}y_i \le f, \\
&&& \mkern-14mu \begin{cases}
a_iy_i+b_i-\lambda_{1i}+\lambda_{2i} \ge 0, \\
0 \le y_i \le Mz_i, \\
0 \le m_i-y_i \le Mw_i,\\
0 \le \lambda_{1i} \le M(1-z_i),\\
0 \le \lambda_{2i} \le M(1-w_i),
\end{cases}
\quad \forall i\in N.
\end{aligned}$$
Details of this approach can be found in [@fortuny1981representation].
Computation time and optimality comparison
------------------------------------------
For simulation purposes, we consider one type of and two types of technologies for the prosumers. We assume that half of the prosumers have and the other half are equipped with . We assign to each prosumer a specific technology of or , randomly. Tables \[table:hp\] and \[table:mchp\] show the data regarding these types and also their corresponding $\abs{b_i}$ parameters. The supplier gas and electricity prices are based on data from [@retailerprice] for the Netherlands and equal to $0.0861 \ \text{\euro}/\mathrm{kWh}$ and $0.1707 \ \text{\euro}/\mathrm{kWh}$, respectively. The price $p$ for both up- and down-regulation is set to $0.6 \ \text{\euro}/\mathrm{kWh}$ based on the settlement price data of TenneT from [@settlementprice] for a period where the is under high stress. It should be noted that the informs the aggregators about this price ex-ante.
Both the convex formulation and the formulation are implemented in MATLAB r2018b and solved by the Gurobi Optimizer [@gurobi]. The simulations were run on four Intel Xeon 2.6 GHz cores and 1024 GB internal memory of the Peregrine high performance computing cluster of the University of Groningen.
The solvers use complicated heuristic methods to find the optimal solution. Moreover, the computation time for computing an optimal solution is highly related to specific parameters of the problem. To find a rough estimate of the optimization run time, we implement a set of 1000 Monte Carlo simulations with uniformly generated random parameters $a_i$, $m_i$ and $f$ for the optimization problem. This is done for different numbers of prosumers. Table \[table:results\] summarizes the run time results for these Monte Carlo scenarios. The last column of this table shows the number of scenarios (out of 1000 Monte Carlo scenarios) where the problem leads to an infeasible solution or an optimal solution with higher cost than the convex problem. We refer to Figure \[fig:runtime\] and Table \[table:results\] for the results.
------- --------------- --------------- ----------------------------------------- --------------- -----------------------------
[Number of scenarios with infeasible]{}
Average (sec) Maximum (sec) Average (sec) Maximum (sec) or non-optimal solution for
10 0.0006 0.0010 0.0016 0.0039 0
100 0.0012 0.0017 0.0032 0.0058 0
1000 0.0038 0.0076 0.0128 0.0371 1
10000 0.0344 0.0484 0.5548 9.0352 11
20000 0.0772 0.1231 3.9498 59.1761 27
30000 0.1161 0.1834 11.1190 161.7937 40
------- --------------- --------------- ----------------------------------------- --------------- -----------------------------
coordinates [ (10,0.0010)(100,0.0017)(1000,0.0076)(10000,0.0484)(20000,0.1231)(30000,0.1834) ]{}; coordinates [ (10,0.0039)(100,0.0058)(1000,0.0371)(10000,9.0352)(20000,59.1761)(30000,161.7937) ]{};
Discussion on the results
-------------------------
The computation time for the convex optimization problem grows approximately linear with respect to the number of prosumers. This can be seen from the average run time in Table \[table:results\] for the convex formulation. If we consider $30000$ as the typical number of prosumers for an aggregator, then the average and the maximum run time are acceptable for a real-time application with $5$-minute time interval. However, this is not the case for an formulation. Figure \[fig:runtime\] and Table \[table:results\] show that the average and maximum computation time of is not suitable for a real-time market since the computational time grows approximately exponentially. Moreover, there are some cases that the formulation with high number of optimization variables does not converge to the global optimal or even to feasible solution.
Conclusions {#sec:con}
===========
Currently, prosumers do not participate in real-time balancing. In this paper, we have developed a market with a , an aggregator and prosumers to address real-time balancing. We modeled the corresponding economic optimization problem of a self-interest aggregator and prosumers as a bilevel optimization problem to represent hierarchy in the market. Generally, bilevel optimization problems are non-convex. We have shown that it suffices to solve a specific convex optimization problem to find the global optimum of the original bilevel optimization problem. In contrast to existing approaches (e.g., ), the convex equivalent of the bilevel optimization problem has very low computation time and is therefore preferable in real-time. Moreover, in this approach the global optimality of the solution is guaranteed.
Low computation time and global optimality are not the only advantages of having a convex equivalent for the bilevel optimization. Centralized aggregator control over the whole community of prosumers can be a difficult task, especially when the number of prosumers is very high. However, having a convex formulation for the balancing problem opens up new horizons in decentralized and distributed control and optimization.
[^1]: K. Shomalzadeh and M. K. Camlibel are with the Jan C. Willems Center for Systems and Control, Bernoulli Institute for Mathematics, Computer Science and Artificial Intelligence, Faculty of Science and Engineering, University of Groningen, Nijenborgh 9, 9747 AG, Groningen, The Netherlands (e-mail: [email protected]; [email protected]).
[^2]: J. M. A. Scherpen is with the Jan C. Willems Center for Systems and Control, Engineering and Technology Institute Groningen, Faculty of Science and Engineering, University of Groningen, Nijenborgh 4, 9747 AG, Groningen, The Netherlands (e-mail: [email protected]).
[^3]: This research was funded by NWO (The Netherlands Organisation for Scientific Research) Energy System Integration project “Hierarchical and distributed optimal control of integrated energy systems” (647.002.002).
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{
"pile_set_name": "ArXiv"
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abstract: 'Standard weak measurement (SWM) has been proved to be a useful ingredient for measuring small longitudinal phase shifts. \[$\emph{Phys. Rev. Lett.}$ $\textbf{111}$, 033604 (2013)\]. In this letter, we show that with specific pre-coupling and postselection, destructive interference can be observed for the two conjugated variables, i.e. time and frequency, of the meter state. Using a broad band source, this conjugated destructive interference (CDI) can be observed in a regime approximately 1 attosecond, while the related spectral shift reaches hundreds of THz. This extreme sensitivity can be used to detect tiny longitudinal phase perturbation. Combined with a frequency-domain analysis, conjugated destructive interference weak measurement (CDIWM) is proved to outperform SWM by two orders of magnitude.'
author:
- 'Zi-Huai Zhang'
- 'Geng Chen$\footnote{email:[email protected]}$'
- 'Xiao-Ye Xu'
- 'Jian-Shun Tang'
- 'Wen-Hao Zhang'
- 'Yong-Jian Han'
- 'Chuan-Feng Li$\footnote{email:[email protected]}$'
- 'Guang-Can Guo'
title: Strengthen Weak Measurement with Conjugated Destructive Interference
---
Introduction
============
High sensitivity detection of longitudinal phase shift is essential in metrology, such as accurate distance measurements, timing synchronization and detection of gravitational waves [@Kim; @Lee; @Abbott]. The standard tool is an interferometer with a balanced homodyne detection [@Caves]. It requires a coherent source and the precision is dominated by the intrinsic quantum noise [@Yurke]. Recently, weak measurement has attracted extensive attention and academic interest due to its practical and potential applications in observing very small physical effects [@Aharonov1; @Aharonov2]. When weak measurements are judiciously combined with preselection and postselection, they lead to a weak-value-amplification phenomenon and give access to an experimental sensitivity beyond the detector’s resolution [@Brunner1; @Brunner2; @Hosten; @Dixon; @Starling1]. Brunner $\emph{et al.}$ compared the standard interferometer proposal with a weak measurement proposal. Their results evidently showed that when the purely imaginary weak value was exploited, weak measurement surpassed the standard interferometer proposal by several orders of magnitude [@Brunner3]. With this method, Xu $\emph{et al.}$ experimentally realized a very precise phase estimation utilizing white light from a commercial light-emitting diode (LED) [@Xu; @Li].
In Xu’s scheme, the working regime is selected to make the coupling strength approximately 0, where postselection only slightly reshapes the wave-package pattern to obtain a larger shift considering the mean value of the spectrum. Here, we propose an improved scheme to measure longitudinal phase shifts. With a fixed pre-coupling of the system and the meter, the joint state becomes entangled. Afterwards with a specific postselection state, destructive interference can be observed in both the time and frequency domains using a broad light source. The phase shift is much smaller than the wave length and it still gets a destructive interference. In this process approximately 1 attosecond, the spectrum shifts a considerable amount decided by the postselection state. Prospectively this effect can be used to perform an ultra-sensitive measurement on tiny longitudinal phase perturbation.
Generation of CDI
=================
We consider a measurement scenario involving a physical quantum state consisting of a system state $\lvert \psi \rangle$ and a meter state $\lvert g(x) \rangle$ where $x$ represents the space coordinate. The Hamiltonian of the system-meter combination is $H=kAP$. $A$ is an obeservable of the system and $P$ is the momentum of the meter. $A$ has two eigenstates $\lvert 0 \rangle$ and $\lvert 1 \rangle$ which satisfy $A\lvert 0 \rangle = \lvert 0 \rangle$ and $A\lvert 1 \rangle = -\lvert 1 \rangle$.
The system-meter evolution due to the interaction $H$ can be described by a unitary operator $U=e^{-iH\Delta t}$ with $\Delta t $ being the duration of the interaction. If the initial system state $\lvert \psi \rangle$ is a superposition of $\lvert 0 \rangle$ and $\lvert 1\rangle$, i.e. $\lvert \psi \rangle= \alpha \lvert 0 \rangle+ \beta \lvert 1\rangle$, the system-meter combination after the interaction is $$\lvert \Psi \rangle= U\lvert \psi \rangle \lvert g(x)\rangle = \alpha \lvert g(x+\tau)\rangle \lvert 0 \rangle +\beta \lvert g(x-\tau)\rangle \lvert 1 \rangle,$$ where $\tau=k\Delta t$. In this letter, we are interested in measuring the ultrasmall $\tau$ precisely.
For clarity, we consider the quantum state to be a Gaussian wave-package. The polarization of the photons serves as the system and the time or the frequency freedom serves as the meter. The interaction $H$ is introduced by a birefringent crystal which has different refractive indexes for the horizontal polarization $\lvert H \rangle$ and the vertical polarization $\lvert V \rangle$. In the scheme by Brunner $\emph{et al.}$, they first derive the wave-package in the time domain and then Fourier transform it into the frequency domain. Here, we will analyze the quantum system in both the time and frequency domains.
In the frequency domain, the wave function of the meter is represented by $f(\omega)=(\pi \delta^2)^{-1/4}exp[-(\omega-\omega_0)^2/2\delta^2]$. Two nearly orthogonal circular polarization states $\lvert \psi \rangle=\frac{1}{\sqrt{2}}(\lvert H \rangle+i\lvert V \rangle)$ and $\lvert \phi \rangle=\frac{1}{\sqrt{2}}(ie^{i\epsilon}\lvert H \rangle+e^{-i\epsilon}\lvert V \rangle)$ are used for the preselection and the postselection processes. Initially, the system-meter combination is a separable state $$\lvert \Phi \rangle = \int d\omega \frac{1}{\sqrt{2}}f(\omega)[\lvert H \rangle+i\lvert V\rangle]\lvert \omega \rangle].$$
After the pre-coupling by inducing an initial delay $\tau$, the polarization and the frequency freedom are entangled $$\label{initial}
\lvert \Psi \rangle=\int d\omega \frac{1}{\sqrt{2}}f(\omega)[e^{i\omega\tau}\lvert H \rangle + ie^{-i\omega\tau}\lvert V\rangle]\lvert \omega \rangle.$$
With a postselection of polarization, the meter state becomes (after normalization) $$\lvert T \rangle=\frac{1}{\sqrt{P}}\int d\omega \frac{i}{2}f(\omega)[e^{i(\omega \tau -\epsilon)}-e^{-i(\omega \tau -\epsilon)}]\lvert \omega \rangle,$$ with the postselection probability $$\label{prob}
P=0.5\{1-exp(-\delta^2 \tau^2)cos[2(\omega_0 \tau-\epsilon)]\}.$$
The postselected wave-package in frequency domain is $f(\omega)sin(\omega \tau -\epsilon)$ (without normalization), so the frequency propability distribution is $$\label{spectrum}
S(\omega)=sin^2(\omega \tau -\epsilon) \lvert f(\omega) \rvert^2.$$
The spectral shift $\Delta \omega$ is calculated as the shift of the mean spectrum $$\label{spectrumshift}
\Delta \omega= \frac{\int S(\omega)\omega d\omega}{\int S(\omega) d \omega}-\omega_0 =\frac{\tau \delta^2}{2P}exp(-\delta^2 \tau^2)sin[2(\omega_0 \tau-\epsilon)].$$
The temporal probability distribution is $T(t)=\lvert F[f(\omega)e^{i(\omega \tau -\epsilon)}-f(\omega)e^{-i(\omega \tau -\epsilon)}] \rvert^2$ where $F[ *]$ denotes the Fourier transform.
The SWM scheme has to satisfy $ \lvert \omega \tau/ \epsilon \rvert \ll 1$ in order to keep the interaction weak. Eq. (\[spectrum\]) is derived to be $$\label{SWMspectrum}
S(\omega)=\epsilon^{2} \lvert f(\omega) \rvert^2.$$
Within the weak interaction range where $\tau$$\rightarrow 0$, there is no destructive interference from Eq. (\[SWMspectrum\]) so the spectral shift can be calculated from the purely imaginary weak value. However, from Eq. (\[spectrum\]), when $\epsilon$$>$0 there is always a small $\tau_{s}$ satisfying $\omega_{0} \tau_{s}-\epsilon \approx 0$ so that destructive interference can be observed around $\tau_{s}$. The wave-package evolution of the time and frequency domains around $\tau_{s}$ are shown in Fig. \[Interaction\]. The initial meter state is normally prepared in a Gaussian superposition with the mean frequency $\omega_0$ to be 2350 THz and the width $\delta$ to be 200 THz, hence $\tau_{s}$ is calculated as 8.5 as. As shown in Fig. \[Interaction\](a), the central part of the time-domain wave package destructs to display a bimodal pattern at first and then recovers to a Gaussian distribution. While for frequency-domain in Fig. \[Interaction\](b), as the extinction point sweeps from the high to the low frequency with increasing $\tau$, the dominant wave package shifts from the low to high frequency range. In the center point around 8.5 as, the spectrum splits into two peaks. As the low frequency peak falls, the high frequency peak rises and they are equivalent at 8.5 as.
![The wave package in both (a) time and (b) frequency domains after postselection in the CDI regime. The pre- and post-selected states are $\lvert \psi \rangle=\frac{1}{\sqrt{2}}(\lvert H \rangle+i\lvert V \rangle)$ and $\lvert \phi \rangle=\frac{1}{\sqrt{2}}(ie^{0.02i}\lvert H \rangle+e^{-0.02i}\lvert V \rangle)$. At the central point of 8.5 as, the central components of both time and frequency domains are extinct due to destructive interference. The red lines in (a) and (b) indicate the central time and spectrum points, respectively.[]{data-label="Interaction"}](PD2.png){width="6in"}
Comparison of two Schemes
=========================
From Fig. \[Interaction\], both the time and the frequency domains destructive interference can be observed in the chosen regime. Considering the current ultimate temporal-resolution which is on the order of several picoseconds, it is difficult to observe the time-domain destructive interference experimentally. In the following analysis we will mainly focus on the measurement in frequency-domain. The peak-position of the start and end time points can be distinguished intuitively in Fig. \[Interaction\](b), indicating a considerable spectral shift during 7.5 to 9.5 as. This means that making use of the CDI effect should give rise to a better resolution when measuring a tiny longitudinal phase change.
It has been proved that the longitudinal phase change can be precisely detected by measuring the spectral shift [@Brunner3]. In Fig. \[SpectrumSlope\], we plot the spectral shifts as well as the shift rates of the two schemes according to Eq. (\[spectrumshift\]). For the SWM scheme, the parameters are set to be identical to Xu’s scheme and the results are consistent with his work. Fig. \[SpectrumSlope\](a) shows the spectral shifts of both the CDIWM scheme and the SWM scheme varying $\tau$. In the CDIWM scheme, the frequency domain wave-package splits into two peaks in the light blue area, either of which the peak position is also shown. It can be seen that the SWM scheme has only a spectral shift of several THz while the CDIWM scheme can reach a shift of several hundreds THz. Even more noteworthy is that there is a very steep spectral shift around 8.5 as, which means an extremely high sensitivity when measuring the phase perturbation.
![(a) Spectral shifts in the CDIWM and the SWM schemes. The dot-dashed blue and long-dashed red lines are the mean spectral shifts in the CDIWM scheme and the SWM scheme respectively. The short-dashed violet and dotted brown lines represent the peak position shifts of the two parts divided by the extinction point in the CDIWM scheme. (b) Spectral shift rates in the CDIWM and the SWM schemes, calculated as the slope of corresponding spectral shift in (a). The dot-dashed blue line represents CDIWM and the long-dashed red line represents SWM. The postselection angle $\epsilon$ is set to be $0.02$ in CDIWM and $-0.02$ in SWM. The mean frequency $\omega_0$ is 2350 THz and the width $\delta$ is 200 THz. The green box and the inset identify the working range of CDIWM.[]{data-label="SpectrumSlope"}](SpecSlopeNN.pdf){width="6in"}
Particularly, when there is a perturbation on the relative phase, we require a change on the meter as large as possible. This can be characterized by the spectral shift rate with respect to the phase perturbation or the equivalent time delay. Fig. \[SpectrumSlope\](b) shows that the spectral shift rate of the CDIWM scheme is far larger than that of the SWM scheme. Working on the most sensitive point, the CDIWM scheme can reach a shift rate beyond the SWM scheme by two orders of magnitude. To obtain the best sensitivity and a stable interference pattern with the CDIWM scheme, the working point should be accurately stabilized in a small time regime (the green box and the inset in Fig. \[SpectrumSlope\](b)). Existing techniques can provide the required accuracy to set the initial work point [@Abbott; @Luis].
Fig. \[setup\] shows the proposed experimental setup based on a Michelson Interferometer (MI). The beam splitter (BS) in the traditional MI is substituted by a polarizing beam splitter (PBS). The specific pre-coupling required by CDIWM is realized by adjusting the length difference of the two arms as well as analyzing the postselected spectrum. Initially, the two arms are set to be equal. As the mirror moves, the postselected spectrum shifts according to Fig. \[Interaction\](b) . The most sensitive point is confirmed when the extinct point is in the middle of the spectrum. Afterwards all the optical elements are locked so that the system is on standby for the measurement.
![The proposed experimental setup for the CDIWM scheme. The sets of polarizers (P), half wave plates (HWP) and quarter wave plates (QWP) are used for the pre- and postselection procedures. A PBS splits the input beam into two orthogonal linear polarization beams propagating along different arms of the interferometer. The QWP together with the mirror rotate the polarization by 90$^\circ$ so as to recombine the two beams at the same PBS. The moveable mirror is used to tune the initial coupling. The fiber collector (FC) collects the output beam whose spectrum is measured using an optical spectrum analyzer (OSA). $\phi$ denotes the tiny longitudinal phase perturbation between different polarization components.[]{data-label="setup"}](setupN.pdf){width="3in"}
The ultimate detectable longitudinal phase perturbation is estimated below. In the SWM scheme, when $ \lvert \omega \tau/ \epsilon \rvert \ll 1$ the spectral shift can be readily obtained from weak value defined as $A_w=\langle \phi \lvert A \rvert \psi \rangle/ \langle \phi \lvert \psi \rangle$. The ordinary weak value is calculated to be $icot(\epsilon)$ so the spectral shift is $$\label{metershift}
\Delta \omega \rightarrow \tau Im[A_w]\delta^2=cot(\epsilon) \delta^2 \Delta\tau \approx \frac{\delta^2 \Delta \tau}{\epsilon}$$ with a longitudinal phase perturbation $\Delta\tau$. The ultimate resolution limit of the SWM scheme is given by [@Brunner3] $$\label{SWMresolution}
\tau>\frac{\lvert \epsilon \rvert \Delta \Omega}{\delta^2},$$ where $\Delta \Omega$ is the resolution of the OSA. (There is a factor of 2 difference compared to the scheme by Brunner $\emph{et al.}$ due to a slight different in the representation of meter.) In the CDIWM scheme, the working regime is restricted to the CDI regime so the spectral shift can be calculated as $$\Delta \omega \rightarrow \frac{d\Delta \omega}{d\tau}\lvert_{\tau \rightarrow \frac{\epsilon}{\omega_0}}\Delta\tau \approx 2\frac{\omega_0^2}{\epsilon}\Delta\tau.$$
The resolution for the CDIWM scheme is similarly analyzed to be $$\label{CDIWMresolution}
\tau>\frac{\lvert \epsilon \rvert \Delta \Omega}{2\omega_0^2}.$$
From the above analysis, CDIWM can achieve a significantly higher resolution. With a visible broad band source, $\omega_0$ is on the order of several thousands THz and $\delta$ is on the order of several hundreds THz so the resolution can be improved by two orders of magnitude. For example, we consider a light beam centering at $\omega_0=2350$ THz ($\lambda \approx 800$ nm) with $\delta =200$ THz ($\delta \lambda \approx 100$ nm) and a OSA having a spectral resolution of 0.01 nm at $\lambda=800$ nm. The achievable resolution is shown in Fig. \[ResolutionSelection\](a) with varying $\epsilon$ calculated from Eq. (\[SWMresolution\]) and Eq. (\[CDIWMresolution\]). When $ \epsilon \rightarrow0$, CDIWM has the ability to detect a time-domain perturbation on the order of $10^{-5}$ as.
However, this improvement is at a cost that much more photons are discarded in the postselection process. In the SWM scheme, the postselection probability $P$ is approximately the probability of simply postselecting $\lvert \psi \rangle$ by $\lvert \phi \rangle$ which leads to $P_{SWM} \approx \epsilon^2$. In the CDIWM scheme, we require $\omega_0 \tau -\epsilon \approx 0$ and the working regime to be $\tau \approx \epsilon/\omega_0$ so the postselection probability now becomes $P_{CDIWM}\approx \delta^2 \epsilon^2/(2\omega_0^2)$. Fig. \[ResolutionSelection\](b) shows the postselection probabilities of the two schemes with respect to different $\epsilon$ calculated from Eq. (\[prob\]).
![The comparison of CDIWM and SWM with respect to the postselection angle $\epsilon$. (a) The resolution of two schemes. (b) Postselection probabilities of two schemes. The dot-dashed blue and long-dashed red lines represent the CDIWM scheme and the SWM scheme, respectively. []{data-label="ResolutionSelection"}](ResolutionSelectionNNN.pdf){width="6in"}
To sum up, by using the CDIWM scheme the resolution is improved by two orders of magnitude with the same orders of decreasing in the postselection probability. Specifically, we can achieve a factor of $2\omega_0^2/\delta^2$ amplification for resolution while decreasing the probability by a factor of $\delta^2/(2\omega_0^2)$ at the same time using the CDIWM scheme instead of the SWM scheme. However, considering the SWM scheme exclusively, an improvement by two orders of magnitude leads to a decreasing of postselection probability by four rather than two orders of magnitude.
Discussion {#Discuss}
==========
By pre-coupling the system and meter and postselecting with specific state, CDI occurs and leads to an improved measurement sensitivity accompanying by a lower postselection probability. This amplification and rejection processes are very similar to weak-value-amplification effect, however, the physical description of CDIWM largely differs from SWM.
In SWM, within the so called weak interaction regime [@Boyd], the probability correction due to the interaction has a linear relationship with the first order weak value. The shift of meter can be calculated according to Eq. (\[metershift\]). However, this is not the case in CDIWM. Fig. \[ProbComp\] shows the postselection probability curves of SWM and CDIWM calculated from Eq. (\[prob\]). The pre-coupling process shifts the working regime to $\tau_{s} = \epsilon/\omega_0$ which is out of the weak interaction regime and reaches the local minimum of postselection probability (the green box and the inset in Fig. \[ProbComp\]).
![The postselection probability curves of SWM and CDIWM with the same parameters in Fig. \[SpectrumSlope\]. The dot-dashed blue line represents CDIWM and the long-dashed red line represents SWM. The green box and the inset indentify the working range of CDIWM. []{data-label="ProbComp"}](ProbCompareNNN.pdf){width="5in"}
On the other hand, this pre-coupling process makes an entangled initial state of the system and the meter according to Eq. (\[initial\]). As a result, the weak value can not be well defined here and the weak interaction approximation $\langle \phi \lvert e^{-iAP\tau}\rvert\psi \rangle \approx \langle \phi \lvert \psi \rangle e^{-iA_w P \tau}$ doesn’t hold anymore in the working regime of CDIWM, where $A_{w} \omega \tau \approx 1$ .
Weak measurement can achieve a better resolution while discarding a large proportion of resources. Considering both the amplification effect and the loss due to postselection, the ultimate precision or signal-to-noise ratio can not outperform the classical metrology [@Combes; @Gauger]. This issue also occurs in the CDIWM scheme because the existence of CDI leads to a further lower postselection. However, some recent proposals have proved that the power recycle technique can be used in weak measurement, thus recovering the inefficiency due to low postselection [@Kwiat1; @Kwiat2]. Adding a partially transmitting mirror to make the interferometer a resonant optical cavity, all the input photons will exit the interferometer with the amplified signal.
Conclusion
==========
We propose an improved weak measurement scheme to detect tiny perturbation to longitudinal phase in this letter. By pre-coupling the system and meter state and selecting a specific postselection state, a novel CDI effect can be observed when a broad band light source is used. From the calculation we find that when CDI occurs, the sensitivity of the spectral shift to longitudinal phase perturbation significantly increases. Respect to the same tiny perturbation on the longitudinal phase, the spectral shift can be amplified by a factor of several hundreds compared to the SWM scheme. Our results also outperform coherent light phase weak measurements [@Starling2] and currently are significantly better than quantum metrology technology measurements using N00N and squeezed states [@Krischek], which are still in the process of solving experimental problems [@Thomas]. Taking advantages of these characteristics, when a physical effect is coupled to a longitudinal phase, it can be precisely estimated through this CDIWM scheme.
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[**Acknowledgments**]{}
This work was supported by the National Basic Research Program of China (Grants No. 2013CB933304), the Strategic Priority Research Program (B) of the Chinese Academy of Sciences (Grant No. XDB01030300), National Natural Science Foundation of China (Grant Nos. 91536219, 61308010), the Fundamental Research Funds for the Central Universities (Grant No. WK2030020019, WK2470000011).
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abstract: 'We present a new approach to gravitational lens massmap reconstruction. Our massmap solutions perfectly reproduce the positions, fluxes, and shears of all multiple images. And each massmap accurately recovers the underlying mass distribution to a resolution limited by the number of multiple images detected. We demonstrate our technique given a mock galaxy cluster similar to Abell 1689 which gravitationally lenses 19 mock background galaxies to produce 93 multiple images. We also explore cases in which as few as four multiple images are observed. Massmap solutions are never unique, and our method makes it possible to explore an extremely flexible range of physical (and unphysical) solutions, all of which perfectly reproduce the data given. Each reconfiguration of the source galaxies produces a new massmap solution. An optimization routine is provided to find those source positions (and redshifts, within uncertainties) which produce the “most physical” massmap solution, according to a new figure of merit developed here. Our method imposes no assumptions about the slope of the radial profile nor mass following light. But unlike “non-parametric” grid-based methods, the number of free parameters we solve for is only as many as the number of observable constraints (or slightly greater if fluxes are constrained). For each set of source positions and redshifts, massmap solutions are obtained “instantly” via direct matrix inversion by smoothly interpolating the deflection field using a recently developed mathematical technique. Our LensPerfect software is straightforward and easy to use and is made publicly available via our website.'
author:
- 'D. Coe'
- 'E. Fuselier'
- 'N. Ben[í]{}tez'
- 'T. Broadhurst'
- 'B. Frye'
- 'H. Ford'
title: 'LensPerfect: Gravitational Lens Massmap Reconstructions Yielding Exact Reproduction of All Multiple Images'
---
Introduction {#intro}
============
Simulations of structure formation in a $\Lambda$CDM universe have provided concrete predictions for the form of Dark Matter halos over a wide range of scales [e.g., @Merritt06; @Bett07; @Diemand07]. These mass distributions, quantified in terms of radial profile, ellipticity, and level of substructure, are among the key predictions of $\Lambda$CDM theory. Uncertainties do persist, especially with regard to baryons, absent from most Dark Matter simulations. Baryons are found to alter the inner profiles and ellipticities of halos, especially on galaxy scales [e.g., @Kazantzidis04; @Gustafsson06; @Dutton07].
Testing these predictions in detail observationally has proven challenging. Gravitational lensing provides us with our most direct tool for mapping the distributions of mass (predominantly Dark Matter) within and surrounding galaxies and galaxy clusters. But massmaps recovered by this method are of much lower resolution than simulated Dark Matter halos. Improvements in imaging quality both from the ground and space now allow for a more definitive measurement of lensing effects, motivating new techniques to take full advantage of this advance.
Deep multi-wavelength high-resolution images of five galaxy clusters (Abell 1689, Abell 2218, Abell 1703, MS1358, and CL0024) have been obtained with ACS, the Advanced Camera for Surveys [@ACS] on-board the Hubble Space Telescope. The deepest of these, taken of A1689, has revealed an unprecedented number of multiple images, with 106 images of 30 source galaxies identified in the original analysis by [@Broadhurst05]. This should allow one to reconstruct a relatively high resolution massmap of the cluster’s Dark Matter halo.
[@Broadhurst05] used a novel massmap parameterization to reproduce the 106 observed multiple images well but not exactly (Fig. \[delens1\]a), with positional offsets of 32 RMS on average in the image plane. Note that these are very significant offsets, as image positions can typically be determined to $0\farcs05$ (the resolution of ACS pixels) or better. Subsequent analyses using similar methods (with different parameterizations for the massmap and revised multiple image lists) improved only marginally on these positional offsets to 25 RMS at best [@Zekser06; @Halkola06; @Halkola07; @Limousin07]. It is possible that cosmic variation of mass density along the multiple sight lines is responsible for some of this scatter. But it is more likely that line of sight variations are an additional source of error (present in any lensing analysis) on top of the models’ inability to properly reproduce all the image positions.
The aforementioned analyses all employ extensions of techniques developed for use with far fewer multiple images. A simple parametric model consisting of an elliptical mass profile plus external shear can accurately reproduce the positions of four multiple images [e.g., @Keeton97]. But such models struggle when confronted with additional multiple images. Multi-component (“halo” + “galaxy”) models are required to obtain decent fits to galaxy cluster lensing [e.g., @Kneib96; @Broadhurst00]. The A1689 multi-component models discussed above don’t contain enough free parameters to perfectly fit the data. Yet they already contain so many parameters that navigating the parameter space proves challenging. It may be impossible to find the best solution with so many free parameters given our current computational capabilities. Advances in computing power may someday allow parametric models the freedom necessary to produce perfect fits. Allowing galaxy subhalos to drift in position and vary individually in mass could dramatically improve the fits while breaking free from the assumption that Dark Matter subhalos strictly follow the light distribution.
The degree to which Dark Matter follows light (and/or gas) is an important outstanding question. The exciting discoveries in this area are currently being made by [*weak*]{} gravitational lensing analyses which make no assumptions about the underlying mass distributions. The “Bullet Cluster” finding [@Clowe06] that gas has been stripped from two colliding Dark Matter halos would not have been possible had the authors assumed mass follows light from the outset. A similar cluster collision observed along the line of sight appears to have left a Dark Matter ring around CL0024. This ring was also detected by weak lensing analysis [@Jee07].
For many years now, we have obtained assumption-free massmap reconstructions from direct inversion of weak gravitational lensing data [@Tyson90; @KaiserSquires93]. Here we present the first method to do the same given [*strong*]{} gravitational lensing data (multiple images). Our method is not entirely assumption free, as a few basic assumptions about the distribution of mass can be helpful in selecting the most “physical” among the possible massmap solutions (§ \[sourcepos\]).
Model-free massmap reconstructions have previously been obtained for Abell 1689 using strong lensing data (though not via direct inversion). As these massmaps are more flexible than model-based solutions, they should produce better fits to the data. But this promise has yet to be fully realized. The SLAP method [@Diego05a] computes massmaps on an adaptive mesh grid, and fits the data to a desired level of scatter. But when this level is set too low, the solutions become “biased” with “a lot of substructure”. Their best solution for A1689 leaves scatters of $\sim 3\arcsec$ in the [*source*]{} plane (@Diego05; Diego 2007, private communication). Meanwhile, [@Saha06] use a method called PixeLens to obtain pixel-based (fixed grid) massmaps which perfectly reproduce all multiple image positions. But computational constraints prevent them from fitting more than 30 images at a time. The massmap we obtain for A1689 (to be presented in an upcoming paper) perfectly reproduces the positions of 135 multiple images (as in Fig. \[delens1\]b) and thus has about twice the resolution as the PixeLens massmap in each spatial direction (as dictated by the density of multiple images).
Massmaps may be further improved by incorporating constraints beyond simple image positions. Images which are resolved and extended should be properly reproduced by the mass model [@WarrenDye03; @Suyu06; @Koopmans06].
Even unresolved images yield the information encoded in their fluxes. A simple mass model which accurately reproduces “quad” image positions may or may not accurately reproduce the image flux ratios. Discrepant cases are known as “flux anomalies”. If other causes (microlensing, time delays, and dust extinction) can be ruled out or accounted for, then small substructure ($\sim 10^6 M_\odot$ subhalos) is generally invoked as the most likely explanation for an observed anomaly.
But this explanation has perhaps been invoked too often. The amount of substructure observed in simulations may not be sufficient to produce flux anomalies as often as observed [@Metcalf04; @Amara06; @MaccioMiranda06; @Diemand07]. One way to resolve this possible discrepancy is to obtain macro mass model solutions which reproduce the observed fluxes without resorting to smaller substructure.
To address this issue, [@EvansWitt03] developed a direct inversion massmap reconstruction method capable of perfectly fitting the observed positions and fluxes of four multiple images. While providing reasonable solutions for the lensed systems Q2237+0305 and PG1115+080, their solution for B1422+231 was clearly unphysical. While it is unclear how exactly to quantify a massmap’s physicality, the authors characterized their B1422+213 model as simply too “wiggly” to be plausible. Developing the method a bit further, [@CongdonKeeton05] produced a more reasonable solution for B1422+231, but were less successful with B2045+265 and B1933+503. For the unsuccessful cases, the authors argue that small scale substructure is the most likely explanation for the observed flux anomalies. But perhaps their models were simply not flexible enough to obtain physical solutions for these systems. We will revisit this question in an upcoming paper.
Our new LensPerfect method uses direct inversion to obtain assumption-free massmap solutions which perfectly reproduce all multiple image positions. Multiple image knots and fluxes may also be perfectly constrained. LensPerfect is made possible by a recent advance in the field of mathematics. The essential tool is a method that produces a curl-free interpolation of vectors given at scattered data points. A set of observed multiple image positions and redshifts along with assumed source positions defines a deflection field at the image positions. Once we interpolate this deflection field across the entire field, we take the gradient, multiply by 2, and “instantly” obtain our perfect massmap solution.
The interpolation of data given at scattered points is a complex problem without a unique solution. One method of attacking such problems involves the use of radial basis functions (RBFs). RBFs have been used to interpolate scattered data since the early 1970’s [@Hardy71] and are used in many applications today. By the 1980’s, RBFs had been applied to interpolate vector fields. And in 1994, a method was developed capable of yielding divergence-free interpolations of scattered vectors [@Narcowich94]. Finally, the curl-free analog of this method was developed last year [@Fuselier06; @Fuselier07]. Here we apply this new method to gravitational lensing analysis.
We describe our method along with the necessary gravitational lensing equations in §\[sec:method\]. In § \[sec:applications\], we demonstrate applications of the method, including the recovery of a known massmap given 93 multiple images (§ \[A1689model\]). We also explore the solutions obtained when only a handful of multiple images are available (§ \[Einstein\]). And we demonstrate the gains made by constraining extra image knots in § \[knots\]. The method for adding flux and/or shear constraints is given in § \[fluxes\]. In § \[discussion\] we discuss the relative merits of model-based and model-free methods, along with the potential of a hybrid method among other techniques that may become possible with LensPerfect. Finally, we provide a summary in § \[summary\]. The LensPerfect software and more information are available at our website.[^1]
Method {#sec:method}
======
Image Deflection
----------------
Image deflection by a gravitational lens is governed by a few simple equations [e.g., @Wambsganss98]. The relativistic bending of light due to a mass $M$ at a distance $R$ away is twice that expected from Newtonian physics: $\tilde \alpha = 4 G M / c^2 R$ given Newton’s gravitational constant $G$, and the speed of light $c$. In a gravitational lens, it is generally safe to assume that all of the deflection occurs in the plane of the lens. (This is known as the thin lens approximation.) Given the projected mass surface density distribution $\kappa(\vec \theta)$ of the lens, we can derive the deflection of light $\vec \alpha (\vec \theta) = \vec \theta - \vec \beta$ from its true position on the sky $\vec \beta$ to that at which we observe it $\vec \theta$ (Fig. \[deflection\]):\
$$\label{deflkappa}
\vec \alpha (\vec \theta) = \frac{1}{\pi} \int d^2 \vec \theta \arcmin
\kappa(\vec \theta \arcmin)
\frac{\vec \theta - \vec \theta \arcmin}
{\left \vert \vec \theta - \vec \theta \arcmin \right \vert ^2},$$\
with the corresponding less-intimidating inverse relation:\
$$\label{div}
\nabla \cdot \vec \alpha = 2 \kappa.$$ The surface density $\kappa = \Sigma / \Sigma_{crit}$ is defined in units of the critical density at the epoch of the lens. The critical density is that generally required for multiple images to be produced. It is a function of source redshift as given by:\
$$\label{E_c}
\Sigma_{crit} = \frac{c^2}{4 \pi G} \frac{D_S}{D_L D_{LS}},$$\
involving a ratio of the angular-diameter distances from observer to source $D_S = D_A(0,z_S)$, observer to lens $D_L = D_A(0,z_L)$, and lens to source $D_{LS} = D_A(z_L,z_S)$. For a flat universe ($\Omega = \Omega_m + \Omega_\Lambda = 1$), angular-diameter distances are calculated as follows [@Fukugita92 filled beam approximation; see also @Hoggcosmo]:\
$$D_A(z_1,z_2) = \frac{c}{1 + z_2} \int_{z_1}^{z_2} \frac{dz^\prime}{H(z^\prime)},$$\
where the Hubble parameter varies with redshift as:\
$$H(z) = H_o \sqrt{\Omega_m (1 + z)^3 + \Omega_\Lambda}.$$ Thus the critical density $\Sigma_{crit}$ is a function of the source redshift. This follows because the deflection angle $\vec \alpha$ is a function of source redshift. As source redshift decreases, the light bend angle $\tilde \alpha$ remains constant, which (imagine moving the galaxy in Fig. \[deflection\] inward along the top blue arrow) requires the image deflection to decrease by the distance ratio:\
$$\label{deflscale}
\vec \alpha = \left( \frac{D_{LS}}{D_S} \right) \vec \alpha_\infty,$$\
where $\vec \alpha_\infty$ is the deflection for a source at infinite redshift.
Thus the problem of massmap reconstruction can be reduced to determining the deflection field with all deflections scaled to a common redshift (e.g., $\vec\alpha_\infty$), at which point we simply take the divergence and divide by 2 to obtain the massmap (Eq. \[div\]). The deflection field $\vec \alpha(\vec \theta) = \vec \theta - \vec \beta$ may be measured at the multiple image positions $\vec \theta$ once source positions $\vec \beta$ are determined (see § \[sourcepos\]). However, in order to take its divergence, the deflection field must be solved for as a continuous function of position (or at least defined on a regular grid). Our interpolated deflection field must also be curl-free:\
$$\label{curlfree}
\nabla \times \vec \alpha = 0.$$ This follows from Equation \[deflkappa\] in the same way that we find an electric field due to a static distribution of charge is also curl-free. One way to demonstrate this is to use the substitution $\nabla \ln \theta = \vec \theta / \theta^2$ to define the lensing potential:\
$$\label{potentialkappa}
\psi (\vec \theta) = \frac{1}{\pi} \int d^2 \vec \theta \arcmin
\kappa(\vec \theta \arcmin)
\ln \left \vert \vec \theta - \vec \theta \arcmin \right \vert,$$\
such that\
$$\vec \alpha = \nabla \psi.$$\
The fact that the deflection field $\vec \alpha (\vec \theta)$ may be written as the gradient of a scalar field $\psi (\vec \theta)$ guarantees that it has no curl.
Curl-Free Vector Interpolation
------------------------------
@Fuselier06 [@Fuselier07] has shown how to obtain a curl-free interpolation of a vector field given on scattered data points. The method is general enough to be applied to an arbitrary number of dimensions, but will be discussed here for the 2-D case. The interpolated vector field is constructed using Radial Basis Functions, or RBFs. An RBF is a positive definite function with radial symmetry $\phi(R/R_o)$. The fact that it is positive definite[^2] guarantees that our interpolation matrix (below) will have a solution [@Wendland05; @Fuselier06]. The scale length $R_o$ is input by the user, and as we show in §\[Einstein\], its freedom is closely related to the classic “mass-sheet” degeneracy.
In our case we must choose smooth RBFs that are at least three times continuously differentiable, or “$C^3$”, this to ensure our final massmap has no discontinuities. Then a set of two curl-free basis vectors are constructed at each data point (image position) by simply taking derivatives of the basis function:\
$$\begin{aligned}
\label{basis}
\vec V_a & = & \phi_{,xx} \hat x + \phi_{,xy} \hat y,\\
\vec V_b & = & \phi_{,yx} \hat x + \phi_{,yy} \hat y,\end{aligned}$$\
where $\phi_{,xx}$ is the 2nd derivative of $\phi(R/R_o)$ with respect to $x$, etc. The vector at that data point is given as a sum over all data points (with coefficients to be solved for):\
$$\label{coeffs}
\vec \alpha = \sum_i^N \left( c_a \vec V_{a_i} + c_b \vec V_{b_i} \right).$$ Rewriting this in matrix form we have:\
$$\left[ \alpha \right] =
\left[ \phi_{,(xy)} \right]
\left[ c \right].$$ For example, in the case of 2 data points, $\vec \alpha (\vec \theta_1)$ and $\vec \alpha (\vec \theta_2)$:\
$$\label{deflmatrix}
\left[
\begin{array}{c}
\alpha_{x_1}\\
\alpha_{y_1}\\
\alpha_{x_2}\\
\alpha_{y_2}
\end{array}
\right]
=
\left[
\begin{array}{cc}
{\phi_{,xx_{11}}} ~~ {\phi_{,yx_{11}}} ~~ {\phi_{,xx_{12}}} ~~ {\phi_{,yx_{12}}}\\
{\phi_{,xy_{11}}} ~~ {\phi_{,yy_{11}}} ~~ {\phi_{,xy_{12}}} ~~ {\phi_{,yy_{12}}}\\
{\phi_{,xx_{21}}} ~~ {\phi_{,yx_{21}}} ~~ {\phi_{,xx_{22}}} ~~ {\phi_{,yx_{22}}}\\
{\phi_{,xy_{21}}} ~~ {\phi_{,yy_{21}}} ~~ {\phi_{,xy_{22}}} ~~ {\phi_{,yy_{22}}}
\end{array}
\right]
\left[
\begin{array}{c}
c_{a_1}\\
c_{b_1}\\
c_{a_2}\\
c_{b_2}
\end{array}
\right],$$\
where, e.g.:\
$$\phi_{,yx_{12}} = \left[ \frac{\partial^2 \phi}{\partial y \partial x} \right]_{R=R_{12}},$$\
with the derivative being evaluated at $R = R_{12} = \left\vert \vec \theta_1 - \vec \theta_2 \right\vert$. We may now simply solve for the coefficients via the matrix inverse:\
$$\left[ c \right] =
\left[ \phi_{,(xy)} \right] ^ {-1}
\left[ \alpha \right].$$ We may guarantee that this matrix inversion yields a solution by selecting a positive definite function for $\phi$. Wendland functions are especially suitable choices, and here we use this “$C^6$”, or 6 times continuously differentiable, Wendland function known as $W_{7,3}(\xi)$:\
$$\phi(\xi) = \left\{
\begin{array}{lc}
(1 - |\xi|)^8 (32 |\xi|^3 + 25|\xi|^2 + 8|\xi| + 1),& |\xi| \leq 1\\
0, & |\xi| > 1
\end{array}\right.
\label{WendlandC6}$$\
where $\xi = R / R_o$. The function is very similar to a Gaussian with a peak of 1 and FWHM $\sim 0.50 R_o$ ($\sigma \sim 0.21 R_o$).[^3]
Of relevance in this work are derivatives of this function which serve as basis functions for our massmaps. Having solved for the deflection field $
\left[ \alpha \right] =
\left[ \phi_{,(xy)} \right]
\left[ c \right],
$ the massmap may then be calculated analytically as:\
$$\begin{aligned}
\label{kappa}
\kappa & = & \onehalf \nabla \cdot \vec \alpha\\
& = & \onehalf \sum_i^N \left[
c_{a_i} ( \phi_{,xxx_i} + \phi_{,xyy_i} ) +
c_{b_i} ( \phi_{,yxx_i} + \phi_{,yyy_i} )
\right].\end{aligned}$$ The basis functions:\
$$\begin{aligned}
\label{KBFab}
\kappa_a & = & \onehalf (\phi_{,xxx} +\phi_{,xyy})\\
\kappa_b & = & \onehalf (\phi_{,yxx} +\phi_{,yyy})\end{aligned}$$\
are shown in Figure \[KBF\]. Different combinations of these two basis functions simply serve to rotate it about its axis and change its amplitude.
\
The basis functions, with amplitudes and orientations solved for, are placed at the positions of the multiple images. The resulting coefficients are generally many orders of magnitude greater than the amplitude of the massmap. These large mass components all cancel out nearly perfectly with the small “residuals” being the massmap solution. Together, these mass components form a very flexible basis set. Although the basis functions are derived from radial basis functions, no symmetry conditions (radial, or otherwise) are imposed on the total massmap solution.
We note that other RBFs may be used in place of $W_{7,3}$, including other Wendland functions, thin plate splines, multiquadrics, power laws, and even Gaussians [e.g., @Fuselier06]. Wendland functions are especially well-suited for use in the interpolation matrix, but other functions may be explored in future work.
Other Observables
-----------------
For reference, we provide here expressions for other observables as functions of our basis function. The lensing potential (Eq. \[potentialkappa\]) is defined as a sum of derivatives of our basis function:\
$$\psi = c_a \phi_{,x} + c_b \phi_{,y}.$$ Other observables may then be derived as derivatives of this potential. The deflection field is given as the gradient of the potential: $$\vec \alpha = \psi_{,x} \hat x + \psi_{,y} \hat y.$$ We also obtain surface mass density\
$$\kappa = \onehalf \left ( \psi_{,xx} + \psi_{,yy} \right ),$$\
shear\
$$\begin{aligned}
\label{shear}
\gamma_+ & = & \onehalf \left ( \psi_{,xx} - \psi_{,yy} \right ),\\
\gamma_\times & = & \psi_{,xy},\\
\gamma^2 & = & \gamma_+^2 + \gamma_\times^2,\end{aligned}$$\
the inverse of the magnification\
$$\begin{aligned}
\label{magnif}
1 / \mu & = & (1 - \kappa)^2 - \gamma^2\\
& = & 1 - \psi_{,xx} - \psi_{,yy} + \psi_{,xx}\psi_{,yy} - \psi_{,xy}^2,\end{aligned}$$\
and time delays\
$$%
\label{timedelay}
\Delta \tau = \frac{(1 + z_L) D_S}{c D_L D_{LS}}
\left [
\onehalf \left \vert \vec \theta - \vec \beta \right \vert ^2 - \psi
\right ].$$
Source Positions and Massmap Physicality {#sourcepos}
----------------------------------------
Image deflections are defined by the combination of image positions, source positions, and redshifts. But source positions are not known in practice and redshift measurements may contain large uncertainties. In our method, these become our free parameters. Each set of source positions and redshifts will redefine the deflection field, yielding a different LensPerfect massmap solution. We iterate to find the most plausible solutions including a single “best” solution, as we describe below. An ideal optimization procedure would lead us directly to the true source positions and redshifts. The optimization procedure we have developed may not be ideal, but we believe it produces an accurate “best” solution (see § \[A1689model\]), similar to that obtained with the true source positions. We describe our optimization procedure in the next subsection (§ \[optimization\]) after first describing our method for rating solutions.
LensPerfect will return a perfect massmap solution for any set of input source positions and redshifts. But only certain sets will yield physical solutions: those with positive mass everywhere within the convex hull, where our solutions are constrained. And some sets will yield “more physical” solutions than others. An extreme example of a “less physical” solution would be a “donut” solution comprised of a high density ring surrounding a lower density center. By developing a method to rate different solutions as more or less physical, we can iterate over possible source positions to find those which yield plausible solutions, including the most plausible or “most physical” massmap solution. Our goal is to discard unreasonable solutions without biasing our result toward our concept of what a massmap should look like (that it should have an NFW profile, for example).
In the PixeLens method, a series of rigid criteria are used to distinguish “physical” from “unphysical” massmaps [@SahaWilliams04]. They define a physical massmap as one that is positive everywhere and satisfies restrictions on maximum pixel-to-pixel variation and direction of the massmap gradient. Additional constraints are optionally imposed: inversion symmetry about the axis and a minimum radial slope. And early incarnations of PixeLens found those massmap solutions which most closely followed the light distribution [as advocated in @SahaWilliams97; @AbdelSalam98a].
After some experimentation, we have developed a new measure of physicality. Rather than imposing rigid arbitrary constraints, we assign a figure of merit to each massmap solution based on the following physicality traits:
1. Positive mass everywhere within the convex hull
2. Low mass scatter in each radial bin, thus preferring azimuthal symmetry
3. No “tunnels”: penalty for individual mass pixels within the convex hull being “too low” relative to others at similar radius
4. Overall smoothness: minimal pixel-to-pixel variation within the convex hull
5. Average mass in radial bins decreases outwards (penalty for increasing outward)
Below, we motivate this physicality list which has been chosen for the purposes of this paper. Note the user has the ability to modify this list rather straightforwardly given the freely-available LensPerfect software.
The first trait is our only rigid constraint, obviously required of any physical massmap. Note that our massmaps are only constrained within the convex hull. Our basis functions do necessarily yield negative mass outside this region, but this is not necessarily a concern. We note that we are able to “get the right answer” inside the convex hull (Figs. \[A1689kLP\], \[A1689kLPopt\]). And if negative mass lies in a symmetric ring outside the convex hull, then it will have no effect on the image positions inside.
What we would like to avoid is one or more large isolated pockets of negative mass off to one side outside the convex hull. These may arise as “corrections” to a solution which is not quite accurate within the convex hull. Large positive mass clumps outside the convex hull are similarly undesirable. Our second physicality trait serves to beat down these clumps, along with other positive mass clumps and underdense pockets within the convex hull. We may worry that this biases against the presence of subhalos. But turning this around, if we are biasing against subhalos, then we can be more confident of any subhalos that do arise in our solutions. And Occam’s razor would dictate that the massmap solution with the fewest subhalos is most likely. Also note we are talking about subhalos in large sub-clumps. Smaller satellites generally cannot be resolved. (Our resolution is limited by the density of multiple images observed in the lens plane.) In the future, we may wish to perform tests to determine exactly how well LensPerfect recovers different amounts of substructure in large subhalos, given different numbers of multiple images.
Underdense pockets incur extra penalty via our third physicality trait. While a small overdense area may be due to substructure, a small underdense area is much less physical. As our massmap is integrated a long the line of sight, an underdense pocket suggests that a “tunnel” has been carved through the entire mass distribution. This is not very plausible, and thus we penalize such tunnels more severely than overdensities.
We further beat lumps and pockets out of our solutions by minimizing pixel-to-pixel variation of mass (our fourth physicality trait).
Finally, we penalize (without forbidding) mass profiles which increase with radius. Of course this might bias against spectacular findings like the ring-like structure in CL0024 reported by [@Jee07]. But we believe that it is perfectly healthy to bias against such spectacular solutions. If a more pedestrian solution may be found which exhibits no ring structure, then it is likely that no ring structure exists. And if such a ring-like structure were to persist in our solution despite this bias, we could be that much more confident of its existence. Additional tests could then be performed to show how well LensPerfect recovers ring structures in known massmaps and whether it might “recover” rings when they are not present. Finally, we note that in CL0024, the ring-like structure is detected (and predicted in simulations of a cluster collision) to lie well outside the strong lensing region.
Note that our physicality traits 2, 3, and 5 assume at least a rough azimuthal symmetry. We have yet to experiment with mass distributions which are strongly multi-modal. While our current method will suffice for many clusters, including Abell 1689, we will probably need to revise our penalty functions to analyze a highly asymmetric cluster such as Abell 2218.
Even having settled on a set of physicality traits, it is unclear how exactly to calculate penalty functions based on them. And once these penalty functions are established, how should they be weighted relative to each other? The goal is for equally “offensive” massmaps to be penalized equally. Again this is highly subjective, but after much experimentation, we have devised a total penalty function which works well. The details are given in the Appendix \[penalty\].
Massmap Optimization {#optimization}
--------------------
With a good penalty function in hand, we may then proceed to find those source positions and redshifts which produce the most physical massmap. Given many lensed galaxies, this is far from trivial. For example, 19 source positions ($x$, $y$) yield 38 parameters which must be optimized over, plus up to 19 redshifts with their associated uncertainties. And for each iteration, we must obtain a low resolution massmap to calculate our penalty function. This takes from less than a second to a few seconds as more images are added. Thus we must choose a scheme which efficiently navigates our 38(+19)-dimensional parameter space, minimizing the number of iterations necessary. Fortunately, we may use a few tricks to converge to good solutions fairly quickly.
The first trick is commonplace in strong lensing massmap reconstruction methods: we build the massmap one galaxy at a time. (That is, we add the multiple images of each galaxy in turn as constraints to our model.) But we take this a step further, tearing our solution down and rebuilding it for every iteration. These rebuilds are extremely quick, as at each step, the deflection field solution need only be calculated at the new image positions.
As we add each galaxy to our model, we place its source position using the average of those predicted from the current solution (Fig. \[source2\]). Each image will delens back to a slightly different position (as these images have not yet been constrained). By taking the average of these (weighted, as discussed below), we can obtain a good initial guess as to its source position.
Given this source position we will obtain a new massmap solution. But we can also perturb this source position and obtain a different massmap solution instead. Thus we iterate and search for that source position which yields the most physical solution according to our penalty function. We perform this 2-dimensional optimization using the [@Powell64] routine[^4] included in the SciPy Python package.[^5] If the redshift of this galaxy is uncertain, we may also optimize the redshift at this time (including a penalty if it drifts too far from its expected value). (For this one-dimensional optimization, we use the simple golden section search method, also included in SciPy.) Once the source position and redshift have been optimized, we may proceed to adding the next galaxy.
Before proceeding to the next galaxy, we may choose to re-optimize all previous source positions and redshifts. This “reshuffling” can improve the overall solution. But once, say, 10 galaxies have been placed, we may worry that our solution has been “locked in”. Attempting to re-optimize the third galaxy position will not be very effective, as it is now tightly constrained by the positions of the other nine galaxies.
Thus we use a flexible parameterization in which the perturbations to the predicted source positions (as described above) constitute our free parameters to be optimized. That is, we maintain a list of source position offsets (one for each galaxy, initially set to $(0,0)$), and it is this list which we optimize, rather than the source positions themselves. Every time we rebuild our massmap solution, we obtain each source position as shown in Fig. \[source2\], but then we offset it according to our list of offsets before placing it and proceeding to add the next galaxy. Note that each offset affects not only the source position of that galaxy, but also (as this modifies the solution) those of all galaxies that follow. In essence, source positions move and adjust with each other.
As mentioned above, we determine each new source position by taking [*weighted*]{} averages of those predicted. We assign more weight to a delensed position for which the image is close to an already established image. For example, in Fig. \[source2\], a red image near any black image position will have its source position given more weight in the average. The idea is that if the deflection field has already been established at a (black) image position, then the deflection field at the nearby (red) point should be similar. We have found this weighting scheme yields better source positions (closer to the true model positions) than straight averaging. We also had some success giving more weight to sources with images at large radii (where there is less mass available to support rapid changes in the deflection field). In the end, the exact scheme is somewhat irrelevant as offsets from these average positions will be perturbed and optimized. But it is possible that a better scheme would yield quicker convergence.
Above we have described all the details of our optimization scheme, except for how we begin! That is, where do we put our first source position? Before any galaxies are placed, we have no massmap solution, and thus no predictions for the first source position. Fortunately just about any choice will do for the first source position, as all yield identical (or nearly so) solutions within the convex hull. Even after additional sources are added, shifting the first source position has little effect on the overall solution. The entire source plane basically shifts along with it, yielding a nearly identical solution within the convex hull. This is a well-known lensing degeneracy, but certain shifts in the source plane do yield more physical solutions than others, as we demonstrate in § \[massstick\]. Correct source positions yield a solution which is more symmetric outside the convex hull.
Constraining Image Fluxes and/or Shears {#fluxes}
---------------------------------------
Image fluxes may provide additional constraints to the mass model. To constrain the fluxes (and shears) of lensed images, previous authors have added terms to their matrix equation relating source positions, lens parameters, and observables [@EvansWitt03; @CongdonKeeton05; @Diego07]. However we prefer not to interfere with our Eq. \[deflmatrix\] which, given proper basis functions, is guaranteed to obtain a perfect interpolated solution of any input deflection vectors. Fortunately, relative image fluxes (magnifications) and shears are determined by local rates of change in the deflection field and thus may be easily and precisely constrained by adding deflection field constraints.
In this subsection we describe how to add flux and/or shear constraints for [*multiple images*]{}. (Also see § \[knots\].) We do not yet have a method for incorporating weak lensing shear or any other constraints from singly-imaged galaxies (though see § \[weaklensing\]).
To constrain the fluxes of multiply-imaged galaxies, we begin by obtaining an initial massmap that reproduces the image positions. Next, we construct a small box around each multiple image and delens each back to the source plane. We measure the area of each delensed box (now a parallelogram in the source plane) and compare to its original area in the image plane. The ratio of these areas yields the magnification for that image, at least as predicted for our initial model.[^6] Now, by modifying the deflection field at the corners of the box, we can adjust the relative sizes of the lensed and unlensed boxes so that they produce the proper (observed) magnifications (Fig. \[fluxcontrol\]). Given these new deflection field inputs, which encode both position and flux constraints for the multiple images, we obtain a new massmap solution that perfectly reproduces the observed positions and fluxes.
We note that image shears could be constrained in a similar manner, if desired, by adjusting the relative shapes of the lensed and delensed boxes. And observational uncertainties may be incorporated by adding multiple realizations of noise to the measurements and finding a perfect solution to each, as done by [@EvansWitt03] and [@CongdonKeeton05].
In practice we find that rather than adding four constraints to the deflection field for each flux measurement, we need only constrain two extra points, one above or below and the other to the left or right of the image. This not only helps reduce computing time, but it also keeps our number of free parameters more in line with the number of observable constraints. We could keep the two numbers exactly equal by adding a single constraint. But this fails to properly constrain the flux, instead squeezing mass out to the sides as when one sits on a balloon.
Computational Efficiency {#compu}
------------------------
Other strong lensing analysis methods (with the exception of PixeLens) face a dilemma over whether to minimize scatter in the source or image plane. The former choice, while less robust and subject to possible biases, is often chosen for computational efficiency. LensPerfect does not have to make this choice as both source and image positions are always perfectly constrained.
And LensPerfect is computationally efficient. Once all source positions and redshifts have been established (along with image positions), our massmap solution coefficients are obtained “instantly” (in a fraction of a second) via direct matrix inversion without need for iterations. Evaluating this massmap solution on a grid, while not quite “instant”, is still very fast. On a Mac Powerbook G4 laptop, given $N_i = 93$ multiple images, the massmap can be evaluated on a $N_p = 2500 = 50 \times 50$ grid in 3 seconds, scaling with $N_i N_p$.
Applications {#sec:applications}
============
Massmap Recovery Test {#A1689model}
---------------------
Here we demonstrate our technique given a mock galaxy cluster with simulated gravitational lensing. More important than the ability to perfectly reproduce all multiple image positions is the accuracy to which we can recover the true massmap distribution.
We would like to test LensPerfect for a case similar to that observed in Abell 1689. Thus we create a mock galaxy cluster “Babell 1689”, which is very similar to Abell 1689. In fact, the massmap of Babell 1689 (Fig. \[A1689kB05\]) is actually a solution obtained by [@Broadhurst05] from their analysis of Abell 1689. But for our purposes, Babell 1689 is just a mock galaxy cluster. We use it to gravitationally lens 19 mock galaxies at redshifts between 1 and 5.5, producing 93 multiple images (Fig. \[multimages\]). This is similar to the number of multiple images (106) identified by [@Broadhurst05].
We stress that we are not analyzing Abell 1689 here. Our analysis of Abell 1689 and its multiple images will be published in an upcoming paper. Here we are analyzing the mock cluster Babell 1689 given its mock multiple images.
The 93 mock multiple image positions along with 19 source positions and redshifts are fed into LensPerfect as input. Fig. \[A1689defl\] shows the input deflection field scaled to a source redshift of infinity along with a LensPerfect curl-free interpolation. The solution was obtained using the Wendland function given in Equation \[WendlandC6\] with a scale factor of $R_o = 700$ in the units plotted. One half the divergence of this deflection field gives the LensPerfect massmap solution (Fig. \[A1689kLP\]). Note that it is basically a low-resolution interpolation of the input massmap (Fig. \[A1689kB05\]). No assumptions were required about the form of the massmap, and yet the general form and significant features of the input massmap are recovered. Were more than 93 input images given, the resolution would increase and finer details would be resolved (§ \[1000\]).
Note that the @Broadhurst05 solution may appear to resolve very fine detail in the Abell 1689 massmap absent from our reconstruction of Babell 1689 presented here. But this is just a result of their assumption that some component of the Dark Matter traces light. LensPerfect makes no such assumption.
In practice, we will not have knowledge of the source positions. So we repeat our analysis, but without providing the source positions as input. Instead we use the source position optimization method outlined in §§ \[sourcepos\] and \[optimization\] and detailed in the Appendix \[penalty\]. The “most physical” massmap solution we find is shown in Fig. \[A1689kLPopt\]. It is very similar to that obtained with the true source positions (Fig. \[A1689kLP\]). In this case the only information we input to LensPerfect were the image positions and redshifts, which we assumed had been measured with perfect precision.[^7]
Radial profiles of the three massmaps (model, recovered with true source positions, and recovered with optimized source positions) are compared in Fig. \[profile\]. Only points within the convex hull are considered. Agreement of the mean mass is excellent except for some deviation in the center and slight deviations toward the outside of the convex hull. The fine structure of the mass peaks is not recovered in these modest-resolution (93 multiple image) massmaps.
Source Position Recovery {#sourcepossec}
------------------------
How well do our optimized source positions match the true source positions? Inaccuracies in the source positions propagate to the deflection field and thus to our massmap. Modest shifts of the entire source plane are acceptable, however. This well-known degeneracy does not affect the solution inside the convex hull (see § \[massstick\]).
With this in mind, we plot (Fig. \[sourceposplot\]) our optimized and true source positions from the previous subsection. A constant shift of a few pixels has been added to the optimized positions to bring them in line (on average) with the true positions. This is justified (not just for our method, but for any method) as shifts in the source plane are a degeneracy in the problem. A massmap solution obtained with one shift is virtually identical and no less accurate inside the convex hull than a solution obtained with a different shift (§ \[massstick\]).
Applying this shift, we find our recovered source positions are offset from the true source positions at the rate of 1.05 pixels on average for the 19 systems. If this were A1689, 1.05 pixels would translate to $0\farcs42$. But this value cannot be directly compared to any published results for A1689 (or any other cluster). Source position offsets are likely inherent to all methods, but they can never be measured in practice since the true source positions are unknown. The only measurable quantity is the scatter of (delensed) source positions within each multiple image system. For LensPerfect, this scatter is zero. In other methods, the errors due to scatter and offsets may be cumulative.
Our optimal source positions are also biased toward a solution with slightly greater magnification than the true solution. This is most likely a consequence of our physicality measure which rewards smoother massmaps. A smoother massmap will have a shallower profile and thus higher magnification. While we have made every effort not to bias profile slope, some small bias may still remain. We will work to reduce both the offsets and this bias in the future. In the meantime, it is a simple exercise for the user to spread out the optimal source positions and explore solutions with lower magnifications. In fact, this should be a part of any comprehensive analysis.
We must keep in mind the ultimate goal in our analysis. The goal is not to obtain the single best massmap, but rather to determine the range of massmaps which produce reasonable solutions. Both of the massmaps presented in § \[A1689model\] are perfectly valid solutions which perfectly reproduce the 93 multiple image positions. We cannot know which is more accurate without knowledge of the source positions. And of course this knowledge is unattainable in practice. This is just an inherent degeneracy in the problem. We are developing methods to thoroughly explore the solution space and will report on them in future work.
1,000 Multiple Images {#1000}
---------------------
The resolution of a LensPerfect massmap is dictated by the density of multiple images detected. Each multiple image samples the deflection field at a given location. In the gaps between these images, the deflection field must be interpolated, and the exact form of the massmap becomes less certain.
To demonstrate this, we consider the massmap solution we may obtain given 1,000 multiple images. Rather than performing mock lensing to produce multiple images as in the previous subsection, here we will simply sample the deflection field at 1,000 points. We restrict these samples to an area of similar size as before (a circle of radius 150 pixels). And we ensure that all samples are separated by 2 pixels or more. Given these 1,000 samples, we then interpolate to find the deflection field elsewhere. Our massmap solution is shown in Fig. \[kappa1000\]. Comparing with the input massmap (Fig. \[A1689kB05\]), we can see that very fine detail is faithfully resolved.
This setup is a bit idealized. The “multiple images” are spread fairly uniformly (randomly) about a circle within the field. In practice we are more likely to find clumps and voids of multiple images due to the magnification pattern, obscuring foreground galaxies, and physically linked background galaxies. Also, we have assumed the equivalent to perfect knowledge of the source positions. Nevertheless, this massmap gives us a rough idea of the resolution we may expect given observations deep enough to reveal 1,000 multiple images.
In practice, we would need to find and optimize a large number of source positions. 1,000 multiple images may be produced by about 300 background galaxies. This saddles us with 600 free parameters. But even in our 93-image system, we find that source positions added toward the end are already fairly well constrained by those added previously. So we can speculate that the next 200+ source positions may similarly “fall into place”, making the computational challenge more manageable.
Fewer Constraints and the Role of $R_o$ {#Einstein}
---------------------------------------
While it is useful that LensPerfect can produce such detailed massmaps when given so many multiple images, what happens when LensPerfect is given far fewer constraints, say a single quadruply-imaged galaxy? Does the sum of four of the oddly-shaped basis functions depicted in Fig. \[KBF\] yield a reasonable massmap solution?
Yes it does. Fig. \[cross\] (left-hand panel) shows the massmap solution obtained given a simple symmetric “Einstein cross” 4-image configuration. The solution is exactly the same (to within $\kappa = 0.01$) if we form a more complete “Einstein ring” with 16 image constraints as shown. But this is just one possible solution, obtained with $R_o$ equal to ten times the Einstein Radius. We begin to explore other solutions by varying the width $R_o$ of our basis function (Eq. \[WendlandC6\]). Radial profiles of these different solutions are shown in the right-hand panel. By changing this single parameter, we neatly demonstrate the “steepness” or “mass-sheet”[^8] degeneracy [originally dubbed the “magnification transformation” in @Gorenstein88], which states that the mass everywhere may be replaced by\
$$\begin{aligned}
\label{masssheet}
1 - \kappa^\prime & = & \lambda (1 - \kappa),\\
\kappa^\prime & = & \lambda \kappa + (1 - \lambda)\end{aligned}$$\
without affecting the observed image positions (unless multiple galaxies of different redshifts have been lensed). Note that the new mass $\kappa^\prime$ is steeper than the previous mass $\kappa$ by a factor of $\lambda$.
The different LensPerfect solutions follow the transformation in Eq. \[masssheet\] very closely, although not exactly. We have rescaled the green $R_o = 100$ curve via this transformation such that its peak aligns with the $R_o = 55$ profile. It now aligns with the full $R_o = 55$ curve very well within the Einstein radius, but not perfectly. (Note that the Einstein radius is also the convex hull in this case. Thus we do not expect to be able to constrain the solution outside.) Also note in the zoomed subplot, that the different profiles attain $\kappa = 1$ at nearly the same radius $R$, but not exactly, as they would if they had resulted from the transformation in Eq. \[masssheet\]. Thus, the degeneracy we probe by simply varying $R_o$ is very similar to but slightly different than the classic and simplest form of the “mass-sheet degeneracy”.
We note that these solutions all have flat $\kappa(R)$ slopes at the center, corresponding to a flat slope in $\rho(r)$, the three-dimensional density, as well. This slope gradually increases outward from the center. This is the same behavior seen in many CDM simulations [@Navarro04; @Merritt05; @Merritt06 and references therein, although see @Diemand05], which suggest that Dark Matter halos may attain the same [@Sersic68] density profiles that we observe in the light profiles of elliptical galaxies. Thus the LensPerfect mass profiles obtained when given only four image constraints do appear to be reasonable. But we stress that these solutions are far from unique, and many other mass profiles are possible.
Source Plane Shifts {#massstick}
-------------------
Theory tells us that given a single source or even multiple sources at the same redshift, the entire source plane can be shifted, and a new massmap solution can be found which does not affect the image positions. The degeneracy was originally named the “prismatic transformation” by [@Gorenstein88], who explained that this shift can be accomplished by adding an infinitely long and thin mass stick to the solution off to one side of the images. Of course such mass sticks are not very physical. Thus, we can overcome this degeneracy by simply finding that solution which does not resort to the addition of mass sticks! This is why we “know” that Einstein rings are produced by an on-axis alignment of the lens and source galaxies. The on-axis configuration requires a simple symmetric massmap, while any off-axis source position would require a less physical massmap, perhaps requiring the addition of a mass stick.
We say “perhaps” because LensPerfect is able to find solutions to such off-axis Einstein rings without resorting to mass sticks. (Fortunately LensPerfect has no mass-stick basis functions to work with.) One such solution is shown in Fig. \[ringoffset\]. This mass distribution, while more plausible than a mass stick, is nevertheless highly improbable. The proper amount of mass must be added off-axis in the correct configuration to refocus the light just enough toward the center of the lens where it then produces the Einstein Ring. It is very unlikely that such a precisely-tuned mass configuration would occur in Nature. Much more likely is the on-axis alignment, in which case any simple symmetric mass configuration will do. Our optimization procedure (§§ \[sourcepos\], \[optimization\]) quickly converges to the on-axis source position as being most likely, as off-axis solutions are penalized for being asymmetric.
But we should not be surprised to find modest source position shifts in our solution either (§ \[sourcepossec\]). The mass within the convex hull is extremely insensitive to these shifts. (A constant shift in the source plane does not affect the divergence of the deflection field at the multiple image positions. Thus the massmap within the enclosed region experiences only negligible changes.)
Constraining Image Shapes and Sizes {#knots}
-----------------------------------
Additional constraints may be derived from gravitationally lensed images by requiring the mass model to correctly reproduce not only image positions, but also their sizes, shapes, orientations, and fluxes. A method to constrain the fluxes (and shears) of unresolved images was discussed in § \[fluxes\]. But observed fluxes (and especially shears) may be uncertain. More precise constraints may be obtained when the multiple images are resolved and extra knots may be identified within them.
Fig. \[MS1358BC\] shows two of the four multiple images of a $z = 4.92$ galaxy produced by the galaxy cluster MS1358 [@Franx97]. Six knots are identified as common in both of these images (B & C), with three of the knots also being identified in image A (not shown). For each knot in turn, all of its images are constrained to originate from the same source position. The deflection model is updated after each knot is added. The final deflection solution yields the source plane images in the right-hand panel (see also Fig. \[MS1358BC2\]). Note the identical alignment of the two images (B & C). At this point, the de-lensed images may be co-added to obtain greater depth, if desired [e.g., @Colley96].
But this capability allows us to do more than simply produce pretty de-lensed images. The extra constraints provided by image knots can add significant detail to the massmap solution. Fig. \[MS1358k\] compares (left) the MS1358 massmap solution obtained when using only a single image position (the brightest knot) for each object and (right) the solution obtained when constraining all knot positions. In the latter, the mass is more stretched toward a second cluster galaxy which proves crucial to traditional parametric mass modeling.
Discussion
==========
We now describe the current limitations of the LensPerfect method and plans for future capabilities LensPerfect may provide us with. We also discuss the relative merits of both “parametric” and “non-parametric” methods, elaborating on points made in the introduction.
Imperfect Perfect Solutions
---------------------------
The “perfect” in LensPerfect refers to the reproduction of multiple image positions attained by its massmap solutions. But these solutions may be perfectly wrong and even unphysical if incorrect source positions and/or redshifts (and/or multiple image identifications) are provided. We have developed a method to find source positions and redshifts which produce reasonable solutions. We do not claim this method is perfect; it appears to work well but may be improved upon in the future.
LensPerfect massmap solutions do not guarantee against predictions of additional multiple images which don’t exist. This is an issue common to many methods, and it has been argued alternately that this should either more or less be a concern. The argument against its importance is that the prediction of additional multiple images is very sensitive to local substructure. Slight modifications might be made to this substructure, it is argued, to make these extra predicted multiple images disappear. LensPerfect may finally give us a tool capable of easily demonstrating this. We may be able to add deflection field constraints that tweak the massmap just enough to eliminate unwanted images. This idea has not yet been tested. In the meantime, we will simply check each solution for the correct number of multiple images. We find our models generally do not produce extra multiple images, as long as reasonable source positions are assumed.
Imperfect Solutions Superior? {#imperfect}
-----------------------------
Would it help to loosen the restriction that our massmap solutions perfectly reproduce all the multiple image positions? That is, might we obtain more accurate solutions by allowing our solutions to be imperfect? There are two parts to this question.
First we consider measurement uncertainties. We have already discussed how we deal with redshift uncertainties in § \[optimization\]. We allow the redshift to vary within the uncertainties as we optimize the massmap. The same could be done for other uncertain measurements. Image position uncertainties are generally very small (about one pixel). But when adding flux / shear constraints, we should certainly include the corresponding uncertainties in our optimization. [@EvansWitt03] and [@CongdonKeeton05] followed this approach in their studies of galaxy lenses. By including uncertainties, they noted modest improvement in their solutions.
The second part of the question is more fundamental. Could a less perfect massmap be more accurate? This does prove true in the SLAP method [@Diego05a]. They purposely leave some scatter in the source plane to avoid solutions which are “biased” with “a lot of substructure”. But this is likely a result of their formulation of the problem. They consider all pixels in each lensed image and find that massmap solution which minimizes the [*sizes*]{} of all the delensed images. Of course the delensed sizes should not be zero, so the delensed image pixels are allowed some “scatter”. (They make no attempt to match internal features as we demonstrate in § \[knots\].) Meanwhile, the proponents of PixeLens do not report such problems with their solutions which perfectly reproduce all image positions [@Saha06].
We performed a test to see what an imperfect solution would look like and if it might be more accurate. We began with our perfect solution to the 93 multiple images in § \[A1689model\] with optimized source positions. We then “reoptimized” the source positions (based on our physicality criteria), one-by-one [*for each individual image*]{}. Source positions were no longer constrained to a single point for each multiple image system. In practice, they drifted by an average of 0.4 pixels. Our resulting imperfect massmap was not extremely different from our perfect massmap but not any more accurate either. Some of the substructure features in the center were erased as the physicality measure guided the optimization toward a smoother solution. The radial profile did not significantly improve nor deteriorate in accuracy.
We believe the “1,000 points of light” massmap in Fig. \[kappa1000\] is a convincing demonstration that perfect is best. When the deflection field is sampled at 1,000 points, we are able to map fine detail in the massmap. This fine structure is encoded in the exact positions of the multiple images. Allowing for an imperfect solution would erase this fine structure and waste information obtained in the observations.
Weak Lensing, Extended Images, and Time Delays {#weaklensing}
----------------------------------------------
It may be possible to directly incorporate measurements of both “weak” lensing and not-so-weak shear into the LensPerfect method. Lacking multiple identifiable image knots, we can constrain galaxy shapes using the same method we use to constrain fluxes (§ \[fluxes\]). Instead of altering the relative sizes of the source and lensed regions, we can alter the shear. Thus we could constrain all delensed images to be round, or perhaps round with some scatter of ellipticity as measured in large scale surveys. We do not develop this idea further here. Such a technique would require three constraints per galaxy, which, given 100 galaxies, would require significant computing time (although at this stage iterations may not be required).
A more practical idea may be to compare measured shears to those predicted from our mass models and include the disagreement as a penalty in our optimization procedure. But note that this would only work well for galaxies within our convex hull as our solution (and thus shears) would be very poorly constrained outside this region.
Beyond simple constraints of flux, shear, and image knots, all of the pixel information in each lensed image may be utilized in the model derivation [e.g., @WarrenDye03; @Suyu06; @Koopmans06]. This is especially desirable in strong galaxy-quasar lensing, in which observable constraints are less plentiful than in cluster lensing. It is unclear how to implement this in LensPerfect, except as another penalty in our optimization routine.
Another constraint often used in strong galaxy-quasar lensing studies is time delays observed among different images. Time delays are not local functions of the deflection field, and as such, we cannot constrain them directly with LensPerfect as we constrain fluxes and shears. But for a given solution, they can be calculated directly from derivatives of our basis functions and compared to those observed, yielding another penalty for our optimization routine.
“Parametric” vs. “Non-parametric” Methods
-----------------------------------------
Methods are often classified as “parametric” or “non-parametric”, depending on whether a clear physical parameterization is used to construct the proposed massmap solutions. “Grid-based” methods are often referred to as “non-parametric”, even though strictly speaking they do have parameters, namely the mass at every pixel on a grid. The real distinction to be made here is between “model-based” and “model-free” methods. The former construct mass halos as physical analytical forms, while the latter do not. To give examples, the fully model-based strong lensing analyses of the A1689 ACS images published to date have been [@Halkola06; @Halkola07] and [@Limousin07]. Meanwhile, [@Diego05], [@Saha06], and [@Leonard07] have obtained model-free massmaps based on these images. The [@Broadhurst05] and [@Zekser06] analyses included both model-based and model-free elements.
LensPerfect, while clearly parametric, is also model-free. The LensPerfect solutions are given as sums of basis functions. But these basis functions have no physical interpretation. And this functional form is practically indiscernible in the final solutions. In fact, the basis function coefficients are generally many orders of magnitude greater than the amplitude of the massmap. These large mass components all cancel out nearly perfectly with the “residuals” being the massmap solution.
But while LensPerfect is model-free, it does not have the large number of free parameters typical of grid-based methods. In fact when fitting image positions only (including extra knots, but not fluxes), the number of free parameters solved for (the coefficients) is exactly equal to the number of constraints. Note that the source positions and redshifts are not solved for directly, but rather must be provided as input. Each flux measurement provides a single constraint but two free parameters are required to constrain it in our models. And a shear measurement provides two constraints, which can be reproduced with an equal number of (two) free parameters.
But which are superior overall, model-based or model-free methods? Individual researchers may have a decided preference for one over the other, but in fact both types of methods have their strengths and each serves a purpose.
Physical model-based massmaps allow us to test whether Dark Matter is distributed in certain ways, according to our assumptions. In principle, they allow for a more straightforward determination of meaningful physical parameters. Cluster mass models attempt to simultaneously constrain the forms of both the overall cluster halo and the individual galaxy halos [e.g., @Halkola07]. But degeneracies are strong between the two additive components.
On the other hand, model-free massmap reconstruction methods allow us to directly test for the presence of Dark Matter free of assumptions about its distribution. In particular, these methods make no assumptions about mass following light. Some of our most important discoveries about Dark Matter have come and are expected to come from cases in which mass does [*not*]{} follow light. Mass peaks offset from light peaks can provide constraints on the collisional nature of Dark Matter particles, as in the Bullet Cluster [@Markevitch04; @Clowe06]. And observations of Dark substructure (not associated with light) may someday vindicate CDM simulations which predict much more halo substructure than is visible [@Diemand07; @Strigari07 and references therein].
Model-free methods are also able to explore a wider range of possible massmap solutions, including (with the advent of LensPerfect) those which perfectly reproduce all 100+ multiple image positions. Of course, exploring the full range of model-free solutions can be a computational challenge. And it is not entirely clear how to sort the physical solutions from those which are “less” physical or aphysical.
Meanwhile, too much model flexibility has been cited a potential problem, as a flexible model may fit incorrect data without any alarms sounding. [@Limousin07] claim that their model-based method and that of [@Halkola06] were unable to fit some multiple image systems incorrectly identified in the initial [@Broadhurst05] work, which was a bit more model-free. But LensPerfect, while more flexible still, may actually be more unforgiving than all previous methods when it comes to incorrect multiple image identifications. The reason is simple. “Imperfect” massmap reconstructions experience some offset in each and every predicted image position. Thus a misidentified multiple image set may have a larger $\chi^2$ than average, but this may be more easily dismissed in the analysis. In LensPerfect, however, each multiple image puts a rigid constraint on the deflection field. Thus a misidentified multiple image is more likely to cause the deflection field to get tangled, leading to a “less physical” (if not aphysical) solution. We stay alert for such ill-fitting multiple image systems as we add each to our models. And, of course, we must take care to avoid misidentifying multiple images from the start whenever possible.
Another objection to model flexibility was raised by [@Kochanek04_review]. He takes exception to the ability of the model-based but flexible [@EvansWitt03] method to produce a mass model that accounts for the flux anomaly observed in the strong galaxy-quasar lens Q2237+0305 even though this flux anomaly has since been shown to have been due to a microlensing event, which has now passed! We would argue that the mass model proposed by [@EvansWitt03] is but one possible solution among several to the observed flux anomaly. All possible solutions should be considered, and in this case, microlensing is proven to be the true solution.
In analyses of galaxy-quasar lensing, model-based methods enjoy even greater appeal than in cluster lensing. Perfect solutions are much easier to come by when there are fewer constraints (4 image positions, for example). And with so few image constraints, a large range of solutions is possible. We demonstrated one way of probing this solution space in § \[Einstein\], and the PixeLens method provides another. But many choose to slash the solution space by making well-motivated model-based assumptions (e.g., an isothermal profile).
A powerful technique would be to combine LensPerfect with a model-based method. We could find good (imperfect) model-based solutions which leave offsets between the observed and predicted image positions, and then perfect these solutions with LensPerfect. The deflection offsets may be interpolated, and this “offset solution” added to the imperfect solution to create a perfect solution. We implemented this idea, but the results appeared unruly. A more creative approach will be required if this capability is to be realized.
We mention one other advantage to our method. That advantage is ease of use. If a simple mass model is required, LensPerfect’s speed is hard to beat. Given a single lensed galaxy, LensPerfect instantly obtains a perfect solution “out of the box”. And if instead given 30+ lensed galaxies as in Abell 1689, LensPerfect still obtains solutions with minimal user input. Parametric methods instead require the user to develop complex models capable of fitting so many multiple images well. Previous studies identified and measured properties of many cluster galaxies in order to model a “galaxy component” which would be added to a separate halo component in their solutions. LensPerfect instead bypasses this parameterization process obtaining a detailed massmap free of strong assumptions.
Summary and Future Work {#summary}
=======================
We have presented a new approach to gravitational lens massmap reconstruction. Given image positions, source positions, and redshifts, a new mathematical technique is used to interpolate the deflection field via direct inversion. The resulting massmap (simply half the divergence of the deflection field) perfectly reproduces all of the observed image positions.
In practice, source positions are unknown. We have devised a method that efficiently optimizes over different possible configurations of the source galaxies. Each configuration produces a different massmap solution which is evaluated for “physicality” based on criteria developed as part of this work. Our criteria make only minimal assumptions about the form of the massmap. Specifically, they make no assumptions about the slope of the radial profile nor mass following light.
We demonstrated our method on mock gravitational lensing data. A known massmap was used to lens 19 mock galaxies to produce 93 multiple images. Our massmap solution perfectly reproduces all the multiple image positions while accurately recovering the known lens mass distribution to a resolution limited by the density of the multiple images. We demonstrate the improved accuracy and fine detail we may expect from a massmap derived from 1,000 multiple images.
We also presented LensPerfect solutions based on far fewer constraints, such as four multiple images of a single galaxy. Image fluxes or extra knots may provide additional constraints which can also be perfectly fit by our models. The number of free parameters solved for is kept nearly equal to the number of observable constraints.
The LensPerfect software is easy to use and is made publicly available at our website (see § \[intro\]).
In subsequent papers, we will apply this method to Abell 1689 and other galaxy clusters. We will also attempt to resolve some of the “flux anomalies” observed in galaxy lenses.
In § \[discussion\], we discussed some possible improvements and extensions of our method, including the incorporation of weak lensing data. We also hope to better quantify the accuracy of our massmap recovery and compare our performance to that of other methods. The uncertanities in our solutions should be well determined by exploring the full range of physical solutions. We will also experiment with other radial basis functions and other recipes for measuring massmap physicality.
We would like to thank many for useful discussions during the development of LensPerfect including Justin Read, Rick White, Marijn Franx, Piero Rosati, Myungkook Jee, Aleksi Halkola, Stella Seitz, Ralf Bender, Jean-Paul Kneib, Bernard Fort, Genevieve Soucail, Tony Tyson, Chris Fassnacht, Leonidas Moustakas, and Maruša Bradač. We thank Arjen van der Wel for comments that helped improve the manuscript. And we especially thank our anonymous referee for a thorough reading of the manuscript and useful scientific contributions. This research is supported by the European Commission Marie Curie International Reintegration Grant 017288-BPZ and the PNAYA grant AYA2005-09413-C02.
Massmap Physicality Penalty Function {#penalty}
====================================
To calculate our penalty function, we first evaluate the massmap solution on a $41 \times 41$ grid. This proves large enough to measure physicality, while taking less than a second to calculate given as many as 50 image positions. (As this evaluation is to be part of our iterative procedure to determine source positions, it should be kept as quick as possible.) Based on this $41 \times 41$ massmap, we calculate the total penalty as follows:
1. The sum of all negative pixels inside the convex hull is multiplied by $-100$. (We could simply assign a penalty of infinity to negative massmaps, but this would create a large region of constant penalty in source coordinate space. Our finite and varying penalty serves better during the optimization to corral the source positions back toward those which yield positive solutions.)
2. The mean mass $\langle \kappa \rangle$ and RMS scatter $\Delta \kappa$ are measured in radial bins of 80 points each. The RMS scatter is totaled and divided by 10. Within the convex hull, the mean and scatter are recalculated in radial bins of 40 points each. This inner scatter is totaled and multiplied by 4. Outside the convex hull, we rebin the massmap, yet again, in bins of 80 points each. Rather than the RMS mass scatter, this time we penalize the peak-to-peak variation (that is, the maximum minus the minimum) in each bin. These variations are totaled and multiplied by 5 and divided by the number of bins.
3. Any outward increase in $\langle \kappa \rangle$ over $R$ is measured and multiplied by 10. This is calculated both for all points and then again for those within the convex hull. Both penalties are added.
4. Given the mean mass $\langle \kappa \rangle$ and RMS scatter $\Delta \kappa$ calculated within the convex hull (see 2 above), we interpolate the 1-$\sigma$ lower limit $\langle \kappa \rangle$ - $\Delta \kappa$ for all points within the convex hull. Deviation below this limit is measured and divided by 2. Note that we certainly expect some points to fall below the 1-$\sigma$ lower bound. (By definition, 16% of the points should fall below.) But the idea is to minimize these deviations. A large deviation below indicates a “tunnel” or unphysical dip in the surface density.
5. Finally, we put a premium on smooth solutions within the convex hull as being the most likely. Numerical derivatives of the massmap are calculated at each pixel ($x$, $y$): $d\kappa / dx = \vert \kappa(x+1, y) - \kappa(x-1, y) \vert$, $d\kappa / dy = \vert \kappa(x, y+1) - \kappa(x, y-1) \vert$ The absolute values of these are totaled the entire sum and divided by 5.
The above penalties and their respective weights were defined after much trial and error. We find that massmaps with lowest total penalty defined as above appear to be the best behaved, or most physical. Again, other weights and penalty schemes are certainly possible.
[^1]: http://www.its.caltech.edu/%7Ecoe/LensPerfect/
[^2]: An $m \times m$ matrix-valued function $\phi$ is [*positive definite*]{} on $\mathbb{R}^n$ if given any finite, distinct set of points $X := \{ x_1, \dots, x_N \} \subset \mathbb{R}^n$ we have $\displaystyle \sum_{j,k} \alpha^T_j \phi(x_j - x_k) \alpha_k \geq 0$ for all $\alpha_1, \dots, \alpha_N$ in $\mathbb{R}^m$.
[^3]: In principle, a Gaussian could be used in place of $W_{7,3}$. However it is well known (to mathematicians) that the interpolation matrix is ill-conditioned when a Gaussian is used. Further, it is easier for a computer to evaluate a polynomial than a Gaussian.
[^4]: More elaborate routines are available, but these may actually be less efficient. Specifically, calculating gradients of the penalty in source position space would require two extra and time-consuming function evaluations for every iteration.
[^5]: http://www.scipy.org
[^6]: Of course we can also calculate the magnification using Eq. \[magnif\].
[^7]: Redshift uncertainties vary greatly from one study to the next. Thus rather than attempting to present a test which includes “typical” redshift uncertainties, we propose that such tests be performed on a case-by-case basis. We note that we have had success in modeling real-life data such as the actual Abell 1689 multiple images complete with their redshift uncertainties (Coe et al., in prep.).
[^8]: As noted by [@SahaWilliams06], the term “mass-sheet” degeneracy may lead to the mistaken belief that a constant mass sheet is what may be added without affecting the images. In fact the mass must also be rescaled, or multiplied, in concert.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We report the detection of a very red source coincident with the gravitational lens 1938+66.6 (Patnaik et al. 1992) in $K'$ ($2.12
\um$), $H$ ($1.6 \um$), $J$ ($1.25 \um$), and Thuan-Gunn $r$ ($0.65
\um$) bands. 1938+66.6 has previously been detected as a partial radio ring indicating lensing. We find $K^\prime=17.1 \pm 0.1$ and $r = 23.9
\pm 0.2$, making it a very red source with $(r-K^\prime)=6.8 \pm 0.25$. We also observed in Thuan-Gunn $g$ band ($0.49 \um$) and found $g>24.5$ at the 90% confidence level. We interpret our observations as a reddened gravitational lens on the basis of its optical-IR color and positional coincidence with the radio source.
author:
- 'James E. Rhoads, Sangeeta Malhotra, & Tomislav Kundić'
title: 'Infrared Counterpart of the Gravitational Lens 1938+66.6'
---
\#1\#2\#3[${#1}^{+#2}_{-#3}$]{} journalname[Astronomical Journal]{} \#1[accptdate[\#1]{}]{} \#1\#2[jourvol[\#1]{}jourdate[\#2]{}]{} \#1\#2[startpage[\#1]{}finishpage[\#2]{}]{}
Introduction
============
There is increasing evidence that quasars and gravitationally lensed quasars sometimes have very red optical-infrared colors. Reddening by intervening dust offers an explanation for this phenomenon (Webster et al. 1995, Lawrence et al. 1995, Larkin et al. 1994). Such large reddenings raise questions about possible incompleteness of optical quasar and lens surveys. This in turn can have important implications for cosmological models, since many constraints on cosmological parameters are derived from the luminosity functions of quasars and from the numbers and seperations of lensed quasars (Turner, Ostriker & Gott 84, Fukugita and Peebles 1993, Fukugita & Turner 1991, Malhotra & Turner 1995, Kochanek 1993, Maoz & Rix 1993). The reddening of gravitationally lensed quasars is also a potential probe of the ISM of the lensing galaxies at high redshift.
One would expect the radio selected gravitational lenses to be free of bias against reddened lenses. Also small seperation lenses, where the optical paths are likely to pass near the center of a single lensing galaxy, are more likely to show substantial amount of reddening.
The radio source 1938+66.6 is one of the small seperation gravitational lenses discovered in the VLA/MERLIN survey of flat spectrum radio sources (Patnaik et al 1992). It has four compact sources and an arc; the maximum seperation between the any of the components is $0''.95$. Patnaik et al report a r=23 object at the location of the lens. We observed this source in the near infrared $K^{\prime}$ ($2.12 \um$), $H$ ($1.6 \um$), and $J$ ($1.25 \um$) bands, and in the optical Thuan-Gunn $r$ ($0.65 \um$) and $g$ ($0.49
\um$) bands. An object was detected at the position of the lens in all bands except $g$. This optical-infrared object will hereafter be referred to as “1938+666(OIR)”; we will argue that it is a counterpart of the radio system.
The observations and data analysis are described in section 2. In section 3 we estimate the probability of the infrared source being a cool star or a high redshift galaxy on the basis of color. Discussion and posssible interpretation of the results are presented in section 4.
Observations
============
We observed 1938+666 in $H$ band on 2 July 1995 (UT), in $J$ and $K'$ bands on 6 August 1995 (UT), and in Thuan-Gunn $g$ and $r$ bands on 4 and 24 July and 5 August 1995 (UT). There was intermittent scattered cloud cover early on the night of 2 July, but the sky cleared shortly before the 1938+666 $H$ band observations. All data were taken with the Apache Point Observatory[^1] 3.5 meter telescope. The near-IR observations used the GRIM II camera, while the $g$ and $r$ band observations used the Double Imaging Spectrograph (DIS) in imaging mode. Seeing was about $1.6''$–$1.8''$ FWHM in all bands. Some parameters of the observations are summarized in table 1.
All data reduction was done using the IRAF package and followed standard procedures. For the near-IR data, individual short exposures were sky subtracted, typically using the 2–4 sky frames nearest in time. A sky flat was made by taking the median of all disregistered exposures in a band, subtracting a mean bias frame, and normalizing the result by its median pixel value. The sky subtracted image frames were then divided by this flat. Finally, the images were registered (using the centroid of a bright star to determine fractional pixel shifts) and a combined image was produced by taking the median of all processed single frames at each pixel. For the $g$ and $r$ bands processing was similar but no sky subtraction step was needed. Final images are shown in figure 1.
On the nights 2 July, 4 July, and 6 August we also observed photometric standards to calibrate our fluxes. For the $H$ band, we used the UKIRT faint standard FS 28; for $J$ and $K'$ bands we used FS 7 (Casali & Hawarden 1992). For the $g$ and $r$ bands, we used the fields Mark A (Jorgensen 1994, Landolt 1992) and F1038-6 (Jorgensen 1994, Stobie et al 1985).
Data Analysis
-------------
We determined the position of the radio source using three HST guide stars that fell within the r band image (epoch 2000.0 coordinates 19 38 14.23 +66 50 23.99, 19 38 45.36 66 48 40.38, and 19 38 01.38 +66 51 14.11). We first used the guide stars to determine the position and orientation of the coordinate grid, and then determined the position of the quasar relative to each guide star. The scatter in the resulting position estimates was $0.7''$ in RA and $0.4''$ in declination.
In the $r$, $J$, $H$, & $K'$ bands, the counterpart to 1938+666 is apparent, and we have done conventional aperture photometry to measure its fluxes. In the $g$ and $r$ bands we have applied additional tests to see if an object is detected and to place upper limits on the counterpart flux. Our results are reported in Table 1.
We measured aperture magnitudes with radii $\sim 0.67 a$ where $a$ is the FWHM (full width at half maximum) of the psf (point spread function) and the coefficient $0.67$ maximizes the signal to noise for aperture photometry of a faint source with a Gaussian psf. To see if the resulting $g$ and $r$ fluxes are significantly above background, we measure fluxes in randomly placed apertures and determine how frequently the flux exceeds that seen for 1938+666. The result is not very sensitive to aperture size for $1.5'' \la a \la
2.0''$. In $g$ band, $89\%$ of randomly placed $1.5''$ apertures had fluxes exceeding that for 1938+666, and we conclude there is no evidence of a $g$ band counterpart to the radio source. In $r$ band, $18\%$ of randomly placed $1.5''$ apertures had fluxes exceeding that for 1938+666. Visual inspection of the final $r$ band image shows a source at the location of the NIR source, and we believe that the weak result of the random aperture test is due to bleeding columns from saturated bright stars and to the substantial density of 23rd magnitude objects on the sky.
By modelling the counts in an aperture of area $A$ pixels as the object flux plus sky noise, we may place an upper limit on the total flux for a point source (modelled as a Gaussian psf) at the H band source location. At confidence levels of $(90\%, 99\%,$ and $99.9\%)$ we find $g > (24.5, 24.0,$ and $23.7) \mag$. Applying the same formalism to the weakly detected $r$ band source gives $r > (23.75, 23.6,$ and $23.5) \mag$.
A source substantially larger than the psf could be brighter than the total magnitude limits quoted above. Additionally, the spectral slope of the putative candidate will influence the $r$ band limit, since our standard star observations suggested a substantial color term ($-0.15
(g-r)$) in the transformation from instrumental to standard $r$ magnitude. No important color term was apparent in $g$ band. We were not able to observe enough NIR standards to allow secure color term corrections in $J$, $H$, and $K'$ bands.
Interpretation
==============
To test the interpretation of 1938+666(OIR) as the counterpart of the radio source 1938+666, we compared its measured colors to those of galactic stars, normal galaxies, quasars, and lensed systems. We first account for Galactic foreground dust. Reddening towards 1938+666 can be estimated at $0.10 \le E(B-V) \le 0.12 \mag$, based on the reddening map of Burstein & Heiles (1982). The normal galactic extinction law (with $R_V \equiv A(V)/E(B-V) = 3.1$) yields $E(r-K) \approx 2.25 E(B-V)$ (cf. Mathis 1990). Thus, Galactic dust does not contribute strongly to the extremely red color of 1938+666(OIR).
Our data indicate $(r-H) = 6.3 \pm 0.25 \mag$, $(r-K') = 6.8 \pm 0.25 \mag$, $(J-H) = 2.1$, and $(H-K') = 0.5$.
For comparison, data in Bessell (1991) plus the color transformations of Jorgensen (1994) give $ (r-H) \approx 6$ for the M6.5 dwarf LHS 3003, and $ (r-H) \approx 6.8$ for an M7.5 dwarf. Thus, only the very coolest stars have colors as red as 1938+666.
Two further tests may be applied to the hypothesis that the object is a very cool star. First, we can see if 1938+666(OIR) is appreciably broader than a point source. In the present $H$ band data, the FWHM of 1938+666(OIR) is $\sim 2.4''$, while a representative stellar FWHM is $\sim 1.8''$. This would be consistent with a source of size $\sim
1''$, but better seeing and a larger signal to noise ratio are required before this test can be considered conclusive. Second, we can use Wainscoat et al’s (1992) model of the near-IR sky to calculate the number density of “M late V” dwarfs with $18 < H < 19$ towards 1938+666; the resulting density is 0.25/arcmin$^2$, giving a probability $\sim 2 \times 10^{-3}$ of finding a suitably red M dwarf within 3 arcseconds of the radio source.
We can also compare the observed color limit to the colors of normal galaxies. The reddest local elliptical galaxies have $(r - H)$ colors around $2.75$ (Persson, Frogel, & Aaronson 1979), while moderate redshift ellipticals ($z$ up to $0.92$) show $(r-H) \le 4$ (Persson 1988). By redshift $z \approx 1.2$, elliptical galaxies can have $R-K$ up to $6$ (Dickinson 1995), and the central component of the $z = 2.016$ radio galaxy 0156$-$252 has $r - K = 7.3$ (McCarthy, Persson, & West 1992), so the color of 1938+666(OIR) is consistent with a lensing galaxy or a lensed radio galaxy at $z > 1$. Spiral galaxy $(r-H)$ colors are not widely published, but an examination of de Jong’s work suggests that $2.5 \la (r-H) \la 3$ is typical while occasional exceptions (e.g. UGC 4256) might be a magnitude redder. (de Jong, 1995).
Taken together, the close positional coincidence and unusually red color are strong circumstantial evidence that 1938+666(OIR) is indeed the gravitational lens system observed in the radio by Patnaik et al. (1992). This interpretation might be confirmed by obtaining an image in sub-arcsecond seeing and comparing the near-IR and radio morphologies; or (preferably) by obtaining a spectrum, and measuring a redshift if the object is indeed a quasar. Either of these tests might determine if the observed infrared source is the lensed object or a high redshift lensing galaxy.
The optical-IR colors of many lensed quasars are found to be as red or redder than observed for 1938+666(OIR) (Lawrence et al. 1994, Larkin et al. 1994, Annis & Luppino 1993, Annis 1992). The most extreme red colors are seen for MG 0414+0534: $(r-H)=7.8$, and for MG1131 +0456: $(J-K) > 4.1$. To compare the reddening directly one needs the redshift of the lens and the dust responsible for the reddening. Optical-IR colors of radio selected quasars can also be quite red. Webster et al. 1995 report $B_J - K \approx 2-7$ for the confirmed quasars among the flat-spectrum radio sources observed with the Parkes telescope, and up to $B_J - K
\approx 8$ for the sources for whom the redshift is not known. The red color of 1938+666(OIR) support its identification as a counterpart to the gravitationally lensed radio ring.
Conclusions
===========
We have detected an infrared object at the location of the radio-detected gravitational lens 1938+666, with very red colors ($r-K^{\prime}=6.7 \mag$). The most natural explanation for these observations is that the we are seeing the lensed object or a lensing galaxy at large redshift ($z \ga 1$). The close positional coincidence observed is quite unlikely to happen for random field stars or galaxies, with the exception of the lensing galaxy. Moreover, empirical measurements show that $r-K^{\prime}$ colors of most stars and low to moderate redshift field galaxies are bluer than our observed limit. On the other hand, a moderate fraction of flat-spectrum radio sources and gravitationally lensed quasars are similarly red. Our interpretation could be tested by imaging 1938+666 with higher spatial resolution, or taking a spectrum of this object.
We wish to thank Edwin L. Turner, James E. Gunn, Michael Strauss, and Robert H. Lupton for useful discussions. This work has been supported in part by NSF Grant AST94-19400.
Annis, J. 1992, 391, 17 Annis, J. & Luppino, G. A. 1993, 407, 69 Bessell, M. S. 1991, 101, 662 Burstein, D., & Heiles, C. 1982, 87, 1165 Casali, M. & Hawarden, T. 1992, JCMT-UKIRT Newsletter 4, 33 de Jong, R. S. 1995, thesis Dickinson, M. 1995, in “Fresh Views on Elliptical Galaxies” Fukugita, M., & Peebles, P. J. E. 1993, Institute for Advanced Study preprint number IASSNS-AST 93/24 Fukugita, M., & Turner, E. L. 1991, 253, 99. Jorgensen, I. 1994, 106, 967 Kochanek, C. S. 1993, 419, 12 Landolt 1992, 104, 340 Larkin, J. E.; Matthews, K.; Lawrence, C. R.; Graham, J. R.; Harrison, W.; Jernigan, G.; Lin, S.; Nelson, J.; Neugebauer, G.; Smith, G 1994 , 420, l9. Lawrence, C.R., Elston, R., Januzzi, B.T., Turner, E.L. 1995, preprint Malhotra, S. & Turner, E. L. 1995, 445, 552 Maoz, D., Rix, H-W, 1993, 416, 425 Mathis, J. S. 1990, 28, 37 McCarthy, P. J., Persson, S. E., & West, S. C. 1992, ApJ, 386, 52 Patnaik, A., Browne, I., King, L., Muxlow, T., Walsh, D., & Wilkinson, P. 1993, in “Sub-Arcsecond Radio Astronomy,” edited by R. Booth & R. J. Davis (Cambridge: Cambridge University Press), 137 Persson, S. E., Frogel, J. A., & Aaronson, M. 1979, 39, 61 Persson, S. E. 1988, in “Towards Understanding Galaxies at Large Redshifts,” eds. R.G. Kron & A. Renzini, 251 Stobie, R. S. et al 1985, A&A Supp, 60, 503 Turner, E. L., Ostriker, J. P., & Gott, J. R. 1984, 284, 1 Wainscoat, R. J. et al, 1992 83, 111 Webster, R.L., Francis, P.J., Peterson, B.A., Drinkwater, M.J., Mascl, F.J. 1995, Nature, 375, 469
[lccccc]{} Filter & central wave- & bandpass & On source & Individual & Magnitude\
&length ($\micron$) & FWHM ($\micron$) &integration time& exposure time&\
$K'$ & 2.12 & 0.35 & 12.3 min & 9 s & $17.1 \pm 0.1$\
$H$ & 1.65 & 0.30 & 20 min & 25 s & $17.6 \pm 0.1$\
$J$ & 1.25 & 0.30 & 12.1 min & 25 s & $19.7 \pm .13$\
$r$ & 0.655 & 0.09 & 48 min & 3, 5 min & $23.9 \pm 0.22$\
$g$ & 0.493 & 0.07 & 48 min & 3, 5 min & $>24.3 (95\%)$\
[^1]: APO is privately owned and operated by the Astrophysical Research Consortium (ARC), consisting of the University of Chicago, Institute for Advanced Study, Johns Hopkins University, New Mexico State University, Princeton University, University of Washington, and Washington State University.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
A long-standing topic of interest in the general theory of relativity is the embedding of curved spacetimes in higher-dimensional flat spacetimes. The main purpose of this paper is to show that the embedding theory can account for the accelerated expansion of the Universe and thereby serve as a model for dark energy. This result is consistent with earlier findings based on noncommutative geometry. A secondary objective is to show that the embedding theory also implies that it is possible, at least in principle, for the accelerated expansion to reverse to become a deceleration.\
author:
- |
Peter K.F. Kuhfittig\* and Vance D. Gladney\*\
[^1] Department of Mathematics, Milwaukee School of Engineering,\
Milwaukee, Wisconsin 53202-3109, USA
title: '[A model for dark energy based on the theory of embedding]{}'
---
**PAC numbers:** 04.20.Jb, 04.20.-q, 04.50.+h\
**Keywords:** dark energy, embedding\
Introduction
============
Embedding theorems have a long history in the general theory of relativity and continue to be a topic of interest since they are able to provide useful connections between the classical theory and higher-dimensional spacetimes. For example, an $n$-dimensional Riemannian space is said to be of embedding class $m$ if $n+m$ is the lowest dimension of the flat space in which the given space can be embedded. In this paper we begin with a generic line element of embedding class two in a cosmological setting. This line element is reduced to class one by a suitable transformation. Our main objective is to show that the resulting model is able to account for the accelerated expansion of the Universe and thereby serve as a model for dark energy. It is also shown that our conclusion is consistent with earlier results based on noncommutative geometry.
A secondary objective is to show that a change in the nature of the embedding space could result in a reversal of the accelerated expansion.
The first embedding {#S:embedding}
===================
The discussion in Ref. [@MG17] begins with the static and spherically symmetric line element $$ds^{2}=e^{\nu(r)}dt^{2}-e^{\lambda(r)}dr^{2}
-r^{2}\left(d\theta^{2}+\sin^{2}\theta \,d\phi^{2}
\right),$$ using units in which $c=G=1$. It is shown in Ref. [@MG17] that this metric of class two can be reduced to a metric of class one by a suitable transformation and can therefore be embedded in the five-dimensional flat spacetime $$ds^{2}=-\left(dz^1\right)^2-\left(dz^2\right)^2
-\left(dz^3\right)^2-\left(dz^4\right)^2
+\left(dz^5\right)^2.$$
In our situation it is more convenient to employ the opposite signature in the above line element: $$\label{E:line1}
ds^{2}=-e^{\nu(r)}dt^{2}+e^{\lambda(r)}dr^{2}
+r^{2}\left(d\theta^{2}+\sin^{2}\theta \,d\phi^{2}
\right).$$ Similarly, the five-dimensional flat spacetime is now written as follows: $$\label{E:line2}
ds^{2}=-\left(dz^1\right)^2+\left(dz^2\right)^2
+\left(dz^3\right)^2+\left(dz^4\right)^2
+\left(dz^5\right)^2.$$ This reduction is accomplished by the following transformation: $z^1=\sqrt{K}\,e^{\frac{\nu}{2}}
\,\text{cosh}{\frac{t}{\sqrt{K}}}$, $z^2=\sqrt{K}
\,e^{\frac{\nu}{2}}\,\text{sinh}{\frac{t}{\sqrt{K}}}$, $z^3=r\,\text{sin}\,\theta\,\text{cos}\,\phi$, $z^4=
r\,\text{sin}\,\theta\,\text{sin}\,\phi$, and $z^5=r\,\text{cos}\,\theta$. The differentials of these components are
$$dz^1=\sqrt{K}\,e^{\frac{\nu}{2}}\,\frac{\nu'}{2}\,
\text{cosh}{\frac{t}{\sqrt{K}}}\,dr + e^{\frac{\nu}{2}}\,
\text{sinh}{\frac{t}{\sqrt{K}}}\,dt,$$
$$dz^2=\sqrt{K}\,e^{\frac{\nu}{2}}\,\frac{\nu'}{2}\,
\text{sinh}{\frac{t}{\sqrt{K}}}\,dr + e^{\frac{\nu}{2}}\,
\text{cosh}{\frac{t}{\sqrt{K}}}\,dt,$$
$$dz^3=dr\,\text{sin}\,\theta\,\text{cos}\,\phi + r\,
\text{cos}\,\theta\,\text{cos}\,\phi\,
d\theta\,-r\,\text{sin}\,\theta\,\text{sin}\,\phi\,d\phi,$$
$$dz^4=dr\,\text{sin}\,\theta\,\text{sin}\,\phi + r\,
\text{cos}\,\theta\,\text{sin}\,\phi\,
d\theta\,+r\,\text{sin}\,\theta\,\text{cos}\,\phi\,d\phi,$$
and $$dz^5=dr\,\text{cos}\,\theta\, - r\,\text{sin}\,\theta\,d\theta,$$ where the prime denotes differentiation with respect to the radial coordinate $r$. To facilitate the substitution into Eq. (\[E:line2\]), we first obtain the expressions for $-\left(dz^1\right)^2+\left(dz^2\right)^2$ and for $\left(dz^3\right)^2+\left(dz^4\right)^2
+\left(dz^5\right)^2$: $$\label{E:partial1}
-\left(dz^1\right)^2+\left(dz^2\right)^2=
-e^{\nu}dt^{2}+\left(\,1+\frac{1}{4}K\,e^{\nu}\,
(\nu')^2\,\right)\,dr^{2}$$ and $$\label{E:partial2}
\left(dz^3\right)^2+\left(dz^4\right)^2
+\left(dz^5\right)^2=dr^2+r^{2}\left(d\theta^{2}
+\sin^{2}\theta\, d\phi^{2} \right).$$ Substituting Eqs. (\[E:partial1\]) and (\[E:partial2\]) into Eq. (\[E:line2\]), we get $$\label{E:line3}
ds^{2}=-e^{\nu}dt^{2}+\left(\,1+\frac{1}{4}K\,e^{\nu}\,
(\nu')^2\,\right)\,dr^{2}+r^{2}\left(d\theta^{2}
+\sin^{2}\theta\, d\phi^{2} \right).$$ Metric (\[E:line3\]) is therefore equivalent to metric (\[E:line1\]) if $$\label{E:lambda1}
e^{\lambda}=1+\frac{1}{4}K\,e^{\nu}\,(\nu')^2,$$ where $K>0$ is a free parameter. The condition is equivalent to the following condition due to Karmarkar [@kK48]: $$R_{1414}=\frac{R_{1212}R_{3434}-R_{1224}R_{1334}}
{R_{2323}},\quad R_{2323}\neq 0.$$ (See Ref. [@pB16] for further details.) Other useful references are [@MM17; @MRG17; @sM17; @MGRD; @MDRK].
The problem
===========
It is well known that Alexander Friedmann proposed in 1922 that our Universe cannot be static, i.e., it must be either expanding or contracting. So if $a(t)$ is the scale factor in the FLRW model, then one of the Friedmann equations is $$\label{E:Friedmann}
\frac{\overset{..}{a}(t)}{a(t)}=
-\frac{4\pi}{3}(\rho +3p),$$ again using units in which $c=G=1$. Since we are assuming that the entire Universe can be embedded in a five-dimensional flat spacetime, we necessarily find ourselves in a cosmological setting. So any point can be used as the origin, since our Universe is a 3-sphere, having neither a center nor an edge. A convenient way to proceed is to use the following generic line element: $$\label{E:line4}
ds^{2}=-e^{\nu(r)}dt^{2}+e^{\lambda(r)}dr^{2}
+r^{2}\left(d\theta^{2}+\sin^{2}\theta \,d\phi^{2}
\right).$$ We also need to assume that the spacetime is asymptotically flat, i.e., $e^{\nu(r)}\
\rightarrow 1$ and $e^{\lambda(r)}
\rightarrow 1$ as $r\rightarrow\infty$.
Finally, let us list the first two of the Einstein field equations [@pK17]: $$\label{E:E1}
8\pi\rho=e^{-\lambda}\left(\frac
{\lambda'}{r}-\frac{1}{r^2}\right)+
\frac{1}{r^2}$$ and $$\label{E:E2}
8\pi p=e^{-\lambda}\left(\frac
{1}{r^2}+\frac{\nu'}{r}\right)-
\frac{1}{r^2}.$$
Our immediate goal is to combine these ideas to show that $\overset{..}{a}(t)$ in Eq. (\[E:Friedmann\]) is positive. That is the topic of the next section.
The solution
============
From Eq. (\[E:lambda1\]) we obtain $$\lambda'=\frac{1}{1+\frac{1}{4}K\,e^{\nu}\,(\nu')^2}
\frac{1}{4}Ke^{\nu}[(\nu')^2+2\nu'\nu''].$$ Then it follows from Eqs. (\[E:E1\]) and (\[E:E2\]) that $$\begin{gathered}
\label{E:F1}
8\pi (\rho +3p)=e^{-\lambda}\left[
\frac{1/r}{1+\frac{1}{4}K\,e^{\nu}\,(\nu')^2}
\frac{1}{4}Ke^{\nu}[(\nu')^3+2\nu'\nu'']
-\frac{1}{r^2}\right]\\+\frac{1}{r^2}
+3e^{-\lambda}\left(\frac{1}{r^2}+
\frac{\nu'}{r}\right)-\frac{3}{r^2}.\end{gathered}$$ Finally, we observe that in view of Eq. (\[E:lambda1\]), Eq. (\[E:F1\]) can also be written as $$\begin{gathered}
\label{E:F2}
8\pi (\rho +3p)=-\frac{2}{r^2}+
\frac{2e^{-\lambda}}{r^2}+e^{-\lambda}
\left[\frac{1/r}{1+\frac{1}{4}K\,
e^{\nu}\,(\nu')^2}\frac{1}{4}Ke^{\nu}
[(\nu')^3+2\nu'\nu'']+\frac{3\nu'}{r}
\right]\\=-\frac{2}{r^2}
\frac{\frac{1}{4}Ke^{\nu}(\nu')^2}
{1+\frac{1}{4}K\,e^{\nu}\,(\nu')^2}
+\frac{1}{1+\frac{1}{4}K\,e^{\nu}\,(\nu')^2}
\left[\frac{1/r}{1+\frac{1}{4}K\,
e^{\nu}\,(\nu')^2}\frac{1}{4}Ke^{\nu}
[(\nu')^3+2\nu'\nu'']+\frac{3\nu'}{r}
\right].\end{gathered}$$ Given that $K$ is a free parameter, $K$ can assume any positive value. In fact, according to Refs. [@MDRK; @pK18], $K$ can be extremely large. So for a sufficiently large $K$, the second term on the right-hand side of Eq. (\[E:F2\]) becomes negligible, leading to $8\pi(\rho +3p)<0$. This shows that $\overset{..}{a}(t)$ in the Friedmann equation is positive, indicating an accelerated expansion.
The second embedding
====================
The idea of an extra spatial dimension had its origin in the Kaluza-Klein theory and was continued in the induced-matter theory in Ref. [@WP92]. One would normally assume that the extra dimension has to be spacelike, resulting in the signature $-++++$ in Eq. (\[E:line2\]). It is proposed in Ref. [@WP92], however, that the signature $--+++$ is in principle allowed, thereby yielding two timelike components. In other words, the line element $$ds^{2}=-e^{2\nu_1(r)}dt_1^{2}-e^{2\nu_2(r)}dt_2^{2}
+e^{2\lambda(r)}dr^{2}
+r^{2}\left(d\theta^{2}+\sin^{2}\theta \,d\phi^{2}
\right)$$ would then be consistent with Einstein’s theory. To show that this is indeed the case, we note that any calculation can be based on the following orthonormal basis: $\theta^0=
e^{\nu_1} dt_1$, $\theta^1=e^{\nu_2}dt_2$, $\theta^2=e^{\lambda}\,dr$, $\theta^3=r\,dr$, $\theta^4=r\,\text{sin}\,\theta\, d\phi$. (See, for example, Ref. [@pK08].) Then the line element takes on the form $$ds^2=\mu_{ij}\theta^i\theta^j,$$ where $\mu_{ij}$ is the usual Minkowski metric $diag(-1,-1,1,1,1)$. In other words, the calculations are not affected by the signature. So while ordinary $4D$ relativity does not allow a second time coordinate, this is not true in $5D$ since there is no conflict with observation.
Returning to line element (\[E:line1\]), we can now consider an embedding in the following five-dimensional flat spacetime: $$ds^{2}=-\left(dz^1\right)^2-\left(dz^2\right)^2
+\left(dz^3\right)^2+\left(dz^4\right)^2
+\left(dz^5\right)^2.$$ Then the transformation formulas for the first two components must be changed to $z^1=\sqrt{K}\,e^{\frac{\nu}{2}}
\,\text{sin}{\frac{t}{\sqrt{K}}}$ and $z^2=\sqrt{K}\,e^{\frac{\nu}{2}}\,\text{cos}
{\frac{t}{\sqrt{K}}}$; thus
$$dz^1=\sqrt{K}\,e^{\frac{\nu}{2}}\,\frac{\nu'}{2}\,
\text{sin}{\frac{t}{\sqrt{K}}}\,dr + e^{\frac{\nu}{2}}\,
\text{cos}{\frac{t}{\sqrt{K}}}\,dt$$
and $$dz^2=\sqrt{K}\,e^{\frac{\nu}{2}}\,\frac{\nu'}{2}\,
\text{cos}{\frac{t}{\sqrt{K}}}\,dr - e^{\frac{\nu}{2}}\,
\text{sin}{\frac{t}{\sqrt{K}}}\,dt.$$ So $$-\left(dz^1\right)^2-\left(dz^2\right)^2=
-\frac{1}{4}Ke^{\nu}(\nu')^2\,dr^2-e^{\nu}\,dt^2.$$ Since Eq. (\[E:partial2\]) remains the same, the resulting line element is $$\label{E:linenew}
ds^{2}=-e^{\nu}dt^{2}+\left(\,1-\frac{1}{4}K\,e^{\nu}\,
(\nu')^2\,\right)\,dr^{2}+r^{2}\left(d\theta^{2}
+\sin^{2}\theta\, d\phi^{2} \right)$$ and $$\label{E:lambda2}
e^{\lambda}=1-\frac{1}{4}K\,e^{\nu}\,(\nu')^2,$$
In view of line element (\[E:linenew\]), $K$ cannot be arbitrarily large. Now repeating the earlier calculation, we arrive at $$\begin{gathered}
8\pi(\rho +3p)=\\\frac{2}{r^2}
\frac{\frac{1}{4}Ke^{\nu}(\nu')^2}
{1-\frac{1}{4}K\,e^{\nu}\,(\nu')^2}
+\frac{1}{1-\frac{1}{4}K\,e^{\nu}\,(\nu')^2}
\left[\frac{-1/r}{1-\frac{1}{4}K\,
e^{\nu}\,(\nu')^2}\frac{1}{4}Ke^{\nu}
[(\nu')^3+2\nu'\nu'']+\frac{3\nu'}{r}
\right].\end{gathered}$$
To draw a conclusion, we need to make another plausible assumption. We know from the asymptotic flatness that $\nu$ and $\nu'$ go to zero as $r\rightarrow\infty$, but we also assume that this occurs smoothly, as shown in Fig. 1. Now, with the Schwarzschild line
{width="80.00000%"}
element in mind, let us assume that $\nu'>0$. Then if $K$ is sufficiently small, we have $8\pi(\rho+3p)>0$ and $\overset{..}{a}(t)<0$, indicating a decelerating expansion. Taken together, these assumptions imply that a reversal of the accelerated expansion is indeed possible.
Astronomy and the fifth dimension
=================================
That the embedding theory in the first part of this paper should yield an accelerated expansion may not be a total surprise in view of the induced-matter theory in Ref. [@WP92]. More precisely, it is noted in Ref. [@pW15] that the field equations for the five-dimensional flat embedding space actually yield the Einstein field equations in four dimensions *containing matter*. In other words, matter and hence energy, including dark energy, come from geometry, thereby echoing John A. Wheeler’s “everything is geometry." Ref. [@pW13] goes on to state that even the equivalence principle may be a direct consequence of the existence of an extra spatial dimension. In fact, our very understanding of physics in four dimensions may be significantly enhanced due to this extra dimension.
Another possible area of agreement is noncommutative geometry, an offshoot of string theory, which also involves extra dimensions. The main idea, discussed in Refs. [@eW96; @SW99], is that coordinates may become noncommuting operators on a $D$-brane. Here the commutator is $[\textbf{x}^{\mu},
\textbf{x}^{\nu}]=i\theta^{\mu\nu}$, where $\theta^{\mu\nu}$ is an antisymmetric matrix. The main idea, discussed in Ref. [@SS03], is that noncommutativity replaces point-like structures by smeared objects. The concentration is therefore on local properties. It is shown in Ref. [@pK17b] that $(-4\pi/3)(\rho+3p)>0$ locally, i.e., in the neighborhood of every point. This suggests but does not necessarily prove that the cumulative effect is an accelerated expansion on a cosmological scale. The present study indicates, however, that this interpretation is actually valid, given that both theories depend on extra dimensions. So noncommutative geometry is another way to account for the accelerated expansion and hence for dark energy.
The accelerated expansion is sometimes referred to as a *phase*. The second part of this paper explores the possibility of just such a reversal due to a change in the signature. That such a change is possible in the first place is shown by the Schwarzschild line element: when crossing the event horizon of a black hole, the first two terms interchange signs. While such a change is not necessitated, it cannot be excluded. The same holds for the form of $\nu(r)$. So a reversal is indeed possible.
*Remark:* Since mass and energy are equivalent, dark energy generates a gravitational field - one that is actually repulsive. If the above change in the signature were to take place, thereby causing the expansion to decelerate, then we may be dealing with a more palatable kind of dark energy - one that is gravitationally attractive.
Conclusion
==========
An $n$-dimensional Riemannian space is said to be of embedding class $m$ if $n+m$ is the lowest dimension of the flat space in which the given space can be embedded. We start with a spherically symmetric line element of embedding class two which is then reduced to a metric of class one by a suitable transformation. This metric may be viewed as a generic line element in a cosmological setting. The assumption of asymptotic flatness then implies that $\nu(r)$ and $\nu'(r)$ in the line element go to zero as $r\rightarrow\infty$. An additional assumption (needed only in the second part) is that the asymptotic behavior occurs smoothly, as shown qualitatively in Fig. 1. The main conclusion is that $(-4\pi/3)(\rho +3p)>0$; so the Friedmann equation implies that $\overset{..}{a}(t)>0$, where $a(t)$ is the scale factor in the FLRW model. The result is an accelerated expansion of the Universe. The embedding theory can therefore serve as a model for dark energy. The results are consistent with earlier findings based on noncommutative geometry.
The second part of the paper suggests that a reversal of the accelerated expansion is possible if the fifth dimension in the embedding space becomes timelike and the function $\nu(r)$ behaves in a manner similar to that of the Schwarzschild line element ($\nu(r)<0$, $\nu'(r)>0$). While the change in the signature is not necessitated, it cannot be ruled out. So it is possible, at least in principle, for the acceleration to become a deceleration.
[20]{}
S.K. Maurya and M. Govender, Generating physically realizable stellar structures via embedding, Eur. Phys. J. C, 77 (2017), Article ID: 347. K.R. Karmarkar, Gravitational metrics of spherical symmetry and class one, Proc. Ind. Acad. Sci., 27 (1948), 56-60. P. Bhar, S.K. Maurya, Y.K. Gupta, and T. Manna, Modelling of anisotropic compact stars of embedding class one, Eur. Phys. J. A, 52 (2016), Article ID: 312. S.K. Maurya and S.D. Maharaj, Anisotropic fluid spheres of embedding class one using Karmarkar condition, Eur. Phys. J. C, 77 (2017), Article ID: 328. S.K. Maurya, B.S. Ratanpal, and M. Govender, Anisotropic stars for spherically symmetric space time satisfying the Karmarkar condition, Ann. Phys., 382 (2017), 36-49. S.K. Maurya, Y.K. Gupta, S. Ray, and D. Deb, A new model for spherically symmetric charged compact stars of embedding class one, Eur. Phys. J. C, 77 (2017), Article ID: 45. S.K. Maurya, Y.K. Gupta, S. Ray, and D. Deb, Generalized model for anisotropic compact stars, Eur. Phys. J. C, 76 (2016), Article ID: 693. S.K. Maurya, D. Deb, S. Ray, and P.K.F. Kuhfittig, Compact stars: a generalized model, arXiv: 1703.08436. P.K.F. Kuhfittig, A note on the cosmological constant in $f(R)$ gravity, J. Appl. Math. Phys., 5 (2017), 933-938. P.K.F. Kuhfittig, Two diverse models of embedding class one, Ann. Phys., 392 (2018), 63-70. P.K.F. Kuhfittig, On a time-dependent extra spatial dimension, Int. J. Pure Appl. Math., 49 (2008), 577-582. P.S. Wesson and J. Ponce de León, Kaluza-Klein equation, Einstein’s equations, and an effective energy-momentum tensor, J. Math. Phys., 33 (1992), Article ID: 3883. P.S. Wesson, The status of modern five-dimensional gravity, Int. J. Mod. Phys. D, 24 (2015), Article ID: 1530001. P.S. Wesson, Astronomy and the fifth dimension, arXiv: 1301.0033. E. Witten, Bound states of strings and $p$-branes, Nucl. Phys. B, 460 (1996), 335-350. N. Seiberg and E. Witten, String theory and noncommutative geometry, J. High Energy Phys., 9909 (1999), Article ID: 032. A. Smailagic and E. Spallucci, Feynman path integral on the non-commutative plane, J. Phys. A, 36 (2003), L-467-L-471. P.K.F. Kuhfittig, Accounting for some aspects of dark matter and dark energy via noncommutative geometry, J. Mod. Phys. 8 (2017), 323-329.
[^1]: [email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Polymers with both soluble and insoluble blocks typically self-assemble into micelles, aggregates of a finite number of polymers where the soluble blocks shield the insoluble ones from contact with the solvent. Upon increasing concentration, these micelles often form gels that exhibit crystalline order in many systems. In this paper, we present a study of both the dynamics and the equilibrium properties of micellar crystals of triblock polymers using molecular dynamics simulations. Our results show that equilibration of single micelle degrees of freedom and crystal formation occurs by polymer transfer between micelles, a process that is described by transition state theory. Near the disorder (or melting) transition, bcc lattices are favored for all triblocks studied. Lattices with fcc ordering are also found, but only at lower kinetic temperatures and for triblocks with short hydrophilic blocks. Our results lead to a number of theoretical considerations and suggest a range of implications to experimental systems with a particular emphasis on Pluronic polymers.'
author:
- 'J.A. Anderson$^{(a)}$, C.D. Lorenz$^{(b)}$ and A. Travesset$^{(a)}$'
title: Micellar Crystals in Solution from Molecular Dynamics Simulations
---
=1
Introduction
============
Micelles of multi-block polymers are finite aggregates, typically around fifty polymers or less, where the insoluble blocks shield the soluble ones from contact with the surrounding solvent. Depending on control variables (temperature, polymer concentration, pH, etc..) micelles may self-assemble into gels that exhibit long-range order such as bcc, fcc, hcp or other, more unusual, crystals. Micellar crystals exhibit a number of unique properties that have made them extremely attractive for fundamental studies as well as for applications [@Alexandridis2000; @Likos2006]. Extensively studied experimental systems include aqueous solutions of Pluronics (also known as Poloxamers), ABA triblocks where A is Polyethylene oxide (PEO) and B is Polypropylene oxide (PPO) [@Wanka1994; @Chu1995; @Mortensen2001] and inverted Pluronics, where the A blocks are PPO and the central block B is PEO [@Mortensen1994; @Mortensen2001] as well as non-aqueous systems such as Polystyrene-Polyisoprene (PS-PI) diblocks in decane [@McConnell1993] and other solvents [@Bang2002; @Lodge2002; @Lodge2004a; @Lodge2004b].
Micelles in solution are highly dynamical entities with polymers continually being absorbed and released through time. Therefore, a micellar crystal has a considerably intricate structure, where the long range order remains stable as the individual polymers are constantly hopping from one micelle to the next. Theoretical approaches such as density functional or mean field theory [@McConnell1996; @vanVlimeren1999; @Ziherl2001; @Lam2003; @Zhang2006; @Likos2006; @Grason2007] directly study the ordered micelles and ignore the dynamical degrees of freedom of the polymers. Studies using molecular dynamics (MD) have the advantage of providing a reasonably realistic description of the dynamics, thus allowing the investigation of the role of single polymer degrees of freedom. In contrast with other approaches, MD also offers the important advantage that no assumptions need to be made about what is the possible thermodynamic state of the system.
The goal of this study is to predict phase diagrams of triblock polymers using MD simulations and to gain an understanding of the dynamics of micellar crystal formation. Because of our ongoing interest in Pluronic systems in aqueous solutions [@Anderson2006], we examine systems of $A_nB_mA_n$ triblocks, where the $A$ beads are hydrophilic and $B$ beads are hydrophobic. Although there have been a number of previous studies of multiblock copolymers in solution using MD, see Ref. [@Ortiz2006] for a recent review, the prediction of crystalline structures presents substantial difficulties. Experimentally, it is well known that the approach to thermodynamic equilibrium is slow in these systems, with time scales of the order of minutes or hours. Therefore, even with suitably coarse-grained models, the long time scales involved provide a considerable challenge for MD simulation studies.
In this paper, we provide a detailed investigation of self assembled micellar crystals using MD. We aim to understand the mechanism by which they form, predict their range of stability and elucidate their static and dynamic properties. We also present an in-depth study on the challenges associated in reaching the thermodynamically stable state using MD and a successful strategy to overcome them.
Model and Simulation Details
============================
Simulation Details
------------------
Systems of polymers are modeled by coarse-grained beads in an implicit solvent. Ref. [@Anderson2006] provides a more detailed justification than the outline provided here. Individual polymers are $A_n B_m A_n$ symmetric triblocks, where $A$ beads are hydrophilic and $B$ beads are hydrophobic. All systems in this work are monodisperse with fixed values for $n$ and $m$. Non-bonded pair potentials consist of a standard attractive Lennard-Jones potential for hydrophobic interactions $$\label{Eq_U_bb}
U_{\mathrm{BB}} = 4\varepsilon \left[ \left(\frac{\sigma}{r}
\right)^{12} - \left(\frac{\sigma}{r}\right)^6 \right] \ ,$$ and a purely repulsive potential for hydrophilic interactions $$\label{Eq_U_ab}
U_{\mathrm{AA,AB}} = 4\varepsilon \left(\frac{\sigma}{r}
\right)^{12} .$$ Pair potentials are cutoff to zero at $r = 3.0\sigma$. All beads have the same mass $M$. $M$, $\varepsilon$ and $\sigma$ are arbitrary and uniquely define the units of all numbers in the simulations. Neighboring atoms in the polymer chain are connected together with a simple harmonic potential. The packing fraction $\phi_P$ of the polymers is given by $$\label{Eq_density}
\phi_P=\frac{\pi N_{\mathrm{poly}} N_{\mathrm{mon}}}{6(L/\sigma)^3} \ ,$$ where $N_{\mathrm{poly}}$ is the number of polymers in a box of linear dimensions $L$ and $N_{\mathrm{mon}}=2n+m$ is the number of beads in each polymer. Simulation boxes are cubic with periodic boundary conditions. For this work, we use a time step of $\Delta t = 0.005
\sqrt{m\sigma^2/\varepsilon}$. All simulations were performed using the LAMMPS software package [@lammps] in the NVT ensemble via Nos[é]{}-Hoover dynamics [@Hoover1985].
The temperature sensitivity of Pluronic systems is a reflection of the underlying strong temperature dependence of the hydrophobic effect. To model temperature dependent phases in such systems, the solvent quality for the $A$ beads must change [@Anderson2006]. In this paper, we keep the solvent quality fixed and vary only the kinetic temperature when needed to equilibrate the systems properly.
Recently, a different implicit solvent model has been developed for describing Pluronic systems [@Bedrov2006], where the pair potentials for the coarse-grained simulation are fitted to results obtained from all-atom MD and quantum chemistry simulations. Although the coarse-graining is different (each monomer $A$,$B$ represents one PEO or PPO monomer) than in our model, the resulting potentials are quite similar to the ones in this work, Equations \[Eq\_U\_bb\] and \[Eq\_U\_ab\], with the only significant difference that the values of $\sigma$ are different between the two $A$ and $B$ monomers, and $U_{\mathrm{AA}}$ shows a minor maximum.
Observables
-----------
A number of observable quantities are monitored for every recorded time step during a simulation run. The micelles themselves are identified by the same algorithm used previously in Ref. [@Anderson2006]. Any hydrophobic beads within a distance of $r_{\mathrm{cut}} = 1.2
\sigma$ of one another are identified as belonging to the same micelle. Identified micelles containing less than 3 polymers are typically free polymers in the process of being transferred from one micelle to another and are removed from further consideration. Observables such as statistics of micelle aggregation number, gyration tensor, center of mass, and micelle lifetime are calculated and examined for every simulation performed in this work. Methods used to calculate these are described in Ref. [@Anderson2006].
The structure factor $S(\vec{q})$ is calculated over the center of mass coordinates $\vec{r}_i$ of all $N_{\mathrm{mic}}$ micelles in the system using the formula $$\label{Eq_Sq}
S(\vec{q})=C_0 \langle \left| \sum_{i=1}^{N_{\mathrm{mic}}} e^{i\vec{q} \cdot \vec{r}_i} \right| ^2\rangle \ ,$$ with the components of ${\vec{q}}$ as multiples of $\frac{2\pi}{L}$ due to the use of periodic boundary conditions. The peaks in $S(\vec{q})$ are then used to reconstruct the full 3D real space lattice basis, if it exists. In this manner, $S(\vec{q})$ is not being used to simulate real scattering intensities as may be obtained by X-ray experiments, but as a mathematical order parameter to discriminate between the different ordered structures that may be present. For convenience, the normalization $C_0$ is chosen so that $S(\vec{q}=0)=1$. A more sophisticated treatment that allows continuum values of ${\vec q}$, suitable for quantitative comparisons with X-ray experiments, has been recently introduced [@Schmidt2007], but we do not use it here.
Individual polymers are constantly hopping from one micelle to another. This is quantified over the entire simulation box as an overall rate of polymer transfer, $r_{\mathrm{PT}}$, by examining contiguous simulation snapshots. Sets of indexed polymers belonging to each micelle are compared between the snapshots to find the number of polymers transferred. The rate $r_{\mathrm{PT}}$ is then expressed as a fraction of the number of polymers in the box transferred per one million time steps. A polymer that is transferred out and back to the same micelle between snapshots will not be counted by this analysis, so snapshots are recorded every 100,000 time steps to minimize undercounting. At $k_B T/\varepsilon = 1.2$, a typical micelle only loses/gains one polymer per ten snapshots recorded.
Radial distributions of the beads surrounding micelles are also of interest. These are calculated by creating a histogram with bin width $dr$ and then counting the number of beads belonging to a micelle $N_\mathrm{count}(r)$ that fall between $r$ and $r+dr$, where $r$ is the distance of the bead from the center of mass of the micelle. Good statistics require averaging this histogram over all micelles in the simulation and over all time steps after the micellar crystal has formed. The average histogram is transformed into a radial density distribution of beads around a micelle by calculating the packing fraction of beads in each bin $$\phi_i(r) = \frac{\pi }{6} \cdot \frac{\langle N_\mathrm{count}(r) \rangle}{4/3 \pi \left( \left(r+dr \right)^3 - r^3\right) / \sigma^3} \ ,$$ where $i$ refers to either $A$ or $B$ beads. We use $dr = 0.2\sigma$ to balance smooth graphs with the need for long simulation runs to obtain detailed statistics.
Micellar crystals studied using MD
==================================
There are two major challenges faced in using MD to determine equilibrium phases. First, the simulation must last longer than any of the relaxation times in the system. Second, the simulation box size must be chosen properly to avoid finite size effects. The first problem is related to the kinetic temperature at which the system is run. The second problem can become particularly severe for simulating crystals with three dimensional order, where the incorrect choice of an even large box size $L$ can force the system into very distorted ordered phases. These two issues are addressed systematically using simulations of the $A_{10}B_7A_{10}$ polymer. It provides a coarse-grained description of one of the most extensively studied Pluronics, F127 [@Wanka1994]. The conclusions of this study lead to a general methodology valid for any other polymer.
![\[fig:polymer\_msd\] Mean squared displacement averaged over all beads in the simulation box. This data was obtained from the initial simulation runs of the $A_{10}B_7A_{10}$ polymer at $N_{\mathrm{poly}} = 500$ and $\phi_P = 0.20$. The origin of the time axis is 30 million time steps which indicates that the recording of these results began after the system reached equilibrium. In the case of $k_B T/\varepsilon = 1.0$ this equilibrium is a metastable state, and the beads are not diffusing. The simulation run performed at $k_B T/\varepsilon = 1.2$ formed a fcc lattice around time step 10 million which persisted until the end of the run at 35 million, and the mean squared displacement shows a characteristic diffusive behavior.](polymer_msd_fig){width="8cm"}
![\[fig:fcc\_500\] Snapshot of a $A_{10}B_7A_{10}$ polymer simulation run at $\phi_P =
0.20$, $k_B T / \varepsilon = 1.2$, $N_{\mathrm{poly}} = 500$, taken after the fcc lattice formed. $A$ beads are represented by orange spheres, and are shown with a reduced radius so they do not obscure important details. Orange lines indicate bonds between these beads. $B$ beads are shown in blue with a radius of $0.6\sigma$. Large yellow spheres are placed on the lattice reconstructed from $S(\vec{q})$. Every yellow sphere is sitting on a micelle, visually confirming a perfect fcc crystal. The $A$ beads have been removed around a single unit cell of the lattice and yellow lines added to guide the eye. All snapshots are generated using PyMol [@pymol]. ](fcc_500){width="8cm"}
![\[fig:n\_mic\_example\] Examples of the behavior of $N_{\mathrm{mic}}$ during a simulation run of the $A_{10}B_7A_{10}$ polymer. Results are shown here for two independent runs with $\phi_P = 0.20$, $k_B T / \varepsilon = 1.2$ and $N_{\mathrm{poly}} = 2134$. The origin of the time axis (t=0) indicates the time step where the entire simulation started from a random initial configuration. Note how both systems eventually plateau at the same state $N_{\mathrm{mic}}=128$, though it occurs at different times. ](n_mic_example_fig){width="8cm"}
Micelle crystallization and kinetic temperature
-----------------------------------------------
Initial simulations are performed with a system size of $N_{\mathrm{poly}} = 500$ at a kinetic temperature $\frac{k_B T}{\varepsilon} = 1.0$ and with concentrations $\phi_P$ of $0.05$, $0.10$, $0.15$, $0.20$ and $0.25$ run over $30$ million time steps each. In all cases, the initially randomly placed $A_{10}B_7A_{10}$ polymers aggregate into micelles quickly in a few thousand time steps.
Visual examination indicates that concentrations at and above $\phi_P = 0.15$ are strong candidates for micellar crystals. Micelles are locked in place and only move a few $\sigma$ from their average positions. The analysis of the order parameter $S(\vec{q})$, however, indicates no long-range ordered structures exist in any of these initial simulations. An inspection of the polymer transfer shows that it remains negligible throughout all simulations. This is confirmed by calculating the mean squared displacement $\langle
(\vec{r}(t) - \vec{r}(0))^2 \rangle$ averaged over all beads in the system. clearly shows non-diffusive behavior at $k_B T/\varepsilon = 1.0$. These results suggest that the individual micelles are quickly frozen in a configuration that is not representative of equilibrium, thus preventing the entire system from reaching thermal equilibrium.
The lack of equilibration of micellar degrees of freedom suggests subsequent runs at a larger kinetic temperature $k_B T/\varepsilon=1.2$. A single simulation run at $\phi_P = 0.20$ formed a textbook fcc lattice after about 10 million steps, and is shown in . Throughout the duration of the run, the rate of polymer transfer was substantial, about $7\%$ of the polymers in the box every million time steps, *even after* equilibrium is reached. Playing a movie of the simulation shows that after the lattice formed, micelles do not appear to move except by vibrating about their average positions. However, while the micelles appear static, polymers are constantly being exchanged among the micelles, so that any individual polymer will eventually explore the entire simulation box. This is independently confirmed by the analysis of the mean squared displacement of beads in , which shows a classic diffusion result at $k_B T/\varepsilon=1.2$.
The behavior of the number of micelles $N_{\mathrm{mic}}$ in the box as a function of simulation time is of particular interest. In simulation runs where micellar crystals are found, the system reaches a plateau where $N_{\mathrm{mic}}$ remains constant, see for an example. Despite their dynamic character even at equilibrium, $N_{\mathrm{mic}}$ then remains constant for the duration of the run. This correlation is a general feature in all simulation runs performed. *Every single* one that leads to a stable plateau in $N_{\mathrm{mic}}$ as a function of time formed a micellar crystal confirmed by peaks in the order parameter $S(\vec{q})$.
![\[fig:distorted\_600\] Snapshot of a $A_{10}B_7A_{10}$ polymer simulation run at $\phi_P = 0.20$, $k_B T / \varepsilon = 1.2$, and $N_{\mathrm{poly}} = 600$ taken after the lattice formed. The resulting lattice in this system is a body-centered tetragonal with $c/a = 1.5$. Coloring conventions are identical to ](distorted_600){width="8cm"}
.
------------- --------------------- -------------------- ---------------- ------------------------------------
No. configs $N_{\mathrm{poly}}$ $N_{\mathrm{mic}}$ Ordering $\langle N_{\mathrm{agg}} \rangle$
1 500 32 *textbook* fcc 15.625
3 500 unstable none
1 600 36 none 16.67
3 600 36 distorted bcc 16.67
5 800 48 distorted bcc 16.67
5 900 54 *textbook* bcc 16.67
1 900 55 distorted bcc 16.36
2 1000 unstable none
4 1000 60 distorted bcc 16.67
------------- --------------------- -------------------- ---------------- ------------------------------------
: \[tab:N\_test\] Summary of results obtained from initial test runs.
No. configs Targeted $N_{\mathrm{poly}}$ $N_{\mathrm{mic}}$ Ordering
------------- ----------------- --------------------- -------------------- ----------------
1 16 micelle bcc 267 16 *textbook* bcc
4 108 micelle fcc 1800 unstable none
4 128 micelle bcc 2134 128 *textbook* bcc
3 250 micelle bcc 4168 unstable none
: \[tab:algo\_test\_A10B7A10\] Summary of simulation results from testing the algorithm on $A_{10}B_7A_{10}$.
### Box size effects {#sec:boxsize_effects}
The influence of the simulation box size choice is assessed by running additional simulations with $N_{\mathrm{poly}} = 500$, $600$, $800$, $900$, and $1000$ with fixed concentration $\phi_P = 0.20$ and $\frac{k_B T}{\varepsilon} = 1.2$. Several different random initial configurations are used at each system size to ensure repeatability. summarizes the results of all these simulation runs.
Interestingly, the additional three simulation runs at $N_{\mathrm{poly}} = 500$ do not exhibit fcc structures. Instead, each of them remains unstable with $N_{\mathrm{mic}}$ never achieving a constant plateau and $S(\vec{q})$ is devoid of peaks. Larger system sizes do reach stable ordered structures with constant $N_{\mathrm{mic}}$ and numerous peaks in $S(\vec{q})$. Most of the structures that occur appear bcc when examined visually, but a detailed analysis of the lattices indicates that many of them are distorted. One obviously distorted structure is depicted in where the lattice is body centered tetragonal with $c/a = 1.5$.
Other distortions include body centered tetragonal with $c/a = 1.054$ for $N_{\mathrm{poly}} = 800$ and a lattice that appears to be almost exactly bcc when $N_{\mathrm{poly}} = 1000$, except that the central micelle in the unit cell is shifted slightly from the true center. Lastly, a textbook bcc lattice is formed in the simulation runs with $N_{\mathrm{poly}} = 900$.
Examining the average aggregation number leads to a very illuminating result. It is found that for almost all simulations these numbers are *identical*. This indicates that the average micelle aggregation number is independent of the box length (at a fixed $\phi_P$) and is only a function of the polymer structure, kinetic temperature and concentration.
Assuming without proof that either the fcc or the bcc lattice represents the real thermodynamic equilibrium of the system, the previous observation then suggests a way to generate *magic numbers* of polymers to simulate equilibrium states free of finite size effects. In a bcc lattice, each cubic unit cell contains $C=2$ micelles, while the fcc lattice contains $C=4$. Therefore, in order to obtain a bcc or fcc lattice with $M$ by $M$ by $M$ unit cells in a cubic simulation box, the number of polymers needed to achieve this is given by $$\label{Eq_MagicNumber} N_{\mathrm{poly}}=C M^3 \langle N_{\mathrm{agg}}\rangle .$$
MD Simulations without finite size effects {#sec:algorithm}
------------------------------------------
An algorithm to simulate micellar crystals without finite size effects follows very naturally from the previous results, and is summarized in the following steps.
1. [*Concentration selection:*]{} The concentration must be chosen high enough so that micelles pack into a potential crystalline state.
2. [*Temperature selection:* ]{} The temperature should be chosen large enough to ensure a significant rate of polymer transfer, but low enough that the micellar crystals are not in a disordered phase. As a rule of thumb, we have been using a polymer transfer around $10\%$ of the polymers in the box every million steps.
3. [*System size selection:* ]{} Calculate the average micelle number via test simulations and use to determine the final system sizes to perform simulations on.
4. [*Ensure reproducibility:* ]{} The formation of micellar crystals is a stochastic process so several simulations with different initial configurations must be run.
The advantage of this algorithm is that steps 1-4 can be accomplished with relatively modest computer resources on small system sizes, leaving the production runs with large polymer numbers as the only computationally intensive calculations.
Micellar crystals of general AnBmAn triblocks
----------------------------------------------
![\[fig:bcc\_2134\] Snapshot of a $A_{10}B_{7}A_{10}$ simulation run at $\phi_P = 0.20$, $k_B T / \varepsilon = 1.2$, and $N_{\mathrm{poly}} = 2134$, taken after the bcc lattice formed. Coloring conventions are identical to ](bcc_2134){width="8cm"}
### A10B7A10
Further simulation runs are carried out on the $A_{10}B_7A_{10}$ polymer system to test the algorithm. These additional simulations target 16, 128 and 250 micelle bcc configurations, along with 32 and 108 micelle fcc ones. The simulations are summarized in . None of the simulation runs targeting fcc phases ever reach a stable number of micelles and correspondingly, $S(\vec{q})$ indicates there is no long-range order. On the other hand, both the 16 and 128 micelle bcc configurations are perfect and completely reproducible with the expected ordering confirmed by the structure factor. shows a snapshot of the 128 micelle bcc phase after it forms. Further discussion on the calculated structure factors are presented below. Larger simulations attempting the formation of a 250 micelle bcc crystal never resulted in a stable $N_{\mathrm{mic}}$ plateau.
### A20B14A20
The $A_{20}B_{14}A_{20}$ polymer is also studied as it provides a closer representation to the real Pluronic F127, since there are twice as many monomers in a polymer and the level of coarse-graining is less, as discussed in Ref. [@Lam2003]. Applying the algorithm developed above to search for micellar crystals, the concentration was selected at $\phi_P = 0.20$ and the kinetic temperature at $k_B T / \varepsilon = 1.9$. Initial simulation runs establish that $\langle N_{\mathrm{agg}}
\rangle = 22.5 $. The results, summarized in , are similar to those for the smaller $A_{10}B_{7}A_{10}$ polymer, with the bcc structure being the most commonly found. The fcc phase again only appears in a single simulation run and is not reproducible.
### A6B7A6
Given the prevalence of bcc lattices so far, a polymer with shorter $A$-blocks ($A_6B_7A_6$), expected to form more *crew-cut* micelles and hence more prone to assemble into a fcc lattice, is also investigated. Applying the algorithm developed above, the concentration is selected at $\phi_P = 0.15$, the kinetic temperature at $k_B T / \varepsilon = 1.1$, and the average aggregation number is found to be $\langle N_{\mathrm{agg}} \rangle = 13.875$. Again, the bcc micellar crystal is most commonly found, as shown in the results in . No fcc phases are found at this kinetic temperature.
However, the rate of polymer transfer at $k_B T / \varepsilon = 1.1$ is larger than $10\%$ per million steps, so some additional simulation runs are performed at a slightly reduced temperature $k_B T / \varepsilon = 1.07$, where polymer transfer is slightly reduced. In this case a completely reproducible fcc structure is found for relatively small system sizes, but no fcc lattices could be stabilized for larger ones as shown in the results in .
No. configs Targeted $N_{\mathrm{poly}}$ $N_{\mathrm{mic}}$ Ordering
------------- ----------------- --------------------- -------------------- ----------------
2 16 micelle bcc 360 16 *textbook* bcc
1 32 micelle fcc 720 32 *textbook* fcc
1 32 micelle fcc 720 35 distorted bcc
2 32 micelle fcc 720 unstable none
1 54 micelle bcc 1215 54 *textbook* bcc
3 54 micelle bcc 1215 55 distorted bcc
2 108 micelle fcc 2430 unstable none
: \[tab:algo\_test\_A20B14A20\] Summary of simulation results from testing the algorithm on $A_{20}B_{14}A_{20}$ at $k_B T/\varepsilon = 1.9$.
No. configs Targeted $N_{\mathrm{poly}}$ $N_{\mathrm{mic}}$ Ordering
------------- ----------------- --------------------- -------------------- ----------------
5 16 micelle bcc 222 16 *textbook* bcc
5 32 micelle fcc 444 unstable none
5 54 micelle bcc 750 54 *textbook* bcc
5 108 micelle fcc 1500 unstable none
3 128 micelle bcc 1778 128 *textbook* bcc
1 128 micelle bcc 1778 136 distorted bcc
3 250 micelle bcc 3472 unstable none
4 432 micelle bcc 6035 unstable none
: \[tab:algo\_test\_A6B7A6\] Summary of simulation results from testing the algorithm on $A_{6}B_{7}A_{6}$ at $k_B T/\varepsilon = 1.1$.
No. configs Targeted $N_{\mathrm{poly}}$ $N_{\mathrm{mic}}$ Ordering
------------- ----------------- --------------------- -------------------- ----------------
4 32 micelle fcc 444 32 *textbook* fcc
1 32 micelle fcc 444 unstable none
2 108 micelle fcc 1500 unstable none
2 108 micelle fcc 1500 103 distorted bcc
1 108 micelle fcc 1500 104 distorted bcc
: \[tab:algo\_test\_A6B7A6\_t07\] Summary of simulation results from testing the algorithm on $A_{6}B_{7}A_{6}$ after cooling to $k_B T / \varepsilon = 1.07$.
Dynamics of crystal formation {#sec:crystal_formation}
=============================
Molecular dynamics simulations not only allow the prediction of equilibrium phase diagram, but also describe the dynamics of micelles as they evolve towards thermal equilibrium. In the simulations presented previously, micelles quickly form from a completely random configuration of polymers, and then after a long simulation time (10 to 20 million time steps) order into a micellar crystal. We now present a quantitative picture of the dynamics of the polymers and micelles as the system approaches equilibrium. All results in this section are obtained from the analysis of the simulations of the $A_{10}B_{7}A_{10}$ polymer.
Polymer transfer is an activated process
----------------------------------------
Polymer transfer plays an important role in achieving the formation of micellar crystals in the simulations discussed above. If there is too little, the single micelle degrees of freedom do not reach equilibrium. Too much pushes the system into a disordered state. Moreover, the rate of polymer transfer is extremely sensitive to the kinetic temperature. An equilibrated $N_{\mathrm{poly}} = 2134$ bcc micellar crystal was taken as an initial configuration for additional simulations that continued with $k_B T/\varepsilon$ ranging from $1.0$ to $1.3$. shows the results. The rate of polymer transfer $r_{\mathrm{PT}}$ starts near 0 at $k_B T/\varepsilon = 1.0$ and increases exponentially, following an Arrhenius form $$\label{Eq:TST_micellar}
r_{\mathrm{PT}}=r_O \exp(-\frac{\Delta G^{\sharp}}{k_B T}) \ .$$ That is to say that polymer transfer is an activated process. Curve fitting, we find $$\label{Eq:G_dagger}
\Delta G^{\sharp} \approx 10 \varepsilon \ .$$
also includes results from simulations performed using a Langevin thermostat [@Frenkel2002]. It controls the temperature by adding an additional force to every particle $\vec{F} = -\gamma \vec{v} + \vec{F}_{\mathrm{rand}}$ where the magnitude of the random force $\vec{F}_{\mathrm{rand}}$ and $\gamma$ set the temperature through the fluctuation dissipation theorem [@Frenkel2002]. The results for two different values of $\gamma$, which are plotted in , show the dependence of the polymer transfer for different drag coefficients.
In , the slope of the lines in the inset plot are universal for both thermostats and both values of $\gamma$. The universality of this value implies that the calculated $\Delta G^{\sharp}$ is the free energy barrier between a polymer in a micelle to a transition state between micelles. While their slopes are universal, the y-intercepts (related to $r_O$) should depend on the diffusion coefficient of the hydrophilic beads, which, from the Einstein relation [@Doi1986] is inversely proportional to the drag coefficient. This is reflected in the offsets of the various plots in , which are clearly different.
![\[fig:poly\_transfer\] Polymer transfer $r_{\mathrm{PT}}$ versus temperature calculated from simulation runs of the $A_{10}B_{7}A_{10}$ polymer at $\phi_P = 0.20$ and $N_{\mathrm{poly}} =
2134$. All simulations start from an already equilibrated bcc phase. Results are included for the Nosé-Hoover thermostat and Langevin thermostat with two different values of $\gamma$. The inset plots the same data as a plot of $\log_{10}(r_{\mathrm{PT}})$ versus $\varepsilon / (k_B T)$ to show that the slopes of the resulting lines ($-\Delta G^{\sharp}$) are universal.](poly_transfer_fig){width="8cm"}
The difference in free energy between a polymer within a micelle and in the transition state entirely surrounded by solvent and hydrophilic beads can be roughly estimated as $\Delta G^{\sharp} \sim m_{exp} \cdot \varepsilon$, where $m_{exp}$ is the number of hydrophobic beads exposed to solvent. If the length of hydrophobic block is low, $m_{exp}\sim m$. Thus for the $A_{10}B_{7}A_{10}$ polymer analyzed here, it is expected that $\Delta G^{\sharp} \sim 7 \varepsilon$, which is consistent with the measured value. Hydrophobic beads in polymers with a longer hydrophobic block are expected to form a globule in the transition state, leading to a slower increase of $\Delta G^{\sharp}$ going as $m_{exp}\sim m^{2/3}$. Assuming that $r_O$ remains constant as $m$ increases, then maintaining the same rate of polymer transfer $r_{\mathrm{PT}}$ will require increasing the kinetic temperature by the same factor. These considerations agree remarkably with the kinetic temperature of $k_B T /\varepsilon = 1.9$ selected for the $A_{20}B_{14}A_{20}$ polymer simulations, $$\left( \frac{ m_{\mathrm{new}} }{ m_{\mathrm{prev}} } \right)^{2/3} \cdot k_B T_{\mathrm{prev}} / \varepsilon = \left( \frac{14}{7} \right)^{2/3} \cdot 1.2 = 1.9$$.
However, as the size of the hydrophobic block grows, the transition state should also have an additional contribution due to the free energy cost of passing a globule through the brush formed by the hydrophilic coronas. So this simple estimate should eventually break down.
Implicit in our arguments is that polymer transfer accounts for the diffusive behavior in . Therefore, the diffusion coefficient can be roughly estimated from the time $\tau_{PT}$ it takes a polymer to travel the nearest neighbor micelle distance $~a_L$. Comparing this estimate with the fit to , a good agreement is obtained: $$\begin{aligned}
\label{Eq:Diffusive_estimate}
\frac{\langle ({\vec r}(t)-{\vec r}(0))^2 \rangle}{t}&=&1.3\cdot 10^{-5} \frac{\sigma^2}{\Delta t}
\\\nonumber
\frac{a_L^2}{\tau_{PT}}=a_L^2r_{PT}&=&1.1\cdot 10^{-5}\frac{\sigma^2}{\Delta t} \ .\end{aligned}$$
![\[fig:n\_ordered\_micelles\] Number of micelles in the ordered phase $N_{\mathrm{ord}}$ as a function of time for a single simulation run of the $A_{10}B_{7}A_{10}$ polymer at $\phi_P=0.20$ and temperature $k_B T / \varepsilon = 1.2$. The origin of the time axis (t=0) on this plot indicates the time step where the simulation was started from a random configuration.](n_ordered_mic_fig){width="8cm"}
Dynamics of micellar crystal formation
--------------------------------------
The structure factor $S(\vec{q})$ is sufficient as an order parameter to determine if the entire system is in an ordered state, but it reveals no information about the dynamics before that final state is formed. For this, we turn to the bond order analysis [@Steinhardt1983] and apply it to all the micelles at every time step in simulation runs performed on the $A_{10}B_{7}A_{10}$ polymer with $N_{\mathrm{poly}} = 2134$. In short, at any given time step where a micellar crystal may have partially formed, the bond order analysis identifies those micelles belonging to the ordered and the disordered phases.
One simple way to examine the results is to count the number of ordered micelles $N_{\mathrm{ord}}$ in the simulation box at every time step. An example from one simulation run is shown in , plots for all other simulation runs performed are qualitatively very similar, though the ordered phase appears at different times.
After starting the simulation shown in from a random configuration, micelles quickly form in only a few thousand time steps and single micelle degrees of freedom are equilibrated before one million time steps have passed. The system then explores configuration space as polymers are transferred and micelles travel anywhere from a few $\sigma$ up to $100 \sigma$ without any micelles appearing ordered until time step 10 million. At this point, the number of ordered micelles grows very quickly over the next one million steps until all 128 micelles are in the bcc lattice and remain there for the duration of the simulation.
During this short time span of $N_{\mathrm{ord}}$ growth, only $7\%$ of the polymers in the box are transferred between micelles, such a small amount that it cannot fully account for the ordering of all the micelles. During the same time interval, a detailed analysis shows that some micelles move significantly (up to $10\sigma$) before the ordered phase finishes forming. This may suggest that micellar crystal formation is a two step process, where first individual micelles are equilibrated by polymer transfer followed by a second step where polymer transfer becomes irrelevant and the actual crystal grows via the movement of micelles.
However, some additional simulations performed to test this hypothesis do not support it. Single micelle degrees of freedom in these tests are first equilibrated at $k_B T / \varepsilon = 1.2$ for 5 million time steps and then the kinetic temperature is quenched to $k_B T / \varepsilon = 1.0$ to significantly reduce polymer transfer and allow micelle movement. None of these simulation runs resulted in the formation of an ordered phase. We therefore conclude that a significant amount of polymer transfer remains a critical component in the actual growth of ordered micellar crystals.
![\[fig:bcc\_2134\_sfactor\] Structure factor calculated after the lattice formed for the $A_{10}B_{7}A_{10}$ polymer simulation run at $\phi_P = 0.20$, $k_B T / \varepsilon = 1.2$, and $N_{\mathrm{poly}} = 2134$. The full 3D $S(\vec{q})$ is plotted as a scatter plot of $S(\vec{q})$ versus $|\vec{q}|$. The multiplicity of the various peaks can be seen. Vertical dotted lines indicate the location of identified peaks, and their positions relative to $q^* = 4\cdot2\pi/L$ are also noted (the factor of 4 is included because there are 4 unit cells along the box length L).](bcc_2134_sfactor_fig){width="8cm"}
![\[fig:lindemann\] Fluctuations in micelle positions vs. temperature calculated for a $A_{10}B_{7}A_{10}$ polymer simulation run at $\phi_P = 0.20$, $k_B T / \varepsilon = 1.2$, and $N_{\mathrm{poly}} = 900$. The micellar crystal was formed at $k_B T/\varepsilon = 1.2$ and then cooled to the target temperature without disrupting the lattice. Simulation runs heated to a higher temperature do disrupt the lattice and $\Delta r^2$ becomes ill-defined. Fluctuations on the y-axis are plotted as a ratio relative to $a_L$, the nearest neighbor distance in the lattice.](lindemann_fig){width="8cm"}
Properties of micellar crystals {#sec:crystal_static}
===============================
Structure factor
----------------
We calculate the structure factor () for the equilibrium state of every simulation run performed and use it as an order parameter to determine if a micellar crystal is present. In those systems where the crystal does form it is a perfect single crystal and, correspondingly, a large number of peaks can be identified in $S(\vec{q})$ as shown in . Peaks occur in reciprocal space at discrete points $\vec{q} = \vec{G}$, where $\vec{G}$ are the reciprocal vectors of the corresponding lattice.
Peaks evident in decrease in magnitude for larger values of $\vec{G}$. This damping is expected to follow a Debye-Waller factor [@Kittel2005] approximately described as $$\label{Eq:DebyeWaller}
S(\vec{q}=\vec{G})\propto \exp(-\langle \Delta r^2 \rangle |\vec{G}|^2/3) \ ,$$ where $\langle \Delta r^2 \rangle$ is the mean square displacement of the micelle center of mass from its ideal lattice position. A curve fit, shown in , is in excellent agreement with . The Lindemann ratio $f_L$ is defined as $$\label{Eq:Lindemann}
\sqrt{\langle \Delta r^2 \rangle}=f_L a_L$$ where $a_L$ is the nearest neighbor distance between micelles. Using the parameters of the curve fit yields $f_L\approx 0.14$.
The Lindemann parameter $f_L$ can alternatively be computed directly by measuring the mean square displacement of micelles about their average lattice positions. The results, shown in agree remarkably well with the estimate from the Debye-Waller factor. Furthermore, it has been established empirically that approximately at $f_L = 0.13$ solids melt into disordered states [@Hansen1976], a result that is also supported from our simulations as we observed no micellar crystals form at kinetic temperatures greater than $k_B T / \varepsilon = 1.2$
![\[fig:mic\_overlap\] Radial density distribution for two nearest neighbor micelles superimposed with a separation of the nearest neighbor distance in the bcc lattice. The y-axis plots the volume fraction of the different beads belonging to a micelle in the local environment around the micelle. The results were calculated from a $A_{10}B_{7}A_{10}$ simulation run at $\phi_P = 0.20$, $k_B T / \varepsilon = 1.2$, $N_{\mathrm{poly}} = 2134$ and averaged over all micelles and time steps after the micellar crystal has formed.](rdf_crystal_fig){width="8cm"}
Structure of the lattice of micelles
------------------------------------
Micelles in the $A_{10}B_{7}A_{10}$ polymer system are arranged with the centers of mass sitting on a bcc lattice with a nearest neighbor spacing of $a_L = 11.53\sigma$. This number remarkably agrees with the length of the polymer if stretched completely into a straight line across the diameter of a circle from the center of one nearest neighbor micelle to the opposite one, $N_{\mathrm{mon}}
\cdot r_0 = 27 \cdot 0.83\sigma = 22.41\sigma \sim 2 a_L$, where $r_0 = 0.83\sigma$ is the equilibrium bond length. This suggests that polymers are maximally stretched, either along the diameter or in a bent configuration. Detailed visual examinations of simulation snapshots indicate that there are a significant number of polymers in the bent configuration, but there are also some in the linear extended configuration. The micellar cores are liquid-like, and over time, a given polymer constantly switches from linear to bent configurations.
The previous arguments also suggest a significant amount of overlap of the coronas between two neighboring micelles in the lattice. To examine this more quantitatively we calculate the micelle density distribution as a function of the radius averaged over all micelles and all time steps in the simulation after the micellar crystal has formed. shows the results, confirming the overlap. Hydrophilic $A$ beads from one micelle explore the solvent until occasionally bumping into the hydrophobic core of one of the nearest neighbor micelles.
A remarkable aspect of micellar crystals found in this work is the stability of the average aggregation number. If the polymers within the micelle are maximally stretched, the core of the micellar radius is $R_c=mr_0\sigma/2$, so the aggregation number can be estimated from $$\label{Eq:Navv_StrongStretched}
\langle N_{agg} \rangle = \frac{4\pi(m r_0/2)^3\sigma^3}{m r_0 \sigma^3}=\frac{\pi}{6} m^2 r^2_0
\ .$$ For the $A_{10}B_{7}A_{10}$ this yields $ \langle N_{agg} \rangle=17.7$, in good agreement with the simulation results in . Aggregation numbers for the other polymers simulated do not agree, implying that the polymers in those systems are not maximally stretched.
Conclusions {#sec:conclusions}
===========
Summary of results
------------------
Cubic micellar crystals of $A_nB_mA_n$ polymers form in MD simulations at sufficiently high concentrations. In order to form the crystals, high enough kinetic temperatures are needed to enable polymer transfer between micelles, which is critical for equilibrating the system. The polymer transfer process is activated and described by transition state theory. It results in an apparent diffusive behavior of the individual polymers while the lattice of micelles remains stable. Excessive polymer transfer at even higher kinetic temperatures triggers a disorder phase transition to a micelle liquid.
In the process of forming the ordered phase, the system spends a long time in a micellar liquid phase equilibration period where no ordered nucleates are present. Eventually, a large nucleation event takes place and the micellar crystal then grows very quickly, filling the entire simulation box in a relatively short time span. During this growth period, polymer transfer and movement of micelles are both crucial in the formation of the final micellar crystal. The preferred ordering near the disordered transition for all triblocks studied in this system is the bcc lattice. Only at lower kinetic temperatures and for polymers with short hydrophilic groups ($A_6B_7A_6$ at $k_B
T/\varepsilon=1.07$) is there some evidence for a stable fcc phase. These results are summarized in .
![\[fig:mic\_phase\_diag\] Summary of the phase diagram encompassing all simulated triblocks $A_nB_mA_n$ (all forming cubic phases with long range order). Near the disorder transition, bcc is always favored and fcc lattices only begin to appear at lower temperatures. At even lower temperatures polymer transfer becomes negligible and MD would require prohibitively long simulations to reach equilibrium. ](phase_diagram_fig){width="8cm"}
All micellar crystals obtained in this work are perfect single crystals displaying a high degree of order. Even in a periodic simulation box with four unit cells along a side, a single additional micelle can disrupt the resulting lattice significantly. The number of micelles is controlled by changing the number of polymers in the box, as the system displays a remarkable stability in the average aggregation number of the micelles that form. In the equilibrium lattice, micelles are closely packed with a significant amount of overlap between the coronas of neighboring micelles. Polymers in the micelles are highly stretched across the liquid hydrophobic core through the solvent and qualitatively well described within the strongly stretched approximation [@Semenov1985; @Grason2007].
Stability of the bcc lattice near the disorder transition
---------------------------------------------------------
Our simulations results show a strong preference for bcc lattices near the disorder transition. A similar result has been experimentally observed for PS-PI diblocks [@Lodge2002], where it has been attributed to the fact that near the disordered phase, micelle aggregation numbers are small. The phase diagram of f-star polymer systems shows that fcc lattices are only stable for large number of arms $f>60$, while bcc lattices are favored when the number of arms is small [@Watzlawek1999]. By considering polymeric micelles as f-star polymers, where the number of arms $f$ is given by $f\sim 2 \langle N_{agg} \rangle$ then bcc lattices are favored when the aggregation numbers are small $\langle N_{agg} \rangle \lesssim 30$. This argument, however, hinges on two key assumptions: the dynamic nature of micelles does not play a significant role and that the hydrophilic blocks are sufficiently long. The first assumption is already somewhat problematic given the importance of polymer transfer found in this work. As for the second assumption, a criteria establishing how long hydrophilic blocks should be has been put forward in Ref. [@McConnell1996], where it is shown that if the size of the corona is $L_c$ and the core radius $R_c$, bcc lattices are favored for $L_c/R_c > 1.5$. Our results for the $A_{10}B_7A_{10}$ and $A_{6}B_7A_{6}$ yield $L_c/R_c \sim 0.9$ and $L_c/R_c \sim 0.7$. It is therefore not possible to attribute the stability of the bcc lattices observed in our simulations as being a consequence of the small aggregation numbers $\langle N_{agg} \rangle \lesssim 30$.
It is tantalizing to interpret the stability of the bcc lattice in terms of the Alexander-McTague (AM) scenario [@Alexander1978], where it was argued that bcc should be generally expected to be the stable phase near a (weakly first order) disorder transition. Subsequent analysis however, showed that cubic lattices other than bcc cannot be ruled out near the disorder transition [@Groh1999; @Klein2001]. In Ref. [@Groh1999; @Klein2001] it is shown that the characteristic property of bcc lattices is that their free energy is closer to the disordered state, thus suggesting that bcc lattices follow an Ostwald step rule [@Ostwald1897], where the solid phase that nucleates first is the one whose free energy is closest to the disordered (or fluid) state. In this case, the complete crystallization process would require an additional step, where after the bcc crystallites are formed, they gradually evolve towards the stable thermodynamic phase. Certainly, it follows from our results that fcc lattices are difficult to obtain by MD for the systems we simulate, but at least in one system ($A_6B_7A_6$ at $k_B T/\varepsilon=1.07$) the fcc lattice has been obtained reproducibly, and did not proceed through an intermediate bcc step. In addition, for this very same system, closer to the disordered state ($k_B T/\varepsilon=1.1$), no fcc structure was found to be stable. Although not completely conclusive, our results are more consistent with the bcc as being a stable thermodynamic phase near the disordered phase.
There are serious limitations in identifying micelles as simple particles, because as the disordered phase is approached micelle aggregation numbers decrease and polymers become essentially free. So the disordered phase is not a simple liquid as it is assumed by AM and all subsequent work. Similar analysis in polymer melts [@Leibler1980], shows that in the vicinity of the spinodal, the only possible phase with cubic symmetry is bcc. Beyond the spinodal, other structures are favored [@Marques1990]. Based on the previous discussion, we attribute the stability of the bcc phases observed in our simulations as a reflection of the admittedly non-rigorous statement that bcc phases are *usually favored* near the disordered transition. We defer to future work to establish this result within a rigorous framework, where all the nuances involved in micelle formation are properly taken into account.
Implications for Pluronic systems
---------------------------------
The $A_{10}B_{7}A_{10}$ and $A_{20}B_{14}A_{20}$ polymers discussed in this paper provide coarse-grained descriptions of Pluronic F127. All simulations have been carried out in very good solvent conditions for the $A$ beads and at total volume fractions of 15–20%. Experimental results in this region of the phase diagram are surprisingly disparate. Simple cubic [@Prud1996], bcc [@Mortensen1995] and fcc [@Liu2000] have all been proposed as the structure in this region. Very recently, on the basis of new experimental data, the situation has been thoroughly reviewed by Li et al. [@Li2006] (although for significantly higher temperatures), see also Ref. [@Pozzo2007], but without clear conclusive results. Our theoretical analysis clearly favors the bcc lattice close to the disorder transition. The comparison of our results with the experimental F127 system is more accurate at low temperatures (at 20-25 C), where the water can be considered as a good solvent for PEO, as discussed in Ref. [@Anderson2006].
Outlook
-------
We have shown that MD allows a detailed investigation of both the dynamics as well as the thermodynamic equilibrium of micellar crystals. Many studies have been performed by modeling micelles as point particles, where the complex structure of the micelles is accommodated through refined two-body potentials, either derived analytically or empirically. While successful in many situations, two-body potentials do not account for the dynamic nature of micelles, which play a critical role in determining the phases of the system, particularly near the disorder transition.
There are a few areas where further work needs to be performed. First, all simulations in this work have been performed in good solvent conditions. MD with implicit solvents of different quality are also of great interest, especially to determine the phases of Pluronic systems over a wide range of temperatures [@Anderson2006]. Next, the largest micellar crystal formed in this work contains 2134 polymers (57,618 beads). We were unable to find any order in larger systems, even after running as many as 50 million time steps. It is possible that significantly longer simulations may be required for larger systems. Also, the range of applicability of MD is restricted to a relatively narrow range near the disorder transition, as schematically shown in . Finally, the role of the solvent will need to be investigated in more detail, as it is found in the Rouse vs. Zimm dynamics for simple homopolymers [@Doi1986]. Future studies will be necessary to completely clarify and expand all these issues.
The relevance of our study goes beyond pure systems. In Ref. [@Enlow2007] for example, Pluronic polymers have been used to template the growth of an inorganic phase of calcium phosphate, aimed at creating new polymer nanocomposites with lightweight/high strength properties or that mimic the structure of real bone. An understanding of the pure systems is clearly a prerequisite for accurate models of polymer nanocomposites. We hope to report more on this topic in the near future.
We acknowledge interest and discussions with S. Mallapragada, K. Schmidt-Rohr and G. Grason. We also thank J. Schmalian for clarifications regarding the Alexander-McTague paper. This work is supported by DOE-BES through the Ames lab under contract no. DE-AC02-07CH11358 and partially supported by the NSF through grant DMR-0426597.
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|
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"pile_set_name": "ArXiv"
}
|
---
abstract: 'Special kinds of rank 2 vector bundles with (possibly irregular) connections on ${{{\mathbb P}^1}}$ are considered. We construct an equivalence between the derived category of quasi-coherent sheaves on the moduli stack of such bundles and the derived category of modules over a TDO ring on certain non-separated curve. We identify this curve with the coarse moduli space of some parabolic bundles on ${{{\mathbb P}^1}}$. Then our equivalence becomes an example of the categorical Langlands correspondence.'
address:
- 'Department of mathematics, University of North Carolina, Chapel Hill, NC'
- |
Mathematics $\&$ Statistics\
Boston University\
111 Cummington St\
Boston, MA
author:
- 'D. Arinkin'
- 'R. Fedorov'
bibliography:
- 'IrregularLanglands.bib'
title: 'An example of the Langlands correspondence for irregular rank two connections on ${{{\mathbb P}^1}}$.'
---
Introduction
============
Let ${\mathcal{C}\mathit{onn}}(X,r)$ be the moduli space of rank $r$ vector bundles with connections on a smooth complex projective curve $X$. Let $\operatorname{\mathcal{B}\mathit{un}}(X,r)$ be the moduli space of rank $r$ vector bundles on $X$. The *categorical Langlands correspondence* for $GL(r)$ is a conjectural equivalence between the derived category of $O$-modules on ${\mathcal{C}\mathit{onn}}(X,r)$ and the derived category of ${\EuScript D}$-modules on $\operatorname{\mathcal{B}\mathit{un}}(X,r)$. We refer the interested reader to [@FrenkelAdvances §6.2].
This correspondence has been proved by one of the authors in the settings of rank two bundles equipped with connections with four simple poles on $X={{{\mathbb P}^1}}$ (cf. [@Arinkin]). In this case, the space $\operatorname{\mathcal{B}\mathit{un}}(X,r)$ should be replaced by the moduli space of bundles with parabolic structures. More precisely, [@Arinkin] works with $SL(2)$-connections and $PGL(2)$-bundles. (See [@FrenkelRamifications] for a discussion of the ramified Langlands program.)
In this paper we extend the results of [@Arinkin] to the case when the ramification divisor still has degree four but we allow higher order poles as long as leading terms are regular semisimple (see Theorems \[Langlands\] and \[MainTh\]). This provides an example of the categorical Langlands correspondence for connections with irregular singularities.
In [@FrenkelGross], Frenkel and Gross present an example of the Langlands correspondence for a different kind of irregular singularities. It is instructive to compare the two settings. Unlike the present paper, the results of Frenkel and Gross apply to arbitrary group $G$, not just $G=GL(2)$. The ramification considered is in a sense the simplest nontrivial: the ramification divisor has degree three. It is proved in [@FrenkelGross] that in these settings, there is a unique up to isomorphism local system with prescribed singularities. In other words, the counterpart of the moduli space ${\mathcal{C}\mathit{onn}}(X,r)$ consists of a single point. In particular, the category of $O$-modules on this space has a unique irreducible object, the structure sheaf of this point. The corresponding category of automorphic ${\EuScript D}$-modules (the counterpart of the category of ${\EuScript D}$-modules on $\operatorname{\mathcal{B}\mathit{un}}(X,r)$) also has a unique irreducible object ([@FrenkelGross] Sections 3, 16). The categorical Langlands transform sends the two irreducible generators into each other.
The present paper studies the ‘next simplest case’: the ramification divisor has degree four. The moduli space of local systems is a surface, and the categorical Langlands transform is an equivalence, similar to the Fourier-Mukai transform.
The techniques used in our argument are similar to that of [@Arinkin] but more conceptual. We hope that the present proof is more suitable for generalizations to divisors of higher order and to the higher genus case.
In positive characteristic, a different approach to Langlands correspondence was discovered by Bezrukavnikov and Braverman. In [@BezrukavnikovBraverman], they construct a version of the categorical Langlands correspondence. In [@Nevins], Nevins uses these ideas for connections with regular singularities.
Our argument requires two steps that may be of independent interest. Firstly, in §\[COMPACTIFICATION\] we prove that the moduli space of connections with possibly irregular singularities has a good moduli space in the sense of [@Alper]; we also construct a modular projective compactification of this space, see Theorems \[GoodModuliSpace\] and \[AmpleBundle\]. This is an extension of Simpson’s results [@SimICM; @Simpson1; @Simpson2].
Secondly, in §\[GENELL\] we study the compactified Jacobians of singular degenerations of elliptic curves, see Proposition \[PicY\], and construct a Fourier-Mukai transform, see Theorem \[FourierMukai\].
Finally, we want to note that our moduli spaces of connections are the moduli spaces of initial conditions of Painlevé equations. More precisely, the case of regular singularities corresponds to Painlevé VI, while the cases of irregular singularities correspond to Painlevé II–Painlevé V, see [@OhyamaOkumuraPainleve].
Conventions
-----------
We work over the ground field of complex numbers, thus ${{{\mathbb P}^1}}$ means $\mathbb{P}^1_{{\mathbb C}}$, a ‘scheme’ means a ‘${{\mathbb C}}$-scheme’ etc. All schemes and stacks are locally of finite type.
Acknowledgments
---------------
We benefited from talks with many mathematicians. The second author wants to especially thank David Ben–Zvi, Roman Bezrukavnikov, Ivan Mirkovic, Emma Previato, and Matthew Szczesny.
The first author is a Sloan Research Fellow, and he is grateful to the Alfred P. Sloan Foundation for the support.
Main Results
============
Let ${\mathfrak D}:=\sum n_ix_i$ be a divisor on ${{{\mathbb P}^1}}=\mathbb{P}^1_{{\mathbb C}}$ with $n_i>0$. Let $L$ be a rank 2 vector bundle on ${{{\mathbb P}^1}}$, $\nabla:L\to L\otimes\Omega_{{{\mathbb P}^1}}({\mathfrak D})$ a connection on $L$ with polar divisor ${\mathfrak D}$. We call such pairs $(L,\nabla)$ *connections* for brevity.
Choosing a formal coordinate $z$ near $x_i$ and a trivialization of $L$ on the formal neighborhood of $x_i$, we can write $\nabla$ near $x_i$ as $${\mathbf d}+a\frac{{\mathbf d}z}{z^{n_i}}+\text{higher order terms},\qquad
a\in\operatorname{\mathfrak{gl}}(2).$$ In the case $n_i>1$ the connection will be called *non-resonant* at $x_i$ if $a$ has distinct eigenvalues; in the case $n_i=1$ the connection will be called non-resonant if the eigenvalues do not differ by an integer. The connection will be called non-resonant if it is non-resonant at all $x_i$. Up to scaling, the conjugacy class of $a$ does not depend on the choices. Thus the notion of non-resonant connections does not depend on the choices.
Moduli stacks {#MODST}
-------------
Let $(L,\nabla)$ be a non-resonant connection, then in a suitable trivialization of $L$ over the formal disc at $x_i$ the connection takes a diagonal form $$\label{fnf}
\nabla={\mathbf d}+\begin{pmatrix} \alpha_i^+ & 0\\ 0&\alpha_i^-
\end{pmatrix},$$ where $\alpha_i^\pm$ are 1-forms on the formal disc. The polar parts of these 1-forms do not depend on the trivialization of $L$, thus we shall call them *the formal type of $\nabla$ at $x_i$*.
Fix ${\mathfrak D}$ and for each $i$ polar parts $\alpha_i^\pm$ of 1-forms at $x_i$. Assume that these polar parts satisfy the following conditions\
[[)]{}]{}\[AlphaI\] The order of the pole of $\alpha_i^\pm$ is at most $n_i$, the order of the pole of $\alpha_i^+-\alpha_i^-$ is exactly $n_i$.\
[[)]{}]{}\[AlphaII\] $d:=-\sum_i\operatorname{res}(\alpha_i^++\alpha_i^-)$ is integer. [[)]{}]{}\[AlphaIII\] $\sum_i\operatorname{res}\alpha_i^\pm\notin{{\mathbb Z}}$. (Here for each $i$ there is exactly one summand $\alpha_i^\pm$, and the choices of signs $+$ and $-$ are independent.)\
[[)]{}]{}\[AlphaIV\] If $n_i=1$, then $\operatorname{res}\alpha_i^+-\operatorname{res}\alpha_i^-\notin{{\mathbb Z}}$.
Let ${{\mathcal M}}={{\mathcal M}}({\mathfrak D},\alpha_i^\pm)$ be the moduli space of connections $(L,\nabla)$ such that $\nabla$ has formal types $\alpha_i^\pm$ at $x_i$. Note that such a connection is non-resonant by (\[AlphaI\]) and . Also, $L$ has degree $d$ by (\[AlphaII\]).
From now on we assume that $\deg{\mathfrak D}=4$.
\[StackProp\] The moduli space ${{\mathcal M}}$ is a smooth connected algebraic stack of dimension $1$. It is a neutral ${{\mathbf{G_m}}}$-gerbe over its coarse moduli space $M$; besides, $M$ is a smooth quasi-projective surface.
This theorem will be proved in §\[MODULISPACES\].
Let $\mathrm{Qcoh}({{\mathcal M}})$ be the category of quasi-coherent sheaves on ${{\mathcal M}}$. Since ${{\mathcal M}}$ is a ${{\mathbf{G_m}}}$-gerbe, we obtain a decomposition $$\mathrm{Qcoh}({{\mathcal M}})=\prod_{i\in{{\mathbb Z}}}\mathrm{Qcoh}({{\mathcal M}})^{(i)},$$ where ${{\mathcal F}}\in\mathrm{Qcoh}({{\mathcal M}})^{(i)}$ if $t\in{{\mathbf{G_m}}}$ acts on ${{\mathcal F}}$ as $t^i$. Let ${{\mathcal D}}^b({{\mathcal M}})$ be the corresponding bounded derived category. By definition, objects of ${{\mathcal D}}^b({{\mathcal M}})$ are complexes of $O_{{\mathcal M}}$-modules with quasi-coherent cohomology. It follows from [@ArinkinBezrukavnikov Claim 2.7] that ${{\mathcal D}}^b({{\mathcal M}})$ is equivalent to the bounded derived category of $\mathrm{Qcoh}({{\mathcal M}})$. Thus we also have a decomposition $${{\mathcal D}}^b({{\mathcal M}})=\prod_{i\in{{\mathbb Z}}}{{\mathcal D}}^b({{\mathcal M}})^{(i)}.$$ It is easy to see that ${{\mathcal F}}\in{{\mathcal D}}^b({{\mathcal M}})^{(i)}$ if and only if $H^\bullet({{\mathcal F}})\in\mathrm{Qcoh}({{\mathcal M}})^{(i)}$.
Twisted differential operators {#TDO}
------------------------------
Denote by $\wp:P\to{{{\mathbb P}^1}}$ the projective line with points $x_i$ doubled. In other words, $P$ is obtained by gluing two copies of ${{{\mathbb P}^1}}$ outside the support of ${\mathfrak D}$. Denote the preimages of $x_i$ by $x_i^-$ and $x_i^+$. Let $j:{{{\mathbb P}^1}}-{\mathfrak D}\hookrightarrow P$ be the natural embedding. This notation will be used throughout the paper.
The main result of the present paper is that ${{\mathcal D}}^b({{\mathcal M}})^{(-1)}$ is equivalent to a category of twisted ${\EuScript D}$-modules on $P$. To give a precise definition of this twist, recall that the isomorphism classes of sheaves of rings of twisted differential operators (TDO) on a smooth (not necessarily separated) curve are classified by the first cohomology group of the sheaf of 1-forms.
\[CohP\] Denote by $\omega_i$ the vector space of polar parts of 1-forms at $x_i\in{{{\mathbb P}^1}}$. Then $$H^1(P,\Omega_P)={{\mathbb C}}\oplus\bigoplus_i\omega_i.$$
Let $D_i^\pm$ be the formal disc at $x_i^\pm$. These discs, together with ${{{\mathbb P}^1}}-{\mathfrak D}$, give a cover of $P$; let us use the corresponding Čech complex. We see that a 1-cocycle is a collection $\beta_i^\pm$ of 1-forms on punctured formal discs, and one easily checks that the map $(\beta_i^\pm)\mapsto(\,\sum_i\operatorname{res}(\beta_i^++\beta_i^-),\beta_i^+-\beta_i^-)$ induces the required isomorphism.
Using this lemma, we define the sheaf of differential operators on $P$ twisted by $(-d,\alpha_i^+-\alpha_i^-)$; denote it by ${\EuScript D}_{P,\alpha}$. In other words, it is given by the 1-cocycle $(\alpha_i^\pm)$.
The integral transform
----------------------
Let $\xi=(L,\nabla)\in{{\mathcal M}}$. Denote by $\xi_\alpha$ the ${\EuScript D}_{P,\alpha}$-module generated by $\wp^*\xi$. More precisely, $\xi_\alpha:=j_{!*}(\xi|_{{{{\mathbb P}^1}}-{\mathfrak D}})$, where $j_{!*}$ is the middle extension for ${\EuScript D}_{P,\alpha}$-modules. Since $\xi|_{{{{\mathbb P}^1}}-{\mathfrak D}}$ is a ${\EuScript D}_{{{{\mathbb P}^1}}-{\mathfrak D}}$-module, and the twist of ${\EuScript D}_{P,\alpha}$ is supported outside of ${{{\mathbb P}^1}}-{\mathfrak D}$, $\xi_\alpha$ is well defined.
\[ForDisc\] Let us describe the restriction of $\xi_\alpha$ to $\wp^{-1}(D_i)$, where $D_i$ is the formal disc centered at $x_i$. Choose 1-forms $\tilde\alpha^\pm$ with polar parts $\alpha_i^\pm$. According to , the restriction of $\xi$ to $D_i$ is isomorphic to $$(O_{D_i},{\mathbf d}+\tilde\alpha^-)\oplus(O_{D_i},{\mathbf d}+\tilde\alpha^+).$$ Now one checks easily that $$\begin{split}
\xi_\alpha|_{D_i^-}&\simeq
(O_{D_i^-},{\mathbf d}+\tilde\alpha^-)\oplus(O_{\dot D},{\mathbf d}+\tilde\alpha^+),\\
\xi_\alpha|_{D_i^+}&\simeq (O_{\dot
D},{\mathbf d}+\tilde\alpha^-)\oplus(O_{D_i^+},{\mathbf d}+\tilde\alpha^+),
\end{split}$$ where $\dot D\subset D_i^\pm$ is the punctured formal disc.
Note that formal normal form exists for families of connections, and it is constant for families in ${{\mathcal M}}$. Thus our middle extension construction still makes sense for families of connections in ${{\mathcal M}}$. Hence we can apply it to the universal family $\xi$ on ${{\mathcal M}}\times{{{\mathbb P}^1}}$, getting an ${{\mathcal M}}$-family $\xi_\alpha$ of ${\EuScript D}_{P,\alpha}$-modules. In other words, $\xi_\alpha$ is an $O_{{\mathcal M}}\boxtimes{\EuScript D}_{P,\alpha}$-module on ${{\mathcal M}}\times P$. Thus $\xi_\alpha$ gives rise to an integral transform from ${{\mathcal D}}^b({{\mathcal M}})$ to the derived category of ${\EuScript D}_{P,\alpha}$-modules.
Denote the natural projections of ${{\mathcal M}}\times P$ to ${{\mathcal M}}$ and $P$ by $p_1$ and $p_2$ respectively.
\[MainTh\] Let $d$ be an odd number. Then the functor $$\Phi_{{{\mathcal M}}\to P}:{{\mathcal F}}\mapsto
Rp_{2,*}\Bigl(\xi_{\alpha}\mathop{\otimes}\limits_{O_{{{\mathcal M}}\times
P}}p_1^*{{\mathcal F}}\Bigr)$$ is an equivalence between ${{\mathcal D}}^b({{\mathcal M}})^{(-1)}$ and the bounded derived category of ${\EuScript D}_{P,\alpha}$-modules.
Theorem \[MainTh\] is the main result of the paper; we prove it in §§\[PRELYM\]–\[LYSENKO\].
\[RemOdd\] [[)]{}]{}It is easy to see that the restriction of $\Phi_{{{\mathcal M}}\to P}$ to ${{\mathcal D}}^b({{\mathcal M}})^{(i)}$ is zero unless $i=-1$.
[[)]{}]{}\[Remodd2\] On the other hand, ${{\mathcal D}}^b({{\mathcal M}})^{(i)}$ depends only on parity of $i$. Indeed, fix $x\in{{{\mathbb P}^1}}-{\mathfrak D}$, and let $\delta_x$ be the line bundle on ${{\mathcal M}}$ whose fiber at $(L,\nabla)$ is equal to $\det L_x$. Then the tensor product with $\delta_x$ provides an equivalence between ${{\mathcal D}}^b({{\mathcal M}})^{(i)}$ and ${{\mathcal D}}^b({{\mathcal M}})^{(i+2)}$.
[[)]{}]{}\[RemOdd3\] Assume that ${\mathfrak D}=\sum n_ix_i$ is not even, that is one of the numbers $n_i$ is odd, then all the categories ${{\mathcal D}}^b({{\mathcal M}})^{(i)}$ are equivalent. Indeed, let $n_i$ be odd, and for $(L,\nabla)\in{{\mathcal M}}$ let $\eta_i$ be a unique level $n_i$ parabolic structure at $x_i$ compatible with $\nabla$ (see Definition \[ParB\] and §\[AFFSTR\]). Tensoring with the line bundle whose fiber at $(L,\nabla)$ is $\det\eta_i$, we get an equivalence between the odd and the even components of the derived category.
[[)]{}]{}In fact our theorem is also valid if $d$ is even but ${\mathfrak D}$ is not even. Indeed, let $n_i$ be odd and define a collection $\beta_i^\pm$ of polar parts of 1-forms by $$\beta_i^+=\alpha_i^++n_i\,\frac{{\mathbf d}z}z,\qquad
\beta_i^-=\alpha_i^-,\qquad
\beta_j^\pm=\alpha_j^\pm\text{ for }i\ne j.$$ Then a modification at $x_i$ provides an isomorphism ${{\mathcal M}}(\alpha)\simeq{{\mathcal M}}(\beta)$, and we can apply the theorem to ${{\mathcal M}}(\beta)$. (See §\[PARBUN\] for the definition of modification.) It remains to notice that the category of ${\EuScript D}_{P,\alpha}$-modules is equivalent to the category of ${\EuScript D}_{P,\beta}$-modules: the equivalence is given by tensoring with a line bundle.
The Langlands Correspondence {#LANGLANDS}
----------------------------
\[ParB\] Let $L$ be a rank 2 vector bundle on ${{{\mathbb P}^1}}$. A *level-${\mathfrak D}$ parabolic structure* on $L$ is a line subbundle $\eta$ in the restriction of $L$ to ${\mathfrak D}$ (we view ${\mathfrak D}$ as a non-reduced subscheme of ${{{\mathbb P}^1}}$). We call a bundle with a parabolic structure *a parabolic bundle*.
Let $\overline\operatorname{\mathcal{B}\mathit{un}}(d^\vee)=\overline\operatorname{\mathcal{B}\mathit{un}}({{{\mathbb P}^1}},2,d^\vee,{\mathfrak D})$ be the moduli stack of rank 2 degree $d^\vee$ vector bundles on ${{{\mathbb P}^1}}$ with level-${\mathfrak D}$ parabolic structures. (We reserve notation $\operatorname{\mathcal{B}\mathit{un}}$ for its open substack of bundles without non-scalar endomorphisms, cf. §\[AFFSTR\].)
Let ${{\mathbb C}}[{\mathfrak D}]$ be the ring of functions on the scheme ${\mathfrak D}$, ${{{\mathbb C}}[{\mathfrak D}]^\times}$ be the group of invertible functions; this is an algebraic group. Choosing local coordinates at the points $x_i$, we get an isomorphism $${{{\mathbb C}}[{\mathfrak D}]^\times}=\prod_i({{\mathbb C}}[z]/z^{n_i})^\times.$$ Let $\pi:\eta_{univ}\to\overline\operatorname{\mathcal{B}\mathit{un}}(d^\vee)$ be the ${{{\mathbb C}}[{\mathfrak D}]^\times}$-torsor whose fiber over $(L,\eta)\in\overline\operatorname{\mathcal{B}\mathit{un}}(d^\vee)$ is $$\{s\in H^0({\mathfrak D},\eta)|\;s(x_i)\ne0\;\text{for all }i\}.$$
The collection $\alpha_i^+$ of polar parts of 1-forms can be viewed as an element of $$(\operatorname{\mathrm{Lie}}({{{\mathbb C}}[{\mathfrak D}]^\times}))^\vee={{\mathbb C}}[{\mathfrak D}]^\vee$$ via the residue pairing. Thus it gives rise to a TDO ring on $\overline\operatorname{\mathcal{B}\mathit{un}}(d^\vee)$ through non-commutative reduction of the sheaf of differential operators on the total space of $\eta_{univ}$ (see §\[TDOEXISTENCE\]). We denote this TDO ring by ${\EuScript D}_{\overline\operatorname{\mathcal{B}\mathit{un}}(d^\vee),\alpha^+}$.
Similarly, we define a ${{{\mathbb C}}[{\mathfrak D}]^\times}$-torsor $\eta'_{univ}$ whose fiber over $(L,\eta)\in\overline\operatorname{\mathcal{B}\mathit{un}}(d^\vee)$ is $$\{s\in H^0({\mathfrak D},(L|_{\mathfrak D})/\eta)|\;s(x_i)\ne0\;
\text{ for all }i\}.$$ Denote the TDO ring corresponding to $\eta'_{univ}$ and the collection $\alpha_i^-$ by ${\EuScript D}_{\overline\operatorname{\mathcal{B}\mathit{un}}(d^\vee),\alpha^-}$. Let ${\EuScript D}_{\overline\operatorname{\mathcal{B}\mathit{un}}(d^\vee),\alpha}$ be the Baer sum of the TDO rings ${\EuScript D}_{\overline\operatorname{\mathcal{B}\mathit{un}}(d^\vee),\alpha^+}$ and ${\EuScript D}_{\overline\operatorname{\mathcal{B}\mathit{un}}(d^\vee),\alpha^-}$.
\[Langlands\] Assume that $d$ is an odd number. Then ${{\mathcal D}}^b({{\mathcal M}})^{(-1)}$ is equivalent to the bounded derived category of ${\EuScript D}_{\overline\operatorname{\mathcal{B}\mathit{un}}(-1),\alpha}$-modules.
Theorem \[Langlands\] is derived from Theorem \[MainTh\] in §\[ProofOfLanglands\].
\[RemEven\] [[)]{}]{}Let us discuss the notion of the derived category of ${\EuScript D}_{\overline\operatorname{\mathcal{B}\mathit{un}}(-1),\alpha}$-modules. Note that $\overline\operatorname{\mathcal{B}\mathit{un}}(-1)$ is a (smooth) algebraic stack, so this notion is not immediate.
As we show in §\[ProofOfLanglands\] the category of ${\EuScript D}_{\overline\operatorname{\mathcal{B}\mathit{un}}(-1),\alpha}$-modules is equivalent to a category of twisted ${\EuScript D}$-modules on $P$ (in fact $P$ is the coarse moduli space of a certain open subset of $\overline\operatorname{\mathcal{B}\mathit{un}}(-1)$). Thus we shall view the derived category of twisted ${\EuScript D}$-modules on $P$ as the definition for the derived category of ${\EuScript D}_{\overline\operatorname{\mathcal{B}\mathit{un}}(-1),\alpha}$-modules. (See also the discussion in §\[ProofOfLanglands\].)
[[)]{}]{}In general, we expect an equivalence of categories between ${{\mathcal D}}^b({{\mathcal M}})^{(d^\vee)}$ and the bounded derived category of ${\EuScript D}_{\overline\operatorname{\mathcal{B}\mathit{un}}(d^\vee),\alpha}$-modules. This statement follows from Theorem \[Langlands\] if $d^\vee$ is odd. Indeed, pick $x\in{{{\mathbb P}^1}}-{\mathfrak D}$, then $(L,\eta)\mapsto(L(x),\eta)$ is an isomorphism between $\overline\operatorname{\mathcal{B}\mathit{un}}(d^\vee)$ and $\overline\operatorname{\mathcal{B}\mathit{un}}(d^\vee+2)$. It remains to use Remark \[RemOdd\](\[Remodd2\]).
[[)]{}]{}We also have the desired equivalence if ${\mathfrak D}$ is not even. Indeed, let $n_i$ be odd. Modification at $x_i$ gives an isomorphism ${\EuScript D}_{\overline\operatorname{\mathcal{B}\mathit{un}}(d^\vee),\alpha}\simeq
{\EuScript D}_{\overline\operatorname{\mathcal{B}\mathit{un}}(d^\vee+n_i),\beta}$, where $\beta$ is obtained from $\alpha$ by swapping $\alpha_i^+$ and $\alpha_i^-$. It remains to use the previous remark, Remark \[RemOdd\](\[RemOdd3\]), and the obvious identification ${{\mathcal M}}(\alpha)={{\mathcal M}}(\beta)$.
The plan of proof of Theorem \[MainTh\] {#PRELYM}
---------------------------------------
Theorem \[MainTh\] reduces to two orthogonality statements.
Let $p_{12}:P\times P\times{{\mathcal M}}\to P\times P$ and $p_{13},p_{23}:P\times P\times{{\mathcal M}}\to P\times{{\mathcal M}}$ be the projections. Let $\xi^\vee$ be the vector bundle on ${{{\mathbb P}^1}}\times{{\mathcal M}}$ dual to $\xi$. Since it has a connection along ${{{\mathbb P}^1}}$, we see that $\xi^\vee_\alpha:=(\xi^\vee)_{-\alpha}$ is a ${\EuScript D}_{P,-\alpha}\boxtimes O_{{\mathcal M}}$-module on $P\times{{\mathcal M}}$.
Set ${{\mathcal F}}_P:=(p_{13}^*\xi_\alpha)\otimes(p_{23}^*\xi_\alpha^\vee)$. Here $p_{13}^*$ and $p_{23}^*$ stand for the $O$-module pullback (from the viewpoint of ${\EuScript D}$-modules, these pullback functors should include a cohomological shift). Note that $\xi_\alpha$ is a flat $O_{P\times{{\mathcal M}}}$-module (see Remark \[ForDisc\]), hence $$(p_{13}^*\xi_\alpha)\otimes(p_{23}^*\xi_\alpha^\vee)=
(p_{13}^*\xi_\alpha)\otimes^L(p_{23}^*\xi_\alpha^\vee).$$
Further, $Rp_{12,*}{{\mathcal F}}_P$ is an object of the derived category of $p_1^{{\bullet}}{\EuScript D}_{P,\alpha}\circledast
p_2^{{\bullet}}{\EuScript D}_{P,-\alpha}$-modules, where $p_1,p_2:P\times P\to
P$ are the projections. Here $p_i^{{\bullet}}$ (resp. $\circledast$) stands for the inverse image (resp. Baer sum) of TDO rings (the corresponding functors on Lie algebroids are described in [@BeilinsonBernstein]).
\[Theorem3\] $Rp_{12,*}{{\mathcal F}}_P=\delta_\Delta[-1]$, where $\Delta\subset
P\times P$ is the diagonal, and $\delta_\Delta$ is the direct image of $O_{\Delta}$ as a ${\EuScript D}_\Delta$-module.
In general, for a map $f:X\to Y$ and a TDO ring ${\EuScript D}_1$ on $Y$, there is a functor $f_+:{{\mathcal D}}^b(f^{{\bullet}}{\EuScript D}_1)\to{{\mathcal D}}^b({\EuScript D}_1)$, where ${{\mathcal D}}^b({\EuScript D}_1)$ is the bounded derived category of ${\EuScript D}_1$-modules. For the embedding $i:\Delta\hookrightarrow
P\times P$, one easily checks that $i^{{\bullet}}(p_1^{{\bullet}}{\EuScript D}_{P,\alpha}\circledast
p_2^{{\bullet}}{\EuScript D}_{P,-\alpha})$ is the (non-twisted) differential operator ring ${\EuScript D}_\Delta$, so $\delta_\Delta:=i_+(O_\Delta)$ is well defined as a $p_1^{{\bullet}}{\EuScript D}_{P,\alpha}\circledast
p_2^{{\bullet}}{\EuScript D}_{P,-\alpha}$-module.
By Theorem \[Theorem3\], $\xi_\alpha$ is an orthogonal $P$-family of vector bundles on ${{\mathcal M}}$. To obtain an equivalence of categories, one should also show that $\xi_\alpha$ is orthogonal as an ${{\mathcal M}}$-family of ${\EuScript D}_{P,\alpha}$-modules. Let us give the precise statement. We follow closely S. Lysenko’s unpublished notes [@Lysenko].
Consider ${{\mathcal F}}_{{\mathcal M}}:=p_{13}^*\xi_\alpha\otimes
p_{23}^*\xi_\alpha^\vee$ (here $p_{13},p_{23}:{{\mathcal M}}\times{{\mathcal M}}\times P\to {{\mathcal M}}\times P$ are the projections).
${{\mathcal F}}_{{\mathcal M}}$ can be viewed as a family of ${\EuScript D}_P$-modules parameterized by ${{\mathcal M}}\times{{\mathcal M}}$. Consider the de Rham complex of ${{\mathcal F}}_{{\mathcal M}}$ in the direction of $P$ $$\operatorname{\mathbb{DR}}({{\mathcal F}}_{{\mathcal M}})=\operatorname{\mathbb{DR}}_P({{\mathcal F}}_{{\mathcal M}}):=({{\mathcal F}}_{{\mathcal M}}\to{{\mathcal F}}_{{\mathcal M}}\otimes
\Omega_{{{\mathcal M}}\times{{\mathcal M}}\times P/{{\mathcal M}}\times{{\mathcal M}}}).$$ Our aim is to compute $Rp_{12,*}\operatorname{\mathbb{DR}}({{\mathcal F}}_{{\mathcal M}})$.
By Theorem \[StackProp\], ${{\mathcal M}}\times{{\mathcal M}}$ is a ${{\mathbf{G_m}}}\times{{\mathbf{G_m}}}$-gerbe over a scheme, so ${{\mathbf{G_m}}}\times{{\mathbf{G_m}}}$ acts on any quasi-coherent sheaf ${{\mathcal F}}$ on ${{\mathcal M}}$. Therefore, ${{\mathcal F}}$ can be decomposed with respect to the characters of ${{\mathbf{G_m}}}\times{{\mathbf{G_m}}}$. Denote by ${{\mathcal F}}^\psi$ the component of ${{\mathcal F}}$ corresponding to the character $\psi:{{\mathbf{G_m}}}\times{{\mathbf{G_m}}}\to{{\mathbf{G_m}}}$ defined by $(t_1,t_2)\mapsto t_1/t_2$.
Let $\operatorname{diag}:{{\mathcal M}}\to{{\mathcal M}}\times{{\mathcal M}}$ be the diagonal morphism.
\[ThLys\] $Rp_{12,*}\operatorname{\mathbb{DR}}({{\mathcal F}}_{{\mathcal M}})=(\operatorname{diag}_*O_{{\mathcal M}})^\psi[-2]$.
\[RemDiag\] Note that ${{\mathcal M}}$ is a ${{\mathbf{G_m}}}$-torsor over $\operatorname{diag}({{\mathcal M}})$. Thus $$\operatorname{diag}_*O_{{\mathcal M}}=\bigoplus_{i\in{{\mathbb Z}}}(\operatorname{diag}_*O_{{\mathcal M}})^{\psi^i}.$$
Objects of ${{\mathcal D}}^b({{\mathcal M}}\times{{\mathcal M}})$ define endofunctors on ${{\mathcal D}}^b({{\mathcal M}})$. Let us consider the functors corresponding to the components of $\operatorname{diag}_*O_{{\mathcal M}}$. Clearly, $${{\mathcal D}}^b({{\mathcal M}})\to{{\mathcal D}}^b({{\mathcal M}}):{{\mathcal F}}\mapsto Rp_{2,*}((\operatorname{diag}_*O_{{\mathcal M}})\otimes p_1^*{{\mathcal F}})$$ is isomorphic to the identity functor. It is easy to see that the functor $${{\mathcal D}}^b({{\mathcal M}})\to{{\mathcal D}}^b({{\mathcal M}}):{{\mathcal F}}\mapsto Rp_{2,*}((\operatorname{diag}_*O_{{\mathcal M}})^{\psi^i}\otimes p_1^*{{\mathcal F}})$$ is isomorphic to the projection ${{\mathcal D}}^b({{\mathcal M}})\to{{\mathcal D}}^b({{\mathcal M}})^{(-i)}$.
The inverse to the functor $\Phi_{{{\mathcal M}}\to P}$ is given by $$\Phi_{P\to{{\mathcal M}}}:{{\mathcal F}}\mapsto
Rp_{1,*}\operatorname{\mathbb{DR}}(\xi_\alpha^\vee
\otimes p_2^*{{\mathcal F}})[2].$$ Indeed, using base change and Theorem \[Theorem3\], one checks that the composition $\Phi_{{{\mathcal M}}\to
P}\circ\Phi_{P\to{{\mathcal M}}}$ is isomorphic to the identity functor. Similarly, it follows from Theorem \[ThLys\] and Remark \[RemDiag\] that the composition $\Phi_{P\to{{\mathcal M}}}\circ\Phi_{{{\mathcal M}}\to P}$ is isomorphic to the projection ${{\mathcal D}}^b({{\mathcal M}})\to{{\mathcal D}}^b({{\mathcal M}})^{(-1)}$.
A compactification of moduli spaces of connections {#COMPACTIFICATION}
==================================================
In this section, we compactify a moduli space of connections with singularities following C. Simpson ([@SimICM; @Simpson1; @Simpson2]).
In [@Simpson2], C. Simpson constructs a natural compactification of the moduli space of vector bundles with connections on a smooth projective variety $X$. We consider the case when $X$ is a smooth projective curve. In this case it is not hard to generalize the result to the case of connections with singularities (we use [@Simpson1]). Then we prove that the compactification is in fact projective (note that for varieties of higher dimension projectivity of the compactification is not known). Our description of the divisor at infinity is also more explicit than in [@Simpson2].
The compactification is constructed in the following generality. Let $X$ be a smooth complex projective curve, $r$ a positive integer, $d$ an integer, and ${\mathfrak D}$ an effective divisor on $X$. Denote by ${{\mathcal N}}={{\mathcal N}}(X,r,d,{\mathfrak D})$ the moduli stack of pairs $(L,\nabla)$, where $L$ is a vector bundle on $X$ of rank $r$ and degree $d$, and $\nabla:L\to
L\otimes\Omega_X({\mathfrak D})$ is a connection on $L$ with the order of poles bounded by ${\mathfrak D}$. Our goal is to construct a compactification of the semistable part of ${{\mathcal N}}$.
Fix $X$, $r$, $d$, and ${\mathfrak D}$.
${\varepsilon}$-connections
---------------------------
The compactification is constructed as a moduli space of P. Deligne’s $\lambda$-connections. Recall the following
\[EpsilonConnections\] Let $L$ be a vector bundle on $X$. For a one-dimensional vector space $E$ and ${\varepsilon}\in E$, an *${\varepsilon}$-connection* on $L$ is a ${{\mathbb C}}$-linear map $\nabla:L\to L\otimes\Omega_X\otimes_{{\mathbb C}}E$ such that $$\nabla(fs)=f\nabla s+s\otimes{\mathbf d}f\otimes{\varepsilon}\;\text{
for all }f\in O_X,s\in L.$$ More generally, an ${\varepsilon}$-connection on $L$ with poles bounded by ${\mathfrak D}$ is a map $\nabla:L\to
L\otimes\Omega_X({\mathfrak D})\otimes_{{\mathbb C}}E$ satisfying the same condition.
Denote by ${{\overline{\mathstrut{{\mathcal N}}}}}={{\overline{\mathstrut{{\mathcal N}}}}}(X,r,d,{\mathfrak D})$ the moduli stack of collections $(L,\nabla;{\varepsilon}\in E)$, where $L$ is a vector bundle on $X$ of rank $r$ and degree $d$, and $\nabla$ is an ${\varepsilon}$-connection on $L$ with poles bounded by ${\mathfrak D}$. This is an algebraic stack, the proof is similar to [@Fedorov Proposition 1].
$(L,\nabla;{\varepsilon}\in E)\in{{\overline{\mathstrut{{\mathcal N}}}}}$ is *semistable* if for any non-zero $\nabla$-invariant subbundle $L_0\subset L$ we have $$\frac{\deg L_0}{\operatorname{rk}L_0}\le\frac{d}{r}.$$ Further, $(L,\nabla;{\varepsilon}\in E)$ is *nilpotent* if ${\varepsilon}=0$ and $\nabla^r=0$. Note that if ${\varepsilon}=0$, $\nabla$ is $O_X$-linear, so $\nabla^r$ makes sense as a map $L\to L\otimes(\Omega_X({\mathfrak D})\otimes_{{\mathbb C}}E)^{\otimes r}$. Equivalently, $(L,\nabla;{\varepsilon}\in E)$ is nilpotent if there is a flag of subbundles $$0=L_0\subset L_1\subset\dots\subset L_k=L$$ with $\nabla(L_i)\subset
L_{i-1}\otimes\Omega_X({\mathfrak D})\otimes_{{\mathbb C}}E$.
Let ${{\overline{\mathstrut{{\mathcal N}}}}}^{ss}\subset{{\overline{\mathstrut{{\mathcal N}}}}}$ be the open substack of semistable ${\varepsilon}$-connections.
Also, let ${{\overline{\mathstrut{{\mathcal N}}}}}^{\,ss,nn}\subset{{\overline{\mathstrut{{\mathcal N}}}}}^{ss}$ be the open substack of semistable ${\varepsilon}$-connections that are not nilpotent.
Taking $E={{\mathbb C}}$, ${\varepsilon}=1$, we see that connections are particular cases of ${\varepsilon}$-connections. Moreover if ${\varepsilon}\ne0$, there is a unique isomorphism $E\to{{\mathbb C}}$ such that ${\varepsilon}\mapsto 1$. It follows that the open substack of ${{\overline{\mathstrut{{\mathcal N}}}}}$ corresponding to ${\varepsilon}$-connections with ${\varepsilon}\ne0$ parameterizes all connections $(L,\nabla)$, where $L$ has rank $r$ and degree $d$, $\nabla$ has poles bounded by ${\mathfrak D}$. Thus this substack can be identified with ${{\mathcal N}}$.
We use the theory of good moduli spaces developed by J. Alper (see [@Alper]). By definition, a quasi-compact map $p:{{\mathcal S}}\to S$ from a stack ${{\mathcal S}}$ to an algebraic space $S$ is a *good moduli space* if the direct image functor $p_*$ is exact on quasi-coherent sheaves and $p_*O_{{\mathcal S}}=O_S$. In particular, this notion reduces to the notion of quotient in the sense of geometric invariant theory when ${{\mathcal S}}$ is the quotient stack of a scheme by an action of an algebraic group (see [@Alper Theorem 13.6]). Note that by [@Alper Theorem 4.16(vi)] $p$ is universal among maps to schemes.
\[GoodModuliSpace\][[)]{}]{}There is a good moduli space [[(]{}-1pt]{}in the sense of [@Alper][[)]{}]{} $p:{{\overline{\mathstrut{{\mathcal N}}}}}^{\,ss,nn}\to{{\overline N}}$ such that ${{\overline N}}$ is a complete scheme.
[[)]{}]{}\[GoodModB\] Set ${{\mathcal N}}^{ss}={{\overline{\mathstrut{{\mathcal N}}}}}^{ss}\cap{{\mathcal N}}={{\overline{\mathstrut{{\mathcal N}}}}}^{\,ss,nn}\cap{{\mathcal N}}$, then ${{\mathcal N}}^{ss}$ has a good moduli space $N$, which is an open subscheme of ${{\overline N}}$ [[(]{}-1pt]{}so $N$ is the moduli space of semistable bundles with connections[[)]{}]{}.
\[SecondFromFirst\] Part of the theorem is clear from the construction of ${{\overline N}}$ in §\[CONSTRUCTION\] but it also easily follows from the first claim and Corollary \[BundlesOnM\]. Indeed, let ${{\mathcal E}}$ be the line bundle on ${{\overline{\mathstrut{{\mathcal N}}}}}^{\,ss,nn}$ whose fiber over $(L,\nabla;{\varepsilon}\in E)$ is $E$. As shown in Corollary \[BundlesOnM\], there is a line bundle ${{\mathcal E}}'$ on ${{\overline N}}$ such that $p^*{{\mathcal E}}'={{\mathcal E}}^{\otimes r!}$. Then by [@Alper Proposition 4.5] we can identify $p_*{{\mathcal E}}^{\otimes r!}$ with ${{\mathcal E}}'$. Thus ${\varepsilon}^{\otimes r!}$ gives a section of ${{\mathcal E}}'$; let $N$ be the complement of its zero locus. Clearly $N$ is open and ${{\mathcal N}}=p^{-1}(N)$. This implies the claim.
Construction of ${{\overline N}}$ {#CONSTRUCTION}
---------------------------------
The moduli of ${\varepsilon}$-connections (on arbitrary projective variety) is constructed by C. Simpson ([@SimICM; @Simpson1; @Simpson2]); it is easy to see that his argument works in the case of ${\varepsilon}$-connections with singularities on curves. Let us quickly recall the construction.
Fix ${\varepsilon}\in{{\mathbb C}}$, and denote by ${{\mathcal N}}_{\varepsilon}$ the moduli stack of ${\varepsilon}$-connections of the form $(L,\nabla;{\varepsilon}\in{{\mathbb C}})$. Equivalently, ${{\mathcal N}}_{\varepsilon}$ is a fiber of the map $${{\overline{\mathstrut{{\mathcal N}}}}}\to{{{\mathbb A}^1}}/{{\mathbf{G_m}}}:(L,\nabla;{\varepsilon}\in E)\mapsto({\varepsilon}\in
E).$$ Here we identify the quotient stack ${{{\mathbb A}^1}}/{{\mathbf{G_m}}}$ with the moduli stack of pairs $({\varepsilon}\in E)$, where $E$ is a one-dimensional vector space.
As ${\varepsilon}$ varies, the stacks ${{\mathcal N}}_{\varepsilon}$ form a family ${{\mathcal N}}_{{\bullet}}\to{{{\mathbb A}^1}}$ (whose fiber over ${\varepsilon}\in{{{\mathbb A}^1}}$ is ${{\mathcal N}}_{\varepsilon}$). The total space ${{\mathcal N}}_{{\bullet}}$ carries an action of ${{\mathbf{G_m}}}$ via $$t\cdot(L,\nabla;{\varepsilon}\in{{\mathbb C}})=(L,t\nabla;t{\varepsilon}\in{{\mathbb C}}),\quad
(L,\nabla;{\varepsilon}\in{{\mathbb C}})\in{{\mathcal N}}_{{\bullet}},\;t\in{{\mathbf{G_m}}}.$$ We can identify ${{\overline{\mathstrut{{\mathcal N}}}}}$ with the quotient stack ${{\mathcal N}}_{{\bullet}}/{{\mathbf{G_m}}}$.
Denote by ${{\mathcal N}}_{\varepsilon}^{ss}\subset{{\mathcal N}}_{\varepsilon}$ (resp. ${{\mathcal N}}^{ss}_{{\bullet}}\subset{{\mathcal N}}_{{\bullet}}$) the open substacks of semistable ${\varepsilon}$-connections.
[[)]{}]{}\[GmsI\] There exists a good moduli space ${{\mathcal N}}_{\varepsilon}^{ss}\to N_{\varepsilon}$;
[[)]{}]{}\[GmsII\] As ${\varepsilon}\in{{\mathbb C}}$ varies, the spaces $N_{\varepsilon}$ form a family $N_{{\bullet}}\to{{{\mathbb A}^1}}$ whose fiber over ${\varepsilon}\in{{{\mathbb A}^1}}$ is $N_{\varepsilon}$. There exists a good moduli space ${{\mathcal N}}^{ss}_{{\bullet}}\to N_{{\bullet}}$.
(\[GmsI\]) This is a particular case of [@Simpson1 Theorem 4.10]. Note that [@Simpson1 Theorem 4.10] applies to the moduli space of semistable modules over a sheaf of split almost polynomial rings of differential operators $\Lambda$ (see [@Simpson1 §2], p. 77, 81 for definition). In our case, $\Lambda$ is the universal enveloping of the Lie algebroid $$\Lambda^{\le1}=(O_X\oplus{{\mathcal T}}_X(-{\mathfrak D}),
[\cdot{\stackrel{{\varepsilon}}{,}}\cdot],\rho),$$ where $\rho$ is the composition of the natural inclusion ${{\mathcal T}}_X(-{\mathfrak D})$ into ${{\mathcal T}}_X$ with multiplication by ${\varepsilon}$, and $$[f_1+\tau_1{\stackrel{{\varepsilon}}{,}}f_2+\tau_2]=
{\varepsilon}(\tau_1(f_2)-\tau_2(f_1)+[\tau_1,\tau_2]).$$
(\[GmsII\]) Consider the sheaf $p_1^*\Lambda$ on $X\times{{{\mathbb A}^1}}$, where $\Lambda$ is the sheaf from (\[GmsI\]) with ${\varepsilon}=1$. Let $\Lambda^R$ be its subsheaf generated by the operators of the form $\sum{\varepsilon}^i\lambda_i$, where $\lambda_i\in\Lambda$ has order at most $i$, ${\varepsilon}$ is the coordinate on ${{{\mathbb A}^1}}$. The family $N^{{\bullet}}$ is constructed by applying Theorem 4.10 to $\Lambda^R$ relative to the projection $X\times{{{\mathbb A}^1}}\to{{{\mathbb A}^1}}$. See [@Simpson1 Section on $\tau$-connections, p. 87].
The action of ${{\mathbf{G_m}}}$ on ${{\mathcal N}}_{{\bullet}}$ induces its action on $N_{{\bullet}}$. (In fact an action of an algebraic group on a stack always induces an action on the good moduli space; for the proof, use the universal property of good moduli spaces and [@Alper Proposition 4.7(i)].) In particular, a point $z\in N_{{\bullet}}$ yields a morphism ${{\mathbf{G_m}}}\to N_{{\bullet}}:t\mapsto t\cdot
z$. If it can be extended to a morphism from ${{{\mathbb A}^1}}\supset{{\mathbf{G_m}}}$ (resp. from ${{{\mathbb P}^1}}-\{0\}\supset{{\mathbf{G_m}}}$), we say that the limit $\lim_{t\to0}t\cdot z$ (resp. $\lim_{t\to\infty}t\cdot z$) exists.
The Hitchin fibration gives the following description of $N_0$. Let $$B:=\prod_{i=1}^r H^0(X,\Omega_X({\mathfrak D})^{\otimes i})$$ be the base of Hitchin fibration. Recall that the Hitchin map sends a Higgs bundle $(L,\nabla;0\in{{\mathbb C}})$ to $(c_1(\nabla),\dots,c_r(\nabla))$, where $c_i(\nabla)$ are coefficients of the characteristic polynomial of $\nabla$. Thus we get a map ${{\mathcal N}}_0\to B$. This map descends to a map $$(c_1,c_2,\ldots,c_r):N_0\to B.$$ Denote by $N_{{\bullet}}^n\subset N_0$ the zero fiber of Hitchin fibration.
Let $z\in N_{{\bullet}}$ correspond to $(L,\nabla;{\varepsilon}\in{{\mathbb C}})\in{{\mathcal N}}^{ss}_{{\bullet}}$. Then $\lim_{t\to\infty}t\cdot z$ exists if and only if $(L,\nabla;{\varepsilon}\in{{\mathbb C}})$ is nilpotent.
Let us start with the ‘only if’ direction. If the limit exists, then $\lim_{t\to\infty}t{\varepsilon}$ exists, so ${\varepsilon}=0$. Also, the coefficients of the characteristic polynomial of $t\nabla$ are equal to $t^i c_i(\nabla)$, and so the limit $\lim_{t\to\infty}t^ic_i(\nabla)$ exists. Therefore, $c_i(\nabla)=0$; in other words, $\nabla$ is nilpotent.
To prove the ‘if’ direction, it suffices to notice that $N_{{\bullet}}^n$ is complete. Indeed, it is the zero fiber of the Hitchin map. This map is proper, the proof is similar to ([@Simpson1], Theorem 6.11). More precisely, one has to repeat the proof of that theorem, changing $T^*$ to $T^*({\mathfrak D})$.
The geometric quotient $(N_{{\bullet}}-N_{{\bullet}}^n)/{{\mathbf{G_m}}}$ exists; the quotient is a complete scheme of finite type, the natural projection $N_{{\bullet}}-N_{{\bullet}}^n\to(N_{{\bullet}}-N_{{\bullet}}^n)/{{\mathbf{G_m}}}$ is an affine map.
This follows from [@Simpson2 Theorem 11.2] and the previous lemma. Indeed, the fixed point set is closed in $N_{{\bullet}}^n$ and thus complete. The fact that $\lim_{t\to0}t\cdot
z$ exists for all $z\in N_{{\bullet}}$ follows from [@Simpson2 Theorem 10.1] (see also Corollary 10.2).
To show that the map is affine, note that [@Simpson2 Theorem 11.2] is derived from [@Simpson2 Theorem 11.1], which in turn uses Proposition 1.9 in [@Mumford], but this proposition also claims that the map is affine.
The geometric quotient ${{\overline N}}=(N_{{\bullet}}-N_{{\bullet}}^n)/{{\mathbf{G_m}}}$ has the properties required in Theorem \[GoodModuliSpace\] because of the following
Let $p:{{\mathcal S}}\to S$ be a good moduli space. Let $G$ be a reductive group acting on ${{\mathcal S}}$. Consider the induced action on $S$ and assume that there is a geometric quotient $S/\!/G$ such that the projection $S\to S/\!/G$ is affine. Then the induced map $\bar p:{{\mathcal S}}/G\to S/\!/G$ is a good moduli space [[(]{}-1pt]{}the quotient in the left hand side is the stacky one[[)]{}]{}.
Let us decompose $\bar p$ as $${{\mathcal S}}/G\xrightarrow{p'}S/G\xrightarrow{p''}S/\!/G.$$ We just have to check that $p'_*$ and $p''_*$ are exact and take the structure sheaves to the structure sheaves. This easily follows from our assumptions.
Projectivity of ${{\overline N}}$
---------------------------------
Let us construct an ample bundle on ${{\overline N}}$. Fix a point $x\in
X$.
\[TrivialAction\] Let $\alpha$ be an automorphism of $(L,\nabla;{\varepsilon}\in
E)\in{{\overline{\mathstrut{{\mathcal N}}}}}^{\,ss,nn}$.\
[[)]{}]{}\[TrivAct1\] The action of $\alpha$ on $E$ is a root of unity of degree at most $r$. In particular, $\alpha$ acts trivially on $E^{\otimes r!}$.\
[[)]{}]{}\[TrivAct2\] If $\alpha$ acts trivially on $E$, then $\alpha$ acts trivially on $$\operatorname{detR\Gamma}(X,L)^{\otimes r}\otimes\det(L_x)^{\otimes(rg-d-r)}.$$ [[)]{}]{}\[TrivAct3\] The automorphism $\alpha$ acts trivially on $$\left(\operatorname{detR\Gamma}(X,L)^{\otimes r}\otimes\det(L_x)^{\otimes
(rg-d-r)}\right)^{\otimes r!}.$$
(\[TrivAct1\]) Note that $\alpha({\varepsilon})={\varepsilon}$, so if ${\varepsilon}\ne0$, then $\alpha$ acts trivially on $E$. If ${\varepsilon}=0$, consider the coefficients of the characteristic polynomial of $\nabla$ $$c_i\in H^0(X,\Omega_X({\mathfrak D})^{\otimes i})\otimes
E^{\otimes i}.$$ Since $\nabla$ is not nilpotent, there is $i$ such that $c_i\ne 0$. Now it suffices to note that $\alpha(c_i)=c_i$.\
(\[TrivAct2\]) We can decompose $L=\oplus_{\lambda\in{{\mathbb C}}}
L^\lambda$, where $\alpha-\lambda$ is nilpotent on $L^\lambda$ (almost all of the summands vanish). Since $\alpha$ acts trivially on $E$, we have $\nabla(L^\lambda)\subset
L^\lambda\otimes\Omega_X({\mathfrak D})\otimes_{{\mathbb C}}E$. By semistability of $L$, $\deg L^\lambda=\frac{d}{r}\operatorname{rk}L^\lambda$. We can then identify $$\operatorname{detR\Gamma}(X,L)\simeq\bigotimes\operatorname{detR\Gamma}(X,L^\lambda)$$ and $\det L_x\simeq\bigotimes\det((L^\lambda)_x)$. Finally, $\alpha$ acts as $\lambda^{\deg L^\lambda-(g-1)\operatorname{rk}L^\lambda}$ (here $g$ is the genus of $X$) on $\operatorname{detR\Gamma}(X,L^\lambda)$ and as $\lambda^{\operatorname{rk}L_\lambda}$ on $\det((L^\lambda)_x)$.\
(\[TrivAct3\]) Follows from (\[TrivAct2\]) applied to $\alpha^{r!}$.
Let us denote by $\delta$ (resp. ${{\mathcal E}}$, resp. $\delta_x$) the line bundle on ${{\overline{\mathstrut{{\mathcal N}}}}}$ whose fiber over $(L,\nabla;{\varepsilon}\in
E)$ equals $\operatorname{detR\Gamma}(X,L)$ (resp. $E$, resp. $\det(L_x)$).
\[BundlesOnM\] The line bundles $${{\mathcal E}}^{\otimes r!}|_{{{\overline{\mathstrut{{\mathcal N}}}}}^{\,ss,nn}}\quad\text{and}\quad
\left.\left(\delta^{\otimes
r}\otimes\delta_x^{\otimes(rg-d-r)}\right)^{\otimes r!}\right|_{{{\overline{\mathstrut{{\mathcal N}}}}}^{\,ss,nn}}$$ are pullbacks of line bundles on ${{\overline N}}$.
By Lemma \[TrivialAction\], automorphisms of any closed point of ${{\overline{\mathstrut{{\mathcal N}}}}}^{\,ss,nn}$ act trivially on the fibers of these two bundles. The statement now follows from [@Alper Theorem 10.3].
Denote the corresponding line bundles on ${{\overline N}}$ by ${{\mathcal E}}'$ and $\delta'$.
\[AmpleBundle\] The line bundle $(\delta')^{-1}\otimes({{\mathcal E}}')^{\otimes k}$ is ample on ${{\overline N}}$ for $k\gg 0$.
Recall that by construction, $(\delta')^{-1}$ is ample on $N$, cf. Remark before [@Simpson1 Theorem 4.10]. Let $N_H$ be ${{\overline N}}-N$ with reduced scheme structure. By construction, $N_H$ is the reduction of the quotient scheme $(N_0-N_{{\bullet}}^n)/{{\mathbf{G_m}}}$. The Hitchin map therefore induces a morphism from $N_H$ to the quotient scheme $(B-\{0\})/{{\mathbf{G_m}}}$, which is a weighted projective space. Recall [@EGAII Definition 4.6.1] that a sheaf ${{\mathcal F}}$ on $Y$ is called relatively ample for a map $f:Y\to Y'$ if for some cover $Y'=\cup Y'_\alpha$ with affine $Y'_\alpha$, the sheaves ${{\mathcal F}}|_{f^{-1}(Y'_\alpha)}$ are ample.
We make the following observations.
[[)]{}]{}$(\delta')^{-1}$ is relatively ample for the Hitchin map $N_0\to B$ (since $(\delta')^{-1}$ is ample on $N_0$);
[[)]{}]{}$(\delta')^{-1}$ is relatively ample for the equivariant Hitchin map $N_H\to(B-\{0\})/{{\mathbf{G_m}}}$. Indeed, the relative ampleness can be proved fiberwise, cf. [@EGAIII Theorem 4.7.1] (We are thankful to Brian Conrad for the reference). On the other hand, a fiber of the equivariant Hitchin map is the categorical quotient of the corresponding fiber of the Hitchin map by the finite stabilizer of the corresponding point in $(B-\{0\})/{{\mathbf{G_m}}}$. It remains to use the following general fact: *the descent of an equivariant ample line bundle to the quotient by a finite group is ample*. This follows from [@Mumford Proposition 1.15, Theorem 1.10];
[[)]{}]{}${{\mathcal E}}'|_{N_H}$ is naturally a pullback of a sheaf on $(B-\{0\})/{{\mathbf{G_m}}}$, which we also denote by ${{\mathcal E}}'$;
[[)]{}]{}${{\mathcal E}}'$ is very ample on $(B-\{0\})/{{\mathbf{G_m}}}$.
Recall from Remark \[SecondFromFirst\] that ${\varepsilon}^{\otimes r!}$ yields a section ${\varepsilon}'\in
H^0({{\overline N}},{{\mathcal E}}')$, whose set-theoretic zero locus is $N_H$. Denote by $N_H'$ the scheme-theoretic zero locus of ${\varepsilon}'$. It is a non-reduced ‘thickening’ of $N_H$.
*Step 1.* For integers $l,k$, consider the line bundle $${{\mathcal L}}={{\mathcal L}}_{l,k}:=(\delta')^{\otimes-l}\otimes{{\mathcal E}}'^{\otimes
k}.$$ There exists $l_0$ such that for all $l>l_0$, there is $k_0=k_0(l)$ such that for all $k>k_0$ the line bundle ${{\mathcal L}}|_{N_H}$ is very ample on $N_H$. This follows from [@EGAII Proposition 4.6.11] and [@EGAII Proposition 4.4.10(ii)].
*Step 2.* For any coherent sheaf ${{\mathcal F}}$ on $N_H$, there exists $l_0$ such that for all $l>l_0$, there is $k_0=k_0(l)$ such that for all $k>k_0$ and all $i>0$, $$H^i(N_H,{{\mathcal F}}\otimes{{\mathcal L}})=0$$ and ${{\mathcal F}}\otimes{{\mathcal L}}$ is generated by global sections. This follows from the fact that the derived functor of global sections on $N_H$ is the composition of the derived direct image to $(B-\{0\})/{{\mathbf{G_m}}}$ and the derived functor of global sections on $(B-\{0\})/{{\mathbf{G_m}}}$.
*Step $2'$.* Same statement as in Step 2 is true with $N_H$ changed to $N'_H$. For the proof, consider a filtration of ${{\mathcal F}}$ with factors supported scheme-theoretically on $N_H$ and use the long exact sequence of cohomology.
*Step $1'$.* Same statement as in Step 1 is true with $N_H$ changed to $N'_H$. Indeed, set ${{\mathcal F}}_i:=O_{N'_H}(-iN_H)$. By Step $2'$ we can assume that ${{\mathcal F}}_1\otimes{{\mathcal L}}$ is generated by global sections and $H^0(N'_H,{{\mathcal L}})$ surjects onto $H^0(N_H,{{\mathcal L}})$. By Step 1 we can assume that ${{\mathcal L}}|_{N_H}$ is very ample. Let us show that ${{\mathcal L}}$ is very ample on $N_H'$.
We are going to use [@Hartshorne Proposition II.7.2]. Take $s\in H^0(N'_H,{{\mathcal L}})$ and let $N'_s$ be the open subset of $N'_H$ defined by $s\ne0$. Then $N_s:=N'_s\cap N_H$ is affine. Therefore, $N'_s$ is also affine. It suffices to show that the set $\{s'/s|\,s'\in H^0(N'_H,{{\mathcal L}})\}$ generates the ring $A_i:=H^0(N'_s,O_{N'_H}/{{\mathcal F}}_i)$ for all $i$. We proceed by induction. For $i=1$ this follows from very ampleness of ${{\mathcal L}}|_{N_H}$. Take $t\in A_i$; using our statement with $i=1$, we can assume that $t\in{{\mathcal F}}_1/{{\mathcal F}}_i$. By assumption it can be written as $\sum\lambda_j(s_j/s)$, where $s_j\in
H^0(N'_H,{{\mathcal F}}_1\otimes{{\mathcal L}})$, $\lambda_j\in A_{i-1}$. It remains to use the inductive hypothesis.
For $i\gg0$ we have ${{\mathcal F}}_i=0$ and we are done.
*Step $3$.* From now on, fix $l$ satisfying the conditions of Steps $1'$ and $2'$, and also such that ${{\mathcal L}}|_N=(\delta')^{\otimes-l}|_N$ is very ample. Then the restriction map $H^0({{\overline N}},{{\mathcal L}})\to H^0(N'_H,{{\mathcal L}})$ is surjective for $k\gg0$.
Indeed, this map fits into the exact sequence $$H^0({{\overline N}},{{\mathcal L}})\to H^0(N'_H,{{\mathcal L}})\to
H^1({{\overline N}},{{\mathcal L}}(-N'_H))\to H^1({{\overline N}},{{\mathcal L}})\to H^1(N'_H,{{\mathcal L}})$$ According to Step $2'$, for $k\gg0$, the rightmost term vanishes, and so the map $H^1({{\overline N}},{{\mathcal L}}(-N'_H))\to H^1({{\overline N}},{{\mathcal L}})$ is surjective. On the other hand, ${{\mathcal L}}(-N'_H)$ equals ${{\mathcal L}}_{l,k-1}$, so $\dim H^1({{\overline N}},{{\mathcal L}})$ decreases as a function of $k$ for $k\gg0$. This dimension is finite by properness of ${{\overline N}}$, and therefore stabilizes. In other words, for $k\gg 0$, the map $H^1({{\overline N}},{{\mathcal L}}(-N'_H))\to H^1({{\overline N}},{{\mathcal L}})$ is an isomorphism.
*Step $4$.* ${{\mathcal L}}={{\mathcal L}}_{l,k}$ is very ample on ${{\overline N}}$ for $k\gg0$ (the choice of $l$ is as in Step $3$).
Recall that $\mathbb{P}(V)$ denotes the projective space of hyperplanes in a vector space $V$. Choose a finite-dimensional vector space $V\subset H^0(N,{{\mathcal L}})$ that defines an embedding $N\hookrightarrow\mathbb{P}(V)$, and for every $k\gg0$ a subspace $W_k\subset H^0(N'_H,{{\mathcal L}})$ that defines an embedding $N'_H\hookrightarrow\mathbb{P}(W_k)$. For $k\gg 0$, the space $V$ is contained in $H^0({{\overline N}},{{\mathcal L}}\otimes({{\mathcal E}}')^{-1})\subset
H^0(N,{{\mathcal L}})$ (because $H^0(N,{{\mathcal L}})$ is the limit of spaces $H^0({{\overline N}},{{\mathcal L}})$ as $k\to\infty$), and $W_k$ can be lifted to a finite-dimensional subspace of $H^0({{\overline N}},{{\mathcal L}})$ (which we still denote by $W_k$) by Step $3$. It follows from [@Hartshorne Proposition II.7.3] that $V{\varepsilon}'+W_k+W_{k-1}{\varepsilon}'$ defines an embedding ${{\overline N}}\hookrightarrow{\mathbb{P}}(V{\varepsilon}'+W_k+W_{k-1}{\varepsilon}')$ for $k\gg0$. Note that this proposition is stated for projective schemes only but it is valid for any proper scheme. Indeed, the projectivity is needed only for applying [@Hartshorne Corollary 5.20], but the corollary is well-known to be true with the weaker assumption.
Properties of ${{\mathcal M}}$ and of its compactification {#MODULISPACES}
==========================================================
In this section we prove Theorem \[StackProp\]. We also prove
\[CohomologicalDimension\] Let ${{\mathcal F}}$ be any quasi-coherent sheaf on ${{\mathcal M}}$. Then $H^i({{\mathcal M}},{{\mathcal F}})=0$ for $i\ge2$.
We construct a compactification ${{\overline{{\mathcal M}}}}={{\mathcal M}}\sqcup{{{{\mathcal M}}_H}}$ (see Proposition \[Cartier\]). We prove that the stacks ${{\overline{{\mathcal M}}}}$ form a flat family as ${\mathfrak D}$ and the local invariants of connections vary (Proposition \[PrFlat\]). We give an explicit description of parabolic bundles underlying bundles with connections (Proposition \[BunP\]). We begin with general statements but starting from Lemma \[Irreducible\] we assume that $X={{{\mathbb P}^1}}$ and $\deg{\mathfrak D}=4$. This assumption continues through the end of the paper.
Connections compatible with parabolic structure
-----------------------------------------------
We start by describing parabolic bundles that possess compatible connections.
Let $X$ be a smooth projective curve of genus $g$. Let $L$ be a vector bundle of degree $d$ on $X$. Denote by $b(L)\in
H^1(X,\operatorname{\mathcal{E}\mathit{nd}}(L)\otimes\Omega_X)$ the class of the Atiyah sequence $$0\to L\otimes\Omega_X\to B(L)\to L\to 0.$$ We have the Serre duality pairing $\langle\cdot,b(L)\rangle:\operatorname{End}(L)\to{{\mathbb C}}$. Recall from [@Atiyah Proposition 18] that $$\label{atiyah}
\begin{split}
\langle A,b(L)\rangle &=0\text{ if $A$ is nilpotent},\\
\langle\operatorname{id}_L,b(L)\rangle &=-d.
\end{split}$$
[[)]{}]{}To match [@Atiyah], the Serre duality pairing should include the factor of $2\pi\sqrt{-1}$.
[[)]{}]{}For every $A\in \operatorname{End}(L)$ the Serre duality pairing $\langle A,b(L)\rangle$ is given by $$\langle A,b(L)\rangle =-\operatorname{tr}(A|R\Gamma(X,L))+\operatorname{tr}(A)\chi(O_X).$$ Here $\operatorname{tr}(A | R\Gamma(X,L))$ is the alternating sum of the traces of maps on $H^i(X,L)$ induced by $A$, and $\chi(O_X)=1-g$. This follows from . Indeed, we can decompose $L$ into a direct sum such that the semisimple part of $A$ is scalar on each summand.
Fix ${\varepsilon}\in{{\mathbb C}}$, distinct points $x_1,\dots,x_k\in X$, and principal parts $$A_i\in\operatorname{\mathcal{E}\mathit{nd}}(L)(\infty\cdot x_i)/\operatorname{\mathcal{E}\mathit{nd}}(L)$$ for the vector bundle $\operatorname{\mathcal{E}\mathit{nd}}(L)$ at $x_i$ for all $i=1,\dots,k$. To these data, we associate the sheaf of ${\varepsilon}$-connections ${{\mathcal C}}={{\mathcal C}}(L,{\varepsilon},\{A_i\}_{i=1}^k)$: its sections over an open subset $U\subset X$ are ${\varepsilon}$-connections $$\nabla:L|_U\to L|_U\otimes\Omega_U\left(\sum_{i=1}^k\infty\cdot x_i\right)$$ such that $\nabla-A_i$ is regular at $x_i$ for all $x_i\in U$. Since ${{\mathcal C}}$ is a torsor over $\operatorname{\mathcal{E}\mathit{nd}}(L)\otimes\Omega_X$, its isomorphism class is given by an element $$c=c\,(L,{\varepsilon},\{A_i\}_{i=1}^k)\in H^1(X,\operatorname{\mathcal{E}\mathit{nd}}(L)\otimes\Omega_X).$$
\[lm:PolarParts\] For every $A\in\operatorname{End}(L)$, $$\langle A,c\rangle={\varepsilon}\langle A,b(L)\rangle+
\sum_{i=1}^k\operatorname{tr}(\operatorname{res}(A\cdot A_i)).$$
The torsor ${{\mathcal C}}$ depends linearly on the collection $(L,{\varepsilon},\{A_i\}_{i=1}^k)$. We can therefore assume that either all $A_i=0$ (and then $c={\varepsilon}b(L)$) or ${\varepsilon}=0$ (this case follows from the definition of Serre pairing).
It is easy to adapt Lemma \[lm:PolarParts\] to the settings of parabolic bundles. For simplicity, we only state it for bundles of rank two. Let us fix a divisor ${\mathfrak D}=\sum
n_ix_i\ge 0$ on $X$. Suppose that $L$ is a rank two vector bundle on $X$, and $\eta$ is a level ${\mathfrak D}$ parabolic structure on $L$, that is, a line subbundle $\eta\subset
L|_{\mathfrak D}$ (cf. Definition \[ParB\]).
Denote by $\operatorname{\mathcal{E}\mathit{nd}}(L,\eta)$ the locally free sheaf of endomorphisms of $L$, preserving $\eta$; let $\operatorname{End}(L,\eta):=H^0({{{\mathbb P}^1}},\operatorname{\mathcal{E}\mathit{nd}}(L,\eta))$ be the corresponding ring of endomorphisms.
\[ConnExist\] Fix ${\varepsilon}\in{{\mathbb C}}$ and principal parts $\alpha^+,\alpha^-\in\Omega_X({\mathfrak D})/\Omega_X$. The following conditions are equivalent:
[[)]{}]{}\[cond:parabolic\] There exists an ${\varepsilon}$-connection $\nabla:L\to L\otimes\Omega_X({\mathfrak D})$ whose ‘polar part’ $L|_{\mathfrak D}\to(L\otimes\Omega_X({\mathfrak D}))|_{\mathfrak D}$ equals to $\alpha^+$ on $\eta$ and induces $\alpha^-$ on $(L|_{\mathfrak D})/\eta$.
[[)]{}]{}For any endomorphism $A\in\operatorname{End}(L,\eta)$, we have $$\operatorname{res}(A_+\alpha_+)+\operatorname{res}(A_-\alpha_-)+{\varepsilon}\langle A, b(L)\rangle=0.$$ Here $A_+, A_-\in {{\mathbb C}}[{\mathfrak D}]$ are the [[(]{}-1pt]{}scalar[[)]{}]{} operators induced by $A$ on $\eta$ and on $(L|_{\mathfrak D})/\eta$ respectively, and the residue functional $\operatorname{res}:\Omega_X({\mathfrak D})/\Omega_X\to{{\mathbb C}}$ is given by $$\operatorname{res}\omega:=\sum_{x\in{\mathfrak D}}\operatorname{res}_x\omega.$$
Denote by $\operatorname{\mathcal{H}\mathit{iggs}}(L,\eta)$ the sheaf of Higgs fields $B:L\to
L\otimes\Omega_X({\mathfrak D})$ whose polar part $L|_{\mathfrak D}\to(L\otimes\Omega_X({\mathfrak D}))|_{\mathfrak D}$ induces $0$ on both $\eta$ and $(L|_{\mathfrak D})/\eta$. In other words, in any $\eta$-compatible local trivialization of $L|_{n_ix_i}$ we have $$B=\begin{pmatrix}
0 & *\\
0 & 0
\end{pmatrix}
+\text{non-singular terms}.$$ Note that $\operatorname{\mathcal{H}\mathit{iggs}}(L,\eta)\simeq\operatorname{\mathcal{E}\mathit{nd}}(L,\eta)^\vee\otimes\Omega_X$. Indeed, the pairing is given by the trace of the product, and one checks in local coordinates that it is perfect.
The sheaf of connections satisfying forms a torsor over $\operatorname{\mathcal{H}\mathit{iggs}}(L,\eta)$; clearly, this torsor is induced from the $\operatorname{\mathcal{E}\mathit{nd}}(L)\otimes\Omega_X$-torsor ${{\mathcal C}}(L,{\varepsilon},\{A_i\}_{i=1}^n)$ for a choice of polar parts $A_i$ compatible with $\eta$ and $\alpha^\pm$. Now the claim follows from Lemma \[lm:PolarParts\].
If $A_i$ has a pole of first order for all $i$, then Corollary \[ConnExist\] becomes Theorem 7.2 in [@Crawley-Boevey], which is a special case of Mihai’s results [@Mihai1; @Mihai2].
Local invariants of connections, revisited {#DEFMB}
------------------------------------------
Let ${{\mathcal N}}={{\mathcal N}}(X,2,d,{\mathfrak D})$ be as in §\[COMPACTIFICATION\]. Let $(L,\nabla;{\varepsilon}\in
E)\in{{\overline{\mathstrut{{\mathcal N}}}}}$ be an ${\varepsilon}$-connection, where $L$ has rank 2. Let $D$ be the formal disc centered at $x_i$. Trivializing $L|_D$, we can write $$\nabla|_D={\varepsilon}{\mathbf d}+a,\qquad a\in\operatorname{\mathfrak{gl}}(2)\otimes
\Omega_D(n_ix_i)\otimes_{{\mathbb C}}E.$$ It is easy to see that $\operatorname{tr}a$ and $\det a$ are well defined (that is independent of the trivialization) as sections of $E\otimes_{{\mathbb C}}(\Omega_X(n_ix_i)/\Omega_X)$ and $E^{\otimes2}\otimes_{{\mathbb C}}(\Omega^{\otimes2}_X(2n_ix_i)/
\Omega^{\otimes2}_X(n_ix_i))$ respectively. Performing this operation at every $x_i$, we get well-defined sections of $E\otimes_{{\mathbb C}}(\Omega_X({\mathfrak D})/\Omega_X)$ and $E^{\otimes2}\otimes_{{\mathbb C}}(\Omega^{\otimes2}_X(2{\mathfrak D})/\Omega^{\otimes2}_X({\mathfrak D}))$, which we denote $[\operatorname{tr}\nabla]$ and $[\det\nabla]$ respectively.
Clearly, in the case of a non-resonant connection $(L,\nabla;1\in{{\mathbb C}})$ we get $$[\operatorname{tr}\nabla]=\nu_1:=(\alpha_i^++\alpha_i^-),\qquad
[\det\nabla]=\nu_2:=(\alpha_i^+\alpha_i^-),$$ where $(\alpha_i^\pm)$ is the formal type of the connection (cf. §\[MODST\]). Thus in this case the data $\bigl([\operatorname{tr}\nabla],[\det\nabla]\bigr)$ is equivalent to the formal type.
Fix $\nu_1\in\Omega_X({\mathfrak D})/\Omega_X$ and $\nu_2\in\Omega^{\otimes2}_X(2{\mathfrak D})/
\Omega^{\otimes2}_X({\mathfrak D})$ and denote by ${{\overline{{\mathcal M}}}}$ the closed substack of ${{\overline{\mathstrut{{\mathcal N}}}}}^{\,ss,nn}$ parameterizing ${\varepsilon}$-connections such that $$\label{hitchincond}
[\operatorname{tr}\nabla]={\varepsilon}\otimes\nu_1,\qquad
[\det\nabla]={\varepsilon}^{\otimes2}\otimes\nu_2.$$
Assume now that $X={{{\mathbb P}^1}}$, $\deg{\mathfrak D}=4$, $d$ is odd. Recall that in §\[MODST\] we defined a moduli stack ${{\mathcal M}}$.
\[Irreducible\] Every connection $(L,\nabla)\in{{\mathcal M}}$ is irreducible.
Assume for a contradiction that $L'\subset L$ is a $\nabla$-invariant line subbundle. One checks that $\operatorname{res}_{x_i}(\nabla|_{L'})=\operatorname{res}\alpha_i^\pm$, where $\alpha_i^\pm$ are defined in §\[MODST\]. This contradicts condition (\[AlphaIII\]) of §\[MODST\] (cf. [@ArinkinLysenko Proposition 1]).
In particular every $(L,\nabla)\in{{\mathcal M}}$ is semistable, and we see that the open substack of ${{\overline{{\mathcal M}}}}$ given by ${\varepsilon}\ne0$ is identified with ${{\mathcal M}}$. Let ${{{{\mathcal M}}_H}}$ be the closed substack of ${{\overline{{\mathcal M}}}}$ defined by ${\varepsilon}=0$. Then ${{\mathcal M}}={{\overline{{\mathcal M}}}}-{{{{\mathcal M}}_H}}$. By Theorem \[GoodModuliSpace\] and [@Alper Lemma 4.14], ${{\overline{{\mathcal M}}}}$ has a good moduli space ${{\overline M}}$. It follows from Theorem \[AmpleBundle\] that ${{\overline M}}$ is projective.
Note that ${{\mathcal M}}$ is a closed substack of ${{\mathcal N}}^{ss}$, so, using again [@Alper Lemma 4.14], we see that $M:=N\cap{{\overline M}}$ is the good moduli space of ${{\mathcal M}}$. Clearly, $M$ is open in ${{\overline M}}$.
\[Cartier\] ${{{{\mathcal M}}_H}}\subset{{\overline{{\mathcal M}}}}$ is a Cartier divisor.
This will be proved below after we prove some properties of ${{\mathcal M}}$.
An affine bundle structure on ${{\mathcal M}}$ {#AFFSTR}
----------------------------------------------
Denote by $\operatorname{\mathcal{B}\mathit{un}}(d)=\operatorname{\mathcal{B}\mathit{un}}({{{\mathbb P}^1}},2,d,{\mathfrak D})$ the moduli stack of level-${\mathfrak D}$ parabolic bundles $(L,\eta)$ such that $L$ has degree $d$ and $\operatorname{End}(L,\eta)={{\mathbb C}}$. By semicontinuity $\operatorname{\mathcal{B}\mathit{un}}(d)$ is an open substack in $\overline\operatorname{\mathcal{B}\mathit{un}}(d)$.
Consider $(L,\nabla)\in{{\mathcal M}}$. The formal classification (\[fnf\]) of connections shows that there is a unique level-${\mathfrak D}$ parabolic structure $\eta$ compatible with $\nabla$ in the following sense: for any $i$ and every section $s$ of $L$ in a neighborhood of $x_i$ such that $s|_{n_ix_i}\in H^0((n_ix_i),\eta)$ we have that $\nabla
s-\alpha_i^+s$ is regular at $x_i$.
\[Affine\] [[)]{}]{}\[Aff0\] If $(L,\nabla)\in{{\mathcal M}}$, then $L\simeq
O_{{{\mathbb P}^1}}(m)\oplus
O_{{{\mathbb P}^1}}(n)$ with $m-n=1$.\
[[)]{}]{}\[Aff1\] Assume that $(L,\eta)$ is the parabolic bundle corresponding to $(L,\nabla)\in{{\mathcal M}}$, then $\operatorname{End}(L,\eta)={{\mathbb C}}$.\
[[)]{}]{}\[Aff2\] The resulting map $\rho:{{\mathcal M}}\to\operatorname{\mathcal{B}\mathit{un}}(d)$ is an affine bundle of rank 1.
(\[Aff0\]) Let us write $L=O_{{{\mathbb P}^1}}(m)\oplus O_{{{\mathbb P}^1}}(n)$ with $m>n$. Assume for a contradiction that $m-n\ge3$. Consider the map $$\overline\nabla:O_{{{\mathbb P}^1}}(m)\hookrightarrow L\xrightarrow{\nabla}
L\otimes\Omega_{{{\mathbb P}^1}}({\mathfrak D})
\twoheadrightarrow O_{{{\mathbb P}^1}}(n)\otimes\Omega_{{{\mathbb P}^1}}({\mathfrak D})\simeq O_{{{\mathbb P}^1}}(n+2).$$ It is easy to see that this map is $O_{{{\mathbb P}^1}}$-linear, thus it is zero. It follows that $O_{{{\mathbb P}^1}}(m)$ is a $\nabla$-invariant subbundle in $L$, which contradicts Lemma \[Irreducible\].
(\[Aff1\]) Assume first that there is $A\in\operatorname{End}(L,\eta)$ such that $A$ has different eigenvalues. Then in exactly the same way as in [@ArinkinLysenko Proposition 3] we come to contradiction with condition (\[AlphaIII\]) of §\[MODST\].
Assume now that $A\in\operatorname{End}(L,\eta)$ is not scalar and it has equal eigenvalues. Then the matrix of $A$ with respect to the decomposition $L=O_{{{\mathbb P}^1}}(m)\oplus O_{{{\mathbb P}^1}}(n)$ is $$A=
\begin{pmatrix}
c & f\\
0 & c
\end{pmatrix},$$ where $c$ is a constant, $f$ is a section of $O_{{{\mathbb P}^1}}(m-n)$, that is a polynomial of degree at most 1.
For every $i$ choose the maximal $m_i$ such that $\eta|_{m_ix_i}=O_{{{\mathbb P}^1}}(m)|_{m_ix_i}$. It is easy to see that $A$ preserves $\eta$ if and only if $f$ vanishes at $x_i$ at least to order $n_i-2m_i$ for all $i$. Hence the existence of non-scalar endomorphism implies $$\label{ineq}
\sum_i n_i-2m_i\le1.$$
Consider again $\overline\nabla:O_{{{\mathbb P}^1}}(m)\to O_{{{\mathbb P}^1}}(n+2)$. It follows from the compatibility that $\overline\nabla$ has zero of order at least $m_i$ at $x_i$. Again, $\overline\nabla\ne0$, so $\sum m_i\le1$. However this inequality together with (\[ineq\]) would imply $\deg{\mathfrak D}\le3$.
(\[Aff2\]) Consider $(L,\eta)\in\operatorname{\mathcal{B}\mathit{un}}(d)$. Combining part , the second formula in , condition of §\[MODST\], and Corollary \[ConnExist\], we see that the fiber of $\rho$ over $(L,\eta)$ is non-empty. Thus it is a torsor over $H^0({{{\mathbb P}^1}},\operatorname{\mathcal{H}\mathit{iggs}}(L,\eta))$. Using the identification $\operatorname{\mathcal{H}\mathit{iggs}}(L,\eta)=\operatorname{\mathcal{E}\mathit{nd}}(L,\eta)^\vee\otimes\Omega_X$, we obtain $$\begin{gathered}
\label{dimcalc}
\dim H^0({{{\mathbb P}^1}},\operatorname{\mathcal{H}\mathit{iggs}}(L,\eta))=\dim H^1({{{\mathbb P}^1}},\operatorname{\mathcal{E}\mathit{nd}}(L,\eta))=1-\chi(\operatorname{\mathcal{E}\mathit{nd}}(L,\eta))=\\
1-\deg(\operatorname{\mathcal{E}\mathit{nd}}(L,\eta))-\operatorname{rk}(\operatorname{\mathcal{E}\mathit{nd}}(L,\eta))=1.\end{gathered}$$ We see that $H^0({{{\mathbb P}^1}},\operatorname{\mathcal{H}\mathit{iggs}}(L,\eta))$ form a vector bundle over $\operatorname{\mathcal{B}\mathit{un}}(d)$, and ${{\mathcal M}}$ is a torsor over this bundle.
Note that, contrary to the case of regular singularities, this proposition is not valid for $n>4$ because the proof of part (\[Aff1\]) is specific for this case.
Parabolic bundles {#PARBUN}
-----------------
Let $(L,\eta)$ be a rank 2 parabolic bundle. We define the *lower modification* of $(L,\eta)$ at $x_i$ as the vector bundle $L_i$ whose sheaf of sections is $$\{s\in L:\,s|_{n_ix_i}\in\eta|_{n_ix_i}\}.$$
Clearly, $\deg L_i=\deg L-n_i$. We shall also use the lower modification $L_\eta$ of $L$ at all $x_i$: its sheaf of sections is $$\{s\in L:\,s|_{\mathfrak D}\in\eta\}.$$ Note that $\eta$ induces parabolic structures on $L_i$ and $L_\eta$. For example in the case of $L_\eta$ we get an exact sequence $$0\to\eta\otimes O_{{{\mathbb P}^1}}(-{\mathfrak D})|_{\mathfrak D}\to
L(-{\mathfrak D})|_{\mathfrak D}\to L_\eta|_{\mathfrak D},$$ thus the image $\eta'$ of $L(-{\mathfrak D})|_{\mathfrak D}$ in $L_\eta|_{\mathfrak D}$ is a parabolic structure on $L_\eta$. Upon choosing local coordinates $z_i$ at each $x_i$, $\eta'$ can be identified with $(L|_{\mathfrak D})/\eta$. It is easy to see that $\operatorname{End}(L,\eta)=\operatorname{End}(L_\eta,\eta')$ and a similar statement is true for $L_i$ with induced parabolic structure.
Recall that $P$ is a projective line doubled at the points of the support of ${\mathfrak D}$.
\[BunP\] $\operatorname{\mathcal{B}\mathit{un}}(d)\simeq P\times B({{\mathbf{G_m}}})$, where $B({{\mathbf{G_m}}}):=pt/{{\mathbf{G_m}}}$ is the classifying stack of ${{\mathbf{G_m}}}$.
The idea of the proof was suggested to the first author by V. Drinfeld.
*Step 1. We can assume $d=-1$.* Let us pick a point $\infty\in{{{\mathbb P}^1}}-{\mathfrak D}$. Then the map $(L,\eta)\mapsto(L\otimes
O_{{{\mathbb P}^1}}(\infty),\eta)$ identifies $\operatorname{\mathcal{B}\mathit{un}}({{{\mathbb P}^1}},2,d,{\mathfrak D})$ with $\operatorname{\mathcal{B}\mathit{un}}({{{\mathbb P}^1}},2,d+2,{\mathfrak D})$. Since $d$ is odd, the statement follows.
*Step 2. $L\simeq O_{{{\mathbb P}^1}}\oplus O_{{{\mathbb P}^1}}(-1)$.* Indeed, it follows from Proposition \[Affine\](\[Aff2\]) that $(L,\eta)$ corresponds to a connection $(L,\nabla)\in{{\mathcal M}}$, thus we can use Proposition \[Affine\](\[Aff0\]).
*Step 3.* The discussion, preceding this proposition, shows that $(L,\eta)\in\operatorname{\mathcal{B}\mathit{un}}(-1)$ implies $(L_\eta\otimes
O_{{{\mathbb P}^1}}(2\infty),\eta')\in\operatorname{\mathcal{B}\mathit{un}}(-1)$ and therefore $L_\eta\simeq
O_{{{\mathbb P}^1}}(-2)\oplus O_{{{\mathbb P}^1}}(-3)$.
Let $\tilde P$ be the moduli stack of collections $(L,\eta,
O_{{{\mathbb P}^1}}\hookrightarrow L, O_{{{\mathbb P}^1}}(-2)\hookrightarrow L_\eta)$, where $(L,\eta)\in\operatorname{\mathcal{B}\mathit{un}}(-1)$. Note that there is a unique up to scalar map $O_{{{\mathbb P}^1}}\to L$ and a unique up to scalar map $O_{{{\mathbb P}^1}}(-2)\to L_\eta$. Thus $\tilde P$ is a principal ${{\mathbf{G_m}}}\times{{\mathbf{G_m}}}$-bundle on $\operatorname{\mathcal{B}\mathit{un}}(-1)$.
*Step 4.* For a point of $\tilde P$ we get a map ${\varphi}:O_{{{\mathbb P}^1}}\oplus O_{{{\mathbb P}^1}}(-2)\to L$. We claim that this map is injective. Indeed, let $m_i$ be as in the proof of Proposition \[Affine\](\[Aff1\]). If the image of ${\varphi}$ is a line subbundle, then $\sum m_i\ge2$. But we saw (again in the proof of Proposition \[Affine\](\[Aff1\])) that this is impossible.
Thus ${\varphi}$ has a simple zero at a single point $q$. Note that $\operatorname{Ker}{\varphi}(q)$ does not coincide with the fiber of $O_{{{\mathbb P}^1}}$ (because $O_{{{\mathbb P}^1}}\to L$ is an embedding of vector bundles). That is, the kernel of ${\varphi}(q)$ is spanned by $(p,1)$, where $p$ is a point in the fiber of $O_{{{\mathbb P}^1}}(2)$ over $q$. (More canonically, $p$ is a homomorphism from the fiber of $O_{{{\mathbb P}^1}}(-2)$ to that of $O_{{{\mathbb P}^1}}$.) The pair $(p,q)$ completely describes $L$ as an upper modification of $O_{{{\mathbb P}^1}}\oplus O_{{{\mathbb P}^1}}(-2)$: the sheaf of sections of $L(-q)$ is $$\label{upper1}
\{(s_1,s_2)\in O_{{{\mathbb P}^1}}\oplus
O_{{{\mathbb P}^1}}(-2)|\,s_1(q)=ps_2(q)\}.$$
*Step 5*. Similarly, we get a map ${\varphi}':O_{{{\mathbb P}^1}}(-{\mathfrak D})\oplus O_{{{\mathbb P}^1}}(-2)\to L_\eta$. It also has exactly one simple zero. Note that $\det{\varphi}=\det{\varphi}'$ (since ${\varphi}$ and ${\varphi}'$ can be identified on ${{{\mathbb P}^1}}-{\mathfrak D}$), so the zero is at the same point $q$. Then $\operatorname{Ker}{\varphi}'(q)$ is spanned by $(1,p')$, where $p'$ is in the fiber of $O_{{{\mathbb P}^1}}(2)$ (more properly, $p'$ is a homomorphism between the fiber of $O_{{{\mathbb P}^1}}(-{\mathfrak D})$ and that of $O_{{{\mathbb P}^1}}(-2)$). Again, $(p',q)$ completely determines $L_\eta$, indeed, the sheaf of sections of $L_\eta(-q)$ is $$\label{upper2}
\{(s_1,s_2)\in O_{{{\mathbb P}^1}}(-{\mathfrak D})\oplus
O_{{{\mathbb P}^1}}(-2)|\,p's_1(q)=s_2(q)\}.$$
*Step 6.* Note that $(p,p',q)$ determines the inclusion $L_\eta\hookrightarrow L$ uniquely as well because it determines it on ${{{\mathbb P}^1}}-{\mathfrak D}$, thus this triple determines a point of $\tilde P$. We must have $L_\eta\subset L$. Looking at (\[upper1\]) and (\[upper2\]) it is easy to see that this condition is exactly $pp'=f(q)$, where $f$ is the canonical section of $O_{{{\mathbb P}^1}}({\mathfrak D})$ (thus the zero locus of $f$ is exactly ${\mathfrak D}$). This makes sense: the product $pp'$ is in the fiber of $O_{{{\mathbb P}^1}}({\mathfrak D})$.
Let $P'$ be the set of triples $(p,p',q)$ as above subject to the condition $pp'=f(q)$. Every such point determines a parabolic bundle $(L,\eta)$ but some of these bundles can have extra automorphisms. In other words, $\tilde P\subset P'$.
Clearly, $P'$ is fibered over ${{{\mathbb P}^1}}$ with coordinate $q$, and the fiber over $x$ is either a hyperbola or a cross, depending on whether $x$ is in ${\mathfrak D}$ or not.
Finally, we need to mod out the embeddings $O_{{{\mathbb P}^1}}\hookrightarrow L$ and $O_{{{\mathbb P}^1}}(-2)\hookrightarrow L_\eta$. If we scale one of them by $a$ and the other by $b$, we get $$(p,p',q)\mapsto((a/b)p,(b/a)p',q).$$
Therefore
- The only points with extra automorphisms are of the form $(0,0,q)$, $q\in{\mathfrak D}$ (the centers of the crosses);
- The stable locus $\tilde P$ is exactly the part of $P'$ that is smooth over ${{{\mathbb P}^1}}$.
- We have $\operatorname{\mathcal{B}\mathit{un}}(-1)=\tilde P/\mathbf{G_m^\mathrm{2}}$. Since the diagonal group $a=b$ acts trivially, this stack is $(\tilde P/{{\mathbf{G_m}}})\times B({{\mathbf{G_m}}})$. Clearly, $\tilde P/{{\mathbf{G_m}}}=P$.
\[RemP\] Note that we can (and shall) view $P$ as the moduli space of collections $$(L,\eta,O_{{{\mathbb P}^1}}(-2)\hookrightarrow L_\eta).$$
By Propositions \[Affine\](\[Aff2\]) and \[BunP\] ${{\mathcal M}}$ is a smooth connected algebraic stack of dimension $1$. To prove that ${{\mathcal M}}=M\times B({{\mathbf{G_m}}})$ consider the moduli stack of triples $(L,\nabla,O_{{{{\mathbb P}^1}}}(-2)\hookrightarrow L_\eta)$, where $(L,\nabla)\in{{\mathcal M}}$. We have ${{\mathcal M}}=M'/{{\mathbf{G_m}}}$, where ${{\mathbf{G_m}}}$ acts by rescaling the embedding $O_{{{{\mathbb P}^1}}}(-2)\hookrightarrow L_\eta$. It is easy to see that this action is trivial, so that ${{\mathcal M}}=M'\times B({{\mathbf{G_m}}})$. On the other hand, Lemma \[Irreducible\] shows that connections in ${{\mathcal M}}$ have only scalar automorphisms, thus $M'$ is an algebraic space. Therefore ${{\mathcal M}}\to M'$ is the good moduli space and $M'=M$ by uniqueness of good moduli spaces. We see that ${{\mathcal M}}$ is a neutral gerbe over $M$. Next, we have a cartesian diagram $$\begin{CD}
M @>>> {{\mathcal M}}\\
@VVV @VV \rho V\\
P @>>> \operatorname{\mathcal{B}\mathit{un}}(d).
\end{CD}$$ Thus $M$ is an affine bundle over $P$. It follows that $M$ is a smooth surface. Finally, $M$ is quasi-projective, since it is open in ${{\overline M}}$.
The map $\rho:{{\mathcal M}}\to\operatorname{\mathcal{B}\mathit{un}}(d)$ is an affine bundle, thus it is an affine morphism. On the other hand, the good moduli space of $\operatorname{\mathcal{B}\mathit{un}}(d)$ is $P$, which is a 1-dimensional scheme.
${{\overline{{\mathcal M}}}}$ is a locally complete intersection
----------------------------------------------------------------
\[mnOne\] If $(L,\nabla;{\varepsilon}\in E)\in{{\overline{{\mathcal M}}}}$, then $L\simeq
O_{{{\mathbb P}^1}}(n)\oplus O_{{{\mathbb P}^1}}(m)$ with $m-n=1$.
For ${\varepsilon}\ne0$ this is Proposition \[Affine\] (\[Aff0\]). Let ${\varepsilon}=0$ and assume that $m-n\ge3$. Then the same argument as in Proposition \[Affine\] (\[Aff0\]) shows that $O_{{{\mathbb P}^1}}(m)$ is $\nabla$-invariant, which contradicts semistability.
\[LocCI\] ${{\overline{{\mathcal M}}}}$ is a locally complete intersection.
Let ${\mathfrak D}'\supset{\mathfrak D}$ be a divisor on $X$. Consider the moduli stack $\widetilde{{\mathcal N}}({\mathfrak D}')\subset{{\overline{\mathstrut{{\mathcal N}}}}}({{{\mathbb P}^1}},2,d,{\mathfrak D}')$ parameterizing $(L,\nabla;{\varepsilon}\in E)$, where $L\simeq
O_{{{\mathbb P}^1}}(n)\oplus O_{{{\mathbb P}^1}}(m)$ with $m-n=1$, $m+n=d$, $\nabla:L\to
L\otimes\Omega_{{{\mathbb P}^1}}({\mathfrak D}')\otimes_{{\mathbb C}}E$ is an ${\varepsilon}$-connection, $(L,\nabla)$ is semistable and non-nilpotent. It is enough to show that if $\deg{\mathfrak D}'$ is big enough, then (i) $\widetilde{{\mathcal N}}({\mathfrak D}')$ is smooth and (ii) ${{\overline{{\mathcal M}}}}$ is defined by $\dim\widetilde{{\mathcal N}}({\mathfrak D}')-\dim{{\overline{{\mathcal M}}}}$ equations in $\widetilde{{\mathcal N}}({\mathfrak D}')$ (note that ${{\overline{{\mathcal M}}}}\subset\widetilde{{\mathcal N}}({\mathfrak D}')$ by Lemma \[mnOne\]).
For (i) it is enough to show that the map $\widetilde{{\mathcal N}}({\mathfrak D}')\to{{{\mathbb A}^1}}/{{\mathbf{G_m}}}$ sending $(L,\nabla;{\varepsilon}\in E)$ to ${\varepsilon}\in E$ is smooth. The relative deformation complex of this map at $(L,\nabla;{\varepsilon}\in E)$ is $${{\mathcal G}}^{{\bullet}}:=
(\operatorname{\mathcal{E}\mathit{nd}}(L)\xrightarrow{\operatorname{ad}\nabla}\operatorname{\mathcal{E}\mathit{nd}}(L)\otimes
\Omega_{{{\mathbb P}^1}}({\mathfrak D}')\otimes_{{\mathbb C}}E),$$ so that the obstruction to smoothness is in $H^1({{{\mathbb P}^1}},\operatorname{\mathcal{E}\mathit{nd}}(L)\otimes\Omega_{{{\mathbb P}^1}}({\mathfrak D}')\otimes_{{\mathbb C}}E)$. This space vanishes for $\deg{\mathfrak D}'$ big enough because $L\simeq O_{{{\mathbb P}^1}}(n)\oplus O_{{{\mathbb P}^1}}(m)$, with $m-n=1$.
For (ii) note that $\dim\widetilde{{\mathcal N}}({\mathfrak D}')=-\chi({{\mathcal G}}^{{\bullet}})=4\deg{\mathfrak D}'-8$.
In Corollary \[MhY\] below, we shall give an explicit description of ${{{{\mathcal M}}_H}}$; this description implies that $\dim{{{{\mathcal M}}_H}}=0$. Combining this with Theorem \[StackProp\], we see that $\dim{{\overline{{\mathcal M}}}}=1$.
Further, let ${{\mathcal L}}$ be the vector bundle on $\widetilde{{\mathcal N}}({\mathfrak D}')$ whose fiber at $(L,\nabla;{\varepsilon}\in E)$ is $\operatorname{\mathcal{E}\mathit{nd}}(L)\otimes
(O_{{{\mathbb P}^1}}({\mathfrak D}')/O_{{{\mathbb P}^1}}({\mathfrak D}))\otimes_{{\mathbb C}}E$. The polar part of $\nabla$ gives rise to a section of ${{\mathcal L}}$ and $\widetilde{{\mathcal N}}({\mathfrak D})\subset\widetilde{{\mathcal N}}({\mathfrak D}')$ is given by the zero locus of this section. Thus $\widetilde{{\mathcal N}}({\mathfrak D})$ is locally cut out by $4(\deg{\mathfrak D}'-\deg{\mathfrak D})$ equations. It follows from the definition of ${{\overline{{\mathcal M}}}}$ (cf. §\[DEFMB\]) that ${{\overline{{\mathcal M}}}}$ is cut out from $\widetilde{{\mathcal N}}({\mathfrak D})$ by $2\deg{\mathfrak D}-1$ equations (note that the sum of residues of $\nabla$ is equal to $-d$).
${{{{\mathcal M}}_H}}$ is given by ${\varepsilon}=0$, so we only need to check that ${\varepsilon}$ is locally not a zero divisor on ${{\overline{{\mathcal M}}}}$. However, if it was the case, ${{{{\mathcal M}}_H}}$ would contain a component of ${{\overline{{\mathcal M}}}}$ (set-theoretically), and we would come to a contradiction with complete intersections having pure dimension.
Universal Moduli Spaces
-----------------------
Recall that ${\mathfrak D}=\sum n_ix_i$. Fix $\infty\in{{{\mathbb P}^1}}\setminus{\mathfrak D}$.
Consider a moduli space ${{\mathcal B}}$, parameterizing local invariants of connections, that is, triples $({\mathfrak D},\nu_1,\nu_2)$, where ${\mathfrak D}$ is a degree 4 divisor on ${{{\mathbb P}^1}}$ such that $\infty\notin\operatorname{\mathrm{supp}}{\mathfrak D}$, $\nu_1\in\Omega_{{{\mathbb P}^1}}({\mathfrak D})/\Omega_{{{\mathbb P}^1}}$, $\nu_2\in\Omega^{\otimes2}_{{{\mathbb P}^1}}(2{\mathfrak D})/\Omega^{\otimes2}_{{{\mathbb P}^1}}({\mathfrak D})$, and the sum of residues of $\nu_1$ equals $-d$, cf. §\[DEFMB\]. We can identify such ${\mathfrak D}$ with roots of degree 4 monic polynomial $p(z)$, then we can write uniquely $$\nu_1=\frac{a_0+a_1z+a_2z^2+dz^3}{p(z)}\,{\mathbf d}z,\qquad
\nu_2=\frac{b_0+b_1z+b_2z^2+b_3z^3}{p(z)^2}\,{\mathbf d}z\otimes{\mathbf d}z.$$ Here $z$ is the standard coordinate on ${{{\mathbb P}^1}}$. Thus ${{\mathcal B}}\simeq{{\mathbb C}}^{11}$. Note that the subset of points in ${{\mathcal B}}$ satisfying conditions of §\[MODST\] is open in *analytic* topology.
As $({\mathfrak D},\nu_1,\nu_2)$ varies, we obtain a family ${{\overline{{\mathcal M}}}}_{univ}\to{{\mathcal B}}$ of moduli stacks. In §\[DEFMB\] we fixed ${\mathfrak D}$, $\nu_1$ and $\nu_2$. Denote the corresponding point of ${{\mathcal B}}$ by $t_0$. Then the fiber of ${{\overline{{\mathcal M}}}}_{univ}\to{{\mathcal B}}$ over $t_0$ is ${{\overline{{\mathcal M}}}}$. Our goal is to prove
\[PrFlat\] The family ${{\overline{{\mathcal M}}}}_{univ}\to{{\mathcal B}}$ is a flat family of stacks in a Zariski neighborhood of $t_0\in{{\mathcal B}}$.
Similarly to the previous subsection we prove that ${{\overline{{\mathcal M}}}}_{univ}$ is a locally complete intersection of dimension $\dim{{\mathcal B}}+1$. It follows that the fibers of ${{\overline{{\mathcal M}}}}_{univ}\to{{\mathcal B}}$ are at least 1-dimensional. By semicontinuity, there is a neighborhood of $t_0$, where fibers are 1-dimensional. It remains to note, that by [@EGAIV Proposition 6.15] a morphism from a locally complete intersection to a smooth scheme with equidimensional fibers is flat.
Generalized line bundles on generalized elliptic curves {#GENELL}
=======================================================
In this section, we provide a version of the Fourier-Mukai transform for singular degenerations of elliptic curves. This generalization is not surprising, and the case of singular reduced irreducible genus one curve (nodal or cuspidal) is well known ([@BK], see also [@BN Theorem 5.2]).
Generalized elliptic curves
---------------------------
For the purposes of this paper, it is important to work with all double covers of ${{{\mathbb P}^1}}$ ramified at four points, including reducible covers (see Remark \[Types\]). We were unable to find a discussion of this case in the literature. However, our argument works in greater generality, and we therefore consider the following class of curves
A projective curve $Y$ is *generalized elliptic* if $H^0(Y,O_Y)={{\mathbb C}}$ (in particular, $Y$ is connected and has no embedded points) and the dualizing sheaf of $Y$ is trivial. In particular, $Y$ is Gorenstein and has arithmetic genus $1$.
\[RemDualizing\] In fact the dualizing sheaf of $Y$ is $O_Y\otimes_{{\mathbb C}}H^1(Y,O_Y)^{\vee}$ rather than $O_Y$. Indeed, the first cohomology group of the dualizing sheaf must be trivialized.
Any plane cubic (reduced or not) is a generalized elliptic curve in this sense.
Note that we do not assume that singularities of $Y$ are planar. For example, an intersection of two space quadrics is a generalized elliptic curve, even if the two quadrics are cones with a common vertex. In this case, the intersection is a union of four lines that meet at the vertex.
Denote by $\Sigma$ the collection of generic points of $Y$ (by definition a point $s\in Y$ is generic if its local ring $O_{Y,s}$ is Artinian). For a sheaf $\ell$ on $Y$ and $s\in
\Sigma$, we denote by $\operatorname{rk}_s\ell$ the length of the stalk $\ell_s$ as a module over the local ring $O_{Y,s}$. In particular, $m(s):=\operatorname{rk}_sO_Y$ is the multiplicity of the corresponding irreducible component.
Fix a weight function $w:\Sigma\to{{\mathbb R}}^{>0}$ and set $$\operatorname{rk}\ell=\operatorname{rk}_w\ell:=\sum_{s\in\Sigma}w(s)\operatorname{rk}_s\ell.$$ We can now use this notion of rank (and the corresponding notion of slope) to define a stability of coherent sheaves on $Y$.
A coherent sheaf $\ell\ne 0$ of pure dimension 1 is said to be *semistable* if for any proper subsheaf $\ell_0\subset\ell$, $\ell_0\ne0,\ell$, we have $$\frac{\chi(\ell)}{\operatorname{rk}\ell}\ge\frac{\chi(\ell_0)}{\operatorname{rk}\ell_0}.$$ If the inequality is strict, $\ell$ is *stable*.
We say that a sheaf $\ell$ is a *generalized line bundle* on $Y$ if it is of pure dimension 1 and its length at all generic points of $Y$ equals to the multiplicity of the corresponding component: $\operatorname{rk}_s\ell=m(s)$ for $s\in\Sigma$.
By definition, $\deg\ell:=\chi(\ell)-\chi(O_Y)=\chi(\ell)$.
Denote by ${\operatorname{\overline{\mathcal{P}\mathit{ic}}}^{\;d}}(Y)$ the stack of generalized line bundles of degree $d$ on $Y$, and let ${\operatorname{\overline{\mathcal{P}\mathit{ic}}}^{\;d}}_s(Y)\subset{\operatorname{\overline{\mathcal{P}\mathit{ic}}}^{\;d}}_{ss}(Y)\subset{\operatorname{\overline{\mathcal{P}\mathit{ic}}}^{\;d}}(Y)$ stand for the open substacks of stable and semistable generalized line bundles, respectively.
Let ${{\mathcal P}}:=O_{Y\times Y}(-\Delta)$ be the ideal sheaf of the diagonal $\Delta\subset Y\times Y$. Note that ${{\mathcal P}}$ is flat over each of the factors, being the kernel of a surjection of flat sheaves, therefore, ${{\mathcal P}}$ is a $Y$-family of degree $-1$ generalized line bundles on $Y$, so it defines a map $Y\to{\operatorname{\overline{\mathcal{P}\mathit{ic}}}^{\;-1}}(Y)$. (Note that ${{\mathcal P}}$ is not in general flat over the product.) The above map naturally extends to a map $$Y\times B({{\mathbf{G_m}}})\to{\operatorname{\overline{\mathcal{P}\mathit{ic}}}^{\;-1}}(Y).\label{picmap}$$ Explicitly, for a test scheme $S$, the map assigns to a morphism $\psi:S\to Y$ and a line bundle $L$ on $S$ (recall that a line bundle on $S$ is the same as a map $S\to B({{\mathbf{G_m}}})$) the sheaf $p_1^*L\otimes O_{S\times
Y}(-\Gamma_\psi)$ on $S\times Y$, viewed as an $S$-family of degree $-1$ generalized line bundles on $Y$, that is, as a morphism $S\to{\operatorname{\overline{\mathcal{P}\mathit{ic}}}^{\;-1}}(Y)$. Here $\Gamma_\psi\subset S\times
Y$ is the graph of $\psi$.
Our goal is to prove the following claims
\[PicY\] The map is an isomorphism $$Y\times B({{\mathbf{G_m}}}){\mathrel{\widetilde\to}}{\operatorname{\overline{\mathcal{P}\mathit{ic}}}^{\;-1}}_s(Y)={\operatorname{\overline{\mathcal{P}\mathit{ic}}}^{\;-1}}_{ss}(Y).$$
In particular, ${\operatorname{\overline{\mathcal{P}\mathit{ic}}}^{\;-1}}_s(Y)={\operatorname{\overline{\mathcal{P}\mathit{ic}}}^{\;-1}}_{ss}(Y)$ does not depend on $w$.
\[FourierMukai\] The Fourier-Mukai transform with kernel ${{\mathcal P}}$ $${{\mathcal D}}^b(Y)\to{{\mathcal D}}^b(Y):{{\mathcal F}}\mapsto Rp_{1,*}({{\mathcal P}}\otimes^Lp_2^*{{\mathcal F}})$$ is an auto-equivalence of the category of ${{\mathcal D}}^b(Y)$ [[(]{}-1pt]{}the bounded derived category of quasi-coherent sheaves on $Y$[[)]{}]{}. Here $p_1,p_2:Y\times Y\to Y$ are projections.
\[Rm:FM\] Proposition \[PicY\] allows us to identify $Y$ and the coarse moduli space of ${\operatorname{\overline{\mathcal{P}\mathit{ic}}}^{\;-1}}_s(Y)$. In fact, the moduli space is fine: the Poincaré sheaf ${{\mathcal P}}$ is a universal sheaf on $Y\times Y$. Theorem \[FourierMukai\] then provides a Fourier-Mukai transform between $Y$ and the coarse moduli space.
Also, consider the Serre dual ${{\mathcal P}}^\vee:=\operatorname{\mathcal{H}\mathit{om}}({{\mathcal P}},O_{Y\times
Y})$ of ${{\mathcal P}}$. Then Corollary \[CEllipticDual\] implies that an analogue of Proposition \[PicY\] holds for ${{\mathcal P}}^\vee$: it provides an isomorphism $Y\times
B({{\mathbf{G_m}}}){\mathrel{\widetilde\to}}{\operatorname{\overline{\mathcal{P}\mathit{ic}}}^{\;1}}_s(Y)={\operatorname{\overline{\mathcal{P}\mathit{ic}}}^{\;1}}_{ss}(Y)$. The Fourier-Mukai transform given by ${{\mathcal P}}^\vee$ is also an equivalence (up to cohomological shift, it is inverse to that given by ${{\mathcal P}}$, see §\[FMproof\]).
Stable generalized line bundles
-------------------------------
Let us prove Proposition \[PicY\]. We start with some remarks about duality on $Y$.
\[EllipticDual\] [[)]{}]{}\[EDa\] Let $\ell$ be a coherent sheaf on $Y$ of pure dimension $1$. Then $\ell$ is Cohen-Macaulay: $\ell^\vee:=R\operatorname{\mathcal{H}\mathit{om}}(\ell,O_Y)=\operatorname{\mathcal{H}\mathit{om}}(\ell,O_Y)$ is a coherent sheaf of pure dimension 1. [[(]{}-1pt]{}Recall that $O_Y$ is the dualizing sheaf on $Y$.[[)]{}]{}
[[)]{}]{}\[EDb\] Let $S$ be a locally Noetherian scheme [[(]{}-1pt]{}or a stack[[)]{}]{} and let $\ell$ be an $S$-family of coherent sheaves of pure dimension $1$ on $Y$; that is, $\ell$ is a coherent sheaf on $S\times Y$ such that $\ell$ is flat over $S$ and its restriction to fibers over $s\in
S$ have pure dimension $1$. Then $\ell^\vee:=R\operatorname{\mathcal{H}\mathit{om}}(\ell,O_{S\times
Y})=\operatorname{\mathcal{H}\mathit{om}}(\ell,O_{S\times Y})$ is a coherent sheaf.
[[)]{}]{}\[EDc\] In the assumptions of part [[(]{}-1pt]{}\[EDb\][[)]{}]{}, $\ell^\vee$ is flat over $S$, and for any point $s\in
S$, we have $(\ell^\vee)|_{\{s\}\times
Y}=(\ell|_{\{s\}\times Y})^\vee$. In other words, duality respects families.
We have $\operatorname{\mathcal{E}\mathit{xt}}^i(\ell,O_Y)=0$ for $i\gg0$ (since the dualizing sheaf has finite injective dimension). Recall that a coherent sheaf with zero fibers vanishes, thus it suffices to check that the derived restriction $L\iota_y^*R\operatorname{\mathcal{H}\mathit{om}}(\ell,O_Y)$ is concentrated in non-positive cohomological degrees for any closed point $\iota_y:\{y\}\hookrightarrow Y$. However, its dual is $R\iota_y^!\ell[1]$ (since Serre duality permutes $L\iota_y^*$ and $R\iota_y^!$), which is concentrated in non-negative degrees as $\ell$ is of pure dimension $1$.
For part , let us show that for any coherent sheaf ${{\mathcal F}}$ on $S$, the sheaf $\operatorname{\mathcal{E}\mathit{xt}}^i(\ell,p_1^*{{\mathcal F}})$ vanishes for $i>0$. The statement is local in $S$, so we may assume that $S$ is an affine Noetherian scheme without loss of generality. If $S$ is a point, the claim follows from part .
For general $S$, the problem is that we do not know [*a priori*]{} that $\operatorname{\mathcal{E}\mathit{xt}}^i$ vanishes for $i\gg0$. To circumvent this problem we proceed by Noetherian induction on $\operatorname{\mathrm{supp}}({{\mathcal F}})$. Fix a generic point $\iota_s:\operatorname{Spec}O_{S,s}\hookrightarrow\operatorname{\mathrm{supp}}({{\mathcal F}})$, and consider the adjunction morphism $${{\mathcal F}}\to\iota_{s,*}\iota_s^*{{\mathcal F}}.$$ Both its kernel and its cokernel are unions of coherent sheaves supported by strictly smaller closed subsets, so the induction hypothesis implies that the induced map $$\operatorname{\mathcal{E}\mathit{xt}}^i(\ell,p_1^*{{\mathcal F}})\to \operatorname{\mathcal{E}\mathit{xt}}^i(\ell,p_1^*\iota_{s,*}\iota_s^*{{\mathcal F}})$$ is an isomorphism for $i>1$. Also, part implies that $$\operatorname{\mathcal{E}\mathit{xt}}^i(\ell,p_1^*\iota_{s,*}\iota_s^*{{\mathcal F}})=0\qquad(i>0),$$ so that $\operatorname{\mathcal{E}\mathit{xt}}^i(\ell,p_1^*{{\mathcal F}})=0$ for $i>1$.
Now note that for any point $\iota_s:\{s\}\hookrightarrow S$, $$L(\iota_s\times\operatorname{id}_Y)^* R\operatorname{\mathcal{H}\mathit{om}}(\ell,p_1^*{{\mathcal F}})=
R\operatorname{\mathcal{H}\mathit{om}}(\ell|_{\{s\}\times Y},L\iota_s^*{{\mathcal F}}\boxtimes O_Y)
\label{dualitybasechange}$$ is concentrated in non-positive cohomological dimensions, thus $\operatorname{\mathcal{E}\mathit{xt}}^1(\ell,p_1^*{{\mathcal F}})=0$. This completes the proof of part .
Part follows by taking ${{\mathcal F}}=O_S$ in .
\[CEllipticDual\] Let $\ell$ be a generalized line bundle on $Y$. Then so is $\ell^\vee$, and $\deg\ell^\vee=-\deg\ell$. Moreover, $\ell$ is [[(]{}-1pt]{}semi[[)]{}]{}stable if and only if $\ell^\vee$ is [[(]{}-1pt]{}semi[[)]{}]{}stable.
For every $d$, the duality $\ell\mapsto\ell^\vee$ defines an isomorphism ${\operatorname{\overline{\mathcal{P}\mathit{ic}}}^{\;d}}(Y){\mathrel{\widetilde\to}}{\operatorname{\overline{\mathcal{P}\mathit{ic}}}^{\;-d}}(Y)$ and also isomorphisms ${\operatorname{\overline{\mathcal{P}\mathit{ic}}}^{\;d}}_{ss}(Y){\mathrel{\widetilde\to}}{\operatorname{\overline{\mathcal{P}\mathit{ic}}}^{\;-d}}_{ss}(Y)$ and ${\operatorname{\overline{\mathcal{P}\mathit{ic}}}^{\;d}}_s(Y){\mathrel{\widetilde\to}}{\operatorname{\overline{\mathcal{P}\mathit{ic}}}^{\;-d}}_s(Y)$.
If $\ell$ is a sheaf of pure dimension 1 on $Y$, then $\ell^\vee$ is also a sheaf, and by Serre duality $\chi(\ell^\vee)=-\chi(\ell)$. The duality for modules over Artinian rings shows that $\ell$ is also a generalized line bundle. Assume that $\ell$ is a (semi)stable generalized line bundle. Let $\ell_0\subset \ell^\vee$ be a subsheaf (necessarily of pure dimension 1). Without loss of generality we can assume that $\ell^\vee/\ell_0$ is also of pure dimension 1. Then by part (\[EDa\]) of the lemma we get a surjection $\ell\twoheadrightarrow\ell_0^\vee$, and we see that the (semi)stability property for $\ell$ implies the similar property for $\ell^\vee$. The remaining claims are obvious.
\[lm:OStable\] $O_Y$ is stable.
By definition, we need to show that $O_Y$ has no proper subsheaves $\ell_0\subset O_Y$ with $\chi(\ell_0)\ge 0$. Assume the converse. Without loss of generality, we may assume that $\ell_1:=O_Y/\ell_0$ has pure dimension 1. By Lemma \[EllipticDual\], its dual $$\ell_1^\vee=\operatorname{\mathcal{H}\mathit{om}}(\ell_1,O_Y)=R\operatorname{\mathcal{H}\mathit{om}}(\ell_1,O_Y)$$ is a sheaf. Since $\ell_1=O_Y/\ell_0$ has a global section (image of $1\in
O_Y$), and $\chi(\ell_1)=-\chi(\ell_0)\le 0$, we see that $H^1(Y,\ell_1)\ne 0$. Then by Serre duality $H^0(Y,\ell_1^\vee)\ne 0$, which is impossible, because $\ell_1^\vee\subset O_Y$ is a proper subsheaf.
Let us prove Proposition \[PicY\] on the level of points.
\[GenLB\] Let $\ell$ be a generalized line bundle of degree $-1$. Then $\ell$ is stable if and only there exists an isomorphism $\ell\simeq O_Y(-y)$, where $O_Y(-y)\subset O_Y$ is the ideal sheaf of a closed point $y\in Y$. The isomorphism is unique up to scaling, and the point $y$ is uniquely determined by $\ell$. Also, stability of $\ell$ is equivalent to semistability.
For a sheaf $\ell$ of degree $-1$, both stability and semistability mean that all quotient sheaves of $\ell$ have non-negative Euler characteristic (excluding $\ell$ itself). In particular, $O_Y(-y)$ is stable; indeed, for any quotient ${{\mathcal F}}$ of $O_Y(-y)$ we have a corresponding quotient ${{\mathcal F}}'$ of $O_Y$ and $\chi({{\mathcal F}}')=\chi({{\mathcal F}})+1$, so we can use Lemma \[lm:OStable\].
Conversely, suppose $\ell$ is stable. Then $H^0(Y,\ell)=0$, because $O_Y$ is stable and its slope is greater than the slope of $\ell$. Therefore, $H^1(Y,\ell)$ is one-dimensional; by Serre duality, this gives a non-zero homomorphism $\kappa:\ell\to O_Y$, unique up to scaling. Since $\ell$ is a generalized line bundle, $\kappa$ cannot be surjective. Thus, by stability of $O_Y$, the image has negative Euler characteristic; together with stability of $\ell$, this implies that $\kappa$ is an embedding.
Once it is proved that $O_Y(-y)$ is stable for all $y\in Y$, Proposition \[GenLB\] can also be derived from Theorem \[FourierMukai\] together with Remark \[Rm:FM\]. More generally, let $L$ be a semistable coherent sheaf of slope $-1$, so that $\deg L=-\operatorname{rk}L$. We claim that the Fourier-Mukai transform of $L^\vee$ is of the form ${{\mathcal F}}[-1]$, where ${{\mathcal F}}$ is a coherent sheaf of finite length that equals $\operatorname{rk}L$, assuming that the weight function $w$ is normalized so that $\operatorname{rk}O_Y=1$ (cf. [@BK Theorem 2.21]). Note that the converse implication is obvious: if the Fourier-Mukai transform of $L^\vee$ is of this form, then $L$ is obtained by successive extension from sheaves of the form $O_Y(-y)$, which implies semistability.
Let us verify the claim. Let ${{\mathcal F}}$ be the Fourier-Mukai transform of $L^\vee$. Then ${{\mathcal F}}$ is concentrated in cohomological degrees $0$ and $1$, and $H^0({{\mathcal F}})$ is of pure dimension 1. Since $L^\vee$ is semistable, $$\begin{split}
\operatorname{Hom}(L^\vee,O_Y(-y)^\vee)=0\quad\text{for all but finitely many }y\in Y,\\
\operatorname{Hom}(O_Y(-y)^\vee,L^\vee)=0\quad\text{for all but finitely many }y\in Y.
\end{split}$$ Indeed, the Homs can be non-zero, only if $O_Y(-y)^\vee$ occurs among the Jordan–Holder factors of $L^\vee$.
Applying the Fourier-Mukai transform, we see that $$\begin{split}
\operatorname{Hom}({{\mathcal F}},O_y[-1])=0\quad\text{for all but finitely many }y\in Y,\\
\operatorname{Hom}(O_y[-1],{{\mathcal F}})=0\quad\text{for all but finitely many }y\in Y.
\end{split}$$ The first statement implies that $H^1({{\mathcal F}})$ is of finite length. Applying the Serre duality to the second statement, we see that $$\operatorname{Hom}({{\mathcal F}},O_y)=0\quad\text{for all but finitely many }y\in Y,$$ hence $H^0({{\mathcal F}})=0$.
In a similar way, Proposition \[FamiliesGenLB\] below can be proved using the Fourier-Mukai transform on $S\times Y$.
\[Chi\] [[)]{}]{}\[Chi1\] Let $\ell\in{\operatorname{\overline{\mathcal{P}\mathit{ic}}}^{\;\pm1}}_s(Y)$, then $\ell$ is a line bundle on the complement to a point.
[[)]{}]{}\[Chi2\] Let $\ell\in{\operatorname{\overline{\mathcal{P}\mathit{ic}}}^{\;-1}}_s(Y)$, then $$\dim H^i(\ell)=\begin{cases}
0\text{ if }i=0,\\
1\text{ if }i=1.
\end{cases}$$ [[)]{}]{}\[Chi3\] Let $\ell\in{\operatorname{\overline{\mathcal{P}\mathit{ic}}}^{\;1}}_s(Y)$, then $$\dim H^i(\ell)=\begin{cases}
1\text{ if }i=0,\\
0\text{ if }i=1.
\end{cases}$$
The first part is clear. We have $H^0(O_Y(-y))=0$ thus $\dim
H^1(O_Y(-y))=-\chi(O_Y(-y))=1$. The last part follows from Serre duality.
We are now ready to prove Proposition \[PicY\]. Note that ${\operatorname{\overline{\mathcal{P}\mathit{ic}}}^{\;-1}}(Y)$ is clearly locally Noetherian (it is of locally finite type over the ground field), so it suffices to prove the following
\[FamiliesGenLB\] Let $S$ be a locally Noetherian scheme or stack, $\ell$ be an $S$-family of degree $-1$ stable generalized line bundles on $Y$ [[(]{}-1pt]{}that is, a coherent sheaf on $S\times Y$, flat over $S$, whose fibers are degree $-1$ stable generalized line bundles[[)]{}]{}.
Then there exist a map $\psi:S\to Y$ and a line bundle $L$ on $S$ such that there exists an isomorphism $$\kappa:\ell{\mathrel{\widetilde\to}}p_1^*L\otimes O_{S\times Y}(-\Gamma_\psi).$$ Moreover, $\psi$ is unique, $L$ is unique up to isomorphism, and $\kappa$ is unique up to scaling by an element of $H^0(S,O_S^\times)$.
By Lemma \[EllipticDual\] and Corollary \[CEllipticDual\], the dual sheaf $\ell^\vee=\operatorname{\mathcal{H}\mathit{om}}(\ell,O_{S\times Y})$ is an $S$-family of stable generalized line bundles on $Y$ of degree $1$. Consider $Rp_{1,*}\ell^\vee$. By base change, we see that for every point $\iota_s:\{s\}\hookrightarrow S$, $$\dim H^i(L\iota_s^*Rp_{1,*}\ell^\vee)=\dim H^i(\{s\}\times Y,\ell^\vee|_{\{s\}\times Y})
=\begin{cases}1&\text{ if }i=0,\\0&\text{ if }i\ne 0,\end{cases}$$ where the last equality follows from Corollary \[Chi\]. Therefore, $p_{1,*}\ell^\vee=Rp_{1,*}\ell^\vee$ is a flat coherent sheaf with one-dimensional fibers, that is, a line bundle (the proof of flatness is as in Lemma \[FlatStacks\]). Let $L:=(p_{1,*}\ell^\vee)^{-1}$ be its dual.
Consider the adjunction map $$f:(p_1^*L)^{-1}\to\ell^\vee.$$ Denote by $U\subset S\times Y$ the open set where $f$ is surjective. Proposition \[GenLB\] implies that $(\{s\}\times
Y)\cap U\ne\emptyset$ for every $s\in S$. It is now easy to see that $f|_U$ is bijective, because $\ell^\vee$ is flat over $S$ and the restriction of $f$ to the fiber $\{s\}\times Y$ is injective for every $s\in S$. In particular $\ell^\vee|_U$ is a line bundle.
Consider the homomorphism $(p_1^*L)^{-1}\otimes\ell\to
O_{S\times Y}$ induced by $f$. It is injective, because $\ell$ has no sections supported by $S\times Y-U$. Therefore, $(p_1^*L)^{-1}\otimes\ell$ is identified with the ideal sheaf of a closed subscheme $S'\subset S\times Y-U$.
We need to show that $p_1$ induces an isomorphism $S'\to S$. Recall that a projective quasi-finite morphism is finite (see [@Hartshorne Exercise III.11.2]). Thus $p_1|_{S'}$ is finite, so it suffices to prove that the natural map $O_S\to
p_{1,*}O_{S'}$ is an isomorphism. Using base change and Proposition \[GenLB\], we see that for every point $\iota_s:\{s\}\hookrightarrow S$, the map induces an isomorphism $$\iota_s^*O_S\to L\iota_s^*p_{1,*}O_{S'}.$$ In other words, $p_{1,*}O_{S'}$ is flat and the map $O_S\to
p_{1,*}O_{S'}$ is an isomorphism on fibers. Therefore, it is an isomorphism. Thus $S'$ is the graph of a map $\psi:S\to Y$.
The uniqueness statements are easy and left to the reader.
Fourier-Mukai transform {#FMproof}
-----------------------
It remains to prove Theorem \[FourierMukai\]. Actually, the argument of I. Burban and B. Kreussler from [@BK] (for the case of an irreducible Weierstrass cubic, nodal or cuspidal) extends without difficulty to our settings, as mentioned in the introduction to [@BK]. We sketch the argument and refer the reader to [@BK] for details.
The key observation is that the structure sheaf $O_Y\in D^b(Y)$ is a *spherical object* in the sense of P. Seidel and R. Thomas [@ST]. A spherical object ${{\mathcal E}}\in D^b(Y)$ defines an equivalence $T_{{\mathcal E}}:D^b(Y)\to D^b(Y)$ called the *twist functor*; roughly speaking, it sends ${{\mathcal F}}\in D^b(Y)$ to the cone of the evaluation morphism $$R\operatorname{Hom}({{\mathcal E}},{{\mathcal F}})\otimes^L{{\mathcal E}}\to{{\mathcal F}}.$$ (One has to replace ${{\mathcal E}}$ by its resolution to ensure that the cone is functorial.) We can then see that for the spherical object ${{\mathcal E}}=O_Y$, the twist functor $T_{{\mathcal E}}$ is the Fourier-Mukai transform of Theorem \[FourierMukai\], cf.[@BK Proposition 2.10].
Let us translate this argument into the language of the Fourier-Mukai kernels, since we shall use Proposition \[FMOrthog\] later. Let $p_{12}$, $p_{13}$, and $p_{23}$ be the usual projections $Y\times Y\times Y\to Y\times
Y$. Set ${{\mathcal F}}_Y:=Rp_{13,*}(p^*_{12}{{\mathcal P}}^\vee\otimes^L
p^*_{23}{{\mathcal P}})$. Note that by Lemma \[EllipticDual\], both ${{\mathcal P}}$ and ${{\mathcal P}}^\vee$ are flat with respect to both projections $Y\times Y\to Y$. In particular, $$p^*_{12}{{\mathcal P}}^\vee\otimes^L p^*_{23}{{\mathcal P}}=p^*_{12}{{\mathcal P}}^\vee\otimes p^*_{23}{{\mathcal P}}$$ is a sheaf, so that ${{\mathcal F}}_Y$ is concentrated in cohomological dimensions $0$ and $1$.
\[FMOrthog\] $${{\mathcal F}}_Y=O_\Delta[-1]\otimes_{{\mathbb C}}H^1(Y,O_Y)\simeq O_\Delta[-1].$$
Note first that $$R\operatorname{\mathcal{H}\mathit{om}}(O_\Delta,O_{Y\times Y})=
R\iota_\Delta^!O_{Y\times Y}=O_\Delta[-1]\otimes_{{\mathbb C}}H^1(Y,O_Y).$$ Here $\iota_\Delta:Y\to Y\times Y$ is the diagonal embedding. The second equality holds because $O_{Y\times Y}$ is the dualizing sheaf up to the second power of $H^1(Y,O_Y)$ (see Remark \[RemDualizing\]) and the $!$-pullback of the dualizing sheaf is the dualizing sheaf up to a cohomological shift.
Applying the duality functor $R\operatorname{\mathcal{H}\mathit{om}}(\cdot\,,O_{Y\times Y})$ to the exact sequence $$0\to{{\mathcal P}}\to O_{Y\times Y}\to O_\Delta\to 0,
\label{PSeq}$$ we obtain an exact sequence $$0\to O_{Y\times Y}\to{{\mathcal P}}^\vee\to O_\Delta\otimes_{{\mathbb C}}H^1(Y,O_Y)\to 0.
\label{PveeSeq}$$ It induces a map $O_Y=p_{1,*} O_{Y\times Y}\to
p_{1,*}{{\mathcal P}}^\vee$; it is easy to see that the map provides an isomorphism $O_Y{\mathrel{\widetilde\to}}Rp_{1,*}{{\mathcal P}}^\vee$ (see Corollary \[Chi\]).
Finally, also yields an exact sequence $$0\to p^*_{12}{{\mathcal P}}^\vee\otimes p^*_{23}{{\mathcal P}}\to p_{12}^*{{\mathcal P}}^\vee\to
p_{12}^*{{\mathcal P}}^\vee\otimes p_{23}^*O_\Delta\to 0.$$ Applying $Rp_{13,*}$, we obtain a distinguished triangle $${{\mathcal F}}_Y\to (Rp_{1,*}{{\mathcal P}}^\vee)\boxtimes O_Y\to{{\mathcal P}}^\vee\to{{\mathcal F}}_Y[1].$$ It remains to notice that the map $(Rp_{1,*}{{\mathcal P}}^\vee)\boxtimes
O_Y\to{{\mathcal P}}^\vee$ becomes the natural embedding $O_{Y\times
Y}\to{{\mathcal P}}^\vee$ from after the identification $Rp_{1,*}{{\mathcal P}}^\vee= O_Y$.
It is easy to see that Proposition \[FMOrthog\] implies Theorem \[FourierMukai\]: indeed, the base change shows, that the functors $${{\mathcal D}}(Y)\to{{\mathcal D}}(Y):{{\mathcal F}}\mapsto Rp_{1,*}({{\mathcal P}}\otimes^Lp_2^*{{\mathcal F}})$$ and $${{\mathcal D}}(Y)\to{{\mathcal D}}(Y):{{\mathcal F}}\mapsto Rp_{2,*}({{\mathcal P}}^\vee\otimes^Lp_1^*{{\mathcal F}})
\otimes_{{\mathbb C}}(H^1(Y,O_Y))^{-1}[1]$$ are mutual inverses.
Geometric description of ${{{{\mathcal M}}_H}}$ {#MH}
===============================================
Recall that our goal is to calculate cohomology of certain natural vector bundle on ${{\mathcal M}}$ (or more precisely, a direct image, see Theorem \[Theorem3\]). In this section we calculate the direct image of the extension of this sheaf to ${{{{\mathcal M}}_H}}$ (see §\[DEFMB\] for the definition of ${{{{\mathcal M}}_H}}$). The main result is Proposition \[MhXxY\]. The calculation is based on explicit identification of ${{{{\mathcal M}}_H}}$, see Corollary \[MhY\], and applying the Fourier–Mukai transform.
We claim that ${{{{\mathcal M}}_H}}$ is the moduli stack of collections $(L,\nabla;E)$, where $L$ is a rank 2 degree $d$ vector bundle on ${{{\mathbb P}^1}}$, $E$ is a one-dimensional vector space, $\nabla:L\to
L\otimes\Omega_{{{\mathbb P}^1}}({\mathfrak D})\otimes_{{\mathbb C}}E$ is an $O_{{{{\mathbb P}^1}}}$-linear morphism, satisfying the following conditions:\
[[)]{}]{}\[NoNilp\] $\nabla$ is not nilpotent, that is $\nabla^2\ne0$.\
[[)]{}]{}\[HitchinTrace\] $\operatorname{tr}\nabla=0$.\
[[)]{}]{}\[HitchinDet\] $\det\nabla$ is a section of $E^{\otimes2}\otimes_{{\mathbb C}}\Omega_{{{\mathbb P}^1}}^{\otimes2}({\mathfrak D})$.\
[[)]{}]{}$(L,\nabla;E)$ is semistable.\
Note that $\operatorname{tr}\nabla$ is a section of $E\otimes_{{\mathbb C}}\Omega_{{{\mathbb P}^1}}({\mathfrak D})$. It follows from (\[hitchincond\]) that $\operatorname{tr}\nabla$ is in fact a section of $E\otimes_{{\mathbb C}}\Omega_{{{\mathbb P}^1}}$, which implies condition (\[HitchinTrace\]). Condition (\[HitchinDet\]) is a condition on the polar part of $\nabla$: a priori $\det\nabla$ is in $E^{\otimes2}\otimes_{{\mathbb C}}\Omega_{{{\mathbb P}^1}}^{\otimes2}(2{\mathfrak D})$. This condition also follows from (\[hitchincond\]).
Note that $\nabla^2=-\det\nabla\otimes\operatorname{id}_L$. Recall that ${{\mathcal E}}$ is the line bundle on ${{\overline{{\mathcal M}}}}$ whose fiber at $(L,\nabla;{\varepsilon}\in E)$ is $E$. For simplicity we write ${{\mathcal E}}$ for ${{\mathcal E}}|_{{{{\mathcal M}}_H}}$. The following statement follows from (\[NoNilp\]) and (\[HitchinDet\]) above
\[Esquare\] $${{\mathcal E}}^{\otimes2}|_{{{{\mathcal M}}_H}}\simeq O_{{{{{\mathcal M}}_H}}}.$$
Let us fix a global section $\mu$ of $\Omega_{{{\mathbb P}^1}}^{\otimes2}({\mathfrak D})\simeq O_{{{{\mathbb P}^1}}}$, $\mu\ne0$. One can choose an isomorphism $E\simeq{{\mathbb C}}$ such that $\det\nabla=\mu$ (there are two choices for such an isomorphism). Denote by ${{\mathcal Y}}$ the moduli stack of pairs $(L,\nabla)$, where $L$ is a rank 2 degree $d$ vector bundle on ${{{\mathbb P}^1}}$, $\nabla\in H^0({{{\mathbb P}^1}},\operatorname{\mathcal{E}\mathit{nd}}(L)\otimes
\Omega_{{{\mathbb P}^1}}({\mathfrak D}))$, $\operatorname{tr}\nabla=0$, $\det\nabla=\mu$, and the pair $(L,\nabla)$ is semistable. We have proved the following
The correspondence $(L,\nabla)\mapsto(L,\nabla;0\in{{\mathbb C}})$ yields a double cover ${{\mathcal Y}}\to{{{{\mathcal M}}_H}}$. Besides, ${{{{\mathcal M}}_H}}$ is identified with the quotient stack $\mu_2\backslash{{\mathcal Y}}$, where $\pm1\in\mu_2$ acts on ${{\mathcal Y}}$ by $(L,\nabla)\mapsto(L,\pm\nabla)$.
It follows directly from the definition of ${{\mathcal Y}}$ that the pullback of ${{\mathcal E}}$ to ${{\mathcal Y}}$ is $O_{{\mathcal Y}}$.
Set ${{\mathcal A}}:=O_{{{\mathbb P}^1}}\oplus\Omega_{{{\mathbb P}^1}}({\mathfrak D})^{-1}$. Then ${{\mathcal A}}$ is a sheaf of $O_{{{\mathbb P}^1}}$-algebras with respect to the multiplication $$(f_1,\tau_1)\times(f_2,\tau_2):=
(f_1f_2-\mu\otimes\tau_1\otimes\tau_2,f_1\tau_2+f_2\tau_1).$$ Set $\pi:Y:= \operatorname{\mathcal{S}\mathit{pec}}({{\mathcal A}})\to{{{\mathbb P}^1}}$. Denote by $y_i\in Y$ the preimage of $x_i\in{{{\mathbb P}^1}}$, and by $\sigma:Y\to Y$ the involution induced by $\sigma^*:{{\mathcal A}}\to{{\mathcal A}}:(f,\tau) \mapsto(f,-\tau)$.
$Y$ is a generalized elliptic curve.
Since $\pi$ is a finite morphism, $Y$ has dimension 1. The dualizing complex of $Y$ is given by $\operatorname{\mathcal{H}\mathit{om}}({{\mathcal A}},\Omega_{{{\mathbb P}^1}})$. Thus we need to show that this sheaf is isomorphic to ${{\mathcal A}}$ as an ${{\mathcal A}}$-module. It is clear on the level of $O_{{{\mathbb P}^1}}$-modules, since ${{\mathcal A}}\simeq O_{{{\mathbb P}^1}}\oplus O_{{{\mathbb P}^1}}(-2)$. Let $\gamma\in\operatorname{\mathcal{H}\mathit{om}}({{\mathcal A}},\Omega_{{{\mathbb P}^1}})$ be the composition of the projection ${{\mathcal A}}\to(\Omega_{{{\mathbb P}^1}}({\mathfrak D}))^{-1}$ and an isomorphism. One checks easily that the map of ${{\mathcal A}}$-modules ${{\mathcal A}}\to\operatorname{\mathcal{H}\mathit{om}}({{\mathcal A}},\Omega_{{{\mathbb P}^1}})$ given by $1\mapsto\gamma$ is injective. Now, an injective map of a vector bundle to an isomorphic one is necessarily an isomorphism.
Also, $H^0(Y,O_Y)=H^0({{{\mathbb P}^1}},{{\mathcal A}})={{\mathbb C}}$, thus $Y$ is generalized elliptic.
\[Types\] Actually $Y$ is always reduced. Precisely, $Y$ is a smooth elliptic curve if ${\mathfrak D}$ has no multiple points; $Y$ is a nodal cubic if ${\mathfrak D}$ has a single multiple point of multiplicity 2; $Y$ is a cuspidal cubic if ${\mathfrak D}=3(x_1)+(x_2)$; $Y$ has two components, isomorphic to ${{{\mathbb P}^1}}$, which intersect transversally at two points if ${\mathfrak D}=2(x_1)+2(x_2)$; and $Y$ has two components, isomorphic to ${{{\mathbb P}^1}}$, which are tangent to each other if ${\mathfrak D}=4(x_1)$.
\[Hitchin\] ${{\mathcal Y}}$ is naturally isomorphic to ${\operatorname{\overline{\mathcal{P}\mathit{ic}}}^{\;d+2}}_sY$, that is the moduli stack of stable generalized line bundles of degree $d+2$ on $Y$.
Let $(L,\nabla)$ be a point of ${{\mathcal Y}}$. Then $L$ is an ${{\mathcal A}}$-module with respect to the multiplication $(f,\tau)s:=fs+\tau\otimes\nabla s$, let us denote the corresponding sheaf on $Y$ by $\ell$. It is a standard fact about the Hitchin system that $\ell$ is a generalized line bundle on $Y$. The inverse construction is given by $\ell\mapsto L:=\pi_*\ell$.
Let the weight function $w$ from §\[GENELL\] be given by the degree of the projection $\pi:Y\to{{{\mathbb P}^1}}$. Then $\operatorname{rk}\pi_*\ell_0=\operatorname{rk}\ell_0$ for any coherent sheaf $\ell_0$ on $Y$.
We would like to show that $\ell$ is stable if and only if $(L,\nabla)$ is stable. Note that $\nabla$-invariant subsheaves of $L$ are in bijection with subsheaves $\ell_0\subset\ell$ via $\ell_0\mapsto\pi_*\ell_0$. Further, $$\label{degree}
\deg\ell_0=\chi(\ell_0)=\chi(\pi_*
\ell_0)=\deg\pi_*\ell_0+\operatorname{rk}\pi_*\ell_0.$$ It follows that the stability condition is the same.
It also follows from (\[degree\]) that the generalized line bundles on $Y$ corresponding to rank 2 degree $d$ bundles on ${{{\mathbb P}^1}}$ have degree $d+2$.
\[SsTrivial\] If ${\mathfrak D}$ is not even, then $Y$ is integral, and for every $\ell_0\subset\ell$, $\ell_0\ne0$ we have $\operatorname{rk}\ell_0=\operatorname{rk}\ell$ thus the semistability condition is trivial.
Fix a degree $(d+3)/2$ line bundle ${\vartheta}$ on ${{{\mathbb P}^1}}$ (recall that $d$ is odd). A Higgs bundle $(L,\nabla)$ is semistable if and only if $(L\otimes{\vartheta},\nabla)$ is. Therefore, Proposition \[PicY\] implies the following
\[MhY\] Consider the map $Y\to{{\mathcal Y}}$ that sends $y\in Y$ to the vector bundle ${\vartheta}\otimes\pi_* O_Y(-y)$ equipped with the natural Higgs field. The map induces an isomorphism $$Y\times B({{\mathbf{G_m}}}){\mathrel{\widetilde\to}}{{\mathcal Y}}.$$ Thus ${{{{\mathcal M}}_H}}$ is the quotient of the generalized elliptic curve $Y$ by the action of $\mu_2\times{{\mathbf{G_m}}}$, where $\mu_2$ acts by $\sigma$, ${{\mathbf{G_m}}}$ acts trivially.
Let us use the isomorphism of Corollary \[MhY\] to describe the universal Higgs bundle on ${{{\mathbb P}^1}}\times{{{{\mathcal M}}_H}}$. Denote this universal Higgs bundle by $\xi$ and its pullback to ${{{\mathbb P}^1}}\times
Y$ by $\tilde\xi:=(\operatorname{id}_{{{\mathbb P}^1}}\times\bar\pi)^*\xi$ (here $\bar\pi$ is the natural composition $Y\to{{\mathcal Y}}\to{{{{\mathcal M}}_H}}$). Recall also that ${{\mathcal P}}:=O_{Y\times Y}(-\Delta)$ is the ideal sheaf of the diagonal.
\[FMcalculation\] We have $\tilde\xi=(\pi\times\operatorname{id}_Y)_*{{\mathcal P}}\otimes p_1^*{\vartheta}$. The action of ${{\mathbf{G_m}}}$ on $\tilde\xi$ is via the identity character $a\mapsto
a$ and the action of $\mu_2$ comes from its action on ${{\mathcal P}}$ [[(]{}-1pt]{}on $Y\times Y$, $-1\in\mu_2$ acts as $\sigma^*\times\sigma^*$[[)]{}]{}.
For the dual bundle, $$\tilde\xi^\vee=(\pi\times\operatorname{id}_Y)_*{{\mathcal P}}^\vee\otimes
p_1^*({\vartheta}^\vee\otimes{{\mathcal T}}_{{{\mathbb P}^1}})\otimes_{{\mathbb C}}H^1(Y,O_Y)^\vee.$$ On this bundle, ${{\mathbf{G_m}}}$ acts via the character $a\mapsto a^{-1}$ and the action of $\mu_2$ comes from its action on ${{\mathcal P}}$ and its action on $H^1(Y,O_Y)$ [[(]{}-1pt]{}by $-1$[[)]{}]{}.
The description of $\tilde\xi$ follows from Proposition \[Hitchin\] and Corollary \[MhY\], and then the description of $\tilde\xi^\vee$ follows from Serre duality.
It is easy to describe the Fourier-Mukai transform of $\tilde\xi$: this is the structure sheaf of the graph of $\pi$ twisted by ${\vartheta}$.
Consider now the sheaf ${{\mathcal F}}_H:=p_{13}^*\xi\otimes
p_{23}^*\xi^\vee$ on ${{{\mathbb P}^1}}\times{{{\mathbb P}^1}}\times{{{{\mathcal M}}_H}}$. The main result of this section is the following
\[MhXxY\] $$Rp_{12,*}({{\mathcal F}}_H\otimes p_3^*{{\mathcal E}}^{\otimes k})\simeq
\begin{cases}
\iota_{\Delta,*}({{\mathcal T}}_{{{\mathbb P}^1}})[-1]&\text{if $k$ is even},\\
\iota_{\Delta,*}({{\mathcal T}}^{\otimes 2}_{{{\mathbb P}^1}}(-{\mathfrak D}))[-1]&\text{if $k$ is odd},
\end{cases}$$ where $\iota_\Delta:{{{\mathbb P}^1}}\to{{{\mathbb P}^1}}\times{{{\mathbb P}^1}}$ is the diagonal embedding.
The pullback $(\operatorname{id}_{{{{\mathbb P}^1}}\times{{{\mathbb P}^1}}}\times\bar\pi)^*(p_{13}^*\xi\otimes
p_{23}^*\xi^\vee)$ to ${{{\mathbb P}^1}}\times{{{\mathbb P}^1}}\times Y$ equals $p_{13}^*\tilde\xi\otimes p_{23}^*\tilde\xi^\vee$. By Corollary \[FMcalculation\] and Proposition \[FMOrthog\], we have the following identity on ${{{\mathbb P}^1}}\times{{{\mathbb P}^1}}$ $$\begin{gathered}
Rp_{12,*}(p_{13}^*\tilde\xi\otimes p_{23}^*\tilde\xi^\vee)=
\\(\pi\times\pi)_*Rp_{12,*}(p_{13}^*{{\mathcal P}}\otimes p_{23}^*{{\mathcal P}}^\vee)
\otimes p_1^*{\vartheta}\otimes
p_2^*({\vartheta}^\vee\otimes{{\mathcal T}}_{{{\mathbb P}^1}})\otimes_{{\mathbb C}}H^1(Y,O_Y)^\vee=\\
\iota_{\Delta,*}(\pi_*O_Y\otimes{{\mathcal T}}_{{{\mathbb P}^1}})[-1].\end{gathered}$$ The action of ${{\mathbf{G_m}}}$ on the right-hand side is trivial. The action of $\mu_2$ on $O_Y$ is the standard action coming from $\sigma:Y\to Y$ (in other words, $-1\in\mu_2$ acts by $\sigma^*$).
Note that $\bar\pi^*{{\mathcal E}}=O_Y$, but $-1\in\mu_2$ acts on $\bar\pi^*{{\mathcal E}}$ as $-\sigma^*$ (and ${{\mathbf{G_m}}}$ acts trivially). Since ${{{{\mathcal M}}_H}}=Y/({{\mathbf{G_m}}}\times\mu_2)$, $$\begin{gathered}
Rp_{12,*}({{\mathcal F}}_H\otimes p_3^*{{\mathcal E}}^{\otimes k})=
\Bigl(Rp_{12,*}((\operatorname{id}_{{{{\mathbb P}^1}}\times{{{\mathbb P}^1}}}\times\bar\pi)^*
({{\mathcal F}}_H\otimes p_3^*{{\mathcal E}}^{\otimes k}))\Bigr)^{{{\mathbf{G_m}}}\times\mu_2}\\
\simeq\begin{cases}(\iota_{\Delta,*}(\pi_*O_Y\otimes{{\mathcal T}}_{{{\mathbb P}^1}}))^{(1)}[-1]&\text{if $k$ is even,}\\
(\iota_{\Delta,*}(\pi_*O_Y\otimes{{\mathcal T}}_{{{\mathbb P}^1}}))^{(-1)}[-1]&\text{if $k$ is
odd}.\end{cases}\end{gathered}$$ Here for a sheaf ${{\mathcal V}}$ with an action of $\mu_2$, we denote by ${{\mathcal V}}^{(1)}$ (resp. ${{\mathcal V}}^{(-1)}$) its eigensheaf on which $-1\in\mu_2$ acts as 1 (resp. $-1$). Finally, $$\pi_*O_Y={{\mathcal A}}=O_{{{\mathbb P}^1}}\oplus\Omega_{{{\mathbb P}^1}}({\mathfrak D})^{-1},$$ and $-1\in\mu_2$ acts on $O_{{{\mathbb P}^1}}$ as $1$ and on $\Omega_{{{\mathbb P}^1}}({\mathfrak D})^{-1}$ as $-1$.
First orthogonality relation {#ORTHOGONALITY}
============================
In this section, we prove Theorem \[Theorem3\].
Recall that ${{\mathcal F}}_P=p_{13}^*\xi_\alpha\otimes
p_{23}^*\xi^\vee_\alpha$ is a quasi-coherent sheaf on $P\times
P\times{{\mathcal M}}$ equipped with an action of ${\EuScript D}_{P,\alpha}$ along the first copy of $P$ and an action of ${\EuScript D}_{P,-\alpha}$ along the second. Accordingly, the direct image $Rp_{12,*}{{\mathcal F}}_P$ is an object of the derived category of $p_1^{{\bullet}}{\EuScript D}_{P,\alpha}\circledast
p_2^{{\bullet}}{\EuScript D}_{P,-\alpha}$-modules on $P\times P$.
Let $\iota_\Delta:P\to P\times P$ be the diagonal embedding. Recall that $\delta_\Delta$ is a ${\EuScript D}_{P,\alpha}\boxtimes{\EuScript D}_{P,-\alpha}$-module given by $\iota_{\Delta,*}O_P$.
\[Shriek\] In the category of ${\EuScript D}_{P,\alpha}\boxtimes{\EuScript D}_{P,-\alpha}$-modules we have $$\delta_\Delta=\iota_{\Delta,*}O_P=\iota_{\Delta,!}O_P.$$
Let $\Delta$ be the diagonal in $P\times P$ and $\overline\Delta$ be its closure. We can decompose $\iota_\Delta$ as $$\Delta\xrightarrow{\iota_1}\overline\Delta\xrightarrow{\iota_2}P\times P.$$ Since $\iota_2$ is a closed embedding, we have $\iota_{2,*}=\iota_{2,!}$. Thus it is enough to show that $$\iota_{1,*}O_P=\iota_{1,!}O_P.$$ Note that $\iota_1$ is an open embedding, $\overline\Delta-\Delta$ consists of 8 points, and twists at these points are given by $\pm(\alpha_i^+-\alpha_i^-)$. Now the statement follows from conditions (\[AlphaI\]) and of §\[MODST\]. Note that $\iota_{\Delta,*}$ and $\iota_{\Delta,!}$ are exact functors, since $\iota_\Delta$ is an affine embedding.
Further, the restriction $(\iota_\Delta\times\operatorname{id}_{{\mathcal M}})^*{{\mathcal F}}_P$ is a quasi-coherent sheaf on $P\times{{\mathcal M}}$ equipped with a structure of a ${\EuScript D}_P$-module. Recall that $\wp:P\to{{{\mathbb P}^1}}$ is the natural projection. It is easy to see that we have a natural inclusion $\wp^*\xi\subset\xi_\alpha$ (see Remark \[ForDisc\]). Thus, the identity automorphism of $\xi$ gives a horizontal section [$$1\in H^0(P\times{{\mathcal M}},
(\iota_\Delta\times\operatorname{id}_{{\mathcal M}})^*(p_{13}^*\wp^*\xi\otimes p_{23}^*\wp^*\xi^\vee))\subset
H^0(P\times{{\mathcal M}},(\iota_\Delta\times\operatorname{id}_{{\mathcal M}})^*{{\mathcal F}}_P).$$]{} We thus obtain a horizontal section of $p_{1,*}(\iota_\Delta\times\operatorname{id}_{{\mathcal M}})^*{{\mathcal F}}_P$, which can be viewed as a morphism $$O_P\to Rp_{1,*}(\iota_\Delta\times\operatorname{id}_{{\mathcal M}})^*{{\mathcal F}}_P=
\iota_\Delta^* Rp_{12,*}{{\mathcal F}}_P$$ in the derived category of ${\EuScript D}_P$-modules (we use base change). Finally, adjunction provides a morphism $${\varphi}:\delta_\Delta[-1]=\iota_{\Delta,!}O_P[-1]
\to Rp_{12,*}{{\mathcal F}}_P$$ (we are using Lemma \[Shriek\]). Note that the appearance of the shift $[-1]$ is due to the fact that our inverse images are $O$-module inverse images; from the point of view of ${\EuScript D}$-modules they should contain shifts.
Theorem \[Theorem3\] claims that ${\varphi}$ is an isomorphism. We derive Theorem \[Theorem3\] from two statements that are proved later in this section.
\[AwayFromDiag\] The direct image $R(\wp\times\wp)_*Rp_{12,*}{{\mathcal F}}_P$ vanishes outside the diagonal in ${{{\mathbb P}^1}}\times{{{\mathbb P}^1}}$.
\[SSupport\] Consider the morphism $$H^1({\varphi}):\delta_\Delta\to R^1p_{12,*}{{\mathcal F}}_P$$ induced by ${\varphi}$. Then its cokernel is such that $(\wp\times\wp)_*\operatorname{Coker}(H^1({\varphi}))$ is coherent.
Note that in Proposition \[SSupport\], we consider naive (not derived) direct image $(\wp\times\wp)_*$. Actually, higher derived images $R^i(\wp\times\wp)_*{{\mathcal G}}$ ($i>0$) vanish for any $p_1^{{\bullet}}{\EuScript D}_{P,\alpha}\circledast
p_2^{{\bullet}}{\EuScript D}_{P,-\alpha}$-module ${{\mathcal G}}$ (see Remark \[cBRem\]).
By Proposition \[CohomologicalDimension\], we see that $R^ip_{12,*}{{\mathcal F}}_P=0$ for all $i\ne 0,1$. Also, $R^0p_{12,*}{{\mathcal F}}_P$ vanishes at the generic point by Proposition \[AwayFromDiag\], which implies $R^0p_{12,*}{{\mathcal F}}_P=0$. Thus $Rp_{12,*}{{\mathcal F}}_P$ is concentrated in cohomological dimension one. It remains to show that $H^1({\varphi})$ is an isomorphism.
By construction, ${\varphi}\ne 0$. Since $\delta_\Delta$ is irreducible as a $p_1^{{\bullet}}{\EuScript D}_{P,\alpha}\circledast
p_2^{{\bullet}}{\EuScript D}_{P,-\alpha}$-module, $H^1({\varphi})$ is injective. Its cokernel ${{\mathcal F}}':=\operatorname{Coker}(H^1({\varphi}))$ is a $p_1^{{\bullet}}{\EuScript D}_{P,\alpha}\circledast
p_2^{{\bullet}}{\EuScript D}_{P,-\alpha}$-module such that $(\wp\times\wp)_*{{\mathcal F}}'$ is a coherent sheaf (by Proposition \[SSupport\]) that vanishes generically (by Proposition \[AwayFromDiag\]). It is now easy to see that ${{\mathcal F}}'=0$.
Indeed, consider a stratification of $P\times P$ by sets of the form $\{(x_i^\pm,x_j^\pm)\}$, $\{x_i^\pm\}\times({{{\mathbb P}^1}}-{\mathfrak D})$, $({{{\mathbb P}^1}}-{\mathfrak D})\times\{x_i^\pm\}$, and $({{{\mathbb P}^1}}-{\mathfrak D})\times
({{{\mathbb P}^1}}-{\mathfrak D})$. We can now show that ${{\mathcal F}}'$ vanishes on all strata by descending induction on the dimension of strata.
Proof of Proposition \[AwayFromDiag\]
-------------------------------------
\[DirectIm\] $$R(\wp\times\operatorname{id}_{{\mathcal M}})_*\xi_\alpha=\xi.$$
The sheaves are obviously identified on $({{{\mathbb P}^1}}-{\mathfrak D})\times{{\mathcal M}}$, so it remains to verify that this identification extends to ${{{\mathbb P}^1}}\times{{\mathcal M}}$. It suffices to check this on $D\times{{\mathcal M}}$, where $D$ is the formal neighborhood of $x_i$. The restriction $\xi|_{D\times{{\mathcal M}}}$ decomposes into a direct sum $\xi^+\oplus\xi^-$ of one-dimensional bundles that are invariant under the connection (that acts in the direction of $D$). This can be viewed as a version of diagonalization .
The preimage $\wp^{-1} D$ is a union of two copies of $D$ glued away from the center, and the restrictions of $\xi_\alpha$ to $\wp^{-1} D$ is of the form $(j_+\times\operatorname{id}_{{\mathcal M}})_*\xi_+\oplus
(j_-\times\operatorname{id}_{{\mathcal M}})_*\xi_-$, where $j_\pm:D\to \wp^{-1} D$ are the embeddings of the two copies (see Remark \[ForDisc\]). Since $\wp\circ j_\pm=\operatorname{id}_D$, the claim follows.
Consider now the sheaf ${{\mathcal F}}_{{{\mathbb P}^1}}:=p_{13}^*\xi\otimes
p_{23}^*\xi^\vee$ on ${{{\mathbb P}^1}}\times{{{\mathbb P}^1}}\times{{\mathcal M}}$.
\[DirectImage\] $$R(\wp\times\wp\times\operatorname{id}_{{\mathcal M}})_*{{\mathcal F}}_P={{\mathcal F}}_{{{\mathbb P}^1}}.$$
This follows from Lemma \[DirectIm\] and a similar statement about $\xi_\alpha^\vee$ upon writing ${{\mathcal F}}_P=\Delta_{24}^*(\xi_\alpha\boxtimes\xi_\alpha^\vee)$, where $\Delta_{24}:P\times P\times{{\mathcal M}}\to P\times{{\mathcal M}}\times
P\times{{\mathcal M}}$ is a partial diagonal.
By Corollary \[DirectImage\], $$\label{dirimage}
R(\wp\times\wp)_*Rp_{12,*}{{\mathcal F}}_P=Rp_{12,*}{{\mathcal F}}_{{{\mathbb P}^1}}.$$ The advantage of working with $\xi$ rather than $\xi_\alpha$ is that $\xi$ is naturally defined as a vector bundle (the universal bundle) on ${{{\mathbb P}^1}}\times{{\overline{{\mathcal M}}}}$. Accordingly, ${{\mathcal F}}_{{{\mathbb P}^1}}$ extends to a vector bundle $\overline{{\mathcal F}}:=p_{13}^*\xi\otimes
p_{23}^*\xi^\vee$ on ${{{\mathbb P}^1}}\times{{{\mathbb P}^1}}\times{{\overline{{\mathcal M}}}}$. Set $${{\mathcal F}}_k:=\overline{{\mathcal F}}(k({{{\mathbb P}^1}}\times {{{\mathbb P}^1}}\times{{{{\mathcal M}}_H}})),\qquad k\in{{\mathbb Z}}.$$ Let $\jmath:{{{\mathbb P}^1}}\times{{{\mathbb P}^1}}\times{{\mathcal M}}\to{{{\mathbb P}^1}}\times{{{\mathbb P}^1}}\times{{\overline{{\mathcal M}}}}$ be the natural embedding. In view of Proposition \[Cartier\], we have a filtration $$\label{filtration}
{{\mathcal F}}_0=\overline{{\mathcal F}}\subset\dots\subset{{\mathcal F}}_k\subset\dots\subset
{{\mathcal F}}_\infty:=\jmath_*{{\mathcal F}}_{{{\mathbb P}^1}}.$$
We shall use notation $\Delta$ for diagonals in $P\times P$ and ${{{\mathbb P}^1}}\times{{{\mathbb P}^1}}$ through the end of this section.
\[Comparison\] For any $k$ and $i$ there is an isomorphism $$(R^i p_{12,*}{{\mathcal F}}_{{{\mathbb P}^1}})|_{{{{\mathbb P}^1}}\times{{{\mathbb P}^1}}-\Delta}\simeq
(R^i p_{12}{{\mathcal F}}_k)|_{{{{\mathbb P}^1}}\times{{{\mathbb P}^1}}-\Delta}.$$
For every $k$ we have the short exact sequence $$0\to{{\mathcal F}}_{k-1}\to{{\mathcal F}}_k\to\iota_*({{\mathcal F}}_k|_{{{{\mathbb P}^1}}\times{{{\mathbb P}^1}}\times{{{{\mathcal M}}_H}}})\to 0,$$ where $\iota:{{{\mathbb P}^1}}\times{{{\mathbb P}^1}}\times{{{{\mathcal M}}_H}}\to{{{\mathbb P}^1}}\times{{{\mathbb P}^1}}\times{{\overline{{\mathcal M}}}}$ is the closed embedding. Since $${{\mathcal F}}_k|_{{{{\mathbb P}^1}}\times{{{\mathbb P}^1}}\times{{{{\mathcal M}}_H}}}={{\mathcal F}}_H\otimes p_3^*{{\mathcal E}}^{\otimes
k},$$ Proposition \[MhXxY\] implies that $Rp_{12,*}({{\mathcal F}}_k/{{\mathcal F}}_{k-1})=0$ away from the diagonal, so $Rp_{12,*}{{\mathcal F}}_k=Rp_{12,*}{{\mathcal F}}_{k-1}$ away from the diagonal. Now the claim follows from the identity $R^i
p_{12,*}{{\mathcal F}}_{{{\mathbb P}^1}}={{\lim\limits_{\longrightarrow}}}R^i p_{12,*}{{\mathcal F}}_k$.
Consider $(x,y)\in{{{\mathbb P}^1}}\times{{{\mathbb P}^1}}-\Delta$. We have $$\label{vanishing1}
H^i({{\overline{{\mathcal M}}}},\overline{{\mathcal F}}|_{(x,y)})=H^i({{\mathcal M}},{{\mathcal F}}_{{{\mathbb P}^1}}|_{(x,y)})=0$$ for $i\ge 2$ by Proposition \[CohomologicalDimension\]. Next, $H^0({{\overline{{\mathcal M}}}},\overline{{\mathcal F}}|_{(x,y)})$ is finite-dimensional because the good moduli space of ${{\overline{{\mathcal M}}}}$ is projective. It follows that $H^0({{\overline{{\mathcal M}}}},{{\mathcal F}}_k|_{(x,y)})=0$ for $k\ll 0$ because ${{\mathcal M}}$ is connected and ${{\mathcal F}}_{{{\mathbb P}^1}}$ is a vector bundle. Therefore Lemma \[Comparison\] implies that $$\label{vanishing2}
H^0({{\overline{{\mathcal M}}}},\overline{{\mathcal F}}|_{(x,y)})=H^0({{\mathcal M}},{{\mathcal F}}_{{{\mathbb P}^1}}|_{(x,y)})=0.$$ It remains to show that ${{\mathcal G}}:=(R^1p_{12,*}{{\mathcal F}}_{{{\mathbb P}^1}})|_{{{{\mathbb P}^1}}\times{{{\mathbb P}^1}}-\Delta}$ vanishes. By Lemma \[Comparison\], ${{\mathcal G}}=R^1p_{12,*}\overline{{\mathcal F}}|_{{{{\mathbb P}^1}}\times{{{\mathbb P}^1}}-\Delta}$. Moreover, and imply that ${{\mathcal G}}$ is a vector bundle on ${{{\mathbb P}^1}}\times{{{\mathbb P}^1}}-\Delta$. Its rank can be computed using the Euler characteristic; it equals $-\chi({{\overline{{\mathcal M}}}},\xi_x\otimes\xi_y^\vee)$ for any $x,y\in{{{\mathbb P}^1}}$. (Here $\xi_x$ is the restriction of $\xi$ to $\{x\}\times{{\overline{{\mathcal M}}}}$.)
Recall from §\[DEFMB\] that the stack ${{\overline{{\mathcal M}}}}$ depends on the divisor ${\mathfrak D}$ and the formal type, which we encode by $\nu_1\in\Omega_{{{\mathbb P}^1}}({\mathfrak D})/\Omega_{{{\mathbb P}^1}}$ and $\nu_2\in\Omega^{\otimes2}_{{{\mathbb P}^1}}(2{\mathfrak D})/\Omega^{\otimes2}_{{{\mathbb P}^1}}({\mathfrak D})$. By Proposition \[PrFlat\], as the parameters vary, stacks ${{\overline{{\mathcal M}}}}$ form a flat family ${{\overline{{\mathcal M}}}}_{univ}$ over a Zariski open subspace in the space of collections $({\mathfrak D},\nu_1,\nu_2)$. The vector bundle $\xi_x\otimes\xi_y^\vee$ makes sense in this family. We shall use
\[FlatStacks\] Let ${{\mathcal X}}\to S$ be a flat family of stacks over a scheme $S$. Let ${{\mathcal F}}$ be a flat sheaf on ${{\mathcal X}}$. For $s\in S$ denote by ${{\mathcal F}}_s$ the fiber over $s$. Assume that there is a good moduli space $p:{{\mathcal X}}\to X$ such that the induced map $X\to S$ is projective.
Then $\chi({{\mathcal F}}_s)$ is locally constant as a function of $s$.
Let us apply this Lemma to $\xi_x\otimes\xi_y^\vee$. A slight generalization of Theorems \[GoodModuliSpace\] and \[AmpleBundle\] shows that ${{\overline{{\mathcal M}}}}_{univ}$ has a good moduli space ${{\overline M}}_{univ}$, which is projective over the space of collections $({\mathfrak D},\nu_1,\nu_2)$. Therefore, $\chi({{\overline{{\mathcal M}}}},\xi_x\otimes\xi_y^\vee)$ does not depend on ${\mathfrak D}$, $\nu_1$, and $\nu_2$. In particular, we may assume that ${\mathfrak D}=x_1+x_2+x_3+x_4$ for distinct $x_i\in{{{\mathbb P}^1}}$ and that $\nu_1$, $\nu_2$ are generic. Using Lemma \[Comparison\], we see that it is enough to prove that $H^i({{\mathcal M}},\xi_x\otimes\xi_y^\vee)=0$ for $x\ne y$ and all $i$ in the case of simple ${\mathfrak D}$ and generic $\nu_1$ and $\nu_2$. This case is treated in [@Arinkin Theorem 2], except for a slight difference that $SL(2)$-bundles are considered there. However, both moduli stacks have the same good moduli space, so the cohomology groups are the same (in fact, our moduli space is $M\times B({{\mathbf{G_m}}})$, while the moduli space in [@Arinkin] is a $\mu_2$-gerbe over $M$.
Note that $p_*{{\mathcal F}}$ is flat on $X$. Indeed, if ${{\mathcal G}}$ is a sheaf on $X$, and ${{\mathcal G}}^{{\bullet}}$ is its resolution by locally free sheaves, we have $$\operatorname{\mathcal{T}\!\mathit{or}}^i({{\mathcal G}},p_*{{\mathcal F}})=H^{-i}({{\mathcal G}}^{{\bullet}}\otimes p_*{{\mathcal F}})=
p_*H^{-i}(p^*{{\mathcal G}}^{{\bullet}}\otimes{{\mathcal F}})=0.$$ We have used the projection formula and the fact that good moduli spaces are cohomologically affine.
By Proposition 4.7(i) of [@Alper], the restriction of $p$ to $s\in S$ is a good moduli space $p_s:{{\mathcal X}}_s\to X_s$ so we have $$\chi({{\mathcal F}}_s)=\chi(p_{s,*}{{\mathcal F}}_s)=\chi((p_*{{\mathcal F}})_s).$$ (We are using a base change).
By Theorem 4.16(ix) of [@Alper], the map $X\to S$ is flat, and we see that $\chi((p_*{{\mathcal F}})_s)$ is locally constant.
Proof of Proposition \[SSupport\]
---------------------------------
It is convenient to replace ${\EuScript D}_{P,\alpha}$-modules with modules over a certain sheaf of algebras on ${{{\mathbb P}^1}}$. Let us make the corresponding definitions. We identify ${\EuScript D}_{P,\alpha}$ with a subsheaf in the pushforward of ${\EuScript D}_{{{{\mathbb P}^1}}-{\mathfrak D}}$ to $P$.
Let $z_i$ be a local coordinate at $x_i$. Let us lift polar parts $\alpha_i^\pm$ to actual 1-forms on formal neighborhoods of $x_i$; we shall denote these 1-forms by the same letters. Consider the open embedding $\jmath:{{{\mathbb P}^1}}-{\mathfrak D}\hookrightarrow{{{\mathbb P}^1}}$.
Define the sheaf of algebras ${{\mathcal B}}={{\mathcal B}}_{\alpha}\subset
\jmath_*{\EuScript D}_{{{{\mathbb P}^1}}-{\mathfrak D}}$ as follows:
- We have ${{\mathcal B}}|_{{{{\mathbb P}^1}}-{\mathfrak D}}={\EuScript D}_{{{{\mathbb P}^1}}-{\mathfrak D}}$;
- Near $x_i\in{{{\mathbb P}^1}}$, ${{\mathcal B}}$ is generated by $O_{{{\mathbb P}^1}}$, $z_i^{n_i}\frac{{\mathbf d}}{{\mathbf d}z_i}$, and $z_i^{-n_i}(z_i^{n_i}\frac{{\mathbf d}-\alpha_i^+}{{\mathbf d}z_i})(z_i^{n_i}\frac{{\mathbf d}-\alpha_i^-}{{\mathbf d}z_i})$.
Clearly, ${{\mathcal B}}$ inherits from $\jmath_*{\EuScript D}_{{{{\mathbb P}^1}}-{\mathfrak D}}$ the filtration by degree of differential operators. We denote by ${{\mathcal B}}^{\le k}\subset{{\mathcal B}}$ the subsheaf of operators of degree at most $k$.
The properties of ${{\mathcal B}}$ are summarized in the following
\[cB\] [[)]{}]{}\[cB1\] ${{\mathcal B}}=\wp_*{\EuScript D}_{P,\alpha}$.\
[[)]{}]{}\[cB2\] Moreover, $R^1\wp_*{\EuScript D}_{P,\alpha}=0$, so that ${{\mathcal B}}=R\wp_*{\EuScript D}_{P,\alpha}$.\
[[)]{}]{}\[cB3\] ${{\mathcal B}}^{\le k}/{{\mathcal B}}^{\le
k-1}={{\mathcal T}}_{{{\mathbb P}^1}}^{\otimes k}(-\left\lceil\frac
k2\right\rceil{\mathfrak D})$. [[(]{}-1pt]{}Here $\lceil\,\rceil$ is the ceiling function.[[)]{}]{}
\[cBRem\][[)]{}]{}The isomorphisms and are naturally normalized by the condition that they become the obvious identifications on ${{{\mathbb P}^1}}-{\mathfrak D}$.\
[[)]{}]{}\[cBRem2\] Let us fix $\mu\in
H^0({{{\mathbb P}^1}},\Omega_{{{\mathbb P}^1}}^{\otimes2}({\mathfrak D}))$, $\mu\ne 0$, as in §\[MH\]. Then can be rewritten as $${{\mathcal B}}^{\le k}/{{\mathcal B}}^{\le k-1}=\begin{cases}
{{\mathcal T}}_{{{\mathbb P}^1}}(-{\mathfrak D}) &\text{if $k$ is odd,}\\
O_{{{\mathbb P}^1}} &\text{if $k$ is even}.
\end{cases}$$ [[)]{}]{}\[cBRem3\] Actually, $\wp:P\to{{{\mathbb P}^1}}$ is affine with respect to ${\EuScript D}_{P,\alpha}$ in the sense that the functor $\wp_*$ is exact on ${\EuScript D}_{P,\alpha}$-modules and provides an equivalence between the category of ${\EuScript D}_{P,\alpha}$-modules and that of ${{\mathcal B}}$-modules. We do not use this claim, so its proof is left to the reader.
As we have already mentioned, the claims are obvious on ${{{\mathbb P}^1}}-{\mathfrak D}$. Therefore, it suffices to consider the formal neighborhood of a point $x_i$. Since we concentrate on a single point, we drop the index $i$ to simplify the notation, so $\alpha^\pm=\alpha^\pm_i$, $z=z_i$, and $n=n_i$.
Let ${\EuScript D}_K:={{\mathbb C}}((z))\langle\frac{{\mathbf d}}{{\mathbf d}z}\rangle$ be the ring of differential operators on the punctured formal neighborhood of $x_i$. Set $$\delta:=z^n\frac{\mathbf d}{{\mathbf d}z},\quad
B:=z^{-n}\left(z^n\frac{{\mathbf d}-\alpha^+}{{\mathbf d}z}\right)
\left(z^n\frac{{\mathbf d}-\alpha^-}{{\mathbf d}z}\right)\in{\EuScript D}_K$$ and $${{\mathcal B}}_O:={{\mathbb C}}[[z]]\left\langle\delta,B\right\rangle\subset{\EuScript D}_K,\qquad
{\EuScript D}_O^\pm:={{\mathbb C}}[[z]]\left\langle\frac{{\mathbf d}-\alpha^\pm}{{\mathbf d}z}\right\rangle\subset{\EuScript D}_K.$$ Then the restriction of ${{\mathcal B}}$ to the formal neighborhood of $x_i$ is ${{\mathcal B}}_O$, the restriction of ${\EuScript D}_{P,\alpha}$ to the formal neighborhoods of $x_i^\pm$ is ${\EuScript D}_O^\pm$, and the restriction of $R^0\wp_*{\EuScript D}_{P,\alpha}$ (resp. $R^1\wp_*{\EuScript D}_{P,\alpha}$) to the formal neighborhood of $x_i$ equals ${\EuScript D}_O^+\cap{\EuScript D}_O^-$ (resp. ${\EuScript D}_K/({\EuScript D}_O^++{\EuScript D}_O^-)$). The proposition thus reduces to the following statements:
1. \[Odin\] ${{\mathcal B}}_O={\EuScript D}_O^+\cap{\EuScript D}_O^-$,
2. \[Dva\] ${\EuScript D}_K={\EuScript D}_O^++{\EuScript D}_O^-$,
3. \[Tri\] The set $\{1,\delta,B,B\delta ,B^2,
B^2\delta,\dots\}$ is a basis of ${{\mathcal B}}_O$ as of a ${{\mathbb C}}[[z]]$-module. (Note that the symbol of $\delta$ (resp. the symbol of $B$) is a section of ${{\mathcal T}}_{{{\mathbb P}^1}}(-{\mathfrak D})$ (resp. of ${{\mathcal T}}_{{{\mathbb P}^1}}^{\otimes2}(-{\mathfrak D})$).
Set $F:={{\mathbb C}}[[z]]\langle\delta\rangle\subset{\EuScript D}_K$, and introduce the filtration $$\dots\subset z F\subset F\subset z^{-1}F\subset\dots\subset{\EuScript D}_K.$$ For an element $C\in{\EuScript D}_K$ denote by $\bar C$ its image in $\operatorname{\mathrm{gr}}{\EuScript D}_K$.
\[lm:Filtration\][[)]{}]{}\[Gr1\] This filtration is exhaustive, separated, and compatible with the ring structure.
[[)]{}]{}\[Gr2\] If $n>1$, then the associated graded ring is isomorphic to ${{\mathbb C}}[\bar z,\bar z^{-1},\bar\delta]$, that is, to the ring of functions on ${{{\mathbb A}^1}}\times({{{\mathbb A}^1}}-0)$.
[[)]{}]{}\[Gr3\] For $n=1$ the associated graded ring is isomorphic to ${{\mathbb C}}[\bar z,\bar
z^{-1}]\langle\bar\delta\rangle/(\bar\delta\bar z-\bar
z\bar\delta-\bar z)$, that is, to the ring of differential operators on ${{{\mathbb A}^1}}-0$.
(\[Gr1\]) Note first that every element of $C\in{\EuScript D}_K$ can be written uniquely as $$\label{present}
C=\sum_{l\ge0} f_l(z)\delta^l,$$ where $f_l(z)\in{{\mathbb C}}((z))$. It follows from commutation relation $$\label{commute}
[\delta,z^k]=kz^{k+n-1}$$ that $C\in F$ if and only if for all $l$ we have $f_l(z)\in{{\mathbb C}}[[z]]$. Thus $C\in z^{-k}F$ if and only if for all $l$ we have $z^kf_l(z)\in{{\mathbb C}}[[z]]$. Hence the filtration is exhaustive and separated.
It follows from commutation relation (\[commute\]) by induction on $l$ that $\delta^lz^k\in z^kF$. Now it is easy to see that the filtration is compatible with the ring structure.
(\[Gr2\]) It follows from (\[commute\]) that $\bar z$ and $\bar\delta$ commute in $\operatorname{\mathrm{gr}}{\EuScript D}_k$ if $n>1$. Thus we get a homomorphism $${{\mathbb C}}[\bar z,\bar z^{-1},\bar\delta]\to\operatorname{\mathrm{gr}}{\EuScript D}_K.$$ Using presentation (\[present\]), we see that it is bijective.
The proof of (\[Gr3\]) is similar to that of (\[Gr2\]).
Denote by $a^\pm$ the leading coefficient of $\alpha^\pm=a^\pm
z^{-n}{\mathbf d}z+\dots$, and define the polynomials $q^\pm_l(t)$ for a non-negative integer $l$ by $$q_l^\pm(t):=\begin{cases}
(t-a^\pm)^l &\text{ if }n>1,\\
\prod_{i=0}^{l-1}(t-a^\pm-i) &\text{ if }n=1.
\end{cases}$$
\[Image\] [[)]{}]{}\[Image1\] The image of $\left(\frac{{\mathbf d}-\alpha^\pm}{{\mathbf d}z}\right)^l$ in $\operatorname{\mathrm{gr}}{\EuScript D}_K$ is $\bar z^{-nl}q_l^\pm(\bar\delta)$.\
[[)]{}]{}\[Image2\] The image of $B^l$ in $\operatorname{\mathrm{gr}}{\EuScript D}_K$ is $\bar z^{-nl}q_l^+(\bar\delta)q_l^-(\bar\delta)$.
(\[Image1\]) The image of $\frac{{\mathbf d}-\alpha^\pm}{{\mathbf d}z}$ in $\operatorname{\mathrm{gr}}{\EuScript D}_K$ is $\bar z^{-n}(\bar\delta-a^\pm)$. If $n>1$, then the statement follows from commutativity of $\operatorname{\mathrm{gr}}{\EuScript D}_K$.
If $n=1$, then we have to move all copies of $\bar z^{-1}$ to the left in $(\bar z^{-1}(\bar\delta-a^\pm))^l$. Now the statement follows from the relation $$\label{commute2}
(\bar\delta-a)\bar z^{-1}=\bar z^{-1}(\bar\delta-a-1),\quad a\in{{\mathbb C}}.$$
(\[Image2\]) We have $\bar B=\bar
z^{-n}(\bar\delta-a^+)(\bar\delta-a^-)$. Now the case $n>1$ is obvious, the case $n=1$ again follows from (\[commute2\]).
By Lemma \[lm:Filtration\] any element of $\operatorname{\mathrm{gr}}_{-k}{\EuScript D}_K$ can be uniquely written as $$\bar z^{-k}p(\bar\delta),\quad p(\bar\delta)\in{{\mathbb C}}[\bar\delta].$$ Denote by $\operatorname{\mathrm{gr}}{\EuScript D}_O^\pm$ the set of images of all elements of ${\EuScript D}_O^\pm$ in $\operatorname{\mathrm{gr}}{\EuScript D}_K$. Define $\operatorname{\mathrm{gr}}{{\mathcal B}}_O$ similarly. Fix $k\in{{\mathbb Z}}$ and set $l:=\lceil\frac kn\rceil$.
$$\begin{aligned}
&(a)\qquad\bar z^{-k}p(\bar\delta)\in\operatorname{\mathrm{gr}}{\EuScript D}_O^\pm &&\text{ if and only if }
k\le 0\text{ or }q_l^\pm(t)|p(t);\\
&(b)\qquad\bar z^{-k}p(\bar\delta)\in\operatorname{\mathrm{gr}}{{\mathcal B}}_O &&\text{ if and only if }
k\le 0\text{ or }q_l^+(t)q_l^-(t)|p(t).\end{aligned}$$
\(a) Consider any element $C\in{\EuScript D}_O^+$, $C\ne0$. It is easy to see that it can be uniquely written as $$\sum_{i,j\ge0}f_{ij}z^j\left(\frac{{\mathbf d}-\alpha^+}{{\mathbf d}z}\right)^i$$ with $f_{ij}\in{{\mathbb C}}$. Let $k$ be the maximum value of the function $(i,j)\mapsto ni-j$ on the set $\{(i,j)|\,f_{ij}\ne0\}$. Then Lemma \[Image\](\[Image1\]) shows that $$C\in\sum_{ni-j=k}f_{ij}z^{-k}q_i^+(\delta)+z^{1-k}F.$$ Since elements $\bar z^{-k}q_i^+(\bar\delta)$ form a basis in $\operatorname{\mathrm{gr}}_{-k}{\EuScript D}_K$, we see that $C\notin z^{1-k}F$ and $$\bar C=\sum_{ni-j=k}f_{ij}\bar z^{-k}q_i^+(\bar\delta).$$ Since $j\ge0$, we see that $i\ge\frac kn$, so $i\ge l$. Thus if $k>0$, then $\bar C=\bar z^{-k}p(\bar\delta)$, where $p$ is divisible by $q_l^+$.
Conversely, given a polynomial $p$ divisible by $q_l^+$ (or any polynomial if $k\le0$), we can write $p=\sum_{i\ge l}f_iq_i^+$ with $f_i\in{{\mathbb C}}$. Set $$C=\sum_if_i z^{ni-k}\left(\frac{{\mathbf d}-\alpha^+}{{\mathbf d}z}\right)^i.$$ Then $C\in{\EuScript D}_O^+$ and $\bar C=\bar z^{-k}p(\bar\delta)$. The case of ${\EuScript D}_O^-$ is completely similar.
\(b) Consider $C\in{{\mathcal B}}_O$ with $\bar C=\bar
z^{-k}p(\bar\delta)$. It is easy to see that ${{\mathcal B}}_O\subset{\EuScript D}_O^+\cap{\EuScript D}_O^-$. Thus it follows from part (a) that $q_l^\pm(t)$ divides $p(t)$. Thus $q_l^+(t)q_l^-(t)$ divides $p(t)$, since $q_l^-(t)$ and $q_l^+(t)$ are coprime. Finally, assume that $q_l^+(t)q_l^-(t)$ divides $p(t)$, we can write $$p(t)=\sum_{i\ge l}f_iq_i^+(t)q_i^-(t)+
\sum_{i\ge l}g_itq_i^+(t)q_i^-(t),\qquad f_i,g_i\in{{\mathbb C}}.$$ Set $$C=\sum_i f_iz^{ni-k}B^i+\sum_i g_i z^{ni-k}B^i\delta.$$ Clearly, $\bar C=\bar z^{-k}p(\bar\delta)$.
We now see that the identities – hold in the associated graded ring of ${\EuScript D}_K$, and hence also in ${\EuScript D}_K$ itself. The proof of part (b) of the lemma shows that every element of $\operatorname{\mathrm{gr}}{{\mathcal B}}_O\cap\operatorname{\mathrm{gr}}_{-k}{\EuScript D}_K$ can be uniquely written as $$\sum_{i\ge k/n} f_i\bar z^{ni-k}\bar B^i+\sum_{i\ge k/n}
g_i \bar z^{ni-k}\bar B^i\bar\delta$$ with $f_i,g_i\in{{\mathbb C}}$, and follows. The proof of Proposition \[cB\] is complete.
Our first goal is to reformulate the proposition as a statement about the cokernel of a map between sheaves on ${{{\mathbb P}^1}}\times{{{\mathbb P}^1}}$.
Note first of all that any $p_1^{{\bullet}}{\EuScript D}_{P,\alpha}\circledast
p_2^{{\bullet}}{\EuScript D}_{P,-\alpha}$-module ${{\mathcal G}}$ on $P\times P$ is, in particular, a $p_1^{-1}{\EuScript D}_{P,\alpha}$-module. Therefore, $(\wp\times\wp)_*{{\mathcal G}}$ has a natural structure of a $p_1^{-1}{{\mathcal B}}$-module coming from the isomorphism of Proposition \[cB\]. (There is also a commuting structure of a $p_2^{-1}{{\mathcal B}}_{-\alpha}$-module that we do not use.)
Thus $$\label{pph1phi}
(\wp\times\wp)_*H^1({\varphi}):(\wp\times\wp)_*\delta_\Delta\to
(\wp\times\wp)_*R^1p_{12,*}{{\mathcal F}}_P$$ is a map of $p_1^{-1}{{\mathcal B}}$-modules.
Consider $\delta_\Delta$ as a $p_1^{-1}{\EuScript D}_{P,\alpha}$-module. It is isomorphic to $\iota_{\Delta,*}({\EuScript D}_{P,\alpha}\otimes_{O_P}{{\mathcal T}}_P$), where $\iota_{\Delta,*}$ is the $O$-module pushforward. By the projection formula, Proposition \[cB\] gives an isomorphism in the derived category of $p_1^{-1}{{\mathcal B}}$-modules $$\label{rpp}
R(\wp\times\wp)_*\delta_\Delta=\iota_{\Delta,*}({{\mathcal B}}\otimes_{O_{{{\mathbb P}^1}}}{{\mathcal T}}_{{{\mathbb P}^1}}).$$ Using this and (\[dirimage\]), we re-write (\[pph1phi\]) as $$\label{pph2phi}
(\wp\times\wp)_*H^1({\varphi}):\iota_{\Delta,*}({{\mathcal B}}\otimes_{O_{{{\mathbb P}^1}}}{{\mathcal T}}_{{{\mathbb P}^1}})\to
R^1p_{12,*}{{\mathcal F}}_{{{\mathbb P}^1}}.$$ As was explained in the proof of Theorem \[Theorem3\], $H^1({\varphi})$ is injective. Also, $R^1(\wp\times\wp)_*\delta_\Delta=0$ by (\[rpp\]). We now see that $$(\wp\times\wp)_*\operatorname{Coker}(H^1({\varphi}))=\operatorname{Coker}((\wp\times\wp)_*H^1({\varphi})).$$ Thus it remains to prove that the cokernel of (\[pph2phi\]) is coherent. Note that (\[pph2phi\]) is an injective maps between $p_1^{-1}{{\mathcal B}}$-modules.
Now recall that ${{\mathcal F}}_{{{\mathbb P}^1}}$ naturally extends to a vector bundle $\overline{{\mathcal F}}:=p_{13}^*\xi\otimes p_{23}^*\xi^\vee$ on ${{{\mathbb P}^1}}\times{{{\mathbb P}^1}}\times{{\overline{{\mathcal M}}}}$, which provides the filtration . By Proposition \[MhXxY\] we obtain an induced filtration $$\label{R1filtration}
R^1p_{12,*}{{\mathcal F}}_0\subset\dots\subset
R^1p_{12,*}{{\mathcal F}}_k\subset\dots\subset
R^1p_{12,*}{{\mathcal F}}_\infty=R^1p_{12,*}{{\mathcal F}}_{{{\mathbb P}^1}}$$ of $R^1p_{12,*}{{\mathcal F}}_{{{\mathbb P}^1}}$ by coherent sheaves on ${{{\mathbb P}^1}}\times{{{\mathbb P}^1}}$.
There is $l\in{{\mathbb Z}}$ such that for $k\gg 0$ the image of $\iota_{\Delta,*}({{\mathcal B}}^{\le k}\otimes_{O_{{{\mathbb P}^1}}}{{\mathcal T}}_{{{\mathbb P}^1}})$ under (\[pph2phi\]) is contained in $R^1p_{12,*}{{\mathcal F}}_{k+l}$ and such that the induced map $$\label{isom}
\iota_{\Delta,*}(({{\mathcal B}}^{\le k+1}/{{\mathcal B}}^{\le k})\otimes_{O_{{{\mathbb P}^1}}}{{\mathcal T}}_{{{\mathbb P}^1}})\to
R^1p_{12,*}{{\mathcal F}}_{k+l+1}/R^1p_{12,*}{{\mathcal F}}_{k+l}.$$ is an isomorphism.
By construction, the filtration agrees with the filtration on ${{\mathcal B}}$, so that $(p_1^{-1}{{\mathcal B}}^{\le
l}){{\mathcal F}}_k\subset{{\mathcal F}}_{k+l}$. Therefore, the filtration also agrees with the filtration on ${{\mathcal B}}$. Using Remark \[cBRem\](\[cBRem2\]) and Proposition \[MhXxY\], we see that for $k\gg 0$ [$$\label{assgraded}
\iota_{\Delta,*}(({{\mathcal B}}^{\le k}/{{\mathcal B}}^{\le k-1})\otimes_{O_{{{\mathbb P}^1}}}{{\mathcal T}}_{{{\mathbb P}^1}})\simeq
R^1p_{12,*}{{\mathcal F}}_k/R^1p_{12,*}{{\mathcal F}}_{k-1}\simeq
\begin{cases}
\iota_{\Delta,*}{{\mathcal T}}^{\otimes 2}_{{{\mathbb P}^1}}(-{\mathfrak D}) &\text{if $k$ is odd,}\\
\iota_{\Delta,*}{{\mathcal T}}_{{{\mathbb P}^1}} &\text{if $k$ is even}.
\end{cases}$$]{}
For each $k$, let $l(k)$ be the smallest index such that the image of $\iota_{\Delta,*}({{\mathcal B}}^{\le k}\otimes_{O_{{{\mathbb P}^1}}}{{\mathcal T}}_{{{\mathbb P}^1}})$ is contained in $R^1p_{12,*}{{\mathcal F}}_{k+l(k)}.$ Since the filtration agrees with the filtration on ${{\mathcal B}}$, we see that $l(k+1)\le l(k)$ for all $k$. Also, injectivity of implies that $$0\le \operatorname{rk}\iota_\Delta^*(R^1p_{12,*}{{\mathcal F}}_{k+l(k)})-\operatorname{rk}({{\mathcal B}}^{\le k}\otimes_{O_{{{\mathbb P}^1}}}{{\mathcal T}}_{{{\mathbb P}^1}})=
(k+l(k)+\operatorname{rk}\iota_\Delta^*(R^1p_{12,*}{{\mathcal F}}_0))-k,$$ and therefore $l(k)\ge -\operatorname{rk}\iota_\Delta^*(R^1p_{12,*}{{\mathcal F}}_0)$. Thus $l(k)$ stabilizes as $k\to\infty$; set $l:=\lim l(k)$.
By the choice of $l$, the map is non-zero for $k\gg 0$. Note that such non-zero map does not exist if $k$ and $l$ are odd. Therefore, $l$ must be even. We now see that for $k\gg0$, the map is a non-zero morphism between isomorphic line bundles on $\Delta$. This implies is an isomorphism.
It follows from the lemma that $R^1p_{12,*}{{\mathcal F}}_k$ maps surjectively onto $\operatorname{Coker}(\wp\times\wp)_*H^1({\varphi})$ for $k\gg0$. This completes the proof of the proposition and of Theorem \[Theorem3\].
Second orthogonality relation. {#LYSENKO}
==============================
In this section we prove Theorem \[ThLys\]. The proof is similar to [@Arinkin] but we want to give some details.
We need to calculate $Rp_{12,*}\operatorname{\mathbb{DR}}({{\mathcal F}}_{{\mathcal M}})$, where $p_{12}:{{\mathcal M}}\times{{\mathcal M}}\times P\to{{\mathcal M}}\times{{\mathcal M}}$. Our first goal is to reduce the problem to a calculation on ${{\mathcal M}}\times{{\mathcal M}}\times{{{\mathbb P}^1}}$. Recall that $\xi$ is the universal bundle on ${{\mathcal M}}\times{{{\mathbb P}^1}}$, set $\xi_{12}:=\operatorname{\mathcal{H}\mathit{om}}(p_{23}^*\xi,
p_{13}^*\xi)$. We have a connection along ${{{\mathbb P}^1}}$ $$\operatorname{ad}\nabla:\xi_{12}\to\xi_{12}\otimes p_3^*\Omega_{{{\mathbb P}^1}}({\mathfrak D}).$$ Its polar part is a well-defined $O$-linear map $$\xi_{12}|_{{{\mathcal M}}\times{{\mathcal M}}\times{\mathfrak D}}\to
(\xi_{12}\otimes p_3^*\Omega_{{{\mathbb P}^1}}({\mathfrak D}))|_{{{\mathcal M}}\times{{\mathcal M}}\times{\mathfrak D}}.$$ Let $\eta_{12}$ be the image of this map. Denote by $\tilde
\xi_{12}$ the modification of $\xi_{12}\otimes
p_3^*\Omega_{{{\mathbb P}^1}}$ whose sheaf of sections is $$\{s\in\xi_{12}\otimes p_3^*\Omega_{{{\mathbb P}^1}}({\mathfrak D})|_{{{\mathcal M}}\times{{\mathcal M}}\times{\mathfrak D}}
:s|_{{{\mathcal M}}\times{{\mathcal M}}\times{\mathfrak D}}\in\eta_{12}\}.$$ As in [@Arinkin Lemmas 12, 13] one proves that $$Rp_{12,*}\operatorname{\mathbb{DR}}({{\mathcal F}}_{{\mathcal M}})=
Rp_{12,*}(\xi_{12}\xrightarrow{\operatorname{ad}\nabla}\tilde\xi_{12}).$$ The next step is to calculate the restriction of the above to a fiber over a point. So consider a closed point $x\in{{\mathcal M}}\times{{\mathcal M}}$ and let ${{\mathcal F}}^{{\bullet}}$ be the restriction of $\xi_{12}\xrightarrow{\operatorname{ad}\nabla}\tilde\xi_{12}$ to $x$.
\[PpLys\] [[)]{}]{}\[PpLys1\] If $x\notin\operatorname{diag}({{\mathcal M}})$, then $\operatorname{{\mathbb H}}^i({{\mathcal F}}^{{\bullet}})=0$ for any $i$;\
[[)]{}]{}\[PpLys2\] If $x\in\operatorname{diag}({{\mathcal M}})$, then $$\dim\operatorname{{\mathbb H}}^i({{\mathcal F}}^{{\bullet}})=\begin{cases}
1\text{ if }i=0,2,\\ 2\mbox{ if }i=1,\\
0\text{ otherwise.}
\end{cases}$$ [[)]{}]{}\[PpLys3\] Suppose $x=((L,\nabla),(L,\nabla))\in{{\mathcal M}}\times{{\mathcal M}}$. Consider the map of complexes $$(O_{{{\mathbb P}^1}}\xrightarrow{{\mathbf d}}\Omega_{{{\mathbb P}^1}})\hookrightarrow{{\mathcal F}}^{{\bullet}}$$ induced by $O_{{{\mathbb P}^1}}\to\xi_{12}|_x:f\mapsto f\operatorname{id}_L$. Then the induced map $$H^i_{DR}({{{\mathbb P}^1}},{{\mathbb C}}):=
\operatorname{{\mathbb H}}^i({{{\mathbb P}^1}},O_{{{\mathbb P}^1}}\xrightarrow{{\mathbf d}}\Omega_{{{\mathbb P}^1}})\to\operatorname{{\mathbb H}}^i({{\mathcal F}}^{{\bullet}})$$ is an isomorphism for $i=0,2$.
The proof is analogous to the proof of [@Arinkin Proposition 10]: one uses irreducibility, duality, and Euler characteristic.
As in [@Arinkin Lemma 14] the duality gives the following
\[RelLys\] Let $S$ be a locally Noetherian scheme, $\iota:S\to{{\mathcal M}}\times{{\mathcal M}}$. Set $${{\mathcal F}}_{(S)}:=Rp_{1,*}((\iota\times\operatorname{id}_{{{\mathbb P}^1}})^*
(\xi_{12}\xrightarrow{\operatorname{ad}\nabla}\tilde\xi_{12}))$$ [[(]{}-1pt]{}here $p_1:S\times{{{\mathbb P}^1}}\to S$[[)]{}]{}. Then $\operatorname{\mathcal{H}\mathit{om}}(H^2({{\mathcal F}}_{(S)}),O_S)$ is isomorphic to a subsheaf of $H^0({{\mathcal F}}_{(S)})$.
Next, $\operatorname{diag}:{{\mathcal M}}\to{{\mathcal M}}\times{{\mathcal M}}$ is a ${{\mathbf{G_m}}}$-torsor over $\operatorname{diag}({{\mathcal M}})$ (cf. Remark \[RemDiag\]). Denote by $\operatorname{Hom}$ the corresponding line bundle. Note that the fiber of $\operatorname{Hom}$ over $((L_1,\nabla_1),(L_2,\nabla_2))$ is $\{A\in\operatorname{Hom}_{O_{{{\mathbb P}^1}}}(L_1,L_2): A\nabla_1=\nabla_2A\}$. Now the following corollary of Proposition \[PpLys\] is obvious.
\[CoLys\] [[)]{}]{}\[CoLys1\] $Rp_{12,*}(\xi_{12}\xrightarrow{\operatorname{ad}\nabla}\tilde\xi_{12})$ vanishes if restricted to ${{\mathcal M}}\times{{\mathcal M}}-\operatorname{diag}({{\mathcal M}})$.
[[)]{}]{}\[CoLys2\] The map $$p_{12}^*\operatorname{Hom}\otimes p_3^*(O_{{{\mathbb P}^1}}\xrightarrow{{\mathbf d}}\Omega_{{{\mathbb P}^1}})
\to(\xi_{12}\xrightarrow{\operatorname{ad}\nabla}\tilde\xi_{12})|_{\operatorname{diag}({{\mathcal M}})}$$ induces an isomorphism $$\operatorname{Hom}=\operatorname{Hom}\otimes\operatorname{{\mathbb H}}^2({{{\mathbb P}^1}},(O_{{{\mathbb P}^1}}\xrightarrow{{\mathbf d}}\Omega_{{{\mathbb P}^1}}))\to
R^2p_{12,*}\left((\xi_{12}\xrightarrow{\operatorname{ad}\nabla}\tilde\xi_{12})|_{\operatorname{diag}({{\mathcal M}})}\right).$$
Let us use the following observation (cf. [@Arinkin Lemma 15] and [@MumfordAbelian Lemma in §13]).
\[LmLys\] Let $Z$ be a locally Noetherian scheme, $Y\subset Z$ a closed subscheme that is locally a complete intersection of pure codimension $n$. Denote by $\iota:Y\hookrightarrow Z$ the natural embedding.
[[)]{}]{}\[LmLys1\] Let ${{\mathcal F}}$ be a quasi-coherent sheaf on $Z$ such that ${{\mathcal F}}|_{Z-Y}=0$, $L_n\iota^*{{\mathcal F}}=0$. Then ${{\mathcal F}}=0$.
[[)]{}]{}\[LmLys2\] Let ${{\mathcal F}}^{{\bullet}}=({{\mathcal F}}^0\to{{\mathcal F}}^1\to\dots)$ be a complex of flat quasi-coherent sheaves on $Z$ such that $H^i({{\mathcal F}}^{{\bullet}})|_{Z-Y}=0$ for all $i<n$. Then $H^i({{\mathcal F}}^{{\bullet}})=0$ for $i<n$.
Clearly, $\operatorname{diag}({{\mathcal M}})=M\times B({{\mathbf{G_m}}}\times{{\mathbf{G_m}}})$ is a complete intersection in ${{\mathcal M}}\times{{\mathcal M}}=M\times M\times B({{\mathbf{G_m}}}\times{{\mathbf{G_m}}})$. Thus Lemma \[LmLys\](\[LmLys2\]) and Corollary \[CoLys\](\[CoLys1\]) imply that $R^ip_{12,*}(\xi_{12}\xrightarrow{\operatorname{ad}\nabla}\tilde\xi_{12})=0$ for $i\ne2$. Set ${{\mathcal F}}^{(2)}:=R^2p_{12,*}(\xi_{12}\xrightarrow{\operatorname{ad}\nabla}\tilde\xi_{12})$. Corollary \[CoLys\](\[CoLys2\]) implies that $\operatorname{Hom}={{\mathcal F}}^{(2)}|_{\operatorname{diag}({{\mathcal M}})}$. It is easy to see that $\operatorname{Hom}$, viewed as a sheaf on ${{\mathcal M}}\times{{\mathcal M}}$, is equal to $(\operatorname{diag}_*O_{{\mathcal M}})^\psi$, where $\psi$ is the character of ${{\mathbf{G_m}}}\times{{\mathbf{G_m}}}$ given by $(t_1,t_2)\mapsto t_1/t_2$ (because a 1-dimensional vector space $E$ can be identified with weight $-1$ functions on $E-\{0\}$).
To complete the proof, it remains to check that ${{\mathcal F}}^{(2)}$ is concentrated (scheme-theoretically) on $\operatorname{diag}({{\mathcal M}})$. Assume for a contradiction that it is not the case. Note that ${{\mathcal F}}^{(2)}$ is coherent and concentrated set-theoretically on $\operatorname{diag}({{\mathcal M}})$.
\[lm:Nilpotents\] Let $Z$ be a locally Noetherian scheme, $Y\subset Z$ be a closed subscheme. Let ${{\mathcal G}}$ be a coherent sheaf on $Z$ concentrated set-theoretically but not scheme-theoretically on $Y$. Then there is a local Artinian scheme $S$ and an $S$-point of $Z$ such that $S^{red}$ factors through $Y$ and such that the restriction of ${{\mathcal G}}$ to $S$ is not concentrated on the scheme-theoretic preimage of $Y$.
We see that there is an $S$-point of ${{\mathcal M}}\times{{\mathcal M}}$ such that the restriction of ${{\mathcal F}}^{(2)}$ to this point is not concentrated on the preimage $S'$ of the diagonal. Let ${{\mathcal F}}_{(S)}$ be as in Lemma \[RelLys\]; using base change we see that $H^2({{\mathcal F}}_{(S)})$ is not concentrated on $S'$. The duality for Artinian rings shows that $\operatorname{\mathcal{H}\mathit{om}}(H^2({{\mathcal F}}_{(S)}),O_S)$ is not concentrated on $S'$ either. But then Lemma \[RelLys\] gives a contradiction, since $H^0({{\mathcal F}}_{(S)})$ is easily seen to be concentrated on $S'$.
We can assume that $Z=\operatorname{Spec}A$, $Y=\operatorname{Spec}A/\mathfrak a$, ${{\mathcal G}}$ corresponds to an $A$-module $M$; by assumption $\mathfrak
aM\ne0$ but $\mathfrak a^nM$=0 for $n$ big enough. Consider a maximal ideal $\mathfrak m$ such that $(\mathfrak
aM)_{\mathfrak m}\ne0$. It follows that $\mathfrak
m\supset\mathfrak a$. By Nakayama’s Lemma $\cap_n\mathfrak
m^nM_{\mathfrak m}=0$ and we can choose $n$ such that $\mathfrak a M\not\subset\mathfrak m^nM$. We can take $S=\operatorname{Spec}(A/\mathfrak m^n)$.
Relation to the Langlands correspondence {#ProofOfLanglands}
========================================
In this section we prove Theorem \[Langlands\]. Let us present the main steps. Recall that ${\mathfrak D}=\sum n_ix_i$. Set $$\beta_i^+:=\alpha_i^++\frac{n_i\lambda}2\frac{{\mathbf d}z_i}{z_i},\qquad
\beta_i^-:=\alpha_i^--\frac{n_i\lambda}2\frac{{\mathbf d}z_i}{z_i},$$ where $\lambda:=\sum_i\operatorname{res}\alpha_i^-$, $z_i$ is a local parameter at $x_i$. Note that the polar parts $\beta_i^\pm$ do not depend on the choice of $z_i$. Denote now the moduli space ${{\mathcal M}}$ defined in §\[MODST\] by ${{\mathcal M}}_\alpha$ to make the choice of formal types explicit. For a sheaf ${{\mathcal R}}$ of rings we denote by ${{\mathcal R}}$-mod the category of left ${{\mathcal R}}$-modules, by ${{\mathcal D}}^b({{\mathcal R}})$ the bounded derived category of ${{\mathcal R}}$-mod. We shall prove first that $${{\EuScript D}_{\overline\operatorname{\mathcal{B}\mathit{un}}(-1),\alpha}\text{-mod}}=
{\EuScript D}_{\operatorname{\mathcal{B}\mathit{un}}(-1),\alpha}\text{-mod}
\simeq{\EuScript D}_{P,\beta}\text{-mod}.$$ It remains to prove the following equivalences $${{\mathcal D}}^b({\EuScript D}_{P,\beta})\simeq {{\mathcal D}}^b({{\mathcal M}}_\beta)^{(-1)}
\simeq{{\mathcal D}}^b({{\mathcal M}}_\alpha)^{(-1)}.$$ Note that if $\alpha_i^\pm$ satisfy the conditions of §\[MODST\], then $\beta_i^\pm$ satisfy these conditions as well ((\[AlphaIII\]) and (\[AlphaIV\]) can be checked case by case). Thus the first equivalence follows from Theorem \[MainTh\]. For the last equivalence we shall prove that ${{\mathcal M}}_\alpha\simeq{{\mathcal M}}_\beta$.
It is well known that the definition of the derived category of ${\EuScript D}$-modules on a stack requires some caution. In this paper, we ignore the difficulty and use the naive definition: the derived category of ${\EuScript D}_{\overline\operatorname{\mathcal{B}\mathit{un}}(-1),\alpha}$-modules is simply the derived category of the abelian category of ${\EuScript D}_{\overline\operatorname{\mathcal{B}\mathit{un}}(-1),\alpha}$-modules.
Twisted ${\EuScript D}$-modules on algebraic stacks {#TDOEXISTENCE}
---------------------------------------------------
Let us summarize the properties of modules over TDO rings on algebraic stacks. We make no attempt to work in most general settings, and consider only smooth stacks, and only twists induced by torsors over an algebraic group. This case is enough for our purposes.
Let $G$ be an algebraic group with Lie algebra $\operatorname{\mathrm{Lie}}(G)$. Fix a $G$-invariant functional $\theta:\operatorname{\mathrm{Lie}}(G)\to{{\mathbb C}}$. First, consider twisted differential operators on a variety.
Let $X$ be a smooth variety and let $p:T\to X$ be a $G$-torsor on $X$. These data determine a TDO ring ${\EuScript D}_{X,T,\theta}$ on $X$, which is obtained by non-commutative reduction of the sheaf of differential operators ${\EuScript D}_T$ on $T$.
Namely, every $\xi\in\operatorname{\mathrm{Lie}}(G)$ gives a first order differential operator $a(\xi)-\theta(\xi)\in{\EuScript D}_T$, where the vector field $a(\xi)$ on $T$ is the action of $\xi$. Let $I$ be the ideal in $p_*{\EuScript D}_T$ generated by these differential operators. It is easy to see that this ideal is $G$-invariant, and we set ${\EuScript D}_{X,T,\theta}:=(p_*{\EuScript D}_T/I)^G$.
The category of quasi-coherent ${\EuScript D}_{X,T,\theta}$-modules can be described using a twisted strong equivariance condition. Let $M$ be a ${\EuScript D}_T$-module equipped with a weak $G$-equivariant structure (that is, $M$ is $G$-equivariant as a quasi-coherent sheaf, and the structure of a ${\EuScript D}_T$-module is $G$-equivariant). We say that $M$ is *strongly equivariant with twist $\theta$* if the action of $\xi\in\operatorname{\mathrm{Lie}}(G)$ on $M$ induced by the $G$-equivariant structure is given by $a(\xi)-\theta(\xi)$.
The sheaves of twisted differential operators have been introduced in [@BeilinsonBernstein]. The correspondence between ${\EuScript D}_{X,T,\theta}$-modules and twisted strongly equivariant modules is a particular case of the formalism of Harish-Chandra algebras from [@BeilinsonBernstein §1.8].
Let now ${{\mathcal X}}$ be an algebraic stack, and let $T\to{{\mathcal X}}$ be a $G$-torsor on ${{\mathcal X}}$. Every smooth morphism $\alpha:X\to{{\mathcal X}}$ from a variety $X$ induces a $G$-torsor $\alpha^*T$ on $X$, and we obtain the TDO ring ${\EuScript D}_{X,\alpha^*T,\theta}$. Such TDO rings form a ${\EuScript D}$-algebra on ${{\mathcal X}}$ in the sense of [@BeilinsonBernstein]. We denote this ${\EuScript D}$-algebra by ${\EuScript D}_{{{\mathcal X}},T,\theta}$. Note that ${\EuScript D}_{{{\mathcal X}},T,\theta}$ is not a sheaf of algebras on ${{\mathcal X}}$.
By definition, a ${\EuScript D}_{{{\mathcal X}},T,\theta}$-module $M$ is given by specifying a ${\EuScript D}_{X,\alpha^*T,\theta}$-module $M_\alpha$ for every smooth morphism $\alpha:X\to{{\mathcal X}}$ and an isomorphism of ${\EuScript D}_{Y,(\alpha\circ f)^*T,\theta}$-modules $f^*M_\alpha\simeq M_{\alpha\circ f}$ for every smooth map $f:Y\to X$ of algebraic varieties; the isomorphisms must be compatible with composition of morphisms $f$. Note in particular that $M$ is a quasi-coherent sheaf on ${{\mathcal X}}$.
\[Ex:ClassStack\] Let ${{\mathcal X}}:=B(H)$ be the classifying stack of an algebraic group $H$. Set $X=\operatorname{Spec}{{\mathbb C}}$. The natural map $\alpha:\operatorname{Spec}{{\mathbb C}}\to{{\mathcal X}}$ is an $H$-torsor (and, in particular, a presentation). For any $G$-torsor $T$ on ${{\mathcal X}}$, the pullback $\alpha^*T$ is isomorphic to the trivial torsor $G\to\operatorname{Spec}{{\mathbb C}}$. Fix a trivialization $\alpha^*T\simeq G$. The group $H$ acts on $\alpha^*T=G$; this is the right action for a homomorphism $\psi:H\to G$. In other words, $T$ is the descent of $G\to\operatorname{Spec}{{\mathbb C}}$, and $\psi$ provides the descent datum.
Let $M$ be a ${\EuScript D}_{{{\mathcal X}},T,\theta}$-module. It is easy to see that the TDO ring ${\EuScript D}_{\operatorname{Spec}{{\mathbb C}},G,\theta}$ is just the field of complex numbers, so the ${\EuScript D}_{\operatorname{Spec}{{\mathbb C}},G,\theta}$-module $\alpha^*M$ is a vector space $V$. Let us view $\alpha^*M$ as a strongly $G$-equivariant ${\EuScript D}_{\alpha^*T}$-module with twist $\theta$. It corresponds to the free $O_G$-module $V\otimes_{{\mathbb C}}O_G$ with the obvious $G$-equivariant structure. The action of ${\EuScript D}_{\alpha^*T}$ is uniquely determined by the twisted strong equivariance condition. On the other hand, $\alpha^*M$ also carries a structure of a strongly $H$-equivariant ${\EuScript D}$-module; this structure is essentially the descent data for $M$. If $V\ne0$, then such structure is provided by a scalar representation of $H$ on $V$ whose derivative is $-\theta\circ{\mathbf d}\psi$.
In particular, suppose that the character $\theta\circ{\mathbf d}\psi:\operatorname{\mathrm{Lie}}(H)\to{{\mathbb C}}$ does not integrate to a representation $H_0\to{{\mathbf{G_m}}}$, where $H_0\subset H$ is the identity component. Then $V=0$ and therefore the only ${\EuScript D}_{{{\mathcal X}},T,\theta}$-module is the zero module.
Step 1: ${\EuScript D}_{\overline\operatorname{\mathcal{B}\mathit{un}}(-1),\alpha}-
\mathrm{mod}={\EuScript D}_{\operatorname{\mathcal{B}\mathit{un}}(-1),\alpha}-\mathrm{mod}$
-------------------------------------------------------------------------------------------
In this section we shall prove
Assume that $(L,\eta)\in\overline\operatorname{\mathcal{B}\mathit{un}}(-1)$ does not correspond to a connection $(L,\nabla)\in{{\mathcal M}}$ in the sense of §\[AFFSTR\]. Then the restriction of any ${\EuScript D}_{\overline\operatorname{\mathcal{B}\mathit{un}}(-1),\alpha}$-module $\xi$ to $(L,\eta)$ is zero.
This proposition and Proposition \[Affine\] imply that the restriction of every ${\EuScript D}_{\overline\operatorname{\mathcal{B}\mathit{un}}(-1),\alpha}$-module to a point $(L,\eta)$ such that $(L,\eta)\notin\operatorname{\mathcal{B}\mathit{un}}(-1)$ is zero. This is valid for not necessarily closed points, and the step follows.
The proof is based on Corollary \[ConnExist\] and Example \[Ex:ClassStack\].
Consider Example \[Ex:ClassStack\] with $H=\operatorname{\mathrm{Aut}}(L,\eta)$. Recall from §\[LANGLANDS\] that the twist is given by the torsor $$T=\eta_{univ}\times_{\overline\operatorname{\mathcal{B}\mathit{un}}(-1)}\eta'_{univ}$$ over $G={{\mathbb C}}[{\mathfrak D}]^\times\times{{\mathbb C}}[{\mathfrak D}]^\times$ and the character $\theta=(\alpha^+,\alpha^-)$. One easily checks that $\psi:\operatorname{\mathrm{Aut}}(L,\eta)\to{{\mathbb C}}[{\mathfrak D}]^\times\times{{\mathbb C}}[{\mathfrak D}]^\times$ is given by the action of automorphisms on $\eta$ and $(L|_{\mathfrak D})/\eta$. Thus, in the notation of Corollary \[ConnExist\], ${\mathbf d}\psi:A\mapsto(A_+,A_-)$ and $$\label{chipsi}
\theta\circ{\mathbf d}\psi:A\mapsto\operatorname{res}(A_+\alpha_+)+\operatorname{res}(A_-\alpha_-).$$ Now assume that $(L,\nabla)$ does not correspond to any connection; our goal is to prove that $\theta\circ{\mathbf d}\psi$ does not integrate to a character of the identity component of $\operatorname{\mathrm{Aut}}(L,\eta)$. It follows from Corollary \[ConnExist\] that there is $A\in\operatorname{End}(L,\eta)$ such that $$\operatorname{res}(A_+\alpha_+)+\operatorname{res}(A_-\alpha_-)+\langle A,b(L)\rangle\ne0.$$ It is enough to consider two cases: $A$ is nilpotent, and $A$ is semisimple. In the first case it follows from and that $\theta\circ{\mathbf d}\psi(A)\ne0$ and $\theta\circ{\mathbf d}\psi$ cannot integrate to a character $H_0\to{{\mathbf{G_m}}}$, so we are done.
Let $A$ be semisimple. It follows from and condition of §\[MODST\] that $$\operatorname{res}((\operatorname{id}_L)_+\alpha_+)+\operatorname{res}((\operatorname{id}_L)_-\alpha_-)+\langle\operatorname{id}_L,b(L)\rangle=0.$$ Thus $A$ is not scalar. Then $(L,\eta)$ decomposes with respect to the eigenvalues of $A$ as $L=L_1\oplus L_2$ (and for every $i$ we have $\eta|_{n_ix_i}=L_1|_{n_ix_i}$ or $\eta|_{n_ix_i}=L_2|_{n_ix_i}$). Let $A'$ be the endomorphism of $(L,\eta)$ that is zero on $L_1$ and the identity on $L_2$. We see that $$\operatorname{res}(A'_+\alpha_+)+\operatorname{res}(A'_-\alpha_-)=\sum_i\operatorname{res}\alpha_i^\pm\notin{{\mathbb Z}}$$ by condition of \[MODST\]. Again, $\theta\circ{\mathbf d}\psi$ does not integrate and we are done.
Step 2: ${\EuScript D}_{\operatorname{\mathcal{B}\mathit{un}}(-1),\alpha}-\mathrm{mod}\simeq
{\EuScript D}_{P,\beta}-\mathrm{mod}$
--------------------------------------------------------------------------------------------
Recall that $\operatorname{\mathcal{B}\mathit{un}}(-1)=P/{{\mathbf{G_m}}}$, where ${{\mathbf{G_m}}}$ acts trivially.
Let $\pi:P\to\operatorname{\mathcal{B}\mathit{un}}(-1)$ be the projection. It follows from the definition of a (strongly) equivariant ${\EuScript D}$-module that ${\EuScript D}_{\operatorname{\mathcal{B}\mathit{un}}(-1),\alpha}-\mathrm{mod}\simeq
\pi^{{\bullet}}{\EuScript D}_{\operatorname{\mathcal{B}\mathit{un}}(-1),\alpha}-\mathrm{mod}$. So all we have to check is that $\pi^{{\bullet}}{\EuScript D}_{\operatorname{\mathcal{B}\mathit{un}}(-1),\alpha}={\EuScript D}_{P,\beta}$.
By Proposition \[BunP\] we have $\operatorname{\mathcal{B}\mathit{un}}(-1)=P/{{\mathbf{G_m}}}$, where $P$ is glued from two copies of ${{{\mathbb P}^1}}$, which we denote now by ${{{\mathbb P}^1_+}}$ and ${{{\mathbb P}^1_-}}$ (so that $x_i^-\in{{{\mathbb P}^1_-}}$). We saw that $P$ can be viewed as the moduli space of triples $(L,\eta,O_{{{\mathbb P}^1}}(-2)\hookrightarrow L_\eta)$ (cf. Remark \[RemP\]). We shall be using the notation from the proof of Proposition \[BunP\]. Let $\rho:\tilde P\to P$ be the projection. We assume that $\rho^{-1}({{{\mathbb P}^1_+}})$ is given by $p'\ne0$, while $\rho^{-1}({{{\mathbb P}^1_-}})$ is given by $p\ne0$.
\[DistingPar\] For all $i$ there exists a unique $(L,\eta)\in\operatorname{\mathcal{B}\mathit{un}}(-1)$ such that $\eta_{x_i}=(O_{{{\mathbb P}^1}})_{x_i}\subset L_{x_i}$ and under the above description of $\operatorname{\mathcal{B}\mathit{un}}(-1)$ this point corresponds to $x_i^-$.
Consider the composition $O_{{{\mathbb P}^1}}(-{\mathfrak D})\hookrightarrow
L(-{\mathfrak D})\hookrightarrow L_\eta$. Clearly, $\eta_{x_i}=(O_{{{\mathbb P}^1}})_{x_i}$ if and only if this composition is zero at $x_i$ . This happens if and only if the rank of ${\varphi}':O_{{{\mathbb P}^1}}(-{\mathfrak D})\oplus O_{{{\mathbb P}^1}}(-2)\rightarrow L_\eta$ drops at $x_i$ with the kernel $O_{{{\mathbb P}^1}}(-{\mathfrak D})_{x_i}$. This is in turn equivalent to $q=x_i$, $p'=0$.
Let $\delta$ be the line bundle on $\operatorname{\mathcal{B}\mathit{un}}(-1)$ whose fiber at $(L,\eta)$ is $\operatorname{detR\Gamma}({{{\mathbb P}^1}},L)$. Let $\delta'$ be the pullback of $\delta$ to $P$. Fix $\infty\in{{{\mathbb P}^1}}-{\mathfrak D}$.
$\delta'\simeq O_P(2(\infty)-\sum n_i x_i^-)$.
Let $t\in{{\mathbf{G_m}}}$ act on $\tilde P$ by $$\label{action}
t\cdot(p,p',q)=(p/t,tp',q).$$ This action gives rise to a ${{\mathbf{G_m}}}$-torsor $\tilde P\to P$. We claim that the corresponding line bundle is $\delta'$.
Indeed, consider the cartesian diagram $$\begin{CD}
\tilde P @>\rho>> P\\
@VVV @VVV\\
P @>>> \operatorname{\mathcal{B}\mathit{un}}(-1).
\end{CD}$$ Here the left hand arrow is the torsor described above. The top arrow corresponds to forgetting the embedding $O_{{{\mathbb P}^1}}(-2)\to
L_\eta$. Thus $P$ on the right parameterizes parabolic bundles with embeddings $O_{{{\mathbb P}^1}}\to L$. However, such an embedding is the same as a non-zero element of $\operatorname{detR\Gamma}({{{\mathbb P}^1}},L)=H^0({{{\mathbb P}^1}},L)$. Hence the torsor on the right is the one corresponding to $\delta$, and the torsor on the left is the one corresponding to $\delta'$.
We have $$\rho^{-1}({{{\mathbb P}^1_+}})=\{(f(q)/p',p',q)\in\tilde P\},$$ where $p'\ne0$ is in the fiber of $O_{{{\mathbb P}^1}}(2)$ over $q$. Thus $\rho^{-1}({{{\mathbb P}^1_+}})$ is the total space of $O_{{{\mathbb P}^1}}(2)$ with the zero section removed, and the action is the standard one. Hence $\delta'|_{{{{\mathbb P}^1_+}}}=O_{{{\mathbb P}^1}}(2)$.
Further, $$\rho^{-1}({{{\mathbb P}^1_-}})=\{(p,f(q)/p,q)\in\tilde P\}$$ is also the total space of $O_{{{\mathbb P}^1}}(2)$ with the zero section removed but the action is the inverse one, so the total space of the corresponding line bundle is obtained by compactifying at infinity and $\delta'|_{{{{\mathbb P}^1_-}}}=O_{{{\mathbb P}^1}}(-2)$. We also see that if a meromorphic section $s$ of $\delta'$ has order $m_i$ at $x_i^+$, then it has order $m_i-n_i$ at $x_i^-$.
Let $s$ be a section of $O_{{{\mathbb P}^1}}(2)=\delta'|_{{{{\mathbb P}^1_+}}}$ with a double zero at $\infty$. We view it as a meromorphic section of $\delta'$. It has no other zeroes on ${{{\mathbb P}^1_+}}$, and, by the previous remark, it has a pole of order $n_i$ at $x_i^-$. Thus the divisor of $s$ is $2(\infty)-\sum n_i x_i^-$.
Denote by $\Delta_-$ and $\Delta_+$ the graphs of the immersions ${\mathfrak D}\hookrightarrow{{{\mathbb P}^1_-}}\hookrightarrow P$ and ${\mathfrak D}\hookrightarrow{{{\mathbb P}^1_+}}\hookrightarrow P$, respectively.
\[Det\] Let ${{\mathcal L}}$ be the universal family on ${{{\mathbb P}^1}}\times P$. Then $$\det{{\mathcal L}}|_{{\mathfrak D}\times P}\simeq
O_{{\mathfrak D}\times P}(\Delta_-+\Delta_+)\otimes p_2^*\delta'.$$
We have a canonical map $p_1^*O_{{{\mathbb P}^1}}(-2)\to{{\mathcal L}}$ (recall the modular description of $P$). On the other hand, we have an adjunction morphism $p_2^*\delta'\to{{\mathcal L}}$ (recall that the fiber of $p_2^*\delta'$ is $H^0({{{\mathbb P}^1}},L)$). These maps give rise to a map $p_1^*O_{{{\mathbb P}^1}}(-2)\otimes p_2^*\delta'\to\det{{\mathcal L}}$, and it vanishes exactly over the graph of $\wp$. Restricting to ${\mathfrak D}\times P$ we obtain the result.
Note that ${{{\mathbb C}}[{\mathfrak D}]^\times}$-torsors on a scheme $Y$ are the same as ${\mathfrak D}$-families of line bundles on $Y$ (i.e., line bundles on ${\mathfrak D}\times Y$). Indeed, a line bundle on ${\mathfrak D}\times Y$ is the same as a rank one locally free module over ${{\mathbb C}}[{\mathfrak D}]\otimes_{{\mathbb C}}O_Y$. Such modules are in one-to-one correspondence with torsors over the sheaf $({{\mathbb C}}[{\mathfrak D}]\otimes_{{\mathbb C}}O_Y)^\times$.
Thus $\eta_{univ}$ and $\eta_{univ}'$ can be viewed as line bundles on ${\mathfrak D}\times\operatorname{\mathcal{B}\mathit{un}}(-1)$. Clearly, $\eta_{univ}$ is a subbundle of ${{\mathcal L}}|_{{\mathfrak D}\times\operatorname{\mathcal{B}\mathit{un}}(-1)}$, and $\eta'_{univ}=({{\mathcal L}}|_{{\mathfrak D}\times\operatorname{\mathcal{B}\mathit{un}}(-1)})/\eta_{univ}$.
[[)]{}]{}\[Pullb1\] The pullback of $\eta_{univ}$ to ${\mathfrak D}\times P$ is $O_{{\mathfrak D}\times P}(\Delta_+)$.
[[)]{}]{}\[Pullb2\] The pullback of $\eta'_{univ}$ to ${\mathfrak D}\times P$ is $p_2^*(O_P(2(\infty)-\sum n_i
x_i^-))\otimes O_{{\mathfrak D}\times P}(\Delta_-)$.
The asymmetry is due to the choice of one of two torsors $P\to\operatorname{\mathcal{B}\mathit{un}}(-1)$.
Note that $\pi^*\eta_{univ}\otimes\pi^*\eta'_{univ}=\det{{\mathcal L}}|_{{\mathfrak D}\times
P}$, thus (\[Pullb1\]) follows from (\[Pullb2\]) and Lemma \[Det\]. Let us prove (\[Pullb2\]). The proof is essentially a family version of Lemma \[DistingPar\].
As in the proof of Lemma \[Det\] we get a map $\bar\delta:=p_2^*\delta'\to{{\mathcal L}}$. Restricting this to ${\mathfrak D}\times P$ and composing with the natural projection we get a map $$\bar\delta|_{{\mathfrak D}\times P}\to\pi^*\eta'_{univ}.$$ We need to show that it vanishes exactly on $\Delta_-$. Clearly, this map vanishes on $S\subset{\mathfrak D}\times P$ if and only if $\bar\delta\to{{\mathcal L}}$ factors through $\eta_{univ}$ over $S$. One checks that this happens if and only if $\bar\delta(-({\mathfrak D}\times P))\to{{\mathcal L}}(-({\mathfrak D}\times
P))\to{{\mathcal L}}_{\eta_{univ}}$ vanishes over $S$. Let us show that $S\subset\Delta_-$ in this case (we leave the converse to the reader).
We see that the rank of ${\varphi}':\bar\delta(-({\mathfrak D}\times
P))\oplus p_1^*O_{{{\mathbb P}^1}}(-2)\rightarrow{{\mathcal L}}_{\eta_{univ}}$ drops on $S$. Recall the modular definition of $\wp$: its graph is given by the scheme, where the rank of ${\varphi}'$ drops (cf. proof of Proposition \[BunP\], Step 5). Thus $S\subset\Delta_-\cup\Delta_+$. But the kernel of ${\varphi}'$ is $\bar\delta(-({\mathfrak D}\times P))$, thus in fact $S\subset\Delta_-$.
Now let us be explicit about what we need to calculate: $\pi^*\eta_{univ}$ and $\pi^*\eta'_{univ}$ correspond to classes $[\pi^*\eta_{univ}],[\pi^*\eta'_{univ}]\in
H^1({\mathfrak D}\times P, O^\times_{{\mathfrak D}\times P})$. There is a natural map $\mathbf{dlog}: O^\times_{{\mathfrak D}\times P}\to
p_2^*\Omega_P: f\mapsto f^{-1}{\mathbf d}_P f$. Applying this map to $[\pi^*\eta_{univ}]$ and $[\pi^*\eta'_{univ}]$ we get elements of $H^1(P,\Omega_P)\otimes_{{\mathbb C}}O_{\mathfrak D}$. The TDO ring $\pi^{{\bullet}}{\EuScript D}_{\operatorname{\mathcal{B}\mathit{un}}(-1),\alpha}$ corresponds to an element of $H^1(P,\Omega_P)$ given by $$\langle\mathbf{dlog}[\pi^*\eta_{univ}],(\alpha_i^+)\rangle+
\langle\mathbf{dlog}[\pi^*\eta'_{univ}],(\alpha_i^-)\rangle,$$ where $$\langle\cdot,\cdot\rangle:H^1(P,\Omega_P)\otimes
O_{\mathfrak D}\otimes O_{\mathfrak D}^\vee\to H^1(P,\Omega_P).$$ Choose local parameters $z_i$ at $x_i\in{{{\mathbb P}^1}}$. Then we obtain an isomorphism $O_{\mathfrak D}=\prod_i{{\mathbb C}}[w_i]/w_i^{n_i}$. Recall the description of $H^1(P,\Omega_P)$ given in Lemma \[CohP\]. An easy calculation shows that $$\begin{split}
\mathbf{dlog}\left(p_2^*(2(\infty)-\sum n_i x_i^-)\right)&=
(0,n_i\,{\mathbf d}z_i/z_i)\otimes1_{\mathfrak D},\\
\mathbf{dlog}(O_{{\mathfrak D}\times P}(\Delta_-))&=
\left(1_{\mathfrak D},-\frac{{\mathbf d}z_i}{z_i-w_i}
\right),\\
\mathbf{dlog}(O_{{\mathfrak D}\times P}(\Delta_+))&=
\left(1_{\mathfrak D},\frac{{\mathbf d}z_i}{z_i-w_i}
\right)
\end{split}$$ ($\frac{{\mathbf d}z_i}{z_i-w_i}$ should be expanded in the powers of $w_i$). Further, $$\begin{split}
\langle(0,n_i\,{\mathbf d}z_i/z_i)\otimes1_{\mathfrak D},(\alpha_i^-)\rangle&=
(0,n_i\lambda\,{\mathbf d}z_i/z_i),\\
\left\langle\left(1_{\mathfrak D},-\frac{{\mathbf d}z_i}{z_i-w_i}
\right),(\alpha_i^-)\right\rangle&=(\lambda,-\alpha_i^-),\\
\left\langle\left(1_{\mathfrak D},\frac{{\mathbf d}z_i}{z_i-w_i}
\right),(\alpha_i^+)\right\rangle&=\left(\sum_i\operatorname{res}\alpha_i^+,\alpha_i^+\right).
\end{split}$$ Note that collections $(\alpha_i^\pm)$ in the left-hand side are viewed as elements of $O_{\mathfrak D}^\vee$, while in the right-hand side they are polar parts of 1-forms.
Applying the previous proposition and recalling that $\lambda+\sum_i\operatorname{res}\alpha_i^+=-d$, we see that the element of $H^1(P,\Omega_P)$ corresponding to $\pi^{{\bullet}}{\EuScript D}_{\operatorname{\mathcal{B}\mathit{un}}(-1),\alpha}$ is $$(-d,\alpha_i^+-\alpha_i^-+n_i\lambda\,{\mathbf d}z_i/z_i).$$ It remains to notice that $\beta_i^\pm$ correspond to the same element of $H^1(P,\Omega_P)$, cf. Lemma \[CohP\].
Step 3: ${{\mathcal M}}_\alpha\simeq{{\mathcal M}}_\beta$.
----------------------------------------------------------
This isomorphism is provided by Katz’s middle convolution. It is defined in [@Katz] in the settings of $l$-adic sheaves; see [@SimpsonRadon] or [@ArinkinFourier] for the settings of de Rham local systems. Here is an explicit description of the isomorphism.
Fix $\infty\in{{{\mathbb P}^1}}-{\mathfrak D}$. There is a unique 1-form $\alpha$ on ${{{\mathbb P}^1}}-{\mathfrak D}-\{\infty\}$ such that $\alpha+\alpha_i^-$ is non-singular at $x_i$ and $\alpha$ has a pole of order one at $\infty$. Similarly, there is a unique 1-form $\beta$ on ${{{\mathbb P}^1}}-{\mathfrak D}-\{\infty\}$ such that $\beta+\beta_i^-$ is non-singular at $x_i$ and $\beta$ has a pole of order one at $\infty$. Note that $\operatorname{res}_\infty\alpha=\sum_i\operatorname{res}\alpha_i^-=\lambda$ and $\operatorname{res}_\infty\beta=-\lambda$.
Fix $(L,\nabla)\in{{\mathcal M}}_\alpha$. The connection $$\nabla+\alpha:L\to L\otimes\Omega_{{{\mathbb P}^1}}({\mathfrak D}+(\infty))$$ has formal type $(0,\alpha_i^+-\alpha_i^-)$ at $x_i$. Let $\tilde L\subset L$ be the largest subsheaf such that $$(\nabla+\alpha)(\tilde L)\subset L\otimes\Omega_{{{\mathbb P}^1}}(\infty).$$ Explicitly, $\tilde L$ is the modification of $L$ with respect to one of two parabolic structures on $L$ induced by $\nabla$. Precisely, this parabolic structure $\eta$ is such that the polar part of $\nabla$ induces multiplication by $\alpha_-$ on $\eta$ (cf. Corollary \[ConnExist\]).
Consider on ${{{\mathbb P}^1}}\times{{{\mathbb P}^1}}$ the differential 1-form $\lambda{\mathbf d}\log(x-y)$, where $x$ and $y$ are the coordinates on the first and second factors, respectively. The preimage $p_1^*L$ carries a flat meromorphic connection $p_1^*\nabla$; let us equip $p_1^*L$ with the flat meromorphic connection $$p_1^*\nabla+p_1^*\alpha+\lambda{\mathbf d}\log(x-y).$$ Denote the ‘horizontal’ and ‘vertical’ parts of this connection by $\nabla_x$ and $\nabla_y$. We then obtain an anti-commutative square [$$\label{twonablas}
\begin{CD}
p_1^*\tilde L @>\nabla_x>> p_1^*L\otimes p_1^*\Omega_{{{\mathbb P}^1}}(\Delta)\\
@V\nabla_yVV @V\nabla_yVV\\
p_1^*\tilde L\otimes p_2^*\Omega_{{{\mathbb P}^1}}(\Delta+{{{\mathbb P}^1}}\times\{\infty\})@>\nabla_x>>
p_1^*L\otimes p_1^*\Omega_{{{\mathbb P}^1}}\otimes p_2^*\Omega_{{{\mathbb P}^1}}(2 \Delta+{{{\mathbb P}^1}}\times\{\infty\}).
\end{CD}$$]{}
Consider the complex $$\begin{CD}
{{\mathcal F}}^{{\bullet}}:=(p_1^*\tilde L @>\nabla_x>> p_1^*L\otimes p_1^*\Omega_{{{\mathbb P}^1}}(\Delta))
\end{CD}$$ of sheaves on ${{{\mathbb P}^1}}\times{{{\mathbb P}^1}}$. The differential $\nabla_x$ is $p_2^{-1}O_{{{\mathbb P}^1}}$-linear, so the direct image $Rp_{2,*}{{\mathcal F}}^{{\bullet}}$ makes sense as an object in the derived category of $O_{{{\mathbb P}^1}}$-modules. It is easy to see that $R^0p_{2,*}{{\mathcal F}}^{{\bullet}}=R^2p_{2,*}{{\mathcal F}}^{{\bullet}}=0$. Now the Euler characteristic argument shows that $Rp_{2,*}{{\mathcal F}}^{{\bullet}}[1]$ is a locally free $O_{{{\mathbb P}^1}}$-module of rank two; let us denote it by $E$.
Similarly, consider the complex $$\begin{CD}
{{\mathcal F}}^{{\bullet}}(\Delta)=(p_1^*\tilde L(\Delta) @>\nabla_x>>
p_1^*L\otimes p_1^*\Omega_{{{\mathbb P}^1}}(2\Delta)).
\end{CD}$$ Then $Rp_{2,*}{{\mathcal F}}^{{\bullet}}(\Delta)[1]$ is a locally free $O_{{{\mathbb P}^1}}$-module of rank two; let us denote it by $\tilde E$.
The natural morphism ${{\mathcal F}}^{{\bullet}}\hookrightarrow{{\mathcal F}}^{{\bullet}}(\Delta)$ induces a homomorphism $\iota:E\to\tilde E$. Recall that $$O_{{{{\mathbb P}^1}}\times{{{\mathbb P}^1}}}(k\Delta)/O_{{{{\mathbb P}^1}}\times{{{\mathbb P}^1}}}((k-1)\Delta)
\approx(\iota_\Delta)_*{{\mathcal T}}_{{{\mathbb P}^1}}^{\otimes k}.$$ Thus we have an exact sequence of complexes $$0\to{{\mathcal F}}^{{\bullet}}\to{{\mathcal F}}^{{\bullet}}(\Delta)\to
(\iota_\Delta)_*(\tilde L\otimes{{\mathcal T}}_{{{\mathbb P}^1}}\to L\otimes{{\mathcal T}}_{{{\mathbb P}^1}})\to0.$$ One checks that the differential in the rightmost complex is induced by the natural inclusion $\tilde L\hookrightarrow L$. Thus $\iota$ is an embedding, and $\operatorname{Coker}(\iota)\simeq
O_{\mathfrak D}$. We can thus identify $\tilde E$ with an upper modification of $E$.
Finally, note that diagram provides a ${{\mathbb C}}$-linear map $\nabla_E:E\to\tilde
E\otimes\Omega_{{{\mathbb P}^1}}(\infty)$. Clearly, $\nabla_E$ satisfies the Leibnitz identity. We view $\nabla_E$ as a connection on $E$ with poles at ${\mathfrak D}\cup\{\infty\}$.
The formal type of $\nabla_E$ at $x_i$ is $(0,\beta_i^+-\beta_i^-)$, and the residue of $\nabla_E$ at $\infty$ is $-\lambda$. In other words, $(E,\nabla_E-\beta)\in
{{\mathcal M}}_\beta$. The correspondence $$(L,\nabla)\mapsto(E,\nabla_E-\beta)$$ is an isomorphism ${{\mathcal M}}_\alpha{\mathrel{\widetilde\to}}{{\mathcal M}}_\beta$.
We shall prove a slightly weaker statement, which is sufficient for our purposes. Namely, we prove that $(E,\nabla_E)$ has the described formal types after a modification.
Let ${\EuScript D}_{{{{\mathbb P}^1}},\lambda}$ be the TDO ring corresponding to $\lambda\in{{\mathbb C}}=H^1({{{\mathbb P}^1}},\Omega_{{{\mathbb P}^1}})$. For $(L,\nabla)\in{{\mathcal M}}_\alpha$ consider the ${\EuScript D}_{{{{\mathbb P}^1}},\lambda}$-module $\jmath_{!*}(L,\nabla+\alpha)$, where $\jmath:{{{\mathbb P}^1}}-({\mathfrak D}\cup\infty)\hookrightarrow{{{\mathbb P}^1}}$ is the natural inclusion. Note that it has no singularity at $\infty$ because of the twist by $\lambda$.
As explained in [@ArinkinFourier §6.3], the Katz–Radon transform gives an equivalence ${{\mathfrak R}}:{\EuScript D}_{{{{\mathbb P}^1}},\lambda}-\mathrm{mod}\to{\EuScript D}_{{{{\mathbb P}^1}},-\lambda}-\mathrm{mod}$ and it is easy to see that it is compatible with our construction in the sense that $${{\mathfrak R}}(\jmath_{!*}\jmath^*(L,\nabla+\alpha))=
\jmath_{!*}\jmath^*(E,\nabla_E).$$ (The proof is similar to [@Arinkin Lemma 13].)
Let $\Phi_{x_i}$ be the functor of vanishing cycles as defined in [@ArinkinFourier]. We get $$\begin{gathered}
\Phi_{x_i}(\jmath_{!*}\jmath^*(E,\nabla_E))=
\Phi_{x_i}{{\mathfrak R}}(\jmath_{!*}\jmath^*(L,\nabla+\alpha))=
{{\mathfrak R}}(x_i,x_i)\Phi_{x_i}\jmath_{!*}\jmath^*(L,\nabla+\alpha)=\\
{{\mathfrak R}}(x_i,x_i)(O_{\dot D},{\mathbf d}+\alpha_i^+-\alpha_i^-)= (O_{\dot
D},{\mathbf d}+\alpha_i^+-\alpha_i^-+n_i\lambda).\end{gathered}$$ Here ${{\mathfrak R}}(x_i,x_i)$ is the local Katz–Radon transform, the second equality is [@ArinkinFourier Corollary 6.11], the last equality is [@ArinkinFourier Theorem C]. Now it is easy to see that $\jmath_{!*}\jmath^*(E,\nabla_E)$ has required singularities.
It follows that the formal type of $\nabla_E$ at $x_i$ is $(m_i,\beta_i^+-\beta_i^-+m_i')$, where $m_i$, and $m_i'$ are integers. Thus $(E,\nabla_E-\beta)$ becomes a connection in ${{\mathcal M}}_\beta$ after a suitable modification.
Note that the construction of $(E,\nabla_E-\beta)$ works in families as well. Since formal normal forms of connections exist in families, after a suitable modification we get a morphism ${{\mathcal M}}_\alpha\to{{\mathcal M}}_\beta$. If we do the same construction with $\alpha$ and $\beta$ switched and use the inverse Katz–Radon transform, we get a morphism ${{\mathcal M}}_\beta\to{{\mathcal M}}_\alpha$. It is easy to see that these morphisms are inverse to each other.
This completes the proof of Theorem \[Langlands\].
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---
abstract: 'Existential rules, long known as tuple-generating dependencies in database theory, have been intensively studied in the last decade as a powerful formalism to represent ontological knowledge in the context of ontology-based query answering. A knowledge base is then composed of an instance that contains incomplete data and a set of existential rules, and answers to queries are logically entailed from the knowledge base. This brought again to light the fundamental chase tool, and its different variants that have been proposed in the literature. It is well-known that the problem of determining, given a chase variant and a set of existential rules, whether the chase will halt on any instance, is undecidable. Hence, a crucial issue is whether it becomes decidable for known subclasses of existential rules. In this work, we consider linear existential rules, a simple yet important subclass of existential rules that generalizes inclusion dependencies. We show the decidability of the *all instance* chase termination problem on linear rules for three main chase variants, namely *semi-oblivious*, *restricted* and *core* chase. To obtain these results, we introduce a novel approach based on so-called derivation trees and a single notion of forbidden pattern. Besides the theoretical interest of a unified approach and new proofs, we provide the first positive decidability results concerning the termination of the restricted chase, proving that chase termination on linear existential rules is decidable for both versions of the problem: Does *every* fair chase sequence terminate? Does *some* fair chase sequence terminate?'
author:
- Michel Leclère
- 'Marie-Laure Mugnier'
- Michaël Thomazo
- Federico Ulliana
bibliography:
- 'bib.bib'
title: A Single Approach to Decide Chase Termination on Linear Existential Rules
---
Introduction
============
The chase procedure is a fundamental tool for solving many issues involving tuple-generating dependencies, such as data integration [@DBLP:conf/pods/Lenzerini02], data-exchange [@DBLP:journals/tcs/FaginKMP05], query answering using views [@DBLP:journals/vldb/Halevy01] or query answering on probabilistic databases [@DBLP:conf/icde/OlteanuHK09]. In the last decade, tuple-generating dependencies raised a renewed interest under the name of *existential rules* for the problem known as ontology-based query answering. In this context, the aim is to query a knowledge base $({\ensuremath{I}}, {\ensuremath{\Sigma}})$, where ${\ensuremath{I}}$ is an instance and ${\ensuremath{\Sigma}}$ is a set of existential rules (see e.g. the survey chapters [@DBLP:books/sp/virgilio09/CaliGL09; @DBLP:conf/rweb/MugnierT14]). In more classical database terms, this problem can be recast as querying an instance ${\ensuremath{I}}$ under incomplete data assumption, provided with a set of constraints ${\ensuremath{\Sigma}}$, which are tuple-generating dependencies. The chase is a fundamental tool to solve dependency-related problems as it allows one to compute a (possibly infinite) *universal model* of $({\ensuremath{I}}, {\ensuremath{\Sigma}})$, *i.e.*, a model that can be homomorphically mapped to any other model of $({\ensuremath{I}}, {\ensuremath{\Sigma}})$. Hence, the answers to a conjunctive query (and more generally to any kind of query closed by homomorphism) over $({\ensuremath{I}}, {\ensuremath{\Sigma}})$ can be defined by considering solely this universal model.
Several variants of the chase have been introduced, and we focus in this paper on the main ones: semi-oblivious [@DBLP:conf/pods/Marnette09] (aka skolem [@DBLP:conf/pods/Marnette09]), restricted [@DBLP:conf/icalp/BeeriV81; @DBLP:journals/tcs/FaginKMP05] (aka standard [@phd/Onet12]) and core [@DBLP:conf/pods/DeutschNR08]. It is well known that all of these produce homomorphically equivalent results but terminate for increasingly larger subclasses of existential rules.
Any chase variant starts from an instance and exhaustively performs a sequence of rule applications according to a redundancy criterion which characterizes the variant itself. The question of whether a chase variant terminates on *all instances* for a given set of existential rules is known to be undecidable when there is no restriction on the kind of rules [@DBLP:journals/ai/BagetLMS11; @DBLP:conf/icalp/GogaczM14]. A number of *sufficient* syntactic conditions for termination have been proposed in the literature for the semi-oblivious chase (see e.g. [@phd/Onet12; @DBLP:journals/jair/GrauHKKMMW13; @DBLP:phd/hal/Rocher16] for syntheses), as well as for the restricted chase [@DBLP:conf/ijcai/CarralDK17] (note that the latter paper also defines a sufficient condition for non-termination). However, only few positive results exist regarding the termination of the chase on specific classes of rules. Decidability was shown for the semi-oblivious chase on guarded-based rules (linear rules, and their extension to (weakly-)guarded rules) [@DBLP:conf/pods/CalauttiGP15]. Decidability of the core chase termination on guarded rules for a fixed instance was shown in [@DBLP:conf/icdt/Hernich12].
In this work, we provide new insights on the chase termination problem for *linear* existential rules, a simple yet important subclass of guarded existential rules, which generalizes inclusion dependencies [@DBLP:journals/tods/Fagin81] and practical ontological languages [@DBLP:journals/ws/CaliGL12]. Precisely, the question of whether a chase variant terminates on all instances for a set of linear existential rules is studied in two fashions:
- does *every* (fair) chase sequence terminate?
- does *some* (fair) chase sequence terminate?
It is well-known that these two questions have the same answer for the semi-oblivious and the core chase variants, but not for the restricted chase. Indeed, this last one may admit both terminating and non-terminating sequences over the same knowledge base. We show that the termination problem is decidable for linear existential rules, whether we consider any version of the problem and any chase variant.
We study chase termination by exploiting in a novel way a graph structure, namely the *derivation tree*, which was originally introduced to solve the ontology-based (conjunctive) query answering problem for the family of greedy-bounded treewidth sets of existential rules [@DBLP:conf/ijcai/BagetMRT11; @DBLP:phd/hal/Thomazo13], a class that generalizes guarded-based rules and in particular linear rules. We first use derivation trees to show the decidability of the termination problem for the semi-oblivious and restricted chase variants, and then generalize them to *entailment trees* to show the decidability of termination for the core chase. For any chase variant we consider, we adopt the same high-level procedure: starting from a finite set of canonical instances (representative of all possible instances), we build a (set of) tree structures for each canonical instance, while forbidding the occurrence of a specific pattern, we call *unbounded-path witness*. The built structures are finite thanks to this forbidden pattern, and this allows us to decide if the chase terminates on the associated canonical instance. By doing so, we obtain a uniform approach to study the termination of several chase variants, that we believe to be of theoretical interest per se. The derivation tree is moreover a simple structure and the algorithms built on it are likely to lead to an effective implementation. Let us also point out that our approach is constructive: if the chase terminates on a given instance, the algorithm that decides termination actually computes the result of the chase (or a superset of it in the case of the core chase), otherwise it pinpoints a forbidden pattern responsible for non-termination.
Besides providing new theoretical tools to study chase termination, we obtain the following results for linear existential rules:
- a new proof of the decidability of the semi-oblivious chase termination, building on different objects than the previous proof provided in [@DBLP:conf/pods/CalauttiGP15]; we show that our algorithm provides the same complexity upper-bound;
- the decidability of the restricted chase termination, for both versions of the problem, i.e., termination of all (fair) chase sequences and termination of some (fair) chase sequence; to the best of our knowledge, these are the first positive results on the decidability of the restricted chase termination;
- a new proof of the decidability of the core chase termination, with different objects than previous work reported in [@DBLP:conf/icdt/Hernich12]; although this latter paper solves the question of the core chase termination given a *single* instance, the results actually allow to infer the decidability of the *all* instance version of the problem, by noticing that only a finite number of instances need to be considered (see the next section).
The paper is organized as follows. After introducing some preliminary notions (Section 2), we define the main components of our framework, namely derivation trees and unbounded-path witnesses (Section 3). We build on these objects to prove the decidability of the semi-oblivious and restricted chase termination (Section 4). Finally, we generalize derivation-trees to entailment trees and use them to prove the decidability of the core chase termination (Section 5). Detailed proofs are provided in the appendix.
Preliminaries {#section-preliminaries}
=============
We consider a logical *vocabulary* composed of a finite set of predicates and an infinite set of constants. An *atom* ${\alpha}$ has the form ${\ensuremath{r}}(t_1,\ldots, t_n)$ where ${\ensuremath{r}}$ is a predicate of arity $n$ and the $t_i$ are terms (i.e., variables or constants). We denote by ${{\ensuremath{\textsl{terms}}}{({\alpha})}}$ (resp. ${{\ensuremath{\textsl{vars}}}{({\alpha})}}$) the set of terms (resp. variables) in ${\alpha}$ and extend the notations to a set of atoms. A *ground* atom does not contain any variable. It is convenient to identify the existential closure of a conjunction of atoms with the set of these atoms. An *instance* is a set of (non-necessarily ground) atoms, which is finite unless otherwise specified. Abusing terminology, we will often see an instance as its isomorphic model.
Given two sets of atoms ${\ensuremath{S}}$ and ${\ensuremath{S}}'$, a *homomorphism* from ${\ensuremath{S}}'$ to ${\ensuremath{S}}$ is a substitution ${\pi}$ of ${{\ensuremath{\textsl{vars}}}{({\ensuremath{S}}')}}$ by ${{\ensuremath{\textsl{terms}}}{({\ensuremath{S}})}}$ such that ${\pi}({\ensuremath{S}}') \subseteq {\ensuremath{S}}$. It holds that ${\ensuremath{S}}\models {\ensuremath{S}}'$ (where $\models$ denotes classical logical entailment) iff there is a homomorphism from ${\ensuremath{S}}'$ to ${\ensuremath{S}}$. An endomorphism of ${\ensuremath{S}}$ is a homomorphism from ${\ensuremath{S}}$ to itself. A set of atoms is a *core* if it admits only injective endomorphisms. Any finite set of atoms is logically equivalent to one of its subsets that is a core, and this core is unique up to isomomorphism (i.e., bijective variable renaming). Given sets of atoms ${\ensuremath{S}}$ and ${\ensuremath{S}}'$ such that ${\ensuremath{S}}\cap {\ensuremath{S}}' \neq \emptyset$, we say that ${\ensuremath{S}}$ *folds* onto ${\ensuremath{S}}'$ if there is a homomorphism ${\pi}$ from ${\ensuremath{S}}$ to ${\ensuremath{S}}'$ such that ${\pi}$ is the identity on ${\ensuremath{S}}\cap {\ensuremath{S}}'$. The homomorphism ${\pi}$ is called a *folding*. In particular, it is well-known that any set of atoms *folds* onto its core.
An existential rule (or simply *rule*) is of the form ${\ensuremath{\sigma}}= \forall{\mathbf{x}}\forall{\mathbf{y}}.[{\ensuremath{\mathrm{body}({\mathbf{x}},{\mathbf{y}})}} \rightarrow \exists {\mathbf{z}}.{\ensuremath{\mathrm{head}({\mathbf{x}},{\mathbf{z}})}}]$ where ${\ensuremath{\mathrm{body}({\mathbf{x}},{\mathbf{y}})}}$ and ${\ensuremath{\mathrm{head}({\mathbf{x}},{\mathbf{z}})}}$ are non-empty conjunctions of atoms on variables, respectively called the *body* and the *head* of the rule, also denoted by ${\ensuremath{\mathrm{body}({\ensuremath{\sigma}})}}$ and ${\ensuremath{\mathrm{head}({\ensuremath{\sigma}})}}$, and ${\mathbf{x}}, {\mathbf{y}}$ and ${\mathbf{z}}$ are pairwise disjoint tuples of variables. The variables of ${\mathbf{z}}$ are called *existential variables*. The variables of ${\mathbf{x}}$ form the *frontier* of ${\ensuremath{\sigma}}$, which is also denoted by ${\ensuremath{\mathrm{fr}({\ensuremath{\sigma}})}}$. For brevity, we will omit universal quantifiers in the examples. A *knowledge base* (KB) is of the form ${\ensuremath{\mathcal{K}}}= {\ensuremath{({\ensuremath{I}},{\ensuremath{\Sigma}})}}$, where ${\ensuremath{I}}$ is an instance and ${\ensuremath{\Sigma}}$ is a finite set of existential rules.
A rule ${\ensuremath{\sigma}}= {\ensuremath{\mathrm{body}({\ensuremath{\sigma}})}} \rightarrow {\ensuremath{\mathrm{head}({\ensuremath{\sigma}})}}$ is *applicable* to an instance ${\ensuremath{I}}$ if there is a homomorphism ${\pi}$ from $ {\ensuremath{\mathrm{body}({\ensuremath{\sigma}})}} $ to ${\ensuremath{I}}$. The pair $({\ensuremath{\sigma}}, {\pi})$ is called a *trigger* for ${\ensuremath{I}}$. The result of the application of ${\ensuremath{\sigma}}$ according to ${\pi}$ on ${\ensuremath{I}}$ is the instance ${\ensuremath{I}}' = {\ensuremath{I}}\cup {\ensuremath{{\pi}^s}}({\ensuremath{\mathrm{head}({\ensuremath{\sigma}})}})$, where ${\ensuremath{{\pi}^s}}$ (here $s$ stands for *safe*) extends ${\pi}$ by assigning a distinct fresh variable (also called a *null*) to each existential variable. We also say that ${\ensuremath{I}}'$ is obtained by *firing* the trigger $({\ensuremath{\sigma}}, {\pi})$ on ${\ensuremath{I}}$. By ${\pi}_{\mid{\ensuremath{\mathrm{fr}({\ensuremath{\sigma}})}}}$ we denote the restriction of ${\pi}$ to the domain ${\ensuremath{\mathrm{fr}({\ensuremath{\sigma}})}}$.
A *${\ensuremath{\Sigma}}$-derivation* (or simply *derivation* when ${\ensuremath{\Sigma}}$ is clear from the context) from an instance ${\ensuremath{I}}= I_0$ to an instance $I_n$ is a sequence $I_0, ({\ensuremath{\sigma}}_1,{\pi}_1), I_1 \ldots, I_{n-1}, ({\ensuremath{\sigma}}_n,{\pi}_n), I_n$, such that for all $1 \leq i \leq n$: ${\ensuremath{\sigma}}_i \in {\ensuremath{\Sigma}}$, $({\ensuremath{\sigma}}_i,{\pi}_i)$ is a trigger for $I_{i-1}$, $I_i$ is obtained by firing $({\ensuremath{\sigma}}_i,{\pi}_i)$ on $I_{i-1}$, and $I_i \neq I_{i-1}$. We may also denote this derivation by the associated sequence of instances $(I_0, \ldots, I_n)$ when the triggers are not needed. The notion of derivation can be naturally extended to an *infinite* sequence.
We briefly introduce below the main chase variants and refer to [@phd/Onet12] for a detailed presentation.
The *semi-oblivious* chase prevents several applications of the same rule through the same mapping of its frontier. Given a derivation from $I_0$ to $I_{i}$, a trigger $({\ensuremath{\sigma}},{\pi})$ for $I_i$ is said to be *active according to the semi-oblivious criterion*, if there is no trigger $({\ensuremath{\sigma}}_j,{\pi}_j)$ in the derivation with ${\ensuremath{\sigma}}= {\ensuremath{\sigma}}_j$ and ${\pi}_{\mid{{\ensuremath{\mathrm{fr}({\ensuremath{\sigma}})}}}} = {\pi}_{j_\mid{{\ensuremath{\mathrm{fr}({\ensuremath{\sigma}}_j)}}}}$. The *restricted* chase performs a rule application only if the added set of atoms is not redundant with respect to the current instance. Given a derivation from $I_0$ to $I_{i}$, a trigger $({\ensuremath{\sigma}},{\pi})$ for $I_i$ is said to be *active according to the restricted criterion* if ${\pi}$ cannot be extended to a homomorphism from $({\ensuremath{\mathrm{body}({\ensuremath{\sigma}})}}\cup{\ensuremath{\mathrm{head}({\ensuremath{\sigma}})}})$ to ${\ensuremath{I}}_{i}$ (equivalently, $ {\pi}^s({\ensuremath{\mathrm{head}({\ensuremath{\sigma}})}})$ does not fold onto ${\ensuremath{I}}_{i}$). A *semi-oblivious (resp. restricted) chase sequence* of ${\ensuremath{I}}$ with ${\ensuremath{\Sigma}}$ is a possibly infinite ${\ensuremath{\Sigma}}$-derivation from ${\ensuremath{I}}$ such that each trigger $({\ensuremath{\sigma}}_i,{\pi}_i)$ in the derivation is active according to the semi-oblivious (resp. restricted) criterion.
Furthermore, a (possibly infinite) chase sequence is required to be *fair*, which means that a possible rule application is not indefinitely delayed. Formally, if some $I_i$ in the derivation admits an active trigger $({\ensuremath{\sigma}},{\pi})$, then there is $j > i$ such that, either $I_j$ is obtained by firing $({\ensuremath{\sigma}},{\pi})$ on $I_{j-1}$, or $({\ensuremath{\sigma}},{\pi})$ is not an active trigger anymore on $I_j$. A *terminating* chase sequence is a finite fair sequence.
In its original definition [[@DBLP:conf/pods/DeutschNR08], the *core* chase proceeds in a breadth-first manner, and, at each step, first fires in parallel all active triggers according to the restricted chase criterion, then computes the core of the result. Alternatively, to bring the definition of the core chase closer to the above definitions of the semi-oblivious and restricted chases,]{} one can define a *core chase sequence* as a possibly infinite sequence $I_0, ({\ensuremath{\sigma}}_1,{\pi}_1), I_1, \ldots$, alternating instances and triggers, such that each instance $I_i$ is obtained from $I_{i-1}$ by first firing the active trigger $({\ensuremath{\sigma}}_i, {\pi}_i)$ according to the restricted criterion, then computing the core of the result. An instance admits a terminating core chase sequence in that sense if and only if the core chase as originally defined terminates on that instance.
For the three chase variants, fair chase sequences compute a (possibly infinite) *universal model* of the KB, but only the core chase stops if and only if the KB has a *finite* universal model.
It is well-known that, for the semi-oblivious and the core chase, if there is a terminating chase sequence from an instance $I$ then all fair sequences from $I$ are terminating. This is not the case for the restricted chase, since the order in which rules are applied has an impact on termination, as illustrated by Example \[ex-intro\].
\[ex-intro\] Let ${\ensuremath{\Sigma}}= \{{\ensuremath{\sigma}}_1, {\ensuremath{\sigma}}_2\}$, with ${\ensuremath{\sigma}}_1 = p(x,y) \rightarrow \exists z ~ p(y,z)$ and ${\ensuremath{\sigma}}_2 = p(x,y) \rightarrow p(y,y)$. Let ${\ensuremath{I}}= p(a,b)$. The KB $({\ensuremath{I}}, {\ensuremath{\Sigma}})$ has a finite universal model, for example, $I^* = \{p(a,b), p(b,b)\}$. The semi-oblivious chase does not terminate on ${\ensuremath{I}}$ as ${\ensuremath{\sigma}}_1$ is applied indefinitely, while the core chase terminates after one breadth-first step and returns $I^*$. The restricted chase has a terminating sequence, for example, $({\ensuremath{\sigma}}_2, \{x \mapsto a, y \mapsto b\})$, which yields $I^*$ as well, but it also has infinite fair sequences, for example, the breadth-first sequence that applies ${\ensuremath{\sigma}}_1$ before ${\ensuremath{\sigma}}_2$ at each step.
We study the following problems for the semi-oblivious, restricted and core chase variants:
- *(All instance) all sequence termination:* Given a set of rules ${\ensuremath{\Sigma}}$, is it true that, for any instance, all fair sequences are terminating?
- *(All instance) one sequence termination:* Given a set of rules ${\ensuremath{\Sigma}}$, is it true that, for any instance, there is a terminating sequence?
Note that, according to the terminology of [@DBLP:journals/fuin/GrahneO18], these problems can be recast as deciding whether, for a chase variant, a given set of rules belongs to the class CT$_{\forall\forall}$ or CT$_{\forall\exists}$, respectively.
An existential rule is called *linear* if its body and its head are both composed of a single atom (e.g., [@DBLP:journals/ws/CaliGL12]). Linear rules generalize *inclusion dependencies* [@DBLP:journals/tods/Fagin81] by allowing several occurrences of the same variable in an atom. They also generalize positive inclusions in the description logic DL-Lite$_\mathcal R$ (the formal basis of the web ontological language OWL2 QL) [@DBLP:journals/jar/CalvaneseGLLR07], which can be seen as inclusion dependencies restricted to unary and binary predicates.
Note that the restriction of existential rules to rules with a single head is often made in the literature, considering that any existential rule with a complex head can be decomposed into several rules with a single head, by introducing a fresh predicate for each rule. However, while this translation preserves the termination of the semi-oblivious chase, it is not the case for the restricted and the core chases. Hence, considering linear rules with a complex head would require to extend the techniques developed in this paper.
To simplify the presentation, we assume in the following that each rule frontier is of size at least one. This assumption is made without loss of generality. [^1]
We first point out that the termination problem on linear rules can be recast by considering solely instances that contain a single atom (as already remarked in several contexts).
\[prop-atomic-instance\] Let ${\ensuremath{\Sigma}}$ be a linear set of rules. The semi-oblivious (resp. restricted, core) chase terminates on all instances if and only if it terminates on all singleton instances.
Obviously, the fact that a chase variant does not halt on an atomic instance implies the fact that it does not terminate on all instances. On the other direction, we can easily see that if the chase does not halt on an instance then it will not halt on one of its atoms. For a chase variant that does not terminate there exists an infinite derivation whose associated chase graph is also infinite. As the arity of the nodes in the chase graph is bounded by the size of the ruleset, the chase graph must contains an infinite path starting from a node of the initial instance. Because the chase graph for linear rules forms a tree it follows that this infinite path is created by a single atom of the initial instance.
We will furthermore rely on the following notion of the type of an atom.
\[definition-type\] The *type of an atom* ${\alpha}= r(t_1,\ldots, t_n)$, denoted by ${{\ensuremath{\textsl{type}}}{({\alpha})}}$, is the pair $(r,\mathcal{P})$ where $\mathcal{P}$ is the partition of $\{1,\ldots,n\}$ induced by term equality (i.e., $i$ and $j$ are in the same class of $\mathcal{P}$ iff ${\ensuremath{t}}_i = {\ensuremath{t}}_j$).
Note that there are finitely (more specifically, exponentially) many types for a given vocabulary.
If two atoms ${\alpha}$ and ${\alpha}'$ have the same type, then there is a *natural mapping* from ${\alpha}$ to ${\alpha}'$, denoted by $\varphi_{{\alpha}\rightarrow {\alpha}'}$, and defined as follows: it is a bijective mapping from ${{\ensuremath{\textsl{terms}}}{({\alpha})}}$ to ${{\ensuremath{\textsl{terms}}}{({\alpha}')}}$, that maps the $i$-th term of ${\alpha}$ to the $i$-th term of ${\alpha}'$. Note that $\varphi_{{\alpha}\rightarrow {\alpha}'}$ may not be an isomorphism, as constants from ${\alpha}$ may not be mapped to themselves. However, if $({\ensuremath{\sigma}},{\pi})$ is a trigger for $\{{\alpha}\}$, then $({\ensuremath{\sigma}},\varphi_{{\alpha}\rightarrow {\alpha}'}\circ{\pi})$ is a trigger for $\{{\alpha}'\}$, as there are no constants in the considered rules.
Together with Proposition \[prop-atomic-instance\], this implies that one can check all instance all sequence termination by checking all sequence termination on a finite set of instances, called *canonical instances*: for each type, there is exactly one canonical instance that has this type.
We will consider different kinds of tree structures, which have in common to be *trees of bags*: these are rooted trees, whose nodes, called *bags*, are labeled by an atom.[^2] We define the following notations for any node ${\ensuremath{B}}$ of a tree of bags ${\ensuremath{\mathcal{T}}}$:
- ${{\ensuremath{\textsl{atom}}}{({\ensuremath{B}})}}$ is the label of ${\ensuremath{B}}$;
- ${{\ensuremath{\textsl{terms}}}{({\ensuremath{B}})}} = {{\ensuremath{\textsl{terms}}}{({{\ensuremath{\textsl{atom}}}{({\ensuremath{B}})}})}}$ is the set of terms of $B$;
- ${{\ensuremath{\textsl{terms}}}{({\ensuremath{B}})}}$ is divided into two sets of terms, those *generated* in ${\ensuremath{B}}$, denoted by ${{\ensuremath{\textsl{generated}}}{({\ensuremath{B}})}}$, and those shared with its parent, denoted by ${{\ensuremath{\textsl{shared}}}{({\ensuremath{B}})}}$; precisely, ${{\ensuremath{\textsl{terms}}}{({\ensuremath{B}})}} = {{\ensuremath{\textsl{shared}}}{({\ensuremath{B}})}} \cup {{\ensuremath{\textsl{generated}}}{({\ensuremath{B}})}}$, ${{\ensuremath{\textsl{shared}}}{({\ensuremath{B}})}} \cap {{\ensuremath{\textsl{generated}}}{({\ensuremath{B}})}} = \emptyset$, and if ${\ensuremath{B}}$ is the root of ${\ensuremath{\mathcal{T}}}$, then ${{\ensuremath{\textsl{generated}}}{({\ensuremath{B}})}} = {{\ensuremath{\textsl{terms}}}{({\ensuremath{B}})}}$ (hence ${{\ensuremath{\textsl{shared}}}{({\ensuremath{B}})}} = \emptyset$), otherwise ${\ensuremath{B}}$ has a parent ${\ensuremath{B}}_p$ and ${{\ensuremath{\textsl{generated}}}{({\ensuremath{B}})}} = {{\ensuremath{\textsl{terms}}}{({\ensuremath{B}})}} \setminus {{\ensuremath{\textsl{terms}}}{({\ensuremath{B}}_p)}}$ (hence, ${{\ensuremath{\textsl{shared}}}{({\ensuremath{B}})}} = {{\ensuremath{\textsl{terms}}}{({\ensuremath{B}}_p)}} \cap {{\ensuremath{\textsl{terms}}}{({\ensuremath{B}})}}$).
We denote by ${{\ensuremath{\textsl{atoms}}}({\ensuremath{\mathcal{T}}})}$ the set of atoms that label the bags in ${\ensuremath{\mathcal{T}}}$.
Finally, we recall some classical mathematical notions. A *subsequence* $S'$ of a sequence $S$ is a sequence that can be obtained from $S$ by deleting some (or no) elements without changing the order of the remaining elements. The *arity* of a tree is the maximal number of children for a node. A *prefix* $T'$ of a tree $T$ is a tree that can be obtained from $T$ by repeatedly deleting some (or no) leaves of $T$.
Derivation Trees
================
A classical tool to reason about the chase is the so-called *chase graph* (see e.g., [@DBLP:journals/ws/CaliGL12]), which is the directed graph consisting of all atoms that appear in the considered derivation, and with an arrow from a node $n_1$ to a node $n_2$ iff $n_2$ is created by a rule application on $n_1$ and possibly other atoms. [^3] In the specific case of KBs of the form $(\{{\alpha}\}, {\ensuremath{\Sigma}})$, where ${\alpha}$ is an atom and ${\ensuremath{\Sigma}}$ is a set of linear rules, the chase graph is a tree. We recall below its definition in this specific case, in order to emphasize its differences with another tree, called *derivation tree*, on which we will actually rely.
\[definition-chase-graph\] Let ${\ensuremath{I}}$ be a singleton instance, ${\ensuremath{\Sigma}}$ be a set of linear rules and ${\ensuremath{I}}= I_0, ({\ensuremath{\sigma}}_1,{\pi}_1), I_1 \ldots, I_{n-1}, ({\ensuremath{\sigma}}_n,{\pi}_n), I_n$ be a semi-oblivious ${\ensuremath{\Sigma}}$-derivation from ${\ensuremath{I}}$. The *chase graph* (also called *chase tree*) assigned to $S$ is a tree of bags built as follows:
- the set of bags is in bijection with ${\ensuremath{I}}_n$ via the labeling function ${{\ensuremath{\textsl{atom}}}{()}}$;
- the set of edges is in bijection with the set of triggers in $S$ and is built as follows: for each trigger $({\ensuremath{\sigma}}_i,{\pi}_i)$ in $S$, there is an edge $({\ensuremath{B}},{\ensuremath{B}}')$ with ${{\ensuremath{\textsl{atom}}}{({\ensuremath{B}})}} = {{\pi}_i({\ensuremath{\mathrm{body}({\ensuremath{\sigma}}_i)}})}$ and ${{\ensuremath{\textsl{atom}}}{({\ensuremath{B}}')}} = {\pi}_i^s({\ensuremath{\mathrm{head}({\ensuremath{\sigma}}_i)}})$.
\[example-chase-graph-not-enough\] Let ${\ensuremath{I}}= q(a)$ and ${\ensuremath{\Sigma}}=\{{\ensuremath{\sigma}}_1,{\ensuremath{\sigma}}_2\}$ where ${\ensuremath{\sigma}}_1 = q(x) \rightarrow \exists y \exists z \exists t ~p(x,y,z,t)$ and ${\ensuremath{\sigma}}_2 = p(x,y,z,t) \rightarrow p(x,z,t,y)$. Let $S = {\ensuremath{I}},({\ensuremath{\sigma}}_1,{\pi}_1),{\ensuremath{I}}_1,({\ensuremath{\sigma}}_2,{\pi}_2),{\ensuremath{I}}_3,({\ensuremath{\sigma}}_2,{\pi}_3),{\ensuremath{I}}_3$ with ${\pi}_1 = \{ x \mapsto a\}$, ${\pi}_1^s({\ensuremath{\mathrm{head}({\ensuremath{\sigma}}_1)}}) = p(a,y_0,z_0,t_0)$, ${\pi}_2 = \{ x \mapsto a, y \mapsto y_0, z \mapsto z_0, t \mapsto t_0\}$ and ${{\pi}_3 = \{ x \mapsto a, y \mapsto z_0,}$ $ z \mapsto t_0, t \mapsto y_0\}$. The chase graph associated with $S$ is a path of four nodes as represented in Figure \[figure-chase-graph-not-enough\].
(126, 38) circle \[x radius= 35, y radius= 20\] ; (126.25, 104.5) circle \[x radius= 55.25, y radius= 24.5\] ; (126.25, 175.5) circle \[x radius= 55.25, y radius= 24.5\] ; (126.25, 246.5) circle \[x radius= 55.25, y radius= 24.5\] ; (126,58) – (126,80) ;
(126,129) – (126,151) ;
(126,200) – (126,222) ;
(366, 37) circle \[x radius= 35, y radius= 20\] ; (366.25, 103.5) circle \[x radius= 55.25, y radius= 24.5\] ; (293.25, 176.5) circle \[x radius= 55.25, y radius= 24.5\] ; (434.25, 176.5) circle \[x radius= 55.25, y radius= 24.5\] ; (366,57) – (366.25,79) ;
(366.25,128) – (293.25,152) ;
(366.25,128) – (434.25,152) ;
(126,37) node [$q( a)$]{}; (124,105) node [$p( a,y_{0} ,z_{0} ,t_{0})$]{}; (125,176) node [$p( a,z_{0} ,t_{0} ,y_{0})$]{}; (126,247) node [$p( a,t_{0} ,y_{0} ,z_{0})$]{}; (76,38) node [$B_{0}$]{}; (56,105) node [$B_{1}$]{}; (54,175) node [$B_{2}$]{}; (57,250) node [$B_{3}$]{}; (117,300) node \[align=left\] [Chase Graph]{}; (367,36) node [$q( a)$]{}; (365,104) node [$p( a,y_{0} ,z_{0} ,t_{0})$]{}; (292,177) node [$p( a,z_{0} ,t_{0} ,y_{0})$]{}; (433,177) node [$p( a,t_{0} ,y_{0} ,z_{0})$]{}; (317,37) node [$B_{0}$]{}; (297,104) node [$B_{1}$]{}; (221,176) node [$B_{2}$]{}; (505,176) node [$B_{3}$]{}; (364,299) node \[align=left\] [Derivation Tree]{};
To check termination of a chase variant on a given KB $(\{{\alpha}\}, {\ensuremath{\Sigma}})$, the general idea is to build a tree of bags associated with the chase on this KB in such a way that the occurrence of some forbidden pattern indicates that a path of unbounded length can be developed, hence the chase does not terminate. The forbidden pattern is composed of two distinct nodes such that one is an ancestor of the other and, intuitively speaking, these nodes “can be extended in similar ways”, which leads to an arbitrarily long path that repeats the pattern.
Two atoms with the same type admit the same rule triggers, however, within a derivation, the same rule applications cannot necessarily be performed on both of them because of the presence of other atoms (this is true already for datalog rules, since the same atom is never produced twice). Hence, on the one hand we will specialize the notion of type, into that of a *sharing type*, and, on the other hand, adopt another tree structure, called a *derivation tree*, in which two nodes with the same sharing type have the required similar behavior.
\[definition-sharing-type\] Given a tree of bags, the *sharing type* of a bag ${\ensuremath{B}}$ is a pair $ ({{\ensuremath{\textsl{type}}}{({{\ensuremath{\textsl{atom}}}{({\ensuremath{B}})}})}},P)$ where $P$ is the set of positions in ${{\ensuremath{\textsl{atom}}}{({\ensuremath{B}})}}$ in which a term of ${{\ensuremath{\textsl{shared}}}{({\ensuremath{B}})}}$ occurs. We denote the fact that two bags ${\ensuremath{B}}$ and ${\ensuremath{B}}'$ have the same sharing type by ${\ensuremath{B}}{\ensuremath{\equiv_{st}}}{\ensuremath{B}}'$. Furthermore, we say that two bags ${\ensuremath{B}}$ and ${\ensuremath{B}}'$ are *twins* if they have the same sharing type, the same parent ${\ensuremath{B}}_p$ and if the natural mapping $\varphi_{{{\ensuremath{\textsl{atom}}}{(B)}}\rightarrow{{\ensuremath{\textsl{atom}}}{(B')}}}$ is the identity on the terms of ${{\ensuremath{\textsl{atom}}}{({\ensuremath{B}}_p)}}$.
We can now specify the forbidden pattern that we will consider: it is a pair of two distinct nodes with the same sharing type, such that one is an ancestor of the other.
An *unbounded-path witness* (UPW) in a derivation tree is a pair of distinct bags $({\ensuremath{B}},{\ensuremath{B}}')$ such that ${\ensuremath{B}}$ and ${\ensuremath{B}}'$ have the same sharing type and ${\ensuremath{B}}$ is an ancestor of ${\ensuremath{B}}'$.
As explained below on Example \[example-chase-graph-not-enough\], the chase graph is not the appropriate tool to use this forbidden pattern as a witness of chase non-termination.
*Example \[example-chase-graph-not-enough\] (cont’d).* $B_1$, $B_2$ and $B_3$ have the same classical type,\
$t = (p, \{\{1\}, \{2\},\{3\},\{4\}\})$. The sharing type of $B_1$ is $(t,\{1\})$, while $B_2$ and $B_3$ have the same sharing type $(t,\{1,2,3,4\})$. $B_2$ and $B_3$ fulfill the condition of the forbidden pattern, however it is easily checked that any derivation that extends this derivation is finite.
Derivation trees were introduced as a tool to define the *greedy bounded treewidth set (gbts)* family of existential rules [@DBLP:conf/ijcai/BagetMRT11; @DBLP:phd/hal/Thomazo13]. A derivation tree is associated with a derivation, however it does not have the same structure as the chase graph. The fundamental reason is that, when a rule ${\ensuremath{\sigma}}$ is applied to an atom ${\alpha}$ via a homomorphism ${\pi}$, the newly created bag is not necessarily attached in the tree as a child of the bag labeled by ${\alpha}$. Instead, it is attached as a child of the *highest* bag in the tree labeled by an atom that contains ${\pi}({\ensuremath{\mathrm{fr}({\ensuremath{\sigma}})}})$, the image by ${\pi}$ of the frontier of ${\ensuremath{\sigma}}$ (note that ${\pi}({\ensuremath{\mathrm{fr}({\ensuremath{\sigma}})}})$ remains the set of terms shared between the new bag and its parent).
In the following definition, a derivation tree is not associated with *any* derivation, but with a semi-oblivious derivation, which has the advantage of yielding trees with bounded arity (Proposition \[prop-bounded-arity\] in the Appendix). This is appropriate to study the termination of the semi-oblivious chase, and later the restricted chase, as a restricted chase sequence is a specific semi-oblivious chase sequence.
Let ${\ensuremath{I}}= \{{\alpha}\}$ be a singleton instance, ${\ensuremath{\Sigma}}$ be a set of linear rules, and ${\ensuremath{S}}= {\ensuremath{I}}_0,({\ensuremath{\sigma}}_1,{\pi}_1),{\ensuremath{I}}_1, \ldots, ({\ensuremath{\sigma}}_n,{\pi}_n),{\ensuremath{I}}_n$ be a semi-oblivious ${\ensuremath{\Sigma}}$-derivation. The *derivation tree* assigned to ${\ensuremath{S}}$ is a tree of bags ${\ensuremath{\mathcal{T}}}$ built as follows:
- the root of the tree, ${\ensuremath{B}}_0$, is such that ${{\ensuremath{\textsl{atom}}}{({\ensuremath{B}}_0)}} = {\alpha}$;
- for each trigger $({\ensuremath{\sigma}}_i,{\pi}_i)$, $0 < i \leq n$, let ${\ensuremath{B}}_i $ be the bag such that ${{\ensuremath{\textsl{atom}}}{({\ensuremath{B}}_i)}} = {\ensuremath{{\pi}^s}}_{i}({\ensuremath{\mathrm{head}({\ensuremath{\sigma}}_{i})}})$. Let $j$ be smallest integer such that ${\pi}_{i}({\ensuremath{\mathrm{fr}({\ensuremath{\sigma}}_{i})}}) \subseteq {{\ensuremath{\textsl{terms}}}{({\ensuremath{B}}_j)}}$: ${\ensuremath{B}}_i$ is added as a child to ${\ensuremath{B}}_j$.
By extension, we say that a derivation tree $ {\ensuremath{\mathcal{T}}}$ is *associated with ${\alpha}$ and ${\ensuremath{\Sigma}}$* if there exists a semi-oblivious ${\ensuremath{\Sigma}}$-derivation ${\ensuremath{S}}$ from ${\alpha}$ such that ${\ensuremath{\mathcal{T}}}$ is assigned to ${\ensuremath{S}}$.
*Example \[example-chase-graph-not-enough\] (cont’d).* The derivation tree associated with $S$ is represented in Figure \[figure-chase-graph-not-enough\]. Bags have the same sharing types in the chase tree and in the derivation tree. However, we can see here that they are not linked in the same way: ${\ensuremath{B}}_3$ was a child of ${\ensuremath{B}}_2$ in the chase tree, it becomes a child of ${\ensuremath{B}}_1$ in the derivation tree. Hence, the forbidden pattern cannot be found anymore in the tree.
Note that every non-root bag ${\ensuremath{B}}$ shares a least one term with its parent (since the rule frontiers are not empty), furthermore this term is generated in its parent (otherwise ${\ensuremath{B}}$ would have been added at a higher level in the tree).
\[prop-bounded-arity\] The arity of a derivation tree is bounded.
We first point out that a bag has a bounded number of twin children. Since we consider semi-oblivious derivations, a bag ${\ensuremath{B}}_p$ cannot have two twin children ${\ensuremath{B}}_{c_1}$ and ${\ensuremath{B}}_{c_2}$, created by applications of the same rule ${\ensuremath{\sigma}}$. Indeed, although these rule applications may map ${\ensuremath{\mathrm{body}({\ensuremath{\sigma}})}}$ to distinct atoms, the associated homomorphisms, say ${\pi}_1$ and ${\pi}_2$, would have the same restriction to the rule frontier, i.e., ${\pi}_1{_{\mid{\ensuremath{\mathrm{fr}({\ensuremath{\sigma}})}}}} = {\pi}_2{_{\mid{\ensuremath{\mathrm{fr}({\ensuremath{\sigma}})}}}}$. Hence, all twin children of a bag come from applications of distinct rules. It follows that the arity of a node is bounded by the number of atom types $\times$ the cardinal of the ruleset.
Semi-Oblivious and Restricted Chase Termination
===============================================
We now use derivation trees and sharing types to characterize the termination of the semi-oblivous chase. The fundamental property of derivation trees that we exploit is that, when two nodes have the same sharing type, the considered (semi-oblivious) derivation can always be extended so that these nodes have the same number of children, and in turn these children have the same sharing type. We first specify the notion of *bag copy*.
Let ${\ensuremath{\mathcal{T}}},{\ensuremath{\mathcal{T}}}'$ be two (possibly equal) trees of bags. Let ${\ensuremath{B}}$ be a bag of ${\ensuremath{\mathcal{T}}}$ and ${\ensuremath{B}}'$ be a bag of ${\ensuremath{\mathcal{T}}}'$ such that ${\ensuremath{B}}{\ensuremath{\equiv_{st}}}{\ensuremath{B}}'$. Let ${\ensuremath{B}}_c$ be a child of ${\ensuremath{B}}$. A *copy* of ${\ensuremath{B}}_c$ *under* ${\ensuremath{B}}'$ is a bag ${\ensuremath{B}}'_c$ such that ${{\ensuremath{\textsl{atom}}}{({\ensuremath{B}}'_c)}} = \varphi^s({{\ensuremath{\textsl{atom}}}{({\ensuremath{B}}_c)}})$, where $\varphi^s$ is a substitution of ${{\ensuremath{\textsl{terms}}}{({\ensuremath{B}}_c)}}$ defined as follows:
- if ${\ensuremath{t}}\in {{\ensuremath{\textsl{shared}}}{({\ensuremath{B}}_c)}}$, then $\varphi^s({\ensuremath{t}}) = \varphi_{{{\ensuremath{\textsl{atom}}}{({\ensuremath{B}})}} \rightarrow {{\ensuremath{\textsl{atom}}}{({\ensuremath{B}}')}}}({\ensuremath{t}})$, where $\varphi_{{{\ensuremath{\textsl{atom}}}{({\ensuremath{B}})}} \rightarrow {{\ensuremath{\textsl{atom}}}{({\ensuremath{B}}')}}}$ is the natural mapping from ${{\ensuremath{\textsl{atom}}}{({\ensuremath{B}})}}$ to ${{\ensuremath{\textsl{atom}}}{({\ensuremath{B}}')}}$;
- if ${\ensuremath{t}}\in {{\ensuremath{\textsl{generated}}}{({\ensuremath{B}}_c)}}$, then $\varphi^s({\ensuremath{t}})$ is a fresh new [variable]{}.
Let ${\ensuremath{\mathcal{T}}}_e$ be obtained from a derivation tree ${\ensuremath{\mathcal{T}}}$ by adding a copy of a bag: strictly speaking, ${\ensuremath{\mathcal{T}}}_e$ may not be a derivation tree in the sense that there may be no derivation to which it can be assigned (intuitively, some rule applications that would allow to produce the copy may be missing). Rather, there is some derivation tree of which ${\ensuremath{\mathcal{T}}}_e$ is a *prefix* (intuitively, one can add bags to ${\ensuremath{\mathcal{T}}}_e$ to obtain a derivation tree). That is why the following proposition considers more generally prefixes of derivation trees.
\[proposition-sharing-type-children-derivation-tree\] Let ${\ensuremath{\mathcal{T}}}$ be a prefix of a derivation tree, ${\ensuremath{B}}$ and ${\ensuremath{B}}'$ be two bags of ${\ensuremath{\mathcal{T}}}$ such that ${\ensuremath{B}}{\ensuremath{\equiv_{st}}}{\ensuremath{B}}'$, and ${\ensuremath{B}}_c$ be a child of ${\ensuremath{B}}$. Then: *(a)* the tree obtained from ${\ensuremath{\mathcal{T}}}$ by adding the copy ${\ensuremath{B}}'_c$ of ${\ensuremath{B}}_c$ under ${\ensuremath{B}}'$ is a prefix of a derivation tree, and *(b)* it holds that ${\ensuremath{B}}_c {\ensuremath{\equiv_{st}}}{\ensuremath{B}}'_c$.
Let ${\ensuremath{B}}$ and ${\ensuremath{B}}'$ be two atoms of ${\ensuremath{\mathcal{T}}}$ having the same sharing type. Let ${\ensuremath{B}}_c$ be a child of ${\ensuremath{B}}$ created by a trigger $({\ensuremath{\sigma}},{\pi})$. By definition of derivation tree, ${\pi}$ maps the rule frontier ${\ensuremath{\mathrm{fr}({\ensuremath{\sigma}})}}$ to ${{\ensuremath{\textsl{terms}}}{({\ensuremath{B}})}}$, without this being possible for the parent of ${\ensuremath{B}}$. Furthermore, we know that ${\pi}$ maps ${\ensuremath{\mathrm{body}({\ensuremath{\sigma}})}}$ to a (possibly strict) descendant of ${\ensuremath{B}}$. We assume that ${\ensuremath{\mathcal{T}}}$ does not already contain the image of ${\ensuremath{\mathrm{head}({\ensuremath{\sigma}})}}$ via ${\pi}$, otherwise the thesis trivially holds. Let ${\ensuremath{S}}$ be the derivation associated with ${\ensuremath{\mathcal{T}}}$ and $\alpha_0,\dots,\alpha_{k}$ be the path of the *chase-graph* associated with ${\ensuremath{S}}$ such that $\alpha_0={{\ensuremath{\textsl{atom}}}{({\ensuremath{B}})}}$ and $\alpha_{k}={{\ensuremath{\textsl{atom}}}{({\ensuremath{B}}_{c})}}$, whose sequence of associated rule applications is $({\ensuremath{\sigma}}_1,{\pi}_1),\dots, ({\ensuremath{\sigma}}_{k},{\pi}_{k})=({\ensuremath{\sigma}},{\pi})$. We define $\hat{\pi}_i^{\mathrm{safe}}(t)=\varphi_{{{\ensuremath{\textsl{atom}}}{({\ensuremath{B}})}}\rightarrow{{\ensuremath{\textsl{atom}}}{({\ensuremath{B}}')}}}\circ{\pi}_i(t)$ whenever ${\pi}_i(t)\in{{\ensuremath{\textsl{terms}}}{({\ensuremath{B}})}}$ and otherwise $\hat{\pi}_i^{\mathrm{safe}}(t)$ to be a fresh new variable consistently used over the rule applications, that is, such that ${\pi}_i^{\mathrm{safe}}(t)={\pi}_j^{\mathrm{safe}}(t)$ if and only if $\hat{\pi}_i^{\mathrm{safe}}(t)=\hat{\pi}_j^{\mathrm{safe}}(t)$. Then, for all $1\leq i \leq k$, we extend ${\ensuremath{S}}$ by adding a trigger $({\ensuremath{\sigma}}_i,\hat{\pi}_i)$[^4] whenever $\hat{\pi}_i^{\mathrm{safe}}({\ensuremath{\mathrm{head}({\ensuremath{\sigma}}_i)}})$ is not an atom already produced by ${\ensuremath{S}}$ thereby obtaining a new derivation ${\ensuremath{S}}'$. Let ${\ensuremath{\mathcal{T}}}'$ be an extension of ${\ensuremath{\mathcal{T}}}$ where a bag labeled with the atom $\hat{\pi}_i^{\mathrm{safe}}({\ensuremath{\mathrm{head}({\ensuremath{\sigma}}_i)}})$ is added for each new trigger in ${\ensuremath{S}}'$ and attached to the highest descendant of ${\ensuremath{B}}'$ whose set of terms contains $\hat{\pi}_i^{}({\ensuremath{\mathrm{fr}({\ensuremath{\sigma}}_i)}})$. Clearly, ${\ensuremath{\mathcal{T}}}'$ is a derivation tree associated with ${\ensuremath{S}}'$. We now show that ${\ensuremath{\mathcal{T}}}'$ contains a node ${\ensuremath{B}}_c'$ which is a copy of ${\ensuremath{B}}_c$ under ${\ensuremath{B}}'$. As ${\ensuremath{B}}$ is the parent of ${\ensuremath{B}}_c$, the image of ${\ensuremath{\mathrm{fr}({\ensuremath{\sigma}})}}$ via ${\pi}$ contains at least one term which is generated in ${\ensuremath{B}}$ (and in general only terms generated by the ancestors of ${\ensuremath{B}}$). Therefore, because ${\ensuremath{B}}$ and ${\ensuremath{B}}'$ have the same sharing type, the image of ${\ensuremath{\mathrm{fr}({\ensuremath{\sigma}})}}$ via $\varphi_{{{\ensuremath{\textsl{atom}}}{({\ensuremath{B}})}}\rightarrow {{\ensuremath{\textsl{atom}}}{({\ensuremath{B}}')}}}\circ{\pi}$ contains at least one term generated in ${\ensuremath{B}}'$ (and in general only terms generated by the ancestors of ${\ensuremath{B}}'$). So, ${\ensuremath{B}}'$ is the only possible parent of ${\ensuremath{B}}'_c$ in ${\ensuremath{\mathcal{T}}}'$. Moreover, it is easy to see that ${\ensuremath{B}}_c {\ensuremath{\equiv_{st}}}{\ensuremath{B}}'_c$. Let ${\ensuremath{\mathcal{T}}}''$ be the extension of ${\ensuremath{\mathcal{T}}}$ with ${\ensuremath{B}}_c'$ under ${\ensuremath{B}}'$. It can be easily verified that ${\ensuremath{\mathcal{T}}}''$ is a prefix of the derivation tree ${\ensuremath{\mathcal{T}}}'$, in the sense that it is a tree of bags which can be obtained by recursively removing some of the leaves of ${\ensuremath{\mathcal{T}}}'$, i.e., those corresponding to the triggers in ${\ensuremath{S}}'\setminus{\ensuremath{S}}$ which are different from $({\ensuremath{\sigma}},{\pi})$.
The size of a derivation tree without UPW is bounded, since its arity is bounded (Proposition \[prop-bounded-arity\] in the Appendix) and its depth is bounded by the number of sharing types. It remains to show that a derivation tree that contains a UPW can be extended to an arbitrarily large derivation tree. We recall that similar property would not hold for the chase tree, as witnessed by Example \[example-chase-graph-not-enough\].
\[proposition-finiteness-derivation-tree\] There exists an arbitrary large derivation tree associated with ${\alpha}$ and ${\ensuremath{\Sigma}}$ if and only if there exists a derivation tree associated with ${\alpha}$ and ${\ensuremath{\Sigma}}$ that contains an unbounded-path witness.
If there is no derivation tree having an unbounded-path witness, then the depth of all derivation trees is upper bounded by the number of sharing types. As derivation trees are of bounded arity, all derivation trees must be of bounded size.
If there is a derivation tree ${\ensuremath{\mathcal{T}}}$ having an unbounded-path witness $({\ensuremath{B}},{\ensuremath{B}}')$, we show that there are arbitrary large derivation trees. We do so by contradiction. Let us assume that $({\ensuremath{B}},{\ensuremath{B}}')$ is a UPW be two such bags such that ${\ensuremath{B}}'$ is of maximal depth among all such pairs and among all trees, which by hypothesis are of bounded size. Let ${\ensuremath{B}}_c$ be the child of ${\ensuremath{B}}$ that is on the shortest path from ${\ensuremath{B}}$ to ${\ensuremath{B}}'$ (possibly ${\ensuremath{B}}_c = {\ensuremath{B}}'$). By Proposition \[proposition-sharing-type-children-derivation-tree\], ${\ensuremath{B}}'$ has a child ${\ensuremath{B}}'_c$ that has the same sharing type as ${\ensuremath{B}}_c$. By Proposition \[proposition-sharing-type-children-derivation-tree\], ${\ensuremath{B}}'$ has a child ${\ensuremath{B}}'_c$ that has the same sharing type as ${\ensuremath{B}}_c$, either in the same tree, or in an extension of this tree, which is in contradiction with the fact that ${\ensuremath{B}}'$ was of maximal depth. Hence there are arbitrary large derivation trees.
The previous proposition yields a characterization of the existence of an infinite semi-oblivious derivation. At this point, one may notice that an infinite semi-oblivious derivation is not necessarily fair. However, from this infinite derivation one can always build a fair derivation by inserting missing triggers. Obviously, this operation has no effect on the termination of the semi-oblivious chase. More precaution will be required for the restricted chase.
One obtains an algorithm to decide termination of the semi-oblivious chase for a given set of rules: for each canonical instance, build a semi-oblivious derivation and the associated derivation tree by applying rules until a UPW is created (in which case the answer is no) or all possible rule applications have been performed; if no instance has returned a negative answer, the answer is yes.
\[corollary-semi-oblivious-finiteness-decidability\] The all-sequence termination problem for the semi-oblivious chase on linear rules is decidable.
This algorithm can be modified to run in polynomial space (which is optimal [@DBLP:conf/pods/CalauttiGP15]), by guessing a canonical instance and a UPW of its derivation tree.
The all-sequence termination problem for the semi-oblivious chase on linear rules is in <span style="font-variant:small-caps;">PSpace</span>.
Let ${\ensuremath{\mathcal{T}}}$ be a derivation tree of root the canonical instance $\{{\alpha}\}$ that contains a UPW $({\ensuremath{B}},{\ensuremath{B}}')$, where the sharing type of both bags is $ST$. We show that there exists a semi-oblivious derivation of length at most exponential whose derivation tree has root $\{{\alpha}\}$ and that contains a UPW $({\ensuremath{B}}_s,{\ensuremath{B}}'_s)$ where the sharing type of both bags is $ST$. First, by Proposition \[proposition-sharing-type-children-derivation-tree\], we conclude that it is not necessary to have twice the same sharing type on the path from the root to ${\ensuremath{B}}'$ in the derivation tree. It is thus enough to show that to generate a child ${\ensuremath{B}}_c$ from its parent ${\ensuremath{B}}_p$, a derivation of length at most exponential is necessary. Let us consider the chase graph of the derivation generating ${{\ensuremath{\textsl{atom}}}{({\ensuremath{B}}_c)}}$ from ${{\ensuremath{\textsl{atom}}}{({\ensuremath{B}}_p)}}$. This chase graph can be assumed w.l.o.g. to be a path. It there are no pairs of atoms having the same sharing type on this path, then the derivation is of length at most exponential. Otherwise, we show that we can build a shorter semi-oblivious derivation that generates ${{\ensuremath{\textsl{atom}}}{({\ensuremath{B}}_c)}}$. Let us thus assume that there is ${\ensuremath{B}}$ and ${\ensuremath{B}}'$ such that both have the same sharing type, and the terms of ${\ensuremath{B}}_p$ that appear in ${\ensuremath{B}}$ appear in the same position in ${\ensuremath{B}}'$, and that ${\ensuremath{B}}'$ is on the path from ${\ensuremath{B}}$ to ${\ensuremath{B}}_c$ in the chase graph. A derivation similar to that applicable after ${\ensuremath{B}}'$ is actually applicable to ${\ensuremath{B}}$, by Proposition \[proposition-sharing-type-children-derivation-tree\]. A copy of ${\ensuremath{B}}_c$ under ${\ensuremath{B}}_p$ is thus generated by this derivation, which proves our claim.
We now describe the algorithm. We guess the canonical instance and the sharing type $ST$ of the UPW. We then check that there is a descendant (not necessarily a child) of that canonical instance that has sharing type $ST$. This can be done by guessing the shortest derivation creating a bag of sharing type $ST$. It is only necessary to remember the sharing type of the “current” bag, as we know that any bag created during a derivation is added as a descendant of the root. We then want to prove that a bag of sharing type $ST$ can have a (strict) descendant of sharing type $ST$. In contrast with the case of the root, a trigger applied below a bag ${\ensuremath{B}}$ does not necessarily create a bag that is as well below ${\ensuremath{B}}$ – it could be added higher up in the tree. We thus have to remember the shared variables of ${\ensuremath{B}}$, and verify at each step that the shared variables of the currently considered bag are not a subset of them. This leads to a <span style="font-variant:small-caps;">PSpace</span> procedure.
We now consider the restricted chase. To this aim, we call *restricted derivation tree* associated with ${\alpha}$ and ${\ensuremath{\Sigma}}$ a derivation tree associated with a restricted ${\ensuremath{\Sigma}}$-derivation from ${\alpha}$. We first point out that Proposition \[proposition-sharing-type-children-derivation-tree\] is not true anymore for a restricted derivation tree, as the order in which rules are applied matters.
Consider a restricted tree that contains bags $B$ and $B'$ with the same sharing type, labeled by atoms $q(t,u)$ and $q(v,w)$ respectively, where the second term is generated. Consider the following rules (the same as in Example \[ex-intro\]):\
${\ensuremath{\sigma}}_1: q(x,y) \rightarrow \exists z ~q(y,z)$\
${\ensuremath{\sigma}}_2: q(x,y) \rightarrow q(y,y)$\
Assume ${\ensuremath{B}}$ has a child ${\ensuremath{B}}_c$ labeled by $q(u,z_0)$ obtained by an application of ${\ensuremath{\sigma}}_1$, and ${\ensuremath{B}}'$ has a child ${\ensuremath{B}}'_1$ labeled by $q(w,w)$ obtained by an application of ${\ensuremath{\sigma}}_2$. It is not possible to extend this tree by copying ${\ensuremath{B}}_c$ under ${\ensuremath{B}}'$. Indeed, the corresponding application of ${\ensuremath{\sigma}}_1$ does not comply with the restricted chase criterion: it would produce an atom of the form $q(w,z_1)$ that folds into $q(w,w)$.
We thus prove a weaker proposition by considering that ${\ensuremath{B}}'$ is a leaf in the restricted derivation tree.
\[proposition-sharing-type-restricted-derivation-tree\] Let ${\ensuremath{\mathcal{T}}}$ be a prefix of a restricted derivation tree, ${\ensuremath{B}}$ and ${\ensuremath{B}}'$ be two bags of ${\ensuremath{\mathcal{T}}}$ such that ${\ensuremath{B}}{\ensuremath{\equiv_{st}}}{\ensuremath{B}}'$ and *${\ensuremath{B}}'$ is a leaf*. Let ${\ensuremath{B}}_c$ be a child of ${\ensuremath{B}}$. Then: *(a)* the tree obtained from ${\ensuremath{\mathcal{T}}}$ by adding the copy ${\ensuremath{B}}'_c$ of ${\ensuremath{B}}_c$ under ${\ensuremath{B}}'$ is a prefix of a restricted derivation tree, and *(b)* it holds that ${\ensuremath{B}}_c {\ensuremath{\equiv_{st}}}{\ensuremath{B}}'_c$.
Let $S$ be the restricted derivation associated with ${\ensuremath{\mathcal{T}}}$. Let $S_c$ be the subsequence of $S$ that starts from ${\ensuremath{B}}$ and produces the strict descendants of ${\ensuremath{B}}$. Obviously, any rule application in $S_c$ is performed on a descendant of ${\ensuremath{B}}$, hence we do not care about rule applications that produce bags that are not descendants of ${\ensuremath{B}}$. We prove the property by induction on the length of $S_c$. If $S_c$ is empty, the property holds with ${\ensuremath{\mathcal{T}}}_e = {\ensuremath{\mathcal{T}}}$. Assume the property is true for $0 \leq |S_c| \leq k$. Let $|S_c| = k + 1$. By induction hypothesis, there is an extension ${\ensuremath{\mathcal{T}}}' $ of ${\ensuremath{\mathcal{T}}}$ such that the subtree of ${\ensuremath{B}}$ restricted to the first $k$ elements of $S_c$ is ‘quasi-isomorphic’ to the subtree rooted in ${\ensuremath{B}}'$ (via a bijective substitution defined by the natural mappings between sharing types, say $\phi$) . Let $({\ensuremath{\sigma}},{\pi})$ be the last trigger of $S_c$, and assume it applies to a bag ${\ensuremath{B}}_d$. In ${\ensuremath{\mathcal{T}}}'$, there is a bag ${\ensuremath{B}}'_d = \phi({\ensuremath{B}}_d)$. Hence, ${\ensuremath{\sigma}}$ can be applied to ${\ensuremath{B}}'_d$ with the homomorphism $\phi \circ {\pi}$. Any folding of the produced bag ${\ensuremath{B}}''$ to a bag in ${\ensuremath{\mathcal{T}}}'$ necessarily maps ${\ensuremath{B}}''$ to a bag in the subtree rooted in ${\ensuremath{B}}'_d$ (because ${\ensuremath{B}}'_d$ and ${\ensuremath{B}}''$ share a term generated in ${\ensuremath{B}}'_d$, that only occurs in the subtree rooted in ${\ensuremath{B}}'_d$ and remains invariant by the folding). Since ${\ensuremath{B}}_d$ and ${\ensuremath{B}}'_d$ have quasi-isomorphic subtrees, and $({\ensuremath{\sigma}},{\pi})$ satisfies the restricted chase criterion, so does $({\ensuremath{\sigma}},\phi \circ {\pi})$. Furthermore, the quasi-isomorphism $\phi$ preserves the sharing types. Hence, ${\ensuremath{B}}'_d$ is added exactly like the bag produced by $({\ensuremath{\sigma}},{\pi})$. We conclude that the property holds true at rank $k+1$.
The previous proposition allows us to obtain a variant of Proposition \[proposition-finiteness-derivation-tree\] adapted to the restricted chase:
\[proposition-finiteness-restricted-derivation-tree\] There exists an arbitrary large restricted derivation tree associated with ${\alpha}$ and ${\ensuremath{\Sigma}}$ if and only if there exists a restricted derivation tree associated with ${\alpha}$ and ${\ensuremath{\Sigma}}$ that contains an unbounded-path witness.
If there is no restricted derivation tree with a UPW, then the size of any restricted derivation tree is bounded since a restricted derivation tree is a derivation tree. We prove the other direction by contradiction. Assume that the size of restricted derivation trees is bounded whereas the forbidden pattern occurs in some of them. Consider a restricted chase sequence $S$ with associated restricted derivation tree ${\ensuremath{\mathcal{T}}}$ that contains a UPW $({\ensuremath{B}},{\ensuremath{B}}')$ of maximal depth among all such pairs and all trees, and such that ${\ensuremath{B}}'$ is a leaf (we can do the latter assumption since the prefix of any restricted derivation is a restricted derivation).
Let ${\ensuremath{B}}_c$ be the child of ${\ensuremath{B}}$ that is on the shortest path from ${\ensuremath{B}}$ to ${\ensuremath{B}}'$ (possibly ${\ensuremath{B}}_c = {\ensuremath{B}}'$). By Proposition \[proposition-sharing-type-restricted-derivation-tree\], there is a restricted derivation tree that extends ${\ensuremath{\mathcal{T}}}$ such that ${\ensuremath{B}}'$ has a child ${\ensuremath{B}}'_c$ of the same sharing type as ${\ensuremath{B}}_c$, hence $({\ensuremath{B}}_c, {\ensuremath{B}}'_c)$ is a UPW of depth strictly greater than $({\ensuremath{B}},{\ensuremath{B}}')$, which contradicts the hypothesis.
It is less obvious than in the case of the semi-oblivious chase that the existence of an infinite derivation entails the existence of an infinite *fair* derivation. However, this property still holds:
For linear rules, every (infinite) non-terminating restricted derivation is a subsequence of a fair restricted derivation.
Let ${\ensuremath{S}}$ be a non-terminating restricted derivation. In particular, there exists a least one infinite branch in the associated derivation tree. Let us consider the following derivation: when the node ${\ensuremath{B}}_k$ of depth $k$ on this branch has been generated, complete the corresponding subsequence by trying to apply (i.e., while respecting the restricted criterion) all currently applicable triggers that add a bag a depth at most $k-1$. These additional rule applications cannot prevent the creation of any bag that is below ${\ensuremath{B}}_k$ in the derivation tree. Indeed, let ${\alpha}_c$ be an atom possibly created by a rule application, whose bag would be attached as a child of a bag ${\ensuremath{B}}$; since ${\alpha}_c$ shares a variable with ${{\ensuremath{\textsl{atom}}}{({\ensuremath{B}})}}$ that is generated in $B$, which thus only occurs in the subtree of $B$, the only possibility for ${\alpha}_c$ to fold into the current instance, is to be mapped to an atom in the subtree of ${\ensuremath{B}}$. By construction, any possible rule application will be performed or inhibited at some point, which implies that the derivation that we build in this fashion is fair.
Similarly to Proposition \[proposition-finiteness-derivation-tree\] for the semi-oblivious chase, Proposition \[proposition-finiteness-restricted-derivation-tree\] provides an algorithm to decide termination of the restricted chase. The difference is that it is not sufficient to build a single derivation for a given canonical instance; instead, all possible restricted derivations from this instance have to be built (note that the associated restricted derivation trees are finite for the same reasons as before, and there is obviously a finite number of them). Hence, we obtain:
\[corollary-restricted-finiteness-decidability\] The all-sequence termination problem for the restricted chase on linear rules is decidable.
A rough analysis of the proposed algorithm provides a <span style="font-variant:small-caps;">co-N2ExpTime</span> upper-bound for the complexity of the problem, by guessing a derivation that is of length at most double exponential, and checking whether there is a UPW in the corresponding derivation tree.
(426, 38) circle \[x radius= 55, y radius= 25\] ;
(426,38) node [$p(x,y)$]{};
(376,67) node [$1$]{};
(426,63) – (346,85) ;
(346, 110) circle \[x radius= 55, y radius= 25\] ;
(346,110) node [$q(y)$]{};
(476,67) node [$3$]{};
(426,63) – (506,85) ;
(506, 110) circle \[x radius= 55, y radius= 25\] ;
(506,110) node [$r(y,x)$]{};
(336,148) node [$2$]{};
(346,135) – (346,157) ;
(346, 182) circle \[x radius= 55, y radius= 25\] ;
(346,182) node [$r(y,z_0)$]{};
(336,220) node [$4$]{};
(346,207) – (346,229) ;
(346, 254) circle \[x radius= 55, y radius= 25\] ;
(346,254) node [$p(z_0,z_1)$]{};
(288,282) node [$5$]{};
(346,279) – (266,299) ;
(256,320) node [$6$]{};
(266,299) – (266,329) ;
(402,282) node [$7$]{};
(346,279) – (426,299) ;
(126, 38) circle \[x radius= 55, y radius= 25\] ;
(126,38) node [$p(x,y)$]{};
(76,67) node [$1$]{};
(126,63) – (46,85) ;
(46, 110) circle \[x radius= 55, y radius= 25\] ;
(46,110) node [$q(y)$]{};
(176,67) node [$2$]{};
(126,63) – (206,85) ;
(206, 110) circle \[x radius= 55, y radius= 25\] ;
(206,110) node [$r(y,x)$]{};
Importantly, the previous algorithm is naturally able to consider solely some type of restricted derivations, i.e., build only derivation trees associated with such derivations, which is of theoretical but also of practical interest. Indeed, implementations of the restricted chase often proceed by building *breadth-first* sequences (which are intrinsically fair), or variants of these. As witnessed by the next example, the termination of all breadth-first sequences is a strictly weaker requirement than the termination of all fair sequences, in the sense that the restricted chase terminates on more sets of rules.
\[ex-bfs-stop\]Consider the following set of rules:\
${\ensuremath{\sigma}}_1 = p(x,y) \rightarrow q(y) \quad \quad \quad ~{\ensuremath{\sigma}}_2 = p(x,y) \rightarrow r(y,x)$\
$ {\ensuremath{\sigma}}_3 = q(y) \rightarrow \exists z ~r(y,z) \quad \quad {\ensuremath{\sigma}}_4 = r(x,y) \rightarrow \exists z ~p(y,z)$\
All breadth-first restricted derivations terminate, whatever the initial instance is. Remark that every application of ${\ensuremath{\sigma}}_1$ is followed by an application of ${\ensuremath{\sigma}}_2$ in the same breadth-first step, which prevents the application of ${\ensuremath{\sigma}}_3$. However, there is a fair restricted derivation that does not terminate (and this is even true for any instance). Indeed, an application of ${\ensuremath{\sigma}}_2$ can always be delayed, so that it comes too late to prevent the application of ${\ensuremath{\sigma}}_3$. See Figure \[figure-infinite-bf-derivation\]: on the left, a finite derivation tree associated with a breadth-first derivation from instance $p(x,y)$; on the right, an infinite derivation tree associated with a (non breadth-first) fair infinite derivation from the same instance. The numbers on edges give the order in which bags are created.
We now prove the decidability of the one-sequence termination problem, building on the same objects as before, but in a different way. Indeed, a (restricted) derivation tree ${\ensuremath{\mathcal{T}}}$ that contains a UPW $(B,B')$ is a witness of the existence of an infinite (restricted fair) derivation, but does not prove that *every* (restricted fair) derivation that extends ${\ensuremath{\mathcal{T}}}$ is infinite. To decide, we will consider trees associated with a *sharing type* instead of a type. A derivation tree associated with a sharing type $T$ has a root bag whose sharing type is $T$, and is built as for usual root bags, except that shared terms are taken into account, i.e., triggers $({\ensuremath{\sigma}}, {\pi})$ such that ${\pi}({\ensuremath{\mathrm{fr}({\ensuremath{\sigma}})}}) \subseteq {{\ensuremath{\textsl{shared}}}{(T)}}$ are simply ignored. The algorithm proceeds as follows:
1. For each sharing type $T$, generate all restricted derivations trees associated with $T$, stopping the construction of a tree when, for each leaf $B_L$, either there is no active trigger on ${{\ensuremath{\textsl{atom}}}{(B_L)}}$ or $B_L$ forms a UPW with one of its ancestors.
2. Mark all the sharing types that have at least one associated tree without UPW.
3. Propagate the marks until stability: if a sharing type $T$ has at least one tree for which all UPWs $(B,B')$ are such that the sharing type of $B$ is marked, then mark $T$.
4. If all sharing types that correspond to instances (i.e., without shared terms) are marked, return *yes*, otherwise return *no*.
The previous algorithm terminates and returns yes if and only if there is a terminating restricted sequence.
(Sketch) Termination follows from the finiteness of the set of sharing types and the bound on the size of a tree. Concerning the correctness of the algorithm, we show that a terminating restricted derivation cannot have a derivation tree that contains an unmarked UPW, i.e., whose associated sharing type is not marked. By contradiction: assume there is a terminating restricted derivation whose derivation tree contains an unmarked UPW; consider such an unmarked UPW $(B,B')$ such that $B'$ is of maximal depth in the tree. The subtree of $B'$ necessarily admits as prefix one of the restricted derivation trees associated with the sharing type of $B'$ built by the algorithm, otherwise the derivation would not be fair. Moreover, since the sharing type of $B'$ is not marked, this prefix contains an unmarked UPW. Hence, the tree contains an unmarked UPW $(B'',B''')$ with $B'''$ of depth strictly greater than the depth of $B'$, which contradicts the hypothesis.
\[corollary-one-sequence-finiteness-decidability\] The one-sequence termination problem for the restricted chase on linear rules is decidable.
By guessing a terminating restricted derivation, which must be of size at most double exponential, and checking that the obtained instance is indeed a universal model, we obtain a <span style="font-variant:small-caps;">N2ExpTime</span> upper bound for the complexity of the one-sequence termination problem.
We conclude this section by noting that the previous Example \[ex-bfs-stop\] may give the (wrong) intuition that, given a set of rules, it is sufficient to consider breadth-first sequences to decide if there exists a terminating sequence. The following example shows that it is not the case: here, no breadth-first sequence is terminating, while there exists a terminating sequence for the given instance.
\[ex-bfs-non-stop-bis\] Let ${\ensuremath{\Sigma}}=\{{\ensuremath{\sigma}}_1,{\ensuremath{\sigma}}_2,{\ensuremath{\sigma}}_3\}$ with ${\ensuremath{\sigma}}_1 = p(x,y) \rightarrow \exists z ~p(y,z) $, ${\ensuremath{\sigma}}_2 = p(x,y) \rightarrow h(y)$, and ${\ensuremath{\sigma}}_3= h(x) \rightarrow ~p(x,x)$. In this case, for every instance, there is a terminating restricted chase sequence, where the application of ${\ensuremath{\sigma}}_2$ and ${\ensuremath{\sigma}}_3$ prevents the indefinite application of ${\ensuremath{\sigma}}_1$. However, starting from ${\ensuremath{I}}= \{p(a,b)\}$, by applying rules in a breadth-first fashion one obtains a non-terminating restricted chase sequence, since ${\ensuremath{\sigma}}_1$ and ${\ensuremath{\sigma}}_2$ are always applied in parallel from the same atom, before applying $ {\ensuremath{\sigma}}_3$.
As for the all-sequence termination problem, the algorithm may restrict the derivations of interest to specific kinds.
Core Chase Termination
======================
We now consider the termination of the core chase of linear rules. Keeping the same approach, we prove that the finiteness of the core chase is equivalent to the existence of a finite tree of bags whose set of atoms is a minimal universal model. We call this a *(finite) complete core*. To bound the size of a complete core, we show that it cannot contain an unbounded-path witness. Note that in the binary case, it would be possible to work again on derivation trees, but this is not true anymore for arbitrary arity. Indeed, as shown in Example \[example-mlm\], there are linear sets of rules for which no derivation tree form a complete core (while it holds for binary rules). We thus introduce a more general tree structure, namely *entailment trees*.
\[example-mlm\] Let us consider the following rules:
-------------------------------------------------- --------------------------------------------
$s(x) \rightarrow \exists y \exists z ~p(y,z,x)$ $p(y,z,x) \rightarrow \exists v ~q(y,v,x)$
$q(y,v,x) \rightarrow p(y,v,x)$
-------------------------------------------------- --------------------------------------------
Let ${\ensuremath{I}}= \{s(a)\}$. The first rule applications yield a derivation tree ${\ensuremath{\mathcal{T}}}$ which is a path of bags $B_0, B_1,B_2,B_3$ respectively labeled by the following atoms:\
$s(a), p(y_0,z_0,a), q(y_0,v_0,a)$ and $p(y_0, v_0,a)$. ${\ensuremath{\mathcal{T}}}$ is represented on the left of Figure \[figure-example-mlm\]. Let $A$ be this set of atoms. First, note that $ A$ is not a core: indeed it is equivalent to its strict subset $A'$ defined by $\{B_0, B_2, B_3\}$ with a homomorphism ${\pi}$ that maps ${{\ensuremath{\textsl{atom}}}{(B_1)}}$ to ${{\ensuremath{\textsl{atom}}}{(B_3)}}$. Trivially, $A'$ is a core since it does not contain two atoms with the same predicate. Second, note that any further rule application on ${\ensuremath{\mathcal{T}}}$ is redundant, i.e., generates a set of atoms equivalent to $A$ (and $A'$). Hence, $A'$ is a complete core, however there is no derivation tree that corresponds to it. There is even no *prefix* of a derivation tree that corresponds to it (which ruins the alternative idea of building a prefix of a derivation tree that would be associated with a complete core). In particular, note that $\{B_0, B_1, B_2\}$ is indeed a core, but it is not complete.
(126, 38) circle \[x radius= 35, y radius= 20\] ; (126.25, 104.5) circle \[x radius= 55.25, y radius= 24.5\] ; (126.25, 175.5) circle \[x radius= 55.25, y radius= 24.5\] ; (126.25, 246.5) circle \[x radius= 55.25, y radius= 24.5\] ; (126,58) – (126,80) ;
(126,129) – (126,151) ;
(126,200) – (126,222) ;
(68.1,229.09) .. controls (6.16,198.84) and (14.56,141.8) .. (70,122) ;
(10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29) ; (375, 76) circle \[x radius= 35, y radius= 20\] ; (375.25, 143.5) circle \[x radius= 55.25, y radius= 24.5\] ; (375, 214.5) circle \[x radius= 55.25, y radius= 24.5\] ; (375,96) – (375,118) ;
(375,168) – (375,190) ;
(126,37) node [$s( a)$]{}; (124,105) node [$p( y_{0} ,z_{0} ,a)$]{}; (125,176) node [$q( y_{0} ,v_{0} ,a)$]{}; (126,247) node [$p( y_{0} ,v_{0} ,a)$]{}; (76,38) node [$B_{0}$]{}; (56,105) node [$B_{1}$]{}; (54,175) node [$B_{2}$]{}; (57,250) node [$B_{3}$]{}; (117,300) node \[align=left\] [Derivation Tree]{}; (364,299) node \[align=left\] [An Entailment Tree]{}; (15,170) node [$\varphi $]{}; (372,75) node [$s( a)$]{}; (374,144) node [$q( y_{0} ,v_{0} ,a)$]{}; (375,215) node [$p( y_{0} ,v_{0} ,a)$]{}; (322,76) node [$B_{0}$]{}; (303,143) node [$B_{2}$]{}; (306,218) node [$B_{3}$]{};
In the following definition of entailment tree, we use the notation ${\alpha}_1 \rightarrow {\alpha}_2$, where ${\alpha}_i$ is an atom, to denote the rule $\forall X ({\alpha}_1 \rightarrow \exists Y ~{\alpha}_2)$ with $X = {{\ensuremath{\textsl{vars}}}{({\alpha}_1)}}$ and $Y = {{\ensuremath{\textsl{vars}}}{({\alpha}_2)}}\setminus X$.
\[definition-entailment-tree\] An *entailment tree* associated with ${\alpha}$ and ${\ensuremath{\Sigma}}$ is a tree of bags ${\ensuremath{\mathcal{T}}}$ such that:
1. ${\ensuremath{B}}_r$, the root of ${\ensuremath{\mathcal{T}}}$, is such that ${\ensuremath{\Sigma}}\models {\alpha}\rightarrow {{\ensuremath{\textsl{atom}}}{({\ensuremath{B}}_r)}}$ and ${\ensuremath{\Sigma}}\models {{\ensuremath{\textsl{atom}}}{({\ensuremath{B}}_r)}} \rightarrow {\alpha}$;
2. 3. there is no pair of twins.
Note that ${\alpha}$ is not necessarily the root of the entailment tree, as it may not belong to the result of the core chase on ${\alpha}$ (hence Point 1).
First note that an entailment tree is independent from any derivation. The main difference with a derivation tree is that it employs a more general parent-child relationship, that relies on entailment rather than on rule application, hence the name entailment tree. Intuitively, with respect to a derivation tree, one is allowed to move a bag $B$ higher in the tree, provided that it contains at least one term generated in its new parent $B_p$; then, the terms of $B$ that are not shared with $B_p$ are freshly renamed. Finally, since the problem of whether an atom is entailed by a linear existential rule knowledge base is decidable (precisely <span style="font-variant:small-caps;">PSpace</span>-complete [@DBLP:books/sp/virgilio09/CaliGL09]), one can actually generate all non-twin children of a bag and keep a tree with bounded arity.
Derivation trees are entailment trees, but not necessarily conversely. A crucial distinction between these two structures is the following statement, which does not hold for derivation trees, as illustrated by Example \[example-mlm\].
\[proposition-entailment-core\] If the core chase associated with ${\alpha}$ and ${\ensuremath{\Sigma}}$ is finite, then there exists an entailment tree ${\ensuremath{\mathcal{T}}}$ such that the set of atoms associated with ${\ensuremath{\mathcal{T}}}$ is a complete core.
*Example \[example-mlm\] (cont’d).* The tree defined by the path of bags $B_0$, $B_2$, $B_3$ is an entailment tree, represented on the right of Figure \[figure-example-mlm\], which defines a complete core.
Let ${\ensuremath{\mathcal{T}}}$ be the derivation tree associated with a derivation containing a core $C$ of ${{\ensuremath{\textsl{chase}}}({\alpha},{\ensuremath{\Sigma}})}$. Let $\varphi$ be an idempotent homomorphism from the atoms of ${\ensuremath{\mathcal{T}}}$ to $C$. We assign to each bag ${\ensuremath{B}}$ of ${\ensuremath{\mathcal{T}}}$ a set of trees $\{T_1,\ldots,T_{n_B}\}$ such that:
1. each tree contains only elements of $C$;
2. the forest assigned to ${\ensuremath{B}}$ contains exactly once the elements of $C$ appearing in the subtree rooted in ${\ensuremath{B}}$;
3. for each pair $({\ensuremath{B}}_p,{\ensuremath{B}}_c)$ of bags in some $T_i$ such that ${\ensuremath{B}}_p$ is a parent of ${\ensuremath{B}}_c$, ${\ensuremath{\Sigma}}\models {{\ensuremath{\textsl{atom}}}{({\ensuremath{B}}_p)}} \rightarrow {{\ensuremath{\textsl{atom}}}{({\ensuremath{B}}_c)}}$;
4. each $T_i$ is a decomposition tree;
5. for each $T_i$, the root of $T_i$ contains all the terms that belong both to $T_i$ and to $C \setminus T_i$;
6. each term ${\ensuremath{t}}$ belonging to distinct $T_i$ and $T_j$ of the forest assigned with a bag ${\ensuremath{B}}$ also belongs to the parent of ${\ensuremath{B}}$.
Moreover, we will show that if $\varphi({\ensuremath{B}})$ is a descendant of ${\ensuremath{B}}$ (including ${\ensuremath{B}}$) in ${\ensuremath{\mathcal{T}}}$, then its associated forest is a tree.
- if ${\ensuremath{B}}$ is a leaf, we consider two cases:
- ${\ensuremath{B}}$ belongs to the core: we assign it a single tree, containing only a root being itself. All conditions are trivial.
- ${\ensuremath{B}}$ does not belong to the core: we assign it an empty forest, and all conditions are trivial.
- if ${\ensuremath{B}}$ is an internal node, let $\{T_1,\ldots,T_n\}$ be the union of the forests assigned to the children of ${\ensuremath{B}}$. We distinguish three cases:
- ${\ensuremath{B}}$ is in the core: we assign to ${\ensuremath{B}}$ the tree $T$ containing ${\ensuremath{B}}$ as root, and having as children the roots of $\{T_1,\ldots,T_n\}$.
- 1\. 2.: holds by induction assumption, the fact that different $T_i$’s cover disjoint subtrees of ${\ensuremath{\mathcal{T}}}$, and the fact that ${\ensuremath{B}}$ belongs to the core
- 3.: it is enough to check this for the pairs (root of $T$, root of $T_i$). The root of $T$ is an ancestor of root of $T_i$ in ${\ensuremath{\mathcal{T}}}$, hence ${\ensuremath{\Sigma}}\models {{\ensuremath{\textsl{atom}}}{(\mathrm{root}(T))}}\rightarrow {{\ensuremath{\textsl{atom}}}{({\ensuremath{B}}_i)}}$, where ${\ensuremath{B}}_i$ is the root of $T_i$
- 4\. if ${\ensuremath{t}}$ appears in $T$ but in no $T_i$, it appears only in ${\ensuremath{B}}$ and the connectivity of the substructure containing ${\ensuremath{t}}$ holds. If it belongs to some $T_i$ and to $C \setminus T_i$, it must belong to the root of $T_i$ by assumption 6.. If ${\ensuremath{t}}$ belongs to $C \setminus T$, it belongs to ${\ensuremath{B}}$ by connectivity of ${\ensuremath{\mathcal{T}}}$. If ${\ensuremath{t}}$ belongs to another $T_j$, we distinguish two cases: $T_j$ is in the same forest as $T_i$, and then by induction assumption 7. on the child of ${\ensuremath{B}}$ to which this forest is associated, ${\ensuremath{t}}$ belongs to ${\ensuremath{B}}$. Or $T_j$ is in the forest of another child of ${\ensuremath{B}}$, and then by connectivity property for ${\ensuremath{t}}$, it belongs to ${\ensuremath{B}}$. Hence the connectivity property for ${\ensuremath{t}}$ in $T$ is fulfilled.
- 5\. By connectivity of ${\ensuremath{\mathcal{T}}}$, as ${\ensuremath{B}}$ is the root of $T$
- 6\. true as there is only one tree
- $\varphi({\ensuremath{B}}) \not = {\ensuremath{B}}$ but is a descendant of ${\ensuremath{B}}$. By induction assumption 2., there exists exactly one tree among the trees associated with children of ${\ensuremath{B}}$ containing $\varphi({\ensuremath{B}})$. Let assume w.l.o.g that it is $T_1$, of root ${\ensuremath{B}}_1$. We build the following tree $T$: for all $T_i \not = T_1$, we add to ${\ensuremath{B}}_1$ a subtree by putting the root of $T_i$ under ${\ensuremath{B}}_1$.
- 1\. No added elements, hence by induction assumption 1.
- 2\. No added elements, hence by induction assumption 2.
- 3\. To check for pairs (${\ensuremath{B}}_1$,${\ensuremath{B}}_i$), where ${\ensuremath{B}}_i$ is the root of $T_i$. ${\ensuremath{\Sigma}}\models {{\ensuremath{\textsl{atom}}}{({\ensuremath{B}}_1)}} \rightarrow {{\ensuremath{\textsl{atom}}}{(\varphi({\ensuremath{B}}))}}$, as $\varphi({\ensuremath{B}})$ is a descendant of ${\ensuremath{B}}_1$ in $T_1$. Moreover, ${\ensuremath{\Sigma}}\models {{\ensuremath{\textsl{atom}}}{(\varphi({\ensuremath{B}}))}} \rightarrow {{\ensuremath{\textsl{atom}}}{({\ensuremath{B}}_i)}}$, as $\varphi({\ensuremath{B}})$ is more specific than ${\ensuremath{B}}$, and $\varphi$ is the identity on shared terms.
- 4\. for all term ${\ensuremath{t}}$ appearing in a single tree, the connectivity property holds by induction assumption 4.. Let ${\ensuremath{t}}$ appearing in two trees. ${\ensuremath{t}}$ appears in the roots of both tree by $6.$, and must appear in ${\ensuremath{B}}$ by connectivity of ${\ensuremath{\mathcal{T}}}$, hence in $\varphi(B)$, and hence in ${\ensuremath{B}}_1$ (by 6.). As ${\ensuremath{B}}_1$ and the roots of both trees are neighbor, this proves the result.
- 5\. let ${\ensuremath{t}}$ belonging to $T$ and to $C \setminus T$. By connectivity of ${\ensuremath{\mathcal{T}}}$, ${\ensuremath{t}}$ belongs to ${\ensuremath{B}}$, hence to $\varphi({\ensuremath{B}})$ (because $\varphi(t) =t$). As ${\ensuremath{t}}$ belongs both to $T_1$ and to $C \setminus T_1$, ${\ensuremath{t}}$ belongs to ${\ensuremath{B}}_1$, and hence to the root of the assigned tree.
- 6\. true as there is only one tree.
- $\varphi({\ensuremath{B}})$ is not a descendant of ${\ensuremath{B}}$. We assign to ${\ensuremath{B}}$ the union of the forests associated to its children.
- 1.-5 By induction assumption
- 6\. let ${\ensuremath{t}}$ belonging to two trees $T_1$ and $T_2$. If $T_1$ and $T_2$ come from forest associated to two different children, ${\ensuremath{t}}$ belongs to ${\ensuremath{B}}$ by connectivity of ${\ensuremath{\mathcal{T}}}$. If $T_1$ and $T_2$ come from the same forest, ${\ensuremath{t}}$ belongs to ${\ensuremath{B}}$ by induction assumption 7. Then ${\ensuremath{t}}$ belongs to ${\ensuremath{B}}$. As ${\ensuremath{t}}$ is in $C$, ${\ensuremath{t}}$ belongs to $\varphi({\ensuremath{B}})$. By connectivity of ${\ensuremath{\mathcal{T}}}$, it belongs to the parent of ${\ensuremath{B}}$, because that parent is on the path from ${\ensuremath{B}}$ to $\varphi({\ensuremath{B}})$, which proves 6.
Finally, we check that the following property is satisfied: for any bag ${\ensuremath{B}}$, if ${\ensuremath{B}}$ is in the core, then a single tree with root ${\ensuremath{B}}$ is assigned to it. If ${\alpha}$ is in the core, we have built such a tree. It remains to obtain an entailment tree: for that, we have to bring up nodes at the highest level with respect to shared terms. We may also have to say something about ’generated’ if it still appear in the definition of an entailment tree.
Differently from the semi-oblivious case, we cannot conclude that the chase does not terminate as soon as a UPW is built, because the associated atoms may later be mapped to other atoms, which would remove the UPW. Instead, starting from the initial bag, we recursively add bags that do not generate a UPW (for instance, we can recursively add all such non-twin children to a leaf). Once the process terminates (the non-twin condition and the absence of UPW ensure that it does), we check that the obtained set of atoms $C$ is complete (i.e., is a model of the KB): for that, it suffices to perform each possible rule application on $C$ and check if the resulting set of atoms is equivalent to $C$. See Algorithm \[algorithm-core-chase\]. The set $C$ may not be a core, but it is complete iff it contains a complete core.
We now focus on the key properties of entailment trees associated with complete cores. We first introduce the notion of *redundant bags*, which captures some cases of bags that cannot appear in a finite core. As witnessed by Example \[example-mlm\], this is not a characterization: $B_1$ is not redundant (according to next Definition \[definition-redundancy\]), but cannot belong to a complete core.
\[definition-redundancy\] Given an entailment tree, a bag ${\ensuremath{B}}_c$ child of ${\ensuremath{B}}$ is redundant if there exists an atom $\beta$ (that may not belong to the tree) with *(i)* ${\ensuremath{\Sigma}}\models {{\ensuremath{\textsl{atom}}}{({\ensuremath{B}})}} \rightarrow \beta$; *(ii)* there is a homomorphism from ${{\ensuremath{\textsl{atom}}}{({\ensuremath{B}}_c)}}$ to $\beta$ that is the identity on ${{\ensuremath{\textsl{shared}}}{({\ensuremath{B}}_c)}}$ *(iii)* $|{{\ensuremath{\textsl{terms}}}{(\beta)}} \setminus {{\ensuremath{\textsl{terms}}}{({\ensuremath{B}})}}| < |{{\ensuremath{\textsl{terms}}}{({\ensuremath{B}}_c)}} \setminus {{\ensuremath{\textsl{terms}}}{({\ensuremath{B}})}}|$.
Note that ${\ensuremath{B}}_c$ may be redundant even if the “cause” for redundancy, i.e., $\beta$, is not in the tree yet.
The role of this notion in the proofs is as follows: we show that if a complete entailment tree contains a UPW then it contains a redundant bag, and that a complete core cannot contain a redundant bag, hence a UPW. To prove this, we rely on next Proposition \[proposition-swissknife-bag-copy\], which is the counterpart for entailment trees of Proposition \[proposition-sharing-type-children-derivation-tree\]: performing a bag copy from an entailment tree results in an entailment tree (the notion of prefix is not needed, since a prefix of an entailment tree is an entailment tree) and keeps the properties of the copied bag.
\[proposition-swissknife-bag-copy\] Let ${\ensuremath{B}}$ be a bag of an entailment tree ${\ensuremath{\mathcal{T}}}$, ${\ensuremath{B}}'$ be a bag of an entailment tree ${\ensuremath{\mathcal{T}}}'$ such that ${\ensuremath{B}}{\ensuremath{\equiv_{st}}}{\ensuremath{B}}'$. Let ${\ensuremath{B}}_c$ be a child of ${\ensuremath{B}}$ and ${\ensuremath{B}}_c'$ be a copy of ${\ensuremath{B}}_c$ under ${\ensuremath{B}}'$. Let ${\ensuremath{\mathcal{T}}}''$ be the extension of ${\ensuremath{\mathcal{T}}}'$ where ${\ensuremath{B}}_c'$ is added as a child of ${\ensuremath{B}}'$. Then *(i)* ${\ensuremath{\mathcal{T}}}''$ is an entailment tree; *(ii)* ${\ensuremath{B}}_c$ and ${\ensuremath{B}}_c'$ have the same sharing type; *(iii)* ${\ensuremath{B}}_c'$ is redundant if and only if ${\ensuremath{B}}_c$ is redundant.
In light of this, the copy of a bag can be naturally extended to the copy of the whole subtree rooted in a bag, which is crucial element in the proof of next Proposition \[proposition-uc-excluded-main-text\]:
Another important property of entailment trees (which is also satisfied by derivation trees) is that its structure provides information on where a bag may be mapped by $\varphi$ if its parent is left invariant by $\varphi$.
\[lemma-locality\] Let ${\ensuremath{\mathcal{T}}}$ be an entailment tree. Let $\varphi$ be a homomorphism from the atoms of ${\ensuremath{\mathcal{T}}}$ to themselves. Let ${\ensuremath{B}}_p$ such that $\varphi_{\mid {{\ensuremath{\textsl{terms}}}{({\ensuremath{B}}_p)}}}$ is the identity. Let ${\ensuremath{B}}_c$ be a child of ${\ensuremath{B}}_p$. Then $\varphi({\ensuremath{B}}_c)$ is in the subtree rooted in $\varphi({\ensuremath{B}}_p) = {\ensuremath{B}}_p$.
${\ensuremath{B}}_c$ is a child of ${\ensuremath{B}}_p$ thus there exists at least one term generated in ${\ensuremath{B}}_p$ that is a term of ${\ensuremath{B}}_c$. As $\varphi$ is the identity on ${\ensuremath{B}}_p$, that term belongs as well to $\varphi({{\ensuremath{\textsl{atom}}}{({\ensuremath{B}}_c)}})$. Thus $\varphi({{\ensuremath{\textsl{atom}}}{({\ensuremath{B}}_c)}})$ should also be in a bag that is in the subtree rooted in ${\ensuremath{B}}_p$.
\[proposition-uc-excluded-main-text\] A complete core cannot contain *(i)* a redundant bag *(ii)* an unbounded-path witness.
\[proposition-strong-redundancy\] A complete core cannot contain a redundant bag.
Let ${\ensuremath{\mathcal{T}}}$ be a complete entailment tree, and let $\hat{{\ensuremath{B}}}$ be a bag that is redundant. We prove that there exists a non-injective endomorphism of ${\ensuremath{\mathcal{T}}}$, showing that ${\ensuremath{\mathcal{T}}}$ cannot be a core. For any entailment tree ${\ensuremath{\mathcal{T}}}_p$ that is a prefix of ${\ensuremath{\mathcal{T}}}$, we build ${\ensuremath{\mathcal{T}}}'_{p}$ and a mapping $\varphi$ from the terms of ${\ensuremath{\mathcal{T}}}_p$ to the terms of ${\ensuremath{\mathcal{T}}}'_{p}$ as follows:
- for any prefix of ${\ensuremath{\mathcal{T}}}_p$ that does not contain $\hat{{\ensuremath{B}}}$, we define ${\ensuremath{\mathcal{T}}}_{p}' = {\ensuremath{\mathcal{T}}}_p$ and $\varphi$ the identity
- for the prefix that contains all nodes of ${\ensuremath{\mathcal{T}}}$, including $\hat{{\ensuremath{B}}}$, except the descendants of $\hat{{\ensuremath{B}}}$, we define ${\ensuremath{\mathcal{T}}}'_p$ as ${\ensuremath{\mathcal{T}}}_p$ to which we add a leaf (if necessary) to the parent of $\hat{{\ensuremath{B}}}$ in ${\ensuremath{\mathcal{T}}}$, which is the witness of the redundancy of $\hat{{\ensuremath{B}}}$. We define $\varphi$ as the identity on any term that is not generated in $\hat{{\ensuremath{B}}}$, and as its image by the $\varphi$ witnessing the redundancy pattern on terms generated in $\hat{{\ensuremath{B}}}$.
- if we have defined ${\ensuremath{\mathcal{T}}}'_{p}$ for ${\ensuremath{\mathcal{T}}}_p$, and we want to define $\varphi$ for ${\ensuremath{\mathcal{T}}}'_n$ for ${\ensuremath{\mathcal{T}}}_n$ which is ${\ensuremath{\mathcal{T}}}_p$ to which a leaf ${\ensuremath{B}}_d$ has been added, we add where it belongs the bag $\varphi({\ensuremath{B}}_d)$, where we extend $\varphi$ to term generated in ${\ensuremath{B}}_d$ by choosing fresh images.
By construction, ${\ensuremath{\mathcal{T}}}'$ is an entailment tree, and $\varphi$ is a homomorphism from ${\ensuremath{\mathcal{T}}}$ to ${\ensuremath{\mathcal{T}}}'$. Moreover, $\varphi$ is not injective: indeed, as $\hat{{\ensuremath{B}}}$ is redundant, $\varphi$ is not injective on the terms of $\hat{{\ensuremath{B}}}$.
As ${\ensuremath{\mathcal{T}}}$ is complete, there exists a homomorphism from ${\ensuremath{\mathcal{T}}}'$ to ${\ensuremath{\mathcal{T}}}$. Hence the composition of the two homomorphisms is a homomorphism from ${\ensuremath{\mathcal{T}}}$ to itself, which is not injective, as $\varphi$ is not. Hence ${\ensuremath{\mathcal{T}}}$ is not a core.
\[proposition-uc-excluded\] A complete core cannot contain any unbounded-path witness.
We prove the result by contradiction. Let us assume that ${\ensuremath{\mathcal{T}}}$ is a complete core containing an unbounded-path witness $({\ensuremath{B}},{\ensuremath{B}}')$. Let us choose $({\ensuremath{B}},{\ensuremath{B}}')$ such that ${\ensuremath{B}}'$ is of maximal depth with respect to its branch, that is, there is no unbounded-path witness $({\ensuremath{B}}''',{\ensuremath{B}}'')$ with ${\ensuremath{B}}''$ a strict descendant of ${\ensuremath{B}}'$.
Let ${\ensuremath{B}}_c$ be the child of ${\ensuremath{B}}$ on the path from ${\ensuremath{B}}$ to ${\ensuremath{B}}'$. Let us denote by ${\ensuremath{\mathcal{T}}}_{{\ensuremath{B}}_c}$ the subtree of ${\ensuremath{\mathcal{T}}}$ which is rooted at ${\ensuremath{B}}_c$ and by ${\ensuremath{\mathcal{T}}}_{{\ensuremath{B}}_c'}$ a copy of ${\ensuremath{\mathcal{T}}}_{{\ensuremath{B}}_c}$ under $B'$ whose root is ${\ensuremath{B}}_c'$. Then, let ${\ensuremath{\mathcal{T}}}'$ be the extension of ${\ensuremath{\mathcal{T}}}$ where ${\ensuremath{\mathcal{T}}}_{{\ensuremath{B}}_c'}$ is added as a child of ${\ensuremath{B}}'$. We want to show that there exists a bag ${\ensuremath{B}}_r'$ child of ${\ensuremath{B}}'$ and a mapping from ${\ensuremath{\mathcal{T}}}_{{\ensuremath{B}}_c'}$ into ${\ensuremath{\mathcal{T}}}_{{\ensuremath{B}}_r'}$, which is the identity on the terms of ${\ensuremath{\mathcal{T}}}$. More precisely, we want to show that for each ${\ensuremath{B}}_d'$ descendant of $ {\ensuremath{B}}_c'$ the following properties hold.
1. the image of ${\ensuremath{B}}_d'$ belongs to ${\ensuremath{\mathcal{T}}}_{{\ensuremath{B}}_r'}$
2. the image of a term generated in ${\ensuremath{B}}_d'$ is a term generated in a bag of ${\ensuremath{\mathcal{T}}}_{{\ensuremath{B}}_r'}$
We do so by induction on $k$ the distance between ${\ensuremath{B}}_d'$ and ${\ensuremath{B}}_c'$ in ${\ensuremath{\mathcal{T}}}$.
- If $k=0$ then ${\ensuremath{B}}_d'={\ensuremath{B}}_c'$. Because ${\ensuremath{\mathcal{T}}}$ is a complete core, there exists a homomorphism from the atoms of $\mathcal{T'} $ to those of $ \mathcal{T}$ which is the identity on the terms of ${\ensuremath{\mathcal{T}}}$.
We show that the image of ${\ensuremath{B}}'_c$ is a strict descendant of a child of ${\ensuremath{B}}'$. Note first that no child of ${\ensuremath{B}}'$ in ${\ensuremath{\mathcal{T}}}$ can be a safe renaming of ${\ensuremath{B}}'_c$. Indeed, by Proposition \[proposition-swissknife-bag-copy\], ${\ensuremath{B}}_c$ and ${\ensuremath{B}}_c'$ have the same sharing type and therefore ${\ensuremath{B}}_c'$ (as well as any safe renaming of its generated terms) cannot be a child of ${\ensuremath{B}}'$ because the couple $({\ensuremath{B}}_c,{\ensuremath{B}}_c')$ would form an unbounded-path witness with ${\ensuremath{B}}_c'$ strictly deeper than ${\ensuremath{B}}'$. Then, if ${\ensuremath{B}}_c'$ maps to ${\ensuremath{B}}'$ then the couple $({\ensuremath{B}}',{\ensuremath{B}}_c')$ is redundant and therefore also $({\ensuremath{B}},{\ensuremath{B}}_c)$ is redundant, by Proposition \[proposition-swissknife-bag-copy\], which in turn implies that ${\ensuremath{\mathcal{T}}}$ is not a core, because of Proposition \[proposition-strong-redundancy\]. Finally, if ${\ensuremath{B}}_c'$ maps to any child of ${\ensuremath{B}}'$ then it does so by specializing the sharing type of ${\ensuremath{B}}_c'$ (as we showed that no safe renaming of ${\ensuremath{B}}_c'$ can be a child of ${\ensuremath{B}}'$), which means that ${\ensuremath{B}}_c'$ is redundant. Therefore, by Proposition \[proposition-swissknife-bag-copy\], ${\ensuremath{B}}_c$ is also redundant and hence, by Proposition \[proposition-strong-redundancy\], ${\ensuremath{\mathcal{T}}}$ is not a core. This proves that the image of ${\ensuremath{B}}_c'$ is a strict descendant of some ${\ensuremath{B}}_r'$ child of ${\ensuremath{B}}'$.
Now, to prove the second point, let $t$ be a term generated in ${\ensuremath{B}}'_c$ and $t'$ its image. It is easy to see that for entailment trees any term that belongs to two bags in ancestor-descendant relationship also belongs to the bags on the shortest path between them. Therefore, if $t'$ is generated by a strict ancestor of ${\ensuremath{B}}_r'$ then $t'$ belongs to the terms of ${\ensuremath{B}}'$. This means that starting from the sharing type of ${\ensuremath{B}}_c'$ one can build a strictly more specific sharing type where the position corresponding to the generated term $t$ becomes shared with ${\ensuremath{B}}'$. From this one can find a node ${\ensuremath{B}}_c''$ which is strictly more specific than ${\ensuremath{B}}_c'$ and that can be added as a child of ${\ensuremath{B}}'$. This means that ${\ensuremath{B}}_c'$ is redundant and by Proposition \[proposition-swissknife-bag-copy\] also ${\ensuremath{B}}_c$ is redundant, so ${\ensuremath{\mathcal{T}}}$ is not a core.
- Let us assume that both properties hold for all bags at distance $k$ from ${\ensuremath{B}}_c'$. We want to show that they still holds for the bags at distance $k+1$.
Let ${\ensuremath{B}}_d'$ be a node at distance $k+1$ from ${\ensuremath{B}}_c'$ whose parent is ${\ensuremath{B}}'_\delta$. By definition, ${\ensuremath{B}}_d'$ contains a term generated by $B'_\delta$ and, by induction, we know that the image of this term is generated in a bag of ${\ensuremath{\mathcal{T}}}_{{\ensuremath{B}}_r'}$. Thus, it follows that the image of ${\ensuremath{B}}_d'$ belongs to ${\ensuremath{\mathcal{T}}}_{{\ensuremath{B}}_r'}$ as required by the first point.
For the second point we reason by contradiction and show that when the property does not hold then ${\ensuremath{\mathcal{T}}}$ admits a non-injective endomorphism and thus it is not a core. We proceed with the following construction. Let ${\ensuremath{\mathcal{T}}}_{{\ensuremath{B}}_r}$ be a copy of ${\ensuremath{\mathcal{T}}}_{{\ensuremath{B}}_r'}$ under ${\ensuremath{B}}$ and ${\ensuremath{\mathcal{T}}}''$ the extension of ${\ensuremath{\mathcal{T}}}$ where ${\ensuremath{\mathcal{T}}}_{{\ensuremath{B}}_r}$ is added as a child of ${\ensuremath{B}}$. We know by induction that there exists a homomorphism from ${\ensuremath{\mathcal{T}}}'$ to ${\ensuremath{\mathcal{T}}}$ mapping all nodes at distance ${k+1}$ from ${\ensuremath{B}}'_c$ to the subtree rooted at ${\ensuremath{B}}'_r$. From this, we can conclude that there exists a homomorphism from ${\ensuremath{\mathcal{T}}}_{(k+1)}$ to ${\ensuremath{\mathcal{T}}}''$, where ${\ensuremath{\mathcal{T}}}_{(k+1)}$ is the prefix of ${\ensuremath{\mathcal{T}}}$ which includes all nodes of ${\ensuremath{\mathcal{T}}}$ except for the descendants of ${\ensuremath{B}}_c$ that are at distance strictly greater than $k+1$ from it. Now, we further extend ${\ensuremath{\mathcal{T}}}''$ by adding an image for all nodes which are at distance strictly greater than $k+1$ from ${\ensuremath{B}}_c$ thereby obtaining a new entailment tree ${\ensuremath{\mathcal{T}}}'''$. It follows that ${\ensuremath{\mathcal{T}}}$ can be mapped to ${\ensuremath{\mathcal{T}}}'''$. Beside, since ${\ensuremath{\mathcal{T}}}$ is complete there exists an homomorphism from ${\ensuremath{\mathcal{T}}}'''$ to ${\ensuremath{\mathcal{T}}}$. So, by composing these two homomorphisms we get a homomorphism from ${\ensuremath{\mathcal{T}}}$ to ${\ensuremath{\mathcal{T}}}$.
We show that the homomorphism from ${\ensuremath{\mathcal{T}}}$ to ${\ensuremath{\mathcal{T}}}'''$ is non-injective. Recall that to construct ${\ensuremath{\mathcal{T}}}'$ the whole subtree rooted at ${\ensuremath{B}}_c'$ has been copied from the subtree rooted at ${\ensuremath{B}}_c$. Let us denote by ${\ensuremath{B}}_d$ the node at distance $k+1$ from ${\ensuremath{B}}_c$ from which ${\ensuremath{B}}_d'$ has been copied under ${\ensuremath{B}}_\delta'$. Let $t$ be a term generated at position $i$ in ${\ensuremath{B}}_d'$. If its image was generated by a strict ancestor of ${\ensuremath{B}}_r'$ then this would also belong to the terms of ${\ensuremath{B}}'$. By Proposition \[proposition-swissknife-bag-copy\], ${\ensuremath{B}}_d$ and ${\ensuremath{B}}_d'$ have the same sharing types, hence the mapping from ${\ensuremath{\mathcal{T}}}_{(k+1)}$ to ${\ensuremath{\mathcal{T}}}''$ (and thus that from ${\ensuremath{\mathcal{T}}}$ to ${\ensuremath{\mathcal{T}}}'''$) maps the generated term at position $i$ of ${\ensuremath{B}}_d$, we call $s$, to a distinct term in ${\ensuremath{B}}$, we call $s'$. Moreover, the homomorphism is the identity on $s'$. Therefore, the homomorphism from ${\ensuremath{\mathcal{T}}}$ to ${\ensuremath{\mathcal{T}}}'''$ is non-injective as both $s'$ and $s$ have the same image.
To finish the proof, we proceed with the following construction. Let ${\ensuremath{\mathcal{T}}}^*$ be an entailment tree derived from ${\ensuremath{\mathcal{T}}}$ where $i)$ the whole subtree rooted at ${\ensuremath{B}}_r'$ has been copied under ${\ensuremath{B}}$ and $ii)$ the subtree rooted at ${\ensuremath{B}}_c$ has been removed. Note that ${\ensuremath{\mathcal{T}}}^*$ is of size strictly smaller than that of ${\ensuremath{\mathcal{T}}}$ because we added a bag for each descendant node of ${\ensuremath{B}}'_r$, which is a strict descendant of bag ${\ensuremath{B}}_c$, and that this last one has been removed. Now, because ${\ensuremath{\mathcal{T}}}_{{\ensuremath{B}}_c'}$ maps to ${\ensuremath{\mathcal{T}}}_{{\ensuremath{B}}_r'}$ it follows that ${\ensuremath{\mathcal{T}}}_{{\ensuremath{B}}_c}$ maps to ${\ensuremath{\mathcal{T}}}_{{\ensuremath{B}}_r}$ and by extending this homomorphism to the identity on all other terms we get that ${\ensuremath{\mathcal{T}}}$ can be mapped to ${\ensuremath{\mathcal{T}}}^*$. Hence, ${\ensuremath{\mathcal{T}}}$ is not a core.
\[corollary-core-finiteness-decidability\] The all-sequence termination problem for the core chase on linear rules is decidable.
`true`
A rough complexity analysis of this algorithm yields a <span style="font-variant:small-caps;">2ExpTime</span> upper bound for the termination problem. Indeed, the exponential number of (sharing) types yields a bound on the number of canonical instances to be checked, the arity of the tree, as well as the length of a path without UPW in the tree, and each edge can be generated with a call to a <span style="font-variant:small-caps;">PSpace</span> oracle.
Concluding remarks
==================
We have shown the decidability of chase termination over linear rules for three main chase variants (semi-oblivious, restricted, core) following a novel approach based on derivation trees, and their generalization to entailment trees, and a single notion of forbidden pattern. As far as we know, these are the first decidability results for the restricted chase, on both versions of the termination problem (i.e., *all sequence* and *one sequence* termination). The simplicity of the structures and algorithms make them subject to implementation.
We leave for future work the study of the precise complexity of the termination problems. A straightforward analysis of the complexity of the algorithms that decide the termination of the restricted and core chases yields upper bounds, however we believe that a finer analysis of the properties of sharing types would provide tighter upper bounds. Future work also includes the extension of the results to more complex classes of existential rules: linear rules with a complex head, which is relevant for the termination of the restricted and core chases, and more expressive classes from the guarded family. Derivation trees were precisely defined to represent derivations with guarded rules and their extensions (i.e., greedy bounded treewidth sets), hence they seem to be a promising tool to study chase termination on that family.
[^1]: For instance, it can always be ensured by adding a position to all predicates, which is filled by the same fresh constant in the initial instance, and by a new frontier variable in each rule.
[^2]: Furthermore the trees we will consider are decomposition trees of the associated set of atoms. That is why we use the classical term of *bag* to denote a node.
[^3]: Note that the chase graph in [@DBLP:conf/pods/DeutschNR08] is a different notion.
[^4]: $\hat{\pi}_i$ is the restriction of $\hat{\pi}_i^{\mathrm{safe}}$ to the variables of the body of ${\ensuremath{\sigma}}_i$.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Measuring a nonlocal observable on a space-like separated quantum system is a resource-hungry and experimentally challenging task. Several theoretical measurement schemes have already been proposed to increase its feasibility, using a shared maximally-entangled ancilla. We present a new approach to this problem, using the language of generalized quantum measurements, to show that it is actually possible to measure a nonlocal spin product observable without necessarily requiring a maximally-entangled ancilla. This approach opens the door to more economical arbitrary-strength nonlocal measurements, with applications ranging from nonlocal weak values to possible new tests of Bell inequalities. The relation between measurement strength and the amount of ancillary entanglement needed is made explicit, bringing a new perspective on the links that tie quantum nonlocality, entanglement and information transmission together.'
author:
- Pierre Vidil
- Keiichi Edamatsu
title: Nonlocal Generalized Quantum Measurements
---
Introduction
============
Almost since its inception, the behavior of space-like separated quantum systems has been at the heart of multiple heated controversies around quatum mechanics [@Einstein1935; @Bell1964; @Laloe2012], as well as the key to some of its most promising technological applications. These include superdense coding [@Bennett1992; @Mattle1996], quantum teleportation [@Bennett1993; @Bouwmeester1997], entanglement swapping [@Zukowski1993; @Pan1998] and device-independent quantum key distribution [@Barrett2005; @Acin2006; @Acin2007] among others. All have in common that they rely on the measurement of an operator that contains information about not just one, but several, possibly entangled, quantum particles.
Sometimes, one might be faced with a situation where those different parts are space-like separated and direct interaction between them is not available. The question of whether or not it is possible to measure such multipartite observables instantaneously in this case was first answered in the negative by Landau and Peierls [@Landau1931] in 1931, on the grounds of locality constraints.
Yet it was proven much later that such *nonlocal measurements* are in fact possible for certain observables, given adequate resources [@Aharonov1981; @Aharonov1986; @Popescu1994]. When the different parts are separated, they are made to strongly interact with an additional maximally-entangled state, a precious resource in quantum information [@Bennett1998; @Horodecki2007], that is used to carry out the measurement and store the result.
This type of measurement scheme is often referred to as a *von Neumann (VN) measurement* [@vonNeumann1935], and the use of a maximally-entangled meter state has been shown to solve the problem of achieving complete Bell State Measurement [@Bennett1996; @Beckman2001], even in linear-optical systems [@Edamatsu2016a]. The interaction between the system and the meter leading to the final result can then be made instantaneous, even though retrieving said result from the entangled meter requires some finite amount of time, as dictated by special relativity. Such a strong VN measurement of nonlocal variables has already been implemented using hyperentangled photonic quantum systems [@Xu2019]. Furthermore, if one is ready to part with the VN approach and discard the final state of the system, all nonlocal observables become measurable [@Groisman2002; @Vaidman2003; @Clark2010] via so-called verification measurements and finite entanglement consumption.
However VN measurements can be more than just strong (projective) measurements [@Wiseman2009], which have been discussed so far. By suitably tailoring the system-meter interaction, as in [Fig. \[fig:one-qubit\]]{}, one can manage to only retrieve part of the information about a quantum state, in order to somewhat preserve it [@Aharonov1987]. This has been successfully applied to local systems for quantum metrology [@Hosten2008; @Dixon2009], or in quantum foundations when one wishes to limit the effects of the measurement back-action via *weak measurements* [@Aharonov1988; @Aharonov2002].
One can naturally wonder if this type of interaction tuning can be extended to the nonlocal case. The quantum erasure scheme, developed by Brodutch and Cohen [@Brodutch2016] and recently implemented by Li and al. [@Li2019], provides a solution by effectively reproducing a nonlocal aribitrary-strength VN interaction. It also extends the class of measurable nonlocal observables, by inserting a probabilistic element that prevents running afoul of causality.
This comes at a price however: on top of a maximally-entangled meter, an extra local meter is necessary to store the result, thus making this method difficult to implement experimentally. The simplest case indeed requires a total of five distinct qubits as can be seen on [Fig. \[sfig:erasure\]]{}.
In this Paper, we present a simpler method that can be used to measure nonlocal spin products, yielding the same post-measurement state evolution and statistics as the quantum erasure method, while using less resources, as shown in [Fig. \[sfig:nmem\]]{}, where the total number of qubits necessary is four. Our approach presents a complemetary point of view to the problem of nonlocal measurements that relies on the language of generalized quantum measurements [@Wiseman2009; @Nielsen2010] applied to spin product observables. We prove that in this particular case, it is possible to reproduce the behavior of an arbitrary-strength nonlocal measurement using a non-maximally-entangled meter, a weaker resource than what was needed in previous schemes. In particular, we show that the optimal amount of meter entanglement necessary is directly related to the desired measurement strength, and that excessive entanglement may on the contrary degrade the purity of the post-measurement system state. One can then achieve a nonlocal weak measurement with only a limited amount of ancillary entanglement, which greatly increases experimental feasibility, notably for linear-optical implementations.
The structure of this Paper is as follows. In Sec. \[sec:one-qubit\], we review one-qubit generalized measurements, which constitute the starting point for the later extension to the nonlocal case. We describe in Sec. \[sec:two-qubit1\] the main result of this Paper, namely how to measure a spin product observable on two qubits using a non-maximally entangled meter state. We then compare it to the quantum erasure scheme in Sec. \[sec:compare\]. In Sec. \[sec:two-qubit2\], we study a possible alternative to the above using a maximally-entangled state and its impact on the post-measurement state of the system. In Sec. \[sec:discussion\], we draw from the previous sections to establish a relation between measurement strength and ancillary entanglement in the two-qubit case. Finally, we conclude in Sec. \[sec:conclusions\] by exposing the advantages and applications of this approach, as well as possible extensions.
Generalized measurement of a single qubit {#sec:one-qubit}
=========================================
A VN generalized quantum measurement consists of an interaction between two quantum states, respectively called the system $S$, initially in the state $\ket{\psi}_S$, and a property of which we wish to measure ; and the meter $M$, prepared in a known initial state, which we will use to measure $S$. The interaction is followed by a projective measurement on $M$, in order to read out the result. By designing an appropriate tunable interaction between the system and the meter, one can actually carry out measurements of different strengths, with much more flexibility than what is allowed by projective measurements.
Several such useful interactions have been proposed in the past for the measurement of single qubits (see [@Baek2008] for instance). We here focus on the one described in [@Lund2010], that can be used to measure the system spin $\sigma_z$, and which is represented in [Fig. \[fig:one-qubit\]]{}. It consists in a local rotation applied to $M$ in order to obtain the following meter state: $$\begin{aligned}
R(\theta)\ket{0}_M=\cos{\theta}\ket{0}_M+\sin{\theta}\ket{1}_M \label{eq:meter_1qubit}\end{aligned}$$ followed by a Controlled-NOT (CNOT) gate between the meter and the system.
![\[fig:one-qubit\]Quantum circuit representation of the one-qubit $\sigma_z$ spin indirect measurement model described in Sec. \[sec:one-qubit\].](Figures/fig1.eps){width="40.00000%"}
After, the result is retrieved via a projective measurement of $\sigma_z^M$ on $M$, the corresponding Positive-Operator Valued Measure (POVM) effects for the whole process are given by $$\begin{aligned}
E_{\pm1}=\frac{1}{2}( \mathbb{1} \pm \underbrace{\cos(2\theta)}_\text{strength} \sigma_{z} ) \label{eq:povm_1qubit}\end{aligned}$$ where $\mathbb{1}$ designates the identity operator.
Computing the statistics associated to this POVM reveals that the $\mathfrak{S}\equiv\cos(2\theta)$ factor acts as the *measurement strength*, with $\mathfrak{S}=1$ corresponding to a strong measurement (perfect meter-system correlation) and $\mathfrak{S}=0$ corresponding to no measurement at all (no correlations): $$\begin{aligned}
\mathfrak{S}&\rightarrow0 & P_{+1}&\rightarrow\frac{1}{2} & P_{-1}&\rightarrow\frac{1}{2}\\
\mathfrak{S}&\rightarrow1 & P_{+1}&\rightarrow\left|\left<0|\psi\right>_S\right|^2 & P_{-1}&\rightarrow\left|\left<1|\psi\right>_S\right|^2 \end{aligned}$$ This generalized measurement scheme for one qubit has the advantage of being implementable using linear optics for polarization qubits [@Pryde2005] and has been used to test experimentally Ozawa’s error-disturbance relations [@Ozawa2003; @Baek2013; @Edamatsu2016b] as well as to measure weak values [@Kaneda2014].
Generalized spin product measurement via a non-maximally entangled meter {#sec:two-qubit1}
========================================================================
We consider a bipartite qubit system where a pair of qubits is distributed between Alice (A) and Bob (B). For clarity, this pair of qubits is initially assumed to be in a pure (possibly entangled) state $\ket{\Psi}_S$. Our goal is to answer the following: is it possible to extend the generalized measurement process of Sec. \[sec:one-qubit\] described by the POVM in Eq. to the case of a two-qubit observable?
Namely, we will now attempt to extend our measurement of $\sigma_z$ to a measurement of the product observable $\sigma_{z_A}\sigma_{z_B}$. It has already been shown that one can carry a projective measurement of $\sigma_{z_A}\sigma_{z_B}$ by using a maximally-entangled meter, e.g. the *Bell state* $\ket{\Phi^+}_M$ [@Edamatsu2016a; @Xu2019]. Following such previous approaches that established maximally-entangled qubit pairs (*ebits*) as the standard resource for nonlocal quantum protocols, one may try to start with a nonlocal meter initialized in the state $\ket{\Phi^+}_M$.
A straightforward generalization of the process described in Sec. \[sec:one-qubit\] would for instance consist in transforming this initial nonlocal meter state $\ket{\Phi^+}_M$ into a superposition of eigenstates associated with different outcomes, analogous to the one in Eq. :
$$\begin{aligned}
&\ket{\Phi^+}_M\rightarrow\ket{\Phi'}_M\equiv\cos{\theta}\underbrace{\ket{\Phi^+}_M}_\text{result $+1$}+\sin{\theta}\underbrace{\ket{\Psi^+}_M}_\text{result $-1$}\label{eq:meter_ideal}\\
&=\frac{1}{\sqrt{2}}\left(\cos{\theta}\ket{00}+\sin{\theta}\ket{01}+\sin{\theta}\ket{10}+\cos{\theta}\ket{11}\right)\label{eq:meter_expanded}\\
&=\cos\alpha\ket{++}+\sin\alpha\ket{--}\label{eq:meter_schmidt}
\end{aligned}$$
where $\ket{\Phi^+}_M$ and $\ket{\Psi^+}_M$ are the usual maximally-entangled Bell states, corresponding to global measurement outcomes $+1$ and $-1$ respectively, $\alpha=\frac{\pi}{4}-\theta$ and $\ket{\pm}\equiv\frac{1}{\sqrt{2}}\left(\ket{0}\pm\ket{1}\right)$.
However, interpreting Eq. as the Schmidt decomposition [@Nielsen2010] for the state $\ket{\Phi'}_M$ suggests that the transformation is not realizable using only local unitary operations. Eq. shows indeed that the meter state $\ket{\Phi'}_M$ is in general not maximally-entangled, hence not accessible from a Bell state via local unitaries [@Hulpke2006].
The state $\ket{\Phi'}_M$ can however be easily obtained from the state $\ket{\Phi^+}_M$ via some non-unitary operation that would discard unwanted amplitudes, in a fashion similar to a filter, in order to achieve the desired imbalance between the Schmidt coefficients of Eq. .
Restricting ourselves to unitary operations, one can implement the transformation probabilistically with a 50% success rate, or deterministically using a classical communication channel between Alice and Bob as guaranteed by Nielsen’s majorization theorem [@Nielsen1999]. An example of such a possible implementation will be presented in Sec. \[sec:compare\].
In general, if one has an entangled qubit pair with known Schmidt coefficients $\lambda_0$ and $\lambda_1$, one can obtain such a state starting from the Schmidt basis and applying a Hadamard gate $H$ on each side.
Description of the measurement scheme {#description-of-the-measurement-scheme .unnumbered}
-------------------------------------
Let us now assume that the non-maximally-entangled meter state $\ket{\Phi'}_M$ has been successfully prepared for some $\theta$ between 0 and $\frac{\pi}{4}$. Alice and Bob can now proceed to couple their qubits with the meter via local CNOT gates, as depicted in [Fig. \[sfig:nmem\]]{}, before each (projectively) measuring their meter qubit. For each of the four possible local outcomes, the final system state is given by the following measurement operators:
\[eq:measop\_1\] $$\begin{aligned}
\begin{split}\label{eq:measop_11}
M_{++}={}&M_{--} \\
=&\frac{1}{\sqrt{2}} \left\{ \cos\theta \left( \Pi_{00} + \Pi_{11} \right) + \sin\theta \left( \Pi_{01} + \Pi_{10} \right)\right\}
\end{split}\\
\begin{split}\label{eq:measop_12}
M_{+-}={}&M_{-+} \\
=&\frac{1}{\sqrt{2}} \left\{ \sin\theta \left( \Pi_{00} + \Pi_{11} \right) + \cos\theta \left( \Pi_{01} + \Pi_{10} \right)\right\}
\end{split}
\end{aligned}$$
where $\Pi_{ij}$ is the projector on $\ket{ij}$, i.e. $\Pi_{ij}=\ket{ij}\bra{ij}$.
From the four different local outcomes, the global outcomes are computed classically by allowing Alice and Bob to share their results. Considering only the global outcomes and discarding any remaining local information, the evolution can be described by two different quantum operations, one for each result (see [Fig. \[fig:instrument\]]{}). The unnormalized post-measurement states of the system are given by the action of the following superoperators on the initial density matrix $\rho=\ket{\psi}\bra{\psi}$:
\[eq:inst\] $$\begin{aligned}
\mathcal{I}_{+1}[\rho]&= M_{++} \rho M^\dagger_{++} + M_{--} \rho M^\dagger_{--} \label{eq:inst1} \\
\mathcal{I}_{-1}[\rho]&= M_{+-} \rho M^\dagger_{+-} + M_{-+} \rho M^\dagger_{-+} \label{eq:inst2}
\end{aligned}$$
These operations form the *quantum instrument* $\mathcal{I}$ [@Davies1970; @Ozawa2004], which fully encapsulates the measurement process as it provides a complete description of both post-measurement states and measurement statistics, as we will see below.
![\[fig:instrument\]Schematic representation of the measurement process. Once Alice’s and Bob’s results are multiplied together and any remaining local information is discarded, the measurement process is described by the quantum instrument $\mathcal{I}$.](Figures/fig3.eps){width="45.00000%"}
The POVM effects can be obtained directly from the quantum instrument $\mathcal{I}$, via the relation $E_r=\mathcal{I}_r^*[\mathbb{1}]$, where \* designates the superoperator adjoint, obtained by taking the adjoints of the measurement operators $M_{ij}$. This yields:
\[eq:povm\] $$\begin{aligned}
E_{+1}&= M^\dagger_{++}M_{++} + M^\dagger_{--}M_{--} \label{eq:povm1} \\
E_{-1}&= M^\dagger_{+-}M_{+-} + M^\dagger_{-+}M_{-+} \label{eq:povm2}
\end{aligned}$$
Substituting with the expressions for the measurement operators , the POVM can be rewritten in the following more compact way: $$\begin{aligned}
E_{\pm1}=\frac{1}{2}\left( \mathbb{1} \pm \cos(2\theta) \sigma_{z_A}\sigma_{z_B} \right) \label{eq:povm3}\end{aligned}$$
This is the desired nonlocal generalization of the POVM of Eq., which yields the statistics expected from a genuine nonlocal measurement.
Moreover, we have $M_{++}=M_{--}$ and $M_{+-}=M_{-+}$, hence for a given global result, the evolution of the system does not depend on the local results. This allows us to rewrite the state evolution in terms of two *effective measurement operators*, one for each global result:
\[eq:effect\_measop\] $$\begin{aligned}
M_+=& \cos\theta \left( \Pi_{00} + \Pi_{11} \right) + \sin\theta \left( \Pi_{01} + \Pi_{10} \right) \label{eq:effect_measop1}\\
M_-=& \sin\theta \left( \Pi_{00} + \Pi_{11} \right) + \cos\theta \left( \Pi_{01} + \Pi_{10} \right) \label{eq:effect_measop2}
\end{aligned}$$
These operators only involve projectors on the two-dimensional eignespaces of the observable being measured, as is to be expected in the case of a degenerate observable, first studied by Luders [@Luders1951]. All eigenstates thus remain unchanged by the measurement and this process is not entanglement-breaking, which are characteristics of an ideal nonlocal measurement.
This is the core result of this Paper: it is possible implement a nonlocal measurement of a spin product using only a meter state that need not be maximally-entangled. This is in sharp contrast with other nonlocal von Neumann measurement schemes developed so far [@Aharonov1986; @Brodutch2016].
Comparison with the quantum erasure method {#sec:compare}
==========================================
![\[fig:prepa\]The entanglement reduction method: starting from a maximally-entangled meter (MEM), one first needs to reduce the entanglement using an additional local qubit before proceeding with the measurement process. Comparing this approach with the quantum erasure method of [Fig. \[sfig:erasure\]]{} shows how the two measurement schemes are complementary in this particular case. ](Figures/fig4.eps){width="47.00000%"}
The method we have just presented is deterministic, once the two parties are allowed to communicate. However if no communication between Alice and Bob is permitted whatsoever, Alice can still teleport her local result to Bob by post-selecting her part of the meter onto a known state. For causality reasons, this can only succeed with probability 50%. The result is then encoded in a single local meter on Bob’s side.
We explained previously how to reduce the meter entanglement using non-unitary operations. In this Section, we will however limit ourselves to unitary operations on each qubits, for comparison purposes with the protocol developed by Brodutch et al. [@Brodutch2016], namely the quantum erasure method. To this end, we consider the case where the two parties share a previously prepared maximally-entangled meter and are not allowed to communicate.
The quantum erasure method consists of four steps (see [Fig. \[sfig:erasure\]]{}): first, a strong coupling between Alice’s and Bob’s systems and their shared maximally-entangled meter (MEM); followed by a post-selection on Alice’s part of the MEM to teleport her result to Bob. Then, Bob realizes a weak coupling between his remaining part of the MEM and an additional local meter. Finally, Bob needs to erase the excess information contained in the MEM by projecting his part on the unbiased state $\ket{+}_{M_B}$.
In our scheme, Alice (or Bob) first implement tranformation to reduce the entanglement of the meter, using for instance an additional ancillary local state (see [Fig. \[fig:prepa\]]{}). They subsequently proceed to strongly couple their systems with the resulting meter state. The result can finally be teleported from one side to the other by post-selecting one part of the meter on a known state, say $\ket{0}_{M_A}$.
We thus show an example of a weak measurement without weak coupling [@Roik2019]: the weak coupling is replaced by a suitably prepared meter, in our case a non-maximally entangled meter. The reduced entanglement guarantees that no excess information is stored in the meter, which makes the erasure step unnecessary.
Generalized spin product measurement via a maximally entangled meter {#sec:two-qubit2}
====================================================================
Before further discussing our results, it is interesting to study what might happen if we try to realize a nonlocal generalized measurement directly using a maximally-entangled meter, for instance the state $\ket{\Phi^+}$. Instead of trying to achieve the transformation , let us consider the meter state resulting from two local rotations implemented on Alice’s and Bob’s sides, of angles $\theta_1$ and $\theta_2$ respectively, as shown on [Fig. \[fig:mem\]]{}.
We obtain (up to a global phase) the following state: $$\begin{aligned}
\ket{\Phi^+}_M\xrightarrow[\, R_B(\theta_2)\, ]{\, R_A(\theta_1)\, }\frac{1}{\sqrt{2}}\big(\begin{aligned}[t]
&\cos{\theta}\ket{00}-\sin{\theta}\ket{01}\\
&+\sin{\theta}\ket{10}+\cos{\theta}\ket{11}\big)
\end{aligned}
\label{eq:meter_afterrotation} \end{aligned}$$ with $\theta\overset{\mathrm{def}}{=}\theta_2-\theta_1$. As expected, this is different from the state ; this will have consequences on the post-measurement system state.
If Alice and Bob locally couple their meter qubits to their system qubits via CNOT gates and locally measure their meters (see [Fig. \[fig:mem\]]{}), the corresponding measurement operators are:
\[eq:measop\_2\] $$\begin{aligned}
M_{++} &= \frac{1}{\sqrt{2}} \left( \cos(\theta) \left( \Pi_{00} + \Pi_{11} \right) + \sin(\theta) \left( \Pi_{01} - \Pi_{10} \right)\right) \label{eq:measop1}\\
M_{+-} &= \frac{1}{\sqrt{2}} \left( \cos(\theta) \left( \Pi_{01} + \Pi_{10} \right) + \sin(\theta) \left( \Pi_{00} - \Pi_{11} \right)\right) \label{eq:measop2} \\
M_{-+} &= \frac{1}{\sqrt{2}} \left( \cos(\theta) \left( \Pi_{01} + \Pi_{10} \right) - \sin(\theta) \left( \Pi_{00} - \Pi_{11} \right)\right) \label{eq:measop3} \\
M_{--} &= \frac{1}{\sqrt{2}} \left( \cos(\theta) \left( \Pi_{00} + \Pi_{11} \right) - \sin(\theta) \left( \Pi_{01} - \Pi_{10} \right)\right) \label{eq:measop4}\end{aligned}$$
![\[fig:mem\] Quantum circuit representation of the measurement described in Sec. \[sec:two-qubit2\]. This time, Alice and Bob each applies a rotation and a CNOT interaction between their qubits and their shared maximally-entangled meter (MEM). ](Figures/fig5.eps){width="45.00000%"}
Using Eq. , we obtain the same POVM as in Sec. \[sec:two-qubit1\]: $$\begin{aligned}
E_{\pm1}=\frac{1}{2}\left( \mathbb{1} \pm \cos(2\theta) \sigma_{z_A}\sigma_{z_B} \right) \label{eq:povm4}\end{aligned}$$
However in this case, since $M_{++}\neq M_{--}$ and $M_{+-}\neq M_{-+}$, we see that a same global result can lead to two different state evolutions. Indeed, some knowledge about the local state of the system can be retreieved from the phase information in the final state. Ignoring the individual outcomes (coarse-graining) thus adds classical noise to the system: the post-measurement state is in general mixed even if the initial state of the system was pure. Such a measurement process is sometimes labeled as an *inefficient quantum measurement* [@Wiseman2009].
The amount of classical noise introduced by the coarse-graining can be evaluated via the difference in purity between the initial and the final states $\Delta\gamma$. It is found to be maximal when the initial state is an equal (in modulus) superposition of states associated with different global results, for instance $\ket{+}_A\ket{+}_B$.
In this case, the purity degradation $\Delta\gamma$ (going from an initially pure state $\gamma=1$ to a mixed state $\gamma<1$) can be related to the measurement strength $\mathfrak{S}$: $$\begin{aligned}
\Delta\gamma=\frac{1-\mathfrak{S}^2}{2}\end{aligned}$$
We see that for a strong measurement ($\mathfrak{S}=1$) the system purity is unaffected, whereas for a weak measurement ($\mathfrak{S}\rightarrow0$), the system purity tends to $\frac{1}{2}$.
Generalization and discussion {#sec:discussion}
=============================
We saw previously that for a nonlocal generalized measurement to be efficient, i.e. without added classical noise, the entanglement of the meter state need to be adjusted in accordance with the desired measurement strength. Hereafter, we shall use the concurrence [@Wootters1998] as our main measure of entanglement, defined as follows for a pure two-qubit state: $$\begin{aligned}
C\equiv2\lambda_0\lambda_1\end{aligned}$$ where $\lambda_0$ and $\lambda_1$ are the Schmidt coefficients.
As was shown in Sec. \[sec:two-qubit1\], for a nonlocal measurement to be efficient, the meter state should be such that coefficients associated to same global outputs should be equal, as in Eq. : $$\begin{aligned}
\frac{1}{\sqrt{2}}\left(\cos{\theta}\ket{00}+\sin{\theta}\ket{01}+\sin{\theta}\ket{10}+\cos{\theta}\ket{11}\right) \end{aligned}$$
It turns out that in this case, the resulting measurement strength $\mathfrak{S}$ is directly equal to the concurrence $C$ of the meter state: $$\begin{aligned}
C=\mathfrak{S}
\label{eq:concurr_strength}\end{aligned}$$
Let us now turn to the case when, as in Sec. \[sec:two-qubit2\], the entanglement $C$ contained in the meter state is higher than the desired measurement strength $\mathfrak{S}$. It is then impossible to generate an ideal meter state, but one can still obtain the desired strength by applying appropriate local unitaries in order to prepare the following state: $$\begin{aligned}
\frac{1}{\sqrt{2}}\left(\cos{\theta}\ket{00}+e^{i\phi}\sin{\theta}\ket{01}+\sin{\theta}\ket{10}+\cos{\theta}\ket{11}\right) \end{aligned}$$ This is a generalized form of Eq. .
The resulting phase $\phi$ is linked to the meter entanglement $C$ and the measurement strength $\mathfrak{S}$ by the relation: $$\begin{aligned}
\cos^2{\left(\frac{\phi}{2}\right)}=\frac{1-C^2}{1-\mathfrak{S}^2}
\label{eq:phi_relation}\end{aligned}$$ The ideal case of Sec. \[sec:two-qubit1\] and the case of Sec. \[sec:two-qubit2\] are recovered by setting $\phi=0$ and $\phi=\pi$, respectively.
The excess entanglement manifests itself through the added phase $\phi$, which in turn is responsible for the purity degradation of the post-measurement system state. As in \[sec:two-qubit2\], this additionnal classical noise is maximal when the system being measured is initially in the state $\ket{+}_A\ket{+}_B$. The purity degradation can then be written as: $$\begin{aligned}
\Delta\gamma=\frac{1}{2}\left\{1-\left(\cos^2{\frac{\phi}{2}}+\mathfrak{S}\sin^2{\frac{\phi}{2}}\right)^2\right\}
\label{eq:gamma_relation}\end{aligned}$$ We recover the efficient measurement case ($\Delta\gamma=0$) by setting $\phi=0$ and the extreme noisy case of Sec. \[sec:two-qubit2\] ($\Delta\gamma=\frac{1}{2}$) by setting $\phi=\pi$.
One can combine relations and to numerically evaluate the noise, as represented in [Fig. \[fig:noise\]]{}.
![\[fig:noise\]Upper bound on the purity degradation $\Delta\gamma$ as a function of the meter concurrence $C$ and the measurement strength $\mathfrak{S}$. The case $C=\mathfrak{S}$ of Eq. is represented as a straight line, and corresponds to an efficient measurement with zero classical noise.](Figures/fig6.eps){width="40.00000%"}
We see that in order to make a measurement of strength $\mathfrak{S}$, one needs at least an amount of entanglement equal to $\mathfrak{S}$. A consequence of this fact is that a nonlocal strong measurement can only be achieved using Bell states. We also notice that the noise increases non-linearly as the measurement strength deviates from the meter entanglement.
Conclusions {#sec:conclusions}
===========
In this Paper, we discussed a new approach to measure nonlocal spin products, using the formalism of generalized quantum measurements. We found that one can achieve an efficient genuine nonlocal generalized measurement using a non-maximally entangled meter state. In particular, we established relations between the desired measurement strength and the necessary entanglement for the measurement to be efficient, that is to say without any additional classical noise. The effect of excessive entanglement was evaluated and found to be detrimental to the purity of the post-measurement state, but not to the overall measurement statistics. Another advantage of this new measurement scheme is that it does not require any quantum erasure step after the interaction. This approach is thus remarkably resource-efficient compared to other already existing schemes [@Brodutch2016; @Kedem2010] and does not involve probabilistic steps. It is also feasible using linear optics, using hyperentangled photon pairs for instance [@Xu2019].
For clarity purposes, we focused our attention on the measurement of the spin product $\sigma_{z_A}\otimes\sigma_{z_B}$, but the proposed scheme can be easily adapted to measure any nonlocal spin product by applying appropriate one-qubit gates. Spin product measurement is a special case of nonlocal measurement as it is one of the few that can be directly measured in the von Neumann paradigm without violating causality. Measuring spin products is crucial in tests of quantum nonlocality, such as testing Bell inequalities. Measuring a spin product as been shown to be equivalent to measuring a modular sum, a relatively easier task. The question of whether or not our approach can be extended to more general observables remains open.
A promising application for this scheme resides in the measurement of weak values [@Aharonov1988][@Laloe2012] in a nonlocal setting, which can be obtained directly as the weak limit of postselected conditioned averages [@Dressel2010]. Measuring nonlocal observables is also important in quantum error correction [@Gottesman1997] and variable measurement strength could be useful quantum computing without strong measurements [@Lund2011].
The authors wish to thank Aharon Brodutch for valuable discussions and Lev Vaidman for pointing us to relevant references. This research was supported in part by JSPS KAKENHI Grant Number JP18J10639 and by MEXT Quantum Leap Flagship Program (MEXT Q-LEAP) Grant Number JPMXS0118067581. P.V. thanks Tohoku University Division for Interdisciplinary Advanced Research and Education for their financial support.
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---
abstract: 'Majorana bound states in topological Josephson junctions induce a $4\pi$ period current-phase relation. Direct detection of the $4\pi$ periodicity is complicated by the quasiparticle poisoning. We reveal that Majorana bound states are also signaled by the anomalous enhancement on the critical current of the junction. We show the landscape of the critical current for a nanowire Josephson junction under a varying Zeeman field, and reveal a sharp step feature at the topological quantum phase transition point, which comes from the anomalous enhancement of the critical current at the topological regime. In multi-band wires, the anomalous enhancement disappears for an even number of bands, where the Majorana bound states fuse into Andreev bound states. This anomalous critical current enhancement directly signals the existence of the Majorana bound states, and also provides a valid signature for the topological quantum phase transition.'
address:
- 'School of Physics, Sun Yat-sen University, Guangzhou 510275, China'
- 'Department of Physics, Shaoxing University, Shaoxing 312000, China'
author:
- Hong Huang
- 'Qi-Feng Liang'
- 'Dao-Xin Yao'
- Zhi Wang
bibliography:
- '<your-bib-database>.bib'
title: Critical Current Anomaly at the Topological Quantum Phase Transition in a Majorana Josephson Junction
---
Majorana bound states ,Critical current ,Topological quantum phase transition ,Topological superconductors ,Josephson effect
Introduction
============
The study of topological superconductivity has seen considerable progress since the theoretical proposal of implementing proximity effect to convert the conventional s-wave superconductivity into topological non-trivial p-wave superconductivity[@fu; @sau; @lutchyn; @oreg; @sato]. In general, topological superconductors contain all superconducting systems with a well defined non-zero topological number[@kanermp; @zhangrmp]. In practice, however, topological superconductors often refer to those superconductors where the non-trivial topology give rise to a type of bizarre boundary states, the Majorana bound states (MBSs)[@read; @kitaevwire; @aliceareview; @beenakkerreview; @franzreview]. MBSs are self-conjugate superconducting quasiparticles, which obey non-Abelian statistics[@ivanov; @teo; @aliceanphy]. Spatially separated MBSs define non-local topological qubits which are immune from local decoherence[@kanermp]. Braiding the MBSs acts as topological gates on these topological qubits[@kanermp; @ivanov]. Therefore, MBSs are considered as a promising cornerstone for realistic quantum computation[@bkreview], despite that these topological gates are not sufficient to make universal quantum operations.
MBSs have been reported in a number of experiments[@mourik; @rokhinson; @das; @yazdani; @albrecht; @jia; @kouwenhoven; @molenkamp; @moler; @lv], which mostly study the hybrid systems consisting of a conventional s-wave superconductor and a spin-orbit coupling nanowire[@mourik; @rokhinson; @albrecht]. The wire becomes topological superconducting with a combination of appropriate Zeeman energy, fine tunned chemical potential, strong spin-orbit coupling, and proximity induced Cooper pairing. Each end of the wire pins one MBS, which brings in unique transport signatures such as the resonant Andreev effect[@law], the crossed Andreev effect[@beenakker08], the quantized thermal conductance[@Akhmerov], and the fractional Josephson effect[@kitaevwire; @jiang]. The fractional Josephson effect draws particular interest since it is a phase coherent signal. It describes the coherent single electron tunneling through the MBS channel, which contributes a $4\pi$ period current-phase relation[@kitaevwire]. Direct measurement of this $4\pi$ period current-phase relation is hindered by the stringent requirement of complete elimination of the quasiparticle poisioning and MBS poisioning[@wangsr].
Quasiparticle poisioning is the external single-electron tunneling into nanowire, causing decoherence in the system. MBS poisioning is the annihilation of two MBSs causes by the strong interaction between them. Up to now, only indirect signals have been discovered experimentally in several systems[@rokhinson; @kouwenhoven; @molenkamp].
![(Color online). Schematic setup of a Majorana Josephson junction, with two conventional superconductors connected by a spin-orbit couping nanwire. A voltage gate is applied in the middle of the wire to produce a tunneling barrier. Four Majorana bound states $\gamma_{1,2,3,4}$ stay at the two sides of the tunneling barrier and the two ends of the wire.[]{data-label="F1"}](Fig1.pdf){width="50.00000%"}
Aside from the current-phase relation, Josephson junctions are also characterized by the critical current, which is the threshold of the phase driving current. For a topological Josephson junction, the critical current consists of two parts: the $4\pi$ period component from the Majorana channel and the $2\pi$ period component carried by the quasiparticle channels[@beenakkerprb]. The former is small in comparison with the latter for bulk systems, since there is only one Majorana channel but many other quasiparticle channels[@grosfeld]. However, the one-dimensional topological Josephson junction behaves differently when the electron tunneling is suppressed by a tunneling barrier. For these systems, the number of quasiparticle channels is restricted. In addition, each quasiparticle channel contributes only a small current for a high tunneling barrier, since it involves a second order perturbation process. In contrast, the Majorana channel contributes a larger Josephson current through a first order perturbation process. Therefore, the MBSs should contribute a much larger critical current than the conventional quasiparticle channels for nanowire topological junctions. If the wire is switched between topological and trivial states by a control parameter such as the Zeeman energy, we expect an anomalous sharp step for the critical current at the topological quantum phase transition (TQPT) point. The TQPT was predicted to be signaled by a quantized thermal conductance in Ref \[29\]. Here, we propose using this critical current anomaly as an alternative signal for the TQPT.
In this work, we show the existence of anomalous critical current enhancement in one-dimensional topological Josephson junctions. For this purpose, we adopt a nanowire hybrid Josephson junction as sketched in Fig. 1. The system consists of two conventional s-wave superconductors and a spin-orbit coupling nanowire. The wire is divided by a tunneling barrier which is produced by a voltage gate. This hybrid junction walks through a TQPT between the trivial phase and the topological superconducting phase when the Zeeman energy is increased from zero. We show that the critical current of the junction increases by orders when the system enters the topological phase, and exhibits a step-like feature at the TQPT point. We also study the behavior of the critical current when the Zeeman field is rotated, and show that the critical current anomaly also appears at the TQPT point. We further reveal that this step-like anomaly disappears for wires with an even number of sub-bands due to the fusion of the MBSs. These anomalous features for the critical current provide a signal for the TQPT and the existence of the MBSs, which must be helpful experimental detection since measuring the critical current is a routine procedure in experiments.
The rest of this paper is organized as follows. We present a toy model and the analytical results based on perturbation calculations in section II. Then we use Bogoliubov-de Gennes (BdG) approach to simulate the critical current of a realistic nanowire junction, and show the results for the increasing and rotating the Zeeman fields in section III. Afterwards we study the critical current for multi-layer systems in section IV. Finally, we give a summary in section V.
Toy Model and Critical Current Enhancement
==========================================
The Majorana Josephson junction is consisting of a superconducting nanowire which is divided into two segments by a voltage gate[@mourik]. We view these two segments as two isolated wires, which are connected by the electron tunneling through the potential barrier. The minimal model for each segment is an one-dimensional chain with spin-orbit coupling, Zeeman energy, and superconducting pairing[@sarmamodel], $$\begin{aligned}
H_\alpha&&=
- t_\alpha \sum_{ \langle i , j \rangle, \alpha, \sigma} c_{i , \alpha, \sigma}^\dagger c_{j , \alpha,\sigma}
+\sum_{i, \alpha} \Delta_\alpha e^{i\phi_\alpha} c_{i, \alpha,\uparrow}^\dag c_{i, \alpha,\downarrow}^\dag
\nonumber\\
&& + \eta_\alpha \sum_{i, \alpha,\sigma, \sigma'}^{n} c_{i+1, \alpha,\sigma}^{\dag} (i\sigma_y)_{\sigma\sigma^\prime} c_{i , \alpha,\sigma'}
- \mu_\alpha \sum_{ i , \alpha,\sigma} c_{i, \alpha,\sigma}^\dagger c_{i, \alpha,\sigma}
\nonumber \\
&& + \sum_{i, \alpha,\sigma\sigma^\prime} c_{i, \alpha, \sigma}^\dag (V_x \sigma_x)_{\sigma\sigma^\prime}
c_{i, \alpha,\sigma'} , \end{aligned}$$ where $\alpha = L, R$ represents the left and the right segments of the wire, $\sigma = \uparrow, \downarrow$ represents the spin of the electron, $t$ is the nearest neighbor hopping in the tight-binding model, $\mu$ is the chemical potential, $\eta$ represents the spin-orbit coupling, $\Delta$ is the superconducting gap from the proximity effect, $\phi$ is the superconducting phase, and $V_{x}$ is the Zeeman energy from the horizontal Zeeman field. For simplicity, we consider identical parameter for the two segments $t_\alpha = t$, $\eta_\alpha = \eta$, $\Delta_\alpha = \Delta$, and $V_\alpha = V$. However, this does not change the physical results. The left chain is connected to the right chain with an electron tunneling Hamiltonian, $$\begin{aligned}
H_T = T \sum_{\sigma} (c_{L, 0 ,\sigma}^{\dagger} c_{R,0,\sigma} +c^\dagger_{R,0,\sigma}c_{L, 0 ,\sigma}) .\end{aligned}$$ where $T$ is the tunneling matrix between the two segments. To avoid quasiparticle poisioning and MBS poisioning, we assume that the system is in open boundary condition and the nanowire is long enough to seperate MBSs. In this paper, we assume each segment has 500 sites.
![(Color online). The critical current of the single-band Josephson junction in response to the Zeeman energy $V_x$. The solid, dashed, and dotted lines are for the case of tunneling matrix $T=0.002t$, $0.005t$, and $0.01t$, respectively. Each curve was normalized to the value of $V_x = 0$ for clarity. We choose the parameters in proportion to $\Delta$, other parameters are taken as $\mu= -43.6\Delta $, $t = 22\Delta$, $\eta=2.68\Delta$. Each current value in $V_x$=0 are $6.4\times10^{-7}\frac{e\Delta}{2\hbar}$, $4.0\times10^{-6}\frac{e\Delta}{2\hbar}$, $1.6\times10^{-5}\frac{e\Delta}{2\hbar}$, respectively.[]{data-label="F2"}](Fig2.pdf){width="0.5\columnwidth"}
The tunneling matrix is well controlled by the voltage gate. When the voltage gate creates a high potential barrier, the tunneling matrix $T$ becomes small and serves as a valid perturbation parameter. In the perturbation approach, two types of tunneling processes contribute to the Josephson current: the second order tunneling process through the quasiparticle channels, and the first order tunneling process through the MBSs[@aliceanphy]. We now make a qualitative estimation on the amplitude these two tunneling processes. We first examine the supercurrent from the quasiparticle channels. For this purpose, we take the trivial limit of zero spin-orbit coupling and Zeeman energy $\eta = V_x = 0$. Then the two segments become two simple one-dimensional s-wave superconductors. In this case, we can calculate the Josephson current with standard Green function technique (see appendix for details). The lowest order contribution to the current is second order in the tunneling matrix. We obtain a $2\pi$ period Josephson current $I = I_1 \sin \theta$, with $\theta= \phi_L - \phi_R$ the phase difference between the two chains. The critical current is $$\begin{aligned}
\label{conventionalcritical}
I_1 = \frac{ e \Delta T^2}{ 2 (1 - \mu^2/ 4 t^2) \hbar t^2},\end{aligned}$$ where $e$ represents the electron charge. We notice that this critical current indeed quadratically depends on the tunneling matrix $T$, which reflects the second order perturbation contribution in the tunneling processes.
We next consider the supercurrent carried by the MBS channel. The tight-binding model described by Eq. (1) enters the topological state in the presence of appropriate spin-orbit coupling and large Zeeman energy $V^2_x > \Delta^2 + (\mu-2t)^2$. Four MBSs appears at the four ends of the two chains. The two MBSs in the junction area, $\gamma_1$ and $\gamma_2$ couple together and contribute to the Josephson current. They support coherent single electron tunneling, which gives a psudo-$4\pi$ periodicity in the current-phase relation (see appendix for details). We obtain a Josephson relation $I = I_m \langle i \gamma_1 \gamma_2 \rangle \sin \frac{\theta}{2} $ with the critical current, $$\begin{aligned}
\label{Majoranacritical}
I_2 = \frac{e \nu T}{2\hbar},\end{aligned}$$ where $\nu = \sum_\sigma \langle \gamma_1 \gamma_2 c_{L, 0 ,\sigma}^{\dagger} c_{R,0,\sigma}\rangle$ is the overlapping between the wave function created by the tunneling Hamiltonian and the wave function created by the two MBSs. The critical current is linearly depending on the tunneling matrix, which is a signature of the degenerate perturbation contribution. The current-phase relation of this Josephson current is unique. It not only depends on the phase difference $\theta$ but also depends on the quantum average Majorana parity state $\langle i \gamma_1 \gamma_2 \rangle$. This gives an extra degree of freedom for the Josephson relation, which leads to three typical scenarios. First, if the quasiparticle poisoning and the Majorana poisoning are totally ignored, the Majorana parity operator $i \gamma_1 \gamma_2$ is a conserved quantity. Then we have an exact $4\pi$ period Josephson relation $I = \pm I_1 \sin \theta /2$. This is the so-called fractional Josephson effect which has been well discussed in literature[@beenakkerprb; @law2]. Second, if the adiabatic processes are considered and all poisoning are included[@kitaevwire], the Josephson relation will reduce to a skewed $2\pi$ period one $I \approx I_2 \frac{\cos \theta /2}{|\cos \theta / 2|}\sin \theta /2$. In experiments, this skewed $2\pi$ period Josephson relation is also used as a signal for MBSs[@moler; @lv]. Finally, if the Majorana parity operator is treated as a quantum psudo-spin, we would have a correlated dynamics of the phase difference and the psudo-spin[@wangLZS]. The former obeys the classical Newton equation and the latter obeys the Shcrödinger equation. This correlated dynamics leads to rich phenomena, and provides methods for controlling the Majorana qubit[@wangLZS].
We focus on the comparison between the critical currents of the Majorana channel and the conventional channel, as presented in Eqs (3) and (4). The Majorana channel gives a Josephson current which linearly depends on the tunneling matrix $T$, while the conventional Josephson current is a quadratic function of $T$. We look at the ratio between these two critical currents, and obtain a ratio of $$\begin{aligned}
R = I_2 / I_1 = \frac{ (1 - \mu^2/ 4 t^2) \nu t^2 }{ \Delta T}.\end{aligned}$$ We find that this ratio $R$ must be a large number for tunneling barrier junction which has a small tunneling matrix. It becomes larger when we reduce the tunneling matrix $T$ by increasing the voltage on the gate. In principle, we have no limit on reducing the tunneling matrix. At small $T$ limit, we face a tremendous increasing of the critical current when the system goes from the trivial phase into the topological phase. If we check the behavior of the critical current as a function of the control parameter, say Zeeman energy, we must find a sharp step-like function at the TQPT point.
Critical Current Steps at the Topological Quantum Phase Transition Point
========================================================================
We have obtained analytic results of the critical current of the junction for two typical parameters. One is for the trivial phase with the critical current shown in Eq. (3) and the other is for the extreme topological phase with the critical current shown in Eq. (4). However, we must access to the critical current for general parameters if we want to study the TQPT, because the TQPT involves a continuous modulation of a control parameter. This task is difficult analytically. Therefore we adopt the numerical BdG approach to obtain Josephson current of the spin-obit coupling chain model shown in Eqs. (1) and (2). In the BdG approach, we calculate the energy spectrum $E_n$ of the total Hamiltonian as a function of the phase difference $\theta$, and obtain the Josephson current with phase derivative of the energy spectrum[@beenakkerprl], $I(\theta)=\frac{e}{\hbar} \sum_n {\partial E_n(\theta)}/ {\partial \theta}$. The maximal value of the Josephson current gives the critical current of the junction. We change the Zeeman energy $V_x$ and calculate the critical current every time. Finally, we obtain the critical current as a function of the Zeeman energy, and show the results in Fig. 2. We study three different tunneling matrix. For clear comparison, we normalized each curve with its value for $V_x =0$. We see that all three critical currents exhibit step-like features around the TQPT point $V_x = \sqrt{\Delta^2 + (\mu-2t)^2}$. The steps become steeper for smaller tunneling matrix due to the increasing of the ratio $R$ in Eq. (5). This anomalous critical current provides a clear marker for the TQPT. We already show that the critical current enhancement directly comes from the Majorana channel. Therefore, the critical current anomaly also gives a valid signal for the existence of MBSs.
{width="0.5\columnwidth"}
\[F4\]
In experiments, another method to achieve TQPT is to rotate the Zeeman field[@mourik]. The study of rotating the Zeeman field on nanowire systems has provided interesting results. Zero energy excitations are found robust within certain Zeemn field directions, which agree with the theoretical predictions. However, the energy gap closing, which should happen at the TQPT points, is not observed[@mourik]. Here, we show that the critical current is a good signal for marking the TQPT when the Zeeman field is rotated. The rotation of the Zeeman energy requires a y-component in the Zeeman energy. We add one more term in the Hamiltonian of the chain in Eq. (1), $$\begin{aligned}
H' = H + \sum_{\bf {r}, \sigma\sigma^\prime} c_{\bf {r},\sigma}^\dag (V_y\sigma_y)_{\sigma\sigma^\prime}
c_{\bf {r},\sigma'},\end{aligned}$$ where $V_y$ represents the Zeeman energy in the y-direction. We fix the amplitude of the Zeeman energy $V = \sqrt{V^2_x + V^2_y }$ and rotate its angle. Then we calculate the critical current of the junction. We first consider a rotation of the magnetic field in the total junction, and show the results in Fig. 3a. We find that The critical current is at the maximum when the direction of the Zeeman field is along the x-axis. This large critical current is mainly carried by the Majorana channel. Then the critical current gradually decreases with the rotation of the Zeeman energy.This should come from the reduction of the MBS wave function overlapping factor $\nu$. When the rotation becomes larger, the TQPT happens and the critical current suddenly drops to a small value. This small critical current is entirely carried by the quasiparticle channel, therefore is not sensitive to the rotation of the Zeeman field. Finally, the Zeeman field is rotated in the reverse direction and the TQPT happens again. The critical current rises and reaches the maximum when the magnetic field rotates to the inverse of the x-axis. We then consider the situation for only rotating the Zeeman field on the right chain, with the results shown in Fig. 3b. We obtain similar results. However, the critical current is asymmetric to the rotation angle, since the MBS wave function overlapping is suppressed when the two chains have opposite Zeeman fields.
The angle dependence of the critical current gives a full landscape. This should be helpful for the experiments. The critical current is easy to measure. Therefore, it could provides information on the TQPT for those systems where the energy gap closing is not observed at the TQPT points.
Multi-band Model
================
Realistic nanowires can be multiband systems, with the number of the sub-bands determined by the width of the wire. We model the multi-band wires by expanding the chain into multi-layers. Therefore, the Hamiltonian in Eq. (1) should be slightly modulated into, $$\begin{aligned}
H_{L,R}&&=
- t \sum_{ \langle \bf {r } , \bf {r'} \rangle, \sigma} c_{\bf {r} , \sigma}^\dag c_{\bf{r'}\sigma}
+ \sum_{\bf {r}, \sigma\sigma^\prime} c_{\bf {r},\sigma}^\dag (V_x\sigma_x)_{\sigma\sigma^\prime}
c_{\bf {r},\sigma'}
\\\nonumber
&& + \frac{\eta}{2} \sum_{\bf{r}, \sigma, \sigma'} (c_{\bf{r+\delta_x}, \sigma}^{\dag} (i\sigma_y)_{\sigma\sigma^\prime} c_{\bf {r},\sigma'} -c_{\bf{r+\delta_y}, \sigma}^{\dag} (i\sigma_x)_{\sigma\sigma^\prime} c_{\bf {r},\sigma'} )
\\\nonumber
&& +\sum_{\bf{r}} (\Delta e^{i\phi_{L,R}} c_{\bf {r},\uparrow}^\dag c_{\bf {r},\downarrow}^\dag+h.c)
- \mu\sum_{ \bf {r } , \sigma} c_{\bf {r} , \sigma}^\dag c_{\bf{r}\sigma},\end{aligned}$$ where $\bf{r}$ is the position of the multi-layer chain, and $\bf{\delta_x}$ is the unit step in the x-direction. The energy spectrum and eigenfunctions of this multi-chain model have been studied[@potter]. The results show that end MBSs fuse and disappear for even number of layers, while MBSs still exist for odd-layer systems. Here we study the critical current of the multi-chain system by adding a tunneling Hamiltonian similar to Eq. (2), where the electron tunneling are restricted to the same layer. We show the critical current for one, two, and three layers in Fig. 3. We see that the critical current anomaly exist for odd number of layers and disappear for even number of layers. This directly links the critical current anomaly with the existence of free MBSs. For even-layer system, all MBSs pair together and forms local Andreev bound states. These Andreev bound states also carry Josephson current, but through second order tunneling processes. Therefore, the critical current enhancement disappears. We also notice that the step become smoother with increasing layers. It comes from the enhancement of the critical current of quasiparticle channels, which increases with increasing layers. If the number of layers is large, the system changes from quasi-1d into quasi-2d. Then the critical current enhancement will entirely disappear.
{width="0.5\columnwidth"}
Conclusion
==========
In summary, we study the critical current in the topological nanowire Josephson junction, where a potential barrier suppresses the tunneling strength. We investigate the tunneling processes which contribute to the Josephson current and give analytical results of the critical current for typical parameters in the topological phase and the trivial phase. We find that the ratio between these two critical currents is linearly depending on the tunneling matrix. Therefor there must be a critical current anomaly between these two phases when the tunneling matrix is small enough. We then use Bogoliubov-de Gennes approach to investigate the critical current under increasing and rotating Zeeman fields. We show that the critical current indeed has a sharp step-like feature at the topological quantum phase transition. We then study multi-band systems, and find that the critical current enhancement disappears for even-layer systems, where the Majorana bound states fuse into Andreev bound states. Our study provides an experimentally accessible signal for the topological quantum phase transition and the existence of Majorana bound states.
Acknowledgement {#acknowledgement .unnumbered}
===============
This work is supported by NSFC-11304400, NSFC-61471401, and SRFDP-20130171120015. Q.F.L is supported by NSFC-11574215. D.X.Y. is supported by NSFC-11074310, NSFC-11275279, SRFDP-20110171110026, and NBRPC-2012CB821400.
Critical Current from Quasiparticle Channel
===========================================
The Josephson current from the quasiparticle channel can be obtained by considering the trivial limit of zero spin-orbit coupling and Zeeman energy, $\eta = V_x = 0$. For this simple parameter, the two superconducting segments become conventional s-wave superconductors, and the tunneling Josephson current is given as[@Mahan], $$\begin{aligned}
I(t)&&= -e \langle \frac{dN_L}{dt} \rangle =- \frac{ei}{\hbar}\langle [ H, { N_L}]\rangle
\\\nonumber
&&= - \frac{ei}{\hbar} \sum_\sigma \langle \psi(t)| T c^{\dagger}_{L,0,\sigma} c_{R,0,\sigma} -h.c |\psi(t)\rangle,\end{aligned}$$ where where $|\psi(t)\rangle$ is the ground state wave function of the total system. We implement periodic boundary conditions for the two lead of the junction. This does not change the results since no edge state exist in the trivial phase. Then we can make Fourier transformations $c^\dagger_{L,j,\sigma} = \frac{1}{\sqrt {N}}\sum_k e^{-ikj} c^\dagger_{k,\sigma}$, and $c_{R,j,\sigma} = \frac{1}{\sqrt {N}}\sum_p e^{ipj} c_{p,\sigma}$, with $N$ the number of sites at each lead. The current is rewritten as, $$\begin{aligned}
I(t ) && = - \frac{e}{N \hbar} {\rm Im} \sum_{{ k},{p},\sigma} \langle\psi(t)| T c^{\dagger}_{{k},\sigma} c_{{ p},\sigma} |\psi(t) \rangle
\\\nonumber
&& = \frac{e}{N \hbar} {\rm Im} \sum_{{ k},{p},\sigma} T \langle\psi_0| e^{-iHt} c^{\dagger}_{{k},\sigma} c_{{ p},\sigma} e^{iHt} |\psi_0\rangle,\end{aligned}$$ where $|\psi_0 \rangle$ is the ground state at the time origin. For [*non-degenerate*]{} system, this ground state evolution is obtained by going to the [*interaction representation*]{}, $$\begin{aligned}
I_S(t )
= \frac{e}{N\hbar} {\rm Im} \sum_{{k},{p},\sigma} T \langle \phi_0 | S^\dagger(t,-\infty) \hat c^{\dagger}_{{k},\sigma} (t) \hat c_{{ p},\sigma}(t) S(t,-\infty) |\phi_0 \rangle, \end{aligned}$$ where $\hat c$ represents the operators in the interaction representation, $\phi_0$ is the ground state wave-function in absence of the tunneling Hamiltonian, $S(t,t')$ is the S-matrix obtained by the evolution operator, which could be expanded to the first order as, $$\begin{aligned}
S(t,-\infty)=\left[1-i\int_{-\infty}^t dt_1 \hat H_T(t_1)\right] + O(T)^2,\end{aligned}$$ We omit higher order perturbations and take the lowest order contributions. The current is expressed as a correlation function, $$\begin{aligned}
I(t)= - \frac{eT^2}{N^2 \hbar} \sum _{k,p,k',p'\sigma}\int_{-\infty}^{\infty} dt' \Theta(t-t') \langle \phi_0| [\hat c^{\dagger}_{{k},\sigma} (t) \hat c_{{ p},\sigma}(t),\hat c^{\dagger}_{{k'},\sigma} (t') \hat c_{{ p'},\sigma}(t')]|\phi_0 \rangle. \nonumber\\\end{aligned}$$ This correlation function can be analytically calculated with the standard [*Green function*]{} technique[@Mahan], where we could draw out the Josephson current part as, $$\begin{aligned}
I = 2 e {\rm Im } [ e^{-2ieVt/\hbar} \Pi_{ret} (eV)],\end{aligned}$$ where the retarded Green function is the analytic continuation of the Matsubara Green function, $$\begin{aligned}
\Pi_{ret} (i\Omega) = 2 T^2 \sum_{k,p,i\omega} \Im^\dagger(k,i\omega) \Im(p, i\omega-i\Omega), \end{aligned}$$ where $\Im^\dagger$ is the off-diagonal Matsubara Green function in superconductors. We finally obtain the dc Josephson current by taking $eV = 0$, $$\begin{aligned}
I = I_1 \sin \theta,\end{aligned}$$ with a critical current, $$\begin{aligned}
I_1 = \frac{e \Delta^2 T^2}{ N^2 \hbar} \sum_{k,p} \frac{1}{E_{ k} E_{ p} (E_{ k}+E_{ p})},\end{aligned}$$ where the energy spectrum of the superconductor is $E_{k} = \sqrt{(-2t \cos k - \mu)^2 + \Delta^2}$. The summation for the critical current changes into an integration in the continuous limit, $$\begin{aligned}
I_1 =\frac{ 4e \Delta^2 T^2 N_L N_R }{N^2 \hbar}\int^\infty_\Delta dE \frac{\rho(E)}{E} \int^\infty_\Delta dE' \frac{\rho(E')}{E'} \frac{1}{E+E'},\end{aligned}$$ where $N_L$ and $N_R$ are the average density of states near the Fermi surface, and $\rho(E) = \frac{E}{\sqrt{E^2 - \Delta^2}}$ is the superconducting density of states. This integral over energy is an elliptic integral, which can be integrated out as, $$\begin{aligned}
I_1 && = \frac{2 \pi^2 e N_L N_R \Delta T^2}{N^2 \hbar}.
\end{aligned}$$ For one-dimensional tight-binding system, the Fermi wave vector is determined by the hopping term and the chemical potential $k_F = \arccos (- \mu/ 2t)$. The density of the states at the Fermi surface is the inverse of the slope of the dispersion function $N_L = N_R = \frac{N} {\pi 2 t \sin k_F} = N / 2 \pi t \sqrt{1 - \mu^2 / 4 t^2}$. Plugging this back into Eq. (A9), we arrive at the formula for the critical current in Eq. (3), $$\begin{aligned}
I_1 = \frac{ e \Delta T^2}{ 2 (1 - \mu^2/ 4 t^2) \hbar t^2}.\end{aligned}$$
Josephson Current From Majorana Channel
=======================================
When the junction enters the topological phase, four isolated MBSs appears in the edges of the wire. Two of them $\gamma'_L$ and $\gamma'_R$ locate at the ends of the wire, while the other two $\gamma_L$ and $\gamma_R$ locate at the two sides of the junction. These four MBSs form the four-fold degenerate ground state for the topological superconductor. This ground state degeneracy prevents the application of the standard interaction picture and S-matrix expansion. However, we can use a degenerate perturbation approach to obtain the current. We illustrate the calculation of the Majorana Josephson current with the simple Kitaev model which grasps the essence of the topological superconductivity. The Hamiltonian for the two segments of the wire writes as, $$\begin{aligned}
H_{{\rm K}} = && \sum_{j} \left[- t c^\dagger_{j} c_{j+1} + e^{i\phi_{L,R}} c_{j} c_{j+1} + h.c. \right] - \mu \sum_j c^\dagger_{j} c_{j},\end{aligned}$$ where $\phi_{L,R}$ represents the superconducting phase for the left and right superconductors, respectively. We take identical parameters for the two superconductors for simplicity. Follow Kitaev, we take the transformation from electron representation to Majorana representation[@kitaevwire], $$\begin{aligned}
&& \gamma_{j,A} = e^{i \phi_{L,R}} c_j + e^{-i \phi_{L,R}} c^\dagger_j
\\\nonumber
&& \gamma_{j,B} = -i e^{i \phi_{L,R}} c_j + ie^{-i \phi_{L,R}} c^\dagger_j.\end{aligned}$$ Then the model is rewritten with Majorana operators,
$$\begin{aligned}
H_{{\rm K}}
= && \frac{ - i {\mu} }{2}\sum_{j} \gamma_{j,A } \gamma_{j,B} + \frac{i(t+ \Delta)}{2}\sum_{j} \gamma_{j,B} \gamma_{j+1,A}
\nonumber\\
&&- \frac{i(t- \Delta)}{2}\sum_{j} \gamma_{j,A} \gamma_{j+1,B} ].\end{aligned}$$
The electron tunneling across the junction connects the left and the right superconductors, which is described by a tunneling Ham $$\begin{aligned}
H_{TK} = T c^{\dagger}_{L,0} c_{R,0} + h.c.\end{aligned}$$ where $T$ is the tunneling matrix. The current through the junction comes from the electron tunneling, which can be expressed as[@Mahan], $$\begin{aligned}
I(t) = - \frac{ieT}{\hbar} \langle \psi(t)| c^{\dagger}_{L,0} c_{R,0} -h.c |\psi(t) \rangle,\end{aligned}$$ We notice that the expression for the current is quite similar to the conventional junction. However, we have a key difference that the ground state $|\psi(t) \rangle$ is now degenerate. Therefore it is impossible to go to the interaction representation and apply Green function technique. We must calculate the current with the degenerate perturbation theory. In the zero’s order degenerate perturbation approach, we restrict the wave function $|\psi(t)\rangle$ into the Hilbert-subspace expanded by the degenerate ground states of the unperturbed Hamiltonian without tunneling $H_{KT}$, and ignore all terms which project the wave function out of this subspace. We remember that the degenerate ground states of Kitaev model are defined by the end Majorana operators, therefore, we massage the formula Eq. (B5) by expanding the electron operators with Majorana operators, $$\begin{aligned}
I (t) = &&- \frac{eT i}{2\hbar} \langle \psi(t)|(\gamma_{L,0,B}\gamma_{R,0,A} -\gamma_{L,0,A}\gamma_{R,0,B} ) \sin\frac{\theta}{2}
\nonumber\\
&&+ (\gamma_{L,0,A}\gamma_{R,0,A}+\gamma_{L,0,B}\gamma_{R,0,B} ) \cos \frac{\theta}{2} |\psi(t) \rangle,\end{aligned}$$ where $\theta = \phi_L - \phi_R$ is the phase difference across the junction. Let us first consider the special parameter of $\mu= 0$ and $t=\Delta$, where the two zero energy Majorana operators $ \gamma_{L,0,B}$ and $\gamma_{R,0,A}$ are the Majorana zero modes which define the unperturbed degenerate ground states. Then we drop all three terms which project $|\psi \rangle$ out of the degenerate ground state subspace, and take the only term which is expressed by the zero energy Majorana operators, $$I_M(t)=
\frac {eT} {2\hbar} \sin({\theta}/{2}) \langle \psi(t)| -i\gamma_{L,0,B}\gamma_{R,0,A} |\psi(t)\rangle,$$ where the ground state wave function $|\psi (t)\rangle$ evolves according to the Schrödinger equation, $$-i \hbar \frac {d} {dt} |\psi(t)\rangle =\left[\frac{T}{2} (-i\gamma_{L,0}\gamma_{R,0}) \cos ({\theta}/{2}) \right] |\psi(t) \rangle.$$
For the general parameters of $\mu \neq 0$ or $t \neq \Delta$, the zero energy MBSs are the combination of the Majorana operators[@kitaevwire], $$\begin{aligned}
\gamma_{L} = \sum_j a_j \gamma_{L,j,B},
\gamma_{R} = \sum_j b_j \gamma_{R,j,A}.\end{aligned}$$ These two MBS are localized near the junction area, that is, $a_0 \approx 1, a_{j \neq 0} \approx 0$, and same for $b_j$. Then the current is expressed as, $$\begin{aligned}
I_M(t) =
\frac { \nu eT} {2 \hbar} \sin({\theta}/{2}) \langle \psi(t)| -i\gamma_{L}\gamma_{R} |\psi(t)\rangle,\end{aligned}$$ where $\nu = a_0 b_0 /4$ comes from the overlapping between the tunneling operator $c^{\dagger}_{L,0} c_{R,0}$ and the MBS operator $i \gamma_L \gamma_R$. We arrive at the critical current shown in Eq. (4), $$\begin{aligned}
I_2 = \frac{e \nu T}{2\hbar}.\end{aligned}$$
For these general parameters, there is a exponentially small but non-zero coupling between the two MBSs at the junction $\gamma_L$ and $\gamma_R$ and the two MBSs at the two ends $\gamma'_L$ and $\gamma'_R$, which provides a small coupling Hamiltonian, $$\begin{aligned}
H_\delta = i\delta_L\gamma'_L\gamma_L + i\delta_R \gamma_R \gamma'_R.\end{aligned}$$ where $\delta_{L,R}$ are exponentially suppressed by the length of the wire. This coupling Hamiltonian is small; however, it qualitative changes the quantum dynamics of the ground state wave function $\psi(t)$, which is now governed by the Schrödinger equation, $$-i \hbar \frac {d} {dt} |\psi(t)\rangle =\left[ \frac{- i \nu T}{2} \gamma_{L}\gamma_{R} \cos ({\theta}/{2}) + H_\delta\right] |\psi(t) \rangle.$$ We see that the $H_\delta$ break the local parity conservation defined by the two MBSs $\gamma_L$ and $\gamma_R$ around the junction, thereby in principle destroys the $4\pi$ period Josephson effect.
BdG Formalism for Josephson Current
===================================
We use BdG equation to get the Josephson current by solving Hamiltonian, $$\begin{aligned}
H=H_L+H_R+H_T,\end{aligned}$$ where $H_L$, $H_R$ and $H_T$ are described in euqtion (1) and (2). Each term in Hamiltonian is bilinear term of $c^{\dag}$ and $c$, with a series of parameters such as t, $\Delta$ and $\theta$. For a series of constant parameters such as $t_0$, $\Delta_0$ and $\theta_0$, we can use BdG method to transform electrons and holes into quasi-particles which can diagonalize the Hamiltonian. The relation between electrons, holes and quasi-particles is BdG equation, $$\begin{aligned}
c_{i\uparrow}= \sum_n u_{n\uparrow}\gamma_n+v_{n\uparrow}^\ast\gamma_n^\dag\\
c_{i\downarrow}= \sum_n u_{n\downarrow}\gamma_n+v_{n\downarrow}^\ast\gamma_n^\dag,\end{aligned}$$ where $\gamma_n^\dag$($\gamma_n$) represents the creation(annihilation) operator of quasi-particles, $u_n$ and $v_n$ are a series of parameters that we choose to diagonalize the Hamiltonian. Then the Hamiltonian is written as $$\begin{aligned}
H_{BdG}=E_g+\sum_n \epsilon_n\gamma_n^\dag\gamma_n,\end{aligned}$$ where $E_g$ represents the ground-state energy, and $\epsilon_n$ is the quasi-particle energy. All the negative quasi-particle wave function are summed up to forms the ground state of the Hamiltonian. The quasi-particle energies $\epsilon_n$ are affected by the value of a series of parameters. To calculate the current-phase relation, we choose phase difference $\theta$ to be changeable. In this way, the quasi-particle energies $\epsilon_n$ are functions of phase difference $\theta$, and the Hamiltonian is the function of phase difference $\theta$ too, $$\begin{aligned}
H_{BdG}(\theta)=E_g+\sum_n\epsilon_n(\theta)\gamma_n^\dag\gamma_n.\end{aligned}$$ Then we take the derivative of all the negative quasi-particle energies to give the current phase relation, $$\begin{aligned}
I(\theta)=\frac{ge}{\hbar}\sum_n\frac{\mathrm{d}}{\mathrm{d}\theta}\epsilon_n(\theta),\end{aligned}$$ where $g$ represents the factor that counts spin and other degeneracies. To simplify the calculation, we make $\frac{ge}{\hbar}=1$.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Frustrated magnets can exhibit many novel forms of order when exposed to high magnetic fields, however, much less is known about materials where frustration occurs in the presence of itinerant electrons. Here we report thermodynamic and transport measurements on micron-sized single crystals of the triangular-lattice metallic antiferromagnet $2H$-AgNiO$_2$, in magnetic fields of up to 90 T and temperatures down to 0.35 K. We observe a cascade of magnetic phase transitions at 13.5, 20, 28 and 39 T in fields applied along the easy axis, and we combine magnetic torque, specific heat and transport data to construct the field-temperature phase diagram. The results are discussed in the context of a frustrated easy-axis Heisenberg model for the localized moments where intermediate applied magnetic fields are predicted to stabilize a magnetic supersolid phase. Deviations in the measured phase diagram from this model predictions are attributed to the role played by the itinerant electrons.'
author:
- 'A. I. Coldea'
- 'L. Seabra'
- 'A. McCollam'
- 'A. Carrington'
- 'L. Malone'
- 'A. F. Bangura'
- 'D. Vignolles'
- 'P. G.[van Rhee]{}'
- 'R. D. McDonald'
- 'T. Sörgel'
- 'M. Jansen'
- 'N. Shannon'
- 'R. Coldea'
title: |
Cascade of field-induced magnetic transitions in a frustrated antiferromagnetic metal-\
possible experimental signature of a magnetic supersolid phase
---
Frustrated magnets have proved a rich source of novel magnetic ground states such as spin liquids on triangular lattices and spin ices on pyrochlore lattices [@Balents2010]. Another intriguing possibility, rooted in work on superfluid Helium, is that a frustrated magnet might realise a magnetic analogue of a supersolid, in which broken translational symmetry co-exists with a superfluid order parameter [@Matsuda1970; @Liu1973]. Most experimental and theoretical studies of frustrated magnetism have focused on insulating systems, however, there are interesting examples of phenomena in metallic systems where frustration is believed to play a crucial role, such as heavy fermion physics in the spinel LiV$_2$O$_4$ [@LiV2O4], and the metallic spin-liquid state in the pyrochlore Pr$_2$Ir$_2$O$_7$ [@Nakatsuji2006]. In this context, the layered delafossites AgNiO$_2$ [@growth] and Ag$_2$NiO$_2$ [@Ag2NiO2] provide model systems for studying the interplay of metallic electrons and local-moment magnetism on a geometrically-frustrated lattice.
Detailed structural studies on the hexagonal, $2H$-polytype, AgNiO$_2$ reveal a charge ordering transition at 365 K, below which one-third of the Ni ions form a triangular lattice of localized Ni$^{2+}$ ($S=1$) magnetic moments, while the remaining Ni$^{3.5+}$ ions form a honeycomb network of itinerant paramagnetic sites [@Wawrzynska2007; @Wawrzynska2008; @Mazin2007; @Pascut2011]. The localized Ni$^{2+}$ spins order magnetically below $T_N=19.5$ K into a collinear antiferromagnetic structure of alternating ferromagnetic stripes in the triangular plane, with spins aligned along the $c$-axis, while the itinerant Ni$^{3.5+}$ sites remain paramagnetic [@Wawrzynska2007; @Wawrzynska2008]. Bandstructure calculations suggest that the Ag [*sp*]{} band is entirely above the Fermi level and that the structure can be visualized as a magnetic insulator formed by Ni$^{2+}$ (like NiO) with a strong tendency to magnetic order, superimposed on a Ni$^{3.5+}$ metal [@Wawrzynska2007]. Here we report thermodynamic and transport measurements on micron-size single crystals of 2$H$-AgNiO$_2$ in high magnetic fields of up to 90 T applied along the $c$-axis. We observe a complex cascade of magnetic phase transitions, and combine magnetic torque, heat capacity and transport measurements to construct the field-temperature phase diagram. Experimental data are compared with the predictions of a frustrated easy-axis Heisenberg model for the localized Ni$^{2+}$ moments, which predicts a field-driven phase transition from the collinear antiferromagnet (CAF) into a magnetic supersolid (SS), which can be viewed as a Bose condensate of magnons of the CAF phase. Deviations from this model compared to the measured phase diagram are attributed to the role played by the itinerant electrons.
![(color online) High magnetic field measurements of micron-size single crystals of $2H$-AgNiO$_2$. Field dependence of (a) torque, and (b) magnetization at constant temperature $T$ for field $\bm{H}$ nearly along the $c$-axis ($\theta \approx 3^{\circ}$). The inset shows magnetization data for a powder sample at 5 K. c) Torque data up to 90 T measured at constant temperatures in pulsed magnetic fields on two different samples (top and bottom panels). Magnetic transitions are indicated by arrows and labels. Top inset shows a typical crystal mounted on a piezolever. (d) Field dependence of interlayer transport at constant temperatures below 30 K when ${\bm H}
\parallel c $ (within $\theta \approx 3^{\circ}$). The arrow indicates the deviation from the linear dependence at $H_{c1}$. The inset shows the low-temperature resistivity and the arrow indicates the position of the magnetic ordering transition at $T_{\rm N}=19.5$ K. In all panels traces at different temperatures are uniformly shifted vertically for clarity. []{data-label="torque"}](fig1ab.eps "fig:"){width="8.5cm"} ![(color online) High magnetic field measurements of micron-size single crystals of $2H$-AgNiO$_2$. Field dependence of (a) torque, and (b) magnetization at constant temperature $T$ for field $\bm{H}$ nearly along the $c$-axis ($\theta \approx 3^{\circ}$). The inset shows magnetization data for a powder sample at 5 K. c) Torque data up to 90 T measured at constant temperatures in pulsed magnetic fields on two different samples (top and bottom panels). Magnetic transitions are indicated by arrows and labels. Top inset shows a typical crystal mounted on a piezolever. (d) Field dependence of interlayer transport at constant temperatures below 30 K when ${\bm H}
\parallel c $ (within $\theta \approx 3^{\circ}$). The arrow indicates the deviation from the linear dependence at $H_{c1}$. The inset shows the low-temperature resistivity and the arrow indicates the position of the magnetic ordering transition at $T_{\rm N}=19.5$ K. In all panels traces at different temperatures are uniformly shifted vertically for clarity. []{data-label="torque"}](fig1cd.eps "fig:"){width="8.5cm"}
For this study we use hexagonal-shaped single crystals (typical size $\sim 70 \times 70 \times 0.1~\mu$m$^3$) grown under high oxygen pressure [@growth]. We performed a series of torque measurements (on more than 10 single crystals) using piezo-resistive, self sensing cantilevers, at low temperatures (0.3 K) both in static magnetic fields (up to 18 T in Oxford and Bristol, 33 T at the HMFL in Nijmegen) and in pulsed fields (up to 55 T at the LNCMP, Toulouse and up to 90 T at NHMFL in Los Alamos). The longitudinal magnetisation was measured by force magnetometry using a highly sensitive magnetometer developed in Nijmegen [@McCollam2011]. Specific heat was measured using a purpose built micro-calorimeter using dc and relaxation techniques. The residual resistivity ratio is up to $\sim 250$, which indicates the high purity of the single crystals.
Magnetic torque in magnetic materials is caused by anisotropy, measuring the misalignment of the magnetization with respect to an uniform applied field. The torque exerted on a sample in an applied magnetic field ${\bm
H}$ is ${\bm \tau}={\bm M} \times \mu_0 {\bm H}$ where ${\bm M}$ is the bulk magnetization. If ${\bm M}$ and ${\bm H}$ lie in the ($ac$) plane, then $\tau=
\mu_0 (M_a H_c- M_c H_a ) = \tfrac{1}{2} \mu_0(\chi_a -\chi_c)
H^2$ $\sin2\theta$, with $\theta = 0$ when ${\bm H} \parallel c$. Thus, torque experiments measure the anisotropy of the magnetization in the $ac$ plane and the torque vanishes in field along the $c$- and $a$-axes (when $\sin2\theta=0$); the longitudinal magnetization provides access to the parallel $M_c$ component of the sample magnetization.
Figs. \[torque\](a) and (b) show the field and temperature dependence of the torque and magnetization respectively, performed with the magnetic field aligned close to the easy axis $c$. At low temperatures and in low magnetic fields, the torque signal varies as $\tau \sim H^2$ implying a constant anisotropy, $\chi_a-\chi_c$, in the CAF phase. By increasing the field, we observe kinks in torque at $\mu_0
H_{c1-4}$=13.5, 20, 28 and 39 T \[see Figs. \[torque\](a) and (c)\], which we attribute to field-induced transitions. No further anomalies are detected at higher fields up to 90 T \[see Fig. \[torque\](c)\], however the torque is finite and increases in absolute magnitude indicating that the magnetization is not yet saturated and the region above $H_{c4}$ is most likely a phase with spontaneous magnetic order. Further evidence for the phase transitions seen in torque data is provided by magnetization measurements shown in Fig. \[torque\](b). At low fields the magnetization has a weak linear field dependence and at $H_{c1}$ the slope suddenly changes, suggesting a linear increase in the $M_c$ component in this phase, followed by a decrease in slope above $H_{c2}$ and a small kink at $H_{c3}$. Experiments of torque and specific heat in constant magnetic field as a function of temperature presented in Fig. \[comparison-field\](a) and (b) also show clearly the anomalies at $T_N$ and $T_{c1}$. Later, we compare in detail these measurements with predictions for a spin Hamiltonian. Another important fact about $2H$-AgNiO$_2$ is that it is a good metal with low residual resistivity (57 $\mu \Omega$ cm) (see Fig.\[torque\](d)) and quantum oscillations have been observed [@Coldea2014]. There is a significant contributions to the density of states at the Fermi level originating from the Ni sites on the honeycomb lattice [@Wawrzynska2007]. Fig.\[torque\](d) shows that that transport measurements also exhibit anomalies at the magnetic phase transitions, showing that the itinerant $d$ electrons are a sensitive probe of the magnetic ground state. There is a significant drop in resistivity below $T_{\rm N}$ (see inset in Fig.\[torque\](d)), which is likely the result of suppression of electronic scattering by low-energy spin fluctuations when a spin gap opens below $T_{\rm N}$ [@Wheeler2008]. Furthermore, magnetoresistance measurements in Fig. \[torque\](d) indicate that in the vicinity of the magnetic transition there is a clear change in slope at $H_{c1}$ that fades away with increasing temperature. In zero field the Ni$^{2+}$ spins order in a collinear antiferromagnetic pattern with spins pointing along the easy $c$-axis, schematically shown in Fig. \[phase\_diagram\](c) [@Wawrzynska2007; @Wheeler2008]. In magnetic fields applied along the easy axis a transition is expected in a field of $\Delta/(g \mu_B)$ that matches the zero-field anisotropy spin gap, $\Delta$. Using the observed value of the first transition field $\mu_0 H_{c1}=13.5$ T in Fig. 1(a) gives $\Delta=1.57$ meV (using $g=2$), in good agreement with the value of $1.7(1)$ meV estimated from inelastic neutron scattering measurements [@Wheeler2008]. For easy-axis antiferromagnets with un-frustrated interactions the transition in field is to a spin-flop phase, signalled by an anomaly in torque [@Nagamiya1955; @Uozaki2000; @Bogdanov2007]. This canted phase is then stable with increasing magnetic field, up to full magnetization saturation. However, for easy-axis triangular lattice antiferromagnets with frustrated interactions, as is believed to be the case for $2H$-AgNiO$_2$, an alternative scenario with a richer phase diagram has been proposed [@Seabra2010], which we describe below.
![ (colour online) (a) The $H-T$ phase diagram of $2H$-AgNiO$_2$ from torque magnetometry (circles), specific heat (triangles) and transport (square). The solid and dashed lines indicate boundaries between different magnetic phases: collinear antiferromagnetic (CAF), field induced phases I-IV and paramagnetic (PM). b) The phase diagram of the classical Heisenberg model on the triangular lattice with first- ($J_1$=$1$) and second-neighbour ($J_2$=$0.15$) in-plane interactions, coupling between layers ($J_\perp$=$-0.15$) and easy-axis anisotropy $D=0.25$, obtained from Monte Carlo simulation [@Seabra2011]. The axes units are scaled to the point (large solid red circle) where the CAF, SS and plateau phases meet with $T^\ast=0.3J_1$, $B^\ast=1.6 J_1$. In the CAF phase (c) the spin excitations are gapped, while in the SS phase (d) electrons can scatter from gapless spin excitations.[]{data-label="phase_diagram"}](fig2.eps){width="8.5cm"}
The magnetic field-temperature phase diagram of $2H$-AgNiO$_2$, based on magnetic torque, transport and specific heat data obtained over a large range of fields and temperatures on different single crystals, is shown in Fig. \[phase\_diagram\](a). Unexpectedly, in this metallic magnet, we observe a cascade of phase transitions suggesting the formation of different magnetic structures with increasing magnetic field, different from typical uniaxial antiferromagnets, see e.g. Refs. [@Becerra1988; @Uozaki2000; @Kawamoto2008; @Toft-Petersen2012], In insulators, field-induced transitions were observed previously in two-dimensional CuFeO$_2$ ($S=5/2$), with a related delafossite-type structure [@Terada2007], and extended models were developed for $S=1$ [@Sengupta2007].
To gain insight into the magnetism of $2H$-AgNiO$_2$ in high magnetic fields, we consider a simple effective spin model describing only the localised $S=1$ (Ni$^{2+}$) spins interacting via a Heisenberg model including first- and second-neighbour antiferromagnetic exchange ($J_1,J_2$) on the triangular lattice, a coupling between layers ($J_{\perp}$) and easy-axis anisotropy ($D$), using parameters obtained from fits to powder inelastic neutron scattering data [@Wheeler2008]. The resulting magnetic phase diagram in easy-axis field obtained from classical Monte Carlo simulations was described in detail in Refs. [@Seabra2010; @Seabra2011] and in the Supplementary Material. Here, we focus on the low-field region of the phase diagram of the spin model in Fig.\[phase\_diagram\](b), and we compare directly the measured and the calculated thermodynamic quantities: torque, magnetization and specific heat. The experimental phase diagram in Fig. \[phase\_diagram\](a) shows that the phase boundary between the CAF phase and phase I have an unusual field-temperature dependence, i.e. the transition field $H_{c1}$ increases strongly with increasing temperature. This behavior is well reproduced by the theoretical phase diagram in Fig. \[phase\_diagram\](b), there the phase transition CAF-SS is understood in terms of Bose-Einstein condensation of magnons within the CAF state [@Seabra2010]; this converts the two-sublattice CAF order into an unusual four-sublattice state, in which two sublattices have spins “up” and the other two have spins “down” and canted away from the easy axis, as illustrated in Fig. 2(d). These canted spin components break spin-rotation symmetry about the magnetic field, and behave like a superfluid order parameter. Meanwhile the components of spin along the magnetic easy axis break the discrete translational symmetry of the lattice, in a way analogous to a solid, therefore, the resulting state is a [*magnetic supersolid*]{} [@Seabra2010; @Seabra2011].
Next, we compare directly the experimental results for magnetic torque, magnetization and specific heat with those predicted by The temperature-dependence of the torque at fixed field has a very similar shape and qualitative form between in the experiment and theoretical model \[see Fig. \[comparison-field\](a) and (b)\]: in both cases upon cooling from high temperatures in the paramagnetic phase the torque changes sign below $T_N$, increases upon decreasing temperature in the CAF phase, then has a peak at $T_{c1}$, identified with the transition into the SS phase. The temperature-dependence of the specific heat data in constant magnetic fields also shows consistent behavior between experiment and theory, see Fig. \[comparison-field\](c) and (d). At low fields a single anomaly is observed at the $T_{\rm N}$ transition. In fields above $H_{c1}$ a second anomaly is observed at low temperatures $T_{c1}$ identified with the transition CAF-SS and this shifts to higher temperatures upon increasing field [@specific_heat]. At those higher fields an additional anomaly (labelled as $T'$) appears near $T_N$, this feature is also present in the theoretical model, where it is associated with the transition to another intermediate-temperature phase (labelled “plateau” in Fig.2(b)).
,(b) ($\theta=5^{\circ}$). In the theoretical model saturation occurs for $H/H^\ast \approx 4.8$ with $T^\ast$ and $H^\ast$ defined in Fig \[phase\_diagram\](b). Traces at different fields (c,d) and different temperatures (e,f) are shifted vertically for clarity. \[comparison-field\] ](fig3ab.eps "fig:"){width="8cm"} ,(b) ($\theta=5^{\circ}$). In the theoretical model saturation occurs for $H/H^\ast \approx 4.8$ with $T^\ast$ and $H^\ast$ defined in Fig \[phase\_diagram\](b). Traces at different fields (c,d) and different temperatures (e,f) are shifted vertically for clarity. \[comparison-field\] ](fig3cd.eps "fig:"){width="8cm"} ,(b) ($\theta=5^{\circ}$). In the theoretical model saturation occurs for $H/H^\ast \approx 4.8$ with $T^\ast$ and $H^\ast$ defined in Fig \[phase\_diagram\](b). Traces at different fields (c,d) and different temperatures (e,f) are shifted vertically for clarity. \[comparison-field\] ](fig3ef.eps "fig:"){width="8cm"}
Following the same approach, we discuss the measured field dependencies of magnetic torque and magnetisation at constant temperature, shown in Fig. \[torque\](a,b), with those predicted by the model in Fig. \[comparison-field\](e,f). At very low temperatures in the CAF phase, the measured and simulated magnetization are both linear in $H$, whereas the torque shows a quadratic dependence, $H^2$, as a function of magnetic field, and displays a kink at the first critical field, $H_{c1}$, that becomes sharper upon lowering temperature. Furthermore, inside the SS phase at fields right above the transition $H_{c1}$, magnetic torque is observed to be almost independent of the applied field (see Fig.3(e)). All these observations are at odds with standard spin-flop transitions, see Refs. [@Nagamiya1955; @Becerra1988; @Uozaki2000; @Bogdanov2007; @Toft-Petersen2012]; in those cases, torque should show a strong divergence at a spin-flop transition field $H_{c}$, then becomes strongly suppressed above $H_{c}$, whereas the magnetization would have an abrupt jump at $H_{c}$, indicative of the first-order nature of this transition. The transition field for a spin-flop transition is usually independent of temperature, since it is determined by a balance of energies between different spin configurations rather than entropy.
While the classical spin model for the localized Ni$^{2+}$ moments can provide a good description of many of the qualitative and quantitative features of the lowest field-induced transition observed at $H_{c1}$, the model cannot capture the full phase diagram and in particular cannot account for the presence of multiple phases spanning relatively narrow field ranges above $H_{c1}$ (labelled as I-III). It may be possible that itinerant electrons, neglected in the spin model, could affect the phase diagram in this region of intermediate fields and potentially lead to additional transitions at $H_{c2}$ and $H_{c3}$ *inside* the SS phase of the classical model. In the SS phase the broken translational symmetry of the [*solid*]{} may lead to reconstruction of the Fermi surface, while the gapless Goldstone modes of the [*superfluid*]{} may lead to the inelastic scattering of electrons and thus an increase of resistivity above $H_{c1}$, as observed in Fig. 1(d). A possible reconstruction of the Fermi surface at the low-field transition may also explain why more entropy is released in experiment than in theory at the SS transition, indicated by the large anomaly in specific heat at $T_{c1}$ in Fig. \[comparison-field\](c) and (d).
The observed $H-T$ phase diagram of $2H$-AgNiO$_2$ reflects the complexity of its magnetic interactions. Since localized magnetic moments are embedded in a metal, they are subject to RKKY interactions, which, unlike superexchange, decays slowly with distance and may provide non-negligible further-neighbour exchange. Furthermore, the band structure calculations show a small, but finite magnetic moment ($m_i \approx 0.1-0.2 \mu_B$) [@Wawrzynska2007] on the itinerant and inherently nonmagnetic Ni sites on the honeycomb (see Fig.\[phase\_diagram\]c). The Hund’s rule coupling on these sites may provide an additional incentive for the localized spins to order and the scale of this interaction given by $I m_i^2$/4 is a few meV (the Stoner factor is $I \approx 0.6 - 0.8 $ eV). By a similar mechanism, the Hund’s energy of induced moments generates ferromagnetism in SrRuO$_3$ [@Mazin1997].
In conclusion, we have probed the magnetic phase diagram of the frustrated antiferromagnetic metal, $2H$-AgNiO$_2$, in strong magnetic fields applied along the easy axis and have observed a cascade of magnetic phase transitions. Thermodynamic measurements have been compared with predictions of an effective localized spin model, which explains part of the phase diagram and identifies a novel magnetic supersolid phase. However, a more realistic model for 2$H$-AgNiO$_2$ needs to consider also the itinerant $d$ electrons on the honeycomb lattice, which may participate in the exchange interactions and may affect the magnetic order of the localized moments. Therefore, the itinerant electrons may be responsible for some of the higher-field transitions observed both in transport and thermodynamic measurements. Further studies will explore how those phase transitions correlate with changes of the Fermi surface topology in this frustrated magnetic metal where $d$ electrons have mixed localized and itinerant character.
We acknowledge and thank J. Analytis, C. Jaudet, P.A. Goddard, Jos Perenboom, M.D. Watson, for technical support during experiments. We thank I.I. Mazin, A. Schofield, I. Vekhter for useful discussions. This work was supported by EPSRC Grants EP/I004475/1, EP/ C539974/1, EP/G031460/1, FCT Grant No. SFRH/BD/27862/2006, and the EuroMagNET II (EU Contract No. 228043). AIC acknowledges an EPSRC Career Acceleration Fellowship (EP/I004475/1). RMcD acknwledges support from BES “Science of 100 T”.
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In the limit of our experimental resolution (specific heat experiments are performed on extremely small samples) as a function of temperature (experiments were made on $70\times70\times10 \mu {\rm{m}}^3$ crystals) we cannot detect clear signatures for the other two transitions clearly observed both in the torque and transport measurements. This could be due to the fact that the scans were performed mainly at constant temperature and transition boundaries are nearly flat and/or that these transitions are related to a reorientation of spins which does not imply a significant change in energy and do not result in a large anomaly in the specific heat.
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{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We consider a vortex line in the B phase of superfluid $^3$He under uniformly precessing magnetization. The magnetization exerts torque on the vortex, causing its order parameter to oscillate. These oscillations generate spin waves, which is analogous to an oscillating charge generating electromagnetic radiation. The spin waves carry energy, causing dissipation in the system. Solving the equations of spin dynamics, we calculate the energy dissipation caused by spin wave radiation for arbitrary tipping angles of the magnetization and directions of the magnetic field, and for both vortex types of $^3$He-B. For the double-core vortex we also consider the anisotropy of the radiation and the dependence of the dissipation on twisting of the half cores. The radiated energy is compared with experiments in the mid-temperature range $T \sim 0.5 T_c$. The dependence of the calculated dissipation on several parameters is in good agreement with the experiments. Combined with numerically calculated vortex structure, the radiation theory produces the order of magnitude of the experimental dissipation. The agreement with the experiments indicates that spin wave radiation is the dominant dissipation mechanism for vortices in superfluid $^3$He-B in the mid-temperature range.'
author:
- 'S. M. Laine'
- 'E. V. Thuneberg'
title: 'Spin wave radiation from vortices in 3He-B'
---
Introduction {#sec:introduction}
============
Superfluid $^3$He is a useful paradigm of an unconventional superfluid or superconductor as it has a spin-triplet, p-wave-pairing order parameter which is precisely known. The B phase of superfluid $^3$He has two well known vortex structures [@Ikkala82]. The [*A-phase-core vortex*]{} has A-phase-like order parameter in the vortex core [@Salomaa85], while the [*double-core vortex*]{} has broken axisymmetry so that the vortex core is split into two half cores [@thuneberg1987]. Major part of the information about the superfluid phases and vortices in superfluid $^3$He has been obtained by nuclear magnetic resonance (NMR). The methods used are linear NMR with small tipping of the magnetization, as well as measurements using large tipping angles. The information about the order parameter is obtained by measuring either the frequency shift of resonance absorption or the amount of absorption, i.e., relaxation.
The purpose of our research is to understand the relaxation seen in NMR experiments on vortices of $^3$He-B. In particular, we consider experiments reported by Kondo et al. [@kondo1991; @dmitriev1990], which are made at intermediate temperatures around $0.5 T_c$ and at large tipping angles. A well known relaxation mechanism was first discussed by Leggett and Takagi [@Leggett77T]. It arises because the dipole-dipole interaction is enhanced by superfluid coherence but affects only the superfluid component, so that the normal and superfluid components of magnetization are driven out of mutual equilibrium. The conversion between the two components then leads to dissipation. This mechanism seems to be most effective at temperatures close to the transition temperature $T_c$. We find [@laine2016] that the Leggett-Takagi relaxation is too weak to explain the relaxation observed by Kondo et al. Another well known relaxation mechanism arises from diffusion of the normal component of the magnetization. It seems, however, that its contribution has to be small in the experiments by Kondo et al. since the observed magnetic field dependence [@dmitriev1990] is opposite to the one expected for spin diffusion.
A third mechanism to cause relaxation was discussed in Ref. [@kondo1991]. It was suggested that the precessing magnetization drags the half cores of the double-core vortex to rotate around themselves. A phenomenological model for the rotation was constructed, and its parameters were fitted to the observed dissipation. More recently, the parameters of the rotational model were calculated based on numerical solution of the vortex structure [@silaev2015]. It was found that, while the rotation of the half cores was confirmed, the friction coefficient for the rotation is so large that the dissipation is negligible. Thus it remained open what causes the major part of the dissipation in the experiments by Kondo et al. [@kondo1991].
In this paper we investigate a fourth relaxation mechanism. The precession of the magnetization makes the order parameter near the vortex to oscillate. These oscillations generate waves in the spin angular momentum, that is, [*spin waves*]{}. The spin waves radiated by the vortex carry energy and thus lead to relaxation of magnetization.
Generally, spin waves are collective excitations in systems possessing magnetic order. First predicted by Felix Bloch almost ninety years ago, they have recently become a subject of intense research in the fields of spintronics and magnonics because of their possible uses, e.g., in data transport and processing [@chumak2016]. In superfluid $^3$He, the spin waves were first detected by Osheroff et al. [@osheroff1977], who saw standing spin wave modes in $^3$He-B. The first observations of vortices were based on the frequency shifts of such standing spin wave modes [@Ikkala82; @Hakonen89]. The radiation of spin waves as a relaxation mechanism was discussed by Ohmi et al. [@Ohmi87] in connection with experiments in Ref. [@Ishikawa89]. The relaxation seen in the Josephson junction arrays of $^3$He has been interpreted in terms of spin wave radiation [@Viljas04]. The relaxation of magnetization by direct and parametric generation of spin waves has been reported by Zavjalov et al. [@Zavjalov2016]. Spin wave radiation from vortices has been pointed out by Volovik [@Volovik].
We study spin wave radiation from a single vortex under precessing magnetization. We consider arbitrary tipping of magnetization, where we need to discuss separately tipping angles smaller and larger than the Leggett angle $\theta_L = \arccos(-1/4) \approx 104^\circ$. Besides a straight vortex, we also consider a vortex that is twisted by the precessing magnetization. By integrating the energy flux tensor around the vortex we calculate the total radiated energy. We find that radiation of spin waves is the dominant relaxation mechanism for vortices at low temperatures. The relaxation seen in the experiments by Kondo et al. [@kondo1991] can be well explained in terms of spin wave radiation.
The paper is organized as follows. In Sec. \[sec:static\_vortex\] we discuss the order parameter structure of static B-phase vortices far from the vortex core. In Sec. \[sec:spin\_dynamics\] we derive the equation of motion for the order parameter. The solution of this equation is discussed in Sec. \[sec:spin\_waves\]. In Sec. \[sec:energy\] we calculate the energy carried by the spin waves. The effect of twisting of the vortex core on energy transport is studied in Sec. \[sec:twisted\]. Finally, we compare the theory with experiments in Sec. \[sec:experiments\].
Static vortex {#sec:static_vortex}
=============
The order parameter of an isolated B-phase vortex far from the vortex axis can be written as [@thuneberg1987; @fogelstrom1995] $$\label{eq:orderParameterStatic}
\mathsf{A} = e^{i \varphi} \Delta_0 \mathsf{R} \left( \theta_0 \bm{\hat{n}} \right) \mathsf{R} \left( \bm \theta \right).$$ Here $\Delta_0$ is the bulk gap and $\mathsf{R} \left( \theta_0 \bm{\hat{n}} \right)$ is a finite rotation by an angle $\theta_0$ about an axis $\bm{\hat{n}}$. These, determined by the bulk, are assumed spatially constants. In the static case $\theta_0$ is fixed at the Leggett angle, $\theta_0 = \theta_L$. The vortex appears through the phase $\varphi$, which equals the azimuthal angle around the vortex axis, and through an additional rotation $\mathsf{R} \left( \bm \theta \right)$ by an angle $ \theta = \left| \bm \theta \right|$ about an axis $ \bm{\hat{\theta}} = \bm \theta / \theta$. The rotation $\bm \theta$ is determined by minimizing the free energy [@Leggett75; @vollhardt1990; @thuneberg2001] $$F = \int_V dV \left( f_D + f_G \right)$$ in the region $V$ excluding the vortex core with appropriate boundary conditions. Here $f_D$ originates from the dipole-dipole interaction between the $^3$He nuclei, $$\label{eq:f_D_static}
f_D = \lambda_D \left( R_{i i} R_{j j} + R_{i j} R_{j i} \right) \approx - \frac{\lambda_D}{2} + \frac{15}{2} \lambda_D \left( \bm{\hat{n}} \cdot \bm \theta \right)^2,$$ while $f_G$ is the gradient energy, $$\label{eq:f_G_static}
\begin{split}
f_G &= \lambda_{G1} \frac{\partial R_{\alpha i}}{\partial r_i} \frac{\partial R_{\alpha j}}{\partial r_j} + \lambda_{G2} \frac{\partial R_{\alpha j}}{\partial r_i} \frac{\partial R_{\alpha j}}{\partial r_i} \\
&\approx 2 \lambda_{G2} \left[ (1 + c) \partial_i \theta_k \partial_i \theta_k - c \partial_i \theta_k \partial_k \theta_i \right].
\end{split}$$ In the above we have denoted $\mathsf{R} = \mathsf{R} \left( \theta_0 \bm{\hat{n}} \right) \mathsf{R} \left( \bm \theta \right)$ and $c = \lambda_{G1} / 2 \lambda_{G2}$. The coefficients $\lambda_D$, $\lambda_{G1}$, and $\lambda_{G2}$ depend on temperature and pressure. We assume $\bm \theta$ to be small so that the energies are well approximated by expressions that are quadratic in $\bm \theta$. The gradient energy dominates the dipole energy when the distance from the vortex core is much less than the dipole length $\xi_D = \sqrt{\lambda_{G2} / \lambda_D}$. If we neglect the dipole energy, the solution describing an isolated vortex is $\bm \theta = \bm \theta_v$, where [@silaev2015; @laine2016] $$\label{eq:theta_v}
\begin{split}
\bm \theta_v (r, \varphi) &= \frac{C_1 \cos \varphi}{r} \left( \frac{\sin \varphi}{1+c} \bm{\hat{r}} + \cos \varphi \bm{\hat{\varphi}} \right) \\
&- \frac{C_2 \sin \varphi}{r} \left( \frac{\cos \varphi}{1+c} \bm{\hat{r}} - \sin \varphi \bm{\hat{\varphi}} \right).
\end{split}$$ Here $\bm{\hat{r}}$, $\bm{\hat{\varphi}}$, and $\bm{\hat{z}}$ are the basis vectors of cylindrical coordinate system with $\bm{\hat{z}}$ oriented along the vortex axis. This is a good approximation at distances $10\, \xi(T) \lesssim r \ll \xi_D$ from the axis, where $\xi(T)$ is the temperature dependent coherence length. Near the core $\theta$ becomes large and the second order expansion of the gradient energy breaks down. The inclusion of the dipole energy causes $\bm \theta$ to vanish more rapidly than $r^{-1}$ at distances greater than $\xi_D$.
The coefficients $C_1$ and $C_2$ depend on the type of the vortex. They can be extracted from the numerical solution of the vortex core structure [@thuneberg1987; @fogelstrom1995; @silaev2015]. Because of axial symmetry, $C_1 = C_2$ for the A-phase-core vortex and thus $\bm \theta_v = C_1 \bm{\hat{\phi}} / r$. This special case of Eq. (\[eq:theta\_v\]) was found by Hasegawa [@hasegawa1985]. For the double-core vortex $C_1 / C_2 \gg 1$.
Since $\bm \theta_v \propto r^{-1}$ to leading order, $r \bm \theta_v$ is independent of $r$. Thus we can visualize $\bm \theta_v$ by plotting $r \bm \theta_v(r, \varphi) \equiv \bm \vartheta_v(\varphi)$ on a circle in the $xy$-plane. This is shown in Fig. \[fig:staticTheta\]. We have used the values $C_1 = C_2 = 1.33 R_0$ for the A-phase-core vortex and $C_1 = 3.00 R_0$, $C_2 = 0.08 R_0$ for the double-core vortex. Here $R_0 = (1 + F_1^s / 3) \xi_0$, $F_1^s$ is a Fermi liquid parameter, $\xi_0 = \hbar v_F / 2 \pi k_B T_c$ is the coherence length, $v_F$ is the Fermi velocity, and $T_c$ is the critical temperature. The values of $C_1$ and $C_2$ correspond to temperature $T = 0.6 T_c$ and pressure $p = 29.3$ bar [@silaev2015]. We have also set $c = 1$ since this is the weak-coupling value assuming vanishing Fermi-liquid parameters $F_1^a$ and $F_3^a$. The structure of the A-phase-core vortex is shown in Fig. \[fig:staticTheta\](a). The structure of the double-core vortex is shown in Fig. \[fig:staticTheta\](b). The half cores are located on the $y$-axis.
Spin dynamics {#sec:spin_dynamics}
=============
We now place the vortex in a static external magnetic field $\bm B$ and study spin dynamics. This is governed by the Leggett theory [@leggett1974]. Within the Leggett theory, the motion of the order parameter in spin space is purely rotational, $\mathsf{A}(t) = \mathsf{R}(t) \mathsf{A}_0$. Here $\mathsf{R}$ is a time-dependent rotation matrix and $\mathsf{A}_0$ is the initial order parameter. Since the order parameter of the vortex is of the correct form, see Eq. (\[eq:orderParameterStatic\]), we can include the time-dependence of $\mathsf{A}$ in variables $\theta_0$, $\bm{\hat{n}}$ and $\bm \theta$. This means that the dynamic order parameter is of the same form as the static one, $$\label{eq:orderParameterDynamic}
\mathsf{A} = e^{i \varphi} \Delta_0 \mathsf{R} \left( \theta_0 \bm{\hat{n}} \right) \mathsf{R} \left( \bm \theta \right) .$$ We study a holonomically constrained problem (see, e.g., [@fetter1980]) where $\theta_0(t)$ and $\bm{\hat{n}}(t)$ are given functions of time. They are determined by the bulk, i.e., they solve the equations of spin dynamics in the absence of the vortex. We take the bulk solution to be the Brinkman-Smith (BS) mode [@Brinkman75b; @Fomin83], where the magnetization precesses uniformly about $\bm B$. Details of the Brinkman-Smith mode are discussed later. The [*system*]{} we want to study is the vortex, described by the field $\bm \theta(\bm r, t)$. The Brinkman-Smith mode then acts as an external [*drive*]{} for the system. In order to maintain the Brinkman-Smith mode in the presence of the vortex, energy is needed from an outside source. Experimentally this is done using a time-dependent magnetic field. In our calculations the energy source is present implicitly through the constraints.
We shall use the following geometry and notation. The $z$-axis of the coordinate system coincides with the vortex axis. The $x$-axis is chosen so that the magnetic field $\bm B$ lies in the $xz$-plane. In addition to the cartesian coordinate system $(x,y,z)$ we shall use the standard cylindrical coordinate system $(r, \varphi, z)$, where $r$ is the distance from the $z$-axis and $\varphi$ is the azimuthal angle, measured anticlockwise from the $x$-axis. The tilting angle of $\bm B$ from the vortex axis is denoted by $\eta$, so that $\bm{\hat{B}} = \cos \eta \bm{\hat{z}} + \sin \eta \bm{\hat{x}} $. The orientation of the vortex core in the $xy$-plane is described by an angle $\zeta$. More specifically, the anisotropy vector $\bm{\hat{b}}$ of the double-core vortex, pointing from one of the half cores to the other, is given by $\bm{\hat{b}} = \cos \zeta \bm{\hat{y}} - \sin \zeta \bm{\hat{x}}$. Since the A-phase-core vortex is cylindrically symmetric, there is no need to define its orientation. Finally, $\beta$ is the tipping angle of the magnetisation $\bm M$, measured from the direction of the magnetic field, so that $\cos \beta = \bm{\hat{M}} \cdot \bm{\hat{B}}$. Figure \[fig:geometry\] shows the definitions of the various quantities in graphical form.
![Definitions of the angles $\beta$, $\zeta$, and $\eta$. The vortex axis coincides with the $z$-axis. $\bm B$ is the external static magnetic field, $\bm{\hat{b}}$ is the anisotropy vector of the double-core vortex, pointing from one of the half cores to the other, and $\bm M$ is the magnetization density. Note that $\bm M$ is not static but precesses uniformly about $\bm B$ with tipping angle $\beta$.[]{data-label="fig:geometry"}](geometry){width="48.00000%"}
As mentioned above, in Brinkman-Smith mode the magnetization precesses uniformly about $\bm B$, $$\bm M_{BS} = M_{BS} \mathsf{R}(\eta \bm{\hat{y}}) \mathsf{R}(\omega_{BS} t \bm{\hat{z}}) \mathsf{R}(\beta \bm{\hat{y}}) \cdot \bm{\hat{z}}.$$ The precession rate is given by $$\label{eq:BrinkmanSmithOmega}
\omega_{BS} = \frac{\omega_L}{2} \left( 1 + \sqrt{ 1 - \frac{16 \Omega^2}{15 \omega_L^2} \left( 1 + 4 \cos \theta_0 \right)} \right) .$$ Here $\omega_L = - \gamma_0 B$ is the Larmor frequency, $\Omega$ is the longitudinal NMR frequency, $\Omega^2 = 15 \mu_0 \gamma_0^2 \lambda_{D} / \chi$, $\mu_0$ is the vacuum permeability, $\gamma_0$ is the gyromagnetic ratio of $^3$He, and $\chi$ is the magnetic susceptibility of the B phase. The rotation angle $\theta_0$ is independent of time. If the magnetization of the sample is tipped by an angle $\beta \leq \theta_L$, then $\theta_0 = \theta_L$ and $\omega_{BS} = \omega_L$. If $\beta > \theta_L$, then $\theta_0$ satisfies $$\cos \beta = \frac{ \omega_{BS} ( \cos \theta_0 - 1) / \omega_L + 1 }{ \sqrt{\omega_{BS}^2 \sin^2 \theta_0 / \omega_L^2 + \left[ \omega_{BS} ( \cos \theta_0 - 1) / \omega_L + 1 \right]^2 }}.$$ This means that $\theta_0 > \theta_L$, and so the precession rate is increased, $\omega_{BS} > \omega_L$. The unit vector $\bm{\hat{n}}$ precesses uniformly about $\bm B$ with the same rate as the magnetization. It can be written as $$\label{eq:nt}
\bm{\hat{n}}(t) = \mathsf{R}(\eta \bm{\hat{y}}) \mathsf{R}(\omega_{BS} t \bm{\hat{z}}) \cdot \bm{\hat{n}}_0,$$ where $$\begin{split}
\bm{\hat{n}}_0 &=
\begin{cases}
\frac{2}{\sqrt{5}}\sqrt{1 - \cos \beta} \bm{\hat{y}} + \frac{1}{\sqrt{5}}\sqrt{1 + 4 \cos \beta} \bm{\hat{z}}, & \beta \leq \theta_L \\
\bm{\hat{y}}, & \beta > \theta_L
\end{cases}.
\end{split}$$
There are different ways to proceed, but a convenient one in our case is the Lagrangian formulation [@maki1975; @maki1977]. As in mechanics, there is an angular velocity $\bm \omega$ related to the rotating motion of the order parameter, defined by $$\label{eq:omega_def}
\dot{R}_{\alpha i} = \varepsilon_{\alpha \beta \gamma} \omega_{\beta} R_{\gamma i}.$$ Here a dot over a letter denotes differentiation with respect to time, $ \varepsilon_{\alpha \beta \gamma}$ is the Levi-Civita symbol, and $\mathsf{R} = \mathsf{R} \left( \theta_0 \bm{\hat{n}} \right) \mathsf{R} \left( \bm \theta \right)$. In terms of $\bm \omega$, the Lagrangian density of the system can be written as $$\label{eq:lagrangian}
\mathcal{L} = \frac{1}{2 \mu_0 \gamma_0^2} \left( \bm \omega - \bm \omega_L \right) \cdot \bm{\stackrel{\leftrightarrow}{\chi}} \cdot \left( \bm \omega - \bm \omega_L \right) - f_D - f_G,$$ where $\bm{\stackrel{\leftrightarrow}{\chi}}$ is the magnetic susceptibility tensor and $\bm \omega_L = - \gamma_0 \bm B = \omega_L \bm{\hat{B}}$ is the Larmor frequency vector. In addition to the angular velocity, one can also define the generalised momentum canonically conjugate to the rotation. This is the spin density $\bm S = \bm M / \gamma_0$. The vectors $\bm \omega$ and $\bm S$ are related by $$\label{eq:spin_density}
\bm S = \frac{\partial \mathcal{L}}{\partial \bm \omega} = \frac{\bm{\stackrel{\leftrightarrow}{\chi}}}{\mu_0 \gamma_0^2} \cdot \left( \bm \omega - \bm \omega_L \right).$$ It follows from Eqs. (\[eq:nt\]) and (\[eq:omega\_def\]) that $$\label{eq:omega_second_order}
\bm \omega \approx \bm \omega_{BS} + \mathsf{R}\left( \theta_0 \bm{\hat{n}} \right) \cdot \left( \bm{\dot{\theta}} + \frac{1}{2} \bm \theta \times \bm{\dot{\theta}} - \bm \omega_{BS} \right),$$ where $\bm \omega_{BS} = \omega_{BS} \bm{\hat{B}}$. Here we have kept again only the two lowest order terms in $\bm \theta$. Since $\theta_0$ is not necessarily equal to $\theta_L$, the second order expansion of the dipole energy is modified from Eq. (\[eq:f\_D\_static\]) to $$\begin{split}
f_D / \lambda_D &\approx 4 \cos \theta_0 (1 + 2 \cos \theta_0) \\
&- 4 \sin \theta_0 (1 + 4 \cos \theta_0)(\bm{\hat{n}} \cdot \bm \theta) \\
&- (1 + \cos \theta_0 )(1 + 4 \cos \theta_0) (\bm \theta \cdot \bm \theta) \\
&+ 3 (1 - \cos \theta_0)(4 \cos \theta_0 + 3)(\bm{\hat{n}} \cdot \bm \theta)^2.
\end{split}$$ The expansion of the gradient energy is still given by Eq. (\[eq:f\_G\_static\]). Substituting these into the Lagrangian density and using the fact that the susceptibility in the B phase is diagonal, $\chi_{\mu \nu} = \chi \delta_{\mu \nu}$, we derive the linearized equation of motion for $\bm \theta$ using the formula familiar from classical field theory [@fetter1980], $$\frac{\partial}{\partial t} \frac{\partial \mathcal{L}}{\partial \dot{\theta_i}} + \partial_j \frac{\partial \mathcal{L}}{\partial \partial_j \theta_i} - \frac{\partial \mathcal{L}}{\partial \theta_i} = 0.$$ As a result we get $$\label{eq:EquationOfMotion}
\begin{split}
\bm{\ddot{\theta}} &- \omega_{BS} \bm w \times \bm{\dot{\theta}} + \Omega^2 \mathsf{L} \cdot \bm \theta \\
&- v^2 \left[(1+c) \nabla^2 \bm{\theta} - c \bm \nabla \left( \bm \nabla \cdot \bm{\theta} \right) \right] = 0.
\end{split}$$ Here $v$ is a characteristic spin wave velocity, defined by $v^2 = 4 \mu_0 \gamma_0^2 \lambda_{G2} / \chi $, $$\bm w = \bm{\hat{B}} - \frac{\omega_{BS} - \omega_L}{\omega_{BS}} \mathsf{R}^T(\theta_0 \bm{\hat{n}}) \cdot \bm{\hat{B}},$$ and $$\begin{split}
\mathsf{L} \cdot \bm \theta &= \frac{2}{15} \sin \theta_0 (1 + 4 \cos \theta_0) \bm{\hat{n}} \times \bm \theta \\
&- \frac{2}{15} (1 + \cos \theta_0 )(1 + 4 \cos \theta_0) \bm \theta \\
&+ \frac{2}{5} (1 - \cos \theta_0)(4 \cos \theta_0 + 3) \bm{\hat{n}} (\bm{\hat{n}} \cdot \bm \theta).
\end{split}$$
In the rest of the paper we shall work with dimensionless quantities, unless stated otherwise. We take the unit of length to be $v / \Omega = 2 \xi_D / \sqrt{15}$ and the unit of time to be $\Omega^{-1}$. We measure the angular frequencies $\omega_L$ and $\omega_{BS}$ in units of $\Omega$ and the coefficients $C_1$ and $C_2$ in units of $R_0$. Finally, we measure $\bm \theta$ in (dimensionless) units of $R_0 \Omega / v$. In these units Eq. (\[eq:EquationOfMotion\]) can be written as $$\label{eq:EquationOfMotionDimensionless}
\begin{split}
&\bm{\ddot{\theta}} - \omega_{BS} \bm w \times \bm{\dot{\theta}} + \mathsf{L} \cdot \bm \theta - (1+c) \nabla^2 \bm{\theta} + c \bm \nabla \left( \bm \nabla \cdot \bm{\theta} \right) = 0.
\end{split}$$
Spin waves {#sec:spin_waves}
==========
In this section we solve the equation of motion (\[eq:EquationOfMotionDimensionless\]) in two different approximations. We shall see that in both cases the solution contains a part representing waves propagating away from the vortex. Since $\bm \theta$ and $\bm S$ are coupled via Eqs. (\[eq:spin\_density\]) and (\[eq:omega\_second\_order\]), this means that the vortex radiates spin waves. Physically this stems from two factors. First, due to the dipole interaction, the rotating $\bm{\hat{n}}$ exerts torque on $\bm \theta$ at each point in space, causing it to oscillate with time. Second, because of the gradient energy, $\bm \theta$ at each point is strongly coupled to its neighbouring points. This means that any disturbances in $\bm \theta$ are propagated in space.
To obtain a solution which properly describes a vortex, we split $\bm \theta$ into two parts, $$\bm \theta (\bm r, t) = \bm \theta_1(\bm r) + \bm \theta_2(\bm r, t).$$ Here $\bm \theta_1(\bm r)$ is a static solution with correct behaviour near the core and $\bm \theta_2(\bm r, t)$ is a time-dependent deviation from the static solution. To ensure that the solution has the correct form near the core, we demand that $\bm \theta_2$ vanishes when $r \to 0$. From Eq. (\[eq:EquationOfMotionDimensionless\]) we then obtain $$\label{eq:EquationOfMotionSource}
\begin{split}
\bm{\ddot{\theta}}_2 &- \omega_{BS} \bm w \times \bm{\dot{\theta}}_2 + \mathsf{L} \cdot \bm \theta_2 \\
&- (1+c) \nabla^2 \bm{\theta}_2 + c \bm \nabla \left( \bm \nabla \cdot \bm{\theta}_2 \right)= \bm \rho \left( \bm r, t \right) ,
\end{split}$$ where $$\label{eq:rho}
\bm \rho \left( \bm r, t \right) = -\mathsf{L} \cdot \bm \theta_1 + (1+c) \nabla^2 \bm{\theta}_1 - c \bm \nabla \left( \bm \nabla \cdot \bm{\theta}_1 \right).$$
Since the gradient energy dominates the dipole energy near the vortex core, we take $\bm \theta_1$ to minimize the gradient free energy. Taking into account the orientation of the vortex we have $$\label{eq:theta_1}
\bm \theta_1\left( r, \varphi \right) = \mathsf{R} ( \zeta \bm{\hat{z}} ) \cdot \bm \theta_v (r, \varphi - \zeta),$$ where $\bm \theta_v (r, \varphi)$ is given by Eq. (\[eq:theta\_v\]). The source term simplifies to $\bm \rho \left( \bm r, t \right) = -\mathsf{L}(t) \cdot \bm \theta_1(\bm r)$. In Sec. \[sec:twisted\] we study the effect of twisting of the vortex core. There we still use Eq. (\[eq:theta\_1\]), but with $\zeta = \zeta(z)$. This means that we have to keep the full expression (\[eq:rho\]).
Written componentwise, Eq. (\[eq:EquationOfMotionSource\]) is a system of three coupled second-order inhomogeneous linear partial differential equations. Solving this is not trivial because of the time-dependent coefficients $\mathsf{L}(t)$ and $\bm w(t)$ and the non-laplacian gradient term $\propto c\bm\nabla\bm\nabla$. While the time dependence of $\mathsf{L}$ and $\bm w$ can be removed by transformation to a frame rotating in the spin space, the non-laplacian operator is complicated there because it is anisotropic in separate spin or orbit space rotations. In the following we study two alternative approximations. In case A we set $c = 0$. This removes the non-laplacian term and allows solution in the rotating frame. In case B we consider the limit of high magnetic field, $\omega_L \gg 1$. In this limit we may neglect the time-dependence of the coefficients $\mathsf{L}(t)$ and $\bm w(t)$ on the left-hand side of Eq. (\[eq:EquationOfMotionSource\]).
When solving the equation of motion, we shall work partly in two dimensional Fourier space. We use the convention $$\begin{aligned}
f(\bm k) &= \iint d^2 r \exp \left( - i \bm k \cdot \bm r \right) f(\bm r), \\
f(\bm r) &= \frac{1}{(2 \pi)^2} \iint d^2 k \exp \left( i \bm k \cdot \bm r \right) f(\bm k),\end{aligned}$$ with $\bm r = x \bm{\hat{x}} + y \bm{\hat{y}} = r \cos \varphi \bm{\hat{x}} + r \sin \varphi \bm{\hat{y}}$ and $\bm k = k_x \bm{\hat{x}} + k_y \bm{\hat{y}} = k \cos \varphi_k \bm{\hat{x}} + k \sin \varphi_k \bm{\hat{y}}$. The equation of motion in the Fourier space is then obtained by making a substitution $\bm \nabla \to i \bm k$. Using the definition above, the Fourier transform of $\bm \theta_v$, Eq. (\[eq:theta\_v\]), is given by $$\label{eq:theta_v_Fourier}
\begin{split}
\bm \theta_v (k, \varphi_k) = &-\frac{2 \pi i}{k} \frac{C_1 - C_2}{2} \sin(2 \varphi_k) \bm{\hat{k}} \\
& -\frac{2 \pi i}{k} \left[ \frac{C_1 + C_2}{2} + \frac{C_1 - C_2}{2(1 + c)} \cos(2 \varphi_k) \right] \bm{\hat{\varphi}}_k .
\end{split}$$ Here $\bm{\hat{k}} = \cos \varphi_k \bm{\hat{x}} + \sin \varphi_k \bm{\hat{y}}$ and $\bm{\hat{\varphi}}_k = -\sin \varphi_k \bm{\hat{x}} + \cos \varphi_k \bm{\hat{y}}$ are the basis vectors of polar coordinate system in Fourier space.
Isotropic approximation
-----------------------
We start by considering Eq. (\[eq:EquationOfMotionSource\]) in the limit $c = 0$. Since the vector $\bm{\hat{n}}$ rotates about $\bm B$ at constant rate, it is convenient to use a basis where $\bm{\hat{n}}$ is constant. Mimicking the form of $\bm{\hat{n}}$ in Eq. (\[eq:nt\]) we define $$\bm \theta_2 \left( \bm r, t \right) = \mathsf{R} ( \eta \bm{\hat{y}} ) \mathsf{R} ( \omega_{BS} t \bm{\hat{z}} ) \cdot \bm \alpha \left( \bm r, t \right).$$ The equation of motion for $\bm \alpha$ is then $$\label{eq:EquationOfMotionRotatingBasis}
\begin{split}
\bm{\ddot{\alpha}} &+ \omega_{BS} \bm{\hat{z}} \times \bm{\dot{\alpha}} - \bm w_0 \times \left( \bm{\dot{\alpha}} + \omega_{BS} \bm{\hat{z}} \times \bm \alpha \right) \\
&+ \mathsf{L}_0 \cdot \bm{\alpha} - \nabla^2 \bm {\alpha} = - \mathsf{R} ( - \omega_{BS} t \bm{\hat{z}} ) \mathsf{R} ( - \eta \bm{\hat{y}} ) \cdot \mathsf{L} \cdot \bm \theta_1,
\end{split}$$ where we have defined $$\begin{split}
\mathsf{L}_0 \cdot \bm \alpha &= \frac{2}{15} \sin \theta_0 (1 + 4 \cos \theta_0) \bm{\hat{n}}_0 \times \bm \alpha \\
&- \frac{2}{15} (1 + \cos \theta_0 )(1 + 4 \cos \theta_0) \bm \alpha \\
&+ \frac{2}{5} (1 - \cos \theta_0)(4 \cos \theta_0 + 3) \bm{\hat{n}}_0 (\bm{\hat{n}}_0 \cdot \bm \alpha)
\end{split}$$ and $$\bm w_0 = \frac{4}{15} \omega_{BS}^{-1} (1 + 4 \cos \theta_0) \mathsf{R}^T(\theta_0 \bm{\hat{n}}_0) \cdot \bm{\hat{z}}.$$ Note that the coefficients on the left-hand side of (\[eq:EquationOfMotionRotatingBasis\]) are independent of time. This happens only when $c = 0$. The source term on the right-hand side of (\[eq:EquationOfMotionRotatingBasis\]) can be written as $$\begin{split}
-\mathsf{R} ( - \omega_{BS} t \bm{\hat{z}} ) \mathsf{R} ( - \eta \bm{\hat{y}} ) \cdot \mathsf{L} \cdot \bm \theta_1 = \Re \left\{ \bm \rho_0(\bm r) + e^{- i \omega_{BS} t} \bm \rho_1 (\bm r) \right\},
\end{split}$$ where $$\begin{aligned}
\bm \rho_0(\bm r) &= -\mathsf{L}_0 \cdot \mathsf{M}_0 \cdot \mathsf{R} ( - \eta \bm{\hat{y}} ) \cdot \bm \theta_1 (\bm r), \\
\bm \rho_1(\bm r) &= -\mathsf{L}_0 \cdot \mathsf{M}_- \cdot \mathsf{R} ( - \eta \bm{\hat{y}} ) \cdot \bm \theta_1 (\bm r),\end{aligned}$$ and $$\begin{aligned}
&\mathsf{M}_0 =
\begin{pmatrix}
0 & 0 & 0 \\
0 & 0 & 0 \\
0 & 0 & 1
\end{pmatrix},
&\mathsf{M}_- =
\begin{pmatrix}
1 & i & 0 \\
-i & 1 & 0 \\
0 & 0 & 0
\end{pmatrix}.\end{aligned}$$
We now make a complex ansatz $$\bm \alpha (r, \varphi) = \bm \alpha_0 (r, \varphi) + e^{-i \omega_{BS} t} \bm \alpha_1 (r, \varphi),$$ the real part of which is the physical solution, and obtain the equations $$\begin{aligned}
\mathsf{K}_0(\bm \nabla) \cdot \bm \alpha_0 (\bm r) &= \bm \rho_0 (\bm r), \\
\mathsf{K}_1(\bm \nabla) \cdot \bm \alpha_1 (\bm r) &= \bm \rho_1 (\bm r),\end{aligned}$$ where $$\begin{aligned}
\mathsf{K}_0(\bm \nabla) &= - \nabla^2 + \mathsf{L}_0 - \omega_{BS} [\bm w_0]_\times \cdot [\bm{\hat{z}}]_\times, \\
\mathsf{K}_1(\bm \nabla) &= \mathsf{K}_0(\bm \nabla) - \omega_{BS}^2 \mathsf{I} - i \omega_{BS} [\omega_{BS} \bm{\hat{z}} - \bm w_0]_\times.\end{aligned}$$ Here $[\bm w]_\times \cdot \bm v \equiv \bm w \times \bm v$ and $\mathsf{I}$ is the identity operator. Since these equations are linear, it is convenient to solve them first in the Fourier space and then transform back to the coordinate space. In Fourier space the two PDEs are transformed into algebraic equations which are easily solved for $\bm \alpha_0 (\bm k)$ and $\bm \alpha_1 (\bm k)$, $$\begin{aligned}
\bm \alpha_0 (\bm k) &= \mathsf{K}^{-1}_0(i \bm k) \cdot \bm \rho_0(\bm k) = \theta_{1x}(\bm k) \sin \eta \frac{\bm D_0}{k^2 - k_0^2}, \label{eq:sol_alpha_kspace_0}\\
\bm \alpha_1 (\bm k) &= \mathsf{K}^{-1}_1(i \bm k) \cdot \bm \rho_1(\bm k) \nonumber \\
&= \left[ \theta_{1y}(\bm k) - i \theta_{1x}(\bm k) \cos \eta \right] \sum_{j = 1}^{3} \frac{\bm D_j}{k^2 - k_j^2} \label{eq:sol_alpha_kspace_1}.\end{aligned}$$ Here $\bm D_m$ and $k_m^2$, $m = 0,1,2,3$, are obtained using partial fraction decomposition with respect to $k^2$. It can be seen that $k_0^2$ is always negative, $k_2^2$ and $k_3^2$ are always positive, and $k_1^2$ is negative at $\beta \lesssim 140^\circ$, changing to positive at larger values of the tipping angle. The explicit expressions of $\bm D_m$ and $k_m^2$ are too cumbersome to be written down here.
Next we take the inverse transform of $\bm \alpha_i(\bm k)$, $$\begin{split}
\bm \alpha_i(\bm r) &= \frac{1}{\left( 2 \pi \right)^2} \int_0^{2 \pi} d \varphi_k \int_0^\infty k dk e^{i k r \cos(\varphi - \varphi_k)} \bm \alpha_i(\bm k) \\
&= \frac{1}{\left( 2 \pi \right)^2} \int_0^{2 \pi} d \varphi_k \int_0^\infty k dk \Bigg\{ J_0(k r) \\
&\hphantom{=}+ 2 \sum_{n = 1}^\infty i^n J_n(k r) \cos \left[ n (\varphi - \varphi_k) \right] \Bigg\} \bm \alpha_i(\bm k).
\end{split}$$ Here we have used the Jacobi-Anger expansion [@abramowitz1965] to expand the exponential. Using $\bm \alpha_i(\bm k)$ from Eqs. (\[eq:sol\_alpha\_kspace\_0\]) and (\[eq:sol\_alpha\_kspace\_1\]), and $\bm \theta_1$ from Eq. (\[eq:theta\_1\]), we have $$\begin{aligned}
\bm \alpha_0 (\bm r) &= \vartheta_{1x}(\varphi) \sin \eta \bm D_0 \int_0^\infty dk \frac{J_1(k r)}{k^2 - k_0^2}, \\
\bm \alpha_1 (\bm r) &= \left[ \vartheta_{1y}(\varphi) - i \vartheta_{1x}(\varphi) \cos \eta \right] \sum_{j = 1}^{3} \bm D_j \int_0^\infty dk \frac{J_1(k r)}{k^2 - k_j^2},\end{aligned}$$ where $\bm \vartheta_1(\varphi) \equiv r \bm \theta_1(r, \varphi)$.
The next step is to evaluate the integral $\int_0^\infty dk J_1(k r) / (k^2 - k_m^2)$. If $k_m^2<0$, the integrand is finite on the positive $k$-axis and can be evaluated analytically [@abramowitz1965]. If $k_m^2>0$, there is a simple pole at $k = k_m$. In this case we use the standard trick and shift the pole slightly away from the real axis by adding a small imaginary part to the denominator, $k_m \to k_m \pm i \varepsilon$, $\varepsilon > 0$. The choice of sign here determines the asymptotic behaviour of the solution. We choose the positive sign since this makes the solution an outward travelling wave. The negative sign would lead to a wave travelling towards the vortex. After evaluating the integral we take the limit $\varepsilon \to 0$. As a result we get $$\begin{aligned}
\bm \alpha_0 (\bm r) &= \vartheta_{1x}(\varphi) \sin \eta \frac{\bm D_0}{k_0} \left[ \frac{i \pi}{2} H_1^{(1)}(k_0 r) -\frac{1}{k_0 r} \right], \\
\bm \alpha_1 (\bm r) &= \left[ \vartheta_{1y}(\varphi) - i \vartheta_{1x}(\varphi) \cos \eta \right] \nonumber \\
&\times \sum_{j = 1}^{3} \frac{\bm D_j}{k_j} \left[ \frac{i \pi}{2} H_1^{(1)}(k_j r) -\frac{1}{k_j r} \right].\end{aligned}$$ Here $H_1^{(1)}(x)$ is a Hankel function of the first kind. Note that both $\bm \alpha_0 (\bm r)$ and $\bm \alpha_1 (\bm r)$ are zero at the origin. This means that the behaviour of $\bm \theta (\bm r)$ near the core is determined by $\bm \theta_1 (\bm r)$, as was claimed earlier. Using the asymptotic expansion of $H_1^{(1)}(x)$ [@abramowitz1965], the leading order approximation of $\bm \alpha$, valid far from the core ($r \gg 1$), is given by $$\label{eq:alpha_asymptotic}
\begin{split}
\bm \alpha (\bm r, t) &\approx \left[ \vartheta_{1y}(\varphi) - i \vartheta_{1x}(\varphi) \cos \eta \right] \\
&\times \sqrt{\frac{\pi}{2 r}} \sum_{j = 1}^{3} \frac{\bm D_j}{k_j^{3/2}} e^{i(k_j r - \omega_{BS} t - \pi / 4)}.
\end{split}$$ This shows that far from the origin the solution indeed consists of waves propagating away from the vortex, as we claimed above.
High-field approximation
------------------------
We shall now consider Eq. (\[eq:EquationOfMotionSource\]) in the limit of high magnetic field, $\omega_L \gg 1$. We are again interested in a solution that is periodic in time. We therefore expand $\bm \theta_2$ in Fourier series as $$\bm \theta_2 \left( \bm r, t \right) = \sum_{n = -\infty}^\infty \bm \beta_n \left( \bm r \right) e^{i n \omega_{BS} t}.$$ Since $\bm \theta_2 \left( \bm r, t \right) $ is real, the coefficients must satisfy the relation $\bm \beta_n(\bm r) = \bm \beta_{-n}^*(\bm r)$. Using the expression of $\bm{\hat{n}}$ from Eq. (\[eq:nt\]), the coefficients $\mathsf{L}(t)$ and $\bm w(t)$ in (\[eq:EquationOfMotionSource\]) can be written as $$\begin{aligned}
\mathsf{L}(t) &= \sum_{n = -2}^{2} \widetilde{\mathsf{L}}_n e^{i n \omega_{BS} t}, \\
\bm w(t) &= \bm{\hat{B}} + \omega_{BS}^{-2} \sum_{n = -1}^{1} \bm{\widetilde{w}}_n e^{i n \omega_{BS} t},\end{aligned}$$ where $\widetilde{\mathsf{L}}_n = \widetilde{\mathsf{L}}_{-n}^*$ and $\bm{\widetilde{w}}_n = \bm{\widetilde{w}}_{-n}^*$. Plugging these into (\[eq:EquationOfMotionSource\]) yields an infinite system of coupled partial differential equations, $$\widetilde{\mathsf{K}}_n(\bm \nabla) \cdot \bm \beta_n (\bm r) + \sum_{m = -\infty}^{\infty} \widetilde{\mathsf{K}}_{n,m} \cdot \bm \beta_m (\bm r) = \bm{\widetilde{\rho}}_n (\bm r),$$ where $$\begin{aligned}
\widetilde{\mathsf{K}}_n(\bm \nabla) &= -(1+c) \nabla^2 + c \bm \nabla \bm \nabla - \omega_{BS}^2 \left( n^2 \mathsf{I} + i n [\bm{\hat{B}}]_\times \right), \\
\widetilde{\mathsf{K}}_{n,m} &= - i m [\bm{\widetilde{w}}_{n-m}]_\times + \widetilde{\mathsf{L}}_{n-m}, \\
\bm{\widetilde{\rho}}_n (\bm r) &= - \widetilde{\mathsf{L}}_n \cdot \bm \theta_1(\bm r).\end{aligned}$$
In the high-field limit we may approximate $\omega_{BS} \approx \omega_L$ for all $\beta$. Furthermore, the constant term in $\widetilde{\mathsf{K}}_n(\bm \nabla)$, proportional to $\omega_{BS}^2$, dominates all the terms $\widetilde{\mathsf{K}}_{n,m}$, except when $n = 0$. We therefore assume that the coupling terms between different $\bm \beta_n$:s may be neglected when $n \neq 0$, and are left with $$\begin{aligned}
\widetilde{\mathsf{K}}_0(\bm \nabla) \cdot \bm \beta_0 (\bm r) + \sum_{m = -\infty}^{\infty} \widetilde{\mathsf{K}}_{0,m} \cdot \bm \beta_m (\bm r) &= \bm{\widetilde{\rho}}_0 (\bm r), \\
\widetilde{\mathsf{K}}_n(\bm \nabla) \cdot \bm \beta_n (\bm r) &= \bm{\widetilde{\rho}}_n (\bm r), & n \neq 0\end{aligned}$$
The time-independent part $\bm \beta_0$ will not carry energy, and so we will ignore it. Because of the symmetry $\bm \beta_n(\bm r) = \bm \beta_{-n}^*(\bm r)$, we will only consider $n < 0$. The equations are again easy to solve in the Fourier space, giving us the solutions of the form $$\label{eq:sol_Highfield_kspace}
\bm \beta_n (\bm k) = \widetilde{\mathsf{K}}^{-1}_n(i \bm k) \cdot \bm{\widetilde{\rho}}_n (\bm k) = \frac{1}{k} \sum_{j = 1}^3 \frac{\bm E_{n,j} \left( \varphi_k \right)}{k^2 - k_{n,j}^2\left( \varphi_k \right)}.$$ Note that the poles $k_{n,j}$ now depend on the angle $\varphi_k$ and so the phase velocities of the waves depend on the direction of propagation.
When taking the inverse Fourier transform we use a different technique than in the case of the isotropic approximation. This is due to the fact the $k_{n,j}$ depend on the angle $\varphi_k$, which makes the exact integration over $\varphi_k$ difficult. We shall here only calculate the asymptotic solution, valid for $r \gg 1$, since this is sufficient to calculate the energy carried by the spin waves.
The inverse Fourier transform of $\bm \beta_n(\bm k)$ is given by $$\begin{split}
\bm \beta_n(\bm r) &= \frac{1}{\left( 2 \pi \right)^2} \int_0^{2 \pi} d \varphi_k \int_0^\infty k dk e^{i k r \cos(\varphi - \varphi_k)} \bm \beta_n(k, \varphi_k) \\
&= \frac{1}{\left( 2 \pi \right)^2} \int_0^{2 \pi} d \varphi_k \int_0^\infty k dk e^{i k r \cos \varphi_k} \bm \beta_n(k, \varphi_k + \varphi),
\end{split}$$ Here we made a change of variables $\varphi_k \to \varphi_k + \varphi$ and used the $2 \pi$-periodicity of the integrand to shift the limits of integration back to the interval $[0,2 \pi]$. The integral over $k$ can be calculated by extending it to the complex plane. First, the poles of $\bm \beta_n$ on the real axis are shifted slightly away from the axis, $k_{n,j} \to k_{n,j} \pm i \varepsilon$, $\varepsilon > 0$. We choose the positive sign, since it produces waves propagating away from the origin when $n < 0$.
If $\cos \varphi_k > 0$, we integrate over the contour $C_+ = [0 , R] \cup C_R^+ \cup [i R, 0]$, where $C_R^+ = \{R e^{i t} | t \in [0, \pi/2]\}$ is an arc of a circle of radius $R>0$ in the first quadrant. If $\cos \varphi_k < 0$, we use the contour $C_- = [0 , R] \cup C_R^- \cup [-i R, 0]$, where $C_R^- = \{R e^{i t} | t \in [0, -\pi/2]\}$ is an arc of a circle of radius $R > 0$ in the fourth quadrant. In the limit $R \to \infty$ the integral over $C_R^\pm$ vanishes due to Jordan’s lemma [@arfken1970]. Furthermore, in the limit $R \to \infty$ and $r \to \infty$ the integral over $[\pm i R, 0]$ tends to zero sufficiently fast as a function of $r$ so that we may neglect it. Thus $\int_{C_\pm} dk \approx \int_0^\infty dk$.
On the other hand, the integral over $C_\pm$ can be calculated using the residue theorem [@arfken1970]. In the limit $\varepsilon \to 0$ and $r \to \infty$ the dominant contribution comes from the poles on the real axis. Thus we obtain $$\bm \beta_n \left( \bm r \right) \approx \frac{i}{4 \pi} \sum_{j} \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} d \varphi_k e^{i k_{n,j} (\tilde{\varphi}_k) r \cos \varphi_k} \frac{\bm E_{n,j} \left( \tilde{\varphi}_k \right)}{k_{n,j} (\tilde{\varphi}_k)},$$ when $n < 0$. Here $\tilde{\varphi}_k = \varphi_k + \varphi$, and the sum is calculated over those values of $j$ for which $k_{n,j}^2 > 0$.
The integral over $\varphi_k$ can be calculated using the stationary phase approximation [@bleistein2010] which states that when $r \gg 1$, the dominant contribution to the integral comes from the points where the derivative of the phase $\Psi_{n,j} \left( \varphi_k \right) = k_{n,j} (\varphi_k + \varphi) \cos \varphi_k$ vanishes. In our case there is only one such stationary point for each $n$ and $j$ in the interval $\left[ -\pi / 2, \pi / 2 \right]$. We denote it by $\Phi_{n,j} \left( \varphi \right)$, so that $\Psi'_{n,j} \left( \Phi_{n,j} \left( \varphi \right) \right)= 0$. Note that the stationary point varies with $\varphi$. As a final result we get $$\bm \beta_n \left( \bm r \right) \approx \frac{i}{\sqrt{8 \pi r}} \sum_{j} \frac{\exp \left\{ i \left[ \Psi_{n,j} r + \frac{\pi}{4} \operatorname{sgn}\left( \Psi''_{n,j} \right) \right] \right\} }{k_{n,j} \sqrt{\left| \Psi''_{n,j} \right|}} \bm E_{n,j},$$ when $n < 0$. Here $\Psi_{n,j} = \Psi_{n,j} (\Phi_{n,j})$, $k_{n,j} = k_{n,j} (\varphi + \Phi_{n,j} )$, $\bm E_{n,j} = \bm E_{n,j} (\varphi + \Phi_{n,j} )$, and $\Psi''_{n,j} = \Psi''_{n,j} (\Phi_{n,j})$ is the second derivative of the phase evaluated at the stationary point.
Energy flux {#sec:energy}
===========
In the preceding section we solved the equation of motion for $\bm \theta$ in two different approximations. In both cases we saw that the asymptotic solution is given by a sum of cylindrical waves propagating away from the vortex axis. In this section we calculate the amount of energy carried by these waves. We take the unit of energy to be $\chi \Omega v R_0^2 / \mu_0 \gamma_0^2$ in our calculations. The units of any related quantities can then be easily determined from the units of energy, length, and time. For example, the unit of power per vortex length is $(\chi \Omega v R_0^2 / \mu_0 \gamma_0^2) \times (1/\Omega)^{-1} \times (v/\Omega)^{-1} = \chi \Omega^3 R_0^2 / \mu_0 \gamma_0^2$.
The amount of energy $E$ stored in the system inside a volume $V$ is given by $$E = \int_V dV \mathcal{H},$$ where $$\mathcal{H} = \bm{\dot{\theta}} \cdot \frac{\partial \mathcal{L}}{\partial \bm{\dot{\theta}}} - \mathcal{L} = \frac{\chi}{2 \mu_0 \gamma_0^2} \left( | \bm{\dot{\theta}}|^2 - \omega_{BS}^2 | \bm w |^2 \right) + f_D + f_G$$ is the Hamiltonian density. The rate of change of energy is then given by $$\label{eq:dE/dt}
\begin{split}
\frac{d E}{dt} &= \int_V dV \frac{\partial \mathcal{H}}{\partial t} = - \int_A \bm{\Sigma} \cdot d\bm A + \int_V dV \, p,
\end{split}$$ where $A$ is the surface of $V$, $$\Sigma_i = - (1+c) \dot{\theta}_k \partial_i \theta_k + c \dot{\theta}_k \partial_k \theta_i,$$ and $$\begin{split}
p &= \frac{2}{5} (1 - \cos \theta_0) (4 \cos \theta_0 + 3) \left( \bm{\hat{n}} \cdot \bm \theta \right) ( \bm{\dot{\hat{n}}} \cdot \bm \theta ) \\
&- \frac{2}{15} \sin \theta_0 (1 + 4 \cos \theta_0) \bm{\dot{\theta}} \cdot \left( \bm{\hat{n}} \times \bm \theta \right) \\
&- \frac{4}{15} \frac{v}{\Omega R_0} \sin \theta_0 (1 + 4 \cos \theta_0) \frac{d}{dt} \left( \bm{\hat{n}} \cdot \bm \theta \right) .
\end{split}$$ We see that two contributions affect the amount of energy inside $V$. The volume integral of $p$ describes the energy pumped into the system by the Brinkman-Smith mode which drives the system. The surface integral of $\bm \Sigma$ gives the energy flow through the surface of $V$. Equation (\[eq:dE/dt\]) expresses the conservation of energy. It is analogous to Poynting’s theorem in electromagnetism [@landau1971], with $\bm \Sigma$ playing the part of the Poynting vector, i.e., the energy flux density vector.
When solving the equation of motion, we assumed $\bm \theta$ to be periodic in time. This means that we study the system in dynamic equilibrium. We therefore expect that the time-averaged power, $\left\langle dE / dt \right\rangle_t$, vanishes. This is indeed so. In dynamic equilibrium the energy absorbed into the system inside the volume $V$ is equal to the energy flux through the surface of $V$.
Because the vortex is uniform in the $z$-direction, we choose the volume $V$ to be a cylinder of radius $r$ with its axis on the vortex axis. Let us denote the amount of energy absorbed into the system inside the cylinder per unit time and vortex length, averaged over time, by $P_{a}(r)$. Similarly, let us denote the time-averaged energy flux per vortex length out of the cylinder by $P_{f}(r)$. Based on the above discussion, these are both equal. We call this common value $P(r)$, so that $$P(r) = P_{a}(r) = P_{f}(r) = \int_0^{2 \pi} d \varphi \sigma_r \left( r, \varphi \right) ,
\label{e.sspow}$$ where we have defined $\sigma_r \left( r, \varphi \right) = r \Sigma_r \left( r, \varphi \right)$. There is no net flow of energy through the upper and lower surfaces of the cylinder because of the uniformity of the vortex along its axis.
In the following we discuss the behaviour of $\sigma_r(r, \varphi)$ and $P(r)$ as a function of different parameters. In the numerical calculations we use the coefficients $C_1 = C_2 = 1.33$ for the A-phase-core vortex and $C_1 = 3.00$, $C_2 = 0.08$ for the double-core vortex. These are the values obtained from numerical calculations at $T = 0.6 T_c$, $p = 29.3$ bar, as we mentioned in Sec. \[sec:static\_vortex\]. In the high-field approximation we set $c = 1$. Finally, if not stated otherwise, we use parameters $\beta = \theta_L$, $\eta = 0$, $\zeta = 0$ and $\omega_L = 2$.
Let us start by considering the dependence of $\sigma_r(r, \varphi)$ and $P(r)$ on $r$. Figure \[fig:Pr\] shows $P(r)$ as a function of $r$ in the case of the double-core vortex for some values of $\omega_L$ and $\beta$ in the isotropic approximation, where we were able to solve the equation of motion for all $r$. The exact form of $P(r)$ depends on the parameters used, but the general trend is clear. $P(r)$ starts from zero at the origin and increases monotonically within the range of a few dipole lengths $\xi_D = \sqrt{15} / 2$. Then, within the next few dipole lengths, there are transient oscillations. Finally, when $r$ is large, $P(r)$ oscillates about some average value. These asymptotic oscillations stem from interference between different wave modes in Eq. (\[eq:alpha\_asymptotic\]). Their amplitude depends on $\omega_L$ and $\beta$, and is at its largest somewhere near $\beta = 90^\circ$, $\omega_L = 1$. There are no asymptotic oscillations when $\beta = \theta_L$ since there is only one wave mode present. The oscillation amplitude approaches zero at large $\omega_L$. This is in accordance with the high-field approximation, which predicts that $P(r)$ is independent of $r$. The behaviour of the A-phase-core vortex is qualitatively similar.
From the form of $P(r)$ we see that most of the energy is absorbed into the system from the region of radius $\sim \xi_D$ around the vortex core. This is smaller than the usual inter-vortex distance in the experiments, which is $\sim 10 \xi_D$. Combining this to the fact that the asymptotic oscillations of $P(r)$ are, at least in most cases, relatively small, we can focus our interest on the average value of $P(r)$ at large $r$. We denote this average value by $$P \equiv \lim_{r \to \infty} \left\langle P(r) \right\rangle_r.$$ Similarly, we denote the average value of $\sigma_r(r,\varphi)$ by $$\sigma_r(\varphi) \equiv \lim_{r \to \infty} \left\langle \sigma_r(r,\varphi) \right\rangle_r.$$ These are related by $$P = \int_0^{2 \pi} d \varphi \sigma_r \left(\varphi \right).$$
The explicit form of $P$ is, in general, inconveniently complicated. One exception is the case $\beta = \theta_L$, $\eta = 0$. In this case we have $$\label{eq:P_c0}
P =\frac{\pi^2}{8} \omega_L \frac{2 \omega_L^2 - 1 + \sqrt{1 + 4 \omega_L^4}}{1 + 4 \omega_L^4} \left( C_1^2 + C_2^2 \right)$$ in the isotropic approximation and $$\label{eq:P_HighField}
\begin{split}
P = \frac{\pi^2}{8} \omega_L^{-1} &\Bigg[ \frac{3 c^2 + 6 c + 4 }{4 \left(1 + c \right)^2} \left( C_1^2 + C_2^2 \right) + \frac{2 c \left( 2 + c \right)}{4 \left(1 + c \right)^2} C_1 C_2 \Bigg]
\end{split}$$ in the high-field approximation. Note that the high-field limit of Eq. (\[eq:P\_c0\]) coincides with Eq. (\[eq:P\_HighField\]) when $c = 0$, as it should. Figure \[fig:Sigmar\] shows the radiation pattern, i.e., the angular dependence of $\sigma_r \left( \varphi \right)$. The A-phase-core vortex radiates symmetrically in both approximations. The pattern of the double-core vortex, on the other hand, is highly anisotropic. Most of the energy flow is in the direction perpendicular to $\bm{\hat{b}}$, with only a small fraction of the flow in the direction of $\bm{\hat{b}}$. We also see that the shape of the pattern is different in the two approximations. This stems from the different values of $c$ used in the approximations.
Figure \[fig:PTippedMagnetisation\] shows the behaviour of $P$ as a function of $\omega_L$ for different tipping angles $\beta$ in the case of the double-core vortex. In the high-field approximation $P \propto \omega_L^{-1}$. In the isotropic approximation $P$ behaves similarly for large $\omega_L$, but the low-field behaviour is different. The power vanishes at $\omega_L = 0$ and has a maximum near $\omega_L = 1$. In both approximations $P$ is an increasing function of $\beta$ up to $\theta_L$, beyond which it starts to decrease. The behaviour of the A-phase-core vortex is qualitatively similar.
Another interesting case to study is the dependence of $P$ on the direction of the magnetic field. Figure \[fig:PTiltedField\] shows $P$ as a function of $\cos^2 \eta$ at three different values of $\zeta$. As noted before, the A-phase-core vortex is symmetric and thus $P$ is independent of $\zeta$. The result for the double-core vortex, on the other hand, is highly dependent on $\zeta$. The susceptibility anisotropy of the double-core vortex favours the orientation $\zeta = \pi / 2$ in tilted field [@thuneberg1987]. Note that in all cases $P\left( \eta \right) = a_0 + a_2 \cos^2 \eta$ with some constants $a_0$ and $a_2$.
Twisted vortex {#sec:twisted}
==============
As mentioned in Sec. \[sec:introduction\], it is possible that the precessing magnetization of the Brinkman-Smith mode can rotate the half cores of the double-core vortex around each other, causing the vortex to twist. In this section we study how the radiation of spin waves is affected by uniform twisting of the core. This can be modelled by assuming that $\zeta$ depends on $z$ as $\zeta (z) = \kappa z$, where $\kappa$ is a dimensionless constant describing the amount of twisting. As a result, $\bm \theta_1$ will also depend on $z$ and there will be a new term in the equation of motion from the derivatives of $\bm \theta_1$ with respect to $z$, see Eqs. (\[eq:rho\]) and (\[eq:theta\_1\]). For simplicity, we shall discuss here only the case $\beta = \theta_L$, $\eta = 0$.
In the isotropic approximation, when there is no twisting, there is only one wave mode present, with wavenumber $k_0$. When twisting increases, the solution is of the form $\bm \alpha \left( \bm r, t \right) = e^{-i \omega_{BS} t}\left[ \bm \beta_1\left( r, \varphi \right) + e^{-2 i \kappa z} \bm \beta_2\left( r, \varphi \right) \right]$. Here $ \bm \beta_1$ describes a wave with the original wave number $k_0$, while $ \bm \beta_2$ describes a wave with a wavenumber $k = \sqrt{ k_0^2 - 4 \kappa^2}$. Thus, when the twisting increases, there is a critical value $\kappa_c = k_0 / 2$ beyond which $k$ becomes imaginary. Since $\bm \beta_1$ is not affected by twisting, it is the only part of the solution that carries energy away from the vortex when $\kappa > \kappa_c$. The power per vortex length is given by $$\label{eq:P_Twisted}
\begin{split}
P / P_0 =
\begin{cases}
1 - \frac{1}{2} \frac{\kappa^2}{\kappa^2_{c}} \frac{ \left( C_1 - C_2 \right)^2}{C_1^2 + C_2^2}, & 0 \leq \kappa \leq \kappa_c \\
\frac{1}{2}\frac{(C_1 + C_2)^2}{C_1^2 + C_2^2}, & \kappa > \kappa_c
\end{cases},
\end{split}$$ where $$P_0 = \frac{\pi^2}{8} \omega_L \frac{2 \omega_L^2 - 1 + \sqrt{1 + 4 \omega_L^4}}{1 + 4 \omega_L^4} \left( C_1^2 + C_2^2 \right)$$ is the value of $P$ for an untwisted vortex and $$\label{eq:kappa_c}
\kappa_c = \sqrt{ \frac{2 \omega_L^2 - 1 + \sqrt{1 + 4 \omega_L^4}}{8} }.$$ First of all we see that $P$ is independent of $\kappa$ in the case of the A-phase-core vortex $\left( C_1 = C_2 \right)$ so only the double-core vortex is affected by twisting. This is again due to the cylindrical symmetry of the A-phase-core vortex. We also see that the result coincides with our earlier result (\[eq:P\_c0\]) when $\kappa = 0$. When $\kappa \leq \kappa_c$, power decreases quadratically with $\kappa$. When $\kappa > \kappa_c$, $P$ is constant. Figure \[fig:Twisting\](a) shows the ratio $P / P_0$ as a function of $\kappa / \kappa_c$ at $\omega_L = 2$. Figure \[fig:Twisting\](b) shows $\kappa_c$ as a function of $\omega_L$.
In the high-field approximation there are more wave modes present when $c \neq 0$. This makes things more complicated. It is, however, easy to calculate what is the maximal effect of twisting. Two of the modes are independent of $\kappa$. The remaining ones all have a critical value $\kappa_c^{(i)}$, so that the $i$:th mode disappears when $\kappa > \kappa_c^{(i)}$. When $\kappa > \kappa_c \equiv \max\{ \kappa_c^{(i)} \}$, the power attains its minimum value $$P_{\min} = \frac{\pi^2}{16} \omega_L^{-1} \left( C_1 + C_2 \right)^2.$$ This is in accordance with the high-field limit of Eq. (\[eq:P\_Twisted\]).
Both approximations therefore show the same qualitative behaviour. Twisting of the vortex core reduces the radiated power up to some saturation point $\kappa_c$. Further twisting has no effect on the power.
Comparison with experiments {#sec:experiments}
===========================
In this section we compare the results above with experimental results from Refs. [@kondo1991] and [@dmitriev1990]. We include only the dissipation by spin wave radiation in the quantitative comparisons, although we know that the Leggett-Takagi relaxation also contributes [@laine2016]. For simplicity, we use the isotropic approximation. For the double-core vortex we use $C_2 / C_1 = 0$, as vortex-structure calculations indicate that $C_2$ is small, and $\zeta = \pi / 2$, which is favored by susceptibility anisotropy. This leaves $C_1$ as the only free parameter. To compare theory with experiments, we first determine $C_1$ that gives the best fit to the measured values. After that, we compare the fitted value of $C_1$ with the one obtained from numerical solution of the vortex structure. Unless otherwise mentioned, the experiments were done using $p = 29.3$ bar, $B = 14.2$ mT, $\eta = 0$, and $\beta = \theta_L$.
Figure \[fig:KondoComparison\] shows the absorption per vortex length as a function of $\cos^2 \eta$. The experimental data is taken from Fig. 2 of Ref. [@kondo1991]. There are three different data sets shown in the figure, one for the A-phase-core vortex at $T = 0.60 T_c$ and two for the double-core vortex at temperatures $0.48 T_c$ and $0.60 T_c$. Each of these would seem to obey the rule $P\left( \eta \right) = a_0 + a_2 \cos^2 \eta$, as noted in [@kondo1991]. This is also predicted by theory. Theoretical curves shown in the figure use parameter $C_1$ fitted to the experimental data. For the A-phase-core vortex we obtain $C_1 = 1.66$. For the double-core vortex we obtain $C_1 = 5.81$ at $T = 0.48 T_c$ and $C_1 = 4.20$ at $T = 0.60 T_c$. Note that since we have assumed $C_2 = 0$ for the double-core vortex, theory predicts that the constant $a_0$ vanishes, which seems to be contrary to the experimental data. Similar problem appears with the Leggett-Takagi relaxation, which also has quadratic dependence on $C_1$ and $C_2$ [@laine2016].
![A comparison between theoretical (lines) and experimental (points) values of $P$ as a function of $\cos^2 \eta$. The experimental data is from Ref. [@kondo1991]. Theoretical curves use $C_1$ fitted to the data. The fitting procedure yields $C_1 = 1.66$ for the A-phase-core vortex at $T = 0.60 T_c$, $C_1 = 5.81$ for the double-core vortex at $T = 0.48 T_c$, and $C_1 = 4.20$ for the double-core vortex at $T = 0.60 T_c$. Theoretical curves are of the form $P\left( \eta \right) = a_0 + a_2 \cos^2 \eta$. The experimental data seems to obey the same formula. Since we have assumed that $C_2 = 0$ for the double-core vortex, theory predicts that $a_0 = 0$. In the experiment $a_0 > 0$. External parameters are given in Sec. \[sec:experiments\].[]{data-label="fig:KondoComparison"}](KondoComparison){width="0.9\columnwidth"}
Figure 1 in Ref. [@dmitriev1990] shows that the measured absorption decreases when the magnetic field is increased from $14.2$ mT to $28.4$ mT. We study the ratio $\varrho = P\left( 28.4 \text{mT} \right)/P\left( 14.2 \text{mT} \right) $ near $T_V$, which is the phase transition temperature between the core structures. The measured values are $\varrho= 0.68$ for both vortex types. Theory predicts $\varrho = 0.54$. As a comparison, the absorption by the Leggett-Takagi relaxation is field-independent, while the absorption by spin diffusion increases quadratically with the field.
Figure \[fig:KondoComparisonTemperature\] shows the absorption as a function of temperature for the double-core vortex. The experimental data is taken from Fig. 1 of Ref. [@kondo1991]. Since the temperature range is quite narrow, $0.48 T_c < T < 0.6 T_c$, we assume that we can approximate $C_1$ by a linear function $C_1(T) = A T / T_c + B$. The coefficients $A$ and $B$ can be calculated using the values $C_1(0.48 T_c) = 5.81$ and $C_1(0.60 T_c) = 4.20$ we obtained above. The theoretical curve shown in the figure uses this linear approximation for $C_1$. The clear temperature dependence is in contrast to the Leggett-Takagi relaxation, where the absorption is essentially temperature-independent [@laine2016].
![A comparison between theoretical (line) and experimental (points) values of $P$ as a function of temperature. The experimental data is from Ref. [@kondo1991]. The theoretical curve is obtained assuming linear dependence of $C_1$ on $T$, $C_1(T) = A T / T_c + B$. The coefficients $A = -13.4$ and $B = 12.2$ were calculated using the values $C_1(0.48 T_c) = 5.81$ and $C_1(0.60 T_c) = 4.20$ that we obtained from fitting to the data as a function of $\cos^2 \eta$, see Fig. \[fig:KondoComparison\]. External parameters are given in Sec. \[sec:experiments\].[]{data-label="fig:KondoComparisonTemperature"}](KondoComparisonTemperature){width="0.9\columnwidth"}
Next we consider the effect of twisting the double-core vortex. According to Ref. [@kondo1991], the measured ratio $\varrho_{\text{twist}}$ between the absorptions in the twisted and untwisted states of the double-core vortex is $\varrho_{\text{twist}} = 0.83$ at $T = 0.5 T_c$ and $\varrho_{\text{twist}} = 0.87$ at $T = 0.6 T_c$. These are in keeping with theory, which predicts that $1/2 \leq \varrho_{\text{twist}} \leq 1$. The measured values of $\varrho_{\text{twist}}$ correspond to $\kappa / \kappa_c = 0.58$ and $\kappa / \kappa_c = 0.51$, or $\kappa = 0.68 $ and $\kappa = 0.66 $, respectively.
Based on the numerical solution of the double-core vortex structure, the value $C_1=3.7$ was obtained in Ref. [@silaev2015] at $T=0.5 T_c$. This is by a factor of $2 / 3$ smaller than the value obtained from the fitting procedure above. Since the absorption is quadratic in $C_1$, approximately one half of the measured absorption is explained by this value of $C_1$. Better agreement is obtained in the temperature dependence. Based on the vortex structure calculation, $C_1(0.6T_c)/C_1(0.5T_c) = 0.81$. This is close to the value $0.75$ obtained above (Fig. \[fig:KondoComparisonTemperature\]).
To summarize, we have compared the theoretical model of dissipation by spin wave radiation with experiments reported in Refs. [@kondo1991] and [@dmitriev1990]. Without any adjustable parameters, it explains the order of magnitude of the absorption. What is more, it accounts well for the dependencies of the absorption on the direction and the magnitude of the magnetic field, on temperature, and on twisting. With the Leggett-Takagi relaxation, only the dependence on the field direction can be understood. Thus it seems that major part of the absorption is explained by radiation of spin waves. Including both the spin wave radiation and the Leggett-Takagi relaxation in the analysis would lead to better agreement with experiments, especially in the magnitude of the absorption. Possible reasons for the remaining problems may be the inadequacy of the weak-coupling theory to calculate the parameters $C_1$ and $C_2$, as well as the omission of spin diffusion and the detailed structure of the vortex core. One task that still remains to be done is detailed comparison of twisting and its dynamics with experiments [@Sonin93; @Krusius93].
Both the present model of spin wave radiation and the Leggett-Takagi relaxation calculated in Ref. [@laine2016] give absorption that is quadratic in the coefficient $C_1$ and $C_2$. These coefficients are solely determined by the structure of the vortex core. For example, the simplest theoretical vortex structure has $C_1=C_2=0$ [@Ohmi83]. This explains the experimental observation that the absorption at large tipping angles is more sensitive to the vortex-core structure than the frequency shift at small tipping [@Ikkala82; @Hakonen89]. The latter is determined by susceptibility anisotropy, which is only partially dependent on the core structure.
Conclusions
===========
We have studied spin dynamics of superfluid $^3$He-B in the presence of an isolated vortex. The vortex perturbs the uniformly precessing magnetization and gives rise to spin waves. These waves carry energy, causing dissipation in the system. We calculated the amount of dissipation and its dependence on several parameters. Good agreement with experiments indicates that spin wave radiation is the dominant dissipation mechanism for vortices in the intermediate-temperature range.
We thank Volodya Eltsov, Matti Krusius, Edouard Sonin, and Grigory Volovik for discussions. This work was financially supported by the Vilho, Yrjö and Kalle Väisälä Foundation, the Jenny and Antti Wihuri Foundation and the Oskar Öflunds Stiftelse sr.
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---
abstract: 'We here provide a distribution-free approach to the random factor analysis model. We show that it leads to the same estimating equations as for the classical ML estimates under normality, but more easily derived, and valid also in the case of more variables than observations ($p>n$). For this case we also advocate a simple iteration method. In an illustration with $p=2000$ and $n=22$ it was seen to lead to convergence after just a few iterations. We show that there is no reason to expect Heywood cases to appear, and that the factor scores will typically be precisely estimated/predicted as soon as $p$ is large. We state as a general conjecture that the nice behaviour is not despite $p>n$, but because $p>n$.'
author:
- |
Rolf Sundberg, Stockholm University\
Uwe Feldmann, University of Saarland
title: 'Distribution-free factor analysis — Estimation theory and applicability to high-dimensional data.'
---
*Key words:* EFA; FA; fixed point iterations; likelihood equations; more variables than observations; SVD.
Introduction
============
In this paper we consider parameter estimation in a distribution-free version of the standard (Gaussian) factor analysis (FA) model, with special emphasis on the case of more variables than observations. The FA model means describing a sample $x_1,\ldots,x_n$ of $p$-dimensional vectors as $$\label{eq:Modelforx}
x_i = \mu + \Lambda f_i + e_i \,, \hspace{5mm} i=1,\ldots,n.$$ Here $\mu$ is the mean value vector, $\Lambda$ is a $p\times k$ coefficients (loadings) matrix, $k<\min(n,p)$, and the $f_i$s are mutually independent latent $k$-vectors (factor scores), standardized to zero mean and unit covariance matrix $I_k$ (for identifiability). The $e_i$s are assumed mutually independent $p$-vectors with uncorrelated components and diagonal covariance matrix $\Psi^2$. Also, $f_i$ and $e_i$ should be mutually independent. In matrix form we write as $X = \mu {\bf1} + F \Lambda^T + E$, with the vectors of as rows.
Usually, normality of $f$ and $e$ in is assumed, and more observations than variables, that is $n>p$. Then Gaussian maximum likelihood methods can be used, and are more or less standard. However, in recent years interest has increased both in more robust methods and in methods for the case of more variables than observations, $p>n$. Among papers having appeared after the comprehensive review by Bartholomew & Knott (1999, ch. 3), we mention Robertson & Symons (2007), who study extension of Gaussian maximum likelihood to the case $p>n$, and a number of papers by Trendafilov and Unkel, in particular Trendafilov & Unkel (2011) and Unkel & Trendafilov (2010a&b), also dealing with the case $p>n$ but proposing alternative models and estimation methods. Trendafilov & Unkel (2011) appear skeptical to the results of Robertson & Symons (2007), and proclaim that when $p>n$ the model assumption of the latter, that $\Psi^2$ is positive definite, is inconsistent with their own model for data. That is certainly right, and we argue below (Sec. 6) that the model for data used by Trendafilov & Unkel is artificial and unrealistic.
Our main aim, however, is to show that the fitting of models of type in the case of large $p$ is not problematic, and that in any case there is no need to assume normality. We will first derive some basic distribution-free properties of model . These are expressed in a normalization of the $x$-components by $\Psi$, shown to be suitable for our purpose. It will turn out without difficulties that these properties lead to estimating equations that are the same as the well-known likelihood equations for $n>p$, thus yielding distribution-free support to the normality-based MLE.
Another well-known technique for dimension reduction is principal components analysis (PCA). PCA aims at describing as much as possible of $\Sigma_{xx}$ by a number of principal components (PCs, linear forms in $x$). There is no model behind PCA, but sometimes the PCs are regarded as representing latent variables in a different, less well-defined way. PCA techniques also have a role in factor analysis. Due to its scale-dependence, the choice of scaling is important.
In the very special case when the error $e$ vanishes, i.e. $\Psi^2=0$ in , $\Lambda \Lambda^T$ can be determined by a PCA on $\Sigma_{xx}$, or estimated by a PCA on the sample covariance matrix $S_{xx}$ (or an SVD on the $x$-data matrix itself). Similarly, if $\Psi^2$ were not zero but regarded as known, we could subtract it from $\Sigma_{xx}$ or $S_{xx}$ and in this way open for use of PCA. This was the basis for the early Principal Factor Analysis method of fitting the FA model: Use some initial $\Psi^2$ to subtract from $S_{xx}$, find PCs yielding an estimate of $\Lambda\Lambda^T$, use this to calculate a new $\Psi^2$, etc. Such methods were found inefficient and unstable, however. In particular they were not scale invariant, in contrast to Gaussian ML (see Bartholomew & Knott, 1999, Sec. 3.17). From the time when ML methods became computationally feasible and attractive (Jöreskog, 1967, Lawley, 1967), ML estimation has widely replaced the principal factor analysis method.
In the present paper a new distribution-free method for FA model fitting is proposed, that utilizes principal components of a naturally *rescaled* instead of *reduced* sample covariance matrix. To our surprise we have not seen this approach in the literature. The methodology has the following properties:
- It yields the same equations as Gaussian ML–FA for $p<n$, and therefore supports the use of these estimation equations even when the Gaussian distribution is questionable;
- It is scale invariant in the sense mentioned above;
- without problems, it allows more variables than observations ($p>n$);
- It yields estimated or predicted factor scores of high precision when $p$ is large.
The basic model properties to be derived in the next section will naturally lead to estimating equations for distribution-free parameter estimation. Different iterative methods to solve these equations are discussed in Section 3. Use of singular value decompositions (SVD) will not only make the computations fast, but also yield some further insight (Sec. 4). The SVD tool is used in Sec. 5 to yield expressions for factor scores and residuals. These are compared in Sec. 6 with the model properties of Trendafilov & Unkel (2011). Finally, in Sec. 7, the recommended iteration method is successfully tried on gene expression data with $p>>n$.
As mentioned above, we assume we have a sample of multivariate $x$-data $x_i$, $i=1,\ldots,n$, $\dim(x)=p$. We will later assume that the $x$-sample is mean-standardized, so we need only consider the sample covariance matrix $S_{xx}=X^T X/(n-1)$ and the corresponding population covariance matrix $\Sigma_{xx}$. In the next section, we concentrate on $\Sigma_{xx}$, so the sample size $n$ and its relation to the dimension $p$ will not yet be a question.
A canonical distribution-free introduction to the FA model
==========================================================
For the FA model , the population covariance matrix $\Sigma_{xx}$ ($p\times p$) is $$\label{eq:Sigmaxx}
\Sigma_{xx} = E(S_{xx}) = \Lambda \Lambda^T + \Psi^2 .$$ There is a rotational ambiguity in the loading parameters of this representation. For uniqueness we will use the same well-known and natural constraint as in the standard Gaussian ML approach: $$\label{eq:diagonal}
\Lambda^T \Psi^{-2} \Lambda \quad \textrm{is diagonal}.$$ This demand will be equivalent with an assumption that the $p\times k$ matrix $\Psi^{-1}\Lambda$ has orthogonal columns. Our motivation to make this particular choice will be clear below.
As mentioned in Sec. 1, classical Principal Factor Analysis requires an initial or current estimate of $\Psi^2$ to be subtracted from $S_{xx}$, so that ideally we would get $\Lambda \Lambda^T$. PCA is now used on the resulting reduced covariance matrix $S_{xx} - \Psi^2$. Below we will instead use a rescaled covariance matrix, that will be demonstrated to have much better properties.
Consider rescaling the vector $x$ to $z=\Psi^{-1} x$, neglecting for a moment the fact that $\Psi$ is unknown (later we will update $\Psi$ iteratively). This will make all observation components have the same error variance. The total covariance matrix $\Sigma_{zz}$ for a $z$-vector is $$\label{eq:Sigmazz}
\Sigma_{zz} = \Psi^{-1} \Sigma_{xx} \Psi^{-1} = \Psi^{-1}\Lambda (\Psi^{-1}\Lambda)^T + I_p = \Lambda_z \Lambda_z^T + I_p,$$ where $I_p$ denotes the $p\times p$ identity matrix and $\Lambda_z=\Psi^{-1}\Lambda$ is $p\times k$, cf. . Because of assumption we know that $\Lambda_z$ has orthogonal columns, and it follows that these columns are eigenvectors of the matrix $\Sigma_{zz}$. In a condensed representation we can write $$\label{eq:eigenrelation}
\Sigma_{zz} \Lambda_z = \Lambda_z \Omega_z$$ where $\Omega_z$ is a diagonal $k \times k$ matrix with the corresponding eigenvalues as diagonal elements, that is $$\label{eq:Omega}
\Omega_z = \Lambda_z^T \Lambda_z + I_k.$$ The sum of these $k$ eigenvalues is $$\label{eq:traceOmega}
{\mathrm{trace}}(\Omega_z) = {\mathrm{trace}}(\Lambda_z \Lambda_z^T) + k = k + {\mathrm{trace}}(\Sigma_{zz} - I_p) = k + \sum_{j=1}^p ((\Sigma_{xx})_{jj}-\psi_j^2)/\psi_j^2.$$ If $k$ latent factors are both necessary and sufficient for the model to hold, precisely these $k$ eigenvalues of $\Sigma_{zz}$ will be $>1$. For a complete set of eigenvectors of $\Sigma_{zz}$, we need to supplement $\Lambda_z$ by $p-k$ vectors spanning the orthogonal complement of the space spanned by $\Lambda_z$. They will all have the eigenvalue 1.
Equation does not specify the length of the eigenvectors in $\Lambda_z$. For that reason we also introduce the corresponding set of normalized eigenvectors $\Phi_z$, $$\Phi_z = \Lambda_z (\Lambda_z^T \Lambda_z)^{-1/2},$$ which is a $p\times k$ matrix of $k<p$ orthonormal eigenvectors. Thus, $\Phi_z^T \Phi_z = I_k$, the $k\times k$ identity matrix. The matrix $\Phi_z$ of course satisfies the same relation as $\Lambda_z$: $$\label{eq:Phi-formula}
\Sigma_{zz} \Phi_z = \Phi_z \Omega_z .$$
Thus, if we knew $\Psi$ and $\Sigma_{xx}$, we could form $\Sigma_{zz}$ and calculate its first (=largest) $k$ eigenvectors $\Phi_z$, with their eigenvalues $\Omega_z$, and solve for the loadings matrix $\Lambda=\Psi \Lambda_z$: $$\label{eq:Lambdaz}
\Lambda_z = \Phi_z (\Lambda_z^T \Lambda_z )^{1/2}= \Phi_z \left( \Omega_z - I_k \right) ^{1/2},$$ and $$\label{eq:Lambda}
\Lambda = \Psi \Lambda _z = \Psi \Phi_z \left( \Omega_z - I_k \right) ^{1/2}.$$ This tells how we can compute $\Lambda$ as a function of $\Psi$ and $\Sigma_{xx}$. In addition, yields a trivially simple formula for the diagonal matrix $\Psi^2$ as a function of $\Lambda$, given $\Sigma_{xx}$: $$\label{eq:diagPsi1}
\mathrm{diag}(\Psi^2) = \mathrm{diag}(\Sigma_{xx} - \Lambda \Lambda^T),$$ where diag stands for the diagonal part of the matrices, as a vector. An equivalent alternative is $$\label{eq:diagPsi2}
\mathrm{diag}(\Psi^{-1} \Sigma_{xx} \Psi^{-1}) = \mathrm{diag}(\Lambda_z \Lambda_z^T + I_p) .$$ Here the left hand side can be obtained by elementwise multiplication of the diagonals of $\Psi^{-2}$ and $\Sigma_{xx}$, or equivalently as $\Psi^{-2}\mathrm{diag}(\Sigma_{xx})$.\
Parameter estimation
====================
For parameter estimation based on data, the formulae above can be used with $S_{xx}$ inserted for $\Sigma_{xx}$: This yields an estimating equation for $\Lambda$ as $$\label{eq:Lambdahat}
\widehat{\Lambda}(\Psi)
= \Psi \, \Phi_{z}(S_{zz}) \, \left(\Omega_{z}(S_{zz}) - I_k \right)^{1/2}.$$ Here it is indicated that $\widehat\Lambda$ from is a function of $\Psi$, and that $\Phi_z$ and $\Omega_{z}$ are obtained from $S_{zz}= \Psi^{-1}S_{xx} \Psi^{-1}$ and not from the theoretical $\Sigma_{zz}$. The other estimating equation is obtained from formula or with $S_{xx}$ for $\Sigma_{xx}$: $$\label{eq:diagPsi}
\mathrm{diag}(\widehat{\Psi}^2) = \mathrm{diag}(S_{xx} - \Lambda \Lambda^T),$$ We thus want a solution of these two estimating equations relating $\Psi$ and $\Lambda$.
When $p>n$, these estimating equations turn out to be identically the same as the Gaussian model likelihood equations. This can be taken either as a robustness argument for the Gaussian ML estimates, or as well as a strong argument for the distribution-free method, at least for large $n$. They are also generally quite intuitive. Formula is an obvious demand, and formula or is a truncated PCA on $\Lambda \Lambda^T$ after a suitable, albeit parameter-dependent rescaling.
There is no explicit solution to the set of equations for $\Lambda$ and $\Psi^2$. Thus we have to use some iterative method, and a partial choice is obvious: Select $\Psi^2$ in some way and use this $\Psi^2$ in a calculation of a corresponding $\Lambda$, to be used to update $\Psi^2$, etc. The step yielding $\Lambda$ will be taken as given in most of the sequel. The question remains how to update $\Psi^2$. Unless some care is used, such equations might yield impossible diagonal elements for $\Psi^2$. We return to this question in the next paragraph.
There are alternative estimation methods to ML proposed in the FA literature. Among unweighted and weighted LS metods, the one denoted $\Delta_2$ in Bartholomew & Knott (1999) appears to be of particular interest in the present context, since it weights data by $\Psi^{-1}$, thus corresponding to our transformation of data. For given $\Psi$, the $\Delta_2$ method yields identically the same estimating equation for $\Lambda$ as the ML method. To estimate $\Psi^2$ by the $\Delta_2$ method is (quoting Bartholomew & Knott) a good deal more complicated. The choice of $\Psi$ should be such that the sum of squared differences from 1 of the $p-k$ smallest eigenvalues of $\Psi^{-1} S_{xx} \Psi^{-1}$ is as small as possible, under the constraint that they are all $\ge 1$. This constraint, however, excludes the case of a singular $S_{xx}$ and in particular the case $p>n$, and the method is therefore of little interest here.
Another type of estimation method are the estimation procedures in for example Trendafilov & Unkel (2011), jointly estimating $F$, $\Lambda$ and $\Psi^2$. They are based on a different model with additional constraints, which are not adequate in the present setting. They will be further commented in [Section \[sec:U&T\]]{}.
Iterative solution of the estimating equation system and
---------------------------------------------------------
The pair of estimating equations and leads naturally to an iterative procedure, where we start with a provisional $\Psi$, calculate $\Lambda$ by , calculate a new $\Psi$ by , etc. Such calculations are simplified by use of SVD on the sample of $z$-vectors, see next section. However, some variants are possible when using the equations for $\Psi$.
The simplest version is to use to express the new $\Psi^2$, in component form $$\label{eq:hatpsi1}
\psi_j^2 =(S_{xx})_{jj} - \left(\Lambda\Lambda^T \right)_{jj}$$ with the current $\Lambda$, based on the previous $\Psi$, on the right hand side. This procedure has a long history, where it turned out to often converge slowly and sometimes to stop before true convergence was achieved. Even worse, the iteration could sometimes yield one or more negative $\Psi^2$ components, known as Generalized Heywood cases. This might be because the best values had not yet been found, but a contributing reason could be the wrong $k$ or an otherwise inadequate model. For these reasons, this iteration procedure for Gaussian ML estimation was abandoned, and replaced by a step of direct likelihood maximization to yield $\Psi$ for given $\Lambda$ (Jöreskog, 1967; Lawley, 1967). Another alternative is to use the EM algorithm (Rubin & Thayer, 1982).
The equivalent formula suggests a different iteration procedure than . Calculate the new $\Psi$ by , with the current $\Lambda_z$ on the right hand side. This yields the iteration step in component form given by $$\label{eq:hatpsi2}
\psi_j^2 =\frac{ (S_{xx})_{jj} }{1 + \left(\Lambda_z\Lambda_z^T \right)_{jj} }.$$ One advantage of this is that it yields a positive $\Psi^2$ whatever is the current $\Lambda_z$. On the other hand, our experiences indicate that it is a slower algorithm, and we do not recommend it.
Theoretical investigation of the rate of convergence of these methods is difficult, due to the updating of eigenvectors involved. On the other hand, we have used the updating formula on data with large $p$ ($p>>n$) without any problems, see further discussion in [Section \[sec:SVD\]]{} and [Section \[sec:genedata\]]{}.
Use of the singular value decomposition (SVD) {#sec:SVD}
=============================================
Let $X$ be the $n\times p$ matrix of column mean-centered $x$-data, and correspondingly $Z=X \Psi^{-1}$ for a provisional $\Psi$. A convenient procedure for carrying out the computations above is to calculate and use the singular value decomposition (SVD) of the matrix $Z$, given $\Psi$: $$Z = U D V^T,$$ where $U$ ($n\times p$ if $p<n$) and $V$ ($p\times p$) have orthonormal columns (the left and right singular vectors), and $D$ is a diagonal $p\times p$ matrix whose diagonal elements, the singular values, are, in decreasing order, the square roots of the eigenvalues of $Z^T Z=V D^2 V^T$. When $p>n$, less than $n$ singular values can be positive (typically $n-1$), and then we let $U$ and $D$ be $n\times n$, and $V$ be $p\times n$.
The right singular vectors forming $V$ are the orthonormal eigenvectors of $Z^T Z$ (or of the covariance matrix $Z^T Z/(n-1)$). Corresponding to the FA model, we truncate the SVD by using only the first $k$ singular vectors, $U_1$ ($n\times k$) and $V_1$ ($p\times k$), say, corresponding to $\Phi_z$. That is, we partition $Z$ as $$Z = U_1 D_1 V_1^T + U_2 D_2 V_2^T,$$ where $U=(U_1,\, U_2)$, etc. Note that it does not affect $U_1 D_1 V_1^T$ whether $p<n$ or $p>n$, but only the second term, where $D_2$ is either $(p-k)\times (p-k)$ or $(n-k)\times (n-k)$, respectively.
Since $V_1$ is formed by the normalized eigenvectors of $(n-1) S_{zz}$ with the $k$ highest eigenvalues, and these are given by the diagonal $D_1^2$, we can identify $V_1= \Phi_z$ and $D_1^2=(n-1)\Omega_z$ from equation . Thus the estimating equation for $\Lambda$ can be expressed in terms of $V_1$ and $D_1$, and for the estimation of $\Lambda$ (given $\Psi$) we will need only $U_1 D_1 V_1^T$. More precisely, $\Lambda= \Psi \Lambda_z$ in combination with $$\label{eq:hatLambda_z}
\widehat{\Lambda}_z = \Phi_z \left(\Omega_z - I_k\right)^{1/2} = V_1 \left(\frac{D_1^2}{n-1} - I_k\right)^{1/2}.$$ Iteration step for $\Psi^2$ takes the following form in terms of $V_1$ and $D_1$: $$\label{eq:DiagPsi2New}
\mathrm{diag}(\Psi_{\mathrm{new}}^2) = \mathrm{diag} \left\{ S_{xx} - \Psi \Lambda_z \Lambda_z^T \Psi \right\} = \mathrm{diag}\left\{S_{xx} - \Psi V_1 \left(\frac{D_1^2}{n-1} - I_k \right) V_1^T \Psi \right\} .$$ The alternative iteration step takes the form $$\psi_j^2 =\frac{ (S_{xx})_{jj} }{1 + \left(V_1(D_1^2/(n-1) - 1)V_1^T \right)_{jj} }.$$
The right hand side of may alternatively be expressed as $$\mathrm{diag} \left\{ \Psi \left(V_1 V_1^T + V_2 {D_2}^2 {V_2}^T /(n-1) \right) \Psi \right\} ,$$ which shows that it is obtained by replacing the first $k$ singular values or eigenvalues in $S_{zz}$ by the value 1. Consequently, the iteration method cannot possibly yield zero or negative values in $\Psi^2$ in any iteration step (presuming start values are positive). What might possibly go wrong, as indicated by , is that $D_1^2/(n-1) - I_k$ is not positive definite. In the case $p>n$, however, we give below some more results about $D_1^2$ and $D_2^2$, showing that we need not worry.
Note first that when $\Psi$ and $\Lambda$ satisfy the estimating equations, all the $p$ diagonal elements of $S_{zz}-\widehat{\Lambda}_{z} \widehat{\Lambda}_{z}^T$ are 1, so its eigenvalues sum to $p$. At the same time, $$\label{eq:residuals1}
S_{zz}-\widehat{\Lambda}_{z} \widehat{\Lambda}_{z}^T= V \frac{D^2}{n-1} V^T - V_1(\Omega_z - I_k) V_1^T = V_1 I_k V_1^T + V_2 \frac{D_2^2}{n-1} V_2^T.$$ Thus, under the same conditions, $$\label{eq:p-k}
{\mathrm{trace}}\left(D_2^2\right)/(n-1) = p-k.$$
If $k$ is not higher than motivated by data, we expect the diagonal matrix $\Omega_z - I_k$ in to have all its diagonal elements positive. When $p<n$, this can fail, and the estimation process too. When $p>n(>k)$, however, the diagonal elements are necessarily positive, at least in a vicinity of the estimation point. To see this, note first that $D_2^2$ contains less than $n-k$ positive values, but has ${\mathrm{trace}}(D_2^2/(n-1))=p-k$. Thus, the average value is at least $(p-k)/(n-k)>1$. Since the $k$ diagonal values in $\Omega_z=D_1^2/(n-1)$ are larger than this, by selection, the corresponding elements of $\Omega_z - I_k$ are necessarily positive, which was to be shown.
In passing, we supplement by an expression for the average of the $k$ first eigenvalues of $S_{zz}$, cf. . This average can be written $${\mathrm{trace}}(\Omega_z)/k = 1 + (\theta - 1) p / k ,$$ where $\theta>1$ is the inverse of the harmonic mean of the $p$ unique factor variance proportions $\widehat{\psi}_j^2/(S_{xx})_{jj}$, $$\theta = \frac{1}{p} \sum_{j=1}^p (S_{xx})_{jj} / \widehat{\psi}_j^2.$$ This is seen by subtracting $p-k$ from ${\mathrm{trace}}(S_{zz})$. Note the proportionality to the dimension $p$ in the second term of ${\mathrm{trace}}(\Omega_z)$, showing the benefit of large $p$. Note also that when $k$ is increased, $\theta$ will also increase.
Factor scores and model residuals {#sec:scores}
=================================
The SVD approach can be used to obtain relatively directly the most common estimates or predictions of the scores $f_i$, or the whole $n\times k$ scores matrix $F$ with the $f$-vectors as rows. As usual in the context of scores estimation/prediction, we provisionally regard the parameters as known (but they are of course estimated). The Bartlett scores, or weighted least squares scores regressing $X$ on $\Lambda$, are given by $$\label{eq:Bartlett1}
\widehat{F} = X \Psi^{-2} \Lambda (\Lambda^T \Psi^{-2} \Lambda)^{-1} = Z \Lambda_z (\Lambda_z^T \Lambda_z)^{-1}$$ so first we can note that with $Z$ as data, Bartlett scores are standard (i.e. equal weights) least squares scores. Continuing from , $$\widehat{F} = U D V^T \Phi_z (\Lambda_z^T \Lambda_z)^{-1/2} = U D V^T V_1 (\Omega_z - I_k)^{-1/2} = U_1 D_1 (\Omega_z - I_k)^{-1/2},$$ using the fact that $\Phi_z = V_1$. This implies that the Bartlett score components are proportional to the SVD vectors $U_1$. More precisely, since $D_1^2= (n-1)\Omega_z$, we achieve the following estimation/prediction formula (two equivalent versions related by ): $$\label{eq:Bartlett2}
\widehat{F} = U_1 \sqrt{n-1} \, \Omega_z^{1/2} (\Omega_z - I_k)^{-1/2} =
U_1 \sqrt{n-1} \left( I_k + (\Lambda_z^T \Lambda_z)^{-1} \right)^{1/2}.$$ To the right of $U_1\sqrt{n-1}$ is a diagonal matrix that scales the $j$th column of $U_1$ by the factor $\sqrt{\omega_j/(\omega_j - 1)}$, $j=1,\ldots,k$. Thus, this is Bartlett’s formula in a disguised but computationally convenient form. Typically, if $p$ is large and $k$ is not too large, all $k$ $\omega$-values will be large (proportionally to $p$, cf. ), and then with good approximation $\widehat{F} \approx U_1 \sqrt{n-1}$.
If we instead predict the scores $F$ by the linear regression of $F$ on the observed $X$-data (or on $Z$), the best linear predictor $\widetilde{F}$ is given by the so called regression or Thomson scores $$\widetilde{F} = U_1 \sqrt{n-1} \, \Omega_z^{-1/2} (\Omega_z - I_k)^{1/2} = U_1 \sqrt{n-1} \left( I_k + (\Lambda_z^T \Lambda_z)^{-1} \right)^{-1/2}.$$ The difference from is the diagonal matrix factor $\Omega_z^{-1}\,(\Omega_z - I_k)$ (cf. Bartholomew & Knott, 1999, sec. 3.24, or Krzanowski & Marriott, 1995, sec. 12.27). Again, if $p$ is large, but not $k$, $\widetilde{F} \approx U_1 \sqrt{n-1}$.
For high dimension $p$ but small or moderate sample size $n$ we cannot expect high precision in the estimation of $\Lambda$ or $\Psi$. Estimation/prediction of the scores $f_i$, however, will be more precise with increasing $p$. More precisely, it can be shown that under mild conditions the variance of the factor estimator/predictor $\widehat{F}$ or $\widetilde{F}$ goes to zero as $p$ increases but $k$ and $n$ are kept constant. To be specific, consider the Bartlett score vector $\widehat{f}$ for an arbitrary observation $i$, $\widehat{f}=(\Lambda_z^T \Lambda_z)^{-1} \Lambda_{z}^T z$.
First, if the difference between $\widehat{\Lambda}_z$ and $\Lambda_z$ is still neglected, formula yields the well-known result $$\label{eq:varf1}
Var(\widehat{f}\,|\,f) = (\Lambda_z^T \Lambda_z)^{-1} \Lambda_z^T I_p \Lambda_z (\Lambda_z^T \Lambda_z)^{-1} = (\Lambda_z^T \Lambda_z)^{-1}.$$ Due to , we may conclude that this diagonal matrix will have small elements when $p$ is large and $k$ is not too large.
The argument above is not justified when $n<p$, however. In that case, let us still regard $\widehat{\Lambda}_z$ as given, but with $\widehat{\Psi}^2$ differing from the right $\Psi^2$. Formula should then replaced by $$\label{eq:varf2}
Var(\widehat{f}\,|\,f) = (\widehat{\Lambda}_z^T \widehat{\Lambda}_z)^{-1} \widehat{\Lambda}_z^T \Psi^2 \widehat{\Psi}^{-2} \widehat{\Lambda}_z (\widehat{\Lambda}_z^T \widehat{\Lambda}_z)^{-1} .$$ This will differ from the corresponding element of by less than a factor $$\max_j \psi_j^2 / \widehat{\psi}_j^{2}.$$ We do not know the true $\psi_j^2$-values, but if there are no components with quite little estimated noise $\widehat{\psi}_j^{2}$, and provided the elements of $(\widehat{\Lambda}_z^T \widehat{\Lambda}_z)^{-1}$ are quite small, we can feel sure the precision in $\widehat{F}$ is high.
When the scores matrix $F$ has been estimated/predicted, we can form the matrix of residuals, for example $\widehat{E}_x = X-\widehat{F} \widehat{\Lambda}^T$. In order to make them all comparable on the same scale, we must variance-standardize to $\widehat{E}_z=Z-\widehat{F} \widehat{\Lambda}_{z}^T$. Now note that $$\widehat{F} \widehat{\Lambda}_{z}^T = U_1 D_1 V_1^T$$ so the standardized residuals matrix is $$\label{eq:residuals2}
\widehat{E}_z = Z-\widehat{F} \widehat{\Lambda}_{z}^T = U_2 D_2 V_2^T .$$ Thus, the sum over $j=1,\ldots,p$ of the mean squared standardized residuals is $V_2 \{D_2^2/(n-1)\} V_2^T$. This may be compared with the result , which tells that the trace of $D_2^2/(n-1)$ is only $p-k$, and not $p$, so the mean squared standardized residuals are “too small”, and must be normalized by $p-k$ instead of $p$ to have the right average size over $j=1,\ldots,p$. This corresponds to the residual degrees of freedom for unbiased variance estimation in a linear model for $Z$, regarding $\Lambda$ as given and the $k(n-1)$ free elements of $F$ as unknowns.
Models with nonrandom common factors, when $p>n$ {#sec:U&T}
================================================
In recent years, methods have been advocated for fitting fixed factor models to data, where also $F$ is regarded as a set of unknown parameters, see the review by Unkel & Trendafilov (2010b). Several papers by those two authors treat the case $p>n$. The methods of Unkel & Trendafilov (2010a) and Trendafilov & Unkel (2011) proceed from a least squares method minimizing a loss function based on the Frobenius norm of data matrices. Quite generally, the fixed model requires more restrictions than the random model, for uniqueness, and when $p>n$. the authors are led to impose special constraints. Let us write $X=F \Lambda^T + \Psi E_z$, so we can let $E_z$ exist also when $\Psi^2$ contains zero variances. The papers referred to above assume the model satisfies the constraints $E_z^T F = 0$, $F^T F \propto I_k$, and (unless $p>n$) $E_z^T E_z \propto I_p$. When $p>n$ they find that $E_z^T E_z = I_p$ cannot be fulfilled, because the rank of $E_z$ can be at most $n$, and conclude that they need to allow at least $p-n$ unique factors to have zero variances, corresponding to a singular $\Psi^2$. In that situation they weaken the constraint $E_z^T E_z = I_p$ to the eigenvector relation $E_z^T E_z \Psi= \Psi$.
On the other hand, a result by Robertson & Symons (2007) states that the Gaussian model likelihood typically (depending on $k$) has a unique global maximum also when $p>n$, and with a *nonsingular* $\Psi$. Trendafilov & Unkel (2011) correctly remark that this result is not consistent with their own model. That the rank of $E_z$ can be at most $n$ (or $n-1$, considering that data are centered) is trivially true for the sample of data, but not for the underlying statistical models assumed by Robertson & Symons (2007) and by us in the present paper. Our conclusion is that their constraints are artificial, and that their method only represents a constrained partitioning of data, and that it does not represent the fitting of a reasonable statistical model.
We shed further light on this situation here by comparing with our distribution-free but ML-related approach as far as it leads to the eigenvector relation for $\Lambda_z$ and the Bartlett scores for estimating the scores matrix $F$, with any given $\Psi$:\
The constraint $E_z^T F = 0$ is satisfied also for the fitted random model and its Bartlett scores $\widehat{F}$, according to [Section \[sec:scores\]]{}.\
The constraint $F^T F \propto I_k$ is not exactly consistent with Bartlett scores but with the large $p$ approximation $\widehat{F} \approx U_1 \sqrt{n-1}$.\
The constraints $E_z^T E_z \propto I_p$ for $n>p$ and $E_z^T E_z \Psi = \Psi$ for $p>n$ are not consistent with our fitted model. and other features of our fitted model, in particular since it does not allow noise outside the diagonal of $E_z^T E_z$.\
Nor is the constraint consistent with Bartlett scores and other features of our model.
As their first illustration, Trendafilov & Unkel (2011) use Thurstone’s 26-variable box data, consisting of a set of $n=20$ boxes and $p=26>20$ variables for each box, representing various aspects of size. When they fit a model with three factors ($k=3$), they get 13 or 14 zero-valued $\psi_j^2$-values (depending on algorithm). When we fit our model we clearly get no more than 6 zeros, and they can be explained by the peculiarities of the data set. In fact, there are only three original variables in the data set: length, width and height. All other variables are constructed as functions of them. In a model with three latent factors, the factors turn out to be precisely length, width and height, and that explains three zeros. Three other variables are linear functions of length, width and height, and that explains the remaining three zeros. So for example adding a little computer-generated random measurement noise to the variables makes the zero variances disappear completely. Thus, all their zero unique factor variances are not really due to $n<p$, but to a combination of their assumed artificial data structure (model) and associated fitting method, and the peculiarities of the data set. An example of more applied relevance is studied in [Section \[sec:genedata\]]{}.
A gene expression example, with $p=2000$ {#sec:genedata}
========================================
We tried the model and the iteration methods on a microarray data set from Alon et al (1999), with 62 tissue samples (a colon cancer sample from each of 40 individuals and non-cancer samples from 22 of these individuals), and $p=2000$ genes (selected by theses authors from a larger set of genes). The data are available on www.bioconductor.org, from where they were fetched. The data have earlier been used for illustrative purposes by McLachlan et al (2003, 2004). The response was taken to be the gene expression on log scale (natural log). Each gene was mean- and variance-standardized, but no other normalization of the data was made. None of the biological structure imposed by the experiment was used in the model, since our aim was not to draw biological conclusions but only to try our methods for model fitting.
We tested the estimation method on the data of all tissue samples ($n=62$), but mostly on the data of only non-cancer tissue ($n=22$). The iteration method was found to be slower and generally inferior to the method . The experiences from running the iteration method were extremely satisfactory. The method converged in about 10 iterations for small $k$ and not more than 20 to 30 iterations for larger $k$, somewhat also depending on the choice of starting values for $\Psi^2$. The time per iteration step seemed to be slowly increasing with $k$, but even with an extremely large $k$, $k=20$ say, iterations did not require more time than a second each, on an ordinary laptop. There was no problem of Heywood type during the iterations. Even if the minimum of the unique factor variances in $\Psi^2$ naturally decreased with $k$, it was in no case estimated to be zero (we tried $k$-values up to 20 for $n=62$, and $k=12$ for $n=22$). After quite few iterations, ${\mathrm{trace}}(D_2^2)/(n-1)$ was reasonably close to $p-k$, cf. . The statements about ${\mathrm{trace}}(D_2^2)/(n-1)$ and about the minimum of the unique factor variances are illustrated in Figures 1 and 2 below, showing how these quantities rapidly converge as the iteration number increases. Both for a small factor dimension ($k=2$) and a moderate ($k=5$) or large such dimension ($k=12$) there are no problems at all, but $k=10$ is also included for the little bump it shows in Figure 1. Starting values were $\psi_j^2=1/2$ for all $j$.
We have thus found substantial support for the conjecture, that the iteration method works so well not *despite* the large $p$-value, but *due to* the large $p$.
Conclusions
===========
Summing up, we have come to the following conclusions from the investigations in this paper.
Distribution-free estimating equations for the parameters of the standard FA model (with random factors), and , are easily derived in a set-up where variables are variance-normalized by their specific factor standard deviations ($\Psi$). This theory extends the Gaussian likelihood equations both to distribution-free settings and to the case $p>n$. The estimating equations are conveniently expressed by use of a singular value decomposition (SVD) under the same normalization.
An iteration scheme that has been much used for MLE computation when $p<n$, but also criticized as unreliable in such cases, is shown to have much stronger properties when $p>n$. The theoretical results are supported empirically in an illustration with $p>>n$, where the method was seen to converge quite rapidly.
Another result for situations of type $p>>n$ is that even though the model parameters cannot be precisely estimated when $n$ is small, the factor scores can be precisely estimated/predicted when $p$ is large.
References {#references .unnumbered}
==========
Alon, U. et al. (1999). Broad patterns of gene expression revealed by clustering analysis of tumor and normal colon tissue probed by oligonucleotide arrays. *Proc. Nat. Acad. Sci. USA* [**96**]{}, 6745–6750.\
Bartholomew, D.J. & Knott, M. (1999). *Latent variable models and factor analysis*, 2nd edn. Arnold, London\
Jöreskog, K.G. (1967). Some contributions to maximum likelihood factor analysis. *Psychometrika* [**32**]{}, 443–482.\
Krzanowski, W.J. & Marriott, F.H.C. (1995). *Multivariate analysis, part 2*. Arnold, London\
Lawley, D.N. (1967). Some new results in maximum likelihood factor analysis. *Proc. Roy. Soc. Edinburgh A* [**67**]{}, 256–264.\
McLachlan, G.J., Peel, D. & Bean, R.W. (2003). Modelling high-dimensional data by mixtures of factor analyzers. *Comp. Stat. & Data Analysis* [**41**]{}, 379–388.\
McLachlan, G.J., Do, K.-A. & Ambroise, C. (2004). *Analyzing microarray gene expression data*. Wiley, Hoboken.\
Rubin, D.B. & Thayer, D.T. (1982). EM algorithms for ML factor analysis. *Psychometrika* [**32**]{}, 443–482.\
Robertson, D. & Symons, J. (2007). Maximum likelihood factor analysis with rank-deficient sample covariance matrices. *Journal of Multivariate Analysis* [**98**]{}, 813–828.\
Trendafilov, N.T. & Unkel, S. (2011). Exploratory factor analysis of data matrices with more variables than observations. *J. Comp. Graph. Stat.* [**20**]{}, 874–891.\
Unkel, S & Trendafilov, N.T. (2010a). A majorization algorithm for simultaneous parameter estimation in robust exploratory factor analysis. *Comp. Stat. & Data Analysis* [**54**]{}, 3348–3358.\
Unkel, S & Trendafilov, N.T. (2010b). Simultaneous parameter estimation in exploratory factor analysis: an expository review. *Int. Stat. Rev.* [**78**]{}, 363–382.\
Addresses:\
Rolf Sundberg, Mathem. statistics, Stockholm University, Sweden, [email protected];\
Uwe Feldmann, Medical biometry, University of Saarland, Germany, [email protected]\
Corresponding author: Rolf Sundberg
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"pile_set_name": "ArXiv"
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---
abstract: 'Amplitude amplification is a central tool used in Grover’s quantum search algorithm and has been used in various forms in numerous quantum algorithms since then. It has been shown to completely eliminate one-sided error of quantum search algorithms where input is accessed in the form of black-box queries. We generalize amplitude amplification for two-sided error quantum algorithm for decision problems in the familiar form where input is accessed in the form of initial states of quantum circuits and where arbitrary projective measurements may be used to ascertain success or failure. This generalization allows us to derive interesting applications of amplitude amplification for distinguishing between two given distributions based on their samples, detection of faults in quantum circuits and eliminating error of one and two-sided quantum algorithms with exact errors.'
author:
- 'Debajyoti Bera [^1]'
title: 'Two-sided Quantum Amplitude Amplification and Exact-Error Algorithms'
---
Introduction {#section:intro}
============
The motivation behind this work is to investigate the characteristics of quantum computation when viewed as randomized algorithms. It is known that quantum amplitude amplification, the key technique underlying Grover’s unordered search algorithm, is able to reduce and even eliminate error of one-sided quantum black-box algorithms for search problems [@BHMT]. We explored that direction further for two-sided error algorithms for decision problems based on the key observation that quantum algorithms appear to be better at distinguishing between two given probability distributions compared to classical randomized algorithms.
Suppose we are given a biased coin whose distribution is either $\mu_1 = \langle 1/3, 2/3 \rangle$ or $\mu_2 = \langle 2/3, 1/3 \rangle$. A classical problem of probabilistic classification is to determine the distribution of the coin by tossing it several times. Various techniques exist like Bayesian classification and maximum likelihood estimation, all of which aim to minimize some kind of error that is inherent in such a probabilistic inference. But it is not believed to be possible to confidently classify a distribution without any error. This is true even if $\mu_1 = \langle 0,1\rangle$ instead.
However, such classification is possible when the distributions come from a [*quantum system*]{}, our definition of a quantum source of random samples. We define a quantum system (QS) as a combination of a quantum circuit $C$, an input to the circuit ${|\psi\rangle}$ and a two-outcome projective measurement operator ${\mathcal{P}}=\langle P_E, I -
P_E \rangle$ (two outcomes will be [*always labeled as $E$ and $F$*]{} for convenience) and denote it by $\langle {|\psi\rangle}, C, {\mathcal{P}}\rangle$. If we are given an actual instance of a QS and we [*apply the circuit on the input followed by measurement using the projective operator*]{}, we will obtain a sample in $\{E,F\}$ from the output probability distribution $\langle p_E, 1-p_E
\rangle$ where $p_E$ denotes the probability of observing outcome $E$ when $C{|\psi\rangle}$ is measured using ${\mathcal{P}}$. The quantum version of the above question of classifying between $\mu_1$ and $\mu_2$ becomes this: given an instance ${\mathcal{Q}}$ which can be either a quantum system ${\mathcal{Q}}_1$ with output distribution $\mu_1$ or QS ${\mathcal{Q}}_2$ with output distribution $\mu_2$, can we confidently figure out if ${\mathcal{Q}}$ is ${\mathcal{Q}}_1$ or ${\mathcal{Q}}_2$ (in other words, determine the actual distribution of ${\mathcal{Q}}$) by using ${\mathcal{Q}}$ in a black-box manner? Assume that both ${\mathcal{Q}}_1$ and${\mathcal{Q}}_2$ involve the same number of qubits and the same set of outcomes ($E$ and $F$). This is analogous to asking if two or more distinct distributions (over same support of two elements) can be distinguished without any probability of error. Even though classical techniques cannot identify the exact distribution from the sample distribution without any error, we show that it is possible to do so for distributions of quantum systems.
\[theorem:application\] Given a quantum system ${\mathcal{Q}}= \langle {|\psi\rangle}, C, {\mathcal{P}}\rangle$ whose output distribution can either be ${\langle \delta, 1-\delta \rangle}$ or ${\langle \epsilon, 1-\epsilon \rangle}$ for some $0 \le \delta
< \epsilon \le 1$, there is a quantum circuit $C'$ which can determine the output distribution of ${\mathcal{Q}}$ without any probability of error. $C'$ takes ${|\psi\rangle}$ as input, makes repeated calls to $C$, $C^\dagger$ and employs gates that depend upon operators of ${\mathcal{P}}$ and ${|\psi\rangle}$.
The core technique is once again, [*quantum amplitude amplification*]{}. It can be thought of as a quantum analog of repeated trials used in randomized algorithms for reducing mis-classification error. It is the workhorse behind Grover’s famous quantum unordered search algorithm [@Grover1996] and was later shown to be also applicable to the Deutsch-Jozsa problem [@BeraDJ2015]. It appears that quantum algorithm designers simply cannot wave it enough; it is applicable to almost any search problem to yield a surprising improvement, usually quadratic, over classical algorithms. Since its inception, amplitude amplification have been used, either directly or in the form of Grover’s search algorithm for a vast range of problems like minimum of an unordered array [@Durr96], minimum spanning tree [@Durr2004] and even clustering [@Aimeur2007]. Nevertheless, we feel that the technique still has a long way to go, especially, when used in a non-blackbox manner.
The most generalized and popular version of this technique was given by Brassard et al.
\[theorem:bhmt\] Consider a Boolean function $\Phi: X \to \{0,1\}$ that partitions a set $X$ between its [*good*]{} (those which $\Phi$ evaluates to 1) and [*bad*]{} (those which evaluate to 0) elements. Consider also a quantum algorithm that uses no measurements and uses oracle gates for computing $\Phi$ such that $C{|0\rangle}$ is quantum superposition of the elements of $X$ and let $a > 0$ denote the success probability that a good element is observed if $C{|0\rangle}$ is measured (in the standard basis). There exists a quantum circuit (that depends upon $a$) which finds a good solution with certainty using at most $\Theta(1/\sqrt{a})$ applications of $C$ and $C^\dagger$.
This theorem is highly versatile as it is. However, for our applications we require further generalizations. For example, we are interested in not only one-sided, but also two-sided error algorithms. We also want to apply it to algorithms which are measured not necessarily in the standard basis. Lastly, we want algorithms which act on non-${|0\rangle}$ input states, specifically, input states that correspond to the input $\Phi$, suitably encoded – this is similar to classical Boolean circuits without oracle gates. Lastly, for the results of this paper we stick to only decision versions of the above theorem (though our results could be extended to circuits that output some solution). The following theorem is our version of Theorem \[theorem:bhmt\] with the constraint that the probability $a$ is fixed for every possible $\Phi$ (condition of [*exactness*]{}).
\[theorem:aa-two-sided\] Consider a Boolean function $\Phi: X \to \{0,1\}$ that partitions a set $X$ between its [*good*]{} (those which $\Phi$ evaluates to 1) and [*bad*]{} (the rest of $X$) elements. Suppose $C$ is a quantum algorithm (or circuit) that uses no measurement and decides $\Phi$ with two-sided exact error $(\delta,\epsilon)$ for some $\delta <
\epsilon$. That is, the probability of error when $C$ is given a good $x \in
X$ is [*exactly*]{} $\epsilon$ and when $x$ is bad is [*exactly*]{} $\delta$. Here success and error is determined upon measurement of the output state of $C$ by any projective measurement with two outcomes. There exists a quantum circuit $C'$ that calls $C$ and $C^\dagger$, uses the same input as that of $C$ (maybe with ancillæ), is measured using an extension of the measurement operator for $C$ and decides $\Phi$ with certainty,
The primary contribution of this paper are a few interesting applications of amplitude amplification. If we have two quantum systems which differ only in their circuit, then we can essentially use their output distribution, after suitably amplifying the systems, to distinguish between those circuits. We show how this can be used to detect faults in quantum circuits.
On the other hand, if we have two systems that differ only in their input states, then we get a way to amplify their probability of acceptance. This is exactly at the core of our proof that quantum classes equivalent to exact two-sided and exact one-sided error classes can be “derandomized”, in the sense that their errors can be completely eliminated.
One of the major, and still open, questions of [*Complexity Theory*]{} is how ¶compares to [$\mathbf{RP}$]{}and [$\mathbf{BPP}$]{}, one-sided and two-sided bounded error polynomial-time complexity classes. The current best results are the obvious inclusions ${\ensuremath{\mathbf{P}}\xspace}\subseteq {\ensuremath{\mathbf{RP}}\xspace}\subseteq {\ensuremath{\mathbf{BPP}}\xspace}$, though there are some evidences of their equivalence. Same question for their quantum analogs is in an equally indeterminate state, i.e., ${\ensuremath{\mathbf{EQP}}\xspace}\subseteq {\ensuremath{\mathbf{RQP}}\xspace}\subseteq {\ensuremath{\mathbf{BQP}}\xspace}$; these are quantum analogs of ¶, ${\ensuremath{\mathbf{RP}}\xspace}$ and ${\ensuremath{\mathbf{BPP}}\xspace}$, respectively. There is not even much evidence that ${\ensuremath{\mathbf{EQP}}\xspace}= {\ensuremath{\mathbf{BQP}}\xspace}$. One approach towards settling this question is studying restricted versions of these classes. Our results show that their exact error versions, ${\ensuremath{\mathbf{ERQP}}\xspace}$ and ${\ensuremath{\mathbf{EBQP}}\xspace}$, are identical to ${\ensuremath{\mathbf{EQP}}\xspace}$ as long as the two(one)-sided errors are fixed for all instances [^2].
[**Organization:**]{} The rest of the paper is organized as follows. We discuss quantum distinguishability of quantum systems in Section \[section:distinguish-qs\]. The proof of our main theorem on distinguishability is given in Section \[section:proof\]. This theorem, even though quite general, is not suitable enough to amplify a collection of quantum systems in a uniform manner; in Section \[section:uniform\] we discuss a uniform version of our main theorem. Section \[section:distinguish-circuits\] contains one of the applications about detection of faults in quantum circuits and in Section \[section:exact-error-classes\] we show that ${\ensuremath{\mathbf{EBQP}}\xspace}={\ensuremath{\mathbf{ERQP}}\xspace}={\ensuremath{\mathbf{EQP}}\xspace}$ and prove Theorem \[theorem:aa-two-sided\] for regular circuits and those with oracle gates.
Distinguishing quantum systems {#section:distinguish-qs}
==============================
We will use $\mu_p$ to denote a distribution ${\langle p, 1-p \rangle}$ over outcomes ${\langle E, F \rangle}$ and $\mu({\mathcal{Q}})$ to denote output distribution of a quantum system ${\mathcal{Q}}$.
As explained earlier, the main problem we are interested in involves a given instance of a quantum system ${\mathcal{Q}}$ which can be either ${\mathcal{Q}}_\delta$ with output distribution $\mu_\delta = {\langle \delta, 1-\delta \rangle}$ or ${\mathcal{Q}}_\epsilon$ with output distribution $\mu_\epsilon = {\langle \epsilon, 1-\epsilon \rangle}$. We want to construct a quantum algorithm, rather a circuit, that can “call ${\mathcal{Q}}$ as a subroutine” and determine if ${\mathcal{Q}}={\mathcal{Q}}_\delta$ or ${\mathcal{Q}}={\mathcal{Q}}_\epsilon$.
We can even extend this to multiple quantum systems ${\mathcal{S}}= \{{\mathcal{Q}}_1, {\mathcal{Q}}_2, \ldots \}$ where output distribution of any ${\mathcal{Q}}_i$ is either $\mu_\delta$ or $\mu_\epsilon$. We use the notation $QD({\mathcal{Q}}_1, {\mathcal{Q}}_2, \ldots)$ or even shorter $QD({\mathcal{S}})$ to refer to the [*quantum distinguishability*]{} problem among quantum systems of ${\mathcal{S}}$.
Our goal is to design a quantum circuit in which we can “embed any given ${\mathcal{Q}}$” as a black-box. This motivated us to define a notion of black-box extension for quantum systems, similar to quantum algorithms with subroutines or quantum circuits with black-box operators, allowing only trivial extensions to inputs states and projection operators. We refer to these as ${\mathcal{B}}$-transforms (${\mathcal{B}}$ standing for “black-box”). A general illustration is given in Figure \[fig:b\_transform\].
\[defn:bb-transform\] A (non-uniform) ${\mathcal{B}}^n_{\delta,\epsilon}$-transform for $n$-qubit systems is a (non-uniform) procedure for extending an $n$-qubit QS ${\mathcal{Q}}_1
= {\big\langle}{|\psi\rangle}, C, {\mathcal{P}}{\big\rangle}$ to a (possibly larger) QS ${\mathcal{Q}}_2 = {\big\langle}{|\psi'\rangle}, C', {\mathcal{P}}' {\big\rangle}$ whose components are black-box extensions of the components of ${\mathcal{Q}}_1$.
- The input in ${\mathcal{Q}}_2$ is an extension of the input in ${\mathcal{Q}}_1$ supplemented by ancillæ qubits initialized to a fixed state (wlog. in state ${|\mathbf{0}\rangle}$), i.e., ${|\psi'\rangle}={|\psi\rangle} \otimes {|00\cdots 00\rangle}$.
- The projection operator in ${\mathcal{Q}}_2$ is an extension of the projection operator in ${\mathcal{Q}}_1$ to include measurement of the ancillæ in a basis independent of ${\mathcal{Q}}_1$, i.e., ${\mathcal{P}}' = {\mathcal{P}}\otimes {\mathcal{P}}_a$.
- The number of ancillæ and the operator ${\mathcal{P}}_a$ are independent of ${\mathcal{Q}}_1$ and depend upon $\delta, \epsilon$.
- The circuit in ${\mathcal{Q}}_2$ calls $C$ and $C^\dagger$ and uses additional gates that depend upon $\delta$ and $\epsilon$.
- $C'$ may also use gates that depend upon ${\mathcal{P}}_E$ and ${|\psi\rangle}$.
We call the transformations that satisfy the final condition as “non-uniform” since the transformed circuit could be using gates that depend upon the input states and measurement operators of the respective quantum system. Note that the non-uniformity is not with respect to $n$, the number of qubits of the quantum system, but with respect to the gates of the transformed circuit. It will be clear from the proof of Theorem \[theorem:main-separable\] that the transformations that will be used in this paper are anyway uniform in $n$. In any case, we will always drop $n$ from the superscript of ${\mathcal{B}}^n_{\delta,\epsilon}$. We will revisit the notion of non-uniformity in Section \[section:uniform\].
We want transformed quantum circuits that solve the quantum distinguishing problem without any error which motives the next definition.
\[defn:solution\] For a set of quantum systems ${\mathcal{S}}= \{{\mathcal{Q}}_1, {\mathcal{Q}}_2, \ldots \}$ with output distributions either $\mu_p$ or $\mu_q$ (for $p < q$), a ${\mathcal{B}}$-transform ${\mathcal{B}}$ is said to solve $QD({\mathcal{S}})$ with error $(\delta,\epsilon)$, in other words ${\mathcal{B}}$ is a $(\delta,\epsilon)$-solution of $QD({\mathcal{S}})$, if the following holds for some $\delta < \epsilon$ and all ${\mathcal{Q}}\in {\mathcal{S}}$.
- If $\mu({\mathcal{Q}}) = \mu_p$, then outcome of ${\mathcal{B}}({\mathcal{Q}})$ is $E$ with probability $\delta$.
- If $\mu({\mathcal{Q}}) = \mu_q$, then outcome of ${\mathcal{B}}({\mathcal{Q}})$ is $E$ with probability $\epsilon$.
$QD({\mathcal{S}})$ is said to have a [*perfect solution*]{} if ${\mathcal{B}}$ is a $(0,1)$-solution of $QD({\mathcal{S}})$.
It can be seen that the identity ${\mathcal{B}}$-transform is a trivial solution of the above $QD({\mathcal{S}})$ with error $(p,q)$. The last part of the above definition is based on the fact that if ${\mathcal{B}}$ is a $(0,1)$-solution of $QD({\mathcal{S}})$, then the outcome of ${\mathcal{Q}}' = {\mathcal{B}}({\mathcal{Q}})$ can be used to correctly infer the output distribution of any given instance ${\mathcal{Q}}\in {\mathcal{S}}$. Let ${\mathcal{Q}}' = {\mathcal{B}}({\mathcal{Q}})$ – which is essentially an extension of the input of ${\mathcal{Q}}$ with some ancillæ, an extension of its measurement operator and a circuit that can call the circuits of ${\mathcal{Q}}$ (and its inverse) in a black-box manner. If the output distribution of ${\mathcal{Q}}$ is $\mu_p$, then the outcome of ${\mathcal{Q}}'$ is never $E$ and otherwise (i.e., if the output distribution of ${\mathcal{Q}}$ is $\mu_q$) the outcome of ${\mathcal{Q}}'$ is always $E$ without any error.
The main theorem of our work is stated next.
\[theorem:main-separable\] Let ${\mathcal{S}}= \{{\mathcal{Q}}_1, {\mathcal{Q}}_2, \ldots \}$ be a collection of quantum systems such that output distribution of any ${\mathcal{Q}}_i \in {\mathcal{S}}$ is either $\mu_\delta$ or $\mu_\epsilon$ for some $\delta < \epsilon$. Then ${\mathcal{S}}$ is perfectly-solvable via some ${\mathcal{B}}$-transition ${\mathcal{B}}_{\delta,\epsilon}$, i.e., any ${\mathcal{Q}}_i \in {\mathcal{S}}$ can be transformed by ${\mathcal{B}}_{\delta,\epsilon}$ to some ${\mathcal{Q}}'_i$ such that:
- if output distribution of ${\mathcal{Q}}_i$ is $\mu_\delta$, then outcome of ${\mathcal{Q}}'_i$ is never $E$ and
- if output distribution of ${\mathcal{Q}}_i$ is $\mu_\epsilon$, then outcome of ${\mathcal{Q}}'_i$ is always $E$.
The proof of this theorem is presented in the next section. Note that, unlike Theorem \[theorem:bhmt\] which only applies to one-sided error algorithms, we prove that two-sided error algorithms can also be “amplified to certainty”. A straight-forward application of this is to exactly distinguish between two QS with known output distributions, such as Theorem \[theorem:application\] (Section \[section:intro\]).
Consider the transformation ${\mathcal{B}}_{\delta,\epsilon}^n$ from Theorem \[theorem:main-separable\]. Given an $n$-qubit ${\mathcal{Q}}= \langle {|\psi\rangle}, C, {\mathcal{P}}\rangle$, construct the transformed QS ${\mathcal{B}}^n_{\delta,\epsilon}({\mathcal{Q}})=\big\langle {|\psi\rangle}\otimes {|\mathbf{00\ldots
0}\rangle}, C', {\mathcal{P}}\otimes {\mathcal{P}}_a \big\rangle$. By Theorem \[theorem:main-separable\], the output state of the transformed circuit $C'$, when given ${|\psi\rangle}$ (along with a few ancillæ in a fixed state), upon measurement by a simple extension of ${\mathcal{P}}$, has outcome either $E$ or $F$, depending upon whether $\mu({\mathcal{Q}})=\mu_\delta$ or $\mu({\mathcal{Q}})=\mu_\epsilon$.
Proof of Theorem \[theorem:main-separable\] {#section:proof}
===========================================
We first state and prove our main technical tool – the [*Separability Lemma*]{} which essentially amplifies amplitudes of one-sided error algorithms. The Lemma can be proven using already known techniques of amplitude amplifications (e.g., see [@BHMT Sec 2.1]). We give an alternative recursive construction that is optimized towards amplifying fixed probabilities.
We use the following notation for the sake of brevity. Given a collection of quantum systems $\{{\mathcal{Q}}_1, {\mathcal{Q}}_2, \ldots \}$ (such collections will be always denoted by ${\mathcal{S}}$), we say that ${\mathcal{S}}$ is $(\delta,\epsilon)$-separable (for some $\delta <\epsilon$) if output distribution of any ${\mathcal{Q}}_i$ in ${\mathcal{S}}$ is either $\mu_\delta$ or $\mu_\epsilon$.
\[lemma:fully-separable\]\[Separability\] For $\delta < \epsilon < 1$ and a collection of quantum systems ${\mathcal{S}}_1$ which is $(\delta,\epsilon)$-separable, there is a ${\mathcal{B}}$-transform ${\mathcal{B}}_\epsilon$ which converts ${\mathcal{S}}_1$ to a $(\delta',1)$-separable collection of quantum systems (for some $\delta \le \delta' < 1$). Additionally, $\delta=\delta'=0$ if and only if $\delta=0$.
Given an instance ${\mathcal{Q}}= \langle {|\psi\rangle}, C, {\mathcal{P}}\rangle$ of some ${\mathcal{Q}}_i \in {\mathcal{S}}_1$, Lemma \[lemma:fully-separable\] gives us a way to determine whether the distribution of ${\mathcal{Q}}$ is ${\langle 0, 1 \rangle}$ or ${\langle \epsilon, 1-\epsilon \rangle}$ by first transforming ${\mathcal{Q}}$ to ${\mathcal{B}}({\mathcal{Q}}) = {\mathcal{Q}}' =
\langle {|\psi'\rangle}, C', {\mathcal{P}}' \rangle$ and then measuring the output of $C'$ on ${|\psi'\rangle}$ (which is a simple extension of the original input state) using measurement operator ${\mathcal{P}}'$ (which is also a simple extension of the original measurement operator).
Grover iterator {#subsection:grover}
---------------
As is usual in all analysis of amplitude amplification, the main operator to study is the Grover iterator [@Grover1996; @BHMT]. Suppose we have a circuit $C$ acting on an input state ${|\psi\rangle}$ and supposed to be measured using a two-output projective measurement operator ${\mathcal{P}}=\langle P_E, I-P_E \rangle$. We consider a generalized version, similar to the one studied by H[ø]{}yer [@Hoyer2000]: $G(C,{|\psi\rangle}, {\mathcal{P}}, \theta,\alpha)
= C S_{{|\psi\rangle}} C^\dagger S_{\mathcal{P}}C$ using these additional gates: $S_{{|\psi\rangle}} = I - (1-e^{{\imath}\theta}){{|\psi\rangle}{\langle \psi |}}$ and $S_{\mathcal{P}}= I -
(1-e^{{\imath}\alpha})P_E$.
Let ${|\psi'\rangle}=C{|\psi\rangle}$ denote the output state, ${|\psi_E\rangle} = P_E
{|\psi'\rangle}$ and $p$ denote ${\langle \psi_E | \psi_E \rangle}$ – the probability of measuring outcome $E$ for this output state. It is easy to see that $C S_{{|\psi\rangle}} C^\dagger
= I - (1-e^{{\imath}\theta}){{|\psi'\rangle}{\langle \psi' |}}$ and $S_{\mathcal{P}}C {|\psi\rangle} = \big(I - (1-e^{{\imath}\alpha})P_E\big){|\psi'\rangle}$. One can then compute ${|\psi''\rangle}=G {|\psi\rangle}$ as $\big( e^{{\imath}\theta} +
(1-e^{{\imath}\alpha})(1-e^{{\imath}\theta})p \big) {|\psi'\rangle} -
(1-e^{{\imath}\alpha}){|\psi_E\rangle}$ and $P_E {|\psi''\rangle} = \big( e^{{\imath}\theta} + e^{{\imath}\alpha} - 1 +
(1-e^{{\imath}\alpha})(1-e^{{\imath}\theta})p \big) {|\psi_E\rangle}$.
We get the following lemma summarizing the relative increase in probability after one application of our Grover iterator. We will use $p'(\theta,\alpha,p)$ to denote the new probability of measuring outcome $E$ on the output state after applying $G$ on input ${|\psi\rangle}$.
\[lemma:grover-iterator\] Given a quantum system ${\mathcal{Q}}_1 = \langle {|\psi\rangle}, C, {\mathcal{P}}\rangle$ and $\alpha,\theta \in [0,\pi]$, let $G$ be the circuit for the Grover iterator $G(C,{|\psi\rangle}, {\mathcal{P}}, \theta,\alpha) =
C S_{{|\psi\rangle}} C^\dagger S_{\mathcal{P}}C$. If $p$ denotes the probability of observing outcome $E$ for ${\mathcal{Q}}_1$ and $p'$ denotes the same probability for the QS $\langle
{|\psi\rangle}, G, {\mathcal{P}}\rangle$, then $p' = p\Delta$ where $\Delta = \left|\big( e^{{\imath}\theta} + e^{{\imath}\alpha} - 1 +
(1-e^{{\imath}\alpha})(1-e^{{\imath}\theta})p \big)\right|^2$.
First, $p=0$ if and only if $p'=0$ which means amplification has no effect on impossible outcomes. On the other hand, if $p > 0$, $p'$ is maximized when $\theta=\alpha$; it can be shown that $\Delta = \big( (1-2p)\cos\theta - 2(1-p) \big)^2 +
\sin^2\theta$ in that case. We will use $\Delta^*_p$ to denote the maximum value of $\Delta$ for any $p$ and using optimal $\theta$ and $\alpha$. The corresponding [*optimal Grover iterator*]{} will be denoted as $G^*_p(C,{|\psi\rangle}, {\mathcal{P}})$; note that $G^*$ increases the probability from $p$ to $p' = p \Delta^*_p$. Table \[table:delta\] summarizes the optimum value of $p'$ and the relative increase for different possible values of initial probability $p$. Details of the relevant calculations are given in Appendix.
The following definition and corollary essentially describes the optimum ${\mathcal{B}}$-transform.
\[defn:opt-bb-transform\] ${\mathcal{B}}_{p}^* : \big\langle {|\psi\rangle}, C, {\mathcal{P}}\big\rangle \longrightarrow
\big\langle {|\psi\rangle}, G^*_p\big(C, {|\psi\rangle}, {\mathcal{P}}\big), {\mathcal{P}}\big\rangle$
\[cor:opt-bb-transform\] If the output distribution of a QS ${\mathcal{Q}}$ is $\mu_\epsilon$, then the output distribution of ${\mathcal{B}}^*_\epsilon({\mathcal{Q}})$ is $\langle \epsilon \Delta^*_\epsilon,
1-\epsilon\Delta^*_\epsilon \rangle$. On the other hand, if the output distribution is $\mu_\delta$ (for some $\delta < \epsilon$), then the output distribution of ${\mathcal{B}}^*_\epsilon({\mathcal{Q}})$ is $\langle \delta', 1-\delta' \rangle$ for some $\delta' \ge \delta$ which can be computed using $\delta$ and $\epsilon$. Furthermore, $\delta =
\delta'$ if and only if $\delta = 0$ (in which case, $\delta' = 0$).
Optimum $\alpha=\theta$ Relative increase $\frac{p'}{p}=\Delta^*_p$
---------------------- ----------------------------------------- --------------------------------------------- --------------------
$p = 0.5$ $\pi/2$ 2 1
$0.25 \le p \le 0.5$ $\arccos \left( 1-\frac{1}{2p} \right)$ $\frac{1}{p}$ 1
$p \le 0.25$ $\pi$ $(3-4p)^2 \ge 4$ $p(3-4p)^2 \ge 4p$
: Optimum Grover iterator for different values of initial probability\[table:delta\]
In the next few subsections, we prove Separability Lemma for different values of $\epsilon$.
${\mathcal{B}}_\epsilon$ for $\epsilon \in [1/4,1/2]$ {#subsection:bb-half}
-----------------------------------------------------
This is the simplest of all cases, to ${\mathcal{B}}$-transform $(\delta,\epsilon)$-separable ${\mathcal{S}}_1$ to a $(\delta',1)$-separable one, for any $1/4 \le \epsilon \le 1/2$ and for some $\delta \le \delta'$. We can clearly use ${\mathcal{B}}_{\epsilon} = {\mathcal{B}}^*_{\epsilon}$ defined in Definition \[defn:opt-bb-transform\]. Separability Lemma immediately follows from Corollary \[cor:opt-bb-transform\] and Table \[table:delta\].
${\mathcal{B}}_{\epsilon}$ for $\epsilon > {\frac{1}{2}}$ {#subsection:bb-half+}
---------------------------------------------------------
We use the idea proposed by Brassard et al. [@BHMT] to first convert ${\mathcal{S}}_1$ to a $(\delta',{\frac{1}{2}})$-separable ${\mathcal{S}}_2$; let ${\mathcal{B}}^+_\epsilon$ denote this transformation which is illustrated in Equation \[eqn:bb-half+\]. This involves an additional qubit in state ${|0\rangle}$ and an additional projective operator ${\mathcal{P}}_\epsilon = \langle P_\epsilon^0, I - P_\epsilon^0 \rangle$, where, $$P_\epsilon^0 = \tfrac{1}{2\epsilon}{{|0\rangle}{\langle 0 |}} +
\sqrt{1-\tfrac{1}{2\epsilon}}\sqrt{\tfrac{1}{2\epsilon}}{{|1\rangle}{\langle 0 |}} +
\sqrt{1-\tfrac{1}{2\epsilon}}\sqrt{\tfrac{1}{2\epsilon}}{{|0\rangle}{\langle 1 |}} +
\left(1-\tfrac{1}{2\epsilon}\right) {{|1\rangle}{\langle 1 |}}$$
Then we convert ${\mathcal{S}}_2$ to a $(\delta'',1)$-separable ${\mathcal{S}}_3$ by using ${\mathcal{B}}_{{\frac{1}{2}}}$ (see Subsection \[subsection:bb-half\]). Combining both of these, we propose the following transformation for ${\mathcal{B}}_\epsilon$. Here ${\mathcal{P}}'$ denotes ${\mathcal{P}}\otimes {\mathcal{P}}_\epsilon$. $$\begin{aligned}
\big\langle {|\psi\rangle}, C, {\mathcal{P}}\big\rangle &
\stackrel{{\mathcal{B}}^+_\epsilon}{\longrightarrow} \big \langle {|\psi\rangle} \otimes {|0\rangle}, C \otimes I, {\mathcal{P}}' \big\rangle
\stackrel{{\mathcal{B}}_{{\frac{1}{2}}}}{\longrightarrow} \big \langle {|\psi\rangle} \otimes {|0\rangle}, G^*_{1/2}\big(C \otimes I,
{|\psi\rangle}\otimes {|0\rangle}, {\mathcal{P}}' \big), {\mathcal{P}}' \big\rangle
\label{eqn:bb-half+}\end{aligned}$$
The transformation from ${\mathcal{S}}_2$ to ${\mathcal{S}}_3$ was shown to be correct in Subsection \[subsection:bb-half\]. Correctness of ${\mathcal{B}}^+_\epsilon$ follows from the fact that the probability of measuring outcome $0$ on the state ${|0\rangle}$ is $\frac{1}{2\epsilon}$ (since ${\frac{1}{2}}<
\epsilon \le 1$, ${\frac{1}{2}}\le \frac{1}{2\epsilon} < 1$). Let $p$ denote the probability of measuring outcome $E$ for some ${\mathcal{Q}}= \langle {|\psi\rangle}, C, {\mathcal{P}}\big\rangle \in {\mathcal{S}}_1$ and let $p'$ denote the same probability for the QS $\langle {|\psi\rangle} \otimes {|0\rangle}, C \otimes I, {\mathcal{P}}\otimes {\mathcal{P}}_\epsilon \rangle$ of ${\mathcal{S}}_2$. Observe that, if $p=0$, then $p'=0$; furthermore, if $p = \epsilon > {\frac{1}{2}}$, then $p'=\epsilon
\frac{1}{2\epsilon} = {\frac{1}{2}}$. Of course, the transformation does not depend upon $\delta$.
${\mathcal{B}}_{\epsilon}$ for $\epsilon < {\frac{1}{4}}$ {#subsection:bb-small}
---------------------------------------------------------
To transform $(\delta,\epsilon)$-separable ${\mathcal{S}}_1$ to $(\delta',1)$-separable one, we first repeatedly apply the optimum Grover iterator enough number of times to amplify $\epsilon$ beyond ${\frac{1}{4}}$ and then apply a suitable ${\mathcal{B}}_{\epsilon_k}$ from Subsection \[subsection:bb-half\].
Suppose $\epsilon < 1/4$. Let $\epsilon_1 = \epsilon \Delta^*_\epsilon$, $\epsilon_2
= \epsilon_1 \Delta^*_{\epsilon_1}$, $\epsilon_3 = \epsilon_2
\Delta^*_{\epsilon_2}, \cdots$. Let $k$ be the smallest integer such that $\epsilon_k \ge 1/4$; clearly, $\epsilon_1, \ldots, \epsilon_{k-1} < 1/4$ and $\epsilon_k \in [1/4,1/2]$. We define ${\mathcal{B}}_\epsilon$ as the $k$ transformations ${\mathcal{B}}_{\epsilon}^*,
{\mathcal{B}}^*_{\epsilon_1}, {\mathcal{B}}^*_{\epsilon_2}, \ldots {\mathcal{B}}^*_{\epsilon_{k-1}}$ applied successively and then followed by ${\mathcal{B}}_{\epsilon_k}$. $$\begin{aligned}
{\mathcal{B}}_\epsilon: \langle {|\psi\rangle}, C, {\mathcal{P}}\rangle & \substack{~{\mathcal{B}}^*_\epsilon~\\\longrightarrow} \langle
{|\psi\rangle}, C_1, {\mathcal{P}}\rangle & \mbox{output
dist.}={\langle \epsilon_1, 1-\epsilon_1 \rangle} \mbox{ \& } C_1 = G^*_\epsilon(C,
{|\psi\rangle},{\mathcal{P}})\\
& \substack{~{\mathcal{B}}^*_{\epsilon_1}~\\\longrightarrow} \langle {|\psi\rangle}, C_2,
{\mathcal{P}}\rangle & \mbox{output dist.}={\langle \epsilon_2, 1-\epsilon_2 \rangle}
\mbox{ \& } C_2 = G^*_{\epsilon_1}(C_1,
{|\psi\rangle},{\mathcal{P}})\\
& \substack{~{\mathcal{B}}^*_{\epsilon_2}~\\\longrightarrow} \cdots & \dots \\
& \substack{~{\mathcal{B}}^*_{\epsilon_{k-1}}~\\\longrightarrow} \langle {|\psi\rangle}, C_k,
{\mathcal{P}}\rangle & \mbox{output dist.}={\langle \epsilon_k, 1-\epsilon_k \rangle}
\mbox{ \& } C_{k} = G^*_{\epsilon_{k-1}}(C_{k-1},
{|\psi\rangle},{\mathcal{P}})\\
& ~\substack{{\mathcal{B}}_{\epsilon_k}\\\longrightarrow} \langle
{|\psi'\rangle}, C_{k+1}, {\mathcal{P}}' \rangle &\end{aligned}$$
Satisfiability Lemma is easily proved by observing that $\epsilon_k \in
[1/4,1/2]$ and so, applying ${\mathcal{B}}_{\epsilon_k}$ (from Subsection \[subsection:bb-half\]) at the last step ensures that the final QS has output distribution ${\langle 1, 0 \rangle}$. It is also easy to check that these output distributions remain unchanged if and only if $\delta = 0$.
Performance Evaluation
----------------------
Even though we propose a recursive approach to reduce error-probability of exact error quantum systems, we show that our approach is essentially same as the existing iterative approaches for amplitude amplification in terms of the number of calls to $C$ and $C^\dagger$.
Take any quantum system $QS =
\langle {|\psi\rangle}, C, {\mathcal{P}}\rangle$. The existing approaches [@BHMT; @Hoyer2000] repeatedly apply the iterative Grover operator ${\mathcal{Q}}= (C S_{{|\psi\rangle}} C^\dagger S_{\mathcal{P}})$ (generalized to act on input encoded as the initial state and output state to be measured by any projective operator) on $C {|\psi\rangle}$. Here $S_{{|\psi\rangle}}$ and $S_{\mathcal{P}}$ modify the phase of certain states by $\theta=\alpha=\pi$ as specified in Subsection \[subsection:grover\].
Let $\epsilon$ denote the probability of observing outcome $E$; let $\beta \in [0,\pi/2]$ be such that $\sin^2 \beta =
\epsilon$. Then, the probability of observing $E$ on repeated applications of ${\mathcal{Q}}$, say $b$ times, on $C {|\psi\rangle}$ (i.e., on the output state of ${\mathcal{Q}}^b C
{|\psi\rangle}$) can be shown to be $\sin^2 \big( (2b+1)\beta \big)$.
As shown in Table \[table:delta\], suitably choice of phases in $S_{{|\psi\rangle}}$ and $S_{\mathcal{P}}$ can amplify any $\epsilon \in [0.25,1]$ to 1 using a ${\mathcal{B}}$-transform that effectively corresponds to one application of ${\mathcal{Q}}$ on $C{|\psi\rangle}$. So, if $\epsilon \ge 0.25$, our recursive method and the iterative approach are identical.
So, we will now analyze ${\mathcal{B}}_{\epsilon}$ for $\epsilon < 0.25$, in fact, for $\epsilon
\ll 0.25$. Let $k$ be the number of ${\mathcal{B}}^*$-transforms required. Recall from Subsection \[subsection:bb-small\] that ${\mathcal{B}}_\epsilon$ keeps the input and the projective operator unchanged and converts $C$ to some $C_{k+1}$ via intermediate circuits $C_1, C_2, \ldots, C_k$ where $C_{j+1} = G^*_{\epsilon_j}(C_j,
{|\psi\rangle}, {\mathcal{P}})$ for $\epsilon < \epsilon_1 < \ldots < \epsilon_k \in
[1/4,1/2]$. The $S_{{|\psi\rangle}}$ and $S_{\mathcal{P}}$ operators in those $G^*$ are defined using phases $\theta=\alpha=\pi$ as per Table \[table:delta\].
[lemma]{}[ckinduction]{}\[lemma:c\_k\_induction\] For any $j \in [1,k]$, $C_j = {\mathcal{Q}}^{\frac{3^j-1}{2}} C$.
This lemma can be easily proved by induction on $k$ (see Appendix). It shows that the final circuit obtained by our recursive approach is identical to that obtained by apply a fixed ${\mathcal{Q}}$ a certain number of times. Therefore, $\epsilon_{k} = \sin^2 \left( 3^k \beta
\right)$ which must be at least $1/4$. This stipulates that $k \ge \log_3
\frac{\pi}{6\beta}$. The total number of calls to $C$ and $C^\dagger$ made by our recursive algorithm to amplify $\epsilon < 0.25$ to some $\epsilon_k > 0.25$ can then be easily shown to be $1+\frac{\pi}{3\beta}$ (rather, the next higher integer) – which is exactly the same as that in ${\mathcal{Q}}^{(3^k-1)/2}C$.
Proof of Theorem \[theorem:main-separable\] {#proof-of-theoremtheoremmain-separable}
-------------------------------------------
We are now ready to prove Theorem \[theorem:main-separable\] using Separability Lemma. We will use the following notation. If ${\mathcal{B}}$ is a transformation for a set of quantum systems ${\mathcal{S}}$, then the set of [*transformed quantum systems*]{} after applying ${\mathcal{B}}$ will be denoted by ${\mathcal{B}}({\mathcal{S}})$.
The given ${\mathcal{S}}$ in the theorem is $(\delta,\epsilon)$-separable. Our required ${\mathcal{B}}_{\delta,\epsilon}$ will be composed of a series of ${\mathcal{B}}$-transforms: ${\mathcal{B}}_\epsilon$, ${\mathcal{B}}_2$ and ${\mathcal{B}}_\delta$.
${\mathcal{B}}_\epsilon$ is chosen such so as to solve $QD({\mathcal{S}})$ with error $(\delta',1)$ for some $\delta < \delta'$. This step can skipped (${\mathcal{B}}_\epsilon$ can be set to identity) if $\epsilon = 1$; on the other hand, if $\epsilon < 1$, we can use ${\mathcal{B}}_\epsilon$ from Lemma \[lemma:fully-separable\], which implies that ${\mathcal{B}}_\epsilon({\mathcal{S}})$ is $(\delta',1)$-separable for some $\delta'$ (that depends on $\delta$ and $\epsilon$). Let ${\mathcal{S}}_1$ denote ${\mathcal{B}}_\epsilon({\mathcal{S}})$.
${\mathcal{B}}_2$ is the following transform: ${\big\langle}{|\psi\rangle}, C, (P_1, P_2)
{\big\rangle}\longrightarrow {\big\langle}{|\psi\rangle}, C, (P_2, P_1) {\big\rangle}$. Let ${\mathcal{S}}_2 = {\mathcal{B}}_2({\mathcal{S}}_1)$. Any $QS \in {\mathcal{S}}_1$ with $\mu(QS)=\mu_{\delta'}$ is transformed to $QS' \in {\mathcal{S}}_2$ with $\mu(QS') = 1-\delta'$ and similarly, if $\mu(QS) = \mu_1$, then $\mu(QS') = \mu_0$. Therefore, ${\mathcal{S}}_2$ is $(0,1-\delta')$-separable.
By property of ${\mathcal{B}}_\epsilon$, $\delta=\delta'=0$ if and only if $\delta=0$ and in that case, we have obtained $(0,1)$-separable ${\mathcal{S}}_2$. On the other hand, if $\delta > 0$, then $\delta' > 0$. Let $\delta''$ denote $1-\delta'$. Since ${\mathcal{S}}_2$ is $(0,\delta'')$-separable, apply Lemma \[lemma:fully-separable\] again to get ${\mathcal{B}}_{\delta}$ such that ${\mathcal{S}}' = {\mathcal{B}}_{\delta}({\mathcal{S}}_2)$ is $(0,1)$-separable.
Our required transform ${\mathcal{B}}$ is a sequential application of ${\mathcal{B}}_\epsilon$ followed by ${\mathcal{B}}_2$ followed by ${\mathcal{B}}_{\delta}$. As explained above, ${\mathcal{B}}_\delta({\mathcal{B}}_2({\mathcal{B}}_\epsilon(\cdot)))$ is a $(0,1)$-solution of $QD({\mathcal{S}})$.
Uniform version of Theorem \[theorem:main-separable\] {#section:uniform}
=====================================================
The non-uniformity in Definition \[defn:bb-transform\] is not very helpful if we wish to obtain a true black-box extension of a quantum system ${\mathcal{Q}}= {\big\langle}{|\psi\rangle},C,{\mathcal{P}}{\big\rangle}$. Note that the extension to the input qubits and the extension to the projective measurement operator is anyway independent of ${\mathcal{Q}}$ and $n$, the gates in $C'$ are uniform in $n$, and furthermore, the transformed circuit $C'$ is allowed to call the original circuit $C$ (and its inverse $C^\dagger$) in a black-box manner; however, some of the gates in $C'$ may additionally depend upon ${|\psi\rangle}$ and operators of ${\mathcal{P}}$. It would be really good to obtain a more uniform conversion which necessitates the following definition.
A ${\mathcal{B}}$-transform for converting multiple QS $\{{\mathcal{Q}}_1, {\mathcal{Q}}_2, \ldots \}$ is said to be [*uniform*]{} if the circuit of ${\mathcal{B}}({\mathcal{Q}}_i)$ is identical for all source ${\mathcal{Q}}_i$ except for the calls to $C$ and $C^\dagger$ corresponding to ${\mathcal{Q}}_i$.
Uniform Grover iterator
-----------------------
We want to study some sufficient conditions for the ${\mathcal{B}}$-transforms to be uniform by constructing a uniform version of Grover iterator.
Since Grover iterator uses ${\mathcal{S}}_{\mathcal{P}}$, it is crucial to have identical measurement operators for all quantum systems. This is, however, not such a major requirement since it is always possible to change measurement operators by extending a quantum circuit with suitable operators.
Except the gates $S_{{|\psi\rangle}} = I - (1-e^{{\imath}\theta}){{|\psi\rangle}{\langle \psi |}}$ which depend upon the corresponding input to the circuit (${|\psi\rangle}$), none of the other gates used in ${\mathcal{B}}$-transforms that are involved in the proof of Theorem \[theorem:main-separable\] depend upon the input state (see Section \[section:proof\]). However, a ${\mathcal{B}}$-transform may still become uniform if all the inputs in ${\mathcal{S}}_1$, and hence all such $S_{{|\psi\rangle}}$ gates, will be identical.
Now consider a second option – all measurement operators are identical and all the input states are not identical but they form an orthonormal set. We show that it is still possible to apply $S_{{|\psi\rangle}}$ in a uniform manner. Recall that this gate changes the phase of any state depending upon whether it is ${|\psi\rangle}$ or not and the main difficulty appears to be the fact that the input state cannot be copied and stored for a later application of the conditional phase gate. So our main idea is to convert ${|\psi\rangle}$ to some state in the standard basis since it is possible to copy and store states in the standard basis using the [*quantum fanout gate*]{} [@Durr1999]. This gate copies a standard basis state to another register: $F_m {|x_1 \ldots x_m\rangle}{|b_1 \ldots b_m\rangle} = {|x_1 \ldots
x_m\rangle}{|(x_1 \oplus b_1) \ldots (x_m \oplus b_m)\rangle}$ for $x_1 \ldots x_m \in
\{0,1\}^m$ and $b_1 \ldots b_m \in \{0,1\}^m$ shows the operation for “copying” $m$-qubits.
[1.2in]{}
[3in]{}
[1.2in]{}
See Figure \[fig:s-psi-uniform\] for a uniform version of ${\mathcal{S}}_{{|\psi\rangle}}$. Figure \[fig:s-psi-uniform-a\] shows $S_{{|\psi\rangle}}$ as a part of an arbitrary quantum circuit, say $C$ that takes as input an $m$-qubit state ${|\psi\rangle}$ (and some ancillæ) and ${\mathcal{S}}_{{|\psi\rangle}}$, on $m$-qubits, is one of its gates Since we are now considering the case that $C$ is applied only on orthogonal input states (suppose denoted by ${|\psi_1\rangle}, {|\psi_2\rangle},
\cdots$), therefore, there exists a one to one mapping between these states and a subset of the $m$-qubit standard basis states ${|1\rangle}, {|2\rangle}, \cdots$. Let $U$ denote the unitary operator for the mapping, i.e., $U{|\psi_v\rangle} =
{|v\rangle}$.
Figure \[fig:s-psi-uniform-b\] illustrates a circuit $C'$ that applies $S_{{|\psi\rangle}}$ without requiring a gate that explicitly depends upon ${|\psi\rangle}$. Apart from the two registers of $C$ (the input ${|\psi\rangle}$ and ancillæ qubits), $C'$ also uses $m$ additional ancillæ qubits in state ${|0\rangle}$. Other than the standard gates ($T$ stands for the unbounded fanout Toffoli and $X$ is the quantum NOT gate), $C'$ uses three additional gates: $F_m$, $P_\theta$ and $S_\theta$. The $F_m$ gate is the quantum fanout gate. $P_\theta$ changes phase of ${|1\rangle}$ by $e^{{\imath}\theta}$: $P_\theta = I - (1-e^{{\imath}\theta}){{|0\rangle}{\langle 0 |}}$. The $S_\theta$ gate uses an additional reusable ancillæ ${|0\rangle}$ and changes the phase by $e^{{\imath}\theta}$ only for the state ${|0^m\rangle}$ (illustrated in Figure \[fig:s-psi-uniform-c\]).
The state of the first two registers after the left dotted box in Figure \[fig:s-psi-uniform-b\] is simply ${|0^m\rangle}{|\psi\rangle} \to
{|v\rangle}{|\psi\rangle}$ where ${|v\rangle}$ is the standard basis vector $U{|\psi\rangle}$. We will next analyze the operator for the right dotted box, say denoted by $U_R$. $S_\theta$ can be written as $I -
(1-e^{{\imath}\theta}){{|0^m\rangle}{\langle 0^m |}}$ and the $F_m$ operator essentially behaves like $F_m {|b_1 \ldots b_m\rangle} \to {|(v_1 \oplus b_1), \ldots (v_m
\oplus b_m)\rangle}$. The following calculation (for only the qubits involved) shows that the operator for the right dotted box is identical with $S_{{|\psi\rangle}}$. $$\begin{aligned}
U_R = & (I \otimes U^\dagger) F_m (I \otimes S_\theta) F_m (I \otimes U)
= (I \otimes U^\dagger) F_m \big(I \otimes (I -
(1-e^{{\imath}\theta}){{|0^m\rangle}{\langle 0^m |}}) \big) F_m (I \otimes U)\\
= & (I \otimes U^\dagger) \big(I \otimes (I -
(1-e^{{\imath}\theta}){{|v\rangle}{\langle v |}}) \big) (I \otimes U)
= I \otimes (I - (1-e^{{\imath}\theta}){{|\psi\rangle}{\langle \psi |}}) = I \otimes
S_{{|\psi\rangle}}\end{aligned}$$
The results of this subsection can be summarized in the following lemma.
\[lemma:uniform-bb-transform\] The ${\mathcal{B}}$-transform in Theorem \[theorem:main-separable\] can be made uniform if all projection operators in the quantum systems of ${\mathcal{S}}$ are identical and all input states in ${\mathcal{S}}$ are either identical or form an orthonormal set of states.
Distinguishing two circuits {#section:distinguish-circuits}
===========================
Suppose we are given a quantum circuit $C$ (as black-box) and two different operators $C_1$ and $C_2$, all acting on the same Hilbert space, and we are told that the operator for $C$ is either $C_1$ or $C_2$. We have to determine $C$ corresponds to which one. We assume that we also have access to its inverse operator $C^\dagger$.
The analogous problem for deterministic (classical) functions is trivial. Two distinct functions must differ at some input which can be determined from their function descriptions (the problem is NP-hard but we are not concerned about feasibility, not efficiency, for this discussion). The output of $C$ on this input will identify whether $C$ is $C_1$ or $C_2$. However, if $C$ is a randomized circuit or algorithm, then except for a few trivial cases, the output of $C$ generates a sample distribution over the output of $C_1$ and $C_2$; the question of determining the correct distribution of ${\mathcal{C}}$ without any error is believed to be hard, if not impossible.
However, it is possible to give a positive answer to the same question for quantum circuits. Select a suitable ${|\phi\rangle}$ and compute the two possible output states ${{|\psi_1\rangle} = C_1{|\phi\rangle}}$, ${{|\psi_2\rangle} = C_2{|\phi\rangle}}$. Choose projective operators ${\mathcal{P}}=\langle I - {{|\psi_1\rangle}{\langle \psi_1 |}},
{{|\psi_1\rangle}{\langle \psi_1 |}} \rangle$ with respective outcomes $E$ and $F$.
Consider these two quantum systems: $\langle {|\phi\rangle},
C_1, {\mathcal{P}}{\big\rangle}$ and $\langle {|\phi\rangle}, C_2, {\mathcal{P}}{\big\rangle}$. The output distribution of the first QS is ${\langle 0, 1 \rangle}$ and that of the second is ${\langle \epsilon, 1-\epsilon \rangle}$ where $\epsilon =
1-|{\langle \psi_1 | \psi_2 \rangle}|^2 > 0$. Now, Theorem \[theorem:main-separable\] can be applied on the QS $\langle
{|\phi\rangle}, C, {\mathcal{P}}\rangle$ which essentially gives us a circuit $C'$ (that calls $C$ and $C^\dagger$) along with suitably extended input and measurement operators, with the property that if the outcome of the QS is $E$, then $C$ is surely $C_1$ and otherwise $C_2$.
It is perfectly okay to use any ${|\phi\rangle}$ as the input state; however, since the size of $C'$ depends inversely upon $\epsilon$ so it makes sense to have the largest possible $\epsilon$. A recent result [@2015BeraATPG] can be used to determine the optimum initial state (details of this is presented in the Appendix).
#### Single-fault detection {#single-fault-detection .unnumbered}
Fault detection is a major step in the workflow of circuit fabrication. It is common in research and industry to assume that practically most faults appear according to a few known fault models. A standard approach to detecting if a circuit is faulty is to generate a set of test patterns (inputs) such that the output of a fault-free circuit would be different from that of a faulty-circuit. This method is known as ATPG (automatic test-pattern generation) and is well-studied for classical circuits and very recently, seeing use even for quantum circuits [@paler2011tomographic].
ATPG is computationally difficult being NP-hard [@IbarraSahniATPG], and even harder for quantum circuits because the measurement output of these circuits is probabilistic, and hence even a single test pattern will generate a distribution over possible outcomes.
However, the technique described earlier in this section can come to our rescue in the special case of only one fault model, i.e., given a circuit $C$ as a black-box unit, we wish to determine if $C$ is fault-free (i.e., $C=C_1$) or $C$ is faulty (with fault model $C_2$). We can reliably answer this question without any chance of error using the approach described above.
Exact Error Algorithms {#section:exact-error-classes}
======================
Usual probabilistic classes like [$\mathbf{RP}$]{}and [$\mathbf{BPP}$]{}are defined in terms of errors that are upper bounded by constants. They are rarely defined in terms of exact error, primarily due to the lack of robustness in definition that accompanies this concept. There is no known technique to show that the class of problems with one-sided error exactly same as $0.3$ remains unchanged if the error is instead $0.301$. Consider, for example, the simplified class [$\mathbf{ERP}$]{}(${\ensuremath{\mathbf{P}}\xspace}\subseteq {\ensuremath{\mathbf{ERP}}\xspace}\subseteq
{\ensuremath{\mathbf{RP}}\xspace}$) whose problems have randomized algorithms similar to those for [$\mathbf{RP}$]{}, but with an additional requirement that the error is same for all “no” instances (of any length). We similarly define [$\mathbf{EBPP}$]{}as the class of problems with exact two-sided error polymomial-time algorithms. Based on what we know, ${\ensuremath{\mathbf{P}}\xspace}\not= {\ensuremath{\mathbf{ERP}}\xspace}\not= {\ensuremath{\mathbf{EBPP}}\xspace}$. However, we were able to prove that the quantum analogs of these classes have identical complexity using our generalization of quantum amplitude amplification.
${\ensuremath{\mathbf{EBQP}}\xspace}_{\delta,\epsilon}$ is the class of languages $L$ for which there exists a uniform family of polynomial-size quantum circuits $\{C_n\}$, a uniform family of states for $a_n$ ancillæ qubits ${|A_n\rangle}$ and a uniform family of two-outcome projective measurement operators $\{{\mathcal{P}}_n\}$ such that $C_n$ and ${\mathcal{P}}_n$ act on a space of $n+a_n$ qubits and the following hold for any $x \in \{0,1\}^n$, $\forall n$:
- if $x \not\in L$, then the output distribution of ${\big\langle}{|x\rangle}
\otimes {|A_n\rangle}, C_n, {\mathcal{P}}_n {\big\rangle}$ is $\mu_\delta$ (i.e., when the output state of $C_{n}$ on input state ${|x\rangle} \otimes {|A_n\rangle}$ is measured using ${\mathcal{P}}_n$, outcome $E$ is observed with probability $\delta$) and
- if $x \in L$, then the output distribution of ${\big\langle}{|x\rangle}
\otimes {|A_n\rangle}, C_n, {\mathcal{P}}_n {\big\rangle}$ is $\mu_\epsilon$ (i.e., outcome $E$ is observed with probability $\epsilon$ upon similar measurement as the above case).
${\ensuremath{\mathbf{ERQP}}\xspace}_\epsilon$ is simply ${\ensuremath{\mathbf{EBQP}}\xspace}_{0,\epsilon}$. Define $\displaystyle{\ensuremath{\mathbf{EBQP}}\xspace}= \bigcup_{\epsilon>\delta\ge 0}
{\ensuremath{\mathbf{EBQP}}\xspace}_{\delta,\epsilon}$ and $\displaystyle{\ensuremath{\mathbf{ERQP}}\xspace}=
\bigcup_{\epsilon>0} {\ensuremath{\mathbf{ERQP}}\xspace}_\epsilon$.
Note that, unlike the usual definitions of probabilistic classes, for these classes it is not even clear if the different classes ${\ensuremath{\mathbf{EBQP}}\xspace}_{\delta,\epsilon}$ for different $\delta$ and $\epsilon$ are identical. However, the following lemma is obvious from these definitions.
${\ensuremath{\mathbf{EQP}}\xspace}= {\ensuremath{\mathbf{EBQP}}\xspace}_{0,1} = {\ensuremath{\mathbf{ERQP}}\xspace}_1$ and ${\ensuremath{\mathbf{EQP}}\xspace}\subseteq {\ensuremath{\mathbf{ERQP}}\xspace}\subseteq {\ensuremath{\mathbf{EBQP}}\xspace}$.
The main result of this section is a simple application of Theorem \[theorem:main-separable\] and Lemma \[lemma:uniform-bb-transform\].
\[theorem:eqp-ebqp\] ${\ensuremath{\mathbf{EQP}}\xspace}= {\ensuremath{\mathbf{ERQP}}\xspace}= {\ensuremath{\mathbf{EBQP}}\xspace}$.
We essentially need to show that ${\ensuremath{\mathbf{EBQP}}\xspace}\subseteq {\ensuremath{\mathbf{EQP}}\xspace}$. To prove this we will show that for any $L$, if $L \in
{\ensuremath{\mathbf{EBQP}}\xspace}_{\delta,\epsilon}$ (for any $\epsilon > \delta \ge 0$), then $L \in
{\ensuremath{\mathbf{EBQP}}\xspace}_{0,1}$.
Fix an arbitrary $n$. For any binary string $x$ of length $n$, define the quantum system ${\mathcal{Q}}_x = {\big\langle}{|x\rangle} \otimes {|A_n\rangle}, C_n, {\mathcal{P}}_n {\big\rangle}$ where ${|A_n\rangle}$, $C_n$ and ${\mathcal{P}}_n$ are obtained from the definition of ${\ensuremath{\mathbf{EBQP}}\xspace}_{\delta,\epsilon}$ and the fact that $L \in {\ensuremath{\mathbf{EBQP}}\xspace}_{\delta,\epsilon}$. Now consider these sets of quantum systems ${\mathcal{S}}_n = \{{\mathcal{Q}}_x ~:~ x \in \{0,1\}^n \}$ for all $n
> 0$. Clearly, there are two possible output distributions of any ${\mathcal{S}}_n$, namely, $\mu_\delta$ and $\mu_\epsilon$. Since the input states in ${\mathcal{S}}_n$ are orthonormal and the projection operators therein are identical, we can therefore apply Theorem \[theorem:main-separable\] and Lemma \[lemma:uniform-bb-transform\] to obtain a uniform transformation ${\mathcal{B}}_{\delta,\epsilon}$ which perfectly solves the problem of $QD({\mathcal{S}}_n)$. Let ${\mathcal{B}}_{\delta,\epsilon}({\mathcal{Q}}_x) =
{\mathcal{Q}}'_x = {\big\langle}{|x\rangle} \otimes{|A_n\rangle} \otimes {|00\ldots 0\rangle}, C'_n, {\mathcal{P}}'_n
{\big\rangle}$ which gives us (i) a circuit $C'_n$ which calls $C_n$ (and $C_n^\dagger$) (ii) a two-outcome projective measurement operator ${\mathcal{P}}'_n$ and a (iii) set of ancillæ qubits in state ${|\mathbf{00\ldots 0}\rangle}$ such that the following holds for the outcome of $C'_n$ on ${|x\rangle} \otimes
{|A_n\rangle} \otimes {|00\ldots 0\rangle}$ when measured using ${\mathcal{P}}'_n$.
- If $x \not\in L$, then the output distribution of ${\mathcal{Q}}'_x$ is $\mu_0$, i.e., the outcome is never $E$.
- If $x \in L$, then the output distribution of ${\mathcal{Q}}'_x$ is $\mu_1$, i.e., the outcome is always $E$.
Therefore, we get a uniform family of circuits $\{C'_n\}$, a uniform family of ancillæ qubits ${|A_n\rangle} \otimes {|\mathbf{00\ldots 0}\rangle}$ and a uniform family of two-outcome projective measurement operator $\{{\mathcal{P}}'_n\}$ such that the outcome of $C'_{|x|}$ on any ${|x\rangle}$, with additional ancillæ qubits in a uniformly generated state, when measured by ${\mathcal{P}}'_{|x|}$ indicates whether $x \in L$ without any probability of error. Since $C_n'$ uses constantly many calls to $C_n$ and $C_n^\dagger$ along with other gates (the constant depends only on $\delta$ and $\epsilon$), this shows that $L \in {\ensuremath{\mathbf{EBQP}}\xspace}_{0,1}$.
We illustrate an application of the above theorem to obtain an error-free circuit for an ${\ensuremath{\mathbf{ERQP}}\xspace}_{1/2}$ language $L$ (see Appendix for an explicit proof). Consider circuit $C$ in Figure \[fig:erqp-half\](a) which can identify if $x \in L$ with one-sided error $0.5$. As is typical in quantum circuits, in this example only one of the output qubits of the circuit is measured in the standard basis ($P_E = {{|0\rangle}{\langle 0 |}} \otimes I$ and ${\mathcal{P}}= \langle P_E, 1-P_E \rangle$); therefore, if $x \not\in L$, then the output qubit is never observed in state ${|0\rangle}$ and if $x \in L$, then the output qubit is observed in states ${|0\rangle}$ or ${|1\rangle}$ with equal probability. The circuit $C'$ shown in Figure \[fig:erqp-half\](b) shows how to remove the probability of error; the same output qubit is measured in the standard basis for outcome and some additional qubits in state ${|0\rangle}$ are used as ancillæ. Apart from calling $C$ and $C^\dagger$, $C'$ uses the n-qubit Fanout gate $F_n$, a conditional phase gate $S_0$ [^3] which changes phase of ${|00\ldots 0\rangle}$ by ${\imath}$, and $P$ does the same to ${|1\rangle}$.
Exact amplitude amplification (Theorem \[theorem:aa-two-sided\])
----------------------------------------------------------------
Let ${\mathcal{P}}$ denote the two-outcome projective measurement operator used in the original two-sided exact error circuit $C$. $C$ can be of two types depending on how it accesses its input. Any input $x \in X$ can be accessed either through the input state ${|x\rangle}$ (along with ancillæ initialized to ${|00\ldots\rangle}$, wlog.) or through an oracle gate $U_x:
{|x,b\rangle} \to {|x, b\oplus \Phi(x)\rangle}$ (for $b \in \{0,1\}$). If $C$ is of the former type, then Theorem \[theorem:aa-two-sided\] is essentially same as Theorem \[theorem:eqp-ebqp\].
Next we focus on circuits with oracle gates. Let $C^{U_x}$ denote this circuit when given $U_x$ as the oracle gate corresponding to an input $x \in X$. The input state to $C^{U_x}$ can be taken to be ${|00\ldots 0\rangle}$, wlog. The proof follows by applying Theorem \[theorem:main-separable\] on this collection of quantum systems: $\left\{ {\big\langle}{|00\ldots\rangle}, C^{U_x}, {\mathcal{P}}{\big\rangle}~:~ x \in X \right\}$.
Observe that this collection satisfies the conditions of Lemma \[lemma:uniform-bb-transform\]. So, the corresponding ${\mathcal{B}}$-transform is uniform which implies that all the transformed circuits for these quantum systems are identical, except for the calls to $C$ and $C^\dagger$. Therefore, we can choose this transformed oracle circuit as our required $C'$ of Theorem \[theorem:aa-two-sided\].
Conclusion
==========
Is there a classical method that can accurately decide the distribution of a random variable $X$ among two given distributions based on multiple samples of $X$? Probably no. On the other hand, if the random variables come from a quantum source, we show that quantum circuits exist that can do the same without any probability of error. A quantum circuit, along with an input state and a measurement operator, can be consider as a quantum source of samples drawn over the distribution of the measurement outcomes.
The underlying technique is a generalization of quantum amplitude amplification to two-sided error and for circuits without oracle gates. We used our amplification technique to distinguish between two circuits, when used as a black box, which has application in fault detection of quantum circuits. We also defined a restricted version of quantum one-sided and two-sided bounded error classes and used generalized amplification to show that those complexity classes collapse to (error-free) quantum polynomial time complexity class.
It would be interesting to investigate if this approach can be used for ATPG with more than one fault models and for amplifying standard bounded-error classes ${\ensuremath{\mathbf{BQP}}\xspace}$ and ${\ensuremath{\mathbf{RQP}}\xspace}$.
Proof of ${\ensuremath{\mathbf{ERQP}}\xspace}_{1/2} \subseteq {\ensuremath{\mathbf{EQP}}\xspace}$
=================================================================================================
\[lemma:rqp\_0.5\_in\_EQP\] If a language $L \in {\ensuremath{\mathbf{ERQP}}\xspace}_{1/2}$, then $L \in {\ensuremath{\mathbf{EQP}}\xspace}$.
We will assume that the algorithms end with a measurement of a specified qubit in the computational basis – this is equivalent most other ways measurement strategies that are commonly applied.
Take any $L \in {\ensuremath{\mathbf{ERQP}}\xspace}_{1/2}$, and consider the corresponding circuit $C$ (illustrated in Figure \[fig:erqp-half\](a)). Suppose $m$ denotes the number of ancillæ qubits used by $C$, and $n$ denotes the length of any input $x$, then $C$ acts on ${\mathcal{H}}^{\tensor n} \tensor {\mathcal{H}}^{\tensor m}$ and its output is given by ${|\psi\rangle} = C
{|x\rangle}{|0^m\rangle}$. Without loss of generality, suppose that the first qubit is specified for measurement, then the projective measurement operator applied is ${|0\rangle}{\langle 0 |} \tensor I$.
We will now a construct an [$\mathbf{EQP}$]{} circuit ${\mathcal{C}}'$ to decide the same language $L$. But first note that, ${|\psi\rangle} = {|0\rangle}{|\psi_0\rangle} + {|1\rangle}{|\psi_1\rangle}$ and that, if $x \not\in L$, ${\langle \psi_1 | \psi_1 \rangle}=0$, and if $x\in L$, ${\langle \psi_1 | \psi_1 \rangle}=1/2$ ($={\langle \psi_0 | \psi_0 \rangle}$). The circuit is constructed as ${\mathcal{C}}' = {\mathcal{A}}S_0 {\mathcal{A}}^{-1} P {\mathcal{A}}$ and described in Figure \[fig:erqp-half\](b). ${\mathcal{C}}'$ acts on ${\mathcal{H}}^{\tensor n} \tensor {\mathcal{H}}^{\tensor n} \tensor {\mathcal{H}}^{\tensor m}$, and we will denote the space as 3 registers $P,Q,R$, respectively, of $n,n,m$ qubits. The gates will be labelled with the registers (as superscripts) they are applied on in the following description.
Besides the circuit $C$, which will be used always on registers $QR$, we will make frequent use of the [*fanout*]{} operator[@Durr1999]. This, and the other components of ${\mathcal{C}}'$, are listed below.
- The fanout operator effectively copies basis states from a control qubit to a target qubit. On two registers of $n$ qubits each, it works as $F_n{|a_1 \ldots a_n\rangle}{|b_1 \ldots b_n\rangle} =
{|a_1 \ldots a_n\rangle}{|(b_1 {\oplus}a_1) \ldots (b_n {\oplus}a_n)\rangle}$. Note that, $F^\dagger_n = F_n$.
- ${\mathcal{A}}= (F_n^{PQ} \tensor I) \tensor (I \tensor C^{QR})$
- $P^{Q} = I - (1-{\imath}){{|0\rangle}{\langle 0 |}}$ is the phase gate $P$ applied on the first qubit of register $Q$. Notice that, the first qubit of register $Q$ is the measurement qubit with respect to $C$.
- $S_0^{QR}=I-(1-{\imath}){{|0^{n+m}\rangle}{\langle 0^{n+m} |}}$ which changes the phase of the basis state in which all qubits are in the state ${|0\rangle}$. Implementation of $S_0$ is shown in Figure \[fig:erqp-half\](c) – it requires one additional qubit initialized to ${|0\rangle}$. However this qubit is in state ${|0\rangle}$ after application of this operator, so this qubit could be reused if required. This extra qubit has been left out in the description of ${\mathcal{C}}'$.
- The input to ${\mathcal{C}}'$ will be ${|x\rangle}{|0^{\tensor n}\rangle}{|0^{\tensor
m}\rangle}$.
- We will measure the first qubit of register $Q$ in the standard basis at the end.
Next, we will describe the operation of ${\mathcal{C}}'$. $$\begin{aligned}
{\mathcal{C}}' {|x\rangle}{|0^n\rangle}{|0^m\rangle}
= & C^{QR} \cdot F_n^{PQ} \cdot S_0^{QR} \cdot F_n^{PQ} \cdot C^{\dagger QR}
\cdot P^{Q} \cdot C^{QR} \cdot F_n^{PQ}
~~ {|x\rangle}{|0^n\rangle}{|0^m\rangle}\\
= & C^{QR} \cdot F_n^{PQ} \cdot S_0^{QR} \cdot F_n^{PQ} \cdot C^{\dagger QR}
\cdot P^{Q} \cdot C^{QR}
~~ {|x\rangle}{|x\rangle}{|0^n\rangle}\\
= & C^{QR} \cdot F_n^{PQ} \cdot S_0^{QR} \cdot F_n^{PQ} \cdot C^{\dagger QR}
\cdot P^{Q}
~~ {|x\rangle} \bigg({|0\rangle}{|\psi_0\rangle} + {|1\rangle}{|\psi_1\rangle}\bigg)\\
= & C^{QR} \cdot F_n^{PQ} \cdot S_0^{QR} \cdot F_n^{PQ} \cdot C^{\dagger QR}
~~ {|x\rangle} \bigg({|0\rangle}{|\psi_0\rangle} + {\imath}{|1\rangle}{|\psi_1\rangle}\bigg)~~~~~(*)\end{aligned}$$
We will now simplify the remaining operator. $$\begin{aligned}
& C^{QR} \cdot F_n^{PQ} \cdot S_0^{QR} \cdot F_n^{PQ} \cdot C^{\dagger QR} \\
= & C^{QR} \cdot F_n^{PQ} \cdot \bigg(I-(1-{\imath})I^P\tensor{{|0^{n+m}\rangle}{\langle 0^{n+m} |}}
\bigg) \cdot F_n^{PQ} \cdot C^{\dagger QR}\\
= & C^{QR} \cdot F_n^{PQ} \cdot \bigg(I-(1-{\imath})\sum_{\text{$n$-bit
$p$}}{{|p,0^{n+m}\rangle}{\langle p,0^{n+m} |}} \bigg) \cdot F_n^{PQ} \cdot C^{\dagger QR}\\
= & C^{QR} \cdot \bigg(I-(1-{\imath})\sum_{\text{$n$-bit
$p$}}F_n^{PQ} {{|p,0^{n+m}\rangle}{\langle p,0^{n+m} |}}F_n^{PQ} \bigg) \cdot C^{\dagger QR}\\
= & C^{QR} \cdot \bigg(I-(1-{\imath})\sum_{\text{$n$-bit
$p$}} {{|p,p,0^{m}\rangle}{\langle p,p,0^{m} |}} \bigg) \cdot C^{\dagger QR}\\
= & I-(1-{\imath})\sum_{\text{$n$-bit
$p$}} {{|p\rangle}{\langle p |}} \tensor (C^{QR}{{|p,0^{m}\rangle}{\langle p,0^{m} |}} C^{\dagger QR}\\\end{aligned}$$
Substituting this simplification in $(*)$ above, $$\begin{aligned}
& {\mathcal{C}}' {|x\rangle}{|0^n\rangle}{|0^m\rangle} \\
= & \bigg( I-(1-{\imath})\sum_{\text{$n$-bit
$p$}} {{|p\rangle}{\langle p |}} \tensor (C^{QR}{{|p,0^{m}\rangle}{\langle p,0^{m} |}} C^{\dagger QR}
\bigg)
~~ {|x\rangle} \bigg({|0\rangle}{|\psi_0\rangle} + {\imath}{|1\rangle}{|\psi_1\rangle}\bigg)\\
= & {|x\rangle} \bigg({|0\rangle}{|\psi_0\rangle} + {\imath}{|1\rangle}{|\psi_1\rangle}\bigg) -\\
& (1-{\imath})\sum_{\text{$n$-bit
$p$}} {|p\rangle}{\langle p | x \rangle} \tensor \bigg(C^{QR}{{|p,0^{m}\rangle}{\langle p,0^{m} |}} C^{\dagger
QR} \bigg)
~~ \bigg({|0\rangle}{|\psi_0\rangle} + {\imath}{|1\rangle}{|\psi_1\rangle}\bigg) \\
= & {|x\rangle} \bigg({|0\rangle}{|\psi_0\rangle} + {\imath}{|1\rangle}{|\psi_1\rangle}\bigg) -
(1-{\imath}) {|x\rangle} \tensor \bigg(C^{QR}{{|x,0^{m}\rangle}{\langle x,0^{m} |}} C^{\dagger
QR} \bigg)
~~ \bigg({|0\rangle}{|\psi_0\rangle} + {\imath}{|1\rangle}{|\psi_1\rangle}\bigg) \\
= & {|x\rangle} \bigg( \big({|0\rangle}{|\psi_0\rangle} + {\imath}{|1\rangle}{|\psi_1\rangle}\big) -
(1-{\imath}) \big({|0\rangle}{|\psi_0\rangle} + {|1\rangle}{|\psi_1\rangle}\big)
\big({\langle 0 |}{\langle \psi_0 |} + {\langle 1 |}{\langle \psi_1 |}\big)
~~ \big({|0\rangle}{|\psi_0\rangle} + {\imath}{|1\rangle}{|\psi_1\rangle}\big)\bigg)\\
= & {|x\rangle} \bigg( \big({|0\rangle}{|\psi_0\rangle} + {\imath}{|1\rangle}{|\psi_1\rangle}\big) -
(1-{\imath}) \big({|0\rangle}{|\psi_0\rangle} + {|1\rangle}{|\psi_1\rangle}\big)
\big( {\langle \psi_0 | \psi_0 \rangle} + {\imath}{\langle \psi_1 | \psi_1 \rangle} \big) \bigg)\\
= & {|x\rangle} \bigg( \big(1 - (1-{\imath})K\big) {|0\rangle}{|\psi_0\rangle} + \big({\imath}-
(1-{\imath})K\big) {|1\rangle}{|\psi_1\rangle} \bigg)~~\text{where,
}K={\langle \psi_0 | \psi_0 \rangle} + {\imath}{\langle \psi_1 | \psi_1 \rangle}\\
= & \left\{
\begin{array}{ll}
{\imath}{|x\rangle}{|0\rangle}{|\psi_0\rangle} & \text{ if, } x \not\in L \text{ i.e., }
{\langle \psi_1 | \psi_1 \rangle}=0,~{\langle \psi_0 | \psi_0 \rangle}=1\\
({\imath}-1){|x\rangle}{|1\rangle}{|\psi_1\rangle} & \text{ if, } x \in L \text{ i.e., }
{\langle \psi_1 | \psi_1 \rangle}={\langle \psi_0 | \psi_0 \rangle}=1/2
\end{array}
\right.$$
Measuring the first qubit of register $Q$ therefore shows ${|1\rangle}$ if and only if $x \in L$.
Optimal values for Grover iterator
==================================
Let $c$ denote $\big( e^{{\imath}\theta} + e^{{\imath}\alpha} - 1 +
(1-e^{{\imath}\alpha})(1-e^{{\imath}\theta})p \big)$. Then, $c^* = -(1-p) + 2(1-p)e^{-{\imath}\theta} + pe^{-2{\imath}\theta}$. Therefore, if $p > 0$, then $\Delta = cc^*$ which we will compute below.
$$\begin{aligned}
& \Delta = c c^* \\
& = (1-p)^2 - 2(1-p)^2 e^{-{\imath}\theta} - p(1-p)e^{-2{\imath}\theta}\\
& - 2(1-p)^2 e^{{\imath}\theta} + 4(1-p)^2 + 2p(1-p)e^{-{\imath}\theta}\\
& -p(1-p)e^{2{\imath}\theta} + 2p(1-p)e^{{\imath}\theta} + p^2\\
& = [(1-p)^2 + 4(1-p)^2 + p^2] + (e^{-{\imath}\theta}+e^{{\imath}\theta})[2p(1-p)-2(1-p)^2]
- (e^{-2{\imath}\theta}+e^{2{\imath}\theta}) p(1-p) \\
& = 6p^2 - 10p + 5 + 4(1-p)(2p-1)\cos\theta - 2p\cos 2\theta + 2p^2\cos
2\theta\\
& = (-10 - 2\cos 2\theta)p + (6 + 2\cos 2\theta)p^2 + (\sin^2\theta +
\cos^2\theta) + 4 + 4(1-p)(2p-1)\cos\theta\\
& = (-8 - 4\cos^2\theta)p + (4+4\cos^2\theta)p^2 + \sin^2\theta +
\cos^2\theta + 4 + 4(1-p)(2p-1)\cos\theta\\
& = \sin^2\theta + (4p^2 -4p + 1)\cos^2\theta + 4 + 4p^2 - 8p +
4(1-p)(2p-1)\cos\theta\\
& = \sin^2\theta + (2p-1)^2\cos^2\theta + 4(1-p)^2 +
4(1-p)(2p-1)\cos\theta\\
& = [(2p-1)\cos\theta + 2(1-p)]^2 + \sin^2\theta\end{aligned}$$
We will give a quick sketch of the proof by induction.
For $k=1$, $C_1 = G^*_\epsilon = C S_{{|\psi\rangle}} C^\dagger S_{\mathcal{P}}C = {\mathcal{Q}}C$ so the claim holds for the base case.
Now, suppose that the claim holds for some $1 \le j < k$. Before discussing the induction case, note that $({\mathcal{Q}}^\dagger)^t = (S_{\mathcal{P}}C S_{{|\psi\rangle}}
C^\dagger)^t = S_{\mathcal{P}}\cdot {\mathcal{Q}}^{t-1} \cdot (C S_{{|\psi\rangle}}
C^\dagger)$ for any $t$.
Then, $C_{j+1} = G^*_{\epsilon_j}(C_j,
{|\psi\rangle}, {\mathcal{P}}) = C_j S_{{|\psi\rangle}} C_j^\dagger S_{\mathcal{P}}C_j$ which, using the induction hypothesis, is ${\mathcal{Q}}^{(3^j-1)/2} C \cdot S_{{|\psi\rangle}} \cdot
C^\dagger ({\mathcal{Q}}^{\dagger})^{(3^j-1)/2} \cdot S_{\mathcal{P}}{\mathcal{Q}}^{(3^j-1)/2} C = $ (using the expression for $({\mathcal{Q}}^\dagger)^t$ above) ${\mathcal{Q}}^{(3^j-1)/2 + 1 + (3^j-1)/2 - 1 + 1
+ (3^j-1)/2} C = {\mathcal{Q}}^{(3^{j+1}-1)/2}C$.
Optimum initial state for distinguishing two circuits
=====================================================
Recall that $| {\langle \psi_1 | \psi_2 \rangle} | = |
{\langle \phi|C_1^\dagger C_2 | \phi \rangle} |$. Denoting $C_1^\dagger C_2$ by $S$, we would like to minimize $| {\langle \phi|S | \phi \rangle} |$ over all possible pure state ${|\phi\rangle}$. Suppose the eigenvalues of $S$ are $e^{i\theta_1},
\ldots$ with corresponding eigenvectors ${|v_1\rangle}, \ldots$. Using a recent result [@2015BeraATPG], the maximum value of $\epsilon$ is obtained by solving the optimization problem $$\min f(\theta_1, \ldots) = \bigg(\sum_j c_j^2 + \sum_{j \not= k} c_j c_k \cos(\theta_j -
\theta_k) \bigg), ~~~\mbox{ where, } \sum_j c_j=1,~~~~ 0 \le c_j \le 1$$
Suppose $f_{OPT}$ denotes the optimal value above and $c_1, \ldots $ denote the corresponding solution. Then, the optimal $\epsilon$ is $1-f_{OPT}^2$ and ${|\phi\rangle}$ can be set to $\sum_j \sqrt{c_j} {|v_j\rangle}$.
[^1]: IIIT-Delhi, New Delhi, India. Email: [[email protected]]{}
[^2]: The same question for classical classes was asked here: <http://cstheory.stackexchange.com/questions/20027/in-what-class-are-randomized-algorithms-that-err-with-exactly-25-chance>.
[^3]: $S_0
{|0 0 \ldots 0\rangle} = {\imath}{|00\ldots 0\rangle}$ and for other states $S_0
{|x_1 x_2 \ldots x_k\rangle} = {|x_1 \ldots x_k\rangle}$ (illustrated in Figure \[fig:erqp-half\](c)).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We investigate $\beta$-interactions of free nucleons and their impact on the electron fraction ($Y_e$) and r-process nucleosynthesis in ejecta characteristic of binary neutron star mergers (BNSMs). For that we employ trajectories from a relativistic BNSM model to represent the density-temperature evolutions in our parametric study. In the high-density environment, positron captures decrease the neutron richness at the high temperatures predicted by the hydrodynamic simulation. Circumventing the complexities of modelling three-dimensional neutrino transport, (anti)neutrino captures are parameterized in terms of prescribed neutrino luminosities and mean energies, guided by published results and assumed as constant in time. Depending sensitively on the adopted $\nu_e$-$\bar\nu_e$ luminosity ratio, neutrino processes increase $Y_e$ to values between 0.25 and 0.40, still allowing for a successful r-process compatible with the observed solar abundance distribution and a significant fraction of the ejecta consisting of r-process nuclei. If the $\nu_e$ luminosities and mean energies are relatively large compared to the $\bar\nu_e$ properties, the mean $Y_e$ might reach values $>$0.40 so that neutrino captures seriously compromise the success of the r-process. In this case, the r-abundances remain compatible with the solar distribution, but the total amount of ejected r-material is reduced to a few percent, because the production of iron-peak elements is favored. Proper neutrino physics, in particular also neutrino absorption, have to be included in BNSM simulations before final conclusions can be drawn concerning r-processing in this environment and concerning observational consequences like kilonovae, whose peak brightness and color temperature are sensitive to the composition-dependent opacity of the ejecta.'
author:
- |
S. Goriely$^1$, A. Bauswein$^2$, O. Just$^{3,5}$, E. Pllumbi$^{3,4}$, and H.-Th. Janka$^3$\
$^1$Institut d’Astronomie et d’Astrophysique, Université Libre de Bruxelles, CP 226, 1050 Brussels, Belgium\
$^2$Department of Physics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece\
$^3$Max-Planck-Institut für Astrophysik, Postfach 1317, 85741 Garching, Germany\
$^4$Physik Department, Technische Universität München, James-Franck-Straße 1, 85748 Garching, Germany\
$^5$Max-Planck/Princeton Center for Plasma Physics (MPPC)
date: Released 2015 Xxxxx XX
title: 'Impact of weak interactions of free nucleons on the r-process in dynamical ejecta from neutron-star mergers'
---
\[firstpage\]
nuclear reactions, nucleosynthesis, abundances – neutrinos – stars: neutron – dense matter – hydrodynamics
Introduction {#sect_intro}
============
The r-process, or rapid neutron-capture process, of stellar nucleosynthesis is invoked to explain the production of the stable (and some long-lived radioactive) neutron-rich nuclides heavier than iron that are observed in stars of various metallicities, as well as in the solar system [for a review, see @arnould07]. Despite important effort to model potential r-process sites, all the proposed scenarios face serious problems and the site(s) of the r-process is (are) not identified yet. Until now, type-II supernovae or $\gamma$-ray bursts models have failed to provide convincing evidence for a successful r-processing that could significantly contribute to the Galactic enrichment in r-material [@wanajo11; @janka12; @burrows13]. Only magneto-rotational supernova explosions with extremely strong pre-collapse magnetic fields and fast rotation seem to provide favourable conditions for r-processing, but such rare events are not expected to be at the origin of the global galactic enrichment in r-process nuclei [@winteler12; @nishimura15; @wehmeyer15].
For this reason, special attention is now being paid to neutron star (NS) mergers following the confirmation by hydrodynamic simulations that a significant amount of r-process enriched material, typically about $10^{-3}$ to a few $10^{-2}$[$M_{\odot}$]{}, can be ejected [@rosswog99; @frei99; @arnould07; @metzger10; @roberts11; @goriely11; @korobkin12; @bauswein13; @goriely13; @wanajo14; @perego14; @just15; @seki15]. Recent nucleosynthesis calculations by @just15 show that the combined contribution of both the dynamical (prompt) ejecta expelled during binary NS or NS-black hole (BH) mergers and the neutrino and viscously driven outflows generated during the post-merger remnant evolution of relic BH-torus systems can lead to the production of r-process elements from mass number $A \ga90$ up to thorium and uranium. The corresponding abundance distribution reproduces the solar distribution extremely well and can also account for the elemental distributions observed in low-metallicity stars [@roederer11; @roederer12]. Furthermore, recent studies [@mat14; @kom14; @mennekens14; @shen14; @voort14; @vangioni14; @wehmeyer15] have reconsidered the galactic or cosmic chemical evolution of r-process elements in different evolutionary contexts. Although they do not converge towards one unique quantitative picture, most of them arrived at the conclusion that double compact star mergers may be the major production sites of r-process elements.
Despite the recent success of nucleosynthesis studies for NS mergers, the possibility of r-processing in these events is still affected by a variety of uncertainties. In particular, the impact of neutrino interactions is not yet studied and understood in detail, the main reason of which is the not yet manageable computational complexity associated with neutrino transport in a generically three-dimensional (3D), highly asymmetric environment with nearly relativistic fluid velocities and rapid changes in time. A computationally simple and much more efficient alternative to solving the time-dependent transport equation for neutrino distributions in the six-dimensional phase space is the use of a neutrino leakage scheme, in which the local neutrino net-emission (i.e., emission minus absorption) rate is estimated by a weighted interpolation between the pure emission rate and an optical-depth dependent diffusive loss term [e.g., @Ruffert1996a; @Rosswog2003]. However, neutrino absorption cannot be straightforwardly included in a self-consistent manner in a leakage scheme because information about the local neutrino densities is missing in such a treatment.
It was long believed that neutrino interactions could not, at least not drastically, affect the initial neutron richness of the ejecta. Such expectations were based on numerical merger models. The temperatures of the merging NSs in these simulations, and therefore the neutrino production rates, remain low until the NSs collide with each other, and they rise in an increasing volume only gradually on a time scale of several milliseconds after the first contact of the two NSs [@Ruffert1999a; @Rosswog2003; @wanajo14]. At this time a significant fraction of the ejecta material is already being expelled with nearly relativistic velocities. Newtonian merger studies suggested that a large part of the ejecta material even from symmetric mergers (i.e., for two equal-mass or close to equal-mass NSs) is thrown out by tidal forces in extended spiral arms forming from matter of the outer faces of the merging objects during their final approach and collision [@rosswog99; @korobkin12]. By the tidal stretching these arms naturally remain unshocked and thus stay cool. Therefore they reach low densities so quickly that electron and positron captures cannot become efficient. Moreover, the ejecta escape to large radii before the neutrino emission of the compact merger remnant becomes sizable and neutrino absorption can affect the electron fraction significantly. Absorption of neutrinos radiated by the massive merger remnant is also diminished because the tidal ejection in Newtonian models happens preferentially in the orbital plane [see e.g. @rosswog99; @korobkin12] while the neutrinos are predominantly emitted perpendicular to this plane [@Rosswog2003; @Dessart2009; @perego14]. However, the described situation applies well only for Newtonian mergers. The situation is different in the relativistic case.
Relativistic simulations of symmetric mergers do not show the development of prominent tidal arms, and the ejection of unshocked matter is therefore not important. Instead, the collision shock that builds up at the interface of the two NSs is typically stronger than in Newtonian conditions [see e.g. @bauswein13], leading to potentially higher temperatures and higher, faster rising neutrino luminosities [@wanajo14; @seki15]. The ejecta are expelled fairly spherically instead of equatorially [@bauswein13; @Hotokezaka2013] and consist of two main components, namely a first one in which very hot material is squeezed out from the collision interface of the two merging bodies, and a second, slightly delayed one that is expelled in waves from a torus-like belt of matter around the high-density core of the merger remnant. This torus is heated by spiral shocks that are sent out from the aspherical, wobbling, and rotating high-density core and that also lead to outward acceleration of parts of the torus matter [@bauswein13; @Hotokezaka2013; @just15; @wanajo14; @seki15].
Indeed, recent relativistic NS-NS merger simulations that took into account neutrino emission by means of a leakage scheme and absorption by an additional approximate transport treatment based on a moment formalism [@wanajo14; @seki15], found that neutrino interactions with free nucleons can significantly increase the electron fraction in the dynamical ejecta for cases in which the collapse of the merger remnant to a black hole is delayed or does not happen. Under such conditions nuclei with mass numbers $A<140$ can also be created in the dynamical ejecta in addition to the heavy r-process elements ($A>140$). Weak interaction processes of free nucleons can consequently affect the strength of the r-process and the emerging abundance distribution.
The accurate inclusion of neutrino interactions in hydrodynamical simulations remains a highly complex task. This motivated us to conduct a simple, parametric study in order to quantify the potential impact of weak interactions on the electron-fraction evolution in merger ejecta and thus to explore the consequences of charged-current neutrino-nucleon reactions for the nucleosynthesis and possible r-processing in these ejecta. More specifically, we investigate the influence of $\beta$-interactions of electron neutrinos ($\nu_e$) and electron antineutrinos ($\bar{\nu}_e$) with free $n$ and $p$ and of their inverse reactions, $$\begin{aligned}
& & \nu_e+ n \rightleftharpoons p + e^- \label{eq:betareac1}\\
& & \bar{\nu}_e + p \rightleftharpoons n + e^+ \quad \,,
\label{eq:betareac2}\end{aligned}$$ on the $Y_e$ distribution and r-process nucleosynthesis at conditions representative of the dynamical ejecta expelled by hydrodynamical forces during NS-NS mergers. These reactions have been neglected in all previous studies of r-process nucleosynthesis for such ejecta except those of @wanajo14, where their quantitative effects may depend on the adopted equation of state [@seki15]. The role of weak interactions for the electron fraction and the corresponding implications for r-process nucleosynthesis, however, demand further exploration in more detail, in particular also by basic, parametric modeling, because a multitude of uncertainties will prevent rigorous, self-consistent solutions of the full problem in the near future. Such uncertainties are associated with, for example, the extreme complexities of 3D energy-dependent neutrino transport in relativistic environments, with the neutrino opacities of dense, potentially highly magnetized matter, and with neutrino-flavor oscillations at rapidly time-variable, largely aspherical conditions of neutrino emission.
The main objective of our present work is a sensitivity study by the use of a parametric approach. It is intended to motivate further explorations of neutrino effects in relativistic NS-NS mergers in more detail and breadth. To this end, we set up a simplified and idealized theoretical framework to test the individual roles of the different weak interaction processes, considering the density and temperature evolution of fluid elements ejected from a prototype hydrodynamical, relativistic NS-NS merger model. For our parameter study we make assumptions about the neutrino emission properties that are guided by data taken from the literature. The electron fractions resulting from the neutrino-processing of the ejecta elements are then used as input for nuclear network calculations, allowing us to immediately link the effects of neutrino interactions to the final heavy-element production.
In Sect. \[sect\_weak\] the employed merger model and our treatment of the weak neutrino reactions with free nucleons are described. The effects of $\beta$-processes on the electron fraction are reported in Sect. \[sect\_ye\], and the subsequent r-process nucleosynthesis is analysed in Sect. \[sect\_rpro\]. Conclusions are drawn in Sect. \[sect\_conc\].
Weak interactions of free nucleons in merger ejecta {#sect_weak}
===================================================
We adopt the density evolution of ejecta fluid elements from a representative NS-NS merger model, namely the symmetric 1.35[$M_{\odot}$]{}-1.35[$M_{\odot}$]{} binary model obtained with the temperature-dependent DD2 equation of state [@hempel10; @typel10] [essentially identical to the one in @bauswein13 but with a higher resolution of $\sim$10$^6$ particles]. This relativistic hydrodynamic simulation also provides the temperature evolution. We do not apply any temperature post-processing as in @goriely11 to disentangle temperature jumps in shocks from artificial heating associated with the use of a numerical viscosity in the smoothed-particle hydrodynamics scheme.
For each trajectory, we follow the expansion starting at a fiducial density of $\rho_\mathrm{eq}=10^{12}$gcm$^{-3}$, where we assume equilibrium to hold between electrons, positrons and neutrinos for the given total lepton number provided by our NS-NS merger model. Below $\rho_\mathrm{eq}=10^{12}$gcm$^{-3}$, electron, positron and electron neutrino and antineutrino captures are systematically included. Reactions of neutrinos on nuclei are, however, neglected. As long as the temperature remains in excess of typically $T>10^{10}$ K, the abundance of heavy nuclei is determined by nuclear statistical equilibrium (NSE) at the given electron fraction, density and temperature. From the density $\rho_\mathrm{eq}$ down to the density $\rho_\mathrm{net}$, at which the temperature reaches 10GK and the full reaction network is initiated, the $\beta$-interactions of free nucleons may affect the electron fraction $Y_e$. If a trajectory stays cooler than 10GK below the density $\rho_\mathrm{eq}$, the network calculation is started at the neutron-drip density $\rho_\mathrm{drip}$, i.e., $\rho_\mathrm{net}=\rho_\mathrm{drip}\simeq 4.2\times 10^{11}$gcm$^{-3}$. The considered $\beta$-reactions involve free nucleons, whose abundance variations are given by $$\begin{aligned}
\frac{dY_n^\mathrm{f}}{dt}&=& -\lambda_+ Y_n^\mathrm{f} + \lambda_- Y_p^\mathrm{f} \nonumber\,, \\
\frac{dY_p^\mathrm{f}}{dt}&=&\lambda_+ Y_n^\mathrm{f} - \lambda_- Y_p^\mathrm{f} \,,
\label{eq_dyq}\end{aligned}$$ where $\lambda_+=\lambda_{\nu_e} + \lambda_{e^+}$ and $\lambda_- =\lambda_{{\bar\nu}_e} + \lambda_{e^-}$. The $\lambda_x$ denote capture rates of species $x\in\{e^-,e^+,\nu_e,\bar\nu_e\}$ onto free nucleons according to the $\beta$-reactions, Eqs. (\[eq:betareac1\],\[eq:betareac2\]). The free neutron and proton numbers are related to $Y_e$ by $$\begin{aligned}
Y_n^\mathrm{f}&=& 1-Y_e-\sum_{Z \ge 2} N~Y(Z,N)\nonumber\,, \\
Y_p^\mathrm{f}&=& Y_e-\sum_{Z \ge 2} Z~Y(Z,N)\,,
\label{eq_yf}\end{aligned}$$ where $Y=X/A$ is the molar fraction (and $X$ the mass fraction) of the nucleus ($Z,N$) of atomic mass $A=Z+N$. Assuming that the NSE molar fractions of nuclei remain constant over the time step $\Delta t$, the time evolution of $Y_e$ can be related to the change of the number of free protons, and written as $$\frac{dY_e}{dt}=-\lambda_{\rm tot}Y_e + \lambda^* \,,
\label{eq_dye}$$ where $\lambda_{\rm tot}=\lambda_+ + \lambda_-$ and $$\lambda^*=\lambda_+ \left[ 1-\sum_{Z\ge 2}NY \right]+\lambda_- \sum_{Z\ge 2}ZY \,.
\label{eq_ls}$$ If only free neutrons and protons are present, $\lambda^*=\lambda_+$. The impact of heavy nuclei is to increase $\lambda^*$ towards $ \lambda_-$ (for matter made of $\alpha$-particles only, $\lambda^*=\lambda_{\rm tot}/2$). The $\alpha$-effect [@mclaughlin96; @meyer98; @pllumbi14], or more generally, the effect of heavy nuclei in binding neutrons and protons inside nuclei, is known to be responsible for driving $Y_e$ towards 0.5 and is included in this term $\lambda^*$. If we assume that $\lambda_{\rm tot}$ and $\lambda^*$ remain constant over the time step $\Delta t$ (or, specifically, that $\Delta t$ is chosen such that $\lambda_{\rm tot}$ and $\lambda^*$ as well as the abundance of nuclei remain essentially constant during the time step), Eq. (\[eq\_dye\]) can be integrated analytically leading to $$Y_e(t+\Delta t)\simeq Y_e(t) e^{(-\lambda_{\rm tot} \Delta t)} +
\frac{\lambda^*}{\lambda_{\rm tot}} \left[ 1-e^{(-\lambda_{\rm tot} \Delta t)} \right] \,,
\label{eq_ye}$$ where $\lambda_{\rm tot}$ and $\lambda^*$ are estimated at time $t$. This equation is used to follow $Y_e$ from the initial density $\rho_\mathrm{eq}$ down to the density $\rho_{\rm net}$, at which the temperature has dropped to 10 GK (or, alternatively, to the drip density if temperatures above 10 GK are not reached for the considered trajectory).
For $t \gg 1/\lambda_{\rm tot}$, $Y_e$ reaches the equilibrium value $Y_e^\infty$ given by $$Y_e^\infty\simeq \frac{\lambda^*}{\lambda_++\lambda_-} \simeq
\frac{\lambda_{\nu_e} + \lambda_{e^+}}{\lambda_{\nu_e} + \lambda_{e^+}+\lambda_{{\bar\nu}_e} +
\lambda_{e^-}} \quad .
\label{eq_yeinf}$$ The electron (anti)neutrino capture rates can be written in terms of the average (anti)neutrino capture cross sections $\langle \sigma_{\nu_e/\bar\nu_e} \rangle$ [@pllumbi14] as $$\begin{aligned}
\lambda_{\nu_e} & \simeq & \frac{L_{\nu_e}}{4\pi r^2 \langle E_{\nu_e}\rangle}
\langle \sigma_{\nu_e} \rangle \,,
\label{eq_lnu}\\ \lambda_{{\bar\nu}_e} & \simeq & \frac{L_{{\bar\nu}_e}}{4\pi r^2 \langle
E_{{\bar\nu}_e}\rangle} \langle \sigma_{{\bar\nu}_e} \rangle \,.
\label{eq_lnub}\end{aligned}$$ Here, the local (anti)neutrino number densities are expressed by the ratios of the global luminosities, $L_{\nu_e/\bar\nu_e}$, and the mean energies of the radiated neutrinos, $\langle E_{\nu_e/\bar\nu_e}\rangle$, multiplied with the spherical surface $4\pi r^2$ that surrounds the central neutrino source at a radial distance $r$ (for every trajectory we adopt the time-dependent radial distance $r$ from our hydrodynamical model). This $r^{-2}$ dilution of the neutrino flux is a crude approximation and holds, at best, far away from the neutrinosphere, provided the emission is isotropic, i.e., if directional variations of the neutrino fluxes do not play a role. Close to and below the neutrinosphere, however, such a description breaks down but can be justified by the fact that at these locations electron and positron captures dominate and their competition enforces a state of weak equilibrium. At large distances the asymptotic electron fraction is determined by neutrino and antineutrino absorptions, for which reason direction-dependent differences of the neutrino exposure of the ejecta would be important for a detailed discussion of neutrino effects on merger ejecta. Nevertheless, despite these shortcomings, we apply the simple ansatz at all radii $r$ where $\rho\le\rho_\mathrm{eq}$ in order to discuss basic aspects of the impact of neutrino processes with nucleons in the merger ejecta in a parametric way.
Similar to Eqs. (\[eq\_lnu\],\[eq\_lnub\]), the electron and positron capture rates are given in terms of the average electron/positron capture cross sections $\langle \sigma_{e^-/e^+} \rangle$ by $$\begin{aligned}
\lambda_{e^+} & = &c~ {\tilde n_{e^+}} \langle \sigma_{e^+} \rangle \,, \\ \lambda_{e^-} & = &c ~
n_{e^-} \langle \sigma_{e^-} \rangle \,,
\label{eq_lep}\end{aligned}$$ where $c$ is the speed of light and $n_{e^-}$ and ${\tilde n_{e^+}}$ the electron and positron densities, as detailed in @pllumbi14.
In turn, the average cross sections for electron neutrino and antineutrino captures, as well as those for electron and positron captures, including the weak magnetism and recoil corrections, are taken from @pllumbi14 [see also @horowitz99]. While the electron and positron capture rates are temperature- and density-dependent only, the (anti)neutrino capture rates depend on the (anti)neutrino luminosities and mean energies, hence require a detailed knowledge of the neutrino properties at each time step.
For the present study, we consider representative (anti)neutrino luminosities and angle-averaged mean energies that are assumed to remain constant in time. For a given mean energy $\langle E_{\nu_e}\rangle$, the electron neutrino temperature $T_{\nu_e}$ is deduced from the relation $$\langle E_{\nu_e}\rangle=k_\mathrm{B}T_{\nu_e} \cdot \frac{F_3(0)}{F_2(0)} \quad ,
\label{eq_tnu}$$ where $k_\mathrm{B}$ is the Boltzmann constant and $F_n$ are the fermi integrals [@takahashi78] of order $n$ for vanishing chemical potential, assuming nondegenerate neutrino spectra. A similar expression holds for the antineutrino temperature. For given (anti)neutrino luminosities and mean energies this allows us to determine all other moments of the (anti)neutrino energy spectra, and consequently the corresponding capture cross sections and rates (Eqs. \[eq\_lnu\],\[eq\_lnub\]).
![(Color online). Time evolution of the electron neutrino and antineutrino luminosities from @wanajo14 and the corresponding $Y_e^{\nu\infty}$ (black solid line) as defined by Eq. (\[eq\_yeinf\_nu\]). The black dotted line gives the $Y_e^{\nu\infty}$ without weak magnetism and recoil corrections, $i.e.$ $f_\nu^\mathrm{mr}=1$ in Eq. (\[eq\_yeinf\_nu\]). The (anti)neutrino mean energies are taken consistently from Fig. 1 (lower panel) of @wanajo14. Asymptotic values of $Y_e^{\nu\infty}$ are calculated only for non-negligible neutrino luminosities, $i.e.$ for times $t\gsimeq 4$ ms. []{data-label="fig_yeinf1"}](fig_yeinf_wan.pdf)
![(Color online). Same as Fig. \[fig\_yeinf1\], but for the electron neutrino and antineutrino luminosities from @ruffert01. The (anti)neutrino mean energies are taken consistently from Fig. 17 (lower panel; Model Bc) of this reference.[]{data-label="fig_yeinf2"}](fig_yeinf_ruf.pdf)
Assuming that electron (anti)neutrino captures dominate over electron and positron captures and the abundance of heavy nuclei is negligible, the asymptotic value of $Y_e$ (Eq. \[eq\_yeinf\]) can be approximated by $$Y_{e}^{\nu\infty}\simeq \frac{L_{\nu_e} \varepsilon_{\nu_e} f_{\nu_e}^\mathrm{mr}}
{L_{\nu_e} \varepsilon_{\nu_e} f_{\nu_e}^\mathrm{mr} + L_{{\bar\nu}_e} \varepsilon_{{\bar\nu}_e}
f_{{\bar\nu}_e}^\mathrm{mr}} \,,
\label{eq_yeinf_nu}$$ because the rates can be expressed as $\lambda_{\nu}\propto L_{\nu}\varepsilon_{\nu}f_{\nu}^\mathrm{mr}$ ($\nu=\nu_e, {\bar\nu}_e$), where $ \varepsilon_{\nu}=\langle E^2_{\nu} \rangle / \langle E_{\nu}\rangle=F_4(0)F_2(0)/F_3^2(0) \times
\langle E_{\nu}\rangle$ and $f_{\nu}^\mathrm{mr}$ corresponds to the weak magnetism and recoil corrections that can be found in @pllumbi14. The corresponding value of $Y_{e}^{\nu\infty}$ is shown in Fig. \[fig\_yeinf1\] for the luminosities and mean energies of @wanajo14 and lies between 0.25 and 0.50 for times $t\gsimeq 5$ms after the merging of the binary NSs. These values may, however, still be modified by effects of heavy nuclei. The weak magnetism and recoil corrections on the (anti)neutrino rates are seen in Fig. \[fig\_yeinf1\] to increase the asymptotic value of $Y_{e}^{\nu\infty}$ by up to 20% because they reduce the antineutrino capture cross section and simultaneously increase the neutrino capture cross section [@horowitz99].
The neutrino properties, and in particular the antineutrino to neutrino luminosity ratio, are found to vary significantly between different hydrodynamical simulations but also depend on the adopted equation of state [@seki15]. Avoiding the complexity of self-consistent neutrino transport in hydrodynamical simulations, we shall restrict ourselves in our sensitivity study to constant luminosities and mean energies taken at selected times from previous simulations [@ruffert01; @wanajo14]. We consider first two representative sets of values for electron (anti)neutrino luminosities and mean energies as obtained by @wanajo14 (see their Fig. 1), namely those corresponding to the instants of 5 and 6ms, i.e.,
- [Case 1:]{} $t\simeq 5$ ms with $L_{\nu_e} =0.6\times 10^{53}$ erg/s; $L_{{\bar\nu}_e}
=1.3\times 10^{53}$ erg/s; $\langle E_{\nu_e}\rangle=12$ MeV; $ \langle E_{{\bar\nu}_e}\rangle = 16$ MeV,
- [Case 2:]{} $t\simeq 6$ ms with $L_{\nu_e} =2.6\times 10^{53}$ erg/s; $L_{{\bar\nu}_e}
=4.0\times 10^{53}$ erg/s; $\langle E_{\nu_e}\rangle=13$ MeV; $ \langle E_{{\bar\nu}_e}\rangle = 16$ MeV.
Note that Case 1 leads to an asymptotic value $Y_{e}^{\nu\infty} \simeq 0.31$, whereas Case 2 yields $Y_{e}^{\nu\infty}\simeq 0.42$ (Fig. \[fig\_yeinf1\]).
In order to test more thoroughly the impact of the neutrino processes on the nucleosynthesis, we also consider here the electron (anti)neutrino properties calculated in the NS merger simulation of @ruffert01 (cf Model Bc in their Fig. 17). The corresponding (anti)neutrino luminosities are shown in Fig. \[fig\_yeinf2\], together with the asymptotic values $Y_{e}^{\nu\infty}$ (with and without the weak magnetism and recoil corrections). Overall, lower luminosities are found in this model in comparison with @wanajo14, but also a relatively higher emission of electron antineutrinos. Consequently, lower values of $Y_{e}^{\nu\infty}$ are predicted for this model. In our present sensitivity analysis we also select two cases for the electron (anti)neutrino luminosities and mean energies from the results of @ruffert01, namely those corresponding to times of 6 and 10ms (and hereafter referred to as Cases 3 and 4, respectively), i.e.,
- [Case 3:]{} $t\simeq 6$ ms with $L_{\nu_e} =0.3\times 10^{53}$ erg/s; $L_{{\bar\nu}_e} =10^{53}$ erg/s; $\langle E_{\nu_e}\rangle=12.5$ MeV; $ \langle E_{{\bar\nu}_e}\rangle = 17.4$ MeV ,
- [Case 4:]{} $t\simeq 10$ ms with $L_{\nu_e} =0.6\times 10^{53}$ erg/s; $L_{{\bar\nu}_e}
=1.6\times 10^{53}$ erg/s; $\langle E_{\nu_e}\rangle=13.5$ MeV; $ \langle E_{{\bar\nu}_e}\rangle =
16.3$ MeV .
Case 3 leads to an asymptotic value $Y_{e}^{\nu\infty} \simeq 0.21$, while Case 4 yields $Y_{e}^{\nu\infty}\simeq 0.29$ (Fig. \[fig\_yeinf2\]).
Finally, it should be mentioned that our assumption of time-independent (anti)neutrino luminosities can be questioned, since the neutrino-ejecta interaction is a highly time-dependent problem, where the relative time between the growth of the neutrino emission and the mass ejection matters. Considering constant luminosities is a very crude but simple approximation, which is sufficiently good to demonstrate the impact of neutrino processes on the time evolution and mass distribution of the electron fraction and the corresponding consequences for the r-process. A predictive assessment of neutrino effects on the nucleosynthesis in merger ejecta would also have to take account variations of the neutrino emission with different directions. Matter expelled towards the polar directions is exposed to different neutrino conditions than matter that leaves the system along equatorial trajectories. Again, a more detailed description of neutrino transport effects is demanded and is beyond the scope of our present parametric study.
Impact of $\beta$-interactions on the electron fraction {#sect_ye}
=======================================================
Time evolution of $Y_e$
-----------------------
To illustrate the impact of electron (anti)neutrino, electron and positron captures on the evolution of $Y_e$, we show in Figs. \[fig\_traj1\] and \[fig\_traj2\] the time evolution for two specific trajectories during the expansion from the initial density $\rho_\mathrm{eq}=10^{12}$gcm$^{-3}$ down to density $\rho_{\rm net}$, where the reaction network calculations are initiated. Both trajectories are studied including (anti)neutrino captures with neutrino properties for Case 1.
![(Color online). [*Upper panel*]{}: Time evolution of temperature, density, radius, and mean atomic mass of nuclei heavier than protons, $\langle A_h\rangle=\sum_{Z\ge 2}AY/\sum_{Z\ge 2}Y$, between $\rho_\mathrm{eq}$ and $\rho_{\rm net}$ for trajectory 400720. [*Lower panel*]{}: Analogue for the electron (anti)neutrino, electron and positron capture rates, and for $\lambda^*$ (Eq. \[eq\_ls\]) and the electron fraction. The (anti)neutrino properties correspond to Case 1. The late-time increase of $Y_e$ is caused by the $\alpha$-effect, which does not asymptote to a terminal value until the network is started. $Y_e$ continues to evolve subsequently during the nucleosynthesis because of $\beta$-decays. []{data-label="fig_traj1"}](fig_traj0400720.pdf)
![(Color online). Same as Fig. \[fig\_traj1\], but for trajectory 486857.[]{data-label="fig_traj2"}](fig_traj0486857.pdf)
For trajectory 400720 (Fig. \[fig\_traj1\]), neutrino and antineutrino captures on free nucleons with rates up to some $10^4~{\rm s^{-1}}$ lead to a rapid increase of $Y_e$ from 0.17 up to 0.31, corresponding to the asymptotic value of $Y_{e}^{\nu\infty}$ in Case 1. The (anti)neutrino rates are found to dominate the electron and positron capture rates already at densities slightly below $\rho_\mathrm{eq}$. This is linked to the slow $1/r^2$ decrease of the (anti)neutrino fluxes, which is more shallow than the steep temperature dependence (roughly like $T^6$) of the electron and positron capture rates. At $t > 4$ms, free neutrons and protons in the expanding matter partially recombine into $\alpha$-particles ($\langle A_h \rangle=\sum_{Z\ge 2} AY / \sum_{Z\ge 2} Y\simeq 4$), thus giving rise to the $\alpha$-effect and therefore a further increase of $Y_e$. The rate $\lambda^*$ becomes larger than $\lambda_+\simeq \lambda_{\nu_e} $ and approaches $(\lambda_{{\bar\nu}_e}+\lambda_{{\nu}_e})/2$ of material dominated by $\alpha$-particles.
For trajectory 486857 (Fig. \[fig\_traj2\]), at the initial density $\rho_\mathrm{eq}$, the mass element is characterised by a low temperature $T < 10^{10}$ K and consequently is composed of heavy nuclei $\langle A_h \rangle \simeq 120$ typical of the low values of $Y_e\simeq 0.1$ in the outer NS crust. The fast $e^-$ and (anti)neutrino capture rates lead to a rapid increase of $Y_e$, but all these rates are comparable and $Y_e$ fluctuates wildly. The mass element is then subject to a new compression phase that sets in at $t\simeq 1.5$ ms, and the corresponding high temperatures ($T\gsimeq 10^{11}$ K) photodissociate the matter into free nucleons. It should be noted that during this recompression episode, the higher densities cause neutrinos to become trapped again, for which reason the application of Eqs. (\[eq\_lnu\],\[eq\_lnub\]) remains problematic and highly schematic. However, at these high-density, high-temperature conditions, electron captures dominate and re-neutronize the material until they are counterbalanced by positron captures, whose rate increases dramatically to achieve weak equilibrium with the electron captures. During the subsequent expansion phase (at $t>2$ms), $Y_e$ rises gradually until (anti)neutrino absorptions take over to push $Y_e$ towards its asymptotic value of $Y_{e}^{\nu\infty}\simeq 0.31$ (at $t\sim$6ms). With decreasing temperatures, the $\alpha$-effect finally becomes responsible for a late increase of $Y_e$ at $t \gsimeq 7.5$ms.
$Y_e$ distributions {#sec:yedistributions}
-------------------
Six different cases are studied to estimate the impact of electron and positron captures as well as electron (anti)neutrino absorption on the ejecta mass distribution as function of $Y_e$. In the first case, weak interactions of free nucleons are not allowed. In the second case, electron and positron captures are turned on, but not (anti)neutrino absorptions. In the remaining four cases, both electron/positron captures as well as (anti)neutrino captures are switched on, with the (anti)neutrino properties being defined by Cases 1–4. For these six cases, the resulting $Y_e$ distributions at density $\rho_{\rm net}$ are shown in Fig. \[fig\_histoye\].
In the first case without weak interactions of free nucleons below $\rho_\mathrm{eq}$ (upper left panel of Fig. \[fig\_histoye\]), the $Y_e$ distribution at $\rho_{\rm net}$ is identical to the one given at the initial density $\rho_\mathrm{eq}$. This case, however, differs from the standard case we considered previously [@goriely11; @bauswein13; @goriely13; @just15] in two aspects: First, we start our calculations of weak interactions at density $\rho_\mathrm{eq}=10^{12}$gcm$^{-3}$ with $\beta$-equilibrium distributions of electrons, positrons and neutrinos. This shifts and broadens the $Y_e$ distribution from previous values of $<$0.1 to a wider range between 0 and $\sim$0.3. Second, no temperature post-processing is performed here. The initial temperatures of the trajectories are significantly higher than those deduced from the post-processing applied in our previous studies.
As visible in Fig. \[fig\_histoye\], lower left panel, the $Y_e$ distribution at density $\rho_\mathrm{net}$ is significantly affected by $e^{\pm}$-captures between $\rho_\mathrm{eq}$ and $\rho_\mathrm{net}$, although some low-$Y_e$ ($\lsimeq 0.2$) ejecta are left. Dominant parts of the ejecta are now found at $Y_e$ values between 0.3 and 0.4. When switching on neutrino absorptions, for any of the Cases 1–4 the asymptotic values of $Y_e^{\nu\infty}$ are approached, and further enhancement by the $\alpha$-effect produces peaks of the mass distributions in a range of $Y_e$ values between 0.2 and 0.5. The peak values depend sensitively on the adopted neutrino properties. Cases 1 and 4 lead to rather similar $Y_e$ distributions, owing to the fact that the neutrino properties are broadly comparable.
These distributions also depend sensitively on the temperature. When the temperature along the trajectory is artificially increased or decreased by factors of 3, rather different results are obtained, as shown in Fig. \[fig\_histoye\_T\], using the (anti)neutrino properties of Case 1. As before we start the expansion evolution at $\rho_\mathrm{eq}$ with the $Y_e$ mass distribution shown in the upper left panel of Fig. \[fig\_histoye\]. Reduced temperatures diminish the presence of positrons and thus favor electron captures compared to positron captures. Moreover, lower temperatures also lead to a faster freeze-out of $e^\pm$ captures during the ejection of the mass elements. Without neutrino and antineutrino absorptions, lower temperatures therefore tend to neutronize the ejecta for $\rho<\rho_\mathrm{eq}$ and the mass distribution becomes more narrow and is shifted to lower values in the range of $0<Y_e\lsimeq 0.1$ (Fig. \[fig\_histoye\_T\], upper left panel). Including neutrino and antineutrino absorptions, reduced temperatures have the opposite effect in pushing the mass distribution to higher values of $Y_e$ (with a peak above 0.4) compared to the standard-temperature result for Case 1 in Fig. \[fig\_histoye\] (upper right panel, with a peak of the distribution between $Y_e = 0.2$ and 0.3). This behavior can be understood by the efficient recombination of free nucleons to $\alpha$ particles and heavy nuclei, which strengthens the heavy-nuclei ($\alpha$) effect so that the peak of the distribution wanders to $Y_e\simeq 0.45 $. On the other side, increased temperatures reduce the electron degeneracy and thus allow for the presence of higher positron densities, thus enhancing positron captures on neutrons. In addition, $e^\pm$ captures continue for a longer period of time along the ejecta trajectories. Without (anti)neutrino absorption, these effects shift the $Y_e$ mass distribution from the initial one at $\rho_\mathrm{eq}$ (upper left panel of Fig. \[fig\_histoye\]) towards higher values of $Y_e$. This shift is stronger for more slowly expanding mass elements and weaker when the expansion is very fast. Correspondingly, the mass distribution versus $Y_e$ at $\rho_\mathrm{net}$ is very broad and stretches from $\sim$0.03 up to a very pronounced maximum close to 0.5, because the $e^\pm$ capture equilibrium at high-entropy conditions favors symmetric conditions with respect to neutrons and protons. Taking into account (anti)neutrino absorption prevents this dramatic shift towards $Y_e\sim 0.5$, because at large distances neutrino captures dominate $e^\pm$ absorptions and therefore $Y_e$ asymptotes to values around $Y_e^{\nu\infty}$ ($\simeq 0.31$ for Case 1, lower right panel of Fig. \[fig\_histoye\_T\]). Since the high temperatures favor nucleons and suppress the early formation of $\alpha$ particles and heavier nuclei, the influence of the $\alpha$ effect is clearly weaker than in the case of reduced temperatures (compare lower and upper right panels of Fig. \[fig\_histoye\_T\]). As mentioned above, our calculations with varied temperatures are not consistent with the initial $Y_e$ distributions used at a density of $\rho_\mathrm{eq}$, because these distributions are calculated for the original ejecta temperatures provided by the hydrodynamic NS-NS merger model. However, Fig. \[fig\_histoye\_T\] demonstrates that asymptotic values determined by neutrino capture equilibrium and influenced by the $\alpha$ (heavy-nuclei) effect are reached for most of the trajectories. The final $Y_e$ values can therefore be expected to mostly have lost the memory of the initial conditions at $\rho_\mathrm{eq}$. Correspondingly, when neutrino absorptions are included, the final mass distributions of $Y_e$ (at $\rho_\mathrm{net}$) are considerably more narrow than the relatively broad distribution of initial $Y_e$ values before the expansion from $\rho_\mathrm{eq}$ to $\rho_\mathrm{net}$ (compare the right panels of Fig. \[fig\_histoye\_T\] with the upper left panel of Fig. \[fig\_histoye\]).
![(Color online). Histograms of fractional mass distributions as functions of $Y_e$ at density $\rho_{\rm net}$ for $e^\pm$ captures exclusively at $\rho<\rho_\mathrm{eq}$ (left) and including electron, positron captures as well as (anti)neutrino captures during the evolution from $\rho_\mathrm{eq}$ to $\rho_{\rm net}$ (Case 1; right), when the temperatures are decreased (upper panels) or increased (lower panels) artificially by factors of 3. []{data-label="fig_histoye_T"}](fig_histoye_T.pdf)
r-process nucleosynthesis {#sect_rpro}
=========================
At densities $\rho<\rho_{\rm net}$, the abundance evolution is followed by a full reaction network [for more details, see @goriely11; @bauswein13; @goriely13; @just15], which in contrast to earlier studies includes now also the weak interactions of free nucleons as detailed in Sect. \[sect\_weak\]. The final abundance distributions are shown in Fig. \[fig\_rpro\] for the same six different cases discussed in Sect. \[sec:yedistributions\] and displayed in Fig. \[fig\_histoye\], namely for a case without weak interactions of free nucleons below $\rho_\mathrm{eq}$, a case including $e^{\pm}$-captures but without (anti)neutrino absorption reactions, and four cases where both $e^{\pm}$- and (anti)neutrino captures are taken into account at $\rho<\rho_\mathrm{eq}$ (Cases 1–4 for the $\nu$ properties).
Because of its low-$Y_e$ distribution, the case neglecting $\beta$-interactions gives rise to an elemental abundance distribution similar to the one obtained in our previous studies [@goriely11; @bauswein13; @goriely13; @just15]. It is characterized by the production of essentially only $A>140$ nuclei through several loops of fission recycling. When $e^\pm$-captures on free nucleons are switched on, some low-$Y_e$ ($\lsimeq 0.2$) material can still lead to nucleosynthesis with fission recycling and a significant production of the third r-process peak, but the higher-$Y_e$ (0.3–0.4) matter can now also contribute to the production of $90 \le A \le 140$ nuclei with a strong second $N=82$ peak.
When (anti)neutrino captures are included, too, nucleosynthesis with fission recycling does not take place any longer, but the final abundance distribution still resembles the one in the solar system fairly well. A significant amount of $50 \le A \le 90$ nuclei can now also be produced in addition to the $A>90$ r-nuclei, especially for (anti)neutrino properties corresponding to Case 2, as shown in Fig. \[fig\_rpro\_nu\]. In this case, important element production around $^{60}$Ni is obtained, originating from ejected mass elements with $Y_e \gsimeq 0.45$.
Without $\beta$-interactions below $\rho_\mathrm{eq}$, about 90% of the ejected material is found to be r-process rich. If $e^\pm$-captures on nucleons are switched on, the material is made of 76% r-process nuclei, and when (anti)neutrino captures are effective, too, we find that 45% of the ejected matter is made of r-nuclei in Case 1, only 1.9% in Case 2, 67% in Case 3 and 40% in Case 4. In the last four cases, a significant part of the material is made of $\alpha$-particles ($\sim 22$% in Cases 1 and 4, 50% in Case 2, and 17% in Case 3), and the remaining part consists of $50 \le A \le 70$ nuclei (see Fig. \[fig\_rpro\_nu\]). Despite the rather robust production of a solar-like distribution of r-nuclei, the absolute and relative amounts of r-material vary strongly from case to case, sensitively depending on the neutrino exposure of the ejecta as determined by the assumed properties of the neutrino emission. Neutrino properties corresponding to Case 2 yield a total amount of ejected r-material that is significantly smaller than for the other three sets of neutrino properties.
The total, mass-averaged nuclear energy-release rate that is available for heating the ejecta per unit of mass, the average temperature of the ejecta, and the average atomic mass number of the ejected abundance yields are plotted as functions of time for our six studied cases in Fig. \[fig\_QTA\]. After some 10s and up to nearly one day, the energy release rates of all cases are fairly similar except for the treatment of neutrinos and $\beta$-interactions according to Case 2. In this case, the large electron fractions of the ejecta favor the production of light elements and only a small amount ($\sim 2\%$) of the ejecta material consists of r-process nuclei (Fig. \[fig\_rpro\_nu\]) and is therefore subject to fast $\beta$-decays. At late times, typically after one day, an additional source of energy is found from the $\alpha$-decay of long-lived heavy nuclei that can only be significantly produced when no $\beta$-interaction of nucleons are included below $\rho_\mathrm{eq}$. Despite the considerable differences of the nuclear energy-release rates at early times, the cooling evolution as measured by the mass-averaged temperatures is rather similar in all cases.
The time evolution of the average nuclear mass number of the ejected material clearly shows that only without $\beta$-interactions of free nucleons a significant amount of fissile nuclei can be produced with $\langle A \rangle$ reaching values up to 170, which is slightly below the value of about 200 obtained when the $Y_e$ distribution of the inner crust of a cold NS is considered as initial state [see, in particular, @goriely11]. The sequence of decreasing values of $\langle A \rangle$ follows basically the hierarchy of increasing values of the mean electron fraction, namely $\langle Y_e \rangle=0.11$ when $\beta$-interactions are ignored, $\langle Y_e \rangle=0.27$ when only $e^{\pm}$-captures are taken into account, and $\langle Y_e \rangle=0.25$, 0.34, 0. 35, 0.45 in the Cases 3, 1, 4 and 2, respectively. The only (slight) inversion is obtained for the calculation with only $e^{\pm}$-captures, because in this case the $Y_e$ distribution is very wide (Fig. \[fig\_histoye\], lower left panel) and the significant amounts of low-$Y_e$ ($\sim 0.05 - 0.15$) material contribute to the production of heavy nuclei which increase the average nuclear mass number $\langle A \rangle$.
![(Color online). Mass fraction as a function of the atomic mass for the matter ejected by our 1.35–1.35[$M_{\odot}$]{} NS merger model, including electron, positron as well as (anti)neutrino captures for Cases 1 and 2 of the neutrino properties. []{data-label="fig_rpro_nu"}](fig_rpro_nu.pdf)
![(Color online). Time evolution of the total radioactive heating rate per unit mass, $\langle Q\rangle$ (red), mass number $\langle A\rangle$ (blue), and temperature $\langle T\rangle$ (black), all mass-averaged over the ejecta, for the 1.35–1.35[$M_{\odot}$]{} NS merger. The solid lines correspond to the case without $\beta$-interactions of free nucleons at $\rho<\rho_\mathrm{eq}$, the dotted lines to the case where only $e^\pm$-captures are taken into account, the dashed lines to the case where $e^\pm$ and (anti)neutrino captures for neutrino properties according to Case 1 are included, the dash-dot lines for neutrinos according to Case 2, the long-dash-dot lines correspond to Case 3, and the double-dash-dot lines to Case 4.[]{data-label="fig_QTA"}](fig_QTA.pdf)
Summary and conclusions {#sect_conc}
=======================
In this paper we reported the results of a parametric study to investigate how $\beta$-interactions of free nucleons can affect the $Y_e$ evolution and mass distribution in NS merger ejecta and the corresponding nucleosynthesis. To this end we used the temperature-density trajectories of a large set of mass elements representing the $\sim 5\times 10^{-3}\,M_\odot$ of matter ejected in a relativistic merger simulation of a symmetric 1.35[$M_{\odot}$]{}-1.35[$M_{\odot}$]{} NS binary with the non-zero temperature DD2 nuclear equation of state. Using the total lepton number provided by our NS-NS merger simulation, we assume matter to be in $\beta$ equilibrium at the temperature of the hydrodynamical model and a fiducial density of $\rho_\mathrm{eq}=10^{12}\,$gcm$^{-3}$. These conditions define the starting points of our post-processing of the composition histories of the considered ejecta elements, for which we take into account electron and positron captures as well as electron neutrino and antineutrino absorption on free neutrons and protons. Below a density $\rho_\mathrm{net}$, where the temperature has decreased to $10^{10}$K (or the neutron drip density, if matter below $\rho_\mathrm{eq}$ remains cooler than $10^{10}$K) the full network calculation is applied instead of nuclear statistical equilibrium. Our description of weak interactions of free nucleons includes weak magnetism and recoil corrections according to @horowitz99 and @pllumbi14. Avoiding the complications of treating neutrino transport, we simply use exemplary data from publications for prescribing the neutrino luminosities and mean energies needed to compute the neutrino absorption rates in our parametric approach (“Cases 1–4”). This elementary prescription also accounts for the still large uncertainties of the model predictions for the neutrino emission and its directional asymmetries in the generically three-dimensional merger scenario.
Our modeling strategy follows the spirit of previous, numerous parametric investigations of nucleosynthesis in the neutrino-driven wind of newly formed neutron stars in supernovae [e.g., @mclaughlin96; @meyer98; @arnould07; @pllumbi14 and references therein] and of accretion tori around black holes as remnants of compact object mergers [e.g., @surman05; @surman08; @wanajo12; @caballero12]. In particular, we track in detail the charged-current $\beta$-interactions of neutrinos with free nucleons, which determine the evolution of the electron fraction outside of the neutrino-trapping regime, and employ a full set of trajectories that characterizes the conditions in dynamical ejecta from the merging phase of a representative binary neutron star. These conditions differ from proto-neutron star and accretion-torus winds not only concerning the range of entropies. The ejecta, especially, possess much faster expansion velocities, which for the bulk of the matter can be 25–50% of the speed of light, for some fraction of the ejecta even faster, whereas neutrino-driven proto-neutron star winds have typical velocities of 3–7% of the speed of light [e.g., @arcones07], neutrino-driven winds from massive neutron stars as relics of NS mergers may achieve expansion velocities up to 8–10% of the speed of light [@perego14], and the main mass of neutrino-driven outflows from BH-accretion tori can reach 10–20% of the speed of light [@just15]. The correspondingly shorter expansion time scales of dynamical merger ejecta can enable a strong r-process even for moderately low $Y_e$.
Our calculations confirm recent results of @wanajo14 that solar-like r-process abundances are produced in dynamical NS merger ejecta even when $\beta$-reactions of free nucleons are taken into account and lead to a significant increase of the average electron fraction. In contrast to previous works, where charged-current neutrino-nucleon interactions were ignored, however, also nuclei with mass numbers $A<140$ are ejected in larger amounts.
In detail, our results can be summarized by the following points:
- Ignoring $\beta$-interactions at densities $\rho < \rho_\mathrm{eq}$, we confirm our previous results of @goriely11 [@bauswein13; @goriely13; @just15] that almost exclusively r-nuclei in the regime $A\gsimeq 140$ are produced, in spite of a moderate increase of the average ejecta $Y_e$ associated with the assumption of $\beta$-equilibrium at density $\rho_\mathrm{eq}$ instead of our previous use of electron fractions of cold neutron star crust matter.
- Positron and electron-neutrino captures, enhanced by weak magnetism corrections (which increase the absorption cross section of $\nu_e$ and reduce that of $\bar\nu_e$) and supported by the $\alpha$ effect, lead to a shift of the average ejecta $Y_e$ towards higher values when matter expands downwards from an initial density of $\rho_\mathrm{eq}=10^{12}$gcm$^{-3}$.
- Captures of $e^\pm$ cause a wide spread of the $Y_e$ mass distribution, reaching from values of $Y_e\ll 0.1$ up to 0.4–0.5. This reflects the wide range of thermodynamic conditions of the ejecta at $\rho_\mathrm{eq}$ with ejecta trajectories that describe cool as well as hot conditions. Absorption processes of $\nu_e$ and $\bar\nu_e$ take over when the temperature of the expanding matter has fallen to low values where $e^\pm$ absorptions become ineffective. These (anti)neutrino captures tend to push $Y_e$ towards the asymptotic value $Y_e^{\nu\infty}$ for $\nu_e$-$\bar\nu_e$ capture equilibrium. The corresponding $Y_e$ mass distributions (at $\rho_\mathrm{net}$, where the full network calculation was started) are rather narrow in all cases, with a spread of $Y_e$ values of only 0.1–0.15. The mean values of the distributions, however, vary between $\sim$0.25 and $\sim$0.45, depending on the relative size of the $\nu_e$ and $\bar\nu_e$ luminosities. The latter are highly uncertain and variable and are sensitive to the binary-parameter dependent merger dynamics, the nuclear equation of state, and the still not well determined directional asymmetries of the neutrino emission.
- In the presence of $\beta$-interactions of free nucleons the temperature also plays an important role for the $Y_e$ evolution of the ejecta. It is another aspect of the hydrodynamical merger models that depends on the system properties and the detailed ejection dynamics (which differ between different ejecta components) and can be numerically problematic, because resolution and numerical/artificial viscosity can have an influence on the accuracy of the determination of the thermal conditions. Lower temperatures reduce the effects of $e^\pm$ captures but increase the importance of the $\alpha$ effect (thus pushing the average $Y_e$ closer to 0.5), whereas higher temperatures enhance $e^\pm$ captures but nevertheless have little influence on the final $Y_e$ distribution at $\rho_\mathrm{net}$, for which the asymptotic value of $\nu_e$-$\bar\nu_e$ capture equilibrium is more relevant than the initially fast $e^\pm$ captures.
- In all investigated model cases, the production of heavy r-process matter with strong second and third abundance peaks and a near-solar distribution in the rare-earth region is a robust outcome. However, in contrast to previous results where weak interactions of free nucleons were ignored, also considerable amounts of matter are synthesized to $A<140$ nuclei. The strength of the production of $A\lsimeq 90$–100 material is sensitive to the neutrino-emission properties that determine the (anti)neutrino absorption. In extreme cases where $Y_e$ gets close to 0.5, significant amounts of iron-group nuclei ($A\sim 50$–60) can be ejected. The relative fraction of heavy r-process matter (from the second peak upward) in the ejecta therefore varies dramatically between the different investigated cases of neutrino-emission conditions and spans a range from more than 75% down to just 2%.
We emphasize that our parametric approach is highly simplified and ignores important neutrino-transport effects like the exact spectral distribution of the neutrino fluxes, the direction dependence of the neutrino emission and corresponding precise radial dilution function, and the time evolution of the neutrino emission relative to the ejection time of the matter. Nevertheless, our results, in support of those of @wanajo14 and @seki15, have important consequences for the further exploration of the nucleosynthesis connected to compact binary mergers and the discussion of astrophysical implications. In fact, they demand a major revision of the current picture of r-process production in such events.
In view of our results it is obvious that a proper treatment of the neutrino physics, in particular of the neutrino irradiation of the ejected material, is essential for making quantitative predictions of the elemental yields and especially of the total mass of r-process material that is thrown out by the dynamical merger ejecta. Approximations like the ones used in our study can be satisfactorily removed only when ultimately detailed, three-dimensional neutrino transport is consistently included in the hydrodynamical simulations.
Our study suggests that the relative contributions of matter with $A\lsimeq 90$, $90\lsimeq A\lsimeq 140$ and $A\gsimeq 140$ are likely to depend strongly on the binary properties and even the direction of mass ejection, in addition to the equation of state dependence that was found for the electron-fraction distribution in the recent work of @seki15. Different from expectations so far, this means that symmetric or nearly symmetric binary NS mergers could exhibit a significantly different ejecta composition than highly asymmetric NS-NS mergers and NS-BH mergers, in which the NS is disrupted before it can be swallowed by the BH. In the last two cases the lower-mass component develops an extended tidal tail, from which considerable amounts of cold, unshocked matter can be centrifugally ejected before the neutrino luminosities rise high and thus before neutrino exposure of these ejecta plays an important role. In such a situation the ejecta will not only be expelled highly anisotropically but will also carry a far dominant fraction of the mass in the form of $A\gsimeq 140$ nuclei as predicted in previous studies [e.g., @goriely11; @bauswein13; @goriely13; @just15]. In contrast, in symmetric or nearly symmetric NS mergers the contribution of $A\lsimeq 140$ material will be higher. If the merger remnant collapses to a BH on a millisecond time scale, neutrino exposure of the ejecta may be avoided, but the broad $Y_e$ distribution caused by $e^\pm$ captures will allow for the production of $A>90$ nuclei with a strong second $N=82$ peak. If the merger remnant remains transiently or permanently stable, neutrino exposure of the dynamically expelled matter becomes important, enabling a higher production of $A\lsimeq 90$ species. For extreme cases of a luminous $\nu_e$ flux, the ejecta might then even be dominated by iron-group nuclei including radioactive nickel isotopes. In particular, however, the exact composition and the relative fraction of high-mass and low-mass species could depend on the direction of the mass ejection. If most of the neutrino flux is emitted to the polar directions due to the rotational deformation of the merger remnant, matter expelled near the equatorial plane will receive less neutrino exposure in addition to its potentially faster escape. This will allow for more neutron-rich conditions close to the equator whereas polar ejecta may contain more proton-rich contributions. Future, more complete merger models with neutrino transport will have to clarify these possibilities.
Since the photon opacity, $\kappa$, of the expanding gaseous ejecta is strongly dependent on the presence of high-opacity, complex ions [the lanthanides; @barnes13; @kasen13; @tanaka13; @tanaka14], the relative contribution of trans-iron elements to the ejecta will have a severe impact on the peak luminosity, $L_\mathrm{peak}\propto \kappa^{-1/2}$, peak time, $t_\mathrm{peak}\propto \kappa^{1/2}$, and the effective peak temperature, $T_\mathrm{peak}\propto \kappa^{-3/8}$, of the electromagnetic transient that is expected from the radioactively heated, dynamical ejecta cloud [“macronova” or “kilonova”, @lipaczynski98; @kulkarni05; @metzger10; @roberts11; @goriely11]. The current picture of merger-type and remnant-type dependent redder (near-infrared) or bluer emission [@metzger14; @perego14] and in particular of the envisioned late-time infra-red radiation component from the dynamical ejecta [@kasen14] might require revision or extension in view of our results.
Finally, it is evident that the strong impact of neutrinos on the neutron-to-proton ratio and the nuclear composition of the dynamic merger ejecta must move neutrino oscillations into the focus of interest. It will be necessary to study the effects of collective neutrino oscillations [see, e.g., @duan10 for a review] with similiar intensity as this subject currently receives in the context of the neutrino emission from newly born neutron stars in supernovae.
Acknowledgments {#acknowledgments .unnumbered}
===============
SG acknowledges financial support from FNRS (Belgium). At Garching, this research was supported by the Max-Planck/Princeton Center for Plasma Physics (MPPC) and by the Deutsche Forschungsgemeinschaft through the Cluster of Excellence EXC 153 “Origin and Structure of the Universe” (http://www.universe-cluster.de). AB is a Marie Curie Intra-European Fellow within the 7th European Community Framework Programme (IEF 331873). We are also grateful for computing time at the Rechenzentrum Garching (RZG).
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\[lastpage\]
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{
"pile_set_name": "ArXiv"
}
|
---
author:
- |
[Ganguk Lee]{}\
KAIST\
[email protected]
- |
[Yeaseul Park]{}\
KAIST\
[email protected]
- |
[Jeongseob Ahn]{}\
Ajou University\
[email protected]
- |
[Youngjin Kwon]{}\
KAIST\
[email protected]
bibliography:
- 'biblio.bib'
title: '**Slicing the IO execution with [ReLayTracer]{}**'
---
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- |
*Benjamin Bahr, MPI für Gravitationsphysik, Albert-Einstein Institut,\
*Am Mühlenberg 1, 14467 Golm, Germany\
*Thomas Thiemann, MPI für Gravitationsphysik, Albert-Einstein Institut,\
*Am Mühlenberg 1, 14467 Golm, Germany;\
*Perimeter Institute for Theoretical Physics,\
*31 Caroline St. N., Waterloo Ontario N2L 2Y5, Canada******
title: 'Gauge-invariant coherent states for Loop Quantum Gravity I: Abelian gauge groups\'
---
Introduction
============
Loop Quantum Gravity (LQG) is a promising candidate for a theory that aims to combine the principles of quantum mechanics and general relativity (see [@INTRO; @ROVELLISBUCH; @INTRO3; @ALLMT] and references therein). The starting point of LQG is the Hamiltonian formulation of general relativity, choosing Ashtekar-variables as phase-space coordinates, which casts GR into a $SU(2)$ gauge theory, leading to the Poisson structure
$$\begin{aligned}
\big\{A_a^I(x)\,,\,A_b^J(y)\big\}\;&=&\;\big\{E_I^a(x)\,,\,E_J^b(y)\big\}\;=\;0\\[5pt]
\big\{A_a^I(x)\,,\,E_J^b(y)\big\}\;&=&\;8\pi
G\b\;\d_{b}^a\,\d_J^I\;\d(x-y).\end{aligned}$$
This system could be canonically quantized with the help of methods well-known from algebraic quantum field theory, which resulted in a representation of the Poisson-algebra on a Hilbert space $\H_{kin}$, which carries the kinematical information of quantum general relativity. One has found recently [@LOST] that this representation is unique up to unitary equivalence if one demands the space-diffeomorphisms to be unitarily implemented.
While the dynamics of classical general relativity is encoded into a set of phase-space functions $G_I,\,D_a,\,H$ that are constrained to vanish, these so-called constraints are, in LQG, promoted to operators that generate gauge-transformations on the kinematical Hilbert space $\H_{kin}$. The physical Hilbert space $\H_{phys}$ is then to be derived as the set of (generalized) vectors being invariant under these gauge-transformations [@HENN-TEITEL].
$$\begin{aligned}
\label{Gl:QuantumConstraints}
\hat G_I|\psi\rangle\;=\;\hat D_a|\psi\rangle\;=\;\hat
H|\psi\rangle\;=\;0.\end{aligned}$$
Although conceptually clear, the actual computation of $\H_{phys}$ is technically quite difficult. This is due to the fact that the constraints $\hat G_I,\,\hat D_a\,\hat H$ act quite non-trivially on $\H_{kin}$. Thus, while the kinematical setting is understood, the physical states of the theory are not known explicitly. It seems that, in its present formulation, LQG is too complicated to be solved analytically.\
While this seems to be discouraging at first, complete solvability is not something one could have expected from the outset. In fact, nearly no theory which realistically describes a part of nature is completely solvable, neither in the quantum, nor in the classical regime. Rather, having the basic equations of a theory as a starting point, one has to develop tools for extracting knowledge about its properties in special cases, reducing the theory to simpler subsectors, approximate some solutions of the theory, or study its behavior via numerical methods. Examples for this range from reducing classical GR to symmetry-reduced situations, which is our main source of understanding the large-scale structure of our cosmos, over particle physics, where perturbative quantum field theory is our access to predict the behavior of elementary particles, to numerical simulations in ordinary quantum mechanics, which allow for computations of atomic and molecular spectra, transition amplitudes or band structures in solid state physics. Although in all of these fields the fundamental equations are well-known, their complete solution is elusive, so one has to rely on approximations and numerics in order to understand the physical processes described by them. In other cases, such as interacting Wightman fields on 4D Minkowski space, not a single example is known to date. On the other hand, the perturbation theory for, say, $SU(N)$-Yang-Mills theory in small couplings is so effective that many particle physicists even regard the perturbative expansion in the coupling parameter as the fundamental theory in itself.\
With these considerations, it seems quite natural to look for a way to gain knowledge about the physical content of LQG by approximation methods. One step into this direction has been done by introducing the complexifier coherent states.
For ordinary quantum mechanics, the well-known harmonic oscillator coherent states (HOCS) $$\begin{aligned}
|z\rangle\;=\;\sum_{n=0}^{\infty}\,\frac{z^n}{\sqrt{n!}}\;|n\rangle\end{aligned}$$
are a major tool for performing analytical calculations and numerical computations. Not only can they be used to approximate quantum propagators [@KECK], they are also the main tool for investigating the transition from quantum to classical behaviour, as well as quantum chaos [@KORSCHCHAOS1; @KORSCHCHAOS2]. They also grant access to the numerical treatment of quantum dynamics for various systems [@KLAUDER; @VAN-VLECK], and their generalization to quantum electrodynamics provides a path to the accurate description of laser light and quantum optics [@GLAUBER].
The complexifier coherent states (CCS), which have been first introduced in [@HALL1; @HALL3], are a natural generalization of the HOCS to quantum mechanics on cotangent bundles over arbitrary compact Lie groups, and the complexifier methods employed to construct these states can also be transferred to other manifolds as well. Furthermore, for the special cases of quantum mechanics on the real line $\R$ and the circle $U(1)$, these states reduce to what has been used as coherent states for quite some time [@KASTRUP; @KRP].
In [@CCS], the complexifier concept has been used to define complexifier coherent states for LQG. They are states on the kinematical Hilbert space $\H_{kin}$ and their properties have been exhibited in [@GCS1; @GCS2]. It was shown that they mimic the HOCS in their semiclassical behavior, in the sense that they describe the quantum system to be close to some point in the corresponding classical phase-space of general relativity, minimizing relative fluctuations. Also, they provide a Bargman-Segal representation of $\H_{kin}$ as holomorphic functions, as well as approximating well quantum observables that correspond to classical phase space variables.
This has indicated that these states are a useful tool for examining the semiclassical limit of LQG. In particular, it has been shown [@TINA1] with the help of the CCS that the constraint operators for LQG, which are defined on $\H_{kin}$ and generate the dynamics of the theory, have the correct classical limit. In particular, CCS that are “concentrated” around a classical solution of GR, are annihilated by the constraint operators up to orders of $\hbar$. This indicates that, at least infinitesimally, LQG has classical GR as semiclassical limit.
On the other hand, since the complexifier coherent states are only defined on $\H_{kin}$, none of them is really physical in the sense of the Dirac quantization programme. That is, while they are peaked on the classical constraint surface, they are not annihilated by the constraint operators, only approximately. Thus, while being a good tool for examining kinematical properties of LQG, it is not clear how well they approximate the dynamical aspects of quantum general relativity.
To do this, it would be desirable to have coherent states at hand that satisfy at least some of (\[Gl:QuantumConstraints\]). We will pursue the first step on this path in this and the following article.\
Some of the constraints (\[Gl:QuantumConstraints\]) are simpler than others. In particular, the easiest ones are the Gauss constraints $\hat G_I$. They are unbounded self-adjoint operators on $\H_{kin}$ and the gauge-transformations generated by them are well understood. The set of vectors being invariant under the Gauss-gauge-transformations (“gauge-invariant” in the following) is a proper subspace of $\H_{kin}$. This space is well known [@SNF], and a basis for it is provided by the gauge-invariant spin network functions, the construction of which involve intertwiners of the corresponding gauge group $SU(2)$. Thus, the straightforward way to construct gauge-invariant coherent states would be to project the CCS to the gauge-invariant Hilbert space. We will do exactly that in this and the following article.
The gauge transformations correspond to gauging the $\mathfrak{su}(2)$-valued Ashtekar connection $A_a^I$ and its canonically conjugate, the electric flux $E_I^a$. Thus, the gauge group $SU(2)$ is involved, and in fact this group plays a prominent role in the construction of the whole kinematical Hilbert space $\H_{kin}$. It is, however, possible to replace $SU(2)$ in this construction by any compact gauge group $G$, arriving at a different kinematical Hilbert space $\H_{kin}^{G}$, which would be the arena for the Hamiltonian formulation of a gauge field theory with gauge group $G$. Of course, one also has to replace the $\hat G_I$ by the corresponding gauge generators. Also the constraints $\hat D_a$ and $\hat H$ can, although nontrivial, be modified to match the new gauge group. Finally, the complexifier method is able to supply corresponding coherent states for each gauge group $G$.
This change of $SU(2)$ into another gauge group has been used frequently. In [@VARA] it has been shown that the quantization of linearized gravity leads to the LQG framework with $U(1)^3$ as gauge group. Furthermore, it has been pointed out [@QFTCST] that changing $SU(2)$ for $U(1)^3$ does not change the qualitative behavior of the theory in the semiclassical limit, and so the $U(1)^3$-CCS have been used widely in order to investigate LQG [@TINA1].\
Before treating the much more complicated case of $G=SU(2)$ in [@GICS-II], in this paper we will, as a warm-up, consider the gauge group $G=U(1)$ and the corresponding CCS. The case $G=U(1)^3$ is then simply obtained by a triple tensor product: Not only the kinematical Hilbert space $$\begin{aligned}
\label{Gl:ThreeTensorProducts}
\H_{kin}^{U(1)^3}\;=\;\H_{kin}^{U(1)}\;\otimes\;\H_{kin}^{U(1)}\;\otimes\;\H_{kin}^{U(1)}\end{aligned}$$
has this simple product structure, but also the respective gauge-invariant subspaces decompose according to (\[Gl:ThreeTensorProducts\]). Also, $U(1)^3$-CCS are obtained by tensoring three $U(1)$-CCS. Due to this simple structure, it is sufficient for our arguments to consider the gauge-invariant coherent states in the case of $G=U(1)$, since all the properties revealed in this article can be carried over straightforwardly to gauge-invariant coherent states for $G=U(1)^3$.\
The plan for this paper is as follows: In chapter \[Ch:KinematicalFramework\], we will shortly repeat the basics of LQG. In particular, the kinematical Hilbert space $\H_{kin}$ for arbitrary gauge group $G$ is defined, the corresponding set of constraints that generate the gauge-transformations are described. In chapter \[Ch:TheCCS\], the complexifier coherent states are defined, where the focus lies on the particular case of $G=U(1)$. A formula for the inner product between two such states is derived, which depends purely on the geometry of the complexification of the gauge group $U(1)^{\C}\simeq\C\backslash\{0\}$. Although this is not of particular importance in this article, we will find a similar formula in [@GICS-II], when we come to the case of $G=SU(2)$. This will hint towards a geometric interpretation of the CCS for arbitrary gauge groups, and we will comment shortly on this at the end of [@GICS-II].
In chapter \[Ch:GICS\] we will apply the projector onto the gauge-invariant subspace of $\H_{kin}$ to the $U(1)$-complexifier coherent states. The involved gauge integrals can be carried out by a special procedure resembling a gauge-fixing. The resulting gauge-invariant states are then investigated, and their properties are displayed. In particular, we will show that they describe semiclassical states peaked at gauge-invariant degrees of freedom.
We will conclude this article with a summary and an outlook to the sequel paper.
The kinematical setting of LQG {#Ch:KinematicalFramework}
==============================
In this section, we will shortly repeat the kinematical framework of LQG.
Loop Quantum Gravity is a quantization of a Hamiltonian formulation of classical GR. This is done by introducing an ADM split of space-time and the introduction of Ashtekar variables [@INTRO]. Thus, GR can be formulated as a constrained SU(2)-gauge theory on a tree-dimensional manifold $\Sig$, which is regarded as space, and is taken to be compact. The quantization for noncompact $\Sig$ can also be carried out, but this requires some more mathematical effort.
On $\Sig$ the Ashtekar $\mathfrak{su}(2)$-connection $A_a^I$ and the electric flux $E_I^a$ are the dynamical variables. They are canonically conjugate to each other:
$$\begin{aligned}
\big\{A_a^I(x)\,,\,A_b^J(y)\big\}\;&=&\;\big\{E_I^a(x)\,,\,E_J^b(y)\big\}\;=\;0\\[5pt]
\big\{A_a^I(x)\,,\,E_J^b(y)\big\}\;&=&\;8\pi
G\b\;\d_{b}^a\,\d_J^I\;\d(x-y).\end{aligned}$$
The fields are not free, but subject to so-called constraints, which are phase-space functions, i.e. functions of $A$ and $E$. They encode the diffeomorphism-invariance of the theory, and the Einstein equations. The reduced phase space consists of all phase space points $A,\,E$ where the constraints vanish. On this set, the constraints act as gauge transformations, and the set of gauge orbits is the physical phase space. The set of constraints is divided into the Gauss constraints $G_I(x)$, the diffeomorphism constraints $D_a(x)$ and the Hamilton constraints $H(x)$. These satisfy the Poisson algebra
$$\begin{aligned}
\nonumber
\Big\{G(s),\,G(t)\Big\}\;&=&\;G(s\wedge t)\\[5pt]\nonumber
\Big\{G(s),\,D(f)\Big\}\;&=&\;\Big\{G(s),\,H(g)\Big\}\;=\;0\\[5pt]\label{Gl:TheConstraints}
\Big\{D(f),\,D(g)\Big\}\;&=&\;D(\mathcal{L}_fg)\\[5pt]\nonumber
\Big\{D(f),\,H(n)\Big\}\;&=&\;H(\mathcal{L}_fn)\\[5pt]\nonumber
\Big\{H(n),\,H(m)\Big\}\;&=&\;D(g^{ab}(n\,m,_b-m\,n,_b))\end{aligned}$$
where $s, t$ are $\mathfrak{su}(2)$-valued functions, $f,g$ are vector fields on $\Sig$, $n,m$ are scalar functions on $\Sig$, the smeared constraints are defined by
$$\begin{aligned}
G(s)\;:=\;\int_{\Sig}G_I(x) s^I(x),\qquad
D(f)\;:=\;\int_{\Sig}D_a(x)\,f^a(x),\qquad
H(n)\;:=\;\int_{\Sig}H(x)\,n(x),\end{aligned}$$
$d$ denotes the exterior derivative on $\Sig$, $\mathcal{L}$ the Lie derivative, and $\flat$ is the isomorphism from one-forms to vector fields provided by the metric. It is this particular occurrence of the metric itself in the Poisson brackets, which makes the algebra structure notoriously difficult.\
The kinematical Hilbert space
-----------------------------
The kinematical Hilbert space $\H_{kin}$ of LQG is computed as a directed limit of Hilbert spaces of functions being cylindrical over a particular graph embedded in $\Sig$. Consider $\g$ to be a graph, consisting of finitely many oriented edges $e_1,\ldots,e_E$ being embedded analytically in $\Sig$, such that the intersection of two edges is either empty or a common endpoint, or vertex $v$. For each such graph $\g$ there is a Hilbert space $H_{\g}$, which consists of all functions being cylindrical over that particular $\g$. In particular, each edge $E$ of the graph defines a function from the set of all connections $$\begin{aligned}
h_e:\;\mathcal{A}\;\longrightarrow\; SU(2)\end{aligned}$$
by setting $h_e(A)$ being the holonomy of the connection $A$ along the edge $e$. Symbolically,
$$\begin{aligned}
h_e(A)\;=\;\P\exp i\int_0^1 dt\;A_a^I(e(t))\frac{\t_I}{2}\,\dot
e^a(t).\end{aligned}$$
A function $f:\mathcal{A}\to\C$ is cylindrical over the graph $\g$, having $E$ edges $e_1,\ldots e_E$ if there is a function $\tilde f:SU(2)^E\to\C$ with
$$\begin{aligned}
\label{Gl:CorrespondingFunction}
f(A)\;=\;\tilde f\Big(h_{e_1}(A),\,\ldots,\,h_{e_E}(A)\Big).\end{aligned}$$
The integration measure in this Hilbert space is just the Haar measure on $SU(2)^E$, which gives the canonical isomorphism
$$\begin{aligned}
\label{Gl:IsomorphismBetweenHGammaAndLTwoOverSU2}
H_{\g}\;\simeq\;L^2\Big(SU(2)^E,\,d\m_H^{\otimes E}\Big).\end{aligned}$$
The set of graphs is a partially ordered set. Let $\g,\,\g'$ be two graphs, then one writes $\g\preceq\g'$, iff there is a subdivision $\g''$ of $\g'$ by inserting additional vertices into the edges, such that $\g$ is a subgraph of $\g''$. Note that, since all graphs consist of analytically embedded edges, this indeed defines a partially ordering, i.e. for any two graphs $\g_1,\g_2$ there is always a $\g_3$ such that $\g_1\preceq\g_3$ and $\g_2\preceq\g_3$.
Each function $f_{\g}$ cylindrical over $\g$ determines a cylindrical function $f_{\g''}$ over $\g''$, simply by defining
$$\begin{aligned}
\tilde f_{\g''}(h_{e_1}(A),\ldots,h_{e_E'}(A))\;:=\;\tilde
f_{\g}(h_{e_{n_1}}(A),\ldots,\,h_{e_{n_E}}(A))\end{aligned}$$
where $e_{n_1},\ldots,\,e_{n_E}$ are the edges in $\g''$ belonging to $\g$. Now, every function cylindrical over $\g''$ is also obviously cylindrical over $\g'$, since $\g''$ is only a refinement of $\g'$. This procedure defines a unitary map
$$\begin{aligned}
U_{\g\g'}\;:\;\H_{\g}\;\longrightarrow\;\H_{\g'}.\end{aligned}$$
One can show that for $\g\preceq\g'\preceq\g''$, one has $U_{\g'\g''}U_{\g\g'}\;=\;U_{\g\g''}$. So, this family of unitary maps defines a projective limit $$\begin{aligned}
\label{Gl:DirectedLimit}
\H_{kin}\;:=\;\lim_{\longrightarrow}\;\H_{\g},\end{aligned}$$
which serves as the kinematical Hilbert space of LQG. Each $\H_{\g}$ has a canonical isometric embedding $U_{\g}$ into $\H_{kin}$, which is compatible with the unitary maps $U_{\g\g'}$ in the following way:
$$\begin{aligned}
U_{\g\g'}\,U_{\g'}\;=\;U_{\g}\qquad\mbox{for all }\g\preceq\g'.\end{aligned}$$
Due to the definition of the inner product in the projective limit, for $\psi_{\g}\in\H_{\g}$ and $\psi_{\g'}\in\H_{\g'}$, where the intersection of $\g$ and $\g'$ is empty, one has that $$\begin{aligned}
\Big\langle U_{\g}\psi_{\g}\Big|U_{\g'}\psi_{\g'}\Big\rangle\;=\;0.\end{aligned}$$
This immediately shows that, since there are uncountably many graphs with mutual empty intersection in $\Sig$, $\H_{kin}$ cannot be separable. On the other hand, since $\H_{kin}$ is built up out of the $\H_{\g}$, we can restrict our considerations to an arbitrary but fixed graph $\g$ for most purposes, dealing only with the Hilbert space $\H_{\g}$, which is separable.\
Note that the whole construction carried out here can be done with an arbitrary compact Lie group $G$. The field $A$ is then a connection on a $\mathfrak{g}$-bundle and $E$ the corresponding electric flux, which is canonically conjugate. Also the definition of the constraints can be adapted to build a theory for arbitrary gauge groups. This is not only a mathematical toy, but in some situations, it is in fact useful to replace the gauge group $SU(2)$ by $U(1)^3$, which can be physically justified [@GCS2; @VARA; @QFTCST]. In particular, we will deal in this article with the complexifier- and gauge-invariant coherent states for the case of $G=U(1)$, which will serve as a warm-up example before coming to the much more difficult (but also more realistic) case of $G=SU(2)$ in [@GICS-II].
Constraint operators and gauge actions
--------------------------------------
In the previous section the kinematical framework for LQG was presented. In this section, we will shortly discuss the constraint operators and the gauge actions they induce on $\H_{kin}$.
Rewriting general relativity in a Hamiltonian formulation using the Ashtekar variables results in the formulation of the Ashtekar connection $A_a^I(x)$ and the electric flux $E^a_I(x)$, which, in the quantized theory, become operators on $\H_{kin}$. One cannot quantize the fields directly, but has to smear them with certain test functions having support on one-dimensional and two-dimensional submanifolds of $\Sig$, respectively. See [@INTRO] for details.
In the classical theory, the dynamics is encoded in the constraints (\[Gl:TheConstraints\]), which in the quantum theory become operators acting on $\H_{kin}$. The physical Hilbert space is determined by the condition that (generalized) states are annihilated by the constraint operators
$$\begin{aligned}
\label{Gl:DefinitionOfAPhysicalState}
\hat D_a\;\psi_{phys}\;=\;\hat G_I\;\psi_{phys}\;=\;\hat
H\;\psi_{phys}\;=\;0.\end{aligned}$$
To implement the Gauss constraints as operators on $\Sig$ is, actually, quite straightforward. Since the kinematical Hilbert space $\H_{kin}$ can be thought of as being built up from $\H_{\g}$ for arbitrary graphs $\g\subset\Sig$ by (\[Gl:DirectedLimit\]), it is sufficient to compute the gauge-transformation generated by the $\hat G_I$.
In particular, the similarity between LQG and a lattice gauge theory on $\g$ is displayed, if one computes the unitary group generated by the constraints $\hat G_I(x)$, which correspond to $SU(2)$-gauge transformations of functions on the graph. In particular, let $k:\Sig\to SU(2)$ be a function and $f$ a cylindrical function over a graph $\g$ with $E$ edges. The action of $k$ on $f$ is given by the induced action of $k$ on the corresponding $\tilde f:SU(2)^E\to \C$ via (\[Gl:CorrespondingFunction\]), to be
$$\begin{aligned}
\label{Gl:ActionOfGaugeGroup}
\a_{k} \tilde f\;\big(h_{e_1},\ldots,h_{e_E})\;:=\;\tilde
f\;\big(k_{b(e_1)}h_{e_1}k_{f(e_1)}^{-1},\ldots,k_{b(e_E)}h_{e_E}k_{f(e_E)}^{-1}\big),\end{aligned}$$
where $b(e_m)$ and $f(e_m)$ are the beginning- and end-point of the edge $e_m$, and $k_x\in SU(2)$ is the value of the map $k$ at $x\in\Sig$. So, the gauge transformations act only at the vertices of a graph.
In particular, one can write down the projector onto the gauge-invariant Hilbert space for functions in $\H_{\g}$:
$$\begin{aligned}
\label{Gl:Projector}
\P
f(h_{e_1},\ldots,h_{e_E})\;&:=&\;\int_{SU(2)^V}d\m_H(k_1,\ldots,k_V)\a_{k_1,\ldots
k_V}\,f(h_{e_1}\ldots,h_{e_E})\\[5pt]\nonumber
&=&\;\int_{SU(2)^V}d\m_H(k_1,\ldots,k_V)f\Big(k_{b(e_1)}h_{e_1}k_{f(e_1)}^{-1},\ldots,k_{b(e_E)}h_{e_E}k_{f(e_E)}^{-1}\Big)\end{aligned}$$
Since there are only finitely many vertices on the graph $\g$, the integral exists and defines a projector $$\begin{aligned}
\P:\;\H_{\g}\;\longrightarrow\;\H_{\g}\end{aligned}$$
onto a sub-Hilbert space of $\H_{\g}$. In particular, the gauge-invariant functions on a graph form a subset of all cylindrical functions on a graph. The gauge-invariant Hilbert spaces can be described using intertwiners between irreducible representations of $SU(2)$, and a basis for the gauge-invariant Hilbert spaces $\P\H_{\g}$ can be written down in terms of gauge-invariant spin network functions [@SNF].\
The diffeomorphism constraints $\hat D$ can, however, not be implemented as operators on $\H_{kin}$ in a straightforward manner. On the classical side, it can be shown that the constraint $D(f)$ is the infinitesimal generator of the one-parameter family of diffeomorphisms defined by the vector field $f$. In particular, a physical state is one that is invariant under diffeomorphisms, which simply reflects the invariance of GR under passive (spatial) diffeomorphisms.
On the quantum side, however, it is straightforward to implement the action of piecewise analytic diffeomorphisms on $\H_{kin}$: Remember that one can think of $\H_{kin}$ as consisting of functions $f:\mathcal{A}\to\C$, which are cylindrical over some graph $\g$. The space of quantum configurations $\mathcal{A}$, i.e. the space of (distributional) connections on $\Sig$ carries a natural action of the diffeomorphism group $\Diff\Sig$. An element $\phi\in\Diff\Sig$ simply acts by $A\to\phi^*A$ on a (distributional) connection $A$. With this, one can simply define the action of $\Diff\Sig$ on $\H_{kin}$ by
$$\begin{aligned}
\a_{\phi}f(A)\;:=\;f(\phi^*A),\end{aligned}$$
where $\phi^*A$ is the pullback of the connection $A$ under the diffeomorphism $\phi$. Note that this definition maps
$$\begin{aligned}
\label{Gl:ActionOfDiffeomorphism}
\a_{\phi}\;\H_{\g}\;\longrightarrow\;\H_{\phi(\g)}.\end{aligned}$$
Here $\phi(\g)$ is the image of $\g$ under $\phi$. This shows that one cannot take arbitrary smooth $\phi$, but has to restrict to analytic diffeomorphisms, since these map a graph consisting of analytic edges into one consisting again of analytic edges.
Note that the action (\[Gl:ActionOfDiffeomorphism\]) is not weakly continuous in $\phi$, since two graphs can be arbitrary “close” to each other, but still not intersecting, which means that their corresponding Hilbert spaces are mutually orthogonal subspaces of $\H_{kin}$. This fits nicely into the picture, since the notion of “being close to each other” only has a meaning on manifolds with metric, and LQG is a quantum theory on a topological manifold only, since the metric itself is a dynamical object, and not something given from the outset.\
The Hamiltonian constraints $H(n)$ could in fact be promoted to operators $\hat H(n)$ on $\H_{kin}$ [@QSD1]. But, the solution of this constraint, i.e. determining the set of (generalized) vectors satisfying $\hat H(n)\psi_{phys}=0$ is still elusive. Also, since these operators exhibit a highly nontrivial bracket structure, it is not clear whether they resemble their classical counterpart (\[Gl:TheConstraints\]). Moreover, these operators cannot be defined on the diffeomorphism-invariant Hilbert space $\H_{\text{diff}}$. To remedy these issues, a modification to the algebra (\[Gl:TheConstraints\]) has been proposed, the so-called master constraint programme. By replacing all $\hat H(n)$ by one operator $\hat M$, one can solve the above issues [@PHOENIX; @QSD8]. Still, the solution of this constraint is quite nontrivial, although some steps into this direction have been undertaken [@TINA1].\
Complexifier coherent states {#Ch:TheCCS}
============================
An important question in LQG is whether the theory contains classical GR in some sort of semiclassical limit [@INTRO; @CCS; @TINA1]. The transition from quantum to classical behavior in the case of, say, a quantum mechanical particle moving in one dimension can be seen best with the help of the harmonic oscillator coherent states (HOSZ) $$\begin{aligned}
|z\rangle\;=\;\sum_{n=0}^{\infty}\,\frac{z^n}{\sqrt{n!}}\;|n\rangle.\end{aligned}$$
They can be seen as minimal uncertainty states, or states that correspond to the system of being in a quantum state close to a classical phase space point. With these states, one can not only investigate the transition from quantum to classical behavior of a system, but one can also try to say something about the dynamics of the quantum system by considering solutions to the classical equations of motion.
This has led people to consider, whether states with equally pleasant properties also exist for LQG. In [@CCS], states in $\H_{kin}$ have been proposed that have been constructed by the so-called complexifier method, first brought up in [@HALL1; @HALL3]. They have been investigated in [@GCS1; @GCS2], and the properties of these states seem to make them ideally suited for the semiclassical analysis of the kinematical sector of LQG [@TINA1].
The complexifier coherent states are defined for each graph $\g\subset\Sig$ separately, and each of these Hilbert spaces is, by (\[Gl:IsomorphismBetweenHGammaAndLTwoOverSU2\]), a tensor product of $L^2(SU(2), d\m_H)$-spaces. Also the complexifier coherent states on $\H_{\g}$ are defined as a tensor product of complexifier coherent states on $L^2(SU(2), d\m_H)$. In fact, the complexifier procedure is quite general and works for every compact Lie group $G$, and is able to define a state on $L^2(G, d\m_H)$. This comes in handy, since Yang-Mills field theory coupled to gravity can be treated at the kinematical level, simply by replacing $SU(2)$ by a compact gauge group $G$ in the whole construction. There are in fact arguments that, in the semiclassical limit, the qualitative behavior of calculations in LQG will not change if one replaces $SU(2)$ by $U(1)^3$. This replacement has been used widely during the investigation of the semiclassical limit of LQG [@TINA1]. The fact that $U(1)^3$ is abelian is a tremendous simplification to the calculations.
Thus, in the following we will give the definition of the complexifier coherent states for arbitrary gauge groups, where the cases of $G=U(1),\, U(1)^3$ and $SU(2)$ are of ultimate interest for the geometry degrees of freedom of LQG.
General gauge groups
--------------------
Consider quantum mechanics on a compact Lie group $G$, which is associated to the Hilbert space $L^2(G,\,d\m_H)$, where $d\m_H$ is the normalized Haar measure on $G$. The classical configuration space is $G$, and the corresponding phase space is
$$\begin{aligned}
T^*G\;\simeq\;G\times\R^{\dim G}\;\simeq\;G^{\C}.\end{aligned}$$
Here, $G^{\C}$ is the complexification of $G$, generated by the complexification of the Lie algebra of $G$, $\mathfrak{g}\otimes\C$. The complexifier coherent states are then defined by $$\begin{aligned}
\label{Gl:DefinitionOfComplexifierCoherntStates}
\psi^t_g(h)\;:=\;\left(e^{\Delta\frac{t}{2}}\;\d_{h'}(h)\right)_{\Big|_{h'\to
g}}.\end{aligned}$$
The $\d_{h'}(h)$ is the delta distribution on $G$ with respect to $d\m_H$, centered around $h'\in G$, $\Delta$ is the Laplacian operator and $h'\to z$ is the analytic continuation from $h'\in G$ to $g\in G^{\C}$. The fact that the spectrum of $\Delta$ grows quadratically for large eigenvalues makes sure that the expression in the brackets is in fact a smooth function on $G$, thus ensuring that $\psi^t_g\in L^2(G,\,d\m_H)$.
These states are named complexifier coherent states, since, instead of $-\Delta$, one could have taken any quantization of a phase space function $C$ (with spectrum bounded from below and spectrum growing at least as $\l^{1+\epsilon}$, in order for the above expression to make sense). The function $C$ is called a complexifier, since it provides an explicit diffeomorphism between $T^*(G)\simeq G^{\C}$, such that the element $g\in G^{\C}$ actually carries a physical interpretation as a point in phase space. This diffeomorphism is, for the complexifier $\hat C=-\Delta$, given by $$\begin{aligned}
T^*G\;\simeq\;G\times\R^{\dim G}\;\ni\;(h,\vec
p)\;\longmapsto\;\exp\left(-i\frac{\t_I}{2}p^I\right)h\;\in\;G^{\C}\end{aligned}$$
which is the inverse of the polar decomposition of elements in $G^{\C}$, while the $\t_I$ are basis elements of $\mathfrak{g}$. A priori, which complexifier $\hat C$ one chooses is not fixed. In the context of LQG, one can, given a graph $\g$, choose a classical function $C$ adapted to this graph, such that its quantization $\hat C$ is - restricted to $\H_{\g}$ - just the Laplacian $-\Delta$ on each edge. See [@CE] for details and a discussion of this operator.\
From (\[Gl:DefinitionOfComplexifierCoherntStates\]) one can deduce a more tractable form of the complexifier coherent states given by
$$\begin{aligned}
\label{Gl:DefinitionOfComplexifierCoherntStates-2}
\psi^t_g(h)\;=\;\sum_{\pi}e^{-\l_{\pi}}d_{\pi}\,\tr\;\pi(g h^{-1})\end{aligned}$$
where the sum runs over all irreducible finite-dimensional representations $\pi$ of $G$. In the specific case of $G=U(1)$ and $G=SU(2)$, the states (\[Gl:DefinitionOfComplexifierCoherntStates-2\]) have been investigated [@CCS; @GCS1; @GCS2], and their properties are known quite well. In particular, they approximate the quantum operators up to small fluctuations, the width of which is proportional $\sqrt t$, which identifies $t$ as the parameter measuring the semiclassicality scale. For kinematical states in LQG being close to some smooth space-time, at the scale of say the LHC $t$ is of the order of $l^2_p/(10^{-18}\text{
cm})^2$, i.e. about $10^{-30}$!\
The states (\[Gl:DefinitionOfComplexifierCoherntStates-2\]) are complexifier coherent states for quantum mechanics on $G$. Technically, this is equivalent to a graph consisting of one edge. For graphs $\g$ being built of many edges $e_1,\ldots e_E$, one can, since $L^2(G,d\m_H)^{\otimes E}\;=\;L^2(G^E,d\m_H^{\otimes E})$, simply construct a state by taking the tensor product over all edges:
$$\begin{aligned}
\label{Gl:TensorProductOfCoherentStates}
\psi^t_{g_1,\ldots, g_E}(h_1,\ldots,
h_E)\;=\;\prod_{m=1}^E\,\psi^t_{g_m}(h_m).\end{aligned}$$
Note that this tensor product contains no information about which edges are connected to each other and which are not.\
The complexifier coherent states on a graph are labeled by elements $g_m\in G^{\C}$. In particular, for the cases of interest for LQG, these spaces are $$\begin{aligned}
U(1)^{\C}\;&\simeq&\;\C\backslash\{0\}\\[5pt]
SU(2)^{\C}\;&\simeq&\;SL(2,\C).\end{aligned}$$
As already stated, the complexified groups $G^{\C}$ are diffeomorphic to the tangent bundle of the groups $T^*G$ themselves. So, the complexifier coherent states are labeled by elements of the classical phase space. A state labeled by $g_1,\ldots, g_E$ corresponds to a state being close to the classical phase space point corresponding to $g_1,\ldots,g_E$. This interpretation is supported by the fact that - as could be shown for the cases $G=U(1)$ and $G=SU(2)$ - the expectation values of quantizations of holonomies and fluxes coincide - up to orders of $\hbar$ - with the classical holonomies and fluxes determined by the phase space point corresponding to $g_1,\ldots, g_E$ [@GCS2]. Furthermore, the overlap between two complexifier coherent states is sharply peaked [@GCS2]:
$$\begin{aligned}
\frac{\Big|\big\langle\psi_{g_1,\ldots,g_E}^t\big|\psi_{h_1,\ldots,h_E}^t\big\rangle\Big|^2}{\big\|\psi_{g_1,\ldots,g_E}^t\big\|^2\;\big\|\psi_{h_1,\ldots,h_E}^t\big\|^2}\;=\;
\left\{\begin{array}{cl}1&\quad g_m=h_m\text{ for all
}m\\\begin{array}{c}\text{ decaying exponentially}\\\text{ as }t\to
0\end{array}& \quad{\rm else}\end{array}\right.\end{aligned}$$
This shows that the complexifier coherent states (\[Gl:DefinitionOfComplexifierCoherntStates-2\]) are suitable to approximate the kinematical operators of LQG quite well. Although the original LQG has been constructed with $G=SU(2)$, it has been shown that in the semiclassical regime, the group $SU(2)$ can be replaced by $U(1)^3$ without changing the qualitative behavior of expectation values or fluctuations. On the other hand, with this trick calculations simplify tremendously, since $U(1)^3$ is an abelian group. Furthermore, $U(1)^3$ is simply the Cartesian product of three copies of $U(1)$, which also completely determines the set of irreducible representations of $U(1)^3$, such that a complexifier coherent state on $U(1)^3$ is nothing but a product of three states on $U(1)$:
$$\begin{aligned}
\psi^t_{(g_1,g_2,g_3)}(h_1,h_2,h_3)\;=\;\psi^t_{g_1}(h_1)\,\psi^t_{g_2}(h_2)\,\psi^t_{g_3}(h_3).\end{aligned}$$
This is, of course, true for any Cartesian product between - not necessarily distinct - compact Lie groups.
Since the properties of complexifier coherent states on $U(1)^3$ can be investigated by considering states on $U(1)$, we will work with the latter from now on.
The case of $G=U(1)$
--------------------
In the last section, the general definition of complexifier coherent states for arbitrary compact Lie groups $G$ has been given. In this section, we will shortly review these states for the simplest case of $G=U(1)$, since we will work with these states in the rest of the article.
From (\[Gl:DefinitionOfComplexifierCoherntStates-2\]), we can immediately deduce the explicit form of the complexifier coherent states, since all irreducible representations of $U(1)$ are known and one-dimensional:
$$\begin{aligned}
\label{Gl:DefinitionOfComplexifierCoherntStatesOnU1}
\psi^t_z(\phi)\;=\;\sum_{n\in\Z}e^{-n^2\frac{t}{2}}\,e^{-in(z-\phi)}\end{aligned}$$
for $g=e^{iz}$ and $h=e^{i\phi}$. With the Poisson summation formula, this expression can be rewritten as $$\begin{aligned}
\psi_g^t(h)\;=\;\sqrt\frac{2\pi}{t}\;\sum_{n\in\Z}\;e^{-\frac{(z\,-\,\phi\,-\,2\pi
n)^2}{2t}}.\end{aligned}$$
The inner product of two of these states is then
$$\begin{aligned}
\label{Gl:InnerProductOfTwoU1States}
\big\langle\psi_g^t\big|\psi_{g'}^t\big\rangle\;=\;\sqrt\frac{\pi}{t}\;\sum_{n\in\Z}\;e^{-\frac{(\bar
z\,-\,z'\,-\,2\pi n)^2}{t}}.\end{aligned}$$
There is a way to interpret (\[Gl:InnerProductOfTwoU1States\]) geometrically. This makes use of the fact that $G^{\C}\;=\;\C\backslash\{0\}$ comes with a pseudo-Riemannian metric provided by the Killing form on its Lie algebra. On arbitrary Lie groups $G$, this metric is denoted, in components, by
$$\begin{aligned}
\label{Gl:ComplexMetric}
h_{IJ}\;=\;-\frac{1}{\dim G}\tr\;\left(g^{-1}\del_Ig\, g^{-1} \del_J
g\right).\end{aligned}$$
Choosing the chart $z\to e^{iz}$ on $\C\backslash\{0\}$, the metric (\[Gl:ComplexMetric\]) simply takes the form $h=1$. Note that the geodesics through $1\in\C\backslash\{0\}$ with respect to this metric are given by
$$\begin{aligned}
t\;\longmapsto\;e^{itz}\end{aligned}$$
for some $z\in\C$, which corresponds to the velocity of the geodesic at $t=0$. Note also that geodesics can be transported via group multiplication, since the metric is defined via group translation. In particular, if $\g(t)$ is a geodesic on $\C\backslash\{0\}$, then $g\g(t)$ is also one for any $g\in\C\backslash\{0\}$.
With $h$ one can define the complex length-square of a geodesic, or any other regular curve $\g$ on $\C\backslash\{0\}$, via
$$\begin{aligned}
l^2(\g)\;:=\;\left(\int dt \sqrt{h(\g(t))\dot \g(t)\dot
\g(t)}\right)^2.\end{aligned}$$
Note that this gives a well-defined complex number, since the square of a complex number is defined up to a sign, and this sign is chosen continuously on the whole curve, which gives a unique choice since the curve is regular, i.e. its velocity vector vanishes nowhere. So, the integral is determined up to a sign, the square of which is then well-defined.
Let $g,\,h\,\in\C\backslash\{0\}$, and $\g:[0,1]\to\C\backslash\{0\}$ be a geodesic from $g$ to $h$. It is straightforward to compute that such a geodesic is not unique, but, for $g=e^{iw}$ and $h=e^{iz}$ (where $z$ and $w$ are determined up to $2\pi n$ for some $n\in\Z$), is given by
$$\begin{aligned}
\label{Gl:Geodesic}
\g(t)\;=\;e^{iw}\,e^{it(z\,-\,w\,-\,2\pi n)}.\end{aligned}$$
for any $n\in\Z$. By changing $n$, one ranges through the set of geodesics from $g$ to $h$. The complex length square of the (\[Gl:Geodesic\]) can easily be computed to be
$$\begin{aligned}
l(\g)^2\;=\;(z\,-\,w\,-2\pi n)^2.\end{aligned}$$
This shows that one can write the inner product between two complexifier coherent states as sum over complex lengths of geodesics:
$$\begin{aligned}
\big\langle\psi^t_g\big|\psi^t_h\big\rangle\;=\;\sum_{\scriptsize\begin{array}{c}\g\text{
geodesic}\\\text{from }g^c \text{ to
}h\end{array}}\;e^{-\frac{l(\g)^2}{t}}\end{aligned}$$
with $g^c:=\bar g^{-1}$.\
Although this seems to be too much effort to rewrite a simple expression like (\[Gl:InnerProductOfTwoU1States\]), we will encounter a similar expression in [@GICS-II] for the case of $SU(2)$-complexifier coherent states. This relates the complexifier coherent states with the geometry of the corresponding group, which is given by the Killing metric (\[Gl:ComplexMetric\]). We will comment on this at the end of [@GICS-II].
Gauge-invariant coherent states with gauge group $G=U(1)$ {#Ch:GICS}
=========================================================
The gauge-invariant sector {#Ch:GaugeInvariantSector}
--------------------------
In the following, we will describe the Hilbert space invariant under the Gauss gauge transformation group. Since this gauge transformation group $\G$ leaves every graph invariant, we can restrict ourselves to the case of one graph, in particular $$\begin{aligned}
\P\,\lim_{\longrightarrow}\;H_{\g}\;=\;\lim_{\longrightarrow}\,\P\H_{\g}.\end{aligned}$$
So we can consider the gauge-invariant cylindrical functions on each graph separately.
The gauge-invariant cylindrical functions on a graph $\g$ with $E$ edges and $V$ vertices can be described in terms of singular cohomology classes with values in the gauge group. In particular, every Hilbert space $\H_{\g}$ is canonically isomorphic to an $L^2$-space: $$\begin{aligned}
H_{\g}\;\simeq\;L^2\left(G^E,\,d\m_{H}^{\otimes E}\right),\end{aligned}$$
where $d\m_H$ is the normalized Haar measure on the compact Lie group $G$. It is known that the gauge-invariant Hilbert space is then canonically isomorphic to an $L^2$-space over the first simplicial cohomology group of $\g$ with values in the gauge group $G$: $$\begin{aligned}
\P H_{\g}\;\simeq\;L^2\left(H^1(\g, G),\,d\m\right),\end{aligned}$$
with a certain measure $d\m$. For abelian gauge groups $G$, first cohomology group of $\g$ with values in $G$ is given by $$\begin{aligned}
H^1(\g, G)\;\simeq\;G^{E-V+1},\end{aligned}$$
and $d\m=d\m_H^{\otimes E-V+1}$ is the $E-V+1$-fold tensor product of the Haar measure on $G$. See appendix \[Ch:Gauge-invariantFunctionsOfU(1)\] for a summary of abelian cohomology groups on graphs and their relation to gauge-invariant functions. For non-abelian gauge groups $G$ a similar result holds, while the definition of the first cohomology class requires more care. This case will be dealt with in [@GICS-II], and we stay with abelian $G$ in this article.
Gauge-invariant coherent states
-------------------------------
We now come to the main part of this article: The computation of the gauge-invariant coherent states. We will derive a closed form for them, revealing the intimate relationship between the gauge-invariant degrees of freedom and the graph topology. From the explicit form we will be able to compute the overlap between two gauge-invariant coherent states, which will allow for an interpretation as semiclassical states for the gauge-invariant sector of the theory.\
The gauge-invariant coherent states are obtained by applying the gauge projector (\[Gl:Projector\]) to the complexifier coherent states on a graph (\[Gl:TensorProductOfCoherentStates\]), (\[Gl:DefinitionOfComplexifierCoherntStatesOnU1\]), i.e. $$\begin{aligned}
\label{Gl:DefinitionGaugeInvariantCoherentState}
\Psi_{[g_1,\ldots,g_E]}^t([h_1,\ldots h_E])\;=\;\P
\psi_{g_1,\ldots,g_E}^t(h_1,\ldots,h_E).\end{aligned}$$
It is known that the set of gauge-invariant functions can be described in terms of functions on the first cohomology class of the graph. See the appendix for details. In particular, if the graph has $E$ edges and $V$ vertices, i.e. the gauge-variant configuration space is diffeomorphic to $U(1)^E$, then the gauge-invariant configuration space is diffeomorphic to $U(1)^{E-V+1}$. This might raise the hope that these states somehow resemble complexifier coherent states on the gauge-invariant configuration space $U(1)^{E-V+1}$. We will see that this is not quite true, but near enough.
The fact that the gauge group is abelean is a great simplification: It allows us to pull back all group multiplications to simple addition on the algebra, simply due to the fact that $\exp\,iz\,\exp\,iw=\exp\,i(z+w)$. This will allow us to explicitly perform the gauge integrals for arbitrary graphs, and obtain a formula for the gauge-invariant coherent states that only depends on gauge-invariant combinations of $h_k=\exp\,i\phi_k$ and $g_k=\exp\,iz_k$, as well as topological information about the graph, in particular its incidence matrix.
Basic graph theory
------------------
In order to be able to deal with the expressions for all graphs, we start with some basics of graph theory. All the material, as well as all the proofs, can be found in [@GRAPH] and the references therein.
Let $\g$ be a directed graph with $V$ vertices and $E$ edges. Let the edges be labeled by numbers $1,\ldots,E$ and the vertices by numbers $1,\ldots,V$. Then the *incidence matrix* $\l\in\text{\emph{Mat}}(E\times V,\Z)$ is defined by the following rule:
$$\begin{aligned}
\l_{kl}\;&:=\;1&\qquad\mbox{if the edge $k$ ends at vertex
$l$}\\[5pt]
\l_{kl}\;&:=\;-1&\qquad\mbox{if the edge $k$ starts at vertex
$l$}\\[5pt]
\l_{kl}\;&:=&\;0\qquad\mbox{else.}\end{aligned}$$
Note in particular that, if edge $k$ starts *and* ends at vertex $l$, i.e. the edge $k$ is a loop, then $\l_{kl}=0$ as well. Since either an edge is a loop or starts at one and ends at some other vertex, every line of the matrix $\l$ is either empty, or contains exactly one $1$ and one $-1$. With the definition $$\begin{aligned}
\label{Gl:DefintionVonU}
u\;:=\;\left(\begin{array}{c}1\\1\\\vdots\\1\end{array}\right)\;\in\;\R^V,\end{aligned}$$
we immediately conclude $$\begin{aligned}
\label{Gl:ULiegtImKernVonLambdaTransponiert}
\l^Tu\;=\;0.\end{aligned}$$
Let $\g'$ be a graph. If $\g'$ contains no loops, then $\g'$ is said to be a *tree*. If $\g'\subset\g$ is a subgraph, then $\g'$ is said to be a *tree in $\g$*. If $\g'\subset\g$ is a subgraph that meets every vertex of $\g$, then $\g'$ is said to be a *maximal tree (in $\g$)*.
Every graph $\g$ has a maximal tree as subgraph. Every tree has $V=E-1$ vertices.
Maximal trees in graphs are not unique. It is quite easy to show that every function cylindrical over a graph $\g$ is gauge equivalent to a function cylindrical over $\g$, which is constant on the edges corresponding to a maximal tree. This will be used later, and by the preceding Lemma we immediately conclude that the number of gauge-invariant degrees of freedoms on a graph with $V$ vertices and $E$ edges is $E-V+1$ for Abelian gauge theories. This will be seen explicitly at the end of this section.
The following theorem relates the numbers of different possible maximal trees to the incidence matrix.
\[Thm:Kirchhoff\] (Kirchhoff) Let $\g$ be a graph and $\l$ its incidence matrix. Then the *Kirchoff-matrix* $K:=\l\l^T$ has nonnegative eigenvalues
$$\begin{aligned}
0\;=\;\m_1\leq\m_2\leq\cdots\leq \m_V.\end{aligned}$$
The lowest eigenvalue is $\m_1=0$, and the degeneracy of $0$ is the number of connected components of the graph $\g$. Furthermore, the product of all nonzero eigenvalues $$\begin{aligned}
G\;:=\;\frac{1}{V}\prod_{\m_k\neq 0}\m_k\end{aligned}$$
is the number of different maximal trees in $\g$.
With this machinery, we will be able to perform the gauge integral for arbitrary graphs. This will include some kind of gauge-fixing procedure, which will make use of a maximal tree.
Gauge-variant coherent states and the gauge integral
----------------------------------------------------
The Abelian nature of the gauge group allows us to pull back the group multiplication to addition on the Lie algebra. This is why throughout this chapter we will, instead of elements $h\in U(1)$, deal with $\phi\in\R$ by $h=\exp \,i\phi$, and instead of elements $g\in\C\backslash\{0\}$, we will work with the corresponding $z\in\C$ such that $g=\exp\,iz$, always having in mind that $\phi$ and $z$ are only defined modulo $2\pi n$ for $n\in\Z$.
We will denote vectors (of any length) as simple letters $z,\phi,\tilde\phi,m,\ldots$ and their various components with indices: $z_k,\phi_k,\tilde\phi_k,\ldots$. The particular range of the indices will be clear from the context, but we will still repeat it occasionally.\
The gauge-variant coherent states on a graph $\g$ with $E$ edges are simply given by the product $$\begin{aligned}
\psi_{z}^t(\phi)\;=\;\prod_{k=1}^E\sum_{m_k\in\Z}e^{-m_k^2\frac{t}{2}}\,e^{im_k(z_k-\phi_k)}\end{aligned}$$
where $z_k=\phi_k-ip_k,\;k=1,\ldots,E$ is labeling the points in phase space where the coherent states are peaked. With the Poisson summation formula one can rewrite this as
$$\begin{aligned}
\label{Gl:EichvarianterKohaerenterZustandAufNemGraphen}
\psi_{z}^t(\phi)\;=\;\sqrt{\frac{2\pi}{t}}^E\;\sum_{m_1,\ldots,m_E\in\Z}\;\exp\left({-\sum_{k=1}^E\frac{(z_k-\phi_k-2\pi
m_k)^2}{2t}}\right)\end{aligned}$$
We will now perform the gauge integral
$$\begin{aligned}
\label{Gl:AusgangsFormel}
\Psi_{[z]}^t(\phi)\;&=&\;\int_Gd\m_H(\tilde
\phi)\;\psi_{\a_{\tilde\phi}z}(\phi)\\[5pt]\nonumber
&=&\;\sqrt{\frac{2\pi}{t}}^E\int_{[0,2\pi]^V}\frac{d\tilde\phi_1}{2\pi}\cdot\ldots\cdot\frac{d\tilde\phi_V}{2\pi}\;
\sum_{m_1,\ldots,m_E\in\Z}\;\exp\left({-\sum_{k=1}^E\frac{(A_k+\l_{ka}\tilde\phi_a-2\pi
m_k)^2}{2t}}\right)\end{aligned}$$
with $A=z-\phi$, and where $\l_{ka}$ are the components of the transpose $\l^T$ of the incidence matrix.
In what follows, we will use the symmetries of this expression, together with a gauge-fixing procedure, to separate the gauge degrees of freedom from the gauge-invariant ones. The integrals will then be performable analytically, and the resulting expression can then be interpreted as states being peaked on gauge-invariant quantities.\
To simplify the notation, we will assume, without loss of generality, that $\g$ is connected. Furthermore choose, once and for all, a maximal tree $\t\subset\g$. Choose the numeration of vertices and edges of $\g$ according to the following scheme:
Start with the maximal tree $\t$. The tree consists of $V$ vertices and $V-1$ edges. Call a vertex that has only one outgoing edge (in $\t$, not necessarily in $\g$) an outer end of $\t$. Remove one outer end and the corresponding edge from $\t$ and obtain a smaller subgraph $\t^1\subset\g$, which is also a tree. Label the removed vertex with the number $1$, and do so with the removed edge as well. So this gives you $v_1$ and $e_1$. From $\t^1$, remove an outer end and the corresponding edge, and label them $v_2$ and $e_2$, and obtain a yet smaller tree $\t^2\subset \t^1\subset \t\subset\g$. Repeat this process until $\t$ has been reduced to $\t^{
(V-1)}$, which is a point. This way, one has obtained $v_1,\ldots,v_{V-1}$ and $e_1,\ldots,e_{E-1}$. Call the last, remaining vertex $v_V$. Label the edges that do not belong to $\t$ by $e_V,e_{V+1},\ldots,e_E$ in any order.
Choosing the numeration of the vertices and the edges in the above manner will help us in rewriting the expression (\[Gl:AusgangsFormel\]). First we note that the first $V-1$ edges and the first $V$ vertices constitute the tree, the last $E-V+1$ edges constitute what is not the tree in $\g$. Furthermore, with this numeration, the edge $e_k$ is starting or ending at vertex $v_k$ for $k=1,\ldots,V-1$. In particular, the diagonal elements of the incidence matrix are all (except maybe the last one) nonzero: $\l_{kk}\neq 0$ for $k=1,\ldots,V-1$.
Let $\g$ be a graph, with vertices $v_1,\ldots v_V$ and edges $e_1,\ldots,e_E$. Between two vertices $v_k$ and $v_l$ there is a unique path in $\t$, since a tree contains no loops. Call $v_k$ being *before* $v_l$, if this path includes $e_k$, otherwise call $v_k$ being *after* $v_l$.
Note that a vertex cannot be both before and after another vertex, but two vertices can both be before or both be after each other.
The numeration we have chosen has the following consequence: For each vertex $v_k$ one has that for all $v_l$ such that $v_k$ is after $v_l$, that $l\leq k$. The converse need not be true. Note further that every vertex is before itself, by this definition. Also, since $e_V$ is not an edge of the graph, it does not even have to be touching $v_V$. So, the question of whether $v_V$ is before or after any other vertex makes no sense in this definition (But note that it does make sense to ask whether any vertex is before or after $v_V$).
We now rewrite formula (\[Gl:AusgangsFormel\]), by replacing the integrals over $[0,2\pi]$ by integrals over $\R$. We do this inductively over the vertices from $v_1$ to $v_{V-1}$. Consider the $E$ terms constituting the sum in the exponent in
$$\begin{aligned}
\Psi_{[z]}^t(\phi)\;&=&\;\int_Gd\m_H(\tilde
\phi)\;\psi_{\a_{\tilde\phi}z}(\phi)\\[5pt]\nonumber
&=&\;\sqrt{\frac{2\pi}{t}}^E\int_{[0,2\pi]^V}\frac{d\tilde\phi_1}{2\pi}\cdot\ldots\cdot\frac{d\tilde\phi_V}{2\pi}\;
\sum_{m_1,\ldots,m_E\in\Z}\;\exp\left({-\sum_{k=1}^E\frac{(A_k+\l_{ka}\tilde\phi_a-2\pi
m_k)^2}{2t}}\right).\end{aligned}$$
In some of them $\tilde\phi_1$ appears, in some of them it does not, precisely if either $\l_{k1}\neq 0$ or $\l_{k1}=0$. Note that $\tilde\phi_1$ definitely appears in the first term, by the above considerations. If $\tilde\phi_1$ appears in the $k$-th term other than $k=1$, shift the infinite sum over $m_k$ by $m_k+\l_{11}\l_{k1}m_1$. The result of this is that, since $\l_{k1}^2=\l_{11}^2=1$ for these $k$, after this shift $\tilde\phi_1$ appears always in the combination $\l_{11}\tilde\phi_1-2\pi m_1$ in all the factors. Now we can employ the formula $$\begin{aligned}
\label{Gl:SuperFormelDieSummenWegmacht}
\int_{[0,2\pi]}\frac{d\tilde\phi}{2\pi}\,\sum_{m\in\Z}\;f(\tilde\phi
\pm 2\pi m)\;=\;\frac{1}{2\pi}\int_{\R}d\tilde\phi\;f(\tilde\phi)\end{aligned}$$
and, regardless of whether $\l_{11}=+1$ or $\l_{11}=-1$, have $$\begin{aligned}
(\ref{Gl:AusgangsFormel})\;&=&\;\sqrt{\frac{2\pi}{t}}^E\int_{\R}\frac{d\tilde\phi_1}{2\pi}\int_{[0,2\pi]^{V-1}}
\frac{d\tilde\phi_2}{2\pi}\cdot\ldots\cdot\frac{d\tilde\phi_V}{2\pi}\;\\[5pt]
&&\qquad\qquad\qquad\times\sum_{m_2,\ldots,m_E\in\Z}\;\exp\left({-\frac{(A_1+\l_{1a}\tilde\phi_a)^2}{2t}-\sum_{k=2}^E\frac{(A_k+\l_{ka}\tilde\phi_a-2\pi
m_k)^2}{2t}}\right).\end{aligned}$$
This being the beginning of the induction, we now describe the induction step from $l$ to $l+1$ by the following technical lemma. By this we will be able to extend all integration ranges over all of $\R$, instead of finite intervals, which will turn out to be very useful.
\[Lem:InduktionsSchrittBeimSummenverschwindenlassen\] Let $\g$ be a graph with $V$ vertices, $E$ edges, and $\l$ be its incidence matrix. Let $A\in\C^E$ and $t>0$, then we have, for $1\leq l\leq V-1$: $$\begin{aligned}
&&\sqrt{\frac{2\pi}{t}}^E\int_{\R^{l-1}}\frac{d\tilde\phi_1}{2\pi}\cdot\ldots\cdot\frac{d\tilde\phi_{l-1}}{2\pi}\int_{[0,2\pi]^{V-l+1}}
\frac{d\tilde\phi_{l}}{2\pi}\cdot\ldots\cdot\frac{d\tilde\phi_V}{2\pi}\;\\[5pt]
&&\qquad\qquad\qquad\times\sum_{m_{l},\ldots,m_E\in\Z}\;\exp\left({-\sum_{k=1}^{l-1}\frac{(A_k+\l_{ka}\tilde\phi_a)^2}{2t}-\sum_{k={l}}^E\frac{(A_k+\l_{ka}\tilde\phi_a-2\pi
m_k)^2}{2t}}\right)\\[5pt]
&=&\sqrt{\frac{2\pi}{t}}^E\int_{\R^{l}}\frac{d\tilde\phi_1}{2\pi}\cdot\ldots\cdot\frac{d\tilde\phi_{l}}{2\pi}\int_{[0,2\pi]^{V-l}}
\frac{d\tilde\phi_{l+1}}{2\pi}\cdot\ldots\cdot\frac{d\tilde\phi_V}{2\pi}\;\\[5pt]
&&\qquad\qquad\qquad\times\sum_{m_{l+1},\ldots,m_E\in\Z}\;\exp\left({-\sum_{k=1}^{l}\frac{(A_k+\l_{ka}\tilde\phi_a)^2}{2t}-\sum_{k={l+1}}^E\frac{(A_k+\l_{ka}\tilde\phi_a-2\pi
m_k)^2}{2t}}\right).\end{aligned}$$
**Proof:** Note that we just proved the formula for $l=1$. In the proof for arbitrary $1\leq l\leq V-1$ we will use the notion of vertices being before and after one another.
Consider all vertices $v_k$ being after $v_l$, other than $v_l$ itself. By construction, for all such $k$, we have $k<l$, so by induction hypothesis, the integration over all these $v_k$ runs over all of $\R$, not over just the interval $[0,2\pi]$ any more. Consequently, the sum over these $m_k$ is not appearing any longer. So we can shift the integration range by $+2\pi\l_{ll}m_l$.
This will affect the terms in the first sum in $$\begin{aligned}
\label{Gl:DieBeidenSummen}
\exp\left(-\sum_{k=1}^{l-1}\frac{(A_k+\l_{ka}\tilde\phi_a)^2}{2t}-\sum_{k={l}}^E\frac{(A_k+\l_{ka}\tilde\phi_a-2\pi
m_k)^2}{2t}\right)\end{aligned}$$
in the following way: Let $k<l$. The edge $e_k$ then belongs to the tree $\t$, and thus $v_l$ is either after both vertices $e_k$ touches, or before both vertices. If $v_l$ is before both, the term does not change at all, since the two $\tilde\phi_a$ in it are not shifted. If $v_l$ is after both and is not itself one of the two vertices, then the term gets changed by $$\begin{aligned}
(A_k+\l_{ka}\tilde\phi_a)^2\;\longrightarrow\;(A_k+\l_{ka}\tilde\phi_a\;\pm2\pi\l_{ll}m_l\;\mp
2\pi\l_{ll}m_l)^2\;=\;(A_k+\l_{ka}\tilde\phi_a)^2,\end{aligned}$$
since the two $\tilde\phi_a$ in a term always appear with opposite sign. So these terms do not change, too. If $v_l$ is after both vertices that touch $e_k$ and is itself one of it (i.e. $e_k$ is an edge adjacent to $e_l$, linked by $v_l$), then the corresponding term changes by
$$\begin{aligned}
(A_k+\l_{ka}\tilde\phi_a)^2\;&=&\;
(A_k+\l_{kl}\tilde\phi_l+\l_{kk}\tilde\phi_k)^2\;=\;(A_k+\l_{kk}(\tilde\phi_k-\tilde\phi_l))^2\\[5pt]
&&\longrightarrow\;(A_k+\l_{kk}(\tilde\phi_k-\tilde\phi_l+2\pi\l_{ll}m_l))^2,$$
where $\l_{kl}=-\l_{ll}$ and $\l_{ll}^2=1$ have been used.
So, after this shift, in all terms in the first sum in (\[Gl:DieBeidenSummen\]) $\tilde\phi_l$ has been replaced by $\tilde\phi_l-2\pi\l_{ll}m_l$. The first term of the second sum reads
$$\begin{aligned}
(A_l+\l_{ll}(\tilde\phi_l-\tilde\phi_a)-2\pi
m_l)^2\;=\;(A_l-\l_{ll}\tilde\phi_a+\l_{ll}(\tilde\phi_l-2\pi\l_{ll}
m_l))^2,\end{aligned}$$
where $v_a$ is the other vertex touching $e_l$, apart from $v_l$. So also in this term $\tilde\phi_l$ and $m_l$ appear in the combination $\tilde\phi_l-2\pi\l_{ll}m_l$.
The terms $k=l+1$ till $k=E-V+1$ remain unchanged, since they all correspond to edges that lie between vertices $v_a$ such that $v_l$ is before both $v_a$, and these $\tilde\phi_a$ are hence not shifted.
The terms $k=E-V+2$ till $k=E$ in (\[Gl:DieBeidenSummen\]), on the other hand, correspond to edges that lie between two vertices such that $v_l$ could be before the one and after the other. This is due to the fact that these edges do not belong to the maximal tree $T$ any longer. So in these terms, a shift by $\pm2\pi\l_{ll}m_l$ could have occurred by the shift of integration range. But in all these terms, there is still a term $-2\pi m_k$ present, and the sum over these $m_k$ is still performed. So, by appropriate shift of these summations, similar to the ones performed in the induction start, one can subsequently produce or erase terms of the form $\pm2\pi\l_{ll}m_l$ in all of the terms corresponding to $k=E-V+2$ till $k=E$. Since there are enough summations left, one has enough freedom to produce a $\pm2\pi\l_{ll}m_l$, where $\tilde\phi_l$ is present, or erase all terms with $m_l$, where $\tilde\phi_l$ is not present.\
Thus, in the end, we again have a function only depending on $\tilde\phi_l-2\pi\l_{ll}m_l$, and thus we can again apply formula (\[Gl:SuperFormelDieSummenWegmacht\]), and, regardless of the sign of $\l_{ll}$, erase the infinite sum over $\m_l$, obtaining an integration range of $\tilde\phi_l$ over all of $\R$:
$$\begin{aligned}
&&\sqrt{\frac{2\pi}{t}}^E\int_{\R^{l-1}}\frac{d\tilde\phi_1}{2\pi}\cdot\ldots\cdot\frac{d\tilde\phi_{l-1}}{2\pi}\int_{[0,2\pi]^{V-l+1}}
\frac{d\tilde\phi_{l}}{2\pi}\cdot\ldots\cdot\frac{d\tilde\phi_V}{2\pi}\;\\[5pt]
&&\qquad\qquad\qquad\times\sum_{m_{l},\ldots,m_E\in\Z}\;\exp\left({-\sum_{k=1}^{l-1}\frac{(A_k+\l_{ka}\tilde\phi_a)^2}{2t}-\sum_{k={l}}^E\frac{(A_k+\l_{ka}\tilde\phi_a-2\pi
m_k)^2}{2t}}\right)\\[5pt]
&=&\sqrt{\frac{2\pi}{t}}^E\int_{\R^{l}}\frac{d\tilde\phi_1}{2\pi}\cdot\ldots\cdot\frac{d\tilde\phi_{l}}{2\pi}\int_{[0,2\pi]^{V-l}}
\frac{d\tilde\phi_{l+1}}{2\pi}\cdot\ldots\cdot\frac{d\tilde\phi_V}{2\pi}\;\\[5pt]
&&\qquad\qquad\qquad\times\sum_{m_{l+1},\ldots,m_E\in\Z}\;\exp\left({-\sum_{k=1}^{l}\frac{(A_k+\l_{ka}\tilde\phi_a)^2}{2t}-\sum_{k={l+1}}^E\frac{(A_k+\l_{ka}\tilde\phi_a-2\pi
m_k)^2}{2t}}\right).\end{aligned}$$
This was the claim of the Lemma.\
An immediate corollary of Lemma \[Lem:InduktionsSchrittBeimSummenverschwindenlassen\] is that
$$\begin{aligned}
\nonumber
&&\sqrt{\frac{2\pi}{t}}^E\int_{[0,2\pi]^V}\frac{d\tilde\phi_1}{2\pi}\cdot\ldots\cdot\frac{d\tilde\phi_V}{2\pi}\;
\sum_{m_1,\ldots,m_E\in\Z}\;\exp\left({-\sum_{k=1}^E\frac{(A_k+\l_{ka}\tilde\phi_a-2\pi
m_k)^2}{2t}}\right)\\[5pt]\label{Gl:AlleIntegrationenGeshiftet}
&=&\;\sqrt{\frac{2\pi}{t}}^E\int_{\R^{V-1}}\frac{d\tilde\phi_1}{2\pi}\cdot\ldots\cdot\frac{d\tilde\phi_{V-1}}{2\pi}\int_0^{2\pi}
\frac{d\tilde\phi_{V}}{2\pi}\\[5pt]\nonumber
&&\qquad\qquad\qquad\times\sum_{m_{V},\ldots,m_E\in\Z}\;\exp\left({-\sum_{k=1}^{V-1}\frac{(A_k+\l_{ka}\tilde\phi_a)^2}{2t}-
\sum_{k={V}}^E\frac{(A_k+\l_{ka}\tilde\phi_a-2\pi
m_k)^2}{2t}}\right).\end{aligned}$$
Note that one cannot perform the induction step with the integration over $\tilde\phi_V$ as well. The reason for this is that for the induction step it is crucial that it does not make sense to define whether $v_V$ is before or after any other vertex, since $e_V$ does not belong to the maximal tree $\t$, in fact it does not even need to touch $v_V$. In particular, the integrand in (\[Gl:AlleIntegrationenGeshiftet\]) does not depend on $\tilde\phi_V$ at all! To see this, one only needs to shift all integrations $\tilde\phi_1,\ldots,\tilde\phi_{V-1}$ by $+\tilde\phi_V$. In all terms, the integration variables appear in the combination $\tilde\phi_a-\tilde\phi_b$ for any two different $a,b=1,\ldots,V$. So either $a$ and $b$ are both not $V$, then nothing changes by this shift of integration, or one of $a$ or $b$ is equal to $V$. In this case the shift of the other one cancels the $\tilde\phi_V$, since both $\tilde\phi_a$ and $\tilde\phi_b$ appear with opposite sign. So, after this shift, $\tilde\phi_V$ occurs nowhere in the formula any more. Thus, we can perform the integration over $\tilde\phi_V$ trivially and obtain $$\begin{aligned}
\label{Gl:FormelMitZuNullGesetzemPhiTildeVau}
(\ref{Gl:AusgangsFormel})\;=\;\sqrt{\frac{2\pi}{t}}^E\int_{\R^{V-1}}\frac{d\tilde\phi_1}{2\pi}\cdot\ldots\cdot\frac{d\tilde\phi_{V-1}}{2\pi}\;
\sum_{m_{V},\ldots,m_E\in\Z}\;\exp\left(-\sum_{k=1}^{E}\frac{(\tilde
A_k+\l_{ka}\tilde\phi_a)^2}{2t}\right)_{\Bigg|_{\tilde\phi_V=0}}\end{aligned}$$
where $$\begin{aligned}
\tilde A_k\;:=\left\{\begin{array}{ll}A_k&1\leq k\leq V-1\\A_k-2\pi
m_k\qquad & V\leq k\leq E\end{array}\right..\end{aligned}$$
To proceed, note that, since in every term in (\[Gl:FormelMitZuNullGesetzemPhiTildeVau\]) the $\tilde\phi_a$ appear as pairs with opposite sign, the integrand is invariant under a simultaneous shift of all variables: $\tilde\phi_a\to\tilde\phi_a+c$. We use this fact to rewrite (\[Gl:FormelMitZuNullGesetzemPhiTildeVau\]), by using the following technical Lemma
\[Lem:WieKommtDieDeltaFunktionInDieFlasche\] Let $f:\R^n\to \C$ be a function with the symmetry $$\begin{aligned}
f(x_1+c,\ldots,x_n+c)\;=\;f(x_1,\ldots,x_n)\qquad\mbox{for all
}c\in\R\end{aligned}$$
such that $x_1,\ldots,x_{n-1}\to f(x_1,\ldots,x_{n-1},0)$ is integrable. Then $$\begin{aligned}
\int_{\R^{n-1}}d^{n-1}x\;f(x_1,\ldots,x_{n-1},0)\;=\;n\int_{\R^n}d^nx\,\d(x_1+\cdots +x_n)\,f(x_1,\ldots,x_n).\end{aligned}$$
**Proof:** The proof is elementary. Write
$$\begin{aligned}
&&\int_{\R^{n-1}}dx_1,\ldots dx_{n-1}\;f(x_1,\ldots,x_{n-1},0)\\[5pt]
\;&=&\;\int_{\R^{n-1}}dx_1,\ldots
dx_{n-1}\;f\left(x_1-\frac{\sum_{k=1}^{n-1}
x_k}{n},\ldots,x_{n-1}-\frac{\sum_{k=1}^{n-1}
x_k}{n},-\frac{\sum_{k=1}^{n-1} x_k}{n}\right)\\[5pt]
\;&=&\;\int_{\R^n}dx_1,\ldots
dx_{n}\;f\left(x_1-\frac{\sum_{k=1}^{n-1}
x_k}{n},\ldots,x_{n-1}-\frac{\sum_{k=1}^{n-1}
x_k}{n},\,x_n\right)\,\d\left(x_n+\frac{\sum_{k=1}^{n-1}x_k}{n}\right)\end{aligned}$$
Now perform a coordinate transformation $$\begin{aligned}
&&\tilde x_k\;:=\;x_k\,-\frac{\sum_{k=1}^{n-1}
x_k}{n},\;\qquad\mbox{
for }k=1,\ldots ,n-1\\[5pt]
&&\tilde x_n\;:=\;x_n.\end{aligned}$$
We have
$$\begin{aligned}
\sum_{n=1}^{n-1}\tilde x_k\;=\;\frac{\sum_{k=1}^{n-1}x_k}{n}\end{aligned}$$
and get $$\begin{aligned}
\nonumber
&&\int_{\R^{n-1}}dx_1,\ldots dx_{n-1}\;f(x_1,\ldots,x_{n-1},0)\\[5pt]\label{Gl:JetztFehltNurNochDieJacobimatrix}
\;&=&\;\frac{1}{J}\int_{\R^n}d^n\tilde x\;f\left(\tilde
x_1,\ldots,\tilde x_{n-1},\,\tilde x_n\right)\;\d(\tilde
x_1+\ldots+\tilde x_n).\end{aligned}$$
Here $J=\det{(\del \tilde x_k/\del x_l)}$ is the Jacobian matrix of the coordinate transform. It is given by $$\begin{aligned}
J\;=\;\det\left[1\,-\,\frac{1}{n}\left(\begin{array}{ccccc}1&1&\cdots&1&0\\1&1&\cdots&1&0\\
\vdots & \vdots&\ddots&\vdots&\vdots\\1&1&\cdots&1&0\\0&0&\cdots
&0&0
\end{array}\right)\right],\end{aligned}$$
the determinant of which can easily computed to be $J=\frac{1}{n}$. Thus, with (\[Gl:JetztFehltNurNochDieJacobimatrix\]), the statement is proven.\
We continue our analysis of the gauge-invariant overlap by using Lemma (\[Lem:WieKommtDieDeltaFunktionInDieFlasche\]) to rewrite (\[Gl:FormelMitZuNullGesetzemPhiTildeVau\]) to obtain
$$\begin{aligned}
(\ref{Gl:AusgangsFormel})\;=\;V\sqrt{\frac{2\pi}{t}}^E\int_{\R^{V}}\frac{d\tilde\phi_1\ldots
d\tilde\phi_{V}}{(2\pi)^{V-1}}\;\d\left(\sum_{a=1}^V\tilde\phi_a\right)
\sum_{m_{V},\ldots,m_E\in\Z}\;\exp\left(-\sum_{k=1}^{E}\frac{(\tilde
A_k+\l_{ka}\tilde\phi_a)^2}{2t}\right).\end{aligned}$$
Now we split the integrations over the $\tilde\phi_a$ from the $\tilde A_k$, in order to perform the integration. Because we are integrating over $\R^V$ and the integrand is holomorphic, we can now shift the $\tilde\phi_a$ also by complex amounts. This is necessary, since the $\tilde A_k$ are generically complex. A generic shift of the $\tilde\phi_a$ by complex numbers $z_a$ looks like $$\begin{aligned}
\nonumber
(\ref{Gl:AusgangsFormel})\;&=&\;V\sqrt{\frac{2\pi}{t}}^E\int_{\R^{V}}\frac{d\tilde\phi_1\ldots
d\tilde\phi_{V}}{(2\pi)^{V-1}}\;\d\left(\sum_{a=1}^V(\tilde\phi_a+z_a)\right)\\[5pt]\nonumber
&&\quad\times
\sum_{m_{V},\ldots,m_E\in\Z}\;\exp\left(-\sum_{k=1}^{E}\frac{(\tilde
A_k+\l_{ka}\tilde\phi_a+\l_{ka}z_a)^2}{2t}\right)\\[5pt]\nonumber
\;&=&\;V\sqrt{\frac{2\pi}{t}}^E\int_{\R^{V}}\frac{d\tilde\phi_1\ldots
d\tilde\phi_{V}}{(2\pi)^{V-1}}\;\d\left(\sum_{a=1}^V(\tilde\phi_a+z_a)\right)\\[5pt]\nonumber
&&\quad\times\sum_{m_{V},\ldots,m_E\in\Z}\;\exp\left[-\sum_{k=1}^{E}\left(\frac{(\l_{ka}\tilde\phi_a)^2}{2t}
\,+\,\frac{\l_{ka}\tilde\phi_a(\l_{ka}z_a+\tilde
A_k)}{t}\,+\,\frac{(\l_{ka}z_a+\tilde
A_k)^2}{2t}\right)\right]\\[5pt]\label{Gl:AllesMitVektorenAusgedrueckt}
\;&=&\;V\sqrt{\frac{2\pi}{t}}^E\int_{\R^{V}}\frac{d\tilde\phi_1\ldots
d\tilde\phi_{V}}{(2\pi)^{V-1}}\;\d\left(u^T\tilde\phi+u^Tz\right)\\[5pt]\nonumber
&&\quad\times\sum_{m_{V},\ldots,m_E\in\Z}\;\exp\left(-\frac{\tilde\phi^T\l\l^T\tilde\phi}{2t}
\,-\,\frac{\tilde\phi^T\l(\l^Tz+\tilde
A)}{t}\,-\,\frac{(\l^Tz+\tilde A)^T(\l^Tz+\tilde A)}{2t}\right).\end{aligned}$$
In (\[Gl:AllesMitVektorenAusgedrueckt\]) we have expressed all variables in terms of vectors and matrices, since this will simplify the handling of the expressions a lot. The vectors $u$, $\tilde\phi$, $z$ have length $V$, the vector $\tilde A$ has length $E$, and $\l$ is the $V\times E$ incidence matrix. The vector $u$ is given by (\[Gl:DefintionVonU\]). The ${}^T$ means transpose.
The following Lemma will help us to simplify this formula.
\[Lem:LoesungenVonGleichungssystemen\] Let $\l$ be the $V\times E$ incidence matrix of a connected graph $\g$ with $E$ edges and $V$ vertices, and $u=(1\,1\,\cdots\,1)^T$ the vector of length $V$ containing only ones. For any vector $\tilde A\in \C^E$ the set of equations $$\begin{aligned}
\l(\l^Tz+\tilde A)\;&=&\;0\\[5pt]
u^Tz\;&=&\;0\end{aligned}$$
has exactly one solution in $\C^V$.
**Proof:** Rewrite the first of these equations as $$\begin{aligned}
\l\l^T\,z\;=\;-\l\tilde A.\end{aligned}$$
Because of (\[Gl:ULiegtImKernVonLambdaTransponiert\]), $-\l\tilde A$ lies in the orthogonal complement of $u$: $-\l\tilde
A\in\{u\}^{\perp}$. Since the graph $\g$ is connected, by Kirchhoff’s theorem (\[Thm:Kirchhoff\]) the Kirchhoff-matrix $\l\l^T$ is positive definite on $\{u\}^{\perp}$, hence invertible on this $V-1$-dimensional subspace of $\C^V$. Define the $V\times
V$-matrix $\s$ to be the inverse of $\l\l^T$ on $\{u\}^{\perp}$, and zero on $u$: $$\begin{aligned}
\s(\l\l^T)v\;=\;(\l\l^T)\s v\;&=&\;v\qquad\mbox{for all }u^Tv=0\\[5pt]
\s u\;&=&\;0.\end{aligned}$$
So, the set of solutions of $\l\l^Tz=-\l\tilde A$ is given by $$\begin{aligned}
z\;=\;-\s\l\tilde A\;+\;\a u\;\qquad\a\in\C.\end{aligned}$$
By the definition of $\s$, this means that $$\begin{aligned}
z\;=\;-\s\l\tilde A\end{aligned}$$
is the unique solution of both equations, which proves the Lemma.\
\[Lem:DasIstJaEinProjektor!\] With the conditions of Lemma \[Lem:LoesungenVonGleichungssystemen\], let $z$ be the unique solution of $\l(\l^Tz+\tilde A)=0$ and $u^Tz=0$, i.e. $z=-\s\l\tilde
A$. Then $$\begin{aligned}
\label{Gl:DasIstJaEinProjektor!}
-\l^T\s\l+{1}_E\;=\;P_{\ker\l},\end{aligned}$$
where ${1}_E$ is the $E\times E$ unit-matrix and $P_{\ker
\l}$ is the orthogonal projector onto the subspace $\ker\l\subset\C^E$. In particular $$\begin{aligned}
\l^Tz+\tilde A\;=\;P_{\ker\l}\tilde A.\end{aligned}$$
**Proof:** Since $$\begin{aligned}
\label{Gl:OrhtogonaleZerlegung}
\ker\l\,\oplus\,\img\l^T\;=\;1_E,\end{aligned}$$
The statement (\[Gl:DasIstJaEinProjektor!\]) can be rephrased as follows: $$\begin{aligned}
\l^T\s\l\;=\;P_{\img\l^T},\end{aligned}$$
which is the projector onto the image of $\l^T$. Let $v\in\img\l^T$, so $v=\l^T w$ for some $w\in\C^V$. Even more, since $\l^Tu=0$, we even can choose $w$ to be orthogonal to $u$: $w\in\{u\}^{\perp}$. Then $$\begin{aligned}
\l^T\s\l\,v\;=\;\l^T\s(\l\l^T)w\;=\;\l^Tw\;=\;v,\end{aligned}$$
since by definition $\s$ is the inverse of $\l\l^T$ on $\{u\}^{\perp}$.
Let, on the other hand, $v\in\{\img\l^T\}^{\perp}=\ker\l$. Then $$\begin{aligned}
\l^T\s\l\,v\;=\;0\end{aligned}$$
trivially. Thus, $\l^T\s\l$ leaves vectors in $\img\l^T$ invariant and annihilates vectors from the orthogonal complement of $\img\l^T$. Hence $\l^T\s\l\;=\;P_{\img\l^T}$, from which it follows that $$\begin{aligned}
-\l^T\s\l+{1}_E\;=\;P_{\ker\l}.\end{aligned}$$
This was the first claim, the second one $$\begin{aligned}
\l^Tz+\tilde A\;=\;P_{\ker\l}\tilde A.\end{aligned}$$
follows immediately.\
The Lemmas \[Lem:LoesungenVonGleichungssystemen\] and \[Lem:DasIstJaEinProjektor!\] enable us to rewrite (\[Gl:AllesMitVektorenAusgedrueckt\]) as $$\begin{aligned}
\label{Gl:JetztKoennenWirIntegrieren!}
(\ref{Gl:AusgangsFormel})\;&=&\;V\sqrt{\frac{2\pi}{t}}^E\int_{\R^{V}}\frac{d\tilde\phi_1\ldots
d\tilde\phi_{V}}{(2\pi)^{V-1}}\;\d\left(u^T\tilde\phi\right)\\[5pt]\nonumber
&&\qquad\qquad\qquad\times
\sum_{m_{V},\ldots,m_E\in\Z}\;\exp\left(-\frac{\tilde\phi^T\l\l^T\tilde\phi}{2t}
\,-\,\frac{\tilde A^TP_{\ker \l}\tilde A}{2t}\right).\end{aligned}$$
We can now finally evaluate the gauge integrals in (\[Gl:JetztKoennenWirIntegrieren!\]) with the help of Kirchhoff’s theorem. Since the delta-function in the integrand of (\[Gl:JetztKoennenWirIntegrieren!\]) assures that we only integrate over the orthogonal complement of $u$, instead of $\R^V$, and Kirchhoff’s theorem \[Thm:Kirchhoff\] assures that the Kirchhoff-matrix $\l\l^T$ is positive definite there, we can immediately evaluate the integral: $$\begin{aligned}
\int_{\R^V}\frac{d\tilde\phi_1\cdots
d\tilde\phi_V}{(2\pi)^{V-1}}\,\d\left(u^T\tilde\phi\right)\;\exp\left(-\frac{\tilde\phi^T\l\l^T\tilde\phi}{2t}\right)\;
&=&\;\sqrt\frac{t}{2\pi}^{V-1}\;\frac{1}{\sqrt{\prod_{a=2}^V\m_a}}\\[5pt]\nonumber
&=&\;\frac{1}{\sqrt{G\,V}}\;\sqrt\frac{t}{2\pi}^{V-1}\end{aligned}$$
where $\m_2,\ldots,\m_V$ are the nonzero eigenvalues of the Kirchhoff-matrix, and $G$ is the number of different possible maximal trees in the graph $\g$. With this, the gauge-invariant coherent state can be written as $$\begin{aligned}
(\ref{Gl:AusgangsFormel})\;=\;\sqrt\frac{V}{G}\sqrt\frac{2\pi}{t}^{E-V+1}\sum_{m_V,\ldots,m_E\in\Z}\exp\left(-\frac{(A-2\pi
m)^TP_{\ker\l}(A-2\pi m)}{2t}\right)\end{aligned}$$
where $A=z-\phi$ is the vector containing $A_k=z_k-\phi_k$ in its $k$-th component, and $m$ being the vector containing $0$ in the first $V-1$ components and $m_V,\ldots,m_E$ in the last $E-V+1$ components.
As already stated, the kernel of $\l$ is $E-V+1$-dimensional. Let $l_1,\ldots,l_{E-V+1}$ be an orthonormal basis of $\ker\l\subset\R^E$. Define $$\begin{aligned}
\label{Gl:EichinvarianteKombinationen}
z_{\n}^{gi}\;:=\;l_{\n}^Tz,\qquad
\phi^{gi}_{\n}\;:=\;l_{\n}^T\phi,\qquad m^{gi}_{\n}\;:=\;l_{\n}^Tm.\end{aligned}$$
With this and $P_{\ker\l}=\sum_{\n=1}^{E-V+1}l_{\n}l_{\n}^T$, we get our final formula $$\begin{aligned}
\label{Gl:FinaleFormel}
\Psi_{[z]}^t(\phi)&&\\[5pt]\nonumber
\;&=&\;\sqrt\frac{V}{G}\sqrt\frac{2\pi}{t}^{E-V+1}\sum_{m_V,\ldots,m_E\in\Z}\exp\left(-\sum_{\n=1}^{E-V+1}\frac{(z^{gi}_{\n}-\phi_{\n}^{gi}-2\pi
m^{gi}_{\n})^2}{2t}\right).\end{aligned}$$
The gauge-invariant coherent state only depends on the $z^{gi}_{\n}$ and $\phi^{gi}_{\n}$, which are gauge-invariant combinations of the $z_k$ and $\phi_k$. That the linear combinations (\[Gl:EichinvarianteKombinationen\]) are gauge-invariant, is clear from the construction, but one can immediately see this from the following: Perform a gauge-transformation, which shifts the $\phi_k$ by $\l_{ka}\tilde\phi_a$. So, in matrices, one has $\phi\to\phi+\l^T\tilde\phi$. Thus, $$\begin{aligned}
\phi^{gi}_{\n}\;=\;l^T_{\n}\phi\;\longrightarrow\;l^T_{\n}(\phi+\l^T\tilde\phi)\;=\;l^T_{\n}\phi\;+\;l^T_{\n}\l^T\tilde\phi\;=\;l^T_{\n}\phi\;=\;\phi^{gi}_{\n},\end{aligned}$$
where $l_{\n}\in\ker\l$ has been used, from which it follows that $\l l_{\n}=0$, so $l_{\n}^T\l^T=0$. Thus, the linear combinations of $\phi$ in $\phi^{gi}_{\n}$ are all gauge-invariant. The same holds true, of course, for the $z_{\n}^{gi}$ and $m_{\n}^{gi}$. So, the coherent states depend only on gauge-invariant combinations of $\phi$, which was clear from the beginning, but can now be seen explicitly. Note that the basis $\{l_{\n}\}_{\n=1}^{N-V+1}$ is, of course, not unique, but can be replaced by any other basis $l'_{\n}=R_{\n\m}l_{\m}$ with $R\in
O(E-V+1)$.\
Compare the formula for the gauge-invariant coherent state (\[Gl:FinaleFormel\]) with the formula for the gauge-variant coherent states on $E$ edges (\[Gl:EichvarianterKohaerenterZustandAufNemGraphen\]). Up to a factor of $(V/G)^{1/2}$, the similarity is striking. One could be led to the conclusion that gauge-invariant coherent states are nothing but gauge-variant coherent states, only depending on gauge-invariant quantities. The fact that the gauge-invariant configuration space is diffeomorphic to $U(1)^{E-V+1}$, supports this guess.
However, this is not true. The reason is that the summation variables $m_V,\ldots,m_E$ are placed in wrong linear combinations in the formula. In particular, a gauge-invariant state is *not* $$\begin{aligned}
\nonumber
\Psi_{[z]}^t(\phi)\;&\neq&\;\sqrt\frac{V}{G}\sqrt\frac{2\pi}{t}^{E-V+1}\sum_{m^{gi}_1,\ldots,m^{gi}_{E-V+1}\in\Z}
\exp\left(-\sum_{\n=1}^{E-V+1}\frac{(z^{gi}_{\n}-\phi_{\n}^{gi}-2\pi
m^{gi}_{\n})^2}{2t}\right)\\[5pt]\label{Gl:SchoenWaers!}
&=&\;\sqrt\frac{V}{G}\psi^t_{z^{gi}}(\phi^{gi}).\end{aligned}$$
Of course, from the form (\[Gl:FinaleFormel\]) one cannot deduce a priori that the $m^{gi}_{\n}$ could not, probably, be reordered in a way, maybe by an intelligent choice of $l_{\n}$ and/or suitable shifting of summations, such that a form like (\[Gl:SchoenWaers!\]), possibly with different $t$ for different variables, could be possible. But already at simple examples like the $3$-bridge graph show that this cannot be done. It could be, if one is lucky (in particular, on the $2$-bridge graph), but generically a gauge-invariant coherent state is no complexifier coherent state depending on gauge-invariant variables.
Peakedness of gauge-invariant coherent states
---------------------------------------------
In this chapter, we will shortly investigate the peakedness properties of the gauge-invariant coherent states. In particular, we will show that they are peaked on gauge-invariant quantities. Let $\g$ be a graph with $E$ edges. Then, a complexifier coherent state is then labeled by $E$ complex numbers $z_1,\ldots,z_E$ and a semiclassicality parameter $t>0$. Such a state is given by $$\begin{aligned}
\psi^t_z(\phi)\;=\;\sqrt\frac{2\pi}{t}^E\sum_{m_1,\ldots,m_E\in\Z}\exp\left(-\sum_{k=1}^E\frac{(z_k-\phi_k-2\pi
m_k)^2}{2t}\right).\end{aligned}$$
The corresponding gauge-invariant coherent states, obtained by applying the projector onto the gauge-invariant sub-Hilbert-space, has, in the last section, been shown to be $$\begin{aligned}
\Psi^t_{[z]}(\phi)\;=\;\sqrt{\frac{V}{G}}\sqrt\frac{2\pi}{t}^{E-V+1}\sum_{m_V,\ldots,m_E\in\Z}\exp\left(-\sum_{\n=1}^{E-V+1}\frac{(z^{gi}_{\n}-\phi^{gi}_{\n}-2\pi
m_{\n}^{gi})^2}{2t}\right).\end{aligned}$$
Here $G$ is the number of different possible maximal trees is the graph $\g$ and $\phi_{\n}^{gi}=l_{\n}^T\phi$, where $l_1,\ldots,l_{E-V+1}$ is an orthonormal base for the kernel $\ker\l\subset \R^E$ of the incidence matrix $\l$ of $\g$. Also, $z_{\n}^{gi}=l_{\n}^Tz$ and $m_{\n}^{gi}=l_{\n}^Tm$, where $m$ is the vector containing zeros in the first $V-1$ entries, and $m_V$ to $m_E$ in the last $E-V+1$ entries.\
The inner product between two gauge-invariant coherent states $\Psi^t_{[w]}$ and $\Psi^t_{[z]}$ is, as one can easily calculate, given by $$\begin{aligned}
\nonumber
\left\langle\Psi^t_{[w]}\Big|\Psi^t_{[z]}\right\rangle\;=\;\sqrt\frac{V}{G}\sqrt\frac{\pi}{t}^{E-V+1}
\sum_{m_V,\ldots,m_E\in\Z}\exp\left(-\sum_{\n=1}^{E-V+1}\frac{(\bar
w^{gi}_{\n}-z^{gi}_{\n}-2\pi m_{\n}^{gi})^2}{t}\right).\\[5pt]\label{Gl:EichinvariantesInneresProdukt}\end{aligned}$$
With $z_k=\phi_k-ip_k$, i.e. by splitting the phase-space points into configuration- and momentum variables, one immediately gets a formula for the norm of a gauge-invariant coherent state:
$$\begin{aligned}
\nonumber
\left\|\Psi^t_{[z]}\right\|^2\;=\;\sqrt\frac{V}{G}\sqrt\frac{\pi}{t}^{E-V+1}
\sum_{m_V,\ldots,m_E\in\Z}\exp\left(4\sum_{\n=1}^{E-V+1}\frac{(p^{gi}_{\n}-\pi
i m_{\n}^{gi})^2}{t}\right).\\[5pt]\label{Gl:NormOFAGaugeInvariantCoherentState}\end{aligned}$$
Note that there is, apart from $m=0$, no combination of $m_V,\ldots,m_E$ such that the corresponding $m_{\n}^{gi}=0$ for all $\n=1,\ldots,E-V+1$. If there is one such combination, there are infinitely many of these combinations, hence infinitely many equally large terms. So, if there were, then the sum in (\[Gl:EichinvariantesInneresProdukt\]) would not exist at all. But we know that the sum in (\[Gl:EichinvariantesInneresProdukt\]) is absolutely convergent, so there is no such combination.
What we just said is equivalent to saying that $$\begin{aligned}
P_{\text{ker}\,\l}\left(\begin{array}{c}0\\\vdots\\0\\m_V\\\vdots\\
m_E\end{array}\right)\;\neq\;0\qquad\mbox{for all }m_V,\ldots
m_E\in\Z,\end{aligned}$$
which is, of course, due to the fact that the last $E-V+1$ components correspond, by construction, to the gauge-invariant directions on $U(1)^E$. In the limit of $t\to 0$, the norm of a gauge-invariant coherent state (\[Gl:NormOFAGaugeInvariantCoherentState\]) can be written as
$$\begin{aligned}
\left\|\Psi^t_{[z]}\right\|^2\;&\leq&\;\sqrt\frac{V}{G}\sqrt\frac{\pi}{t}^{E-V+1}
\sum_{m_V,\ldots,m_E\in\Z}\exp\left(4\sum_{\n=1}^{E-V+1}\frac{(p^{gi}_{\n})^2-\pi^2
(m_{\n}^{gi})^2}{t}\right)\\[5pt]\nonumber
&=&\;\sqrt\frac{V}{G}\sqrt\frac{\pi}{t}^{E-V+1}\,\exp\left({4\sum_{\n=1}^{E-V+1}\frac{(p_{\n}^{gi})^2}{t}}\right)
\sum_{m_V,\ldots,m_E\in\Z}\;\exp\left(-4\pi^2\sum_{\n=1}^{E-V+1}\frac{
m^TP_{\text
{ker}\l}m}{t}\right)$$
Define $$\begin{aligned}
K\;:=\;\min_{\|m\|=1}\left\|P_{\text {ker}}m\right\|\;>\;0.\end{aligned}$$
With this, $m^TP_{\text {ker}}m\;\geq\,K^2\|m\|^2$, so we get $$\begin{aligned}
\nonumber
\sum_{m_V,\ldots,m_E\in\Z}\;\exp\left(-4\pi^2\sum_{\n=1}^{E-V+1}\frac{
m^TP_{\text
{ker}\l}m}{t}\right)\;&\leq&\;\sum_{m_V,\ldots,m_E\in\Z}\exp\left(-4\pi^2K^2\frac{\|m\|^2}{t}\right)\\[5pt]\label{Gl:RechnungWarumManDieMsWeglassenKann}
&=&\;\left[\sum_{n\in\Z}\exp\left(\frac{-4\pi^2K^2}{t}n^2\right)\right]^{E-V+1}\\[5pt]\nonumber
&=&\;1\,+\,O(t^{\infty}).\end{aligned}$$
Thus, we can write $$\begin{aligned}
\label{Gl:NormDerEichinvariantenZustaende}
\left\|\Psi^t_{[z]}\right\|^2\;=\;\sqrt\frac{V}{G}\sqrt\frac{\pi}{t}^{E-V+1}
\sum_{m_V,\ldots,m_E\in\Z}\exp\left(4\sum_{\n=1}^{E-V+1}\frac{(p^{gi}_{\n})^2}{t}\right)(1+O(t^{\infty})).\end{aligned}$$
For the inner product between complexifier coherent states, one has $$\begin{aligned}
\label{Gl:ShiftenDerArgumenteDerKomplexifiziererZustaende}
\left\langle\psi_w^t\Big|\psi^t_z\right\rangle\;=\;\left\langle\psi_0^t\Big|\psi^t_{z-\bar
w}\right\rangle,\end{aligned}$$
as can be readily deduced from the explicit formula of the inner product between two complexifier coherent states. This is also true for the gauge-invariant coherent states, which have $$\begin{aligned}
\left\langle\Psi_{[w]}^t\Big|\Psi^t_{[z]}\right\rangle\;=\;\left\langle\Psi_{[0]}^t\Big|\Psi^t_{[z-\bar
w]}\right\rangle.\end{aligned}$$
This can either be deduced by applying the gauge-projector onto (\[Gl:ShiftenDerArgumenteDerKomplexifiziererZustaende\]), or directly from formula (\[Gl:EichinvariantesInneresProdukt\]).
So, in order to show that the overlap of two gauge-invariant coherent states, labeled by $[z]$ and $[w]$, is peaked at $[z]=[w]$, one only has to show that the overlap between a state labeled by $[z]$ and $\Psi^t_{[0]}$ is peaked at $[z]=[0]$. With (\[Gl:NormDerEichinvariantenZustaende\]) and $z=\phi-ip$, we get
$$\begin{aligned}
\frac{\left\langle\Psi_{[0]}^t\Big|\Psi^t_{[z]}\right\rangle}{\left\|\Psi^t_{[0]}\right\|\;\left\|\Psi^t_{[z]}\right\|}
\;&=&\;\sum_{m_V,\ldots,m_E\in\Z}\exp\left(-\sum_{\n=1}^{E-V+1}\frac{(\phi^{gi}_{\n}-ip^{gi}{\n}+2\pi
m_{\n}^{gi})^2}{t}\,-\,\sum_{\n=1}^{E-V+1}\frac{2(p^{gi}_{\n})^2}{t}\right)\\[5pt]
&&\qquad\times(1+O(t^{\infty}))\\[5pt]
&=&\;\sum_{m_V,\ldots,m_E\in\Z}\exp\left(-\sum_{\n=1}^{E-V+1}\frac{(\phi^{gi}_{\n}-2\pi
m_{\n}^{gi})^2+(p_{\n}^{gi})^2}{t}+2i\frac{p_{\n}^{gi}(\phi^{gi}_{\n}-2\pi
m_{\n}^{gi})}{t}\right)\\[5pt]
&&\qquad\times(1+O(t^{\infty})).\end{aligned}$$
If the $\phi^{gi}_{\n}$ are close to zero, then the term with all $m^{gi}_{\n}=0$, which corresponds to all $m_k=0$, is significantly larger than the other terms. So this can, with similar arguments as in (\[Gl:RechnungWarumManDieMsWeglassenKann\]), be further simplified into $$\begin{aligned}
\frac{\left\langle\Psi_{[0]}^t\Big|\Psi^t_{[z]}\right\rangle}{\left\|\Psi^t_{[0]}\right\|\;\left\|\Psi^t_{[z]}\right\|}
\;&=&\;\exp\left(-\sum_{\n=1}^{E-V+1}\frac{(\phi^{gi}_{\n})^2+(p_{\n}^{gi})^2}{t}+2i\frac{p_{\n}^{gi}\phi^{gi}_{\n}}{t}\right)(1+O(t^{\infty})).\end{aligned}$$
This approaches $1$ if the gauge-invariant quantities $\phi^{gi}$ and $p^{gi}$ are close to zero, but as soon as the gauge-invariant quantities are away from zero, the expression becomes tiny, due to the tiny $t$. It follows that the overlap is peaked at gauge-invariant quantities.\
Summary and conclusion
======================
This is the first of two articles concerning the gauge-invariant coherent states for Loop Quantum Gravity. In this one, we have replaced the gauge-group $G=SU(2)$ of LQG by the much simpler $G=U(1)$, the case $G=U(1)^3$, which is also of interest for LQG, follows immediately. We have investigated the gauge-invariant coherent states, in particular we have computed their explicit form and their overlap. The results found are very encouraging: While the complexifier coherent states are peaked on points in the kinematical phase space, which contains gauge information, the gauge-invariant coherent states, which are labeled by gauge-equivalence classes, are also sharply peaked on these. In particular, the overlap between two gauge-invariant coherent states labeled with different gauge orbits tends to zero exponentially fast as the semiclassicality parameter $t$ tends to zero. Even more, it could be shown that the overlap is actually a Gaussian in the gauge-invariant variables.
This shows the good semiclassical properties of these states: As $t$ tends to zero, different states become approximately orthogonal very quickly, suppressing the quantum fluctuations between them. Also, the expectation values of operators corresponding to gauge-invariant kinematical observables (such as volume or area) are approximated well, which immediately follows from the corresponding properties of the gauge-variant CCS states.
This shows that the gauge-invariant coherent states are in fact useful for the semiclassical analysis of the gauge-invariant sector of LQG, and is the first step on the road to *physical* coherent states.
Apart from the nice semiclassical properties, the computation has revealed an explicit connection between the gauge-invariant sector and the graph topology. In particular, the formula for the gauge-invariant coherent states on a graph $\g$ contains the incidence matrix $\l$ of $\g$. In contrast, the CCS are simply a product of states on each edge of the graph, hence have no notion of which edges are connected to each other and which are not, while the gauge-invariant coherent states explicitly contain information about the graph topology. This is simply due to the fact that the set of gauge-invariant degrees of freedom depend on the graph topology and can be computed by graph-theoretic methods.\
While the results for $G=U(1)$ are quite encouraging, the case of ultimate interest for LQG is $G=SU(2)$, which is much more complicated. We will address this topic in the following article, which will deal with this issue and try to establish as much results as possible from $U(1)$, where the problem could be solved completely and analytically, also for $SU(2)$.
Acknowledgements {#acknowledgements .unnumbered}
================
BB would like to thank Hendryk Pfeiffer for the discussions about gauge-invariant functions and cohomology. Research at the Perimeter Institute for Theoretical Physics is supported by the Government of Canada through NSERC and by the Province of Ontario.
Cohomology with values in abelian groups {#Ch:Gauge-invariantFunctionsOfU(1)}
========================================
In the following we will write down the definition for singular cohomology with values in an abelian group. This will allow for a compact notation of the gauge-invariant Hilbert space. In particular, we will characterize the cohomology spaces in question to arrive at a better understanding what to expect, when computing the gauge-invariant coherent states on graphs and their overlaps. Note that we will employ, for brevity, the notation
$$\begin{aligned}
A^B\;:=\;\{f:B\to A\text{ any map}\}\end{aligned}$$
for the set of maps from any set $B$ to any set $A$.
Consider a CW complex $K$, i.e. a topological space that is successively built up of $n$-cells (n-dimensional closed balls), such that the intersection of two cells is a collection of lower-dimensional sub-cells, and around each point there is a neigbourhood that contains finitely many cells. In particular, any graph in $\Sigma$ is a CW complex of dimension 1, i.e. consisting only of 1-cells (the edges), that intersect at the 0-cells (the vertices).
Let $K^n$ be the set of all $n$-cells in the CW complex $K$. Let $G$ be an abelian group, then define $C^n(K, G)$ to be the set $G^{K^n}$, i.e. the set of all maps from $K^n$ to $G$. Then $C^n(K,G)$ is obviously an abelian group, simply by defining the group multiplication pointwise. This group is obviously homomorphic to $G^{|K^n|}$.
We then define a chain by $$\begin{aligned}
\label{Gl:Kettenkomplex}
\{1\}\;\stackrel{\d}{\longrightarrow}\;C^0(K,G)\;\stackrel{\d}{\longrightarrow}\;C^1(K,G)\;
\stackrel{\d}{\longrightarrow}\;C^2(K,G)\;\stackrel{\d}{\longrightarrow}\;\ldots\end{aligned}$$
where $\d: C^n(K,G)\to C^{n+1}(K,G)$ is defined by the following rule: Let $f:K^n\to G $ be an element of $C^n(K,G)$. Then, for an $n+1$-cell $c$ we define $$\begin{aligned}
\label{Gl:DefinitionBoundaryOperator}
\d f(c)\;:=\;f(v_1)^{\s_1}\cdot\ldots\cdot
f(v_{|K^n|})^{\s_{|K^n|}},\end{aligned}$$
where $v_1,\ldots v_{|K^n|}$ are all $n$-cells and the factors $\sigma_k$ are defined to be $+1$ if $v_k$ is part of the boundary of $c$ and the orientation of $v_k$ is the same as the induced one from $c$, $-1$ if $v_k$ is in the boundary of $c$ but the induced orientation from $c$ and the given one on $v_k$ differ by a sign, $0$ if $v_k$ is not part of the boundary of $c$.
Note that $\d$ is a group homomorphism, which follows from the abeliness of $G$. Hence, for each $n$, both sets $\ker\;\d:C^n(K,G)\to C^{n+1}(K,G)$ and $\text{img}\,\d:C^{n-1}(K,G)\to C^n(K,G)$ are subgroups of $C^n(K,G)$, where the kernel of a group homomorphism is defined to be the set of all elements being mapped to the unit element.
One can explicitly check that with this definition, that the map $$\begin{aligned}
\d\d:C^n(K,G)\;\longrightarrow\;C^{n+2}(K,G)\end{aligned}$$
maps every $C^n(K,G)$ to the unit element in $C^{n+2}(K,G)$. It follows that even $\text{img}\,\d:C^{n-1}(K,G)\to
C^n(K,G)$ is a subgroup of $\ker\;\d:C^n(K,G)\to C^{n+1}(K,G)$. Thus, one can define the quotients $$\begin{aligned}
H^n(K,G)\;:=\;\frac{\ker\;\d:C^n(K,G)\to
C^{n+1}(K,G)}{\text{img}\,\d:C^{n-1}(K,G)\to C^n(K,G)},\end{aligned}$$
which is called the *$n$-th cohomology group of $K$ with values in $G$*. As the name suggests, this is of course also an abelian group.\
The definition above is fairly general, but we will now see what it means for the specific case of the CW complex being an oriented graph $\g$ (with the orientations of the vertices all being set to the number $+1$). We keep the abelian group $G$ arbitrary for the moment, having in mind the application to $G=U(1)$ or $G=U(1)^3$ lateron.\
Let us consider a graph $\g$, consisting of a set of edges $E(\g)$ and vertices $V(\g)$. The chain in (\[Gl:Kettenkomplex\]) is then simply $$\begin{aligned}
\{1\}\;\stackrel{\d}{\longmapsto}\;G^{V(\g)}\;\stackrel{\d}{\longmapsto}\;G^{E(\g)}\;\stackrel{\d}{\longmapsto};\{1\}.\end{aligned}$$
The only nontrivial map is $\d:G^{V(\g)}\to G^{E(\g)}$. For every edge $e\in E(\g)$, $b(e)$ and $f(e)$ are called the beginning- and endpoint of the edge, and are by construction both elements of $V(\g)$. So let $k:V(\g)\to G$ be an element of $G^{V(\g)}$. Then the definition of $\d$ given above implies $$\begin{aligned}
\label{Gl:ActionOfCoboundaryOperator}
(\d k)_e\;=\;k_{b(e)}\,k_{f(e)}^{-1},\end{aligned}$$
so $\d k $ is a map from $E(\g)$ to $G$, that is an element of $G^{E(\g)}$. The only nontrivial cohomology groups we can form are then $$\begin{aligned}
\label{Gl:DefinitionZerothCohomology}
H^0(\g,G)\;&=&\;\frac{\ker\;\d:G^{V(\g)}\to
G^{E(\g)}}{\text{img}\,\d:\{1\}\to
G^{V(\g)}}\;=\;\ker\;\d:G^{V(\g)}\to
G^{E(\g)},\\[5pt]\label{Gl:DefinitionFirstCohomology}
H^1(\g,G)\;&=&\;\frac{\ker\;\d:G^{E(\g)}\to
\{1\}}{\text{img}\,\d:G^{V(\g)}\to
G^{E(\g)}}\;=\;\frac{G^{E(\g)}}{\text{img}\,\d:G^{V(\g)}\to
G^{E(\g)}}.\end{aligned}$$
These two groups have nice interpretations in terms of the graph topology, which are stated by the following lemma:
\[Lem:TopologyInterpretationOfCohomologyGroups\] Let $\g$ be a graph (connected, oriented, finitely many edges, embedded in a 3-manifold $\Sigma$). Then, for any abelian group $G$, we have $$\begin{aligned}
\label{Gl:MeaningOfCohomologyGroups}
H^0(\g,G)\;&\simeq&\;G,\\ H^1(\g, G)\;&\simeq&\;\text{Hom
}(\pi_1(\g),G),\end{aligned}$$
Loosely speaking, $H^0(\g,G)$ counts the connected parts of $\g$, and $H^1(\g, G)$ counts the numbers of “holes” in $\g$.
**Proof:** The proof is quite standard, but we will still repeat it here.
By (\[Gl:ActionOfCoboundaryOperator\]) and (\[Gl:DefinitionZerothCohomology\]), we see that $H^0(\g,G)$ consists of all maps $k$ from $V(\g)$ to $G$, such that, for every edge $e$, $k_{b(e)}\,k_{f(e)}^{-1}=1$. Since the graph is connected, this is equivalent to saying that the map $k$ assigns to each vertex $v\in V(\g)$ the same element in $G$: $$\begin{aligned}
k_v\;=\; h \quad\mbox{for some } h\in G\mbox{ and all }v\in V(\g).\end{aligned}$$
The group of all such maps is then clearly equivalent to the group $G$ itself, since the graph $\g$ is connected. So we have $$\begin{aligned}
H^0(\g,G)\;\simeq\;G.\end{aligned}$$
To show the second part of (\[Gl:MeaningOfCohomologyGroups\]), consider a maximal tree $\t$ in the graph $\g$. A maximal tree is a subgraph such that each vertex of $\g$ is also contained in $\t$ (i.e. $V(\g)=V(\t)$), and the graph $\t$ contains no closed loops. Call all edges in $\g$ that are not in $\t$ leaves. Maximal trees exist for all graphs, although they are far from unique. The number of leaves in a graph, though, is independent from the choice of $\t$.
To compute $H^1(\g,G)$, we have to compute the orbits of the subgroup $\d(G^{V(\g)})\subset G^{E(\g)}$. We do this by showing that, to each $h\in G^{E(\g)}$, we can apply an element of $\d(G^{V(\g)}$, such that the result is an element $\tilde h\in
G^{E(\g)}$ such that $\tilde h_e=1$ for all $e\in E(\t)$. In short, we show that one can gauge fix the group elements on the edges belonging to the tree $\t$ to 1. The remaining distribution of elements $\tilde h_e$ for leaves $e$ is unique, due to the fact that the group $G$ is abelian.
Consider an element $h$ of $G^{E(\g)}=C^1(\g,G)$, i.e. a distribution $(h_{e_1},\ldots,h_{e_E})$ of elements in $G$ among the edges in $E(\g)$. Construct an element $k\in V(\g)$ by the following method: Choose a vertex $v$ in $V(\g)$. For each other vertex $w\in
V(\g)$, there is a unique path from $w$ to $v$ along edges in $\t$, since $\t$ contains no loops. So, in order to get from $w$ to $v$, one has to go, say, along edges $e_1,\ldots e_n$, either parallel or antiparallel to the orientation of the $e_i$. Define the $k_w$ to be the product $$\begin{aligned}
\label{Gl:GaugeFixingProcedure}
k_w\;=\;h_{e_1}^{\pm 1}h_{e_{2}}^{\pm 1}\cdots h_{e_n}^{\pm 1},\end{aligned}$$
where the element $h_{e_i}$ is contained in the product, if the path from $w$ to $v$ is parallel to the orientation of $e_i$. If the path is antiparallel, then take $h_{e_i}^{-1}$ instead.
Thus, an element $k\in G^{V(\g)}$ is defined. Now consider the product $\tilde h:=\d k\cdot h$. It is quite easy to see that the element $\tilde h$ assigns $1\in G$ to each $e\in E(\t)$: consider an $e\in E(\t)$. The path from $f(e)$ to $v$ passes through $b(e)$, or the other way round. Assume the first to be the case, the other case works analogously. We have then $$\begin{aligned}
k_{f(e)}\;=\;h_e\,k_{b(e)},\end{aligned}$$
since the path from $f(e)$ goes against the orientation of $e$ to $b(e)$, and then is identical to the way from $b(e)$ to $v$, since $\t$ contains no loops. So $$\begin{aligned}
\tilde h_e\;=\;k_{b(e)}\;h_e\;k_{f(e)}^{-1}\;=\;1.\end{aligned}$$
Thus, we have shown, the orbit of each element $h\in
G^{E(\g)}$ under the action of the subgroup $\d(G^{V(\g)})$ contains an element $\tilde h$ such that only the elements assigned to the leaves in $\g$ are potentially different from $1\in G$. One can also see that the only element in $\d(G^{V(\g)})$ that leaves the distribution of $1$ along the edges of $E(\t)$ unchanged, is an element $k\in \ker \d$, i.e. such that $k_v=h$ for some $h\in G$ and all $v\in V(\g)$. The multiplication with $\d k$ leaves $G^{E(\g)}$ invariant, since $G$ is abelian. We thus see that the element of $\tilde h$ is unique for each $h\in
E(\g)$, hence does not depend on the choice of the vertex $v$. This shows that each orbit in $G^{E(\g)}$ under the action of $\d(G^{V(\g)})$ determines uniquely a distribution of group elements in $G$ among the leaves of $\g$.
Since $\t$ contains no loops, it is contractible. Consider the flower graph $\tilde \g$ that one obtains by contracting the tree $\t$ to a point. This graph contains just one vertex $V$ and a number of edges, all starting and ending at $V$, corresponding to the number of leaves of $\g$. Note that $H^1(\tilde \g,G)=E(\tilde
\g)^G$. From this and our considerations above it follows that there is a natural group isomorphism between $H^1(\g,G)\simeq
H^1(\tilde\g,G)$. It is clear that the first fundamental group $\pi_1(\tilde \g)$ is freely generated by the elements of $E(\tilde\g)$. In particular, $H^1(\tilde\g,G)\simeq\text{
Hom}(\pi_1(\tilde\g),G)$.
Since $\t$ contains no loops, the tree is contractible, hence $\tilde \g$ is a retraction of $\g$. In particular, both graphs are homotopy equivalent. Since the first fundamental group is a homotopy invariant, we conclude $$\begin{aligned}
H^1(\g,G)\;\simeq\;\text{Hom}(\pi_1(\g),G).\end{aligned}$$
In particular, $H^1(\g,G)\simeq G^L$, where $L$ is the number of leaves in $\g$ (which is independent of the choice of the maximal tree $\t$).
Gauge-invariant functions
-------------------------
The notion of gauge-invariant cylindrical functions fits nicely into the framework of cohomology. Remember that a gauge-variant function on a graph $\g$ is determined via (\[Gl:CorrespondingFunction\]) by a function of a number of copies of the gauge group $G$: $$\begin{aligned}
\tilde f:\underbrace{G\times\cdots\times G}_{\mbox{one for each edge
in }E(\g)}\;\to\;\C\end{aligned}$$
that is square-integrable with respect to the product Haar-measure $d\m_H^{\otimes|E(\g)|}$. These function constitute the Hilbert space $\H_{\g}$, and with the notions of the previous sections, we identify this space to be $$\begin{aligned}
\H_{\g}\;\simeq\;L^2\big(C^1(\g,G),\;d\m_H^{\otimes|E(\g)|}\big).\end{aligned}$$
The gauge transformed $\tilde f$ is determined by letting the gauge group $G$ act on every vertex $v\in V(\g)$ via (\[Gl:ActionOfGaugeGroup\]): $$\begin{aligned}
\a_{k_{v_1},\ldots,k_{v_V}}\tilde
f\;\big(h_{e_1},\ldots,h_{e_E}\big)\;=\;\tilde
f\big(k_{b(e_1)}^{-1}h_{e_1}
k_{f(e_1)},\;\ldots\;,k_{b(e_E)}^{-1}h_{e_E} k_{f(e_E)}\big),\end{aligned}$$
where $b(e)$ and $f(e)$ are the vertices sitting at the beginning and the end of the edge $e$ respectively.\
Not only do we recognize the gauge transformation group as the space $G^{V(\g)}=C^0(\g,G)$ from the previous section, one can see readily the connection between the gauge transformation $\a$ and the coboundary operator $\d$: $$\begin{aligned}
(\a_{g_1,\ldots,g_V}\tilde f)(h_1,\ldots,h_E)\;=\;\tilde
f\big(\d(g_1,\ldots,g_V)\cdot (h_1,\ldots,h_E)\big),\end{aligned}$$
where $\cdot$ means group multiplication in $C^1(\g,G)=G^{E(\g)}$. So, the gauge-invariant functions on the graph $\g$ are just the functions on the group $G^{E(\g)}$ that are invariant under the action of $\d(G^{V(\g)})$. We conclude that the gauge-invariant functions coincide with the functions on the first cohomology class
$$\begin{aligned}
\P\H_{\g}\;\simeq\;L^2\big(H^1(\g,G),\;d\m\big),\end{aligned}$$
where the measure $d\m$ is the quotient measure of $d\m_H^{\otimes|E(\g)|}$ under the action of the gauge transformation group $G^{V(\g)}$, which, since $H^1(\g,G)$ is a group for abelian $G$, can be identified with the normalized Haar measure on $H^1(\g,G)$.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We derive an exact duality transformation for pure non-Abelian gauge theory regularized on a lattice. The duality transformation can be applied to gauge theory with an arbitrary compact Lie group $G$ as the gauge group and on Euclidean space-time lattices of dimension $d\geq 2$. It maps the partition function as well as the expectation values of generalized non-Abelian Wilson loops (spin networks) to expressions involving only finite-dimensional unitary representations, intertwiners and characters of $G$. In particular, all group integrations are explicitly performed. The transformation maps the strong coupling regime of non-Abelian gauge theory to the weak coupling regime of the dual model. This dual model is a system in statistical mechanics whose configurations are spin foams on the lattice.'
address: |
Department of Applied Mathematics and Theoretical Physics,\
Centre for Mathematical Sciences, Cambridge CB3 0WA, UK
author:
- 'Robert Oeckl[^1] and Hendryk Pfeiffer[^2]'
title: |
The dual of pure non-Abelian lattice gauge theory\
as a spin foam model
---
Introduction
============
Besides the electric-magnetic duality of the vacuum Maxwell equations, the first example of a duality transformation relating the strong coupling regime of one field theoretic system with the weak coupling regime of the same or another system was probably the Kramers-Wannier transformation for the Ising model [@KrWa41a]. This transformation was generalized to a wide class of Abelian lattice models (spin models, gauge theories and their higher rank tensor generalizations) in any Euclidean space-time dimension. For a review see [@Sa80].
In this paper, we generalize the transformation to the case of lattice gauge theory with a non-Abelian gauge group $G$ (our proofs are valid for compact Lie groups and finite groups). The resulting dual model generalizes the well-known results for the Abelian case and is described in terms of the finite-dimensional unitary representations of $G$, their representation morphisms (intertwiners) and the character expansion of the Boltzmann weight. In particular, all group integrations are explicitly performed.
The method we use is the Peter-Weyl decomposition of the algebra $\Calg(G)$ of representation functions of $G$, the ‘algebraic functions’ on $G$. This decomposition can be viewed as a generalization of Fourier transformation to functions on a compact non-Abelian Lie group. It is convenient to exploit the Hopf algebra structure of $\Calg(G)$ and to employ a purely algebraic description of the Haar measure. The duality transformation follows the lines of the well-known Abelian case, but some attention and geometric intuition is necessary to make the generalized gauge constraint appear in a local form in the dual model.
The duality transformation establishes the equality of the partition function and the expectation value of the non-Abelian Wilson loop (the generic gauge invariant expression which is given by a spin network) with their corresponding purely algebraic expressions in the dual model. The dual model is found to have a Boltzmann weight of such a form that the strong coupling regime of non-Abelian lattice gauge theory is mapped to the weak coupling regime of the dual model. In addition, the strong coupling expansion can be applied to this reformulation of non-Abelian gauge theory in a systematic way.
The duality relation is stated in Theorems \[thm\_dualpartition1\] and \[thm\_dualwilson1\] which form the main result of this paper. The dual model is a system in statistical mechanics whose configurations are spin foams on the lattice. These configurations are assigned Boltzmann weights and are subject to certain constraints. Spin foams have been introduced in the study of quantum gravity, see [@Re94; @Iw95; @ReRo97; @Ba98] and the recent introductory article [@Ba99]. The configurations of our dual model are closed spin foams on the lattice according to the definition given in [@Ba98]. The transformation thus provides an explicit example for the relation of lattice gauge theory with a particular spin foam model.
The dual model reduces to the known results for non-Abelian gauge theory in $2$ dimensions where the partition function is particularly simple, as well as to the known results for Abelian lattice gauge theory in arbitrary dimension, in the cases $G=U(1)$, $\Z$ and $\Z_n$.
Whereas the duality transformation in the Abelian case is known to work in a similar way for spin models, gauge theories and higher rank tensor models on the lattice [@Sa80], this is not the case for the non-Abelian generalization. Even though it can be applied to spin models in a straight forward way, the resulting ‘gauge’ constraint cannot be cast in a local form.
The motivation for deriving a dual description of non-Abelian lattice gauge theory arises from conceptual issues such as the ‘dual superconductor’ picture of confinement and from rigorous studies in the framework of constructive quantum field theory as well as from technical and numerical problems in lattice gauge theory.
At present only a few ways of studying gauge theories in their strong coupling regime are known — in particular if there are no additional symmetries like supersymmetry. In the famous paper by Wilson [@Wi74], the lattice formulation of $U(1)$ gauge theory was used in conjunction with the high temperature expansion which is known from statistical mechanics and which plays the role of a strong coupling expansion.
The duality transformation for Abelian lattice gauge theories (see [@Sa80]) can be seen on the one hand as a result of attempts to make this strong coupling expansion systematic. On the other hand there is the picture of ‘dual superconductivity’ as an explanation of confinement going back to ideas of t’Hooft and Mandelstam, see [@BaMy77; @Pe78]. Whereas in a superconductor electrically charged quasi-particles condense and force the magnetic flux into quantized tubes, the picture is that in a gauge theory with confinement, magnetic monopoles condense. This leads to the formation of electric flux tubes which are responsible for the linearity of the static potential between opposite external electric charges.
In this picture the magnetic monopoles appear as collective excitations of the gauge theory. They are found to be quantized topological defects [@FrMa86; @FrMa87]. The Abelian duality transformation allows one to spot these topological degrees of freedom. All group integrations are performed, and the partition function of Abelian lattice gauge theory is rewritten in new variables such that the topological degrees of freedom have the form of ordinary expectation values of the dual fields.
In the Abelian case, the lattice approach to gauge theories in conjunction with the duality transformation and related techniques has lead to a number of remarkable results. We mention the existence of a phase transition in $U(1)$ lattice gauge theory in $d=4$ [@Gu80; @FrSp82], the existence of world-lines of magnetic monopoles in the same model which behave like infra-particles [@FrMa86; @FrMa87], the fact that these are responsible for the phase transition by condensation of magnetic monopoles, and finally the absence of a deconfinement phase transition in $d=3$ [@GoMa82]. The picture of dual superconductivity leading to confinement in the Abelian case is well established, see the study of the monopole degrees of freedom in Monte-Carlo simulations [@PoWi91] and their properties at the deconfining phase transition [@JeNe99] of $U(1)$ lattice gauge theory in $d=4$.
Although there are strong conjectures both from lattice studies of QCD and from results in supersymmetric Yang-Mills theory that confinement in non-Abelian gauge theories can be described in a similar way by dual superconductivity, no analogous approach is available for non-Abelian gauge theory. The exact duality transformation presented in this paper is meant to be a first step in this direction.
Finally, we comment on the relations of this work with other approaches. Firstly, in the Abelian case, the dual model is again a gauge theory if certain cohomologies of the space-time lattice are trivial. The case of general topology is studied in [@Ja99]. Furthermore, for $SU(2)$ lattice gauge theory, there are results by explicit computations which relate this theory to certain simplex models of gravity [@AnCh93; @DiPe99].
Furthermore, there has been an interesting categorial approach to models which could be dual to non-Abelian gauge theory [@GrSc98]. It uses the same colouring of links and plaquettes of the lattice with representations and intertwiners, respectively, as our dual model does on the dual lattice in $d=3$ and seems to enjoy many close similarities. However, the gauge degrees of freedom proposed in [@GrSc98] are not in a straight forward way a symmetry of the dually transformed model as derived in this paper. Nevertheless, the ideas developed there might prove useful for a further study of the dual model.
We thank M.B. Halpern for bringing the paper [@Ba82b] to our attention. In [@Ba82b] a plaquette formulation of lattice gauge theory is derived emphasizing the non-Abelian Bianchi identity. Using the mathematical methods that we describe in Sections \[sect\_prelimhopf\] to \[sect\_prelimhaar\], it can be shown that the approach of [@Ba82b] can be extended to a full duality transformation similar to the treatment in our Section \[sect\_transform\]. The result would be the same as given in Theorem \[thm\_dualpartition1\]. Finally, we thank a referee for drawing our attention to the paper [@HoFa96] where a duality transformation of classical Yang-Mills theory in continuous space-time based on the loop approach is suggested.
This paper is organized as follows. In Section \[sect\_prelim\], we recall all definitions and structures which are needed to present the duality transformation. In particular these are the structure of the Hopf algebra of representation functions $\Calg(G)$ of $G$ and the lattice formulation of gauge theories.
In Section \[sect\_transform\], we present the duality transformation in detail first for the partition function, and then for the non-Abelian generalizations of the Wilson loop. We specialize our result to the known cases of non-Abelian lattice gauge theory in $2$ dimensions and to Abelian lattice gauge theory in arbitrary dimension in Section \[sect\_special\].
Finally, in Section \[sect\_outlook\] we indicate how the dual formulation can be used in conjunction with strong coupling expansion techniques and discuss open questions and directions for further research.
Preliminaries {#sect_prelim}
=============
The Hopf algebra of representation functions {#sect_prelimhopf}
--------------------------------------------
In this section we collect definitions and basic statements related to the algebra of representation functions $\Calg(G)$ of $G$. These and the results presented in the next section about the Peter-Weyl decomposition and the Peter-Weyl theorem are basically text book knowledge, see [@BrDi85; @CaSe95]. We recall the basic facts to fix our notation and present a purely algebraic treatment of the relevant results which we have not found elsewhere in this form.
Let $G$ be a finite group or a compact Lie group. We denote finite-dimensional complex vector spaces on which $G$ is represented by $V_\rho$ and by $\rho\colon G\to\Aut V_\rho$ the corresponding group homomorphisms. Since each finite-dimensional complex representation of $G$ is equivalent to a unitary representation, we select a set $\Rep$ containing one unitary representation of $G$ for each equivalence class of finite-dimensional representations. The tensor product, the direct sum and taking the dual are supposed to be closed operations on this set. This amounts to a particular choice of representation isomorphisms $\rho_1\otimes\rho_2\leftrightarrow\rho_3$ , $\rho_j\in\Rep$, which is implicit in our formulas. We furthermore denote by $\Irrep\subseteq\Rep$ the subset of irreducible representations.
If $\rho\in\Rep$, we write $\rho^\ast$ for the dual representation and denote the dual vector space of $V_\rho$ by $V_\rho^\ast$. The dual representation is given by $\rho^\ast\colon G\mapsto \Aut
V_\rho^\ast$, where $$\label{eq_dualrep}
\rho^\ast(g)\colon V_\rho^\ast\to V_\rho^\ast,\quad
\eta\mapsto \eta\circ\rho(g^{-1}),$$ $(\rho^\ast(g)\eta)(v)=\eta(\rho(g^{-1})v)$ for all $v\in
V_\rho$.
For the unitary representations $V_\rho$, $\rho\in\Rep$, we have standard (sesquilinear) scalar products $\left<\cdot;\cdot\right>$ and orthonormal bases $(v_j)$ in such a way that the basis $(v_j)$ of $V_\rho$ is dual to the basis $(\eta^j)$ of $V_\rho^\ast$, $\eta^j(v_k)=\delta_{jk}$. This means that duality is given by the scalar product, $$\left<v_j;v_k\right>=\eta^j(v_k),\qquad
\bigl<\eta^j;\eta^k\bigr>=\eta^k(v_j),
\qquad 1\leq j,k\leq\dim V_\rho.$$ There exists a one-dimensional ’trivial’ representation of $G$ which we denote by $V_{[1]}\cong\C$.
The functions $$t_{\eta,v}^{(\rho)}\colon G\to\C,\quad g\mapsto\eta(\rho(g)v),$$ where $\rho\in\Rep$, $v\in V_\rho$ and $\eta\in V_\rho^\ast$, are called the *representation functions* of $G$. They form a commutative and associative unital algebra over $\C$, $$\Calg(G) := \{\,t_{\eta,v}^{(\rho)}\colon\quad
\rho\in\Rep, v\in V_\rho, \eta\in V_\rho^\ast\,\},$$ whose operations are given by
$$\begin{aligned}
\label{eq_operationsum}
(t_{\eta,v}^{(\rho)} + t_{\theta,w}^{(\sigma)})(g)
&:=& t_{\eta+\theta,v+w}^{(\rho\oplus\sigma)}(g),\\
\label{eq_operationprod}
(t_{\eta,v}^{(\rho)}\cdot t_{\theta,w}^{(\sigma)})(g)
&:=& t_{\eta\otimes\theta,v\otimes w}^{(\rho\otimes\sigma)}(g),\end{aligned}$$
where $\rho,\sigma\in\Rep$ and $v\in V_\rho$, $w\in V_\sigma$, $\eta\in V_\rho^\ast$, $\theta\in V_\sigma^\ast$ and $g\in G$. The zero element of $\Calg(G)$ is given by $t_{0,0}^{[1]}(g)=0$ and its unit element by $t_{\eta,v}^{[1]}(g)=1$ where we have normalized $\eta(v)=1$.
The algebra $\Calg(G)$ is furthermore equipped with a Hopf algebra structure with the coproduct $\Delta\colon\Calg(G)\to\Calg(G)\otimes\Calg(G)\cong\Calg(G\times G)$, the co-unit $\epsilon\colon\Calg(G)\to\C$ and the antipode $S\colon\Calg(G)\to\Calg(G)$ which are defined by
$$\begin{aligned}
\label{eq_matrixcopro}
\Delta t_{\eta,v}^{(\rho)} (g,h)
&:=& t_{\eta,v}^{(\rho)}(g\cdot h),\\
\epsilon t_{\eta,v}^{(\rho)}
&:=& t_{\eta,v}^{(\rho)}(1),\\
\label{eq_matrixanti}
S t_{\eta,v}^{(\rho)} (g)
&:=& t_{\eta,v}^{(\rho)}(g^{-1}),\end{aligned}$$
where $\rho\in\Rep$ and $v\in V_\rho$, $\eta\in V_\rho^\ast$ and $g,h\in G$.
Since $G$ is a finite group or a compact Lie group, all finite-dimensional representations of $G$ are completely reducible. Moreover, all representations of $G\times G$ are tensor products of representations of $G$ such that we have an isomorphism of algebras $\Calg(G\times G)\cong\Calg(G)\otimes\Calg(G)$ which is used in the definition of the coproduct. This tensor product is algebraic, and there is no need for a topology or a completion of the tensor product at this point.
In the standard orthonormal bases, the representation functions are given by the coefficients of representation matrices, $$t_{mn}^{(\rho)}(g) := t_{\eta^m,v_n}^{(\rho)}(g)
= \eta^m(\rho(g)v_n) = \left<v_m;\rho(g)v_n\right> = {\rho(g)}_{mn},$$ such that the coproduct corresponds to the matrix product, $$ \Delta t_{mn}^{(\rho)}(g,h)
= \sum_{j=1}^{\dim V_\rho} t_{mj}^{(\rho)}(g)t_{jn}^{(\rho)}(h),$$ while the antipode refers to the inverse matrix, $S
t_{mn}^{(\rho)}(g)={({\rho(g)}^{-1})}_{mn}$, and the co-unit describes the coefficients of the unit matrix, $\epsilon
t_{mn}^{(\rho)}=\delta_{mn}$. Furthermore, the antipode relates a representation with its dual, $$\begin{aligned}
\label{eq_antipode}
S t_{mn}^{(\rho)}(g) = \eta^m({\rho(g)}^{-1}v_n)
= (\rho^\ast(g)\eta^m)(v_n)
= \left<\eta^n;\rho^\ast(g)\eta^m\right>
= t_{nm}^{(\rho^\ast)}(g),\end{aligned}$$ which is just the conjugate representation because on the other hand $$S t_{mn}^{(\rho)}(g) = \left<v_m;\rho(g^{-1})v_n\right>
= \left<\rho(g)v_m;v_n\right>
= \overline{\left<v_n;\rho(g)v_m\right>}
= \overline{t_{nm}^{(\rho)}(g)}.$$ Here the bar denotes complex conjugation.
Peter-Weyl decomposition and theorem
------------------------------------
The structure of the algebra $\Calg(G)$ can be understood if $\Calg(G)$ is considered as a representation of $G\times G$ by combined left and right translation of the function argument, $$(G\times G)\times\Calg(G)\to\Calg(G),\quad
(g_1,g_2,f)\mapsto (h\mapsto f(g_1^{-1}hg_2)).$$ It can then be decomposed into its irreducible components as a representation of $G\times G$:
Let $G$ be a finite group or a compact Lie group.
There is an isomorphism $$\label{eq_structure_calg}
\Calg(G)\cong_{G\times G}
\bigoplus_{\rho\in\Irrep}(V_\rho^\ast\otimes V_\rho),$$ of representations of $G\times G$. Here the direct sum is over one unitary representative of each equivalence class of finite-dimensional irreducible representations of $G$. The direct summands $V_\rho^\ast\otimes V_\rho$ are irreducible as representations of $G\times G$.
The direct sum in is orthogonal with respect to the $L^2$ scalar product on $\Calg(G)$ which is formed using the Haar measure on $G$ on the left hand side, and using the standard scalar products on the right hand side, namely $$\label{eq_l2measure}
{\bigl<t_{\eta,v}^{(\rho)};t_{\theta,w}^{(\sigma)}\bigr>}_{L^2}
= \int_G\overline{t_{\eta,v}^{(\rho)}(g)}\cdot t_{\theta,w}^{(\sigma)}(g)\,dg
= \frac{1}{\dim V_{\rho}}\delta_{\rho\sigma}
\overline{\left<\eta;\theta\right>}\left<v;w\right>,$$ where $\rho,\sigma\in\Irrep$ are irreducible. The Haar measure is denoted by $\int_G\,dg$ and normalized such that $\int_G\,dg=1$.
If $G$ is finite, the Haar measure coincides with the normalized summation over all group elements.
The decomposition directly corresponds to our notation of the representation functions $t^{(\rho)}_{nm}(g)$ if $\rho\in\Irrep$ is irreducible.
Each representation function $f\in\Calg(G)$ has a decomposition according to , $$\label{eq_peterweyl_series}
f = \sum_{\rho\in\Irrep}f_\rho,$$ such that we find for the $L^2$-norm $${||f||}^2_{L^2} = \sum_{\rho\in\Irrep}\frac{1}{\dim V_\rho}{||f_\rho||}^2,$$ where $f_\rho\in V_\rho^\ast\otimes V_\rho$, $\rho\in\Irrep$, and all except finitely many $f_\rho$ are zero. Here ${||f_\rho||}^2$ is the trace norm for the finite-dimensional space $V_\rho^\ast\otimes V_\rho\cong\End V_\rho$.
The analytical aspects of $\Calg(G)$ can now be stated.
Let $G$ be a compact Lie group. Then $\Calg(G)$ is dense in $L^2(G)$.
We use the Peter-Weyl theorem to complete $\Calg(G)$ with respect to the $L^2$ norm to $L^2(G)$. Functions $f\in L^2(G)$ then correspond to square summable series in . These series are thus invariant under a reordering of summands, and their limits commute with group integrations. We will make use of these invariances in the duality transformation.
If $G$ is a finite group, $\Calg(G)$ is a finite-dimensional vector space such that the corresponding results hold trivially.
Character decomposition
-----------------------
If $G$ is a finite group or a compact Lie group, the characters $\chi^{(\rho)}\colon G\to\C$ associated with the finite-dimensional unitary representations $\rho\in\Rep$ of $G$ are obtained from the representation functions by $$\chi^{(\rho)} := \sum_{j=1}^{\dim V_\rho} t_{jj}^{(\rho)}.$$ Each class function $f\in\Calg(g)$ has a character decomposition $$\label{eq_chardecomp}
f(g) = \sum_{\rho\in\Irrep}\chi^{(\rho)}(g)\,\hat f_\rho,
\qquad\mbox{where}\qquad
\hat f_\rho = \dim V_\rho\,\int_G\overline{\chi^{(\rho)}(g)}f(g)\,dg.$$ The completion of $\Calg(G)$ to $L^2(G)$ is compatible with this decomposition.
Projector description of the Haar measure {#sect_prelimhaar}
-----------------------------------------
For the duality transformation, it is important to understand the Haar measure on $G$ in the picture of the Peter-Weyl decomposition . We describe the Haar measure in terms of projectors.
Let $G$ be a finite group or a compact Lie group and $\rho\in\Rep$ be a finite-dimensional unitary representation of $G$ with the orthogonal decomposition $$V_\rho\cong\bigoplus_{j=1}^k V_{\tau_j},\qquad \tau_j\in\Irrep, k\in\N,$$ into irreducible components $\tau_j$. Let $P^{(j)}\colon V_\rho\to
V_{\tau_j}\subseteq V_\rho$ be the $G$-invariant orthogonal projectors associated with the above decomposition. Assume that precisely the first $\ell$ components $\tau_1,\ldots,\tau_\ell$, $0\leq\ell\leq k$, are equivalent with the trivial representation. Then the Haar measure of a representation function $t_{mn}^{(\rho)}$, $1\leq m,n\leq\dim
V_\rho$, is given by $$\label{eq_haaralg}
\int_G t_{mn}^{(\rho)}(g)\,dg = \sum_{j=1}^\ell\bigl<v_m;P^{(j)}v_n\bigr>
= \sum_{j=1}^\ell\bigl<P^{(j)}v_m;P^{(j)}v_n\bigr>
= \sum_{j=1}^\ell P^{(j)}_mP^{(j)}_n,$$ where $P^{(j)}_m=\eta(P^{(j)}v_m)$ denotes the matrix elements of the $j$-th projector. Here $\eta\in V_\rho^\ast$ is the normalized linear form which is zero everywhere except on the one-dimensional sub-spaces $V_{\tau_j}\subseteq V_\rho$, $1\leq j\leq\ell$.
The representation function is Peter-Weyl decomposed by inserting $\openone=\sum_{j=1}^k P^{(j)}$ twice into the right hand side of $t_{mn}^{(\rho)}(g)=\left<v_m;\rho(g)v_n\right>$. We use hermiticity ${P^{(j)}}^\dagger=P^{(j)}$, $G$-invariance $[P^{(j)},\rho(g)]=0$ and transversality $P^{(i)}P^{(j)}=\delta_{ij}P^{(j)}$ to obtain $$\int_G t_{mn}^{(\rho)}(g)\,dg =
\sum_{j=1}^k\int_G\bigl<P^{(j)}v_m;\rho(g)P^{(j)}v_n\bigr>\,dg.$$ Since the Haar measure is bi-invariant, all terms vanish except those corresponding to the $\tau_j$, $1\leq j\leq\ell$, which are equivalent to the trivial representation.
The Lattice formulation of non-Abelian gauge theories
-----------------------------------------------------
The purpose of this section is to fix a notation in which we can write down the partition function and expectation values of non-Abelian lattice gauge theory and which is suitable to formulate the duality transformation. For all other issues we refer the reader to the standard text books on lattice gauge theory, [@Ro92; @MoMu94], and references therein.
We consider a regular hyper-cubic lattice corresponding to an Euclidean space-time of dimension $d\geq 2$. The *lattice points* (vertices) are denoted by tuples of integer numbers $$\Lambda^0 := \{\,(i_1,\ldots,i_d)\in\Z^d\colon\quad
i_\mu\in\{1,\ldots,N_\mu\}\,\},$$ where the lattice is of size $N_\mu$ in the $\mu$-th dimension. We denote the unit vectors along the lattice axes by $$\hat\mu:=(0,\ldots,0,\underbrace{1}_\mu,0,\ldots,0),\qquad 1\leq\mu\leq d.$$ Thus $i+\hat\mu$ refers to the neighbour of the point $i$ in the direction $\mu$. We choose periodic ( toroidal) boundary conditions and identify $i\pm N_\mu\cdot\hat\mu\equiv i$ for all $\mu\in\{1,\ldots,d\}$. It is crucial for the existence of various integrals that we work on a finite lattice. Periodic boundary conditions are the most convenient choice for our purpose. The non-trivial homologies introduced by the periodic boundary conditions do not play any role for the duality transformation in the form presented below.
The set of all *links* (edges) is called $\Lambda^1$, the set of all *plaquettes* (faces, squares) $\Lambda^2$, and more generally the set of all $k$-*cells* $\Lambda^k$. These are specified by $$\Lambda^k := \{\,(i,\mu_1,\ldots,\mu_k)\colon\quad i\in\Lambda^0,
1\leq\mu_1<\cdots<\mu_k\leq d\,\}.$$ In particular, the sets $\Lambda^k$, $0\leq k\leq d$, are all finite. The $k$-cells are considered unoriented, we do not want to distinguish the plaquette $(i,\mu,\nu)$ from $(i,\nu,\mu)$. In our notation both are represented in the standard way $(i,\mu,\nu)$ where $\mu<\nu$.
The *configurations* of lattice gauge theory are the maps $$g\colon\Lambda^1\to G,\quad (i,\mu)\mapsto g_{i\mu},$$ which assign a group element $g_{i\mu}\in G$ to each link $(i,\mu)\in\Lambda^1$ of the lattice. These group elements correspond to the parallel transports of the gauge connection along the links.
The path integral measure depends on these configurations only via the *plaquette product* (see also Figure \[fig\_plaquette\]), $$\label{eq_plaquette}
dg\colon\Lambda^2\to G,\quad
(i,\mu,\nu)\mapsto dg_{i\mu\nu}:=
g_{i\mu}\cdot g_{i+\hat\mu,\nu}\cdot g_{i+\hat\nu,\mu}^{-1}\cdot g_{i,\nu}^{-1},$$ which is the path ordered product of the link variables around a given plaquette $(i,\mu,\nu)\in\Lambda^2$.
For arbitrary *generating functions* $\phi\colon\Lambda^0\to G$, any class function of $G$, evaluated on $dg_{i\mu\nu}$, is invariant under the *gauge transformation* $$\label{eq_regauge}
g_{i\mu}\mapsto g_{i\mu}^\prime:=\phi_i\cdot g_{i\mu}\cdot\phi_{i+\hat\mu}^{-1}.$$ Let $G$ be a compact Lie group (or a finite group) and $\int_G\,dg$ denote the Haar measure. The path integral integrates over all configurations, it consists of one integration over $G$ for each link. We denote this integration by $$\int\sym{D}g = \Bigl(\prod_{(i,\mu)\in\Lambda^1}\int_G\,dg_{i\mu}\Bigr)
:= \underbrace{\int_G\,dg_{i\mu}\cdots\int_G\,dg_{i\mu}}_{
\mbox{one $\int_G$ for each link}}.$$ The path integral measure of lattice gauge theory is this integration together with a Boltzmann weight $\exp(-S(dg))$. Here the (local) *action* $S$ is given by a sum over all plaquettes, $$S(dg) := \sum_{(i,\mu,\nu)\in\Lambda^2} s(dg_{i\mu\nu}),$$ where $s\colon G\to\R$ is an $L^2$ integrable class function on $G$ which is bounded from below. The action is thus manifestly gauge invariant.
The full *partition function* of lattice gauge theory with gauge group $G$ finally reads $$\label{eq_partition}
Z = \int\sym{D}g\,\exp(-S(dg))
= \Bigl(\prod_{(i,\mu)\in\Lambda^1}\int_G\,dg_{i\mu}\Bigr)\,
\prod_{(i,\mu,\nu)\in\Lambda^2}f(dg_{i\mu\nu}),$$ where $f(g)=\exp(-s(g))$ is a positive real and $L^2$ integrable class function on $G$. The standard example for the action is the *Wilson action*, $$s(g) = -\frac{\beta}{2\dim V_\rho}(\chi^{(\rho)}(g)+\overline{\chi^{(\rho)}(g)}),$$ where $\chi^{(\rho)}\colon G\to\C$ is the character of a unitary matrix representation $\rho$ of $G$, usually the fundamental representation. The *inverse temperature* $\beta$ encodes the coupling constant.
In and in the following we are careful not to waste letters of the alphabet for dummy indices. The indices $i,\mu$ of the first product sign are just there to indicate that group integration is performed for each link of the lattice. We adopt the convention that $i$ and $\mu$ in this case do not have any meaning outside the enclosing brackets. So we can use the same letters again after the closing bracket.
Gauge invariant quantities
--------------------------
Of course, Wilson loops are gauge invariant expressions. However, in non-Abelian lattice gauge theory, not all gauge invariant expressions are given by Wilson loops. The generic gauge invariant expressions are so-called *spin networks* which include branchings of the Wilson lines. This is familiar, for example, from the expression which is used to determine the static three-quark potential.
Here we formalize this generalization and give the following slightly technical definition:
\[def\_spinnetwork\] Let $\tau\colon\Lambda^1\to\Irrep,(j,\kappa)\mapsto\tau_{j\kappa}$, associate a finite-dimensional irreducible unitary representation of $G$ with each link of the lattice. Choose furthermore for each lattice point $j\in\Lambda^0$ an intertwiner $$\label{eq_spinnetmap}
Q^{(j)}\colon\bigotimes_{\mu=1}^d\tau_{j-\hat\mu,\mu}\to
\bigotimes_{\mu=1}^d\tau_{j\mu},$$ which maps from the tensor product of the representations of the $d$ ‘incoming’ links to the tensor product of the $d$ ‘outgoing’ links at the point $j\in\Lambda^0$.
The *non-Abelian Wilson loop* (or *spin network*) associated with $\tau$ and $Q^{(j)}$ is defined by $$\begin{aligned}
\label{eq_spinnetwork}
\Wloop
&:=& \Bigl(\prod_{(j,\kappa)\in\Lambda^1}\sum_{a_{j\kappa},b_{j\kappa}}\Bigr)\,
\Bigl(\prod_{(j,\kappa)\in\Lambda^1}
t_{a_{j\kappa}b_{j\kappa}}^{(\tau_{j\kappa})}(g_{j\kappa})\Bigr)\\
&\times&\Bigl(\prod_{j\in\Lambda^0}
Q^{(j)}_{(b_{j-\hat1,1}\ldots b_{j-\hat d,d}),(a_{j1}\ldots a_{jd})}\Bigr).\nn\end{aligned}$$ The indices $(b_{j-\hat1,1}\ldots b_{j-\hat d,d})$ of $Q^{(j)}$ refer to the tensor factors of the domain of $Q^{(j)}$ (‘incoming’) while the $(a_{j1}\ldots a_{jd})$ refer to the image (‘outgoing’), .
In we have abbreviated the summation of the vector indices $a_{j\kappa}$ and $b_{j\kappa}$ for all links by $$\Bigl(\prod_{(j,\kappa)\in\Lambda^1}\sum_{a_{j\kappa},b_{j\kappa}}\Bigr)
= \underbrace{\sum_{a_{j\kappa},b_{j\kappa}=1}^{\dim V_{\tau_{j\kappa}}}
\cdots\sum_{a_{j\kappa},b_{j\kappa}=1}^{\dim V_{\tau_{j\kappa}}}}_{
\mbox{one $\sum$ for each link}}.$$ This notation is frequently used in the duality transformation in Section \[sect\_transform\].
The expression $\Wloop$ of the non-Abelian Wilson loop in is gauge invariant under the transformation .
Consider an arbitrary lattice point $j\in\Lambda^0$. The gauge transformation multiplies all ‘incoming’ links by $\phi_j^{-1}$ and all ‘outgoing’ links by $\phi_j$. Since $Q^{(j)}$ in is an intertwiner, $\Wloop$ is unchanged. This holds for all lattice points $j\in\Lambda^0$.
Depending on the representations $\tau_{j\kappa}$, there are situations in which for some $j\in\Lambda^0$ the only choice is $Q^{(j)}=0$ and thus $\Wloop=0$. This is the case if a would-be Wilson loop is not properly closed.
All links $(j,\kappa)$ for which $\tau_{j\kappa}\cong V_{[1]}$ is the trivial representation, disappear from the expression . For an ordinary Wilson loop, for example, all links are labelled with the trivial representation except those links which are part of the loop. These are labelled with the fundamental representation of $G$. The intertwiners $Q^{(j)}$ (if non-vanishing) are in this case uniquely determined up to normalization.
In Definition \[def\_spinnetwork\], the requirement that the $\tau_{j\kappa}$ be irreducible, and that there be only one factor $t_{ab}^{(\tau)}$ per link, can be imposed without loss of generality. Otherwise the expression $\Wloop$ would decompose into similar expressions involving only irreducible representations and only one function $t_{ab}^{(\tau)}$ per link, and .
If $G$ is Abelian, then $\Irrep\cong\Z$, and all irreducible representations are one-dimensional. Thus $\Wloop$ can be decomposed into a sum of products of Abelian Wilson loops.
The normalized expectation value of a non-Abelian Wilson loop finally reads $$\label{eq_expect}
\bigl<\Wloop\bigr>=\frac{1}{Z}\int\sym{D}g\,\Wloop\,\exp(-S(dg)),$$ and .
The duality transformation {#sect_transform}
==========================
In this section we present the duality transformation in detail. We start with the transformation of the partition function and then turn to the expectation value of the non-Abelian Wilson loop.
The dual of the partition function {#sect_transformpart}
----------------------------------
We start with the partition function of non-Abelian lattice gauge theory . Since the Boltzmann weight function $f(g)$ is an $L^2$ class function, we can insert its character decomposition, see : $$\label{eq_charexp}
f(g) = \sum_{\rho\in\Irrep}\hat f_\rho\chi^{(\rho)}(g)
= \sum_{\rho\in\Irrep}\hat f_\rho\sum_{n=1}^{\dim V_\rho}t_{nn}^{(\rho)}(g).$$ The partition function thus reads $$Z = \Bigl(\prod_{(i,\mu)\in\Lambda^1}\int_G\,dg_{i\mu}\Bigr)\,
\prod_{(i,\mu,\nu)\in\Lambda^2}\Bigl(\sum_{\rho_{i\mu\nu}\in\Irrep}
\hat f_{\rho_{i\mu\nu}}\sum_{n_{i\mu\nu}=1}^{\dim V_{\rho_{i\mu\nu}}}
t_{n_{i\mu\nu}n_{i\mu\nu}}^{(\rho_{i\mu\nu})}
(g_{i\mu}g_{i+\hat\mu,\nu}g_{i+\hat\nu,\mu}^{-1}g_{i,\nu}^{-1})\Bigr),$$ where each plaquette $(i,\mu,\nu)\in\Lambda^2$ is coloured with an irreducible representation $\rho_{i\mu\nu}\in\Irrep$, and the indices $n_{i\mu\nu}$ which originate from the traces, are summed once for each plaquette.
The next step is to employ the coproduct and the antipode, see and , in order to remove all group products and inverses from the arguments of the representation functions. Furthermore, we reorganize the summations: $$\begin{aligned}
\label{eq_step2}
Z&=&\Bigl(\prod_{(i,\mu)\in\Lambda^1}\int_G\,dg_{i\mu}\Bigr)\,
\Bigl(\prod_{(i,\mu,\nu)\in\Lambda^2}\sum_{\rho_{i\mu\nu}\in\Irrep}\Bigr)\,
\prod_{(i,\mu,\nu)\in\Lambda^2}\Biggl[\hat f_{\rho_{i\mu\nu}}\,\times\Biggr.\\
&\times&\Biggl.\Bigl(\sum_{n_{i\mu\nu},m_{i\mu\nu},p_{i\mu\nu},q_{i\mu\nu}}
t_{n_{i\mu\nu}m_{i\mu\nu}}^{(\rho_{i\mu\nu})}(g_{i\mu})
t_{m_{i\mu\nu}p_{i\mu\nu}}^{(\rho_{i\mu\nu})}(g_{i+\hat\mu,\nu})
St_{p_{i\mu\nu}q_{i\mu\nu}}^{(\rho_{i\mu\nu})}(g_{i+\hat\nu,\mu})
St_{q_{i\mu\nu}n_{i\mu\nu}}^{(\rho_{i\mu\nu})}(g_{i\nu})\Bigr)\Biggr].\nn\end{aligned}$$ In all places where we have applied the coproduct, written schematically as $$t_{nn}(g_1g_2g_3g_4)=\sum_{m,p,q}t_{nm}(g_1)t_{mp}(g_2)t_{pq}(g_3)t_{qn}(g_4),$$ new vector indices have entered which are associated with the plaquette $(i,\mu,\nu)\in\Lambda^2$ and denoted by $m_{i\mu\nu}$, $p_{i\mu\nu}$ and $q_{i\mu\nu}$. They are summed over the range $1\ldots\dim V_{\rho_{i\mu\nu}}$.
In order to perform the group integrations, we have to reorganize the product in such that all representation functions whose argument refers to the same link are grouped together. Therefore we have to find all plaquettes which contain a given link $(i,\mu)\in\Lambda^1$ in their boundary, all plaquettes which *cobound* the link. Figure \[fig\_linkplaqu\] illustrates this situation for $d=3$. The reorganized product reads $$\begin{aligned}
\label{eq_reordering}
Z&=&\Bigl(\prod_{(i,\mu,\nu)\in\Lambda^2}\sum_{\rho_{i\mu\nu}\in\Irrep}\Bigr)\,
\Bigl(\prod_{(i,\mu,\nu)\in\Lambda^2}\hat
f_{\rho_{i\mu\nu}}\Bigr)\,
\Bigl(\prod_{(i,\mu,\nu)\in\Lambda^2}
\sum_{n_{i\mu\nu},m_{i\mu\nu},p_{i\mu\nu},q_{i\mu\nu}}\Bigr)\\
&\times&\prod_{(i,\mu)\in\Lambda^1}\Biggl\{\int_G\,dg_{i\mu}\,\Biggl[
\prod_{\lambda=1}^{\mu-1}\biggl(
t_{m_{i-\hat\lambda,\lambda,\mu}p_{i-\hat\lambda,\lambda,\mu}}^{
(\rho_{i-\hat\lambda,\lambda,\mu})}(g_{i\mu})\cdot
St_{q_{i\lambda\mu}n_{i\lambda\mu}}^{(\rho_{i\lambda\mu})}(g_{i\mu})\biggr)
\Biggr.\Biggr.\nn\\
&&\qquad\Biggl.\Biggl.\times\prod_{\nu=\mu+1}^d\biggl(
St_{p_{i-\hat\nu,\mu,\nu}q_{i-\hat\nu,\mu,\nu}}^{(\rho_{i-\hat\nu,\mu,\nu})}(g_{i\mu})
\cdot t_{n_{i\mu\nu}m_{i\mu\nu}}^{(\rho_{i\mu\nu})}(g_{i\mu})\biggr)
\Biggr]\Biggr\}.\nn\end{aligned}$$
The expression means that each plaquette is coloured with an irreducible representation. There is furthermore a (dual Boltzmann) weight factor $\hat f_\rho$ per plaquette. Since we have reorganized the product of the representation functions $t_{ij}(g)$ such that those whose argument refers to the same link $(i,\mu)\in\Lambda^1$ are placed next to each other, the group integrations in are performed for each link separately. In the integrand, the two products $\prod_{\lambda}$ and $\prod_{\nu}$ enumerate all plaquettes cobounding the link $(i,\mu)$ in arbitrary dimension $d$ and are such that $1\leq\lambda<\mu<\nu\leq
d$ always. In dimension $d$, there are $2(d-1)$ factors for each link direction $\mu$.
Next we eliminate the antipodes using . The group integrals in thus read $$\begin{aligned}
\label{eq_delantipode}
\int_G\,dg_{i\mu}\Biggl[
\prod_{\lambda=1}^{\mu-1}\biggl(
t_{m_{i-\hat\lambda,\lambda,\mu}p_{i-\hat\lambda,\lambda,\mu}}^{
(\rho_{i-\hat\lambda,\lambda,\mu})}(g_{i\mu})\cdot
t_{n_{i\lambda\mu}q_{i\lambda\mu}}^{(\rho_{i\lambda\mu}^\ast)}(g_{i\mu})\biggr)
\Biggr.\nn\\
\qquad\Biggl.\times\prod_{\nu=\mu+1}^d\biggl(
t_{q_{i-\hat\nu,\mu,\nu}p_{i-\hat\nu,\mu,\nu}}^{(\rho_{i-\hat\nu,\mu,\nu}^\ast)}(g_{i\mu})
\cdot t_{n_{i\mu\nu}m_{i\mu\nu}}^{(\rho_{i\mu\nu})}(g_{i\mu})\biggr)
\Biggr].\end{aligned}$$ Now the group integrations can be performed using the expression in terms of projectors. We obtain $$\begin{aligned}
\label{eq_step4}
&\displaystyle\int_G\,dg_{i\mu}\biggl[\cdots\biggr]&\nn\\
= &\displaystyle\sum_{P\in\sym{P}_{i\mu}}
P_{(\underbrace{m_{i-\hat\lambda,\lambda,\mu}n_{i\lambda\mu}\ldots}_{
\lambda\in\{1,\ldots,\mu-1\}})
(\underbrace{q_{i-\hat\nu,\mu,\nu}n_{i\mu\nu}\ldots}_{
\nu\in\{\mu+1,\ldots,d\}})}\cdot
P_{(\underbrace{p_{i-\hat\lambda,\lambda,\mu}q_{i\lambda\mu}\ldots}_{
\lambda\in\{1,\ldots,\mu-1\}})
(\underbrace{p_{i-\hat\nu,\mu,\nu}m_{i\mu\nu}\ldots}_{
\nu\in\{\mu+1,\ldots,d\}})}.&\end{aligned}$$ Here the sum is over a complete set $\sym{P}_{i\mu}$ of inequivalent orthogonal projectors onto the trivial one-dimensional components in the decomposition of $$\label{eq_step5}
\underbrace{
(\rho_{i-\hat\lambda,\lambda,\mu}\otimes\rho_{i\lambda\mu}^\ast)\otimes\cdots}_{
\lambda\in\{1,\ldots,\mu-1\}}\otimes
\underbrace{
(\rho_{i-\hat\nu,\mu,\nu}^\ast\otimes\rho_{i\mu\nu})\otimes\cdots}_{
\nu\in\{\mu+1,\ldots,d\}}$$ into irreducible components. The dots “$\cdots$” indicate that there are pairs $\rho\otimes\rho^\ast$ of tensor factors for all $\lambda\in\{1,\ldots,\mu-1\}$ and pairs $\rho^\ast\otimes\rho$ for all $\nu\in\{\mu+1,\ldots,d\}$. This gives the correct result in arbitrary dimension and takes into account the orientation of the link $(i,\mu)$ in the boundary of the given plaquette. Opposite orientations of the link correspond to dual representations. Similarly, the dots “$\ldots$” in indicate that there is one pair of indices for each pair $\rho\otimes\rho^\ast$ resp. $\rho^\ast\otimes\rho$ which appears in .
With this step we have evaluated all group integrations over the links. As new degrees of freedom for the dual path integral, the colourings of all plaquettes with irreducible representations of $G$ have emerged: $$\begin{aligned}
\label{eq_step6}
Z&=&\Bigl(\underbrace{\prod_{(i,\mu,\nu)\in\Lambda^2}\sum_{\rho_{i\mu\nu}\in\Irrep}}_{
{\genfrac{}{}{0pt}{2}{\scriptstyle \mbox{part of the}}{\scriptstyle \mbox{dual path integral}}}}\Bigr)\,
\Bigl(\underbrace{\prod_{(i,\mu,\nu)\in\Lambda^2}\hat
f_{\rho_{i\mu\nu}}}_{\mbox{dual Boltzmann weight}}\Bigr)\,
\Bigl(\underbrace{\prod_{(i,\mu,\nu)\in\Lambda^2}
\sum_{n_{i,\mu,\nu},m_{i\mu\nu},p_{i\mu\nu},q_{i\mu\nu}}}_{
\mbox{vector index summations}}\Bigr)\nn\\
&\times&\prod_{(i,\mu)\in\Lambda^1}\biggl(
\underbrace{\sum_{P\in\sym{P}_{i\mu}} P_{(\cdots)(\cdots)}\cdot
P_{(\cdots)(\cdots)}}_{\mbox{one constraint for each link}}\biggr).\end{aligned}$$ The last sum has the form as in for each link $(i,\mu)$ appearing in the product.
There are still the summations over the vector indices $n_{i\mu\nu},\ldots,q_{i\mu\nu}$ for all plaquettes. We call the factors arising from them and from the projectors *gauge constraints* because they generalize the conditions which ensure in the Abelian case that the integer $2$-form appearing in the dual model, is co-closed. They seem to form a number of complicated non-local constraints. However, the summations can again be reordered in a suitable way so that the constraints appear only locally in a certain sense. In order to see this, some geometric intuition is necessary.
The projectors which have appeared in the group integration $\int_G\,dg_{i\mu}$ for the link $(i,\mu)\in\Lambda^1$ and their indices are of the form $$\label{eq_projsplit}
P_{(m_{i-\hat\lambda,\lambda,\mu}n_{i\lambda\mu}\ldots)
(q_{i-\hat\nu,\mu,\nu}n_{i\mu\nu}\ldots)}\cdot
P_{(p_{i-\hat\lambda,\lambda,\mu}q_{i\lambda\mu}\ldots)
(p_{i-\hat\nu,\mu,\nu}m_{i\mu\nu}\ldots)},$$ the indices correspond to the plaquettes located at $i-\hat\lambda$, $i$, $\ldots$, $i-\hat\nu$, $i$, $\ldots$ for the first projector and similarly for the second projector.
The crucial geometrical observation (Figure \[fig\_linkplaqu\]) is that all vector indices $m_{i-\hat\lambda,\lambda,\mu},n_{i\lambda\mu},\ldots$, $q_{i-\hat\nu,\mu,\nu},n_{i\mu\nu},\ldots$ which appear at the first projector correspond to the lattice point $i$ whereas all vector indices $p_{i-\hat\lambda,\lambda,\mu},q_{i\lambda\mu},\ldots$, $p_{i-\hat\nu,\mu,\nu},m_{i\mu\nu},\ldots$ of the second projector correspond to the lattice point $i+\hat\mu$. Recall that the enumeration of the indices $n,m,p,q$ for a given plaquette $(i,\mu,\nu)\in\Lambda^2$ was done starting with $n$ at the point $i$, then proceeding counter-clockwise in the $(\mu,\nu)$-plane. It is thus possible to associate the summations over the $n,m,p,q$ with the lattice points $i\in\Lambda^0$. For each link $(i,\mu)\in\Lambda^1$, one of the two projectors in then belongs to $i$, the other to $i+\hat\mu$. However, the projectors can be separated only if the summation over the projectors $\sym{P}_{i\mu}$ which is associated with a link rather than a point, can be removed from these expressions.
In the partition function, there is in total one such summation over projectors for each link of the lattice. It is thus natural to consider these summations as part of the dual path integral. If this is done, we can reorganize the expressions such that all vector index summations and projectors are associated with the lattice points $i\in\Lambda^0$ and such that they involve only data from the neighbouring links and plaquettes. This is what is meant by ’locality’. In Figure \[fig\_indices2\] we show the lattice point $i\in\Lambda^0$ together with the $2d$ links and $2d(d-1)$ plaquettes which contain $i$. The partition function finally reads $$\label{eq_step7}
Z=\underbrace{\Bigl(\prod_{(i,\mu,\nu)\in\Lambda^2}\sum_{\rho_{i\mu\nu}\in\Irrep}\Bigr)\,
\Bigl(\prod_{(i,\mu)\in\Lambda^1}\sum_{P^{(i\mu)}\in\sym{P}_{i\mu}}\Bigr)}_{
\mbox{dual path integral}}\,
\Bigl(\underbrace{\prod_{(i,\mu,\nu)\in\Lambda^2}\hat
f_{\rho_{i\mu\nu}}}_{\mbox{dual Boltzmann weight}}\Bigr)\,
\prod_{i\in\Lambda^0}\,C(i).$$ where $C(i)$ encompasses the vector index summations and projectors associated with the lattice point $i\in\Lambda^0$: $$\begin{aligned}
\label{eq_constraint}
C(i) &=& \Bigl(\prod_{1\leq\mu<\nu\leq d}
\sum_{p_{i-\hat\mu-\hat\nu,\mu,\nu}=1}^{\dim V_{\rho_{i-\hat\mu-\hat\nu,\mu,\nu}}}
\sum_{q_{i-\hat\nu,\mu,\nu}=1}^{\dim V_{\rho_{i-\hat\nu,\mu,\nu}}}
\sum_{m_{i-\hat\mu,\mu,\nu}=1}^{\dim V_{\rho_{i-\hat\mu,\mu,\nu}}}
\sum_{n_{i\mu\nu}=1}^{\dim V_{\rho_{i\mu\nu}}}\Bigr)\\
&&\prod_{\mu=1}^d
P^{(i\mu)}_{(\underbrace{m_{i-\hat\lambda,\lambda,\mu}n_{i\lambda\mu}\ldots}_{
\lambda\in\{1,\ldots,\mu-1\}})
(\underbrace{q_{i-\hat\nu,\mu,\nu}n_{i\mu\nu}\ldots}_{
\nu\in\{\mu+1,\ldots,d\}})}\nn\\
&&\qquad\cdot P^{(i-\hat\mu,\mu)}_{(\underbrace{p_{i-\hat\mu-\hat\lambda,\lambda,\mu}
q_{i-\hat\mu,\lambda,\mu}\ldots}_{
\lambda\in\{1,\ldots,\mu-1\}})
(\underbrace{p_{i-\hat\mu-\hat\nu,\mu,\nu}m_{i-\hat\mu,\mu,\nu}\ldots}_{
\nu\in\{\mu+1,\ldots,d\}})}.\nn\end{aligned}$$ The first product parameterizes all possible planes by $\mu<\nu$. The four sums are then associated with the four plaquettes $(i-\hat\mu-\hat\nu,\mu,\nu)$, $(i-\hat\nu,\mu,\nu)$, $(i-\hat\mu,\mu,\nu)$ and $(i,\mu,\nu)$ in the $(\mu,\nu)$-plane which contain the point $i$ (Figure \[fig\_indices2\]). The last product enumerates the ‘outgoing’ and ‘incoming’ links and contains the projectors associated with this link and with the point $i$. Note that the projectors in for a given lattice point $i\in\Lambda^0$ involve only vector indices whose summation is part of the same $C(i)$.
The partition function consists now of a sum over irreducible representations for all plaquettes and a sum over the projectors onto the trivial components in the tensor product for all links. This tensor product involves the representations of all plaquettes which cobound the given link. In particular, if the tensor product does not contain a trivial component, this sum over projectors is empty and does not contribute to the path integral.
These results are summarized in the following theorem:
\[thm\_dualpartition1\] Let $G$ be a compact Lie group or a finite group. The partition function of lattice gauge theory with the gauge group $G$ on a $d$-dimensional finite lattice with periodic boundary conditions is equal to the expression $$\label{eq_dualpartition1}
Z=\underbrace{\Bigl(\prod_{(i,\mu,\nu)\in\Lambda^2}\sum_{\rho_{i\mu\nu}\in\Irrep}\Bigr)\,
\Bigl(\prod_{(i,\mu)\in\Lambda^1}\sum_{P^{(i\mu)}\in\sym{P}_{i\mu}}\Bigr)}_{
\mbox{dual path integral}}\,
\Bigl(\underbrace{\prod_{(i,\mu,\nu)\in\Lambda^2}\hat
f_{\rho_{i\mu\nu}}}_{\mbox{dual Boltzmann weight}}\Bigr)\,
\prod_{i\in\Lambda^0}\,C(i).$$ Here $\Irrep$ denotes a set containing one unitary representation for each equivalence class of finite-dimensional irreducible representations of $G$. $\sym{P}_{i\mu}$ denotes the set of all projectors onto the different one-dimensional trivial components in the decomposition of the tensor product into its irreducible components. $C(i)$ describes a *gauge constraint factor* for each lattice point $i\in\Lambda^0$ which is given by . The coefficients $\hat f_{\rho_{i\mu\nu}}$ are defined by the character decomposition of the original Boltzmann weight $\exp(-s(g))$, $$\label{eq_dualboltzmann}
\hat f_{\rho_{i\mu\nu}} =\dim V_{\rho_{i\mu\nu}}\,\int_G\,
\overline{\chi^{(\rho_{i\mu\nu})}(g)}\,\exp(-s(g))\,dg.$$
The dual partition function can be described in words as follows: Colour all plaquettes with finite-dimensional irreducible representations of $G$ in all possible ways. Colour all links with projectors onto the trivial components in the tensor products (if there are any). The partition function contains a (local) dual Boltzmann weight factor which is the coefficient of the character expansion of the original Boltzmann weight for each plaquette. The partition function contains furthermore a (local) gauge constraint factor $C(i)$, see , for each lattice point.
The two main differences to the Abelian case (see [@Sa80]) are the following: Firstly, in the Abelian case only objects on a single level, namely the plaquettes, are coloured with integer numbers (which characterize the finite-dimensional irreducible unitary representations). In the non-Abelian case we have to colour the plaquettes with representations and the links with intertwiners. The configurations of the dual model are thus spin foams. Note that the choice of projectors $\sym{P}_{i\mu}$ in and agrees up to canonical isomorphisms with the choice of intertwiners in the definition of a closed spin foam as given in [@Ba98]. Here the assignment of ‘incoming’ and ‘outgoing’ faces has to be made according to our standard orientations of plaquettes and links.
Secondly, the integrand is not just a Boltzmann weight, but contains in addition the factor $C(i)$ for each lattice point. In the Abelian case, this factor together with the sum over projectors enforces co-closedness of the integer $2$-form. The dual of Abelian gauge theory is again a gauge theory because if this $2$-form is also co-exact, it can be integrated and gauge degrees of freedom appear. In the non-Abelian case there is no obvious integration which would introduce gauge degrees of freedom.
The dual of the non-Abelian Wilson loop
---------------------------------------
The duality transformation of the expectation value of the non-Abelian Wilson loop proceeds along the same lines. However, the expressions become slightly more complicated due to the presence of the additional integrand. In the following description of the transformation, we often refer to the calculations for the partition function in Section \[sect\_transformpart\].
The expectation value of the non-Abelian Wilson loop is given by $$\begin{aligned}
\bigl<\Wloop\bigr> &=& \frac{1}{Z}\,
\Bigl(\prod_{(i,\mu)\in\Lambda^1}\int_G\,dg_{i\mu}\Bigr)\,
\Bigl(\prod_{(i,\mu,\nu)\in\Lambda^2}f(dg_{i\mu\nu})\Bigr)\,
\Bigl(\prod_{(j,\kappa)\in\Lambda^1}\sum_{a_{j\kappa}b_{j\kappa}}\Bigr)\nn\\
&\times& \Bigl(\prod_{(j,\kappa)\in\Lambda^1}
t_{a_{j\kappa}b_{j\kappa}}^{(\tau_{j\kappa})}(g_{j\kappa})\Bigr)\,
\Bigl(\prod_{j\in\Lambda^0}
Q^{(j)}_{(b_{j-\hat1,1}\ldots b_{j-\hat d,d}),(a_{j1}\ldots a_{jd})}\Bigr).\end{aligned}$$ We insert the character decomposition , employ the coproduct and the antipode and reorganize the factors just as in the calculation for the partition function. The result generalizes in which has been inserted: $$\begin{aligned}
\bigl<\Wloop\bigr> &=& \frac{1}{Z}\,
\Bigl(\prod_{(i,\mu,\nu)\in\Lambda^2}\sum_{\rho_{i\mu\nu}\in\Irrep}\Bigr)\,
\Bigl(\prod_{(i,\mu,\nu)\in\Lambda^2}\hat f_{\rho_{i\mu\nu}}\Bigr)\,
\Bigl(\prod_{(j,\kappa)\in\Lambda^1}\sum_{a_{j\kappa}b_{j\kappa}}\Bigr)\\
&&\Bigl(\prod_{(i,\mu,\nu)\in\Lambda^2}\sum_{n_{i\mu\nu},m_{i\mu\nu},p_{i\mu\nu},q_{i\mu\nu}}\Bigr)\,
\Bigl(\prod_{j\in\Lambda^0}
Q^{(j)}_{(b_{j-\hat1,1}\ldots b_{j-\hat d,d}),(a_{j1}\ldots a_{jd})}\Bigr)\nn\\
&\times&\prod_{(i,\mu)\in\Lambda^2}\Biggl\{\int_G\,dg_{i\mu}\,\Biggl[
\prod_{\lambda=1}^{\mu-1}\biggl(
t_{m_{i-\hat\lambda,\lambda,\mu}p_{i-\hat\lambda,\lambda,\mu}}^{
(\rho_{i-\hat\lambda,\lambda,\mu})}(g_{i\mu})\cdot
t_{n_{i\lambda\mu}q_{i\lambda\mu}}^{(\rho_{i\lambda\mu}^\ast)}(g_{i\mu})\biggr)
\Biggr.\Biggr.\nn\\
&&\Biggl.\Biggl.\times\prod_{\nu=\mu+1}^d\biggl(
t_{q_{i-\hat\nu,\mu,\nu}p_{i-\hat\nu,\mu,\nu}}^{
(\rho_{i-\hat\nu,\mu,\nu}^\ast)}(g_{i\mu})
\cdot t_{n_{i\mu\nu}m_{i\mu\nu}}^{(\rho_{i\mu\nu})}(g_{i\mu})\biggr)
\cdot\underbrace{t_{a_{i\mu}b_{i\mu}}^{(\tau_{i\mu})}(g_{i\mu})}_{
\mbox{new factor}}
\Biggr]\Biggr\}.\nn\end{aligned}$$ The features which are new compared with and are the summations over $a_{j\kappa}$ and $b_{j\kappa}$ for each link, the product over the intertwiners $Q^{(j)}$ for each lattice point and the additional factor $\tau_{a_{i\mu}b_{i\mu}}^{(\tau_{i\mu})}(g_{i\mu})$ in the integrand for each link $(i,\mu)$.
Now we can apply the projector expression for the Haar measure . Compared with , the additional factor in the integrand produces additional indices $a_{i\mu}$ and $b_{i\mu}$ of the projectors, $$\begin{aligned}
&&\int_G\,dg_{i\mu}\biggl[\cdots\biggr]\nn\\
&=& \sum_{P\in\sym{P}^\prime_{i\mu}}
P_{(\underbrace{m_{i-\hat\lambda,\lambda,\mu}n_{i\lambda\mu}\ldots}_{
\lambda\in\{1,\ldots,\mu-1\}})
(\underbrace{q_{i-\hat\nu,\mu,\nu}n_{i\mu\nu}\ldots}_{
\nu\in\{\mu+1,\ldots,d\}})\underbrace{a_{i\mu}}_{\mbox{new}}}\nn\\
&&\quad\cdot
P_{(\underbrace{p_{i-\hat\lambda,\lambda,\mu}q_{i\lambda\mu}\ldots}_{
\lambda\in\{1,\ldots,\mu-1\}})
(\underbrace{p_{i-\hat\nu,\mu,\nu}m_{i\mu\nu}\ldots}_{
\nu\in\{\mu+1,\ldots,d\}})\underbrace{b_{i\mu}}_{\mbox{new}}}.\end{aligned}$$ These indices correspond to the additional tensor factor in the following decomposition: The orthogonal projectors $P\in\sym{P}^\prime_{i\mu}$ project onto the distinct one-dimensional trivial components in the decomposition of $$\label{eq_tensorwilson}
\underbrace{
(\rho_{i-\hat\lambda,\lambda,\mu}\otimes\rho_{i\lambda\mu}^\ast)\otimes\cdots}_{
\lambda\in\{1,\ldots,\mu-1\}}\otimes
\underbrace{
(\rho_{i-\hat\nu,\mu,\nu}^\ast\otimes\rho_{i\mu\nu})\otimes\cdots}_{
\nu\in\{\mu+1,\ldots,d\}}\otimes\underbrace{\tau_{i\mu}}_{\mbox{new}}$$ into its irreducible components, .
Similar to the calculation for the partition function, the vector index summations over the $n_{i\mu\nu},\ldots,q_{i\mu\nu}$ as well as over the $a_{j\kappa}$ and $b_{j\kappa}$, the projectors $P_{(\cdots)(\cdots)}$ and the intertwiners $Q^{(j)}$ can be reorganized to form local expressions. This construction is entirely analogous to the derivation of and . We obtain the following result which generalizes Theorem \[thm\_dualpartition1\]:
\[thm\_dualwilson1\] Let $G$ be a compact Lie group or a finite group and consider lattice gauge theory with gauge group $G$ on a $d$-dimensional finite lattice with periodic boundary conditions. The normalized expectation value of the non-Abelian Wilson loop is equal to the expression $$\begin{aligned}
\label{eq_dualwilson1}
\bigl<\Wloop\bigr> &=& \frac{1}{Z}\,
\underbrace{\Bigl(\prod_{(i,\mu,\nu)\in\Lambda^2}\sum_{\rho_{i\mu\nu}\in\Irrep}\Bigr)\,
\Bigl(\prod_{(i,\mu)\in\Lambda^1}\sum_{P^{(i\mu)}\in\sym{P}^\prime_{i\mu}}\Bigr)}_{
\mbox{dual path integral}}\,
\Bigl(\underbrace{\prod_{(i,\mu,\nu)\in\Lambda^2}\hat
f_{\rho_{i\mu\nu}}}_{\mbox{dual Boltzmann weight}}\Bigr)\nn\\
&\times&
\prod_{i\in\Lambda^0}\,\Biggl[
\Bigl(\prod_{\mu=1}^d\sum_{a_{i\mu}=1}^{\dim V_{\tau_{i\mu}}}
\sum_{b_{i-\hat\mu,\mu}=1}^{\dim V_{\tau_{i-\hat\mu,\mu}}}\Bigr)\,
Q^{(i)}_{(b_{i-\hat1,1}\ldots b_{i-\hat d,d}),(a_{i1}\ldots a_{id})}\,
\tilde C(i)\Biggr].\end{aligned}$$ Here $\sym{P}^\prime_{i\mu}$ denotes the set of all projectors onto the different trivial components in the decomposition of the tensor product into its irreducible components. $\tilde C(i)$ describes a *gauge constraint factor* for each lattice point $i\in\Lambda^0$ which is given by $$\begin{aligned}
\label{eq_constraint2}
\tilde C(i) &=& \Bigl(\prod_{1\leq\mu<\nu\leq d}
\sum_{p_{i-\hat\mu-\hat\nu,\mu,\nu}=1}^{\dim V_{\rho_{i-\hat\mu-\hat\nu,\mu,\nu}}}
\sum_{q_{i-\hat\nu,\mu,\nu}=1}^{\dim V_{\rho_{i-\hat\nu,\mu,\nu}}}
\sum_{m_{i-\hat\mu,\mu,\nu}=1}^{\dim V_{\rho_{i-\hat\mu,\mu,\nu}}}
\sum_{n_{i\mu\nu}=1}^{\dim V_{\rho_{i\mu\nu}}}\Bigr)\\
&&\prod_{\mu=1}^d
P^{(i\mu)}_{(\underbrace{m_{i-\hat\lambda,\lambda,\mu}n_{i\lambda\mu}\ldots}_{
\lambda\in\{1,\ldots,\mu-1\}})
(\underbrace{q_{i-\hat\nu,\mu,\nu}n_{i\mu\nu}\ldots}_{
\nu\in\{\mu+1,\ldots,d\}})\displaystyle a_{i\mu}}\nn\\
&&\qquad\cdot P^{(i-\hat\mu,\mu)}_{(\underbrace{p_{i-\hat\mu-\hat\lambda,\lambda,\mu}q_{i-\hat\mu,\lambda,\mu}\ldots}_{
\lambda\in\{1,\ldots,\mu-1\}})
(\underbrace{p_{i-\hat\mu-\hat\nu,\mu,\nu}m_{i-\hat\mu,\mu,\nu}\ldots}_{
\nu\in\{\mu+1,\ldots,d\}})\displaystyle b_{i-\hat\mu,\mu}}.\nn\end{aligned}$$
The dual of the non-Abelian Wilson loop can be described in words as follows: Just as in the Abelian case, it is not an expectation value under the dual partition function, but looks rather like a modified partition function. In addition to the dual partition function, there are summations over the vector indices $a_{j\kappa}$ and $b_{j\kappa}$ which are necessary to multiply the representation matrices which form the non-Abelian Wilson loop. The loop enters in two places. First the intertwiners $Q^{(i)}$ appear for each lattice point $i\in\Lambda^0$. Furthermore, the representations $\tau_{j\kappa}$ on the links which form the non-Abelian Wilson loop, enter the tensor product , and the corresponding indices $a_{j\kappa}$ and $b_{j\kappa}$ thus appear in . The fact that the presence of the Wilson loop changes the constraint factors $C(i)$ is familiar from the Abelian case. There it occurs in the expressions before the co-closed $2$-form is integrated.
The constraints $C(i)$ on the dual lattice
------------------------------------------
In the expression of the dual partition function and , the factors $C(i)$ look very complicated. They can be understood most easily on the dual lattice. We explain this idea for the case $d=3$ where the relevant pictures can be drawn. Analogous constructions can be made for arbitrary $d\geq 2$.
We construct the dual lattice in the standard way which is illustrated in Figure \[fig\_dual\] for the case $d=3$. To each $k$-cell $(i,\mu_1,\ldots,\mu_k)$, $1\leq\mu_1<\cdots<\mu_k\leq d$, of the original lattice, there corresponds a $(d-k)$-cell $(i,\nu_1,\ldots,\nu_{d-k})$, $1\leq\nu_1<\cdots<\nu_{d-k}\leq d$, of the dual lattice such that $$\{\mu_1,\ldots,\mu_k\}\cup\{\nu_1,\ldots,\nu_{d-k}\}
= \{1,\ldots,d\}.$$
In the dual partition function on the original lattice, the plaquettes are coloured with irreducible representations. The links are assigned projectors in a certain tensor product whose factors are given by the representations belonging to the plaquettes that cobound the link. Conversely, on the dual lattice in $d=3$, the plaquettes cobounding a given link correspond to the links in the boundary of a plaquette (see Figure \[fig\_dualcobound\]). Thus we have to colour the links of the dual lattice with irreducible representations. The plaquettes of the dual lattice are then assigned projectors onto the trivial components of some tensor product. This tensor product is the product of the representations belonging to the links in the boundary of the plaquette.
Instead of the projectors onto trivial components, schematically $$P\colon \rho_1\otimes\rho_2\otimes\rho_3^\ast\otimes\rho_4^\ast
\to\C$$ we now write intertwiners $$F\colon \rho_1\otimes\rho_2\to\rho_3\otimes\rho_4,$$ using the isomorphisms of $G$-modules $\Hom_G(V_\rho^\ast\otimes
V_\tau,\C)\cong_G\Hom_G(V_\tau,V_\rho)$. The intertwiners $F$ thus map from two links of a given plaquette to the other two links. Note that the $F$ inherit a normalization from the $P$ coming from the implicit inclusion $\C\subseteq\rho_1\otimes\rho_2\otimes\rho_3^\ast\otimes\rho_4^\ast$.
The factors $C(i)$ of are associated with the cubes of the dual lattice. The expression , interpreted on the dual lattice in $d=3$, contains one intertwiner per face of the cube as indicated in Figure \[fig\_cube\]. In this figure, the intertwiners $F$ are represented by double arrows leading from two links of each plaquette to the other two links. The arrows illustrate how the intertwiners have to be composed to account for the contraction of the indices in . The remaining indices are then summed over.
A similar visualization is straight forward for the factors $\tilde
C(i)$ in .
Special cases {#sect_special}
=============
Abelian gauge theory in arbitrary dimension
-------------------------------------------
In this section we show how Theorem \[thm\_dualpartition1\] reduces to the well-known results for $G=U(1)$. Similar calculations are available for $\Z$ or $\Z_n$ (Note that the transformation is applicable to $\Z$ although $\Z$ is neither compact nor finite. This is because $\Z$ gauge theory is dual to $U(1)$ gauge theory).
We start with the dual partition function . The unitary finite-dimensional irreducible representations of $U(1)$ are all one-dimensional. They are given by homomorphisms $g\mapsto g^k$ for $g\in U(1)$ and are characterized by integer numbers $k\in\Z$, $\Irrep\cong\Z$. The dual representation is then given by $g\mapsto g^{-k}$.
Consider the tensor product and specify the representations by integer numbers $k_{i\mu\nu}\in\Z$. Since all irreducible representations are one-dimensional, so are their tensor products. The question is therefore just whether or not the tensor product is equivalent to the trivial representation. This is the case if and only if $$\label{eq_tensoru1}
\sum_{\lambda=1}^{\mu-1}(k_{i-\hat\lambda,\lambda,\mu}-k_{i\lambda\mu})
+\sum_{\nu=\mu+1}^d(-k_{i-\hat\nu,\mu,\nu}+k_{i\mu\nu})=0.$$ In this case, there is exactly one projector onto a trivial component of the tensor product which is the identity map. If does not hold, there is no such projector.
Furthermore, since all irreducible representations are one-dimensional, the summations in the constraints $C(i)$, see , disappear. Moreover, the projectors $P^{(i\mu)}$ do not have indices and are all equal to $1$ if they exist. Thus $C(i)=1$ if holds.
The partition function therefore reads for $G=U(1)$, $$\begin{aligned}
\label{eq_zgauge}
Z&=&\Bigl(\prod_{(i,\mu,\nu)\in\Lambda^2}\sum_{k_{i\mu\nu}\in\Z}\Bigr)\,
\Bigl(\prod_{(i,\mu,\nu)\in\Lambda^2}\hat f_{k_{i\mu\nu}}\Bigr)\\
&\times&\Bigl(\prod_{(i,\mu)\in\Lambda^1}\delta\bigl(
\sum_{\lambda=1}^{\mu-1}(k_{i-\hat\lambda,\lambda,\mu}-k_{i\lambda\mu})
+\sum_{\nu=\mu+1}^d(-k_{i-\hat\nu,\mu,\nu}+k_{i\mu\nu})\bigr)\Bigr).\nn\end{aligned}$$ Here we have used the notation $\delta(n)=\delta_{0,n}$ for $n\in\Z$.
The dual path integral thus reduces to the summation over the integer numbers for each plaquette while the dual Boltzmann weight is again given by the character decomposition of the original Boltzmann weight, $$\hat f_{k_{i\mu\nu}}
= \frac{1}{2\pi}\,\int_0^{2\pi}e^{-ik_{i\mu\nu}\phi}\exp\bigl(-s(e^{i\phi})\bigr)\,d\phi.$$ The $\delta$-constraint ensures that the integer $2$-form $k_{i\mu\nu}$ is co-closed. This condition provides the dual model with the properties of a gauge theory. As is well-known, the partition function describes the dual of $U(1)$ lattice gauge theory, the so-called $\Z$ gauge theory [@Sa80; @PoWi91] and is here presented on the original rather than on the dual lattice.
Non-Abelian gauge theory in two dimensions
------------------------------------------
In this section we demonstrate how Theorem \[thm\_dualpartition1\] reduces to the familiar result for non-Abelian lattice gauge theory in $d=2$. In this case the partition function is particularly simple.
We start again with the dual partition function . In $d=2$, there are only two plaquettes which cobound a given link $(i,\mu)\in\Lambda^1$. Imagine the situation of Figure \[fig\_linkplaqu\] in $d=2$. The tensor product therefore consists of only two factors. It reads for links $(i,1)\in\Lambda^1$ in the $1$-direction, $$\label{eq_tensord2a}
\rho_{i-\hat 2}^\ast\otimes\rho_i,$$ and for links $(i,2)$ in the $2$ direction, $$\label{eq_tensord2b}
\rho_{i-\hat 1}\otimes\rho_i^\ast.$$ Here we have suppressed the last two indices of $\rho_{i\mu\nu}$ which are always $\mu=1$ and $\nu=2$. In both cases and , there are trivial components in the tensor product if and only if $$\rho_{i-\hat 2}\cong\rho_i\qquad\mbox{resp.}\qquad
\rho_{i-\hat 1}\cong\rho_i.$$ Since this holds for all $i\in\Lambda^0$, the only contributions to the partition function are given by configurations which assign the same representation to all plaquettes. We thus have $$Z = \sum_{\rho\in\Irrep}{(\hat f_\rho)}^{|\Lambda^2|}\,
\prod_{i\in\Lambda^0} C(i).$$ Observe further that the projectors onto the trivial component in the tensor products and are both given by the trace, $P^{(i\mu)}_{ab}=\frac{1}{\dim
V_\rho}\delta_{ab}$. The constraint $C(i)$ can be easily calculated: $$\begin{aligned}
C(i) &=& \sum_{p_{i-\hat1-\hat2}=1}^{\dim V_{\rho}}
\sum_{q_{i-\hat2}=1}^{\dim V_{\rho}}
\sum_{m_{i-\hat1}=1}^{\dim V_{\rho}}
\sum_{n_{i}=1}^{\dim V_{\rho}}\,
P^{(i,1)}_{q_{i-\hat2}n_i}\cdot P^{(i-\hat1,1)}_{p_{i-\hat1-\hat2}m_{i-\hat 1}}
\cdot P^{(i,2)}_{m_{i-\hat1}n_i}\cdot P^{(i-\hat2,2)}_{p_{i-\hat1-\hat2}q_{i-\hat2}}\nn\\
&=& \frac{1}{{(\dim V_\rho)}^3}.\end{aligned}$$ Therefore the partition function reads $$Z = \sum_{\rho\in\Irrep}{(\hat f_\rho)}^{|\Lambda^2|}\,
{(\dim V_\rho)}^{-3|\Lambda^0|}.$$ This is the well-known result for lattice gauge theory in two dimensions, see [@DrZu83].
Discussion {#sect_outlook}
==========
The duality transformation given in Theorems \[thm\_dualpartition1\] and \[thm\_dualwilson1\] is a strong-weak duality. For example, the character decomposition of the Boltzmann weight reads for the Wilson action of $G=U(1)$, $$\hat f_k=I_k(\beta),\qquad k\in\Z,$$ and for the Wilson action of $G=SU(2)$ using the fundamental representation, $$\hat f_j=2(2j+1)\,I_{2j+1}(\beta)/\beta,\qquad 2j\in\N_0.$$ Here the representations are parameterized by integers $k$ resp.non-negative half-integers $j$, and $I_n(x)$ denote the modified Bessel functions. The coefficients $\hat f_k$ resp. $\hat f_j$ are positive and can thus be written $\hat f_k=\exp(-s^\ast(k))$ resp.$\hat f_j=\exp(-s^\ast(j))$. The $\beta$-dependence of the dual action $s^\ast$ is such that high and low temperature regimes are exchanged or, in the language of gauge theory, strong and weak coupling (see [@Sa80; @DrZu83]). However, the coupling constant does not occur as a prefactor of the interaction terms of the dual model because its interactions do not arise from the dual Boltzmann weight but rather from the selection of projectors $\sym{P}_{i\mu}$ and from the factors $C(i)$, . For details about the character decompositions of the various common actions in lattice gauge theory, see [@MoMu94; @DrZu83] and references therein.
Of course, it is also possible to define the action of non-Abelian lattice gauge theory in terms of the character decomposition of its Boltzmann weight. For example, the *heat kernel action* (or generalized Villain action) is given by the choice $$\hat f_\rho = \dim V_\rho\cdot\exp(-C_\rho/\beta),$$ which makes the strong-weak duality manifest. Here $C_\rho$ denotes the eigenvalue of the quadratic Casimir operator (in a certain normalization) on the irreducible representation $\rho$ of $G$. Since $C_\rho$ is essentially quadratic in the highest weight of the representation $\rho$, it is apparent that higher representations are exponentially suppressed in the dual path integral. The smaller $\beta$ is chosen, the more pronounced is the suppression. The dual expressions and can therefore serve as generating functions for the strong coupling expansion. For details about strong coupling expansion techniques, see [@DrZu83].
Since the duality transformation for non-Abelian lattice gauge theory constructed in this paper generalizes the Abelian case in the form written with an explicit gauge constraint rather than in the form which is integrated and exhibits gauge degrees of freedom, there is no immediate answer to the question whether the dual model has any gauge invariance and how these degrees of freedom could be parametrized.
A non-Abelian generalization of the integration of a closed (and exact) $k$-cocycle to the coboundary of a $(k-1)$-cocycle up to a gauge freedom is not easy to find. If it exists, this paper might help to assemble information on how a non-Abelian generalization of cohomology might look like. Interesting in this context are the ideas developed in [@GrSc98].
Degrees of freedom in the dual model which are always present, are related to the choice of unitary representatives $\rho\in\Rep$ of each class of equivalent irreducible representations of $G$. The Clebsh-Gordan coefficients which enter the analysis extensively as the coefficients of the various projectors, depend on these choices.
In any case, the dual model can be expected to be the appropriate starting point to search for the non-Abelian magnetic degrees of freedom which generalize the magnetic monopoles of $U(1)$ lattice gauge theory, and to provide a framework for a rigorous treatment of their properties.
As far as the strong coupling expansion is concerned, the crucial question is to what extent the limit of this expansion is compatible with the continuum limit, whether properties derived via the strong coupling expansion hold (at least qualitatively) for all couplings. Recall that the continuum limit of lattice QCD consists of a combination of sending the inverse temperature $\beta\to\infty$ and the lattice spacing $a\to0$.
Both authors are grateful to DAAD for their scholarships. R.O. furthermore acknowledges support by EPSRC. We would like to thank in particular I. Drummond, J. Jers[á]{}k, A. J. Macfarlane, S. Majid, T. Neuhaus and K.-G. Schlesinger for valuable discussions, comments on the manuscript and for drawing our attention to relevant literature.
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[^1]: e-mail: [email protected]
[^2]: e-mail: [email protected]
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We study the performance of neural network models on random geometric transformations and adversarial perturbations. Invariance means that the model’s prediction remains unchanged when a geometric transformation is applied to an input. Adversarial robustness means that the model’s prediction remains unchanged after small adversarial perturbations of an input. In this paper, we show a quantitative trade-off between rotation invariance and robustness. We empirically study the following two cases: (a) change in adversarial robustness as we improve only the invariance of equivariant models via training augmentation, (b) change in invariance as we improve only the adversarial robustness using adversarial training. We observe that the rotation invariance of equivariant models (StdCNNs and GCNNs) improves by training augmentation with progressively larger random rotations but while doing so, their adversarial robustness drops progressively, and very significantly on MNIST. We take adversarially trained LeNet and ResNet models which have good $L_\infty$ adversarial robustness on MNIST and CIFAR-10, respectively, and observe that adversarial training with progressively larger perturbations results in a progressive drop in their rotation invariance profiles. Similar to the trade-off between accuracy and robustness known in previous work, we give a theoretical justification for the invariance vs. robustness trade-off observed in our experiments.'
author:
- Sandesh Kamath$^1$
- Amit Deshpande$^2$
- |
K V Subrahmanyam$^{1}$ $^1$Chennai Mathematical Institute,Chennai\
$^2$Microsoft Research, India\
[email protected], [email protected], [email protected]
bibliography:
- 'robustness.bib'
title: 'Invariance vs. Robustness of Neural Networks'
---
Introduction
============
Neural networks achieve state of the art accuracy on several standard datasets used in image classification. However, their performance in the wild depends on how well they can handle natural or non-adversarial transformations of input seen in real-world data as well as known deliberate, adversarial attacks created to fool the model.
Natural or non-adversarial transformations seen in real-world images include translations, rotations, and scaling. Convolutional Neural Networks (CNNs) are translation-invariant or shift-invariant by design. Invariance to other symmetries, and especially rotations, have received much attention recently, e.g., Harmonic Networks (H-Nets) [@Worrall16], cyclic slicing and pooling [@Dieleman16], Transformation-Invariant Pooling (TI-Pooling) [@Laptev16], Group-equivariant Convolutional Neural Networks (GCNNs) [@Cohen16], Steerable CNNs [@Cohen17], Deep Rotation Equivariant Networks (DREN) [@Li17], Rotation Equivariant Vector Field Networks (RotEqNet) [@Marcos17], and Polar Transformer Networks (PTN) [@Esteves18]. For a given symmetry group $G$, a $G$-equivariant network learns a representation or feature map at every intermediate layer such that any transformation $g \in G$ applied to an input corresponds to an equivalent transformation of its representations. Any model can improve its invariance to a given group of symmetries through sufficient training augmentation. Equivariant models use efficient weight sharing [@Kondor18] and require smaller sample complexity to achieve better invariance. Equivariant models such as CNNs and GCNNs too generalize well to progressively larger random rotations, but only when their training data is augmented similarly. Adversarial attacks on neural network models are certain, deliberate changes to inputs that fool a highly accurate model but are unlikely to fool humans. Given any neural network model, Szegedy et al. show how to change the pixel values of images only slightly so that the change is almost imperceptible to human eye but makes highly accurate models misclassify. They find these adversarial pixel-wise perturbations of small magnitude by maximizing the prediction error of a given model using box-constrained L-BFGS. Goodfellow et al. propose Fast Gradient Sign Method (FGSM) that adversarially perturbs $x$ to $x' = x + \epsilon~ \text{sign}\left(\nabla_{x} J(\theta, x, y)\right)$. Here $J(\theta, x, y)$ is the loss function used to train the network, $x$ is the input and $y$ is the target label and $\theta$ are the model parameters. Goodfellow et al. propose adversarial training, or training augmented with points $(x', y)$, as a way to improve adversarial robustness of a model.
Subsequent work introduced multi-step variants of FGSM. Kurakin et el. use an iterative method to produce an attack vector. Madry et al. proposed the Projected Gradient Descent (PGD) attack. Given any model, these attacks produce adversarial perturbation for every test image $x$ from a small $\ell_{\infty}$-ball around it, namely, each pixel value $x_{i}$ is perturbed within $[x_{i} -\epsilon, x_{i} + \epsilon]$. PGD attack does so by solving an inner optimization by projected gradient descent over $\ell_{\infty}$-ball of radius $\epsilon$ around $x$, to approximate the optimal perturbation. Adversarial training with PGD perturbations improves the adversarial robustness of models and it is one of the best known defenses to make models robust to perturbations of bounded $\ell_{\infty}$ norm on MNIST and CIFAR-10 datasets [@Madry18; @Athalye2018obfuscated].
Recent work has looked at simultaneous robustness to multiple adversarial attacks. Engstrom et al. show that adversarial training with PGD makes CNNs robust against perturbations of bounded $\ell_{\infty}$ norm but an adversarially chosen combination of a small rotation and a translation can nevertheless still fool these models. In Schott et al. , the authors shows that PGD adversarial training is a good defense against perturbations of bounded $\ell_{\infty}$ norm but can be broken with adversarial perturbations of small $\ell_{0}$ or $\ell_{2}$ norm that are also imperceptible to humans or have little semantic meaning for humans. Schott et al. show how to build models for MNIST dataset that are simultaneously robust to perturbations of small $\ell_{0}$, $\ell_{2}$ and $\ell_{\infty}$ norms.
Problem formulation and our results
-----------------------------------
Let $f$ be a neural network classifier trained on a training set of labeled images. The accuracy of $f$ is the fraction of test inputs $x$ for which the predicted label $f(x)$ matches the true label $y$. Similarly, for a given adversarial attack $\mathcal{A}$, the *fooling rate* of $\mathcal{A}$ on $f$ is the fraction of test inputs $x+\mathcal{A}(x)$ for which $f(x+\mathcal{A}(x)) \neq f(x)$. For a given transformation $T$ of the image space, $f$ is said to be $T$-invariant if the predicted label remains unchanged after the transformation $T$ for all inputs, i.e., $f(Tx) = f(x)$, for all inputs $x$. If $f$ is $T$-invariant and $f(x+\mathcal{A}(x)) \neq f(x)$, for some input $x$ and its adversarial perturbation $\mathcal{A}(x)$ with $\|\mathcal{A}(x)\|_{p} \leq \epsilon$, then by invariance $f(Tx +T\mathcal{A}(x)) = f(x+\mathcal{A}(x)) \neq (f(x) = f(Tx))$. Let $T$ be a translation, rotation, or, more generally, a permutation of coordinates as considered by Tramer and Boneh . We have $\|T\mathcal{A}(x)\|_{p} = \|\mathcal{A}(x)\|_{p} \leq \epsilon$. Hence, $T\mathcal{A}(x)$ is an adversarial perturbation for input $Tx$ of small $\ell_p$ norm. When a change of variables maps $x$ to $Tx$ by a permutation of coordinates, then the gradient and $T$ operators can be swapped. In other words, $\operatorname{grad}(f)$ at $Tx$ is the same as $T$ applied to $\operatorname{grad}(f)$ at $x$. This gives a 1-1 correspondence between FGSM (and PGD) perturbations of $x$ and $Tx$, respectively, with $\ell_p$ norm at most $\epsilon$. As a corollary, the $\ell_{p}$ fooling rate for any $T$-invariant classifier $f$ on the transformed data $\{Tx \;:\; x \in \mathcal{X}\}$ must be equal to its $\ell_{p}$ fooling rate on the original data $\mathcal{X}$.
A subtlety kicks in when $f$ is not truly invariant, that is, $f(Tx) = f(x)$, for most inputs $x$ but not all $x$. Define the *rate of invariance* of a classifier $f$ to a transformation $T$ as the fraction of test images whose predicted labels remain unchanged when $T$ is applied to $x$, i.e., $f(Tx) = f(x)$. For a class of transformations, e.g., rotations upto degree $[-\theta^{\circ}, +\theta^{\circ}]$, we define the *rate of invariance* as the average rate of invariance over transformations $T$ in this class. The rate of invariance is $100 \%$ if the model $f$ is truly invariant. When $f$ is not truly invariant, the interplay between the invariance under transformations and robustness under adversarial perturbations of small $\ell_{p}$-norm is subtle. *This interplay is exactly what we investigate.*
In this paper, we study neural network models and the simultaneous interplay between their rate of invariance for random rotations between $[- \theta^{\circ}, + \theta^{\circ}]$, and their adversarial robustness to pixel-wise perturbations of $\ell_{\infty}$ norm at most $\epsilon$. Measuring the robustness of a model to adversarial perturbations of $\ell_{p}$ norm at most $\epsilon$ is NP-hard [@KatzBDJK2017; @SinhaND2018]. Athalye et al. compare most of the known adversarial attacks and argue that PGD is among the strongest. Therefore, we use a models accuracy on test data adversarially perturbed using PGD as a proxy for its adversarial robustness.
Unlike previous studies by Engstrom et al. and Tramer and Boneh , we do not fix the magnitude of pixel-wise adversarial perturbations (e.g., say $\epsilon = 0.3$), nor do we limit ourselves to small rotations up to $\pm 30^{\circ}$. Another important difference is we consider random rotations instead of adversarial rotations. We compute the rate of invariance of a given model on inputs rotated by a random angle between $[- \theta^{\circ}, + \theta^{\circ}]$, for $\theta$ varying in the range $[0, 180]$. Similarly, we normalize the underlying dataset, and compute the accuracy of a given model on test inputs adversarially perturbed using PGD attack of $\ell_{\infty}$ norm at most $\epsilon$, for $\epsilon$ varying in the range $[0, 1]$.
We empirically study the following:
change in $\ell_{\infty}$ adversarial robustness as we improve only the rate of rotation invariance using training augmentation with progressively larger rotations,
change in invariance as we improve only adversarial robustness using PGD adversarial training with progressively larger $\ell_{\infty}$-norm of pixel-wise perturbations.
We study equivariant models, StdCNNs and GCNNs, as well as LeNet and ResNet models used by Madry et al. . Equivariant models, especially GCNNs, when trained with random rotation augmentations come very close to being truly rotation invariant [@Cohen16; @Cohen17; @Cohen18]. StdCNNs are translation-equivariant by design and GCNNs are rotation-equivariant by design through clever weight sharing [@Kondor18]. However, these models do not have high rate of invariance if their training data is not sufficiently augmented. This appears to be folklore so we do not elaborate on this. LeNet and ResNet models adversarially trained with PGD are among the best known $\ell_{\infty}$ adversarially robust models on MNIST and CIFAR-10, respectively, as shown by Madry et al. , and reconfirmed by Athalye et al. . In other words, equivariant models with training augmentation and PGD-trained LeNet and ResNet models essentially represent the two separate solutions known currently for achieving invariance and adversarial robustness, respectively.
**Our two main observations are as follows.**
- Equivariant models (StdCNNs and GCNNs) progressively improve their rate of rotation invariance when their training is augmented with progressively larger random rotations but while doing so, their $\ell_{\infty}$ adversarial robustness drops progressively. This drop or trade-off is very significant on MNIST.
- LeNet and ResNet models adversarially trained using progressively larger $\ell_{\infty}$-norm attacks improve their adversarial robustness but while doing so, their rate of invariance to random rotations upto $\pm \theta^{\circ}$ drops progressively.
We give a theoretical justification for the invariance vs. robustness trade-off observed in our experiments (see Theorem \[theorem:proof\]) by building upon the ideas in previous work on accuracy vs robustness trade-off [@Tsipras2019odds; @TramerBoneh2019].
#### Related Work.
Schott et al. study simultaneous robustness to adversarial perturbations of small $\ell_{0}$, $\ell_{2}$, and $\ell_{\infty}$-norm. Tramer and Boneh show an impossibility result by exhibiting a data distribution where no binary classifier can have substantially better-than-random accuracy simultaneously against both $\ell_{\infty}$ and $\ell_{1}$ perturbations. They consider a spatial perturbation that permutes a small number of coordinates of the input to model a combination of a small translation and a small rotation. They also construct a distribution and show that no model can have good accuracy simultaneously against both $\ell_\infty$ perturbations and spatial perturbations. They empirically validate this claim on MNIST and CIFAR-10 datasets for simultaneous robustness against $\ell_{\infty}$ adversarial perturbation and an adversarially chosen combination of translations upto $\pm 3$ pixels and rotations upto $\pm 30^{\circ}$. Intuitively and theoretically, it has been argued by Tsipras et al. and Tramer and Boneh that *small, adversarial pixel-wise perturbations* and *small, adversarial geometric transformations* are essentially dissimilar attacks focusing on different features, due to which a simultaneous solution to both may be inherently difficult. They do not postulate any gradual trade-off between invariance and robustness. They do not consider group-equivariant models such as GCNNs, a natural choice for invariance to geometric transformations.
Rotation invariance vs. $\ell_{\infty}$ adversarial robustness {#sec:invar-robust}
==============================================================
In this section, we present our main result about the interplay between rotation invariance and $\ell_{\infty}$ adversarial robustness of models on MNIST and CIFAR-10 data. In Subsection \[subsec:first-invar-then-robust\], we take StdCNN and GCNN models (see details in Section \[sec:experiment-arch\]) and study their rotation invariance and $\ell_{\infty}$ adversarial robustness, as we train them with random rotations of progressively larger degree. In Subsection \[subsec:first-robust-then-invar\], we take LeNet and ResNet models [@Madry18] and study their rotation invariance and $\ell_{\infty}$ adversarial robustness, as we do PGD adversarial training with progressively larger $\ell_{\infty}$ norms.
We present our experimental results as rotation invariance profiles and adversarial robustness profiles explained below.
Rotation invariance means that the predicted labels of an image and any of its rotations should be the same. Since most datasets are centered, we restrict our attention to rotations about the center of each image. We quantify rotation invariance by measuring the rate of invariance or the fraction of test images whose predicted label remains the same after rotation by a random angle between $[-\theta^{\circ}, \theta^{\circ}]$. As $\theta$ varies from $0$ to $180$, we plot the rate of invariance. We call this the *rotation invariance profile* of a given model.
Adversarial robustness means that the predicted labels of an image and its adversarial perturbation should be the same. We quantify the $\ell_{\infty}$ adversarial robustness of a given model to a fixed adversarial attack (e.g., PGD) and a fixed $\ell_{\infty}$ norm $\epsilon \in [0, 1]$ by (1 - fooling rate), i.e., the fraction of test inputs for which their predicted label does not change after adversarial perturbation. We plot this for $\epsilon$ varying from $0$ to $1$. We call this the *robustness profile* of a given model.
#### Convention used in the legends of our figures.
We use the following convention in the legends of some plots. A coloured line labeled $A/B$ indicates that the training data is augmented with random rotations from $[-A^{\circ}, A^{\circ}]$ and the test data is augmented with random rotations from $[-B^{\circ}, B^{\circ}]$. If $A$ (resp. $B$) is zero it means the training data (resp. test data) is unrotated. If the model is trained with random rotations from $[-A^{\circ}, A^{\circ}]$ and the test data is randomly rotated with varying $B$ to draw the plot, we only mention $A$ and not $B$, which is self-explanatory.
Effect of rotation invariance on $l_{\infty}$ adversarial robustness {#subsec:first-invar-then-robust}
--------------------------------------------------------------------
For any fixed $\theta \in [0, 180]$, we take an equivariant model, namely, StdCNN or GCNN, and augment its training data by random rotations from $[-\theta^{\circ}, +\theta^{\circ}]$. Figure \[stdcnn-mnist-inv-robust-fig\](left), Figure \[stdcnn-cifar10-inv-robust-fig\](left) shows how the robustness profile of StdCNN change on MNIST and CIFAR-10 respectively, as we increase the degree $\theta$ used in training augmentation of the model. We use PGD adversarial attack for MNIST and CIFAR-10. Figure \[stdcnn-mnist-inv-robust-fig\](right) and Figure \[stdcnn-cifar10-inv-robust-fig\](right) show the rotation invariance profile of the same models on MNIST, CIFAR-10 respectively.
The black line in Figure \[stdcnn-mnist-inv-robust-fig\] (left), shows that the adversarial robustness of a StdCNN which is trained to handle rotations up to $\pm 180$ degrees on MNIST, drops by more than 50%, even when the $\epsilon$ budget for PGD attack on unrotated MNIST is only $0.1$. The black line in Figure \[stdcnn-mnist-inv-robust-fig\] (right), shows this models rotation invariance profile - this model is invariant to larger rotations in the test data. This can be contrasted with the model depicted by the red line - this StdCNN is trained to handle rotations up to $60$ degrees. The rotation invariance profile of this model is below that of the model depicted by the black line and so it is not invariant to large rotations. However this model can handle adversarial $\ell_{\infty}$-perturbations up to 0.3 on unrotated data, with an accuracy more than 80$\%$ - this can be seen from the red line in Figure \[stdcnn-mnist-inv-robust-fig\] (left).
From these plots it is clear that [*the rotation invariance of these models improves by training augmentation but at the cost of their adversarial robustness, indicating a trade-off between invariance to rotations and adversarial robustness.*]{} The above observations, of there being a trade-off between handling larger rotations with training augmentation and handling larger adversarial perturbations is seen in GCNNs also. This can be seen from Figures \[gcnn-mnist-inv-robust-fig\] and Figure \[gcnn-cifar10-inv-robust-fig\]. The plots are very similar to what we observe with StdCNNs.
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Effect of $\ell_{\infty}$ adversarial training on rotation invariance {#subsec:first-robust-then-invar}
---------------------------------------------------------------------
The most common approach to improve adversarial robustness is adversarial training, i.e., training the model on adversarially perturbed training data. Adversarial training with PGD attack is one of the strongest known defenses on MNIST and CIFAR-10 datasets (see [@Athalye2018obfuscated]).
For any fixed $\epsilon \in [0, 1]$ we adversarially train our models, LeNet and ResNet, as done by Madry et al. with PGD adversarial perturbations with $\ell_{\infty}$ budget $\epsilon$. As in Madry et al. we use the LeNet model for MNIST and the ResNet model for CIFAR-10. We then plot their rotation invariance profiles and robustness profiles. Each colored line in Figure \[mnist-robustprof-fig-1\] and Figure \[mnist-robustprof-fig\] corresponds to a model adversarially trained with a different value of $\epsilon$.
On MNIST, adversarial training with PGD with larger $\epsilon$ results in a drop in the invariance profile of LeNet based model - in Figure \[mnist-robustprof-fig-1\] (left), the yellow line (PGD with $\epsilon=0.4$) is below the light blue line (PGD with $\epsilon=0.1$). Similar qualitative drop holds for the ResNet based model too on CIFAR-10, as can be seen from Figure \[mnist-robustprof-fig\] (left). In other words, [*adversarial training with progressively larger $\epsilon$ leads to the drop in the rate of invariance on the test data.*]{}
To complete this picture we plot the robustness profile curves of the LeNet and ResNet based model for MNIST and CIFAR-10, respectively. It is known that as these models are trained with PGD using larger $\epsilon$ budget their adversarial robustness increases. The robustness profile curves of the LeNet model trained with larger PGD budget dominates the robustness profile curve of the same model trained with a smaller PGD budget - the red line in Figure \[mnist-robustprof-fig-1\] (right), dominates the light blue line. This is true of the ResNet based model too, as can be seen from Figure \[mnist-robustprof-fig\] (right).
Invariance vs. robustness trade-off proof
=========================================
In this section, we give theoretical demonstration of an invariance-vs-robustness trade-off similar to our experiments. We consider an $\ell_{\infty}$ adversarial perturbation $\mathcal{A}(x)$ that perturbs the coordinates of any input $x$ by a small value. Observe that rotation by $0^{\circ}$ leaves any input $x$ unchanged whereas rotation by $180^{\circ}$ keeps the center pixel fixed and makes pairwise swaps for all other pixels radially opposite to each other around the center. We consider a random permutation of coordinates $r(x)$ to mimic picking a uniformly random rotation from $\{0^{\circ}, 180^{\circ}\}$. We consider a joint distribution on input-label pairs such that there exists a classifier of high accuracy. However, we prove that no classifier can have high accuracy (w.r.t. true labels) after the random transformation $r(x)$ as well as the adversarial perturbation $\mathcal{A}(x)$, simultaneously. Even though our theorem is about accuracy after $r(x)$ instead of the rate of invariance, and accuracy after $\mathcal{A}(x)$ instead of (1 - fooling rate), these values are close for any classifier that has high accuracy on the original data distribution.
Consider a random input-label pair $(X, Y)$ with $X = (X_{0}, X_{1}, \dotsc, X_{2d})$ taking values in $\R^{2d+1}$ and $Y$ taking values in $\{-1, 1\}$ generated as follows. The class label $Y$ takes value $\pm 1$ with probability $1/2$ each. $X_{0} \given Y=y$ takes value $y$ with probability $p$ and $-y$ with probability $1-p$, for some $p \geq 1/2$. The remaining coordinates are independent and normally distributed with $X_{2t-1} \given Y=y$ as $N(3y/\sqrt{d}, 1)$ and $X_{2t} \given Y=y$ as $N(-3y/\sqrt{d}, 1)$, for $1 \leq t \leq d$. First of all, there exists a classifier $f^{*}(x) = {\operatorname{sign}\left(\sum_{t=1}^{d} x_{2t-1}\right)}$ with high accuracy ${\operatorname{Pr}\left({\operatorname{sign}\left(\sum_{i=1}^{d} X_{2t-1}\right)} = Y\right)} > 99\%$ for this data distribution. The proof of this follows from the three-sigma rule for normal distributions, and is similar to the equation (4) in Subsection 2.1 of Tsipras et al. .
Let $\mathcal{A}(x)$ denote an adversarial perturbation for $(x, y)$ given by $(\mathcal{A}(x))_{0} = 0$, and $(\mathcal{A}(x))_{2t-1} = -6y/\sqrt{d}$, $(\mathcal{A}(x))_{2t} = 6y/\sqrt{d}$, for $1 \leq t \leq d$. Note that ${\left\|\mathcal{A}(x)\right\|}_{\infty} = 6/\sqrt{d}$, for all $x \in \R^{2d+1}$.
Given any input $x \in \R^{2d+1}$, let $r(x)$ be a random transformation of $x$ that leaves $x$ unchanged as $r(x) = x$ with probability $1/2$, and swaps successive odd-even coordinate-pairs $(x_{2t-1}, x_{2t})$ for $1 \leq t \leq d$ to get $(x_{0}, x_{2}, x_{1}, \dotsc, x_{2d}, x_{2d-1})$, with probability $1/2$.
\[theorem:proof\] Given input distributions defined above with $1/2 \leq p < 1-\delta$, for any classifier $f: \R^{2d+1} \rightarrow \{-1, 1\}$, both ${\operatorname{Pr}\left(f(r(X)) = Y\right)}$ and ${\operatorname{Pr}\left(f(X + \mathcal{A}(X)) = Y\right)}$ cannot be more than $1-\delta$ simultaneously.
The random transformation $r(X)$ leaves $X$ unchanged with probability $1/2$ but with the remaining $1/2$ probability, it makes the data distribution of $r(X)$ the same as $X + \mathcal{A}(X)$. Thus, $$\begin{split}
{\operatorname{Pr}\left(f(r(X)) = Y\right)} \\
& \hspace{-0.6in} = \frac{1}{2}~ {\operatorname{Pr}\left(f(X) = Y\right)} + \frac{1}{2}~ {\operatorname{Pr}\left(f(X + \mathcal{A}(X)) = Y\right)},\\
\end{split}$$ where the LHS probability is over $r, X, Y$ while the RHS probabilities are only over $X, Y$. Let $G_{y}$ denote $(X_{1}, \dotsc, X_{2d}) \given Y=y$. The adversarial perturbation $X + \mathcal{A}(X)$ turns $G_{y}$ into $G_{-y}$. $$\begin{split}
{\operatorname{Pr}\left(f(X + \mathcal{A}(X)) \neq Y\right)} \\
& \hspace{-1.4in} = \frac{1}{2} \sum_{y \in \{-1, 1\}} {\operatorname{Pr}\left(f((X_{0}, G_{-y})) = -y\right)} \\
& \hspace{-1.4in} = \frac{1}{2} \big \{\sum_{y \in \{-1, 1\}} p~ {\operatorname{Pr}\left(f((y, G_{-y})) = -y\right)} \\
& \hspace{-0.4in} + (1-p)~ {\operatorname{Pr}\left(f((-y, G_{-y})) = -y\right)} \big \}\\
& \hspace{-1.4in} \geq \frac{1-p}{2p} \big \{\sum_{y \in \{-1, 1\}} (1-p)~ {\operatorname{Pr}\left(f((y, G_{-y})) = -y\right)} \\
& \hspace{-0.4in} + p~ {\operatorname{Pr}\left(f((-y, G_{-y})) = -y\right)} \big \} \\
& \hspace{-1.4in} = \frac{1-p}{2p} \big \{\sum_{y \in \{-1, 1\}} (1-p)~ {\operatorname{Pr}\left(f((-y, G_{y})) = y\right)} \\
& \hspace{-0.4in} + p~ {\operatorname{Pr}\left(f((y, G_{y})) = y\right)} \big \} \\
& \hspace{-1.4in} = \frac{1-p}{p}~ {\operatorname{Pr}\left(f(X) = Y\right)}.
\end{split}$$ Plugging this in the above expression of ${\operatorname{Pr}\left(f(r(X)) = Y\right)}$ we get $$\begin{split}
{\operatorname{Pr}\left(f(r(X)) = Y\right)} + \frac{2p-1}{2(1-p)}~ {\operatorname{Pr}\left(f(X + \mathcal{A}(X)) = Y\right)} \\
& \hspace{-1.0 in} \leq \frac{p}{2(1-p)}.
\end{split}$$ If both ${\operatorname{Pr}\left(f(r(X)) = Y\right)}$ and ${\operatorname{Pr}\left(f(X + \mathcal{A}(X)) = Y\right)}$ are at least $1-\delta$, then the above inequality implies $1-\delta \leq p$. Thus, if $p < 1 - \delta$, we get a contradiction.
Details of experiments {#sec:experiment-arch}
======================
All experiments performed on neural network-based models were done using MNIST and CIFAR-10 datasets with appropriate augmentations applied to the train/validation/test set.
#### Data sets.
MNIST dataset consists of $70,000$ images of $28 \times 28$ size, divided into $10$ classes. $55,000$ used for training, $5,000$ for validation and $10,000$ for testing. CIFAR-10 dataset consists of $60,000$ images of $32 \times 32$ size, divided into $10$ classes. $40,000$ used for training, $10,000$ for validation and $10,000$ for testing.
#### Equivariant Model Architectures.
For the MNIST based experiments we use the network architecture of GCNN as given in Cohen and Welling . The StdCNN architecture is similar to the GCNN except that the operations are as per CNNs. Refer to Table \[gcnn-table\] for details. For the CIFAR-10 based experiments we use the VGG16 architecture as given in Simonyan and Zisserman and its GCNN equivalent is obtained replacing the various layer operations with equivalent GCNN operations as given in Cohen and Welling . This is similar to how we obtained a GCNN architecture from StdCNN for the MNIST based experiments. Input training data was augmented with random cropping and random horizontal flips.
#### Adversarially Robust Model Architectures.
For the adversarial training experiments we used the LeNet based architecture for MNIST and the ResNet architecture for CIFAR-10. Both these models are exactly as given in Madry et al. .
[ll]{} &\
\
Conv(10,3,3) + Relu & P4ConvZ2(10,3,3) + Relu\
Conv(10,3,3) + Relu & P4ConvP4(10,3,3) + Relu\
Max Pooling(2,2) & Group Spatial Max Pooling(2,2)\
Conv(20,3,3) + Relu & P4ConvP4(20,3,3) + Relu\
Conv(20,3,3) + Relu & P4ConvP4(20,3,3) + Relu\
Max Pooling(2,2) & Group Spatial Max Pooling(2,2)\
FC(50) + Relu & FC(50) + Relu\
Dropout(0.5) & Dropout(0.5)\
FC(10) + Softmax & FC(10) + Softmax\
Conclusion
==========
We observe that as equivariant models (StdCNNs and GCNNs) are trained with progressively larger rotations their rotation invariance improves but at the cost of their adversarial robustness. Adversarial training with perturbations of progressively increasing norms improves the robustness of LeNet and ResNet models, but with a resulting drop in their rate of invariance. We give a theoretical justification for the invariance-vs-robustness trade-off observed in our experiments.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We revisit our recent contribution [@AloLo1] and give two simpler proofs of the so-called Haff’s law for granular gases (with non-necessarily constant restitution coefficient). The first proof is based upon the use of entropy and asserts that Haff’s law holds whenever the initial datum is of finite entropy. The second proof uses only the moments of the solutions and holds in some weakly inelasticity regime which has to be clearly defined whenever the restitution coefficient is non-constant.\
<span style="font-variant:small-caps;">Keywords:</span> Boltzmann equation, inelastic hard spheres, granular gas, cooling rate, Haff’s law.\
<span style="font-variant:small-caps;">AMS subject classification:</span> 76P05, 76P05, 47G10, 82B40, 35Q70, 35Q82.
address:
- '**Ricardo J. Alonso**, Department of Computational and Applied Mathematics, Rice University, Houston, TX 77005-1892.'
- '**Bertrand Lods**, Dipartimento di Statistica e Matematica Applicata, Collegio Carlo Alberto, Università degli Studi di Torino, Corso Unione Sovietica, 218/bis, 10134 Torino, Italy.'
author:
- 'Ricardo J. Alonso & Bertrand Lods'
title: '[Two proofs of Haff’s law for dissipative gases: the use of entropy and the weakly inelastic regime]{}'
---
Introduction {#intro}
============
The main objective of the present paper is to revisit our recent contribution [@AloLo1] and give two simpler proofs of the so-called Haff’s law for granular gases (with non-necessarily constant restitution coefficient). The first proof is based upon the use of Boltzmann’s entropy and asserts that Haff’s law holds whenever the initial datum is of finite entropy. The second proof uses only the moments of the solutions and shows that Haff’s law holds in some weakly inelasticity regime (see Theorem \[main\] for a precise definition in the case of non-constant restitution coefficient) for initial datum with finite energy.
Motivation
----------
We consider in this paper freely cooling granular gases governed by the spatially homogeneous Boltzmann equation $$\label{cauch}\begin{cases}
\partial_t f(t,v)&=\Q_{e}(f,f)(t,v) \qquad \qquad t >0, \; v \in \mathbb{R}^3\\
f(0,v)&=f_0(v), \qquad \qquad v \in \mathbb{R}^3
\end{cases}$$ where the initial datum $f_0$ is a nonnegative velocity distribution such that $$\label{initial}\IR f_0(v)\d v=1, \quad \IR f_0(v) v \d v =0 \quad \text{ and } \quad \IR f_0(v)|v|^2 \d v < \infty.$$ The operator $\Q_e(f,f)$ is the inelastic Boltzmann collision operator, expressing the effect of binary collisions of particles. We assume here that the granular particles are perfectly smooth hard-spheres of mass $m=1$. The inelasticity of the collision mechanism is characterized by a single scalar parameter known as the coefficient of normal restitution $0 \leq
e \leq 1$. Indeed, if $v$ and $\vb$ denote the velocities of two particles before they collide, their respective velocities $v'$ and $\vb'$ after collisions are such that $$\label{coef}
(u'\cdot \n)=-(u\cdot \n) \,e,$$ where $\n \in \mathbb{S}^2$ determines the impact direction, i.e. $\n$ stands for the unit vector that points from the $v$-particle center to the $\vb$-particle center at the instant of impact. Here above $$u=v-\vb \quad \text{and}\quad u'=v'-\vb',$$ denote respectively the relative velocity before and after collision. In this work, the restitution coefficient $e$ is assumed to be a function of the impact velocity, i.e. $$e:=e(|u \cdot \n|).$$ In virtue of and the conservation of momentum, the post-collision velocities $(v',\vb')$ are given by $$\label{transfpre}
v'=v-\frac{1+e}{2}\,(u\cdot \n)\n,
\qquad \vb'=\vb+\frac{1+e}{2}\,(u\cdot \n)\n.$$ The main assumptions on the function $e(\cdot)$ are listed here after (see Proposition \[HYP1\]) and ensure that the Jacobian of the above transformation is given by $$J_e(|u\cdot \n|)=\left|\dfrac{\partial v'\partial \vb'}{\partial v \partial \vb}\right|=e(|u\cdot \n|)+ |u\cdot \n|e'(|u\cdot \n|)=:\vartheta'_e(|u\cdot \n|)$$ where $e'(\cdot)$ and $\vartheta_e'(\cdot)$ denote the derivative of $r \mapsto e(r)$ and $r \mapsto \vartheta_e(r)$ respectively (this prime symbol should not be confused with the one we have chosen for the post-collisional velocity). We refer the reader to [@AlonsoIumj] for more details. The main examples of restitution coefficient we shall deal within this paper are the following:
1. The first fundamental example is the one of a ***constant restitution coefficient*** for which $e(r)=\e \in (0,1]$ for any $r \geq 0.$
2. The most physically relevant variable restitution coefficient is the one corresponding to the so-called ***viscoelastic hard-spheres*** [@BrPo]. For such a model, the properties of the restitution coefficient have been derived in [@BrPo; @PoSc] and it can be shown that $e(z)$ is defined implicitly by the following $$\label{visco}e(r) + a r^{1/5} e(r )^{3/5}=1 \qquad \qquad \forall r \geq 0$$ where $a>0$ is a suitable positive constant depending on the material viscosity.
With the above notations, the Boltzmann collision operator is given, in weak form, by the following equation $$\begin{gathered}
\label{weakn}
\IR \Q_e(f,f)\psi(v)\d v\\=\dfrac{1}{2\pi}\int_{\R^3 \times \R^3 \times \S} |u \cdot \n|f(v)f(\vb)\big(\psi(v')+\psi(\vb')-\psi(v)-\psi(\vb)\big)\d v \d\vb \d\n\end{gathered}$$ for any smooth test-function $\psi(v)$. The strong form of $\Q_e$ can be recovered easily (see [@AlonsoIumj]). Notice that an alternative parametrization of the post-collision velocities would lead to a slighty different weak formulation of the collision operator (this alternative formulation was preferred in [@AloLo1]).
As explained in [@AloLo1], in absence of any heating source, the granular temperature $$\E(t)=\IR f(t,v)|v|^2\d v, \qquad \quad t \geq 0$$ is continuously decreasing and tending to zero as time goes to infinity, expressing the *cooling of the granular gases*. The precise cooling rate of the temperature is the main concern of this note. It was proven in [@AloLo1], and predicted by the physics literature long ago, that the cooling rate is strongly depending on the choice of the restitution coefficient. Note that using the weak form with the test function $\psi(v)=|v|^2$, the evolution of $\E(t)$ is governed by the following relation $$\label{temperature}\dfrac{\d }{\d t}\E(t)=- \IRR f(t,v)f(t,\vb)\mathbf{\Psi}_e(|u|^2)\d v\d\vb,$$ where the dissipation energy potential associated to $e(\cdot)$ is given by $$\label{Psie}
\mathbf{\Psi}_e(r)=\frac{r^{3/2}}{2} \int_0^{1} \left(1-e^2(\sqrt{r}z) \right) z^3\d z \qquad \qquad r >0.$$ We refer to [@AloLo1] for technical details. In the *op. cit.* we introduced the following general assumptions:
\[HYP1\] The restitution coefficient $e(\cdot)$ is such that the following hold:
1. The mapping $r \in \mathbb{R}_+ \mapsto e(r) \in (0,1]$ is absolutely continuous.
2. The mapping $r\in\mathbb{R}^{+} \mapsto \vartheta(r):=r\;e(r)$ is strictly increasing.
3. $\limsup_{r \to \infty} e(r)=e_0 < 1.$
4. The function $x>0 \longmapsto \mathbf{\Psi}_e(x)$ defined in is strictly increasing and convex over $(0,+\infty)$.
These assumptions are fulfilled by the two examples described above and, more generally, they hold whenever the restitution coefficient $r \mapsto e(r)$ is an absolutely continuous and non-increasing mapping (see [@AloLo1 Appendix]). Notice that the first two assumptions are exactly those needed in order to compute the Jacobian of the transformation . The last two are needed in order to get the following proposition which is based on Jensen’s inequality.
\[prop:cool\] Let $f_0$ be a nonnegative velocity distribution satisfying and let $f(t,v)$ be the associated solution to the Cauchy problem where the variable restitution coefficient satisfies Assumptions \[HYP1\]. Then, $$\dfrac{\d }{\d t}\E(t) \leq -\mathbf{\Psi}_e(\E(t)) \qquad \qquad \forall t \geq 0$$ and, as a consequence, $\lim_{t \to \infty}\E(t)=0.$ Moreover, if one assumes that there exist $\alpha >0$ and $\gamma \geq 0$ such that $$\label{smallez}e(z) \simeq 1-\alpha\, z^\gamma \quad \text{ for } \quad z \simeq 0$$ then there exist $C >0$ and $t_0 >0$ such that $$\E(t) \leq C\left(1+t\right)^{-\frac{2}{1+\gamma}} \qquad \qquad \forall t \geq t_0.$$
The well-posedness of the Cauchy problem has been proved in [@MMR].
Notice that the above assumption is equivalent to assume that $$\label{ell}\ell_\gamma(e)=\sup_{r >0} \dfrac{1-e(r)}{r^\gamma} < \infty$$ since, for large value of $r >0$, $(1-e(r))/r^\gamma$ is clearly finite for any $\gamma >0$. In particular, for a constant restitution coefficient $e(r)=e_0$, one has $\ell_{0}(e) = 1-e_0$ which means that holds. For the model of viscoelastic hard-spheres given by , the restitution coefficient $e(\cdot)$ is such that $\ell_{1/5}(e)= a$. Furthermore, if $\ell_\gamma(e) < \infty$ for some $\gamma >0$, then $\ell_\alpha(e)=\infty$ for any $\alpha \neq \gamma$. Indeed, the parameter $\gamma$ is exactly the one that prescribes the behavior of $e(r)$ for ***small values*** of $r$.
Proposition \[prop:cool\] illustrates the fact that the decay of the temperature is governed by the behavior of the restitution coefficient $e(r)$ for small value of $r$. Now, in order to match the precise cooling rate of the temperature and prove the so-called generalized Haff’s law, one needs to prove that, under Assumptions \[HYP1\] and , there exists $c>0$ such that $$\label{converseE}
\E(t) \geq c(1+t)^{-\frac{2}{1+\gamma}}\qquad \forall t \geq 0.$$ This was precisely the main objective in [@AloLo1] and, as far as the cooling rate is concerned, the main result of the *op. cit.* can be formulate as
\[haff1\] For any initial distribution velocity $f_0 \geq 0$ satisfying the conditions given by with $f_0 \in L^{p}(\R^3)$ for some $1 < p < \infty$, the solution $f(t,v)$ to the associated Boltzmann equation satisfies the generalized Haff’s law for variable restitution coefficient $e(\cdot)$ fulfilling Assumptions \[HYP1\] and : $$\label{Haff's}
c (1+t)^{-\frac{2}{1+\gamma}} \leq \E(t) \leq C (1+t)^{-\frac{2}{1+\gamma}}, \qquad t \geq 0$$ where $c,C$ are positive constants.
An additional assumption was required in theorem \[haff1\] on the restitution coefficient $e(\cdot)$, see [@AloLo1 Assumption 4.10]. We do not insist on this point since we believe such assumption is only of technical nature and likely unnecessary.
For constant restitution coefficient, Haff’s law has been proved in [@MiMo]. This approach was generalized in [@AloLo1] leading to Theorem \[haff1\]. The proof is based on the following steps:
1. The study of the moments of solutions to the Boltzmann equation using a generalization of the Povzner’s lemma developed in [@BoGaPa].
2. Precise $L^p$ estimates, in the same spirit of [@MiMo], of the solution to the Boltzmann equation for $1<p<\infty$.
3. A study of the problem in self-similar variable (that is, for suitable *rescaled solutions*).
Because of the method of proof, step $(2)$ hereabove, the above result requires strong integrability assumption on the initial density $f_0$ which has to belong to some $L^{p}$ space with $p >1.$ The main purpose of this paper is to remove the unphysical assumption $$\label{f0p}
f_0 \in L^p \qquad \text{ for some } p >1,$$ and prove that the generalized Haff’s law still holds under less restrictive assumptions.
Main results
------------
We present two independent treatments of the above problem:
1. We prove that Haff’s law still holds if we replace by the less restrictive constraint $$\H(f_0)=\IR f_0(v)\log f_0(v)\d v < \infty$$ and $e(\cdot)$ satisfying for some $\gamma >0$.
2. Using only finiteness of mass and energy on the initial datum, we prove that Haff’s law holds true in some weakly inelastic regime defined in the sequel. We notice that for constant restitution coefficient, the proof of Haff’s law given in [@MiMo] required the assumption $f_0 \in L^p$ with $p >1$. However, it was observed in [@AloLo1] that, for this case, Haff’s law holds assuming only that the restitution coefficient is sufficiently close to one. We give a complete proof of this fact in the sequel.
More precisely, the main results of the present paper can be stated in the following theorems.
\[main0\] Let $e(\cdot)$ be a non-constant restitution coefficient that satisfies Assumptions \[HYP2\] below. Furthermore, assume that the initial distribution $f_0$ satisfies together with $\H(f_0) < \infty$, and let $f(t,v)$ be the unique solution to . Then, the generalized Haff’s law holds true.
For constant restitution coefficient, our result is weaker, however, we give a qualitative version of Haff’s law in this case indicating an algebraic rate of decrease for the temperature, see Theorem \[entrHaff1\].
Referring to the point $(2)$ we state two different results, distinguishing the constant and non-constant cases.
\[mainc\] Let $f_0$ be a nonnegative velocity distribution satisfying and let $f(t,v)$ be the associated solution to . Assume that the restitution coefficient $e(\cdot)$ is constant $e(r)=\e \in (0,1)$ and such that $$\label{small*}\frac{3(1-\e^2)}{8} < 1-\kappa_{3/2}$$ where the expression of $\kappa_{3/2}$ is given in Prop. \[povzner\]. Then the generalized Haff’s law holds true.
For non-constant restitution coefficient, the situation is different and the condition on the restitution coefficient will depend on the initial datum. One can formulate our result as follows (see Theorem \[mainnon\] for a more precise statement).
\[main\] Let $f_0$ be a nonnegative velocity distribution satisfying and let $f(t,v)$ be the associated solution to the Cauchy problem . For any $\gamma>0$, there exists some explicit $\ell_0:=\ell_{0}(f_0,\gamma) >0$ such that, if the restitution coefficient $e(\cdot)$ satisfies Assumptions \[HYP1\] and with $\ell_\gamma(e) < \ell_0$, then the generalized Haff’s law hold true.
The proof of Theorem \[main0\] is much simpler than the proof of [@AloLo1] under the assumption on the initial datum. In particular, it does not requires the introduction of self-similar variables. It is based essentially on the fact that entropy of the solution $f(t,v)$ to grows at most logarithmically, namely, there exists $K_0 >0$ such that $$\H(f(t)) \leq K_0 \log (1+t) \qquad \forall t \geq 0.$$ Then, using some estimates which allow to relate the energy $\E(t)$ to the entropy, we can deduce from such logarithmic growth that the decreasing of the energy $\E(t)$ is *at most* algebraic, that is, there exists some finite $\lambda >0$ such that $\inf_{t\geq0}(1+t)^\lambda \E(t)>0$. It is known from [@AloLo1] (see also Proposition \[lambda\]) that for non-constant restitution coefficient, this is enough to conclude the Haff’s law . The proof of Theorem \[main0\] is given in Section 3 (see Theorem \[entrHaff\]) while several inequalities relating energy and entropy are given in the Appendix.\
Concerning Theorems \[mainc\] and \[main\], their proofs are surprisingly simple and rely only on a careful study of the various moments of the solution to the Cauchy problem . They will be the object of Section 4.
Some known results
==================
We briefly recall some known estimates on the moments of the solution to the Cauchy problem. In this section, we will assume that the restitution coefficient $e(\cdot)$ satisfies Assumptions \[HYP1\] and that the initial datum $f_0$ satisfies . We denote then by $f(t,v)$ the associated solution to the Cauchy problem . For any $t \geq 0$ and any $p \geq 1$ we define $$\label{defmp}m_p(t):=\IR f(t,v)|v|^{2p}\d v$$ with the convention of notation $\E(t)=m_1(t)$. Then, one has the following proposition, see [@AloLo1].
\[povzner\] For any real $p \geq 1$, one has $$\label{QepSp}
\dfrac{\d}{\d t}m_p(t)=\IR \Q_{e}(f,f)(t,v)|v|^{2p}\d v\leq-(1-\kappa_{p})m_{p+1/2}(t)+\kappa_{p}\;S_{p}(t),$$ where, $$S_{p}(t)=\sum^{[\frac{p+1}{2}]}_{k=1}\left(
\begin{array}{c}
p\\k
\end{array}
\right)\left(m_{k+1/2}(t)\;m_{p-k}(t)+m_{k}(t)\;m_{p-k+1/2}(t)\right),$$ $[\frac{p+1}{2}]$ denoting the integer part of $\frac{p+1}{2}$ and $$\kappa_p=\sup_{\widehat{U} \in \mathbb{S}^2}\int_{\widehat{U}\cdot \sigma \geq 0} \left(\dfrac{3+\widehat{U}\cdot \sigma}{4}\right)^p + \left(\dfrac{1-\widehat{U}\cdot\sigma}{4}\right)^p\dfrac{\d\sigma}{2\pi}=\int_0^1 \left(\frac{3+t}{4}\right)^p + \left(\frac{1-t}{4}\right)^p \d t >0$$ is an explicit constant such that $\kappa_p < 1$ for any $p > 1.$
A simple consequence of the above is the following, [@AloLo1 Corollary 3.6]: For any $p \geq 1$, there exists some constant $K_p >0$ such that $$m_p(0) < \infty \implies m_p(t) \leq K_p(1+t)^{-\frac{2p}{1+\gamma}} \qquad \forall t \geq 0.$$ Furthermore, since we are dealing with hard spheres, the phenomenon of appearance of moments occurs in the same way as in the classical elastic Boltzmann [@desvillettes; @We99]. Thus, as soon as $\E(0) < \infty$, the higher moments satisfy $\sup_{t\geq t_0}m_p(t) < \infty$ for any $t_0 >0$. In particular, one can rephrase [@AloLo1 Corollary 3.6].
\[moments\] Let $f_0$ be a nonnegative velocity distribution satisfying and let $f(t,v)$ be the associated solution to the Cauchy problem where the variable restitution coefficient satisfies Assumptions \[HYP1\] and . For any $t_0 >0$ and any $p \geq 0$, there exists $K_p >0$ such that $$\label{kp}
m_p(t) \leq K_p \left(1+t\right)^{-\frac{2p}{1+\gamma}} \qquad \forall t \geq t_0.$$
Observe that in order to prove that the second part of Haff’s law , it is enough to control $m_{\frac{3+\gamma}{2}}(t)$ in terms of $\E(t)^{\frac{3+\gamma}{2}}$. Indeed, recall that $$-\dfrac{\d}{\d t}\E(t)=\IRR f(t,v)f(t,\vb)\mathbf{\Psi}_e(|u|^2)\d v \d \vb \qquad \forall t \geq 0.$$ Since $\ell_\gamma(e) < \infty$, one has $$1-e(r) \leq \ell_\gamma(e)r^\gamma \qquad \forall r >0.$$ Plugging this estimate in the definition of $\mathbf{\Psi}_e(r^2)$ and using the fact that $1-e^2(r) \leq 2(1-e(r))$ for any $r >0$, we get that $$\mathbf{\Psi}_e(|u|^2) \leq \ell_\gamma(e)|u|^{3+\gamma}\int_0^1 z^{3+\gamma}\d z=\dfrac{\ell_\gamma(e)}{4+\gamma} |u|^{3+\gamma} \qquad \forall u\in\mathbb{R}^{3}.$$ Since $|u|^{3+\gamma} \leq 2^{2+\gamma}\left(|v|^{3+\gamma}+|\vb|^{3+\gamma}\right)$, one gets $$\begin{split}\label{Egamma}
- \dfrac{\d}{\d t}\E(t) &\leq \dfrac{2^{2+\gamma}\ell_\gamma(e)}{4+\gamma} \IRR f(t,v)f(t,\vb)\left(|v|^{3+\gamma}+|\vb|^{3+\gamma}\right)\d v\d\vb\\
&=\dfrac{2^{3+\gamma}\ell_\gamma(e)}{4+\gamma} m_{\frac{3+\gamma}{2}}(t).\end{split}$$ Therefore, if there exists some constant $K>0$ such that $$\label{controlg} m_{\frac{3+\gamma}{2}}(t) \leq K \E(t)^{\frac{3+\gamma}{2}}(t) \qquad \forall t \geq t_0>0,$$ setting $C_\gamma=\frac{2^{3+\gamma}K}{4+\gamma}\ell_\gamma(e)$, we obtain from that $$- \dfrac{\d}{\d t}\E(t) \leq C_\gamma\,\E(t)^{\frac{3+\gamma}{2}} \qquad \forall t \geq t_0.$$ A simple integration of this inequality yields implies, $$\E(t)\geq \frac{\E(t_0)}{\left(1+\frac{1+\gamma}{2}\E(t_0)^{\frac{1+\gamma}{2}}C_{\gamma}\;(t-t_0)\right)^{\frac{2}{1+\gamma}}}\quad\forall t\geq t_0.$$ which implies . An additional simplification in the arguments comes with the following proposition which has already been used implicitly in [@AloLo1]. We give a complete proof of it for the sake of clarity.
\[propstrat\] Assume that the restitution coefficient $e(\cdot)$ satisfy Assumptions \[HYP1\] and is such that $\ell_\gamma(e) < \infty$ for some $\gamma \geq 0$. Assume that there exists some constant $C>0$ such that $$\label{controlm}
m_{\frac{3}{2}}(t) \leq C \E(t)^{\frac{3}{2}}(t) \qquad \forall t \geq t_0>0.$$ Then, for any $p \geq 3/2$, there exists a constant $K_p >0$ such that $$\label{kp} m_p(t) \leq K_p \E(t)^p \qquad \forall t \geq t_0.$$ In particular, the generalized Haff’s law holds.
Let $t_0 > 0$ be fixed. First observe that using classical interpolation, it suffices to prove the result for any $p \geq 3/2$ such that $2p\in \mathbb{N}$. Argue by induction assuming that for any integer $j$ such that $2j\in\mathbb{N}$, and $1 \leq j \leq p-1/2$ there exists $K_j >0$ such that $m_j(t) \leq K_j \E(t)^j$ for $t\geq t_0$.\
Recall that, according to Proposition \[povzner\] $$\frac{\d }{\d t}m_p(t) \leq-(1-\kappa_{p})m_{p+1/2}(t)+\kappa_{p}\;S_{p}(t),$$ where $$S_{p}(t)=\sum^{[\frac{p+1}{2}]}_{k=1}\left(
\begin{array}{c}
p\\k
\end{array}
\right)\left(m_{k+1/2}(t)\;m_{p-k}(t)+m_{k}(t)\;m_{p-k+1/2}(t)\right).$$ The crucial point is that, for $p \geq 2$, the above expression $S_p(t)$ involves moments of order less than $p-1/2$ except for $p=3/2$ which explains its peculiar role. The induction hypothesis implies therefore that there exists a constant $C_p>0$ such that $$S_p(t) \leq C_p\, \E(t)^{p+1/2} \qquad \forall t \geq t_0,$$ where $C_p$ can be taken as $$C_p=\sum^{[\frac{p+1}{2}]}_{k=1}\left(
\begin{array}{c}
p\\k
\end{array}
\right)\left(K_{k+1/2}\;K_{p-k}+K_{k}\;K_{p-k+1/2}\right).$$ Furthermore, according to Jensen’s inequality $m_{p+1/2}(t)\geq m_{p}^{1+1/2p}(t)$, therefore we obtain $$\label{difmp}
\dfrac{\d }{\d t}m_p(t) \leq -(1-\kappa_{p})m_{p}^{1+1/2p}(t)+ \kappa_p\;C_p \,\E(t)^{p+1/2} \qquad \forall t \geq t_0.$$ Additionally, according to and since $e(r) \leq 1$ for any $r \geq 0$, one has clearly $\mathbf{\Psi}_e(|u|^2) \leq \frac{|u|^3}{8}$ for any $u \in \R^3.$ Thus, using , $$\begin{aligned}
\label{dEmm32}
-\dfrac{\d}{\d t}\E(t) &\leq \dfrac{1}{8} \IRR |u|^3 f(t,v)f(t,\vb)\d v\d\vb \nonumber\\
&\leq \IR f(t,v)|v|^3\d v=m_{3/2}(t) \qquad \forall t \geq t_0.\end{aligned}$$ Let $K>$ be conveniently chosen later and define $U_{p}(t):=m_{p}(t)- K\E(t)^{p}$. Then, combining and , $$\begin{split}
\dfrac{\d}{\d t}U_p(t)&=\dfrac{\d}{\d t}m_p(t)-pK\E(t)^{p-1}\dfrac{\d}{\d t}\E(t)\\
&\leq -(1-\kappa_{p})m_{p}^{1+1/2p}(t)+ \kappa_p\;C_p \,\E(t)^{p+1/2} +pKm_{3/2}(t)\E(t)^{p-1} \qquad \forall t \geq t_0.
\end{split}$$ Therefore, using we obtain $$\dfrac{\d}{\d t}U_p(t) \leq -(1-\kappa_{p})m_{p}^{1+1/2p}(t)+ \kappa_p\;C_p \,\E(t)^{p+1/2} +pK\E(t)^{p+1/2} \qquad \forall t\geq t_0.$$ This is enough to prove for $K=K_p$ large enough. Indeed, pick $K$ so that $m_{p}(t_0)<K \E(t_0)^{p}$. Then, by time-continuity in the moments, the estimate follows at least for some finite subsequent time. Assume that there exists a time $t_{\star} > t_0$ such that $m_{p}(t_{\star})=K \E(t_{\star})^{p}$, then the above inequality implies $$\dfrac{\d U_{p}}{\d t}(t_\star)\leq \left(-(1-\kappa_{p})K^{1+1/2p}+\kappa_p\,C_p + pK\right)\E(t_\star)^{p+1/2} <0$$ whenever $K$ is large enough. This proves that holds for any $p \geq 3/2$.
For *non-constant restitution coefficient*, an interesting result holds: in order to prove the lower bound , it is enough to prove that the cooling of $\E(t)$ is *at most algebraic* with arbitrary rate. More precisely, we have [@AloLo1 Theorem 3.7] the following proposition which follows from Proposition \[propstrat\].
\[lambda\] Assume that the restitution coefficient $e(\cdot)$ satisfy Assumptions \[HYP1\] and with $\gamma >0$. If there exist $C_0 >0$ and $\lambda >0$ such that $$\label{lam}
\E(t) \geq C_0\,(1+t)^{-\lambda} \qquad \forall t >0,$$ then there exists $C >0$ such that $\E(t) \geq C\,(1+t)^{-\tfrac{2}{1+\gamma}}$ for any $t \geq 0$, i.e. the generalized Haff’s law holds true.
Entropy-based proof of Haff’s law
=================================
The aim of this section is to prove Theorem \[main0\]. We begin computing the entropy production associated to the Boltzmann equation for granular gases .
Entropy production functional
-----------------------------
For any nonnegative $f$, one can use the weak form with the test function $\psi(v)=\log f(v)$ to compute the production of entropy $$\mathcal{S}_e(f):=\IR \Q_e(f,f)\log f \d v.$$ More precisely, $$\begin{split}
\mathcal{S}_{e}(f)& =\dfrac{1}{2\pi} \int_{\R^6 \times \S} |u \cdot \n|f(v)f(\vb)\log \left(\frac{f(v')f(\vb')}{f(v)f(\vb)}\right)\d v \d\vb \d\n\\
&=\dfrac{1}{2\pi} \int_{\R^6 \times \S} |u \cdot \n|f(v)f(\vb)\left(\log \left(\frac{f(v')f(\vb')}{f(v)f(\vb)}\right)-\frac{f(v')f(\vb')}{f(v)f(\vb)}+1\right)\d v \d\vb \d\n\\
&\phantom{++++}+\dfrac{1}{2\pi} \int_{\R^6 \times \S} |u \cdot \n|\left(f(v')f(\vb')-{f(v)f(\vb)}\right)\d v\d\vb\d\n.
\end{split}$$ Define, $$\label{Def}
\mathcal{D}_{e}(f):=-\dfrac{1}{2\pi} \int_{\R^6 \times \S} |u \cdot \n|f(v)f(\vb)\left(\log \left(\frac{f(v')f(\vb')}{f(v)f(\vb)}\right)-\frac{f(v')f(\vb')}{f(v)f(\vb)}+1\right)\d v \d\vb \d\n$$ which is a non-negative quantity since $\log x \leq x-1$ for any $x >0.$ Notice that, if $e=1$, then $\mathcal{D}_{e}$ is the classical entropy production functional and $\mathcal{S}_{e}(f)=-\mathcal{D}_e(f) \leq 0$, which means that the entropy production $\mathcal{S}_e$ is non-positive.\
For inelastic collisions, the entropy production functional is more intricate, $$\mathcal{S}_{e}(f)=-\mathcal{D}_e(f)+ \dfrac{1}{2\pi} \int_{\R^6 \times \S} |u \cdot \n|\left(f(v')f(\vb')-{f(v)f(\vb)}\right)\d v\d\vb\d\n$$ which means that the entropy production splits into a dissipative part $(-\mathcal{D}_e)$ and a non-negative part. Let us compute more precisely this last term. Since $\vartheta_e(\cdot)$ is strictly increasing, it is bijective. Moreover, $|u'\cdot \n|=\vartheta_e(|u\cdot \n|))$, thus, one can write $|u\cdot\n|=\vartheta_e^{-1}(|u'\cdot\n|)$. Then, using the change of variables $(v',\vb') \to (v,\vb)$ we obtain $$\begin{split}
\int_{\R^6 \times \S} |u \cdot \n| f(v')f(\vb')\d v\d\vb\d\n&=\int_{\R^6 \times \S}\vartheta_e^{-1}(|u'\cdot\n|) f(v')f(\vb')\d v\d\vb\d\n\\
&=\int_{\R^6 \times \S}\vartheta_e^{-1}(|u \cdot\n|) f(v )f(\vb )\dfrac{\d v\d\vb\d\n}{J_e(\vartheta_e^{-1}(|u\cdot \n|))}.
\end{split}$$ We used that $$\d v'\d\vb'=J_e(|u\cdot\n|)\d v\d\vb=J_e(\vartheta_e^{-1}(|u'\cdot\n|)\d v\d\vb.$$ It is easy to see that $\vartheta_e^{-1}(|u\cdot\n|)=\frac{|u\cdot\n|}{e(\vartheta_e^{-1}(|u\cdot\n|)}$. Then, we deduce that $$\begin{gathered}
\dfrac{1}{2\pi}\int_{\R^6 \times \S} |u \cdot \n|\left(f(v')f(\vb')-{f(v)f(\vb)}\right)\d v\d\vb\d\n\\
=\dfrac{1}{2\pi}\int_{\R^6 \times \S} |u \cdot \n|{f(v)f(\vb)}\left(\dfrac{1}{e (\vartheta_e^{-1}(|u\cdot\n|)) J_e(\vartheta_e^{-1}(|u\cdot\n|))}-1\right)\d v\d\vb\d\n\\
=\IRR |u|f(v)f(\vb)\mathbf{\Phi}_e(|u|)\d v\d\vb.\end{gathered}$$ For any fixed $v,\vb$, we have defined $$\mathbf{\Phi}_e(|u|):=\dfrac{1}{2\pi}\int_{\S} |\widehat{u} \cdot \n|\left(\dfrac{1}{e (\vartheta_e^{-1}(|u\cdot\n|)) J_e(\vartheta_e^{-1}(|u\cdot\n|))}-1\right) \d\n.$$ After some minor computations, $$\mathbf{\Phi}_e(|u|)=\frac{2}{|u|^2}\int_0^{|u|} \left(\dfrac{1}{e (\vartheta_e^{-1}(z))\, J_e(\vartheta_e^{-1}(z))}-1\right)z\d z.$$ Setting then $r=\vartheta_e^{-1}(z)$ and recalling that $J_e(y)=\vartheta'_e(y)$, we easily get that $$\mathbf{\Phi}_e(|u|)=\dfrac{2}{|u|^2}\int_0^{\vartheta_e^{-1}(|u|)} \left(r-\vartheta_e(r)\,\vartheta_e'(r)\right)\d r$$ where we also used that $\vartheta_e^{-1}(0)=0.$ We just proved the following proposition.
Assume that the restitution coefficient $e(\cdot)$ satisfies Assumption \[HYP1\], items (1) and (2). Then, for any non-negative distribution function $f(v)$ $$\label{dissip}
\mathcal{S}_{e}(f)=\IR \Q_e(f,f)\log f \d v=-\mathcal{D}_e(f) + \IRR |u|f(v)f(\vb)\mathbf{\Phi}_e(|u|)\d v\d\vb$$ where $\mathcal{D}_e(f) \geq 0$ is given by while $\mathbf{\Phi}_e(\cdot)$ is defined by $$\label{Phie}
\mathbf{\Phi}_e(\varrho)=\dfrac{2}{\varrho^2}\int_0^{\vartheta_e^{-1}(\varrho)} \left(r-\vartheta_e(r)\,\vartheta_e'(r)\right)\d r, \qquad \forall \varrho >0.$$
Additional qualitative properties of $\mathbf{\Phi}_e$ are given in the following lemma.
\[lemPhie\] Assume that the restitution coefficient $e(\cdot)$ satisfies Assumption \[HYP1\]. Assume moreover that is satisfied for some $\alpha >0$ and $\gamma > 0$ and that there exist two positive constants $C >0$ and $m \geq 1$ such that $$\label{large}\vartheta_e^{-1}(y) \leq C y^m \qquad \text{ for large } y.$$ Then, $\mathbf{\Phi}_e(|u|) \leq C |u|^{2(m-1)}$ for large $|u|$, and $\mathbf{\Phi}_e(|u|) \simeq 2\alpha |u|^\gamma$ for small $|u| \simeq 0$.
Since $\vartheta_e(\cdot)$ is assumed to be increasing, one clearly has $r-\vartheta_e(r)\vartheta_e'(r) \leq r$ for any $r \geq 0$. Therefore, $$\label{boundPhie}\mathbf{\Phi}_e(|u|) \leq \left(\dfrac{\vartheta^{-1}_e(|u|)}{|u|}\right)^2 \qquad \forall u \in \R^3$$ and the first part of the Lemma follows from . Moreover, if $e(r) \simeq 1-\alpha r^\gamma$ for $r \simeq 0$ and $\gamma >0$, then $r-\vartheta_e(r)\,\vartheta_e'(r) \simeq \alpha(2+\gamma)r^{\gamma+1}$ for $r \simeq 0$. Since $\vartheta_e^{-1}(r) \simeq r$ for small $r$, one gets easily the second part of the result.
The case of a constant restitution coefficient is included in the previous lemma, however, in this case $\mathbf{\Phi}_e$ is explicit, we refer to [@MiMo; @GaPaVi] for previous uses of the entropy production functional in the constant case.
\[**Constant restitution coefficient**\]\[constante\] If $e(z)=\e \in (0,1]$ for any $z \geq 0$, then $J_e=\e$ and $$\mathbf{\Phi}_e(\varrho)=\dfrac{2(1-\e^2)}{\varrho^2}\int_0^{\varrho/\e}r\d r= \dfrac{1-\e^2}{\e^2}.$$
If $e(\cdot)$ is the restitution coefficient for visco-elastic hard-spheres, there exists $a >0$ such that $$e(r)+ar^{\tfrac{1}{5}}\,e^{\tfrac{3}{5}}(r)=1 \qquad \forall r >0.$$ One checks without difficulty that $e(r) \simeq a^{-\tfrac{5}{3}}r^{-\tfrac{1}{3}}$ as $r \to \infty.$ In particular, $$\vartheta_e(r) \simeq a^{-\tfrac{5}{3}}r^{\tfrac{2}{3}} \qquad \text{ as } r \to \infty.$$ Consequently, there exists some positive constant $C_a >0$ such that $\vartheta_e^{-1}(y) \leq C_0 y^{\tfrac{3}{2}}$ for large $y >0.$ Therefore, the assumption of the above Lemma is fulfilled with $m=3/2$, and one obtains that there exists some positive constant $C >0$ such that $$\mathbf{\Phi}_e(|u|) \leq C |u| \qquad \text{ for large } \: u \in \R^3.$$ Since is known to hold with $\gamma=1/5$ and $\alpha=a$, we also have $$\mathbf{\Phi}_e(|u|) \simeq 2a|u|^{1/5} \qquad \text{ as } \qquad |u| \simeq 0.$$
Evolution of the entropy and the temperature: Haff’s law
--------------------------------------------------------
In all this section, we shall assume the following additional conditions on the restitution coefficient.
\[HYP2\] Assume that the restitution coefficient fulfills Assumptions \[HYP1\]. Moreover, assume that and holds, that is,
1. There exist $\alpha >0$ and $\gamma \geq 0$ such that $1-e(r) \simeq \alpha r^\gamma$ as $r\simeq 0$.
2. There exist $m \geq 1+\gamma/2$ and $C >0$ such that $\vartheta_e^{-1}(y) \leq Cy^m$ for large $y.$
Note that the assumption $m \geq 1+\gamma/2$ is no restrictive since the condition $(2)$ concerns large values of $y$.
Under this conditions, the growth of the entropy of the solution to is at most logarithmic.
\[logarithm\] Assume that the restitution coefficient $e(\cdot)$ satisfies Assumptions \[HYP2\]. In addition, assume that the initial distribution $f_0$ satisfies together with $\H(f_0) < \infty$ and let $f(t,v)$ be the solution to . Then, there exists a constant $C_0 >0$ such that the entropy $\H(f(t))$ of $f(t,v)$ satisfies $$\H(f(t)) \leq \H(f_0) + C_0 \log (1+t) \qquad \forall t \geq 0.$$
From the results of previous section, the evolution of the entropy $\H(f(t))$ is governed by $$\label{dHH}
\dfrac{\d}{\d t}\H(f(t))= -\mathcal{D}_e(f(t)) + \IRR |u|f(t,v)f(t,\vb)\mathbf{\Phi}_e(|u|)\d v\d\vb \qquad \forall t \geq 0.$$ Under Assumption \[HYP2\], Lemma \[lemPhie\] implies that we have $\mathbf{\Phi}_e(|u|) \leq C |u|^{2(m-1)}$ for large $|u|$ while $\mathbf{\Phi}_e(|u|) \simeq 2\alpha |u|^\gamma$ for small $ |u| \simeq 0$. In particular, there are two positive constants $A$ and $B$ such that $$\mathbf{\Phi}_e(|u|) \leq A|u|^\gamma+B|u|^{2(m-1)} \qquad \forall u \in \R^3.$$ From and since $-\mathcal{D}_e(f(t)) \geq 0$, $$\dfrac{\d}{\d t}\H(f(t)) \leq A\IRR |u|^{\gamma+1} f(t,v)f(t,\vb)\d v\d\vb + B\IRR |u|^{2m-1} f(t,v)f(t,\vb)\d v\d\vb.$$ Moreover, since $|u|^{\gamma+1} \leq 2^\gamma \left(|v|^{\gamma+1}+|\vb|^{\gamma+1}\right)$ one has $$\IRR |u|^{\gamma+1} f(t,v)f(t,\vb)\d v\d\vb \leq 2^{\gamma+1}\IRR |v|^{\gamma+1} f(t,v)\d v=2^{\gamma+1}m_{\frac{\gamma+1}{2}}(t).$$ Moreover, setting $p=m-1/2$, one notices that there is a constant $c_p$ depending only on $p$ such that $$\IRR |u|^{2m-1} f(t,v)f(t,\vb)\d v\d\vb \leq m_p(t)$$ where the $m_p(t)$ terms are the $p^{\text{th}}$ order moments defined in . Using Proposition \[prop:cool\] together with Proposition \[moments\], one concludes that there exist two positive constants $C_1$ and $C_2$ such that $$\dfrac{\d}{\d t}\H(t) \leq C_1 (1+t)^{-1} + C_2 \left(1+t\right)^{-\frac{2p}{1+\gamma}} \qquad \forall t > 0.$$ Since $p=m-1/2$ with $m \geq 1+\gamma/2$ one has $\tfrac{2p}{1+\gamma} \geq 1$ and, setting $C_0=C_1+C_2$, we get $$\dfrac{\d}{\d t}\H(t) \leq C_0 (1+t)^{-1} \qquad \forall t > 0$$ which yields the conclusion.
The above logarithmic growth is exactly what we need to prove Haff’s law. Indeed, the following general result allows to control from below the temperature using the entropy. The proof of the following proposition is given in the Appendix.
\[propcontrol\] Let $\mathcal{C}$ denote the class of nonnegative velocity distributions $f=f(v)$ with unit mass, finite energy and finite entropy $$\int_{\R^3} f(v)\d v=1 \qquad \qquad \int_{\R^3} |v|^2 f(v)\d v < \infty, \qquad \H(f)=\int_{\R^3} f(v)\log f(v)\d v < \infty.$$ Define $$\bH(f)=\IR f(v)|\log f(v)|\d v, \qquad f \in \mathcal{C}.$$ Then, there is some constant $c >0$ such that $$\bH(f) \leq \H(f) + c \left(\IR f(v)\,|v|^2\d v\right)^{5/3},$$ and some other constant $C >0$ such that $$\label{control}\IR f(v)|v|^2\d v \geq C \exp\left(-\tfrac{4}{3} \bH(f)\right) \qquad \forall f \in \mathcal{C}.$$
Proposition \[propcontrol\] combined with the logarithmic growth of $\H(f(t))$ prove the Haff’s law for non-constant restitution coefficient.
\[entrHaff\] Assume that the restitution coefficient $e(\cdot)$ satisfies Assumptions \[HYP2\] with $\gamma >0$. Let the initial distribution $f_0$ satisfies and $\H(f_0) < \infty$, and let $f(t,v)$ be the unique solution to . Then, there is a constant $c >0$ such that $$\E(t) \geq c\left(1+t\right
)^{-\frac{2}{1+\gamma}} \qquad \forall t \geq 0.$$ In particular, the generalized Haff’s law holds true.
With the notations of the above Proposition \[propcontrol\], there is some constant $C >0$ independent of time such that $$\bH(f(t)) \leq \H(f(t)) + C\E(t)^{5/3} \qquad \forall t \geq 0.$$ Since $\E(t)$ is bounded, one can find constants $k_0, K_0 >0$ such that $$\label{K0}
\bH(f(t)) \leq k_0+ K_0 \log (1+t) \qquad \forall t \geq 0.$$ Proposition \[propcontrol\] also implies that there exists a constant $c_1 >0$ such that $$\E(t) \geq c_1 \exp\left(-\frac{4}{3}\bH(f(t))\right) \geq c_2 \exp\left(-\frac{4K_0}{3}\log (1+t)\right) \qquad \forall t \geq 0,$$ with $c_2=c_1 \exp(-\frac{4k_0}{3}).$ Therefore, setting $\lambda_0=\frac{2K_0}{3}>0$ we get that $$\label{lambda0} \E(t) \geq c_2 (1+t)^{-2\lambda_0} \qquad \forall t \geq 0.$$ Therefore, according to Proposition \[lambda\], the estimate is enough to prove the second part of Haff’s law .
For constant restitution coefficient, our result is less precise, however, we prove an *integrated version of Haff’s law* in the next theorem.
\[entrHaff1\] Let $\e \in (0,1)$ be a constant restitution coefficient and let the initial distribution $f_0$ satisfies with $\H(f_0) < \infty$ , and let $f(t,v)$ be the unique solution to . Then, there are two positive constants $a,b >0$ such that $$\label{intEt}
\int_0^t \sqrt{\E(s)}\d s \geq \dfrac{1}{b}\log\left(1+ab\, t\right) \qquad \forall t \geq 0.$$ Consequently, $$\sup\left\{\lambda \geq 0\,,\,\sup_{t \geq 0}(1+t)^{2\lambda}\E(t) < \infty\right\}= \inf\left\{\lambda >0\,;\,\limsup_{t \to \infty} (1+t)^{2\lambda}\,\E(t) >0\right\}=1.$$
Recall that, for constant restitution coefficient $\e \in (0,1)$ the evolution of the entropy is given by $$\dfrac{\d}{\d t}\H(f(t))=- \mathcal{D}_e(f(t)) + \dfrac{1-\e^2}{\e^2} \IRR |u|f(t,v)f(t,\vb) \d v\d\vb \qquad \forall t \geq 0$$ where we used Eq. and Example \[constante\]. Arguing as in the previous proof, we see that $$\dfrac{\d}{\d t}\H(f(t)) \leq 2\dfrac{1-\e^2}{\e^2}m_{1/2}(t) \leq 2\dfrac{1-\e^2}{\e^2}\sqrt{\E(t)} \qquad \forall t \geq 0.$$ Integrating this inequality yields $$\H(f(t)) \leq \H(f_0) + 2\dfrac{1-\e^2}{\e^2}\int_0^t \sqrt{\E(s)}\d s \qquad \forall t \geq 0.$$ Consequently, there exists a positive constant $K_1 >0$ such that $$\bH(f(t)) \leq K_1+ 2\dfrac{1-\e^2}{\e^2} \int_0^t \sqrt{\E(s)}\d s \qquad \forall t \geq 0.$$ Then, from Proposition \[propcontrol\], $$\E(t) \geq C \exp\left(-\frac{4K_1}{3} -\frac{8(1-\e^2)}{3\e^2}\int_0^t \sqrt{\E(s)}\d s \right) \qquad \forall t \geq 0,$$ i.e. there are two positive constants $a=\sqrt{C}\exp(-\tfrac{2K_1}{3})$ and $b=\tfrac{4(1-\e^2)}{3\e^2}$ such that $$\sqrt{\E(t)} \geq a\exp\left(-b\int_0^t \sqrt{\E(s)}\d s\right) \qquad \forall t \geq 0$$ which yields . Now, setting $$\mathcal{A}=\{\lambda \geq 0\,;\, \sup_{t \geq 0} (1+t)^{2\lambda} \E(t) < \infty\}$$ we know from Proposition \[prop:cool\] that $1 \in \mathcal{A}$, i.e. $\mathcal{A} \neq \varnothing$. Then, it follows from that $\sup \mathcal{A}=1$. Now, let us define $$\mathcal{B}=\{\lambda \geq 0\,;\,\limsup_{t \to \infty} (1+t)^{2\lambda}\E(t) >0\}.$$ Notice that inequality holds for constant restitution coefficient. In particular, it proves that $\mathcal{B}\neq \varnothing$. Notice also that if $\lambda_1 \in \mathcal{B}$, then any $\lambda_2 \geq \lambda_1$ belongs to $\mathcal{B}$. We argue by contradiction to prove that $\inf \mathcal{B} =1$. Otherwise, from the previous observation, one would have $\inf \mathcal{B} =\lambda_{\mathcal{B}} > 1$. Pick $\lambda\in(1,\lambda_{\mathcal{B}})$, it follows that $\lambda \notin \mathcal{B}$, that is, $\limsup_{t \to \infty} (1+t)^{2\lambda}\E(t)=0$ and, in particular, $\lambda \in \mathcal{A}$. This is impossible since $\sup \mathcal{A} =1$, and thus, $\inf \mathcal{B}=1$.
Though less precise that the converse inequality , the above integrated version of Haff’s law asserts that $(1+t)^{-2}$ is the only possible [ **algebraic rate**]{} for the cooling of the temperature $\E(t)$. Notice that $\mathbf{v}_T(t)=\sqrt{\E(t)}$ is proportional to the so-called thermal velocity [@BrPo] and we may wonder what is the physical relevance of the above identity . Finally, we recall that it is expected the existence of a self-similar profile $\Phi_H(\cdot)$, an ***homogeneous cooling state***, such that $f(t,v)= \mathbf{v}_T(t)^{-3}{\Phi}_H\big(\frac{v}{\mathbf{v}_T(t)}\big)$ is a solution to . The existence of homogeneous cooling state has been proven in [@MiMo] (with a slightly different definition where $\mathbf{v}_T(t)$ was replaced by $(1+t)^{-1}$) and where the self-similar profile $\Phi_H$ is, by construction, satisfying $\Phi_H \in L^p$ for some $p >1$. We conjecture that the existence of such an homogeneous cooling state can be obtained using only entropy estimates, that is $\H(\Phi_H) < \infty$.
Notice that, since $\E(t) \leq C(1+t)^{-\tfrac{2}{1+\gamma}}$, it follows from that there is some constant $K >0$ such that $$\bH(f(t)) \leq K\log(1+t) \qquad \forall t \geq 0$$ which means that the logarithmic growth obtained in Proposition \[logarithm\] is optimal.
Haff’s law in the weakly inelastic regime
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This section is devoted to the proof of Theorems \[mainc\] and \[main\]. Let us now explain briefly the strategy of proof to get a precise version of generalized Haff’s law. Note that due to Proposition \[prop:cool\], one only has to prove a lower bound of the type $$\E(t) \geq c(1+t)^{-\frac{2}{1+\gamma}}$$ for some positive constant $c >0$ independent of time. The following approach uses only the evolution of some moments of the solution $f(t,v)$ with the particular use of Proposition \[propstrat\]. We will distinguish between the case of a constant restitution coefficient $\gamma=0$ and the non-constant case $\gamma>0$ since the two results are different.
The case of a constant restitution coefficient {#constant}
----------------------------------------------
We assume here that the restitution coefficient $e(\cdot)$ is constant: $e(r)=\e$ for for any $r \in \R_+$. In this case, $$\mathbf{\Psi}_e(r)=\dfrac{1-\e^2}{8}r^{3/2} \qquad \forall r >0$$ and the evolution of the temperature is given by $$\dfrac{\d}{\d t}\E(t)=-\dfrac{1-\e^2}{8}\int_{\R^3 \times \R^3} f(t,v)f(t,\vb)|v-\vb|^3\d v\d\vb, \qquad t \geq 0.$$ Since $|v-\vb|^3 \leq (|v|+|\vb|)^3=|v|^3+|\vb|^3+3|v|^2|\vb|+3|v||\vb|^2,$ one deduces that $$\label{Em32}-\dfrac{\d }{\d t}\E(t)\leq \frac{1-\e^2}{4}m_{3/2}(t)+ 3\frac{1-\e^2}{4}\,\E(t)m_{1/2}(t) \qquad \forall t \geq 0.$$ This observation will help us in proving the following theorem.
\[momentse\] Let $f_0$ be a nonnegative velocity distribution satisfying and let $f(t,v)$ be the associated solution to . Assume that the constant restitution coefficient $\e$ is such that $$\label{small}\frac{3(1-\e^2)}{8} < 1-\kappa_{3/2}.$$ Then, for any $t_0 >0$, there is an explicit positive constant $C_0 >0$ such that $$\label{k32}
m_{3/2}(t) \leq C_0\, \E(t)^{3/2} \qquad \forall t \geq t_0.$$ Consequently, the second part of Haff’s law holds $$\E(t) \geq \dfrac{\E(0)}{\left(1+C_0\,\sqrt{\E(0)}(1-\e^2)t\right)^2} \qquad \forall t \geq 0.$$
Let $t_0 >0$ be fixed. According to Proposition \[povzner\] $$\dfrac{\d }{\d t}m_{3/2}(t)\leq-(1-\kappa_{3/2})m_{2}(t)+\kappa_{3/2}\;S_{3/2}(t).$$ From the expression of $S_{3/2}(t)$ one gets $$\label{m3/2}
\dfrac{\d }{\d t}m_{3/2}(t) \leq -(1-\kappa_{3/2})m_{2}(t)+m_{3/2}(t)m_{1/2}(t)+\E^{2}(t) \qquad \forall t \geq t_0$$ where we used the fact that $\kappa_{3/2} < 1.$ Let $K $ be a positive number to be chosen later and define $$U_{3/2}(t):=m_{3/2}(t)- K\E(t)^{3/2}.$$ From Holder’s inequality, $$\label{m3/2E}
m_{3/2}(t)\leq \sqrt{\E(t)} \sqrt{m_{2}(t)} \quad \text{ and } \quad m_{1/2}(t)\leq \sqrt{\E(t)} \qquad \forall t \geq 0,$$ so that $$\dfrac{\d U_{3/2}}{\d t}(t) \leq -(1-\kappa_{3/2}) \dfrac{m_{3/2}^2(t)}{\E(t)}
+\sqrt{\E(t)} m_{3/2}(t)+ \E^2 (t)-\frac{3}{2}K\frac{\d \E(t)}{\d t}\E(t)^{1/2}.$$ Now, using , one gets $$\begin{gathered}
\label{p3/2}\dfrac{\d U_{3/2}}{\d t}(t)\leq -(1-\kappa_{3/2}) \dfrac{m_{3/2}^2(t)}{\E(t)}
+\sqrt{\E(t)} m_{3/2}(t)+ \E^2 (t)\\
+ \frac{3(1-\e^2)}{8}Km_{3/2}(t)\E(t)^{1/2}+\frac{9(1-\e^2)}{8}K\E(t)^{2}. \qquad \forall t >t_0.\end{gathered}$$ This last inequality, together with the smallness assumption imply the result provided $K$ is large enough. Indeed, choose $K$ so that $m_{3/2}(t_0)<K \E^{3/2}(t_0)$. Then, by time-continuity of the moments, the result follows at least for some finite time. Assume that there exists a time $t_{\star}>t_0$ such that $m_{3/2}(t_{\star})=K \E^{3/2}(t_{\star})$, then implies $$\dfrac{\d U_{3/2}}{\d t}(t_\star)\leq \left(-aK^2+1+\left(1+\frac{9(1-\e^2)}{8}\right)K\right)\E^2(t_\star)$$ where $a:=(1-\kappa_{3/2})-\frac{3}{8}(1-\e^2) >0$ by . Choosing $K=C_0$ large enough such that $\frac{\d U_{3/2}}{\d t}(t_\star)\leq 0$ for any $t_{\star}$ the conclusion holds.
We just proved that, under the sole assumption , the solution $f(t,v)$ to the Boltzmann equation satisfies the generalized Haff’s law $$\label{Haff's}
c (1+t)^{-2} \leq \E(t) \leq C (1+t)^{-2}, \qquad t \geq 0$$ where $c,C$ are positive constants depending only on $\E(0)$.
Using the explicit expression of $\kappa_{3/2}$ given by Proposition \[povzner\], one sees that amounts to $\e \geq \sqrt{\tfrac{8\kappa_{3/2}}{3}-\tfrac{5}{3}} \simeq 0.809.$
Non-constant case $\gamma>0$ {#sec:haff}
----------------------------
We consider in this section the non-constant case $\gamma>0$ with restitution coefficient $e(\cdot)$ satisfying Assumption \[HYP1\] with $$\ell_\gamma(e)=\sup_r \dfrac{1-e(r)}{r^\gamma} < \infty \qquad \text{ for some } \gamma > 0.$$ The proof is more involved but still based on Proposition \[propstrat\]. We therefore need only to estimate $m_{3/2}(t)$. We will work with the following class of initial datum: Let $\E_0$, $\varrho_0$ be two fixed positive constants. Define $\mathscr{F}(\E_0, \varrho_0)$ as the set of nonnegative distributions $g \in L^1_{3}$ such that $$\IR g(v)\d v =1, \qquad \quad \IR g(v) v \d v =0, \qquad \IR |v|^2 g(v)\d v=\E_0$$ and $$\IR g(v)|v|^3 \d v \leq \varrho_0 \left(\IR g(v)|v|^2 \d v\right)^{3/2}.$$ With this definitions we have the following result.
\[mainnon\] Let $f_0$ be a nonnegative velocity distribution satisfying with initial energy $\E_0$. Then, there exists $\ell_0:=\ell_0(\E_0,\gamma) >0$ such that, if the restitution coefficient satisfies $\ell_\gamma(e) < \ell_0$, then solution $f(t,v)$ to fulfills the generalized Haff’s law .
Throughout the proof, we simply denote $\ell_\gamma(e)=\lambda$ since both $\gamma$ and $e(\cdot)$ are fixed. Fix a time $t_0>0$, due to appearance of moments, there exists a $e$-independent constant $C:=C(t_0,\E_0,\gamma)>0$ such that $\sup_{t\geq t_0}m_{2\gamma}(t)\leq C$. Note that $f(t_0,v)\in\mathscr{F}(\E_{t_0}, \varrho_{t_0})$ where $$\frac{m_{3/2}(t_0)}{m_{1}(t_0)^{3/2}}<\varrho_{t_0}<\infty.$$ Additionally, recall that $$\label{m3/2bis}
\dfrac{\d }{\d t}m_{3/2}(t) \leq -2\alpha m_{2}(t)+m_{3/2}(t)m_{1/2}(t)+\E^{2}(t) \qquad \forall t \geq t_0,$$ where we have set $2\alpha=(1-\kappa_{3/2}) >0$. Now, using $$\begin{aligned}
-\E(t)^{1/2}\frac{\d}{\d t}\E(t)&\leq \lambda\, c_\gamma \E(t)^{1/2}m_{(3+\gamma)/2}(t)\\&\leq \lambda\, c_\gamma \left(\frac{1}{4}\E(t)^{2}+\frac{3}{4}m^{4/3}_{(3+\gamma)/2}(t)\right)\nonumber\\
&\leq \lambda\, c_\gamma\left(\frac{1}{4}\E(t)^{2}+\frac{3}{4}m^{1/3}_{2\gamma}(t)m_2(t)\right)\quad \forall t\geq t_0.\end{aligned}$$ For $t\geq t_0$ the quantity $m_{2\gamma}(t)^{1/3}$ is controlled uniformly by $A:=C^{1/3}$, therefore, $$\label{lbe2}
-\E(t)^{1/2}\frac{\d}{\d t}\E(t) \leq \lambda\, \frac{c_\gamma}{4}\left(\E(t)^{2} + 3A m_2(t)\right) \qquad \forall t \geq t_0.$$ Set $U_{3/2}(t):=m_{3/2}(t)-K\E(t)^{3/2}$ where the constant $K >0$ will be suitable determined later on. Use estimates and to get for all $t\geq t_0$ $$\begin{split}
\dfrac{\d }{\d t}U_{3/2}(t) &\leq -2\alpha m_{2}(t)+m_{3/2}(t)m_{1/2}(t)+\E^{2}(t) + \frac{3c_\gamma}{8}\lambda\,K \E(t)^{2}\\
&\hspace{7cm} + \frac{9c_\gamma}{8}\lambda\,K A m_2(t)\\
&\leq -\alpha m_{2}(t)+m_{3/2}(t)m_{1/2}(t)+ \left(1+\lambda\,K \frac{3c_\gamma}{8}\right)\E(t)^2\\
&\hspace{6cm} +\left(-\alpha + \lambda\,K \frac{9c_\gamma}{8}A \right)m_2(t).
\end{split}$$ For the first term in the left-hand side, one uses the same estimate as in Theorem \[moments\] $$-\alpha m_{2}(t)+m_{3/2}(t)m_{1/2}(t) \leq -\alpha \E(t)^{-1}m_{3/2}^2(t) + m_{3/2}(t)\E(t)^{1/2},$$ thus, $$\begin{gathered}
\label{tt}
\dfrac{\d }{\d t}U_{3/2}(t) \leq -\alpha \E(t)^{-1}m_{3/2}^2(t) + m_{3/2}(t)\E(t)^{1/2} + \left(1+ 3K c_\gamma\right)\E(t)^2\\
+\left(-\alpha + \lambda\,K \frac{9c_\gamma}{8}A \right)m_2(t) \quad\forall t\geq t_0.\end{gathered}$$ Without loss of generality, we have assumed $\lambda < 8$ (any other positive number would have worked). Now, let $K_0 >0$ be the positive root of $-\alpha X^2+(1+3c_\gamma)X+1=0$. Then, if $K =\max(K_0, \varrho_{t_0})$, one gets that $$\lambda < \ell_0 \implies m_{3/2}(t) \leq K\E^{3/2}(t)\quad \forall t\geq t_0$$ where $\ell_0=\frac{8\alpha}{9Ac_\gamma\,K}.$ Indeed, since $U_{3/2}(t_0) < 0$, by a continuity argument, then $U_{3/2}(t)$ remains nonpositive at least for some finite time. Assume that there exists a time $t_{\star}>t_0$ such that $U_{3/2}(t_\star)=0$, that is, $m_{3/2}(t_{\star})=K \E^{3/2}(t_{\star})$. Then, from $$\dfrac{\d}{\d t}U_{3/2}(t_\star) \leq \left(-\alpha K^2 + (1+3c_\gamma)\,K+1\right)\E(t_\star)^2
+\left(-\alpha + \lambda\,K \frac{9c_\gamma}{8}A \right)m_2(t_\star).$$ Since $K \geq K_0$, the first term on the right-hand side is negative while, by choice of $\lambda < \ell_0:=\ell_0(t_0,\E_0,\gamma)$, the last term is negative as well. In other words, $\dfrac{\d}{\d t}U_{3/2}(t_\star) < 0$ from which we deduce that $U_{3/2}(t)$ will remain non-positive for all $t\geq t_0$. As explained in the paragraph precedent to Proposition \[propstrat\], this yields Theorem \[mainnon\].
It is important to notice that the smallness condition is depending on the initial datum $f_0$. This is the major difference with respect to the constant case where the smallness assumption is universal.
Appendix: Functional inequalities relating moments and entropy {#appendix-functional-inequalities-relating-moments-and-entropy .unnumbered}
==============================================================
In this section, we present some functional inequalities that relate moments and entropy. We present the results in ${{\mathbb{R}^n}}$ with $n \geq 1$ regardless we only use them in dimension $n=3$. In this section, for any measurable subset $E \subset {{\mathbb{R}^n}}$, $|E|$ will stand for the Lebesgue measure of $E$. Let $f=f(v)$ be a nonnegative distribution and denote $$\H(f)=\int_{{{\mathbb{R}^n}}} f(v)\log f(v)\d v, \qquad \mathbf{H}(f)=\int_{{{\mathbb{R}^n}}} f(v)|\log f(v)|\d v,$$ while, for any $k \geq 0$ we set $$M_k(f) =\int_{{{\mathbb{R}^n}}} f( v)|v|^k\d v.$$ First recall the following simple estimate which can be traced back to [@diperna]:
If there is $k \in (0,n)$ such that $M_k(f) < \infty$ and $\H(f) < \infty$ then $\bH(f) < \infty.$ More precisely, for any $k \in (0,n)$ there exists some positive constant $c_{n,k} >0$ such that $$\label{estimHfk}0 \leq \mathbf{H}(f) \leq \H(f) +c_{n,k} \,M_k(f)^{\tfrac{n}{n+k}}.$$
Although it is almost explicitly stated in [@diperna], we provide the complete proof of the above estimate. Define $A=\{v \in {{\mathbb{R}^n}}\,,\,f(v) < 1\}$, thus $$\begin{split}
\mathbf{H}(f)&=\int_{{{\mathbb{R}^n}}\setminus A} f(v)\log f( v)\d v - \int_{A} f( v)\log f( v)\d v=\H(f) -2\int_{A} f(v)\log f(v)\d v\\
&=\H(f) + 2 \int_{A} f(v)\log \left(\frac{1}{f(v)}\right)\d v.
\end{split}$$ For any $a >0$, let $B =\{v \in {{\mathbb{R}^n}}\,;\,f( v) \geq \exp(-a|v|^k)\}$ and $B^c={{\mathbb{R}^n}}\setminus B.$ If $v \in A \cap B$ then $\log(\tfrac{1}{f(v)}) \leq a|v|^k$ and $$\begin{split}
\mathbf{H}(f)&\leq \H(f) + 2a\int_{A \cap B} f(v)|v|^k\d v + 2 \int_{B^c \cap A} f(v)\log \left(\frac{1}{f(v)}\right)\d v\\
&\leq \H(f) + 2aM_k(f)+ 2 \int_{B^c} f( v)\log \left(\frac{1}{f( v)}\right)\d v.\end{split}$$ Since $x\log(1/x) \leq M \sqrt{x}$ for any $x \in (0,1)$, where $M= 2\exp(-1)$, we get $$\int_{B^c} f(v)\log \left(\frac{1}{f(v)}\right)\d v \leq M\int_{B^c} \sqrt{f(v)}\d v \leq M \int_{{{\mathbb{R}^n}}} \exp\left(-a\frac{|v|^k}{2}\right)\d v.$$ Set $$J_{n,k}(a)=\int_{{{\mathbb{R}^n}}} \exp\left(-\frac{a}{2}|v|^k\right)\d v=\dfrac{|\mathbb{S}^{n-1}|}{k}\left(\dfrac{2}{a}\right)^{\tfrac{n}{k}}\Gamma\left(\frac{n}{k}\right)$$ where $\Gamma(\cdot)$ is the Gamma function. We have $\mathbf{H}(f) \leq \H(f) + 2aM_k(f) + 2J_{n,k}(a)$ for any $a >0.$ Therefore, for any $k \in (0,n)$, there exists some positive constant $C_{n,k} >0$ such that $$\mathbf{H}(f) \leq \H(f)+2aM_k + C_{n,k} a^{-\tfrac{n}{k}} \qquad \forall a >0.$$ Optimizing the parameter $a >0$ yields for some explicit constant $c_{n,k}.$
We now give a general estimate which allow to control $M_k(f)$ from below in terms of $\H(f)$. It is likely that such an estimate is well-known by specialists. We include a proof below.
\[entropy-moment\] Let $f \geq 0$ be such that $\bH(f) < \infty$ with $\int_{{{\mathbb{R}^n}}} f(v)\d v=1.$ Then, for any $k \geq 0$ and any $\varepsilon >0$, there exists $C(n,k,\varepsilon) >0$ independent of $f$ such that $$\label{mkfbhf}M_k(f) \geq C(n,k,\varepsilon) \exp\left(-\frac{k}{n(1-\varepsilon)}\mathbf{H}(f)\right).$$
For any $R >0$, let $B_R$ denote the ball with center in the origin and radius $R$, and let $B_R^c$ be its complement. Then, $$\begin{gathered}
\label{BR}
M_k(f)=\int_{B_R} f(v)|v|^k\d v + \int_{B_R^c}f(v)|v|^k\d v \\
\geq R^k \int_{B^c_R} f(v)\d v=R^k\left(1-\int_{B_R}f(v) \d v\right).\end{gathered}$$ Therefore, in order to control $M_k(f)$ from below it is enough to control the mass of $f$ on a suitable ball. Let us fix $R >0$ and recall the generalized Young’s inequality $$xy \leq x\log x -x+\exp(y) \qquad \forall x >0, \qquad \forall y \in \mathbb{R}$$ or, equivalently, $$\label{young}xy \leq \frac{x}{\lambda}\log \frac{x}{\lambda} -\frac{x}{\lambda}+\exp(\lambda\,y) \qquad \forall x >0, \lambda >0, \qquad \forall y \in \mathbb{R}.$$ Using this inequality with $x=f(v)$ and $y=1$ and integrating the inequality over $B_R$ we obtain $$\int_{B_R} f(v)\d v \leq \dfrac{1}{\lambda}\int_{B_R} f(v)\log f(v)\d v -\dfrac{1}{\lambda} \left(\log \lambda + 1\right)\int_{B_R} f(v)\d v + \exp(\lambda) |B_R| \qquad \forall \lambda >0.$$ Consequently, since $\int_{B_R} f(v)\log f(v)\d v \leq \bH(f)$, we get $$\dfrac{1}{\lambda}\left(\log \lambda + \lambda+ 1\right)\int_{B_R} f(v)\d v \leq \dfrac{1}{\lambda} \mathbf{H}(f) + \exp(\lambda) |B_R| \qquad \forall \lambda >0,$$ that is, $$\left(\log (\lambda \exp(\lambda))+1\right)\int_{B_R} f(v)\d v \leq \mathbf{H}(f) + \lambda \exp(\lambda) |B_R|\qquad \forall \lambda >0.$$ Set $x=\log(\lambda \exp(\lambda))$ and assume $x >-1$. The last inequality becomes $$\int_{B_R} f(v)\d v \leq \dfrac{\mathbf{H}(f)+\exp(x)|B_R|}{1+x} \qquad \forall x > -1.$$ We optimize the parameter $x >-1$ by noticing that the right-hand side reaches its minimal value for $x_0$ such that $$x_0 \exp(x_0)|B_R|=\mathbf{H}(f).$$ The mapping $x \mapsto x\exp(x)$ is strictly increasing over $(-1,\infty)$ and we define $W$ its inverse (Lambert function). We get then $x_0=W \left(\mathbf{H}(f)/|B_R|\right)$ and $$\int_{B_R} f(v)\d v \leq \dfrac{\mathbf{ H}(f)+\exp(x_0)|B_R|}{1+x_0}=\exp(x_0)|B_R|=\exp\left(W \left(\frac{\mathbf{H}(f)}{|B_R|}\right)\right)|B_R|.$$ Combining this with we get $$M_k \geq R^k \left(1-\exp\left(W \left(\frac{\mathbf{H}(f)}{|B_R|}\right)\right)|B_R|\right) \qquad \forall R >0$$ and we still have to optimize the parameter $R$. For simplicity, set $Y= \mathbf{H}(f)$ and $X=|B_R|$. For any $\varepsilon \in (0,1)$, $$1-\exp(W(Y/X))X=\varepsilon \Longleftrightarrow W(Y/X)=\log\left(\frac{1-\varepsilon}{X}\right)\Longleftrightarrow Y/X=\frac{1-\varepsilon}{X}\log\left(\frac{1-\varepsilon}{X}\right),$$ where we used that $W^{-1}(t)=t\exp(t)$. Thus, $$1-\exp(W(Y/X))X=\varepsilon \Longleftrightarrow \frac{Y}{1-\varepsilon}=\log\left(\frac{1-\varepsilon}{X}\right) \Longleftrightarrow X=(1-\varepsilon)\exp\left(-\frac{Y}{1-\varepsilon}\right).$$ To summarize, for any $\varepsilon >0$, if $R_0(\varepsilon) >0$ is such that $|B_{R_0}|=(1-\varepsilon)\exp\left(-\frac{\mathbf{ H}(f)}{1-\varepsilon}\right)$, then $$M_k \geq \varepsilon R_0^k.$$ Since $|B_{R_0}|=\tfrac{|\mathbb{S}^{n-1}|}{n} R_0^n$ we get our conclusion with $C(n,k,\varepsilon)=\varepsilon\left(\frac{n(1-\varepsilon)}{|\mathbb{S}^{n-1}|}\right)^\frac{k}{n}.$
In dimension $n=3$ with $k=2$ and $\varepsilon=1/2$, one sees that there is some constant $C >0$ such that $$\label{estimateH}\int_{\mathbb{R}^3} f(v)|v|^2 \d v \geq \frac{1}{2}\left(\frac{3}{8\pi}\right)^{2/3}\exp(-4\mathbf{H}(f)/3)$$ for any nonnegative distribution function $f(v) >0$ with $\IR f(v)\d v=1.$
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{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'A shape-function independent relation is derived between the partial $\bar B\to X_u\,l^-\bar\nu$ decay rate with a cut on $P_+=E_X-|\vec P_X|\le\Delta$ and a weighted integral over the normalized $\bar B\to X_s\gamma$ photon-energy spectrum. The leading-power contribution to the weight function is calculated at next-to-next-to-leading order in renormalization-group improved perturbation theory, including exact two-loop matching corrections at the scale $\mu_i\sim\sqrt{m_b\Lambda_{\rm QCD}}$. The overall normalization of the weight function is obtained up to yet unknown corrections of order $\alpha_s^2(m_b)$. Power corrections from phase-space factors are included exactly, while the remaining subleading contributions are included at first order in $\Lambda_{\rm QCD}/m_b$. At this level unavoidable hadronic uncertainties enter, which are estimated in a conservative way. The combined theoretical accuracy in the extraction of $|V_{ub}|$ is at the level of 5% if a value of $\Delta$ near the charm threshold can be achieved experimentally.'
---
CLNS 05/1922\
MIT-CTP 3668\
[hep-ph/0508178]{}\
August 16, 2005
**A two-loop relation between inclusive radiative and semileptonic $B$-decay spectra**
[Björn O. Lange$^a$, Matthias Neubert$^{b,c}$, and Gil Paz$^b$]{}\
Introduction
============
In the recent past, much progress has been made in the theoretical understanding of inclusive charmless $B$ decays near the kinematic endpoint of small $P_+=E_X-|\vec P_X|$, where $E_X$ and $\vec P_X$ are the energy and momentum of the final-state hadronic system in the $B$-meson rest frame. In $\bar B\to X_s\gamma$ decays the $P_+$ variable is related to the $B$-meson mass and the photon energy, $P_+=M_B-2E_\gamma$, and the measurement of its spectrum leads directly to the extraction of the leading hadronic structure function, called the shape function [@Neubert:1993ch; @Neubert:1993um; @Bigi:1993ex]. The $P_+$ spectrum in semileptonic $\bar B\to X_u\,l^-\bar\nu$ decays, on the other hand, enables us to determine $|V_{ub}|$ [@Mannel:1999gs; @Aglietti:2002md; @Bosch:2004bt], but this requires a precise knowledge of the shape function. One approach for measuring $|V_{ub}|$ is to first extract the shape function from the $\bar B\to X_s\gamma$ photon spectrum, and then to use this information for predictions of event distributions in $\bar B\to X_u\,l^-\bar\nu$. A comprehensive description of this program has been presented in [@Lange:2005yw]. Equivalently, it is possible to eliminate the shape function in $\bar B\to X_u\,l^-\bar\nu$ decay rates in favor of the $\bar B\to X_s\gamma$ photon-energy spectrum. This idea was first put forward in [@Neubert:1993um] and later refined in [@Leibovich:1999xf; @Leibovich:2000ey; @Neubert:2001sk; @Hoang:2005pj]. Partial $\bar B\to X_u\,l^-\bar\nu$ decay rates are then given as weighted integrals over the $\bar B\to X_s\gamma$ photon-energy spectrum, $$\label{eq:relation}
\Gamma_u(\Delta)
= \underbrace{\int_0^\Delta\!dP_+\,
\frac{d\Gamma_u}{dP_+}}_{\hbox{\footnotesize exp.\ input}}
= |V_{ub}|^2 \int_0^\Delta\!dP_+
\underbrace{\phantom{\frac{1}{\Gamma_s(E_*)}}\hspace{-12mm}
W(\Delta,P_+)}_{\hbox{\footnotesize theory}}\,
\underbrace{\frac{1}{\Gamma_s(E_*)}\,
\frac{d\Gamma_s}{dP_+}}_{\hbox{\footnotesize exp.\ input}} \,,$$ where the weight function $W(\Delta,P_+)$ is perturbatively calculable at leading power in $\Lambda_{\rm QCD}/m_b$. A comparison of both sides of the equation determines the CKM matrix element $|V_{ub}|$ directly. For the measurement of the left-hand side to be free of charm background, $\Delta$ must be less than $M_D^2/M_B\approx 0.66$GeV. However, the $P_+$ spectrum in $\bar B\to X_u\,l^-\bar\nu$ decays displays many of the features of the charged-lepton energy spectrum, so that it is not inconceivable that the cut can be further relaxed for the same reasons that experimenters are able to relax the lepton cut beyond the charm threshold. We stress that for an application of relation (\[eq:relation\]) a measurement of the $\bar B\to X_s\gamma$ photon spectrum is needed only for $E_\gamma\ge\frac12(M_B-\Delta)\approx 2.3$GeV (or slightly lower, if the cut is relaxed into the charm region). This high-energy part of the spectrum has already been measured with good precision.
Previous authors [@Neubert:1993um; @Leibovich:1999xf; @Leibovich:2000ey; @Neubert:2001sk; @Hoang:2005pj] have considered relations such as (\[eq:relation\]) in the slightly different form $$\label{oldrelation}
\underbrace{\int_0^\Delta\!dP_+\,
\frac{d\Gamma_u}{dP_+}}_{\hbox{\footnotesize exp.\ input}}
= \frac{|V_{ub}|^2}{|V_{tb} V_{ts}^*|^2} \int_0^\Delta\!dP_+
\underbrace{\phantom{\frac{1}{\Gamma_s(E_*)}}\hspace{-12mm}
\widetilde W(\Delta,P_+)}_{\hbox{\footnotesize theory}}\!
\underbrace{\frac{d\Gamma_s}{dP_+}}_{\hbox{\footnotesize exp.\ input}}
\hspace{-0.3cm} .$$ Normalizing the photon spectrum by the total[^1] rate $\Gamma_s(E_*)$ as done in (\[eq:relation\]) has several advantages. Firstly, it is a known fact that event fractions in $\bar B\to X_s\gamma$ decay can be calculated with better accuracy than partial decay rates (see [@Neubert:2004dd] for a recent discussion), and likewise the normalized rate does not suffer from the relatively large experimental error on the total branching ratio. Secondly, relation (\[eq:relation\]) is independent of the CKM factor $|V_{tb} V_{ts}^*|$. Thirdly, unlike the total $\bar B\to X_s\gamma$ decay rate, the shape of the photon spectrum is rather insensitive to possible New Physics contributions [@Kagan:1998ym], which could distort the outcome of a $|V_{ub}|$ measurement via relation (\[oldrelation\]). Lastly, as we will see below, the weight function $W(\Delta,P_+)$ possesses a much better perturbative expansion than the function $\widetilde W(\Delta,P_+)=|V_{tb} V_{ts}^*|^2\,W(\Delta,P_+)/\Gamma_s(E_*)$. This last point can be traced back to the fact that most of the very large contribution from the $O_1 - O_{7\gamma}$ operator mixing in the effective weak Hamiltonian cancels in the theoretical expression for the normalized photon spectrum.
In principle, any partial $\bar B\to X_u\,l^-\bar\nu$ decay rate can be brought into the form (\[eq:relation\]), with complicated weight functions. The relation between the two $P_+$ spectra is particularly simple, because the leading-power weight function is a constant at tree level. Experiments typically reject semileptonic $B$-decay events with very low lepton energy. The effect of such an additional cut can be determined from [@Lange:2005yw]. Alternatively, it is possible to modify the weight function so as to account for a lepton cut, however at the expense of a significant increase in complexity. We will not pursue this option in the present work.
The weight function depends on the kinematical variable $P_+$ and on the size $\Delta$ of the integration domain. It possesses integrable singularities of the form $\alpha_s^n(\mu) \ln^k[m_b(\Delta-P_+)/\mu^2]$, with $k\le n$, in perturbation theory. Different strategies can be found in the literature concerning these logarithms. Leibovich et al. resummed them by identifying $\mu^2$ with $m_b(\Delta-P_+)$ [@Leibovich:1999xf; @Leibovich:2000ey]. The $P_+$ dependence of the weight function then enters via the running coupling $\alpha_s(\sqrt{m_b(\Delta-P_+)})$. This is a legitimate choice of scale as long as $(\Delta-P_+)$ has a generic value of order $\Lambda_{\rm QCD}$; however, it is [*not*]{} a valid choice in the small region where $m_b(\Delta-P_+)\sim\Lambda_{\rm QCD}^2$. A key result underlying relation (\[eq:relation\]) is that, by construction, the weight function is insensitive to soft physics. Quark-hadron duality ensures that the region near the point $P_+=\Delta$, which is without any physical significance, does not require special consideration after integration over $P_+$. In the approach of [@Leibovich:1999xf; @Leibovich:2000ey], the attempt to resum the above logarithms near the endpoint of the $P_+$ integral leads to integrals over unphysical Landau singularities of the running coupling in the nonperturbative domain. Hoang et al. chose to calculate the weight function in fixed-order perturbation theory at the scale $\mu=m_b$ [@Hoang:2005pj]. This leads to parametrically large logarithms, since $\Delta-P_+\ll m_b$. In the present paper we separate physics effects from two parametrically distinct scales, a hard scale $\mu_h\sim m_b$ and an intermediate scale $\mu_i\sim \sqrt{m_b\Lambda_{\rm QCD}}$, so that we neither encounter Landau singularities nor introduce parametrically large logarithms. The shape of the weight function is then governed by a perturbative expansion at the intermediate scale, $\alpha_s^n(\mu_i)
\ln^k[m_b(\Delta-P_+)/\mu_i^2]$. As will be explained later, the coefficients in this series, as well as the overall normalization, possess themselves an expansion in $\alpha_s(\mu_h)$.
The calculation of the weight function starts with the theoretical expressions for the $P_+$ spectra in $\bar B\to X_u\,l^-\bar\nu$ and $\bar B\to X_s\gamma$ decays, which are given as [@Lange:2005yw] $$\begin{aligned}
\frac{d\Gamma_u}{dP_+}
&=& \frac{G_F^2 |V_{ub}|^2}{96\pi^3}\,U(\mu_h,\mu_i)\,(M_B-P_+)^5
\label{eq:gammaU} \\
&&\times \int_0^1\!dy\,y^{2-2a_\Gamma(\mu_h,\mu_i)}
\Big\{ (3-2y)\,\F_1(P_+,y) + 6(1-y)\,\F_2(P_+,y) + y\,\F_3(P_+,y) \Big\}
\,, \nonumber\\
\frac{d\Gamma_s}{dP_+}
&=& \frac{\alpha G_F^2 |V_{tb}V^*_{ts}|^2}{32\pi^4}\,U(\mu_h,\mu_i)\,
(M_B-P_+)^3\,\overline{m}_b^2(\mu_h)\,[C_{7\gamma}^{\rm eff}(\mu_h)]^2\,
\F_\gamma(P_+) \,, \label{eq:gammaS}\end{aligned}$$ where $$\label{eq:y}
y = \frac{P_- -P_+}{M_B-P_+}$$ with $P_-=E_X+|\vec{P}_X|$ is a kinematical variable that is integrated over the available phase space. Expressions for the structure functions $\F_i$ valid at next-to-leading order (NLO) in renormalization-group (RG) improved perturbation theory and including first- and second-order power corrections can be found in [@Lange:2005yw] (see also [@Neubert:2004dd; @Bauer:2003pi; @Bosch:2004th]). Symbolically, they are written as $H(\mu_h)\cdot J(\mu_i)\otimes S(\mu_i)$, where $H(\mu_h)$ contains matching corrections at the hard scale $\mu_h$. The jet function $J(\mu_i)$, which is a perturbative quantity at the intermediate scale $\mu_i$, is convoluted with a non-perturbative shape function renormalized at that same scale. Separation of the two scales $\mu_h$ and $\mu_i$ allows for the logarithms in matching corrections to be small, while logarithms of the form $\ln\mu_h/\mu_i$, which appear at every order in perturbation theory, are resummed in a systematic fashion and give rise to the RG evolution functions $U(\mu_h,\mu_i)$ and $a_\Gamma(\mu_h,\mu_i)$ [@Bosch:2004th].
The leading-power jet function $J(p^2,\mu_i)$ entering the expressions for $\F_i$ is universal and has been computed at one-loop order in [@Bauer:2003pi; @Bosch:2004th]. More recently, the two-loop expression for $J$ has been obtained apart from a single unknown constant [@Neubert:2005nt], which is the two-loop coefficient of the local $\delta(p^2)$ term. This constant does not enter in the two-loop result for the weight function $W(\Delta,P_+)$ in (\[eq:relation\]). Due to the universality of the leading-power jet function, it is possible to calculate the complete $O(\alpha_s^2(\mu_i))$ corrections to the weight function. However, the extraction of hard corrections at two-loop order would require multi-loop calculations for both decay processes, which are unavailable at present. As a result, we will be able to predict the $\Delta$ and $P_+$ dependence of the weight function $W(\Delta,P_+)$ at next-to-next-to-leading order (NNLO) in RG-improved perturbation theory, including exact two-loop matching contributions and three-loop running effects. However, the overall normalization of the weight function will have an uncertainty of $O(\alpha_s^2(\mu_h))$ from yet unknown hard matching corrections.
The total rate $\Gamma_s(E_*)$ has been calculated in a local operator product expansion and reads (including only the leading non-perturbative corrections) [@Neubert:2004dd] $$\label{eq:totalBtoS}
\Gamma_s(E_*) = \frac{\alpha G_F^2 |V_{tb}V^*_{ts}|^2}{32\pi^4}\,
m_b^3\,\overline{m}_b^2(\mu_h)\,[C_{7\gamma}^{\rm eff}(\mu_h)]^2\,
|H_s(\mu_h)|^2\,H_\Gamma(\mu_h) \left[ 1 - \frac{\lambda_2}{9m_c^2}\,
\frac{C_1(\mu_h)}{C_{7\gamma}^{\rm eff}(\mu_h)} \right] ,$$ where $H_s$ is the hard function of $\F_\gamma$, and $H_\Gamma$ contains the remaining radiative corrections. We will present an explicit expression for this quantity at the end of Section \[sec:results\] below. The hadronic correction proportional to $\lambda_2/m_c^2$ cancels against an identical term in $\F_\gamma$. Apart from two powers of the running $b$-quark mass defined in the $\overline{\rm MS}$ scheme, which is part of the electromagnetic dipole operator $O_{7\gamma}$ in the effective weak Hamiltonian, three more powers of $m_b$ emerge from phase-space integrations. To avoid the renormalon ambiguities of the pole scheme we use a low-scale subtracted quark-mass definition for $m_b$. Specifically, we adopt the shape-function mass $m_b^{\rm SF}(\mu_*,\mu_*)$ [@Bosch:2004th; @Neubert:2004sp] defined at a subtraction scale $\mu_*=1.5$GeV, which relates to the pole mass as $$\label{eq:SFmassToPole}
m_b^{\rm pole} = m_b^{\rm SF}(\mu_*,\mu_*)
+ \mu_*\,\frac{C_F \alpha_s(\mu_h)}{\pi} + \ldots \,.$$ Throughout this paper we will use $m_b^{\rm SF}(\mu_*,\mu_*)$ as the $b$-quark mass and refer to it as $m_b$ for brevity. The present value of this parameter is $m_b=(4.61\pm 0.06)$GeV [@Neubert:2005nt].
Calculation of the weight function
==================================
Leading power
-------------
The key strategy for the calculation of the weight function is to make use of QCD factorization theorems for the decay distributions on both sides of (\[eq:relation\]) and to arrange the resulting, factorized expressions such that they are both given as integrals over the shape function, $\int d\hat\omega\,\hat S(\hat\omega)\,g_i(\hat\omega)$, with different functions $g_i$ for the left-hand and right-hand sides. Relation (\[eq:relation\]) can then be enforced by matching $g_{\rm LHS}$ to $g_{\rm RHS}$. Following this procedure, we find for the integrated $P_+$ spectrum in $\bar B\to X_u\,l^-\bar\nu$ decay after a series of integration interchanges $$\begin{aligned}
\label{eq:uSwitch}
\int\limits_0^\Delta dP_+\,\frac{d\Gamma_u}{dP_+}
&\propto& \int\limits_0^1\!dy\,y^{-2a} H_u(y)
\int\limits_0^\Delta\!dP_+\,(M_B-P_+)^5
\int\limits_0^{P_+}\!d\hat\omega\,y m_b\,J(ym_b(P_+ -\hat\omega))\,
\hat S(\hat\omega) \nonumber \\
&=& \int\limits_0^\Delta\!d\hat\omega\,\hat S(\hat\omega)
\int\limits_0^{M_B-\hat\omega}\!dq\,5q^4
\int\limits_0^1\!dy\,y^{-2a} H_u(y)\,
j\left(\ln \frac{m_b(\Delta_q-\hat\omega)}{\mu_i^2} + \ln y \right) ,\end{aligned}$$ where $\Delta_q=\mbox{min}(\Delta,M_B-q)$, and [@Neubert:2005nt] $$\label{jdef}
j\left(\ln \frac{Q^2}{\mu_i^2},\mu_i \right)
= \int_0^{Q^2}\!dp^2\,J(p^2,\mu_i)$$ is the integral over the jet function. For the sake of transparency, we often suppress the explicit dependence on $\mu_i$ and $\mu_h$ when it is clear at which scales the relevant quantities are defined. The function $H_u$ is a linear combination of the hard functions entering the structures $\F_i$ in (\[eq:gammaU\]), which in the notation of [@Lange:2005yw] is given by $$H_u(y,\mu_h) = 2y^2(3-2y)\,H_{u1}(y,\mu_h) + 12y^2(1-y)\,H_{u2}(y,\mu_h)
+ 2y^3\,H_{u3}(y,\mu_h) \,.$$ RG resummation effects build up the factor $y^{-2a}$ in (\[eq:uSwitch\]), where $a\equiv a_\Gamma(\mu_h,\mu_i)$ is the value of the RG-evolution function $$\label{adef}
a_\Gamma(\mu_h,\mu_i)
= \int_{\mu_i}^{\mu_h}\!\frac{d\mu}{\mu}\,\Gamma_{\rm cusp}(\alpha_s(\mu))
= - \int_{\alpha_s(\mu_h)}^{\alpha_s(\mu_i)}\!d\alpha\,
\frac{\Gamma_{\rm cusp}(\alpha)}{\beta(\alpha)} \,,$$ which depends only on the cusp anomalous dimension [@Korchemsky:wg; @Korchemskaya:1992je]. The quantity $a$ has its origin in the geometry of time-like and light-like Wilson lines underlying the kinematics of inclusive $B$ decays into light particles. Our definition is such that $a$ is a positive number for $\mu_h>\mu_i$ and vanishes in the limit $\mu_h\to\mu_i$. We find it convenient to treat the function $a_\Gamma(\mu_h,\mu_i)$ as a running “physical” quantity, much like $\alpha_s(\mu)$ or $\overline{m}_b(\mu)$. Since the cusp anomalous dimension is known to three-loop order [@Moch:2004pa], the value of $a$ can be determined very accurately. Note that three-loop accuracy in $a$ (as well as in the running coupling $\alpha_s$) is required for a consistent calculation of the weight function at NNLO. The corresponding expression is $$\begin{aligned}
a = a_\Gamma(\mu_h,\mu_i) &=& \frac{\Gamma_0}{2\beta_0}\,\Bigg\{
\ln\frac{\alpha_s(\mu_i)}{\alpha_s(\mu_h)}
+ \left( \frac{\Gamma_1}{\Gamma_0} - \frac{\beta_1}{\beta_0} \right)
\frac{\alpha_s(\mu_i) - \alpha_s(\mu_h)}{4\pi} \nonumber\\
&&\mbox{}+ \left[ \frac{\Gamma_2}{\Gamma_0} - \frac{\beta_2}{\beta_0}
- \frac{\beta_1}{\beta_0}
\left( \frac{\Gamma_1}{\Gamma_0} - \frac{\beta_1}{\beta_0} \right) \right]
\frac{\alpha_s^2(\mu_i) - \alpha_s^2(\mu_h)}{32\pi^2} + \dots \Bigg\} \,,\end{aligned}$$ where the expansion coefficients $\Gamma_n$ and $\beta_n$ of the cusp anomalous dimension and $\beta$-function can be found, e.g., in [@Neubert:2004dd].
Instead of the jet function $J$ itself, we need its integral $j(\ln Q^2/\mu_i^2,\mu_i)$ in the second line of (\[eq:uSwitch\]). Since the jet function has a perturbative expansion in terms of “star distributions”, which are logarithmically sensitive to the upper limit of integration [@DeFazio:1999sv], it follows that $j(L,\mu_i)$ is a simple polynomial in $L$ at each order in perturbation theory. The two-loop result for this quantity has recently been computed by solving the integro-differential evolution equation for the jet function [@Neubert:2005nt]. An unknown integration constant of $O(\alpha_s^2 L^0)$ does not enter the expression for the weight function.
We now turn to the right-hand side of (\[eq:relation\]) and follow the same steps that lead to (\[eq:uSwitch\]). It is helpful to make an ansatz for the leading-power contribution to the weight function, $W^{(0)}(\Delta,P_+)$, where the dependence on $\Delta$ is solely given via an upper limit of integration. To this end, we define a function $f(k)$ through $$\label{eq:fdef}
W^{(0)}(\Delta,P_+)\propto \frac{1}{(M_B-P_+)^3}
\int_0^{\Delta-P_+}\!dk\,f(k)\,(M_B-P_+-k)^5 \,.$$ This allows us to express the weighted integral over the $\bar B\to X_s\gamma$ photon spectrum as $$\begin{aligned}
\label{eq:sSwitch}
\int\limits_0^\Delta\!dP_+\,\frac{d\Gamma_s}{dP_+}\,W^{(0)}(\Delta,P_+)
&\propto& \int\limits_0^\Delta\!dP_+\,(M_B-P_+)^3\,W^{(0)}(\Delta,P_+)
\int\limits_0^{P_+}\!d\hat\omega\,m_b\,J(m_b(P_+ -\hat\omega))\,
\hat S(\hat\omega) \nonumber \\
&\propto& \int\limits_0^\Delta\!d\hat\omega\,\hat S(\hat\omega)
\int\limits_0^{M_B-\hat\omega}\!dq\,5q^4
\int\limits_0^{\Delta_q-\hat\omega}\!dk\,f(k)\,
j\left(\ln\frac{m_b(\Delta_q-\hat\omega-k)}{\mu_i^2} \right) . \qquad\end{aligned}$$ Note that the jet function $J$ (and with it $j$) is the same in semileptonic and radiative decays. The difference is that the argument of the jet function in (\[eq:uSwitch\]) contains an extra factor of $y$, which is absent in (\[eq:sSwitch\]). Comparing these two relations leads us to the matching condition $$\label{eq:fmatch}
\int_0^1\!dy\, y^{-2a}\,H_u(y)\,
j\left(\ln\frac{m_b\Omega}{\mu_i^2}+\ln y \right)
\stackrel{!}{=} \int_0^\Omega\!dk\,f(k)\,
j\left(\ln\frac{m_b(\Omega-k)}{\mu_i^2} \right) ,$$ which holds to all orders in perturbation theory and allows for the calculation of $W^{(0)}(\Delta,P_+)$ via (\[eq:fdef\]). The main feature of this important relation is that the particular value of $\Omega$ is irrelevant for the determination of $f(k)$. It follows that, as was the case for the jet function $J$, the perturbative expansion of $f(k)$ in $\alpha_s(\mu_i)$ at the intermediate scale involves star distributions, and $W^{(0)}(\Delta,P_+)$ depends logarithmically on $(\Delta-P_+)$. At two-loop order it suffices to make the ansatz $$\begin{aligned}
\label{eq:fexpansion}
f(k) &\propto& \delta(k) + C_F\frac{\alpha_s(\mu_i)}{4\pi}
\left[ c_0^{(1)}\,\delta(k)
+ c_1^{(1)} \left( \frac{1}{k} \right)_*^{[\mu_i^2/m_b]} \right] \\
&&\mbox{}+ C_F\left( \frac{\alpha_s(\mu_i)}{4\pi} \right)^2
\left[ c_0^{(2)}\,\delta(k)
+ c_1^{(2)} \left( \frac{1}{k} \right)_*^{[\mu_i^2/m_b]}
+ 2c_2^{(2)} \left( \frac{1}{k}\ln\frac{m_b k}{\mu_i^2}
\right)_*^{[\mu_i^2/m_b]} \right] + \dots \,, \nonumber \end{aligned}$$ where the star distributions have the following effect when integrated with some smooth function $\phi(k)$ over an interval $\Omega$: $$\begin{aligned}
\label{stardistris}
\int_0^\Omega\!dk \left( \frac{1}{k} \right)_*^{[\mu_i^2/m_b]} \phi(k)
&=& \int_0^{\Omega}\!dk\,\frac{\phi(k)-\phi(0)}{k}
+ \phi(0)\,\ln\frac{m_b\Omega}{\mu_i^2} \,, \nonumber\\
\int_0^\Omega\!dk\,\left( \frac{1}{k}\ln\frac{m_b k}{\mu_i^2}
\right)_*^{[\mu_i^2/m_b]} \phi(k)
&=& \int_0^{\Omega}\!dk\,\frac{\phi(k)-\phi(0)}{k}\,
\ln\frac{m_b k}{\mu_i^2} \,
+ \frac{\phi(0)}{2}\,\ln^2\frac{m_b\Omega}{\mu_i^2} \,.\end{aligned}$$
A sensitivity to the hard scale $\mu_h$ enters into $f(k)$ via the appearance of $H_u(y,\mu_h)$ in (\[eq:fmatch\]). Because of the polynomial nature of $j(L,\mu_i)$, all we ever need are moments of the hard function with respect to $\ln y$. We thus define the master integrals $$\label{eq:Tn}
T_n(a,\mu_h)\equiv \int_0^1\!dy\,y^{-2a}\,H_u(y,\mu_h)\,\ln^n y \,; \qquad
h_n(a,\mu_h) = \frac{T_n(a,\mu_h)}{T_0(a,\mu_h)} \,,$$ which can be calculated order by order in $\alpha_s(\mu_h)$. Therefore, the coefficients $c_k^{(n)}$ of the perturbative expansion in (\[eq:fexpansion\]) at the intermediate scale have the (somewhat unusual) feature that they possess themselves an expansion in $\alpha_s(\mu_h)$. This is a consequence of the fact that, unlike the differential decay rates (\[eq:gammaU\]) and (\[eq:gammaS\]), the weight function itself does not obey a simple factorization formula, in which the hard correction can be factored out. Rather, as can be seen from (\[eq:fmatch\]), it is a convolution of the type $W=H(\mu_h)\otimes J(\mu_i)$. To one-loop accuracy, the hard function $H_u$ reads $$\begin{aligned}
\label{eq:Hu}
H_u(y,\mu_h)
&=& 2y^2(3-2y)\,\Bigg[ 1 + \frac{C_F\alpha_s(\mu_h)}{4\pi}\,
\bigg( -4\ln^2\frac{ym_b}{\mu_h} + 10\ln\frac{ym_b}{\mu_h} - 4\ln y
\nonumber\\
&&\hspace{2.3cm}\mbox{}- 4 L_2(1-y) - \frac{\pi^2}{6} - 12 \bigg) \Bigg]
- \frac{C_F\alpha_s(\mu_h)}{\pi}\,3y^2\ln y \,.\end{aligned}$$ Explicit expressions for the quantities $T_0$, $c_k^{(n)}$, and $h_n$ entering the distribution function $f(k)$ will be given below.
Subleading power {#sec:corr}
----------------
Power corrections to the weight function can be extracted from the corresponding contributions to the two $P_+$ spectra in (\[eq:gammaU\]) and (\[eq:gammaS\]). There exists a class of power corrections associated with the phase-space prefactors $(M_B-P_+)^n$ in these relations, whose effects are treated exactly in our approach, see e.g. (\[eq:fdef\]). This is important, because these phase-space corrections increase in magnitude as the kinematical range $\Delta$ over which the two spectra are integrated is enlarged. One wants to make $\Delta$ as large as experimentally possible so as to increase statistics and justify the assumption of quark-hadron duality, which underlies the theory of inclusive $B$ decays.
The remaining power corrections fall into two distinct classes: kinematical corrections that start at order $\alpha_s$ and come with the leading shape function [@Kagan:1998ym; @DeFazio:1999sv], and hadronic power corrections that start at tree level and involve new, subleading shape functions [@Bauer:2001mh; @Leibovich:2002ys; @Bauer:2002yu; @Neubert:2002yx; @Burrell:2003cf; @Lee:2004ja; @Bosch:2004cb; @Beneke:2004in]. Because different combinations of these hadronic functions enter in $\bar B\to X_u\,l^-\bar\nu$ and $\bar B\to X_s\gamma$ decays, it is impossible to eliminate their contributions in relations such as (\[eq:relation\]). As a result, at $O(\Lambda_{\rm QCD}/m_b)$ there are non-perturbative hadronic uncertainties in the calculation of the weight function $W(\Delta,P_+)$, which need to be estimated before a reliable extraction of $|V_{ub}|$ can be performed. For the case of the charged-lepton energy spectrum and the hadronic invariant mass spectrum, this aspect has been discussed previously in [@Bauer:2002yu; @Neubert:2002yx] and [@Burrell:2003cf], respectively.
Below, we will include power corrections to first order in $\Lambda_{\rm QCD}/m_b$. Schematically, the subleading corrections to the right-hand side of (\[eq:relation\]) are computed according to $\Gamma_u^{(1)}\sim W^{(0)}\otimes d\Gamma_s^{(1)}/dP_++W^{(1)}\otimes d\Gamma_s^{(0)}/dP_+$, where the superscripts indicate the order in $1/m_b$ power counting. The power corrections to the weight function, denoted by $W^{(1)}$, are derived from the mismatch in the power corrections to the two decay spectra. The kinematical power corrections to the two spectra are known at $O(\alpha_s)$, without scale separation. We assign a coupling $\alpha_s(\bar\mu)$ to these terms, where the scale $\bar\mu$ will be chosen of order the intermediate scale [@Lange:2005yw]. At first subleading power the leading shape function is convoluted with either a constant or a single logarithm of the form $\ln[(P_+ -\hat\omega)/(M_B-P_+)]$, and we have (with $n=0,1$) $$\begin{aligned}
\int_0^\Delta\!dP_+\,\frac{d\Gamma_u}{dP_+}
&\ni& \alpha_s(\bar\mu) \int_0^\Delta\!dP_+\,(M_B-P_+)^4
\int_0^{P_+}\!d\hat\omega\,\hat S(\hat\omega)\,
\ln^n\frac{P_+ -\hat\omega}{M_B-P_+} \nonumber \\
&=& \alpha_s(\bar\mu) \int_0^\Delta\!d\hat\omega\,\hat S(\hat\omega)
\int_0^{\Delta-\hat\omega}\!dk\,(M_B-\hat\omega-k)^4
\ln^n\frac{k}{M_B-\hat\omega-k} \,,\end{aligned}$$ and similarly for the photon spectrum. On the other hand, the weighted integral in (\[eq:relation\]) also contains terms where the photon spectrum is of leading power and the weight function of subleading power, $$\int_0^\Delta\!dP_+\,\frac{d\Gamma_s}{dP_+}\,W^{\rm kin(1)}(\Delta,P_+)
\ni \int_0^\Delta\!dP_+\,(M_B-P_+)^3\,\hat S(P_+)\,
W^{\rm kin(1)}(\Delta,P_+) \,.$$ Therefore the kinematical corrections to the weight function must have the form $$\label{eq:wkin1}
W^{\rm kin(1)}(\Delta,P_+)\propto \frac{\alpha_s(\bar\mu)}{(M_B-P_+)^3}
\int_0^{\Delta-P_+}\!dk\,(M_B-P_+ -k)^4
\left( A + B\,\ln\frac{k}{M_B-P_+ -k} \right) ,$$ and a straightforward calculation determines the coefficients $A$ and $B$. The hadronic power corrections to the weight function, $W^{\rm hadr(1)}$, can be expressed in terms of the subleading shape functions $\hat t(\hat\omega)$, $\hat u(\hat\omega)$, and $\hat v(\hat\omega)$ defined in [@Bosch:2004cb]. These terms are known at tree level only, and at this order their contribution to the weight function can be derived using the results of [@Lange:2005yw].
Results {#sec:results}
=======
Including the first-order power corrections and the exact phase-space factors, the weight function takes the form $$\begin{aligned}
\label{eq:masterform}
W(\Delta,P_+)
&=& \frac{G_F^2 m_b^3}{192\pi^3}\,T_0(a,\mu_h)\,H_\Gamma(\mu_h)\,
(M_B-P_+)^2 \nonumber\\
&\times& \Bigg\{ 1 + \frac{C_F\alpha_s(\mu_i)}{4\pi}
\left[ c_0^{(1)} + c_1^{(1)} \left( \ln \frac{m_b(\Delta-P_+)}{\mu_i^2}
- p_1(\delta) \right) \right] \nonumber\\
&&\hspace{0.55cm}\mbox{}+ C_F \left( \frac{\alpha_s(\mu_i)}{4\pi} \right)^2
\left[ c_0^{(2)} + c_1^{(2)} \left( \ln \frac{m_b(\Delta-P_+)}{\mu_i^2}
- p_1(\delta) \right) \right. \nonumber\\
&&\hspace{1.6cm}\left.
\mbox{}+ c_2^{(2)} \left( \ln^2\frac{m_b(\Delta-P_+)}{\mu_i^2}
- 2 p_1(\delta) \ln\frac{m_b(\Delta-P_+)}{\mu_i^2}
+ 2 p_2(\delta) \right) \right] \nonumber\\
&&\mbox{}+ \frac{\Delta-P_+}{M_B-P_+}\,\frac{C_F\alpha_s(\bar\mu)}{4\pi}
\left[ A(a,\mu_h)\,I_A(\delta) + B(a,\mu_h)\,I_B(\delta) \right] \\
&&\mbox{}+ \frac{1}{M_B-P_+}\,\frac{1}{2(1-a)(3-a)}\,
\bigg[ 4(1-a)(\bar\Lambda-P_+) + 2(4-3a)\,\frac{\hat t(P_+)}{\hat S(P_+)}
\nonumber\\
&&\mbox{}+ (4-a)\,\frac{\hat u(P_+)}{\hat S(P_+)}
+ (8-13a+4a^2)\,\frac{\hat v(P_+)}{\hat S(P_+)} \bigg]
+ \frac{m_s^2}{M_B-P_+}\,\frac{\hat S'(P_+)}{\hat S(P_+)}
+ \dots \Bigg\} \,, \nonumber\end{aligned}$$ where $\bar\Lambda=M_B-m_b$ is the familiar mass parameter of heavy-quark effective theory, and $\delta=(\Delta-P_+)/(M_B-P_+)$. The first line denotes an overall normalization, the next three lines contain the leading-power contributions, and the remaining expressions enter at subleading power. The different terms in this result will be discussed in the remainder of this section.
For the leading-power terms in the above result we have accomplished a complete separation of hard and intermediate (hard-collinear) contributions to the weight function in a way consistent with the factorization formula $W=H\otimes J$ mentioned in the previous section. The universality of the shape function, which encodes the soft physics in both $\bar B\to X_s\gamma$ and $\bar B\to X_u\,l^-\bar\nu$ decays, implies that the weight function is insensitive to physics below the intermediate scale $\mu_i\sim\sqrt{m_b\Lambda_{\rm QCD}}$. In particular, quark-hadron duality ensures that the small region in phase space where the argument $m_b(\Delta-P_+)$ of the logarithms scales as $\Lambda_{\rm QCD}^2$ or smaller does not need special consideration. At a technical level, this can be seen by noting that the jet function is the discontinuity of the collinear quark propagator in soft-collinear effective theory [@Bosch:2004th; @Bauer:2001yt], and so the $P_+$ integrals can be rewritten as a contour integral in the complex $p^2$ plane along a circle of radius $m_b\Delta\sim\mu_i^2$.
Leading power
-------------
The leading-power corrections in the curly brackets in (\[eq:masterform\]) are determined completely at NNLO in RG-improved perturbation theory, including three-loop running effects via the quantity $a$ in (\[adef\]), and two-loop matching corrections at the scale $\mu_i$ as indicated above. To this end we need expressions for the one-loop coefficients $c_n^{(1)}$ including terms of $O(\alpha_s(\mu_h))$, while the two-loop coefficients $c_n^{(2)}$ are needed at leading order only. We find $$\label{eq:1-loopCoeffs}
c_0^{(1)} = -3 h_1(a,\mu_h)
\left[ 1 - \frac{C_F\alpha_s(\mu_h)}{\pi}\,\frac{4\mu_*}{3m_b} \right]
+ 2 h_2(a,\mu_h) \,, \qquad
c_1^{(1)} = 4 h_1(a,\mu_h) \,,$$ and $$\begin{aligned}
\label{eq:2-loopCoeffs}
c_0^{(2)}
&=& C_F\,\Bigg[ \left( -\frac32 + 2\pi^2 - 24\zeta_3 \right) h_1(a,\mu_h)
+ \left( \frac92 - \frac{4\pi^2}{3} \right) h_2(a,\mu_h)
- 6 h_3(a,\mu_h) + 2 h_4(a,\mu_h) \Bigg] \nonumber\\
&&\mbox{}+ C_A \left[ \left( -\frac{73}{9} + 40\zeta_3 \right) h_1(a,\mu_h)
+ \left( \frac83 - \frac{2\pi^2}{3} \right) h_2(a,\mu_h) \right]
\nonumber\\
&&\mbox{}+ \beta_0 \left[
\left( -\frac{247}{18} + \frac{2\pi^2}{3} \right) h_1(a,\mu_h)
+ \frac{29}{6}\,h_2(a,\mu_h) - \frac23\,h_3(a,\mu_h) \right] , \nonumber\\
c_1^{(2)}
&=& C_F\,\Big[ -12 h_2(a,\mu_h) + 8 h_3(a,\mu_h) \Big]
+ C_A \left[ \left( \frac{16}{3} - \frac{4\pi^2}{3} \right) h_1(a,\mu_h)
\right] \nonumber\\
&&\mbox{}+ \beta_0 \left[ \frac{29}{3}\,h_1(a,\mu_h) - 2 h_2(a,\mu_h)
\right] , \nonumber \\
c_2^{(2)} &=& 8 C_F\,h_2(a,\mu_h) - 2\beta_0\,h_1(a,\mu_h) \,.\end{aligned}$$ As always $C_F=4/3$, $C_A=3$, and $\beta_0=11-2 n_f/3$ is the first coefficient of the QCD $\beta$-function. The term proportional to $\mu_*$ in the expression for $c_0^{(1)}$ arises because of the elimination of the pole mass in favor of the shape-function mass, see (\[eq:SFmassToPole\]). Since the logarithms $\ln[m_b(\Delta-P_+)/\mu_i^2]$ in (\[eq:masterform\]) contain $m_b$, all coefficients except $c_n^{(n)}$ receive such contributions. However, to two-loop order only $c_0^{(1)}$ is affected.
Next, the corresponding expressions for the hard matching coefficients $h_i$ are calculated from (\[eq:Tn\]). To the required order they read $$\begin{aligned}
h_1(a,\mu_h)
&=& - \frac{15-12a+2a^2}{2(2-a)(3-a)(3-2a)} \nonumber\\
&&\mbox{}+ \frac{C_F\alpha_s(\mu_h)}{4\pi}\,\Bigg[
- \frac{2(189-318a+192a^2-48a^3+4a^4)}{(2-a)^2(3-a)^2(3-2a)^2}\,
\ln\frac{m_b}{\mu_h} \nonumber\\
&&\quad\mbox{}+
\frac{2331-5844a+5849a^2-2919a^3+726a^4-72a^5}{(2-a)^3(3-a)^2(3-2a)^3}
- 4\psi^{(2)}(3-2a) \Bigg] + \dots \,, \nonumber
$$ $$\begin{aligned}
h_2(a,\mu_h)
&=& \frac{69-90a+36a^2-4a^3}{2(2-a)^2(3-a)(3-2a)^2} \nonumber\\
&&\mbox{}+ \frac{C_F\alpha_s(\mu_h)}{4\pi}\,\Bigg[
\frac{2(1692-3699a+3138a^2-1272a^3+240a^4-16a^5)}{(2-a)^3(3-a)^2(3-2a)^3}
\,\ln\frac{m_b}{\mu_h} \nonumber\\
&&\quad\mbox{}-
\frac{46521-140064a+175479a^2-117026a^3+43788a^4-8712a^5+720a^6}
{2(2-a)^4(3-a)^2(3-2a)^4} \nonumber\\
&&\quad\mbox{}+ \frac{4(15-12a+2a^2)}{(2-a)(3-a)(3-2a)}\,\psi^{(2)}(3-2a)
- 4\psi^{(3)}(3-2a) \Bigg] + \dots \,, \nonumber\\
h_3(a,\mu_h)
&=& - \frac{3(303-552a+360a^2-96a^3+8a^4)}{4(2-a)^3(3-a)(3-2a)^3} + \dots
\,, \nonumber\\
h_4(a,\mu_h)
&=& \frac{3(1293-3030a+2760a^2-1200a^3+240a^4-16a^5)}{2(2-a)^4(3-a)(3-2a)^4}
+ \dots \,,
\phantom{aaaaaaaaaaaaaaaa}\end{aligned}$$ where $\psi^{(n)}(x)$ is the $n$-th derivative of the polygamma function. Because of the exact treatment of the phase space there are corrections to the logarithms in (\[eq:masterform\]), which are finite-order polynomials in the small ratio $\delta=(\Delta-P_+)/(M_B-P_+)$. Explicitly, $$\begin{aligned}
p_1(\delta) &=& 5\delta - 5\delta^2 + \frac{10}{3}\,\delta^3
- \frac54\,\delta^4 + \frac15\,\delta^5 \,, \nonumber\\
p_2(\delta) &=& 5\delta - \frac52\,\delta^2 + \frac{10}{9}\,\delta^3
- \frac{5}{16}\,\delta^4 + \frac{1}{25}\,\delta^5 \,.\end{aligned}$$ This concludes the discussion of the leading-power expression for the weight function.
Subleading power {#sec:power}
----------------
The procedure for obtaining the kinematical power corrections to the weight function has been discussed in Section \[sec:corr\]. For the coefficients $A$ and $B$ in (\[eq:masterform\]) we find $$\begin{aligned}
A(a,\mu_h)
&=& \frac{-388+702a-429a^2+123a^3-34a^4+8a^5}{2(1-a)^2(2-a)(3-a)(3-2a)}
+ \left(\frac13- \frac49\,\ln\frac{m_b}{m_s} \right)
\frac{[C_{8g}^{\rm eff}(\mu_h)]^2}{[C^{\rm eff}_{7\gamma}(\mu_h)]^2}
\nonumber\\
&&- \frac{10}{3}\,
\frac{C_{8g}^{\rm eff}(\mu_h)}{C^{\rm eff}_{7\gamma}(\mu_h)}
+ \frac83 \left(
\frac{C_1(\mu_h)}{C^{\rm eff}_{7\gamma}(\mu_h)}
- \frac13\,\frac{C_1(\mu_h)\,C_{8g}^{\rm eff}(\mu_h)}
{[C^{\rm eff}_{7\gamma}(\mu_h)]^2}\,\right) g_1(z)
- \frac{16}{9}\,\frac{[C_1(\mu_h)]^2}{[C^{\rm eff}_{7\gamma}(\mu_h)]^2}\,
g_2(z) \,, \nonumber\\
B(a,\mu_h)
&=& - \frac{2(8+a)}{(1-a)(3-a)}
- \frac29\,\frac{[C_{8g}^{\rm eff}(\mu_h)]^2}
{[C^{\rm eff}_{7\gamma}(\mu_h)]^2} \,.\end{aligned}$$ Here $C_i(\mu_h)$ denote the (effective) Wilson coefficients of the relevant operators in the effective weak Hamiltonian, which are real functions in the Standard Model. The variable $z=(m_c/m_b)^2$ enters via charm-loop penguin contributions to the hard function of the $\bar B\to X_s\gamma$ photon spectrum [@Kagan:1998ym], and $$g_1(z) = \int_0^1\!dx\,x\,\mbox{Re} \left[\,
\frac{z}{x}\,G\!\left(\frac{x}{z}\right) + \frac12 \,\right] , \qquad
g_2(z) = \int_0^1\!dx\,(1-x) \left|\,\frac{z}{x}\,
G\!\left(\frac{x}{z}\right) + \frac12\,\right|^2 ,$$ with $$G(t) = \left\{ \begin{array}{ll}
-2\arctan^2\!\sqrt{t/(4-t)} & ;~ t<4 \,, \\[0.1cm]
2 \left( \ln\!\Big[(\sqrt{t}+\sqrt{t-4})/2\Big]
- \displaystyle\frac{i\pi}{2} \right)^2 & ;~ t\ge 4 \,.
\end{array} \right.$$ Furthermore we need the integrals over $k$ in (\[eq:wkin1\]), which encode the phase-space corrections. They give rise to the functions $$\begin{aligned}
I_A(\delta) &=& 1 - 2\delta + 2\delta^2 - \delta^3 + \frac15\,\delta^4
\,, \nonumber\\
I_B(\delta) &=& I_A(\delta)\,\ln\frac{\delta}{1-\delta}
+ \frac{\ln(1-\delta)}{5\delta} - \frac45 + \frac35\,\delta
- \frac{4}{15}\,\delta^2 + \frac{1}{20}\,\delta^3 \,.\end{aligned}$$
The hadronic power corrections come from subleading shape functions in the theoretical expressions for the two decay rates. We give their tree-level contributions to the weight function in the last two lines of (\[eq:masterform\]), where $\hat S$ denotes the leading shape function, and $\hat t, \hat u,\hat v$ are subleading shape functions as defined in [@Bosch:2004cb]. For completeness, we also include a contribution proportional to $m_s^2$ resulting from finite-mass effects in the strange-quark propagator in $\bar B\to X_s\gamma$ decays. For $m_s=O(\Lambda_{\rm QCD})$ these effects are formally of the same order as other subleading shape-function contributions [@Chay:2005ck], although numerically they are strongly suppressed. The appearance of subleading shape functions introduces an irreducible hadronic uncertainty to a $|V_{ub}|$ determination via (\[eq:relation\]). In practice, this uncertainty can be estimated by adopting different models for the subleading shape functions. This will be discussed in detail in Section \[sec:SSF\] below. Until then, let us use a “default model”, in which we assume the functional forms of the subleading shape functions $\hat t(\hat\omega)$, $\hat u(\hat\omega)$, and $\hat v(\hat\omega)$ to be particular linear combinations of the functions $\hat S'(\hat\omega)$ and $(\bar\Lambda-\hat\omega)\,\hat S(\hat\omega)$. These combinations are chosen in such a way that the results satisfy the moment relations derived in [@Bosch:2004cb], and that all terms involving the parameter $\bar\Lambda$ cancel in the expression (\[eq:masterform\]) for the weight function for any value of $a$. These requirements yield $$\label{defaultmodel}
\hat t\to-\frac34\,(\bar\Lambda-\hat\omega)\,\hat S
- \left( \lambda_2 + \frac{\lambda_1}{4} \right) \hat S' \,, \quad
\hat u\to\frac12\,(\bar\Lambda-\hat\omega)\,\hat S
+ \frac{5\lambda_1}{6}\,\hat S' \,, \quad
\hat v\to\lambda_2\,\hat S' \,,$$ and the last two lines inside the large bracket in the expression (\[eq:masterform\]) for the weight function simplify to $$\label{eq:SSFdefault}
- \frac{\Lambda_{\rm SSF}^2(a)}{M_B-P_+}\,
\frac{\hat S'(P_+)}{\hat S(P_+)}
~\widehat{=} - \frac{\Lambda_{\rm SSF}^2(a)}{M_B-P_+}\,\delta(P_+ -\Delta)
- \frac{4\Lambda_{\rm SSF}^2(a)}{(M_B-P_+)^2} \,,$$ where $$\Lambda_{\rm SSF}^2(a)\equiv - \frac{(2+a)\,\lambda_1}{3(1-a)(3-a)}
+ \frac{a(7-4a)\,\lambda_2}{2(1-a)(3-a)} - m_s^2 \,.$$ Here $\lambda_1$ and $\lambda_2=\frac14(M_{B^*}^2-M_B^2)$ are hadronic parameters describing certain $B$-meson matrix elements in heavy-quark effective theory [@Falk:1992wt]. The strange-quark mass is a running mass evaluated at a scale typical for the final-state hadronic jet, for which we take 1.5GeV. As mentioned above, the numerical effect of the strange-quark mass correction is small. For typical values of the parameters, it reduces the result for $\Lambda_{\rm SSF}^2$ by about 10% or less. The expression on the right-hand side in (\[eq:SSFdefault\]) is equivalent to that on the left-hand side after the integration with the photon spectrum in (\[eq:relation\]) has been performed. It has been derived using the fact that the normalized photon spectrum is proportional to the shape function $\hat S(P_+)$ at leading order. Note that the second term in the final formula is power suppressed with respect to the first one. It results from our exact treatment of phase-space factors and thus is kept for consistency.
Normalization
-------------
Finally, let us present explicit formulae for the overall normalization factor in (\[eq:masterform\]). The new ingredient here is the factor $T_0$, which is defined in (\[eq:Tn\]). At one-loop order we find $$\begin{aligned}
T_0(a,\mu_h)
&=& \frac{2(3-a)}{(2-a)(3-2a)}\,\Bigg\{ 1
- \frac{C_F\alpha_s(\mu_h)}{4\pi} \Bigg[ 4 \ln^2 \frac{m_b}{\mu_h}
- \frac{2(120-159a+69a^2-10a^3)}{(2-a)(3-a)(3-2a)}\,\ln \frac{m_b}{\mu_h}
\nonumber \\
&&\mbox{}+
\frac{1539-3845a+3842a^2-1920a^3+480a^4-48a^5}{(2-a)^2(3-a)(3-2a)^2}
+ 4 \psi^{(1)}(3-2a) + \frac{\pi^2}{6} \Bigg] \Bigg\} \,. \nonumber\\\end{aligned}$$ When the product of $T_0$ with the quantity [@Neubert:2004dd] $$\begin{aligned}
\label{Hgamma}
H_\Gamma(\mu_h)
&=& 1 + \frac{C_F\alpha_s(\mu_h)}{4\pi}\,
\Bigg[ 4\ln^2\frac{m_b}{\mu_h} - 10\ln\frac{m_b}{\mu_h}
+ 7 - \frac{7\pi^2}{6} + \frac{12\mu_*}{m_b} \nonumber \\
&&\mbox{}- 2\ln^2\delta_* - (7+4\delta_*-\delta_*^2)\,\ln\delta_*
+ 10\delta_* + \delta_*^2 - \frac23\,\delta_*^3 \nonumber \\
&&\mbox{}+ \frac{[C_1(\mu_h)]^2}{[C^{\rm eff}_{7\gamma}(\mu_h)]^2}\,
\hat f_{11}(\delta_*)
+ \frac{C_1(\mu_h)}{C^{\rm eff}_{7\gamma}(\mu_h)}\,\hat f_{17}(\delta_*)
+ \frac{C_1(\mu_h)\,C_{8g}^{\rm eff}(\mu_h)} {[C^{\rm eff}_{7\gamma}(\mu_h)]^2}\,\hat f_{18}(\delta_*)
\nonumber\\
&&\mbox{}+ \frac{C_{8g}^{\rm eff}(\mu_h)}{C^{\rm eff}_{7\gamma}(\mu_h)}\,
\hat f_{78}(\delta_*)
+ \frac{[C_{8g}^{\rm eff}(\mu_h)]^2}{[C^{\rm eff}_{7\gamma}(\mu_h)]^2}\,
\hat f_{88}(\delta_*) \Bigg]\end{aligned}$$ from the total $\bar B\to X_s\gamma$ decay rate is consistently expanded to $O(\alpha_s(\mu_h))$, the double logarithm cancels out. Here $\delta_*=1-2E_*/m_b=0.9$, and the functions $\hat f_{ij}(\delta_*)$ capture effects from operator mixing.
Numerical results {#sec:num}
=================
We are now in a position to explore the phenomenological implications of our results. We need as inputs the heavy-quark parameters $\lambda_2=0.12$GeV$^2$, $\lambda_1=(-0.25\pm 0.10)$GeV$^2$, and the quark masses $m_b=(4.61\pm 0.06)$GeV [@Neubert:2005nt], $m_s=(90\pm 25)$MeV [@Gamiz:2004ar; @Aubin:2004ck], and $m_c/m_b=0.222\pm 0.027$ [@Neubert:2004dd]. Here $m_b$ is defined in the shape-function scheme at a scale $\mu_*=1.5$GeV, $m_s$ is the running mass in the $\overline{\rm MS}$ scheme evaluated at 1.5GeV, and $m_c/m_b$ is a scale invariant ratio of running masses. Throughout, we use the 3-loop running coupling normalized to $\alpha_s(M_Z)=0.1187$, matched to a 4-flavor theory at 4.25GeV. For the matching scales, we pick the default values $\mu_h^{\rm def}=m_b/\sqrt 2$ and $\mu_i^{\rm def}=\bar\mu^{\rm def}=1.5$GeV, which are motivated by the underlying dynamics of inclusive processes in the shape-function region [@Lange:2005yw; @Bosch:2004th].
In the remainder of this section we present results for the partial decay rate $\Gamma_u(\Delta)$ computed by evaluating the right-hand side of relation (\[eq:relation\]). This is more informative than to focus on the value of the weight function for a particular choice of $P_+$. For the purpose of our discussion we use a simple model for the normalized photon spectrum that describes the experimental data reasonably well, namely $$\label{model}
\frac{1}{\Gamma_s}\,\frac{d\Gamma_s}{dP_+}
= \frac{b^b}{\Gamma(b)\Lambda^b}\,(P_+)^{b-1}
\exp\left( -b\,\frac{P_+}{\Lambda} \right)$$ with $\Lambda=0.77$GeV and $b=2.5$.
Studies of the perturbative expansion {#sec:pert}
-------------------------------------
The purpose of this section is to investigate the individual contributions to $\Gamma_u(\Delta)$ that result from the corresponding terms in the weight function, as well as their residual dependence on the matching scales. For $\Delta=0.65$GeV we find numerically $$\begin{aligned}
\label{eq:breakup}
\frac{\Gamma_u(0.65\,{\rm GeV})}{|V_{ub}|^2\,{\rm ps}^{-1}}
&=& 43.5\,\big( 1 + 0.158\,\hbox{\scriptsize $[\alpha_s(\mu_i)]$}
- 0.095\,\hbox{\scriptsize $[\alpha_s(\mu_h)]$}
+ 0.076\,\hbox{\scriptsize $[\alpha_s^2(\mu_i)]$} \nonumber\\[-0.2cm]
&&\hspace{1.2cm}
\mbox{}- 0.037\,\hbox{\scriptsize $[\alpha_s(\mu_i)\alpha_s(\mu_h)]$}
+ 0.009\,\hbox{\scriptsize [kin]}
- 0.043\,\hbox{\scriptsize [hadr]} \big) = 46.5 \,.\end{aligned}$$ The terms in parenthesis correspond to the contributions to the weight function arising at different orders in perturbation theory and in the $1/m_b$ expansion, as indicated by the subscripts. Note that the perturbative contributions from the intermediate scale are typically twice as large as the ones from the hard scale, which is also the naive expectation. Indeed, the two-loop $\alpha_s^2(\mu_i)$ correction is numerically of comparable size to the one-loop $\alpha_s(\mu_h)$ contribution. This confirms the importance of separating the scales $\mu_i$ and $\mu_h$. The contributions from kinematical and hadronic power corrections turn out to be numerically small, comparable to the two-loop corrections.
The weight function (\[eq:masterform\]) is formally independent of the matching scales $\mu_h$, $\mu_i$, and $\bar\mu$. In Figure \[fig:scales\] we plot the residual scale dependence resulting from the truncation of the perturbative series. Each of the three scales is varied independently by a factor between $1/\sqrt2$ and $\sqrt2$ about its default value. The scale variation of $\mu_i$ is still as significant as the variation of $\mu_h$, even though the former is known at NNLO and the latter only at NLO. We have checked analytically that the result (\[eq:masterform\]) is independent of $\mu_i$ through two-loop order, i.e. the residual scale dependence is an $O(\alpha_s^3(\mu_i))$ effect. In order to obtain a conservative estimate of the perturbative uncertainty in our predictions we add the individual scale dependencies in quadrature. This gives the gray band shown in the figure.
Figure \[fig:onetwo\] displays the result for $\Gamma_u(0.65\,\mbox{GeV})$ at different orders in RG-improved perturbation theory. At LO, we dismiss all $\alpha_s$ terms including the kinematical power corrections; however, leading logarithms are still resummed and give rise to a non-trivial dependence of $T_0$ on the coefficient $a$. At NLO, we include the $O(\alpha_s(\mu_h))$, $O(\alpha_s(\mu_i))$, and $O(\alpha_s(\bar\mu))$ contributions, but drop terms of order $\alpha_s^2(\mu_i)$ or $\alpha_s(\mu_i)\,\alpha_s(\mu_h)$. At NNLO, we include all terms shown in (\[eq:masterform\]). In studying the different perturbative approximations we vary the matching scales simultaneously (and in a correlated way) about their default values. Compared with Figure \[fig:scales\] this leads to a reduced scale variation. The gray bands in Figure \[fig:onetwo\] show the total perturbative uncertainty as determined above. While the two-loop NNLO contributions are sizable, we observe a good convergence of the perturbative expansion and a reduction of the scale sensitivity in higher orders. The right-hand plot in the figure contrasts these findings with the corresponding results in fixed-order perturbation theory, which are obtained from (\[eq:masterform\]) by setting $\mu_h=\mu_i=\bar\mu=\mu$ and truncating the series at $O(\alpha_s(\mu))$ for consistency. We see that the fixed-order results are also rather insensitive to the value of $\mu$ unless this scale is chosen to be small; yet, the predicted values for $\Gamma_u$ are significantly below those obtained in RG-improved perturbation theory. We conclude that the small scale dependence observed in the fixed-order calculation does not provide a reliable estimator of the true perturbative uncertainty. In our opinion, a fixed-order calculation at a high scale is not only inappropriate in terms of the underlying dynamics of inclusive decay processes in the shape-function region, it is also misleading as a basis for estimating higher-order terms in the perturbative expansion.
Comments on the normalization of the photon spectrum
----------------------------------------------------
We mentioned in the Introduction that the use of the normalized photon spectrum is advantageous because event fractions in $\bar B\to X_s\gamma$ decay can be calculated more reliably than partial decay rates. In this section we point out another important advantage, namely that the perturbative series for the weight function $W(\Delta,P_+)$ is much better behaved than that for $\widetilde W(\Delta,P_+)$. The difference of the two weight functions lies in their normalizations, which are $$\label{hardfunx}
W(\Delta,P_+)\propto m_b^3\, T_0(a,\mu_h) H_\Gamma(\mu_h) \,, \qquad
\widetilde W(\Delta,P_+)\propto
\frac{T_0(a,\mu_h)}{[C_{7\gamma}^{\rm eff}(\mu_h)]^2\, \overline{m}_b^2(\mu_h)\,|H_s(\mu_h)|^2} \,.$$ Here $H_s$ is the hard function in the factorized expression for the structure function $\F_\gamma$ in (\[eq:gammaS\]), which has been derived in [@Neubert:2004dd]. Note that the two weight functions have a different dependence on the $b$-quark mass. In the case of $W$, three powers of $m_b$ enter through phase-space integrations in the total decay rate $\Gamma_s(E_*)$, and it is therefore appropriate to use a low-scale subtracted quark-mass definition, such as the shape-function mass. In the case of $\widetilde W$, on the other hand, two powers of the running quark mass $\overline{m}_b(\mu_h)$ enter through the definition of the dipole operator $O_{7\gamma}$, and it is appropriate to use a short-distance mass definition such as that provided by the $\overline{\rm MS}$ scheme. In practice, we write $\overline{m}_b(\mu_h)$ as $\overline{m}_b(m_b)$ times a perturbative series in $\alpha_s(\mu_h)$.
The most pronounced effect of the difference in normalization is that the weight function $\widetilde W$ receives very large radiative corrections at order $\alpha_s(\mu_h)$, which range between $-68\%$ and $-43\%$ when the scale $\mu_h$ is varied between $m_b$ and $m_b/2$. This contrasts the well-behaved perturbative expansion of the weight function $W$, for which the corresponding corrections vary between $-11\%$ and $-7\%$. In other words, the hard matching corrections for $\widetilde W$ are about six times larger than those for $W$. Indeed, these corrections are so large that in our opinion relation (\[oldrelation\]) should not be used for phenomenological purposes.
The different perturbative behavior of the hard matching corrections to the weight functions is mostly due to the mixing of the dipole operator $O_{7\gamma}$ with other operators in the effective weak Hamiltonian for $\bar B\to X_s\gamma$ decay. In order to illustrate this fact, consider the one-loop hard matching coefficients defined as $$W\propto 1 + k\,\frac{\alpha_s(\mu_h)}{\pi} + \dots \,, \qquad
\widetilde W\propto 1 + \widetilde k\,\frac{\alpha_s(\mu_h)}{\pi}
+ \dots \,.$$ With our default scale choices we have $k=-2.32+1.13=-1.19$, where the second contribution ($+1.13$) comes from operator mixing, which gives rise to the terms in the last two lines in (\[Hgamma\]). For the weight function $W$, this contribution has the opposite sign than the other terms, so that the combined value of $k$ is rather small. For the weight function $\widetilde W$, on the other hand, we find $\widetilde k=-1.58-5.52=-7.10$. Here the contribution from operator mixing is dominant and has the same sign as the remaining terms, thus yielding a very large value of $\widetilde k$. Such a large $O(\alpha_s)$ correction was not observed in [@Hoang:2005pj], because these authors chose to omit the contribution from operators mixing. Note that at a higher scale $\mu_h=m_b$, as was adopted in this reference, the situation is even worse. In that case we find $k=-2.81+1.19=-1.62$ and $\widetilde k=-0.66-9.13=-9.79$.
A visualization of the perturbative uncertainty is depicted in Figure \[fig:band\], where predictions for $\Gamma_u(\Delta)$ are shown using either (\[eq:relation\]) or (\[oldrelation\]). In each case, the error band is obtained by varying the different scales about their default values, $\mu_n \in[\mu_n^{\rm def}/\sqrt2,\mu_n^{\rm def}\sqrt2]$, and adding the resulting uncertainties in quadrature. The dark-gray band bordered by solid lines denotes the perturbative uncertainty of predictions when using the normalized photon spectrum, as in (\[eq:relation\]). (At the point $\Delta=0.65$GeV this uncertainty is identical to the gray band depicted in Figures \[fig:scales\] and \[fig:onetwo\]). The light-gray band bordered by dashed lines corresponds to the use of the absolute photon spectrum, as in (\[oldrelation\]). The difference in precision between the two methods would be even more pronounced if we used the higher default value $\mu_h=m_b$ for the hard matching scale. Obviously, the use of the normalized photon spectrum will result in a more precise determination of $|V_{ub}|$.
Comments on $\beta_0\alpha_s^2$ terms and scale separation
----------------------------------------------------------
The separation of different momentum scales using RG techniques, which is one of the key ingredients of our approach, is well motivated by the dynamics of charmless inclusive $B$ decays in the shape-function region. Factorized expressions for the $B$-decay spectra involve hard functions renormalized at $\mu_h$ multiplied by jet and shape functions defined at a lower scale. While physics at or below the intermediate scale is very similar for $\bar B\to X_u\,l^-\bar\nu$ and $\bar B\to X_s\gamma$ (as is manifested by the fact that the leading shape and jet functions are universal), the physics at the hard scale in $\bar B\to X_s\gamma$ decay is considerably more complicated than in semileptonic decay, and it might even contain effects of New Physics. Therefore it is natural to respect the hierarchy $\mu_h\gg\mu_i$ and disentangle the various contributions, as done in the present work. In fact, our ability to calculate the dominant two-loop corrections is a direct result of this scale separation. Nevertheless, at a technical level we can reproduce the results of a fixed-order calculation by simply setting all matching scales equal to a common scale, $\mu_h=\mu_i=\bar\mu=\mu$. In this limit, the expressions derived in this work smoothly reduce to those obtained in conventional perturbation theory. While factorized expressions for the decay rates are superior to fixed-order results whenever there are widely separated scales in the problem, they remain valid in the limit where the different scales become of the same order.
In a recent publication, the $O(\beta_0\alpha_s^2)$ BLM corrections [@Brodsky:1982gc] to the weight function $\widetilde W(\Delta,P_+)$ in (\[oldrelation\]) were calculated in fixed-order perturbation theory [@Hoang:2005pj]. For simplicity, only the contribution of the operator $O_{7\gamma}$ to the $\bar B\to X_s\gamma$ decay rate was included in this work. We note that without the contributions from other operators the expression for $\widetilde W$ is not renormalization-scale and -scheme invariant. Neglecting operator mixing in the calculation of $\widetilde W$ is therefore not a theoretically consistent approximation. However, having calculated the exact NNLO corrections at the intermediate scale allows us to examine some of the terms proportional to $\beta_0\alpha_s^2(\mu_i)$ and compare them to the findings of [@Hoang:2005pj]. In this way we confirm their results for the coefficients multiplying the logarithms $\ln^n[m_b(\Delta-P_+)/\mu_i^2]$ with $n=1,2$ in (\[eq:masterform\]). While the $\beta_0\alpha_s^2$ terms approximate the full two-loop coefficients of these logarithms arguably well, we stress that the two-loop constant at the intermediate scale is not dominated by terms proportional to $\beta_0$. Numerically we find $$c^{(2)}_0 = - 47.4 + 39.6\,\frac{\beta_0}{25/3}
+ a \left[ - 31.8 + 38.8\,\frac{\beta_0}{25/3} \right] + O(a^2) \,,$$ which means that the approximation of keeping only the BLM terms would overestimate this coefficient by almost an order of magnitude and give the wrong sign. This shows the importance of a complete two-loop calculation, as performed in the present work.
We believe that the perturbative approximations adopted in our paper, i.e. working to NNLO at the intermediate scale and to NLO at the hard scale, are sufficient for practical purposes in the sense that the residual perturbative uncertainty is smaller than other uncertainties encountered in the application of relation (\[eq:relation\]). Still, one may ask what calculations would be required to determine the missing $\alpha_s^2(\mu_h)$ terms in the normalization of the weight function in (\[eq:masterform\]), or at least the terms of order $\beta_0\alpha_s^2(\mu_h)$. For the case of $\bar B\to X_s\gamma$ decay, the contribution of the operator $O_{7\gamma}$ to the normalized photon spectrum was recently calculated at two-loop order [@Melnikov:2005bx], while the contributions from other operators are known to $O(\beta_0\alpha_s^2)$ [@Ligeti:1999ea]. What is still needed are the two-loop corrections to the double differential (in $P_+$ and $y$) $\bar B\to X_u\,l^-\bar\nu$ decay rate in (\[eq:gammaU\]).
Subleading corrections from hadronic structures {#sec:SSF}
-----------------------------------------------
Due to the fact that different linear combinations of the subleading shape functions $\hat t(\hat\omega)$, $\hat u(\hat\omega)$, and $\hat v(\hat\omega)$ enter the theoretical description of radiative and semileptonic decays starting at order $\Lambda_{\rm QCD}/m_b$, the weight function cannot be free of such hadronic structure functions. Consequently, we found in (\[eq:masterform\]) all of the above subleading shape functions, divided by the leading shape function $\hat S(\hat\omega)$. Our default model (\[defaultmodel\]) for the subleading shape functions was chosen such that the combined effect of all hadronic power corrections could be absorbed into a single hadronic parameter $\Lambda_{\rm SSF}^2$. More generally, we define a function $\delta_{\rm hadr}(\Delta)$ via (a factor 2 is inserted for later convenience) $$\Gamma_u(\Delta) = [\Gamma_u(\Delta)]_{\rm def}
\left[ 1 + 2\delta_{\rm hadr}(\Delta) \right] ,$$ where $[\Gamma_u(\Delta)]_{\rm def}$ denotes the result obtained with the default model for the subleading shape functions. From (\[eq:masterform\]), one finds that $$\label{epsrela}
\delta_{\rm hadr}(\Delta)
= \frac{\int_0^\Delta\!dP_+\,(M_B-P_+)^4 \left[ 2(4-3a)\,h_t(P_+)
+ (4-a)\,h_u(P_+) + (8-13a+4a^2)\,h_v(P_+) \right]} {4(1-a)(3-a)\int_0^\Delta\!dP_+\,(M_B-P_+)^5\,\hat S(P_+)} \,,$$ where we have used that, at leading order in $\alpha_s$ and $\Lambda_{\rm QCD}/m_b$, the $\bar B\to X_s\gamma$ photon spectrum is proportional to $(M_B-P_+)^3\,\hat S(P_+)$. In the relation above, $h_t(\hat\omega)\equiv\hat t(\hat\omega)-[\hat t(\hat\omega)]_{\rm def}$ etc. denote the differences between the true subleading shape functions and the functions adopted in our default model. By construction, these are functions with vanishing normalization and first moment.
The above expression for $\delta_{\rm hadr}(\Delta)$ is exact to the order we are working; however, in practice we do not know the precise form of the functions $h_i(\hat\omega)$. Our goal is then to find a conservative bound, $|\delta_{\rm hadr}(\Delta)|<\epsilon_{\rm hadr}(\Delta)$, and to interpret the function $\epsilon_{\rm hadr}(\Delta)$ as the relative hadronic uncertainty on the value of $|V_{ub}|$ extracted using relation (\[eq:relation\]). To obtain the bound we scan over a large set of realistic models for the subleading shape functions. In [@Lange:2005yw], four different functions $h_i(\hat\omega)$ were suggested, which can be added or subtracted (in different combinations) to each of the subleading shape functions. Together, this provides a large set of different models for these functions. To be conservative, we pick from this set the model which leads to the largest value of $|\delta_{\rm hadr}(\Delta)|$. The integrand in the numerator in (\[epsrela\]) is maximized if all three $h_i(\hat\omega)$ functions are equal to a single function, whose choice depends on the value of $\Delta$. In the denominator, we find it convenient to eliminate the shape function $\hat S(P_+)$ in favor of the normalized photon spectrum. Working consistently to leading order, we then obtain $$\epsilon_{\rm hadr}(\Delta)
= \frac{5-5a+a^2}{(1-a)(3-a)}\,\frac{U(\mu_h,\mu_i)}{m_b^3}\,
\max_i
\frac{\displaystyle\left|\int_0^\Delta\!dP_+\,(M_B-P_+)^4\,h_i(P_+)\right|} {\displaystyle\int_0^\Delta\!dP_+\,(M_B-P_+)^2\,
\frac{1}{\Gamma_s(E_*)}\,\frac{d\Gamma_s}{dP_+}} \,,$$ where as before $a=a_\Gamma(\mu_h,\mu_i)\approx 0.12$ for the default choice of matching scales, and $U(\mu_h,\mu_i)\approx 1.11$ [@Lange:2005yw].
The result for the function $\epsilon_{\rm hadr}(\Delta)$ obtained this way is shown in Figure \[fig:sublSF\]. We set the matching scales to their default values and use the model (\[model\]) for the photon spectrum, which is a good enough approximation for our purposes. From this estimate it is apparent that the effects of subleading shape functions are negligible for large values $\Delta\gg\Lambda_{\rm QCD}$ and moderate for $\Delta\sim\Lambda_{\rm QCD}$, which is the region of interest for the determination of $|V_{ub}|$. In the region $\Delta<0.3$GeV the accuracy of the calculation deteriorates. For example, we find $\epsilon_{\rm hadr}=2.0\%$ for $\Delta=0.65$GeV, $\epsilon_{\rm hadr}=4.8\%$ for $\Delta=0.5$GeV, and $\epsilon_{\rm hadr}=8.2\%$ for $\Delta=0.4$GeV.
Conclusions
===========
Model-independent relations between weighted integrals of $\bar B\to X_s\gamma$ and $\bar B\to X_u\,l^-\bar\nu$ decay distributions, in which all reference to the leading non-perturbative shape function is avoided, offer one of the most promising avenues to a high-precision determination of the CKM matrix element $|V_{ub}|$. In order to achieve a theoretical precision of better than 10%, it is necessary to include higher-order corrections in $\alpha_s$ and $\Lambda_{\rm QCD}/m_b$ in this approach.
In the present work, we have calculated the weight function $W(\Delta,P_+)$ in the relation between the hadronic $P_+=E_X-|\vec{P}_X|$ spectra in the two processes, integrated over the interval $0\le P_+\le\Delta$. Based on QCD factorization theorems for the differential decay rates, we have derived an exact formula (\[eq:fmatch\]) that allows for the calculation of the leading-power weight function to any order in perturbation theory. We have calculated the $\Delta$- and $P_+$-dependent terms in the weight function exactly at next-to-next-to-leading order (NNLO) in renormalization-group improved perturbation theory, including two-loop matching corrections at the intermediate scale $\mu_i\sim\sqrt{m_b\Lambda_{\rm QCD}}$ and three-loop running between the intermediate scale and the hard scale $\mu_h\sim m_b$. The only piece missing for a complete prediction at NNLO is the two-loop hard matching correction to the overall normalization of the weight function. A calculation of the $\alpha_s^2(\mu_h)$ term would require the knowledge of both decay spectra at two-loop order, which is currently still lacking. We also include various sources of power corrections. Power corrections from phase-space factors are treated exactly. The remaining hadronic and kinematical power corrections are given to first order in $\Lambda_{\rm QCD}/m_b$ and to the order in perturbation theory to which they are known.
A dedicated study of the perturbative behavior of our result for the weight function has been performed for the partial $\bar B\to X_u\,l^-\bar\nu$ decay rate $\Gamma_u(\Delta)$ as obtained from the right-hand side of relation (\[eq:relation\]). It exhibits good convergence of the expansion and reduced scale sensitivity in higher orders. We find that corrections of order $\alpha_s^2(\mu_i)$ at the intermediate scale are typically as important as first-order $\alpha_s(\mu_h)$ corrections at the hard scale. We have also seen that fixed-order perturbation theory significantly underestimates the value of $\Gamma_u(\Delta)$, even though the apparent stability with respect to scale variations would suggest a good perturbative convergence. In order to obtain a well-behaved expansion in powers of $\alpha_s(\mu_h)$, it is important to use the normalized photon spectrum in relation (\[eq:relation\]). A similar relation involving the differential $\bar B\to X_s\gamma$ decay rate receives uncontrollably large matching corrections at the hard scale and is thus not suitable for phenomenological applications. At next-to-leading order in the $1/m_b$ expansion, the weight function receives terms involving non-perturbative subleading shape functions, which cannot be eliminated. Our current ignorance about the functional forms of these functions leads to a hadronic uncertainty, which we have estimated by scanning over a large set of models. We believe that a reasonable estimate of the corresponding relative uncertainty $\epsilon_{\rm hadr}$ on $|V_{ub}|$ is given by the solid line in Figure \[fig:sublSF\].
Let us summarize our main result for the partial $\bar B\to X_u\,l^-\bar\nu$ decay rate with a cut $P_+\le 0.65$GeV, which is close to the charm threshold $M_D^2/M_B$, and present a detailed list of the various sources of theoretical uncertainties. We find $$\begin{aligned}
\Gamma_u(0.65\,{\rm GeV})
&=& \left( 46.5\pm 1.4\,\hbox{\scriptsize [pert]}\,
\pm 1.8\, \hbox{\scriptsize [hadr]}\,
\pm 1.8\, \hbox{\scriptsize [$m_b$]}\,
\pm 0.8\, \hbox{\scriptsize [pars]}\,
\pm 2.8\, \hbox{\scriptsize [norm]}\,
\right) |V_{ub}|^2\,{\rm ps}^{-1} \nonumber\\
&=& (46.5\pm 4.1)\,|V_{ub}|^2\,{\rm ps}^{-1}\,,\end{aligned}$$ where the central value is derived assuming that the $\bar B\to X_s\gamma$ photon spectrum can be accurately described by the function (\[model\]). The errors refer to the perturbative uncertainty as estimated in Section \[sec:pert\], the uncertainty due to the ignorance about subleading shape functions as discussed in Section \[sec:SSF\], the error in the value of the $b$-quark mass, other parametric uncertainties from variations of $m_c$, $m_s$, and $\lambda_1$, and finally a 6% uncertainty in the calculation of the normalization of the photon spectrum [@Neubert:2004dd]. To a good approximation the errors scale with the central value. The above numbers translate into a combined theoretical uncertainty of 4.4% on $|V_{ub}|$ when added in quadrature.
[*Acknowledgments:*]{} We thank the Institute of Nuclear Theory at the University of Washington, where part of this research has been performed. The work of M.N. was supported in part by a Research Award of the Alexander von Humboldt Foundation. The work of B.O.L. was supported in part by funds provided by the U.S. Department of Energy under cooperative research agreement DE-FC02-94ER40818. The research of M.N. and G.P. was supported by the National Science Foundation under Grant PHY-0355005.
[99]{}
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[^1]: Due to an unphysical soft-photon singularity, the total decay rate is commonly defined to include all events with photon energies above $E_*=m_b/20$ [@Kagan:1998ym].
|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- |
<span style="font-variant:small-caps;">J. Andersson</span>\
[*Max Planck Institute for Mathematics in the Sciences ,*]{}\
[*Inselstr. 22, D-04103 Leipzig, Germany*]{}\
<span style="font-variant:small-caps;">G.S. Weiss</span>\
[*Graduate School of Mathematical Sciences,*]{}\
[*University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153-8914 Japan,*]{}\
[*Guest of the Max Planck Institute for Mathematics in the Sciences,*]{}\
[*Inselstr. 22, D-04103 Leipzig, Germany*]{} [^1]\
bibliography:
- 'anderssonweiss050810.bib'
title: |
[**Cross-shaped and Degenerate Singularities in an\
Unstable Elliptic Free Boundary Problem**]{}
---
We investigate singular and degenerate behavior of solutions of the unstable free boundary problem $$\Delta u = -\chi_{\{u>0\}}\; .$$ First, we construct a solution that is not of class $C^{1,1}$ and whose free boundary consists of four arcs meeting in a [*cross*]{}-shaped singularity. This solution is completely unstable/repulsive from above and below which would make it hard to get by the usual methods, and even numerics is non-trivial.\
We also show existence of a degenerate solution. This answers two of the open questions in the paper [@MW] by R. Monneau-G.S. Weiss.
Introduction
============
We will investigate singular and degenerate behavior of solutions of the unstable elliptic free boundary problem $$\label{eq}
\Delta u = -\chi_{\{ u>0\}}\;\quad \hbox{ in } \Omega\; .$$ The problem (\[eq\]) is related to traveling wave solutions in solid combustion with ignition temperature (see the introduction of [@MW] for more details).\
An equation similar to (\[eq\]) arises in the composite membrane problem (see [@chanillo1], [@chanillo2], [@blank]). Another application is the shape of self-gravitating rotating fluids describing stars (see [@stars equation (1.26)]).\
This problem has been investigated by R. Monneau-G.S. Weiss in [@MW]. Their main result is that local minimisers of the energy $$\int_{\Omega}|\nabla u|^2-2\max(u,0)$$ are $C^{1,1}$ and that their free boundaries are locally analytic. They also establish partial regularity for *second order non-degenerate* solutions of (\[eq\]) (cf. Definition \[nondeg\]). More precisely they show that the singular set has Hausdorff dimension less than or equal to $n-2$, and that in two dimensions the free boundary consists close to singular points of four Lipschitz graphs meeting at right angles. However they left open the question of the existence of cross-shaped singular points and of degenerate singularities (cf. [@MW Section 9 and 10]).
In this paper we will construct both singular points where the free boundary consists of four arcs meeting in a cross (see Corollary \[cross\] and Figure \[crossfig\])
and solutions that are degenerate of second order at a free boundary point (see Corollary \[ast\]). At this time we do not know whether the shape of the singularity is that of an asterisk or a product of even higher disconnectivity (see Figure \[astfig\]).
\
In particular, the cross-example is a counter-example to regularity of the solution since the solution is not of class $C^{1,1}$.\
In [@MW] it has been shown that the second variation of the energy takes the value $-\infty$ at the function $x_1^2-x_2^2$. That means that the cross-solution is completely unstable/repulsive. Moreover it cannot be approximated from above or below. This makes it hard to construct it by methods like the implicit function theorem or comparison methods.\
Our approach is simple. We construct an operator $T$ such that each fixed point of $T$, [*when adding a certain constant*]{}, satisfies equation (\[eq\]) [*and the origin is a point of the $0$-level set!*]{} By reflection and results from [@MW] it is then possible to show that origin is non-degenerate of second order and to obtain the cross.\
The construction of degenerate solutions is similar but simpler.\
[**Acknowledgement:**]{} We thank Carlos Kenig, Herbert Koch and Régis Monneau for discussions.
Notation
========
Throughout this article $\R^n$ will be equipped with the Euclidean inner product $x\cdot y$ and the induced norm $\vert x \vert\> .$ We define $e_i$ as the $i$-th unit vector in $\R^n\> ,$ and $B_r(x^0)$ will denote the open $n$-dimensional ball of center $x^0\> ,$ radius $r$ and volume $r^n\> \omega_n\> .$ When not specified, $x^0$ is assumed to be $0$. We shall often use abbreviations for inverse images like $\{u>0\} :=
\{x\in \Omega\> : \> u(x)>0\}\> , \> \{x_n>0\} :=
\{x \in \R^n \> : \> x_n > 0\}$ etc. and occasionally we shall employ the decomposition $x=(x_1,\dots,x_n)$ of a vector $x\in \R^n\> .$ When considering a set $A\> ,$ $\chi_A$ shall stand for the characteristic function of $A\> ,$ while $\nu$ shall typically denote the outward normal to a given boundary.
Preliminaries
=============
In this section we state some of the definitions and tools from [@MW].
\[nondeg\] Let $u$ be a solution of (\[eq\]) in $\Omega,$ satisfying at $x^0\in \Omega$ $$\label{ndeg}
\liminf_{r\to 0} r^{-2}\left(r^{1-n}\int_{\partial B_{r}(x^0)} u^2\> d{\cal H}^{n-1}
\right)^{1\over 2}>0\; .$$ Then we call $u$ “non-degenerate of second order at $x^0$”. We call $u$ “non-degenerate of second order” if it is non-degenerate of second order at each point in $\Omega$.
In [@MW Section 3] it has been shown that the maximal solution and each local energy minimiser are non-degenerate of second order.
A powerful tool, that we will use in Corollary \[cross\], is the monotonicity formula introduced in [@cpde] by one of the authors for a class of semilinear free boundary problems. For the sake of completeness let us state the unstable case here:
\[mon\] Suppose that $u$ is a solution of (\[eq\]) in $\Omega$ and that $B_\delta(x^0)\subset \Omega\> .$ Then for all $0<\rho<\sigma<\delta$ the function $$\Phi_{x^0}(r) := r^{-n-2} \int_{B_r(x^0)} \left(
{\vert \nabla u \vert}^2 \> -\> 2\max(u,0)
\right)$$$$- \; 2 \> r^{-n-3}\> \int_{\partial B_r(x^0)}
u^2 \> d{\cal H}^{n-1}\; ,$$ defined in $(0,\delta)\> ,$ satisfies the monotonicity formula $$\Phi_{x^0}(\sigma)\> -\> \Phi_{x^0}(\rho) \; = \;
\int_\rho^\sigma r^{-n-2}\;
\int_{\partial B_r(x^0)} 2 \left(\nabla u \cdot \nu - 2 \>
{u \over r}\right)^2 \; d{\cal H}^{n-1} \> dr \; \ge 0 \; \; .$$
The following proposition has been proven in [@MW Section 5].
\[fixedcenter\] Let $u$ be a solution of (\[eq\]) in $\Omega$ and let us consider a point $x^0\in \Omega\cap \{ u=0\}\cap\{ \nabla u =0\}.$\
1) In the case $\Phi_{x^0}(0+)=-\infty$, $\lim_{r\to 0} r^{-3-n}\int_{\partial B_r(x^0)} u^2 \> d{\cal H}^{n-1}
= +\infty$, and for $S(x^0,r) := \left(r^{1-n}\int_{\partial B_{r}(x^0)} u^2\> d{\cal H}^{n-1}
\right)^{1\over 2}$ each limit of $$\frac{u(x^0+r x)}{S(x^0,r)}$$ as $r\to 0$ is a homogeneous harmonic polynomial of degree $2$.\
2) In the case $\Phi_{x^0}(0+)\in (-\infty,0)$, $$u_r(x) := \frac{u(x^0+r x)}{r^2}$$ is bounded in $W^{1,2}(B_1(0))$, and each limit as $r\to 0$ is a homogeneous solution of degree $2$.\
3) Else $\Phi_{x^0}(0+)=0$, and $$\frac{u(x^0+r x)}{r^2}\to 0\hbox{ in } W^{1,2}(B_1(0)) \hbox{ as } r\to 0\; .$$
\[int\] 1) As shown in [@MW Lemma 5.2], the case 2) is not possible in two dimensions.\
2) Case 3) is equivalent to $u$ being degenerate of second order at $x^0$.
Main Results
============
Let $\pi/{\phi_0}\in \N$ and let us define the disk sector $K=K_{\phi_0}=\{r(\cos \phi,\sin\phi):\, 0<r<1, 0<\phi<
\phi_0\}$. For $g\in C^{\alpha}(\partial B_1\cap \partial K)$, $C^{\alpha}_g(\bar K)$ will denote the subspace of $C^{\alpha}(\bar K)$ consisting of all the functions with boundary values $g$ on $\partial B_1\cap \partial K$.\
Consider now the operator $T=T_{\epsilon,g}:C^{\alpha}_g(\bar K)\rightarrow C^{\alpha}_g(\bar K)$ defined by $$\begin{array}{ll}
\Delta T(u)=-f_{\epsilon}(u-u(0)) & \textrm{in } K \> ,\\
T(u)=g & \textrm{on } \partial B_1\cap \partial K\> ,\\
\frac{\partial (T(u))}{\partial \nu}=0 & \textrm{on } \partial K-\partial B_1\> ;
\end{array}$$ here $f_\epsilon\in C^\infty(\R),f_\epsilon(z)\ge
\chi_{\{ z>0\}}$ in $\R$ and $f_\epsilon
\downarrow \chi_{\{ z>0\}}$ as $\epsilon\downarrow 0.$
Since there exists for $F\in L^\infty(K)$ a $W^{1,2}(K)$-solution $v$ of $$\begin{array}{ll}
\Delta v = F & \hbox{in } K\; ,\\
v=g & \textrm{on } \partial B_1\cap \partial K\> ,\\
\frac{\partial v}{\partial \nu}=0 & \textrm{on } \partial K-\partial B_1\> ,
\end{array}$$ we obtain after reflection a $W^{1,2}(B_1)$-function that solves $\Delta v = F$ in $B_1-\{ 0\}$, where $F$ means the reflected function defined on $B_1$. As the origin is a set of vanishing capacity, $v$ is a weak solution of $\Delta v = F$ in $B_1$. Applying the regularity theory for elliptic equations (see for example [@GT Lemma 9.29]), we see that $T$ is for small $\alpha$ a continuous compact operator from $C^{\alpha}_g(\bar K)$ into itself, and that $$\|T_{\epsilon,g}(u)\|_{C^{\alpha}(\bar K)}\le C\; ,$$ where $C$ is a constant depending only on $g$.
From Schauder’s fixed point theorem (see for example [@GT Chapter 11]) we infer that $T_{\epsilon,g}$ has a fixed point $u_{\epsilon}\in
C^{\alpha}_g(\bar K)\cap \{\|u\|_{C^{\alpha}(\bar K)}\le C\}$. Alternatively, we could also show existence of a fixed point in a class of symmetric functions.\
Reflecting and applying $L^p$-estimates we obtain a sequence $\epsilon_m\to 0$ such that the reflected $u_{\epsilon_m}-u_{\epsilon_m}(0)\to u$ strongly in $C^{1,\beta}(\overline{B_{1-\delta}})$ and weakly in $W^{2,p}(B_{1-\delta})$ for each $\delta \in (0,1)$ as $m\to \infty$. At a.e. point of $\{ u>0\} \cup \{ u<0\}$, $u$ satisfies the equation $\Delta u= -\chi_{\{ u>0\}}$. At a.e. point of $\{ u=0\}$, the weak second derivatives of the $W^{2,2}$-function $u$ are $0$, so that we obtain:
\[pro:pro\] For each $g\in C^{\alpha}(\partial B_1\cap\partial K)$ there exists a constant $\kappa$ such that the boundary value problem $$\begin{array}{ll}
\Delta u=-\chi_{\{u>0\}}& \textrm{in } K \\
u=g-\kappa & \textrm{on } \partial B_1\cap \partial K\> ,\\
\frac{\partial u}{\partial \nu}=0 &
\textrm{on } \partial K-\partial B_1
\end{array}$$ has a solution $u\in \bigcap_{\delta \in (0,1)}C^{1,\beta}(\bar K\cap \overline{B_{1-\delta}})$ such that $u(0)=0.$
We will use Proposition \[pro:pro\] to prove the existence of singular and degenerate solutions:
\[cross\] There exists a solution $u$ of $$\Delta u= -\chi_{\{u>0\}} \quad \textrm{in } B_1$$ that is not of class $C^{1,1}$, such that each limit of $$\frac{u(r x)}{S(0,r)}$$ as $r\to 0$ is after rotation the function $(x_1^2-x_2^2)/\Vert x_1^2-x_2^2\Vert_{L^2(\partial B_1(0))}$.
*Proof:* By Proposition \[pro:pro\] there exists for each $M\in \R-\{ 0\}$ a constant $\kappa \in \R$ and a solution in $K_{\pi/2}$ with boundary values $g=M(x_1^2-x_2^2)-\kappa$ on $\partial B_1\cap \partial K_{\pi/2}$ satisfying the homogeneous Neumann boundary condition on $\partial K_{\pi/2}-\partial B_1$. Using the homogeneous Neumann boundary condition and the fact that $u\in C^{1,\beta}(\overline{K_{\pi/2}\cap B_{1-\delta}})$ we can reflect this solution twice at the coordinate axes to obtain a solution in the unit ball $B_1$, called again $u$.
Also by Proposition \[pro:pro\], we know that $u(0)=0$. Thus $u(0)=0$ and $\nabla u(0)=0$ so that Proposition \[fixedcenter\] applies. What remains to be done is to exclude case 3) of Proposition \[fixedcenter\] (see Remark \[int\] 1)). That done, it follows from the statement in case 1) that $u$ is not of class $C^{1,1}$.\
To this end we use the monotonicity formula Theorem \[mon\]. If $\lim_{r\rightarrow 0}\Phi_0(r)=0$, then $\Phi_0(r)\ge 0$ for all $r>0$. Therefore we only need to show that $\Phi_0(1)< 0$:
For $h=M(x_1^2-x_2^2)$ and $g=h$ let us write $u=v+h-\kappa$: The function $v$ satisfies $$\begin{array}{ll}
\Delta v = \Delta u & \textrm{in } B_1 \hbox{ and}\\
v=0 & \textrm{on } \partial B_1.
\end{array}$$ Notice that $-1\le \Delta v \le 0$ implies that $0<v<C_1$ and $|\nabla v|<C_1$ where $C_1$ is a universal constant. In particular $C_1$ is independent of $M$. We also know that $\kappa=v(0)\in (0,C_1)$ since $u(0)=0$. Now we calculate the energy $\Phi_0(1)$ of $u$. $$\begin{array}{l}
\Phi_0(1)=\int_{B_1}|\nabla u|^2 - 2u^+-2\int_{\partial B_1}u^2\> d{\cal H}^{n-1} \\
= \int_{B_1}|\nabla (v+h)|^2 - 2(v+h-\kappa)^+-2\int_{\partial B_1}
(v+h-\kappa)^2\> d{\cal H}^{n-1} \\
= \int_{B_1}|\nabla v|^2+2\nabla v \cdot \nabla h +|\nabla h|^2 - 2(v+h-\kappa)^+-2\int_{\partial B_1}(h-\kappa)^2\> d{\cal H}^{n-1}\; ,
\end{array}$$ where we have used that $\kappa$ is a constant and that $v=0$ on $\partial B_1$. Integrating by parts and using the specific form of $h$ shows that $$\begin{array}{l}
\Phi_0(1)=\int_{B_1}|\nabla v|^2-2(v+h-\kappa)^+-2\int_{\partial B_1}\kappa^2\> d{\cal H}^{n-1} \\
< \int_{B_1}|\nabla v|^2- 2(v+h-\kappa)^+< \int_{B_1}C_1^2-2(h-C_1)^+ \\
=\int_{B_1}C_1^2-2(M(x_1^2-x_2^2)-C_1)^+.
\end{array}$$ The last integral is negative if $M$ is large. We have thus shown that $\Phi_0(1)<0$ for sufficiently large $M$.
To calculate the just obtained solution numerically would – because of the severe instability – not be easy.
The next corollary establishes the existence of degenerate solutions of second order:
\[ast\] There exists a non-trivial solution $u$ of $$\Delta u= -\chi_{\{u>0\}} \quad \textrm{in } B_1$$ that is degenerate of second order at the origin.
*Proof:* This is also a direct consequence of Proposition \[pro:pro\]. The proposition yields a solution in $K_{\pi/4}$ with boundary data $\cos(4\phi)-\kappa$ on $\partial K_{\pi/4}\cap \partial B_1$. Let us reflect this solution three times to get a solution $u$ in the unit ball $B_1$. As in the previous corollary $0=u(0)=|\nabla u(0)|$. We only have to show that $u$ is degenerate of second order. Suppose towards a contradiction that this is not true: then by Remark \[int\] 1), case 1) of Proposition \[fixedcenter\] has to apply. We obtain after a rotation a blow-up limit of the form $(x_1^2-x_2^2)/\Vert x_1^2-x_2^2\Vert_{L^2(\partial B_1(0))}$. But there is no rotation for which that blow-up limit could be symmetric with respect to the two axes $x_1=0$ and $x_1=x_2$, yielding a contradiction.
Open Questions
==============
Concerning the set of degenerate singular points there remains the question whether [*large*]{} degenerate singular sets are possible. Also it would be nice to know the precise shape of isolated degenerate singularities, and whether infinite order vanishing is possible or not.
[^1]: G.S. Weiss has been partially supported by the Grant-in-Aid 15740100 of the Japanese Ministry of Education and partially supported by a fellowship of the Max Planck Society. Both authors thank the Max Planck Institute for Mathematics in the Sciences for the hospitality during their stay in Leipzig.
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{
"pile_set_name": "ArXiv"
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---
abstract: 'An approach to the modelling of financial return series using a class of uniformity-preserving transforms for uniform random variables is proposed. V-transforms describe the relationship between quantiles of the return distribution and quantiles of the distribution of a predictable volatility proxy variable constructed as a function of the return. V-transforms can be represented as copulas and permit the formulation and estimation of models that combine arbitrary marginal distributions with linear or non-linear time series models for the dynamics of the volatility proxy. The idea is illustrated using a transformed Gaussian ARMA process for volatility, yielding the class of VT-ARMA copula models. These can replicate many of the stylized facts of financial return series and facilitate the calculation of marginal and conditional characteristics of the model including quantile measures of risk. Estimation of models is carried out by adapting the exact maximum likelihood approach to the estimation of ARMA processes.'
author:
- 'Alexander J. McNeil[[^1]]{}'
date: 27th March 2020
title: 'Modelling volatility with v-transforms'
---
[*JEL*]{} Codes: C52; G21; G28; G32\
[*Keywords*]{}:time series; volatility; probability-integral transform; ARMA model; copula\
Introduction {#sec:intro}
============
In this paper we propose a class of transforms for uniform random variables that may be used to construct some new discrete-time models for volatile financial time series, such as log-returns on asset prices. Although the existing literature on volatility modelling is vast, the resulting models have some attractive features. In particular, they are copula-based models, which means that marginal and dependence characteristics can be easily separated in the construction and estimation of models. Moreover, both the marginal and conditional distributions of the underlying stationary process are accessible, permitting the calculation of static and dynamic variances, quantiles and other measures of risk.
A distinction is commonly made between genuine stochastic volatility models (in discrete or continuous time) where an unobservable process describes the volatility at any time point, and GARCH models where volatility is a function of observable information describing the past behaviour of the process; see the review articles by [@bib:shephard-96] and @bib:andersen-benzoni-10. The models of this paper have more in common with the GARCH class [see @bib:engle-82; @bib:bollerslev-86; @bib:ding-engle-granger-93; @bib:glosten-jagannathan-runkle-93; @bib:bollerslev-engle-nelson-94 among others]. However, there are some notable differences.
In GARCH modelling the marginal distribution of a stationary process is inextricably linked to the dynamics of the process as well as the distribution of the driving innovations; if the dynamics are altered, the marginal distribution is also changed. Moreover, the marginal distributions of most models in the GARCH family have no simple closed parametric form, although the behaviour of tail indices and higher moments such as kurtosis is well understood [@bib:mikosch-starica-00].
In practical applications in financial risk management, it is convenient to be able to make exact calculations of marginal and conditional measures of risk, such as value-at-risk (VaR), and this is a primary motivation for this paper. A model of the type we propose is used in @bib:gordy-mcneil-18 to conduct a simulation study in a risk-model backtesting context.
Let $X_1,\ldots,X_n$ be a series of financial return data sampled at (say) daily frequency and assume that the data are modelled by a strictly stationary stochastic process $(X_t)$ with marginal distribution function (cdf) $F_X$. To match the stylized facts of financial return data [@bib:campbell-lo-mackinlay-97; @bib:cont-01], it is generally agreed that $(X_t)$ should have limited serial correlation, but the squared or absolute processes $(X_t^2)$ and $(|X_t|)$ should have significant and persistent positive serial correlation to describe the effects of volatility clustering.
In this paper we refer to transformed series like $(|X_t|)$, in which volatility is revealed through serial correlation, as *volatility proxy series*. More generally, a volatility proxy series $(T(X_t))$ is obtained by applying a transformation $T:\R
\mapsto \R$ which (i) depends on a change point $\mu_T$ which may be zero, (ii) is increasing in $X_t-\mu_t$ for $X_t \geq \mu_T$ and (iii) is increasing in $\mu_t - X_t$ for $X_t \leq \mu_t$. Our approach in this paper is to model the probability-integral transform (PIT) series $(V_t)$ of volatility proxy series. This is defined by $V_t = F_{T(X)}(T(X_t))$ for all $t$, where $F_{T(X)}$ denotes the cdf of $T(X_t)$. If $(U_t)$ is the PIT series of the original process $(X_t)$, defined by $U_t = F_X(X_t)$ for all $t$, then a *v-transform* is a function describing the relationship between the terms of $(V_t)$ and the terms of $(U_t)$.
Equivalently, a v-transform describes the relationship between quantiles of the distribution of $X_t$ and the distribution of the volatility proxy $T(X_t)$. Alternatively, it characterizes the dependence structure or copula of the pair of variables $(X_t, T(X_t))$. In this paper we show how to derive flexible, parametric families of v-transforms for practical modelling purposes.
To gain insight into the typical form of a v-transform, let $\hat{U}_1,\ldots, \hat{U}_n$ and $\hat{V}_1,\ldots, \hat{V}_n$ be the samples obtained by applying the transformations $\hat{V}_t =F^{(|X|)}_n(|X_t|) $ and $\hat{U}_t =
F^{(X)}_n(X_t)$, where $F_n^{(X)}(x)=\frac{1}{n+1}\sum_{t=1}^n \indicator{X_t \leq x}$ and $F_n^{(|X|)}(x)=\frac{1}{n+1}\sum_{t=1}^n \indicator{|X_t| \leq x}$ denote scaled versions of the empirical distribution functions of the $X_t$ and $|X_t|$ samples respectively. The graph of $\hat{V}_t$ against $\hat{U}_t$ gives an empirical estimate of the v-transform for $(X_t,|X_t|)$. In the left-hand plot of Figure \[fig:1\] we show the relationship for a sample of $n=1000$ daily log-returns of the S&P 500 index for the period from 3 January 2007 to 21 December 2010. Note how the empirical v-transform takes the form of a slightly asymmetric ‘V’.
![\[fig:1\] Scatterplot of $\hat{V}_t$ against $\hat{U}_t$ (left) and sample acf of $\hat{Z}_t =\Phi^{-1}(\hat{V}_t)$ (right). Data are defined by $\hat{V}_t =F^{(|X|)}_n(|X_t|) $ and $\hat{U}_t =
F^{(X)}_n(X_t)$ where $F_n^{(X)}$ and $F_n^{(|X|)}$ denote versions of the empirical distribution function of the $X_t$ and $|X_t|$ values respectively. The sample size is $n=1000$ and the data are daily log-returns of the S&P index for the period from 3 January 2007 to 21 December 2010.](R_exploratory_plots.pdf){width="16cm" height="9cm"}
To construct a volatility model for $(X_t)$ using v-transforms we need to specify a process for $(V_t)$. In principle any model for a series of serially dependent uniform variables can be applied to $(V_t)$. In this paper we illustrate ideas using the Gaussian copula model implied by the standard ARMA dependence structure. We apply the inverse-normal transformation $Z_t = \Phi^{-1}(V_t)$ to obtain a series of standard normal variables $(Z_t)$ and then model these using a Gaussian ARMA process. The right-hand plot of Figure \[fig:1\] shows the sample autocorrelation function (acf) of the data $\hat{Z}_t =
\Phi^{-1}(\hat{V}_t)$ and reveals a pattern of persistent positive serial correlation.
Although copulas play a role in describing the class of models based on v-transforms, the models of this paper are distinct from other copula time series models proposed in the econometrics literature; see, for example, the review papers by @bib:patton-12 and @bib:fan-patton-14. However, some of the models in these papers would be viable alternatives for modelling the volatility PIT process $(V_t)$. One possibility would be the first-order Markov copula models investigated in @bib:chen-fan-06b, [@bib:chen-wu-yi-09] and [@bib:domma-giordano-perri-09]. The underlying theory of Markov copula models is explored in [@bib:darsow-nguyen-olsen-92] and [@bib:beare-10] while higher-order Markov extensions are treated in [@bib:ibragimov-09].
The paper is structured as follows. In Section \[sec:simple-symm-model\] we provide motivation for the paper by constructing a symmetric model using the simplest example of a v-transform. The general theory of v-transforms is developed in Section \[sec:v-transforms\] and is used to construct the class of VT-ARMA processes and analyse their properties in Section \[sec:properties-model\]. Section \[sec:estimation-model\] treats estimation and statistical inference for VT-ARMA processes and provides examples of their application to data; Section \[sec:conclusion\] concludes. Proofs may be found in the Appendix.
A motivating model and GARCH comparison {#sec:simple-symm-model}
=======================================
Given a probability space $(\Omega,\mathcal{F},\P)$, we construct a symmetric, strictly stationary process $(X_t)_{t\in\N\setminus\{0\}}$ such that, under the even transformation $T(x)=|x|$, the serial dependence in the volatility proxy series $(T(X_t))$ is of ARMA type. We assume that the marginal cdf $F_X$ of $(X_t)$ is absolutely continuous and the density $f_X$ satisfies $f_X(x) = f_X(-x)$ for all $x >0$. Since $F_X$ and $F_{|X|}$ are both continuous the properties of the probability-integral (PIT) transform imply that the series $(U_t)$ and $(V_t)$ given by $U_t = F_X(X_t)$ and $V_t = F_{|X|} (|X_t|)$ both have standard uniform marginal distributions. Henceforth we refer to $(V_t)$ as the *volatility PIT process* and $(U_t)$ as the *series PIT process*.
Any other volatility proxy series that can be obtained by a continuous and strictly increasing transformation of the terms of $(|X_t|)$, such as $(X_t^2)$, yields exactly the same volatility PIT process. For example, if $\tilde{V}_t=
F_{X^2}(X_t^2)$, then it follows from the fact that $F_{X^2}(x) =
F_{|X|}(\sqrt[+]{x})$ for $x\geq 0$ that $\tilde{V}_t=F_{X^2}(X_t^2) = F_{|X|} (|X_t|)=V_t$. In this sense we can think of classes of equivalent volatility proxies, such as $(|X_t|)$, $(X_t^2)$, $(\exp|X_t|)$ and $(\ln(1+|X_t|))$. In fact $(V_t)$ is itself an equivalent volatility proxy to $(|X_t|)$ since $F_{|X|}$ is a continuous and strictly increasing transformation.
The symmetry of $f_X$ implies that $F_{|X|}(x) = 2F_X(x)-1 = 1-2F_X(-x)$ for $x\geq 0$. Hence we find that $$V_t = F_{|X|} (|X_t|) =
\begin{cases}
\begin{aligned}
F_{|X|}(-X_t) &= 1-2F_X(X_t) =1 -2U_t, &\text{if $X_t <0$}\\
F_{|X|}(X_t) &= 2F_X(X_t)-1 = 2U_t -1, &\text{if $X_t \geq 0$}
\end{aligned}
\end{cases}$$ which implies that the relationship between the volatility PIT process $(V_t)$ and the series PIT process $(U_t)$ is given by $$\label{eq:13}
V_t = \vtrans(U_t) =|2 U_t - 1|$$ where $\vtrans(u) =
|2u-1|$ is a perfectly symmetric v-shaped function that maps values of $U_t$ close to 0 or 1 to values of $V_t$ close to 1, and values close to $0.5$ to values close to 0. $\mathcal{V}$ is the canonical example of a v-transform. It is related to the so-called tent-map transformation $\mathcal{T}(u)=2\min(u,1-u)$ by $\mathcal{V}(u)=1-\mathcal{T}(u)$.
Given $(V_t)$ let the process $(Z_t)$ be defined by setting $Z_t =
\Phi^{-1}(V_t)$ so that we have the following chain of transformations $$\label{eq:23}
\begin{tikzcd}
X_t \ar{r}{F_X} & U_t \ar{r}{\vtrans} & V_t
\ar{r}{\Phi^{-1}} & Z_t\quad .
\end{tikzcd}$$ We refer to $(Z_t)$ as a *normalized volatility proxy series*. Let us assume that $(Z_t)$ is *exactly* a Gaussian ARMA process with mean zero and variance one. Our aim is to construct a process $(X_t)$ such that under the chain of transformations in we obtain $(Z_t)$.
The transformation $\vtrans$ is not an injection and, for any $V_t>0$, there are two possible inverse values, $\tfrac{1}{2}(1-V_t)$ and $\tfrac{1}{2}(1+V_t)$. However, by randomly choosing between these values, we can ‘stochastically invert’ $\vtrans$ to construct a random variable $U_t$ such that $\vtrans(U_t)
= V_t$, This is summarized in Lemma \[lemma:reconstructU\], which is a special case of a more general result in Proposition \[prop:resconstructU\].
\[lemma:reconstructU\] Let $V$ be a standard uniform variable. If $V=0$ set $U =
\tfrac{1}{2}$. Otherwise let $U = \tfrac{1}{2}(1-V)$ with probability 0.5 and $U = \tfrac{1}{2}(1+V)$ with probability 0.5. Then $U$ is uniformly distributed and $\vtrans(U) = V$.
This simple result suggests the following algorithm for constructing a process $(X_t)$ with symmetric marginal density $f_X$ such that the corresponding normalized volatility proxy process $(Z_t)$ under the absolute value transformation (or continuous and strictly increasing functions thereof) is an ARMA process. We describe the resulting model as a VT-ARMA process.
\[algo1\]
1. Generate $(Z_t)$ as a causal and invertible Gaussian ARMA process of order $(p,q)$ with mean zero and variance one.
2. Form the volatility PIT process $(V_t)$ where $V_t =\Phi(Z_t)$ for all $t$.
3. Generate a process of iid Bernoulli variables $(Y_t)$ such that $\P(Y_t=1) = 0.5$.
4. Form the PIT process $(U_t)$ using the transformation $ U_t =0.5 (1-V_t)^{\indicator{Y_t = 0}} (1+V_t)^{\indicator{Y_t =
1}}$.
5. Form the process $(X_t)$ by setting $X_t = F_X^{-1}(U_t)$.
It is important to state that the use of the Gaussian process $(Z_t)$ as the fundamental building block of the VT-ARMA process in Algorithm \[algo1\] has no effect on the marginal distribution of $(X_t)$, which is $F_X$ as specified in the final step of the algorithm. The process $(Z_t)$ is exploited *only for its serial dependence structure*, which is described by a family of finite-dimensional Gaussian copulas; this dependence structure is applied to the volatility proxy process. The main drawbacks of this model choice are the radial symmetry of the Gaussian copula and its lack of extremal dependence, rather than the symmetry and thin tails of the normal distribution. The radial symmetry of the copula implies a similar dependence structure for jointly large and jointly small values of the volatility proxy variable which may not be realistic. Nevertheless the model that results from Algorithm \[algo1\] provides a useful archetype that can reproduce many of the stylized facts of financial time series and indicate directions for further research.
It is instructive to compare the symmetric VT-ARMA process with a symmetric GARCH process to see the similarities and differences between the models. Let $(Z_t)$ follow the causal stationary and invertible ARMA(1,1) model given by $Z_t = \alpha_1 Z_{t-1} + \beta_1 \epsilon_{t-1} + \epsilon_t$ for some iid innovation series $(\epsilon_t)$ such that $\var(Z_t) =1$ for all $t$. Recall that $V_t$ can be written as $V_t = F_{X^2} (X_t^2)$. It follows that we may write the model as $$\label{eq:VT-ARMA-compare-GARCH}
\Phi^{-1} \circ F_{X^2}\left( X_t^2\right) = \alpha_1 \Big( \Phi^{-1} \circ F_{X^2}\left(X_{t-1}^2\right)\Big) + \beta_1 \epsilon_{t-1} + \epsilon_t$$ in terms of the composite transformation $\Phi^{-1} \circ F_{X^2}$. Thus we may consider a VT-ARMA(1,1) model to be an ARMA(1,1) model applied to a transformation of the squared data (or the absolute data).
Now consider a GARCH(1,1) model taking the form $X_t = \sqrt{h_t} \omega_t$ where $(\omega_t)$ denotes an iid series with mean zero and variance one and the conditional variance series $(h_t)$ satisfies the equations $ h_t = a_0 + a_1 X_{t-1}^2 + b_1 h_{t-1}$ for parameters $a_0>0$, $a_1\geq 0$ and $b_1 \geq 0$ with $a_1 + b_1 < 1$ (the condition for covariance stationarity). This process may be written $$\label{eq:GARCH-squares}
X_t^2 - \mu_{X^2} = (a_1 + b_1) \left(X_{t-1}^2 - \mu_{X^2} \right) - b_1 \varepsilon_{t-1} + \varepsilon_t$$ where $\varepsilon_t = h_t(\omega_t^2 -1)$ and $\mu_{X^2} =\E(X_t^2) =
a_0/(1-a_1-b_1)$. The process $(\varepsilon_t)$ is not iid but it is a stationary process with the martingale difference property $\E(\varepsilon_t \mid \mathcal{F}_{t-1})=0$. It may be regarded as the innovation process for the squared value process. While has the structure of an ARMA(1,1) process for $(X_t^2)$ it only fulfills the usual definition of such a process if $(\varepsilon_t)$ has finite variance, for which a necessary and sufficient condition is $\E((a_1\omega_t^2 + b_1)^2) < 1$; see, for example, @bib:mcneil-frey-embrechts-15 (Section 4.2) for more details.
Equations and allow the models to be compared and highlight two major differences between the models. The first is the form of the transformation applied to the squares $(X_t^2)$: the dynamic equation in the VT-ARMA model changes their distribution to standard Gaussian; the GARCH model simply centres them to have mean zero. The second is in the form of the innovations in equations and : the VT-ARMA model has iid Gaussian innovations $(\epsilon_t)$; the GARCH model has innovations $(\varepsilon_t)$ which are not independent and which have a skewed distribution determined by both the choice of innovation distribution for $(\omega_t)$ and the dynamics implicit in $(h_t)$.
This comparison also gives clues concerning parameter values to mimic the features of real financial return data. In a GARCH model a choice of values like $a_1 = 0.1$ and $b_1 = 0.85$ might produce realistic patterns of volatility. Corresponding values in the VT-ARMA model might be $\alpha_1 = a_1+b_1 = 0.95$ and $\beta_1 = - b_1 = -0.85$, as illustrated in Figure \[fig:3A\].
![\[fig:3A\] Realizations of length $n=500$ of $(X_t)$ and $(Z_t)$ for a VT-ARMA(1,1) process with a marginal Student t distribution with $\nu=3$ degrees of freedom and ARMA paramaters $\alpha=0.95$ and $\beta =-0.85$. ACF plots for $(X_t)$ and $(|X_t|)$ are also shown.](R_simulation_exampleA.pdf){width="14cm" height="10cm"}
V-transforms {#sec:v-transforms}
============
To generalize the class of v-transforms we admit two forms of asymmetry in the construction described in Section \[sec:simple-symm-model\]: we allow the density $f_X$ to be skewed; we introduce an asymmetric volatility proxy.
Let $T_1$ and $T_2$ be strictly increasing, continuous and differentiable functions on $\R^+=[0,\infty)$ such that $T_1(0) = T_2(0)$. Let $\mu_T \in \R$. Any transformation $T:\R \to \R$ of the form $$\label{eq:9}
T(x) =
\begin{cases}
T_1(\mu_T-x)&\quad x \leq \mu_T\\
T_2(x-\mu_T) &\quad x > \mu_T
\end{cases}$$ is a volatility proxy transformation. The parameter $\mu_T$ is the *change point* of $T$ and the associated function $g_T:\R^+\to\R^+$, $g_T(x) = T_2^{-1} \circ
T_1(x)$ is the *profile function* of $T$.
By introducing $\mu_T$ we allow the possibility that the natural change point may not be identical to zero. By introducing different functions $T_1$ and $T_2$ for returns on either side of the change point, we allow the possibility that one or other may contribute more to the volatility proxy. This has a similar economic motivation to the *leverage* effects in GARCH models [@bib:ding-engle-granger-93]; falls in equity prices increase a firm’s leverage and increase the volatility of the share price.
Clearly the profile function of a volatility proxy transformation is a strictly increasing, continuous and differentiable function on $\R^+$ such that $g_T(x) =
0$. In conjunction with $\mu_T$, the profile contains all the information about $T$ that is relevant for constructing v-transforms. In the case of a volatility proxy transformation that is symmetric about $\mu_T$, the profile satisfies $g_T(x) = x$.
The following result shows how v-transforms $V = \vtrans(U)$ can be obtained by considering different continuous distributions $F_X$ and different volatility proxy transformations $T$ of type (\[eq:9\]).
\[prop:model-with-skewness\] Let $X$ be a random variable with absolutely continuous and strictly increasing cdf $F_X$ on $\R$ and let $T$ be a volatility proxy transformation. Let $U=
F_X(X)$ and $V= F_{T(X)}(T(X))$. Then $V$ and $U$ are related by $$\label{eq:4}
V = \vtrans(U) =
\begin{cases}
F_X\left(\mu_T + g_T\left(\mu_T-F_X^{-1}(U)\right)\right) -
U, &U \leq F_X(\mu_T) \\
U - F_X\left(\mu_T- g_T^{-1}\left( F_X^{-1}(U) -\mu_T \right)\right) ,& U > F_X(\mu_T)\,.
\end{cases}$$
The result implies that any two volatility proxy transformations $T$ and $\tilde{T}$ which have the same change point $\mu_T$ and profile function $g_T$ belong to an equivalence class with respect to the resulting v-transform. This generalizes the idea that $T(x) =|x|$ and $T(x) =x^2$ give the same v-transform in the symmetric case of Section \[sec:simple-symm-model\]. Note also that the volatility proxy transformations $T^{(V)}$ and $T^{(Z)}$ defined by $T^{(V)}(x) =
F_{T(X)}(T(x))$ and $T^{(Z)}(x) = \Phi^{-1}(T^{(V)}(x))$ are in the same equivalence class as $T$ since they share the same change point and profile function as $T$.
\[def:v-transforms-1\] Any transformation $\vtrans$ that can be obtained from equation (\[eq:4\]) by choosing an absolutely continuous and strictly increasing cdf $F_X$ on $\R$ and a volatility proxy transformation $T$ is a v-transform. The value $\downprob = F_X(\mu_T)$ is the *fulcrum* of the v-transform.
A flexible parametric family
----------------------------
In this section we derive a family of v-transforms using construction (\[eq:4\]) by taking a tractable asymmetric model for $F_X$ from the family proposed by @bib:fernandez-steel-98 and by setting $\mu_T=0$ and $g_T(x) = k x^\xi$ for $k>0$ and $\xi>0$. This profile function contains the identity profile $g_T(x) = x$ (corresponding to the symmetric volatility proxy transformation) as a special case, but allows cases where negative or positive returns contribute more to the volatility proxy. The choices we make may at first sight seem rather arbitrary, but the resulting family can in fact assume many of the shapes that are permissable for v-transforms, as we will argue.
Let $f_0$ be a density that is symmetric about the origin and let $\gamma>0$ be a scalar parameter. Fernandez and Steel suggested the model $$\label{eq:10}
\ f_X(x ;\gamma) =
\begin{cases}
\frac{2\gamma}{1+\gamma^2}\; f_0(\gamma x) &\quad x \leq 0 \\
\frac{2\gamma}{1+ \gamma^2}\; f_0\left(\frac{x}{\gamma}\right)&\quad x > 0\,.
\end{cases}$$ This model is often used to obtain skewed normal and skewed Student distributions for use as innovation distributions in econometric models. A model with $\gamma > 1$ is skewed to the right while a model with $\gamma < 1$ is skewed to the left, as might be expected for asset returns. We consider the particular case of a double exponential distribution $f_0(x) = 0.5
\exp(-|x|)$ which leads to particularly tractable expressions.
\[prop:parametric-family\] Let $F_X(x;\gamma)$ be the cdf of the density when $f_0(x) = 0.5
\exp(-|x|)$. Set $\mu_T=0$ and let $g_T(x) = k x^\xi$ for $k,\xi>0$. The v-transform (\[eq:4\]) is given by $$\label{eq:11}
\vtrans_{\downprob,\kappa,\xi}(u) =
\begin{cases}
1-u -
(1-\downprob)\exp\left(-\kappa \left( -\ln\left(\frac{u}{\downprob}\right) \right)^\xi
\right)&\quad u
\leq \downprob,\\
u -
\downprob \exp\left( - \kappa^{-1/\xi} \left( -\ln\left(\frac{1-u}{1-\downprob} \right) \right)^{1/\xi}\right)& \quad u > \downprob,
\end{cases}$$ where $\downprob = F_X(0) =(1+\gamma^2)^{-1} \in (0,1)$ and $\kappa =
k/\gamma^{\xi+1} > 0$.
It is remarkable that is a uniformity-preserving transformation. If we set $\xi=1$ we get the two-parameter v-transform $\vtrans_{\downprob,\kappa}$ and when, in addition, $\kappa=1$ we get $$\label{eq:1}
\vtrans_\downprob(u) =
\begin{cases}
(\downprob-u)/\downprob& \quad u \leq \downprob,\\
(u-\downprob)/(1-\downprob)& \quad u > \downprob
\end{cases}$$ which obviously includes the symmetric model $\vtrans_{0.5}(u) = |2u -1|$. The v-transform $\vtrans_\downprob(u)$ in is a very convenient special case and we refer to it as the *linear* v-transform.
In Figure \[fig:99\] we show the v-transform $\vtrans_{\downprob,\kappa,\xi}$ when $\downprob = 0.55$, $\kappa =
1.4$ and $\xi = 0.65$. We will use this particular v-transform to illustrate further properties of v-transforms and find a characterization.
![\[fig:99\] An asymmetric v-transform from the family defined in (\[eq:11\]). For any v-transform, if $v =\vtrans(u)$ and $u^*$ is the dual of $u$, then the points $(u,0)$, $(u,v)$, $(\udual,0)$ and $(\udual,v)$ form the vertices of a square. For the given fulcrum $\downprob$, a v-transform can never enter the gray shaded area of the plot. ](R_figure_vtransform.pdf){width="12cm" height="12cm"}
Characterizing v-transforms
---------------------------
It is easily verified that any v-transform obtained from (\[eq:4\]) consists of two arms or branches, described by continuous and strictly monotonic functions; the left arm is decreasing and the right arm increasing. See Figure \[fig:99\] for an illustration. At the fulcrum $\downprob$ we have $\vtrans(\downprob) = 0$. Every point $u \in [0,1]\setminus \{\downprob\}$ has a *dual point* $\udual$ on the opposite side of the fulcrum such that $\vtrans(\udual) =\vtrans(u)$. Dual points can be interpreted as the quantile probability levels of the distribution of $X$ that give rise to the same level of volatility.
We collect these properties together in the following lemma and add one further important property that we refer to as the *square property* of a v-transform; this property places constraints on the shape that v-transforms can take and is illustrated in Figure \[fig:99\].
\[cor:vtranstions\] A v-transform is a mapping $\vtrans:[0,1] \to [0,1]$ with the following properties:
1. $\vtrans(0) = \vtrans(1) = 1$;
2. There exists a point $\downprob$ known as the fulcrum such that $0 < \downprob < 1$ and $\vtrans(\downprob) = 0$;
3. $\vtrans$ is continuous;
4. $\vtrans$ is strictly decreasing on $[0,\downprob]$ and strictly increasing on $[\downprob, 1]$;
5. Every point $u \in [0,1]\setminus \{\downprob\}$ has a dual point $u^*$ on the opposite side of the fulcrum satisfying $\vtrans(u) = \vtrans(u^*)$ and $|u^* - u | = \vtrans(u)$ (square property).
It is instructive to see why the square property must hold. Consider Figure \[fig:99\] and fix a point $u \in [0,1]
\setminus \{\downprob\}$ with $\vtrans(u) = v$. Let $U\sim U(0,1)$ and let $V =\vtrans(U)$. The events $\left\{V \leq v\right\}$ and $\left\{ \min(u,\udual) \leq U \leq \max(u,\udual)\right\}
$ are the same and hence the uniformity of $V$ under a v-transform implies that $$\label{eq:3}
v = \P(V \leq v) = \P \left(\min(u,\udual) \leq U \leq
\max(u,\udual) \right) =
|\udual -u| \,.$$
The properties in Lemma \[cor:vtranstions\] could be taken as the basis of an alternative definition of a v-transform. In view of it is clear that any mapping $\vtrans$ that has these properties is a uniformity-preserving transformation. We can characterize the mappings $\vtrans$ that have these properties as follows.
\[theorem:v-characterization\] A mapping $\vtrans: [0,1] \to [0,1]$ has the properties listed in Lemma \[cor:vtranstions\] if and only if it takes the form $$\label{eq:2}
\vtrans(u) =
\begin{cases}
(1-u) - (1-\downprob) \Psi \left( \frac{u}{\downprob} \right) & u \leq
\downprob, \\
u - \downprob \Psi^{-1}\left( \frac{1-u}{1-\downprob} \right) & u > \downprob,
\end{cases}$$ where $\Psi$ is a continuous and strictly increasing distribution function on $[0,1]$.
Our arguments so far show that every v-transform must have the form . It remains to verify that every uniformity-preserving transformation of the form can be obtained from construction (\[eq:4\]) and this is the purpose of the final result of this section. This allows us to view Definition \[def:v-transforms-1\], Lemma \[cor:vtranstions\] and the characterization as three equivalent approaches to the definition of v-transforms.
\[prop:reconstructT\] Let $\vtrans$ be a uniformity-preserving transformation of the form and $F_X$ a continuous distribution function. Then $\vtrans$ can be obtained from construction using any volatility proxy transformation with change point $\mu_T = F_X^{-1}(\delta)$ and profile $$\label{eq:volprofile}
g_T(x) = F_X^{-1}\left( F_X(\mu_T-x) + \vtrans\left( F_X(\mu_T-x) \right)\right) -\mu_T, \quad x \geq 0.$$
Henceforth we can view as the general equation of a v-transform. Distribution functions $\Psi$ on $[0,1]$ can be thought of as *generators* of v-transforms. Comparing with (\[eq:11\]) we see that our parametric family $\vtrans_{\downprob,\kappa,\xi}$ is generated by $\Psi(x) = \exp(-\kappa(-
(\ln x)^\xi))$. This is a 2-parameter distribution whose density can assume many different shapes on the unit interval including increasing, decreasing, unimodal and bathtub-shaped forms. In this respect it is quite similar to the beta distribution which would yield an alternative family of v-transforms. The uniform distribution function $\Psi(x) = x$ gives the family of linear v-transforms $\vtrans_\downprob$.
In applications we construct models starting from the building blocks of a tractable v-transform $\vtrans$ such as and a distribution $F_X$; from these we can always infer an implied profile function $g_T$ using . The alternative approach of starting from $g_T$ and $F_X$ and constructing $\vtrans$ via can lead to v-transforms that are cumbersome and computationally expensive to evaluate. For example, for applications to asset return modelling, we might choose a marginal model from the generalized hyperbolic family [@bib:barndorff-nielsen-78; @bib:barndorff-nielsen-blaesild-81; @bib:eberlein-10]. In this case the inversion of the cumulative distribution function requires numerical integration of the density and numerical root finding, which makes the evaluation of $\vtrans$ in very slow.
V-transforms and copulas
------------------------
If two uniform random variables are linked by the v-transform $V =
\vtrans(U)$ then the joint distribution function of $(U,V)$ is a special kind of copula. In this section we derive the form of the copula, which facilitates the construction of stochastic processes using v-transforms.
To state the main result we use the notation $\vtrans^{-1}$ and $\vtrans^\prime$ for the the inverse function and the gradient function of a v-transform $\vtrans$. Although there is no unique inverse $\vtrans^{-1}(v)$ (except when $v=0$) the fact that the two branches of a v-transform mutually determine each other allows us to define $\vtrans^{-1}(v)$ to be the inverse of the left branch of the v-transform given by $\vtrans^{-1}: [0,1] \to [0,\delta], \;
\vtrans^{-1}(v) = \inf\{u :\vtrans(u) = v\}$. The gradient $\vtrans^\prime(u)$ is defined for all points $u \in [0,1] \setminus \{\downprob\}$ and we adopt the convention that $\vtrans^\prime(\downprob)$ is the left derivative as $u \to \downprob$.
\[prop:copula-vtransform\] Let $V$ and $U$ be random variables related by the v-transform $V=\vtrans(U)$.
1. The joint distribution function of $(U, V)$ is given by the copula $$\label{eq:6}
C(u,v) = \P\left(U \leq u, V \leq v\right) =
\begin{cases}
0 & u < \vtrans^{-1}(v) \\
u - \vtrans^{-1}(v) & \vtrans^{-1}(v) \leq u < \vtrans^{-1}(v)+v \\
v & u \geq \vtrans^{-1}(v)+v\,.
\end{cases}$$
2. Conditional on $V = v$ the distribution of $U$ is given by $$\label{eq:35}
U =
\begin{cases}
\vtrans^{-1}(v) & \text{with probability $\Downprob(v)$ if $v
\neq 0$} \\
\vtrans^{-1}(v)+v & \text{with probability $1-\Downprob(v)$ if $v
\neq 0$} \\
\downprob & \text{if $v=0$}
\end{cases}$$ where $$\label{eq:7}
\Downprob(v) = - \frac{1}{\vtrans^\prime(\vtrans^{-1}(v))} \,.$$
3. $\E\left(\Downprob(V)\right) = \downprob$.
In the case of the symmetric v-transform $\vtrans(u)=|1-2u|$ the copula in takes the form $C(u,v) = \max(\min(u+\frac{v}{2}-\frac{1}{2},v),0)$. We note that this copula is related to a special case of the tent map copula family $C^{\mathcal{T}}_\theta$ in @bib:remillard-13 by $C(u,v) = u - C^{\mathcal{T}}_1(u,v)$.
For the linear v-transform family the conditional probability $\Downprob(v)$ in (\[eq:7\]) satisfies $\Downprob(v) = \downprob$ for $v \neq 0$. This implies that the value of $V$ contains no information about whether $U$ is likely to be below or above the fulcrum; the probability is always the same regardless of $V$. In general this is not the case and the value of $V$ does contain information about whether $U$ is large or small. Part (2) of Theorem \[prop:copula-vtransform\] is the key to stochastically inverting a v-transform in the general case. Based on this result we define the concept of stochastic inversion of a v-transform. We refer to the function $\Downprob$ as the *conditional down probability* of $\mathcal{V}$.
\[def:stochinverse\] Let $\vtrans$ be a v-transform with conditional down probability $\Delta$. The two-place function $\bm{\vtrans}^{-1} : [0,1] \times [0,1] \to [0,1]$ defined by $$\label{eq:1}
\bm{\vtrans}^{-1}(v,w) =
\begin{cases}
\vtrans^{-1}(v) & \text{if $w \leq \Delta(v)$} \\
v + \vtrans^{-1}(v) & \text{if $w > \Delta(v)$.}
\end{cases}$$ is the stochastic inversion function of $\vtrans$.
The following proposition, which generalizes Lemma \[lemma:reconstructU\], allows us to construct general asymmetric processes that generalize the process of Algorithm \[algo1\].
\[prop:resconstructU\] Let $V$ and $W$ be iid $U(0,1)$ variables and let $\vtrans$ be a v-transform with stochastic inversion function $\bm{\vtrans}$. If $U =
\bm{\vtrans}^{-1}(V,W)$, then $\vtrans(U) = V$ and $U \sim U(0,1)$.
In Section \[sec:properties-model\] we apply v-transforms and their stochastic inverses to the terms of time series models. To understand the effect this has on the serial dependencies between random variables, we need to consider multivariate componentwise v-transforms of random vectors with uniform marginal distributions and these can also be represented in terms of copulas. We now give a result which forms the basis for the analysis of serial dependence properties. The first part of the result shows the relationship between copula densities under componentwise v-transforms. The second part shows the relationship under the componentwise stochastic inversion of a v-transform; in this case we assume that the stochastic inversion of each term takes place independently given $\bm{V}$ so that all serial dependence comes from $\bm{V}$.
\[theorem:multivariate-vtransform\] Let $\vtrans$ be a v-transform and let $\bm{U} =
(U_1,\ldots,U_d)^\prime$ and $\bm{V} = (V_1,\ldots,V_d)^\prime$ be vectors of uniform random variables with copula densities $c_{\bm{U}}$ and $c_{\bm{V}}$ respectively.
1. If $\bm{V} = (\vtrans(U_1),\ldots,\vtrans(U_d))^\prime$ then $$\label{eq:16}
c_{\bm{V}}(v_1, \ldots, v_d) = \sum_{j_1=1}^2 \cdots \sum_{j_d=1}^2
c_{\bm{U}}(u_{1j_1}, \ldots,u_{d j_d}) \prod_{i=1}^d
\Delta(v_i)^{\indicator{j_i=1}} \left(1- \Delta(v_i)\right)^{\indicator{j_i = 2}}$$ where $u_{i1} = \vtrans^{-1}(v_i)$ and $u_{i2} = \vtrans^{-1}(v_i) + v_i$ for all $i\in\{1,\ldots,d\}$.
2. If $\bm{U} =
(\bm{\vtrans}^{-1}(V_1 , W_1),\ldots,\bm{\vtrans}^{-1}(V_d, W_d))^\prime$ where $W_1,\ldots,W_d$ are iid uniform random variables that are also independent of $V_1,\ldots,V_d$, then $$\label{eq:15}
c_{\bm{U}}(u_1,\ldots,u_d) = c_{\bm{V}}(\vtrans(u_1),\ldots,\vtrans(u_d)).$$
VT-ARMA copula models {#sec:properties-model}
=====================
In this section we study some properties of the class of time series models obtained by the following algorithm, which generalizes Algorithm \[algo1\]. The models obtained are described as VT-ARMA processes since they are stationary time series constructed using the fundamental building blocks of a v-transform $\vtrans$ and an ARMA process.
\[algo2\]
1. Generate $(Z_t)$ as a causal and invertible Gaussian ARMA process of order $(p,q)$ with mean zero and variance one.
2. Form the volatility PIT process $(V_t)$ where $V_t =\Phi(Z_t)$ for all $t$.
3. Generate iid $U(0,1)$ random variables $(W_t)$.
4. Form the series PIT process $(U_t)$ by taking the stochastic inverses $U_t = \bm{\vtrans}^{-1}(V_t, W_t)$.
5. Form the process $(X_t)$ by setting $X_t = F_X^{-1}(U_t)$ for some continuous cdf $F_X$.
We can add any marginal behaviour in the final step and this allows for an infinitely rich choice. We can, for instance, even impose an infinite-variance or an infinite-mean distribution, such as the Cauchy distribution, and still obtain a strictly stationary process for $(X_t)$. We make the following definitions.
\[def:svpit-process\] Any stochastic process $(X_t)$ that can be generated using Algorithm \[algo2\] by choosing an underlying ARMA process with mean zero and variance one, a v-transform $\vtrans$ and and a continuous distribution function $F_X$ is a VT-ARMA process. The process $(U_t)$ obtained at the penultimate step of the algorithm is a VT-ARMA copula process.
Figure \[fig:3\] gives an example of a simulated process using Algorithm \[algo2\] and the v-transform $\vtrans_{\downprob,\kappa,\xi}$ in with $\kappa=0.9$ and MA parameter $\xi =1.1$. The marginal distribution is a heavy-tailed skewed Student distribution of type with degrees-of-freedom $\nu=3$ and skewness $\gamma=0.8$, which gives rise to more large negative returns than large positive returns. The underlying time series model is an ARMA(1,1) model with AR parameter $\alpha=0.95$ and MA parameter $\beta =-0.85$. See caption of figure for full details of parameters.
In the remainder of this section we concentrate on the properties of VT-ARMA copula processes $(U_t)$ from which related properties of VT-ARMA processes $(X_t)$ may be easily inferred.
Stationary distribution {#sec:uncond-distr}
-----------------------
The VT-ARMA copula process $(U_t)$ of Definition \[def:svpit-process\] is a strictly stationary process since the joint distribution of $(U_{t_1},\ldots,U_{t_k})$ for any set of indices $t_1< \cdots < t_k$ is invariant under time shifts. This property follows easily from the strict stationarity of the underlying ARMA process $(Z_t)$ according to the following result, which uses Theorem \[theorem:multivariate-vtransform\].
\[theorem:uncond-copula\] Let $(U_t)$ follow a VT-ARMA copula process with, v-transform $\vtrans$ and an underlying ARMA($p$,$q$) structure with autocorrelation function $\rho(k)$. The random vector $(U_{t_1},\ldots,U_{t_k})$ for $k \in \N$ has joint density $c^{\text{Ga}}_{P(t_1,\ldots,t_k)}(\vtrans(u_{1}),\ldots,\vtrans(u_{k}))$ where $c^{\text{Ga}}_{P(t_1,\ldots,t_k)}$ denotes the density of the Gaussian copula $C^{\text{Ga}}_{P(t_1,\ldots,t_k)}$ and $P(t_1,\dots,t_k)$ is a correlation matrix with $(i,j)$ element given by $\rho(| t_j - t_i|)$.
An expression for the joint density facilitates the calculation of a number of dependence measures for the bivariate marginal distribution of $(U_t,
U_{t+k})$. In the bivariate case the correlation matrix of the underlying Gaussian copula $C^{\text{Ga}}_{P(t,t+k)}$ contains a single off-diagonal value $\rho(k)$ and we simply write $C^{\text{Ga}}_{\rho(k)}$. The Pearson correlation of $(U_t, U_{t+k})$ is given by $$\begin{aligned}
\label{eq:29}
\rho(U_t, U_{t+k}) &= 12 \int_0^1 \int_0^1 u_1 u_2 c^{\text{Ga}}_{\rho(k)}\left(\vtrans(u_1),
\vtrans(u_2)\right) \rd u_1 \rd u_2 -3 \;.\end{aligned}$$ This value is also the value of the Spearman rank correlation $\rho_S(X_t, X_{t+k})$ for a VT-ARMA process $(X_t)$ with copula process $(U_t)$ (since the Spearman’s rank correlation of a pair of continuous random variables is the Pearson correlation of their copula). The calculation of typically requires numerical integration. However, in the special case of the linear v-transform $\vtrans_\downprob$ in we can get a simpler expression as shown in the following result.
\[prop:ARMA-dependence\] Let $(U_t)$ be a VT-ARMA copula process satisfying the assumptions of Proposition \[theorem:uncond-copula\] with linear v-transform $\vtrans_\downprob$. Let $(Z_t)$ denote the underlying Gaussian ARMA process. Then $$\begin{aligned}
\rho(U_t, U_{t+k}) & = & (2\downprob-1)^2\rho_S(Z_t,Z_{t+k}) = \frac{6 (2\downprob-1)^2
\arcsin\left(\frac{\rho(k)}{2}\right)}{\pi}\;.
\label{eq:31}
\end{aligned}$$
For the symmetric v-transform $\vtrans_{0.5}$, equation obviously yields a correlation of zero so that, in this case, the VT-ARMA copula process $(U_t)$ is a white noise with an autocorrelation function that is zero, except at lag zero. However even a very asymmetric model with $\downprob=0.4$ or $\downprob=0.6$ gives $ \rho(U_t, U_{t+k}) =0.04 \rho_S(Z_t,
Z_{t+k})$ so that serial correlations tend to be very weak.
When we add a marginal distribution, the resulting process $(X_t)$ has a different auto-correlation function to $(U_t)$, but the same rank autocorrelation function. The symmetric model of Section \[sec:simple-symm-model\] is a white noise process. General asymmetric processes $(X_t)$ are not perfect white noise processes but have only weak serial correlation.
Conditional distribution {#sec:cond-distr}
------------------------
To derive the conditional distribution of a VT-ARMA copula process we use the vector notation $\bm{U}_t = (U_1,\ldots,U_t)^\prime$ and $\bm{Z}_t =
(Z_1,\ldots,Z_t)^\prime$ to denote the history of processes up to time point $t$ and $\bm{u}_t$ and $\bm{z}_t$ for realizations. These vectors are related by the componentwise transformation $\bm{Z}_{t}=
\Phi^{-1}(\vtrans(\bm{U}_{t})) $. We assume all processes have time index set given by $t \in \{1,2,\ldots\}$.
\[theorem:cond-density\] For $t > 1$ the conditional density $f_{U_t \mid \bm{U}_{t-1}}(u \mid \bm{u}_{t-1})$ is given by $$f_{U_t \mid \bm{U}_{t-1}}(u \mid \bm{u}_{t-1}) =
\frac{\phi\left(
\frac{\Phi^{-1}\left(\vtrans(u)\right) -\mu_t}{\sigma_\epsilon}
\right)}{\sigma_\epsilon \phi\left(\Phi^{-1}(\vtrans(u))\right)}
\label{eq:22}$$ where $\mu_t= \E(Z_t \mid
\bm{Z}_{t-1} = \Phi^{-1}(\vtrans(\bm{u}_{t-1})) )$ and $\sigma_\epsilon$ is the standard deviation of the innovation process for the ARMA model followed by $(Z_t)$.
When $(Z_t)$ is iid white noise $\mu_t = 0$, $\sigma_\epsilon = 1$ and reduces to the uniform density $f_{U_t \mid \bm{U}_{t-1}}(u \mid \bm{u}_{t-1}) = 1$ as expected. In the case of the first-order Markov AR(1) model $Z_t = \alpha_1 Z_{t-1} + \epsilon_t$ the conditional mean of $Z_t$ is $\mu_t = \alpha_1 \Phi^{-1}\left(\vtrans(u_{t-1})\right)$ and $\sigma_\epsilon^2 = 1-\alpha_1^2$. The conditional density can be easily shown to simplify to $f_{U_t \mid U_{t-1}}(u \mid u_{t-1}) = c_{\alpha_1}^{\text{Ga}}\left(\vtrans\left(u\right),\vtrans\left(u_{t-1}\right)\right)$ where $c_{\alpha_1}^{\text{Ga}}\left(\vtrans\left(u_1\right),\vtrans\left(u_2\right)\right)$ denotes the copula density derived in Proposition \[theorem:uncond-copula\]. In this special case the VT-ARMA model falls within the class of first-order Markov copula models considered by @bib:chen-fan-06b, although the copula is new.
If we add a marginal distribution $F_X$ to the VT-ARMA copula model to obtain a model for $(X_t)$ and use similar notational conventions as above, the resulting VT-ARMA model has conditional density $$\label{eq:501}
f_{X_t \mid \bm{X}_{t-1}}(x \mid \bm{x}_{t-1}) = f_X(x) f_{U_t \mid \bm{U}_{t-1}}(F_X(x) \mid F_X(\bm{x}_{t-1}))$$ with $f_{U_t \mid \bm{U}_{t-1}}$ as in . An interesting property of the VT-ARMA process is that the conditional density can have a pronounced bimodality for values of $\mu_t$ in excess of zero, that is in high volatility situations where the conditional mean of $Z_t$ is higher than the marginal mean value of zero; in low volatility situations the conditional density appears more concentrated around zero. This phenomenon is illustrated in Figure \[fig:3\]. The bimodality in high volatility situations makes sense: in such cases it is likely that the next return will be large in absolute value and relatively less likely that it will be close to zero.
![\[fig:3\] Top left: realization of length $n=500$ of $(X_t)$ for a process with a marginal skewed Student distribution (parameters: $\nu=3$, $\gamma=0.8$, $\mu=0.3$, $\sigma=1$) a v-transform of the form (parameters: $\downprob=0.50$, $\kappa=0.9$, $\xi=1.1$) and an underlying ARMA process ($\alpha=0.95$, $\beta =-0.85$, $\sigma_\epsilon =0.95$). Top right: the underlying ARMA process $(Z_t)$ in gray with the conditional mean $(\mu_t)$ superimposed in black; horizontal lines at $\mu_t = 0.5$ (a high value) and $\mu_t = -0.5$ (a low value). The corresponding conditional densities are shown in the bottom figures with the marginal density as a dashed line.](R_simulation_example.pdf){width="16cm" height="12cm"}
The conditional distribution function of $(X_t)$ is $F_{X_t \mid \bm{X}_{t-1}}(x \mid \bm{x}_{t-1}) = F_{U_t \mid \bm{U}_{t-1}}(F_X(x) \mid F_X(\bm{x}_{t-1}))$ and hence the $\psi$-quantile $x_{\psi,t}$ of $F_{X_t\mid\bm{X}_{t-1}}$ can be obtained by solving $$\label{eq:VaR}
\psi = F_{U_t\mid\bm{U}_{t-1}}(F_X(x_{\psi,t})\mid F_X(\bm{x}_{t-1}))\,.$$ For $\psi < 0.5$ the negative of this value is the conditional $(1-\psi)$-VaR at time $t$. Note that the conditional distribution function $F_{U_t \mid \bm{U}_{t-1}}$ does not have a simple closed form in general, so numerical integration is necessary.
Statistical inference {#sec:estimation-model}
=====================
In the copula approach to dependence modelling, the copula is the object of central interest and marginal distributions are often of secondary importance. A number of different approaches to estimation are found in the literature. Suppose we have a dataset $x_1,\ldots,x_n$ representing realization of variables $X_1,\ldots,X_n$ from the time series process $(X_t)$.
The semi-parametric approach developed by @bib:genest-ghoudi-rivest-95 is very widely used in copula inference and has been applied by @bib:chen-fan-06b to first-order Markov copula models in the time series context. In this approach the marginal distribution $F_X$ is first estimated non-parametrically using the scaled empirical distribution function $F_n^{(X)}$ (see definition in Section \[sec:intro\]) and the data are transformed onto the $(0,1)$ scale This has the effect of creating pseudo-copula data $u_t = \text{rank}(x_t)/(n+1)$ where $\text{rank}(x_t)$ denotes the rank of $x_t$ within the sample. The copula is fitted to the pseudo-copula data by maximum likelihood (ML).
The inference-functions-for-margins (IFM) approach of @bib:joe-15 is also a two-step method although in this case a parametric model $\hat{F}_X$ is estimated in the first step and the copula is fitted to the data $u_t = \hat{F}_X(x_t)$ in the second step. Semi-parametric marginal models that combine the empirical distribution function in the centre of the distribution with tail models suggested by extreme value theory can also be applied [@bib:mcneil-frey-00].
The marginal distribution $F_X$ and the copula process can be estimated jointly by maximum likelihood in a single step, although badly chosen marginal distributions can lead to poor estimates of the copula. We first consider the estimation of the VT-ARMA copula process for a sample of data $u_1,\ldots,u_n$ and then consider joint estimation of copula and marginal distribution as a simple extension.
Maximum likelihood estimation of VT-ARMA copula process {#sec:MLestimation}
-------------------------------------------------------
Let $\bm{\theta}^{(V)}$ and $\bm{\theta}^{(A)}$ denote the parameters of the v-transform and ARMA model respectively. It follows from Theorem \[theorem:multivariate-vtransform\] (part 2) and Proposition \[theorem:uncond-copula\] that the log-likelihood for the sample $u_1,\ldots,u_n$ is simply the log density of the Gaussian copula under componentwise inverse v-transformation. This is given by $$\label{eq:loglik}
\begin{split}
L (\bm{\theta}^{(V)}, \bm{\theta}^{(A)} \mid u_1,\ldots,u_n) &=
L^*(\bm{\theta}^{(A)}\mid \Phi^{-1}(\vtrans_{\bm{\theta}^{(V)}}(u_1)),\ldots,
\Phi^{-1}(\vtrans_{\bm{\theta}^{(V)}}(u_n))) \\
& \hspace{6cm} - \sum_{t=1}^n \ln \phi\left(\Phi^{-1}\left( \vtrans_{\bm{\theta}^{(V)}}(u_t) \right) \right)
\end{split}$$ where the first term $L^*$ is the log-likelihood for an ARMA model with a standard N(0,1) marginal distribution. Both terms in the log-likelihood are relatively straightforward to evaluate.
The evaluation of the ARMA likelihood $L^*(\bm{\theta}^{(A)}
\mid z_1,\ldots,z_n) $ for parameters $\bm{\theta}^{(A)}$ and data $z_1,\ldots,z_n$ can be accomplished using the Kalman filter. However, it is important to note that the assumption that the data $z_1,\ldots,z_n$ are standard normal requires a bespoke implementation of the Kalman filter, since standard software always treats the error variance $\sigma^2_\epsilon$ as a free parameter in the ARMA model. In our case we need to constrain $\sigma^2_\epsilon$ to be a function of the ARMA parameters so that $\var(Z_t) =1$. For example, in the case of an ARMA(1,1) model with AR parameter $\alpha_1$ and MA parameter $\beta_1$, this means that $\sigma_\epsilon^2 = \sigma_\epsilon^{2} (\alpha_1,\beta_1) =
(1-\alpha_1^2)/(1 + 2\alpha_1\beta_1 + \beta_1^2)$. The constraint on $\sigma^2_\epsilon$ must be incorporated into the state-space representation of the ARMA model.
Model validation tests for the VT-ARMA copula can be based on residuals $$\label{eq:residuals}
r_t = z_t - \widehat{\mu}_t,\quad z_t = \Phi^{-1}(\vtrans_{\widehat{\bm{\theta}}^{(V)}}(u_t)))$$ where $z_t$ denotes the implied realization of the normalized volatility proxy variable and where an estimate $\widehat{\mu}_t$ of the conditional mean $\mu_t = \E(Z_t \mid \bm{Z}_{t-1}=\bm{z}_t)$ may be obtained as an output of the Kalman filter. The residuals should behave like an iid sample from a normal distribution. Standardized residuals can also be obtained by dividing by the implied estimate of $\sigma_\epsilon$ and then comparing to standard normal.
Using the estimated model, it is straightforward to implement a likelihood-ratio (LR) test for the presence of stochastic volatility in the data. Under the null hypothesis that $\bm{\theta}^{(A)} =
\bm{0}$ the log-likelihood (\[eq:loglik\]) is identically equal to zero. Thus the size of the maximized log-likelihood $L(\widehat{\bm{\theta}}^{(V)},
\widehat{\bm{\theta}}^{(A)}\,;\, u_1,\ldots,u_n)$ provides a measure of the evidence for the presence of stochastic volatility.
Adding a marginal model
-----------------------
Suppose we have data $x_1,\ldots,x_n$ representing realisations of random variables $X_1,\ldots,X_n$ from a VT-ARMA process $(X_t)$ with marginal distribution function and density $F_X$ and $f_X$ and with parameters $\bm{\theta}^{(M)}$. As noted, we can either estimate the model in two steps following the IFM approach of @bib:joe-15 or estimate all parameters $\bm{\theta}$ jointly. Generally, a two-step estimation is a sensible prelude to joint estimation to make sure that both components of the model are reasonable. For joint estimation the log-likelihood is simply $$\begin{gathered}
\label{eq:likelihood-step1}
L^{\text{full}}(\bm{\theta} \mid x_1,\ldots,x_n) = \sum_{t=1}^n \ln f_X(x_t \,;\,
\bm{\theta}^{(M)}) \\
+
L\left(\bm{\theta}^{(V)},\bm{\theta}^{(A)} \mid F_X(x_1 \,;\,
\bm{\theta}^{(M)}),\ldots, F_X(x_n \,;\,
\bm{\theta}^{(M)}) \right)\end{gathered}$$ where the first term is the log-likelihood for a sample of iid data from the marginal distribution $F_X$ and the second term is .
To validate the fitted marginal model the usual suite of graphical and numerical goodness-of-fit tests for comparing $x_1,\ldots,x_n$ with the model $\widehat{F}_X(x) = F_X(x; \widehat{\bm{\theta}}^{(M)})$ is available, for example QQplots and $\chi$-squared and Kolmogorov-Smirnov goodness-of-fit tests.
When a marginal model is added we can recover the implied form of the volatility proxy transformation using Proposition \[prop:reconstructT\]. If $\widehat{\downprob}$ is the estimated fulcrum parameter of the v-transform then the estimated change point is $ \widehat{\mu}_T = \widehat{F}_X^{-1}(\widehat{\downprob})$ and the implied profile function is $$\begin{aligned}
~\label{eq:proxy-profile}
\widehat{g}_T(x) & = & \widehat{F}_X^{-1}\left( \widehat{F}_X(\widehat{\mu}_T-x) - \vtrans_{\widehat{\bm{\theta}}^{(V)}}\left( \widehat{F}_X(\widehat{\mu}_T-x) \right)\right) - \widehat{\mu}_T\;\;.
\end{aligned}$$ This expression can also be used with a non-parametric estimator of $F_X$ to obtain a non-parametric estimate of $g_T$.
Note that is is possible to force the change point to be zero in a joint estimation of marginal model and copula by imposing the constraint $F_X(0; \bm{\theta}^{(M)}) = \downprob$ on the fulcrum and marginal parameters during the optimization. However, in practice superior fits can often be obtained when these parameters are unconstrained.
Examples {#sec:examples}
--------
To indicate what is possible with the methods of this paper, we analyse two excerpts of $n=1000$ daily log-returns from the S&P index; values are multiplied by 100 to give approximate percentage returns. The first excerpt covers the time period from 3 January 2007 to 21 December 2010 containing the financial crisis of 2007-09; the second covers the less turbulent period from 3 January 2012 to 22 December 2015.
We first apply the method of estimating margins with the scaled empirical distribution function and fitting VT-ARMA copula models to the standardized ranks of the time series observations. The log-likelihood surface (\[eq:loglik\]) can sometimes have local maxima for sample sizes of order $n = 1000$, although this issue diminishes for larger samples. As a first stage in the analysis, we find it useful to plot the profile likelihood function for the key fulcrum parameter $\downprob$. Figure shows the results for the first data excerpt when the copula model consists of the linear v-transform (\[eq:1\]) together with underlying ARMA(1,1) and ARMA(2,1) models.
![\[fig:profile\] Profile likelihood for $\downprob$ plotted at 101 equally spaced values when v-transform is linear and underlying ARMA model is ARMA(2,1) (solid line) or ARMA(1,1) (dashed line; vertical line gives empirical probability of a negative log-return.](R_profile_fulcrum.pdf){width="12cm" height="8cm"}
The picture clearly shows the presence of local maxima as well as the fact that the global maxima are to the right of the empirical probability of a negative return (vertical line) in both cases.
In the second stage we use the global maxima from the profile likelihood analysis to choose starting values and attempt to improve the fit by introducing further parameters to the v-transform. Results for the two excerpts are contained in Tables \[table1\] and \[table2\]. To refer to models we use the mnemonic VTARMA($n$, $p$, $q$) where $(p,q)$ refers to the ARMA model and $n$ indexes the v-transform: 1 is the linear v-transform $\vtrans_\downprob$ in ; 2 and 3 are the 2-parameter and 3-parameter versions of $\vtrans_{\downprob,\kappa,\xi}$ in .
The column marked $L$ gives the value of the maximized log-likelihood. All values are large and positive showing strong evidence of stochastic volatility in all cases, but comparison of the two tables shows that the weight of evidence is much higher for the first excerpt containing the crisis than the second. The model VTARMA(1,1,0) is a first-order Markov model with linear v-transform. The fit of this model is noticeably poorer than the others indicating that Markov models are insufficient to capture stochastic volatility, as would be expected given the persistence of typical volatility clustering.
The column marked SW contains the p-value for a Shapiro-Wilks test of normality applied to the residuals from the VT-ARMA copula model. The null hypothesis is rejected for the first dataset for all of the copula models except for the two Markov models, suggesting imperfections in the fit. It is particularly challenging to model this series with a single stationary process given the dramatic regime shift that took place at the time of the financial crisis; it may be noted that the fitted models are very close to integrated with estimates of $\alpha_1$ just less than one in the ARMA(1,1) models. The null hypothesis of normality is not rejected for the second dataset, which appears better modelled by the VT-ARMA process.
The non-significant results of the Shapiro-Wilks test for the two Markov models applied to the first dataset are potentially misleading. These models do a poor job of explaining the serial dependence in the data and the estimated AR coefficients are small. This has the effect that the estimated conditional mean values $\widehat{\mu}_t$ are small and the residuals $r_t$ in are close to the implied values of the normalized volatility proxy $z_t$, which are normal by design. Although the residuals remain relatively normal, they are strongly serially correlated for these models.
![\[fig:5\] Plots for a VTARMA(1,2,1) model fitted to the S&P return data from 3 January 2007 to 21 December 2010: QQplot of the residuals against normal (upper left); ACF of the residuals (upper right); ACF of the absolute residuals (lower left); estimated conditional mean process $(\mu_t)$ (lower right).](R_SPa_plot.pdf){width="16cm" height="12cm"}
![\[fig:6\] Plots for a VTARMA(1,2,1) model fitted to the S&P return data from 3 January 2012 to 22 December 2015: QQplot of the residuals against normal (upper left); acf of the residuals (upper right); acf of the absolute residuals (lower left); estimated conditional mean process $(\mu_t)$ (lower right).](R_SPb_plot.pdf){width="16cm" height="12cm"}
For both datasets the models based on an ARMA(2,1) generally offer a better fit than those based on ARMA(1,1); the improvement is particularly significant for the first series. We experimented with higher order ARMA processes but this did not lead to further significant improvements. According to the AIC values, the VTARMA(1,2,1) model incorporating the linear v-transform is generally a sufficient model and the non-linear v-transforms add relatively little for these data. Note that the fulcrums are off-centre with $\widehat{\downprob} = 0.579$ for the first series and $\widehat{\downprob} = 0.587$ for the second.
Figures \[fig:5\] and \[fig:6\] provide some more details of the fit of the VTARMA(1,2,1) model to the two series. The pictures in the panels show the QQplot of the residuals against normal, acf plots of the residuals and squared residuals and the estimated conditional mean process $(\widehat{\mu}_t)$, which can be taken as an indicator of high and low volatility. The QQplot clearly shows why the Shapiro-Wilks rejects normality of the residuals for the first excerpt; the plot for the second excerpt is more linear.
The residuals and absolute residuals show very little evidence of serial correlation suggesting that the ARMA filter has been successful in explaining much of the serial dependence structure of the normalized volatility proxy process. The estimated conditional mean process takes its maximum values at the height of the 2008-09 crisis for the first dataset; for the second dataset the highest values occur during August 2015 when concerns about the Chinese economy led to a stock market sell-off.
We now add a marginal distribution to the VT-ARMA copula model and estimate all parameters of the model jointly. We have experimented with a number of marginal distributions all of which can be described by four parameters: a location $\mu$, a scale parameter $\sigma$ a skewness parameter $\gamma$ and a shape parameter $\eta$. In particular we have compared the skewed Student t distribution in the family of @bib:fernandez-steel-98, the asymmetric Student t distribution in the generalized hyperbolic (GH) family and the normal inverse-Gaussian (NIG) distribution in the GH family. Of these the NIG yields the best marginal fit in the majority of cases and we present results for that distribution.
Results are shown in Table \[table3\] for both datasets. We only give results for the ARMA(2,1) model, which we again find to be superior to ARMA(1,1) in analyses that are not presented. This is combined with the linear and 2-parameter v-transforms. The estimates of the parameters of the VT-ARMA copula process change a little when the parametric marginal model is added. The Shapiro-Wilks test for the normality of the residuals in the first model improves and, while still significant at the 5% level, is no longer significant at the 1% level. As before, on the basis of the Akaike values there is no evidence that the 2-parameter transform offers any significant improvement over the linear transform for these two datasets.
Figure \[fig:7\] shows some aspects of the joint fit for the second dataset and the model VTARMA(1,2,1). A QQplot of the data against the fitted NIG distribution suggests that the latter is a reasonable marginal model. Using the implied volatility proxy profile function $g_T$ can be constructed and is found to lie just below the line $y=x$ as shown in the upper-right panel. The change point $\mu_T$ is estimated to be $\widehat{F}_X^{-1}(\widehat{\downprob}) = 0.19$. Interestingly, this value is not zero; the implication is that market volatility is at its lowest when log-returns take modest positive values.
We can also infer an implied volatility proxy transformation $T$, although there is flexibility in the exact member of the equivalence class defined by $g_T$ that we pick. Natural ones to consider are the uniformized volatility proxy transformation $T^{(V)}(x) =
\vtrans_{\widehat{\bm{\theta}}^{(V)}}(F_X( x ;
\widehat{\bm{\theta}}^{(M)})$ and the normalized volatility proxy transformation $T^{(Z)}(x) = \Phi^{-1}(T^{(V)}(x))$.
In the lower-left panel of Figure \[fig:7\] we show the empirical v-transform formed from the data $(X_t,T(X_t))$ for any choice of the implied transformation $T$ together with the fitted parametric v-transform. The empirical v-transform is the plot $(\hat{U}_t,\hat{V}_t)$ where $\hat{U}_t =
F^{(X)}_n(X_t)$ and $\hat{V}_t =
F^{(T(X))}_n(T(X_t))$, as in Figure \[fig:1\]. The empirical v-transform and the fitted parametric v-transform should correspond, as they clearly do.
The lower-right panel of Figure \[fig:7\] shows the standardized volatility proxy transformation $x \mapsto T^{(Z)}(x)$ as a curve. This is superimposed on the points $(X_t,
\Phi^{-1}(\hat{V}_t))$ to show how it corresponds to the underlying data. Using the curve we can compare the effects of, for example, a log-return ($\times$ 100) of -2 and a log-return of 2. For the fitted model these are 1.96 and 1.88 showing that the down movement is associated with higher volatility.
![\[fig:7\] Plots for a VTARMA(1,2,1) model combined with a normal inverse-Gaussian (NIG) marginal distribution fitted to the S&P return data from 3 January 2012 to 22 December 2015: QQplot of the data against fitted NIG model (upper left); estimated volatility proxy profile function $g_T$ (upper right); estimated v-transform (lower left); implied relationship between data and volatility proxy variable (lower right).](R_jointplots2.pdf){width="16cm" height="12cm"}
The final application we consider is estimation of a conditional value-at-risk (VaR) using equation . Figure \[fig:8\] shows the 95% conditional VaR estimate for the first time period based on the VTARMA(1,2,1) model. For comparison a dashed line shows the corresponding estimate for a GARCH(1,1) model with skewed Student t innovations. There is clearly a good deal of correspondence between the two estimates indicating that VT-ARMA models give VaR estimates that are broadly in line with standard methods.
![\[fig:8\] Plot of estimated 95% value-at-risk (VaR) for S&P index for the time period from 3 January 2007 to 21 December 2010 superimposed on log returns. Solid line shows VaR estimated using the VTARMA(1,2,1) model combined with an NIG marginal distribution; the dashed line shows VaR estimated using a GARCH(1,1) model with skewed t innovation distribution.](R_VaR_plot.pdf){width="16cm" height="12cm"}
Conclusion {#sec:conclusion}
==========
We have shown how v-transforms may be used to model volatile financial time series, such as asset returns. V-transforms describe the relationships between quantiles of the return distribution and quantiles of the distribution of a predictable volatility proxy variable. The volatility proxy variable is a function of the return which measures the magnitude of movement with respect to some central change point and which may take different forms according to whether returns lie below or above the change point. We have characterized v-transforms mathematically and shown how the stochastic inverse of a v-transform may be used to construct stationary models for return series where arbitrary marginal distributions may be coupled with arbitrary dynamic models for the serial dependence in the volatility proxy.
The construction was illustrated using the serial dependence model implied by a Gaussian ARMA process. The resulting class of VT-ARMA processes is able to capture serial dependence features of financial return series including near-zero serial correlation (white noise behaviour) and volatility clustering. Moreover, since the models are copula-based, they can match any marginal behaviour, including infinite-variance and infinite-mean behaviour, and therefore capture the very heavy tails that are typical of some return series.
The VT-ARMA models are relatively straightforward to estimate building on the classical maximum-likelihood estimation of an ARMA model using the Kalman filter. This can be accomplished in the stepwise manner that is typical in copula modelling or through joint modelling of marginal and copula process. The resulting models yield insights into the way that volatility responds to returns of different magnitude and sign and can give estimates of unconditional and conditional quantiles (VaR) for practical risk measurement purposes.
There are many possible uses for VT-ARMA copula processes. Because we have complete control over the marginal distribution they are very natural candidates for the innovation distribution in other time series model. For example, they could be applied to the innovations of an ARMA model to obtain ARMA models with VT-ARMA errors; this might be particularly appropriate for longer interval returns, such as weekly or monthly returns, where some serial dependence is likely to be present in the raw return data. To extend the class of VT copula processes and improve their fit to empirical data we need to look beyond the Gaussian ARMA process. Changing the choice of v-transform family has very little impact on the models since v-transforms are relatively constrained in the forms they may take. In unreported analyses we verified that changing from our 3-parameter family $\vtrans_{\downprob,\kappa,\xi}$ to a 3-parameter family based on the beta distribution had negligible effect on our conclusions.
Moving away from Gaussian ARMA processes could have a much larger effect. The radial symmetry of the underlying Gaussian copula means that the serial dependence between large values of the volatility proxy must mirror the serial dependence between small values. Moreover this copula does not admit tail dependence in either tail and it seems plausible that very large values of the volatility proxy might have a tendency to occur in particularly rapid succession. To extend the class of models based on v-transforms we should look for models for the volatility PIT process $(V_t)$ with higher dimensional marginal distributions given by asymmetric copulas with upper tail dependence. First-order Markov copula models as developed in @bib:chen-fan-06b can give asymmetry and tail dependence, but they cannot model the dependencies at longer lags that we find in empirical data. Higher-order Markov copula models may be more successful. Note that making simple distributional changes to ARMA processes such as using heavy-tailed, asymmetric innovations does not provide a simple solution, because we then lose our knowledge of the exact stationary distribution of the resulting ARMA process which is an essential part of the model construction. Further applications of v-transforms in time series modelling is a topic for future research.
Acknowledgements {#acknowledgements .unnumbered}
================
The author is grateful for valuable input from a number of researchers including Hansjoerg Albrecher, Martin Bladt, Valérie Chavez-Demoulin, Alexandra Dias, Christian Genest, Michael Gordy, Yen Hsiao Lok, Johanna Nešlehová and Ruodu Wang.
Proofs {#sec:proofs}
======
Proof of Proposition \[prop:model-with-skewness\] {#sec:proof-model-with-skewness}
-------------------------------------------------
We observe that for $x \geq 0$ $$F_{T(X)}(x) = \P(\mu_T-T_1^{-1}(x) \leq X_t \leq \mu_T + T_2^{-1}(x)) = F_X(\mu_T+ T_2^{-1}(x)) -
F_X(\mu_T-T_1^{-1}(x)).$$ $\{ X_t \leq \mu_T\} \iff \{U \leq F_X(\mu_T)\}$ and in this case $$\begin{aligned}
V = F_{T(X)}(T(X_t)) = F_{T(X)}(T_1(\mu_T-X_t)) &= F_X(\mu_T + T_2^{-1}(T_1(\mu_T-X_t))) -
F_X(X_t) \\
&= F_X\left(\mu_T + g_T\left(\mu_T-F_X^{-1}(U)\right)\right) - U.\end{aligned}$$ $ \{ X_t > \mu_T\} \iff \{U > F_X(\mu_T)\}$ and in this case $$\begin{aligned}
V = F_{T(X)}(T(X_t)) = F_{T(X)}(T_2(X_t -\mu_T)) &= F_X(X_t) -
F_X(\mu_T-T_1^{-1}(T_2( X_t -\mu_T) ) ) \\
&= U - F_X\left(\mu_T- g_T^{-1}\left( F_X^{-1}(U) -\mu_T\right) \right).\end{aligned}$$
Proof of Proposition \[prop:parametric-family\] {#prop:proof-parametric-family}
-----------------------------------------------
The cumulative distribution function $F_0(x)$ of the double exponential distribution is equal to $0.5e^x$ for $x \leq 0$ and $1 - 0.5e^{-x}$ if $x>0$. It is straightforward to verify that $$F_X(x;\gamma) =
\begin{cases}
\downprob e^{\gamma x} & x\leq 0 \\
1 - (1-\downprob)e^{-\frac{x}{\gamma}} & x > 0
\end{cases}
\quad\text{and}\quad
F_X^{-1}(u;\gamma) =
\begin{cases}
\frac{1}{\gamma} \ln\left(\frac{u}{\downprob}\right) & u\leq \downprob \\
-\gamma \ln \left( \frac{1-u}{1-\downprob}\right) & u > \downprob\,.
\end{cases}$$ When $g_T(x) = k x^\xi$ we obtain for $u \leq \downprob$ that $$\begin{aligned}
\vtrans_{\downprob,\kappa,\xi}(u) = F_X\left(\frac{k}{\gamma^\xi} \left(
\ln \left(\frac{
\downprob}{u} \right)^\xi\right) ;\gamma \right) - u
&= 1-u -
(1-\downprob)\exp\left(- \frac{k}{\gamma^{\xi+1}}\left(-\ln\left(\frac{u}{\downprob}\right)\right)^\xi
\right)\;.
$$ For $u > \downprob$ we make a similar calculation.
Proof of Theorem \[theorem:v-characterization\] {#proof:v-characterization}
-----------------------------------------------
It is easy to check that equation fulfills the list of properties in Lemma \[cor:vtranstions\]. We concentrate on showing that a function that has these properties must be of the form . It helps to consider the picture of a v-transform in Figure \[fig:99\]. Consider the lines $v = 1 - u$ and $v = \downprob - u$ for $u \in
[0,\downprob]$. The areas above the former and below the latter are shaded gray.
The left branch of the v-transform must start at $(0,1)$, end at $(\downprob, 0)$ and lie strictly between these lines in $(0,\downprob)$. Suppose, to the contrary, that $v =\vtrans(u) \leq \downprob - u$ for $u \in
(0,\downprob)$. This would imply that the dual point $u^*$ given by $u^* = u +v$ satisfies $u^* \leq \downprob$ which contradicts the requirement that $u^*$ must be on the opposite side of the fulcrum. Similarly, if $v =\vtrans(u) \geq 1 - u$ for $u \in
(0,\downprob)$ then $u^* \geq 1$ and this is also not possible; if $u^*=1$ then $u=0$ which is a contradiction.
Thus the curve that links $(0,1)$ and $(\downprob, 0)$ must take the form $$\vtrans(u) =
(\downprob -u) \Psi\left(\frac{u}{\downprob}\right) + (1-u) \left(1 -
\Psi\left(\frac{u}{\downprob}\right) \right) =
(1-u) - (1-\downprob) \Psi \left( \frac{u}{\downprob} \right)$$ where $\Psi(0) =0$, $\Psi(1)=1$ and $0 < \Psi(x) < 1$ for $x \in
(0,1)$. Clearly $\Psi$ must be continuous to satisfy the conditions of the v-transform. It must also be strictly increasing. If it were not then the derivative would satisfy $\vtrans^\prime(u) \geq -1$ which is not possible: if at any point $u \in
(0,\downprob)$ we have $\vtrans^\prime (u) = -1$ then the opposite branch of the v-transform would have to jump vertically at the dual point $u^*$, contradicting continuity; if $\vtrans^\prime (u) > -1$ then $\vtrans$ would have to be a decreasing function at $u^*$, which is also a contradiction.
Thus $\Psi$ fulfills the conditions of a continuous, strictly increasing distribution function on $[0,1]$ and we have established the necessary form for the left branch equation. To find the value of the right branch equation at $u >
\downprob$ we invoke the square property. Since $\vtrans(u) =
\vtrans(\udual) = \vtrans(u - \vtrans(u))$ we need to solve the equation $x = \vtrans(u-x)$ for $x\in [0,1]$ using the formula for the left branch equation of $\vtrans$. Thus we solve $x = 1- u + x - (1-\downprob) \Psi(\tfrac{u-x}{\downprob})$ for $x$ and this yields the right branch equation as asserted.
Proof of Proposition \[prop:reconstructT\] {#propr:proof-reconstructT}
------------------------------------------
Let $g_T(x)$ be as given in and let $u(x) = F_X(\mu_T-x)$. For $x \in \R^+$, $u(x)$ is a continuous, strictly decreasing function of $x$ starting at $u(0) = \downprob$ and decreasing to $0$. Since $\Psi$ is a cumulative distribution function, it follows that $$\udual(x) = u(x) + \vtrans \left(u(x)\right) = 1 - (1-\downprob)\Psi\left(\frac{u(x)}{\downprob}\right)$$ is a continuous, strictly increasing function starting at $\udual(0) =
\downprob$ and increasing to $1$. Hence $g_T(x) = F_X^{-1}(\udual(x)) - \mu_T$ is continuous and strictly increasing on $\R^+$ with $g_T(0) = 0$ as required of the profile function of a volatility proxy transformation. It remains to check that if we insert in we recover $\vtrans(u)$, which is straightforward.
Proof of Theorem \[prop:copula-vtransform\] {#prop:proof-copula-vtransform}
-------------------------------------------
1. For any $0 \leq v \leq 1$ the event $\{U \leq u, V \leq v\}$ has zero probability for $u < \vtrans^{-1}(v)$. For $u \geq \vtrans^{-1}(v)$ we have $$\{U \leq u, V \leq v\} = \{\vtrans^{-1}(v) \leq U \leq \min(u,\vtrans^{-1}(v)+v)\}$$ and hence $\P\left(U \leq u, V \leq v\right) =
\min(u,\vtrans^{-1}(v)+v) - \vtrans^{-1}(v) $ and follows.
2. We can write $\P\left( U \leq u, V \leq v\right ) =
C(u,v)$ where $C$ is the copula given by . It follows from the basic properties of a copula that $$\P\left( U \leq u, V = v\right ) = \frac{\rd}{\rd v}
C(u,v) =
\begin{cases}
0 & u < \vtrans^{-1}(v) \\
- \frac{\rd}{\rd v} \vtrans^{-1}(v) & \vtrans^{-1}(v) \leq u < \vtrans^{-1}(v)+v \\
1 & u \geq \vtrans^{-1}(v)+v
\end{cases}$$ This is the distribution function of a binomial distribution and it must be the case that $ \Downprob(v) = - \frac{\rd}{\rd v}
\vtrans^{-1}(v)$. Equation follows by differentiating the inverse.
3. Finally, $\E\left(\Downprob(V)\right) = \downprob$ is easily verified by making the substitution $x =
\vtrans^{-1}(v)$ in the integral $ \E\left(\Downprob(V)\right)
= - \int_0^1
\frac{1}{\vtrans^\prime(\vtrans^{-1}(v))}
\rd v $.
Proof of Proposition \[prop:resconstructU\] {#propr:proof-resconstructU}
-------------------------------------------
It is obviously true that $\vtrans(\bm{\vtrans}^{-1}(v,W)) = v$ for any $W$. Hence $\vtrans(U) = \vtrans(\bm{\vtrans}^{-1}(V,W)) = V$. The uniformity of $U$ follows from the fact that $$\P\left(\bm{\vtrans}^{-1}(V,W) = \vtrans^{-1}(v) \mid V = v \right) =
\P\left(W \leq \Delta(v) \mid V = v \right) = \P(W \leq \Delta(v) )= \Delta(v)\;.$$ Hence the pair of random variables $(U,V)$ has the conditional distribution (\[eq:35\]) and is distributed according to the copula $C$ in (\[eq:6\]).
Proof of Theorem \[theorem:multivariate-vtransform\] {#theorem:proof-multivariate-vtransform}
----------------------------------------------------
1. Since the event $\{V_i \leq v_i\}$ is equal to the event $\{\vtrans^{-1}(v_i) \leq U_i \leq \vtrans^{-1}(v_i) + v_i \}$ we first compute the probability of a box $[a_1,b_1] \times \cdots \times [a_d, b_d]$ where $a_i = \vtrans^{-1}(v_i) \leq \vtrans^{-1}(v_i) + v_i =
b_i$. The standard formula for such probabilities implies that the copulas $C_{\bm{V}}$ and $C_{\bm{U}}$ are related by $$C_{\bm{V}}(v_1, \ldots, v_d) = \sum_{j_1=1}^2 \cdots \sum_{j_d=1}^2 (-1)^{j_1+ \cdots + j_d} C_{\bm{U}}(u_{1j_1}, \ldots,u_{d j_d})\;;$$ see, for example, [@bib:mcneil-frey-embrechts-15], page 221. Thus the copula densities are related by $$c_{\bm{V}}(v_1, \ldots, v_d) = \sum_{j_1=1}^2 \cdots \sum_{j_d=1}^2
c_{\bm{U}}(u_{1j_1}, \ldots,u_{d j_d})
\prod_{i=1}^d \frac{\rd }{\rd v_i} (-1)^{j_i} u_{i j_i}$$ and the result follows if we use (\[eq:7\]) to calculate that $$\frac{\rd}{\rd v_i} (-1)^j u_{i j} =
\begin{cases}
\frac{\rd}{\rd v_i}
\left(- \vtrans^{-1}(v_i) \right)
= \Downprob(v_i) & \text{if $j=1$,}\\
\frac{\rd}{\rd v_i}
\left(v_i + \vtrans^{-1}(v_i)\right)
= 1-\Downprob(v_i) & \text{if $j=2$.}
\end{cases}$$
2. For the point $(u_1,\ldots,u_d) \in [0,1]^d$ we consider the set of events $A_i(u_i)$ defined by $$A_i(u_i) =
\begin{cases}
\left\{ U_i \leq u_i\right\}& \text{if $u_i \leq \downprob$} \\
\left\{ U_i > u_i\right\}& \text{if $u_i > \downprob$}
\end{cases}$$ The probability $\P(A_1(u_1),\ldots,A_d(u_d))$ is the probability of an orthant defined by the point $(u_1,\ldots,u_d)$ and the copula density at this point is given by $$c_{\bm{U}}(u_1,\ldots,u_d) = (-1)^{\sum_{i=1}^d \indicator{u_i > \downprob}}\frac{\rd^d}{\rd u_1 \cdots \rd u_d} \P\left(\bigcap_{i=1}^d A_i(u_i)\right)\;\;.$$ The event $A_i(u_i)$ can be written $$A_i(u_i) =
\begin{cases}
\left\{ V_i \geq \vtrans(u_i), W_i \leq \Downprob(V_i) \right\}& \text{if $u_i \leq \downprob$} \\
\left\{ V_i > \vtrans(u_i), W_i > \Downprob(V_i) \right\}& \text{if $u_i > \downprob$}
\end{cases}$$ and hence we can use Theorem \[prop:copula-vtransform\] to write $$\P\left( \bigcap_{i=1}^d A_i(u_i) \right) = \int_{\vtrans(u_1)}^1 \cdots \int_{\vtrans(u_d)}^1 c_{\bm{V}}(v_1,\ldots,v_d) \prod_{i=1}^d \Downprob(v_i)^{\indicator{u_i \leq \downprob}}(1-\Downprob(v_i))^{\indicator{u_i > \downprob}}
\rd v_1\cdots \rd v_d\;.$$ The derivative is given by $$\frac{\rd^d}{\rd u_1 \cdots \rd u_d} \\P\left( \bigcap_{i=1}^d A_i(u_i) \right) = (-1)^d c_{\bm{V}}(\vtrans(u_1),\ldots,\vtrans(u_d)) \prod_{i=1}^d p(u_i)^{\indicator{u_i \leq \downprob}}(1-p(u_i))^{\indicator{u_i > \downprob}} \vtrans^\prime(u_i)$$ where $p(u_i) = \Downprob(\vtrans(u_i))$ and hence we obtain $$c_{\bm{U}}(u_1,\ldots,u_d) = c_{\bm{V}}(\vtrans(u_1),\ldots,\vtrans(u_d)) \prod_{i=1}^d (-p(u_i))^{\indicator{u_i \leq \downprob}}(1-p(u_i))^{\indicator{u_i > \downprob}} \vtrans^\prime(u_i).$$
It remains to verify that each of the terms in the product is identically equal to 1. For $u_i \leq \downprob$ this follows easily from since $-p(u_i) = -\Downprob(\vtrans(u_i)) = 1/\vtrans^\prime(u_i)$. For $u_i >\downprob$ we need an expression for the derivative of the right branch equation. Since $\vtrans(u_i) = \vtrans(u_i - \vtrans(u_i))$ we obtain $$\vtrans^\prime(u_i) = \vtrans^\prime(u_i - \vtrans(u_i))(1 - \vtrans^\prime(u_i)) = \vtrans^\prime(\udual_i)(1 - \vtrans^\prime(u_i))
\Longrightarrow \vtrans^\prime(u_i) = \frac{\vtrans^\prime(\udual_i)}{1+\vtrans^\prime(\udual_i)}$$ implying that $$1-p(u_i ) = 1-\Downprob(\vtrans(u_i)) = 1-\Downprob(\vtrans(\udual_i)) = 1 + \frac{1}{\vtrans^\prime(\udual_i)} = \frac{1+\vtrans^\prime(\udual_i)}{\vtrans^\prime(\udual_i)} = \frac{1}{\vtrans^\prime(u_i)}\;.$$
Proof of Proposition \[theorem:uncond-copula\] {#theorem:proof-uncond-copula}
----------------------------------------------
Let $V_t = \vtrans(U_t)$ and $Z_t = \Phi^{-1}(V_t)$ as usual. The process $(Z_t)$ is an ARMA process with acf $\rho(k)$ and hence $(Z_{t_1},\ldots,Z_{t_k})$ are jointly standard normally distributed with correlation matrix $P(t_1,\ldots,t_k)$. This implies that the joint distribution function of $(V_{t_1},\ldots,V_{t_k})$ is the Gaussian copula with density $c^{\text{Ga}}_{P(t_1,\ldots,t_k)}$ and hence by Part 2 of Theorem \[theorem:multivariate-vtransform\] the joint distribution function of $(U_{t_1},\ldots,U_{t_k})$ is the copula with density $c^{\text{Ga}}_{P(t_1,\ldots,t_k)}(\vtrans(u_{1}),\ldots,\vtrans(u_{k}))$.
Proof of Proposition \[prop:ARMA-dependence\] {#prop:proof-ARMA-dependence}
---------------------------------------------
We split the integral in (\[eq:29\]) into four parts. First observe that by making the substitutions $v_1 = \vtrans(u_1) = 1-u_1/\downprob$ and $v_2
= \vtrans(u_2) = 1-u_2/\downprob$ on $[0,\downprob] \times [0,\downprob]$ we get $$\begin{aligned}
\int_0^\downprob \int_0^\downprob u_1 u_2 c^{\text{Ga}}_{\rho(k)} \left(\vtrans(u_1),
\vtrans(u_2)\right) \rd u_1 \rd u_2
&= \downprob^4\int_0^1 \int_0^1 (1-v_1)(1-v_2)
c^{\text{Ga}}_{\rho(k)}\left(v_1,v_2
\right) \rd v_1 \rd v_2 \\
&= \downprob^4 \E( (1-V_{t})(1-V_{t+k}) ) \\
&= \downprob^4 \left( 1 - \E(V_t) - \E(V_{t+k}) + \E(V_{t}V_{t+k}) \right) = \downprob^4 \E(V_{t}V_{t+k})\end{aligned}$$ where $(V_t,V_{t+k})$ is a random pair with joint distribution given by the Gaussian copula $C^{\text{Ga}}_{\rho(k)}$. Similarly by making the substitutions $v_1 = \vtrans(u_1) = 1-u_1/\downprob$ and $v_2
=\vtrans(u_2)=(u_2-\downprob)/(1-\downprob)$ on $[0,\downprob] \times [\downprob,1]$ we get $$\begin{aligned}
\lefteqn{\int_0^\downprob \int_\downprob^1 u_1 u_2 c^{\text{Ga}}_{\rho(k)}\left(\vtrans(u_1),
\vtrans(u_2)\right) \rd u_1 \rd u_2} \\ &= \int_0^1 \int_0^1
\downprob^2(1-\downprob)(1-v_1)
\Big(\downprob + (1-\downprob)v_2 \Big)
c^{\text{Ga}}_{\rho(k)}\left(v_1,v_2
\right) \rd v_1 \rd v_2 \\
&= \downprob^3 (1-\downprob)\E(1-V_t) + \downprob^2
(1-\downprob)^2\E\left((1-V_t)V_{t+k}\right)\\
&= \frac{\downprob^2(1-\downprob)}{2} - \downprob^2(1-\downprob)^2 \E(V_t V_{t+k})
\end{aligned}$$ and the same value is obtained on the quadrant $[\downprob,1] \times
[0,\downprob]$. Finally making the substitutions $v_1 = \vtrans(u_1) = (u_1-\downprob)/(1-\downprob)$ and $v_2
=\vtrans(u_2)=(u_2-\downprob)/(1-\downprob)$ on $[\downprob,1] \times [\downprob,1]$ we get $$\begin{aligned}
\lefteqn{\int_\downprob^1 \int_\downprob^1 u_1 u_2 c^{\text{Ga}}_{\rho(k)}\left(\vtrans(u_1),
\vtrans(u_2)\right) \rd u_1 \rd u_2}\\
&= \int_0^1 \int_0^1 (1-\downprob)^2
\Big(\downprob + (1-\downprob)v_1 \Big) \Big(\downprob + (1-\downprob)v_2 \Big)
c^{\text{Ga}}_{\rho(k)}\left(v_1,v_2
\right) \rd v_1 \rd v_2
\\
&= \int_0^1 \int_0^1
(1-\downprob)^2
\Big( \downprob^2 + \downprob(1-\downprob) v_1 + \downprob(1-\downprob) v_2 + (1-\downprob)^2
v_1 v_2\Big)
c^{\text{Ga}}_{\rho(k)}\left(v_1,v_2
\right) \rd v_1 \rd v_2 \\
&=\downprob^2(1-\downprob)^2 + \downprob (1-\downprob)^3 \E(V_t) + \downprob(1-\downprob)^3
\E(V_{t+k}) + (1-\downprob)^4 \E(V_t V_{t+k}) \\
&= \downprob(1-\downprob)^2 + (1-\downprob)^4 \E(V_t V_{t+k})
\end{aligned}$$ Collecting all of these terms together yields $$\begin{aligned}
\int_0^1 \int_0^1 u_1 u_2 c^{\text{Ga}}_{\rho(k)}\left(\vtrans(u_1),
\vtrans(u_2)\right) \rd u_1 \rd u_2
&= \downprob(1-\downprob) + (2\downprob-1)^2 \E(V_t
V_{t+k})\end{aligned}$$ and since $\rho_S(Z_t ,Z_{t+k}) = 12 \E(V_t V_{t+k}) -3$ it follows that $$\begin{aligned}
\rho(U_t, U_{t+k}) =12\E(U_t U_{t+k}) -3&= 12 \int_0^1 \int_0^1 u_1 u_2 c^{\text{Ga}}_{\rho(k)}\left(\vtrans(u_1),
\vtrans(u_2)\right) \rd u_1 \rd u_2 -3 \\
&= 12 \downprob(1-\downprob) + 12 (2\downprob-1)^2 \E(V_t
V_{t+k}) -3 \\ &=
12 \downprob(1-\downprob) + (2\downprob-1)^2 \left( \rho_S(Z_t ,Z_{t+k}) +3
\right) -3 \\
&= (2\downprob-1)^2 \rho_S(Z_t ,Z_{t+k})\,.\end{aligned}$$ The value of Spearman’s rho $ \rho_S(Z_t ,Z_{t+k})$ for the bivariate Gaussian distribution is well known; see for example [@bib:mcneil-frey-embrechts-15].
Proof of Proposition \[theorem:cond-density\] {#theorem:proof-cond-density}
---------------------------------------------
The conditional density satisfies $$\begin{aligned}
f_{U_t \mid \bm{U}_{t-1}}(u \mid \bm{u}_{t-1}) & = &
\frac{c_{\bm{U}_t}(u_1,\ldots,u_{t-1},u)}{c_{\bm{U}_{t-1}}(u_1,\ldots,u_{t-1})}
=
\frac{ c^{\text{Ga}}_{P(1,\ldots,t)}(\vtrans(u_1),\ldots,\vtrans(u_{t-1}),\vtrans(u))}{c^{\text{Ga}}_{P(1,\ldots,t-1)}(\vtrans(u_1),\ldots,\vtrans(u_{t-1}))}\;.\end{aligned}$$ The Gaussian copula density is given in general by $$c^{\text{Ga}}_P(v_1,\ldots,v_d) =
\frac{f_{\bm{Z}}\big(\Phi^{-1}(v_1),\ldots,\Phi^{-1}(v_d)\big)}{\prod_{i=1}^d \phi\big(\Phi^{-1}(v_i)\big)}$$ where $\bm{Z}$ is a multivariate Gaussian vector with standard normal margins and correlation matrix $P$. Hence it follows that we can write $$\begin{aligned}
f_{U_t \mid \bm{U}_{t-1}}(u \mid \bm{u}_{t-1}) & = &
\frac{f_{\bm{Z}_t}\Big( \Phi^{-1}\big(\vtrans(u_1)\big), \ldots,
\Phi^{-1}\big(\vtrans(u_{t-1})\big), \Phi^{-1}\big(\vtrans(u)\big)
\Big)}{
f_{\bm{Z}_{t-1}}\Big( \Phi^{-1}\big(\vtrans(u_1)\big), \ldots,
\Phi^{-1}\big(\vtrans(u_{t-1})\big)
\Big)
\phi\big(
\Phi^{-1}\big(\vtrans(u)\big)
\big)} \\
& = & \frac{f_{Z_t
\mid \bm{Z}_{t-1}}\Big( \Phi^{-1}\big(\vtrans(u)\big) \mid \Phi^{-1}\big(\vtrans(\bm{u}_{t-1})\big) \Big)}{ \phi\big(\Phi^{-1}\big(\vtrans(u)\big)\big) }\end{aligned}$$ where $f_{Z_t
\mid \bm{Z}_{t-1}}$ is the conditional density of the ARMA process, from which follows easily.
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[^1]: Address correspondence to Alexander J. McNeil, The York Management School, University of York, Freboys Lane, York YO10 5GD, UK, +44 (0) 1904 325307, `[email protected]`.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We consider the problem of characterising the spatial extent of a composite light source using the superresolution imaging technique when the centroid of the source is not known precisely. We show that the essential features of this problem can be mapped onto a simple qubit model for joint estimation of a phase shift and a dephasing strength.'
address:
- |
Faculty of Physics, University of Warsaw,\
ul. Pasteura 5, 02-093 Warszawa, Poland
- |
Centre of New Technologies, University of Warsaw\
ul. Banacha 2c, 02-097 Warszawa, Poland\
$^\[email protected]
author:
- 'ANDRZEJ CHROSTOWSKI and RAFA[Ł]{} DEMKOWICZ-DOBRZAŃSKI'
- 'MARCIN JARZYNA and KONRAD BANASZEK$^\ast$'
title: ON SUPERRESOLUTION IMAGING AS A MULTIPARAMETER ESTIMATION PROBLEM
---
\#1[[\#1:]{}]{}
Introduction
============
The promise of superresolution imaging is to determine characteristics of a spatially extended light source with precision better than that defined by the diffraction limit [@TsangNairPRX2016; @NairTsangPRL2016]. The diffraction limit is a consequence of a finite aperture of the optical instrument used for observation combined with the conventional measurement of the spatial distribution of light intensity in the image plane.[@BornWolfPrinciplesofOptics] At the fundamental level, an intensity measurement consists of registering a finite number of counts generated by incident photons. Generation of each photocount is an inherently random process that needs to be described in statistical terms.[@MandelWolfOpticalCoherence] The randomness of locations where individual photons are registered effectively masks features of a composite light source whose extent is below the diffraction limit. The reconstruction of such features by conventional means would require collection of an immense amount of data to suppress the effects of statistical uncertainty.[@BobroffRSI1986]
As proposed by Tsang and Nair[@TsangNairPRX2016; @NairTsangPRL2016] and demonstrated in a series of proof-of-principle experiments,[@PaurStoklasaOPTICA2016; @YangTashilinaOPTICA2016] the above difficulty can be overcome by detecting the incident photons after separating them in a carefully selected basis of spatial modes. This technique, called mode demultiplexing, requires additional [*a priori*]{} knowledge of the centroid of the light source. The most straightforward approach to deal with this issue is to perform a standard spatially resolved measurement on a fraction of available photons and to estimate the centroid as the average position.[@TsangNairPRX2016] In this contribution we consider the problem of simultaneous determination of the centroid and the spatial extent of a composite light source. We show that when these two parameters are well below the diffraction limit, the problem can be modelled with the help of an elementary qubit system in which the parameters of interest correspond respectively to a rotation and a contraction of the Bloch vector. This observation links supperresolution imaging to quantum multiparameter estimation which has been addressed in several recent works.[@PerezDelgadoPearcePRL2012; @VidrighinDonatiNCOMM2014; @RagyJarzynaPRA2016; @PearceCampbellQuantum2017]
This paper is organised as follows. In Sec. \[Sec:Demultiplexing\] we review the spatial mode demultiplexing technique for the determination of the spatial extent of a composite light source when its characteristic size is well below the diffraction limit. The measurement error is discussed in Sec. \[Sec:Error\]. The qubit model for the underlying estimation problem is presented in Sec. \[Sec:Qubitmodel\]. Finally, Sec. \[Sec:Conclusions\] concludes the paper.
Spatial mode demultiplexing {#Sec:Demultiplexing}
===========================
Consider an ensemble of mutually incoherent point sources labelled with an index $j$ characterised by relative strengths $w_j$, $\sum_j w_j = 1$. For simplicity we will discuss image formation using one spatial dimension. We will also assume that the probability of detecting more than one photon in a given observation time interval is negligibly small and hence we will think of image formation as a series of repeated single photon detection events. In the image plane each source generates a coherent field distribution described by an amplitude transfer function $u(x-x_j)$ with its centre located at $x_j$. We will assume that the transfer function $u(x)$ is real and even, i.e. $u(x) = u^\ast(x) = u(-x)$, and furthermore that its square is normalized to one, $\int_{-\infty}^{\infty} {\textrm{d}}x \,[u(x)]^2 = 1$. The objective is to determine the spatial extent of the sources in a scenario when they are spread over a range much smaller than the characteristic width of the transfer function, corresponding to the diffraction limit. In the standard spatial intensity distribution measurement, the probability density for detecting a photon at a given point $x$ is given by a weighted sum $$p(x) = \sum_j w_j [u(x-x_j)]^2.
\label{Eq:p(x)}$$ In the limit of small spreads, this distribution can hardly be distinguished from the one corresponding to a single point source located at the centroid of the source $x_C = \sum_{j}w_j x_j$, in which case it would read $p(x)=[u(x-x_C)]^2$. This is illustrated in Fig. \[Fig:Principle\](a) with the example of two equally weighted point sources separated by a distance $2d$ much smaller than the width $\sigma$ of the transfer function, assumed to have the Gaussian form $$u(x) = \frac{1}{\sqrt[4]{2\pi\sigma^2}} \exp\left( -\frac{x^2}{4\sigma^2} \right).
\label{Eq:u(x)Gaussian}$$ This difficulty leads to an intuitive expectation—the Rayleigh’s criterion—that estimating the separation between the sources in this regime is subject to large uncertainty. Using methods of parameter estimation theory that provide general quantitative bounds on how precisely a parameter can be estimated within a given probabilistic model, this intuition can be formulated in a rigorous way confirming that in the case of a direct measurement of the spatial intensity distribution, the precision indeed deteriorates significantly in the small separation regime irrespectively of what inference strategy one pursues. [@Bettens1999; @TsangNairPRX2016; @Tsang2016]. In particular, for the given example of two sources separated by distance $2d$ with the Gaussian transfer function (\[Eq:u(x)Gaussian\]), the precision behaves as $\Delta d \approx \frac{\sigma^2}{d}\sqrt{\frac{2}{N}}$, where $N$ is the number of registered photons. This formula clearly shows divergence in the limit of small separations $d\to 0$.
![(a) Spatial intensity distribution produced by a pair of incoherent point sources separated by $2d=0.01 $ with centroid located at $x_C = 0.025 $ with respect to the reference position $x_R=0$. Each individual source produces a Gaussian distribution characterised by standard deviation $\sigma=1$. The problem of characterising the extended source can be described with two dimensionless parameters $\varepsilon = d/\sigma$ and $\theta = (x_C- x_R)/\sigma$. (b) The amplitude transfer function $u(x)$ and its normalised derivative $v(x)$ used in the spatial mode demultiplexing technique.[]{data-label="Fig:Principle"}](fig1)
The basic idea of superresolution imaging based on mode demultiplexing is to measure the intensity of incoming radiation in a basis of spatial modes that provides a signal more sensitive to the extent of the ensemble. In the simplest model valid for small extents, it is sufficient to consider a mode function proportional to the derivative of the transfer function $$v(x) = - 2 \sigma \frac{{\textrm{d}} u}{{\textrm{d}} x}.
\label{Eq:vdef}$$ The multiplicative factor warrants the normalisation of $v(x)$. To satisfy this condition for a general transfer function that is not necessarily Gaussian, the parameter $\sigma$ should be taken as $$\sigma = \frac{1}{2} \left( \int_{-\infty}^{\infty} {\textrm{d}} x \left( \frac{{\textrm{d}} u}{{\textrm{d}} x} \right)^2\right)^{-1/2}.$$ For a real transfer function $u(x)$ the functions $u(x)$ and $v(x)$ are mutually orthogonal, i.e.$\int_{-\infty}^{\infty} {\textrm{d}} x \, v(x) u(x)=0$. An exemplary set of mode functions $u(x)$ and $v(x)$, assuming a Gaussian shape for the former given explicitly by Eq. (\[Eq:u(x)Gaussian\]) is shown in Fig. \[Fig:Principle\](b). The functions are analogous to the ground state and the first excited state of a quantum mechanical harmonic oscillator in the position representation. The modes $u(x)$ and $v(x)$ can be separated using integrated optics structures[@TsangNairPRX2016] or a spatial light modulator.[@PaurStoklasaOPTICA2016]
Suppose now that instead of conventional spatially resolved detection, the incoming light is demultiplexed in the basis of spatial modes $u(x-x_R)$ and $v(x-x_R)$ centred at a reference point $x_R$ and that the intensity of individual components is measured. The probability that an incoming photon is detected in the mode $v(x)$ is given by $$I(x_R) = \sum_j w_j \left( \int_{-\infty}^{\infty} {\textrm{d}} x \, v(x-x_R) u(x-x_j) \right)^2 \approx \frac{1}{(2\sigma)^2} \sum_j w_j (x_j - x_R)^2,
\label{Eq:Intensityv}$$ where in the second step we have expanded $u(x-x_j) \approx u(x-x_R) + \frac{1}{2\sigma} (x_j-x_R) v(x-x_R)$ up to the linear term and the orthogonality of the mode functions $u(x)$ and $v(x)$ has been used. Note that $I(x_R)$ is the fraction of the source intensity directed to the mode $v(x-x_R)$. Intrinsic properties of the composite source can be determined if the above measurement is performed at the centroid of the system $x_C$, given by the weighted sum $x_C = \sum_j w_j x_j$, in which case $$I(x_C) = I_C = \frac{1}{(2\sigma)^2} \sum_{j} w_j d_j^2,
\label{Eq:IC}$$ where $d_j = x_j - x_C$ denote relative distances of individual sources from the system centroid and satisfy $\sum_{j} w_j d_j = 0$. The right hand side of the above formula has a simple interpretation as the second moment for the distribution of point sources with respect to the centroid of the system, expressed in units $(2\sigma)^2$.
Measurement error {#Sec:Error}
=================
As a concrete example, suppose that the system comprises two point sources of equal brightness separated by the distance $2d$, which implies that $I_C = d^2/(2\sigma)^2$. Let us consider a situation where there are $N$ incoming photons in total. If the centroid is known perfectly, they are detected in the mode basis $u(x-x_C)$, $v(x-x_C)$ with respective probabilities $p_{u} \approx 1-I_C$, $p_{v} \approx I_C$. A general result on the asymptotic efficiency of maximum likelihood estimation [@kay1993fundamentals] implies that in the limit of large $N$ the optimal precision of estimating $d$ is given by $$\Delta d = \frac{1}{\sqrt{N F}},\quad F = \sum_{i=u,v} \frac{1}{p_i} \left(\frac{\partial p_i}{\partial d}\right)^2,$$ where $F$ is the Fisher information. In our case $F = [\sigma^{2}(1-\frac{d^2}{4\sigma^2})]^{-1}$, and hence the parameter $d$ can be estimated with uncertainty equal to:[@TsangNairPRX2016] $$\label{eq:separation}
\Delta d= \frac{\sigma}{\sqrt{N}}\left(1-\frac{d^2}{4\sigma^2}\right)^{1/2}.$$ This provides a huge advantage over the spatial intensity distribution measurement and allows to circumvent the Rayleigh criterion since the estimation uncertainty does not diverge even when taking the limit ${d}/{\sigma} \rightarrow 0$. Note that the above derivation is valid only in the regime $d/\sigma \ll 1$.
What happens if one does not know exactly the location of the centroid? If the measurement is performed with respect to a general position $x_R$ a straightforward calculation using the approximate expression derived in Eq. (\[Eq:Intensityv\]) yields the probability of detection equal to $$I(x_R) = I_C + \frac{1}{(2\sigma)^2} (x_R - x_C)^2.
\label{Eq:Steiner}$$ Note that this formula is analogous to Steiner’s theorem for the mass moment of inertia. The second term in Eq. (\[Eq:Steiner\]), originating from the imperfect knowledge of the centroid, constitutes a systematic error. In the standard approach, the location $x_R$ is determined via spatially resolved detection of a certain number photons characterized by the position distribution given in Eq. (\[Eq:p(x)\]). The variance of this distribution is $$\label{eq:var}
\text{Var}(x) = \int_{-\infty}^{\infty} {\textrm{d}} x \, (x-x_C)^2 p(x) = \sum_j w_j d_j^2 + \int_{-\infty}^{\infty} {\textrm{d}}x \, x^2 [u(x)]^2 . $$ In the regime when the spatial extent of the source is well below the diffraction limit, the second term in the above expression, equal to the second moment of the squared transfer function, dominates the variance. Therefore for the Gaussian model considered here we have $\text{Var}(x) \approx \sigma^2$. By sacrificing $n$ photons to perform estimation of the centroid we may estimate its position with precision $\sigma/\sqrt{n}$. For $n$ large enough, more precisely $n^2 \gg I_C$, we may therefore make the second term in Eq. (\[Eq:Steiner\]) arbitrary small compared to the first one and estimate the parameter $d$ as before. More formally, having the total of $N$ photons at our disposal, we can sacrifice $n \propto N^{\alpha}$, $0 < \alpha < 1$ to estimate the centroid, and keep the rest $N - N^{\alpha}$ for separation estimation. In the asymptotic limit of $N \rightarrow \infty$, since $N - N^\alpha \approx N$, we should recover precision scaling as given in . For any finite $N$ one needs to resort to numerical means to find the optimal partition of the photons used to measure the centroid and the extent of the source.
An elementary method to deal with the problem of biasedness in determining $I_C$ would be to perform measurements at several locations $x_R$ in the vicinity of the centroid and then to fit a parabolic curve described by the right hand side of Eq. (\[Eq:Steiner\]) with $I_C$ and $x_C$ taken as free parameters. An exemplary numerical simulation of this procedure with the same parameters as those used in Fig. \[Fig:Principle\](a) is presented in Fig. \[Fig:Simulation\](a). In order to evaluate the accuracy of the procedure, we have performed $20000$ repetitions of the numerical simulation, which yielded the value $I_C = 0.6197(1) \times 10^{-3}$ as the average result. This is lower than $0.625 \times 10^{-3}$ calculated using Eq. (\[Eq:IC\]), but for a fair comparison one should use the exact integral expression given in Eq. (\[Eq:Intensityv\]), which yields the figure $0.6246 \times 10^{-3}$ that remains higher than $I_C$ obtained from numerical simulations. Thus the quadratic fit method seems to produce an estimate that on average is slightly biased below the actual value of $I_C$ and in particular would underestimate the separation parameter $d$ for a pair of point sources.
![(a) Simulation of the measurement of the intensity $I(x_R)$ in the mode $v(x-x_R)$ with respect to a reference point $x_R$ assuming $N=10^5$ photons received from the source (crosses with vertical error bars) and the fitted parabolic function with the obtained values of free parameters $I_C$ and $x_C$. Actual parameters of the source are the same as those used in Fig. \[Fig:Principle\]. (b) A histogram of the values of $I_C$ obtained from 20000 repetitions of a simulation presented in the panel (a). The solid line is a Gaussian fit with the mean $0.6197(1) \times 10^{-3}$.[]{data-label="Fig:Simulation"}](fig2)
Qubit model {#Sec:Qubitmodel}
===========
The essential features of the problem discussed in the preceding section can be distilled by considering a two-dimensional subspace spanned by the mode functions $u(x-x_R)$ and $v(x-x_R)$ that effectively defines a qubit. In order to keep the formulas concise, we will now switch to Dirac notation and use the following kets $$u(x-x_R) \equiv {{|0\rangle_{}}} , \qquad v(x-x_R) \equiv {{|1\rangle_{}}}.$$ The field from a source at location $x_j$ can be now written as: $$u(x-x_j) \approx u(x-x_R) + \frac{1}{2\sigma} (x_j-x_R) v(x-x_R) \equiv {{|0\rangle_{}}} + \frac{1}{2\sigma} (x_j-x_R) {{|1\rangle_{}}}.
\label{Eq:u(x-x_j)qubit}$$ Because the sources are mutually incoherent, the radiation needs to be described by the first-order coherence function[@MandelWolfOpticalCoherence] which in the Dirac notation takes the form of an operator: $$\begin{gathered}
\sum_j w_j u(x-x_j)u^\ast (x'-x_j) \\ \cong {{{|0\rangle_{}}}_{}{{}_{}\langle 0|}} + \frac{1}{2\sigma}(x_C-x_R) \bigl( {{|1\rangle_{}}}{{}_{}\langle 0|} + {{|0\rangle_{}}} {{}_{}\langle 1|} \bigr)
+ \frac{1}{(2\sigma)^2} \left( (x_C-x_R)^2 + \sum_j w_j d_j^2 \right){{{|1\rangle_{}}}_{}{{}_{}\langle 1|}}\end{gathered}$$ It will be convenient to introduce two dimensionless parameters that express relevant lengths in the units of $\sigma$: the location of the centroid $\theta = (x_C-x_R)/\sigma$ and the effective radius of the source $\varepsilon = \left( \sum_j w_j (d_j/\sigma)^2 \right)^{1/2}$. The normalised density matrix written in the basis ${{|0\rangle_{}}}, {{|1\rangle_{}}}$ takes the form $$\hat{\varrho} = \frac{1}{1 + \frac{\varepsilon^2}{4} + \frac{\theta^2}{4}} \begin{pmatrix} 1 & \frac{\theta}{2} \\ \frac{\theta}{2} & \frac{\varepsilon^2}{4} + \frac{\theta^2}{4} \end{pmatrix} .$$ The role of the parameters $\varepsilon$ and $\theta$ can be most easily understood by considering the Bloch representation of the qubit state, $\hat{\varrho} = \frac{1}{2}\left({\mathbbm 1}+ \sum_{i=1}^3 s_i \hat{\sigma}_i \right)$, where $\hat{\sigma}$ denote Pauli matrices. The three components of the Bloch vector are given by $$\begin{aligned}
s_1 & = \frac{\theta}{1 + \frac{\varepsilon^2}{4} + \frac{\theta^2}{4}} \approx \left(1-\frac{\varepsilon^2}{2}\right) \sin \theta\nonumber \\
s_2 & = 0 \\
s_3 & = \frac{1 - \frac{\varepsilon^2}{4} - \frac{\theta^2}{4}}{1 + \frac{\varepsilon^2}{4} + \frac{\theta^2}{4}} \approx \left(1-\frac{\varepsilon^2}{2}\right) \cos \theta . \nonumber\end{aligned}$$ The second approximate expressions are correct up to the quadratic order in $\varepsilon$ and $\theta$. They offer a simple interpretation of our problem illustrated with Fig. \[Fig:QubitModel\]. In the great circle located in the plane $s_1, s_3$ the parameter $\theta$ corresponds to a rotation of the Bloch sphere about the axis $s_2$, whereas $\varepsilon$ is responsible for the contraction of the Bloch vector. This signals a connection of the studied problem with joint estimation of a phase shift and a dephasing strength.[@VidrighinDonatiNCOMM2014]
![Bloch sphere representation of the qubit model for superresolution imaging at an uncertain location. The phase shift $\theta$ corresponds to the location of the source centroid with respect to the reference system and the reduction of the Bloch vector length by $\varepsilon^2/2$ can be related to the spatial extent od the source.[]{data-label="Fig:QubitModel"}](fig3)
Using the introduced model, we may now address the problem of estimating the parameters $\varepsilon$ and $\theta$ without specifying a priori any particular measurement. Instead, we will calculate the quantum Fisher information (QFI) [@helstrom1976quantum] on the state $\hat{\varrho}$, which for a single parameter estimation problem equals the value of Fisher information corresponding to the most informative measurement performed on the state. Specifically, the QFI for the estimation of a single parameter $\varepsilon$ can be calculated using the following formula: $$F_Q = {\textrm{Tr}}(\hat{\varrho} \hat{L}_{\varepsilon}^2),$$ where $\hat{L}_{\varepsilon}$, called the symmetric logarithmic derivative, is given implicitly by $\partial_{\varepsilon} \hat{\varrho} = (\hat{L}_{\varepsilon} \hat{\varrho} + \hat{\varrho} \hat{L}_{\varepsilon})/2$. As a result, the fundamental precision bound on estimating parameter $\varepsilon$, irrespectively of what measurement was performed, reads $\Delta \varepsilon \geq 1/\sqrt{N F_Q}$, where $N$ is the number of repetitions of the experiment. In case of multiparameter estimation, one needs to define the QFI matrix, $(F_Q)_{\mu\nu} = {\textrm{Tr}}(\hat{\varrho} \hat{L}_\mu \hat{L}_\nu)$, where $\hat{L}_\mu$ is the symmetric logarithmic derivative corresponding to the parameter $\mu$. The inverse of the QFI matrix multiplied by the number of repetitions $(NF_Q)^{-1}$ provides a lower bound on the covariance matrix of the estimated parameters. The QFI matrix corresponding to the two-parameter estimation problem reads: $$F_Q = \begin{pmatrix}
(F_Q)_{\varepsilon\varepsilon} & (F_Q)_{\varepsilon\theta}\\
(F_Q)_{\theta\varepsilon} & (F_Q)_{\theta\theta} \end{pmatrix}
=
\begin{pmatrix}
1 + \frac{\varepsilon^2}{4} & 0 \\
0 & 1 - \varepsilon^2
\end{pmatrix},$$ where the elements are specified up to the second order in the parameters $\varepsilon$ and $\theta$. As a result, we obtain the following bounds on the estimation precision in the leading order of $\varepsilon$: $$\Delta \varepsilon \geq \frac{1}{\sqrt{N}}\left( 1- \frac{\varepsilon^2}{4} \right)^{1/2}, \qquad \Delta \theta \geq \frac{(1 + \varepsilon^2)^{1/2}}{\sqrt{N}}.$$ Recalling the relation between $\varepsilon, \theta$ and the dimensional parameters $d, x_C-x_R, \sigma$, we see that the bound on $\Delta \varepsilon$ corresponds exactly to the uncertainty formula for $d$ given in , while the bound $\Delta \theta$ corresponds exactly to the estimation precision of the centroid as discussed below equation , when the first term in equal to $\sigma^2 \varepsilon^2$ is not neglected. This shows that the measurements considered in Sec. \[Sec:Error\] are indeed optimal for the determination of one of the parameters $\varepsilon$ or $\theta$. Unfortunately, these measurements are not compatible and cannot be performed jointly. This can also be seen explicitly in the qubit model considered here. In general, the optimal measurements that maximize Fisher information with respect to a given parameter are projection measurements in the eigenbasis of the respective symmetric logarithmic derivative operator. At the operating point $\varepsilon = \theta = 0$ the corresponding eigenbases for $\hat{L}_{\varepsilon}$ and $\hat{L}_\theta$ are ${{|0\rangle_{}}},{{|1\rangle_{}}}$ and ${{|\pm\rangle_{}}} = ({{|0\rangle_{}}} \pm {{|1\rangle_{}}})/\sqrt{2}$ respectively. It is seen that these two measurements do not commute with each other. Hence, if one insists on using the optimal projective measurements, they need to be performed separately on different subsets of the input ensemble as discussed in Sec. \[Sec:Error\]. It is worth mentioning that while the symmetric logarithmic derivatives do not commute, their commutator yields zero when traced over the state $\hat{\varrho}$. This implies that the simultaneous measurement reaching the optimal values of precision is possible provided collective measurements on many probes are performed [@RagyJarzynaPRA2016].
Conclusions {#Sec:Conclusions}
===========
We have presented an elementary discussion of the superresolution imaging technique based on spatial mode demultiplexing. In the regime when the spatial extent of a composite light source is much less than the diffraction limit, the demultiplexing method enables one to determine the second moment of the distribution of the constituent point sources. Intrinsic properties of the source are given by this second moment with respect to the centroid. If the exact position of the centroid is not known, it is necessary to adopt a multiparameter estimation approach. We have described a simple quantum mechanical model in which the spatial extent and the centroid of the source are analogues of the phase shift and the dephasing strength of a qubit. Interestingly, optimal projective measurements for estimating individually these parameters are mutually incompatible. The qubit model can provide insights in other imaging-related scenarios, e.g. involving hypothesis testing.[@arXiv:1609.00684]
Acknowledgments {#acknowledgments .unnumbered}
===============
We acknowledge insightful disscussions with Saikat Guha and Wojciech Wasilewski. This work is part of the project “Quantum Optical Communication Systems” carried out within the TEAM programme of the Foundation for Polish Science cofinanced by the European Union under the European Regional Development Fund. R.D.-D. acknowledges support of National Science Center (Poland) grant No. 2016/22/E/ST2/00559.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In this work we study an extension of the commonly used 5F scheme, where $b$ quarks are treated as massless partons, in which full mass effects are retained in both the initial and in the final state. We name this scheme 5F massive scheme (5FMS). We implement this scheme in the [S]{}Monte Carlo event generator at [MP@N]{}accuracy, and we compare it for two relevant cases for the [L]{}: $b\bar{b} \rightarrow H$ and $pp\rightarrow Zb$.'
author:
- |
Davide Napoletano$^{1,2}$\
[$^1$IPhT, CEA Saclay, CNRS UMR 3681, F-91191, Gif-Sur-Yvette, France]{}\
[$^2$Institute for Particle Physics Phenomenology, Durham University, Durham DH1 3LE, UK]{}\
title: '$b$ quark mass effects in associated production[^1]'
---
Introduction
============
Processes with heavy quarks in the initial state, in particular associated production processes, have seen in recent years a renewed interest [@Krauss:2016orf; @Napoletano:2017czh; @Forte:2015hba; @Forte:2016sja; @Maltoni:2012pa; @Lim:2016wjo; @Bonvini:2015pxa; @Bonvini:2016fgf; @Bertone:2017djs]. From the theoretical point of view, they are interesting applications of multiscale processes with largely different scales. Ratio of these large scales, can give rise to large logarithms which might spoil the convergence of the perturbative series. To avoid this, one can consider the $b$ as a massless parton, and construct a $b$-PDF which resums this potentially large collinear logarithms, at the price of neglecting mass effects. An alternative point of view can be that of treating the $b$-quark as a massive, decoupled particle, which is only produced in the final state, or treating the $b$-quark as a massless parton on the same footing as the other, thus contributing to the QCD evolution. In this way one is able to retain full mass effects at the price of keeping the aforementioned possibly large collinear logs.
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![In the plot is shown the error that is made when taking and $\alpha_s$ and a gluon PDF in the 4FS with respect to the 5FS baseline. As it can be seen the two effects partially mitigate each other, although this is true only for processes that start at a low enough power of $\alpha_S$, and have a large gluon contribution.[]{data-label="Fig:alphapdf"}](alpha.pdf "fig:"){width="50.00000%"} ![In the plot is shown the error that is made when taking and $\alpha_s$ and a gluon PDF in the 4FS with respect to the 5FS baseline. As it can be seen the two effects partially mitigate each other, although this is true only for processes that start at a low enough power of $\alpha_S$, and have a large gluon contribution.[]{data-label="Fig:alphapdf"}](pdfs.pdf "fig:"){width="50.00000%"}
--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
The former of these two approaches is called [*five-flavour*]{} (5F) scheme and would schematically corresponds to the right hand side plot of Fig.\[Fig:4Fvs5F\], while the latter is refered to as [*four-flavour*]{} (4F) scheme and is represented in the left plot of Fig.\[Fig:4Fvs5F\]. These two approaches have generally been used in a complementary, with the old way of saying being:
> “ [*use the 4FS for exclusive observables,\
> and the 5FS for inclusive observables*]{} "
Many studies have however now shown that the 5FS scheme performs generally better both when compared to data, [@Krauss:2016orf], or when comparing it with a matched calculation [@Forte:2016sja; @Bonvini:2016fgf], although this too is only true up to a certain extent. There are, in fact regions of phase space where one might still want to include exact mass effects, which would in principle require the use of the 4FS.
In this work we investigate the possibility of using a scheme, built upon the 5FS, with exact mass dependence. We name this scheme five-flavour-massive-scheme (5FMS). We implement the necessary ingredients to perform calculations in this scheme in the [S]{}Monte Carlo event generator [@Gleisberg:2008ta], at [M@N]{}accuracy [@Gehrmann:2012yg; @Hoeche:2012yf]. A detailed description of this scheme and its implementation can be found in [@Krauss:2017wmx].
Including mass effects
======================
Fixed order
-----------
In order to study the effects introduced by this new scheme, we take an explicit example: $b\bar{b} \rightarrow H$. Reference diagrams that contribute to the [*next-to-leading*]{} order are shown in Fig.\[Fig:bvr\]. At the level of partonic matrix elements, the only difference between the 5FS and the 5FMS is that in the latter full mass dependence is retained, including in the initial state. As the infrared divergent structure is modified by the presence of the $b$ mass, that acts as a collinear regulator, a modification of the standard Catani-Seymour subtraction is required [@Krauss:2017wmx]. With this in place, we can generate [*fixed-order*]{} events, Fig.\[Fig:5F5FMS\_fo\]. As an example observable, we focus on the $p_T$ of the produced $H$ boson.
We know that mass effects contribute only a few percent to the total cross section for this process. In addition, we know that they are power suppressed and we expect them to scale like $m_b^2/p_T^2$. This is, indeed, roughly the behaviour shown in Fig.\[Fig:5F5FMS\_fo\].
[M@N]{}
-------
We now want to study what happens when this scheme is matched to the parton shower. Since we don’t have a theoretical reference here, we use $pp\rightarrow Z b$ data [@Aaboud:2017xsd] from [[A]{}]{}. In particular we replicate the set-up used in [@Krauss:2016orf], and we compare with the 5FS [MP@N]{}line referenced therein, see Fig.\[fig:mcatnlo\]. The difference with respect to that set-up is that we have [M@N]{}accuracy only for the core $pp\rightarrow Z$ processes, while extra jet contributions that are merged on top of that only come at leading order accuracy. Strictly speaking thus, we should compare the 5FMS [MP@N]{}here with the 5F [MP@L]{}prediction of [@Krauss:2016orf], however we expect some mass effects to make up for some of the differences in accuracy.
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![We show prediction obtained in the 5FS, massless, at [MP@N]{}accuracy, with up to 2 jets at NLO plus up to three jets at leading order. The 5FMS prediction on the other hand includes only the 0 jet contribution at NLO, while the 1,2 and 3 jets contributions are merged with LO accuracy.[]{data-label="fig:mcatnlo"}](d03-x01-y01.pdf "fig:"){width="40.00000%"} ![We show prediction obtained in the 5FS, massless, at [MP@N]{}accuracy, with up to 2 jets at NLO plus up to three jets at leading order. The 5FMS prediction on the other hand includes only the 0 jet contribution at NLO, while the 1,2 and 3 jets contributions are merged with LO accuracy.[]{data-label="fig:mcatnlo"}](d05-x01-y01.pdf "fig:"){width="40.00000%"}
![We show prediction obtained in the 5FS, massless, at [MP@N]{}accuracy, with up to 2 jets at NLO plus up to three jets at leading order. The 5FMS prediction on the other hand includes only the 0 jet contribution at NLO, while the 1,2 and 3 jets contributions are merged with LO accuracy.[]{data-label="fig:mcatnlo"}](d15-x01-y01.pdf "fig:"){width="40.00000%"} ![We show prediction obtained in the 5FS, massless, at [MP@N]{}accuracy, with up to 2 jets at NLO plus up to three jets at leading order. The 5FMS prediction on the other hand includes only the 0 jet contribution at NLO, while the 1,2 and 3 jets contributions are merged with LO accuracy.[]{data-label="fig:mcatnlo"}](d17-x01-y01.pdf "fig:"){width="40.00000%"}
-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
As our aim is to investigate mass effects, in $b$-initiated processes, we look at events in which at least one jet containing a $b$ is tagged, and we plot distributions for the leading $b$-jet and the $Z$ boson $p_T$ and $y$ against data. These plots are reported in Fig.\[fig:mcatnlo\]. As it can be seen, this new scheme performs rather well, and, indeed, it shows the same type of compatibility with data of the 5FS [MP@N]{}prediction, which is reassuring.
Further details and studies on this new scheme can be found in [@Krauss:2017wmx]
Acknowledgements {#acknowledgements .unnumbered}
================
We want to thank our colleagues from the [S]{}collaboration for fruitful discussions and technical support. We acknowledge financial support from the EU research networks funded by the Research Executive Agency (REA) of the European Union under Grant Agreements PITN-GA2012-316704 (“HiggsTools”) and PITN-GA-2012-315877 (“MCnetITN”), by the ERC Advanced Grant MC@NNLO (340983), and from BMBF under contracts 05H12MG5 and 05H15MGCAA,
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S. H[ö]{}che, F. Krauss, M. Sch[ö]{}nherr and F. Siegert, *[A critical appraisal of NLO+PS matching methods]{}*, JHEP **09** (2012), [049](http://inspirehep.net/record/944643), \[[[arXiv:1111.1220]{}](http://arXiv.org/pdf/1111.1220) \[hep-ph\]\] S. H[ö]{}che, F. Krauss, S. Schumann and F. Siegert, *[QCD matrix elements and truncated showers]{}*, JHEP **05** (2009), [053](http://www.slac.stanford.edu/spires/find/hep/www?eprint=arXiv:0903.1219), \[[[arXiv:0903.1219]{}](http://arXiv.org/pdf/0903.1219) \[hep-ph\]\] T. Gehrmann, S. H[ö]{}che, F. Krauss, M. Sch[ö]{}nherr and F. Siegert, *[NLO QCD matrix elements + parton showers in $e^+e^-\to$ hadrons]{}*, [[arXiv:1207.5031]{}](http://arXiv.org/pdf/1207.5031) \[hep-ph\] S. H[ö]{}che, F. Krauss, M. Sch[ö]{}nherr and F. Siegert, *[QCD matrix elements + parton showers: The NLO case]{}*, JHEP **1304** (2013), [027](http://www.slac.stanford.edu/spires/find/hep/www?eprint=1207.5030), \[[[arXiv:1207.5030]{}](http://arXiv.org/pdf/1207.5030) \[hep-ph\]\] G. Aad et al., The ATLAS Collaboration collaboration, *[Measurement of differential production cross-sections for a $Z$ boson in association with $b$-jets in 7 TeV proton-proton collisions with the ATLAS detector]{}*, JHEP **1410** (2014), [141](http://www.slac.stanford.edu/spires/find/hep/www?eprint=1407.3643), \[[[arXiv:1407.3643]{}](http://arXiv.org/pdf/1407.3643) \[hep-ex\]\] S. Chatrchyan et al., The CMS Collaboration collaboration, *[Measurement of the cross section and angular correlations for associated production of a Z boson with b hadrons in pp collisions at $\sqrt{s} =$ 7 TeV]{}*, JHEP **1312** (2013), [039](http://www.slac.stanford.edu/spires/find/hep/www?eprint=1310.1349), \[[[arXiv:1310.1349]{}](http://arXiv.org/pdf/1310.1349) \[hep-ex\]\]
[^1]: Presented at the HiggsTools Final Meeting, Durham
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The purpose of this benchmark is to evaluate the planning and control aspects of robotic in-hand manipulation systems. The goal is to assess the system’s ability to change the pose of a hand-held object by either using the fingers, environment or a combination of both. Given an object surface mesh from the YCB data-set, we provide examples of initial and goal states (i.e. static object poses and fingertip locations) for various in-hand manipulation tasks. We further propose metrics that measure the error in reaching the goal state from a specific initial state, which, when aggregated across all tasks, also serves as a measure of the system’s in-hand manipulation capability. We provide supporting software, task examples, and evaluation results associated with the benchmark.'
author:
- |
Silvia Cruciani\*$^1$, Balakumar Sundaralingam\*$^2$, Kaiyu Hang$^3$, Vikash Kumar$^4$,\
Tucker Hermans$^{2,5}$, and Danica Kragic$^1$ [^1] [^2] [^3] [^4] [^5] [^6] [^7] [^8] [^9]
bibliography:
- 'In-Hand.bib'
- 'Benchmarks.bib'
title: 'Benchmarking In-Hand Manipulation'
---
Performance Evaluation and Benchmarking; Dexterous Manipulation.
Introduction
============
Protocol Design {#sec:protocol_design}
===============
Demonstration {#sec:demo}
=============
We benchmark methods for level I and level III tasks in Sec. \[sec:level1\] and Sec. \[sec:level3\] respectively. Since the level III task demonstrate the metrics used for level II tasks, we do not explicitly demonstrate a separate method for level II. We refer readers to [@sundaralingam-icra2018-finger-gaiting] for an example approach that could be easily extended for evaluation using the level II protocol.
Conclusion {#sec:conclusion}
==========
We proposed a benchmarking scheme for quantifying in-hand manipulation capabilities in a robotic system. We designed tasks for in-hand manipulation systems using the widely available YCB objects set, and we provided suggestions for adapting these tasks given the constraints of the hardware used for the evaluation. We have shown example results to demonstrate the outcome of the proposed benchmarking scheme. These results also serve as baselines for comparison with different methods in the future. By using this standardized evaluation we enable a comparison between different in-hand manipulation techniques that also considers different kinds of hardware platforms.
[^1]: Manuscript received: August, 15, 2019; Revised November, 18, 2019; Accepted December, 1, 2019.
[^2]: This paper was recommended for publication by Editor Han Ding upon evaluation of the Associate Editor and Reviewers’ comments.
[^3]: S. Cruciani was supported by Swedish Foundation for Strategic Research project GMT14-0082 FACT and B. Sundaralingam was supported by NSF Award \#1846341
[^4]: These two authors contributed equally.
[^5]: $^{1}$Division of Robotics, Perception & Learning, EECS, KTH Royal Institute of Technology, Stockholm, Sweden
[^6]: $^{2}$Robotics Center & School of Computing, University of Utah, UT USA.
[^7]: $^{3}$Dept. of Mechanical Engineering & Material Science, Yale University, New Haven, CT, USA.
[^8]: $^{4}$Google AI
[^9]: $^{5}$NVIDIA Research
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'A commonly studied means of parameterizing graph problems is the deletion distance from triviality [@guo_structural_2004], which counts vertices that need to be deleted from a graph to place it in some class for which efficient algorithms are known. In the context of graph isomorphism, we define triviality to mean a graph with maximum degree bounded by a constant, as such graph classes admit polynomial-time isomorphism tests. We generalise deletion distance to a measure we call elimination distance to triviality, based on elimination trees or tree-depth decompositions. We establish that graph canonisation, and thus graph isomorphism, is $\FPT$ when parameterized by elimination distance to bounded degree, extending results of Bouland et al. [@bouland_tractable_2012].'
author:
- Jannis Bulian
- Anuj Dawar
bibliography:
- 'isomorphism\_elim\_distance.bib'
title: 'Graph Isomorphism Parameterized by Elimination Distance to Bounded Degree[^1]'
---
Introduction
============
The *graph isomorphism problem* ($\GI$) is the problem of determining, given a pair of graphs $G$ and $H$, whether they are isomorphic. This problem has an unusual status in complexity theory as it is neither known to be in $\P$ nor known to be $\NP$-complete, one of the few natural problems for which this is the case. Polynomial-time algorithms are known for a variety of special classes of graphs. Many of these lead to natural parameterizations of $\GI$ by means of structural parameters of the graphs which can be used to study the problem from the point of view of parameterized complexity. For instance, it is known that $\GI$ is in $\XP$ parameterized by the genus of the graph, [@Miller:1980iq; @Filotti:1980eg], by maximum degree [@luks_isomorphism_1982; @babai_canonical_1983] and by the size of the smallest excluded minor [@Ponomarenko:1991bm], or more generally, the smallest excluded topological minor [@GroheMarx2012]. For each of these parameters, it remains an open question whether the problem is $\FPT$. On the other hand, $\GI$ has been shown to be $\FPT$ when parameterized by eigenvalue multiplicity [@Evdokimov:hy], tree distance width [@Yamazaki:dt], the maximum size of a simplical component [@Toda:2006bb; @Uehara:2005cx] and minimum feedback vertex set [@Kratsch:ke]. Bouland et al. [@bouland_tractable_2012] showed that the problem is $\FPT$ when parameterized by the tree depth of a graph and extended this result to a parameter they termed *generalised tree depth*. In a recent advance on this, Lokshtanov et al. [@LokshtanovPPS14] have announced that graph isomorphism is also $\FPT$ parameterized by *tree width*.
Our main result extends the results of Bouland et al. and is incomparable with that of Lokshtanov et al. We show that graph canonisation is $\FPT$ parameterized by *elimination distance to degree $d$*, for any constant $d$. The structural graph parameter we introduce is an instance of what Guo et al. [@guo_structural_2004] call *distance to triviality* and it may be of interest in the context of other graph problems.
To put this parameter in context, consider the simplest notion of distance to triviality for a graph $G$: the number $k$ of vertices of $G$ that must be deleted to obtain a graph with no edges. This is, of course, just the size of a minimal vertex cover in $G$ and is a parameter that has been much studied (see for instance [@FellowsLMRS08]). Indeed, it is also quite straightforward to see that $\GI$ is $\FPT$ when parameterized by vertex cover number. Consider two ways this observation might be strengthened. The first is to relax the notion of what we consider to be “trivial”. For instance, as there is, for each $d$, a polynomial time algorithm deciding $\GI$ among graphs with maximum degree $d$, we may take this as our trivial base case. We then parameterize $G$ by the number $k$ of vertices that must be deleted to obtain a subgraph of $G$ with maximum degree $d$. This yields the parameter *deletion distance to bounded degree*, which we consider in Section \[S:deletion\_distance\] below. Alternatively, we relax the notion of “distance” so that rather than considering the sequential deletion of $k$ vertices, we consider the recursive deletion of vertices in a tree-like fashion. To be precise, say that a graph $G$ has *elimination distance* $k+1$ from triviality if, in each connected component of $G$ we can delete a vertex so that the resulting graph has distance $k$ to triviality. If triviality is understood to mean the empty graph, this just yields a definition of the tree depth of $G$. In our main result, we combine these two approaches by parameterizing $G$ by the elimination distance to triviality, where a graph is trivial if it has maximum degree $d$. We show that, for any fixed $d$, this gives a structural parameter on graphs for which graph canonisation is $\FPT$. Along the way, we establish a number of characterisations of the parameter that may be interesting in themselves. The key idea in the proof is the separation, in a canonical way. of any graph of elimination distance $k$ to degree $d$ into two subgraphs, one of which has degree bounded by $d$ and the other tree-depth bounded by a function of $k$ and $d$. It should be noted that the parameter termed *generalised tree depth* in [@bouland_tractable_2012] can be seen as a special case of elimination distance to degree 2.
A central technique used in the proof is to construct, from a graph $G$, a term (or equivalently a labelled, ordered tree) $T_G$ that is an isomorphism invariant of the graph $G$. It should be noted that this general method is widely deployed in practical isomorphism tests such as McKay’s graph isomorphism testing program “nauty” [@McKay:1981ug; @McKay:2014uw]. The recent advance by Lokshtanov et al. [@LokshtanovPPS14] is also based on such an approach.
In Section \[S:prelim\] we recall some definitions from graph theory and parameterized complexity theory. Section \[S:deletion\_distance\] introduces the notion of deletion distance to bounded degree and presents a kernelisation procedure that allows us to decide isomorphism. In Section \[S:elimination\_distance\] we introduce the main parameter of our paper, elimination distance to bounded degree, and establish its key properties. The main result on $\FPT$ graph canonisation is established in Section \[S:elimination\_distance\_alg\].
Preliminaries {#S:prelim}
=============
Parameterized complexity theory is a two-dimensional approach to the study of the complexity of computational problems. A *language* (or *problem*) $L$ is a set of strings $L \subseteq \Sigma^*$ over a finite alphabet $\Sigma$. A *parameterization* is a function $\kappa :
\Sigma^* \to \mathbb{N}$. We say that $L$ is *fixed-parameter tractable* with respect to $\kappa$ if we can decide whether an input $x \in \Sigma^*$ is in $L$ in time $O(f(\kappa(x)) \cdot |x|^c)$, where $c$ is a constant and $f$ is some computable function. For a thorough discussion of the subject we refer to the books by Downey and Fellows [@Downey:2012vk], Flum and Grohe [@Flum:2006vj] and Niedermeier [@Niedermeier:2006ei].
A *graph* $G$ is a set of vertices $V(G)$ and a set of edges $E(G)
\subseteq V(G) \times V(G)$. We will usually assume that graphs are loop-free and undirected, i.e. that $E$ is irreflexive and symmetric. If $E$ is not symmetric, we call $G$ a *directed graph*. We mostly follow the notation in Diestel [@Diestel:2000vm].
If $v \in G$ and $S \subseteq V(G)$, we write $E_G(v, S)$ for the set of edges $\{vw \mid w \in S\}$ between $v$ and $S$.
The *neighbourhood* of a vertex $v$ is $N_G(v) := \{w \in V(G) \mid vw
\in E(G)\}$. The *degree* of a vertex $v$ is the size of its neighbourhood $\deg_G(v) := |N_G(v)|$. For a set of vertices $S \subseteq V(G)$ its neighbourhood is defined to be $N_G(S) := \bigcup_{v \in S} N_G(v)$. The *degree* of a graph $G$ is the maximum degree of its vertices $\Delta(G) := \max\{\deg_G(v) \mid v \in V(G)\}$. If it is clear from the context what the graph is, we will sometimes omit the subscript.
A subgraph $H$ of $G$ is a graph with vertices $V(H) \subseteq V(G)$ and edges $E(H) \subseteq (V(H) \times V(H)) \cap E(G)$. If $A \subseteq V(G)$ is a set of vertices of $G$, we write $G[A]$ for the subgraph *induced* by $A$, i.e. $V(G[A]) = A$ and $E(G[A]) = E(G)
\cap (A \times A)$. If $A$ is a subset of $V(G)$, we write $G
\setminus A$ for $G[V(G) \setminus A]$. For a vertex $v \in V(G)$, we write $G \setminus v$ for $G \setminus \{v\}$.
A vertex $v$ is said to be *reachable* from a vertex $w$ in $G$ if $v=w$ or if there is a sequence of edges $a_1a_2, \dots,
a_{s-1}a_s\in E(V)$ with the $a_i$ pairwise distinct and $w=a_1$ and $v=a_s$. We call the subgraph $P$ of $G$ with vertices $V(P) = \{a_1, \dots, a_s\}$ and edges $E(P) = \{a_1a_2, \dots, a_{s-1}a_s\}$ a *path from $w$ to $v$*.
Let $H$ be a subgraph of $G$ and $v, w \in V(G)$. A *path through $H$ from $w$ to $v$* is a path $P$ from $w$ to $v$ in $G$ with all vertices, except possibly the endpoints, in $V(H)$, i.e. $(V(P) \setminus \{v, w\}) \subseteq V(H)$.
It is easy to see that for undirected graphs reachability defines an equivalence relation on the vertices of $G$. A subgraph of an undirected graph induced by a reachability class is called a *component*.
Two graphs $G$, $G'$ are *isomorphic* if there is a bijection $\phi : V(G) \to V(G')$ such that for all $v, w \in V(G)$ we have that $vw \in E(G)$ if and only if $\phi(v)\phi(w) \in E(G')$. We write $G
\cong G'$ if $G$ and $G'$ are isomorphic. We write $\GI$ to denote the problem of deciding, given $G$ and $G'$ whether $G\cong G'$.
A *(k-)colouring* of a graph $G$ is a map $c : V(G) \to \{1, \dots,
k\}$ for some $k \in \NAT$. We call a graph together with a colouring a *coloured* graph. Two coloured graphs $G, G'$ with respective colourings $c : V(G) \to
\{1, \dots, k\}, c' : V(G') \to \{1, \dots, k\}$ are *isomorphic* if there is a bijection $\phi : V(G) \to V(G')$
such that:
- for all $v, w \in V(G)$ we have that $vw \in E(G)$ if and only if $\phi(v)\phi(w) \in E(G')$;
- for all $v \in V(G)$, we have that $c(v) = c'(\phi(v))$.
Note that we require the colour classes to match exactly, and do not allow a permutation of the colour classes.
Let $\C$ be a class of (coloured) graphs closed under isomorphism. A *canonical form for $\C$* is a function $F : \C \to \C$ such that
- for all $G \in \C$, we have that $F(G) \cong G$;
- for all $G, H \in \C$, we have that $G \cong H$ if, and only if, $F(G) = F(H)$.
Recall that a *partial order* is a binary relation $\leq$ on a set $S$ which is reflexive, antisymmetric and transitive. If $\leq$ is a partial order on $S$, and for each element $a \in S$, the set $\{b \in S \mid b \leq
a\}$ is totally ordered by $\leq$, we say $\leq$ is a *tree order*. (Note that the covering relation of a tree order is not necessarily a tree, but may be a forest.)
\[def:elimination-order\] An *elimination order* $\leq$ is a tree order on the vertices of a graph $G$, such that for each edge $uv \in E(G)$ we have either $u
\leq v$ or $v \leq u$.
We say that an order has *height* $k$ if the length of the longest chain in it is $k$.
We write $\td{G}$ for the *tree-depth* of $G$, which is defined as follows $$\td(G) := \begin{cases}
0,
& \text{if }V(G) = \emptyset; \\
1 + \min\{\td(G \setminus v) \mid v \in V(G)\},
& \text{if $G$ is connected;} \\
\max\{\td(H) \mid H \text{ a component of $G$}\},
& \text{otherwise.}
\end{cases}$$ Note that there is an elimination order $\leq$ of height $k$ for a graph $G$ if, and only if, $\td(G) \leq k$.
Isomorphism on bounded-degree graphs
====================================
In this section we collect some well known results about isomorphism tests and canonisation of bounded degree graphs that we will use. Luks [@luks_isomorphism_1982] shows that isomorphism of bounded-degree graphs is decidable in polynomial time. This result extends, by an easy reduction, to *coloured* graphs of bounded-degree. For completeness, we present this reduction explicitly.
The isomorphism problem for coloured graphs can be reduced to $\GI$ in polynomial time.
Let $G, G'$ be graphs and let $c, c' : V(G) \to \{1, \dots, k\}$ be colourings of $G, G'$ respectively for some $k \in \NAT$.
We define $H$ to be the graph whose vertices include $V(G)$ and, additionally, for each $v \in V(G)$, $c(v)+1$ new vertices $u^v_1,\ldots,u^v_{c(v)+1}$. The edges of $H$ are the edges $E(G)$ plus additional edges so that the vertices $v$ and $u^v_1,\ldots,u^v_{c(v)+1}$ form a simple cycle of length $c(v)+2$. We obtain $H'$ in a similar way from $G'$.
We claim that $G \cong G'$ if, and only if, $H \cong H'$. Clearly, if $G \cong G'$ and $\phi$ is an isomorphism witnessing this, it can be extended to an isomorphism from $H$ to $H'$ by mapping $u^v_i$ to $u^{\phi(v)}_i$. For the converse, suppose $H \cong H'$ and let $\phi : H \to H'$ be an isomorphism. We use it to define an isomorphism $\phi'$ from $G$ to $G'$. Note that, if $v \in V(G)$ is not an isolated vertex of $G$, then it has degree at least 3 in $H$. Since $\phi(v)$ has the same degree, it is in $V(G')$, and we let $\phi'(v) = \phi(v)$. If $v$ is an isolated vertex of $G$, then its component in $H$ is a simple cycle of length $c(v)+2$. The image of this component under $\phi$ is a simple cycle of $H'$ which must contain exactly one vertex $v'$ of $V(G')$. We let $\phi'(v)=v'$. It is easy to see that there is an edge between $v_1,v_2$ in $G$ if, and only if, there is an edge between $\phi'(v_1)$ and $\phi'(v_2)$ in $G'$. To see that $\phi'$ also preserves colours, note that $\phi$ must map the cycle containing $u^v_{c(v)}$ to the cycle containing $u^{\phi'(v)}_{c(\phi'(v))}$ and therefore $c(v) = c(\phi'(v))$.
Note that the construction in the proof increases the degree of each vertex by $2$, so if $G$ and $G'$ are graphs of degree $d$, then $H,
H'$ are graphs of degree $d+2$.
As Luks [@luks_isomorphism_1982] proves that isomorphism of bounded degree graphs can be decided in polynomial time, we have the following:
\[T:bdd\_iso\] We can test in polynomial time whether two (coloured) graphs with maximal degree bounded by a constant are isomorphic.
Babai and Luks [@babai_canonical_1983] give a polynomial time canonisation algorithm for bounded degree graphs. Just as above we can reduce canonisation of coloured bounded degree graphs to the bounded degree graph canonisation problem.
\[T:bdd\_canon\] Let $\C$ be a class of (coloured) bounded degree graphs closed under isomorphism. Then there is a canonical form $F$ for $\C$ that allows us to compute $F(G)$ in polynomial time.
Deletion distance to bounded degree {#S:deletion_distance}
===================================
We first study the notion of deletion distance to bounded degree and establish in this section that graph isomorphism is FPT with this parameter. Though the result in this section is subsumed by the more general one in Section \[S:elimination\_distance\_alg\], it provides a useful warm-up and a tighter, polynomial kernel. In the present warm-up we only give an algorithm for the graph isomorphism problem, though the result easily holds for canonisation as well (and this follows from the more general result in Section \[S:elimination\_distance\_alg\]). The notion of deletion distance to bounded degree is a particular instance of the general notion of distance to triviality introduced by Guo et al. [@guo_structural_2004]. In the context of graph isomorphism, we have chosen triviality to mean graphs of bounded degree.
A graph $G$ has *deletion distance $k$ to degree $d$* if there are $k$ vertices $v_1, \dots, v_k \in V(G)$ such that $G
\setminus \{v_1, \dots, v_k\}$ has degree $d$. We call the set $\{v_1, \dots, v_k\}$ a *$d$-deletion set*.
To say that $G$ has deletion distance $0$ from degree $d$ is just to say that $G$ has maximum degree $d$. Also note that if $d=0$, then the $d$-deletion set is just a vertex cover and the minimum deletion distance the vertex cover number of $G$.
We show that isomorphism is fixed-parameter tractable on such graphs parameterized by $k$ with fixed degree $d$; in particular we give a procedure that computes a polynomial kernel for the deletion set in linear time.
\[T:deldistance\_kernel\] For any graph G and integers $d,k > 0$, we can identify in linear time a subgraph $G'$ of $G$, a set of vertices $U \subseteq V(G')$ with $|U| = O(k(k+d)^2)$ and a $k' \leq k$ such that: $G$ has deletion distance $k$ to degree $d$ if and only if $G'$ has deletion distance $k'$ to $d$ and, moreover, if $G'$ has deletion distance at most $k'$, then any minimum size $d$-deletion set for $G'$ is contained in $U$.
Let $H := \{v \in V(G) \mid \deg(v) > k + d\}$. Now, if $R$ is a minimum size $d$-deletion set for $G$ and $G$ has deletion distance at most $k$ to degree $d$, then $|R| \leq k$ and the vertices in $V(G \setminus R)$ have degree at most $k + d$ in $G$. So $H \subseteq R$. This means that if $|H| > k$, then $G$ must have deletion distance greater than $k$ to degree $d$ and in that case we let $G' := G, k' := k$ and $U
= \emptyset$.
Otherwise let $G' := G \setminus H$ and $k' := k - |H|$. We have shown that every $d$-deletion set of size at most $k$ must contain $H$. Thus $G$ has deletion distance $k$ to degree $d$ if and only if $G'$ has deletion distance $k'$ to degree $d$.
Let $S := \{v \in V(G') \mid \deg_{G'}(v) > d\}$ and $U := S \cup
N_{G'}(S)$. Let $R' \subseteq V(G')$ be a minimum size $d$-deletion set for $G'$. We show that $R' \subseteq U$. Let $v \not\in U$. Then by the definition of $U$ we know that $\deg_{G'}(v) \leq d$ and all of the neighbours of $v$ have degree at most $d$ in $G'$. So if $v
\in R'$, then $G \setminus (R' \setminus \{v\})$ also has maximal degree $d$, which contradicts the assumption that $R'$ is of minimum size. Thus $v \not\in R'$.
Note that the vertices in $G' \setminus (R' \cup N(R'))$ have the same degree in $G'$ as in $G$ and thus all have degree at most $d$. So $S \subseteq R' \cup N(R')$ and thus $|U|
\leq k' + k'(k+d) + k'(k+d)^2 = O(k(k+d)^2)$.
Finally, the sets $H$ and $U$ defined as above can be found in linear time, and $G', k'$ can be computed from $H$ in linear time.
Note that if $U = \emptyset$ and $k' > 0$, then there are no $d$-deletion sets of size at most $k'$.
Next we see how the kernel $U$ can be used to determine whether two graphs with deletion distance $k$ to degree $d$ are isomorphic by reducing the problem to isomorphism of coloured graphs of degree at most $d$.
Suppose we are given two graphs $G$ and $H$ with $d$-deletion sets $S
= \{v_1, \dots, v_k\}$ and $T = \{w_1, \dots, w_k\}$ respectively. Further suppose that the map $v_i \mapsto w_i$ is an isomorphism on the induced subgraphs $G[S]$ and $H[T]$. We can then test if this map can be extended to an isomorphism from $G$ to $H$ using Theorem \[T:bdd\_canon\]. To be precise, we define the coloured graphs $G'$ and $H'$ which are obtained from $G\setminus S$ and $H\setminus
T$ respectively, by colouring vertices. A vertex $u \in V(G')$ gets the colour $\{i \mid v_i \in N_G(u)\}$, i.e. the set of indices of its neighbours in $S$. Vertices in $H'$ are similarly coloured by the sets of indices of their neighbours in $T$. It is clear that $G'$ and $H'$ are isomorphic if, and only if, there is an isomorphism between $G$ and $H$, extending the fixed map between $S$ and $T$. The coloured graphs $G'$ and $H'$ have degree bounded by $d$, so Theorem \[T:bdd\_canon\] gives us a polynomial-time isomorphism test on these graphs.
Now, given a pair of graphs $G$ and $H$ which have deletion distance $k$ to degree $d$, let $A$ and $B$ be the sets of vertices of degree greater than $k+d$ in the two graphs respectively. Also, let $U$ and $V$ be the two kernels in the graphs obtained from Theorem \[T:deldistance\_kernel\]. Thus, any $d$-deletion set in $G$ contains $A$ and is contained in $A \cup U$ and similarly, any $d$-deletion set for $H$ contains $B$ and is contained in $B \cup V$. Therefore to test $G$ and $H$ for isomorphism, it suffices to consider all $k$-element subsets $S$ of $A \cup U$ containing $A$ and all $k$-element subsets $T$ of $B\cup V$ containing $B$, and if they are $d$-deletion sets for $G$ and $H$, check for all $k!$ maps between them whether the map can be extended to an isomorphism from $G$ to $H$. As $d$ is constant this takes time $O^*\left({{k^3}\choose{k}}^2 \cdot k!\right)$, which is $O^*\left(2^{7k\log k}\right)$.
Elimination distance to bounded degree {#S:elimination_distance}
======================================
In this section we introduce a new structural parameter for graphs. We generalise the idea of deletion distance to triviality by recursively allowing deletions from each component of the graph. This generalises the idea of elimination height or tree-depth, and is equivalent to it when the notion of triviality is the empty graph. In the context of graph isomorphism and canonisation we again define triviality to mean bounded degree, so we look at the elimination distance to bounded degree.
The *elimination distance to degree $d$* of a graph $G$ is defined as follows: [ $$\textstyle{\edd(G)} :=
\begin{cases}
0,
& \text{if }\Delta(G) \leq d; \\
1 + \min \{\edd(G \setminus v) \mid v \in V(G)\},
& \text{if $\Delta(G) > d$ and $G$ is connected;} \\
\max\{\edd(H) \mid H \text{ a connected component of $G$}\},
& \text{otherwise.}
\end{cases}$$ ]{}
We first introduce other equivalent characterisations of this parameter. If $G$ is a graph that has elimination distance $k$ to degree $d$, then we can associate a certain tree order $\leq$ with it:
\[D:elim\_order\_to\_deg\] A tree order $\leq$ on $V(G)$ is an *elimination order to degree $d$* for $G$ if for each $v \in V(G)$ the set $$S_v := \{u \in V(G) \mid uv \in E(G) \text{ and } u \not\leq v \text{ and } v \not\leq u\}$$ satisfies either:
- $S_v = \emptyset$; or
- $v$ is ${\leq}$-maximal, $|S_v| \leq d$, and for all $u \in
S_v$, we have $\{w \mid w < u\} = \{w \mid w < v\}$.
Note that if $S_v = \emptyset$ for all $v \in V(G)$, then an elimination order to degree $d$ is just an elimination order, in the sense of Definition \[def:elimination-order\].
\[prop:tree-depth-order\] A graph $G$ has $\edd(G) \leq k$ if, and only if, there is an elimination order to degree $d$ of height $k$ for $G$.
Let $S_v$ be as in Definition \[D:elim\_order\_to\_deg\]. We prove the proposition by induction on $k$. If $k = 0$, then the graph has no vertex of degree larger than $d$ and we define the elimination order $\leq$ to be the identity relation on $V(G)$. Then every $v \in V(G)$ is maximal, we have $|S_v| \leq d$, and for all $u \in S_v$ we have $\{w \mid w < u\} = \emptyset = \{w \mid w < v\}$.
Suppose $k>0$ and the statement is true for smaller values. If $G$ is not connected, we apply the following argument to each component. So in the following we assume that $G$ is connected.
Suppose $\edd(G) = k$. Then there is a vertex $a \in V(G)$ such that the components $C_1, \dots, C_r$ of $G \setminus a$ all have $\edd(C_i) \leq k-1$. So by the induction hypothesis each $C_i$ has a tree order $\leq_i$ to degree $d$ of height at most $k-1$ with the properties in Definition \[D:elim\_order\_to\_deg\]. For each $v \in V(C_i)$ define $$S_v^i := \{u \in V(C_i) \mid uv \in E(G) \text{ and } u \not\leq v \text{ and } v \not\leq u\}.$$
Let $${\leq} := \{(a, w) \mid w \in V(G)\} \cup \bigcup_i \leq_i.$$ Then $\leq$ is clearly a tree order for $G$. Note that $S_a =
\emptyset$. Let $v \in V(G) \setminus a$ be a vertex different from $a$, say $v \in V(C_i)$. Note $S_v^i = S_v$. If $S_v \neq \emptyset$, then $v$ is $\leq_i$-maximal, and thus also $\leq$-maximal. Moreover, $|S_v^i| =
|S_v| \leq d$. Lastly for any $u \in S_v$: $$\{w \mid w < u\} = \{a\} \cup \{w \mid w <_i u\} = \{a\} \cup \{w \mid w <_i v\} = \{w \mid w < v\}.$$
Conversely assume there is an elimination order $\leq$ to degree $d$ of height $k$ for $G$. There is a single minimal element $v$ of $\leq$ because $G$ is connected and $k > 0$. Note that $\leq$ restricted to a component $C$ of $G \setminus
v$ has height $k-1$ and thus by the induction assumption we have that $\edd(C) \leq k-1$.
We can split a graph with an elimination order to degree $d$ in two parts: one of low degree, and one with an elimination order defined on it. So if $G$ is a graph that has elimination distance $k$ to degree $d$, we can associate an elimination order $\leq$ for a subgraph $H$ of $G$ of height $k$ with $G$, so that each component of $G \setminus
V(H)$ has degree at most $d$ and is connected to $H$ along just one branch (this is defined more formally below).
\[P:elim\_order\_char2\] Let $G$ be a graph and $\leq$ an elimination order to degree $d$ for $G$ of height $k$. If $A$ is the set of vertices in $V(G)$ that are not $\leq$-maximal, then:
1. $\leq$ restricted to $A$ is an elimination order of height $k-1$ of $G[A]$; and
2. $G \setminus A$ has degree at most $d$;
3. if $C$ is the vertex set of a component of $G \setminus A$, and $u, v \in A$ are $\leq$-incomparable, then either $E(u, C) = \emptyset$ or $E(v, C) = \emptyset$.
As any $v\in A$ is non-maximal, by Definition \[D:elim\_order\_to\_deg\], $S_v = \emptyset$. Hence if there is an edge between $u,v \in A$, either $u < v$ or $v< u$, and (1) follows.
Since $G\setminus A$ contains the $\leq$-maximal elements, they are all incomparable. By definition of an elimination order to degree $d$, this means that each vertex in $G\setminus A$ has at most $d$ neighbours in $G\setminus A$, so this graph has degree at most $d$, establishing (2).
To show (3), let $C$ be the vertex set of a component of $G \setminus
A$ and let $u, v \in A$ be such that $E(u, C) \neq \emptyset$ and $E(v, C) \neq \emptyset$. Then there are $a, b \in C$ such that $au, bv \in E(G)$. By Definition \[D:elim\_order\_to\_deg\], $u<a$ and $v<b$. Moreover, there is a path from $a$ to $b$ through $C$ and as all vertices along this path $\leq$-maximal, if $(a',b')$ is an edge in the path, it must be that $\{w \mid w < a'\} = \{w \mid w < b'\}$. By transitivity, $\{w \mid w < a\} = \{w \mid w < b\}$, and so $u < b$ and $v < a$. Since $\leq$ is a tree-order, the set $\{w \mid w < a\}$ is linearly orderd and we conclude that $u$ and $v$ are comparable.
We also have a converse to the above in the following sense.
\[P:char2-converse\] Suppose $G$ is a graph with $A \subseteq V(G)$ a set of vertices and $\leq_A$ an elimination order of $G[A]$ of height $k$, such that:
1. $G \setminus A$ has degree at most $d$;
2. if $C$ is the vertex set of a component of $G \setminus A$, and $u, v \in A$ are incomparable, then either $E(u, C) = \emptyset$ or $E(v, C) = \emptyset$.
Then, $\leq_A$ can be extended to an elimination order to degree $d$ for $G$ of height $k+1$.
Let $$\begin{aligned}
{\leq} := &{\leq_A} \cup \{(v, v) \mid v \in (V(G) \setminus A)\} \\
&\cup
\{(u, v) \mid
u \in A, v \in C,
\text{$C$ a component of $G \setminus A$}, E(w, C)\neq \emptyset
\text{ for some $u \leq w$}\}.\end{aligned}$$ Then it is easily seen that $\leq$ is a tree order on $G$. Indeed, $\leq_A$ is, by assumption, a tree order on $A$ and for any $v \in
V(G)\setminus A$, assumption 2 guarantees that $\{w \mid w \leq v\}$ is linearly ordered.
Let $v \in V(G)$ and let $S_v$ be as in Definition \[D:elim\_order\_to\_deg\]. Suppose $S_v \neq \emptyset$. Then $v \in (V(G) \setminus A)$ and has degree at most $d$ in $G \setminus A$. By the construction $v$ is $\leq$-maximal. Let $u \in S_v$. Then there is a component $C$ of $G
\setminus A$ that contains both $u$ and $v$ and thus $\{w \mid w < u\}
= \{w \mid w < v\}$.
In the following, given a graph $G$ and an elimination order to degree $d$, $\leq$, we call the subgraph of $V(G)$ induced by the non-maximal elements of the order $\leq$ the *non-maximal subgraph of $G$ under $\leq$*.
In the proof of Proposition \[P:char2-converse\] above, a suitable tree order on a subset $A$ of $V(G)$ is extended to an elimination order to degree $d$ of $G$ by making all vertices not in $A$ maximal in the order. This is a form of construction we use repeatedly below.
The alternative characterisations of elimination order to degree $d$ established above are very useful. In the next section, we use them to construct a *canonical* elimination order to degree $d$ of $G$, based on an elimination order of a graph we call the *torso* of $G$, which consists of the high-degree vertices of $G$, along with some additional edges.
Canonical Elimination Order to Bounded Degree
=============================================
The aim of this section is to show that if a graph $G$ has elimination distance $k$ to degree $d$, then there is an elimination order to degree $d$ whose height is still bounded by a function of $d$ and $k$ and which is *canonical*. To be precise, we identify a graph which we call the $d$-degree torso of $G$, which contains all the vertices of $G$ of degree more than $k$ and has additional edges to represent paths between these vertices that go through the rest of $G$. We show that this torso necessarily has tree-depth bounded by a function of $k$ and $d$ and the canonical elimination order witnessing this can be extended to an elimination order to degree $d$ of $G$. The result is established through a series of lemmas. A pattern of construction that is repeatedly used here is that we define a certain set $A$ of vertices of $G$ and construct an elimination order of $G[A]$. It is then shown that extending the order by making all vertices in $V(G)\setminus A$ maximal yields an elimination order to degree $d$ of $G$. Necessarily, in this extended order, all the non-maximal elements are in $A$.
The following lemma establishes that if $G$ has elimination distance $k$ to degree $d$ and moreover the degree of $G$ is at most $k+d$, then we can construct an alternative elimination order on $G$ in which all the vertices of degree greater than $d$ are included in the non-maximal subgraph and the height of the new elimination order is still bounded by a function of $k$ and $d$.
\[L:adding\_vertices\] Let $G$ be a graph with maximal degree $\Delta(G) \leq k+d$. Let $\leq$ be an elimination order to degree $d$ of height $k$ of $G$ with non-maximal subgraph $H$, and let $A = V(H) \cup \{v \in V(G) \mid \deg_G(v) > d\}$. Then $G$ has an elimination order $\sqsubseteq$ of height at most $k(k+d+1)$ for which the non-maximal elements are in $A$.
Let $G, H, A$ and $\leq$ be as in the statement of the lemma. We will adapt $\leq$ to an elimination order $\sqsubseteq$ of $G[A]$.
Let $B$ be the set of $\leq$-maximal elements in $V(H)$. For each $w \in A \setminus V(H)$ let $C_w$ be the component of $G
\setminus V(H)$ that contains $w$. Note that $N(C_w) \neq \emptyset$, because $\deg(w) > d$, so at least one vertex in $H$ must be adjacent to $w$. By Definition \[D:elim\_order\_to\_deg\], all vertices in $N(C_w)$ are $\leq$-comparable, so they are linearly ordered and there is a unique $b \in B$ such that $b \geq a$ for all $a \in
N(C_w)$. We write $b(w)$ to denote this element of $B$ associated with every $w \in A \setminus V(H)$. For each $b \in B$, let $W_b := \{w \in A \setminus V(H) \mid b(w) =
b\}$, and let $\sqsubseteq_b$ be an arbitrary linear order on $W_b$.
For any $u, v \in V(G)$, define $u \sqsubseteq v$ if one of the following holds:
- $u = v$;
- $u, v \in H$ and $u \leq v$;
- $u \in H$, $v \in G \setminus A$ and $u \leq v$;
- $u \in H$, $v \in A\setminus V(H)$ and $u \leq b(v)$;
- $u \in A \setminus V(H)$, $v \in G \setminus A$ and $b(u)
\leq v$;
- $u, v \in H'\setminus V(H)$, $b(u)= b(v)$ and $u \sqsubseteq_b v$.
It follows from the construction that $\sqsubseteq$ restricted to $A$ is an elimination order of $G[A]$, and that $\sqsubseteq$ is an elimination order to degree $d$ of $G$.
For each $b \in B$, the set $\{v \in H \mid v \leq b\}$ has at most $k$ elements, by the assumption on the height of the order $\leq$. Since $G$ has maximum degree $k+d$ and $W_b \subseteq
N(\{v \in H \mid v \leq b\})$, we have that $W_b$ has at most $k(k+d)$ vertices. Since the height of any $\sqsubseteq$ chain is at most the height of a $\leq$-chain plus $|W_b|$, we conclude that the height of $\sqsubseteq$ is at most $k(k+d+1)$.
The lemma above allows us to re-arrange the elimination order so that it includes all vertices of large degree. In contrast, the next lemma gives us a means to re-arrange the elimination order so that all vertices of small degree are made maximal in the order. This is again done achieved while keeping the height of the elimination order bounded by a function of $k$ and $d$.
\[L:removing\_vertices\] Let $G$ be a graph. Let $\leq$ be an elimination order to degree $d$ of $G$ of height $k$ with non-maximal subgraph $H$, such that $H$ contains all vertices of degree greater than $d$, and let $A = \{v \in V(H) \mid \deg_G(v) > d\}$. Then, there is an elimination order to degree $d$ of $G$ of height at most $k((k+1)d)^{2^k}+1$ for which all the non-maximal elements are in $A$.
Let $G, H, A$ and $\leq$ be as in the statement of the lemma. We assume that $G$ is connected – if not, we can apply the argument to each component of $G$. We construct an elimination order $\sqsubseteq$ of $G[A]$ from $\leq$, making sure that it has height at most $k((k+1)d)^{2^k}$. This extends to an elimination order to degree $d$ of $G$ by making all vertices not in $A$ maximal, as in the Proposition \[P:char2-converse\].
Let $J := H \setminus A$.
For $v \in V(J)$, let $K_v$ be the set of vertices $w \in A$ such that:
1. $v \leq w$;
2. there is a path from $v$ to $w$ through $G \setminus A$; and
3. for any $u$ with $u < v$, there is no path from $u$ to $w$ through $G \setminus A$.
Note that because $\leq$ is a tree order and the third condition, the sets $K_v$ are pairwise disjoint. Let $\overline K := A \setminus (\bigcup_{v \in V(J)}
K_v)$ be the set of vertices in $A$ that are not contained in $K_v$ for any $v$.
For each $v \in V(J)$, let $\sqsubseteq_v$ be an arbitrary linear order on $K_v$. The idea behind the construction below is that we replace $v$ in the elimination order by $K_v$, ordered by $\sqsubseteq_v$. Formally, for any $u, w \in V(G)$, define $u \sqsubseteq w$ if one of the following holds:
- $u = w$;
- $u \in K_v$, $w \in G \setminus A$ and $v \leq w$;
- $u \in \overline K$, $w \in G \setminus A$ and $u \leq w$;
- $u, w \in K_v$ and $u \sqsubseteq_v w$;
- $u \in K_v$, $w \in K_{v'}$ and $v < v'$;
- $u \in \overline K$, $w \in K_v$ and $u \leq v$;
- $u \in K_v$, $w \in \overline K$ and $v \leq w$;
- $u, w \in \overline K$ and $u \leq w$.
We first show that $\sqsubseteq$ is an elimination order for $G[A]$. The construction ensures $\sqsubseteq$ is a tree order. Let $u, w \in V(H')$. We show that if $u \leq w$, then either $u
\sqsubseteq w$ or $w \sqsubseteq u$. We go through all possible cases: If $u = w$, we have $u \sqsubseteq w$. If there is some $v \in V(J)$ such that $u, w \in K_v$, then $u \sqsubseteq w$ or $w
\sqsubseteq u$. If $u \in K_v$, $w \in K_{v'}$ for two different $v,
v' \in V(J)$, then $v \leq u \leq w$ and $v' \leq w$, so $v' \leq v$ and thus $w \sqsubseteq u$. If $u \in \overline K$ and $w \in K_v$, then both $u, v \leq w$, so either $u \leq v$ or $v \leq u$, and thus either $u \sqsubseteq w$ or $w \sqsubseteq u$. The case where $u \in K_v$, $w \in \overline K$ is symmetric. Finally, if both $u,
w \in \overline K$, then $u \sqsubseteq w$. Thus if $uw \in E(H')$, we have $u \leq w$ or $w \leq u$ and therefore $u \sqsubseteq w$ or $w \sqsubseteq u$. Hence $\sqsubseteq$ is an elimination order for $G[A]$.
Let $Z$ be a component of $G \setminus A$. We assumed that $H$ contains all vertices of degree greater than $d$, and by the construction $A$ also contains all those vertices. Thus $Z$ has maximum degree $d$.
Suppose $u, v \in A$ are two vertices that are connected to $Z$, i.e. $E_G(u, V(Z)) \neq \emptyset \neq E_G(v, V(Z))$. We show that either $u \sqsubseteq v$ or $v \sqsubseteq u$. Note that there is a path $P$ through $Z \subseteq G \setminus A$ from $u$ to $v$, i.e. all vertices in $P$, except for the endpoints, lie outside of $A$. If $P$ contains no vertices from $J$, then the connected component $Z'$ of $G \setminus V(H)$ containing $P \setminus
\{u,v\}$ satisfies $E_G(u, V(Z')) \neq \emptyset \neq E_G(v, V(Z'))$ and thus $u \leq v$ or $v \leq u$, and therefore by the above $u
\sqsubseteq v$ or $v \sqsubseteq u$.
Otherwise, $P$ contains vertices from $J$. Let $w$ be a $\leq$-minimal vertex in $V(P) \cap V(J)$. Then there is a path outside of $A$ from $w$ to $u$, and also to $v$ (both part of $P$). Moreover, if neither $u \leq v$ nor $v \leq u$, then $w \leq u$ and $w \leq v$. Thus $u$ and $v$ are in $K_w$ (or in $K_{w'}$ for some $w' < w$), and therefore $u \sqsubseteq v$ or $v \sqsubseteq u$.
It remains to show that the size of $K_v$ is bounded by $k((k+1)d)^{2^k}$ for all $v \in V(J)$. Let $G'$ be the graph obtained from $G$ by adding an edge between two vertices $s,t \in V(J)$ whenever there is a path through $G \setminus V(H)$ between $s$ and $t$. This increases the degree of vertices in $V(J)$ by at most $kd$, because each of these vertices is connected to at most $d$ components of $G \setminus V(H)$ and each of these is connected to at most $k$ vertices in $H$. Now there is a path between two vertices $s,t \in J$ in $G$ outside of $A$ if and only if there is a path between $s$ and $t$ in $G'[V(J)]$. Moreover, $\leq$ is also an elimination order for $G'[V(J)]$. So, as $G'[V(J)]$ has tree-depth at most $k$ it does not contain a path of length more than $2^k$. Since each vertex on the path has degree at most $(k+1)d$, we can reach at most $((k+1)d)^{2^k}$ vertices in $A$ on paths only containing vertices outside of $A$. Thus $|K_v| \leq ((k+1)d)^{2^k}$ and the height of $\sqsubseteq$ is bounded by $k|K_v| \leq k((k+1)d)^{2^k}$.
Next we introduce the notion of $d$-degree torso and prove that it captures the properties that we require of an elimination tree to degree $d$.
Let $G$ be a graph, let $d > 0$ and let $H$ be the induced subgraph of $G$ containing the vertices of degree larger than $d$. The *$d$-degree torso of $G$* is the graph $C$ obtained from $H$ by adding an edge between two vertices $u,v \in H$ if there is a path through $G \setminus V(H)$ from $u$ to $v$ in $G$.
The next lemma establishes an upper bound on the tree-depth of the torso of a graph when the maximum degree is bounded.
\[L:torso\_and\_tree-depth\] Let $G$ be a graph and let $C$ be the $d$-degree torso of $G$. Let $H
= G[V(C)]$ and let ${\leq}$ be an elimination order for $H$. Then ${\leq}$ is an elimination order for $C$ of height $h$ if, and only if, ${\leq}$ can be extended to an elimination order to degree $d$ for $G$ of height $h+1$.
Let $G, C, H$ and $\leq$ be as above.
Suppose $\leq$ is an elimination order for $C$. Since $C$ is a supergraph of $H$, this means that $\leq$ is an elimination order for $H$. Let $Z$ be a component of $G \setminus V(H)$. Since $C$ contains all vertices of degree greater than $d$, $Z$ has maximal degree $d$. If $E(Z, u) \neq \emptyset$ and $E(Z, v) \neq \emptyset$ for two vertices $u, v \in H$, then there is a path through $Z \subseteq G
\setminus V(H)$ connecting $u$ and $v$, so by the definition of the $d$-degree torso $uv \in E(C)$ and thus $u, v$ are $\leq$-comparable. We can extend $\leq$ to a tree order $\leq'$ on $V(G)$ where all the vertices from $V(G) \setminus V(H)$ are maximal.
Conversely assume that $\leq$ can be extended to an elimination order to degree $d$ for $G$. Let $uv \in E(C)$. If $uv \in E(H)$, then $u$ and $v$ must be $\leq$-comparable. Otherwise $uv \not\in E(H)$, so there is a path through $G \setminus V(H)$ from $u$ to $v$ in $G$, i.e. both $u$ and $v$ are connected to a component $Z$ of $G \setminus V(H)$ and thus comparable. Therefore $\leq$ is an elimination order for $C$.
\[L:bounded\_case\] Let $G$ be a graph with elimination distance $k$ to degree $d$ and maximum degree $\Delta(G) \leq k + d$. Let $C$ be the $d$-degree torso of $G$ and let $\leq$ be a minimum height elimination order for $C$. Then $\leq$ has height at most $k(k+d+1)((k(k+d+1)+1)d)^{2^{k(k+d+1)}}$.
Let $\sqsubseteq$ be a minimum height elimination order to degree $d$ of $G$. Since $G$ has elimination distance to degree $d$ at most $k$, the height of $\sqsubseteq$ is at most $k$. Let $H$ be the non-maximal subgraph of $G$ under $\sqsubseteq$ and define $$A = V(H) \cup \{v \in V(G) \mid \deg_G(v) > d\}.$$ By Lemma \[L:adding\_vertices\], the graph $G[A]$ has an elimination order $\preceq$ of height at most $k(k+d+1)$ that can be extended to an elimination order to degree $d$ for $G$.
Let $A' = \{v \in V(H') \mid \deg_G(v) > d\}$. By Lemma \[L:removing\_vertices\], the graph $A'$ has an elimination order $\leq$ of height at most $k(k+d+1)((k(k+d+1)+1)d)^{2^{k(k+d+1)}}$ that can be extended to an elimination order to degree $d$ for $G$.
Lastly note that $A' = V(C)$, so that by Lemma \[L:torso\_and\_tree-depth\], $\leq$ is an elimination order for $C$.
We are now ready to prove the main result:
Let $G$ be a graph that has elimination distance $k$ to degree $d$. Let $\leq$ be a minimum height elimination order of the $d$-degree torso $G$. Then $\leq$ can be extended to an elimination order to degree $d$ of $G$ of height at most $$k((k+1)(k+d))^{2^{k}} + k(1+k+d)(k(1+k+2d))^{2^{k(1+k+d)}}+1.$$
We show that the $d$-degree torso of $G$ has an elimination order of height at most $ k((k+1)(k+d))^{2^{k}} + k(1+k+d)(k(1+k+2d))^{2^{k(1+k+d)}}$. The Theorem then follows by Lemma \[L:torso\_and\_tree-depth\].
Let $C$ be the $(k+d)$-degree torso of $G$. We first show that the tree-depth of $C$ is bounded by $k((k+1)(k+d))^{2^k}$. To see this, let $\sqsubseteq$ be an elimination order to degree $d$ of $G$ of minimum height with non-maximal subgraph $H$. Note that $H$ contains all vertices of degree greater than $k+d$, because vertices in $G \setminus V(H)$ are adjacent to at most $k$ vertices in $H$.
Let $A = \{v \in V(H) \mid \deg_G(v) > k+d\}$. By Lemma \[L:removing\_vertices\], the graph $G[A]$ has an elimination order $\preceq$ of depth at most $h:= k((k+1)(k+d))^{2^k}$ that can be extended to an elimination order to degree $k+d$ of $G$ of height $h+1$. Note that $A = V(C)$, so by Lemma \[L:torso\_and\_tree-depth\], the order $\preceq$ is an elimination order for $C$. Let $\preceq'$ denote its extension to $G$.
Let $Z$ be a component of $G \setminus A$ and let $C_Z$ be the $d$-degree torso of $Z$. By Lemma \[L:bounded\_case\], there is an elimination order $\preceq_Z$ for $C_Z$ of height at most $k(k+d+1)((k(k+d+1)+1)d)^{2^{k(k+d+1)}}$. Let $v_Z$ be the $\preceq$-maximal element in $C$ such that there is a $w \in C_Z$ with $v_Z \preceq'
w$. Define $$\begin{aligned}
\leq' := &\preceq \cup \bigcup_Z \preceq_Z
\cup \bigcup_Z \{(v,w) \mid v \preceq' v_Z, w \in C_Z \}.
\end{aligned}$$ Observe that $C \cup \bigcup_{Z} C_Z$ is a subgraph of the $d$-degree torso of $G$. Thus $\leq'$ is an elimination order for the $d$-degree torso of $G$. The height of $\leq'$ is bounded by $$\begin{aligned}
\td(C) + \max \{\td(C_Z)\}_Z
&\leq k((k+1)(k+d))^{2^{k}} + k(1+k+d)(k(1+k+2d))^{2^{k(1+k+d)}}.
\end{aligned}$$
Canonisation parameterized by elimination distance to bounded degree {#S:elimination_distance_alg}
====================================================================
In this section we show that graph canonisation, and thus graph isomorphism, is $\FPT$ parameterized by elimination distance to bounded degree. The main idea is to construct a labelled directed tree $T_G$ from a graph $G$ (of elimination distance $k$ to degree $d$) that is an isomorphism invariant for $G$. From the labelled tree $T_G$ we obtain a canonical labelled tree using the tree canonisation algorithm from Lindell [@lindell_logspace_1992]. In the last step we construct a canonical form of $G$ from the canonical labelled tree.
The tree $T_G$ is obtained from $G$ by taking a tree-depth decomposition of the $d$-degree torso of $G$ and labelling the nodes with the isomorphism types of the low-degree components that attach to them. The tree-depth decomposition of a graph is just the elimination order in tree form. We formally define it as follows:
\[def:tree-depth-decomp\] Given a graph $H$ and an elimination order $\leq$ on $H$, the *tree-depth decomposition* associated with $\leq$ is the directed tree with nodes $V(H)$ and an arc $a\rightarrow b$ if, and only if, $a
< b$ and there is no $c$ such that $a < c < b$.
The tree-depth decomposition corresponding to an elimination order is what, in the language of partial orders, is known as its covering relation.
Note that, in general, the tree-depth decomposition of a graph that is not connected may be a forest. By results of Bouland *et. al* [@bouland_tractable_2012], we can construct a canonical tree-depth decomposition of an $n$-vertex graph of tree-depth $k$ in time $f(k) \cdot n^c$ for some comuptable $f$ and constant $c$.
Before defining $T_G$ formally, we need one piece of terminology.
Let $G$ be a graph and let $\leq$ be a tree order for $G$. The *level* of a vertex $v \in V(G)$ is the length of the chain $\{w \in V(G) \mid w \leq v\}$. We denote the level of $v$ by $\level_\leq(v)$.
Given a graph $G$ of elimination distance $k$ to degree $d$, let $C$ be the $d$-degree torso of $G$, let $T$ be a canonical tree-depth decomposition of $C$ and $\leq$ the corresponding elimination order. Let $Z$ be a component of $G\setminus C$. We let $Z^C$ denote the coloured graph that is obtained by colouring each vertex $v$ in $Z$ by the colour $\{ i \mid uv \in E(G) \text{ for some } u \in C \text{
with } \level_\leq(u) = i\}$. We write $F(Z^C)$ for the canonical form of this coloured graph given by Theorem \[T:bdd\_canon\]. Note that, by the definition of elimination distance, there is, for each $Z$ and $i$ at most one vertex $u \in C$ with $\level_\leq(u) = i$ which is in $N_G(Z)$.
We are now ready to define the labelled tree $T_G$. The nodes of $T_G$ are the nodes of $T$ together with a new node $r$, and the arcs are the arcs of $T$ along with new arcs from $r$ to the root of each tree in $T$. Define, for each node $u$ of $T_G$, $\mathcal{Z}_u$ to be the set $\{Z \mid Z \text{ is a component
of } G\setminus C \text{ with } u \leq\text{-maximal in } C \cap
N_G(Z) \}$ (if $u \neq r)$ and $\{Z \mid Z \text{ is a component of }
G\setminus C \text{ with } C \cap N_G(Z)= \emptyset \}$ (if $u=r$). Each node $u$ in $T$ carries a label consisting of two parts:
- $L_w := \{level(w) \mid w < u \text{ and } uw \in E(G)\}$; and
- the multiset $\{F(Z^C) \mid Z \in \mathcal{Z}_u \}$.
\[prop:canonical\_tree\] For any graphs $G$ and $G'$, $T_G$ and $T_{G'}$ are isomorphic labelled trees if, and only if, $G \cong G'$.
If $G \cong G'$ then, by construction, their $d$-degree torsos induce isomorphic graphs. The canonical tree-depth decomposition of Bouland et al. then produces isomorphic directed trees and the isomorphism must preserve the labels that encode the rest of the graphs $G$ and $G'$ respectively.
For the converse direction, suppose we have an isomorphism $\phi$ between the labelled trees $T_G$ and $T_{G'}$. Since the label $L_u$ of any node $u$ encodes all ancestors of $u$ which are neighbours, $\phi$ must preserve all edges and non-edges in the $d$-degree torso $C$ of $G$. To extend $\phi$ to all of $G$, for each node $u$ in $T_G$, let $\beta_u$ be a bijection from $\mathcal{Z}_u$ to the corresponding set $\mathcal{Z}_{\phi(u)}$ of components of $G'\setminus C'$, such that $F(Z^C) =
F(\beta_u(Z)^{C'})$ (such a bijection exists as $u$ and $\phi(u)$ carry the same label). Thus, in particular, there is an isomorphism between $Z^C$ and $\beta_u(Z)^{C'}$, since they have the same canonical form. We define, for each $v \in
V(G)\setminus C$, $\phi(v)$ to be the image of $v$ under the isomorphism taking the component $Z$ containing $v$ to $\beta_u(Z)$. Note that this gives a well-defined function on $V(G)$, because for each such $v$, there is exactly one node $u$ of $T_G$ such that the component containing $v$ is in $\mathcal{Z}_u$. We claim that $\phi$ is now an isomorphism from $G$ to $G'$. Let $vw$ be an edge of $G$. If both $v$ and $w$ are in $C$, then either $v< w$ or $w
< v$. Assume, without loss of generality, that it is the former. Then, $\level(v) \in L_w$ is in the label of $w$ in $T_G$ and since $\phi$ is a label-preserving isomorphism from $T_G$ to $T_{G'}$, $\phi(v)\phi(w)$ is an edge in $G'$. If both $v$ and $w$ are in $G
\setminus C$, then there is some component $Z$ of $G \setminus C$ that contains them both. Since $\phi$ maps $Z$ to an isomorphic component of $G' \setminus C'$, $\phi(v)\phi(w) \in E(G')$. Finally, suppose $v$ is in $C$ and $w$ in $G\setminus C$ and let $Z$ be the component containing $w$. Then $i := \level(v)$ is part of the colour of $w$ in $Z^C$ and hence part of the colour of $\phi(w)$ in the corresponding component of $G'\setminus C'$. Moreover, if $u$ is the $\leq$-maximal element in $C \cap N_G(Z)$, then we must have $v \leq u$. Thus $\phi(v)$ is the unique element of level $i$ in $C' \cap
N_{G'}(\beta_u(Z))$ and we conclude that $\phi(v)\phi(w) \in E(G')$. By a symmetric argument, we have that for any edge $vw \in E(G')$, $\phi^{-1}(v)\phi^{-1}(w) \in E(G)$ and we conclude that $\phi$ is an isomorphism.
With this, we are able to establish our main result.
Graph Canonisation is $\FPT$ parameterized by elimination distance to bounded degree.
Suppose we are given a graph $G$ with $|V(G)| = n$. We first compute the $d$-degree torso $C$ of $G$ in $O(n^4)$ time. Using the result from Bouland *et. al* [@bouland_tractable_2012 Theorem 11], we can find a canonical tree-depth decomposition for $C$ in time $O(h(k)n^3log(n))$ for some computable function $h$. To compute the labels of the nodes in the trees (and hence obtain) $T_G$, we determine, for each $u \in C$, the set $\{level(w) \mid w
< u \text{ and } uw \in E(G)\}$. This can be done in time $O(n^2)$. Then, we find the components of $G\setminus C$, and colour the vertices with the levels of their neighbours in $C$. This can be done in $O(n^2)$ time. Finally, we compute for each coloured component $Z^C$ the canonical representative $F(Z^C)$ which, by Theorem \[T:bdd\_canon\] can be done in polynomial time (where the degree of the polynomial depends on $d$).
Having obtained $T_G$, we compute the canonical form $T_G'$ in linear time using Lindell’s canonisation algorithm [@lindell_logspace_1992]. Using the labels of $T_G'$ one can, in linear time, construct a graph $G'$ such that $T(G')=T_G'$. By Proposition \[prop:canonical\_tree\], this is a canonical form $G'$ of $G$.
Graph Isomorphism is $\FPT$ parameterized by elimination distance to bounded degree.
Conclusion {#S:conclusion}
==========
We introduce a new way of parameterizing graphs by their distance to triviality, i.e. by elimination distance. In the particular case of graph canonisation, and thus also graph isomorphism, taking triviality to mean graphs of bounded degree, we show that the problem is $\FPT$.
A natural question that arises is what happens when we take other classes of graphs for which graph isomorphism is known to be tractable as our “trivial” classes. For instance, what can we say about $\GI$ when parameterized by elimination distance to planar graphs? Unfortunately techniques such as those deployed in the present paper are unlikely to work in this case. Our techniques rely on identifying a canonical subgraph which defines an elimination tree into the trivial class. In the case of planar graphs, consider graphs which are subdivisions of $K_5$, each of which is deletion distance 1 away from planarity. However the deletion of *any* vertex yields a planar graph and it is therefore not possible to identify a canonical such vertex.
More generally, the notion of elimination distance to triviality seems to offer promise for defining tractable parameterizations for many graph problems other than isomorphism. This is a direction that bears further investigation.
It is easy to see that if a class of graphs $\C$ is characterised by a finite set of excluded minors, that the class $\hat\C$ of graphs with bounded elimination distance to $\C$ is characterised by a finite set of excluded minors as well. An interesting question is whether we can, given the set of excluded minors for $\C$, compute the excluded minors for $\hat\C$ as well?
[^1]: Research supported in part by EPSRC grant EP/H026835, DAAD grant A/13/05456, and DFG project *Logik, Struktur und das Graphenisomorphieproblem*.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'The spin structure of wave functions is reflected in the magnetic structure of the one-particle density matrix. Indeed, for single determinants we can use either one to determine the other. In this work we discuss how one can simply examine the one-particle density matrix to faithfully determine whether the spin magnetization density vector field is collinear, coplanar, or noncoplanar. For single determinants, this test suffices to distinguish collinear determinants which are eigenfunctions of $\hat{S}_{\hat{n}}$ from noncollinear determinants which are not. We also point out the close relationship between noncoplanar magnetism on the one hand and complex conjugation symmetry breaking on the other. Finally, we use these ideas to classify the various ways single determinant wave functions break and respect symmetries of the Hamiltonian in terms of their one-particle density matrix.'
author:
- 'Thomas M. Henderson'
- 'Carlos A. Jiménez-Hoyos'
- 'Gustavo E. Scuseria'
bibliography:
- 'GHF.bib'
title: On the Magnetic Structure of Density Matrices
---
Introduction
============
Magnetic structures are ubiquitous in nature and are of significant technological importance. At the microscopic level, we associate magnetism with electronic or nuclear spin: the spin structure of the electronic wave function yields information about observed magnetic properties. At the mean-field level, electronic magnetism is frequently associated with spin symmetry breaking, simply because most spin eigenfunctions cannot be described by a mean-field wave function. We should note, however, that restricted open-shell wave functions can be spin eigenfunctions and yet have magnetic character.
On the one hand, the symmetry breaking of Hartree-Fock is certainly artificial: for finite systems, the exact solution does not break symmetries. On the other hand, this symmetry breaking is not entirely unphysical, either. For example, consider the dissociation of the H$_2$ molecule. For large bond lengths, Hartree-Fock breaks spin symmetry, localizing the $\uparrow$-spin electron on one atom and the $\downarrow$-spin electron on the other. While the exact solution is entangled and does not have broken spin symmetry, it is also true that the exact solution, unlike the symmetry-adapted Hartree-Fock, always has one electron on one atom and the other electron on the other atom (at infinite separation). Thus, the broken spin symmetry has a certain degree of physical correctness: both the Hartree-Fock solution and the exact solutions display antiferromagnetism. What the broken-symmetry mean-field lacks is entanglement; it gives a sort of classical picture of the dissociated limit.
The story, somewhat unfortunately, is slightly more complicated than that. In addition to breaking $\hat{S}^2$ spin symmetry, Hartree-Fock can also break $\hat{S}_z$ spin symmetry, in what is known as generalized Hartree-Fock (GHF)[@fukutome1981; @stuber2003; @jimenezhoyos2011]. But not all GHF solutions are alike. Some may actually have an axis of spin quantization – for example, the wave function may be an eigenstate of $\hat{S}_x$. Though this would appear to be a GHF-type wave function, it is actually just an unrestricted Hartree-Fock (UHF) determinant with a rotated axis of spin quantization. We can always create such a solution by acting a spin rotation operator on a UHF determinant. But while these kinds of “GHF” solutions have a collinear (i.e. ferromagnetic or antiferromagnetic) structure, other kinds of GHF solutions may have coplanar but noncollinear magnetic structure, or even a general noncoplanar ordering, and when we refer to a GHF determinant we are really interested in one which is fully noncollinear. These noncollinear GHF states are particularly prevalent in systems which exhibit spin frustration[@Yamaguchi1999].
How are we to distinguish these various kinds of magnetic orderings of broken spin symmetry wave functions? Conceptually this seems easy enough: one could simply plot the magnetization vector field defined in Eqn. \[Def:MagnetizationVector\] below and examine it. But this is by no means a practical solution except, perhaps, for lattice Hamiltonians.
An important step was provided by Small, Sundstrom, and Head-Gordon (SSHG)[@Small2015]. They pointed out that because a collinear wave function is necessarily an eigenfunction of some spin operator $\hat{S}_{\hat{n}}$, where $\hat{n}$ is some spatial direction, one can determine whether a wave function is collinear or not by looking for the direction $\hat{n}$ which minimizes the fluctuation $\langle \hat{S}_{\hat{n}}^2 \rangle - \langle \hat{S}_{\hat{n}} \rangle^2$. If in some direction the fluctuation vanishes, the wave function must be an $\hat{S}_{\hat{n}}$ eigenfunction. This leads to the SSHG test to determine collinearity: a wave function is collinear if and only if the lowest eigenvalue of a matrix $\mathbf{A}$ vanishes, where the elements of $\mathbf{A}$ are $$A_{ij} = \langle \hat{S}_i \, \hat{S}_j \rangle - \langle \hat{S}_i \rangle \, \langle \hat{S}_j \rangle
\label{Def:SSHGMatrix}$$ and where $i$ and $j$ run over $x$, $y$, and $z$. Note that this test in general requires the two-particle density matrix. Moreover, it tests the spin structure of the wave function where we are more interested in examining the magnetic structure of the electronic density. Obviously the former determines the latter, but testing the latter is, as we shall see, perhaps somewhat simpler in that it does not require the two-particle density matrix. Finally, the SSHG test does not distinguish between coplanar and noncoplanar magnetizations, which would appear to arise from wave functions which break $\hat{S}_{\hat{n}}$ symmetry in different ways.
In this work, we seek to do several things. First, we provide a test for the collinearity or noncollinearity of the magnetization density, based on the structure of the spinorbital one-particle density matrix. This test is equivalent to the SSHG test for single determinant wave functions, though we provide an alternative conceptual motivation. We also show how to distinguish between coplanar and noncoplanar magnetization densities; this test is motivated by the observation that a noncoplanar magnetization density requires a complex wave function, and is novel. Finally, we note that while testing the magnetic structure of the one-particle density matrix does not allow us to infer too much about the spin characteristics of a general correlated wave function, they do allow us to determine whether a single determinant is collinear (i.e. an eigenfunction of $\hat{S}_{\hat{n}}$ for some $\hat{n}$) or not. Since our test and that of SSHG are equivalent for single determinants, this is none too surprising, but it allows us to extend the work of Fukutome[@fukutome1981] and of Stuber and Paldus[@stuber2003], who classified Hartree-Fock solutions in terms of the occupied molecular orbital coefficients. We show the equivalent classifications in terms of the one-particle density matrix. We demonstrate our ideas for a handful of systems for which GHF solutions can be found.
The Spinorbital One-Particle Density Matrix and the Magnetization Density
=========================================================================
Before we can discuss collinearity tests, we will require some preparatory material.
Let us begin, then, by considering the full spinorbital one-particle density matrix associated with a normalized state $|\Psi\rangle$, which may or may not be a single determinant and which we write as $$\gamma^{\eta\xi}_{\mu \nu} = \langle \Psi| c_{\nu_\xi}^\dagger \, c_{\mu_\eta} |\Psi\rangle$$ where $\nu$ and $\mu$ index spatial basis functions and $\eta$ and $\xi$ are spin indices. Quite generally, in a spin-orbital basis in which the first block index corresponds to $\uparrow$ spin and the second block index corresponds to $\downarrow$ spin, we have $$\bm{\gamma} = \begin{pmatrix} \bm{\gamma}^{\uparrow \uparrow} & \bm{\gamma}^{\uparrow \downarrow} \\ \bm{\gamma}^{\downarrow \uparrow} & \bm{\gamma}^{\downarrow \downarrow} \end{pmatrix}.$$
For our purposes, it is more convenient to decompose the density matrix into a charge component $\mathbf{P}$ and spin components $\vec{\mathbf{M}}$ as $$\bm{\gamma} = \begin{pmatrix} \mathbf{P} + \mathbf{M}^z & \mathbf{M}^x - \mathrm{i} \, \mathbf{M}^y \\ \mathbf{M}^x + \mathrm{i} \, \mathbf{M}^y & \mathbf{P} - \mathbf{M}^z \end{pmatrix} = \mathbf{P} \otimes \bm{1} + \vec{\mathbf{M}} \otimes \vec{\bm{\sigma}}.$$ Here, $\bm{1}$ is the identity matrix in spinor space, $\vec{\bm{\sigma}}$ is the vector of Pauli matrices, and $\otimes$ denotes the Kronecker product; $\mathbf{P}$ is the charge density matrix and $\vec{\mathbf{M}}$ is the vector of spin density matrices: $$\vec{\mathbf{M}} = \left( \mathbf{M}^x, \mathbf{M}^y, \mathbf{M}^z \right).$$ The individual component matrices are
$$\begin{aligned}
\mathbf{P} &= \frac{1}{2} \, \left(\bm{\gamma}^{\uparrow\uparrow} + \bm{\gamma}^{\downarrow\downarrow}\right),
\\
\mathbf{M}^x &= \frac{1}{2} \, \left(\bm{\gamma}^{\downarrow\uparrow} + \bm{\gamma}^{\uparrow\downarrow}\right),
\\
\mathbf{M}^y &= \frac{1}{2 \, \mathrm{i}} \, \left(\bm{\gamma}^{\downarrow\uparrow} - \bm{\gamma}^{\uparrow\downarrow}\right),
\\
\mathbf{M}^z &= \frac{1}{2} \, \left(\bm{\gamma}^{\uparrow\uparrow} - \bm{\gamma}^{\downarrow\downarrow}\right),\end{aligned}$$
and can be extracted from
$$\begin{aligned}
P_{\mu\nu} &= \frac{1}{2} \, \langle \Psi| c_{\nu_\uparrow}^\dagger \, c_{\mu_\uparrow} + c_{\nu_\downarrow}^\dagger \ c_{\mu_\downarrow} |\Psi\rangle \equiv \langle \Psi| \hat{P}_{\mu\nu} |\Psi\rangle,
\\
M^x_{\mu\nu} &= \frac{1}{2} \, \langle \Psi| c_{\nu_\uparrow}^\dagger \, c_{\mu_\downarrow} + c_{\nu_\downarrow}^\dagger \ c_{\mu_\uparrow} |\Psi\rangle \equiv \langle \Psi| \hat{M}^x_{\mu\nu} |\Psi\rangle,
\\
M^y_{\mu\nu} &= \frac{1}{2 \, \mathrm{i}} \, \langle \Psi| c_{\nu_\uparrow}^\dagger \, c_{\mu_\downarrow} - c_{\nu_\downarrow}^\dagger \ c_{\mu_\uparrow} |\Psi\rangle \equiv \langle \Psi| \hat{M}^y_{\mu\nu} |\Psi\rangle,
\\
M^z_{\mu\nu} &= \frac{1}{2} \, \langle \Psi| c_{\nu_\uparrow}^\dagger \, c_{\mu_\uparrow} - c_{\nu_\downarrow}^\dagger \ c_{\mu_\downarrow} |\Psi\rangle \equiv \langle \Psi| \hat{M}^z_{\mu\nu} |\Psi\rangle.\end{aligned}$$
Having defined these magnetization density matrices, we can now define the magnetization vector field or, if one prefers, the spin density vector field. Choosing our basis to be real as we can do without loss of generality, the magnetization vector at a point in space is simply $$\vec{m}(\vec{r}) = \sum_{\mu,\nu} \chi_\mu(\vec{r}) \, \chi_\nu(\vec{r}) \, \vec{\mathbf{M}}_{\mu\nu}.
\label{Def:MagnetizationVector}$$ Note that only the symmetric part of $\vec{\mathbf{M}}$ contributes to the magnetization vector. Because $\vec{\mathbf{M}}$ is Hermitian, its symmetric part is its real part. If the density matrix $\bm{\gamma}$ is real, then $\mathbf{M}^y$ is purely imaginary, hence $m_y(\vec{r})$ vanishes identically and the magnetization density $\vec{m}(\vec{r})$ is coplanar. In other words, a real wave function has coplanar magnetism. The converse is not necessarily true: coplanar magnetism does not necessarily imply a real wave function. Conceivably, we could have, for example, $\mathbf{M}^y$ purely imaginary with complex $\mathbf{M}^x$ and $\mathbf{M}^z$. We should note that the imaginary parts of of $\vec{\mathbf{M}}$ do contribute to the spin current density[@fukutome1981]. We shall have more to say on the spin current density later.
Testing Magnetic Structure
==========================
Suppose that a wave function is an eigenfunction of $\hat{S}_z$ (and of the total number operator). Then that wave function has a definite number of $\uparrow$-spin and of $\downarrow$-spin electrons. For such a wave function, $\gamma^{\uparrow\downarrow}$ and $\gamma^{\downarrow\uparrow}$ must be identically zero, because the operator $c^\dagger_\uparrow \, c_\downarrow$ changes the number of electrons of each spin direction when acting on that wave function. Thus, an $\hat{S}_z$ eigenfunction has a block diagonal spinorbital density matrix, and if the spinorbital density matrix is not block diagonal, the wave function is not an eigenfunction of $\hat{S}_z$.
Note that if the spinorbital density matrix is block diagonal, we cannot guarantee that the underlying wave function is an eigenfunction of $\hat{S}_z$ unless the wave function is a single determinant. If the wave function is a single determinant, diagonalizing the spinorbital density matrix allows us to obtain the occupied orbitals; if the density matrix is block diagonal, the occupied orbitals can be chosen to be $\hat{S}_z$ eigenfunctions, and if the occupied orbitals are $\hat{S}_z$ eigenfunctions, so is the determinant. To test the spin structure of a general wave function, we require the two-particle density matrix, as SSHG pointed out.
We have discussed the special case of magnetization aligned along the $z$ axis, but of course nothing privileges that axis. Quite generally, if a wave function is an $\hat{S}_{\hat{n}}$ eigenfunction then the spinorbital density matrix is block diagonal in spin blocks where $\uparrow$ and $\downarrow$ are defined relative to $\hat{n}$. If the density matrix cannot be brought to this form, the magnetization is noncollinear and the wave function is not an eigenfunction of $\hat{S}_{\hat{n}}$ for any direction $\hat{n}$; if the density matrix can be brought to this form, the magnetization vector field is collinear and the wave function, if a single determinant, is guaranteed to be an eigenfunction of $\hat{S}_{\hat{n}}$.
Spin Rotation Operators
-----------------------
To test this possibility, we must consider spin rotation operators. We define a unitary spin rotation operator[@PHF] $$R(\Omega) = \mathrm{e}^{\mathrm{i} \, \gamma \, \hat{S}_z} \, \mathrm{e}^{\mathrm{i} \, \beta \, \hat{S}_y} \, \mathrm{e}^{\mathrm{i} \, \alpha \, \hat{S}_z}.
\label{Def:RotationOperators}$$ where $\Omega$ stands for the collection of rotation angles $\left(\alpha,\beta,\gamma\right).$ The three angles $\alpha$, $\beta$, and $\gamma$ are Euler angles and cover the sphere. With this operator we can define a rotated state $$|\tilde{\Psi}_\Omega\rangle = \hat{R}(\Omega) |\Psi\rangle.$$ Note that if the Hamiltonian commutes with the spin operators then $|\Psi\rangle$ and $|\tilde{\Psi}_\Omega\rangle$ are degenerate $$\langle \tilde{\Psi}_\Omega | \hat{H} | \tilde{\Psi}_\Omega \rangle
= \langle \Psi | \hat{R}^\dagger(\Omega) \, \hat{H} \, \hat{R}(\Omega) |\Psi\rangle
= \langle \Psi| \hat{H} |\Psi\rangle$$ where we have used the fact that $\hat{H}$ commutes with $\hat{R}$ and that $\hat{R}^\dagger \, \hat{R} = 1$. Note also that if $|\Psi\rangle$ is a single determinant, so too is $|\tilde{\Psi}(\Omega)\rangle$, because $\hat{R}$ is a series of exponentials of one-body operators, i.e. it is a Thouless transformation[@Thouless1960]. Together, these observations imply that if $|\Psi\rangle$ is a solution of the Hartree-Fock equations, then so too is $|\tilde{\Psi}(\Omega)\rangle$. In fact, the collection of states $|\tilde{\Psi}(\Omega)\rangle$ forms a manifold known as the Goldstone manifold and is used in spin symmetry projection[@Ring80; @Blaizot85; @PHF].
Let us now consider the rotated density matrix. Generically, we will have
$$\begin{aligned}
\tilde{\gamma}^{\eta\xi}_{\mu \nu}
&= \langle \tilde{\Psi}| c_{\nu_\xi}^\dagger \, c_{\mu_\eta} |\tilde{\Psi}\rangle
\\
&= \langle \Psi| \hat{R}^\dagger \, c_{\nu_\xi}^\dagger \, c_{\mu_\eta} \, \hat{R}|\Psi\rangle
\\
&= \langle \Psi| \hat{R}^\dagger \, c_{\nu_\xi}^\dagger \, \hat{R} \, \hat{R}^\dagger \, c_{\mu_\eta} \, \hat{R}|\Psi\rangle
\\
&= \langle \Psi| \tilde{c}_{\nu_\xi}^\dagger \, \tilde{c}_{\mu_\eta} |\Psi\rangle\end{aligned}$$
where $\tilde{c}$ is the rotated annihilation operator. The first line shows a sort of active rotation perspective: the rotation operator is understood as rotating the wave function, and we consider the density matrix expressed in terms of the original spin coordinates. We see, however, from the last line that this is equivalent to a passive rotation perspective: the wave function is left alone and the underlying basis is rotated. This latter perspective is more helpful for our purposes: if by such a rotation we can eliminate $\mathbf{M}^x$ and $\mathbf{M}^y$, the magnetization density is collinear.
Using the representation of $\hat{S}_z$ in terms of fermionic creation and annihilation operators, $$\hat{S}_z = \frac{1}{2} \, \left(c_\uparrow^\dagger \, c_\uparrow - c_\downarrow^\dagger \, c_\downarrow\right),$$ we see that
$$\begin{aligned}
[c_\uparrow^\dagger, \hat{S}_z] &= -\frac{1}{2} \, c_\uparrow^\dagger,
\\
[c_\downarrow^\dagger, \hat{S}_z] &= \frac{1}{2} \, c_\downarrow^\dagger.\end{aligned}$$
Then we can resum the commutator expansion analytically, and one can show that
$$\begin{aligned}
\mathrm{e}^{-\mathrm{i} \, \theta \, \hat{S}_z} \, c_\uparrow^\dagger \, \mathrm{e}^{\mathrm{i} \, \theta \, \hat{S}_z}
&= \mathrm{e}^{-\frac{1}{2} \, \mathrm{i} \, \theta} \, c_\uparrow^\dagger,
\\
\mathrm{e}^{-\mathrm{i} \, \theta \, \hat{S}_z} \, c_\downarrow^\dagger \, \mathrm{e}^{\mathrm{i} \, \theta \, \hat{S}_z}
&= \mathrm{e}^{\frac{1}{2} \, \mathrm{i} \, \theta} \, c_\downarrow^\dagger.\end{aligned}$$
This is turn implies that
$$\begin{aligned}
\tilde{\mathbf{P}} &= \mathbf{P},
\\
\tilde{\mathbf{M}}^x &= \cos(\theta) \, \mathbf{M}^x + \sin(\theta) \, \mathbf{M}^y,
\\
\tilde{\mathbf{M}}^y &= \cos(\theta) \, \mathbf{M}^y - \sin(\theta) \, \mathbf{M}^x,
\\
\tilde{\mathbf{M}}^z &= \mathbf{M}^z.\end{aligned}$$
where $\tilde{\mathbf{P}}$ and $\tilde{\mathbf{M}}^i$ are the components of the rotated density matrix $\tilde{\bm{\gamma}}$. We can express this concisely as
$$\begin{aligned}
\tilde{\mathbf{P}} &= \mathbf{P},
\\
\tilde{\mathbf{M}} &= \mathbf{R}_z(\theta) \, \mathbf{M}.\end{aligned}$$
Here, $\mathbf{R}_z(\theta)$ is the rotation matrix corresponding to rotation by angle $\theta$ about the $z$ axis and $\tilde{\mathbf{M}}$ and $\mathbf{M}$ are written as column vectors. One finds equivalent results for $\hat{S}_x$ and $\hat{S}_y$ spin rotations. Spin rotations of the wave function manifest as spatial rotations of the magnetization density matrices.
Note that we are using the word “spatial” here in a somewhat cavalier sense: the directions $x$, $y$, and $z$ in the magnetization density matrices are not physically significant in the absence of an external field.
It may prove useful to note the spatial rotation matrices corresponding to the spin rotation operator of Eqn. \[Def:RotationOperators\]. We have
$$\begin{aligned}
\mathbf{R}_x(\theta) &= \begin{pmatrix} 1 & 0 & 0 \\ 0 & \cos(\theta) & \sin(\theta) \\ 0 & -\sin(\theta) & \cos(\theta) \end{pmatrix},
\\
\mathbf{R}_y(\theta) &= \begin{pmatrix} \cos(\theta) & 0 & -\sin(\theta) \\ 0 & 1 & 0 \\ \sin(\theta) & 0 & \cos(\theta) \end{pmatrix},
\\
\mathbf{R}_z(\theta) &= \begin{pmatrix} \cos(\theta) & \sin(\theta) & 0 \\ -\sin(\theta) & \cos(\theta) & 0 \\ 0 & 0 & 1\end{pmatrix}.\end{aligned}$$
Note that this is opposite the usual convention for passive rotations. We have included $\mathbf{R}_x(\theta)$ for completeness.
Let us make one final observation. While spin rotations cannot convert a collinear density matrix into a noncollinear density matrix or a coplanar density matrix into a noncoplanar one, they can convert a real density matrix into a complex density matrix. The spin rotation operator, that is, does not commute with the complex conjugation operator defined below. A complex conjugation eigenfunction, upon spin rotation, may cease to be a complex conjugation eigenfunction.
Testing Collinearity and Coplanarity
------------------------------------
We can take advantage of the correspondence between spin rotations of $|\Psi\rangle$ and spatial rotations of $\vec{\mathbf{M}}$ to test the magnetic structure of the density matrix. We note the following:
- If the spin density matrices $\vec{\mathbf{M}}$ are all identically zero, then the magnetization density vanishes. If the wave function is a single determinant, it is an eigenfunction of $\hat{S}^2$ with eigenvalue zero, and is also an eigenfunction of $\hat{S}_{\hat{n}}$ with eigenvalue 0 for all directions $\hat{n}$. This is the case for RHF.
- If the spin density matrices can be rotated so that $\mathbf{M}^z$ is nonzero but both $\mathbf{M}^x$ and $\mathbf{M}^y$ are zero, then the magnetization density is collinear. The underlying wave function is not a singlet (but may be an eigenfunction of $\hat{S}^2$). If the wave function is a single determinant, it is definitely an eigenfunction of $\hat{S}_{\hat{n}}$ for some direction $\hat{n}$. This is the case of UHF and also of rotated UHF solutions.
- Otherwise the magnetization is noncollinear and the wave function is not an eigenfunction of $\hat{S}_{\hat{n}}$ for any direction $\hat{n}$, whether the wave function is a single determinant or not. If the wave function is a single determinant, it is not an eigenfunction of $\hat{S}^2$. This is the case of GHF.
In other words, if the wave function yields a nonzero spin density matrix, it is not a singlet; if the density matrix can be rotated to have the UHF structure, then the magnetization vector field is collinear and the wave function, if a single determinant, is definitely an eigenfunction of $\hat{S}_{\hat{n}}$; if the density matrix cannot be rotated to have the UHF structure then the magnetization vector field is noncollinear and the wave function is not an eigenfunction of $\hat{S}_{\hat{n}}$.
To see whether a spin density matrix vanishes or not, it is simplest to test its Frobenius norm. Recall that the (square of the) Frobenius norm of a matrix $\mathbf{X}$ is $$\| \mathbf{X}\|^2 = \sum_{pq} X_{pq} \, X_{pq}^\star = \mathrm{Tr}(\mathbf{X} \, \mathbf{X}^\dagger).$$ Our matrices are Hermitian, so $$\| \mathbf{X} \|^2 = \mathrm{Tr}(\mathbf{X}^2).$$
We wish to maximize the norm of one component of $\vec{\mathbf{M}}$. To do so, we can diagonalize a matrix $\mathbf{T}$ given by $$T_{ij} = \mathrm{Tr}(\mathbf{M}^i \, \mathbf{M}^j).
\label{Eqn:DefTMatrix}$$ This matrix is real and symmetric. Its diagonal components are the norms of the various magnetization density matrices. Its off-diagonal components can be brought to zero by a sequence of rotations or, more correctly, we can bring $\mathbf{T}$ to diagonal form using spin rotation operators of the sort given in Eqn. \[Def:RotationOperators\]. Diagonalizing $\mathbf{T}$ is tantamount to finding the spin rotation which maximizes the norm of the largest component of $\vec{\mathbf{M}}$ and minimizes the norm of the smallest compotent; in other words, the diagonal elements of $\mathbf{T}$ cannot be rotated to be larger than the largest eigenvalue of $\mathbf{T}$ or smaller than the smallest eigenvalue of $\mathbf{T}$.
Our procedure in full is thus simple. We build the matrix $\mathbf{T}$ and diagonalize it. If $\mathbf{T}$ has three zero eigenvalues, then $\vec{\mathbf{M}}$ vanishes and the wave function, if a single determinant, has the RHF structure. If $\mathbf{T}$ has two zero eigenvalues, the magnetization was collinear. Otherwise it was noncollinear. Depending on the outcome of the test and on whether the wave function is a single determinant or not, we may or may not be able to say whether the wave function itself is an eigenfunction of $\hat{S}^2$ or of $\hat{S}_{\hat{n}}$. For single determinants, the test is equivalent to the test of SSHG (see below); for multideterminantal wave functions it is not.
If the magnetization is noncollinear, we can repeat the test but with a modified matrix $$\mathcal{T}_{ij} = \mathrm{Tr}[\mathrm{Re}{(\mathbf{M}^i)} \, \mathrm{Re}{(\mathbf{M}^j)}].$$ If there are any zero eigenvalues, the magnetization was coplanar because we could rotate to make one of the components of $\vec{\mathbf{M}}$ purely imaginary. Note that this coplanarity test is new.
In a non-orthonormal basis, we have $$T_{ij} = \mathrm{Tr}(\mathbf{M}_i \, \mathbf{S} \, \mathbf{M}_j \, \mathbf{S})$$ where $\mathbf{S}$ is the overlap matrix of spatial orbitals, and analogously for $\mathcal{T}_{ij}$.
Let us make a few caveats. First, it is possible in principle for $\mathbf{T}$ to have one zero eigenvalue, which means we can bring $\vec{\mathbf{M}}$ to the form $\vec{\mathbf{M}} = (\mathbf{M}^x,\mathbf{0},\mathbf{M}^z)$. This would of course correspond to the coplanar case, but where a typical coplanar magnetic structure has coplanar spin density but may have noncoplanar spin density matrices, this case corresponds to coplanar spin density matrices. Second, we must point out the existence of paired UHF and paired GHF solutions (see below). In these cases, $\vec{m}$ vanishes, and $\bm{\mathcal{T}} = \mathbf{0}$, yet $\mathbf{T}$ may have one or more nonzero eigenvalues. Lastly, after diagonalization of $\mathbf{T}$ we choose directions such that $\| \mathbf{M}^y \| \le \| \mathbf{M}^x \| \le \| \mathbf{M}^z\|$, which we can always do as a matter of convenience.
### Collinear Spin Densities
Let us make a quick comment on the collinear case. If after the final rotation the density matrices are $$\vec{\mathbf{M}}^{\prime\prime\prime} = (\mathbf{0},\mathbf{0},\mathbf{Z}),$$ then before that final rotation (i.e. after the second rotation) the density matrices must also have been $$\vec{\mathbf{M}}^{\prime\prime} = (\mathbf{0},\mathbf{0},\mathbf{Z}).$$ This in turn means that before the $y$ rotation (and therefore after the first $z$ rotation) the density matrices were $$\vec{\mathbf{M}}^\prime = (\mathbf{Z} \, \sin(\beta), \mathbf{0}, \mathbf{Z} \, \cos(\beta)).$$ And lastly, this in turn means the initial unrotated density matrices were $$\vec{\mathbf{M}} = (\mathbf{Z} \, \sin(\beta) \, \cos(\alpha),-\mathbf{Z} \, \sin(\beta) \, \sin(\alpha),\mathbf{Z} \, \cos(\beta)).$$ A collinear solution, in other words, is characterized by a density matrix vector which is a spatial unit vector times a single matrix: $$\vec{\mathbf{M}} = \hat{n} \, \mathbf{Z}$$ where $$\hat{n} = (\sin(\beta) \, \cos(\alpha), -\sin(\beta) \, \sin(\alpha), \cos(\beta)).$$ An alternative test for whether a set of spin density matrices is collinear, then, is simply to see whether the components $\mathbf{M}^x$, $\mathbf{M}^y$, and $\mathbf{M}^z$ are all multiples of the same matrix $\mathbf{M}$.
### The SSHG Test
The collinearity test of SSHG in general looks at eigenvalues of the matrix $\mathbf{A}$ defined in Eqn. \[Def:SSHGMatrix\]. Let us take a moment to rewrite this matrix for the case of a single determinant, using the language of the previous section. We will assume an orthonormal basis for simplicity.
In their paper, SSHG note that for a single determinant, one finds $$A_{ij} = -\mathrm{Tr}(\mathbf{O}_i \, \mathbf{O}_j) + \frac{1}{4} \, \delta_{ij} \, N,$$ where $N$ is the number of electrons and $$\mathbf{O}_i = \frac{1}{2} \, \mathbf{C}_\mathrm{occ}^\dagger \, \left(\bm{1} \otimes \bm{\sigma}_i\right) \, \mathbf{C}_\mathrm{occ}.$$ Here, $\mathbf{C}_\mathrm{occ}$ is the matrix of occupied orbital coefficients.
Using the cyclic property of traces, we can equivalently write $$\begin{aligned}
A_{ij} = -\frac{1}{4} \, \mathrm{Tr}[&\mathbf{C}_\mathrm{occ} \, \mathbf{C}_\mathrm{occ}^\dagger \, \left(\bm{1} \otimes \bm{\sigma}_i\right)
\\
& \times \mathbf{C}_\mathrm{occ} \, \mathbf{C}_\mathrm{occ}^\dagger \, \left(\bm{1} \otimes \bm{\sigma}_j\right)] + \frac{1}{4} \, \delta_{ij} \, N.
\nonumber\end{aligned}$$ One can recognize the density matrix $\bm{\gamma} = \mathbf{C}_\mathrm{occ} \, \mathbf{C}_\mathrm{occ}^\dagger$, and note that $$N = \mathrm{Tr}(\bm{\gamma}) = 2 \, \mathrm{Tr}(\mathbf{P}).$$ Then one has $$A_{ij} = -\frac{1}{4} \, \mathrm{Tr}[\bm{\gamma} \, \left(\bm{1} \otimes \bm{\sigma}_i\right) \, \bm{\gamma} \, \left(\bm{1} \otimes \bm{\sigma}_j\right)] + \frac{1}{2} \, \delta_{ij} \, \mathrm{Tr}(\mathbf{P}).$$
Inserting our decomposition of $\bm{\gamma}$, one finds that the components of $\mathbf{A}$ are $$A_{ij} = -\mathrm{Tr}(\mathbf{M}^i \, \mathbf{M}^j) + \frac{1}{2} \, \delta_{ij} \, \mathrm{Tr}[\mathbf{P} - \mathbf{P}^2 + \vec{\mathbf{M}} \cdot \vec{\mathbf{M}}].$$
For a single determinant, $\bm{\gamma}$ is idempotent. We have $$\bm{\gamma}^2 = \left(\mathbf{P} \otimes \bm{1} + \mathbf{M}^i \otimes \bm{\sigma}^i\right)^2 = \mathbf{P} \otimes \bm{1} + \mathbf{M}^i \otimes \bm{\sigma}^i$$ where here we employ the summation convention. Using $$\bm{\sigma}^i \, \bm{\sigma}^j = \delta_{ij} \, \bm{1} + \mathrm{i} \, \epsilon_{ijk} \, \bm{\sigma}^k,$$ we see that the portion of $\bm{\gamma}^2$ which is proportional to the identity in spin space is simply $\mathbf{P}^2 + \mathbf{M}^i \, \mathbf{M}^i$. Idempotency of the one-particle density matrix implies that $$\mathbf{P} - \mathbf{P}^2 = \vec{\mathbf{M}} \cdot \vec{\mathbf{M}}.$$ We can thus write the matrix $\mathbf{A}$ simply as $$A_{ij} =- T_{ij} + \delta_{ij} \, \mathrm{Tr}(\mathbf{P} - \mathbf{P}^2).$$ For a single determinant, the SSHG test can be reformulated in terms of diagonalization of the simple matrix $\mathbf{T}$.
For the sake of completeness, we reiterate the three possibilities for a single determinant here, in terms of $\mathbf{T}$ and of $\mathbf{A}$:
- If the determinant is a singlet, then $\mathbf{A} = \mathbf{T} = \mathbf{0}$; both matrices of course have three zero eigenvalues.
- If the determinant is collinear, then $\mathbf{T}$ has one non-zero eigenvalue $\lambda = \mathrm{Tr}(\mathbf{P} - \mathbf{P}^2)$ and two zero eigenvalues; the eigenvalues of $\mathbf{A}$ are $(0,\lambda,\lambda)$.
- If the determinant is noncollinear, then $\mathbf{T}$ has no more than one zero eigenvalue (and usually has none); $\mathbf{A}$ has no zero eigenvalues.
------------- --------------- ----------------------------------------------------- ---------------------------------------------------------------------- --------------------------------------------------------------------------------------------------
Fukutome Stuber-Paldus Symmetries Structure of Occupied Orbital Structure of
Designation Designation Preserved Coefficient Matrix $\mathbf{C}_\mathrm{occ}$ Density Matrices
TICS[^1] Real RHF $\hat{S}^2$, $\hat{S}_z$, $\hat{K}$, $\hat{\Theta}$ $\begin{pmatrix} $\vec{\mathbf{M}} = \vec{\bm{0}}$, $\mathbf{P} \in \mathbb{R}$
\mathbf{C}_{\sigma\sigma} & \bm{0} \\
\bm{0} & \mathbf{C}_{\sigma\sigma}
\end{pmatrix},
\mathbf{C}_\mathrm{occ} \in \mathbb{R}$
CCW[^2] Complex RHF $\hat{S}^2$, $\hat{S}_z$ $\begin{pmatrix} $\vec{\mathbf{M}} = \vec{\bm{0}}$
\mathbf{C}_{\sigma\sigma} & \bm{0} \\
\bm{0} & \mathbf{C}_{\sigma\sigma}
\end{pmatrix}$
ASCW[^3] Paired UHF $\hat{S}_z$, $\hat{\Theta}$ $\begin{pmatrix} $\mathbf{M}^x = \mathbf{M}^y = \bm{0}$, $(\mathbf{P},\mathrm{i} \, \mathbf{M}^z) \in \mathbb{R}$
\mathbf{C}_{\sigma\sigma} & \bm{0} \\
\bm{0} & \mathbf{C}_{\sigma\sigma}^\star
\end{pmatrix}$
ASDW[^4] Real UHF $\hat{S}_z$, $\hat{K}$ $\begin{pmatrix} $\mathbf{M}^x = \mathbf{M}^y = \bm{0}$, $(\mathbf{P},\mathbf{M}^z) \in \mathbb{R}$
\mathbf{C}_{\sigma\sigma} & \bm{0} \\
\bm{0} & \mathbf{C}_{\sigma'\sigma'}
\end{pmatrix},
\mathbf{C}_\mathrm{occ} \in \mathbb{R}$
ASW[^5] Complex UHF $\hat{S}_z$ $\begin{pmatrix} $\mathbf{M}^x = \mathbf{M}^y = \bm{0}$
\mathbf{C}_{\sigma\sigma} & \bm{0} \\
\bm{0} & \mathbf{C}_{\sigma'\sigma'}
\end{pmatrix}$
TSCW[^6] Paired GHF $\hat{\Theta}$ $\begin{pmatrix} $(\mathbf{P},\mathrm{i} \vec{\mathbf{M}}) \in \mathbb{R}$
\mathbf{C}_{\sigma \sigma} & \mathbf{C}_{\sigma \sigma'} \\
-\mathbf{C}_{\sigma\sigma'}^\star & \mathbf{C}_{\sigma\sigma}^\star
\end{pmatrix}$
TSDW[^7] Real GHF $\hat{K}$ $\begin{pmatrix} $(\mathbf{P},\vec{\mathbf{M}}) \in \mathbb{R}$
\mathbf{C}_{\sigma\sigma} & \mathbf{C}_{\sigma\sigma'} \\
\mathbf{C}_{\sigma'\sigma} & \mathbf{C}_{\sigma'\sigma'}
\end{pmatrix},
\mathbf{C}_\mathrm{occ} \in \mathbb{R}$
TSW[^8] Complex GHF $\begin{pmatrix}
\mathbf{C}_{\sigma\sigma} & \mathbf{C}_{\sigma\sigma'} \\
\mathbf{C}_{\sigma'\sigma} & \mathbf{C}_{\sigma'\sigma'}
\end{pmatrix}$
------------- --------------- ----------------------------------------------------- ---------------------------------------------------------------------- --------------------------------------------------------------------------------------------------
Classification of Hartree-Fock Solutions
========================================
We have seen that the collinearity test informs us about the symmetries of single determinants. Let us therefore take a moment to revisit the classification of Hartree-Fock solutions in terms of symmetries, first proposed by Fukutome[@fukutome1981] and later analyzed by Stuber and Paldus[@stuber2003]. The various classifications are presented in Tab. \[Tab:Symmetries\]. In addition to the spin operators $\hat{S}^2$ and $\hat{S}_z$ we also have the complex conjugation operator $\hat{K}$ and the time reversal operator $\hat{\Theta}$. For our purposes it is enough to define $\hat{K}$ and $\hat{\Theta}$ by their action on a single determinant. Suppose a determinant $\Phi$ is specified by a matrix of occupied molecular orbital coefficients $$\mathbf{C}_\mathrm{occ}(\Phi) = \begin{pmatrix} \mathbf{C}_\mathrm{occ}^\uparrow \\ \mathbf{C}_\mathrm{occ}^\downarrow \end{pmatrix}.$$ Then the determinants $\hat{K} \Phi$ and $\hat{\Theta} \Phi$ are specified by matrices of occupied molecular orbital coefficients which are respectively
$$\begin{aligned}
\mathbf{C}_\mathrm{occ}(\hat{K} \Phi)
&= \begin{pmatrix} \left(\mathbf{C}_\mathrm{occ}^\uparrow\right)^\star \\ \left(\mathbf{C}_\mathrm{occ}^\downarrow\right)^\star \end{pmatrix},
\\
\mathbf{C}_\mathrm{occ}(\hat{\Theta} \Phi)
&= \begin{pmatrix} -\left(\mathbf{C}_\mathrm{occ}^\downarrow\right)^\star \\ \left(\mathbf{C}_\mathrm{occ}^\uparrow\right)^\star \end{pmatrix}.\end{aligned}$$
Thus, we have $$\hat{\Theta} = -\mathrm{i} \, \bm{\sigma}^y \, \hat{K}.$$ The classifications in Tab. \[Tab:Symmetries\] were presented originally in terms of occupied molecular orbital coefficients; here, we list the corresponding constraints on the density matrix components, which were also discussed earlier by Weiner and Trickey[@Weiner1998].
For the most part the constraints on the density matrix are obvious. We must spend a few moments to consider the density matrices of paired UHF and paired GHF. Note that paired UHF requires an equal number of $\uparrow$-spin and $\downarrow$-spin electrons, while paired GHF requires an even number of electrons.
The paired UHF molecular orbital coefficients satisfy $$\mathbf{C}_\mathrm{occ} = \begin{pmatrix} \mathbf{A} & \mathbf{0} \\ \mathbf{0} & \mathbf{A}^\star \end{pmatrix}$$ so the density matrix is $$\begin{aligned}
\bm{\gamma}
&= \mathbf{C}_\mathrm{occ} \, \mathbf{C}^\dagger_\mathrm{occ}
\\
&= \begin{pmatrix} \mathbf{A} \, \mathbf{A}^\dagger & \mathbf{0} \\ \mathbf{0} & \mathbf{A}^\star \, \mathbf{A}^\mathsf{T} \end{pmatrix}.
\nonumber\end{aligned}$$ Then
$$\begin{aligned}
\mathbf{P}
&= \frac{1}{2} \, \left(\mathbf{A} \, \mathbf{A}^\dagger + \mathbf{A}^\star \, \mathbf{A}^\mathsf{T}\right),
\\
\mathbf{M}^z
&= \frac{1}{2} \, \left(\mathbf{A} \, \mathbf{A}^\dagger - \mathbf{A}^\star \, \mathbf{A}^\mathsf{T}\right).\end{aligned}$$
Clearly, $\mathbf{P}$ is real and $\mathbf{M}^z$ is purely imaginary.
Similarly, for paired GHF the orbital coefficients are $$\mathbf{C}_\mathrm{occ} = \begin{pmatrix} \mathbf{A} & \mathbf{B} \\ -\mathbf{B}^\star & \mathbf{A}^\star \end{pmatrix}$$ so that the density matrix is $$\bm{\gamma} = \begin{pmatrix} \mathbf{A} \, \mathbf{A}^\dagger + \mathbf{B} \, \mathbf{B}^\dagger & -\mathbf{A} \, \mathbf{B}^\mathsf{T} + \mathbf{B} \, \mathbf{A}^\mathsf{T} \\ -\mathbf{B}^\star \, \mathbf{A}^\dagger + \mathbf{A}^\star \, \mathbf{B}^\dagger & \mathbf{A}^\star \, \mathbf{A}^\mathsf{T} + \mathbf{B} \, \mathbf{B}^\mathsf{T} \end{pmatrix}.$$ Then the charge and spin density matrices are
$$\begin{aligned}
\mathbf{P}
&= \frac{1}{2} \, \left(\mathbf{A} \, \mathbf{A}^\dagger + \mathbf{B} \, \mathbf{B}^\dagger + \mathbf{A}^\star \, \mathbf{A}^\mathsf{T} + \mathbf{B} \, \mathbf{B}^\mathsf{T}\right),
\\
\mathbf{M}^x
&= \frac{1}{2} \, \left(-\mathbf{B}^\star \, \mathbf{A}^\dagger + \mathbf{A}^\star \, \mathbf{B}^\dagger -\mathbf{A} \, \mathbf{B}^\mathsf{T} + \mathbf{B} \, \mathbf{A}^\mathsf{T}\right),
\\
\mathbf{M}^x
&= \frac{1}{2 \, \mathrm{i}} \, \left(-\mathbf{B}^\star \, \mathbf{A}^\dagger + \mathbf{A}^\star \, \mathbf{B}^\dagger + \mathbf{A} \, \mathbf{B}^\mathsf{T} - \mathbf{B} \, \mathbf{A}^\mathsf{T}\right),
\\
\mathbf{M}^z
&= \frac{1}{2} \, \left(\mathbf{A} \, \mathbf{A}^\dagger + \mathbf{B} \, \mathbf{B}^\dagger - \mathbf{A}^\star \, \mathbf{A}^\mathsf{T} - \mathbf{B} \, \mathbf{B}^\mathsf{T}\right).\end{aligned}$$
Again, it is clear that $\mathbf{P}$ is real and $\vec{\mathbf{M}}$ is purely imaginary.
Recall from our earlier discussions that if a density matrix component is purely imaginary, the corresponding magnetization density vector field component vanishes. We thus see that paired UHF and paired GHF both have $\vec{m}(\vec{r}) = \vec{0}$. This is physically sensible, in that paired UHF and paired GHF remain time-reversal invariant. Only mean-field wave functions which break time-reversal symmetry can have non-zero magnetization density vector fields,[^9] just as only those which break complex conjugation symmetry can have non-coplanar magnetization density vector fields.
Type of Determinant $\vec{j}(\vec{r})$ $\vec{m}(\vec{r})$ $\vec{J}^x(\vec{r})$ $\vec{J}^y(\vec{r})$ $\vec{J}^z(\vec{r})$
--------------------- -------------------- ------------------------- ---------------------- ---------------------- ----------------------
Real RHF – – – – –
Complex RHF $\checkmark$ – – – –
Real UHF – $m(\vec{r}) \, \hat{z}$ – – –
Paired UHF $\checkmark$ – – – $\checkmark$
Complex UHF $\checkmark$ $m(\vec{r}) \, \hat{z}$ – – $\checkmark$
Real GHF – $m_y(\vec{r}) = 0$ – $\checkmark$ –
Paired GHF $\checkmark$ – $\checkmark$ $\checkmark$ $\checkmark$
Complex GHF $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$
: Constraints on densities and current densities for various kinds of single determinants. If the entry is – then the corresponding vector must vanish; if the entry is $\checkmark$ then the corresponding vector is not constrained. \[Tab:Classification2\]
Let us take one more brief digression. In addition to the magnetization density vector field $\vec{m}(\vec{r})$ we can define three other relevant densities. There is of course the familiar charge density $$n(\vec{r}) = \sum_{\mu,\nu} \chi_\mu(\vec{r}) \, \chi_\nu(\vec{r}) \, P_{\mu\nu}.$$ For purposes of classifying determinants, it is uninteresting. We can also define the charge current density (see, e.g., Ref. ), $$\vec{j}(\vec{r}) = -\mathrm{i} \sum_{\mu,\nu} \left[\chi_\nu(\vec{r}) \, \nabla \chi_\mu(\vec{r}) - \chi_\mu(\vec{r}) \, \nabla \chi_\nu(\vec{r})\right] \, P_{\mu\nu}.$$ Only the antisymmetric (and hence imaginary) component of $\mathbf{P}$ contributes to $\vec{j}$. And we can define the spin current density $$\vec{J}^k(\vec{r}) = -\mathrm{i} \sum_{\mu,\nu} \left[\chi_\nu(\vec{r}) \, \nabla \chi_\mu(\vec{r}) - \chi_\mu(\vec{r}) \, \nabla \chi_\nu(\vec{r})\right] \, M^k_{\mu\nu},$$ where here $k$ indexes $x$, $y$, or $z$. Again, only the imaginary components of $\mathbf{M}^k$ contribute to $\vec{J}^k$. Table \[Tab:Classification2\] relates the different types of determinants to different restrictions on the current density $\vec{j}$, magnetization density $\vec{m}$, and spin current density $\vec{J}^k$.
{width="90.00000%"}
Applications
============
Let us examine the basic idea with a few examples.
Tetrahedral H$_4$
-----------------
Consider a uniform stretching of tetrahedral H$_4$. We use the cc-pVDZ basis for simplicity. Calculations are carried out in in-house code. As the exact ground state is of singlet character for which $\langle \hat{\vec{S}} \rangle = \vec{0}$, we limit our discussion to Hartree-Fock solutions which satisfy this constraint. We find a real RHF solution, a real UHF solution, a real GHF solution which following Ref. we will denote “$\mathrm{rGHF}$”, and a complex GHF solution which we denote “$\mathrm{cGHF}$.” In fact, there are three distinct degenerate UHF solutions, three distinct degenerate real GHF solutions, and two distinct degenerate complex GHF solutions, the basic structures of which are shown in Fig. \[Fig:H4Spins\]. By “distinct solutions” we mean solutions which cannot be transformed into one another merely by spin rotation.
To initialize the GHF solutions, we add a Fermi contact term to the Hamiltonian and gradually turn off the strength of this perturbation. Our Fermi contact perturbations are motivated by vibronic distortions of the electronic structure. Tetrahedral H$_4$ is Jahn-Teller active; distorting the orbitals along the Jahn-Teller active modes[@Bersuker] without displacing the nuclei is thus likely to lead to lower energy solutions. We should note that a global rotation of the Fermi contact bias is associated with a global rotation of the spin magnetization in the GHF solution. Accordingly, the physically relevant quantity is the relative orientations of vectors on different atoms, but not the global orientation.
![Dissociation energies of tetrahedral H$_4$ in the cc-pVDZ basis set when uniformly stretched. The RHF curve is far too high in energy to see on the plot. \[Fig:H4Energy\]](H4Energy){width="0.95\columnwidth"}
In Fig. \[Fig:H4Energy\] we show dissociation energies for the Hartree-Fock solutions as well as from coupled cluster with singles and doubles[@CCSD] (CCSD) based on these determinants. We also show the full configuration interaction (FCI) curve as a reference. Note that we have excluded RHF from the plot as RHF dissociates to the wrong limit and would not fit on our plot. The other three Hartree-Fock solutions all dissociate correctly. The real GHF is never more than a few milliHartree below the UHF, and the complex GHF is never more than a few milliHartree below the real GHF, so it is not easy to distinguish the various solutions on the plot. Interestingly, the RHF-based CCSD is perhaps the best of the CCSD curves at large bond lengths, and the CCSD based on complex GHF is the worst of the lot.
We think it is valuable to understand the origin of the near-degeneracy between different spin arrangements for tetrahedral H$_4$. At long atomic separations, the Hamiltonian reduces to a Heisenberg Hamiltonian with an anti-ferromagnetic $J$. Interestingly, the frustration inherent in the tetrahedral arrangement yields an exact degeneracy in the HF solution to the Heisenberg Hamiltonian:[^10] the UHF solution with two spin-up and two spin-down electrons, the square planar arrangement, and the tetrahedral arrangement of spins all have the same energy. While in H$_4$ there are deviations from this degeneracy, they remain small and reflect discrepancies between the molecular Hamiltonian and the corresponding Heisenberg Hamiltonian which arise from our use of a finite interatomic separation.
![Norms of the various magnetization density matrices for the three magnetic solutions in the symmetric dissociation of tetrahedral H$_4$. Recall that, for example, $T_{xx} = \| \mathbf{M}^x \|^2 = \mathrm{Tr}(\mathbf{M}^x \, \mathbf{M}^x)$. In this case, after rotation the real GHF has $\| \mathbf{M}^x \| = \| \mathbf{M}^z \|$ while for the complex GHF, all three components of $\vec{\mathbf{M}}$ have equal norms. \[Fig:H4Trace\]](H4Traces){width="0.95\columnwidth"}
We are not, however, particularly interested in the total energies. In Fig. \[Fig:H4Trace\] we show the norms of the various components of $\vec{\mathbf{M}}$ after rotation for the three magnetic Hartree-Fock solutions. Note that due to the high symmetry of the problem, all three components of $\vec{\mathbf{M}}$ have the same norm in the noncoplanar complex GHF, while $\mathbf{M}^x$ and $\mathbf{M}^z$ have the same norm for the coplanar real GHF. We can tell that the complex GHF is noncoplanar by, for example, noticing that all three components of $\vec{\mathbf{M}}$ have non-zero real parts, or simply by checking $\bm{\mathcal{T}}$, which has three non-zero eigenvalues. The noncoplanar GHF solution yields forces on the nuclei that respect the tetrahedral geometry, while the UHF and coplanar GHF solutions, in contract, are susceptible to a tetragonal Jahn-Teller distortion that can lower the energy.
{width="0.95\columnwidth"} {width="0.95\columnwidth"}
Hydrogen Rings
--------------
We next consider another artificial hydrogen system. This time, we place five hydrogen atoms equally spaced around a circle such that the distance between nearest neighbor atoms is 3 Bohr. For large interatomic separation, the hydrogen atoms should be coupled antiferromagnetically. When the rings have an odd number of atoms, this leads to spin frustration and a GHF ground state[@Goings2015]. We use the STO-3G basis set for maximal simplicity, and employ a Fermi contact term which directs the spin on each atom to be at an angle of $144^\circ$ from that on its neighbors. We expect to converge to a coplanar GHF (and do; see Fig. \[Fig:H5Plot\]).
To complicate things, and to showcase our coplanarity test, we do a global spin rotation of the Fermi contact term with arbitrary parameters $\alpha$, $\beta$, and $\gamma$ (see Eqn. \[Def:RotationOperators\]). The resulting wave function is complex (and in fact breaks complex conjugation symmetry). After diagonalization, we find $T_{yy} = 0.156$ and $T_{xx} = T_{zz} = 1.713$, indicating a noncollinear solution. Because the solution is noncollinear, we test for coplanarity, and after diagonalization we find $\mathcal{T}_{yy} = 0$ and $\mathcal{T}_{xx} = \mathcal{T}_{zz} = 1.713$, indicating coplanarity. We do not generally expect the non-zero eigenvalues of $\mathbf{T}$ and of $\bm{\mathcal{T}}$ to be the same, but they are the same here because our determinant is just a spin rotation of a real GHF wave function for which $\mathbf{T}$ and $\bm{\mathcal{T}}$ have the same non-zero eigenvalues. The spins in Fig. \[Fig:H5Plot\] have been rotated back into the molecular plane.
{width="0.95\columnwidth"} {width="0.95\columnwidth"}
{width="0.95\columnwidth"} {width="0.95\columnwidth"}
Fullerenes
----------
In previous work[@JimenezHoyos2014], we have pointed out that there are non-collinear HF solutions for fullerene molecules. In simple terms, fused aromatic rings display a strong tendency towards anti-ferromagnetism [@Rivero2013]. In fullerenes, the presence of pentagon rings leads to frustration that is relieved by arranging the corresponding spins in non-collinear arrangements. Here, we choose to discuss two representative cases, namely C$_{36}$ and C$_{60}$.
The structure of C$_{36}$ (with $D_{6h}$ symmetry) can be thought of as an hexacene ring capped by an additional hexagon on the top and bottom. The GHF solution has all magnetic moments lying on the same plane, as illustrated in Fig. \[Fig:C36\]. There is full antiferromagnetic arrangement between carbon atoms in the hexacene ring related by a mirror plane perpendicular to the $C_6$ axis of the molecule. In the case of C$_{60}$, the spin arrangement coincides with the one obtained by Coffey and Tugman[@Coffey1992] on the basis of the Heisenberg Hamiltonian. All atomic magnetic moments corresponding to the same pentagon are coplanar, but the planes corresponding to different pentagons are not parallel (left panel of Fig. \[Fig:C60\]). There is exact antiferromagnetic arrangement along hexagon-hexagon edges (right panel of Fig. \[Fig:C60\]).
While the magnetic structures discussed above were obtained by visual inspection, our collinearity tests fully confirms this picture, as seen in Tab. \[Tab:Fullerenes\]. In C$_{36}$ we have a coplanar solution (with $T_{zz} > T_{xx}$), while in C$_{60}$ we have a three-dimensional spin structure (with $T_{xx} = T_{yy} = T_{zz}$, leading to $\langle \hat{S}_x \rangle = \langle \hat{S}_y \rangle = \langle \hat{S}_z \rangle = 0$).
Eigenvalue C$_{36}$ C$_{60}$
-------------------- ---------- ----------
$T_{xx}$ 6.164 7.076
$T_{yy}$ 0.621 7.076
$T_{zz}$ 9.989 7.076
$\mathcal{T}_{xx}$ 6.164 6.761
$\mathcal{T}_{yy}$ 0.000 6.761
$\mathcal{T}_{zz}$ 9.989 6.761
: Eigenvalues of $\mathbf{T}$ and of $\bm{\mathcal{T}}$ for GHF solutions in C$_{36}$ and C$_{60}$. From plotting magnetization densities we know that the magnetic structure in C$_{36}$ is noncollinear but coplanar, while in C$_{60}$ it is fully noncoplanar, as confirmed by our tests. \[Tab:Fullerenes\]
Conclusions
===========
Describing magnetic phenomena at a first-principles level is not always straightforward, even in the absence of an applied external magnetic field. For correlated wave functions, magnetic ordering can be discerned by examining the two-particle density matrix or even higher-body density matrices. The situation is simpler at the mean-field level, where the one-particle density matrix suffices.
Particularly at the mean-field level, the description of magnetism is frequently related to symmetry breaking. Unfortunately, spin symmetry can break in manifold ways, and we would like a simple way to determine the form of symmetry breaking. In general this requires considering the two-particle density matrix[@Small2015], though again the one-particle density matrix is enough to understand the precise form of symmetry breaking for mean-field wave functions. Even for correlated broken symmetry wave functions, there may be a significant amount of information to be gleaned from single-particle properties.
There are three main messages of this work. The first is that noncoplanar magnetism requires an underlying complex conjugation symmetry breaking, just as non-zero spin magnetization density requires an underlying time-reversal symmetry breaking. Second, a relatively straightforward examination of the one-particle density matrix can provide complete information about the magnetic structure of a single-determinant wave function and useful, albeit incomplete, information about the magnetic structure of a correlated wave function. The noncollinearity test discussed here is equivalent to that of SSHG for single determinants; the coplanarity test is novel. Finally, we want to reiterate that because the one-particle density matrix encapsulates all relevant information about single-determinant wave functions, one can readily see which symmetries a mean-field wave function has broken simply by looking at the density matrix without resort to orbital coefficients; indeed, the density matrix is perhaps a better place to look because unlike orbital coefficients, it is invariant to any orbital rotation which changes the wave function by no more than an overall phase.
This work was supported by the National Science Foundation, under award CHE-1462434. GES is a Welch Foundation Chair (C-0036). CAJH acknowledges support by start-up funding from Wesleyan University.
Complex Coplanar GHF
====================
We have said that coplanar spin densities do not necessarily correspond to real GHF determinants or even to those which can be rotated to be real. Here we wish to provide a few simple examples showing that a coplanar spin density cannot necessarily be made to correspond to a real GHF determinant.
For a density matrix to correspond to a single determinant, it merely needs to be Hermitian and idempotent; as a consequence of the latter, it traces to the integer particle number[@Coleman1963].
Consider, then, the density matrix $\bm{\gamma}$ with components
$$\begin{aligned}
\mathbf{P}
&= \begin{pmatrix} \frac{1}{2} & 0 & \mathrm{i} \, x \\ 0 & \frac{1}{2} & 0 \\ -\mathrm{i} \, x & 0 & \frac{1}{2} \end{pmatrix},
\\
\mathbf{M}^x
&= \begin{pmatrix} -\frac{1}{4} & 0 & 0 \\ 0 & x & 0 \\ 0 & 0 & \frac{1}{4} \end{pmatrix},
\\
\mathbf{M}^y
&= \mathbf{0},
\\
\mathbf{M}^z
&= \begin{pmatrix} \frac{1}{4} & 0 & 0 \\ 0 & x & 0 \\ 0 & 0 & -\frac{1}{4} \end{pmatrix}.\end{aligned}$$
This density matrix is Hermitian and for $x = \frac{1}{\sqrt{8}}$ it is also idempotent; it therefore corresponds to some single determinant. Because $\mathbf{M}^y = \mathbf{0}$ it is clearly coplanar, and it is noncollinear since $\mathbf{M}^x$ and $\mathbf{M}^z$ are not multiples of one another. Because $\mathbf{P}$ is complex and spin rotations do not change $\mathbf{P}$, it is clear that $\bm{\gamma}$ corresponds to an intrinsically complex coplanar GHF.
While it is clear that a complex charge density matrix guarantees a complex GHF, one can have an intrinsically complex coplanar GHF even when $\mathbf{P}$ is real. Consider the density matrix $\bm{\gamma}$ with components
$$\begin{aligned}
\mathbf{P} &= \begin{pmatrix} \frac{1}{2} + \lambda & 0 \\ 0 & \frac{1}{2} - \lambda \end{pmatrix} = \frac{1}{2} \mathbf{1} + \lambda \, \bm{\sigma}^z,
\\
\mathbf{M}^x &= \begin{pmatrix} \lambda & 0 \\ 0 & \lambda \end{pmatrix} = \lambda \, \mathbf{1},
\\
\mathbf{M}^y &= \begin{pmatrix} 0 & -\mathrm{i} \lambda \\ \mathrm{i} \lambda & 0 \end{pmatrix} = \lambda \, \bm{\sigma}^y,
\\
\mathbf{M}^z &= \begin{pmatrix} 0 & \left(-1-\mathrm{i}\right) \lambda \\ \left(-1 + \mathrm{i}\right)\lambda & 0 \end{pmatrix} = \lambda \, \left(\bm{\sigma}^y - \bm{\sigma}^x\right).\end{aligned}$$
Again, $\bm{\gamma}$ is Hermitian and for $\lambda^2 = \frac{1}{20}$ it is idempotent and thus corresponds to a single determinant. Because $\mathbf{M}^y$ is purely imaginary, the magnetization density is coplanar. One can readily verify that $\vec{\mathbf{M}}$ is noncollinear, most easily by noting that $\mathbf{M}^x$, $\mathbf{M}^y$, and $\mathbf{M}^z$ are not all multiples of the same matrix $\mathbf{Z}$.
Spin rotations can change $\bm{\gamma}$ to $\tilde{\bm{\gamma}}$ and components $\mathbf{M}^k$ to $\tilde{\mathbf{M}}^k$ via an orthogonal transformation. If $\tilde{\bm{\gamma}}$ is to be purely real, then $\tilde{\mathbf{M}}^y$ must be purely imaginary. Since $\mathbf{M}^x$ and $\mathbf{M}^z$ have real parts on different matrix elements, a purely imaginary $\tilde{\mathbf{M}}^y$ can have no contributions from $\mathbf{M}^x$ or $\mathbf{M}^z$.
Because orthogonal transformations preserve the angles between vectors, if neither $\mathbf{M}^x$ nor $\mathbf{M}^z$ contributes to $\tilde{\mathbf{M}}^y$, then $\mathbf{M}^y$ contributes to neither $\tilde{\mathbf{M}}^x$ nor $\tilde{\mathbf{M}}^z$. For $\tilde{\bm{\gamma}}$ to be real, both these matrices must be real, which means neither can have any contribution from $\mathbf{M}^z$. But if no part of $\tilde{\bm{\gamma}}$ has any contribution from $\mathbf{M}^z$, the transformation could not have been invertible, let alone orthogonal.
In short, then, there is no spin rotation which can make $\tilde{\bm{\gamma}}$ real, yet the magnetization vector field $\vec{m}(\vec{r})$ is clearly coplanar. A coplanar $\vec{m}(\vec{r})$ can arise from a density matrix $\bm{\gamma}$ which cannot be transformed to a real GHF by spin rotations.
[^1]: Time-reversal Invariant Closed-Shell
[^2]: Charge Current Wave
[^3]: Axial Spin Current Wave
[^4]: Axial Spin Density Wave
[^5]: Axial Spin Wave
[^6]: Torsional Spin Current Wave
[^7]: Torsional Spin Density Wave
[^8]: Torsional Spin Wave
[^9]: Note that even a restricted open-shell determinant, which remains a spin eigenfunctions, breaks time-reversal invariance.
[^10]: The exact ground state for tetrahedral H$_4$ and the equivalent Heisenberg Hamiltonian is also doubly degenerate.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We report x-ray synchrotron experiments on epitaxial films of uranium, deposited on niobium and tungsten seed layers. Despite similar lattice parameters for these refractory metals, the uranium epitaxial arrangements are different and the strains propagated along the *a*-axis of the uranium layers are of opposite sign. At low temperatures these changes in epitaxy result in dramatic modifications to the behavior of the charge-density wave in uranium. The differences are explained with the current theory for the electron-phonon coupling in the uranium lattice. Our results emphasize the intriguing possibilities of producing epitaxial films of elements that have complex structures like the light actinides uranium to plutonium.'
author:
- 'R. Springell'
- 'R. C. C. Ward'
- 'J. Bouchet'
- 'J. Chivall'
- 'D. Wermeille'
- 'P. S. Normile'
- 'S. Langridge'
- 'S. W. Zochowski'
- 'G. H. Lander'
bibliography:
- 'U\_malleability.bib'
title: 'The malleability of uranium: manipulating the charge-density wave in epitaxial films'
---
Introduction
============
Almost all metallic elements have simple crystal structures (*fcc*, *bcc*, *hcp*, *dhcp*) at ambient pressure and temperature [@Donohue]. There are exceptions, of course, such as Mn and Hg, but the most exotic structures are found with the early actinides Pa, U, Np, and Pu. These latter four elements all adopt complicated structures, most have many allotropes before melting, and the ambient structures are all different [@Moore]. This structural diversity arises because of the interplay between the partially occupied 5*f* and 6*d* states [@Soderlind] and at the nanoscale, these elements may well prove more malleable in forming unexpected epitaxial structures than the more conventional elements.
Epitaxial engineering [@Bland] (i.e. the production of thin films in single-crystal form, on atomically ordered substrates) has been practiced now for almost 50 years via various processes, and has illustrated in many ways how the elements can be manipulated, principally through “lattice matching”, thus reducing the interfacial strain between substrate and film. However, the simplicity of the atomic structures of most elements constrains the available options. Chromium, for example, has fascinating properties, such as the spin-density wave (SDW). Changes in the SDW, induced by epitaxial engineering, are significant [@Zabel], but are restricted by the isotropy and robustness of the underlying *bcc* lattice of Cr. With the light actinides, however, the structural diversity implies the possibility of many new effects and structures that are not observed in the bulk. We have shown earlier [@Springell] that *hcp*-U films can be stabilized, a structure that cannot be found in the bulk phase diagram. At elevated temperature (above 1050K) uranium exists in a *bcc* structure - if this could be stabilized at low temperature it might order magnetically, as a consequence of the large inter U spacing. Similarly, plutonium exhibits both the *fcc* ($\delta$-phase) and *bcc* ($\varepsilon$-phase) above $\sim$580K and $\sim$870K, respectively. To our knowledge, epitaxial films of such transuranic elements are yet to be synthesized, and this represents a challenge for the future. Our results form the first step in such a task.
Alpha-uranium (the stable crystal structure at ambient pressure and temperature) is famous for being the only element that spontaneously exhibits a charge-density wave (CDW), which occurs at $T_{0}$=43K [@Lander]. Recently, the CDW has been investigated in more detail, both theoretically [@Bouchet] and experimentally [@Raymond], emphasizing the importance of the strong electron-phonon coupling along the \[100\] axis. The results show that the length of this \[100\] $a_{U}$-axis is the key parameter in determining the behavior of the CDW. Furthermore, as the CDW is suppressed by pressure, the temperature at which uranium becomes superconducting increases [@Lander], demonstrating the link between the two phemonena, as shown recently in high-$T_{C}$ materials [@Chang], and placing uranium in the context of such materials of interest from a fundamental perspective.
Earlier, we reported geometric relationships at room temperature for the orthorhombic (space group Cmcm) $\alpha$-U structure with the \[110\] growth axis on Nb, and the \[001\] axis on W [@Ward]. In the present work we demonstrate how the malleability of uranium allows it to form different epitaxial structures with these two commonly used buffer materials, Nb and W, and that the strains produced for the two orientations on the important \[100\] uranium axis gives rise to very different behaviors of the resulting CDW’s.
Epitaxy conditions
==================
Figure \[fig:1\] shows the epitaxial relationships reported by Ward *et al.* [@Ward] for U grown on the refractory metal buffers Nb(110) and W(110), deposited on (11.0) plane sapphire substrates. Although the difference between the lattice parameters of Nb and W is only 4.3%, the orientations that $\alpha$-U adopts for epitaxy on these two elements are different.
On Nb(110), $\alpha$-U grows in a (110) orientation and the epitaxy is governed by the fit between the U\[1-10\] and Nb\[001\] rows of atoms in the interfacial plane, i.e. the horizontal atomic rows in Fig. \[fig:1\](a). The calculated misfit ($\mathrm{\Delta=½(s_{U}-s_{Nb})/(s_{U}+s_{Nb})}$) is -1.1% at room temperature, and increases slightly (-1.4%) at the growth temperature (T$_{d}$) of 450$^{\circ}$C . Note that the misfit in the perpendicular in-plane direction (\[001\] of $\alpha$-U) is much larger (+6.2%), but this is a common feature of metal epitaxy, where a match in one direction between parallel, close-packed rows of atoms at the interface is often the governing factor.
In the case of W(110) the corresponding misfit for the epitaxy of Fig. \[fig:1\](a) is +2.8%. This is too large to be acceptable, and instead the $\alpha$-U prefers to grow in the (001) orientation, as shown in Fig. \[fig:1\](b). In this case the in-plane parallel rows of atoms are U\[-110\] and W\[1-11\], which have a misfit in spacing of only +0.2% at the growth temperature. By comparison, the corresponding misfit for U/Nb would be -4.1% which is unacceptably large. A feature of low-symmetry structures such as orthorhombic $\alpha$-U is that there exist many more optional orientations available for epitaxy, and the lowest-energy relationships are often difficult to predict.
[![Epitaxial relationship for $\alpha$-U (blue circles) on (a) Nb ($a_{0}=3.300\,\AA$) and (b) on W ($a_{0}=3.165\,\AA$). For the (a) orientation the governing factor is the distance in the horizontal plane between rows of U atoms, $s_{U}=\frac{1}{2}(a_{U}^{2}+b_{U}^{2})^{\frac{1}{2}}$=3.264Å. For the orientation, (b), the rows of U and W atoms must be within register so it is necessary that the uranium $s_{U}=d(110)_{U}$ must be close to the tungsten $s_{W}=2\times d(112)_{W}$.[]{data-label="fig:1"}](Fig1a.pdf "fig:"){width="40.00000%"}]{}\
[![Epitaxial relationship for $\alpha$-U (blue circles) on (a) Nb ($a_{0}=3.300\,\AA$) and (b) on W ($a_{0}=3.165\,\AA$). For the (a) orientation the governing factor is the distance in the horizontal plane between rows of U atoms, $s_{U}=\frac{1}{2}(a_{U}^{2}+b_{U}^{2})^{\frac{1}{2}}$=3.264Å. For the orientation, (b), the rows of U and W atoms must be within register so it is necessary that the uranium $s_{U}=d(110)_{U}$ must be close to the tungsten $s_{W}=2\times d(112)_{W}$.[]{data-label="fig:1"}](Fig1b.pdf "fig:"){width="40.00000%"}]{}
The final strains found in the U layers depend not only on lattice mismatch but also the substrate clamping effect due to the different thermal expansion coefficients of substrate and layers. This latter effect is particularly significant in our case because of the large and anisotropic linear thermal-expansion coefficients ($\alpha$) of uranium [@Lloyd]. At T$_{d}$ the values are $\alpha_{U}$\[100\]=+33, $\alpha_{U}$\[010\]=-6.1, $\alpha_{U}$\[001\]=+30.6 (all in units $\times10^{-6}$K$^{-1}$). In contrast, for the refractory *bcc* metals and the sapphire substrate, the $\alpha$ coefficients show little temperature dependence and are all between 5 and 9 $\times10^{-6}$K$^{-1}$. The substrate clamping effect introduces a crucial difference in the state of strain of the U(001) and U(110) layers. In the case of U(001), both $a_{U}$ and $b_{U}$ are in-plane and are restrained by the substrate from contracting ($a_{U}$) or expanding ($b_{U}$), as they would like on cooling to room temperature. We therefore expect $a_{U}$ to be in tension and $b_{U}$ to be in compression for U/W. Along the growth direction, $c_{U}$ is free to respond to the strained in-plane cell parameters, and is expected to change to preserve the unit-cell volume. On the other hand, for U(110) as in U/Nb only $c_{U}$ is in-plane and will be in tension after cooling; $a_{U}$ is the axis closest to the surface normal and would therefore be expected to be in compression to maintain the atomic volume.
Thus the different U orientations found on the two refractory metal buffers, a feature of the low symmetry of $\alpha$-U, together with its anomalous thermal expansion coefficients, result in the $a_{U}$ axis being in compression on Nb and in tension on W. Because of the importance of $a_{U}$ to the CDW transition, these strains are anticipated to lead to a different behavior of the CDW at low temperature between the U/Nb and U/W samples.
Experimental results
====================
All experiments have been performed, using a monochromated beam of 10keV x-rays at the XMaS beamline (BM28) [@XMaS] at the European Synchrotron Radiation Facility, Grenoble. All samples were grown, using a dedicated uranium deposition system, developed at Oxford University and now housed at the University of Bristol [@Springell; @Ward].
Case of U/Nb
------------
In the case of U/Nb, as already discussed [@Springell] for a 5000Å sample, the CDW appears at approximately the same $T_{0}$ as in the bulk (43K) and with the same wave-vector components [@Lander]. We have examined a large number of epitaxial samples, ranging from 70 to 2000Å[@Chivall], and in all cases the CDW appears at a similar $T_{0}$, with similar components, to those reported in Ref. \[6\]. A comparison to measurements on bulk samples [@Lander], shows (by normalizing to a lattice peak) that the CDW in U/Nb epitaxial films is reduced in intensity compared to the bulk, and the domain population is heavily biased, unlike in the bulk.
[![(a) Plane of the film for U/Nb. The growth direction is \[110\], with the $a_{U}$ axis at 25.8$^{\circ}$ to this \[110\] direction. The four CDW domains that can be readily accessed are $\textbf{Q}_{\textbf{1}}\mathrm{(+q_{x} +q_{y} +q_{z})}$, $\textbf{Q}_{\textbf{2}}\mathrm{(+q_{x} -q_{y} +q_{z})}$, $\textbf{Q}_{\textbf{3}}\mathrm{(+q_{x} +q_{y} -q_{z})}$, and $\textbf{Q}_{\textbf{4}}\mathrm{(+q_{x} -q_{y} -q_{z})}$ where $\mathrm{q_{x}=0.5\,a^{\ast}}$, $\mathrm{q_{y}=0.22\,b^{\ast}}$ and $\mathrm{q_{z}=0.167\,c^{\ast}}$ are symmetrically spaced at 17$^{\circ}$ away from the $a_{U}$ axis \[100\]. (b) Lattice spacing $d(220)_{\mathrm{U}}$ as a function of temperature for both a 250Å-thick U/Nb film (solid circles) and a bulk sample (open circles). The temperature at which the CDW develops ($T_{0}$) is 45K, as in bulk $\alpha$-U.[]{data-label="fig:2"}](Fig2a.pdf "fig:"){width="45.00000%"}]{}\
[![(a) Plane of the film for U/Nb. The growth direction is \[110\], with the $a_{U}$ axis at 25.8$^{\circ}$ to this \[110\] direction. The four CDW domains that can be readily accessed are $\textbf{Q}_{\textbf{1}}\mathrm{(+q_{x} +q_{y} +q_{z})}$, $\textbf{Q}_{\textbf{2}}\mathrm{(+q_{x} -q_{y} +q_{z})}$, $\textbf{Q}_{\textbf{3}}\mathrm{(+q_{x} +q_{y} -q_{z})}$, and $\textbf{Q}_{\textbf{4}}\mathrm{(+q_{x} -q_{y} -q_{z})}$ where $\mathrm{q_{x}=0.5\,a^{\ast}}$, $\mathrm{q_{y}=0.22\,b^{\ast}}$ and $\mathrm{q_{z}=0.167\,c^{\ast}}$ are symmetrically spaced at 17$^{\circ}$ away from the $a_{U}$ axis \[100\]. (b) Lattice spacing $d(220)_{\mathrm{U}}$ as a function of temperature for both a 250Å-thick U/Nb film (solid circles) and a bulk sample (open circles). The temperature at which the CDW develops ($T_{0}$) is 45K, as in bulk $\alpha$-U.[]{data-label="fig:2"}](Fig2b.pdf "fig:"){width="45.00000%"}]{}
In these films, as shown in Fig. \[fig:2\](a), the $a_{U}$ axis, \[100\], is marked, as are the four CDW wave-vectors, $\textbf{Q}_{\textbf{1}}$, $\textbf{Q}_{\textbf{2}}$, $\textbf{Q}_{\textbf{3}}$, and $\textbf{Q}_{\textbf{4}}$. Since $\pm$$q_{z}$ are equal displacements with respect to the film, we expect the domain intensities of $\textbf{Q}_{\textbf{1}}$, and $\textbf{Q}_{\textbf{3}}$, on one hand, and $\textbf{Q}_{\textbf{2}}$, and $\textbf{Q}_{\textbf{4}}$, on the other, to be equivalent. This is experimentally found - see Fig. 2 of Ref. \[6\] - in all films. However, domain $\textbf{Q}_{\textbf{1}}$ is found to have at least 100 times the intensity of domain $\textbf{Q}_{\textbf{2}}$. This imbalance appears because $\textbf{Q}_{\textbf{2}}$ has a larger component in the plane of the film than for $\textbf{Q}_{\textbf{1}}$. The CDW thus favors domain $\textbf{Q}_{\textbf{1}}$ as the in-plane ($a_{U}$ and $b_{U}$) axes are subject to less clamping from the buffer and substrate than in domain $\textbf{Q}_{\textbf{2}}$.
Although the CDW satellite peaks give directly the ordering temperature $T_{0}$ (and periodicity) of the CDW, it is also instructive to examine the lattice peaks as a function of temperature. In Fig. \[fig:2\](b) we show the d(220)$_{U}$ plane spacing from a 250Å-thick epitaxial film of U/Nb. Similar figures exist for all samples. These measure the spacing of the atomic planes perpendicular to the \[110\] growth direction. The bulk values are taken from Barrett *et al.* [@Barrett]. The film value is slightly smaller than the bulk one, consistent with the compression, as discussed above in the U/Nb configuration, and the relative change of the d(220)$_{U}$ plane spacing below $T_{0}$ is far less than that found in the bulk. All these features, as well as the domain imbalance, are consistent with the compression of the $a_{U}$ axis in the U/Nb films.
Case of U/W
-----------
The epitaxy of U/W is as shown in Fig. \[fig:1\](b) with the $a_{U}$ and $b_{U}$ axes in the plane of the film, and the growth direction \[001\]. A 1500Å film exhibits a rocking curve (full width at half maximum) of 0.35$^{\circ}$. The only difference in the epitaxial relationship of this film with those discussed in Ref. \[11\] is that we have deposited a thin (100Å) seed layer of Nb on top of the sapphire substrate before depositing the 250Å buffer of W. This reduces the number of domains of the W buffer, from two to one. When the uranium is deposited, the number of domains is then reduced from four to two (B1 and B4 in Fig. 7 of Ref. \[11\]), whereas in these earlier studies [@Ward] up to six domains were reported.
![The curves, which represent interpolation between data points, show the high-temperature lattice parameter (red) varying only slowly with temperature for 300$>$T$>$150K, and then the emergence of a new (smaller) $c_{U}$ lattice parameter for T$<$150K (blue). The indexing is normalized to L=4 at base temperature. The curves can be fitted with identical widths ($\Delta$L/L) for the two different lattice parameters at all temperatures.\[figure3\]](Fig3.pdf){width="40.00000%"}
Since the change in $c_{U}$ can be gauged directly from the position of the (004) reflection, we show this in Fig. \[figure3\] as a function of temperature. Curiously, the widths of the (004) reflection, which are a measure of the correlations across the thickness of the film in the \[001\] direction, remain independent of temperature. Thus, the domains in the \[001\] direction are transforming from one lattice parameter to the other; they do not co-exist in the same domain, as this would give rise to a broadening of the peaks. The widths are the same for the high- and low-temperature phases, both reflecting the finite thickness of the film. The width of off-specular Bragg reflections with a greater in-plane component, shows a consistent broadening, indicating that the lateral dimension of the structural domains is even smaller than the film thickness of 1500Å.
![$a_{U}$, $b_{U}$, and $c_{U}$ lattice parameters, and the atomic volume as a function of temperature for the U/W film of 1500Å. The open stars give the bulk values [@Barrett]. The red points represent the high-temperature phase and the blue, the low temperature phase, in which the CDW appears. The size of the points is an indication of the volume of the sample that has such a lattice parameter. \[figure4\]](Fig4.pdf){width="45.00000%"}
The temperature dependence of the $a_{U}$, $b_{U}$, and $c_{U}$ lattice parameters are shown in Fig. \[figure4\]. Recall that the growth direction is \[001\] $c_{U}$, so that the changes in this parameter act to preserve the atomic volume. The largest change (of almost 1%) is in $a_{U}$. Initially, from ambient to $\sim$150K, contraction over this temperature range of both buffer and substrate are unobservable on this scale. However, as $a_{U}$ starts to contract at lower temperatures, a new, larger, $a_{U}$ emerges, and by the lowest temperature the complete volume of the film exhibits this new $a_{U}$. The atomic volume (see bottom panel of Fig. \[figure4\]) of the U/W film matches that of the bulk at ambient temperature, but at low-temperature is 0.5% larger, reflecting mainly the expansion of the $a_{U}$ axis.
Accompanying the large change in $a_{U}$ at low temperature is the appearance of a new set of satellites ($q_{x}=0.5, q_{y}=q_{z}=0$), and the temperature dependence of the (1.5 0 3) reflection is shown in Fig. \[figure5\], together with its full-width at half maximum. Since this satellite has no $q_{y}$ or $q_{z}$ components, the CDW is different from that found in bulk $\alpha$-U, but closely related. It is the so-called $\alpha_{1}$-phase, as discussed in Ref. \[8\], and incorporates the principal physics of the CDW in terms of the strong electron-phonon interaction, which is known [@Raymond] to have its maximum amplitude at the position (0.5 0 0) in the Brillouin zone. $T_{0}$ is now 120K, rather than the bulk value of 43K, an almost three-fold increase. We observe diffuse scattering corresponding to the soft phonon at this position (see below) up to $\sim$180K, and the width of this scattering increases as a function of $\Delta$T from $T_{0}$, as expected for a phonon-mediated phase transition [@Raymond; @Marmeggi]. An estimate of the $\beta$-value for the growth of the intensity of the CDW peak gives 0.53$\pm$0.03, consistent with a simple Landau-type order parameter, as suggested by earlier work on the soft-phonon that drives this transition [@Marmeggi]. The width of the CDW peak, measured in the \[001\] direction, is approximately 0.009 r.l.u, which corresponds closely to that found for the (004) charge peak, as in Fig. \[figure3\]. Thus, the CDW extends across the whole film thickness, however, above $T_{0}$ much shorter-range correlations exist.
![The integrated intensity of the (1.5 0 3) CDW reflection as a function of temperature for the 1500Åfilm is represented by the solid black points and the left-hand y-axis scale. The width of the (1.5 0 3) reflection is represented by the open magenta triangles and the right-hand y-axis scale. \[figure5\]](Fig5.pdf){width="50.00000%"}
At a lower temperature ($\sim$45K) we observe the small incommensurate satellites along the $b_{U}^{\ast}$ and $c_{U}^{\ast}$ reciprocal axes corresponding to the $\textbf{Q}_{\textbf{CDW}}$ described for the U/Nb samples (see ref. \[6\] and above) and found in the bulk [@Lander], but they are very weak ($<$1% of the main satellites) and almost certainly arise from strain effects [@Lander]. What is unique about this U/W epitaxy is that the CDW is formed at much higher temperatures than in the bulk, and it appears with just the $q_{x}=0.5$ component. The peak at (1.5 0 3) is intense ($\sim$5% of the strong (202) charge reflection) and corresponds to a displacement of the U atoms by $\sim$0.07Å from their equilibrium positions, which is more than twice as large as that found in the bulk [@Lander].
Theory
======
We now turn to an understanding of the development of the CDW as a function of the electronic structure of uranium. Our *ab-initio* calculations have been performed following Ref. \[8\] and Ref. \[9\]. As shown in these previous works, there is an intrinsic soft-mode in the $\alpha$-U structure that is a result of the electron-phonon interaction along the \[100\] direction, peaked at $h$=0.5, and this drives the formation of the CDW. The $\alpha$-U structure is not stable at T=0K, as demonstrated by the results for the bulk (solid blue line) in Figure \[figure5\]. Instead, the stable structure is the $\alpha_{1}$ structure with a doubling of the $a_{U}$ axis. Similarly, calculations using cell parameters corresponding to the film (solid red line) show, as expected, that the $\alpha$-U structure is even more unstable in the film, and it is not surprising that the $\alpha_{1}$ structure is formed at a higher $T_{0}$ than found in the bulk. The inset shows the changes in the soft-phonon energy as a function of changes in the $a_{U}$ parameter.
![Theoretical results using the same calculations as reported in Ref. \[8\]. The experimental points at room temperature for key modes are shown - see Ref. \[7\]. The $\alpha$-U structure is unstable at T=0K as shown by the soft phonon at $h$=0.5 having an imaginary frequency. The inset shows how this frequency changes as $a_{U}$ is either compressed or expanded. The thin film refers to U/W.\[figure6\]](Fig6.pdf){width="45.00000%"}
In contrast, with increasing pressure, i.e. a compression of the $a_{U}$ axis, the anomaly in the phonons is suppressed [@Bouchet], as experimentally verified in the bulk [@Raymond]. Thus, the CDW becomes weaker in the U/Nb films, as observed.
Furthermore, these anomalies result in a failure of the density-functional theory with quasi-harmonic thermodynamics to accurately predict the equation of state of $\alpha$-U at ambient conditions [@Dewaele].
Conclusions
===========
Our experiments have revealed that with uranium there are surprising differences in epitaxial relationships with different substrates and these drastically affect the subsequent behavior of the CDW at low temperature. It is for that reason we refer to the malleability of uranium. In the case of U/Nb films the $a_{U}$ axis is in slight compression and this leads to a reduction of the CDW amplitude as compared to the bulk, although little change in the transition temperature T0. In the case of U/W, where the epitaxial relationship is different from that found in U/Nb, the $a_{U}$ axis is in tension and, compared to the bulk, this increases (by almost a factor of three) the transition temperature of the CDW, as well as increasing its magnitude, and changing its form. These changes are consistent with the theory presented previously [@Bouchet; @Raymond] for bulk $\alpha$-U, and emphasize the importance of the electron-phonon interaction.
Since the CDW is intimately connected to the superconductivity in bulk uranium [@Lander; @Raymond], we anticipate some interesting behavior in the U-films. In particular, for the U/W film the superconducting temperature should be suppressed for ambient pressure, but likely to be greater than the maximum of 2.3$\,$K found for bulk $\alpha$-U under pressure [@Lander; @smith], when the CDW is suppressed. Of course, such experiments under pressure with thin films are challenging [@Park], and it is unknown at what pressure the strong CDW will be suppressed in such films.
The results reported here emphasize that new behavior may be expected when complex crystal structures (different from the well-known simple structures such as *bcc*, *fcc*, *hcp*, and *dhcp*) are used in epitaxial engineering. The only complex structure that has been examined previously is that of $\alpha$-Mn [@Grigorov], but in this case numerous domains complicated the elucidation of new physics. No doubt the domain behavior can be complex, but by understanding the epitaxial relationships, such problems may be minimized. In the case of U/Nb we have one domain, and with U/W we were able to obtain two equally populated domains, both of which show identical behavior.
The light actinide elements (Pa, U, Np, and Pu) present new physics with their strong mixing of the 5*f* and conduction states, and it seems likely that if simple crystal structures can be made by epitaxial engineering, then other consequences of the strong electron-phonon coupling, that should be intrinsically present in all these materials, may be found.
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{
"pile_set_name": "ArXiv"
}
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\
Subir Ghosh\
.5cm Physics and Applied Mathematics Unit\
Indian Statistical Institute\
203 B. T. Road, Kolkata 700108, India
.3cm [**[Abstract:]{}**]{} Wave packets for the Quantum Non-Linear Oscillator are considered in the Generalized Coherent State framerwork. To first order in the non-linearity parameter the Coherent State behaves very similarly to its classical counterpart. The position expectation value oscillates in a simple harmonic manner. The energy-momentum uncertainty relation is time independent as in a harmonic oscillator. Various features, (such as the Squeezed State nature), of the Coherent State have been discussed.
.5cm
In this paper we apply a recently developed scheme [@sp] of constructing Generalized Coherent States (GCS) [@kl],[@gcs] to a widely studied model: Quantum Non-linear Harmonic Oscillator (QNHO) [@ml], with interesting consequences. Its classical analysis reveals periodic solutions. One can exploit its shape invariance property to generate exactly the energy spectrum and eigen-functions [@car]. It has also been analyzed as a Harmonic Oscillator (HO) with position dependent mass [@br]. In certain limits similar models have appeared [@sg1] in oscillator models compatible with a non-commutative $\kappa$-Minkowski spacetime. Quite interestingy, it was shown that although the coordinate undergoes a “simple” harmonic motion, actually the full Hamiltonian operator appears in the equation of motion in place of the frequency that appears in conventional harmonic oscillator.
Our scheme is computationally very simple. It is perturbative in nature. We restrict ourselves to $O(\lambda )$ results where for $\lambda =0$ the Non-linear HO reduces to simple HO. We will exploit the factorization property [@car] to derive a Darboux-like transformation to rewrite the QNHO in terms of canonical creation-annihilation operators in a HO Fock basis. We will explicitly demonstrate that the GCS behave in a very interesting manner and the classical behaviour is qualitatively preserved. At various stages we will compare and contrast features of QNHO with the GCS [@sp] constructed for Non-Commutative HO compatible with the Generalized Uncertainty Principle [@gup]
[**[I. 1-D Quantum Non-linear Harmonic Oscillator:]{}**]{} The one-dimensional Lagrangean model of the QNHO is [@ml; @car; @br] $$L=\frac{1}{2}(\frac{1}{1+\lambda x^2})(\dot x^2-\alpha^2x^2).
\label{l}$$ For $\lambda =0$ we get back HO. With $$p=(\partial L)/(\partial \dot x)=\dot
x/(1+\lambda x^2)$$ one obtains the Hamiltonian $$H=p\dot x-L=\frac{1}{2}((1+\lambda x^2)p^2+\frac{\alpha^2x^2}{1+\lambda x^2}).
\label{h1}$$ It has been shown [@car] that the quantum Hamiltonian operator corresponding to (\[h1\]) admits a factorization $H'=H-\beta /2~,~H'=A^\dagger
A$ where, $$A=\frac{1}{\sqrt{2}}(i{\sqrt{1-\lambda x^2}}p+\frac{\beta}{{\sqrt{1-\lambda
x^2}}}x),$$$$
A^\dagger=\frac{1}{\sqrt{2}}(-i{\sqrt{1-\lambda
x^2}}p+\frac{\beta}{{\sqrt{1-\lambda x^2}}}x)
\label{A}$$ provided $$\alpha^2=\beta(\beta +\lambda ).
\label{al}$$ The energy eigenvalues are, $$<n\mid A^\dagger A\mid n>=\beta n-\frac{\lambda }{2}n^2,
\label{en}$$ $$\frac{1}{2}<n\mid A^\dagger A +AA^\dagger \mid n>=\beta
(n+\frac{1}{2})-\frac{\lambda }{2}n^2.
\label{een}$$ In the above $n$ is an integer. We drop the zero point energy and consider the energy to be, $$<n\mid A^\dagger A\mid n>=\beta n-\frac{\lambda }{2}n^2.
\label{en1}$$ $\lambda $ can be both positive and negative. Hence for $\lambda \le 0$ $n$ is unrestricted but for $\lambda \ge 0$ the allowed integer values of $n$ are restricted by $n\leq (2\beta )/\lambda $.\
[**[II. Canonical map of $x,p$ in terms of $a,a^\dagger $:]{}**]{} It will be convenient for our purpose to express $x,p$ in terms of canonical creation-annihilation operators $a,a^\dagger $. For $\lambda =0,~A_{\lambda =0}=\sqrt \beta a,~A^\dagger_{\lambda =0}=\beta a
a^\dagger$ where $a$ is the canonical annihilation operator written in terms $x,p$ which can in turn be inverted to express $x,p$ in terms of $a,a^\dagger $. In the present case we wish to do the same for non-zero $\lambda$: express $x,p$ in terms of $a,a^\dagger $ to $O(\lambda
)$. Since $$[\frac{1}{\sqrt 2}(ip+\beta x),\frac{1}{\sqrt 2}(-ip+\beta x)]=\beta
\label{xp}$$ we have $$a\equiv \frac{1}{\sqrt {2\beta}}(ip+\beta x)~,~~a^\dagger \equiv \frac{1}{\sqrt
{2\beta }}(-ip+\beta x)
\label{a}$$ and furthermore $$x=\frac{1}{\sqrt {2\beta}}(a+a^\dagger
)~,~~p=-i\sqrt{\frac{\beta}{2}}(a-a^\dagger ).
\label{xxp}$$ Quite clearly the above constitute the $\lambda =0$ relations. Now to $O(\lambda
)$ $$A\approx \frac{1}{\sqrt{2}}(i{\sqrt{1-\lambda x^2}}p+\beta x(1-\frac{\lambda
}{2})$$$$
\approx\frac{1}{\sqrt{2}}((ip+\beta x)+\frac{\lambda }{2}x^2(ip-\beta x)),
\label{A1}$$ $$A^\dagger
\approx\frac{1}{\sqrt{2}}((-ip+\beta x)+\frac{\lambda }{2}x^2(-ip-\beta x)),
\label{A1}$$ Now we need to be careful since operator ordering is involved[[^1]]{} . We take care of this below when we write $A,A^\dagger $ in terms of $a,a^\dagger $ and invoke Weyl ordering. $$A={\sqrt {\beta }}(a-\frac{\lambda }{4\beta}(a+a^\dagger)^2a^\dagger
)_{WO}~,$$$$
A^\dagger ={\sqrt {\beta }}(a^\dagger -\frac{\lambda }{4\beta}(a+a^\dagger)^2a
)_{WO}
\label{AA}$$ From the combination using the exact relations (\[al\]), $$(A+A^\dagger )=\frac{\sqrt {2}\beta x}{{\sqrt{1+\lambda x^2}}}
\label{Ax}$$ we obtain $$x=\frac{1}{\sqrt 2\beta}((A+A^\dagger )+\frac{\lambda }{4\beta }(a+a^\dagger
)^3)_{WO} $$$$
=\frac{1}{\sqrt {2\beta}}(a+a^\dagger ).
\label{xa}$$ This simple algebra leads us to the cherished expressions, $$x=\frac{1}{\sqrt {2\beta}}(a+a^\dagger
)~,~~p=-i\sqrt{\frac{\beta}{2}}(a-a^\dagger ).
\label{xxp}$$ It is somewhat unexpected to find out that [*[to the first non-trivial order in]{}*]{} $\lambda$, $x,p$ [*[retain their canonical form when expressed in terms of]{}*]{} $a,a^\dagger$. The HO Fock space is $$a\mid n>={\sqrt{n}}\mid n-1>~,~~a^\dagger\mid n>={\sqrt{n+1}}\mid n+1>.
\label{n}$$ From here on the computations are straightforward. The Hamiltonian to $O(\lambda
)$ is obtained from $H=A^\dagger A+\frac{\beta}{2}$ and (\[AA\]) with the necessary Weyl ordering. In HO Fock space representation we find $$H=\frac{1}{2}[\beta (aa^\dagger +a^\dagger
a)+\frac{\lambda}{2}\{a^2+(a^\dagger)^2+aa^\dagger +a^\dagger a )$$$$-(a^4+
(a^\dagger)^4+a^2(a^\dagger)^2+(a^\dagger)^2a^2
+\frac{(a^\dagger)3a}{2}+\frac{
a^\dagger a^3}{2}+\frac{a(a^\dagger)^3}{2}+\frac{a^3a^\dagger}{2}
$$$$+\frac{a^\dagger a(a^\dagger)^2}{2}+\frac{a^2a^\dagger
a}{2}+\frac{(a^\dagger)^2aa^\dagger}{2}
+\frac{aa^\dagger a^2}{2} )\}]+\frac{\beta}{2}.
\label{ha}$$ Since we are considering first order perturbation theory only terms with equal number of $a$ and $a^\dagger $ will contribute to the energy expectation value in state $\mid n>$, $$H\mid n>=(\beta n-\frac{\lambda}{2}n^2+\frac{1}{2}(2\beta
-\frac{\lambda}{2})\mid n>.
\label{hn}$$ One can compare the above energy expression with the exact value [@car], with (\[en\],\[een\]). Note that the $n$-dependent terms in the exact energy (\[een\]) and our first order corrected value (\[hn\]) are identical. In fact for our GCS construction the constant shift in energy is unimportant and henceforth will be ignored.
[**[III. Generalized Coherent States for NQHM:]{}**]{} After these preliminaries we are now ready for the main piece of our work: construction of the GCS. We follow the notation of [@gcs] and the GCS $\mid
J,\gamma >$ is the following wave packet, $$\mid J,\gamma >=\frac{1}{N^2(J)}\sum_{n\geq 1}\frac{J^{n/2}e^{-i\gamma
e_n}}{{\sqrt{\rho_n}}}\mid n>~,~~\rho_n =e_1e_2...e_n .
\label{J}$$ The parameter $\gamma $ is proportional to $\beta $ and $J$ is related to $<H>$ for the GCS. In the above $e_n$ is defined as $$E_n=\beta n(1-\frac{\lambda}{2\beta}n)=\beta
n(1-\frac{\lambda '}{2}n)=\beta e_n,~~ \lambda '=\frac{\lambda }{\beta}.
\label{en1}$$
[**[IV. Properties of the Generalized Coherent States for NQHM:]{}**]{} We start with the Revival time analysis. Recalling the energy expression as $E_n=\beta n-(\lambda /2) n^2$ we find that there are two time scales involved: the Classical time $T_{c}=(2\pi)/ \beta $ and the Revival time $T_r=(4\pi )/\lambda $ with the condition that $\lambda /(2\beta )$ an integer. Hence the motion with period $T_c$ will be modulated by $T_r$. Since we consider $\lambda $ to be small, $T_r\>>T_c$. Full revival of the wave packet will occur at each multiple or $T_r$. In between $T_r$ there will be fractional revival where the wave packet collapses into subsidiary packets that evolve with period $T_c$. This fraction revival phenomena and the Revival time scale is a manifestation of the non-linearity in the system, showing up in the non-linear energy spectrum.
Next we come to the features of the GCS. The building blocks for further analysis is $<a>,<a^\dagger >$, expectation values of $a,a^\dagger $ in the GCS. We find $$<a>\equiv < J,\gamma \mid a\mid J,\gamma >=\frac{1}{N^2(J)}\sum_{n\geq
1}\frac{J^{n-\frac{1}{2}}e^{-i\gamma (e_{n-1}-e_n)}}{{\sqrt{\rho_{n-1}\rho_n}}}
\label{<a>}$$ The phase turns out to be $$e_{n-1}-e_n=-(n+\frac{\lambda '}{2}(1-2n)),
\label{ee}$$ and the exponential can be expanded as a power series in $\lambda '$. Hence we find $$<a>\approx \frac{\sqrt J}{N^2}e^{i\gamma}\sum_{n\geq
1}\frac{J^n(1-\frac{n}{2}(1+2n))}{\rho_n(1-\frac{\lambda ' n}{2})}.
\label{aap}$$ A simple algebra leads to, $$<a>={\sqrt {J}}e^{i\gamma}(1-\frac{\lambda '}{2}(1+J))~,~~<a^\dagger >={\sqrt
{J}}e^{-i\gamma}(1-\frac{\lambda '}{2}(1+J)).
\label{aj}$$ This immediately yields the cherished expressions for the quantum behaviour of position and momentum in coherent states, $$<x>={\sqrt{\frac{2J}{\beta}}}(1-\frac{\lambda '}{2}(1+J))cos\gamma ,
\label{x}$$ $$<p>={\sqrt{2J\beta}}(1-\frac{\lambda '}{2}(1+J))sin\gamma .
\label{p}$$ It is very interesting to note that, for the GCS, non-linearity affects only the amplitude, keeping the oscillatory motion intact. This is one of our major observations.
At this point it is worthwhile to compare this feature of NQHO with GCS of another recently studied HO with non-linear deformation [@sp]. The latter system is an extension of HO in a noncanonical phase space that is compatible with the Generalized Uncertainty Principle [@gup]. For the latter system, compared to (\[en1\]) $$e^{GUP}_n \approx n(1+\lambda (1+n)),
\label{gn}$$ for which $$<x> \approx [cos \gamma -\lambda
((1+\frac{J}{2})cos\gamma +2(1+J)\gamma sin\gamma )],
\label{j1}$$ $$<p> \approx [-sin\gamma
+\lambda (\frac{1}{6}(2+J)sin\gamma -2\gamma (1+J)cos\gamma +\frac{J}{3}sin
(3\gamma ))].
\label{j3}$$ Notice that for the GUP HO the time dependence is much more involved with higher frequencies coming in to play.
Following the same procedure, although more complicated, one can compute the dispersions, $$(\Delta x)^2=<x^2>-(<x>)^2=\frac{1}{\beta}[\frac{1}{2}+\lambda '
J(\frac{3}{2}(1+J)$$$$+(\frac{7}{4}+\frac{3}{2}J)cos(2\gamma) +2(1+J)\gamma
sin(2\gamma))],
\label{dx}$$
$$(\Delta p)^2=<p^2>-(<p>)^2=\beta[\frac{1}{2}+\lambda '
J(\frac{3}{2}(1+J)$$$$-(\frac{7}{4}+\frac{3}{2}J)cos(2\gamma) -2(1+J)\gamma
sin(2\gamma))].
\label{dp}$$
Here we find a marked qualitative difference from the HO behaviour since the non-linearity introduces a $\gamma $ or time-dependent oscillatory motion. [*[However it is remarkable that this time dependence disappears in the uncertainty relation to give]{}*]{} $$(\Delta x)^2(\Delta p)^2=[\frac{1}{4}+\frac{3}{2}\lambda ' J(1+J)].
\label{dxp}$$ This is another of our interesting observations. We point out that for the GUP HO [@sp] the time independent behavior of the Uncertainty Product of $x$ and $p$ is not maintained.
It is worthwhile to point out that the GCS is an example of a Squeezed State. Consider the instant $\gamma =0$ in the oscillating variances $(\Delta x)^2$ and $(\Delta p)^2$ in (\[dx\]),(\[dp\]), $$(\Delta x)^2=\frac{1}{\beta}[\frac{1}{2}+\lambda '
J(\frac{13}{2}+3J)]~, ~
(\Delta p)^2=\beta[\frac{1}{2}-\frac{1}{4}\lambda '
J].
\label{sq}$$ Clearly $(\Delta p)^2\leq\beta[\frac{1}{2}$ showing that the GCS is a Squeezed State and that $(\Delta p)^2$ and $(\Delta x)^2$ attains their minimum and maximum values respectively. The opposite happens for $\gamma =\pi /2$. However it is not a minimum uncertainty Squeezed State since $(\Delta x)^2(\Delta p)^2\geq\frac{1}{4}$.
The Weyl ordered Hamiltonian yields the GCS energy expectation value, $$<H>=\frac{\beta}{2}(1+2J)+\lambda '
[\frac{\beta}{2}J(1+J)+\frac{J^2}{4}-\frac{1}{4}(1+J+2Jcos(2\gamma ))^2].
\label{h}$$ An interesting point is to note that for non-zero $\lambda ' $, $<H>$ has a $cos
(2\gamma )$ dependence indicating that there is a little fuzziness in energy of the packet. Since $\gamma \sim \beta $ the $cos$-term averages out for $\beta t >>1$. From the condition (\[al\]) $$\beta ^2+\beta\lambda ' -\alpha ^2=0~\rightarrow \beta \approx (\lambda \pm
2\alpha
)/2,
\label{eav}$$ we find the oscillatory behavior can be ignored for $t>>\beta ^{-1}\approx
\alpha^{-1}(1+\frac{\lambda '}{2}\alpha ^{-1})$.
A direct way to ascertain the non-classical behavior is to construct the Mandel parameter $Q$ out of the dispersion in number operator, $$Q=(\Delta n)^2/<n> -1
\label{q0}$$ where $(\Delta n)^2=<N^2>-<N^2>$. For the present problem we find, $$<N>=<a^\dagger a>=J[1+\frac{\lambda ' }{2}(1+J)],
\label{n}$$ $$<N^2>=<a^\dagger aa^\dagger a >=J[1+J+\lambda ' (\frac{1}{2}+1+J^2)],
\label{n^2}$$ leading to $$(\Delta n)^2=<N^2>-<N^2>=J+2\lambda ' J(1+J)[-\frac{1}{2}+J+J^2].
\label{dn2}$$ The Mandel parameter follows, $$Q=\frac{\lambda ' }{2}(1+J)(4J^2+3J-3).
\label{q}$$ One denotes $Q=0$ as the Poissionian statistics and $Q\ge 0 $ ($Q\leq 0
$) as Super-Poissionian (Sub-Poissionian) statistics. The distribution will be Super-Poissionian ($Q\geq 0$) for $J\geq 0.5$ and Sub-Poissionian ($Q\leq
0$) for $0.5\geq J\geq 0$. Poission statistics is recovere for $J=(\sqrt{57}-3)/8$.
The remaining task is to check up on the status of the Correspondence Principle. First we discuss the quantum equation of motion by directly exploiting the Heisenberg equation of motion, $$\dot B =i[H,B]~\rightarrow ~<\dot B> =i<[H,B]>,
\label{dot}$$ for a generic observable $B$. In the present case we obtain the following operator equations, $$\dot a=-\frac{i}{2}[2\beta a +\lambda (a^\dagger +a-2(a^\dagger
)^3-a^2a^\dagger-a^\dagger a^2$$$$-(a^\dagger )^2a
-a^3-a(a^\dagger )^2-a^\dagger a a^\dagger )],
\label{dota}$$ $$\ddot a=-\beta ^2a+\lambda \beta (-a+a^2a^\dagger+a^\dagger
a^2-2(a^\dagger)^3+2a^3 ).
\label{ddota}$$ Expectation values of the above along with their hermitian conjugates lead to the equation of motion $$<\ddot x>=\frac{1}{{\sqrt{2\beta }}}<\ddot a+\ddot (a^\dagger
)>={\sqrt{2J\beta}}[-\beta +\lambda ' (-1+2J+(1+J)\frac{\beta}{2})]cos\gamma .
\label{xx}$$ Let us now consider the classical equation of motion. The Hamiltonian equations of motion yield, $$\dot x=p+\lambda x^2p~,~~\dot p=-\alpha^2x-\lambda (p^2x-2\alpha^2x^3) $$$$
\rightarrow ~\ddot x=-\alpha^2x+2\lambda x\frac{p^2+\alpha
^2x^2}{2}=-\alpha^2x+2\lambda xH +O(\lambda ^2).
\label{xcl}$$ Now we compute the expectation value keeping in mind the operating ordering, $$<\ddot x>=<(-\beta ^2 x+\lambda (-\beta x+4\frac{xH+Hx}{2}))>$$$$
={\sqrt{2J\beta}}[-\beta +\lambda ' (-1+(4+\frac{\beta}{2})(1+J)]cos\gamma.
\label{xxcl}$$ One immediately observes the striking similarity between the equation of motion obtained from Heisenberg (quantum) and Hamiltonian (classical) formalisms.\
[**[V. Conclusion:]{}**]{} We have constructed wave packets as Generalized Coherent States for the Quantum Non-Linear Harmonic Oscillator to the first non-trivial order in the non-linearity parameter. Quite remarkably the wave packet closely mimics the behavior of position variable in classical simple harmonic oscillator. The oscillatory motion of the Coherent State in the quantum Non-linear oscillator only has a modified amplitude. The uncertainty relation for the Coherent State is time independent with only its magnitude modified by non-linearity parameter. Once again the similarity with quantum harmonic oscillator striking. Furthermore the Coherent State has the charecteristics of a Squeezed State. The Correspondence Principle is also very nearly maintained. Mandel parameter analysis shows that departures from the Poissionian behavior is possible depending on Coherent State parameters and Poission statistics is recovered for a particular value of the parameter.\
[**[ Acknowledgement:]{}**]{} I thank Bikashkali Midya for discussions.
[99]{}
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[^1]: For a rigorous and mathematical discussion on the issue of operator ordering in the present problem see [@car]. We, on the other hand, take a more naive approach, one reason being that we are considering only $O(\lambda )$ extension and this might clash with the exact analysis provided in [@car].
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'Given their 2.2 $\mu$S lifetime, muons must be accelerated fairly rapidly for a neutrino factory or muon collider. Muon bunches tend to be large. Progress in fixed field, alternating gradient (FFAG) lattices to meet this challenge is reviewed. FFAG magnets are naturally wide; low momentum muons move from the low field side of a gradient magnet to the high field side as they gain energy. This can be exploited to do double duty and allow a large beam admittance without unduly increasing the magnetic field volume. If the amount of RF must be reduced to optimize cost, an FFAG ring can accommodate extra orbits. I describe both scaling FFAGs in which the bends in each magnet are energy independent and non-scaling FFAGs in which the bends in each magnet do vary with muon energy. In all FFAG designs the sum of the bends in groups of magnets are constant; otherwise orbits would not close. Ways of keeping the accelerating beam in phase with the RF are described. Finally, a 1 MeV [*proof of principle*]{} scaling FFAG has been built at KEK and began accelerating protons in June 2000 with a 1 kHz repetition rate.'
address: |
Department of Physics and Astronomy, University of Mississippi–Oxford\
University, MS 38677, USA
author:
- 'D. J. SUMMERS[^1]'
title: 'MUON ACCELERATION USING FIXED FIELD, ALTERNATING GRADIENT (FFAG) RINGS'
---
Introduction
============
Scaling FFAG rings were proposed independently a half century ago by Ohkawa,$^1$ Symon,$^2$ and Kolomensky.$^3$ The Mid-Western Universities Research Association (MURA) built both radial-sector (1957) and spiral-sector (1960) models and tested them with electrons. However, the serious development of FFAGs ceased with the ascendancy of ramping synchrotrons, which allowed smaller diameter, smaller bore rings for a given energy and magnetic field. Because the voltage needed to quickly ramp synchrotrons$\,^4$ filled with wide bunches of low energy muons is rather large, FFAGs have recently experienced a renaissance.$^{5,6}$ The FFAG design permits multiple passages of muons though both RF cavities and magnet arcs for reduced cost.
One reason FFAG rings are of interest today is because they offer economical muon acceleration for a neutrino factory$\,^{7,8}$ or a muon collider.$^9$ At a neutrino factory accelerated muons are stored in a racetrack to produce neutrino beams ($\mu^- \to e^- \, {\overline{\nu}}_e \, \nu_{\mu}$ and $\mu^+ \to e^+ \, \nu_e \, {\overline{\nu}}_{\mu}$). Neutrino oscillations have been observed.$^{10}$ Further exploration at a neutrino factory could reveal CP violation in the lepton sector,$^{11}$ and is particularly useful if the coupling between $\nu_e$ and $\nu_{\tau}$, $\theta_{13}$, is small.$^8$ A muon collider can do s-channel scans to separate the $H^0$ and $A^0$ Higgs doublet.$^{12}$ Above the ILC’s 800 GeV there are a large array of supersymmetric particles that might be produced,$^{13}$ as well as mini black holes,$^{14}$ if large extra dimensions exist. Note that the energy resolution of a muon collider is not smeared by beamstrahlung.
A cyclotron has a large volume magnetic field which is constant in time. Particle orbits move from the center to the edge of the cyclotron as they accelerate. A synchrotron has a small magnetic field volume. The $B$ field increases with time. Particle orbit radii do not change as a particle accelerates. An FFAG ring is in between a cyclotron and a synchrotron in its design. As particles accelerate they move a small distance in gradient magnets which can accommodate higher energy orbits at slightly different radii. FFAG magnetic fields are fixed in time and their volume is larger than a synchrotron but smaller than a cyclotron.
A Neutrino Factory Design using Two Non-Scaling FFAG Rings
==========================================================
The most recent neutrino factory design$^8$ incorporates 5 $\to$ 10 and 10 $\to$ 20 GeV non-scaling FFAG rings. Acceleration up to 5 GeV uses a linac and a dogbone recycling linac.$^{8,15}$ A layout appears in Fig. 1 and parameters in Table 1. The 20 GeV ring is almost five times larger than a synchrotron with 5.5 T magnets. The ratio of focusing–to–bending in an FFAG ring is high. Normally resonances are a problem in non-scaling FFAGs, but the fast muon acceleration cycle can prevent them from building up as can highly symmetric lattice designs. Each cell uses a FDF triplet of superconducting magnets as shown in Fig. 2. Much work has gone into the lattice design to keep the beam size and hence the magnetic apertures relatively small. The idea is to control cost by reducing the magnetic field volume and by using superconducting magnets with moderate fields.
Superconducting RF (fixed 201 MHz, 10 MV/m) is used for acceleration. A niobium coated copper cavity running at 201 MHz has recently achieved a gradient of 11 MV/m and prototypes may reach 15 MV/m.$^{16}$ At 201 MHz, ${1\over{4}} \lambda$ = 37 cm, on the same order as the phase difference just due to the muons increasing in speed as shown in the last row of Table 1. Its hard to change the RF phase itself quickly. An advantage of the non-scaling FFAG is the additional control over the physical path length muons follow. Path lengths dominate speed increases in determining muon phase with respect to the RF. Fig. 3 notes the parabolic time of flight (TOF) relation that can be achieved. Muons cross the RF crest three times during the acceleration cycle. Staying closer to crest minimizes the amount and cost of RF that is needed. In a scaling FFAG, TOF increases monotonically.
-- --
-- --
Scaling FFAG Rings Being Built in Japan
=======================================
FFAGs are being built for muon phase rotation, radiation therapy, CT scanning, and accelerator–driven sub–critical nuclear reactor operation in Japan. A 1 MeV scaling FFAG with 8 DFD sectors has been built at KEK and has accelerated protons with a 1 kHz repetition rate.$^{5,17}$ A 150 MeV scaling FFAG with 12 DFD sectors is nearing completion. Beam has been accelerated to 150 MeV. Orbits shift from a radius of 4.4 to 5.5 m during the acceleration cycle. In these scaling FFAGs, orbit shapes and magnet focal lengths are energy independent. See Fig. 2 of Ref. 5 for a nice drawing of particle paths in scaling and non-scaling FFAGs.
Acknowledgments {#acknowledgments .unnumbered}
===============
Many thanks to J. Gallardo, S. Berg, C. Johnstone, R. Palmer, and Y. Torun.
[99]{}
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[^1]: [email protected] Supported by DE–FG02–91ER40622.
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{
"pile_set_name": "ArXiv"
}
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---
abstract: |
The hyper Zagreb index is a kind of extensions of Zagreb index, used for predicting physicochemical properties of organic compounds. Given a graph $G= (V(G), E(G))$, the first hyper-Zagreb index is the sum of the square of edge degree over edge set $E(G)$ and defined as $HM_1(G)=\sum_{e=uv\in
E(G)}d(e)^2$, where $d(e)=d(u)+d(v)$ is the edge degree. In this work we define the second hyper-Zagreb index on the adjacent edges as $HM_2(G)=\sum_{e\sim f}d(e)d(f)$, where $e\sim f$ represents the adjacent edges of $G$. By inequalities, we explore some upper and lower bounds of these hyper-Zagreb indices, and provide the relation between Zagreb indices and hyper Zagreb indices.\
\
Accepted by MATHEMATICAL REPORTS.
[**MSC:**]{} 05C12; 05C90 [**Keywords:**]{} Degree, Minimum degree, Maximum degree, Zagreb indices, Hyper-Zagreb indices.
author:
- |
SHAOHUI WANG$^{1,}$[^1], WEI GAO$^{2}$, MUHAMMAD K. JAMIL$^3$,\
MOHAMMAD R. FARAHANI$^4$, JIA-BAO LIU$^{5,*}$\
*1. Department of Mathematics and Computer Science, Adelphi University, Garden City, NY, USA.*\
\
*2. School of Information Science and Technologys, Yunnan Normal University, Kunming, PR China.*\
\
*3. Abdus Salam School of Mathematical Sciences, Government College University, Lahore, Pakistan.*\
\
*4. Department of Applied Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran.*\
\
*5. School of Mathematics and Physics, Anhui Jianzhu University, Hefei, PR China*
title: BOUNDS OF ZAGREB INDICES AND HYPER ZAGREB INDICES
---
Introduction {#intro}
============
The graphs $G=(V(G),E(G))$ considered in this paper are finite, loopless and contain no multiple edges. Given a graph $G= (V, E)$, $V$ and $E$ represent the set of vertices and the set of edges with $n = |V|$ vertices and $m=|E|$ edges, respectively. For a vertex $u \in V$, the number of vertices adjacent with $u$ is called its degree $d(u)$. In a graph $G$, $\triangle$ and $\delta$ represent the maximum and the minimum degree, respectively.
In 1947, Harold Wiener introduced famous Wiener index, a most widely known topological descriptor [@16]. The Winner index is the oldest and one of the most popular molecular structure descriptors, well correlated with many physical and chemical properties of a variety of classes of chemical compounds. Based on the success on the Wiener index, many topological indices have been introduced. Almost forty years ago, Gutman et al. defined the important degree-based topological indices: the first and second Zagreb indices [@8]. These are defined as $$M_1(G)=\sum_{v\in V(G)} d(v)^2,
M_2(G)=\sum_{uv\in E(G)} d(u)d(v).$$ In 2004, Mili$\breve{c}$evi$\acute{c}$ [@13] reformulated these Zagreb indices in terms of edge degrees, $d(e)=d(u)+d(v)-2$, for $e=uv$ and defined reformulated Zagreb indices, $$EM_1(G)=\sum_{e\in E(G)}d(e)^2,
EM_2(G)=\sum_{e\sim f} d(e)d(f).$$ In 2013, Shirdel et al. [@15] defined the [*first hyper Zagreb index*]{} as follows, $$HM_1(G)=\sum_{e\in E(G)}d(e)^2,$$ where $d(e)=d(u)+d(v)$. In 2016, Jamil et al. [@7] improved and extended the Shirdel’s results. Based on this definition of edge degree, we define the [*second hyper Zagreb index*]{} as follows, $$HM_2(G)=\sum_{e\sim f}d(e)d(f),$$ where $e\sim f$ represents the adjacent edges of $G$. Furthermore, $G$ is called regular if every vertex has the same degree and edge degree regular if every edge has the same degree, respectively.
These graph invariants, based on vertex-degrees and edge-degrees of a graph, are widely used in theoretical chemistry. For applications of Zagreb indices in QSPR/QSAR and latest results, refer to [@1; @17; @2; @5; @6; @10; @01; @02; @03; @04; @14; @18; @w2015; @w2016; @w1; @w2; @w3; @w4; @z1; @z2].
As a fundamental dynamical processing system, the basis of graph structure has received considerable interest from the scientific community. Recent work shows that the key quantity-degree-based topological indices to a given graph class on uncorrelated random scale-free networks is qualitatively reliant on the heterogeneity of network structure. However, in addition to the transformations of these graph basis, most real system models (topological indices) are also characterized by degree correlations. In this paper, we explore some properties of hyper Zagreb indices in terms of the number of vertices $n$, the number of edges $m$, maximum and minimum degree $\triangle,\delta$, respectively. Also we provide the relation between hyper Zagreb indices and first Zagreb index $M_1(G)$.
Preliminaries and main results
==============================
After introducing the construction and structural properties of degree-based topological indices, we will provide our main results by presenting their inequalities.
Let G be a graph with n number of vertices and m number of edges, then $$\delta^2\le \frac{HM_1(G)}{4m}\le \triangle^2,$$ the left and right equalities hold if and only if G is $\delta$-regular and $\triangle$-regular, respectively.
Note that $\delta\le d(v_i)\le \triangle, ~i=1,2,\cdots,n$. Then $$2\delta\le d(e_j)\le 2\triangle,~j=1,2,\cdots,m.$$ By the definition of the first hyper Zagreb index, we have $$\delta^2\le \frac{HM_1(G)}{4m}\le \triangle^2.$$ Clearly, the equalities hold if and only if G is $\delta$-regular and $\triangle$-regular. In particular, if $G$ is general regular connected graph, then $\delta(G) = 2$ and $\triangle(G) =n-1$.
Let G be a graph with m edges, then $$HM_1(G)\ge \frac{M_1(G)^2}{m},$$ the equality holds if and only if G is edge degree regular.
Let $d(e_i)$ be the edge degree of $G$. By Cauchy-Schwartz inequality, we obtain $$[d(e_1)^2+d(e_2)^2+\cdots+d(e_m)^2][1^2+1^2+\cdots+1^2]\ge [d(e_1)\cdot1+d(e_2)\cdot1+\cdots+d(e_m)\cdot1]^2.$$ Note that $\sum_{e \in E(G)} d(e) = \sum_{v \in V(G)} d(v)^2$. By the concept of $M_1(G)$, we obtain the relation between $HM_1(G)$ and $M_1(G)$ below. $$HM_1(G)\cdot m\ge M_1(G)^2,$$ that is, $$HM_1(G)\ge \frac{M_1(G)^2}{m}.$$ Clearly, the equality holds if and only if every edge has the same degree, that is, $G$ is edge degree regular.
\[thm3\] Let G be a graph with n vertices and m edges, then $$HM_1(G)\le M_1(G)(m+2\delta-1)-2m(m-1)\delta,$$ the equality holds if and only if G is regular.
We keep the same notations as [@4]. Let $d(e_i)\mu_i$ be the sum of degrees of the edges adjacent to the edge $e_i$. We have $$d(e_i)\mu_i=\sum_{e_i\sim e_j}d(e_j)\le \sum_{i=1}^m d(e_i)-d(e_i)-(m-1-d(e_i))2\delta.$$ Thus, $$\begin{aligned}
HM_1(G)&=\sum_{e_i\in E(G)}d(e_i)^2=\sum_{i=1}^m d(e_i)\mu_i\\
&\le \sum_{i=1}^m[\sum_{i=1}^m d(e_i)-d(e_i)-(m-d(e_i))2\delta]\\
&= M_1(G)(m+2\delta-1)-2m(m-1)\delta.\end{aligned}$$ Clearly, the equality holds if and only if $G$ is regular.
By the results of [@9] that $M_1(G)\le
2m(\triangle+\delta)-n\triangle \delta$, where the equality holds if and only if $G$ is regular, we have the following corollary.
Let G be a graph with n vertices and m edges, then $$HM_1(G)\le (2m(\triangle+\delta)-n\triangle \delta)(m+2\delta-1)-2m(m-1)(\delta-1),$$ where the equality holds if and only if G is regular.
\[a\] Let G be a graph with n vertices, m edges and minimum degree $\delta\ge 2$, then $$HM_1(G)\le\frac{(\triangle+\delta)^2}{4m\triangle\delta}M_1(G)^2,$$ the equality holds if and only if G is a regular graph, or there are exactly $\frac{m\delta}{\triangle+\delta}$ edges of degree $2\triangle$ and $\frac{m\triangle}{\triangle+\delta}$ edges of degree $2\delta$ such that $(\triangle+\delta)$ divides $m\delta$.
If $a,a_1,a_2,\cdots,a_m$ and $b,b_1,b_2,\cdots,b_m$ are positive real numbers such that $a\le a_i\le A$, $b\le b_i\le B$ for $1\le
i\le m$ with $a<A$ and $b<B$, by P$\acute{o}$lya-Szeg$\acute{o}$ Inequality[@20], we have $$\sum_{i=1}^m {a_i}^2\cdot \sum_{i=1}^m {b_i}^2\le \frac{1}{4}\Big(\sqrt{\frac{AB}{ab}}+\sqrt{\frac{ab}{AB}}\Big)^2\cdot\Big(\sum_{i=1}^m {a_ib_i}\Big)^2,$$ and the equality holds if and only if the numbers $$k=\frac{\frac{A}{a}}{\frac{A}{a}+\frac{B}{b}},l=\frac{\frac{B}{b}}{\frac{A}{a}+\frac{B}{b}}$$ are integers, $a=a_1=a_2=\cdots=a_k$; $A=a_{k+1}=a_{k+2}=\cdots=a_m$ and $B=b_1=b_2=\cdots = b_l$; $b=b_{l+1}=b_{l+2}=\cdots=b_m$. If we allow $a=A$ or $b=B$, the equality holds if $AB=ab$, i.e., $A=a=a_1=a_2=\cdots=a_m$ and $B=b=b_1=b_2=\cdots,b_m$. By setting the values $a_i=1$ and $b_i=d(e_i)$ for $i=1,2,\cdots,m$, we obtain $$\sum_{i=1}^m {1}^2\cdot \sum_{i=1}^m {d(e_i)}^2\le \frac{(AB+ab)^2}{4ABab}\cdot\Big(\sum_{i=1}^m {d(e_i)}\Big)^2.$$ So, $$mHM_1(G)\le \frac{(AB+ab)^2}{4ABab}\cdot M_1(G)^2.$$ Now since $a\le a_i\le A$, we have $a=A=1$ and since $b\le b_i\le
B$, we have $b=2\delta$ and $B=2\triangle$. Hence, $$HM_1(G)\le\frac{(2\triangle+2\delta)^2}{16\triangle\delta}M_1(G)^2,$$ which is the expected result. In the last expression, the equality holds if and only if G is a regular graph, or there are exactly $\frac{m\delta}{\triangle+\delta}$ edges of degree $2\triangle$ and $\frac{m\triangle}{\triangle+\delta}$ edges of degree $2\delta$ such that $(\triangle+\delta)$ divides $m\delta$.
\[cor2\] Let G be a graph with n vertices, m edges and minimum degree $\delta\ge 2$, then $$HM_1(G)\le \frac{(n+1)^2}{8m(n-1)}M_1(G)^2,$$ the equality holds if G has exactly $\frac{m}{n-1}$ edges of degree 2(n-2) and $\frac{m(n-2)}{n-1}$ edges of degree 2 such that n-1 divides m.
Note that $$\frac{(\triangle+\delta)^2}{\triangle\delta}=\frac{\triangle}{\delta}+\frac{\delta}{\triangle}+2.$$ By Theorem \[a\], we have $$HM_1(G)\le\Big[\frac{\triangle}{\delta}+\frac{\delta}{\triangle}+2\Big]M_1(G)^2.$$ As the function $f(x)=x+\frac{1}{x}$ is increasing for $x\ge1$, so $\Big[\frac{\triangle}{\delta}+\frac{\delta}{\triangle}+2\Big]$ is increasing for $\frac{\triangle}{\delta}\ge1$. Now for $\delta\ge2$, $1\le \frac{\triangle}{\delta}\le \frac{n-1}{2}$. So, $\Big[\frac{\triangle}{\delta}+\frac{\delta}{\triangle}+2\Big]\le \frac{(n+1)^2}{2(n-1)}$. So, $$HM_1(G)\le \frac{(n+1)^2}{8m(n-1)}M_1(G)^2,$$ the equality holds if $G$ has exactly $\frac{m}{n-1}$ edges of degree 2(n-2) and $\frac{m(n-2)}{n-1}$ edges of degree $2$ such that $n-1$ divides $m$.
Let G be a graph with n vertices and m edges, then $$HM_1(G)\le \frac{m^3(n+1)^6}{16n^2(n-1)^2},$$ the equality holds if and only if $G\cong K_3$.
Note that [@9] $M_1(G)\le
\frac{m^2(n+1)^2}{2n(n-1)}$, for $\delta \ge 2$ with the equality holds if and only if $G\cong K_3$. Thus, Corollary \[cor2\] yields the result.
\[thm5\] Let G be a graph with n vertices and m edges, then $$HM_1(G)\le 2(\triangle+\delta)M_1(G)-4m\triangle\delta,$$ the equality holds if and only if G is a regular graph.
Suppose $a_i$, $b_i$, $p$ and $P$ are real numbers such that $pa_i\le b_i\le Pa_i$ for $i=1,2,\cdots,m$, then we have Diaz-Metcalf inequality[@19], $$\sum_{i=1}^mb_i^2+pP\sum_{i=1}^ma_i^2\le (p+P)\sum_{i=1}^ma_ib_i,$$ and the equality holds if and only if $b_i=pa_i$ or $b_i=Pa_i$ for every $i=1,2,\cdots,m$. By setting $a_i=1$ and $b_i=d(e_i)$, for $i=1,2,\cdots,m$, from the above inequality we obtain $$\sum_{i=1}^md(e_i)^2+2\triangle\cdot 2\delta\sum_{i=1}^m1^2\le2(\triangle+\delta)\sum_{i=1}^md(e_i).$$ and $$HM_1(G)\le 2(\triangle+\delta)M_1(G)-4m\triangle\delta.$$ Thus, the equality holds if and only if $G$ is a regular graph.
By the results of [@9] we have, $M_1(G)\le
2m(\triangle+\delta)-n\triangle\delta$, with the equality holds if and only if $G$ is regular. So, we have the following result
Let G be a graph with n vertices and m edges, then $$HM_1(G)\le 4m(\triangle+\delta)^2-\triangle\delta(n+4m).$$
Let G be a graph with n vertices and m edges, then $$\delta^2\le \frac{HM_2(G)}{2(M_1(G)-2m)}\le \triangle^2,$$ the equality holds if and only G is a regular graph.
The number of pairs of edges which have a common end point is $\sum_{i=1}^n \left( \begin{array}{c} d_i \\ 2 \end{array} \right) =\frac{1}{2}M_1(G)-2m$. Also, $2\delta\le d(e_j)\le
2\triangle$, for $j=1,2,\cdots,m$. So, from the definition of second hyper Zagreb index, we have $$4\Big(\frac{1}{2}M_1(G)-m\Big)\delta^2\le HM_2(G)\le 4\Big(\frac{1}{2}M_1(G)-m\Big)\triangle^2,$$ and $$\delta^2\le \frac{HM_2(G)}{2(M_1(G)-2m)}\le \triangle^2,$$ the equality holds if and only if $G$ is a regular graph.
Let G be a graph with n vertices and m edges, then $$HM_2(G)\ge \frac{M_1(G)^3}{2m^2},$$ the equality holds if and only if G is regular.
For arithmetic and geometric mean inequality, $$\frac{1}{N}\sum_{e_i\sim e_j}d(e_i)d(e_j)\ge \Big[\prod_{e_i\sim e_j}d(e_i)d(e_j)\Big]^{\frac{1}{N}}=\Big[\prod_{i=1}^md(e_i)^{d(e_i)}\Big]^{\frac{1}{N}},$$ where $N=\frac{1}{2}M_1(G)$. Suppose that $L=\prod_{i=1}^md(e_i)^{d(e_i)}$. Taking natural logarithm on both sides, we obtain $$ln\,\, L=\sum_{i-1}^md(e_i)ln\,\, d(e_i)\ge \sum_{i=1}^md(e_i)ln\,\, \frac{1}{m}\sum_{i=1}^md(e_i),$$ and $$L\ge \Big(\frac{M_1(G)}{m}\Big)^{M_1(G)}.$$ Hence, $$\begin{aligned}
HM_2(G)&\ge N\Big[\frac{M_1(G)}{m}\Big]^{\frac{M_1(G)}{N}}\\
&= \frac{M_1(G)^3}{2m^2}.\end{aligned}$$ Clearly, the equality holds if and only if $G$ is a regular graph.
\[thm8\] Let G be a graph with n vertices and m edges, then $$HM_2(G)\le \frac{1}{2}M_1(G)^2-\delta(m-1)M_1(G)+(\delta-\frac{1}{2})HM_1(G),$$ the equality holds if and only if G is regular.
By the result of Theorem 3, we have $$d(e_i)\mu_i=\sum_{e_i\sim e_j}d(e_j)\le \sum_{i=1}^m d(e_i)-d(e_i)-(m-1-d(e_i))2\delta.$$ Thus $$\begin{aligned}
HM_2(G)&=\sum_{e_i\sim e_j}d(e_i)d(e_j)=\frac{1}{2}\sum_{i=1}^md(e_i)^2\mu_i=\frac{1}{2}\sum_{i=1}^md(e_i)\Big(\sum_{e_i\sim e_j}d(e_j)\Big)\\
&\le\frac{1}{2}\sum_{i=1}^md(e_i)\Big(\sum_{i=1}^md(e_i)-d(e_i)-2(m-1-d(e_i)\delta)\Big)\\
&=\frac{1}{2}M_1(G)^2-\frac{1}{2}HM_1(G)-\delta(m-1)M_1(G)+\delta
HM_1(G).\end{aligned}$$ The expected result is obtained from the above proof process.
Clearly, the equality holds when the graph $G$ is regular.
Let G be a graph with n vertices, m edges and $\delta$ minimum degree, then $$HM_2(G)\le \frac{1}{2}K^2-\Big(\delta(m-1)+(\delta-\frac{1}{2})(m+2\delta-1)\Big)K-m(m-1)(2\delta-1)\delta,$$ where $K=M_1(G)$ or $K=2m(\triangle+\delta-1)-n\triangle\delta$ with the equality if and only if G is regular.
Using Theorem \[thm3\] and Theorem \[thm8\], we obtain the expected result with $K=M_1(G)$. Moreover, we have $M_1(G)\le
2m(\triangle+\delta-1)+2m-n\triangle\delta$ [@9] with the equality holds if and only if $G$ is regular, so the expected result clearly follows for $K=2m(\triangle+\delta-1)-n\triangle\delta$.
[**Acknowledgments**]{}\
The authors would like to express their sincere gratitude to the anonymous referees and the editor for many friendly and helpful suggestions, which led to great deal of improvement of the original manuscript.
This work is partially supported by National Natural Science Foundation of China (nos. 11601006, 11471016, 11401004, Anhui Provincial Natural Science Foundation (nos. KJ2015A331, KJ2013B105).
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*1. Department of Mathematics and Computer Science, Adelphi University, Garden City, NY, USA.*\
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*2. School of Information Science and Technologys, Yunnan Normal University, Kunming, PR China.*\
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*3. Abdus Salam School of Mathematical Sciences, Government College University, Lahore, Pakistan.*\
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*4. Department of Applied Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran.*\
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*5. School of Mathematics and Physics, Anhui Jianzhu University, Hefei, PR China*
[^1]: Corresponding authors. Emails: S. Wang (e-mail: [email protected], [email protected]), W. Gao([email protected]), M.K. Jamil([email protected]), M.R. Farahani ([email protected]), J.-B. Liu([email protected]).
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{
"pile_set_name": "ArXiv"
}
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---
abstract: 'Options are financial instruments that depend on the underlying stock. We explain their non-Gaussian fluctuations using the nonextensive thermodynamics parameter $q$. A generalized form of the Black-Scholes (B-S) partial differential equation, and some closed-form solutions are obtained. The standard B-S equation ($q=1$) which is used by economists to calculate option prices requires multiple values of the stock volatility (known as the volatility smile). Using $q=1.5$ which well models the empirical distribution of returns, we get a good description of option prices using a single volatility.'
author:
- |
Lisa Borland\
Iris Financial Engineering and Systems\
456 Montgomery Street, Suite 800\
San Francisco, CA 94104, USA
title: 'Option Pricing Formulas based on a non-Gaussian Stock Price Model '
---
Although empirical stock price returns clearly do not follow the log-normal distribution, many of the most famous results of mathematical finance are based on that distribution. For example, Black and Scholes (B-S) were able to derive the prices of options and other derivatives of the underlying stock based on such a model. An option is the right to buy or sell the underlying stock at some set price (called the strike) at some time in the future. While of great importance and widely used, such theoretical option prices do not quite match the observed ones. In particular, the B-S model underestimates the prices of options in situations when the stock price at the time of exercise is different from the strike. In order to match the observed market values, the B-S model would need to use a different value of the volatility for each value of the strike. Such “implied volatilities” of options of various strike prices form a convex function known as the “volatility smile”.
Indeed, attempts have been made to modify the B-S model in ways that can correct for the smile effect (cf [@Hull] or more recently [@hyperbolic; @bouchaud]). However, those approaches are often very complicated or rather ad-hoc, and do not result in managable closed form solutions, which is the forte of the B-S approach. In this paper we do however succeed in developing a theory of non-Gaussian option pricing which allows for closed form solutions for European options, which are such that can be exercised exclusively on a fixed day of expiration and not before (as is the case for American options).
Our approach uses stochastic processes with statistical feedback [@pdependent] as a model for stock prices. Such processes were recently developed within the Tsallis generalized thermostatistics [@Tsallis]. The driving noise can be interpreted as a generalized Wiener process governed by a Tsallis distribution of entropic index $q$. In the limit $q \rightarrow 1$ the standard model is recovered. For $q \approx 1.5$, this model closely fits the empirically observed distribution for many financial time series, such as stock prices [@Osorioetal] (Figure 1), SP500 index,[@Osorioetal]), FX rates, etc. This is consistent with a cumulutive distribution having power tails of index 3 [@gopik]. We derive closed form option pricing formulas, reproducing prices which, relative to the standard B-S model, exhibit volatility smiles very close to those observed empirically (Figure 4). Note that $q=1.5$ well models hydrodynamic turbulence on small scales [@beck], reinforcing notions of a possible analogy between these two systems.
The standard model for stock prices is $
S = S_0 e^{Y(t)}
$ where $Y(t) = \ln(S(t+ t_0)/\ln S(t_0) $ follows $$\label{eq:lanstand}
d Y = \mu dt + \sigma d \omega$$ The drift $\mu$ is the mean rate of return and $\sigma^2$ is the variance of the stock logarithmic return. The noise $\omega$ is a Brownian motion defined with respect to a probability measure $F$. It is a Wiener process and satisfies $
E^F[d\omega(t)d\omega(t')] = dtdt'\delta(t-t')
$ where the notation $E^F[]$ means the expectation value with respect to the measure $F$. This model yields a Gaussian distribution for $Y$ resulting in a log-normal distribution for $S$. Within this framework, Black and Scholes were able to establish a pricing model to obtain the fair value of options on the underlying stock $S$.
In this paper we assume that the log returns $Y(t) = \ln S(t + t_0) /\ln S(t_0)$ follow $$\label{eq:langen}
d Y = \mu dt + \sigma d \Omega$$ with respect to the timescale $t$. Here $\Omega$ evolves according to the statistical feedback process[@pdependent] $$\label{eq:nomega}
d \Omega = P(\Omega)^{\frac{1-q }{2}} d \omega$$ The probability distribution $P$ satisfies the nonlinear Fokker-Planck equation $$\label{eq:nlfpcond}
\frac{ \partial}{\partial t} P(\Omega, t \mid \Omega ', t') = \frac{1}{2} \frac{ \partial }{\partial \Omega ^2} P^{2-q}(\Omega, t \mid \Omega ', t')$$ Explicit solutions for $P$ are given by Tsallis distributions $$\label{eq:ptsalliscond}
P_q(\Omega, t \mid \Omega(0), 0) = \frac{1}{Z(t)} \{ 1 - \beta(t) (1-q)[\Omega(t) - \Omega(0)] ^2 \}^{\frac{1}{1-q}}$$ Choosing $
\beta(t) = c^{\frac{1-q}{3-q}}((2-q) (3-q) t)^{-2/(3-q)} $ and $ Z(t) = ( (2-q) (3-q) c t )^{\frac{1}{3-q}}
$ ensures that the initial condition $P_q = \delta(\Omega(t) - \Omega(0)) $ is satisfied. The $q$-dependent constant $c$ is given by $
c = \beta Z^2 $ with $Z = \int_{- \infty}^{\infty} ( 1 - (1-q)\beta \Omega^2)^{\frac{1}{1-q}} d \Omega
$ for any $\beta$. With $\Omega(0) = 0$, we obtain a generalized Wiener process, distributed according to a zero-mean Tsallis distribution In the limit $q \rightarrow 1$ the standard theory Eq(\[eq:lanstand\]) is recovered, and $P_q$ becomes a Gaussian. We are concerned with the range $1 \le q < 5/3$ in which positive tails and finite variances are found [@Levy]. The distribution for $\ln S$ becomes $$\label{eq:ptsallisnonzero}
P_q(\ln S(t+t_0),t+t_0 \mid \ln S(t_0), t_0) = \frac{1}{Z(t)} \{1 - \tilde{\beta}(t) (1-q)[\ln \frac{S(t+t_0)}{S(t_0)} - \mu t ] ^2 \}^{\frac{1}{1-q}}$$ with $\tilde{\beta} = \beta(t)/ \sigma ^2 $. This implies that log-returns $\ln [S(t +t_0)/S(t_0)]$ over the timescale $ t$ follow a Tsallis distribution, consistent with empirical evidence for several markets, e.g. S&P500 (Figure 1 [@Osorioetal]) , with $q \approx 1.5$.
Our model exhibits a feedback from the macroscopic level characterised by $P$, to the microscopic level characterised by $\Omega$. We can imagine that this is really due to the interactions of many individual traders whose actions all contribute to shocks to the stock price which keep it in equilibrium. Their collective behaviour yields a nonhomogenous reaction to returns: rare events (i.e. extreme returns) will be accompanied by large reactions, and will tend to be followed by large returns in either direction.
Using Ito calculus [@Gardiner; @Risken], the equation for $S$ follows from Eq(\[eq:langen\]) as $$\label{eq:langenS}
d S = \tilde{\mu} Sdt + \sigma S d \Omega$$ where $
\tilde{\mu} = \mu + \frac{\sigma^2}{2} P_q^{1-q}. $ The term $\frac{\sigma^2}{2} P_q^{1-q}$ is the noise induced drift. Remember that $P_q$ is a function of $\Omega$ with $$\label{eq:omegaands}
\Omega(t) = \frac{{\ln S(t)}/{\ln S(0)} - \mu t}{\sigma}$$ (with $t_0=0$ for simplicity.) As in the standard case (cf [@Hull]), the noise term driving $S$ is the same as that driving the price $f(S)$ of a derivative of the underlying stock. It should be possible to invest one’s wealth in a portfolio of shares and derivatives such that the noise terms cancel each other, yielding a risk-free portfolio, the return on which must be the risk-free rate $r$. This results in a generalized B-S PDE $$\label{eq:bsgen}
\frac{\partial f}{\partial t} + rS \frac{\partial f}{\partial S} + \frac{1}{2} \frac{\partial^2f}{\partial S^2} \sigma^2
S^2 P_q^{1-q}
= rf$$ where $P_q( \Omega(t))$ evolves according to Eq(\[eq:nlfpcond\]). For $q \rightarrow 1$ the standard case is recovered. This PDE depends explicitly only on the risk-free rate and the variance, not on $\mu$, but it does depend implicitly on $\mu$ through $P_q(\Omega)$, with $\Omega$ given by Eq(\[eq:omegaands\]). Therefore, to be consistent with risk-free pricing theory, we should first transform our original stochastic equation for $S$ into a martingale before we apply the above analysis. This will not affect our results other than that $\tilde{\mu}$ will be replaced by the risk-free rate $r$, ultimately eliminating the dependency on $\mu$. We now show how this is done.
The discounted stock price $G = e^{-rt}S$ follows $
dG = (\tilde{\mu} -r)Gdt + \sigma G d \Omega
$ where $d \Omega$ follows Eq(\[eq:nomega\]). For there to be no arbitrage opportunities, risk-free asset pricing theory requires that this process be a martingale, which it is not due to the drift term $(\tilde{\mu} - r)G dt$. One can however define an alternative driving noise $z$ associated with an equivalent probability measure $Q$ so that, with respect to the new noise measure, the discounted stock price has zero drift and is thereby a martingale. Explicitly, $$\label{eq:dGomega}
dG = (\tilde{\mu} -r)Gdt + \sigma G P^{\frac{1-q}{2}} d \omega$$ Here, $P$ is a non-vanishing bounded function of $\Omega$. With respect to the initial noise $\omega$, $\Omega$ relates to $S$ via Eq(\[eq:omegaands\]). That is why for all means and purposes, $P$ in Eq(\[eq:dGomega\]) is simply a function of $S$ (or $G$), and the stochastic process can be seen as a standard state-dependent Brownian one. As a consequence, both the Girsanov theorem (which specifies the conditions under which we can transform from the measure $F$ to $Q$) and the Radon-Nikodym theorem (which relates the measure $F$ to $Q$) are valid, and we can formulate equivalent martingale measures much as in the standard case . We rewrite Eq(\[eq:dGomega\]) as $$\begin{aligned}
\label{eq:dGmart}
dG & = & \sigma G P^{\frac{1-q}{2}} d z\end{aligned}$$ where the new driving noise term $z$ is related to $\omega$ through $$\label{eq:dz}
dz = \frac{(\tilde{\mu} -r)}{\sigma P_q^{\frac{1-q}{2}}} dt + d \omega$$ With respect to $z$, we thus obtain $
dG = \sigma G d\Omega
$ with $
d \Omega = P_q^{\frac{1-q}{2}} dz
$ which is non other than a zero-mean Tsallis distributed generalized Wiener process, completely analogous to the one defined in Eq(\[eq:nomega\]). Transforming back to $S$ yields $
dS = r dt + \sigma S d\Omega.
$ Compared with Eq(\[eq:langenS\]), the rate of return $\tilde{\mu}$ has been replaced with the risk-free rate $r$. This recovers the same result as in the standard asset pricing theory. Consequently, in the risk-free representation, Eq(\[eq:omegaands\]) becomes $$\Omega(t) = \frac{1}{\sigma}(\frac{\ln S(t)}{\ln S(0)} - r t + \frac{\sigma^2}{2}\int_{0}^t P_q^{1-q}( \Omega(s)) ds)$$ This eliminates the dependency on $\mu$ which we alluded to in the discussion of Eq(\[eq:bsgen\]). As discussed later on, by standardizing the distributions $P_q(\Omega(s))$ we can explicitly solve for $\Omega(t)$ as a function of $S(t)$ and $r$.
Suppose that we have a European option $C$ which depends on $S(t)$, whose price $f$ is given by its expectation value in a risk-free (martingale) world as $
f(C) = E^Q [e^{-rT} C]
$. We assume the payoff on this option depends on the stock price at time $T$ so that $
C = h(S(T))
$. After stochastic integration of Eq(\[eq:dGmart\]) to obtain $S(T)$ we get $$\label{eq:fprice1}
f = e^{-rT} E^Q\left[h \left( S(0) \exp \left( \int_0^T \sigma P_q^{\frac{1-q}{2}} dz_s + \int_0^T(r - \frac{\sigma^2}{2} P_q^{1-q}) ds \right) \right) \right]$$ The key point is that the random variable $
\int_0^T P_q^{\frac{1-q}{2}} dz_s = \int_0^T d \Omega(s) = \Omega(T)
%\end{equation}
$ follows the Tsallis distribution Eq(\[eq:ptsalliscond\]). This gives $$\begin{aligned}
\label{eq:generalformula}
f &=& \frac{e^{-rT}}{Z(T)} \int_R h\left[ S(0)\exp(\sigma \Omega(T) +rT - \frac{\sigma^2}{2} \alpha T^{\frac{2}{3-q}}
+ (1-q) \alpha T^{\frac{2}{3-q}} \frac{\beta(T)}{2} \sigma^2 \Omega^2(T) ) \right]
\nonumber \\ & & (1-{\beta}(T)(1-q)\Omega(T)^2)^{\frac{1}{1-q}} d\Omega_T\end{aligned}$$ with $
\alpha = \frac{1}{2}(3-q) ((2-q)(3-q))c)^{\frac{q-1}{3-q}}.
$ We have utilized the fact that each of the distributions $P(\Omega(s))$ occuring in the latter term of Eq(\[eq:fprice1\]) can be mapped onto the distribution of $\Omega(T)$ at time $T$ via the appropriate variable transformations $
\Omega(s) = \sqrt{ {\beta(T)}/{\beta(s)} } \Omega(T)
$. A major difference to the standard case is the $\Omega^2(T)$-term which is a result of the noise induced drift. With $q= 1$, the standard option price is recovered [@Oksendal].
Eq(\[eq:generalformula\]) is valid for an arbitrary payoff $h$. We shall evaluate it explicitly for a European call option, which gives the holder the right to buy the stock $S$ at the strike price $K$, on the day of expiration $T$. The payoff is $
C = \max [ S(T) - K, 0]
$. Only if $S(T) > K$ will the option have value at expiration $T$ (it will be in-the-money). The price $c$ of such an option becomes $$\begin{aligned}
\label{eq:optionprice}
c & = & E^Q [e^{-rT} C]
= E^Q [ e^{-rT} S(T)]_D - E^Q [ e^{-rT} K]_D
= J_1 - J_2 \label{eq:cprice}\end{aligned}$$ where the subscript $D$ stands for the set $\{ S(T) > K \}$. This condition is met if $
-\frac{\sigma^2}{2} \alpha T^{\frac{2}{3-q}} +
(1-q)\alpha T^{\frac{2}{3-q}} \frac{\beta(T)}{2} \sigma^2 \Omega^2 + \sigma \Omega + rT > \ln {K}/{S(0)},
%\end{equation}
$ which is satisfied for $\Omega$ between the two roots $s_1$ and $s_2$ of the corresponding quadratic equation. This is a very different situation from the standard case, where the inequality is linear and the condition $S(T) > K$ is satisfied for all values of the random variable greater than a threshold. In our case, due to the noise induced drift, values of $S(T) $ in the risk-neutral world are not monotonically increasing as a function of the noise. As $q \rightarrow 1$, the larger root $s_2$ goes toward $\infty$, recovering the standard case. But as $q$ increases, the tails of the noise distribution get larger, as does the noise induced drift which tends to pull the system back. As a result we obtain $$\begin{aligned}
J_1
\label{eq:j1}
& = & S(0) \frac{1}{Z(T)} \int_{s_1}^{s_2} \exp( \sigma \Omega -
\frac{\sigma^2}{2} \alpha T^{\frac{2}{3-q}}
- (1-q) \alpha T^{\frac{2}{3-q}} \frac{\beta(T)}{2} \sigma^2 \Omega^2 ) \nonumber \\
& & (1 - (1-q) \beta(T) \Omega^2)^{\frac{1}{1-q}} d\Omega \\
\label{eq:j2}
J_2 & = & e^{-rT} K \frac{1}{Z(T)} \int_{s_1}^{s_2} (1 - (1-q) {\beta}(T) \Omega^2)^{\frac{1}{1-q}} d\Omega \end{aligned}$$
The equation Eq(\[eq:cprice\]) with Eq(\[eq:j1\]) and Eq(\[eq:j2\]) constititutes a closed form expression for the price of a European call. We calculated option prices for different values of the index $q$, and studied their properties as a function of the relevant variables such as the current stock price $S(0)$, the strike price $K$, time to expiration $T$, the risk free rate $r$ and $\sigma$. The results obtained by our closed form pricing formula were confirmed both by implicitly solving the generalized B-S PDE Eq(\[eq:bsgen\]) and via Monte Carlo simulations of Eq(\[eq:dGmart\]). Note that American option prices can be solved numerically via Eq(\[eq:bsgen\]).
We compare results of the standard model ($q = 1$) with those obtained for $q= 1.5$, which fits well to real stock returns. Figure 2 shows the difference in call price. In Figure 3, the B-S implied volatilities (which make the $q=1$ model match the $q=1.5$ results) are plotted as a function of $K$. The assymetric smile shape, which is more pronounced for shorter times, reproduces well-known systematic features of the “volatility smile” that appears when using the standard $q=1$ model to price real options. In Figure 4, the volatility smiles for actual traded options on BP and S&P 500 futures is shown together with those resulting from our model using $q=1.5$. These results are encouraging, and we are currently studying a larger sample of options data. Empirical work is required to see if arbitrage opportunities can be uncovered that do not appear when the standard model is used. Another potential application will be with respect to option replication and hedging.
Acknowledgements: Fruitful discussions with Roberto Osorio and Jeremy Evnine are gratefully acknowledged.
[99]{}
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We obtain constraints on parameters of the Yukawa-type corrections to Newton’s gravitational law from measurements of the gradient of the Casimir force between surfaces coated with ferromagnetic metal Ni and from measurements of the Casimir force between Au-coated sinusoidally corrugated surfaces at various angles between corrugations. It is shown that constraints following from the experiment with magnetic surfaces are slightly weaker than currently available strongest constraints, but benefit from increased reliability and independence of systematic effects. The constraints derived from the experiment with corrugated surfaces within the interaction region from 11.6 to 29.2nm are stronger up to a factor of 4 than the strongest constraints derived from other experiments. The possibility of further strengthening of constraints on non-Newtonian gravity by using the configurations with corrugated boundaries is proposed.'
author:
- 'G. L. Klimchitskaya'
- 'U. Mohideen'
- 'V. M. Mostepanenko'
title: Constraints on corrections to Newtonian gravity from two recent measurements of the Casimir interaction between metallic surfaces
---
Introduction
============
In the last few decades corrections to Newton’s law of gravitation and constraints on them have become the subject of considerable study (see the monograph [@1] and reviews [@2; @3; @4; @5]). From the experimental standpoint, it is of most importance that at separations between the test bodies below 0.1mm Newton’s law is not confirmed by measurements with sufficient precision. Theoretically, many extensions of the Standard Model of elementary particles and their interactions predict corrections to the Newton law of power- and Yukawa-type due to exchange of light and massless elementary particles [@6; @7; @8; @9; @10]. On the other hand, similar corrections are predicted by the extra-dimensional physics with a low-energy compactification scale [@11; @12; @13; @14; @15]. This makes the search for such corrections, or at least constraining their parameters, interesting for the problems of dark matter and unification of gravitation with other fundamental interactions.
A lot of successful work has been done on constraining the power- and Yukawa-type corrections to Newton’s law of gravitation from gravitational experiments of Eötvos- and Cavendish-type [@1; @2; @3; @4; @16; @17]. Although the most strong constraints on the power-type corrections were obtained in this way, it was found that the resulting constraints on the Yukawa-type corrections become much weaker in the interaction range below a few micrometers. This is explained by the fact that at sufficiently small separations between the test bodies the van der Waals [@18] and Casimir force [@19] becomes the dominant background force in place of gravitation. The two names belong to a single force of quantum origin caused by the zero-point and thermal fluctuations of the electromagnetic field. They are usually used at short and relatively large separations, respectively, where the effects of relativistic retardation are immaterial or, on the contrary, are influential and should be taken into account.
The possibility to constrain corrections to Newton’s law from the van der Waals and Casimir force was proposed long ago [@20; @21] for the cases of Yukawa-type and power-type corrections, respectively. At that time, however, reasonably precise measurements of the van der Waals and Casimir force were not available. Things have changed during the last 15 years when a lot of more precise experiments on measuring the Casimir force between metallic, dielectric and semiconductor bodies have been performed (see reviews [@22; @23; @24; @25]). The measure of agreement between the measurement data of these experiments and theoretical description of the Casimir force in the framework of the Lifshitz theory resulted in the strengthening of previously known constraints on the parameters of Yukawa-type corrections up to a factor of $2.4\times 10^7$ [@5; @19; @26]. Using different experiments on measurement of the Casimir force, the strongest constraints on the corrections to Newton’s law were obtained over a wide interaction region from about 1nm to a few micrometers. Note that for shorter interaction regions the strongest constraints on the Yukawa-type corrections follow from precise atomic physics [@27], whereas starting from a few micrometers the gravitational experiments [@1; @2; @3; @4; @16; @17] remain the most reliable source of such constraints.
In this paper we obtain constraints on the Yukawa-type corrections to Newton’s gravitational law from two recently performed experiments on measuring the Casimir interaction by means of an atomic force microscope (AFM). Each of these experiments is highly significant, as compared with all earlier measurements of the Casimir interaction. In the first [@28] the dynamic AFM was used to measure the gradient of the Casimir force between a plate and a sphere both coated with the ferromagnetic metal Ni with no spontaneous magnetization. As a result, the predictions of the Lifshitz theory generalized for the case of magnetic materials more than 40 years ago [@29] were experimentally confirmed. The distinguished feature of the experiment with two magnetic surfaces is also that it allows to shed light on the role of some important systematic effects (see Sec. II for details) and, thus, remove any doubt in the reliability of constraints obtained.
In the second experiment of our interest here [@30] the static AFM was used to measure the Casimir force between a plate and a sphere both with sinusoidally corrugated surfaces coated with nonmagnetic metal Au. The unusual feature of this experiment, as compared with earlier performed experiments with corrugated surfaces, is that the Casimir force was measured at various angles between the longitudinal corrugations on both bodies. This introduced into the problem an additional parameter (the angle between corrugations) that can be chosen to obtain the most strong constraints from the measure of agreement between the experimental data and theory of the Casimir force for corrugated surfaces based on the derivative expansion [@31; @32; @33].
The constraints on corrections to Newton’s law obtained by us from the experiment with magnetic surfaces are in agreement with those obtained earlier [@34] for smooth Au surfaces by means of dynamic AFM [@35], but slightly weaker due to different densities of Au and Ni. The advantage of constraints following from the experiment with Ni surfaces is that they are not only fully justified on their own, but add substantiation to the constraints obtained from the experiments with nonmagnetic metal surfaces. As to constraints obtained from the experiment with corrugated surfaces, they are stronger up to a factor of 4 than the most strong constraints reported so far [@26; @36; @37] in the interaction region from 11.6 to 29.2nm.
The paper is organized as follows. In Sec. II we present the exact expression for the Yukawa-type interaction in the experimental configuration of Ref. [@28] and derive the respective constraints on corrections to Newton’s gravitational law. The advantages of using magnetic materials are elucidated. Section III is devoted to the experiment with corrugated surfaces [@30]. Here, we derive an expression for the Yukawa-type force in configurations with different angles between corrugations. The obtained expression is used to find the stronger constraints on corrections to Newton’s law. Some modifications in the setup are proposed allowing further strengthening of the constraints in configurations with corrugated surfaces. In Sec. IV the reader will find our conclusions and discussion.
Constraints from the gradient of the Casimir force [\
]{} between two magnetic surfaces
=====================================================
We begin with the standard parametrization of the spin-independent Yukawa-type correction to Newton’s gravitational law [@1; @2; @3; @4; @5] (for spin-dependent corrections see Refs. [@38; @39]). The total gravitational potential between the two point-like masses $m_1$ and $m_2$ spaced at a separation $r$ takes the form $$V(r)=V_N(r)+V_{\rm Yu}(r)=-\frac{Gm_1m_2}{r}\left(
1+\alpha e^{-r/\lambda}\right),
\label{eq1}$$ where $V_N(r)$ and $V_{\rm Yu}(r)$ are the Newtonian part and the Yukawa-type correction, respectively. Here, $G$ is the Newtonian gravitational constant, and $\alpha$ and $\lambda$ are the strength and interaction range of the Yukawa-type correction. Similar to Ref. [@40] it can be shown that at separations below a few micrometers the Newtonian gravitational force between the test bodies $V_1$ and $V_2$ in experiments under consideration is much smaller than the error in measurements of the Casimir force. Because of this, in all subsequent calculations the Newtonian potential can be neglected, and the Yukawa-type addition to it is considered on the background of the measured Casimir force. Then the gravitational force acting between the test bodies at short separations can be obtained by the integration of the Yukawa-type correction $V_{\rm Yu}(r)$ defined in Eq. (\[eq1\]) over the volumes of both bodies $$V_{\rm Yu}(a)=-G\alpha\int_{V_1}d^3r_1
\rho_1(\mbox{\boldmath$r$}_1)\int_{V_2}d^3r_2
\rho_2(\mbox{\boldmath$r$}_2)
\frac{e^{-|{\scriptsize{\mbox{\boldmath$r$}_1-
\mbox{\boldmath$r$}_2}}|/\lambda}}{|\mbox{\boldmath$r$}_1-
\mbox{\boldmath$r$}_2|}.
\label{eq2}$$ Here, $a$ is the closest separation between the test bodies and $\rho_1(\mbox{\boldmath$r$}_1)$ and $\rho_2(\mbox{\boldmath$r$}_2)$ are the respective mass densities (note that $\rho_1$ and $\rho_2$ are not constant because in the experiments used below each test body consists of several homogeneous layers of different densities). The gravitational force due to the Yukawa-type correction and its gradient are given by $$F_{\rm Yu}(a)=-\frac{\partial V_{\rm Yu}(a)}{\partial a},
\qquad
\frac{\partial F_{\rm Yu}(a)}{\partial a}
=-\frac{\partial^2 V_{\rm Yu}(a)}{\partial a^2}.
\label{eq3}$$
In the experiment [@28] the gradient of the Casimir force was measured between a Ni-coated plate and a Ni-coated hollow microsphere attached to the tip of an AFM cantilever operated in the dynamic regime [@19; @22]. The silicon plate ($V_1$) of a few millimeter diameter and thickness can be considered as having an infinitely large area and an infinitely large thickness when we have to deal with the submicrometer region of $\lambda$. The density of Si is $\rho_{\,\rm Si}=2.33\times 10^3\,\mbox{kg/m}^{3}$. For technological purposes the Si plate was coated first with a layer of Cr having a thickness $\Delta_{\rm Cr}^{\!(1)}=10\,$nm and density $\rho_{\,\rm Cr}=7.15\times 10^3\,\mbox{kg/m}^{3}$ and then with a layer of Al having a thickness $\Delta_{\rm Al}^{\!(1)}=40\,$nm and density $\rho_{\,\rm Al}=2.7\times 10^3\,\mbox{kg/m}^{3}$. Finally the plate was coated with an outer layer of magnetic metal Ni with a thickness $\Delta_{\rm Ni}^{\!(1)}=250\,$nm and density $\rho_{\,\rm Ni}=8.9\times 10^3\,\mbox{kg/m}^{3}$. The hollow microsphere ($V_2$) was made of glass with density $\rho_{g}=2.5\times 10^3\,\mbox{kg/m}^{3}$. The thickness of the spherical envelope was $\Delta_{g}^{\!(2)}=5\,\mu$m. The sphere was also coated with successive layers of Cr, Al and Ni having the thicknesses $\Delta_{\rm Cr}^{\!(2)}=\Delta_{\rm Cr}^{\!(1)}$, $\Delta_{\rm Al}^{\!(2)}=\Delta_{\rm Al}^{\!(1)}$, and $\Delta_{\rm Ni}^{\!(2)}=210\,$nm. The outer radius of the sphere with all the coatings included is $R=61.7\,\mu$m.
The exact integration over the volumes of a plate and a sphere in Eq.(\[eq2\]) with account of their layer structure can be performed like in Ref. [@41]. Then, substituting the obtained result in Eq. (\[eq3\]), we arrive at $$\frac{\partial F_{\rm Yu}(a)}{\partial a}=
4\pi^2G\alpha\lambda^2e^{-a/\lambda}X^{(1)}(\lambda)X^{(2)}(\lambda),
\label{eq4}$$ where the following notations are introduced: $$\begin{aligned}
&&
X^{(1)}(\lambda)=\rho_{\,\rm Ni}-(\rho_{\,\rm Ni}-\rho_{\,\rm Al})
e^{-\Delta_{\rm Ni}^{\!(1)}/\lambda}
\nonumber \\
&&~~~~
-(\rho_{\,\rm Al}-\rho_{\,\rm Cr})
e^{-(\Delta_{\rm Ni}^{\!(1)}+\Delta_{\rm Al}^{\!(1)})/\lambda}
\nonumber \\
&&~~~~
-(\rho_{\,\rm Al}-\rho_{\,\rm Si})
e^{-(\Delta_{\rm Ni}^{\!(1)}+\Delta_{\rm Al}^{\!(1)}
+\Delta_{\rm Cr}^{\!(1)})/\lambda},
\label{eq5}\\
&&
X^{(2)}(\lambda)=\rho_{\,\rm Ni}\Phi(R,\lambda)-
(\rho_{\,\rm Ni}-\rho_{\,\rm Al})\Phi(R-\Delta_{\rm Ni}^{\!(2)},\lambda)
e^{-\Delta_{\rm Ni}^{\!(2)}/\lambda}
\nonumber \\
&&~~
-(\rho_{\,\rm Al}-\rho_{\,\rm Cr})\Phi(R-\Delta_{\rm Ni}^{\!(2)}-\Delta_{\rm Al}^{\!(2)},\lambda)
e^{-(\Delta_{\rm Ni}^{\!(2)}+\Delta_{\rm Al}^{\!(2)})/\lambda}
\nonumber \\
&&~~
-(\rho_{\,\rm Cr}-\rho_{g})
\Phi(R-\Delta_{\rm Ni}^{\!(2)}-\Delta_{\rm Al}^{\!(2)}-\Delta_{\rm Cr}^{\!(2)},\lambda)
e^{-(\Delta_{\rm Ni}^{\!(2)}+\Delta_{\rm Al}^{\!(2)}+\Delta_{\rm Cr}^{\!(2)})/\lambda}
\nonumber \\
&&~~
-\rho_{g}\Phi(R-\Delta_{\rm Ni}^{\!(2)}-\Delta_{\rm Al}^{\!(2)}
-\Delta_{\rm Cr}^{\!(2)}-\Delta_{g}^{\!(2)},\lambda)
e^{-(\Delta_{\rm Ni}^{\!(2)}+
\Delta_{\rm Al}^{\!(2)}+\Delta_{\rm Cr}^{\!(2)}+\Delta_g^{\!(2)})/\lambda},
\nonumber\end{aligned}$$ and the following notation is introduced $$\Phi(r,\lambda)=r-\lambda+(r+\lambda)e^{-2r/\lambda}.
\label{eq6}$$
The constraints on the parameters ($\lambda,\alpha$), which are often referred to as the parameters of [*non-Newtonian gravity*]{}, can be obtained from the comparison between the measurement data for the gradient of the Casimir force $F_{C}^{\prime}(a)$ and respective theory. In Ref. [@28] it was found that within the entire separation region from 223 to 550nm there is an excellent agreement between the data and theoretical predictions of the Lifshitz theory of the van der Waals and Casimir force [@18; @19] with omitted relaxation properties of conduction electrons (the so-called [*plasma model approach*]{}). The predictions of the Lifshitz theory with included relaxation properties of free charge carriers (the so-called [*Drude model approach*]{}) were excluded by the measurement data at a 95% confidence level within the separation region from 223 to 350nm (in the end of this section we provide a brief discussion of different approaches to the Lifshitz theory which is essential for obtaining constraints on non-Newtonian gravity). The measure of agreement with the adequate theory is characterized by the total experimental error $\Delta_{F_C^{\prime}}(a)$ in the measured gradient of the Casimir force determined at a 67% confidence level [@28]. Keeping in mind that within the limits of this error no additional interaction of Yukawa-type was observed, the constraints on the parameters $\lambda$ and $\alpha$ can be obtained from the inequality $$\left|\frac{\partial F_{\rm Yu}(a)}{\partial a}\right|
\leq\Delta_{F_C^{\prime}}(a).
\label{eq7}$$
We have substituted Eqs. (\[eq4\])–(\[eq6\]) in Eq. (\[eq7\]) and analyzed the resulting inequality at different separations. It was found that for $\lambda\lesssim 200\,$nm the strongest constraints are determined at the shortest separation $a=223\,$nm where $\Delta_{F_C^{\prime}}=1.2\,\mu$N/m [@28]. For $200\,\mbox{nm}\lesssim\lambda\lesssim 315\,$nm and $315\,\mbox{nm}\lesssim\lambda\lesssim 630\,$nm the strongest constraints follow at $a=250\,$ and 300nm, respectively (with respective $\Delta_{F_C^{\prime}}=1.05$ and $0.89\,\mu$N/m). Finally, at $\lambda>630\,$nm the strongest constraints are obtained at $a=350\,$nm ($\Delta_{F_C^{\prime}}=0.81\,\mu$N/m). The resulting constraints are shown by the solid line in Fig. \[fg1\]. Here and below the region of ($\lambda,\alpha$) plane above each line is prohibited and below is allowed by the results of respective experiment. In the same figure by the dashed line we show the constraints obtained in Ref. [@34] from measurements of the gradient of the Casimir force between two Au-coated surfaces by means of dynamic AFM [@35]. The dotted line shows the constraints obtained [@34] from the experiment on measuring gradient of the Casimir force between an Au-coated sphere and a Ni-coated plate [@42] using the same setup. As can be seen in Fig. \[fg1\], the constraints indicated by the solid line are slightly weaker than those shown by the dashed and dotted lines. This is caused by the fact that density of Ni is smaller than density of Au and by different experimental errors. Note also that our constraints shown by the solid line can be obtained in a simpler way by using the proximity force approximation [@19; @22] $$F_{\rm Yu}(a)=2\pi R E_{\rm Yu}(a),
\label{eq7a}$$ to calculate the gradient of the Yukawa-type force, where $E_{\rm Yu}(a)$ is the energy per unit area of Yukawa-type interaction between two plane-parallel plates having the same layer structure as our test bodies. According to the results of Refs. [@41; @43], this is possible under the conditions $$\frac{\lambda}{R}\ll 1, \qquad
\frac{\Delta_{\rm Au}^{\!(2)}+\Delta_{\rm Al}^{\!(2)}+
\Delta_{\rm Cr}^{\!(2)}+\Delta_g^{\!(2)}}{R}\ll 1,
\label{eq8}$$ which are satisfied in our experimental configuration with a wide safety margin. In this case the function $\Phi(r,\lambda)$ with any argument $r$ can be approximately replaced with $R$.
It would be interesting also to compare the constraints on non-Newtonian gravity obtained here from the experiment with two Ni surfaces (solid line in Fig. \[fg1\]) with the strongest constraints obtained so far using the alternative setups. For this purpose in Fig. \[fg2\] we reproduce the solid line of Fig. \[fg1\] as the solid line 1. The solid line 2 in Fig. \[fg2\] was obtained [@26] from measurements of the thermal Casimir-Polder force between ${}^{87}$Pb atoms belonging to the Bose-Einstein condensate and a SiO${}_2$ plate [@44], and the solid line 3 was obtained [@45] from measurements of gradient of the Casimir force between an Au sphere and a rectangular corrugated semiconductor (Si) plate by means of a micromachined oscillator [@46]. Next, the solid line 4 in Fig. \[fg2\] was found from an effective measurement of the Casimir pressure between two parallel Au plates by means of a micromachined oscillator [@36; @37], and the dashed line was obtained from the Casimir-less experiment [@47]. As can be seen in Fig. \[fg2\], various constraints obtained using quite different setups are consistent with the constraints of line 1 obtained from the most recent experiment with two magnetic surfaces.
In the end of this section it is pertinent to note that the experiment with two magnetic surfaces [@28] plays the key role in the test of validity of the Lifshitz theory. Keeping in mind that constraints on non-Newtonian gravity are derived from the measure of agreement between the measurement data and theory, this experiment is also important to validate the reliability of constraints obtained. As mentioned above, the Lifshitz theory is in agreement with the plasma model approach to the Casimir force, which disregards the relaxation properties of free charge carriers, and excludes the Drude model approach taking these properties into account (see the experiments of Refs. [@35; @36; @37] and earlier experiments reviewed in Refs. [@19; @22]). This is against expectations of many and gave rise to the search of some systematic effects which could reverse the situation. After several unsuccessful attempts (see Ref. [@48] for a review) the influence of large surface patches was selected as the most probable systematic effect which could bring the data in agreement with the Drude model approach [@49]. In two experiments on measuring the Casimir force between Au surfaces [@50; @51] hypothetical large patches were described by models with free fitting parameters and used in respective fitting procedures. In these experiments, which are not independent measurements of the Casimir force, an agreement of the data with the Drude model approach has been claimed (see Refs. [@52; @53; @54; @55; @56] for a critical discussion).
The crucial point to underline here is that for nonmagnetic metals the Drude model approach leads to smaller gradients of the Casimir force than the plasma model approach [@19; @22; @35; @36; @37]. Thus, the effect of large patch potentials (which leads to an attraction similar to the Casimir force) is added to the predictions of the Drude model approach and might make the total theoretical force compatible with the measurement data [@49]. By contrast, for two magnetic metals the Drude model approach leads to larger gradients of the Casimir force than the plasma model approach [@28; @57; @58]. Thus, if the effect of patches were important in this case, it would further increase the disagreement between the predictions of the Drude model approach and the measurement data observed in Ref. [@28]. This confirms that surface patches do not play an important role in precise experiments on measuring the Casimir force in accordance with the model of patches [@59] predicting a negligibly small effect from patches [@19; @22]. Recently the patches on Au samples used in measurements of the Casimir force were investigated by means of Kelvin probe microscopy [@60]. The force originating from them was found to be too small to affect the conclusions following from precise measurements of the Casimir force. It is the matter of fact that the experimental data of all independent measurements of the Casimir interaction between both nonmagnetic and magnetic metals are in excellent agreement with the predictions of the Lifshitz theory combined with the plasma model approach and exclude the Drude model approach. Although the fundamental reasons behind this fact have not yet been finally understood, the constraints on non-Newtonian gravity obtained on this basis can be already considered as reliable enough.
Constraints from the Casimir force between two corrugated surfaces
==================================================================
In Sec. II we have used the most recent measurement of the Casimir interaction where the material dependence played a major role in theory-experiment comparison. Another recent experiment [@30] is of quite a different nature. In Ref. [@30] the normal Casimir force acting perpendicular to the surface was measured between the sinusoidally corrugated surfaces of a sphere and a plate. The corrugated boundary surfaces have long been used in measurements of the Casimir force (see Refs. [@22; @23] for a review). For example, the normal Casimir force between a rectangular corrugated semiconductor (Si) plate and a smooth Au sphere has been measured by means of a micromachined oscillator and compared with theory based on the exact scattering approach [@46]. The obtained constraints on non-Newtonian gravity are discussed in Sec. II (see solid line 3 in Fig. \[fg2\]). A further example is the lateral Casimir force between a sinusoidally corrugated plate and a sinusoidally corrugated sphere, both coated with Au, which was measured and compared with exact theory in Refs. [@61; @62]. This experiment resulted in the maximum strengthening of constraints on non-Newtonian gravity from the Casimir effect by a factor of $2.4\times 10^{7}$ discussed in Sec. I. In experiments with corrugated surfaces the nontrivial geometry plays a major role in the theory-experiment comparison whereas different approaches to the description of material properties cannot be differentiated due to the lower experimental precision.
The specific feature of the experiment of Ref. [@30] is that the normal Casimir force between a sinusoidally corrugated Au-coated plate and a sinusoidally corrugated Au-coated sphere was measured at various angles between corrugations using an AFM. The plate in this experiment is the diffraction grating with uniaxial sinusoidal corrugations of period $\Lambda=570.5\,$nm and amplitude $A_1=40.2\,$nm. The grating was made of hard epoxy with density $\rho_e=1.08\times 10^3\,\mbox{kg/m}^3$ and coated with an Au layer of thickness $\Delta_{\rm Au}^{\!(1)}=300\,$nm. The corrugated plate was used as a template for the pressure imprinting of the corrugations on the bottom surface of a sphere. The polystyrene sphere has a density $\rho_p=1.06\times 10^3\,\mbox{kg/m}^3$. It was coated with a layer of Cr of thickness $\Delta_{\rm Cr}^{\!(2)}=10\,$nm, then with a layer of Al of thickness $\Delta_{\rm Al}^{\!(2)}=20\,$nm and finally with a layer of Au of thickness $\Delta_{\rm Au}^{\!(2)}=110\,$nm. The outer radius of the coated sphere is $R=99.6\,\mu$m. The imprinted corrugations on the sphere have the same period as on the plate and the amplitude $A_2=14.6\,$nm. The size of an imprint area was measured to be $L_x\approx L_y\approx 14\,\mu$m, i.e., it is much larger than $\Lambda$. In Ref. [@30] the Casimir force between the sphere and the plate was measured at the following angles between the axes of corrugations on both bodies: $\theta=0^{\circ}$, $1.2^{\circ}$, $1.8^{\circ}$, and $2.4^{\circ}$.
Now we calculate the Yukawa-type force in the experimental configuration of Ref. [@30]. For this purpose we first consider the Yukawa-type energy per unit area in the configuration of two plane-parallel plates spaced at a separation $a$ having the same layer structure as a plate and a sphere in the experiment. The result is [@41] $$E_{\rm Yu}(a)=
-2\pi G\alpha\lambda^3e^{-a/\lambda}X^{(1)}(\lambda)X^{(2)}(\lambda),
\label{eq9}$$ where now $$\begin{aligned}
&&
X^{(1)}(\lambda)=\rho_{\,\rm Au}-
(\rho_{\,\rm Au}-\rho_{e})
e^{-\Delta_{\rm Au}^{\!(1)}/\lambda},
\label{eq10} \\
&&
X^{(2)}(\lambda)=\rho_{\,\rm Au}-
(\rho_{\,\rm Au}-\rho_{\,\rm Al})
e^{-\Delta_{\rm Au}^{\!(2)}/\lambda}
\nonumber \\
&&~~
-(\rho_{\,\rm Al}-\rho_{\,\rm Cr})
e^{-(\Delta_{\rm Au}^{\!(2)}+\Delta_{\rm Al}^{\!(2)})/\lambda}
\nonumber \\
&&~~
-(\rho_{\,\rm Cr}-\rho_{p})
e^{-(\Delta_{\rm Au}^{\!(2)}+\Delta_{\rm Al}^{\!(2)}+
\Delta_{\rm Cr}^{\!(2)})/\lambda}.
\nonumber\end{aligned}$$ Next, we introduce corrugations at an angle $\theta$ on the parallel plates and find their effect by means of the geometrical averaging [@19; @22] $$E_{\rm Yu}^{\rm corr}(a)=\frac{1}{L_xL_y}
\int_{-L_x/2}^{L_x/2}\!\!dx
\int_{-L_y/2}^{L_y/2}\!\!dy\,E_{\rm Yu}\Big(z(a,x,y)\Big).
\label{eq11}$$ Here, $E_{\rm Yu}$ is the energy per unit area defined in Eq. (\[eq9\]) calculated at different separations $z$ between the corrugated plates which are assumed parallel to the ($x,y$) plane $$z(a,x,y)=a+A_1\cos\frac{2\pi x}{\Lambda}-
A_2\cos\frac{2\pi x^{\prime}}{\Lambda}.
\label{eq12}$$ Note that there is no phase shift between the corrugations on both plates, so that $x^{\prime}=x\cos\theta-y\sin\theta$.
Finally, to obtain the Yukawa-type force between a corrugated plate and a corrugated sphere, we apply the proximity force approximation (\[eq7a\]) taking into account different radii of separate spherical layers. After an easy calculation using Eqs. (\[eq9\])–(\[eq12\]), the Yukawa-type force between a corrugated plate and a corrugated sphere takes the form $$F_{\rm Yu}^{\rm corr}(a)=
-4\pi^2G\alpha\lambda^3e^{-a/\lambda}X^{(1)}(\lambda)
\tilde{X}^{(2)}(\lambda)X(\lambda,\theta),
\label{eq13}$$ where $$\begin{aligned}
&&
\tilde{X}^{(2)}(\lambda)=R\rho_{\,\rm Au}-
(\rho_{\,\rm Au}-\rho_{\,\rm Al})(R-\Delta_{\rm Au}^{\!(2)})
e^{-\Delta_{\rm Au}^{\!(2)}/\lambda}
\nonumber \\
&&~~
-(\rho_{\,\rm Al}-\rho_{\,\rm Cr})
(R-\Delta_{\rm Au}^{\!(2)}-\Delta_{\rm Al}^{\!(2)})
e^{-(\Delta_{\rm Au}^{\!(2)}+\Delta_{\rm Al}^{\!(2)})/\lambda}
\label{eq14} \\
&&~~
-(\rho_{\,\rm Cr}-\rho_{p})
(R-\Delta_{\rm Au}^{\!(2)}-\Delta_{\rm Al}^{\!(2)}-\Delta_{\rm Cr}^{\!(2)})
e^{-(\Delta_{\rm Au}^{\!(2)}+\Delta_{\rm Al}^{\!(2)}+\Delta_{\rm Cr}^{\!(2)})/\lambda}
\nonumber\end{aligned}$$ and the function $X(\lambda,\theta)$ is defined as $$X(\lambda,\theta)=\frac{1}{L_xL_y}
\int_{-L_x/2}^{L_x/2}\!\!dx
\int_{-L_y/2}^{L_y/2}\!\!dy\,
e^{-[A_1\cos(2\pi x/\Lambda)-
A_2\cos(2\pi x^{\prime}/\Lambda)]/\lambda}.
\label{eq15}$$
For zero angle between corrugations at both surfaces ($\theta=0$) one arrives to a more simple representation $$X(\lambda,0)=\frac{1}{L_x}
\int_{-L_x/2}^{L_x/2}\!\!dx
e^{-[(A_1-A_2)\cos(2\pi x/\Lambda)]/\lambda}.
\label{eq16}$$ The integral in Eq. (\[eq16\]) can be evaluated analytically using the formula 2.5.10(3) in Ref. [@63] if there is an integer $n$ such that $n\Lambda=L_x$. In this case $$X(\lambda,0)=I_0\left(\frac{A_1-A_2}{\lambda}\right),
\label{eq17}$$ where $I_0(z)$ is the Bessel function of imaginary argument. If $n\Lambda+\eta=L_x$ where $0<\eta<\Lambda$, Eq. (\[eq17\]) is satisfied only approximately. If in the interaction region of our interest (see Fig. \[fg4\] below) it occurs $(A_1-A_2)/\lambda\gg 1$, the maximum error arising from the use of Eq. (\[eq17\]) achieves 5%. In the case $(A_1-A_2)/\lambda\sim 1$ this error is equal to $\approx 2$%. In the general case of an arbitrary $\theta$ the quantity $X(\lambda,\theta)$ can be computed numerically. In Fig. \[fg3\] the computational results are plotted by the solid lines as functions of $\lambda$ at $\theta=0^{\circ}$, $1.2^{\circ}$, $1.8^{\circ}$, and $2.4^{\circ}$ used in the experiment of Ref. [@30] from bottom to top, respectively, to a logarithmic scale.
The measurement data of Ref. [@30] for the normal Casimir force between corrugated surfaces were compared with the results of numerical computations based on the derivative expansion approach [@31; @32; @33] and a good agreement was found within the limits of the experimental errors $\Delta_{F_C}(a)$ determined at the 67% confidence level (minor disagreement at the shortest separations in Fig. 3 of Ref. [@30] comes from the use of an oscillator model in place of the optical data for the complex index of refraction). Then the constraints on the parameters $\lambda$ and $\alpha$ of the corrections to Newton’s law were found from the inequality $$|F_{\rm Yu}(a)|\leq \Delta_{F_C}(a),
\label{eq18}$$ where $F_{\rm Yu}(a)$ is given by Eq. (\[eq13\]) with the notations in Eqs. (\[eq10\]), (\[eq14\]) and (\[eq15\]). We have numerically analyzed Eq. (\[eq18\]) at different separations $a$ and with different values of the angle $\theta$ between corrugations. The strongest constraints were obtained at the shortest separation $a=127\,$nm where $\Delta_{F_C}=0.94\,$pN. They are shown by the solid lines in Figs. \[fg4\](a)–\[fg4\](d) at the values of $\theta=0^{\circ}$, $1.2^{\circ}$, $1.8^{\circ}$, and $2.4^{\circ}$, respectively. For comparison purposes, the dashed lines 1 and 2 in Figs. \[fg4\](a)–\[fg4\](d) show the strongest constraints obtained earlier [@26] within this interaction region from measurements of the lateral Casimir force between sinusoidally corrugated surfaces [@61; @62] and from effective measurements of the Casimir pressure between metallic plates by means of a micromachined oscillator [@36; @37]. As can be seen in Fig. \[fg4\], at any $\theta$ measurements of the normal Casimir force between sinusoidally corrugated surfaces result in stronger constraints within some interaction region than were known so far. Thus at $\theta=0^{\circ}$ the strengthening of previously available constraints up to a factor 1.8 holds within the interaction region $14.3\,\mbox{nm}\leq\lambda\leq 19.5\,$nm with the largest strengthening achieved at $\lambda= 17.2\,$nm \[see Fig. \[fg4\](a)\]. At $\theta=1.2^{\circ}$ and $1.8^{\circ}$ the strengthening up to factors 2.8 and 3.5 occurs for $13.8\,\mbox{nm}\leq\lambda\leq 25.1\,$nm and $12.9\,\mbox{nm}\leq\lambda\leq 27.5\,$nm , respectively. The maximum strengthening up to a factor 4 (achieved at $\lambda= 17.2\,$nm) within the interaction region $11.6\,\mbox{nm}\leq\lambda\leq 29.2\,$nm takes place at the angle between corrugations $\theta=2.4^{\circ}$.
The obtained stronger constraints following from measurements of the normal Casimir force between sinusoidally corrugated surfaces can be further strengthened at the expense of some modification of the experimental setup. Thus, it would be useful to switch from a static AFM mode used in this experiment to the dynamic mode used in Refs. [@28; @35; @42]. This results in a higher experimental precision though makes it necessary to perform measurements at larger separation distances. As an example, we calculate the prospective constraints on $\lambda,\alpha$ which can be obtained from dynamic measurements of the gradient of the Casimir force between corrugated surfaces at $a=170\,$nm. In so doing we assume that the total experimental error obtainable at this experiment is $\Delta_{F_C^{\prime}}=0.62\,\mu$N/m. For the sake of simplicity we consider the case $\theta=0^{\circ}$ which does not lead to the maximum strengthening of the respective constraints. Then from Eq. (\[eq13\]) one obtains $$\frac{\partial F_{\rm Yu}^{\rm corr}(a)}{\partial a}=
4\pi^2G\alpha\lambda^2e^{-a/\lambda}X^{(1)}(\lambda)
\tilde{X}^{(2)}(\lambda)X(\lambda,0).
\label{eq19}$$ Substituting Eq. (\[eq19\]) in the left-hand side of Eq. (\[eq7\]) adapted for the case of corrugated surfaces, we arrive at the prospective constraints shown by the dotted line in Fig. \[fg5\]. In the same figure the strongest constraints obtained [@26] from measurements of the lateral Casimir force between sinusoidally corrugated surfaces [@61; @62], from effective measurements of the Casimir pressure between metallic plates by means of a micromachined oscillator [@36; @37], and from the Casimir-less experiment [@47] are indicated by the dashed lines 1, 2, and 3, respectively. As can be seen in Fig. \[fg5\], the prospective constraints shown by the dotted line are stronger than the strongest current constraints over a wide interaction region from 12 to 160nm. At the moment three different experiments are used to constrain the Yukawa-type corrections to Newton’s law within this interaction region. The maximum strengthening up to a factor of 12.6 occurs at $\lambda= 17.2\,$nm.
Conclusions and discussion
==========================
In the foregoing we have obtained constraints on the parameters of Yukawa-type corrections to Newton’s law of gravitation following from two recent experiments on measuring the Casimir interaction. Each of these experiments is of particular interest, as compared with all previous work in the field. The experiment of Ref. [@28] pioneered measuring the gradient of the Casimir force between two magnetic surfaces and confirmed the predictions of the Lifshitz theory combined with the plasma model approach. In this way it was demonstrated that magnetic properties of the material boundaries influence the Casimir force. The outstanding property of magnetic materials is that the force gradients predicted by the Drude model approach are larger than those predicted by the plasma model approach (just opposite to the case of nonmagnetic metals). Thus, it was confirmed that such a widely discussed systematic effect as the patch potentials cannot be used for the reconciliation of the measurement data with the Drude model approach leading to further support of constraints on non-Newtonian gravity obtained from the measure of agreement between experiment and theory. Although constraints following from the experiment with magnetic surfaces are slightly weaker than the previously known ones (this is due to smaller density of Ni as compared to Au), the increased reliability can be considered as an advantage.
The experiment of Ref. [@30] pioneered measurements of the normal Casimir force between metallized sinusoidally corrugated surfaces at various angles between corrugations. It was demonstrated that the Casimir force depends on these angles in accordance with theory using the derivative expansion. We have calculated the Yukawa-type force in the experimental configuration with corrugated surfaces and obtained the respective constraints on its parameters. It was shown that the strength of constraints increases with increasing angle between corrugations. The maximum strengthening up to a factor of 4, as compared to the strongest previously known constraints, was shown to occur within the interaction range from 11.6 to 29.2nm. We have also proposed some modification in the measurement scheme allowing strengthening of the previously known constraints up to a factor of 12.6 within a wide interaction region presently covered using the results of three different experiments. This means that measurements of the Casimir interaction retain considerable potential for further strengthening of constraints on the Yukawa-type corrections to Newton’s gravitational law in submicrometer interaction region.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by the NSF Grant No. PHY0970161 and DOE grant DEF010204ER46131 (U.M.).
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![\[fg1\] Constraints on the parameters of Yukawa-type corrections to Newton’s gravitational law obtained in this work from measurement of the gradient of the Casimir force between two Ni surfaces (solid line), between two Au surfaces (dashed line) and between an Au and a Ni surfaces (dotted line). Here and in Figs. \[fg2\],\[fg4\] the regions of $(\lambda,\alpha)$ plane below each line are allowed and above each line are prohibited (see text for further discussion). ](figYu-1.ps)
![\[fg2\] Constraints on the parameters of Yukawa-type corrections to Newton’s gravitational law obtained in this work (solid line 1), in Ref. [@26] from measurements of the thermal Casimir-Polder force [@44] (solid line 2), in Ref. [@45] from measurements of the gradient of the Casimir force between metallic and corrugated semiconductor surfaces [@46] (solid line 3), in Refs. [@36; @37] from measurements of the gradient of the Casimir force between two metallic surfaces (solid line 4), and in Ref. [@47] from the Casimir-less experiment (dashed line). ](figYu-2.ps)
![\[fg3\] The quantity $X(\lambda,\theta)$ defined in Eq. (\[eq15\]) is plotted by the solid lines as a function of $\lambda$ at $\theta=0^{\circ}$, $1.2^{\circ}$, $1.8^{\circ}$, and $2.4^{\circ}$ from bottom to top, respectively. ](figYu-3.ps)
![\[fg4\] Constraints on the parameters of Yukawa-type corrections to Newton’s gravitational law obtained in this work (solid line), in Ref. [@26] from measurements of the lateral Casimir force between sinusoidally corrugated surfaces [@61; @62] (dashed line 1), and from effective measurements of the Casimir pressure between metallic plates by means of a micromachined oscillator [@36; @37] (dashed line 2). The angle between the axes of corrugations is equal to (a) $\theta=0^{\circ}$, (b) $\theta=1.2^{\circ}$, (c) $\theta=1.8^{\circ}$, and (d) $\theta=2.4^{\circ}$. ](figYu-4.ps)
![\[fg5\] Prospective constraints on the parameters of Yukawa-type corrections to Newton’s gravitational law which can be obtained from dynamic measurement of the gradient of the Casimir force between sinusoidally corrugated surfaces are shown by the dotted line. For comparison purposes the dashed lines 1, 2, and 3 indicate the strongest current constraints obtained in Ref. [@26] from measurements of the lateral Casimir force between sinusoidally corrugated surfaces [@61; @62], from effective measurements of the Casimir pressure between metallic plates by means of a micromachined oscillator [@36; @37], and in Ref. [@47] from the Casimir-less experiment, respectively. ](figYu-5.ps)
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
We show that simultaneous precision measurements of the $CP$-violating phase in time-dependent $B_s \to J/\psi\phi$ study and the $B_s \to \mu^+\mu^-$ rate, together with measuring $m_{t'}$ by direct search at the LHC, would determine $V_{t's}^*V_{t'b}$ and therefore the $b\to s$ quadrangle in the four-generation standard model. The forward–backward asymmetry in $B\to K^*\ell^+\ell^-$ provides further discrimination.
PACS numbers
: 14.65.Jk 12.15.Hh 11.30.Er 13.20.He
author:
- 'Wei-Shu Hou$^{a,b}$, Masaya Kohda$^{a}$, and Fanrong Xu$^{a}$'
title: 'Measuring the Fourth Generation $b \to s$ Quadrangle at the LHC'
---
\[sec:Intro\]INTRODUCTION\
==========================
Much like the completion of the three-generation “$b\to d$ triangle" in 2001 by the B factories, we may be at the dawn of measuring the “$b\to s$ quadrangle" at the LHC, [*if*]{} a fourth generation of quarks should exist.
Measurement of the time-dependent $CP$-violating (CPV) phase $\sin2\beta/\phi_1$ in $B_d \to J/\psi K^0$ decays by the BaBar and Belle experiments confirmed [@PDG] the Kobayashi–Maskawa [@Kobayashi:1973fv] mechanism of the standard model with three generations of quarks (SM3). Here, $\sin2\beta = \sin2\phi_1 \equiv \sin2\Phi_{B_d}$ is the CPV phase of the $\bar B_d \to B_d$ mixing amplitude. With the continuous run of the Large Hadron Collider (LHC) throughout 2011-2012, the LHCb experiment will measure $\sin2\Phi_{B_s}$, the CPV phase of $\bar B_s \to B_s$ mixing, via time-dependent study of $B_s\to J/\psi\phi$ and similar decays. We point out that, together with the measurement of $B_s\to \mu^+\mu^-$ rate, which is accessible not only by LHCb, but by the CMS experiment (and eventually, ATLAS) as well, combined with the direct search program of fourth-generation quarks, one may determine the Cabibbo–Kobayashi–Maskawa (CKM) mixing matrix element [@Kobayashi:1973fv; @Cabibbo:1963yz; @Glashow:1970gm] product $V_{t's}V_{t'b}^*$, thereby complete the SM4 quadrangle of $$V_{us}V_{ub}^* + V_{cs}V_{cb}^* + V_{ts}V_{tb}^* + V_{t's}V_{t'b}^* = 0.
\label{bsQuad}$$ Much progress has been made in summer 2011 on the above, so let us retrace how we reached the present.
Interest in the fourth generation renewed with the “$B\to K\pi$ direct CPV (DCPV) difference" puzzle: DCPV in $B^+ \to K^+\pi^0$ and $B^0 \to K^+\pi^-$ appeared opposite in sign [@BelleNature; @PDG], even though they proceed by the same spectator diagrams. The effect could be due to [@HNS] the nondecoupling of the heavy SM4 $t'$ quark in the $bsZ$ penguin, which brings in a new CPV phase in $V_{t's}^*V_{t'b}$. But hadronic effects make the $B\to K\pi$ DCPV measurements less amenable to interpretation.
However, an SM4 effect in the $b\to s$ $Z$-penguin loop should give a correlated effect in the $b\bar s \to s\bar b$ box diagram, making $\sin2\Phi_{B_s}$ large and negative [@HNS; @Hou:2005yb], in contrast with $-0.04$ in SM3. After the 2006 measurement [@PDG] of $B_s$ mixing, i.e., $\Delta m_{B_s}$, by the CDF experiment at the Tevatron, the “prediction" was strengthened [@Hou:2006mx] to “$\sin2\Phi_{B_s} = -0.5$ to $-0.7$ for $m_{t'} = 300$ GeV." Interestingly, by 2008, both the CDF and D0 experiments reported [@PDG] hints for negative $\sin2\Phi_{B_s}$ (called respectively $-\sin2\beta_s$ and $\sin\phi_s$). Although weakening in 2010, the measurement [@LHCb10] by LHCb using just the 2010 data of 36 pb$^{-1}$ showed a $\sin\phi_s$ that deviated from SM3 by $1.2\sigma$, *i.e., in same direction as CDF and D0!* So, there was much anticipation for LHCb to unveil their result with 10 times the data. To one’s surprise, however, analyzing 0.34 fb$^{-1}$ data, the LHCb experiment found [@Raven] $$\phi_s \equiv 2\Phi_{B_s} = 0.03 \pm 0.16 \pm 0.07,\ \ \
{\rm (LHCb\ 0.34\ fb}^{-1})
\label{phis1108}$$ which is consistent with zero (hence SM3). In fact, $B_s\to J/\psi\phi$ alone gave $0.13 \pm 0.18 \pm 0.07$, while Eq. (\[phis1108\]) is the combined result with $B_s\to J/\psi f_0(980)$.
There was another development that aroused the interest in the fourth generation in the past few years, namely the realization [@Kribs:2007nz; @Holdom:2009rf] in 2007 that electroweak precision tests did not firmly rule out a fourth generation, but rather indicated that the $t'$, $b'$ quarks be heavy, split in mass — but not by too much — while the Higgs mass bound would loosen. The direct search for $t'$ and $b'$ at the Tevatron had in any case been ongoing. At the LHC, the limit [@Chatrchyan:2011em] of $m_{b'} > 361$ GeV (95% C.L.) was reached with 2010 data alone, and became 495 (450) GeV for $b'$ ($t'$) by [@DeRoeck] summer 2011. We are already at the doorstep of the unitarity bound (UB) of 500–550 GeV [@Chanowitz:1978uj].
It is difficult to enhance $B_s \to \mu^+\mu^-$ in SM4 by more than a factor of 2, because it is constrained by $B \to X_s\ell^+\ell^-$, which is consistent with SM3 in rate. Hence, this mode appeared less relevant for SM4, until recently. Based on 2010 data, the competitive limit [@Aaij:2011rj] by LHCb was already within 20 times the SM3 expectation of $3.2 \times 10^{-9}$ [@Buras:2010pi]. Since 2010, the progress is significant, both at the Tevatron and the LHC (see **Note Added**), and a measurement of $B_s \to \mu^+\mu^-$ at the SM3 level now seems possible with 2011-2012 LHC data. With the signal of two charged tracks from a displaced vertex, the CMS experiment has demonstrated its competitiveness, in part due to an advantage in luminosity. The combined result [@comboBsmumu] of LHCb and CMS gives $${\cal B}(B_s\to \mu^+\mu^-) < 11 \times 10^{-9},\ \
{\rm (LHCb+CMS,\,2011)}
\label{Bsmumu1107}$$ at 95% CL, which is only 3.5 times the SM3 level.
[{width="70mm"} {width="70mm"} ]{}
While $B_s \to \mu^+\mu^-$ has been considered in recent SM4 studies [@Buras:2010pi; @Soni:2010xh; @Eberhardt:2010bm; @Golowich:2011cx], what we point out is that, together with the measurements of $\sin2\Phi_{B_s}$ and $m_{t'}$, the CKM element product $V_{t's}^*V_{t'b}$ can be determined. Since $V_{us}^*V_{ub}$ and $V_{cs}^*V_{cb}$ are known from tree processes, a measurement of $V_{t's}^*V_{t'b}$ would already complete the $b\to s$ quadrangle of Eq. (1), assuming that one has only SM4 and no other new physics. This quadrangle could be relevant for [@Hou:2008xd] the baryon asymmetry of our Universe (BAU). We will discuss the issue of the Higgs boson at the end.
\[sec:II\]Impact of $\sin2\Phi_{B_s}$ and $B_s \to \mu^+\mu^-$\
===============================================================
The $\bar B_s$–$B_s$ mixing amplitude is well-known, $$\begin{aligned}
M_{12}^s &=& \frac{G_F^2M_W^2}{12\pi^2}m_{B_s}f_{B_s}^2\hat B_{B_s}\eta_B
\Bigl[\left(\lambda_t^{\rm\scriptsize SM}\right)^2 S_0(t,t) \nonumber\\
&& \ \ \ \ \ \ \ \;\;
+ 2\lambda_t^{\rm\scriptsize SM}\lambda_{t'}\Delta S_0^{(1)}
+ \lambda_{t'}^2\Delta S_0^{(2)}
\Bigr],
\label{M12s}\end{aligned}$$ where $\lambda_q \equiv V_{qs}^*V_{qb}$ hence $-\lambda_t^{\rm\scriptsize SM} = \lambda_c + \lambda_u$, and we have approximated by factoring out a common short distance QCD factor $\eta_B$. With $S_0$ and $\Delta S_0^{(i)}$ as defined in Ref. [@Hou:2006mx], Eq. (\[M12s\]) manifestly respects the Glashow-Iliopoulos-Maiani (GIM) mechanism [@Glashow:1970gm].
The mass difference $\Delta m_{B_s} \equiv 2|M_{12}^s|$ depends on the hadronic parameter $f_{B_s}^2\hat B_{B_s}$, hence it is not useful for extracting short distance information. However, defining $\Delta_{12}^s = [\; \ldots\; ]$ in Eq. (\[M12s\]), the CPV phase $$2\Phi_{B_s} \equiv \arg M_{12}^s = \arg \Delta_{12}^s,
\label{argM12s}$$ depends only on $m_{t'}$ and $\lambda_{t'} = V_{t's}^*V_{t'b}$. Note that $\lambda_t^{\rm\scriptsize SM} \cong -0.04 -V_{us}^*V_{ub}$, and we will take the current best fit value for $V_{us}^*V_{ub}$ from PDG [@PDG]. Note that $V_{us}^*V_{ub}$ can be directly measured via tree processes at LHCb.
We plot, in Fig. 1(a), the contours for $\sin2\Phi_{B_s}$ in the $\phi_{sb} \equiv \arg V_{t's}^*V_{t'b}$, $r_{sb} \equiv |V_{t's}^*V_{t'b}|$ plane for $m_{t'} = 550$ GeV. This $m_{t'}$ value is chosen because 500 GeV is almost ruled out, while going beyond 550 GeV, one is no longer sure of the numerical accuracy of Eq. (\[M12s\]). That is, above the UB, the perturbative computation of the functions $\Delta S_0^{(i)}$ would no longer be valid. However, some form like Eq. (\[M12s\]) should continue to hold even above the UB. We have checked that our results do not change qualitatively if we straightforwardly apply $m_{t'} = 650$ GeV.
At first sight, the $B_s \to \mu^+\mu^-$ decay rate is also proportional to $f_{B_s}^2$, bringing in large hadronic uncertainties. However, this can largely be mitigated [@Buras:2003td] by taking the ratio with $\Delta m_{B_s}/\Delta m_{B_s}|^{\rm exp}$, namely $$\mathcal{B}(B_s\to \bar\mu\mu)
= C\frac{\tau_{B_s}\eta_Y^2}{\hat{B}_{B_s}\eta_B}
\frac{|\lambda_t^{\rm\scriptsize SM}Y_0(x_t)+\lambda_{t'}\Delta Y_0|^2}
{|\Delta_{12}^s|/\Delta m_{B_s}|^{\rm exp}},
\label{BrBsmumu}$$ where $C = 3g_W^4m_{\mu}^2/2^7\pi^3M_{W}^2$, and $\eta_Y= \eta_Y(x_t) = \eta_Y(x_{t'})$ is taken. Hadronic dependence is now only in the better-known “bag parameter," $\hat B_{B_s}$. Furthermore, stronger $t'$ dependence is brought in through the short distance function $|\Delta_{12}^s|$ that enters $\Delta m_{B_s}$. We plot the contours for ${\cal B}(B_s \to \mu^+\mu^-)$ in the $\phi_{sb}$–$r_{sb}$ plane for $m_{t'} = 550$ in Fig. 1(b).
[{width="70mm"} {width="70mm"} ]{} -0.35cm
To anticipate the progress with full 2011 data, and towards 2012, we project possible values for $\sin2\Phi_{B_s}$ and ${\cal B}(B_s \to \mu^+\mu^-)$. The LHCb result of Eq. (\[phis1108\]) is at some odds with earlier results. A study [@Hou:2010mm] of high mass $m_{t'} = 500$ GeV case considering all relevant data, as compared with $m_{t'} = 300$ GeV (now ruled out) case studied earlier [@Hou:2005yb], suggested a smaller $\sin2\Phi_{B_s}$ value of order $-0.3$. This value is still within 2$\sigma$ of Eq. (\[phis1108\]). Given the surprise shift from a hint of a large and negative central value prior to 2011, the next update could possibly shift back. Thus, we shall take two possible values $$\sin2\Phi_{B_s} = -0.3 \pm 0.1;\ -0.04 \pm 0.1 \ \ ({\rm LHCb} > 1\; {\rm fb}^{-1})
\label{sinphis}$$ where the first is more aggressive but reflects the past trend, while the second follows Eq. (\[phis1108\]).
An enhanced $\sin2\Phi_{B_s}$ implies the same for $B_s \to \mu^+\mu^-$, so we should entertain the possibility that ${\cal B}(B_s \to \mu^+\mu^-)$ is larger than the SM3 value of $3.2 \times 10^{-9}$. On the other hand, given that $\sin2\Phi_{B_s}$ is now suitably consistent with SM3, one should consider not only the possibility that ${\cal B}(B_s \to \mu^+\mu^-)$ is consistent with SM3, but entertain even the possibility that ${\cal B}(B_s \to \mu^+\mu^-)$ might be found to be *less* than the SM3 expectation. Following the reasoning of Ref. [@Akeroyd:2011kd] for how the luminosity, hence errors, might scale for the combination of LHCb and CMS results, we adopt the two values of $$10^{9} \, {\cal B}(B_s \to \mu^+\mu^-) = 5.0 \pm 1.5;\ 2.0 \pm 1.5
\ \ \, ({\rm 2012})
\label{Bsmumu}$$ to project into 2012. We have chosen two adjacent regions of somewhat enhanced vs somewhat suppressed $B_s \to \mu^+\mu^-$, which contains the SM3 case in intersection. In the following, we will illustrate with the errors as in Eqs. (\[sinphis\]) and (\[Bsmumu\]), as well as half the error, anticipating further progress with data.
We illustrate in Fig. 2(a) for $m_{t'} = 550$ GeV the overlap of the contours for $\sin2\Phi_{B_s}$ and ${\cal B}(B_s \to \mu^+\mu^-)$ when both take larger than SM3 values in Eqs. (\[sinphis\]) and (\[Bsmumu\]). We denote this as Case A. The light shaded overlap region correspond to the 1$\sigma$ range in Eqs. (\[sinphis\]) and (\[Bsmumu\]). Reducing errors by half, one gets the dark shaded area by the overlap of the two sets of solid contours. Roughly speaking, the overlap region extends from $(r_{sb},\; \phi_{sb}) \sim (0.011, 40^\circ)$ to $(0.004, 130^\circ)$.
Figure 2(b) shows the cases when $\sin2\Phi_{B_s} = -0.04 \pm 0.10$ in Eq. (\[sinphis\]), but ${\cal B}(B_s \to \mu^+\mu^-)$ is either higher (Case B) or lower (Case C) than SM3 expectations in Eq. (\[Bsmumu\]). The shadings are the same as Fig. 2(a). The two values in Eq. (\[Bsmumu\]) complement each other, as can be seen from Fig. 2(b). Taken together, Cases B+C complement Case A of Fig. 2(a), where both $\sin2\Phi_{B_s}$ and ${\cal B}(B_s \to \mu^+\mu^-)$ are on the high side. A remaining Case D is the small region chipped off from Fig. 2(a) that lies between Case A and Cases B+C. We do not discuss this case further, as it can be inferred from Cases A–C.
Inspecting the overlap regions for Cases B and C, both allow large $r_{sb}$ solutions for $|\phi_{sb}| \lesssim 40^\circ$, with $r_{sb}$ ranging around 0.013 (0.011) for Case B (C). There is, however, a low $r_{sb} \lesssim 0.004$ overlap region for all $\phi_{sb}$, with Cases B and C complementing each other, with Case B ranging between $90^\circ$ to $270^\circ$. When $r_{sb}$ is small, in general $\sin2\Phi_{B_s}$ would become close to the SM3 value and become small. The full domain of $\phi_{sb}$ is allowed, which in turn has different implications for ${\cal B}(B_s \to \mu^+\mu^-)$. Note that the contour line of ${\cal B}(B_s \to \mu^+\mu^-) = 3.5 \times 10^{-9}$ is very close to the SM3 contour of $3.2 \times 10^{-9}$ (the dashed curves in Fig. 1(b)). Thus, to the left of $90^\circ$ (and to the right of $270^\circ$) for low $r_{sb}$, ${\cal B}(B_s \to \mu^+\mu^-)$ is suppressed compared to SM3 (compare Fig. 1(b)), which is precisely Case C. This is a case that still might emerge at the LHC, even when $\sin2\Phi_{B_s}$ is found consistent with SM3. The small $\sin2\Phi_{B_s}$ value can of course turn out to deviate from SM3 when very high precision is reached.
[ {width="70mm"} {width="70mm"} ]{}
\[sec:AFB\] Utility of $A_{\rm FB}(B^0 \to K^{*0}\mu^+\mu^-)$\
==============================================================
We have focused so far on $\sin2\Phi_{B_s}$ and ${\cal B}(B_s \to \mu^+\mu^-)$, the two $B$ physics trump cards in the quest for new physics at the LHC. But a third measurable can be done well by LHCb: the forward-backward asymmetry in $B^0 \to K^{*0}\mu^+\mu^-$. Earlier measurements [@PDG] by the B factories, and by CDF, found no indication of a zero crossing. However, the summer 2011 result [@Patel] of LHCb once again turned out in support of SM3. This has implications on the overlap regions of Fig. 2.
The zero crossing point $s_0 \equiv q^2|_{A_{\rm FB} = 0}$ is insensitive to form factors, hence an important probe of possible new physics. It has been found generally [@Buras:2010pi; @Soni:2010xh] that, once other flavor and CPV data are taken into account, the variation in $s_0$ for SM4 probably cannot be distinguished from SM3 within experimental resolution. But to investigate the power of LHC data alone, we plot in Fig. 3(a) the contours of constant $s_0$ in the $\phi_{sb}$–$r_{sb}$ plane for $m_{t'} = 550$, overlaid with the overlap regions of Fig. 2. We will now show that the consistency of the summer 2011 $A_{\rm FB}(B \to K^*\mu^+\mu^-)$ result of LHCb [@Patel] with SM3 rules out the low $\phi_{sb}$, high $r_{sb}$ region, as well as the upper tip of allowed region for Case A.
We take sample points from the overlap regions, illustrated as small ellipses in Fig. 3(a), and plot the corresponding $dA_{\rm FB}/dq^2$ vs $q^2 \equiv m_{\mu^+\mu^-}^2$ in Fig. 3(b), where the black solid curve is for SM3. For the more interesting Case A, i.e., $\sin2\Phi_{B_s} = -0.3\pm 0.1$ and ${\cal B}(B_s \to \mu^+\mu^-) = (5.0 \pm 1.5) \times 10^{-9}$ both enhanced over SM3 values, we take $$V_{t's}^*V_{t'b} \equiv r_{sb}\, e^{i\phi_{sb}} \simeq 0.0065\, e^{i70^\circ},
\label{Vt'sVt'b}$$ which lies near the center of the allowed region for Case A (third small ellipse from left in Fig. 3(a)), and is close to the $s_0 \simeq 4$ GeV$^2$ contour. This is plotted as the red dashed curve in Fig. 3(b), where we have used the form factor model of Ref. [@Ball:2004rg] within QCD factorization framework. Indeed, the zero crossing lies lower than the black solid SM3 curve, with $A_{\rm FB}$ weaker than SM3 below the zero crossing. But away from the zero crossing point, form factor model dependence would set in, hence we deem the vicinity of this region in $\phi_{sb}$–$r_{sb}$ as allowed by $A_{\rm FB}$. If one moves to the lower right tip of Case A, one moves closer to $s_0 \simeq 4.4$ GeV$^2$ contour of Fig. 3(a), hence $A_{\rm FB}$ would be even harder to distinguish from SM3. This is illustrated by the green (light grey) solid curve in Fig. 3(b) for the sample point of $V_{t's}^*V_{t'b} = 0.004\, e^{i130^\circ}$ (see Fig. 3(a)), which is indeed hard to distinguish from the SM3 curve. In fact, it is easily checked that for all points with $r_{sb} \lesssim 0.004$, $A_{\rm FB}$ would appear SM3-like.
The opposite is true for large $r_{sb}$ case. Within Case A, let us take the sample point of $V_{t's}^*V_{t'b} = 0.0085\, e^{i55^\circ}$, which roughly sits on the $s_0 \simeq 3$ GeV$^2$ contour of Fig. 3(a) (second small ellipse from left), and is in the upper, darker shaded region for Case A. This point is plotted as the purple dotdashed line in Fig. 3(b), with indeed $s_0 \simeq 3$ GeV$^2$. But now the $A_{\rm FB}$ value is so low for all $q^2 < 6$ GeV$^2$, LHCb could probably tell it apart, even with form factor uncertainties. However, low $A_{\rm FB}$ values would make the precise determination of $s_0$ harder. As an extreme case, we take $V_{t's}^*V_{t'b} = 0.01\, e^{i15^\circ}$ (first small ellipse from left in Fig. 3(a)), which is plotted as the blue dotted curve in Fig. 3(b). This $\phi_{sb}$–$r_{sb}$ combination falls on the $s_0 \simeq 6$ GeV$^2$ contour in Fig. 3(a), as we can see also from the $dA_{\rm FB}/dq^2$ plot. However, $A_{\rm FB}$ now has the *wrong sign* as compared with data, hence this region is *ruled out*. This in fact applies to the whole region to the left of, roughly (to be determined fully by experiment) the $s_0 \simeq 0.5$ GeV$^2$ contour. Together with the previous point that $s_0 \simeq 3$ GeV$^2$ probably would involve $A_{\rm FB}$ values that are too small, practically all $r_{sb} \gtrsim 0.008$ regions are ruled out, or disfavored, by $A_{\rm FB}$ measurement.
A little further explanation can shed light on the $A_{\rm FB}$ behavior. The differential $dA_{\rm FB}/dq^2$ is proportional to the strength of the Wilson coefficient $C_{10}$, while the $B_s \to \mu^+\mu^-$ amplitude is proportional to $C_{10}$. The point of convergence of the $s_0$ contours in Fig. 3(a) for $\phi_{sb} = 0$ corresponds to the vanishing point for ${\cal B}(B_s \to \mu^+\mu^-)$. $C_{10}$ crosses through zero at this point, and has opposite sign above and below. This explains the sign of the blue dotted curve in Fig. 3(b). There is a second convergence point for the $s_0$ contours in Fig. 3(a), and one could see ellipse shaped contours, e.g. for $s_0 = 5$ GeV$^2$. This is because $dA_{\rm FB}/dq^2$ is a quadratic function of $r_{sb}\,e^{i\phi_{sb}}$. One has similar behavior that the upper part of the $s_0 = 5$ GeV$^2$ ellipse give the wrong sign for $A_{\rm FB}$.
[{width="115mm"} ]{}
\[sec:Implication\] Implications and Discussion\
================================================
We would like to give some interpretation of the impact of this possible future extraction of $\phi_{sb}$ and $r_{sb}$. We illustrate with the relatively aggressive value of Eq. (\[Vt’sVt’b\]), which corresponds to $\sin2\Phi_{B_s} = -0.3\pm 0.1$ and ${\cal B}(B_s \to \mu^+\mu^-) = (5.0 \pm 1.5) \times 10^{-9}$ both enhanced over SM3 values, and $m_{t'} = 550$ GeV. We note that Eq. (\[Vt’sVt’b\]) is consistent with the finding of Ref. [@Hou:2010mm], but if it emerged in 2012, *the information would be purely from these two measurements from the LHC*, rather than from “global" considerations [@Buras:2010pi; @Soni:2010xh; @Eberhardt:2010bm; @Hou:2010mm].
A measurement like Eq. (\[Vt’sVt’b\]) would complete the unitarity quadrangle of Eq. (\[bsQuad\]), assuming, of course that one only established SM4 but no further new physics. Let us start by drawing the familiar SM3 $b\to d$ triangle, $V_{ud}V_{ub}^* + V_{cd}V_{cb}^* + V_{td}V_{tb}^* = 0$, in Fig. 4. By standard convention [@PDG], $-V_{cd}V_{cb}^*$ is real and positive, $V_{ud}V_{ub}^*$ points above the real axis, while $V_{td}V_{tb}^*$ points from $V_{ud}V_{ub}^*$ to $-V_{cd}V_{cb}^*$, giving the familiar apex angle $\beta/\phi_1$, as indicated. Switching from $b\to d$ to $b\to s$, $V_{us}V_{ub}^*$ shrinks by $|V_{us}/V_{ud}| \simeq 0.23$ in length, but it is in the same direction as $V_{ud}V_{ub}^*$. The real and positive $V_{cs}V_{cb}^*$ extends parallel to the real axis from $V_{us}V_{ub}^*$ (most presentations by the experiments misrepresent this), but it is $|V_{cs}/V_{cd}| \simeq 1/0.22$ times longer than $-V_{cd}V_{cb}^*$. If $V_{t's}V_{t'b}^* = 0$, then $V_{ts}V_{tb}^*$ brings one straight back to the origin (dashed line in Fig. 4), i.e. $V_{us}V_{ub}^* + V_{cs}V_{cb}^* \equiv -V_{ts}V_{tb}^*|^{\rm SM3}$: one has a rather squashed SM3 $b\to s$ triangle with tiny $\Phi_{B_s}|^{\rm SM}$, but the same area as the $b\to d$ triangle.
But with $V_{t's}V_{t'b}^*$ finite as in Eq. (\[Vt’sVt’b\]), $V_{ts}V_{tb}^*$ would now differ from $V_{ts}V_{tb}^*|^{\rm SM3}$, and carry a larger CPV phase itself. The quadrangle of Eq. (1), as shown in Fig. 4, would be larger in area than the $b\to d$ or $b\to s$ triangles in SM3 by a factor $|V_{t's}V_{t'b}^*|/|V_{us}V_{ub}^*| \sim 0.0065/0.00088
\simeq 7$, as the strength of phase angle is similar.
Equation (\[Vt’sVt’b\]) corresponds to $\sin2\Phi_{B_s}$ that is $\sim 2\sigma$ away from the current LHCb central value of Eq. (\[phis1108\]), and may not be realized. Equation (\[phis1108\]) prefers a small $\sin2\Phi_{B_s}$ value. With the large $r_{sb}$ possibilities ruled out by $A_{\rm FB}$ as discussed, one is left with $r_{sb} \equiv |V_{t's}V_{t'b}^*| \lesssim 0.004$, with $\phi_{sb}$ practically unconstrained at present. One can picture this in Fig. 4 by reducing the length of $|V_{t's}V_{t'b}^*|$ by 60%, and with the full 360$^\circ$ $\phi_{sb}$ area allowed. This would probably need more data than 2011-2012 to measure.
The LHCb result [@Raven] for $B_s\to J/\psi\phi$ alone gave a positive central value of $\sin\phi_s = 0.13$. If this situation is borne out, we note from Fig. 2(b) that the branch for small and positive $\phi_{sb}$ is ruled out by $A_{\rm FB}$. But, depending on what ${\cal B}(B_s\to \mu^+\mu^-)$ value turns up, there is a strip of allowed domain for $\phi_{sb} \in (200^\circ,\ 330^\circ)$. Following roughly the $\sin2\Phi_{B_s} = +0.06$ dashed line on the righthand side of Fig. 2(b), the region above Case B and C (see also Fig. 3(a)) would be inferred. Larger $r_{sb}$ values for $\phi_{sb} \simeq 320^\circ$–$330^\circ$ would again be ruled out by $A_{\rm FB}$, but otherwise $A_{\rm FB}$ for this region would be quite consistent with SM3. The $b\to s$ quadrangle could again be easily drawn, with $r_{sb}$ typically in 0.004 to 0.005 range.
We note here a curiosity. In Fig. 1(a), the dashed curves correspond to SM3 contours, in the presence of $t'$. Comparing with Fig. 3, the upper left and right curves are ruled out by $A_{\rm FB}$. The two vertical dashed lines in Fig. 1(a) corresponds to $V_{t's}V_{t'b}^*$ being “parallel" to $V_{ts}V_{tb}^*|^{\rm SM3}$. The quadrangle of Eq. (\[bsQuad\]) would then become degenerate with SM3 hence have the same area.
We now offer a few points for further discussion.
The importance of measuring the SM4 $b\to s$ quadrangle cannot be overemphasized. It not only reflects possible new physics discoveries in $\sin2\Phi_{B_s}$ and $B_s\to \mu^+\mu^-$, but interpreting via Fig. 4 may relate [@Hou:2008xd] the measurement to BAU. Following the steps of Ref. [@Huet:1994jb], assuming a first-order phase transition, the generated BAU seems to be in the right ballpark [@HKK11]. Of course, Ref. [@Huet:1994jb] may not apply to heavy $m_{t'}$, but the nontrivial step of extending the computation into strong Yukawa coupling may address the other questionable assumption of order of phase transition. The problem is too important to be brushed aside just because of current inadequacies. We have also checked [@HHX11] that the neutron electric dipole moment could get enhanced to $10^{-31}$ $e\,$cm order, but it seems safely below the $10^{-28}$ $e\,$cm reach of the new generation of experiments, even with hadronic enhancement. As for the same-sign dilepton asymmetry uncovered by D0, although SM4 can give large and negative $\sin2\Phi_{B_s}$, it cannot affect $b\to c\bar cs$ decay, and here we await the cross-check by LHCb.
A recent “global fit" (in contrast to others [@Buras:2010pi; @Soni:2010xh; @Eberhardt:2010bm; @Hou:2010mm]) of SM4 parameters found a rather small $|V_{t's}^*V_{t'b}| < 10^{-3}$ [@Alok:2010zj]. This could be due to two inputs: allowing the central value of 1.04 (which violates unitarity) for $|V_{cs}|$, with an error of 0.06, may have inadvertently overconstrained $|V_{t's}|$; holding to the 2% lattice error for $\xi \equiv f_{B_s}^2 \hat B_{B_s} / f_{B_d}^2 \hat B_{B_d}$ (with $\Delta m_{B_s}/\Delta m_{B_d}$ precisely measured) in their fit, but not allowing the larger values of Eqs. (\[sinphis\]) and (\[Bsmumu\]) as possible *future* input, may be too strong a bias. We should add that the authors of Ref. [@Alok:2010zj] did not include the hints for sizable $\sin2\Phi_{B_s}$ into their fits. In any event, looking at Table III of Ref. [@Alok:2010zj], it seems unreasonable that $|V_{t's}^*V_{t'b}| < 10^{-3}$, while $|V_{t'd}^*V_{t'b}| > 10^{-3}$ is allowed, especially when we are just entering the era for major progress in $b\to s$ measurements. A small $|V_{t's}^*V_{t'b}|$ is certainly possible, but the three measurements stressed in this work would soon dominate the determination.
Why do we retain the SM3 $b\to d$ triangle, even when we extend to the SM4 $b\to s$ quadrangle? This point was addressed in the semiglobal analysis of Ref. [@Hou:2005yb]. When considering kaon constraints on $V_{t'd}^*V_{t's}$, a CKM unitarity approach showed that $V_{t'd}V_{t'b}^*$ and $V_{td}V_{tb}^*$ are relatively colinear with $V_{td}V_{tb}^*|^{\rm SM3}$, and cannot be easily distinguished by the $\sin2\phi_1/\beta$ measurement. This, in fact, predated the subsequent realization of some tension in $B_d$ mixing and/or $\epsilon_K$ [@Lunghi:2008aa], and would require Super B factory and kaon studies to disentangle.
We have used $m_{t'} = 550$ GeV, which is at the unitarity bound, for our discussion. This value can be uncovered by direct search by 2012. If, however, the $t'$ and $b'$ quarks are above the UB, i.e $m_{t'} \gtrsim 550$ GeV, then the 14 TeV run would be necessary. However, with the Yukawa coupling turned nonperturbative, the phenomenology may change [@Enkhbat:2011vp]. On the other hand, we would definitely learn in the next two years whether $\sin2\Phi_{B_s}$ and $B_s\to \mu^+\mu^-$ are beyond SM3 expectations.
Finally, we should mentioned that if a Higgs boson with SM3-like cross section and properties emerge at the LHC, indications of which could appear by end of 2011, SM4 alone would be in great difficulty [@Djouadi]. One would have to extend beyond simple SM4, even if SM-like $t'$ and $b'$ quarks are found. On the other hand, the standard Higgs of SM3 itself, with mass below 600 GeV or so, might get ruled out by 2012. If such is the case, then we might enter the heavy–Higgs, heavy–quark world of SM4 [@Enkhbat:2011vp]. We are in exciting times indeed.
\[sec:Conclusion\] Conclusion\
==============================
In conclusion, although once again SM3 seems to hold sway, whether time-dependent CPV in $B_s \to J/\psi \phi$ is considerably stronger than SM3 expectations will be conclusively settled with the full 2011–2012 data at LHCb, while one could discover that $B_s \to \mu^+\mu^-$ is mildly enhanced. If such is the case, we have shown that the fourth generation $b\to s$ unitarity quadrangle would become measured, which could have a bearing on the matter-antimatter asymmetry of the Universe. The main thrusts in this quest at the LHC are $\sin2\Phi_{B_s}$, ${\cal B}(B_s \to \mu^+\mu^-)$ and $A_{\rm FB}(B^0 \to K^{*0}\mu^+\mu^-)$.
0.3cm [**Acknowledgement**]{}. WSH thanks the National Science Council for an Academic Summit grant, NSC 100-2745-M-002-002-ASP, while MK and FX are supported under the NTU grant 10R40044 and the Laurel program.
0.2cm **Note Added**. Immediately after submission of our work, we learned that CDF measured [@CDFmumu11] ${\cal B}(B_s\to \mu^+\mu^-) = (18^{+11}_{\;-9})\times 10^{-9}$, which was countered by lower values from LHCb [@LHCb-mumu11] and CMS [@CMSmumu11] *within a week*. The subsequent rapid unfolding of the LHCb results of $A_{\rm FB}(B^0 \to K^{*0}\mu^+\mu^-)$ at EPS-HEP 2011, and $\sin\phi_s$ at LP 2011 was both exhilarating and somewhat disappointing, and resulted in major revision of this paper.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We study the Lebesgue-Bochner discretization property of Banach spaces $Y$, which ensures that the Bourgain’s discretization modulus for $Y$ has a good lower estimate. We prove that there exist spaces that do not have the Lebesgue-Bochner discretization property, and we give a class of examples of spaces that enjoy this property.'
author:
- 'Mikhail I. Ostrovskii and Beata Randrianantoanina'
title: 'Bourgain discretization using Lebesgue-Bochner spaces'
---
[*This paper is dedicated to the memory of our friend Joe Diestel (1943–2017). Lebesgue-Bochner spaces were one of the main passions of Joe. He started to work in this direction in his Ph.D. thesis [@Die68], and devoted to Lebesgue-Bochner spaces a large part of his most popular, classical, Dunford-Schwartz-style monograph [@DU77], joint with Jerry Uhl.*]{}
[[**2010 Mathematics Subject Classification.**]{} Primary: 46B85; Secondary: 46B06, 46B07, 46E40.]{}
[[**Keywords.**]{} Bourgain discretization theorem, distortion of an embedding, Lebesgue-Bochner space]{}
Introduction
============
We denote by $c_Y(X)$ the greatest lower bound of distortions of bilipschitz embeddings of a metric space $(X,d_X)$ into a metric space $(Y,d_Y)$, that is, the greatest lower bound of the numbers $C$ for which there is a map $f:X\to Y$ and a real number $r>0$ such that $$\forall u,v\in X\quad r d_X(u,v)\le d_Y(f(u),f(v))\le r
Cd_X(u,v).$$ See [@Mat02], [@Nao18], and [@Ost13] for background on this notion. Let $X$ be a finite-dimensional Banach space and $Y$ be an infinite-dimensional Banach space.
For ${\varepsilon}\in (0,1)$ let $\delta_{X\hookrightarrow Y}({\varepsilon})$ be the supremum of those $\delta\in (0,1)$ for which every $\delta$-net $\mathcal N_\delta$ in $B_X$ satisfies $c_Y(\mathcal{N}_\delta)\ge (1-{\varepsilon})c_Y(X)$. The function $\delta_{X\hookrightarrow Y}({\varepsilon})$ is called the *discretization modulus for embeddings of $X$ into $Y$*.
It is not immediate that the discretization modulus is defined for any ${\varepsilon}\in(0,1)$, but this can be derived using the argument of [@Rib76] and [@HM82] (see [@GNS12 Introduction]). Giving a new proof of the Ribe theorem [@Rib76] Bourgain proved the following remarkable result [@Bou87] (we state it in a stronger form which was proved in [@GNS12]):
\[T:BourgainDiscr\] There exists $C\in (0,\infty)$ such that for every two Banach spaces $X,Y$ with $\dim X=n<\infty$ and $\dim Y=\infty$, and every ${\varepsilon}\in (0,1)$, we have $$\label{E:BourgainDiscrImpr}
\delta_{X\hookrightarrow Y}({\varepsilon})\ge e^{-(c_Y(X)/{\varepsilon})^{Cn}}.$$
Bourgain’s discretization theorem and the described below result of [@GNS12] on improved estimates in the case of $L_p$ spaces have important consequences for quantitative estimates of $L_1$-distortion of the metric space consisting of finite subsets (of equal cardinality) in the plane with the minimum weight matching distance, see [@NS07 Theorem 1.2].
The proof of Bourgain’s discretization theorem was clarified and simplified in [@Beg99] and [@GNS12] (see also its presentation in [@Ost13 Section 9.2]). Different approaches to proving Bourgain’s discretization theorem in special cases were found in [@LN13], [@HLN16], and [@HN16+]. However these approaches do not improve the order of estimates for the discretization modulus.
On the other hand the paper [@GNS12] contains a proof with much better estimates in the case where $Y=L_p$. The approach of [@GNS12] is based on the following result (whose proof uses methods of [@JMS09]; origins of this approach can be found in [@GK03]).
\[T:LBdiscr\] There exists a universal constant $\kappa\in (0,\infty)$ with the following property. Assume that $\delta,{\varepsilon}\in (0,1)$ and $D\in
[1,\infty)$ satisfy $\delta\le \kappa{\varepsilon}^2/(n^2D)$. Let $X,Y$ be Banach spaces with $\dim X=n<\infty$, and let $\mathcal N_\delta$ be a $\delta$-net in $B_X$. Assume that $c_Y(\mathcal N_\delta)\le
D$. Then there exists a separable probability space $(\Omega,\nu)$, a finite dimensional linear subspace $Z\subseteq
Y$, and a linear operator $T:X\to L_\infty(\nu,Z)$ satisfying $$\label{E:LBfactor}
\forall x\in X,\quad \frac{1-{\varepsilon}}{D}\|x\|_X\le
\|Tx\|_{L_1(\nu,Z)}\le \|Tx\|_{L_\infty(\nu,Z)}\le
(1+{\varepsilon})\|x\|_X.$$
As is noted in [@GNS12], since $(\Omega,\nu)$ is a probability measure, we have $$\|\cdot\|_{L_1(\nu,Z)}\le \|\cdot\|_{L_p(\nu,Z)}\le \|\cdot\|_{L_\infty(\nu,Z)}$$ for every $p\in[1,\infty]$, therefore implies that $X$ admits an embedding into ${L_p(\nu,Z)}$ with distortion $\le\displaystyle{\frac{D(1+{\varepsilon})}{1-{\varepsilon}}}$. Since, by the well-known Carathéodory theorem, $L_p(\nu,L_p)$ is isometric to $L_p$ (see [@Lac74 §14]) we get that if $Z$ is a subspace of $L_p$, then $L_p(\nu,Z)$ is also a subspace of $L_p$, and, as explained in [@GNS12], it follows that the Bourgain’s discretization modulus for the case of $Y=L_p$ satisfies a much better estimate $${\delta}_{X\hookrightarrow L_p}({\varepsilon})\ge \frac{\kappa {\varepsilon}^2}{n^{5/2}}$$ (since for all spaces $X, Y$ and all ${\delta}>0$, $c_Y(\mathcal N_\delta)\le\sqrt{n}$, see [@GNS12]).
To generalize this approach to a wider class of spaces it is natural to introduce the following definition.
\[D:LBDiscr\] We say that a Banach space $Y$ has [*the Lebesgue-Bochner discretization property*]{} if for any separable probability measure $\mu$, there exists a function $f:[1,\infty)\to [1,\infty)$ so that for any $C\ge 1$ and any finite dimensional subspace $Z\subset Y$, if $W$ is any finite-dimensional subspace of $L_\infty(\mu,Z)$ such that for all $w\in W$ $$\label{E:LBdp}\|w\|_{L_\infty(\mu,Z)} \le
C\|w\|_{L_1(\mu,Z)},$$ then $W$ is $f(C)$-embeddable into $Y$.
The following is a corollary of Theorem \[T:LBdiscr\].
Let $Y$ be a Banach space with the Lebesgue-Bochner discretization property, $\delta\le
\kappa{\varepsilon}^2/(n^{5/2})$, where $\kappa$ is the constant of Theorem \[T:LBdiscr\], and $\mathcal{N}_{\delta}$ be a $\delta$-net in an $n$-dimensional Banach space $X$. Then $$\label{gdisc}
c_Y(X)\le g\left(\frac{1+{\varepsilon}}{1-{\varepsilon}}c_Y(\mathcal{N}_{\delta})\right),$$ where $g(t):=tf(t)$ and $f$ is the function of Definition \[D:LBDiscr\]. Thus if, for an increasing function $g$, we define $\delta^g_{X\hookrightarrow Y}({\varepsilon})$ as the supremum of ${\delta}$ so that is satisfied for all ${\delta}$-nets $\mathcal{N}_{\delta}$ of $B_X$, we have that $\delta^g_{X\hookrightarrow Y}({\varepsilon})\ge \kappa{\varepsilon}^2/(n^{5/2})$.
By Theorem \[T:LBdiscr\], there exists a finite dimensional subspace $Z\subset Y$ and a finite-dimensional subspace $W \subset
L_\infty(\nu,Z)$ (the image of the operator $T$) so that $W$ satisfies with $C=\frac{1+{\varepsilon}}{1-{\varepsilon}}c_Y(\mathcal{N}_{\delta})$. Thus by the Lebesgue-Bochner discretization property of $Y$, $c_Y(W)\le
f(\frac{1+{\varepsilon}}{1-{\varepsilon}}c_Y(\mathcal{N}_{\delta}))$, and we obtain $$c_Y(X)\le\frac{1+{\varepsilon}}{1-{\varepsilon}}\ c_Y(\mathcal{N}_{\delta})f\left(\frac{1+{\varepsilon}}{1-{\varepsilon}}c_Y(\mathcal{N}_{\delta})\right).\qedhere$$
\[P:LBdiscr\] Characterize Banach spaces with the Lebesgue-Bochner discretization property.
At the meeting of the Simons Foundation (New York City, February 20, 2015) Assaf Naor mentioned that at that time no examples of Banach spaces which do not have the Lebesgue-Bochner discretization property were known although people who were working on this (Assaf Naor and Gideon Schechtman) believed that such examples should exist.
We note that the based on the Fubini and Carathéodory theorems argument showing that $L_p(L_p)$ is isometric to $L_p$ (for suitable measure spaces) fails for other functions spaces even in a certain ‘isomorphic’ form (see [@BBS02 Appendix]). For some spaces a very strong opposite of the situation in the $L_p$-case happens: Raynaud [@Ray89] proved that when $L_{\varphi}([0,1],\mu)$ is an Orlicz space that is not isomorphic to some $L_p$ and does not contain $c_0$ or $\ell_1$, then, for any $r\in[1,\infty)$ the space $\ell_r(L_{\varphi})$ (and thus also $L_{\varphi}([0,1],\mu,L_{\varphi})$) not only does not embed in $L_{\varphi}([0,1],\mu)$, but is not even crudely finitely representable in it.
In general, if $E$ is a Banach function space on a measure space $({\Omega},\mu)$, the structure of the $E$-valued Bochner space $E({\Omega},\mu,E)$ can be very different from the structure of the space $E$, see [@Rea90], [@BBS02], [@FPP08]. We refer the reader to [@BBS02] for a detailed discussion and history of related results.
In this paper we show (Proposition \[P:NonSQNonLB\]) that there is a class of Banach spaces which do not have the Lebesgue-Bochner discretization property and observe that this class contains the space constructed by Figiel [@Fig72].
We also find some examples, besides $L_p$, of Banach spaces that have the Lebesgue-Bochner discretization property. An easy observation is that the Lebesgue-Bochner spaces $L_p(E)$, where $E$ is any Banach space, have the Lebesgue-Bochner discretization property. It is interesting that even the finite direct sums of such spaces have the Lebesgue-Bochner discretization property, see Proposition \[propLpk\]. We would like to mention that many well-known and important spaces are of the form $L_p(E)$. In particular, the mixed norm Lebesgue spaces $L^P$ introduced in [@BP61] are such and thus have the Lebesgue-Bochner discretization property. For $P=(p_1,\dots,p_m)\in[1,\infty)^m$, the space $L^P$ consists of measurable functions $f$ on ${\Omega}=\prod_{i=1}^m ({\Omega}_j,\mu_j)$, the norm defined by $$\|f\|_P:=
\left(\int\dots\left(\int \left(\int
|f(t_1,\dots,t_n)|^{p_1}d\mu_1\right)^{p_2/p_1}d\mu_2\right)^{p_3/p_2}\dots
d\mu_m\right)^{1/p_m}.$$ Mixed norm spaces of this type arise naturally in harmonic and functional analysis. Such norms (and their generalizations that use other function space norms in place of the $L_{p_j}$-norms) are used for example to study Fourier and Sobolev inequalities and embeddings of Sobolev spaces. The properties and applications of mixed norm spaces are extensively studied in the literature, see e.g. [@GS16; @CS16; @DPS10] and their references.
Finitely squarable Banach spaces
================================
An infinite-dimensional Banach space $Y$ is called [*finitely squarable*]{} if there exists a constant $C$ such that for every finite-dimensional subspace $Z\subset Y$ the direct sum $Z\oplus_\infty Z$ admits a linear embedding into $Y$ with distortion bounded by $C$.
The first examples of Banach spaces which are not finitely squarable were constructed by Figiel [@Fig72]. An easy observation is that a Banach space $Y$ which is isomorphic to $Y\oplus Y$, is finitely squarable. The converse it false. In fact, both of the earliest examples of Banach spaces which are not isomorphic to their squares, the James [@Jam50] quasireflexive space $J$ [@BP60] and $c(\omega_1)$ [@Sem60] are finitely squarable, and for very simple reason: they have trivial cotype. For the James space this was proved in [@GJ73], for $c(\omega_1)$ this is obvious. Modern Banach space theory provides much more sophisticated examples of finitely squarable spaces which are not isomorphic to their squares, for example, the Argyros-Haydon space [@AH11].
\[P:NonSQNonLB\] Any space which is not finitely squarable does not have the Lebesgue-Bochner discretization property.
Let $Z$ be a subspace of $Y$ for which $Z\oplus_\infty Z$ is “very far” from a subspace of $Y$.
We introduce the following subspace $W\subset L_\infty ([0,1],Z)$: it consists of all $Z$-valued functions which are constant on the first half and constant on the second half, but these constants can be different vectors of $Z$. It is clear that this space is isometric to $Z\oplus_\infty Z$. It is also clear that the $L_1([0,1],Z)$ norm on this subspace is $2$-equivalent to the $L_\infty$-norm. The conclusion follows.
This proposition makes the following problem important:
\[P:FinNonSq\] Does there exist a finitely squarable space which does not have the Lebesgue-Bochner discretization property?
We conjecture that the answer to Problem \[P:FinNonSq\] is positive.
Examples of spaces with the Lebesgue-Bochner discretization property
====================================================================
In this section we provide some examples of spaces having the Lebesgue-Bochner discretization property. In all proofs below we use the notation of Definition \[D:LBDiscr\], that is: $Y$ is a Banach space, $C>0$, $Z\subset Y$ is a finite dimensional subspace of $Y$. Since we consider separable probability measures, by the Carathéodory theorem [@Lac74] we may assume that $W$ is a finite-dimensional subspace of $L_\infty([0,1],\mu,Z)$, such that for all $w\in W$ $${\label}{normequiv}
\frac1C \|w\|_{L_\infty([0,1],\mu,Z)} \le\|w\|_{L_1([0,1],\mu,Z)}\le
\|w\|_{L_\infty([0,1],\mu,Z)}.$$
Since $W$ is finite dimensional, for any ${\varepsilon}>0$, there exists a subspace $\tilde{W}\subseteq L_\infty([0,1],\mu,Z)$ with Banach-Mazur distance from $W$ less than $1+{\varepsilon}$, so that $\tilde{W}$ is spanned by simple functions and all $w\in \tilde{W}$ satisfy with $C$ replaced by $(1+{\varepsilon})C$. Thus, [without loss of generality]{}, we may assume that $W$ is spanned by simple functions which are constant on elements $\{\Delta_i\}_{i=1}^n$ of some partition of $[0,1]$ into sets of measure $\frac1n$. Thus we can denote elements $w\in W$ as $$w=(w_1,\dots,w_n),$$ meaning that $w=\sum_{i=1}^n{\mathbf{1}}_{\Delta_i}\otimes w_i$. For all $w\in W$ we have $\|w\|_{L_\infty([0,1],\mu,Z)}=\max_{1\le i\le n}
\|w_i\|_Z$, and for all $p$, $1\le p<\infty$, we have $$\|w\|_{L_p([0,1],\mu,Z)}
=\Big(\frac 1n \sum_{i=1}^n \|w_i\|_Z^p\Big)^{\frac1p}.$$
Given any $p\in[1,\infty]$, $k\in {\mathbb{N}}$, and any Banach spaces $E_1,\dots, E_k$, by $L_p^k(E_1,\dots, E_k)$ we denote the Banach space of all $k$-tuples $(a_1,\dots,a_k)$ such that $a_j\in E_j$ for all $j\in[k]$, endowed with the norm $$\|(a_1,\dots,a_k)\|_{L_p^k(E_1,\dots, E_k)}:=
\Big(\frac 1k \sum_{i=1}^k \|a_i\|_Z^p\Big)^{\frac1p}, \hbox{ if
}p<\infty,$$ $$\|(a_1,\dots,a_k)\|_{L_\infty^k(E_1,\dots, E_k)}:=\max_{1\le
i\le k}\|a_i\|_Z.$$ If the spaces $E_1,\dots,E_k$ are equal to the same space $E$, we denote $L_p^k(E_1,\dots, E_k)$ by $L_p^k(E)$.
\[propLpk\] Let $k\in {\mathbb{N}}$, $p,q_1,\dots,q_k\ge 1$, $X_1, \dots,X_k$ be any Banach spaces, and for each $j\in[k]$ let $({\Omega}_j,\mu_j)$ be any nonatomic separable measure space, with finite or infinite measure, or ${\Omega}_j={\mathbb{N}}$ and $\mu_j$ is the counting measure. Then the space $$Y=L_p^k(L_{q_1}({\Omega}_1,\mu_1,X_1), L_{q_2}({\Omega}_2,\mu_2,X_2),\dots,L_{q_k}({\Omega}_k,\mu_k,X_k))$$ has the Lebesgue-Bochner discretization property with $f(C)\le
k^{2-1/p}C$.
Note that since the constant $f(C)$ in Definition \[D:LBDiscr\] can depend on $k$, the fact that $Y$ is an $L_p^k$-sum is not essential, essential is the fact that $Y$ is a finite direct sum.
To simplify notation we will omit the measure spaces when writing the symbol for a Lebesgue-Bochner space, i.e. we will write $L_{q_j}(X_j)$ instead of $L_{q_j}({\Omega}_j,\mu_j,X_j)$ with the understanding that for all $j\in[k]$, the measure spaces are those fixed in the statement of the proposition.
Since for any $p\ge 1$, the space $L_p^k(L_{q_1}(X_1),
L_{q_2}(X_2),\dots,L_{q_k}(X_k))$ is $k^{1-1/p}$-isomorphic to $L_1^k(L_{q_1}(X_1), L_{q_2}(X_2),\dots,L_{q_k}(X_k))$, it is enough to prove that in the case where $p=1$ we have $f(C)\le kC$. Note that if at least one of $q_j$ is equal to $\infty$, the space $Y$ has trivial cotype and thus has the Lebesgue-Bochner discretization property. In the following we assume that $q_j<\infty$ for all $j\in[k]$.
Using the discussion and notation preceding Proposition \[propLpk\], we see that it suffices to prove that any subspace $W\subseteq L_\infty^n(Y)$ satisfying $$\label{normequiv2}
\forall w\in W\quad \quad \frac1C \|w\|_{L_\infty^n(Y)} \le
\|w\|_{L_1^n(Y)}$$ admits a $kC$-isomorphic embedding into $Y$.
Let $n\in{\mathbb{N}}$, $w=(w_1,\dots,w_n)\in W\subseteq L^n_\infty(Y)$ and, for $i\in[n]$, $w_i=(w_{ij})_{j=1}^k\in L_1^k(L_{q_1}(X_1), L_{q_2}(X_2),\dots,L_{q_k}(X_k))$, where, for all $i\in[n]$, $j\in[k]$, $w_{ij}\in L_{q_j}(X_j)$. We will define a map ${\varphi}$ from $L^n_\infty(Y)$ to $Y$ so that for all $w\in L^n_\infty(Y)$ we have $${\label}{normvfw}
\begin{split}
\|{\varphi}(w)\|_{Y}=\frac 1k\sum_{j=1}^k \Big(\frac 1n\sum_{i=1}^n \|w_{ij}\|^{q_j}_{q_j}\Big)^{\frac1{q_j}}.
\end{split}$$
For each $j\in[k]$, we select $n$ mutually disjoint subsets $\{{\Omega}_{j\nu}\}_{\nu=1}^n$ of ${\Omega}_j$ so that for each $\nu\in[n]$ there exists a constant $a_{j\nu}>0$ and a surjective isometry $T_{j\nu}:L_{q_j}({\Omega}_j,\mu_j)\to L_{q_j}({\Omega}_{j\nu},a_{j\nu}\mu_j)$. It is well-known that when $({\Omega}_j,\mu_j)$ is nonatomic or equal to ${\mathbb{N}}$ with the counting measure then such choices are possible, and that the isometry $T_{j\nu}$ can be naturally extended to the isometry $\bar{T}_{j\nu}$ from the Lebesgue-Bochner space $L_{q_j}({\Omega}_j,\mu_j,X_j)$ onto $L_{q_j}({\Omega}_{j\nu},a_{j\nu}\mu_j,X_j)$, cf. e.g. [@DU77].
We define the map ${\varphi}_j: L^n_\infty(L_{q_j}(X_j),\dots,
L_{q_j}(X_j))\to L_{q_j}(X_j)$ by setting for all $(x_1,\dots,x_n)\in L^n_\infty(L_{q_j}(X_j),\dots,L_{q_j}(X_j))$ $$\begin{split}
{\varphi}_j(x_1,\dots,x_n):=\sum_{\nu=1}^n \left(\frac
{a_{j\nu}}{n}\right)^{\frac1{q_j}}\bar{T}_{j\nu}x_\nu.
\end{split}$$
Since the sets $\{{\Omega}_{j\nu}\}_{\nu=1}^n$ are mutually disjoint, we get $${\label}{normvfj}
\begin{split}
\|{\varphi}_j(x_1,\dots,x_n)\|_{q_j}=\Big(\frac 1n\sum_{\nu=1}^n
a_{j\nu}\|\bar{T}_{j\nu}x_\nu\|_{q_j}^{q_j}\Big)^{\frac1{q_j}}=
\Big(\frac 1n\sum_{\nu=1}^n \|x_\nu\|_{q_j}^{q_j}\Big)^{\frac1{q_j}}.
\end{split}$$
Next, given $w=(w_1,\dots,w_n)\in L^n_\infty(Y)$ where, for $i\in[n]$, $w_i=(w_{ij})_{j=1}^k\in Y= L_1^k(L_{q_1}(X_1), L_{q_2}(X_2),\dots,L_{q_k}(X_k))$, we define ${\varphi}(w)\in Y$ by setting $$\begin{split}
{\varphi}((w_i)_{i=1}^n)&=\Big({\varphi}_j\big((w_{i j})_{i=1}^n\big)\Big)_{j=1}^k.
\end{split}$$
By , equality is satisfied.
We will show that for all $w\in W\subseteq L^n_\infty(Y)$, we have $${\label}{goal}
\begin{split}
\|w\|_{L^n_1(Y)}\le \|{\varphi}(w)\|_{Y}\le k\|w\|_{L^n_\infty(Y)}.
\end{split}$$
To prove the leftmost inequality we write $$\begin{split}
\|w\|_{L^n_1(Y)}&=\frac 1n\sum_{i=1}^n \|w_i\|_{Y} =\frac 1n\sum_{i=1}^n \Big(\frac 1k\sum_{j=1}^k \|w_{ij}\|_{q_j}\Big)\\
&=\frac 1k\sum_{j=1}^k \Big(\frac 1n\sum_{i=1}^n \|w_{ij}\|_{q_j}\Big)
\le\frac 1k\sum_{j=1}^k \Big(\frac 1n\sum_{i=1}^n \|w_{ij}\|^{q_j}_{q_j}\Big)^{\frac1{q_j}}\\
&=\|{\varphi}(w)\|_{Y},
\end{split}$$ where the inequality follows from the classical theorem on averages ([@HLP52 Theorem 16]) applied to each of the summands with the corresponding exponent $q_j$, for $1\le j\le k$, respectively.
To prove the rightmost inequality, for each $j\in[k]$, let $i_j\in[n]$ be such that $\|w_{i_jj}\|_{q_j}=\max_{1\le i\le n}
\|w_{ij}\|_{q_j}$. Then $$\begin{split}
\|{\varphi}(w)\|_{Y}&=\frac 1k\sum_{j=1}^k \Big(\frac 1n\sum_{i=1}^n \|w_{ij}\|^{q_j}_{q_j}\Big)^{\frac1{q_j}}\le \frac 1k\sum_{j=1}^k \Big(\max_{1\le i\le n} \|w_{ij}\|_{q_j}\Big) \\
&= \frac 1k\sum_{j=1}^k \|w_{i_jj}\|_{q_j}\le \frac 1k\sum_{j=1}^k \Big( \sum_{l=1}^k \|w_{i_jl}\|_{q_j}\Big)\\
&= \sum_{j=1}^k \Big(\frac 1k \sum_{l=1}^k \|w_{i_jl}\|_{q_j}\Big)
=\sum_{j=1}^k \|w_{i_j}\|_{Y}\\
&\le k \max_{1\le i\le n} \|w_{i}\|_{Y}=k\|w\|_{L^n_\infty(Y)}.
\end{split}$$
Thus, holds, and therefore, by , $$ \begin{split}
\frac1C\|w\|_{L^n_\infty(Y)}\le \|{\varphi}(w)\|_{Y}\le
k\|w\|_{L^n_\infty(Y)}.\qedhere
\end{split}$$
[^1]
[BNSWW12]{}
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<span style="font-variant:small-caps;">Department of Mathematics and Computer Science, St. John’s University, 8000 Utopia Parkway, Queens, NY 11439, USA</span>
*E-mail address*: `[email protected]`
<span style="font-variant:small-caps;">Department of Mathematics, Miami University, Oxford, OH 45056, USA</span>
*E-mail address*: `[email protected]`
[^1]: **Acknowledgements:** We would like to thank Assaf Naor for suggesting the problem, and Florence Lancien for valuable discussions.
The first named author was supported by the National Science Foundation under Grant Number DMS–1700176.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We present a simple and accurate method for computing analytically the regeneration probability of solar neutrinos in the Earth. We apply this method to the calculation of several solar model independent quantities than can be measured by the SuperKamiokande and Sudbury Neutrino Observatory experiments.'
address: |
Dipartimento di Fisica and Sezione INFN di Bari,\
Via Amendola 173, I-70126 Bari, Italy
author:
- Eligio Lisi and Daniele Montanino
title: |
Earth regeneration effect in solar neutrino oscillations:\
an analytic approach
---
Introduction
============
The Mikheyev-Smirnov-Wolfenstein (MSW) mechanism of neutrino oscillations in matter [@MSWm] represents a fascinating solution to the long-standing solar neutrino problem [@NuAs]. The possible observation of the $\nu_e$ regeneration effect in the Earth [@olds] would be a spectacular, solar model independent confirmation of this theory (for reviews, see [@Ku89; @Mi89]).
The available data from the real-time solar neutrino experiment at Kamioka [@Hi91; @Fu96] are consistent with no Earth regeneration effect within the quoted uncertainties. This information can be used to exclude a region of the neutrino mass-mixing parameters in fits to the solar neutrino data [@Ha93; @Fi94; @Fo96].
A larger region of oscillation parameters relevant for the Earth effect will be probed by the new generation of solar neutrino experiments (as shown, e.g., in [@Ba94; @Kr96]). In particular, the SuperKamiokande experiment [@To95] (running) and the Sudbury Neutrino Observatory (SNO) experiment [@Do96] (in construction) are expected to probe possible day-night modulations of the solar neutrino flux with unprecedented statistics and accuracy. A correct interpretation of the forthcoming high-quality data will demand precision calculations of the Earth-related observables.
The calculation of the Sun-Earth $\nu_e$ survival probability $P_{SE}(\nu_e)$ is based on the relation (Mikheyev and Smirnov, in [@olds]) $$P_{SE}(\nu_e)=P_S(\nu_e)+
\frac
{[2P_S(\nu_e)-1]\,[\sin^2\theta-P_E(\nu_2\to\nu_e)]}
% ----------------------------------------------------
{\cos2\theta}
\ ,
\label{eq:P_SE}$$ where $P_S(\nu_e)$ is the $\nu_e$ survival probability at the Earth surface (or daytime probability), and $P_E(\nu_2\to\nu_e)$ is the probability of the transition from the mass state $\nu_2$ to $\nu_e$ along the neutrino path in the Earth.[^1]
The calculation of $P_E$ is notoriously difficult. Since the electron density in the Earth is not a simple function of the radius, the MSW equations have to be integrated numerically, unless step-wise approximations are adopted at the price of lower precision. Moreover, $P_E$ must be averaged over given intervals of time, $$\langle P_E\rangle=
\frac
{\displaystyle\int^{\tau_{d_2}}_{\tau_{d_1}}d\tau_d
\int^{\tau_{h_2}(\tau_d)}_{\tau_{h_1}(\tau_d)}d\tau_h\,
P_E(\eta(\tau_d,\,\tau_h))}
% --------------------------------------------------------
{\displaystyle\int^{\tau_{d_2}}_{\tau_{d_1}}d\tau_d
\int^{\tau_{h_2}(\tau_d)}_{\tau_{h_1}(\tau_d)}d\tau_h}
\ ,
\label{eq:PDH}$$ where $\tau_d$ and $\tau_h$ are the daily and hourly times, respectively, and $\eta$ is the nadir angle of the sun at the detector site. In typical applications, the interval $[\tau_{d_1},\,\tau_{d_2}]$ covers one year and the intervals $[\tau_{h_1}(\tau_d),\,\tau_{h_2}(\tau_d)]$ cover the nights, but other choices are possible.
The integration in Eq. (\[eq:PDH\]) is time-consuming. For instance, the authors of Refs. [@Ha93] and [@Fi94] quote a grid of about $30\times 30$ integration points in the ${\rm (year)}\times{\rm (night)}$ domain, which requires massive calculations for spanning the relevant region of neutrino mass and mixing parameters with acceptable precision. The issue of numerical accuracy and stability is not secondary, since coarser integrations may generate fuzzy and misleading results (see, e.g., Fig. 5 of [@Ga95]).
A faster and more elegant method for averaging $P_E$ consists in transforming the double integral of Eq. (\[eq:PDH\]) into a single integral of the form $$\langle P_E\rangle =
\displaystyle\int^{\eta_2}_{\eta_1}d\eta\,W(\eta)\,P_E(\eta)
\ ,
\label{eq:PETA}$$ where the weight function $W(\eta)$ represents the “solar exposure” of the trajectory corresponding to the nadir angle $\eta$. This method was used by Cherry and Lande [@olds] for calculating the day-night asymmetry at the Homestake site. We have used this approach in our previous works [@Fo96] by computing numerically the Jacobian $d\tau/d\eta$ required to transform Eq. (\[eq:PDH\]) into Eq. (\[eq:PETA\]).
In this paper we show that, actually, the weight function $W(\eta)$ can be calculated analytically in several cases of practical interest. Moreover, we show that $P_E$ can also be calculated analytically through a simple approximation which is more accurate than is required by the present (imperfect) knowledge of the Earth’s interior. Within this approach, we work out the calculation of several solar model independent observables for the SuperKamiokande and SNO experiments in a two-family oscillation scenario.
Our work is organized as follows. In Sec. II we present the parametrization of the electron density. In Secs. III and IV we discuss the analytic calculation of $P_E$ and $W$, respectively. In Sec. V we apply these calculations to the SuperKamiokande and SNO experiments. We draw our conclusions in Sec. VI. In order to make this work as self-contained and useful as possible, we organize in Appendixes A–C the relevant mathematical proofs. The reader interested mainly in the final results for SuperKamiokande and SNO may skip Secs. II–IV and the appendixes, and read only Sec. V.
Parametrizing the Earth electron density
========================================
In solar neutrino physics, the “standard electroweak model” of particle physics and the “standard solar model” of astrophysics must be supplemented by a “standard Earth model” of geophysics, such as the Preliminary Earth Reference Model (PREM) of Anderson and Dziewonsky [@PREM]. This seemingly “preliminary” model elaborated in 1981 still represents the standard framework for the interpretation of seismological data,[^2] as far as possible shell asphericities are neglected [@ASPH].
Eight shells are identified in the PREM model, but for any practical purpose related to solar neutrinos the four outer shells can be grouped into a single one (the “upper mantle”). The Earth matter density profile $\rho(r)$ is given in detail in Table I of [@PREM].
We have derived the electron density profile $N(r)$ from $\rho(r)$ by assuming the following chemical compositions (in weight): (1) Mantle, SiO$_2$ (45.0%), Al$_2$O$_3$ (3.2%), FeO (15.7%), MgO (32.7%), and CaO (3.4%) [@MANT]; (2) Core, Fe (96%) and Ni (4%) [@CORE]. It follows that $$%
N/\rho =
\left\{\begin{array}{cl}
0.494 &,\;{\rm(mantle)}\ ,\\
0.466 &,\;{\rm(core)}\ .
\end{array}\right.$$
Figure \[F:1\] shows the five relevant Earth shells (in scale) and the electron density $N(r)$, together with the basic geometry that will be used in the following sections. For each shell $j$, we use a polynomial fit that approximates accurately the true radial density, $$%
N_j(r) = \alpha_j + \beta_j r^2 + \gamma_j r^4
\ ,
\label{eq:Nj(r)}$$ where the coefficients $\alpha_j$, $\beta_j$, and $\gamma_j$ are given in Table \[tb:N\].
The functional form in Eq. (\[eq:Nj(r)\]) is invariant for nonradial $(\eta\neq 0)$ neutrino trajectories: $$%
N_j(x) = \alpha'_j + \beta'_j x^2 + \gamma'_j x^4
\ ,
\label{eq:Nj(x)}$$ where
$$\begin{aligned}
%
\alpha'_j &=& \alpha_j + \beta_j \sin^2\eta + \gamma_j \sin^4\eta\ ,\\
\beta'_j &=& \beta_j + 2 \gamma_j \sin^2\eta \ ,\\
\gamma'_j &=& \gamma_j
\ ,
\label{eq:abc}\end{aligned}$$
with the trajectory coordinate $x$ and the nadir angle $\eta$ defined as in Fig. \[F:1\].
For later purposes it is useful to split the density (in each shell and for each trajectory) as $$%
N_j(x)={\overline N}_j+\delta N_j(x)
\ ,
\label{eq:Nsplit}$$ where $\overline N$ is the ($\eta$-dependent) average density along the shell chord, $$%
{\overline N}_j=\int_{x_{j-1}}^{x_j}dx\,N_j(x)\bigg/(x_j-x_{j-1})
\ ,
\label{eq:Nbar}$$ and $\delta N_j(x)$ is the residual density variation. It will be seen that the above parametrization of $N(x)$ plays a basic role in the analytic calculation of the neutrino probability $P_E$.
We end this section with an estimate of the likely uncertainties affecting $N(x)$. The core, which is usually assumed to be iron-dominated, could contain a large fraction of lighter elements without necessarily conflicting with the seismological data. An example is given by a model of core made of Fe (55%) and FeO (45%) [@Re95], which would increase $N$ by 0.65%. Concerning the mantle, alternative chemical compositions (see Table 4 in [@MANT]) typically reduce $N$ by 1–2 %. We will evaluate the effect of representative density uncertainties by varying $N$ by $+1$% in the core and by $-1.5$% in the mantle. However, these error estimates might be optimistic, according to Birch’s old admonition [@Bi52].
Calculating $P_E$ with elementary functions
===========================================
In this section we show that the probability $P_E(\nu_2\to\nu_e)$ can be accurately approximated by elementary analytic expressions. We start by observing that $P_E$ can be expressed as: $$%
P_E = |{\cal U}_{ee}\sin\theta+{\cal U}_{e\mu}\cos\theta|^2
\ ,
\label{eq:PSEU}$$ where ${\cal U}$ is the neutrino evolution operator in the $(\nu_e,\,\nu_\mu)$ flavor basis.
In the same flavor basis, the MSW Hamiltonian ${\cal H}_j(x)$ along the $j$th shell chord traversed by the neutrino is given by $${\cal H}_j(x) = \frac{1}{2}
\left(\begin{array}{cc}
%
\sqrt{2} G_F N_j(x) - k\cos2\theta & k\sin2\theta
\\
k\sin2\theta & k\cos2\theta - \sqrt{2} G_F N_j(x)
%
\end{array}\right)
\ ,
\label{eq:H(x)}$$ where $k=\delta m^2/2E_\nu$ is the vacuum oscillation wave number, $N_j(x)$ is the electron density as in Eq. (\[eq:Nj(x)\]), and $\delta m^2$, $\theta$, and $E_\nu$ are the neutrino mass square difference, mixing angle, and energy, respectively.
Following Eq. (\[eq:Nsplit\]), we split the Hamiltonian into a constant matrix plus a perturbation, $$%
{\cal H}_j(x)={\overline{\cal H}}_j+\delta{\cal H}_j(x)
\ ,
\label{eq:Hsplit}$$ where ${\overline{\cal H}}_j={\cal H}_j\big|_{N\to\overline N}$, and $\delta{\cal H}_j(x)={G_F}/\sqrt{2}\,{\rm diag}
[\delta N_j(x),\,-\delta N_j(x)]$. Notice that the unperturbed Hamiltonian ${\overline{\cal H}}_j$ depends on the nadir angle $\eta$ through ${\overline{N}}_j$.
We have worked out explicitly, at the first perturbative order, the evolution operator ${\cal U}_j$ for the $j$th shell chord in the flavor basis. The result is: $$\begin{aligned}
{\cal U}_j(x_j,\,x_{j-1})
&=& e^{-i{\overline{\cal H}}_j(x_j-x_{j-1})}
-i\int_{x_{j-1}}^{x_j}dx\,
e^{-i{\overline{\cal H}}_j(x_j-x)}
\delta{\cal H}_j(x)
e^{-i{\overline{\cal H}}_j(x-x_{j-1})}
+ {\cal O}(\delta{\cal H}_j^2)\\
&=& \left(\begin{array}{cc}
%
c_j+is_j\cos2\bar\theta_m & -is_j\sin2\bar\theta_m \\
%
-is_j\sin2\bar\theta_m & c_j-is_j\cos2\bar\theta_m
%
\end{array}\right)-\frac{i}{2}\sin 2\bar\theta_m\nonumber\\
&\times&\left(\begin{array}{cc}
%
C_j \sin2\bar\theta_m & C_j\cos2\bar\theta_m - i S_j\\
%
C_j\cos2\bar\theta_m + iS_j & -C_j \sin2\bar\theta_m
%
\end{array}\right) + {\cal O}(\delta{\cal H}_j^2)
\ ,
\label{eq:U}\end{aligned}$$ where $\bar\theta_m$ is the average mixing angle in matter, $$\sin2\bar\theta_m/\sin2\theta =
\left[(\cos2\theta-\sqrt{2}G_F{\overline N}_j/k)^2+\sin^22\theta
\right]^{-\frac{1}{2}}
\ ,
\label{eq:sinm}$$ and
$$\begin{aligned}
c_j &=& \cos[\bar k_m(x_j-x_{j-1})/2]\ , \\
s_j &=& \sin[\bar k_m(x_j-x_{j-1})/2]\ , \\
C_j &=& \sqrt{2}G_F\int^{x_j}_{x_{j-1}}dx\,\delta N_j(x)
\cos \bar k_m(x-\bar x)\ ,\\
S_j &=& \sqrt{2}G_F\int^{x_j}_{x_{j-1}}dx\,\delta N_j(x)
\sin \bar k_m(x-\bar x)
\ ,\end{aligned}$$
\[eq:cjsj\]
with $\bar k_m=k\sin2\theta/\sin2\bar\theta_m$ (average matter oscillation wave number) and $\bar x=(x_j+x_{j-1})/2$ (shell chord midpoint).
The integrals in Eqs. (\[eq:cjsj\]c) and (\[eq:cjsj\]d) are elementary, $\delta N_j$ being a (biquadratic) polynomial in $x$ (see Sec. II). The property $\int_{x_{j-1}}^{x_j}dx\,\delta N_j(x)=0$, which follows from Eqs. (\[eq:Nsplit\]) and (\[eq:Nbar\]), is crucial for obtaining the compact expression of ${\cal U}_j$ in Eq. (\[eq:U\]).
The evolution operator along the total neutrino path $\overline{IF}$ (see Fig. \[F:1\]) is simply given by the ordered product of the partial evolution operators along the shell chords, ${\cal U}(x_F,\,x_I)=\prod_j {\cal U}(x_{j},\,x_{j-1})$. Actually, due the symmetry of the electron density with respect to the trajectory midpoint $M$, one needs only to calculate the evolution operator from $x_M(=0)$ to $x_F(=-x_I)$, $$\begin{aligned}
{\cal U}(x_F,\,x_I) &\equiv & {\cal U}(x_F,\,0)\cdot {\cal U}(0,-x_F)
\nonumber\\
&=& {\cal U}(x_F,\,0) \cdot {\cal U}^T(x_F,\,0)
\ .
\label{eq:Ufact}\end{aligned}$$ The proof of the above property is given in Appendix B.
So far we have solved analytically the MSW equations in the Earth at first order in perturbation theory, by expressing the total evolution operator $\cal U$ in the flavor basis as a product of matrices (one for each shell traversed in a semitrajectory) involving only elementary functions. The desired probabilities $P_E$ and $P_{SE}$ are then given by Eqs. (\[eq:PSEU\]) and (\[eq:P\_SE\]), respectively. Now we discuss the accuracy of such first-order approximation.
Figure \[F:2\] shows, for two representative mass-mixing scenarios and for diametral crossing, the results of various approximations of $P_{SE}$ as a function of the neutrino energy. Figures \[F:2\](a) and \[F:2\](b) refer to the small mixing angle solution to the solar neutrino problem, corresponding to $(\delta m^2/{\rm eV}^2,\,\sin^2 2\theta)\simeq(5.2\times 10^{-6},\,8.1\times
10^{-3})$ [@Fo96]. In Fig. \[F:2\](a), the probability $P_S$ (dotted line) is calculated semianalytically [@Fo96] and averaged over the $^8$B production region in the Sun [@Ba95] (as required for applications to SuperKamiokande and SNO). The probability $P_{SE}$ in Fig. \[F:2\](a) (thick, solid line) has been obtained by integrating the MSW equations in the Earth with the highest possible accuracy, i.e., with a Runge–Kutta method and with the true (PREM) electron density. In Fig. \[F:2\](b) we show the residuals $\Delta P_{SE}$ of different calculations with respect to the “Runge–Kutta” $P_{SE}$. The solid curve in Fig. \[F:2\](b) refers to the first-order perturbative approach discussed in this section. The dotted curve is obtained by using the simple zeroth-order approximation (i.e., average density shells). The dashed curve shows the variations of $P_{SE}$ induced by plausible uncertainties in the electron density $N$ (as discussed at the end of Sec. II). It can be seen that the effect of the latter uncertainties is comparable to the errors associated to the zeroth-order approximation, and is much larger than the errors of the first-order approximation.
Figures \[F:2\](c) and \[F:2\](d) are the analogous of Figs. \[F:2\](a) and \[F:2\](b) for the large mixing angle solution to the solar neutrino problem, corresponding to $(\delta m^2/{\rm eV}^2,\,\sin^2 2\theta)\simeq(1.5\times 10^{-5},\,0.64)$ [@Fo96]. It can be seen that the errors associated to the first-order approximation are generally smaller than the effect of the $N$ uncertainties, which are in turn much smaller than the errors of the zeroth-order approximation.
The results of Fig. \[F:2\] and of many other checks that we have performed for different values of $(\delta m^2,\,\sin^2 2\theta)$ and $\eta$ show that the analytic (first-order) solution discussed in this section represents a very good approximation to the true electron survival probability in the Earth, with an accuracy better (often much better) than is required by the likely uncertainties affecting the Earth electron density.
Finally, we point out that our analytic approximation for the neutrino evolution operator in the Earth matter can be applied also in the analysis of atmospheric neutrinos, and that its computer evaluation is much faster (about two orders of magnitude) than typical Runge–Kutta numerical integrations.
Weighting neutrino trajectories
===============================
As anticipated in the Introduction, the time average of the neutrino regeneration probability in the Earth \[Eq. (\[eq:PDH\])\] can be transformed into a (more manageable) weighted average over the trajectory nadir angle $\eta$ \[Eq. (\[eq:PETA\])\], with a weight function $W(\eta)$ having a compact, analytic form in several cases of practical interest. In this section we describe the results for the important case of annual averages during (a fraction of) night. We refer the reader to Appendix C for mathematical proofs and for a discussion of other cases.
The weight function $W(\eta)$ for annual averages is presented in Table \[tb:W\]. In different ranges of the detector latitude $\lambda$ and of the nadir angle $\eta$, $W(\eta)$ takes different functional forms, involving the calculation of a complete elliptic integral of the first kind [@Gr94; @Ab72] (which is coded in many computer libraries; see, e.g., [@CERN]).
In Fig. \[F:3\] the function $W(\eta)$ is plotted for the SuperKamiokande and SNO latitudes. The area under each curve is equal to 1. We show $W(\eta)$ also for the Gran Sasso site, relevant for several proposed solar neutrino projects such as the Borexino experiment [@BORE], the Imaging of Cosmic And Rare Underground Signals (ICARUS) experiment [@ICAR], the permanent Gallium Neutrino Observatory (GNO) [@PGNO], and the Helium at Liquid Azote temperature (HELLAZ) detector [@HELZ]. Finally, the dotted line in Fig. \[F:3\] represents the weight function for a hypothetical detector located at the equator, where the Earth regeneration effect would be more sizeable [@EQUA]. The divergence of $W(\eta)$ is logarithmic and thus it is integrated out by binning in $\eta$. Several methods exist for dealing with the numerical quadrature of divergent integrands [@QUAD].
By using $W(\eta)$ as given in Table \[tb:W\] (or in Fig. \[F:3\]), the average probability during night simply reads $$\langle P_E \rangle_{\rm night} =
\int^{\pi/2}_0 d\eta\,W(\eta)\,P_E(\eta)
\ .
\label{eq:Pnight}$$
The annual average during the fractions of night in which the Earth core is crossed has also a particular relevance as emphasized, e.g., in [@Ba94; @EQUA]. With the weight method, it can be easily calculated as $$\langle P_E \rangle_{\rm core} =
\frac{\displaystyle
\int^{\eta_{\rm core}}_0 d\eta\,W(\eta)\,P_E(\eta)}
{\displaystyle % ---------------------------------------------------
\int^{\eta_{\rm core}}_0 d\eta\,W(\eta)}
\ ,
\label{eq:Pcore}$$ where $\eta_{\rm core}(=0.577$ rad) is the nadir angle subtending the Earth (inner and outer) core.
A final remark is in order. In the expression for the time average \[Eq. (\[eq:PDH\])\] we have not included the geometric factor $L^{-2}$ accounting for the neutrino flux variations with the Earth-Sun distance $L$. We have implicitly assumed that the data from the real-time SuperKamiokande and SNO experiments will be corrected for this factor in any period of data taking. The effect of dropping this assumption is examined in Appendix D. We anticipate that, for annual averages, the effect is less than 1%.
Calculating solar model independent observables
===============================================
The SuperKamiokande and SNO experiments are sensitive only to the high-energy part of the solar neutrino spectrum, namely, to $^8$B neutrinos. Since the estimated uncertainty of the theoretical $^8$B neutrino flux $\Phi_B$ is relatively large ($\sim 16\%$ at $1\sigma$ [@Ba95]), it is important to focus on observables that do not depend on the absolute value of $\Phi_B$, but are sensitive only to the [*shape*]{} of the $^8$B energy spectrum (which is rather well known [@Al96]). Important examples of these quantities are the night-day rate asymmetry, the shape distortions of the angular spectrum, and the shape distortions of the recoil electron energy spectrum. The SNO experiment can measure, in addition, the charged-to-neutral current event ratio, which is also solar model independent. In this Section we calculate the annual averages of several such observables, by including the Earth effect with the method described in the previous sections.
We take from [@Ba96; @TABL] the neutrino interaction cross sections for SuperKamiokande. These cross sections already include the effect of the detector energy resolution and threshold (see Table I of [@Ba96] for the detector technical specifications). Concerning the distortions of the electron energy spectra due to neutrino oscillations, we adopt, as in [@Ba96], the approach in terms of the first two moments of the electron energy distribution, namely, the average electron kinetic energy $\langle T\rangle$ and the variance $\sigma^2$ of the energy spectrum. The reader is referred to [@Ba96] for an extensive discussion of the spectral moments and for an estimate of their likely uncertainties.
SuperKamiokande
---------------
Figure \[F:4\] shows the nadir angle ($\eta$) distribution of events expected at SuperKamiokande. We use the same format (five bins in $\cos\eta$) as the Kamiokande experiment [@Hi91]. The solid line is the distribution expected in the absence of oscillations, which is simply obtained by integrating the weight function of Fig. \[F:3\] in each bin of $\cos\eta$. The dashed and dotted histograms refer to the (best-fit) small-mixing and large-mixing solutions, respectively. All histograms are normalized to the same number of events in order to make the relative deviations independent of the absolute neutrino flux. Assuming a statistics of 10000 nighttime events, the small mixing angle case appears to be separated by $\sim 3\sigma$ (statistical errors only) from the no oscillation case in the last bin, which collects neutrinos crossing the Earth core. In fact, in the small mixing angle case there is a strong regeneration effect in the core (see, e.g., [@Ba94]). In the (best-fit) large mixing angle case, instead, the sudden variations of $P_{SE}$ with $\eta$ [@Ba94] happen to be smeared by binning and the net deviations are smaller (the effect, however, is very sensitive to the specific mass-mixing parameters chosen).
Figure \[F:5\] shows the night-day asymmetry of neutrino rates, which is perhaps the most popular characterization of the Earth effect. The 90% C.L. regions corresponding to the small and large mixing angle solution [@Fo96] are superposed to curves of equal values of the asymmetry. Similar results have been obtained by Krastev in [@Kr96]. Notice that asymmetry measurements at the percent level would allow a complete (partial) exploration of the large (small) mixing angle solution.
Figure \[F:6\] shows the fractional deviations in the first two moments of the electron energy distribution ($\langle T\rangle$ and $\sigma^2$) with respect to their no-oscillation values ($\langle T\rangle_0$ and $\sigma^2_0$). These deviations represent a useful characterization of the spectral distortions [@Ba96]. The deviations expected for the small mixing angle solution [@Ba96], although significant, are only slightly affected by the Earth effect. The deviations for the large angle solution are very small. In the region of “intermediate” mixing there could be strong, Earth-related deformations of the spectrum. Calculations of $\langle T\rangle$ including the Earth effect were first presented in [@Fi94]. A comparison of their Fig. 5 [@Fi94] with our Fig. \[F:6\] shows once again that the accuracy and stability of numerical calculations of the Earth effect are important issues. We obtain results very similar to those in Ref. [@Ba96] when the Earth effect is switched off.[^3]
Figure \[F:7\] shows the night-day variation of the spectral moments at SuperKamiokande. The relative deviations of the nighttime $(N)$ and daytime $(D)$ values of $\langle T\rangle$ and $\sigma^2$ characterize the daily deformations of the electron spectrum due to neutrino oscillations in the Earth matter (averaged over the year). For $\delta m^2\gtrsim 3\times 10^{-6}$ eV$^2$ ($\delta m^2\lesssim 3\times 10^{-6}$ eV$^2$) the Earth effect tend to increase the rate in the high-energy (low-energy) part of the electron spectrum. This explains the sign of the night-day spectral deviations in Fig. \[F:7\]. Therefore, if a significant Earth effect were observed, the sign of these deviations could provide an additional handle for discriminating the value of $\delta m^2$.
SNO
---
The results of our calculation of the angular distribution, day-night asymmetry, and spectral deviations for the SNO experiments are presented in Figs. \[F:8\]–\[F:11\]. These figures are analogous to Figs. \[F:4\]–\[F:7\] for SuperKamiokande, and similar comments apply. We just add that, in general, the various Earth-related effects appear to be more significant in SNO than in SuperKamiokande, as a result of the intrinsically higher correlation between the (observed) electron energy and the (unknown) neutrino energy.
In addition, the SNO experiment will separate events produced in charged current (CC) interactions of $\nu_e$’s from events produced in neutral current (NC) interactions of neutrinos of all flavors. The ratio of the CC and NC rates is perhaps the most crucial, solar model independent observable that will be measured in the next few years. Curves of the CC/NC ratio, including the Earth effect, are shown in Fig. \[F:12\]. The value expected for no oscillation (indicated in the left, lower corner) agrees with the value given in [@Ba96]. Notice that we have taken the efficiencies for detecting CC and NC events ($\varepsilon_{\rm CC}$ and $\varepsilon_{\rm NC}$, respectively) equal to 100%. When the true experimental efficiencies will be known, the values in Fig. \[F:12\] should be multiplied by $\varepsilon_{\rm CC}/\varepsilon_{\rm NC}$.
Conclusions
===========
The observation of solar neutrino oscillations enhanced by the Earth matter would be a spectacular confirmation of the MSW theory. The new generation of solar neutrino experiments can probe this possibility with unprecedented accuracy. In particular, the interpretation of the forthcoming, high-quality data from the SuperKamiokande and SNO experiments demands precision calculations of the Earth effect in solar neutrino oscillations.
We have presented an analytic method for approximating the $\nu_e$ regeneration probability in the Earth, based on a first-order perturbative expansion of the MSW Hamiltonian and on a convenient parametrization of the Earth electron density. We have also shown how time averages of the $\nu_e$ survival probability can be transformed into weighted averages over the nadir angle, with weights that can be calculated analytically in several relevant cases. Mathematical proofs and final results are described in detail, especially for the case of annual averages.
We have then calculated accurately the following solar model independent observables for the SuperKamiokande and SNO experiments: (1) the angular distribution of events; (2) the night-day asymmetry of the neutrino rates; (3) the fractional deviations of the first two spectral moments of the electron energy distribution; (4) the night-day fractional variations of such moments; and (5) the charged-to-neutral current event ratio for SNO.
The approach to the Earth effect presented in this paper allows simpler, faster, and more versatile calculations than brute-force integration methods. We hope that these advantages may lead more people to try a do-it-yourself analysis of the Earth regeneration effect in solar neutrino oscillations.
We thank Professor G. L. Fogli for useful suggestions and for careful reading of the manuscript. We thank P. I. Krastev for fruitful discussions. The work of D.M. was supported in part by Ministero dell’Università e della Ricerca Scientifica and in part by INFN.
The Sun-Earth survival probability $P_{SE}$
===========================================
In this appendix we report, for the sake of completeness, the derivation of Eq. (\[eq:P\_SE\]) (Mikheyev and Smirnov, in [@olds]). A solar neutrino arriving at Earth in the flavor state $\nu_\alpha$ is an incoherent mixture of vacuum mass states $\nu_i$. The corresponding probabilities $P_S(\nu_\alpha)$ and $P_S(\nu_i)$ are then given by $$\left[\begin{array}{c}
P_{S}(\nu_e)\\
P_{S}(\nu_\mu)
\end{array}\right] =
\left[\begin{array}{cc}
\cos^2\theta& \sin^2\theta \\
\sin^2\theta & \cos^2\theta
\end{array}\right]
\left[\begin{array}{c}
P_{S}(\nu_1)\\
P_{S}(\nu_2)
\end{array}\right]
\ .
\label{eq:PS}$$
The probability $P_{SE}(\nu_\alpha)$ that a solar neutrino has flavor $\alpha$ after traversing the Earth can be expressed as $$\left[\begin{array}{c}
P_{SE}(\nu_e)\\
P_{SE}(\nu_\mu)
\end{array}\right] =
\left[\begin{array}{cc}
1-P_E(\nu_2\to\nu_e) & P_E(\nu_2\to\nu_e) \\
P_E(\nu_2\to\nu_e) & 1-P_E(\nu_2\to\nu_e)
\end{array}\right]
\left[\begin{array}{c}
P_{S}(\nu_1)\\
P_{S}(\nu_2)
\end{array}\right]
\ ,
\label{eq:PSE}$$ where $P_E$ is the probability of the $\nu_2\to\nu_e$ transition in the Earth. Equation (\[eq:P\_SE\]) follows then from Eqs. (\[eq:PS\]) and (\[eq:PSE\]).
The incoherence of neutrino mass state components in Eqs. (\[eq:PS\]) and (\[eq:PSE\]) is guaranteed by at least three facts: (1) the neutrino production region in the Sun is an order of magnitude larger than the Earth radius; (2) for typical values of neutrino mass and mixing parameters, solar neutrinos oscillate many times in their Sun-Earth path, with final wavepacket divergences larger than the oscillation wavelength; and (3) any detection process implies some energy smearing. A hypothetical coherent mixture would give rather different numerical results for $P_{SE}$ [@Br89].
Proof of the property ${\cal U}(0,\,-{\lowercase{x}})={\cal U}^T({\lowercase{x}},\,0)$
======================================================================================
Let us consider the Schr[ö]{}dinger equation $$i\frac{d\psi(x)}{dx}={\cal H}(x)\psi(x)
\label{eq:Sch}$$ and its formal solution $$\psi(x) = {\cal U}(x,\,0)\psi(0)
\ ,
\label{eq:psix}$$ where $\cal U$ is the evolution operator (${\cal U}^\dagger{\cal U}=\openone$).
If the Hamiltonian is real (${\cal H}={\cal H}^*$) and obeys the symmetry ${\cal H}(x)={\cal H}(-x)$, then $\psi^*(-x)$ is also a solution of Eq. (\[eq:Sch\]), $$\psi^*(-x) = {\cal U}(x,\,0)\psi^*(0)
\ ,
\label{eq:psi-x}$$ that is $$\begin{aligned}
\psi(0) &=& [{\cal U}^*(x,\,0)]^{-1}\psi(-x)
\nonumber\\
&=& {\cal U}^T(x,\,0)\psi(-x)
\ ,
\label{eq:psi}\end{aligned}$$ which implies that ${\cal U}(0,\,-x)={\cal U}^T(x,\,0)$.
Changing integration measure, $\int\!\lowercase{d}\tau_{\lowercase{d}}\int\!{\lowercase{d}}\tau_{\lowercase{h}}
\to \int\! \lowercase{d}\eta$
================================================================================================================
In this appendix we show how to transform an integral of the kind $\int\! d\tau_d\int\! d\tau_h$ into an integral of the kind $\int\!d\eta$. In particular, Eq. (\[eq:Pnight\]) is explicitly derived for a detector latitude between the Tropic and the Polar Circle (see Table \[tb:W\]). Other cases are discussed at the end of this appendix.
The daily and hourly times are conventionally normalized to the interval $[0,\,2\pi]$: $$\begin{aligned}
\tau_d & = & \frac{\rm day}{365}\,2\pi
\ , \\
\tau_h & = & \frac{\rm hour}{24}\,2\pi
\ ,\end{aligned}$$ with $\tau_d=0$ at the winter solstice and $\tau_h=0$ at midnight. The nadir angle $\eta$, the daily time $\tau_d$, and the hourly time $\tau_h$, are linked by the relations $$\begin{aligned}
\cos\eta &=&\cos\lambda\cos\tau_h\cos\delta_S-\sin\lambda\sin\delta_S\ ,
\label{eq:time}\\
\sin\delta_S&=&-\sin i\cos\tau_d
\ ,
\label{eq:decl}\end{aligned}$$ where $\lambda$ is the detector latitude (in radiants), $i$ is the Earth inclination ($i=0.4091$ rad), and $\delta_S$ is the Sun declination. The sunrise (sr) and sunset (ss) times (corresponding to $\eta=\pm\pi/2$) are then given by $\tau_h^{\rm sr}=\arccos(\tan\lambda\tan\delta_S)$ and $\tau_h^{\rm ss}=-\tau_h^{\rm sr}$, respectively.
The annual average during nights can be restricted, for symmetry, to half year and to half night (midnight–sunrise interval), $$\langle P_E \rangle_{\rm night}=
\frac{\displaystyle \int^\pi_0 d\tau_d\int^{\tau_h^{\rm sr}(\tau_d)}_0
d\tau_h\,P_E(\eta(\tau_d,\,\tau_h))}
% --------------------------------------------------
{\displaystyle \int^\pi_0 d\tau_d\int^{\tau_h^{\rm sr}(\tau_d)}_0
d\tau_h}
\ .
\label{eq:Phalf}$$
The integral at the denominator in Eq. (\[eq:Phalf\]) is trivial and gives $\pi^2/2$. The integral at the numerator in Eq. (\[eq:Phalf\]) can be transformed as $$\begin{aligned}
\lefteqn{
\displaystyle
\int_0^{\pi}d\tau_d\int_{\lambda+\delta_S}^{\pi/2}
d\eta\, \frac{d\tau_h}{d\eta}\,(\tau_d,\,\eta)\,P_E(\eta)}
\\&=&\displaystyle
\int_{\lambda-i}^{\pi/2}d\eta\,P_E(\eta)\int_0^{\hat{\tau}_d
(\eta)}d\tau_d\,\frac{d\tau_h}{d\eta}(\tau_d,\,\eta)
\label{eq:obscure}\\&=&
\frac{\pi^2}{2}\displaystyle\int_0^{\pi/2}d\eta\,P_E(\eta)
\,W(\eta)
\ ,\end{aligned}$$ where $$\hat\tau_d(\eta)=\left\{\begin{array}{cl}
0
&,\;0\leq\eta<\lambda-i\ ,\\
\noalign{\smallskip}
\arccos\left(\frac{\sin(\lambda-\eta)}{\sin i}\right)
&,\;\lambda-i\leq\eta<\lambda+i\ ,\\
\noalign{\smallskip}
\pi &,\;\lambda+i\leq\eta\leq\pi/2
\ ,
\end{array}\right.
\label{eq:hat}$$ and $W(\eta)$ is defined as $$W(\eta) = \frac{2}{\pi^2}\left\{\begin{array}{cl}
0 &,\;0\leq\eta<\lambda-i\ ,\\
\noalign{\smallskip}\displaystyle
\int^{\hat\tau_d(\eta)}_0 d\tau_d\,\frac{d\tau_h}{d\eta}(\tau_d,\,\eta)
&,\;\lambda-i\leq\eta\leq \pi/2
\ .
\end{array}\right.
\label{eq:WW}$$
The interchange of integration variables in Eq. (\[eq:obscure\]) and the definition in Eq. (\[eq:hat\]) can be understood by drawing the integration domain in the $(\tau_d,\,\eta)$ plane (not shown) for a detector latitude between the Tropic and the Polar Circle $(i<\lambda\leq \pi/2-i)$.
From Eqs. (\[eq:time\]) and (\[eq:decl\]) one derives, after some algebra, $$\displaystyle
\int_0^{\hat\tau_d(\eta)}d\tau_d\frac{d\tau_h}{d\eta}(\tau_d,\,\eta)
=
\frac{\sin\eta}{\sin i}\int^1_{\max\{p,\,-1\}}
\frac
{d\xi}
% -------------------------------------
{\sqrt{(\xi+1)(\xi-1)(\xi-p)(\xi-q)}}
\ ,
\label{eq:preK}$$ where
$$\begin{aligned}
p &=& \sin(\lambda-\eta)/\sin i\ ,\\
q &=& \sin(\lambda+\eta)/\sin i\ ,\\
\xi &=& \cos\tau_d
\ .\end{aligned}$$
The r.h.s. of Eq. (\[eq:preK\]) can be expressed in terms of the complete elliptic integral of the first kind, defined as $$K(x)=\int_0^1 {\frac{ds}{\sqrt{(1-s^2)(1-x^2s^2)}}}\ .$$ (see [@Gr94], pp. 241–243). The results in the third column of Table \[tb:W\] are finally obtained with the positions
$$\begin{aligned}
z &=& \sin i\,\sqrt{(q-p)/2}\ ,\\
y &=& \sin i\,\sqrt{(1-p)(1+q)/4}
\ ,\end{aligned}$$
that can be easily shown to coincide with the definitions in the bottom row of Table \[tb:W\].
The other cases reported in Table \[tb:W\] (nearly equatorial or polar detector latitudes) can be derived analogously, the only difference being the shape of the integration domain in the $(\tau_d,\,\eta)$ rectangle. The above calculation can be specialized to annual averages during specific fractions of night, such as the period in which the Earth core is crossed \[see Eq. (\[eq:Pcore\]) and related comments\].
As concerns the case in which the time average is taken over a fraction of year (e.g., a season), we only mention that the weight function can still be expressed analytically, but the generic integration limits for $\tau_d$ require the calculation of the [*incomplete*]{} elliptic integral of the first kind (see [@CERN] for its numerical evaluation). The possible cases for the functional form of $W(\eta)$ acquire an additional dependence on the fraction of year considered for the average, and are not discussed in this paper.
Effect of Earth-Sun distance variations
=======================================
Throughout this work, the Earth-Sun distance $L$ has been taken constant $(L=L_0)$, in the hypothesis that the trivial $1/L^2$ geometrical variations of the solar neutrino signal will be factorized out in real-time experiments. However, such a continuous correction of the data implies a real-time subtraction of the background and thus requires a difficult, daily task of monitoring background, efficiencies, and calibrations (which are instead better defined over large periods of time). Therefore, we consider also the effect of dropping the assumption of a real-time, geometric correction of the signal, for the relevant case of annual averages during (a fraction of) nighttime.
Given the orbital equation $$L(\tau_d)=L_0(1-\epsilon \cos(\tau_d-\tau_d^p))
+ {\cal O}(\epsilon^2)
\ ,$$ where $\tau_d^p=0.24$ corresponds to the perihelion and $\epsilon=0.0167$ is the Earth orbit eccentricity, the time-averaged probability reads $$\langle P_E \rangle'_{\rm night}=
\frac{\displaystyle \int^{2\pi}_0 d\tau_d\,L^{-2}(\tau_d)
\int^{\tau_h^{\rm sr}(\tau_d)}_0
d\tau_h\,P_E(\eta(\tau_d,\,\tau_h))}
{\displaystyle% -------------------------------------
\int^{2\pi}_0 d\tau_d\, L_0^{-2}
\int^{\tau_h^{\rm sr}(\tau_d)}_0
d\tau_h}
\ .
\label{eq:P'half}$$ We give without proof the final results for detector latitudes between the Tropic and the Polar Circle (see also Table \[tb:W\]): $$\langle P_E \rangle_{\rm night}' =
\int^{\pi/2}_0 d\eta\,W'(\eta)\,P_E(\eta)
\ ,$$ $$W'(\eta) = W(\eta) \pm \epsilon\, Y(\eta)
\ ,$$ where the upper (lower) sign refer to the northern (southern) hemisphere, $W(\eta)$ is given in Table \[tb:W\] (first and third column), and the function $Y(\eta)$ is defined as $$Y(\eta) =
\frac{4}{\pi^2}\cos\tau_d^p\sin\eta\cdot\left\{
\begin{array}{cl}
0 & ,\;\ 0\leq\eta<\lambda-i\ ,
\\ \noalign{\bigskip}
\frac{1}{z}\left[q\,K\left(\frac{y}{z}\right)-(q-1)\,
\Pi\left(\frac{1-p}{q-p},\,\frac{y}{z}\right)\right]
& ,\;\ \lambda-i\leq\eta<\lambda+i\ ,
\\ \noalign{\bigskip}
\frac{1}{y}\left[q\,K\left(\frac{z}{y}\right)-(q-1)\,
\Pi\left(\frac{2}{q+1},\,\frac{z}{y}\right)\right]
& ,\;\ \lambda+i<\eta\leq\pi/2
\ .
\label{eq:Pi}
\end{array}\right.$$
In Eq. (\[eq:Pi\]) the variables $p$, $q$, $y$, and $z$, are defined as in Appendix C, and $\Pi$ is the complete elliptic integral of the third kind [@Gr94; @Ab72], $$\Pi(r,\,x)=\displaystyle\int^1_0
\frac{ds}{(1-r s^2)\sqrt{(1-s^2)(1-x^2 s^2)}}$$ (see [@CERN] for its numerical evaluation).
The “eccentricity correction” $\pm \epsilon Y(\eta)$ is small. At latitudes of interest, the difference between annual averages with and without this term is less than 1%: $$\left|\langle P_E\rangle'_{\rm night}-\langle P_E\rangle_{\rm night}\right|
\leq \epsilon\int_0^{\pi/2}d\eta\,|Y(\eta)| =
\left\{ \begin{array}{cl}
0.82\% & {\rm\ (Kamioka)}\ ,
\\ \noalign{\medskip}
0.95\% & {\rm\ (Sudbury)}\ ,
\\ \noalign{\medskip}
0.90\% & {\rm\ (Gran\ Sasso)}
\ .
\end{array}
\right.$$
Finally, we mention that, for averages over fractions of year, the eccentricity correction involves the evaluation of [*incomplete*]{} elliptic integrals of the third kind.
[clcddd]{} $j$& Shell & $[r_{j-1},\,r_j]$ &$\alpha_j$ &$\beta_j$ &$\gamma_j$\
1 & Inner core & $[0,\,0.192]$ & 6.099 & $-$4.119 & 0.000\
2 & Outer core & $[0.192,\,0.546]$ & 5.803 & $-$3.653 & $-$1.086\
3 & Lower mantle & $[0.546,\,0.895]$ & 3.156 & $-$1.459 & 0.280\
4 & Transition Zone & $[0.895,\,0.937]$ &$-$5.376 & 19.210 &$-$12.520\
5 & Upper mantle & $[0.937,\,1]$ & 11.540 &$-$20.280 & 10.410
\[tb:N\]
[cccc]{} Weight function &\
$W(\eta)$ &Equator to Tropic &Tropic to Polar Circle &Polar Circle to Pole\
for annual averages &$(0\leq\lambda\leq i)$ &$(i<\lambda\leq\pi/2-i)$ &$(\pi/2-i<\lambda\leq\pi/2)$\
0 & — & $0\leq\eta<\lambda-i$ & $0\leq\eta\leq\lambda-i$\
\
$\displaystyle\frac{2\sin\eta}{\pi^2 z}\,K(y/z)$ &$i-\lambda<\eta<i+\lambda$ & $\lambda-i\leq\eta<\lambda+i$ &$\lambda-i\leq\eta<\pi-\lambda-i$\
\
$\displaystyle\frac{2\sin\eta}{\pi^2 y}\,K(z/y)$ & $0\leq\eta<i-\lambda$ , & $\lambda+i<\eta\leq\pi/2$ & $\pi-\lambda-i<\eta\leq\pi/2$\
&or $i+\lambda<\eta\leq\pi/2$ & &\
Definitions:&
\[tb:W\]
[99]{}
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[FIG. 1. Section of the Earth showing the relevant shells (in scale) and the electron density profile $N(r)$. The geometric definitions used in the text are also displayed.]{} [FIG. 2. Comparison of different calculations of $P_{SE}(E_\nu)$ for $^8$B neutrinos crossing the Earth diameter. (a) Calculation of $P_{SE}$ with Runge-Kutta integration for the small mixing angle case (solid line). Also shown is the function $P_S$ (dotted line). (b) Variations of $P_{SE}$ induced by representative density shifts (dashed line), and by the first-order and zeroth order approximations discussed in the text (solid and dotted line, respectively). Panels (c) and (d) are analogous to (a) and (b), but refer to the large mixing angle case.]{} [FIG. 3. Annual solar exposure (weight) of the trajectory at nadir angle $\eta$ for representative values of the latitude $\lambda$. See the text for details.]{} [FIG. 4. Nadir angle distribution of nighttime events at SuperKamiokande. Error bars are statistical only.]{} [FIG. 5. Night-day asymmetry of neutrino rates at SuperKamiokande. The best-fit points and the 90% C.L. regions for the small and large mixing angle solutions are superposed.]{} [FIG. 6. Fractional deviations of the first two moments of the SuperKamiokande electron energy distribution ($\langle T\rangle$ and $\sigma^2$) from their no-oscillation values ($\langle T\rangle_0$ and $\sigma^2_0$).]{} [FIG. 7. Night-day fractional variations of the spectral moments at SuperKamiokande.]{} [FIG. 8. Nadir angle distribution of nighttime events at SNO. Error bars are statistical only.]{} [FIG. 9. Night-day asymmetry of neutrino rates at SNO.]{} [FIG. 10. Fractional deviations of the first two moments of the SNO electron energy distribution ($\langle T\rangle$ and $\sigma^2$) from their no-oscillation values.]{} [FIG. 11. Night-day fractional variation of the spectral moments at SNO.]{} [FIG. 12. Ratio of charged current to neutral current neutrino interactions at SNO.]{}
[^1]: The derivation of Eq. (\[eq:P\_SE\]) is reported in Appendix A for completeness.
[^2]: For a review of recent progresses in the study of the Earth interior, see also [@Re95].
[^3]: It is worth mentioning that the computer codes used in this work are independent from those used in [@Ba96]).
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We have completed a measurement of the $(6s^26p^2)\, ^3\!P_0 \rightarrow \, ^3\!P_2$ 939 nm electric quadrupole ($E2$) transition amplitude in atomic lead. Using a Faraday rotation spectroscopy technique and a sensitive polarimeter, we have measured this very weak $E2$ transition for the first time, and determined its amplitude to be $\langle ^3\!P_2 || Q || ^3\!P_0 \rangle$ = 8.91(9) a.u.. We also present an *ab initio* theoretical calculation of this matrix element, determining its value to be 8.86(5) a.u., which is in excellent agreement with the experimental result. We heat a quartz vapor cell containing $^{208}$Pb to between 800 and , apply a $\sim \! 10 \, {\rm G}$ longitudinal magnetic field, and use polarization modulation/lock-in detection to measure optical rotation amplitudes of order 1 mrad with noise near . We compare the Faraday rotation amplitude of the $E2$ transition to that of the $^3\!P_0 -\, ^3\!P_1$ 1279 nm magnetic dipole ($M1$) transition under identical sample conditions.'
author:
- 'Daniel L. Maser'
- Eli Hoenig
- 'B.-Y. Wang'
- 'P. M. Rupasinghe'
- 'S. G. Porsev'
- 'M. S. Safronova'
- 'P. K. Majumder'
title: 'High-precision measurement and *ab initio* calculation of the $(6s^26p^2)\,^3\!P_0 \rightarrow \, ^3\!P_2$ electric quadrupole transition amplitude in $^{208}$Pb'
---
Introduction {#intro}
============
Atoms have long served as testbeds for precision measurements and low-energy tests of fundamental physics. Searches for new physics, including potential candidate particles for dark matter, are ongoing using, for example, the technology of atomic magnetometers [@Budker2014], atomic clocks [@Nicholson2015; @Derevianko2014], and atom interferometers [@Hamilton2015]. A comprehensive recent review of the role of atoms and molecules in these searches can be found in [@Safronova2018].
A particular class of these atomic physics experiments has exploited the symmetry-violating properties of the weak interaction to study atomic parity nonconservation (PNC), and thus potentially probe both electroweak Standard Model physics and potential new physics. A number of these measurements have reached the 1% level of experimental accuracy [@Antypas2019; @Wood1997; @Vetter1994; @Meekhof1993]. Since electroweak effects in neutral atoms scale rapidly with the atomic number, $Z$, such atomic-physics-based tests have focused on heavy atoms, and require independent theoretical wavefunction calculations in the relevant atomic systems to link measured experimental observables to fundamental parameters [@Khriplovich].
Cesium, the heaviest stable alkali element, is an example of an atomic system where very high-precision *ab initio* atomic theory [@Porsev2009; @Dzuba2012] has come together with precise experimental efforts [@Wood1997] to provide an important low-energy test of electroweak physics. More recently, significant progress has been made in *ab initio* calculational techniques for multi-valence atomic systems [@Safronova2008; @Safronova2009]. In the trivalent thallium system, an existing high-precision PNC measurement [@Vetter1994], coupled with high-precision calculations [@Kozlov2001], has yielded another atomic-physics-based electroweak test. Current theory accuracy lags that of experiment by roughly a factor of two, so that modest further improvements in multi-valence theory will have a significant impact. In a close experiment/theory collaboration, we have completed a series of precise measurements of atomic properties of thallium and its trivalent cousin indium [@Ranjit2013; @Augenbraun2016; @Vilas2018], which have served as benchmarks for ongoing calculational efforts [@Safronova2013]. In particular, by comparing a series of excited-state polarizability measurements in indium to theoretical predictions from two complementary calculational approaches, we were able to show that a configuration interaction (CI) approach, combined with the coupled-cluster (CC), all-orders method to the three-valence system gave better agreement with experiment than the pure CC method [@Vilas2018].
Recently, Porsev *et al.* have undertaken a new *ab initio* calculation of the atomic structure of tetravalent lead [@Porsev2016]. Two high-precision parity nonconservation optical rotation experiments were completed in the 1990s [@Meekhof1993; @Phipp1996], but the atomic theory accuracy at that time in this complicated system was estimated to have an uncertainty near 10%, limiting the potential impact of the measurements on testing electroweak parameters. The 2016 theory work [@Porsev2016] improves the precision of the PNC calculation by better than a factor of two. Testing the accuracy of this new calculation and guiding forward further improvements will require a similar suite of benchmark measurements in lead. Beyond some energy level measurements and hyperfine structure measurements in $^{207}$Pb [@Bouazza2001; @Persson2018], measurements of atomic properties such as transition amplitudes and polarizabilities at the 1% level of accuracy do not exist for this element.
Here we present a new measurement and accompanying *ab initio* calculation of the lead ground-state $^3\!P_0 \rightarrow \,^3\!P_2$ electric quadrupole ($E2$) transition amplitude. We intend to follow up this result with future measurements of lead excited-state polarizability within the $6s^2 6p 7s$ manifold (see Fig. \[fig\_structure\]) using similar techniques and apparatus used for our earlier polarizability work in thallium and indium. Thus, we also include relevant *ab initio* polarizability calculations in Sec. \[theory\].
In the present transition amplitude work, we measure the ratio of the $E2$ amplitude to the that of the ground-state $^3\!P_0 \rightarrow \, ^3\!P_1$ magnetic dipole ($M1$) transition amplitude. This allows us experimentally to eliminate a number of common factors responsible for measured absorptivity of both transitions and extract a ratio of quantum-mechanical amplitudes. Because the $M1$ amplitude is precisely calculable without detailed wavefunction knowledge [@Porsev2016], we ultimately can determine the $E2$ amplitude (proportional to the transition quadrupole moment) from our experimental ratio measurement. Comparative absorptivity measurements have been completed recently [@Rafac1998; @Antypas2013; @Damitz2019] in Cs, producing high-precision determinations of transition amplitude ratios for electric dipole (E1) transitions, but to our knowledge this is the first such measurement using E1-forbidden transitions.
The $E2$ transition linestrength is roughly a factor of 30 weaker than that of the already-weak $M1$ transition. In this work, a highly sensitive optical polarimetry technique [@Meekhof1995; @Majumder1999; @Kerckhoff2005] was used to measure the Faraday rotation signals of the two transitions in an identical longitudinal magnetic field. An analogous precision measurement of the $E2$/$M1$ amplitude ratio within the Tl $6p_{1/2} \rightarrow 6p_{3/2}$ transition was completed in our laboratory using a similar technique some years ago [@Majumder1999]. In Sec. \[background\], we outline the atomic structure details involved with extracting transition amplitude information from the observed Faraday rotation lineshapes. Secs. \[experiment\] and \[results\] include a description of the experimental apparatus, method, and data analysis. Sec. \[theory\] outlines the *ab initio* theoretical calculation of the electric quadrupole matrix element, and also the atomic polarizability of several relevant excited states of lead. We conclude with a comparison of experiment to theory.
Atomic Structure and Faraday Rotation Lineshape {#background}
===============================================
![Low-lying energy levels of $^{208}$Pb, with the $M1$ and $E2$ transitions shown in red and blue, respectively.[]{data-label="fig_structure"}](structure){width="0.85\columnwidth"}
For these spectroscopic studies, we made use of an isotopically enriched (99.9%) sample of $^{208}$Pb ($I=0$), providing us with a simple, single-feature spectroscopic lineshape for both transitions studied. Fig. \[fig\_structure\] shows an energy level diagram for the relevant states. Due to the intrinsically weak nature of the $E2$ transition, there is no detectable direct absorption feature, even at the highest sample temperature and density we can achieve. We therefore choose to focus on the real, rather than imaginary, part of the refractive index, and measure the milliradian-sized Faraday rotation lineshape induced by a small longitudinal magnetic field. The observed optical rotation results from the difference in the Zeeman-shifted refractive indices, $n_\pm$, for right and left-circularly-polarized electric field components driving $\Delta m = \pm 1$ transitions originating from the $|^3\!P_0, \, m = 0 \rangle$ ground state. The Faraday rotation signal can be written $$\label{eq1}
\Phi_F(\omega)=\frac{\omega \ell}{2c} (n_+(\omega) - n_-(\omega)),$$ where $\ell$ is the interaction path length through the optically active medium, $\omega$ is the laser frequency, $c$ is the speed of light, and $n_{\pm}$ represents the dispersive real part of the refractive index for a given circular polarization.
The application of a small magnetic field, ${\bf B} = B_0\hat{z}$, parallel to the laser propagation direction causes equal and opposite Zeeman shifts to the resonant frequency of the circular polarization components, $\omega \rightarrow \omega_0 \pm \frac{\mu_B g_J B_0}{\hbar}$, where $\mu_B=\frac{|e| \hbar}{2m_e}$ is the Bohr magnetion, $e$ is the electron charge, $m_e$ is the electron mass, and $g_J$ is the Landé $g$-factor for a given transition. When the Zeeman shift is small compared to the linewidth, we can approximate $$\label{eqderivapprox}
n_+(\omega) - n_-(\omega) \approx \frac{\mathrm{d} n(\omega)}{\mathrm{d}\omega} \; \left(\frac{2 \mu_B g_J B_0}{\hbar}\right).$$ In Sec. \[errors\], we explore the differences between the derivative approximation and the (exact) difference forms of the resonance lineshape in order to assess potential systematics associated with the lineshape model. According to Eq. (\[eqderivapprox\]), the Faraday rotation lineshape follows a symmetric derivative-of-dispersion shape. Its amplitude is also proportional to the atomic density, $N$, and the appropriate quantum mechanical linestrength factor, $\langle T \rangle^2$, $$\label{eq3}
n(\omega) \propto N \langle T \rangle^2 \; \frac{2 \mu_B g_J B_0}{\hbar} \; \frac{\text{d}}{\text{d}\omega}\left( \frac{ \omega - \omega_0}{(\omega - \omega_0)^2 + \Gamma^2/4}\right),$$ where $\Gamma$ is the homogeneous linewidth (due here to collisional broadening). Finally, we must convolve this function with a normalized Gaussian, accounting for the velocity distribution of the atomic ensemble. We define the convolved lineshape, $\mathcal{L}$, as follows: $$\begin{aligned}
\label{eqL}
\mathcal{L} (\omega,\omega_0, \Gamma, \sigma) &\equiv \frac{\mathcal{C}}{\sigma\sqrt{2\pi}} \int_{-\infty}^{+\infty} \frac{\text{d}}{\text{d}\omega}\left( \frac{ \omega - \omega^\prime}{(\omega - \omega^\prime)^2 + \Gamma^2/4}\right) \nonumber \\
&\times \exp{\left[\frac{-(\omega^\prime - \omega_0)^2}{2\sigma^2}\right]} d\omega^\prime,\end{aligned}$$ where $\sigma$, the Doppler width, is proportional to the laser frequency and the root-mean-square velocity of the hot atoms. Our experimental optical rotation spectra are carefully calibrated in terms of radians. We fit our spectra to a lineshape of the form of Eq. (\[eqL\]) (see Sec. \[results\]) allowing the amplitude scaling factor $\mathcal{C}$ to link the numerical value of the integrand on resonance (which itself is a function of the component widths) to the peak value of the experimental spectrum. Making use of Eqs. (\[eq1\]–\[eq3\]) and ignoring a number of common numerical factors and fundamental constants, we find the following expression for the ratio of Faraday rotation amplitude factors: $$\label{Eqratio}
\frac{\mathcal{C}_{\mathrm E2}}{\mathcal{C}_{\mathrm M1}} = \frac{\omega_{\mathrm E2}}{\omega_{\mathrm M1}} \frac{(B_0 \ell N)_{\mathrm E2} \; g_J^{\mathrm{E2}}}{(B_0 \ell N)_{\mathrm{M1}} \; g_J^{\mathrm M1}}
\frac{\lvert \langle ^3\!P_2,m=1 | E2 | ^3\!P_0 \rangle \rvert^2}{\lvert \langle ^3\!P_1,m=1 | M1 | ^3\!P_0 \rangle \rvert^2}.$$
Here $\omega_{\mathrm E2}$ ($\omega_{\mathrm M1}$) is the resonant frequency for the () transition.
To find the matrix elements in [Eq. (\[Eqratio\])]{} for many-electron states, we define the electric quadrupole and magnetic dipole moment operators, $Q_\nu$ and ${\bm \mu}$ as the sum of one-particle operators, $$\begin{aligned}
Q_{\nu} &=& -|e| \sum_{i=1}^{N_e} \left[ r_i^2\,C_{2\nu}({\bf n}_i) \right], \nonumber \\
{\bm \mu} &=& -\frac{\mu_B}{c} \sum_{i=1}^{N_e} [{\bf j}_i + {\bf s}_i] ,
\label{eq:Qdef}\end{aligned}$$ where $N_e$ is the number of the electrons in the atom, ${\bf n}_i \equiv {\bf r}_i/r_i$, and $r_i$ is the radial position of the $i$th electron. ${\bf j}_i$ and ${\bf s}_i$ are the unitless total angular momentum and spin of the $i$th electron, as defined in [@Sobelman1979], and $C_{2\nu}({\bf n}_i)$ are the normalized spherical harmonics [@Varshalovich]. While the sums in Eq. (\[eq:Qdef\]) extend over all electrons, in practice the valence $p$ electrons provide the main contribution to the matrix elements for the case of Pb.
Though, in general, each amplitude factor is proportional to the interaction length, we work hard to ensure that both laser beams traverse nearly identical physical paths through the cell. We also alternate scans in a sequence that minimizes drift-related systematic errors associated with density and magnetic field changes (see Sec. \[results\]). We have inserted into Eq. (\[Eqratio\]) matrix elements for the $E2$ and $M1$ transitions that reflect the $|\Delta m| = 1$ selection rule appropriate to the transitions we study. We make use of the fact that the matrix elements are the same for $\Delta m = +1$ and $\Delta m = -1$ for both the $E2$ and $M1$ transitions. It is possible, when the laser beam propagation direction is not precisely collinear with the B-field axis, for the $E2$ transition to exhibit small $\Delta m = \pm 2$ components, and potential consequences of this are discussed below in Sec. \[errors\].
Assuming then that the relevant path length, atomic density, and magnetic field are identical for sequential laser scans for the two transitions, so that $(B_0 \ell N)_{\rm E2} = (B_0 \ell N)_{\rm M1}$, we arrive at an expression for the (unitless) quantum mechanical transition amplitude ratio, $\chi$, in terms of experimental amplitudes, resonant frequencies, and $g$-factors: $$\chi \equiv \left| \frac{ \langle ^3\!P_2,m=1 | E2\, | ^3\!P_0 \rangle}{\langle ^3\!P_1,m=1 | M1\, | ^3\!P_0 \rangle} \right|=
\sqrt{\frac{\mathcal{C}_{\mathrm{E2}} \; \omega_{\mathrm{M1}} \; g_J^{\mathrm{M1}} }{\mathcal{C}_{\mathrm{M1}} \; \omega_{\mathrm{E2}} \; g_J^{\mathrm{E2}}}}.
\label{Eqfinal}$$ A comparison of this expression with the theory prediction will be presented below in Sec. \[discussion\].
The $g$-factors are well known [@Porsev2016], so that the statistical uncertainty in our ratio, $\chi$, is entirely determined by the results of our lineshape fits which determine $\mathcal{C}_{\mathrm{E2}}$ and $\mathcal{C}_{\mathrm{M1}}$.
Experimental Details {#experiment}
====================
Furnace and Vapor Cell {#furnace}
----------------------
A schematic of the experimental layout is shown in Fig. \[fig\_schematic\]. The centerpiece of the experiment is the furnace, in the middle of which sits a 1-inch-diameter, 6-inch-long evacuated quartz vapor cell, containing a small quantity of isotopically enriched $^{208}$Pb (99.9% purity). The quartz cell windows are welded to the body at angles to eliminate the possibility of etalon effects in the optical path. Because of the inherent low vapor pressure of lead and the weak transition amplitudes being studied, we focus on temperatures in the 800–940 range where the density is sufficiently high for easily detectable optical rotation signals. This is achieved using four ceramic clamshell heaters, which surround a meter-long ceramic tube that contains the cell. The tube is sealed at both ends with endcaps that include fused silica windows, and is evacuated and backfilled with of argon in order to minimize optical beampath fluctuations due to convective air currents. A function generator operating at 10 kHz drives four audio amplifiers, which in turn drive the heaters. The frequency is sufficiently high that it does not interfere with the lock-in detection and signal analysis described below. Two thermocouple probes are positioned at the center of the vapor cell and one of the edges, which provide a temperature estimate, as well as a measure of temperature uniformity. A software *p-i-d* servo loop controls the amplitude of the function generator signal, allowing us to set and stabilize the oven temperature. The furnace contains a pair of Helmholtz coils to apply the magnetic field used to create the Faraday rotation signal, and the entire assemply is enclosed in $\mu$-metal magnetic shielding. The roughly 100-fold reduction in ambient field afforded by the shielding is sufficient to bring magnetic field fluctuations to a negligible level, especially since we take the difference between sequential magnetic field-on and field-off laser scans.
![Schematic of the experimental setup. Two commercial external cavity diode lasers (ECDLs) are scanned across the transitions’ center frequencies in a sequence determined by computer-controlled shutters. The laser scans are monitored using a pair of Fabry-Pérot cavities. A calcite prism linearly polarizes the light before the furnace, after which the polarization is modulated and analyzed using a second calcite prism. The transmitted light is separated using a diffraction grating prior to detection. See text for further details.[]{data-label="fig_schematic"}](schematic){width="\columnwidth"}
Optical Setup {#setup}
-------------
Two commercial external cavity diode lasers (ECDLs) at ($E2$) and ($M1$) (Toptica DL pro series and Sacher Lasertechnik Lynx series, respectively) pass through optical isolators before a small fraction of each is directed into one of two Fabry-Pérot (FP) cavities which monitor the frequency scan range and linearity. The confocal FP cavities (finesse near 30) are constructed with invar spacers, and contained inside insulated boxes for passive thermal stabilization. The cavity free spectral ranges for the $E2$ and $M1$ lasers were independently calibrated and measured to be and , respectively. A pair of shutters allow measurements of the two transition to be made in quick succession. The beam paths are combined using a dichroic filter, and directed first through a calcite prism polarizer, then into the furnace and through the vapor cell interaction region.
Upon exiting the furnace, the laser beams pass through a 1-cm-diameter, 5-cm-long glass rod with a large Verdet constant (“Faraday glass”) which is contained within a solenoid to which we can apply AC and DC currents, thus either modulating or tilting the laser polarization. We typically drive the solenoid with of AC current at $\omega = 2\pi \times 500~$, which results in a polarization modulation amplitude of a few milliradians. The laser beams then traverse a second, crossed calcite polarizer. Our polarizer pair in isolation has a finite extinction ratio of better than $10^{-6}$, but the presence of the furnace, vapor cell windows, and Faraday glass limit the effective extinction ratio of our polarimeter to about $2\times 10^{-5}$. The polarizers are each housed in a rotational lever mount actuated with a differential micrometer. Given the geometry of our mount and the resolution of the differential micrometer, we can reliably set and control the polarizer tilt angle at the level.
The light is then incident on a diffraction grating, which separates the two laser beam paths. With the aid of collimators and lenses, we focus each laser beam onto a high-gain, low-noise photodiode detector. This arrangement also allows us to reject nearly all of the substantial (but incoherent) blackbody radiation emanating from the furnace. This is important given that the coherent laser radiation reaching our detector after exiting the polarimeter is never more than about .
Modulation, Lock-in Detection, and Calibration {#detection}
----------------------------------------------
The detection scheme, similar to that described in [@Meekhof1995], uses the modulator combined with a pair of lock-in amplifiers for each wavelength in order to extract the optical rotation signal. After passing through the atomic vapor, the laser intensity is $I(f)$, reflecting the absorption lineshape, and there is also a frequency-dependent rotation of $\Phi_{\text F}(f)$, due to the atomic Faraday effect. We also account for a small frequency-dependent optical birefringence, $\Phi_{\text{br}}(f)$, unrelated to the atoms. The Faraday modulator introduces an additional sinusoidal rotation of $\Phi_{\mathrm{rot}} \cos(\omega t)$. The resulting intensity through the second polarizer is thus (using the small angle approximation):
$$\begin{aligned}
I_{\text{out}} &= I(f) \sin^2 \left[ \Phi_{\text{Pb}}(f) + \Phi_{\text{br}}(f) + \Phi_{\text{rot}} \cos(\omega t) \right] \nonumber\\
&\approx I(f) \left[ \Phi_{\text{Pb}}^2(f) + \Phi_{\text{br}}^2(f) + 2\Phi_{\text{Pb}}(f)\Phi_{\text{br}}(f) \right. \\
&+ \left. 2 \Phi_{\text{rot}} \cos(\omega t)(\Phi_{\text{Pb}}(f) + \Phi_{\text{br}}(f)) + \Phi^2_{\text{rot}}\cos^2(\omega t) \right], \nonumber\end{aligned}$$
where we have ignored the small constant transmission component from the finite polarimeter extinction. This expansion results in three important components: a constant term, one oscillating at $\omega$, and another oscillating at $2\omega$. Lock-in detection at $\omega$ and $2\omega$ removes the DC term; the $2\omega$ term is only dependent on the transmitted intensity, whereas the $1\omega$ term is proportional to the Faraday optical rotation times the transmission. Thus, the ratio of the two signals $S_{1\omega}/S_{2\omega}$ yields a signal proportional to the optical rotation only. Four lock-in amplifiers (Stanford Research Systems SRS 810) are set to the fundamental and second harmonic of the modulation frequency for the two lasers, and the extracted signals from the four are collected using a data acquisition board.
The size of the lock-in signal we detect is also proportional to the the amplitude of the modulation, $\Phi_{\mathrm{rot}}$. However, we know that the Verdet constant of our Faraday glass is substantially different at our two laser frequencies. To account for this in our calibration procedure, we first perform the following off-line exercise for each laser in turn. We fix the laser frequency at a value away from the atomic resonance. While still modulating the magnetic field, we add a stepwise series of increasing DC currents to the solenoid. At each step, we use the micrometer controlling the second polarizer to ‘re-cross’ the polarimeter by noting when the $1\omega$ lock-in output reaches exactly zero. In this way, we can accurately find the ratio of the rotatory effects of the Faraday glass for our two laser frequencies. Repeated calibration exercises such as these were performed over the one-month period of data collection to study reproducibility upon laser beam and polarimeter realignment. With these measurements in hand, we can, as noted below, incorporate a second procedure into our data collection sequence in which we apply a large, discrete DC current step to the solenoid, and, while directing both lasers through the cell (at fixed frequencies), detect the corresponding step-size changes in the lock-in outputs. When we include the results of both calibration procedures, we can then convert the units of the experimental signal of interest (ratio of lock-in outputs) to absolute radians for each transition.
Data Acquisition Procedure {#acquisition}
--------------------------
Data acquisition was performed at a range of temperatures ( – ) and with a range of applied currents to the Helmholtz coils ( – ). Acquisition was done in three steps: an initial calibration sequence, the main measurement sequence, and a final calibration sequence. The main measurement sequence has eight components and is typically looped five times. Table \[sequence\] summarizes the data collection sequence. We refer to this as a ‘run.’ The goal of the sequence is to examine possible sources of systematic error by measuring each transition’s rotation with a background scan without an applied magnetic field either immediately before or immediately after the field is applied and the rotation is measured. We acquire field-on / field-off scans, and also $E2$ and $M1$ scans in an “ABBA” sequence configuration to allow us to study and minimize temporal drift-related systematic errors. Such a collection sequence typically required one hour to complete.
**Data** Cal. Cal.
-------------------------- ----------- ------ ------ ------ ------ ------ ------ ------ ------ -----------
$\mathbf{\lambda}$ $E2$/$M1$ $E2$ $E2$ $M1$ $M1$ $M1$ $E2$ $E2$ $M1$ $E2$/$M1$
$\mathbf{B_{\text ext}}$ x x x x
: Data acquisition sequence. Each individual up/down scan pair takes 15 seconds. The ‘x’ in the $\mathbf{B_{\text ext}}$ row reflects application of the longitudinal magnetic field to the atoms.[]{data-label="sequence"}
An individual scan is based upon a triangle wave applied to a laser’s intracavity piezoelectric transducer (PZT), changing its frequency and scanning across the transition’s linecenter, which typically requires to complete. The atomic spectral features of interest extend over roughly , and a typical laser scan extended over . We separately analyzed the frequency-increasing portion of the scan (“upscan”) as well as the portion with a downward slope (“downscan”). For each run with a particular laser, a data acquisition computer recorded the triangle voltage wave, the transmission of the Fabry-Pérot cavity the $1\omega$ lock-in amplifier signal, and the $2\omega$ lock-in amplifier signal.
At each temperature, we acquired between 4 and 6 runs, between which optical realignments, changes of laser beam powers, and changes in laser sweep characteristics were applied. In all, roughly of data were collected, representing 800 distinct $E2$/$M1$ amplitude ratio measurements. The temperature range over which we worked corresponds to more than an order of magnitude change in lead vapor density. The corresponding $M1$ Faraday rotation amplitudes range from to , while the $E2$ amplitudes were in the to range.
Data Analysis and Results {#results}
=========================
Data Analysis Procedure {#analysis}
-----------------------
The first step in data analysis involves using the Fabry-Pérot transmission data to linearize and calibrate the frequency scans. Using the FP peak locations, we model the frequency as a fourth-order polynomial function of scan point number to account for small nonlinearity in the PZT voltage response. We found that higher-order polynomials did not improve the statistical quality of the FP peak fits. Using this frequency axis, we construct the unitless ratio of the $1\omega$ to $2\omega$ lock-in outputs, and then apply the calibration factors described above to convert this ratio to units of radians. In each case, we use the average step calibration values obtained by the pre- and post-calibration scans for that particular data run. This procedure is applied to both the $M1$ and $E2$ scans for both the field-on and field-off configurations. We next subtract the field-off scans proximate to the associated field-on scan, removing background features unrelated to the atoms that are typically a few percent of the field-on Faraday signals.
The subtracted lineshape is then fitted using a standard nonlinear least squares algorithm to the convolution function described in Eq. (\[eqL\]). With two thermocouple temperature monitors near the cell, we have a fairly accurate estimate of the temperature. We choose, then, to fix the Doppler width to a calculated value for the case of each laser scan. Below we discuss our exploration of lineshape changes and associated systematic amplitude errors resulting from our estimated temperature uncertainty. We note that, since ultimately we determine the ratio of the $E2$ to $M1$ amplitudes, overall temperature uncertainty largely cancels in this ratio, since the ratio of Doppler widths is temperature-independent. We therefore analyzed our Faraday lineshapes by fitting to two key parameters: the Lorentz width, $\Gamma$, due here to lead-lead collisional broadening, and the amplitude parameter $\mathcal{C}$ introduced in Sec. \[background\], connecting our convolution lineshape to the experimental peak height. We find this homogeneous linewidth component to be roughly ten times smaller than the Doppler width for the case of both transitions. In order to account for imperfect background subtraction, we also add constant and linear background parameters to the fit, which are always quite small, and, in the case of the linear term, often statistically unresolved. Examples of single background-subtracted scans of each transition at (near the low end of our temperature range) are shown in Figs. \[fig:M1\_sample\] and \[fig:E2\_sample\], along with the residuals of the fits. Each scan shown represents about of data collection. As one can see, the residual RMS optical rotation noise is at the few level in both cases. Because of its much larger amplitude, the $M1$ scan exhibits a baseline signal-to-noise ratio of more than 1000:1. Interestingly, in this case there is a significant increase in the size of the residuals near linecenter. In fact, this can be easily modeled as an effective amplitude noise induced by short-term frequency jitter of the diode laser as it scans across the transition — something that would manifest in the regions of the lineshape where the slope is steepest. The dashed envelope included in the lower box of Fig, \[fig:M1\_sample\] shows the expected amplitude noise from a frequency jitter of — something quite typical of ECDL systems such as ours.
![Sample data from $M1$ Faraday rotation signal (black dots, every fifth point shown) and fit result (red line). Residuals, expanded by a factor of 20, are shown below; solid blue shows the unweighted residual, while the dashed black line shows the envelope of the noise expected from a model that includes laser frequency jitter (see text).[]{data-label="fig:M1_sample"}](M1_residual){width=".9\columnwidth"}
![Sample data from $E2$ Faraday rotation signal (black dots, every fifth point shown) and fit result (red line). Expanded residuals are shown below.[]{data-label="fig:E2_sample"}](E2_residual){width=".9\columnwidth"}
Fit results are organized by laser scan direction and order of field-on / field-off sequencing. We scale our amplitude fit parameters using the calibration factors discussed above. The difference between the pre- and post-calibration scans within a data run yields a measure of calibration uncertainty, which can be combined with the error bar generated by the fit procedure to arrive at a final uncertainty for the corrected fit amplitude. We then construct the ratio of the fit amplitudes for the two transitions, $\mathcal{C}_{\mathrm{E2}}/\mathcal{C}_{\mathrm{M1}}$, for each set of consecutive $E2$ and $M1$ scans. Inserting the values for the ratios of the frequencies and $g$-factors of these transitions, we finally obtain experimental values for $\chi$ as defined in Eq. (\[Eqfinal\]). We accumulated statistics on all amplitude ratios taken at a given temperature. In some cases, the scatter between the weighted mean value for different data runs at a given temperature slightly exceeded their respective standard errors, due, for example, to small changes in experimental conditions, thermal drift, or relative beam path changes of the two lasers due to purposeful optical realignment. In each case, we expanded our error bars to account for this measured variance. We also took the approach of generating a histogram for all values at a given temperature and fitting this distribution to a Gaussian (see Fig. \[fig:histogram\]). The mean values arrived at by these two methods agreed very well within statistical uncertainties. Fig. \[fig:ratio\] shows the complete data set for our measured values of $\chi$ plotted as function of temperature, and corresponding $M1$ absorptive optical depth. Final weighted mean and $1\sigma$ statistical uncertainty are indicated in blue. This range of temperatures corresponds to a roughly a factor of 15 in vapor density, and as such is associated with changes in amplitude, and component spectral widths of the Faraday lineshape. Below , the $E2$ amplitudes were too small to achieve reliable fit results. At the upper end of our temperature range, where relatively large rotation amplitudes should have provided the best statistical precision, we observed a large scan-to-scan and run-to-run variation in ratios, likely due to much larger thermal drifts and optical birefringence effects at this temperature. Ultimately, while the mean value at agrees well with other data sets, the uncertainty is significantly higher due to this increased scatter, and we did not seek to increase the temperature further. Our final value and $1\sigma$ statistical uncertainty in the measured ratio is $\chi =0.1496(7)_{\text{stat}}$.
![Distribution of $\chi$ measurements from all 95 individual scan ratios taken at . Main figure: $\chi$ and corresponding error bars, with mean and standard deviation (solid blue) shown. Inset: histogram of $\chi$ (red bar plot) and a fitted Gaussian (thick blue curve). Intrinsic precision of $\chi$ values varies for subsets of these data depending, for example, on magnetic field employed for a given run.[]{data-label="fig:histogram"}](histogram){width=".9\columnwidth"}
![The amplitude ratio $\chi$ as a function of optical depth (bottom axis) and temperature (top axis). The mean and standard error are shown in solid and dashed blue lines, respectively.[]{data-label="fig:ratio"}](ratio){width=".9\columnwidth"}
Exploration of Systematic Errors {#errors}
--------------------------------
Potential systematic errors in our experimental value for $\chi$ were studied extensively, and the results are summarized in Table \[error\_table\], where systematic error contributions to the unitless ratio are expressed in percentages. Potential error sources which did not show statistically resolved effects are listed with a dash. In many cases, the fact that we are taking the ratio of two amplitudes tends to reduce systematic error impact (such as for temperature uncertainties, or magnetic field inhomogeneities). In addition, since $\chi$ is proportional to the square root of the amplitude ratio, the size of potential errors in $\chi$ associated with extracting Faraday signal amplitude are immediately reduced by a factor of two. Further, $1/f$-type noise associated with thermal, mechanical, or magnetic field drifts occurring on time scales comparable to our scan sequence would tend to show up as increased scatter between measurements rather than systematic bias, especially given our choice of sequencing repeated measurements in an “ABBA” pattern.
As can be seen in Fig. \[fig:ratio\], we importantly do not see a resolved systematic trend in our measured ratio as a function of temperature/density. In addition to comparing results at a number of different temperatures, we compared results for laser scan direction, different scan speeds and scan ranges, different temporal order of field-on/field-off scans, and different temporal order of $E2$ vs. $M1$ scans. Occasionally, we saw comparisons of subsets of data that differed by 1.5 to 2.0$\sigma$, where $\sigma$ is the combined error of the data subsets, and these contributions to the net systematic uncertainty are included in the table.
**Source** **Error in $\chi$ (%)**
------------------------------------ -------------------------
**Statistical error** 0.48
**Fitting**
Frequency Linearization 0.02
Fixing vs. Floating Lorentz Widths 0.37
Linear Background 0.27
Lineshape Weighting 0.32
Incorrect Doppler Widths 0.10
Include / Discount Scan Wings 0.13
**Signal Modeling**
Derivative vs. Difference 0.19
Magnetic Field Dependence –
**Geometry**
$E2$ $\Delta m = 2$ Transitions 0.35
**Laser Scanning Properties**
Scan Direction 0.22
Scan Speed / Width –
**Data Collection**
Field-On / Field Off Order –
$E2$ / $M1$ Order 0.13
**Angle Calibration**
Off-Line Calibration 0.28
Pre/Post Variance 0.29
**Other**
Isotopic Purity 0.02
**TOTAL:** **0.98%**
: Summary of error contributions and sources, expressed as percentage errors in the experimental ratio $\chi$. Horizontal line entries reflect the lack of a resolved systematic error contribution.[]{data-label="error_table"}
### Calibration
Errors in any aspect of our calibration procedure would directly impact our amplitude ratio measurement, and these were explored as follows. We completed several of the off-line calibration exercises over the course of our data-collection period, and compared the ratio of calibration factors obtained in these procedures. We assign a systematic error component based on the variation of these measurements (most likely due to small changes in the relative optical paths of the lasers, or possibly small thermal drifts over the time scale of the measurements). Also, for all of our data collection runs, we studied the differences between the pre- and post-calibration scans to estimate the potential errors associated with using their average to calibrate all scans in that run. An estimate of the systematic error associated with taking our approach of calibrating all runs based on the average of the two calibration values is also included in Table \[error\_table\].
### Fitting Methods
We explored a number of alternative approaches to fitting our Faraday rotation spectra to quantify systematic effects associated with lineshape analysis. First, as noted above, we explored different polynomial orders for parametrization of the ECDL scan nonlinearity, finding that beyond fourth order, no statistically significant changes to the fitted amplitude were seen. Our nominal method for fitting our spectra involved equal weighting of all points in the scan. We explored two alternatives. First, we explored a model that weighted data points according to a model that accounted for the frequency noise and associated fluctuations as noted in Fig. \[fig:M1\_sample\] and discussed in the previous section. Second, we explored truncating our fit ranges to exclude portions of the scan farther away from the resonant lineshape. We saw small changes in our fitted amplitude results, always well below the 1% level, and include small error contributions for these in Table \[error\_table\].
We studied the reliability of our Lorentzian and Gaussian (Doppler) width determinations in detail. Possible errors in these parameters impact the peak value of the lineshape convolution function defined in Eq. (\[eqL\]), and thus directly affect the fitted amplitude parameters, $\mathcal{C}$, from which we determine $\chi$. As noted, since the ratio of the Doppler widths for the two transitions is temperature-independent, a potential systematic error in $\chi$ due to temperature error could only come from the secondary effect of producing associated changes in other fit parameters that would affect the two transitions lineshapes in different ways. We explored this by systematically choosing a temperature (and hence Doppler widths) over a $\pm$20-degree range centered on the nominal temperature (which is taken to be the average of our two thermocouple readings). We then fit both experimental lineshapes, extracting the Lorentzian width, $\Gamma$, and peak amplitude factor, $\mathcal{C}$, in our usual fashion. Even using this relatively large temperature range, roughly equal to the difference in our thermocouple readings, we saw changes in the value of $\chi$ only at the $\pm 0.1\%$ level, and have included this in our error table.
![Exploration of a potential systematic error from a fixed, miscalculated Lorentzian width. The amplitude ratio, $\chi$, is plotted with black dots and dashes on the left y-axis. The corresponding fit error for those values and Lorentz widths are plotting with red plusses, on the right y-axis. The orange star indicates the Lorentz width and $\chi$ of a floated Lorentz width. Further details are provided in the text.[]{data-label="fig:gamma_sys"}](lorentz){width=".9\columnwidth"}
Of more concern is the accuracy of our Lorentz width determinations. These widths are an order of magnitude smaller than the Doppler widths, and thus more challenging to extract. However, their value clearly affects the amplitude of our lineshape function (Eq. (\[eqL\])). Our standard analysis method starts with fixed Doppler widths and optimizes the Lorentz width parameter in the fit process. In order to explore the effect of potential errors in Lorentz width values on our ratio $\chi$, we proceeded as follows. Since the $E2$ Faraday amplitudes have substantially lower signal-to-noise ratio, we assumed, for the purpose of this exercise, that the standard $M1$ fit procedure is able to extract the ‘correct’ Lorentz width, $\Gamma_{\mathrm{M1}}$. Then, we fit $E2$ lineshapes using a modified procedure where instead we fix the Lorentz width (in addition to the Doppler width) to a series of values above and below the apparent ‘best fit’ value, and allow only the peak amplitude factor to be optimized (this optimized value is clearly correlated with the choice of $\Gamma_{\mathrm{E2}}$). We then recorded the summed chi-squared value of the overall lineshape fit for each fixed choice of $\Gamma_{\mathrm{E2}}$. Figure \[fig:gamma\_sys\] summarizes this exploration for the case of all the data runs taken at one temperature (here ). The red curve indicates the changing ‘quality of fit’ for the entire collection of fits at at each fixed choice of $\Gamma_{\mathrm{E2}}$. The black line simply maps out the correlation between $\chi$ and $\Gamma_{\mathrm{E2}}$ , assuming that $\Gamma_{\mathrm{M1}}$ remains constant. The orange ‘star’ shows the average Lorentz width parameter generated by our standard fitting procedure, in which $\Gamma_{\mathrm{E2}}$ is ‘floated.’ The excellent agreement between the two methods in terms of finding the optimal value $\Gamma_{\mathrm{E2}} \approx$ is reassuring, and we can see even a very large fractional change in $\Gamma_{\mathrm{E2}}$ of $\pm$ that yields a change in $\chi$ of only $\pm 1\%$. A more extensive analysis of data sets at all temperatures allows us to place a $\pm 0.4\%$ systematic error based on our estimated uncertainty in the the extracted Lorentz widths.
### Lineshape Model
We also considered the systematic error associated with using the derivative approximation to the Faraday lineshape. First, for a series of Zeeman splittings in our experimental range, we generated theoretical lineshapes with typical values for component widths using the difference (rather than the derivative) of the dispersive real part of the refractive index lineshapes. We then proceeded to fit these lineshapes using our standard (derivative approximation) fitting function and studied the changes in fitted amplitude as a function of the Zeeman splitting. Since the $g$-factors and component widths of the two transitions are different, this would impact the two transitions differently, and hence would produce a systematic error in $\chi$. From this investigation, we put a limit of the potential systematic error of our derivative approximation at the 0.2% level. As a second experimental check, we studied the correlation of $\chi$ with the current applied to the Helmholtz coils for the data we collected. This showed no statistically resolved trend over the $\approx$ 3– range of magnetic fields that we explored.
### Geometrical Misalignment
Finally, we note that our analysis assumes that the laser beam paths are exactly collinear with the magnetic field axis within the vapor cell interaction region. This effectively allows us to view the electric quadrupole interaction as an operator proportional to the $\ell = 2, m = 1$ spherical harmonic (see Sec. \[discussion\] below). For small deviations from collinearity, $\delta\theta$, one can show that $\Delta m = \pm 2$ transitions are possible, and that the size of these components relative to the dominant $\Delta m=\pm 1$ transitions is proportional to $|\sin(\delta\theta)|$ [@Roos2000]. Given our apparatus geometry and laser beam collimation, we estimate that $|\sin(\delta\theta)| \le 2^{\circ}$. We were able to explore the implications of this by generating simulated Faraday rotation spectra with small $\Delta m = \pm2$ components, and then analyzing these modified lineshapes using our standard fitting routine. By studying the impact of this non-ideal geometry on the fitted lineshape amplitudes, we can place a limit on its potential systematic error contribution to $\chi$, which is included in Table \[error\_table\]. We note that, even with perfect collinearity, small stray magnetic fields, either from external sources or mu-metal remanence, would ultimately produce a small systematic geometric uncertainty. For the experimental fields employed here, we estimate this contribution to misalignment to be several times smaller than the current optical collinearity contribution.
We lastly mention that such geometrical misalignment also produces more complicated magneto-optical effects, including the so-called ‘Voigt’ effect. As discussed in detail in [@Edwards1995], the size of these additional components, given the estimated size of our misalignment, would produce changes to our Faraday lineshape that are well below our level of statistical sensitivity.
### Isotopic Purity
Given the quoted isotopic purity of the vapor cell (99.9%), we generated realistic simulated lineshapes and fit these using our standard analysis procedure to produce the systematic relevant error estimate in Table \[error\_table\].
### Final Experimental Ratio
Combining all of the systematic error contributions in quadrature gives an uncertainty roughly twice that of the statistical error. Combining these leads to a final experimental value for our unitless amplitude ratio: $\chi = 0.1496 \pm 0.0015$. In Sec. \[discussion\], we establish the connection between this ratio and the reduced electric quadrupole matrix element, the *ab initio* theoretical derivation for which we present next.
Theory
======
We evaluated the reduced matrix elements (MEs) of the $6p^2\,\,^3\!P_0 -\, 6p^2\,\,^3\!P_2$ and $6p^2\,\,^3\!P_0 -\, 6p^2\,\,^1\!D_2$ $E2$ transitions as well as the static scalar and tensor polarizabilities of the $6p^2\,\,^3\!P_1$ and $6p7s\,\,^3\!P_0^o$ states of Pb using the high-precision relativistic CI+all-order method [@Safronova2009]. This method was adopted by us for calculating the PNC amplitude for the $6p^2\,\,^3\!P_0 -\, 6p^2\,\,^3\!P_1$ transition [@Porsev2016].
We consider Pb as a four-valence atom. The basis set was constructed using a $V^{N-2}$ approximation in the framework of the Dirac-Fock-Sturm approach (see Ref. [@Porsev2016] for more details). In this calculation, we use the wave functions obtained in [@Porsev2016] in the CI+MBPT [@Dzuba1996] and CI+all-order approximations. We carry out calculations in both approximations considering the CI+all-order results as the recommended ones. Atomic units ($\hbar=|e|=m=1$) are used throughout unless stated otherwise.
$E2$ Transitions {#theory_E2}
----------------
Using the expression for the electric quadrupole moment operator, given by Eq. (\[eq:Qdef\]), we obtain for the $E2$ $6p^2\,\,^3\!P_0 -\, 6p^2\,\,^3\!P_2$ transition, $$\begin{aligned}
|\langle ^3\!P_0 ||Q|| ^3\!P_2 \rangle| &\approx& 8.91\,\, {\rm a.u.}\,\, ({\rm CI+MBPT}), \nonumber \\
&\approx& 8.86\,\, {\rm a.u.}\,\, ({\rm CI+all-order}) .\end{aligned}$$
Inclusion of the Breit interaction correction increases the absolute value of the matrix element (ME) by 0.02 a.u.. The quantum-electrodynamic (QED) correction is negligible at the current level of calculation accuracy. The difference of the values obtained at the CI+MBPT and CI+all-order stages gives us an estimate of the uncertainty. Thus, the final recommended value is: $$\label{eq:Qfinal}
|\langle ^3\!P_2 ||Q|| ^3\!P_0 \rangle| = 8.88(5)~\text{a.u.}.$$ We have also estimated the reduced ME of the electric quadrupole $6p^2\,\,^3\!P_0 -\, 6p^2\,\,^1\!D_2$ transition. This is an intercombination transition (the initial and final states have different total spin $S$). As a result, it is an order of magnitude smaller than $|\langle ^3\!P_2 ||Q|| ^3\!P_0 \rangle|$. We find $$|\langle ^1\!D_2 ||Q|| ^3\!P_0 \rangle| \approx 0.63\,\, {\rm a.u.}.$$
Polarizabilities {#theory_polar}
----------------
[cccrrcc]{} & & & & & &\
\
\[-0.3pc\] $6p^2\,\,^3\!P_1$ & $6p7s\,\,^3\!P_0^o$ & 27207 & 27141 & 1.92 & 6.6 & 6.6\
& $6p7s\,\,^3\!P_1^o$ & 27533 & 27468 & 1.41 & 3.5 & 3.5\
& $6p6d\,\,^3\!F_2^o$ & 38222 & 37624 & 0.08 & 0.01 & 0.01\
& $6p6d\,\,^3\!D_2^o$ & 39046 & 38242 & 3.45 & 14.9 & 15.2\
& $6p6d\,\,^3\!D_1^o$ & 39110 & 38249 & 0.63 & 0.5 & 0.5\
& $6p7s\,\,^3\!P_2^o$ & 40572 & 40370 & 0.78 & 0.7 & 0.7\
& $6p8s\,\,^3\!P_1^o$ & 41737 & 40868 & 1.13 & 1.5 & 1.5\
& $6p8s\,\,^3\!P_0^o$ & 42275 & 40907 & 0.65 & 0.5 & 0.5\
& $6p7s\,\,^1\!P_1^o$ & 42670 & 41621 & 0.20 & 0.04 & 0.05\
& Other & & & & 25.9 & 25.9\
& Total val. & & & & 54.2 & 54.6\
& Core + Vc & & & & 3.8 & 3.8\
& Total & & & & 58.0 & 58.4\
\
\[-0.5pc\] $6p7s\,\,^3\!P_0^o$ & $6p^2\,\,^3\!P_1$ &-27207 & -27141 & 1.92 & -20 & -20\
& $6p7p\,\,^3\!P_1$ & 7837 & 7959 & 3.99 & 298 & 293\
& $6p7p\,\,^3\!D_1$ & 9605 & 9715 & 5.43 & 450 & 445\
& $6p8p\,\,^3\!P_1$ & 17800 & 16361 & 0.17 & 0.2 & 0.2\
& $6p8p\,\,^3\!D_1$ & 18336 & 16957 & 1.04 & 8.6 & 9.4\
& Other & & & & 19 & 19\
& Total val. & & & & 756 & 747\
& Core +Vc & & & & 4.1 & 4.1\
& Total & & & & 760 & 751
[cccccc]{} State & & CI+MBPT & CI+all-order & diff(%) & Recom.\
\
\[-0.5pc\] $6p^2\,\,^3\!P_1$ & $\alpha_0$ & 58.7 & 58.0 & 1.2 & 58.0(7)\
& $\alpha_2$ & -5.8 & -5.7 & 1.5 & -5.7(1)\
$6p7s\,\,^3\!P_0^o$ & $\alpha_0$ & 752 & 760 & 1.1 & 760(8)
The scalar dynamic polarizability $\alpha(\omega)$ can be separated into three parts: $$\alpha(\omega) = \alpha_v(\omega) + \alpha_c(\omega) + \alpha_{vc}(\omega),
\label{alpha}$$ Where $alpha_v$ is the valence polarizability and $\alpha_c$ is the ionic core polarizability. A small term, $\alpha_{vc}$, is included due to the presence of the four valence electrons and possible excitation of a core electron to the occupied shell. Thus, $\alpha_{vc}$ serves to restore the Pauli principle and slightly modifies the core polarizability [@Porsev2002].
The valence part of the a.c. electric dipole polarizability of the $|\Phi_0 \rangle$ state can be written in the following form: $$\begin{aligned}
\alpha_v (\omega) &=& 2\, \sum_k \frac { \left( E_k-E_0 \right)
|\langle \Phi_0 |D_0| \Phi_k \rangle|^2 }
{ \left( E_k-E_0 \right)^2 - \omega^2 } \nonumber \\
&=& \sum_k \left[ \frac {|\langle \Phi_0 |D_0| \Phi_k \rangle|^2 } {E_k - E_0 + \omega}
+\frac {|\langle \Phi_0 |D_0| \Phi_k \rangle|^2 } {E_k - E_0 - \omega} \right]\!,
\label{Eqn_alpha}\end{aligned}$$ where $D_0$ is the $z$-component of the effective electric dipole operator ${\bf D}$, defined (in a.u.) as ${\bf D} = -{\bf r}$. By the effective (or “dressed”) electric dipole operator, we mean that the operator also includes the random-phase approximation (RPA) corrections [@Dzuba1998].
To account for intermediate high-lying discrete states and the continuum, we calculated $\alpha_v(\omega)$ by solving the inhomogeneous equation in valence space. We use the Sternheimer [@Sternheimer1950] or Dalgarno-Lewis [@Dalgarno1955] method implemented in the CI+all-order approach [@Kozlov1999]. Given the $\Phi_0$ wave function and energy $E_0$ of the $|\Phi_0 \rangle$ state, we find intermediate-state wave functions $\delta \psi_{\pm}$ from an inhomogeneous equation, $$\begin{aligned}
|\delta \psi_{\pm} \rangle & = & \frac{1}{H_{\rm eff} - E_0 \pm \omega}\,
\sum_k | \Phi_k \rangle \langle \Phi_k | D_0 | \Phi_0 \rangle \nonumber \\
&=& \frac{1}{H_{\rm eff}- E_0 \pm \omega} \, D_0 | \Phi_0 \rangle .
\label{delpsi}\end{aligned}$$ Using Eq. (\[Eqn\_alpha\]) and $\delta \psi_{\pm}$ introduced above, we obtain: $$\alpha_v (\omega ) = \langle \Phi_0 |D_0| \delta \psi_+ \rangle
+ \langle \Phi_0 |D_0| \delta \psi_- \rangle \, ,
\label{alpha2}$$ where the subscript $v$ emphasizes that only excitations of the valence electrons are included in the intermediate-state wave functions $\delta \psi_{\pm}$ due to the presence of $H_{\rm eff}$.
The $\alpha_{c}$ and $\alpha_{vc}$ terms were evaluated in the RPA. The small $\alpha_{vc}$ term was calculated by adding $\alpha_{vc}$ contributions from the individual electrons. For example, for the $6s^2 6p^2 \,\, ^3\!P_1$ state, we find $\alpha_{vc} = 2 \alpha_{vc}(6s)+ \alpha_{vc}(6p_{1/2}) + \alpha_{vc}(6p_{3/2})$.
For the case of static polarizabilities, where $\omega=0$, Eq. (\[Eqn\_alpha\]) is written as: $$\begin{aligned}
\alpha_v (0) &=& 2\, \sum_k \frac {|\langle \Phi_0 |D_0| \Phi_k \rangle|^2} {E_k-E_0}.
\label{stat_alpha}\end{aligned}$$
To establish the dominant contributions to the valence polarizabilities, we combine the electric-dipole matrix elements and energies according to the sum-over-states formula given by Eq. (\[stat\_alpha\]). We have carried out two calculations of the dominant contributions of the intermediate states to the polarizabilities using our theoretical and experimental energies. In Table \[tab\_scal\], we present results obtained in the CI+all-order approximation. The absolute [*ab initio*]{} values of the corresponding reduced electric-dipole matrix elements are listed (in a.u.) in column labeled “$D$.” The theoretical and experimental [@NIST] transition energies are given in columns $\Delta E_{\rm th}$ and $\Delta E_{\rm expt}$. The remaining valence contributions are given in rows labeled “Other.” The contributions from the core and $\alpha_{vc}$ terms are listed together in the row labeled “Core + Vc.” The dominant contributions to $\alpha_0$, listed in columns $\alpha_0[\text{A}]$ and $\alpha_0[\text{B}]$, are calculated with CI+all-order+RPA matrix elements and theoretical \[A\] and experimental \[B\] energies [@NIST], respectively. The results listed in the column $\alpha_0[\text{A}]$ are the recommended ones.
The results obtained in the CI+MBPT and CI+all-order approximations, their differences, and the recommended values are presented in Table \[tab\_polar\].
Comparison of Experiment to Theory {#discussion}
==================================
We turn now to the connection between our unitless $E2$/$M1$ amplitude ratio, $\chi$, and the theoretical expressions for the respective matrix elements. It is helpful to recall that both the $M1$ and $E2$ matrix element components emerge from the same term in the expansion of the interaction Hamiltonian. Following a standard textbook derivation of these higher-order terms [@Fitzpatrick], we find that both the $M1$ and $E2$ transition amplitudes originate from a matrix element, $T_{fi}$, containing both the position and momentum operators, $$\label{eqmultipole}
T _{fi \; (\rm{M1, E2})} \propto \langle f | (\hat{k}\cdot\ {\bf r})\;(\hat{\epsilon}\cdot\ {\bf p} ) | i \rangle,$$ where $\hat{k}~(\hat{z}$ in our case) is the laser propagation direction, and $\hat{\epsilon}~(\hat{x}$ in our case) is the laser polarization axis. We can ignore overall multiplicative factors since they will cancel in the eventual $E2$/$M1$ amplitude ratio.
After some vector algebra and use of a commutator to re-express the momentum operator in terms of position [@Fitzpatrick], we can separate the $M1$ (vector) and $E2$ (second-rank tensor) components of the matrix element in Eq. (\[eqmultipole\]). We note that this process introduces a factor of $\omega_{\mathrm{E2}}/c$ into the $E2$ component. In our case, the $M1$ final state of interest is $|^3\!P_1, m=1 \rangle$, where as for the $E2$ component it will be $|^3\!P_2, m=1\rangle$.
According to the Wigner-Eckart theorem, for the case of the $|J=0\rangle \rightarrow |J_f, m=1\rangle$ transitions, the multiplicative factor connecting the $|\Delta m| = 1$ matrix elements that we measure with the associated [*reduced*]{} matrix element is $1/\sqrt{2J_f+1}$. Given our geometry, the operator for the $E2$ term is proportional to $\langle xz \rangle$. This can then be rewritten in terms of the operator $\langle r^2 C_{21}\rangle$ as introduced in Sec. \[background\].
Assembling a theoretical expression that is equivalent to the (unitless) experimental amplitude ratio $\chi$, given by Eq. (\[Eqfinal\]), we arrive at $$\chi = \frac{1}{2\sqrt{5}} \frac{\omega_{E2}}{c} \, \frac{\langle ^3\!P_2 ||Q|| ^3\!P_0 \rangle}{\langle ^3\!P_1 ||\mu|| ^3\!P_0 \rangle}.
\label{chi_frac}$$
Here the reduced ME $\langle ^3\!P_2 ||Q|| ^3\!P_0 \rangle$ is expressed in $|e| a_B^2$ (where $a_B$ is the Bohr radius; note that for this ME $1 \, {\rm a.u.} = 1\, |e| a_B^2$) and $\langle ^3\!P_1 ||\mu|| ^3\!P_0 \rangle$ is expressed in $\mu_B/c$.
Inserting our experimental value, $\chi = 0.1496(15)$, as well as the (highly accurate) theoretical value for the $M1$ reduced matrix element $\langle ^3\!P_1 ||\mu|| ^3\!P_0 \rangle = 1.293(1)\,\mu_B/c$ [@Porsev2016], we can compute an experimentally-derived value for the reduced quadrupole matrix element: $\langle ^3\!P_2 ||Q|| ^3\!P_0\rangle_{\mathrm{exp}} = 8.91(9)$ a.u.. This is in excellent agreement with, and of comparable precision to, the recommended *ab initio* theory value from Eq. (\[eq:Qfinal\]) in Sec. \[theory\]: $\langle ^3\!P_2 ||Q|| ^3\!P_0\rangle_{\mathrm{th}} = 8.88(5)$ a.u.. Together, we have demonstrated consistency between experiment and theory for this lead E2 transition amplitude at the 1.2% level of accuracy.
Concluding Remarks {#conclusion}
==================
We have completed a precise measurement of the electric quadrupole $^3\!P_0 \rightarrow \, ^3\!P_2$ transition amplitude within the $6s^2 6p^2$ configuration in atomic lead. This result is in excellent agreement with a precise *ab initio* calculation of this amplitude, which has also been presented here. The calculation builds upon on recent theoretical work in the four-valence lead system aimed at improving PNC calculations in this element [@Porsev2016]. The experimental work relies critically on a high-precision polarimetry technique used previously to measure PNC optical rotation in Pb and Tl [@Meekhof1993; @Vetter1994], and has allowed direct measurement of this forbidden $E2$ transition for the first time. We have also presented *ab initio* calculations of the static polarizability of several low-lying states in lead. This now provides additional opportunities to test the accuracy and further guide the refinement of theory through precise atomic-beam-based measurements of Stark shifts in this element, employing experimental techniques analogous to those used by our group in recent indium and thallium polarizability measurements [@Doret2002; @Ranjit2013; @Augenbraun2016; @Vilas2018].
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors thank Gabriel Patenotte and Sameer Khanbhai for their assistance in the construction and testing of the experimental apparatus. We are grateful for valuable conversations with David DeMille, and thank Charles Doret for helpful comments on the manuscript. The experimental work described here was completed with the support of the National Science Foundation RUI program, through Grant No. PHY-1404206. The theoretical work was supported in part by NSF Grant No. PHY-1620687. S. P. acknowledges support by the Russian Science Foundation under Grant No. 19-12-00157.
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1.truecm
**2D FRUSTRATED ISING MODEL WITH FOUR PHASES**
1.4truecm
M. Pasquini and M. Serva
.5truecm
*Dipartimento di Matematica and Istituto Nazionale Fisica della Materia, Università dell’Aquila*
*I-67010 Coppito, L’Aquila, Italy*
1.6truecm
ABSTRACT
.4truecm In this paper we consider a $d=2$ random Ising system on a square lattice with nearest neighbour interactions. The disorder is short range correlated and asymmetry between the vertical and the horizontal direction is admitted. More precisely, the vertical bonds are supposed to be non random while the horizontal bonds alternate: one row of all non random horizontal bonds is followed by one row where they are independent dichotomic random variables. We solve the model using an approximate approach that replace the quenched average with an annealed average under the constraint that the number of frustrated plaquettes is keep fixed and equals that of the true system. The surprising fact is that for some choices of the parameters of the model there are three second order phase transitions separating four different phases: antiferromagnetic, glassy-like, ferromagnetic and paramagnetic.
PACS NUMBERS: 05.50.+q, 02.50.+s
Mean field spin glasses models have been studied and deeply understood both from a static and a dynamic point of view and key words like replica symmetry breaking, aging and ultrametricity, have become of very wide use in statistical mechanics of disordered systems . The reason why spin glasses have attracted so much attention is probably more a consequence of the many successful applications to biological modeling (neural networks, immune system, adaptive evolution) then to their original scope limited to the description of disordered materials. For this reason and may be for objective technical difficulties most of the typical features which are very well established for the mean field models have not been found out for short range spin glasses. For example, it is commonly believed that a finite temperature glassy phase only exists for $d \ge 3$ spin glasses while in $d=2$ one has only the paramagnetic phase. This is an almost surely true statement if one consider $d=2$ spin system with independent bonds and with vertical-horizontal symmetry but may be it is a false statement if one consider $d=2$ spin asymmetric systems with correlated disorder. For example, in models with layered disorder the existence of a low temperature phase seems to be an established fact , nevertheless, one may think that these models are pathological since layered disorder is somehow a long range correlated disorder.
In this paper we consider a $d=2$ Ising system where there is both a short range correlation of the disorder and an asymmetry between vertical and horizontal direction. The specific interaction we chose is not motivated by a deep physical insight but it is merely dictated by technical reasons. Nevertheless, the model is not very artificial and the disorder correlation is limited to the fact that frustrated plaquettes always are present in near couples while the asymmetry only lies in a difference of strength of vertical and horizontal bonds.
We solve the model using an approximate approach that replaces the quenched average with an annealed average under the constraint that the number of frustrated plaquettes is keep fixed and equals that of the true system. The surprising fact is that for some choices of the parameters of the models one can find four different phases.
The paper is organized as follows.
In section 2, after a brief overlook of the constrained annealing, we introduce our model with a particular attention to the concept of frustration; then we write the partition function with constrained frustration and the relative free energy.
In section 3 we derive the solution of the model, obtaining an expression for the free energy that can be computed via numerical methods. Moreover, the ground state energy is exactly found.
In section 4 the conditions that yield to second order phase transitions are derived.
In section 5 we describe the various behaviours of the model, showing a total of four distinct phases, three almost conventional (high temperature paramagnetic phase, ferromagnetic and antiferromagnetic phases at low temperature) and a fourth paramagnetic phase that we guess to have a ’glassy’ nature.
In section 6 we present our conclusions.
The model is defined on a square $d=2$ lattice and the interaction is supposed to be effective only between nearest neighbours. The number of spins is $N=LM$ where $M$ is the number of columns of the lattice and $L$ is the number of rows.
The vertical bonds are supposed to be non random and one can assume without loss of generality that they equal $1$ while the horizontal bonds alternate; one row of all non random horizontal equal $1$ bonds is followed by one row where they are independent dichotomic random variables which equal $1$ with probability $p$ and equal the negative value $-\g$ with probability $1-p$ (see ).
It follows that the Hamiltonian of our model can be written as: H\_N=-\_[i=1]{}\^L \_[j=1]{}\^M (\_[i,j]{} \_[i+1,j]{} + J\_[i,j]{}\_[i,j]{} \_[i,j+1]{} ) where $\s_{i,j}=\pm 1$ is the spin in the site located by the $i$-th row and the $j$-th column while the $J_{i,j}$ are the horizontal bonds which equal $1$ when $i$ is even and are defined by
when $i$ is odd.
The model is parameterized by $\g>0$ and $p$ and, in general, it is random, except in the two limit cases $p \to 0$ and $p \to 1$. In the first limit case $p=0$ all the couplings equals the unity and, therefore, we have the pure $d=2$ Ising model . In the second limit case $p=1$ the model is also not random, but while all the vertical couplings equal the unity, the horizontal couplings alternate one row in which they are all positive and equal the unity to one row in which they are all negative and equal $-\g$. In this second limit the model can be solved by standard transfer matrix methods and it shows a low temperature magnetic phase; for $\g<1$ this low temperature phase is ferromagnetic while for $\g>1$ there is horizontal antiferromagnetic order and vertical ferromagnetic order between the spins. For the sake of simplicity we will call hereafter this complicated magnetic phase simply the antiferromagnetic phase. Finally, the special choice $p=1$, $\g=1$ corresponds the so-called ’fully frustrated model’ which is also not random and has a transition only at $T=0$
In order to explain the nature of our approximation, let us first recall that the elementary unit for frustration is the plaquette. If the product of the signs of the bonds around a plaquette is negative the plaquette is frustrated, otherwise, the plaquette is unfrustrated. In our Ising model, only the sign of the random variable $J_{i,j}$ with odd $i$ can be negative, therefore the two square plaquettes which share this bond are frustrated if this bond is effectively negative (e.g. ’a’ plaquettes in ) and they are unfrustrated if it is positive (’b’ plaquettes). As a consequence of this definition of elementary frustration, we may define the total frustration of the system $\phi_N$ as the rate of frustrated plaquettes. In our model This quantity equals $p$ in average, furthermore the strong law of large numbers assures that $\phi_N \to p$ with probability $1$ in the thermodynamic limit.
We are far from being able to solve the quenched model, nevertheless we think that the qualitative behaviour of the system is captured by the above definition of total frustration ( , for a more general definition see ). Therefore, our proposal is to consider an annealed approximation where $\phi_N$ is constrained to coincide, in the thermodynamic limit, with the quenched total frustration $p$. This model corresponds to averaging $Z$ only over the realizations of the disorder with total frustration $p$. We not only believe that the approximated model has the same qualitative features of the quenched one, but it is also in good quantitative agreement with it. In fact, our experience is that constrained annealing is a really powerful tool for estimating the free energy of disordered systems . We would like also to stress that the fixed frustration model can be also seen as an independent model where the bonds as well as the spins are allowed to arrange themselves in order to minimize the free energy provided they satisfy the global frustration constraint.
In order to obtain the free energy of the fixed frustration model we follow the general method (, ). We must first define the generalized partition function where $\be={1\over T}$ is the inverse temperature, and the average is over all realizations of the couplings $J_{i,j}$, than we obtain the free energy of the constrained annealed model as where the $N\to \infty$ limit means that both $M$ and $L$ must tend to the same limit. In fact, the minimization over $\mu$ automatically selects the realizations of the disorder for which $\phi_N = p$ in the thermodynamic limit.
The generalized partition function is a sum of a product of randomly independent variables, therefore, we can write $$Z_N (\be,\g,\mu)=
\sum_{\sigma} \prod_{i=1}^L \prod_{j=1}^M
\avd{ \exp
\left[ \be \s_{i,j} \s_{i+1,j} +
\be J_{i,j} \s_{i,j} \s_{i,j+1}
+\mu (2{1-J_{i,j} \over 1+\g }-p)
\right] }$$ The average can be now easily performed, obtaining: where we have introduced the effective hamiltonian $$\tilde{H}_N=-\sum_{i=1}^L \sum_{j=1}^M \left( \s_{i,j} \s_{i+1,j}
+\tilde J_i \s_{i,j} \s_{i,j+1} \right)$$ The new effective horizontal bonds $ \tilde{J}_i$ are not random, they are all equal in the same row, and they alternate two possible values in different rows; in fact, one has $ \tilde{J}_i = 1$ when $i$ is even and $ \tilde{J}_i = {b\over 2\be}$ when $i$ is odd. The constants $a$ and $b$ are It is possible to show that b is a monotonic decreasing function of $\mu$ with $-2\g \be \le b \le 2 \be$, so that we can directly use $b$ as variational parameter in order to realize the minimum in .
The effective hamiltonian $\tilde{H}_N$ is indeed associated to a pure $2d$ Ising model with unitary strength couplings along the vertical bonds, and with alternated rows of unitary and $b\over{2\be}$ strength couplings. This model can be solved by trivially generalizing the Onsager solution and it is mapped into the problem of diagonalizing a collection of $2 \times 2$ matrices. In the thermodynamic limit $N \to \infty$ the total free energy reads $$f(\be,\g)=-{\g+p(1-\g)\over2}-{1\over{2\be}}\ln\left[{4 p^p (1-p)^{1-p} }
\sinh(\be(1+\g)) \sinh(2\be) \right] +$$ $$-\min_b\left[ {b\over{4\be}}(1-2p)-{1-p\over{2\be}}
\ln (\e{b+2\be\g}-1) -{p\over{2\be}} \ln (\e{2\be-b}-1) \right. +$$ where $\lambda(q,b)$ indicates the maximum eigenvalue in modulus of the product of the two matrices $$\left\{ \eqalign{
& {\bf T}_\be (q)=\exp\left[\be^* ({\bf \tau}_z \cos q+{\bf \tau}_x \sin q)
\right]
\exp(-2\be{\bf \tau}_z)
\cr
& {\bf \tilde T}_b (q)=\exp\left[\be^* ({\bf \tau}_z \cos q+{\bf \tau}_x
\sin q) \right]
\exp(-b{\bf \tau}_z)
}\right.$$ where $\be^*=-\ln\tanh\be$, and ${\bf \tau}_x$, ${\bf \tau}_z$ are Pauli matrices. After some trivial algebra one gets: with: $$t(q,b)=2 \cos^2 q \ {\sinh b\over\sinh2\be} \
-\ 2 \cos q \ \cosh2\be \ {\sinh(2\be{+}b)\over\sinh^2 2\be} \ +$$
The minimum of is realized for $b=b^*$ and it is achieved by looking for the zero of its derivative. One has the self-consistent equation for $b$: When $p\to 0$ the model has to reduce to the standard Ising model and in fact, the previous formula leads to $b^* \to 2\be$, while in other limit case $p\to 1$ one has $b^* \to -2\g\be$.
Equation is an ordinary equation in $b$, nevertheless, it cannot be explicitly solved so that we are not able to give a compact expression for $b^*$ in terms of $T$, $\g$ and $p$. However, and then can be numerically computed with the necessary precision in order to fully investigate the model. Furthermore, at $ T=0$ while we don’t have the complete solution of we are able to derive the leading terms of $b^*$ and to compute the ground state energy $U_0$. We find out that $U_0$ has different linear behaviours in $p$ depending on $\g$ $$U_0= -2 + ( 1 - {|1-\g| \over2} ) p$$
The $T=0$ entropy $S_0$ can only be computed numerically and it is shown in . Unlike the energy $U_0$, $S_0$ equals a constant function of $p$ for any of the three choices of $\g$. It is always zero for $\g<1$ and $S_0\ge0$ for $\g=1$. In the case $\g>1$ one can show that the entropy becomes positive for $p>\tilde{p}$, where $\tilde{p}<{3\over4}$, and $S_0(p={3\over4})\simeq 0.01$.
Let us stress again that formulae - represent the solution of the model, and, in principle, all the informations about it can be derived from them. Fortunately, even if we are unable to give an explicit expression of $b^*$ starting from , we can easily obtain some analytic results. For instance, in this section we find the conditions that yield to a second order phase transition.
Looking carefully at - , one can realize that the mechanism of the usual Onsager transition is preserved: the discontinuity occurs when $b^*$, the zero of , nullifies also the argument of the square root in , i.e. when (the case $t(q,b^*)=-1$ is not possible since $t(q,b)\ge1$ for $\forall q$ and $\forall b$). In other terms, one has to find out the solution $b^*$ of a system of two equation, and . Obviously this solution can exist only for certain values of $\be$, $\g$ and $p$.
A direct inspection of shows that can be satisfied only for the specific choices $q=0$ or $q=\pi$, and it determines the existence of two distinct transition lines in the $p-T$ phase diagram at fixed $\g$, (see ). The first line ends on the $p=0$ (pure Ising model) axis at the Onsager critical temperature, so that in the following we will refer to this transition line as the ferromagnetic one. The second line exists only for $\g\ge1$ and it ends on the $p=1$ axis in correspondence of the critical temperature separating the antiferromagnetic phase from the paramagnetic one; for this reason we will call it the antiferromagnetic line.
After some trivial algebra reduces to where the sign $+$ correspond to the ferromagnetic line ($q=0$), and the sign $-$ to the antiferromagnetic one ($q=\pi$).
In the limit case $p=0$ ($b^*=2\be$), the recovers the well-known result $\sinh(2\be)=1$, while in the other limit case $p=1$ ($b^*=-2\be\g$) the transition is present when $\sinh (2\be(1-\g))=-2{\cosh 2\be \over \sinh^2 2\be}$, i.e. at finite temperature when $\g>1$ and at zero temperature for the ’fully frustrated model’ ($\g=1$).
From a practical point of view, in order to compute numerically the transition lines which are showen in , it is convenient to solve with respect to $b^*$ Then, keeping $\g$ fixed and substituting $b^*$ into , one obtains the two transition lines $p(T)$, which can be easily computed by standard numerical algorithms.
In the previous section we have seen that gives the known critical temperatures of the non random models ($p=0$ or $p=1$). Another preliminary information about the behaviour of the model comes from the observation that the right hand side of goes to zero in the limit $T\to0$, so that the ferromagnetic and the antiferromagnetic lines must coincide when they end on the $T=0$ axis. Studying the leading terms of close to $T=0$, one finds that this coinciding point is at $p=1$ for $\g =1$ and at $p={3\over4}$ for $\g>1$.
The full description of the different behaviours can be derived computing the transition lines in the $p - T$ phase diagram at varying $\g$, as seen in the previous section. The following four scenarios listed in , are obtained.
For $\g<1$ (see a, where $\g=0.8$) only the ferromagnetic line is present, separating two well-known phases: a ferromagnetic phase at low temperature, and a paramagnetic phase at high temperature, exactly as for the Onsager non random model. In fact the antiferromagnetic random couplings are too weak with respect to the ferromagnetic ones, so that they are not able to change the structure of the phases of the pure model.
When $\g=1$ (b) the scenario is quite similar to the previous one, apart from the fact that the ferromagnetic line reaches the axis $T=0$ at $p=1$, in correspondence of the $T=0$ transition of the fully frustrated model. Notice that the antiferromagnetic transition line is still absent. In fact, the antiferromagnetic couplings have the same strength of the others but, for any $p<1$, their number is lower than the number of horizontal ferromagnetic couplings so that the ferromagnetic order prevails at low temperature.
The most interesting situation corresponds to the choice $1<\g<2$ (c, where $\g=1.2$). First of all notice that both the transition lines are present. The first line starts on the $p=0$ axis and it ends at $p={3\over4}$ on the $T=0$ axis and it delimitates the low temperature ferromagnetic region. The second line starts on the $T=0$ axis at $p={3\over4}$ ending on the $p=1$ axis, and it delimitates the antiferromagnetic phase. Outside this two regions there is a non-magnetic phase, but notice that this one has a narrow tongue dividing the magnetic regions and reaching the $T=0$ axis at $p={3\over4}$. As a consequence, if one fixes the probability between ${3\over4} < p < {3\over4}+\delta p(\g)$, where $\delta p(\g)$ is a small but finite number depending on $\g$, one can observe three different second order phase transitions varying the temperature $T$. The transitions separate four phases; starting from low temperatures, the first is ferromagnetic, the third is antiferromagnetic, the fourth is an ordinary paramagnetic phase while the second is a low temperature paramagnetic phase. In it is shown the specific heat $C$ as a function of $T$ at fixed $\g=1.2$ and $p=0.82$, computed starting from the numerical solution of and . In this case the specific heat $C$ exhibits three distinct peaks next to $T\simeq0.299$, $T\simeq0.349$ and $T\simeq0.545$. Indeed we need to magnify the picture since it is necessary to compute with a great precision in order to show a certain growth of $C$ around its discontinuity.
The appearance of a low temperature paramagnetic phase between the antiferro and the ferromagnetic ones represents an interesting peculiarity of this model. In particular, it happens at relatively low temperature and with an extremely narrow width. These are the main features that persuade us to guess a glassy nature for this paramagnetic phase. Moreover in our constrained annealed model, as seen at the end of section 3, the region at low temperature with an unphysical solution (negative zero temperature entropy $S_0$) do not reach the critical transition point ($p={3\over4}$ , $T=0$) where the ’glassy’ paramagnetic phase ends.
The description of the different scenarios is completed with the case $\g>2$ (d, where $\g=3$). The structure of the phase diagram is similar to the previous one with the difference that the narrow tongue between the ferro and the antiferro phases is suppressed. As a consequence for $p<{3\over4}$ we only have the ferro and the para phases while for $p>{3\over4}$ we only have the antiferro and the para phases. When $\g\to\infty$ the temperature of end point on the $p=1$ axis goes to infinite.
The surprising feature of our model is that for some choices of the parameters $\g$ and $p$ the magnetic phases are separated by a low temperature paramagnetic phase. We do not expect any long distance magnetic order in this phase i.e. $\avd{ {<}\sigma_{i,j} \sigma_{i,j+k} {>} }
=\avd{ {<}\sigma_{i,j} \sigma_{i+k,j} {>} }=0$ in the limit $k\to \infty$ but we expect that $\avd{ {<} \sigma_{i,j} \sigma_{i,j+k} {>}^2 } >0$ and $\avd{ {<} \sigma_{i,j} \sigma_{i+k,j} {>}^2 } >0$ in the same limit. Our proposal is that these last two quantities properly characterize the glassy-like paramagnetic phase and they should vanish in the high temperature paramagnetic phase. Unfortunately, it is well known that in two dimensional models the computation of long distance correlation is a difficult task and, in our case, the computation also involves averages over the disorder making the situation even more complicated. We think that some work can be made in this direction but it will demand much technical effort so that our claim that the low temperature paramagnetic phase is somehow a glassy phase is, at this point, more a conjecture than an established fact.
Two more questions remain to be answered. The first is the most relevant: what is the role of the annealed approximation in the qualitative features of the phase diagram? Or better, is the new phase a mere consequence of the annealed approximation? In this case, our fixed frustration model would have an interest in itself but it would not be a good approximation of the quenched one. We think that this question can only be answered by direct Montecarlo simulation.
The second question is: in this model the frustrated plaquettes appears only in couples, what is the role of this special correlation? To be more specific, a model in which all vertical bonds equals the unity while the horizontal are independent random variables which take two possible values of opposite sign would have the same qualitative behaviour? We cannot approximate such a model by our fixed frustration technique so that also this question can only be answered by Montecarlo simulation, nevertheless, we are convinced that the answer should be positive. In fact, the special nature of correlation between plaquettes is short ranged and there is not reason why it should affect the long range behavior of the system.
In conclusion we would like to stress that in spite of the very partial results contained in this paper and of the many unsolved questions this work sheds some light on the very important point of the existence of a finite temperature glassy phase for $d=2$ frustrated systems. In fact, in the most restrictive interpretation of our result we can still affirm that the fixed frustration annealed $d=2$ model has low temperature glassy-like phase, while in the most generous one we can say that a true glassy phase exists for quenched random $d=2$ systems.
[**Acknowledgements**]{}
We acknowledge the financial support of the I.N.F.N., National Laboratories of Gran Sasso ([*Iniziativa Specifica*]{} FI11). We thank Roberto Baviera for useful discussions.
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---
abstract: 'The equivalence theorem, between the longitudinal gauge bosons and the states eaten up by them in the process of symmetry breaking, is shown to be valid in a class of models where the details of dynamical symmetry breaking makes it obscure.'
author:
- |
Palash B. Pal\
*Center for Particle Physics, University of Texas, Austin, TX 78712, USA*
title: 'August 1993 DOE-ER40757-023 / CPP-93-23'
---
=-0.5in =9.7in =2ex =3ex =-.25in =-.25in plus.3em minus.5em plus.3em minus.5em plus.2em minus.4em plus.2em minus.4em =10000
In a recent paper [@DoTa93], Donoghue and Tandean have made an intriguing point regarding the validity of the equivalence theorem [@CLT74; @BQT77; @ChGa85; @Wil88; @BaSc90; @Vel90] in a class of gauge models which exhibit dynamical symmetry breaking. As a paradigm, they considered the pedagogical model [@Wei79; @Sus79] where the electroweak gauge symmetry is dynamically broken by quark condensates which are also responsible for the chiral symmetry breaking: $$\begin{aligned}
\langle \overline u_L u_R \rangle =
\langle \overline d_L d_R \rangle \neq 0 \,.
\end{aligned}$$ The pions, $\pi^\pm\pi^0$, which would have been the Goldstone bosons for the chiral symmetry breaking $SU(2)_L\times SU(2)_R \to SU(2)_V$, are eaten up by the electroweak gauge bosons in this model in absence of any fundamental Higgs bosons. Donoghue and Tandean [@DoTa93] then calculate the amplitude of the process $e^+e^-\to Z\gamma$ at energies larger than $M_Z$ mediated by the triangle diagram. Both quarks and leptons can appear in the triangle, and from the condition of vanishing of gauge anomalies, they obtain that the amplitude vanishes. Since $\pi^0$ constitutes the longitudinal part of the $Z$ in this model, they then compare it with the process $e^+e^-\to \pi^0\gamma$, mediated through the triangle diagram. However, since the pion consists of quarks but no leptons, only quarks circulate in the loop now, and therefore the amplitude does not vanish. This, they claim, is a violation of the equivalence theorem. The purpose of this article is to examine this claim.
In Higgs models of symmetry breaking, one can calculate the couplings of the unphysical Higgs bosons (which are eaten up by the gauge bosons) from those of the gauge bosons without having to make any assumption about the Higgs content of the model. This can be done by demanding that, if one uses the $R_\xi$ gauges to calculate any amplitude, the $\xi$-dependent poles must cancel [@FLS72; @CBLP84]. Thus, for two fermions $a$ and $b$ which are represented by their field operators $\psi_a$ and $\psi_b$, if the coupling to any gauge boson $V$ is given by $$\begin{aligned}
\overline \psi_a \gamma^\mu \left( {\cal G}_{ab} + {\cal G}'_{ab}
\gamma_5 \right) \psi_b V_\mu \,,
\label{Vcoupling}
\end{aligned}$$ this requirement demands that the corresponding unphysical Higgs boson, $S$, will have the coupling $$\begin{aligned}
{1\over M_V} \overline \psi_a \left[ (m_a - m_b) {\cal G}_{ab} +
(m_a + m_b) {\cal G}'_{ab} \gamma_5 \right] \psi_b S \,,
\label{Scoupling}
\end{aligned}$$ where $M_V$ is the mass of the gauge boson $V$ after symmetry breaking. From this requirement alone, one can verify the equivalence theorem for any amplitude.
However, the equivalence theorem is more deep-rooted than the Higgs mechanism of symmetry breaking for the following reason. In any gauge symmetry breaking, some gauge bosons obtain masses $M_V$ whose exact values depend on some parameters of the theory. For any $M_V \neq 0$, the longitudinal components of the massive gauge bosons are physical states. The “Nambu-Goldstone” modes, the states absorbed by the gauge bosons, are unphysical. On the other hand, for $M_V=0$, the Nambu-Goldstone modes are physical states but the longitudinal components are not, since the symmetry is not broken. The equivalence theorem then merely states that all observables are continuous in the limit $M_V\to 0$. In other words, in that limit, the amplitudes with any process with longitudinal component of a vector boson is the same (apart from a phase maybe) with the amplitudes for the corresponding processes where the longitudinal gauge bosons are replaced by the states that are eaten up by them in the process of symmetry breaking. Stated this way, the equivalence theorem seems to be a statement of continuity of certain parameters of the theory, and hence is expected to be valid for any model with symmetry breaking. In light of this, the claim of Donoghue and Tandean [@DoTa93] is indeed surprising.
To examine their claim, let us use their paradigm of QCD condensates breaking the electroweak gauge symmetry [@Wei79; @Sus79], and for the sake of definiteness, let us talk about processes involving $Z$ bosons only. Obviously, one can make similar arguments for processes involving $W$ bosons. Since the $Z$-boson does not provide any flavor changing neutral current with the standard model fermions, the indices $a$ and $b$ in Eq. (\[Vcoupling\]) have to be equal, and we denote the couplings ${\cal G}$ and ${\cal G}'$ in this case with a single index. From Eq. (\[Scoupling\]) now, we see that proving the equivalence theorem is tantamount to proving that the coupling of the $\pi^0$ to the fermion field $\psi_a$ is given by $$\begin{aligned}
{2m_a \over M_Z}\, {\cal G}'_a \; \overline \psi_a \gamma_5 \psi_a
\pi^0 \,.
\label{zcoupling}
\end{aligned}$$
(85,36)(-5,32) (15,50)[(-1,1)[15]{}]{} (7,58)[(1,-1)[2]{}]{} (4,62)[$a(p)$]{} (15,50)[(-1,-1)[15]{}]{} (7,42)[(-1,-1)[2]{}]{} (4,35)[$a(p')$]{} (16,50)(4,0)[10]{}[(2,2)\[t\]]{} (18,50)(4,0)[10]{}[(2,2)\[b\]]{} (35,48)[(0,0)[(1,0)[5]{}]{}]{} (35,54)[(0,0)\[b\][$Z^\mu(q)$]{}]{} (55,50) (55,50)(3,0)[10]{}[(1,0)[2]{}]{} (70,52)[(0,0)\[b\][$\pi^0(q)$]{}]{}
If the fermion $a$ in question is the electron, for example, it might naively seem that such a coupling with the pion cannot exist since the pion wave function does not have any electron. However, this is not true, as can be seen from the diagram of Fig. \[f:coupling\]. Two important points need to be made before we calculate this coupling. First, the intermediate line can only be $Z$. The diagram with intermediate photon line cannot contribute since the photon couplings are vectorial, whereas the pion couples only to the axial vector current through the relation $$\begin{aligned}
\left< 0 \left| \overline Q \gamma^\mu \gamma_5 {\tau^I\over 2} Q
\right| \pi^J (q) \right> = f_\pi q^\mu \delta^{IJ} \,,
\label{fpi}
\end{aligned}$$ where $|0\rangle$ is the hadronic vacuum, $I,J$ indices run over the adjoint representation of the isospin symmetry SU(2)$_V$, and $$\begin{aligned}
Q \equiv \left( \begin{array}{c} u \\ d \end{array} \right) \,.
\end{aligned}$$ Second, the diagram must be calculated in the unitary gauge where unphysical degrees of freedom cannot appear in the intermediate state. Otherwise, the pions can appear even as intermediate states and it will be impossible to calculate the diagram. In the unitary gauge, the propagator of the gauge boson is given by $$\begin{aligned}
-i D^{\mu\nu} (q) = {-i (g^{\mu\nu} - q^\mu q^\nu/M_Z^2) \over
q^2-M_Z^2} \,.
\label{Zpropag}
\end{aligned}$$ Thus the effective interaction between the fermions and the pion derived from this diagram is given by[^1] $$\begin{aligned}
i {\cal L}_{\rm eff} =
{\bf \overline u_{\em a} (p')} \, i \gamma_\mu
\left( {\cal G}_a + {\cal G}'_a \gamma_5
\right) {\bf u_{\em a} (p)} \cdot \left[ -i D^{\mu\nu} (q) \right] \cdot
i\left< 0 \left| J_\nu^{(Z)} \right| \pi^0 (q) \right> \,,
\label{picoup}
\end{aligned}$$ where $J_\nu^{(Z)}$ is the current that couples to the $Z$ boson: $$\begin{aligned}
J_\nu^{(Z)} = -\, {g \over 4\cos \theta_W} \left(
\overline u \gamma_\nu \gamma_5 u - \overline d \gamma_\nu \gamma_5 d
+ \cdots \right) \,,
\label{jz}
\end{aligned}$$ where $g$ is the weak SU(2) gauge coupling, $\sin\theta_W=e/g$, and the dots signify vector currents as well as currents of fermions other than the up and the down quarks which are of no interest for us. From Eqs. (\[fpi\]), (\[Zpropag\]) and (\[jz\]), we obtain $$\begin{aligned}
D^{\mu\nu} (q) \; \left< 0 \left| J_\nu^{(Z)} \right| \pi^0 (q)
\right> = {q^\mu \over M_Z} \,,
\label{DJ}
\end{aligned}$$ using the relations for the gauge boson masses obtained in this model, viz., $$\begin{aligned}
M_W = M_Z \cos \theta_W = {1\over 2} gf_\pi \,.
\label{MWMZ}
\end{aligned}$$ Putting Eq. (\[DJ\]) in Eq. (\[picoup\]), it is straightforward to show that the amplitude is $$\begin{aligned}
{2m_a \over M_Z} \, {\cal G}'_a {\bf \overline u_{\em a} (p')}
\gamma_5 {\bf u_{\em a} (p)} \,,
\label{finalcoup}
\end{aligned}$$ where we have used the spinor definitions $$\begin{aligned}
\rlap/p {\bf u (p)} = m {\bf u (p)} \,, \qquad
{\bf\overline u (p)} \rlap/p = m {\bf\overline u (p)} \,.
\label{spinor}
\end{aligned}$$ Obviously, Eq. (\[finalcoup\]) is equivalent to the interaction of Eq. (\[zcoupling\]). Since this coupling is obtained in this model, it is now easy to verify that the equivalence theorem is valid for any amplitude, as we argued before.
(70,30)(0,35) (5,60)[(1,0)[30]{}]{} (15,60)[(1,0)[1]{}]{} (15,62)[(0,0)\[b\][$e^-(p)$]{}]{} (5,40)[(1,0)[30]{}]{} (15,40)[(1,0)[1]{}]{} (15,38)[(0,0)\[t\][$e^+(p')$]{}]{} (35,40)[(0,1)[20]{}]{} (35,50)[(0,-1)[1]{}]{} (37,50)[(0,0)\[l\][$e^-(p-q)$]{}]{} (36,60)(4,0)[7]{}[(2,2)\[t\]]{} (38,60)(4,0)[6]{}[(2,2)\[b\]]{} (62,60)[(2,2)\[bl\]]{} (62,59)[(1,0)[2]{}]{} (65,60)[(0,0)\[l\][$Z(q)$]{}]{} (36,40)(4,0)[7]{}[(2,2)\[t\]]{} (38,40)(4,0)[6]{}[(2,2)\[b\]]{} (62,40)[(2,2)\[bl\]]{} (62,39)[(1,0)[2]{}]{} (65,40)[(0,0)\[l\][$\gamma(k)$]{}]{}
(70,30)(0,35) (5,60)[(1,0)[30]{}]{} (15,60)[(1,0)[1]{}]{} (15,62)[(0,0)\[b\][$e^-(p)$]{}]{} (5,40)[(1,0)[30]{}]{} (15,40)[(1,0)[1]{}]{} (15,38)[(0,0)\[t\][$e^+(p')$]{}]{} (35,40)[(0,1)[20]{}]{} (35,50)[(0,-1)[1]{}]{} (37,50)[(0,0)\[l\][$e^-(p-k)$]{}]{} (36,60)(4,0)[7]{}[(2,2)\[t\]]{} (38,60)(4,0)[6]{}[(2,2)\[b\]]{} (62,60)[(2,2)\[bl\]]{} (62,59)[(1,0)[2]{}]{} (65,60)[(0,0)\[l\][$\gamma(k)$]{}]{} (36,40)(4,0)[7]{}[(2,2)\[t\]]{} (38,40)(4,0)[6]{}[(2,2)\[b\]]{} (62,40)[(2,2)\[bl\]]{} (62,39)[(1,0)[2]{}]{} (65,40)[(0,0)\[l\][$Z(q)$]{}]{}
The skeptic in us may wonder whether our proof is valid for processes where $Z$ or $\pi^0$ couples to internal fermion lines, given that we have used the on-shell condition for the spinors, Eq. (\[spinor\]), to derive Eq. (\[finalcoup\]).[^2] We put to rest such doubts by explicitly calculating the process $e^+e^-\to Z\gamma$, which is the process calculated by Donoghue and Tandean [@DoTa93]. However, we note that the triangle diagrams considered by them are not the lowest order diagrams for this process. There are tree diagrams, given in Fig. \[f:Zgam\], which contribute. The amplitude for this diagram, ${\cal A}_Z$, can be written as $$\begin{aligned}
i{\cal A}_Z = ie \varepsilon^\mu (q) \epsilon^\nu(k) {\bf \overline v
(p')} \Gamma_{\mu\nu} {\bf u (p)} \,,
\label{AZ}
\end{aligned}$$ where $\varepsilon$ and $\epsilon$ represent the polarization vectors of the $Z$ and the photon respectively, and $$\begin{aligned}
\Gamma_{\mu\nu} = \gamma_\nu {\rlap/p - \rlap/q + m_e \over (p-q)^2 -
m_e^2} \gamma_\mu \left( {\cal G}_e + {\cal G}'_e \gamma_5 \right) +
\gamma_\mu \left( {\cal G}_e + {\cal G}'_e \gamma_5 \right)
{\rlap/p - \rlap/k + m_e \over (p-k)^2 - m_e^2} \gamma_\nu \,.
\end{aligned}$$
The diagrams for $e^+e^-\to \pi^0\gamma$, on the other hand, are obtained if the $Z$-boson lines of Fig. \[f:Zgam\] couple to the pion wavefunction in the manner shown in Fig. \[f:coupling\]. The amplitude of such diagrams is given by $$\begin{aligned}
i{\cal A}_\pi &=& ie \epsilon^\nu(k) {\bf \overline v
(p')} \Gamma_{\mu\nu} {\bf u (p)} \cdot
\left[ -i D^{\mu\rho} (q) \right] \cdot
i\left< 0 \left| J_\rho^{(Z)} \right| \pi^0 (q) \right> \nonumber\\
&=& ie \epsilon^\nu(k) {\bf \overline v
(p')} \Gamma_{\mu\nu} {\bf u (p)} \, {q^\mu \over M_Z} \,,
\label{Api}
\end{aligned}$$ using Eq. (\[DJ\]) in the last step. In general, the amplitudes in Eqs. (\[AZ\]) and (\[Api\]) are not equal, even in magnitude. However, if we consider a longitudinal polarized $Z$ boson in Eq.(\[AZ\]) whose 4-momentum is given by $(E,\kappa\hat n)$ for some unit 3-vector $\hat n$, the polarization vector will be given by $\varepsilon^\mu_{\rm long} (q) \equiv (\kappa,E\hat n)/M_Z$. In the limit $M_Z\to 0$ or equivalently $E/ M_Z\to \infty$, since $\kappa\approx
E$, we obtain $\varepsilon^\mu_{\rm long} (q) = q^\mu/M_Z$. Thus, in this limit, the amplitudes for $e^+e^-\to Z_{\rm long}\gamma$ and $e^+e^-\to \pi^0\gamma$ are indeed equal, as seen from Eqs.(\[AZ\]) and (\[Api\]). This is the verification of the equivalence theorem to this order.
It is now easy to see that the proof can be extended to any process, e.g., with any number of $Z$-bosons in the initial and final states. If we have a diagram with an external $Z_{\rm
long}$ line, we obtain a factor $\varepsilon^\mu_{\rm long} (q)$ in the amplitude. On the other hand, if we let the $Z$-boson to couple to the $\pi^0$ wavefunction, we will obtain the factors $\left[ -i D^{\mu\rho} (q) \right] \cdot
i\left< 0 \left| J_\rho^{(Z)} \right| \pi^0 (q) \right>$, coming from the $Z$ propagator and the matrix element involving the pion wavefunction. However, as shown in Eq. (\[DJ\]), this equals $q^\mu/M_Z$, which is $\varepsilon^\mu_{\rm long} (q)$ in the limit $M_Z\to 0$. Thus, equivalence theorem is valid for any process.
(16,10)(-2,47) (1,50)(4,0)[3]{}[(2,2)\[t\]]{} (3,50)(4,0)[3]{}[(2,2)\[b\]]{} (6,53)[(0,0)\[b\][$Z$]{}]{}
$\null + \null$
(42,10)(-2,47) (1,50)(4,0)[3]{}[(2,2)\[t\]]{} (3,50)(4,0)[3]{}[(2,2)\[b\]]{} (6,53)[(0,0)\[b\][$Z$]{}]{} (12,50)(2.5,0)[6]{}[(1,0)[1.5]{}]{} (19,53)[(0,0)\[b\][$\Pi^0$]{}]{} (27,50)(4,0)[3]{}[(2,2)\[t\]]{} (29,50)(4,0)[3]{}[(2,2)\[b\]]{} (32,53)[(0,0)\[b\][$Z$]{}]{}
$\null + \null$
(68,10)(-2,47) (1,50)(4,0)[3]{}[(2,2)\[t\]]{} (3,50)(4,0)[3]{}[(2,2)\[b\]]{} (6,53)[(0,0)\[b\][$Z$]{}]{} (12,50)(2.5,0)[6]{}[(1,0)[1.5]{}]{} (19,53)[(0,0)\[b\][$\Pi^0$]{}]{} (27,50)(4,0)[3]{}[(2,2)\[t\]]{} (29,50)(4,0)[3]{}[(2,2)\[b\]]{} (32,53)[(0,0)\[b\][$Z$]{}]{} (38,50)(2.5,0)[6]{}[(1,0)[1.5]{}]{} (45,53)[(0,0)\[b\][$\Pi^0$]{}]{} (53,50)(4,0)[3]{}[(2,2)\[t\]]{} (55,50)(4,0)[3]{}[(2,2)\[b\]]{} (58,53)[(0,0)\[b\][$Z$]{}]{}
$\null + \cdots$
And in fact, it is also easy to see that the proof can be easily extended to any model of dynamical symmetry breaking. In general, let us denote the spin-0 state eaten up by the $Z$-boson by $\Pi^0$. Since the $Z$ mass is generated by the series of diagrams given in Fig.\[f:Zpole\], it is easy to see that one requires $$\begin{aligned}
\left< 0 \left| J_\nu^{(Z)} \right| \Pi^0 (q)
\right> = - M_Z q_\nu \,,
\end{aligned}$$ no matter how $M_Z$ is related to the parameters of the unbroken theory.[^3] This is all one needs to verify Eq. (\[DJ\]), and thereby the equivalence theorem.
Earlier, we said that the equivalence theorem is the statement of continuity of physical observables in the limit $M_V\to 0$. Since this limit can be realized as $g\to 0$, it is obvious that one expects the equivalence at only the lowest non-trivial order in the gauge coupling constant [@BaSc90], to all orders in other couplings in the model. And we have already proved the theorem to this order for the process $e^+e^- \to
Z_{\rm long}\gamma$. The diagram discussed by Donoghue and Tandean [@DoTa93] is higher order in gauge coupling constant and hence is not relevant for the validity of the equivalence theorem for the process $e^+e^-\to Z\gamma$.
One can of course consider some other process for which tree diagrams do not exist [@prcom]. Take, for example, the process $\nu\bar\nu \to Z_{\rm long}\gamma$. Here, even the non-triangle diagrams are fourth order in gauge coupling constants, and so is the triangle-mediated diagram shown in Fig.\[f:triangle\]. To examine the validity of the equivalence theorem for this process, we will have to get into a detailed analysis of the triangle part of the diagram. This has been done in earlier papers [@DPO90; @PhPh90; @Hik90] in a different context. We mainly follow the notation and analysis of Hikasa [@Hik90] in what follows.
(85,36)(-5,32) (15,50)[(-1,1)[15]{}]{} (7,58)[(1,-1)[2]{}]{} (15,50)[(-1,-1)[15]{}]{} (7,42)[(1,1)[2]{}]{} (16,50)(4,0)[7]{}[(2,2)\[t\]]{} (18,50)(4,0)[7]{}[(2,2)\[b\]]{} (30,48)[(0,0)[(-1,0)[5]{}]{}]{} (30,52)[(0,0)\[b\][$Z^\beta(K)$]{}]{} (43,50)[(1,1)[12]{}]{} (43,50)[(1,-1)[12]{}]{} (55,38)[(0,1)[24]{}]{} (55,50)[(0,0)[(0,1)[5]{}]{}]{} (55,62)(3,0)[10]{}[(1,0)[2]{}]{} (70,60)[(0,0)\[t\][$\pi^0(q)$]{}]{} (83,62)[(1,0)[3]{}]{} (56,38)(4,0)[7]{}[(2,2)\[t\]]{} (58,38)(4,0)[7]{}[(2,2)\[b\]]{} (74,42)[(0,0)[(1,0)[5]{}]{}]{} (74,42)[(0,0)\[b\][$\gamma^\alpha(k)$]{}]{}
The triangle part gives a $Z\gamma\pi$ effective coupling. Since only the vector part of the $Z$ coupling matters, we can consider it as a $\gamma^*\gamma\pi$ coupling apart from some irrelevant differences in the coupling constants, $\gamma^*$ being a photon which is not necessarily on shell. The matrix element for $\pi^0\to \gamma^*\gamma$ transition is given by $$\begin{aligned}
\left< \left. \gamma^* (K) \gamma (k) \right| \pi^0 (q) \right> &=&
\lim_{q^2 \to m_\pi^2} (m_\pi^2 - q^2)
\left< \gamma^* (K) \gamma (k) \left| \phi_\pi \right| 0 \right> \,.
\label{lsz}
\end{aligned}$$ where $\phi_\pi$ is the interpolating field for the pion. We now employ the PCAC relation: $$\begin{aligned}
\partial_\mu J_5^\mu = f_\pi m_\pi^2 \phi_\pi + {e^2 \over 16\pi^2}
F_{\mu\nu} \tilde F^{\mu\nu} \,,
\label{pcac}
\end{aligned}$$ where $J_5^\mu$ is the axial vector current of the quarks, and $F_{\mu\nu}$ is the electromagnetic field strength tensor. Using this, we can rewrite the matrix element of Eq. (\[lsz\]) as $$\begin{aligned}
\left< \gamma^* (K) \gamma (k) | \pi^0 (q) \right> &=&
{m_\pi^2 - q^2 \over f_\pi m_\pi^2}
\left\{ -e^2 \left< 0 \left| {\cal T} (\partial^\mu J_{5\mu} J_\alpha J_\beta)
\right| 0 \right> - {e^2\over 4\pi^2} [kK]_{\alpha\beta} \right\}
\nonumber\\
&=& e^2 \, {m_\pi^2 - q^2 \over f_\pi m_\pi^2} \,
\left\{ q^\mu T_{\mu\alpha\beta} -
{1\over 4\pi^2} [kK]_{\alpha\beta} \right\} \,, \label{pigg}
\end{aligned}$$ where $T_{\mu\alpha\beta}$ is the 3-point function with the axial current and two vector currents, and $$\begin{aligned}
[kK]_{\alpha\beta}
&\equiv& \varepsilon_{\alpha\beta\lambda\rho}
k^\lambda K^\rho \,.
\end{aligned}$$ Now, from the requirements of vector current conservation and Lorentz invariance, one can write down the most general form for $T_{\mu\alpha\beta}$ as follows: $$\begin{aligned}
T_{\mu\alpha\beta} &=& \phantom{+}
\left\{ k^2 \varepsilon_{\mu\alpha\beta\rho}
K^\rho + k_\alpha [kK]_{\mu\beta} \right\} F_1 \nonumber\\
&&+ \left\{ K^2 \varepsilon_{\mu\alpha\beta\rho}
k^\rho + K_\beta [kK]_{\mu\alpha} \right\} F_2 \nonumber\\
&&+ (k+K)_\mu [kK]_{\alpha\beta} F_3 + (k-K)_\mu [kK]_{\alpha\beta}
F_4\,,
\end{aligned}$$ where $F_i\; (i=1\cdots4)$ are form factors[^4]. Thus, $$\begin{aligned}
q^\mu T_{\mu\alpha\beta} &=&
\left\{ k^2 F_1 - K^2 F_2 - q^2 F_3 + (k^2-K^2) F_4 \right\}
[kK]_{\alpha\beta} \,.
\label{qT}
\end{aligned}$$ We are obviously interested in the case $k^2=0$ since one photon is on shell, and $q^2=0$ since the pion is a Goldstone boson in this model. Thus, only the form factors $F_2$ and $F_4$ are relevant, and from the general formulas given by Hikasa [@Hik90], we obtain $$\begin{aligned}
F_2 &=& {1\over 2\pi^2} \int_0^1 dz \int_0^{1-z} dz'
{zz' \over m^2 - zz' K^2} \,,
\label{F2} \\
F_4 &=& 0 \,,
\end{aligned}$$ where $m$ is the mass of the fermions in the loop. Now, the fermions in the loop are obviously the $u$ and the $d$ quarks. Since the pion mass is zero, and $m_\pi^2 \propto m_u+m_d$, the up and the down quarks must be massless in this model. Thus, putting $m=0$ in Eq. (\[F2\]), we obtain $F_2=-(4\pi^2K^2)^{-1}$, so that from Eqs. (\[qT\]) and (\[pigg\]), we find that the amplitude of the triangular loop actually vanishes in this case.
Two comments should be made here. First, the vanishing of this amplitude has nothing to do with the decay $\pi^0\to \gamma\gamma$ in the real world where the pion has a small mass [@DPO90; @PhPh90; @Hik90]. For that case, in the soft pion limit one is interested in the limit $k^2=K^2=0$ and $q^2$ small, so that the form factor $F_3$ is important in that case. Moreover, since $F_3=(48\pi^2m^2)^{-1}$, Eq. (\[qT\]) shows that the term $q^\mu
T_{\mu\alpha\beta}$ vanishes in Eq. (\[pigg\]) for $q^2=0$, which is the famous Sutherland-Veltman theorem [@Sut67; @Vel67]. Second, in the present limit this implies that the contribution of the diagram in Fig. \[f:triangle\] is actually zero no matter which fermion-antifermion pair is considered at the outer lines. This is in complete agreement with the equivalence theorem. The result can be, and has been [@DPO90], obtained by using a $\sigma$-model which incorporates the anomalous contributions in a straightforward way.
We can summarize as follows. In models exhibiting dynamical symmetry breaking, verification of the equivalence theorem may not be obvious, but is nevertheless possible. And we believe that the equivalence theorem is always valid because it is based only on the requirement that physical observables are continuous in the values of certain parameters of the theory.
#### Note added : {#note-added .unnumbered}
After the paper was submitted for publication, I was made aware of a paper by Zhang [@Zha89] which also addresses the issue of the Equivalence theorem in models where symmetry is broken by fermion condensates. The conclusions of that paper is similar to the present paper.
#### Acknowledgements : {#acknowledgements .unnumbered}
The work was supported by the Department of Energy, USA. I am indebted to D. A. Dicus for long discussions and to J. F. Donoghue for constructive criticism. Conversations with D. Bowser-Chao, A. El-Khadra, J. Gunion, F. Olness, S. Weinberg and S. Willenbrock are also gratefully acknowledged.
[WW]{}
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[^1]: The up-quark field $u$ (in italics) is not to be confused with the positive energy spinor $\bf u$ (in boldface).
[^2]: In fact, one can ask the same question about Higgs models of symmetry breaking since Eq.(\[spinor\]) is used to derive Eq. (\[Scoupling\]) as well.
[^3]: Actually, the right side can have an arbitrary phase, which will appear as an overall phase of all couplings of $\Pi^0$. But this phase does not affect any physics, including the equivalence theorem.
[^4]: Note that $F_4=0$ from Bose symmetry. But we do not impose Bose symmetry at this level, since we want to use our results in the case where the two vector particles are the photon and the $Z$
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'In this work, we consider the problem of placing replicas in a data center or storage area network, represented as a digraph, so as to lexico-minimize a previously proposed reliability measure which minimizes the impact of all failure events in the model in decreasing order of severity. Prior work focuses on the special case in which the digraph is an arborescence. In this work, we consider the broader class of multitrees: digraphs in which the subgraph induced by vertices reachable from a fixed node forms a tree. We parameterize multitrees by their number of “roots” (nodes with in-degree zero), and rule out membership in the class of fixed-parameter tractable problems (FPT) by showing that finding optimal replica placements in multitrees with 3 roots is NP-hard. On the positive side, we show that the problem of finding optimal replica placements in the class of *untangled* multitrees is FPT, as parameterized by the replication factor $\rho$ and the number of roots $k$. Our approach combines dynamic programming (DP) with a novel tree decomposition to find an optimal placement of $\rho$ replicas on the leaves of a multitree with $n$ nodes and $k$ roots in $O(n^2\rho^{2k+3})$ time.'
author:
- 'K. Alex Mills'
- 'R. Chandrasekaran'
- Neeraj Mittal
- Firstname Lastname
- Firstname Lastname
bibliography:
- 'cocoa2017.bib'
title: 'Lexico-minimum Replica Placement in Multitrees'
---
Reliable replica placement, discrete lexicographic optimization, multitrees, tree decomposition, dynamic programming
Introduction
============
As data centers become larger, ensuring reliable access to the data they store becomes a greater concern. Each piece of hardware introduces a new point of failure – the more hardware, the more likely it is that failure will occur. Moreover, to keep large-scale data centers cost-effective, they are typically built using commodity hardware, further increasing the likelihood of a failure event. Ensuring the availability and responsiveness of data center operations in such environments has been a subject of recent interest.
Many availability problems are solved through the use of replication: placing identical copies of data or tasks across multiple machines to ensure the survival of one replica in case of failure. While this approach has been known for decades, researchers have recently begun to cast the specific problem of replica placement as an optimization problem in which the dependencies among failure events are modeled [@Korupolu2016; @Mills2015]. To date, these approaches have relied on the simplifying assumption that the failure event model is hierarchically arranged. While such models coincide with some real-world systems [@Parallels; @VMWare], providing optimal replica placements for more general models remains an interesting problem.
Of special interest is the measurement used to score the reliability of a placement. One standard approach involves assigning to each failure event its likelihood of occurrence. But this approach is subject to the following critiques. First, measurements or estimations of failure probability may themselves be unreliable – thereby providing an unreliable basis for optimization. Second, even a perfect measurement of failure based on historical behavior cannot account for a failure pattern which has never occurred before, and therefore could not have been measured. In other words: “past performance is not an indicator of future results”.
In light of these concerns, we have proposed in [@Mills2015] a multi-criteria reliability measure which minimizes the impact of failure events in the aggregate. Specifically, we introduce a reliability metric which places failure events into buckets based on their *impact* – the number of replicas which they cause to become unavailable. We then minimize the number of events in each bucket in decreasing order of impact. As a result, the placements we obtain achieve the minimum number of events which cause all replicas to fail (i.e. the number of events with maximum impact). Subject to this quantity being minimized, we then minimize the number of events which cause all but one replica to fail, followed by minimizing the number of events that cause all but two replicas to fail, and so on.
This goal is achieved by minimizing a vector quantity called the *failure aggregate* in the lexicographic order. Our past work investigates minimizing failure aggregates of replicas placed on the leaves of a tree. For this problem an $O(n + \rho \log \rho)$ algorithm can be achieved, where $n$ is the number of nodes in the tree, and $\rho$ is the number of replicas to be placed [@Mills2015]. We have also investigated simultaneously minimizing *multiple* placements on the leaves of a tree [@Mills2017]. Our current solution to this problem runs in polynomial time when the *skew* is constant. The skew is defined as the maximum absolute difference in number of replicas placed among all pairs of placements. For a skew of $\delta$, we present an algorithm to place $m$ groups of replicas on the leaves of a tree with $n$ nodes in $\tilde{O}(n\rho^3\delta^3m^\Delta / \Delta!)$ time where $\rho$ is the maximum number of replicas placed among all $m$ groups, and $\Delta = O(\delta^2)$ [@Mills2017].
While some commercially available storage area networks use failure domains modeled by trees [@Parallels; @VMWare], extensions to more general failure domain models are an important research goal. In this work, we initiate the parameterized study of the problem of lexico-minimum replica placement in multitrees, as parameterized by the number of its roots. A *multitree* is defined as a directed acyclic graph (DAG) in which, for any fixed vertex $v$, the set of vertices reachable from $v$ forms a tree as an induced subgraph. The *roots* and *leaves* of a multitree are defined as nodes with in-degree zero and out-degree zero respectively. We emphasize the parameter by referring to a multitree with $k$ roots as a $k$-multitree. Our goal is to place $\rho$ replicas on the leaves of a $k$-multitree so that the failure aggregate is minimized in the lexicographic order.
We show that lexico-minimum replica placement is NP-hard even in $3$-multitrees, ruling out fixed-parameter tractability for this parameterization. The proof we present relies on the Four Color Theorem [@Appel1977] to exploit a disparity in hardness of two well-known problems restricted to cubic planar bridgeless graphs. In such graphs, finding a 3-edge-coloring can be done in polynomial time, while solving <span style="font-variant:small-caps;">independent set</span> remains NP-hard. To circumvent this hardness result, we define *untangled* multitrees, a class of multitrees for which we exhibit membership in FPT. We develop a FPT algorithm based on the tree decomposition approach. Since multitrees are a special case of directed acyclic graphs, standard decomposition approaches such as treewidth [@Bodlaender2008], pathwidth [@Andreica2008], and DAG-width [@Berwanger2012] do not apply. Instead, we provide a novel decomposition technique tailored to our problem.
Our algorithm works in two successive phases, a *decomposition phase* and an *optimization phase*. The decomposition phase produces a specialized *decomposition tree*, a full[^1] binary tree in which each node is associated with an induced subgraph of the input multitree. The optimization phase then runs a bottom-up dynamic programming algorithm over the nodes of the decomposition tree. While the overall process is similar to FPT algorithms for graphs with restricted treewidth, our decomposition technique and application are both novel. Our algorithm for untangled $k$-multitrees runs in $O(n^2\rho^{2k+3})$ time, thus demonstrating that lexico-minimum replica placement on untangled $k$-multitrees is in FPT, as parameterized by $\rho$ *and* $k$.
Modeling Reliable Replica Placement in Multitrees
=================================================
In this section we formalize the model presented in the introduction. We model the failure domains of a data center as a *multitree*, a directed acyclic graph (DAG) whose formal definition we defer to the next paragraph. Non-leaf vertices represent failure events which are typically associated with the failure of a physical hardware component, but may instead be associated with abstract events such as network maintenance or software failures. Leaf vertices represent servers on which replicas of data may be placed. A directed edge between two failure events $u$ and $v$ indicates that the failure of event $u$ may trigger failure event $v$.
A *multitree* is a directed acyclic graph (DAG) in which the set of vertices reachable from any vertex forms an arboresence (see Fig. 1(a)). In the context of graph $G$, let $u \rightsquigarrow v$ denote the assertion “there is a path from $u$ to $v$ in $G$”, and $u \rightarrow v$ denote the assertion “there is an edge from $u$ to $v$ in G”. Then a multitree is equivalently defined as a *diamond-free* DAG [@Furnas1994]. See Fig. 1(b) for a depiction of the forbidden subgraphs used to define diamond-free DAGs below.
A *multitree* $M = (V,E)$ is a DAG in which there are no diamonds (i.e. a DAG which is *diamond-free*). A *diamond* is either
a set of three vertices $a,b,c \in V$ in which $a \rightarrow b \rightsquigarrow c$, and, even when the edge $(a,b)$ is removed, $a \rightsquigarrow c$, or
a set of four vertices $a,b,c,d \in V$ in which $a \rightsquigarrow b \rightsquigarrow d$ and $a \rightsquigarrow c \rightsquigarrow d$, while there is no path from $b$ to $c$ or vice versa.
[0.38]{} ![() A multitree in which red highlights depict an induced subgraph forming an arboresence, () Forbidden subgraphs in which squiggles depict an arbitrary path.[]{data-label="f:multitree-fig"}](Butterfly_multitree "fig:") \[f:multitree-subfig\]
[0.5]{}
\[f:diamonds-forbidden-subgraphs\]
A $k$-multitree is a multitree with $k$ roots. In context of a multitree $M = (V,E)$ we denote the set of leaves of $M$ by $L \subseteq V$. In context of our problem we seek a subset of leaves on which to place replicas of data. To this end, we define a *placement* of $\rho$ replicas as a subset[^2] of leaves $P \subseteq L$ with size $|P| = \rho$.
Given a placement $P$, we associate to each failure event its *failure number*: the number of replicas from $P$ which can be made unavailable should the event occur. The failure number of $u$ is equal to the number of nodes in $P$ which are reachable from $u$, which we denote as $f(u,P) := |\{x \in P : u \rightsquigarrow x \}|$.
To aggregate the failure numbers across all failure events into a single vector-valued quantity, we denote the *failure aggregate* by ${\boldsymbol{f}}(P) = \langle p_0, p_1, ..., p_{\rho}\rangle$, where $p_i = |\{u \in V : f(u,P) = \rho - i\}|$. Intuitively, the $i^{th}$ entry of ${\boldsymbol{f}}(P)$ contains the number of events whose failure leaves $i$ replicas surviving.
Our optimization goal is to minimize the failure aggregate in the lexicographic order, which was motivated in the introduction. The (strict) lexicographic order $<_L$ between vectors ${\boldsymbol{x}} = \langle x_0,...,x_n \rangle$ and ${\boldsymbol{y}} = \langle y_0,...,y_n \rangle$ is defined via the formula $$x <_L y \iff \exists j \in [0, n] : (x_j < y_j \wedge \forall i < j [x_i = y_i]),$$ while the weak lexicographic order $\leq_L$ is defined by extending $<_L$ in the usual way. We use the short-hand “lexico-minimum” and “lexico-minimizes” to mean “minimum” and “minimizes” in the lexicographic order respectively.
With these definitions in hand, we provide the formal definition of the parameterized optimization problem we consider in the remainder of this paper.
[problemtitle[<span style="font-variant:small-caps;">Lexico-minimum Single-block Placement in $k$-Multitrees</span> ($k$-LSP)]{}]{} [probleminput[A $k$-multitree, $M=(V,E)$; the set of leaves $L \subseteq V$; a positive integer $\rho < |L|$]{}]{} [problemoutput[A placement $P \subseteq L$ with $|P| = \rho$ such that ${\boldsymbol{f}}(P)$ is lexico-minimum among all placements $P \subseteq L$ with $|P| = \rho$.]{}]{}
NP-hardness of 3-LSP
====================
In this section, we concern ourselves with how the hardness of $k$-LSP depends on the parameter $k$. Prior work has shown that $1$-LSP can be solved in polynomial time [@Mills2015], since a $1$-multitree is just an arboresence. In this section we show that $3$-LSP is NP-hard, thereby ruling out a fixed-parameter tractable algorithm parameterized by the number of roots.
Specifically, we show hardness of the following decision problem.
[problemtitle[<span style="font-variant:small-caps;">Lexicographic Replica Placement in $3$-multitrees</span> ($3$-LSP)]{}]{} [probleminput[A $3$-multitree, $M=(V,E)$ with leaves $L \subseteq V$; a positive integer $\rho$; and a vector ${\boldsymbol{w}} \in \mathbb{N}^{\rho+1}$]{}]{} [problemquestion[Is there a placement $P \subseteq L$ with $|P| = \rho$ such that ${\boldsymbol{f}}(P) \leq_L {\boldsymbol{w}}$?]{}]{}
We will prove that this problem is NP-hard by reduction from <span style="font-variant:small-caps;">Independent Set</span> restricted to cubic planar bridgeless graphs. Cubic planar bridgeless graphs are guaranteed to have a 3-edge-coloring [@Goemans2012]. Moreover, 3-coloring the edges of such graphs is equivalent to 4-coloring their faces [@Tait1880b]. The faces of such graphs correspond to the vertices of a planar graph, and, as a consequence of the Four Color Theorem, finding a 4-vertex-coloring of a planar graph may be done in $O(n^2)$ time [@Cole2008]. On the other hand, finding an independent set in such graphs is NP-hard, as was shown in [@Mohar2001]. We exploit the disparity in the hardness of these two problems to show that $3$-LSP is NP-hard, by reduction from the following problem.
[problemtitle[<span style="font-variant:small-caps;">Restricted Independent Set</span> (RIS)]{}]{} [probleminput[An undirected cubic planar bridgeless graph $G = (V,E)$; a positive integer $k$.]{}]{} [problemquestion[Does $G$ admit an independent set of size exactly $k$?]{}]{}
\[t:3-multitree-np-hard\] RIS reduces to $3$-LSP in polynomial time. Thus, $3$-LSP is NP-hard.
Given a cubic planar bridgeless graph $G=(V,E)$, we can form a $3$-multitree, $H$, as follows. Let $H = (V',E')$. Add a vertex to $H$ for every edge in $E$ and for every vertex in $V$. Let the vertices of $H$ that represent vertices of $G$ be denoted by $H(V)$ and let the vertices of $H$ that represent edges of $G$ be denoted by $H(E)$. Next, for every edge $e = (u,v)$ of $G$, add directed edges $(e,u)$ and $(e,v)$ to $H$. Next, we partition $H(E)$ into three sets, $S_1, S_2, S_3$, such that no node in $H(V)$ has two neighbors in the same set. This partition corresponds to finding a $3$-edge-coloring of $G$, which may be done in $O(n^2)$ time [@Cole2008]. We then add three special nodes $\alpha, \beta$ and $\gamma$ to $H$, and add edges $(\alpha,s_1), (\beta,s_2), (\gamma,s_3)$ for all $s_1 \in S_1, s_2 \in S_2 $ and $s_3 \in S_3$.
[0.38]{}
[0.5]{}
We claim that $H$ is a 3-multitree. $H$ clearly has only three nodes with in-degree zero, so it suffices to show that no diamond is formed. Three-node diamonds are clearly impossible by construction. Instead suppose that there are vertices $a,b,c,d$ of $H$ which form a four-node diamond (i.e., $(a,b)(a,c)(b,c)(c,d) \in E'$). By construction, $d$ must be a node in $H(V)$, thus $b$ and $c$ must be nodes in $H(E)$, and $a = \chi$ for some $\chi \in \{\alpha,\beta,\gamma\}$, all of which follows from our construction. But then $d$ is a vertex in $H(V)$ which has two of its neighbors connected to the same root node $\chi$, a contradiction. Hence no diamond is created and $H$ is a $3$-multitree.
Since each node in $G$ must be adjacent to an edge from each color class, every node in $H(V)$ must have $\alpha,\beta$ and $\gamma$ as ancestors. Thus, each of $\alpha,\beta$ and $\gamma$ have failure number $\rho$ in any placement of size $\rho$ on the leaves of $H$. Finally, we complete the reduction by showing that $H$ has a placement $P\subseteq H(V)$ with $|P| = k$ for which ${\boldsymbol{f}}(P) \leq_L \langle 3,0,...,0,\infty,\infty \rangle$ if and only if $G$ has an independent set of size $k$. This portion of the proof is straight-forward, and has been moved to Appendix \[a:omitted-proofs\].
Since it shows that, $k$-LSP is NP-hard even for a *fixed* value of the parameter $k$, Theorem \[t:3-multitree-np-hard\] rules out the existence of an FPT algorithm for $k$-multitrees as parameterized by the number of roots. Thus, $k$-LSP falls no lower in the $W$-hierarchy than $W[1]$. While a polynomial time algorithm for $1$-LSP was shown in [@Mills2015], the complexity of $2$-LSP is open.
Untangling Multitrees
=====================
On the positive side, we show how a tree decomposition approach may be employed to yield an FPT algorithm for the subclass of *untangled* $k$-multitrees. We use the term *connectors* to refer to vertices of a multitree which have in-degree strictly greater than 1. An untangled multitree is a multitree with additional requirements placed on the ancestry of connectors. Roughly speaking, we require that an untangled multitree may be split into two subgraphs such that a) the descendants of each non-root node fall into the same subgraph, and b) each connector is present in only *one* of the two subgraphs. This property allows us to perform a decomposition of each multitree into two subgraphs. To make this idea precise, we employ the following modified notion of laminarity which we call a *laminar pair* of set families.
Two set families $\mathcal{F}, \mathcal{F}' \subseteq 2^X$ on the same ground set $X$ form a *laminar pair* when, for all $U \in \mathcal{F}$, $V \in \mathcal{F}'$, either $U \subseteq V, U \supseteq V$, or $U \cap V = \emptyset$.
To ensure the decomposability of a multitree $M=(V,E)$ into subgraphs $M_1$ and $M_2$, we require that for every child $c$ of each root, the set of connectors which are descendants of $c$ all lie in either $M_1$ or $M_2$. To formalize this idea, we define the *connector shadow* as follows.
Given a vertex $u \in V$, the *connector shadow of $u$*, denoted $Sh(u)$, is the set of connectors of $M$ which are descendants of $u$.
Given a vertex $u \in V$, with children $c_1,...,c_m$, the *child shadows* of $u$ is the set family defined as $\mathcal{C}(u) := \{Sh(c_1), ..., Sh(c_m)\}$.
Multitree $M = (V,E)$ is said to be *untangled* if, for every pair of vertices $u,v \in V$ where $u$ is not reachable from $v$ and vice versa, $\mathcal{C}(u)$ and $\mathcal{C}(v)$ are laminar pairs.
Being untangled is easily seen to be a hereditary graph property[^3].
While the class of untangled multitrees may appear to be highly specialized, it is in fact general enough to capture any directed acyclic graph. Any directed acyclic graph $G = (V,E)$ with leaves $L$ can be converted to a *canonical placement model*, $H = (V,E')$, where $$E' = \{(u,v) : u \in V \setminus L, v \in L, \text{ and } v \text{ is reachable from $u$ in $G$.}\}.$$ See for an example. By definition, the canonical placement model $H$ has the same reachability relation as the original graph $G$. This further implies that the failure numbers of placements on the leaves of $H$ have the same failure aggregate as their counterparts in $G$. Thus, a lexico-minimum placement in $H$ is also lexico-minimum in $G$. Furthermore, $H$ is easily seen to be a multitree, but also an *untangled* multitree, since the set of child shadows for any vertex in $H$ is a family only containing singleton sets, and any pair of families of singleton sets trivially forms a laminar pair.
[0.45]{}
$\implies$
[0.45]{}
Decomposing $k$-multitrees
==========================
As previously discussed, our algorithm runs in two sequential phases: a decomposition phase and an optimization phase. The decomposition phase of our algorithm takes as input a (weakly-connected) untangled $k$-multitree $M =(V,E)$ and produces as output a *decomposition tree*. A decomposition tree is a full binary tree in which each node $u$ is associated with a subset of vertices of $M$ we call a *subproblem*, denoted by $\Gamma_u \subseteq V$.
\[d:subproblem-tree\] A *decomposition tree* $\tau$ is a binary tree in which each node $u$ is associated with a subproblem $\Gamma_u \subseteq V$.
\[d:trivial\] A subproblem $\Gamma_u$ is said to be *trivial* if $\Gamma_u$ contains no leaf nodes.
To ensure that our decomposition preserves optimal substructure, we define the notion of an *admissible* subproblem. In every decomposition tree produced by our procedure, internal nodes are associated with admissible subproblems.
\[d:child-descendant-complete\] A subproblem $\Gamma_u \subseteq V$ is *child-descendant complete* if, for each node $v$ which is a child of a root of $M[\Gamma_u]$, each descendant of $v$ is present in $\Gamma_u$.
\[d:connector-complete\] A subproblem $\Gamma_u \subseteq V$ of multitree $M = (V,E)$ is *connector complete* if, for every connector $c \in V$, if one parent of $v$ is contained in $\Gamma_u$, then *all* parents of $v$ are contained in $\Gamma_u$. Formally, if any node $v \in \Gamma_u$ is connected to $c$ by an edge $(v,c)$, then for every node $v \in V$ such that $(v,c) \in E$, $v$ is also in $\Gamma_u$.
\[d:admissible\] A subproblem $\Gamma_u \subseteq V$ is *admissible* if it is both connector complete and child-descendant complete.
[0.3]{}
\[f:non-admissible-1\]
[0.3]{}
\[f:non-admissible-2\]
[0.3]{}
\[f:admissible-example\]
Examples of admissible and non-admissible subproblems are shown in . Notice that, according to Definition \[d:admissible\], $V$ forms an admissible subproblem. This “sub”-problem forms the root of the decomposition tree we will construct. Our decomposition procedure decomposes each admissible subproblem into two subproblems each of which is either 1) trivial, 2) base, or 3) admissible. The decomposition is continued on admissible subproblems, while trivial and base subproblems form the leaves of the decomposition tree we will construct.
\[d:k-multitree-base-subproblem\] A subproblem $\Gamma_u \subseteq V$ is said to be *base* if $M[\Gamma_u$\] forms either a $j$-multitree where $j < k$, or a trivial graph[^4] on $k$ nodes.
Base subproblems which form $j$-multitrees for $j > 1$ are decomposed inductively by a decomposition procedure for $j$-multitrees. Base subproblems which are $1$-multitrees are not decomposed any further. In the optimization phase, base subproblems which are $1$-multitree subproblems will be solved via the algorithm for LSP in trees presented in [@Mills2015].
Each subproblem $\Gamma_u$ is associated with a set of *local roots*, which are roots of the subgraph induced by $M[\Gamma_u]$. Let $R(\Gamma_u)$ be the set of local roots of $\Gamma_u$. Our decomposition procedure works by applying one of five cases based on the structure of the local roots and their adjacent nodes. Given a non-base, non-trivial admissible subproblem, $\Gamma_u$, the decomposition procedure uses the following recursive cases to construct a decomposition tree $\tau$.
- (UP): If some local root $r \in R(\Gamma_u)$ has a single child which is not a connector, we can remove $r$ from $R(\Gamma_u)$ to form an admissible subproblem,[^5] while $\{r\}$ forms a trivial subproblem.
- (OUT): If some local root $r \in R(\Gamma_u)$ has a child $c$ which has no connectors as descendants, removing $c$ and all of its descendants from $\Gamma_u$ forms an admissible subproblem.[@xdefthefnmark[\[ft:child-desc\]]{}footnotemark]{} Moreover, the set containing node $c$ along with its descendants forms a base subproblem.[@xdefthefnmark[\[ft:child-desc\]]{}footnotemark]{}
- (INCLUDE): If local roots in set $Q \subseteq R(\Gamma_u)$ each share a child $c$, which is the only child of each root in $Q$ and, moreover, every parent of $c$ is contained in $Q$, then we can remove the set of local roots $Q$ to form an admissible subproblem[@xdefthefnmark[\[ft:child-desc\]]{}footnotemark]{} $\Gamma_u \setminus Q$, while $Q$ forms a trivial subproblem.
- (MERGE): If every local root has one or more children and at least one local root has at least two children, then we shall show how to partition the children of each local root node along with their descendants to form two admissible subproblems $\Gamma'$ and $\Gamma''$.
To each admissible subproblem we attempt to apply each of the above cases in the order given. Only when one case does not apply are the following cases checked. The UP, OUT, and INCLUDE cases are each used to peel off the “easy” portions of the subproblem. The MERGE case is the workhorse of the decomposition, and requires additional discussion.
To partition the children of local roots in the MERGE case, we find maximal connected components in a certain hypergraph. Algorithms for finding maximal connected components in a (directed[^6]) hypergraph in $O(\alpha(N)N)$ time are known [@Allamigeon2014], where $N$ is the size of the description of the hypergraph, and $\alpha(N)$ is the inverse Ackermann function. We will therefore constrain ourselves to discussing the hypergraph and its connection to the decomposition procedure.
In order to preserve admissibility in the MERGE case, we require that each connector from $\Gamma_u$ lie in $\Gamma'$ or $\Gamma''$ and not both. To ensure this, we form a hypergraph $H$ which has as vertices the connectors present in $\Gamma_u$, denoted by $\kappa(\Gamma_u) \subseteq \Gamma_u$. The hyperedges of $H$ are formed by the child shadows of all local roots of $\Gamma_u$. Formally, $H$ is defined via $$\label{eq:H-def}
H := \Big(\kappa(\Gamma_u), \bigcup_{r \in R(\Gamma_u)} \mathcal{C}(r) \Big).$$ Thus, each hyperedge of $H$ is associated with a child of some local root of $\Gamma_u$. This association between hyperedges of $H$ and children of nodes in $R(\Gamma_u)$ is employed to further associate a subset of children of $R(\Gamma_u)$ to each strongly connected component of $H$. We form the subproblems $\Gamma'$ and $\Gamma''$ by partitioning children of $R(\Gamma_u)$ to ensure that children which fall into the same connected component of $H$ lie in the same subproblem, either $\Gamma'$ or $\Gamma''$. For example, in , the children $a,b,c$ and $d$ are each associated with one maximal connected component of $H$, while the child $e$ is associated with another.
[0.48]{}
[0.48]{}
To ensure that this decomposition may be repeated as needed on the subproblems $\Gamma'$ and $\Gamma''$ we must establish a few properties of $H$.
\[l:hypergraph-props\] A hypergraph $H$ as defined via may be decomposed into maximal connected components $H_1 = (V_1, \mathcal{E}_1), ..., H_t = (V_t, \mathcal{E}_t)$ for which the following properties hold.
i) for all $i$, $V_i \in \mathcal{E}_i$, (i.e. each maximal connected component is covered by a single edge.)
ii) for all $i \neq j$, $V_i \cap V_j = \emptyset$, (i.e. no connector lies in two maximal connected components.)
iii) for all $r, r' \in R(\Gamma_u)$ and $i \in {1,...,t}$: $\mathcal{C}(r) \cap \mathcal{E}_i$ and $\mathcal{C}(r') \cap \mathcal{E}_i$ form a laminar pair.
Statement $(iii)$ ensures that this lemma continues to hold in the subproblems $\Gamma'$ and $\Gamma''$. A proof of Lemma \[l:hypergraph-props\] can be found in Appendix \[a:omitted-proofs\].
It remains to show that *any* $k$-multitree may be decomposed according to this procedure. The proof we present here focuses on the more involved MERGE case and only sketches the argument for the INCLUDE case. A full proof appears in Appendix \[a:omitted-proofs\].
\[t:k-multitree-structure-theorem\] Any untangled $k$-multitree $M = (V,E)$ can be decomposed into a decomposition tree $\tau$ in which:
1) all leaves of $\tau$ are associated either with base or trivial subproblems and,
2) at each internal node $u \in V$, one of the UP, OUT, INCLUDE, or MERGE cases can be applied to the subproblem $\Gamma_u$ to obtain the subproblems associated with the children of $u$.
Given an untangled $k$-multitree $M=(V,E)$, we first note that $V$ is an admissible subproblem of $G$. We proceed to show that if $\Gamma_u$ is a non-base admissible subproblem of $M$, that $\Gamma_u$ can be decomposed into at most two admissible subproblems of $G$. Since $G$ is finite, this process cannot proceed indefinitely, and thus must terminate, yielding $\tau$.
If any local root $r \in R(\Gamma_u)$ has a single child which is not a connector, the UP case can be applied to yield subproblem $\Gamma_u \setminus \{r\}$. This is easily seen to be an admissible subproblem, since the child of $r$ is not a connector and $\Gamma_u$ is child descendant complete.
If some root has a child $c$ with no connectors as descendants, the OUT case can be applied as follows. The set $D$ containing $c$ and all $c$’s descendants forms a base subproblem. Thus, $\Gamma_u \setminus D$ is easily seen to be admissible.
If neither the UP nor OUT case can be applied, it is clear that
if any local root of $\Gamma_u$ has only a single child, it must be a connector, and
every local root has at least one connector as a descendant.
Then let $c_{max}$ be the child with the maximum number of connectors as descendants. We split into two cases.
[ **Case 1)** Every connector in $\Gamma_u$ is a descendant of $c_{max}$. ]{}
We can argue that each parent of $c_{max}$ is a local root of $\Gamma_u$ since otherwise, we can exhibit a cycle or a diamond, contradicting that $M$ is a multitree (see Appendix \[a:omitted-proofs\]). Moreover, $c_{max}$ must have in-degree strictly greater than 1. Otherwise, it has only one parent, which implies that the UP case could be applied (a contradiction). Since the UP case cannot be applied, if $c_{max}$ has only one parent then $c_{max}$ must be a connector, which implies that $c_{max}$ has in-degree strictly greater than 1, as required.
Let $Q \subseteq R(\Gamma_u)$ be the subset of local roots which are parents of $c_{max}$. Then $Q$ is a trivial subproblem while $\Gamma_u \setminus Q$ is easily seen to be an admissible subproblem on which the INCLUDE case may be applied.
[ **Case 2)** Some connector in $\Gamma_u$ is not a descendant of $c_{max}$. ]{}
In this case we apply the MERGE case by forming the hypergraph $H$ as defined in . By Lemma \[l:hypergraph-props\], we can form maximal connected components $H_1,...,H_t$ where $H_i = (C_i, \mathcal{E}_i)$, with $C_i \cap C_j = \emptyset$ for all $i \neq j$. To apply the MERGE case we require at least two maximal connected components, which we argue as follows.
Suppose there is a single maximal connected component, $H_1 = (C_1, \mathcal{E}_1)$. By Lemma \[l:hypergraph-props\](i) $C_1$ is a hyperedge, which implies that there must be some child of $R(\Gamma_u)$ which covers all connectors of $\Gamma_u$. But this child must be $c_{max}$, which contradicts that some connector is *not* a descendant of $c_{max}$.
We can then form two admissible subproblems $\Gamma'$ and $\Gamma''$ as follows. For each local root $r \in R(\Gamma_u)$, let $X_r$ be the set of children of $r$, and let $$X_r' := \{u \in X_r : Sh(u) \in \mathcal{E}_1 \}, ~~~~~
X_r'' := \{u \in X_r : Sh(u) \in \mathcal{E}_2 \cup ... \cup \mathcal{E}_t \}.$$ As before, since each child has at least one connector, each child is in one of $X_r'$ or $X_r''$ for some $r \in R(\Gamma_u)$. We form $\Gamma'$ and $\Gamma''$ as follows $$\begin{aligned}
\Gamma' &:= \{u \in \Gamma_u : u \text{ is a descendant of a node in } \bigcup_{r \in R(\Gamma_u)} X_r' \} && \hspace{-2em}\cup R(\Gamma_u);\\
\Gamma'' &:= \{u \in \Gamma_u : u \text{ is a descendant of a node in } \bigcup_{r \in R(\Gamma_u)} X_r'' \} &&\hspace{-2em}\cup R(\Gamma_u).
\end{aligned}$$
We must show that each of $\Gamma'$ and $\Gamma''$ is an admissible subproblem. Both $\Gamma'$ and $\Gamma''$ are clearly child-descendant complete, having been formed by taking all descendants of a set of children of each root.
To see that $\Gamma'$ is connector complete, we will examine an arbitrary connector $c \in \Gamma'$.
Since $c \in \Gamma'$, $c \in C_1$, and by Lemma \[l:hypergraph-props\](i), $C_1 \in \mathcal{E}_1$, which implies that there must be some node $v \in \Gamma_u$ which is a child of a local root of $\Gamma_u$ such that $Sh(v) = C_1$. Let $r \in R(\Gamma_u)$ be the local root which is a parent of $v$. Since $c$ is a connector, it must have at least two local roots as ancestors. Then let $r' \in R(\Gamma_u)$ be an arbitrary local root which is an ancestor of $c$ such that $r \neq r'$. Let $w$ be the child on the path from $r'$ to $c$. Since $M$ is untangled, and $(C_1, \mathcal{E}_1)$ is a *maximal* connected component, we must have that $Sh(w) \subseteq Sh(v)$. Thus both $v$ and $w$ are in the set $\bigcup_{r \in R(\Gamma_u)} X_r'$, which implies that all of $v$ and $w$’s descendants are in $\Gamma'$, including $c$ and the two of $c$’s parents which are descendants of $v$ and $w$. Moreover, since $r'$ was chosen arbitrarily, this argument can be repeated for all $r' \in R(\Gamma_u)$ such that $r \neq r'$ to show that every parent of $c$ is contained in $\Gamma'$.
A similar argument shows that $\Gamma''$ is connector complete, ending Case 2.
Finally, the decomposition terminates since each subproblem created by this process is *strictly smaller* than the subproblem from which it was formed.
Optimizing LSP Over a Decomposition Tree
========================================
Once the decomposition tree $\tau$ is formed via the procedure from the prior section, we can apply a recurrence bottom-up to solve $k$-LSP.
Let $\Gamma_u$ be a subproblem in decomposition tree $\tau$ which has local roots denoted by $q_1,...,q_k$. To each placement $P$ on the leaves of $M[\Gamma_u]$ we associate an *ancestry signature*: a $k$-tuple in $\mathbb{N}^k$ whose $i^{th}$ entry contains the number of replicas of $P$ which have $q_i$ as an ancestor. We denote the ancestry signature of $P$ by ${\boldsymbol{\alpha}}(P) = \langle \alpha_1, ..., \alpha_k\rangle$.
We use the ancestry signature to index our DP recurrence, along with the number of replicas placed on a given node. We use the $F(\Gamma_u, r, {\boldsymbol{\alpha}})$ to denote the lexico-minimum failure aggregate obtained by any placement on the leaves of $M[\Gamma_u]$ which has size $r$ and ancestry signature equal to ${\boldsymbol{\alpha}}$. Since they store failure aggregates, values of $F$ are non-negative integer vectors of size $\rho + 1$. We set $F(\Gamma_u, r, {\boldsymbol{\alpha}}) = \infty$ when $\Gamma_u$ is a trivial subproblem, or when $M[\Gamma_u]$ does not admit any placement of size $r$ with ancestry signature ${\boldsymbol{\alpha}}$. We consider $\infty$ to be lexicographically larger than any vector.
Our goal is to describe $F(\Gamma_u, r, {\boldsymbol{\alpha}})$ in terms of values of $F$ taken the children of $u$ in subproblem tree $\tau$. Let $u$ have children $v$ and $w$. The DP recurrence we present has four cases depending on the case which was applied to $u$ to obtain $v$ and $w$. Each case of the recurrence is a sum of terms involving $\Gamma_v$ and $\Gamma_w$ along with a correction factor. This correction factor increments or decrements the number of nodes with a given failure number. Incrementing or decrementing the number of nodes with failure number $i$, is achieved by adding or subtracting ${\boldsymbol{e}}(i) = \langle 0,...,0,1,0,...,0\rangle$ where the 1 appears in the $(\rho - i)^{th}$ index. As we shall see, the only nodes whose failure numbers must be corrected are the local roots of subproblem $\Gamma_u$.
In the UP case, the value of $F(\Gamma_u, r, {\boldsymbol{\alpha}})$ must be updated to include the failure number of the new local root $q_i$. This is achieved by adding ${\boldsymbol{e}}(\alpha_i)$, yielding: $$F(\Gamma_u, r, {\boldsymbol{\alpha}}) = F(\Gamma_v, r, {\boldsymbol{\alpha}}) + {\boldsymbol{e}}(\alpha_i)
~~~~~~~~~ \text{(UP at root $q_i$)}.$$
[0.48]{}
[0.48]{}
Consider next the OUT case at local root $q_i$ (see ). Allow $\Gamma_w$ to represent the subproblem with no connectors and recall that $M[\Gamma_w]$ forms a tree. Thus, we may use the algorithm for trees developed previously [@Mills2015] to find ${\boldsymbol{T}}(\Gamma_w, x)$ the lexico-minimum failure aggregate attainable in $M[\Gamma_w]$ using $x$. To attain the optimal value overall, we take the minimum over all possible ways to split replicas which are descendants of $q_i$ among leaves of $M[\Gamma_w]$ and $M[\Gamma_v]$. $$F(\Gamma_u, r, {\boldsymbol{\alpha}}) =
\displaystyle\min_{\substack{\alpha_i' + x = \alpha_i\\r'+x=r}}\Big[F(\Gamma_v, r', {\boldsymbol{\alpha}}') + {\boldsymbol{T}}(\Gamma_w, x) + {\boldsymbol{e}}(\alpha_i) - {\boldsymbol{e}}(\alpha_i') \Big]
~~~~~~~~~ \text{(OUT at root $q_i$)}.$$ where ${\boldsymbol{\alpha}}' := \langle \alpha_1, ...,\alpha_{i-1}, \alpha_i', \alpha_{i+1}, ..., \alpha_k\rangle$. The corrective factor of ${\boldsymbol{e}}(\alpha_i) - {\boldsymbol{e}}(\alpha_i')$ adjusts the failure number of root $q_i$ from its previous value of $\alpha_i'$ (which is included from $F(\Gamma_v, r', {\boldsymbol{\alpha}}')$) to its new value of $\alpha_i$.
In the MERGE case we consider subproblems $\Gamma_v$ and $\Gamma_w$ which share only the $k$ local roots among them. Thus, as in the previous case, the leaves of $M[\Gamma_v]$ and $M[\Gamma_w]$ are disjoint. Taking the lexico-minimum over all ways to split the ancestry signature ${\boldsymbol{\alpha}}$ into ${\boldsymbol{\alpha}}'$ and ${\boldsymbol{\alpha}}''$ yields the optimal value overall, as shown below. $$F(\Gamma_u, r, {\boldsymbol{\alpha}}) =
\displaystyle\min_{\substack{{\boldsymbol{\alpha}}' + {\boldsymbol{\alpha}}'' = {\boldsymbol{\alpha}}\\r' + r'' = r}}\Big[F(\Gamma_v, r', {\boldsymbol{\alpha}}') + F(\Gamma_w, r'', {\boldsymbol{\alpha}}'') + correct_k({\boldsymbol{\alpha'}}, {\boldsymbol{\alpha}}'')\Big]
~~~~~~~~~ \text{(MERGE)}$$ where the corrective factor $correct_k({\boldsymbol{\alpha}}',{\boldsymbol{\alpha}}'') := \sum_{i=1}^k {\boldsymbol{e}}(\alpha_i) - {\boldsymbol{e}}(\alpha_i') - {\boldsymbol{e}}(\alpha_i'')$ for and ${\boldsymbol{\alpha}}'' = \langle \alpha_1'',...,\alpha_k'' \rangle$. The $i^{th}$ term in the corrective factor adjusts the failure number of root $q_i$ by replacing the contributions of ${\boldsymbol{e}}(\alpha_i')$ and ${\boldsymbol{e}}(\alpha_i'')$ (which were included from $F(\Gamma_v, r', {\boldsymbol{\alpha}}')$ and $F(\Gamma_w, r'', {\boldsymbol{\alpha}}'')$ respectively) with the corrected value of ${\boldsymbol{e}}(\alpha_i)$.
The INCLUDE case requires special consideration since $\Gamma_v$ has strictly fewer local roots than $\Gamma_u$. Thus placements on the leaves of $M[\Gamma_v]$ will have ancestry signatures with length $j$, whereas the parent subproblem $\Gamma_u$ requires ancestry signatures of length $k$. These signatures will need to be appropriately mapped onto one another. Moreover, not all values of ${\boldsymbol{\alpha}}$ are valid as ancestry signatures of $\Gamma_u$, since local roots in $Q$ must all *share* the same failure number (see ). Thus, our recurrence will only be computed at values of ${\boldsymbol{\alpha}}$ for which this is true. To address these details, we employ a mapping ${\boldsymbol{h}} : \mathbb{N}^j \to \mathbb{N}^k$ which maps ancestry signatures of $\Gamma_v$ to their corresponding signature in $\Gamma_u$. A formal definition of ${\boldsymbol{h}}$ can be found in Appendix \[a:omitted-proofs\].
With the mapping ${\boldsymbol{h}}$ in hand we can describe the optimal value of $F(\Gamma_u, r, {\boldsymbol{\alpha}}({\boldsymbol{\beta}}))$ as follows. Let $\Gamma_v$ be the base subproblem which forms a $j$-multitree, and which has local roots $s_1,...,s_j$. Moreover, $\Gamma_v$ has a distinguished local root, $s_\ell$, whose parents all lie in the set $Q \subseteq \{q_1,...,q_k\}$. Given values of $F(\Gamma_v, r, {\boldsymbol{\beta}})$, we can compute the recurrence as follows $$F(\Gamma_u, r, {\boldsymbol{h}}({\boldsymbol{\beta}})) =
F(\Gamma_v, r, {\boldsymbol{\beta}}) + |Q|\cdot{\boldsymbol{e}}(\beta_\ell) ~~~~~~~~~ \text{(INCLUDE where $s_\ell$ has parents in $Q$)}$$ In the above equation, the term $|Q|\cdot{\boldsymbol{e}}(\beta_\ell)$ corrects for the addition of all $|Q|$ local roots in $Q$. Each such local root will have a failure number matching that of $s_\ell$. For all values of ${\boldsymbol{\alpha}}$ which do not match ${\boldsymbol{h}}({\boldsymbol{\beta}})$ for some ${\boldsymbol{\beta}}$, we set $F(\Gamma_u, r, {\boldsymbol{\alpha}})) = \infty$.
Time Analysis and Conclusion
============================
In both phases, the time required to compute the MERGE case dominates the remaining cases. To bound the time taken to run the decomposition phase, notice that the number of edges in any $k$-multitree is no more than $kn$, where $|V| = n$. Thus, the size of a description of the connector-shadow hypergraph $H$ may be no more than $O(kn)$, and therefore maximal connected components of $H$ may be found in $O(\alpha(kn)kn)$ time per application of the MERGE case. Since each application of a MERGE separates at least one connector from the rest, there may only be $O(c)$ MERGE cases, where $c$ is the number of connectors in $M$.
For the optimization phase, $O(\rho^k)$ is an upper bound on both a) the number of ways to split an ancestry signature ${\boldsymbol{\alpha}}$ into ${\boldsymbol{\alpha}}'$ and ${\boldsymbol{\alpha}}''$ and b) the number of values of ${\boldsymbol{\alpha}}$ for which $F(\Gamma_u, r, {\boldsymbol{\alpha}})$ must be computed. Moreover, there are $O(\rho)$ values of $r$, $O(\rho)$ ways to split values of $r$ into $r'$ and $x$, and an additional factor of $O(\rho)$ must be included for summing vector values of $F$. Overall, any MERGE phase is bounded by $O(n\rho^{2k+3})$, since each subproblem is split into two strictly smaller subproblems at each step, and this may be done only $n$ times. Notice that base subproblems considered in the INCLUDE case have strictly less than $k$ roots, so their running times are each bounded by $O(n\rho^{2j+3})$ where $j < k$. Since in practice $c$ may be either $O(n)$ or $o(\rho^{2k+3})$, we report the total running time as $O(n\rho^{2k+3} + \alpha(kn)ckn)$. A looser, somewhat snappier bound is $O(n^2\rho^{2k+3})$. Either bound suffices to establish fixed-parameter tractability of *untangled* $k$-LSP.
At the end of Section 4 we briefly described how an optimal placement algorithm for untangled $k$-multitrees suffices to solve the problem in *canonical placement models* and thus in DAGs. However, in the general case, the number of roots may be large, making optimization prohibitively expensive. Thus, a procedure for minimizing the number of roots in a canonical placement model would be a useful future contribution. Other directions for future work include approximation algorithms and algorithms based upon alternative parameterizations, particularly output-sensitive parameterizations based upon the failure aggregate.
Omitted / Truncated Proofs {#a:omitted-proofs}
==========================
We complete the reduction by showing that $H$ has a placement $P\subseteq H(V)$ with $|P| = k$ for which ${\boldsymbol{f}}(P) \leq_L \langle 3,0,...,0,\infty,\infty \rangle$ if and only if $G$ has an independent set of size $k$.
$``\implies"$ Suppose $H$ has a placement $P \subseteq H(V)$ with $|P|=k$ and ${\boldsymbol{f}}(P) \leq \langle 3,0,...,0,\infty,\infty \rangle$. Nodes $\alpha, \beta$ and $\gamma$ each have failure number $k$, since every node in $P$ has each of $\alpha,\beta$ and $\gamma$ as an ancestor. Thus, the upper bound on ${\boldsymbol{f}}(P)$ implies that all other nodes in $H$ have a failure number of at most 1. Thus, no node of $H(E)$ has failure number 2, which further implies that $P$ is a subset of $k$ nodes of $H(V)$ such that no node in $H(E)$ is connected to two or more nodes of $P$. Thus, every node in $H(E)$ is connected to at most one node of $P$, which implies that no two nodes of $P$ are adjacent as vertices of $G$. Thus, $P$ corresponds to an independent set of size $|P|=k$ in $G$.
$``\impliedby"$ Suppose instead that $G$ has an independent set $I$ of size $k$. Then $I$ corresponds to a subset $P \subseteq H(V)$ of size $k$ in which no two vertices of $P$ are adjacent to the same node in $H(E)$. But this implies that every node in $H(E)$ has a failure number of at most 1. Moreover, no vertex in $H(V)$ can have failure number greater than $1$, and, as we have shown, each node $\alpha, \beta$ and $\gamma$ has failure number exactly $k$. Therefore, ${\boldsymbol{f}}(P) \leq_L \langle 3,0,...,0,\infty,\infty \rangle$, and so $P \subseteq H(V)$ is a placement of size $k$ with the required upper bound on ${\boldsymbol{f}}(P)$.
For convenience, Lemma \[l:hypergraph-props\] is restated below.
[\[l:hypergraph-props\]]{} Given an admissible subproblem $\Gamma_u$, the hypergraph $H$ defined via $$H := \Big(\kappa(\Gamma_u), \bigcup_{r \in R(\Gamma_u)} \mathcal{C}(r) \Big)$$ may be decomposed into maximal connected components $H_1 = (V_1, \mathcal{E}_1), ..., H_t = (V_t, \mathcal{E}_t)$ for which the following properties hold.
i) for all $i$, $V_i \in \mathcal{E}_i$, (i.e. each maximal connected component is covered by a single edge.)
ii) for all $i \neq j$, $V_i \cap V_j = \emptyset$, (i.e. no connector lies in two maximal connected components.)
iii) for all $r, r' \in R(\Gamma_u)$ and $i \in {1,...,t}$: $\mathcal{C}(r) \cap \mathcal{E}_i$ and $\mathcal{C}(r') \cap \mathcal{E}_i$ form a laminar pair.
Recall from the main body of the paper that for each $r \in R(\Gamma_u)$, any two hyperedges $X,Y \in \mathcal{C}(r)$ are disjoint. Moreover, since $M[\Gamma_u]$ is an untangled multitree, $\mathcal{C}(r)$ and $\mathcal{C}(r')$ form a laminar pair by definition. Each property may be proven as follows.
1. Suppose that $V_i \notin \mathcal{E}_i$. Let $E$ be the largest hyperedge of $\mathcal{E}_i$, and note that $|E| < |V_i|$, since otherwise $E = V_i$. Since $H_i$ is connected, the vertices in $V_i \setminus E$ must be reachable from the vertices in $E$. These vertices can only be reached via a hyperedge of $H_i$, and since $\mathcal{E}_i$ is laminar, this vertex must entirely contain $E$, thereby contradicting that $E$ is the largest hyperedge of $H_i$.
2. Suppose that $V_i \cap V_j \neq \emptyset$ for some $i \neq j$. Since $V_i$ is an edge of $H_i$ it must also be a hyperedge of $H$, and likewise for $V_j$. The hyperedges of $H$ are easily seen to be laminar. Thus all hyperedges of $H$ are either disjoint, or one is a subset of another. If $V_i \subseteq V_j$, then since $H_i$ is a *maximal* connected component, we must have the $V_j \in \mathcal{E}_i$, and thus $V_j$ is part of the same connected component of as $V_i$, implying $i=j$, a contradiction.
3. Since $\mathcal{E}_i$ and $\mathcal{E}_j$ form a laminar pair, and any pair of subsets of a laminar pair forms a laminar pair, in particular $\mathcal{C}(r) \cap \mathcal{E}_k$ and $\mathcal{C}(r') \cap \mathcal{E}_k$ form a laminar pair.
In the proof of Theorem \[t:k-multitree-structure-theorem\], we argued in Case 1 that each parent of $c_{max}$ must be a local root of $\Gamma_u$, since otherwise, we can exhibit a cycle or a diamond, both of which are forbidden structures. We now provide the justification for this claim.
We claim that the choices for parent of $c_{max}$ are limited to nodes in $R(\Gamma_u)$. Suppose instead that some other node $v \notin R(\Gamma_u)$ is a parent of $c_{max}$. Then $v$ must have an ancestor $a \in R(\Gamma_u)$. Let $x$ be the child of $a$ on the path from $a$ to $c_{max}$. It is clear that $x$ is not a connector, as we now show. If $x$ is a connector then there exists a cycle from $c_{max} \rightsquigarrow x \rightsquigarrow c_{max}$, contradicting that $M$ is acyclic.
So $x$ is not a connector. Thus, since the UP case could not be applied, root $a$ must have multiple children. Let $x' \neq x$ be another of the children of $a$. Since the OUT case(s) could not be applied, $x'$ must have a connector $y$ as a descendant. But since in Case 1 all connectors are descendants of $c_{max}$ this forms a diamond from $a \rightsquigarrow c_{max} \rightsquigarrow y$ and $a \rightsquigarrow x' \rightsquigarrow y$. Thus choices for parents of $c_{max}$ are limited to nodes in $R(\Gamma_u)$ as claimed.
Mapping Ancestry Signatures in the INCLUDE Case
-----------------------------------------------
Let $\Gamma_v$ be the base subproblem which forms a $j$-multitree, and which has local roots $s_1,...,s_j$. Moreover, $\Gamma_v$ has a distinguished local root, $s_\ell$, whose parents all lie in the set $Q \subseteq \{q_1,...,q_k\}$. Given values of $F(\Gamma_v, r, {\boldsymbol{\beta}})$, we wish to compute the optimal value of $F(\Gamma_u, r, {\boldsymbol{\alpha}})$ for appropriate values of ${\boldsymbol{\alpha}}$.
Observe that in the INCLUDE case, not all values of ${\boldsymbol{\alpha}}$ are valid as ancestry signatures of $\Gamma_u$, since local roots in $Q$ must all *share* the same failure number. Thus, our recurrence will only be defined for values of ${\boldsymbol{\alpha}}$ for which this is true. To describe this formally, we employ a mapping ${\boldsymbol{h}} : \mathbb{N}^j \to \mathbb{N}^k$ which maps ancestry signatures of $\Gamma_v$ to their corresponding signature in $\Gamma_u$.
To define ${\boldsymbol{h}} : \mathbb{N}^j \to \mathbb{N}^k$ we employ a one-to-one mapping to capture which local roots of $\Gamma_v$ are also local roots of $\Gamma_u$. Recall that $\Gamma_v$ has local roots $s_1,...,s_j$ while $\Gamma_u$ has local roots $q_1,...,q_k$, and these roots are not necessarily distinct. Then there exists a one-to-one mapping $\pi : \mathbb{N} \to \mathbb{N}$ such that $q_i = s_{\pi(i)}$ for any local root $q_i \notin Q$.
The mapping $\pi$ allows us to formally define ${\boldsymbol{h}}({\boldsymbol{\beta}})$ as follows. Let ${\boldsymbol{\beta}} = \langle \beta_1, ..., \beta_j \rangle$ be the ancestry signature of a placement on the leaves of $M[\Gamma_v]$. For each such ${\boldsymbol{\beta}}$ there is one valid value of ${\boldsymbol{h}}({\boldsymbol{\beta}})$, defined as $${\boldsymbol{h}}({\boldsymbol{\beta}}) = \langle h_1,...,h_k \rangle \text{ where } h_i = \begin{cases}
\beta_\ell & \text{ if $q_i \in Q$}, \\
\beta_{\pi(i)} & \text{ if $q_i \notin Q$}.
\end{cases}$$ A concrete example depicting how $\pi$ works together with the definitions of ${\boldsymbol{\beta}}$ and ${\boldsymbol{h}}({\boldsymbol{\beta}})$ can be seen in .
With the mapping ${\boldsymbol{h}}$ in hand we can describe the optimal value of $F(\Gamma_u, r, {\boldsymbol{h}}({\boldsymbol{\beta}}))$ by means of the recurrence $$F(\Gamma_u, r, {\boldsymbol{h}}({\boldsymbol{\beta}})) =
F(\Gamma_v, r, {\boldsymbol{\beta}}) + |Q|\cdot{\boldsymbol{e}}(\beta_\ell) ~~~~~~~~~ \text{(INCLUDE where $s_\ell$ has parents in $Q$)}$$ where the term $|Q|\cdot{\boldsymbol{e}}(\beta_\ell)$ corrects for the addition of all $|Q|$ local roots in $Q$. Each such local root will have a failure number matching that of $s_\ell$.
In case an ancestry signature ${\boldsymbol{\alpha}}'$ is not in the image of ${\boldsymbol{h}}$, the value of $F(\Gamma_u, r, {\boldsymbol{\alpha}}')$ remains $\infty$.
[^1]: Recall that in a full binary tree every node has 0 or 2 children.
[^2]: Using a subset as opposed to a multiset rules out the possibility of placing multiple replicas on the same server, which would defeat the purpose of replication.
[^3]: That is, if $M$ is an untangled multitree, then for every $U \subseteq V$, the vertex-induced subgraph $M[U] = (U, (U \times U) \cap E)$ is also an untangled multitree.
[^4]: Recall that a *trivial graph* is a graph with no edges.
[^5]: \[ft:child-desc\]Where admissibility follows by child-descendant completeness of $\Gamma_u$.
[^6]: An algorithm for undirected hypergraphs with the same running time exists. In any case, undirected hypergraphs can be handled via [@Allamigeon2014] by adding an extra hyperedge going in the reverse direction.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Through desingularization of Clifford torus, we prove the existence of a sequence of nondegenerate (in the sense of Duyckaerts-Kenig-Merle ([@DKM])) nodal nonradial solutions to the critical Yamabe problem $$-\Delta u=\frac{n(n-2)}{4}|u|^{\frac{4}{n-2}}u,\qquad u\in{{\mathscrD}}^{1,2}({{\mathbb R}}^n).$$ The case $n=4$ is the first example in the literature of a solution with [*maximal rank*]{} ${{{\mathscrN}}}=2n+1+\frac{n(n-1)}{2}$.'
address:
- 'Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna 4860, Santiago, Chile'
- 'Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna 4860, Santiago, Chile'
- 'Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2'
author:
- Maria Medina
- Monica Musso
- Juncheng Wei
bibliography:
- 'refs.bib'
title: Desingularization of Clifford Torus and Nonradial Solutions to the Yamabe Problem with Maximal Rank
---
[^1]
Consider the problem $$\label{prob}
-\Delta u=\gamma |u|^{p-1}u\hbox{ in }\mathbb{R}^n,\qquad \gamma:=\frac{n(n-2)}{4},\qquad u\in{{\mathscrD}}^{1,2}({{\mathbb R}}^n),$$ where $n{\geqslant}4$, $p=\frac{n+2}{n-2}$ and ${{\mathscrD}}^{1,2}({{\mathbb R}}^n)$ is the completion of $C_0^\infty({{\mathbb R}}^n)$ with the norm $\|\nabla u\|_{L^2({{\mathbb R}}^n)}$.
When $u>0$, problem (\[prob\]) arises in the classical Yamabe problem or extremal equation for Sobolev inequality. For positive or sign-changing $u$ Problem (\[prob\]) corresponds to the steady state of the energy-critical focusing nonlinear wave equation $$\label{2m}
\partial_t^2 u-\Delta u- |u|^{\frac{4}{n-2}} u=0, \ (t,x)\in {{\mathbb R}}\times {{\mathbb R}}^n.$$ These are classical problems that have attracted the attention of many researchers ([@DKM2; @DKM3; @KM1; @KM2; @KST]). The study of (\[2m\]) naturally relies on the complete classification of the set of non-zero finite energy solutions to Problem (\[prob\]), which is defined by $$\label{3m}
\Sigma:= \left\{ Q \in {{{\mathscrD}}}^{1,2} ({{\mathbb R}}^n) \backslash \{0\}: \ -\Delta Q= \frac{n(n-2)}{4} |Q|^{\frac{4}{n-2}} Q\right \}.$$
By the classical work of Caffarelli-Gidas-Spruck [@CGS] all positive solutions to (\[prob\]) are given by $$\label{bubble}
U(y)=\left(\frac{2}{1+|y|^2}\right)^{\frac{n-2}{2}},$$ and all its translations and dilations $$\label{bubble1}
U_{\alpha,\bar{y}}:=\alpha^{-\frac{n-2}{2}}U\left(\frac{y-\bar{y}}{\alpha}\right),\qquad \alpha>0,\; \bar{y}\in\mathbb{R}^n.$$
For sign-changing solutions much less is known. A direct application of Pohozaev’s identity gives that all sign-changing solutions to Problem (\[prob\]) are nonradial. The existence of elements of $\Sigma$ that are nonradial, sign-changing, and with arbitrary large energy was first proved by Ding [@D] using Ljusternik-Schnirelman category theory. However no other qualitative properties are known for Ding’s solutions. Recently more explicit constructions of sign-changing solutions to Problem (\[prob\]) have been obtained by del Pino-Musso-Pacard-Pistoia [@dPMPP; @dmpp2]. In [@MW], the second and the third authors established the rigidity of the solutions constructed in [@dPMPP] by showing that they are [*nondegenerate*]{} in the sense of Duyckaerts-Kenig-Merle ([@DKM], see definitions below).
The purpose of this work is to give a positive answer to an open question formulated in the work of M. Musso and J. Wei ([@MW]): whether there exists a solution that, apart from nondegenerate, is [*maximal*]{}. To properly explain this framework, let us denote by $$\Sigma:=\left\{Q\in {{\mathscrD}}^{1,2}({{\mathbb R}}^n)\setminus\{0\}:-\Delta Q=\gamma |Q|^{p-1}Q\right\}$$ the set of nontrivial finite energy solutions of . It can be seen that the equation in is invariant under four transformations: translation, dilation, orthogonal transformation and Kelvin transform. More precisely, if $Q\in\Sigma$, then:
- $Q(y+a)\in \Sigma$ for every $a\in{{\mathbb R}}^n$;
- $\lambda^{\frac{n-2}{2}}Q(\lambda y)\in\Sigma$ for every $\lambda>0$;
- $Q(Py)\in\Sigma$ for every $P\in {{\mathscrO}}_n$, where ${{\mathscrO}}_n$ denotes the classical orthogonal group;
- $|y|^{2-n}Q(|y|^{-2}y)\in\Sigma$.
Denote by ${{\mathscrM}}$ the group of isometries of ${{\mathscrD}}^{1,2}({{\mathbb R}}^n)$ generated by these transformations. Then, ${{\mathscrM}}$ derives a family of transformations in a neighborhood of the identity (see [@DKM Lemma 3.8]) of dimension $$\label{defN}
{{{\mathscrN}}}:=2n+1+\frac{n(n-1)}{2}.$$ In particular, ${{\mathscrM}}$ generates the vector space $$\tilde{{{\mathscrI}}}_Q=\mbox{span}\left\{
\begin{array}{c}
(2-n)y_\alpha Q+|y|^2\partial_{y_\alpha}Q-2y_\alpha y\cdot\nabla Q,\;\;\partial_{y_\alpha}Q,\;\;1{\leqslant}\alpha{\leqslant}n,\\
(y_\alpha \partial_{y_\beta}-y_\beta\partial_{y_\alpha})Q,\;\;1{\leqslant}\alpha<\beta{\leqslant}n,\;\;\frac{n-2}{2}Q+y\cdot Q
\end{array}
\right\}.$$ Consider the associated linearized operator around $Q\in\Sigma$, i.e., $$L_Q:=-\Delta-\gamma p |Q|^{p-2}Q,$$ and its kernel $${{\mathscrI}}_Q:=\{f\in{{\mathscrD}}^{1,2}({{\mathbb R}}^n):\,L_Qf=0\}.$$ Clearly $\tilde{{{\mathscrI}}}_Q\subseteq{{\mathscrI}}_Q$ and, following the work of T. Duyckaerts, C. Kenig and F. Merle ([@DKM]), we can define the notion of [*nondegeneracy*]{}.
\[defNondeg\] $Q\in\Sigma$ is said to be [*nondegenerate*]{} if ${{\mathscrI}}_Q=\tilde{{{\mathscrI}}}_Q$.
Let $Q$ be nondegenerate. Its [*rank*]{} is defined as the dimension of $\tilde{{{\mathscrI}}}_Q$, which is at most ${{\mathscrN}}$. Actually, the positive solutions $Q=W$ can be proved to be nondegenerate as a consequence of the radial symmetry, and $\tilde{{{\mathscrI}}}_W$, which is $$\tilde{{{\mathscrI}}}_W=\left\{\frac{n-2}{2}W+y\cdot\nabla W,\;\;\partial_{y_\alpha}W,\;\;1{\leqslant}\alpha{\leqslant}n\right\},$$ has rank $n+1$ ([@Rey]). In this case, the rank is strictly less than ${{{\mathscrN}}}$.
In [@MW], the authors give the first example of nodal nonradial sign-changing solution satisfying the nondegeneracy condition. Indeed, they consider the solution $u_k$ of built in [@dPMPP Theorem 1] given by $$u_k(y)=U(y)-\sum_{j=1}^k \mu_k^{-\frac{n-2}{2}}U(\mu_k^{-1}(y-\xi_j))+o(1),$$ where $$\mu_k:=\frac{c_n}{k^2}, \qquad \xi_j:=(e^{\frac{2j\pi i}{k}},0,\ldots),\qquad U(y):=\left(\frac{2}{1+|y|^2}\right)^{\frac{n-2}{2}},$$ and they prove that $\tilde{{{\mathscrI}}}_{u_k}={{\mathscrI}}_{u_k}$, where the dimension of these vector spaces is $3n$, i.e. the rank is $3n$. Also in this case, the rank is strictly less than ${{\mathscrN}}$.
The purpose of this work is to provide the first example in the literature of a nondegenerate solution $u$ to which has the [*maximal rank*]{} ${{{\mathscrN}}}$.
\[defMaximal\] A nondegenerate solution $Q\in\Sigma$ is said to be [*maximal*]{} if $$\mbox{dim}(\tilde{{{\mathscrI}}}_Q)=\mbox{dim}({{\mathscrI}}_Q)={{{\mathscrN}}},$$ where ${{{\mathscrN}}}$ was defined in .
Thus, our main result can be formulated as follows.
\[teounico\] Let $n{\geqslant}4$. Then, there exists a sequence of nodal solutions to , with arbitrarily large energy, which are nondegenerate according to Definition \[defNondeg\]. If $n=4$ these solutions are maximal in the sense of Definition \[defMaximal\].
To prove this result, we will build a solution in the following way: let $k$ and $h$ be two large positive integers (not necessarily equal), and $$\label{deltaeps}
\mu:=\frac{\delta^{\frac{2}{n-2}}}{k^2},\qquad \lambda:=\frac{\varepsilon^{\frac{2}{n-2}}}{h^2},$$ where $\delta$ and $\varepsilon$ are positive parameters so that $$c_1<\delta<c_1^{-1},\qquad c_2<\varepsilon<c_2^{-1},$$ for some constants $c_1,c_2>0$ which are independent of $k$ and $h$ as they tend to infinity. Consider now the points $$\begin{split}\label{points}
\xi_j&:=\sqrt{1-\mu^2}(e^{\frac{2\pi i(j-1)}{k}},0,\ldots,0)\in \mathbb{R}^2\times \mathbb{R}^{n-2}, j=1,\ldots,k,\\
\eta_l&:=\sqrt{1-\lambda^2}(0,0,e^{\frac{2\pi i(l-1)}{h}},0,\ldots,0)\in \mathbb{R}^2\times\mathbb{R}^2\times \mathbb{R}^{n-4}, l=1,\ldots,h,
\end{split}$$ which satisfy $$\label{mod1}
|\xi_j|^2+\mu^2=1,\qquad |\eta_l|^2+\lambda^2=1.$$ Consider $$\label{nodalSol}
u(y)=U(y)-\sum_{j=1}^k U_{\mu,\xi_j}(y)-\sum_{l=1}^h U_{\lambda,\eta_l}(y)+\phi(y)$$ where $U$ is defined in , $$\label{transBubbles}
U_{\mu,\xi_j}(y):=\mu^{-\frac{n-2}{2}}U\left(\frac{y-\xi_j}{\mu}\right),\qquad U_{\lambda,\eta_l}(y):=\lambda^{-\frac{n-2}{2}}U\left(\frac{y-\eta_l}{\lambda}\right),$$ and $\phi$ is a small function when compared with the other terms (for the sake of simplicity we do not make explicit the dependence of $u$ in $k$ and $h$).
Notice that functions $U$, $U_{\mu,\xi_j}$ and $U_{\lambda,\eta_l}$ are invariant under rotation of angle $\frac{2\pi}{k}$ in the $(y_1,y_2)$ plane and of angle $\frac{2\pi}{h}$ in the $(y_3,y_4)$ angle. Furthermore, they are even in the $y_\alpha$-coordinates, for $\alpha=2,4,5,\ldots,n$ and invariant under Kelvin’s transform (due to ). Assume that $\phi$ also satisfies these properties (we will prove this in Part \[Existence\]).
Consider the following set of functions: $$\label{z0}
z_0(y):=\frac{n-2}{2}u(y)+\nabla u(y)\cdot y,$$ $$\label{z1}
z_\alpha(y):=\frac{\partial}{\partial y_\alpha}u(y),\qquad \alpha=1,\ldots,n,$$ $$\begin{split}\label{z2}
z_{n+1}(y):=&-y_2\frac{\partial}{\partial y_1}u(y)+y_1\frac{\partial}{\partial y_2}u(y),\\
z_{n+2}(y):=&-y_4\frac{\partial}{\partial y_3}u(y)+y_3\frac{\partial}{\partial y_4}u(y),
\end{split}$$ $$\begin{split}\label{z3}
z_{n+\alpha+2}(y)&:=-2y_\alpha z_0(y)+|y|^2z_\alpha(y), \quad \alpha = 1, 2, 3, 4,
\end{split}$$ $$\begin{split}\label{z4}
z_{n+\alpha+4}(y)&:=-y_\alpha z_1(y)+y_1z_\alpha(y), \qquad \alpha=3,\ldots,n,\\
z_{2n+\alpha+2}(y)&:=-y_\alpha z_2(y)+y_2z_\alpha(y), \qquad \alpha=3,\ldots,n,
\end{split}$$ and $$\begin{split}\label{z5}
z_{3n+\alpha-2}(y)&:=-y_\alpha z_3(y)+y_3z_\alpha(y), \qquad \alpha=5,\ldots,n,\\
z_{4n+\alpha-6}(y)&:=-y_\alpha z_4(y)+y_4z_\alpha(y), \qquad \alpha=5,\ldots,n.
\end{split}$$ Functions and are related to the invariance of problem under dilations and translations respectively, and to the rotation in the $(y_1,y_2)$ and $(y_3,y_4)$ planes. Likewise, arises from the invariance under Kelvin transform, and , from the rotation in the planes $(y_1, y_\alpha)$, $(y_2,y_\alpha)$, for $\alpha=3,\ldots,n$, and $(y_3, y_\alpha)$, $(y_4,y_\alpha)$ for $\alpha=5,\ldots,n$. If we denote by $L$ the linearized operator around $u$ associated to , i.e., $$\label{lin}
L(\varphi):=\Delta \varphi+p\gamma|u|^{p-2}u\varphi,$$ - provide $N_0:=5(n-1)$ elements of the kernel of $L$.
We will prove that these are indeed all the elements in the kernel, i.e., solution is a second example of nodal nondegenerate solution of . But what is more remarkable here is that if $n=4$, then $N_0={{{\mathscrN}}}$, that is, the solution is maximal in the sense of Definition \[defMaximal\], which is the first example of a nondegenerate maximal solution in the literature, and answers the open question formulated in [@MW].
When $\mu \not =\lambda, h \not = k$, our solution is different from the ones constructed in [@dPMPP; @dmpp2]. In [@dmpp2] the symmetric case $\mu=\lambda, h=k$ is considered, which corresponds to the [*Clifford torus*]{}. In this case the solution has an additional symmetry which reduces the problem to one dimensional. Because of this symmetry the rank of the solutions constructed in [@dmpp2] can be shown to be strictly less than ${{{\mathscrN}}}$. Thus our solutions are [*new*]{}. Our construction can be considered as a sort of [*desingularization*]{} of Clifford torus. For geometric application of desingularization of Clifford torus, we refer to the recent papers [@BreKa; @Ka] and the references therein.
The construction can be extended to higher even dimensions, that is, one can anagolously set bubbles in the $(y_5,y_6)$, $(y_7, y_8)$, $\ldots$, planes, in such a way that the solution is expected to be nondegenerate and the elements corresponding to the invariances generate a space of dimension exactly ${{{\mathscrN}}}$. Therefore, this type of construction presumably provides a sequence of nodal nondegenerate and maximal rank solutions of for any even dimension $n{\geqslant}4$. The existence of a maximal solution for odd dimensions is still an open question.
Nondegenerate solutions to (\[prob\]) play an important role in the analysis of possible singularity formations in energy-critical wave equations. We refer to [@DKM; @DKM2; @DKM3; @KM1; @KM2; @KST] and the references therein.
The existence of sign-changing solutions for critical exponents in other contexts has been studied in [@HV; @RV1; @RV2].
There exist many works on the uniqueness and nondegeneracy of [*positive solutions*]{} to semilinear equations, whether or not for classical nonlinear Schrodinger equations [@K] or for nonlinear fractional equations [@FL; @FLS]. The rank of the positive solutions is at most $n+1$. For sign-changing solutions the nondegeneracy question is in general quite difficult without knowing the precise behavior of the solution. Our result is the first of the type for sign-changing solutions with maximal rank.
Along the work we will denote points $y\in\mathbb{R}^n$, $n{\geqslant}4$, as $$y=(\overline{y},\hat{y}),\;\overline{y}:=(y_1,y_2),\;\hat{y}:=(y_3,y_4), \mbox{ if } n=4,$$ $$y=(\overline{y},\hat{y},y'),\;\overline{y}:=(y_1,y_2),\;\hat{y}:=(y_3,y_4),\;y':=(y_5,\ldots,y_n),\mbox{ if } n{\geqslant}5,$$ and we will work with the norms $$\label{normStarStar}
\|h\|_{**}:=\|(1+|y|)^{n+2-\frac{2n}{q}}h\|_{L^q(\mathbb{R}^n)}, \qquad \|\phi\|_*:=\|(1+|y|^{n-2})\phi\|_{L^\infty(\mathbb{R}^n)},$$ where $\frac{n}{2}<q<n$ is a fixed number.
Part \[Existence\] of the paper is devoted to prove that solves , and Part \[nondegeneracyProof\] concerns the proof of its nondegeneracy.
\[Existence\]
To prove that is a solution of we use a Lyapunov-Schmidt reduction method, following the ideas of [@dPMPP]. We linearize the equation around a first approximation and take advantage of the invertibility tools available for this setting. Then, performing a careful analysis of the error of the approximation and of the non linear terms we solve the problem by a fixed point argument. Let us point out that the precise scaling of the parameters $\mu$ and $\lambda$ plays a fundamental role here.
Recalling the definitions given in , and , the main result of this part can be stated as follows.
\[mainThm\] Let $n{\geqslant}4$, and let $k, h$ be positive integers so that $k=O(h)$. Then, for sufficiently large $k$ and $h$ there is a finite energy solution of the form $$u_{k,h}(x)=U(x)-\sum_{j=1}^k U_{\mu,\xi_j}(x)-\sum_{l=1}^h U_{\lambda,\eta_l}(x)+o_k(1)+o_h(1),$$ where $o_k(1)$ and $o_h(1)$ denote quantities that tend to zero when $k$ and $h$ tend to infinity respectively.
Error of the approximation {#error}
==========================
Denote $$U_*(y):=U(y)-\sum_{j=1}^k U_{\mu,\xi_j}(y)-\sum_{l=1}^h U_{\lambda,\eta_l}(y),$$ and suppose that the solution $u$ we are looking for has the form $$u=U_*+\phi,$$ where $\phi$ is a small function when compared with $U_*$. Then solving equation is equivalent to find $\phi$ such that $$\label{probPhi}
\Delta\phi+p\gamma |U_*|^{p-1}\phi +E+\gamma N(\phi)=0,$$ where $$\begin{split}
E&:=\Delta U_*+\gamma|U_*|^{p-1}U_*,\\
N(\phi)&:=|U_*+\phi|^{p-1}(U_*+\phi)-|U_*|^{p-1}U_*-p|U_*|^{p-1}\phi.
\end{split}$$ In this section we try to estimate the error term $E$. In particular, $$\begin{split}
\gamma^{-1}E=&\left|U-\sum_{j=1}^kU_{\mu,\xi_j}-\sum_{l=1}^h U_{\lambda,\eta_l}\right|^{p-1}\left(U-\sum_{j=1}^kU_{\mu,\xi_j}-\sum_{l=1}^h U_{\lambda,\eta_l}\right)\\
&-U^p-\sum_{j=1}^kU_{\mu,\xi_j}^p-\sum_{l=1}^h U_{\lambda,\eta_l}^p
\end{split}$$ We divide the study of the error in three diffentent regions. Roughly speaking, we will estimate first the $\|\cdot\|_{**}$ norm of the error far from the points $\xi_j$ and $\eta_l$, then around $\xi_j$, and finally around $\eta_l$, for any $j=1,\ldots,k$ and $l=1,\ldots,h$. Indeed, let ${\overline{\alpha}}$ and ${\hat{\alpha}}$ be positive numbers independent of $k$ and $h$.
[*Exterior region:*]{} $y\in \{\cap_{j=1}^k\{|y-\xi_j|>\frac{{\overline{\alpha}}}{k}\}\}\cap\{\cap_{l=1}^h\{|y-\eta_l|>\frac{{\hat{\alpha}}}{h}\}\}.$
For $y$ in this region we can estimate $$\begin{split}\label{errExtEst}
|E|{\leqslant}&\,C\left[\frac{1}{(1+|y|^2)^2}+\bigg|\sum_{j=1}^k\frac{\mu^{\frac{n-2}{2}}}{|y-\xi_j|^{n-2}}\bigg|^{\frac{4}{n-2}}+\bigg|\sum_{l=1}^h\frac{\lambda^{\frac{n-2}{2}}}{|y-\eta_l|^{n-2}}\bigg|^{\frac{4}{n-2}}\right]\\
&\,\,\cdot\left(\sum_{j=1}^k\frac{\mu^{\frac{n-2}{2}}}{|y-\xi_j|^{n-2}}+\sum_{l=1}^h\frac{\lambda^{\frac{n-2}{2}}}{|y-\eta_l|^{n-2}}\right)\\
{\leqslant}& \,\frac{C}{(1+|y|^2)^2}\left[\sum_{j=1}^k\frac{\mu^{\frac{n-2}{2}}}{|y-\xi_j|^{n-2}}+\sum_{l=1}^h\frac{\lambda^{\frac{n-2}{2}}}{|y-\eta_l|^{n-2}}\right],
\end{split}$$ where in the last inequality we have used that here $$\sum_{j=1}^k\frac{\mu^{\frac{n-2}{2}}}{|y-\xi_j|^{n-2}}{\leqslant}C\left(1-\frac{k-1}{k^{n-2}}\right),\qquad \sum_{l=1}^h\frac{\lambda^{\frac{n-2}{2}}}{|y-\eta_l|^{n-2}} {\leqslant}C\left(1-\frac{h-1}{h^{n-2}}\right).$$ Thus $$\begin{split}\label{errExt}
\|(1+|y|)^{n+2-\frac{2n}{q}}E&\|_{L^q(\{\cap_{j=1}^k\{|y-\xi_j|>\frac{{\overline{\alpha}}}{k}\}\}\cap\{\cap_{l=1}^h\{|y-\eta_l|>\frac{{\hat{\alpha}}}{h}\}\})}\\
&{\leqslant}C\left(\frac{\mu^{\frac{n-2}{2}}k^{n-2}}{k^{\frac{n}{q}-1}}\textcolor{black}{+\frac{\lambda^{\frac{n-2}{2}}h^{n-2}}{h^{\frac{n}{q}-1}}}\right)\\
&{\leqslant}C(k^{1-\frac{n}{q}}\textcolor{black}{+h^{1-\frac{n}{q}}}).
\end{split}$$
[*Interior regions around $\xi_j$*]{}: $y\in \{|y-\xi_j|<\frac{{\overline{\alpha}}}{k}\}$ for some $j=1,\ldots,k$.
Let $j$ be fixed. For some $s\in (0,1)$ we have $$\begin{split}
\gamma^{-1}E=&\,p(U_{\mu,\xi_j}+s(-\sum_{i\neq j} U_{\mu,\xi_i}+U-\sum_{l=1}^h U_{\lambda,\eta_l}))^{p-1}(-\sum_{i\neq j}U_{\mu,\xi_i}+U-\sum_{l=1}^h U_{\lambda,\eta_l})\\
&-U^p-\sum_{i\neq j}U_{\mu,\xi_i}^p-\sum_{l=1}^h U_{\lambda,\eta_l}^p.
\end{split}$$ Let us define $${\overline{E}}_j(y):=\mu^{\frac{n+2}{2}}E(\xi_j+\mu y), \qquad |y|<\frac{{\overline{\alpha}}}{\mu k}.$$ Thus, $$\begin{split}\label{ErrIntXi}
\gamma^{-1}{\overline{E}}_j(y)=&\,p\left(-U(y)+s(-\sum_{i\neq j}U(y-\mu^{-1}(\xi_j-\xi_i))+\mu^{\frac{n-2}{2}}U(\xi_j+\mu y)\right.\\
&\left. -\sum_{l=1}^h\mu^{\frac{n-2}{2}}\lambda^{-\frac{n-2}{2}}U(\lambda^{-1}(\xi_j+\mu y-\eta_l)\right)^{p-1}(-\sum_{i\neq j}U(y-\mu^{-1}(\xi_j-\xi_i))\\
&+\mu^{\frac{n-2}{2}}U(\xi_j+\mu y)-\sum_{l=1}^h\mu^{\frac{n-2}{2}}\lambda^{-\frac{n-2}{2}}U(\lambda^{-1}(\xi_j+\mu y-\eta_l))\\
&-\mu^{\frac{n+2}{2}}U^p(\xi_j+\mu y)-\sum_{i\neq j} U^p(y-\mu^{-1}(\xi_i-\xi_j))\\
&-\sum_{l=1}^h\mu^{\frac{n+2}{2}}\lambda^{-\frac{n+2}{2}}U^p(\lambda^{-1}(\xi_j+\mu y-\eta_l),
\end{split}$$ and consequently $$\begin{split}\label{errIntEst}
|\overline{E}_j(y)|&{\leqslant}C\left[\frac{(k\mu)^{n-2}+(\mu\lambda)^{\frac{n-2}{2}}h}{1+|y|^4}+\mu^{\frac{n+2}{2}}+(\mu\lambda)^{\frac{n+2}{2}}h^{\frac{n+2}{n-2}}\right]\\
&{\leqslant}C\left[\frac{\mu^{\frac{n-2}{2}}}{1+|y|^4}+\mu^{\frac{n+2}{2}}\right].
\end{split}$$ Noticing that $h=O(k)$ we can compute $$\label{errInt1}
\|(1+|y|)^{n+2-\frac{2n}{q}}\overline{E}_j\|_{L^q(|y|<\frac{{\overline{\alpha}}}{\mu k})}{\leqslant}C\mu^{\frac{n}{2q}}{\leqslant}Ck^{-\frac{n}{q}}.$$
[*Interior regions around $\eta_l$*]{}: $y\in \{|y-\eta_l|<\frac{{\hat{\alpha}}}{h}\}$ for some $l=1,\ldots,h$.
The estimates in this region follow analogously to the previous case, but interchanging the role of $\mu$, $k$ and $\lambda$, $h$. Thus, considering $$\hat{E}_l(y):=\lambda^{\frac{n+2}{2}}E(\eta_l+\mu y), \qquad |y|<\frac{{\hat{\alpha}}}{\lambda h},$$ we get $$\begin{split}\label{intErrEst2}
|\hat{E}_l(y)|&{\leqslant}C\left[\frac{(h\lambda)^{n-2}+(\mu\lambda)^{\frac{n-2}{2}}k}{1+|y|^4}+\lambda^{\frac{n+2}{2}}+(\mu\lambda)^{\frac{n+2}{2}}k^{\frac{n+2}{n-2}}\right]\\
&{\leqslant}C\left[ \frac{\lambda^{\frac{n-2}{2}}}{1+|y|^4}+\lambda^{\frac{n+2}{2}}\right]
\end{split}$$ and therefore $$\label{errInt2}
\|(1+|y|)^{n+2-\frac{2n}{q}}\hat{E}_l\|_{L^q(|y|<\frac{{\hat{\alpha}}}{\lambda h})}{\leqslant}C\lambda^{\frac{n}{2q}}{\leqslant}Ch^{-\frac{n}{q}}.$$
Building the solution {#sec3}
=====================
Recall from Section \[error\] that to find a solution to we will prove the existence of a function $\phi$ that solves . We will try to build this function in a special form.
Let $\zeta(s)$ be a smooth function such that $\zeta(s)=1$ for $s>1$ and $\zeta(s)=0$ for $s>2$, and let $\overline{\alpha}, \hat{\alpha}>0$ be fixed numbers independent of $k$ and $h$. Define $$\begin{split}
\overline{\zeta}_j(y)&:=\begin{cases}
\zeta(k\overline{\alpha}^{-1}|y|^{-2}|y-\xi_j|y|^2|)\hbox{ if }|y|>1,\\
\zeta(k\overline{\alpha}^{-1}|y-\xi_j|)\hbox{ if }|y|{\leqslant}1,
\end{cases}\\
\hat{\zeta}_l(y)&:=\begin{cases}
\zeta(h\hat{\alpha}^{-1}|y|^{-2}|y-\eta_l|y|^2|)\hbox{ if }|y|>1,\\
\zeta(h\hat{\alpha}^{-1}|y-\eta_l|)\hbox{ if }|y|{\leqslant}1.
\end{cases}\end{split}$$ A function of the form $$\label{defPhi}
\phi=\sum_{j=1}^k\overline{\phi}_j+\sum_{l=1}^h\hat{\phi}_l+\psi$$ is a solution of if we solve the system $$\label{system1}
\Delta {\overline{\phi}}_j+p\gamma |U_*|^{p-1}{\overline{\zeta}}_j{\overline{\phi}}_j+{\overline{\zeta}}_j\left[p\gamma |U_*|^{p-1}\psi+E+\gamma N(\phi)\right]=0, \; ยบ;j=1,\ldots,k,$$ $$\label{system2}
\Delta {\hat{\phi}}_l+p\gamma |U_*|^{p-1}{\hat{\zeta}}_l{\hat{\phi}}_l+{\hat{\zeta}}_l\left[p\gamma |U_*|^{p-1}\psi+E+\gamma N(\phi)\right]=0, \; \;l=1,\ldots,h,$$ $$\label{system3}\begin{split}
\Delta\psi&+p\gamma U^{p-1}\psi+[p\gamma(|U_*|^{p-1}-U^{p-1})(1-\sum_{j=1}^k{\overline{\zeta}}_j-\sum_{l=1}^h{\hat{\zeta}}_l)\\
&+p\gamma U^{p-1}(\sum_{j=1}^k{\overline{\zeta}}_j+\sum_{l=1}^h{\hat{\zeta}}_l)]\psi+ p\gamma |U_*|^{p-1}\sum_{j=1}^k(1-{\overline{\zeta}}_j){\overline{\phi}}_j\\
&+p\gamma |U_*|^{p-1}\sum_{l=1}^h(1-{\hat{\zeta}}_l){\hat{\phi}}_l+(1-\sum_{j=1}^k{\overline{\zeta}}_j-\sum_{l=1}^h{\hat{\zeta}}_l)(E+\gamma N(\phi))=0.
\end{split}$$ We assume in addition the following symmetry properties on ${\overline{\phi}}_j$ and ${\hat{\phi}}_l$, $$\label{cond1}
{\overline{\phi}}_j(\overline{y},\hat{y},y')={\overline{\phi}}_1(e^{-\frac{2\pi \textcolor{black}{(j-1)}}{k}i} \overline{y},\hat{y},y'),\qquad j=1,\ldots\,\textcolor{black}{k},$$ where $$\begin{split}\label{cond2}
&{\overline{\phi}}_1(y_1,\ldots,y_j,\ldots,y_n)={\overline{\phi}}_1(y_1,\ldots,-y_j,\ldots,y_n),\qquad j=2,4,5,\ldots,n,\\
&{\overline{\phi}}_1(y)=|y|^{2-n}{\overline{\phi}}_1(|y|^{-2}y),\\
&{\overline{\phi}}_1(\overline{y},\hat{y},y')={\overline{\phi}}_1(\overline{y},e^{\frac{2\pi (l-1)}{h}i}\hat{y},y'),\qquad \l=1,\ldots,h,
\end{split}$$ and $$\label{cond3}
{\hat{\phi}}_j(\overline{y},\hat{y},y')={\hat{\phi}}_1(\overline{y},e^{-\frac{2\pi (l-1)}{h}i} \hat{y},y'),\qquad l=1,\ldots\,h,$$ where $$\begin{split}\label{cond4}
&{\hat{\phi}}_1(y_1,\ldots,y_j,\ldots,y_n)={\hat{\phi}}_1(y_1,\ldots,-y_j,\ldots,y_n),\qquad j=2,4,5,\ldots,n,\\
&{\hat{\phi}}_1(y)=|y|^{2-n}{\hat{\phi}}_1(|y|^{-2}y),\\
&{\hat{\phi}}_1(\overline{y},\hat{y},y')={\hat{\phi}}_1(e^{\frac{2\pi (j-1)}{k}i}\overline{y},\hat{y},y'),\qquad j=1,\ldots,k.
\end{split}$$
Assume in addition that $$\begin{split}\label{cond5}
&\|\overline{{\overline{\phi}}}_1\|_*{\leqslant}\rho, \qquad \overline{{\overline{\phi}}}_1(y):=\mu^{\frac{n-2}{2}}{\overline{\phi}}_1(\xi_1+\mu y),\\
&\|\hat{{\hat{\phi}}}_1\|_*{\leqslant}\rho, \qquad \hat{{\hat{\phi}}}_1(y):=\lambda^{\frac{n-2}{2}}{\hat{\phi}}_1(\eta_1+\lambda y),
\end{split}$$ for $\rho>0$ small.
There exist constants $k_0,h_0,C,\rho_0$ such that, for all $k{\geqslant}k_0$ and $h{\geqslant}h_0$, if ${\overline{\phi}}_j$, $j=1,\ldots,k$, and ${\hat{\phi}}_l$, $l=1,\ldots,h$ satisfy conditions - with $\rho<\rho_0$ then there exists a unique solution $\psi=\Psi({\overline{\overline{\phi}}}_1,{\hat{\hat{\phi}}}_1)$ to equation , that satisfies the symmetries $$\label{condpsi1}
\psi(y_1,\ldots,y_\alpha,\ldots)=\psi(y_1,\ldots,-y_\alpha,\ldots),\;\; \alpha=5,\ldots,n,$$ $$\label{condpsi2}
\psi(\overline{y},\hat{y},y')=\psi(e^{\frac{2\pi (j-1)}{k}i}\overline{y},\hat{y},y'),\;\; j=1,\ldots,k,$$ $$\label{condpsi3}
\psi(\overline{y},\hat{y},y')=\psi(\overline{y},e^{\frac{2\pi (l-1)}{h}i}\hat{y},y'),\;\; l=1,\ldots,h,$$ $$\label{condpsi4}
\psi(y)=|y|^{2-n}\psi(|y|^{-2}y),$$ and such that $$\label{opPsi}
\|\psi\|_*{\leqslant}C\left[\|{\overline{\overline{\phi}}}_1\|_*+\|{\hat{\hat{\phi}}}_1\|_*+k^{1-\frac{n}{q}}+h^{1-\frac{n}{q}}\right].$$ Moreover, the operator $\Psi$ satisfies $$\label{lipPsi}
\|\Psi({\overline{\overline{\phi}}}_1^1,{\hat{\hat{\phi}}}_1^1)-\Psi({\overline{\overline{\phi}}}_1^2,{\hat{\hat{\phi}}}_1^2)\|_*{\leqslant}C(\|{\overline{\overline{\phi}}}_1^1-{\overline{\overline{\phi}}}_1^2\|_*+\|{\hat{\hat{\phi}}}_1^1-{\hat{\hat{\phi}}}_1^2\|_*).$$
We write equation as $$\label{probPsi}
\Delta\psi+p\gamma U^{p-1}\psi+V(y)\psi+ p\gamma |U_*|^{p-1}(\sum_{j=1}^k(1-{\overline{\zeta}}_j){\overline{\phi}}_j+\sum_{l=1}^h(1-{\hat{\zeta}}_l){\hat{\phi}}_l)+M(\psi)=0,$$ where $$\begin{split}
V(y)&:=p\gamma(|U_*|^{p-1}-U^{p-1})(1-\sum_{j=1}^k{\overline{\zeta}}_j-\sum_{l=1}^h{\hat{\zeta}}_l)+p\gamma U^{p-1}(\sum_{j=1}^k{\overline{\zeta}}_j+\sum_{l=1}^h{\hat{\zeta}}_l),\\
&=:V_1(y)+V_2(y),
\end{split}$$ and $$M(\psi):=(1-\sum_{j=1}^k{\overline{\zeta}}_j-\sum_{l=1}^h{\hat{\zeta}}_l)(E+\gamma N(\phi))=0.$$ Consider first the problem $$\label{linh}
\Delta\psi+p\gamma U^{p-1}\psi=h,$$ where $h$ is a function satisfying -, $\|h\|_{**}<+\infty$ and $$\label{propH}
h(y)=|y|^{-n-2}h(|y|^{-2}y).$$ Let $$\label{defZ}
Z_\alpha:=\partial_{y_\alpha}U,\;\;\alpha=1,\cdots,n\;\;\hbox{and}\;\;Z_{0}=y\cdot\nabla U+\frac{n-2}{2}U.$$ Due to the oddness of $Z_\alpha$ and assumption on $h$ it yields $$\int_{\mathbb{R}^n}Z_\alpha h=0\hbox{ for all }\alpha=5,\cdots,n.$$ The cases $\alpha=0,1,2,3,4$ also vanish proceeding as in the proof of [@dPMPP Lemma 4.1] as a consequence of -. Thus we can apply the linear existence result [@dPMPP Lemma 3.1] to ensure the existence of a unique solution $\psi$ to such that $$\int_{\mathbb{R}^n}U^{p-1}Z_\alpha\psi=0\hbox{ for all }\alpha=0,1,\ldots,n,$$ and $\|\psi\|_*{\leqslant}C\|h\|_{**}$. Notice in addition that the functions $$\begin{split}
\psi_\alpha(y)&:=\psi(\overline{y},\hat{y},\ldots,-y_i,\ldots,y_n),\;\; \alpha=5,\cdots,n,\\
\psi_{12j}(y)&:=\psi(e^{\frac{2\pi (j-1)}{k}i}\overline{y},\hat{y},y'),\;\; j=1,\ldots,k,\\
\psi_{34l}(y)&:=\psi(\overline{y},e^{\frac{2\pi (l-1)}{h}i}\hat{y},y'),\;\; l=1,\ldots,h,\\
\psi_{n+1}(y)&:=|y|^{2-n}\psi(|y|^{-2}y),
\end{split}$$ also satisfy and thus, by the uniqueness, $$\psi=\psi_\alpha=\psi_{12j}=\psi_{34l}=\psi_{n+1}$$ for all $\alpha=5,\ldots,n$, $j=1,\ldots,k$, $l=1,\ldots,h$, i.e., $\psi$ satisfies -. Therefore, has a unique bounded solution $\psi=T(h)$ satisfying symmetries - and $$\|\psi\|_*{\leqslant}C\|h\|_{**}$$ for a constant depending only on $q$ and $n$.
We will solve by means of a fixed point argument, writting $$\psi=-T(V\psi+ p\gamma |U_*|^{p-1}(\sum_{j=1}^k(1-{\overline{\zeta}}_j){\overline{\phi}}_j+\sum_{l=1}^h(1-{\hat{\zeta}}_l){\hat{\phi}}_l)+M(\psi))=:{{\mathscrM}}(\psi),$$ $\psi\in X,$ where $X$ is the space of continuous functions $\psi$ with $\|\psi\|_*<+\infty$ satisfying -. Thanks to the special form of $U_*$ and to the symmetry assumptions on ${\overline{\phi}}_j$ and ${\hat{\phi}}_l$, $$V\psi+ p\gamma |U_*|^{p-1}(\sum_{j=1}^k(1-{\overline{\zeta}}_j){\overline{\phi}}_j+\sum_{l=1}^h(1-{\hat{\zeta}}_l){\hat{\phi}}_l)+M(\psi)$$ satisfies - and if $\psi\in X$, and ${{\mathscrM}}$ is well defined. Actually, we claim that ${{\mathscrM}}$ is a contraction mapping in the $\| \|_*$ norm in a small ball around the origin in $X$. Indeed, $$\begin{split}
|V_1(y)|&{\leqslant}\gamma p(p-1)\bigg|U-s(\sum_{j=1}^k U_{\mu,\xi_j}+\sum_{l=1}^h U_{\lambda,\eta_l})\bigg|^{p-2}\left(\sum_{j=1}^k U_{\mu,\xi_j}+\sum_{l=1}^h U_{\lambda,\eta_l}\right)\\
&{\leqslant}CU^{p-2}(y)\left(\sum_{j=1}^k\frac{\mu^{\frac{n-2}{2}}}{|y-\xi_j|^{n-2}}+\sum_{l=1}^h\frac{\lambda^{\frac{n-2}{2}}}{|y-\eta_l|^{n-2}}\right),
\end{split}$$ and thus $$|V_1\psi(y)|{\leqslant}C\|\psi\|_*U^{p-1}(y)\left(\sum_{j=1}^k\frac{\mu^{\frac{n-2}{2}}}{|y-\xi_j|^{n-2}}+\sum_{l=1}^h\frac{\lambda^{\frac{n-2}{2}}}{|y-\eta_l|^{n-2}}\right).$$ Proceeding as in we get $$\label{V1}
\|V_1\psi\|_{**}{\leqslant}C\|\psi\|_*(k^{1-\frac{n}{q}}+h^{1-\frac{n}{q}}).$$ On the other hand, $$\begin{split}\label{V2}
\|V_2\psi\|_{**}&{\leqslant}\sum_{j=1}^k\|pU^{p-1}\psi{\overline{\zeta}}_j\|_{**}+\sum_{l=1}^h\|pU^{p-1}\psi{\hat{\zeta}}_l\|_{**}\\
&{\leqslant}C\|\psi\|_*(k^{1-\frac{n}{q}}+h^{1-\frac{n}{q}}),
\end{split}$$ and putting together and we conclude $$\label{V}
\|V\psi\|_{**}{\leqslant}C\|\psi\|_*(k^{1-\frac{n}{q}}+h^{1-\frac{n}{q}}).$$ Assume $|y-\xi_j|>\frac{{\overline{\alpha}}}{2k}$ and $|y-\eta_l|>\frac{{\hat{\alpha}}}{2h}$ for all $j$ and $l$. We knew that in this region $$\|E\|_{**}{\leqslant}C(k^{1-\frac{n}{q}}+h^{1-\frac{n}{q}}).$$ Moreover, $$\bigg|N\left(\sum_{j=1}^k{\overline{\phi}}_j+\sum_{l=1}^h{\hat{\phi}}_l+\psi\right)\bigg|{\leqslant}C U^{p-2}\left(\bigg|\sum_{j=1}^k{\overline{\phi}}_j\bigg|^2+\bigg|\sum_{l=1}^h{\hat{\phi}}_l\bigg|^2+|\psi|^2\right).$$ From , we get $$|{\overline{\phi}}_j(y)| {\leqslant}C \|{\overline{\overline{\phi}}}_1\|_*\frac{\mu^\frac{n-2}{2}}{\mu^{n-2} + |y-\xi_j|^{n-2}} , \quad
|{\hat{\phi}}_l(y)| {\leqslant}C\|{\hat{\hat{\phi}}}_1\|_*\frac{\lambda^\frac{n-2}{2}}{\lambda^{n-2} + |y-\eta_l|^{n-2}}.$$ Moreover, $$U^{p-2}|\psi|^2{\leqslant}U^p\|\psi\|_*^2.$$ Hence, proceeding again as in , we obtain $$\label{M}
\|M(\psi)\|_{**}{\leqslant}Ck^{1-\frac{n}{q}}(1+\|{\overline{\overline{\phi}}}_1\|_*^2)+Ch^{1-\frac{n}{q}}(1+\|{\hat{\hat{\phi}}}_1\|_*^2)+C\|\psi\|_*^2.$$ Likewise, if $|y-\xi_j|>\frac{{\overline{\alpha}}}{2k}$ and $|y-\eta_l|>\frac{{\hat{\alpha}}}{2h}$, $$\label{Ustar}
\||U_*|^{p-1}(\sum_{j=1}^k{\overline{\phi}}_j+\sum_{l=1}^h{\hat{\phi}}_l)\|_{**}{\leqslant}Ck^{1-\frac{n}{q}}\|{\overline{\overline{\phi}}}_1\|_*+Ch^{1-\frac{n}{q}}\|{\hat{\hat{\phi}}}_1\|_*.$$ Moreover, for $\psi_1,\psi_2$ satisfying $\|\psi_1\|<\rho$, $\|\psi_2\|<\rho$ it follows $$\|M(\psi_1)-M(\psi_2)\|_{**}{\leqslant}C\rho \|\psi_1-\psi_2\|_*.$$ Joining , and , we see that for $\rho$ small enough the operator ${{\mathscrM}}$ defines a contraction map in the set of functions $\psi\in X$ with $$\|\psi\|_*{\leqslant}C[\|{\overline{\overline{\phi}}}_1\|_*+k^{1-\frac{n}{q}}+\|{\hat{\hat{\phi}}}_1\|_*+h^{1-\frac{n}{q}}],\;\;\|\overline{{\overline{\phi}}}_1\|_*<\rho,\;\;\|\hat{{\hat{\phi}}}_1\|_*<\rho,$$ for $\rho$ small. Therefore, there exists a solution of satisfying conditions -. The Lipschitz condition easily follows.
Consider the operator $\Psi({\overline{\overline{\phi}}}_1,{\hat{\hat{\phi}}}_1)$. Equations and reduce to solve one of each, for example for ${\overline{\phi}}_1$ and ${\hat{\phi}}_1$.
We try to solve first $$\Delta {\overline{\phi}}_1+p\gamma |U_*|^{p-1}{\overline{\zeta}}_1{\overline{\phi}}_1+{\overline{\zeta}}_1\left[p\gamma |U_*|^{p-1}\Psi({\overline{\overline{\phi}}}_1,{\hat{\hat{\phi}}}_1)+E+\gamma N(\phi)\right]=0\hbox{ in }\mathbb{R}^n,$$ or equivalently, $$\label{eq1}
\Delta {\overline{\phi}}_1+p\gamma |U_{\mu,\xi_1}|^{p-1}{\overline{\phi}}_1+{\overline{\zeta}}_1 E+\gamma\overline{{{\mathscrN}}}(\overline{{\overline{\phi}}}_1,\hat{{\hat{\phi}}}_1)=0,$$ where $$\overline{{{\mathscrN}}}(\overline{{\overline{\phi}}}_1,\hat{{\hat{\phi}}}_1):=p(|U_*|^{p-1}{\overline{\zeta}}_1-|U_{\mu,\xi_1}|^{p-1}){\overline{\phi}}_1+{\overline{\zeta}}_1\left[p|U_*|^{p-1}\Psi({\overline{\overline{\phi}}}_1,{\hat{\hat{\phi}}}_1)+N(\phi)\right].$$ Consider first a general function $\overline{h}$ and the problem $$\label{projProb}
\Delta{\overline{\phi}}+p\gamma U_{\mu,\xi_1}^{p-1}{\overline{\phi}}+\overline{h}=\overline{c}_{0}U_{\mu,\xi_1}^{p-1}\overline{Z}_{0}\hbox{ in }\mathbb{R}^n,$$ where $$\overline{Z}_{0}(y):=\mu^{-\frac{n-2}{2}}Z_{0}\left(\frac{y-\xi_1}{\mu}\right),\qquad \overline{c}_{0}:=\frac{\int_{\mathbb{R}^n}\overline{h}\,\overline{Z}_{0}}{\int_{\mathbb{R}^n}U_{\mu,\xi_1}^{p-1}\overline{Z}_{0}^2},$$ with $Z_0$ defined in .
\[proj1\] Suppose that $\overline{h}$ is even with respect to each of the variables $y_2,y_4,y_5,\ldots,y_n$ and such that $$\label{hypRot}
\overline{h}(y)=|y|^{-n-2}{\overline{h}}(|y|^{-2}y),\qquad {\overline{h}}(y)={\overline{h}}(\overline{y},e^{\frac{2\pi(l-1)}{h}}\hat{y},y'), \;\;l=1,\ldots,h.$$ Assume that $$h(y):=\mu^{\frac{n+2}{2}}\overline{h}(\xi_1+\mu y)$$ satisfies $\|h\|_{**}<+\infty$. Then problem has a unique solution ${\overline{\phi}}:=\overline{T}(\overline{h})$ that is even with respect to the variables $y_2,y_4,y_5,\ldots,y_n$, invariant under Kelvin’s tranform, i.e., $${\overline{\phi}}(y)=|y|^{2-n}{\overline{\phi}}(|y|^{-2}y),$$ and with ${\overline{\overline{\phi}}}(y):=\mu^{\frac{n-2}{2}}{\overline{\phi}}(\xi_1+\mu y)$ satisfying $$\int_{\mathbb{R}^n}{\overline{\overline{\phi}}}U^{p-1}Z_{0}=0,\qquad \|{\overline{\overline{\phi}}}\|_{*}{\leqslant}C\|h\|_{**}.$$
We assume with no loss of generality that $$\int_{\mathbb{R}^n}{\overline{h}}\,\overline{Z}_{0}=0,\qquad \hbox{ i.e., }\qquad\overline{c}_{0}=0.$$ Thus, equation is equivalent to $$\Delta {\overline{\overline{\phi}}}+p\gamma|U|^{p-1}{\overline{\overline{\phi}}}=-h \qquad \hbox{in }\mathbb{R}^n.$$ Due to the evenness of $h$ we know that $$\label{vanZ}
\int_{\mathbb{R}^n}h Z_\alpha=0, \qquad \alpha=2,4,5,\cdots,n.$$ The proof of $$\int_{\mathbb{R}^n}h Z_1=0$$ follows exactly as in the proof of [@dPMPP Lemma 4.2], so we focus on the case $\alpha=3$.
Indeed, denote by $w_{\mu}(y):=\mu^{-\frac{n-2}{2}}U(\mu^{-1}y)$, and $J(t):=\int_{\mathbb{R}^n}w_{\mu}(y-\xi_1+te_3){\overline{h}}(y)\,dy $. Notice first that $$\label{der1}
\frac{d}{dt} J(t)\bigg|_{t=0}=\int_{\mathbb{R}^n}\partial_{y_3}w_{\mu}(y-\xi_1){\overline{h}}(y)\,dy=\int_{\mathbb{R}^n}hZ_3.$$ On the other hand, defining $\tilde{y}:=(\overline{y},e^{\frac{2\pi(l-1)}{h}}\hat{y},y')$ for some $l=2,3,\ldots,h$, it can be checked that $$|\tilde{y}-\xi_1+te_3|^2=|y-\xi_1+t\tilde{e}|^2,\qquad t\in{{\mathbb R}}^n,$$ where $\tilde{e}:=(0,0,\cos(\frac{2\pi(l-1)}{h}),-\sin(\frac{2\pi(l-1)}{h}),0,\ldots)$. Thus, after a change of variables, by , $$\begin{split}
J(t)&=\int_{\mathbb{R}^n}w_{\mu}(\tilde{y}-\xi_1+te_3){\overline{h}}(\tilde{y})\,d\tilde{y}
=\int_{\mathbb{R}^n}w_{\mu}(y-\xi_1+t\tilde{e}){\overline{h}}(y)\,dy.
\end{split}$$ Differentiating here, $$\begin{split}
\frac{d}{dt}J(t)\bigg|_{t=0}&=\cos\left(\frac{2\pi(l-1)}{h}\right)\int_{\mathbb{R}^n}\partial_{y_3}w_{\mu}(y-\xi_1){\overline{h}}(y)\,dy\\
&\;\;\;-\sin\left(\frac{2\pi(l-1)}{h}\right)\int_{\mathbb{R}^n}\partial_{y_4}w_{\mu}(y-\xi_1){\overline{h}}(y)\,dy\\
&=\cos\left(\frac{2\pi(l-1)}{h}\right)\int_{\mathbb{R}^n}hZ_3\,dy-\sin\left(\frac{2\pi(l-1)}{h}\right)\int_{\mathbb{R}^n}hZ_4\,dy.
\end{split}$$ Applying and we conclude that necessarily $\int_{\mathbb{R}^n}h Z_3=0$. Thus, by [@dPMPP Lemma 3.1] there exists a unique solution ${\overline{\overline{\phi}}}$ satisfying $$\|{\overline{\overline{\phi}}}\|_*{\leqslant}C\|h\|_{**},\qquad \int_{\mathbb{R}^n}{\overline{\overline{\phi}}}U^{p-1}Z_{n+1}=0.$$ The invariance under Kelvin transform and the symmetries are obtained as a consequence of the uniqueness.
Likewise, we rewrite $$\Delta {\hat{\phi}}_1+p\gamma |U_*|^{p-1}{\hat{\zeta}}_1{\hat{\phi}}_1+{\hat{\zeta}}_1\left[p\gamma |U_*|^{p-1}\Psi({\overline{\overline{\phi}}}_1,{\hat{\hat{\phi}}}_1)+E+\gamma N(\phi)\right]=0\hbox{ in }\mathbb{R}^n,$$ as $$\label{eq2}
\Delta {\hat{\phi}}_1+p\gamma |U_{\lambda,\eta_1}|^{p-1}{\hat{\phi}}_1+{\hat{\zeta}}_1 E+\gamma\hat{{{\mathscrN}}}(\overline{{\overline{\phi}}}_1,\hat{{\hat{\phi}}}_1)=0,$$ with $$\hat{{{\mathscrN}}}(\overline{{\overline{\phi}}}_1,\hat{{\hat{\phi}}}_1):=p(|U_*|^{p-1}{\hat{\zeta}}_1-|U_{\lambda,\eta_1}|^{p-1}){\hat{\phi}}_1+{\hat{\zeta}}_1\left[p|U_*|^{p-1}\Psi({\overline{\overline{\phi}}}_1,{\hat{\hat{\phi}}}_1)+N(\phi)\right],$$ and we consider the problem $$\label{projProb2}
\Delta{\hat{\phi}}+p\gamma U_{\lambda,\eta_1}^{p-1}{\hat{\phi}}+\hat{h}=\hat{c}_{0}U_{\lambda,\eta_1}^{p-1}\hat{Z}_{0}\hbox{ in }\mathbb{R}^n,$$ where $\hat{h}$ is a general function and $$\hat{Z}_{0}(y):=\lambda^{-\frac{n-2}{2}}Z_{0}\left(\frac{y-\eta_1}{\lambda}\right),\qquad \hat{c}_{0}:=\frac{\int_{\mathbb{R}^n}\hat{h}\hat{Z}_{0}}{\int_{\mathbb{R}^n}U_{\lambda,\eta_1}^{p-1}\hat{Z}_{0}^2}.$$
\[proj2\] Suppose that $\hat{h}$ is even with respect to each of the variables $y_2,y_4,y_5,\ldots,y_n$ and such that $$\hat{h}(y)=|y|^{-n-2}{\hat{h}}(|y|^{-2}y),\qquad \hat{h}(y)=\hat{h}(e^{\frac{2\pi(j-1)}{k}}\overline{y},\hat{y},y'),\;\;j=1,\ldots,k.$$ Assume that $$h(y):=\lambda^{\frac{n+2}{2}}\hat{h}(\eta_1+\lambda y)$$ satisfies $\|h\|_{**}<+\infty$. Then problem has a unique solution ${\hat{\phi}}:=\hat{T}(\hat{h})$ that is even with respect to the variables $y_2,y_4,y_5,\ldots,y_n$, invariant under Kelvin’s tranform, i.e., $${\hat{\phi}}(y)=|y|^{2-n}{\hat{\phi}}(|y|^{-2}y),$$ and with ${\hat{\hat{\phi}}}(y):=\lambda^{\frac{n-2}{2}}{\hat{\phi}}(\eta_1+\lambda y)$ satisfying $$\int_{\mathbb{R}^n}{\hat{\hat{\phi}}}U^{p-1}Z_{0}=0,\qquad \|{\hat{\hat{\phi}}}\|_{*}{\leqslant}C\|h\|_{**}.$$
The proof of this result is analogous to the one for Lemma \[proj1\], interchanging the roles of $\mu$ and $\xi_1$ with $\lambda$ and $\eta_1$, so we skip it.
We use these lemmas to solve the projected versions of and , that is, $$\begin{split}\label{projEq}
&\Delta {\overline{\phi}}_1+p\gamma |U_{\mu,\xi_1}|^{p-1}{\overline{\phi}}_1+{\overline{\zeta}}_1 E+\gamma\overline{{{\mathscrN}}}(\overline{{\overline{\phi}}}_1,\hat{{\hat{\phi}}}_1)=\overline{c}_{0}U_{\mu,\xi_1}^{p-1}\overline{Z}_{0},\\
&\Delta {\hat{\phi}}_1+p\gamma |U_{\lambda,\eta_1}|^{p-1}{\hat{\phi}}_1+{\hat{\zeta}}_1 E+\gamma\hat{{{\mathscrN}}}(\overline{{\overline{\phi}}}_1,\hat{{\hat{\phi}}}_1)=\hat{c}_{0}U_{\lambda,\eta_1}^{p-1}\hat{Z}_{0},
\end{split}$$ in $\mathbb{R}^n$, with $$\overline{c}_{0}:=\frac{\int_{\mathbb{R}^n}({\overline{\zeta}}_1 E+\gamma\overline{{{\mathscrN}}}(\overline{{\overline{\phi}}}_1,\hat{{\hat{\phi}}}_1))\overline{Z}_{0}}{\int_{\mathbb{R}^n}U_{\mu,\xi_1}^{p-1}\overline{Z}_{0}^2},\qquad\hat{c}_{0}:=\frac{\int_{\mathbb{R}^n}({\hat{\zeta}}_1 E+\gamma\hat{{{\mathscrN}}}(\overline{{\overline{\phi}}}_1,\hat{{\hat{\phi}}}_1))\hat{Z}_{0}}{\int_{\mathbb{R}^n}U_{\lambda,\eta_1}^{p-1}\hat{Z}_{0}^2}.$$
\[existProjProb\] There exists a unique solution $\phi_1=(\overline{{\overline{\phi}}}_1,\hat{{\hat{\phi}}}_1)=(\overline{{\overline{\phi}}}_1(\delta),\hat{{\hat{\phi}}}_1(\varepsilon))$ to such that $$\|\phi_1\|_*:=\|\overline{{\overline{\phi}}}_1\|_*+\|\hat{{\hat{\phi}}}_1\|_*{\leqslant}C(k^{-\frac{n}{q}}+h^{-\frac{n}{q}}),$$ and $$\label{estN}
\|\overline{{{\mathscrN}}}(\overline{{\overline{\phi}}}_1,\hat{{\hat{\phi}}}_1)\|_{**}{\leqslant}C k^{-\frac{2n}{q}},\qquad \|\hat{{{\mathscrN}}}(\overline{{\overline{\phi}}}_1,\hat{{\hat{\phi}}}_1)\|_{**}{\leqslant}Ch^{-\frac{2n}{q}}.$$
Denote by ${\overline{T}}$ and ${\hat{T}}$ the linear operators predicted by Lemma \[proj1\] and Lemma \[proj2\] respectively. Thus, solving is equivalent to solve the fixed point problem $$\phi_1=
\left(\begin{array}{c}
{\overline{\phi}}_1\\
{\hat{\phi}}_1
\end{array}
\right)
=\left(
\begin{array}{c}
{\overline{T}}({\overline{\zeta}}_1E+\gamma\overline{{{\mathscrN}}}(\overline{{\overline{\phi}}}_1,\hat{{\hat{\phi}}}_1))\\
{\hat{T}}({\hat{\zeta}}_1E+\gamma\hat{{{\mathscrN}}}(\overline{{\overline{\phi}}}_1,\hat{{\hat{\phi}}}_1))
\end{array}
\right)
=\left(
\begin{array}{c}
\overline{{{\mathscrM}}}(\overline{{\overline{\phi}}}_1,\hat{{\hat{\phi}}}_1)\\
\hat{{{\mathscrM}}}(\overline{{\overline{\phi}}}_1,\hat{{\hat{\phi}}}_1)
\end{array}
\right)
=:{{\mathscrM}}(\phi_1).$$ Let us focus first on $\overline{{{\mathscrM}}}(\overline{{\overline{\phi}}}_1,\hat{{\hat{\phi}}}_1):={\overline{T}}({\overline{\zeta}}_1E+\gamma\overline{{{\mathscrN}}}(\overline{{\overline{\phi}}}_1,\hat{{\hat{\phi}}}_1))$. Recall that $$\overline{{{\mathscrN}}}(\overline{{\overline{\phi}}}_1,\hat{{\hat{\phi}}}_1):=p(|U_*|^{p-1}{\overline{\zeta}}_1-|U_{\mu,\xi_1}|^{p-1}){\overline{\phi}}_1+{\overline{\zeta}}_1\left[p|U_*|^{p-1}\Psi({\overline{\overline{\phi}}}_1,{\hat{\hat{\phi}}}_1)+N(\phi)\right].$$ Denote in general $$\tilde{f}(y)=\mu^{\frac{n+2}{2}}f(\xi_1+\mu y).$$ Consider $$f_1(y):=p{\overline{\zeta}}_1(|U_*|^{p-1}-|U_{\mu,\xi_1}|^{p-1}){\overline{\phi}}_1.$$ For $|y|<\frac{{\overline{\alpha}}}{\mu k}$, $$\begin{split}
|\tilde{f}_1(y)|=\bigg|&p((U(y)+\sum_{j=2}^kU(y+\mu^{-1}(\xi_1-\xi_j))\\
&+\sum_{l=1}^h\mu^{\frac{n-2}{2}}\lambda^{-\frac{n-2}{2}}U(\lambda^{-1}(\xi_1+\mu y-\eta_l))\\
&-\mu^{\frac{n-2}{2}}U(\xi_1+\mu y))^{p-1}-U^{p-1}(y))\overline{{\overline{\phi}}}_1(y)\bigg|.
\end{split}$$ Noticing that $$\label{u1}
\sum_{j=2}^kU(y+\mu^{-1}(\xi_1-\xi_j)){\leqslant}C\mu^{n-2}k^{n-2}\sum_{j=1}^k\frac{1}{j^{n-2}}{\leqslant}C\mu^{\frac{n-2}{2}},$$ and $$\begin{split}\label{u2}
\sum_{l=1}^h\mu^{\frac{n-2}{2}}\lambda^{-\frac{n-2}{2}}&U(\lambda^{-1}(\xi_1+\mu y-\eta_l))\\
&{\leqslant}C\sum_{l=1}^h\mu^{\frac{n-2}{2}}\lambda^{-\frac{n-2}{2}}{\leqslant}C\mu^{\frac{n-2}{2}}\lambda^{-\frac{n-2}{2}}h{\leqslant}C\mu^{\frac{n-2}{2}},
\end{split}$$ doing a Taylor expansion we get $$\label{tilf1}
|\tilde{f}_1(y)|{\leqslant}C \mu^{\frac{n-2}{2}}U^{p-2}(y)|\overline{{\overline{\phi}}}_1(y)|{\leqslant}C\mu^{\frac{n-2}{2}}U^{p-1}(y)\|\overline{{\overline{\phi}}}_1\|_*$$ and, proceeding as in the computations for the interior error, $$\|\tilde{f}_1\|_{**}{\leqslant}C\mu^{\frac{n}{2q}}\|\overline{{\overline{\phi}}}_1\|_*.$$ For the term $$f_2:=({\overline{\zeta}}_1-1)U_{\mu,\xi_1}^{p-1}{\overline{\phi}}_1,$$ we have $$\label{tilf2}
|\tilde{f}_2(y)|{\leqslant}U^p(y)\|\overline{{\overline{\phi}}}_1\|_*,\qquad |y|>\frac{{\overline{\alpha}}}{\mu k},\qquad \|\tilde{f}_2\|_{**}{\leqslant}C\mu^{\frac{n}{2q}}\|\overline{{\overline{\phi}}}_1\|_*.$$ Consider now $$f_3:={\overline{\zeta}}_1p|U_*|^{p-1}\Psi({\overline{\phi}}_1,{\hat{\phi}}_1).$$ Using , and we get that, for $|y|{\leqslant}\frac{{\overline{\alpha}}}{\mu k}$, $$\begin{split}\label{tilf3}
|\tilde{f}_3(y)|&{\leqslant}CU^{p-1}\mu^{\frac{n-2}{2}}\|\Psi({\overline{\phi}}_1,{\hat{\phi}}_1)\|_{L^\infty({{\mathbb R}}^n)}\\
&{\leqslant}CU^{p-1}\mu^{\frac{n-2}{2}}(\|{\overline{\overline{\phi}}}_1\|_*+\|{\hat{\hat{\phi}}}_1\|_*+k^{1-\frac{n}{q}}+h^{1-\frac{n}{q}}),
\end{split}$$ $$\|\tilde{f}_3\|_{**}{\leqslant}C\mu^{\frac{n}{2q}}(\|{\overline{\overline{\phi}}}_1\|_*+\|{\hat{\hat{\phi}}}\|_*+k^{1-\frac{n}{q}}+h^{1-\frac{n}{q}}).$$ Denote $$f_4:={\overline{\zeta}}_1N(\phi),\qquad f_5:={\overline{\zeta}}_1 E.$$ Notice that $$\tilde{N}(\phi)=|V_*+\tilde{\phi}_1|^{p-1}(V_*+\tilde{\phi}_1)-|V_*|^{p-1}V_*-p|V_*|^{p-1}\tilde{\phi}_1,$$ where $\tilde{\phi}_1(y):=\mu^{\frac{n-2}{2}}\phi(\xi_1+\mu y)$, and $$\begin{split}
V_*(y):=&-U(y)-\sum_{j=2}^kU(y+\mu^{-1}(\xi_1-\xi_j))\\
&-\sum_{l=1}^h\mu^{\frac{n-2}{2}}\lambda^{-\frac{n-2}{2}}U(\lambda^{-1}(\xi_1+\mu y-\eta_l))\\
&+\mu^{\frac{n-2}{2}}U(\xi_1+\mu y).
\end{split}$$ Hence, for $$\phi={\overline{\phi}}_1+\sum_{j=2}^k{\overline{\phi}}_j+\sum_{l=1}^h{\hat{\phi}}_l+\Psi({\overline{\overline{\phi}}}_1,{\hat{\hat{\phi}}}_1),$$ one has $$|\tilde{\phi}_1|{\leqslant}C\mu^{\frac{n-2}{2}}(\|{\overline{\overline{\phi}}}_1\|_*+\|{\hat{\hat{\phi}}}_1\|_*)+\mu^{\frac{n-2}{2}}\|\Psi({\overline{\overline{\phi}}}_1,{\hat{\hat{\phi}}}_1)\|_{L^\infty({{\mathbb R}}^n)}.$$ Furthermore, in the region $|y|<\frac{{\overline{\alpha}}}{\mu k}$ it holds $U(y)\sim \mu^{\frac{n-2}{2}}$ and thus, after a second order Taylor expansion one has $$\begin{split}\label{tilf4}
|\tilde{f}_4(y)|&{\leqslant}C |V_*|^{p-2}|\tilde{\phi}_1|^2{\leqslant}C U^{p-2} \mu^{\frac{n-2}{2}}|\tilde{\phi}_1|\\
&{\leqslant}CU^{p-1}\mu^{\frac{n-2}{2}}(\|{\overline{\overline{\phi}}}_1\|_*+\|{\hat{\hat{\phi}}}_1\|_*+\|\Psi\|_*),
\end{split}$$ and $$\|\tilde{f}_4\|_{**}{\leqslant}C\mu^{\frac{n}{2q}}(\|{\overline{\overline{\phi}}}_1\|_*+\|{\hat{\hat{\phi}}}_1\|_*+k^{1-\frac{n}{q}}+h^{1-\frac{n}{q}}).$$ Finally, by , we know $$\label{tilf5}
\|\tilde{f}_5\|_{**}{\leqslant}C\mu^{\frac{n}{2q}}.$$ Likewise, one can obtain analogous estimates for $${\hat{T}}({\hat{\zeta}}_1E+\gamma\hat{{{\mathscrN}}}({\overline{\overline{\phi}}}_1,{\hat{\hat{\phi}}}_1))$$ to conclude that ${{\mathscrM}}$ maps functions $\phi_1$ with $\|\phi_1\|_*{\leqslant}C(\mu^{\frac{n}{2q}}+\lambda^{\frac{n}{2q}})$ into the same class of functions. Besides, one can prove that the map is indeed a contraction, and thus we conclude the existence of a unique solution to the system .
The symmetry conditions and follow straightforward as consequence of the uniqueness.
Proof of Theorem \[mainThm\]
============================
Thanks to Proposition \[existProjProb\] we have ${\overline{\phi}}_1$ and ${\hat{\phi}}_1$ solutions to . Thus, if we find $\delta$ and $\varepsilon$ in so that $\overline{c}_{0}(\delta,\varepsilon)=\hat{c}_{0}(\delta,\varepsilon)=0$ they actually solve and . Repeating this argument for every $j=1,\ldots,k-1$ and $l=1,\ldots,h-1$ we conclude that $$u=U_*+\phi$$ with $\phi$ defined in is the solution to problem we were looking for. Thus, we want to prove the existence of $\delta$ and $\varepsilon$ so that (we keep the names in an abuse of notation) $$\begin{split}
\overline{c}_{0}(\delta,\varepsilon)&=\int_{\mathbb{R}^n}({\overline{\zeta}}_1E+\gamma\overline{{{\mathscrN}}}({\overline{\overline{\phi}}}_1,{\hat{\hat{\phi}}}_1)){\overline{Z}}_{0}=0,\\
\hat{c}_{0}(\delta,\varepsilon)&=\int_{\mathbb{R}^n}({\hat{\zeta}}_1E+\gamma\hat{{{\mathscrN}}}({\overline{\overline{\phi}}}_1,{\hat{\hat{\phi}}}_1)){\hat{Z}}_{0}=0.
\end{split}$$ Indeed, we will prove that $$\begin{split}\label{eqsc}
\overline{c}_{0}(\delta,\varepsilon)&=-A_n\frac{\delta}{k^{n-2}}[\delta a^1_{n,k} -a^2_{n,k} ]+\frac{1}{k^{n-1}} \Theta_{k,h}(\delta,\varepsilon),\\
\hat{c}_{0}(\delta,\varepsilon)&=-A_n\frac{\varepsilon}{h^{n-2}}[\varepsilon b^1_{n,h} -b^2_{n,h}]+\frac{1}{h^{n-1}}\Theta_{k,h}(\delta,\varepsilon).
\end{split}$$ Here $A_n$ is a fixed positive constant that depends on $n$, while for $i=1,2$, $ a^i_{n,k}$, $b^i_{n,h}$ are positive constants, of the form $a^i_{n,k} = a^i_n + O({1\over k}) $, $b^i_{n,h} = b^i_n + O({1\over h}) $, as $k,h \to \infty$, with $a_n^i$ and $b_n^i$ positive constants. Furthermore, $\Theta_{k,h}(\delta,\varepsilon)$ denotes a generic function, which is smooth in its variables, and it is uniformly bounded, together with its first derivatives, in $\delta$ and $\varepsilon$ satisfying the bounds , when $k\rightarrow \infty$ and $h\rightarrow\infty$. By a fixed point argument one can prove the existence of a solution $(\delta,\varepsilon)$ to the system $$\begin{split}
\bar c_0 (\delta , \varepsilon ) = \hat c_0 (\delta , \varepsilon ) = 0.
\end{split}$$ Thus, if we prove that holds, we conclude the proof of Theorem \[mainThm\].
Both estimates in follow in the same way, so let us prove the first one. We write $$\label{threeTerms}
\overline{c}_{0}(\delta,\varepsilon)=\int_{\mathbb{R}^n}E{\overline{Z}}_{0}+\int_{\mathbb{R}^n}({\overline{\zeta}}_1-1)E{\overline{Z}}_{0}+\gamma\int_{\mathbb{R}^n}\overline{{{\mathscrN}}}({\overline{\overline{\phi}}}_1,{\hat{\hat{\phi}}}_1){\overline{Z}}_{0}.$$ and we analyze every term independently. For $\delta $ and $\varepsilon$ satisfying , we have that
[**Claim 1:**]{} $$\label{claim1}
\int_{\mathbb{R}^n}E{\overline{Z}}_{0}=-A_n\frac{\delta}{k^{n-2}}[\delta a^1_{n,k} -a^2_{n,k} ]+\frac{1}{k^{n-1}} \Theta_{k,h}(\delta,\varepsilon).$$
[**Claim 2:**]{} $$\label{claim2}
\int_{\mathbb{R}^n}({\overline{\zeta}}_1-1)E{\overline{Z}}_{0}=\frac{1}{k^{n-1}} \Theta_{k,h}(\delta,\varepsilon).$$
[**Claim 3:**]{} $$\label{claim3}
\int_{\mathbb{R}^n}\overline{{{\mathscrN}}}({\overline{\overline{\phi}}}_1,{\hat{\hat{\phi}}}_1){\overline{Z}}_{0}=\frac{1}{k^{n-1}} \Theta_{k,h}(\delta,\varepsilon),$$ as $k$ and $h \to \infty$. It is clear that these claims imply the validity of the first equation in .
[**Proof of Claim 1.**]{} Let us denote $$Ext:=\{\cap_{j=1}^k\{|y-\xi_j|>\frac{{\overline{\alpha}}}{k}\}\}\cap\{\cap_{l=1}^h\{|y-\eta_l|>\frac{{\hat{\alpha}}}{h}\}\}.$$ For ${\overline{\alpha}}>0$ independent of $k$ we can write the first term as $$\label{three}
\int_{\mathbb{R}^n}E{\overline{Z}}_{0}=\int_{B(\xi_1,\frac{{\overline{\alpha}}}{k})}E{\overline{Z}}_{0}+\int_{Ext}E{\overline{Z}}_{0}+\sum_{j\neq 1}\int_{B(\xi_j,\frac{{\overline{\alpha}}}{k})}E{\overline{Z}}_{0}+\sum_{l= 1}^h\int_{B(\eta_l,\frac{{\hat{\alpha}}}{h})}E{\overline{Z}}_{0}.$$ Considering $\overline{E}_1(y)=\mu^{\frac{n+2}{2}}E(\xi_1+\mu y)$ and using we obtain $$\begin{split}
\int_{B(\xi_1,\frac{{\overline{\alpha}}}{k})}E{\overline{Z}}_{0}=&\int_{B(0,\frac{{\overline{\alpha}}}{\mu k})}\overline{E}_1(y)Z_{0}(y)\,dy\\
=&\textcolor{black}{-\gamma p \sum_{j\neq 1}\int_{B(0,\frac{{\overline{\alpha}}}{\mu k})}U^{p-1}U(y-\mu^{-1}(\xi_j-\xi_1))Z_{0}\,dy}\\
&\textcolor{black}{+\gamma p \mu^{\frac{n-2}{2}}\int_{B(0,\frac{{\overline{\alpha}}}{\mu k})}U^{p-1}U(\xi_1+\mu y)Z_{0}\,dy}\\
&\textcolor{black}{-\gamma p \sum_{l=1}^h\mu^{\frac{n-2}{2}}\lambda^{-\frac{n-2}{2}}\int_{B(0,\frac{{\overline{\alpha}}}{\mu k})}U^{p-1}U(\lambda^{-1}(\xi_1+\mu y-\eta_l))Z_{0}\,dy}\\
&\textcolor{black}{+\gamma p\int_{B(0,\frac{{\overline{\alpha}}}{\mu k})}[(U(y)+sV)^{p-1}-U^{p-1}]V(y)Z_{0}\,dy}\\
&-\mu^{\frac{n+2}{2}}\int_{B(0,\frac{{\overline{\alpha}}}{\mu k})}U^p(\xi_1+\mu y)Z_{0}\,dy\\
&-\sum_{j\neq 1} \int_{B(0,\frac{{\overline{\alpha}}}{\mu k})}U^p(y-\mu^{-1}(\xi_j-\xi_1))Z_{0}\,dy\\
&\textcolor{black}{-\sum_{l=1}^h\mu^{\frac{n+2}{2}}\lambda^{-\frac{n+2}{2}}\int_{B(0,\frac{{\overline{\alpha}}}{\mu k})}U^p(\lambda^{-1}(\xi_1+\mu y-\eta_l))Z_{0}\,dy},
\end{split}$$ where $$\begin{split}
V(y):=&-\sum_{j\neq 1}U(y-\mu^{-1}(\xi_j-\xi_1))+\mu^{\frac{n-2}{2}}U(\xi_1+\mu y)\\
&-\sum_{l=1}^h\mu^{\frac{n-2}{2}}\lambda^{-\frac{n-2}{2}}U(\lambda^{-1}(\xi_1+\mu y-\eta_l)).
\end{split}$$
Doing a Taylor expansion, for $j\neq 1$ there holds $$\int_{B(0,\frac{{\overline{\alpha}}}{\mu k})}U^{p-1}U(y-\mu^{-1}(\xi_j-\xi_1))Z_{0}\,dy=\frac{c_1 \mu^{n-2}}{|{\xi}_j-{\xi}_1|^{n-2}}(1+{\mu^2
\over |\xi_j - \xi_1|^2} \Theta_{k,h}(\delta,\varepsilon) ),$$ and $$\int_{B(0,\frac{{\overline{\alpha}}}{\mu k})}U^{p-1}U(\lambda^{-1}(\xi_1+\mu y-\eta_l))Z_{0}\,dy=\frac{c_1 \lambda^{n-2}}{|\xi_1-\eta_l|^{n-2}}(1+{\lambda^2
\over |\xi_j - \xi_1|^2} \Theta_{k,h}(\delta,\varepsilon) ),$$ where $c_1$ is some positive constant, and, as before, $\Theta_{k,h}(\delta,\varepsilon)$ denotes a generic function, which is smooth in its variables, and it is uniformly bounded, together with its first derivatives, in $\delta$ and $\varepsilon$ satisfying the bounds , when $k\rightarrow \infty$ and $h\rightarrow\infty$. Proceeding in a similar way, $$\mu^{\frac{n-2}{2}}\int_{B(0,\frac{{\overline{\alpha}}}{\mu k})}U^{p-1}U(\xi_1+\mu y)Z_{0}\,dy=c_2\mu^{\frac{n-2}{2}}(1+(\mu k)^2\Theta_{k,h}(\delta,\varepsilon) ),$$ for some positive constant $c_2$. On the other hand, $$\label{est1}
\bigg|\mu^{\frac{n+2}{2}}\int_{B(0,\frac{{\overline{\alpha}}}{\mu k})}U^p(\xi_1+\mu y)Z_{0}\,dy\bigg|{\leqslant}C\mu^{\frac{n+2}{2}}\int_{B(0,\frac{{\overline{\alpha}}}{\mu k})}\frac{1}{(1+|y|)^{n-2}}{\leqslant}C\mu^{\frac{n-2}{2}}k^{-2},$$ $$\begin{split}\label{est2}
\bigg|\sum_{j\neq 1} \int_{B(0,\frac{{\overline{\alpha}}}{\mu k})}U^p(y-&\mu^{-1}(\xi_j-\xi_1))Z_{0}\,dy\bigg|\\&{\leqslant}\sum_{j\neq 1}\frac{\mu^{n+2}}{|{\xi}_j-{\xi}_1|^{n+2}}\int_{B(0,\frac{{\overline{\alpha}}}{\mu k})}\frac{1}{(1+|y|)^{n-2}}\\
&{\leqslant}C(\mu k)^{-2}\sum_{j\neq 1}\frac{\mu^{n+2}}{|{\xi}_j-{\xi}_1|^{n+2}},
\end{split}$$ and $$\begin{split}\label{est3}
\bigg|\sum_{l=1}^h\mu^{\frac{n-2}{2}}\lambda^{-\frac{n+2}{2}}&\int_{B(0,\frac{{\overline{\alpha}}}{\mu k})}U^p(\lambda^{-1}(\xi_1+\mu y-\eta_l))Z_{0}\bigg|\\
&{\leqslant}C\sum_{l=1}^h\mu^{\frac{n-2}{2}}\lambda^{\frac{n+2}{2}}\int_{B(0,\frac{{\overline{\alpha}}}{\mu k})}\frac{1}{(1+|y|)^{n-2}}\\
&{\leqslant}C\mu^{\frac{n+2}{2}}\lambda^{\frac{n+2}{2}}hk^2.
\end{split}$$ Finally, putting together , and , $$\begin{split}
\bigg|\int_{B(0,\frac{{\overline{\alpha}}}{\mu k})}&[(U(y)+sV)^{p-1}-U^{p-1}]V(y)Z_{0}\,dy\bigg|\\
&{\leqslant}C\left(\mu^{\frac{n-2}{2}}k^{-2}+(\mu k)^{-2}\sum_{j\neq 1}\frac{\mu^{n+2}}{|{\xi}_j-{\xi}_1|^{n+2}}+\mu^{\frac{n+2}{2}}\lambda^{\frac{n+2}{2}}hk^2\right).
\end{split}$$ To estimate the second term in we apply Hölder inequality to get $$\begin{split}\label{Zfuera}
\bigg|\int_{Ext}E&{\overline{Z}}_{0}\bigg|{\leqslant}C\|(1+|y|)^{n+2-\frac{2n}{q}}E\|_{L^q(Ext)}\|(1+|y|)^{-n-2+\frac{2n}{q}}{\overline{Z}}_{0}\|_{L^{\frac{q}{q-1}}(Ext)}.
\end{split}$$ Proceeding as in [@dPMPP] we have $$\|(1+|y|)^{-n-2+\frac{2n}{q}}{\overline{Z}}_{0}\|_{L^{\frac{q}{q-1}}(Ext)}{\leqslant}C\mu^{\frac{n-2}{2}}k^{n-2}k^{\frac{n}{q}-n},$$ and using the estimates obtained in Section \[error\] we see that $$\begin{split}
\|(1+|y|)^{n+2-\frac{2n}{q}}&E\|_{L^q(Ext)}
{\leqslant}\,C(\mu^{\frac{n-2}{2}}k^{n-2}k^{1-\frac{n}{q}}+\lambda^{\frac{n-2}{2}}h^{n-2}h^{1-\frac{n}{q}}),
\end{split}$$ and thus, substituting in , $$\label{errExt1}
\bigg|\int_{Ext}E{\overline{Z}}_{0}\bigg|{\leqslant}C\left(\frac{\mu^{n-2}k^{2(n-2)}}{k^{n-1}}+\frac{\mu^{\frac{n-2}{2}}\lambda^{\frac{n-2}{2}}k^{n-2}h^{n-2}}{k^{n-\frac{n}{q}}h^{\frac{n}{q}-1}}\right).$$ Arguing as in [@dPMPP] the third term in can be estimated as $$\label{errExt2}
\bigg|\sum_{j\neq 1}\int_{B(\xi_j,\frac{{\overline{\alpha}}}{k})}E{\overline{Z}}_{0}\bigg| {\leqslant}\frac{\mu^{\frac{n-2}{2}}}{(\mu k)^{n-4}}\left[\mu^{n-2}\sum_{j\neq 1}\frac{1}{|\xi_j-\xi_1|^{n-2}}\right].$$ Likewise, $$\begin{split}
\bigg|\int_{B(\eta_l,\frac{{\hat{\alpha}}}{h})}&E{\overline{Z}}_0\bigg|=\bigg|\lambda^{\frac{n-2}{2}}\int_{B(0,\frac{{\hat{\alpha}}}{\lambda h})}\hat{E}_l(y){\overline{Z}}_0(\lambda y+\eta_l)\bigg|\\
&{\leqslant}\lambda^{\frac{n-2}{2}}\|(1+|y|)^{n+2-\frac{2n}{q}}\hat{E}_l\|_{L^q(|y|<\frac{{\hat{\alpha}}}{\lambda h})}\\
&\;\;\cdot\|(1+|y|)^{-n-2+\frac{2n}{q}}\mu^{-\frac{n-2}{2}}Z_0(\mu^{-1}(\lambda y+\eta_l-\xi_1)\|_{L^{\frac{q}{q-1}}(|y|<\frac{{\hat{\alpha}}}{\lambda h})}.
\end{split}$$ Noticing that $$\begin{split}
\|(1+|y|)^{-n-2+\frac{2n}{q}}&\mu^{-\frac{n-2}{2}}Z_0(\mu^{-1}(\lambda y+\eta_l-\xi_1))\|_{L^{\frac{q}{q-1}}({{\mathbb R}}^n)}\\
&{\leqslant}C\mu^{\frac{n-2}{2}}\left(\int_1^\frac{{\hat{\alpha}}}{\lambda h}\frac{t^{n-1}\,dt}{t^{(n+2-\frac{2n}{q})\frac{q}{q-1}}}\right)^{\frac{q-1}{q}}{\leqslant}C\mu^\frac{n-2}{2}(\lambda h)^{2-\frac{n}{q}},
\end{split}$$ by we conclude that $$\label{errExt3}
\bigg|\sum_{l=1}^h\int_{B(\eta_l,\frac{{\hat{\alpha}}}{h})}E{\overline{Z}}_0\bigg|{\leqslant}C \lambda^{\frac{n-2}{2}}h^{-\frac{n}{q}}\mu^{\frac{n-2}{2}}(\lambda h)^{2-\frac{n}{q}}h$$ Claim 1 follows from these estimates applying the fact that $h=O(k)$.
[**Proof of Claim 2.**]{} Let us estimate the second term of . Notice first that $$\bigg|\int_{\mathbb{R}^n}({\overline{\zeta}}_1-1)E{\overline{Z}}_{0}\bigg|{\leqslant}C\bigg|\int_{\{|y-\xi_1|>\frac{{\overline{\alpha}}}{k}\}}E{\overline{Z}}_{0}\bigg|,$$ and we separate this integral as $$\int_{\{|y-\xi_1|>\frac{{\overline{\alpha}}}{k}\}}E{\overline{Z}}_{0}=\int_{Ext}E{\overline{Z}}_{0}+\sum_{j=2}^k\int_{\{|y-\xi_j|{\leqslant}\frac{{\overline{\alpha}}}{k}\}}E{\overline{Z}}_{0}+\sum_{l=1}^h\int_{\{|y-\eta_l|{\leqslant}\frac{{\hat{\alpha}}}{h}\}}E{\overline{Z}}_{0}.$$ Thus, Claim 2 follows from , and .
[**Proof of Claim 3.**]{} Notice that $$\int_{\mathbb{R}^n}\overline{{{\mathscrN}}}({\overline{\overline{\phi}}}_1,{\hat{\hat{\phi}}}_1){\overline{Z}}_{0}=\mu^{\frac{n+2}{2}}\int_{\mathbb{R}^n}\overline{{{\mathscrN}}}({\overline{\overline{\phi}}}_1,{\hat{\hat{\phi}}}_1)(\xi_1+\mu y)Z_{0}(y),$$ and thus, from estimates $\eqref{tilf1}-\eqref{tilf4}$ and the fact $h=O(k)$ we conclude $$\int_{\mathbb{R}^n}\overline{{{\mathscrN}}}({\overline{\overline{\phi}}}_1,{\hat{\hat{\phi}}}_1){\overline{Z}}_{0}{\leqslant}C k^{3-n-\frac{n}{q}}\int_{\mathbb{R}^n}U^{p-1}|Z_{0}|,$$ and the claim follows.
\[nondegeneracyProof\] As stated before, the goal of this part is to prove the nondegeneracy (see Definition \[defNondeg\]) of the solution $u$ provided by Theorem \[mainThm\] (we drop the dependence on $k$ and $h$ by simplicity). Recalling the functions $z_\alpha$ defined in - we can formulate the result in Theorem \[teounico\] as follows.
\[nondeg\] There exists a sequence of solutions $u$ to Problem among the ones constructed in Theorem \[mainThm\] for which all bounded solutions to the equation $$\label{ll11}-\Delta\varphi-\gamma p |u|^{p-2}u \varphi=0$$ are linear combination of the functions $z_\alpha$ for $\alpha=0,\ldots,N_0-1.$ (Recall that $N_0 := 5 (n-1)$.)
For later simplification, we introduce the following functions $$\begin{split}\label{newzalpha}
{\bf z}_\beta := z_\beta, \quad &{\mbox {if}} \quad \beta \not= n+3, n+4, n+5, n+6, \\
{\bf z}_{n+2+\alpha} := { z_\alpha -z_{n+2+\alpha} \over 2} , \quad &{\mbox {if}} \quad \alpha = 1, 2, 3, 4.
\end{split}$$ Since ${\bf z}_\beta$ are linear combinations of the original functions $z_\beta$, the statement of Theorem \[nondeg\] is equivalent to say that there exists a sequence of solutions among the ones constructed in Theorem \[mainThm\] for which all bounded solutions to are linear combinations of ${\bf z}_\beta$, for $\beta = 0, \ldots , N_0-1$.
Thus, let $\varphi$ be a bounded solution of , namely $L(\varphi)=0$, with $L$ defined in . We decompose $\varphi$ as $$\label{defphitilde}
\varphi(y)=\sum_{\beta=0}^{N_0-1}a_\beta {\bf z}_\beta(y)+\tilde{\varphi}(y),$$ where $a_\beta$ are chosen so that $$\label{ortCond}
\int_{\mathbb{R}^n}|u|^{p-1}{\bf z}_\beta\tilde{\varphi}=0$$ holds. Notice that, since ${\bf z}_\beta\in \ker\{L\}$, one has $L(\tilde{\varphi})=0$ and thus our goal will be to prove that actually $\tilde{\varphi}\equiv 0$.
Recall that our solution $u$ has the form $$\label{finalform}
u(y)=U(y)-\sum_{j=1}^k U_{\mu,\xi_j}(y)-\sum_{l=1}^h U_{\lambda,\eta_l}(y)+\phi(y),$$ where $\phi$ is defined in as $
\phi=\sum_{j=1}^k\overline{\phi}_j+\sum_{l=1}^h\hat{\phi}_l+\psi$.
We introduce the following functions $$\begin{split}\label{Z0big}
Z_{00}(y)&:=\frac{n-2}{2}\left[ U + \psi \right] (y)+\nabla [U +\psi ] (y)\cdot y , \\
Z_{\alpha 0} (y) &:= \frac{\partial}{\partial y_\alpha}U(y) + \frac{\partial}{\partial y_\alpha}\psi (y) \qquad \alpha=1,\ldots,n.
\end{split}$$ For $j$ fixed in $\{ 1, \ldots , k\}$, we define $$\begin{split}
{\overline{Z}}_{0j}(y)&:=\frac{n-2}{2}[ U_{\mu,\xi_j} + \bar \phi_j] (y)+\nabla [ U_{\mu,\xi_j} + \bar \phi_j] (y)\cdot(y-\xi_j)\\
{\overline{Z}}_{1j}(y)&:= \xi_j \cdot \nabla_y [ U_{\mu , \xi_j} + \bar \phi_j ] (y), \qquad {\overline{Z}}_{2j}(y):= \xi_j^\perp \cdot \nabla_y [U_{\mu , \xi_j} + \bar \phi_j] (y),\\
{\overline{Z}}_{\alpha j}(y)&:=\frac{\partial}{\partial y_\alpha}[ U_{\mu,\xi_j}(y) + \bar \phi_j ],\qquad \alpha=3,\ldots,n.
\end{split}$$ For $l$ fixed in $\{ 1, \ldots , h\}$, we define $$\begin{split}
{\hat{Z}}_{0j}(y)&:=\frac{n-2}{2}[ U_{\lambda,\eta_l} + \hat \phi_l] (y)+\nabla [ U_{\lambda,\eta_l} + \hat \phi_l] (y)\cdot(y-\eta_l)\\
{\hat{Z}}_{3l}(y)&:= \eta_l \cdot \nabla_y [ U_{\lambda , \eta_l} + \hat \phi_l ] (y), \qquad {\hat{Z}}_{4l}(y):= \eta_l^\perp \cdot \nabla_y [U_{\lambda , \eta_l} + \hat \phi_l] (y),\\
{\hat{Z}}_{\alpha l}(y)&:=\frac{\partial}{\partial y_\alpha}[ U_{\lambda,\eta_l}(y) + \hat \phi_l ],\qquad \alpha=1, 2, 5, 6,\ldots,n.
\end{split}$$ In Appendix \[appe1\] we provide the expressions of the functions ${\bf z}_\beta$, $\beta = 0 , \ldots , N_0 -1$, in terms of the functions $Z_{\alpha 0}$, $\overline Z_{\alpha j}$, $j=1, \ldots , k$, and $\hat Z_{\alpha , l} $, $l=1, \ldots , h$, for any $\alpha =0, \ldots , n$. These relations will be useful in other parts of our argument.
We rearrange the functions above in $(n+1)$ vector fields as $$\label{Pibig}
\Pi_\alpha:=\left[Z_{\alpha0},{\overline{Z}}_{\alpha 1},\ldots,{\overline{Z}}_{\alpha k},{\hat{Z}}_{\alpha 1},\ldots,{\hat{Z}}_{\alpha h}\right]^T, \quad \alpha = 0 , 1 , \ldots , n,$$ and, for any given vector $d=\left[d_0,{\overline{d}}_1,\ldots,{\overline{d}}_k,{\hat{d}}_1,\ldots,{\hat{d}}_h\right]^T\in\mathbb{R}^{1+k+h}$ we use the notation $$d\cdot\Pi_\alpha:=d_0Z_{\alpha 0}+\sum_{j=1}^k{\overline{d}}_j{\overline{Z}}_{\alpha j}+\sum_{l=1}^h{\hat{d}}_l{\hat{Z}}_{\alpha l}.$$ With this in mind, we write the function $\tilde{\varphi}$ in as $$\label{tildeVarphi}
\tilde{\varphi}(y)=\sum_{\alpha=0}^n c_\alpha\cdot \Pi_\alpha(y)+\varphi^\perp(y),$$ where $$c_\alpha:=\left[c_{\alpha 0},{\overline{c}}_{\alpha 1},\ldots,{\overline{c}}_{\alpha k},{\hat{c}}_{\alpha 1},\ldots,{\hat{c}}_{\alpha h}\right]^T,\qquad \alpha=0,\ldots, n,$$ are $(n+1)$ vectors in $\mathbb{R}^{k+h+1}$ chosen so that $$\begin{split}
&\int_{\mathbb{R}^n}U^{p-1}Z_{\alpha 0}\varphi^\perp =0,\qquad \alpha=0,1,\ldots,n,\\
&\int_{\mathbb{R}^n}U^{p-1}_{\mu,\xi_j}{\overline{Z}}_{\alpha j}\varphi^\perp =0,\qquad j=1,\ldots,k,\;\;\alpha=0,1,\ldots,n,\\
&\int_{\mathbb{R}^n}U^{p-1}_{\lambda,\eta_l}{\hat{Z}}_{\alpha l}\varphi^\perp =0,\qquad l=1,\ldots,h,\;\;\alpha=0,1,\ldots,n.
\end{split}$$ Hence, to prove that $\tilde{\varphi}\equiv 0$ we have to see that $c_\alpha=0$ for every $\alpha$ and $\varphi^\perp\equiv 0$. This will be consequence of the following three facts.
[**Fact 1:**]{} Since $L(\tilde{\varphi})=0$, one has that $$\label{eqPi}
\sum_{\alpha=0}^n c_\alpha\cdot L(\Pi_\alpha)=-L(\varphi^\perp),$$ with $L$ defined in . We write $\varphi^\perp=\varphi_0^\perp+\sum_{j=1}^k{\overline{\varphi}}_j^\perp+\sum_{l=1}^h{\hat{\varphi}}_l^\perp$, where $$\begin{split}
-L(\varphi_0^\perp)&=\sum_{\alpha=0}^n c_{\alpha 0}L(Z_{\alpha 0}),\\
-L({\overline{\varphi}}_j^\perp)&=\sum_{\alpha=0}^n {\overline{c}}_{\alpha j} L({\overline{Z}}_{\alpha j}), \qquad j=1,\ldots, k,\\
-L({\hat{\varphi}}_l^\perp)&=\sum_{\alpha=0}^n {\hat{c}}_{\alpha l} L({\hat{Z}}_{\alpha l}), \qquad l=1,\ldots, h.
\end{split}$$ Furthermore, let us define $${\overline{\overline{\varphi}}}_j^\perp(y):=\mu^{\frac{n-2}{2}}{\overline{\varphi}}_j^\perp(\mu y+\xi_j),\qquad {\hat{\hat{\varphi}}}_l^\perp(y):=\lambda^{\frac{n-2}{2}}{\hat{\varphi}}_l^\perp(\lambda y+\eta_l),$$ and $$\|\varphi^\perp\|:=\|\varphi_0^\perp\|_*+\sum_{j=1}^k\|{\overline{\overline{\varphi}}}_j^\perp\|_*+\sum_{l=1}^h\|{\hat{\hat{\varphi}}}_l^\perp\|_*.$$ Thus, as we will prove in Section \[fact2\], there exists a positive constant $C$ such that $$\label{f2}
\|\varphi^\perp\|{\leqslant}C k^{-2 + {2\over n}} \, \sum_{\alpha=0}^n\|c_\alpha\|.$$
[**Fact 2:**]{} Condition is equivalent to $$\begin{split}\label{systemFact1}
\sum_{\alpha=0}^nc_\alpha\cdot\int_{\mathbb{R}^n}\Pi_\alpha |u|^{p-1}{\bf z}_\beta=&\sum_{\alpha=0}^n\left[c_{\alpha 0}\int_{\mathbb{R}^n}Z_{\alpha 0}|u|^{p-1}{\bf z}_\beta+\sum_{j=1}^k{\overline{c}}_{\alpha j}\int_{\mathbb{R}^n}{\overline{Z}}_{\alpha j}|u|^{p-1}{\bf z}_\beta\right.\\
&\left.+\sum_{l=1}^h{\hat{c}}_{\alpha l}\int_{\mathbb{R}^n}{\hat{Z}}_{\alpha l}|u|^{p-1}{\bf z}_\beta\right]\\
=&-\int_{\mathbb{R}^n}\varphi^\perp |u|^{p-1}{\bf z}_\beta,\qquad \beta=0,\ldots,N_0-1.
\end{split}$$
Let us denote $$\begin{split}\label{cossin}
& {\bf{{\overline{\cos}}}}:=\left[
\begin{array}{c}
1\\\cos{\overline{\theta}}_2\\\ldots\\\cos{\overline{\theta}}_{k-1}
\end{array}
\right], \;
{\bf{ {\overline{\sin}}}}:=\left[
\begin{array}{c}
0\\\sin{\overline{\theta}}_2\\\ldots\\\sin{\overline{\theta}}_{k-1}
\end{array}
\right], \qquad {\overline{\theta}}_j:=\frac{2\pi}{k}(j-1), \\
&{\bf {\hat{\cos}}}:=\left[
\begin{array}{c}
1\\\cos{\hat{\theta}}_2\\\ldots\\\cos{\hat{\theta}}_{h-1}
\end{array}
\right], \;
{\bf {\hat{\sin}}}:=\left[
\begin{array}{c}
0\\\sin{\hat{\theta}}_2\\\ldots\\\sin{\hat{\theta}}_{h-1}
\end{array}
\right], \qquad {\hat{\theta}}_l:=\frac{2\pi}{h}(l-1),
\end{split}$$ and $$\label{onezero}
{\overline{1}}:=\left[
\begin{array}{c}
1\\1\\\ldots\\1
\end{array}
\right], \qquad
{\hat{1}}:=\left[
\begin{array}{c}
1\\1\\\ldots\\1
\end{array}
\right], \qquad
{\overline{0}}:=\left[
\begin{array}{c}
0\\0\\\ldots\\0
\end{array}
\right], \qquad
{\hat{0}}:=\left[
\begin{array}{c}
0\\0\\\ldots\\0
\end{array}
\right],$$ where ${\overline{1}}$ and ${\overline{0}}$ are $k$-dimensional vectors, and ${\hat{1}}$ and ${\hat{0}}$ are vectors of dimension $h$. Likewise, define $$\tilde{c}_0:=\left[
\begin{array}{c}
c_{00}\\\ldots\\c_{n0}
\end{array}
\right]\in {{\mathbb R}}^{n+1}, \qquad
\overline{c}_\alpha:=\left[
\begin{array}{c}
\overline{c}_{\alpha 1}\\\ldots\\\overline{c}_{\alpha k}
\end{array}
\right]\in {{\mathbb R}}^k, \qquad
\hat{c}_\alpha:=\left[
\begin{array}{c}
\hat{c}_{\alpha 1}\\\ldots\\\hat{c}_{\alpha h}
\end{array}
\right]\in {{\mathbb R}}^h,$$ and $$\overline{c}:=\left[
\begin{array}{c}
\overline{c}_0\\\overline{c}_1\\\ldots\\\overline{c}_{n+1}
\end{array}
\right]\in{{\mathbb R}}^{(n+1)k}, \qquad
\hat{c}:=\left[
\begin{array}{c}
\hat{c}_0\\\hat{c}_1\\\ldots\\\hat{c}_n
\end{array}
\right]\in {{\mathbb R}}^{(n+1)h}.$$ Thus,
\[systC\]Solving system is equivalent to solve $$\begin{split}\label{t0}
c_0
\cdot \left[
\begin{array}{c}
1\\-{\overline{1}}\\{\hat{1}}\end{array}
\right]
&
+c_1\cdot \left[
\begin{array}{c}
0\\-{\overline{1}}\\{\hat{0}}\end{array}
\right]
+c_3\cdot \left[
\begin{array}{c}
0\\{\overline{0}}\\-{\hat{1}}\end{array}
\right]
=\,t_0\\
&+R_{h,k} [ c_0 , c_1 , \ldots c_n],
\end{split}$$
$$\begin{split}\label{t1}
c_1\cdot \left[
\begin{array}{c}
1\\-{\overline{\cos}}\\-{\hat{1}}\end{array}
\right]
&+c_2\cdot \left[
\begin{array}{c}
0\\{\overline{\sin}}\\{\hat{0}}\end{array}
\right]
=\,t_1+ R_{h,k} [ c_0 , c_1 , \ldots c_n],
\end{split}$$
$$\begin{split}\label{t2}
c_1\cdot \left[
\begin{array}{c}
0\\-{\overline{\sin}}\\{\hat{0}}\end{array}
\right]
&+c_2\cdot \left[
\begin{array}{c}
1\\-{\overline{\cos}}\\-{\hat{1}}\end{array}
\right]
=\,t_2+R_{h,k} [ c_0 , c_1 , \ldots c_n],
\end{split}$$
$$\begin{split}\label{t3}
c_3\cdot \left[
\begin{array}{c}
1\\-{\overline{1}}\\-{\hat{\cos}}\end{array}
\right]
&
+ c_4\cdot \left[
\begin{array}{c}
0\\{\overline{0}}\\{\hat{\sin}}\end{array}
\right]
=\,t_3+R_{h,k} [ c_0 , c_1 , \ldots c_n],
\end{split}$$
$$\begin{split}\label{t4}
c_3\cdot \left[
\begin{array}{c}
0\\{\overline{0}}\\-{\hat{\sin}}\end{array}
\right]
&+c_4\cdot \left[
\begin{array}{c}
1\\-{\overline{1}}\\-{\hat{\cos}}\end{array}
\right]
=\,t_4+R_{h,k} [ c_0 , c_1 , \ldots c_n],
\end{split}$$
$$\begin{split}\label{zalpha}
c_\alpha\cdot \left[
\begin{array}{c}
1\\-{\overline{1}}\\-{\hat{1}}\end{array}
\right]
=t_\alpha+R_{h,k} [ c_0 , c_1 , \ldots c_n],
\end{split}$$
for $\alpha=5,\ldots,n$, $$\begin{split}\label{nmas1}
c_2\cdot \left[
\begin{array}{c}
0\\{\overline{1}}\\{\hat{0}}\end{array}
\right]
=t_{n+1}+R_{h,k} [ c_0 , c_1 , \ldots c_n],
\end{split}$$
$$\begin{split}\label{nmas2}
c_4 \cdot \left[
\begin{array}{c}
0\\{\overline{0}}\\{\hat{1}}\end{array}
\right]
=t_{n+2}+R_{h,k} [ c_0 , c_1 , \ldots c_n],
\end{split}$$
$$\begin{split}\label{nmas3}
c_0\cdot \left[
\begin{array}{c}
0\\{\overline{\cos}}\\\hat 0
\end{array}
\right] -c_1\cdot \left[
\begin{array}{c}
0\\{\overline{\cos}}\\\hat 0
\end{array}
\right]
=t_{n+3}+R_{h,k} [ c_0 , c_1 , \ldots c_n],
\end{split}$$
$$\begin{split}\label{nmas4}
c_0\cdot \left[
\begin{array}{c}
0\\{\overline{\sin}}\\ \hat 0
\end{array}
\right] - c_1\cdot \left[
\begin{array}{c}
0\\{\overline{\sin}}\\\hat 0
\end{array}
\right]
=t_{n+4}+R_{h,k} [ c_0 , c_1 , \ldots c_n],
\end{split}$$
$$\begin{split}\label{nmas5}
c_0\cdot \left[
\begin{array}{c}
0\\ \overline 0 \\ {\hat{\cos}}\end{array}
\right] - c_3\cdot \left[
\begin{array}{c}
0\\ \overline 0 \\ {\hat{\cos}}\end{array}
\right]
=t_{n+5}+R_{h,k} [ c_0 , c_1 , \ldots c_n],
\end{split}$$
$$\begin{split}\label{nmas6}
c_0\cdot \left[
\begin{array}{c}
0\\ \overline 0 \\ {\hat{\sin}}\end{array}
\right] - c_3\cdot \left[
\begin{array}{c}
0\\ \overline 0 \\ {\hat{\sin}}\end{array}
\right]
=t_{n+6}+R_{h,k} [ c_0 , c_1 , \ldots c_n],
\end{split}$$
$$\begin{split}\label{nmas7}
c_1\cdot \left[
\begin{array}{c}
0\\{\overline{0}}\\{\hat{\cos}}\end{array}
\right]+
c_3\cdot \left[
\begin{array}{c}
0\\-{\overline{\cos}}\\{\hat{0}}\end{array}
\right]
=t_{n+7}+R_{h,k} [ c_0 , c_1 , \ldots c_n],
\end{split}$$
$$\begin{split}\label{nmas8}
c_1\cdot \left[
\begin{array}{c}
0\\{\overline{0}}\\{\hat{\sin}}\end{array}
\right]+
c_4\cdot \left[
\begin{array}{c}
0\\-{\overline{\cos}}\\{\hat{0}}\end{array}
\right]
=t_{n+8}+R_{h,k} [ c_0 , c_1 , \ldots c_n],
\end{split}$$
$$\begin{split}\label{nmasalphamas4}
c_\alpha\cdot \left[
\begin{array}{c}
0\\-{\overline{\cos}}\\{\hat{0}}\end{array}
\right]
=t_{n+\alpha+4}+R_{h,k} [ c_0 , c_1 , \ldots c_n],
\end{split}$$
for $\alpha=5,\ldots,n$, $$\begin{split}\label{2nmas5}
c_2\cdot \left[
\begin{array}{c}
0\\{\overline{0}}\\{\hat{\cos}}\end{array}
\right]+
c_3\cdot \left[
\begin{array}{c}
0\\-{\overline{\sin}}\\{\hat{0}}\end{array}
\right]
=t_{2n+5} +R_{h,k} [ c_0 , c_1 , \ldots c_n] ,
\end{split}$$
$$\begin{split}\label{2nmas6}
c_2\cdot \left[
\begin{array}{c}
0\\{\overline{0}}\\{\hat{\sin}}\end{array}
\right]+c_4\cdot \left[
\begin{array}{c}
0\\-{\overline{\sin}}\\{\hat{0}}\end{array}
\right]
=t_{2n+6} +R_{h,k} [ c_0 , c_1 , \ldots c_n],
\end{split}$$
and, for $\alpha=5,\ldots,n$, $$\begin{split}\label{2nmasalphamas2}
c_\alpha\cdot \left[
\begin{array}{c}
0\\-{\overline{\sin}}\\{\hat{0}}\end{array}
\right]
=t_{2n+\alpha+2}+R_{h,k} [ c_0 , c_1 , \ldots c_n],
\end{split}$$ $$\begin{split}\label{3nmasalphamenos2}
c_\alpha\cdot \left[
\begin{array}{c}
0\\{\overline{0}}\\-{\hat{\sin}}\end{array}
\right]
=t_{3n+\alpha-2} +R_{h,k} [ c_0 , c_1 , \ldots c_n],
\end{split}$$ $$\begin{split}\label{4nmasalphamenos6}
c_\alpha\cdot \left[
\begin{array}{c}
0\\{\overline{0}}\\-{\hat{\cos}}\end{array}
\right]
=t_{4n+\alpha-6}+R_{h,k} [ c_0 , c_1 , \ldots c_n].
\end{split}$$ Here $t_i$, $i=0,\ldots, 5n-6$, are fixed numbers such that $$\|t_i\|{\leqslant}C\|\varphi^\perp\|.$$ Moreover, $R_{h,k} [ c_0 , c_1 , \ldots c_n]$ stands for a function, whose specific definition changes from line to line, which can be described as follows: $$\begin{split}
R_{h,k} [ c_0 , c_1 , \ldots c_n] &= \Theta_{k,h} {{\mathscrL}} \left(
c_{00}, \ldots , c_{n0}
\right) + \overline \Theta_{k,h} \overline{{{\mathscrL}}} \left( \overline c_1 , \ldots , \overline c_k \right) \\
&+\hat \Theta_{k,h} \hat{{{\mathscrL}}} \left( \hat c_1 , \ldots , \hat c_h \right)
\end{split}$$ and ${{\mathscrL}}:{{\mathbb R}}^{n+1}\rightarrow {{{\mathbb R}}}$, $\overline{{{\mathscrL}}}: {{\mathbb R}}^{k(n+1)}\rightarrow {{{\mathbb R}}}$, $\hat{{{\mathscrL}}}: {{\mathbb R}}^{h(n+1)}\rightarrow {{{\mathbb R}}}$ are linear functions uniformly bounded when $k,h\rightarrow\infty$, and $$\Theta_{k,h}= O (k^{1-\frac{n}{q}}),\qquad \overline \Theta_{k,h} =O ( k^{-\frac{n}{q}}),\qquad \hat \Theta_{k,h} =O( k^{-\frac{n}{q}} ),$$ where $O(1)$ denotes a quantity uniformly bounded when $k,h\rightarrow \infty$, and $\frac{n}{2}<q<n$ is the number fixed in .
We will prove this result in Section \[prop31\].
[**Fact 3:**]{} Multiplying for every $Z_{\alpha 0}$, ${\overline{Z}}_{\alpha j}$, ${\hat{Z}}_{\alpha l}$, $\alpha=0,1,\ldots, n$, $j=1,\ldots, k$ and $l=1,\ldots, h$ and integrating in $\mathbb{R}^n$ we get a system of the form $$\label{syst}
M
\left[\begin{array}{c}
c_0\\c_1\\\vdots\\c_n
\end{array}\right]=
-\left[\begin{array}{c}
r_0\\r_1\\\vdots\\r_n
\end{array}\right]\qquad \hbox{ with }\qquad r_\alpha:=\left[\begin{array}{c}
\int_{\mathbb{R}^n}L(\varphi^\perp)Z_{\alpha 0}\\
\int_{\mathbb{R}^n}L(\varphi^\perp){\overline{Z}}_{\alpha 1}\\
\vdots\\
\int_{\mathbb{R}^n}L(\varphi^\perp){\overline{Z}}_{\alpha k}\\
\int_{\mathbb{R}^n}L(\varphi^\perp){\hat{Z}}_{\alpha 1}\\
\vdots\\
\int_{\mathbb{R}^n}L(\varphi^\perp){\hat{Z}}_{\alpha h}
\end{array}\right].$$ Due to the symmetries, the matrix $M$ has the form $$M=
\left[\begin{array}{cc}
M_1 & 0\\
0 & M_2
\end{array}\right],$$ where $M_1$ and $M_2$ are square matrices of dimensions $(5\times(k+h+1))^2$ and $((n-4)\times(k+h+1))^2$ of the form $$M_1=
\left[\begin{array}{ccccc}
\tilde{A} & \tilde{B} & \tilde{C} & \tilde{D} & \tilde{E}\\
\tilde{B}^T & \tilde{F} & \tilde{G} & \tilde{H} & \tilde{I}\\
\tilde{C}^T & \tilde{G}^T & \tilde{J} & \tilde{K} & \tilde{L}\\
\tilde{D}^T & \tilde{H}^T & \tilde{K}^T & \tilde{M} & \tilde{N}\\
\tilde{E}^T & \tilde{I}^T & \tilde{L}^T & \tilde{N}^T & \tilde{P}
\end{array}\right],\qquad
M_2=
\left[\begin{array}{cccc}
\tilde{H}_5 & 0 & 0 & 0 \\
0 & \tilde{H}_6 & 0 & 0\\
\ldots & \ldots & \ldots & \ldots\\
0 & 0 & 0 & \tilde{H}_n
\end{array}\right],$$ with $$\label{Halpha}
\tilde{H}_\alpha=
\left[\begin{array}{ccc}
\int L(Z_{\alpha 0})Z_{\alpha 0} & \left(\int L(Z_{\alpha 0}){\overline{Z}}_{\alpha j}\right)_{j} & \left(\int L(Z_{\alpha 0}){\hat{Z}}_{\alpha l}\right)_{l} \\
\left(\int L({\overline{Z}}_{\alpha i})Z_{\alpha 0}\right)_{i} & \left(\int L({\overline{Z}}_{\alpha i}){\overline{Z}}_{\alpha j}\right)_{i,j} & \left(\int L({\overline{Z}}_{\alpha i}){\hat{Z}}_{\alpha l}\right)_{i,l}\\
\left(\int L({\hat{Z}}_{\alpha m})Z_{\alpha 0}\right)_{m} & \left(\int L({\hat{Z}}_{\alpha m}){\overline{Z}}_{\alpha j}\right)_{m,j} & \left(\int L({\hat{Z}}_{\alpha m}){\hat{Z}}_{\alpha l}\right)_{m,l}
\end{array}\right],$$ for $i,j=1,\ldots,k$ and $m,l=1,\ldots,h$. Thus, solving is equivalent to find a solution of $$\label{finalSyst}
M_1\left[
\begin{array}{c}
c_0\\
c_1\\
c_2\\
c_3\\
c_4
\end{array}
\right]=
\left[
\begin{array}{c}
r_0\\
r_1\\
r_2\\
r_3\\
r_4
\end{array}
\right]
,\qquad
\tilde{H}_\alpha c_\alpha=r_\alpha\hbox{ for }\alpha=5,\ldots,n,$$ with $r_\alpha$ defined in .
\[nonso\] There exists $k_0,h_0$ such that, for all $k>k_0$, $h>h_0$, system is solvable. Moreover, the solution has the form $$\begin{split}
&c_0
=\;
v_0+
t_0\left[
\begin{array}{c}
1\\-{\overline{1}}\\-{\hat{1}}\end{array}
\right]+t_1\left[
\begin{array}{c}
0\\{\overline{0}}\\{\hat{0}}\end{array}
\right]+t_2\left[
\begin{array}{c}
0\\{\overline{0}}\\{\hat{0}}\end{array}
\right]
+t_3\left[
\begin{array}{c}
0\\{\overline{0}}\\{\hat{0}}\end{array}
\right]+t_4\left[
\begin{array}{c}
0\\{\overline{0}}\\{\hat{0}}\end{array}
\right]\\
&\qquad+
\overline{t}_0\left[
\begin{array}{c}
0\\ {\overline{0}}\\ {\hat{0}}\\
\end{array}\right]+\overline{t}_1 \left[
\begin{array}{c}
0\\ {\overline{\cos}}\\ {\hat{0}}\end{array}\right]
+\overline{t}_2 \left[
\begin{array}{c}
0\\ {\overline{\sin}}\\ {\hat{0}}\end{array}\right]
+
\hat{t}_0 \left[
\begin{array}{c}
0\\ {\overline{0}}\\ {\hat{0}}\end{array}\right]+\hat{t}_1 \left[
\begin{array}{c}
0\\ {\overline{0}}\\ {\hat{\cos}}\end{array}\right]+\hat{t}_2 \left[
\begin{array}{c}
0\\ {\overline{0}}\\ {\hat{\sin}}\end{array}\right],\\
\end{split}$$ $$\begin{split}
&c_1
=\;
v_1+
t_0\left[
\begin{array}{c}
0\\-{\overline{1}}\\{\hat{0}}\end{array}
\right]+t_1\left[
\begin{array}{c}
1\\-\frac{1}{\sqrt{1-\mu^2}}{\overline{\cos}}\\-{\hat{1}}\end{array}
\right]+t_2\left[
\begin{array}{c}
0\\-\frac{1}{\sqrt{1-\mu^2}}{\overline{\sin}}\\{\hat{0}}\end{array}
\right]+t_3\left[
\begin{array}{c}
0\\{\overline{0}}\\{\hat{0}}\end{array}
\right]+t_4\left[
\begin{array}{c}
0\\{\overline{0}}\\{\hat{0}}\end{array}
\right]\\
&\qquad
+
\overline{t}_0\left[
\begin{array}{c}
0\\ {\overline{0}}\\ {\hat{0}}\\
\end{array}\right]+\overline{t}_1 \left[
\begin{array}{c}
0\\ -{\overline{\cos}}\\ {\hat{0}}\\
\end{array}\right]
+\overline{t}_2 \left[
\begin{array}{c}
0\\ -{\overline{\sin}}\\ {\hat{0}}\\
\end{array}\right]
+
\hat{t}_0 \left[
\begin{array}{c}
0\\ {\overline{0}}\\ {\hat{0}}\\
\end{array}\right]+\hat{t}_1 \left[
\begin{array}{c}
0\\ {\overline{0}}\\ {\hat{0}}\\
\end{array}\right]+\hat{t}_2 \left[
\begin{array}{c}
0\\ {\overline{0}}\\ {\hat{0}}\end{array}\right],
\end{split}$$ $$\begin{split}
&c_2
=\;
v_2+
t_0\left[
\begin{array}{c}
0\\{\overline{0}}\\{\hat{0}}\end{array}
\right]+t_1\left[
\begin{array}{c}
0\\\frac{1}{\sqrt{1-\mu^2}}{\overline{\sin}}\\{\hat{0}}\end{array}
\right]+t_2\left[
\begin{array}{c}
1\\-\frac{1}{\sqrt{1-\mu^2}}{\overline{\cos}}\\-{\hat{1}}\end{array}
\right]+t_3\left[
\begin{array}{c}
0\\{\overline{0}}\\{\hat{0}}\end{array}
\right]+t_4\left[
\begin{array}{c}
0\\{\overline{0}}\\{\hat{0}}\end{array}
\right]\\
&\qquad
+
\overline{t}_0\left[
\begin{array}{c}
0\\ {\overline{1}}\\ {\hat{0}}\\
\end{array}\right]+\overline{t}_1 \left[
\begin{array}{c}
0\\ {\overline{0}}\\ {\hat{0}}\\
\end{array}\right]
+\overline{t}_2 \left[
\begin{array}{c}
0\\ {\overline{0}}\\ {\hat{0}}\\
\end{array}\right]+
\hat{t}_0 \left[
\begin{array}{c}
0\\ {\overline{0}}\\ {\hat{0}}\\
\end{array}\right]
+\hat{t}_1 \left[
\begin{array}{c}
0\\ {\overline{0}}\\ {\hat{0}}\\
\end{array}\right]+\hat{t}_2 \left[
\begin{array}{c}
0\\ {\overline{0}}\\ {\hat{0}}\\
\end{array}\right],
\end{split}$$ $$\begin{split}
&c_3
=\;
v_3+
t_0\left[
\begin{array}{c}
0\\{\overline{0}}\\-{\hat{1}}\end{array}
\right]+t_1\left[
\begin{array}{c}
0\\{\overline{0}}\\{\hat{0}}\end{array}
\right]+t_2\left[
\begin{array}{c}
0\\{\overline{0}}\\{\hat{0}}\\
\end{array}
\right]+t_3\left[
\begin{array}{c}
1\\-{\overline{1}}\\-\frac{1}{\sqrt{1-\lambda^2}}{\hat{\cos}}\end{array}
\right]+t_4\left[
\begin{array}{c}
0\\{\overline{0}}\\-\frac{1}{\sqrt{1-\lambda^2}}{\hat{\sin}}\end{array}
\right]\\
&\qquad
+
\overline{t}_0\left[
\begin{array}{c}
0\\ {\overline{0}}\\ {\hat{0}}\end{array}\right]+\overline{t}_1 \left[
\begin{array}{c}
0\\ {\overline{0}}\\ {\hat{0}}\end{array}\right]
+\overline{t}_2 \left[
\begin{array}{c}
0\\ {\overline{0}}\\ {\hat{0}}\end{array}\right]
+
\hat{t}_0 \left[
\begin{array}{c}
0\\ {\overline{0}}\\ {\hat{0}}\end{array}\right]+\hat{t}_1 \left[
\begin{array}{c}
0\\ {\overline{0}}\\ -{\hat{\cos}}\end{array}\right]+\hat{t}_2 \left[
\begin{array}{c}
0\\ {\overline{0}}\\ -{\hat{\sin}}\end{array}\right],
\end{split}$$ $$\begin{split}
&c_4
= \;
v_4+
t_0\left[
\begin{array}{c}
0\\{\overline{0}}\\{\hat{0}}\end{array}
\right]+t_1\left[
\begin{array}{c}
0\\{\overline{0}}\\{\hat{0}}\end{array}
\right]+t_2\left[
\begin{array}{c}
0\\{\overline{0}}\\{\hat{0}}\end{array}
\right]+t_3\left[
\begin{array}{c}
0\\{\overline{0}}\\\frac{1}{\sqrt{1-\lambda^2}}{\hat{\sin}}\end{array}
\right]+t_4\left[
\begin{array}{c}
1\\-{\overline{1}}\\-\frac{1}{\sqrt{1-\lambda^2}}{\hat{\cos}}\end{array}
\right]\\
&\qquad
+
\overline{t}_0\left[
\begin{array}{c}
0\\ {\overline{0}}\\ {\hat{0}}\end{array}\right]+\overline{t}_1 \left[
\begin{array}{c}
0\\ {\overline{0}}\\ {\hat{0}}\end{array}\right]+\overline{t}_2 \left[
\begin{array}{c}
0\\ {\overline{0}}\\ {\hat{0}}\end{array}\right]
+
\hat{t}_0 \left[
\begin{array}{c}
0\\ {\overline{0}}\\ {\hat{1}}\end{array}\right]+\hat{t}_1 \left[
\begin{array}{c}
0\\ {\overline{0}}\\ {\hat{0}}\end{array}\right]+\hat{t}_2 \left[
\begin{array}{c}
0\\ {\overline{0}}\\ {\hat{0}}\end{array}\right],
\end{split}$$ and, for $\alpha=5,\ldots,n$, $$\begin{split}
c_\alpha&=v_\alpha
+t_{\alpha }\left[\begin{array}{c}
1\\-{\overline{1}}\\-{\hat{1}}\end{array}\right] + \overline \nu_{\alpha 1} \left[\begin{array}{c}
0\\{\overline{\cos}}\\ {\hat{0}}\end{array}\right] + \overline \nu_{\alpha 2} \left[\begin{array}{c}
0\\{\overline{\sin}}\\ {\hat{0}}\end{array}\right] + \hat \nu_{\alpha 1} \left[\begin{array}{c}
0\\{\overline{0}}\\ {\hat{\cos}}\end{array}\right] +\hat \nu_{\alpha 2} \left[\begin{array}{c}
0\\{\overline{0}}\\ {\hat{\sin}}\end{array}\right],
\end{split}$$ for any $t_0,t_1,t_2,t_3,t_4$, $\overline{t}_0,\overline{t}_1,\overline{t}_2$,$\hat{t}_0,\hat{t}_1,\hat{t}_2$, and $t_{\alpha } , \overline \nu_{\alpha 1} , \overline \nu_{\alpha 2} , \hat \nu_{\alpha 1} , \hat \nu_{\alpha 2}$ real parameters. The vectors $v_\alpha\in {{\mathbb R}}^{k+h+1}$ are fixed and satisfy $$\|v_\alpha\|{\leqslant}C\|\varphi^\perp\|, \;\alpha=0,1,\ldots, n.$$
Proceeding as in [@MW Proposition 6.1] it can be checked that, for any $\alpha=0,\ldots,n$, $$\|\overline{r}_\alpha\|{\leqslant}C\mu^{\frac{n-2}{2}}\|\varphi^\perp\|,\qquad \|\hat{r}_\alpha\|{\leqslant}C\lambda^{\frac{n-2}{2}}\|\varphi^\perp\|,$$ and combining this estimate with Lemma \[uno\] and Lemma \[dos\] we obtain the result.
We shall use the following notations: for any $\alpha=0,1,\ldots,n,$ $$\label{vectorsC}
\tilde{c}_\alpha:=\left[\begin{array}{c}
{\overline{c}}_\alpha\\{\hat{c}}_{\alpha}
\end{array}\right] \in {{\mathbb R}}^{k+h} ,\;\;{\overline{c}}_\alpha:=\left[\begin{array}{c}{\overline{c}}_{\alpha 1}\\\ldots\\{\overline{c}}_{\alpha k}\end{array}\right]\in {{\mathbb R}}^k ,\;\;{\hat{c}}_\alpha:=\left[\begin{array}{c}{\hat{c}}_{\alpha 1}\\\ldots\\{\hat{c}}_{\alpha h}\end{array}\right]\in {{\mathbb R}}^h,$$ $$\tilde{r}_\alpha:=\left[\begin{array}{c}
{\overline{r}}_\alpha\\{\hat{r}}_\alpha
\end{array}\right]\in {{\mathbb R}}^{k+h},\;\;$$ where $${\overline{r}}_\alpha:=\left[\begin{array}{c}
\int_{\mathbb{R}^n}L(\varphi^\perp){\overline{Z}}_{\alpha 1}\\
\vdots\\
\int_{\mathbb{R}^n}L(\varphi^\perp){\overline{Z}}_{\alpha k}\end{array}\right]\in {{\mathbb R}}^k,\;\;
{\hat{r}}_\alpha:=\left[\begin{array}{c}
\int_{\mathbb{R}^n}L(\varphi^\perp){\hat{Z}}_{\alpha 1}\\
\vdots\\
\int_{\mathbb{R}^n}L(\varphi^\perp){\hat{Z}}_{\alpha h}
\end{array}\right]\in {{\mathbb R}}^h.$$
Solving the second system in
=============================
Let $\alpha $ be fixed in $\{ 5, \ldots , n \}$. This section is devoted to solve $$\label{semplice}
\tilde{H}_\alpha c_\alpha=r_\alpha,$$ where $r_\alpha$ is the vector defined in .
Using and and the fact that $L({\bf z}_\alpha)=0$ it follows that $$\hbox{row}_1(\tilde{H}_\alpha)=\sum_{l=2}^{k+h+1}\hbox{row}_l(\tilde{H}_\alpha).$$ As a consequence, $\left[\begin{array}{c}
1\\
-{\overline{1}}\\
-{\hat{1}}\end{array}\right]\in\ker(\tilde{H}_\alpha),$ and hence $\tilde{H}_\alpha c_\alpha=r_\alpha$ has a solution only if $r_\alpha\cdot\left[\begin{array}{c}
1\\
-{\overline{1}}\\
-{\hat{1}}\end{array}\right]=0$. This last orthogonality condition is indeed fulfilled since one has $$\label{rowalpha}
\hbox{row}_1(r_\alpha)=\sum_{j=2}^{k+1}\hbox{row}_j(r_\alpha)+\sum_{l=k+2}^{h+k+1}\hbox{row}_l(r_\alpha),$$ again as consequence of the fact that $L({\bf z}_\alpha)=0$. Thus, the general solution to has the form $$c_\alpha=\left[\begin{array}{c}
0\\\tilde{c}_\alpha
\end{array}\right]
+t\left[\begin{array}{c}
1\\-{\overline{1}}\\-{\hat{1}}\end{array}\right],\qquad t\in{{\mathbb R}},$$ where $\tilde c_\alpha$ solves $$\label{semplice1}
H_\alpha \tilde c_\alpha = \tilde r_\alpha,$$ with $$H_\alpha=\left[
\begin{array}{cc}
\overline{H}_\alpha & \gamma_\alpha \mathds{1}_{k\times h}\\
\gamma_\alpha \mathds{1}_{h\times k} & \hat{H}_\alpha
\end{array}
\right],\qquad \alpha=5,\ldots,n,$$ being $\bar H_\alpha$ and $\hat H_\alpha$ square matrices of dimensions $k\times k$ and $h\times h$ respectively, defined by $$\bar H_\alpha := \left( \int L(\bar Z_{\alpha , i} ) \bar Z_{\alpha ,j} \, dy \right)_{i,j=1, \ldots k}, \quad
\hat H_\alpha := \left( \int L(\hat Z_{\alpha , l} ) \hat Z_{\alpha ,m} \, dy \right)_{l,m=1, \ldots h}$$ and $$\gamma_\alpha:=\int_{{{\mathbb R}}^n}L({\overline{Z}}_{\alpha 1}){\hat{Z}}_{\alpha 1}.$$ By $\mathds{1}_{s\times t}$ we mean a $s\times t$-dimensional matrix whose entries are all 1. Observe that $$\label{ss0}
\left| \gamma_\alpha \right| {\leqslant}C k^{4-2n},$$ for some fixed constant $C$. Arguing as in [@MW], one can show that $\bar H_\alpha$ and $\hat H_\alpha$ are circulant matrices of dimensions $(k\times k)$ and $(h\times h)$ respectively (see [@KS] for properties). Moreover, [@MW Proposition 5.1] ensures that $$\label{semplice3}
\overline{H}_\alpha[{\overline{c}}_\alpha]={\overline{s}}_\alpha,\;\; \hat{H}_\alpha[{\hat{c}}_\alpha]={\hat{s}}_\alpha,$$ has a solution if $${\overline{s}}_\alpha\cdot{\overline{\cos}}= {\overline{s}}_\alpha\cdot{\overline{\sin}}= 0\;\hbox{ and }\;{\hat{s}}_\alpha\cdot{\hat{\cos}}= {\hat{s}}_\alpha\cdot{\hat{\sin}}= 0.$$ Actually, if a solution to exists, it has the form $$\begin{split}
&{\overline{c}}_\alpha={\overline{w}}_\alpha+\overline \nu_1{\overline{\cos}}+\overline \nu_2{\overline{\sin}}, \quad
{\hat{c}}_\alpha={\hat{w}}_\alpha+\hat \nu_1{\hat{\cos}}+\hat \nu_2{\hat{\sin}},
\end{split}$$ for all $\overline \nu_1,\,\overline \nu_2,\, \hat \nu_1,\, \hat \nu_2\in{{\mathbb R}}$, where ${\overline{w}}_\alpha,\, {\hat{w}}_\alpha$ are the unique solutions to $$\overline H_\alpha {\overline{w}}_\alpha = {\overline{s}}_\alpha , \quad {\overline{w}}_\alpha \cdot {\overline{\cos}}= {\overline{w}}_\alpha \cdot {\overline{\sin}}= 0$$ and $$\hat H_\alpha {\hat{w}}_\alpha = {\hat{s}}_\alpha , \quad {\hat{w}}_\alpha \cdot {\hat{\cos}}= {\hat{w}}_\alpha \cdot {\hat{\sin}}= 0.$$ Furthermore, in [@MW Proposition 5.1] it is proved that there exists a constant $C$ independent of $k$ so that, for all $k $ large $$\|{\overline{w}}_\alpha \|{\leqslant}C k^{n-4} \|{\overline{s}}_\alpha \|\;\;\hbox{and}\;\;\|{\hat{w}}_\alpha\|{\leqslant}C k^{n-4} \|{\hat{s}}_\alpha\|.$$
We start with the observation that system is solvable. Indeed, since $L({\bf z}_{n+\alpha + 4} )= L ({\bf z}_{2n+\alpha+2} ) = 0$, one has that $\overline r_\alpha \cdot {\overline{\cos}}=
\overline r_\alpha \cdot {\overline{\sin}}=0$. Similarly, one gets that $\hat r_\alpha \cdot {\hat{\cos}}=
\hat r_\alpha \cdot {\hat{\sin}}=0$, as consequence of the fact that $L({\bf z}_{3n+\alpha -2} )= L ({\bf z}_{4n+\alpha-6} ) = 0$. Moreover, the vector $\mathds{1}_{k\times h} \hat c_\alpha$ is a multiple of ${\overline{1}}$, and $\mathds{1}_{h\times k} \overline c_\alpha$ is a multiple of ${\hat{1}}$. Thus $\mathds{1}_{k\times h} \hat c_\alpha \cdot \cos = \mathds{1}_{k\times h} \hat c_\alpha \cdot \sin =0$, and $\mathds{1}_{h\times k} \overline c_\alpha \cdot {\hat{\cos}}= \mathds{1}_{h\times k} \overline c_\alpha\cdot {\hat{\sin}}=0$. Now we observe that the solution of system has the form $$\begin{split} \label{ss1}
\overline c_\alpha &= {\overline{w}}_\alpha + \overline \nu_1 {\overline{\cos}}+ \overline \nu_2 {\overline{\sin}}\\
\hat c_\alpha &= {\hat{w}}_\alpha + \hat \nu_1 {\overline{\cos}}+ \hat \nu_2 {\overline{\sin}},
\end{split}$$ for any value for $\overline \nu_1,\,\overline \nu_2,\, \hat \nu_1,\, \hat \nu_2\in{{\mathbb R}}$, where ${\overline{w}}_\alpha$ and ${\hat{w}}_\alpha$ are the unique solutions to $$\begin{split} \label{ss2}
\overline H_\alpha {\overline{w}}_\alpha &= \overline r_\alpha -\gamma_\alpha \mathds{1}_{k\times h} \hat w_\alpha, \quad {\overline{w}}_\alpha \cdot {\overline{\cos}}= {\overline{w}}_\alpha \cdot {\overline{\sin}}= 0, \\
\hat H_\alpha {\hat{w}}_\alpha &= \hat r_\alpha -\gamma_\alpha \mathds{1}_{h\times k} \overline w_\alpha, \quad {\hat{w}}_\alpha \cdot {\hat{\cos}}= {\hat{w}}_\alpha \cdot {\hat{\sin}}= 0.
\end{split}$$ Moreover, there exists a constant $C$ so that $$\label{ss3}
\|{\overline{w}}_\alpha \|{\leqslant}C k^{n-4} \|\overline r_\alpha \|\;\;\hbox{and}\;\;\|{\hat{w}}_\alpha\|{\leqslant}C k^{n-4} \|\hat r_\alpha\|.$$ Existence and uniqueness of solutions to satisfying follows from a contraction map argument. Indeed, $\left[\begin{array}{c}
{\overline{w}}_\alpha \\{\hat{w}}_\alpha
\end{array}\right]$ is a solution if and only if it is a fixed point to $A_\alpha \left[\begin{array}{c}
{\overline{w}}_\alpha \\{\hat{w}}_\alpha
\end{array}\right] := T_\alpha^{-1} \left( \left[\begin{array}{c}
\overline r_\alpha -\gamma_\alpha \mathds{1}_{k\times h} \hat w_\alpha \\ \hat r_\alpha -\gamma_\alpha \mathds{1}_{h\times k} \overline w_\alpha
\end{array}\right] \right)$, where we denote by $T_\alpha$ the linear map $T_\alpha \left( \left[\begin{array}{c}
{\overline{w}}_\alpha \\{\hat{w}}_\alpha
\end{array}\right] \right) = \left( \left[\begin{array}{c}
\overline H_\alpha {\overline{w}}_\alpha \\ \hat H_\alpha {\hat{w}}_\alpha
\end{array}\right] \right)$, which is invertible for vectors that are orthogonal to ${\overline{\cos}}$, ${\overline{\sin}}$, in their first components, and to ${\hat{\cos}}$, ${\hat{\sin}}$ in their second components. Let $$B_r := \{ \left[\begin{array}{c}
{\overline{w}}_\alpha \\{\hat{w}}_\alpha
\end{array}\right] \in K_\alpha \, : \, \|{\overline{w}}_\alpha \|{\leqslant}r k^{n-4} \|\overline r_\alpha \|\;\;\hbox{and}\;\;\|{\hat{w}}_\alpha\|{\leqslant}r k^{n-4} \|\hat r_\alpha\| \},$$ where $K_\alpha := \{ \left[\begin{array}{c}
{\overline{w}}_\alpha \\{\hat{w}}_\alpha
\end{array}\right] \in {{\mathbb R}}^{k+h} \, : \, {\overline{w}}_\alpha \cdot {\overline{\cos}}= {\overline{w}}_\alpha \cdot {\overline{\sin}}=0, {\hat{w}}_\alpha \cdot {\hat{\cos}}= {\hat{w}}_\alpha \cdot {\hat{\sin}}=0 \}$. Then, choosing $r$ large but fixed, and thanks to , one has that $A_\alpha$ is a contraction in $B_r$. This gives the existence of solutions to , satisfying .
Summarizing the above arguments, we have
\[uno\] Let $\alpha \in \{5, \ldots ,n \}$ be fixed. Then system is solvable, and the solution has the form $$\begin{split} \label{solsemplice}
c_\alpha&=\left[\begin{array}{c}
0\\{\overline{w}}_\alpha \\ {\hat{w}}_\alpha
\end{array}\right]
+t\left[\begin{array}{c}
1\\-{\overline{1}}\\-{\hat{1}}\end{array}\right] + \overline \nu_1 \left[\begin{array}{c}
0\\{\overline{\cos}}\\ {\hat{0}}\end{array}\right] + \overline \nu_2 \left[\begin{array}{c}
0\\{\overline{\sin}}\\ {\hat{0}}\end{array}\right] + \hat \nu_1 \left[\begin{array}{c}
0\\{\overline{0}}\\ {\hat{\cos}}\end{array}\right] +\hat \nu_2 \left[\begin{array}{c}
0\\{\overline{0}}\\ {\hat{\sin}}\end{array}\right],
\end{split}$$ for any values of $t , \overline \nu_1 , \overline \nu_2 , \hat \nu_1 , \hat \nu_2 \in {{\mathbb R}}$. In the above formula $\left[\begin{array}{c}
{\overline{w}}_\alpha\\ {\hat{w}}_\alpha
\end{array}\right]$ is the unique solution to , and satisfies .
Solving the first system in
============================
This section is devoted to solve the first system in , namely $$\label{difficile}
M_1\left[
\begin{array}{c}
c_0\\
c_1\\
c_2\\
c_3\\
c_4
\end{array}
\right]=
\left[
\begin{array}{c}
r_0\\
r_1\\
r_2\\
r_3\\
r_4
\end{array}
\right].$$ Using and and the fact that $L({\bf z}_\alpha)=0$ for every $\alpha=0,\ldots,4$, together with the result in Section \[appe1\], we observe that $$\begin{split}
\hbox{row}_1(M_1)=&\sum_{i=1}^{k}\hbox{row}_{1+i}(M_1)+\hbox{row}_{k+h+2+i}(M_1)+\sum_{i=1}^{h}\hbox{row}_{k+1+i}(M_1)+\hbox{row}_{4k+3h+4+i}(M_1),
\end{split}$$ $$\begin{split}
\hbox{row}_{k+h+2}(M_1)=&\frac{1}{\sqrt{1-\mu^2}}\left[\sum_{i=1}^{k}\cos{\overline{\theta}}_i\hbox{row}_{k+h+2+i}(M_1)-\sin{\overline{\theta}}_i \hbox{row}_{2k+2h+3+i}(M_1)\right]\\
&+\sum_{i=1}^{h}\hbox{row}_{2k+h+2+i}(M_1),
\end{split}$$
$$\begin{split}
\hbox{row}_{2k+2h+3}(M_1)=&\frac{1}{\sqrt{1-\mu^2}}\left[\sum_{i=1}^{k}\sin{\overline{\theta}}_i\hbox{row}_{k+h+2+i}(M_1)+\cos{\overline{\theta}}_i \hbox{row}_{2k+2h+3+i}(M_1)\right]\\
&+\sum_{i=1}^{h}\hbox{row}_{3k+2h+3+i}(M_1),
\end{split}$$
$$\begin{split}
&\hbox{row}_{3k+3h+4}(M_1)=\sum_{i=1}^{k}\hbox{row}_{3k+3h+4+i}(M_1)\\
&\qquad+\frac{1}{\sqrt{1-\lambda^2}}\left[\sum_{i=1}^{h}\cos{\hat{\theta}}_i\hbox{row}_{4k+3h+4+i}(M_1)-\sin{\hat{\theta}}_i \hbox{row}_{5k+4h+5+i}(M_1)\right],
\end{split}$$
$$\begin{split}
&\hbox{row}_{4k+4h+5}(M_1)=\sum_{i=1}^{k}\hbox{row}_{4k+4h+5+i}(M_1)\\
&\qquad+\frac{1}{\sqrt{1-\lambda^2}}\left[\sum_{i=1}^{h}\sin{\hat{\theta}}_i\hbox{row}_{4k+3h+4+i}(M_1)+\cos{\hat{\theta}}_i \hbox{row}_{5k+4h+5+i}(M_1)\right].
\end{split}$$
From these facts we deduce that system is solvable only if $$\label{ortw}
\left[
\begin{array}{c}
r_0\\r_1\\r_2\\r_3\\r_4
\end{array}
\right]\cdot w_j=0,\qquad j=0,1,\ldots,4,$$ where (recall definitions and ) $$\label{vectors11}
w_0:=
\left[
\begin{array}{c}
1\\-{\overline{1}}\\-{\hat{1}}\\0\\-{\overline{1}}\\{\hat{0}}\\0\\{\overline{0}}\\{\hat{0}}\\0\\{\overline{0}}\\-{\hat{1}}\\0\\{\overline{0}}\\{\hat{0}}\end{array}
\right],\;\;
w_1:=\left[
\begin{array}{c}
0\\{\overline{0}}\\{\hat{0}}\\1\\-\frac{1}{\sqrt{1-\mu^2}}{\overline{\cos}}\\-{\hat{1}}\\0\\\frac{1}{\sqrt{1-\mu^2}}{\overline{\sin}}\\{\hat{0}}\\0\\{\overline{0}}\\{\hat{0}}\\0\\{\overline{0}}\\{\hat{0}}\end{array}
\right],\;\;
w_2:=\left[
\begin{array}{c}
0\\{\overline{0}}\\{\hat{0}}\\0\\-\frac{1}{\sqrt{1-\mu^2}}{\overline{\sin}}\\{\hat{0}}\\1\\-\frac{1}{\sqrt{1-\mu^2}}{\overline{\cos}}\\-{\hat{1}}\\0\\{\overline{0}}\\{\hat{0}}\\0\\{\overline{0}}\\{\hat{0}}\end{array}
\right],$$ $$\label{vectors12}
w_3:=\left[
\begin{array}{c}
0\\{\overline{0}}\\{\hat{0}}\\0\\{\overline{0}}\\{\hat{0}}\\0\\{\overline{0}}\\{\hat{0}}\\1\\-{\overline{1}}\\-\frac{1}{\sqrt{1-\lambda^2}}{\hat{\cos}}\\0\\{\overline{0}}\\\frac{1}{\sqrt{1-\lambda^2}}{\hat{\sin}}\end{array}
\right],\;\;
w_4:=\left[
\begin{array}{c}
0\\{\overline{0}}\\{\hat{0}}\\0\\{\overline{0}}\\{\hat{0}}\\0\\{\overline{0}}\\{\hat{0}}\\0\\{\overline{0}}\\-\frac{1}{\sqrt{1-\lambda^2}}{\hat{\sin}}\\1\\-{\overline{1}}\\-\frac{1}{\sqrt{1-\lambda^2}}{\hat{\cos}}\end{array}
\right],$$ which belong all to $\ker(M_1)$. On the other hand, using again that $L({\bf z}_\alpha)=0$ for every $\alpha=0,\ldots,4$ one sees that the vectors $r_\alpha$ satisfy the following relations $$\begin{split}
\hbox{row}_1(r_0)=&\sum_{j=2}^{k+1}\left[\hbox{row}_j(r_0)+\hbox{row}_j(r_1)\right]+\sum_{l=k+2}^{h+k+1}\left[\hbox{row}_l(r_0)+\hbox{row}_l(r_3)\right]\\
\hbox{row}_1(r_1)=&\frac{1}{\sqrt{1-\mu^2}}\sum_{j=2}^{k+1}\left[\cos{\overline{\theta}}_{j-1}\hbox{row}_j(r_1)-\sin{\overline{\theta}}_{j-1}\hbox{row}_j(r_2)\right]+\sum_{l=k+2}^{h+k+1}\hbox{row}_l(r_1),\\
\hbox{row}_1(r_2)=&\frac{1}{\sqrt{1-\mu^2}}\sum_{j=2}^{k+1}\left[\sin{\overline{\theta}}_{j-1}\hbox{row}_j(r_1)+\cos{\overline{\theta}}_{j-1}\hbox{row}_j(r_2)\right]+\sum_{l=k+2}^{h+k+1}\hbox{row}_l(r_2),\\
\hbox{row}_1(r_3)=&\sum_{j=2}^{k+1}\hbox{row}_j(r_3)+\frac{1}{\sqrt{1-\lambda^2}}\sum_{l=k+2}^{h+k+1}\left[\cos{\hat{\theta}}_{l-1}\hbox{row}_l(r_3)-\sin{\hat{\theta}}_{l-1}\hbox{row}_l(r_4)\right],\\
\hbox{row}_1(r_4)=&\sum_{j=2}^{k+1}\hbox{row}_j(r_4)+\frac{1}{\sqrt{1-\lambda^2}}\sum_{l=k+2}^{h+k+1}\left[\sin{\hat{\theta}}_{l-1}\hbox{row}_l(r_3)+\cos{\hat{\theta}}_{l-1}\hbox{row}_l(r_4)\right].
\end{split}$$ These facts imply that the orthogonality conditions are satisfied, and thus is solvable. The solution to has the form $$\left[
\begin{array}{c}
c_0\\c_1\\c_2\\c_3\\c_4
\end{array}
\right]=\left[
\begin{array}{c}
0\\\tilde{c}_0\\0\\\tilde{c}_1\\0\\\tilde{c}_2\\0\\\tilde{c}_3\\0\\\tilde{c}_4
\end{array}\right]
+tw_0+sw_1+rw_2+uw_3+vw_4,$$ for any values of $t,s,r,u,v\in{{\mathbb R}},$ where $\tilde{c}_\alpha :=\left[\begin{array}{c}
{\overline{c}}_\alpha\\{\hat{c}}_{\alpha}
\end{array}\right]$ are solutions of $$\label{systemQ}
Q\left[
\begin{array}{c}
\tilde{c}_0\\\tilde{c}_1\\\tilde{c}_2\\\tilde{c}_3\\\tilde{c}_4
\end{array}\right]=
\left[
\begin{array}{c}
\tilde{r}_0\\\tilde{r}_1\\\tilde{r}_2\\\tilde{r}_3\\\tilde{r}_4
\end{array}\right], \quad \tilde r_\alpha :=\left[\begin{array}{c}
\overline r_\alpha\\\hat r_{\alpha}
\end{array}\right]$$ Here $Q$ is the square matrix of dimension $[5 (k+h)\times (k+h)]^2$ defined as $$Q:=\left[\begin{array}{ccccc}
A & B & C & D & E\\
B^T & F & G & H & I\\
C^T & G^T & J & K & L\\
D^T & H^T & K^T & M & N\\
E^T & I^T & L^T & N^T & P
\end{array}\right],$$ where every submatrix of $Q$ has dimension $(k+h)\times (k+h)$ and entries of the form $$\int_{{{\mathbb R}}^n}L(V)W,$$ where
- In $A$: $V,W\in \{({\overline{Z}}_{0j})_{j=1,\ldots,k},({\hat{Z}}_{0l})_{l=1,\ldots,h}\}$.
- In $B$: $V\in \{({\overline{Z}}_{0j})_{j=1,\ldots,k},({\hat{Z}}_{0l})_{l=1,\ldots,h}\}$, $W\in \{({\overline{Z}}_{1j})_{j=1,\ldots,k},({\hat{Z}}_{1l})_{l=1,\ldots,h}\}$.
- In $C$: $V\in \{({\overline{Z}}_{0j})_{j=1,\ldots,k},({\hat{Z}}_{0l})_{l=1,\ldots,h}\}$, $W\in \{({\overline{Z}}_{2j})_{j=1,\ldots,k},({\hat{Z}}_{2l})_{l=1,\ldots,h}\}$.
- In $D$: $V\in \{({\overline{Z}}_{0j})_{j=1,\ldots,k},({\hat{Z}}_{0l})_{l=1,\ldots,h}\}$, $W\in \{({\overline{Z}}_{3j})_{j=1,\ldots,k},({\hat{Z}}_{3l})_{l=1,\ldots,h}\}$.
- In $E$: $V\in \{({\overline{Z}}_{0j})_{j=1,\ldots,k},({\hat{Z}}_{0l})_{l=1,\ldots,h}\}$, $W\in \{({\overline{Z}}_{4j})_{j=1,\ldots,k},({\hat{Z}}_{4l})_{l=1,\ldots,h}\}$.
- In $F$: $V,W\in \{({\overline{Z}}_{1j})_{j=1,\ldots,k},({\hat{Z}}_{1l})_{l=1,\ldots,h}\}$.
- In $G$: $V\in \{({\overline{Z}}_{1j})_{j=1,\ldots,k},({\hat{Z}}_{1l})_{l=1,\ldots,h}\}$, $W\in \{({\overline{Z}}_{2j})_{j=1,\ldots,k},({\hat{Z}}_{2l})_{l=1,\ldots,h}\}$.
- In $H$: $V\in \{({\overline{Z}}_{1j})_{j=1,\ldots,k},({\hat{Z}}_{1l})_{l=1,\ldots,h}\}$, $W\in \{({\overline{Z}}_{3j})_{j=1,\ldots,k},({\hat{Z}}_{3l})_{l=1,\ldots,h}\}$.
- In $I$: $V\in \{({\overline{Z}}_{1j})_{j=1,\ldots,k},({\hat{Z}}_{1l})_{l=1,\ldots,h}\}$, $W\in \{({\overline{Z}}_{4j})_{j=1,\ldots,k},({\hat{Z}}_{4l})_{l=1,\ldots,h}\}$.
- In $J$: $V,W\in \{({\overline{Z}}_{2j})_{j=1,\ldots,k},({\hat{Z}}_{2l})_{l=1,\ldots,h}\}$.
- In $K$: $V\in \{({\overline{Z}}_{2j})_{j=1,\ldots,k},({\hat{Z}}_{2l})_{l=1,\ldots,h}\}$, $W\in \{({\overline{Z}}_{3j})_{j=1,\ldots,k},({\hat{Z}}_{3l})_{l=1,\ldots,h}\}$.
- In $L$: $V\in \{({\overline{Z}}_{2j})_{j=1,\ldots,k},({\hat{Z}}_{2l})_{l=1,\ldots,h}\}$, $W\in \{({\overline{Z}}_{4j})_{j=1,\ldots,k},({\hat{Z}}_{4l})_{l=1,\ldots,h}\}$.
- In $M$: $V,W\in \{({\overline{Z}}_{3j})_{j=1,\ldots,k},({\hat{Z}}_{3l})_{l=1,\ldots,h}\}$.
- In $N$: $V\in \{({\overline{Z}}_{3j})_{j=1,\ldots,k},({\hat{Z}}_{3l})_{l=1,\ldots,h}\}$, $W\in \{({\overline{Z}}_{4j})_{j=1,\ldots,k},({\hat{Z}}_{4l})_{l=1,\ldots,h}\}$.
- In $P$: $V,W\in \{({\overline{Z}}_{4j})_{j=1,\ldots,k},({\hat{Z}}_{4l})_{l=1,\ldots,h}\}$.
Let us analize the structure of every matrix, where we will make use of the notation $$\label{defBeta}
\beta_{\alpha_1,\alpha_2}:=\int L({\overline{Z}}_{\alpha_11}){\hat{Z}}_{\alpha_2 1},$$ $${\overline{\theta}}_j:=\frac{2\pi}{k}(j-1)\qquad\mbox{ and }\qquad {\hat{\theta}}_l:=\frac{2\pi}{h}(l-1).$$
[**Matrix A.**]{} Due to the invariance properties one can check that for $i, j=1,\ldots,k$ and $l,m=1,\ldots, h$, $$\begin{split}
\int_{{{\mathbb R}}^n}L({\overline{Z}}_{0i}){\overline{Z}}_{0j}&=\int_{{{\mathbb R}}^n}L({\overline{Z}}_{01}){\overline{Z}}_{0,|i-j|+1},\\
\int_{{{\mathbb R}}^n}L({\hat{Z}}_{0l}){\hat{Z}}_{0m}&=\int_{{{\mathbb R}}^n}L({\hat{Z}}_{01}){\hat{Z}}_{0,|l-m|+1},\\
\int_{{{\mathbb R}}^n}L({\overline{Z}}_{0j}){\hat{Z}}_{0l}&=\int_{{{\mathbb R}}^n}L({\hat{Z}}_{0l}){\overline{Z}}_{0j}=\int_{{{\mathbb R}}^n}L({\overline{Z}}_{01}){\hat{Z}}_{01}=\beta_{00}.
\end{split}$$ Thus we can write $$A=\left[
\begin{array}{cc}
\overline{A} & A_1:=\beta_{00} \mathds{1}_{k\times h}\\
A_2:=\beta_{00} \mathds{1}_{h\times k} & \hat{A}
\end{array}
\right],$$ where $\overline{A}$ and $\hat{A}$ are circulant matrices of dimensions $(k\times k)$ and $(h\times h)$ (see [@KS]).
[**Matrix B.**]{} Applying the invariance properties like in matrix A we obtain $$B=\left[
\begin{array}{cc}
\overline{B} & B_1:=\left(\int L({\overline{Z}}_{0j}){\hat{Z}}_{1l}\right)_{j=1,\ldots,k,\, l=1,\ldots,h}\\
B_2:=\left(\int L({\hat{Z}}_{0l}){\overline{Z}}_{1j}\right)_{l=1,\ldots,h,\, j=1,\ldots,k} & 0
\end{array}
\right],$$ where $\overline{B}$ is a $(k\times k)$ circulant matrix. Rotating in the $(y_1,y_2)$ and $(y_3,y_4)$ one obtains $$\begin{split}
\int L({\overline{Z}}_{0j}){\hat{Z}}_{1l}&=\cos {\overline{\theta}}_j\beta_{01}-\sin{\overline{\theta}}_j\beta_{02}=\cos {\overline{\theta}}_j\beta_{01},\\
\int L({\hat{Z}}_{0l}){\overline{Z}}_{1j}&=\cos {\overline{\theta}}_j\beta_{10}-\sin{\overline{\theta}}_j\beta_{20}=\cos {\overline{\theta}}_j\beta_{10},
\end{split}$$ for $j=1,\ldots,k,\, l=1,\ldots,h$, since $\beta_{02}=\beta_{20}=0$ due to the symmetry properties. Notice that both expressions are independent of $l$.
[**Matrix C.**]{} Likewise, $$C=\left[
\begin{array}{cc}
\overline{C} & C_1:=\left(\int L({\overline{Z}}_{0j}){\hat{Z}}_{2l}\right)_{j=1,\ldots,k,\, l=1,\ldots,h}\\
C_2:=\left(\int L({\hat{Z}}_{0l}){\overline{Z}}_{2j}\right)_{l=1,\ldots,h,\, j=1,\ldots,k} & 0
\end{array}
\right],$$ whith $\overline{C}$ being a $(k\times k)$ circulant matrix and $$\begin{split}
\int L({\overline{Z}}_{0j}){\hat{Z}}_{2l}&=\sin {\overline{\theta}}_j\beta_{01},\qquad j=1,\ldots,k,\, l=1,\ldots,h,\\
\int L({\hat{Z}}_{0l}){\overline{Z}}_{2j}&=\sin {\overline{\theta}}_j\beta_{10},\qquad j=1,\ldots,k,\, l=1,\ldots,h.
\end{split}$$
[**Matrix D.**]{} $$D=\left[
\begin{array}{cc}
0 & D_1:=\left(\int L({\overline{Z}}_{0j}){\hat{Z}}_{3l}\right)_{j=1,\ldots,k,\, l=1,\ldots,h}\\
D_2:=\left(\int L({\hat{Z}}_{0l}){\overline{Z}}_{3j}\right)_{l=1,\ldots,h,\, j=1,\ldots,k} & \hat{D}
\end{array}
\right],$$ whith $\hat{D}$ being a $(h\times h)$ circulant matrix and $$\begin{split}
\int L({\overline{Z}}_{0j}){\hat{Z}}_{3l}&=\cos {\hat{\theta}}_l\beta_{03},\qquad j=1,\ldots,k,\, l=1,\ldots,h,\\
\int L({\hat{Z}}_{0l}){\overline{Z}}_{3j}&=\cos{\hat{\theta}}_l\beta_{30},\qquad j=1,\ldots,k,\, l=1,\ldots,h,
\end{split}$$ where we have used that $\beta_{04}=\beta_{40}=0$.
[**Matrix E.**]{} $$E=\left[
\begin{array}{cc}
0 & E_1:=\left(\int L({\overline{Z}}_{0j}){\hat{Z}}_{4l}\right)_{j=1,\ldots,k,\, l=1,\ldots,h}\\
E_2:=\left(\int L({\hat{Z}}_{0l}){\overline{Z}}_{4j}\right)_{l=1,\ldots,h,\, j=1,\ldots,k} & \hat{E}
\end{array}
\right],$$ whith $\hat{E}$ being a $(h\times h)$ circulant matrix and $$\begin{split}
\int L({\overline{Z}}_{0j}){\hat{Z}}_{4l}&=\sin {\hat{\theta}}_l\beta_{03},\qquad j=1,\ldots,k,\, l=1,\ldots,h,\\
\int L({\hat{Z}}_{0l}){\overline{Z}}_{4j}&=\sin {\hat{\theta}}_l\beta_{30},\qquad j=1,\ldots,k,\, l=1,\ldots,h.
\end{split}$$
[**Matrix F.**]{} $$F=\left[
\begin{array}{cc}
\overline{F} & F_1:=\left(\int L({\overline{Z}}_{1j}){\hat{Z}}_{1l}\right)_{j=1,\ldots,k,\, l=1,\ldots,h}\\
F_2:=\left(\int L({\hat{Z}}_{1l}){\overline{Z}}_{1j}\right)_{l=1,\ldots,h,\, j=1,\ldots,k} & \hat{F}
\end{array}
\right],$$ where $\overline{F}$ and $\hat{F}$ are $(k\times k)$ and $(h\times h)$ circulant matrices respectively and $$\begin{split}
\int L({\overline{Z}}_{1j}){\hat{Z}}_{1l}&=\int L({\hat{Z}}_{1l}){\overline{Z}}_{1j}= \cos^2{\overline{\theta}}_j\beta_{11}+\sin^2{\overline{\theta}}_j\beta_{22},
\end{split}$$ for $j=1,\ldots,k,\, l=1,\ldots,h,$ using that $\beta_{12}=\beta_{21}=0$.
[**Matrix G.**]{} $$G=\left[
\begin{array}{cc}
\overline{G} & G_1:=\left(\int L({\overline{Z}}_{1j}){\hat{Z}}_{2l}\right)_{j=1,\ldots,k,\, l=1,\ldots,h}\\
G_2:=\left(\int L({\hat{Z}}_{1l}){\overline{Z}}_{2j}\right)_{l=1,\ldots,h,\, j=1,\ldots,k} & 0
\end{array}
\right],$$ where $\overline{G}$ is a $(k\times k)$ circulant matrix and $$\begin{split}
\int L({\overline{Z}}_{1j}){\hat{Z}}_{2l}&= \int L({\hat{Z}}_{1l}){\overline{Z}}_{2j}= \cos{\overline{\theta}}_j\sin{\overline{\theta}}_j\beta_{11}-\sin{\overline{\theta}}_j\cos{\overline{\theta}}_j\beta_{22},
\end{split}$$ for $j=1,\ldots,k,\, l=1,\ldots,h.$
[**Matrix H.**]{} $$H=\left[
\begin{array}{cc}
0 & H_1:=\left(\int L({\overline{Z}}_{1j}){\hat{Z}}_{3l}\right)_{j=1,\ldots,k,\, l=1,\ldots,h}\\
H_2:=\left(\int L({\hat{Z}}_{1l}){\overline{Z}}_{3j}\right)_{l=1,\ldots,h,\, j=1,\ldots,k} & 0
\end{array}
\right],$$ where, since $\beta_{14}=\beta_{41}=\beta_{23}=\beta_{32}=\beta_{24}=\beta_{42}=0$, $$\begin{split}
\int L({\overline{Z}}_{1j}){\hat{Z}}_{3l}&= \cos{\overline{\theta}}_j\cos{\hat{\theta}}_l\beta_{13},\\
\int L({\hat{Z}}_{1l}){\overline{Z}}_{3j}&= \cos{\overline{\theta}}_j\cos{\hat{\theta}}_l\beta_{31},
\end{split}$$ for $j=1,\ldots,k,\, l=1,\ldots,h.$
[**Matrix I.**]{} $$I=\left[
\begin{array}{cc}
0 & I_1:=\left(\int L({\overline{Z}}_{1j}){\hat{Z}}_{4l}\right)_{j=1,\ldots,k,\, l=1,\ldots,h}\\
I_2:=\left(\int L({\hat{Z}}_{1l}){\overline{Z}}_{4j}\right)_{l=1,\ldots,h,\, j=1,\ldots,k} & 0
\end{array}
\right],$$ where $$\begin{split}
\int L({\overline{Z}}_{1j}){\hat{Z}}_{4l}&= \cos{\overline{\theta}}_j\sin{\hat{\theta}}_l\beta_{13},\\
\int L({\hat{Z}}_{1l}){\overline{Z}}_{4j}&= \cos{\overline{\theta}}_j\sin{\hat{\theta}}_l\beta_{31},
\end{split}$$ for $j=1,\ldots,k,\, l=1,\ldots,h.$
[**Matrix J.**]{} $$J=\left[
\begin{array}{cc}
\overline{J} & J_1:=\left(\int L({\overline{Z}}_{2j}){\hat{Z}}_{2l}\right)_{j=1,\ldots,k,\, l=1,\ldots,h}\\
J_2:=\left(\int L({\hat{Z}}_{2l}){\overline{Z}}_{2j}\right)_{l=1,\ldots,h,\, j=1,\ldots,k} & \hat{J}
\end{array}
\right],$$ where $\overline{J}$ and $\hat{J}$ are $(k\times k)$ and $(h\times h)$ circulant matrices respectively and $$\begin{split}
\int L({\overline{Z}}_{2j}){\hat{Z}}_{2l}&=\int L({\hat{Z}}_{2l}){\overline{Z}}_{2j}= \sin^2{\overline{\theta}}_j\beta_{11}+\cos^2{\overline{\theta}}_j\beta_{22},
\end{split}$$ for $j=1,\ldots,k,\, l=1,\ldots,h$.
[**Matrix K.**]{} $$K=\left[
\begin{array}{cc}
0 & K_1:=\left(\int L({\overline{Z}}_{2j}){\hat{Z}}_{3l}\right)_{j=1,\ldots,k,\, l=1,\ldots,h}\\
K_2:=\left(\int L({\hat{Z}}_{2l}){\overline{Z}}_{3j}\right)_{l=1,\ldots,h,\, j=1,\ldots,k} & 0
\end{array}
\right],$$ where $$\begin{split}
\int L({\overline{Z}}_{2j}){\hat{Z}}_{3l}&= \sin{\overline{\theta}}_j\cos{\hat{\theta}}_l\beta_{13},\\
\int L({\hat{Z}}_{2l}){\overline{Z}}_{3j}&= \sin{\overline{\theta}}_j\cos{\hat{\theta}}_l\beta_{31},
\end{split}$$ for $j=1,\ldots,k,\, l=1,\ldots,h.$
[**Matrix L.**]{} $$L=\left[
\begin{array}{cc}
0 & L_1:=\left(\int L({\overline{Z}}_{2j}){\hat{Z}}_{4l}\right)_{j=1,\ldots,k,\, l=1,\ldots,h}\\
L_2:=\left(\int L({\hat{Z}}_{2l}){\overline{Z}}_{4j}\right)_{l=1,\ldots,h,\, j=1,\ldots,k} & 0
\end{array}
\right],$$ where $$\begin{split}
\int L({\overline{Z}}_{2j}){\hat{Z}}_{4l}&= \sin{\overline{\theta}}_j\sin{\hat{\theta}}_l\beta_{13},\\
\int L({\hat{Z}}_{2l}){\overline{Z}}_{4j}&= \sin{\overline{\theta}}_j\sin{\hat{\theta}}_l\beta_{31},
\end{split}$$ for $j=1,\ldots,k,\, l=1,\ldots,h.$
[**Matrix M.**]{} $$M=\left[
\begin{array}{cc}
\overline{M} & M_1:=\left(\int L({\overline{Z}}_{3j}){\hat{Z}}_{3l}\right)_{j=1,\ldots,k,\, l=1,\ldots,h}\\
M_2:=\left(\int L({\hat{Z}}_{3l}){\overline{Z}}_{3j}\right)_{l=1,\ldots,h,\, j=1,\ldots,k} & \hat{M}
\end{array}
\right],$$ where $\overline{M}$ and $\hat{M}$ are $(k\times k)$ and $(h\times h)$ circulant matrices respectively and, since $\beta_{34}=\beta_{43}=0$, $$\begin{split}
\int L({\overline{Z}}_{3j}){\hat{Z}}_{3l}&=\int L({\hat{Z}}_{3l}){\overline{Z}}_{3j}= \cos^2{\hat{\theta}}_l\beta_{33}+\sin^2{\hat{\theta}}_l\beta_{44},
\end{split}$$ for $j=1,\ldots,k,\, l=1,\ldots,h.$
[**Matrix N.**]{} $$N=\left[
\begin{array}{cc}
0 & N_1:=\left(\int L({\overline{Z}}_{3j}){\hat{Z}}_{4l}\right)_{j=1,\ldots,k,\, l=1,\ldots,h}\\
N_2:=\left(\int L({\hat{Z}}_{3l}){\overline{Z}}_{4j}\right)_{l=1,\ldots,h,\, j=1,\ldots,k} & \hat{N}
\end{array}
\right],$$ where $\hat{N}$ is a $(h\times h)$ circulant matrix and $$\begin{split}
\int L({\overline{Z}}_{3j}){\hat{Z}}_{4l}&= \int L({\hat{Z}}_{3l}){\overline{Z}}_{4j}= \cos{\hat{\theta}}_l\sin{\hat{\theta}}_l\beta_{33}-\sin{\hat{\theta}}_l\cos{\hat{\theta}}_l\beta_{44},
\end{split}$$ for $j=1,\ldots,k,\, l=1,\ldots,h.$
[**Matrix P.**]{} $$P=\left[
\begin{array}{cc}
\overline{P} & P_1:=\left(\int L({\overline{Z}}_{4j}){\hat{Z}}_{4l}\right)_{j=1,\ldots,k,\, l=1,\ldots,h}\\
P_2:=\left(\int L({\hat{Z}}_{4l}){\overline{Z}}_{4j}\right)_{l=1,\ldots,h,\, j=1,\ldots,k} & \hat{P}
\end{array}
\right],$$ where $\overline{P}$ and $\hat{P}$ are $(k\times k)$ and $(h\times h)$ circulant matrices respectively and $$\begin{split}
\int L({\overline{Z}}_{4j}){\hat{Z}}_{4l}&=\int L({\hat{Z}}_{4l}){\overline{Z}}_{4j}= \sin^2{\hat{\theta}}_l\beta_{33}+\cos^2{\hat{\theta}}_l\beta_{44},
\end{split}$$ for $j=1,\ldots,k,\, l=1,\ldots,h.$
Notice that $$\hat{F}=\hat{J}\qquad\mbox{and}\qquad\overline{M}=\overline{P}.$$
Henceforth, system can be decomposed in two different systems in the following way, $$\label{system1}
\left[
\begin{array}{ccccc}
\overline{A} & \overline{B} & \overline{C} & 0 & 0 \\
\overline{B}^T & \overline{F} & \overline{G} & 0 & 0\\
\overline{C}^T & \overline{G}^T & \overline{J} & 0 & 0 \\
0 & 0 & 0 & \overline{M} & 0 \\
0 & 0 & 0 & 0 & \overline{P} \\
\end{array}\right]
\left[
\begin{array}{c}
\overline{c}_0\\ \overline{c}_1\\ \overline{c}_2\\ \overline{c}_3\\ \overline{c}_4
\end{array}\right]
=
\left[
\begin{array}{c}
\overline{r}_0\\ \overline{r}_1\\ \overline{r}_2\\ \overline{r}_3\\ \overline{r}_4
\end{array}\right]
-
\left[
\begin{array}{ccccc}
A_1 & B_1 & C_1 & D_1 & E_1 \\
B_2^T & F_1 & G_1 & H_1 & I_1\\
C_2^T & G_2^T & J_1 & K_1 & L_1 \\
D_2^T & H_2^T & K_2^T & M_1 & N_1 \\
E_2^T & I_2^T & L_2^T & N_2^T & P_1 \\
\end{array}\right]
\left[
\begin{array}{c}
\hat{c}_0\\ \hat{c}_1\\ \hat{c}_2\\ \hat{c}_3\\ \hat{c}_4
\end{array}\right],$$ $$\label{system2}
\left[
\begin{array}{ccccc}
\hat{A} & 0 & 0 & \hat{D} & \hat{E}\\
0 & \hat{F} & 0 & 0 &0 \\
0 & 0 & \hat{J} & 0 & 0 \\
\hat{D}^T & 0 & 0 & \hat{M} & \hat{N}\\
\hat{E}^T & 0 & 0 & \hat{N}^T & \hat{P}
\end{array}\right]
\left[
\begin{array}{c}
\hat{c}_0\\ \hat{c}_1\\ \hat{c}_2\\ \hat{c}_3\\ \hat{c}_4
\end{array}\right]
=
\left[
\begin{array}{c}
\hat{r}_0\\ \hat{r}_1\\ \hat{r}_2\\ \hat{r}_3\\ \hat{r}_4
\end{array}\right]
-
\left[
\begin{array}{ccccc}
A_2 & B_2 & C_2 & D_2 & E_2 \\
B_1^T & F_2 & G_2 & H_2 & I_2\\
C_1^T & G_1^T & J_2 & K_2 & L_2 \\
D_1^T & H_1^T & K_1^T & M_2 & N_2 \\
E_1^T & I_1^T & L_1^T & N_1^T & P_2 \\
\end{array}\right]
\left[
\begin{array}{c}
\overline{c}_0\\ \overline{c}_1\\ \overline{c}_2\\ \overline{c}_3\\ \overline{c}_4
\end{array}\right].$$ By [@MW Proposition 5.1] we know that the systems $$\left[
\begin{array}{ccccc}
\overline{A} & \overline{B} & \overline{C} & 0 & 0 \\
\overline{B}^T & \overline{F} & \overline{G} & 0 & 0\\
\overline{C}^T & \overline{G}^T & \overline{J} & 0 & 0 \\
0 & 0 & 0 & \overline{M} & 0 \\
0 & 0 & 0 & 0 & \overline{P} \\
\end{array}\right]
\left[
\begin{array}{c}
\overline{c}_0\\ \overline{c}_1\\ \overline{c}_2\\ \overline{c}_3\\ \overline{c}_4
\end{array}\right]
=
\left[
\begin{array}{c}
\overline{s}_0\\ \overline{s}_1\\ \overline{s}_2\\ \overline{s}_3\\ \overline{s}_4
\end{array}\right],$$ $$\left[
\begin{array}{ccccc}
\hat{A} & 0 & 0 & \hat{D} & \hat{E}\\
0 & \hat{F} & 0 & 0 &0 \\
0 & 0 & \hat{J} & 0 & 0 \\
\hat{D}^T & 0 & 0 & \hat{M} & \hat{N}\\
\hat{E}^T & 0 & 0 &\hat{N}^T & \hat{P}
\end{array}\right]
\left[
\begin{array}{c}
\hat{c}_0\\ \hat{c}_1\\ \hat{c}_2\\ \hat{c}_3\\ \hat{c}_4
\end{array}\right]
=
\left[
\begin{array}{c}
\hat{s}_0\\ \hat{s}_1\\ \hat{s}_2\\ \hat{s}_3\\ \hat{s}_4
\end{array}\right]$$ are solvable if the orthogonality conditions $$\begin{split}\label{GeneralConditions}
&{\overline{s}}_2\cdot{\overline{1}}=({\overline{s}}_0+{\overline{s}}_1)\cdot{\overline{\cos}}=({\overline{s}}_0+{\overline{s}}_1)\cdot{\overline{\sin}}= 0,\\
&{\overline{s}}_3\cdot{\overline{\cos}}={\overline{s}}_3\cdot{\overline{\sin}}=0,\\
&{\overline{s}}_4\cdot{\overline{\cos}}={\overline{s}}_4\cdot{\overline{\sin}}=0,\\
&{\hat{s}}_4\cdot{\hat{1}}=({\hat{s}}_0+{\hat{s}}_3)\cdot{\hat{\cos}}=({\hat{s}}_0+{\hat{s}}_3)\cdot{\hat{\sin}}= 0,\\
&{\hat{s}}_1\cdot{\hat{\cos}}={\hat{s}}_1\cdot{\hat{\sin}}=0,\\
&{\hat{s}}_2\cdot{\hat{\cos}}={\hat{s}}_2\cdot{\hat{\sin}}=0,
\end{split}$$ hold. Moreover, $$\label{formCalpha}\begin{split}
&\left[\begin{array}{c}
{\overline{c}}_0\\{\overline{c}}_1\\{\overline{c}}_2
\end{array}\right]=
\left[\begin{array}{c}
{\overline{w}}_0\\{\overline{w}}_1\\{\overline{w}}_2
\end{array}\right]+{\overline{t}}_1
\left[\begin{array}{c}
0\\0\\{\overline{1}}\end{array}\right]+{\overline{t}}_2
\left[\begin{array}{c}
{\overline{\cos}}\\-{\overline{\cos}}\\0
\end{array}\right]+{\overline{t}}_3
\left[\begin{array}{c}
{\overline{\sin}}\\-{\overline{\sin}}\\0
\end{array}\right],\;\;\;\forall \,{\overline{t}}_1,{\overline{t}}_2,{\overline{t}}_3\in{{\mathbb R}},\\
&{\overline{c}}_3={\overline{w}}_3+{\overline{t}}_4{\overline{\cos}}+{\overline{t}}_5{\overline{\sin}}\;\;\forall\,{\overline{t}}_4,{\overline{t}}_5\in{{\mathbb R}},\\
&{\overline{c}}_4={\overline{w}}_4+{\overline{t}}_6{\overline{\cos}}+{\overline{t}}_7{\overline{\sin}},\;\;\forall\,{\overline{t}}_6,{\overline{t}}_7\in{{\mathbb R}},\\
&{\hat{c}}_1={\hat{w}}_1+{\hat{t}}_1{\hat{\cos}}+{\hat{t}}_2{\hat{\sin}}\;\;\forall\,{\hat{t}}_1,{\hat{t}}_2\in{{\mathbb R}},\\
&{\hat{c}}_2={\hat{w}}_2+{\hat{t}}_3{\hat{\cos}}+{\hat{t}}_4{\hat{\sin}},\;\;\forall\,{\hat{t}}_3,{\hat{t}}_4\in{{\mathbb R}},\\
&\left[\begin{array}{c}
{\hat{c}}_0\\{\hat{c}}_3\\{\hat{c}}_4
\end{array}\right]=
\left[\begin{array}{c}
{\hat{w}}_0\\{\hat{w}}_3\\{\hat{w}}_4
\end{array}\right]+{\hat{t}}_5
\left[\begin{array}{c}
0\\0\\{\hat{1}}\end{array}\right]+{\hat{t}}_6
\left[\begin{array}{c}
{\hat{\cos}}\\-{\hat{\cos}}\\0
\end{array}\right]+{\hat{t}}_7
\left[\begin{array}{c}
{\hat{\sin}}\\-{\hat{\sin}}\\0
\end{array}\right],\;\;\;\forall \,{\hat{t}}_5,{\hat{t}}_6,{\hat{t}}_7\in{{\mathbb R}},
\end{split}$$ with $\left[\begin{array}{c}{\overline{w}}_0\\\ldots \\{\overline{w}}_4\end{array}\right],\,\left[\begin{array}{c}{\hat{w}}_0\\\ldots \\{\hat{w}}_4\end{array}\right]$ fixed vectors such that $$\label{boundsW}
\|\left[\begin{array}{c}{\overline{w}}_0\\\ldots\\ {\overline{w}}_4\end{array}\right]\|{\leqslant}\frac{C}{k^n\mu^{n-2}}\|\left[\begin{array}{c}{\overline{s}}_0\\\ldots\\ {\overline{s}}_4\end{array}\right]\|,\;\;\;\;
\|\left[\begin{array}{c}{\hat{w}}_0\\\ldots\\ {\hat{w}}_4\end{array}\right]\|{\leqslant}\frac{C}{h^n\lambda^{n-2}}\|\left[\begin{array}{c}{\hat{s}}_0\\\ldots\\ {\hat{s}}_4\end{array}\right]\|.$$ We will prove that and have a solution $\left[\begin{array}{c}{\overline{c}}_0\\{\hat{c}}_0\\\ldots\\{\overline{c}}_4\\{\hat{c}}_4\end{array}\right]$ in the space $$X:=\left\{\left[\begin{array}{c}{\overline{c}}_0\\{\hat{c}}_0\\\ldots\\{\overline{c}}_4\\{\hat{c}}_4\end{array}\right]:\,
\begin{array}{cc}
{\overline{c}}_3\cdot{\overline{\cos}}={\overline{c}}_3\cdot{\overline{\sin}}=0, & {\overline{c}}_4\cdot{\overline{\cos}}={\overline{c}}_4\cdot{\overline{\sin}}=0,\\
{\hat{c}}_1\cdot{\hat{\cos}}={\hat{c}}_1\cdot{\hat{\sin}}=0, & {\hat{c}}_2\cdot{\hat{\cos}}={\hat{c}}_2\cdot{\hat{\sin}}=0.
\end{array}\right\}.$$ We need the following auxiliar result, whose proof follows straightforward using the same argument as in Lemma \[proj1\] so we skip it.
\[AuxLemma\] Let $h, g$ be functions in ${{\mathbb R}}^n$ such that $h(y)=h(e^{\frac{2\pi}{k}(j-1)}\overline{y},\hat{y},y')$ for all $j=1,\ldots, k$ and $g(y)=g(\overline{y},e^{\frac{2\pi}{h}(l-1)}\hat{y},y')$ for all $l=1,\ldots, h$. Then, $$\int_{{{\mathbb R}}^n}\hat{Z}_{1l}(y)h(y)\,dy=\int_{{{\mathbb R}}^n}\hat{Z}_{2l}(y)h(y)\,dy=0,\; \forall \,l=1,\ldots, h,$$ $$\int_{{{\mathbb R}}^n}\overline{Z}_{3j}(y)g(y)\,dy=\int_{{{\mathbb R}}^n}\overline{Z}_{4j}(y)g(y)\,dy=0,\; \forall \,j=1,\ldots, k.$$
Let us focus on . By , $$\begin{split}
\overline{r}_2\cdot {\overline{1}}&=\sum_{j=1}^k\int L(\varphi^\perp){\overline{Z}}_{2j}\\
&=\sum_{j=1}^k\left[\int L(\sum_{\alpha=0}^n\sum_{i=1}^k{\overline{Z}}_{\alpha i}){\overline{Z}}_{2j}+\int L(\sum_{\alpha=0}^n\sum_{l=1}^h{\hat{Z}}_{\alpha l}){\overline{Z}}_{2j}\right].
\end{split}$$ Notice that the second term vanishes due to Lemma \[AuxLemma\] since, by the symmetry properties of the functions, $$\sum_{j=1}^k\int L(\sum_{\alpha=0}^n\sum_{l=1}^h{\hat{Z}}_{\alpha l}){\overline{Z}}_{2j}=\sum_{j=1}^k\int L(\sum_{l=1}^h{\hat{Z}}_{2 l}){\overline{Z}}_{2j}=\sum_{l=1}^h\int {\hat{Z}}_{2l}L(\sum_{j=1}^k {\overline{Z}}_{2j}),$$ and $L(\sum_{j=1}^k {\overline{Z}}_{2j})$ is invariant under rotation of angle $\frac{2\pi}{k}(j-1)$ in the $(y_1,y_2)$ plane. Therefore, $$\begin{split}\label{ortOR2}
\overline{r}_2\cdot {\overline{1}}&=\left(\int L(\sum_{\alpha=0}^n\sum_{i=1}^k {\overline{Z}}_{\alpha i}){\overline{Z}}_{21}\right) \sum_{j=1}^k \sin{\overline{\theta}}_j+\left(\int L(\sum_{\alpha=0}^k\sum_{i=1}^k {\overline{Z}}_{\alpha i}){\overline{Z}}_{22}\right) \sum_{j=1}^k \cos{\overline{\theta}}_j=0.
\end{split}$$ On the other hand, as a consequence of [@MW Lemma 6.1], $$(\overline{r}_0+\overline{r}_1)\cdot{\overline{\cos}}=(\overline{r}_0+\overline{r}_1)\cdot{\overline{\sin}}=0.$$ Using the invariances under rotation in the planes $(y_1,y_2)$ and $(y_3,y_4)$ and Lemma \[AuxLemma\] we get $$\begin{split}
\int L(\varphi^\perp){\overline{Z}}_{3j}&=\int L(\sum_{\alpha=1}^n\sum_{i=1}^k{\overline{Z}}_{\alpha i}){\overline{Z}}_{3j}+\int L(\sum_{\alpha=1}^n\sum_{l=1}^h{\hat{Z}}_{\alpha l}){\overline{Z}}_{3j}\\
&=\int L(\sum_{\alpha=1}^n\sum_{i=1}^k{\overline{Z}}_{\alpha i}){\overline{Z}}_{31},
\end{split}$$ and thus $$\label{ortOR3}
\overline{r}_3\cdot{\overline{\cos}}= \int L(\sum_{\alpha=1}^n\sum_{i=1}^k{\overline{Z}}_{\alpha i}){\overline{Z}}_{31}\left(\sum_{j=1}^k\cos{\theta_j}\right)=0.$$ Analogously, $$\label{ortOR4}
\overline{r}_3\cdot{\overline{\sin}}= \overline{r}_4\cdot{\overline{\cos}}= \overline{r}_4\cdot{\overline{\sin}}= 0.$$ Let us now check the last term in . We expect $$\begin{aligned}
&&(C_2^T[{\hat{c}}_0]+G_2^T[{\hat{c}}_1]+J_1[{\hat{c}}_2]+K_1[{\hat{c}}_3]+L_1[{\hat{c}}_4])\cdot{\overline{1}}=0,\label{cond1}\\
&&(A_1[{\hat{c}}_0]+B_1[{\hat{c}}_1]+C_1[{\hat{c}}_2]+D_1[{\hat{c}}_3]+E_1[{\hat{c}}_4]\nonumber\\
&&\;\; +B_2^T[{\hat{c}}_0]+F_1[{\hat{c}}_1]+G_1[{\hat{c}}_2]+H_1[{\hat{c}}_3]+I_1[{\hat{c}}_4])\cdot {\overline{\cos}}=0,\label{cond2}\\
&&(A_1[{\hat{c}}_0]+B_1[{\hat{c}}_1]+C_1[{\hat{c}}_2]+D_1[{\hat{c}}_3]+E_1[{\hat{c}}_4]\nonumber\\
&&\;\; +B_2^T[{\hat{c}}_0]+F_1[{\hat{c}}_1]+G_1[{\hat{c}}_2]+H_1[{\hat{c}}_3]+I_1[{\hat{c}}_4])\cdot {\overline{\sin}}=0,\label{cond3}\\
&&(D_2^T[{\hat{c}}_0]+H_2^T[{\hat{c}}_1]+K_2^T[{\hat{c}}_2]+M_1[{\hat{c}}_3]+N_1[{\hat{c}}_4])\cdot{\overline{\cos}}=0,\label{cond4}\\
&&(D_2^T[{\hat{c}}_0]+H_2^T[{\hat{c}}_1]+K_2^T[{\hat{c}}_2]+M_1[{\hat{c}}_3]+N_1[{\hat{c}}_4])\cdot{\overline{\sin}}=0,\label{cond5}\\
&&(E_2^T[{\hat{c}}_0]+I_2^T[{\hat{c}}_1]+L_2^T[{\hat{c}}_2]+N_2^T[{\hat{c}}_3]+P_1[{\hat{c}}_4])\cdot{\overline{\cos}}=0,\label{cond6}\\
&&(E_2^T[{\hat{c}}_0]+I_2^T[{\hat{c}}_1]+L_2^T[{\hat{c}}_2]+N_2^T[{\hat{c}}_3]+P_1[{\hat{c}}_4])\cdot{\overline{\sin}}=0,\label{cond7}\end{aligned}$$ where ${\hat{c}}_0,\ldots,{\hat{c}}_4$ satisfy $$\begin{split}\label{GeneralConditionsHat}
&{\hat{c}}_1\cdot{\hat{\cos}}={\hat{c}}_1\cdot{\hat{\sin}}=0,\\
&{\hat{c}}_2\cdot{\hat{\cos}}={\hat{c}}_2\cdot{\hat{\sin}}=0.
\end{split}$$ Notice that, since $C_2$ and $G_2$ have all their rows identical, $$C_2^T[{\hat{c}}_0]\cdot{\overline{1}}= \beta_{10}\left(\sum_{l=1}^h {\hat{c}}_{0l}\right)\left(\sum_{j=1}^k \sin{\overline{\theta}}_j\right) =0,$$ $$G_2^T[{\hat{c}}_1]\cdot{\overline{1}}= (\beta_{11}-\beta_{22})\left(\sum_{l=1}^h {\hat{c}}_{1l}\right)\left(\sum_{j=1}^k \sin{\overline{\theta}}_j\cos{\overline{\theta}}_j\right) =0.$$ Likewise, using the definition of $J_1$ and Lemma \[AuxLemma\], $$J_1[{\hat{c}}_2]\cdot{\overline{1}}= \left(\sum_{l=1}^h {\hat{c}}_{2l}\right) \int L\left(\sum_{j=1}^k {\overline{Z}}_{2j}\right){\hat{Z}}_{11}=0,$$ since $L\left(\sum_{j=1}^k {\overline{Z}}_{2j}\right)$ is invariant under rotation of angle $\theta_j$, and $$K_1[{\hat{c}}_3]\cdot{\overline{1}}=\beta_{13}({\hat{c}}_3\cdot{\hat{\cos}})\left(\sum_{j=1}^k\sin{\overline{\theta}}_j\right)=0,$$ $$L_1[{\hat{c}}_4]\cdot{\overline{1}}=\beta_{13}({\hat{c}}_4\cdot{\hat{\sin}})\left(\sum_{j=1}^k\sin{\overline{\theta}}_j\right)=0.$$ Thus, follows. Furthermore, again by [@MW Lemma 6.1], $$(A_1+B_2^T)_{jl}=\int L({\hat{Z}}_{0l})({\overline{Z}}_{0j}+{\overline{Z}}_{1j})=0,$$ $$(B_1+F_1)_{jl}=\int L({\hat{Z}}_{1l})({\overline{Z}}_{0j}+{\overline{Z}}_{1j})=0,$$ $$(C_1+G_1)_{jl}=\int L({\hat{Z}}_{2l})({\overline{Z}}_{0j}+{\overline{Z}}_{1j})=0,$$ and thus $(A_1+B_2^T)[{\hat{c}}_0]=(B_1+F_1)[{\hat{c}}_1]=(C_1+G_1)[{\hat{c}}_2]=0$. Likewise, $$(D_1+H_1)_{jl}=\int L({\hat{Z}}_{3l})({\overline{Z}}_{0j}+{\overline{Z}}_{1j})=0,$$ $$(E_1+I_1)_{jl}=\int L({\hat{Z}}_{4l})({\overline{Z}}_{0j}+{\overline{Z}}_{1j})=0,$$ and therefore $(D_1+H_1)[{\hat{c}}_3]=(E_1+I_1)[{\hat{c}}_4]=0$. can be analogously proved. Furthermore, $$D_2^T[{\hat{c}}_0]\cdot{\overline{\cos}}=\beta_{31}({\hat{c}}_0\cdot{\hat{\cos}})\left(\sum_{j=1}^k\cos{\overline{\theta}}_j\right)=0,$$ $$M_1[{\hat{c}}_3]\cdot{\overline{\cos}}=\left[\beta_{33}\left(\sum_{l=1}^h{\hat{c}}_{3l}\cos^2{\hat{\theta}}_l\right)+\beta_{44}\left(\sum_{l=1}^h{\hat{c}}_{3l}\sin^2{\hat{\theta}}_l\right)\right]\left(\sum_{j=1}^k \cos{\overline{\theta}}_j\right)=0,$$ $$N_1[{\hat{c}}_4]\cdot{\overline{\cos}}=(\beta_{33}-\beta_{44})\left(\sum_{l=1}^h{\hat{c}}_{4l}\sin{\hat{\theta}}_l\cos{\hat{\theta}}_l\right)\left(\sum_{j=1}^k \cos{\overline{\theta}}_j\right)=0,$$ and, due to , $$H_2^T[{\hat{c}}_1]\cdot{\overline{\cos}}=\beta_{31}({\hat{c}}_1\cdot{\hat{\cos}})\left(\sum_{j=1}^k\cos^2{\overline{\theta}}_j\right)=0,$$ $$K_2^T[{\hat{c}}_2]\cdot{\overline{\cos}}=\beta_{31}({\hat{c}}_2\cdot{\hat{\cos}})\left(\sum_{j=1}^k\sin{\overline{\theta}}_j\cos{\overline{\theta}}_j\right)=0,$$ so holds. Identities - can be obtained in a similar way, and thus is solvable. An analogous reasoning proves the solvability of . Thus, the systems and have a solution in $X$ with the form , where $[{\overline{w}}_0,\ldots,{\overline{w}}_4,{\hat{w}}_0,\ldots,{\hat{w}}_4]$ satisfies . It can be checked that $$\begin{split}\label{boundsBeta}
|\beta_{\alpha \alpha}|{\leqslant}Ck^{-2n+4},\;\alpha=0,2,4,&\qquad |\beta_{\alpha \alpha}|{\leqslant}Ck^{-2n+6},\;\alpha=1,3,\\
|\beta_{\alpha_1 \alpha_2}|{\leqslant}Ck^{-2n+6},\;\alpha_1,&\;\alpha_2=0,1,3,\;\alpha_1\neq\alpha_2.
\end{split}$$ where $\beta_{\alpha_1,\alpha_2}$ was defined in , and henceforth, by and recalling that $h=O(k)$, $$\|{\overline{w}}_\alpha\|{\leqslant}Ck^{n-4}\|\overline{r}_\alpha\|,\qquad \|{\hat{w}}_\alpha\|{\leqslant}Ck^{n-4}\|\hat{r}_\alpha\|,\qquad \alpha=0, 1, \ldots, 4.$$ As it was done in the case $\alpha{\geqslant}5$, we will solve the systems by means of a fixed point argument. If we denote $$\overline{M}_1:=\left[
\begin{array}{ccccc}
\overline{A} & \overline{B} & \overline{C} & 0 & 0 \\
\overline{B}^T & \overline{F} & \overline{G} & 0 & 0\\
\overline{C}^T & \overline{G}^T & \overline{J} & 0 & 0 \\
0 & 0 & 0 & \overline{M} & 0 \\
0 & 0 & 0 & 0 & \overline{P} \\
\end{array}\right],\;\overline{\overline{M}}_1:=\left[
\begin{array}{ccccc}
A_1 & B_1 & C_1 & D_1 & E_1 \\
B_2^T & F_1 & G_1 & H_1 & I_1\\
C_2^T & G_2^T & J_1 & K_1 & L_1 \\
D_2^T & H_2^T & K_2^T & M_1 & N_1 \\
E_2^T & I_2^T & L_2^T & N_2^T & P_1 \\
\end{array}\right],$$ $$\hat{M}_1:=\left[
\begin{array}{ccccc}
\hat{A} & 0 & 0 & \hat{D} & \hat{E}\\
0 & \hat{F} & 0 & 0 &0 \\
0 & 0 & \hat{J} & 0 & 0 \\
\hat{D}^T & 0 & 0 & \hat{M} & \hat{N}\\
\hat{E}^T & 0 & 0 & \hat{N}^T & \hat{P}
\end{array}\right],\;\hat{\hat{M}}_1:=\left[
\begin{array}{ccccc}
A_2 & B_2 & C_2 & D_2 & E_2 \\
B_1^T & F_2 & G_2 & H_2 & I_2\\
C_1^T & G_1^T & J_2 & K_2 & L_2 \\
D_1^T & H_1^T & K_1^T & M_2 & N_2 \\
E_1^T & I_1^T & L_1^T & N_1^T & P_2 \\
\end{array}\right],$$ then $[{\overline{w}}_0,\ldots,{\overline{w}}_4,{\hat{w}}_0,\ldots,{\hat{w}}_4]$ is a solution of - if and only if it is a fixed point of $$F\left[
\begin{array}{c}
{\overline{w}}_0\\ \ldots\\ {\overline{w}}_4\\ {\hat{w}}_0\\\ldots\\ {\hat{w}}_4
\end{array}\right]:=S^{-1}\left(
\left[
\begin{array}{c}
\overline{r}_0\\ \overline{r}_1\\ \overline{r}_2\\ \overline{r}_3\\ \overline{r}_4
\end{array}\right]
-
\overline{\overline{M}}_1
\left[
\begin{array}{c}
\hat{w}_0\\ \hat{w}_1\\ \hat{w}_2\\ \hat{w}_3\\ \hat{w}_4
\end{array}\right],
\left[
\begin{array}{c}
\hat{r}_0\\ \hat{r}_1\\ \hat{r}_2\\ \hat{r}_3\\ \hat{r}_4
\end{array}\right]
-
\hat{\hat{M}}_1
\left[
\begin{array}{c}
\overline{w}_0\\ \overline{w}_1\\ \overline{w}_2\\ \overline{w}_3\\ \overline{w}_4
\end{array}\right]
\right),$$ where $$S\left(\left[
\begin{array}{c}
\overline{w}_0\\ \overline{w}_1\\ \overline{w}_2\\ \overline{w}_3\\ \overline{w}_4
\end{array}\right], \left[
\begin{array}{c}
\hat{w}_0\\ \hat{w}_1\\ \hat{w}_2\\ \hat{w}_3\\ \hat{w}_4
\end{array}\right] \right):=\left(\overline{M}_1\left[
\begin{array}{c}
\overline{w}_0\\ \overline{w}_1\\ \overline{w}_2\\ \overline{w}_3\\ \overline{w}_4
\end{array}\right], \hat{M}_1\left[
\begin{array}{c}
\hat{w}_0\\ \hat{w}_1\\ \hat{w}_2\\ \hat{w}_3\\ \hat{w}_4
\end{array}\right]\right).$$ Notice that $S$ is a linear map which is invertible for vectors satisfying the orthogonality conditions . Let $$\begin{split}
B_r:=\{\left[\begin{array}{c}
\overline{w}_0\\ \ldots \\ \overline{w}_4
\end{array}\right], &
\left[
\begin{array}{c}
\hat{w}_0\\ \ldots\\ \hat{w}_4
\end{array}\right]\in K : \|\left[\begin{array}{c}
\overline{w}_0\\ \ldots \\ \overline{w}_4
\end{array}\right]\|{\leqslant}rk^{n-4}\|\left[\begin{array}{c}
\overline{r}_0\\ \ldots \\ \overline{r}_4
\end{array}\right]\|,\\
&\|\left[
\begin{array}{c}
\hat{w}_0\\ \ldots\\ \hat{w}_4
\end{array}\right]\|{\leqslant}rk^{n-4}\|\left[
\begin{array}{c}
\hat{r}_0\\ \ldots\\ \hat{r}_4\end{array}\right]\},
\end{split}$$ for some fixed $r$ large, where $$K:=\{\left[\begin{array}{c}
\overline{w}_0\\ \ldots \\ \overline{w}_4
\end{array}\right]\in{{\mathbb R}}^{5\times k},
\left[
\begin{array}{c}
\hat{w}_0\\ \ldots\\ \hat{w}_4
\end{array}\right]\in {{\mathbb R}}^{5\times h}\mbox{ satisfying }\eqref{GeneralConditions}.\}.$$ Thanks to the particular form of the matrices $\overline{\overline{M}}_1$, $\hat{\hat{M}}$ (all their submatrices are combinations of sinus and cosinus multiplied by a term $\beta_{\alpha_1,\alpha_2}$) and it can be checked that $F$ is a contraction mapping that sends $B_r$ into $B_r$. This finishes the proof of the existence of a solution to - satisfying .
Define the vectors $$\label{vectors21}
\overline{u}_0:=
\left[
\begin{array}{c}
0\\ {\overline{0}}\\ {\hat{0}}\\
0\\ {\overline{0}}\\ {\hat{0}}\\
0\\ {\overline{1}}\\ {\hat{0}}\\
0\\ {\overline{0}}\\ {\hat{0}}\\
0\\ {\overline{0}}\\ {\hat{0}}\end{array}\right],\,
\overline{u}_1:=
\left[
\begin{array}{c}
0\\ {\overline{\cos}}\\ {\hat{0}}\\
0\\ -{\overline{\cos}}\\ {\hat{0}}\\
0\\ {\overline{0}}\\ {\hat{0}}\\
0\\ {\overline{0}}\\ {\hat{0}}\\
0\\ {\overline{0}}\\ {\hat{0}}\end{array}\right],\,
\overline{u}_2:=
\left[
\begin{array}{c}
0\\ {\overline{\sin}}\\ {\hat{0}}\\
0\\ -{\overline{\sin}}\\ {\hat{0}}\\
0\\ {\overline{0}}\\ {\hat{0}}\\
0\\ {\overline{0}}\\ {\hat{0}}\\
0\\ {\overline{0}}\\ {\hat{0}}\end{array}\right],\,
$$ $$\label{vectors23}
\hat{u}_0:=
\left[
\begin{array}{c}
0\\ {\overline{0}}\\ {\hat{0}}\\
0\\ {\overline{0}}\\ {\hat{0}}\\
0\\ {\overline{0}}\\ {\hat{0}}\\
0\\ {\overline{0}}\\ {\hat{0}}\\
0\\ {\overline{0}}\\ {\hat{1}}\end{array}\right],\,
\hat{u}_1:=
\left[
\begin{array}{c}
0\\ {\overline{0}}\\ {\hat{\cos}}\\
0\\ {\overline{0}}\\ {\hat{0}}\\
0\\ {\overline{0}}\\ {\hat{0}}\\
0\\ {\overline{0}}\\ -{\hat{\cos}}\\
0\\ {\overline{0}}\\ {\hat{0}}\end{array}\right],\,
\hat{u}_2:=
\left[
\begin{array}{c}
0\\ {\overline{0}}\\ {\hat{\sin}}\\
0\\ {\overline{0}}\\ {\hat{0}}\\
0\\ {\overline{0}}\\ {\hat{0}}\\
0\\ {\overline{0}}\\ -{\hat{\sin}}\\
0\\ {\overline{0}}\\ {\hat{0}}\end{array}\right].\,
$$ We can summarize this section in the following lemma.
\[dos\] System is solvable, and the solution has the form $$\begin{split}
\left[
\begin{array}{c}
\tilde{c}_0\\\ldots\\\tilde{c}_4
\end{array}
\right]=&
\,w+t_0w_0+t_1w_1+t_2w_2+t_3w_3+t_4w_4\\
&+
\overline{t}_0 \overline{u}_0+\overline{t}_1 \overline{u}_1+\overline{t}_2 \overline{u}_2+
\hat{t}_0 \hat{u}_0+\hat{t}_1 \hat{u}_1+\hat{t}_2 \hat{u}_2,
\end{split}$$ where $$w:=\left[
\begin{array}{ccccccccccccccc}
0 &{\overline{w}}_0 &{\hat{w}}_0 & 0 & {\overline{w}}_1&{\hat{w}}_1& 0 & {\overline{w}}_2&{\hat{w}}_2 & 0 & {\overline{w}}_3&{\hat{w}}_3 & 0 & {\overline{w}}_4&{\hat{w}}_4
\end{array}
\right]^T$$ (${\overline{w}}_0,\ldots, {\overline{w}}_4,{\hat{w}}_0,\ldots,{\hat{w}}_4$ being the unique solution to the system - satisfying and ), $t_0,\ldots,t_4$, $\overline{t}_0,\overline{t}_1,\overline{t}_2$, $\hat{t}_0,\hat{t}_1,\hat{t}_2$ $\in {{\mathbb R}}$ and $\tilde{c}_0,\ldots,\tilde{c}_4$, $w_0,\ldots,w_4$, $\overline{u}_0,\overline{u}_1,\overline{u}_2$, $\hat{u}_0,\hat{u}_1,\hat{u}_2$ are defined in , , , and .
Proof of Proposition \[systC\] {#prop31}
==============================
With this in mind, we observe that Proposition \[systC\] is a consequence of the following estimates: if $k, \, h \to \infty$ $$\begin{split}\label{int1}
\int |u|^{p-1}Z_{\alpha 0} Z_{\beta 0} & = \int U^{p-1}Z_{00}^2 +O(\mu^{\frac{n-2}{2}}+\lambda^{\frac{n-2}{2}})\qquad \mbox{if } \alpha=\beta=0,\\
& = \int U^{p-1}Z_{10}^2 +O(\mu^{\frac{n-2}{2}}+\lambda^{\frac{n-2}{2}})\qquad \mbox{if } \alpha=\beta\neq 0,\\
& = O(\mu^{\frac{n-2}{2}}+\lambda^{\frac{n-2}{2}})\qquad\mbox{otherwise},
\end{split}$$
$$\begin{split}\label{int2}
\int |u|^{p-1}{\overline{Z}}_{\alpha i} {\overline{Z}}_{\beta j} & = \int U^{p-1}Z_{00}^2 +O(\mu^{\frac{n-2}{2}})\qquad \mbox{if } \alpha=\beta=0,\, i=j,\\
& = \int U^{p-1}Z_{10}^2 +O(\mu^{\frac{n-2}{2}})\qquad \mbox{if } \alpha=\beta\neq 0,\, i=j\\
& = O(\mu^{\frac{n-2}{2}}) \qquad\mbox{otherwise},
\end{split}$$
$$\begin{split}\label{int3}
\int |u|^{p-1}{\hat{Z}}_{\alpha l} {\hat{Z}}_{\beta m} & = \int U^{p-1}Z_{00}^2 +O(\lambda^{\frac{n-2}{2}})\qquad \mbox{if } \alpha=\beta=0,\,l=m,\\
& = \int U^{p-1}Z_{10}^2 +O(\lambda^{\frac{n-2}{2}})\qquad \mbox{if } \alpha=\beta\neq 0,\,l=m,\\
& = O(\lambda^{\frac{n-2}{2}}) \qquad\mbox{otherwise},
\end{split}$$
$$\begin{split}\label{int4}
\int |u|^{p-1}{\overline{Z}}_{\alpha i} Z_{\beta 0} & = O(\mu^{\frac{n-2}{2}}), \quad \int |u|^{p-1}{\hat{Z}}_{\alpha l} Z_{\beta 0} = O(\lambda^{\frac{n-2}{2}}),
\end{split}$$
$$\begin{split}\label{int6}
\int |u|^{p-1}{\overline{Z}}_{\alpha i} {\hat{Z}}_{\beta l} & = O(\mu^{\frac{n-2}{2}}\lambda^{\frac{n-2}{2}}).
\end{split}$$
In the formulas above, $i,j=1,\ldots,k,$ $l,m=1,\ldots, h$, $\alpha,\beta =0,\ldots, n$.
The proof of and follows like (8.3) in [@MW], and , are obtained analogously. Let us prove .
The key point here is to notice that if $y\in B(\xi_i,\frac{{\overline{\alpha}}}{k})$ then $|y-\eta_l|{\geqslant}C$, with $C$ independent of $i$, $l$ and $k$, and $|y-\xi_i|{\geqslant}C$ whenever $y\in B(\eta_l,\frac{{\hat{\alpha}}}{h})$, where $C$ is independent of $i$, $l$ and $h$. Consider the case $\alpha=\beta=0$. We split the integral into four parts. $$\begin{split}
\int |u|^{p-1}{\overline{Z}}_{0 i} {\hat{Z}}_{0 l} =& \int_{B(\xi_i,\frac{{\overline{\alpha}}}{k})} |u|^{p-1}{\overline{Z}}_{0 i} {\hat{Z}}_{0 j}+\int_{B(\eta_l,\frac{{\hat{\alpha}}}{h})} |u|^{p-1}{\overline{Z}}_{0 i} {\hat{Z}}_{0 j}\\
& +\int_{{{\mathbb R}}^n\setminus B(0,2)} |u|^{p-1}{\overline{Z}}_{0 i} {\hat{Z}}_{0 j}\\
&+\int_{B(0,2)\setminus (B(\xi_i,\frac{{\overline{\alpha}}}{k})\cup B(\eta_l,\frac{{\hat{\alpha}}}{h}))} |u|^{p-1}{\overline{Z}}_{0 i} {\hat{Z}}_{0 j}\\
=:&\, i_1+i_2+i_3+i_4,
\end{split}$$ Firstly, using the definition of ${\overline{Z}}_{0i}$ and ${\hat{Z}}_{0l}$, $$\begin{split}
i_1{\leqslant}& \,C\int_{B(\xi_i,\frac{{\overline{\alpha}}}{k})} |u|^{p-1}{\overline{Z}}_{0 i}\frac{\lambda^{\frac{n-2}{2}}}{|y-\eta_l|^{n-2}}{\leqslant}C\mu^{\frac{n-2}{2}}\lambda^{\frac{n-2}{2}}\int_{B(0,\frac{{\overline{\alpha}}}{\mu k})}U^{p-1}Z_{00}\\
{\leqslant}&\, C\mu^{\frac{n-2}{2}}\lambda^{\frac{n-2}{2}},
\end{split}$$ where the second inequality follows by the change of variable $x=\xi_i+\mu y$. $i_2$ follows in the same way only by translating to $x=\eta_l+\lambda y$. On the other hand, we have that $$i_3{\leqslant}C\mu^{\frac{n-2}{2}}\lambda^{\frac{n-2}{2}}\int_{{{\mathbb R}}^n\setminus B(0,2)}\frac{1}{|y|^4|y|^{n-2}|y|^{n-2}}\,dy{\leqslant}C\mu^{\frac{n-2}{2}}\lambda^{\frac{n-2}{2}}.$$ To estimate $i_4$ we take into account that, since $\xi_i$ and $\eta_l$ are separated, $|y-\xi_i|^{-(n-2)}$ and $|y-\eta_l|^{-(n-2)}$ cannot be singular at the same time, so the behavior of the integral comes determined by the singularity of only one of them. That is, $$\begin{split}
i_4{\leqslant}&\, C\mu^{\frac{n-2}{2}}\lambda^{\frac{n-2}{2}}\int_{B(0,2)\setminus (B(\xi_i,\frac{{\overline{\alpha}}}{k})\cup B(\eta_l,\frac{{\hat{\alpha}}}{h}))} |u|^{p-1}\frac{1}{|y-\xi_i|^{n-2}}\frac{1}{|y-\eta_l|^{n-2}}\,dy\\
{\leqslant}&\,C\mu^{\frac{n-2}{2}}\lambda^{\frac{n-2}{2}}\int_{B(0,2)\setminus (B(\xi_i,\frac{{\overline{\alpha}}}{k})\cup B(\eta_l,\frac{{\hat{\alpha}}}{h}))}\left(\frac{1}{|y-\xi_i|^{n-2}}+\frac{1}{|y-\eta_l|^{n-2}}\right)\,dy\\
{\leqslant}& \, C \mu^{\frac{n-2}{2}}\lambda^{\frac{n-2}{2}}.
\end{split}$$ The case $\alpha,\beta\neq 0$ follows analogously just by noticing that $${\overline{Z}}_{\alpha i}\sim \frac{\mu^{\frac{n-2}{2}}}{|y-\xi_i|^{n-1}},\qquad {\hat{Z}}_{\beta l}\sim \frac{\lambda^{\frac{n-2}{2}}}{|y-\eta_l|^{n-1}},$$ and hence is proved.
We now need the following result.
\[behaviorPi\] The functions $\pi_\alpha$ can be decomposed as $$\pi_\alpha(y)=\sum_{j=1}^k{\overline{\pi}}_{\alpha j}(y)+\sum_{l=1}^h{\hat{\pi}}_{\alpha l}+\tilde{\pi}_\alpha(y),$$ where $${\overline{\pi}}_{\alpha j}(y)={\overline{\pi}}_{\alpha 1}(e^{-\frac{2\pi(j-1)}{k}}\overline{y},\hat{y},y'),\qquad {\hat{\pi}}_{\alpha l}(y)={\hat{\pi}}_{\alpha 1}(\overline{y}, e^{-\frac{2\pi(l-1)}{h}}\hat{y},y').$$ Furthermore, there exists a positive constant $C$ such that $$\|\tilde{\pi}_\alpha\|_*{\leqslant}C(k^{1-\frac{n}{q}}+h^{1-\frac{n}{q}}),\qquad \alpha=0,1,\ldots,n,$$ $$\|\overline{{\overline{\pi}}}_{\alpha 1}\|_*{\leqslant}C k^{-\frac{n}{q}}, \;\alpha=0,\ldots,n,\;\;\mbox{ where }\;\;\overline{{\overline{\pi}}}_{\alpha 1}:=\mu^{\frac{n-2}{2}}{\overline{\pi}}_{\alpha 1}(\xi_1+\mu y),$$ $$\|\hat{{\hat{\pi}}}_{\alpha 1}\|_*{\leqslant}C h^{-\frac{n}{q}}, \;\alpha=0,\ldots,n,\;\;\mbox{ where }\;\;\hat{{\hat{\pi}}}_{\alpha 1}:=\lambda^{\frac{n-2}{2}}{\hat{\pi}}_{\alpha 1}(\eta_1+\lambda y).$$
We omit the proof of this result.
Thanks to Proposition \[behaviorPi\], we get $$\label{intpi1}
\bigg|\int |u|^{p-1}Z_{\alpha 0}\pi_\beta\bigg|{\leqslant}C\|\tilde{\pi}_\beta\|_*,$$ $$\label{intpi2}
\bigg|\int |u|^{p-1}{\overline{Z}}_{\alpha i}\pi_\beta\bigg|{\leqslant}C\|{\overline{\pi}}_{\beta 1}\|_*, \qquad \bigg|\int |u|^{p-1}{\hat{Z}}_{\alpha l}\pi_\beta\bigg|{\leqslant}C\|{\hat{\pi}}_{\beta 1}\|_*.$$ Notice next (see (8.5) in [@MW] for a proof) that $$\int U^{p-1}Z_{00}^2=\int U^{p-1}Z_{10}^2>0,$$ and thus we can define $$t_\beta:=-\frac{1}{\int U^{p-1}Z_{00}^2}\int \varphi^\perp |u|^{p-1}{\bf z}_\beta,$$ which satisfies $|t_\beta|{\leqslant}C\|\varphi^\perp\|_*$, with $C$ independent of $k$ and $h$.
Consider for $\beta=0$. Thus, by using the definition of $z_0$, -, , and Proposition \[behaviorPi\] we get $$\begin{split}
\sum_{\alpha=0}^n&\left[c_{\alpha 0}\int_{\mathbb{R}^n}Z_{\alpha 0}|u|^{p-1}z_0+\sum_{j=1}^k{\overline{c}}_{\alpha j}\int_{\mathbb{R}^n}{\overline{Z}}_{\alpha j}|u|^{p-1}z_0+\sum_{l=1}^h{\hat{c}}_{\alpha l}\int_{\mathbb{R}^n}{\hat{Z}}_{\alpha l}|u|^{p-1}z_0\right]\\
&=\,c_{00}\int U^{p-1}Z_{00}^2-\sum_{j=1}^k{\overline{c}}_{0j}\int U^{p-1}Z_{00}^2-\sum_{l=1}^h{\hat{c}}_{0l}\int U^{p-1}Z_{00}^2\\
&\;\;-\sum_{j=1}^k{\overline{c}}_{1j}\int U^{p-1}Z_{00}^2-\sum_{l=1}^h{\hat{c}}_{3l}\int U^{p-1}Z_{00}^2+O(k^{1-\frac{n}{q}}+h^{1-\frac{n}{q}}){{\mathscrL}}\left[\begin{array}{c} c_{00}\\\ldots\\c_{n0}\end{array}\right]\\
&\;\;+O(k^{-\frac{n}{q}})\overline{{{\mathscrL}}}\left[\begin{array}{c} {\overline{c}}_0\\\ldots\\{\overline{c}}_{n}\end{array}\right]+O(h^{-\frac{n}{q}})\hat{{{\mathscrL}}}\left[\begin{array}{c} {\hat{c}}_0\\\ldots\\{\hat{c}}_{n}\end{array}\right],
\end{split}$$ where ${{\mathscrL}}$, $\overline{{{\mathscrL}}}$ and $\hat{{{\mathscrL}}}$ are linear functions with coefficients uniformly bounded in $k$ and $h$. Identity follows straightforward from here. - are obtained in the same way.
Proof of {#fact2}
=========
We proceed as in [@MW Section 9]. Indeed, we decompose $\varphi_0^\perp$ as $$\varphi_0^\perp=\sum_{\alpha=0}^n c_{\alpha 0} \varphi_{\alpha 0}^\perp,\qquad\mbox{with}\qquad L(\varphi_{\alpha 0}^\perp)=-L(Z_{\alpha 0}),$$ which is equivalent to $$\label{VarphiZero}
\Delta(\varphi_{\alpha 0}^\perp)+p\gamma U^{p-1}(\varphi_{\alpha 0}^\perp)+a_0(y)\varphi_{\alpha 0}^\perp=-L(Z_{\alpha 0}),$$ where $a_0(y):=p\gamma(|u|^{p-1}-U^{p-1})$. Adapting the arguments of [@MW] it can be seen that $a\in L^{\frac{n}{2}}({{\mathbb R}}^n)$, $$\label{kz00}
|y|^{-n-2}L(Z_{00})(|y|^{-2}y)=-L(Z_{00})(y),$$ $$\label{kzAlpha0}
|y|^{-n-2}L(Z_{\alpha 0})(|y|^{-2}y)=L(Z_{\alpha 0})(y),\;\alpha=1,\ldots,n,$$ and $$\label{acero}\|L(Z_{\alpha 0})\|_{L^{\frac{2n}{n+2}}({{\mathbb R}}^n)}{\leqslant}C(\mu^{\frac{n-1}{n}}+\lambda^{\frac{n-1}{n}}).$$ We will solve as a fixed point problem. Let us consider the problem $$L_0(\varphi)=h-a_0(y)\phi,$$ where $L_0(\varphi):= \Delta \varphi+p\gamma U^{p-1}\varphi$ and $h\in L^{\frac{2n}{n+2}}({{\mathbb R}}^n)$ satisfies $$h(y)=|y|^{-n-2}h(|y|^{-2}y).$$ Let $T$ be the operator that associates to every $\phi$ the solution $\varphi$ to this problem, that is, $$\varphi=T(h-a_0(y)\phi).$$ Naming $A(\phi):=T(h-a_0(y)\phi)$ we are going to see that this operator is a contraction and that maps the ball $$B:=\{\phi\in{{\mathscrD}}^{1,2}({{\mathbb R}}^n): \|\phi\|_*{\leqslant}C(\mu^{\frac{n-1}{n}}+\lambda^{\frac{n-1}{n}}), \phi(y)=|y|^{-n+2}\phi(|y|^{-2}y)\},$$ into herself. Indeed, assume $\phi\in B$. Thus, $$a_0(y)\phi(y)=|y|^{-n-2}a_0(|y|^{-2}y)\phi(|y|^{-2}y),$$ and, by [@MW Proposition 9.1], we know that $$\|\varphi\|_*{\leqslant}C\|h-a_0(y)\phi\|_{L^{\frac{2n}{n+2}}({{\mathbb R}}^n)}{\leqslant}C\left(\|h\|_{L^{\frac{2n}{n+2}}({{\mathbb R}}^n)}+\|a_0(y)\phi\|_{L^{\frac{2n}{n+2}}({{\mathbb R}}^n)}\right),$$ and $\varphi(y)=|y|^{2-n}\varphi(|y|^{-2}y)$.
We study the last term in two different regions. First, in $${{\mathbb R}}^n\setminus (\{\cup_{j=1}^k B(\xi_j,\frac{{\overline{\alpha}}}{k})\}\cup \{\cup_{l=1}^h B(\eta_l, \frac{{\hat{\alpha}}}{h})\})$$ we can estimate $a_0$ as $$|a_0(y)|{\leqslant}C U^{p-2}\left[\sum_{j=1}^k\frac{\mu^{\frac{n-2}{2}}}{|y-\xi_j|^{n-2}}+\sum_{l=1}^h\frac{\lambda^{\frac{n-2}{2}}}{|y-\eta_l|^{n-2}}\right],$$ and consequently, $$\begin{split}\label{a0first}
\int_{{{\mathbb R}}^n\setminus \left(\{\cup_{j=1}^k B(\xi_j,\frac{{\overline{\alpha}}}{k})\}\cup \{\cup_{l=1}^h B(\eta_l, \frac{{\hat{\alpha}}}{h})\}\right)}|a_0(y)|^{\frac{2n}{n+2}}\,dy{\leqslant}C\left(k^{-(n-1)}+h^{-(n-1)}\right).
\end{split}$$ Consider now $j\in\{1,\ldots,k\}$ and the ball $B(\xi_j,\frac{{\overline{\alpha}}}{k})$. Here $$|a_0(y)|{\leqslant}C|U_{\mu,\xi_j}(y)|^{p-1},$$ and thus $$\begin{split}\label{a0sec}
\sum_{j=1}^k\int_{B(\xi_j,\frac{{\overline{\alpha}}}{k})} |a_0(y)|^{\frac{2n}{n+2}}\,dy&{\leqslant}C\sum_{j=1}^k\int_{B(\xi_j,\frac{{\overline{\alpha}}}{k})}\left[\frac{\mu^{-\frac{n-2}{2}}}{1+|y-\xi_j|^{n-2}}\right]^{(p-1)\frac{2n}{n+2}}\\
&{\leqslant}C k \mu^{-2\frac{2n}{n+2}}\mu^n\int_{B(0,\frac{1}{\mu k})}\left[\frac{1}{1+|y|^{n-2}}\right]^{(p-1)\frac{2n}{n+2}}\,dy\\
&{\leqslant}C k^{-(n-1)}.
\end{split}$$ Likewise, $$\label{a0third}
\sum_{l=1}^k\int_{B(\eta_l,\frac{{\hat{\alpha}}}{h})} |a_0(y)|^{\frac{2n}{n+2}}\,dy{\leqslant}Ch^{-(n-1)}.$$ Putting , and together we conclude that if $\phi\in B$, then $$\|a_0(y)\phi\|_{L^{\frac{2n}{n+2}}({{\mathbb R}}^n)}{\leqslant}\|\phi\|_{L^\infty({{\mathbb R}}^n)}\|a_0(y)\|_{L^{\frac{2n}{n+2}}({{\mathbb R}}^n)}{\leqslant}C(\mu^{\frac{n-1}{n}}+\lambda^{\frac{n-1}{n}}).$$ Furthermore $$\begin{split}
\|A(\phi_1)-A(\phi_2)\|_*&{\leqslant}C \|a_0(y)(\phi_1-\phi_2)\|_{L^{\frac{2n}{n+2}}({{\mathbb R}}^n)}\\
&{\leqslant}C \|a_0(y)\|_{L^{\frac{2n}{n+2}}({{\mathbb R}}^n)}\|\phi_1-\phi_2\|_*\\
&=o(1)\|\phi_1-\phi_2\|_*,
\end{split}$$ where $o(1)$ denotes a quantity which goes to zero when $k$, $h$ tend to infinity. Thus, $A$ defines a contraction mapping whenever $$\|h\|_{L^{\frac{2n}{n+2}}}{\leqslant}C(\mu^{\frac{n-1}{n}}+\lambda^{\frac{n-1}{n}}).$$ Hence, considering $h=L(Z_{\alpha 0})$, by we conclude the existence of a solution to satisfying $$\|\varphi_{\alpha 0}^\perp\|_*{\leqslant}C(\mu^{\frac{n-1}{n}}+\lambda^{\frac{n-1}{n}}).$$
Consider now $j\in\{1,\ldots,k\}$, $l\in\{1,\ldots, h\}$, and let us write, $${\overline{\varphi}}_j^\perp=\sum_{\alpha=0}^n c_{\alpha j} {\overline{\varphi}}_{\alpha j}^\perp,\qquad\mbox{with}\qquad L({\overline{\varphi}}_{\alpha j}^\perp)=-L({\overline{Z}}_{\alpha j}),$$ $${\hat{\varphi}}_l^\perp=\sum_{\alpha=0}^n c_{\alpha l} {\hat{\varphi}}_{\alpha l}^\perp,\qquad\mbox{with}\qquad L({\hat{\varphi}}_{\alpha l}^\perp)=-L({\hat{Z}}_{\alpha l}).$$ Performing the change of variables $$\overline{{\overline{\varphi}}}_j^\perp(y):=\mu^{\frac{n-2}{2}}{\overline{\varphi}}_{\alpha j}^\perp(\mu y +\xi_j),\qquad \hat{{\hat{\varphi}}}_l^\perp(y):=\lambda^{\frac{n-2}{2}}{\hat{\varphi}}_{\alpha l}^\perp(\lambda y +\eta_l),$$ the previous equations turn into $$\Delta(\overline{{\overline{\varphi}}}_j^\perp)+p\gamma U^{p-1}(\overline{{\overline{\varphi}}}_j^\perp)+p\gamma \overline{a}_j(y)\overline{{\overline{\varphi}}}_j^\perp={\overline{h}}_j(y),$$ $$\Delta(\hat{{\hat{\varphi}}}_l^\perp)+p\gamma U^{p-1}(\hat{{\hat{\varphi}}}_l^\perp)+p\gamma \hat{a}_l(y)\hat{{\hat{\varphi}}}_l^\perp={\hat{h}}_l(y),$$ where $$\overline{a}_j(y):=p\gamma[(\mu^{-\frac{n-2}{2}}|u|(\mu y +\xi_j))^{p-1}-U^{p-1}],\;\; {\overline{h}}_j(y):=-\mu^{\frac{n+2}{2}}L({\overline{Z}}_{\alpha j})(\mu y+\xi_j),$$ $$\hat{a}_l(y):=p\gamma[(\lambda^{-\frac{n-2}{2}}|u|(\lambda y +\eta_l))^{p-1}-U^{p-1}],\;\; {\hat{h}}_l(y):=-\lambda^{\frac{n+2}{2}}L({\hat{Z}}_{\alpha l})(\lambda y+\eta_l).$$ Performing an analogous fixed point argument we conclude .
Final argument {#final}
==============
Let $\left[\begin{array}{c} c_0\\ c_1\\\ldots\\c_n\end{array}\right]$ be the solution to provided by Proposition \[nonso\], and let $t_0,t_1,t_2,t_3,t_4$, $\overline{t}_0,\overline{t}_1,\overline{t}_2$, $\hat{t}_0,\hat{t}_1,\hat{t}_2,$ and $t_\alpha,\overline{\nu}_{\alpha 1},\overline{\nu}_{\alpha 2},\hat{\nu}_{\alpha 1}, \hat{\nu}_{\alpha 2}$, $\alpha=5,\ldots,n$, be the associated parameters. Thus, it follows straightforward the existence of a unique vector of parameters $$(t_0^*,\ldots,t_4^*,\overline{t}_0^*,\overline{t}_1^*,\overline{t}_2^*,\hat{t}_0^*,\hat{t}_1^*,\hat{t}_2^*,t_5^*,\overline{\nu}_{5 1}^*,\overline{\nu}_{5 2}^*,\hat{\nu}_{5 1}^*, \hat{\nu}_{5 2}^*,\ldots,t_n^*,\overline{\nu}_{n 1}^*,\overline{\nu}_{n 2}^*,\hat{\nu}_{n 1}^*, \hat{\nu}_{n 2}^*)$$ such that $\left[\begin{array}{c} c_0\\ c_1\\\ldots\\c_n\end{array}\right]$ solves the system in Proposition \[systC\] and, equivalently, . Moreover, $$\begin{split}
\|(t_0^*,t_1^*,t_2^*,t_3^*,t_4^*,\overline{t}_0^*,\overline{t}_1^*,\overline{t}_2^*,\hat{t}_0^*,\hat{t}_1^*,\hat{t}_2^*,&t_5^*,\overline{\nu}_{5 1}^*,\overline{\nu}_{5 2}^*,\hat{\nu}_{5 1}^*, \hat{\nu}_{5 2}^*,\ldots,t_n^*,\overline{\nu}_{n 1}^*,\overline{\nu}_{n 2}^*,\hat{\nu}_{n 1}^*, \hat{\nu}_{n 2}^*)\|{\leqslant}C\|\varphi^\perp\|,
\end{split}$$ and therefore $$\|\left[\begin{array}{c} c_0\\ c_1\\\ldots\\c_n\end{array}\right]\|{\leqslant}C\|\varphi^\perp\|.$$ This estimate, together with , allows us to conclude $$c_\alpha =0\qquad\forall\,\alpha=0,\ldots,n,$$ and thus $\varphi^\perp\equiv 0$. Replacing this in the proof of Theorem \[nondeg\] is complete.
Appendix {#appe1}
========
According to their definitions, see – and , it is convenient to rewrite the functions $z_\alpha$ as $$\begin{split}\label{z0small}
z_0(y)=&Z_{00}(y)-\sum_{j=1}^k\left[{\overline{Z}}_{0j}(y)+{\overline{Z}}_{1j}(y)\right]-\sum_{l=1}^h\left[{\hat{Z}}_{0l}(y)+{\hat{Z}}_{3l}(y)\right],
\end{split}$$ $$\begin{split}
z_1(y)=&Z_{10}(y)-\sum_{j=1}^k\frac{\cos{{\overline{\theta}}_j}{\overline{Z}}_{1j}(y)-\sin{{\overline{\theta}}_j}{\overline{Z}}_{2j}(y)}{\sqrt{1-\mu^2}}-\sum_{l=1}^h{\hat{Z}}_{1l}(y),\\
z_2(y)=&Z_{20}(y)-\sum_{j=1}^k\frac{\sin{{\overline{\theta}}_j}{\overline{Z}}_{1j}(y)+\cos{{\overline{\theta}}_j}{\overline{Z}}_{2j}(y)}{\sqrt{1-\mu^2}}-\sum_{l=1}^h{\hat{Z}}_{2l}(y),
\\
z_3(y)=&Z_{30}(y)-\sum_{j=1}^k{\overline{Z}}_{3j}(y)-\sum_{l=1}^h\frac{\cos{{\hat{\theta}}_l}{\hat{Z}}_{3l}(y)-\sin{{\hat{\theta}}_l}{\hat{Z}}_{4l}(y)}{\sqrt{1-\lambda^2}},\\
z_4(y)=&Z_{40}(y)-\sum_{j=1}^k{\overline{Z}}_{4j}(y)-\sum_{l=1}^h\frac{\sin{{\hat{\theta}}_l}{\hat{Z}}_{3l}(y)+\cos{{\hat{\theta}}_l}{\hat{Z}}_{4l}(y)}{\sqrt{1-\lambda^2}},\\
z_\alpha(y)=&Z_{\alpha 0}(y)-\sum_{j=1}^k{\overline{Z}}_{\alpha j}(y)-\sum_{l=1}^h{\hat{Z}}_{\alpha l}(y),\quad \alpha=5,\ldots,n \\
z_{n+1}(y)&=-\sum_{j=1}^k{\overline{Z}}_{2j}(y),\quad
z_{n+2}(y)=-\sum_{l=1}^h{\hat{Z}}_{4l}(y),\\
\end{split}$$ $$\begin{split}
z_{n+7}(y)&=-\sqrt{1-\mu^2}\sum_{j=1}^k\cos{{\overline{\theta}}_j}{\overline{Z}}_{3j}(y)+\sqrt{1-\lambda^2}\sum_{l=1}^h\cos{{\hat{\theta}}_l}{\hat{Z}}_{1l}(y),\\
z_{n+8}(y)&=-\sqrt{1-\mu^2}\sum_{j=1}^k\cos{{\overline{\theta}}_j}{\overline{Z}}_{4j}(y)+\sqrt{1-\lambda^2}\sum_{l=1}^h\sin{{\hat{\theta}}_l}{\hat{Z}}_{1l}(y),\\
z_{n+\alpha+4}(y)&=-\sqrt{1-\mu^2}\sum_{j=1}^k\cos{{\overline{\theta}}_j}{\overline{Z}}_{\alpha j}(y), \quad \alpha=5,\ldots,n,\\
z_{2n+5}(y)&=-\sqrt{1-\mu^2}\sum_{j=1}^k\sin{{\overline{\theta}}_j}{\overline{Z}}_{3j}+\sqrt{1-\lambda^2}\sum_{l=1}^h\cos{{\hat{\theta}}_l}{\hat{Z}}_{2l}\\
z_{2n+6}(y)&=-\sqrt{1-\mu^2}\sum_{j=1}^k\sin{{\overline{\theta}}_j}{\overline{Z}}_{4j}+\sqrt{1-\lambda^2}\sum_{l=1}^h\sin{{\hat{\theta}}_l}{\hat{Z}}_{2l}\\
\end{split}$$ and, for $\alpha=5,\ldots,n$, $$\begin{split}
z_{2n+\alpha+2}(y)=-\sqrt{1-\mu^2}&\sum_{j=1}^k\sin{{\overline{\theta}}_j}{\overline{Z}}_{\alpha j},\quad
z_{3n+\alpha-2}(y)=-\sqrt{1-\lambda^2}\sum_{l=1}^h\sin{{\hat{\theta}}_l}{\hat{Z}}_{3l},\\
z_{4n+\alpha-6}(y)&=-\sqrt{1-\lambda^2}\sum_{l=1}^h\cos{{\hat{\theta}}_l}{\hat{Z}}_{4l}.
\end{split}$$ The proof of the above identities follows from straightforward computations, and the symmetry properties for $U (y) + \psi (y)$, for $ U_{\mu , \xi_j} (y) + \overline \phi_j (y) $ and $U_{\lambda , \eta_l} (y) + \hat \phi_l (y)$ respectively.
A less straightforward computation gives that, $\alpha = n+3, n+4, n+5, n+6$, we have $$\begin{split} \label{soloquesto}
{\bf z}_{n+3}(y)&=z_1 - 2 \sqrt{1-\mu^2}\sum_{j=1}^k \cos{\overline{\theta}}_j\left[ {\overline{Z}}_{0j}(y) + {\overline{Z}}_{1j} \right] ,\\
{\bf z}_{n+4}(y)&=z_2 - 2 \sqrt{1-\mu^2}\sum_{j=1}^k \sin{\overline{\theta}}_j \left[ {\overline{Z}}_{0j}(y) + {\overline{Z}}_{1j} \right], \\
{\bf z}_{n+5}(y)&=z_1 - 2 \sqrt{1-\lambda^2} \sum_{l=1}^h \cos{\hat{\theta}}_l \left[ {\hat{Z}}_{0l}(y) + {\hat{Z}}_{3l} \right], \\
{\bf z}_{n+6}(y)&=z_1 - 2 \sqrt{1-\lambda^2} \sum_{j=1}^h \sin{\hat{\theta}}_l\left[ {\hat{Z}}_{0l}(y) + {\hat{Z}}_{4l} \right] .\\
\end{split}$$ We shall prove the validity of the first identity in . The proofs of the validity of the the other expressions in are similar. We write $${\bf z}_{n+3} = z_1 + T (u), \qquad T (u) := (|y|^2 -1 ) {\partial u \over \partial y_1} - 2y_1 ({n-2 \over 2} u (y) + \nabla u (y) \cdot y ).$$ Thus follows from and from $$\label{newsoloquesto}
T(u) = -2\sum_{j=1}^k \xi_{j1} \left[ {\overline{Z}}_{0j} +\nabla (U_{\mu , \xi_j} + \overline \phi_j ) (y) \cdot \xi_j \right]
.$$ From the explicit expression of $u$ in , we get $$T(u) = T(U +\psi ) - \sum_{j=1}^k T (U_{\mu , \xi_j} + \overline \phi_j ) - \sum_{l=1}^h T(U_{\lambda , \eta_l} +\hat \phi_l ) .$$ We shall first show that $ T(U +\psi ) (y) \equiv 0$, and $T(U_{\lambda , \eta_l} +\hat \phi_l ) (y) \equiv 0$ for any $l=1, \ldots , h$.
Observe that, if $v$ is any smooth function and if we define $h(z) := {\partial \over \partial z_1} \left( |z|^{2-n} v({z \over |z|^2} ) \right)$, then we have $$\begin{split}
h(z) &= -{2 z_1 \over |z|^n} \left[ {n-2 \over 2} v \left( {z \over |z|^2} \right) + \nabla v \left( {z \over |z|^2} \right) \cdot \left( {z \over |z|^2} \right) \right] \\
&+ {1\over |z|^n} {\partial v \over \partial z_1} \left( {z \over |z|^2} \right),
\end{split}$$ and $g(y):= {1\over |y|^{n-2} } h ({y \over |y|^2} )$ takes the form $$g(y) = -2 y_1 \left[ {n-2 \over 2} v(y) + \nabla v (y) \cdot y \right] + |y|^2 {\partial v \over \partial y_1} (y).$$ With this is mind, one gets that if $v$ is Kelvin invariant $
v(y ) = |y|^{n-2} v \left( {y \over |y|^2} \right), $ then $$\label{gino}{1\over |y|^{n-2}} {\partial v \over \partial y_1} \left( {y\over |y|^2} \right) =
-2 y_1 \left[ {n-2 \over 2} v(y) + \nabla v (y) \cdot y \right] + |y|^2 {\partial v \over \partial y_1} (y).$$ On the other hand, if $v$ is Kelvin invariant (with respect to the origin) and even in $y_1$, then also the function ${\partial v \over \partial y_1}$ is Kelvin invariant, that is $
{\partial v \over \partial y_1}(y ) = |y|^{n-2} {\partial v \over \partial y_1} \left( {y \over |y|^2} \right). $ By , we get that any function $v$ which is invariant under Kelvin transform (with respect to the origin) and even in the $y_1$ direction, one that $T(v) (y) \equiv 0$. Since the functions $ (U +\psi ) (y) $, and $(U_{\lambda , \eta_l} +\hat \phi_l ) (y)$ for any $l=1, \ldots , h$ are invariant under Kelvin transform and even in $y_1$, we get the proof of our claim.
Let us fix $j \in \{1 , \ldots , k\}$. We write, for $v(y) = ( U_{\mu , \xi_j} + \overline \phi_j) (y)$, $$\begin{split}
T (U_{\mu , \xi_j} + \overline \phi_j ) (y) &=\underbrace{(|y|^2 -1) {\partial v \over \partial y_1 } (y) - 2 (y-\xi_j )_1 [ {n-2 \over 2} v (y) + \nabla v (y) \cdot y] }_{=:T_j (v)} \\
&-2 (\xi_j)_1\left[ {\overline{Z}}_{0j} (y) + \nabla v(y) \cdot \xi_j \right] .
\end{split}$$ We claim that $T_j (U_{\mu , \xi_j} + \overline \phi_j ) (y) \equiv 0$. To prove this fact, we recall that $$v(y):= (U_{\mu , \xi_j} + \overline \phi_j ) (y) = \mu^{-{n-2 \over 2} } (U + \overline{{\overline{\phi}}}_1 ) \left({y-\xi_j \over \mu} \right),$$ see Section \[sec3\]. Also, $\mu$ and $|\xi|$ are related so that $U_{\mu , \xi_j} + \overline \phi_j $ is invariant under Kelvin transform. Thus, from , we get $$T_j (v) (y) = {1\over |y|^{n-2}} {\partial v \over \partial y_1} \left({y \over |y|^2} \right)- {\partial v \over \partial y_1} (y) + 2 (\xi_j)_1 \left[ {n-2 \over 2} v(y) + \nabla v (y) \cdot y\right].$$ We note that, in this case, $U_{\mu , \xi_j} + \overline \phi_j $ is not even in the $y_1$ variable, so that one gets $$\begin{split}
{1\over |y|^{n-2}} {\partial v \over \partial y_1} \left({y \over |y|^2} \right)&= {\partial v \over \partial y_1} (y) - (\xi_j)_1 \left[ (n-2) v(y) +2 \nabla v (y) \cdot y\right].
\end{split}$$ This concludes the proof of .
[1]{}
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[^1]: The first author is supported by the grant FONDECYT Postdoctorado, No. 3160077, CONICYT (Chile) and grant MTM2013-40846-P, MINECO (Spain). The second author is supported by FONDECYT Grant 1160135 and Millennium Nucleus Center for Analysis of PDE, NC130017. The third author is supported by NSERC of Canada.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Let $X$ be a complex projective manifold and $f$ a dominating rational map from $X$ onto $X$. We show that the topological entropy ${{\rm h}}(f)$ of $f$ is bounded from above by the logarithm of its maximal dynamical degree.'
author:
- 'Tien-Cuong Dinh et Nessim Sibony'
title: 'Une borne supérieure pour l’entropie topologique d’une application rationnelle'
---
Soit $X$ une variété projective complexe de dimension $k\geq
2$, munie d’une forme de Kähler $\omega$ normalisée par $\int_X\omega^k=1$. Soit $f:X\longrightarrow X$ une application rationnelle dominante, [[*c.-à-d.* ]{}]{}localement ouverte en un point générique de $X$. Notons $\lambda_l(f)$ le degré dynamique d’ordre $l$ de $f$, $1\leq l\leq k$, et ${{\rm h}}(f)$ l’entropie topologique de $f$. Ils sont définis plus loin. Il s’agit de montrer que ${{\rm h}}(f)\leq \max_{1\leq l\leq k}
\log\lambda_l(f)$. Un intermédiaire utile est ${{\rm lov}}(f)$, c’est un indicateur de la croissance du volume des graphes des itérés de $f$. Plus précisément, soit $\Gamma_n$ le graphe de l’application $(f,f^2,\ldots,f^n)$ où $f^i:=f\circ\cdots\circ f$ ($i$ fois). On pose $${{\rm lov}}(f):=\limsup_{n\rightarrow \infty}
\frac{1}{n}\log{{\rm vol}}(\Gamma_n).$$
Utilisant une inégalité de Lelong, Gromov [@Gromov1] a montré que ${{\rm h}}(f)\leq {{\rm lov}}(f)$ lorsque $f$ est holomorphe. Sa preuve reste valable pour les applications rationnelles. Nous allons montrer que ${{\rm lov}}(f) = \max_{1\leq l\leq k}\log\lambda_l(f)$ et en déduire le théorème suivant.
\
[**Théorème 1**]{} [*Soit $X$ une variété projective complexe de dimension $k\geq 2$ et soit $f:X\longrightarrow X$ une application rationnelle dominante. Alors $${{\rm h}}(f)\leq {{\rm lov}}(f)=\max_{1\leq l\leq k} \log\lambda_l(f).$$*]{}
Ce résultat a été annoncé dans [@Friedland]. L’auteur a utilisé l’inégalité $(f\circ
f)^*\leq f^*\circ f^*$ [@Friedland lemma 3] qui n’est pas valable lorsque ces opérateurs agissent sur les cycles analytiques. Un contre-exemple est donné dans [@Guedj]. Dans notre approche, on ne définit pas $f^*$ sur tous les courants ni même sur les cycles. Le produit des courants n’est considéré que là où ils sont lisses.
Rappelons quelques notions ([[*voir* ]{}]{}par exemple [@Bowen; @Sibony2; @Guedj; @DinhSibony1]). Notons $\Gamma$ le graphe de $f$ dans $X\times X$. C’est un sous-ensemble analytique irréductible de dimension $k$. Lelong a montré que l’intégration sur la partie régulière de $\Gamma$ définit un courant positif fermé $[\Gamma]$ de bidimension $(k,k)$. Soient $\pi_1$ et $\pi_2$ les projections de $X\times X$ sur le premier et le second facteur. Pour tout ensemble $Y\subset X$, posons $$f(Y):=\pi_2(\pi_1^{-1}(Y)\cap \Gamma) \ \mbox{ et }\
f^{-1}(Y):=\pi_1(\pi_2^{-1}(Y)\cap \Gamma).$$
Notons $I_f$ l’ensemble des points $x\in X$ tels que $\dim \pi_1^{-1}(x)\cap\Gamma\geq 1$. C’est [*l’ensemble des points d’indétermination de $f$*]{}. Il est de codimension au moins 2. On a $\dim \pi_1^{-1}(I_f)\cap\Gamma \leq k-1$. Pour toute forme $\varphi$ lisse de bidegré $(l,l)$ posons $$\begin{aligned}
f^*(\varphi) & := & (\pi_1)_*(\pi_2^*(\varphi)\wedge [\Gamma])\end{aligned}$$ Le courant $f^*(\varphi)$ est lisse sur $X\setminus I_f$. Puisqu’il est de masse finie et sans masse sur $I_f$, ses coefficients sont dans ${{{\rm L}^1}}$. En particulier, $f^*(\varphi)$ ne charge pas les sous-ensembles analytiques propres de $X$. L’opérateur $f^*$ est continu de l’espace des formes lisses dans l’espace des formes à coefficients dans ${{{\rm L}^1}}$.
Posons $\delta_l(f):=\int_X f^*(\omega^l)\wedge
\omega^{k-l}$ pour $1\leq l\leq k$. On définit [*le degré dynamique d’ordre $l$*]{} de $f$ par $$\lambda_l(f):=\limsup_{n\rightarrow\infty} [\delta_l(f^n)]^{1/n}.$$ On verra que la suite $[\delta_l(f^n)]^{1/n}$ est en toujours convergente (corollaire 7). [*Le degré topologique*]{} $d_t:=\lambda_k(f)$ est égal au nombre de préimages par $f$ d’un point générique de $X$.
Soit $$\Omega_f:=X\setminus \cup_{n\in{\mathbb{Z}}} f^n(I_f).$$ C’est un ensemble invariant par $f$ et $f^{-1}$. On dira qu’une famille $F\subset\Omega_f$ est [*$(n,\epsilon)$-séparée*]{}, $\epsilon>0$, si $$\max_{0\leq i\leq n-1} {{\rm dist}}(f^i(x),f^i(y))\geq \epsilon
\ \mbox{ pour }\ x,y\in F \mbox{ distincts}.$$ [*L’entropie topologique*]{} ([[*voir* ]{}]{}[@Bowen]) ${{\rm h}}(f)$ est définie par $${{\rm h}}(f):=\sup_{\epsilon>0}\left(\limsup_{n\rightarrow\infty}
\frac{1}{n}
\log \max\big\{\#F, \ F\
(n,\epsilon)\mbox{-s\'epar\'ee}\big\}\right).$$
Notons $\Gamma_n$ l’adhérence dans $X^n$ de l’ensemble des points $$(x,f(x),\ldots, f^{n-1}(x)),\ \ \ x\in\Omega_f.$$ C’est un sous-ensemble analytique de dimension $k$ de $X^n$. Soient $\Pi_i$ les projections de $X^n$ sur ses facteurs. On munit $X^n$ de la forme de Kähler $\omega_n:=\sum
\Pi_i^*(\omega)$. On a $${{\rm lov}}(f):=\limsup_{n\rightarrow\infty} \frac{1}{n}
\log({{\rm vol}}(\Gamma_n)):=
\limsup_{n\rightarrow\infty}\frac{1}{n}\log
\left(\int_{\Gamma_n}\omega_n^k\right).$$ On verra plus loin que la suite $\big(\frac{1}{n}\log\int_{\Gamma_n}\omega_n^k\big)$ est toujours convergente. Utilisant une inégalité de Lelong, Gromov [@Gromov1] a montré que ${{\rm h}}(f)\leq {{\rm lov}}(f)$. Dans la suite, nous montrons que ${{\rm lov}}(f)=\max \log \lambda_l(f)$.
Pour tout courant positif fermé de bidegré $(l,l)$ sur $X$, notons $\|S\|:=\int_X S\wedge \omega^{k-l}$ [*la masse*]{} de $S$. Puisqu’on a supposé $\int_X\omega^k=1$, les courants $\omega^l$ sont de masse 1. Pour les résultats fondamentaux sur les courants positifs fermés nous renvoyons à Lelong [@Lelong] et Demailly [@Demailly]. Notre outil principal est le lemme suivant.\
\
[**Lemme 2.**]{}
*Il existe une constante $c>0$, qui ne dépend que de $X$, telle que pour tout courant positif fermé $S$ de bidegré $(l,l)$ sur $X$ on puisse trouver une suite de courants positifs fermés lisses $(S_m)_{m\geq 1}$, de bidegré $(l,l)$, vérifiant les propriétés suivantes*
1. La suite $(S_m)$ converge vers un courant positif fermé $S'$.
2. $S'\geq S$, c’est-à-dire que le courant $S'-S$ est positif.
3. On a $\|S_m\|\leq c\|S\|$ pour tout $m\geq 1$.
Soit $\omega_{{\rm FS}}$ la forme de Fubini-Study de ${\mathbb{P}}^k$ normalisée par $\int_{{\mathbb{P}}^k}\omega_{{\rm FS}}^k=1$. Rappelons que les groupes de cohomologie de Dolbeault ${{\cal H}}^{l,l}({\mathbb{P}}^k,{\mathbb{R}})$ sont de dimension $1$. En particulier, tout courant positif fermé $R$ de bidegré $(l,l)$ sur ${\mathbb{P}}^k$ est cohomologue à $\|R\|\omega_{{\rm FS}}^l$.
Puisque $X$ est projective, on peut choisir une famille finie d’applications holomorphes surjectives $\Psi_i$, $1\leq i\leq s$, de $X$ dans ${\mathbb{P}}^k$ telles qu’en tout point $x\in X$ au moins l’une des applications $\Psi_i$ soit de rang maximal. Il suffit de plonger $X$ dans un ${\mathbb{P}}^N$ et de prendre une famille de projections sur ${\mathbb{P}}^k$. Posons $T_i:=(\Psi_i)_*(S)$. L’opérateur $(\Psi_i)_*$ étant continu, il existe une constante $c_1>0$ indépendante de $S$ telle que $\|T_i\|\leq c_1\|S\|$.
La variété ${\mathbb{P}}^k$ étant homogène, si $R$ est un courant positif fermé dans ${\mathbb{P}}^k$, il existe des courants positifs fermés lisses $(R_m)$ tendant vers $R$ avec $\|R_m\|=\|R\|$. Donc il existe des courants positifs fermés lisses $T_{i,m}$ de bidegré $(l,l)$ sur ${\mathbb{P}}^k$ qui convergent faiblement vers $T_i$ et qui vérifient $\|T_{i,m}\|=\|T_i\|$. On a donc $\|T_{i,m}\|\leq c_1\|S\|$.
Posons $S_m:=\sum_{i=1}^s (\Psi_i)^*(T_{i,m})$. Estimons la masse de $(\Psi_i)^*(T_{i,m})$: $$\begin{aligned}
\|(\Psi_i)^*(T_{i,m})\| & = & \int_X (\Psi_i)^*(T_{i,m})\wedge
\omega^{k-l}\\
& = & \|T_{i,m}\|\int_X (\Psi_i)^*(\omega_{{\rm FS}}^l)
\wedge\omega^{k-l}\\
& \leq & c_1\|S\|\int_X (\Psi_i)^*(\omega_{{\rm FS}}^l)
\wedge\omega^{k-l}\\
& \leq & c_2\|S\|\end{aligned}$$ pour une constante $c_2>0$ indépendante de $S$. Donc la masse de $S_m$ est majorée par $c\|S\|$ avec $c:=sc_2$. Quitte à extraire une sous-suite, on peut supposer que la suite $(S_m)$ tend faiblement vers un courant $S'$. Au voisinage de chaque point $x\in X$, on vérifie, puisque l’un des $\Psi_i$ est un biholomorphisme local, que $S'-S$ est positif. On peut bien sûr choisir les constantes $c_1$, $c_2$ et $c$ indépendantes de $l$, $1\leq l\leq k$.
[**Remarque 3.**]{} L’ensemble des classes de courants positif fermés de bidegré $(l,l)$ et de masse 1 est borné dans ${{\cal H}}^{l,l}(X,{\mathbb{R}})$. Il existe donc une constante $\alpha_X>0$ telle que la classe de $\alpha_X\omega^l -T$ soit représentée par une forme lisse positive pour tout courant positif fermé $T$ de bidegré $(l,l)$ et de masse plus petite ou égale \` a 1. On dira que $T$ est [*cohomologiquement dominé*]{} par $\alpha_X\omega^l$. La propriété ci-dessus est valable pour toute variété kählérienne compacte. Dans le lemme 2, les courants $S_m$ sont cohomologiquement dominés par $c_X\|S\|\omega^l$ où $c_X:=c\alpha_X$.\
Notons ${\cal C}_f$ l’ensemble des points au voisinage desquels $f$ n’est pas une application holomorphe localement inversible. Posons $\Omega_{1,f}:=
X\setminus {\cal C}_f$. C’est un ouvert de Zariski de $X$. Nous allons définir $f^*(S)$ lorsque $S$ est un courant positif fermé de bidegré $(l,l)$ sur $X$. Le courant $f^*(S)$ est bien défini sur $\Omega_{1,f}$. Si sa masse sur $\Omega_{1,f}$, qui est définie par $\|f^*(S)\|:=\int_{\Omega_{1,f}} f^*(S)\wedge
\omega^{k-l}$, est finie, d’après Skoda [@Skoda], son prolongement trivial $\widetilde{f^*(S)}$ est un courant positif fermé sur $X$. Le lemme suivant montre que c’est le cas. Nous utilisons par la suite cette extension par 0.\
\
[**Lemme 4.**]{} [*Soit $S$ un courant positif fermé de bidegré $(l,l)$ sur $X$. Alors $\|f^*(S)\|\leq c_X\delta_l(f)\|S\|$. En particulier, $\widetilde{f^*(S)}$ est positif fermé dans $X$ et sa masse est bornée par $c_X\delta_l(f)\|S\|$.* ]{}
Soit $(S_m)$ la suite de courants lisses vérifiant le lemme 2 (appliqué au courant $S$). D’après la remarque 3, ces courants sont cohomologiquement dominés par $c_X\|S\|\omega^l$. On en déduit que la masse de $f^*(S_m)$, qui se calcule cohomologiquement, est majorée par $c_X \delta_l(f)\|S\|$. Plus précisément, on a pour tout compact $K\subset\Omega_{1,f}$ $$\begin{aligned}
\int_K f^*(S)\wedge \omega^{k-l} & \leq &
\int_K f^*(S')\wedge \omega^{k-l}\leq
\lim_{m\rightarrow\infty} \int_X f^*(S_m)\wedge
\omega^{k-l} \\
& \leq & c_X\|S\|\int_X
f^*(\omega^l)\wedge\omega^{k-l}
= c_X\delta_l(f)\|S\|.\end{aligned}$$ Ceci implique le lemme.
\
[**Lemme 5.**]{} [*Soit $\epsilon>0$. Il existe une constante $c_\epsilon>0$ telle qu’on ait $$\int_{\Omega_f} (f^{n_1})^*\omega\wedge \ldots\wedge
(f^{n_k})^*\omega \leq c_\epsilon
(\max_{1\leq l\leq k}\lambda_l+\epsilon)^{n_1}$$ pour tous les entiers naturels $n_1,\ldots,
n_k$ vérifiant $n_1\geq\cdots\geq n_k\geq 0$.* ]{}\
Posons $\lambda_\epsilon:=\max_{1\leq l\leq
k}\lambda_l+\epsilon$. Soit $c>0$ une constante telle que $\delta_l(f^n)\leq c\lambda_\epsilon^n$ pour tout $n\geq 0$ et pour tout $l$ avec $1\leq l\leq k$. Soit $\Omega_{n,f}:=X\setminus \cup_{0\leq i\leq n-1} f^{-i}({\cal C}_f)$. C’est un ouvert de Zariski de $X$. Montrons par récurrence sur $s$, $0\leq s\leq k$, que pour tous $n_1,\ldots,
n_s$ vérifiant $n_1\geq\cdots\geq n_s\geq 0$ on a $\|T_s\|\leq c^s c_X^s \lambda_\epsilon^{n_1}$, $c_X$ étant la constante de la remarque 3 et $$T_s:=(f^{n_1})^*\omega\wedge \ldots\wedge (f^{n_s})^*\omega,
\ \ \ \ T_0:=1.$$
C’est clair au rang $s=0$. Supposons le au rang $s-1$, $1\leq s\leq k$. On a $\|T'_{s-1}\|\leq c^{s-1} c_X^{s-1} \lambda_\epsilon^{n_1-n_s}$ où $$T'_{s-1}:=(f^{n_1-n_s})^*\omega\wedge
\ldots\wedge (f^{n_{s-1}-n_s})^*\omega.$$ Le courant $T'_{s-1}$ étant de masse finie sur $\Omega_{n_1-n_s,f}$, d’après le théorème de Skoda [@Skoda], son prolongement trivial $\widetilde{T_{s-1}'}$ dans $X$ est un courant positif fermé dont la masse est majorée par $c^{s-1} c_X^{s-1} \lambda_\epsilon^{n_1-n_s}$. Utilisant le lemme 4 appliqué au courant $S=\widetilde{T'_{s-1}}\wedge\omega$ et à l’application $f^{n_s}$, on obtient $$\|T_s\|=\|(f^{n_s})^*(T'_{s-1}\wedge\omega)\|\leq c_X
\delta_l(f^{n_s}) \|T'_{s-1}\|\leq
c^sc_X^s\lambda_\epsilon^{n_1}.$$ Ceci termine la récurrence. Pour $s=k$, on obtient le lemme avec $c_\epsilon:=c^kc_X^k$.
[**Fin de la démonstration du théorème 1.**]{} Il nous faut estimer ${{\rm lov}}(f)$. D’après le lemme 5, on a pour tout $\epsilon>0$ $$\begin{aligned}
{{\rm vol}}(\Gamma_n) & = & \sum_{0\leq i_1,\ldots i_k\leq n-1}
\int_{\Omega_f}
(f^{i_1})^*\omega \wedge \ldots \wedge (f^{i_k})^*\omega\\
& \leq & c_\epsilon n^k (\max_{1\leq l\leq
k}\lambda_l(f)+\epsilon)^{n-1}.\end{aligned}$$ D’où ${{\rm lov}}(f)\leq \max_{1\leq l\leq k} \log\lambda_l(f)$. On a aussi ${{\rm lov}}(f)\geq \max_{1\leq l\leq k} \log\lambda_l(f)$ car $${{\rm vol}}(\Gamma_n)\geq \int_{\Omega_f}
(f^{n-1})^*\omega^l \wedge \omega^{k-l} =\delta_l(f^{n-1}).$$
$\square$\
\
[**Proposition 6.**]{} [*Soient $f$ et $g$ deux applications rationnelles de $X$ dans $X$. On a $$\delta_l(f\circ g)\leq c_X\delta_l(f)\delta_l(g).$$* ]{}
Par définition de $\delta_l(f)$, on a $\|f^*\omega^l\|=\delta_l(f)$. Le courant $(f\circ g)^*\omega^l$, qui ne charge pas les sous-ensembles analytiques propres de $X$, est égal à $g^*(\widetilde{f^*\omega^l})$ sur $\Omega_{1,g}\cap\Omega_{1,f\circ g}$. Le lemme 4, appliqué à $g$ et au courant $\widetilde{f^*\omega^l}$, entraîne que $$\delta_l(f\circ g)=\|g^*(\widetilde{f^*\omega^l})
\|\leq c_X\delta_l(g)\|f^*\omega^l\| =
c_X\delta_l(f)\delta_l(g).$$
\
[**Corollaire 7.**]{} [*La suite $[\delta_l(f^n)]^{1/n}$ est convergente. Les degrés dynamiques $\lambda_l$ de $f$ sont des invariants birationnels.* ]{}
D’après la proposition 6, on a $\delta_l(f^{m+n}) \leq c_X \delta_l(f^m)\delta_l(f^n)$ pour tous $m,n\geq 1$. Ceci implique que la suite $[\delta_l(f^n)]^{1/n}$ converge vers $\inf_{n\geq 1}[\delta_l(f^n)]^{1/n}$.
Soit $g$ une application birationnelle de $X$ dans $X$. Posons $h:=g\circ f\circ g^{-1}$. On a $$\delta_l(h^n)=\delta_l(g\circ f^n\circ g^{-1})\leq c_X^2
\delta_l(g)\delta_l(g^{-1})\delta_l(f^n).$$ Donc $\lambda_l(f)\leq \lambda_l(h)$. Puisque $f=g^{-1}\circ h\circ g$, on a aussi $\lambda_l(h)\leq \lambda_l(f)$.
\
[**Remarques 8.**]{} [**a.**]{} Russakovskii et Shiffman [@RussakovskiiShiffman] ont montré l’inégalité $\delta_l(f\circ g)\leq \delta_l(f)\delta_l(g)$ lorsque $X={\mathbb{P}}^k$. Diller et Favre [@DillerFavre] ont décrit précisément la croissance de $\delta_1(f^l)$ lorsque $f$ est une application biméromorphe sur une surface complexe. Dans le cas de dimension $k\leq 3$ et dans le cas des variétés homogènes, les résultats ci-dessus ont été démontrés par Vincent Guedj [@Guedj]. Il a alors prouvé l’existence d’une unique mesure invariante d’entropie maximale $\log d_t(f)$ pour toute application rationnelle $f$ vérifiant $d_t(f)>
\lambda_l(f)$, $1\leq l\leq k-1$ (pour la méthode [[*voir* ]{}]{}également Briend-Duval [@BriendDuval] ainsi que [@DinhSibony1; @DinhSibony3]). Le théorème 1 permet d’étendre ce résultat au cas d’une variété projective quelconque.
[**b.**]{} D’après Iskovkikh-Manin [@IskovkikhManin], il existe des variétés $X$ lisses non rationnelles de dimension 3 dans ${\mathbb{P}}^4$ qui sont unirationelles, [[*c.-à-d.* ]{}]{}pour lesquelles il existe une application rationnelle de rang maximal $f:{\mathbb{P}}^3\longrightarrow X$. On peut donc composer une projection holomorphe $g$ de $X$ sur ${\mathbb{P}}^3$ avec $f$ pour obtenir beaucoup d’applications rationnelles d’entropie positive sur $X$. Nous remercions F. Campana et N. Mok qui nous ont indiqué ces exemples.
Il est facile cependant de construire des correspondances sur les variétés projectives ([[*voir* ]{}]{}par exemple [@ClozelUllmo; @Voisin; @Dinh; @DinhSibony2]). La proposition 6 et le corollaire 7 restent valables pour les correspondances. Rappelons qu’une correspondance sur $X$ est la donnée d’un ensemble analytique $\Gamma\subset X\times X$ de dimension $k$ dont les images de chaque composante par $\pi_1$ et $\pi_2$ sont égales à $X$. On peut poser $f:=\pi_2\circ(\pi_{1|\Gamma})^{-1}$. L’image réciproque d’une forme lisse est définie par l’équation (1). Les degrés dynamiques sont définis de façon analogue que pour les applications rationnelles.
[**c.**]{} Soit $\pi:Y\longrightarrow X$ une application holomorphe surjective où $X$ et $Y$ sont des variétés projectives complexes. Soit $T$ un courant positif fermé sur $X$. L’images réciproque $\pi^*(T)$ de $T$ par $\pi$ est bien définie sur un ouvert de Zariski de $Y$ (là où $f$ est une submersion locale). Le lemme 2 permet de prolonger $\pi^*(T)$ en courant positif fermé $\widetilde{\pi^*(T)}$ dans $Y$. L’opérateur $T\mapsto \widetilde{\pi^*(T)}$ est semi-continu inférieurement. Plus précisément, si $T_n\rightarrow T$, $\lim
\widetilde{\pi^*(T_n)}\geq \widetilde{\pi^*T}$. Cette définition est utile dans le cadre des courants dynamiques. Nous reviendrons sur cette question dans un prochain travail. Notons que Méo [@Meo] a donné un exemple qui montre qu’on ne peut pas toujours définir $\pi^*(T)$ dans le cas où $X$ et $Y$ ne sont pas compactes. Il a aussi donné une définition de $\pi^*(T)$ dans le cas local et lorsque $\pi$ est une application à fibres discrètes. On ne sait pas si sa définition est indépendante des coordonnées.
[11]{} *R. Bowen*, Topological entropy for non compact sets, *Trans. A.M.S.*, **184** (1973), 125-136. *J.-Y. Briend et J. Duval*, Deux caractérisations de la mesure d’équilibre d’un endomorphisme de ${\mathbb{P}}^k({\mathbb{C}})$, *Publ. Math. Inst. Hautes Etudes Sci.*, **93** (2001), 145-159. *L. Clozel et E. Ullmo*, Correspondances modulaires et mesures invariantes, à paraître dans [*Journal für die reine und angew. Math.*]{} *J.P. Demailly*, Monge-Ampère Operators, Lelong numbers and Intersection theory in Complex Analysis and Geometry, *Plemum Press* (1993), 115-193, *(V. Ancona and A. Silva editors)*. *J. Diller and C. Favre*, Dynamics of bimeromorphic maps of surfaces, *Amer. J. Math.* **123** (2001), no. 6, 1135-1169. *T.C. Dinh*, Distribution des préimages et des points périodiques d’une correspondance polynomiale, [*Bull. Soc. Math. France*]{}, à paraître. *T.C. Dinh et N. Sibony*, Dynamique des applications d’allure polynomiale, [*J. Math. Pures et Appl.*]{}, **82** (2003), 367-423. *T.C. Dinh et N. Sibony*, Dynamique des applications polynomiales semi-régulières, [*Arkiv för Mahematik*]{}, [**42**]{} (2004), 61-85. *T.C. Dinh et N. Sibony*, Distribution des valeurs de transformations méromorphes et applications, *prépublication* (2003).\
arxiv.org/abs/math.DS/0306095. *S. Friedland*, Entropy of polynomial and rational maps, *Ann. of Math.* **133** (1991), 359-368. *M. Gromov*, On the entropy of holomorphic maps, *Enseignement Math.* **49** (2003), 217-235. [*Manuscript*]{} (1977). *V. Guedj*, Ergodic properties of rational mappings with large topological degree, *Ann. Math.*, to appear. *V.A. Iskovkikh and Yu.I. Manin*, Three dimensional quartics and counterexamples to the Lüroth problem, *Math. USSR Sb.*, **15** (1971), 141-166. *P. Lelong*, Fonctions plurisousharmoniques et formes différentielles positives, Dunod Paris 1968. *M. Méo*, Image inverse d’un courant positif fermé par une application surjective, *C.R.A.S.*, **322** (1996), 1141-1144. *A. Russakovski and B. Shiffman*, Value distribution for sequences of rational mappings and complex dynamics, *Indiana Univ. Math. J.*, **46** (1997). *N. Sibony*, Dynamique des applications rationnelles de $\mathbb{P}^k$, *Panoramas et Synthèses*, (1999), 97-185. *H. Skoda*, Prolongement des courants positifs, fermés de masse finie, *Invent. Math.*, **66** (1982), 361-376. *C. Voisin*, Intrinsic pseudovolume forms and $K$-correspondences, *preprint*.
Tien-Cuong Dinh et Nessim Sibony,\
Mathématique - Bât. 425, UMR 8628, Université Paris-Sud, 91405 Orsay, France.\
E-mails: [email protected] et [email protected].
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Double white dwarf binaries in the Galaxy dominate the gravitational wave sky and would be detectable for an instrument such as LISA. Most studies have calculated the expected gravitational wave signal under the assumption that the binary white dwarf system can be represented by two point masses in orbit. We discuss the accuracy of this approximation for real astrophysical systems. For non-relativistic binaries in circular orbit the gravitational wave signal can easily be calculated. We show that for these systems the point mass approximation is completely justified when the individual stars are axisymmetric irrespective of their size. We find that the signal obtained from Smoothed-Particle Hydrodynamics simulations of tidally deformed, Roche-lobe filling white dwarfs, including one case when an accretion disc is present, is consistent with the point mass approximation. The difference is typically at the level of one per cent or less in realistic cases, yielding small errors in the inferred parameters of the binaries.'
author:
- |
D. van den Broek$^{1}$[^1], G. Nelemans$^{1,2,3}$, M. Dan$^{4}$ and S. Rosswog$^{4}$\
$^{1}$Department of Astrophysics/IMAPP, Radboud University Nijmegen, P.O. Box 9010, NL-6500 GL, The Netherlands\
$^{2}$Institute for Astronomy, KU Leuven, Celestijnenlaan 200D, 3001 Leuven, Belgium\
$^{3}$Nikhef, Science Park 105, 1098 XG Amsterdam, The Netherlands\
$^{4}$School of Engineering and Science, Jacobs University Bremen, Campus Ring 1, 28759 Bremen, Germany
title: On the point mass approximation to calculate the gravitational wave signal from white dwarf binaries
---
\[firstpage\]
stars: white dwarfs – gravitational waves
Introduction
============
At low frequencies (mHz), millions of double white dwarf binaries in the Galaxy are expected to dominate the gravitational wave (GW) sky. At the lowest frequencies they form an unresolved foreground, while at frequencies above several mHz, thousands of sources would be individually detectable for an instrument such as the Laser Interferometer Space Antenna, LISA (@e1, @l1, @h1, @nyp01 [@nyp03]) or eLISA/NGO (@eLISA). These binaries come in two flavours: detached systems and semi-detached (mass-transferring) systems that are know as AM CVn systems (see @solheim2010, @marsh2011 for reviews). Several known binaries should be detected by LISA within the first weeks of operation and are known as verification binaries (@sv2006, @rgb+07 [@roelofs2010], @brown2011). By measuring their gravitational wave amplitude and frequency (evolution), the type of astrophysical source and its parameters can be determined (e.g. @cutler1994, @littenberg2011, @blaut2011). In all these calculations the gravitational wave signal was determined under the assumption that the binary white dwarf system can be represented by two point masses in orbit, even for the tidally deformed stars in semi-detached binaries. The goal of this study is to determine the accuracy of this assumption. In section \[section gravwaves\] we will discuss our method of calculating the GW signal. In section \[section baloons\] we will give an algebraic view on the assumption of using point masses and in section \[section SPH\] we will calculate the GW signal from smoothed-particle hydrodynamics (SPH) simulations of AM CVn stars. In section \[section conclusions\] we discuss the conclusions of this study.
Gravitational waves from arbitrary sources in circular motion {#section gravwaves}
=============================================================
We calculate the GW wave signal from a collection of point particles with arbitrary coordinates and masses, all rotating about a fixed point with the same angular speed $\omega$. Since the stars do not move at highly relativistic speeds we use linearised general relativity. In linearised general relativity the trace reversed metric for any non-relativistic, far away source is given by [e.g. @rindler] $$\overline{h}_{ij} = \frac{-2G}{c^4R}\int \frac{d^2}{dt^2}\rho \bmath{x_ix_j}dV = \frac{-2G}{c^4R}\sum_\alpha \frac{d^2}{dt^2}m_\alpha \bmath{x_{i\alpha} x_{j\alpha}},$$ where R is the distance from observer to the source taken as an average over the distance to all source points. $m_\alpha$ and $\bmath{x_{i\alpha}}$ are the mass and coordinates of points in the source. $\sum_\alpha m_\alpha \bmath{x_{i\alpha} x_{j\alpha}}$ is the quadrupole moment of the source. In the transverse-traceless (TT) gauge for a wave travelling in the z-direction, the wave has only two degrees of freedom left. They manifest themselves as so called $+$ and $\times$ polarisations that can be measured by a detector. The wave metric then becomes [@rindler] $$h_{ij}=\
\left(
\begin{array}{ccc}
h_{+} & h_{\times} & 0\\
h_{\times} & -h_{+} & 0\\
0 & 0 & 0
\end{array}
\right).$$ $h_+$ and $h_\times$ can be obtained by [@p1] $$h_+ = \frac{1}{2}(\overline{h}_{xx}-\overline{h}_{yy})\\
h_\times = \overline{h}_{xy}.$$ If we now take $N$ particles and assign them masses $m_\alpha$ and positions in polar coordinated $r_\alpha$ and $\phi_\alpha = \omega
t+\theta_\alpha$, we can derive a general expression for $h_+$ and $h_\times$. Also taking into account the inclination angle $i$ for the relative orientation of the source, we get: $$\label{h+}
h_+ = \frac{-4\omega^2G}{c^4 R}(1+\cos^2 i) \left(S_1\cos2\omega t -S_2\sin2\omega t \right)$$ $$\label{hx}
h_\times = \frac{-4\omega^2G}{c^4 R} \cos i \left(S_2\cos2\omega t -S_1\sin2\omega t \right),$$ where $$\label{S1}
S_1 \equiv \sum_\alpha m_\alpha r^2_\alpha \cos2\theta_\alpha$$ $$\label{S2}
S_2 \equiv \sum_\alpha m_\alpha r^2_\alpha \sin2\theta_\alpha.$$ These expressions depend only on the given initial coordinates and can easily be calculated numerically. Because $h_+$ and $h_\times$ have a cosine and a sine term with different amplitudes it is useful to define the average strain amplitude as: $$\label{strainamp}
h \equiv \sqrt{\frac{1}{2}(h_{+max}^2+h_{\times max}^2)}.$$ As we will later determine the accuracy of the point mass approximation, it is worth estimating the error we make by using the linearised theory for these systems. The first Post-Newtonian correction is proportional to $(v/c)^2$ (e.g. @1973grav.book.....M,@blanchet1995) which for double white dwarf systems with orbital period of several minutes is of order $10^{-5}$.
Parallel axis theorem for gravitational waves {#section baloons}
=============================================
Using equations (\[h+\]) and (\[hx\]) we look at what we can say about the two point mass approximation algebraically. The parts of the equations that depend on the configuration of the system are $S_1$ and $S_2$. For these expressions something very similar to the parallel axis theorem for moment of inertia can be expressed. Consider a body in the $xy$-plane with its centre of mass at the origin. Its mass distribution can be described by point masses with coordinates $x_\alpha$,$y_\alpha$ and mass $m_\alpha$ with respect to its own centre of mass. So $S_1$ and $S_2$ are obtained from Eqs. \[S1\],\[S2\]. We call these $S_{\rm 1, spinning}$ and $S_{\rm 2, spinning}$, because these are the $S$’s that would arise if the body was spinning around its centre of mass. We now change the coordinates of our origin to another point in the plane such that all point masses get new coordinates $x'_\alpha = x_\alpha - x_{\rm CM}$ and $y'_\alpha = y_\alpha - y_{\rm CM}$. All coordinates denoted with subscript CM refer to the location of the body’s centre of mass in the new frame and we assume the body is in circular motion around the new origin.
To do the translation we first transform to Cartesian coordinates, translate and then transform back to polar coordinates. For the transform we use $$\frac{y_\alpha}{x_\alpha} = \left\{ \begin{array}{l l}
\tan(\theta_\alpha) & \text{for} \quad x_\alpha\geq0\\
\tan(\theta_\alpha+\pi) &\text{for} \quad x_\alpha<0\\
\end{array} \right \}.\\$$ The mirror symmetry, $\cos(2\theta) = \cos(2(\theta+\pi))$ and $\sin(2\theta) = \sin(2(\theta+\pi))$ , relieves us from having to split the sum over particles into separate sums for positive and negative $x_\alpha$, allowing the new $S$ to be written as the old $S$ plus an additional term. The resulting $S$’s that determine the GW radiation are: $$\label{S_1 translated}
S_1 = S_{\rm 1, spinning}+\sum_\alpha m_\alpha r^2_{\rm CM}\cos2\theta_{\rm CM}$$ $$\label{S_2 translated}
S_2 = S_{\rm 2, spinning}+\sum_\alpha m_\alpha r^2_{\rm CM}\sin2\theta_{\rm CM},$$ i.e. a first term denoting the original $S_{\rm 1, spinning}$ and $S_{\rm 2, spinning}$ and a second displacement term that sums all mass in one point, so the change of coordinates has contributed only the effect of a point mass at the body’s centre of mass.
So far we have considered the case in which there is one body rotating around an arbitrary point. In the case of a binary system, the procedure can be repeated for the second star and the $x_{\rm CM}, y_{\rm CM}$ for each star now refer to the distance of the centres of the two stars to the system barycentre.
We thus can conclude that, when determining $h$, every body with $S_{\rm 1, spinning} = S_{\rm 2, spinning} = 0$ can be considered as a point mass without implications to the result. These are bodies that do not radiate GW if they were only spinning around an axis through their own CM. In other words: bodies without a quadrupole moment, like axisymmetric spheres or disks. This result even holds when the body overlaps with the point it is rotating about, if that is physically possible. So only asymmetries in the stars that form a binary GW source may lead to deviations from the result as calculated using a source consisting only of two point masses.
Numerical calculation of gravitational waves from SPH simulations {#section SPH}
=================================================================
For semi-detached binaries, we know they do not consist of spherical stars. The contribution of the accretion disk will depend on if it is circular or not and on how the mass is distributed over the disk. The contribution of the donor will also depend on the mass distribution over its shape. We therefore need to look at the GW signals of more realistic mass distributions.
The simulations
---------------
Roche lobe filling stars and accretion disks are not completely symmetric around their centres of mass. Using equations (\[h+\]) and (\[hx\]) we can calculate the contribution this has to the gravitational wave signal if we know their mass distributions. For this study we use a set of SPH simulations of double white dwarfs at the onset of mass transfer [taken from @Dan2011; @Dan2012], hereafter referred to as RocheSPH. We sample the different mass combinations as shown in Table \[SPHtable\]. An example of one star of an SPH simulation is shown in Fig. \[rocheplot\] in such a way that the non-axisymmetric SPH particles are shown more prominently. In addition we have used an SPH simulation of an accretion disc in an ultra-compact binary, kindly provided to us by Prof. Matt Wood (based on @Wood2009). The latter is a $M_{\rm
donor}/M_{\rm accretor} =q=1/10$ system with an accretion disk around the accretor formed by adding mass at the inner Lagrange point of the Roche potential with the two stars treated as point masses (Fig. \[diskplot\]), from here onwards referred to as the DiskSPH. We assume the mass in the disk is $10^{-5}$ of the mass of the accretor.
All simulation results are snapshots at some point in the evolution of the binary. Because of this, we can not take all movement of the system into account, so we assumed that these mass distributions rotate with fixed angular speeds around their centres of mass. Now equations \[h+\], \[hx\] and \[strainamp\] can be used to calculate the GW strain amplitude.
The results
-----------
The differences of GW strain amplitude for the DiskSPH and all RocheSPH systems compared to two point systems are listed in Table \[SPHtable\]. The DiskSPH results shows a deviation at the $10^{-5}$ level, while the RocheSPH results, depending on the mass of the deformed donor star and the mass ratio $q$, range between 0.2 and 1.3 per cent. The results of the RocheSPH calculations as function of mass ratio $q$ and donor mass are shown graphically in Fig. \[rochevary\]. The deviations are largest for equal mass systems, but more interesting, our results clearly show the fact that lower donor masses are more deformed than more massive donors at the same mass ratio.
---------------- ----------------- -------------------- ------- ----------------------------------
SPH simulation $M_{\rm donor}$ $M_{\rm accretor}$ $q$ $\frac{h_{\rm SPH}}{h_{\rm PM}}$
$(M_\odot)$ $(M_\odot)$
DiskSPH 0.1 1.000010
RocheSPH 0.2 0.2 1.0 1.013418
RocheSPH 0.2 0.25 0.8 1.008075
RocheSPH 0.2 0.3 0.667 1.006612
RocheSPH 0.2 0.4 0.5 1.004254
RocheSPH 0.2 0.6 0.33 1.003183
RocheSPH 0.2 1.2 0.167 1.002072
RocheSPH 0.6 0.6 1.0 1.010717
RocheSPH 0.6 0.8 0.75 1.004861
RocheSPH 0.6 0.9 0.667 1.004226
RocheSPH 0.6 1.2 0.5 1.003254
RocheSPH 0.8 0.8 1.0 1.009355
RocheSPH 0.8 1.0 0.8 1.004264
RocheSPH 0.8 1.2 0.667 1.003461
---------------- ----------------- -------------------- ------- ----------------------------------
: \[SPHtable\]Ratio of GW strain amplitude of SPH systems and two point mass systems. The mass of the accretion disk in DiskSPH was taken to be $10^{-5}M_{\rm accretor}$.
To explore the influence of an accretion disk in somewhat more detail, we varied the mass of the accretion disk compared to the mass in the two stars (Fig. \[diskvary\]). As can be expected, the deviation from the point mass approximation scales linearly with the disk mass, which has to be unrealistically high (1 per cent of the total mass) in order to get near the effect of the deformed donor star discussed above. The deviation for a system with Roche-lobe filling donor plus an accretion disk is approximately the sum of the RocheSPH and DiskSPH result (see above) and thus for realistic disk masses is dominated by the deformation of the donor.
Conclusions {#section conclusions}
===========
We have calculated the deviation of the gravitational wave signal of finite size and non-spherical white dwarfs to that of point masses, which is usually assumed. For any finite size, non-relativistic axisymmetric body in circular orbit, the result is exactly the same. For semi-detached white dwarf binaries, we find that an accretion disk of reasonable mass changes the gravitational wave signal at the level of $10^{-4}$ to $10^{-3}$, small but still significantly larger than errors due to the neglect of post-Newtonian corrections in the calculation of the signal. Deformations due to filling the Roche lobe of semi-detached binaries increase for mass ratios closer to unity for fixed donor mass and are distinctly stronger at fixed mass ratio for lower mass donors and can in the most extreme cases be of order 1 per cent. This in principle will change the frequency evolution and the accuracy with which the parameters can be determined (e.g. @blanchet1995, @cutler1994), so calculations in which accuracies of better than one per cent are needed should take the finite size into account. Also, the different strength of the gravitational wave angular momentum losses will affect the mass transfer rate and stability of the mass transfer in semi-detached systems (e.g. @marsh04). However, the level of deviation we found shows that the calculations presented in the literature on the expected signals of (verification) binaries for LISA, are essentially unaffected: for monochromatic sources, the amplitude of the signal will be slightly different, but indistinguishable from sources with slightly higher (chirp) masses and/or smaller distances, properties that are much more uncertain than one per cent. For systems with measurable period derivatives, the degeneracy between (chirp) mass and distance is broken, but the deviations we found can still only be detected if there is independent extremely accurate measurement of the (chirp) mass of the binary. For a LISA-like detector such independent mass estimates, if available, are typically accurate at the 10 per cent level at best (e.g. @littenberg2011). The detailed evolution (and possible merger of the system) will be affected as well, but the calculations of this phase (e.g. @Dan2012) already take the finite size of the stars into account. We therefore conclude that in the majority of cases, the use of the point mass approximation is well justified.
Acknowledgments {#acknowledgments .unnumbered}
===============
We thank Matt Wood for sharing the results of his SPH simulations with us and the anonymous referee for comments that greatly improved the paper.
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\[lastpage\]
[^1]: E-mail: [email protected]; [email protected]
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'This note points out some bounds for the number of negative eigenvalues of Schrödinger operators with Hardy-type potentials, which follow from a simple coordinate transformation, and could prove useful in a spectral analysis of certain supersymmetric quantum mechanical models.'
author:
- 'Douglas Lundholm[^1]'
title: 'Some spectral bounds for Schrödinger operators with Hardy-type potentials'
---
Introduction
============
In a recent approach [@weighted-toy] to study the spectrum of a class of quantum mechanical models, called supersymmetric matrix models and described by matrix-valued Schrödinger operators (see e.g. [@Taylor; @octonionic]), it is relevant to consider the negative spectrum of Schrödinger operators with critical Hardy terms, i.e. operators of the form $$\label{Hardy-operator}
H = -\Delta_{\mathbb{R}^d} - \frac{(d-2)^2}{4|x|^2} + V(x),$$ where $V$ is a real- or operator-valued potential. This approach has so far only been applied to a simplified model, where a bound for the number of negative eigenvalues of a one-dimensional Schrödinger operator with Hardy term, following from a simple coordinate transformation, turned out to be very important. The aim of this note is to extend this transformation to higher dimensions and derive corresponding bounds which could be useful in an extension of the technique to the higher-dimensional matrix models. It also allows for generalizations of some statements in [@Birman-Laptev; @Weidl; @Ekholm-Frank; @Ekholm-Frank-halfline] regarding the one-dimensional, and higher-dimensional, operators. After searching the literature, we found that the transformation we use and some of its consequences have been considered before (see e.g. [@Seto; @Egorov-Kondratev] for the one-dimensional case, and [@Chadan_et_al] for higher dimensions), however, we are not aware of any reference stating these explicit bounds. In Section 2 we recall some Hardy-type inequalities, while the essential coordinate transformation is considered in Section 3, and the bounds for the negative eigenvalues are stated and proved in Section 4.
Some Hardy-type inequalities
============================
In the following we will denote by $\bar{B}_r(x)$ the closed ball of radius $r \ge 0$ at $x \in \mathbb{R}^d$. We also use the conventions $\mathbb{R}_+ := (0,\infty)$ and $\mathbb{N} := \{0,1,2,\ldots\}$.
For $x \in \mathbb{R}^d \smallsetminus \{0\}$, $d=1,2,3,\ldots$, let $$\begin{aligned}
\Psi_d(x) &:=& |x|^{-(d-2)},\quad \textrm{and} \\
\Phi(x) &:=& \ln |x|.\end{aligned}$$ These are the fundamental solutions of the Laplace operator on $\mathbb{R}^{d \neq 2}$ and $\mathbb{R}^2$, respectively, since in the sense of distributions $$\begin{aligned}
-\Delta_{\mathbb{R}^d}\Psi_d(x) &=& c_d \delta(x),\quad \textrm{and} \\
-\Delta_{\mathbb{R}^2}\Phi(x) &=& c_2 \delta(x),\end{aligned}$$ for some constants $c_d$ and $c_2$. By considering the square root of these functions, we can prove the following Hardy-type inequalities.
\[prop\_Hardy\_type\_ineq\] We have $$\label{normal_Hardy}
-\Delta_{\mathbb{R}^d} - \frac{(d-2)^2}{4|x|^2} \quad \ge \quad 0,$$ considered as a quadratic form on $C_0^{\infty}(\mathbb{R}^d \smallsetminus \{0\})$, and $$\label{extended_Hardy}
-\Delta_{\mathbb{R}^d} - \frac{(d-2)^2}{4|x|^2} - \frac{1}{4|x|^2(\ln |x|)^2} \quad \ge \quad 0,$$ considered as a quadratic form on $C_0^{\infty}(\mathbb{R}^d \smallsetminus \bar{B}_{1}(0))$. In other words, $$\label{normal_Hardy_ineq}
\frac{(d-2)^2}{4} \int_{\mathbb{R}^d} \frac{|u|^2}{|x|^2} dx \quad \le \quad
\int_{\mathbb{R}^d} |\nabla u|^2 dx,$$ (the standard Hardy inequality in $L^2(\mathbb{R}^d)$) for all $u \in C_0^{\infty}(\mathbb{R}^d \smallsetminus \{0\})$, and $$\label{extended_Hardy_ineq}
\frac{(d-2)^2}{4} \int_{\mathbb{R}^d} \frac{|u|^2}{|x|^2} dx
+ \frac{1}{4} \int_{\mathbb{R}^d} \frac{|u|^2}{|x|^2(\ln |x|)^2} dx \quad \le \quad
\int_{\mathbb{R}^d} |\nabla u|^2 dx,$$ for all $u \in C_0^{\infty}(\mathbb{R}^d \smallsetminus \bar{B}_{1}(0))$.
Let us consider the first inequaliy . It is straightforward to check that $$\nabla \ln \Psi_d(x)^{\frac{1}{2}}
= \frac{1}{2} \Psi_d(x)^{-1} (\nabla \Psi_d(x))
= -\frac{d-2}{2} \frac{x}{|x|^2},$$ and $$\Delta \ln \Psi_d(x)^{\frac{1}{2}}
= -\frac{(d-2)^2}{2|x|^2}.$$ Now, define the vector-valued operator $$Q := \nabla + \dot{\nabla} \ln \Psi_d(\dot{x})^{\frac{1}{2}}
= \nabla + \frac{1}{2} \Psi_d(x)^{-1} \dot{\nabla} \Psi_d(\dot{x}).$$ Then, considered in the sense of quadratic forms on $C_0^{\infty}(\mathbb{R}^d \smallsetminus \{0\})$, we have $$\begin{aligned}
0 &\le& Q \cdot Q^\dagger
= (\nabla + \dot{\nabla} \ln \Psi_d(\dot{x})^{\frac{1}{2}}) \cdot (-\nabla + \dot{\nabla} \ln \Psi_d(\dot{x})^{\frac{1}{2}}) \\
&=& -\nabla \cdot \nabla + \nabla \cdot \dot{\nabla} \ln \Psi_d(\dot{x})^{\frac{1}{2}}
- \dot{\nabla} \ln \Psi_d(\dot{x})^{\frac{1}{2}} \cdot \nabla + |\dot{\nabla} \ln \Psi_d(\dot{x})^{\frac{1}{2}}|^2 \\
&=& -\Delta + \dot{\Delta} \ln \Psi_d(\dot{x})^{\frac{1}{2}} + |\dot{\nabla} \ln \Psi_d(\dot{x})^{\frac{1}{2}}|^2 \\
&=& -\Delta - \frac{(d-2)^2}{2|x|^2} + \frac{(d-2)^2}{4|x|^2},
\end{aligned}$$ which gives .
For the second inequality , we observe that $$\nabla \ln \Phi(x)^{\frac{1}{2}}
= \frac{1}{2} \Phi(x)^{-1} (\nabla \Phi(x))
= \frac{1}{2(\ln |x|)} \frac{x}{|x|^2},$$ and $$\Delta \ln \Phi(x)^{\frac{1}{2}}
= \nabla \cdot \frac{1}{2(\ln |x|)} \frac{x}{|x|^2}
= -\frac{1}{2|x|^2(\ln |x|)^2} + \frac{d-2}{2|x|^2 \ln |x|}.$$ Hence, defining $$\tilde{Q} := \nabla + \dot{\nabla} \ln (\Psi_d(\dot{x}) \Phi(\dot{x}))^{\frac{1}{2}}
= \nabla + \frac{1}{2} \Psi_d(x)^{-1} \dot{\nabla} \Psi_d(\dot{x}) + \frac{1}{2} \Phi(x)^{-1} \dot{\nabla} \Phi(\dot{x}),$$ we obtain, in the sense of quadratic forms on $C_0^{\infty}(\mathbb{R}^d \smallsetminus \bar{B}_1(0))$, $$\begin{aligned}
0 &\le& \tilde{Q} \cdot \tilde{Q}^\dagger \\
&=& (\nabla + \dot{\nabla} \ln \Psi_d(\dot{x})^{\frac{1}{2}} + \dot{\nabla} \ln \Phi(\dot{x})^{\frac{1}{2}})
\cdot (-\nabla + \dot{\nabla} \ln \Psi_d(\dot{x})^{\frac{1}{2}} + \dot{\nabla} \ln \Phi(\dot{x})^{\frac{1}{2}}) \\
&=& -\Delta
+ \dot{\Delta} \ln \Psi_d(\dot{x})^{\frac{1}{2}} + \dot{\Delta} \ln \Phi(\dot{x})^{\frac{1}{2}}
+ 2 \dot{\nabla} \ln \Psi_d(\dot{x})^{\frac{1}{2}} \cdot \dot{\nabla} \ln \Phi(\dot{x})^{\frac{1}{2}} \\
&& +\ |\dot{\nabla} \ln \Psi_d(\dot{x})^{\frac{1}{2}}|^2 + |\dot{\nabla} \ln \Phi(\dot{x})^{\frac{1}{2}}|^2 \\
&=& -\Delta - \frac{(d-2)^2}{2|x|^2} - \frac{1}{2|x|^2(\ln |x|)^2} + \frac{d-2}{2|x|^2 \ln |x|} - 2 \frac{d-2}{2|x|} \frac{1}{2|x| \ln |x|} \\
&& +\ \frac{(d-2)^2}{4|x|^2} + \frac{1}{4(\ln |x|)^2|x|^2},
\end{aligned}$$ which proves .
Note that if $\tilde{Q}$ is considered as taking values in the grade-one part of $\mathcal{G}(\mathbb{R}^d)$, the Clifford algebra over $\mathbb{R}^d$, (hence a Dirac-type operator) then also the Clifford product $\tilde{Q} \tilde{Q}^\dagger = \tilde{Q} \cdot \tilde{Q}^\dagger + \tilde{Q} \wedge \tilde{Q}^\dagger$ (decomposed in terms of inner and outer products) is a non-negative operator on e.g. $C_0^\infty(\mathbb{R}^d \smallsetminus \bar{B}_1(0)) \otimes \mathcal{S}$, where $\mathcal{S}$ denotes a representation of $\mathcal{G}(\mathbb{R}^d)$.
Transformation of quadratic forms
=================================
Combining the so-called ground state representation of the operator (which is implicitly used in Proposition \[prop\_Hardy\_type\_ineq\]), with a coordinate transformation, we can relate a Schrödinger operator with a Hardy term defined on a domain in $\mathbb{R}^d$ to a corresponding operator without the term on a transformed domain. More precisely, denoting $B_R^c := \mathbb{R}^d \smallsetminus \bar{B}_R(0)$, for $R \ge 0$, and for $R < 0$ generalizing this to denote the cone parametrized by $(r,\omega) \in (R,\infty) \times S^{d-1}$ (for $d=1$ we write $B_R^c := (R,\infty)$), we have the following simple result.
\[lem\_transf\_Schroedinger\] For any $u \in C_0^\infty(B_R^c)$, $R \ge 0$ for $d$ odd, $R \ge 1$ for $d$ even, we have $$\begin{aligned}
\lefteqn{
\left\langle u, \left( -\Delta_{\mathbb{R}^d} - \frac{(d-2)^2}{4|x|^2} + V(x) \right) u \right\rangle_{L^2(B_R^c)} } \\
&=& \left\langle \psi, \left( -\Delta_{\mathbb{R}^d} - \left(1 - \frac{1}{|x|^2}\right)\Delta_{S^{d-1}} - \frac{(d-1)(d-3)}{4|x|^2} + e^{2r}V(e^{r}\omega) \right) \psi \right\rangle_{L^2(B_{\ln R}^c)}
\end{aligned}$$ where $\psi(r\omega) := r^{-\frac{d-1}{2}} e^{\frac{d-2}{2} r} u(e^r \omega)$, and $(r,\omega) \in (\ln R, \infty) \times S^{d-1}$.
In spherical coordinates, the l.h.s., denote it $I$, is $$\int_R^\infty \int_{\omega \in S^{d-1}} \overline{u(r,\omega)} \left(
-\frac{1}{r^{d-1}} \partial_r r^{d-1} \partial_r - \frac{1}{r^2}\Delta_{\omega} - \frac{(d-2)^2}{4r^2} + V(r\omega)
\right) u(r,\omega) r^{d-1} dr d\omega.$$ First, put $u(x) =: \Psi_d(x)^{\frac{1}{2}} v(x)$, i.e $v(r,\omega) := r^{\frac{d-2}{2}}u(r,\omega)$, and arrive via partial integration at the ground state representation, $$\begin{aligned}
I &=& \int_R^\infty \int_{\omega \in S^{d-1}} \left(
|\partial_r u(r,\omega)|^2 - \frac{(d-2)^2}{4r^2} |u|^2 + \overline{u} \left(-\frac{1}{r^2}\Delta_{\omega} + V(r\omega)\right) u
\right) r^{d-1} dr d\omega \\
&=& \int_R^\infty \int_{S^{d-1}} \left(
|\partial_r v(r,\omega)|^2 + \overline{v} \left(-\frac{1}{r^2}\Delta_{\omega} + V(r\omega)\right) v
\right) r dr d\omega.
\end{aligned}$$ Because of the form of the integral measure here, this expression actually possesses two-dimensional features, which explains why the function $\Phi$ enters in the proof of Proposition \[prop\_Hardy\_type\_ineq\]. Next, change variables, $r =: e^s$, $dr = rds$, $w(s,\omega) := v(e^s,\omega)$, which in a sense lowers the dimension by one: $$\begin{aligned}
I &=& \int_{\ln R}^\infty \int_{S^{d-1}} \left(
|\partial_s w(s,\omega)|^2 + \overline{w} \left( -\Delta_{\omega} + e^{2s}V(e^s\omega) \right) w
\right) ds d\omega \nonumber \\
&=& \left\langle w, \left(
-\partial_s^2 - \Delta_{S^{d-1}} + e^{2s}V(e^s\omega)
\right) w\right\rangle_{L^2((\ln R,\infty)) \otimes L^2(S^{d-1})}. \label{one-dim_groundstate_rep}
\end{aligned}$$ Finally, transform back from this corresponding ground state representation, by taking $\psi(s,\omega) := s^{-\frac{d-1}{2}} w(s,\omega)$, resulting in $$\begin{aligned}
I &=& \int_{\ln R}^\infty \int_{S^{d-1}} \overline{\psi} \left(
-\frac{1}{s^{d-1}} \partial_s s^{d-1} \partial_s - \frac{(d-1)(d-3)}{4s^2} - \Delta_{\omega} + e^{2s}V(e^s\omega)
\right) \psi \ s^{d-1} ds d\omega \\
&=& \left\langle \psi, \left(
-\Delta_{\mathbb{R}^d} + \left( \frac{1}{s^2} - 1 \right) \Delta_{\omega} - \frac{(d-1)(d-3)}{4s^2} + e^{2s}V(e^s\omega)
\right) \psi \right\rangle_{L^2(B_{\ln R}^c)},
\end{aligned}$$ which is the r.h.s. of the claimed identity.
In particular, we have the following special cases and consequences.
\[prop\_transf\_1d\] Consider $d=1$. For all $u \in C_0^\infty(\mathbb{R}_+)$, we have $$\left\langle u, \left( -\frac{d^2}{dx^2} - \frac{1}{4x^2} + V(x) \right) u \right\rangle_{L^2(\mathbb{R}_+)}
= \left\langle \psi, \left( -\frac{d^2}{dx^2} + e^{2x}V(e^{x}) \right) \psi \right\rangle_{L^2(\mathbb{R})}.$$ Furthermore, if $V(x) = -\frac{1}{4x^2(\ln x)^2} + W(x)$ then we have for all $u \in C_0^\infty((1,\infty))$ $$\begin{aligned}
\lefteqn{
\left\langle u, \left( -\frac{d^2}{dx^2} - \frac{1}{4x^2} - \frac{1}{4x^2(\ln x)^2} + W(x) \right) u \right\rangle_{L^2((1,\infty))} } \\
&& \quad = \left\langle \phi, \left( -\frac{d^2}{dx^2} + e^{2x}e^{2e^x}W(e^{e^x}) \right) \phi \right\rangle_{L^2(\mathbb{R})},
\end{aligned}$$ with $\phi(x) = e^{-x/2}\psi(e^x) = e^{-x/2}e^{-e^x/2}u(e^{e^x})$. This procedure can be iterated further to the interval $(e,\infty)$, and so on.
\[prop\_transf\_2d\] Consider $d=2$, with polar coordinates $(r,\varphi)$. For all $u \in C_0^\infty(B_1^c)$, we have $$\begin{aligned}
\lefteqn{
\left\langle u, \left( -\Delta_{\mathbb{R}^2} + V(x) \right) u \right\rangle_{L^2(B_1^c)} } \\
&& \quad = \left\langle \psi, \left( -\Delta_{\mathbb{R}^2} -(1 - r^{-2})\frac{d^2}{d\varphi^2} + \frac{1}{4r^2} + e^{2r}V(e^{r},\varphi) \right) \psi \right\rangle_{L^2(\mathbb{R}^2)},
\end{aligned}$$ so that, with $V(x) = -\frac{1}{4|x|^2(\ln |x|)^2} + W(x)$ we have for all $u \in C_0^\infty(B_1^c)$ $$\begin{aligned}
\lefteqn{
\left\langle u, \left( -\Delta_{\mathbb{R}^2} - \frac{1}{4r^2(\ln r)^2} + W(x) \right) u \right\rangle_{L^2(B_1^c)} } \\
&=& \left\langle \psi, \left( -\Delta_{\mathbb{R}^2} - (1 - r^{-2})\frac{d^2}{d\varphi^2} + e^{2r}W(e^r,\varphi) \right) \psi \right\rangle_{L^2(\mathbb{R}^2)}.
\end{aligned}$$
\[prop\_transf\_any-d\] For general $d=1,2,3,\ldots$, we have $$\begin{aligned}
\lefteqn{
\left\langle u, \left( -\Delta_{\mathbb{R}^d} - \frac{(d-2)^2}{4|x|^2} - \frac{1}{4|x|^2(\ln |x|)^2} + V(x) \right) u \right\rangle_{L^2(B_1^c)} } \\
&=& \left\langle \psi, \left( -\Delta_{\mathbb{R}^d} - \left(1 - \frac{1}{|x|^2}\right)\Delta_{S^{d-1}} - \frac{(d-2)^2}{4|x|^2} + e^{2r}V(e^{r}\omega) \right) \psi \right\rangle_{L^2(\mathbb{R}^d)}
\end{aligned}$$ for all $u \in C_0^\infty(B_1^c)$, where $\psi(r\omega) = r^{-\frac{d-1}{2}} e^{\frac{d-2}{2} r} u(e^r \omega)$.
This follows immediately from Lemma \[lem\_transf\_Schroedinger\] because $(d-1)(d-3) + 1 = (d-2)^2$.
The above transformations all extend to the case when $V$ is operator-valued (cp. [@Hundertmark]).
In the following, denote $$\ln^{(n)} x := \underbrace{\ln \circ \ln \circ \ldots \circ \ln}_{\textrm{$n$ factors}} (x)
\quad \textrm{and} \quad
\exp^{(n)} x := \underbrace{\exp \circ \exp \circ \ldots \circ \exp}_{\textrm{$n$ factors}} (x).$$ Then we also obtain by iteration of Lemma \[lem\_transf\_Schroedinger\] the following generalization of Proposition \[prop\_Hardy\_type\_ineq\]:
\[prop\_general\_Hardy\] For general $d=1,2,3,\ldots$, we have $$\begin{aligned}
-\Delta_{\mathbb{R}^d} - \frac{(d-2)^2}{4|x|^2} - \frac{1}{4|x|^2(\ln |x|)^2} - \ldots - \frac{1}{4|x|^2(\ln |x|)^2 \ldots (\ln^{(n)} |x|)^2} \quad \ge 0
\end{aligned}$$ in the sense of quadratic forms on $C_0^\infty \left( B_{\exp^{(n)} 0}^c \right)$.
Bounds for the number of negative eigenvalues
=============================================
Denote by $N(A)$ the rank of the spectral projection on $(-\infty,0)$ of a self-adjoint operator $A$, and by $V_{\pm}$ the positive/negative parts of a function $V$. In the one-dimensional case we have the following (cp. e.g. Proposition 3.2 in [@Ekholm-Frank] and Theorem 9 in Chapter 8 of [@Egorov-Kondratev]):
\[CLR-bound\_one-dim\] Let $n \in \mathbb{N}$, and $V$ be a real-valued potential such that\
$x^2(\ln x)^2 \ldots (\ln^{(n)} x)^2 V(x)$ is bounded from below. Then the self-adjoint operator $$H_{1,n} := -\frac{d^2}{dx^2} - \frac{1}{4x^2} - \ldots - \frac{1}{4x^2(\ln x)^2 \ldots (\ln^{(n)} x)^2} + V(x),$$ defined by Friedrichs extension on $C_0^\infty \left( (\exp^{(n)} 0, \infty) \right)$, has at least one negative eigenvalue for all negative (non-zero) potentials $V$. Furthermore, the number of negative eigenvalues is bounded by $$N(H_{1,n}) \le 1 + \int_{\exp^{(n)} 0}^\infty |V(x)_-| |x| |\ln x| \ldots |\ln^{(n+1)} x| \thinspace dx.$$ On the other hand, $H_{1,n}$ defined by Friedrichs extension on $C_0^\infty \left( (\exp^{(n)} 1, \infty) \right)$ satisfies the bound $$N(H_{1,n}) \le \int_{\exp^{(n)} 1}^\infty |V(x)_-| |x| |\ln x| \ldots |\ln^{(n+1)} x| \thinspace dx.$$
For a different version of the latter bound (for $n=0$) in the case of operator-valued potentials, and an application, see [@weighted-toy].
We will use that Bargmann’s bound in three dimensions, together with Dirichlet boundary conditions, implies (see e.g. [@Reed-Simon]) $$N\left( -\frac{d^2}{dx^2}\big|_\mathbb{R} + V(x) \right) \le 1 + \int_{-\infty}^\infty |V(x)_-| |x| \thinspace dx.$$ By Proposition \[prop\_transf\_1d\], we have for any $u \in C_0^\infty((\exp^{(n)} 0, \infty))$ $$\label{one-dim_quad_form_relation}
\langle u, H_{1,n} u \rangle_{L^2((\exp^{(n)} 0,\infty))}
= \left\langle \phi, \left(
-\frac{d^2}{dx^2}\big|_\mathbb{R} + e^{2x} \ldots e^{2\exp^{(n)}x} V(\exp^{(n+1)}x)
\right) \phi \right\rangle_{L^2(\mathbb{R})},$$ for some $\phi \in C_0^\infty(\mathbb{R})$. From this expression one immediately obtains the first statement of the theorem by relating to the case for a one-dimensional Schrödinger operator. Furthermore, linearly independent sets of such functions $u$ correspond to linearly independent sets of $\phi$. Hence, since $N(H_{1,n})$ is equal to the maximal dimension of a subspace of functions $u \in C_0^\infty((\exp^{(n)} 0, \infty))$ s.t. $\langle u,H_{1,n}u \rangle < 0$, and correspondingly for the operator on the r.h.s. of , we have $$\begin{aligned}
N(H_{1,n}) &=& N\left( -\frac{d^2}{dx^2}\big|_\mathbb{R} + e^{2x} \ldots e^{2\exp^{(n)} x} V(\exp^{(n+1)}x) \right) \\
&\le& 1 + \int_{-\infty}^{\infty} e^{2x} \ldots e^{2\exp^{(n)}x} |V(\exp^{(n+1)}x)_-| |x| \thinspace dx \\
&=& 1 + \int_{\exp^{(n)} 0}^\infty |V(y)_-| |y| |\ln y| \ldots |\ln^{(n+1)} y| \thinspace dy.
\end{aligned}$$ The second bound is proved analogously, using that $$N\left( -\frac{d^2}{dx^2}\big|_{\mathbb{R}_+} + V(x) \right) \le \int_{0}^\infty |V(x)_-| |x| \thinspace dx. \qedhere$$
For higher dimensions we have instead the following version of the above bounds:
\[CLR-bound\_higher-dim\] Let $n \in \mathbb{N}$, $V$ be real-valued and s.t. $|x|^2(\ln |x|)^2 \ldots (\ln^{(n)} |x|)^2 V(x)$ is bounded from below, and let $$H_{d,n} := -\Delta_{\mathbb{R}^d} - \frac{(d-2)^2}{4|x|^2} - \ldots - \frac{1}{4|x|^2(\ln |x|)^2 \ldots (\ln^{(n)} |x|)^2} + V(x)$$ be defined as a self-adjoint operator by Friedrichs extension on $C_0^\infty(B_{\exp^{(n+2)} 0}^c)$. For $d \ge 3$, and some universal positive constant $C_d$, we have the following bound for the number of negative eigenvalues: $$\begin{aligned}
N(H_{d,n}) &\le& C_d \int\limits_{|x| > \exp^{(n+2)} 0}
\left( \frac{(d-1)(d-3)}{4|x|^2(\ln |x|)^2 \ldots (\ln^{(n+1)} |x|)^2} - V(x) \right)_+^{\frac{d}{2}} \\
&& \qquad \qquad \cdot
(\ln |x|)^{d-1} \ldots (\ln^{(n+1)} |x|)^{d-1} \thinspace dx.
\end{aligned}$$ On the other hand, $H_{d,n}$ defined by Friedrichs extension on $C_0^\infty(B_{\exp^{(n+2)} 1}^c)$ satisfies the bound $$\begin{aligned}
N(H_{d,n}) &\le& C_d \int\limits_{|x| > \exp^{(n+2)} 1}
\left( \frac{ (d-1)(d-3) - (\ln^{(n+2)} |x|)^2 }{4|x|^2(\ln |x|)^2 \ldots (\ln^{(n+2)} |x|)^2} - V(x) \right)_+^{\frac{d}{2}} \\
&& \qquad \qquad \cdot
(\ln |x|)^{d-1} \ldots (\ln^{(n+2)} |x|)^{d-1} \thinspace dx.
\end{aligned}$$
These bounds also extend to operator-valued potentials according to [@Hundertmark], where $(\ )_+^{\frac{d}{2}}$ is replaced by $\operatorname{tr}\thinspace (\ )_+^{\frac{d}{2}}$, and $C_d$ is slightly larger. Also, by a monotonicity argument (see e.g. Remark 2.2 in [@Hundertmark]), they imply corresponding Lieb-Thirring inequalities for non-zero moments of the eigenvalues (cp. [@Ekholm-Frank]).
Note that there is always an extra contribution to the above bound for the number of negative eigenvalues of $H_{d,n}$ for all $d \ge 4$, but not so in the case of $d=1$ (Theorem \[CLR-bound\_one-dim\]) and $d=3$. This is quite interesting when related with the fact that supersymmetric matrix models, split into coordinates of $\mathbb{R}^d \times \mathbb{R}^2$ (cp. [@octonionic]), are conjectured to have zero energy states for $d=7$, but not for $d=0,1,3$.
Here we apply the Cwikel-Lieb-Rozenblum bound for $d \ge 3$ (see e.g. [@Reed-Simon; @Egorov-Kondratev]): $$N\left( -\Delta_{\mathbb{R}^d} + V(x) \right) \le C_d \int_{\mathbb{R}^d} |V(x)_-|^{\frac{d}{2}} dx.$$ By iterating the bound obtained from Lemma \[lem\_transf\_Schroedinger\], $$\begin{aligned}
\lefteqn{
\left\langle u, \left( -\Delta_{\mathbb{R}^d} - \frac{(d-2)^2}{4|x|^2} + V(x) \right) u \right\rangle_{L^2(B_e^c)} } \\
&\ge& \left\langle \psi, \left( -\Delta_{\mathbb{R}^d} - \frac{(d-1)(d-3)}{4|x|^2} + e^{2|x|}V(e^{|x|}\omega) \right) \psi \right\rangle_{L^2(B_1^c)},
\end{aligned}$$ with $\psi(r\omega) = r^{-\frac{d-1}{2}} e^{\frac{d-2}{2} r} u(e^r \omega)$, we have as in the one-dimensional case $$\begin{aligned}
N(H_{d,n}) &\le& N\left( -\Delta_{B_1^c} - \frac{(d-1)(d-3)}{4|x|^2} + e^{2|x|} \ldots e^{2 \exp^{(n)} |x|} V\left( (\exp^{(n+1)} |x|)\omega \right) \right) \\
&\le& C_d \int_{B_1^c}
\left( \frac{(d-1)(d-3)}{4|x|^2} - e^{2|x|} \ldots e^{2 \exp^{(n)} |x|} V\left( (\exp^{(n+1)} |x|)\omega \right) \right)_+^{\frac{d}{2}} dx \\
&=& C_d \int_{B_{\exp^{(n+2)} 0}^c}
\left( \frac{(d-1)(d-3)}{4|x|^2(\ln |x|)^2 \ldots (\ln^{(n+1)} |x|)^2} - V(x) \right)_+^{\frac{d}{2}} \\
&& \qquad \qquad \cdot (\ln |x|)^{d-1} \ldots (\ln^{(n+1)} |x|)^{d-1} \thinspace dx.
\end{aligned}$$ For the operator on the domain $B_{\exp^{n+2} 1}^c$, we can add and subtract a term $1/(4|x|^2)$ and iterate one step further to obtain $$\begin{aligned}
N(H_{d,n})
&\le& C_d \int_{B_1^c}
\left( \frac{(d-1)(d-3)}{4|x|^2} - \frac{1}{4} - e^{2|x|} \ldots e^{2 \exp^{(n+1)} |x|} V\left( (\exp^{(n+2)} |x|)\omega \right) \right)_+^{\frac{d}{2}} dx.
\end{aligned}$$ The stated bound then follows as above.
We expect that it is possible to find analogous bounds on the larger domains $B_{\exp^{(n)} 0}^c$ and $B_{\exp^{(n)} 1}^c$. Indeed, for central potentials we have the following:
If $V(x) = \tilde{V}(|x|)$ is a central potential s.t.\
$r^2(\ln r)^2 \ldots (\ln^{(n)} r)^2 \tilde{V}(r)$ is bounded from below, then for $H_{d,n}$ defined on the domain $B_{\exp^{(n)} 0}^c$ $$\begin{aligned}
N(H_{d,n}) &\le& \sum_{l=0}^{l_\textup{max}} D_{d,l} \left(
1 + \int_{\exp^{(n)} 0}^\infty \left( -\frac{l(l+d-2)}{r^2} - \tilde{V}(r) \right)_+ |r| |\ln r| \ldots |\ln^{(n+1)} r| \thinspace dr
\right),
\end{aligned}$$ while on $B_{\exp^{(n)} 1}^c$ $$\begin{aligned}
N(H_{d,n}) &\le& \sum_{l=0}^{l_\textup{max}} D_{d,l}
\int_{\exp^{(n)} 1}^\infty \left( -\frac{l(l+d-2)}{r^2} - \tilde{V}(r) \right)_+ r |\ln r| \ldots |\ln^{(n+1)} r| \thinspace dr,
\end{aligned}$$ where $$D_{d,l} := \frac{(2l+d-2)\Gamma(d+l-2)}{\Gamma(d-1)\Gamma(l+1)},$$ and $l_\textup{max}$ is the maximal integer $l \ge 0$ s.t. the negative part of $\frac{l(l+d-2)}{r^2} + \tilde{V}(r)$ is non-zero on the respective domain.
For central potentials, we can split the Hilbert space $\mathcal{H} = \bigoplus_{l=0}^{\infty} \mathcal{H}_l$ into eigenspaces of the the angular laplacian, where $
-\Delta_{S^{d-1}}|_{\mathcal{H}_l} = l(l+d-2)
$ with degeneracy $D_{d,l}$ (see e.g. [@Vilenkin]; cp. [@Seto]). Using and iterating, we have for $u = \tilde{u} \otimes \psi \in
C_0^\infty((\exp^{(n)} 0,\infty)) \otimes L^2(S^{d-1}) \cap \mathcal{H}_l$ $$\begin{aligned}
\lefteqn{ \langle u, H_{d,n} u \rangle_{L^2(\mathbb{R}^d)} }\\
&=& \left\langle w, \left(
-\partial_s^2 + e^{2s} \ldots e^{2\exp^{(n)} s} \left( \frac{l(l+d-2)}{(\exp^{(n+1)} s)^2} + \tilde{V}(\exp^{(n+1)} s) \right)
\right) w\right\rangle_{L^2(\mathbb{R})} \\
&& \quad \cdot \|\psi\|_{L^2(S^{d-1})}^2,
\end{aligned}$$ with $w \in C_0^\infty(\mathbb{R})$. Hence, by reasoning as in the proof of Theorem \[CLR-bound\_one-dim\], $$\begin{aligned}
N(H_{d,n}|_{\mathcal{H}_l})
&\le& D_{d,l}
\left( 1 + \int_{-\infty}^\infty e^{2s} \ldots e^{2 \exp^{(n)} s}
\left( -\frac{l(l+d-2)}{(\exp^{(n+1)} s)^2} - \tilde{V}(\exp^{(n+1)} s) \right)_+ |s| \thinspace ds
\right)
\end{aligned}$$ if $l \le l_\textup{max}$, and $N(H_{d,n}|_{\mathcal{H}_l}) = 0$ otherwise. The first statement of the theorem then follows by a change of variables, and similar reasoning gives the second statement.
### Acknowledgements {#acknowledgements .unnumbered}
I am most grateful to Oleg Safronov for many useful discussions and for pointing out to me the one-dimensional case (Proposition \[prop\_transf\_1d\] and a version of Theorem \[CLR-bound\_one-dim\]). I would also like to thank Jens Hoppe for bringing us together, as well as Ari Laptev for useful discussions. This work was supported by the Swedish Research Council, the Knut and Alice Wallenberg Foundation (grant KAW 2005.0098), and the European Science Foundation activity MISGAM.
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[^1]: e-mail: [email protected]
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'I present a simple model of the time dependence of the contact area between solid bodies, assuming either a totally uncorrelated surface topography, or a self affine surface roughness. Time dependent deformation due to “creep" processes is incorporated using a recently proposed model, and produces the time increase of the contact area $A(t)$. For an uncorrelated surface topography, $A(t)$ is numerically found to be well fitted by expressions of the form \[$A(\infty)-A(t)]\sim (t+t_0)^{-q}$, where the exponent $q$ depends on the normal load $F_N$ as $q\sim F_N^{\beta}$, with $\beta$ close to 0.5. In particular, when the contact area is much lower than the nominal area I obtain $A(t)/A(0) \sim 1+C\ln(t/t_0+1)$, i.e., a logarithmic time increase of the contact area, in accordance with experimental observations. The logarithmic increase for low loads is also obtained analytically in this case. For the more realistic case of self affine surfaces, the results are qualitatively similar.'
address: 'Centro Atómico Bariloche, Comisión Nacional de Energía Atómica, (8400) Bariloche, Argentina'
author:
- 'E. A. Jagla'
title: Towards a modeling of the time dependence of contact area between solid bodies
---
Introduction
============
The contact area between solid bodies is in general only a tiny fraction of the apparent (or nominal) macroscopic contact area, and is essentially proportional to the normal force between the bodies[@persson; @bt]. This fact is at the core of many important characteristics of friction phenomena. In particular, it gives a natural explanation of the independence of the friction force with the nominal contact area, and the linear increase of friction force with normal load. However, contact area, even under static external condition is a time dependent quantity, and this produces time dependent effects in friction. For instance, the logarithmic time increase of static friction coefficient with the contact time, that is well described in many materials[@mudet; @marone0], is mostly related to a corresponding logarithmic increase in time of the contact area[@caroli]. Direct evidence of this geometric aging effect was provided by Dieterich and Kilgore[@dk]. The time increase of contact area is also related with the phenomenon of velocity weakening during sliding, namely the fact that the friction force between sliding surfaces can decrease when the relative velocity is increased. The velocity weakening effect is extremely important in relation with the dynamics of the friction process. In fact, it can be shown that this effect is at the base of the process of earthquake generation during the relative motion of tectonic plates[@scholz], and also plays an important role in the stick-slip dynamics of many sliding systems[@persson]. These examples highlight the central role that the time increase of contact area has in the description of many frictional phenomena. An understanding of the time dependence of contact area is thus a very basic aim of any friction theory.
It is generally accepted that the physical mechanisms by which contact area increases in time are associated with plastic phenomena occurring in the materials. For our purposes, plastic processes can roughly be classified into two qualitatively different groups[@plasticity; @lakes]. Rapid plastic effects occur when the imposed deformation conditions produce the overpassing of the yield stress of the material. This behavior is typically referred to as plastic flow. For lower applied stresses, in cases in which the yield stress is not overpassed, there still exists the possibility of thermally activated reacommodations in the system that tend to reduce gradually its free energy. These processes are much slower than those of plastic flow, and are strongly temperature dependent. They are refereed to as creep phenomena. When two solid bodies are pressed together, the real contact occurs only in a tiny part of the surface, and the local stresses are correspondingly very high. This may produce local plastic flows that increase the contact area until the local stresses decay below the material yield stress. After this initial stage, the contact area typically increases slowly in time due to creep processes. The time increase of contact area in this stage is seen to be logarithmic in time.
Creep processes (that in other contexts are referred to as aging effects) produce in the contacting bodies a tendency to reach progressively more stable configurations. The meaning of “more stable" here, is that the atomic rearrangements implied by a creep process must produce on average a decrease in the total free energy of the system. This stabilizing tendency originates in the fact that energy barriers to jump onto a lower energy state are in general lower than those to go to higher energy states, so on average creep produces always an energy reduction in the system. Along these lines, the phenomenon of contact area increase has been qualitatively described in some idealized geometries[@brechet; @caroli], and the experimental logarithmic increase law has been justified for these cases. Crucial in this analysis is the assumption of a phenomenological creep law in which some generic strain rate $\dot s$ is exponentially related to an appropriate stress $\sigma$, i.e., $\dot s\sim \exp(\sigma/S)$, where $S$ is called the strain rate sensitivity and is temperature dependent [@caroli]. The modeling of the phenomenon in more realistic cases is prevented by the complication to describe creep in the materials in a sensible and analytically (or even numerically) tractable way. As far as I know there is no statistical model that, based on well defined microscopic evolution laws, is able to predict the logarithmic increase of contact area with time.
Recently, a way to incorporate creep effects in the dynamics of sliding friction has been proposed in [@uno; @dos]. In particular this has been applied to models (the Burridge-Knopoff[@bk] and Olami-Feder-Christensen[@ofc] models) used to describe seismic phenomena. This kind of models describe the friction phenomena occurring at the flat (on average) contact surface between two solid bodies that are sheared against each other. The kind of modeling proposed in [@uno; @dos] consists in defining some “plastic" degrees of freedom, and arguing that they evolve with a tendency to minimize the total energy of the system, which is a generic realization of creep processes as described in the previous paragraph. It was shown that the proposed relaxation mechanism is able to generate realistic sequences of earthquakes, a goal that had not been obtained previously without this kind of modification. Also, realistic frictional properties are well reproduced using this relaxation mechanism. In particular, a logarithmic increase of the static friction coefficient with contact time, and an approximately logarithmic decrease of the average friction force as a function of relative velocity has been obtained.
In view of the standard interpretation of macroscopic friction features in terms of contact area, and since the structural relaxation mechanism in [@uno; @dos] is consistent with macroscopic friction properties, the question arises if there is a way to use that relaxation mechanism to model the time increase of contact area of solid bodies under static contact. Such a modeling, if successful, would give further support to the structural relaxation mechanism, and would provide an appropriate framework to study the phenomenon of time increase of contact area in greater detail than the rather qualitative descriptions available up to now. I attempt this particular goal here. In order to do this I explain a variation of the models presented in [@uno; @dos] that allows to define the contact area, and at the same time incorporates the structural relaxation mechanism. I show that the main phenomenology associated to the contact area is re-obtained, particularly, its logarithmic time increase.
The Model
=========
Typically, the surfaces of solid bodies in contact are at the same time rough and elastic (or elasto-plastic), and this is important when studying sliding friction. In the context of static contact however, some simplifying assumption can be made. I will consider the case of an elastic surface that is perfectly flat down to atomic scale in the absence of external forces, and an opposing surface that is atomically rough, but strictly rigid. The two surfaces are oriented horizontally. The underlying rigid rough surface is described by a random variable defined over the plane $\xi({\bf r})$. In the numerical simulations the values of ${\bf r}$ will be restricted to lie on a two dimensional square mesh. Two cases will be considered separately: one in which the $\xi({\bf r})$ are drawn from a unique Gaussian distribution independently for each value of ${\bf r}$, and a second case in which the $\xi({\bf r})$ is spatially correlated in order to model a self-affine surface (see below).
A realistic modeling of the upper elastic surface should consist in principle in determining the equilibrium values of a three dimensional vector displacement field ${\bf u}$ depending upon the two horizontal coordinates, under the action of the surface forces, taking into account the elastic response of the surface, measured by an appropriate response function. This is what a full contact mechanics calculation aims to. In the present case, and already foreseeing the further inclusion of creep effect, some drastic simplifications have to be made.
The simplified description of the elastic surface (sketched in Fig. 1) consists of a collection of scalar coordinates $u({\bf r})$, representing the vertical positions of the elastic surface at point ${\bf r}$, which are coupled via elastic springs connecting nearest neighbor mesh points, and elastic interactions with the bulk of the material, defined by a set of coordinates $u^0({\bf r})$. Note that the only degrees of freedom I allow are vertical displacements, and along this direction all the springs act. This modeling of the elastic surface is highly simplistic, but at this point this is necessary in order to have a solvable model. Note in particular that the consideration of only vertical displacements means that we are dealing with a material with “zero Poisson ratio". Also, it can be seen that a localized force onto this surface generates a distortion that decays exponentially with distance, whereas the true response of a semi-infinite elastic body is known to decay as $\sim r^{-1}$. The limitations and some unrealistic features of the elastic model I am using are described in more detail in the last section of the paper.
The restriction imposed by the contact geometry is that $u({\bf r})\geq \xi({\bf r})$. Given a set of values $u^0({\bf r})$, the elastic energy of the system is $$E=\frac{k_0}{2}\sum_{({\bf r},{\bf r}')} [u({\bf r})-u({\bf r}')]^2 +\frac{k_1}{2}\sum_{{\bf r}}[u^0({\bf r})-u({\bf r})]^2
\label{e}$$ where $({\bf r},{\bf r}')$ stands for pairs of neighbor sites on the numerical mesh. In all cases periodic boundary conditions will be used. The equilibrium values of $u$’s are thus found by solving the set of equations obtained by minimizing Equation (\[e\]), namely $$k_0(\nabla^2 u)({\bf r})+k_1[u^0({\bf r})-u({\bf r})]=0
\label{us}$$ (where $\nabla^2$ is the discrete Laplacian operator on the square lattice, lattice parameter is taken as unit of length) with the constraint imposed by the contact condition $u({\bf r})\geq \xi({\bf r})$. The number of points for which $u({\bf r})= \xi({\bf r})$ is the number of contact points, and I will take this number as a measure of the contact area $A$ in the model. The global vertical position of the elastic surface can be measured by the mean value $\overline {u^0}$ of the $u^0$’s. Note also that $\overline {u^0}$ can be interpreted as a measure of the nominal distance between the two bodies and that the normal force $F_N$ is the sum of the forces of all vertical springs, i.e, $F_N=k_1\sum_{\bf r} (u({\bf r})-u^0({\bf r}))$.
Conceptually in the same manner as in previous work[@uno; @dos], plastic relaxation is incorporated through a time dependence of the values of $u^0({\bf r})$. The form of the evolution will be obtained from the prescription that the energy of the system $E$ tends to be reduced during relaxation. Concretely, I use $$\frac{du^0({\bf r})}{dt}=R\nabla^2\frac{\delta E}{\delta u^0({\bf r})}=k_1 R\nabla^2 (u^0({\bf r})-u({\bf r}))
\label{u0s}$$ i.e, a standard relaxation equation that tends to reduce the value of $E$ over time as much as possible. The time scale for relaxation (controlled by $R^{-1}$) will be assumed to be much larger than the elastic time scale in which the elastic variables $u$ accommodate to satisfy Eq. (\[us\]). Note that the meaning of Eq. (\[u0s\]) is that relaxation makes the force exerted by the vertical springs tend to an uniform value when $t\to\infty$.
![One dimensional sketch of the model. (a) A flat and elastic surface (defined by the black dots) on top of a rigid rough one (defined by the top of the vertical segments), before contact.(b) The situation after contact. Dots \[with vertical coordinates $u({\bf r})$\] are in equilibrium under the action of vertical and horizontal springs, and eventually the force exerted by the rigid rough surface defined by $\xi({\bf r})$. The coordinates $u^0({\bf r})$ evolve in time according to the relaxation equation (\[u0s\]). The external control parameter is the mean value $\overline{u^0}$. []{data-label="f1"}](contact_f1.eps){width=".5\textwidth"}
The variable that is assumed we can control is $\overline {u^0}$. In this respect, note that this variable is not modified by the time evolution \[Eq. (\[u0s\])\]. Actually, this is one of the reasons to use a conserving dynamics, in which the Laplacian operator is introduced in Eq. (\[u0s\]), instead of a non-conserving one, in which the Laplacian is absent[@chl] (in connection with this choice, see also the final Section of the paper). To describe a possible experimental situation, I first assume (Fig. \[f1\](a)) that $\overline {u^0}$ is very large, in such a way that there is no contact between the two surfaces at any point. Allowing infinite time to relax under this condition, the system reaches the uniform state in which $u^0({\bf r})=\overline{u^0}$ and $u({\bf r})=\overline{u^0}$ everywhere. This is in fact the most relaxed configuration since the elastic energy of every spring is zero. I then place $\overline {u^0}$ at some value in which contact occurs at some positions (Fig. \[f1\](b)), and solve Equations (\[us\]) and (\[u0s\]) as a function of time. Numerically, the procedure consists in advancing the solution of Eq. (\[u0s\]) for $u^0$ by one time step, solve Eq. (\[us\]) using a standard relaxation algorithm[@numrec] for the new values of $u$, and iterate the process.
Results for uncorrelated roughness
==================================
I first show results in the case in which the variables $\xi({\bf r})$ that describe the roughness of the underlying surface are taken independently at each site, from a Gaussian distribution of zero mean and unitary variance. Although this case is not very realistic, we will see that in addition to the possibility of accurate numerical simulation, it allows for very insightful analytical treatment. Results corresponding to $R=0$ are presented in Fig. 2, where the contact area $A$ (i.e., the number of points in contact) is plotted against $F_N$. Curves for different values of the ratio $k_1/k_0$ are shown. Two main regimes are observed. For low normal load the contact area is essentially proportional to the load, whereas if load is too high, we reach a regime of full contact. The crossover between partial contact and full contact occurs at a value of $F_N$ that depends on the elastic constants of the model. If one of the spring constant dominates over the other the crossover value of $F_N$ is proportional to the dominating spring constant. For instance, if $k_0$ is negligible compared to $k_1$, the elastic surface becomes a collection of independent springs (in other contexts this kind of description of an elastic surface is described as the Winkler model[@winkler]), and the crossover to full contact occurs for a normal load per spring of order $k_1\sigma$, where $\sigma$ is the typical roughness of the surface, that is taken as 1 in the present simulations.
Given the equilibrium configuration for some value of $F_N$, we can set a finite value of $R$ and follow the evolution of the contact area in time. In this process it has to be taken into account that if we keep $\overline {u^0}$ fixed, the value of $F_N$ will change in time. Since experiments are usually done at constant normal force instead of constant relative distance, I implemented a feedback loop in the simulation that allows to keep the value of $F_N$ as constant by changing $\overline{u^0}$.
The asymptotic ($t\to \infty$) value of the contact area is determined (according to Eq. \[u0s\]) by the condition that forces on all springs $k_1$ are equal[@notita]. In this situation the actual value of $k_1$ is irrelevant, and the fully relaxed contact area becomes a function of $F_N/k_0$ (Fig. 1, open symbols). Note that this asymptotic value of the contact area is not in general equal to the nominal area, i.e., the system does not reach full contact unless $F_N$ overpasses a crossover value that is proportional to $k_0$. In particular this means that if we consider the simplest case $k_0=0$ we obtain the unrealistic result that the contact area tends to the nominal value as $t\to\infty$. This is the reason that makes mandatory the use of a lateral spring $k_0$ in the model. In view of standard experimental conditions, below I will concentrate in cases in which $F_N$ is such that the contact area is only a small fraction of the nominal one. I found that the dependence of the asymptotic contact area with normal force at low loads follows a power law $A(t\to \infty)\sim F_N^p$, with an exponent $p\sim 0.8$.
![ Contact area as a function of normal load, for different values of $k_1/k_0$ without relaxation (full symbols), and the fully relaxed state (open symbols), in which all the forces on the $k_1$ springs are equal, in a system of 200 $\times$ 200 elements. Note that the relaxed state has always a contact area larger than unrelaxed states with the same normal force. For reference, a dotted line with a slope of 1 is also plotted.[]{data-label="f2"}](contact_f2.eps){width=".5\textwidth"}
The curves in Fig. \[f2\] correspond either to “instantaneous" or to “fully relaxed" values of the contact area. As a function of the contact time, the values of the contact area must evolve between these two limits. First of all, note from Fig. \[f2\] that the fully relaxed contact area is always larger than the unrelaxed configuration with the same normal force. This means that contact area will increase in time when relaxation is acting, which is the expected result. The full temporal evolution of the contact area obtained by solving Eqs. (\[us\]) and (\[u0s\]) is shown in Fig. 3 for different values of the normal force $F_N$. I have found that all curves of Fig. 3 are very well described by expressions of the form $$A(t)/A(0)=a-\frac{a-1}{(t/t_0+1)^q},
\label{a}$$ i.e., they indicate a saturation towards the asymptotic value with the form of a power law. The values of $a$, $q$, and $t_0$ depend on $F_N$. In particular, the dependence of the exponent $q$ on $F_N$ is seen in an inset in Fig. \[f3\]. It is remarkable that $q$ has a dependence on $F_N$ of the form $q\sim F_N^{\beta}$ with $\beta$ being close to $0.5$. This means in particular that for $F_N\to 0$, we can approximate Eq. (\[a\]) by $$A(t)/A(0)=1+C\ln(t/t_0+1) ~~~~~{\mbox {for}}~~ F_N\to 0
\label{log}$$ with $C=(a-1)q$. It is numerically found that the value of $C$ is not singular in this limit. Since experimentally the values of $F_N$ are typically tiny compared to those necessary to reach full contact, we can say that Eq. (\[log\]) shows that in general we must expect a logarithmic increase of the contact area in time.
![Time evolution of the contact area, for different values of $F_N$, and two values of the ratio $k_1/k_0$, for a system of 200$\times$200 sites. The corresponding fitting with the expression given in the text (Eq. \[a\]) is shown by thin continuous lines. The exponent $q$ in the fitting function is plotted as a function of $F_N$ in the inset, where it is seen that $q$ goes to zero with $F_N$ as a power law (dotted line in the inset has a slope of 0.5). []{data-label="f3"}](contact_f3.eps){width=".5\textwidth"}
Results for Self-Affine Surfaces
================================
The results of the previous section were obtained using uncorrelated asperities. Numerically, there is no additional complication in trying more realistic distribution of surface roughness (although this will preclude analytic treatments as that of the next Section). Real surfaces are in fact better described as self affine fractals[@fract], and are characterized in terms of its Hurst exponent $H$. This exponent measures the decaying in wave vector of the spectral distribution of surface roughness. In this section I present numerical results using a self affine surface, and show that the results are qualitatively similar to those of the previous section.
![(a) Topography of one realization of a self affine surface ($H=1/2$) in a a system of size 256 $\times$ 256 sites and $k_1/k_0=10$. Surface height goes form -0.3 to 0.3 from back to white. (b) Contact surface as a function of load, in the absence of relaxation. []{data-label="f51"}](contact_f5.eps){width=".5\textwidth"}
I construct the self affine rough surface using the successive random mean point algorithm of Voss [@voss; @robbins]. The surface is characterized by its Hurst exponent $H$ and its small scale rms roughness $\Delta$. The algorithm for the definition of the self-affine rough surface proceeds as follows[@robbins]: given a mesh of size $L\times L$ (for convenience $L$ is chosen to be a power of 2) the central point of the mesh is given a value of $\xi$ chosen at random from a Gaussian distribution with zero mean and width ${\ell}^H\Delta$, where ${\ell}=L/\sqrt{2}$ is the distance from the center to the corners. This center now becomes a corner of four new squares rotated by 45 $^\circ$ and with a new center-to-corner distance ${\ell}$ smaller by a factor $\sqrt{2}$. The value of $\xi$ at the center of each new square is obtained as the average of $\xi$ at the four corners plus a random value chosen from a Gaussian of width ${\ell}^H\Delta$. The process is iterated down to ${\ell}=1$. This algorithm produces a surface that is self-affine in the spatial scale between the mesh size and the full system size. A Hurst exponent $H=0.5$ will be used (tests using other values of $H$ show no qualitative differences). I also use the value $\Delta=0.01$. An example of the kind of surface obtained by this method is presented in Fig. \[f51\](a). Once the underlying rough surface $\xi({\bf r})$ is defined in this way, the contact with the elastic surface is numerically evaluated exactly by the same methods used in the previous Section. The actual contact area in the absence of relaxation and for different values of $F_N$ can be seen in \[f51\](b). This looks qualitatively similar to the results obtained using more realistic elastic surfaces, as that made in Ref. [@robbins]. In Fig. \[f4\] I present results for the dependence of the contact area on $F_N$ in the present model. In the absence of relaxation, the dependence can be fitted by an expression of the form $A \sim F_N^q$, with $q\simeq 0.8$, i.e, is a slightly sub-linear dependence. Other numerical analysis of this problem, like those in Ref. [@robbins] have obtained a linear dependence. I attribute this slight discrepancy to the somewhat artificial description of the elastic surface in the present approach, compared with the more precise, truly three dimensional description in Ref. [@robbins].
![Contact area for a self affine surface with Hurst exponent $H=1/2$, using $k_1/k_0=10$. Solid symbols: Contact area as a function of normal load without relaxation, for two system sizes. A power law with exponent 0.8 is plotted as a reference. Note the independence of the contact area on system size in the region away from full contact. Open symbols: corresponding fully relaxed states for the two system sizes analyzed.[]{data-label="f4"}](contact_f4.eps){width=".5\textwidth"}
![Time evolution of the contact area in the presence of relaxation for a self affine surface with $H=1/2$ (a) Un-normalized values for different normal forces and system sizes. (b) Results in units of the initial area for two different values of $k_1/k_0$. The results in (b) are well approximated by a logarithmic time increase with a slope that is independent of normal force, and roughly proportional to the value of $k_1/k_0$. []{data-label="f6"}](contact_f6.eps){width=".5\textwidth"}
Now I will consider the effect of relaxation. The value of the limit contact area, i.e, the contact area at infinite time as a function of the normal force can be seen also in Fig. \[f4\]. This curve, as in the uncorrelated case, was obtained in a simulation in which a constant force on each of the vertical springs is imposed. The results for the increase of contact area with time are presented in Fig. \[f6\] (note that the values of $F_N/k_0$ correspond, according to Fig. \[f4\] to cases in which the contact area is a small fraction of the nominal one). We see that qualitatively the behavior is very similar to the previous case. In particular, for the case of very light loads and except for very long times, a very good fitting is provided by an expression as that given by Eq. (\[log\]), with $C$ being independent of the external load.
![Time evolution of the contact surface for $F_N/k_0=10$, $k_1/k_0=10$. System size is 256 $\times$ 256. The final configuration in the last panel ($t\to \infty$) is obtained from an independent simulation, as explained in the text. []{data-label="f52"}](contact_f7.eps){width=".5\textwidth"}
We can see the actual contact surface at different times in Fig. \[f52\]. We observe how the increase of contact area involves both an increase of the area of individual contact spots, and the appearance of new ones. This trend is very similar to the experimental findings of Dieterich and Kilgore[@dk]. The asymptotic surface of contact (which cannot be typically accessed experimentally), shown in the last plot in Fig. \[f52\], displays a uniformly scattered distribution of contact points. In particular, points that were at contact in the first stages of the process may become detached at very long times, due to the stress redistribution that occurs due to relaxation.
Analytical Results
==================
In the limit of very light loads, the contact between surface and substrate occurs only in very few points. If we are considering the case of uncorrelated surface roughness, these contact point will typically be well separated spatially. This allows for an analytical solution of the model in this limit. We will confirm in this way the logarithmic increase of contact area in time. The analytical treatment I present turns out to be formally similar to that made in the viscoelastic Greenwood-Williamson model [@caroli; @hui; @gauss-exp; @ronsin]. In fact, the creep phenomena I am modeling through the relaxation mechanism is a kind of viscoelastic relaxation [@lakes; @fischer-c1; @fischer-c2]. I will come back to this point by the end of the paper.
The strategy to find the solution in this limiting case is to calculate the response of the system in an “indentation hardness test"[@fischer-c2], and then exploit the linearity of Eqs. (\[us\]-\[u0s\]) to find the full solution. In fact, due to their linearity, Eqs. (\[us\]-\[u0s\]) can be solved by Fourier decomposition. In the case there is a single contact point between the two surfaces (supposed to be the coordinate origin) on which the force is zero for $t<0$, and takes some constant value $f_0$ for $t>0$, a direct calculation of the Fourier modes $\tilde u_q$ gives:
$$\tilde u_q=f_0 \left [ \frac{1}{k_0q^2}+\left (\frac{1}{k_1+k_0q^2}-\frac{1}{k_0q^2} \right)\exp[-z_qt]\right]
\label{uq}$$
where $$z_q=\frac{Rq^4k_0k_1}{k_1+q^2k_0},$$ and the initial condition $u^0({\bf r})=0$ at $t=0$ (corresponding to a totally relaxed configuration [*before*]{} contact) has been used.
![The dimensionless $G(x_1,x_2)$ function (Eq. (\[u\])) as a function of $x_2$, for different values of $x_1$. Note the logarithmic increase for large $x_2$, independent of the value of $x_1$. Dotted line is the function $a_0 \ln (x_2)+a_1$, with $a_0=0.04$, $a_1=0.3$. []{data-label="funciong"}](contact_f8.eps){width=".5\textwidth"}
By Fourier inverting this expression we can obtain the time evolution of the surface under this indentation condition. Of particular importance will be the time evolution of the variable $u$ at the contact point, i.e, $u({\bf r}=0,t)$, and its velocity $v_0(t)\equiv du({\bf r}=0,t)/dt$. The structure of Eq. (\[uq\]) allows to write the solution in the form
$$u({\bf r}=0,t)=\frac{f_0}{k_0}G\left(\frac{k_1}{k_0},k_0Rt\right)
\label{u}$$
and $$v_0(t)=f_0RG'\left(\frac{k_1}{k_0},k_0Rt\right)
\label{v0}$$ where $G(x_1,x_2)$ is a dimensionless function of two variables, and $G'(x_1,x_2)\equiv dG/dx_2$. In Fig. \[funciong\] I show the values of $G$ as a function of $x_2$, for a few different values of $x_1$. We see that for $x_2$ sufficiently large (this means according to Eq. (\[u\]), for $t$ sufficiently large) the function has a logarithmic increase, with a slope that is independent of other parameters of the model. The form of the function $G$ is all we need to get a full solution of the contact problem on the assumption of few well separated contacts.
In the contact geometry, the value of $u({\bf r}=0,t)$ must be constant and equal to $\xi({\bf r}=0)$. In order to maintain the contact force equal to $f_0$, we have to adapt $\overline {u^0}$ accordingly, namely we must have $$\frac{d\overline {u^0}}{dt}=-v_0(t)
\label{du0}$$ To solve the problem of many contact points (in the assumption that the distance among them is large compared to the size of the distortion that any contact exerts onto the surface) we first need to generalize the form of $v_0(t)$ for a constant force $f_0$, to a new function $v(t)$ for an arbitrary time dependent form of the contact force $f(t)$. This can be done from the previous solution because of the linearity of the equations. The result is
$$v(t)=\int_{-\infty}^{t} d\tau R\frac{df(\tau)}{d\tau} G'\left(\frac{k_1}{k_0}, k_0R(t-\tau)\right)
\label{vt}$$
To generalize Eq. (\[du0\]) to the case of many contacts, under the action of a total normal force $F_N$, I use the fact that this normal force has to be distributed among all contacts at any time. We obtain
$$\frac{d\overline {u^0}}{dt}=\frac{\sum_{m} v_m(t)}{N(t)}$$
where $N(t)$ is the number of points at contact at time $t$, $m$ labels the contact points, and $v_m(t)$ is expression (\[vt\]) calculated with the particular (still unknown) form of $f_m(t)$ for the contact force at point $m$. Using Eq. (\[vt\]) and the fact that all forces must sum up to $F_N$, we obtain the simple result $$\frac{d\overline {u^0}}{dt}= \frac{v_0(t)}{N(t)}$$ where $v_0(t)$ is given in Eq. (\[v0\]). Namely, the change of the control variable is equal to a prescribed function of time, divided by the actual number of contacts at that time. From here, and assuming a generic distribution of asperities given by a probability distribution $P(\xi)$, the following equation for the time evolution of the number of contacts can be derived:
$$\frac{dN(t)}{dt}=N_0P(\overline {u^0}(t))\frac{d\overline {u^0}}{dt}=\frac{N_0P(\overline {u^0}(t))v_0(t)}{N(t)}$$
where $N_0$ is the total number of mesh points in the system. Integrating, one obtains $$\int_{N(t=0)}^{N(t)} \frac{N(t')}{P(\overline {u^0}(t'))}dN(t')=N_0\int_0^t v_0(t')dt'$$ The integral on the left of this equation is explicit, but it is not analytic if the $P$ distribution is taken to be a Gaussian. We can replace the Gaussian form of $P$ by an exponential to obtain a closed form, i.e., assuming $P(\xi)=\gamma \exp(-\gamma \xi)$, we obtain $$N(t)=N(t=0)\left (1+k_1\int_0^t v_0(t')dt' \right).
\label{ndet}$$ where $N(t=0)$ is the number of contacts at $t=0$, and is given (for exponentially distributed asperities) by $N(t=0)=\gamma f_0/k_1$. Using the asymptotic form of $v_0$ (Eqs. (\[u\]),(\[v0\]), and Fig. \[funciong\]), we obtain for large $t$ $$\frac{N(t)}{N(t=0)}\simeq\frac{a_0 k_1}{k_0}\ln \left(k_0Rt\right).
\label{ndet2}$$ with $a_0\simeq 0.04$. As numerically shown by Greenwood and Williamson[@gauss-exp] the consideration of a Gaussian distribution of asperities does not alter this result in the load range in which $1\ll N(t)\ll N_0$. In this way, we obtain analytically a result that is nicely compatible with the numerical results: for small loads, the contact area increases logarithmically in time, and once scaled with the value of the area at $t=0$, the result is independent of the precise value of the applied force. Note also that the numerical results (Figs. \[f3\] and \[f6\]) show an increase of the logarithm prefactor when $k_1/k_0$ is increased, compatible with Eq. (\[ndet2\]). The main hypothesis to derive Eq. (\[ndet2\]) have been the uncorrelated distribution of asperities, and the fact that elastic distortions of the surface at the contact points do not influence other contact points, and this means that the contact points have to be sufficiently away from each other. I will analyze this expression further in the last section of the paper.
Summary and discussion
======================
In this paper, a simplified model for the time evolution of the contact area between solid bodies in contact has been presented. The goals of the model are twofold. On one side, for very light loads and in the case of a totally uncorrelated surface roughness, the model can be worked out analytically, and it can be shown that a logarithmic dependence of the contact area with time emerges. This means that we are able to go all the way from a well defined statistical model to its solution, and obtain logarithmic aging. Secondly, numerical simulations can be done in cases where the applied load is not necessarily small, and systematic dependences of the contact area in time others than logarithmic have been found in this case. Also, the model allows to study more realistic cases in which the roughness of the surface is assumed to be correlated. I now focus on a discussion of the process of logarithmic increase in time of contact area, that is the most directly comparable with available experimental results.
The analytical results of the previous Section provide the clearest understanding of the origin of such a logarithmic increase within the framework of the present model. In fact, this logarithmic increase is originated in the form of the surface response to a localized and constant applied load. This can be rephrased in the following form: If we push the surface of (our model of) an elastic body with a constant force at a single point, and look for the deformation this force produces, the indentation increases logarithmically in time due to the relaxation processes considered by the model. In this respect, I notice that the logarithmic increase is crucially dependent on the dimensionality of the surface. For instance, in the case of a line (i.e., the border of a half plane) the same relaxation mechanism would produce a displacement that grows with time as $t^{1/4}$ (this dependence appearing when summing up the $q$ modes of Eq. (\[uq\]) in a one-dimensional geometry), which is (at least in principle) discernible from a logarithmic increase. It thus remains to be seen if there are experimental realizations on this confined geometry configuration that allows to test this prediction.
I now want to discuss in more detail the analytical expression in Eq. (\[ndet2\]), that we saw is well verified in the simulations, and compare it with available experimental results. One of the most detailed experimental studies of temporal effects in friction measurement have been provided by Baumberger and co-workers in a series of papers [@caroli; @berthoud; @baum1; @baum2]. One important parameter they consider, is the coefficient defined as the derivative of the static friction coefficient $\mu_s$ with respect to the logarithm of the hold time[@caroli; @marone0], namely $B\equiv \frac{d \mu_s}{d\ln(t)}$. This hold time is the time during which the two surfaces are left in rest contact, before the friction force necessary to start sliding is measured. This coefficient (that is independent of the time unit chosen) is typically found to have a conserved value for a variety of materials, in a range close to $10^{-2}$.
Following Tabor[@bt], the friction force $F_{fr}$ between two solids can be written as $F_{fr}=\sigma_S A$, where $\sigma_S$ is the so-called shear strength of the interface and $A$ the real area of contact. Using also the standard expression $F_{fr}=\mu_s F_N$, we can write $\mu_s=A\sigma_S/F_N$, where we see that the friction coefficient is directly related to the contact area. In particular, if we assume that $\sigma_S/F_N$ takes some constant value, we can write $$\frac{1}{\mu_s}\frac{d\mu_s}{d\ln t}=\frac{1}{A}\frac{dA}{d\ln t}$$ and since $\mu_s$ is typically of order one, we can roughly write $$B\simeq \frac{1}{A}\frac{dA}{d\ln t}\simeq \frac{d(A/A(t=0))}{d\ln t}$$ This form gives a direct access to the $B$ coefficient as predicted by the model presented in this paper. We see first of all (from Eq. \[ndet2\]) that $B$ does not depend on the relaxation coefficient $R$. This seems a bit surprising but is not contradictory: although the contact area increase in time is produced by a non-zero value of $R$, this coefficient enters directly in setting the time scale (as it is obvious from Eq. (\[u0s\])) but the logarithmic derivative of contact area is independent of it. Once this fact is recognized, we may ask to what extent the value of $B$ in our model can be considered to be constant, independent of other parameters, and if this is the case, if this value is in the experimentally observed range $B\sim 10^{-2}$. In this respect, it seems that the answer is negative, since the coefficient is directly related to the ratio $k_1/k_0$ (Eq. (\[ndet2\])), which can in principle be set arbitrarily. However, I already stressed the fact that the description of our elastic surface is not very accurate. In fact, the elastic properties of the surface of an elastic body, that I consider isotropic for simplicity, can be characterized by the values of one elastic constant (the Young modulus $Y$, for instance) and are only weakly dependent of a second parameter, namely the Poisson ratio $\nu$. The dependence of $B$ on the dimensionless ratio $k_1/k_0$ is an indication that in a hypothetical more accurate description of the elastic body, $B$ cannot depend explicitly on the value of $Y$. Only a dependence on $\nu$ can exist. Then I expect in fact a rather weak dependence of $B$ on the elastic parameters of the body, and then a conserved value of $B$ for different materials. Whether this conserved value is compatible with the experimental value $\sim 10^{-2}$ has to be answered once a more realistic description of the elastic surface of the body is done.
Going a step further we must discuss the effect of temperature. The Brechet-Estrin analysis [@brechet] predicts that the $B$ coefficient must be temperature dependent, with values that increase with increasing temperature. Berthoud [*et al.*]{}[@berthoud] were able to observe systematic variations of this coefficient with temperature in different experimental situations, that are roughly compatible with the predictions of Brechet and Estrin. Typically, an increase of $B$ (up to a factor of roughly ten) was observed when the glass temperature of the material was approached. In the present model it is not obvious where a temperature dependence can enter that alter the value of the $B$ coefficient. The only obvious temperature dependent parameter is the relaxation coefficient $R$, but we have already seen that $B$ is independent of $R$.
![Depth of an indentation experiment on the model, under a constant load $f_0$, as a function of time, for $k_1/k_0=3$, in the presence of creep relaxation and viscoelastic relaxation, measured respectively by the coefficients $R$ and $\tilde R$. (see Eq. (\[dosrs\])). Note the increase in the slope of the asymptotic logarithmic behavior, as $\tilde R/R$ is increased. []{data-label="indent"}](contact_f9.eps){width=".5\textwidth"}
A possible way out to this situation may involve to consider the possibility of other mechanisms of relaxation, in addition to the one considered in Eq. (\[u0s\]). For instance, in addition to the present mechanism that responds to the fluctuations of the forces on the $k_1$ springs, we can add a term that is directly dependent of the force itself. This would generalize Eq. (\[u0s\]) to an equation of the type
$$\frac{du^0({\bf r})}{dt}=R\nabla^2\frac{\delta E}{\delta u^0({\bf r})}-\tilde R\frac{\delta E}{\delta u^0({\bf r})}
\label{dosrs}$$
One of the main qualitative difference caused by the inclusion of the last term is that under the action of a constant value of $\overline {u^0}$, the value of $F_N$ goes to zero at very large times (contrary to the finite value reached in the presence of the first term alone). In this sense the last term represents a “viscous" relaxation in the system, that may have a progressively larger effect as the temperature increases towards the glass temperature of the system. This suggests that $\tilde R/R$ may be considered to be an increasing function of temperature.
The effect of this term can be seen in Fig. \[indent\]. There I show the time dependence of the indentation depth for a point contact on the surface of the elastic body for increasing values of $\tilde R/R$. These results are obtained by solving analytically the system equations for $u_q$ as I did in Section V, using Eq. (\[dosrs\]) instead of (\[u0s\]), and Fourier inverting numerically the result. We see that the logarithmic increase of $u$ in time is conserved in the presence of the $\tilde R$ term, and the slope increases as $\tilde R$ increases. This slope measures directly the value of $B$ in the model. It can be easily shown that $B$ doubles its value as $\tilde R/R$ goes from zero to large values. This means that the consideration of alternative relaxation mechanisms can justify a variation of the $B$ coefficient with temperature. Whether the change of the $B$ coefficient observed by Berthoud [*et al.*]{}[@berthoud] is related to a change of relaxation mechanism or not remains to be investigated further, both theoretically and experimentally.
This research was financially supported by Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Argentina. Partial support from PICT 32859/2005 (ANPCyT, Argentina) is also acknowledged.\
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{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Density functional theory (DFT) and many body perturbation theory at the G$_0$W$_0$ level are employed to study the electronic properties of polythiophene (PT) adsorbed on graphene surface. Analysis of charge density difference shows the substrate-adsorbate interaction leading to a strong physisorption and interfacial electric dipole moment formation. The electrostatic potential displays a -0.19 eV shift in the graphene work function from its initial value of 4.53 eV, as the result of the interaction. The LDA band gap of the polymer does not show any change, however the energy level lineshapes are modified by the orbital hybridization. The interfacial polarization effects on the band gap and levels alignment are investigated within G$_0$W$_0$ level and shows notable reduction of PT band gap compared to that of the isolated chain.'
author:
- 'F. Marsusi'
- 'I. A. Fedorov'
- 'S. Gerivani'
date: 'August 10, 2017'
title: 'Improving the Performance of Polythiophene-based Electronic devices by Controlling the Band Gap in the Presence of Graphene'
---
INTRODUCTION
============
Polythiophene (PT) and its derivatives are the most important stable and ease of preparation $\pi$–conjugated semiconductors with a broad spectrum of applications in organic electronics. The applications range from the organic light-emitting diodes (OLED) and displays to solar cells and sensors [@Li; @Reynolds; @Perzon; @Ho; @Mwaura; @Zou]. However, PT–derivatives based electronic devices suffer from relatively large electronic band gap and low carrier mobility which reduce the corresponding quantum efficiency [@Kaloni; @Zhou].\
On the other hand, due to its ballistic charge transport and high electron mobility, graphene expected to be an outstanding new material for the electronic applications. These exceptional features suggest that graphene can be utilized to improve the electronic characteristic and charge transport of polymer–based organic semiconductors. Single and few multilayers graphene are transparent. This property is also of high importance for those devices that the light should enter their active layer. Recently, organic PT-derivatives, poly (3-hexylthiophene (P3HT) and poly(3-octylthiophene) (P3OT) photovoltaic as electron donors and an organic functionalized graphene as electron-accepter has been fabricated [@Liu]. It was shown that this composite works well due to the interaction between graphene and polymers [@Liu]. Recently, the crystallization of highly regular P3HT on single layer graphene is also reported [@Skrypnychuk]. The P3HT polymer film deposited on graphene reported to have a very different distribution of crystallite orientations. Also much higher degree of $\pi-\pi$ stacking perpendicular to the plane of the graphene film, compared to deposition on silicon surface was exhibited [@Skrypnychuk]. Graphene is reported as an ideal template for growing ultra-flat organic films with a face-on orientation [@Wang]. These preference features are accounted to obtain higher charge transports and efficiencies from PT derivatives [@Skrypnychuk].\
Therefore, this paper presents a theoretical attempt to unveil and understand theoretically the influence of graphene at the electronic properties of PT. To this end, first the electronic structure of isolated graphene and PT are investigated. Next, the adsorption mechanism and the modification of the electronic structure when PT adsorbed on the graphene surface are inspected. In order to obtain a comprehensive view, we use three different approaches: (i) local density approximation (LDA), (ii) the Hubbard corrected LDA+U functional and (iii) the many-body perturbation theory at the $G_0W_0$ level. The outcomes are compared with the previous available theoretical and experimental data. As our main purpose, PT is brought close to the graphene surface in face-on orientation. Then it is inspected that how the electronic states of isolated PT are perturbed by the graphene surface using the three mentioned approaches. Van der Waals (vdW) interaction are considered when the binding energy are calculated by semi-local PBE-GGA functional. Our results show that the most stable geometry occurs in the adsorption distance of about 3.4-3.5 Å. Since a typical adsorbate-substrate spacing in a physisorption is more than 3 Å, the obtained value is an evidence of a physisorption. Charge density analysis shows no charge is transferred between PT and graphene. However, results show a clear charge density distortion close to the graphene surface, which is a sign of electric dipole formation. Previous many body ab initio studies have clarified that the polarization, even for a physisorbed and weakly coupled interaction, can considerably influence the size of the adsorbate gap [@Fu; @Puschnig]. It is also well known that local (semi-local) nature of density functional theory (DFT) functional cannot contribute for the polarization effect in the interface, due to self-interaction errors. By ignoring the polarization effects, the LDA PT gap is independent of the underlying substrate, and the only connection with the graphene is done through a weak interaction with slightly change in the electronic states through orbital hybridizations. It is well known that the self-interaction error which appears in the occupied states in the standard DFT with local (or semi-local) exchange-correlation functionals over delocalizes the highest occupied state and pushes it up, therefore reduces the band gap [@Puschnig; @Lanzillo; @Garcia; @Neaton]. Subsequently, when a molecule is brought close to the substrate, local (semi-local) functionals exaggerate the density distribution for the added electron or hole. The consequent of the delocalization error is incorrect convex behavior (instead of piecewise constant) of the energy with respect to the number of electrons . As a result, the derivatives of the energy with respect to fractional charge produce incorrect ionization energy and electron affinity and smaller band gap [@Mori].\
DFT+U inspired by the Hubbard model as a typical approach may be used to correct the DFT unphysical curvature of the total energy. Our outcomes show that as for many typical conjugated polymers, DFT+U does not improve the band gaps of isolated PT and the adsorbed PT on graphene significantly [@Baeriswyl]. In order to describe the dynamical polarization effect of the surrounding electrons, we employed many-body perturbation theory implemented through G$_0$W$_0$ approximation on the top of the DFT calculations. The self-energy $\Sigma$ is given as the product of the non-interacting single-particle Green function $G_0$, and the dynamically screened Coulomb interaction $W_0$, calculated within the plasmon pole approximation. It is well-known that including dynamical electronic correlation in G$_0$W$_0$ approximation mimics the long-range image potential effects of electrons near the interface [@Inkson]. Using G$_0$W$_0$, we obtained a relatively strong renormalization of the PT band gap, once it physisorbed on the graphene surface.
COMPUTATIONAL DETAILS {#sec: computational}
=====================
The formation of a regular PT and graphene layer will be considered, when the lateral PT-PT interactions to be less attractive than the corresponding interaction between PT and the graphene surface. To this end, we should perform the calculations in a supercell with the size large enough to prevent the interactions between PT and its images. In addition, the thiophene oligomer in the supercell must arranged in a way that forming the polymer chain, when the cell subjected to the periodic boundary condition (PBC). At the same time, from the repetition of this unit cell in both two coordinates (x, y), the graphene sheet must be constructed. However, the main axes of the graphene lattice do not commensurate at all with the periodicity of PT. Therefore, constructing PT in the face-on orientation is not a trivial commission. In practice, there are many possible on-face arrangement for the combined system, but regarding the computational limitation was explained, we found that the concurrent assembly can be best done when the oligomer is placed along the base-diameter of the monoclinic supercell shown in Fig. \[fig1\](a).
![\[fig1\] [(Color online) (a) Top view showing the monoclinic supercell model describing PT polymer adsorbed on top of the graphene monolayer. For clarity the cell is duplicated in both in-plane directions. (b) Top view showing the unit cell vectors of the hexagonal graphene primitive cell $\textbf{a}_1$ and $\textbf{a}_2$ (blue arrows), as well as the monoclinic supercell basis vectors $\textbf{a}'_1$ and $\textbf{a}'_2$ (red arrows). (c) Illustration of the supercell Brillouin zone along with the high-symmetry points that are selected to define **k**-vector paths.]{}](fig1)
This arrangement provides us an opportunity to study and analyze the electronic behaviour of PT adsorbed on graphene surface, at the cost of increasing the number of carbon atoms in the graphene cell up to 72 atoms. If we denote the graphene primitive vectors by $\textbf{a}_1$ and $\textbf{a}_2$ ($|\textbf{a}_1|$=$|\textbf{a}_2|$=a (1.73 Å)), the supercell vectors assembled in this work can be described by $\textbf{a$^\prime$}_1$ ($|\textbf{a$^\prime$}_1|=\frac{12a}{\sqrt{3}}=17.15 $ Å) and $\textbf{a$^\prime$}_2$ ($|\textbf{a$^\prime$}_2|$=$\frac{9a}{\sqrt{3}}$=12.86 Å). A vacuum region of 14 Å along the perpendicular direction is imposed to guarantee a vanishing interaction between periodically repeated images. By this value the total energy versus z–component of the supercell is converged to less than 2 meV.\
By repetition of the supercell along the two in-plane vectors, both graphene sheet and PT are built, as shown in Fig. \[fig1\](b), while the supercell is large enough to prevent the interaction between PT-PT themselves, which are at the distance of 12.8 Å from each other. The first Brillouin zone (BZ) corresponding to the supercell is shown in Fig. \[fig1\](c). The two paths through the three symmetric points of the BZ are shown in Fig. \[fig1\](c). These points are selected to investigate the electronic behaviours of the isolated components and combined system. Optimization procedure has been performed within LDA frame work by using plane-wave pseudopotential as implemented in the ABINIT code [@ABINIT]. To converge the total energy to within 1 meV a plane-wave cutoff energy of 30 a.u. is required. The Brillouin zone (BZ) are sampled by $4\times4\times1$ Monkhorst-Pack **k**-vectors [@Monkhorst]. LDA Troullier–-Martins (TM) pseudopotentials are used in our calculations [@Troullier]. Both volume of the supercell and the position of PT atoms inside it are relaxed within the forces less than 5 meV/Å. The structure of isolated graphene sheet are relaxed within the same criteria.\
The U correction are adopted to the p–orbitals of all atoms in the cell within the projector augmented-wave (PAW) method. The PAW kinetic energy cut-off is converged by 20 a.u. To determine the U and J parameters, we follow a semi-empirical strategy by looking for the optimum values when the band gap is increased to its experimental value.\
![\[fig2\] [(Color online) (a) The optimized geometry model of PT oligomer used in the supercell explained in the text. Atoms in a hexagonal ring are labelled according to parameters shown Table I.]{}](fig2)
parameter LDA $^a$LDA
--------------------- -------- ---------
C6–C3 1.42 1.40
C3=C1 1.38 1.38
C1–C2 1.40
C1–H1 1.09
C4–C5 1.42 1.42
S1–C4 1.72
$\angle$(C3-C1-C2) 113.31 113.3
$\angle$ (C2-C4-C5) 128.89 128.9
: \[tab1\] Calculated optimized bond lenghts and angles of polythiophene according to atoms labeled in Fig. \[fig2\]. The previous available LDA predictions are listed for comparison. All lengths are in Å and angles in degree.
\
$^{a}$Data taken from from Ref. \[23\]
The G$_0$W$_0$ approximation [@Ondia; @Hedin] provided by the YAMBO code [@Marini] using TM pseudopotentials. The dielectric function is calculated using the plasmon-pole approximation, while the convergence in the self-energy and the dielectric matrix with respect to the number of bands are considered. The calculations were done with up to 200 empty bands corresponding to a maximal band energy of 0.44 a.u. (12 eV), as well as a cut-off for the dielectric matrix up to 2.5 a.u.
RESULTS and DISCUSSIONS
=======================
Isolated polymer
----------------
Before considering the electronic properties of PT-absorbed on the graphene surface, we present the LDA band structure of isolated PT chain computed in the supercell shown in Fig \[fig1\](b). The polymer structure is constructed from the oligomer in the cell, when subjected to the PBC in both two in-plane directions, as shown in Fig. \[fig1\](a). The oligomer including four thiophene rings, as shown in Fig. \[fig2\], and is oriented parallel to the supercell base-diameter. The LDA optimized structure and geometry parameters are shown in Fig.s \[fig1\](b) and \[fig2\], respectively. The value of the corresponding parameters are listed in Table \[tab1\]. The obtained values are also compared with the previous theoretical LDA data, and the agreements support our technical process. By subjecting the supercell to the PBC, in fact we are neglecting the possible torsion angle between oligomers. By this assumption, the alignment of the $\pi$–orbitals will be maximized[@Scherf], which is an appropriate model for the chains including more than 10 monomers [@Pesant].\
The LDA electronic band structure of the isolated PT are shown in Fig 3. Both the maximum of the valence and the minimum of the conduction bands are happen at the $\Gamma$ point. These two bands are stemming from the inter-ring $\pi$–bonding and $\pi^*$–antibonding states occur at -3.65 and -2.55 eV, respectively, and therefore, a direct gap of 1.10 eV is opened at the $\Gamma$ point. Depending on the pseudopotential and other calculation parameters given in Sec. \[sec: computational\], this gap is about 0.12-0.26 eV smaller than the previous LDA reported gaps [@Horst; @Pesant]. The LDA valence and conduction bandwidth between point $\Gamma$ to the point A are found relatively small, as shown in Fig. \[fig3\], with value of 0.76 and 0.60 eV, respectively, and reflecting the wave functions are localized on individual chain.
![(Color online) DFT-LDA (top) and G$_0$W$_0$ (down) band structure of polythiophne (PT). The Fermi energies are set to zero.[]{data-label="fig3"}](fig3)
The LDA band gaps obtained in this study and the previous efforts display relatively large deviation from experimental value (2.1 eV) [@Kobayashi]. One of the reasons is the spurious self-interaction error (SIE) related to the delocalization error of DFT exchange-correlation (xc) functionals stemming from dominating Coulomb term that pushes electrons apart [@Mori]. In some cases, hybrid functionals can partially correct for the SIE by including a fraction of exact exchange term adjusted to cancel the SIE [@Pesant]. However, using hybrid functionals is not a general solution and for a large number of atoms is computationally quite expensive. Here, we have performed DFT+U as an alternative method to cure the SIE by applying the U correction term only to the p orbitals. To determine parameters U and J, we follow a semi-empirical strategy and controlling the band gap. By increasing U and J to respectively 6 and 0.6 eV, the band gap is raised up and saturated at about 1.40 eV, which is not a significant improvement. Therefore, applying DFT+U, inspired by the Hubbard model, cannot provide a realistic model to correct the PT band gap, as has been proved in past for many conjugated polymers with delocalized interring $\pi$ electrons [@Baeriswyl].\
On the other hand, it has been shown that many body perturbations theory based on ab initio calculations of the quasiparticle (QP) band structure with the GW approximation overcomes the problematic effect of self-energy on polymer band gap [@Puschnig; @Sun; @Lanzillo; @Samsonidze]. Therefore, we constructed GW calculations in a non-self-consistently manner from the LDA orbitals and eigenvalues, which is referred as the G$_0$W$_0$ method. The G$_0$W$_0$ approach is computationally so expensive for the present supercell introduced in Fig. \[fig1\](b). The sever limitation is the large number of atoms necessary to study the adsorption of PT on graphene surface. Therefore, we had to calculate electron self-energy by summing over computationally reasonable number of empty states. The initial calculation was performed with 100 number of unoccupied orbitals. The SIE corrected PT gap improved from LDA predicted 1.10 eV to G$_0$W$_0$ 2.96 eV at point $\Gamma$, as seen in Fig. \[fig3\]. By increasing the number of empty states up to the 200 states, the energy of states are modified, however the initial G$_0$W$_0$ gap improves only by about 0.03 eV to 2.99 eV. Indeed, compared to LDA, the G$_0$W$_0$ predict larger bandwidth for the valence and conduction bands with value of 0.96 and 0.74 eV, respectively. Further increasing of the empty states in our G$_0$W$_0$ calculations imposed computationally sever challenges, especially when the adsorption of PT on graphene is considered. The previous G$_0$W$_0$ gap computed within the generalized plasmon pole model and by summing self-energy over 1592 empty states within a smaller supercell including two thiophene rings is 3.10 eV, which is in satisfactory agreement with our outcomes driven from 200 empty states [@Samsonidze]. Here we point that the band gap of 3.59 eV is also predicted by a previous higher level calculation based on the self-consistent GW calculations [@Horst]. Moreover, the computed G$_0$W$_0$ quasiparticle gaps computed in this and the previous works are about 1 eV larger than the experimental value. The disagreement has been mostly cured by including electron-hole interaction for the optical response, and inter-chain interactions in bulk polymer [@Horst; @Samsonidze].
Uncovered graphene {#sec: Uncovered graphene}
------------------
At first, all carbon atoms in the graphene layer are relaxed to the ground state minimum potential energy point. The LDA predicts C-C bond length of 1.429 Å for graphene. Fig. \[fig4\] shows the LDA predicted band structure of pristine graphene deduced from the monoclinic supercell illustrated in Fig. \[fig1\](a) including 36 primitive cells and 72 carbons. Two of the six Dirac points at the corners of the hexagonal BZ are folded to point $\Gamma$ inside the BZ shown in Fig. \[fig1\](c), so that there are four-fold degeneracy at this point with the energy of -2.13 eV, forming two pairs of touching cones.
![\[fig4\] [(Color online) DFT-LDA band structure of isolated graphene calculated in the supercell described in the text. The Fermi energy is set to zero. The four touching bands forming two Dirac cones at $\Gamma$ are specified by red colors.]{}](fig4)
PT adsorbed on graphene {#sec: Uncovered graphene }
-----------------------
The optimum distance between PT and graphene plane is determined by the value which minimized the total energy of the compound. PT adsorption energy on graphene surface as a function of adsorption distance, calculated by using different functionals are depicted in Fig. \[fig5\]. In agreement with the previous report for poly(para-phenylene) (PPP) adsorbed on graphene [@Puschnig], we also observe that adsorption energy and distance are sensitive to the choice of the applied functional. Moreover, each of LDA and GGA functionals exhibits different picture. According to LDA, the most stable geometry occurs in the adsorption distance of z$_0$ =3.4 Å with binding energy of 0.77 eV. GGA-PBE functional predicts a shallow potential well at 4.2 Å with binding energy of 0.17 eV. Since a typical substrate-adsorbate spacing in a physisorption is more than 3 Å, these results suggest a physisorption of PT on graphene. The GGA-PBE small binding energy is a result of DFT deficient description of the long-range dispersion forces. The calculated binding energy based on LDA may not be reliable, since LDA does not include the long-ranged vdW interactions [@Nabok]. In fact, the over binding effect observed in the LDA functional compensates this missing term, and LDA benefits from the cancellation of errors [@Nabok]. For GGA-PBE, one should contribute a small nonlocal vdW dispersion force, which almost is the binding force for the most of the organic materials [@Dion]. Including DFT-D2 correction within Grimme empirical scheme [@Grimme], as implemented in the ABINIT code, improves PBE binding energy to 1.16 eV at the adsorption distance of z$_0$= 3.5 Å. Therefore, the contribution of vdW interactions to the binding energy amounts to 1 eV and indicates a strong physisorption.
![\[fig5\] [(Color online) Adsorption energy of PT on graphene surface obtained by using different functionals: GGA-PBE, LDA, and GGA-D2, which the latter includes a semi-empirical long-range dispersion correction, based on the Grimme$^\prime$s method, in terms of the adsorption distance. The zero line indicates the two independent components with no binding energy]{}](fig5)
The orbital overlap and vdW interaction rearrange the charge density, which in turn should influence the valance and conduction energy levels of the polymer, and hence the band gap. Normally, the degree of the substrate-adsorbate vdW interaction depends on the adsorbed distance, the polarizability of the substrate and the number and type of the atoms involved. The reported vdW corrected GGA binding energy of PPP on the graphene surface is about 0.39 eV [@Puschnig], which is so much smaller than what we obtained here for PT.\
To get more insight into the PT physisorption mechanism on graphene, we calculated the charge density rearrangement at the substrate-adsorbate interface by computing the in plane-averaged charge density of combined system relative to the charge densities of isolated components: $\Delta\rho=\rho$(G+PT)-($\rho$(PT)+$\rho$(G)). Here, $\rho$(G+PT) represents the charge density of the combined system, and $\rho$(PT), $\rho$(G) are the densities of the seperated polymer and uncovered graphene, respectively. The total plane-averaged and difference charge density, as well as the electrostatic potential are represented in Fig. \[fig6\]. As shown in the top panel of Fig. \[fig6\], there is almost no change in the charge density of the polymer, suggesting no charge transfer between graphene and PT. The calculated charge density difference near the graphene surface obtained in this study is about one order of magnitude larger than for the PPP on graphene, reported in Ref. \[13\]. This result indicates the key role of the sulphur atom in this study, and explains the origin of the strong physisorbtion of PT on graphene. By considering $\Delta\rho$, we find electron charge density of graphene is attracted toward PT and makes a region of charge accumulation ($\Delta\rho >0$) and a region of charge depletion ($\Delta\rho >0$). Therefore the interaction between polymer and graphene gives rise to an interfacial electric dipole moment formation near the graphene surface in the region between two components. By PT absorption, the Fermi energy of the graphene exhibits a 1.09 eV downward shift, and the interfacial electric dipole moment makes a modification to graphene work function by a -0.19 eV shift down from its initial value of 4.53 eV (4.34 eV).\
![\[fig6\] [(Color online) Top panel: plane-averaged total charge density (red solid line) and charge density difference (black dashed line) of the combined system (G+PT) along the perpendicular direction to the graphene plane. Small charge modification are observed in the region between the graphene and polymer. Down panel: electrostatic potential energy along the perpendicular direction to the graphene plane inside the supercell. The work function ($\phi$) of the combined system and uncovered graphene (G) are determined from the corresponding Fermi energies. . Work function at the bottom ( $\phi_b$) and the top sides ($\phi_t$) of the cell are illustrated. Black dashed line shows the electrostatic potential energy of the uncovered graphene. The dotted-black line indicates the Fermi energy of the uncovered graphene. The maximum value of the electrostatic potential energy of the combined system is set to zero.]{}](fig6)
LDA band structure of PT at the 3.4 Å above the graphene sheet is shown in Fig. \[fig7\]. To identify the PT valence and conduction bands within the states of the combined system, the corresponding states of the isolated PT are appended to this figure. We see that according to LDA, PT band gap is slightly renormalized by a value of about 50 meV. The conduction band at point $\Gamma$ experiences only a small downward shift of 40 meV, which is the result of the weak charge density variation in PT after adsorption. As the result of the orbital hybridization, from the $\Gamma$ point toward point A, energy of the PT states are influenced by the graphene orbitals.\
![\[fig7\] [(Color online) Top panel: LDA-predicted band structure of the combined system at an adsorption height of 3.4 Å. The LDA valence and conduction bands of the isolated PT (blue-dashed lines) are shown to identify the corresponding states of the polymer (red-solid line) in the combined system. Zero energy is set to the Fermi energy of the combined system. Down panel: G$_0$W$_0$-predicted band structure of the combined system at an adsorption height of 3.4 Å. The G$_0$W$_0$ valence and conduction bands of the isolated PT (blue-dashed line) are shown to identify the corresponding states of the polymer (red-solid line) in the combined system. The G$_0$W$_0$ calculations were done in the three points shown by the square mark in each state. Zero energy is set to the energy of the graphene Dirac point (black-solid line at $\Gamma$ point) in the combined system. 3.29 eV. The Fermi energy of the combined system is illustrated in dotted line of the lower panel. The Fermi energy of the uncovered graphene (at -2.13 eV) is not shown for clarity purpose and obtained 3.50 eV ]{}](fig7)
While many physical effects can influence the energy level of a molecule near a polarizable substrate, LDA (DFT) is not a successful tool to describe them. An important effect in a physisorption is the Coulomb interaction between the surface and the electron or hole corresponding to the occupied or unoccupied states of the adsorbed molecule. The result of this interaction may cause the surface to polarize. The additional interaction between the polarized surface and the molecule influences the energy levels of the molecule and may strongly renormalize its gap.\
In classical picture, this subject can be described by a point charge *q* located at the position $z_0$ above a polarizable surface with dielectric $\epsilon$ filling the half space $z<z_0$. The electrostatic interaction potential is given by [@White]\
$$\label{eq.1}
V=\frac{qq^\prime}{4(z-z_0)},$$ where the size of the image charge is $q^\prime =q(1-\epsilon)/(1+\epsilon)$. As a result, the energy of the unoccupied states should shift downward, since they experience an attractive potential according to Eq. \[eq.1\]. For the occupied states the situation is inverse. Since the interaction with the positive image charge reduces the binding of the electron to the molecular core, these states are shifted upward. In contrast to LDA, long-range image static potential effects, for an electron near an interface can be well described by GW approximation. GW studies confirm that the electronic energy levels of a molecule outside a surface would also obey the image potential of Eq. \[eq.1\], and the response of the surface to an added electron or hole is reproduced within the G$_0$W$_0$ approach by means the screened Coulomb potential W$_0$. [@Puschnig; @Fu; @Neaton; @Garcia; @Rohlfing]. The interaction with the image charge shift the unoccupied levels downward, whereas the occupied levels upward and therefore, reduces the band gap.\
The band structure of the combined PT-graphene system at the G$_0$W$_0$ level and adsorption distance of 3.4 Å is shown in Fig. \[fig7\]. For clarity, the valence and conduction bands of the combined system are specified in the red-solid lines and black squares. The corresponding isolated PT bands are also integrated to this figure and depicted by blue-dashed lines and red squares. In contrast to the LDA prediction, image charges induced in the graphene layer significantly perturb PT electronic states, so that the PT band gap reduces from 2.99 eV into 1.60 eV ($\Delta$E=1.39 eV). Four fold degeneracy of the graphene layer at the $\Gamma$ point is split by the vdW interaction, so that a gap of 60 meV is opened at this point.\
![\[fig8\] [(Color online) The electronic energy-level alignments of PT polymer obtained from DFT-LDA and G$_0$W$_0$ calculations near the combined graphene–PT interface in two different adsorption distances (4 and 3.4 Å). The interface position is shown by red rectangular and the energy levels of polymer by blue line segments. The positions of each valence and conduction bands at points $"\Gamma"$ and $"A"$ are shown in eV. The vacuum level is set to 0 eV, and distances between line segments are scaled arbitrarily.]{}](fig8)
We also study the dependence of the band alignment at the interface on the adsorption distance . Due to computational cost, our calculations are limited only to two cases with 3.4 Å and 4 Å distances from graphene layer. The results are shown in Fig. \[fig8\], which we plot the LDA and G$_0$W$_0$ energies of the valence and conduction bands of PT at $\Gamma$ and A points of BZ. Since the LDA respective valence and conduction positions of PT shift slightly in the same direction, the change in the gap is so small (30 meV at $\Gamma$ point). At the G$_0$W$_0$ level and from 4 into 3.4 Å, the valence band moves up by 70 meV, whereas the conduction band moves down by 200 meV. Consequently, the band gap is reduced by 270 meV. Therefore, G$_0$W$_0$ not only can describe the band gap renormalization upon the dynamic polarization of the substrate, but also it can describe the effect of the substrate-adsorbate distance.
Conclusion
==========
In summary, by local density approximation within DFT frame work and many-body perturbation theory at the G$_0$W$_0$ level, we investigate electronic properties and possible modifications of the electronic band structure of polythiophene by adsorbtion on graphene surface. The calculated charge density difference predicts the formation of an interface electric dipole near graphene surface, however we found that the charge transfer between them is negligible. Analysis of the electrostatic potential along the direction perpendicular to the graphene plane shows a -0.19 eV shift in the graphene work function from its initial value of 4.53 eV. According to LDA, the adsorption of PT on graphene does not alter the PT band gap, but orbital overlapping of PT and graphene modify the PT energy levels lineshapin far from the $\Gamma$ point. By reducing the substrate-adsorbate distance, the LDA valence and conduction bands move slightly in the same direction leading to a small change in the gap. However, by the G$_0$W$_0$ the valence band moves up and the conduction band moves down, results in a significant band reduction effect. According to our finding in this study, we predict a significant change of the polythiophene electronic behaviours in the presence of graphene.
Acknowledgement
===============
F. Marsusi appreciates the computer assistance provided by Mrs Z. Zeinali in the Department of Energy Engineering and Physics at Amirkabir University of Technology. I. Fedorov gratefully acknowledge the Center for collective use “High Performance Parallel Computing” of the Kemerovo State University for providing the computational facilities.
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|
{
"pile_set_name": "ArXiv"
}
|
---
author:
- |
Luis N. Epele and Esteban Roulet\
Depto. de Física, Universidad Nacional de La Plata\
CC67, 1900, La Plata, Argentina\
title: On the propagation of the highest energy cosmic ray nuclei
---
Introduction
============
The cosmic ray (CR) spectrum is known to extend up to energies beyond $10^{20}$ eV, with the highest energy air showers observed having energies of 2–3$\times 10^{20}$ eV. The origin and nature of these ultra-high energy (UHE) events are one of the pressing unsolved problems defying us today. The CR spectrum has the overall shape of a leg, and is well fitted by power laws, whose index increases (spectrum steepening) for energies above the ‘knee’ ($E\sim 3\times 10^{15}$ eV), flattening again above the ‘ankle’ (at $E\sim 5\times 10^{18}$ eV). The CR composition becomes heavier for increasing energies around the knee, and the CR are probably mostly of galactic origin up to the ankle. Approaching the ankle, the CR composition seems to become lighter again, and the increasing rigidity of CRs does not allow anymore their confinement into the Galaxy. Hence, CR fluxes are most probably of extragalactic origin above the ankle. There have been studies suggesting that the arrival direction of the highest energy events may be indicating that their origin lies in the local supercluster, but they are not conclusive. The small anisotropies observed may also be compatible with a cosmological origin of the highest energy events.
The big difficulty which appears is that CR protons with $E\ga 5\times
10^{19}$ eV, i.e. relativistic $\gamma$ factors $\ga 5\times 10^{10}$, are not able to propagate more than $\sim 100$ Mpc due to their energy losses by photopion production off the cosmic microwave background (CMB) photons, giving rise to the well known GZK cutoff [@gzk]. At energies $2\times 10^{20}$ eV their mean free path is already only 30 Mpc. Heavy nuclei with smaller $\gamma$ factors, but comparable energies, also get attenuated but mainly by photodisintegrations off the intergalactic infrared (IR) background and off CMB photons, as well as by pair creation losses to a lesser extent [@st69; @tk75; @pu76; @el95].
A detailed study of the propagation of UHECR Fe nuclei, including all the relevant energy loss mechanisms, was performed more than twenty years ago by Puget, Stecker and Bredekamp [@pu76]. However, the estimates of the density of IR photons employed then were about an order of magnitude larger than the new empirically based estimates obtained using the measured emissivity of IRAS galaxies [@ma98]. In the light of the lower IR background densities inferred recently, it was suggested that UHECR nuclei may propagate much longer distances unattenuated [@st98], so that the events with energies 2–3$\times 10^{20}$ eV could have possibly originated as Fe nuclei produced at distances up to 100 Mpc, and hence in particular in the whole local supercluster[^1]. However, as we showed in a recent letter [@ep98], at energies larger than $10^{20}$ eV it is photodisintegration off CMB photons (and not off IR ones) which dominates the opacity for Fe disintegration. This implies that the maximum energies with which the surviving fragments can reach the Earth are not significantly changed (for distances below a few hundred Mpc) with the new estimates of the IR density. In particular, for sources at distances of 100 Mpc the maximum energies of the surviving fragments do not exceed $\sim 10^{20}$ eV, and can arrive to $2\times 10^{20}$ eV only for distances below 10 Mpc.
The aim of the present paper is to re-evaluate in detail the propagation of heavy nuclei, following the photodisintegration histories by means of a Monte Carlo which includes all relevant processes, much in the spirit of the original Puget et al. paper [@pu76]. From this we can establish all the effects resulting from the new estimates of the IR background density. In particular, we obtain the final mass composition and energy as a function of the distance to the source, as well as the possible fluctuations in these quantities which may arise from the particular way in which the photodisintegration takes place in each case. We also study the effects of pair creation losses, which turn out to be important in some cases for the determination of the final mass composition.
The propagation of heavy nuclei
===============================
As we said before, CR with energies above the ankle are most probably extragalactic. This means that in their journey they may be attenuated by the interactions with the photon background. This background consists essentially of the microwave photons of the 2.7$^\circ$K cosmic background radiation and, at larger energies, of the intergalactic background of IR photons emitted by galaxies. The background of optical radiation turns out to be of no relevance for UHECR propagation.
Although the CMB density is well known, the intergalactic IR one cannot be measured directly and has to be estimated from the observation of the spatial distribution, IR spectra and emissivity of the galaxies which are sources for this IR emission. This was done recently by Malkan and Stecker [@ma98], who obtained a result which is about an order of magnitude smaller than previous estimates. We will then adopt for the spectral density of the IR background $${{\rm d}n\over {\rm d}\epsilon}=1.1\times 10^{-4}\left({\epsilon\over {\rm eV}}
\right)^{-2.5}\ {\rm cm^{-3}eV^{-1}}$$ for photon energies $\epsilon$ in the range between $2\times 10^{-3}$ eV and 0.8 eV. This is a factor of 10 smaller than the “high infrared (HIR)” density adopted in ref. [@pu76], and is in the upper range of the recent estimates. In order to quantify the possible effects of an optical intergalactic background, we just modeled this last with a Planckian distribution with $T=5000^\circ$K and a dilution factor of $1.2\times 10^{-15}$, as in [@pu76].
UHECR nuclei propagating through these photon backgrounds will loose energy mainly by two processes:
$i)$ photopair production, which has a threshold corresponding to photon energies in the rest frame of the nucleus of $2m_ec^2
\simeq 1$ MeV. This process was studied in detail by Blumenthal [@bl70], and the main contribution arises from interactions with CMB photons. We used for the energy loss rate the expressions given in ref. [@ch92].
$ii)$ photodisintegration losses, for which the rate of emission of $i$ nucleons from a nucleus of mass $A$ (with cross section $\sigma_{A,i}$) is given by $$R_{A,i}={1\over 2\gamma^2}\int_0^\infty {{\rm d}\epsilon \over
\epsilon^2}{{\rm d}n\over {\rm d}\epsilon}\int_0^{2\gamma \epsilon}
{\rm d}\epsilon' \epsilon'\sigma_{A,i}(\epsilon'),$$ where $\gamma$ is the relativistic factor of the nucleus, $\epsilon$ the photon energy in the observer’s system and $\epsilon'$ its energy in the rest frame of the CR nucleus.
The cross sections for photodisintegration $\sigma_{A,i}(\epsilon')$ contain essentially two regimes. At $\epsilon'<30$ MeV there is the domain of the giant resonance and the disintegration proceeds mainly by the emission of one or two nucleons. At higher energies, the cross section is dominated by multi-nucleon emission for heavy nuclei and is approximately flat up to $\epsilon'\sim 150$ MeV. We fitted the various $\sigma_{A,i}$ with the parameters in Table I and II of ref. [@pu76]. A useful quantity to estimate the energy loss rate by photodisintegration is given by the effective rate $$R_{eff,A}={{\rm d}A\over {\rm d}t}=\sum_i iR_{A,i}.$$
Since neglecting pair creation processes one has that photodisintegrations alone lead to $E^{-1}$d$E/$d$t=A^{-1}$d$A/$d$t$, the energy loss time for photodisintegration is then $A/R_{eff,A}$. The different contributions to this quantity are plotted in Fig. 1. We show separately the contributions to the disintegration from CMB, IR and optical photons for Fe nuclei, together with the total one (solid line) and the photopair creation energy loss rate[^2].
It is apparent that the optical background has no relevant effect, that the IR one dominates the photodisintegration processes below $10^{20}$ eV and the CMB dominates above $10^{20}$ eV. The pair creation rate is relevant for Fe energies $4\times 10^{19}$ eV–$2\times
10^{20}$ eV (i.e. $\gamma$ factors $\sim 1$–$4\times 10^9$), for which the typical CMB photon energy in the rest frame of the nucleus is above threshold ($>1$ MeV) but still well below the peak of the giant resonance ($\sim 10$–20 MeV). The effect of pair creation losses is to reduce the $\gamma$ factor of the nucleus, obviously leaving $A$ unchanged.
Results
=======
Using the rates just discussed, we performed a Monte Carlo simulation in order to follow the possible disintegration histories of Fe nuclei. In figures 2 and 3 we plot the final mass (i.e. $A$) and energy $E$ of the heaviest fragment surviving from the disintegration process. We show in these figures the results of simulations with initial values of the relativistic factor $\gamma_0=1\times 10^{10}$, $4\times 10^9$ and $1\times 10^9$. The curves shown for each value of $\gamma_0$ correspond, in Fig. 2, to the average value $\langle A\rangle$ from all the simulations (solid line) and the region (between the two dashed lines) including 95% of the simulations[^3]. This gives a clear idea of the range of values which can result from fluctuations from the average behaviour.
To further understand the relevance of the different processes and the impact of the new determinations of the IR density, we also plot the results for $\langle A\rangle$ obtained in simulations which do not include pair creation processes (dotted lines) and also the results we would obtain (dot-dashed line) with an IR density a factor of ten larger (i.e. the HIR density of ref. [@pu76]). Figure 3 is similar but for the final values of the energy.
From these figures we can draw the following conclusions:
$i)$ For large initial energies ($E_0>2.5\times 10^{20}$ eV, i.e. $\gamma_0>5\times 10^9$), both the effects of the IR photons and of pair creation processes are of no relevance along the whole journey of the nucleus, and the energy losses are essentially due to photodisintegration off CMB photons alone.
$ii)$ At $\gamma_0<5\times 10^9$ the pair creation losses start to be relevant, reducing the value of $\gamma$ significantly as the nucleus propagates distances $O$(100 Mpc). The effect is maximum for $\gamma_0\simeq 4\times 10^9$ but becomes small again for $\gamma_0\la 1\times 10^9$, for which appreciable effects only appear for cosmological distances ($>10^3$ Mpc), as can be simply understood from Fig. 1. The effect of neglecting pair creation losses translates into keeping $\gamma=\gamma_0$ constant during the propagation, and this enhances the photodisintegration rates and then reduces $\langle A\rangle$ more rapidly.
$iii)$ Also for $\gamma_0<5\times 10^9$ the reduction in the IR density adopted has sizeable effects. In this respect point $ii)$ is relevant, since pair creation losses shift the values of $\gamma$ towards a domain where IR photons become increasingly important with respect to CMB ones. With the new values of the IR density the effects of photodisintegrations become small already for $\gamma_0\simeq 1\times 10^9$ if we consider propagation distances below $10^3$ Mpc (i.e. for $t<10^{17}$ s).
$iv)$ The effects of neglecting pair creation losses are less pronounced in Fig. 3. For instance, for $\gamma_0=4\times 10^9$ the average energies with and without pair creation processes are similar up to $t\simeq 10^{16}$ s while the $\langle A\rangle$ values differ sizeably already for $t\simeq 3\times 10^{15}$ s. This is due to a partial cancellation between the effects of the evolution of $\gamma$ and of $A$ in the values of the final energy ($E=m_p\gamma A$), since neglecting pair creation losses does not allow $\gamma$ to decrease but makes instead $A$ to drop faster[^4].
$v)$ The effects of fluctuations due to different photodisintegration histories are not negligible. They give a spread in $A$ (and $E$) of the order of 10% (considering the 95% probability range) for $\langle A\rangle \simeq 40$ but relatively larger for smaller $\langle A\rangle$, since variations $\Delta A\sim 10$–15 at a given time $t$ can appear between different simulations.
In figures 4 and 5 we plot more sample values of $\langle A\rangle$ and $\langle E\rangle$, for values of $\gamma_0=2\times 10^{10}$, $1
\times 10^{10}$ and (8, 6, 4, 2, 1)$\times 10^9$. Looking at Fig. 5 it is easy to infer the maximum energies which can be obtained from Fe nuclei injected at any fixed distance $d$. In particular, for $d=100$ Mpc ($t=10^{16}$ s) the maximum average energy is $E_{max}\simeq 8\times 10^{19}$ eV and originates from $\gamma_0\simeq 2$–4$\times 10^9$. Comparing with Fig. 4 we see that these maximum energy events would correspond to fragments with masses $A(E_{max})\simeq 30$–50, i.e. a rather heavy composition.
Fluctuations from the average behaviour can only slightly increase the maximum attainable energies, and this is illustrated with the dashed line, which represents the upper boundary of the 95% CL ranges (i.e. 97.5% of the simulations are below this curve) for any initial value of $\gamma_0$.
For source distances $d= 10$ Mpc, average energies up to $E_{max}\simeq 2\times 10^{20}$ eV can result, and an interesting pile-up effect is observed since a broad range of initial energies (with $\gamma_0\sim 4$–$8\times 10^9$) lead to approximately the same final energy ($\sim E_{max}$). This can produce a bump in the spectrum from sources at these distances if indeed the highest energy events originate from heavy nuclei. Due to the spread in values of $\gamma_0$ at $E_{max}$, we see from a comparison with Fig. 4 that there will also be a wide spread in the final composition, with $A(E_{max})\simeq 10$–45. Events with energies 2–$3\times 10^{20}$ eV may appear as low probability fluctuations from the mean behaviour if $d\simeq 5$–8, having initially $\gamma_0>10^{10}$ and a low mass final composition ($A<10$).
For smaller source distances ($d\sim$ few Mpc), events with $E\simeq
3\times 10^{20}$ eV could be heavier nuclei with smaller initial energies ($\gamma_0>6\times 10^9$).
For very large values of $\gamma_0$ ($\gamma_0>2\times 10^{10}$), the heavy nuclei completely disintegrate in less than 10 Mpc, and the photopion production (not included here) becomes the main attenuation process for the secondary nucleons, which are then subject to the usual GZK cutoff.
In conclusion, the main implication of the lower values of the IR density recently estimated is to increase the mean free path of the heavy nuclei with initial $\gamma$ factors below $\sim 5\times 10^9$, for which most CMB photons are below the peak of the giant resonance for photodisintegration. Due to the fragmentation of the nuclei by photodisintegration and the pair creation energy losses, the final energies of the fragments are typically below $2\times 10^{20}$ eV for travel distances $\sim 10$ Mpc, and below $10^{20}$ eV for distances $\sim 100$ Mpc. The new value of the IR density is then of little help in the attempts to understand the highest energy events observed ($E\sim 2$–$3\times
10^{20}$ eV), which could not have originated as heavy nuclei at distances beyond $\sim 10$ Mpc. The lack of obvious candidate sources at closer distances [@el95] leave the nature and origin of these events still a mistery.
Work partially supported by CONICET, Argentina. We thank Luis Anchordoqui and M. Teresa Dova for discussions.
[100]{}
K. Greisen, ; G. T. Zatsepin and V. A. Kuz’min, [*Sov. Phys. JETP Lett.* ]{}[**4**]{} (1968) 78.
F. Stecker, .
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J. W. Elbert and P. Sommers, [ *Astrophys. J.* ]{} [**441**]{} (1995) 151.
M. A. Malkan and F. W. Stecker, [ *Astrophys. J.* ]{} [**496**]{} (1998) 13.
F. W. Stecker, .
L. N. Epele and E. Roulet, .
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[^1]: It has to be stressed that Fe nuclei are good candidates for UHECRs, due to their high abundance in supernova environments and their large value of $Z$, which enhances the energy achievable in the acceleration process.
[^2]: Looking at this and the following figures, it is important to keep in mind that 1 Mpc$=1.03\times 10^{14}$ s.
[^3]: Only 2.5% of the simulations are below the lower curves and 2.5% are above the upper ones.
[^4]: This in particular shows that the inclusion of pair creation losses does not modify the maximum attainable energies computed in [@ep98].
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'The hierarchy of the integrable nonlinear equations associated with the quadratic bundle is considered. The expressions for the solution of the linearization of these equations and their conservation law in the terms of the solutions of the corresponding Lax pairs are found. It is shown for the first member of the hierarchy that the conservation law is connected with the solution of the linearized equation due to the Noether’s theorem. The local hierarchy and three nonlocal ones of the infinitesimal symmetries and the conservation laws that are explicitly expressed through the variables of the nonlinear equations are derived.'
author:
- |
N.V. Ustinov\
*Quantum Field Theory Department, Tomsk State University,\
*36 Lenin Avenue, Tomsk, 634050, Russia**
date:
title: 'Infinitesimal symmetries and conservation laws of the DNLSE hierarchy and the Noether’s theorem'
---
Introduction
============
One of the most effective tools of studying the nonlinear phenomena is the inverse scattering transformation (IST) method [@NMPZ; @AS]. This method reduces the solution of the Cauchy initial value problem of the nonlinear partial differential equations (PDE’s), which admit a representation as the compatibility condition of the overdetermined linear system (Lax pair), to solving linear singular integral equations. It is of a special significance that many of the PDE’s playing an important role in different branches of physics can be investigated in its frameworks. For example, the derivative nonlinear Schrödinger equation (DNLSE) that was originally deduced for the Alfvén waves of finite amplitude [@MOMT; @M] and the equations of massive Thirring model (MTM) [@Th] belong to the class of the PDE’s integrable with the help of the IST method for the quadratic bundle \[6–8\]. It was revealed that DNLSE describes also the behavior of drifting filamentations in nonlinear electrostatic waves of magnetized plasmas [@SShY], light pulses in the optical fibers \[10–12\], magnetic holes of space plasmas [@B] and large-amplitude magnetohydrodynamics waves [@R]. The MTM equations were recently shown to appear in the coherent optics and nonlinear acoustics [@Z1] as a limiting case of the system of long/short-wave coupling (see [@SU] and references therein).
The integrable nonlinear PDE’s are well known to possess the infinite hierarchies of infinitesimal symmetries and conservation laws. An existence of them was proposed as the integrability test to characterize the equations solvable by IST (see, e.g., [@Fokas; @MShS]). There are different methods of obtaining the infinitesimal symmetries and conservation laws, which originate from the study of KdV equation [@Miura]. Given a set of the scattering data (namely, the time-invariant part of them), infinite hierarchies of the conserved densities are constructed \[1, 2, 6, 20–22\]. The Bäcklund transformation (BT) of the integrable equation was used to generate the hierarchy of its conservation laws [@K; @DB]. The approach that exploits the Noether’s theorem was applied in [@St_1; @St_2] for the derivation of the conservation laws of sine–Gordon and KdV equations. To produce the corresponding hierarchy of infinitesimal symmetries, the implicit expressions for the solutions of the linearized equations, which are obtained by means of infinitesimal BT, were expanded in the power series on the parameter of this BT. The infinitesimal version of the dressing method was suggested in [@OSch] to construct the infinitesimal symmetries of integrable PDE’s. Similar expressions for the perturbations of some nonlinear PDE’s and their Lax pairs were presented in [@S]. The geometrical approaches that utilize the projective transformations or treat the soliton equations as descriptions of pseudospherical surfaces were developed for nonlinear PDE’s associated with matrix Lax pairs of the second order in [@S_1; @S_2] (see also [@KSI]) and [@CT], respectively. The hierarchies of local and nonlocal conservation laws for DNLSE were found by means of these methods [@S_2; @W]. The method based on the theory of $\tau$-functions was applied to scalar and two-component KP hierarchies [@MaSS].
Although the methods mentioned above appeal to underlying Lax pair to produce the hierarchies of infinitesimal symmetries and conservation laws, they do not entirely cover the class of PDE’s representable as the compatibility condition. This concerns especially the cases of reductions of the nonlinear PDE’s [@Mikh] and their integrable deformations (see, e.g., [@BZ; @Z1; @Z2]), which are most interesting from physical point of view. The knowledge of the infinitesimal symmetries and conservation densities of the hierarchy allows one to make sure that the PDE given belongs to it. The approach applicable to all integrable nonlinear equations can be based, for instance, on explicit expressions for the solution of linearized equation and the conservation law in the terms of the solutions of corresponding Lax pairs. In the present report, we construct the infinite hierarchies of local and nonlocal infinitesimal symmetries and conservation laws for the DNLSE hierarchy using such the approach.
The paper is organized as follows. The nonlinear equations of the DNLSE hierarchy and their Lax pairs are presented in Sec.II. The formulas for the expansions in series on the spectral parameter powers of the solutions of the Lax pairs are also given there. The solution of the linearization of the nonlinear PDE’s, which is expressed in the terms of the solutions of corresponding Lax pairs, is obtained by means of the infinitesimal version of the binary Darboux transformation (DT) [@MS] in Sec.III. This technique has been applied to the DNLSE \[38–40\] and to obtain the infinitesimal symmetries of the nonlinear PDE’s \[41–43\]. To generate the hierarchies of local and nonlocal infinitesimal symmetries explicitly expressed through the variables of the nonlinear equations under consideration, the expansions in series of the solutions of the Lax pairs or the recursion operator of the hierarchy can be used. In Sec.IV the conservation law for the DNLSE hierarchy is derived. Substitution of the expansion in series of the Lax pairs solutions into this formula yields the hierarchies of local and nonlocal conservation laws. The connection due to the Noether’s theorem of the infinitesimal symmetries and conservation laws we found is shown in this section for the DNLSE case.
The DNLSE hierarchy
===================
Let us consider (direct) Lax pair $$\psi_x=U(\lambda)\,\psi,
\label{psi_x}$$ $$\psi_t=V(\lambda)\,\psi,
\label{psi_t}$$ where $\psi=\psi(x,t,\lambda)=(\psi_1,\psi_2)^T$ is the vector-column solution; $\lambda$ is complex parameter referred to as the spectral parameter in the IST theory; $U(\lambda)=U(x,t,\lambda)$ and $V(\lambda)=V(x,t,\lambda)$ are $2\times2$ matrix coefficients. The compatibility condition of the overdetermined system (\[psi\_x\],\[psi\_t\]) is $$U(\lambda)_t-V(\lambda)_x+[\,U(\lambda)\,,V(\lambda)\,]=0.
\label{cc}$$
We suppose in what follows that $$U(\lambda)=\lambda^2U^{(2)}+\lambda\,U^{(1)}
\label{U}$$ (i.e., Eq.(\[psi\_x\]) is the quadratic bundle) and $$U^{(2)}=
\left(
\begin{array}{cc}
-i&0\\
0&i
\end{array}
\right),\quad
U^{(1)}=
\left(
\begin{array}{cc}
0&q\\
r&0
\end{array}
\right).
\label{U's}$$ If $V(\lambda)$ is chosen in the next form $$V(\lambda)=\sum_{j=1}^{2m}\lambda^jV^{(j)},
\label{V}$$ then Eq.(\[cc\]) gives the expressions for the matrix coefficients of $V(\lambda)$: $$V^{(2m-2j)}=v^{(2m-2j)}
\left(
\begin{array}{cc}
1&0\\
0&-1
\end{array}
\right),\quad
V^{(2m-2j-1)}=
\left(
\begin{array}{cc}
0&v^{(2m-2j-1)}_{12}\\
v^{(2m-2j-1)}_{21}&0
\end{array}
\right),
\label{V's}$$ ($j=0,...,m-1$), where $$v^{(2m-2j)}=\partial_x^{-1}(qv^{(2m-2j-1)}_{21}-rv^{(2m-2j-1)}_{12})\,,\quad
\label{v's1}$$ $$\left(
\begin{array}{c}
{v^{(2m-2j-1)}_{12}}_{\mathstrut}\\
{v^{(2m-2j-1)}_{21}}^{\mathstrut}
\end{array}
\right)=\widehat{R}^{j+1}\left(
\begin{array}{c}
{0_{\mathstrut}}_{\mathstrut}\\
{0^{\mathstrut}}^{\mathstrut}
\end{array}
\right),
\label{v's2}$$ $$\widehat{R}=\frac12
\left(
\begin{array}{cc}
i\partial_x+q\partial_x^{-1}r\partial_x&q\partial_x^{-1}q\partial_x\\
r\partial_x^{-1}r\partial_x&-i\partial_x+r\partial_x^{-1}q\partial_x
\end{array}
\right),
\label{Rh}$$ and system of nonlinear equations $$\left(
\begin{array}{c}
q_t\\
r_t
\end{array}
\right)=\partial_x\widehat{R}^m\left(
\begin{array}{c}
0\\
0
\end{array}
\right).
\label{h}$$ (Note that operators $\partial_x^{-1}$ in Eqs.(\[v’s1\],\[v’s2\]) for equal $j$’s add the same time–dependent functions as the constants of integration.) To obtain these formulas we make use the identities $$R\partial_x=\partial_x\widehat{R},$$ $$R^{-1}=2
\left(
\begin{array}{cc}
-i+q\partial_x^{-1}r&-q\partial_x^{-1}q\\
-r\partial_x^{-1}r&i+r\partial_x^{-1}q
\end{array}
\right)
\left(
\begin{array}{cc}
\partial_x^{-1}&0\\
0&\partial_x^{-1}
\end{array}
\right)
\label{invR}$$ with operator $R$ being defined in the following manner $$R=\frac12
\left(
\begin{array}{cc}
i\partial_x+\partial_xq\partial_x^{-1}r&\partial_xq\partial_x^{-1}q\\
\partial_xr\partial_x^{-1}r&-i\partial_x+\partial_xr\partial_x^{-1}q
\end{array}
\right).
\label{R}$$ As it will be seen in the next section, $\widehat{R}$ and its adjoint $R$ are the squared eigenfunction operator and the recursion one [@Fokas] of the hierarchy considered.
The hierarchy of nonlinear equations (\[h\]) was found in [@GIK]. It admits under appropriate choice of the constants of integration the next reduction $$r=\pm q^*.
\label{rc}$$ In this case, the first nontrivial equation of the hierarchy is reduced after rescaling to DNLSE $$iq_t+q_{xx}\mp i(|q|^2q)_x=0.
\label{DNLSE}$$
Let us consider the expansions of the solutions of Eqs.(\[psi\_x\],\[psi\_t\]) in the series on the spectral parameter powers. In the neighborhood of point $\lambda=\infty$, the vector solutions of the Lax pairs of nonlinear equations (\[h\]) are represented as $$\psi=\sum\limits_{k=0}^{\infty}\lambda^{-k}A^{(k)}\Lambda\,|a\rangle.
\label{psi_inf}$$ Here $|a\rangle$ is a constant vector-column, $$\Lambda=\left(
\begin{array}{cc}
\mbox{e}^{\displaystyle-i\lambda^2x+\lambda^{2m}v^{(2m)}t}&0\\
0&\mbox{e}^{\displaystyle i\lambda^2x-\lambda^{2m}v^{(2m)}t}
\end{array}
\right)$$ and coefficients $A^{(k)}$ solve system of equations $$\left\{
\begin{array}{l}
[A^{(k)},U^{(2)}]+A_x^{(k-2)}=U^{(1)}A^{(k-1)}\\
\mbox{}[A^{(k)},V^{(2m)}]+A_t^{(k-2m)}=
\sum\limits_{j=1}^{2m-1}V^{(2m-j)}A^{(k-j)}
\end{array}
\right..$$ An expansion in series of the solutions of Lax pairs considered in the neighborhood of point $\lambda=0$ has form $$\psi=\sum\limits_{k=0}^{\infty}\lambda^kB^{(k)}\,|a\rangle,
\label{psi_0}$$ where $B^{(0)}=E$ and coefficients $B^{(k)}$ ($k\ge1$) are determined from equations $$\left\{
\begin{array}{l}
B_x^{(k)}=U^{(2)}B^{(k-2)}+U^{(1)}B^{(k-1)}\\
B_t^{(k)}=\sum\limits_{j=1}^{2m}V^{(j)}B^{(k-j)}
\end{array}
\right..$$ It is seen that the first coefficients of the expansions are $$A^{(0)}=\left(
\begin{array}{cc}
w&0\\
0&w^{-1}
\end{array}
\right),\quad
A^{(1)}=\frac{i}{2}\left(
\begin{array}{cc}
0&-qw^{-1}\\
rw&0
\end{array}
\right),$$ $$A^{(2)}=\frac14\left(
\begin{array}{cc}
\displaystyle w\!\int\limits^{\,\,x}\!(qr_x+iq^2r^2/2)\,dx&0\\
0&\displaystyle w^{-1}\!\int\limits^{\,\,x}\!(q_xr-iq^2r^2/2)\,dx
\end{array}
\right),$$ $$B^{(1)}=\left(
\begin{array}{cc}
0&u\\
v&0
\end{array}
\right),\quad
B^{(2)}=\left(
\begin{array}{cc}
\displaystyle -ix+\!\int\limits^{\,\,x}\!qv\,dx&0\\
0&\displaystyle ix+\!\int\limits^{\,\,x}\!ru\,dx
\end{array}
\right),$$ where $$w=\mbox{exp}\Bigl(i\!\int\limits^{\,\,x}\!qr/2\,dx\Bigr),\quad
u=\int\limits^{\,\,x}\!q\,dx,\quad v=\int\limits^{\,\,x}\!r\,dx.$$
Darboux transformation and infinitesimal symmetries
===================================================
Hierarchy of nonlinear equations (\[h\]) follows also from the compatibility condition of dual Lax pair $$\xi_x=-\xi\,U(\symbol{26}),
\label{xi_x}$$ $$\xi_t=-\xi\,V(\symbol{26}).
\label{xi_t}$$ Here $\xi\equiv\xi(x,t,\symbol{26})$ is a vector-row solution, $\symbol{26}$ is the spectral parameter of the dual pair. Since matrix coefficients $U(\lambda)$ and $V(\lambda)$ defined by Eqs.(\[U\]–\[V’s\]) satisfy conditions $$\sigma_1U(-\lambda)+U(\lambda)^T\sigma_1=0,\quad
\sigma_1V(-\lambda)+V(\lambda)^T\sigma_1=0,
\label{cond1}$$ where $\sigma_1$ is Pauli matrix $$\sigma_1=
\left(
\begin{array}{cc}
0&1\\
1&0
\end{array}
\right),$$ the next connection between the solutions of systems (\[psi\_x\],\[psi\_t\]) and (\[xi\_x\],\[xi\_t\]) exists: $$\xi=\psi^T\sigma_1,\quad\symbol{26}=-\lambda.
\label{conn1}$$ In the case of reduction (\[rc\]) the solutions with complex conjugate spectral parameters are also connected. For instance, $(\psi_2^*,\pm\psi_1^*)^T$ is a solution of direct Lax pair (\[psi\_x\],\[psi\_t\]) with spectral parameter $\lambda^*$.
Let vector-column $\varphi=(\varphi_1,\varphi_2)^T$ and vector-row $\chi=(\chi_1,\chi_2)$ are solutions of Lax pairs (\[psi\_x\],\[psi\_t\]) and (\[xi\_x\],\[xi\_t\]) with spectral parameters $\mu$ and $\nu$, respectively. The Lax pairs are covariant with respect to ”turned” binary Darboux transformation (BDT) $\{\psi,\xi,U(\lambda),V(\lambda)\}\to
\{\psi[1],\xi[1],U(\lambda)[1],V(\lambda)[1]\}$ of the form $$\psi[1]=gT(\lambda)\psi,\quad
\xi[1]=\xi T(\symbol{26})^{-1}g^{-1},
\label{t_psi_xi}$$ $$U(\lambda)[1]=\lambda^2U^{(2)}[1]+\lambda\,U^{(1)}[1],\quad
V(\lambda)[1]=\sum_{j=1}^{2m}\lambda^jV^{(j)}[1],
\label{t_U_V}$$ where $$T(\lambda)=E-\frac{\mu-\nu}{\lambda-\nu}P=
\Bigl(1-\frac{\nu-\mu}{\lambda-\mu}P\Bigr)^{-1},\quad
P=\frac{\varphi\chi}{\chi\varphi},\quad g=\sigma_1T(0)^{-1}$$ and $$\vphantom{\sum\limits_{k=j+1}^{2m}}
U^{(2)}[1]=gU^{(2)}g^{-1},\quad V^{(2m)}[1]=gV^{(2m)}g^{-1},
\label{t_U2}$$ $$\vphantom{\sum\limits_{k=j+1}^{2m}}
U^{(1)}[1]=g\Bigl(U^{(1)}+(\mu-\nu)[U^{(2)},P]\Bigr)g^{-1},
\label{t_U1}$$ $$V^{(j)}[1]=gV^{(j)}g^{-1}+(\mu-\nu)\sum\limits_{k=j+1}^{2m}\nu^{k-j-1}
\Bigl(V^{(k)}[1]gP-gP\,V^{(k)}\Bigr)g^{-1}
\label{t_V}$$ ($j=1,...,2m-1$). We call this transformation as ”turned” because formulas (\[t\_psi\_xi\]–\[t\_V\]) is a product of usual BDT [@U_1; @LU] and additional gauge transformation performed with the help of matrix $g$. This additional transformation allows us to avoid an appearance of the terms at the zero power of $\lambda$ in the expressions for $U(\lambda)[1]$ and $V(\lambda)[1]$.
Conditions (\[cond1\],\[conn1\]) are fulfilled for transformed matrix coefficients $U(\lambda)[1]$, $V(\lambda)[1]$ and solutions $\psi[1]$, $\xi[1]$ of the transformed Lax pairs if we impose restriction $$\chi=\varphi^T\sigma_1,\quad\nu=-\mu.$$ In this case, we have $$U^{(2)}[1]=U^{(2)},$$ $$V^{(2m)}[1]=V^{(2m)}.$$ Then, Eq.(\[t\_U1\]) gives us expressions for new (transformed) solutions of hierarchy of nonlinear equations (\[h\]): $$q[1]=r-\frac{1}{\mu}\left(\frac{\varphi_2}{\varphi_1}\right)_{\!x},$$ $$r[1]=q-\frac{1}{\mu}\left(\frac{\varphi_1}{\varphi_2}\right)_{\!x}.$$ The second iteration of the BDT (\[t\_psi\_xi\]–\[t\_V\]) keeping conditions (\[cond1\],\[conn1\]) yields the following formulas $$q[2]=q-\frac{\mu_1^2-\mu_2^2}{\mu_1\mu_2}
\left(\frac{\varphi_1^{(1)}\varphi_1^{(2)}}
{\mu_1\varphi_1^{(1)}\varphi_2^{(2)}-\mu_2\varphi_2^{(1)}\varphi_1^{(2)}}
\right)_{\!x},
\label{t_q_2}$$ $$r[2]=r+\frac{\mu_1^2-\mu_2^2}{\mu_1\mu_2}
\left(\frac{\varphi_2^{(1)}\varphi_2^{(2)}}
{\mu_2\varphi_1^{(1)}\varphi_2^{(2)}-\mu_1\varphi_2^{(1)}\varphi_1^{(2)}}
\right)_{\!x},
\label{t_r_2}$$ where $\varphi^{(k)}_1$ and $\varphi^{(k)}_2$ are the components of vector solution $\varphi^{(k)}$ of the direct Lax pair with spectral parameters $\mu_k$ ($k=1,2$). If we put here $\varphi^{(2)}=({\varphi^{(1)}_2}^*,\pm{\varphi^{(1)}_1}^*)^T$ and $\mu_2=\mu_1^*$, then $$r[2]=\pm q[2]^*.$$ This way we come to DT for the DNLSE hierarchy. The compact form of $N$-th iteration of this transformation is presented in [@St_3].
Considering limits $\mu_1\to\mu$ and $\mu_2\to\mu$ in Eqs.(\[t\_q\_2\],\[t\_r\_2\]), one obtains the next expressions (up to a multiplier) for solution of the linearization of system (\[h\]): $$\delta q=\left(\varphi_1^{(1)}\varphi_1^{(2)}\right)_{\!x},
\label{iq}$$ $$\delta r=-\left(\varphi_2^{(1)}\varphi_2^{(2)}\right)_{\!x}.
\label{ir}$$ It is checked by straightforward calculation that $$R\left(\begin{array}{c}\delta q\\ \delta r\end{array}\right)
=\mu^2\left(\begin{array}{c}\delta q\\ \delta r\end{array}\right).
\label{Rd}$$ This identity allows us to define in a recurrent manner the coefficients of expansions of the right–hand sides of Eqs.(\[iq\],\[ir\]) in the power series on the spectral parameter at a neighborhood of the points $\mu=\infty$ and $\mu=0$. The coefficients of these expansions $$\delta q=\sum\limits_{k=0}^{\infty}\mu^{-2k}\delta q^{(k)},\quad
\delta r=\sum\limits_{k=0}^{\infty}\mu^{-2k}\delta r^{(k)}$$ and $$\delta q_j=\sum\limits_{k=0}^{\infty}\mu^{2k}\delta q_j^{(k)},\quad
\delta r_j=\sum\limits_{k=0}^{\infty}\mu^{2k}\delta r_j^{(k)}$$ form the infinite hierarchies of infinitesimal symmetries. Operator $R$ satisfying (\[Rd\]) is nothing but the recursion operator of the hierarchy (\[h\]). In the case of point $\mu=0$, there exist three hierarchies of nonlocal infinitesimal symmetries $\delta q_j^{(k)}$, $\delta r_j^{(k)}$ ($j=1,2,3$, $k=0,1,...$) that correspond to different choices of the constants of integration in operator $R^{-1}$ (see Eq.(\[invR\])). The first nontrivial members of the hierarchies for the points $\mu=\infty$ and $\mu=0$, respectively, are $$\left\{
\begin{array}{l}
\delta q^{(1)}=q_x\\
\delta r^{(1)}=r_x
\end{array}
\right.\!,\quad
\left\{
\begin{array}{l}
\delta q^{(2)}=q_t/2\\
\delta r^{(2)}=r_t/2
\end{array}
\right.\!,\quad
\left\{
\begin{array}{l}
\delta q^{(3)}=(-q_{xx}+3iq_xqr+3q^3r^2/2)_x/4\\
\delta r^{(3)}=(-r_{xx}-3ir_xqr+3q^2r^3/2)_x/4
\end{array}
\right.\!,$$ $$\left\{
\begin{array}{l}
\delta q_1^{(1)}=q\\
\delta r_1^{(1)}=-r
\end{array}
\right.\!,\quad
\left\{
\begin{array}{l}
\delta q_2^{(1)}=2(qv-i)\\
\delta r_2^{(1)}=-2rv
\end{array}
\right.\!,\quad
\left\{
\begin{array}{l}
\delta q_3^{(1)}=2qu\\
\delta r_3^{(1)}=-2(ru+i)
\end{array}
\right.\!.$$ It is seen from these formulas that $\delta q^{(k)}\sim v_{12,x}^{(2m-2k+1)}$, $\delta r^{(k)}\sim v_{21,x}^{(2m-2k+1)}$ and the infinite hierarchy corresponding to the point $\mu=\infty$ is local. Another way of producing the hierarchies of the infinitesimal symmetries is to substitute expansions (\[psi\_inf\]) and (\[psi\_0\]) into Eqs.(\[iq\],\[ir\]).
Conservation laws and Noether’s theorem
=======================================
Let us consider identity $$(\xi\psi)_{xt}=(\xi\psi)_{tx}.$$ Excluding the derivatives of $\psi$ and $\xi$ on $x$ in the left-hand side and the derivatives on $t$ in the right-hand side with the help of Eqs.(\[psi\_x\],\[psi\_t\]) and (\[xi\_x\],\[xi\_t\]), respectively, and dividing the relation obtained on $\lambda-\symbol{26}$, we come to the conservation law of the DNLSE hierarchy $$T_t+X_x=0,$$ where $$T=\xi\Bigl((\lambda+\symbol{26})U^{(2)}+U^{(1)}\Bigr)\psi,
\label{T}$$ $$X=-\xi\sum_{k=1}^{2m}\sum_{j=0}^{k-1}\lambda^{k-j-1}\symbol{26}^jV^{(k)}\psi.
\label{X}$$ If we put $\lambda=\symbol{26}=\mu$, $\psi=\varphi^{(1)}$ and $\xi=(\varphi_2^{(2)},-\varphi_1^{(2)})$, where vectors $\varphi^{(k)}=(\varphi_1^{(k)},\varphi_2^{(k)})^T$ ($k=1,2$), as it was supposed at the end of the previous section, are solutions of Lax pair (\[psi\_x\],\[psi\_t\]) with spectral parameter $\mu$, then expressions (\[T\],\[X\]) are rewritten in the next manner $$T=-2i\mu(\varphi_1^{(1)}\varphi_2^{(2)}+\varphi_2^{(1)}\varphi_1^{(2)})
+q\varphi_2^{(1)}\varphi_2^{(2)}-r\varphi_1^{(1)}\varphi_1^{(2)},
\label{T_phi}$$ $$\begin{array}{c}
\displaystyle X=-2\sum_{k=1}^{m}k\mu^{2k-1}v^{(2k)}
(\varphi_1^{(1)}\varphi_2^{(2)}+\varphi_2^{(1)}\varphi_1^{(2)})+{}\\
\displaystyle{}+\sum_{k=1}^{m}(2k-1)\mu^{2k-2}
\left(v_{21}^{(2k-1)}\varphi_1^{(1)}\varphi_1^{(2)}-
v_{12}^{(2k-1)}\varphi_2^{(1)}\varphi_2^{(2)}\right).
\end{array}
\label{X_phi}$$ Substitution of expansions (\[psi\_inf\]) and (\[psi\_0\]) of the Lax pair solutions at the neighborhood of points $\mu=\infty$ and $\mu=0$ into these formulas leads to the hierarchies of the conservation laws expressed explicitly through the solutions of nonlinear equations (\[h\]). In the case of point $\mu=0$, for example, we have three infinite hierarchies $T_{j,t}^{(k)}+X_{j,x}^{(k)}=0$ ($j=1,2,3$, $k=0,1,...$), whose first conserved densities and currents are $$T_1^{(0)}=q,\quad X_1^{(0)}=-v_{12}^{(1)},\quad
T_2^{(0)}=r,\quad X_2^{(0)}=-v_{21}^{(1)},
\label{TX_0a}$$ $$T_3^{(1)}=qv-ur,\quad X_3^{(1)}=uv_{21}^{(1)}-vv_{12}^{(1)}-2v^{(2)}.
\label{TX_0b}$$ The first two conservation laws are immediate consequence of the divergent form of Eqs.(\[h\]).
Let us discuss the connection between solutions (\[iq\],\[ir\]) of the linearized equations and the conservation laws found in the case of system of nonlinear equations $$iq_t+q_{xx}-i(q^2r)_x=0,
\label{q_t}$$ $$ir_t-r_{xx}-i(qr^2)_x=0.
\label{r_t}$$ Coefficients of the second equation of Lax pair (\[psi\_x\],\[psi\_t\]) of the system under consideration are $$V^{(4)}=2U^{(2)},\quad V^{(3)}=2U^{(1)},\quad V^{(2)}=qrU^{(2)},\quad
V^{(1)}=
\left(
\begin{array}{cc}
0&iq_x+q^2r\\
-ir_x+qr^2&0
\end{array}
\right)$$ DNLSE (\[DNLSE\]) follows these equations by imposing condition $r=\pm q^*$.
In the terms of potentials $u$ and $v$ the Lagrangian of Eqs.(\[q\_t\],\[r\_t\]) reads as $${\cal L}=i(u_xv_t+v_xu_t)+u_{xx}v_x-v_{xx}u_x-iu_x^2v_x^2\,.$$ Using notations for the Euler–Lagrange equations $$\Lambda(u)\equiv-\left(\frac{\partial\cal L}{\partial u_t}\right)_t
-\left(\frac{\partial\cal L}{\partial u_x}\right)_x
+\left(\frac{\partial\cal L}{\partial u_{xx}}\right)_{xx}=
-2(ir_t-r_{xx}-i(qr^2)_x)=0,$$ $$\Lambda(v)\equiv-\left(\frac{\partial\cal L}{\partial v_t}\right)_t
-\left(\frac{\partial\cal L}{\partial v_x}\right)_x
+\left(\frac{\partial\cal L}{\partial v_{xx}}\right)_{xx}=
-2(iq_t+q_{xx}-i(q^2r)_x)=0,$$ the variation of the Lagrangian, which is caused by the infinitesimal transformations of potentials $u\to u+\varepsilon\delta u$ and $v\to v+\varepsilon\delta v$, is written in a form of Noether’s identity $$\delta{\cal L}=\varepsilon(A_t+B_x+\Lambda(u)\delta u+\Lambda(v)\delta v).
\label{Ni}$$ Here $$A=\frac{\partial\cal L}{\partial u_t}\,\delta u+
\frac{\partial\cal L}{\partial v_t}\,\delta v=i(q\,\delta v+r\,\delta u),$$ $$B=\left(\frac{\partial\cal L}{\partial u_x}-
\left(\frac{\partial\cal L}{\partial u_{xx}}\right)_x\,\right)\delta u+
\frac{\partial\cal L}{\partial u_{xx}}\,\delta u_x+
\left(\frac{\partial\cal L}{\partial v_x}-
\left(\frac{\partial\cal L}{\partial v_{xx}}\right)_x\,\right)\delta v+
\frac{\partial\cal L}{\partial v_{xx}}\,\delta v_x=$$ $$=q_x\,\delta v-r_x\,\delta u+r\,\delta q-q\,\delta r-iqr^2\,\delta u-
iq^2r\,\delta v.$$ Given a symmetry of Eqs.(\[q\_t\],\[r\_t\]), a conservation law is derived from Eq.(\[Ni\]) due to the Noether’s theorem. Few examples of the symmetries and associated conservation densities and currents are listed below:
[**1)**]{} $\delta u=1$, $\delta v=0$: $$\delta {\cal L}=0,$$ $$T_1=ir,\quad X_1=iv_t-2r_x-2iqr^2.
\label{TX_1}$$ [**2)**]{} $\delta u=0$, $\delta v=1$: $$\delta {\cal L}=0,$$ $$T_2=iq,\quad X_2=iu_t+2q_x-2iq^2r.
\label{TX_2}$$ [**3)**]{} $u\to u\mbox{e}^{i\varepsilon}$, $v\to v\mbox{e}^{-i\varepsilon}$, $\delta u=iu$, $\delta v=-iv$: $$\delta {\cal L}=0,$$ $$T_3=qv-ur,\quad X_3=u_tv-uv_t-2i(q_xv-qr+ur_x)-2(qv-ur)qr.
\label{TX_3}$$ [**4)**]{} $x\to x+\varepsilon$, $\delta u=q$, $\delta v=r$: $$\delta {\cal L}=\varepsilon{\cal L}_x,$$ $$T_4=2iqr,\quad X_4=2(q_xr-qr_x)-3iq^2r^2.
\label{TX_4}$$ [**5)**]{} $t\to t+\varepsilon$, $\delta u=u_t$, $\delta v=v_t$: $$\delta {\cal L}=\varepsilon{\cal L}_t,$$ $$T_5=-q_xr+qr_x+iq^2r^2,\quad X_5=i(q_{xx}r-2q_xr_x+qr_{xx}-2q^3r^3)+
3(q_xr-qr_x)qr.
\label{TX_5}$$ Conservation laws that arise in the first and second cases are trivial. The symmetries of the potentials in the third, fourth and fifth cases correspond, respectively, to infinitesimal symmetries $\delta q_1^{(1)}$, $\delta r_1^{(1)}$, $\delta q^{(1)}$, $\delta r^{(1)}$ and $\delta q^{(2)}$, $\delta r^{(2)}$ presented at the end of previous section. Conserved density $T_5$ is proportional to the Hamiltonian density of DNLSE [@GIK]. The Noether’s theorem was applied in [@DF] to obtain $T_3$ and $X_3$, which are nothing but $T_3^{(1)}$ and $X_3^{(1)}$ (\[TX\_0b\]). Hence, $\delta r_1^{(1)}$ and $\delta q^{(1)}$ are connected by the Noether’s theorem with $T_3^{(1)}$ and $X_3^{(1)}$. It will be proven in the sequel that this is valid for all members of the hierarchies of infinitesimal symmetries and conservation laws.
Formulas (\[iq\],\[ir\]) give us solutions of the linearized equations on potentials $$\delta u=\varphi_1^{(1)}\varphi_1^{(2)},$$ $$\delta v=-\varphi_2^{(1)}\varphi_2^{(2)}.$$ It is remarkable that we are able to put the corresponding variation of Lagrangian in divergent form: $$\delta{\cal L}=\left(iu\,\delta r+iv\,\delta q+
4\mu(\varphi_1^{(1)}\varphi_2^{(2)}+\varphi_2^{(1)}\varphi_1^{(2)})\right)_t+$$ $$+\left(\vphantom{\varphi_1^{(1)}}
v\,\delta q_{{}\,x}-u\,\delta r_{{}\,x}+q\,\delta r-r\,\delta q-
i(ur+2qv)r\,\delta q-i(qv+2ur)q\,\delta r+
\right.$$ $$\left.
+8i\mu^2(r\delta u+q\delta v)
-16\mu^3(\varphi_1^{(1)}\varphi_2^{(2)}+\varphi_2^{(1)}\varphi_1^{(2)})
\right)_x\,.$$ Combining this expression with Eq.(\[Ni\]), we come after a cancellation of the terms with potentials $u$ and $v$ to the conservation law, whose conserved density $\tilde T$ and current $\tilde X$ are defined in the following manner $$\tilde T=4\mu(\varphi_1^{(1)}\varphi_2^{(2)}+\varphi_2^{(1)}\varphi_1^{(2)})+
2i(q\varphi_2^{(1)}\varphi_2^{(2)}-r\varphi_1^{(1)}\varphi_1^{(2)}),$$ $$\tilde X=-(16\mu^3+4\mu qr)
(\varphi_1^{(1)}\varphi_2^{(2)}+\varphi_2^{(1)}\varphi_1^{(2)})+
12i\mu^2(r\varphi_1^{(1)}\varphi_1^{(2)}-q\varphi_2^{(1)}\varphi_2^{(2)})+$$ $$+2(r_x+iqr^2)\varphi_1^{(1)}\varphi_1^{(2)}
+2(q_x-iq^2r)\varphi_2^{(1)}\varphi_2^{(2)}).$$ These expressions are proportional to ones given by Eqs.(\[T\_phi\],\[X\_phi\]). This way, we show that solutions (\[iq\],\[ir\]) of the linearized equations and conserved densities (\[T\_phi\]) and currents (\[X\_phi\]) are connected in the case of DNLSE in accordance with the Noether’s theorem. This connection takes also place between the infinite hierarchies of infinitesimal symmetries $\delta q^{(k)}$, $\delta r^{(k)}$ and $\delta q_j^{(k)}$, $\delta r_j^{(k)}$ ($j=1,2,3$, $k=0,1,...$) and the hierarchies of conservation laws obtained by expansion in formulas (\[T\_phi\],\[X\_phi\]) of the Lax pair solutions on the spectral parameter powers. First terms of expansions (\[psi\_inf\]) and (\[psi\_0\]) lead to the conservation laws determined by formulas (\[TX\_4\],\[TX\_5\]) and (\[TX\_0a\],\[TX\_0b\]), respectively, that coincide with ones presented in [@GIK; @S_1; @W].
Conclusion
==========
In the present report, we have found the expressions for the solution of the linearization of the DNLSE hierarchy equations and their conservation law in the terms of the solutions of associated Lax pairs. The approach exploited is based on the Darboux transformation technique. It is shown in the DNLSE case that the conservation law is connected with the solution of the linearized equation accordingly to the Noether’s theorem. The local hierarchy and three nonlocal ones of the infinitesimal symmetries and the conservation laws that are explicitly expressed through the variables of the nonlinear equations are produced using the recursion operator and/or expanding the Lax pair solutions in the series on the spectral parameter powers.
The explicit form of the infinitesimal symmetries and the conservation laws of various hierarchies is useful to determine an integrability of the nonlinear PDE’s given. This is especially important for the cases interesting from the physical point of view, such as the reductions of the PDE’s and their deformations. Recently, it was revealed that some deformations of the well-known nonlinear integrable equations, which have the physical meaning, are also integrable [@Z1; @Z2]. This opens the problems of a description of the classes of the deformations keeping the integrability and an extension to them of the methods having been developed in the IST theory. The approach suggested here is not specific for the hierarchy considered and can be applied to other integrable hierarchies and their integrable deformations. An investigation of the hierarchy of the deformed nonlinear equations, which is associated with the quadratic bundle and contains as a particular case the following integrable deformation of the DNLSE equation $$iq_t+\alpha q_x^*+q_{xx}\pm i(|q|^2q)_x=0,$$ where $\alpha$ is an arbitrary parameter, is a subject of the future work.
Acknowledgements
==================
I am grateful Dr. Heinz Steudel for stimulating discussions and hospitality. I thank Gottlieb Daimler- und Karl Benz-Stiftung for financial support.
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|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: |
Существует широкий спектр задач посвященных возможности обхода лабиринта конечными автоматоми. Они могут отличаться как типом лабиринта(это может быть любой граф, даже бесконечный), так и самими автоматами или их количеством. В частности у конечного автомата может быть память(магазин) или генератор случайных битов. В дальнейшем будем считать, что робот — это конечный автомат с генератором случайных битов, если не сказано иное. Кроме того в этой системе могут быть камни-объект, который конечный автомат может переносить по графу, и флажки- объект, наличие которого конечный автомат может только “наблюдать”. Эта тема представляет интерес в связм с тем, что некоторые из этих задач тесно связаны с задачами из теории вероятности и сложности вычислений.
В данной работе продолжают решаться некоторые открытые вопросы, поставленные в диссертации Аджанса: обход роботом с генератором случайных битов целочисленных пространств при наличии камня и подпространства флажков [@And1]. Подобные задачи помогают развить математический аппарат в данной области, кроме того в этой работе мы исследуем практически не изученное поведение робота с генератором случайных чисел. Представляется чрезвычайно важным перенос комбинаторных методов, разработанных А. М. Райгородским в задачах этой тематики.
Данная работа посвящена обходу лабиринта конечным автоматом с генератором случайных битов. Эта задача является частью активно развивающейся темы обхода лабиринта различными конечными автоматами или их коллективами, которая тесно связана с задачами из теории сложности вычислений и теории вероятности. В данной работе показано, при каких размерностях робот с генератором случайных битов и камнем может обойти целочисленное пространство с подпостранством флажков. В данной работе будет изучено поведение конечного автомата с генератором случайных битов на целочисленных пространствах. В частности доказано, что робот обходит ${\ensuremath{\mathbb{Z}}}^2$ и не может обойти ${\ensuremath{\mathbb{Z}}}^3$; робот c камнем обходит ${\ensuremath{\mathbb{Z}}}^4$ и не может обойти ${\ensuremath{\mathbb{Z}}}^5$; робот c камнем и флажком обходит ${\ensuremath{\mathbb{Z}}}^6$ и не может обойти ${\ensuremath{\mathbb{Z}}}^7$; робот c камнем и плоскостью флажков обходит ${\ensuremath{\mathbb{Z}}}^8$ и не может обойти ${\ensuremath{\mathbb{Z}}}^9$.
Работа поддержана Российским Научным Фондом, грант №17-11-01377.
author:
- 'Е.Г. Кондакова, МФТИ, А. Я. Канель-Белов'
title: '**Вероятностные методы обхода лабиринта с использованием камней и датчика случайных чисел**'
---
Введение {#введение .unnumbered}
========
Данная работа связана с задачей о поведении роботов в лабиринтах. Вопросы этой темы представляют большой интерес. Это связано с тем, что продвижения в некоторых важных задачах теоретической Computer Sсince могут быть получены из области поведения роботов в лабиринтах. Такое положение дел делает актуальными задачи данной тематики. Роботы в лабиринтах очень важны, в частности, им посвящено много литературы [@Kilib1; @Kilib3; @Kilib4]. В данной работе продолжают решаться некоторые открытые вопросы, поставленные в диссертации Аджанса: обход роботом с генератором случайных битов целочисленных пространств при наличии камня и подпространства флажков [@And1]. Подобные задачи помогают развить математический аппарат в данной области, кроме того в этой работе мы исследуем практически не изученное поведение робота с генератором случайных чисел. Представляется чрезвычайно важным перенос комбинаторных методов, разработанных А. М. Райгородским в задачах этой тематики.
Задачи этой темы могут принимать разные виды, но есть несколько общих частей. Основным элементом является робот или коллектив роботов [@Kilib5]. Робот — это некоторый конечный автомат, но у него может быть генератор случайных битов или память в какой-то форме. Он перемещается в некоторой среде [@Kilib2]. Её называют лабиринтом. В этой среде могут быть камни, которые робот может переносить, кроме того эта среда может быть раскрашена. Множество флагов в лабиринте, которые робот может увидеть, но не может переносить, по сути является двухцветной раскраской. В дальнейшем будем рассматривать n-цветные раскраски пространств как (n-1)-цветную раскраску подмножества флажков. Робот решает разные задачи такие как: встречи двух роботов, распознавание типа лабиринта и его обход, что фактически является решением нахождения клетки выхода из него.
Имеющиеся результаты для робота {#имеющиеся-результаты-для-робота .unnumbered}
-------------------------------
В задачах ниже робот считается обходящим пространство, если для любой клетки пространства вероятность в ней побывать равна единице.
В моей бакалаврской работе были доказаны простейшие случаи возможности обхода роботом с генератором случайных чисел целочисленного пространства:
- Робот обходит ${\ensuremath{\mathbb{Z}}}^2$ и не может обойти ${\ensuremath{\mathbb{Z}}}^3$,
- Робот c камнем обходит ${\ensuremath{\mathbb{Z}}}^4$ и не может обойти ${\ensuremath{\mathbb{Z}}}^5$,
- Робот c камнем и флажком обходит ${\ensuremath{\mathbb{Z}}}^6$ и не может обойти ${\ensuremath{\mathbb{Z}}}^7$,
- Робот c камнем и плоскостью флажков обходит ${\ensuremath{\mathbb{Z}}}^8$ и не может обойти ${\ensuremath{\mathbb{Z}}}^9$.
Аналогично решенным задачам можно показать, что робот c камнем и подпространством флажков размерности ${\ensuremath{\mathbb{Z}}}^n$ не может обойти ${\ensuremath{\mathbb{Z}}}^n+7$
Увеличение количества камней также не представляет интереса, так как робот с двумя камнями может работать, как машина Минского (это будет показано в данной работе). А машина Минского может обойти любое ${\ensuremath{\mathbb{Z}}}^n$.
Базовые определения нашей модели {#chapt1}
================================
[Базовые определения системы робот-лабиринт]{} Сначала дадим основные определения, которые нам понадобятся для понимания теоретической основы разбираемой темы. В рамках данной работы средой, в которой движется робот, будем считать некоторую группу. Тогда нашу модель можно описать с помощью следующих определений. Часть из этих терминов даны в урезанном варианте и рассматривается только с точки зрения решаемых задач.
[*Системой лабиринт-робот*]{} называется $L = (G,R,n,D,M)$, где $G$ — некоторая группа, $R$ — конечный автомат специального вида, $n$ — неотрицательное целое число, $D$ — подмножество $G$, $M$ — подмножество $G$. ([@Kilib6]).
Конечный автомат с генератором случайных битов $R=(Q,q_{0},\delta,\xi=(\xi _{1},\dots,\xi _{k}, \dots))$ называется [*автоматом робота*]{}, где $Q$ — множество состояний автомата, $q_{0}\in Q$ — начальное состояние автомата, $\delta \subset Q\times {0,1}^{(n+2)}\rightarrow Q$ — функция переходов в автомате, $\xi =(\xi_{1},\dots,\xi_{k}, \dots)$ —последовательность независимых одинаково распределенных случайных величин, $P(\xi_{i}=0)=P(\xi_{i}=1)=\frac{1}{2}$.
Переходы в автомат $R=(Q,q_{0},\delta,\xi)$ осуществляются по битовым векторам длины $(n+2)$
Состоянием системы лабиринт-робот называется набор $(a,s_{1},\dots,s_{n},k,q)$, где $a\in Q$, $s _{i}\in Q$, $q\in Q$, $k\in \mathbb{N}$. Будем называть $a$ — расположением робота, $s _{i}$ — расположениями камней, $k$ — номером хода робота, соответствующим случайной величине $\xi _{k}$, которая дает случайный бит. Начальным состоянием системы лабиринт-робот будет $(e,e,e,\dots,e,1,q _{0})$, где $e$ — нейтральный элемент группы G.
$(d_1,d_2,\dots,d_m) = D$ называются [*переходными элементами*]{} системы лабиринт-робот.
Ходом в состоянии $q\in Q$ называется пара (d,p), где $d\in D$, $p\in {\{0,1\}}^{n}$.
По сути ход соответствует перемещению робота в $G$ и множеству камней, которые он перенесёт с собой. Каждому элементу $Q\times {\{0,1\}}^{n+2}$ соответствует свой ход.
[*Результатом хода робота*]{} в состоянии системы $(a, s_l,s_2,\dots, s_n, k, q)$ будет состояние $(a', s_i, s'_2,\dots,s'_n,k + l, q')$ со следующими свойствами. Обозначим через $(d,p)$ ход соответствующий состоянию $q$, тогда
- $a' = ad$;
- $s'_i = s_id$ если $i$-ый бит $p$ равен 1 и $s_i = a$, иначе $s'_i = s_i$;
- $ w\in {\{0,1\}}^{n+2}$, где:
- $i$-ый бит $w$ равен 1 если $s_i = a$, иначе 0;
- $n + 1$-ый бит равен 1, если $a \in M$, иначе 0;
- $n + 2$-ой бит получается из $\xi_k$
- $q'$ получается из состояния $q$ в автомате $R$ по вектору $w$.
В целом, система лабиринт-робот $L$ работает так. Начинаем с начального состояния, и поочерёдно изменяем состояние согласно ходам робота. То есть, можно сказать, что данная система генерирует последовательность состояний $l_k$ согласно вышеописанным правилам. Расположение робота на $k$-ом ходу обозначим через $a_k$. Важно отметить, что вероятности перемещений не зависят от номера хода.
Для задач обхода G равносильно рассматривать существование конечного автомата с генератором случайных чисел и существование недетерминированного автомата с рациональными вероятностями перехода (мы действуем в предположении, что существует $d_id_j = e$).
Если существует автомат с генератором случайных чисел, то существует недетерминированный автомат с рациональными вероятностями перехода, так как, по сути, автомат с генератором случайных чисел — частный случай недетерминированного автомата с рациональными вероятностями перехода.
Теперь покажем, что утверждение верно и в обратную сторону. Построим автомат с генератором случайных чисел. Там будут состояния, аналогичные состояниям недетерминированного автомата. Если переходы из вершины имели вероятности $(\frac{p_1}{q_1},\frac{p_2}{q_2},\dots,\frac{p_k}{q_k})$, тогда приведём их к общему знаменателю $$(\frac{p'_1}{q},\frac{p'_2}{q},\dots,\frac{p'_k}{q}).$$ Построим переходы с промежуточным состояниями таким образом, чтобы были аналогичные переходы с вероятностью $$(\frac{p'_1}{2^{2q}},\frac{p'_2}{2^{2q}},\dots,\frac{p'_k}{2^{2q}})$$ и возврат в исходную вершину с тем же состоянием с вероятностью $$\frac{2^{2q}-\sum p'_i}{2^{2q}}.$$ Для этого возьмём сбалансированное бинарное дерево из исходной вершины с $2^{2q}$ листами. Ход между ними имеет вид $(d_i,p)$ из вершины нечётного уровня и $(d_j,p)$ из вершины чётного уровня, где $p={\{0\}}^{n}$, а переходы зависят только от случайного бита. Из ${p'}_i$ листовых вершин переход в вершину аналогичной той, в которую был переход с вероятностью $\frac{p_i}{q_i}$ с таким же ходом. Из оставшихся листов однозначные переходы в ещё одну добавленную вершину с ходом $(d_i,p)$, а из нее переход в исходную вершину с ходом $(d_j,p)$. Так как$\frac{2^{2q}-\sum p'_i}{2^{2q}}<1$, то с вероятностью 1 робот в какой-то момент перейдёт в одно из состояний соответствующих переходам недетерминированного автомата из рассматриваемой вершины.
Общий смысл этих определений в том, что у нас есть робот, который является недетерминированным конечным автоматом $R$, $n$ камней, лабиринт $G$ и множество флагов на нем $M$, Робот итерационно переходит по своим состояниям и в соответствии этим переходам ходит по лабиринту. Кроме того, он может носить с собой камни, если находится с ними в одной клетке. По сути, робот является программой с конечной памятью и с возможностью получать случайные биты, что мы используем далее для более удобной демонстрации возможности обхода некоторых лабиринтов. Равносильность этих утверждений расписывать не будем, но задача написать программу по роботу или построить робота по программе не составляет особого труда.
[Случайные блуждания]{}
[Базовые понятия случайных блужданий]{} Рассмотрим некоторые необходимые понятия случайных блужданий:
Простое дискретное случайное блуждание в ${\ensuremath{\mathbb{Z}}}^k$ — это случайный процесс ${\{Y_n\}}_{n\geq0}$ с дискретным временем, имеющий вид
- $Y_n=Y_0+\mathlarger{\mathlarger{\sum}}_{i=1}^{n}X_i$, где $Y_0$ — начальное состояние ${\{0\}}^k$;
- $P(X_i=e_j)=(X_i=-e_j)=\frac{1}{2k}$, где $e_1,\dots e_k$ -вектора естественного ортогонального базиса
- Случайные величины $X_i,\ i = 1, 2,\dots$ совместно независимы.
Для случайного блуждания в ${\ensuremath{\mathbb{Z}}}^k$ равносильны следующие свойства:
- $P$ ($Y_i={\{0\}}^k$ бесконечное число раз$)=1$, иначе говоря
$$\forall l \ P(x:\exists i_1,i_2,\dots,i_l:\ i_1<i_2<\dots<i_l, \forall j \ Y_{i_j}(x)={\{0\}}^k)=1.$$
Будем писать $P(x:\exists i_1,i_2,\dots,i_l:\ i_1<i_2<\dots<i_l, \forall j \ Y_{i_j}(x)={\{0\}}^k)=1$), как $P(Y_i={\{0\}}^k$ хотя бы $k$ раз$)$,
- $P(x:\exists i > 0:\ Y_i(x)={\{0\}}^k)=1$. Где $x$ элемент вероятностного пространства, на котором задано случайное блуждание). Назовём это свойство [*возвратностью*]{} случайного блуждания.
- $\forall \vec{x} $ Если $\exists i: P(Y_i=\vec{x})> 0$, то $P(Y_i=\vec{x})=1; P(\exists i>0: Y_i={\{O\}}^k)$
Покажем несколько следствий между этими свойствами,
- $1\Rightarrow2$
Очевидно, так как получается подставлением $l=1$
- $2\Rightarrow1$
Продемонстрируем верность этого факта индукцией по $l$. В качестве базы будет свойство 2.
[**Переход.**]{} Пусть верно для $l$, докажем для $l+1$.
$$Q(l,m_1)=P(\exists i_1,\dots,i_l:\ i_1<i_2<\dots<i_l<m_1, \forall j \ Y_{i_j}(x)={\{0\}}^k) \underset{m\rightarrow \infty}{\longrightarrow}1$$
$$Q(1,m_2)=P(\exists i:\ i<m_2, \ Y_{i}(x)={\{0\}}^k) \underset{m\rightarrow \infty}{\longrightarrow}1$$
Из независимости случайных величин можно вывести, что
$$Q(l+1,m_1+m_2)\geq Q(l,m_1)Q(1,m_2)$$
Значит, он тоже стремится к 1.
- $3\Rightarrow2$ Берём $\vec{x}={\{0\}}^k$
- $1\Rightarrow3$
$$P(Y_i = {\{0\}}^k\ \mbox{бесконечное\ число\ раз})=1$$
Тогда, если $\exists i:\ P(Y_i=\vec{x})>0$, то $P(y:\exists i>0:Y_i(y)=\overrightarrow{-x})=1$ Аналогичные для $\overrightarrow{-x}$ получаем $P(y:\exists i>0:Y_i(y)=\vec{x})=1$.
Из этого следует, что возвратность для простого случайного блуждания эквивалентна гарантированному обходу достижимого пространства [@TV3].
$P(x:\exists i > 0:\ Y_i(x)={\{0\}}^k)=1$ эквивалентно расходимости ряда $\mathlarger{\mathlarger{\sum}}_{i=1}^{\infty}P(Y_i={\{0\}}^k)$.
Взаимно однозначное отображение $ \pi= (\pi_1,\pi_2,\dots)$ множества (1, 2,…) в себя назовём конечной перестановкой, если $pi_n=n$ для всех $n$, за исключением, быть может, конечного числа.
Если $\xi=(\xi_1,\xi_2,\dots)$ — некоторая последовательность случайных величин, то через $\pi(\xi)$ будем обозначать последовательность $\xi=(\xi_{\pi_1},\xi_{\pi_2},\dots)$. Обозначим $\mathscr{F}_n^{\infty}=\sigma(\xi_n,\xi_{n+1},\dots)$ — $\sigma$-алгебру, порожденную случайными величинами $\xi_1,\xi_2,\dots$. И пусть $\mathscr{F}'=\mathlarger{\mathlarger{\bigcup}}_{n=1}^{\infty}\mathscr{F}_n^{\infty}$. Поскольку пересечение $\sigma$-алгебр есть $\sigma$-алгебра, то $\mathscr{F}'$ — это $\sigma$-алгебра. Она называется «хвостовой» или «остаточной».
Событие $A=\{\xi \in B\}, B \in \mathscr{B}(\mathbb{R}^{\infty})$, то через $\pi(A)$ обозначим событие $$\{\pi(\xi \in B)\},B \in \mathscr{B}(\mathbb{R}^{\infty}).$$
Событие $A=\{\xi \in B\}, B \in \mathscr{B}(\mathbb{R}^{\infty})$, называется [*перестановочным*]{}, если для любой конечной перестановки $\pi$ событие $\pi(A)$ совпадает с $A$.
Закон «0 или 1» Хьюитта и Сэвиджа [@TV1]. \[theo:hyusev\]
Пусть $\xi=(\xi_1,\xi_2,\dots)$ - последовательность независимых одинаково распределенных случайных величин и $A = \{\xi \in B\}$-перестановочное событие. Тогда вероятность $P(A)$ может принимать лишь два значения: нуль или единица.
Заметим, что свойство [*блуждания $Y_i$ c шагом $X_i\ A=(Y_i=\{o\}^k)$ бесконечное число раз*]{} является перестановочным событием случайных величин $X_y$. Значит по Теореме \[theo:hyusev\] $P(A)$ принимает значение $0$ или $1$,
[Невозвратность случайного блуждания с тремя некомпланарными векторами на ${\ensuremath{\mathbb{Z}}}^k$]{} Доказательство невозвратности случайного блуждания с тремя некомпланарными векторами на ${\ensuremath{\mathbb{Z}}}^k$ разобьём на три части:
- Невозвратность простого случайного блуждания на ${\ensuremath{\mathbb{Z}}}^3$;
- Смесь двух случайных блужданий, одно из которых эквивалентно простому случайному блужданию на ${\ensuremath{\mathbb{Z}}}^3$ невозвратно;
- Невозвратность случайного блуждания с тремя некомпланарными векторами на ${\ensuremath{\mathbb{Z}}}^k$.
[Невозвратность простого случайного блуждания на ${\ensuremath{\mathbb{Z}}}^3$.]{}
Докажем, что $\mathlarger{\mathlarger{\sum}}_{i=1}^{\infty}P(Y_i={\{0\}}^3)$ конечна, так как из этого следует, что
$$P(\exists i > 0:\ Y_i(x)={\{0\}}^3)=\varepsilon<1.$$
$P(Y_{2n+1}={\{0\}}^k)=0$, потому что мы не можем вернуться в исходную клетку за нечётное число шагов.
$$P(Y_{2n}={\{0\}}^3)=
\mathlarger{\mathlarger{\sum}}_{\substack{0\leq i_1, i_2, i_3\leq n\\ i_1+i_2+i_3=n}}\Big(\frac{(2n)!}{{(i_1!i_2!i_3!)}^2}{\big(\frac{1}{6}\big)}^{2n}\Big)=$$
$$=2^{-2n}\cdot C^{n}_{2n} \cdot
\mathlarger{\mathlarger{\sum}}_{\substack{0\leq i_1, i_2, i_3\leq n\\ i_1+i_2+i_3=n}}{\Big(\frac{n!}{i_1!i_2!i_3!}{\big(\frac{1}{3}\big)}^{n}\Big)}^2\leq$$
$$\leq 2^{-2n}\cdot C^{n}_{2n} \cdot C_{n} \cdot
\mathlarger{\mathlarger{\sum}}_{\substack{0\leq i_1, i_2, i_3\leq n\\ i_1+i_2+i_3=n}}\frac{n!}{i_1!i_2!i_3!}{\big(\frac{1}{3}\big)}^{2n}$$
где $C_n=\underset{\substack{0\leq i_1, i_2, i_3\leq n\\ i_1+i_2+i_3=n}}{\max}\frac{n!}{i_1!i_2!i_3!}$, тогда
$$P(Y_{2n}={\{0\}}^3)\leq 2^{-2n}\cdot C^{n}_{2n} \cdot C_{n} \cdot 3^{-n}\cdot
\mathlarger{\mathlarger{\sum}}_{\substack{0\leq i_1, i_2, i_3\leq n\\ i_1+i_2+i_3=n}}\frac{n!}{i_1!i_2!i_3!}{\big(\frac{1}{3}\big)}^{n}=2^{-2n}\cdot C^{n}_{2n} \cdot C_{n} \cdot 3^{-n}$$
Покажем, что $$C_n\sim \frac{n!}{\lceil {\frac{n}{3}+1}\rceil!^3} \sim \frac{n!}{\lfloor \frac{n}{3}!\rfloor^3}.$$ Ясно, что $$C_n \geq \frac{n!}{\lfloor \frac{n}{3}!\rfloor^3}$$ (возьмём $i_j\geq \lfloor \frac{n}{3}\rfloor$). Кроме того, при $i_1>\lfloor \frac{n}{3}\rfloor>i_2$ верно, что $i_1!\cdot i_2!>(i-1)! \cdot (j+1)!$. Увеличивая подобным образом $\frac{n!}{i_1!i_2!i_3!}$ мы остановимся, когда все $i_j$ будут равны $\lfloor \frac{n}{3}\rfloor$ или $\lceil \frac{n}{3}\rceil.$
Применив к $$2^{-2n}\cdot C^{n}_{2n} \cdot C_{n} \cdot 3^{-n}$$ формулу Стирлинга, получим $\frac{3\sqrt{3}}{2 \cdot \pi^{1.5} \cdot n^{1.5}}$.
Сумма $\sum \frac{3\sqrt{3}}{2 \cdot \pi^{1.5} \cdot n^{1.5}}$ сходится, значит $P(Y_{2n}={\{0\}}^3)$ ограничена функцией, сумма ряда которой сходится. Но, тогда она сама сходится, и блуждание является невозвратным.
$$P(Y_n(\vec{x})\leq 6^3 \cdot (P(Y_n={\{0\}}^3)+P(Y_{n+1}={\{0\}}^3)+P(Y_{n+2}={\{0\}}^3)+P(Y_{n+3}={\{0\}}^3))$$
Обозначим координаты $\vec{x}=(x_1,x_2,x_3)$, $x'_i=x_i mod 2$, $x'=\mathlarger{\mathlarger{\sum}}_{i=1}^{3} x'_i$. Без ограничения общности можно считать, что $x_i \geq 0$. Тогда
$$P(Y_{2n+x_1+x_2+x_3}=\vec{x})=$$
$$=\mathlarger{\mathlarger{\sum}}_{\substack{0\leq i_1, i_2, i_3\leq n\\ i_1+i_2+i_3=n}}\Big(\frac{(2n+{\mathlarger{\sum}}_{j=1}^{j \leq 3}x_j)!}{{\mathlarger{\prod}}_{j=1}^{j \leq 3}(i_j!\cdot(i_j+x_j)!)}{\big(\frac{1}{6}\big)}^{2n+{\mathlarger{\sum}}_{j=1}^{j \leq 3}x_j}\Big)\leq$$
$$\leq 6^3 \cdot \mathlarger{\mathlarger{\sum}}_{\substack{0\leq i_1, i_2, i_3\leq n\\ i_1+i_2+i_3=n}}\Big(\frac{(2n+x'+{\mathlarger{\sum}}_{j=1}^{j \leq 3}x_j)!}{{\mathlarger{\prod}}_{j=1}^{j \leq 3}(i_j!\cdot(i_j+x_j+x'_j)!)}{\big(\frac{1}{6}\big)}^{2n+x'+{\mathlarger{\sum}}_{j=1}^{j \leq 3}x_j}\Big)\leq$$
$$\leq 6^3 \cdot \mathlarger{\mathlarger{\sum}}_{\substack{0\leq i_1, i_2, i_3\leq n\\ i_1+i_2+i_3=n}}\Big(\frac{(2n+x'+{\mathlarger{\sum}}_{j=1}^{j \leq 3}x_j)!}{{\big({\mathlarger{\prod}}_{j=1}^{j \leq 3}(i_j+(x_j+x'_j)/2)!\big)}^2}{\big(\frac{1}{6}\big)}^{2n+x'+{\mathlarger{\sum}}_{j=1}^{j \leq 3}x_j}\Big)\leq$$
$$\leq 6^3 \cdot
\mathlarger{\mathlarger{\sum}}_{\substack{0\leq i_1, i_2, i_3\leq n'\\ i_1+i_2+i_3=n'}}\Big(\frac{(2n')!}{{(i_1!i_2!i_3!)}^2}{\big(\frac{1}{6}\big)}^{2n'}\Big)=6^3\cdot P(Y_{n'}={\{0\}}^3)$$
Где $n' = n + (\mathlarger{\mathlarger{\sum}}_{j=1}^{j \leq 3}(x_j+x'j))/2$. Тогда $2n' - 2n - (x_1+ x_2 + x_3 ) =$ 0, 1,2 или 3.
Из этого следует, что $P(Y_n =\overrightarrow{-x})<\frac{c}{n\sqrt{n}}$, где $c$ — некоторая константа, одинаковая для всех $\vec{x}$.
[Смесь двух блужданий, одно из которых эквивалентно случайному блужданию на ${\ensuremath{\mathbb{Z}}}^3$ невозвратно.]{} Смесь блужданий $X$ и $Y$, с вероятностями $p, q$ $(p+q=1; p, q>0)$, где $X$ – эквивалентен простому случайному блужданию, обозначим через $Z$, Тогда
$$P(Z_n=\vec{z})=\sum_{\vec{x}+\vec{y}=\vec{z}}\sum_{i=0}^{n}P(X_i=\vec{x})\cdot P(Y_{n-i}=\vec{y})\cdot p^i \cdot q^{n-i} \cdot C_n^i .$$
Мы хотим доказать, что $P(Z_n = \vec{z}) < \frac{b}{n\sqrt{n}}$ для некоторого $b$. Обозначим за $\varepsilon = \min(p, q)^2$ и докажем, что такие константы b есть для
$$\sum_{\vec{x}+\vec{y}=\vec{z}}\sum_{i=0}^{i<\varepsilon n}P(X_i=\vec{x})\cdot P(Y_{n-i}=\vec{y})\cdot p^i \cdot q^{n-i} \cdot C_n^i$$
и
$$\sum_{\vec{x}+\vec{y}=\vec{z}}\sum_{i\geq \varepsilon n}^{n}P(X_i=\vec{x})\cdot P(Y_{n-i}=\vec{y})\cdot p^i \cdot q^{n-i} \cdot C_n^i.$$
Начнём cо второго случая
$$\sum_{\vec{x}+\vec{y}=\vec{z}}\sum_{i\geq \varepsilon n}^{n}P(X_i=\vec{x})\cdot P(Y_{n-i}=\vec{y})\cdot p^i \cdot q^{n-i} \cdot C_n^i\leq$$
$$\leq \max_{i\geq\varepsilon n,\vec{x}}(P(X_i=\vec{x}))\sum_{\vec{x}+\vec{y}=\vec{z}}\sum_{i\geq \varepsilon n}^{n}\cdot P(Y_{n-i}=\vec{y})\cdot p^i \cdot q^{n-i} \cdot C_n^i=$$
$$= \max_{i\geq\varepsilon n,\vec{x}}(P(X_i=\vec{x}))\sum_{i\geq \varepsilon n}^{n}\sum_{\vec{x}+\vec{y}=\vec{z}} P(Y_{n-i}=\vec{y})\cdot p^i \cdot q^{n-i} \cdot C_n^i=$$
$$= \max_{i\geq\varepsilon n,\vec{x}}(P(X_i=\vec{x}))\sum_{i\geq \varepsilon n}^{n} p^i \cdot q^{n-i} \cdot C_n^i \cdot \sum_{\vec{x}+\vec{y}=\vec{z}} P(Y_{n-i}=\vec{y})\leq$$
$$\leq \max_{i\geq\varepsilon n,\vec{x}}(P(X_i=\vec{x}))\sum_{i\geq \varepsilon n}^{n} p^i \cdot q^{n-i} \cdot C_n^i\leq$$
$$\leq \max_{i\geq\varepsilon n,\vec{x}}(P(X_i=\vec{x}))\cdot {p+q}^n=\max_{i\geq\varepsilon n,\vec{x}}(P(X_i=\vec{x}))\leq \frac{c}{\varepsilon n \sqrt{\varepsilon n}}=\frac{c_2}{n\sqrt{n}}.$$
Теперь разберём первый случай
$$\sum_{\vec{x}+\vec{y}=\vec{z}}\sum_{i=0}^{i<\varepsilon n}P(X_i=\vec{x})\cdot P(Y_{n-i}=\vec{y})\cdot p^i \cdot q^{n-i} \cdot C_n^i.$$
Обозначим через $p'=\min(p,q),q'=1-p'$. Тогда, так как $\varepsilon \leq \frac{1}{4}$, то
$$\sum_{\vec{x}+\vec{y}=\vec{z}}\sum_{i=0}^{i<\varepsilon n}P(X_i=\vec{x})\cdot P(Y_{n-i}=\vec{y})\cdot p^i \cdot q^{n-i} \cdot C_n^i\leq$$
$$\leq\sum_{\vec{x}+\vec{y}=\vec{z}}\sum_{i=0}^{i<\varepsilon n}P(X_i=\vec{x})\cdot P(Y_{n-i}=\vec{y})\cdot p'^i \cdot q'^{n-i} \cdot C_n^i$$
Зададим $f(i)=p'^i\cdot q'^{n-i}\cdot c^n_i$. Тогда $f(i)$ возрастает на $(0,\varepsilon n)$.
$$\frac{f(i+1)}{f(i)}=\frac{p'}{q'}\cdot\frac{n-i}{i+1}\geq\frac{p'}{q'}\cdot\frac{n-\varepsilon n}{\varepsilon n+1}\geq \frac{p' \cdot (1-p'^2)}{p'^2 \cdot (1-p')}=1+\frac{1}{p'}$$
Значит, с какого-то $n$ можно сказать, что $f(i)$ возрастает на $(0,\varepsilon n)$. Тогда
$$\sum_{\vec{x}+\vec{y}=\vec{z}}\sum_{i=0}^{i<\varepsilon n}P(X_i=\vec{x})\cdot P(Y_{n-i}=\vec{y})\cdot p'^i \cdot q'^{n-i} \cdot C_n^i\leq$$
$$\leq f(\lfloor\varepsilon n\rfloor)\cdot\sum_{i=0}^{i<\varepsilon n} \sum_{\vec{x}+\vec{y}=\vec{z}}P(X_i=\vec{x})\cdot P(Y_{n-i}=\vec{y})\leq$$
$$\leq f(\lfloor\varepsilon n\rfloor)\cdot\varepsilon n$$
Так как $\sum_{\vec{x}+\vec{y}=\vec{z}}P(X_i=\vec{x})\cdot P(Y_{n-i}=\vec{y})\leq\sum_{\vec{x}}P(X_i=\vec{x})\cdot \sum_{\vec{y}}P(Y_i=\vec{y})=1$.
$$f(\lfloor\varepsilon n\rfloor)\cdot\varepsilon n=p'^2 \cdot n\cdot C^{n\cdot p'^2}_n \cdot {p'}^{np'^2}\cdot(1-p')^{n(1-p'^2)}$$
Применив сюда формулу Стирлинга, получим
$$f(\lfloor\varepsilon n\rfloor)\cdot\varepsilon n\sim const \cdot \frac{n}{\sqrt{n}}\cdot\Big(\frac{p'^{p'^2}\cdot (1-p')^{1-p'^2}}{(p'^2)^{p'^2}\cdot (1-p'^2)^{1-p'^2}}\Big)^n$$
Если мы докажем, что $d=\frac{p'^{p'^2}\cdot(1-p')^{1-p'^2}}{(p'^2)^{p'^2}\cdot (1-p'^2)^{1-p'^2}}<1$, тогда так как верно, что $\forall c$$\forall d'<0$$ \exists n_0 \forall n>n_0$
$$const \sqrt{n}\cdot d'^n\leq c \cdot \frac{1}{n \sqrt{n}}$$
Мы получим, что $\exists n_0,c': \forall n>n_0$
$$\sum_{\vec{x}+\vec{y}=\vec{z}}\sum_{i=0}^{i<\varepsilon n}P(X_i=\vec{x})\cdot P(Y_{n-i}=\vec{y})\cdot p^i \cdot q^{n-i} \cdot C_n^i\leq \frac{c'}{n\sqrt{n}}$$
Так как эта сумма сама по себе не превосходит единицу, то $\exists n_0,c_1: \forall n>0$
$$\sum_{\vec{x}+\vec{y}=\vec{z}}\sum_{i=0}^{i<\varepsilon n}P(X_i=\vec{x})\cdot P(Y_{n-i}=\vec{y})\cdot p^i \cdot q^{n-i} \cdot C_n^i\leq \frac{c_1+n_0\sqrt{n_0}}{n\sqrt{n}}$$
Осталось проверить утверждение про d: $$d<1\Leftrightarrow \frac{p'^{p'^2}\cdot(1-p')^{1-p'^2}}{(p'^2)^{p'^2}\cdot (1-p'^2)^{1-p'^2}}<1\Leftrightarrow$$
$$\Leftrightarrow p'^{p'^2}\cdot(1-p')^{1-p'^2}<(p'^2)^{p'^2}\cdot (1-p'^2)^{1-p'^2}\Leftrightarrow p'^{(-p'^2)}<(1+p')^{1-p'^2}\Leftrightarrow$$
$$\Leftrightarrow
\big(1+\frac{1}{p'}\big)^{(p'^2)}<1+p'\Leftrightarrow
p'^2\cdot \ln\big(1+\frac{1}{p'}\big)<\ln(1+p')\Leftrightarrow$$
$$\Leftrightarrow
\frac{\ln(1+p')}{p'}<p'\cdot{\ln\big(1+\frac{1}{p'}\big)}$$
Заметим, что последнее это $g(1+p')>g\big(1+\frac{1}{p'}\big)$, где $g(x)=\frac{\ln(1+x)}{x}$. Нам достаточно показать, что $g(x)$ убывающая функция при $x>0$.
$$g'(x)<0
\Leftrightarrow
\frac{\frac{x}{1+x}-\ln(x+1)}{x^2}<0
\Leftrightarrow
\frac{x}{1+x}-\ln(x+1)<0
\Leftrightarrow$$
$$g'(x)<0
\Leftrightarrow
0<\ln(x+1)-1+\frac{1}{1+x}=h(x)$$
$h(0)=0$ и $h'(x)=\frac{1}{1+x}-\frac{1}{(1+x)^2}=\frac{x}{(1+x)^2}$, что больше нуля при $x>0$, тогда $h(x)>0$.
Из разбора этих двух случаев становится ясно, что $P(Z_n=\vec{z})\leq \frac{c_1+c_2+n_0\vec{n_0}}{n\sqrt{n}}$.
[Невозвратность случайного блуждания с тремя некомпланарными векторами.]{}
Докажем это утверждение на ${\ensuremath{\mathbb{Z}}}^k$.
Если бы это было не так, то из любой достижимой клетки можно было бы вернуться в стартовую с положительной вероятностью.
Если блуждание $Y_i$ построенное на шагах $X_i$ возвратное, то $Y^{0}_i$ которое строится на нем с шагами равным $n$ обычных ходов $X'_i=\sum_{j=n\cdot (i-1)+1}^{n\cdot i}X_i$, то есть $Y^{0}_i=Y_{ni}$ тоже возвратное.
Если бы это было не так, то $P(Y^{0}_i ={\{0\}}^k )$ бесконечное число раз$)=0$ (по закону 0 и 1), кроме того, рассмотрим блуждания $Y^{l}_i=Y_{ni+l}$ , при $l<n$. Какое-то из них посещает ${\{0\}}^k$ бесконечное число раз, но тогда рассмотрим его после первого посещения ${\{0\}}^k$. Оно будет эквивалентно $P(Y^{0}_i ={\{0\}}^k $ бесконечное число раз$)$. Противоречие.
Пусть случайное блуждания $Y_i$ с тремя некомпланарными векторами на ${\ensuremath{\mathbb{Z}}}^k$ возвратно (обозначим эти вектора $\vec{a_1}, \vec{a_2}, \vec{a_3}$). Тогда $\exists n_1>1: P(Y_{n_1-1}=\vec{-a_1})>0$, кроме того $P(Y_{n_1-1}=(n_1-1)\cdot \vec{a_1})>0$ ($n_1-1$ шаг по $-\vec{a}$). Из этого следует, что для $m_1=n_1\cdot(n_1-1)^2,\ l_1=(n_1-1)\cdot n_1$ верно, что $P(Y_{m_1}=-l_1\cdot\vec{a_1})>0$ (делаем $n_1\cdot(n_1-1)$ раз шаг за $n_1-1$ ходов дающий $\overrightarrow{-a}$ ) и $P(Y_{m_1}=l_1\cdot\vec{a_1})>0$ ( делаем $(n_1-2)\cdot(n_1-1)$ раз шаг за $n_1-1$ ходов дающий $\overrightarrow{-a_1}$ и $2(n_1-1)^2 $ шагов по вектору $\vec{a_1}$). Проведя аналогичные действия получим $Y'_i=Y_{m_1\cdot m_2\cdot m_3}$ с положительными вероятностями переходов по векторам $$l_1\cdot m_2 \cdot m_3 \cdot \vec{a_1}, -l_1\cdot m_2 \cdot m_3 \cdot \vec{a_1},
m_1\cdot l_2 \cdot m_3 \cdot \vec{a_2}, -m_1\cdot l_2 \cdot m_3 \cdot \vec{a_2},
m_1\cdot m_2 \cdot l_3 \cdot \vec{a_3}, -m_1\cdot m_2 \cdot l_3 \cdot \vec{a_3}.$$ Но тогда $Y'_i$ — смесь блуждания эквивалентного простому в ${\ensuremath{\mathbb{Z}}}^3$ c коэффициентом минимальной вероятности одного из этих 6 шагов( а она положительная) умноженным на 6 и еще некоторого блуждания. Значит оно невозвратно. Противоречие.
Для случайного блуждания с тремя некомпланарными векторами (и ненулевой вероятностью вернуться в стартовую клетку из любой достигаемой) $Y_i$ $\forall \varepsilon>0 \exists$ конечное число $\vec{x}: P(\exists i: Y_i={\vec{x}})>\varepsilon$.
Это легко доказать, показав, что для $$Y_i\ \exists c\ \forall \vec{x}\ \forall n \ P(Y_n=\vec{x})<\frac{c}{n\sqrt{n}}.$$ Из этого следует, что $$\exists n_0\ \forall \vec{x} : P(\exists n>n_0 :Y_n=\vec{x})<\frac{\varepsilon}{2},$$ но количество $\vec{x}:P(\exists n\leq n_0 :Y_n=\vec{x})>=\frac{\varepsilon}{2}$ не больше $ \frac{2n_0}{\varepsilon}$. Значит, количество клеток, которые будут посещены с вероятностью хотя бы $\varepsilon$ не больше $\frac{2n_0}{\varepsilon}$.
Обход пространства размерностью меньше 10.
==========================================
[Робот с генератором случайных битов.]{}
Разберём для начала случай, когда есть только робот, и покажем, что он может обойти ${\ensuremath{\mathbb{Z}}}^2$ и не может ${\ensuremath{\mathbb{Z}}}^3$ . В этом случае $M=\{\mbox{\o}\}, n=0$. Множество $D$ в пространстве ${\ensuremath{\mathbb{Z}}}^k$ имеет вид $(\vec{e_1},-\vec{e_1},\vec{e_2},-\vec{e_2},\dots,\vec{e_k},-\vec{e_k})$. Робот считается обходящим пространство, если для любой ее клетки он посещает её с вероятностью один.
[Обход роботом ${\ensuremath{\mathbb{Z}}}^2$.]{} Написать программу для такого робота довольно легко. Он просто должен эмулировать случайное блуждание. А так как оно обходит плоскость, то и робот ее обойдёт [@TV2].
[Невозможность обойти ${\ensuremath{\mathbb{Z}}}^3$.]{} Докажем это для любого робота. Рассмотрим граф, соответствующий недетерминированному конечному автомату. Обозначим его размер через $m$. Будем считать, что вероятность каждого ребра больше нуля и все вершины достижимы из начальной (кроме, возможно, её самой). Иначе их можно просто выкинуть и это никак не повлияет на поведение робота. Возьмём компоненты сильной связности графа. Назовём компоненту листовой, если из неё нет рёбер в другие компоненты (такая обязательно есть). Клетками будем называть элементы лабиринта (в данном случае элементы пространства ${\ensuremath{\mathbb{Z}}}^k$), вершинами состояния робота. Расстояние между клетками будет считать по метрике расстояния городских кварталов. Плоскостями будем называть множество клеток вида $\vec{x_0}+a_1\vec{x_1}+a_2\vec{x_2}$ при фиксированных $\vec{x_0},\ \vec{x_1},\ \vec{x_2}$ и целых $a_1,\ a_2$. Предположим, что есть робот, обходящий пространство.
Робот с недетерминированным графом, соответствующим листовой компоненте сильной связанности робота, обходящего пространство, тоже должен обходить пространство.
Обозначим какую-то из его вершин за новое начальное состояние $q'_0$, а новый автомат, построенный на этой листовой компоненте за $R'_0(q'_0)$, размер компоненты обозначим через $m'$. Тогда новый робот должен обходить все пространство, кроме не более чем $m$ клеток. Это связано с тем, что от $q_0$ до $q'_0$ есть путь не более, чем за $m$ шагов (обозначим за $\vec{x_1}(q'_0)$ клетку в пространстве в которой мы оказались дойдя до $(q'_0)$). Тогда продолжение этого пути с вероятностью 1 должно побывать во всех клетках пространства, кроме тех, в которой он уже был, а их не более $m$. Так как они последовательны, то эти клетки помещаются в сферу с центром в ${\{0\}}^k$ (где $k$ — размерность пространства, в котором мы работаем) и радиусом $m$, и в сферу с центром $\vec{x_1}(q'_0)$ и радиусом $m$. Это верно при выборе любого $q'_0$ из нашей листовой компоненты. С вероятностью один мы когда-нибудь попадём в клетку $\vec{x_2}$ на расстоянии $2m+1$ от ${\{0\}}^k$. Обозначим состояние, в котором мы оказались в этой клетке за $q'_1$ (Это состояние принадлежит нашей листовой компоненте). С этого момента робот ведёт себя как робот $R'(q'_1)$, находящийся в клетке $\vec{x_2}$ в изначальном состоянии, а он обходит все пространство, кроме сферы с центром в $\vec{x_2}$ и радиусом $m$. Значит $R'(q'_0)$ обходит с одной стороны все клетки кроме сферы с центром в ${\{0\}}^k$ и радиусом $m$, с другой стороны все клетки кроме сферы с центром в $\vec{x_2}$ и радиусом $m$. Эти сферы не пересекаются, поэтому $R'_0(q'_0)$ обходит все пространство. Кроме того, перемещения этого робота являются возвратными.
Теперь разберёмся, почему недетерминированный автомат, ориентированный граф которого является компонентой сильной связности, не может обходить пространство (начальная вершина — $q_0$, размер графа $m$). Обозначим за $p_{i,j}$ вероятность попасть в состояние $q_j$ из состояния $q_i$ за количество шагов, не превосходящее $m$ (при $i=j$ считаем, что нужен хотя бы один шаг). Так как это компонента сильной связности c m вершинами $p_{i,j}>0$. Обозначим за p = $\mathlarger{\min_{i,j}}\ p_{i,j}$. Тогда вероятность не попасть в течение $l\cdot m$ шагов в $j$-ую вершину меньше ${(1 - p)}^l$. А оно стремится к нулю при $l\rightarrow\infty$. Значит, с вероятностью 1 мы побываем в состоянии $q_j$ из чего следует, что мы побываем там бесконечное число раз.
Мы уже получили, что перемещения такого робота возвратны, то есть он побывает в клетке ${\{0\}}^k$ бесконечное число раз. Рассмотрим вероятности попасть в клетку в состояние $q_j$, если прошлый раз мы были в ней в состояние $q_i$. Обозначим эту вероятность через $p_{i,j}$ ($p_{i,j}$ не зависит от клетки в которой мы начинаем). $\forall i \sum_{j=1}^{m} p_{i,j}= 1$. Рассмотрим эти переходы по состояниям, как некоторый другой ориентированный граф, по которому мы гуляем бесконечно долго. В нем можно выбрать несколько состояний $q_{i,1},q_{i,2},\dots,q_{i,l}$ таких, что мы гарантировано побываем хотя бы в одном из них, они все достижимы из $q_0$ и $q_{i,j_1}$ не достижимо из $q_{i,j_2}$ (будем брать по одной вершине из листовых компонент сильной связности нового ориентированного графа). Мы побываем в одном из этих состояний бесконечное число раз. Обозначим его через $q'$.
Для любой клетки, где робот был в состояние $q'$, он побывал там бесконечное число раз.
Теперь рассмотрим случайное блуждание в пространстве с векторами переходов и их вероятностями такими, что пара (вектор, вероятность) соответствуют паре (вектор, вероятность) перемещения робота между двумя состояниями $q'$. Как сказано выше, это случайное блуждание возвратно, значит оно не может содержать некомпланарных векторов. Тогда после первого попадания в $q'$ множество клеток, где мы можем быть в состояние $q'$ является элементом какой-то плоскости. Но мы гарантировано побываем в любой клетке, а в сфере с центром в ней и радиусом $m$ есть вероятность побывать в состояние $q'$. Тогда если мы возьмём клетку на расстояние больше $m+1$ от этой плоскости, то мы не сможем в неё попасть. Противоречие.
[Робот с генератором случайных чисел и камнем]{} Теперь рассмотрим случай робота с камнем. Изначальное положение его и камня в клетке с нулевыми координатами.
[Обход ${\ensuremath{\mathbb{Z}}}^4$]{} Довольно легко описать программу, в соответствии с которой будет перемещаться робот. Он отходит от камня, случайно блуждает вдоль координат $x_1$, $x_2$, пока не вернётся к камню, а потом делает шаг в случайном направлении вдоль $x_3$, $x_4$ вместе с камнем. Повторяет. Так как случайное блуждание на плоскости возвратно, робот все время возвращается к камню. Тогда, ходя вместе с камнем, робот обходит всю плоскость $x_3$, $x_4$. так как перемещения с камнем в случайном направлении тоже является случайным блужданием. Оно возвратно, поэтому робот побывает с камнем во всех клетках плоскости $x_3$, $x_4$ бесконечное число раз. Тогда для любой из этих клеток мы бесконечное число раз блуждали от камня вдоль $x_1$, $x_2$. Рассмотрев только эти ходы получим простое случайное блуждания из клетки плоскости $x_3$, $x_4$ вдоль $x_1$, $x_2$, а таким образом можно получить все клетки пространства.
[Невозможность обойти ${\ensuremath{\mathbb{Z}}}^5$]{} Разберёмся, как ходит робот. Разделим его перемещения на два типа:
- Перемещение с камнем,
- Перемещение без камня.
[Перемещение без камня.]{}
Если робот, обходящий пространство, уходит от камня, то он должен к нему вернуться с вероятностью один
Робот, который с какого-то момента имеет вероятность не вернуться, с этого момента соответствует какому-то роботу без камня. А про них уже доказано, что они обходят в лучшем случае плоскость и клетки, удалённые от неё не более, чем на $m$(где $m$-количество состояний робота). А это — явно не всё пространство.
Покажем, что если робот возвращается к камню с вероятностью один, то множество клеток, которые он может посетить, отойдя от камня, является конечным объединением плоскостей. Пусть мы отходим от камня в состоянии $q_1$. Тогда возьмём недетерминированный конечный автомат $R'$ с состояниями, аналогичными $R$ и новым начальным состоянием $q'_1$; вероятностям перехода между состояниями, соответствующими $R$ совпадают с аналогичными в $R$ при условии отсутствия камней в клетке; переход из $q'_1$ соответствуют переходам из $q_1$ при условие наличия камня в клетке. Обозначим количество состояний в $R'$ через $m$. Новый робот имеет те же вероятности положения в пространстве, начиная с отхода от камня в состояние $q'_1$ до возвращения к нему.
Рассмотрим его ориентированный граф компонент сильной связности. Если какой-нибудь лист не достижим при блуждании в пространстве до первого возвращения в изначальную клетку, то мы можем его выкинуть из графа, так как мы все равно в него не попадаем. Выкинув таким образом все недостижимые компоненты, возьмём какое-нибудь состояние $q\neq q'_1$.
Множество клеток, в которых мы можем оказаться в состоянии $q$, будет подмножеством объединения конечного числа плоскостей с одинаковыми образующими векторами.
Этот факт очевиден для случая, когда таких клеток конечно, так как их можно покрыть конечным количеством плоскостей (по плоскости на каждую клетку), так что будем рассматривать только $q$, для которых существует сколь угодно далёкая клетка с вероятностью её посетить в состоянии $q$.
Докажем это сначала для случая, где $q$ находится в листовой компоненте. Возьмём случайное блуждание, построенное на переходах в пространстве робота между двумя состояниями $q$ (между ними могут быть только состояния отличные от $q$).
Если в нём нет трёх некомпланарных векторов, то все клетки, куда робот может попасть в этом состоянии лежат в какой-то одной плоскости (даже без условия остановки на камне). Кроме того, от остальных состояний он может дойти до этого за не более, чем $m$ ходов. Тогда множество клеток, где робот может оказаться в состояние $q$ удалено от стартовой клетки не больше, чем на $m$. Значит, состояние $q$ достижимо роботом только в клетках являющимися параллельными плоскостями, удалёнными от стартовой клетки не больше, чем на $m$, а таких конечно.
Для блуждания с тремя некомпланарными векторами $\forall \varepsilon>0$ существует конечное число клеток, достижимых с вероятностью хотя бы $\varepsilon$. Тогда $\exists r(\varepsilon):$ для клеток удалённых от места нынешнего нахождения хотя бы на $r(\varepsilon)$ вероятность в них оказаться меньше $\varepsilon$. Обозначим за $p$ минимальную вероятность дойти из состояния того же листа $q'$ до $q$ за не более, чем $m$ шагов. Значит, есть клетка($\vec{x_1}$), достижимая блужданием на расстоянии не больше $m$ от изначальной клетки с вероятностью хотя бы $p$, но тогда она достижима из любой клетки в состоянии $q$ с вероятностью хотя бы $p$. Кроме того, есть клетка($\vec{x_2}$), в которой робот может оказаться в состоянии $q$ на расстоянии от изначальной клетки($\vec{x_0}$) большем, чем $m+r(p)$. Тогда расстояние между ($\vec{x_2}$) и ($\vec{x_1}$) хотя бы $r(p)$, но в этом случае вероятность попасть в ($\vec{x_1}$) из ($\vec{x_2}$) меньше $p$. Противоречие.
Если компонента с состоянием $q$ не листовая, то из неё за не более, чем $m$ шагов можно дойти до состояния $q'$. Тогда клетки, где мы можем оказаться в состоянии $q$ находятся на расстоянии не больше $m$ от подмножества клеток объединения конечного числа плоскостей, что тоже является подмножеством клеток объединения конечного числа плоскостей.
Возьмём множество плоскостей $L_i$, соответствующих плоскостям состояний $q$. Кроме того, разрешим $L_i$ совпадать, чтобы плоскость учитывалась отдельно для каждого промежуточного состояния $q$ и $q_1$ из которого мы стартовали от камня. Их количество обозначим за $l$.
[Перемещения с камнем.]{} Осталось разобраться, как устроены перемещения с камнем. Построим автомат, соответствующий тасканию камня изначального робота. Для этого состояние в котором мы уходим из клетки оставив камень будем рассматривать, как переход из этого состояния в состояния, в которых мы могли бы вернуться к камню, с вероятностями, с которыми это могло бы произойти. Чтобы мы не оставляли камень, добавим по одному состоянию на каждый такой переход, чтобы сделать шаг вверх с камнем и шаг вниз с ним же. Так как этот автомат все время таскает камень, то он ведёт себя, как просто робот без камня.
Обозначим множество клеток, где побывает этот робот за $B$. Рассмотрим какую-нибудь из его листовых компонент связности. Мы можем в неё попасть за какое-то конечное число ходов робота. Обозначим $B'$ множество клеток, где побывает листовой робот. Рассмотрим вероятности побывать в клетках ${\ensuremath{\mathbb{Z}}}^5$ в состояние $q$ на плоскости $L_i$.
Покажем, что $\forall \varepsilon>0$ множество клеток $x_j$ для которых $P (\exists i:$листовой робот попал в $L_i(x_j)$ в состояние $q)>\varepsilon$ имеет вид конечного объединения подпространств размерности четыре, где за подпространство размерности четыре берём множество клеток представимых в виде $\vec{x_0}+a_1\vec{x_1}+a_2\vec{x_2}+a_3\vec{x_3}+a_4\vec{x_4}$ при фиксированных линейных независимых $\vec{x_0},\ \vec{x_1},\ \vec{x_2},\ \vec{x_3},\ \vec{x_4}$ и целых $a_1,\ a_2,\ a_3,\ a_4$.
Давайте посмотрим на множество клеток, в которые он попадёт при блуждание по плоскостям $L_i$ с вероятностью хотя бы $\frac{\varepsilon}{l}$. Для этого введём на ${\ensuremath{\mathbb{Z}}}^5$ отношение эквивалентности $\vec{x_1}\sim\vec{x_2}$, которое верно при $L_i(\vec{x_1})=L_i(\vec{x_2})$. Посмотрим, что произойдёт с блужданием на склейке. Обозначим новый лабиринт через $Z'$, а его случайное блуждание через $X'$. Если $X'$ возвратное, то достижимые клетки задаются не более чем двумя векторами, но тогда $L_i(B)$ уже подпространство размерности четыре. Из невозвратности $X'$ следует, что в $Z'$ конечное число клеток с вероятностью попадания хотя бы $\frac{\varepsilon}{l}$. Развернув отображение обратно получим, что множество клеток ${\ensuremath{\mathbb{Z}}}^5$, в которые робот попадёт при блуждание от камня соответствущее плоскости $L_i$ с вероятностью хотя бы $\frac{\varepsilon}{l}$ будет конечным объединением плоскостей. Так как вероятность робота попасть способом, соответствующим $L_i$, в прообраз не превосходит вероятности из образа.
Чтобы $P (\exists i:$листовой робот попал в $L_i(x_j)$ в состоянии $q)>\varepsilon$ надо, чтобы $\mathlarger\sum_{i}P($листовой робот попал в $L_i(x_j)$ в состоянии $q)>\varepsilon$, но тогда верно, что $\exists i:P($листовой робот попал в $L_i(x_j)$ в состоянии $q)>\frac{\varepsilon}{l}$. А множество таких $x_j$ конечно. Значит клеток, которые гарантированно посетит робот с вероятностью хотя бы $\varepsilon$ не больше конечного объединения подпространств четвёртой степени. Проведя аналогичные рассуждения для других состояний этой листовой компоненты получим, что множество клеток, в которые попадает листовой робот с вероятностью хотя бы $m\cdot\varepsilon$ конечно. Тогда взяв $\varepsilon<\frac{1}{m}$ получим, что листовой робот гарантировано побывает в множестве клеток, покрываемым конечным объединением подпространств четвёртой степени, то есть изначальный робот не может обойти ${\ensuremath{\mathbb{Z}}}^5$
[Робот с генератором случайных чисел, камнем и флажком.]{}
Перейдём к случаю робота с камнем и флажком. Изначальное положение камня и робота в клетке с нулевыми координатами, там же находится единственный элемент из $M$.
[Обход ${\ensuremath{\mathbb{Z}}}^6$]{} Для начала опишем, как, имея флажок и камень, побывать камнем на ${\ensuremath{\mathbb{Z}}}^4$, Робот ходит от камня без него по случайным векторам из множества $(\vec{e_1},\ -\vec{e_1},\ \vec{e_2},\ -\vec{e_2})$. Если он возвращается к камню не попав в процессе на клетку с флажком, то он перемещается с камнем по случайному вектору из $(\vec{e_3},\ -\vec{e_3},\ \vec{e_4},\ -\vec{e_4})$, иначе он перемещается с камнем по случайному вектору из $(\vec{e_1},\ -\vec{e_1},\ \vec{e_2},\ -\vec{e_2})$, а потом вместе с камнем переходит по случайному вектору из $(\vec{e_3},\ -\vec{e_3},\ \vec{e_4},\ -\vec{e_4})$. Из-за того, что случайное блуждание возвратно и обходит всю плоскость, робот с камнем гарантировано вернётся на плоскость $\vec{e_1},\ \vec{e_2}$. Так как он окажется там бесконечное число раз, то с вероятностью один робот в какой-то момент дойдёт до флага и обратно. Тогда камень гарантированно побывает во всех клетках плоскости $\vec{e_1},\ \vec{e_2}$ бесконечное число раз и из каждой клетки плоскости проблуждает вдоль $\vec{e_3},\ \vec{e_4}$. Добавив случайное блуждание от камня и обратно по векторам из множества $(\vec{e_5},\ -\vec{e_5},\ \vec{e_6},\ -\vec{e_6})$ после каждого перемещения камня получим обход ${\ensuremath{\mathbb{Z}}}^6$.
[Невозможность обхода ${\ensuremath{\mathbb{Z}}}^7$]{} Рассмотрим блуждания робота. Они могут иметь следующие виды:
- Перемещение с камнем,
- Перемещение от камня и обратно,
- Перемещение от флага до камня или от камня до флага.
- Перемещение от флага до флага; этот вид можно не рассматривать, как отдельный тип, т.к. оно может обойти только множество клеток, являющихся подмножеством конечного числа плоскостей.
Заметим, что чтобы обходить ${\ensuremath{\mathbb{Z}}}^7$ мы всегда должны с вероятностью один возвращаться к флагу, иначе мы и ${\ensuremath{\mathbb{Z}}}^5$ не сможем обойти, и с вероятностью один возвращаться к камню, так как флаг — это камень, который мы не можем таскать, а робот с камнем не может обойти даже ${\ensuremath{\mathbb{Z}}}^5$.
Посмотрим на множество клеток, где может располагаться камень, чтобы от него можно было дойти до флага.
Оно совпадает с множеством клеток до которых можно дойти от камня, а это множество является подмножеством объединения конечного числа плоскостей(из-за того, что блуждание робота без камня возвратное).
Обозначим его через $L=\mathlarger\bigcup_{i}L_i$, где $L_i$-плоскости покрывающие перемещения робота от камня до камня или флага, мощность этого множества обозначим за $l$. Так как мы возвращаемся к флагу, то камень в какой-то момент должен возвращаться в $L({\{0\}}^k)$. Посмотрим, как себя может вести робот с камнем между встречами с флажком. Пусть мы в первый раз подойдём к камню в состоянии $q_1$. Тогда покажем, какой робот без камня соответствует перемещениям робота с камнем с момента первой встречи камня после флага, до последнего отхода перед флагом. Недетерминированный конечный автомат $R'(q'_1)$:
- Состояния аналогичны $R$,
- Начальное состояние соответствует состоянию $q_1$,
- Вероятностям перехода между состояниями, соответствующими $R$, совпадают с реализуемыми вероятностями перехода между изначальными состояния с камнем до следующего состояния с камнем. В случае, если при этом камень оставался на месте, к состояниям $R'$ добавляется дополнительные для прохода с камнем вверх и вниз (вероятность посещения клеток новым роботом от этого может только увеличиться),
- Обозначим количество состояний в $R'$ через $m$.
Новый робот имеет те же вероятности положения в пространстве с камнем, начиная с отхода от камня в состояние $q'_1$ до возвращения к нему. Кроме того для клеток, где может оказаться $R$ с камнем верно, что блуждая из этой клетки $R'$ попадёт в множество клеток $L({\{0\}}^7)$ с вероятностью 1. Покажем, что тогда множество клеток, которые он может посетить является конечным объединением подпространств размерности четыре. Рассмотрим его ориентированный граф компонент сильной связности. Возьмём какое-нибудь из состояние $q'$.м Этот факт очевиден для случая, когда таких клеток конечно, так их можно покрыть конечным количеством подпространств, так что будем рассматривать только $q'$ для которых существует сколь угодно далёкая клетка с вероятностью её посетить в состояние $q'$.
Докажем это сначала для случая, где $q'$ находится в листовой компоненте. Возьмём случайное блуждание, построенное на переходах в пространстве робота между двумя состояниями $q'$ (между ними могут быть только состояния отличные от $q'$).
Обозначим за $p$ минимальную вероятность дойти из состояния того же листа $q'_2$ до $q'$ за не более, чем $m$ шагов. Так как на блуждание не наложены ограничения остановки, то из любой клетки, где $R'$ может побывать в состояние $q'$(обозначим их через $B$) можно с вероятностью хотя бы $p$ попасть в клетку из $L'=\mathlarger\bigcup_{d(\vec{x},{\{0\}}^7)\leq m} L(\vec{x})$ в состояние $q'$. Возьмём конечное множество подпространств размерности четыре $L'_i: L'=\mathlarger\bigcup_{i} L'_i$, их количество $l'$. Обозначим за $B_i\subset B:$ блуждая из них можно с вероятностью хотя бы $\frac{p}{l'}$ попасть в клетку из $L'_i$. Тогда $B=\mathlarger\bigcup_{i=1}^{l'} B_i$.
Докажем, что множество клеток, из которых случайным блужданием можно попасть в $L'_i$ с вероятностью хотя бы $\frac{p}{l'}$ является подмножеством конечного объединения подпространств размерности четыре, тогда будет верно и то, что $B$ является подмножеством конечного объединения подпространств размерности четыре. Для этого введём на ${\ensuremath{\mathbb{Z}}}^7$ отношение эквивалентности $\vec{x_1}\sim\vec{x_2}$, которое верно при $L'_i(\vec{x_1})=L'_i(\vec{x_2})$. Посмотрим, что произойдёт с блужданием на склейке. Обозначим новый лабиринт через $Z'$, а его случайное блуждание через $X'$. Если $X'$ возвратное, то достижимые клетки задаются не более чем двумя векторами, но тогда $L'_i(B_i)$ уже подпространство размерности четыре. Из невозвратности $X'$ следует, что в $Z'$ конечное число клеток с вероятностью попадания хотя бы $\frac{p}{l'}$. Развернув отображение обратно получим, что множество клеток ${\ensuremath{\mathbb{Z}}}^7$, в которые робот попадёт при блуждание от камня соответствущее плоскости $L'_i$ с вероятностью хотя бы $\frac{p}{l'}$ будет подмножеством конечного объединения плоскостей. Так как вероятность робота попасть способом, соответствующим $L'_i$, в прообраз не превосходит вероятности из образа.
Если компонента с состоянием $q$ не листовая, то из неё за не более, чем $m$ шагов можно бы было дойти до состояния $q'$. Тогда клетки, где мы можем оказаться в состояние $q$ находятся на расстояние не больше $m$ от подмножества клеток объединения конечного числа подпространств размерности четыре, что тоже является подмножеством клеток объединения конечного числа подпространств размерности четыре.
Количество разных $R'$ конечно, в каждом из них конечное число состояний для каждого из которых верно, что множество клеток в котором он побывает является подмножеством объединения конечного числа подпространств размерности четыре. Значит робот $R$ побывает с камнем в множестве клеток, являющемся подмножеством объединения конечного числа подпространств размерности четыре. Но так как от камня робот уходит не дальше $L$, то множество клеток, где он побывает является подмножеством объединения конечного числа подпространств размерности шесть. Но тогда он не обходит ${\ensuremath{\mathbb{Z}}}^7$.
[Робот с генератором случайных чисел, камнем и плоскостью флажком]{}
Осталось рассмотреть случай робота с камнем и плоскостью флажков. Изначальное положение камня и робота в клетке с нулевыми координатами. Из неё же выходит плоскость флажков с координатами $(0,0,0,0,0,0,a,b)$.
[Обход ${\ensuremath{\mathbb{Z}}}^8$]{}
Чтобы построить обход ${\ensuremath{\mathbb{Z}}}^8$ просто слегка модернизируем программу робота, обходящего ${\ensuremath{\mathbb{Z}}}^8$ с камнем и флажком. Оказавшись на флаге робот должен сделать случайный, равновероятный ход по одному из векторов $(\vec{e_7},\ \vec{-e_7},\ \vec{e_8},\ \vec{-e_8}, 0)$. Переход на $\vec{0}$ делаем ходом по вектору $\vec{e_7}$, а потом обратно. Из-за того, что это блуждание побывает во всех клетках плоскости бесконечное число раз, а изначальный робот был во всех клетках ${\ensuremath{\mathbb{Z}}}^6$ (и находился на клетке с флагом и камнем бесконечное число раз), то теперь робот побывает в каждой клетке пространстве ${\ensuremath{\mathbb{Z}}}^8$,
[Невозможность обхода ${\ensuremath{\mathbb{Z}}}^9$]{} Этот случай не требует долгих доказательств, так как может быть доказан по аналогии со случем одного флажка. Множество клеток, где может оказаться камень будет являться объединением конечного числа “подпространств” размерности 6. То есть клетки, которые гарантировано посетит робот, являются объединением конечного числа “подпространств” размерности 8. Значит робот не обойдёт ${\ensuremath{\mathbb{Z}}}^9$.
[99]{}
Г. Килибарда, В. Б. Кудрявцев, Ш. М. Ушчумлич. *Независимые системы автоматов в лабиринтах.* Дискрет. матем., 1987г.
Г. Килибарда. *О сложности автоматного обхода лабиринтов.* Дискрет. матем., 1993г.
Г. Килибарда, В. Б. Кудрявцев, Ш. М. Ушчумлич. *Независимые системы автоматов в лабиринтах.* Дискрет. матем., 2003г.
Анджанс А. В. *Поведение детерминированных и вероятностных автоматов в лабиринтах:Дисс. канд. физ.-мат. наук* Рига, 1987г.
Г. Килибарда, В. Б. Кудрявцев, Ш. М. Ушчумлич. *Коллективы автоматов в лабиринтах.* Дискрет. матем., 2003г.
Г. Килибарда, Ш. М. Ушчумлич. *О лабиринтах-ловушках для коллективов автоматов.* Дискрет. матем., 1993г.
Г. Килибарда, В. Б. Кудрявцев, Ш. М. Ушчумлич. *Системы автоматов в лабиринтах.* Дискрет. матем., 2006г.
F. Spitzer. *Principles of Random Walks.* Van Nostrand, 1964г.
А. Н. Ширяев. *Вероятность: в 2-х кн. Вероятность-1, Вероятность-2.* МЦНМО, 2004г.
Дынкин Е.Б., Юшкевич А.А. *Теоремы и задачи о процессах Маркова* Наука, 1967г.
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{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'We examine the string cosmology equations with a dilaton potential in the context of the Pre-Big Bang Scenario with the desired scale factor duality, and give a generic algorithm for obtaining solutions with appropriate evolutionary properties. This enables us to find pre-big bang type solutions with suitable dilaton behaviour that are regular at $t=0$, thereby solving the graceful exit problem. However to avoid fine tuning of initial data, an ‘exotic’ equation of state is needed that relates the fluid properties to the dilaton field. We discuss why such an equation of state should be required for reliable dilaton behaviour at late times.'
author:
- |
G F R Ellis[^1], D C Roberts[^2], D Solomons[^3], and P K S Dunsby[^4]\
Mathematics Department, University of Cape Town,\
Rondebosch, Cape Town 7701, South Africa[^5]
date: 3 March 2000
title: |
A Solution to the Graceful Exit Problem in\
Pre-Big Bang Cosmology
---
1.5cm
Introduction
============
In this paper, we investigate the equations of string cosmology [@Gasperini], [@Lidsey] in the string frame, allowing for a dilaton potential $V(\phi)$. The Pre-Big Bang Scenario is motivated by the search for cosmological solutions with an $a(t)\rightarrow 1/a(-t)$ symmetry in the scale factor $a(t),$ which implements an analogue of the T-duality symmetry of M-theory. However one must distinguish between symmetries of the equations and those of their solutions. We look at cases in which the [ *equations* ]{} have such a scale factor symmetry, when solutions may or may not exhibit the same symmetry, and at cases in which the [*solutions*]{} obey the scale factor symmetry, even if the equations do not. In the latter case we obtain some solutions that seem to have most of the properties desired in the Pre-Big Bang scenario, in that they have the desired scale factor symmetry, the desired evolution of the dilaton field, and continuity at $
t=0$ of $a(t),\phi (t),\dot{\phi}(t)$, and the Hubble parameter $H(t)\equiv
\dot{a}(t)/a(t)\ $ (but allowing a discontinuity in $\dot{H}(t)$ and $\ddot{\phi}(t)$ there, implying a corresponding discontinuity in $\partial V/\partial \phi $), thus providing a solution to the graceful exit problem [@nogo; @nogo1]. However, to obtain the desired dilaton behaviour at recent times, we need to employ an ‘exotic’ equation of state as discussed below.
There are ‘no-go theorems’ that exclude such regular transitions in the presence of a perfect fluid and Kalb-Ramond sources. A ‘lowest order’ Einstein frame analysis by [@nogo1] discusses graceful exit in generalized phase-space, and derives a set of necessary conditions for transition from a classical dilaton-driven inflationary pre-big bang phase to a radiation-dominated era, joined at $t=0$ in a Planck epoch of maximal finite curvature $\dot{H}(t)$. They show that a successful exit requires violation of the null energy condition (NEC). Classical sources tend to obey NEC, but various new kinds of effective sources generating non-singular evolution have been considered that do not. Thus failing invocation of higher order curvature terms, some kind of exotic behaviour of matter is necessary in order to obtain a graceful exit from the pre-big bang phase.
In this paper we follow Gasperini et al [@Gasperini] by working in the string frame. The relation to the Einstein frame is left for another paper. It should be made clear from the start that our solutions are rather special in the spectrum of pre-big bang models; those we concentrate on in the main show an exact scale factor duality in the solutions, and thus we do not consider here the more exciting possibility of a phase of early kinetic-dilaton dominated inflation which leads to an early phase which is not radiation dual but is genuinely stringy inflationary vacuum. Nevertheless the set of solutions investigated here help to understand the spectrum of possibilities available within the broad Pre-Big Bang set of ideas.
String Cosmology Equations {#streqns}
==========================
One can determine the general equations of string cosmology by extremizing the lowest order effective action of dilaton gravity, $$S=-\frac 1{2\lambda _s^{d-1}}\int d^{d+1}x\sqrt{|g|}e^{-\phi }\left[
R+(\nabla \phi )^2-\frac 1{12}H^2+V(\phi )\right] \linebreak[4]+\int
d^{d+1}x \sqrt{|g|}L_m,$$ where $\phi $ is the scalar dilaton, $H=dB$ (antisymmetric tensor field strength), $V(\phi )$ is the dilaton potential, $\lambda _s$ is the fundamental string length scale, and $L_m$ is the Langrangian density of other matter sources. To derive string cosmology equations for the $d=3$, homogeneous, isotropic, conformally flat background we will follow Gasperini [@Gasperini] in assuming $B=0$, a perfect fluid minimally coupled to the dilaton, and a Bianchi I type metric (see Appendix C of [@Gasperini] for details). Unlike Gasperini we assume $V(\phi )\ne 0,$ to obtain the string cosmology equations in the following canonical form: $$H^2=\frac{e^\phi }6\rho +H\dot{\phi}+\frac V6-\frac{\dot{\phi}^2}6,
\label{friedman}$$ $$\dot{H}+H^2=e^\phi (\frac p2-\frac \rho 3)-H\dot{\phi}+\frac{\dot{\phi}^2}3-
\frac{V^{\prime }}2-\frac V3, \label{ray}$$ $$\ddot{\phi}=-3H\dot{\phi}+\dot{\phi}^2-V-V^{\prime }+e^\phi (\frac{3p}
2-\frac \rho 2), \label{dilaton}$$ where $V^{\prime }=\frac{\partial V}{\partial \phi }$. When combined, these imply the standard energy conservation equation: $$\dot{\rho}=-3H(\rho +p)\;. \label{energy}$$ In a relationship analogous to that between the classical Friedmann equation and Raychaudhuri equation, $$Eq.(\ref{friedman})~is ~the~ first~ integral~ of~ eq.(\ref{ray})~ provided
~that ~eq.(\ref{dilaton})~ and~ eq.(\ref{energy})~ hold. \label{cons}$$ These four equations will be the basis for the analysis in this paper.
One of the primary motivations for the pre-big bang scenario [@Veneziano] is that when $V(\phi )=0$, these equations are invariant under the following transformation: $$a(t)\rightarrow \hat{a}(t)=a^{-1}(t) \label{a}$$ provided that the dilaton transforms as $\phi \rightarrow \hat{\phi}=\phi -6\ln a$ and the energy density and pressure as $\rho \rightarrow \rho ^{\prime
}=a^6\rho ,\;p\rightarrow p^{\prime }=-pa^{-6}$. Thus if $a(t)$ is a solution, so is $\hat{a}(t)$ for suitable $\phi ,\rho ,p.$ Since the string cosmology equations are also invariant under time reversal symmetry,
$$a(t)\rightarrow \overline{a}(t)=a(-t) \label{time_rev}$$
the deceleration associated with standard post-big bang cosmology can be associated with an accelerated evolution prior to the big bang by the generalized transformation $$a(t)\rightarrow \tilde{a}(t)=a^{-1}(-t). \label{dual}$$ where $\tilde{a}(t)$ is a solution for suitable $\phi ,\rho ,p$ because $
a(t) $ is. The solution has T-duality symmetry if for each $t,$
$$a(t)=\tilde{a}(t)=a^{-1}(-t). \label{duality}$$
However, if we assume $V(\phi )\ne 0$ as in eqs.(\[friedman\]-\[dilaton\] ), then in general the equations are not invariant under the symmetry eq.( \[duality\]) even if the solutions are. We will look at both cases in what follows, but generically allowing a potential that does not preserve the symmetry. Note that if we assume matter with [*the same* ]{} equation of state before and after $t=0,$ then the matter equations also will not be invariant under the scale factor symmetry. One has to decide what is more physically meaningful: matter with a universal equation of state applicable at all times, or that has a discontinuous equation of state that preserves this symmetry. In what follows, we adopt the first option. We return to discuss this choice in the conclusion.
Flat Dilaton Potential with Exotic Equation of State
----------------------------------------------------
To obtain equations of motion preserving the scale factor symmetry eq.(\[duality\]), we assume the simplest potential, namely a flat potential: $$V(\phi )=\kappa \label{flat}$$ where $\kappa $ is a constant, and then investigate the behaviour of the universe. In order to reliably obtain proper limiting behaviour of the dilaton, we assume that the equation of state $$p=\frac \rho 3+\frac 23e^{-\phi }\kappa \label{eqnofstate}$$ holds at all times (this choice, which is not invariant under the duality symmetry, is discussed further in the following sections). One can immediately see that at late times if $\phi \rightarrow $ constant, as we will show follows from this choice, then this equation of state simply reduces to radiation plus a constant.
We are interested in getting satisfactory dynamics for $H(t)$ and $\phi (t)$ , or equivalently for $\chi (t)\equiv \dot{\phi}$. To see when this occurs, we manipulate the string cosmology equations (\[friedman\]-\[dilaton\]) subject to eqs.(\[flat\],\[eqnofstate\]) to obtain the two-dimensional phase space with coordinates ($\chi $,$H$) governed by the following equations: $$\dot{H}=\frac{\chi ^2}6-2H^2+\frac \kappa 6,
\label{dotH}$$ $$\dot{\chi}=\chi (\chi -3H),
\label{dotchi}$$
the latter following because of choice (\[eqnofstate\]). Having chosen the constant $\kappa $, we can set initial conditions $(\chi _0,H_0)$ at $t=0$, and then extend the solution to positive and negative values of $t$ by use of these equations. For $\kappa <0$, there are no fixed points in the phase plane, and on every trajectory both $H$ and $\chi $ diverge as $
|t|\rightarrow \infty $. For $\kappa =0$, i.e. no dilaton potential, there is one fixed point at the origin, but for any initial condition (set at $t=0$ ), $\chi $ and $H$ will diverge either as you run time forwards or run time backwards.
The interesting dynamics is obtained when $\kappa >0$. There are then fixed points at $A_{+}$: $(0,\sqrt{\kappa /12})$ (a source), $A_{-}$: $(0,\sqrt{
\kappa /12})$ (a sink), $B_{+}$: $(\sqrt{3\kappa },\sqrt{\kappa /3})$ (a saddle point), and $B_{-}$:$\;(-\sqrt{3\kappa },-\sqrt{\kappa /3})$ (a saddle point). In the phase plane depicted in Figure 1 above, we claim the initial conditions in the region I bounded by $A_{+}$, $B_{+}$ and $A_{-}$ and the separatrixes joining them give satisfactory dynamics of both $H$ and $\chi $ which include $\chi \rightarrow 0$ as $
|t|\rightarrow \infty $, $\;\chi >0$ for all times so $\phi (t)$ is monotonic, $H$ remains finite, and a “bounce” occurs that avoids the initial singularity. Since region I is bounded by fixed points that have coordinates proportional to $\sqrt{\kappa }$, increasing $\kappa $ will give one a larger region of initial conditions that lead to a nonsingular universe with proper dilaton dynamics. We can obtain a solution on the boundary of region I (evolving along the line joining $A_{-}$ to $A_{+},$ which does not lie in I) that is invariant under symmetry (\[time\_rev\]) by setting $\chi _0=0,\;H_0=0$ at $t=0,$ but this solution, given explicitly by $a(t)=\cosh ^{1/2}(\sqrt{\kappa /3}t),$ is not invariant under the symmetry (\[duality\]). A drawback of all these models is that inflation will not stop at $t>0$, but as discussed in the conclusion, the string cosmology equations derived in section \[streqns\] do not apply to the present cosmological regime without modification, so it is possible that a radiation dominated evolution started after the time when these equations no longer apply. In any case this gives a specific family of solutions where the equations display the desired symmetry (\[duality\]) but the solutions do not - which is not very surprising, given the prevalence of broken symmetries in physics.
Obtaining Desired Dynamics From a Dilaton Potential
===================================================
In this section, we generalize the method introduced by Ellis and Madsen [@Ellis] through which they obtain a classical scalar potential associated with a specified $a(t)$ in the standard gravitational equations. No field has been observed that coincides with a dilaton potential $V(\phi )$ , so we assume that it is a freely disposable function. We show that by suitable choice of $V(\phi )$ one can obtain almost any behaviour for $a(t),$ or alternatively for $\phi (t)$. We first present an algorithm for determining $V(\phi )$ from a desired $a(t)$ or a desired $\phi (t)$, and then present an analytically smooth solvable example. This solution illustrates our main point, but has little physical relevance (although it does satisfy the symmetry (\[duality\])). In the following section we use these methods to obtain two solutions that resemble the standard “pre-big bang scenario”, but with continuity of $a(t)$ and $H(t)$ and with satisfactory dynamics of $\phi (t)$. The associated dilaton potentials are [*ad hoc*]{} because they are derived from the desired behaviour of the universe, rather than from a field theory model; as discussed in many inflationary and quintessence studies, see e.g. [@Lidsey1],[@Saini].
The Algorithms {#algorithm}
--------------
We proceed by providing the following [*general algorithm* ]{}for determining a dilaton potential $V(\phi )$ that produces a desired $a(t)$:
[**1)**]{} Specify a desired monotonic function for the scale parameter $a(t)$ , consequently determining $H(t)$ and $\dot{H}(t)$,
[**2)**]{} Choose an equation of state and solve for $\rho (a)$ from eq.(\[energy\])[^6]; as $a(t)$ is known, this determines $
\rho (t)$.
[**3)**]{} Eliminate $V$ and $V^{\prime }$ from eq.(\[dilaton\]) by use of eqns.(\[friedman\]) and (\[ray\]) to obtain a differential equation relating $H(t)$, $\phi(t)$, $\rho(t)$, and their time derivatives.
[**4)**]{} Solve the equation obtained in step 3) for $\phi (t).$
[**5)**]{} Substitute the now known functions $\phi (t)$, $\rho (t)$, and $
H(t) $ and $a(t)$ into the rearranged version of eq.(\[friedman\]) $$V(t)=6H^2-e^\phi \rho -6H\dot{\phi}+\dot{\phi}^2 \label{V}$$ to obtain $V(t)$.
[**6)**]{} Invert $\phi (t)$ to obtain $t(\phi )$, and
[**7)**]{} Transform $V(t)$ as follows: $V(t)=V(t(\phi ))\Rightarrow V(\phi )$ . This is possible for each range of $t$ on which $\phi (t)$ is monotonic (if it is not monotonic on some range of $t$, in general $V(\phi )$ will not be well-defined because it will not be single valued for the corresponding values of $\phi $).
Thus, provided $\phi (t)$ determined from step 3) is monotonic, we find a $V(\phi )$ that corresponds to a given monotonic function $a(t)$. Because we have now satisfied eqs.(\[energy\]), (\[friedman\]) and the equation obtained in step 3), the latter depending essentially on eq.( \[dilaton\]), it follows from statement (\[cons\]) that eq.(\[ray\]) will be true also, so we have satisfied all the equations of the theory for this matter description (c.f. [@Ellis]); hence we have a solution of the desired form.
Alternatively, we can give an algorithm for determining a dilaton potential $
V(\phi )$ that produces a desired dilaton evolution $\phi (t)$[^7] by proceeding in the same way as above, except for minor changes: replace step [**1)**]{} by
[**1’)**]{} specify the desired monotonic function for the dilaton, $\phi (t),$\
in step [**2)**]{}, leave $\rho $ in the form $\rho (a),$ and replace step [**4)**]{} by
[**4’)**]{} solve the equation obtained in step 3) for $a(t)$ (or for $H(t)$).\
The rest of the algorithm is as before.
Finally, note that we can carry out these procedures piecewise: for example we can specify $a(t)$ for some range of $t$ and $\phi(t)$ for some adjoining range of $t$, or different behaviours for $a(t)$ for different ranges of $t$ , then join the solutions together, ensuring that $a(t)$, $H(t)$, $
\phi(t) $ and $\chi(t)$ are continuous where these ranges meet.
Exponential Scale Factor Behaviour with No Matter
-------------------------------------------------
To demonstrate the procedure, we give a simple analytically solvable example with a pure scalar field, i.e. $\rho =p=0$. Consider an exponential expansion as in classical inflation, $$a(t)=e^{wt}\Rightarrow H=w, ~\dot{H}=0,$$ where $w$ is a positive constant. This solution has the desired symmetry ( \[duality\]).
In this case the differential equation for $\phi (t)$ takes the form: $$\ddot{\phi}=H\dot{\phi} \label{phi1}$$ Using eq.(\[phi1\]) and eq.(\[V\]) we obtain $$\phi (t)=\phi _0+\frac{\dot{\phi}_0}w(e^{wt}-1), \label{phi(t)}$$ a monotonic function as required, and $$V(t)=6w^2-6\dot{\phi}_0we^{wt}+\dot{\phi}_0^2e^{2wt}. \label{V(t)}$$ After inserting the inverted eq.(\[phi(t)\]), $$t(\phi )=\frac 1w\log \left[ \frac w{\dot{\phi}_0}\left( \phi -\phi _0+\frac{
\dot{\phi}_0}w\right) \right] ,$$ into eq.(\[V(t)\]), one obtains $$V(\phi )=w^2(\phi -3-\phi _0+\frac{\dot{\phi}_0}w)^2-3w^2$$ which is simply a quadratic potential plus a constant. Clearly the behaviour for $\phi (t)$ is unphysical since $\phi (t)\rightarrow \infty $ instead of asymptoting to a constant. However, this gives a transparent example where even though the scale factor symmetry (\[duality\]) is broken in the equations (because $V(\phi )$ is not constant), the solution obeys that symmetry.
“Pre-big Bang” Behaviour
========================
In this section we try to use the methods just explained to obtain solutions that resemble the “pre-big bang scenario” but with satisfactory dynamics of $\phi (t)$ and a continuous transition from the pre-big bang to post-big bang phases. In these examples, we seek solutions that evolve from a string perturbative vacuum, i.e. $H\rightarrow 0$ and $e^\phi \rightarrow 0$ (no interactions), to the present scenario where $e^\phi $, which acts as the coupling constant, asymptotes to a constant. We will assume the following behaviour of the universe: $$a(t)=(t+1)^{\frac 12},\;t\geq 0\Rightarrow H(t)=\frac 1{2(t+1)} \label{t}$$ determining $a(t)$ for $t\geq 0,$ and by the symmetry (\[duality\]) $$a(t)=(-t+1)^{-\frac 12},\;t\leq 0\Rightarrow H(t)=\frac 1{2(-t+1)}
\label{t-}$$ determining $a(t)$ for $t\leq 0$. Both $a(t)$ and $H(t)$ are continuous at $
t=0$ with $a(0)=1,\;H(0)=1/2$, but $\dot{H}(t)$ is not continuous there.
This behaviour, which is essentially radiation dominated evolution of the universe for positive times and power-law inflation for negative times, is motivated by the “pre-big bang” scenario introduced in [@Veneziano], and exactly obeys the scale factor symmetry (\[duality\]). Note that we have shifted the origin of time in each branch from that customarily used, in order to get a smooth evolution through $t=0$; this of course makes no difference to the desired physical behaviour, for we can choose the origin of time to be wherever we want (and the equations are invariant under time translation $t\rightarrow t^{\prime }=t+c).$ Although the power law inflation ends with the scale-factor value $a(0)=1,$ required by continuity together with the symmetry (\[duality\]), the solution has sufficient inflation for any purpose because it involves an infinite number of e-foldings (it starts with the asymptotical value $a=0$ as $t\rightarrow
-\infty ).$
Pre-Big Bang behaviour with radiation equation of state
-------------------------------------------------------
First we assume the radiation equation of state holds at all times, that is, $$p=\frac \rho 3, \label{rad}$$ which, using eqs.(\[energy\]) and (\[t\],\[t-\]), implies $$\rho (\pm t)=\rho _0(\pm t+1)^{\mp 2} \label{rho}$$ where $\rho _0$ is a positive constant and ‘$+t$’ refers to the post-big bang era, ‘$-t$’ to the pre-big bang era. Notice that both $\rho $ and $\dot{
\rho}$ are continuous at $t=0$.
The equation for $\phi $ now takes the form $$\ddot{\phi}=\frac 23e^\phi \rho +H\dot{\phi}+2\dot{H}. \label{phi2}$$ Substituting in eqs. (\[t\]) and (\[rho\]), we could not find an analytical solution to eq.(\[phi2\]), so we investigate the three dimensional phase space with coordinates $(t,\phi ,\chi ),$ given from eqs. (\[phi2\],\[t\],\[t-\],\[rho\]) by $$\dot{\phi}=\chi ,~~~\dot{\chi}=\frac 23e^\phi \rho _0(\pm t+1)^{\mp 2}+\frac
\chi {2(\pm t+1)}\mp \frac 1{(\pm t+1)^2}, \label{ev}$$ where the top sign holds for $t>0$ and the bottom sign for $t<0.$ We can set initial data at $t=0$, and then investigate the phase plane orbits as we run the trajectory forward and backwards in time in such a way that $\chi $ and $
\phi $ are continuous through $t=0$. Then $\dot{\chi}$ is discontinuous there, but we have no problem in joining the solutions for $t>0$ and $t<0$.
For $t>0$, there is an exceptional integral curve $\gamma (t)$ given by $(t,
\tilde{\phi}_0$,$0$), where $\tilde{\phi}_0\equiv $ ln$(\frac 3{2\rho _0})$; this is the only integral curve with a fixed value of $\phi $ and $\chi $. Note that setting $\phi _0$ and $\chi _0$ determines the initial point in the phase space, and specifying $\rho _0$ determines the location of this exceptional curve. In the 2-dimensional sub-spaces $t=const$ with coordinates ($\phi ,\chi ),$ the curve $\gamma (t)$ has coordinates $(\tilde{
\phi}_0,0)$ for all $t,$ and represents a set of saddle points parametrised by $t$. To get exactly the desired dilaton dynamics in the future ($\chi >0$ , $e^\phi \rightarrow $ constant $\Rightarrow \chi \rightarrow 0$ as $
t\rightarrow \infty $), one must restrict the initial conditions ($\phi _0$,$
\chi _0$) to start precisely on the stable branch of these saddle points, which intersects the surface $t=0$ in a single curve $(0,\gamma _{+}(\chi
),\chi )$ passing through the exceptional point $\gamma _0=(0,\tilde{\phi}
_0,0)$ (for more details, see Appendix A). However there is actually slightly more freedom than this in finding physically relevant initial conditions because if a trajectory starts close enough to the stable branch (but not exactly on it), then the trajectory will stay close to the fixed point for an arbitrarily long period of time before $\phi $ and $\chi $ diverge, and this may suffice for physical purposes even if the solution eventually diverges (cf. the discussion of intermediate isotropisation in [@wain]). Nevertheless, the physically relevant set of solutions is very unstable and requires very precise fine tuning, in order to obtain the desired dilaton dynamics, lying in a small open neighbourhood ${\cal D_{+}}$ of the curve $\phi _0=\gamma _{+}(\chi _0)$ in the initial data set at $t=0$. Indeed we have found it very difficult to obtain numerical solutions with the desired behaviour because of this instability.
For $t<0$ there are no points with a fixed value of $\phi$ and $\chi$ (because we assume $\rho _0>0)$. To get the desired dilaton dynamics in the past ($\chi >0$, $e^\phi \rightarrow 0$ as $t\rightarrow -\infty $) one must further restrict the initial conditions, the problem being that eq.(\[ev\] ) is an inhomogeneous equation for $\chi $ with a time-varying source function (albeit a source function that decays away as $t\rightarrow \pm
\infty )$. We can obtain the desired behaviour if $\;y_0=\frac
23e^{\phi_0}\rho _0\ll 1,$ i.e. $\phi_0\ll \ln (\frac 3{2\rho _0})$ (details are given in Appendix A). This is a sufficient condition; there will be a wider domain ${\cal D_{-}}$ of initial data at $t=0$, containing this set, that will ensure that at early enough times the desired behaviour is attained.
To get a satisfactory solution for all time, for a given choice of $\rho _0,$ one must set the initial conditions to lie in both ${\cal D}_{{\cal +}}$ and ${\cal D_{-}}$, so the crucial issue is whether they intersect or not. We have not attained finality on this point. It may be that the ‘no-go’ theorems with a potential [@nogo] imply they do not intersect, but this implication is not entirely clear, as the conditions of those theorems may not correspond precisely to the conditions we contemplate here. If they do intersect, we can attain the desired behaviour $\chi \rightarrow 0$ and $
e^\phi \rightarrow 0$ when time runs backwards as well as $e^\phi
\rightarrow const$ as time runs forward and in principle, one can obtain a continuous $V(\phi )$ associated with the unstable solution described above because every function is continuous on the right hand side of eq.(\[V\]). Furthermore, $\phi (t)$ is monotonic and continuous, and therefore invertible, so one can complete Step 6 of the algorithm set out in section \[algorithm\]. However attaining such solutions will require extreme fine-tuning of the initial data, and this is very difficult to do because one does not know where the stable branch of the saddle point intersects $t=0$. Thus if such solutions do exist, the extreme fine tuning required for their initial data make them seem impracticable as cosmologies despite their other desirable properties.
“Pre-big Bang” Behaviour with Exotic Equation of State
------------------------------------------------------
Finally, we assume the identical “pre-big bang” behaviour of the last example (eqs.(\[t\],\[t-\])), but we obtain a stable solution with a different equation of state. The instability in the last example arises because of our choice of the equation of state, as can be seen by inspection of eq.(\[dilaton\]), which we write now as $$\dot{\chi}=-3H\chi +\chi ^2+\beta , \label{dilaton2}$$ where $$\beta \equiv -V-V^{\prime }+e^\phi (\frac{3p}2-\frac \rho 2). \label{beta}$$
As mentioned before, we want to obtain $e^\phi \rightarrow $ constant, i.e. $
\chi \rightarrow 0$, at late times, which implies $\beta \rightarrow 0$ in eq.(\[dilaton2\]). If we choose the radiation equation of state as in the last example, (eq.(\[rad\])), then $\beta =-V-V^{\prime }$. Therefore, requiring $\beta \rightarrow 0$ as $t\rightarrow \infty $ puts a heavy restriction on the dilaton potential, namely $V\rightarrow e^{-\phi }$ at late times. Consequently, there is a fine-tuning problem if you use the radiation equation of state.
In the present example, we assume $\beta =0$ for all times, which from eq.( \[dilaton2\]) demands the exotic equation of state, $$p=\frac \rho 3+\frac 23e^{-\phi }(V+V^{\prime }), \label{exotic}$$ at all times (note that eq.(\[eqnofstate\]) is the special case resulting when $V^{\prime }=0$). Using this equation of state implies, $$\rho (t)=\int 2He^{-\phi }(12H\dot{\phi}+6\dot{H}-3\dot{\phi}^2)dt
\label{ccons}$$ which allows the density to go through zero and become negative. We discuss this equation further in the [*Conclusion*]{}.
The differential equation that relates $H(t)$ to $\phi (t)$ is simply eq.( \[dilaton2\]) with $\beta =0$, $$\ddot{\phi}=-3H\dot{\phi}+\dot{\phi}^2,$$ For arbitrary $a(t),$ this can be solved (with $a_0^{\ }=1$ and $\chi
_0\equiv \dot{\varphi}_0$) by
$$\exp (\phi _0-\phi (t))=1-\chi _0\int_0^ta^{-3}(t)dt. \label{soln}$$
For the specific case given by eqs.(\[t\],\[t-\]) we obtain from this the analytical solution $$\phi (t)=+\phi _0-\ln \left| 1-2\chi _0[1-(1+t)^{-{\frac 12}}]\;\right|
\label{phi+}$$ for $t>0$ and $$\phi (t)=+\phi _0-\ln \left| 1-{\frac{2\chi _0}5[1-}(1-t)^{{\frac 52}
}]\;\right| \label{phi-}$$ for $t<0$. Inverting eq.(\[phi+\]) we obtain $$t(\phi )=\frac{4\chi _0^2e^{2(\phi -\phi _0)}}{[(1-2\chi _0)e^{\phi -\phi
_0}-1]^2}-1 \label{solnst}$$ and inverting eq.(\[phi-\]) we obtain $$t(\phi )=1-\left[ \frac{5e^{\phi -\phi _0}-5+2\chi _0}{2\chi _0}\right]
^{2/5} \label{solnend}$$ Now we can solve eq.(\[ccons\]) to obtain $\rho (a)$ and so $\rho (t)$ (see Appendix B for one particular case), and substitute our results into eq.(\[V\]) to obtain the dilaton potential $V(\phi )$ that is associated with our specified “pre-big bang” behaviour. This is straightforward but tedious, and results in very complex analytic expressions (the real complexity coming through the expressions for $\rho (t)$ that occur consequent on the choice of the exotic equation of state). Rather than giving these analytic expressions, we give a graph of the potential for one particular case below.\
To discuss the relevant initial conditions, it is instructive to look at the phase plane (Figure 2 above) with coordinates ($t,\chi$ ), where $\chi =\dot{\phi}$ is governed by the equation: $$\dot{\chi}=-\frac 3{2(\pm t+1)}\chi +\chi ^2 \label{chi}$$ where we again use $+$ to represent $t>0$ and $-$ to represent $t<0$. One can easily see that $\chi =0$ $(\Rightarrow \dot{\chi}=0)$ is an attractor, and represents a physically uninteresting solution with $\phi =const$. Also $
\chi =\frac 3{2(\pm t+1)}$is a nullcline, characterising the other points where $\dot{\chi}=0$. This curve starts at $(0,\frac32)$ and drops symmetrically away to zero as $t\rightarrow \pm \infty .$ Now we can solve eq.(\[chi\]) analytically for $t>0,$ finding
$$\chi =\frac 1{2(t+1)(1+C_{+}\sqrt{t+1})} \label{chi1}$$
where $C_{+}=(\frac 1{2\chi _0}-1)$ is positive iff $\;\chi _0<1/2.$[** **]{} The separatrix between the solutions that diverge and those that go asymptotically to zero as $t\rightarrow \infty $ is the special solution with $C_{+}=0$ which goes through $(0,\frac12),$ that is,
$$\chi =\frac 1{2(t+1)} \label{chi2}$$
which itself goes to zero as $t\rightarrow \infty .$ If we specify the initial conditions at $t=0$ such that $\phi _0$ is free and $0<\chi _0<\frac
12\Leftrightarrow C_{+}>0$, then as we run the trajectories forward in time $
\chi \rightarrow 0.$ In this case, for large positive values of $t$, eq.(\[chi1\]) will be approximately
$$\chi =\frac 1{2C_{+}\;t\sqrt{2t}\ }\ >0 \label{chi1+}$$
(note that $\phi (t)$ is monotonic for $t>0$ because $\chi >0$ on these trajectories). Let $T_{+}$ be such that eq.(\[chi1+\]) is valid for all $
t>T_{+}>0$. Then for $t>T_{+},$ $$\phi (t)\simeq \int_{T_{+}}^t\frac 1{2C_{+}t\sqrt{2t}\ }dt+\phi
_{T_{+}}=\frac 1{C_{+}\sqrt{2}\ }[T_{+}^{-1/2}-t^{-1/2}]+\phi _{T_{+}}.$$ Thus as $t\rightarrow \infty $, for all $\chi _{_0},$ $\phi (t)\rightarrow $ a constant value, say $\phi _\infty ,$ and $\exp \phi (t)\rightarrow \exp
(\phi _\infty) $. (Note that it is essential to check this result even though $\chi \rightarrow 0$, cf. the discussion below of what happens as $
t\rightarrow -\infty ).$ If we specify the initial conditions at $t=0$ such that $\phi _0$ is free and $\frac 12<\chi _0\Leftrightarrow C_{+}<0$, as we run the trajectories forward in time then $\chi \rightarrow \infty $ as $
t\rightarrow t_0$ given by $1+C_{+}\sqrt{t_0+1}=0,$ that is $t_0=\frac{
(2\chi _0)^2-1}{(2\chi _0-1)^2}.$ In this case for large values of $\chi ,$ eq.(\[chi\]) can be approximated as follows:
$$\chi \gg \frac 3{2(t+1)}\Rightarrow \dot{\chi}\simeq \chi ^2\Rightarrow \chi
\simeq 1/(t-t_0). \label{chismall}$$
The solution diverges as $t\rightarrow t_0$ and the approximation eq.(\[chi1+\]) never applies. This behaviour conforms to that implied by eq.(\[phi+\]), and may be seen clearly on the phase plane.
If we run the trajectories backward in time, starting from initial data with $\chi _0>0$, they will cross the nullcline and then drop to zero, never becoming negative because $\chi =0$ is an exceptional solution of the equations. Then $\phi (t)$ is monotonic for $t<0$ also because $\chi >0$ on these trajectories. Solving eq.(\[chi\]) analytically for $t<0$ gives
$$\chi =-\frac 52\frac{(t-1)\sqrt{-t+1}}{(t^2-2t+1)\sqrt{-t+1}+C_{-}}
\label{neg}$$
where $C_{-}=(\frac 5{2\chi _{_0}}-1).$ This expression goes to zero for all $C_{-}>-1$, corresponding to $\chi _{_0}>\;0\ $ (note that it does not matter if $C_{-}$ is positive or negative). For large negative $t$ its value, for all $C_{-},$ will be approximately $$\chi =d\phi /dt\simeq -\frac 5{2t\ }. \label{neg1}$$ Let $T_{-}$ be such that eq.(\[neg1\]) is valid for $t<T_{-}<0.$ Then for $
t<T_{-}$, $$\phi (t)\simeq \int_{T_{-}}^t(-\frac 5{2t\ })dt+\phi_{T_{-}} =\frac 52\ln (
\frac{T_{-}}t)+\phi_{T_{-}}-~~ \Rightarrow ~~\exp \phi (t)\propto (\frac{
T_{-}} t)^{5/2}$$ Thus as $t\rightarrow -\infty $, for all $\chi _{_0},$ $\phi
(t)\rightarrow -\infty $ even though $\chi \rightarrow 0,$ and $\exp \phi
(t)\rightarrow 0,$ which is the dynamics we desire [@Gasperini], and indeed is indicated already by eq.(\[phi-\]). The value of $C_{-}$ corresponding to the separatrix eq.(\[chi2\]) is $C_{-}=4,$ which does not give any special behaviour for $t<0.$
Typical results of the integrations for this case are given in Figures 3-5 below.
fig3.tex
fig4.tex
In summary, one gets a stable solution for $0<\chi _0<1/2$, as one can see from the phase plane, with good “pre-big bang” behaviour and the desired dynamics for $
\phi (t){\bf \ }$ for both large and small $t.$ The shape of the potential is a bit unusual, but results directly from the specific requested ‘pre-big bang’ evolution eqs.(\[t\],\[t-\]) and the chosen initial conditions. Smoothing out that behaviour at $t=0,$ so that the solution departs from the ‘radiation’ form eq.(\[t\]) at very early times while preserving the symmetry (\[duality\]), will result in a smoothed out potential $V(\phi )$ ; we can choose $a(t)$ in this way so that $H(t)$ and hence $V(\phi )$ are continuous at $t=0.$ Initial conditions can be set so that the matter has the desired late time behaviour: $p/\rho \rightarrow 1/3,\rho \rightarrow 0;$ however it then has unusual behaviour at early times in that both $\rho $ and $h\equiv \rho +p$ go negative for some values of $t<0$. It is unclear if this should be regarded as a serious defect of the model or not, remembering that with the unusual equation of state adopted, the properties of matter are different than usual, and in particular the speed of sound will no longer be given by the usual expression. This needs further investigation. What is clear is that these solutions are not physically reliable as $t\rightarrow +\infty $ (see below), and they will have to be joined on to some other solution to give an adequate model of the universe with ordinary matter behaviour at late times. However, as discussed below, that problem occurs in the entire family of pre-big bang models, and so is not restricted to the models considered here.
fig5.tex
Discussion
==========
We have given examples making very clear the distinction between the equations and the solution having the desired ‘pre-big bang’ symmetry. We have given a broad method of attaining desired string cosmology solutions when there is a dilaton potential $V$ not equal to zero, and used it to obtain ‘Pre-Big Bang’ solutions that seem to have close to the desired properties. In the first case considered, choice of the exact radiation equation of state (\[rad\]) at all times leads to a very unstable situation where extreme fine-tuning of initial conditions is required to attain the desired results, and indeed there may be no initial data leading to the desired behaviour in both the forward and backwards directions of time. In the second case we impose an ‘exotic’ equation of state (\[exotic\]) that links the fluid behaviour to the potential in a way that generalises the perfect fluid equation of state, and we obtain solutions of the desired type without the need for fine tuning the initial data set at $t=0.$
This equation of state looks strange, and the resulting matter behaviour is certainly unusual, but we have no solid handle to use in restricting equations of state in this early era; and we suggest that [*it is essential to choose such an equation if one wants the solution to reliably tend to the ‘classical’ form at late times*]{}. This is because of the form of the equation for $\ddot{\phi}$; if we do not set $\beta =0$, where $\beta $ is defined by eq.(\[beta\]) then almost always that desired classical state will not be attained, because of eq.(\[dilaton2\]); but setting $
\beta =0$, which leads to the desired behaviour, leads immediately to our ‘exotic’ equation of state. Insofar as that equation of state and resulting behaviour is unsatisfactory, this indicates that [*there is a problem with the form of the equation for*]{} $\ddot{\phi}$, which comes directly from the standard variational principle employed in the context of the pre-big bang scenario. The remedy probably lies in finding other scenarios with alternative forms of the variational principle, leading to other equations for $\ddot{\phi}$.
This is also indicated because the present form of the equations does not accommodate ordinary matter, the point being that the above analysis applies even if there is no dilaton potential. Suppose $V=0;$ then eq.(\[dilaton2\] ) remains true, but now $$\beta =e^\phi (\frac{3p}2-\frac \rho 2), \label{beta1}$$ so a reliable approach of the dilaton to a classical solution at late times, requiring $\beta =0,$ demands the radiation equation of state (\[rad\]); a baryon dominated epoch is not allowed[^8]. This is usually dealt with by stating that eqs.(\[friedman\]-\[energy\]) don’t apply at late times in the history of the universe - a different set of equations are to be used then, and the solutions for early times obtained from eqs.( \[friedman\]-\[energy\]) must be suitably joined on to that late time evolution. However given the vision of M-theory as representing the fundamental theory of gravity, it should be able to describe that epoch too; this apparently requires some modified scenario and associated variational principle (note that although we have discussed the issue in the string frame, it also arises in essentially the same form in the Einstein frame). In any case, whether one accepts this argument or not, given the standard variational principle and equations, we argue that the ‘exotic’ equation of state implied by setting $\beta =0$ is [*necessary*]{} to give the desired behaviour; when adopted, it enables obtaining that behaviour reliably (i.e it eliminates the need for extreme fine-tuning of data set at $t=0$).
However one should note here that we have perhaps been somewhat extreme in imposing this equation of state at all times. It is only really needed, on our approach, near the time of the turnaround, and one could obtain far more general behaviours by modifying what we have here in that light; what is required is that the quantity $\beta$ must go to zero in the period when the dilaton is stabilised. It has also been pointed out to us that it is not clear why the deviation from its vanishing point should be absorbed completely in the pressure, and then promoted into the conservation equation; other models of the transition [@trans1; @trans2; @trans3] successfully stabilise the dilaton at late times without this requirement, with suggestions for classical and quantum corrections in the effective action taking the place of the exotic fluid. Hence our proposal must just be seen as one of a range of possibilities in this regard.
Because we have not made the usual separation of our solution into a ‘+’ and a ‘-’ branch, it is not immediately clear why these solutions are not ruled out by the ‘no-go’ theorems involving a dilaton potential [@nogo]; this is presumably because those theorems exclude fluids with the equation of state we have assumed. We also have not examined the relation of these string-frame solutions to the corresponding Einstein-frame versions. These issues await investigation.
We thank M Gasperini, A Coley, R Tavakol, and the referee for helpful comments, and particularly J Lidsey for helpful discussions. DR wishes to thank Elaine Kuok for her patience in checking many of the calculations in this paper. We thank the NRF (South Africa) and Queen Mary College, London, for financial support.
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Ellis, G F R, and Madsen, M S. 1991. Class. Quantum Grav. 8, 667-676.
Wainright J and Ellis G F R. 1997. [*Dynamical Systems in Cosmology*]{}. Cambridge University Press.
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Appendix A: Pre-Big Bang Evolution for Radiation {#appendix-a-pre-big-bang-evolution-for-radiation .unnumbered}
================================================
For given $\rho _0$, it is convenient to define $y=\frac 23e^\phi \rho _0\ $ and change variables to $(t,y,\chi ).$ The equations (\[ev\]) for $t>0$ become
$$\dot{y}=\chi y,\;\dot{\chi}=\frac{y-1}{(t+1)^2}+\frac \chi {2(t+1)},
\label{evplus}$$
In the 2-dimensional sub-spaces $t=const$ with coordinates ($y,\chi ),$ the curve $\gamma (t)$ has coordinates $(1,0)$ for all $t,$ and represents a set of saddle points parametrised by $t$. To get exactly the desired dilaton dynamics in the future ($\chi >0$, $e^\phi \rightarrow $ constant $
\Rightarrow \chi \rightarrow 0$ as $t\rightarrow \infty $), one must restrict the initial conditions ($y_0$,$\chi _0$) to start precisely on the [** **]{}stable branch of these saddle points, which intersects the surface $
t=0$ in a curve $(0,\gamma _{+}(\chi ),\chi )$ passing through the exceptional point $\gamma _0=(0,1,0)$. One can obtain approximate solutions by rewriting the second of eqs.(\[evplus\]) in the form $\;$ $$\left( \frac \chi {(1+t)^{1/2}}\right) ^{.}=\frac{y-1}{(t+1)^{5/2}}$$ Suppose $y$ is almost constant for $t>T_{+},$ implying $\chi $ is close to zero then. Then we can integrate to get
$$t>T_+ \Rightarrow \chi =-\frac 23\frac{y-1}{1+t}+C_{+}\sqrt{1+t}$$ where $C_{+}$ determines the magnitude of $\chi $ at time $T_{+.}$ The first part decays away as desired, but the second part grows with time unless $
C_{+}=0;$ this is the fine-tuning required to attain the desired behaviour of $\chi .$
To investigate $t<0$, it is again convenient to define $y=\frac 23e^\phi
\rho _0$ and change variables to $(t,y,\chi ).$ The equations for $t<0$ become
$$\dot{y}=\chi y,\;\dot{\chi}=y(-t+1)^2+\frac \chi {2(-t+1)}+\frac 1{(-t+1)^2},
\label{evminus}$$
implying that $\dot{\chi}>0$ for all $t<0$; hence $\chi $ necessarily decreases at all times in the past. The problem is that it can become negative, because $\chi =0$ is not an invariant set of the equation. We want a solution where $\chi $ remains positive for all time so that $\phi $ decreases for all time; this means we need $\chi $ to go to a positive value or zero, but not to become negative, and $y$ to go to zero. . As in the previous case one can obtain approximate solutions by rewriting the second of eqs.(\[evminus\]) in the form $\;$ $$\left( \chi (1-t)^{1/2}\right) ^{.}=\frac 1{(-t+1)^{3/2}}(1+y(-t+1)^4).$$ Suppose $$y(-t+1)^4\ll 1~~{\rm for}~~t<T_{-}. \label{inequ}$$ Then we can ignore the second term on the right and integrate to get
$$t<T_{-}\Rightarrow \chi =\frac 2{1-t}+\frac{C_{-}}{\sqrt{1-t}},~~y=y_0\frac
1{(1-t)^2}\exp (-2C_{-}\sqrt{1-t})$$ where $C_{-},\;y_0$ represent the magnitude of $\chi ,\;y$ at time $T_{-\ }.$ This decays away as desired, and consistently preserves the inequality (\[inequ\]) for all earlier times because the exponential always dominates the power law terms. The question then is whether for suitable initial conditions we can attain this inequality at some time $T_{-},$ requiring $
\;y(T_{-})\ll (1-T_{-})^{-4}.$ We can satisfy this with $T_{-}=0$ if $
\;y_0=\frac 23e^{\phi_0}\rho _0\ll 1,$ i.e. $\phi_0\ll \ln (\frac 3{2\rho
_0}).$
Appendix B: Density evolution with exotic equation of state {#appendix-b-density-evolution-with-exotic-equation-of-state .unnumbered}
===========================================================
The ‘pre-big bang’ evolution (\[t\],\[t-\]) implies $H$ and $\dot{H}$ in terms of $a$:
$$\begin{aligned}
\;t \geq 0:&H(a)=\frac 1{2a^2},\;\dot{H}(a)=\frac{-1}{2a^4},\; \label{aa} \\
t \leq 0:&H(a)=\frac{a^2}2,\;\dot{H}(a)=\frac{a^4}2.\end{aligned}$$
Assuming the exotic equation of state (\[exotic\]) implied by setting $
\beta =0$ at all times, from (\[soln\]) we find $\varphi $ in terms of $a$: $$\begin{aligned}
t \geq 0:&\exp (\phi (a))=\exp (\phi _0)\frac a{a(1-2\chi _0)+2\chi _0},\;
\label{soln1} \\
t \leq 0:&\exp (\phi (a))=\exp (\phi _0)\frac{a^{5/2}}{a^{5/2}(1-\frac
25\chi _0)+\frac 25\chi _0\ },\end{aligned}$$ and from (\[chi1\],\[neg\]) we find $\chi $ in terms of $a$: $$\begin{aligned}
t \geq 0:&\chi (a)=\frac{\chi _0}{a^2\left( 2\chi _0+(1-2\chi _0)\ a\right)
}, \label{chi11} \\
t \leq 0:&\;\chi (a)=-\ \frac{5a\chi _{_0}}{2\chi _{_0}+(5-2\chi
_{_0})a^{5/2}}.\end{aligned}$$ A particularly simple case occurs when $\chi _0=\frac 14$. Then $$\begin{aligned}
t \geq 0: &\exp (\phi (a))=\exp (\phi _0)\frac{2a}{a+1},\;\\
t \leq 0:&\exp (\phi
(a))=\exp (\phi _0)\frac{10a^{5/2}}{9a^{5/2}+1\ }. \label{phi2a}\end{aligned}$$ and $$\begin{aligned}
t \geq 0:&\chi (a)=\frac 1{2a^2\left( 1\ +\ a\right) },\\
\;t \leq 0:&\chi (a)=-\
\frac{5a\ }{2(1\ +9a^{5/2})}. \label{chi2a}\end{aligned}$$
Now $\rho (t)$ is determined by (\[ccons\]); using the above expressions, for $t>0$ and $\chi _0=\frac 14$ this becomes $$\frac{d\rho }{da}=-\frac 3{a^5}-\frac 3{4a^6(1+a)}$$ which can be solved to give $$\rho (a)=C+\frac 3{20a^5}+\frac 9{16a^4}+\frac 1{4a^3}-\frac 3{8a^2}+\frac
3{4a}+\frac 34\ln (\frac a{1+a}).$$ This implies $\rho (t)\rightarrow C+\frac{107}{80}-\frac 34\ln 2=C+0.81764...
$ as $t\rightarrow 0_{+}$ and $\rho (t)\rightarrow C$ as $t\rightarrow
\infty ;$ hence choosing $C=0,$ $\rho (t)\rightarrow 0.81764...$ as $
t\rightarrow 0_{+}$ and $\rho (t)\rightarrow 0$ as $t\rightarrow \infty .$ Also $p/\rho \rightarrow 1/3$ as $t\rightarrow \infty .$ The expression for $
V(\varphi )$ in this case follows on putting this into(\[V\]) and using ( \[solnst\]), (\[t\]), and the various expressions above. Similar (more complicated) expressions can be obtained for $t<0.$
[^1]: email: [email protected]
[^2]: email: [email protected]
[^3]: email: [email protected]
[^4]: email: [email protected]
[^5]: Present address of D Roberts: Department of Physics, University of Oxford, Oxford, UK.
[^6]: If the equation of state is a function of $V$ or $V^{\prime}$, then you will have to eliminate these quantities using eqs.(\[friedman\]) and (\[ray\] ) before solving eq.(\[energy\]).
[^7]: It is important to note that one has freedom to choose only $a(t)$ or $
\phi(t)$, not both.
[^8]: Although of course by the algorithm given above, we can simulate a matter dominated phase by suitable choice of the potential $V.$
|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'Edge and vertex connectivity are fundamental concepts in graph theory. While they have been thoroughly studied in the case of undirected graphs, surprisingly not much has been investigated for directed graphs. In this paper we study $2$-edge connectivity problems in directed graphs and, in particular, we consider the computation of the following natural relation: We say that two vertices $v$ and $w$ are $2$-*edge-connected* if there are two edge-disjoint paths from $v$ to $w$ and two edge-disjoint paths from $w$ to $v$. This relation partitions the vertices into blocks such that all vertices in the same block are $2$-edge-connected. Differently from the undirected case, those blocks do not correspond to the $2$-edge-connected components of the graph. We show how to compute this relation in linear time so that we can report in constant time if two vertices are $2$-edge-connected. We also show how to compute in linear time a sparse certificate for this relation, i.e., a subgraph of the input graph that has $O(n)$ edges and maintains the same $2$-edge-connected blocks as the input graph, where $n$ is the number of vertices.'
author:
- 'Loukas Georgiadis$^{1}$'
- 'Giuseppe F. Italiano$^{2}$'
- 'Luigi Laura$^{3}$'
- 'Nikos Parotsidis$^{1}$'
title: '**2-Edge Connectivity in Directed Graphs**'
---
[10]{}
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A. L. Buchsbaum, L. Georgiadis, H. Kaplan, A. Rogers, R. E. Tarjan, and J. R. Westbrook. Linear-time algorithms for dominators and other path-evaluation problems. , 38(4):1533–1573, 2008.
A. L. Buchsbaum, H. Kaplan, A. Rogers, and J. R. Westbrook. A new, simpler linear-time dominators algorithm. , 20(6):1265–96, 1998. Corrigendum in 27(3):383-7, 2005.
T. H. Cormen, C. E. Leiserson, and R. L. Rivest. . The MIT Electrical Engineering and Computer Science Series. MIT Press, Cambridge, MA, 1991.
Ya. M. Erusalimskii and G. G. Svetlov. Bijoin points, bibridges, and biblocks of directed graphs. , 16(1):41–44, 1980.
D. Firmani, G. F. Italiano, L. Laura, A. Orlandi, and F. Santaroni. Computing strong articulation points and strong bridges in large scale graphs. In [*Proc. 10th Int’l. Symp. on Experimental Algorithms*]{}, pages 195–207, 2012.
W. Fraczak, L. Georgiadis, A. Miller, and R. E. Tarjan. Finding dominators via disjoint set union. , 23:2–20, 2013.
H. N. Gabow. A poset approach to dominator computation. Unpublished manuscript 2010, revised unpublished manuscript, 2013.
H. N. Gabow and R. E. Tarjan. A linear-time algorithm for a special case of disjoint set union. , 30(2):209–21, 1985.
L. Georgiadis, L. Laura, N. Parotsidis, and R. E. Tarjan. Loop nesting forests, dominators, and applications. In [*Proc. 13th Int’l. Symp. on Experimental Algorithms*]{}, pages 174–186, 2014.
L. Georgiadis and R. E. Tarjan. Finding dominators revisited. In [*Proc. 15th ACM-SIAM Symp. on Discrete Algorithms*]{}, pages 862–871, 2004.
L. Georgiadis and R. E. Tarjan. Dominator tree certification and independent spanning trees. , abs/1210.8303, 2012.
Y. Guo, F. Kuipers, and P. Van Mieghem. Link-disjoint paths for reliable qos routing. , 16(9):779–798, 2003.
A. Itai and M. Rodeh. The multi-tree approach to reliability in distributed networks. , 79(1):43–59, 1988.
G. F. Italiano, L. Laura, and F. Santaroni. Finding strong bridges and strong articulation points in linear time. , 447(0):74–84, 2012.
R. Jaberi. On computing the $2$-vertex-connected components of directed graphs. , abs/1401.6000, 2014.
T. Lengauer and R. E. Tarjan. A fast algorithm for finding dominators in a flowgraph. , 1(1):121–41, 1979.
K. Menger. Zur allgemeinen kurventheorie. , 10:96–115, 1927.
H. Nagamochi and T. Ibaraki. A linear-time algorithm for finding a sparse $k$-connected spanning subgraph of a $k$-connected graph. , 7:583–596, 1992.
H. Nagamochi and T. Ibaraki. . Cambridge University Press. 1st edition, 2008.
R. E. Tarjan. Depth-first search and linear graph algorithms. , 1(2):146–160, 1972.
R. E. Tarjan. Edge-disjoint spanning trees, dominators, and depth-first search. Technical report, Stanford University, Stanford, CA, USA, 1974.
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|
{
"pile_set_name": "ArXiv"
}
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---
abstract: 'We present a spectroscopic redshift of $z=1.675$ for the well-known multiply lensed system of arcs seen in the $z=0.39$ cluster Cl0024+16. In contrast to earlier work, we find that the lensed images are accurately reproduced by a projected mass distribution which traces the locations of the brightest cluster ellipticals, suggesting that the most significant minima of the cluster potential are not fully erased. The averaged mass profile is shallow and consistent with predictions of recent numerical simulations. The source redshift enables us to determine an enclosed cluster mass of M($<$100kpc/h)=$1.11\pm0.03\times 10^{14}h^{-1}$M$_{\odot}$ and a mass-to-light ratio of M/L$_B$($<$100kpc/h)=320h(M/L$_B$)$_{\odot}$, after correction for passive stellar evolution. The arc spectrum contains many ionized absorption lines and closely resembles that of the local Wolf-Rayet galaxy NGC4217. Our lens model predicts a high magnification ($\simeq$20) for each image and identifies a new pair of multiple images at a predicted redshift of z=1.3.'
author:
- 'Tom Broadhurst, Xiaosheng Huang, Brenda Frye, Richard Ellis'
title: 'A Spectroscopic Redshift for the Cl0024+16 Multiple Arc System: Implications for the Central Mass Distribution'
---
Introduction
============
The $z=0.39$ rich cluster Cl0024+16 (Zwicky 1959) displays one of the finest examples of gravitational lensing. Four clearly related images are identified around the tangential critical curve in HST WF/PC-1 data (e.g. Smail 1996). A further radially directed image of the same source was later found in a refurbished HST WFPC-2 image by Colley (1996). These arcs have been used by Colley (1996) to construct an image of the source and by Tyson (1998) to examine details of the mass distribution.
For many years the redshift of this lensed source has eluded identification despite long exposures on large telescopes. The blue colour and lack of optical emission lines suggests a redshift $1<$z$<$2 (Mellier 1991). The importance of the redshift for lensing studies lies primarily in measuring the central mass and mass-to-light ratio of the lensing cluster, and in the case of Cl0024+16 these quantities can be measured particularly accurately for this cluster because of the symmetric arrangement of the images. The results will depend on the source redshift through its effect on the ratio of lens to source separations, $d_{LS}/d_S$. For a nearby lens this ratio saturates very quickly with increasing source redshift, but for a more distant lens like Cl0024+16 there is a larger range of $d_{LS}/d_S$ and hence a larger uncertainty in the mass, depending on the source redshift.
Observations
============
The HST imaging data used here is that obtained by Colley (1996). The archival data was first reduced for the purposes of selecting lensed targets for mulislit spectroscopy. Briefly, the images comprise seven exposures totaling 8400s in the F450W band and six exposures totaling 6600s in the F814W band. Images were aligned to the nearest integer pixel and cosmic-ray rejected to obtain an average flux. With multislits on LRIS at the 10m Keck II telescope a 15slit was centered on a high surface-brightness feature of the largest, and hence most magnified arc (upper end of arc A, see Fig 2a) to maximize the detection of spectral features. The total exposure time was 80 minutes in 0.8 arcsec seeing, using the 300 line grating blazed at 5000Å, providing a useful wavelength range of 4500Å-9500Å. The resulting spectrum is shown in Fig 1, revealing many absorption lines, with a redshift of z=1.675. The spectrum, observed at an average resolution of 6Å in the restframe matches closely that of the nearby starburst galaxy NGC4214 (Leitherer 1996), showing a similar continuum slope and common absorption lines of SiII 1527Å, CIV 1550Å, FeII 1608Å,2344Å,2382Å, 2600Å, AlII 1671Å, AlIII 1859Å. We also detect a foreground $z$=0.18 blue dwarf galaxy (Fig 2a) which appeared initially to be related to the lensed arcs but is difficult to reproduce in the lens model (Fig 2b).
Modelling the Cluster Lens
==========================
Lens models for Cl0024+16 have generally improved with higher quality imaging. Using ground based data, Kassiola (1994) and Wallington (1995) reproduced fits to the close triplet of arcs (A,B,C of Figure 2) but considered arc D an unlikely counter image. Subsequently HST images revealed that A,B,C and D are morphologically similar in detail (Smail 1996) and that a further radially directed arc in the cluster center, E, is another complete image of the same source (Colley 1996). Recently a 512 parameter fit to the resolved imaging data for the arc system has been presented by Tyson (1998). This solution required the inclusion of a number of small dark deflecting ‘mascons’ around each of the images to offset the symmetry of a dominant central potential (see Fig 2 of Tyson 1998).
To investigate the uncertainties in mass we revisit the mass model using a simple approach which we nonetheless find sufficient to reproduce the basic properties of the image configuration. We start by assigning profiles to the brightest cluster members using the form advocated by NFW (Navarro, Frenk and White 1995), which has a characteristic scale, $r_s$, and dimensionless normalization relative to the cosmological critical density, $\delta_c=\rho_s/\rho_{crit}$, and allows a wide range of mass concentrations. Integrating the mass along a column, z, where $r^2=({\xi_r}{r_s})^2+z^2$ gives $$\rm{M}(\xi_r)={\rho_s}{r_s^3}(\xi_r)\int_o^{\xi_{r}}d^2\xi_r
\int_{-\infty}^{\infty}
{\frac{1}{(r/r_s)(1+r/r_s)^2}}{dz\over{r_s}}$$ resulting in a deflection angle $\alpha(\theta)=\frac{4GM(<\theta)}{c^2\theta{d_L}}
\frac{d_{LS}}{d_S}$ in the image plane at position $\vec{\theta}=\vec\xi_r{r_s}/d_L$.
Only the brightest 8 cluster members need be included to generate an accurate fit (all cD galaxies, see Fig 2a) with $r_s$ and $\delta_c$ unconstrained, corresponding to a simple vector addition of deflection fields. Inclusion of many fainter members produces noise as the fit rapidly becomes unconstrained for the small number of independent constraints (5 images). The fit is achieved with the “downhill simplex” algorithm (Press ) by minimizing the difference between the model predicted locations of the three obvious features (HII regions) common to the 5 main images of the source i.e. $$\chi^2=\sum_{k}\sum_{i,j,i>j}\left(({\vec\theta_{i,k}}-{\vec\alpha{(\vec\theta_{i,k})}})
-({\vec\theta_{j,k}}-{\vec\alpha{(\vec\theta_{j,k})}})\right)^2$$ a sum over all k points of all images.
The projected (2-D) mass density contours shown in fig 2 are modulated by the cluster members despite the projection. The fit favours an overall shallow profile centered on the central tight clump of luminous galaxies (fig 2a,b). The two outer ellipticals are seen to be responsible for the largest departure from symmetry generating the triple images A,B,& C. The solution although good is not of course unique, since a set of discrete profiles generates a fairly smooth potential (see Fig 2b) and for this reason we do not need to introduce additional unknown parameters to describe a diffuse component. We can convert the the azimuthally-averaged projected model slope of $\theta^{-0.55}$ in the vicinity of 100kpc/h for comparison with an NFW profile. The conversion to a real space slope is $\gamma=-1-2\frac{\xi_r}{1+\xi_r}$, or $\gamma=-1.26$ at the critical radius. This is very close to the NFW expectation for massive clusters which have a predicted slope of $\sim-1.3$ at $\sim$100kpc/h (NFW, Ghingna 1998) corresponding to a combination of $r_s\sim400$kpc/h and $\delta_c\sim8000$.
Mass/Light Ratios
=================
The rest-frame luminosities of the cluster galaxies are converted from data numbers to an ST-magnitude using header information. The instrumental colours of the bright ellipticals are very similar with a mean of $V_{450W}-I_{814W}$=1.62 (2.818 in the AB system or 3.334 normalised to Vega) which in the Johnson system $B-I$=3.7 (Holzman 1996) and corresponds well to a passively evolved old stellar population which at the observed redshift $z=0.39$, with $z_f=3$ and $\tau=.01$ (Bruzual & Charlot 1995) predicts colours of $B-I$=3.825 (3.727) for $\Omega=0.1$($\Omega=1$) and h=0.5 (solar metallicity). Correction to the present requires removal of $0.^m49$–$0.^m52$ of passive evolution.
The arcs define a convenient radial aperture for comparing mass and light with a radius of 30.5 arcsec radius intersecting the four bright tangential images. Using the observed lens and source redshifts, we normalise the mass distribution by the ratio $d_{LS}/d_S=0.61$ (virtually independent of curvature). Integrating over our model mass distribution yields $M(\theta)=1.28\times10^{14}h^{-1}M_{\odot}$ for $\Omega=0.1$, corresponding to a metric radius of $\sim106$kpc/h at the lens. Note $\Omega$ enters only via the angular-diameter distance of the lens, so that for $\Omega=1$ the mass is lower by 9% for the same angular aperture. The model mass is close to the Einstein mass of a symmetric lens, $M(\theta<30.5'')_{crit}=1.33\times10^{14}M_{\odot}$, ($\Omega=0.1$ and independent of profile) as expected given the near circular arrangement of the images about the cluster which means we have confidence in the mass to an accuracy of less than $\sim$2%.
The integrated I-band light in this aperture is $I^{ST}_{814W}=16.2$ or $L_B(\theta)=3.94\times10^{11}h^{-2}L_{\odot}$ ($\Omega=0.1$) after subtracting passive evolution above and hence, the central a mass-light ratio is, $M/L_B(\theta)=324h(M/L_B)_{\odot}$ at $z$=0 (using $M_{B\odot}=5.48$). This is slightly higher than other lensing clusters (Kneib 1996, Natarajan 1998), but note that neglecting evolution reduces $M/L_B$ by 50%, and should be allowed for in accurate comparisons between clusters. This value is short of that necessary for closure estimated from local redshift surveys in the same passband estimated requires $(M/L_B)_{crit}=1500^{+700}_{-400}h(M/L)_{B\odot}$ (Efstathiou, Ellis & Peterson 1988) corresponding to a larger volume and a later mean galaxy type.
The true size of the source galaxy after subtraction of the deflection field of image radius, is found to be $r\sim$0.25, $\sim 20$ times fainter than the tangential images , with an unlensed luminosity $M_B=-20.85\pm0.7-5\log{h}$ and an apparent magnitude of $I\sim24.8$, typical of what may be expected for a galaxy at the measured redshift (see Figure 18 of Bouwens, Broadhurst & Silk 1998).
New Multiple Images
===================
New multiply lensed images may be sought with our lens model to check and improve upon the model. The unknown redshift introduces an additional free parameter from the distance dependence in the bend angle. Hence to search for new multiple images we need only take the deflection field $\vec \alpha(\vec \theta)$ defined for the five arc system at $z_1=1.67$ with $(d_{LS}/{d_S})_{z_1}$ and multiply by a scalar, $f={(d_{LS}/d_S)_{z_2}}/{(d_{LS}/d_{S})}_{z_1}$, mimicking the effect of a change in the source redshift.
In practice it is difficult to securely identify new images, mainly because galaxies are similarly blue and numerous so that a unique identification based on only 2 passbands is difficult. Furthermore arcs which are obviously radial or tangential lie close or straddle the critical curves and therefore much more magnified than their counterimages which consequently may be too faint to detect. Furthermore, for a given mass distribution a source must lie above some minimum redshift to generate multiple images, corresponding to z$>$1.0 on average for sources within the einstein ring of Cl0024+16. Here we claim a secure identification of one new pair of arcs as shown on figure 2. The relative deflection is 90% of that of the main arcs corresponding to a predicted redshift of z=1.31,z=1.34,z=1.33 for $\Omega$=0.05,$\Omega$=1,$\Omega+\Lambda$=$0.3+0.7$. In principle then a sufficiently accurate lens model can produce a geometric constraint on the cosmological curvature, however many more multiple images of higher redshift sources need to be identified to make this practical.
Discussion and Conclusions
==========================
It is clear from the above modeling that some degree of mass substructure is required in Cl0024+16, contrary to the conclusions of Tyson (1998). The large mass-to-light ratio we assign to the location of the luminous ellipticals is far in excess of isolated elliptical galaxies, meaning these galaxies simply trace well local minima of a general potential. This finding is consistent with the substructure apparent in all carefully studied lensing clusters, notably, A2218, A370, AC114, A2390, Cl0939+47 (Kneib 1996, Smail 1996, Abdelsalam 1998, Pierre 1996, Natarajan 1997, Seitz 1996). It is not clear if this level of substructure is in excess of N-body predictions which, as Ghigna (1997) point out are certainly underestimates within the central 50Kpc/h where the problem of “overmerging” is still significant despite their superior dynamic range.
Our azimuthally averaged mass profile is shallow, in good agreement with recent high-resolution CDM N-body simulations, corresponding to an NFW profile with $r_s\sim$400kpc/h which is four times greater than the observed einstein ring radius and leads to large central image magnifications. Although our model does not explicitly incorporate a separate diffuse component it can be seen in Fig 2, that the sum of the eight profiles forms a generally smooth mass distribution illustrating the degeneracy of this sort of modeling and hence the redundancy of a separate provision for diffuse matter.
We have successfully used our lens model to find new multiply lensed galaxies. It is clear that with more color information further systems will be distinguished in an iterative procedure where the mass model is successively refined with the incorporation of the new images. Relative distance predictions can be made this way for comparison with measured redshifts allowing, in principle, a geometric measure of the cosmological curvature.
We thank Rychard Bouwens and Ben Moore for useful conversations. TJB acknowledges NASA grant AR07522.01-96A.
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{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'Extremely fast pattern recognition capabilities are necessary to find and fit billions of tracks at the hardware trigger level produced every second anticipated at high luminosity LHC (HL-LHC) running conditions. Associative Memory (AM) based approaches for fast pattern recognition have been proposed as a potential solution to the tracking trigger. However, at the HL-LHC, there is much less time available and speed performance must be improved over previous systems while maintaining a comparable number of patterns. The Vertically Integrated Pattern Recognition Associative Memory (VIPRAM) Project aims to achieve the target pattern density and performance goal using 3DIC technology. The first step taken in the VIPRAM work was the development of a 2D prototype ([[protoVIPRAM00]{}]{}) in which the associative memory building blocks were designed to be compatible with the 3D integration. In this paper, we present the results from extensive performance studies of the [[protoVIPRAM00]{}]{} chip in both realistic HL-LHC and extreme conditions. Results indicate that the chip operates at the design frequency of 100 MHz with perfect correctness in realistic conditions and conclude that the building blocks are ready for 3D stacking. We also present performance boundary characterization of the chip under extreme conditions.'
author:
-
-
title: 'Performance Study of the First 2D Prototype of Vertically Integrated Pattern Recognition Associative Memory (VIPRAM)'
---
3DIC, associative memory, IC testing, real time pattern recognition
Introduction {#sec:intro}
============
Associative memory (AM)-based pattern recognition is a powerful approach to solving complex combinatorics for fast triggering on particle tracks [@Dell'Orso:1988zz]. It has been successfully used in previous high-energy physics experiments such as the CDF Silicon Vertex Trigger (SVT) at the Fermilab Tevatron [@2004NIMPA.518..532A] and is currently being implemented at the ATLAS FastTracKer (FTK) at the CERN LHC [@ftkTDR]. This massively parallel architecture is ideally suited to tackle the intrinsic combinatorics of track finding algorithms, avoiding the typical power-law dependence of execution time on occupancy and solving the pattern recognition in times roughly proportional to the number of hits. This is of crucial importance given the large occupancies typical of hadronic collisions and low latency requirements.
There will be a much higher hit occupancy than ever seen before in proton-proton collisions at the anticipated high luminosity Large Hadron Collider (HL-LHC) at CERN. This effect, typically called pileup, comes from multiple (140-200) collisions per proton bunch collision. In order to keep the rate of events manageable for the experiments, a track trigger system will be required [@cmsTP]. The latency requirements of such a track trigger system require it to be operated at speeds much faster than previous systems. At the same time, the HL-LHC detectors are being designed to have a much larger number of channels in their tracking volume than previous LHC and Tevatron detectors, and thus, there is an enormous challenge in implementing fast pattern recognition for a track trigger.
The rigorous technical requirements of a silicon-based hardware tracking trigger push the limits of Pattern Recognition Associative Memories (PRAM) in pattern density, speed, and power density. It is estimated that the number of patterns needed for a tracking trigger system at CMS [@cmsTP] for the HL-LHC is two orders of magnitude greater than the CDF system [@2004NIMPA.518..532A] requiring a greatly increased pattern density. For a similar system, the latency estimate is at least an order of magnitude less than the ATLAS FTK system [@ftkTDR] requiring greatly increased speed performance.
The associative memory approach involves using content addressable memories (CAMs) and a majority logic (ML) to find matching detector hits from different detector layers to form track candidates with extremely low latencies. Approaches to this goal in simple 2D VLSI, which were previously used, are limited. Reducing the feature size of the technology node while scaling up the number of patterns is an option [@amchip05; @amchip06; @wetext], but can present its own design challenges. Given these new challenges, a new concept to use emerging 3D technology has been proposed [@Liu:2011zzw]. Design in 3D vertical integration is, in a sense, the logical partitioning of functionality into a third dimension. The PRAM structure is intrinsically adaptable to the 3rd dimension from the full pattern level down to the individual CAM level.
The VIPRAM (Vertically Integrated PRAM) approach is to divide the PRAM structure among 3D VLSI tiers to reduce the area consumed by a single pattern, to reduce the parasitic capacitance of long traces, to increase the effective number of routing layers, and to increase the readout speed significantly. The essence of VIPRAM is to divide this approach up into different tiers, maximizing pattern density while minimizing critical lengths and parasitics and therefore the power density. In Fig. \[fig:stack\](a), a PRAM element is laid out in 2D VLSI and in Fig. \[fig:stack\](b), 3D VLSI on the bottom. In Fig. \[fig:stack\](c), a charged particle which represents a pattern is illustrated where each location in the detector can be associated with a CAM cell. From the figure, one can see that the pattern density directly depends on the cross-sectional area of one of the CAM cells and is greatly increased by stacking each of the PRAM components. The lines from the CAM to the majority logic (ML) cell, which contains logic to assert a match, are long in the conventional 2D implementation. They are now implemented vertically and are therefore shorter in 3D because each tier will be thinned down to about 10 $\mu m$ during the 3D stacking process. Due to the high number of repetitive structures in associative memories, 3D integration can have a significant impact on performance. The vertical integration also provides flexibility in layout optimization of the building blocks, and therefore chip performance.
We present details related to design and testing of a 2D prototype of the 3D VIPRAM concept, which we refer to as [[protoVIPRAM00]{}]{}. Because vertical integration is an emerging technology, we have studied first the basic building blocks of the 3D concept laid out in 2D to verify their performance in simple 2D VLSI. Then, in the next version of the chip which includes 3D integration, we will be able to directly compare and quantify the expected performance gains with respect to the 2D layout.
Previous discussion on the chip details, design, simulation, and initial test results can be found in [@Liu:2015oca]. The main focus of this paper is detailed testing results. In Section \[sec:design\], we describe the design and functionality of the [[protoVIPRAM00]{}]{} from the single PRAM design to the full prototype chip layout. In Section \[sec:testbench\], the testing setup for the [[protoVIPRAM00]{}]{} chips is detailed. This is comprised of a simple setup to test the generic functionality of the chip and also a more sophisticated setup for testing realistic scenarios and boundary conditions. In Section \[sec:results\], the results of the [[protoVIPRAM00]{}]{} testing are presented. First, we describe the tests of the basic functionality of the chip as a generic tool for pattern recognition. Second, scenarios based on realistic HL-LHC simulations are presented. Third, we detail tests which are designed to test extreme boundary conditions of the chip. Finally, for the various tests performed, we examine the corresponding power consumption which will be important in contrasting and benchmarking future versions of the chip. In Section \[sec:concl\], we summarize the paper and provide an outlook on the project status.
[0.35]{} ![Pattern Recognition Associative Memory laid out in 2D (a) and stacked in 3D (b). An illustration of the signal for a charged particle in detector layers is given in (c).[]{data-label="fig:stack"}](figures/PRAM2D.png "fig:"){width="\textwidth"}
\
[0.2]{} ![Pattern Recognition Associative Memory laid out in 2D (a) and stacked in 3D (b). An illustration of the signal for a charged particle in detector layers is given in (c).[]{data-label="fig:stack"}](figures/PRAM3D.png "fig:"){width="\textwidth"}
[0.2]{} ![Pattern Recognition Associative Memory laid out in 2D (a) and stacked in 3D (b). An illustration of the signal for a charged particle in detector layers is given in (c).[]{data-label="fig:stack"}](figures/track.png "fig:"){width="\textwidth"}
Associative memory building blocks {#sec:design}
==================================
Since 3D Vertical Integration is an emerging technology and the requirements of the hardware track trigger have themselves been evolving, the first logical step is to test the two basic building blocks, the CAM cell and the majority logic cell, through a simple 2D prototype run. This will provide verification of their functionality in preparation for the 3D stacking and low latency readout developments in the near future. The associative memory building blocks were laid out as if this was a 3D design. Space was reserved for as yet non-existent through silicon vias (TSV) and routing was performed to avoid these areas. The readout circuitry of the PRAM array is deliberately simplified to allow for direct performance studies of the CAM and control cells. The 2D prototype run also serves as a benchmark to understand the performance improvements that can be gained by 3D stacking.
The [[protoVIPRAM00]{}]{} was designed and fabricated in a 130 nm Low Power CMOS process that has been used previously in HEP 3D designs. Fabrication is performed by Global Foundaries. The size of the chip is 5.46 mm $\times$ 5.46 mm. The layout was implemented such that, in future 3D designs, the basic building blocks can be directly reused and placed on different 3D tiers. The prototype chip has 4096 patterns distributed in 128 rows and 32 columns.
Single PRAM design
------------------
In keeping with the design philosophy of testability, each PRAM pattern consists of four identical CAM cells and a control cell resulting in the ability to recognize 4-layer pattern matches. The choice of four CAM cells is made for simplicity though realistic systems will require more where current designs consider eight. Each CAM cell is a 15-bit address where the 15-bits are comprised of 4 NAND cells, 8 NOR cells, and 3 Ternary bits with a 4-bit selective pre-charge [@Liu:2011zzw]. Figure \[fig:PRAM\](a) shows the floor plan of the CAM cell including the space in the middle reserved for TSVs. The [*Matchline*]{}, indicated in Fig. \[fig:PRAM\](a), is the single signal that connects the different bits in the CAM cell and its parasitic impedance impacts the chip performance. The selective pre-charge is made with four NAND cells which, when matched, allow the Matchline to be charged. The choice of the number of pre-charge bits was made to optimize performance and is a balance between less power consumption (more pre-charge bits) and increased clock frequency (fewer pre-charge bits).
The Majority Logic cell, shown in Fig. \[fig:PRAM\](b), is designed to have the same footprint as the CAM cell. The Majority Logic uses Pass Transistor Logic to produce a 3-bit code indicative of the possible match conditions: All Layer Match, One Missing Layer Match, Two Missing Layers Match, and First Layer Match. The Match Processing Logic compares the output of the Majority Logic with the user-supplied threshold and, if met, asserts a matched pattern.
Each of the majority logic and CAM cells are 25$\mu$m $\times$ 25$\mu$m in size and there is an additional 10$\mu$m $\times$ 125$\mu$m of space left for pattern routing such that a PRAM is 35$\mu$m $\times$ 125$\mu$m.
[0.35]{} ![CAM Cell schematic (a) and Majority Logic Cell (b).[]{data-label="fig:PRAM"}](figures/CAM.png "fig:"){width="\textwidth"}
[0.35]{} ![CAM Cell schematic (a) and Majority Logic Cell (b).[]{data-label="fig:PRAM"}](figures/CAMML.png "fig:"){width="\textwidth"}
Full [[protoVIPRAM00]{}]{} design {#sec:fullchip}
---------------------------------
The chip operates in two modes: [*load mode*]{} and [*run mode*]{}. In load mode, patterns are stored in each PRAM, one at a time. High speed performance is not required in load mode as this typically happens between running conditions with respect to the pattern matching during run mode. This pre-defined set of patterns is determined from offline simulation (and can be determined from real tracks in data in the future) and is often referred to as a [*pattern bank*]{}. In run mode, incoming data is compared to the stored patterns and matched pattern flags are generated based on the match threshold conditions asserted. At the end of the event, an “event re-arm” signal can be asserted to clear all the matches and data, and the next event can be inputted. There is a 32-bit output which reads out the matched pattern flags, corresponding to each of the columns of a selected row. The readout implementation is kept simple to allow for easy testing of the pattern matching performance. In Fig. \[fig:protovipram\](a), a block diagram of the [[protoVIPRAM00]{}]{} functionality is shown. In addition to the four 15-bit inputs and 32-bit output, 5(7)-bit inputs are used to designate the column (row) and there are inputs related to the clock signals and power inputs. There is also one more bit available for each of the four inputs which we designate as “data valid” bit. This denotes whether or not to ignore the incoming data and allows us to invalidate certain inputs due to dead detector elements or to handle variations in the number of inputs per layer.
The chip has a multi-VDD design, which allowed us to study the power behavior of the chip in great detail. The power inputs of the chip are $V_{\rm DVDD}$, $V_{\rm DVD}$, and $V_{\rm charge}$. $V_{\rm DVDD}$ drives the input and clock buffers, $V_{\rm DVD}$ supplies the majority logic and the SRAM storage cells inside the CAM cells, and $V_{\rm charge}$ charges the matchline inside the CAM cells.
Figure \[fig:protovipram\](b) shows a picture of the actual [[protoVIPRAM00]{}]{} wire-bonded to a standard 144-pin Thin Pin Grid Array (PGA). The design has been thoroughly simulated at all levels with timing, signal dispersion, and power consumption. Further details on design and on the simulation checks on the design can be found in [@Liu:2015oca].
[0.40]{} ![Block diagram of the [[protoVIPRAM00]{}]{} (a) and picture of [[protoVIPRAM00]{}]{} (b).[]{data-label="fig:protovipram"}](figures/protovipram00-block.png "fig:"){width="\textwidth"}
[0.35]{} ![Block diagram of the [[protoVIPRAM00]{}]{} (a) and picture of [[protoVIPRAM00]{}]{} (b).[]{data-label="fig:protovipram"}](figures/protovipram00-pic.png "fig:"){width="\textwidth"}
Test bench for performance studies {#sec:testbench}
==================================
The [[protoVIPRAM00]{}]{} chips, wire-bonded in a socket, are mounted on a test mezzanine card, see Fig. \[fig:mezz\](a). The FMC Test Mezzanine card features a Xilinx Kintex XC7K160T FPGA, 4 SFP+ optical transceivers, 128MB DDR3, and a 144 pin socket used for testing custom ASIC chips which are indicated in Fig. \[fig:mezz\](a). The power supply provides 1.5 V to each of the three power inputs of the chip, $V_{\rm DVDD}$, $V_{\rm DVD}$, and $V_{\rm charge}$. We can configure each of these power inputs separately. The mezzanine is connected to a PC running the Linux SL6 operating system. JTAG Communication with the FPGA is done using Xilinx Design Suite. Gigabit Ethernet communication is provided via the SFP+ optical transceivers.
Testing proceeds in two ways: [*basic functional validation*]{} and [*automated testing*]{}. Basic functional validation stores test vector data in internal FPGA blockRAMs and the chip output is analyzed via internal FPGA logic analyzer, ChipScope Pro. Studies are limited by the size of the FPGA memory and interpreting results from the logic analyzer output.
The automated testing data flow is illustrated in Fig. \[fig:mezz\](b). Automated testing stores test vector input and output in internal FPGA blockRAMs and reads them in and out via an optical Gigabit Ethernet connection based on simple UDP packet transfers (the IP-bus protocol [@ipbus]). The software needed to initialize, control, and transfer data to and from the FPGA and to analyze the data is custom written in Python. The clock frequency is dynamically controlled in the software by dividing the internal Kintex-7 FPGA 1 GHz clock by integers. Therefore, allowable testable frequencies are, $f = 1000/n~{\rm MHz}$ where $n$ is an integer. Functional tests were performed in the frequency range from 2 to 166 MHz. Additional current monitors are available on the mezzanine to monitor the voltage sources to [[protoVIPRAM00]{}]{} provide measurements of the chip power consumption. The sampling frequency of the current monitors is approximately $\sim$1 kHZ which is slower than the operational frequency and so when measuring power consumption we repeat functional tests serially to get a consistent current measurement. Measurements from the current monitors are sampled until the power consumption asymptotes so that we can obtain reliable power measurements. The generated input patterns and the sampled outputs from the chip are verified against functional simulations done on the full chip design using Cadence NC-Sim.
There are additional factors which can affect the performance of the chip and which should be considered in the testing. First, because of the way that the CAM cell is designed with a 4-bit selective pre-charge, the order of the bits of the patterns injected can change the chip performance. Current flows through the CAM cell matchline when the 4-bit selective pre-charge is matched, so putting often-matched bits in the logic cells can consume more power in the chip overall. Second, the power supply voltage supplied to the [[protoVIPRAM00]{}]{} can affect the performance. Our multi-VDD CAM chip can operate on a wide range of voltages for the three power inputs, which can be optimized to get better performance at lower power consumption [@multivdd]. For the studies presented here, we nominally run at 1.5V. Details will be discussed further below. Finally, the [[protoVIPRAM00]{}]{} clock is supplied by two signals which we designate $MC_A$ and $MC_B$. The clock signal is defined logically by ‘$MC_A$ && !$MC_B$’ and the signals are offset by a phase delay. The offset determines the discharge time of the Matchline. If the Matchline is not given sufficient time to discharge, this would result in testing results giving “false positives”, e.g. matches when no match exists. The first two are subdominant effects though they are worth quantifying. The final factor can result in large “false positive” matches and should be carefully considered. These factors will be discussed further in the testing results.
[0.47]{} ![Mezzanine card with [[protoVIPRAM00]{}]{} mounted (a) and block diagram of testbench data flow (b).[]{data-label="fig:mezz"}](figures/mezz-lo.png "fig:"){width="\textwidth"}
[0.47]{} ![Mezzanine card with [[protoVIPRAM00]{}]{} mounted (a) and block diagram of testbench data flow (b).[]{data-label="fig:mezz"}](figures/2017-09-19_protovipram_testing.png "fig:"){width="\textwidth"}
Results {#sec:results}
=======
Functional validation
---------------------
In functional testing, a simple analysis can be done using the logic analyzer output to test very basic chip functionality as a generic pattern matching device. An example is given in Fig. \[fig:basic\]. The green text boxes indicate the clocking in of a dummy pattern into a PRAM. Loading a pattern into a PRAM consists of a primary load step and a secondary load step to verify the pattern. The red text boxes indicate the searching of the memory for the input dummy pattern where the final step is the appearance of the found pattern on the output bits. Although the level of sophistication of analysis is limited, further tests using the basic testing setup are performed including checks of each PRAM’s functionality.
With the functional testing, we can verify that each pattern within a chip is working properly. In this test, we “walk” through each column in the chip and check that each row element in the chip can match a pattern. Many variations on the pattern validation are performed. For example, we try a number of different input patterns or varying the patterns and we also maximally vary the inputs and the patterns flipping all the bits in a pattern. No issues in functionality are observed. Generally, we find that no chips have partial functionality, they are fully functional. We check other functional behavior such as the data valid bit. We verify that setting the data valid bit to 0 ignores the input. We also check the various majority logic for functionality and we find no issues with any of the possible logic states: All Layer Match, One Missing Layer Match, Two Missing Layers Match, and First Layer Match. For the remainder of the following test, we typically stay with “All Layer Match” patterns for consistency.
Finally, during the functional testing, we lay out the timing parameters of the different chip operational modes. We find that for [*load mode*]{}, we are able to load the patterns into memory into the chip at a frequency of 10 MHz. Operating load mode at a higher frequency can sometimes cause a pattern to not be properly loaded. However, as was discussed in Sec. \[sec:design\], the operational frequency of load mode is not an important parameter for performance of the chip as timing constraints are not strong when loading patterns. Alternatively, the operational frequency in run mode is of paramount importance. We study this in great detail below and understand chip performance against a number of parameters. The remaining important timing parameter is output latency onto the output bits. We generally find, although a pattern match is found based on the run mode frequency, it may show up on the output bus with a typical delay of 5 ns. We hypothesize that this comes from pushing a signal through the low speed chip carrier PGA package although the final configuration is for the chip to be bonded directly to the mezzanine. The chip is not designed for fast readout and we study this in greater detail for future chips, as mentioned above, and we note this simply as an issue when designing our tests.
![[[protoVIPRAM00]{}]{} I/O waveforms captured by the Chipscope embedded logic analyzer firmware[]{data-label="fig:basic"}](figures/basictesting.png "fig:"){width="3.5in"}\
Realistic HL-LHC scenarios
--------------------------
After verifying functional performance of the [[protoVIPRAM00]{}]{}, we go on to test the chip performance using realistic HL-LHC scenarios. Automated testing allows us to perform large scale tests of the entire chip, both by providing test vectors larger than the size of the internal FPGA RAM and analyzing output with ASCII output, without having the re-program the FPGA. The data flow is from the custom software through SFP+ optical transceiver to the Kintex7 FPGA from which the test vectors are pushed to the [[protoVIPRAM00]{}]{} where the output is written back to the FPGA and then back to the software package. We are able to scan in operational frequency and monitor chip power consumption in real-time. Input vectors are created and sent via UDP packet transfers to the blockRAMs which are then sent to the chip. Output signals are captured and stored in blockRAM to be read by the analysis software. This is done iteratively if the size of the test vectors exceeds the blockRAM of the FPGA, which fits $2 \times 10^{15}$ clock cycles of test vectors.
We use simulated data from high occupancy HL-LHC collisions and input the data as if it was coming from the collisions to the triggering system. As we introduced in Sec. \[sec:intro\], the associative memory technique is used for identifying charged particle tracks in the detector, creating a set of known detector patterns. The challenge is to identify these patterns with an extremely low latency in with a lot of noise (uninteresting hits) from additional low energy collisions. Therefore, in order to benchmark the [[protoVIPRAM00]{}]{} performance in realistic scenarios, we use simulation of from the CMS experiment in the HL-LHC collision environments. The simulation provides for us a set of allowed patterns in the detector ([*pattern bank*]{}). Of course, we are not yet testing results for an entire detector which requires millions of patterns. The inputs for our realistic tests are defined as a unique set of patterns taken as a subset of a full pattern bank from the CMS experiment HL-LHC simulation. A [*trigger tower*]{} is defined as a regional partitioning of the entirety of the detector hits into various detector regions; for example, we simulate the CMS detector split into 48 trigger towers. In our realistic tests, we emulate a part of one such trigger tower. We note that the parameters below we use for testing are not near the requirements for the final system but instead simply a test benchmark based on our best knowledge at the time.
The set of allowed hits are then as the hits within our mock pattern bank which is part of a trigger tower. The data from the HL-LHC detector comes as charged particle hits on the various layers of the detector. The hits on the various layers come randomly ordered. The number of hits per layer is taken from HL-LHC simulation of the CMS detector with high pileup and four top quark events. From simulation studies, we take the benchmarks for the average number of hits per layer as:
- Layer 1: 90 hits
- Layer 2: 60 hits
- Layer 3: 45 hits
- Layer 4: 35 hits
From those sets of hits, we randomize them and send them to the [[protoVIPRAM00]{}]{}. Additionally, from simulation, we determine that a typical event has 5 true tracks for our mock trigger system. The hits of those true tracks are also mixed in with the random hits and sent to the chip. The threshold for success for a given event is to find the patterns for those 5 true tracks. Note that the fraction of true matches is extremely small ($\sim5/4096$) with respect to the number of total patterns in realistic HL-LHC scenarios. We define the matching efficiency for real tracks in these realistic tests as: $\varepsilon_{\rm match} = N_{\rm found}/N_{\rm expected}$. We also look for [*false positives*]{} which are fake matches despite no expected real track, and we find the contribution from false positives in our current settings, to be negligible. We discuss this more below.
Results of realistic testing are shown in Table \[tab:table1\] as the matching efficiency as a function of the operational frequency, the speed at which we introduce test vectors to the chip and search for patterns. The [[protoVIPRAM00]{}]{} shows 100% matching efficiency up to the target operational frequency of 100 MHz, which was the design goal of the prototype chip. Further optimization of the pattern bank can improve the performance of the chip. If we have a full match of the 4-bit selective pre-charge for a given CAM cell, then power is driven through that cell. By distributing evenly how likely the 4-bit selective pre-charge is matched across the entire pattern bank, we can reduce power consumption across the chip and moderately improve the operational frequency of the chip. This can be seen by comparing the second and third columns of Table \[tab:table1\], where “re-ordered” is the optimized pattern bank. We also verify two other factors: the effect of input supply voltages and the effect of clock phase on the Matchline discharge time. We vary the voltages supplied to $V_{\rm DVDD}$, $V_{\rm VDD}$, and $V_{\rm charge}$ from 1.4V to 1.6V. $V_{\rm DVDD}$ and $V_{\rm VDD}$ have a very small effect on the performance of the chip, which is dominated by the $V_{\rm charge}$ supply. We run with the default recommended settings of 1.5 V although we find that scanning through each of the other voltages could yield approximately a 10% improvement in operational frequency. Details on the effects and optimization of the multi-VDD supplies on CAMs and the power modeling can be found in [@Li:ICCD2015; @multivdd]. We also vary the phase of the two clock signals. By decreasing the time per clock cycle devoted to discharging the Matchline, we increase the available time for doing pattern matching. We ultimately find that the discharge time must be $\geq 1~{\rm ns}$ or else we begin to observe false positives in our testing results.
Freq (MHz) $\varepsilon_{\rm match}$ (default) $\varepsilon_{\rm match}$ (re-ordered)
------------ ------------------------------------- ----------------------------------------
50 100% 100%
60 100% 100%
71 100% 100%
76 100% 100%
83 100% 100%
90 100% 100%
100 100% 100%
111 99.76% 100%
125 95.3% 98.8%
: Matching efficiency of the [[protoVIPRAM00]{}]{} chip as a function of operational frequency. []{data-label="tab:table1"}
We perform realistic tests of 12 wire-bonded [[protoVIPRAM00]{}]{} chips and show consistent performance across all chips tested.
Extreme boundary conditions
---------------------------
To understand the bounds of the chip performance, we use dummy data to test the [[protoVIPRAM00]{}]{} in scenarios far exceeding what we expect in realistic scenarios. These extreme boundary condition tests are used to benchmark the limits of the chip performance and understand systematically its limitations and breakdown points in terms of match occupancy and operational frequency. Additionally, performing a detailed study of the power consumption with these tests will guide us to finding any improvements for future chip designs. Many tests are performed to test the performance of the [[protoVIPRAM00]{}]{} and here we describe the most complete set of tests. The typical match occupancy for a realistic system is $< 1\%$ of patterns matching within a given event and matches do not happen simultaneously in time because the hits arrive at the chip randomly and matches occur throughout the entire event. In these extreme boundary condition tests we force matches within a given event to occur in the same clock cycle, e.g. there is only one clock cycle in the event. Further, we require $\gg1\%$ of the chip to match at the same time, scanning a fraction, $f_{\rm ext}$, of 10% to 100% of the chip simultaneously expected to match. Here, the subscript “ext" refers to the extreme fractional occupancy of the chip. We do this by filling the initial pattern bank with only 2 unique patterns occupying $f_{\rm ext}$ and $1 - f_{\rm ext}$ of the chip, respectively. They are distributed evenly through the chip geometrically in order to not bias the tests based on the location of the patterns in the chip layout[^1]. We then send the pattern which constitutes $f_{\rm ext}$ of the pattern bank and check to see how many of them matched. This test is performed many times to get a large set of statistics from which to compute the matching efficiency. We perform this test at various frequencies and determine when the chip performance begins to degrade. The results of the extreme boundary condition tests can be seen in Fig. \[fig:stress\].
![Results of extreme boundary condition testing on [[protoVIPRAM00]{}]{}[]{data-label="fig:stress"}](figures/funk.png "fig:"){width="3.6in"}\
One can see that as the operational frequency of the chip is increased, the performance begins to degrade. The frequency at which the performance begins to degrades decreases as we increase $f_{\rm ext}$. This behavior is expected. As the operational frequency increases the majority logic voltage has less time to give a matched signal. As the fraction of the chip that is matched increases, $f_{\rm ext}$, the signal propagation throughout the chip becomes more delayed with increasing matches and the chip begins to miss expected matches. The testing results quantify where chip degradation occurs and where the chip ultimately can still perform in the most extreme scenarios.
Power Consumption
-----------------
CAMs are very attractive for pattern recognition applications due to their high speed performance, however they incur significant power and area overheads. The primary reason for this is the massively parallel operation of the CAMs. The massively parallel structure in a CAM needs a large amount of driver circuits to multiply and drive the input data signal to each of the CAM cells. Therefore, it is important to measure the power consumption of the chip and properly model it so that we can estimate requirements for larger scale systems. The multi-VDD design of our chip, with a separate supply for the major functional blocks, allowed us to study the power behavior of the chip in detail and model it. This will help us to understand the absolute scale of power consumption for an ultimate track trigger system; validate the breakdown in performance of the chip and extrapolate its performance to 3D. Recall from Sec. \[sec:fullchip\] that the power inputs are $V_{\rm DVDD}$, $V_{\rm VDD}$, and $V_{\rm charge}$ where $V_{\rm DVDD}$ drives the input signals, $V_{\rm VDD}$ primarily drives the majority logic, and $V_{\rm charge}$ charges the Matchline.
Programmable voltage regulators on the mezzanine card support current readback for the chip. The dominant source of power consumption in [[protoVIPRAM00]{}]{} comes from the input line drivers, $V_{\rm DVDD}$, and for an operational frequency of 100 MHz, the typical power consumption of the chip is approximately 250 mW. Scaling the power consumption of the pattern matching is non-trivial for the 4096 pattern chip but in the given prototype it is not the dominant contribution.
In unrealistic extreme boundary conditions, we monitor all power lines and can use it to verify the performance of the chip. For example, we can monitor $V_{\rm VDD}$ in our extreme boundary condition tests. This is shown in Fig. \[fig:pwerstress\]. The $V_{\rm VDD}$ shows the matched pattern power consumption and as we increase $f_{\rm ext}$ and the operational frequency and the chip performance begins to degrade, we can see that the power of $V_{\rm VDD}$ also correspondingly begins to degrade. This provides us with an excellent validation of our understanding of the internal chip functionality.
![Power results of stress testing on [[protoVIPRAM00]{}]{}[]{data-label="fig:pwerstress"}](figures/pow.png "fig:"){width="3.6in"}\
Beyond measurements of the power consumption, a methodology for characterizing the power consumption of the chip has also been developed. It shows excellent agreement with the testing results. Chip behavior can be well-modeled for a variety of different factors including varying the types of patterns and selective pre-charge as well as the dependence on the voltages. Overall, our model can predict the average total power consumption of the chip to within 4% of the actual measurement. Many more details about the modeling of the chip power consumption and results can be found in [@Li:ICCD2015; @multivdd].
Conclusion and Outlook {#sec:concl}
======================
Fast triggering on particle tracks is vital to the physics program of the HL-LHC at CERN. Associative Memory-based pattern recognition provides a powerful approach to solve the complex combinatorics inherent to this challenge. The PRAM (pattern recognition associate memory) devices that are at the core of its concept are well-suited to modern 3D integration. Emerging 3D technologies provide an opportunity to improve in pattern density while simultaneously improving speed capabilities and reducing power consumption. The first [[protoVIPRAM00]{}]{} chip was designed and fabricated in a 130 nm Low Power CMOS process. The layout was deliberately implemented in 2D so that the basic associative memory building blocks can be directly re-used for 3D stacking. The design has been successfully tested both for functionality and performance using a custom test setup. Results indicate that the chip operates at the design frequency of 100 MHz with perfect correctness in realistic conditions and conclude that the building blocks are ready for 3D stacking.
The set of testing results above span the possible parameters of the chip we could conceive. We checked all the basic functionalities of the chip outlining generic performance parameters. Then we performed tests using realistic HL-LHC scenarios and also devised scenarios to understand the boundary conditions of the [[protoVIPRAM00]{}]{}. We varied different parameters which could affect the chip performance from the clock phase, input voltages, pattern order, and operational frequency. We tested a number of chips and found consistent performance across the set. In all cases, we find the chip to behave within design specifications and gained a large amount of experience in how to test the chip, troubleshooting various issues, and defining a baseline for the type of tests that should be performed when studying future AM chips. The testing results show that the basic associative memory building blocks, the CAM and the control cell that comprise [[protoVIPRAM00]{}]{} are ready for 3D vertical integration for a proof-of-principle demonstration of the VIPRAM concept.
Following successful performance evaluation of the [[protoVIPRAM00]{}]{} presented in this paper, two new chips have been developed to split the development path towards different goals. The [VIPRAM\_3D]{} [@Liu:2011zzw] is meant as a verification of multi-tier 3D stacking and is identical to the [[protoVIPRAM00]{}]{} in all relevant design choices except that it will be stacked in 3D. It is fully pin-compatible with the 2D [[protoVIPRAM00]{}]{} and can provide a direct diagnostic of the 3D integration process. The [VIPRAM\_L1CMS]{} [@vipram_l1cms] focuses on bringing the system interface to maturity including pipelined operation and sparsified readout and includes 3D integration of 2 layers: the IO tier and the PRAM tier. The design of the next generation VIPRAM chips have been completed, the wafers have been fabricated, and they are now in 3D processing.
Acknowledgment {#acknowledgment .unnumbered}
==============
Operated by Fermi Research Alliance, LLC under Contract No. De-AC02-07CH11359 with the United States Department of Energy.
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[^1]: Other tests were made to determine which location of the chip was most likely to fail with a moderate preference for the middle of the chip where signals take the longest to propagate.
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{
"pile_set_name": "ArXiv"
}
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[**Mass and Free Energy of Lovelock Black Holes**]{} 0.5cm 5.mm [**David Kastor${}^{a1}$, Sourya Ray${}^{b2}$ and Jennie Traschen${}^{a3}$**]{}\
0.5cm
${}^a$ Department of Physics, University of Massachusetts, Amherst, MA 01003\
${}^b$ Centro de Estudios Cientõficos (CECS), Casilla 1469, Valdivia, Chile\
${}^1$`[email protected],` ${}^2$ `[email protected],` ${}^3$ `[email protected]`
0.5in
[**Abstract**]{}
> An explicit formula for the ADM mass of an asymptotically AdS black hole in a generic Lovelock gravity theory is presented, identical in form to that in Einstein gravity, but multiplied by a function of the Lovelock coupling constants and the AdS curvature radius. A Gauss’ law type formula relates the mass, which is an integral at infinity, to an expression depending instead on the horizon radius. This and other thermodynamic quantities, such as the free energy, are then analyzed in the limits of small and large horizon radius, yielding results that are independent of the detailed choice of Lovelock couplings. In even dimensions, the temperature diverges in both limits, implying the existence of a minimum temperature for black holes. The negative free energy of sufficiently large black holes implies the existence of a Hawking-Page transition. In odd dimensions the temperature still diverges for large black holes, which again have negative free energy. However, the temperature vanishes as the horizon radius tends to zero and sufficiently small black holes have positive specific heat. 2.mm
Introduction
============
Many interesting, stationary black holes are not known in analytic form. The list includes charged rotating black holes in $D>4$, black rings in $D>5$, localized Kaluza-Klein black holes and rotating black holes in Lovelock gravity theories. Results have been obtained using a variety of approximate techniques, including [*e.g.*]{} perturbative expansions in the slowly rotating limit [@Aliev:2005npa; @Aliev:2006yk; @Kim:2007iw; @Zou:2010dx; @Yue:2011et] or effective field theory methods [@Emparan:2009cs; @Emparan:2009at; @Caldarelli:2010xz]. However, one may also ask whether in the absence of analytic solutions, it might still be possible to at least obtain certain properties of such spacetimes, such as thermodynamic ones, exactly.
In this paper we will address this question for a class of spacetimes that might best be called “semi-unknown", namely static Lovelock black holes. It is known [@Wheeler:1985qd] that these spacetimes are Schwarzschild-like, in the sense that the metric written in the general static, spherically symmetric form $$\label{generalmetric}
ds^2 = - \phi(r) dt^2 + {dr^2\over f(r)} +r^2d\Omega_{D-2}^2,$$ additionally has $\phi(r)=f(r)$. For a generic Lovelock theory, the function $f(r)$ must satisfy an algebraic equation of order $\left[ {D-1\over 2}\right]$, where $D$ is the spacetime dimension and the closed brackets denote taking the integer part. The coefficients of the polynomial are the coupling constants of the higher curvature terms in the Lovelock Lagrangian. Since the roots of a generic polynomial equation can be found in terms of radicals only up to order $4$, it follows that in spacetimes dimensions $D>10$ the general solution for a static Lovelock black hole cannot be written down in a simple closed form and hence remains unknown[^1].
We make use of results from references [@Kastor:2008xb; @Kastor:2009wy; @Kastor:2010gq], derived using the Hamiltonian formulation [@lovelock-hamiltonian], to obtain exact properties of Lovelock black holes, without requiring the explicit (and in general unknown) form of the metric function $f(r)$. In fact with future applications to other, even more unknown, black holes in mind, we will “forget" that we know that $\phi(r)=f(r)$ and take the general static, spherically symmetric form (\[generalmetric\]) as our starting point. Our focus throughout will be on black holes with asymptotically AdS boundary conditions.
Further motivation comes from recent work on the CFT duals of asymptotically AdS solutions to higher curvature gravities. Significant work has been done, for example, on the relation between CFT plasmas and their gravitational duals, [@Dias:2009iu; @Caldarelli:2008mv; @Aharony:2005bm; @Lahiri:2007ae] and on implications for the CFT of causality and stability in the bulk using holography [@Kovtun:2004de; @Kats:2007mq; @Brigante:2008gz; @Ge:2008ni; @Ge:2009eh; @Ge:2009ac; @Buchel:2009sk; @Buchel:2009tt; @deBoer:2009gx; @Hofman:2009ug; @Shu:2009ax; @deBoer:2009pn; @Camanho:2009vw; @Camanho:2009hu; @Ge:2010aa]. As the correspondence between the higher curvature bulk theory and the boundary CFT becomes better understood, it is interesting to ask whether the constructs we will make use of below, the Killing-Lovelock potentials and the associated Gauss’ law relations that connect behavior at the horizon to behavior at infinity, have an analogue in the dual CFT.
The paper will proceed as follows. In section (\[lovelocksection\]) we recall the basic elements of Lovelock gravity theories [@Lovelock:1971yv] that will subsequently be used. In section (\[masssection\]) we will derive the expression for the ADM mass of an asymptotically AdS Lovelock black hole in terms of its far field behavior. The ADM boundary integral receives contributions from each higher curvature term as well as from the Einstein term. The formula for the ADM mass is implicit in the work of [@Jacobson:1993xs] which computed the entropy in Lovelock theories and established the first law. However significant additional steps are required to make the result of [@Jacobson:1993xs] fully explicit. The expression we find ultimately takes the simple form of the standard ADM mass integral multiplied by a function of the Lovelock couplings and the asymptotic AdS curvature radius. This formula for the mass constitutes a new result.
In section (\[kl\_potentials\]) we review certain elements from our previous work [@Kastor:2008xb; @Kastor:2009wy; @Kastor:2010gq] that we will be making use of. This includes the Killing-Lovelock potentials which allow for the derivation of a Gauss’ law type expression for the Hamiltonian constraint, which will be our main tool. This expression relates certain boundary integrals at infinity to surface integrals on the horizon, without requiring the explicit form of the metric in between. For reference, we also present the Smarr formula [@Kastor:2010gq] which gives the mass in terms of the entropy and additional thermodynamic quantities that arise in the Lovelock theory.
The bulk of our new results appear in sections (\[mainsection\]) and (\[freesection\]). In section (\[mainsection\]) we use the Gauss’ law formula to obtain an expression for the mass, originally given by the far field behavior of the metric, in terms of the horizon radius of the black hole. With the goal in mind of providing a simple expression for the free energy, we also present formulas for the entropy, surface gravity and other thermodynamic quantities in terms of the horizon radius. The detailed behavior of these expressions over the full range of horizon radii depends on the choice of Lovelock coupling constants. However, we find that they have generic behavior in the limits of small and large horizon radii. There are qualitative differences between even and odd dimensions that arise in the small black hole regime. In particular, one finds that in odd dimensions there exists a “mass gap", in the sense that the mass tends to a finite value as the horizon radius tends to zero. The Hawking temperature also vanishes in this limit and sufficiently small black holes have positive specific heat. For even dimensions, however, the mass tends to zero for vanishing horizon radius and the specific heat for small black holes is negative.
In section (\[freesection\]) we assemble ingredients to construct a relatively simple expression for the free energy of black holes in generic Lovelock theories. The free energy had previously been computed only up to inclusion of the quadratic Gauss-Bonnet term [@Cai:2001dz; @Cvetic:2001bk] in the Lovelock Lagrangian. The free energy for a stationary black hole solution is generally obtained by computing its Euclidean action. We emphasize that our method does not require the explicit functional form of the metric. We analyze the free energy and also the specific heat in the small and large black hole limits and comment upon the Hawking-Page phase transition for generic AdS-Lovelock black holes. In section (\[discussion\]) we briefly summarize our results and offer some directions for further related work.
Lovelock gravity {#lovelocksection}
================
The Lagrangian of a Lovelock gravity theory in $D$ spacetime dimensions is given by $$\label{lovelocklagrangian}
{\cal L}={1\over 16 \pi G} \sum_{k= 0}^{\kmax}b_k{\cal L}^{(k)}$$ where $\kmax = [(D-1)/2]$ and the $b_k$ are real-valued coupling constants. The symbol $\callk$ stands for the contraction of $k$ powers of the Riemann tensor given by $$\label{lovelagran}
\call^{(k)} ={1\over 2^k } \delta ^{a_1 b_1...a_k b_k } _{c_1 d_1 ....c_k d_k }
R_{a_1 b_1}{}^{c_1 d_1 }\dots R_{a_k b_k}{}^{c_k d_k }.$$ where the $\delta$-symbol is the totally anti-symmetrized product normalized so that it takes nonzero values $\pm 1$. The term $\call^{(0)}$ gives the cosmological constant term in the Lagrangian, while $\call^{(1)}$ gives the Einstein-Hilbert term and $\call^{(2)}$ the quadratic Gauss-Bonnet term. The upper bound in the sum (\[lovelocklagrangian\]) comes about because $\call^{(k)}$ vanishes identically for $D<2k$ and turns out to make no contribution to the equations of motion in $D=2k$.
The equations of motion for Lovelock gravity can be written as $\calg^a{}_b= 0$ where $$\calg^a{}_b= \sum_{k=0}^{\kmax} b_k{\cal G}^{(k)a}{}_b.$$ We will not need the explicit expressions for these quantities. However, it is crucial in what follows that each of the tensors in this sum satisfies a conservation law $\nabla _a {\cal G}^{(k)a}{} _b =0$.
Depending on the values of the coupling constants $b_k$ the theory may have anywhere from zero up to $\kmax$ distinct constant curvature vacuum solutions. Because we will be focusing on asymptotically AdS black holes below, our considerations will implicitly be limited to the subset of theories admitting at least one constant negative curvature vacuum. We will denote the curvature radius of this AdS vacuum by $l$.
We will also need the Hamiltonian formulation of Lovelock gravity which was developed in [@lovelock-hamiltonian]. As usual in the Hamiltonian picture the spacetime metric is split according to $$\label{metricsplit}
g_{ab}=-n_a n_b +s_{ab}$$ where $n^a$ is the unit timelike normal to a spatial slice $\Sigma$ with induced metric $s_{ab}$ and these satisfy the orthogonality relation $s_{ab} n^b =0$. As in Einstein gravity the time-time and time-space components of the field equations act as constraints on initial data. In Lovelock gravity the Hamiltonian and momentum constraint operators $H = -2n^an^b\,\calg_{ab}$ and $H_a = -2s_a{}^b\,n^c\, \calg_{bc}$ are given by the sums $$H= \sum_{k=0}^{\kmax} b_k\, H^{(k)},\qquad H_a= \sum_{k=0}^{\kmax} b_k\, H_a^{(k)}$$ with $H^{(k)}=-2n^an^b\,\calg^{(k)}_{ab}$ and $H_a^{(k)}=-2s_a{}^b\,n^c\, \calg^{(k)}_{bc}$. We will particularly need the expression for $$\label{hperp}
H^{(k)} = -\, {1\over 2^k}\, \tilde\delta ^{a_1 b_1...a_k b_k } _{c_1 d_1 ....c_k d_k }\,
R_{a_1 b_1}{}^{c_1 d_1 }\dots R_{a_k b_k}{}^{c_k d_k }$$ where the tilde on the $\delta$-symbol indicates that its indices are projected with the spatial metric $s^a{}_b$. The actual Hamiltonian function for evolution of initial data with respect to a vector field $\xi^a$ is then given by ${\cal H}_\xi =F\,H +F^a\,H_a$, where the lapse and shift $(F,F^a)$ are the components of the vector field $\xi^a$ normal to and along the spatial slice, so that $\xi ^a = Fn^a +F ^a$. As in Einstein gravity the lapse and shift are Lagrange multipliers.
Finally, for use below we define the sums $$\label{sums}
s_{(n)} = \sum_{k=0}^{\kmax}(-1)^{k+n}{k!\,\,\bhat_k\over (k-n)!\,\, l^{2k-2}}$$ where the rescaled coefficients $\bhat_k$ are given by $\bhat_k = (D-1)!\, b_k/(D-2k-1)!$. Note that the combinations $\bhat_k/l^{2(k-1)}$ are dimensionless. The condition $s_{(0)}=0$, in fact, determines the allowed constant curvature vacua of the theory, while the sums $s_{(1)}$ and $s_{(2)}$ will turn up in our results below.
ADM Mass {#masssection}
========
An expression for the ADM mass of an asymptotically AdS Lovelock black hole can be obtained using the methods of Regge and Teitelboim [@Regge:1974zd]. We find a simple, explicit formula which is similar to that for the ADM mass. As in Einstein gravity, a boundary term must be included in the Lovelock Hamiltonian to ensure that the Hamiltonian variational principle correctly yields the equations of motion. The variation of this boundary term cancels another boundary term arising via integration by parts from the variation of the bulk Hamiltonian. The ADM mass is defined to be the value of the Hamiltonian divided by a factor of $16 \pi G$. Since the bulk Hamiltonian vanishes on solutions, the ADM mass is simply proportional to the value of the Hamiltonian boundary term. Implicit here is that the vector $\xi^a$ determining the direction of Hamiltonian evolution should be asymptotic to the time translation Killing vector of AdS.
This procedure was formally carried out in reference [@Jacobson:1993xs] as part of establishing the first law for Lovelock black holes. Each higher curvature term in the Lovelock theory makes a contribution to the Hamiltonian boundary term. For asymptotically flat solutions, possible if $b_0=0$ in the Lagrangian (\[lovelocklagrangian\]), because of the fall-off of the curvature tensor, only the boundary term corresponding to the Einstein term is nonzero at infinity. The formula for the mass is then the same as in Einstein gravity. However, the non-zero asymptotic value of the curvature tensor for AdS Lovelock black holes leads to contributions from all the higher curvature boundary terms. We will see, in fact, that the result is given by the usual ADM integral multiplied by a function of the Lovelock couplings and the background AdS curvature. To our knowledge this expression has not been derived in the literature[^2].
The Regge-Teitelboim type construction of the ADM mass in [@Jacobson:1993xs] yields an expression of the form $$\label{lovelockmass}
M = - {1\over 16\pi G}\sum_{k=0}^{\kmax} b_k\int_\infty da_c B^{(k)c}.$$ We will determine the boundary integrands $B^{(k)c}$ explicitly for asymptotically AdS boundary conditions. Because our interest is focused on static black holes, we will assume that the momentum and the shift vector vanish sufficiently rapidly in the asymptotic region that they do not play a role in this construction. It is then sufficient to retain only the term $\calh_\xi=F H$ in the Hamiltonian and to take the curvature tensor in (\[hperp\]) to be that of the spatial metric $s_{ab}$. Now, assume that the spatial metric $s_{ab}$ solves the Lovelock constraint equations with asymptotically AdS boundary conditions and add to it an arbitrary perturbation $h_{ab}$ that also respects these boundary conditions. To first order, the perturbation to the Riemann tensor is then $\delta R_{ab}^{\ \ cd} = {R}_{ab}^{\ \ e[c}h_e ^{\ d]} -2 D _{[a}D ^{[c} h_{b]} ^{\ \ d]}$, where $D_a$ is the covariant derivative operator for the spatial metric $s_{ab}$. Plugging this in to the variation of (\[hperp\]) yields the perturbation of the $k$th Hamiltonian $$\label{deltahk}
\delta H^{(k)} = -{k\over 2^k}\, \tilde\delta ^{a_1 b_1...a_k b_k } _{c_1 d_1 ....c_k d_k }\,
{R}_{a_1 b_1}{}^{c_1 d_1 }\dots {R}_{a_{k-1} b_{k-1}}{}^{c_{k-1} d_{k-1} }
\left( {R}_{a_k b_k }{}^{ e c_k }h_e{}^{d_k} -2 D _{a_k} D ^{c_k} h_{b_k}{}^{d_k} \right)$$ Combining these, multiplying by the lapse function $F$, integrating by parts, and making use of the Bianchi identity for the Riemann tensor then gives an overall expression of the form $$F\delta H = \delta H^* \cdot F +\sum_k b_k D_c B^{(k)c}.$$ Here the first term represents an adjoint differential operator acting on the lapse function $F$ and the vectors $B^{(k)c}$ are given by $$\label{btk}
B^{(k)c} = {k\over 2^k}\, \tilde\delta ^{cd m_1 n_1...m_k n_k } _{ab e_1 f_1 ....e_k f_k }\,
{R}_{m_1 n_1}{}^{e_1 f_1 }\dots {R}_{m_k n_{k-1}}{}^{e_{k-1} f_{k-1} }
\left( F D^a h_d {}^b -h_d{} ^b D^a F \right)$$ These are the vectors appearing in the expression for the ADM mass given above. The quantity $h_{ab}$ is then the deviation of the static black hole metric from the asymptotic AdS background, while the minus sign in (\[lovelockmass\]) arises from the cancellation of boundary terms required in the Regge-Teitelboim construction. The expression can be made more fully explicit by noting that near infinity only the leading order background AdS curvature $R_{ab }{}^{cd}
=(1/ l^2) \delta _{ab} ^{cd}$ contributes. This leads to the result $$\label{finalmass}
M =
{s_{(1)}\over (D-1)(D-2)}\cdot \left( {-1\over 16\pi G}
\int_\infty da_c\left\{ F(D^ch-D_bh^{cb}) - hD^cF +h^{cb}D_b F\right\}\right).$$ The part of the formula in parenthesis is the usual expression for the ADM mass. The sum $s_{(1)}$, defined in (\[sums\]), depends on the Lovelock coupling constants and the curvature radius $l$ of the asymptotic AdS vacuum. For Einstein gravity with a cosmological constant the prefactor outside the parenthesis reduces to one. The asymptotic form of the lapse function is $F=\sqrt{1+ r^2 / l^2}$, and therefore terms in (\[finalmass\]) involving the derivative of the lapse can make non-trivial contributions to the mass. In the asymptotically flat case, these terms fall off too quickly to contribute and the integrand reduces to that of the ADM mass in the asymptotically flat case.
Killing-Lovelock Potentials {#kl_potentials}
===========================
Killing-Lovelock potentials [@Kastor:2008xb; @Kastor:2010gq] allow the Lovelock Hamiltonian constraint equations for a spacetime with a Killing vector to be written in a Gauss’ law form. In reference [@Kastor:2010gq] this property was used to derive an extended first law including variation of the Lovelock coupling constants and also a related Smarr formula for Lovelock black holes. This will also be our primary tool below.
For a spacetime, such as a stationary black hole, with a Killing vector $\xi^a$ the Killing-Lovelock potentials were defined in [@Kastor:2008xb] to be antisymmetric tensors $ \beta^{(k)ab}$ satisfying the relations $$\label{potdef}
\nabla_a \beta^{(k)ab} = -2 {\cal G}^{(k)a}{} _b\xi^b.$$ Their existence is guaranteed by the vanishing divergence of the right hand side. However, they are not uniquely determined, since a divergenceless tensor such as the divergence of an arbitrary $3$-index antisymmetric tensor may always be added. This ambiguity does not affect results such as the extended first law or Smarr formula.
Now consider Hamiltonian evolution with respect to the Killing vector, so that the lapse and shift are given by the decomposition $\xi ^a =Fn^a +F^a$. The definition of the Killing-Lovelock potentials allows us to write the $k$th Hamiltonian function as a total divergence, $$\label{kthhamdiv}
\calh^{(k)}= F \hk +F^a \hk _a
= -2 {\cal G}^{(k)d} _c \xi ^c n_d = D_c ( \beta^{(k)cd} n_d ).$$ The Hamiltonian constraint equation $\calh=0$ can then be written in the form of Gauss’s law. It ultimately follows from this (see [@Kastor:2010gq] for how this works) that the Killing-Lovelock potentials make contributions to the thermodynamics of AdS-Lovelock black holes that are similar in form to that of the electrostatic potential in the case of a charged black hole. For example, one contribution to the change in energy in the extended first law of [@Kastor:2010gq] is proportional to the difference in the integrals of Killing potential over the boundaries of a spatial slice at infinity and at the black hole horizon. The AdS boundary conditions require a subtraction at infinity of the Killing-Lovelock potential $\beta _{AdS} ^{(k)ab}$ for the asymptotic AdS spacetime resulting in thermodynamic potentials $$\label{theta}
\Theta ^{(k)} =- \left( \int _\infty da r_c (\beta^{(k)cd}
- \beta _{AdS} ^{(k)cd} )n_d - \int _{h} da r_c \beta^{(k)cd} n_d \right).$$ that multiply variations $\delta b_k$ of the Lovelock couplings in the extended first law. As a consequence of the constraint equations these thermodynamic potentials satisfy the sum rule $$\label{potsum}
\sum _{k=0}^{\kmax} b_k \Theta ^{(k)} =0.$$
The Smarr formula relates the mass of a black hole which is defined in terms of the behavior of the metric near infinity to the entropy which comes from the behavior at the horizon. For static Lovelock black holes the Smarr formula [@Kastor:2010gq], which we will also make use of below, is given by $$\label{smarr}
\left( (D-3) +2{s_{(2)} \over s_{(1)} }\right) M ={\kappa \over 2\pi} \left[ (D-2) S - S^\prime\right]
-\Theta$$ The entropy $S$ [@Jacobson:1993xs] has contributions from the higher curvature Lovelock terms and is given by $S=\hat A/4G$ with $\hat A=\sum_k b_k A_k$ and $$A_k =k\, \int_{h} d^{D-2}x \sqrt{\gamma}\, \call^{(k-1)}(\gamma_{ab})$$ where $\gamma_{ab}$ is the induced metric on the boundary of the spatial slice at the horizon. There is also an additional entropy-like contribution at the horizon as well as the net contribution Lovelock thermodynamic potentials which are given respectively by $$\label{extras}
S^\prime={1\over 2G}\sum _{k=0}^{\kmax} (k-1)b_k A^{(k)} , \qquad \Theta \equiv {1\over 8\pi G}\sum_{k=0}^{\kmax} (k-1) b_k \Theta^{(k)}.$$ The Smarr formula is derived in [@Kastor:2010gq] via an overall scaling argument from an extended form of the first law in which variations of the dimensionful Lovelock coupling constants are taken into account. The contribution of $S^\prime$ to the Smarr formula arises from the explicit dependence of the entropy on the Lovelock couplings, and the second term in the parenthesis on the left in (\[smarr\]) similarly arises from the dependence of the mass (\[lovelockmass\]) on the couplings $b_k$.
For Einstein gravity with vanishing cosmological constant one finds that $s_{(2)} =S^\prime=\Theta=0$ and one recovers Smarr formula for static, asymptotically flat black holes. The case of Einstein gravity with non-zero cosmological constant was discussed in [@Kastor:2009wy]. In this case, one still has $s_{(2)} =S^\prime=0$. However, there is a non-trivial contribution to $\Theta$ from the product[^3] $\Lambda\Theta^{(0)}$. In [@Kastor:2009wy] it was argued that $\Theta^{(0)}$ can be regarded as minus an effective volume behind the black hole horizon. Since the cosmological constant is proportional to minus the background pressure, one sees that the $k=0$ term in the Smarr formula has the form $pV$ which is familiar from classical thermodynamics. The mass $M$ should then be regarded as the spacetime enthalpy. This interpretation was explored further in [@Dolan:2010ha; @Cvetic:2010jb].
Lovelock black holes {#mainsection}
====================
In this section we will derive formulas for the basic thermodynamic properties of static, spherically symmetric AdS Lovelock black holes in terms of the horizon radius $r_h$ and the Lovelock parameters $b_k$. We consider static, spherical symmetric, asymptotically AdS spacetimes with AdS curvature radius $l$. The metric can then be taken to have the form (\[generalmetric\]), where at large radius the metric functions have the asymptotic forms $$\label{falloff}
\phi (r)\sim 1+ { r^2\over l^2} -{c_t \over r^{D-3}}\ , \quad f (r)\sim 1+ { r^2\over l^2} -{c_r\over r^{D-3}}$$ for some constants $c_t$ and $c_r$. We assume that there is a Killing horizon at $r=r_h$, where $\phi (r_h ) =0$. Static spherically symmetric black hole solutions of Lovelock gravity theories have been known for some time, starting with the work of [@Boulware:1985wk; @Wheeler:1985nh] in the Gauss-Bonnet case and [@Wheeler:1985qd] in the general Lovelock case (see also [@Myers:1988ze]). These solutions all have $\phi(r)=f(r)$ and the field equations reduce to the requirement that $f(r)$ solve a certain polynomial equation of order $\kmax$, with coefficients determined by the Lovelock coupling constants[^4]. Except in certain special cases, such as Einstein or Gauss-Bonnet gravity or when the Lovelock couplings are tuned such that the polynomial has a unique degenerate root [@Crisostomo:2000bb], the solutions for $f(r)$ are generally not known explicitly.
Here we will use the general relations given in the previous sections to derive expressions for the mass $M$, surface gravity $\kappa$, entropy $S$ and free energy $F$ of Lovelock black holes in terms of $r_h$ without requiring the explicit solution of the field equation. This approach provides a geometrical understanding of the formulae. As noted in the introduction, with the application to other even less explicitly known black hole spacetimes in mind, we will take the general form of the metric (\[generalmetric\]) as our starting point and “forget" that we know that solutions to the field equations will have $\phi(r)=f(r)$.
Constant curvature vacua and Killing potentials
-----------------------------------------------
Working in the Hamiltonian picture, we assume that for a spherically symmetric static black hole the spatial metric has the form $$\label{spatialmetric}
s_{ab} dx^a dx^b = { dr^2 \over f(r) } +r^2 d\Omega ^2 _{D-2},$$ that the extrinsic curvature $K_{ab}$ vanishes, and that the Hamiltonian evolution is carried out along the static Killing field $\xi ^a = F n^a$. The function $F^2$ must then have the same large $r$ fall off conditions as the metric function $\phi$ in (\[falloff\]) and satisfy $F^2(r_h )=0$. Substituting the spatial metric into the Lovelock Hamiltonian functions $H^{(k)}$ in (\[hperp\]) we find that we find for these metrics that $$\label{hkform}
H^{(k) } = - \gamma _k {1\over r^{(D-2)} }
{\partial\over \partial r}\left( r^{(D-1-2k)}(1-f)^k \right)$$ where $ \gamma _k ={(D-2)!\over (D-2k-1)! } $. The constraint equation is $$\label{constreq}
H =\sum _{k=0}^{\kmax} b_k H^{(k)} =0$$ First consider the background AdS metric with radius of curvature $l$ and no black hole. The AdS functions are $F_{AdS}^2 =f_{AdS} =1+r^2 /l^2 $. Since the metric satisfies the constraint equation with $f_{AdS}$, substituting into (\[constreq\]) gives a relation between the couplings and $l^2$, $$\label{ldef}
\sum _{k=0}^{\kmax} (-1)^k {\bhat _k \over l^{2(k-1)} } = 0$$ As noted above, this is the condition that the sum $s_{(0)}$ defined above in (\[sums\]) vanish in a constant curvature vacuum. Whether, or not, there are real positive solutions for $l^2$ depends on the values of the Lovelock couplings. We will assume that we are working with a Lovelock gravity theory that has at least one real, positive solution $l^2$.
Next we find the projected Killing-Lovelock potentials $\beta^{(k)rd}n_d $, which are solutions to equation (\[kthhamdiv\]) with vanishing shift vector $F^a$. Plugging in the form of the $k$th Hamiltonian (\[hkform\]) this becomes $$\label{tosolve}
\sqrt{f } {\partial\over \partial r}\left({ r^{(D-2)} \over \sqrt{f}} \beta^{(k)rd}n_d \right)
= - F \gamma _k
{\partial\over \partial r}\left( r^{(D-1-2k)}(1-f)^k \right)$$ Now, recall that the existence of the Killing-Lovelock potentials depends on the the existence of a Killing vector $\xi ^a $, but not on the metric being a solution to the field equations. The statement that the thermodynamic potentials $\Theta^{(k)}$ sum to zero in (\[potsum\]), however, does assume that the constraint equation is satisfied. Since we will want to make use of (\[potsum\]) our current task is to solve for the Killing-Lovelock potentials assuming that $f$ solves the constraint equation (\[constreq\]). Note also that the differential equation (\[tosolve\]) includes a factor of the lapse function $F$. In the Hamiltonian picture the lapse is a gauge choice that may be freely specified[^5]. We can choose any function $F$ consistent with $-F^2$ being the norm of the static Killing field for a spherically symmetric AdS black hole. Specifically, we can choose any $F$ that depends only on $r$ with $F (r_h )=0$ and satisfying the large $r$ fall off conditions in (\[falloff\]).
On the other hand, inspection shows that equation (\[tosolve\]) is easy to solve if we can choose $F = \sqrt{f}$. This choice has the correct fall off conditions at large $r$, since the function $f(r)$ does. Therefore, we only need to worry about the condition that $F$ vanishes at the horizon, or equivalently whether the function $f(r)$ necessarily vanishes at the horizon? Of course it is well known that the Schwarzchild solution in Einstein gravity has $F =\sqrt{f}$, with $f(r)$ vanishing at the horizon. As noted above the static Lovelock black holes continue to have this Schwarzchild form [@Wheeler:1985qd]. However, one can also demonstrate that $f(r_h ) =0$ under weaker conditions than requiring that the full set of field equations be solved. Let us assume that the surface gravity for the black hole metric (\[generalmetric\]) is finite and non-zero, and that the constraint equation is satisfied at the horizon radius $r_h$. We show in the Appendix that these conditions imply that $f \sim F^2 $ near the horizon, and therefore that $f(r_h) =0$. Hence we can choose $F=\sqrt{f}$ in (\[tosolve\]) and the solution for the projected Killing potentials are $$\label{betasol}
\beta^{(k)rd}n_d = -\gamma _k r^{1-2k} \sqrt{f} (1-f)^k$$ In the next section we make use of these solutions to express the mass in terms of the horizon radius $r_h$.
Dependence of mass on horizon radius {#mass}
------------------------------------
It is straightforward to evaluate the boundary integral for the ADM mass (\[finalmass\]) in terms of the coefficient $c_r$ that characterizes the far field behavior of the spatial metric. With the AdS boundary conditions (\[falloff\]), one finds that the terms in (\[finalmass\]) that depend on the derivative of the lapse $F$ cancel, and the integral becomes $$\label{admmasscr}
M= {\Omega _{D-2}s_{(1)} c_r \over 16\pi G (D-1) } ,$$ where we have taken $b_1 =1$. We next turn to the task of finding the fall-off coefficient $c_r$ and hence also the mass in terms of the horizon radius $r_h$ by applying the results (\[theta\]) and (\[potsum\]). The integrand at large $r$ in (\[theta\]) becomes $$r ^{( D-2k -1)} [ (1-f)^k - (1-f_{AdS})^k ] \simeq (-1)^{k-1} {k c_r\over l^{2(k-1)}} ,$$ while the boundary term on the horizon is easily evaluated using the condition $f(r_h )=0$. Assembling these pieces one arrives at the expression for the thermodynamic potentials $$\label{nicetheta}
\Theta ^{(k)} =- \gamma _k \Omega _{D-2} \left( r_h ^{D-2k -1} +
(-1)^k { k c_r \over l^{2(k-1)} } \right)$$ where $\Omega _{D-2}$ is the area of a unit $D-2$ sphere. For $k=0$ this reproduces the result of [@Kastor:2009wy] that $\Theta ^{(0)}$ is given by minus an effective volume of the region behind the horizon. Finally, requiring that the sum rule (\[potsum\]) be satisfied yields the relation $$\label{crequals}
c_r
= \left( {r_h ^{D-3}\over s_{(1)}}\right) \sum _{k=0}^{\kmax} { \hat{b}_k \over r_h ^{2(k-1)} }$$ This is a key result since it relates the far field behavior of the black hole solution to the horizon radius, without making use of an exact analytic expression for $f(r)$ in the region between.
There are two important points to be made about the expression for $c_r$. First, we have noted that the solutions for static Lovelock black holes are known to be specified in terms of solutions to a polynomial equation of order $\kmax$ in the function $f(r)$ [@Wheeler:1985qd]. This equation arises from integrating the Hamiltonian constraint and $c_r$ is a constant of integration. Equation (\[crequals\]) may also be obtained by evaluating this polynomial at $r_h$ [@Cai:2003kt]. What we have learned from our more general treatment is that the relation (\[crequals\]) expresses the fact that the Hamiltonian constraint is a total divergence when defined with respect to evolution along a Killing field. Put differently, if a metric does not have a Killing field we have no reason to expect that the mass is simply a function of data on the horizon, but in general will depend on volume integrals as well.
Second, we can ask whether there a computational advantage to working with the Loveloock potentials and the sum rule (\[potsum\]) compared to simply writing out the field equations and analyzing them? In the spherically symmetric case this is likely a matter of taste. The analysis in terms of the Killing-Lovelock potentials is more complicated, but exposes an underlying geometrical structure. For more complicated spacetimes, such as rotating black holes, it may be that using the Killing potentials allows one to find geometrical relations of interest more simply than through a brute force analysis of the field equations.
We can now substitute in for $c_r$ using (\[crequals\]) and obtain the sought after, result for the mass of a static Lovelock black hole in terms of its horizon radius $$\label{admmassrh}
M= { \Omega _{D-2} r_h ^{D-3}\over 16\pi G (D-1)} \sum _k { \hat{b}_k \over r_h ^{2(k-1)} }.$$ This generalizes to all Lovelock theories the result for Gauss-Bonnet black holes given in [@Cai:2001dz]. Such a generalization was assumed to hold in [@Cai:2003kt] and we have now shown that this is indeed the case.
To get an idea of how this formula behaves, let us examine the dependence of the mass on $r_h$ in different limiting regimes. The behavior of the mass as $r_h \rightarrow 0$ turns out to be interesting in that it differs between even and odd spacetime dimensions. Since the small black holes are dominated by the highest curvature terms, behavior in this limit depends on whether the order $\kmax$ term actually appears in the Lagrangian with non-zero coupling constant. We will assume that this is the case. Note that this includes Gauss-Bonnet gravity in $D=5$ and in $D=6$, for which $\kmax=2$, but not in higher dimension.
By a “small" black hole will be one such that the horizon radius is sufficiently small that $r_h^{2(\kmax-k)}\ll |\hat b_{\kmax}/\hat b_k|$ for all $k<\kmax$. After noting that $\kmax =(D-1)/2$, and $2\kmax =(D-2)/2$ respectively in odd and even dimensions, one finds that in the small black hole regime the mass depends on $r_h$ as $$\begin{aligned}
&M\approx {\Omega _{D-2}\over 16\pi G (D-1)}\bhat _{\kmax} r_h\qquad & D\,\, even\\
&M\approx {\Omega _{D-2}\over 16\pi G (D-1)}\left( \bhat _{\kmax} + \bhat _{{\kmax} -1 }r_h ^2
\right)\qquad & D\,\, odd\end{aligned}$$ However, for $D$ odd the mass goes to a nonzero value, We see that for even dimensions, the mass goes smoothly to zero with the horizon radius as it does for Schwarzschild black holes. However, for odd dimensions there is a minimum mass for black holes that is proportional to $\hat b_{\kmax}$. This minimum mass, or mass gap, has been discussed previously for $D=5$ Gauss-Bonnet black holes in [@Cai:2001dz] and for Chern-Simons-Lovelock theories, which have a unique constant curvature vacuum, in [@Crisostomo:2000bb].
So long as the coefficient $b_0$ of the cosmological constant term in the Lovelock Lagrangian is non-zero, black holes in the opposite regime of large $r_h$ are dominated by the cosmological constant and look qualitatively the same in all dimensions. A“large" large black hole will be one such that the horizon radius satisfies $r_h^{2k}\gg |\hat b_k/\hat b_0|$ for all $k>0$. For large black holes one finds that the mass depends on the horizon radius as $$\label{largemass}
M\approx {\Omega _{D-2}\over 16\pi G (D-1)}b_0 r_h ^{D-1} .$$
Surface gravity of Lovelock black holes {#surfacegrav}
---------------------------------------
For a metric of the general form (\[generalmetric\]) the surface gravity is given by $\kappa =(1/2) \phi ' (r_h )$. In the appendix we argue that near the horizon of a black hole the metric functions must satisfy $g_{tt} \approx - 1/g_{rr} $, with equality for the functions and their first derivatives at the horizon. Hence $f(r_h )=0$ and $ f' (r_h )= 2\kappa $. Evaluating the Hamiltonian function $\hk$ in (\[hkform\]) at the horizon and applying the sum rule (\[potsum\]) yields a general relation between the surface gravity and the horizon radius without the need for the explicit form of the metric function $f(r)$, $$\label{surfgrav}
\kappa ( \sum _{k=0}^{\kmax} {k\,\hat{b}_k \over r_h ^{2(k-1)} } ) =
{1\over 2 r_h } \sum _{k=0}^{\kmax} { (D-2k-1)\hat{b}_k \over r_h ^{2(k -1)}}$$ For Gauss-Bonnet gravity this agrees with the expressions for the surface gravity obtained from the explicit solutions in [@Cai:2001dz; @Cvetic:2001bk; @Cai:2003kt] using an approach similar to the one here.
Let us examine at the behavior of the surface gravity in various limits regimes starting with small black holes as defined above. In this regime, one finds that $$\begin{aligned}
\label{smallkappaone}
& \kappa \approx {1\over 2\kmax\, r_h } ,
\qquad & D\ even \\ \label{smallkappatwo}
& \kappa \approx {\bhat _{\kmax}r_h \over \kmax\bhat _{({\kmax }-1)} }
,\qquad &D\ odd
\end{aligned}$$ Again, we see a qualitative difference between even and odd dimensions. In even dimensions the surface gravity diverges in the limit of vanishing horizon radius, as it does for Schwarzschild black holes in $D=4$. However for generic Lovelock theories in odd dimensions, [*i.e.*]{} those in which $b_{\kmax}$ is nonvanishing, the surface gravity goes smoothly to zero with the horizon radius. This change in behavior was noted for Gauss Bonnet gravity in $D=5,6$ in [@Cai:2001dz]. Here we note, in agreement with the observations of [@Cai:2003kt], that this is characteristic of Lovelock black holes in general.
On the other hand, the surface gravity of large black holes has qualitatively the same behavior for all AdS-Lovelock black holes. One finds that $$\label{largekappa}
\kappa \approx {b_0 r_h \over 2(D-2)}$$ where we have set $b_1 =1$, so that $\bhat _1 =(D-1)(D-2)$. In any dimension, therefore, there exists a large black hole solution for sufficiently high temperature. In even dimensions also always exists a small black hole at suficiently high temperatures. Hence for $D$ even, and so long as the Lovelock couplings are such that the surface gravity stays positive for all horizon radii, there will be a minimum temperature at which static black holes exist, while in odd dimensions there is no minimum temperature. In addition, there may be local extrema of the temperature depending on the choices of the $b_k$.
Entropy, thermodynamic potential and Smarr relation
---------------------------------------------------
The expression for the entropy [@Jacobson:1993xs] given in section (\[kl\_potentials\]) is a sum of integrals of Lovelock invariants constructed from the induced metric $\gamma_{ab}$ on the horizon cross section. For a spherically symmetric black hole $\gamma _{ab}$ is the metric of a round $D-2$ dimensional sphere of radius $r_h$ and the Riemann tensor is given simply by $R_{ab}{}^{cd}=(1/r_h^2)\, \delta_{ab}^{cd}$. Evaluating the various Lovelock terms explicitly gives $\call^{(k-1)} (\gamma_{ab}) = {(D-2)! \over (D-2k)! }\, r_h ^{-2(k-1)}$. The entropy for a static, spherically symmetric Lovelock black hole can then be written as $$\label{sphentropy}
S
= {\Omega _{D-2} r_h ^{D-2} \over 4(D-1)G}\,\,
\sum_{k=0}^{\kmax} {k\, \bhat_k \over (D-2k)\, r_h ^{2(k-1)}}.$$ We can also compute the quantity $S^\prime$ defined in (\[extras\]) which appears in the Smarr formula. It is given in terms of the horizon radius by $$\label{sprime}
S^\prime
= {\Omega _{D-2} r_h ^{D-2} \over 4(D-1)G}\,\,
\sum_{k=0}^{\kmax} {2k(k-1)\, \bhat_k \over (D-2k)\, r_h ^{2(k-1)}}.$$ Our goal in this section has been to develop expressions for the thermodynamic properties of AdS-Lovelock black holes purely in terms of the horizon radius. The final element of the Smarr relation (\[smarr\]) is the overall thermodynamic potential $\Theta$, which we will now compute. With this in hand, we can check all of our results are consistent with the Smarr relation. Substituting our result (\[crequals\]) for $c_r$ into the expression (\[nicetheta\]) for the $\theta^{(k)}$ and performing the sum over $k$ in (\[extras\]) gives the result $$\label{sumthetatwo}
\Theta = - {\Omega _{D-2} r_h ^{D-3}\over 8\pi G (D-1)} \sum_{k=0}^{\kmax}\left( k -1 + {s_{(2)} \over s_{(1)} } \right)
{ \hat{b}_k \over r_h ^{2(k-1)} }$$ Combining the results for the mass (\[admmassrh\]), surface gravity (\[surfgrav\]), entropy (\[sphentropy\]), $S^\prime$ in (\[sprime\]) and thermodynamic potential (\[sumthetatwo\]), it is now straightforward to check that the validity of the Smarr formula (\[smarr\]) for the AdS-Lovelock black holes.
Free energy and phase transitions {#freesection}
=================================
In this section, we will make use of our results to give a general expression for the free energy of AdS-Lovelock black holes. We will discuss the behavior of this expression in the limits of large and small horizon radius, in which it is independent of the detailed choice of Lovelock couplings.
The free energy of asymptotically AdS black holes has been a topic of interest since the work of Hawking and Page [@Hawking:1982dh] in Einstein gravity. They found that there exists a minimum temperature $T_0$ for such black holes which occurs for a horizon radius $r_0$. The temperature diverges both in the limit of large black holes and in the limit of zero horizon radius. Black holes with horizon $r_h<r_0$, like asymptotically flat black holes, have negative specific heat and cannot be in stable equilibrium with a thermal bath of radiation. However, solutions with $r_h>r_0$ have positive specific heat and can be in stable equilibrium. For large black holes with temperatures just above $T_0$, the free energy is positive and thermal AdS space, with zero free energy, represents that globally preferred thermodynamic state. However, the free energy of large black holes becomes negative above a critical temperature $T_1>T_0$ (or correspondingly for black holes with radii exceeding a certain threshold $r_1$) and the black hole is then the globally preferred state.
Hawking and Page found the free energy by computing the Euclidean action, which requires the analytic form of the AdS-Schwarzschild spacetimes. Similar computations of the free energy have been carried out in Gauss-Bonnet gravity [@Cai:2001dz; @Cvetic:2001bk; @Nojiri:2001aj; @Cho:2002hq], where the explicit solutions are also known [@Boulware:1985wk; @Wheeler:1985nh]. An expresssion for free energy in general Lovelock gravity is given in [@Maeda:2011ii] which is calculated using a generalized quasi-local mass defined for spherically (plane or hyperbolic) symmetric spacetimes. However, theses computations are considerably more complicated. Our method offer a much a simpler route to the result and offers some degree of physical interpretation to the different terms in the result. It also yields the answer in the general Lovelock case, where do to the absence of explicit analytic solutions, computation of the Euclidean action may not be practical. Moreover as stated above, we envision further applications to even less well understood solutions, such as rotating Lovelock black holes.
It is worth noting explicitly that the equality of the free energy with the Euclidean action times the temperature continues to hold generally in Lovelock gravity theories. Hawking and Horowitz [@Hawking:1995fd] used the Hamiltonian framework for Einstein gravity to demonstrate that $I_E =\beta M -S$ where $\beta$ is the Euclidean period. A similar construction works in Lovelock gravity, the basic steps being as follows.
Write the volume term of the Euclidean action for a static black hole in Hamiltonian variables, and then directly derive the boundary term for the action by varying this expression. The variation of the volume term is the same calculation that is done to find the mass (see [@Crisostomo:2000bb; @Kastor:2010gq]) with an additional integration over Euclidean time. Evaluated on static solutions, the volume term of the action reduces to a sum of the constraints and hence vanishes. The value of the action is then given by the boundary terms. At infinity the boundary term is simply $M/T$. The Euclidean metric does not have a horizon. However, Hamiltonian evolution with respect to Euclidean time fails to be well defined at $r_h$. In order to compensate for this, one introduces an inner boundary at $r=r_h +\epsilon$. In the limit $\epsilon \rightarrow 0$ the boundary term is equal to the entropy $S$. One then has the result that the free energy defined by $F=TI_E$ coincides with the thermodynamic free energy $F=M-TS$. Note also that the free energy has been defined such that it vanishes for AdS[^6].
Let us see how the Hawking-Page transition arises in the present framework. In Lovelock gravity the cosmological constant is given by $\Lambda=-b_0/2$, so that $b_0$ is positive for $\Lambda$ negative. Starting from $F=M-TS$, let us use the Smarr formula (\[smarr\]) to eliminate the mass. For Einstein gravity the quantities $s_{(2)}$ and $S^\prime=0$ in the Smarr relation vanish, while as shown in [@Kastor:2009wy] the overall Lovelock thermodynamic potential is given by $\Theta = - V_{bh} $ where $V_{bh}$ is an effective volume for the black hole given by the flat (or AdS) spacetime volume of a sphere of radius $r_h$. One then arrives at an expression for the free energy $$\begin{aligned}
\label{hpfree}
F&=&{1\over 8\pi G(D-3)} \left(\kappa A -b_0 V_{bh}\right)\\
&=& { \Omega _{D-2} r_h^{D-2} \over 8\pi G (D-3) } \left( \kappa - { b_0 r_h \over (D-1) } \right) \end{aligned}$$ The important observation is that the apparent contribution of $\kappa A$ to the free energy has changed sign in (\[hpfree\]) relative to the original expression $F=M-TS$. This is because the mass receives a positive contribution from $\kappa A$ in the Smarr formula and the overall net coefficient is always positive. For an asymptotically flat black hole, [*i.e.*]{} with vanishing $b_0$, this is the only term and the free energy is always positive. With a negative cosmological constant there is a negative definite contribution that takes the form of a cosmological pressure $b_0$ times an effective volume of the black hole $V_{bh}$. Hence the Hawking-Page phase transition at which the free energy changes sign may be thought of as arising from the $\Theta$ term in the Smarr relation, which is itself analogous to a $PV$-type contribution in classical thermodynamics.
To determine whether the positive or negative term dominates $F$ one needs to use know how the surface gravity behaves. Hawking and Page used the analytic solutions to compute $\kappa$ and found that the free energy is positive for small black holes as in Schwarzchild. However, they found that large black holes have negative free energy. Indeed, substituting $\kappa$ from equation (\[surfgrav\]) one recovers these results. However, rather than recall this case in more detail, we turn to an analysis of the free energy for general Lovelock black holes, which includes the case of Einstein gravity. We will see that the behavior of the free energy for large black holes is qualitatively the same as the Hawking-Page case. On the other hand, the free energy of small black holes differs between even and odd dimensions due to the alternating behavior of the surface gravity.
Thermodynamic stability and phase transitions for Lovelock black holes
----------------------------------------------------------------------
There are a number of contributions to the general Lovelock free energy and the simplification made above in the Einstein case using the Smarr formula no longer yields an easily interpretable expression. Instead we simply substitute into the free energy the formulas for the mass (\[admmassrh\]) and the entropy (\[sphentropy\]) to obtain $$\label{morefree}
F = { \Omega _{D-2} r_h^{D-3} \over 16\pi G (D-1) }
\sum_{k=0}^{\kmax} {\bhat_k \over r_h ^{2(k-1)} }
\left( 1- \kappa r_h { 2k \over (D-2k)} \right)$$ where $\kappa$ can also be regarded as in (\[surfgrav\]) as a function of the horizon radius and the Lovelock couplings. This expression for $F$ agrees with that of reference [@Cai:2001dz] in the case of Gauss-Bonnet gravity and [@Maeda:2011ii] in general Lovelock gravity, but disagrees[^7] with that given in reference [@Cvetic:2001bk].
Let us start by examining the behavior of the free energy in the large black hole limit. Form (\[largekappa\]), we see that the surface gravity grows like $b_0\, r_h$ for large $r_h$ in all dimensions. One then finds that there are both positive and negative contributions at leading order growing like $b_0 r_h^{D-1}$. The net result for the free energy in the large black hole limit turns out to be negative, $$\label{largef}
F \approx -{ \Omega _{D-2} b_0 r_h ^{D-1} \over 16\pi G (D-1)(D-2) }$$ This result was also pointed out in [@Camanho:2011rj] (see also [@Cai:2006pq; @Cai:2009de]) and is not surprising since the behavior in this regime is dominated by the cosmological constant and Einstein terms.
The behavior of the free energy (\[morefree\]) in the small black hole regime, on the other hand, is dominated by the highest curvature terms and differs between even and odd dimensions. In even dimensions both the mass $M$ and the product $\kappa S$ scale like $r_h$ in the limit of small horizon radius. One finds in this case that the positive contribution to the free energy coming from the mass dominates, giving $$\label{smallevenf}
F \approx { \Omega _{D-2}\,\hat b_{\kmax} r_h\over 32\pi G (D-1) } , \qquad
D\,\, even$$ In odd dimensions both the surface gravity and entropy vanish as $r_h\rightarrow 0$ (see equation (\[smallkappatwo\])), while the mass has a finite positive limiting value, giving $$\label{smalloddf}
F\approx { \Omega _{D-2} \,\hat b_{\kmax} r_h\over 16\pi G (D-1) } , \qquad D\,\, odd$$ Hence $F$ is positive for small black holes in all dimensions. For $D$ even $F$ goes to zero, while for $D$ odd $F$ goes to a nonzero positive value.
It is also straightforward to compute the specific heat in the large and small black hole limits. Using the expressions for the mass and the surface gravity in section (\[mainsection\]) we find for large black holes $$\label{heatlarge}
{\partial M \over \partial T} \approx {(D-2)\Omega _{D-2}\, r_h ^{D-2} \over 4G}$$ and for small black holes $$\begin{aligned}
\label{heatsmalleven}
{\partial M\over \partial T} &\approx -{ \kmax\,\Omega _{D-2}\, {\hat b}_{\kmax} r_h ^2\over 4G (D-1)}
\qquad &D\,\, even\\
\label{heatsmallodd}
{\partial M\over \partial T}& \approx {\kmax\, \Omega _{D-2}\, {\hat b}^2 _{(\kmax - 1)} \, r_h \over 4G (D-1){\hat b}_{\kmax}}
\qquad &D\,\, odd\end{aligned}$$ where we have assumed that the couplings $b_0$, $b_{\kmax}$ and $b_{\kmax -1}$ are all nonzero.
Let us now summarize these results. While a detailed understanding of the global and local thermodynamic stability of AdS-Lovelock black holes throughout the entire range of horizon radii would require specifying the entire set of Lovelock couplings, the behavior in the large and small black hole regimes is generic. In dimensions, we find that the behavior of AdS-Lovelock black holes in these regimes is similar to that found by Hawking and Page in Einstein gravity [@Hawking:1982dh]. Black holes become arbitrarily hot in both limits. Small black holes in even dimensions exhibit negative specific heat and have positive free energy, indicating instability to both perturbative and non-perturbative fluctuations. Large black holes, on the other hand, have positive specific heat and negative free energy, both indicating their stability to small thermal fluctuations and that they are the thermodynamically globally preferred state. If one assumes that the Lovelock couplings are such that the temperature stays positive for all $r_h$, then there must exist a minimum temperature $T_{0}$ below which no black hole solutions exist. As a consequence of these similarities, we can expect that even dimensional Lovelock black holes will at least have a simple Hawking-Page phase transition and possibly a more complicated structure of phase transitions, depending on the detailed behavior of the temperature and free energy over the whole range of horizon radii.
In odd dimensions only large black holes exist at very high temperatures. They have positive specific heat and negative free energy as in even dimensions and are therefore thermodynamically stable. The odd dimensional small black holes, on the other hand, have positive positive free energy and also have positive specific heat. So they are stable to small, but not large, thermal fluctuations, and unlike small black holes is even dimensions can be in stable equilibrium with a thermal bath.
In odd dimensions the low temperature picture is different. This was studied in $D=5$ Gauss-Bonnet gravity in references [@Cai:2001dz; @Cho:2002hq] . Black holes exist for [@Cai:2001dz] temperatures down to $T=0$, and the low temperature black holes have positive specific heat, so there is a locally stable alternative to the gas state. At these low temperatures the pure gas state has lower free energy since $F_{gas}\rightarrow 0$ as $T\rightarrow 0$, while the black hole starts with $F$ of order $b_{\kmax} $. Still, a small black hole can exist in equilibrium with a low temperature gas, unlike the situation in even dimensions. So the behavior of small black holes in $D=5$ Gauss-Bonnet gravity continues in odd dimensions as long as the highest curvature Lovelock term is included. For very high temperatures there is only the one large black hole state. As in even dimensions this is both globally and locally preferred.
Discussion
==========
In this paper we started by deriving a fully explicit formula for the ADM mass of an asymptotically AdS spacetime in a generic Lovelock gravity theory, via the Hamiltonian methods of Regge & Teitelboim [@Regge:1974zd]. We then proceeded to study various thermodynamic properties of AdS-Lovelock black holes. In particular, we made use of the Killing-Lovelock potentials that exist in these spacetimes in order to evaluate the mass in terms of the horizon radius and the Lovelock couplings. After finding expressions for the surface gravity and entropy, these ingredients were assembled to give the free energy. All of these expressions are quite general, assuming only that solutions exist with the prescribed asymptotic forms.
As mentioned in the introduction, we envision further applications of these techniques to stationary solutions that are even “more unknown", such as higher dimensional rotating charged black holes, or rotating Lovelock black holes. Another possible direction for future work would be to look at black holes/branes in AdS with planar horizons. In this case the asymptotic boundary of a spatial slice is a plane rather than a sphere. If one of these directions is compact with length $L$, then it would be necessary to further extend the first law to include an appropriate $\delta L$ term as in the asymptotically flat Kaluza-Klein case [@Kastor:2006ti].
Acknowledgements {#acknowledgements .unnumbered}
----------------
The work of DK and JT was supported by NSF grant PHY-0555304. DK and JT also acknowledge the hospitality of the Centro de Ciencias de Benasque, Spain where this work was begun. The work of SR was funded by FONDECYT grant 3095018 and by the CONICYT grant Southern Theoretical Physics Laboratory ACT-91. Centro de Estudios Científicos (CECS) is funded by the Chilean Government through the Millennium Science Initiative and the Centers of Excellence Base Financing Program of CONICYT. CECS is also supported by a group of private companies which at present includes Antofagasta Minerals, Arauco, Empresas CMPC, Indura, Naviera Ultragas and Telefónica del Sur. CIN is funded by CONICYT and the Gobierno Regional de Los Ríos.
Appendix - Near horizon behavior of $g_{rr}$ {#appendix---near-horizon-behavior-of-g_rr .unnumbered}
============================================
The horizon of a static black hole occurs where the norm of the static Killing field is zero. In the coordinates of the metric (\[generalmetric\]) this is at $r_h$ such that $\phi (r_h )=0$. In this appendix we show that if one assumes that the surface gravity is finite and that the Hamiltonian constraint equation is satisfied at $r_h$, then near the horizon it follows that $f(r)\sim \phi (r )$, and in particular that $f(r_h )=0 $ , $f' (r_h )= 2\kappa $.
We point out that for the spherically symmetric black hole metric (\[generalmetric\]) it has been shown [@Wheeler:1985nh] [@Wheeler:1985qd] that the vacuum field equations imply that $- g_{tt}= 1/g_{rr}$, or $\phi (r) =f(r) $ everywhere. Clearly this is a stronger result than the result shown here. However, to derive the expressions for $M$, $\kappa$, and the $\Theta ^{(k)}$ we only need to use the values of $f$ and $f'$ at $r_h$, which can be found with correspondingly less work, as follows.
Assume that the metric function $\phi$ in (\[generalmetric\]) goes to zero like a power law as $r$ approaches $r_h$, [*i.e.*]{} that near the horizon $$\label{phih}
\phi \simeq \phi _1 (r-r_h ) ^ p$$ where $\phi_1$ is a constant. The surface gravity, given by $\kappa ^2 =-{1\over 2}( \nabla _a \xi _b) \nabla ^a \xi ^ b$, then becomes $$\label{sg}
\kappa ^2 =
{1\over 2} p^2 \lim_{r\rightarrow r_h}(r-r _H )^{p-2} f(r)$$ In order for $\kappa^2$ to be finite, it must be that the metric function $f(r)$ behaves like $$\label{fh}
f \simeq f _1 (r-r_h ) ^ {2-p}$$ with $f_1$ another constant. Let us rewrite the expression (\[hkform\]) for $H^{(k)}_\perp$ as $$H ^{(k)} = r^{1-2k} \left[(D-2k-1)(1-f)/r -k(1-f)^k f' \right]$$ and consider the constraint equation (\[constreq\]). If the power law index $p>1$, we see from line (\[fh\]) that $f' $ diverges at the horizon, and inspection shows that $H=0$ cannot be satisfied at $r_h$. On the other hand, if the power law index $p<1$, then $f(r_h ) = f'(r_h ) =0$ and again the constraint cannot be satisfied[^8] at $r_h$. Hence, in order to have finite, non-zero surface gravity, the power law index in (\[sg\]) and (\[fh\]) must be $p=1$, so that $\phi \sim \phi _1 (r-r_h )$ and $f \sim f_1 (r-r_h ) $, which gives $\kappa ^2 ={1\over 2} \phi _1 f_1$. We see that the overall numerical scale of $\kappa $ is not fixed by this argument, which makes sense because the normalization of $\kappa$ is fixed by the norm of the time-translation Killing vector at infinity, which requires knowing the function $\phi (r)$ throughout the spacetime. However, one can fix the scale by requiring that it gives the right answer for Schwarzchild.
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[^1]: Given that the solutions to generic cubic and quartic equations are quite cumbersome, in practice the general solution is only known in useful form for $D\le 6$, although certain non-generic solutions such as those for Gauss-Bonnet gravity [@Boulware:1985wk; @Wheeler:1985nh; @Cai:2001dz; @Cai:2003gr], or Chern-Simons gravity [@Cai:1998vy; @Crisostomo:2000bb] are known in all dimensions. Note also that an analytic solution for a rotating black hole in $D=5$ Gauss-Bonnet gravity has also been found in the case of Chern-Simons couplings [@Anabalon:2009kq] (see also [@Anabalon:2010ns]).
[^2]: A number of related results have been established in the literature. The Hamiltonian approach of [@Regge:1974zd] has been used to obtain an expression for the mass in pure Lovelock gravity theories [@Crisostomo:2000bb] with only a single term in Lagrangian. Significant work has also been done in the case of Gauss-Bonnet gravity. Deser and Tekin have extended the formalism of Abbott and Deser [@Abbott:1981ff] to general quadratic theories of gravity [@Deser:2002rt; @Deser:2002jk]. The mass in Gauss-Bonnet gravity has also been found by means of a Noether’s current construction by Deruelle et. al. [@Deruelle:2003ps] and using the Palatini formalism by Katz and Livshits [@katz] The mass of asymptotically AdS black holes in general higher curvature theories has also been addressed in [@Okuyama:2005fg].
Padilla [@Padilla:2003qi] uses the Cartan formalism to derive a general expression for the mass in Gauss-Bonnet gravity by transforming the trace $K$ boundary term in the action into Hamiltonian variables. By assuming that a generalization of this result holds in general Lovelock theories, the mass of static Lovelock black hole is identified by Cai [@Cai:2003kt] with a constant of integration arising from integration of the Hamiltonian constraint. Our general derivation of the mass in Lovelock theories demonstrates the correctness of this assumption.
[^3]: One has $\Lambda=-2b_0$ in this case.
[^4]: Specifically, if one defines the function $F$ according to $f=1-r^2 F$, then $F$ must satisfy the polynomial equation $ \sum_{k=0}^{\kmax} {\hat b}_k F^k = {\omega\over r^{D-1}}$ for some constant $\omega$. Constant curvature vacua of Lovelock theories solve this equation with $\omega=0$.
[^5]: Note that if one arbitrarily specifies $F$ at each subsequent time in the evolution, then in general the evolution will not be along a Killing field. To keep the evolution along the Killing vector, one imposes $\dot{s}_{ab} = \dot{K}_{ab} =0$, which implies a set of differential equations that $F$ must satisfy.
[^6]: However, because of the presence of a mass gap for odd dimensional AdS-Lovelock black holes, the limit $r_h\rightarrow 0$ will not vanish in our subsequent expressions for $F$ given below.
[^7]: The expression for the free energy in [@Cvetic:2001bk] is obtained by computing the volume term in the Euclidean action, apparently without the inclusion of a boundary term. A subtraction of the action for pure $AdS$ at large radius is used to regularize the result. There are then several differences with our calculation. First, we implicitly use the Euclidean action with the boundary term that gives a well defined variational principal in the Hamiltonian variables. As described above the action on solutions is then given entirely by the boundary term and is equal to the finite quanity $M-TS$, without a need for regularization.
[^8]: If the spacetime is not vaccuum, the the gravitational constraint is $H=-16\pi\rho$, and so powers $p<1$ are not ruled out.
|
{
"pile_set_name": "ArXiv"
}
|
---
abstract: 'DeConvNet, Guided BackProp, LRP, were invented to better understand deep neural networks. We show that these methods do not produce the theoretically correct explanation for a linear model. Yet they are used on multi-layer networks with millions of parameters. This is a cause for concern since linear models are simple neural networks. We argue that explanation methods for neural nets should work reliably in the limit of simplicity, the linear models. Based on our analysis of linear models we propose a generalization that yields two explanation techniques (PatternNet and PatternAttribution) that are theoretically sound for linear models and produce improved explanations for deep networks.'
author:
- |
Pieter-Jan Kindermans[^1]\
Google Brain\
`[email protected]`\
Kristof T. Schütt & Maximilian Alber\
TU Berlin\
`{maximilian.alber,kristof.schuett}@tu-berlin.de`\
Klaus-Robert Müller[^2]\
TU Berlin\
`[email protected]`\
Dumitru Erhan & Been Kim\
Google Brain\
`{dumitru,beenkim}@google.com` Sven Dähne[^3]\
TU Berlin\
`[email protected]`
bibliography:
- 'iclr2018\_conference.bib'
title: |
Learning how to explain neural networks:\
PatternNet and PatternAttribution
---
Introduction
============
Deep learning made a huge impact on a wide variety of applications [@Lecun2015; @Schmidhuber2015; @krizhevsky2012imagenet; @mnih2015human; @Silver2016; @Sutskever2014] and recent neural network classifiers have become extremely good at detecting relevant *signals* (say, the presence of a cat) contained in input data points such as images by filtering out all other, non-relevant and *distracting* components also present in the data. This separation of signal and distractors is achieved by passing the input through many layers with millions of parameters and nonlinear activation functions in between until finally at the output layer, these models yield a highly condensed version of the signal, e.g. a single number indicating the probability of a cat being in the image.
While deep neural networks learn efficient and powerful representations, they are often considered a ‘black-box’. In order to better understand classifier decisions and to gain insight into how these models operate, a variety techniques have been proposed [@Simonyan2013; @Yosinski2015; @Nguyen2016; @Baehrens2010; @Bach2015; @Montavon2017; @Zeiler2014; @Springenberg2014; @Zintgraf2017; @Mukund2017; @Smilkov2017].
The aforementioned methods for explaining classifier decisions operate under the assumption that it is possible to propagate the condensed output signal back through the classifier to arrive at something that shows how the relevant signal was encoded in the input and thereby explains the classifier decision. Simply put, if the classifier detected a cat, the visualization should point to the cat-relevant aspects of the input image from the perspective of the network. Techniques that are based on this principle include saliency maps from network gradients [@Baehrens2010; @Simonyan2013], DeConvNet [@Zeiler2014 DCN], Guided BackProp [@Springenberg2014 GBP], Layer-wise Relevance Propagation [@Bach2015 LRP] and the Deep Taylor Decomposition [@Montavon2017 DTD], Integrated Gradients [@Mukund2017] and SmoothGrad [@Smilkov2017].
The merit of explanation methods is often proven by applying them to state-of-the-art deep learning models in the context of high dimensional real world data, such as ImageNet. Here we begin with a different approach. We first take a step back and analyze explanation methods in the context of the simplest neural network setting: a purely linear model and data stemming from a linear generative model. We chose this simplified setup because it allows us to (i) fully control how signal and distractor components are encoded in the input data and (ii) analytically track how the resulting explanation relates to the known signal component. This analysis allows us to highlight shortcomings of current explanation approaches that carry over to non-linear models as well.
On the basis of our findings, we then propose PatternNet and PatternAttribution, which alleviate these flaws. Finally we apply our methods to practically relevant networks and datasets, and show that our approach produces qualitatively improved signal visualizations and attributions (see [ Fig. \[fig:fig1\]]{} and [ Fig. \[fig:fig5\]]{}). In addition to the qualitative evaluation, we also experimentally verify whether our proposed theoretical model holds up empirically (see [ Fig. \[fig:fig3\]]{}).
The remainder of the paper is structured as follows: visualization in linear models is analyzed in section 2. Section 3 relates this analysis to existing approaches for neural network visualization. Section 4 introduces PatternNet and PatternAttribution, which then is evaluated in section 5 before concluding.
{width="\textwidth"}
#### Notation and scope
Scalars are lowercase letters ($i$), column vectors are bold (${\boldsymbol{u}}$), element-wise multiplication is ($\odot$). The covariance between ${\boldsymbol{u}}$ and ${\boldsymbol{v}}$ is ${\textrm{cov}[{\boldsymbol{u}},{\boldsymbol{v}}]}$, the covariance of ${\boldsymbol{u}}$ and $i$ is ${\textrm{cov}[{\boldsymbol{u}},i]}$. The variance of a scalar random variable $i$ is $\sigma^2_{i}$. Estimates of random variables will have a hat ($\hat{{\boldsymbol{u}}}$). We analyze neural networks excluding the final soft-max output layer. To allow for analytical treatment, we only consider networks with linear neurons optionally followed by a rectified linear unit (ReLU), max-pooling or soft-max. We analyze linear neurons and nonlinearities independently such that every neuron has its own weight vector. These restrictions are similar to those in the saliency map [@Simonyan2013], DCN [@Zeiler2014], GBP [@Springenberg2014], LRP [@Bach2015] and DTD [@Montavon2017]. Without loss of generality, biases are considered constant neurons to enhance clarity.
Understanding linear models {#sec:linearmodels}
===========================
{width="\textwidth"}
Before moving to deep networks, we analyze the behavior of a linear model (see [ Fig. \[fig:fig1\]]{}).Consider the following toy example where we generate data ${\boldsymbol{x}}$ as: $$\begin{aligned}
{\boldsymbol{x}}&= {\boldsymbol{s}}+ {\boldsymbol{d}}&
{\boldsymbol{s}}&= {\boldsymbol{a}}_s y, &\textrm{with } {\boldsymbol{a}}_s & =\left(1,0\right)^T, \,~~~~~~~ y \in \left[-1,1\right] \\
& & {\boldsymbol{d}}&= {\boldsymbol{a}}_d \epsilon, &\textrm{with } {\boldsymbol{a}}_d & =\left(1,1\right)^T, \,~~~~~~~ \epsilon \sim \mathcal{N}\left(\mu,\sigma^2\right).\end{aligned}$$ We train a linear regression model to extract $y$ from ${\boldsymbol{x}}$. By construction, ${\boldsymbol{s}}$ is the [*signal*]{} in our data, i.e., the part of ${\boldsymbol{x}}$ containing information about $y$. Using the terminology of @Haufe2014 the [*distractor*]{} ${\boldsymbol{d}}$ obfuscates the signal making the detection task more difficult. To optimally extract $y$, our model has to be able to filter out the distractor ${\boldsymbol{d}}$. This is why the weight vector is also called the [*filter*]{}. In the example, ${\boldsymbol{w}}=\left[1,-1\right]^T$ fulfills this task.
From this simple example, we can make several observations: The optimal weight vector ${\boldsymbol{w}}$ does *not* align, in general, with the signal direction ${\boldsymbol{a}}_s$, but tries to filter the contribution of the distractor (see [ Fig. \[fig:fig1\]]{}). This optimally solved when the weight vector is orthogonal to the distractor ${\boldsymbol{w}}^T {\boldsymbol{d}}= 0$. Therefore, when the direction of the distractor ${\boldsymbol{a}}_d$ changes, ${\boldsymbol{w}}$ must follow, as illustrated on the right hand side of the figure. On the other hand, a change in signal direction ${\boldsymbol{a}}_s$ can be compensated for by a change in sign and magnitude of ${\boldsymbol{w}}$ such that ${\boldsymbol{w}}^T{\boldsymbol{a}}_s=1$, but the direction stays constant.
This implies that in the situation where we have a signal and distractor(s), the direction of the weight vector in a linear model is largely determined by the distractor. This is essential to understand how linear models operate. This also indicates that given only the weight vector, we cannot know what part of the input produces the output $y$. This is the direction ${\boldsymbol{a}}_s$ and must be learned from data. Please note that the linear problem above is convex, therefore a weight vector obtained by optimizing the model would converge to the analytical solution defined above.
Now assume that we have no distractor but instead we have additive isotropic Gaussian noise. It is easy to verify that the mean of the noise is of little importance since it can be compensated for with a bias change. Therefore, we only have to consider case where the noise is zero mean. Because isotropic Gaussian noise does not contain any correlations or structure, the only way to cancel it out is by averaging over different measurements. It is *not* possible to cancel it out effectively by using a well-chosen weight vector. However, it is well known that adding Gaussian noise shrinks the weight vector and corresponds to l2 regularization. In the absence of a structured distractor, the smallest weight vector ${\boldsymbol{w}}$ such that ${\boldsymbol{w}}^T{\boldsymbol{a}}_s=1$ is the one in the direction of the signal. Therefore in practice both these effects influence the actual weight vector.
Considering the reasoning above, we have to wonder under which conditions we are working in a deep neural network. Especially since DeConvNet and Guided BackProp produce crisp visualizations using (modified) gradients. For this reason we will perform the following quantitative and qualitative experiments which indicate that our theory also holds for a deep network:
- In [ Fig. \[fig:fig3\]]{} we have evaluated how well the weight vector or a learned direction captures the information content in the input of every single neuron in VGG16. This experiment empirically shows that a learned direction captures more information than the direction defined by the weight vector. This indicates that we are working (largely) in the distractor-regime.
- This experiment is confirmed by an image degradation experiment in [ Fig. \[fig:fig4\]]{}.
- It is also corroborated by the qualitative inspection of the visualizations in [ Fig. \[fig:fig2\]]{}, [ Fig. \[fig:fig5\]]{} and [ Fig. \[fig:fig6\]]{}.
Finally, there is also an intuitive argument. Neural networks are considered layer-wise feature extractors that add more invariances as we move through the layers. Since cancelling out a distractor is adding an invariance, the proposed theory fits this interpretation well.
Before moving on to the discussion of interpretability methods, we would like to remind the reader of the terminology that is used throughout this manuscript: The [*filter*]{} ${\boldsymbol{w}}$ tells us how to extract the output $y$ optimally from data ${\boldsymbol{x}}$. The *pattern* ${\boldsymbol{a}}_s$ is the direction in the data along which the desired output $y$ varies. Both constitute the *signal* ${\boldsymbol{s}}={\boldsymbol{a}}_s y$, i.e., the contributing part of ${\boldsymbol{x}}$. The [*distractor*]{} ${\boldsymbol{d}}$ is the component of the data that does not contain information about the desired output.
Overview of explanation approaches and their behavior {#sec:methods}
=====================================================
In this section, we take a look at a subset of explanation methods for individual classifier decisions and discuss how they are connected to our analysis of linear models in the previous section. [ Fig. \[fig:fig2\]]{} gives an overview of the different types of explanation methods which can be divided into function, signal and attribution visualizations. These three groups all present different information about the network and complement each other.
#### Functions – gradients, saliency map
Explaining the function in input space corresponds to describing the operations the model uses to extract $y$ from ${\boldsymbol{x}}$. Since deep neural networks are highly nonlinear, this can only be approximated. The saliency map estimates how moving along a particular direction in input space influences $y$ (i.e., sensitivity analysis) where the direction is given by the model gradient [@Baehrens2010; @Simonyan2013]. In case of a linear model $y = {\boldsymbol{w}}^T {\boldsymbol{x}}$, the saliency map reduces to analyzing the weights $\partial y / \partial {\boldsymbol{x}}= {\boldsymbol{w}}$. Since it is mostly determined by the distractor, as demonstrated above , it is not representing the signal. It tells us how to extract the signal, not what the signal is in a deep neural network.
#### Signal – DeConvNet, Guided BackProp, PatternNet
The signal ${\boldsymbol{s}}$ detected by the neural network is the component of the data that caused the networks activations. @Zeiler2014 formulated the goal of these methods as “\[...\] to map these activities back to the input pixel space, showing what input pattern originally caused a given activation in the feature maps”.
In a linear model, the signal corresponds to ${\boldsymbol{s}}={\boldsymbol{a}}_s y$. The pattern ${\boldsymbol{a}}_s$ contains the signal direction, i.e., it tells us where a change of the output variable is expected to be measurable in the input [@Haufe2014]. Attempts to visualize the signal for deep neural networks were made using DeConvNet [@Zeiler2014] and Guided BackProp [@Springenberg2014]. These use the same algorithm as the saliency map, but treat the rectifiers differently (see [ Fig. \[fig:fig2\]]{}): DeConvNet leaves out the rectifiers from the forward pass, but adds additional ReLUs after each deconvolution, while Guided BackProp uses the ReLUs from the forward pass as well as additional ones. The back-projections for the linear components of the network correspond to a superposition of what are assumed to be the signal directions of each neuron. For this reason, these projections must be seen as an approximation of the features that activated the higher layer neuron. It is not a reconstruction in input space [@Zeiler2014].
For the simplest of neural networks – the linear model – these visualizations reduce to the gradient[^4]. They show the filter ${\boldsymbol{w}}$ and *neither* the pattern ${\boldsymbol{a}}_s$, *nor* the signal ${\boldsymbol{s}}$. Hence, DeConvNet and Guided BackProp do not guarantee to produce the detected signal for a linear model, which is proven by our toy example in [ Fig. \[fig:fig1\]]{}. Since they do produce compelling visualizations, we will later investigate whether the direction of the filter ${\boldsymbol{w}}$ coincides with the direction of the signal ${\boldsymbol{s}}$. We will show that this is *not* the case and propose a new approach, PatternNet (see [ Fig. \[fig:fig2\]]{}), to estimate the correct direction that improves upon the DeConvNet and Guided BackProp visualizations.
#### Attribution – LRP, Deep Taylor Decomposition, PatternAttribution
Finally, we can look at how much the signal dimensions contribute to the output through the layers. This will be referred to as the *attribution*. For a linear model, the optimal attribution would be obtained by element-wise multiplying the signal with the weight vector: $
{\boldsymbol{r}}^{input} = {\boldsymbol{w}}\odot {\boldsymbol{a}}y,
$ with $\odot$ the element-wise multiplication. [@Bach2015] introduced *layer-wise relevance propagation* (LRP) as a decomposition of pixel-wise contributions (called *relevances*). [@Montavon2017] extended this idea and proposed the deep Taylor decomposition (DTD). The key idea of DTD is to decompose the activation of a neuron in terms of contributions from its inputs. This is achieved using a first-order Taylor expansion around a root point ${\boldsymbol{x}}_0$ with ${\boldsymbol{w}}^T{\boldsymbol{x}}_0=0$. The relevance of the selected output neuron $i$ is initialized with its output from the forward pass. The relevance from neuron $i$ in layer $l$ is re-distributed towards its input as: $$r^{output}_i=y,~~~~~~~~~~~~r^{output}_{j\neq i} = 0,~~~~~~~~~~~~{\boldsymbol{r}}^{l-1}_i=\frac{{\boldsymbol{w}}\odot\left({\boldsymbol{x}}-{\boldsymbol{x}}_0\right)}{{\boldsymbol{w}}^T{\boldsymbol{x}}}r^l_i.$$ Here we can safely assume that ${\boldsymbol{w}}^T{\boldsymbol{x}}>0$ because a non-active ReLU unit from the forward pass stops the re-distribution in the backward pass. This is identical to how a ReLU stops the propagation of the gradient. The difficulty in the application of the deep Taylor decomposition is the choice of the root point ${\boldsymbol{x}}_0$, for which many options are available. It is important to recognize at this point that selecting a root point for the DTD corresponds to estimating the distractor ${\boldsymbol{x}}_0 = {\boldsymbol{d}}$ and, by that, the signal $\hat{{\boldsymbol{s}}}={\boldsymbol{x}}-{\boldsymbol{x}}_0$. PatternAttribution is a DTD extension that learns from data how to set the root point.
Summarizing, the **function** extracts the **signal** from the data by removing the distractor. The **attribution** of output values to input dimensions shows how much an individual component of the signal contributes to the output, which is what LRP calls *relevance*.
Learning to estimate the signal
===============================
Visualizing the function has proven to be straightforward [@Baehrens2010; @Simonyan2013]. In contrast, visualizing the signal [@Haufe2014; @Zeiler2014; @Springenberg2014] and the attribution [@Bach2015; @Montavon2017; @Mukund2017] is more difficult. It requires a good estimate of what is the signal and what is the distractor. In the following section we first propose a quality measure for neuron-wise signal estimators. This allows us to evaluate existing approaches and, finally, derive signal estimators that optimize this criterion. These estimators will then be used to explain the signal (PatternNet) and the attribution (PatternAttribution). All mentioned techniques as well as our proposed signal estimators treat neurons independently, i.e., the full explanation will be a superposition of neuron-wise explanations.
Quality criterion for signal estimators {#sec:measure}
---------------------------------------
Recall that the input data ${\boldsymbol{x}}$ comprises both signal and distractor: ${\boldsymbol{x}}= {\boldsymbol{s}}+ {\boldsymbol{d}}, $ and that the signal contributes to the output but the distractor does not. Assuming the filter ${\boldsymbol{w}}$ has been trained sufficiently well to extract $y$, we have $${\boldsymbol{w}}^T {\boldsymbol{x}}= y,~~~~~{\boldsymbol{w}}^T {\boldsymbol{s}}= y,~~~~~~{\boldsymbol{w}}^T {\boldsymbol{d}}= 0.$$ Note that estimating the signal based on these conditions alone is an ill-posed problem. We could limit ourselves to linear estimators of the form $\hat{{\boldsymbol{s}}} = {\boldsymbol{u}}({\boldsymbol{w}}^T{\boldsymbol{u}})^{-1} y$, with ${\boldsymbol{u}}$ a random vector such that ${\boldsymbol{w}}^T{\boldsymbol{u}}\neq0$. For such an estimator, the signal estimate $\hat{{\boldsymbol{s}}}={\boldsymbol{u}}\left({\boldsymbol{w}}^T{\boldsymbol{u}}\right)^{-1} y$ satisfies ${\boldsymbol{w}}^T \hat{{\boldsymbol{s}}}= y$. This implies the existence of an infinite number of possible rules for the DTD as well as infinitely many back-projections for the DeConvNet family.
To alleviate this issue, we introduce the following quality measure $\rho$ for a signal estimator $S({\boldsymbol{x}})=\hat{{\boldsymbol{s}}}$ that will be written with explicit variances and covariances using the shorthands $\hat{{\boldsymbol{d}}}={\boldsymbol{x}}-S({\boldsymbol{x}})$ and $y={\boldsymbol{w}}^T{\boldsymbol{x}}$: $$\rho(S) = 1 - \max_{{\boldsymbol{v}}} corr\left({\boldsymbol{w}}^T{\boldsymbol{x}},{\boldsymbol{v}}^T\left({\boldsymbol{x}}-S({{\boldsymbol{x}}})\right)\right) = 1 - \max_{{\boldsymbol{v}}} \frac{{\boldsymbol{v}}^T{\textrm{cov}[\hat{{\boldsymbol{d}}},y]}}{\sqrt{\sigma^2_{{\boldsymbol{v}}^T\hat{{\boldsymbol{d}}}} \sigma^2_{y}}}. \label{eq:corr}$$ This criterion introduces an additional constraint by measuring how much information about $y$ can be reconstructed from the residuals ${\boldsymbol{x}}- \hat{{\boldsymbol{s}}}$ using a linear projection. The best signal estimators remove most of the information in the residuals and thus yield large $\rho(S)$. Since the correlation is invariant to scaling, we constrain ${\boldsymbol{v}}^T\hat{{\boldsymbol{d}}}$ to have variance $\sigma^2_{{\boldsymbol{v}}^T\hat{{\boldsymbol{d}}}} =\sigma^2_{y}$. Finding the optimal ${\boldsymbol{v}}$ for a fixed $S({\boldsymbol{x}})$ amounts to a least-squares regression from $\hat{{\boldsymbol{d}}}$ to $y$. This enables us to assess the quality of signal estimators efficiently.
Existing Signal Estimators
--------------------------
Let us now discuss two signal estimators that have been used in previous approaches.
#### $S_{{\boldsymbol{x}}}$ – the identity estimator
The naive approach to signal estimation is to assume the entire data is signal and there are no distractors: $$S_{{\boldsymbol{x}}}({\boldsymbol{x}})={\boldsymbol{x}}.$$ With this being plugged into the deep Taylor framework, we obtain the $z$-rule [@Montavon2017] which is equivalent to LRP [@Bach2015]. For a linear model, this corresponds to ${\boldsymbol{r}}= {\boldsymbol{w}}\odot {\boldsymbol{x}}$ as the attribution. It can be shown that for ReLU and max-pooling networks, the $z$-rule reduces to the element-wise multiplication of the input and the saliency map [@LRPGRAD16; @Kindermans2016]. This means that for a whole network, the assumed signal is simply the original input image. It also implies that, if there are distractors present in the data, they are included in the attribution: $${\boldsymbol{r}}= {\boldsymbol{w}}\odot {\boldsymbol{x}}= {\boldsymbol{w}}\odot {\boldsymbol{s}}+ {\boldsymbol{w}}\odot {\boldsymbol{d}}.$$ When moving through the layers by applying the filters ${\boldsymbol{w}}$ during the forward pass, the contributions from the distractor ${\boldsymbol{d}}$ are cancelled out. However, they cannot be cancelled in the backward pass by the element-wise multiplication. The distractor contributions ${\boldsymbol{w}}\odot {\boldsymbol{d}}$ that are included in the LRP explanation cause the noisy nature of the visualizations based on the $z$-rule.
#### $S_{{\boldsymbol{w}}}$ – the filter based estimator
The implicit assumption made by DeConvNet and Guided BackProp is that the detected signal varies in the direction of the weight vector ${\boldsymbol{w}}$. This weight vector has to be normalized in order to be a valid signal estimator. In the deep Taylor decomposition framework this corresponds to the ${\boldsymbol{w}}^2$-rule and results in the following signal estimator: $$S_{{\boldsymbol{w}}}({\boldsymbol{x}}) = \frac{{\boldsymbol{w}}}{{\boldsymbol{w}}^T{\boldsymbol{w}}}{\boldsymbol{w}}^T{\boldsymbol{x}}.$$ For a linear model, this produces an attribution of the form $\frac{{\boldsymbol{w}}\odot {\boldsymbol{w}}}{{\boldsymbol{w}}^T{\boldsymbol{w}}}y$. This estimator does not reconstruct the proper signal in the toy example of section \[sec:linearmodels\]. Empirically it is also sub-optimal in our experiment in [ Fig. \[fig:fig3\]]{}.
PatternNet and PatternAttribution
---------------------------------
We suggest to learn the signal estimator $S$ from data by optimizing the previously established criterion. A signal estimator $S$ is optimal with respect to Eq. if the correlation is zero for all possible ${\boldsymbol{v}}$: $\forall {\boldsymbol{v}}, {\textrm{cov}[y,\hat{{\boldsymbol{d}}}]}{\boldsymbol{v}}= \boldsymbol{0}$. This is the case when there is no covariance between $y$ and $\hat{{\boldsymbol{d}}}$. Because of linearity of the covariance and since $\hat{{\boldsymbol{d}}}={\boldsymbol{x}}-S({\boldsymbol{x}})$ the above condition leads to $${\textrm{cov}[y,\hat{{\boldsymbol{d}}}]}=\boldsymbol{0} \Rightarrow {\textrm{cov}[{\boldsymbol{x}},y]}={\textrm{cov}[S({\boldsymbol{x}}),y]}.\label{eq:cov_equal}$$ It is important to recognize that the covariance is a summarizing statistic and consequently the problem can still be solved in multiple ways. We will present two possible solutions to this problem. Note that when optimizing the estimator, the contribution from the bias neuron will be considered $0$ since it does not covary with the output $y$.
#### $S_{{\boldsymbol{a}}}$ – The linear estimator
A linear neuron can only extract linear signals ${\boldsymbol{s}}$ from its input ${\boldsymbol{x}}$. Therefore, we could assume a linear dependency between ${\boldsymbol{s}}$ and $y$, yielding a signal estimator: $$S_{{\boldsymbol{a}}}({\boldsymbol{x}})={{\boldsymbol{a}}}{\boldsymbol{w}}^T{\boldsymbol{x}}.\label{eq:linear_estimator}$$ Plugging this into [Eq. ]{} and optimising for ${{\boldsymbol{a}}}$ yields $$\notag
{\textrm{cov}[{\boldsymbol{x}},y]}={\textrm{cov}[{{\boldsymbol{a}}} {\boldsymbol{w}}^T{\boldsymbol{x}},y]}={{\boldsymbol{a}}} {\textrm{cov}[y,y]} \Rightarrow
{{\boldsymbol{a}}} =\frac{{\textrm{cov}[{\boldsymbol{x}},y]}}{\sigma^2_y}.\label{eq:closed_a}$$ Note that this solution is equivalent to the approach commonly used in neuro-imaging [@Haufe2014] despite different derivation. With this approach we can recover the signal of our toy example in section \[sec:linearmodels\]. It is equivalent to the filter-based approach only if the distractors are orthogonal to the signal. We found that the linear estimator works well for the convolutional layers. However, when using this signal estimator with ReLUs in the dense layers, there is still a considerable correlation left in the distractor component (see [ Fig. \[fig:fig3\]]{}).
#### $S_{{\boldsymbol{a}}_{+-}}$ – The two-component estimator
To move beyond the linear signal estimator, it is crucial to understand how the rectifier influences the training. Since the gate of the ReLU closes for negative activations, the weights only need to filter the distractor component of neurons with $y > 0$. Since this allows the neural network to apply filters locally, we cannot assume a global distractor component. We rather need to distinguish between the positive and negative regime: $${\boldsymbol{x}}=
\begin{cases}
{\boldsymbol{s}}_+ + {\boldsymbol{d}}_+ & \text{if } y>0\\
{\boldsymbol{s}}_- + {\boldsymbol{d}}_- & \text{otherwise}
\end{cases}$$ Even though signal and distractor of the negative regime are canceled by the following ReLU, we still need to make this distinction in order to approximate the signal. Otherwise, information about whether a neuron fired would be retained in the distractor. Thus, we propose the two-component signal estimator: $$\label{eq:twocomponent}
S_{{\boldsymbol{a}}+-}({\boldsymbol{x}})=\begin{cases}
{\boldsymbol{a}}_+{\boldsymbol{w}}^T{\boldsymbol{x}},~~~~\textrm{if}~{\boldsymbol{w}}^T{\boldsymbol{x}}>0\\
{\boldsymbol{a}}_-{\boldsymbol{w}}^T{\boldsymbol{x}},~~~~\textrm{otherwise}
\end{cases}$$
Next, we derive expressions for the patterns ${\boldsymbol{a}}_+$ and ${\boldsymbol{a}}_-$. We denote expectations over ${\boldsymbol{x}}$ within the positive and negative regime with $\operatorname{\mathbb{E}}_+\left[{\boldsymbol{x}}\right]$ and $\operatorname{\mathbb{E}}_-\left[{\boldsymbol{x}}\right]$, respectively. Let $\pi_+$ be the expected ratio of inputs ${\boldsymbol{x}}$ with ${\boldsymbol{w}}^T{\boldsymbol{x}}>0$. The covariance of data/signal and output become: $$\begin{aligned}
{\textrm{cov}[{\boldsymbol{x}},y]}&=&\pi_+\left(\operatorname{\mathbb{E}}_+ \left[{\boldsymbol{x}}y\right]-\operatorname{\mathbb{E}}_+ \left[ {\boldsymbol{x}}\right]\operatorname{\mathbb{E}}\left[y\right]\right)
&~~+&\left(1-\pi_+\right)\left(\operatorname{\mathbb{E}}_- \left[{\boldsymbol{x}}y\right]-\operatorname{\mathbb{E}}_- \left[ {\boldsymbol{x}}\right]\operatorname{\mathbb{E}}\left[y\right]\right)\\
{\textrm{cov}[{\boldsymbol{s}},y]}&=&\pi_+\left(\operatorname{\mathbb{E}}_+ \left[{\boldsymbol{s}}y\right]-\operatorname{\mathbb{E}}_+ \left[ {\boldsymbol{s}}\right]\operatorname{\mathbb{E}}\left[y\right]\right)
&~~+&\left(1-\pi_+\right)\left(\operatorname{\mathbb{E}}_- \left[{\boldsymbol{s}}y\right]-\operatorname{\mathbb{E}}_- \left[ {\boldsymbol{s}}\right]\operatorname{\mathbb{E}}\left[y\right]\right)\end{aligned}$$ Assuming both covariances are equal, we can treat the positive and negative regime separately using Eq. to optimize the signal estimator: $$\begin{aligned}
\operatorname{\mathbb{E}}_+ \left[{\boldsymbol{x}}y\right]-\operatorname{\mathbb{E}}_+ \left[ {\boldsymbol{x}}\right]\operatorname{\mathbb{E}}\left[y\right]&=&\operatorname{\mathbb{E}}_+ \left[{\boldsymbol{s}}y\right]-\operatorname{\mathbb{E}}_+ \left[ {\boldsymbol{s}}\right]\operatorname{\mathbb{E}}\left[y\right]\end{aligned}$$ Plugging in Eq. and solving for ${\boldsymbol{a}}_+$ yields the required parameter (${\boldsymbol{a}}_-$ analogous). $$\begin{aligned}
{\boldsymbol{a}}_+&=&\frac{\operatorname{\mathbb{E}}_+ \left[{\boldsymbol{x}}y\right]-\operatorname{\mathbb{E}}_+ \left[ {\boldsymbol{x}}\right]\operatorname{\mathbb{E}}\left[y\right]}{{\boldsymbol{w}}^T\operatorname{\mathbb{E}}_+ \left[{\boldsymbol{x}}y\right]-{\boldsymbol{w}}^T\operatorname{\mathbb{E}}_+ \left[ {\boldsymbol{x}}\right]\operatorname{\mathbb{E}}\left[y\right]}\label{eq:closed_a+-}\end{aligned}$$ The solution for $S_{{\boldsymbol{a}}+-}$ reduces to the linear estimator when the relation between input and output is linear. Therefore, it solves our introductory linear example correctly.
#### PatternNet and PatternAttribution
Based on the presented analysis, we propose PatternNet and PatternAttribution as illustrated in [ Fig. \[fig:fig2\]]{}. [*PatternNet*]{} yields a layer-wise back-projection of the estimated signal to input space. The signal estimator is approximated as a superposition of neuron-wise, nonlinear signal estimators $S_{{\boldsymbol{a}}+-}$ in each layer. It is equal to the computation of the gradient where during the backward pass the weights of the network are replaced by the informative directions. In [ Fig. \[fig:fig2\]]{}, a visual improvement over DeConvNet and Guided Backprop is apparent.
[*PatternAttribution*]{} exposes the attribution ${\boldsymbol{w}}\odot {\boldsymbol{a}}_+$ and improves upon the layer-wise relevance propagation (LRP) framework [@Bach2015]. It can be seen as a root point estimator for the Deep-Taylor Decomposition (DTD). Here, the explanation consists of neuron-wise contributions of the estimated *signal* to the classification score. By ignoring the distractor, PatternAttribution can reduce the noise and produces much clearer heat maps. By working out the back-projection steps in the Deep-Taylor Decomposition with the proposed root point selection method, it becomes obvious that PatternAttribution is also analogous to the backpropagation operation. In this case, the weights are replaced during the backward pass by ${\boldsymbol{w}}\odot{\boldsymbol{a}}_+$.
Experiments and discussion
==========================
To evaluate the quality of the explanations, we focus on the task of image classification. Nevertheless, our method is not restricted to networks operating on image inputs. We used Theano [@Bergstra2010] and Lasagne [@Dieleman2015] for our implementation. We restrict the analysis to the well-known ImageNet dataset [@Imagenet2015] using the pre-trained VGG-16 model [@Simonyan2014]. Images were rescaled and cropped to 224x224 pixels. The signal estimators are trained on the first half of the training dataset.
The vector ${\boldsymbol{v}}$, used to measure the quality of the signal estimator $\rho({\boldsymbol{x}})$ in [Eq. ]{}, is optimized on the second half of the training dataset. This enables us to test the signal estimators for generalization. All the results presented here were obtained using the official validation set of 50000 samples. The validation set was not used for training the signal estimators, nor for training the vector ${\boldsymbol{v}}$ to measure the quality. Consequently our results are obtained on previously unseen data.
The linear and the two component signal estimators are obtained by solving their respective closed form solutions ([Eq. ]{} and [Eq. ]{}). With a highly parallelized implementation using 4 GPUs this could be done in 3-4 hours. This can be considered reasonable given that several days are required to train the actual network. The quality of a signal estimator is assessed with [Eq. ]{}. Solving it with the closed form solution is computationally prohibitive since it must be repeated for every single weight vector in the network. Therefore we optimize the equivalent least-squares problem using stochastic mini-batch gradient descent with ADAM [@Kingma2014] until convergence. This was implemented on a NVIDIA Tesla K40 and took about 24 hours per optimized signal estimator.
After learning to explain, individual explanations are computationally cheap since they can be implemented as a back-propagation pass with a modified weight vector. As a result, our method produces explanations at least as fast as the work by [@Dabkowski2017] on real time saliency. However, our method has the advantage that it is not only applicable to image models but is a generalization of the theory commonly used in neuroimaging [@Haufe2014].
[0.48]{}
[0.48]{}
#### Measuring the quality of signal estimators
In [ Fig. \[fig:fig3\]]{} we present the results from the correlation measure $\rho({\boldsymbol{x}})$, where higher values are better. We use random directions as baseline signal estimators. Clearly, this approach removes almost no correlation. The filter-based estimator $S_{{\boldsymbol{w}}}$ succeeds in removing some of the information in the first layer. This indicates that the filters are similar to the patterns in this layer. However, the gradient removes much less information in the higher layers. Overall, it does not perform much better than the random estimator. [*This implies that the weights do not correspond to the detected stimulus in a neural network.*]{} Hence the implicit assumptions about the signal made by DeConvNet and Guided BackProp is not valid. The optimized estimators remove much more of the correlations across the board. For convolutional layers, $S_{{\boldsymbol{a}}}$ and $S_{{\boldsymbol{a}}+-}$ perform comparably in all but one layer. The two component estimator $S_{{\boldsymbol{a}}+-}$ is best in the dense layers.
#### Image degradation
The first experiment was a direct measurement of the quality of the signal estimators of individual neurons. The second one is an indirect measurement of the quality, but it considers the whole network. We measure how the prediction (after the soft-max) for the initially selected class changes as a function of corrupting more and more patches based on the ordering assigned by the attribution [see @Samek2016]. This is also related to the work by @Zintgraf2017. In this experiment, we split the image in non-overlapping patches of 9x9 pixels. We compute the attribution and sum all the values within a patch. We sort the patches in decreasing order based on the aggregate heat map value. In step $n=1..100$ we replace the first $n$ patches with the their mean per color channel to remove the information in this patch. Then, we measure how this influences the classifiers output. We use the estimators from the previous experiment to obtain the function-signal attribution heat maps for evaluation. A steeper decay indicates a better heat map.
Results are shown in [ Fig. \[fig:fig4\]]{}. The baseline, in which the patches are randomly ordered, performs worst. The linear optimized estimator $S_{{\boldsymbol{a}}}$ performs quite poorly, followed by the filter-based estimator $S_{{\boldsymbol{w}}}$. The trivial signal estimator $S_{{\boldsymbol{x}}}$ performs just slightly better. However, the two component model $S_{{\boldsymbol{a}}+-}$ leads to the fastest decrease in confidence in the original prediction by a large margin. Its excellent quantitative performance is also backed up by the visualizations discussed next.
#### Qualitative evaluation
In [ Fig. \[fig:fig5\]]{}, we compare all signal estimators on a single input image. For the trivial estimator $S_{{\boldsymbol{x}}}$, the signal is by definition the original input image and, thus, includes the distractor. Therefore, its noisy attribution heat map shows contributions that cancel each other in the neural network. The $S_{{\boldsymbol{w}}}$ estimator captures some of the structure. The optimized estimator $S_{{\boldsymbol{a}}}$ results in slightly more structure but struggles on color information and produces dense heat maps. The two component model $S_{{\boldsymbol{a}}+-}$ on the right captures the original input during signal estimation and produces a crisp heat map of the attribution.
[ Fig. \[fig:fig6\]]{} shows the visualizations for six randomly selected images from ImageNet. PatternNet is able to recover a signal close to the original without having to resort to the inclusion of additional rectifiers in contrast to DeConvNet and Guided BackProp. We argue that this is due to the fact that the optimization of the pattern allows for capturing the important directions in input space. This contrasts with the commonly used methods DeConvNet, Guided BackProp, LRP and DTD, for which the correlation experiment indicates that their implicit signal estimator cannot capture the true signal in the data. Overall, the proposed approach produces the most crisp visualization in addition to being measurably better, as shown in the previous section.
#### Relation to previous methods
Our method can be thought of as a generalization of the work by [@Haufe2014], making it applicable on deep neural networks. Remarkably, our proposed approach can solve the toy example in section \[sec:linearmodels\] optimally while none of the previously published methods for deep learning are able to solve this [@Bach2015; @Montavon2017; @Smilkov2017; @Mukund2017; @Zintgraf2017; @Dabkowski2017; @Zeiler2014; @Springenberg2014]. Our method shares the idea that to explain a model properly one has to learn how to explain it with @Zintgraf2017 and @Dabkowski2017. Furthermore, since our approach is after training just as expensive as a single back-propagation step, it can be applied in a real-time context, which is also possible for the work done by @Dabkowski2017 but not for @Zintgraf2017.
Conclusion
==========
Understanding and explaining nonlinear methods is an important challenge in machine learning. Algorithms for visualizing nonlinear models have emerged but theoretical contributions are scarce. We have shown that the direction of the model gradient does not necessarily provide an estimate for the signal in the data. Instead it reflects the relation between the signal direction and the distracting noise contributions ([ Fig. \[fig:fig1\]]{}). This implies that popular explanation approaches for neural networks (DeConvNet, Guided BackProp, LRP) do not provide the correct explanation, even for a simple linear model. Our reasoning can be extended to nonlinear models. We have proposed an objective function for neuron-wise explanations. This can be optimized to correct the signal visualizations (PatternNet) and the decomposition methods (PatternAttribution) by taking the data distribution into account. We have demonstrated that our methods constitute a theoretical, qualitative and quantitative improvement towards understanding deep neural networks.
### Acknowledgments {#acknowledgments .unnumbered}
This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement NO 657679, the BMBF for the Berlin Big Data Center BBDC (01IS14013A), a hardware donation from NVIDIA. We thank Sander Dieleman, Jonas Degraeve, Ira Korshunova, Stefan Chmiela, Malte Esders, Sarah Hooker, Vincent Vanhoucke for their comments to improve this manuscript. We are grateful to Chris Olah and Gregoire Montavon for the valuable discussions.
[^1]: Part of this work was done at TU Berlin, part of the work was part of the Google Brain Residency program.
[^2]: KRM is also with Korea University and Max Planck Institute for Informatics, Saarbrücken, Germany
[^3]: Sven Dähne is now at Amazon
[^4]: In tensorflow terminoloy: linear model on MNIST can be seen as a convolutional neural network with VALID padding and a 28 by 28 filter size.
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